Properties

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

$q$-expansion

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

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:

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.r 1
3.b odd 2 1 9702.2.a.g 1
7.b odd 2 1 3234.2.a.u yes 1
21.c even 2 1 9702.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.r 1 1.a even 1 1 trivial
3234.2.a.u yes 1 7.b odd 2 1
9702.2.a.g 1 3.b odd 2 1
9702.2.a.q 1 21.c even 2 1

Atkin-Lehner signs

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

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

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