Properties

Label 108.6.b.c
Level 108
Weight 6
Character orbit 108.b
Analytic conductor 17.321
Analytic rank 0
Dimension 20
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 108.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.3214525398\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{50}\cdot 3^{40} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{15} q^{5} -\beta_{5} q^{7} + ( \beta_{4} + \beta_{17} ) q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{15} q^{5} -\beta_{5} q^{7} + ( \beta_{4} + \beta_{17} ) q^{8} + ( 9 - \beta_{1} ) q^{10} + ( \beta_{4} + \beta_{11} ) q^{11} + ( -6 + \beta_{1} - \beta_{8} ) q^{13} + ( -\beta_{4} - \beta_{9} - 2 \beta_{15} ) q^{14} + ( -208 + 2 \beta_{2} + 2 \beta_{5} - \beta_{7} + \beta_{8} ) q^{16} + ( 25 \beta_{4} - \beta_{9} - 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{17} + ( 2 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{10} ) q^{19} + ( 10 \beta_{4} + \beta_{9} + 4 \beta_{11} + 3 \beta_{15} + \beta_{16} + \beta_{19} ) q^{20} + ( 34 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} ) q^{22} + ( -15 \beta_{4} - \beta_{9} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + 7 \beta_{17} + \beta_{18} ) q^{23} + ( -761 - 2 \beta_{1} - 3 \beta_{2} - \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{12} ) q^{25} + ( -\beta_{4} - \beta_{9} + 2 \beta_{13} + 19 \beta_{15} - 2 \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{26} + ( -240 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 5 \beta_{5} + 2 \beta_{6} - \beta_{7} - 6 \beta_{8} + \beta_{10} + \beta_{12} ) q^{28} + ( 95 \beta_{4} + 2 \beta_{9} + 3 \beta_{14} + 8 \beta_{15} + 6 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{29} + ( -4 - 10 \beta_{1} + 24 \beta_{2} + 2 \beta_{5} - 4 \beta_{8} + 3 \beta_{10} + 4 \beta_{12} ) q^{31} + ( -205 \beta_{4} + 3 \beta_{9} - 4 \beta_{11} - 2 \beta_{13} + 3 \beta_{14} + 13 \beta_{15} - 3 \beta_{16} + 5 \beta_{19} ) q^{32} + ( -827 + 3 \beta_{1} + 22 \beta_{2} - 8 \beta_{3} - 32 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{34} + ( 222 \beta_{4} + 4 \beta_{9} - \beta_{11} - 7 \beta_{14} + 6 \beta_{16} - 20 \beta_{17} + 6 \beta_{18} - 2 \beta_{19} ) q^{35} + ( -326 - 11 \beta_{1} + 73 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{10} - 6 \beta_{12} ) q^{37} + ( -16 \beta_{4} - 2 \beta_{9} - 16 \beta_{11} + 4 \beta_{13} - 5 \beta_{14} - 55 \beta_{15} - 8 \beta_{17} - \beta_{18} ) q^{38} + ( 74 - 4 \beta_{1} - 10 \beta_{2} + 22 \beta_{3} - 42 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{10} - 6 \beta_{12} ) q^{40} + ( -347 \beta_{4} + 2 \beta_{9} - 8 \beta_{13} + 5 \beta_{14} - 4 \beta_{16} - 42 \beta_{17} - 6 \beta_{18} + 2 \beta_{19} ) q^{41} + ( 10 \beta_{1} + 128 \beta_{2} + 4 \beta_{3} - 6 \beta_{5} - 8 \beta_{6} + 5 \beta_{10} ) q^{43} + ( -2 \beta_{4} + 3 \beta_{9} - 4 \beta_{11} + 8 \beta_{13} + 4 \beta_{14} - 127 \beta_{15} - \beta_{16} - 4 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{44} + ( -466 + 3 \beta_{1} + 13 \beta_{2} - 29 \beta_{3} + 33 \beta_{5} + 15 \beta_{6} - 4 \beta_{7} + \beta_{8} + 3 \beta_{10} + 5 \beta_{12} ) q^{46} + ( -90 \beta_{4} - 5 \beta_{9} - 15 \beta_{11} - \beta_{13} - 4 \beta_{14} - 9 \beta_{15} + 5 \beta_{16} - 21 \beta_{17} + 15 \beta_{18} - 10 \beta_{19} ) q^{47} + ( -2218 + 30 \beta_{1} - 54 \beta_{2} - 8 \beta_{3} + 8 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 6 \beta_{8} - 2 \beta_{10} + 4 \beta_{12} ) q^{49} + ( -789 \beta_{4} + 8 \beta_{9} + 16 \beta_{11} - 4 \beta_{13} + 18 \beta_{14} - 72 \beta_{15} - 16 \beta_{16} - 8 \beta_{17} - 4 \beta_{18} + 16 \beta_{19} ) q^{50} + ( 405 - 12 \beta_{1} - 3 \beta_{2} + 46 \beta_{3} + 114 \beta_{5} - 14 \beta_{6} + 4 \beta_{7} - 10 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} ) q^{52} + ( 419 \beta_{4} + 4 \beta_{9} + 5 \beta_{14} + 22 \beta_{15} + 10 \beta_{16} + 52 \beta_{17} - 14 \beta_{18} - 14 \beta_{19} ) q^{53} + ( 4 + 14 \beta_{1} - 494 \beta_{2} + 6 \beta_{3} - 30 \beta_{5} + 24 \beta_{6} + 4 \beta_{8} - \beta_{10} - 4 \beta_{12} ) q^{55} + ( -190 \beta_{4} - 9 \beta_{9} - 4 \beta_{11} + 14 \beta_{13} - 29 \beta_{14} + 193 \beta_{15} - 11 \beta_{16} - \beta_{17} - 4 \beta_{18} - 3 \beta_{19} ) q^{56} + ( -2930 - 6 \beta_{1} + 148 \beta_{2} - 72 \beta_{3} + 88 \beta_{5} - 12 \beta_{6} - 8 \beta_{7} + 16 \beta_{8} - 8 \beta_{10} - 8 \beta_{12} ) q^{58} + ( 639 \beta_{4} + 10 \beta_{9} + 15 \beta_{11} - 22 \beta_{13} - 42 \beta_{14} - 6 \beta_{15} + 2 \beta_{16} + 90 \beta_{17} + 30 \beta_{18} + 8 \beta_{19} ) q^{59} + ( 4220 - 5 \beta_{1} - 350 \beta_{2} + 8 \beta_{3} - 8 \beta_{5} - 26 \beta_{6} + 24 \beta_{7} - 19 \beta_{8} + 6 \beta_{10} + 4 \beta_{12} ) q^{61} + ( 110 \beta_{4} + 8 \beta_{9} + 16 \beta_{11} + 12 \beta_{13} + 16 \beta_{14} + 390 \beta_{15} + 16 \beta_{16} + 8 \beta_{17} - 2 \beta_{18} + 16 \beta_{19} ) q^{62} + ( -318 - 28 \beta_{1} - 246 \beta_{2} + 94 \beta_{3} - 58 \beta_{5} + 46 \beta_{6} + 2 \beta_{7} + 24 \beta_{8} - 6 \beta_{10} + 2 \beta_{12} ) q^{64} + ( 2486 \beta_{4} - 17 \beta_{9} - 8 \beta_{13} + 71 \beta_{14} + 16 \beta_{15} + 27 \beta_{16} + 25 \beta_{17} - 18 \beta_{18} - 10 \beta_{19} ) q^{65} + ( 8 + 315 \beta_{2} - 19 \beta_{3} - 20 \beta_{5} - 16 \beta_{6} + 8 \beta_{8} - 16 \beta_{10} - 8 \beta_{12} ) q^{67} + ( -810 \beta_{4} - 31 \beta_{9} + 4 \beta_{11} + 8 \beta_{13} + 52 \beta_{14} + 147 \beta_{15} - 23 \beta_{16} - 16 \beta_{18} + 9 \beta_{19} ) q^{68} + ( 6934 - 17 \beta_{1} + 231 \beta_{2} - 137 \beta_{3} - 155 \beta_{5} - 39 \beta_{6} + 32 \beta_{7} + 9 \beta_{8} + 7 \beta_{10} + 9 \beta_{12} ) q^{70} + ( 1492 \beta_{4} - 34 \beta_{9} + 16 \beta_{11} - 2 \beta_{13} - 66 \beta_{14} - 34 \beta_{15} - 2 \beta_{16} - 18 \beta_{17} + 34 \beta_{18} - 32 \beta_{19} ) q^{71} + ( 4289 + 22 \beta_{1} + 852 \beta_{2} - 16 \beta_{3} + 16 \beta_{5} + 76 \beta_{6} + 16 \beta_{7} + 26 \beta_{8} + 4 \beta_{10} + 24 \beta_{12} ) q^{73} + ( -451 \beta_{4} + 17 \beta_{9} + 80 \beta_{11} + 10 \beta_{13} - 78 \beta_{14} - 403 \beta_{15} - 14 \beta_{16} + 88 \beta_{17} - 11 \beta_{18} + 18 \beta_{19} ) q^{74} + ( 4474 + 40 \beta_{1} - 121 \beta_{2} + 161 \beta_{3} - 75 \beta_{5} - 12 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} - 13 \beta_{10} + 19 \beta_{12} ) q^{76} + ( -1403 \beta_{4} - 18 \beta_{9} - 16 \beta_{13} - 39 \beta_{14} + 137 \beta_{15} + 24 \beta_{16} - 22 \beta_{17} - 22 \beta_{18} - 6 \beta_{19} ) q^{77} + ( 4 - 42 \beta_{1} + 1000 \beta_{2} - 8 \beta_{3} + 91 \beta_{5} - 48 \beta_{6} + 4 \beta_{8} - 29 \beta_{10} - 4 \beta_{12} ) q^{79} + ( -182 \beta_{4} - 30 \beta_{9} + 8 \beta_{11} + 12 \beta_{13} + 30 \beta_{14} - 834 \beta_{15} - 2 \beta_{16} - 12 \beta_{17} + 8 \beta_{18} + 14 \beta_{19} ) q^{80} + ( 11054 + 10 \beta_{1} - 236 \beta_{2} - 248 \beta_{3} + 8 \beta_{5} + 84 \beta_{6} + 24 \beta_{7} - 32 \beta_{8} - 24 \beta_{10} - 24 \beta_{12} ) q^{82} + ( -3935 \beta_{4} + 46 \beta_{9} + 94 \beta_{11} - 22 \beta_{13} + 107 \beta_{14} + 10 \beta_{15} + 32 \beta_{16} + 6 \beta_{17} + 44 \beta_{18} + 14 \beta_{19} ) q^{83} + ( 9006 - 146 \beta_{1} - 579 \beta_{2} + 28 \beta_{3} - 28 \beta_{5} - 49 \beta_{6} + 44 \beta_{7} + 62 \beta_{8} + 11 \beta_{10} - 6 \beta_{12} ) q^{85} + ( -86 \beta_{4} - 16 \beta_{9} - 80 \beta_{11} + 20 \beta_{13} + 104 \beta_{14} - 286 \beta_{15} + 88 \beta_{17} - 6 \beta_{18} ) q^{86} + ( -4258 + 116 \beta_{1} - 150 \beta_{2} + 306 \beta_{3} + 58 \beta_{5} - 66 \beta_{6} + 6 \beta_{7} + 16 \beta_{8} - 10 \beta_{10} - 2 \beta_{12} ) q^{88} + ( -2083 \beta_{4} + 15 \beta_{9} - 16 \beta_{13} - 36 \beta_{14} + 238 \beta_{15} - 7 \beta_{16} - 47 \beta_{17} - 24 \beta_{18} - 8 \beta_{19} ) q^{89} + ( -8 - 78 \beta_{1} - 2203 \beta_{2} - 65 \beta_{3} + 236 \beta_{5} + 128 \beta_{6} - 8 \beta_{8} - 23 \beta_{10} + 8 \beta_{12} ) q^{91} + ( -306 \beta_{4} + 29 \beta_{9} - 60 \beta_{11} - 176 \beta_{14} + 559 \beta_{15} - 7 \beta_{16} + 4 \beta_{17} - 28 \beta_{18} + 25 \beta_{19} ) q^{92} + ( -3062 + 5 \beta_{1} + 87 \beta_{2} - 403 \beta_{3} - \beta_{5} - 27 \beta_{6} + 60 \beta_{7} - 41 \beta_{8} + 29 \beta_{10} + 59 \beta_{12} ) q^{94} + ( 6571 \beta_{4} - 43 \beta_{9} - 93 \beta_{11} + 29 \beta_{13} - 197 \beta_{14} - 35 \beta_{15} + 17 \beta_{16} - 227 \beta_{17} + 23 \beta_{18} - 60 \beta_{19} ) q^{95} + ( -10949 + 48 \beta_{1} - 1093 \beta_{2} - 40 \beta_{3} + 40 \beta_{5} - 71 \beta_{6} - 28 \beta_{7} + 72 \beta_{8} - 7 \beta_{10} + 26 \beta_{12} ) q^{97} + ( -2000 \beta_{4} - 18 \beta_{9} - 96 \beta_{11} - 4 \beta_{13} + 100 \beta_{14} + 822 \beta_{15} - 68 \beta_{16} - 80 \beta_{17} - 10 \beta_{18} - 4 \beta_{19} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{4} + O(q^{10}) \) \( 20q + 20q^{4} + 184q^{10} - 116q^{13} - 4168q^{16} + 696q^{22} - 15228q^{25} - 4764q^{28} - 16520q^{34} - 6452q^{37} + 1504q^{40} - 9336q^{46} - 44464q^{49} + 8236q^{52} - 58736q^{58} + 84604q^{61} - 6496q^{64} + 138696q^{70} + 85420q^{73} + 89172q^{76} + 221200q^{82} + 180320q^{85} - 85824q^{88} - 60936q^{94} - 219908q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 94 x^{18} + 5872 x^{16} - 207192 x^{14} + 5271952 x^{12} - 76648960 x^{10} + 792478720 x^{8} - 4371873792 x^{6} + 17152147456 x^{4} - 32033996800 x^{2} + 41943040000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-460704143658979 \nu^{18} + 124695093054722826 \nu^{16} - 10164131794348609488 \nu^{14} + 561544881192713335368 \nu^{12} - 18463980174518540834608 \nu^{10} + 436065653865726069987840 \nu^{8} - 5689545685597236396218880 \nu^{6} + 51785393359182419604000768 \nu^{4} - 189772925723537704456781824 \nu^{2} + 439870119641170416623616000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(2025108364759321 \nu^{18} - 183262754924726374 \nu^{16} + 11308789133078963712 \nu^{14} - 384903352930491537432 \nu^{12} + 9553121267554200960592 \nu^{10} - 127782126513936579967360 \nu^{8} + 1247906014504122939294720 \nu^{6} - 5681695433800747191877632 \nu^{4} + 23647024687404932521394176 \nu^{2} - 28667103365530265593446400\)\()/ \)\(74\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-7941903885677141 \nu^{18} + 733274058936664694 \nu^{16} - 45381030604901708592 \nu^{14} + 1567661237940170446392 \nu^{12} - 39250868738170123231952 \nu^{10} + 545361211061027871157760 \nu^{8} - 5529503216977116344816640 \nu^{6} + 27152472936530627512369152 \nu^{4} - 121264667895439192203001856 \nu^{2} + 154189188359109663417958400\)\()/ \)\(11\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(4192319258278379 \nu^{19} - 379387069912772986 \nu^{17} + 23299976696837590128 \nu^{15} - 787107289106915922888 \nu^{13} + 19388985433357370257328 \nu^{11} - 254489496003440123157760 \nu^{9} + 2485112904381882576122880 \nu^{7} - 9542973426203545805242368 \nu^{5} + 34845537509257593866092544 \nu^{3} + 89344298184646199858954240 \nu\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(61625285938470617 \nu^{18} - 6130587419414591598 \nu^{16} + 392866261187449047024 \nu^{14} - 14576644984472235296664 \nu^{12} + 382328218486498064397584 \nu^{10} - 5976558030002683083025920 \nu^{8} + 61622892449857918224983040 \nu^{6} - 354378996184623310087716864 \nu^{4} + 957023457909163150321713152 \nu^{2} - 1388429937591980365892812800\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-13641952209501573 \nu^{18} + 1279409015937634862 \nu^{16} - 79740725406807488256 \nu^{14} + 2774385162156881033016 \nu^{12} - 70060081847236445319696 \nu^{10} + 987371754973002724983680 \nu^{8} - 10624025903430190912680960 \nu^{6} + 57364487060500990556553216 \nu^{4} - 276047473570523411661324288 \nu^{2} + 367366853240306023134003200\)\()/ \)\(74\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-15905468877639401 \nu^{18} + 1477429698694240814 \nu^{16} - 91324711104549049392 \nu^{14} + 3149394774974613497112 \nu^{12} - 77883159619719210263312 \nu^{10} + 1054197067445216151434240 \nu^{8} - 9903873045831916038435840 \nu^{6} + 43578229693424334370701312 \nu^{4} - 121039028103645076144848896 \nu^{2} + 284028569628335523957309440\)\()/ \)\(19\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-32486376825862229 \nu^{18} + 2978855553562335026 \nu^{16} - 184242350698656670488 \nu^{14} + 6351095236524049482168 \nu^{12} - 158533033014290818045808 \nu^{10} + 2179592158143438899560640 \nu^{8} - 21097084694441573659476480 \nu^{6} + 94474763181588007097991168 \nu^{4} - 270930587848745482637901824 \nu^{2} + 289009175764533947629568000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(296140778013868963 \nu^{19} - 56122483251424608122 \nu^{17} + 4337505881504821589136 \nu^{15} - 222704715610570791225096 \nu^{13} + 7047486266576057020140976 \nu^{11} - 157328332369309697333319680 \nu^{9} + 1943711264583270197928975360 \nu^{7} - 16494233561207933471416221696 \nu^{5} + 48104762132440963649897955328 \nu^{3} - 100763358600177342252843008000 \nu\)\()/ \)\(17\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-261672442786614989 \nu^{18} + 24212996175671896966 \nu^{16} - 1491149387095871333808 \nu^{14} + 51453187329670068864888 \nu^{12} - 1276540646916351692072528 \nu^{10} + 17582157036383928723197440 \nu^{8} - 169241014213551154091258880 \nu^{6} + 755783144048294747563327488 \nu^{4} - 2218449507213069766752272384 \nu^{2} + 1620210423900006551348838400\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-1316774314364016479 \nu^{19} + 93157928672259273826 \nu^{17} - 5061036889336440687888 \nu^{15} + 110355805969689325958568 \nu^{13} - 1501642290738750145386608 \nu^{11} - 33671715356070043593474560 \nu^{9} + 760877001976682070713349120 \nu^{7} - 11591294702943107087050309632 \nu^{5} + 49724173917447559662930558976 \nu^{3} - 162470415726723975407992832000 \nu\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(752127443307319439 \nu^{18} - 69713474123729962466 \nu^{16} + 4305341034047363657808 \nu^{14} - 148452649381825960168488 \nu^{12} + 3682014564931620880009328 \nu^{10} - 50244064973842798438538240 \nu^{8} + 483075551387847498874874880 \nu^{6} - 2139556160199309930199154688 \nu^{4} + 6251824341744479320088182784 \nu^{2} - 3569033370967527187546112000\)\()/ \)\(29\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-756793047030795379 \nu^{19} + 65897032096562553626 \nu^{17} - 4020747321644898025488 \nu^{15} + 129317861195078557893768 \nu^{13} - 3118267856886664485957808 \nu^{11} + 35965357219709673795875840 \nu^{9} - 379992038767265442888698880 \nu^{7} + 1259536381805670460331655168 \nu^{5} - 9127646138547512794563149824 \nu^{3} - 10218742690210858804235468800 \nu\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(465796903166270941 \nu^{19} - 43083919648627432054 \nu^{17} + 2674136849514918119952 \nu^{15} - 92785623979163830042872 \nu^{13} + 2332238598416184864974032 \nu^{11} - 32704476126711124094114560 \nu^{9} + 331803001324651264689515520 \nu^{7} - 1723162956029532392119222272 \nu^{5} + 7047778075795297723263287296 \nu^{3} - 10979771483804891803706982400 \nu\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(3125632685218440679 \nu^{19} - 292125345344028096626 \nu^{17} + 18167751214708956990288 \nu^{15} - 635003702150453758372968 \nu^{13} + 15963767484362811213953008 \nu^{11} - 225040565417909525894973440 \nu^{9} + 2214759879064174771154426880 \nu^{7} - 10672824521023414011965472768 \nu^{5} + 32323145228769849036646580224 \nu^{3} - 30659206373021256903557120000 \nu\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-1164061941134453351 \nu^{19} + 96585129775921626994 \nu^{17} - 5551207103660730357072 \nu^{15} + 159806179974063223948392 \nu^{13} - 3171176425616956251783152 \nu^{11} + 13584240261507792135976960 \nu^{9} + 200061353458647835426145280 \nu^{7} - 5848011391945437684901675008 \nu^{5} + 35464499172972805890239954944 \nu^{3} - 51026519319868498045514547200 \nu\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(601958164253216691 \nu^{19} - 55466663205400824554 \nu^{17} + 3431597071085307731952 \nu^{15} - 118577386109700393844872 \nu^{13} + 2967450794426208289450032 \nu^{11} - 41242896450633547164642560 \nu^{9} + 412132060209942501570155520 \nu^{7} - 2015002924382901125188534272 \nu^{5} + 7671849751813606585297207296 \nu^{3} - 11321637789915819457799782400 \nu\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-8057086085426941181 \nu^{19} + 738193912261409478214 \nu^{17} - 45665226331029802027632 \nu^{15} + 1569303499218541805980152 \nu^{13} - 39247300335610394478277712 \nu^{11} + 540440217153288547690332160 \nu^{9} - 5502225037934077972589368320 \nu^{7} + 27703012288536208735984484352 \nu^{5} - 118197303196246893038790311936 \nu^{3} + 195500596722247115770691584000 \nu\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-10236740974709912603 \nu^{19} + 922342217862151387882 \nu^{17} - 56331171112785337601616 \nu^{15} + 1887119557670251546735176 \nu^{13} - 45824324154795137177297456 \nu^{11} + 582737984606009358574497280 \nu^{9} - 5315832330660693500155100160 \nu^{7} + 17619046547903180866679832576 \nu^{5} - 41328829749474104982579642368 \nu^{3} - 109995232184479052687369830400 \nu\)\()/ \)\(17\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{19} + 12 \beta_{18} + 2 \beta_{16} - 8 \beta_{15} + 29 \beta_{14} - 2 \beta_{13} + 51 \beta_{4}\)\()/2592\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{12} + 2 \beta_{10} - 3 \beta_{8} - 4 \beta_{7} + 6 \beta_{6} - 5 \beta_{5} - 83 \beta_{3} - 136 \beta_{2} + 6089\)\()/648\)
\(\nu^{3}\)\(=\)\((\)\(34 \beta_{19} + 56 \beta_{18} - 96 \beta_{17} - 82 \beta_{16} - 352 \beta_{15} + 579 \beta_{14} + 22 \beta_{13} + 48 \beta_{9} + 16965 \beta_{4}\)\()/1296\)
\(\nu^{4}\)\(=\)\((\)\(31 \beta_{12} + 30 \beta_{10} + \beta_{8} + 24 \beta_{7} + 96 \beta_{6} - 53 \beta_{5} - 443 \beta_{3} - 446 \beta_{2} - 16 \beta_{1} - 31415\)\()/108\)
\(\nu^{5}\)\(=\)\((\)\(170 \beta_{19} - 824 \beta_{18} - 2000 \beta_{17} - 362 \beta_{16} - 840 \beta_{15} + 799 \beta_{14} + 550 \beta_{13} + 32 \beta_{11} + 184 \beta_{9} + 104361 \beta_{4}\)\()/216\)
\(\nu^{6}\)\(=\)\((\)\(1759 \beta_{12} + 634 \beta_{10} + 2025 \beta_{8} + 2536 \beta_{7} + 3714 \beta_{6} + 491 \beta_{5} - 491 \beta_{3} + 38996 \beta_{2} - 552 \beta_{1} - 3299855\)\()/162\)
\(\nu^{7}\)\(=\)\((\)\(-1110 \beta_{19} - 60424 \beta_{18} - 77952 \beta_{17} + 11766 \beta_{16} + 91960 \beta_{15} - 146549 \beta_{14} + 25686 \beta_{13} + 2208 \beta_{11} - 11496 \beta_{9} + 77229 \beta_{4}\)\()/324\)
\(\nu^{8}\)\(=\)\((\)\(-1983 \beta_{12} - 21522 \beta_{10} + 37467 \beta_{8} + 22884 \beta_{7} - 31176 \beta_{6} + 82749 \beta_{5} + 413387 \beta_{3} + 1163926 \beta_{2} + 12600 \beta_{1} - 29846433\)\()/81\)
\(\nu^{9}\)\(=\)\((\)\(-167546 \beta_{19} - 255784 \beta_{18} + 732960 \beta_{17} + 583274 \beta_{16} + 2652416 \beta_{15} - 3609675 \beta_{14} - 88238 \beta_{13} - 415728 \beta_{9} - 101562429 \beta_{4}\)\()/162\)
\(\nu^{10}\)\(=\)\((\)\(-1267466 \beta_{12} - 1224500 \beta_{10} - 42966 \beta_{8} - 836528 \beta_{7} - 3753900 \beta_{6} + 2755310 \beta_{5} + 15463026 \beta_{3} + 15085656 \beta_{2} + 871776 \beta_{1} + 1092980314\)\()/81\)
\(\nu^{11}\)\(=\)\((\)\(-2606246 \beta_{19} + 15694200 \beta_{18} + 40762128 \beta_{17} + 6613574 \beta_{16} + 17368024 \beta_{15} - 17706373 \beta_{14} - 10458938 \beta_{13} - 687456 \beta_{11} - 3683400 \beta_{9} - 1867847859 \beta_{4}\)\()/81\)
\(\nu^{12}\)\(=\)\((\)\(-14478644 \beta_{12} - 5122296 \beta_{10} - 17613356 \beta_{8} - 20489184 \beta_{7} - 32420208 \beta_{6} - 4234052 \beta_{5} + 4234052 \beta_{3} - 344835432 \beta_{2} + 4911200 \beta_{1} + 26727703156\)\()/27\)
\(\nu^{13}\)\(=\)\((\)\(5784084 \beta_{19} + 249255888 \beta_{18} + 332180544 \beta_{17} - 51139988 \beta_{16} - 386955408 \beta_{15} + 597007198 \beta_{14} - 109330356 \beta_{13} - 7567552 \beta_{11} + 49028208 \beta_{9} + 50202002 \beta_{4}\)\()/27\)
\(\nu^{14}\)\(=\)\((\)\(124343864 \beta_{12} + 1107810320 \beta_{10} - 1889345688 \beta_{8} - 1131258016 \beta_{7} + 1494468240 \beta_{6} - 4171454120 \beta_{5} - 20395314328 \beta_{3} - 58252907552 \beta_{2} - 633743808 \beta_{1} + 1472401899464\)\()/81\)
\(\nu^{15}\)\(=\)\((\)\(1395291160 \beta_{19} + 2075597728 \beta_{18} - 6328900352 \beta_{17} - 4864253144 \beta_{16} - 21973194496 \beta_{15} + 30056156164 \beta_{14} + 680306568 \beta_{13} + 3468961984 \beta_{9} + 842754549724 \beta_{4}\)\()/27\)
\(\nu^{16}\)\(=\)\((\)\(63228608976 \beta_{12} + 61786455456 \beta_{10} + 1442153520 \beta_{8} + 41667848832 \beta_{7} + 186990034752 \beta_{6} - 135179610096 \beta_{5} - 766925516048 \beta_{3} - 730999317856 \beta_{2} - 43092543744 \beta_{1} - 54201041207376\)\()/81\)
\(\nu^{17}\)\(=\)\((\)\(131214426224 \beta_{19} - 783186283072 \beta_{18} - 2050461281664 \beta_{17} - 328139997296 \beta_{16} - 852039882432 \beta_{15} + 900932735720 \beta_{14} + 519267977744 \beta_{13} + 23074456320 \beta_{11} + 185330389824 \beta_{9} + 92570416336344 \beta_{4}\)\()/81\)
\(\nu^{18}\)\(=\)\((\)\(2150512607200 \beta_{12} + 767566959424 \beta_{10} + 2587457624352 \beta_{8} + 3070267837696 \beta_{7} + 4880456912832 \beta_{6} + 615378688352 \beta_{5} - 615378688352 \beta_{3} + 52077998142464 \beta_{2} - 741321559296 \beta_{1} - 3983574929503712\)\()/81\)
\(\nu^{19}\)\(=\)\((\)\(-870395542112 \beta_{19} - 37222217409152 \beta_{18} - 49312297651200 \beta_{17} + 7708514779232 \beta_{16} + 57331839506304 \beta_{15} - 88932476718416 \beta_{14} + 16414338227296 \beta_{13} + 705225890304 \beta_{11} - 7199484984960 \beta_{9} - 20574644736816 \beta_{4}\)\()/81\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
3.14287 1.81454i
3.14287 + 1.81454i
5.24560 3.02855i
5.24560 + 3.02855i
−1.31722 0.760496i
−1.31722 + 0.760496i
−1.93085 1.11477i
−1.93085 + 1.11477i
5.25764 3.03550i
5.25764 + 3.03550i
−5.25764 3.03550i
−5.25764 + 3.03550i
1.93085 1.11477i
1.93085 + 1.11477i
1.31722 0.760496i
1.31722 + 0.760496i
−5.24560 3.02855i
−5.24560 + 3.02855i
−3.14287 1.81454i
−3.14287 + 1.81454i
−5.64474 0.370087i 0 31.7261 + 4.17809i 38.9183i 0 132.403i −177.539 35.3256i 0 −14.4032 + 219.683i
107.2 −5.64474 + 0.370087i 0 31.7261 4.17809i 38.9183i 0 132.403i −177.539 + 35.3256i 0 −14.4032 219.683i
107.3 −4.45658 3.48409i 0 7.72226 + 31.0543i 40.4806i 0 148.504i 73.7809 165.301i 0 141.038 180.405i
107.4 −4.45658 + 3.48409i 0 7.72226 31.0543i 40.4806i 0 148.504i 73.7809 + 165.301i 0 141.038 + 180.405i
107.5 −4.10571 3.89142i 0 1.71366 + 31.9541i 98.6190i 0 67.7266i 117.311 137.863i 0 −383.768 + 404.901i
107.6 −4.10571 + 3.89142i 0 1.71366 31.9541i 98.6190i 0 67.7266i 117.311 + 137.863i 0 −383.768 404.901i
107.7 −3.63934 4.33073i 0 −5.51036 + 31.5220i 80.5384i 0 216.819i 156.567 90.8555i 0 348.790 293.107i
107.8 −3.63934 + 4.33073i 0 −5.51036 31.5220i 80.5384i 0 216.819i 156.567 + 90.8555i 0 348.790 + 293.107i
107.9 −0.821088 5.59695i 0 −30.6516 + 9.19117i 8.15731i 0 63.0026i 76.6102 + 164.009i 0 −45.6560 + 6.69787i
107.10 −0.821088 + 5.59695i 0 −30.6516 9.19117i 8.15731i 0 63.0026i 76.6102 164.009i 0 −45.6560 6.69787i
107.11 0.821088 5.59695i 0 −30.6516 9.19117i 8.15731i 0 63.0026i −76.6102 + 164.009i 0 −45.6560 6.69787i
107.12 0.821088 + 5.59695i 0 −30.6516 + 9.19117i 8.15731i 0 63.0026i −76.6102 164.009i 0 −45.6560 + 6.69787i
107.13 3.63934 4.33073i 0 −5.51036 31.5220i 80.5384i 0 216.819i −156.567 90.8555i 0 348.790 + 293.107i
107.14 3.63934 + 4.33073i 0 −5.51036 + 31.5220i 80.5384i 0 216.819i −156.567 + 90.8555i 0 348.790 293.107i
107.15 4.10571 3.89142i 0 1.71366 31.9541i 98.6190i 0 67.7266i −117.311 137.863i 0 −383.768 404.901i
107.16 4.10571 + 3.89142i 0 1.71366 + 31.9541i 98.6190i 0 67.7266i −117.311 + 137.863i 0 −383.768 + 404.901i
107.17 4.45658 3.48409i 0 7.72226 31.0543i 40.4806i 0 148.504i −73.7809 165.301i 0 141.038 + 180.405i
107.18 4.45658 + 3.48409i 0 7.72226 + 31.0543i 40.4806i 0 148.504i −73.7809 + 165.301i 0 141.038 180.405i
107.19 5.64474 0.370087i 0 31.7261 4.17809i 38.9183i 0 132.403i 177.539 35.3256i 0 −14.4032 219.683i
107.20 5.64474 + 0.370087i 0 31.7261 + 4.17809i 38.9183i 0 132.403i 177.539 + 35.3256i 0 −14.4032 + 219.683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.b.c 20
3.b odd 2 1 inner 108.6.b.c 20
4.b odd 2 1 inner 108.6.b.c 20
12.b even 2 1 inner 108.6.b.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.b.c 20 1.a even 1 1 trivial
108.6.b.c 20 3.b odd 2 1 inner
108.6.b.c 20 4.b odd 2 1 inner
108.6.b.c 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 19432 T_{5}^{8} + 117977728 T_{5}^{6} + 246930301952 T_{5}^{4} + \)\(17\!\cdots\!68\)\( T_{5}^{2} + \)\(10\!\cdots\!88\)\( \) acting on \(S_{6}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T^{2} + 1092 T^{4} - 9504 T^{6} - 1286400 T^{8} - 761856 T^{10} - 1317273600 T^{12} - 9965666304 T^{14} + 1172526071808 T^{16} - 10995116277760 T^{18} + 1125899906842624 T^{20} \)
$3$ 1
$5$ \( ( 1 - 11818 T^{2} + 71630853 T^{4} - 313823973048 T^{6} + 1185932779192818 T^{8} - 3986228637264265212 T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(66\!\cdots\!25\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(88\!\cdots\!25\)\( T^{20} )^{2} \)
$7$ \( ( 1 - 72919 T^{2} + 2924383791 T^{4} - 81340986295962 T^{6} + 1774418094261308301 T^{8} - \)\(32\!\cdots\!57\)\( T^{10} + \)\(50\!\cdots\!49\)\( T^{12} - \)\(64\!\cdots\!62\)\( T^{14} + \)\(65\!\cdots\!59\)\( T^{16} - \)\(46\!\cdots\!19\)\( T^{18} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \)
$11$ \( ( 1 + 822230 T^{2} + 360495679845 T^{4} + 108431998645925448 T^{6} + \)\(24\!\cdots\!70\)\( T^{8} + \)\(44\!\cdots\!92\)\( T^{10} + \)\(64\!\cdots\!70\)\( T^{12} + \)\(72\!\cdots\!48\)\( T^{14} + \)\(62\!\cdots\!45\)\( T^{16} + \)\(37\!\cdots\!30\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \)
$13$ \( ( 1 + 29 T + 791115 T^{2} + 39777006 T^{3} + 380980388709 T^{4} + 35518566649227 T^{5} + 141455351464930737 T^{6} + 5483598057428624094 T^{7} + \)\(40\!\cdots\!55\)\( T^{8} + \)\(55\!\cdots\!29\)\( T^{9} + \)\(70\!\cdots\!93\)\( T^{10} )^{4} \)
$17$ \( ( 1 - 6202690 T^{2} + 18689060069517 T^{4} - 34705007722821565080 T^{6} + \)\(46\!\cdots\!02\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(94\!\cdots\!98\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!33\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( ( 1 - 13782439 T^{2} + 98755621335207 T^{4} - \)\(47\!\cdots\!14\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(48\!\cdots\!01\)\( T^{10} + \)\(10\!\cdots\!01\)\( T^{12} - \)\(17\!\cdots\!14\)\( T^{14} + \)\(22\!\cdots\!07\)\( T^{16} - \)\(19\!\cdots\!39\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \)
$23$ \( ( 1 + 42721646 T^{2} + 918206452664445 T^{4} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{8} + \)\(95\!\cdots\!68\)\( T^{10} + \)\(53\!\cdots\!66\)\( T^{12} + \)\(22\!\cdots\!28\)\( T^{14} + \)\(65\!\cdots\!05\)\( T^{16} + \)\(12\!\cdots\!46\)\( T^{18} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( ( 1 - 142578466 T^{2} + 9766751584325109 T^{4} - \)\(42\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!16\)\( T^{10} + \)\(56\!\cdots\!22\)\( T^{12} - \)\(75\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!09\)\( T^{16} - \)\(44\!\cdots\!66\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \)
$31$ \( ( 1 - 53059450 T^{2} + 1830387109253997 T^{4} - \)\(54\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!98\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{10} + \)\(14\!\cdots\!98\)\( T^{12} - \)\(36\!\cdots\!08\)\( T^{14} + \)\(10\!\cdots\!97\)\( T^{16} - \)\(23\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( ( 1 + 1613 T + 123026979 T^{2} + 343711884654 T^{3} + 5803261980890325 T^{4} + 31670451235762477275 T^{5} + \)\(40\!\cdots\!25\)\( T^{6} + \)\(16\!\cdots\!46\)\( T^{7} + \)\(41\!\cdots\!47\)\( T^{8} + \)\(37\!\cdots\!13\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{4} \)
$41$ \( ( 1 - 204662074 T^{2} + 6421654936070589 T^{4} + \)\(14\!\cdots\!28\)\( T^{6} - \)\(52\!\cdots\!06\)\( T^{8} - \)\(96\!\cdots\!76\)\( T^{10} - \)\(70\!\cdots\!06\)\( T^{12} + \)\(26\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!89\)\( T^{16} - \)\(66\!\cdots\!74\)\( T^{18} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( ( 1 - 1038109906 T^{2} + 527969585590823397 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{8} - \)\(67\!\cdots\!12\)\( T^{10} + \)\(85\!\cdots\!98\)\( T^{12} - \)\(80\!\cdots\!40\)\( T^{14} + \)\(53\!\cdots\!53\)\( T^{16} - \)\(22\!\cdots\!06\)\( T^{18} + \)\(47\!\cdots\!49\)\( T^{20} )^{2} \)
$47$ \( ( 1 + 920931614 T^{2} + 499070896097327373 T^{4} + \)\(18\!\cdots\!24\)\( T^{6} + \)\(53\!\cdots\!58\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(27\!\cdots\!42\)\( T^{12} + \)\(50\!\cdots\!24\)\( T^{14} + \)\(72\!\cdots\!77\)\( T^{16} + \)\(70\!\cdots\!14\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \)
$53$ \( ( 1 - 1301597650 T^{2} + 1197727639041193893 T^{4} - \)\(82\!\cdots\!04\)\( T^{6} + \)\(45\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{10} + \)\(79\!\cdots\!82\)\( T^{12} - \)\(25\!\cdots\!04\)\( T^{14} + \)\(64\!\cdots\!57\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \)
$59$ \( ( 1 + 1806503990 T^{2} + 2218805640623321733 T^{4} + \)\(21\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{10} + \)\(87\!\cdots\!70\)\( T^{12} + \)\(55\!\cdots\!92\)\( T^{14} + \)\(29\!\cdots\!33\)\( T^{16} + \)\(12\!\cdots\!90\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \)
$61$ \( ( 1 - 21151 T + 2606330283 T^{2} - 19051394797770 T^{3} + 2604033901923348357 T^{4} - \)\(93\!\cdots\!13\)\( T^{5} + \)\(21\!\cdots\!57\)\( T^{6} - \)\(13\!\cdots\!70\)\( T^{7} + \)\(15\!\cdots\!83\)\( T^{8} - \)\(10\!\cdots\!51\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \)
$67$ \( ( 1 - 10598201455 T^{2} + 53515912648993384359 T^{4} - \)\(16\!\cdots\!10\)\( T^{6} + \)\(36\!\cdots\!89\)\( T^{8} - \)\(58\!\cdots\!01\)\( T^{10} + \)\(67\!\cdots\!61\)\( T^{12} - \)\(56\!\cdots\!10\)\( T^{14} + \)\(32\!\cdots\!91\)\( T^{16} - \)\(11\!\cdots\!55\)\( T^{18} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \)
$71$ \( ( 1 + 6724308230 T^{2} + 20388189580420161501 T^{4} + \)\(45\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(22\!\cdots\!68\)\( T^{10} + \)\(34\!\cdots\!26\)\( T^{12} + \)\(48\!\cdots\!64\)\( T^{14} + \)\(70\!\cdots\!01\)\( T^{16} + \)\(75\!\cdots\!30\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 - 21355 T + 2385405975 T^{2} - 91039324043106 T^{3} + 7775807385428270205 T^{4} - \)\(26\!\cdots\!29\)\( T^{5} + \)\(16\!\cdots\!65\)\( T^{6} - \)\(39\!\cdots\!94\)\( T^{7} + \)\(21\!\cdots\!75\)\( T^{8} - \)\(39\!\cdots\!55\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{4} \)
$79$ \( ( 1 - 13863408967 T^{2} + 96370465581362357535 T^{4} - \)\(46\!\cdots\!74\)\( T^{6} + \)\(17\!\cdots\!65\)\( T^{8} - \)\(59\!\cdots\!89\)\( T^{10} + \)\(16\!\cdots\!65\)\( T^{12} - \)\(41\!\cdots\!74\)\( T^{14} + \)\(81\!\cdots\!35\)\( T^{16} - \)\(11\!\cdots\!67\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( ( 1 + 15102812222 T^{2} + \)\(16\!\cdots\!49\)\( T^{4} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(64\!\cdots\!90\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!10\)\( T^{12} + \)\(27\!\cdots\!68\)\( T^{14} + \)\(60\!\cdots\!01\)\( T^{16} + \)\(87\!\cdots\!22\)\( T^{18} + \)\(89\!\cdots\!49\)\( T^{20} )^{2} \)
$89$ \( ( 1 - 46714027858 T^{2} + \)\(10\!\cdots\!53\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{8} - \)\(83\!\cdots\!68\)\( T^{10} + \)\(39\!\cdots\!82\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!53\)\( T^{16} - \)\(44\!\cdots\!58\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 + 54977 T + 25395595455 T^{2} + 1194992001523350 T^{3} + \)\(29\!\cdots\!17\)\( T^{4} + \)\(12\!\cdots\!15\)\( T^{5} + \)\(24\!\cdots\!69\)\( T^{6} + \)\(88\!\cdots\!50\)\( T^{7} + \)\(16\!\cdots\!15\)\( T^{8} + \)\(29\!\cdots\!77\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} )^{4} \)
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