Properties

Label 6160.2.a.bj
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + q^{5} - q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + q^{5} - q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} - q^{11} + \beta_{2} q^{13} -\beta_{2} q^{15} + ( 2 - \beta_{2} ) q^{17} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} + \beta_{2} q^{21} + ( -1 + \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( 2 + 2 \beta_{2} ) q^{27} + ( 2 - 4 \beta_{1} ) q^{29} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{31} + \beta_{2} q^{33} - q^{35} + ( -5 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( -3 + \beta_{1} + \beta_{2} ) q^{39} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{41} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{45} + ( -2 - 4 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{51} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{53} - q^{55} + ( -4 + 6 \beta_{2} ) q^{57} + ( -3 + \beta_{1} ) q^{59} + ( -1 + 3 \beta_{1} - 4 \beta_{2} ) q^{61} + ( \beta_{1} + \beta_{2} ) q^{63} + \beta_{2} q^{65} + ( 1 - \beta_{1} + \beta_{2} ) q^{67} + ( -2 + 2 \beta_{2} ) q^{69} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -10 + \beta_{2} ) q^{73} -\beta_{2} q^{75} + q^{77} + ( -1 - \beta_{1} + \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( 4 - 6 \beta_{1} ) q^{83} + ( 2 - \beta_{2} ) q^{85} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 4 - 6 \beta_{2} ) q^{89} -\beta_{2} q^{91} + ( -5 + \beta_{1} + 5 \beta_{2} ) q^{93} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -8 - 6 \beta_{2} ) q^{97} + ( \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - q^{9} + O(q^{10}) \) \( 3 q + 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99} + O(q^{100}) \)

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
−1.48119
2.17009
0.311108
0 −1.67513 0 1.00000 0 −1.00000 0 −0.193937 0
1.2 0 −0.539189 0 1.00000 0 −1.00000 0 −2.70928 0
1.3 0 2.21432 0 1.00000 0 −1.00000 0 1.90321 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.bj 3
4.b odd 2 1 385.2.a.g 3
12.b even 2 1 3465.2.a.ba 3
20.d odd 2 1 1925.2.a.u 3
20.e even 4 2 1925.2.b.o 6
28.d even 2 1 2695.2.a.i 3
44.c even 2 1 4235.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.g 3 4.b odd 2 1
1925.2.a.u 3 20.d odd 2 1
1925.2.b.o 6 20.e even 4 2
2695.2.a.i 3 28.d even 2 1
3465.2.a.ba 3 12.b even 2 1
4235.2.a.o 3 44.c even 2 1
6160.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} - 4 T_{3} - 2 \)
\( T_{13}^{3} - 4 T_{13} + 2 \)
\( T_{17}^{3} - 6 T_{17}^{2} + 8 T_{17} - 2 \)
\( T_{19}^{3} + 10 T_{19}^{2} + 12 T_{19} - 40 \)
\( T_{23}^{3} + 2 T_{23}^{2} - 4 T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -2 - 4 T + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 2 - 4 T + T^{3} \)
$17$ \( -2 + 8 T - 6 T^{2} + T^{3} \)
$19$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$23$ \( -4 - 4 T + 2 T^{2} + T^{3} \)
$29$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$31$ \( -2 + 6 T + 8 T^{2} + T^{3} \)
$37$ \( -100 + 20 T + 12 T^{2} + T^{3} \)
$41$ \( 334 - 26 T - 10 T^{2} + T^{3} \)
$43$ \( 148 - 28 T - 6 T^{2} + T^{3} \)
$47$ \( -26 - 16 T + 10 T^{2} + T^{3} \)
$53$ \( -268 - 84 T + T^{3} \)
$59$ \( 10 + 18 T + 8 T^{2} + T^{3} \)
$61$ \( 358 - 118 T + T^{3} \)
$67$ \( -4 - 8 T - 2 T^{2} + T^{3} \)
$71$ \( 64 - 80 T - 4 T^{2} + T^{3} \)
$73$ \( 962 + 296 T + 30 T^{2} + T^{3} \)
$79$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$83$ \( 248 - 108 T - 6 T^{2} + T^{3} \)
$89$ \( 80 - 96 T - 12 T^{2} + T^{3} \)
$97$ \( -1072 + 48 T + 24 T^{2} + T^{3} \)
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