Properties

Label 378.2.a.b
Level $378$
Weight $2$
Character orbit 378.a
Self dual yes
Analytic conductor $3.018$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} + 3 q^{11} - 4 q^{13} - q^{14} + q^{16} - 6 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} - 3 q^{23} + 4 q^{25} + 4 q^{26} + q^{28} + 5 q^{31} - q^{32} + 6 q^{34} - 3 q^{35} - 7 q^{37} + 7 q^{38} + 3 q^{40} - 9 q^{41} - 10 q^{43} + 3 q^{44} + 3 q^{46} + 6 q^{47} + q^{49} - 4 q^{50} - 4 q^{52} + 12 q^{53} - 9 q^{55} - q^{56} - 6 q^{59} + 8 q^{61} - 5 q^{62} + q^{64} + 12 q^{65} - 4 q^{67} - 6 q^{68} + 3 q^{70} + 9 q^{71} + 2 q^{73} + 7 q^{74} - 7 q^{76} + 3 q^{77} - 10 q^{79} - 3 q^{80} + 9 q^{82} + 18 q^{85} + 10 q^{86} - 3 q^{88} + 15 q^{89} - 4 q^{91} - 3 q^{92} - 6 q^{94} + 21 q^{95} + 8 q^{97} - q^{98}+O(q^{100}) \) 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 0 1.00000 −3.00000 0 1.00000 −1.00000 0 3.00000
\(n\): 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.b 1
3.b odd 2 1 378.2.a.g yes 1
4.b odd 2 1 3024.2.a.c 1
5.b even 2 1 9450.2.a.cu 1
7.b odd 2 1 2646.2.a.n 1
9.c even 3 2 1134.2.f.o 2
9.d odd 6 2 1134.2.f.b 2
12.b even 2 1 3024.2.a.bb 1
15.d odd 2 1 9450.2.a.h 1
21.c even 2 1 2646.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.b 1 1.a even 1 1 trivial
378.2.a.g yes 1 3.b odd 2 1
1134.2.f.b 2 9.d odd 6 2
1134.2.f.o 2 9.c even 3 2
2646.2.a.n 1 7.b odd 2 1
2646.2.a.q 1 21.c even 2 1
3024.2.a.c 1 4.b odd 2 1
3024.2.a.bb 1 12.b even 2 1
9450.2.a.h 1 15.d odd 2 1
9450.2.a.cu 1 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\):

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T + 7 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T + 10 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 9 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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