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Properties

Label 980.1.p.c
Level 980980
Weight 11
Character orbit 980.p
Analytic conductor 0.4890.489
Analytic rank 00
Dimension 88
Projective image D4D_{4}
CM discriminant -4
Inner twists 1616

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,1,Mod(79,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 980=22⋅5⋅72 980 = 2^{2} \cdot 5 \cdot 7^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 980.p (of order 66, degree 22, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 0.4890837123800.489083712380
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ24)\Q(\zeta_{24})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x4+1 x^{8} - x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D4D_{4}
Projective field: Galois closure of 4.2.137200.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == q−ζ2410q2−ζ248q4−ζ247q5−ζ246q8+ζ244q9−ζ245q10+(ζ249+ζ243)q13−ζ244q16+⋯+(−ζ249−ζ243)q97+O(q100) q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} - \zeta_{24}^{5} q^{10} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{13} - \zeta_{24}^{4} q^{16} + \cdots + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{97} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+4q4+4q9−4q16+8q36−8q50−8q64+4q65−8q74−4q81−8q85+O(q100) 8 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 8 q^{36} - 8 q^{50} - 8 q^{64} + 4 q^{65} - 8 q^{74} - 4 q^{81} - 8 q^{85}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 101101 197197 491491
χ(n)\chi(n) −ζ244-\zeta_{24}^{4} −1-1 −1-1

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
79.1
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.965926 + 0.258819i 0 0 − 1.00000i 0.500000 − 0.866025i 0.965926 + 0.258819i
79.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.965926 − 0.258819i 0 0 − 1.00000i 0.500000 − 0.866025i −0.965926 − 0.258819i
79.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.258819 − 0.965926i 0 0 1.00000i 0.500000 − 0.866025i 0.258819 − 0.965926i
79.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.258819 + 0.965926i 0 0 1.00000i 0.500000 − 0.866025i −0.258819 + 0.965926i
459.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.965926 − 0.258819i 0 0 1.00000i 0.500000 + 0.866025i 0.965926 − 0.258819i
459.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.965926 + 0.258819i 0 0 1.00000i 0.500000 + 0.866025i −0.965926 + 0.258819i
459.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.258819 + 0.965926i 0 0 − 1.00000i 0.500000 + 0.866025i 0.258819 + 0.965926i
459.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.258819 − 0.965926i 0 0 − 1.00000i 0.500000 + 0.866025i −0.258819 − 0.965926i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by Q(−1)\Q(\sqrt{-1})
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
140.c even 2 1 inner
140.p odd 6 1 inner
140.s even 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.c 8
4.b odd 2 1 CM 980.1.p.c 8
5.b even 2 1 inner 980.1.p.c 8
7.b odd 2 1 inner 980.1.p.c 8
7.c even 3 1 980.1.f.e ✓ 4
7.c even 3 1 inner 980.1.p.c 8
7.d odd 6 1 980.1.f.e ✓ 4
7.d odd 6 1 inner 980.1.p.c 8
20.d odd 2 1 inner 980.1.p.c 8
28.d even 2 1 inner 980.1.p.c 8
28.f even 6 1 980.1.f.e ✓ 4
28.f even 6 1 inner 980.1.p.c 8
28.g odd 6 1 980.1.f.e ✓ 4
28.g odd 6 1 inner 980.1.p.c 8
35.c odd 2 1 inner 980.1.p.c 8
35.i odd 6 1 980.1.f.e ✓ 4
35.i odd 6 1 inner 980.1.p.c 8
35.j even 6 1 980.1.f.e ✓ 4
35.j even 6 1 inner 980.1.p.c 8
140.c even 2 1 inner 980.1.p.c 8
140.p odd 6 1 980.1.f.e ✓ 4
140.p odd 6 1 inner 980.1.p.c 8
140.s even 6 1 980.1.f.e ✓ 4
140.s even 6 1 inner 980.1.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.f.e ✓ 4 7.c even 3 1
980.1.f.e ✓ 4 7.d odd 6 1
980.1.f.e ✓ 4 28.f even 6 1
980.1.f.e ✓ 4 28.g odd 6 1
980.1.f.e ✓ 4 35.i odd 6 1
980.1.f.e ✓ 4 35.j even 6 1
980.1.f.e ✓ 4 140.p odd 6 1
980.1.f.e ✓ 4 140.s even 6 1
980.1.p.c 8 1.a even 1 1 trivial
980.1.p.c 8 4.b odd 2 1 CM
980.1.p.c 8 5.b even 2 1 inner
980.1.p.c 8 7.b odd 2 1 inner
980.1.p.c 8 7.c even 3 1 inner
980.1.p.c 8 7.d odd 6 1 inner
980.1.p.c 8 20.d odd 2 1 inner
980.1.p.c 8 28.d even 2 1 inner
980.1.p.c 8 28.f even 6 1 inner
980.1.p.c 8 28.g odd 6 1 inner
980.1.p.c 8 35.c odd 2 1 inner
980.1.p.c 8 35.i odd 6 1 inner
980.1.p.c 8 35.j even 6 1 inner
980.1.p.c 8 140.c even 2 1 inner
980.1.p.c 8 140.p odd 6 1 inner
980.1.p.c 8 140.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3 T_{3} acting on S1new(980,[χ])S_{1}^{\mathrm{new}}(980, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8−T4+1 T^{8} - T^{4} + 1 Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 (T2+2)4 (T^{2} + 2)^{4} Copy content Toggle raw display
1717 (T4−2T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
1919 T8 T^{8} Copy content Toggle raw display
2323 T8 T^{8} Copy content Toggle raw display
2929 T8 T^{8} Copy content Toggle raw display
3131 T8 T^{8} Copy content Toggle raw display
3737 (T4−4T2+16)2 (T^{4} - 4 T^{2} + 16)^{2} Copy content Toggle raw display
4141 (T2−2)4 (T^{2} - 2)^{4} Copy content Toggle raw display
4343 T8 T^{8} Copy content Toggle raw display
4747 T8 T^{8} Copy content Toggle raw display
5353 T8 T^{8} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
6767 T8 T^{8} Copy content Toggle raw display
7171 T8 T^{8} Copy content Toggle raw display
7373 (T4−2T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
7979 T8 T^{8} Copy content Toggle raw display
8383 T8 T^{8} Copy content Toggle raw display
8989 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
9797 (T2+2)4 (T^{2} + 2)^{4} Copy content Toggle raw display
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