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Properties

Label 3380.1.v.b
Level 33803380
Weight 11
Character orbit 3380.v
Analytic conductor 1.6871.687
Analytic rank 00
Dimension 44
Projective image D6D_{6}
CM discriminant -4
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,1,Mod(2219,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.2219");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 3380=22⋅5⋅132 3380 = 2^{2} \cdot 5 \cdot 13^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 3380.v (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: 1.686839742701.68683974270
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]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 260)
Projective image: D6D_{6}
Projective field: Galois closure of 6.0.2970344000.2

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+ζ12q2+ζ122q4−ζ12q5+ζ123q8+ζ122q9−ζ122q10+ζ124q16+(−ζ124+1)q17+ζ123q18+⋯−ζ125q98+O(q100) q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12} q^{5} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{2} q^{10} + \zeta_{12}^{4} q^{16} + ( - \zeta_{12}^{4} + 1) q^{17} + \zeta_{12}^{3} q^{18} + \cdots - \zeta_{12}^{5} q^{98} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q+2q4+2q9−2q10−2q16+6q17+2q25−2q29−2q36+2q40+2q49−2q61−4q64+6q68+2q74−2q81−6q82+2q90+O(q100) 4 q + 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{16} + 6 q^{17} + 2 q^{25} - 2 q^{29} - 2 q^{36} + 2 q^{40} + 2 q^{49} - 2 q^{61} - 4 q^{64} + 6 q^{68} + 2 q^{74} - 2 q^{81} - 6 q^{82} + 2 q^{90}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3380Z)×\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times.

nn 677677 16911691 18611861
χ(n)\chi(n) −1-1 −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}
2219.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.866025 + 0.500000i 0 0 − 1.00000i 0.500000 + 0.866025i −0.500000 − 0.866025i
2219.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.866025 − 0.500000i 0 0 1.00000i 0.500000 + 0.866025i −0.500000 − 0.866025i
3019.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.866025 − 0.500000i 0 0 1.00000i 0.500000 − 0.866025i −0.500000 + 0.866025i
3019.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.866025 + 0.500000i 0 0 − 1.00000i 0.500000 − 0.866025i −0.500000 + 0.866025i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by Q(−1)\Q(\sqrt{-1})
13.b even 2 1 inner
52.b odd 2 1 inner
65.l even 6 1 inner
65.n even 6 1 inner
260.v odd 6 1 inner
260.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.v.b 4
4.b odd 2 1 CM 3380.1.v.b 4
5.b even 2 1 3380.1.v.c 4
13.b even 2 1 inner 3380.1.v.b 4
13.c even 3 1 3380.1.h.f 4
13.c even 3 1 3380.1.v.c 4
13.d odd 4 1 260.1.w.a ✓ 2
13.d odd 4 1 3380.1.w.c 2
13.e even 6 1 3380.1.h.f 4
13.e even 6 1 3380.1.v.c 4
13.f odd 12 1 260.1.w.b yes 2
13.f odd 12 1 3380.1.g.a 2
13.f odd 12 1 3380.1.g.b 2
13.f odd 12 1 3380.1.w.b 2
20.d odd 2 1 3380.1.v.c 4
39.f even 4 1 2340.1.dd.b 2
39.k even 12 1 2340.1.dd.a 2
52.b odd 2 1 inner 3380.1.v.b 4
52.f even 4 1 260.1.w.a ✓ 2
52.f even 4 1 3380.1.w.c 2
52.i odd 6 1 3380.1.h.f 4
52.i odd 6 1 3380.1.v.c 4
52.j odd 6 1 3380.1.h.f 4
52.j odd 6 1 3380.1.v.c 4
52.l even 12 1 260.1.w.b yes 2
52.l even 12 1 3380.1.g.a 2
52.l even 12 1 3380.1.g.b 2
52.l even 12 1 3380.1.w.b 2
65.d even 2 1 3380.1.v.c 4
65.f even 4 1 1300.1.z.a 4
65.g odd 4 1 260.1.w.b yes 2
65.g odd 4 1 3380.1.w.b 2
65.k even 4 1 1300.1.z.a 4
65.l even 6 1 3380.1.h.f 4
65.l even 6 1 inner 3380.1.v.b 4
65.n even 6 1 3380.1.h.f 4
65.n even 6 1 inner 3380.1.v.b 4
65.o even 12 1 1300.1.z.a 4
65.s odd 12 1 260.1.w.a ✓ 2
65.s odd 12 1 3380.1.g.a 2
65.s odd 12 1 3380.1.g.b 2
65.s odd 12 1 3380.1.w.c 2
65.t even 12 1 1300.1.z.a 4
156.l odd 4 1 2340.1.dd.b 2
156.v odd 12 1 2340.1.dd.a 2
195.n even 4 1 2340.1.dd.a 2
195.bh even 12 1 2340.1.dd.b 2
260.g odd 2 1 3380.1.v.c 4
260.l odd 4 1 1300.1.z.a 4
260.s odd 4 1 1300.1.z.a 4
260.u even 4 1 260.1.w.b yes 2
260.u even 4 1 3380.1.w.b 2
260.v odd 6 1 3380.1.h.f 4
260.v odd 6 1 inner 3380.1.v.b 4
260.w odd 6 1 3380.1.h.f 4
260.w odd 6 1 inner 3380.1.v.b 4
260.bc even 12 1 260.1.w.a ✓ 2
260.bc even 12 1 3380.1.g.a 2
260.bc even 12 1 3380.1.g.b 2
260.bc even 12 1 3380.1.w.c 2
260.be odd 12 1 1300.1.z.a 4
260.bl odd 12 1 1300.1.z.a 4
780.bb odd 4 1 2340.1.dd.a 2
780.cr odd 12 1 2340.1.dd.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.w.a ✓ 2 13.d odd 4 1
260.1.w.a ✓ 2 52.f even 4 1
260.1.w.a ✓ 2 65.s odd 12 1
260.1.w.a ✓ 2 260.bc even 12 1
260.1.w.b yes 2 13.f odd 12 1
260.1.w.b yes 2 52.l even 12 1
260.1.w.b yes 2 65.g odd 4 1
260.1.w.b yes 2 260.u even 4 1
1300.1.z.a 4 65.f even 4 1
1300.1.z.a 4 65.k even 4 1
1300.1.z.a 4 65.o even 12 1
1300.1.z.a 4 65.t even 12 1
1300.1.z.a 4 260.l odd 4 1
1300.1.z.a 4 260.s odd 4 1
1300.1.z.a 4 260.be odd 12 1
1300.1.z.a 4 260.bl odd 12 1
2340.1.dd.a 2 39.k even 12 1
2340.1.dd.a 2 156.v odd 12 1
2340.1.dd.a 2 195.n even 4 1
2340.1.dd.a 2 780.bb odd 4 1
2340.1.dd.b 2 39.f even 4 1
2340.1.dd.b 2 156.l odd 4 1
2340.1.dd.b 2 195.bh even 12 1
2340.1.dd.b 2 780.cr odd 12 1
3380.1.g.a 2 13.f odd 12 1
3380.1.g.a 2 52.l even 12 1
3380.1.g.a 2 65.s odd 12 1
3380.1.g.a 2 260.bc even 12 1
3380.1.g.b 2 13.f odd 12 1
3380.1.g.b 2 52.l even 12 1
3380.1.g.b 2 65.s odd 12 1
3380.1.g.b 2 260.bc even 12 1
3380.1.h.f 4 13.c even 3 1
3380.1.h.f 4 13.e even 6 1
3380.1.h.f 4 52.i odd 6 1
3380.1.h.f 4 52.j odd 6 1
3380.1.h.f 4 65.l even 6 1
3380.1.h.f 4 65.n even 6 1
3380.1.h.f 4 260.v odd 6 1
3380.1.h.f 4 260.w odd 6 1
3380.1.v.b 4 1.a even 1 1 trivial
3380.1.v.b 4 4.b odd 2 1 CM
3380.1.v.b 4 13.b even 2 1 inner
3380.1.v.b 4 52.b odd 2 1 inner
3380.1.v.b 4 65.l even 6 1 inner
3380.1.v.b 4 65.n even 6 1 inner
3380.1.v.b 4 260.v odd 6 1 inner
3380.1.v.b 4 260.w odd 6 1 inner
3380.1.v.c 4 5.b even 2 1
3380.1.v.c 4 13.c even 3 1
3380.1.v.c 4 13.e even 6 1
3380.1.v.c 4 20.d odd 2 1
3380.1.v.c 4 52.i odd 6 1
3380.1.v.c 4 52.j odd 6 1
3380.1.v.c 4 65.d even 2 1
3380.1.v.c 4 260.g odd 2 1
3380.1.w.b 2 13.f odd 12 1
3380.1.w.b 2 52.l even 12 1
3380.1.w.b 2 65.g odd 4 1
3380.1.w.b 2 260.u even 4 1
3380.1.w.c 2 13.d odd 4 1
3380.1.w.c 2 52.f even 4 1
3380.1.w.c 2 65.s odd 12 1
3380.1.w.c 2 260.bc even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(3380,[χ])S_{1}^{\mathrm{new}}(3380, [\chi]):

T3 T_{3} Copy content Toggle raw display
T7 T_{7} Copy content Toggle raw display
T172−3T17+3 T_{17}^{2} - 3T_{17} + 3 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 T4−T2+1 T^{4} - T^{2} + 1 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 (T2−3T+3)2 (T^{2} - 3 T + 3)^{2} Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
3131 T4 T^{4} Copy content Toggle raw display
3737 T4−T2+1 T^{4} - T^{2} + 1 Copy content Toggle raw display
4141 T4+3T2+9 T^{4} + 3T^{2} + 9 Copy content Toggle raw display
4343 T4 T^{4} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 (T2+3)2 (T^{2} + 3)^{2} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 (T2+T+1)2 (T^{2} + T + 1)^{2} Copy content Toggle raw display
6767 T4 T^{4} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 T4 T^{4} Copy content Toggle raw display
8989 T4 T^{4} Copy content Toggle raw display
9797 T4−4T2+16 T^{4} - 4T^{2} + 16 Copy content Toggle raw display
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