Properties

Label 370.2.e.f
Level $370$
Weight $2$
Character orbit 370.e
Analytic conductor $2.954$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} + 18 x^{2} - 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} + 8 \nu - 40 \)\()/82\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 10 \nu^{4} + 9 \nu^{3} - 36 \nu^{2} + 16 \nu - 121 \)\()/41\)
\(\beta_{4}\)\(=\)\((\)\( -10 \nu^{5} + 9 \nu^{4} - 45 \nu^{3} - 25 \nu^{2} - 162 \nu + 72 \)\()/82\)
\(\beta_{5}\)\(=\)\((\)\( -26 \nu^{5} + 7 \nu^{4} - 117 \nu^{3} - 65 \nu^{2} - 454 \nu - 26 \)\()/82\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 4 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 13 \beta_{4} - 2 \beta_{1} - 13\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 18 \beta_{2} - 18 \beta_{1}\)

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.235342 + 0.407624i
−0.906803 1.57063i
1.17146 + 2.02903i
0.235342 0.407624i
−0.906803 + 1.57063i
1.17146 2.02903i
−0.500000 + 0.866025i −1.38923 2.40621i −0.500000 0.866025i 0.500000 + 0.866025i 2.77846 −1.65389 2.86462i 1.00000 −2.35991 + 4.08749i −1.00000
121.2 −0.500000 + 0.866025i 0.144584 + 0.250427i −0.500000 0.866025i 0.500000 + 0.866025i −0.289169 −1.26222 2.18623i 1.00000 1.45819 2.52566i −1.00000
121.3 −0.500000 + 0.866025i 1.24464 + 2.15579i −0.500000 0.866025i 0.500000 + 0.866025i −2.48929 1.91611 + 3.31879i 1.00000 −1.59828 + 2.76830i −1.00000
211.1 −0.500000 0.866025i −1.38923 + 2.40621i −0.500000 + 0.866025i 0.500000 0.866025i 2.77846 −1.65389 + 2.86462i 1.00000 −2.35991 4.08749i −1.00000
211.2 −0.500000 0.866025i 0.144584 0.250427i −0.500000 + 0.866025i 0.500000 0.866025i −0.289169 −1.26222 + 2.18623i 1.00000 1.45819 + 2.52566i −1.00000
211.3 −0.500000 0.866025i 1.24464 2.15579i −0.500000 + 0.866025i 0.500000 0.866025i −2.48929 1.91611 3.31879i 1.00000 −1.59828 2.76830i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.e.f 6
37.c even 3 1 inner 370.2.e.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.f 6 1.a even 1 1 trivial
370.2.e.f 6 37.c even 3 1 inner

Hecke kernels

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

\( T_{3}^{6} + 7 T_{3}^{4} - 4 T_{3}^{3} + 49 T_{3}^{2} - 14 T_{3} + 4 \)
\( T_{7}^{6} + 2 T_{7}^{5} + 18 T_{7}^{4} + 36 T_{7}^{3} + 260 T_{7}^{2} + 448 T_{7} + 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( 4 - 14 T + 49 T^{2} - 4 T^{3} + 7 T^{4} + T^{6} \)
$5$ \( ( 1 - T + T^{2} )^{3} \)
$7$ \( 1024 + 448 T + 260 T^{2} + 36 T^{3} + 18 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( ( 2 + 4 T - 5 T^{2} + T^{3} )^{2} \)
$13$ \( 11881 + 3379 T + 1288 T^{2} + 125 T^{3} + 40 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( 30976 - 9504 T + 3268 T^{2} - 244 T^{3} + 58 T^{4} - 2 T^{5} + T^{6} \)
$19$ \( 16 - 80 T + 364 T^{2} - 172 T^{3} + 61 T^{4} - 9 T^{5} + T^{6} \)
$23$ \( ( 34 + 36 T + 11 T^{2} + T^{3} )^{2} \)
$29$ \( ( 8 - 28 T + 2 T^{2} + T^{3} )^{2} \)
$31$ \( ( -108 + 9 T + 12 T^{2} + T^{3} )^{2} \)
$37$ \( 50653 - 12321 T + 2664 T^{2} - 403 T^{3} + 72 T^{4} - 9 T^{5} + T^{6} \)
$41$ \( 1156 + 1394 T + 1273 T^{2} + 424 T^{3} + 103 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( ( -2 - 11 T + 8 T^{2} + T^{3} )^{2} \)
$47$ \( ( 722 - 17 T^{2} + T^{3} )^{2} \)
$53$ \( 1156 + 1054 T + 1165 T^{2} - 118 T^{3} + 67 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( 1156 + 544 T + 698 T^{2} - 276 T^{3} + 153 T^{4} - 13 T^{5} + T^{6} \)
$61$ \( 179776 - 83104 T + 27392 T^{2} - 4248 T^{3} + 480 T^{4} - 26 T^{5} + T^{6} \)
$67$ \( 719104 + 91584 T + 18448 T^{2} + 832 T^{3} + 172 T^{4} + 8 T^{5} + T^{6} \)
$71$ \( 952576 + 124928 T + 26144 T^{2} + 672 T^{3} + 228 T^{4} + 10 T^{5} + T^{6} \)
$73$ \( ( 176 - 54 T - 2 T^{2} + T^{3} )^{2} \)
$79$ \( 4096 + 4352 T + 3600 T^{2} + 960 T^{3} + 188 T^{4} + 16 T^{5} + T^{6} \)
$83$ \( 123904 - 11264 T + 4544 T^{2} - 384 T^{3} + 132 T^{4} - 10 T^{5} + T^{6} \)
$89$ \( 46656 - 38880 T + 26568 T^{2} - 4428 T^{3} + 549 T^{4} - 27 T^{5} + T^{6} \)
$97$ \( ( 872 - 124 T - 6 T^{2} + T^{3} )^{2} \)
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