Properties

Label 966.2.g.e
Level $966$
Weight $2$
Character orbit 966.g
Analytic conductor $7.714$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 326 x^{12} + 27081 x^{8} + 96196 x^{4} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 326 x^{12} + 27081 x^{8} + 96196 x^{4} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 30 \nu^{12} + 10275 \nu^{8} + 897461 \nu^{4} + 2736336 \)\()/364028\)
\(\beta_{2}\)\(=\)\((\)\( 425 \nu^{14} + 139062 \nu^{10} + 11684785 \nu^{6} + 59306340 \nu^{2} \)\()/46595584\)
\(\beta_{3}\)\(=\)\((\)\( 1255 \nu^{15} + 19584 \nu^{13} + 332330 \nu^{11} + 5875456 \nu^{9} + 14339167 \nu^{7} + 429268096 \nu^{5} - 998571556 \nu^{3} - 1324973568 \nu \)\()/ 745529344 \)
\(\beta_{4}\)\(=\)\((\)\( -731 \nu^{14} - 1328 \nu^{12} - 230866 \nu^{10} - 350832 \nu^{8} - 18392099 \nu^{6} - 22094784 \nu^{4} - 38603628 \nu^{2} - 40917504 \)\()/23297792\)
\(\beta_{5}\)\(=\)\((\)\( 1255 \nu^{15} - 19584 \nu^{13} + 332330 \nu^{11} - 5875456 \nu^{9} + 14339167 \nu^{7} - 429268096 \nu^{5} - 998571556 \nu^{3} + 1324973568 \nu \)\()/ 745529344 \)
\(\beta_{6}\)\(=\)\((\)\( -1211 \nu^{14} - 395266 \nu^{10} - 32751475 \nu^{6} - 94033900 \nu^{2} \)\()/11648896\)
\(\beta_{7}\)\(=\)\((\)\( -1019 \nu^{15} - 3328 \nu^{13} - 329506 \nu^{11} - 1139840 \nu^{9} - 25842835 \nu^{7} - 96451968 \nu^{5} + 111025876 \nu^{3} + 41256448 \nu \)\()/ 186382336 \)
\(\beta_{8}\)\(=\)\((\)\( -1019 \nu^{15} + 3328 \nu^{13} - 329506 \nu^{11} + 1139840 \nu^{9} - 25842835 \nu^{7} + 96451968 \nu^{5} + 111025876 \nu^{3} - 41256448 \nu \)\()/ 186382336 \)
\(\beta_{9}\)\(=\)\((\)\( 40007 \nu^{15} + 50304 \nu^{13} + 12980842 \nu^{11} + 16397056 \nu^{9} + 1062386367 \nu^{7} + 1348268160 \nu^{5} + 2010513244 \nu^{3} + 2968093184 \nu \)\()/ 745529344 \)
\(\beta_{10}\)\(=\)\((\)\( 40007 \nu^{15} - 50304 \nu^{13} + 12980842 \nu^{11} - 16397056 \nu^{9} + 1062386367 \nu^{7} - 1348268160 \nu^{5} + 2010513244 \nu^{3} - 2968093184 \nu \)\()/ 745529344 \)
\(\beta_{11}\)\(=\)\((\)\( -46807 \nu^{15} - 50304 \nu^{13} - 15205834 \nu^{11} - 16397056 \nu^{9} - 1249342927 \nu^{7} - 1348268160 \nu^{5} - 2959414684 \nu^{3} - 2222563840 \nu \)\()/ 745529344 \)
\(\beta_{12}\)\(=\)\((\)\( 46807 \nu^{15} - 50304 \nu^{13} + 15205834 \nu^{11} - 16397056 \nu^{9} + 1249342927 \nu^{7} - 1348268160 \nu^{5} + 2959414684 \nu^{3} - 2222563840 \nu \)\()/ 745529344 \)
\(\beta_{13}\)\(=\)\((\)\(-46807 \nu^{15} + 50304 \nu^{13} + 53248 \nu^{12} - 15205834 \nu^{11} + 16397056 \nu^{9} + 18237440 \nu^{8} - 1249342927 \nu^{7} + 1348268160 \nu^{5} + 1543231488 \nu^{4} - 2959414684 \nu^{3} + 2968093184 \nu + 830955520\)\()/ 745529344 \)
\(\beta_{14}\)\(=\)\((\)\(-46807 \nu^{15} + 23392 \nu^{14} + 50304 \nu^{13} - 157184 \nu^{12} - 15205834 \nu^{11} + 7387712 \nu^{10} + 16397056 \nu^{9} - 50507264 \nu^{8} - 1249342927 \nu^{7} + 588547168 \nu^{6} + 1348268160 \nu^{5} - 4088264704 \nu^{4} - 2959414684 \nu^{3} + 1235316096 \nu^{2} + 2968093184 \nu - 6998802432\)\()/ 745529344 \)
\(\beta_{15}\)\(=\)\((\)\(-46807 \nu^{15} + 29904 \nu^{14} + 50304 \nu^{13} - 15205834 \nu^{11} + 9722080 \nu^{10} + 16397056 \nu^{9} - 1249342927 \nu^{7} + 787398480 \nu^{6} + 1348268160 \nu^{5} - 2959414684 \nu^{3} + 121388864 \nu^{2} + 2968093184 \nu\)\()/ 745529344 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{15} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{6} + 14 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{12} - 9 \beta_{11} - 11 \beta_{10} - 11 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-30 \beta_{13} - 15 \beta_{12} + 15 \beta_{11} - 15 \beta_{10} + 15 \beta_{9} + 26 \beta_{1} - 162\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-163 \beta_{12} - 163 \beta_{11} + 137 \beta_{10} - 137 \beta_{9} - 60 \beta_{8} + 60 \beta_{7} + 26 \beta_{5} - 26 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-394 \beta_{15} + 8 \beta_{14} + 8 \beta_{13} - 189 \beta_{12} + 189 \beta_{11} - 189 \beta_{10} + 189 \beta_{9} - 326 \beta_{6} - 8 \beta_{4} - 2038 \beta_{2} + 8 \beta_{1} - 8\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-1413 \beta_{12} + 1413 \beta_{11} + 1747 \beta_{10} + 1747 \beta_{9} + 828 \beta_{8} + 828 \beta_{7} - 302 \beta_{5} - 302 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(224 \beta_{14} + 5310 \beta_{13} + 2767 \beta_{12} - 2767 \beta_{11} + 2767 \beta_{10} - 2767 \beta_{9} + 224 \beta_{4} - 3874 \beta_{1} + 25698\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(25795 \beta_{12} + 25795 \beta_{11} - 21473 \beta_{10} + 21473 \beta_{9} + 11292 \beta_{8} - 11292 \beta_{7} - 3426 \beta_{5} + 3426 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(65530 \beta_{15} - 4440 \beta_{14} - 4440 \beta_{13} + 28325 \beta_{12} - 28325 \beta_{11} + 28325 \beta_{10} - 28325 \beta_{9} + 51590 \beta_{6} + 4440 \beta_{4} + 330374 \beta_{2} - 4440 \beta_{1} + 4440\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(230717 \beta_{12} - 230717 \beta_{11} - 286747 \beta_{10} - 286747 \beta_{9} - 153260 \beta_{8} - 153260 \beta_{7} + 38270 \beta_{5} + 38270 \beta_{3}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-76720 \beta_{14} - 921214 \beta_{13} - 498967 \beta_{12} + 498967 \beta_{11} - 498967 \beta_{10} + 498967 \beta_{9} - 76720 \beta_{4} + 573314 \beta_{1} - 4137698\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-4098331 \beta_{12} - 4098331 \beta_{11} + 3371577 \beta_{10} - 3371577 \beta_{9} - 2072588 \beta_{8} + 2072588 \beta_{7} + 419874 \beta_{5} - 419874 \beta_{3}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-10888330 \beta_{15} + 1232840 \beta_{14} + 1232840 \beta_{13} - 4211325 \beta_{12} + 4211325 \beta_{11} - 4211325 \beta_{10} + 4211325 \beta_{9} - 8196662 \beta_{6} - 1232840 \beta_{4} - 53802294 \beta_{2} + 1232840 \beta_{1} - 1232840\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-37789477 \beta_{12} + 37789477 \beta_{11} + 47218979 \beta_{10} + 47218979 \beta_{9} + 27940860 \beta_{8} + 27940860 \beta_{7} - 4498142 \beta_{5} - 4498142 \beta_{3}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1
2.55383 2.55383i
−0.912244 + 0.912244i
−2.48351 + 2.48351i
−0.691340 + 0.691340i
0.691340 0.691340i
2.48351 2.48351i
0.912244 0.912244i
−2.55383 + 2.55383i
2.55383 + 2.55383i
−0.912244 0.912244i
−2.48351 2.48351i
−0.691340 0.691340i
0.691340 + 0.691340i
2.48351 + 2.48351i
0.912244 + 0.912244i
−2.55383 2.55383i
1.00000 1.00000i 1.00000 −4.19800 1.00000i 2.55383 + 0.691340i 1.00000 −1.00000 −4.19800
643.2 1.00000 1.00000i 1.00000 −2.93440 1.00000i −0.912244 + 2.48351i 1.00000 −1.00000 −2.93440
643.3 1.00000 1.00000i 1.00000 −1.36314 1.00000i −2.48351 + 0.912244i 1.00000 −1.00000 −1.36314
643.4 1.00000 1.00000i 1.00000 −0.952834 1.00000i −0.691340 2.55383i 1.00000 −1.00000 −0.952834
643.5 1.00000 1.00000i 1.00000 0.952834 1.00000i 0.691340 + 2.55383i 1.00000 −1.00000 0.952834
643.6 1.00000 1.00000i 1.00000 1.36314 1.00000i 2.48351 0.912244i 1.00000 −1.00000 1.36314
643.7 1.00000 1.00000i 1.00000 2.93440 1.00000i 0.912244 2.48351i 1.00000 −1.00000 2.93440
643.8 1.00000 1.00000i 1.00000 4.19800 1.00000i −2.55383 0.691340i 1.00000 −1.00000 4.19800
643.9 1.00000 1.00000i 1.00000 −4.19800 1.00000i 2.55383 0.691340i 1.00000 −1.00000 −4.19800
643.10 1.00000 1.00000i 1.00000 −2.93440 1.00000i −0.912244 2.48351i 1.00000 −1.00000 −2.93440
643.11 1.00000 1.00000i 1.00000 −1.36314 1.00000i −2.48351 0.912244i 1.00000 −1.00000 −1.36314
643.12 1.00000 1.00000i 1.00000 −0.952834 1.00000i −0.691340 + 2.55383i 1.00000 −1.00000 −0.952834
643.13 1.00000 1.00000i 1.00000 0.952834 1.00000i 0.691340 2.55383i 1.00000 −1.00000 0.952834
643.14 1.00000 1.00000i 1.00000 1.36314 1.00000i 2.48351 + 0.912244i 1.00000 −1.00000 1.36314
643.15 1.00000 1.00000i 1.00000 2.93440 1.00000i 0.912244 + 2.48351i 1.00000 −1.00000 2.93440
643.16 1.00000 1.00000i 1.00000 4.19800 1.00000i −2.55383 + 0.691340i 1.00000 −1.00000 4.19800
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.g.e 16
3.b odd 2 1 2898.2.g.j 16
7.b odd 2 1 inner 966.2.g.e 16
21.c even 2 1 2898.2.g.j 16
23.b odd 2 1 inner 966.2.g.e 16
69.c even 2 1 2898.2.g.j 16
161.c even 2 1 inner 966.2.g.e 16
483.c odd 2 1 2898.2.g.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.e 16 1.a even 1 1 trivial
966.2.g.e 16 7.b odd 2 1 inner
966.2.g.e 16 23.b odd 2 1 inner
966.2.g.e 16 161.c even 2 1 inner
2898.2.g.j 16 3.b odd 2 1
2898.2.g.j 16 21.c even 2 1
2898.2.g.j 16 69.c even 2 1
2898.2.g.j 16 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{8} - 29 T_{5}^{6} + 226 T_{5}^{4} - 464 T_{5}^{2} + 256 \)
\( T_{11}^{8} + 90 T_{11}^{6} + 2680 T_{11}^{4} + 27200 T_{11}^{2} + 25600 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{16} \)
$3$ \( ( 1 + T^{2} )^{8} \)
$5$ \( ( 256 - 464 T^{2} + 226 T^{4} - 29 T^{6} + T^{8} )^{2} \)
$7$ \( 5764801 - 153664 T^{4} + 5566 T^{8} - 64 T^{12} + T^{16} \)
$11$ \( ( 25600 + 27200 T^{2} + 2680 T^{4} + 90 T^{6} + T^{8} )^{2} \)
$13$ \( ( 256 + 784 T^{2} + 616 T^{4} + 49 T^{6} + T^{8} )^{2} \)
$17$ \( ( 4096 - 3696 T^{2} + 916 T^{4} - 56 T^{6} + T^{8} )^{2} \)
$19$ \( ( 25600 - 27200 T^{2} + 2680 T^{4} - 90 T^{6} + T^{8} )^{2} \)
$23$ \( ( 529 + 46 T - 18 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$29$ \( ( -400 + 500 T - 80 T^{2} - 5 T^{3} + T^{4} )^{4} \)
$31$ \( T^{16} \)
$37$ \( ( 256 + 1616 T^{2} + 1266 T^{4} + 101 T^{6} + T^{8} )^{2} \)
$41$ \( ( 25600 + 19600 T^{2} + 2680 T^{4} + 105 T^{6} + T^{8} )^{2} \)
$43$ \( ( 602176 + 98024 T^{2} + 5566 T^{4} + 129 T^{6} + T^{8} )^{2} \)
$47$ \( ( 65536 + 179456 T^{2} + 10116 T^{4} + 181 T^{6} + T^{8} )^{2} \)
$53$ \( ( 15241216 + 1110144 T^{2} + 27016 T^{4} + 274 T^{6} + T^{8} )^{2} \)
$59$ \( ( 4096 + 49664 T^{2} + 17296 T^{4} + 264 T^{6} + T^{8} )^{2} \)
$61$ \( ( 4096 - 50176 T^{2} + 4816 T^{4} - 126 T^{6} + T^{8} )^{2} \)
$67$ \( ( 38738176 + 2802016 T^{2} + 55216 T^{4} + 406 T^{6} + T^{8} )^{2} \)
$71$ \( ( -1024 - 672 T - 76 T^{2} + 12 T^{3} + T^{4} )^{4} \)
$73$ \( ( 1048576 + 282624 T^{2} + 14656 T^{4} + 244 T^{6} + T^{8} )^{2} \)
$79$ \( ( 2383936 + 392864 T^{2} + 18036 T^{4} + 244 T^{6} + T^{8} )^{2} \)
$83$ \( ( 65536 - 41344 T^{2} + 6216 T^{4} - 194 T^{6} + T^{8} )^{2} \)
$89$ \( ( 409600 - 305600 T^{2} + 26980 T^{4} - 380 T^{6} + T^{8} )^{2} \)
$97$ \( ( 157351936 - 11465216 T^{2} + 156136 T^{4} - 711 T^{6} + T^{8} )^{2} \)
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