Properties

Label 2151.2.a.h
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} - 479 x^{3} + 129 x^{2} + 88 x - 31\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} - 479 x^{3} + 129 x^{2} + 88 x - 31\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{10} - 13 \nu^{9} + 26 \nu^{8} + 45 \nu^{7} - 103 \nu^{6} + 3 \nu^{5} + 118 \nu^{4} - 143 \nu^{3} - 18 \nu^{2} + 51 \nu - 5 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} - 12 \nu^{9} + 40 \nu^{8} + 35 \nu^{7} - 162 \nu^{6} + 23 \nu^{5} + 177 \nu^{4} - 114 \nu^{3} + 34 \nu^{2} + 23 \nu - 18 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 50 \nu^{8} + 35 \nu^{7} + 173 \nu^{6} - 331 \nu^{5} - 68 \nu^{4} + 541 \nu^{3} - 234 \nu^{2} - 103 \nu + 59 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{11} + 4 \nu^{10} + 49 \nu^{9} - 58 \nu^{8} - 279 \nu^{7} + 291 \nu^{6} + 655 \nu^{5} - 604 \nu^{4} - 581 \nu^{3} + 510 \nu^{2} + 103 \nu - 95 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{11} + 14 \nu^{10} + 85 \nu^{9} - 222 \nu^{8} - 533 \nu^{7} + 1215 \nu^{6} + 1509 \nu^{5} - 2662 \nu^{4} - 1805 \nu^{3} + 2046 \nu^{2} + 409 \nu - 371 \)\()/8\)
\(\beta_{9}\)\(=\)\( \nu^{11} - 3 \nu^{10} - 15 \nu^{9} + 45 \nu^{8} + 78 \nu^{7} - 227 \nu^{6} - 172 \nu^{5} + 437 \nu^{4} + 169 \nu^{3} - 277 \nu^{2} - 33 \nu + 43 \)
\(\beta_{10}\)\(=\)\((\)\( 7 \nu^{11} - 10 \nu^{10} - 127 \nu^{9} + 162 \nu^{8} + 839 \nu^{7} - 933 \nu^{6} - 2423 \nu^{5} + 2242 \nu^{4} + 2775 \nu^{3} - 1954 \nu^{2} - 619 \nu + 345 \)\()/8\)
\(\beta_{11}\)\(=\)\((\)\( -9 \nu^{11} + 20 \nu^{10} + 151 \nu^{9} - 310 \nu^{8} - 917 \nu^{7} + 1653 \nu^{6} + 2453 \nu^{5} - 3508 \nu^{4} - 2723 \nu^{3} + 2594 \nu^{2} + 581 \nu - 429 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 9 \beta_{3} + 28 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(-12 \beta_{11} - 23 \beta_{10} - 10 \beta_{9} + 3 \beta_{8} - 11 \beta_{7} + \beta_{6} - 12 \beta_{4} + 14 \beta_{3} + 34 \beta_{2} + 12 \beta_{1} + 85\)
\(\nu^{7}\)\(=\)\(-4 \beta_{11} - 19 \beta_{10} + 10 \beta_{9} + 15 \beta_{8} - 16 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} - 19 \beta_{4} + 75 \beta_{3} + 169 \beta_{1} + 12\)
\(\nu^{8}\)\(=\)\(-111 \beta_{11} - 208 \beta_{10} - 78 \beta_{9} + 48 \beta_{8} - 98 \beta_{7} + 15 \beta_{6} - 2 \beta_{5} - 114 \beta_{4} + 146 \beta_{3} + 191 \beta_{2} + 117 \beta_{1} + 509\)
\(\nu^{9}\)\(=\)\(-69 \beta_{11} - 229 \beta_{10} + 78 \beta_{9} + 165 \beta_{8} - 176 \beta_{7} + 203 \beta_{6} + 90 \beta_{5} - 224 \beta_{4} + 611 \beta_{3} - 2 \beta_{2} + 1077 \beta_{1} + 123\)
\(\nu^{10}\)\(=\)\(-934 \beta_{11} - 1732 \beta_{10} - 563 \beta_{9} + 530 \beta_{8} - 818 \beta_{7} + 164 \beta_{6} - 30 \beta_{5} - 997 \beta_{4} + 1347 \beta_{3} + 1072 \beta_{2} + 1053 \beta_{1} + 3162\)
\(\nu^{11}\)\(=\)\(-817 \beta_{11} - 2308 \beta_{10} + 551 \beta_{9} + 1588 \beta_{8} - 1668 \beta_{7} + 1639 \beta_{6} + 664 \beta_{5} - 2198 \beta_{4} + 4906 \beta_{3} - 36 \beta_{2} + 7158 \beta_{1} + 1190\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.78161
2.34937
2.33213
1.86392
0.719502
0.650058
0.411990
−0.496586
−1.40336
−1.42188
−2.27963
−2.50712
−2.78161 0 5.73734 1.06781 0 2.18960 −10.3958 0 −2.97024
1.2 −2.34937 0 3.51954 −1.00390 0 −3.03107 −3.56996 0 2.35853
1.3 −2.33213 0 3.43882 −3.08269 0 0.766491 −3.35552 0 7.18923
1.4 −1.86392 0 1.47419 3.80019 0 4.40195 0.980069 0 −7.08324
1.5 −0.719502 0 −1.48232 1.41968 0 −0.0123010 2.50553 0 −1.02146
1.6 −0.650058 0 −1.57742 −3.22288 0 3.97632 2.32553 0 2.09506
1.7 −0.411990 0 −1.83026 3.83303 0 −3.43834 1.57803 0 −1.57917
1.8 0.496586 0 −1.75340 −1.64136 0 −3.34365 −1.86389 0 −0.815074
1.9 1.40336 0 −0.0305942 0.939648 0 3.20775 −2.84964 0 1.31866
1.10 1.42188 0 0.0217417 −3.65064 0 2.24006 −2.81285 0 −5.19077
1.11 2.27963 0 3.19672 2.95582 0 0.183248 2.72808 0 6.73817
1.12 2.50712 0 4.28565 −0.414699 0 3.85993 5.73041 0 −1.03970
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(239\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.2.a.h 12
3.b odd 2 1 717.2.a.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.g 12 3.b odd 2 1
2151.2.a.h 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\(T_{2}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -31 - 88 T + 129 T^{2} + 479 T^{3} + 23 T^{4} - 571 T^{5} - 134 T^{6} + 256 T^{7} + 75 T^{8} - 47 T^{9} - 15 T^{10} + 3 T^{11} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( 1520 + 2064 T - 6360 T^{2} - 3384 T^{3} + 7563 T^{4} + 1629 T^{5} - 3247 T^{6} - 339 T^{7} + 546 T^{8} + 31 T^{9} - 39 T^{10} - T^{11} + T^{12} \)
$7$ \( 64 + 4704 T - 39648 T^{2} + 73152 T^{3} - 38700 T^{4} - 7552 T^{5} + 11866 T^{6} - 1814 T^{7} - 949 T^{8} + 293 T^{9} + 9 T^{10} - 11 T^{11} + T^{12} \)
$11$ \( -15536 - 36176 T + 91708 T^{2} + 26660 T^{3} - 64379 T^{4} - 7714 T^{5} + 15264 T^{6} + 2075 T^{7} - 1502 T^{8} - 300 T^{9} + 44 T^{10} + 15 T^{11} + T^{12} \)
$13$ \( 1280 - 6784 T - 1344 T^{2} + 41696 T^{3} - 31072 T^{4} - 34728 T^{5} + 44508 T^{6} - 14406 T^{7} + 115 T^{8} + 655 T^{9} - 69 T^{10} - 7 T^{11} + T^{12} \)
$17$ \( -2880080 + 4078736 T - 547944 T^{2} - 1282552 T^{3} + 355979 T^{4} + 155057 T^{5} - 50427 T^{6} - 9117 T^{7} + 3138 T^{8} + 263 T^{9} - 91 T^{10} - 3 T^{11} + T^{12} \)
$19$ \( -4568000 + 2520960 T + 3042496 T^{2} - 1580224 T^{3} - 611132 T^{4} + 331584 T^{5} + 35862 T^{6} - 28482 T^{7} + 557 T^{8} + 929 T^{9} - 68 T^{10} - 10 T^{11} + T^{12} \)
$23$ \( -520192 + 421888 T + 701184 T^{2} - 354432 T^{3} - 383408 T^{4} + 60736 T^{5} + 84040 T^{6} + 5502 T^{7} - 5233 T^{8} - 793 T^{9} + 72 T^{10} + 20 T^{11} + T^{12} \)
$29$ \( -27648560 - 43505216 T - 10417024 T^{2} + 10882288 T^{3} + 3772803 T^{4} - 971704 T^{5} - 344609 T^{6} + 35330 T^{7} + 12766 T^{8} - 476 T^{9} - 193 T^{10} + 2 T^{11} + T^{12} \)
$31$ \( 622739200 + 506057088 T - 350399408 T^{2} - 102328896 T^{3} + 31354993 T^{4} + 6439341 T^{5} - 1199278 T^{6} - 176008 T^{7} + 23726 T^{8} + 2181 T^{9} - 242 T^{10} - 10 T^{11} + T^{12} \)
$37$ \( -6555065600 - 10002574720 T + 3926212288 T^{2} + 434644576 T^{3} - 271004352 T^{4} + 6713768 T^{5} + 6441508 T^{6} - 519942 T^{7} - 53391 T^{8} + 7195 T^{9} + 22 T^{10} - 30 T^{11} + T^{12} \)
$41$ \( -605102912 - 191198176 T + 172084848 T^{2} + 33884552 T^{3} - 21277660 T^{4} - 1321720 T^{5} + 1244778 T^{6} - 50978 T^{7} - 26929 T^{8} + 2724 T^{9} + 133 T^{10} - 28 T^{11} + T^{12} \)
$43$ \( -142624448 + 632881920 T - 491235392 T^{2} + 33058368 T^{3} + 55074868 T^{4} - 12072792 T^{5} - 1027122 T^{6} + 548202 T^{7} - 42115 T^{8} - 3877 T^{9} + 810 T^{10} - 48 T^{11} + T^{12} \)
$47$ \( 2011774976 + 1927790592 T + 423889408 T^{2} - 139632768 T^{3} - 75083056 T^{4} - 7650080 T^{5} + 1436288 T^{6} + 332052 T^{7} + 2935 T^{8} - 3749 T^{9} - 203 T^{10} + 13 T^{11} + T^{12} \)
$53$ \( 649713088 + 387585952 T - 342895696 T^{2} - 111519512 T^{3} + 47632508 T^{4} + 7450820 T^{5} - 2374370 T^{6} - 159392 T^{7} + 47831 T^{8} + 1073 T^{9} - 378 T^{10} - 2 T^{11} + T^{12} \)
$59$ \( 13769595904 - 376613376 T - 2945852800 T^{2} - 246177920 T^{3} + 158278928 T^{4} + 16325488 T^{5} - 3815752 T^{6} - 381182 T^{7} + 48323 T^{8} + 3824 T^{9} - 329 T^{10} - 14 T^{11} + T^{12} \)
$61$ \( 15680924560 + 12748208304 T - 1061957948 T^{2} - 1821208560 T^{3} + 123889883 T^{4} + 66562241 T^{5} - 4742914 T^{6} - 967832 T^{7} + 69584 T^{8} + 6109 T^{9} - 438 T^{10} - 14 T^{11} + T^{12} \)
$67$ \( -862161920 - 821496832 T + 1492108160 T^{2} - 696534848 T^{3} + 121115472 T^{4} + 5651848 T^{5} - 5555440 T^{6} + 795860 T^{7} - 13791 T^{8} - 8424 T^{9} + 1038 T^{10} - 52 T^{11} + T^{12} \)
$71$ \( 55275520 - 220711936 T + 191008512 T^{2} - 12712896 T^{3} - 40160272 T^{4} + 13521072 T^{5} - 335100 T^{6} - 365828 T^{7} + 31847 T^{8} + 2903 T^{9} - 335 T^{10} - 7 T^{11} + T^{12} \)
$73$ \( 1080955648 - 2911910912 T + 2003475968 T^{2} - 67533216 T^{3} - 186711936 T^{4} + 29220752 T^{5} + 3794240 T^{6} - 1058038 T^{7} + 27661 T^{8} + 7321 T^{9} - 406 T^{10} - 14 T^{11} + T^{12} \)
$79$ \( 35608000 + 74194560 T + 17679264 T^{2} - 29891736 T^{3} - 4048276 T^{4} + 3958400 T^{5} + 186474 T^{6} - 210206 T^{7} + 2517 T^{8} + 4158 T^{9} - 230 T^{10} - 15 T^{11} + T^{12} \)
$83$ \( 112487344 - 157055216 T - 28789948 T^{2} + 46203204 T^{3} + 1685743 T^{4} - 4905132 T^{5} + 72922 T^{6} + 223649 T^{7} - 6956 T^{8} - 4070 T^{9} + 30 T^{10} + 29 T^{11} + T^{12} \)
$89$ \( -8473904128 - 2004426496 T + 13521419072 T^{2} - 8070607664 T^{3} + 1613773164 T^{4} - 32701132 T^{5} - 32193426 T^{6} + 4509256 T^{7} - 146357 T^{8} - 16864 T^{9} + 1858 T^{10} - 71 T^{11} + T^{12} \)
$97$ \( -108705900800 - 67016156416 T - 5692105984 T^{2} + 3506915936 T^{3} + 592744576 T^{4} - 59016944 T^{5} - 14327824 T^{6} + 371490 T^{7} + 145957 T^{8} - 644 T^{9} - 649 T^{10} + T^{12} \)
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