Properties

Label 3330.2.d.p
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 51 x^{6} - 124 x^{5} + 154 x^{4} - 46 x^{3} + x^{2} + 4 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{7} -\beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} - q^{4} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{7} -\beta_{2} q^{8} + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{10} + ( -1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{13} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{14} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{17} + ( -2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{19} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} ) q^{20} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{22} + ( -\beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{23} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} ) q^{26} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{28} + ( 3 + \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{29} + ( 3 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{31} + \beta_{2} q^{32} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{34} + ( -2 - 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{35} -\beta_{2} q^{37} + ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{38} + ( \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{40} + ( 1 + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{41} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{43} + ( 1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{44} + ( -\beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{9} ) q^{46} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} - 2 \beta_{9} ) q^{47} + ( -6 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{49} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{50} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{52} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} ) q^{53} + ( 5 - \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{8} ) q^{55} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{56} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{9} ) q^{58} + ( 4 - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{61} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{62} - q^{64} + ( 2 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{65} + ( 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} ) q^{68} + ( -1 - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{70} + ( 2 - 4 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} + 4 \beta_{9} ) q^{71} + ( -4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + \beta_{6} - \beta_{8} ) q^{73} + q^{74} + ( 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} ) q^{76} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{9} ) q^{77} + ( -4 + \beta_{1} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{79} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} ) q^{80} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{9} ) q^{82} + ( -4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{83} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 4 \beta_{8} - 3 \beta_{9} ) q^{85} + ( 5 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{86} + ( \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{88} + ( 4 - 2 \beta_{1} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{89} + ( 8 - 2 \beta_{1} + 2 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 8 \beta_{9} ) q^{91} + ( \beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{92} + ( -4 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} ) q^{94} + ( 2 \beta_{3} + 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{95} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{97} + ( 3 \beta_{1} - 6 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} - 6 q^{5} + O(q^{10}) \) \( 10 q - 10 q^{4} - 6 q^{5} + 2 q^{10} - 6 q^{11} - 2 q^{14} + 10 q^{16} - 8 q^{19} + 6 q^{20} + 4 q^{25} + 12 q^{26} + 22 q^{29} + 46 q^{31} - 18 q^{34} - 32 q^{35} - 2 q^{40} + 14 q^{41} + 6 q^{44} + 12 q^{46} - 60 q^{49} - 8 q^{50} + 42 q^{55} + 2 q^{56} + 40 q^{59} - 18 q^{61} - 10 q^{64} - 4 q^{65} - 6 q^{70} - 12 q^{71} + 10 q^{74} + 8 q^{76} - 40 q^{79} - 6 q^{80} + 36 q^{85} + 34 q^{86} + 24 q^{89} + 32 q^{91} - 24 q^{94} - 12 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 51 x^{6} - 124 x^{5} + 154 x^{4} - 46 x^{3} + x^{2} + 4 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -8189 \nu^{9} - 27007 \nu^{8} + 63073 \nu^{7} - 48258 \nu^{6} - 261048 \nu^{5} - 1171720 \nu^{4} + 3781077 \nu^{3} - 5967211 \nu^{2} + 989985 \nu + 81466 \)\()/1402866\)
\(\beta_{3}\)\(=\)\((\)\( -9773 \nu^{9} + 29441 \nu^{8} + 10432 \nu^{7} - 56736 \nu^{6} - 566796 \nu^{5} + 1401884 \nu^{4} - 260445 \nu^{3} - 3989713 \nu^{2} + 2894160 \nu - 972956 \)\()/1402866\)
\(\beta_{4}\)\(=\)\((\)\( 11204 \nu^{9} - 16670 \nu^{8} + 30482 \nu^{7} - 71109 \nu^{6} + 540120 \nu^{5} - 1213433 \nu^{4} + 2027820 \nu^{3} - 1662629 \nu^{2} + 926646 \nu - 346156 \)\()/467622\)
\(\beta_{5}\)\(=\)\((\)\( -76468 \nu^{9} + 174376 \nu^{8} - 151780 \nu^{7} + 373635 \nu^{6} - 3969330 \nu^{5} + 10297489 \nu^{4} - 12786054 \nu^{3} + 6038593 \nu^{2} - 1180074 \nu - 912988 \)\()/1402866\)
\(\beta_{6}\)\(=\)\((\)\( -93757 \nu^{9} + 280360 \nu^{8} - 307876 \nu^{7} + 400662 \nu^{6} - 5058726 \nu^{5} + 16117294 \nu^{4} - 22392627 \nu^{3} + 9536722 \nu^{2} + 4399206 \nu - 1174924 \)\()/1402866\)
\(\beta_{7}\)\(=\)\((\)\( -109244 \nu^{9} + 194594 \nu^{8} - 195959 \nu^{7} + 396594 \nu^{6} - 5584197 \nu^{5} + 12501560 \nu^{4} - 14749095 \nu^{3} + 3241250 \nu^{2} - 3152958 \nu + 2825248 \)\()/1402866\)
\(\beta_{8}\)\(=\)\((\)\( -116740 \nu^{9} + 142891 \nu^{8} - 66016 \nu^{7} + 264366 \nu^{6} - 5536122 \nu^{5} + 10028530 \nu^{4} - 7007874 \nu^{3} - 8939543 \nu^{2} + 5464986 \nu + 123896 \)\()/1402866\)
\(\beta_{9}\)\(=\)\((\)\( -150175 \nu^{9} + 250360 \nu^{8} - 304354 \nu^{7} + 635400 \nu^{6} - 7542114 \nu^{5} + 16479244 \nu^{4} - 21830091 \nu^{3} + 8386744 \nu^{2} - 5693844 \nu - 1400668 \)\()/1402866\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{6} + \beta_{3} - 4 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{6} + 6 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-6 \beta_{9} + 6 \beta_{7} - \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 2 \beta_{1} - 24\)
\(\nu^{5}\)\(=\)\(3 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} - 31 \beta_{1} + 24\)
\(\nu^{6}\)\(=\)\(2 \beta_{9} - 39 \beta_{8} + 2 \beta_{7} + 39 \beta_{6} + 9 \beta_{5} + 27 \beta_{4} - 73 \beta_{3} + 160 \beta_{2} + 34 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-82 \beta_{9} + 36 \beta_{8} + 36 \beta_{7} - 82 \beta_{6} + 14 \beta_{5} - 290 \beta_{4} + 290 \beta_{3} - 218 \beta_{2} + 7 \beta_{1} - 218\)
\(\nu^{8}\)\(=\)\(276 \beta_{9} + 29 \beta_{8} - 276 \beta_{7} + 29 \beta_{6} + 60 \beta_{5} + 576 \beta_{4} - 269 \beta_{3} - 240 \beta_{1} + 1132\)
\(\nu^{9}\)\(=\)\(-329 \beta_{9} - 636 \beta_{8} + 636 \beta_{7} + 329 \beta_{6} - 298 \beta_{4} - 298 \beta_{3} + 1810 \beta_{2} + 1468 \beta_{1} - 1810\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
1.24331 + 1.24331i
−1.95884 1.95884i
−0.282359 0.282359i
0.478560 + 0.478560i
1.51933 + 1.51933i
1.24331 1.24331i
−1.95884 + 1.95884i
−0.282359 + 0.282359i
0.478560 0.478560i
1.51933 1.51933i
1.00000i 0 −1.00000 −2.07757 + 0.826871i 0 4.67211i 1.00000i 0 0.826871 + 2.07757i
1999.2 1.00000i 0 −1.00000 −1.74265 + 1.40113i 0 3.20984i 1.00000i 0 1.40113 + 1.74265i
1999.3 1.00000i 0 −1.00000 −1.70518 1.44650i 0 1.83227i 1.00000i 0 −1.44650 + 1.70518i
1999.4 1.00000i 0 −1.00000 1.10390 + 1.94458i 0 3.51336i 1.00000i 0 1.94458 1.10390i
1999.5 1.00000i 0 −1.00000 1.42149 1.72608i 0 4.14336i 1.00000i 0 −1.72608 1.42149i
1999.6 1.00000i 0 −1.00000 −2.07757 0.826871i 0 4.67211i 1.00000i 0 0.826871 2.07757i
1999.7 1.00000i 0 −1.00000 −1.74265 1.40113i 0 3.20984i 1.00000i 0 1.40113 1.74265i
1999.8 1.00000i 0 −1.00000 −1.70518 + 1.44650i 0 1.83227i 1.00000i 0 −1.44650 1.70518i
1999.9 1.00000i 0 −1.00000 1.10390 1.94458i 0 3.51336i 1.00000i 0 1.94458 + 1.10390i
1999.10 1.00000i 0 −1.00000 1.42149 + 1.72608i 0 4.14336i 1.00000i 0 −1.72608 + 1.42149i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.d.p 10
3.b odd 2 1 370.2.b.d 10
5.b even 2 1 inner 3330.2.d.p 10
15.d odd 2 1 370.2.b.d 10
15.e even 4 1 1850.2.a.bd 5
15.e even 4 1 1850.2.a.be 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.d 10 3.b odd 2 1
370.2.b.d 10 15.d odd 2 1
1850.2.a.bd 5 15.e even 4 1
1850.2.a.be 5 15.e even 4 1
3330.2.d.p 10 1.a even 1 1 trivial
3330.2.d.p 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{10} + 65 T_{7}^{8} + 1592 T_{7}^{6} + 18096 T_{7}^{4} + 92800 T_{7}^{2} + 160000 \)
\( T_{11}^{5} + 3 T_{11}^{4} - 23 T_{11}^{3} - 71 T_{11}^{2} + 19 T_{11} + 1 \)
\( T_{17}^{10} + 89 T_{17}^{8} + 2752 T_{17}^{6} + 35424 T_{17}^{4} + 172800 T_{17}^{2} + 135424 \)
\( T_{29}^{5} - 11 T_{29}^{4} - 9 T_{29}^{3} + 251 T_{29}^{2} + 5 T_{29} - 335 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{5} \)
$3$ \( T^{10} \)
$5$ \( 3125 + 3750 T + 2000 T^{2} + 850 T^{3} + 600 T^{4} + 350 T^{5} + 120 T^{6} + 34 T^{7} + 16 T^{8} + 6 T^{9} + T^{10} \)
$7$ \( 160000 + 92800 T^{2} + 18096 T^{4} + 1592 T^{6} + 65 T^{8} + T^{10} \)
$11$ \( ( 1 + 19 T - 71 T^{2} - 23 T^{3} + 3 T^{4} + T^{5} )^{2} \)
$13$ \( 64 + 3425 T^{2} + 22014 T^{4} + 2627 T^{6} + 94 T^{8} + T^{10} \)
$17$ \( 135424 + 172800 T^{2} + 35424 T^{4} + 2752 T^{6} + 89 T^{8} + T^{10} \)
$19$ \( ( 640 + 80 T - 176 T^{2} - 40 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$23$ \( 82944 + 71793 T^{2} + 20518 T^{4} + 2299 T^{6} + 86 T^{8} + T^{10} \)
$29$ \( ( -335 + 5 T + 251 T^{2} - 9 T^{3} - 11 T^{4} + T^{5} )^{2} \)
$31$ \( ( -10511 - 3199 T + 655 T^{2} + 107 T^{3} - 23 T^{4} + T^{5} )^{2} \)
$37$ \( ( 1 + T^{2} )^{5} \)
$41$ \( ( -5821 - 1377 T + 1151 T^{2} - 119 T^{3} - 7 T^{4} + T^{5} )^{2} \)
$43$ \( 9339136 + 2985856 T^{2} + 280112 T^{4} + 11016 T^{6} + 185 T^{8} + T^{10} \)
$47$ \( 64000000 + 10400000 T^{2} + 636800 T^{4} + 18096 T^{6} + 232 T^{8} + T^{10} \)
$53$ \( 1076889856 + 101468416 T^{2} + 3458784 T^{4} + 53216 T^{6} + 377 T^{8} + T^{10} \)
$59$ \( ( -3136 - 2352 T + 672 T^{2} + 52 T^{3} - 20 T^{4} + T^{5} )^{2} \)
$61$ \( ( -2323 - 2939 T - 993 T^{2} - 81 T^{3} + 9 T^{4} + T^{5} )^{2} \)
$67$ \( 153664 + 220745 T^{2} + 76322 T^{4} + 7971 T^{6} + 202 T^{8} + T^{10} \)
$71$ \( ( 13792 + 3792 T - 912 T^{2} - 184 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$73$ \( 216855076 + 84519713 T^{2} + 3812318 T^{4} + 63147 T^{6} + 430 T^{8} + T^{10} \)
$79$ \( ( 8168 - 4891 T - 1620 T^{2} - 13 T^{3} + 20 T^{4} + T^{5} )^{2} \)
$83$ \( 7573504 + 11272448 T^{2} + 1969408 T^{4} + 47200 T^{6} + 384 T^{8} + T^{10} \)
$89$ \( ( -55552 + 4032 T + 1792 T^{2} - 144 T^{3} - 12 T^{4} + T^{5} )^{2} \)
$97$ \( 1016064 + 17724672 T^{2} + 2503648 T^{4} + 71968 T^{6} + 497 T^{8} + T^{10} \)
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