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Properties

Label 1764.1.y.c
Level 17641764
Weight 11
Character orbit 1764.y
Analytic conductor 0.8800.880
Analytic rank 00
Dimension 44
Projective image D2D_{2}
CM/RM discs -7, -84, 12
Inner twists 1616

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,1,Mod(667,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.667");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1764=22⋅32⋅72 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1764.y (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: 0.8803506822850.880350682285
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4−x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D2D_{2}
Projective field: Galois closure of Q(3,−7)\Q(\sqrt{3}, \sqrt{-7})
Artin image: D4:C6D_4:C_6
Artin field: Galois closure of Q[x]/(x24−⋯ )\mathbb{Q}[x]/(x^{24} - \cdots)

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−ζ12q2+ζ122q4−ζ123q8+2ζ125q11+ζ124q16+2q22+2ζ12q23+ζ122q25−ζ125q32+2ζ124q37+⋯+2ζ123q92+O(q100) q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{5} q^{11} + \zeta_{12}^{4} q^{16} + 2 q^{22} + 2 \zeta_{12} q^{23} + \zeta_{12}^{2} q^{25} - \zeta_{12}^{5} q^{32} + 2 \zeta_{12}^{4} q^{37} + \cdots + 2 \zeta_{12}^{3} q^{92} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q+2q4−2q16+8q22+2q25−4q37−4q46−4q64+4q88+O(q100) 4 q + 2 q^{4} - 2 q^{16} + 8 q^{22} + 2 q^{25} - 4 q^{37} - 4 q^{46} - 4 q^{64} + 4 q^{88}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1764Z)×\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times.

nn 785785 883883 10811081
χ(n)\chi(n) 11 −1-1 ζ124\zeta_{12}^{4}

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}
667.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 0 − 1.00000i 0 0
667.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0 1.00000i 0 0
1243.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 0 1.00000i 0 0
1243.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 0 − 1.00000i 0 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by Q(−7)\Q(\sqrt{-7})
12.b even 2 1 RM by Q(3)\Q(\sqrt{3})
84.h odd 2 1 CM by Q(−21)\Q(\sqrt{-21})
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.y.c 4
3.b odd 2 1 inner 1764.1.y.c 4
4.b odd 2 1 inner 1764.1.y.c 4
7.b odd 2 1 CM 1764.1.y.c 4
7.c even 3 1 1764.1.g.c ✓ 2
7.c even 3 1 inner 1764.1.y.c 4
7.d odd 6 1 1764.1.g.c ✓ 2
7.d odd 6 1 inner 1764.1.y.c 4
12.b even 2 1 RM 1764.1.y.c 4
21.c even 2 1 inner 1764.1.y.c 4
21.g even 6 1 1764.1.g.c ✓ 2
21.g even 6 1 inner 1764.1.y.c 4
21.h odd 6 1 1764.1.g.c ✓ 2
21.h odd 6 1 inner 1764.1.y.c 4
28.d even 2 1 inner 1764.1.y.c 4
28.f even 6 1 1764.1.g.c ✓ 2
28.f even 6 1 inner 1764.1.y.c 4
28.g odd 6 1 1764.1.g.c ✓ 2
28.g odd 6 1 inner 1764.1.y.c 4
84.h odd 2 1 CM 1764.1.y.c 4
84.j odd 6 1 1764.1.g.c ✓ 2
84.j odd 6 1 inner 1764.1.y.c 4
84.n even 6 1 1764.1.g.c ✓ 2
84.n even 6 1 inner 1764.1.y.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.c ✓ 2 7.c even 3 1
1764.1.g.c ✓ 2 7.d odd 6 1
1764.1.g.c ✓ 2 21.g even 6 1
1764.1.g.c ✓ 2 21.h odd 6 1
1764.1.g.c ✓ 2 28.f even 6 1
1764.1.g.c ✓ 2 28.g odd 6 1
1764.1.g.c ✓ 2 84.j odd 6 1
1764.1.g.c ✓ 2 84.n even 6 1
1764.1.y.c 4 1.a even 1 1 trivial
1764.1.y.c 4 3.b odd 2 1 inner
1764.1.y.c 4 4.b odd 2 1 inner
1764.1.y.c 4 7.b odd 2 1 CM
1764.1.y.c 4 7.c even 3 1 inner
1764.1.y.c 4 7.d odd 6 1 inner
1764.1.y.c 4 12.b even 2 1 RM
1764.1.y.c 4 21.c even 2 1 inner
1764.1.y.c 4 21.g even 6 1 inner
1764.1.y.c 4 21.h odd 6 1 inner
1764.1.y.c 4 28.d even 2 1 inner
1764.1.y.c 4 28.f even 6 1 inner
1764.1.y.c 4 28.g odd 6 1 inner
1764.1.y.c 4 84.h odd 2 1 CM
1764.1.y.c 4 84.j odd 6 1 inner
1764.1.y.c 4 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(1764,[χ])S_{1}^{\mathrm{new}}(1764, [\chi]):

T5 T_{5} Copy content Toggle raw display
T114−4T112+16 T_{11}^{4} - 4T_{11}^{2} + 16 Copy content Toggle raw display
T29 T_{29} Copy content Toggle raw display

Hecke characteristic polynomials

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