Properties

Label 3381.2.a.d
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3q^{8} + q^{9} - q^{10} - 2q^{11} - q^{12} + 7q^{13} + q^{15} - q^{16} - 3q^{17} - q^{18} - 8q^{19} - q^{20} + 2q^{22} + q^{23} + 3q^{24} - 4q^{25} - 7q^{26} + q^{27} - 8q^{29} - q^{30} - 6q^{31} - 5q^{32} - 2q^{33} + 3q^{34} - q^{36} + 6q^{37} + 8q^{38} + 7q^{39} + 3q^{40} - 4q^{41} - 4q^{43} + 2q^{44} + q^{45} - q^{46} - 9q^{47} - q^{48} + 4q^{50} - 3q^{51} - 7q^{52} + 3q^{53} - q^{54} - 2q^{55} - 8q^{57} + 8q^{58} + 12q^{59} - q^{60} - 2q^{61} + 6q^{62} + 7q^{64} + 7q^{65} + 2q^{66} + q^{67} + 3q^{68} + q^{69} + 9q^{71} + 3q^{72} + 3q^{73} - 6q^{74} - 4q^{75} + 8q^{76} - 7q^{78} - 4q^{79} - q^{80} + q^{81} + 4q^{82} - 18q^{83} - 3q^{85} + 4q^{86} - 8q^{87} - 6q^{88} + 10q^{89} - q^{90} - q^{92} - 6q^{93} + 9q^{94} - 8q^{95} - 5q^{96} - 2q^{99} + O(q^{100}) \)

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
−1.00000 1.00000 −1.00000 1.00000 −1.00000 0 3.00000 1.00000 −1.00000
\(n\): 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.d 1
7.b odd 2 1 3381.2.a.a 1
7.c even 3 2 483.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.c 2 7.c even 3 2
3381.2.a.a 1 7.b odd 2 1
3381.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2} + 1 \)
\( T_{5} - 1 \)
\( T_{11} + 2 \)
\( T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( -7 + T \)
$17$ \( 3 + T \)
$19$ \( 8 + T \)
$23$ \( -1 + T \)
$29$ \( 8 + T \)
$31$ \( 6 + T \)
$37$ \( -6 + T \)
$41$ \( 4 + T \)
$43$ \( 4 + T \)
$47$ \( 9 + T \)
$53$ \( -3 + T \)
$59$ \( -12 + T \)
$61$ \( 2 + T \)
$67$ \( -1 + T \)
$71$ \( -9 + T \)
$73$ \( -3 + T \)
$79$ \( 4 + T \)
$83$ \( 18 + T \)
$89$ \( -10 + T \)
$97$ \( T \)
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