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Properties

Label 280.2.s.c
Level $280$
Weight $2$
Character orbit 280.s
Analytic conductor $2.236$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(13,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.s (of order \(4\), degree \(2\), 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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{2} - 4 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{2} - 4 q^{7} - 16 q^{8} - 8 q^{15} + 40 q^{16} - 40 q^{18} + 4 q^{22} + 40 q^{23} - 8 q^{25} - 4 q^{28} - 44 q^{30} - 24 q^{32} - 72 q^{36} + 40 q^{42} + 16 q^{46} - 60 q^{50} + 32 q^{56} + 32 q^{57} - 68 q^{58} + 64 q^{60} - 100 q^{63} - 56 q^{65} + 16 q^{70} - 80 q^{71} + 16 q^{72} - 4 q^{78} + 72 q^{81} - 72 q^{86} + 48 q^{88} + 72 q^{92} + 80 q^{95} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.39086 − 0.255964i −1.92461 − 1.92461i 1.86896 + 0.712018i 2.22709 + 0.200181i 2.18423 + 3.16949i −0.516406 + 2.59487i −2.41721 − 1.46870i 4.40827i −3.04632 − 0.848477i
13.2 −1.39086 − 0.255964i 1.92461 + 1.92461i 1.86896 + 0.712018i −2.22709 − 0.200181i −2.18423 − 3.16949i −2.59487 + 0.516406i −2.41721 − 1.46870i 4.40827i 3.04632 + 0.848477i
13.3 −1.39041 + 0.258373i −1.88603 − 1.88603i 1.86649 − 0.718491i −1.79374 − 1.33510i 3.10966 + 2.13506i −2.30054 − 1.30671i −2.40954 + 1.48125i 4.11423i 2.83899 + 1.39289i
13.4 −1.39041 + 0.258373i 1.88603 + 1.88603i 1.86649 − 0.718491i 1.79374 + 1.33510i −3.10966 − 2.13506i 1.30671 + 2.30054i −2.40954 + 1.48125i 4.11423i −2.83899 − 1.39289i
13.5 −1.37024 + 0.349904i −0.152811 − 0.152811i 1.75513 − 0.958907i −2.05351 + 0.884920i 0.262858 + 0.155920i 2.63870 + 0.192988i −2.06944 + 1.92806i − 2.95330i 2.50418 − 1.93109i
13.6 −1.37024 + 0.349904i 0.152811 + 0.152811i 1.75513 − 0.958907i 2.05351 − 0.884920i −0.262858 − 0.155920i −0.192988 − 2.63870i −2.06944 + 1.92806i − 2.95330i −2.50418 + 1.93109i
13.7 −1.25643 − 0.649134i −0.639680 − 0.639680i 1.15725 + 1.63119i −1.45663 + 1.69653i 0.388478 + 1.21895i 0.0280541 + 2.64560i −0.395151 − 2.80069i − 2.18162i 2.93144 − 1.18603i
13.8 −1.25643 − 0.649134i 0.639680 + 0.639680i 1.15725 + 1.63119i 1.45663 − 1.69653i −0.388478 − 1.21895i −2.64560 − 0.0280541i −0.395151 − 2.80069i − 2.18162i −2.93144 + 1.18603i
13.9 −1.19418 − 0.757577i −1.10062 − 1.10062i 0.852154 + 1.80937i −0.557471 − 2.16546i 0.480540 + 2.14815i 2.20653 − 1.45987i 0.353110 − 2.80630i − 0.577263i −0.974781 + 3.00829i
13.10 −1.19418 − 0.757577i 1.10062 + 1.10062i 0.852154 + 1.80937i 0.557471 + 2.16546i −0.480540 − 2.14815i 1.45987 − 2.20653i 0.353110 − 2.80630i − 0.577263i 0.974781 − 3.00829i
13.11 −0.788440 − 1.17404i −1.96562 − 1.96562i −0.756725 + 1.85131i −0.686018 + 2.12823i −0.757936 + 3.85748i −1.91879 − 1.82161i 2.77014 − 0.571227i 4.72730i 3.03951 − 0.872574i
13.12 −0.788440 − 1.17404i 1.96562 + 1.96562i −0.756725 + 1.85131i 0.686018 − 2.12823i 0.757936 − 3.85748i 1.82161 + 1.91879i 2.77014 − 0.571227i 4.72730i −3.03951 + 0.872574i
13.13 −0.482209 − 1.32946i −0.851048 − 0.851048i −1.53495 + 1.28216i 2.18999 + 0.451591i −0.721055 + 1.54182i 2.50752 + 0.844017i 2.44475 + 1.42239i − 1.55144i −0.455660 − 3.12928i
13.14 −0.482209 − 1.32946i 0.851048 + 0.851048i −1.53495 + 1.28216i −2.18999 − 0.451591i 0.721055 − 1.54182i −0.844017 − 2.50752i 2.44475 + 1.42239i − 1.55144i 0.455660 + 3.12928i
13.15 −0.349904 + 1.37024i −0.152811 − 0.152811i −1.75513 − 0.958907i −2.05351 + 0.884920i 0.262858 − 0.155920i −0.192988 − 2.63870i 1.92806 − 2.06944i − 2.95330i −0.494024 − 3.12345i
13.16 −0.349904 + 1.37024i 0.152811 + 0.152811i −1.75513 − 0.958907i 2.05351 − 0.884920i −0.262858 + 0.155920i 2.63870 + 0.192988i 1.92806 − 2.06944i − 2.95330i 0.494024 + 3.12345i
13.17 −0.258373 + 1.39041i −1.88603 − 1.88603i −1.86649 − 0.718491i −1.79374 − 1.33510i 3.10966 − 2.13506i 1.30671 + 2.30054i 1.48125 − 2.40954i 4.11423i 2.31980 − 2.14908i
13.18 −0.258373 + 1.39041i 1.88603 + 1.88603i −1.86649 − 0.718491i 1.79374 + 1.33510i −3.10966 + 2.13506i −2.30054 − 1.30671i 1.48125 − 2.40954i 4.11423i −2.31980 + 2.14908i
13.19 0.00402547 − 1.41421i −1.09655 − 1.09655i −1.99997 − 0.0113857i 0.721255 − 2.11655i −1.55517 + 1.54634i −2.63247 + 0.264726i −0.0241525 + 2.82832i − 0.595144i −2.99034 − 1.02852i
13.20 0.00402547 − 1.41421i 1.09655 + 1.09655i −1.99997 − 0.0113857i −0.721255 + 2.11655i 1.55517 − 1.54634i −0.264726 + 2.63247i −0.0241525 + 2.82832i − 0.595144i 2.99034 + 1.02852i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
56.h odd 2 1 inner
280.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.s.c ✓ 72
4.b odd 2 1 1120.2.w.c 72
5.c odd 4 1 inner 280.2.s.c ✓ 72
7.b odd 2 1 inner 280.2.s.c ✓ 72
8.b even 2 1 inner 280.2.s.c ✓ 72
8.d odd 2 1 1120.2.w.c 72
20.e even 4 1 1120.2.w.c 72
28.d even 2 1 1120.2.w.c 72
35.f even 4 1 inner 280.2.s.c ✓ 72
40.i odd 4 1 inner 280.2.s.c ✓ 72
40.k even 4 1 1120.2.w.c 72
56.e even 2 1 1120.2.w.c 72
56.h odd 2 1 inner 280.2.s.c ✓ 72
140.j odd 4 1 1120.2.w.c 72
280.s even 4 1 inner 280.2.s.c ✓ 72
280.y odd 4 1 1120.2.w.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.c ✓ 72 1.a even 1 1 trivial
280.2.s.c ✓ 72 5.c odd 4 1 inner
280.2.s.c ✓ 72 7.b odd 2 1 inner
280.2.s.c ✓ 72 8.b even 2 1 inner
280.2.s.c ✓ 72 35.f even 4 1 inner
280.2.s.c ✓ 72 40.i odd 4 1 inner
280.2.s.c ✓ 72 56.h odd 2 1 inner
280.2.s.c ✓ 72 280.s even 4 1 inner
1120.2.w.c 72 4.b odd 2 1
1120.2.w.c 72 8.d odd 2 1
1120.2.w.c 72 20.e even 4 1
1120.2.w.c 72 28.d even 2 1
1120.2.w.c 72 40.k even 4 1
1120.2.w.c 72 56.e even 2 1
1120.2.w.c 72 140.j odd 4 1
1120.2.w.c 72 280.y odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 180 T_{3}^{32} + 11594 T_{3}^{28} + 312340 T_{3}^{24} + 3138761 T_{3}^{20} + 13351184 T_{3}^{16} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display