Properties

Label 370.2.a.a
Level $370$
Weight $2$
Character orbit 370.a
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 −2.00000 1.00000 −1.00000 2.00000 −1.00000 −1.00000 1.00000 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.a 1
3.b odd 2 1 3330.2.a.v 1
4.b odd 2 1 2960.2.a.j 1
5.b even 2 1 1850.2.a.o 1
5.c odd 4 2 1850.2.b.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.a 1 1.a even 1 1 trivial
1850.2.a.o 1 5.b even 2 1
1850.2.b.g 2 5.c odd 4 2
2960.2.a.j 1 4.b odd 2 1
3330.2.a.v 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\):

\( T_{3} + 2 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 2 + T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( -3 + T \)
$13$ \( 4 + T \)
$17$ \( -3 + T \)
$19$ \( -2 + T \)
$23$ \( -6 + T \)
$29$ \( -3 + T \)
$31$ \( -5 + T \)
$37$ \( -1 + T \)
$41$ \( -3 + T \)
$43$ \( 1 + T \)
$47$ \( -12 + T \)
$53$ \( -3 + T \)
$59$ \( T \)
$61$ \( 1 + T \)
$67$ \( 4 + T \)
$71$ \( -6 + T \)
$73$ \( 16 + T \)
$79$ \( -8 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( -17 + T \)
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