Properties

Label 975.1.j.a
Level 975975
Weight 11
Character orbit 975.j
Analytic conductor 0.4870.487
Analytic rank 00
Dimension 44
Projective image D4D_{4}
CM discriminant -3
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,1,Mod(593,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.593");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 975=35213 975 = 3 \cdot 5^{2} \cdot 13
Weight: k k == 1 1
Character orbit: [χ][\chi] == 975.j (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: 0.4865883873170.486588387317
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,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D4D_{4}
Projective field: Galois closure of 4.2.164775.1

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) == qζ8q3+q4+(ζ83+ζ8)q7+ζ82q9ζ8q12+ζ8q13+q16+(ζ821)q19+(ζ82+1)q21ζ83q27++(ζ83ζ8)q97+O(q100) q - \zeta_{8} q^{3} + q^{4} + (\zeta_{8}^{3} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{9} - \zeta_{8} q^{12} + \zeta_{8} q^{13} + q^{16} + (\zeta_{8}^{2} - 1) q^{19} + ( - \zeta_{8}^{2} + 1) q^{21} - \zeta_{8}^{3} q^{27} + \cdots + (\zeta_{8}^{3} - \zeta_{8}) q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q4+4q164q19+4q214q314q49+4q644q764q81+4q844q91+O(q100) 4 q + 4 q^{4} + 4 q^{16} - 4 q^{19} + 4 q^{21} - 4 q^{31} - 4 q^{49} + 4 q^{64} - 4 q^{76} - 4 q^{81} + 4 q^{84} - 4 q^{91}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/975Z)×\left(\mathbb{Z}/975\mathbb{Z}\right)^\times.

nn 301301 326326 352352
χ(n)\chi(n) ζ82\zeta_{8}^{2} 1-1 ζ82-\zeta_{8}^{2}

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}
593.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 1.00000 0 0 1.41421i 0 1.00000i 0
593.2 0 0.707107 + 0.707107i 1.00000 0 0 1.41421i 0 1.00000i 0
707.1 0 −0.707107 + 0.707107i 1.00000 0 0 1.41421i 0 1.00000i 0
707.2 0 0.707107 0.707107i 1.00000 0 0 1.41421i 0 1.00000i 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
3.b odd 2 1 CM by Q(3)\Q(\sqrt{-3})
5.b even 2 1 inner
15.d odd 2 1 inner
65.f even 4 1 inner
65.k even 4 1 inner
195.j odd 4 1 inner
195.u odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.j.a 4
3.b odd 2 1 CM 975.1.j.a 4
5.b even 2 1 inner 975.1.j.a 4
5.c odd 4 2 975.1.u.a yes 4
13.d odd 4 1 975.1.u.a yes 4
15.d odd 2 1 inner 975.1.j.a 4
15.e even 4 2 975.1.u.a yes 4
39.f even 4 1 975.1.u.a yes 4
65.f even 4 1 inner 975.1.j.a 4
65.g odd 4 1 975.1.u.a yes 4
65.k even 4 1 inner 975.1.j.a 4
195.j odd 4 1 inner 975.1.j.a 4
195.n even 4 1 975.1.u.a yes 4
195.u odd 4 1 inner 975.1.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.j.a 4 1.a even 1 1 trivial
975.1.j.a 4 3.b odd 2 1 CM
975.1.j.a 4 5.b even 2 1 inner
975.1.j.a 4 15.d odd 2 1 inner
975.1.j.a 4 65.f even 4 1 inner
975.1.j.a 4 65.k even 4 1 inner
975.1.j.a 4 195.j odd 4 1 inner
975.1.j.a 4 195.u odd 4 1 inner
975.1.u.a yes 4 5.c odd 4 2
975.1.u.a yes 4 13.d odd 4 1
975.1.u.a yes 4 15.e even 4 2
975.1.u.a yes 4 39.f even 4 1
975.1.u.a yes 4 65.g odd 4 1
975.1.u.a yes 4 195.n even 4 1

Hecke kernels

This newform subspace is the entire newspace S1new(975,[χ])S_{1}^{\mathrm{new}}(975, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4+1 T^{4} + 1 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 T4+1 T^{4} + 1 Copy content Toggle raw display
1717 T4 T^{4} Copy content Toggle raw display
1919 (T2+2T+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 (T2+2T+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
3737 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
4141 T4 T^{4} Copy content Toggle raw display
4343 T4+16 T^{4} + 16 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 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 (T22)2 (T^{2} - 2)^{2} 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 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
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