Properties

Label 3234.2.a.u
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
\(n\): 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.u yes 1
3.b odd 2 1 9702.2.a.q 1
7.b odd 2 1 3234.2.a.r 1
21.c even 2 1 9702.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.r 1 7.b odd 2 1
3234.2.a.u yes 1 1.a even 1 1 trivial
9702.2.a.g 1 21.c even 2 1
9702.2.a.q 1 3.b odd 2 1

Hecke kernels

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

\( T_{5} + 2 \)
\( T_{13} + 4 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

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