Properties

Label 1680.2.f.h
Level 16801680
Weight 22
Character orbit 1680.f
Analytic conductor 13.41513.415
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(881,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1680=24357 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1680.f (of order 22, degree 11, 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: 13.414867539613.4148675396
Analytic rank: 00
Dimension: 44
Coefficient field: Q(3,11)\Q(\sqrt{-3}, \sqrt{-11})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x32x23x+9 x^{4} - x^{3} - 2x^{2} - 3x + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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 basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3q5+(β2+2)q7+(β3+β2+1)q9+(β3β1)q11+(3β3+2β2+3β1)q13β1q15+(β3+β12)q17++(2β3β2+β14)q99+O(q100) q + \beta_1 q^{3} - q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_{3} + \beta_{2} + 1) q^{9} + ( - \beta_{3} - \beta_1) q^{11} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{13} - \beta_1 q^{15} + ( - \beta_{3} + \beta_1 - 2) q^{17}+ \cdots + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+q34q5+8q7+5q9q156q17q21+4q25+16q27+7q338q354q3715q39+24q41+4q435q4518q47+4q49+15q51+13q99+O(q100) 4 q + q^{3} - 4 q^{5} + 8 q^{7} + 5 q^{9} - q^{15} - 6 q^{17} - q^{21} + 4 q^{25} + 16 q^{27} + 7 q^{33} - 8 q^{35} - 4 q^{37} - 15 q^{39} + 24 q^{41} + 4 q^{43} - 5 q^{45} - 18 q^{47} + 4 q^{49} + 15 q^{51}+ \cdots - 13 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x32x23x+9 x^{4} - x^{3} - 2x^{2} - 3x + 9 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν3+2ν22ν6)/3 ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 Copy content Toggle raw display
β3\beta_{3}== (ν3ν22ν3)/3 ( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+β2+1 -\beta_{3} + \beta_{2} + 1 Copy content Toggle raw display
ν3\nu^{3}== 2β3+β2+2β1+4 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1680Z)×\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times.

nn 241241 337337 421421 11211121 14711471
χ(n)\chi(n) 1-1 11 11 1-1 11

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}
881.1
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
1.68614 + 0.396143i
0 −1.18614 1.26217i 0 −1.00000 0 2.00000 1.73205i 0 −0.186141 + 2.99422i 0
881.2 0 −1.18614 + 1.26217i 0 −1.00000 0 2.00000 + 1.73205i 0 −0.186141 2.99422i 0
881.3 0 1.68614 0.396143i 0 −1.00000 0 2.00000 + 1.73205i 0 2.68614 1.33591i 0
881.4 0 1.68614 + 0.396143i 0 −1.00000 0 2.00000 1.73205i 0 2.68614 + 1.33591i 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.h 4
3.b odd 2 1 1680.2.f.g 4
4.b odd 2 1 105.2.b.c 4
7.b odd 2 1 1680.2.f.g 4
12.b even 2 1 105.2.b.d yes 4
20.d odd 2 1 525.2.b.g 4
20.e even 4 2 525.2.g.e 8
21.c even 2 1 inner 1680.2.f.h 4
28.d even 2 1 105.2.b.d yes 4
28.f even 6 1 735.2.s.g 4
28.f even 6 1 735.2.s.j 4
28.g odd 6 1 735.2.s.h 4
28.g odd 6 1 735.2.s.i 4
60.h even 2 1 525.2.b.e 4
60.l odd 4 2 525.2.g.d 8
84.h odd 2 1 105.2.b.c 4
84.j odd 6 1 735.2.s.h 4
84.j odd 6 1 735.2.s.i 4
84.n even 6 1 735.2.s.g 4
84.n even 6 1 735.2.s.j 4
140.c even 2 1 525.2.b.e 4
140.j odd 4 2 525.2.g.d 8
420.o odd 2 1 525.2.b.g 4
420.w even 4 2 525.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 4.b odd 2 1
105.2.b.c 4 84.h odd 2 1
105.2.b.d yes 4 12.b even 2 1
105.2.b.d yes 4 28.d even 2 1
525.2.b.e 4 60.h even 2 1
525.2.b.e 4 140.c even 2 1
525.2.b.g 4 20.d odd 2 1
525.2.b.g 4 420.o odd 2 1
525.2.g.d 8 60.l odd 4 2
525.2.g.d 8 140.j odd 4 2
525.2.g.e 8 20.e even 4 2
525.2.g.e 8 420.w even 4 2
735.2.s.g 4 28.f even 6 1
735.2.s.g 4 84.n even 6 1
735.2.s.h 4 28.g odd 6 1
735.2.s.h 4 84.j odd 6 1
735.2.s.i 4 28.g odd 6 1
735.2.s.i 4 84.j odd 6 1
735.2.s.j 4 28.f even 6 1
735.2.s.j 4 84.n even 6 1
1680.2.f.g 4 3.b odd 2 1
1680.2.f.g 4 7.b odd 2 1
1680.2.f.h 4 1.a even 1 1 trivial
1680.2.f.h 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1680,[χ])S_{2}^{\mathrm{new}}(1680, [\chi]):

T114+7T112+4 T_{11}^{4} + 7T_{11}^{2} + 4 Copy content Toggle raw display
T172+3T176 T_{17}^{2} + 3T_{17} - 6 Copy content Toggle raw display
T416 T_{41} - 6 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4T32T2++9 T^{4} - T^{3} - 2 T^{2} + \cdots + 9 Copy content Toggle raw display
55 (T+1)4 (T + 1)^{4} Copy content Toggle raw display
77 (T24T+7)2 (T^{2} - 4 T + 7)^{2} Copy content Toggle raw display
1111 T4+7T2+4 T^{4} + 7T^{2} + 4 Copy content Toggle raw display
1313 T4+51T2+576 T^{4} + 51T^{2} + 576 Copy content Toggle raw display
1717 (T2+3T6)2 (T^{2} + 3 T - 6)^{2} Copy content Toggle raw display
1919 (T2+12)2 (T^{2} + 12)^{2} Copy content Toggle raw display
2323 T4+76T2+256 T^{4} + 76T^{2} + 256 Copy content Toggle raw display
2929 T4+19T2+16 T^{4} + 19T^{2} + 16 Copy content Toggle raw display
3131 (T2+12)2 (T^{2} + 12)^{2} Copy content Toggle raw display
3737 (T2+2T32)2 (T^{2} + 2 T - 32)^{2} Copy content Toggle raw display
4141 (T6)4 (T - 6)^{4} Copy content Toggle raw display
4343 (T22T32)2 (T^{2} - 2 T - 32)^{2} Copy content Toggle raw display
4747 (T2+9T+12)2 (T^{2} + 9 T + 12)^{2} Copy content Toggle raw display
5353 T4+76T2+256 T^{4} + 76T^{2} + 256 Copy content Toggle raw display
5959 (T2+6T24)2 (T^{2} + 6 T - 24)^{2} Copy content Toggle raw display
6161 (T2+48)2 (T^{2} + 48)^{2} Copy content Toggle raw display
6767 (T22T32)2 (T^{2} - 2 T - 32)^{2} Copy content Toggle raw display
7171 T4+184T2+16 T^{4} + 184T^{2} + 16 Copy content Toggle raw display
7373 (T2+48)2 (T^{2} + 48)^{2} Copy content Toggle raw display
7979 (T2+T8)2 (T^{2} + T - 8)^{2} Copy content Toggle raw display
8383 (T2+12T96)2 (T^{2} + 12 T - 96)^{2} Copy content Toggle raw display
8989 (T2+18T+48)2 (T^{2} + 18 T + 48)^{2} Copy content Toggle raw display
9797 T4+123T2+144 T^{4} + 123T^{2} + 144 Copy content Toggle raw display
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