Properties

Label 3675.1.u.f
Level 36753675
Weight 11
Character orbit 3675.u
Analytic conductor 1.8341.834
Analytic rank 00
Dimension 88
Projective image D4D_{4}
CM discriminant -15
Inner twists 1616

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,1,Mod(851,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3675.851");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 3675=35272 3675 = 3 \cdot 5^{2} \cdot 7^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 3675.u (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: 1.834063921431.83406392143
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q(ζ24)\Q(\zeta_{24})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8x4+1 x^{8} - x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 735)
Projective image: D4D_{4}
Projective field: Galois closure of 4.2.15435.1
Artin image: C12×D8C_{12}\times D_8
Artin field: Galois closure of Q[x]/(x96)\mathbb{Q}[x]/(x^{96} - \cdots)

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) == q+(ζ247ζ24)q2ζ242q3ζ248q4+(ζ249+ζ243)q6+ζ244q9+ζ2410q12+ζ244q16++(ζ247+ζ24)q96+O(q100) q + (\zeta_{24}^{7} - \zeta_{24}) q^{2} - \zeta_{24}^{2} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{6} + \zeta_{24}^{4} q^{9} + \zeta_{24}^{10} q^{12} + \zeta_{24}^{4} q^{16} + \cdots + (\zeta_{24}^{7} + \zeta_{24}) q^{96} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+4q4+4q9+4q16+8q36+8q46+8q644q81+O(q100) 8 q + 4 q^{4} + 4 q^{9} + 4 q^{16} + 8 q^{36} + 8 q^{46} + 8 q^{64} - 4 q^{81}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3675Z)×\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times.

nn 11771177 12261226 25512551
χ(n)\chi(n) 11 1-1 ζ244-\zeta_{24}^{4}

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}
851.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 0.965926i
−1.22474 + 0.707107i −0.866025 0.500000i 0.500000 0.866025i 0 1.41421 0 0 0.500000 + 0.866025i 0
851.2 −1.22474 + 0.707107i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.41421 0 0 0.500000 + 0.866025i 0
851.3 1.22474 0.707107i −0.866025 0.500000i 0.500000 0.866025i 0 −1.41421 0 0 0.500000 + 0.866025i 0
851.4 1.22474 0.707107i 0.866025 + 0.500000i 0.500000 0.866025i 0 1.41421 0 0 0.500000 + 0.866025i 0
1451.1 −1.22474 0.707107i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.41421 0 0 0.500000 0.866025i 0
1451.2 −1.22474 0.707107i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.41421 0 0 0.500000 0.866025i 0
1451.3 1.22474 + 0.707107i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.41421 0 0 0.500000 0.866025i 0
1451.4 1.22474 + 0.707107i 0.866025 0.500000i 0.500000 + 0.866025i 0 1.41421 0 0 0.500000 0.866025i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 851.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by Q(15)\Q(\sqrt{-15})
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
105.g even 2 1 inner
105.o odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.1.u.f 8
3.b odd 2 1 inner 3675.1.u.f 8
5.b even 2 1 inner 3675.1.u.f 8
5.c odd 4 1 735.1.o.c 4
5.c odd 4 1 735.1.o.d 4
7.b odd 2 1 inner 3675.1.u.f 8
7.c even 3 1 3675.1.c.f 4
7.c even 3 1 inner 3675.1.u.f 8
7.d odd 6 1 3675.1.c.f 4
7.d odd 6 1 inner 3675.1.u.f 8
15.d odd 2 1 CM 3675.1.u.f 8
15.e even 4 1 735.1.o.c 4
15.e even 4 1 735.1.o.d 4
21.c even 2 1 inner 3675.1.u.f 8
21.g even 6 1 3675.1.c.f 4
21.g even 6 1 inner 3675.1.u.f 8
21.h odd 6 1 3675.1.c.f 4
21.h odd 6 1 inner 3675.1.u.f 8
35.c odd 2 1 inner 3675.1.u.f 8
35.f even 4 1 735.1.o.c 4
35.f even 4 1 735.1.o.d 4
35.i odd 6 1 3675.1.c.f 4
35.i odd 6 1 inner 3675.1.u.f 8
35.j even 6 1 3675.1.c.f 4
35.j even 6 1 inner 3675.1.u.f 8
35.k even 12 1 735.1.f.c 2
35.k even 12 1 735.1.f.d yes 2
35.k even 12 1 735.1.o.c 4
35.k even 12 1 735.1.o.d 4
35.l odd 12 1 735.1.f.c 2
35.l odd 12 1 735.1.f.d yes 2
35.l odd 12 1 735.1.o.c 4
35.l odd 12 1 735.1.o.d 4
105.g even 2 1 inner 3675.1.u.f 8
105.k odd 4 1 735.1.o.c 4
105.k odd 4 1 735.1.o.d 4
105.o odd 6 1 3675.1.c.f 4
105.o odd 6 1 inner 3675.1.u.f 8
105.p even 6 1 3675.1.c.f 4
105.p even 6 1 inner 3675.1.u.f 8
105.w odd 12 1 735.1.f.c 2
105.w odd 12 1 735.1.f.d yes 2
105.w odd 12 1 735.1.o.c 4
105.w odd 12 1 735.1.o.d 4
105.x even 12 1 735.1.f.c 2
105.x even 12 1 735.1.f.d yes 2
105.x even 12 1 735.1.o.c 4
105.x even 12 1 735.1.o.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.1.f.c 2 35.k even 12 1
735.1.f.c 2 35.l odd 12 1
735.1.f.c 2 105.w odd 12 1
735.1.f.c 2 105.x even 12 1
735.1.f.d yes 2 35.k even 12 1
735.1.f.d yes 2 35.l odd 12 1
735.1.f.d yes 2 105.w odd 12 1
735.1.f.d yes 2 105.x even 12 1
735.1.o.c 4 5.c odd 4 1
735.1.o.c 4 15.e even 4 1
735.1.o.c 4 35.f even 4 1
735.1.o.c 4 35.k even 12 1
735.1.o.c 4 35.l odd 12 1
735.1.o.c 4 105.k odd 4 1
735.1.o.c 4 105.w odd 12 1
735.1.o.c 4 105.x even 12 1
735.1.o.d 4 5.c odd 4 1
735.1.o.d 4 15.e even 4 1
735.1.o.d 4 35.f even 4 1
735.1.o.d 4 35.k even 12 1
735.1.o.d 4 35.l odd 12 1
735.1.o.d 4 105.k odd 4 1
735.1.o.d 4 105.w odd 12 1
735.1.o.d 4 105.x even 12 1
3675.1.c.f 4 7.c even 3 1
3675.1.c.f 4 7.d odd 6 1
3675.1.c.f 4 21.g even 6 1
3675.1.c.f 4 21.h odd 6 1
3675.1.c.f 4 35.i odd 6 1
3675.1.c.f 4 35.j even 6 1
3675.1.c.f 4 105.o odd 6 1
3675.1.c.f 4 105.p even 6 1
3675.1.u.f 8 1.a even 1 1 trivial
3675.1.u.f 8 3.b odd 2 1 inner
3675.1.u.f 8 5.b even 2 1 inner
3675.1.u.f 8 7.b odd 2 1 inner
3675.1.u.f 8 7.c even 3 1 inner
3675.1.u.f 8 7.d odd 6 1 inner
3675.1.u.f 8 15.d odd 2 1 CM
3675.1.u.f 8 21.c even 2 1 inner
3675.1.u.f 8 21.g even 6 1 inner
3675.1.u.f 8 21.h odd 6 1 inner
3675.1.u.f 8 35.c odd 2 1 inner
3675.1.u.f 8 35.i odd 6 1 inner
3675.1.u.f 8 35.j even 6 1 inner
3675.1.u.f 8 105.g even 2 1 inner
3675.1.u.f 8 105.o odd 6 1 inner
3675.1.u.f 8 105.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(3675,[χ])S_{1}^{\mathrm{new}}(3675, [\chi]):

T242T22+4 T_{2}^{4} - 2T_{2}^{2} + 4 Copy content Toggle raw display
T13 T_{13} Copy content Toggle raw display
T37 T_{37} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T42T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
33 (T4T2+1)2 (T^{4} - T^{2} + 1)^{2} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8 T^{8} Copy content Toggle raw display
1717 T8 T^{8} Copy content Toggle raw display
1919 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
2323 (T42T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
2929 T8 T^{8} Copy content Toggle raw display
3131 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
3737 T8 T^{8} Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 T8 T^{8} Copy content Toggle raw display
4747 T8 T^{8} Copy content Toggle raw display
5353 (T42T2+4)2 (T^{4} - 2 T^{2} + 4)^{2} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T4+2T2+4)2 (T^{4} + 2 T^{2} + 4)^{2} Copy content Toggle raw display
6767 T8 T^{8} Copy content Toggle raw display
7171 T8 T^{8} Copy content Toggle raw display
7373 T8 T^{8} Copy content Toggle raw display
7979 T8 T^{8} Copy content Toggle raw display
8383 T8 T^{8} Copy content Toggle raw display
8989 T8 T^{8} Copy content Toggle raw display
9797 T8 T^{8} Copy content Toggle raw display
show more
show less