Properties

Label 980.2.a.h
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 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{5} + q^{9} - 2q^{13} + 2q^{15} + 6q^{17} + 4q^{19} + 6q^{23} + q^{25} - 4q^{27} + 6q^{29} + 4q^{31} + 2q^{37} - 4q^{39} - 6q^{41} - 10q^{43} + q^{45} + 6q^{47} + 12q^{51} - 6q^{53} + 8q^{57} - 12q^{59} - 2q^{61} - 2q^{65} + 2q^{67} + 12q^{69} - 12q^{71} - 2q^{73} + 2q^{75} + 8q^{79} - 11q^{81} - 6q^{83} + 6q^{85} + 12q^{87} + 6q^{89} + 8q^{93} + 4q^{95} - 2q^{97} + O(q^{100}) \)

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 2.00000 0 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.h 1
3.b odd 2 1 8820.2.a.g 1
4.b odd 2 1 3920.2.a.h 1
5.b even 2 1 4900.2.a.e 1
5.c odd 4 2 4900.2.e.f 2
7.b odd 2 1 20.2.a.a 1
7.c even 3 2 980.2.i.c 2
7.d odd 6 2 980.2.i.i 2
21.c even 2 1 180.2.a.a 1
28.d even 2 1 80.2.a.b 1
35.c odd 2 1 100.2.a.a 1
35.f even 4 2 100.2.c.a 2
56.e even 2 1 320.2.a.a 1
56.h odd 2 1 320.2.a.f 1
63.l odd 6 2 1620.2.i.h 2
63.o even 6 2 1620.2.i.b 2
77.b even 2 1 2420.2.a.a 1
84.h odd 2 1 720.2.a.h 1
91.b odd 2 1 3380.2.a.c 1
91.i even 4 2 3380.2.f.b 2
105.g even 2 1 900.2.a.b 1
105.k odd 4 2 900.2.d.c 2
112.j even 4 2 1280.2.d.g 2
112.l odd 4 2 1280.2.d.c 2
119.d odd 2 1 5780.2.a.f 1
119.f odd 4 2 5780.2.c.a 2
133.c even 2 1 7220.2.a.f 1
140.c even 2 1 400.2.a.c 1
140.j odd 4 2 400.2.c.b 2
168.e odd 2 1 2880.2.a.f 1
168.i even 2 1 2880.2.a.m 1
280.c odd 2 1 1600.2.a.c 1
280.n even 2 1 1600.2.a.w 1
280.s even 4 2 1600.2.c.d 2
280.y odd 4 2 1600.2.c.e 2
308.g odd 2 1 9680.2.a.ba 1
420.o odd 2 1 3600.2.a.be 1
420.w even 4 2 3600.2.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 7.b odd 2 1
80.2.a.b 1 28.d even 2 1
100.2.a.a 1 35.c odd 2 1
100.2.c.a 2 35.f even 4 2
180.2.a.a 1 21.c even 2 1
320.2.a.a 1 56.e even 2 1
320.2.a.f 1 56.h odd 2 1
400.2.a.c 1 140.c even 2 1
400.2.c.b 2 140.j odd 4 2
720.2.a.h 1 84.h odd 2 1
900.2.a.b 1 105.g even 2 1
900.2.d.c 2 105.k odd 4 2
980.2.a.h 1 1.a even 1 1 trivial
980.2.i.c 2 7.c even 3 2
980.2.i.i 2 7.d odd 6 2
1280.2.d.c 2 112.l odd 4 2
1280.2.d.g 2 112.j even 4 2
1600.2.a.c 1 280.c odd 2 1
1600.2.a.w 1 280.n even 2 1
1600.2.c.d 2 280.s even 4 2
1600.2.c.e 2 280.y odd 4 2
1620.2.i.b 2 63.o even 6 2
1620.2.i.h 2 63.l odd 6 2
2420.2.a.a 1 77.b even 2 1
2880.2.a.f 1 168.e odd 2 1
2880.2.a.m 1 168.i even 2 1
3380.2.a.c 1 91.b odd 2 1
3380.2.f.b 2 91.i even 4 2
3600.2.a.be 1 420.o odd 2 1
3600.2.f.j 2 420.w even 4 2
3920.2.a.h 1 4.b odd 2 1
4900.2.a.e 1 5.b even 2 1
4900.2.e.f 2 5.c odd 4 2
5780.2.a.f 1 119.d odd 2 1
5780.2.c.a 2 119.f odd 4 2
7220.2.a.f 1 133.c even 2 1
8820.2.a.g 1 3.b odd 2 1
9680.2.a.ba 1 308.g odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\):

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

Hecke characteristic polynomials

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