Properties

Label 4680.2.a.bl
Level $4680$
Weight $2$
Character orbit 4680.a
Self dual yes
Analytic conductor $37.370$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1849.1
Defining polynomial: \(x^{3} - x^{2} - 14 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( 2 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 2 - \beta_{1} ) q^{7} + ( -1 - \beta_{2} ) q^{11} + q^{13} -\beta_{1} q^{17} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} + ( -2 + \beta_{1} ) q^{23} + q^{25} + ( 1 + \beta_{1} + \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} ) q^{31} + ( 2 - \beta_{1} ) q^{35} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{41} + 4 q^{43} -2 \beta_{1} q^{47} + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{49} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + ( 7 + \beta_{2} ) q^{61} + q^{65} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( 9 + \beta_{1} + \beta_{2} ) q^{73} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{77} + ( -7 + \beta_{2} ) q^{79} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} -\beta_{1} q^{85} + ( -5 + 4 \beta_{1} + \beta_{2} ) q^{89} + ( 2 - \beta_{1} ) q^{91} + ( 1 + \beta_{1} - \beta_{2} ) q^{95} + ( 8 + \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{5} + 5q^{7} + O(q^{10}) \) \( 3q + 3q^{5} + 5q^{7} - 3q^{11} + 3q^{13} - q^{17} + 4q^{19} - 5q^{23} + 3q^{25} + 4q^{29} + 10q^{31} + 5q^{35} + 5q^{37} - 5q^{41} + 12q^{43} - 2q^{47} + 16q^{49} + 13q^{53} - 3q^{55} + 21q^{61} + 3q^{65} + 8q^{67} + q^{71} + 28q^{73} - 5q^{77} - 21q^{79} - 4q^{83} - q^{85} - 11q^{89} + 5q^{91} + 4q^{95} + 25q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 14 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 9\)

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
4.50331
−0.615072
−2.88824
0 0 0 1.00000 0 −2.50331 0 0 0
1.2 0 0 0 1.00000 0 2.61507 0 0 0
1.3 0 0 0 1.00000 0 4.88824 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 4680.2.a.bl 3
3.b odd 2 1 1560.2.a.p 3
4.b odd 2 1 9360.2.a.db 3
12.b even 2 1 3120.2.a.bh 3
15.d odd 2 1 7800.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.p 3 3.b odd 2 1
3120.2.a.bh 3 12.b even 2 1
4680.2.a.bl 3 1.a even 1 1 trivial
7800.2.a.bf 3 15.d odd 2 1
9360.2.a.db 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\):

\( T_{7}^{3} - 5 T_{7}^{2} - 6 T_{7} + 32 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 40 T_{11} - 128 \)
\( T_{17}^{3} + T_{17}^{2} - 14 T_{17} + 8 \)
\( T_{19}^{3} - 4 T_{19}^{2} - 52 T_{19} + 176 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 32 - 6 T - 5 T^{2} + T^{3} \)
$11$ \( -128 - 40 T + 3 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( 8 - 14 T + T^{2} + T^{3} \)
$19$ \( 176 - 52 T - 4 T^{2} + T^{3} \)
$23$ \( -32 - 6 T + 5 T^{2} + T^{3} \)
$29$ \( 176 - 52 T - 4 T^{2} + T^{3} \)
$31$ \( 256 - 24 T - 10 T^{2} + T^{3} \)
$37$ \( 548 - 92 T - 5 T^{2} + T^{3} \)
$41$ \( -548 - 92 T + 5 T^{2} + T^{3} \)
$43$ \( ( -4 + T )^{3} \)
$47$ \( 64 - 56 T + 2 T^{2} + T^{3} \)
$53$ \( 484 - 44 T - 13 T^{2} + T^{3} \)
$59$ \( T^{3} \)
$61$ \( 44 + 104 T - 21 T^{2} + T^{3} \)
$67$ \( 1408 - 208 T - 8 T^{2} + T^{3} \)
$71$ \( -352 - 100 T - T^{2} + T^{3} \)
$73$ \( -176 + 204 T - 28 T^{2} + T^{3} \)
$79$ \( 128 + 104 T + 21 T^{2} + T^{3} \)
$83$ \( 512 - 224 T + 4 T^{2} + T^{3} \)
$89$ \( -2596 - 232 T + 11 T^{2} + T^{3} \)
$97$ \( -472 + 194 T - 25 T^{2} + T^{3} \)
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