Properties

Label 980.2.a.a
Level $980$
Weight $2$
Character orbit 980.a
Self dual yes
Analytic conductor $7.825$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - q^{5} + 6q^{9} + O(q^{10}) \) \( q - 3q^{3} - q^{5} + 6q^{9} - 2q^{11} - 6q^{13} + 3q^{15} + 2q^{17} - 9q^{23} + q^{25} - 9q^{27} + 3q^{29} + 2q^{31} + 6q^{33} + 8q^{37} + 18q^{39} + 5q^{41} + q^{43} - 6q^{45} + 8q^{47} - 6q^{51} + 4q^{53} + 2q^{55} - 8q^{59} + 7q^{61} + 6q^{65} - 3q^{67} + 27q^{69} + 8q^{71} + 14q^{73} - 3q^{75} + 4q^{79} + 9q^{81} - q^{83} - 2q^{85} - 9q^{87} + 13q^{89} - 6q^{93} - 10q^{97} - 12q^{99} + O(q^{100}) \)

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 0 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 980.2.a.a 1
3.b odd 2 1 8820.2.a.w 1
4.b odd 2 1 3920.2.a.bi 1
5.b even 2 1 4900.2.a.v 1
5.c odd 4 2 4900.2.e.c 2
7.b odd 2 1 980.2.a.i 1
7.c even 3 2 140.2.i.b 2
7.d odd 6 2 980.2.i.a 2
21.c even 2 1 8820.2.a.k 1
21.h odd 6 2 1260.2.s.b 2
28.d even 2 1 3920.2.a.d 1
28.g odd 6 2 560.2.q.a 2
35.c odd 2 1 4900.2.a.a 1
35.f even 4 2 4900.2.e.b 2
35.j even 6 2 700.2.i.a 2
35.l odd 12 4 700.2.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 7.c even 3 2
560.2.q.a 2 28.g odd 6 2
700.2.i.a 2 35.j even 6 2
700.2.r.c 4 35.l odd 12 4
980.2.a.a 1 1.a even 1 1 trivial
980.2.a.i 1 7.b odd 2 1
980.2.i.a 2 7.d odd 6 2
1260.2.s.b 2 21.h odd 6 2
3920.2.a.d 1 28.d even 2 1
3920.2.a.bi 1 4.b odd 2 1
4900.2.a.a 1 35.c odd 2 1
4900.2.a.v 1 5.b even 2 1
4900.2.e.b 2 35.f even 4 2
4900.2.e.c 2 5.c odd 4 2
8820.2.a.k 1 21.c even 2 1
8820.2.a.w 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\):

\( T_{3} + 3 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( 6 + T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( 9 + T \)
$29$ \( -3 + T \)
$31$ \( -2 + T \)
$37$ \( -8 + T \)
$41$ \( -5 + T \)
$43$ \( -1 + T \)
$47$ \( -8 + T \)
$53$ \( -4 + T \)
$59$ \( 8 + T \)
$61$ \( -7 + T \)
$67$ \( 3 + T \)
$71$ \( -8 + T \)
$73$ \( -14 + T \)
$79$ \( -4 + T \)
$83$ \( 1 + T \)
$89$ \( -13 + T \)
$97$ \( 10 + T \)
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