Properties

Label 6160.2.a.bb
Level $6160$
Weight $2$
Character orbit 6160.a
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + q^{5} - q^{7} - q^{9} + q^{11} + ( -2 - \beta ) q^{13} + \beta q^{15} + ( -2 + 3 \beta ) q^{17} -4 \beta q^{19} -\beta q^{21} + 2 q^{23} + q^{25} -4 \beta q^{27} + ( -6 + 2 \beta ) q^{29} + ( 6 - \beta ) q^{31} + \beta q^{33} - q^{35} -2 \beta q^{37} + ( -2 - 2 \beta ) q^{39} + 3 \beta q^{41} + 2 q^{43} - q^{45} + ( 4 + 3 \beta ) q^{47} + q^{49} + ( 6 - 2 \beta ) q^{51} + ( 4 - 6 \beta ) q^{53} + q^{55} -8 q^{57} + ( -6 + \beta ) q^{59} + ( -8 - 3 \beta ) q^{61} + q^{63} + ( -2 - \beta ) q^{65} + ( 2 - 8 \beta ) q^{67} + 2 \beta q^{69} + ( -4 + 6 \beta ) q^{71} + ( -6 - 3 \beta ) q^{73} + \beta q^{75} - q^{77} + ( -10 + 2 \beta ) q^{79} -5 q^{81} + ( -12 - 4 \beta ) q^{83} + ( -2 + 3 \beta ) q^{85} + ( 4 - 6 \beta ) q^{87} + ( -6 + 2 \beta ) q^{89} + ( 2 + \beta ) q^{91} + ( -2 + 6 \beta ) q^{93} -4 \beta q^{95} + ( 2 - 8 \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{23} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 2 q^{35} - 4 q^{39} + 4 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 12 q^{51} + 8 q^{53} + 2 q^{55} - 16 q^{57} - 12 q^{59} - 16 q^{61} + 2 q^{63} - 4 q^{65} + 4 q^{67} - 8 q^{71} - 12 q^{73} - 2 q^{77} - 20 q^{79} - 10 q^{81} - 24 q^{83} - 4 q^{85} + 8 q^{87} - 12 q^{89} + 4 q^{91} - 4 q^{93} + 4 q^{97} - 2 q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −1.00000 0 −1.00000 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.bb 2
4.b odd 2 1 3080.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.i 2 4.b odd 2 1
6160.2.a.bb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6160))\):

\( T_{3}^{2} - 2 \)
\( T_{13}^{2} + 4 T_{13} + 2 \)
\( T_{17}^{2} + 4 T_{17} - 14 \)
\( T_{19}^{2} - 32 \)
\( T_{23} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 2 + 4 T + T^{2} \)
$17$ \( -14 + 4 T + T^{2} \)
$19$ \( -32 + T^{2} \)
$23$ \( ( -2 + T )^{2} \)
$29$ \( 28 + 12 T + T^{2} \)
$31$ \( 34 - 12 T + T^{2} \)
$37$ \( -8 + T^{2} \)
$41$ \( -18 + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( -2 - 8 T + T^{2} \)
$53$ \( -56 - 8 T + T^{2} \)
$59$ \( 34 + 12 T + T^{2} \)
$61$ \( 46 + 16 T + T^{2} \)
$67$ \( -124 - 4 T + T^{2} \)
$71$ \( -56 + 8 T + T^{2} \)
$73$ \( 18 + 12 T + T^{2} \)
$79$ \( 92 + 20 T + T^{2} \)
$83$ \( 112 + 24 T + T^{2} \)
$89$ \( 28 + 12 T + T^{2} \)
$97$ \( -124 - 4 T + T^{2} \)
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