Properties

Label 3381.2.a.k
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + 4q^{11} + q^{12} + 6q^{13} - q^{16} - 4q^{17} + q^{18} - 2q^{19} + 4q^{22} - q^{23} + 3q^{24} - 5q^{25} + 6q^{26} - q^{27} + 2q^{29} - 4q^{31} + 5q^{32} - 4q^{33} - 4q^{34} - q^{36} + 2q^{37} - 2q^{38} - 6q^{39} - 2q^{41} + 10q^{43} - 4q^{44} - q^{46} + q^{48} - 5q^{50} + 4q^{51} - 6q^{52} - 12q^{53} - q^{54} + 2q^{57} + 2q^{58} + 12q^{59} + 6q^{61} - 4q^{62} + 7q^{64} - 4q^{66} - 10q^{67} + 4q^{68} + q^{69} + 8q^{71} - 3q^{72} + 14q^{73} + 2q^{74} + 5q^{75} + 2q^{76} - 6q^{78} + 10q^{79} + q^{81} - 2q^{82} - 12q^{83} + 10q^{86} - 2q^{87} - 12q^{88} + 16q^{89} + q^{92} + 4q^{93} - 5q^{96} + 10q^{97} + 4q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 −1.00000 0 −1.00000 0 −3.00000 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.k 1
7.b odd 2 1 69.2.a.a 1
21.c even 2 1 207.2.a.a 1
28.d even 2 1 1104.2.a.c 1
35.c odd 2 1 1725.2.a.e 1
35.f even 4 2 1725.2.b.g 2
56.e even 2 1 4416.2.a.x 1
56.h odd 2 1 4416.2.a.f 1
77.b even 2 1 8349.2.a.a 1
84.h odd 2 1 3312.2.a.k 1
105.g even 2 1 5175.2.a.v 1
161.c even 2 1 1587.2.a.e 1
483.c odd 2 1 4761.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.a.a 1 7.b odd 2 1
207.2.a.a 1 21.c even 2 1
1104.2.a.c 1 28.d even 2 1
1587.2.a.e 1 161.c even 2 1
1725.2.a.e 1 35.c odd 2 1
1725.2.b.g 2 35.f even 4 2
3312.2.a.k 1 84.h odd 2 1
3381.2.a.k 1 1.a even 1 1 trivial
4416.2.a.f 1 56.h odd 2 1
4416.2.a.x 1 56.e even 2 1
4761.2.a.b 1 483.c odd 2 1
5175.2.a.v 1 105.g even 2 1
8349.2.a.a 1 77.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2} - 1 \)
\( T_{5} \)
\( T_{11} - 4 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

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