Properties

Label 15.5.c.a
Level 15
Weight 5
Character orbit 15.c
Analytic conductor 1.551
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( -8 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{3} q^{5} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6} + ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8} + ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( -8 + \beta_{2} + \beta_{4} ) q^{4} + \beta_{3} q^{5} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6} + ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8} + ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9} + ( 7 - 5 \beta_{2} - \beta_{4} + \beta_{5} ) q^{10} + ( 4 - 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{5} ) q^{11} + ( -73 + 3 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{12} + ( -70 + 2 \beta_{2} + 2 \beta_{4} ) q^{13} + ( -2 + 36 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{14} + ( 9 - 15 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} ) q^{15} + ( 126 - 31 \beta_{2} + \beta_{4} + 8 \beta_{5} ) q^{16} + ( -8 - 20 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} - 8 \beta_{5} ) q^{17} + ( 194 + 63 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{18} + ( -34 + 26 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} ) q^{19} + ( 5 + 35 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + 5 \beta_{5} ) q^{20} + ( -150 + 21 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{21} + ( 64 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{22} + ( 7 - 72 \beta_{1} + 14 \beta_{2} - 48 \beta_{3} + 7 \beta_{5} ) q^{23} + ( -132 - 144 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} ) q^{24} -125 q^{25} + ( -2 - 104 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} ) q^{26} + ( -70 + 18 \beta_{1} - 29 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} - 5 \beta_{5} ) q^{27} + ( -622 + 22 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} ) q^{28} + ( -2 + 146 \beta_{1} - 4 \beta_{2} - 48 \beta_{3} - 2 \beta_{5} ) q^{29} + ( 365 - 30 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 15 \beta_{4} ) q^{30} + ( 636 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{31} + ( 15 + 53 \beta_{1} + 30 \beta_{2} - 42 \beta_{3} + 15 \beta_{5} ) q^{32} + ( 744 + 102 \beta_{1} + 12 \beta_{2} + 42 \beta_{3} + 36 \beta_{4} + 10 \beta_{5} ) q^{33} + ( 490 - 114 \beta_{2} - 26 \beta_{4} + 22 \beta_{5} ) q^{34} + ( -15 + 20 \beta_{1} - 30 \beta_{2} + 20 \beta_{3} - 15 \beta_{5} ) q^{35} + ( -1252 + 126 \beta_{1} + 73 \beta_{2} - 42 \beta_{3} + 21 \beta_{4} - 26 \beta_{5} ) q^{36} + ( 306 - 58 \beta_{2} + 54 \beta_{4} + 28 \beta_{5} ) q^{37} + ( -26 - 134 \beta_{1} - 52 \beta_{2} + 120 \beta_{3} - 26 \beta_{5} ) q^{38} + ( -200 + 6 \beta_{1} + 74 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} ) q^{39} + ( -764 + 35 \beta_{2} + 27 \beta_{4} - 2 \beta_{5} ) q^{40} + ( -14 + 182 \beta_{1} - 28 \beta_{2} - 90 \beta_{3} - 14 \beta_{5} ) q^{41} + ( -318 - 36 \beta_{1} - 6 \beta_{2} - 126 \beta_{3} + 42 \beta_{5} ) q^{42} + ( -1277 - 168 \beta_{2} - 52 \beta_{4} + 29 \beta_{5} ) q^{43} + ( 34 - 22 \beta_{1} + 68 \beta_{2} + 132 \beta_{3} + 34 \beta_{5} ) q^{44} + ( 310 - 45 \beta_{1} + 5 \beta_{2} + 24 \beta_{3} + 15 \beta_{4} - 25 \beta_{5} ) q^{45} + ( 1420 + 224 \beta_{2} - 24 \beta_{4} - 62 \beta_{5} ) q^{46} + ( 31 - 212 \beta_{1} + 62 \beta_{2} + 72 \beta_{3} + 31 \beta_{5} ) q^{47} + ( 2405 - 213 \beta_{1} - 92 \beta_{2} - 78 \beta_{3} - 66 \beta_{4} - 27 \beta_{5} ) q^{48} + ( -187 + 126 \beta_{2} + 14 \beta_{4} - 28 \beta_{5} ) q^{49} -125 \beta_{1} q^{50} + ( -1434 - 366 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} - 54 \beta_{4} - 32 \beta_{5} ) q^{51} + ( 1452 - 148 \beta_{2} - 84 \beta_{4} + 16 \beta_{5} ) q^{52} + ( 48 + 340 \beta_{1} + 96 \beta_{2} + 6 \beta_{3} + 48 \beta_{5} ) q^{53} + ( -92 + 333 \beta_{1} - 214 \beta_{2} - 129 \beta_{3} - 30 \beta_{4} + 77 \beta_{5} ) q^{54} + ( -210 + 50 \beta_{2} - 70 \beta_{4} - 30 \beta_{5} ) q^{55} + ( -54 - 78 \beta_{1} - 108 \beta_{2} + 192 \beta_{3} - 54 \beta_{5} ) q^{56} + ( -1922 + 168 \beta_{1} + 14 \beta_{2} + 42 \beta_{3} + 66 \beta_{4} + 12 \beta_{5} ) q^{57} + ( -3848 + 370 \beta_{2} + 194 \beta_{4} - 44 \beta_{5} ) q^{58} + ( -54 - 758 \beta_{1} - 108 \beta_{2} - 96 \beta_{3} - 54 \beta_{5} ) q^{59} + ( 858 + 360 \beta_{1} + 15 \beta_{2} - 60 \beta_{3} + 21 \beta_{4} + 34 \beta_{5} ) q^{60} + ( 1048 - 150 \beta_{2} + 122 \beta_{4} + 68 \beta_{5} ) q^{61} + ( -30 + 582 \beta_{1} - 60 \beta_{2} + 132 \beta_{3} - 30 \beta_{5} ) q^{62} + ( 2547 - 342 \beta_{1} + 180 \beta_{2} - 72 \beta_{3} + 18 \beta_{4} + 63 \beta_{5} ) q^{63} + ( 510 - 113 \beta_{2} + 111 \beta_{4} + 56 \beta_{5} ) q^{64} + ( 10 + 70 \beta_{1} + 20 \beta_{2} - 70 \beta_{3} + 10 \beta_{5} ) q^{65} + ( -2150 + 210 \beta_{1} - 100 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{66} + ( 2299 + 32 \beta_{2} - 140 \beta_{4} - 43 \beta_{5} ) q^{67} + ( -14 + 854 \beta_{1} - 28 \beta_{2} - 456 \beta_{3} - 14 \beta_{5} ) q^{68} + ( 870 + 693 \beta_{1} + 21 \beta_{2} + 279 \beta_{3} - 81 \beta_{4} - 147 \beta_{5} ) q^{69} + ( -400 - 200 \beta_{2} + 50 \beta_{5} ) q^{70} + ( 78 + 236 \beta_{1} + 156 \beta_{2} + 120 \beta_{3} + 78 \beta_{5} ) q^{71} + ( -339 - 783 \beta_{1} + 150 \beta_{2} + 342 \beta_{3} + 72 \beta_{4} - 147 \beta_{5} ) q^{72} + ( 6 - 308 \beta_{2} - 196 \beta_{4} + 28 \beta_{5} ) q^{73} + ( 58 - 304 \beta_{1} + 116 \beta_{2} - 180 \beta_{3} + 58 \beta_{5} ) q^{74} + ( -125 + 125 \beta_{2} ) q^{75} + ( 3408 - 526 \beta_{2} - 222 \beta_{4} + 76 \beta_{5} ) q^{76} + ( -68 - 576 \beta_{1} - 136 \beta_{2} - 516 \beta_{3} - 68 \beta_{5} ) q^{77} + ( -264 - 546 \beta_{1} - 6 \beta_{2} + 294 \beta_{3} + 18 \beta_{4} - 86 \beta_{5} ) q^{78} + ( -2594 + 386 \beta_{2} - 38 \beta_{4} - 106 \beta_{5} ) q^{79} + ( 45 - 685 \beta_{1} + 90 \beta_{2} + 70 \beta_{3} + 45 \beta_{5} ) q^{80} + ( 907 - 828 \beta_{1} + 2 \beta_{2} + 60 \beta_{3} - 30 \beta_{4} + 176 \beta_{5} ) q^{81} + ( -5054 + 520 \beta_{2} + 272 \beta_{4} - 62 \beta_{5} ) q^{82} + ( -21 - 288 \beta_{1} - 42 \beta_{2} + 408 \beta_{3} - 21 \beta_{5} ) q^{83} + ( -2250 + 156 \beta_{1} + 594 \beta_{2} + 66 \beta_{3} + 42 \beta_{4} + 24 \beta_{5} ) q^{84} + ( -498 + 20 \beta_{2} + 164 \beta_{4} + 36 \beta_{5} ) q^{85} + ( 168 - 74 \beta_{1} + 336 \beta_{2} - 834 \beta_{3} + 168 \beta_{5} ) q^{86} + ( -804 + 1104 \beta_{1} - 6 \beta_{2} - 456 \beta_{3} - 162 \beta_{4} + 44 \beta_{5} ) q^{87} + ( 2612 + 70 \beta_{2} - 186 \beta_{4} - 64 \beta_{5} ) q^{88} + ( -156 - 12 \beta_{1} - 312 \beta_{2} + 396 \beta_{3} - 156 \beta_{5} ) q^{89} + ( 1133 - 395 \beta_{2} + 210 \beta_{3} - 69 \beta_{4} + 19 \beta_{5} ) q^{90} + ( -1946 - 64 \beta_{2} + 104 \beta_{4} + 42 \beta_{5} ) q^{91} + ( -112 - 6 \beta_{1} - 224 \beta_{2} + 204 \beta_{3} - 112 \beta_{5} ) q^{92} + ( -1578 + 210 \beta_{1} - 672 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{93} + ( 5716 - 324 \beta_{2} - 284 \beta_{4} + 10 \beta_{5} ) q^{94} + ( -20 + 610 \beta_{1} - 40 \beta_{2} + 8 \beta_{3} - 20 \beta_{5} ) q^{95} + ( 2412 + 1194 \beta_{1} + 45 \beta_{2} - 24 \beta_{3} + 9 \beta_{4} + 14 \beta_{5} ) q^{96} + ( 1822 + 712 \beta_{2} + 216 \beta_{4} - 124 \beta_{5} ) q^{97} + ( -126 - 733 \beta_{1} - 252 \beta_{2} + 588 \beta_{3} - 126 \beta_{5} ) q^{98} + ( 1424 - 342 \beta_{1} - 632 \beta_{2} - 564 \beta_{3} + 300 \beta_{4} + 64 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 8q^{3} - 50q^{4} - 2q^{6} + 76q^{7} + 118q^{9} + O(q^{10}) \) \( 6q + 8q^{3} - 50q^{4} - 2q^{6} + 76q^{7} + 118q^{9} + 50q^{10} - 452q^{12} - 424q^{13} + 50q^{15} + 802q^{16} + 1160q^{18} - 244q^{19} - 876q^{21} + 340q^{22} - 786q^{24} - 750q^{25} - 352q^{27} - 3764q^{28} + 2200q^{30} + 3772q^{31} + 4420q^{33} + 3124q^{34} - 7606q^{36} + 1896q^{37} - 1336q^{39} - 4650q^{40} - 1980q^{42} - 7384q^{43} + 1900q^{45} + 8196q^{46} + 14668q^{48} - 1318q^{49} - 8492q^{51} + 8976q^{52} - 278q^{54} - 1300q^{55} - 11584q^{57} - 23740q^{58} + 5050q^{60} + 6452q^{61} + 14796q^{63} + 3174q^{64} - 12760q^{66} + 13816q^{67} + 5472q^{69} - 2100q^{70} - 2040q^{72} + 596q^{73} - 1000q^{75} + 21348q^{76} - 1400q^{78} - 16124q^{79} + 5086q^{81} - 31240q^{82} - 14736q^{84} - 3100q^{85} - 4900q^{87} + 15660q^{88} + 7550q^{90} - 11632q^{91} - 8184q^{93} + 34924q^{94} + 14354q^{96} + 9756q^{97} + 9680q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 73 x^{4} + 1096 x^{2} + 180\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 71 \nu^{3} - 47 \nu^{2} + 1002 \nu - 114 \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 79 \nu^{3} + 1330 \nu \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 71 \nu^{3} + 95 \nu^{2} - 1002 \nu + 1266 \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} + \nu^{4} + 142 \nu^{3} + 47 \nu^{2} + 2004 \nu + 90 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2} - 24\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + 6 \beta_{3} - 2 \beta_{2} - 41 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(8 \beta_{5} - 47 \beta_{4} - 79 \beta_{2} + 1022\)
\(\nu^{5}\)\(=\)\(79 \beta_{5} - 426 \beta_{3} + 158 \beta_{2} + 1909 \beta_{1} + 79\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
7.20990i
4.56632i
0.407512i
0.407512i
4.56632i
7.20990i
7.20990i 8.77108 + 2.01697i −35.9827 11.1803i 14.5422 63.2386i 23.3388 144.073i 72.8637 + 35.3820i 80.6091
11.2 4.56632i −7.98405 4.15391i −4.85128 11.1803i −18.9681 + 36.4577i 61.6068 50.9086i 46.4900 + 66.3301i −51.0530
11.3 0.407512i 3.21297 + 8.40695i 15.8339 11.1803i 3.42594 1.30932i −46.9457 12.9727i −60.3537 + 54.0225i −4.55613
11.4 0.407512i 3.21297 8.40695i 15.8339 11.1803i 3.42594 + 1.30932i −46.9457 12.9727i −60.3537 54.0225i −4.55613
11.5 4.56632i −7.98405 + 4.15391i −4.85128 11.1803i −18.9681 36.4577i 61.6068 50.9086i 46.4900 66.3301i −51.0530
11.6 7.20990i 8.77108 2.01697i −35.9827 11.1803i 14.5422 + 63.2386i 23.3388 144.073i 72.8637 35.3820i 80.6091
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.c.a 6
3.b odd 2 1 inner 15.5.c.a 6
4.b odd 2 1 240.5.l.d 6
5.b even 2 1 75.5.c.i 6
5.c odd 4 2 75.5.d.d 12
12.b even 2 1 240.5.l.d 6
15.d odd 2 1 75.5.c.i 6
15.e even 4 2 75.5.d.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.c.a 6 1.a even 1 1 trivial
15.5.c.a 6 3.b odd 2 1 inner
75.5.c.i 6 5.b even 2 1
75.5.c.i 6 15.d odd 2 1
75.5.d.d 12 5.c odd 4 2
75.5.d.d 12 15.e even 4 2
240.5.l.d 6 4.b odd 2 1
240.5.l.d 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 23 T^{2} + 264 T^{4} - 4684 T^{6} + 67584 T^{8} - 1507328 T^{10} + 16777216 T^{12} \)
$3$ \( 1 - 8 T - 27 T^{2} + 504 T^{3} - 2187 T^{4} - 52488 T^{5} + 531441 T^{6} \)
$5$ \( ( 1 + 125 T^{2} )^{3} \)
$7$ \( ( 1 - 38 T + 4653 T^{2} - 114976 T^{3} + 11171853 T^{4} - 219062438 T^{5} + 13841287201 T^{6} )^{2} \)
$11$ \( 1 - 34466 T^{2} + 627812895 T^{4} - 9331838759740 T^{6} + 134577269649570495 T^{8} - \)\(15\!\cdots\!26\)\( T^{10} + \)\(98\!\cdots\!41\)\( T^{12} \)
$13$ \( ( 1 + 212 T + 97943 T^{2} + 12289064 T^{3} + 2797350023 T^{4} + 172934912852 T^{5} + 23298085122481 T^{6} )^{2} \)
$17$ \( 1 - 256778 T^{2} + 40830133839 T^{4} - 4057376310317644 T^{6} + \)\(28\!\cdots\!99\)\( T^{8} - \)\(12\!\cdots\!18\)\( T^{10} + \)\(33\!\cdots\!21\)\( T^{12} \)
$19$ \( ( 1 + 122 T + 284939 T^{2} + 25213796 T^{3} + 37133535419 T^{4} + 2071994691002 T^{5} + 2213314919066161 T^{6} )^{2} \)
$23$ \( 1 - 541878 T^{2} - 37441234461 T^{4} + 62508481606281076 T^{6} - \)\(29\!\cdots\!41\)\( T^{8} - \)\(33\!\cdots\!58\)\( T^{10} + \)\(48\!\cdots\!41\)\( T^{12} \)
$29$ \( 1 - 1427666 T^{2} + 445361323935 T^{4} + 117229967968878500 T^{6} + \)\(22\!\cdots\!35\)\( T^{8} - \)\(35\!\cdots\!86\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \)
$31$ \( ( 1 - 1886 T + 3804195 T^{2} - 3654810940 T^{3} + 3513253970595 T^{4} - 1608552496613726 T^{5} + 787662783788549761 T^{6} )^{2} \)
$37$ \( ( 1 - 948 T + 3630423 T^{2} - 3404430056 T^{3} + 6803997200103 T^{4} - 3329830522317108 T^{5} + 6582952005840035281 T^{6} )^{2} \)
$41$ \( 1 - 9622286 T^{2} + 49511941482495 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!95\)\( T^{8} - \)\(61\!\cdots\!26\)\( T^{10} + \)\(50\!\cdots\!61\)\( T^{12} \)
$43$ \( ( 1 + 3692 T + 9012053 T^{2} + 15407946584 T^{3} + 30810415808453 T^{4} + 43152835424902892 T^{5} + 39959630797262576401 T^{6} )^{2} \)
$47$ \( 1 - 19358678 T^{2} + 191449450821219 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(45\!\cdots\!59\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!81\)\( T^{12} \)
$53$ \( 1 - 31471178 T^{2} + 496434526631919 T^{4} - \)\(48\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!59\)\( T^{8} - \)\(12\!\cdots\!38\)\( T^{10} + \)\(24\!\cdots\!81\)\( T^{12} \)
$59$ \( 1 - 20221586 T^{2} + 518070122195295 T^{4} - \)\(58\!\cdots\!40\)\( T^{6} + \)\(76\!\cdots\!95\)\( T^{8} - \)\(43\!\cdots\!26\)\( T^{10} + \)\(31\!\cdots\!61\)\( T^{12} \)
$61$ \( ( 1 - 3226 T + 32346515 T^{2} - 81211143220 T^{3} + 447864703594115 T^{4} - 618447791729228506 T^{5} + \)\(26\!\cdots\!21\)\( T^{6} )^{2} \)
$67$ \( ( 1 - 6908 T + 65136693 T^{2} - 250302748936 T^{3} + 1312577382182853 T^{4} - 2805115516561276028 T^{5} + \)\(81\!\cdots\!61\)\( T^{6} )^{2} \)
$71$ \( 1 - 121682966 T^{2} + 6612859983587535 T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!35\)\( T^{8} - \)\(50\!\cdots\!86\)\( T^{10} + \)\(26\!\cdots\!81\)\( T^{12} \)
$73$ \( ( 1 - 298 T + 50688863 T^{2} - 79767435436 T^{3} + 1439474547489983 T^{4} - 240325107384436138 T^{5} + \)\(22\!\cdots\!21\)\( T^{6} )^{2} \)
$79$ \( ( 1 + 8062 T + 112728699 T^{2} + 533077126316 T^{3} + 4390791957074619 T^{4} + 12230931225466694782 T^{5} + \)\(59\!\cdots\!41\)\( T^{6} )^{2} \)
$83$ \( 1 - 211578198 T^{2} + 21276155528463939 T^{4} - \)\(12\!\cdots\!04\)\( T^{6} + \)\(47\!\cdots\!99\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{12} \)
$89$ \( 1 - 255353766 T^{2} + 32750513998015695 T^{4} - \)\(25\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(39\!\cdots\!26\)\( T^{10} + \)\(61\!\cdots\!41\)\( T^{12} \)
$97$ \( ( 1 - 4878 T + 170499663 T^{2} - 362588117636 T^{3} + 15094212576132303 T^{4} - 38231001073370815758 T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \)
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