Properties

Label 672.4.i.c
Level 672672
Weight 44
Character orbit 672.i
Analytic conductor 39.64939.649
Analytic rank 00
Dimension 8080
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,4,Mod(209,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 672=2537 672 = 2^{5} \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 672.i (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: 39.649283523939.6492835239
Analytic rank: 00
Dimension: 8080
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 80q64q7+104q9+8q15976q25568q394048q491448q57+2152q634992q79+1568q81+O(q100) 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) Copy content Toggle raw display

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   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}
209.1 0 −5.19433 0.137767i 0 6.83655i 0 −13.6972 + 12.4654i 0 26.9620 + 1.43121i 0
209.2 0 −5.19433 0.137767i 0 6.83655i 0 −13.6972 12.4654i 0 26.9620 + 1.43121i 0
209.3 0 −5.19433 + 0.137767i 0 6.83655i 0 −13.6972 12.4654i 0 26.9620 1.43121i 0
209.4 0 −5.19433 + 0.137767i 0 6.83655i 0 −13.6972 + 12.4654i 0 26.9620 1.43121i 0
209.5 0 −5.10794 0.953394i 0 17.0023i 0 5.04990 17.8185i 0 25.1821 + 9.73975i 0
209.6 0 −5.10794 0.953394i 0 17.0023i 0 5.04990 + 17.8185i 0 25.1821 + 9.73975i 0
209.7 0 −5.10794 + 0.953394i 0 17.0023i 0 5.04990 + 17.8185i 0 25.1821 9.73975i 0
209.8 0 −5.10794 + 0.953394i 0 17.0023i 0 5.04990 17.8185i 0 25.1821 9.73975i 0
209.9 0 −4.74822 2.11055i 0 15.5059i 0 14.6158 + 11.3745i 0 18.0912 + 20.0427i 0
209.10 0 −4.74822 2.11055i 0 15.5059i 0 14.6158 11.3745i 0 18.0912 + 20.0427i 0
209.11 0 −4.74822 + 2.11055i 0 15.5059i 0 14.6158 11.3745i 0 18.0912 20.0427i 0
209.12 0 −4.74822 + 2.11055i 0 15.5059i 0 14.6158 + 11.3745i 0 18.0912 20.0427i 0
209.13 0 −4.72061 2.17161i 0 2.27823i 0 13.7109 12.4503i 0 17.5682 + 20.5026i 0
209.14 0 −4.72061 2.17161i 0 2.27823i 0 13.7109 + 12.4503i 0 17.5682 + 20.5026i 0
209.15 0 −4.72061 + 2.17161i 0 2.27823i 0 13.7109 + 12.4503i 0 17.5682 20.5026i 0
209.16 0 −4.72061 + 2.17161i 0 2.27823i 0 13.7109 12.4503i 0 17.5682 20.5026i 0
209.17 0 −4.05635 3.24747i 0 13.2397i 0 −18.4748 1.29729i 0 5.90788 + 26.3457i 0
209.18 0 −4.05635 3.24747i 0 13.2397i 0 −18.4748 + 1.29729i 0 5.90788 + 26.3457i 0
209.19 0 −4.05635 + 3.24747i 0 13.2397i 0 −18.4748 + 1.29729i 0 5.90788 26.3457i 0
209.20 0 −4.05635 + 3.24747i 0 13.2397i 0 −18.4748 1.29729i 0 5.90788 26.3457i 0
See all 80 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.80
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.i.c 80
3.b odd 2 1 inner 672.4.i.c 80
4.b odd 2 1 168.4.i.c 80
7.b odd 2 1 inner 672.4.i.c 80
8.b even 2 1 inner 672.4.i.c 80
8.d odd 2 1 168.4.i.c 80
12.b even 2 1 168.4.i.c 80
21.c even 2 1 inner 672.4.i.c 80
24.f even 2 1 168.4.i.c 80
24.h odd 2 1 inner 672.4.i.c 80
28.d even 2 1 168.4.i.c 80
56.e even 2 1 168.4.i.c 80
56.h odd 2 1 inner 672.4.i.c 80
84.h odd 2 1 168.4.i.c 80
168.e odd 2 1 168.4.i.c 80
168.i even 2 1 inner 672.4.i.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.i.c 80 4.b odd 2 1
168.4.i.c 80 8.d odd 2 1
168.4.i.c 80 12.b even 2 1
168.4.i.c 80 24.f even 2 1
168.4.i.c 80 28.d even 2 1
168.4.i.c 80 56.e even 2 1
168.4.i.c 80 84.h odd 2 1
168.4.i.c 80 168.e odd 2 1
672.4.i.c 80 1.a even 1 1 trivial
672.4.i.c 80 3.b odd 2 1 inner
672.4.i.c 80 7.b odd 2 1 inner
672.4.i.c 80 8.b even 2 1 inner
672.4.i.c 80 21.c even 2 1 inner
672.4.i.c 80 24.h odd 2 1 inner
672.4.i.c 80 56.h odd 2 1 inner
672.4.i.c 80 168.i even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T520+1372T518+782932T516+239880544T514+42479046160T512++39 ⁣ ⁣12 T_{5}^{20} + 1372 T_{5}^{18} + 782932 T_{5}^{16} + 239880544 T_{5}^{14} + 42479046160 T_{5}^{12} + \cdots + 39\!\cdots\!12 acting on S4new(672,[χ])S_{4}^{\mathrm{new}}(672, [\chi]). Copy content Toggle raw display