Properties

Label 2548.2.g.a
Level 25482548
Weight 22
Character orbit 2548.g
Analytic conductor 20.34620.346
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(2157,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 2548=227213 2548 = 2^{2} \cdot 7^{2} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2548.g (of order 22, degree 11, 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: 20.345882435020.3458824350
Analytic rank: 00
Dimension: 44
Coefficient field: 4.0.65712.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+7x2+3 x^{4} + 7x^{2} + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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 basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qq3β1q52q9β1q11β2q13+β1q15+(β3+β2+1)q17+(β3β2β1)q19+3q23+(β3β22)q25++2β1q99+O(q100) q - q^{3} - \beta_1 q^{5} - 2 q^{9} - \beta_1 q^{11} - \beta_{2} q^{13} + \beta_1 q^{15} + ( - \beta_{3} + \beta_{2} + 1) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} + 3 q^{23} + (\beta_{3} - \beta_{2} - 2) q^{25}+ \cdots + 2 \beta_1 q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q38q92q13+8q17+12q2312q25+20q27+2q3912q438q51+16q5332q55+2q6512q69+12q7516q79+4q8128q95+O(q100) 4 q - 4 q^{3} - 8 q^{9} - 2 q^{13} + 8 q^{17} + 12 q^{23} - 12 q^{25} + 20 q^{27} + 2 q^{39} - 12 q^{43} - 8 q^{51} + 16 q^{53} - 32 q^{55} + 2 q^{65} - 12 q^{69} + 12 q^{75} - 16 q^{79} + 4 q^{81} - 28 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+7x2+3 x^{4} + 7x^{2} + 3 : Copy content Toggle raw display

β1\beta_{1}== ν3+6ν \nu^{3} + 6\nu Copy content Toggle raw display
β2\beta_{2}== ν2+ν+4 \nu^{2} + \nu + 4 Copy content Toggle raw display
β3\beta_{3}== ν2+ν4 -\nu^{2} + \nu - 4 Copy content Toggle raw display
ν\nu== (β3+β2)/2 ( \beta_{3} + \beta_{2} ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β3+β28)/2 ( -\beta_{3} + \beta_{2} - 8 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== 3β33β2+β1 -3\beta_{3} - 3\beta_{2} + \beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/2548Z)×\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times.

nn 197197 885885 12751275
χ(n)\chi(n) 1-1 11 11

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}
2157.1
0.677214i
2.55761i
2.55761i
0.677214i
0 −1.00000 0 3.75270i 0 0 0 −2.00000 0
2157.2 0 −1.00000 0 1.38464i 0 0 0 −2.00000 0
2157.3 0 −1.00000 0 1.38464i 0 0 0 −2.00000 0
2157.4 0 −1.00000 0 3.75270i 0 0 0 −2.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.g.a 4
7.b odd 2 1 2548.2.g.d 4
7.c even 3 2 364.2.y.a 8
7.d odd 6 2 2548.2.y.a 8
13.b even 2 1 inner 2548.2.g.a 4
21.h odd 6 2 3276.2.gv.d 8
91.b odd 2 1 2548.2.g.d 4
91.r even 6 2 364.2.y.a 8
91.s odd 6 2 2548.2.y.a 8
273.w odd 6 2 3276.2.gv.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.y.a 8 7.c even 3 2
364.2.y.a 8 91.r even 6 2
2548.2.g.a 4 1.a even 1 1 trivial
2548.2.g.a 4 13.b even 2 1 inner
2548.2.g.d 4 7.b odd 2 1
2548.2.g.d 4 91.b odd 2 1
2548.2.y.a 8 7.d odd 6 2
2548.2.y.a 8 91.s odd 6 2
3276.2.gv.d 8 21.h odd 6 2
3276.2.gv.d 8 273.w odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3+1 T_{3} + 1 acting on S2new(2548,[χ])S_{2}^{\mathrm{new}}(2548, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
55 T4+16T2+27 T^{4} + 16T^{2} + 27 Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T4+16T2+27 T^{4} + 16T^{2} + 27 Copy content Toggle raw display
1313 T4+2T3++169 T^{4} + 2 T^{3} + \cdots + 169 Copy content Toggle raw display
1717 (T24T33)2 (T^{2} - 4 T - 33)^{2} Copy content Toggle raw display
1919 T4+40T2+363 T^{4} + 40T^{2} + 363 Copy content Toggle raw display
2323 (T3)4 (T - 3)^{4} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 T4+40T2+363 T^{4} + 40T^{2} + 363 Copy content Toggle raw display
3737 T4+48T2+243 T^{4} + 48T^{2} + 243 Copy content Toggle raw display
4141 T4+64T2+432 T^{4} + 64T^{2} + 432 Copy content Toggle raw display
4343 (T2+6T28)2 (T^{2} + 6 T - 28)^{2} Copy content Toggle raw display
4747 T4+16T2+27 T^{4} + 16T^{2} + 27 Copy content Toggle raw display
5353 (T28T21)2 (T^{2} - 8 T - 21)^{2} Copy content Toggle raw display
5959 T4+144T2+2187 T^{4} + 144T^{2} + 2187 Copy content Toggle raw display
6161 (T237)2 (T^{2} - 37)^{2} Copy content Toggle raw display
6767 T4+120T2+3267 T^{4} + 120T^{2} + 3267 Copy content Toggle raw display
7171 T4+252T2+3888 T^{4} + 252T^{2} + 3888 Copy content Toggle raw display
7373 T4+280T2+15123 T^{4} + 280 T^{2} + 15123 Copy content Toggle raw display
7979 (T2+8T21)2 (T^{2} + 8 T - 21)^{2} Copy content Toggle raw display
8383 T4+252T2+3888 T^{4} + 252T^{2} + 3888 Copy content Toggle raw display
8989 T4+144T2+2187 T^{4} + 144T^{2} + 2187 Copy content Toggle raw display
9797 T4+292T2+21168 T^{4} + 292 T^{2} + 21168 Copy content Toggle raw display
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