Properties

Label 370.2.a.g
Level $370$
Weight $2$
Character orbit 370.a
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{2} q^{3} + q^{4} - q^{5} - \beta_{2} q^{6} - \beta_1 q^{7} + q^{8} + (2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{2} q^{3} + q^{4} - q^{5} - \beta_{2} q^{6} - \beta_1 q^{7} + q^{8} + (2 \beta_1 + 3) q^{9} - q^{10} + (\beta_{2} - \beta_1 + 4) q^{11} - \beta_{2} q^{12} + 2 \beta_{2} q^{13} - \beta_1 q^{14} + \beta_{2} q^{15} + q^{16} + ( - \beta_{2} + \beta_1) q^{17} + (2 \beta_1 + 3) q^{18} - \beta_{2} q^{19} - q^{20} + (2 \beta_{2} + 2) q^{21} + (\beta_{2} - \beta_1 + 4) q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} - \beta_{2} q^{24} + q^{25} + 2 \beta_{2} q^{26} + ( - 4 \beta_{2} - 4) q^{27} - \beta_1 q^{28} + (\beta_{2} + \beta_1 - 2) q^{29} + \beta_{2} q^{30} + 3 \beta_1 q^{31} + q^{32} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + \beta_1 q^{35} + (2 \beta_1 + 3) q^{36} - q^{37} - \beta_{2} q^{38} + ( - 4 \beta_1 - 12) q^{39} - q^{40} + (\beta_{2} - 3 \beta_1 - 2) q^{41} + (2 \beta_{2} + 2) q^{42} + (\beta_{2} + \beta_1 - 4) q^{43} + (\beta_{2} - \beta_1 + 4) q^{44} + ( - 2 \beta_1 - 3) q^{45} + (2 \beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{2} + 2 \beta_1) q^{47} - \beta_{2} q^{48} + (\beta_{2} - \beta_1 - 1) q^{49} + q^{50} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{51} + 2 \beta_{2} q^{52} + ( - \beta_{2} + 3 \beta_1 + 6) q^{53} + ( - 4 \beta_{2} - 4) q^{54} + ( - \beta_{2} + \beta_1 - 4) q^{55} - \beta_1 q^{56} + (2 \beta_1 + 6) q^{57} + (\beta_{2} + \beta_1 - 2) q^{58} + ( - \beta_{2} + 2 \beta_1 + 4) q^{59} + \beta_{2} q^{60} + (\beta_{2} + \beta_1 + 2) q^{61} + 3 \beta_1 q^{62} + ( - 2 \beta_{2} - \beta_1 - 12) q^{63} + q^{64} - 2 \beta_{2} q^{65} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{66} + ( - \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{68} + (4 \beta_{2} - 4 \beta_1 - 8) q^{69} + \beta_1 q^{70} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{71} + (2 \beta_1 + 3) q^{72} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{73} - q^{74} - \beta_{2} q^{75} - \beta_{2} q^{76} + ( - \beta_{2} - 5 \beta_1 + 4) q^{77} + ( - 4 \beta_1 - 12) q^{78} + ( - \beta_{2} - 2 \beta_1 - 8) q^{79} - q^{80} + (4 \beta_{2} + 2 \beta_1 + 15) q^{81} + (\beta_{2} - 3 \beta_1 - 2) q^{82} + (3 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{2} + 2) q^{84} + (\beta_{2} - \beta_1) q^{85} + (\beta_{2} + \beta_1 - 4) q^{86} + ( - 2 \beta_1 - 8) q^{87} + (\beta_{2} - \beta_1 + 4) q^{88} + 6 q^{89} + ( - 2 \beta_1 - 3) q^{90} + ( - 4 \beta_{2} - 4) q^{91} + (2 \beta_{2} - 2 \beta_1) q^{92} + ( - 6 \beta_{2} - 6) q^{93} + ( - \beta_{2} + 2 \beta_1) q^{94} + \beta_{2} q^{95} - \beta_{2} q^{96} + (\beta_{2} - 3 \beta_1 - 6) q^{97} + (\beta_{2} - \beta_1 - 1) q^{98} + (5 \beta_{2} + 7 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 11 q^{11} - q^{14} + 3 q^{16} + q^{17} + 11 q^{18} - 3 q^{20} + 6 q^{21} + 11 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{27} - q^{28} - 5 q^{29} + 3 q^{31} + 3 q^{32} - 14 q^{33} + q^{34} + q^{35} + 11 q^{36} - 3 q^{37} - 40 q^{39} - 3 q^{40} - 9 q^{41} + 6 q^{42} - 11 q^{43} + 11 q^{44} - 11 q^{45} - 2 q^{46} + 2 q^{47} - 4 q^{49} + 3 q^{50} + 14 q^{51} + 21 q^{53} - 12 q^{54} - 11 q^{55} - q^{56} + 20 q^{57} - 5 q^{58} + 14 q^{59} + 7 q^{61} + 3 q^{62} - 37 q^{63} + 3 q^{64} - 14 q^{66} + 2 q^{67} + q^{68} - 28 q^{69} + q^{70} + 22 q^{71} + 11 q^{72} - 16 q^{73} - 3 q^{74} + 7 q^{77} - 40 q^{78} - 26 q^{79} - 3 q^{80} + 47 q^{81} - 9 q^{82} + 2 q^{83} + 6 q^{84} - q^{85} - 11 q^{86} - 26 q^{87} + 11 q^{88} + 18 q^{89} - 11 q^{90} - 12 q^{91} - 2 q^{92} - 18 q^{93} + 2 q^{94} - 21 q^{97} - 4 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 8x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 6 \) Copy content Toggle raw display

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
2.59774
−2.91729
1.31955
1.00000 −3.34596 1.00000 −1.00000 −3.34596 −2.59774 1.00000 8.19547 −1.00000
1.2 1.00000 0.406728 1.00000 −1.00000 0.406728 2.91729 1.00000 −2.83457 −1.00000
1.3 1.00000 2.93923 1.00000 −1.00000 2.93923 −1.31955 1.00000 5.63910 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 370.2.a.g 3
3.b odd 2 1 3330.2.a.bg 3
4.b odd 2 1 2960.2.a.u 3
5.b even 2 1 1850.2.a.z 3
5.c odd 4 2 1850.2.b.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 1.a even 1 1 trivial
1850.2.a.z 3 5.b even 2 1
1850.2.b.o 6 5.c odd 4 2
2960.2.a.u 3 4.b odd 2 1
3330.2.a.bg 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\):

\( T_{3}^{3} - 10T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 8T_{7} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 10T + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 8T - 10 \) Copy content Toggle raw display
$11$ \( T^{3} - 11 T^{2} + 28 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 40T - 32 \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 12T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 10T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 16 T - 76 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 72 T + 270 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 9 T^{2} - 40 T - 364 \) Copy content Toggle raw display
$43$ \( T^{3} + 11 T^{2} + 16 T - 80 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 30 T + 56 \) Copy content Toggle raw display
$53$ \( T^{3} - 21 T^{2} + 80 T + 316 \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} + 34 T + 80 \) Copy content Toggle raw display
$61$ \( T^{3} - 7 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 30 T + 56 \) Copy content Toggle raw display
$71$ \( T^{3} - 22 T^{2} + 64 T + 640 \) Copy content Toggle raw display
$73$ \( T^{3} + 16 T^{2} + 36 T - 208 \) Copy content Toggle raw display
$79$ \( T^{3} + 26 T^{2} + 170 T + 224 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 158 T - 664 \) Copy content Toggle raw display
$89$ \( (T - 6)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 21 T^{2} + 80 T - 316 \) Copy content Toggle raw display
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