Properties

Label 1040.2.dh.a
Level $1040$
Weight $2$
Character orbit 1040.dh
Analytic conductor $8.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.dh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{10} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{10} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{7} ) q^{9} + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} ) q^{11} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} ) q^{13} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{15} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{17} + ( -2 - \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{19} + ( 1 - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{21} + ( \beta_{1} - \beta_{10} ) q^{23} + ( -1 - \beta_{2} - \beta_{4} - \beta_{8} - \beta_{11} ) q^{25} + ( \beta_{2} + \beta_{11} ) q^{27} -3 \beta_{5} q^{29} + ( 2 - \beta_{2} + 2 \beta_{4} + \beta_{10} - \beta_{11} ) q^{31} + ( 2 \beta_{1} + \beta_{2} ) q^{33} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{9} ) q^{35} + ( -1 + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{39} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{5} + \beta_{9} ) q^{45} + ( -2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{47} + ( -1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{49} + ( -1 + \beta_{6} + \beta_{7} + \beta_{8} ) q^{51} + ( \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{55} + ( \beta_{2} + \beta_{11} ) q^{57} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - 2 \beta_{9} ) q^{59} + ( 1 - 2 \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{61} + ( 2 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{63} + ( \beta_{1} - \beta_{3} - 5 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{65} + ( -\beta_{1} + \beta_{10} - 2 \beta_{11} ) q^{67} + ( -4 + \beta_{3} - 4 \beta_{5} + \beta_{7} ) q^{69} + ( 2 - \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{71} + ( -2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{73} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{75} + ( -2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} ) q^{77} + ( 8 + \beta_{2} - 2 \beta_{4} - \beta_{10} + \beta_{11} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{81} + ( \beta_{2} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{83} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{85} -3 \beta_{1} q^{87} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{89} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{91} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{93} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{95} + ( -2 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( -8 - \beta_{2} + 2 \beta_{4} + \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{5} + 6q^{9} + O(q^{10}) \) \( 12q - 6q^{5} + 6q^{9} + 4q^{15} - 12q^{19} - 8q^{21} - 2q^{25} + 18q^{29} + 16q^{31} - 10q^{35} + 32q^{39} + 14q^{41} - 29q^{45} + 6q^{49} - 24q^{51} + 26q^{55} + 4q^{59} + 6q^{61} + 23q^{65} - 24q^{69} + 12q^{71} - 2q^{75} + 104q^{79} + 14q^{81} + 21q^{85} + 20q^{89} + 44q^{91} - 20q^{95} - 104q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 108 \nu^{11} - 729 \nu^{9} + 4768 \nu^{7} - 1242 \nu^{5} + 135 \nu^{3} + 9934 \nu \)\()/1222\)
\(\beta_{2}\)\(=\)\((\)\( 108 \nu^{11} - 729 \nu^{9} + 4768 \nu^{7} - 1242 \nu^{5} + 135 \nu^{3} + 12378 \nu \)\()/1222\)
\(\beta_{3}\)\(=\)\((\)\( -92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045 \)\()/2444\)
\(\beta_{4}\)\(=\)\((\)\( 135 \nu^{11} - 16 \nu^{10} - 1064 \nu^{9} + 108 \nu^{8} + 7182 \nu^{7} - 729 \nu^{6} - 9801 \nu^{5} + 184 \nu^{4} + 12236 \nu^{3} - 20 \nu^{2} - 108 \nu - 3531 \)\()/1222\)
\(\beta_{5}\)\(=\)\((\)\( -135 \nu^{10} + 1064 \nu^{8} - 7182 \nu^{6} + 9801 \nu^{4} - 12236 \nu^{2} + 108 \)\()/1222\)
\(\beta_{6}\)\(=\)\((\)\( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524 \)\()/2444\)
\(\beta_{7}\)\(=\)\((\)\( 92 \nu^{11} + 563 \nu^{10} - 621 \nu^{9} - 4564 \nu^{8} + 4039 \nu^{7} + 30807 \nu^{6} - 1058 \nu^{5} - 46495 \nu^{4} + 115 \nu^{3} + 52486 \nu^{2} + 5181 \nu - 5705 \)\()/2444\)
\(\beta_{8}\)\(=\)\((\)\( -655 \nu^{11} - 92 \nu^{10} + 5185 \nu^{9} + 621 \nu^{8} - 34846 \nu^{7} - 4039 \nu^{6} + 47553 \nu^{5} + 1058 \nu^{4} - 52601 \nu^{3} - 115 \nu^{2} + 524 \nu - 5181 \)\()/2444\)
\(\beta_{9}\)\(=\)\((\)\( 389 \nu^{10} - 3084 \nu^{8} + 20817 \nu^{6} - 29219 \nu^{4} + 35466 \nu^{2} - 1222 \nu - 3855 \)\()/1222\)
\(\beta_{10}\)\(=\)\((\)\( 1195 \nu^{11} - 9441 \nu^{9} + 63574 \nu^{7} - 86757 \nu^{5} + 100323 \nu^{3} - 956 \nu \)\()/1222\)
\(\beta_{11}\)\(=\)\((\)\( 1357 \nu^{11} - 10840 \nu^{9} + 73170 \nu^{7} - 105117 \nu^{5} + 124660 \nu^{3} - 13550 \nu \)\()/1222\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} + \beta_{10} - 2 \beta_{9} - 6 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{11} - 3 \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{2}\)
\(\nu^{4}\)\(=\)\((\)\(-14 \beta_{9} + 2 \beta_{7} - 34 \beta_{5} + 2 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} - 34\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(23 \beta_{11} - 37 \beta_{10} - 16 \beta_{8} + 16 \beta_{6} - 16 \beta_{5} + 16 \beta_{3} + 37 \beta_{1} - 16\)\()/2\)
\(\nu^{6}\)\(=\)\(23 \beta_{11} - 23 \beta_{10} + 8 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} - 46 \beta_{4} + 23 \beta_{2} - 99\)
\(\nu^{7}\)\(=\)\((\)\(-108 \beta_{7} - 108 \beta_{5} + 108 \beta_{3} - 145 \beta_{2} + 237 \beta_{1} - 108\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(299 \beta_{11} - 299 \beta_{10} + 598 \beta_{9} + 108 \beta_{8} + 108 \beta_{6} + 1378 \beta_{5} - 598 \beta_{4} - 108 \beta_{3} + 598 \beta_{2} - 299 \beta_{1} + 108\)\()/2\)
\(\nu^{9}\)\(=\)\(-467 \beta_{11} + 766 \beta_{10} + 353 \beta_{8} - 353 \beta_{7} - 353 \beta_{6} - 467 \beta_{2}\)
\(\nu^{10}\)\(=\)\((\)\(3878 \beta_{9} - 706 \beta_{7} + 8918 \beta_{5} - 706 \beta_{3} + 1939 \beta_{2} - 1939 \beta_{1} + 8918\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-6045 \beta_{11} + 9923 \beta_{10} + 4584 \beta_{8} - 4584 \beta_{6} + 4584 \beta_{5} - 4584 \beta_{3} - 9923 \beta_{1} + 4584\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.286513 + 0.165418i
−2.20467 1.27287i
−1.02826 0.593667i
1.02826 + 0.593667i
2.20467 + 1.27287i
−0.286513 0.165418i
0.286513 0.165418i
−2.20467 + 1.27287i
−1.02826 + 0.593667i
1.02826 0.593667i
2.20467 1.27287i
−0.286513 + 0.165418i
0 −2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 2.90420 + 1.67674i 0 2.12291 3.67698i 0
289.2 0 −1.86449 + 1.07646i 0 −0.817544 2.08125i 0 −2.54486 1.46928i 0 0.817544 1.41603i 0
289.3 0 −0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 −1.75765 1.01478i 0 −1.44045 + 2.49493i 0
289.4 0 0.298874 0.172555i 0 1.44045 1.71029i 0 1.75765 + 1.01478i 0 −1.44045 + 2.49493i 0
289.5 0 1.86449 1.07646i 0 −0.817544 + 2.08125i 0 2.54486 + 1.46928i 0 0.817544 1.41603i 0
289.6 0 2.33117 1.34590i 0 −2.12291 0.702335i 0 −2.90420 1.67674i 0 2.12291 3.67698i 0
529.1 0 −2.33117 1.34590i 0 −2.12291 0.702335i 0 2.90420 1.67674i 0 2.12291 + 3.67698i 0
529.2 0 −1.86449 1.07646i 0 −0.817544 + 2.08125i 0 −2.54486 + 1.46928i 0 0.817544 + 1.41603i 0
529.3 0 −0.298874 0.172555i 0 1.44045 1.71029i 0 −1.75765 + 1.01478i 0 −1.44045 2.49493i 0
529.4 0 0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 1.75765 1.01478i 0 −1.44045 2.49493i 0
529.5 0 1.86449 + 1.07646i 0 −0.817544 2.08125i 0 2.54486 1.46928i 0 0.817544 + 1.41603i 0
529.6 0 2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 −2.90420 + 1.67674i 0 2.12291 + 3.67698i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.dh.a 12
4.b odd 2 1 65.2.n.a 12
5.b even 2 1 inner 1040.2.dh.a 12
12.b even 2 1 585.2.bs.a 12
13.c even 3 1 inner 1040.2.dh.a 12
20.d odd 2 1 65.2.n.a 12
20.e even 4 2 325.2.e.e 12
52.b odd 2 1 845.2.n.e 12
52.f even 4 2 845.2.l.f 24
52.i odd 6 1 845.2.b.e 6
52.i odd 6 1 845.2.n.e 12
52.j odd 6 1 65.2.n.a 12
52.j odd 6 1 845.2.b.d 6
52.l even 12 2 845.2.d.d 12
52.l even 12 2 845.2.l.f 24
60.h even 2 1 585.2.bs.a 12
65.n even 6 1 inner 1040.2.dh.a 12
156.p even 6 1 585.2.bs.a 12
260.g odd 2 1 845.2.n.e 12
260.u even 4 2 845.2.l.f 24
260.v odd 6 1 65.2.n.a 12
260.v odd 6 1 845.2.b.d 6
260.w odd 6 1 845.2.b.e 6
260.w odd 6 1 845.2.n.e 12
260.bc even 12 2 845.2.d.d 12
260.bc even 12 2 845.2.l.f 24
260.bg even 12 2 4225.2.a.bq 6
260.bj even 12 2 325.2.e.e 12
260.bj even 12 2 4225.2.a.br 6
780.br even 6 1 585.2.bs.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 4.b odd 2 1
65.2.n.a 12 20.d odd 2 1
65.2.n.a 12 52.j odd 6 1
65.2.n.a 12 260.v odd 6 1
325.2.e.e 12 20.e even 4 2
325.2.e.e 12 260.bj even 12 2
585.2.bs.a 12 12.b even 2 1
585.2.bs.a 12 60.h even 2 1
585.2.bs.a 12 156.p even 6 1
585.2.bs.a 12 780.br even 6 1
845.2.b.d 6 52.j odd 6 1
845.2.b.d 6 260.v odd 6 1
845.2.b.e 6 52.i odd 6 1
845.2.b.e 6 260.w odd 6 1
845.2.d.d 12 52.l even 12 2
845.2.d.d 12 260.bc even 12 2
845.2.l.f 24 52.f even 4 2
845.2.l.f 24 52.l even 12 2
845.2.l.f 24 260.u even 4 2
845.2.l.f 24 260.bc even 12 2
845.2.n.e 12 52.b odd 2 1
845.2.n.e 12 52.i odd 6 1
845.2.n.e 12 260.g odd 2 1
845.2.n.e 12 260.w odd 6 1
1040.2.dh.a 12 1.a even 1 1 trivial
1040.2.dh.a 12 5.b even 2 1 inner
1040.2.dh.a 12 13.c even 3 1 inner
1040.2.dh.a 12 65.n even 6 1 inner
4225.2.a.bq 6 260.bg even 12 2
4225.2.a.br 6 260.bj even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 12 T_{3}^{10} + 109 T_{3}^{8} - 412 T_{3}^{6} + 1177 T_{3}^{4} - 140 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 16 - 140 T^{2} + 1177 T^{4} - 412 T^{6} + 109 T^{8} - 12 T^{10} + T^{12} \)
$5$ \( ( 125 + 75 T + 25 T^{2} + 10 T^{3} + 5 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$7$ \( 160000 - 71600 T^{2} + 22441 T^{4} - 3496 T^{6} + 397 T^{8} - 24 T^{10} + T^{12} \)
$11$ \( ( 64 + 104 T + 169 T^{2} + 16 T^{3} + 13 T^{4} + T^{6} )^{2} \)
$13$ \( 4826809 - 428415 T^{2} + 6591 T^{4} - 322 T^{6} + 39 T^{8} - 15 T^{10} + T^{12} \)
$17$ \( 28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12} \)
$19$ \( ( 100 + 10 T + 61 T^{2} + 14 T^{3} + 37 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$23$ \( 16 - 140 T^{2} + 1177 T^{4} - 412 T^{6} + 109 T^{8} - 12 T^{10} + T^{12} \)
$29$ \( ( 9 - 3 T + T^{2} )^{6} \)
$31$ \( ( -40 - 40 T - 4 T^{2} + T^{3} )^{4} \)
$37$ \( 28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12} \)
$41$ \( ( 25 + 145 T + 806 T^{2} + 213 T^{3} + 78 T^{4} - 7 T^{5} + T^{6} )^{2} \)
$43$ \( 65536 - 72448 T^{2} + 59609 T^{4} - 22128 T^{6} + 6117 T^{8} - 80 T^{10} + T^{12} \)
$47$ \( ( 270400 + 14640 T^{2} + 236 T^{4} + T^{6} )^{2} \)
$53$ \( ( 400 + 1040 T^{2} + 171 T^{4} + T^{6} )^{2} \)
$59$ \( ( 18496 - 7480 T + 3297 T^{2} - 162 T^{3} + 59 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$61$ \( ( 13225 - 5635 T + 2746 T^{2} - 83 T^{3} + 58 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$67$ \( 406586896 - 52486892 T^{2} + 4759209 T^{4} - 219972 T^{6} + 7397 T^{8} - 100 T^{10} + T^{12} \)
$71$ \( ( 676 - 26 T + 157 T^{2} - 46 T^{3} + 37 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$73$ \( ( 250000 + 13900 T^{2} + 215 T^{4} + T^{6} )^{2} \)
$79$ \( ( -160 + 180 T - 26 T^{2} + T^{3} )^{4} \)
$83$ \( ( 640000 + 23600 T^{2} + 276 T^{4} + T^{6} )^{2} \)
$89$ \( ( 2515396 - 249002 T + 40509 T^{2} - 1602 T^{3} + 257 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$97$ \( 41740124416 - 2934418352 T^{2} + 149090649 T^{4} - 3613032 T^{6} + 64037 T^{8} - 280 T^{10} + T^{12} \)
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