Properties

Label 231.2.i.f
Level 231
Weight 2
Character orbit 231.i
Analytic conductor 1.845
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -2 - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{8} ) q^{5} + \beta_{6} q^{6} + ( -\beta_{4} - \beta_{5} + \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -2 - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{8} ) q^{5} + \beta_{6} q^{6} + ( -\beta_{4} - \beta_{5} + \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{8} + \beta_{2} q^{9} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{10} + ( -1 - \beta_{2} ) q^{11} + ( -2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{12} + ( 2 - \beta_{4} - \beta_{9} ) q^{13} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{7} + 2 \beta_{8} ) q^{14} + ( 1 - \beta_{1} ) q^{15} + ( -\beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{16} + ( \beta_{3} + \beta_{6} ) q^{17} + ( \beta_{3} + \beta_{6} ) q^{18} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{20} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{21} -\beta_{6} q^{22} + ( -\beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{23} + ( -\beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{24} + ( -1 - \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} ) q^{25} + ( 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + 3 \beta_{8} ) q^{26} - q^{27} + ( 2 - \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} - 4 \beta_{8} ) q^{28} + ( 3 \beta_{1} - \beta_{4} - \beta_{5} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{30} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{31} + ( \beta_{3} + \beta_{6} + 2 \beta_{8} ) q^{32} -\beta_{2} q^{33} + ( 4 + \beta_{5} - \beta_{6} - \beta_{9} ) q^{34} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{35} + ( 2 + \beta_{5} - \beta_{6} - \beta_{9} ) q^{36} + ( \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{8} ) q^{37} + ( -1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{38} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{8} - 2 \beta_{9} ) q^{39} + ( -4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{7} - 5 \beta_{8} + \beta_{9} ) q^{40} + ( -4 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( -3 - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{42} + ( 2 + \beta_{1} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{9} ) q^{43} + ( 2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{44} + ( 1 + \beta_{2} + \beta_{8} ) q^{45} + ( 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{46} + ( -2 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{7} ) q^{47} + ( -3 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{48} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{49} + ( 8 - 2 \beta_{1} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{9} ) q^{50} + \beta_{3} q^{51} + ( -12 - 3 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - 7 \beta_{8} + 5 \beta_{9} ) q^{52} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{53} + \beta_{3} q^{54} + ( -1 + \beta_{1} ) q^{55} + ( 4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{8} - \beta_{9} ) q^{56} + ( \beta_{4} + \beta_{6} + \beta_{9} ) q^{57} + ( -3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{58} + ( -3 - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{60} + ( -3 \beta_{4} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} ) q^{61} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{62} + ( \beta_{1} - \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{63} + ( -2 + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{64} + ( -2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{8} + 2 \beta_{9} ) q^{65} + ( -\beta_{3} - \beta_{6} ) q^{66} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{67} + ( -3 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{68} + ( -3 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{69} + ( 7 + 9 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{70} + ( 4 - 3 \beta_{5} + \beta_{6} + 3 \beta_{9} ) q^{71} + ( -\beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{72} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{9} ) q^{73} + ( 8 - \beta_{1} + 8 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{74} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{8} ) q^{75} + ( 8 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{76} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{77} + ( 3 + 3 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{78} + ( 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{7} + 4 \beta_{8} - \beta_{9} ) q^{79} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} ) q^{82} + ( -2 + 2 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{9} ) q^{83} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{84} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 5 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} ) q^{86} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{87} + ( \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{89} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{90} + ( -3 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + \beta_{4} - 6 \beta_{6} - \beta_{7} + \beta_{8} ) q^{91} + ( 12 + 2 \beta_{1} + 2 \beta_{4} + 7 \beta_{5} - 5 \beta_{6} - 5 \beta_{9} ) q^{92} + ( -\beta_{2} + \beta_{4} + \beta_{7} ) q^{93} + ( -16 + \beta_{1} - 16 \beta_{2} - \beta_{3} - 4 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{94} + ( 4 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} + 5 \beta_{8} - 5 \beta_{9} ) q^{95} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{8} ) q^{96} + ( -4 - 2 \beta_{1} + 4 \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{9} ) q^{97} + ( 2 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2q^{2} + 5q^{3} - 10q^{4} + 4q^{5} - 4q^{6} - q^{7} + 12q^{8} - 5q^{9} + O(q^{10}) \) \( 10q - 2q^{2} + 5q^{3} - 10q^{4} + 4q^{5} - 4q^{6} - q^{7} + 12q^{8} - 5q^{9} - 2q^{10} - 5q^{11} + 10q^{12} + 10q^{13} - 10q^{14} + 8q^{15} - 16q^{16} - 2q^{17} - 2q^{18} + 3q^{19} - 16q^{20} - 2q^{21} + 4q^{22} - 16q^{23} + 6q^{24} - 7q^{25} + 10q^{26} - 10q^{27} + 4q^{28} + 2q^{30} - 5q^{31} - 4q^{32} + 5q^{33} + 40q^{34} + 26q^{35} + 20q^{36} - 15q^{37} - 6q^{38} + 5q^{39} + 6q^{40} - 44q^{41} - 14q^{42} + 6q^{43} - 10q^{44} + 4q^{45} - 16q^{46} + 2q^{47} - 32q^{48} + 31q^{49} + 68q^{50} + 2q^{51} - 40q^{52} - 6q^{53} + 2q^{54} - 8q^{55} - 12q^{56} + 6q^{57} - 12q^{58} - 16q^{59} - 8q^{60} - 12q^{61} - 8q^{62} - q^{63} - 8q^{64} + 28q^{65} + 2q^{66} - 7q^{67} - 10q^{68} - 32q^{69} + 32q^{70} + 48q^{71} - 6q^{72} - 17q^{73} + 36q^{74} + 7q^{75} + 60q^{76} + 2q^{77} + 20q^{78} - 7q^{79} - 16q^{80} - 5q^{81} - 8q^{82} - 24q^{83} - 28q^{84} + 4q^{85} + 18q^{86} - 6q^{88} + 6q^{89} + 4q^{90} + 11q^{91} + 136q^{92} + 5q^{93} - 82q^{94} + 18q^{95} + 4q^{96} - 28q^{97} - 38q^{98} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 15 x^{8} + 72 x^{6} + 120 x^{4} + 72 x^{2} + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} + 7 \nu^{2} + 5 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + 15 \nu^{7} + 70 \nu^{5} + 102 \nu^{3} + 36 \nu - 4 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 2 \nu^{8} + 13 \nu^{7} + 28 \nu^{6} + 44 \nu^{5} + 118 \nu^{4} + 6 \nu^{3} + 140 \nu^{2} - 36 \nu + 40 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} + \nu^{8} + 14 \nu^{7} + 15 \nu^{6} + 59 \nu^{5} + 70 \nu^{4} + 70 \nu^{3} + 100 \nu^{2} + 18 \nu + 32 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} - \nu^{8} - 14 \nu^{7} - 13 \nu^{6} - 59 \nu^{5} - 48 \nu^{4} - 70 \nu^{3} - 36 \nu^{2} - 18 \nu + 4 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{8} - 14 \nu^{6} - 59 \nu^{4} - 70 \nu^{2} - 20 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{8} + 15 \nu^{7} - 30 \nu^{6} + 74 \nu^{5} - 140 \nu^{4} + 130 \nu^{3} - 204 \nu^{2} + 60 \nu - 76 \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{9} - 43 \nu^{7} - 188 \nu^{5} - 4 \nu^{4} - 246 \nu^{3} - 28 \nu^{2} - 96 \nu - 20 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{9} + \nu^{8} - 14 \nu^{7} + 15 \nu^{6} - 59 \nu^{5} + 70 \nu^{4} - 70 \nu^{3} + 100 \nu^{2} - 18 \nu + 32 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\(-\beta_{9} - \beta_{6} + \beta_{5} - 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} - 2 \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{9} + 7 \beta_{6} - 7 \beta_{5} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(17 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} - \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 24 \beta_{3} - 12 \beta_{2} - 5 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(-45 \beta_{9} - 45 \beta_{6} + 47 \beta_{5} + 2 \beta_{4} - 11 \beta_{1} - 98\)
\(\nu^{7}\)\(=\)\(-101 \beta_{9} + 58 \beta_{8} - 170 \beta_{7} + 11 \beta_{6} - 85 \beta_{5} + 16 \beta_{4} - 148 \beta_{3} + 88 \beta_{2} + 29 \beta_{1} + 44\)
\(\nu^{8}\)\(=\)\(287 \beta_{9} + 285 \beta_{6} - 315 \beta_{5} - 28 \beta_{4} + 95 \beta_{1} + 618\)
\(\nu^{9}\)\(=\)\(607 \beta_{9} - 350 \beta_{8} + 1114 \beta_{7} - 95 \beta_{6} + 557 \beta_{5} - 50 \beta_{4} + 924 \beta_{3} - 652 \beta_{2} - 175 \beta_{1} - 326\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.518255i
1.21103i
2.57330i
2.42024i
0.886226i
0.518255i
1.21103i
2.57330i
2.42024i
0.886226i
−1.29725 2.24690i 0.500000 0.866025i −2.36571 + 4.09752i −1.09601 1.89835i −2.59450 −2.61329 + 0.413178i 7.08664 −0.500000 0.866025i −2.84359 + 4.92525i
67.2 −1.17616 2.03717i 0.500000 0.866025i −1.76671 + 3.06003i 2.05761 + 3.56389i −2.35232 2.51585 0.818848i 3.60708 −0.500000 0.866025i 4.84017 8.38342i
67.3 −0.307468 0.532550i 0.500000 0.866025i 0.810927 1.40457i −0.747986 1.29555i −0.614936 2.59895 + 0.495442i −2.22721 −0.500000 0.866025i −0.459963 + 0.796680i
67.4 0.534421 + 0.925645i 0.500000 0.866025i 0.428788 0.742682i 1.34592 + 2.33120i 1.06884 −0.855706 2.50355i 3.05430 −0.500000 0.866025i −1.43858 + 2.49169i
67.5 1.24646 + 2.15892i 0.500000 0.866025i −2.10730 + 3.64995i 0.440463 + 0.762904i 2.49291 −2.14580 + 1.54775i −5.52081 −0.500000 0.866025i −1.09804 + 1.90185i
100.1 −1.29725 + 2.24690i 0.500000 + 0.866025i −2.36571 4.09752i −1.09601 + 1.89835i −2.59450 −2.61329 0.413178i 7.08664 −0.500000 + 0.866025i −2.84359 4.92525i
100.2 −1.17616 + 2.03717i 0.500000 + 0.866025i −1.76671 3.06003i 2.05761 3.56389i −2.35232 2.51585 + 0.818848i 3.60708 −0.500000 + 0.866025i 4.84017 + 8.38342i
100.3 −0.307468 + 0.532550i 0.500000 + 0.866025i 0.810927 + 1.40457i −0.747986 + 1.29555i −0.614936 2.59895 0.495442i −2.22721 −0.500000 + 0.866025i −0.459963 0.796680i
100.4 0.534421 0.925645i 0.500000 + 0.866025i 0.428788 + 0.742682i 1.34592 2.33120i 1.06884 −0.855706 + 2.50355i 3.05430 −0.500000 + 0.866025i −1.43858 2.49169i
100.5 1.24646 2.15892i 0.500000 + 0.866025i −2.10730 3.64995i 0.440463 0.762904i 2.49291 −2.14580 1.54775i −5.52081 −0.500000 + 0.866025i −1.09804 1.90185i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.i.f 10
3.b odd 2 1 693.2.i.j 10
7.c even 3 1 inner 231.2.i.f 10
7.c even 3 1 1617.2.a.ba 5
7.d odd 6 1 1617.2.a.bb 5
21.g even 6 1 4851.2.a.bz 5
21.h odd 6 1 693.2.i.j 10
21.h odd 6 1 4851.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.f 10 1.a even 1 1 trivial
231.2.i.f 10 7.c even 3 1 inner
693.2.i.j 10 3.b odd 2 1
693.2.i.j 10 21.h odd 6 1
1617.2.a.ba 5 7.c even 3 1
1617.2.a.bb 5 7.d odd 6 1
4851.2.a.bz 5 21.g even 6 1
4851.2.a.ca 5 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} - 3 T^{4} - 6 T^{5} + 6 T^{6} + 18 T^{7} + 9 T^{8} - 34 T^{9} - 62 T^{10} - 68 T^{11} + 36 T^{12} + 144 T^{13} + 96 T^{14} - 192 T^{15} - 192 T^{16} + 512 T^{18} + 1024 T^{19} + 1024 T^{20} \)
$3$ \( ( 1 - T + T^{2} )^{5} \)
$5$ \( 1 - 4 T - T^{2} + 36 T^{3} - 69 T^{4} - 36 T^{5} + 246 T^{6} - 132 T^{7} - 279 T^{8} - 400 T^{9} + 2929 T^{10} - 2000 T^{11} - 6975 T^{12} - 16500 T^{13} + 153750 T^{14} - 112500 T^{15} - 1078125 T^{16} + 2812500 T^{17} - 390625 T^{18} - 7812500 T^{19} + 9765625 T^{20} \)
$7$ \( 1 + T - 15 T^{2} - 36 T^{3} + 60 T^{4} + 402 T^{5} + 420 T^{6} - 1764 T^{7} - 5145 T^{8} + 2401 T^{9} + 16807 T^{10} \)
$11$ \( ( 1 + T + T^{2} )^{5} \)
$13$ \( ( 1 - 5 T + 27 T^{2} - 180 T^{3} + 732 T^{4} - 2514 T^{5} + 9516 T^{6} - 30420 T^{7} + 59319 T^{8} - 142805 T^{9} + 371293 T^{10} )^{2} \)
$17$ \( 1 + 2 T - 73 T^{2} - 90 T^{3} + 3192 T^{4} + 2424 T^{5} - 98319 T^{6} - 38142 T^{7} + 2348199 T^{8} + 280406 T^{9} - 44440892 T^{10} + 4766902 T^{11} + 678629511 T^{12} - 187391646 T^{13} - 8211701199 T^{14} + 3441733368 T^{15} + 77047120248 T^{16} - 36930480570 T^{17} - 509230293193 T^{18} + 237175752994 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 3 T - 44 T^{2} + 81 T^{3} + 990 T^{4} - 603 T^{5} - 16476 T^{6} + 16119 T^{7} + 196221 T^{8} - 324198 T^{9} - 1776456 T^{10} - 6159762 T^{11} + 70835781 T^{12} + 110560221 T^{13} - 2147168796 T^{14} - 1493087697 T^{15} + 46575422190 T^{16} + 72403610859 T^{17} - 747276773804 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 + 16 T + 71 T^{2} - 72 T^{3} + 24 T^{4} + 14832 T^{5} + 74109 T^{6} + 71904 T^{7} + 239211 T^{8} + 5124568 T^{9} + 32231056 T^{10} + 117865064 T^{11} + 126542619 T^{12} + 874855968 T^{13} + 20738736669 T^{14} + 95463839376 T^{15} + 3552861336 T^{16} - 245147432184 T^{17} + 5560079954951 T^{18} + 28818442583408 T^{19} + 41426511213649 T^{20} \)
$29$ \( ( 1 + 49 T^{2} + 24 T^{3} + 1765 T^{4} + 102 T^{5} + 51185 T^{6} + 20184 T^{7} + 1195061 T^{8} + 20511149 T^{10} )^{2} \)
$31$ \( 1 + 5 T - 122 T^{2} - 441 T^{3} + 9747 T^{4} + 24042 T^{5} - 546756 T^{6} - 692730 T^{7} + 24442245 T^{8} + 9807311 T^{9} - 844500050 T^{10} + 304026641 T^{11} + 23488997445 T^{12} - 20637119430 T^{13} - 504940647876 T^{14} + 688302048342 T^{15} + 8650498378707 T^{16} - 12133062822951 T^{17} - 104052706567802 T^{18} + 132198110803355 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 15 T + 10 T^{2} - 603 T^{3} + 1314 T^{4} + 25845 T^{5} - 126420 T^{6} - 536415 T^{7} + 9212421 T^{8} + 1856814 T^{9} - 461695068 T^{10} + 68702118 T^{11} + 12611804349 T^{12} - 27171028995 T^{13} - 236931433620 T^{14} + 1792194568665 T^{15} + 3371364501426 T^{16} - 57243921911199 T^{17} + 35124794539210 T^{18} + 1949426096926155 T^{19} + 4808584372417849 T^{20} \)
$41$ \( ( 1 + 22 T + 329 T^{2} + 3256 T^{3} + 27018 T^{4} + 181492 T^{5} + 1107738 T^{6} + 5473336 T^{7} + 22675009 T^{8} + 62166742 T^{9} + 115856201 T^{10} )^{2} \)
$43$ \( ( 1 - 3 T + 107 T^{2} - 114 T^{3} + 5299 T^{4} - 1683 T^{5} + 227857 T^{6} - 210786 T^{7} + 8507249 T^{8} - 10256403 T^{9} + 147008443 T^{10} )^{2} \)
$47$ \( 1 - 2 T - 37 T^{2} - 354 T^{3} - 1920 T^{4} + 14694 T^{5} + 211017 T^{6} + 985758 T^{7} + 2945283 T^{8} - 50876312 T^{9} - 605046656 T^{10} - 2391186664 T^{11} + 6506130147 T^{12} + 102344352834 T^{13} + 1029695645577 T^{14} + 3369995532858 T^{15} - 20696093431680 T^{16} - 179344584643902 T^{17} - 881017606485157 T^{18} - 2238260946205534 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 + 6 T - 133 T^{2} - 354 T^{3} + 10587 T^{4} - 5544 T^{5} - 677754 T^{6} + 507696 T^{7} + 35054169 T^{8} + 3665190 T^{9} - 1656194019 T^{10} + 194255070 T^{11} + 98467160721 T^{12} + 75584257392 T^{13} - 5347805059674 T^{14} - 2318475813192 T^{15} + 234654091272723 T^{16} - 415847743502298 T^{17} - 8280538824711013 T^{18} + 19798581550812798 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 + 16 T - 109 T^{2} - 1800 T^{3} + 25080 T^{4} + 228672 T^{5} - 2449527 T^{6} - 10271040 T^{7} + 253584315 T^{8} + 417790408 T^{9} - 15321919856 T^{10} + 24649634072 T^{11} + 882727000515 T^{12} - 2109455924160 T^{13} - 29681802938247 T^{14} + 163483169300928 T^{15} + 1057887783716280 T^{16} - 4479572672674200 T^{17} - 16004517698870989 T^{18} + 138607933098479024 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 12 T + 19 T^{2} - 900 T^{3} - 12441 T^{4} - 72288 T^{5} + 130650 T^{6} + 4443912 T^{7} + 34513281 T^{8} + 10060452 T^{9} - 1324169655 T^{10} + 613687572 T^{11} + 128423918601 T^{12} + 1008683589672 T^{13} + 1808959126650 T^{14} - 61054177406688 T^{15} - 640964977425201 T^{16} - 2828468552418900 T^{17} + 3642438946948339 T^{18} + 140329753114009692 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 7 T - 194 T^{2} - 1203 T^{3} + 21423 T^{4} + 103134 T^{5} - 1828416 T^{6} - 6223746 T^{7} + 129539373 T^{8} + 171080449 T^{9} - 8649139910 T^{10} + 11462390083 T^{11} + 581502245397 T^{12} - 1871872518198 T^{13} - 36844632054336 T^{14} + 139243802785338 T^{15} + 1937889921206487 T^{16} - 7291036061203569 T^{17} - 78777129445988354 T^{18} + 190445740774064629 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( ( 1 - 24 T + 421 T^{2} - 5436 T^{3} + 57793 T^{4} - 524712 T^{5} + 4103303 T^{6} - 27402876 T^{7} + 150680531 T^{8} - 609880344 T^{9} + 1804229351 T^{10} )^{2} \)
$73$ \( 1 + 17 T + 22 T^{2} + 51 T^{3} + 12429 T^{4} + 38334 T^{5} + 91686 T^{6} + 7567200 T^{7} + 6227061 T^{8} + 177295901 T^{9} + 7596523564 T^{10} + 12942600773 T^{11} + 33184008069 T^{12} + 2943769442400 T^{13} + 2603721124326 T^{14} + 79469126446062 T^{15} + 1880933098545981 T^{16} + 563417324473947 T^{17} + 17742122021669782 T^{18} + 1000816974040554521 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 + 7 T - 92 T^{2} + 1035 T^{3} + 8313 T^{4} - 163986 T^{5} + 174174 T^{6} + 5014596 T^{7} - 64520235 T^{8} + 8158399 T^{9} + 3747754354 T^{10} + 644513521 T^{11} - 402670786635 T^{12} + 2472391397244 T^{13} + 6784091408094 T^{14} - 504594170646414 T^{15} + 2020786017746073 T^{16} + 19876045800674565 T^{17} - 139574010511403612 T^{18} + 838961171878328233 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( ( 1 + 12 T + 295 T^{2} + 3168 T^{3} + 40450 T^{4} + 363240 T^{5} + 3357350 T^{6} + 21824352 T^{7} + 168677165 T^{8} + 569499852 T^{9} + 3939040643 T^{10} )^{2} \)
$89$ \( 1 - 6 T - 241 T^{2} + 978 T^{3} + 28551 T^{4} - 40212 T^{5} - 3189150 T^{6} + 2254020 T^{7} + 350296557 T^{8} - 203364690 T^{9} - 32844883695 T^{10} - 18099457410 T^{11} + 2774699027997 T^{12} + 1589014225380 T^{13} - 200094417885150 T^{14} - 224546198563188 T^{15} + 14189312838227511 T^{16} + 43258245527827362 T^{17} - 948717902174201521 T^{18} - 2102138422244911254 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( ( 1 + 14 T + 159 T^{2} + 2856 T^{3} + 33045 T^{4} + 251706 T^{5} + 3205365 T^{6} + 26872104 T^{7} + 145115007 T^{8} + 1239409934 T^{9} + 8587340257 T^{10} )^{2} \)
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