Properties

Label 3381.2.a.h
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{4} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{4} + q^{9} - 2q^{11} - 2q^{12} - 3q^{13} + 4q^{16} + 2q^{17} + 3q^{19} + q^{23} - 5q^{25} + q^{27} + 6q^{29} - 3q^{31} - 2q^{33} - 2q^{36} - 3q^{37} - 3q^{39} + 3q^{43} + 4q^{44} - 10q^{47} + 4q^{48} + 2q^{51} + 6q^{52} - 4q^{53} + 3q^{57} - 6q^{59} - 2q^{61} - 8q^{64} + q^{67} - 4q^{68} + q^{69} - 6q^{71} + q^{73} - 5q^{75} - 6q^{76} - 7q^{79} + q^{81} + 4q^{83} + 6q^{87} - 14q^{89} - 2q^{92} - 3q^{93} - 10q^{97} - 2q^{99} + O(q^{100}) \)

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
0 1.00000 −2.00000 0 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 3381.2.a.h 1
7.b odd 2 1 3381.2.a.e 1
7.c even 3 2 483.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.a 2 7.c even 3 2
3381.2.a.e 1 7.b odd 2 1
3381.2.a.h 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2} \)
\( T_{5} \)
\( T_{11} + 2 \)
\( T_{13} + 3 \)

Hecke characteristic polynomials

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