Properties

Label 6160.2.a.bp
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 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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + q^{5} - q^{7} + ( 3 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + q^{5} - q^{7} + ( 3 \beta_{1} + \beta_{2} ) q^{9} - q^{11} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 2 + 2 \beta_{2} ) q^{19} + ( -1 - \beta_{1} ) q^{21} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{23} + q^{25} + ( 2 + 4 \beta_{1} + 4 \beta_{2} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( -1 - \beta_{1} ) q^{33} - q^{35} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( -1 - 5 \beta_{1} - 3 \beta_{2} ) q^{39} + ( 2 \beta_{1} + \beta_{2} ) q^{41} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{43} + ( 3 \beta_{1} + \beta_{2} ) q^{45} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{47} + q^{49} + ( 1 + 5 \beta_{1} + 5 \beta_{2} ) q^{51} + ( 1 + \beta_{1} - 5 \beta_{2} ) q^{53} - q^{55} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 4 - 4 \beta_{1} - \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{61} + ( -3 \beta_{1} - \beta_{2} ) q^{63} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{65} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{67} + ( 10 + 4 \beta_{1} ) q^{69} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 1 + \beta_{1} ) q^{75} + q^{77} + ( -3 - 3 \beta_{1} - 7 \beta_{2} ) q^{79} + ( 6 + 5 \beta_{1} + 5 \beta_{2} ) q^{81} + ( 6 + 4 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -2 - 6 \beta_{1} - 4 \beta_{2} ) q^{87} + ( 8 - 4 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{91} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{93} + ( 2 + 2 \beta_{2} ) q^{95} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -3 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 4 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 4 q^{13} + 4 q^{15} - 6 q^{17} + 6 q^{19} - 4 q^{21} + 6 q^{23} + 3 q^{25} + 10 q^{27} - 2 q^{29} + 16 q^{31} - 4 q^{33} - 3 q^{35} - 8 q^{39} + 2 q^{41} - 2 q^{43} + 3 q^{45} + 14 q^{47} + 3 q^{49} + 8 q^{51} + 4 q^{53} - 3 q^{55} + 4 q^{57} + 8 q^{59} - 4 q^{61} - 3 q^{63} - 4 q^{65} + 10 q^{67} + 34 q^{69} - 6 q^{73} + 4 q^{75} + 3 q^{77} - 12 q^{79} + 23 q^{81} + 22 q^{83} - 6 q^{85} - 12 q^{87} + 20 q^{89} + 4 q^{91} + 10 q^{93} + 6 q^{95} - 4 q^{97} - 3 q^{99} + O(q^{100}) \)

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

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

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
0.311108
2.17009
0 −0.481194 0 1.00000 0 −1.00000 0 −2.76845 0
1.2 0 1.31111 0 1.00000 0 −1.00000 0 −1.28100 0
1.3 0 3.17009 0 1.00000 0 −1.00000 0 7.04945 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.bp 3
4.b odd 2 1 3080.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.j 3 4.b odd 2 1
6160.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{3} - 4 T_{3}^{2} + 2 T_{3} + 2 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 10 T_{13} - 38 \)
\( T_{17}^{3} + 6 T_{17}^{2} - 22 T_{17} - 122 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 4 T_{19} + 40 \)
\( T_{23}^{3} - 6 T_{23}^{2} - 72 T_{23} + 428 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 2 + 2 T - 4 T^{2} + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -38 - 10 T + 4 T^{2} + T^{3} \)
$17$ \( -122 - 22 T + 6 T^{2} + T^{3} \)
$19$ \( 40 - 4 T - 6 T^{2} + T^{3} \)
$23$ \( 428 - 72 T - 6 T^{2} + T^{3} \)
$29$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$31$ \( -62 + 72 T - 16 T^{2} + T^{3} \)
$37$ \( -52 - 28 T + T^{3} \)
$41$ \( -10 - 12 T - 2 T^{2} + T^{3} \)
$43$ \( -20 - 44 T + 2 T^{2} + T^{3} \)
$47$ \( 274 - 10 T - 14 T^{2} + T^{3} \)
$53$ \( 52 - 108 T - 4 T^{2} + T^{3} \)
$59$ \( 214 - 28 T - 8 T^{2} + T^{3} \)
$61$ \( -34 - 8 T + 4 T^{2} + T^{3} \)
$67$ \( 124 - 12 T - 10 T^{2} + T^{3} \)
$71$ \( -608 - 160 T + T^{3} \)
$73$ \( -278 - 46 T + 6 T^{2} + T^{3} \)
$79$ \( -1580 - 136 T + 12 T^{2} + T^{3} \)
$83$ \( 184 + 76 T - 22 T^{2} + T^{3} \)
$89$ \( 320 + 48 T - 20 T^{2} + T^{3} \)
$97$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
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