Properties

Label 1050.2.b.e
Level 1050
Weight 2
Character orbit 1050.b
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.b (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.68439 0.403509i
−1.06864 1.36309i
−0.721683 + 1.57454i
0.721683 1.57454i
1.06864 + 1.36309i
1.68439 + 0.403509i
−1.68439 + 0.403509i
−1.06864 + 1.36309i
−0.721683 1.57454i
0.721683 + 1.57454i
1.06864 1.36309i
1.68439 0.403509i
1.00000i −1.68439 + 0.403509i −1.00000 0 0.403509 + 1.68439i 2.31502 + 1.28088i 1.00000i 2.67436 1.35934i 0
251.2 1.00000i −1.06864 + 1.36309i −1.00000 0 1.36309 + 1.06864i −2.62932 0.294447i 1.00000i −0.716015 2.91330i 0
251.3 1.00000i −0.721683 1.57454i −1.00000 0 −1.57454 + 0.721683i 1.31429 + 2.29622i 1.00000i −1.95835 + 2.27264i 0
251.4 1.00000i 0.721683 + 1.57454i −1.00000 0 1.57454 0.721683i 1.31429 2.29622i 1.00000i −1.95835 + 2.27264i 0
251.5 1.00000i 1.06864 1.36309i −1.00000 0 −1.36309 1.06864i −2.62932 + 0.294447i 1.00000i −0.716015 2.91330i 0
251.6 1.00000i 1.68439 0.403509i −1.00000 0 −0.403509 1.68439i 2.31502 1.28088i 1.00000i 2.67436 1.35934i 0
251.7 1.00000i −1.68439 0.403509i −1.00000 0 0.403509 1.68439i 2.31502 1.28088i 1.00000i 2.67436 + 1.35934i 0
251.8 1.00000i −1.06864 1.36309i −1.00000 0 1.36309 1.06864i −2.62932 + 0.294447i 1.00000i −0.716015 + 2.91330i 0
251.9 1.00000i −0.721683 + 1.57454i −1.00000 0 −1.57454 0.721683i 1.31429 2.29622i 1.00000i −1.95835 2.27264i 0
251.10 1.00000i 0.721683 1.57454i −1.00000 0 1.57454 + 0.721683i 1.31429 + 2.29622i 1.00000i −1.95835 2.27264i 0
251.11 1.00000i 1.06864 + 1.36309i −1.00000 0 −1.36309 + 1.06864i −2.62932 0.294447i 1.00000i −0.716015 + 2.91330i 0
251.12 1.00000i 1.68439 + 0.403509i −1.00000 0 −0.403509 + 1.68439i 2.31502 + 1.28088i 1.00000i 2.67436 + 1.35934i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{11}^{6} + 46 T_{11}^{4} + 529 T_{11}^{2} + 900 \)
\( T_{17}^{6} - 73 T_{17}^{4} + 1123 T_{17}^{2} - 4563 \)
\( T_{37}^{3} - 2 T_{37}^{2} - 74 T_{37} - 152 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{6} \)
$3$ \( 1 + 4 T^{4} - 30 T^{6} + 36 T^{8} + 729 T^{12} \)
$5$ 1
$7$ \( ( 1 - 2 T - 5 T^{2} + 36 T^{3} - 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 - 15 T^{2} + 559 T^{4} - 5110 T^{6} + 94471 T^{8} - 428415 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 + 29 T^{2} + 494 T^{4} + 5297 T^{6} + 142766 T^{8} + 2422109 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 73 T^{2} + 2415 T^{4} - 58154 T^{6} + 1277535 T^{8} - 20428393 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} - 300 T^{3} + 1369 T^{4} - 2738 T^{5} + 50653 T^{6} )^{4} \)
$41$ \( ( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 + 3 T + 109 T^{2} + 266 T^{3} + 4687 T^{4} + 5547 T^{5} + 79507 T^{6} )^{4} \)
$47$ \( ( 1 + 78 T^{2} + 7315 T^{4} + 322124 T^{6} + 16158835 T^{8} + 380615118 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( ( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 38244535 T^{8} - 1301929365 T^{10} + 22164361129 T^{12} )^{2} \)
$59$ \( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2} \)
$61$ \( ( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 16 T + 258 T^{2} - 2108 T^{3} + 17286 T^{4} - 71824 T^{5} + 300763 T^{6} )^{4} \)
$71$ \( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 - 332 T^{2} + 52356 T^{4} - 4849846 T^{6} + 279005124 T^{8} - 9428216012 T^{10} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 - 14 T + 193 T^{2} - 2172 T^{3} + 15247 T^{4} - 87374 T^{5} + 493039 T^{6} )^{4} \)
$83$ \( ( 1 + 309 T^{2} + 44338 T^{4} + 4238477 T^{6} + 305444482 T^{8} + 14664621189 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 - 158 T^{2} + 21375 T^{4} - 2986564 T^{6} + 201117375 T^{8} - 13987626398 T^{10} + 832972004929 T^{12} )^{2} \)
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