Properties

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

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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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} - q^{13} + 2 q^{14} + q^{15} + q^{16} + 8 q^{17} + q^{18} - 6 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 6 q^{23} - q^{24} + q^{25} - q^{26} - q^{27} + 2 q^{28} - 4 q^{29} + q^{30} + q^{32} - 4 q^{33} + 8 q^{34} - 2 q^{35} + q^{36} - 2 q^{37} - 6 q^{38} + q^{39} - q^{40} - 2 q^{41} - 2 q^{42} - 4 q^{43} + 4 q^{44} - q^{45} + 6 q^{46} - q^{48} - 3 q^{49} + q^{50} - 8 q^{51} - q^{52} - 10 q^{53} - q^{54} - 4 q^{55} + 2 q^{56} + 6 q^{57} - 4 q^{58} + 4 q^{59} + q^{60} - 10 q^{61} + 2 q^{63} + q^{64} + q^{65} - 4 q^{66} + 12 q^{67} + 8 q^{68} - 6 q^{69} - 2 q^{70} - 8 q^{71} + q^{72} - 8 q^{73} - 2 q^{74} - q^{75} - 6 q^{76} + 8 q^{77} + q^{78} + 8 q^{79} - q^{80} + q^{81} - 2 q^{82} + 12 q^{83} - 2 q^{84} - 8 q^{85} - 4 q^{86} + 4 q^{87} + 4 q^{88} - 14 q^{89} - q^{90} - 2 q^{91} + 6 q^{92} + 6 q^{95} - q^{96} - 16 q^{97} - 3 q^{98} + 4 q^{99}+O(q^{100}) \) 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
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.e 1
3.b odd 2 1 1170.2.a.e 1
4.b odd 2 1 3120.2.a.o 1
5.b even 2 1 1950.2.a.h 1
5.c odd 4 2 1950.2.e.f 2
12.b even 2 1 9360.2.a.bh 1
13.b even 2 1 5070.2.a.e 1
13.d odd 4 2 5070.2.b.e 2
15.d odd 2 1 5850.2.a.bi 1
15.e even 4 2 5850.2.e.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 1.a even 1 1 trivial
1170.2.a.e 1 3.b odd 2 1
1950.2.a.h 1 5.b even 2 1
1950.2.e.f 2 5.c odd 4 2
3120.2.a.o 1 4.b odd 2 1
5070.2.a.e 1 13.b even 2 1
5070.2.b.e 2 13.d odd 4 2
5850.2.a.bi 1 15.d odd 2 1
5850.2.e.i 2 15.e even 4 2
9360.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(390))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 8 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 8 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T + 16 \) Copy content Toggle raw display
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