Properties

Label 1050.3.p.b
Level $1050$
Weight $3$
Character orbit 1050.p
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
Defining polynomial: \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( \beta_{2} + 2 \beta_{4} ) q^{6} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + 3 \beta_{3} q^{9} + ( -1 - 2 \beta_{1} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{11} + ( 4 - 2 \beta_{3} ) q^{12} + ( -2 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( -6 + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{14} -4 \beta_{3} q^{16} + ( -7 + 2 \beta_{2} - 7 \beta_{3} + \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{17} -3 \beta_{4} q^{18} + ( 18 - 2 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{19} + ( -\beta_{1} - \beta_{2} + 4 \beta_{4} + 5 \beta_{6} ) q^{21} + ( 6 - 4 \beta_{1} + \beta_{2} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{22} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{23} + ( -4 \beta_{2} - 2 \beta_{4} ) q^{24} + ( -16 + 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{26} + ( 3 - 6 \beta_{3} ) q^{27} + ( 6 \beta_{1} + 6 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{28} + ( 9 - \beta_{1} + 13 \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -11 - 2 \beta_{2} - 11 \beta_{3} - \beta_{4} - 2 \beta_{6} + 5 \beta_{7} ) q^{31} + 4 \beta_{4} q^{32} + ( 2 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 6 \beta_{5} ) q^{33} + ( 2 + 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{34} -6 q^{36} + ( 5 \beta_{1} - 2 \beta_{2} + 24 \beta_{3} - 2 \beta_{4} - \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 14 - 18 \beta_{2} + 14 \beta_{3} - 9 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( 6 - 2 \beta_{1} - 6 \beta_{3} + 12 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -3 + 19 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} + 4 \beta_{7} ) q^{41} + ( 8 + 2 \beta_{3} + 5 \beta_{5} + 4 \beta_{7} ) q^{42} + ( 14 - 9 \beta_{1} + 5 \beta_{2} + 4 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{44} + ( 2 - 8 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{46} + ( 4 + 8 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -4 + 8 \beta_{3} ) q^{48} + ( 15 + 9 \beta_{3} - 16 \beta_{5} - 10 \beta_{7} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{51} + ( -4 + 16 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -8 - 8 \beta_{1} + 8 \beta_{3} + 26 \beta_{4} - 16 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{53} + ( -3 \beta_{2} + 3 \beta_{4} ) q^{54} + ( 8 - 12 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} ) q^{56} + ( -27 + 2 \beta_{1} + 21 \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{57} + ( -4 \beta_{1} - 9 \beta_{2} - 26 \beta_{3} - 9 \beta_{4} - \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 11 + 14 \beta_{2} + 11 \beta_{3} + 7 \beta_{4} - 21 \beta_{6} - 8 \beta_{7} ) q^{59} + ( 16 - 16 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 16 \beta_{5} ) q^{61} + ( -2 - 10 \beta_{1} + 11 \beta_{2} + 4 \beta_{3} + 22 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 2 \beta_{7} ) q^{62} + ( 6 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} - 9 \beta_{6} ) q^{63} + 8 q^{64} + ( -6 - 2 \beta_{2} - 6 \beta_{3} - \beta_{4} + 12 \beta_{6} + 3 \beta_{7} ) q^{66} + ( 30 + 6 \beta_{1} - 30 \beta_{3} + 9 \beta_{4} - 16 \beta_{5} - 3 \beta_{6} - 8 \beta_{7} ) q^{67} + ( 28 + 6 \beta_{1} - 2 \beta_{2} - 14 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{68} + ( -3 + 9 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{69} + ( 1 - 7 \beta_{1} + 29 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} ) q^{71} + ( 6 \beta_{2} + 6 \beta_{4} ) q^{72} + ( -2 + 20 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + 25 \beta_{6} - 15 \beta_{7} ) q^{73} + ( -4 + 4 \beta_{1} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} ) q^{74} + ( -18 + 4 \beta_{1} - 14 \beta_{2} + 36 \beta_{3} - 28 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{76} + ( 31 + 7 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} + 42 \beta_{4} + 8 \beta_{5} - 21 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 24 - 2 \beta_{1} - 6 \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{78} + ( 2 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} - 38 \beta_{7} ) q^{79} + ( -9 + 9 \beta_{3} ) q^{81} + ( -4 - 8 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 19 \beta_{5} ) q^{82} + ( 3 - 7 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} + 30 \beta_{4} - 34 \beta_{5} + 7 \beta_{6} - 34 \beta_{7} ) q^{83} + ( -8 \beta_{1} - 8 \beta_{2} - 10 \beta_{4} - 2 \beta_{6} ) q^{84} + ( 8 \beta_{1} - 14 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} - 9 \beta_{5} - 16 \beta_{6} - 18 \beta_{7} ) q^{86} + ( -9 - 26 \beta_{2} - 9 \beta_{3} - 13 \beta_{4} + 3 \beta_{6} - 6 \beta_{7} ) q^{87} + ( -12 + 16 \beta_{1} + 12 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} ) q^{88} + ( 82 - 19 \beta_{1} + 11 \beta_{2} - 41 \beta_{3} - 11 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -42 + 12 \beta_{1} + 5 \beta_{2} + 7 \beta_{3} - 13 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} ) q^{91} + ( 6 - 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{92} + ( -2 \beta_{1} + 3 \beta_{2} + 33 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} ) q^{93} + ( 40 - 4 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} + 8 \beta_{7} ) q^{94} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{96} + ( -36 - 8 \beta_{1} - 20 \beta_{2} + 72 \beta_{3} - 40 \beta_{4} - 18 \beta_{5} + 8 \beta_{6} - 18 \beta_{7} ) q^{97} + ( 20 \beta_{1} - 15 \beta_{2} - 24 \beta_{4} + 12 \beta_{6} ) q^{98} + ( -3 - 3 \beta_{1} - 9 \beta_{2} + 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} - 4 q^{11} + 24 q^{12} - 40 q^{14} - 16 q^{16} - 84 q^{17} + 108 q^{19} + 48 q^{22} - 12 q^{23} - 96 q^{26} + 72 q^{29} - 132 q^{31} + 12 q^{33} - 48 q^{36} + 96 q^{37} + 168 q^{38} + 24 q^{39} + 72 q^{42} + 112 q^{43} - 8 q^{44} + 8 q^{46} + 24 q^{47} + 156 q^{49} + 84 q^{51} - 48 q^{52} - 32 q^{53} + 16 q^{56} - 216 q^{57} - 104 q^{58} + 132 q^{59} + 96 q^{61} + 64 q^{64} - 72 q^{66} + 120 q^{67} + 168 q^{68} + 8 q^{71} - 24 q^{73} - 16 q^{74} + 216 q^{77} + 192 q^{78} + 12 q^{79} - 36 q^{81} - 24 q^{82} - 40 q^{86} - 108 q^{87} - 48 q^{88} + 492 q^{89} - 308 q^{91} + 48 q^{92} + 132 q^{93} + 480 q^{94} - 24 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36 \)\()/63\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu \)\()/189\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162 \)\()/63\)
\(\beta_{7}\)\(=\)\((\)\( 19 \nu^{7} - 49 \nu^{5} + 133 \nu^{3} - 684 \nu \)\()/189\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{5} + 7 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 19 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(-7 \beta_{6} + 7 \beta_{1} + 22\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.72286 0.178197i
−1.01575 + 1.40294i
−1.72286 + 0.178197i
1.01575 1.40294i
1.72286 + 0.178197i
−1.01575 1.40294i
−1.72286 0.178197i
1.01575 + 1.40294i
−0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −5.10237 4.79227i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.51658 2.55620i 2.82843 1.50000 + 2.59808i 0
451.3 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.51658 + 2.55620i −2.82843 1.50000 + 2.59808i 0
451.4 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 5.10237 + 4.79227i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −5.10237 + 4.79227i 2.82843 1.50000 2.59808i 0
901.2 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 6.51658 + 2.55620i 2.82843 1.50000 2.59808i 0
901.3 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −6.51658 2.55620i −2.82843 1.50000 2.59808i 0
901.4 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 5.10237 4.79227i −2.82843 1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.b 8
5.b even 2 1 210.3.o.a 8
5.c odd 4 2 1050.3.q.c 16
7.d odd 6 1 inner 1050.3.p.b 8
15.d odd 2 1 630.3.v.b 8
35.i odd 6 1 210.3.o.a 8
35.i odd 6 1 1470.3.f.a 8
35.j even 6 1 1470.3.f.a 8
35.k even 12 2 1050.3.q.c 16
105.p even 6 1 630.3.v.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 5.b even 2 1
210.3.o.a 8 35.i odd 6 1
630.3.v.b 8 15.d odd 2 1
630.3.v.b 8 105.p even 6 1
1050.3.p.b 8 1.a even 1 1 trivial
1050.3.p.b 8 7.d odd 6 1 inner
1050.3.q.c 16 5.c odd 4 2
1050.3.q.c 16 35.k even 12 2
1470.3.f.a 8 35.i odd 6 1
1470.3.f.a 8 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\(T_{11}^{8} + \cdots\)
\(T_{17}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( ( 3 + 3 T + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 5764801 - 187278 T^{2} + 5243 T^{4} - 78 T^{6} + T^{8} \)
$11$ \( 22505536 - 9715712 T + 5617504 T^{2} + 576448 T^{3} + 93448 T^{4} + 2896 T^{5} + 316 T^{6} + 4 T^{7} + T^{8} \)
$13$ \( 33189121 + 2396652 T^{2} + 56678 T^{4} + 492 T^{6} + T^{8} \)
$17$ \( 15808576 + 24428544 T + 15366112 T^{2} + 4300800 T^{3} + 658056 T^{4} + 58800 T^{5} + 3052 T^{6} + 84 T^{7} + T^{8} \)
$19$ \( 3571138081 - 723561972 T + 463098 T^{2} + 9807480 T^{3} + 279971 T^{4} - 87480 T^{5} + 4698 T^{6} - 108 T^{7} + T^{8} \)
$23$ \( 1032256 + 1560576 T + 1891936 T^{2} + 730944 T^{3} + 231048 T^{4} - 2448 T^{5} + 604 T^{6} + 12 T^{7} + T^{8} \)
$29$ \( ( 20104 + 7872 T - 460 T^{2} - 36 T^{3} + T^{4} )^{2} \)
$31$ \( 56085121 + 103977276 T + 76432266 T^{2} + 22575384 T^{3} + 3247283 T^{4} + 214632 T^{5} + 7434 T^{6} + 132 T^{7} + T^{8} \)
$37$ \( 686911681 - 473020032 T + 256800634 T^{2} - 42434112 T^{3} + 5158083 T^{4} - 216384 T^{5} + 6586 T^{6} - 96 T^{7} + T^{8} \)
$41$ \( 6360766467136 + 21661234752 T^{2} + 21449888 T^{4} + 7992 T^{6} + T^{8} \)
$43$ \( ( -877199 + 129944 T - 2314 T^{2} - 56 T^{3} + T^{4} )^{2} \)
$47$ \( 17622397993216 - 45538862592 T - 22830954624 T^{2} + 59099904 T^{3} + 25569584 T^{4} + 130752 T^{5} - 5256 T^{6} - 24 T^{7} + T^{8} \)
$53$ \( 1469648896 - 3425091584 T + 7727799296 T^{2} - 590790656 T^{3} + 41268928 T^{4} - 391168 T^{5} + 7664 T^{6} + 32 T^{7} + T^{8} \)
$59$ \( 19263180552256 - 2847011029248 T + 158253288928 T^{2} - 2659555200 T^{3} - 7342584 T^{4} + 541200 T^{5} + 1708 T^{6} - 132 T^{7} + T^{8} \)
$61$ \( 981652934656 + 227515711488 T + 11061556224 T^{2} - 1510060032 T^{3} + 34904768 T^{4} + 631296 T^{5} - 3504 T^{6} - 96 T^{7} + T^{8} \)
$67$ \( 848699720001 + 97836643800 T + 10388513466 T^{2} + 323688960 T^{3} + 12755907 T^{4} - 328320 T^{5} + 13434 T^{6} - 120 T^{7} + T^{8} \)
$71$ \( ( -2872184 - 278336 T - 6988 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$73$ \( 52818460352161 + 7580953107672 T + 449019135866 T^{2} + 12390084336 T^{3} + 140009619 T^{4} - 285072 T^{5} - 11686 T^{6} + 24 T^{7} + T^{8} \)
$79$ \( 11859027266503921 + 1749791718948 T + 2439817185346 T^{2} + 2253624528 T^{3} + 393143259 T^{4} + 236688 T^{5} + 22546 T^{6} - 12 T^{7} + T^{8} \)
$83$ \( 8967746052897856 + 5541444773952 T^{2} + 905020064 T^{4} + 52728 T^{6} + T^{8} \)
$89$ \( 91315148362816 + 9140634983424 T + 67279672416 T^{2} - 23794988544 T^{3} + 785244488 T^{4} - 12238992 T^{5} + 105564 T^{6} - 492 T^{7} + T^{8} \)
$97$ \( 3470767019462656 + 2182950729728 T^{2} + 469171584 T^{4} + 39392 T^{6} + T^{8} \)
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