Properties

Label 9680.2.a.bu
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
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: \(0\)
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: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - \beta q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} - \beta q^{7} - 2 q^{9} - 2 \beta q^{13} - q^{15} + 4 \beta q^{17} - 2 \beta q^{19} - \beta q^{21} + q^{25} - 5 q^{27} + 8 q^{31} + \beta q^{35} - 8 q^{37} - 2 \beta q^{39} - 7 \beta q^{41} + 5 \beta q^{43} + 2 q^{45} - 9 q^{47} - 4 q^{49} + 4 \beta q^{51} + 6 q^{53} - 2 \beta q^{57} + 12 q^{59} - 5 \beta q^{61} + 2 \beta q^{63} + 2 \beta q^{65} + 5 q^{67} + 12 q^{71} + q^{75} - 6 \beta q^{79} + q^{81} + 2 \beta q^{83} - 4 \beta q^{85} + 3 q^{89} + 6 q^{91} + 8 q^{93} + 2 \beta q^{95} - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9} - 2 q^{15} + 2 q^{25} - 10 q^{27} + 16 q^{31} - 16 q^{37} + 4 q^{45} - 18 q^{47} - 8 q^{49} + 12 q^{53} + 24 q^{59} + 10 q^{67} + 24 q^{71} + 2 q^{75} + 2 q^{81} + 6 q^{89} + 12 q^{91} + 16 q^{93} - 20 q^{97}+O(q^{100}) \) 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 1.00000 0 −1.00000 0 −1.73205 0 −2.00000 0
1.2 0 1.00000 0 −1.00000 0 1.73205 0 −2.00000 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.bu 2
4.b odd 2 1 605.2.a.e 2
11.b odd 2 1 inner 9680.2.a.bu 2
12.b even 2 1 5445.2.a.u 2
20.d odd 2 1 3025.2.a.l 2
44.c even 2 1 605.2.a.e 2
44.g even 10 4 605.2.g.i 8
44.h odd 10 4 605.2.g.i 8
132.d odd 2 1 5445.2.a.u 2
220.g even 2 1 3025.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.e 2 4.b odd 2 1
605.2.a.e 2 44.c even 2 1
605.2.g.i 8 44.g even 10 4
605.2.g.i 8 44.h odd 10 4
3025.2.a.l 2 20.d odd 2 1
3025.2.a.l 2 220.g even 2 1
5445.2.a.u 2 12.b even 2 1
5445.2.a.u 2 132.d odd 2 1
9680.2.a.bu 2 1.a even 1 1 trivial
9680.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} - 12 \) 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
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 147 \) Copy content Toggle raw display
$43$ \( T^{2} - 75 \) Copy content Toggle raw display
$47$ \( (T + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 75 \) Copy content Toggle raw display
$67$ \( (T - 5)^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 108 \) Copy content Toggle raw display
$83$ \( T^{2} - 12 \) Copy content Toggle raw display
$89$ \( (T - 3)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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