Properties

Label 6048.2.c.e
Level 6048
Weight 2
Character orbit 6048.c
Analytic conductor 48.294
Analytic rank 0
Dimension 20
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{5} + q^{7} +O(q^{10})\) \( q + \beta_{12} q^{5} + q^{7} + ( -\beta_{4} - \beta_{10} + \beta_{12} ) q^{11} + \beta_{5} q^{13} -\beta_{1} q^{17} + \beta_{2} q^{19} + \beta_{11} q^{23} + ( -1 + \beta_{8} ) q^{25} + ( -\beta_{4} + \beta_{12} - \beta_{16} ) q^{29} + ( -2 - \beta_{7} ) q^{31} + \beta_{12} q^{35} + ( -\beta_{5} - \beta_{9} - \beta_{13} ) q^{37} -\beta_{17} q^{41} + ( \beta_{3} - \beta_{5} - \beta_{9} ) q^{43} + ( -\beta_{11} - \beta_{15} - \beta_{19} ) q^{47} + q^{49} + ( -\beta_{4} + \beta_{12} - \beta_{14} ) q^{53} + ( -1 - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{18} ) q^{55} + ( -\beta_{4} + \beta_{16} ) q^{59} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{13} ) q^{61} + ( \beta_{1} + \beta_{19} ) q^{65} + \beta_{9} q^{67} + ( \beta_{1} - \beta_{15} - \beta_{17} ) q^{71} + ( -2 \beta_{7} - \beta_{18} ) q^{73} + ( -\beta_{4} - \beta_{10} + \beta_{12} ) q^{77} + ( -3 - \beta_{6} ) q^{79} + ( -2 \beta_{10} + \beta_{12} + \beta_{14} - 2 \beta_{16} ) q^{83} + ( \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{85} + ( \beta_{1} + \beta_{15} - \beta_{17} ) q^{89} + \beta_{5} q^{91} + ( \beta_{11} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{95} + ( 3 - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + x^{18} + 4 x^{16} + 8 x^{12} + 4 x^{10} + 32 x^{8} + 256 x^{4} + 256 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{17} + \nu^{15} - 2 \nu^{13} - 8 \nu^{11} + 8 \nu^{9} - 20 \nu^{7} + 40 \nu^{5} - 192 \nu \)\()/128\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{18} - 17 \nu^{16} + 4 \nu^{14} + 8 \nu^{12} + 8 \nu^{10} + 124 \nu^{8} + 160 \nu^{6} + 96 \nu^{4} - 64 \nu^{2} - 3072 \)\()/896\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{18} + 4 \nu^{16} - 3 \nu^{14} + 8 \nu^{12} + 64 \nu^{10} + 68 \nu^{8} + 244 \nu^{6} + 208 \nu^{4} + 160 \nu^{2} + 960 \)\()/448\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{19} - 3 \nu^{17} + 4 \nu^{15} - 6 \nu^{13} + 8 \nu^{11} - 100 \nu^{9} - 120 \nu^{7} + 40 \nu^{5} - 288 \nu^{3} - 832 \nu \)\()/896\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{18} + \nu^{16} + 4 \nu^{14} + 8 \nu^{10} + 4 \nu^{8} + 32 \nu^{6} + 512 \nu^{2} + 256 \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{18} + 3 \nu^{16} - 16 \nu^{12} - 40 \nu^{10} + 28 \nu^{8} + 112 \nu^{6} + 128 \nu^{4} - 384 \nu^{2} + 256 \)\()/256\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{18} - \nu^{16} - 4 \nu^{14} - 8 \nu^{10} - 4 \nu^{8} - 32 \nu^{6} - 256 \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{18} - 3 \nu^{16} - 16 \nu^{12} + 8 \nu^{10} - 28 \nu^{8} + 16 \nu^{6} + 128 \nu^{4} + 256 \nu^{2} - 512 \)\()/256\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{18} + 3 \nu^{16} + 16 \nu^{12} - 8 \nu^{10} + 28 \nu^{8} - 16 \nu^{6} + 384 \nu^{4} - 256 \nu^{2} + 768 \)\()/256\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{19} + 3 \nu^{17} + 16 \nu^{13} - 8 \nu^{11} + 28 \nu^{9} - 16 \nu^{7} + 128 \nu^{5} + 256 \nu^{3} + 768 \nu \)\()/512\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{19} - 3 \nu^{17} - 16 \nu^{13} + 8 \nu^{11} - 28 \nu^{9} + 16 \nu^{7} - 128 \nu^{5} + 768 \nu^{3} - 768 \nu \)\()/512\)
\(\beta_{12}\)\(=\)\((\)\( 9 \nu^{19} - 15 \nu^{17} + 20 \nu^{15} + 40 \nu^{13} + 40 \nu^{11} - 220 \nu^{9} + 128 \nu^{7} + 32 \nu^{5} + 2816 \nu^{3} - 1024 \nu \)\()/3584\)
\(\beta_{13}\)\(=\)\((\)\( -15 \nu^{18} - 31 \nu^{16} + 4 \nu^{14} - 48 \nu^{12} + 8 \nu^{10} - 828 \nu^{8} + 608 \nu^{6} - 1472 \nu^{4} - 2304 \nu^{2} - 10240 \)\()/1792\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{19} - 2 \nu^{17} + 5 \nu^{15} - 4 \nu^{13} + 80 \nu^{11} + 92 \nu^{9} + 228 \nu^{7} + 736 \nu^{5} + 32 \nu^{3} + 640 \nu \)\()/896\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{19} - \nu^{17} - 4 \nu^{15} - 8 \nu^{11} - 4 \nu^{9} - 32 \nu^{7} - 256 \nu^{3} + 256 \nu \)\()/256\)
\(\beta_{16}\)\(=\)\((\)\( \nu^{19} + \nu^{17} + 4 \nu^{15} + 8 \nu^{11} + 4 \nu^{9} + 32 \nu^{7} + 256 \nu^{3} + 768 \nu \)\()/256\)
\(\beta_{17}\)\(=\)\((\)\( -3 \nu^{19} - 11 \nu^{17} + 4 \nu^{15} - 8 \nu^{13} + 8 \nu^{11} - 12 \nu^{9} - 64 \nu^{7} - 416 \nu^{5} - 256 \nu^{3} - 2560 \nu \)\()/512\)
\(\beta_{18}\)\(=\)\((\)\( \nu^{18} - \nu^{14} - 4 \nu^{8} + 28 \nu^{6} + 16 \nu^{4} + 224 \nu^{2} + 128 \)\()/64\)
\(\beta_{19}\)\(=\)\((\)\( -\nu^{19} - \nu^{17} + 2 \nu^{15} - 2 \nu^{13} - 8 \nu^{11} - 20 \nu^{9} + 48 \nu^{7} - 40 \nu^{5} - 160 \nu^{3} - 576 \nu \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{16} + \beta_{15}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{5}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + \beta_{10}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{9} + \beta_{8} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} + \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{18} + \beta_{13} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(2 \beta_{19} - \beta_{17} + \beta_{15} + \beta_{14} + 3 \beta_{12} - 2 \beta_{10} - 5 \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-\beta_{18} - 3 \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 3 \beta_{2} - 3\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-2 \beta_{19} + 3 \beta_{17} - 3 \beta_{15} + \beta_{14} - 5 \beta_{12} - 2 \beta_{11} - 9 \beta_{4} + 3 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-3 \beta_{18} - \beta_{13} - 2 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + 7 \beta_{3} - 3 \beta_{2} - 5\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-6 \beta_{19} + 5 \beta_{17} - 2 \beta_{16} - 7 \beta_{15} + 7 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} + 9 \beta_{4} - 11 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(7 \beta_{18} + 5 \beta_{13} + 10 \beta_{9} - 17 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 3 \beta_{3} + 7 \beta_{2} - 15\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-2 \beta_{19} + 7 \beta_{17} - 14 \beta_{16} - 5 \beta_{15} - 3 \beta_{14} + 47 \beta_{12} - 26 \beta_{11} + 16 \beta_{10} - 13 \beta_{4} - 9 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-19 \beta_{18} + 7 \beta_{13} + 10 \beta_{9} + \beta_{8} - 49 \beta_{7} + 3 \beta_{6} + 48 \beta_{5} - 17 \beta_{3} + 13 \beta_{2} + 31\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(10 \beta_{19} + 17 \beta_{17} + 58 \beta_{16} - 39 \beta_{15} - 5 \beta_{14} - 31 \beta_{12} - 30 \beta_{11} + 16 \beta_{10} + 45 \beta_{4} + 25 \beta_{1}\)\()/2\)
\(\nu^{16}\)\(=\)\((\)\(-5 \beta_{18} - 7 \beta_{13} - 2 \beta_{9} - 17 \beta_{8} - 7 \beta_{7} + 13 \beta_{6} + 16 \beta_{5} + 17 \beta_{3} - 69 \beta_{2} - 367\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(6 \beta_{19} - 33 \beta_{17} - 34 \beta_{16} - 113 \beta_{15} - 27 \beta_{14} - 161 \beta_{12} + 22 \beta_{11} + 56 \beta_{10} + 67 \beta_{4} - 41 \beta_{1}\)\()/2\)
\(\nu^{18}\)\(=\)\((\)\(77 \beta_{18} - 33 \beta_{13} - 14 \beta_{9} + 9 \beta_{8} - 241 \beta_{7} - 21 \beta_{6} - 176 \beta_{5} - 41 \beta_{3} - 3 \beta_{2} - 249\)\()/2\)
\(\nu^{19}\)\(=\)\((\)\(-54 \beta_{19} - 55 \beta_{17} - 54 \beta_{16} - 79 \beta_{15} - 45 \beta_{14} + 233 \beta_{12} - 134 \beta_{11} - 280 \beta_{10} - 123 \beta_{4} + 49 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3025.1
−0.885915 + 1.10234i
0.885915 + 1.10234i
1.19566 0.755240i
−1.19566 0.755240i
0.725842 1.21374i
−0.725842 1.21374i
0.328272 1.37559i
−0.328272 1.37559i
1.37874 0.314750i
−1.37874 0.314750i
−1.37874 + 0.314750i
1.37874 + 0.314750i
−0.328272 + 1.37559i
0.328272 + 1.37559i
−0.725842 + 1.21374i
0.725842 + 1.21374i
−1.19566 + 0.755240i
1.19566 + 0.755240i
0.885915 1.10234i
−0.885915 1.10234i
0 0 0 3.50133i 0 1.00000 0 0 0
3025.2 0 0 0 3.50133i 0 1.00000 0 0 0
3025.3 0 0 0 3.16969i 0 1.00000 0 0 0
3025.4 0 0 0 3.16969i 0 1.00000 0 0 0
3025.5 0 0 0 3.06888i 0 1.00000 0 0 0
3025.6 0 0 0 3.06888i 0 1.00000 0 0 0
3025.7 0 0 0 0.512447i 0 1.00000 0 0 0
3025.8 0 0 0 0.512447i 0 1.00000 0 0 0
3025.9 0 0 0 0.114591i 0 1.00000 0 0 0
3025.10 0 0 0 0.114591i 0 1.00000 0 0 0
3025.11 0 0 0 0.114591i 0 1.00000 0 0 0
3025.12 0 0 0 0.114591i 0 1.00000 0 0 0
3025.13 0 0 0 0.512447i 0 1.00000 0 0 0
3025.14 0 0 0 0.512447i 0 1.00000 0 0 0
3025.15 0 0 0 3.06888i 0 1.00000 0 0 0
3025.16 0 0 0 3.06888i 0 1.00000 0 0 0
3025.17 0 0 0 3.16969i 0 1.00000 0 0 0
3025.18 0 0 0 3.16969i 0 1.00000 0 0 0
3025.19 0 0 0 3.50133i 0 1.00000 0 0 0
3025.20 0 0 0 3.50133i 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3025.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.c.e 20
3.b odd 2 1 inner 6048.2.c.e 20
4.b odd 2 1 1512.2.c.e 20
8.b even 2 1 inner 6048.2.c.e 20
8.d odd 2 1 1512.2.c.e 20
12.b even 2 1 1512.2.c.e 20
24.f even 2 1 1512.2.c.e 20
24.h odd 2 1 inner 6048.2.c.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.e 20 4.b odd 2 1
1512.2.c.e 20 8.d odd 2 1
1512.2.c.e 20 12.b even 2 1
1512.2.c.e 20 24.f even 2 1
6048.2.c.e 20 1.a even 1 1 trivial
6048.2.c.e 20 3.b odd 2 1 inner
6048.2.c.e 20 8.b even 2 1 inner
6048.2.c.e 20 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{10} + 32 T_{5}^{8} + 342 T_{5}^{6} + 1252 T_{5}^{4} + 321 T_{5}^{2} + 4 \)
\( T_{17}^{10} - 131 T_{17}^{8} + 5266 T_{17}^{6} - 69022 T_{17}^{4} + 319757 T_{17}^{2} - 395839 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 269525 T^{12} - 1005000 T^{14} + 2921875 T^{16} - 7031250 T^{18} + 9765625 T^{20} )^{2} \)
$7$ \( ( 1 - T )^{20} \)
$11$ \( ( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 14790314 T^{12} - 124038552 T^{14} + 1043449429 T^{16} - 7288201954 T^{18} + 25937424601 T^{20} )^{2} \)
$13$ \( ( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 333899722 T^{12} - 2814857916 T^{14} + 17053116197 T^{16} - 67705649843 T^{18} + 137858491849 T^{20} )^{2} \)
$17$ \( ( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 13868243 T^{12} - 198947022 T^{14} + 10982593895 T^{16} + 272054540199 T^{18} + 2015993900449 T^{20} )^{2} \)
$19$ \( ( 1 - 70 T^{2} + 3325 T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 1019138378 T^{12} - 14002730808 T^{14} + 156427554325 T^{16} - 1188849412870 T^{18} + 6131066257801 T^{20} )^{2} \)
$23$ \( ( 1 + 139 T^{2} + 10081 T^{4} + 485484 T^{6} + 16965806 T^{8} + 447075730 T^{10} + 8974911374 T^{12} + 135858328044 T^{14} + 1492349797009 T^{16} + 10885226954059 T^{18} + 41426511213649 T^{20} )^{2} \)
$29$ \( ( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 35597657934 T^{12} - 707006574972 T^{14} + 10198245838545 T^{16} - 94546572049629 T^{18} + 420707233300201 T^{20} )^{2} \)
$31$ \( ( 1 + 9 T + 171 T^{2} + 1092 T^{3} + 11058 T^{4} + 50402 T^{5} + 342798 T^{6} + 1049412 T^{7} + 5094261 T^{8} + 8311689 T^{9} + 28629151 T^{10} )^{4} \)
$37$ \( ( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 720703205 T^{12} + 26088321120 T^{14} + 4543901470339 T^{16} - 77274547986262 T^{18} + 4808584372417849 T^{20} )^{2} \)
$41$ \( ( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 61031645069 T^{12} + 2543049263472 T^{14} + 79093985716891 T^{16} + 1517135793532990 T^{18} + 13422659310152401 T^{20} )^{2} \)
$43$ \( ( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 11009265877 T^{12} - 318502338762 T^{14} + 35494453520135 T^{16} - 1531154236365731 T^{18} + 21611482313284249 T^{20} )^{2} \)
$47$ \( ( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 105520282941 T^{12} + 3621347901168 T^{14} + 120587081885523 T^{16} + 2619241532793710 T^{18} + 52599132235830049 T^{20} )^{2} \)
$53$ \( ( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 361191675098 T^{12} - 14346504116124 T^{14} + 455632771728853 T^{16} - 12016120249392673 T^{18} + 174887470365513049 T^{20} )^{2} \)
$59$ \( ( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 4288879046741 T^{12} - 158168858287158 T^{14} + 4067848483804399 T^{16} - 65926866484340129 T^{18} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 11500636220 T^{10} + 651652694538 T^{12} - 29798354399832 T^{14} + 1211604643847637 T^{16} - 35657560217494266 T^{18} + 713342911662882601 T^{20} )^{2} \)
$67$ \( ( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 8919512756154 T^{12} - 372548202129636 T^{14} + 10863418489821717 T^{16} - 201003500390537295 T^{18} + 1822837804551761449 T^{20} )^{2} \)
$71$ \( ( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 1558433185354 T^{12} + 122029839916996 T^{14} + 6739740237935573 T^{16} + 221493461217296023 T^{18} + 3255243551009881201 T^{20} )^{2} \)
$73$ \( ( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 1051930 T^{6} + 2302128 T^{7} + 57963533 T^{8} + 2073071593 T^{10} )^{4} \)
$79$ \( ( 1 + 16 T + 335 T^{2} + 3202 T^{3} + 40117 T^{4} + 296322 T^{5} + 3169243 T^{6} + 19983682 T^{7} + 165168065 T^{8} + 623201296 T^{9} + 3077056399 T^{10} )^{4} \)
$83$ \( ( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 82503245 T^{8} - 3517947618 T^{10} + 568364854805 T^{12} - 55706197523232 T^{14} + 4455216467899363 T^{16} - 310816328035187658 T^{18} + 15516041187205853449 T^{20} )^{2} \)
$89$ \( ( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 30321779070362 T^{12} + 1782720139460436 T^{14} + 76140018681680005 T^{16} + 2153314076719038307 T^{18} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( ( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 5144686 T^{6} - 32743320 T^{7} + 311221493 T^{8} - 1239409934 T^{9} + 8587340257 T^{10} )^{4} \)
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