Properties

Label 6160.2.a.x
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} - q^{7} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{5} - q^{7} + 5 q^{9} + q^{11} + ( 2 - \beta ) q^{13} -\beta q^{15} + ( 2 + \beta ) q^{17} -\beta q^{21} + 4 q^{23} + q^{25} + 2 \beta q^{27} -2 q^{29} + ( -4 + \beta ) q^{31} + \beta q^{33} + q^{35} + ( 6 - 2 \beta ) q^{37} + ( -8 + 2 \beta ) q^{39} + ( 6 - \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} -5 q^{45} -\beta q^{47} + q^{49} + ( 8 + 2 \beta ) q^{51} + ( -2 - 2 \beta ) q^{53} - q^{55} + ( 4 - \beta ) q^{59} + ( 6 + \beta ) q^{61} -5 q^{63} + ( -2 + \beta ) q^{65} + 2 \beta q^{67} + 4 \beta q^{69} + 2 \beta q^{71} + ( -6 - \beta ) q^{73} + \beta q^{75} - q^{77} + q^{81} + 2 \beta q^{83} + ( -2 - \beta ) q^{85} -2 \beta q^{87} + ( -6 + 4 \beta ) q^{89} + ( -2 + \beta ) q^{91} + ( 8 - 4 \beta ) q^{93} + ( 2 + 4 \beta ) q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + 10 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} + 10 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 2 q^{35} + 12 q^{37} - 16 q^{39} + 12 q^{41} + 8 q^{43} - 10 q^{45} + 2 q^{49} + 16 q^{51} - 4 q^{53} - 2 q^{55} + 8 q^{59} + 12 q^{61} - 10 q^{63} - 4 q^{65} - 12 q^{73} - 2 q^{77} + 2 q^{81} - 4 q^{85} - 12 q^{89} - 4 q^{91} + 16 q^{93} + 4 q^{97} + 10 q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −2.82843 0 −1.00000 0 −1.00000 0 5.00000 0
1.2 0 2.82843 0 −1.00000 0 −1.00000 0 5.00000 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.x 2
4.b odd 2 1 3080.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.g 2 4.b odd 2 1
6160.2.a.x 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} - 8 \)
\( T_{13}^{2} - 4 T_{13} - 4 \)
\( T_{17}^{2} - 4 T_{17} - 4 \)
\( T_{19} \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -4 - 4 T + T^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( 4 - 12 T + T^{2} \)
$41$ \( 28 - 12 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( -28 + 4 T + T^{2} \)
$59$ \( 8 - 8 T + T^{2} \)
$61$ \( 28 - 12 T + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( 28 + 12 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -32 + T^{2} \)
$89$ \( -92 + 12 T + T^{2} \)
$97$ \( -124 - 4 T + T^{2} \)
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