Properties

Label 980.1.p.a
Level $980$
Weight $1$
Character orbit 980.p
Analytic conductor $0.489$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 0 1.00000 0 0.500000 + 0.866025i
459.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 1.00000 0 0.500000 0.866025i
\(n\): 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 \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
140.p odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\).

Hecke characteristic polynomials

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