Properties

Label 6080.2.a.bs
Level $6080$
Weight $2$
Character orbit 6080.a
Self dual yes
Analytic conductor $48.549$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.5490444289\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_1 - 2) q^{11} + ( - 2 \beta_{2} - \beta_1 - 2) q^{13} - \beta_1 q^{15} + ( - \beta_{2} - 4) q^{17} - q^{19} + (\beta_{2} - 2 \beta_1 + 2) q^{21} + ( - \beta_{2} + 2 \beta_1 + 2) q^{23} + q^{25} + ( - \beta_{2} - 2 \beta_1 - 6) q^{27} + 3 \beta_{2} q^{29} + ( - 2 \beta_{2} + 4) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{33} + (\beta_{2} + 2) q^{35} + (\beta_{2} - \beta_1 - 4) q^{37} + ( - \beta_{2} + 4 \beta_1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{43} + (\beta_{2} + 2 \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 4 \beta_1 - 2) q^{51} + ( - 2 \beta_{2} - \beta_1 + 2) q^{53} + (2 \beta_1 - 2) q^{55} + \beta_1 q^{57} + (\beta_{2} - 2) q^{59} + (4 \beta_1 - 8) q^{61} + (2 \beta_1 + 4) q^{63} + ( - 2 \beta_{2} - \beta_1 - 2) q^{65} + (4 \beta_{2} + 3 \beta_1) q^{67} + ( - 3 \beta_{2} - 6 \beta_1 - 10) q^{69} + ( - 4 \beta_1 + 4) q^{71} + ( - 3 \beta_{2} - 4 \beta_1 - 4) q^{73} - \beta_1 q^{75} + ( - 4 \beta_{2} + 4 \beta_1 - 8) q^{77} + (4 \beta_{2} + 2 \beta_1 - 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + (2 \beta_{2} - 8) q^{83} + ( - \beta_{2} - 4) q^{85} + (3 \beta_{2} + 6) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 3 \beta_{2} + 2 \beta_1 - 14) q^{91} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{93} - q^{95} + (\beta_{2} + 7 \beta_1 - 4) q^{97} + (6 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} + 5 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{5} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} - q^{15} - 11 q^{17} - 3 q^{19} + 3 q^{21} + 9 q^{23} + 3 q^{25} - 19 q^{27} - 3 q^{29} + 14 q^{31} - 24 q^{33} + 5 q^{35} - 14 q^{37} + 5 q^{39} - 10 q^{41} - 10 q^{43} + 4 q^{45} + 4 q^{49} - q^{51} + 7 q^{53} - 4 q^{55} + q^{57} - 7 q^{59} - 20 q^{61} + 14 q^{63} - 5 q^{65} - q^{67} - 33 q^{69} + 8 q^{71} - 13 q^{73} - q^{75} - 16 q^{77} - 14 q^{79} + 15 q^{81} - 26 q^{83} - 11 q^{85} + 15 q^{87} + 18 q^{89} - 37 q^{91} - 14 q^{93} - 3 q^{95} - 6 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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
3.12489
−0.363328
−1.76156
0 −3.12489 0 1.00000 0 1.51514 0 6.76491 0
1.2 0 0.363328 0 1.00000 0 −1.14134 0 −2.86799 0
1.3 0 1.76156 0 1.00000 0 4.62620 0 0.103084 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 6080.2.a.bs 3
4.b odd 2 1 6080.2.a.bw 3
8.b even 2 1 1520.2.a.r 3
8.d odd 2 1 760.2.a.h 3
24.f even 2 1 6840.2.a.bj 3
40.e odd 2 1 3800.2.a.y 3
40.f even 2 1 7600.2.a.bo 3
40.k even 4 2 3800.2.d.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 8.d odd 2 1
1520.2.a.r 3 8.b even 2 1
3800.2.a.y 3 40.e odd 2 1
3800.2.d.k 6 40.k even 4 2
6080.2.a.bs 3 1.a even 1 1 trivial
6080.2.a.bw 3 4.b odd 2 1
6840.2.a.bj 3 24.f even 2 1
7600.2.a.bo 3 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6080))\):

\( T_{3}^{3} + T_{3}^{2} - 6T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 20T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 6T + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 5T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 20 T - 64 \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} - 22 T - 106 \) Copy content Toggle raw display
$17$ \( T^{3} + 11 T^{2} + 32 T + 20 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 9 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$29$ \( T^{3} + 3 T^{2} - 72 T - 108 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$37$ \( T^{3} + 14 T^{2} + 46 T - 20 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} - 44 T - 472 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 88 T - 848 \) Copy content Toggle raw display
$47$ \( T^{3} - 40T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 7 T^{2} - 14 T - 2 \) Copy content Toggle raw display
$59$ \( T^{3} + 7 T^{2} + 8 T - 8 \) Copy content Toggle raw display
$61$ \( T^{3} + 20 T^{2} + 32 T - 640 \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 134 T + 530 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 80 T + 512 \) Copy content Toggle raw display
$73$ \( T^{3} + 13 T^{2} - 64 T - 500 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} - 56 T + 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 26 T^{2} + 192 T + 352 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + 68 T - 40 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 274 T - 2308 \) Copy content Toggle raw display
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