Properties

Label 6160.2.a.bh
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 1540)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} - q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} - q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} - q^{11} -\beta_{2} q^{13} + \beta_{2} q^{15} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{17} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{19} -\beta_{2} q^{21} + ( 3 - 5 \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( 2 + 2 \beta_{2} ) q^{27} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -3 + 3 \beta_{1} ) q^{31} + \beta_{2} q^{33} - q^{35} + ( -1 - 5 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 3 - \beta_{1} - \beta_{2} ) q^{39} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( -6 + 4 \beta_{1} + 3 \beta_{2} ) q^{47} + q^{49} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -7 + \beta_{1} + \beta_{2} ) q^{53} + q^{55} + ( -10 + 2 \beta_{1} + 4 \beta_{2} ) q^{57} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( 3 + 5 \beta_{1} ) q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + \beta_{2} q^{65} + ( 7 + 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -2 - 4 \beta_{1} - 5 \beta_{2} ) q^{73} -\beta_{2} q^{75} - q^{77} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -4 + 6 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{85} + ( 4 + 2 \beta_{2} ) q^{87} + ( -8 + 4 \beta_{2} ) q^{89} -\beta_{2} q^{91} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( -4 + 8 \beta_{1} + 4 \beta_{2} ) q^{97} + ( \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7} - q^{9} + O(q^{10}) \) \( 3 q - 3 q^{5} + 3 q^{7} - q^{9} - 3 q^{11} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99} + O(q^{100}) \)

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.bh 3
4.b odd 2 1 1540.2.a.g 3
20.d odd 2 1 7700.2.a.x 3
20.e even 4 2 7700.2.e.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.g 3 4.b odd 2 1
6160.2.a.bh 3 1.a even 1 1 trivial
7700.2.a.x 3 20.d odd 2 1
7700.2.e.r 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} - 4 T_{3} - 2 \)
\( T_{13}^{3} - 4 T_{13} - 2 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 48 T_{17} - 134 \)
\( T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200 \)
\( T_{23}^{3} - 4 T_{23}^{2} - 72 T_{23} + 268 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -2 - 4 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -2 - 4 T + T^{3} \)
$17$ \( -134 - 48 T + 2 T^{2} + T^{3} \)
$19$ \( 200 - 60 T - 2 T^{2} + T^{3} \)
$23$ \( 268 - 72 T - 4 T^{2} + T^{3} \)
$29$ \( 8 + 44 T + 14 T^{2} + T^{3} \)
$31$ \( -54 - 18 T + 6 T^{2} + T^{3} \)
$37$ \( -1076 - 128 T + 8 T^{2} + T^{3} \)
$41$ \( -338 - 10 T + 12 T^{2} + T^{3} \)
$43$ \( -76 + 128 T - 22 T^{2} + T^{3} \)
$47$ \( -338 + 14 T^{2} + T^{3} \)
$53$ \( 260 + 128 T + 20 T^{2} + T^{3} \)
$59$ \( -50 + 50 T - 14 T^{2} + T^{3} \)
$61$ \( 278 - 18 T - 14 T^{2} + T^{3} \)
$67$ \( 428 + 108 T - 24 T^{2} + T^{3} \)
$71$ \( -32 - 32 T - 4 T^{2} + T^{3} \)
$73$ \( -466 - 80 T + 10 T^{2} + T^{3} \)
$79$ \( 4 + 20 T + 12 T^{2} + T^{3} \)
$83$ \( 296 - 148 T + 6 T^{2} + T^{3} \)
$89$ \( 128 + 128 T + 24 T^{2} + T^{3} \)
$97$ \( -1472 - 208 T + 4 T^{2} + T^{3} \)
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