Properties

Label 245.2.j.c
Level 245245
Weight 22
Character orbit 245.j
Analytic conductor 1.9561.956
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] = [245,2,Mod(79,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
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("245.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 245=572 245 = 5 \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 245.j (of order 66, 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: 1.956334849521.95633484952
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 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 35)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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 ζ12\zeta_{12}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+ζ12q2+(ζ123+ζ12)q3ζ122q4+(ζ122+2ζ12+1)q5+q63ζ123q8+(2ζ1222)q9+(ζ123+2ζ122+ζ12)q10+16ζ123q97+O(q100) q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + q^{6} - 3 \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 2) q^{9} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{10} + \cdots - 16 \zeta_{12}^{3} q^{97} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q2q4+2q5+4q64q9+4q10+8q15+2q1612q194q206q24+6q25+4q2628q29+2q30+4q31+8q34+8q364q39+12q40+10q96+O(q100) 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{9} + 4 q^{10} + 8 q^{15} + 2 q^{16} - 12 q^{19} - 4 q^{20} - 6 q^{24} + 6 q^{25} + 4 q^{26} - 28 q^{29} + 2 q^{30} + 4 q^{31} + 8 q^{34} + 8 q^{36} - 4 q^{39} + 12 q^{40}+ \cdots - 10 q^{96}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/245Z)×\left(\mathbb{Z}/245\mathbb{Z}\right)^\times.

nn 101101 197197
χ(n)\chi(n) 1+ζ122-1 + \zeta_{12}^{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}
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i −1.23205 1.86603i 1.00000 0 3.00000i −1.00000 + 1.73205i 0.133975 + 2.23205i
79.2 0.866025 + 0.500000i 0.866025 0.500000i −0.500000 0.866025i 2.23205 + 0.133975i 1.00000 0 3.00000i −1.00000 + 1.73205i 1.86603 + 1.23205i
214.1 −0.866025 + 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i −1.23205 + 1.86603i 1.00000 0 3.00000i −1.00000 1.73205i 0.133975 2.23205i
214.2 0.866025 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.23205 0.133975i 1.00000 0 3.00000i −1.00000 1.73205i 1.86603 1.23205i
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.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.j.c 4
5.b even 2 1 inner 245.2.j.c 4
7.b odd 2 1 35.2.j.a 4
7.c even 3 1 245.2.b.b 2
7.c even 3 1 inner 245.2.j.c 4
7.d odd 6 1 35.2.j.a 4
7.d odd 6 1 245.2.b.c 2
21.c even 2 1 315.2.bf.a 4
21.g even 6 1 315.2.bf.a 4
21.g even 6 1 2205.2.d.d 2
21.h odd 6 1 2205.2.d.e 2
28.d even 2 1 560.2.bw.b 4
28.f even 6 1 560.2.bw.b 4
35.c odd 2 1 35.2.j.a 4
35.f even 4 1 175.2.e.a 2
35.f even 4 1 175.2.e.b 2
35.i odd 6 1 35.2.j.a 4
35.i odd 6 1 245.2.b.c 2
35.j even 6 1 245.2.b.b 2
35.j even 6 1 inner 245.2.j.c 4
35.k even 12 1 175.2.e.a 2
35.k even 12 1 175.2.e.b 2
35.k even 12 1 1225.2.a.d 1
35.k even 12 1 1225.2.a.f 1
35.l odd 12 1 1225.2.a.b 1
35.l odd 12 1 1225.2.a.g 1
105.g even 2 1 315.2.bf.a 4
105.o odd 6 1 2205.2.d.e 2
105.p even 6 1 315.2.bf.a 4
105.p even 6 1 2205.2.d.d 2
140.c even 2 1 560.2.bw.b 4
140.s even 6 1 560.2.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 7.b odd 2 1
35.2.j.a 4 7.d odd 6 1
35.2.j.a 4 35.c odd 2 1
35.2.j.a 4 35.i odd 6 1
175.2.e.a 2 35.f even 4 1
175.2.e.a 2 35.k even 12 1
175.2.e.b 2 35.f even 4 1
175.2.e.b 2 35.k even 12 1
245.2.b.b 2 7.c even 3 1
245.2.b.b 2 35.j even 6 1
245.2.b.c 2 7.d odd 6 1
245.2.b.c 2 35.i odd 6 1
245.2.j.c 4 1.a even 1 1 trivial
245.2.j.c 4 5.b even 2 1 inner
245.2.j.c 4 7.c even 3 1 inner
245.2.j.c 4 35.j even 6 1 inner
315.2.bf.a 4 21.c even 2 1
315.2.bf.a 4 21.g even 6 1
315.2.bf.a 4 105.g even 2 1
315.2.bf.a 4 105.p even 6 1
560.2.bw.b 4 28.d even 2 1
560.2.bw.b 4 28.f even 6 1
560.2.bw.b 4 140.c even 2 1
560.2.bw.b 4 140.s even 6 1
1225.2.a.b 1 35.l odd 12 1
1225.2.a.d 1 35.k even 12 1
1225.2.a.f 1 35.k even 12 1
1225.2.a.g 1 35.l odd 12 1
2205.2.d.d 2 21.g even 6 1
2205.2.d.d 2 105.p even 6 1
2205.2.d.e 2 21.h odd 6 1
2205.2.d.e 2 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(245,[χ])S_{2}^{\mathrm{new}}(245, [\chi]):

T24T22+1 T_{2}^{4} - T_{2}^{2} + 1 Copy content Toggle raw display
T34T32+1 T_{3}^{4} - T_{3}^{2} + 1 Copy content Toggle raw display
T3122T31+4 T_{31}^{2} - 2T_{31} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
33 T4T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
55 T42T3++25 T^{4} - 2 T^{3} + \cdots + 25 Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
1717 T44T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
1919 (T2+6T+36)2 (T^{2} + 6 T + 36)^{2} Copy content Toggle raw display
2323 T49T2+81 T^{4} - 9T^{2} + 81 Copy content Toggle raw display
2929 (T+7)4 (T + 7)^{4} Copy content Toggle raw display
3131 (T22T+4)2 (T^{2} - 2 T + 4)^{2} Copy content Toggle raw display
3737 T464T2+4096 T^{4} - 64T^{2} + 4096 Copy content Toggle raw display
4141 (T+5)4 (T + 5)^{4} Copy content Toggle raw display
4343 (T2+49)2 (T^{2} + 49)^{2} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 T436T2+1296 T^{4} - 36T^{2} + 1296 Copy content Toggle raw display
5959 (T2+10T+100)2 (T^{2} + 10 T + 100)^{2} Copy content Toggle raw display
6161 (T27T+49)2 (T^{2} - 7 T + 49)^{2} Copy content Toggle raw display
6767 T425T2+625 T^{4} - 25T^{2} + 625 Copy content Toggle raw display
7171 (T+2)4 (T + 2)^{4} Copy content Toggle raw display
7373 T436T2+1296 T^{4} - 36T^{2} + 1296 Copy content Toggle raw display
7979 (T2+2T+4)2 (T^{2} + 2 T + 4)^{2} Copy content Toggle raw display
8383 (T2+121)2 (T^{2} + 121)^{2} Copy content Toggle raw display
8989 (T2+9T+81)2 (T^{2} + 9 T + 81)^{2} Copy content Toggle raw display
9797 (T2+256)2 (T^{2} + 256)^{2} Copy content Toggle raw display
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