Properties

Label 3381.2.a.m
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

\(f(q)\) \(=\) \( q + 2q^{2} - q^{3} + 2q^{4} - 2q^{6} + q^{9} + O(q^{10}) \) \( q + 2q^{2} - q^{3} + 2q^{4} - 2q^{6} + q^{9} + q^{11} - 2q^{12} - 2q^{13} - 4q^{16} - 4q^{17} + 2q^{18} + 3q^{19} + 2q^{22} + q^{23} - 5q^{25} - 4q^{26} - q^{27} - 6q^{29} + 2q^{31} - 8q^{32} - q^{33} - 8q^{34} + 2q^{36} - 2q^{37} + 6q^{38} + 2q^{39} - q^{41} - 8q^{43} + 2q^{44} + 2q^{46} + 5q^{47} + 4q^{48} - 10q^{50} + 4q^{51} - 4q^{52} + 3q^{53} - 2q^{54} - 3q^{57} - 12q^{58} - 5q^{59} - 13q^{61} + 4q^{62} - 8q^{64} - 2q^{66} - 8q^{68} - q^{69} + 16q^{73} - 4q^{74} + 5q^{75} + 6q^{76} + 4q^{78} - 2q^{79} + q^{81} - 2q^{82} - 6q^{83} - 16q^{86} + 6q^{87} - 6q^{89} + 2q^{92} - 2q^{93} + 10q^{94} + 8q^{96} - 10q^{97} + q^{99} + O(q^{100}) \)

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
2.00000 −1.00000 2.00000 0 −2.00000 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.m 1
7.b odd 2 1 483.2.a.a 1
21.c even 2 1 1449.2.a.c 1
28.d even 2 1 7728.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.a 1 7.b odd 2 1
1449.2.a.c 1 21.c even 2 1
3381.2.a.m 1 1.a even 1 1 trivial
7728.2.a.e 1 28.d even 2 1

Hecke kernels

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

\( T_{2} - 2 \)
\( T_{5} \)
\( T_{11} - 1 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

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