Properties

Label 845.2.d.d
Level $845$
Weight $2$
Character orbit 845.d
Analytic conductor $6.747$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 22 x^{10} + 147 x^{8} + 390 x^{6} + 413 x^{4} + 128 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{3} - \beta_{11} ) q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{9} q^{5} + ( -2 \beta_{4} - \beta_{10} ) q^{6} + ( -\beta_{1} - \beta_{6} ) q^{7} + ( \beta_{1} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{8} + ( -1 - \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{3} - \beta_{11} ) q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{9} q^{5} + ( -2 \beta_{4} - \beta_{10} ) q^{6} + ( -\beta_{1} - \beta_{6} ) q^{7} + ( \beta_{1} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{8} + ( -1 - \beta_{7} - \beta_{8} ) q^{9} + ( 1 - \beta_{2} + \beta_{8} ) q^{10} + ( \beta_{4} - \beta_{5} + \beta_{9} ) q^{11} + ( 1 - 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{12} + ( -4 + \beta_{2} ) q^{14} + ( \beta_{1} - \beta_{4} - 2 \beta_{6} - \beta_{10} ) q^{15} + ( 3 - \beta_{2} + \beta_{7} + \beta_{8} ) q^{16} + ( 1 + \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{17} + ( \beta_{5} + \beta_{9} ) q^{18} + ( 3 \beta_{4} - \beta_{5} + \beta_{9} ) q^{19} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{20} + ( \beta_{4} - \beta_{5} + \beta_{9} + 2 \beta_{10} ) q^{21} + ( 1 + \beta_{3} + \beta_{7} - \beta_{8} ) q^{22} + ( -\beta_{3} - \beta_{11} ) q^{23} + ( -5 \beta_{4} - \beta_{5} + \beta_{9} - 2 \beta_{10} ) q^{24} + ( 1 - \beta_{2} + 2 \beta_{3} - \beta_{8} - \beta_{11} ) q^{25} + ( -\beta_{3} + \beta_{11} ) q^{27} + ( -4 \beta_{1} + \beta_{5} + 3 \beta_{6} + \beta_{9} ) q^{28} -3 q^{29} + ( 2 - \beta_{2} - 3 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{30} + ( 2 \beta_{4} + 2 \beta_{10} ) q^{31} + ( 2 \beta_{1} + \beta_{6} ) q^{32} + ( -\beta_{1} + 3 \beta_{6} ) q^{33} + ( 2 \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} ) q^{34} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{8} + 2 \beta_{11} ) q^{35} + ( -1 + 2 \beta_{2} + \beta_{7} + \beta_{8} ) q^{36} + ( -\beta_{1} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{37} + ( 1 + 3 \beta_{3} + \beta_{7} - \beta_{8} ) q^{38} + ( 7 - 2 \beta_{2} + 3 \beta_{3} - \beta_{8} + \beta_{11} ) q^{40} + ( 3 \beta_{4} + 2 \beta_{10} ) q^{41} + ( -1 + 5 \beta_{3} - \beta_{7} + \beta_{8} + 2 \beta_{11} ) q^{42} + ( \beta_{3} + 3 \beta_{11} ) q^{43} + ( -\beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} ) q^{44} + ( -2 \beta_{1} - 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{45} + ( -2 \beta_{4} - \beta_{10} ) q^{46} + ( -6 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{47} + ( 1 - 5 \beta_{3} + \beta_{7} - \beta_{8} - 4 \beta_{11} ) q^{48} + ( 1 + \beta_{7} + \beta_{8} ) q^{49} + ( 3 \beta_{1} + 6 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{50} + ( -2 - \beta_{7} - \beta_{8} ) q^{51} + ( -1 - 2 \beta_{3} - \beta_{7} + \beta_{8} - 4 \beta_{11} ) q^{53} + ( -4 \beta_{4} - \beta_{10} ) q^{54} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{55} + ( -4 + 4 \beta_{2} - \beta_{7} - \beta_{8} ) q^{56} + ( \beta_{1} + \beta_{6} ) q^{57} -3 \beta_{1} q^{58} + ( \beta_{4} - \beta_{5} + \beta_{9} - 2 \beta_{10} ) q^{59} + ( \beta_{1} - 8 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{10} ) q^{60} + ( -1 - 2 \beta_{7} - 2 \beta_{8} ) q^{61} + ( -2 + 6 \beta_{3} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{62} + ( -2 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{9} ) q^{63} + ( 1 - 2 \beta_{7} - 2 \beta_{8} ) q^{64} + \beta_{2} q^{66} + ( -3 \beta_{1} - \beta_{6} ) q^{67} + ( -2 \beta_{3} + \beta_{11} ) q^{68} + ( -4 - \beta_{7} - \beta_{8} ) q^{69} + ( -3 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{9} - \beta_{10} ) q^{70} + ( -\beta_{4} - \beta_{5} + \beta_{9} ) q^{71} + ( -6 \beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{72} + ( 6 \beta_{1} + \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{73} + ( -1 - \beta_{2} + \beta_{7} + \beta_{8} ) q^{74} + ( 2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{8} + 3 \beta_{11} ) q^{75} + ( \beta_{4} + \beta_{5} - \beta_{9} + \beta_{10} ) q^{76} + ( -2 - 3 \beta_{3} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{11} ) q^{77} + ( -8 - 2 \beta_{2} ) q^{79} + ( 5 \beta_{1} + 5 \beta_{4} - \beta_{9} ) q^{80} + ( -3 + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{8} ) q^{81} + ( -2 + 7 \beta_{3} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{9} ) q^{83} + ( 13 \beta_{4} + 3 \beta_{5} - 3 \beta_{9} + 3 \beta_{10} ) q^{84} + ( 3 \beta_{1} - 5 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{85} + \beta_{10} q^{86} + ( 3 \beta_{3} + 3 \beta_{11} ) q^{87} + ( -2 - 5 \beta_{3} - 2 \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{88} + ( -3 \beta_{4} + \beta_{5} - \beta_{9} + 4 \beta_{10} ) q^{89} + ( -6 + \beta_{2} - 2 \beta_{3} + \beta_{8} + \beta_{11} ) q^{90} + ( 1 - 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{92} + ( 4 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{93} + ( -10 + 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} ) q^{94} + ( 1 + \beta_{2} - 2 \beta_{3} - 3 \beta_{7} + \beta_{8} + \beta_{11} ) q^{95} + ( -3 \beta_{4} + \beta_{5} - \beta_{9} - 3 \beta_{10} ) q^{96} + ( 3 \beta_{1} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{9} ) q^{97} + ( -\beta_{5} - \beta_{9} ) q^{98} + ( 8 \beta_{4} - 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 8q^{4} - 12q^{9} + O(q^{10}) \) \( 12q + 8q^{4} - 12q^{9} + 14q^{10} - 44q^{14} + 32q^{16} + 2q^{25} - 36q^{29} + 8q^{30} - 20q^{35} - 4q^{36} + 70q^{40} + 12q^{49} - 24q^{51} + 52q^{55} - 32q^{56} - 12q^{61} + 12q^{64} + 4q^{66} - 48q^{69} - 16q^{74} + 4q^{75} - 104q^{79} - 28q^{81} - 62q^{90} - 112q^{94} + 40q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 22 x^{10} + 147 x^{8} + 390 x^{6} + 413 x^{4} + 128 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4 \nu^{10} - 77 \nu^{8} - 372 \nu^{6} - 452 \nu^{4} + 67 \nu^{2} + 124 \)\()/102\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{10} + 77 \nu^{8} + 372 \nu^{6} + 452 \nu^{4} - 16 \nu^{2} + 80 \)\()/51\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{11} + 250 \nu^{9} + 1771 \nu^{7} + 5034 \nu^{5} + 5447 \nu^{3} + 1478 \nu \)\()/204\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{11} + 250 \nu^{9} + 1771 \nu^{7} + 5034 \nu^{5} + 5447 \nu^{3} + 1274 \nu \)\()/204\)
\(\beta_{5}\)\(=\)\((\)\( -25 \nu^{11} + 30 \nu^{10} - 528 \nu^{9} + 620 \nu^{8} - 3243 \nu^{7} + 3606 \nu^{6} - 7568 \nu^{5} + 7368 \nu^{4} - 7465 \nu^{3} + 5218 \nu^{2} - 3390 \nu + 872 \)\()/408\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{10} - 248 \nu^{8} - 1439 \nu^{6} - 2869 \nu^{4} - 1635 \nu^{2} + 32 \)\()/102\)
\(\beta_{7}\)\(=\)\((\)\( 33 \nu^{11} - 8 \nu^{10} + 716 \nu^{9} - 188 \nu^{8} + 4667 \nu^{7} - 1424 \nu^{6} + 12076 \nu^{5} - 4508 \nu^{4} + 12873 \nu^{3} - 5408 \nu^{2} + 4162 \nu - 1180 \)\()/408\)
\(\beta_{8}\)\(=\)\((\)\( -33 \nu^{11} - 8 \nu^{10} - 716 \nu^{9} - 188 \nu^{8} - 4667 \nu^{7} - 1424 \nu^{6} - 12076 \nu^{5} - 4508 \nu^{4} - 12873 \nu^{3} - 5408 \nu^{2} - 4162 \nu - 772 \)\()/408\)
\(\beta_{9}\)\(=\)\((\)\( 25 \nu^{11} + 30 \nu^{10} + 528 \nu^{9} + 620 \nu^{8} + 3243 \nu^{7} + 3606 \nu^{6} + 7568 \nu^{5} + 7368 \nu^{4} + 7465 \nu^{3} + 5218 \nu^{2} + 3390 \nu + 872 \)\()/408\)
\(\beta_{10}\)\(=\)\((\)\( -15 \nu^{11} - 327 \nu^{9} - 2143 \nu^{7} - 5486 \nu^{5} - 5380 \nu^{3} - 1048 \nu \)\()/102\)
\(\beta_{11}\)\(=\)\((\)\( -35 \nu^{11} - 746 \nu^{9} - 4649 \nu^{7} - 10772 \nu^{5} - 8717 \nu^{3} - 1210 \nu \)\()/204\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{4} + \beta_{3}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 3 \beta_{10} - \beta_{8} + \beta_{7} + 10 \beta_{4} - 8 \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(4 \beta_{9} + \beta_{8} + \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 13 \beta_{2} - 24 \beta_{1} + 36\)
\(\nu^{5}\)\(=\)\(17 \beta_{11} - 45 \beta_{10} + 5 \beta_{9} + 18 \beta_{8} - 18 \beta_{7} - 5 \beta_{5} - 111 \beta_{4} + 85 \beta_{3} - 18\)
\(\nu^{6}\)\(=\)\(-68 \beta_{9} - 23 \beta_{8} - 23 \beta_{7} - 62 \beta_{6} - 68 \beta_{5} + 166 \beta_{2} + 286 \beta_{1} - 408\)
\(\nu^{7}\)\(=\)\(-228 \beta_{11} + 588 \beta_{10} - 91 \beta_{9} - 257 \beta_{8} + 257 \beta_{7} + 91 \beta_{5} + 1317 \beta_{4} - 1003 \beta_{3} + 257\)
\(\nu^{8}\)\(=\)\(936 \beta_{9} + 348 \beta_{8} + 348 \beta_{7} + 816 \beta_{6} + 936 \beta_{5} - 2105 \beta_{2} - 3496 \beta_{1} + 4960\)
\(\nu^{9}\)\(=\)\(2921 \beta_{11} - 7473 \beta_{10} + 1284 \beta_{9} + 3389 \beta_{8} - 3389 \beta_{7} - 1284 \beta_{5} - 16156 \beta_{4} + 12318 \beta_{3} - 3389\)
\(\nu^{10}\)\(=\)\(-12146 \beta_{9} - 4673 \beta_{8} - 4673 \beta_{7} - 10394 \beta_{6} - 12146 \beta_{5} + 26569 \beta_{2} + 43420 \beta_{1} - 61640\)
\(\nu^{11}\)\(=\)\(-36963 \beta_{11} + 94281 \beta_{10} - 16819 \beta_{9} - 43388 \beta_{8} + 43388 \beta_{7} + 16819 \beta_{5} + 201125 \beta_{4} - 153525 \beta_{3} + 43388\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
844.1
3.54574i
3.54574i
2.18733i
2.18733i
1.33084i
1.33084i
0.669163i
0.669163i
0.187333i
0.187333i
1.54574i
1.54574i
−2.54574 2.15293i 4.48079 −2.08125 0.817544i 5.48079i 2.93855 −6.31544 −1.63509 5.29833 + 2.08125i
844.2 −2.54574 2.15293i 4.48079 −2.08125 + 0.817544i 5.48079i 2.93855 −6.31544 −1.63509 5.29833 2.08125i
844.3 −1.18733 0.345110i −0.590239 1.71029 + 1.44045i 0.409761i 2.02956 3.07548 2.88090 −2.03069 1.71029i
844.4 −1.18733 0.345110i −0.590239 1.71029 1.44045i 0.409761i 2.02956 3.07548 2.88090 −2.03069 + 1.71029i
844.5 −0.330837 2.69180i −1.89055 −0.702335 + 2.12291i 0.890547i 3.35348 1.28714 −4.24581 0.232358 0.702335i
844.6 −0.330837 2.69180i −1.89055 −0.702335 2.12291i 0.890547i 3.35348 1.28714 −4.24581 0.232358 + 0.702335i
844.7 0.330837 2.69180i −1.89055 0.702335 2.12291i 0.890547i −3.35348 −1.28714 −4.24581 0.232358 0.702335i
844.8 0.330837 2.69180i −1.89055 0.702335 + 2.12291i 0.890547i −3.35348 −1.28714 −4.24581 0.232358 + 0.702335i
844.9 1.18733 0.345110i −0.590239 −1.71029 1.44045i 0.409761i −2.02956 −3.07548 2.88090 −2.03069 1.71029i
844.10 1.18733 0.345110i −0.590239 −1.71029 + 1.44045i 0.409761i −2.02956 −3.07548 2.88090 −2.03069 + 1.71029i
844.11 2.54574 2.15293i 4.48079 2.08125 + 0.817544i 5.48079i −2.93855 6.31544 −1.63509 5.29833 + 2.08125i
844.12 2.54574 2.15293i 4.48079 2.08125 0.817544i 5.48079i −2.93855 6.31544 −1.63509 5.29833 2.08125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 844.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.d.d 12
5.b even 2 1 inner 845.2.d.d 12
13.b even 2 1 inner 845.2.d.d 12
13.c even 3 2 845.2.l.f 24
13.d odd 4 1 845.2.b.d 6
13.d odd 4 1 845.2.b.e 6
13.e even 6 2 845.2.l.f 24
13.f odd 12 2 65.2.n.a 12
13.f odd 12 2 845.2.n.e 12
39.k even 12 2 585.2.bs.a 12
52.l even 12 2 1040.2.dh.a 12
65.d even 2 1 inner 845.2.d.d 12
65.f even 4 1 4225.2.a.bq 6
65.f even 4 1 4225.2.a.br 6
65.g odd 4 1 845.2.b.d 6
65.g odd 4 1 845.2.b.e 6
65.k even 4 1 4225.2.a.bq 6
65.k even 4 1 4225.2.a.br 6
65.l even 6 2 845.2.l.f 24
65.n even 6 2 845.2.l.f 24
65.o even 12 2 325.2.e.e 12
65.s odd 12 2 65.2.n.a 12
65.s odd 12 2 845.2.n.e 12
65.t even 12 2 325.2.e.e 12
195.bh even 12 2 585.2.bs.a 12
260.bc even 12 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 13.f odd 12 2
65.2.n.a 12 65.s odd 12 2
325.2.e.e 12 65.o even 12 2
325.2.e.e 12 65.t even 12 2
585.2.bs.a 12 39.k even 12 2
585.2.bs.a 12 195.bh even 12 2
845.2.b.d 6 13.d odd 4 1
845.2.b.d 6 65.g odd 4 1
845.2.b.e 6 13.d odd 4 1
845.2.b.e 6 65.g odd 4 1
845.2.d.d 12 1.a even 1 1 trivial
845.2.d.d 12 5.b even 2 1 inner
845.2.d.d 12 13.b even 2 1 inner
845.2.d.d 12 65.d even 2 1 inner
845.2.l.f 24 13.c even 3 2
845.2.l.f 24 13.e even 6 2
845.2.l.f 24 65.l even 6 2
845.2.l.f 24 65.n even 6 2
845.2.n.e 12 13.f odd 12 2
845.2.n.e 12 65.s odd 12 2
1040.2.dh.a 12 52.l even 12 2
1040.2.dh.a 12 260.bc even 12 2
4225.2.a.bq 6 65.f even 4 1
4225.2.a.bq 6 65.k even 4 1
4225.2.a.br 6 65.f even 4 1
4225.2.a.br 6 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 8 T_{2}^{4} + 10 T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + 10 T^{2} - 8 T^{4} + T^{6} )^{2} \)
$3$ \( ( 4 + 35 T^{2} + 12 T^{4} + T^{6} )^{2} \)
$5$ \( 15625 - 625 T^{2} + 375 T^{4} + 50 T^{6} + 15 T^{8} - T^{10} + T^{12} \)
$7$ \( ( -400 + 179 T^{2} - 24 T^{4} + T^{6} )^{2} \)
$11$ \( ( 64 + 169 T^{2} + 26 T^{4} + T^{6} )^{2} \)
$13$ \( T^{12} \)
$17$ \( ( 169 + 163 T^{2} + 35 T^{4} + T^{6} )^{2} \)
$19$ \( ( 100 + 121 T^{2} + 38 T^{4} + T^{6} )^{2} \)
$23$ \( ( 4 + 35 T^{2} + 12 T^{4} + T^{6} )^{2} \)
$29$ \( ( 3 + T )^{12} \)
$31$ \( ( 1600 + 1280 T^{2} + 96 T^{4} + T^{6} )^{2} \)
$37$ \( ( -169 + 163 T^{2} - 35 T^{4} + T^{6} )^{2} \)
$41$ \( ( 25 + 771 T^{2} + 107 T^{4} + T^{6} )^{2} \)
$43$ \( ( 256 + 283 T^{2} + 80 T^{4} + T^{6} )^{2} \)
$47$ \( ( -270400 + 14640 T^{2} - 236 T^{4} + T^{6} )^{2} \)
$53$ \( ( 400 + 1040 T^{2} + 171 T^{4} + T^{6} )^{2} \)
$59$ \( ( 18496 + 3569 T^{2} + 114 T^{4} + T^{6} )^{2} \)
$61$ \( ( -115 - 49 T + 3 T^{2} + T^{3} )^{4} \)
$67$ \( ( -20164 + 2603 T^{2} - 100 T^{4} + T^{6} )^{2} \)
$71$ \( ( 676 + 313 T^{2} + 38 T^{4} + T^{6} )^{2} \)
$73$ \( ( -250000 + 13900 T^{2} - 215 T^{4} + T^{6} )^{2} \)
$79$ \( ( 160 + 180 T + 26 T^{2} + T^{3} )^{4} \)
$83$ \( ( -640000 + 23600 T^{2} - 276 T^{4} + T^{6} )^{2} \)
$89$ \( ( 2515396 + 56369 T^{2} + 414 T^{4} + T^{6} )^{2} \)
$97$ \( ( -204304 + 14363 T^{2} - 280 T^{4} + T^{6} )^{2} \)
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