MathieuCPrime[MathieuCharacteristicA[1, q], q, z] \[Proportional] -Sin[z] + (3/8) Sin[3 z] q + (Sin[z]/128 + (3/64) Sin[3 z] - (5/192) Sin[5 z]) q^2 + (Sin[z]/512 - Sin[3 z]/1024 - (5 Sin[5 z])/1152 + (7 Sin[7 z])/9216) q^3 + ((37 Sin[z])/294912 - (49 Sin[3 z])/24576 + (7 Sin[7 z])/49152 - Sin[9 z]/81920) q^4 + (-((121 Sin[z])/1769472) - (317 Sin[3 z])/786432 + (205 Sin[5 z])/1179648 + (7 Sin[7 z])/5898240 - Sin[9 z]/409600 + (11 Sin[11 z])/88473600) q^5 + (-((8105 Sin[z])/339738624) + (103 Sin[3 z])/18874368 + (731 Sin[5 z])/18874368 - (2653 Sin[7 z])/471859200 - Sin[9 z]/31457280 + (11 Sin[11 z])/424673280 - (13 Sin[13 z])/14863564800) q^6 + (-((481 Sin[z])/226492416) + (102547 Sin[3 z])/4529848320 + (659 Sin[5 z])/2264924160 - (88739 Sin[7 z])/67947724800 + (181 Sin[9 z])/1887436800 + (11 Sin[11 z])/26424115200 - (13 Sin[13 z])/69363302400 + Sin[15 z]/221962567680) q^7 + ((1237783 Sin[z])/1449551462400 + (940781 Sin[3 z])/181193932800 - (322897 Sin[5 z])/163074539520 - (65891 Sin[7 z])/3261490790400 + (2143 Sin[9 z])/93952409600 - (13519 Sin[11 z])/13317754060800 - (13 Sin[13 z])/3805072588800 + Sin[15 z]/1014686023680 - (17 Sin[17 z])/958878292377600) q^8 + ((11221967 Sin[z])/32614907904000 - (506831 Sin[3 z])/34789235097600 - (1550665 Sin[5 z])/3131031158784 + (5039101 Sin[7 z])/78275778969600 + (16013 Sin[9 z])/35514010828800 - (34837 Sin[11 z])/142056043315200 + (6253 Sin[13 z])/852336259891200 + Sin[15 z]/51140175593472 - (17 Sin[17 z])/4314952315699200 + (19 Sin[19 z])/345196185255936000) q^9 + ((4539285691 Sin[z])/125241246351360000 - (1282939901 Sin[3 z])/4174708211712000 - (10304813 Sin[5 z])/ 1168918299279360 + (48740801 Sin[7 z])/2922295748198400 - (7516703 Sin[9 z])/6818690079129600 - (338503 Sin[11 z])/ 61368210712166400 + (329927 Sin[13 z])/184104632136499200 - (2839 Sin[15 z])/73641852854599680 - (17 Sin[17 z])/204560702373888000 + (19 Sin[19 z])/1534205267804160000 - Sin[21 z]/7232681976791040000) q^10 + SeriesData[$CellContext`q, 0, {}, 0, 11, 1]

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sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 128 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 64 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 192 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 512 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 1024 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 1152 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 9216 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> q </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 37 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 294912 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 49 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 24576 </mn> </mfrac> 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<cn type='rational'> 11221967 <sep /> 32614907904000 </cn> <apply> <sin /> <ci> $CellContext`z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> -506831 <sep /> 34789235097600 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -1550665 <sep /> 3131031158784 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 5039101 <sep /> 78275778969600 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 16013 <sep /> 35514010828800 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -34837 <sep /> 142056043315200 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 6253 <sep /> 852336259891200 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 13 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 51140175593472 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 15 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> -17 <sep /> 4314952315699200 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 17 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 19 <sep /> 345196185255936000 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 19 </cn> <ci> $CellContext`z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 4539285691 <sep /> 125241246351360000 </cn> <apply> <sin /> <ci> $CellContext`z </ci> </apply> 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type='integer'> 11 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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