Properties

Label 6160.2.a.bf
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: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + q^{5} + q^{7} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + q^{5} + q^{7} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} - q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + ( -1 - \beta_{2} ) q^{15} -2 \beta_{2} q^{17} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( -3 + \beta_{2} ) q^{23} + q^{25} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -1 - 3 \beta_{2} ) q^{29} + ( 4 - \beta_{1} - \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + q^{35} + ( 1 - \beta_{2} ) q^{37} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 10 - 2 \beta_{1} ) q^{51} + ( 5 + 3 \beta_{2} ) q^{53} - q^{55} + ( -6 + 3 \beta_{1} - 7 \beta_{2} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 3 - \beta_{1} + \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} ) q^{65} + 4 \beta_{2} q^{67} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{69} + ( 10 - 2 \beta_{1} ) q^{71} + ( -4 + 2 \beta_{2} ) q^{73} + ( -1 - \beta_{2} ) q^{75} - q^{77} + ( -7 + 2 \beta_{1} - \beta_{2} ) q^{79} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{81} + ( 2 - 2 \beta_{2} ) q^{83} -2 \beta_{2} q^{85} + ( 16 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( 2 - 6 \beta_{2} ) q^{93} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{95} + ( -1 + 2 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} + 3q^{5} + 3q^{7} + 7q^{9} + O(q^{10}) \) \( 3q - 2q^{3} + 3q^{5} + 3q^{7} + 7q^{9} - 3q^{11} + 2q^{13} - 2q^{15} + 2q^{17} + 6q^{19} - 2q^{21} - 10q^{23} + 3q^{25} - 8q^{27} + 12q^{31} + 2q^{33} + 3q^{35} + 4q^{37} + 8q^{39} + 4q^{41} - 8q^{43} + 7q^{45} + 3q^{49} + 28q^{51} + 12q^{53} - 3q^{55} - 8q^{57} + 4q^{59} - 6q^{61} + 7q^{63} + 2q^{65} - 4q^{67} - 8q^{69} + 28q^{71} - 14q^{73} - 2q^{75} - 3q^{77} - 18q^{79} + 3q^{81} + 8q^{83} + 2q^{85} + 44q^{87} + 18q^{89} + 2q^{91} + 12q^{93} + 6q^{95} - 4q^{97} - 7q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
−1.81361
2.34292
0.470683
0 −3.10278 0 1.00000 0 1.00000 0 6.62721 0
1.2 0 −1.14637 0 1.00000 0 1.00000 0 −1.68585 0
1.3 0 2.24914 0 1.00000 0 1.00000 0 2.05863 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.bf 3
4.b odd 2 1 770.2.a.m 3
12.b even 2 1 6930.2.a.ce 3
20.d odd 2 1 3850.2.a.bt 3
20.e even 4 2 3850.2.c.ba 6
28.d even 2 1 5390.2.a.ca 3
44.c even 2 1 8470.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 4.b odd 2 1
3850.2.a.bt 3 20.d odd 2 1
3850.2.c.ba 6 20.e even 4 2
5390.2.a.ca 3 28.d even 2 1
6160.2.a.bf 3 1.a even 1 1 trivial
6930.2.a.ce 3 12.b even 2 1
8470.2.a.ci 3 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 6 T_{3} - 8 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 16 T_{13} + 16 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 28 T_{17} - 8 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 46 T_{19} + 232 \)
\( T_{23}^{3} + 10 T_{23}^{2} + 26 T_{23} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -8 - 6 T + 2 T^{2} + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 16 - 16 T - 2 T^{2} + T^{3} \)
$17$ \( -8 - 28 T - 2 T^{2} + T^{3} \)
$19$ \( 232 - 46 T - 6 T^{2} + T^{3} \)
$23$ \( 16 + 26 T + 10 T^{2} + T^{3} \)
$29$ \( -92 - 66 T + T^{3} \)
$31$ \( 32 + 20 T - 12 T^{2} + T^{3} \)
$37$ \( 4 - 2 T - 4 T^{2} + T^{3} \)
$41$ \( -164 - 90 T - 4 T^{2} + T^{3} \)
$43$ \( -848 - 108 T + 8 T^{2} + T^{3} \)
$47$ \( 128 - 112 T + T^{3} \)
$53$ \( 292 - 18 T - 12 T^{2} + T^{3} \)
$59$ \( 128 - 64 T - 4 T^{2} + T^{3} \)
$61$ \( -344 - 100 T + 6 T^{2} + T^{3} \)
$67$ \( 64 - 112 T + 4 T^{2} + T^{3} \)
$71$ \( -64 + 200 T - 28 T^{2} + T^{3} \)
$73$ \( -8 + 36 T + 14 T^{2} + T^{3} \)
$79$ \( -256 + 50 T + 18 T^{2} + T^{3} \)
$83$ \( 32 - 8 T - 8 T^{2} + T^{3} \)
$89$ \( 1208 - 28 T - 18 T^{2} + T^{3} \)
$97$ \( 316 - 154 T + 4 T^{2} + T^{3} \)
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