Properties

Label 950.2.f.a
Level 950950
Weight 22
Character orbit 950.f
Analytic conductor 7.5867.586
Analytic rank 00
Dimension 44
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(493,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 950=25219 950 = 2 \cdot 5^{2} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 950.f (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: 7.585788192027.58578819202
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: no (minimal twist has level 190)
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ8\zeta_{8}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+ζ83q2ζ82q4+(ζ82+1)q7+ζ8q83ζ82q9+2q116ζ8q13+(ζ83ζ8)q14q16+(3ζ823)q17+6ζ82q99+O(q100) q + \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + (\zeta_{8}^{2} + 1) q^{7} + \zeta_{8} q^{8} - 3 \zeta_{8}^{2} q^{9} + 2 q^{11} - 6 \zeta_{8} q^{13} + (\zeta_{8}^{3} - \zeta_{8}) q^{14} - q^{16} + ( - 3 \zeta_{8}^{2} - 3) q^{17} + \cdots - 6 \zeta_{8}^{2} q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q7+8q114q1612q174q23+24q26+4q2812q36+12q38+20q4328q47+32q6124q62+12q6312q68+4q734q76+8q77++4q92+O(q100) 4 q + 4 q^{7} + 8 q^{11} - 4 q^{16} - 12 q^{17} - 4 q^{23} + 24 q^{26} + 4 q^{28} - 12 q^{36} + 12 q^{38} + 20 q^{43} - 28 q^{47} + 32 q^{61} - 24 q^{62} + 12 q^{63} - 12 q^{68} + 4 q^{73} - 4 q^{76} + 8 q^{77}+ \cdots + 4 q^{92}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/950Z)×\left(\mathbb{Z}/950\mathbb{Z}\right)^\times.

nn 7777 401401
χ(n)\chi(n) 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}
493.1
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 1.00000i 0.707107 0.707107i 3.00000i 0
493.2 0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 1.00000i −0.707107 + 0.707107i 3.00000i 0
607.1 −0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i 0.707107 + 0.707107i 3.00000i 0
607.2 0.707107 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i −0.707107 0.707107i 3.00000i 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
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.a 4
5.b even 2 1 190.2.f.a 4
5.c odd 4 1 190.2.f.a 4
5.c odd 4 1 inner 950.2.f.a 4
15.d odd 2 1 1710.2.p.a 4
15.e even 4 1 1710.2.p.a 4
19.b odd 2 1 inner 950.2.f.a 4
95.d odd 2 1 190.2.f.a 4
95.g even 4 1 190.2.f.a 4
95.g even 4 1 inner 950.2.f.a 4
285.b even 2 1 1710.2.p.a 4
285.j odd 4 1 1710.2.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.f.a 4 5.b even 2 1
190.2.f.a 4 5.c odd 4 1
190.2.f.a 4 95.d odd 2 1
190.2.f.a 4 95.g even 4 1
950.2.f.a 4 1.a even 1 1 trivial
950.2.f.a 4 5.c odd 4 1 inner
950.2.f.a 4 19.b odd 2 1 inner
950.2.f.a 4 95.g even 4 1 inner
1710.2.p.a 4 15.d odd 2 1
1710.2.p.a 4 15.e even 4 1
1710.2.p.a 4 285.b even 2 1
1710.2.p.a 4 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3 T_{3} acting on S2new(950,[χ])S_{2}^{\mathrm{new}}(950, [\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 (T22T+2)2 (T^{2} - 2 T + 2)^{2} Copy content Toggle raw display
1111 (T2)4 (T - 2)^{4} Copy content Toggle raw display
1313 T4+1296 T^{4} + 1296 Copy content Toggle raw display
1717 (T2+6T+18)2 (T^{2} + 6 T + 18)^{2} Copy content Toggle raw display
1919 T434T2+361 T^{4} - 34T^{2} + 361 Copy content Toggle raw display
2323 (T2+2T+2)2 (T^{2} + 2 T + 2)^{2} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 (T2+72)2 (T^{2} + 72)^{2} Copy content Toggle raw display
3737 T4+1296 T^{4} + 1296 Copy content Toggle raw display
4141 (T2+72)2 (T^{2} + 72)^{2} Copy content Toggle raw display
4343 (T210T+50)2 (T^{2} - 10 T + 50)^{2} Copy content Toggle raw display
4747 (T2+14T+98)2 (T^{2} + 14 T + 98)^{2} Copy content Toggle raw display
5353 T4+1296 T^{4} + 1296 Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 (T8)4 (T - 8)^{4} Copy content Toggle raw display
6767 T4+20736 T^{4} + 20736 Copy content Toggle raw display
7171 (T2+72)2 (T^{2} + 72)^{2} Copy content Toggle raw display
7373 (T22T+2)2 (T^{2} - 2 T + 2)^{2} Copy content Toggle raw display
7979 (T272)2 (T^{2} - 72)^{2} Copy content Toggle raw display
8383 (T2+18T+162)2 (T^{2} + 18 T + 162)^{2} Copy content Toggle raw display
8989 (T272)2 (T^{2} - 72)^{2} Copy content Toggle raw display
9797 T4+1296 T^{4} + 1296 Copy content Toggle raw display
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