Properties

Label 3249.2.a.g
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.9433956167\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 2 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + 2 q^{5} - 3 q^{8} + 2 q^{10} - 6 q^{13} - q^{16} + 6 q^{17} - 2 q^{20} - 4 q^{23} - q^{25} - 6 q^{26} + 2 q^{29} - 8 q^{31} + 5 q^{32} + 6 q^{34} + 10 q^{37} - 6 q^{40} - 2 q^{41} - 4 q^{43} - 4 q^{46} - 12 q^{47} - 7 q^{49} - q^{50} + 6 q^{52} - 6 q^{53} + 2 q^{58} - 12 q^{59} - 2 q^{61} - 8 q^{62} + 7 q^{64} - 12 q^{65} + 4 q^{67} - 6 q^{68} + 10 q^{73} + 10 q^{74} - 2 q^{80} - 2 q^{82} - 16 q^{83} + 12 q^{85} - 4 q^{86} - 2 q^{89} + 4 q^{92} - 12 q^{94} - 10 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 0 −1.00000 2.00000 0 0 −3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)

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 3249.2.a.g 1
3.b odd 2 1 1083.2.a.a 1
19.b odd 2 1 171.2.a.a 1
57.d even 2 1 57.2.a.c 1
76.d even 2 1 2736.2.a.s 1
95.d odd 2 1 4275.2.a.m 1
133.c even 2 1 8379.2.a.e 1
228.b odd 2 1 912.2.a.b 1
285.b even 2 1 1425.2.a.a 1
285.j odd 4 2 1425.2.c.g 2
399.h odd 2 1 2793.2.a.i 1
456.l odd 2 1 3648.2.a.bf 1
456.p even 2 1 3648.2.a.o 1
627.b odd 2 1 6897.2.a.a 1
741.d even 2 1 9633.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.c 1 57.d even 2 1
171.2.a.a 1 19.b odd 2 1
912.2.a.b 1 228.b odd 2 1
1083.2.a.a 1 3.b odd 2 1
1425.2.a.a 1 285.b even 2 1
1425.2.c.g 2 285.j odd 4 2
2736.2.a.s 1 76.d even 2 1
2793.2.a.i 1 399.h odd 2 1
3249.2.a.g 1 1.a even 1 1 trivial
3648.2.a.o 1 456.p even 2 1
3648.2.a.bf 1 456.l odd 2 1
4275.2.a.m 1 95.d odd 2 1
6897.2.a.a 1 627.b odd 2 1
8379.2.a.e 1 133.c even 2 1
9633.2.a.h 1 741.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 16 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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