Properties

Label 6160.2.a.s
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1540)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} - q^{5} - q^{7} + ( 1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} - q^{5} - q^{7} + ( 1 - 2 \beta ) q^{9} - q^{11} + ( -1 - \beta ) q^{13} + ( 1 - \beta ) q^{15} + ( 3 - \beta ) q^{17} + ( -2 - 2 \beta ) q^{19} + ( 1 - \beta ) q^{21} + q^{25} -4 q^{27} -2 \beta q^{29} + ( 1 + 3 \beta ) q^{31} + ( 1 - \beta ) q^{33} + q^{35} + ( 2 + 2 \beta ) q^{37} -2 q^{39} + ( 3 + 3 \beta ) q^{41} -2 q^{43} + ( -1 + 2 \beta ) q^{45} + ( -3 - 5 \beta ) q^{47} + q^{49} + ( -6 + 4 \beta ) q^{51} + ( 6 + 2 \beta ) q^{53} + q^{55} -4 q^{57} + ( -9 + \beta ) q^{59} + ( -7 + \beta ) q^{61} + ( -1 + 2 \beta ) q^{63} + ( 1 + \beta ) q^{65} + 4 q^{67} + ( 6 + 2 \beta ) q^{71} + ( 5 - 3 \beta ) q^{73} + ( -1 + \beta ) q^{75} + q^{77} -2 q^{79} + ( 1 + 2 \beta ) q^{81} + ( -6 + 6 \beta ) q^{83} + ( -3 + \beta ) q^{85} + ( -6 + 2 \beta ) q^{87} + ( 6 + 4 \beta ) q^{89} + ( 1 + \beta ) q^{91} + ( 8 - 2 \beta ) q^{93} + ( 2 + 2 \beta ) q^{95} + ( 8 + 6 \beta ) q^{97} + ( -1 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} + 2 q^{25} - 8 q^{27} + 2 q^{31} + 2 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{47} + 2 q^{49} - 12 q^{51} + 12 q^{53} + 2 q^{55} - 8 q^{57} - 18 q^{59} - 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 12 q^{71} + 10 q^{73} - 2 q^{75} + 2 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{89} + 2 q^{91} + 16 q^{93} + 4 q^{95} + 16 q^{97} - 2 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −2.73205 0 −1.00000 0 −1.00000 0 4.46410 0
1.2 0 0.732051 0 −1.00000 0 −1.00000 0 −2.46410 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.s 2
4.b odd 2 1 1540.2.a.e 2
20.d odd 2 1 7700.2.a.n 2
20.e even 4 2 7700.2.e.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.e 2 4.b odd 2 1
6160.2.a.s 2 1.a even 1 1 trivial
7700.2.a.n 2 20.d odd 2 1
7700.2.e.l 4 20.e even 4 2

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{13}^{2} + 2 T_{13} - 2 \)
\( T_{17}^{2} - 6 T_{17} + 6 \)
\( T_{19}^{2} + 4 T_{19} - 8 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -2 + 2 T + T^{2} \)
$17$ \( 6 - 6 T + T^{2} \)
$19$ \( -8 + 4 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -26 - 2 T + T^{2} \)
$37$ \( -8 - 4 T + T^{2} \)
$41$ \( -18 - 6 T + T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( -66 + 6 T + T^{2} \)
$53$ \( 24 - 12 T + T^{2} \)
$59$ \( 78 + 18 T + T^{2} \)
$61$ \( 46 + 14 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( 24 - 12 T + T^{2} \)
$73$ \( -2 - 10 T + T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( -72 + 12 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( -44 - 16 T + T^{2} \)
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