Properties

Label 770.2.n.f
Level $770$
Weight $2$
Character orbit 770.n
Analytic conductor $6.148$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
Defining polynomial: \(x^{8} + x^{6} + 16 x^{4} + 66 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{7} ) q^{3} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( \beta_{2} + \beta_{6} ) q^{6} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{7} ) q^{3} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( \beta_{2} + \beta_{6} ) q^{6} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{9} - q^{10} + ( \beta_{1} + 2 \beta_{4} - \beta_{6} ) q^{11} + ( -1 + \beta_{4} ) q^{12} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + \beta_{3} q^{14} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{15} + \beta_{2} q^{16} + ( 3 \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{18} + ( 1 - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{19} -\beta_{5} q^{20} + ( 1 - \beta_{4} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{6} - \beta_{7} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{23} + ( \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{24} + \beta_{3} q^{25} + ( 4 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{26} + ( -3 - 7 \beta_{2} - 3 \beta_{3} - \beta_{6} - \beta_{7} ) q^{27} -\beta_{2} q^{28} + ( -5 \beta_{3} - \beta_{4} + 5 \beta_{5} + \beta_{6} ) q^{29} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{30} + ( 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} - q^{32} + ( -2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{33} + ( -3 + \beta_{1} - \beta_{4} - \beta_{7} ) q^{34} + \beta_{5} q^{35} + ( -1 + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{36} + ( -2 - 2 \beta_{2} - 7 \beta_{3} - \beta_{4} + 7 \beta_{5} + \beta_{6} ) q^{37} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{38} + ( 1 - 7 \beta_{2} + \beta_{3} - 5 \beta_{6} ) q^{39} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{40} + ( \beta_{1} + 6 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{42} + ( 3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{7} ) q^{43} + ( \beta_{1} + \beta_{7} ) q^{44} + ( 2 + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{45} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{46} + ( -4 - 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{48} + \beta_{2} q^{49} -\beta_{2} q^{50} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{51} + ( -1 + \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{52} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 7 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 7 + \beta_{1} + 3 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{54} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{55} + q^{56} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{57} + ( -5 - \beta_{1} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{58} + ( 1 - 2 \beta_{1} + \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{59} + ( -\beta_{2} - \beta_{6} ) q^{60} + ( 3 + 5 \beta_{2} + 3 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{61} + ( -6 + 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{62} + ( 1 - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{63} -\beta_{5} q^{64} + ( 4 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{65} + ( -1 - \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{66} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{67} + ( \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{68} + ( -6 - 2 \beta_{1} - 4 \beta_{3} + 6 \beta_{5} + 2 \beta_{6} ) q^{69} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{70} + ( 3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{6} - \beta_{7} ) q^{71} + ( -1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{72} + ( -11 - 5 \beta_{1} - 11 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} ) q^{73} + ( -5 - \beta_{1} - 7 \beta_{3} + \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{74} + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{75} + ( -1 - \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{76} + ( -\beta_{1} - \beta_{7} ) q^{77} + ( 7 - \beta_{2} - 5 \beta_{4} + \beta_{5} ) q^{78} + ( -\beta_{1} + 6 \beta_{2} + 6 \beta_{3} - \beta_{4} - 5 \beta_{5} ) q^{79} + \beta_{3} q^{80} + ( -6 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{81} + ( -6 \beta_{2} + \beta_{6} + \beta_{7} ) q^{82} + ( -6 - \beta_{2} - 6 \beta_{3} + 3 \beta_{6} ) q^{83} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{84} + ( \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{85} + ( \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{86} + ( 5 + 6 \beta_{1} + 9 \beta_{2} - \beta_{4} - 9 \beta_{5} - 6 \beta_{7} ) q^{87} + ( -\beta_{1} + \beta_{4} + 2 \beta_{7} ) q^{88} + ( -4 - 3 \beta_{1} - 6 \beta_{2} - \beta_{4} + 6 \beta_{5} + 3 \beta_{7} ) q^{89} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{90} + ( 1 - \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{92} + ( 11 + 5 \beta_{2} + 11 \beta_{3} + 5 \beta_{6} + 6 \beta_{7} ) q^{93} + ( -4 - 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{94} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{96} + ( -5 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} - q^{98} + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} - 8q^{10} - 8q^{12} - 2q^{13} - 2q^{14} + 2q^{15} - 2q^{16} - 6q^{17} + 8q^{18} + 6q^{19} - 2q^{20} + 8q^{21} - 2q^{24} - 2q^{25} + 12q^{26} - 4q^{27} + 2q^{28} + 20q^{29} - 2q^{30} - 6q^{31} - 8q^{32} - 8q^{33} - 24q^{34} + 2q^{35} - 8q^{36} + 16q^{37} + 4q^{38} + 20q^{39} + 2q^{40} - 12q^{41} + 2q^{42} + 20q^{43} + 12q^{45} - 16q^{47} + 2q^{48} - 2q^{49} + 2q^{50} - 20q^{51} - 12q^{52} - 30q^{53} + 44q^{54} + 8q^{56} - 20q^{58} - 18q^{59} + 2q^{60} + 8q^{61} - 24q^{62} + 8q^{63} - 2q^{64} + 28q^{65} + 18q^{66} - 6q^{68} - 28q^{69} - 2q^{70} + 22q^{71} - 2q^{72} - 50q^{73} - 16q^{74} + 2q^{75} - 4q^{76} + 60q^{78} - 34q^{79} - 2q^{80} - 28q^{81} + 12q^{82} - 34q^{83} - 2q^{84} - 6q^{85} + 4q^{87} - 8q^{89} - 2q^{90} + 12q^{91} + 56q^{93} - 24q^{94} + 6q^{95} - 2q^{96} - 8q^{98} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 16 x^{4} + 66 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} - 37 \nu^{4} + 629 \nu^{2} - 363 \)\()/1991\)
\(\beta_{3}\)\(=\)\((\)\( -28 \nu^{6} + 148 \nu^{4} - 525 \nu^{2} - 539 \)\()/1991\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{7} + 148 \nu^{5} - 525 \nu^{3} - 539 \nu \)\()/1991\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{6} + 73 \nu^{4} + 750 \nu^{2} + 2761 \)\()/1991\)
\(\beta_{6}\)\(=\)\((\)\( -61 \nu^{7} + 38 \nu^{5} - 646 \nu^{3} - 1672 \nu \)\()/1991\)
\(\beta_{7}\)\(=\)\((\)\( 68 \nu^{7} - 75 \nu^{5} + 1275 \nu^{3} + 3300 \nu \)\()/1991\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{7} + 4 \beta_{6} + \beta_{4} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 10 \beta_{3} - 7\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} + 17 \beta_{4} - 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(37 \beta_{5} - 37 \beta_{3} - 75 \beta_{2} - 75\)
\(\nu^{7}\)\(=\)\(-38 \beta_{7} - 75 \beta_{6}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1 - \beta_{2} - \beta_{3} + \beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−1.73855 + 1.26313i
1.73855 1.26313i
−1.73855 1.26313i
1.73855 + 1.26313i
−0.476925 + 1.46782i
0.476925 1.46782i
−0.476925 1.46782i
0.476925 + 1.46782i
−0.309017 0.951057i −0.929529 0.675342i −0.809017 + 0.587785i 0.309017 0.951057i −0.355049 + 1.09273i 0.809017 0.587785i 0.809017 + 0.587785i −0.519114 1.59767i −1.00000
71.2 −0.309017 0.951057i 2.54756 + 1.85091i −0.809017 + 0.587785i 0.309017 0.951057i 0.973083 2.99484i 0.809017 0.587785i 0.809017 + 0.587785i 2.13715 + 6.57747i −1.00000
141.1 −0.309017 + 0.951057i −0.929529 + 0.675342i −0.809017 0.587785i 0.309017 + 0.951057i −0.355049 1.09273i 0.809017 + 0.587785i 0.809017 0.587785i −0.519114 + 1.59767i −1.00000
141.2 −0.309017 + 0.951057i 2.54756 1.85091i −0.809017 0.587785i 0.309017 + 0.951057i 0.973083 + 2.99484i 0.809017 + 0.587785i 0.809017 0.587785i 2.13715 6.57747i −1.00000
421.1 0.809017 0.587785i −0.785942 2.41888i 0.309017 0.951057i −0.809017 0.587785i −2.05762 1.49495i −0.309017 + 0.951057i −0.309017 0.951057i −2.80623 + 2.03884i −1.00000
421.2 0.809017 0.587785i 0.167908 + 0.516768i 0.309017 0.951057i −0.809017 0.587785i 0.439589 + 0.319380i −0.309017 + 0.951057i −0.309017 0.951057i 2.18820 1.58982i −1.00000
631.1 0.809017 + 0.587785i −0.785942 + 2.41888i 0.309017 + 0.951057i −0.809017 + 0.587785i −2.05762 + 1.49495i −0.309017 0.951057i −0.309017 + 0.951057i −2.80623 2.03884i −1.00000
631.2 0.809017 + 0.587785i 0.167908 0.516768i 0.309017 + 0.951057i −0.809017 + 0.587785i 0.439589 0.319380i −0.309017 0.951057i −0.309017 + 0.951057i 2.18820 + 1.58982i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 631.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.f 8
11.c even 5 1 inner 770.2.n.f 8
11.c even 5 1 8470.2.a.co 4
11.d odd 10 1 8470.2.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.f 8 1.a even 1 1 trivial
770.2.n.f 8 11.c even 5 1 inner
8470.2.a.co 4 11.c even 5 1
8470.2.a.cs 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2 T_{3}^{7} + 4 T_{3}^{6} - 8 T_{3}^{5} + 46 T_{3}^{4} + 80 T_{3}^{3} + 65 T_{3}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( 25 + 65 T^{2} + 80 T^{3} + 46 T^{4} - 8 T^{5} + 4 T^{6} - 2 T^{7} + T^{8} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$11$ \( 14641 - 2299 T^{2} + 231 T^{4} - 19 T^{6} + T^{8} \)
$13$ \( 48400 - 48400 T + 22440 T^{2} - 3520 T^{3} + 936 T^{4} + 108 T^{5} + 24 T^{6} + 2 T^{7} + T^{8} \)
$17$ \( 625 + 1500 T + 3975 T^{2} + 3540 T^{3} + 1576 T^{4} + 336 T^{5} + 46 T^{6} + 6 T^{7} + T^{8} \)
$19$ \( 21025 + 8700 T + 4740 T^{2} + 730 T^{3} + 246 T^{4} + 4 T^{5} + 21 T^{6} - 6 T^{7} + T^{8} \)
$23$ \( ( 176 - 32 T^{2} + T^{4} )^{2} \)
$29$ \( 26896 + 101680 T + 137184 T^{2} - 61360 T^{3} + 16136 T^{4} - 2560 T^{5} + 294 T^{6} - 20 T^{7} + T^{8} \)
$31$ \( 21939856 + 4665264 T + 1000712 T^{2} + 70272 T^{3} + 3880 T^{4} - 132 T^{5} + 72 T^{6} + 6 T^{7} + T^{8} \)
$37$ \( 1210000 + 242000 T - 39600 T^{2} - 25520 T^{3} + 13776 T^{4} - 2056 T^{5} + 286 T^{6} - 16 T^{7} + T^{8} \)
$41$ \( 703921 + 422856 T + 124553 T^{2} + 18840 T^{3} + 7156 T^{4} + 720 T^{5} + 102 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( ( 971 + 280 T - 53 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$47$ \( 2611456 + 51712 T + 82752 T^{2} - 10240 T^{3} + 2816 T^{4} + 800 T^{5} + 168 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 7862416 + 2860080 T + 878096 T^{2} + 175560 T^{3} + 29296 T^{4} + 4080 T^{5} + 446 T^{6} + 30 T^{7} + T^{8} \)
$59$ \( 24025 - 55800 T + 42560 T^{2} + 28530 T^{3} + 11266 T^{4} + 2412 T^{5} + 329 T^{6} + 18 T^{7} + T^{8} \)
$61$ \( 80656 - 31808 T + 18088 T^{2} - 296 T^{3} - 420 T^{4} - 76 T^{5} + 138 T^{6} - 8 T^{7} + T^{8} \)
$67$ \( ( -29 - 50 T - 23 T^{2} + T^{4} )^{2} \)
$71$ \( 309136 - 153456 T + 75368 T^{2} - 23360 T^{3} + 6376 T^{4} - 1420 T^{5} + 232 T^{6} - 22 T^{7} + T^{8} \)
$73$ \( 689010001 + 143057050 T + 23027261 T^{2} + 2704620 T^{3} + 266756 T^{4} + 21030 T^{5} + 1286 T^{6} + 50 T^{7} + T^{8} \)
$79$ \( 2016400 + 1391600 T + 646640 T^{2} + 190200 T^{3} + 39296 T^{4} + 5664 T^{5} + 566 T^{6} + 34 T^{7} + T^{8} \)
$83$ \( 1113025 + 242650 T + 308940 T^{2} + 173810 T^{3} + 49526 T^{4} + 6764 T^{5} + 621 T^{6} + 34 T^{7} + T^{8} \)
$89$ \( ( -2749 - 1396 T - 169 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$97$ \( 2042041 + 785950 T + 397546 T^{2} + 65260 T^{3} + 12316 T^{4} + 140 T^{5} - 49 T^{6} + T^{8} \)
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