Properties

Label 3312.2.a.n
Level $3312$
Weight $2$
Character orbit 3312.a
Self dual yes
Analytic conductor $26.446$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( q + 2 q^{5} + 2 q^{7} - 6 q^{11} - 2 q^{13} - q^{23} - q^{25} - 6 q^{29} - 8 q^{31} + 4 q^{35} - 10 q^{41} + 12 q^{43} - 8 q^{47} - 3 q^{49} - 2 q^{53} - 12 q^{55} - 12 q^{59} + 4 q^{61} - 4 q^{65} + 12 q^{67} - 10 q^{73} - 12 q^{77} + 6 q^{79} + 14 q^{83} - 4 q^{91} - 6 q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)

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 3312.2.a.n 1
3.b odd 2 1 1104.2.a.e 1
4.b odd 2 1 414.2.a.d 1
12.b even 2 1 138.2.a.a 1
24.f even 2 1 4416.2.a.z 1
24.h odd 2 1 4416.2.a.m 1
60.h even 2 1 3450.2.a.y 1
60.l odd 4 2 3450.2.d.j 2
84.h odd 2 1 6762.2.a.q 1
92.b even 2 1 9522.2.a.i 1
276.h odd 2 1 3174.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.a 1 12.b even 2 1
414.2.a.d 1 4.b odd 2 1
1104.2.a.e 1 3.b odd 2 1
3174.2.a.b 1 276.h odd 2 1
3312.2.a.n 1 1.a even 1 1 trivial
3450.2.a.y 1 60.h even 2 1
3450.2.d.j 2 60.l odd 4 2
4416.2.a.m 1 24.h odd 2 1
4416.2.a.z 1 24.f even 2 1
6762.2.a.q 1 84.h odd 2 1
9522.2.a.i 1 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3312))\):

\( T_{5} - 2 \)
\( T_{7} - 2 \)
\( T_{11} + 6 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

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