Properties

Label 342.2.a.d
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 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 4q^{5} + 3q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} + 4q^{5} + 3q^{7} - q^{8} - 4q^{10} - 2q^{11} - q^{13} - 3q^{14} + q^{16} - 3q^{17} - q^{19} + 4q^{20} + 2q^{22} + q^{23} + 11q^{25} + q^{26} + 3q^{28} + 5q^{29} - 8q^{31} - q^{32} + 3q^{34} + 12q^{35} - 2q^{37} + q^{38} - 4q^{40} + 8q^{41} + 4q^{43} - 2q^{44} - q^{46} - 8q^{47} + 2q^{49} - 11q^{50} - q^{52} + q^{53} - 8q^{55} - 3q^{56} - 5q^{58} - 15q^{59} + 2q^{61} + 8q^{62} + q^{64} - 4q^{65} + 3q^{67} - 3q^{68} - 12q^{70} - 2q^{71} + 9q^{73} + 2q^{74} - q^{76} - 6q^{77} - 10q^{79} + 4q^{80} - 8q^{82} + 6q^{83} - 12q^{85} - 4q^{86} + 2q^{88} - 3q^{91} + q^{92} + 8q^{94} - 4q^{95} - 2q^{97} - 2q^{98} + O(q^{100}) \)

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 4.00000 0 3.00000 −1.00000 0 −4.00000
\(n\): 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.d 1
3.b odd 2 1 38.2.a.b 1
4.b odd 2 1 2736.2.a.w 1
5.b even 2 1 8550.2.a.u 1
12.b even 2 1 304.2.a.d 1
15.d odd 2 1 950.2.a.b 1
15.e even 4 2 950.2.b.c 2
19.b odd 2 1 6498.2.a.y 1
21.c even 2 1 1862.2.a.f 1
24.f even 2 1 1216.2.a.g 1
24.h odd 2 1 1216.2.a.n 1
33.d even 2 1 4598.2.a.a 1
39.d odd 2 1 6422.2.a.b 1
57.d even 2 1 722.2.a.b 1
57.f even 6 2 722.2.c.f 2
57.h odd 6 2 722.2.c.d 2
57.j even 18 6 722.2.e.d 6
57.l odd 18 6 722.2.e.c 6
60.h even 2 1 7600.2.a.h 1
228.b odd 2 1 5776.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 3.b odd 2 1
304.2.a.d 1 12.b even 2 1
342.2.a.d 1 1.a even 1 1 trivial
722.2.a.b 1 57.d even 2 1
722.2.c.d 2 57.h odd 6 2
722.2.c.f 2 57.f even 6 2
722.2.e.c 6 57.l odd 18 6
722.2.e.d 6 57.j even 18 6
950.2.a.b 1 15.d odd 2 1
950.2.b.c 2 15.e even 4 2
1216.2.a.g 1 24.f even 2 1
1216.2.a.n 1 24.h odd 2 1
1862.2.a.f 1 21.c even 2 1
2736.2.a.w 1 4.b odd 2 1
4598.2.a.a 1 33.d even 2 1
5776.2.a.d 1 228.b odd 2 1
6422.2.a.b 1 39.d odd 2 1
6498.2.a.y 1 19.b odd 2 1
7600.2.a.h 1 60.h even 2 1
8550.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(342))\):

\( T_{5} - 4 \)
\( T_{7} - 3 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( -3 + T \)
$11$ \( 2 + T \)
$13$ \( 1 + T \)
$17$ \( 3 + T \)
$19$ \( 1 + T \)
$23$ \( -1 + T \)
$29$ \( -5 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( -8 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( -1 + T \)
$59$ \( 15 + T \)
$61$ \( -2 + T \)
$67$ \( -3 + T \)
$71$ \( 2 + T \)
$73$ \( -9 + T \)
$79$ \( 10 + T \)
$83$ \( -6 + T \)
$89$ \( T \)
$97$ \( 2 + T \)
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