Properties

Label 6160.2.a.bn
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
2.17009
0.311108
−1.48119
0 −1.17009 0 −1.00000 0 1.00000 0 −1.63090 0
1.2 0 0.688892 0 −1.00000 0 1.00000 0 −2.52543 0
1.3 0 2.48119 0 −1.00000 0 1.00000 0 3.15633 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.bn 3
4.b odd 2 1 385.2.a.f 3
12.b even 2 1 3465.2.a.bh 3
20.d odd 2 1 1925.2.a.v 3
20.e even 4 2 1925.2.b.n 6
28.d even 2 1 2695.2.a.g 3
44.c even 2 1 4235.2.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.f 3 4.b odd 2 1
1925.2.a.v 3 20.d odd 2 1
1925.2.b.n 6 20.e even 4 2
2695.2.a.g 3 28.d even 2 1
3465.2.a.bh 3 12.b even 2 1
4235.2.a.q 3 44.c even 2 1
6160.2.a.bn 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 2 T_{3} + 2 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 22 T_{13} - 2 \)
\( T_{17}^{3} - 30 T_{17} + 2 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 8 \)
\( T_{23}^{3} - 10 T_{23}^{2} + 28 T_{23} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 2 - 2 T - 2 T^{2} + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -2 - 22 T - 2 T^{2} + T^{3} \)
$17$ \( 2 - 30 T + T^{3} \)
$19$ \( -8 - 4 T + 6 T^{2} + T^{3} \)
$23$ \( -20 + 28 T - 10 T^{2} + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( 26 + 20 T - 10 T^{2} + T^{3} \)
$37$ \( -100 + 52 T + 16 T^{2} + T^{3} \)
$41$ \( -54 - 36 T + T^{3} \)
$43$ \( -268 - 92 T - 2 T^{2} + T^{3} \)
$47$ \( -158 + 110 T - 20 T^{2} + T^{3} \)
$53$ \( 4 + 20 T + 12 T^{2} + T^{3} \)
$59$ \( -74 + 14 T^{2} + T^{3} \)
$61$ \( -62 - 60 T - 10 T^{2} + T^{3} \)
$67$ \( 172 - 112 T - 2 T^{2} + T^{3} \)
$71$ \( 800 + 80 T - 24 T^{2} + T^{3} \)
$73$ \( 190 - 158 T - 4 T^{2} + T^{3} \)
$79$ \( -244 - 112 T + 8 T^{2} + T^{3} \)
$83$ \( 1096 - 116 T - 10 T^{2} + T^{3} \)
$89$ \( 320 + 48 T - 20 T^{2} + T^{3} \)
$97$ \( -160 - 192 T + T^{3} \)
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