Properties

Label 980.2.a.g
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

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} - 2q^{9} + 6q^{11} + 2q^{13} - q^{15} - 6q^{17} + 8q^{19} + 3q^{23} + q^{25} - 5q^{27} + 3q^{29} + 2q^{31} + 6q^{33} + 8q^{37} + 2q^{39} - 3q^{41} + 5q^{43} + 2q^{45} - 6q^{51} + 12q^{53} - 6q^{55} + 8q^{57} - q^{61} - 2q^{65} - 7q^{67} + 3q^{69} - 10q^{73} + q^{75} - 4q^{79} + q^{81} + 3q^{83} + 6q^{85} + 3q^{87} - 3q^{89} + 2q^{93} - 8q^{95} - 10q^{97} - 12q^{99} + O(q^{100}) \)

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 1.00000 0 −1.00000 0 0 0 −2.00000 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.g 1
3.b odd 2 1 8820.2.a.p 1
4.b odd 2 1 3920.2.a.k 1
5.b even 2 1 4900.2.a.i 1
5.c odd 4 2 4900.2.e.m 2
7.b odd 2 1 980.2.a.e 1
7.c even 3 2 140.2.i.a 2
7.d odd 6 2 980.2.i.f 2
21.c even 2 1 8820.2.a.a 1
21.h odd 6 2 1260.2.s.c 2
28.d even 2 1 3920.2.a.w 1
28.g odd 6 2 560.2.q.f 2
35.c odd 2 1 4900.2.a.q 1
35.f even 4 2 4900.2.e.n 2
35.j even 6 2 700.2.i.b 2
35.l odd 12 4 700.2.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 7.c even 3 2
560.2.q.f 2 28.g odd 6 2
700.2.i.b 2 35.j even 6 2
700.2.r.a 4 35.l odd 12 4
980.2.a.e 1 7.b odd 2 1
980.2.a.g 1 1.a even 1 1 trivial
980.2.i.f 2 7.d odd 6 2
1260.2.s.c 2 21.h odd 6 2
3920.2.a.k 1 4.b odd 2 1
3920.2.a.w 1 28.d even 2 1
4900.2.a.i 1 5.b even 2 1
4900.2.a.q 1 35.c odd 2 1
4900.2.e.m 2 5.c odd 4 2
4900.2.e.n 2 35.f even 4 2
8820.2.a.a 1 21.c even 2 1
8820.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\):

\( T_{3} - 1 \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -6 + T \)
$13$ \( -2 + T \)
$17$ \( 6 + T \)
$19$ \( -8 + T \)
$23$ \( -3 + T \)
$29$ \( -3 + T \)
$31$ \( -2 + T \)
$37$ \( -8 + T \)
$41$ \( 3 + T \)
$43$ \( -5 + T \)
$47$ \( T \)
$53$ \( -12 + T \)
$59$ \( T \)
$61$ \( 1 + T \)
$67$ \( 7 + T \)
$71$ \( T \)
$73$ \( 10 + T \)
$79$ \( 4 + T \)
$83$ \( -3 + T \)
$89$ \( 3 + T \)
$97$ \( 10 + T \)
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