Properties

Label 980.1.f.a
Level $980$
Weight $1$
Character orbit 980.f
Self dual yes
Analytic conductor $0.489$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -20
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.489083712380\)
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)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.980.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.26891200.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{10} - q^{12} + q^{15} + q^{16} - q^{20} + q^{23} + q^{24} + q^{25} + q^{27} - q^{29} - q^{30} - q^{32} + q^{40} + q^{41} + q^{43} - q^{46} + 2q^{47} - q^{48} - q^{50} - q^{54} + q^{58} + q^{60} + q^{61} + q^{64} + q^{67} - q^{69} - q^{75} - q^{80} - q^{81} - q^{82} - q^{83} - q^{86} + q^{87} + q^{89} + q^{92} - 2q^{94} + q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0
−1.00000 −1.00000 1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.f.a 1
4.b odd 2 1 980.1.f.d 1
5.b even 2 1 980.1.f.d 1
7.b odd 2 1 980.1.f.b 1
7.c even 3 2 980.1.p.b 2
7.d odd 6 2 140.1.p.b yes 2
20.d odd 2 1 CM 980.1.f.a 1
21.g even 6 2 1260.1.ci.a 2
28.d even 2 1 980.1.f.c 1
28.f even 6 2 140.1.p.a 2
28.g odd 6 2 980.1.p.a 2
35.c odd 2 1 980.1.f.c 1
35.i odd 6 2 140.1.p.a 2
35.j even 6 2 980.1.p.a 2
35.k even 12 4 700.1.u.a 4
56.j odd 6 2 2240.1.bt.b 2
56.m even 6 2 2240.1.bt.a 2
84.j odd 6 2 1260.1.ci.b 2
105.p even 6 2 1260.1.ci.b 2
140.c even 2 1 980.1.f.b 1
140.p odd 6 2 980.1.p.b 2
140.s even 6 2 140.1.p.b yes 2
140.x odd 12 4 700.1.u.a 4
280.ba even 6 2 2240.1.bt.b 2
280.bk odd 6 2 2240.1.bt.a 2
420.be odd 6 2 1260.1.ci.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 28.f even 6 2
140.1.p.a 2 35.i odd 6 2
140.1.p.b yes 2 7.d odd 6 2
140.1.p.b yes 2 140.s even 6 2
700.1.u.a 4 35.k even 12 4
700.1.u.a 4 140.x odd 12 4
980.1.f.a 1 1.a even 1 1 trivial
980.1.f.a 1 20.d odd 2 1 CM
980.1.f.b 1 7.b odd 2 1
980.1.f.b 1 140.c even 2 1
980.1.f.c 1 28.d even 2 1
980.1.f.c 1 35.c odd 2 1
980.1.f.d 1 4.b odd 2 1
980.1.f.d 1 5.b even 2 1
980.1.p.a 2 28.g odd 6 2
980.1.p.a 2 35.j even 6 2
980.1.p.b 2 7.c even 3 2
980.1.p.b 2 140.p odd 6 2
1260.1.ci.a 2 21.g even 6 2
1260.1.ci.a 2 420.be odd 6 2
1260.1.ci.b 2 84.j odd 6 2
1260.1.ci.b 2 105.p even 6 2
2240.1.bt.a 2 56.m even 6 2
2240.1.bt.a 2 280.bk odd 6 2
2240.1.bt.b 2 56.j odd 6 2
2240.1.bt.b 2 280.ba even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\):

\( T_{3} + 1 \)
\( T_{23} - 1 \)

Hecke characteristic polynomials

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