Properties

Label 525.2.i.i
Level 525
Weight 2
Character orbit 525.i
Analytic conductor 4.192
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{5} q^{3} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{4} -\beta_{3} q^{6} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{5} q^{3} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{4} -\beta_{3} q^{6} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{6} + \beta_{7} ) q^{8} + ( -1 - \beta_{5} ) q^{9} + ( -\beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{11} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{12} + ( -1 - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{14} + ( -5 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{3} ) q^{18} + ( -3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{2} - \beta_{3} ) q^{21} + ( -3 - \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{6} + \beta_{7} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{24} + ( 4 + 3 \beta_{1} + 4 \beta_{5} ) q^{26} - q^{27} + ( -1 + \beta_{1} - \beta_{2} + 4 \beta_{3} - 5 \beta_{5} + \beta_{6} ) q^{28} + 4 q^{29} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 4 \beta_{5} + 2 \beta_{7} ) q^{32} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{33} + ( 5 - 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{34} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{36} + ( -3 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 7 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{38} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{41} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{42} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{43} + ( -7 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{44} + ( -4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{46} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{47} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{48} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{51} + ( -2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{52} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{53} + \beta_{1} q^{54} + ( -8 + \beta_{1} - 5 \beta_{3} + 2 \beta_{4} - 12 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} ) q^{57} -4 \beta_{1} q^{58} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{59} + ( 5 + 5 \beta_{5} ) q^{61} + ( 12 - 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{62} -\beta_{4} q^{63} + ( 4 - \beta_{1} - \beta_{2} + 8 \beta_{3} - \beta_{4} + \beta_{6} ) q^{64} + ( -3 \beta_{1} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{66} + ( -\beta_{2} + \beta_{3} - \beta_{7} ) q^{67} + ( -1 - 6 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{68} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{69} + ( 6 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{71} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{73} + ( 7 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{74} + ( -7 + 6 \beta_{3} + \beta_{4} + \beta_{7} ) q^{76} + ( 3 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{77} + ( 4 + 3 \beta_{3} ) q^{78} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{79} + \beta_{5} q^{81} + ( 3 - 6 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{82} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{83} + ( -5 - 2 \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{84} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{86} -4 \beta_{5} q^{87} + ( 9 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - \beta_{4} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{88} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{89} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} ) q^{91} + ( -18 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{92} + ( 2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{93} + ( 5 \beta_{1} - 5 \beta_{3} - \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{94} + ( 4 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{96} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( 4 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 8 \beta_{5} - 2 \beta_{7} ) q^{98} + ( 3 - \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 4q^{3} - 7q^{4} - 2q^{6} + q^{7} + 6q^{8} - 4q^{9} + O(q^{10}) \) \( 8q - q^{2} + 4q^{3} - 7q^{4} - 2q^{6} + q^{7} + 6q^{8} - 4q^{9} - 8q^{11} + 7q^{12} - 14q^{13} - 12q^{14} - 17q^{16} + 6q^{17} - q^{18} - 3q^{19} + 2q^{21} - 24q^{22} - 2q^{23} + 3q^{24} + 19q^{26} - 8q^{27} + 15q^{28} + 32q^{29} - 9q^{31} - 17q^{32} + 8q^{33} + 28q^{34} + 14q^{36} - 8q^{37} - 27q^{38} - 7q^{39} + 8q^{41} + 3q^{42} + 10q^{43} - 26q^{44} + 6q^{46} - 6q^{47} - 34q^{48} - 21q^{49} - 6q^{51} + 35q^{52} + 6q^{53} + q^{54} - 21q^{56} - 6q^{57} - 4q^{58} - 10q^{59} + 20q^{61} + 88q^{62} + q^{63} + 42q^{64} - 12q^{66} - q^{67} - 8q^{68} - 4q^{69} + 44q^{71} - 3q^{72} - 4q^{73} + 21q^{74} - 46q^{76} + 32q^{77} + 38q^{78} - 8q^{79} - 4q^{81} + 8q^{82} + 4q^{83} - 18q^{84} - 12q^{86} + 16q^{87} - 28q^{88} + 10q^{89} + 21q^{91} - 132q^{92} + 9q^{93} - 22q^{94} + 17q^{96} - 24q^{97} + 67q^{98} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 8 x^{6} - 3 x^{5} + 50 x^{4} - 27 x^{3} + 53 x^{2} + 20 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 689 \nu^{7} - 7573 \nu^{6} + 14556 \nu^{5} - 51899 \nu^{4} + 65994 \nu^{3} - 301543 \nu^{2} + 311369 \nu - 57732 \)\()/75828\)
\(\beta_{3}\)\(=\)\((\)\( -329 \nu^{7} + 21 \nu^{6} - 2209 \nu^{5} - 1081 \nu^{4} - 17462 \nu^{3} - 3431 \nu^{2} - 1692 \nu - 8164 \)\()/18957\)
\(\beta_{4}\)\(=\)\((\)\( 1653 \nu^{7} + 3659 \nu^{6} + 10196 \nu^{5} + 37929 \nu^{4} + 67606 \nu^{3} + 217641 \nu^{2} - 19483 \nu + 208424 \)\()/75828\)
\(\beta_{5}\)\(=\)\((\)\( -2041 \nu^{7} + 3357 \nu^{6} - 16412 \nu^{5} + 14959 \nu^{4} - 97726 \nu^{3} + 124955 \nu^{2} - 94449 \nu - 34052 \)\()/75828\)
\(\beta_{6}\)\(=\)\((\)\( 1871 \nu^{7} - 1195 \nu^{6} + 17076 \nu^{5} - 4685 \nu^{4} + 112020 \nu^{3} - 34651 \nu^{2} + 187457 \nu - 33126 \)\()/37914\)
\(\beta_{7}\)\(=\)\((\)\( 6159 \nu^{7} - 5771 \nu^{6} + 42256 \nu^{5} - 12261 \nu^{4} + 255062 \nu^{3} - 136173 \nu^{2} + 59659 \nu + 161284 \)\()/75828\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + 4 \beta_{5} - \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - \beta_{6} - 7 \beta_{3} + \beta_{2} + \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-8 \beta_{7} - 7 \beta_{6} - 25 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} + \beta_{2} + 7 \beta_{1} - 25\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 8 \beta_{6} - 12 \beta_{5} + 8 \beta_{4} + 41 \beta_{3} - 2 \beta_{2} - 39 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{7} + 57 \beta_{6} - 47 \beta_{4} + 120 \beta_{3} - 57 \beta_{2} - 57 \beta_{1} + 166\)
\(\nu^{7}\)\(=\)\(83 \beta_{7} + 26 \beta_{6} + 121 \beta_{5} - 83 \beta_{4} + 83 \beta_{3} - 57 \beta_{2} + 177 \beta_{1} + 121\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.35143 + 2.34074i
0.614340 + 1.06407i
−0.247087 0.427967i
−1.21868 2.11082i
1.35143 2.34074i
0.614340 1.06407i
−0.247087 + 0.427967i
−1.21868 + 2.11082i
−1.35143 2.34074i 0.500000 0.866025i −2.65271 + 4.59463i 0 −2.70285 −1.65271 2.06605i 8.93406 −0.500000 0.866025i 0
151.2 −0.614340 1.06407i 0.500000 0.866025i 0.245174 0.424653i 0 −1.22868 1.24517 + 2.33443i −3.05984 −0.500000 0.866025i 0
151.3 0.247087 + 0.427967i 0.500000 0.866025i 0.877896 1.52056i 0 0.494173 1.87790 1.86373i 1.85601 −0.500000 0.866025i 0
151.4 1.21868 + 2.11082i 0.500000 0.866025i −1.97036 + 3.41277i 0 2.43736 −0.970361 + 2.46138i −4.73024 −0.500000 0.866025i 0
226.1 −1.35143 + 2.34074i 0.500000 + 0.866025i −2.65271 4.59463i 0 −2.70285 −1.65271 + 2.06605i 8.93406 −0.500000 + 0.866025i 0
226.2 −0.614340 + 1.06407i 0.500000 + 0.866025i 0.245174 + 0.424653i 0 −1.22868 1.24517 2.33443i −3.05984 −0.500000 + 0.866025i 0
226.3 0.247087 0.427967i 0.500000 + 0.866025i 0.877896 + 1.52056i 0 0.494173 1.87790 + 1.86373i 1.85601 −0.500000 + 0.866025i 0
226.4 1.21868 2.11082i 0.500000 + 0.866025i −1.97036 3.41277i 0 2.43736 −0.970361 2.46138i −4.73024 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.i 8
5.b even 2 1 525.2.i.j yes 8
5.c odd 4 2 525.2.r.h 16
7.c even 3 1 inner 525.2.i.i 8
7.c even 3 1 3675.2.a.bw 4
7.d odd 6 1 3675.2.a.bx 4
35.i odd 6 1 3675.2.a.bq 4
35.j even 6 1 525.2.i.j yes 8
35.j even 6 1 3675.2.a.br 4
35.l odd 12 2 525.2.r.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i 8 1.a even 1 1 trivial
525.2.i.i 8 7.c even 3 1 inner
525.2.i.j yes 8 5.b even 2 1
525.2.i.j yes 8 35.j even 6 1
525.2.r.h 16 5.c odd 4 2
525.2.r.h 16 35.l odd 12 2
3675.2.a.bq 4 35.i odd 6 1
3675.2.a.br 4 35.j even 6 1
3675.2.a.bw 4 7.c even 3 1
3675.2.a.bx 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{3} + 3 T^{5} + 7 T^{6} + 8 T^{7} - 6 T^{8} + 16 T^{9} + 28 T^{10} + 24 T^{11} - 32 T^{13} + 128 T^{15} + 256 T^{16} \)
$3$ \( ( 1 - T + T^{2} )^{4} \)
$5$ \( \)
$7$ \( 1 - T + 11 T^{2} - 12 T^{3} + 116 T^{4} - 84 T^{5} + 539 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 396 T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 26224 T^{9} + 46464 T^{10} + 527076 T^{11} + 5885682 T^{12} + 19326120 T^{13} + 49603708 T^{14} + 155897368 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + 7 T + 33 T^{2} + 94 T^{3} + 326 T^{4} + 1222 T^{5} + 5577 T^{6} + 15379 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 6 T - 12 T^{2} + 108 T^{3} + 22 T^{4} + 258 T^{5} - 5760 T^{6} - 6186 T^{7} + 134727 T^{8} - 105162 T^{9} - 1664640 T^{10} + 1267554 T^{11} + 1837462 T^{12} + 153344556 T^{13} - 289650828 T^{14} - 2462032038 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 3 T - 30 T^{2} + 125 T^{3} + 969 T^{4} - 3616 T^{5} + 6406 T^{6} + 66486 T^{7} - 238148 T^{8} + 1263234 T^{9} + 2312566 T^{10} - 24802144 T^{11} + 126281049 T^{12} + 309512375 T^{13} - 1411376430 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 2 T - 20 T^{2} - 84 T^{3} - 562 T^{4} - 922 T^{5} + 2568 T^{6} + 43258 T^{7} + 424639 T^{8} + 994934 T^{9} + 1358472 T^{10} - 11217974 T^{11} - 157270642 T^{12} - 540652812 T^{13} - 2960717780 T^{14} + 6809650894 T^{15} + 78310985281 T^{16} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{8} \)
$31$ \( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 4778991 T^{9} + 21363030 T^{10} - 48529539 T^{11} + 691717229 T^{12} + 2576623590 T^{13} + 11537547853 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 8 T - 22 T^{2} - 800 T^{3} - 2955 T^{4} + 21128 T^{5} + 190466 T^{6} - 114864 T^{7} - 6645332 T^{8} - 4249968 T^{9} + 260747954 T^{10} + 1070196584 T^{11} - 5538145755 T^{12} - 55475165600 T^{13} - 56445980998 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} \)
$41$ \( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 10496 T^{5} + 174824 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 5 T + 120 T^{2} - 677 T^{3} + 6782 T^{4} - 29111 T^{5} + 221880 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 6 T - 88 T^{2} + 52 T^{3} + 6918 T^{4} - 17418 T^{5} - 221392 T^{6} + 441630 T^{7} + 3733263 T^{8} + 20756610 T^{9} - 489054928 T^{10} - 1808389014 T^{11} + 33757633158 T^{12} + 11925940364 T^{13} - 948570948952 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 6 T - 100 T^{2} + 508 T^{3} + 4614 T^{4} - 6198 T^{5} - 355984 T^{6} - 202434 T^{7} + 25854567 T^{8} - 10729002 T^{9} - 999959056 T^{10} - 922739646 T^{11} + 36406679334 T^{12} + 212443310444 T^{13} - 2216436112900 T^{14} - 7048266839022 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 10 T - 96 T^{2} - 1492 T^{3} + 4998 T^{4} + 116970 T^{5} + 158320 T^{6} - 3153670 T^{7} - 19452177 T^{8} - 186066530 T^{9} + 551111920 T^{10} + 24023181630 T^{11} + 60562570278 T^{12} - 1066667054108 T^{13} - 4049331229536 T^{14} + 24886514848190 T^{15} + 146830437604321 T^{16} \)
$61$ \( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 + T - 250 T^{2} - 133 T^{3} + 38093 T^{4} + 11624 T^{5} - 3926238 T^{6} - 284754 T^{7} + 306049684 T^{8} - 19078518 T^{9} - 17624882382 T^{10} + 3496069112 T^{11} + 767616652253 T^{12} - 179566639231 T^{13} - 22614595542250 T^{14} + 6060711605323 T^{15} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 128226 T^{5} + 1270332 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( 1 + 4 T - 154 T^{2} + 776 T^{3} + 16497 T^{4} - 112424 T^{5} - 312970 T^{6} + 5766588 T^{7} - 11491964 T^{8} + 420960924 T^{9} - 1667817130 T^{10} - 43734847208 T^{11} + 468485781777 T^{12} + 1608703556168 T^{13} - 23305470848506 T^{14} + 44189594076388 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 8 T - 182 T^{2} - 1560 T^{3} + 19865 T^{4} + 148676 T^{5} - 1461174 T^{6} - 5521756 T^{7} + 104973092 T^{8} - 436218724 T^{9} - 9119186934 T^{10} + 73303066364 T^{11} + 773743359065 T^{12} - 4800207982440 T^{13} - 44241916904822 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \)
$83$ \( ( 1 - 2 T + 220 T^{2} - 894 T^{3} + 22430 T^{4} - 74202 T^{5} + 1515580 T^{6} - 1143574 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( 1 - 10 T - 260 T^{2} + 1572 T^{3} + 55478 T^{4} - 195874 T^{5} - 7285344 T^{6} + 5340058 T^{7} + 785732359 T^{8} + 475265162 T^{9} - 57707209824 T^{10} - 138085097906 T^{11} + 3480814046198 T^{12} + 8778141453828 T^{13} - 129215135649860 T^{14} - 442313348955290 T^{15} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 12 T + 330 T^{2} + 3296 T^{3} + 45123 T^{4} + 319712 T^{5} + 3104970 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{2} \)
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