Properties

Label 1050.2.d.g
Level $1050$
Weight $2$
Character orbit 1050.d
Analytic conductor $8.384$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 4 x^{8} - 30 x^{6} + 36 x^{4} + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{10} q^{3} + q^{4} -\beta_{10} q^{6} + ( -\beta_{1} + \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{7} - q^{8} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{10} q^{3} + q^{4} -\beta_{10} q^{6} + ( -\beta_{1} + \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{7} - q^{8} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} + ( -\beta_{5} - \beta_{6} ) q^{11} + \beta_{10} q^{12} + ( -\beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{13} + ( \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{14} + q^{16} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{17} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{18} + ( \beta_{1} + \beta_{9} ) q^{19} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{21} + ( \beta_{5} + \beta_{6} ) q^{22} + ( -1 + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{23} -\beta_{10} q^{24} + ( \beta_{1} - \beta_{2} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{26} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{27} + ( -\beta_{1} + \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{28} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{29} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{31} - q^{32} + ( -\beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{10} - \beta_{11} ) q^{33} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{34} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{36} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{37} + ( -\beta_{1} - \beta_{9} ) q^{38} + ( \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{39} + ( -2 \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{9} + \beta_{11} ) q^{41} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{42} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{43} + ( -\beta_{5} - \beta_{6} ) q^{44} + ( 1 - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{46} + ( -3 \beta_{1} - \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{47} + \beta_{10} q^{48} + ( -3 + \beta_{1} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{49} + ( 2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{51} + ( -\beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{52} + ( -1 + 2 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{53} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{54} + ( \beta_{1} - \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{56} + ( 1 - 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{57} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{58} + ( -2 \beta_{1} + \beta_{2} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{59} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{61} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{62} + ( -3 + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{63} + q^{64} + ( \beta_{1} - \beta_{2} - \beta_{7} + \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{66} + ( \beta_{1} - 2 \beta_{3} - 6 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{67} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{68} + ( -\beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{69} + ( \beta_{1} - 2 \beta_{3} - 8 \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{71} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{72} + ( \beta_{1} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{73} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{74} + ( \beta_{1} + \beta_{9} ) q^{76} + ( 4 + \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{77} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{78} + ( -6 - 2 \beta_{1} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{79} + ( -2 - 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{81} + ( 2 \beta_{1} + \beta_{2} + \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{82} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{83} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{84} + ( \beta_{4} - \beta_{5} - \beta_{6} ) q^{86} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{87} + ( \beta_{5} + \beta_{6} ) q^{88} + ( -2 \beta_{1} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{89} + ( -1 - 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} ) q^{91} + ( -1 + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{92} + ( 3 - 3 \beta_{1} + \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{93} + ( 3 \beta_{1} + \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{94} -\beta_{10} q^{96} + ( -4 \beta_{1} - 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} ) q^{97} + ( 3 - \beta_{1} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{98} + ( -3 + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{4} - 12q^{8} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{4} - 12q^{8} + 12q^{16} + 14q^{21} - 20q^{23} - 12q^{32} - 12q^{39} - 14q^{42} + 20q^{46} - 28q^{49} + 28q^{51} - 20q^{53} + 8q^{57} - 30q^{63} + 12q^{64} + 44q^{77} + 12q^{78} - 56q^{79} - 16q^{81} + 14q^{84} - 20q^{91} - 20q^{92} + 48q^{93} + 28q^{98} - 48q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 4 x^{8} - 30 x^{6} + 36 x^{4} + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + 9 \nu^{8} + 50 \nu^{7} + 54 \nu^{6} + 111 \nu^{5} - 126 \nu^{4} - 306 \nu^{3} + 189 \nu^{2} - 324 \nu - 972 \)\()/972\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} + 9 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 51 \nu^{5} + 207 \nu^{4} - 72 \nu^{3} - 405 \nu^{2} + 891 \nu - 486 \)\()/972\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} - 3 \nu^{8} + 5 \nu^{6} - 9 \nu^{4} + 9 \nu^{2} - 243 \)\()/324\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} - 6 \nu^{10} - 23 \nu^{7} + 57 \nu^{6} + 51 \nu^{5} + 180 \nu^{4} - 72 \nu^{3} + 594 \nu^{2} + 891 \nu - 1215 \)\()/972\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 6 \nu^{10} + 23 \nu^{7} + 105 \nu^{6} - 51 \nu^{5} - 180 \nu^{4} + 72 \nu^{3} + 54 \nu^{2} - 891 \nu - 1215 \)\()/972\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} + 3 \nu^{10} - 27 \nu^{8} - 23 \nu^{7} + 12 \nu^{6} + 51 \nu^{5} + 45 \nu^{4} - 72 \nu^{3} + 675 \nu^{2} + 891 \nu \)\()/972\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{11} + 3 \nu^{9} + 5 \nu^{7} + 15 \nu^{5} - 9 \nu^{3} - 27 \nu \)\()/324\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{11} + 4 \nu^{7} - 30 \nu^{5} + 36 \nu^{3} \)\()/243\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{11} - 3 \nu^{9} + 5 \nu^{7} - 9 \nu^{5} + 9 \nu^{3} - 243 \nu \)\()/324\)
\(\beta_{11}\)\(=\)\((\)\( 4 \nu^{11} + 3 \nu^{10} - 18 \nu^{9} - 18 \nu^{8} + 43 \nu^{7} + 66 \nu^{6} - 30 \nu^{5} - 81 \nu^{4} + 792 \nu^{3} + 864 \nu^{2} - 1053 \nu - 972 \)\()/972\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - \beta_{6} - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} - 4 \beta_{10} - 5 \beta_{9} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(6 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 15\)
\(\nu^{7}\)\(=\)\(2 \beta_{11} + 10 \beta_{10} + 16 \beta_{9} + 14 \beta_{8} + 4 \beta_{7} - 6 \beta_{6} - 6 \beta_{5} + 6 \beta_{3} + 10 \beta_{2} + 5 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-12 \beta_{10} - 12 \beta_{9} - 12 \beta_{8} - 26 \beta_{7} + 4 \beta_{6} + 21 \beta_{5} - 45 \beta_{4} - 15 \beta_{3} + 12 \beta_{1} - 10\)
\(\nu^{9}\)\(=\)\(-5 \beta_{11} - 41 \beta_{10} + 17 \beta_{9} + 57 \beta_{8} - 3 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - 8 \beta_{3} - 11 \beta_{2} - 25 \beta_{1}\)
\(\nu^{10}\)\(=\)\(9 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + 60 \beta_{7} + 27 \beta_{6} - 44 \beta_{5} - 151 \beta_{4} + 29 \beta_{3} - 9 \beta_{1} - 120\)
\(\nu^{11}\)\(=\)\(16 \beta_{11} - 124 \beta_{10} + 65 \beta_{9} - 92 \beta_{8} + 20 \beta_{7} - 36 \beta_{6} - 36 \beta_{5} + 36 \beta_{3} + 56 \beta_{2} - 116 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
0.721683 1.57454i
0.721683 + 1.57454i
−1.06864 + 1.36309i
−1.06864 1.36309i
−1.68439 + 0.403509i
−1.68439 0.403509i
1.68439 + 0.403509i
1.68439 0.403509i
1.06864 + 1.36309i
1.06864 1.36309i
−0.721683 1.57454i
−0.721683 + 1.57454i
−1.00000 −1.57454 0.721683i 1.00000 0 1.57454 + 0.721683i −2.29622 + 1.31429i −1.00000 1.95835 + 2.27264i 0
1049.2 −1.00000 −1.57454 + 0.721683i 1.00000 0 1.57454 0.721683i −2.29622 1.31429i −1.00000 1.95835 2.27264i 0
1049.3 −1.00000 −1.36309 1.06864i 1.00000 0 1.36309 + 1.06864i −0.294447 + 2.62932i −1.00000 0.716015 + 2.91330i 0
1049.4 −1.00000 −1.36309 + 1.06864i 1.00000 0 1.36309 1.06864i −0.294447 2.62932i −1.00000 0.716015 2.91330i 0
1049.5 −1.00000 −0.403509 1.68439i 1.00000 0 0.403509 + 1.68439i 1.28088 2.31502i −1.00000 −2.67436 + 1.35934i 0
1049.6 −1.00000 −0.403509 + 1.68439i 1.00000 0 0.403509 1.68439i 1.28088 + 2.31502i −1.00000 −2.67436 1.35934i 0
1049.7 −1.00000 0.403509 1.68439i 1.00000 0 −0.403509 + 1.68439i −1.28088 + 2.31502i −1.00000 −2.67436 1.35934i 0
1049.8 −1.00000 0.403509 + 1.68439i 1.00000 0 −0.403509 1.68439i −1.28088 2.31502i −1.00000 −2.67436 + 1.35934i 0
1049.9 −1.00000 1.36309 1.06864i 1.00000 0 −1.36309 + 1.06864i 0.294447 2.62932i −1.00000 0.716015 2.91330i 0
1049.10 −1.00000 1.36309 + 1.06864i 1.00000 0 −1.36309 1.06864i 0.294447 + 2.62932i −1.00000 0.716015 + 2.91330i 0
1049.11 −1.00000 1.57454 0.721683i 1.00000 0 −1.57454 + 0.721683i 2.29622 1.31429i −1.00000 1.95835 2.27264i 0
1049.12 −1.00000 1.57454 + 0.721683i 1.00000 0 −1.57454 0.721683i 2.29622 + 1.31429i −1.00000 1.95835 + 2.27264i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1049.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.d.g 12
3.b odd 2 1 1050.2.d.h 12
5.b even 2 1 1050.2.d.h 12
5.c odd 4 1 1050.2.b.d 12
5.c odd 4 1 1050.2.b.e yes 12
7.b odd 2 1 inner 1050.2.d.g 12
15.d odd 2 1 inner 1050.2.d.g 12
15.e even 4 1 1050.2.b.d 12
15.e even 4 1 1050.2.b.e yes 12
21.c even 2 1 1050.2.d.h 12
35.c odd 2 1 1050.2.d.h 12
35.f even 4 1 1050.2.b.d 12
35.f even 4 1 1050.2.b.e yes 12
105.g even 2 1 inner 1050.2.d.g 12
105.k odd 4 1 1050.2.b.d 12
105.k odd 4 1 1050.2.b.e yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.b.d 12 5.c odd 4 1
1050.2.b.d 12 15.e even 4 1
1050.2.b.d 12 35.f even 4 1
1050.2.b.d 12 105.k odd 4 1
1050.2.b.e yes 12 5.c odd 4 1
1050.2.b.e yes 12 15.e even 4 1
1050.2.b.e yes 12 35.f even 4 1
1050.2.b.e yes 12 105.k odd 4 1
1050.2.d.g 12 1.a even 1 1 trivial
1050.2.d.g 12 7.b odd 2 1 inner
1050.2.d.g 12 15.d odd 2 1 inner
1050.2.d.g 12 105.g even 2 1 inner
1050.2.d.h 12 3.b odd 2 1
1050.2.d.h 12 5.b even 2 1
1050.2.d.h 12 21.c even 2 1
1050.2.d.h 12 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{6} + 46 T_{11}^{4} + 529 T_{11}^{2} + 900 \)
\( T_{13}^{6} - 63 T_{13}^{4} + 1300 T_{13}^{2} - 8748 \)
\( T_{23}^{3} + 5 T_{23}^{2} - 20 T_{23} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( 1 + 4 T^{4} + 30 T^{6} + 36 T^{8} + 729 T^{12} \)
$5$ 1
$7$ \( 1 + 14 T^{2} + 99 T^{4} + 652 T^{6} + 4851 T^{8} + 33614 T^{10} + 117649 T^{12} \)
$11$ \( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 15 T^{2} + 559 T^{4} + 5110 T^{6} + 94471 T^{8} + 428415 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 - 29 T^{2} + 494 T^{4} - 5297 T^{6} + 142766 T^{8} - 2422109 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 + 5 T + 49 T^{2} + 224 T^{3} + 1127 T^{4} + 2645 T^{5} + 12167 T^{6} )^{4} \)
$29$ \( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 70 T^{2} + 2907 T^{4} - 101660 T^{6} + 3979683 T^{8} - 131191270 T^{10} + 2565726409 T^{12} )^{2} \)
$41$ \( ( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 209 T^{2} + 19659 T^{4} - 1076742 T^{6} + 36349491 T^{8} - 714529409 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 - 78 T^{2} + 7315 T^{4} - 322124 T^{6} + 16158835 T^{8} - 380615118 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 + 5 T + 95 T^{2} + 548 T^{3} + 5035 T^{4} + 14045 T^{5} + 148877 T^{6} )^{4} \)
$59$ \( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 260 T^{2} + 33680 T^{4} - 2779070 T^{6} + 151189520 T^{8} - 5239291460 T^{10} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 + 332 T^{2} + 52356 T^{4} + 4849846 T^{6} + 279005124 T^{8} + 9428216012 T^{10} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 + 14 T + 193 T^{2} + 2172 T^{3} + 15247 T^{4} + 87374 T^{5} + 493039 T^{6} )^{4} \)
$83$ \( ( 1 - 309 T^{2} + 44338 T^{4} - 4238477 T^{6} + 305444482 T^{8} - 14664621189 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 158 T^{2} + 21375 T^{4} + 2986564 T^{6} + 201117375 T^{8} + 13987626398 T^{10} + 832972004929 T^{12} )^{2} \)
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