Properties

Label 2548.2.j.q
Level 25482548
Weight 22
Character orbit 2548.j
Analytic conductor 20.34620.346
Analytic rank 00
Dimension 88
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(1145,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.1145");
 
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.j (of order 33, 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: 20.345882435020.3458824350
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+11x6+36x4+32x2+4 x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 32 3^{2}
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β4+β1)q3β6q5+(β62β4+2β2+3)q9+(β7β6β5+β1)q11+q13+(2β5β32)q15++(β54β33β2+1)q99+O(q100) q + ( - \beta_{4} + \beta_1) q^{3} - \beta_{6} q^{5} + (\beta_{6} - 2 \beta_{4} + 2 \beta_{2} + \cdots - 3) q^{9} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{11} + q^{13} + (2 \beta_{5} - \beta_{3} - 2) q^{15}+ \cdots + (\beta_{5} - 4 \beta_{3} - 3 \beta_{2} + 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+3q3q59q94q11+8q1318q1511q235q2554q27+22q295q3115q3327q37+3q3912q41+8q43+6q45+4q47+6q99+O(q100) 8 q + 3 q^{3} - q^{5} - 9 q^{9} - 4 q^{11} + 8 q^{13} - 18 q^{15} - 11 q^{23} - 5 q^{25} - 54 q^{27} + 22 q^{29} - 5 q^{31} - 15 q^{33} - 27 q^{37} + 3 q^{39} - 12 q^{41} + 8 q^{43} + 6 q^{45} + 4 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+11x6+36x4+32x2+4 x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== (ν5+7ν3+10ν+2)/4 ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 Copy content Toggle raw display
β2\beta_{2}== ν2+3 \nu^{2} + 3 Copy content Toggle raw display
β3\beta_{3}== ν46ν24 -\nu^{4} - 6\nu^{2} - 4 Copy content Toggle raw display
β4\beta_{4}== (ν7+10ν5+31ν3+2ν2+30ν+6)/4 ( \nu^{7} + 10\nu^{5} + 31\nu^{3} + 2\nu^{2} + 30\nu + 6 ) / 4 Copy content Toggle raw display
β5\beta_{5}== (ν69ν422ν210)/2 ( -\nu^{6} - 9\nu^{4} - 22\nu^{2} - 10 ) / 2 Copy content Toggle raw display
β6\beta_{6}== (ν7+10ν5ν4+28ν36ν2+18ν4)/2 ( \nu^{7} + 10\nu^{5} - \nu^{4} + 28\nu^{3} - 6\nu^{2} + 18\nu - 4 ) / 2 Copy content Toggle raw display
β7\beta_{7}== (2ν7ν6+21ν59ν4+63ν322ν2+40ν10)/4 ( 2\nu^{7} - \nu^{6} + 21\nu^{5} - 9\nu^{4} + 63\nu^{3} - 22\nu^{2} + 40\nu - 10 ) / 4 Copy content Toggle raw display
ν\nu== (2β7+2β6+β5β3+2β11)/3 ( -2\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{3} + 2\beta _1 - 1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== β23 \beta_{2} - 3 Copy content Toggle raw display
ν3\nu^{3}== (8β710β64β5+4β4+5β32β28β1+4)/3 ( 8\beta_{7} - 10\beta_{6} - 4\beta_{5} + 4\beta_{4} + 5\beta_{3} - 2\beta_{2} - 8\beta _1 + 4 ) / 3 Copy content Toggle raw display
ν4\nu^{4}== β36β2+14 -\beta_{3} - 6\beta_{2} + 14 Copy content Toggle raw display
ν5\nu^{5}== (36β7+50β6+18β528β425β3+14β2+48β124)/3 ( -36\beta_{7} + 50\beta_{6} + 18\beta_{5} - 28\beta_{4} - 25\beta_{3} + 14\beta_{2} + 48\beta _1 - 24 ) / 3 Copy content Toggle raw display
ν6\nu^{6}== 2β5+9β3+32β270 -2\beta_{5} + 9\beta_{3} + 32\beta_{2} - 70 Copy content Toggle raw display
ν7\nu^{7}== (172β7250β686β5+168β4+125β384β2292β1+146)/3 ( 172\beta_{7} - 250\beta_{6} - 86\beta_{5} + 168\beta_{4} + 125\beta_{3} - 84\beta_{2} - 292\beta _1 + 146 ) / 3 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) 11 1+β1-1 + \beta_{1} 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}
1145.1
0.385731i
1.07834i
2.06288i
2.33086i
0.385731i
1.07834i
2.06288i
2.33086i
0 −0.925606 + 1.60320i 0 1.56470 + 2.71015i 0 0 0 −0.213493 0.369780i 0
1145.2 0 −0.418594 + 0.725026i 0 −0.812371 1.40707i 0 0 0 1.14956 + 1.99109i 0
1145.3 0 1.12774 1.95330i 0 −1.71189 2.96508i 0 0 0 −1.04359 1.80755i 0
1145.4 0 1.71646 2.97300i 0 0.459555 + 0.795973i 0 0 0 −4.39248 7.60799i 0
1353.1 0 −0.925606 1.60320i 0 1.56470 2.71015i 0 0 0 −0.213493 + 0.369780i 0
1353.2 0 −0.418594 0.725026i 0 −0.812371 + 1.40707i 0 0 0 1.14956 1.99109i 0
1353.3 0 1.12774 + 1.95330i 0 −1.71189 + 2.96508i 0 0 0 −1.04359 + 1.80755i 0
1353.4 0 1.71646 + 2.97300i 0 0.459555 0.795973i 0 0 0 −4.39248 + 7.60799i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.j.q 8
7.b odd 2 1 364.2.j.e 8
7.c even 3 1 2548.2.a.p 4
7.c even 3 1 inner 2548.2.j.q 8
7.d odd 6 1 364.2.j.e 8
7.d odd 6 1 2548.2.a.q 4
21.c even 2 1 3276.2.r.j 8
21.g even 6 1 3276.2.r.j 8
28.d even 2 1 1456.2.r.o 8
28.f even 6 1 1456.2.r.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.j.e 8 7.b odd 2 1
364.2.j.e 8 7.d odd 6 1
1456.2.r.o 8 28.d even 2 1
1456.2.r.o 8 28.f even 6 1
2548.2.a.p 4 7.c even 3 1
2548.2.a.q 4 7.d odd 6 1
2548.2.j.q 8 1.a even 1 1 trivial
2548.2.j.q 8 7.c even 3 1 inner
3276.2.r.j 8 21.c even 2 1
3276.2.r.j 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(2548,[χ])S_{2}^{\mathrm{new}}(2548, [\chi]):

T383T37+15T366T35+60T34+216T32+144T3+144 T_{3}^{8} - 3T_{3}^{7} + 15T_{3}^{6} - 6T_{3}^{5} + 60T_{3}^{4} + 216T_{3}^{2} + 144T_{3} + 144 Copy content Toggle raw display
T58+T57+13T56+4T55+136T54+64T53+256T52128T5+256 T_{5}^{8} + T_{5}^{7} + 13T_{5}^{6} + 4T_{5}^{5} + 136T_{5}^{4} + 64T_{5}^{3} + 256T_{5}^{2} - 128T_{5} + 256 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T83T7++144 T^{8} - 3 T^{7} + \cdots + 144 Copy content Toggle raw display
55 T8+T7++256 T^{8} + T^{7} + \cdots + 256 Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8+4T7++10201 T^{8} + 4 T^{7} + \cdots + 10201 Copy content Toggle raw display
1313 (T1)8 (T - 1)^{8} Copy content Toggle raw display
1717 T8+42T6++8649 T^{8} + 42 T^{6} + \cdots + 8649 Copy content Toggle raw display
1919 T8+12T6++441 T^{8} + 12 T^{6} + \cdots + 441 Copy content Toggle raw display
2323 T8+11T7++16 T^{8} + 11 T^{7} + \cdots + 16 Copy content Toggle raw display
2929 (T411T3+746)2 (T^{4} - 11 T^{3} + \cdots - 746)^{2} Copy content Toggle raw display
3131 T8+5T7++71824 T^{8} + 5 T^{7} + \cdots + 71824 Copy content Toggle raw display
3737 T8+27T7++2782224 T^{8} + 27 T^{7} + \cdots + 2782224 Copy content Toggle raw display
4141 (T4+6T336T24)2 (T^{4} + 6 T^{3} - 36 T - 24)^{2} Copy content Toggle raw display
4343 (T44T324T2++16)2 (T^{4} - 4 T^{3} - 24 T^{2} + \cdots + 16)^{2} Copy content Toggle raw display
4747 T84T7++337561 T^{8} - 4 T^{7} + \cdots + 337561 Copy content Toggle raw display
5353 T8+18T7++1580049 T^{8} + 18 T^{7} + \cdots + 1580049 Copy content Toggle raw display
5959 T8+16T7++6436369 T^{8} + 16 T^{7} + \cdots + 6436369 Copy content Toggle raw display
6161 T810T7++528529 T^{8} - 10 T^{7} + \cdots + 528529 Copy content Toggle raw display
6767 T82T7++2809 T^{8} - 2 T^{7} + \cdots + 2809 Copy content Toggle raw display
7171 (T4+5T3++12208)2 (T^{4} + 5 T^{3} + \cdots + 12208)^{2} Copy content Toggle raw display
7373 T8+7T7++15178816 T^{8} + 7 T^{7} + \cdots + 15178816 Copy content Toggle raw display
7979 T8+17T7++817216 T^{8} + 17 T^{7} + \cdots + 817216 Copy content Toggle raw display
8383 (T46T3++1296)2 (T^{4} - 6 T^{3} + \cdots + 1296)^{2} Copy content Toggle raw display
8989 T8+21T7++747913104 T^{8} + 21 T^{7} + \cdots + 747913104 Copy content Toggle raw display
9797 (T4+8T3+248)2 (T^{4} + 8 T^{3} + \cdots - 248)^{2} Copy content Toggle raw display
show more
show less