Properties

Label 378.2.a.f
Level $378$
Weight $2$
Character orbit 378.a
Self dual yes
Analytic conductor $3.018$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.01834519640\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + 5q^{11} - q^{14} + q^{16} + 2q^{17} - q^{19} + q^{20} + 5q^{22} - q^{23} - 4q^{25} - q^{28} + 4q^{29} - 9q^{31} + q^{32} + 2q^{34} - q^{35} + 5q^{37} - q^{38} + q^{40} - 9q^{41} - 10q^{43} + 5q^{44} - q^{46} + 6q^{47} + q^{49} - 4q^{50} + 12q^{53} + 5q^{55} - q^{56} + 4q^{58} - 14q^{59} - 9q^{62} + q^{64} - 8q^{67} + 2q^{68} - q^{70} - 13q^{71} - 2q^{73} + 5q^{74} - q^{76} - 5q^{77} + 6q^{79} + q^{80} - 9q^{82} - 4q^{83} + 2q^{85} - 10q^{86} + 5q^{88} - 9q^{89} - q^{92} + 6q^{94} - q^{95} + 16q^{97} + q^{98} + O(q^{100}) \)

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 0 1.00000 1.00000 0 −1.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 378.2.a.f yes 1
3.b odd 2 1 378.2.a.c 1
4.b odd 2 1 3024.2.a.t 1
5.b even 2 1 9450.2.a.bx 1
7.b odd 2 1 2646.2.a.v 1
9.c even 3 2 1134.2.f.c 2
9.d odd 6 2 1134.2.f.n 2
12.b even 2 1 3024.2.a.m 1
15.d odd 2 1 9450.2.a.dc 1
21.c even 2 1 2646.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 3.b odd 2 1
378.2.a.f yes 1 1.a even 1 1 trivial
1134.2.f.c 2 9.c even 3 2
1134.2.f.n 2 9.d odd 6 2
2646.2.a.i 1 21.c even 2 1
2646.2.a.v 1 7.b odd 2 1
3024.2.a.m 1 12.b even 2 1
3024.2.a.t 1 4.b odd 2 1
9450.2.a.bx 1 5.b even 2 1
9450.2.a.dc 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\):

\( T_{5} - 1 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( -5 + T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( 1 + T \)
$29$ \( -4 + T \)
$31$ \( 9 + T \)
$37$ \( -5 + T \)
$41$ \( 9 + T \)
$43$ \( 10 + T \)
$47$ \( -6 + T \)
$53$ \( -12 + T \)
$59$ \( 14 + T \)
$61$ \( T \)
$67$ \( 8 + T \)
$71$ \( 13 + T \)
$73$ \( 2 + T \)
$79$ \( -6 + T \)
$83$ \( 4 + T \)
$89$ \( 9 + T \)
$97$ \( -16 + T \)
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