Properties

Label 6080.2.a.bm
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: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 3040)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - \beta_1 - 3) q^{11} + (\beta_{2} - 2 \beta_1 + 2) q^{13} + (\beta_1 + 1) q^{15} + 2 q^{17} + q^{19} - 2 q^{21} + (\beta_{2} - \beta_1 + 1) q^{23} + q^{25} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + (4 \beta_{2} + 2 \beta_1 + 4) q^{29} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{31} + (2 \beta_{2} + 6 \beta_1 + 4) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + (\beta_{2} - 2 \beta_1) q^{37} + (\beta_{2} + \beta_1 + 3) q^{39} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{41} + (3 \beta_{2} + 7 \beta_1 - 7) q^{43} + ( - \beta_{2} - 3 \beta_1) q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - 2 \beta_1 - 2) q^{51} + ( - \beta_{2} + 2 \beta_1 + 4) q^{53} + (\beta_{2} + \beta_1 + 3) q^{55} + ( - \beta_1 - 1) q^{57} + (2 \beta_{2} + 2 \beta_1 - 4) q^{59} + ( - 7 \beta_{2} - \beta_1 - 3) q^{61} + (3 \beta_{2} - \beta_1 + 5) q^{63} + ( - \beta_{2} + 2 \beta_1 - 2) q^{65} + (4 \beta_{2} + 3 \beta_1 + 1) q^{67} + 2 q^{69} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{71} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - \beta_1 - 1) q^{75} + (2 \beta_{2} - 4 \beta_1 + 4) q^{77} + (6 \beta_{2} + 2 \beta_1 + 4) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + ( - 3 \beta_{2} + 5 \beta_1 - 1) q^{83} - 2 q^{85} + ( - 6 \beta_{2} - 12 \beta_1 - 4) q^{87} + (4 \beta_{2} + 2 \beta_1) q^{89} + ( - 4 \beta_{2} + 6 \beta_1 - 12) q^{91} + (6 \beta_{2} + 8 \beta_1 + 4) q^{93} - q^{95} + (3 \beta_{2} + 4 \beta_1 + 2) q^{97} + ( - 5 \beta_{2} - 15 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 10 q^{11} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 2 q^{23} + 3 q^{25} - 10 q^{27} + 14 q^{29} + 2 q^{31} + 18 q^{33} + 2 q^{35} - 2 q^{37} + 10 q^{39} + 4 q^{41} - 14 q^{43} - 3 q^{45} - 2 q^{47} - q^{49} - 8 q^{51} + 14 q^{53} + 10 q^{55} - 4 q^{57} - 10 q^{59} - 10 q^{61} + 14 q^{63} - 4 q^{65} + 6 q^{67} + 6 q^{69} + 2 q^{71} + 10 q^{73} - 4 q^{75} + 8 q^{77} + 14 q^{79} + 23 q^{81} + 2 q^{83} - 6 q^{85} - 24 q^{87} + 2 q^{89} - 30 q^{91} + 20 q^{93} - 3 q^{95} + 10 q^{97} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
2.17009
0.311108
−1.48119
0 −3.17009 0 −1.00000 0 0.630898 0 7.04945 0
1.2 0 −1.31111 0 −1.00000 0 1.52543 0 −1.28100 0
1.3 0 0.481194 0 −1.00000 0 −4.15633 0 −2.76845 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.bm 3
4.b odd 2 1 6080.2.a.ca 3
8.b even 2 1 3040.2.a.p yes 3
8.d odd 2 1 3040.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.j 3 8.d odd 2 1
3040.2.a.p yes 3 8.b even 2 1
6080.2.a.bm 3 1.a even 1 1 trivial
6080.2.a.ca 3 4.b odd 2 1

Hecke kernels

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

\( T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 10T_{11}^{2} + 28T_{11} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 10 T^{2} + 28 T + 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 16 T - 10 \) Copy content Toggle raw display
$17$ \( (T - 2)^{3} \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 14 T^{2} + 4 T + 344 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 20 T - 50 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 56 T - 80 \) Copy content Toggle raw display
$43$ \( T^{3} + 14 T^{2} - 92 T - 1388 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} + 44 T + 34 \) Copy content Toggle raw display
$59$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} - 152 T - 1444 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} - 58 T + 218 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$79$ \( T^{3} - 14 T^{2} - 68 T + 1112 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 148 T + 796 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 60 T + 200 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 32 T + 46 \) Copy content Toggle raw display
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