Properties

Label 370.2.n.f
Level $370$
Weight $2$
Character orbit 370.n
Analytic conductor $2.954$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
Defining polynomial: \(x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + 144 \nu^{3} - 48 \nu^{2} + 48 \nu + 748 \)\()/460\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 \)\()/460\)
\(\beta_{3}\)\(=\)\((\)\( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1370 \nu^{2} + 12 \nu - 12 \)\()/460\)
\(\beta_{4}\)\(=\)\((\)\( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 \)\()/230\)
\(\beta_{5}\)\(=\)\((\)\( -12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} - 110 \nu^{4} + 50 \nu^{3} + 1140 \nu^{2} - 12 \nu + 12 \)\()/230\)
\(\beta_{6}\)\(=\)\((\)\( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + 278 \nu^{4} - 118 \nu^{3} - 1982 \nu^{2} + 32 \nu - 32 \)\()/460\)
\(\beta_{7}\)\(=\)\((\)\( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + 1614 \nu^{4} + 3424 \nu^{3} + 484 \nu^{2} - 484 \nu - 420 \)\()/460\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 \)\()/20\)
\(\beta_{9}\)\(=\)\((\)\( -101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 \)\()/460\)
\(\beta_{10}\)\(=\)\((\)\( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + 2510 \nu^{4} + 2790 \nu^{3} - 1294 \nu^{2} + 1060 \nu - 1060 \)\()/460\)
\(\beta_{11}\)\(=\)\((\)\( -60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 \)\()/230\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{5} - 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{9} + \beta_{7} + 2 \beta_{4} + \beta_{2} - 2 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{11} + \beta_{10} + 7 \beta_{8} + \beta_{2} - 5 \beta_{1}\)
\(\nu^{5}\)\(=\)\(8 \beta_{11} + 3 \beta_{10} + 9 \beta_{8} - 3 \beta_{6} - 8 \beta_{5} - 9 \beta_{3} - 9\)
\(\nu^{6}\)\(=\)\(22 \beta_{9} + 6 \beta_{7} + 28 \beta_{4}\)
\(\nu^{7}\)\(=\)\(33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + 39 \beta_{3} + 11 \beta_{2} - 33 \beta_{1}\)
\(\nu^{8}\)\(=\)\(94 \beta_{11} + 28 \beta_{10} + 116 \beta_{8} - 116\)
\(\nu^{9}\)\(=\)\(138 \beta_{9} + 44 \beta_{7} + 166 \beta_{4} - 44 \beta_{2} + 138 \beta_{1} - 166\)
\(\nu^{10}\)\(=\)\(398 \beta_{9} + 122 \beta_{7} + 122 \beta_{6} + 398 \beta_{5} + 486 \beta_{4} + 486 \beta_{3}\)
\(\nu^{11}\)\(=\)\(580 \beta_{11} + 182 \beta_{10} + 702 \beta_{8} + 182 \beta_{6} + 580 \beta_{5} + 702 \beta_{3} - 702\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.98293 + 0.531325i
−1.16746 0.312819i
0.550552 + 0.147520i
−0.147520 + 0.550552i
0.312819 1.16746i
−0.531325 + 1.98293i
1.98293 0.531325i
−1.16746 + 0.312819i
0.550552 0.147520i
−0.147520 0.550552i
0.312819 + 1.16746i
−0.531325 1.98293i
−0.866025 0.500000i −2.18708 + 1.26271i 0.500000 + 0.866025i −1.37659 1.76210i 2.52543 −1.91766 + 1.10716i 1.00000i 1.68889 2.92525i 0.311108 + 2.21432i
269.2 −0.866025 0.500000i −1.41240 + 0.815449i 0.500000 + 0.866025i −1.60976 + 1.55199i 1.63090 0.466951 0.269594i 1.00000i −0.170086 + 0.294598i 2.17009 0.539189i
269.3 −0.866025 0.500000i 2.73346 1.57816i 0.500000 + 0.866025i 2.12032 + 0.710109i −3.15633 1.45071 0.837565i 1.00000i 3.48119 6.02961i −1.48119 1.67513i
269.4 0.866025 + 0.500000i −2.73346 + 1.57816i 0.500000 + 0.866025i −0.445186 + 2.19130i −3.15633 −1.45071 + 0.837565i 1.00000i 3.48119 6.02961i −1.48119 + 1.67513i
269.5 0.866025 + 0.500000i 1.41240 0.815449i 0.500000 + 0.866025i 2.14894 0.618092i 1.63090 −0.466951 + 0.269594i 1.00000i −0.170086 + 0.294598i 2.17009 + 0.539189i
269.6 0.866025 + 0.500000i 2.18708 1.26271i 0.500000 + 0.866025i −0.837733 2.07321i 2.52543 1.91766 1.10716i 1.00000i 1.68889 2.92525i 0.311108 2.21432i
359.1 −0.866025 + 0.500000i −2.18708 1.26271i 0.500000 0.866025i −1.37659 + 1.76210i 2.52543 −1.91766 1.10716i 1.00000i 1.68889 + 2.92525i 0.311108 2.21432i
359.2 −0.866025 + 0.500000i −1.41240 0.815449i 0.500000 0.866025i −1.60976 1.55199i 1.63090 0.466951 + 0.269594i 1.00000i −0.170086 0.294598i 2.17009 + 0.539189i
359.3 −0.866025 + 0.500000i 2.73346 + 1.57816i 0.500000 0.866025i 2.12032 0.710109i −3.15633 1.45071 + 0.837565i 1.00000i 3.48119 + 6.02961i −1.48119 + 1.67513i
359.4 0.866025 0.500000i −2.73346 1.57816i 0.500000 0.866025i −0.445186 2.19130i −3.15633 −1.45071 0.837565i 1.00000i 3.48119 + 6.02961i −1.48119 1.67513i
359.5 0.866025 0.500000i 1.41240 + 0.815449i 0.500000 0.866025i 2.14894 + 0.618092i 1.63090 −0.466951 0.269594i 1.00000i −0.170086 0.294598i 2.17009 0.539189i
359.6 0.866025 0.500000i 2.18708 + 1.26271i 0.500000 0.866025i −0.837733 + 2.07321i 2.52543 1.91766 + 1.10716i 1.00000i 1.68889 + 2.92525i 0.311108 + 2.21432i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 359.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.n.f 12
5.b even 2 1 inner 370.2.n.f 12
37.c even 3 1 inner 370.2.n.f 12
185.n even 6 1 inner 370.2.n.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.n.f 12 1.a even 1 1 trivial
370.2.n.f 12 5.b even 2 1 inner
370.2.n.f 12 37.c even 3 1 inner
370.2.n.f 12 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{12} - 19 T_{3}^{10} + 254 T_{3}^{8} - 1695 T_{3}^{6} + 8238 T_{3}^{4} - 18083 T_{3}^{2} + 28561 \)
\( T_{7}^{12} - 8 T_{7}^{10} + 48 T_{7}^{8} - 120 T_{7}^{6} + 224 T_{7}^{4} - 64 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$3$ \( 28561 - 18083 T^{2} + 8238 T^{4} - 1695 T^{6} + 254 T^{8} - 19 T^{10} + T^{12} \)
$5$ \( 15625 + 625 T^{2} - 4000 T^{3} + 150 T^{4} - 80 T^{5} + 501 T^{6} - 16 T^{7} + 6 T^{8} - 32 T^{9} + T^{10} + T^{12} \)
$7$ \( 16 - 64 T^{2} + 224 T^{4} - 120 T^{6} + 48 T^{8} - 8 T^{10} + T^{12} \)
$11$ \( ( 10 + 18 T + 8 T^{2} + T^{3} )^{4} \)
$13$ \( 390625 - 361875 T^{2} + 305866 T^{4} - 25963 T^{6} + 1630 T^{8} - 47 T^{10} + T^{12} \)
$17$ \( 16 - 48 T^{2} + 112 T^{4} - 88 T^{6} + 52 T^{8} - 8 T^{10} + T^{12} \)
$19$ \( ( 33856 - 9568 T + 3072 T^{2} - 264 T^{3} + 56 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$23$ \( ( 4 + 16 T^{2} + 8 T^{4} + T^{6} )^{2} \)
$29$ \( ( -928 - 112 T + 8 T^{2} + T^{3} )^{4} \)
$31$ \( ( 59 - 7 T - 7 T^{2} + T^{3} )^{4} \)
$37$ \( 2565726409 - 5622483 T^{2} - 1519590 T^{4} - 9583 T^{6} - 1110 T^{8} - 3 T^{10} + T^{12} \)
$41$ \( ( 25 - 55 T + 166 T^{2} + 109 T^{3} + 70 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$43$ \( ( 841 + 587 T^{2} + 51 T^{4} + T^{6} )^{2} \)
$47$ \( ( 5476 + 1184 T^{2} + 64 T^{4} + T^{6} )^{2} \)
$53$ \( 28561 - 49855 T^{2} + 81110 T^{4} - 9987 T^{6} + 930 T^{8} - 35 T^{10} + T^{12} \)
$59$ \( ( 100 - 420 T + 1624 T^{2} - 568 T^{3} + 154 T^{4} - 14 T^{5} + T^{6} )^{2} \)
$61$ \( ( 18496 - 8160 T + 4960 T^{2} + 328 T^{3} + 160 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$67$ \( 33556377856 - 5132082944 T^{2} + 724811904 T^{4} - 8822880 T^{6} + 79568 T^{8} - 328 T^{10} + T^{12} \)
$71$ \( ( 1024 + 2048 T + 3584 T^{2} + 960 T^{3} + 192 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$73$ \( ( 289444 + 16124 T^{2} + 256 T^{4} + T^{6} )^{2} \)
$79$ \( ( 67600 + 24960 T + 9736 T^{2} + 328 T^{3} + 100 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$83$ \( 268435456 - 67108864 T^{2} + 14680064 T^{4} - 491520 T^{6} + 12288 T^{8} - 128 T^{10} + T^{12} \)
$89$ \( ( 5776 + 9728 T + 14712 T^{2} + 2664 T^{3} + 356 T^{4} + 22 T^{5} + T^{6} )^{2} \)
$97$ \( ( 952576 + 31168 T^{2} + 320 T^{4} + T^{6} )^{2} \)
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