Properties

Label 105.3.t.a
Level 105
Weight 3
Character orbit 105.t
Analytic conductor 2.861
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 105.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{2} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{3} + ( 3 - 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} ) q^{4} -\beta_{2} q^{5} + ( 4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{6} + 7 q^{7} + ( 3 \beta_{2} - 9 \beta_{4} + 3 \beta_{6} ) q^{8} + ( 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{2} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{3} + ( 3 - 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} ) q^{4} -\beta_{2} q^{5} + ( 4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{6} + 7 q^{7} + ( 3 \beta_{2} - 9 \beta_{4} + 3 \beta_{6} ) q^{8} + ( 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{9} + ( -5 + 5 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{10} + ( -3 \beta_{5} + 3 \beta_{6} ) q^{11} + ( -7 \beta_{1} + \beta_{2} + 13 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{12} + ( 1 + 10 \beta_{1} + 5 \beta_{4} + 10 \beta_{5} + 10 \beta_{7} ) q^{13} + ( 7 \beta_{2} - 7 \beta_{4} - 7 \beta_{5} ) q^{14} + ( -5 - \beta_{1} + \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{15} + ( 8 \beta_{1} - 21 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{16} + ( -\beta_{5} + 8 \beta_{6} ) q^{17} + ( -13 + 13 \beta_{3} + 13 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} ) q^{18} + ( -14 \beta_{1} + 9 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{19} + ( -3 \beta_{2} + 10 \beta_{4} - 3 \beta_{6} ) q^{20} + ( -7 + 7 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} ) q^{21} -9 q^{22} + ( -\beta_{2} + 16 \beta_{4} + 16 \beta_{5} ) q^{23} + ( 6 - 6 \beta_{3} - 24 \beta_{5} - 12 \beta_{6} - 15 \beta_{7} ) q^{24} + ( 5 - 5 \beta_{3} ) q^{25} + ( -9 \beta_{2} + 24 \beta_{4} + 24 \beta_{5} ) q^{26} + ( 7 + 5 \beta_{1} + \beta_{2} - 15 \beta_{4} + 5 \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{27} + ( 21 - 21 \beta_{3} - 14 \beta_{5} - 28 \beta_{7} ) q^{28} + ( 8 \beta_{2} - 5 \beta_{4} + 8 \beta_{6} ) q^{29} + ( 7 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{30} + ( 6 - 6 \beta_{3} - 7 \beta_{5} - 14 \beta_{7} ) q^{31} + ( 5 \beta_{5} + 17 \beta_{6} ) q^{32} + ( -9 \beta_{1} - 18 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} ) q^{33} + ( -38 + 14 \beta_{1} + 7 \beta_{4} + 14 \beta_{5} + 14 \beta_{7} ) q^{34} -7 \beta_{2} q^{35} + ( -34 + 4 \beta_{1} - 19 \beta_{2} + 24 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 4 \beta_{7} ) q^{36} + ( -26 \beta_{1} - 3 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} ) q^{37} + ( -44 \beta_{5} - 23 \beta_{6} ) q^{38} + ( 24 - 24 \beta_{3} + 21 \beta_{5} - 6 \beta_{6} - 9 \beta_{7} ) q^{39} + ( -18 \beta_{1} + 15 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} ) q^{40} + ( 9 \beta_{2} + 9 \beta_{6} ) q^{41} + ( 28 - 7 \beta_{1} - 14 \beta_{2} + 21 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} - 7 \beta_{7} ) q^{42} + ( 6 - 10 \beta_{1} - 5 \beta_{4} - 10 \beta_{5} - 10 \beta_{7} ) q^{43} + ( 3 \beta_{2} + 21 \beta_{4} + 21 \beta_{5} ) q^{44} + ( 5 - 5 \beta_{3} - 14 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} ) q^{45} + ( -37 + 37 \beta_{3} + 17 \beta_{5} + 34 \beta_{7} ) q^{46} + ( -13 \beta_{2} - 26 \beta_{4} - 26 \beta_{5} ) q^{47} + ( 41 - 29 \beta_{1} + 17 \beta_{2} - 24 \beta_{4} - 29 \beta_{5} + 17 \beta_{6} - 29 \beta_{7} ) q^{48} + 49 q^{49} + ( 5 \beta_{2} - 5 \beta_{4} + 5 \beta_{6} ) q^{50} + ( -10 \beta_{1} + 7 \beta_{2} - 41 \beta_{3} - 24 \beta_{4} - 24 \beta_{5} ) q^{51} + ( -97 + 97 \beta_{3} + 13 \beta_{5} + 26 \beta_{7} ) q^{52} + ( -29 \beta_{5} - 17 \beta_{6} ) q^{53} + ( 37 \beta_{1} + 2 \beta_{2} - 40 \beta_{3} + 24 \beta_{4} + 24 \beta_{5} ) q^{54} + ( 15 + 6 \beta_{1} + 3 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} ) q^{55} + ( 21 \beta_{2} - 63 \beta_{4} + 21 \beta_{6} ) q^{56} + ( -44 + 23 \beta_{1} - 2 \beta_{2} + 42 \beta_{4} + 23 \beta_{5} - 2 \beta_{6} + 23 \beta_{7} ) q^{57} + ( 26 \beta_{1} - 50 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} ) q^{58} + ( 58 \beta_{5} - 2 \beta_{6} ) q^{59} + ( -5 + 5 \beta_{3} + 26 \beta_{5} + 13 \beta_{6} + 17 \beta_{7} ) q^{60} + ( 28 \beta_{1} + 66 \beta_{3} + 14 \beta_{4} + 14 \beta_{5} ) q^{61} + ( 20 \beta_{2} - 41 \beta_{4} + 20 \beta_{6} ) q^{62} + ( 28 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} + 42 \beta_{4} + 42 \beta_{5} ) q^{63} + ( -11 + 12 \beta_{1} + 6 \beta_{4} + 12 \beta_{5} + 12 \beta_{7} ) q^{64} + ( -\beta_{2} - 25 \beta_{4} - 25 \beta_{5} ) q^{65} + ( 9 - 9 \beta_{3} - 9 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{66} + ( 72 - 72 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} ) q^{67} + ( -20 \beta_{2} + 77 \beta_{4} + 77 \beta_{5} ) q^{68} + ( 11 + 31 \beta_{1} + 17 \beta_{2} - 3 \beta_{4} + 31 \beta_{5} + 17 \beta_{6} + 31 \beta_{7} ) q^{69} + ( -35 + 35 \beta_{3} + 7 \beta_{5} + 14 \beta_{7} ) q^{70} + ( 12 \beta_{2} + 15 \beta_{4} + 12 \beta_{6} ) q^{71} + ( -42 \beta_{1} - 30 \beta_{2} + 87 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} ) q^{72} + ( -72 + 72 \beta_{3} - 7 \beta_{5} - 14 \beta_{7} ) q^{73} + ( -62 \beta_{5} - 23 \beta_{6} ) q^{74} + ( -5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{75} + ( 167 - 78 \beta_{1} - 39 \beta_{4} - 78 \beta_{5} - 78 \beta_{7} ) q^{76} + ( -21 \beta_{5} + 21 \beta_{6} ) q^{77} + ( -21 + 39 \beta_{1} + 33 \beta_{2} - 27 \beta_{4} + 39 \beta_{5} + 33 \beta_{6} + 39 \beta_{7} ) q^{78} + ( -6 \beta_{1} - 86 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{79} + ( -20 \beta_{5} - 21 \beta_{6} ) q^{80} + ( -7 + 7 \beta_{3} - 20 \beta_{5} - 28 \beta_{6} - 32 \beta_{7} ) q^{81} + ( 18 \beta_{1} - 45 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{82} + ( -26 \beta_{2} - 40 \beta_{4} - 26 \beta_{6} ) q^{83} + ( -49 \beta_{1} + 7 \beta_{2} + 91 \beta_{3} - 84 \beta_{4} - 84 \beta_{5} ) q^{84} + ( 40 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{85} + ( 16 \beta_{2} - 31 \beta_{4} - 31 \beta_{5} ) q^{86} + ( 35 - 35 \beta_{3} - 26 \beta_{5} - 13 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 9 - 9 \beta_{3} + 18 \beta_{5} + 36 \beta_{7} ) q^{88} + ( 12 \beta_{2} + 18 \beta_{4} + 18 \beta_{5} ) q^{89} + ( 10 - 16 \beta_{1} + 13 \beta_{2} - 33 \beta_{4} - 16 \beta_{5} + 13 \beta_{6} - 16 \beta_{7} ) q^{90} + ( 7 + 70 \beta_{1} + 35 \beta_{4} + 70 \beta_{5} + 70 \beta_{7} ) q^{91} + ( -67 \beta_{2} + 58 \beta_{4} - 67 \beta_{6} ) q^{92} + ( -20 \beta_{1} - \beta_{2} + 41 \beta_{3} - 42 \beta_{4} - 42 \beta_{5} ) q^{93} + ( -13 + 13 \beta_{3} - 13 \beta_{5} - 26 \beta_{7} ) q^{94} + ( 35 \beta_{5} + 9 \beta_{6} ) q^{95} + ( -7 \beta_{1} + 22 \beta_{2} - 80 \beta_{3} - 51 \beta_{4} - 51 \beta_{5} ) q^{96} + ( -56 + 10 \beta_{1} + 5 \beta_{4} + 10 \beta_{5} + 10 \beta_{7} ) q^{97} + ( 49 \beta_{2} - 49 \beta_{4} - 49 \beta_{5} ) q^{98} + ( -9 - 18 \beta_{1} + 18 \beta_{2} + 27 \beta_{4} - 18 \beta_{5} + 18 \beta_{6} - 18 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 12q^{4} + 32q^{6} + 56q^{7} + 8q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 12q^{4} + 32q^{6} + 56q^{7} + 8q^{9} - 20q^{10} + 52q^{12} + 8q^{13} - 40q^{15} - 84q^{16} - 52q^{18} + 36q^{19} - 28q^{21} - 72q^{22} + 24q^{24} + 20q^{25} + 56q^{27} + 84q^{28} - 40q^{30} + 24q^{31} - 72q^{33} - 304q^{34} - 272q^{36} - 12q^{37} + 96q^{39} + 60q^{40} + 224q^{42} + 48q^{43} + 20q^{45} - 148q^{46} + 328q^{48} + 392q^{49} - 164q^{51} - 388q^{52} - 160q^{54} + 120q^{55} - 352q^{57} - 200q^{58} - 20q^{60} + 264q^{61} + 56q^{63} - 88q^{64} + 36q^{66} + 288q^{67} + 88q^{69} - 140q^{70} + 348q^{72} - 288q^{73} + 20q^{75} + 1336q^{76} - 168q^{78} - 344q^{79} - 28q^{81} - 180q^{82} + 364q^{84} + 320q^{85} + 140q^{87} + 36q^{88} + 80q^{90} + 56q^{91} + 164q^{93} - 52q^{94} - 320q^{96} - 448q^{97} - 72q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 4 x^{6} + 7 x^{4} + 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 14 \nu^{4} - 7 \nu^{2} - 36 \)\()/63\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{6} + 7 \nu^{4} + 28 \nu^{2} + 144 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{7} - 7 \nu^{5} + 35 \nu^{3} - 81 \nu \)\()/189\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 7 \nu^{5} - 35 \nu^{3} - 180 \nu \)\()/189\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{6} - 14 \nu^{4} + 7 \nu^{2} - 162 \)\()/63\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} - 7 \nu^{3} - 36 \nu \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 4 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-5 \beta_{7} + 7 \beta_{5}\)
\(\nu^{6}\)\(=\)\(-7 \beta_{6} - 7 \beta_{2} - 22\)
\(\nu^{7}\)\(=\)\(-21 \beta_{5} - 21 \beta_{4} - 29 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(-1\)

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
−1.40294 1.01575i
1.40294 + 1.01575i
0.178197 + 1.72286i
−0.178197 1.72286i
−1.40294 + 1.01575i
1.40294 1.01575i
0.178197 1.72286i
−0.178197 + 1.72286i
−3.16124 + 1.82514i −2.61469 + 1.47085i 4.66228 8.07530i 1.93649 1.11803i 5.58114 9.42188i 7.00000 19.4361i 4.67319 7.69164i −4.08114 + 7.06874i
11.2 −0.711747 + 0.410927i −2.25829 1.97487i −1.66228 + 2.87915i 1.93649 1.11803i 2.41886 + 0.477612i 7.00000 6.01972i 1.19979 + 8.91967i −0.918861 + 1.59151i
11.3 0.711747 0.410927i 2.83943 + 0.968306i −1.66228 + 2.87915i −1.93649 + 1.11803i 2.41886 0.477612i 7.00000 6.01972i 7.12477 + 5.49888i −0.918861 + 1.59151i
11.4 3.16124 1.82514i 0.0335498 + 2.99981i 4.66228 8.07530i −1.93649 + 1.11803i 5.58114 + 9.42188i 7.00000 19.4361i −8.99775 + 0.201286i −4.08114 + 7.06874i
86.1 −3.16124 1.82514i −2.61469 1.47085i 4.66228 + 8.07530i 1.93649 + 1.11803i 5.58114 + 9.42188i 7.00000 19.4361i 4.67319 + 7.69164i −4.08114 7.06874i
86.2 −0.711747 0.410927i −2.25829 + 1.97487i −1.66228 2.87915i 1.93649 + 1.11803i 2.41886 0.477612i 7.00000 6.01972i 1.19979 8.91967i −0.918861 1.59151i
86.3 0.711747 + 0.410927i 2.83943 0.968306i −1.66228 2.87915i −1.93649 1.11803i 2.41886 + 0.477612i 7.00000 6.01972i 7.12477 5.49888i −0.918861 1.59151i
86.4 3.16124 + 1.82514i 0.0335498 2.99981i 4.66228 + 8.07530i −1.93649 1.11803i 5.58114 9.42188i 7.00000 19.4361i −8.99775 0.201286i −4.08114 7.06874i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.t.a 8
3.b odd 2 1 inner 105.3.t.a 8
7.c even 3 1 inner 105.3.t.a 8
21.h odd 6 1 inner 105.3.t.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.t.a 8 1.a even 1 1 trivial
105.3.t.a 8 3.b odd 2 1 inner
105.3.t.a 8 7.c even 3 1 inner
105.3.t.a 8 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 14 T_{2}^{6} + 187 T_{2}^{4} - 126 T_{2}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 11 T^{4} - 78 T^{6} - 271 T^{8} - 1248 T^{10} + 2816 T^{12} + 8192 T^{14} + 65536 T^{16} \)
$3$ \( 1 + 4 T + 4 T^{2} - 24 T^{3} - 81 T^{4} - 216 T^{5} + 324 T^{6} + 2916 T^{7} + 6561 T^{8} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T )^{8} \)
$11$ \( 1 + 358 T^{2} + 70081 T^{4} + 10310758 T^{6} + 1282826884 T^{8} + 150959807878 T^{10} + 15022484739361 T^{12} + 1123557358866118 T^{14} + 45949729863572161 T^{16} \)
$13$ \( ( 1 - 2 T + 89 T^{2} - 338 T^{3} + 28561 T^{4} )^{4} \)
$17$ \( 1 + 512 T^{2} + 32126 T^{4} + 32243712 T^{6} + 24037993859 T^{8} + 2693027069952 T^{10} + 224103183549566 T^{12} + 298302585461637632 T^{14} + 48661191875666868481 T^{16} \)
$19$ \( ( 1 - 18 T + 11 T^{2} + 7362 T^{3} - 149316 T^{4} + 2657682 T^{5} + 1433531 T^{6} - 846825858 T^{7} + 16983563041 T^{8} )^{2} \)
$23$ \( 1 + 1082 T^{2} + 328601 T^{4} + 305601162 T^{6} + 303243586964 T^{8} + 85519734775242 T^{10} + 25733068074321881 T^{12} + 23711623635445987322 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( ( 1 - 2624 T^{2} + 3071906 T^{4} - 1855905344 T^{6} + 500246412961 T^{8} )^{2} \)
$31$ \( ( 1 - 12 T - 1324 T^{2} + 5448 T^{3} + 1093119 T^{4} + 5235528 T^{5} - 1222741804 T^{6} - 10650044172 T^{7} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 + 6 T - 1021 T^{2} - 10086 T^{3} - 806196 T^{4} - 13807734 T^{5} - 1913518381 T^{6} + 15394358454 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( ( 1 - 2957 T^{2} + 2825761 T^{4} )^{4} \)
$43$ \( ( 1 - 12 T + 3484 T^{2} - 22188 T^{3} + 3418801 T^{4} )^{4} \)
$47$ \( 1 + 4442 T^{2} + 9608921 T^{4} + 1612805802 T^{6} - 17630196388396 T^{8} + 7869977828709162 T^{10} + \)\(22\!\cdots\!81\)\( T^{12} + \)\(51\!\cdots\!22\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 + 4982 T^{2} + 12556241 T^{4} - 17521091178 T^{6} - 107696253516316 T^{8} - 138249837039276618 T^{10} + \)\(78\!\cdots\!01\)\( T^{12} + \)\(24\!\cdots\!62\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 + 428 T^{2} - 23559094 T^{4} - 210766032 T^{6} + 414645620367539 T^{8} - 2553928096281552 T^{10} - \)\(34\!\cdots\!74\)\( T^{12} + \)\(76\!\cdots\!68\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( ( 1 - 132 T + 7586 T^{2} - 316272 T^{3} + 18105699 T^{4} - 1176848112 T^{5} + 105034549826 T^{6} - 6800689415652 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 - 144 T + 6734 T^{2} - 723456 T^{3} + 82820979 T^{4} - 3247593984 T^{5} + 135697648814 T^{6} - 13026007032336 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 17824 T^{2} + 128951106 T^{4} - 452937802144 T^{6} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 + 144 T + 5384 T^{2} + 675936 T^{3} + 96783519 T^{4} + 3602062944 T^{5} + 152896129544 T^{6} + 21792128585616 T^{7} + 806460091894081 T^{8} )^{2} \)
$79$ \( ( 1 + 172 T + 9796 T^{2} + 1256632 T^{3} + 167981119 T^{4} + 7842640312 T^{5} + 381554993476 T^{6} + 41811042349612 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( ( 1 - 14396 T^{2} + 103463846 T^{4} - 683209989116 T^{6} + 2252292232139041 T^{8} )^{2} \)
$89$ \( 1 + 28948 T^{2} + 504871786 T^{4} + 6010485861328 T^{6} + 54451796051280979 T^{8} + \)\(37\!\cdots\!48\)\( T^{10} + \)\(19\!\cdots\!66\)\( T^{12} + \)\(71\!\cdots\!08\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( ( 1 + 112 T + 21704 T^{2} + 1053808 T^{3} + 88529281 T^{4} )^{4} \)
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