Properties

Label 3234.2.a.a
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

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

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

\( T_{5} + 3 \)
\( T_{13} + 6 \)
\( T_{17} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + 3 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + T \)
$13$ \( 1 + 6 T + 13 T^{2} \)
$17$ \( 1 - 5 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 - 5 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 5 T + 41 T^{2} \)
$43$ \( 1 + 10 T + 43 T^{2} \)
$47$ \( 1 + 9 T + 47 T^{2} \)
$53$ \( 1 - 2 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 - 5 T + 61 T^{2} \)
$67$ \( 1 - 5 T + 67 T^{2} \)
$71$ \( 1 - 4 T + 71 T^{2} \)
$73$ \( 1 + 12 T + 73 T^{2} \)
$79$ \( 1 + T + 79 T^{2} \)
$83$ \( 1 + T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 9 T + 97 T^{2} \)
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