Properties

Label 348.2.a.c
Level $348$
Weight $2$
Character orbit 348.a
Self dual yes
Analytic conductor $2.779$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 348.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.77879399034\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 4q^{5} - 3q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 4q^{5} - 3q^{7} + q^{9} - q^{11} - 3q^{13} - 4q^{15} - 5q^{17} + 4q^{19} - 3q^{21} - 6q^{23} + 11q^{25} + q^{27} - q^{29} + 2q^{31} - q^{33} + 12q^{35} + 6q^{37} - 3q^{39} + 6q^{41} - 12q^{43} - 4q^{45} + 7q^{47} + 2q^{49} - 5q^{51} - 12q^{53} + 4q^{55} + 4q^{57} - 10q^{59} + 10q^{61} - 3q^{63} + 12q^{65} - 13q^{67} - 6q^{69} - 2q^{71} + 14q^{73} + 11q^{75} + 3q^{77} - 8q^{79} + q^{81} + 6q^{83} + 20q^{85} - q^{87} + 5q^{89} + 9q^{91} + 2q^{93} - 16q^{95} - q^{99} + O(q^{100}) \)

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
0 1.00000 0 −4.00000 0 −3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(29\) \(1\)

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 348.2.a.c 1
3.b odd 2 1 1044.2.a.j 1
4.b odd 2 1 1392.2.a.a 1
5.b even 2 1 8700.2.a.i 1
5.c odd 4 2 8700.2.g.h 2
8.b even 2 1 5568.2.a.r 1
8.d odd 2 1 5568.2.a.bj 1
12.b even 2 1 4176.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.2.a.c 1 1.a even 1 1 trivial
1044.2.a.j 1 3.b odd 2 1
1392.2.a.a 1 4.b odd 2 1
4176.2.a.bk 1 12.b even 2 1
5568.2.a.r 1 8.b even 2 1
5568.2.a.bj 1 8.d odd 2 1
8700.2.a.i 1 5.b even 2 1
8700.2.g.h 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(348))\).

Hecke characteristic polynomials

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