Properties

Label 6160.2.a.bs
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.111028.1
Defining polynomial: \(x^{4} - 10 x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} - q^{11} + ( 1 - \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( 2 - \beta_{1} ) q^{17} + ( -1 + \beta_{1} - \beta_{3} ) q^{19} + \beta_{1} q^{21} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + q^{25} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{27} + 2 q^{29} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} -\beta_{1} q^{33} + q^{35} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( 3 - \beta_{1} - \beta_{2} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{45} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} - q^{55} + ( 4 - 2 \beta_{1} ) q^{57} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{59} + ( 6 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 2 + \beta_{2} ) q^{63} + ( 1 - \beta_{3} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{67} + ( -5 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{69} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 6 + \beta_{1} ) q^{73} + \beta_{1} q^{75} - q^{77} + ( -3 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{79} + ( 10 + 2 \beta_{1} + \beta_{2} ) q^{81} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + 2 \beta_{1} q^{87} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{3} ) q^{91} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{93} + ( -1 + \beta_{1} - \beta_{3} ) q^{95} + ( -1 + \beta_{1} + \beta_{3} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 9 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 9 \beta_{1} + 1\)

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
−2.97964
−0.766757
0.547280
3.19911
0 −2.97964 0 1.00000 0 1.00000 0 5.87824 0
1.2 0 −0.766757 0 1.00000 0 1.00000 0 −2.41208 0
1.3 0 0.547280 0 1.00000 0 1.00000 0 −2.70049 0
1.4 0 3.19911 0 1.00000 0 1.00000 0 7.23433 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.bs 4
4.b odd 2 1 1540.2.a.i 4
20.d odd 2 1 7700.2.a.bb 4
20.e even 4 2 7700.2.e.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.i 4 4.b odd 2 1
6160.2.a.bs 4 1.a even 1 1 trivial
7700.2.a.bb 4 20.d odd 2 1
7700.2.e.t 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{4} - 10 T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{13}^{4} - 2 T_{13}^{3} - 34 T_{13}^{2} - 2 T_{13} + 96 \)
\( T_{17}^{4} - 8 T_{17}^{3} + 14 T_{17}^{2} + 10 T_{17} - 24 \)
\( T_{19}^{4} + 6 T_{19}^{3} - 28 T_{19}^{2} - 152 T_{19} - 96 \)
\( T_{23}^{4} - 6 T_{23}^{3} - 56 T_{23}^{2} + 236 T_{23} - 184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 - 2 T - 10 T^{2} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 96 - 2 T - 34 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( -24 + 10 T + 14 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( -96 - 152 T - 28 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( -184 + 236 T - 56 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( 176 - 134 T - 40 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( -264 + 244 T - 32 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( -668 + 370 T - 4 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( -144 + 196 T - 72 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 1044 + 158 T - 70 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 1608 + 364 T - 144 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( -1448 - 810 T - 108 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( -5636 + 738 T + 140 T^{2} - 26 T^{3} + T^{4} \)
$67$ \( -8 + 36 T - 36 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( 8704 - 384 T - 192 T^{2} + 4 T^{3} + T^{4} \)
$73$ \( 952 - 746 T + 206 T^{2} - 24 T^{3} + T^{4} \)
$79$ \( 10528 - 1908 T - 280 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 3936 + 488 T - 260 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 2592 - 352 T - 160 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 544 - 48 T - 48 T^{2} + 2 T^{3} + T^{4} \)
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