Properties

Label 6160.2.a.bx
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.15785648.1
Defining polynomial: \(x^{5} - 2 x^{4} - 12 x^{3} + 22 x^{2} + 24 x - 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} - q^{7} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} - q^{7} + ( 3 + \beta_{2} ) q^{9} - q^{11} + ( 2 + \beta_{4} ) q^{13} -\beta_{1} q^{15} + \beta_{4} q^{17} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{19} + \beta_{1} q^{21} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{23} + q^{25} + ( -2 \beta_{1} - \beta_{3} ) q^{27} + ( \beta_{1} + \beta_{4} ) q^{29} + ( \beta_{2} + \beta_{4} ) q^{31} + \beta_{1} q^{33} - q^{35} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{37} + ( 4 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{39} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( 2 + \beta_{2} + \beta_{3} ) q^{43} + ( 3 + \beta_{2} ) q^{45} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + q^{49} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{51} + ( 4 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{53} - q^{55} + ( -2 - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{57} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( -\beta_{2} - \beta_{4} ) q^{61} + ( -3 - \beta_{2} ) q^{63} + ( 2 + \beta_{4} ) q^{65} + ( -2 \beta_{1} + \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} - 2 \beta_{4} ) q^{69} + ( 8 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{71} + ( 4 - 4 \beta_{1} - \beta_{4} ) q^{73} -\beta_{1} q^{75} + q^{77} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + ( 3 + \beta_{2} + 2 \beta_{4} ) q^{81} + ( 4 + \beta_{3} + 2 \beta_{4} ) q^{83} + \beta_{4} q^{85} + ( -2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{87} + ( 6 + \beta_{3} - 2 \beta_{4} ) q^{89} + ( -2 - \beta_{4} ) q^{91} + ( 4 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{93} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{95} + ( \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{97} + ( -3 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} + O(q^{10}) \) \( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 13 q^{9} - 5 q^{11} + 12 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 4 q^{23} + 5 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{33} - 5 q^{35} + 18 q^{37} + 8 q^{39} + 4 q^{41} + 6 q^{43} + 13 q^{45} - 6 q^{47} + 5 q^{49} + 12 q^{51} + 14 q^{53} - 5 q^{55} - 12 q^{57} + 4 q^{59} - 13 q^{63} + 12 q^{65} - 6 q^{67} - 10 q^{69} + 28 q^{71} + 10 q^{73} - 2 q^{75} + 5 q^{77} + 14 q^{79} + 17 q^{81} + 22 q^{83} + 2 q^{85} - 16 q^{87} + 24 q^{89} - 12 q^{91} + 10 q^{93} - 8 q^{95} + 10 q^{97} - 13 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 12 x^{3} + 22 x^{2} + 24 x - 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 8 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 10 \nu^{2} + 12 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 10 \beta_{2} + 48\)

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
3.04127
2.57873
0.840204
−1.37916
−3.08104
0 −3.04127 0 1.00000 0 −1.00000 0 6.24933 0
1.2 0 −2.57873 0 1.00000 0 −1.00000 0 3.64983 0
1.3 0 −0.840204 0 1.00000 0 −1.00000 0 −2.29406 0
1.4 0 1.37916 0 1.00000 0 −1.00000 0 −1.09791 0
1.5 0 3.08104 0 1.00000 0 −1.00000 0 6.49280 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bx 5
4.b odd 2 1 3080.2.a.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.u 5 4.b odd 2 1
6160.2.a.bx 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{5} + 2 T_{3}^{4} - 12 T_{3}^{3} - 22 T_{3}^{2} + 24 T_{3} + 28 \)
\( T_{13}^{5} - 12 T_{13}^{4} + 30 T_{13}^{3} + 102 T_{13}^{2} - 408 T_{13} + 112 \)
\( T_{17}^{5} - 2 T_{17}^{4} - 26 T_{17}^{3} + 74 T_{17}^{2} + 56 T_{17} - 216 \)
\( T_{19}^{5} + 8 T_{19}^{4} - 62 T_{19}^{3} - 532 T_{19}^{2} + 392 T_{19} + 5488 \)
\( T_{23}^{5} + 4 T_{23}^{4} - 42 T_{23}^{3} - 208 T_{23}^{2} - 104 T_{23} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 28 + 24 T - 22 T^{2} - 12 T^{3} + 2 T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( 112 - 408 T + 102 T^{2} + 30 T^{3} - 12 T^{4} + T^{5} \)
$17$ \( -216 + 56 T + 74 T^{2} - 26 T^{3} - 2 T^{4} + T^{5} \)
$19$ \( 5488 + 392 T - 532 T^{2} - 62 T^{3} + 8 T^{4} + T^{5} \)
$23$ \( 56 - 104 T - 208 T^{2} - 42 T^{3} + 4 T^{4} + T^{5} \)
$29$ \( -80 + 136 T + 52 T^{2} - 22 T^{3} - 4 T^{4} + T^{5} \)
$31$ \( 2744 + 1288 T - 82 T^{2} - 72 T^{3} + T^{5} \)
$37$ \( -8 - 1824 T + 412 T^{2} + 54 T^{3} - 18 T^{4} + T^{5} \)
$41$ \( -6428 + 1012 T + 574 T^{2} - 106 T^{3} - 4 T^{4} + T^{5} \)
$43$ \( -400 + 32 T + 548 T^{2} - 88 T^{3} - 6 T^{4} + T^{5} \)
$47$ \( 17280 + 6464 T - 1146 T^{2} - 202 T^{3} + 6 T^{4} + T^{5} \)
$53$ \( 1224 - 1232 T + 284 T^{2} + 30 T^{3} - 14 T^{4} + T^{5} \)
$59$ \( 5248 + 3680 T + 10 T^{2} - 136 T^{3} - 4 T^{4} + T^{5} \)
$61$ \( -2744 + 1288 T + 82 T^{2} - 72 T^{3} + T^{5} \)
$67$ \( -1152 - 1568 T - 628 T^{2} - 68 T^{3} + 6 T^{4} + T^{5} \)
$71$ \( 39168 - 19712 T + 2272 T^{2} + 120 T^{3} - 28 T^{4} + T^{5} \)
$73$ \( 3736 + 4696 T + 1126 T^{2} - 154 T^{3} - 10 T^{4} + T^{5} \)
$79$ \( -11480 + 344 T + 828 T^{2} - 38 T^{3} - 14 T^{4} + T^{5} \)
$83$ \( 24640 - 13312 T + 1832 T^{2} + 44 T^{3} - 22 T^{4} + T^{5} \)
$89$ \( -34816 - 256 T + 2016 T^{2} + 4 T^{3} - 24 T^{4} + T^{5} \)
$97$ \( -176960 + 21216 T + 3008 T^{2} - 318 T^{3} - 10 T^{4} + T^{5} \)
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