Properties

Label 3234.2.a.k
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - q^{11} + q^{12} + 2q^{13} + q^{16} + 4q^{17} - q^{18} - 6q^{19} + q^{22} - 4q^{23} - q^{24} - 5q^{25} - 2q^{26} + q^{27} - 10q^{29} - 6q^{31} - q^{32} - q^{33} - 4q^{34} + q^{36} - 6q^{37} + 6q^{38} + 2q^{39} + 12q^{41} - 8q^{43} - q^{44} + 4q^{46} - 2q^{47} + q^{48} + 5q^{50} + 4q^{51} + 2q^{52} + 6q^{53} - q^{54} - 6q^{57} + 10q^{58} + 8q^{59} - 6q^{61} + 6q^{62} + q^{64} + q^{66} - 4q^{67} + 4q^{68} - 4q^{69} - q^{72} + 12q^{73} + 6q^{74} - 5q^{75} - 6q^{76} - 2q^{78} + q^{81} - 12q^{82} - 14q^{83} + 8q^{86} - 10q^{87} + q^{88} - 10q^{89} - 4q^{92} - 6q^{93} + 2q^{94} - q^{96} - 10q^{97} - q^{99} + O(q^{100}) \)

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:

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.k 1
3.b odd 2 1 9702.2.a.bt 1
7.b odd 2 1 462.2.a.b 1
21.c even 2 1 1386.2.a.i 1
28.d even 2 1 3696.2.a.y 1
77.b even 2 1 5082.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.b 1 7.b odd 2 1
1386.2.a.i 1 21.c even 2 1
3234.2.a.k 1 1.a even 1 1 trivial
3696.2.a.y 1 28.d even 2 1
5082.2.a.s 1 77.b even 2 1
9702.2.a.bt 1 3.b odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} \)
\( T_{13} - 2 \)
\( T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ \( 1 + 5 T^{2} \)
$7$ 1
$11$ \( 1 + T \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 - 4 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 + 10 T + 29 T^{2} \)
$31$ \( 1 + 6 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 - 12 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 + 2 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 - 8 T + 59 T^{2} \)
$61$ \( 1 + 6 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 12 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 14 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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