↑

Properties

Label 3675.1.u.c
Level 36753675
Weight 11
Character orbit 3675.u
Analytic conductor 1.8341.834
Analytic rank 00
Dimension 44
Projective image D2D_{2}
CM/RM discs -15, -35, 21
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=3⋅52⋅72 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: 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: x4−x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 735)
Projective image: D2D_{2}
Projective field: Galois closure of Q(−15,21)\Q(\sqrt{-15}, \sqrt{21})
Artin image: D4:C6D_4:C_6
Artin field: Galois closure of Q[x]/(x24−⋯ )\mathbb{Q}[x]/(x^{24} - \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−ζ12q3+ζ124q4+ζ122q9−ζ125q12−ζ122q16−2ζ12q17−ζ123q27−q36−2ζ125q47+ζ123q48+⋯−2ζ123q83+O(q100) q - \zeta_{12} q^{3} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{5} q^{12} - \zeta_{12}^{2} q^{16} - 2 \zeta_{12} q^{17} - \zeta_{12}^{3} q^{27} - q^{36} - 2 \zeta_{12}^{5} q^{47} + \zeta_{12}^{3} q^{48} + \cdots - 2 \zeta_{12}^{3} q^{83} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q−2q4+2q9−2q16−4q36+4q51+4q64−4q79−2q81+O(q100) 4 q - 2 q^{4} + 2 q^{9} - 2 q^{16} - 4 q^{36} + 4 q^{51} + 4 q^{64} - 4 q^{79} - 2 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 −ζ122-\zeta_{12}^{2}

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
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 0.500000i −0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
851.2 0 0.866025 + 0.500000i −0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 0
1451.1 0 −0.866025 + 0.500000i −0.500000 − 0.866025i 0 0 0 0 0.500000 − 0.866025i 0
1451.2 0 0.866025 − 0.500000i −0.500000 − 0.866025i 0 0 0 0 0.500000 − 0.866025i 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
15.d odd 2 1 CM by Q(−15)\Q(\sqrt{-15})
21.c even 2 1 RM by Q(21)\Q(\sqrt{21})
35.c odd 2 1 CM by Q(−35)\Q(\sqrt{-35})
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.g even 6 1 inner
21.h odd 6 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.c 4
3.b odd 2 1 inner 3675.1.u.c 4
5.b even 2 1 inner 3675.1.u.c 4
5.c odd 4 1 735.1.o.a 2
5.c odd 4 1 735.1.o.b 2
7.b odd 2 1 inner 3675.1.u.c 4
7.c even 3 1 3675.1.c.e 2
7.c even 3 1 inner 3675.1.u.c 4
7.d odd 6 1 3675.1.c.e 2
7.d odd 6 1 inner 3675.1.u.c 4
15.d odd 2 1 CM 3675.1.u.c 4
15.e even 4 1 735.1.o.a 2
15.e even 4 1 735.1.o.b 2
21.c even 2 1 RM 3675.1.u.c 4
21.g even 6 1 3675.1.c.e 2
21.g even 6 1 inner 3675.1.u.c 4
21.h odd 6 1 3675.1.c.e 2
21.h odd 6 1 inner 3675.1.u.c 4
35.c odd 2 1 CM 3675.1.u.c 4
35.f even 4 1 735.1.o.a 2
35.f even 4 1 735.1.o.b 2
35.i odd 6 1 3675.1.c.e 2
35.i odd 6 1 inner 3675.1.u.c 4
35.j even 6 1 3675.1.c.e 2
35.j even 6 1 inner 3675.1.u.c 4
35.k even 12 1 735.1.f.a ✓ 1
35.k even 12 1 735.1.f.b yes 1
35.k even 12 1 735.1.o.a 2
35.k even 12 1 735.1.o.b 2
35.l odd 12 1 735.1.f.a ✓ 1
35.l odd 12 1 735.1.f.b yes 1
35.l odd 12 1 735.1.o.a 2
35.l odd 12 1 735.1.o.b 2
105.g even 2 1 inner 3675.1.u.c 4
105.k odd 4 1 735.1.o.a 2
105.k odd 4 1 735.1.o.b 2
105.o odd 6 1 3675.1.c.e 2
105.o odd 6 1 inner 3675.1.u.c 4
105.p even 6 1 3675.1.c.e 2
105.p even 6 1 inner 3675.1.u.c 4
105.w odd 12 1 735.1.f.a ✓ 1
105.w odd 12 1 735.1.f.b yes 1
105.w odd 12 1 735.1.o.a 2
105.w odd 12 1 735.1.o.b 2
105.x even 12 1 735.1.f.a ✓ 1
105.x even 12 1 735.1.f.b yes 1
105.x even 12 1 735.1.o.a 2
105.x even 12 1 735.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.1.f.a ✓ 1 35.k even 12 1
735.1.f.a ✓ 1 35.l odd 12 1
735.1.f.a ✓ 1 105.w odd 12 1
735.1.f.a ✓ 1 105.x even 12 1
735.1.f.b yes 1 35.k even 12 1
735.1.f.b yes 1 35.l odd 12 1
735.1.f.b yes 1 105.w odd 12 1
735.1.f.b yes 1 105.x even 12 1
735.1.o.a 2 5.c odd 4 1
735.1.o.a 2 15.e even 4 1
735.1.o.a 2 35.f even 4 1
735.1.o.a 2 35.k even 12 1
735.1.o.a 2 35.l odd 12 1
735.1.o.a 2 105.k odd 4 1
735.1.o.a 2 105.w odd 12 1
735.1.o.a 2 105.x even 12 1
735.1.o.b 2 5.c odd 4 1
735.1.o.b 2 15.e even 4 1
735.1.o.b 2 35.f even 4 1
735.1.o.b 2 35.k even 12 1
735.1.o.b 2 35.l odd 12 1
735.1.o.b 2 105.k odd 4 1
735.1.o.b 2 105.w odd 12 1
735.1.o.b 2 105.x even 12 1
3675.1.c.e 2 7.c even 3 1
3675.1.c.e 2 7.d odd 6 1
3675.1.c.e 2 21.g even 6 1
3675.1.c.e 2 21.h odd 6 1
3675.1.c.e 2 35.i odd 6 1
3675.1.c.e 2 35.j even 6 1
3675.1.c.e 2 105.o odd 6 1
3675.1.c.e 2 105.p even 6 1
3675.1.u.c 4 1.a even 1 1 trivial
3675.1.u.c 4 3.b odd 2 1 inner
3675.1.u.c 4 5.b even 2 1 inner
3675.1.u.c 4 7.b odd 2 1 inner
3675.1.u.c 4 7.c even 3 1 inner
3675.1.u.c 4 7.d odd 6 1 inner
3675.1.u.c 4 15.d odd 2 1 CM
3675.1.u.c 4 21.c even 2 1 RM
3675.1.u.c 4 21.g even 6 1 inner
3675.1.u.c 4 21.h odd 6 1 inner
3675.1.u.c 4 35.c odd 2 1 CM
3675.1.u.c 4 35.i odd 6 1 inner
3675.1.u.c 4 35.j even 6 1 inner
3675.1.u.c 4 105.g even 2 1 inner
3675.1.u.c 4 105.o odd 6 1 inner
3675.1.u.c 4 105.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]):

T2 T_{2} 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 T4 T^{4} Copy content Toggle raw display
33 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 T4−4T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 T4 T^{4} Copy content Toggle raw display
3737 T4 T^{4} Copy content Toggle raw display
4141 T4 T^{4} Copy content Toggle raw display
4343 T4 T^{4} Copy content Toggle raw display
4747 T4−4T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
5353 T4 T^{4} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 T4 T^{4} Copy content Toggle raw display
6767 T4 T^{4} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 T4 T^{4} Copy content Toggle raw display
7979 (T2+2T+4)2 (T^{2} + 2 T + 4)^{2} Copy content Toggle raw display
8383 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
8989 T4 T^{4} Copy content Toggle raw display
9797 T4 T^{4} Copy content Toggle raw display
show more
show less