Properties

Label 3234.2.a.j
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - q^{11} + q^{12} - 4q^{13} + q^{16} + q^{17} - q^{18} + 3q^{19} + q^{22} - q^{23} - q^{24} - 5q^{25} + 4q^{26} + q^{27} - q^{29} - 6q^{31} - q^{32} - q^{33} - q^{34} + q^{36} - 3q^{37} - 3q^{38} - 4q^{39} + 6q^{41} + q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 5q^{50} + q^{51} - 4q^{52} - q^{54} + 3q^{57} + q^{58} - 7q^{59} - 6q^{61} + 6q^{62} + q^{64} + q^{66} - 4q^{67} + q^{68} - q^{69} - 15q^{71} - q^{72} - 12q^{73} + 3q^{74} - 5q^{75} + 3q^{76} + 4q^{78} + q^{81} - 6q^{82} + 16q^{83} - q^{86} - q^{87} + q^{88} + 8q^{89} - q^{92} - 6q^{93} - q^{94} - q^{96} - 7q^{97} - q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 3234.2.a.j 1
3.b odd 2 1 9702.2.a.bq 1
7.b odd 2 1 3234.2.a.c 1
7.d odd 6 2 462.2.i.d 2
21.c even 2 1 9702.2.a.bv 1
21.g even 6 2 1386.2.k.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.d 2 7.d odd 6 2
1386.2.k.f 2 21.g even 6 2
3234.2.a.c 1 7.b odd 2 1
3234.2.a.j 1 1.a even 1 1 trivial
9702.2.a.bq 1 3.b odd 2 1
9702.2.a.bv 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} \)
\( T_{13} + 4 \)
\( T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( 4 + T \)
$17$ \( -1 + T \)
$19$ \( -3 + T \)
$23$ \( 1 + T \)
$29$ \( 1 + T \)
$31$ \( 6 + T \)
$37$ \( 3 + T \)
$41$ \( -6 + T \)
$43$ \( -1 + T \)
$47$ \( -1 + T \)
$53$ \( T \)
$59$ \( 7 + T \)
$61$ \( 6 + T \)
$67$ \( 4 + T \)
$71$ \( 15 + T \)
$73$ \( 12 + T \)
$79$ \( T \)
$83$ \( -16 + T \)
$89$ \( -8 + T \)
$97$ \( 7 + T \)
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