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Properties

Label 3380.1.w.a
Level 33803380
Weight 11
Character orbit 3380.w
Analytic conductor 1.6871.687
Analytic rank 00
Dimension 22
Projective image D2D_{2}
CM/RM discs -4, -260, 65
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(699,3380)]
 
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, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.699");
 
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.w (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: 22
Coefficient field: Q(ζ6)\Q(\zeta_{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x+1 x^{2} - x + 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: D2D_{2}
Projective field: Galois closure of Q(i,65)\Q(i, \sqrt{65})
Artin image: C3×D4C_3\times D_4
Artin field: Galois closure of Q[x]/(x12−⋯ )\mathbb{Q}[x]/(x^{12} - \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+ζ62q2−ζ6q4−q5+q8+ζ6q9−ζ62q10+ζ62q16−q18+ζ6q20+q25−2ζ62q29−ζ6q32+⋯−ζ6q98+O(q100) q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{5} + q^{8} + \zeta_{6} q^{9} - \zeta_{6}^{2} q^{10} + \zeta_{6}^{2} q^{16} - q^{18} + \zeta_{6} q^{20} + q^{25} - 2 \zeta_{6}^{2} q^{29} - \zeta_{6} q^{32} + \cdots - \zeta_{6} q^{98} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−q2−q4−2q5+2q8+q9+q10−q16−2q18+q20+2q25+2q29−q32+q36−2q37−2q40−q45−q49−q50+2q58+2q61+⋯−q98+O(q100) 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} + q^{9} + q^{10} - q^{16} - 2 q^{18} + q^{20} + 2 q^{25} + 2 q^{29} - q^{32} + q^{36} - 2 q^{37} - 2 q^{40} - q^{45} - q^{49} - q^{50} + 2 q^{58} + 2 q^{61}+ \cdots - q^{98}+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 ζ6\zeta_{6}

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}
699.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0 1.00000 0.500000 − 0.866025i 0.500000 + 0.866025i
1499.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.00000 0 0 1.00000 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})
65.d even 2 1 RM by Q(65)\Q(\sqrt{65})
260.g odd 2 1 CM by Q(−65)\Q(\sqrt{-65})
13.c even 3 1 inner
52.j odd 6 1 inner
65.l even 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.w.a 2
4.b odd 2 1 CM 3380.1.w.a 2
5.b even 2 1 3380.1.w.d 2
13.b even 2 1 3380.1.w.d 2
13.c even 3 1 260.1.g.b yes 1
13.c even 3 1 inner 3380.1.w.a 2
13.d odd 4 2 3380.1.v.a 4
13.e even 6 1 260.1.g.a ✓ 1
13.e even 6 1 3380.1.w.d 2
13.f odd 12 2 3380.1.h.a 2
13.f odd 12 2 3380.1.v.a 4
20.d odd 2 1 3380.1.w.d 2
39.h odd 6 1 2340.1.i.b 1
39.i odd 6 1 2340.1.i.a 1
52.b odd 2 1 3380.1.w.d 2
52.f even 4 2 3380.1.v.a 4
52.i odd 6 1 260.1.g.a ✓ 1
52.i odd 6 1 3380.1.w.d 2
52.j odd 6 1 260.1.g.b yes 1
52.j odd 6 1 inner 3380.1.w.a 2
52.l even 12 2 3380.1.h.a 2
52.l even 12 2 3380.1.v.a 4
65.d even 2 1 RM 3380.1.w.a 2
65.g odd 4 2 3380.1.v.a 4
65.l even 6 1 260.1.g.b yes 1
65.l even 6 1 inner 3380.1.w.a 2
65.n even 6 1 260.1.g.a ✓ 1
65.n even 6 1 3380.1.w.d 2
65.q odd 12 2 1300.1.e.e 2
65.r odd 12 2 1300.1.e.e 2
65.s odd 12 2 3380.1.h.a 2
65.s odd 12 2 3380.1.v.a 4
156.p even 6 1 2340.1.i.a 1
156.r even 6 1 2340.1.i.b 1
195.x odd 6 1 2340.1.i.b 1
195.y odd 6 1 2340.1.i.a 1
260.g odd 2 1 CM 3380.1.w.a 2
260.u even 4 2 3380.1.v.a 4
260.v odd 6 1 260.1.g.a ✓ 1
260.v odd 6 1 3380.1.w.d 2
260.w odd 6 1 260.1.g.b yes 1
260.w odd 6 1 inner 3380.1.w.a 2
260.bc even 12 2 3380.1.h.a 2
260.bc even 12 2 3380.1.v.a 4
260.bg even 12 2 1300.1.e.e 2
260.bj even 12 2 1300.1.e.e 2
780.br even 6 1 2340.1.i.b 1
780.cb even 6 1 2340.1.i.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.g.a ✓ 1 13.e even 6 1
260.1.g.a ✓ 1 52.i odd 6 1
260.1.g.a ✓ 1 65.n even 6 1
260.1.g.a ✓ 1 260.v odd 6 1
260.1.g.b yes 1 13.c even 3 1
260.1.g.b yes 1 52.j odd 6 1
260.1.g.b yes 1 65.l even 6 1
260.1.g.b yes 1 260.w odd 6 1
1300.1.e.e 2 65.q odd 12 2
1300.1.e.e 2 65.r odd 12 2
1300.1.e.e 2 260.bg even 12 2
1300.1.e.e 2 260.bj even 12 2
2340.1.i.a 1 39.i odd 6 1
2340.1.i.a 1 156.p even 6 1
2340.1.i.a 1 195.y odd 6 1
2340.1.i.a 1 780.cb even 6 1
2340.1.i.b 1 39.h odd 6 1
2340.1.i.b 1 156.r even 6 1
2340.1.i.b 1 195.x odd 6 1
2340.1.i.b 1 780.br even 6 1
3380.1.h.a 2 13.f odd 12 2
3380.1.h.a 2 52.l even 12 2
3380.1.h.a 2 65.s odd 12 2
3380.1.h.a 2 260.bc even 12 2
3380.1.v.a 4 13.d odd 4 2
3380.1.v.a 4 13.f odd 12 2
3380.1.v.a 4 52.f even 4 2
3380.1.v.a 4 52.l even 12 2
3380.1.v.a 4 65.g odd 4 2
3380.1.v.a 4 65.s odd 12 2
3380.1.v.a 4 260.u even 4 2
3380.1.v.a 4 260.bc even 12 2
3380.1.w.a 2 1.a even 1 1 trivial
3380.1.w.a 2 4.b odd 2 1 CM
3380.1.w.a 2 13.c even 3 1 inner
3380.1.w.a 2 52.j odd 6 1 inner
3380.1.w.a 2 65.d even 2 1 RM
3380.1.w.a 2 65.l even 6 1 inner
3380.1.w.a 2 260.g odd 2 1 CM
3380.1.w.a 2 260.w odd 6 1 inner
3380.1.w.d 2 5.b even 2 1
3380.1.w.d 2 13.b even 2 1
3380.1.w.d 2 13.e even 6 1
3380.1.w.d 2 20.d odd 2 1
3380.1.w.d 2 52.b odd 2 1
3380.1.w.d 2 52.i odd 6 1
3380.1.w.d 2 65.n even 6 1
3380.1.w.d 2 260.v odd 6 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
T17 T_{17} Copy content Toggle raw display
T372+2T37+4 T_{37}^{2} + 2T_{37} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

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