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: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 15 x^{8} + 72 x^{6} + 120 x^{4} + 72 x^{2} + 12\)
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

$p$ $F_p(T)$
$2$ \( 100 + 110 T + 261 T^{2} + 6 T^{3} + 264 T^{4} + 78 T^{5} + 81 T^{6} + 12 T^{7} + 12 T^{8} + 2 T^{9} + T^{10} \)
$3$ \( ( 1 - T + T^{2} )^{5} \)
$5$ \( 1024 - 640 T + 1296 T^{2} + 48 T^{3} + 816 T^{4} - 96 T^{5} + 156 T^{6} - 24 T^{7} + 24 T^{8} - 4 T^{9} + T^{10} \)
$7$ \( 16807 + 2401 T - 5145 T^{2} - 1764 T^{3} + 420 T^{4} + 402 T^{5} + 60 T^{6} - 36 T^{7} - 15 T^{8} + T^{9} + T^{10} \)
$11$ \( ( 1 + T + T^{2} )^{5} \)
$13$ \( ( 476 + 524 T + 80 T^{2} - 38 T^{3} - 5 T^{4} + T^{5} )^{2} \)
$17$ \( 100 + 110 T + 261 T^{2} + 6 T^{3} + 264 T^{4} + 78 T^{5} + 81 T^{6} + 12 T^{7} + 12 T^{8} + 2 T^{9} + T^{10} \)
$19$ \( 363609 - 146529 T + 124173 T^{2} - 24408 T^{3} + 20061 T^{4} - 3681 T^{5} + 1845 T^{6} - 90 T^{7} + 51 T^{8} - 3 T^{9} + T^{10} \)
$23$ \( 38416 + 34300 T + 39249 T^{2} + 19740 T^{3} + 17322 T^{4} + 8484 T^{5} + 4371 T^{6} + 1032 T^{7} + 186 T^{8} + 16 T^{9} + T^{10} \)
$29$ \( ( -1290 + 1707 T + 24 T^{2} - 96 T^{3} + T^{5} )^{2} \)
$31$ \( 400 - 1120 T + 2496 T^{2} - 2112 T^{3} + 1572 T^{4} - 324 T^{5} + 168 T^{6} + 24 T^{7} + 33 T^{8} + 5 T^{9} + T^{10} \)
$37$ \( 40401 + 164619 T + 609255 T^{2} + 262674 T^{3} + 121221 T^{4} + 15189 T^{5} + 6309 T^{6} + 1062 T^{7} + 195 T^{8} + 15 T^{9} + T^{10} \)
$41$ \( ( -11536 - 5044 T - 352 T^{2} + 124 T^{3} + 22 T^{4} + T^{5} )^{2} \)
$43$ \( ( -2973 + 741 T + 402 T^{2} - 108 T^{3} - 3 T^{4} + T^{5} )^{2} \)
$47$ \( 783664036 - 255781178 T + 97817697 T^{2} - 6183528 T^{3} + 1978734 T^{4} - 90774 T^{5} + 29523 T^{6} - 636 T^{7} + 198 T^{8} - 2 T^{9} + T^{10} \)
$53$ \( 42302016 + 6790176 T + 4914288 T^{2} + 634896 T^{3} + 406944 T^{4} + 50424 T^{5} + 11700 T^{6} + 600 T^{7} + 132 T^{8} + 6 T^{9} + T^{10} \)
$59$ \( 38416 + 34300 T + 39249 T^{2} + 19740 T^{3} + 17322 T^{4} + 8484 T^{5} + 4371 T^{6} + 1032 T^{7} + 186 T^{8} + 16 T^{9} + T^{10} \)
$61$ \( 2361960000 + 456645600 T + 172265616 T^{2} + 1259712 T^{3} + 4094064 T^{4} + 134136 T^{5} + 43740 T^{6} + 1296 T^{7} + 324 T^{8} + 12 T^{9} + T^{10} \)
$67$ \( 9265936 + 3238816 T + 2422752 T^{2} + 108960 T^{3} + 256356 T^{4} + 27156 T^{5} + 10368 T^{6} + 204 T^{7} + 141 T^{8} + 7 T^{9} + T^{10} \)
$71$ \( ( 5232 - 6675 T + 1380 T^{2} + 66 T^{3} - 24 T^{4} + T^{5} )^{2} \)
$73$ \( 9265936 + 33837104 T + 115285776 T^{2} + 29638896 T^{3} + 6360780 T^{4} + 641460 T^{5} + 66960 T^{6} + 3774 T^{7} + 387 T^{8} + 17 T^{9} + T^{10} \)
$79$ \( 8999937424 + 903522832 T + 302831424 T^{2} + 26897280 T^{3} + 6754716 T^{4} + 529476 T^{5} + 70644 T^{6} + 2694 T^{7} + 303 T^{8} + 7 T^{9} + T^{10} \)
$83$ \( ( 2688 + 1440 T - 816 T^{2} - 120 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$89$ \( 315417600 - 104855040 T + 49349376 T^{2} - 1149696 T^{3} + 1551168 T^{4} - 84000 T^{5} + 27216 T^{6} - 624 T^{7} + 204 T^{8} - 6 T^{9} + T^{10} \)
$97$ \( ( -38906 + 33821 T - 2576 T^{2} - 326 T^{3} + 14 T^{4} + T^{5} )^{2} \)
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