Properties

Label 930.2.h.d
Level $930$
Weight $2$
Character orbit 930.h
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{14} + 6 x^{12} + 36 x^{10} - 142 x^{8} + 324 x^{6} + 486 x^{4} - 2916 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{14} + 6 x^{12} + 36 x^{10} - 142 x^{8} + 324 x^{6} + 486 x^{4} - 2916 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{15} - 101 \nu^{13} + 111 \nu^{11} - 63 \nu^{9} - 4355 \nu^{7} + 2187 \nu^{5} - 22113 \nu^{3} - 57591 \nu \)\()/69984\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{14} - 101 \nu^{12} + 111 \nu^{10} - 63 \nu^{8} - 4355 \nu^{6} + 2187 \nu^{4} + 1215 \nu^{2} - 80919 \)\()/23328\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{13} + 6 \nu^{11} + 36 \nu^{9} - 142 \nu^{7} + 324 \nu^{5} + 486 \nu^{3} - 2916 \nu \)\()/2187\)
\(\beta_{6}\)\(=\)\((\)\( 14 \nu^{14} - 65 \nu^{12} - 42 \nu^{10} + 369 \nu^{8} - 1826 \nu^{6} + 711 \nu^{4} + 5022 \nu^{2} - 45927 \)\()/11664\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{14} + 4 \nu^{12} - 6 \nu^{10} - 36 \nu^{8} + 142 \nu^{6} - 324 \nu^{4} - 486 \nu^{2} + 2916 \)\()/729\)
\(\beta_{8}\)\(=\)\((\)\( -23 \nu^{14} - 61 \nu^{12} - 93 \nu^{10} - 207 \nu^{8} - 1999 \nu^{6} - 2493 \nu^{4} - 10125 \nu^{2} - 57591 \)\()/11664\)
\(\beta_{9}\)\(=\)\((\)\( -19 \nu^{14} + 22 \nu^{12} + 21 \nu^{10} - 684 \nu^{8} + 997 \nu^{6} + 54 \nu^{4} - 14499 \nu^{2} + 20412 \)\()/5832\)
\(\beta_{10}\)\(=\)\((\)\( -14 \nu^{15} + 65 \nu^{13} + 42 \nu^{11} - 369 \nu^{9} + 1826 \nu^{7} - 711 \nu^{5} - 5022 \nu^{3} + 45927 \nu \)\()/11664\)
\(\beta_{11}\)\(=\)\((\)\( -61 \nu^{15} - 17 \nu^{13} - 51 \nu^{11} - 1575 \nu^{9} - 5 \nu^{7} - 2385 \nu^{5} - 39123 \nu^{3} - 5103 \nu \)\()/34992\)
\(\beta_{12}\)\(=\)\((\)\( 7 \nu^{15} - 14 \nu^{13} - 23 \nu^{11} + 210 \nu^{9} - 625 \nu^{7} + 442 \nu^{5} + 4113 \nu^{3} - 15390 \nu \)\()/3888\)
\(\beta_{13}\)\(=\)\((\)\( 85 \nu^{15} - 168 \nu^{14} - 43 \nu^{13} - 192 \nu^{12} - 273 \nu^{11} + 504 \nu^{10} + 3951 \nu^{9} - 3456 \nu^{8} - 4051 \nu^{7} - 1416 \nu^{6} + 2133 \nu^{5} - 1728 \nu^{4} + 64719 \nu^{3} - 138024 \nu^{2} - 117369 \nu \)\()/69984\)
\(\beta_{14}\)\(=\)\((\)\( -179 \nu^{15} - 168 \nu^{14} + 635 \nu^{13} - 192 \nu^{12} + 303 \nu^{11} + 504 \nu^{10} - 6039 \nu^{9} - 3456 \nu^{8} + 23717 \nu^{7} - 1416 \nu^{6} - 10773 \nu^{5} - 1728 \nu^{4} - 120609 \nu^{3} - 138024 \nu^{2} + 508113 \nu \)\()/69984\)
\(\beta_{15}\)\(=\)\((\)\( -179 \nu^{15} + 168 \nu^{14} + 635 \nu^{13} + 192 \nu^{12} + 303 \nu^{11} - 504 \nu^{10} - 6039 \nu^{9} + 3456 \nu^{8} + 23717 \nu^{7} + 1416 \nu^{6} - 10773 \nu^{5} + 1728 \nu^{4} - 120609 \nu^{3} + 138024 \nu^{2} + 508113 \nu \)\()/69984\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - \beta_{13} + \beta_{10} - 2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{15} - \beta_{14} + 2 \beta_{9} + \beta_{8} - 3 \beta_{7} - 2 \beta_{4} + 3\)
\(\nu^{5}\)\(=\)\(\beta_{15} + 2 \beta_{14} - \beta_{13} + 5 \beta_{12} + 2 \beta_{11} + 8 \beta_{5} - 4 \beta_{3} + 4 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-3 \beta_{15} + 3 \beta_{14} - 4 \beta_{8} + \beta_{7} + 8 \beta_{6} - 8 \beta_{4} + 6 \beta_{2} - 20\)
\(\nu^{7}\)\(=\)\(5 \beta_{15} + \beta_{14} + 4 \beta_{13} - 8 \beta_{12} - 2 \beta_{11} - 16 \beta_{10} + 7 \beta_{5} - 14 \beta_{3} - 8 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-3 \beta_{15} + 3 \beta_{14} - 12 \beta_{9} + 8 \beta_{8} - 10 \beta_{7} - 16 \beta_{6} - 16 \beta_{4} - 10 \beta_{2} + 3\)
\(\nu^{9}\)\(=\)\(29 \beta_{15} - 23 \beta_{14} + 52 \beta_{13} - 32 \beta_{12} + 10 \beta_{11} - 16 \beta_{10} + 16 \beta_{5} + 10 \beta_{3} + 11 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-39 \beta_{15} + 39 \beta_{14} - 48 \beta_{9} - 52 \beta_{8} + 26 \beta_{7} - 40 \beta_{6} + 56 \beta_{4} + 21 \beta_{2} - 156\)
\(\nu^{11}\)\(=\)\(-38 \beta_{15} - 35 \beta_{14} - 3 \beta_{13} - 152 \beta_{12} - 74 \beta_{11} + 39 \beta_{10} + 4 \beta_{5} + 92 \beta_{3} - 160 \beta_{1}\)
\(\nu^{12}\)\(=\)\(112 \beta_{15} - 112 \beta_{14} + 2 \beta_{9} + 111 \beta_{8} - 55 \beta_{7} - 352 \beta_{6} + 162 \beta_{4} - 234 \beta_{2} - 63\)
\(\nu^{13}\)\(=\)\(-40 \beta_{15} + 3 \beta_{14} - 43 \beta_{13} + 339 \beta_{12} + 504 \beta_{10} - 168 \beta_{5} + 406 \beta_{3} - 336 \beta_{1}\)
\(\nu^{14}\)\(=\)\(40 \beta_{15} - 40 \beta_{14} + 80 \beta_{9} - 424 \beta_{8} + 369 \beta_{7} + 544 \beta_{6} + 400 \beta_{4} - 336 \beta_{2} - 320\)
\(\nu^{15}\)\(=\)\(-104 \beta_{15} + 544 \beta_{14} - 648 \beta_{13} + 664 \beta_{12} - 848 \beta_{11} - 400 \beta_{10} - 683 \beta_{5} + 992 \beta_{3} - 296 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
0.206738 + 1.71967i
−1.29725 + 1.14767i
1.60627 + 0.647994i
−1.64143 + 0.552909i
1.64143 0.552909i
−1.60627 0.647994i
1.29725 1.14767i
−0.206738 1.71967i
0.206738 1.71967i
−1.29725 1.14767i
1.60627 0.647994i
−1.64143 0.552909i
1.64143 + 0.552909i
−1.60627 + 0.647994i
1.29725 + 1.14767i
−0.206738 + 1.71967i
1.00000i −1.71967 0.206738i −1.00000 1.00000i −0.206738 + 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.2 1.00000i −1.14767 + 1.29725i −1.00000 1.00000i 1.29725 + 1.14767i 3.69457 1.00000i −0.365730 2.97762i 1.00000
371.3 1.00000i −0.647994 1.60627i −1.00000 1.00000i −1.60627 + 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.4 1.00000i −0.552909 + 1.64143i −1.00000 1.00000i 1.64143 + 0.552909i −2.30718 1.00000i −2.38858 1.81512i 1.00000
371.5 1.00000i 0.552909 1.64143i −1.00000 1.00000i −1.64143 0.552909i −2.30718 1.00000i −2.38858 1.81512i 1.00000
371.6 1.00000i 0.647994 + 1.60627i −1.00000 1.00000i 1.60627 0.647994i 0.794255 1.00000i −2.16021 + 2.08171i 1.00000
371.7 1.00000i 1.14767 1.29725i −1.00000 1.00000i −1.29725 1.14767i 3.69457 1.00000i −0.365730 2.97762i 1.00000
371.8 1.00000i 1.71967 + 0.206738i −1.00000 1.00000i 0.206738 1.71967i −1.18164 1.00000i 2.91452 + 0.711040i 1.00000
371.9 1.00000i −1.71967 + 0.206738i −1.00000 1.00000i −0.206738 1.71967i −1.18164 1.00000i 2.91452 0.711040i 1.00000
371.10 1.00000i −1.14767 1.29725i −1.00000 1.00000i 1.29725 1.14767i 3.69457 1.00000i −0.365730 + 2.97762i 1.00000
371.11 1.00000i −0.647994 + 1.60627i −1.00000 1.00000i −1.60627 0.647994i 0.794255 1.00000i −2.16021 2.08171i 1.00000
371.12 1.00000i −0.552909 1.64143i −1.00000 1.00000i 1.64143 0.552909i −2.30718 1.00000i −2.38858 + 1.81512i 1.00000
371.13 1.00000i 0.552909 + 1.64143i −1.00000 1.00000i −1.64143 + 0.552909i −2.30718 1.00000i −2.38858 + 1.81512i 1.00000
371.14 1.00000i 0.647994 1.60627i −1.00000 1.00000i 1.60627 + 0.647994i 0.794255 1.00000i −2.16021 2.08171i 1.00000
371.15 1.00000i 1.14767 + 1.29725i −1.00000 1.00000i −1.29725 + 1.14767i 3.69457 1.00000i −0.365730 + 2.97762i 1.00000
371.16 1.00000i 1.71967 0.206738i −1.00000 1.00000i 0.206738 + 1.71967i −1.18164 1.00000i 2.91452 0.711040i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 371.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.d 16
3.b odd 2 1 inner 930.2.h.d 16
31.b odd 2 1 inner 930.2.h.d 16
93.c even 2 1 inner 930.2.h.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.d 16 1.a even 1 1 trivial
930.2.h.d 16 3.b odd 2 1 inner
930.2.h.d 16 31.b odd 2 1 inner
930.2.h.d 16 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{3} - 10 T_{7}^{2} - 2 T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( 6561 + 2916 T^{2} + 486 T^{4} - 324 T^{6} - 142 T^{8} - 36 T^{10} + 6 T^{12} + 4 T^{14} + T^{16} \)
$5$ \( ( 1 + T^{2} )^{8} \)
$7$ \( ( 8 - 2 T - 10 T^{2} - T^{3} + T^{4} )^{4} \)
$11$ \( ( 8 - 86 T^{2} + 134 T^{4} - 43 T^{6} + T^{8} )^{2} \)
$13$ \( ( 128 + 216 T^{2} + 114 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$17$ \( ( 8192 - 8064 T^{2} + 1184 T^{4} - 60 T^{6} + T^{8} )^{2} \)
$19$ \( ( 256 + 128 T - 36 T^{2} - 5 T^{3} + T^{4} )^{4} \)
$23$ \( ( 512 - 640 T^{2} + 228 T^{4} - 27 T^{6} + T^{8} )^{2} \)
$29$ \( ( 61952 - 98752 T^{2} + 7154 T^{4} - 152 T^{6} + T^{8} )^{2} \)
$31$ \( ( 923521 + 119164 T - 53816 T^{2} - 372 T^{3} + 2798 T^{4} - 12 T^{5} - 56 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$37$ \( ( 3463712 + 623856 T^{2} + 22258 T^{4} + 264 T^{6} + T^{8} )^{2} \)
$41$ \( ( 1024 + 7808 T^{2} + 2340 T^{4} + 100 T^{6} + T^{8} )^{2} \)
$43$ \( ( 209952 + 91854 T^{2} + 5994 T^{4} + 135 T^{6} + T^{8} )^{2} \)
$47$ \( ( 135424 + 76368 T^{2} + 6580 T^{4} + 176 T^{6} + T^{8} )^{2} \)
$53$ \( ( 4892192 - 614926 T^{2} + 21810 T^{4} - 279 T^{6} + T^{8} )^{2} \)
$59$ \( ( 4734976 + 653312 T^{2} + 22336 T^{4} + 272 T^{6} + T^{8} )^{2} \)
$61$ \( ( 247808 + 84320 T^{2} + 7842 T^{4} + 192 T^{6} + T^{8} )^{2} \)
$67$ \( ( -64 - 336 T - 128 T^{2} + 6 T^{3} + T^{4} )^{4} \)
$71$ \( ( 7744 + 852272 T^{2} + 33388 T^{4} + 357 T^{6} + T^{8} )^{2} \)
$73$ \( ( 2048 + 12160 T^{2} + 6352 T^{4} + 287 T^{6} + T^{8} )^{2} \)
$79$ \( ( 59710592 + 3278944 T^{2} + 59332 T^{4} + 419 T^{6} + T^{8} )^{2} \)
$83$ \( ( 31490048 - 2882104 T^{2} + 57714 T^{4} - 420 T^{6} + T^{8} )^{2} \)
$89$ \( ( 1131008 - 178560 T^{2} + 9348 T^{4} - 179 T^{6} + T^{8} )^{2} \)
$97$ \( ( 592 + 20 T - 74 T^{2} + 2 T^{3} + T^{4} )^{4} \)
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