Properties

Label 370.2.a.d
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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( -1 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( -6 + T \)
$19$ \( -2 + T \)
$23$ \( T \)
$29$ \( -6 + T \)
$31$ \( 10 + T \)
$37$ \( -1 + T \)
$41$ \( 6 + T \)
$43$ \( 4 + T \)
$47$ \( 6 + T \)
$53$ \( -6 + T \)
$59$ \( 6 + T \)
$61$ \( 10 + T \)
$67$ \( -2 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( 10 + T \)
$83$ \( 6 + T \)
$89$ \( 6 + T \)
$97$ \( -2 + T \)
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