Properties

Label 6160.2.a.bd
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: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} - q^{5} + q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} - q^{5} + q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + q^{11} + ( -1 + \beta_{1} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{17} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( -\beta_{1} - \beta_{2} ) q^{23} + q^{25} -2 \beta_{2} q^{27} -2 q^{29} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( -1 + \beta_{1} ) q^{33} - q^{35} + ( -2 + \beta_{1} + \beta_{2} ) q^{37} + ( 4 - \beta_{1} + \beta_{2} ) q^{39} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{45} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{47} + q^{49} + ( 2 - 5 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{53} - q^{55} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{57} + ( 1 - 5 \beta_{2} ) q^{59} + ( -7 - 4 \beta_{1} + 3 \beta_{2} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{63} + ( 1 - \beta_{1} ) q^{65} + ( 2 + 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -4 - 2 \beta_{1} ) q^{69} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -7 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( -1 + \beta_{1} ) q^{75} + q^{77} + ( 10 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( -5 - \beta_{1} - \beta_{2} ) q^{81} + ( -4 - 2 \beta_{2} ) q^{83} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 - 2 \beta_{1} ) q^{87} + ( 2 - 2 \beta_{1} ) q^{89} + ( -1 + \beta_{1} ) q^{91} + ( -10 + \beta_{1} - \beta_{2} ) q^{93} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 2 - 6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

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

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
−1.65544
−0.210756
2.86620
0 −2.65544 0 −1.00000 0 1.00000 0 4.05137 0
1.2 0 −1.21076 0 −1.00000 0 1.00000 0 −1.53407 0
1.3 0 1.86620 0 −1.00000 0 1.00000 0 0.482696 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.bd 3
4.b odd 2 1 3080.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.m 3 4.b odd 2 1
6160.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 6 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 4 T_{13} - 6 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 20 T_{17} - 66 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 20 T_{19} - 88 \)
\( T_{23}^{3} + 2 T_{23}^{2} - 16 T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -6 - 4 T + 2 T^{2} + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -6 - 4 T + 2 T^{2} + T^{3} \)
$17$ \( -66 - 20 T + 4 T^{2} + T^{3} \)
$19$ \( -88 - 20 T + 6 T^{2} + T^{3} \)
$23$ \( 4 - 16 T + 2 T^{2} + T^{3} \)
$29$ \( ( 2 + T )^{3} \)
$31$ \( 154 - 26 T - 6 T^{2} + T^{3} \)
$37$ \( -36 - 12 T + 4 T^{2} + T^{3} \)
$41$ \( -2 + 10 T + 12 T^{2} + T^{3} \)
$43$ \( 348 - 52 T - 10 T^{2} + T^{3} \)
$47$ \( -118 + 4 T + 12 T^{2} + T^{3} \)
$53$ \( 148 - 20 T - 8 T^{2} + T^{3} \)
$59$ \( -946 - 182 T + 2 T^{2} + T^{3} \)
$61$ \( -626 + 66 T + 22 T^{2} + T^{3} \)
$67$ \( 244 - 60 T - 6 T^{2} + T^{3} \)
$71$ \( 192 - 80 T - 12 T^{2} + T^{3} \)
$73$ \( 162 + 108 T + 20 T^{2} + T^{3} \)
$79$ \( -716 + 300 T - 32 T^{2} + T^{3} \)
$83$ \( -88 + 36 T + 14 T^{2} + T^{3} \)
$89$ \( 48 - 16 T - 4 T^{2} + T^{3} \)
$97$ \( -784 - 192 T - 4 T^{2} + T^{3} \)
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