Properties

Label 3234.2.a.t
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 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} - 2q^{10} - q^{11} + q^{12} + 6q^{13} - 2q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} - 2q^{20} - q^{22} + 4q^{23} + q^{24} - q^{25} + 6q^{26} + q^{27} + 6q^{29} - 2q^{30} + q^{32} - q^{33} - 2q^{34} + q^{36} + 6q^{37} - 4q^{38} + 6q^{39} - 2q^{40} + 6q^{41} + 4q^{43} - q^{44} - 2q^{45} + 4q^{46} + 12q^{47} + q^{48} - q^{50} - 2q^{51} + 6q^{52} + 2q^{53} + q^{54} + 2q^{55} - 4q^{57} + 6q^{58} - 12q^{59} - 2q^{60} + 14q^{61} + q^{64} - 12q^{65} - q^{66} + 4q^{67} - 2q^{68} + 4q^{69} - 12q^{71} + q^{72} + 6q^{73} + 6q^{74} - q^{75} - 4q^{76} + 6q^{78} - 4q^{79} - 2q^{80} + q^{81} + 6q^{82} - 4q^{83} + 4q^{85} + 4q^{86} + 6q^{87} - q^{88} - 10q^{89} - 2q^{90} + 4q^{92} + 12q^{94} + 8q^{95} + q^{96} + 14q^{97} - q^{99} + O(q^{100}) \)

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 −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.t 1
3.b odd 2 1 9702.2.a.x 1
7.b odd 2 1 66.2.a.b 1
21.c even 2 1 198.2.a.a 1
28.d even 2 1 528.2.a.j 1
35.c odd 2 1 1650.2.a.k 1
35.f even 4 2 1650.2.c.e 2
56.e even 2 1 2112.2.a.e 1
56.h odd 2 1 2112.2.a.r 1
63.l odd 6 2 1782.2.e.e 2
63.o even 6 2 1782.2.e.v 2
77.b even 2 1 726.2.a.c 1
77.j odd 10 4 726.2.e.g 4
77.l even 10 4 726.2.e.o 4
84.h odd 2 1 1584.2.a.f 1
105.g even 2 1 4950.2.a.bu 1
105.k odd 4 2 4950.2.c.p 2
168.e odd 2 1 6336.2.a.cj 1
168.i even 2 1 6336.2.a.bw 1
231.h odd 2 1 2178.2.a.g 1
308.g odd 2 1 5808.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 7.b odd 2 1
198.2.a.a 1 21.c even 2 1
528.2.a.j 1 28.d even 2 1
726.2.a.c 1 77.b even 2 1
726.2.e.g 4 77.j odd 10 4
726.2.e.o 4 77.l even 10 4
1584.2.a.f 1 84.h odd 2 1
1650.2.a.k 1 35.c odd 2 1
1650.2.c.e 2 35.f even 4 2
1782.2.e.e 2 63.l odd 6 2
1782.2.e.v 2 63.o even 6 2
2112.2.a.e 1 56.e even 2 1
2112.2.a.r 1 56.h odd 2 1
2178.2.a.g 1 231.h odd 2 1
3234.2.a.t 1 1.a even 1 1 trivial
4950.2.a.bu 1 105.g even 2 1
4950.2.c.p 2 105.k odd 4 2
5808.2.a.bc 1 308.g odd 2 1
6336.2.a.bw 1 168.i even 2 1
6336.2.a.cj 1 168.e odd 2 1
9702.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

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

Hecke characteristic polynomials

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