Properties

Label 9680.2.a.bf
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + q^{5} + 3q^{7} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} + q^{5} + 3q^{7} + 6q^{9} + 4q^{13} + 3q^{15} - 4q^{19} + 9q^{21} + 8q^{23} + q^{25} + 9q^{27} + 6q^{29} + 2q^{31} + 3q^{35} - 8q^{37} + 12q^{39} - 5q^{41} - 5q^{43} + 6q^{45} + 3q^{47} + 2q^{49} + 4q^{53} - 12q^{57} + 2q^{59} - 11q^{61} + 18q^{63} + 4q^{65} + 13q^{67} + 24q^{69} - 2q^{71} - 8q^{73} + 3q^{75} - 10q^{79} + 9q^{81} - 4q^{83} + 18q^{87} + q^{89} + 12q^{91} + 6q^{93} - 4q^{95} - 8q^{97} + O(q^{100}) \)

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
0
0 3.00000 0 1.00000 0 3.00000 0 6.00000 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

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 9680.2.a.bf 1
4.b odd 2 1 605.2.a.a 1
11.b odd 2 1 9680.2.a.be 1
12.b even 2 1 5445.2.a.h 1
20.d odd 2 1 3025.2.a.g 1
44.c even 2 1 605.2.a.c yes 1
44.g even 10 4 605.2.g.b 4
44.h odd 10 4 605.2.g.d 4
132.d odd 2 1 5445.2.a.d 1
220.g even 2 1 3025.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.a 1 4.b odd 2 1
605.2.a.c yes 1 44.c even 2 1
605.2.g.b 4 44.g even 10 4
605.2.g.d 4 44.h odd 10 4
3025.2.a.c 1 220.g even 2 1
3025.2.a.g 1 20.d odd 2 1
5445.2.a.d 1 132.d odd 2 1
5445.2.a.h 1 12.b even 2 1
9680.2.a.be 1 11.b odd 2 1
9680.2.a.bf 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3} - 3 \)
\( T_{7} - 3 \)
\( T_{13} - 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -1 + T \)
$7$ \( -3 + T \)
$11$ \( T \)
$13$ \( -4 + T \)
$17$ \( T \)
$19$ \( 4 + T \)
$23$ \( -8 + T \)
$29$ \( -6 + T \)
$31$ \( -2 + T \)
$37$ \( 8 + T \)
$41$ \( 5 + T \)
$43$ \( 5 + T \)
$47$ \( -3 + T \)
$53$ \( -4 + T \)
$59$ \( -2 + T \)
$61$ \( 11 + T \)
$67$ \( -13 + T \)
$71$ \( 2 + T \)
$73$ \( 8 + T \)
$79$ \( 10 + T \)
$83$ \( 4 + T \)
$89$ \( -1 + T \)
$97$ \( 8 + T \)
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