Properties

Label 3234.2.a.g
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} - q^{8} + q^{9} - 3 q^{10} - q^{11} - q^{12} - 2 q^{13} - 3 q^{15} + q^{16} - 3 q^{17} - q^{18} - 2 q^{19} + 3 q^{20} + q^{22} + 3 q^{23} + q^{24} + 4 q^{25} + 2 q^{26} - q^{27} - 6 q^{29} + 3 q^{30} + 4 q^{31} - q^{32} + q^{33} + 3 q^{34} + q^{36} + 2 q^{37} + 2 q^{38} + 2 q^{39} - 3 q^{40} + 3 q^{41} + 2 q^{43} - q^{44} + 3 q^{45} - 3 q^{46} + 9 q^{47} - q^{48} - 4 q^{50} + 3 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} - 3 q^{55} + 2 q^{57} + 6 q^{58} + 12 q^{59} - 3 q^{60} - 5 q^{61} - 4 q^{62} + q^{64} - 6 q^{65} - q^{66} + 5 q^{67} - 3 q^{68} - 3 q^{69} + 12 q^{71} - q^{72} + 16 q^{73} - 2 q^{74} - 4 q^{75} - 2 q^{76} - 2 q^{78} + 17 q^{79} + 3 q^{80} + q^{81} - 3 q^{82} - 9 q^{83} - 9 q^{85} - 2 q^{86} + 6 q^{87} + q^{88} + 6 q^{89} - 3 q^{90} + 3 q^{92} - 4 q^{93} - 9 q^{94} - 6 q^{95} + q^{96} - 17 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 3.00000 1.00000 0 −1.00000 1.00000 −3.00000
\(n\): 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.g 1
3.b odd 2 1 9702.2.a.bd 1
7.b odd 2 1 3234.2.a.h 1
7.d odd 6 2 462.2.i.c 2
21.c even 2 1 9702.2.a.cf 1
21.g even 6 2 1386.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.c 2 7.d odd 6 2
1386.2.k.c 2 21.g even 6 2
3234.2.a.g 1 1.a even 1 1 trivial
3234.2.a.h 1 7.b odd 2 1
9702.2.a.bd 1 3.b odd 2 1
9702.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T - 9 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 5 \) Copy content Toggle raw display
$67$ \( T - 5 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 16 \) Copy content Toggle raw display
$79$ \( T - 17 \) Copy content Toggle raw display
$83$ \( T + 9 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 17 \) Copy content Toggle raw display
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