Properties

Label 6160.2.a.a
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} - q^{5} - q^{7} + q^{9} + q^{11} + 2q^{13} + 2q^{15} + 2q^{17} - 6q^{19} + 2q^{21} - 6q^{23} + q^{25} + 4q^{27} + 4q^{29} - 2q^{33} + q^{35} + 8q^{37} - 4q^{39} - 4q^{43} - q^{45} + 4q^{47} + q^{49} - 4q^{51} - 12q^{53} - q^{55} + 12q^{57} + 2q^{61} - q^{63} - 2q^{65} + 8q^{67} + 12q^{69} + 12q^{71} - 6q^{73} - 2q^{75} - q^{77} - 10q^{79} - 11q^{81} + 12q^{83} - 2q^{85} - 8q^{87} + 14q^{89} - 2q^{91} + 6q^{95} + 4q^{97} + q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 6160.2.a.a 1
4.b odd 2 1 770.2.a.e 1
12.b even 2 1 6930.2.a.bk 1
20.d odd 2 1 3850.2.a.m 1
20.e even 4 2 3850.2.c.c 2
28.d even 2 1 5390.2.a.c 1
44.c even 2 1 8470.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.e 1 4.b odd 2 1
3850.2.a.m 1 20.d odd 2 1
3850.2.c.c 2 20.e even 4 2
5390.2.a.c 1 28.d even 2 1
6160.2.a.a 1 1.a even 1 1 trivial
6930.2.a.bk 1 12.b even 2 1
8470.2.a.bg 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3} + 2 \)
\( T_{13} - 2 \)
\( T_{17} - 2 \)
\( T_{19} + 6 \)
\( T_{23} + 6 \)

Hecke characteristic polynomials

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