Properties

Label 387.5.b.c
Level 387
Weight 5
Character orbit 387.b
Analytic conductor 40.004
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 387.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.0041757134\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -8 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{9} ) q^{5} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{10} ) q^{7} + ( -8 \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -8 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{9} ) q^{5} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{10} ) q^{7} + ( -8 \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{8} + ( 17 - \beta_{2} - 3 \beta_{7} + 3 \beta_{8} ) q^{10} + ( 14 + 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{11} + ( -19 + 4 \beta_{2} + 4 \beta_{3} - 3 \beta_{5} - 4 \beta_{7} + 2 \beta_{8} ) q^{13} + ( -65 + 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} - \beta_{8} ) q^{14} + ( 96 - 6 \beta_{2} + \beta_{3} + 3 \beta_{5} - 7 \beta_{7} - 4 \beta_{8} ) q^{16} + ( -55 - 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{5} - 6 \beta_{7} - 2 \beta_{8} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{4} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{19} + ( 14 \beta_{1} - 11 \beta_{4} - \beta_{6} + 4 \beta_{9} + 3 \beta_{10} ) q^{20} + ( -35 \beta_{1} + 17 \beta_{4} + 4 \beta_{6} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{22} + ( -120 - 14 \beta_{2} - 15 \beta_{3} - 2 \beta_{5} - 8 \beta_{7} + 9 \beta_{8} ) q^{23} + ( -15 - 13 \beta_{2} - \beta_{3} + 10 \beta_{7} - 26 \beta_{8} ) q^{25} + ( -78 \beta_{1} + 40 \beta_{4} + 5 \beta_{6} - 8 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{26} + ( -74 \beta_{1} + 15 \beta_{4} + 7 \beta_{6} + 13 \beta_{9} + 5 \beta_{10} + 3 \beta_{11} ) q^{28} + ( 11 \beta_{1} - 13 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{29} + ( 492 + 2 \beta_{2} - 32 \beta_{3} - 3 \beta_{5} - 25 \beta_{7} - \beta_{8} ) q^{31} + ( 108 \beta_{1} - 29 \beta_{4} - 4 \beta_{6} - 5 \beta_{9} + 9 \beta_{10} - 3 \beta_{11} ) q^{32} + ( 21 \beta_{1} + 22 \beta_{4} - 8 \beta_{6} + \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{34} + ( -72 - 46 \beta_{2} + 11 \beta_{3} + 15 \beta_{7} + 48 \beta_{8} ) q^{35} + ( -30 \beta_{1} + 33 \beta_{4} - 8 \beta_{6} + 30 \beta_{9} - 15 \beta_{10} + 4 \beta_{11} ) q^{37} + ( -104 + 3 \beta_{2} - 15 \beta_{3} - 33 \beta_{5} - 18 \beta_{7} - 10 \beta_{8} ) q^{38} + ( -208 + 15 \beta_{2} + 11 \beta_{3} - 15 \beta_{5} - 44 \beta_{7} + 58 \beta_{8} ) q^{40} + ( -415 + 19 \beta_{2} - 57 \beta_{3} + 16 \beta_{5} + 34 \beta_{7} - 26 \beta_{8} ) q^{41} + ( -74 - 32 \beta_{1} - 11 \beta_{2} + 20 \beta_{3} + 61 \beta_{4} + 15 \beta_{5} + 8 \beta_{6} - 26 \beta_{7} + 49 \beta_{8} + 17 \beta_{9} - 13 \beta_{10} - 3 \beta_{11} ) q^{43} + ( 1302 - 43 \beta_{2} + 19 \beta_{3} + 37 \beta_{5} - 29 \beta_{7} + 6 \beta_{8} ) q^{44} + ( 108 \beta_{1} + 8 \beta_{4} - 11 \beta_{6} - 49 \beta_{9} + 21 \beta_{10} + 2 \beta_{11} ) q^{46} + ( 492 - 51 \beta_{2} + 21 \beta_{3} + 18 \beta_{5} - 24 \beta_{7} + 72 \beta_{8} ) q^{47} + ( -693 - 96 \beta_{2} - 33 \beta_{3} - 24 \beta_{5} + 21 \beta_{7} + 42 \beta_{8} ) q^{49} + ( 284 \beta_{1} + 38 \beta_{4} - 29 \beta_{6} + 71 \beta_{9} - 9 \beta_{10} ) q^{50} + ( 2084 - 87 \beta_{2} + 69 \beta_{3} + 93 \beta_{5} - 57 \beta_{7} + 36 \beta_{8} ) q^{52} + ( -95 + 100 \beta_{2} - 41 \beta_{3} - 15 \beta_{5} - 75 \beta_{7} - 28 \beta_{8} ) q^{53} + ( -88 \beta_{1} - 64 \beta_{4} - 14 \beta_{6} + 76 \beta_{9} + \beta_{10} ) q^{55} + ( 890 - 130 \beta_{2} + 114 \beta_{3} + 62 \beta_{5} + 2 \beta_{7} + 22 \beta_{8} ) q^{56} + ( -413 + 63 \beta_{2} + \beta_{3} + 75 \beta_{5} + 56 \beta_{7} + 17 \beta_{8} ) q^{58} + ( -1182 + 52 \beta_{2} + 77 \beta_{3} + 2 \beta_{5} - 31 \beta_{7} + 60 \beta_{8} ) q^{59} + ( 289 \beta_{1} + 54 \beta_{4} - 38 \beta_{6} + 88 \beta_{9} + 44 \beta_{10} - 3 \beta_{11} ) q^{61} + ( 600 \beta_{1} + 93 \beta_{4} - 21 \beta_{6} - 80 \beta_{9} + 54 \beta_{10} + 3 \beta_{11} ) q^{62} + ( -1320 + 50 \beta_{2} + 63 \beta_{3} - 87 \beta_{5} - 69 \beta_{7} - 96 \beta_{8} ) q^{64} + ( -231 \beta_{1} - 198 \beta_{4} - 30 \beta_{6} + 74 \beta_{9} + 3 \beta_{11} ) q^{65} + ( -52 - 57 \beta_{2} - 193 \beta_{3} - 45 \beta_{5} - 5 \beta_{7} - 59 \beta_{8} ) q^{67} + ( -1274 + 80 \beta_{2} + 123 \beta_{3} + 65 \beta_{5} - 34 \beta_{7} + 56 \beta_{8} ) q^{68} + ( 353 \beta_{1} - 195 \beta_{4} + 17 \beta_{6} - 55 \beta_{9} - 26 \beta_{10} ) q^{70} + ( -352 \beta_{1} + 90 \beta_{4} - 12 \beta_{6} - 162 \beta_{9} - 30 \beta_{10} + 12 \beta_{11} ) q^{71} + ( 473 \beta_{1} + 230 \beta_{4} + 14 \beta_{6} - 122 \beta_{9} + 46 \beta_{10} + \beta_{11} ) q^{73} + ( 698 + 91 \beta_{2} - 76 \beta_{3} + 164 \beta_{5} - 43 \beta_{7} + 184 \beta_{8} ) q^{74} + ( 50 \beta_{1} + 502 \beta_{4} + 8 \beta_{6} - 15 \beta_{9} - 32 \beta_{10} + 17 \beta_{11} ) q^{76} + ( -425 \beta_{1} - 98 \beta_{4} + 32 \beta_{6} + 4 \beta_{9} + 42 \beta_{10} + 15 \beta_{11} ) q^{77} + ( 2042 - 191 \beta_{2} + 284 \beta_{3} + 3 \beta_{5} + \beta_{7} + 190 \beta_{8} ) q^{79} + ( -334 \beta_{1} - 88 \beta_{4} + 28 \beta_{6} - 129 \beta_{9} + 66 \beta_{10} + 15 \beta_{11} ) q^{80} + ( -684 \beta_{1} - 18 \beta_{4} + 11 \beta_{6} + 63 \beta_{9} + 39 \beta_{10} - 16 \beta_{11} ) q^{82} + ( 628 + 35 \beta_{2} + 42 \beta_{3} - 121 \beta_{5} - 202 \beta_{7} + 203 \beta_{8} ) q^{83} + ( -109 \beta_{1} - 377 \beta_{4} + 14 \beta_{6} - 177 \beta_{9} + 13 \beta_{10} + 14 \beta_{11} ) q^{85} + ( 1350 - 74 \beta_{1} - 184 \beta_{2} - 29 \beta_{3} - 426 \beta_{4} - 11 \beta_{5} - 3 \beta_{6} - 173 \beta_{7} + 30 \beta_{8} - 130 \beta_{9} + 21 \beta_{10} - 15 \beta_{11} ) q^{86} + ( 1460 \beta_{1} - 485 \beta_{4} - 39 \beta_{6} - 99 \beta_{9} + 31 \beta_{10} - 21 \beta_{11} ) q^{88} + ( -131 \beta_{1} - 206 \beta_{4} + 20 \beta_{6} + 190 \beta_{9} - 30 \beta_{10} + 15 \beta_{11} ) q^{89} + ( -1190 \beta_{1} - 378 \beta_{4} + 56 \beta_{6} + 4 \beta_{9} + 5 \beta_{10} + 34 \beta_{11} ) q^{91} + ( -4096 - 20 \beta_{2} - 80 \beta_{3} + 6 \beta_{5} + 177 \beta_{7} + 106 \beta_{8} ) q^{92} + ( 1005 \beta_{1} - 600 \beta_{4} - 21 \beta_{6} - 171 \beta_{9} + 21 \beta_{10} - 18 \beta_{11} ) q^{94} + ( -16 + 75 \beta_{2} + 46 \beta_{3} + 20 \beta_{5} - 121 \beta_{7} - 98 \beta_{8} ) q^{95} + ( -489 + 311 \beta_{2} - 18 \beta_{3} + 48 \beta_{5} - 165 \beta_{7} + 6 \beta_{8} ) q^{97} + ( 606 \beta_{1} + 195 \beta_{4} - 9 \beta_{6} - 75 \beta_{9} - 12 \beta_{10} + 24 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 92q^{4} + O(q^{10}) \) \( 12q - 92q^{4} + 182q^{10} + 180q^{11} - 216q^{13} - 732q^{14} + 1076q^{16} - 678q^{17} - 1566q^{23} - 174q^{25} + 5710q^{31} - 936q^{35} - 1242q^{38} - 2618q^{40} - 4878q^{41} - 1108q^{43} + 15168q^{44} + 5526q^{47} - 8544q^{49} + 24084q^{52} - 1212q^{53} + 10152q^{56} - 4666q^{58} - 14016q^{59} - 15580q^{64} - 1088q^{67} - 15186q^{68} + 7674q^{74} + 24302q^{79} + 7032q^{83} + 14412q^{86} - 48354q^{92} - 606q^{95} - 5842q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 142 x^{10} + 7173 x^{8} + 157368 x^{6} + 1510016 x^{4} + 5098688 x^{2} + 90352\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 24 \)
\(\beta_{3}\)\(=\)\((\)\( -1013 \nu^{10} - 51724 \nu^{8} + 3751835 \nu^{6} + 249791190 \nu^{4} + 3190508236 \nu^{2} + 4858167448 \)\()/ 218680368 \)
\(\beta_{4}\)\(=\)\((\)\( 7373 \nu^{11} + 1060068 \nu^{9} + 54814785 \nu^{7} + 1256312182 \nu^{5} + 12979904372 \nu^{3} + 48750171768 \nu \)\()/ 437360736 \)
\(\beta_{5}\)\(=\)\((\)\( -8029 \nu^{10} - 1003616 \nu^{8} - 42059193 \nu^{6} - 669406178 \nu^{4} - 3108074228 \nu^{2} + 2072740888 \)\()/ 218680368 \)
\(\beta_{6}\)\(=\)\((\)\( -7373 \nu^{11} - 1060068 \nu^{9} - 54814785 \nu^{7} - 1256312182 \nu^{5} - 12761224004 \nu^{3} - 40002957048 \nu \)\()/ 218680368 \)
\(\beta_{7}\)\(=\)\((\)\( -3124 \nu^{10} - 397873 \nu^{8} - 16704679 \nu^{6} - 262452874 \nu^{4} - 1457233216 \nu^{2} - 1347066320 \)\()/54670092\)
\(\beta_{8}\)\(=\)\((\)\( 15593 \nu^{10} + 2019468 \nu^{8} + 86326317 \nu^{6} + 1342893190 \nu^{4} + 5715018932 \nu^{2} - 4421146200 \)\()/ 218680368 \)
\(\beta_{9}\)\(=\)\((\)\( -10113 \nu^{11} - 1379868 \nu^{9} - 65318629 \nu^{7} - 1285172518 \nu^{5} - 10485382436 \nu^{3} - 28475128152 \nu \)\()/ 145786912 \)
\(\beta_{10}\)\(=\)\((\)\( -43059 \nu^{11} - 6156292 \nu^{9} - 313089127 \nu^{7} - 6867724298 \nu^{5} - 63796972524 \nu^{3} - 193079423624 \nu \)\()/ 437360736 \)
\(\beta_{11}\)\(=\)\((\)\( -130223 \nu^{11} - 18990012 \nu^{9} - 996377731 \nu^{7} - 23117282490 \nu^{5} - 239736846012 \nu^{3} - 897659100328 \nu \)\()/ 437360736 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 24\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 2 \beta_{4} - 40 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-4 \beta_{8} - 7 \beta_{7} + 3 \beta_{5} + \beta_{3} - 54 \beta_{2} + 992\)
\(\nu^{5}\)\(=\)\(-3 \beta_{11} + 9 \beta_{10} - 5 \beta_{9} - 68 \beta_{6} - 157 \beta_{4} + 1900 \beta_{1}\)
\(\nu^{6}\)\(=\)\(224 \beta_{8} + 491 \beta_{7} - 327 \beta_{5} - 17 \beta_{3} + 2834 \beta_{2} - 47912\)
\(\nu^{7}\)\(=\)\(327 \beta_{11} - 801 \beta_{10} + 517 \beta_{9} + 3876 \beta_{6} + 10977 \beta_{4} - 95996 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-9240 \beta_{8} - 28143 \beta_{7} + 25791 \beta_{5} + 513 \beta_{3} - 149594 \beta_{2} + 2454152\)
\(\nu^{9}\)\(=\)\(-25791 \beta_{11} + 53421 \beta_{10} - 37293 \beta_{9} - 212768 \beta_{6} - 722533 \beta_{4} + 4980124 \beta_{1}\)
\(\nu^{10}\)\(=\)\(315080 \beta_{8} + 1529399 \beta_{7} - 1788243 \beta_{5} - 58445 \beta_{3} + 7968494 \beta_{2} - 128941408\)
\(\nu^{11}\)\(=\)\(1788243 \beta_{11} - 3259197 \beta_{10} + 2370193 \beta_{9} + 11601216 \beta_{6} + 45565005 \beta_{4} - 262283620 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
7.49282i
6.72223i
4.43775i
3.65497i
2.75662i
0.133471i
0.133471i
2.75662i
3.65497i
4.43775i
6.72223i
7.49282i
7.49282i 0 −40.1424 9.40180i 0 53.5326i 180.895i 0 70.4460
343.2 6.72223i 0 −29.1884 1.48242i 0 13.7963i 88.6554i 0 9.96518
343.3 4.43775i 0 −3.69363 22.3554i 0 51.9837i 54.6126i 0 −99.2077
343.4 3.65497i 0 2.64117 45.6695i 0 34.3337i 68.1330i 0 166.921
343.5 2.75662i 0 8.40105 21.9831i 0 63.3272i 67.2644i 0 −60.5989
343.6 0.133471i 0 15.9822 26.0324i 0 87.9232i 4.26869i 0 3.47456
343.7 0.133471i 0 15.9822 26.0324i 0 87.9232i 4.26869i 0 3.47456
343.8 2.75662i 0 8.40105 21.9831i 0 63.3272i 67.2644i 0 −60.5989
343.9 3.65497i 0 2.64117 45.6695i 0 34.3337i 68.1330i 0 166.921
343.10 4.43775i 0 −3.69363 22.3554i 0 51.9837i 54.6126i 0 −99.2077
343.11 6.72223i 0 −29.1884 1.48242i 0 13.7963i 88.6554i 0 9.96518
343.12 7.49282i 0 −40.1424 9.40180i 0 53.5326i 180.895i 0 70.4460
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.5.b.c 12
3.b odd 2 1 43.5.b.b 12
12.b even 2 1 688.5.b.d 12
43.b odd 2 1 inner 387.5.b.c 12
129.d even 2 1 43.5.b.b 12
516.h odd 2 1 688.5.b.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.b.b 12 3.b odd 2 1
43.5.b.b 12 129.d even 2 1
387.5.b.c 12 1.a even 1 1 trivial
387.5.b.c 12 43.b odd 2 1 inner
688.5.b.d 12 12.b even 2 1
688.5.b.d 12 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(387, [\chi])\):

\( T_{2}^{12} + 142 T_{2}^{10} + 7173 T_{2}^{8} + 157368 T_{2}^{6} + 1510016 T_{2}^{4} + 5098688 T_{2}^{2} + 90352 \)
\( T_{11}^{6} - 90 T_{11}^{5} - 18220 T_{11}^{4} + 2312718 T_{11}^{3} - 63695785 T_{11}^{2} + 475717044 T_{11} - 473184052 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 50 T^{2} + 1349 T^{4} - 26056 T^{6} + 463232 T^{8} - 8751936 T^{10} + 150915312 T^{12} - 2240495616 T^{14} + 30358372352 T^{16} - 437147140096 T^{18} + 5793910882304 T^{20} - 54975581388800 T^{22} + 281474976710656 T^{24} \)
$3$ 1
$5$ \( 1 - 3663 T^{2} + 6511015 T^{4} - 7365160765 T^{6} + 5973224140031 T^{8} - 3902236317085676 T^{10} + 2399998678381934162 T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(91\!\cdots\!75\)\( T^{16} - \)\(43\!\cdots\!25\)\( T^{18} + \)\(15\!\cdots\!75\)\( T^{20} - \)\(33\!\cdots\!75\)\( T^{22} + \)\(35\!\cdots\!25\)\( T^{24} \)
$7$ \( 1 - 10134 T^{2} + 60060006 T^{4} - 249160434566 T^{6} + 812915964019071 T^{8} - 2246049549704995236 T^{10} + \)\(56\!\cdots\!48\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(27\!\cdots\!71\)\( T^{16} - \)\(47\!\cdots\!66\)\( T^{18} + \)\(66\!\cdots\!06\)\( T^{20} - \)\(64\!\cdots\!34\)\( T^{22} + \)\(36\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 - 90 T + 69626 T^{2} - 4275732 T^{3} + 2084651350 T^{4} - 90865763142 T^{5} + 37469241503078 T^{6} - 1330365638162022 T^{7} + 446863530661139350 T^{8} - 13419078640054034772 T^{9} + \)\(31\!\cdots\!86\)\( T^{10} - \)\(60\!\cdots\!90\)\( T^{11} + \)\(98\!\cdots\!41\)\( T^{12} )^{2} \)
$13$ \( ( 1 + 108 T + 78130 T^{2} + 14124934 T^{3} + 3639591614 T^{4} + 705010114760 T^{5} + 121575713991590 T^{6} + 20135793887660360 T^{7} + 2968926691433773694 T^{8} + \)\(32\!\cdots\!54\)\( T^{9} + \)\(51\!\cdots\!30\)\( T^{10} + \)\(20\!\cdots\!08\)\( T^{11} + \)\(54\!\cdots\!61\)\( T^{12} )^{2} \)
$17$ \( ( 1 + 339 T + 345843 T^{2} + 103717833 T^{3} + 57770879546 T^{4} + 15179725016055 T^{5} + 6016834256618483 T^{6} + 1267825813065929655 T^{7} + \)\(40\!\cdots\!86\)\( T^{8} + \)\(60\!\cdots\!13\)\( T^{9} + \)\(16\!\cdots\!83\)\( T^{10} + \)\(13\!\cdots\!39\)\( T^{11} + \)\(33\!\cdots\!21\)\( T^{12} )^{2} \)
$19$ \( 1 - 1175657 T^{2} + 671010023409 T^{4} - 244979577528319381 T^{6} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!98\)\( T^{12} - \)\(20\!\cdots\!10\)\( T^{14} + \)\(18\!\cdots\!03\)\( T^{16} - \)\(12\!\cdots\!01\)\( T^{18} + \)\(55\!\cdots\!49\)\( T^{20} - \)\(16\!\cdots\!57\)\( T^{22} + \)\(23\!\cdots\!41\)\( T^{24} \)
$23$ \( ( 1 + 783 T + 1163077 T^{2} + 843582057 T^{3} + 713091466102 T^{4} + 397049506101027 T^{5} + 259574077696956889 T^{6} + \)\(11\!\cdots\!07\)\( T^{7} + \)\(55\!\cdots\!62\)\( T^{8} + \)\(18\!\cdots\!97\)\( T^{9} + \)\(71\!\cdots\!97\)\( T^{10} + \)\(13\!\cdots\!83\)\( T^{11} + \)\(48\!\cdots\!41\)\( T^{12} )^{2} \)
$29$ \( 1 - 5231219 T^{2} + 13592412986871 T^{4} - 23367533349246758977 T^{6} + \)\(29\!\cdots\!35\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!82\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(74\!\cdots\!35\)\( T^{16} - \)\(29\!\cdots\!37\)\( T^{18} + \)\(85\!\cdots\!11\)\( T^{20} - \)\(16\!\cdots\!19\)\( T^{22} + \)\(15\!\cdots\!61\)\( T^{24} \)
$31$ \( ( 1 - 2855 T + 6518969 T^{2} - 8600011177 T^{3} + 10228398490358 T^{4} - 8682474860253963 T^{5} + 8850492205571119245 T^{6} - \)\(80\!\cdots\!23\)\( T^{7} + \)\(87\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!97\)\( T^{9} + \)\(47\!\cdots\!89\)\( T^{10} - \)\(19\!\cdots\!55\)\( T^{11} + \)\(62\!\cdots\!21\)\( T^{12} )^{2} \)
$37$ \( 1 - 6978249 T^{2} + 25643197308697 T^{4} - 70260487375765593853 T^{6} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(39\!\cdots\!70\)\( T^{10} + \)\(80\!\cdots\!70\)\( T^{12} - \)\(13\!\cdots\!70\)\( T^{14} + \)\(21\!\cdots\!19\)\( T^{16} - \)\(30\!\cdots\!33\)\( T^{18} + \)\(39\!\cdots\!57\)\( T^{20} - \)\(37\!\cdots\!49\)\( T^{22} + \)\(18\!\cdots\!21\)\( T^{24} \)
$41$ \( ( 1 + 2439 T + 13822603 T^{2} + 19943096547 T^{3} + 69051390319336 T^{4} + 65595635499999207 T^{5} + \)\(21\!\cdots\!21\)\( T^{6} + \)\(18\!\cdots\!27\)\( T^{7} + \)\(55\!\cdots\!56\)\( T^{8} + \)\(44\!\cdots\!07\)\( T^{9} + \)\(88\!\cdots\!23\)\( T^{10} + \)\(43\!\cdots\!39\)\( T^{11} + \)\(50\!\cdots\!61\)\( T^{12} )^{2} \)
$43$ \( 1 + 1108 T + 1897246 T^{2} - 4681128092 T^{3} + 681017924063 T^{4} - 846563366867960 T^{5} + 64607141521498794692 T^{6} - \)\(28\!\cdots\!60\)\( T^{7} + \)\(79\!\cdots\!63\)\( T^{8} - \)\(18\!\cdots\!92\)\( T^{9} + \)\(25\!\cdots\!46\)\( T^{10} + \)\(51\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!01\)\( T^{12} \)
$47$ \( ( 1 - 2763 T + 21754149 T^{2} - 55851382377 T^{3} + 234662825346711 T^{4} - 481604898621884868 T^{5} + \)\(14\!\cdots\!34\)\( T^{6} - \)\(23\!\cdots\!08\)\( T^{7} + \)\(55\!\cdots\!71\)\( T^{8} - \)\(64\!\cdots\!57\)\( T^{9} + \)\(12\!\cdots\!29\)\( T^{10} - \)\(76\!\cdots\!63\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \)
$53$ \( ( 1 + 606 T + 25718024 T^{2} + 24674633094 T^{3} + 346581142619392 T^{4} + 336085601563787574 T^{5} + \)\(32\!\cdots\!82\)\( T^{6} + \)\(26\!\cdots\!94\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(12\!\cdots\!54\)\( T^{9} + \)\(99\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!06\)\( T^{11} + \)\(24\!\cdots\!81\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 7008 T + 80137258 T^{2} + 394451068512 T^{3} + 2532059909380847 T^{4} + 9247600606403540352 T^{5} + \)\(41\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!72\)\( T^{7} + \)\(37\!\cdots\!87\)\( T^{8} + \)\(70\!\cdots\!72\)\( T^{9} + \)\(17\!\cdots\!78\)\( T^{10} + \)\(18\!\cdots\!08\)\( T^{11} + \)\(31\!\cdots\!61\)\( T^{12} )^{2} \)
$61$ \( 1 - 32665486 T^{2} + 1118892201421590 T^{4} - \)\(24\!\cdots\!14\)\( T^{6} + \)\(52\!\cdots\!07\)\( T^{8} - \)\(83\!\cdots\!88\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{12} - \)\(16\!\cdots\!28\)\( T^{14} + \)\(19\!\cdots\!27\)\( T^{16} - \)\(17\!\cdots\!74\)\( T^{18} + \)\(15\!\cdots\!90\)\( T^{20} - \)\(84\!\cdots\!86\)\( T^{22} + \)\(49\!\cdots\!81\)\( T^{24} \)
$67$ \( ( 1 + 544 T + 66684038 T^{2} + 18315837698 T^{3} + 2452178722840730 T^{4} + 587634295199545788 T^{5} + \)\(60\!\cdots\!70\)\( T^{6} + \)\(11\!\cdots\!48\)\( T^{7} + \)\(99\!\cdots\!30\)\( T^{8} + \)\(14\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!78\)\( T^{10} + \)\(18\!\cdots\!44\)\( T^{11} + \)\(66\!\cdots\!21\)\( T^{12} )^{2} \)
$71$ \( 1 - 101562020 T^{2} + 5784818387848562 T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(47\!\cdots\!67\)\( T^{8} - \)\(75\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(48\!\cdots\!24\)\( T^{14} + \)\(20\!\cdots\!07\)\( T^{16} - \)\(54\!\cdots\!84\)\( T^{18} + \)\(10\!\cdots\!42\)\( T^{20} - \)\(11\!\cdots\!20\)\( T^{22} + \)\(72\!\cdots\!61\)\( T^{24} \)
$73$ \( 1 - 193491698 T^{2} + 19000747654703030 T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!56\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(18\!\cdots\!36\)\( T^{14} + \)\(39\!\cdots\!75\)\( T^{16} - \)\(65\!\cdots\!38\)\( T^{18} + \)\(80\!\cdots\!30\)\( T^{20} - \)\(66\!\cdots\!98\)\( T^{22} + \)\(27\!\cdots\!81\)\( T^{24} \)
$79$ \( ( 1 - 12151 T + 155715789 T^{2} - 800821687889 T^{3} + 4128429862975343 T^{4} + 5685482156720652576 T^{5} - \)\(30\!\cdots\!42\)\( T^{6} + \)\(22\!\cdots\!56\)\( T^{7} + \)\(62\!\cdots\!23\)\( T^{8} - \)\(47\!\cdots\!49\)\( T^{9} + \)\(35\!\cdots\!69\)\( T^{10} - \)\(10\!\cdots\!51\)\( T^{11} + \)\(34\!\cdots\!81\)\( T^{12} )^{2} \)
$83$ \( ( 1 - 3516 T + 117162268 T^{2} - 863865543048 T^{3} + 7914887456622244 T^{4} - 75598926847021341108 T^{5} + \)\(40\!\cdots\!22\)\( T^{6} - \)\(35\!\cdots\!68\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} - \)\(92\!\cdots\!28\)\( T^{9} + \)\(59\!\cdots\!08\)\( T^{10} - \)\(84\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!21\)\( T^{12} )^{2} \)
$89$ \( 1 - 474457834 T^{2} + 111655192487117862 T^{4} - \)\(17\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(68\!\cdots\!52\)\( T^{14} + \)\(30\!\cdots\!71\)\( T^{16} - \)\(10\!\cdots\!06\)\( T^{18} + \)\(26\!\cdots\!02\)\( T^{20} - \)\(44\!\cdots\!34\)\( T^{22} + \)\(37\!\cdots\!81\)\( T^{24} \)
$97$ \( ( 1 + 2921 T + 384124299 T^{2} + 837647682043 T^{3} + 69794294829141482 T^{4} + \)\(11\!\cdots\!33\)\( T^{5} + \)\(76\!\cdots\!95\)\( T^{6} + \)\(10\!\cdots\!73\)\( T^{7} + \)\(54\!\cdots\!02\)\( T^{8} + \)\(58\!\cdots\!63\)\( T^{9} + \)\(23\!\cdots\!79\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{11} + \)\(48\!\cdots\!81\)\( T^{12} )^{2} \)
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