Properties

Label 390.2.a.g
Level $390$
Weight $2$
Character orbit 390.a
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.11416567883\)
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

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

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 1.00000 1.00000 −1.00000 1.00000 2.00000 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 390.2.a.g 1
3.b odd 2 1 1170.2.a.g 1
4.b odd 2 1 3120.2.a.b 1
5.b even 2 1 1950.2.a.b 1
5.c odd 4 2 1950.2.e.k 2
12.b even 2 1 9360.2.a.bg 1
13.b even 2 1 5070.2.a.k 1
13.d odd 4 2 5070.2.b.n 2
15.d odd 2 1 5850.2.a.bk 1
15.e even 4 2 5850.2.e.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.g 1 1.a even 1 1 trivial
1170.2.a.g 1 3.b odd 2 1
1950.2.a.b 1 5.b even 2 1
1950.2.e.k 2 5.c odd 4 2
3120.2.a.b 1 4.b odd 2 1
5070.2.a.k 1 13.b even 2 1
5070.2.b.n 2 13.d odd 4 2
5850.2.a.bk 1 15.d odd 2 1
5850.2.e.r 2 15.e even 4 2
9360.2.a.bg 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(390))\):

\( T_{7} - 2 \)
\( T_{11} \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( T \)
$19$ \( -2 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( 4 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( -8 + T \)
$79$ \( -8 + T \)
$83$ \( 12 + T \)
$89$ \( -6 + T \)
$97$ \( -8 + T \)
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