Properties

Label 1680.2.f.a
Level 16801680
Weight 22
Character orbit 1680.f
Analytic conductor 13.41513.415
Analytic rank 11
Dimension 22
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: 11
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 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 420)
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 primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ62)q3+q5+(2ζ61)q7+(3ζ6+3)q9+(6ζ6+3)q11+(2ζ6+1)q13+(ζ62)q153q17+(4ζ62)q19++(9ζ69)q99+O(q100) q + (\zeta_{6} - 2) q^{3} + q^{5} + ( - 2 \zeta_{6} - 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 6 \zeta_{6} + 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{13} + (\zeta_{6} - 2) q^{15} - 3 q^{17} + (4 \zeta_{6} - 2) q^{19}+ \cdots + ( - 9 \zeta_{6} - 9) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q3q3+2q54q7+3q93q156q17+9q21+2q25+9q334q3516q37+3q3912q4120q43+3q45+6q47+2q49+9q516q57+27q99+O(q100) 2 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9} - 3 q^{15} - 6 q^{17} + 9 q^{21} + 2 q^{25} + 9 q^{33} - 4 q^{35} - 16 q^{37} + 3 q^{39} - 12 q^{41} - 20 q^{43} + 3 q^{45} + 6 q^{47} + 2 q^{49} + 9 q^{51} - 6 q^{57}+ \cdots - 27 q^{99}+O(q^{100}) 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
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 0.866025i 0 1.00000 0 −2.00000 + 1.73205i 0 1.50000 + 2.59808i 0
881.2 0 −1.50000 + 0.866025i 0 1.00000 0 −2.00000 1.73205i 0 1.50000 2.59808i 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.a 2
3.b odd 2 1 1680.2.f.d 2
4.b odd 2 1 420.2.d.b yes 2
7.b odd 2 1 1680.2.f.d 2
12.b even 2 1 420.2.d.a 2
20.d odd 2 1 2100.2.d.a 2
20.e even 4 2 2100.2.f.d 4
21.c even 2 1 inner 1680.2.f.a 2
28.d even 2 1 420.2.d.a 2
60.h even 2 1 2100.2.d.e 2
60.l odd 4 2 2100.2.f.c 4
84.h odd 2 1 420.2.d.b yes 2
140.c even 2 1 2100.2.d.e 2
140.j odd 4 2 2100.2.f.c 4
420.o odd 2 1 2100.2.d.a 2
420.w even 4 2 2100.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.a 2 12.b even 2 1
420.2.d.a 2 28.d even 2 1
420.2.d.b yes 2 4.b odd 2 1
420.2.d.b yes 2 84.h odd 2 1
1680.2.f.a 2 1.a even 1 1 trivial
1680.2.f.a 2 21.c even 2 1 inner
1680.2.f.d 2 3.b odd 2 1
1680.2.f.d 2 7.b odd 2 1
2100.2.d.a 2 20.d odd 2 1
2100.2.d.a 2 420.o odd 2 1
2100.2.d.e 2 60.h even 2 1
2100.2.d.e 2 140.c even 2 1
2100.2.f.c 4 60.l odd 4 2
2100.2.f.c 4 140.j odd 4 2
2100.2.f.d 4 20.e even 4 2
2100.2.f.d 4 420.w even 4 2

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]):

T112+27 T_{11}^{2} + 27 Copy content Toggle raw display
T17+3 T_{17} + 3 Copy content Toggle raw display
T41+6 T_{41} + 6 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 T2+3T+3 T^{2} + 3T + 3 Copy content Toggle raw display
55 (T1)2 (T - 1)^{2} Copy content Toggle raw display
77 T2+4T+7 T^{2} + 4T + 7 Copy content Toggle raw display
1111 T2+27 T^{2} + 27 Copy content Toggle raw display
1313 T2+3 T^{2} + 3 Copy content Toggle raw display
1717 (T+3)2 (T + 3)^{2} Copy content Toggle raw display
1919 T2+12 T^{2} + 12 Copy content Toggle raw display
2323 T2 T^{2} Copy content Toggle raw display
2929 T2+27 T^{2} + 27 Copy content Toggle raw display
3131 T2+108 T^{2} + 108 Copy content Toggle raw display
3737 (T+8)2 (T + 8)^{2} Copy content Toggle raw display
4141 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
4343 (T+10)2 (T + 10)^{2} Copy content Toggle raw display
4747 (T3)2 (T - 3)^{2} Copy content Toggle raw display
5353 T2+108 T^{2} + 108 Copy content Toggle raw display
5959 (T6)2 (T - 6)^{2} Copy content Toggle raw display
6161 T2+48 T^{2} + 48 Copy content Toggle raw display
6767 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
7171 T2+108 T^{2} + 108 Copy content Toggle raw display
7373 T2+48 T^{2} + 48 Copy content Toggle raw display
7979 (T13)2 (T - 13)^{2} Copy content Toggle raw display
8383 (T+12)2 (T + 12)^{2} Copy content Toggle raw display
8989 T2 T^{2} Copy content Toggle raw display
9797 T2+3 T^{2} + 3 Copy content Toggle raw display
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