Properties

Label 342.2.a.f
Level $342$
Weight $2$
Character orbit 342.a
Self dual yes
Analytic conductor $2.731$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 4q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 4q^{7} + q^{8} - 4q^{11} + 4q^{14} + q^{16} + 2q^{17} + q^{19} - 4q^{22} + 2q^{23} - 5q^{25} + 4q^{28} + 6q^{29} + 6q^{31} + q^{32} + 2q^{34} - 8q^{37} + q^{38} - 10q^{41} - 12q^{43} - 4q^{44} + 2q^{46} - 10q^{47} + 9q^{49} - 5q^{50} - 2q^{53} + 4q^{56} + 6q^{58} - 4q^{59} - 10q^{61} + 6q^{62} + q^{64} + 2q^{68} + 16q^{71} - 2q^{73} - 8q^{74} + q^{76} - 16q^{77} + 10q^{79} - 10q^{82} + 16q^{83} - 12q^{86} - 4q^{88} + 2q^{89} + 2q^{92} - 10q^{94} - 10q^{97} + 9q^{98} + O(q^{100}) \)

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 0 1.00000 0 0 4.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)

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 342.2.a.f 1
3.b odd 2 1 114.2.a.a 1
4.b odd 2 1 2736.2.a.j 1
5.b even 2 1 8550.2.a.a 1
12.b even 2 1 912.2.a.h 1
15.d odd 2 1 2850.2.a.x 1
15.e even 4 2 2850.2.d.s 2
19.b odd 2 1 6498.2.a.h 1
21.c even 2 1 5586.2.a.p 1
24.f even 2 1 3648.2.a.j 1
24.h odd 2 1 3648.2.a.bb 1
57.d even 2 1 2166.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.a 1 3.b odd 2 1
342.2.a.f 1 1.a even 1 1 trivial
912.2.a.h 1 12.b even 2 1
2166.2.a.i 1 57.d even 2 1
2736.2.a.j 1 4.b odd 2 1
2850.2.a.x 1 15.d odd 2 1
2850.2.d.s 2 15.e even 4 2
3648.2.a.j 1 24.f even 2 1
3648.2.a.bb 1 24.h odd 2 1
5586.2.a.p 1 21.c even 2 1
6498.2.a.h 1 19.b odd 2 1
8550.2.a.a 1 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(342))\):

\( T_{5} \)
\( T_{7} - 4 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

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