Properties

Label 1008.1.y.a
Level 10081008
Weight 11
Character orbit 1008.y
Analytic conductor 0.5030.503
Analytic rank 00
Dimension 44
Projective image D4D_{4}
CM discriminant -7
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,1,Mod(251,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 1008=24327 1008 = 2^{4} \cdot 3^{2} \cdot 7
Weight: k k == 1 1
Character orbit: [χ][\chi] == 1008.y (of order 44, degree 22, 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: 0.5030575327340.503057532734
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 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: D4D_{4}
Projective field: Galois closure of 4.2.2709504.2

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) == qζ8q2+ζ82q4q7ζ83q82ζ8q11+ζ8q14q16+2ζ82q22+(ζ83ζ8)q23ζ82q25+ζ8q98+O(q100) q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - q^{7} - \zeta_{8}^{3} q^{8} - 2 \zeta_{8} q^{11} + \zeta_{8} q^{14} - q^{16} + 2 \zeta_{8}^{2} q^{22} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23} - \zeta_{8}^{2} q^{25} + \cdots - \zeta_{8} q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q74q164q37+4q434q46+4q494q678q88+O(q100) 4 q - 4 q^{7} - 4 q^{16} - 4 q^{37} + 4 q^{43} - 4 q^{46} + 4 q^{49} - 4 q^{67} - 8 q^{88}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1008Z)×\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times.

nn 127127 577577 757757 785785
χ(n)\chi(n) 1-1 1-1 ζ82\zeta_{8}^{2} 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}
251.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 −1.00000 0.707107 0.707107i 0 0
251.2 0.707107 + 0.707107i 0 1.00000i 0 0 −1.00000 −0.707107 + 0.707107i 0 0
755.1 −0.707107 + 0.707107i 0 1.00000i 0 0 −1.00000 0.707107 + 0.707107i 0 0
755.2 0.707107 0.707107i 0 1.00000i 0 0 −1.00000 −0.707107 0.707107i 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})
3.b odd 2 1 inner
16.f odd 4 1 inner
21.c even 2 1 inner
48.k even 4 1 inner
112.j even 4 1 inner
336.v odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.1.y.a 4
3.b odd 2 1 inner 1008.1.y.a 4
7.b odd 2 1 CM 1008.1.y.a 4
16.f odd 4 1 inner 1008.1.y.a 4
21.c even 2 1 inner 1008.1.y.a 4
48.k even 4 1 inner 1008.1.y.a 4
112.j even 4 1 inner 1008.1.y.a 4
336.v odd 4 1 inner 1008.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.1.y.a 4 1.a even 1 1 trivial
1008.1.y.a 4 3.b odd 2 1 inner
1008.1.y.a 4 7.b odd 2 1 CM
1008.1.y.a 4 16.f odd 4 1 inner
1008.1.y.a 4 21.c even 2 1 inner
1008.1.y.a 4 48.k even 4 1 inner
1008.1.y.a 4 112.j even 4 1 inner
1008.1.y.a 4 336.v odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T114+16 T_{11}^{4} + 16 acting on S1new(1008,[χ])S_{1}^{\mathrm{new}}(1008, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+1 T^{4} + 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 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
1111 T4+16 T^{4} + 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 (T2+2)2 (T^{2} + 2)^{2} 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+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
4141 T4 T^{4} Copy content Toggle raw display
4343 (T22T+2)2 (T^{2} - 2 T + 2)^{2} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 T4+16 T^{4} + 16 Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 T4 T^{4} Copy content Toggle raw display
6767 (T2+2T+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
7171 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
7373 T4 T^{4} Copy content Toggle raw display
7979 (T2+4)2 (T^{2} + 4)^{2} 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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