Properties

Label 9680.2.a.bg
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} + q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + q^{5} + \beta q^{7} + q^{9} - 2 q^{15} + 2 \beta q^{17} - 2 \beta q^{19} - 2 \beta q^{21} - 6 q^{23} + q^{25} + 4 q^{27} - 4 q^{31} + \beta q^{35} + 10 q^{37} - 2 \beta q^{41} - \beta q^{43} + q^{45} + 6 q^{47} + 5 q^{49} - 4 \beta q^{51} - 6 q^{53} + 4 \beta q^{57} - 2 \beta q^{61} + \beta q^{63} - 10 q^{67} + 12 q^{69} + 2 \beta q^{73} - 2 q^{75} - 2 \beta q^{79} - 11 q^{81} - 5 \beta q^{83} + 2 \beta q^{85} - 6 q^{89} + 8 q^{93} - 2 \beta q^{95} - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{5} + 2 q^{9} - 4 q^{15} - 12 q^{23} + 2 q^{25} + 8 q^{27} - 8 q^{31} + 20 q^{37} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 12 q^{53} - 20 q^{67} + 24 q^{69} - 4 q^{75} - 22 q^{81} - 12 q^{89} + 16 q^{93} - 20 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 −2.00000 0 1.00000 0 −3.46410 0 1.00000 0
1.2 0 −2.00000 0 1.00000 0 3.46410 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.bg 2
4.b odd 2 1 605.2.a.f 2
11.b odd 2 1 inner 9680.2.a.bg 2
12.b even 2 1 5445.2.a.r 2
20.d odd 2 1 3025.2.a.j 2
44.c even 2 1 605.2.a.f 2
44.g even 10 4 605.2.g.h 8
44.h odd 10 4 605.2.g.h 8
132.d odd 2 1 5445.2.a.r 2
220.g even 2 1 3025.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.f 2 4.b odd 2 1
605.2.a.f 2 44.c even 2 1
605.2.g.h 8 44.g even 10 4
605.2.g.h 8 44.h odd 10 4
3025.2.a.j 2 20.d odd 2 1
3025.2.a.j 2 220.g even 2 1
5445.2.a.r 2 12.b even 2 1
5445.2.a.r 2 132.d odd 2 1
9680.2.a.bg 2 1.a even 1 1 trivial
9680.2.a.bg 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 48 \) Copy content Toggle raw display
$43$ \( T^{2} - 12 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 48 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 48 \) Copy content Toggle raw display
$79$ \( T^{2} - 48 \) Copy content Toggle raw display
$83$ \( T^{2} - 300 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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