Properties

Label 475.2.j.b
Level $475$
Weight $2$
Character orbit 475.j
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.50712647503417344.1
Defining polynomial: \(x^{12} - 13 x^{10} + 119 x^{8} - 552 x^{6} + 1863 x^{4} - 2450 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{11} q^{3} + ( 2 - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{4} + ( 1 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{6} + ( \beta_{8} + \beta_{10} + \beta_{11} ) q^{7} + ( \beta_{1} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} + ( 1 + \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{11} q^{3} + ( 2 - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{4} + ( 1 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{6} + ( \beta_{8} + \beta_{10} + \beta_{11} ) q^{7} + ( \beta_{1} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{8} + ( 1 + \beta_{3} + \beta_{5} ) q^{9} + ( -1 + \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( 2 \beta_{1} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{12} + ( 5 \beta_{7} - 5 \beta_{8} ) q^{13} + ( 2 \beta_{2} - 3 \beta_{3} ) q^{14} + ( -\beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{16} + ( -3 \beta_{1} - 2 \beta_{7} - 2 \beta_{11} ) q^{17} + ( -5 \beta_{8} - \beta_{10} - \beta_{11} ) q^{18} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{19} + ( \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{21} + ( -\beta_{1} + 3 \beta_{7} + 2 \beta_{11} ) q^{22} + ( -3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{23} + 7 \beta_{3} q^{24} + ( -5 \beta_{5} + 5 \beta_{6} ) q^{26} + ( -\beta_{1} - 3 \beta_{8} + \beta_{9} ) q^{27} + ( -4 \beta_{7} + 4 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{28} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{29} + ( -\beta_{5} + \beta_{6} ) q^{31} + ( -3 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{32} + ( -\beta_{1} + 4 \beta_{7} + 2 \beta_{11} ) q^{33} + ( -10 - \beta_{2} - 9 \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{34} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{36} + ( -2 \beta_{1} - 3 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{37} + ( -8 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} ) q^{38} + ( 5 + 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} ) q^{39} + ( -\beta_{2} - \beta_{3} + 2 \beta_{6} ) q^{41} + ( -8 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{42} + ( 3 \beta_{1} + \beta_{7} + \beta_{11} ) q^{43} + ( -4 + 3 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{44} + ( -10 - \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{46} + ( \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{47} + ( -2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{48} + ( 4 + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{49} + ( 3 + 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{51} + ( 5 \beta_{1} + 15 \beta_{7} + 5 \beta_{11} ) q^{52} + ( -2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 4 \beta_{10} ) q^{53} + ( \beta_{2} - 5 \beta_{3} + 4 \beta_{6} ) q^{54} + ( 4 - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{56} + ( 3 \beta_{1} - 7 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{57} + ( 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{58} + ( -\beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{59} + ( 3 - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{61} + ( \beta_{1} + 5 \beta_{7} + \beta_{11} ) q^{62} + ( -4 \beta_{7} + 4 \beta_{8} + 3 \beta_{10} ) q^{63} + ( -5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{64} + ( -6 + 5 \beta_{2} - 11 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} + 5 \beta_{6} ) q^{66} + ( -8 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{67} + ( -7 \beta_{1} - 14 \beta_{8} + 7 \beta_{9} - \beta_{10} - \beta_{11} ) q^{68} + ( 2 + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{69} + ( 3 \beta_{2} - 11 \beta_{3} + \beta_{6} ) q^{71} + ( 6 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} ) q^{72} + ( -3 \beta_{1} - 9 \beta_{7} - 2 \beta_{11} ) q^{73} + ( -8 \beta_{2} + 5 \beta_{3} ) q^{74} + ( -3 - 2 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{76} + ( \beta_{1} - 6 \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{77} + ( 5 \beta_{1} + 15 \beta_{7} + 10 \beta_{11} ) q^{78} + ( 2 \beta_{2} + 6 \beta_{3} + 4 \beta_{6} ) q^{79} + ( -\beta_{2} - 6 \beta_{3} - 2 \beta_{6} ) q^{81} + ( 13 \beta_{7} - 13 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} ) q^{82} + ( 3 \beta_{1} + 2 \beta_{8} - 3 \beta_{9} ) q^{83} + ( 5 - 6 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{84} + ( 11 - \beta_{2} + 12 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{86} + ( -\beta_{1} + 4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{87} + ( -2 \beta_{1} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{88} + ( -6 - 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} ) q^{89} + ( -5 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{91} + ( -8 \beta_{1} - 17 \beta_{7} - 2 \beta_{11} ) q^{92} + ( \beta_{1} + 3 \beta_{7} + 2 \beta_{11} ) q^{93} -7 q^{94} + ( 4 - \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{96} + ( 4 \beta_{1} + 3 \beta_{11} ) q^{97} + ( 5 \beta_{1} + 11 \beta_{7} + 5 \beta_{11} ) q^{98} + ( -2 + 3 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 14 q^{4} + 12 q^{6} + 8 q^{9} + O(q^{10}) \) \( 12 q + 14 q^{4} + 12 q^{6} + 8 q^{9} - 20 q^{11} + 14 q^{14} - 6 q^{16} + 24 q^{21} - 42 q^{24} - 20 q^{26} - 4 q^{29} - 4 q^{31} - 50 q^{34} - 6 q^{36} + 20 q^{39} + 4 q^{41} - 36 q^{44} - 96 q^{46} + 28 q^{49} + 12 q^{51} + 20 q^{54} + 60 q^{56} + 12 q^{59} + 18 q^{61} + 32 q^{64} - 58 q^{66} + 20 q^{69} + 58 q^{71} - 14 q^{74} - 38 q^{76} - 48 q^{79} + 42 q^{81} + 112 q^{84} + 64 q^{86} - 28 q^{89} + 20 q^{91} - 84 q^{94} + 68 q^{96} - 26 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 13 x^{10} + 119 x^{8} - 552 x^{6} + 1863 x^{4} - 2450 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 736 \nu^{10} - 31772 \nu^{8} + 290836 \nu^{6} - 1755683 \nu^{4} + 4553172 \nu^{2} - 5987800 \)\()/1328929\)
\(\beta_{3}\)\(=\)\((\)\( 850 \nu^{10} - 9867 \nu^{8} + 90321 \nu^{6} - 370073 \nu^{4} + 1414017 \nu^{2} - 1859550 \)\()/1328929\)
\(\beta_{4}\)\(=\)\((\)\( 169 \nu^{10} - 1547 \nu^{8} + 14161 \nu^{6} - 24219 \nu^{4} + 31850 \nu^{2} + 467838 \)\()/189847\)
\(\beta_{5}\)\(=\)\((\)\( -200 \nu^{10} + 3917 \nu^{8} - 21252 \nu^{6} + 87076 \nu^{4} + 37412 \nu^{2} + 124852 \)\()/189847\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{10} - 26 \nu^{8} + 238 \nu^{6} + 289 \nu^{4} + 3726 \nu^{2} - 4900 \)\()/5131\)
\(\beta_{7}\)\(=\)\((\)\( 52 \nu^{11} - 476 \nu^{9} + 2271 \nu^{7} - 7452 \nu^{5} + 9800 \nu^{3} - 164812 \nu \)\()/189847\)
\(\beta_{8}\)\(=\)\((\)\( -100 \nu^{11} + 859 \nu^{9} - 10626 \nu^{7} + 43538 \nu^{5} - 200461 \nu^{3} + 62426 \nu \)\()/251419\)
\(\beta_{9}\)\(=\)\((\)\( -850 \nu^{11} + 9867 \nu^{9} - 90321 \nu^{7} + 370073 \nu^{5} - 1414017 \nu^{3} + 1859550 \nu \)\()/1328929\)
\(\beta_{10}\)\(=\)\((\)\( 8973 \nu^{11} - 159328 \nu^{9} + 1458464 \nu^{7} - 7813716 \nu^{5} + 22832928 \nu^{3} - 30027200 \nu \)\()/9302503\)
\(\beta_{11}\)\(=\)\((\)\( 273 \nu^{11} - 2499 \nu^{9} + 18703 \nu^{7} - 39123 \nu^{5} + 51450 \nu^{3} + 328061 \nu \)\()/189847\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} - \beta_{4} + 5 \beta_{3} - \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{10} - 5 \beta_{9} + 2 \beta_{8} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} + 28 \beta_{3} - 7 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-6 \beta_{10} - 29 \beta_{9} + 19 \beta_{8} - 19 \beta_{7}\)
\(\nu^{6}\)\(=\)\(-13 \beta_{6} + 13 \beta_{5} + 41 \beta_{4} - 122\)
\(\nu^{7}\)\(=\)\(28 \beta_{11} - 147 \beta_{7} - 176 \beta_{1}\)
\(\nu^{8}\)\(=\)\(232 \beta_{6} + 119 \beta_{5} + 232 \beta_{4} - 964 \beta_{3} + 232 \beta_{2} - 732\)
\(\nu^{9}\)\(=\)\(113 \beta_{11} + 113 \beta_{10} + 1083 \beta_{9} - 1059 \beta_{8} - 1083 \beta_{1}\)
\(\nu^{10}\)\(=\)\(2255 \beta_{6} - 5754 \beta_{3} + 1309 \beta_{2}\)
\(\nu^{11}\)\(=\)\(363 \beta_{10} + 6700 \beta_{9} - 7348 \beta_{8} + 7348 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.17114 1.25351i
−1.97899 1.14257i
−1.05818 0.610938i
1.05818 + 0.610938i
1.97899 + 1.14257i
2.17114 + 1.25351i
−2.17114 + 1.25351i
−1.97899 + 1.14257i
−1.05818 + 0.610938i
1.05818 0.610938i
1.97899 1.14257i
2.17114 1.25351i
−2.17114 1.25351i −1.05818 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i 0.221876i 5.72889i −0.753509 1.30512i 0
49.2 −1.97899 1.14257i −2.17114 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i 3.50702i 2.79216i 1.64257 + 2.84502i 0
49.3 −1.05818 0.610938i 1.97899 + 1.14257i −0.253509 0.439091i 0 −1.39608 2.41808i 1.28514i 3.06327i 1.11094 + 1.92420i 0
49.4 1.05818 + 0.610938i −1.97899 1.14257i −0.253509 0.439091i 0 −1.39608 2.41808i 1.28514i 3.06327i 1.11094 + 1.92420i 0
49.5 1.97899 + 1.14257i 2.17114 + 1.25351i 1.61094 + 2.79023i 0 2.86445 + 4.96137i 3.50702i 2.79216i 1.64257 + 2.84502i 0
49.6 2.17114 + 1.25351i 1.05818 + 0.610938i 2.14257 + 3.71104i 0 1.53163 + 2.65287i 0.221876i 5.72889i −0.753509 1.30512i 0
349.1 −2.17114 + 1.25351i −1.05818 + 0.610938i 2.14257 3.71104i 0 1.53163 2.65287i 0.221876i 5.72889i −0.753509 + 1.30512i 0
349.2 −1.97899 + 1.14257i −2.17114 + 1.25351i 1.61094 2.79023i 0 2.86445 4.96137i 3.50702i 2.79216i 1.64257 2.84502i 0
349.3 −1.05818 + 0.610938i 1.97899 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i 1.28514i 3.06327i 1.11094 1.92420i 0
349.4 1.05818 0.610938i −1.97899 + 1.14257i −0.253509 + 0.439091i 0 −1.39608 + 2.41808i 1.28514i 3.06327i 1.11094 1.92420i 0
349.5 1.97899 1.14257i 2.17114 1.25351i 1.61094 2.79023i 0 2.86445 4.96137i 3.50702i 2.79216i 1.64257 2.84502i 0
349.6 2.17114 1.25351i 1.05818 0.610938i 2.14257 3.71104i 0 1.53163 2.65287i 0.221876i 5.72889i −0.753509 + 1.30512i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.b 12
5.b even 2 1 inner 475.2.j.b 12
5.c odd 4 1 95.2.e.b 6
5.c odd 4 1 475.2.e.d 6
15.e even 4 1 855.2.k.g 6
19.c even 3 1 inner 475.2.j.b 12
20.e even 4 1 1520.2.q.j 6
95.i even 6 1 inner 475.2.j.b 12
95.l even 12 1 1805.2.a.g 3
95.l even 12 1 9025.2.a.ba 3
95.m odd 12 1 95.2.e.b 6
95.m odd 12 1 475.2.e.d 6
95.m odd 12 1 1805.2.a.h 3
95.m odd 12 1 9025.2.a.z 3
285.v even 12 1 855.2.k.g 6
380.v even 12 1 1520.2.q.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 5.c odd 4 1
95.2.e.b 6 95.m odd 12 1
475.2.e.d 6 5.c odd 4 1
475.2.e.d 6 95.m odd 12 1
475.2.j.b 12 1.a even 1 1 trivial
475.2.j.b 12 5.b even 2 1 inner
475.2.j.b 12 19.c even 3 1 inner
475.2.j.b 12 95.i even 6 1 inner
855.2.k.g 6 15.e even 4 1
855.2.k.g 6 285.v even 12 1
1520.2.q.j 6 20.e even 4 1
1520.2.q.j 6 380.v even 12 1
1805.2.a.g 3 95.l even 12 1
1805.2.a.h 3 95.m odd 12 1
9025.2.a.z 3 95.m odd 12 1
9025.2.a.ba 3 95.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 13 T_{2}^{10} + 119 T_{2}^{8} - 552 T_{2}^{6} + 1863 T_{2}^{4} - 2450 T_{2}^{2} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2401 - 2450 T^{2} + 1863 T^{4} - 552 T^{6} + 119 T^{8} - 13 T^{10} + T^{12} \)
$3$ \( 2401 - 2450 T^{2} + 1863 T^{4} - 552 T^{6} + 119 T^{8} - 13 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 1 + 21 T^{2} + 14 T^{4} + T^{6} )^{2} \)
$11$ \( ( -1 + 2 T + 5 T^{2} + T^{3} )^{4} \)
$13$ \( ( 625 - 25 T^{2} + T^{4} )^{3} \)
$17$ \( 2401 - 95550 T^{2} + 3798139 T^{4} - 173452 T^{6} + 5971 T^{8} - 89 T^{10} + T^{12} \)
$19$ \( ( 6859 - 133 T^{3} + T^{6} )^{2} \)
$23$ \( 5764801 - 4593113 T^{2} + 3433875 T^{4} - 175020 T^{6} + 6923 T^{8} - 94 T^{10} + T^{12} \)
$29$ \( ( 1 - 5 T + 23 T^{2} - 12 T^{3} + 9 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$31$ \( ( -7 - 6 T + T^{2} + T^{3} )^{4} \)
$37$ \( ( 51529 + 15069 T^{2} + 242 T^{4} + T^{6} )^{2} \)
$41$ \( ( 1369 - 1591 T + 1923 T^{2} + 12 T^{3} + 47 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$43$ \( 214358881 - 31888098 T^{2} + 3440635 T^{4} - 164560 T^{6} + 5743 T^{8} - 89 T^{10} + T^{12} \)
$47$ \( 5764801 - 1529437 T^{2} + 285719 T^{4} - 27048 T^{6} + 1863 T^{8} - 50 T^{10} + T^{12} \)
$53$ \( 9354951841 - 832380926 T^{2} + 54235431 T^{4} - 1570788 T^{6} + 33419 T^{8} - 205 T^{10} + T^{12} \)
$59$ \( ( 2401 - 343 T + 343 T^{2} - 56 T^{3} + 43 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$61$ \( ( 2401 - 2401 T + 2842 T^{2} + 343 T^{3} + 130 T^{4} - 9 T^{5} + T^{6} )^{2} \)
$67$ \( 59969536 - 63067136 T^{2} + 64899840 T^{4} - 1483008 T^{6} + 25712 T^{8} - 184 T^{10} + T^{12} \)
$71$ \( ( 218089 - 110212 T + 42153 T^{2} - 5910 T^{3} + 605 T^{4} - 29 T^{5} + T^{6} )^{2} \)
$73$ \( 35153041 - 61074629 T^{2} + 104628351 T^{4} - 2563392 T^{6} + 52199 T^{8} - 250 T^{10} + T^{12} \)
$79$ \( ( 61504 - 28768 T + 19408 T^{2} + 3280 T^{3} + 460 T^{4} + 24 T^{5} + T^{6} )^{2} \)
$83$ \( ( 5929 + 2454 T^{2} + 117 T^{4} + T^{6} )^{2} \)
$89$ \( ( 3136 + 2016 T + 2080 T^{2} - 392 T^{3} + 232 T^{4} + 14 T^{5} + T^{6} )^{2} \)
$97$ \( 214358881 - 88578050 T^{2} + 33952479 T^{4} - 1065768 T^{6} + 26711 T^{8} - 181 T^{10} + T^{12} \)
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