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Properties

Label 280.2.bv.e
Level $280$
Weight $2$
Character orbit 280.bv
Analytic conductor $2.236$
Analytic rank $0$
Dimension $160$
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(117,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.bv (of order \(12\), degree \(4\), 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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{2} + 12 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{2} + 12 q^{7} + 4 q^{8} - 6 q^{10} + 6 q^{12} - 8 q^{15} + 4 q^{16} - 12 q^{17} - 28 q^{18} - 24 q^{22} - 16 q^{23} + 20 q^{25} - 12 q^{26} - 46 q^{28} + 32 q^{30} + 48 q^{31} + 18 q^{32} - 12 q^{33} - 32 q^{36} - 48 q^{38} + 54 q^{40} + 6 q^{42} - 64 q^{46} - 132 q^{47} - 12 q^{50} - 20 q^{56} - 88 q^{57} + 6 q^{58} + 34 q^{60} - 32 q^{63} - 28 q^{65} - 180 q^{66} + 60 q^{68} - 108 q^{70} - 160 q^{71} + 52 q^{72} + 84 q^{73} + 48 q^{78} - 48 q^{80} + 16 q^{81} - 90 q^{82} - 84 q^{86} - 12 q^{87} + 44 q^{88} + 36 q^{92} - 20 q^{95} - 48 q^{96} - 94 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −1.41101 − 0.0951125i −2.31786 − 0.621070i 1.98191 + 0.268410i 1.20230 + 1.88533i 3.21146 + 1.09679i −0.485979 − 2.60074i −2.77096 − 0.567234i 2.38869 + 1.37911i −1.51714 − 2.77458i
117.2 −1.40415 + 0.168429i −2.01733 − 0.540542i 1.94326 − 0.472997i −1.27480 − 1.83708i 2.92367 + 0.419225i 2.57779 − 0.595830i −2.64896 + 0.991460i 1.17936 + 0.680902i 2.09943 + 2.36482i
117.3 −1.39835 + 0.211218i 1.14290 + 0.306239i 1.91077 − 0.590715i −2.15054 + 0.612514i −1.66286 − 0.186828i −0.757877 + 2.53488i −2.54716 + 1.22962i −1.38564 − 0.800000i 2.87784 − 1.31074i
117.4 −1.31662 + 0.516255i −1.14290 − 0.306239i 1.46696 − 1.35942i 2.15054 − 0.612514i 1.66286 − 0.186828i −0.757877 + 2.53488i −1.22962 + 2.54716i −1.38564 − 0.800000i −2.51523 + 1.91668i
117.5 −1.30024 + 0.556211i 2.01733 + 0.540542i 1.38126 − 1.44642i 1.27480 + 1.83708i −2.92367 + 0.419225i 2.57779 − 0.595830i −0.991460 + 2.64896i 1.17936 + 0.680902i −2.67936 − 1.67960i
117.6 −1.29882 − 0.559518i 1.21391 + 0.325266i 1.37388 + 1.45343i 2.11050 − 0.738764i −1.39466 − 1.10167i 2.63915 − 0.186831i −0.971208 − 2.65646i −1.23029 − 0.710311i −3.15452 − 0.221340i
117.7 −1.24982 − 0.661775i 3.03007 + 0.811904i 1.12411 + 1.65420i 0.203394 + 2.22680i −3.24974 − 3.01996i −2.64044 − 0.167503i −0.310224 − 2.81136i 5.92404 + 3.42025i 1.21943 − 2.91770i
117.8 −1.17442 + 0.787876i 2.31786 + 0.621070i 0.758504 − 1.85059i −1.20230 − 1.88533i −3.21146 + 1.09679i −0.485979 − 2.60074i 0.567234 + 2.77096i 2.38869 + 1.37911i 2.89741 + 1.26690i
117.9 −1.16877 − 0.796228i −2.48518 − 0.665902i 0.732043 + 1.86121i −2.21132 + 0.331769i 2.37439 + 2.75706i −2.18569 + 1.49090i 0.626360 − 2.75820i 3.13462 + 1.80978i 2.84868 + 1.37295i
117.10 −0.910407 − 1.08220i −0.0340321 − 0.00911887i −0.342320 + 1.97049i −1.86153 + 1.23883i 0.0211146 + 0.0451315i 1.05182 − 2.42769i 2.44411 − 1.42349i −2.59700 − 1.49938i 3.03542 + 0.886711i
117.11 −0.880343 − 1.10680i 1.90798 + 0.511242i −0.449991 + 1.94872i −1.79531 − 1.33299i −1.11384 − 2.56181i 1.57290 + 2.12743i 2.55298 − 1.21749i 0.780947 + 0.450880i 0.105136 + 3.16053i
117.12 −0.868191 − 1.11635i −2.84558 − 0.762471i −0.492489 + 1.93842i 1.93059 − 1.12820i 1.61932 + 3.83864i 2.58960 + 0.542205i 2.59153 − 1.13312i 4.91789 + 2.83934i −2.93558 − 1.17573i
117.13 −0.845055 + 1.13397i −1.21391 − 0.325266i −0.571766 − 1.91653i −2.11050 + 0.738764i 1.39466 − 1.10167i 2.63915 − 0.186831i 2.65646 + 0.971208i −1.23029 − 0.710311i 0.945756 − 3.01754i
117.14 −0.751490 + 1.19802i −3.03007 − 0.811904i −0.870527 − 1.80061i −0.203394 − 2.22680i 3.24974 − 3.01996i −2.64044 − 0.167503i 2.81136 + 0.310224i 5.92404 + 3.42025i 2.82061 + 1.42975i
117.15 −0.614070 + 1.27394i 2.48518 + 0.665902i −1.24584 − 1.56457i 2.21132 − 0.331769i −2.37439 + 2.75706i −2.18569 + 1.49090i 2.75820 − 0.626360i 3.13462 + 1.80978i −0.935251 + 3.02081i
117.16 −0.464402 − 1.33579i −0.0539369 − 0.0144524i −1.56866 + 1.24069i 1.91563 + 1.15341i 0.00574315 + 0.0787600i −1.56007 − 2.13686i 2.38578 + 1.51922i −2.59538 − 1.49844i 0.651085 − 3.09453i
117.17 −0.279851 − 1.38625i 2.92962 + 0.784989i −1.84337 + 0.775886i 1.27091 − 1.83978i 0.268333 − 4.28086i 0.0664755 − 2.64492i 1.59144 + 2.33823i 5.36839 + 3.09944i −2.90605 − 1.24694i
117.18 −0.247335 + 1.39242i 0.0340321 + 0.00911887i −1.87765 − 0.688786i 1.86153 − 1.23883i −0.0211146 + 0.0451315i 1.05182 − 2.42769i 1.42349 − 2.44411i −2.59700 − 1.49938i 1.26455 + 2.89843i
117.19 −0.240080 − 1.39369i −0.931735 − 0.249658i −1.88472 + 0.669192i −0.0962936 − 2.23399i −0.124254 + 1.35848i −1.59310 + 2.11235i 1.38513 + 2.46605i −1.79228 − 1.03477i −3.09037 + 0.670540i
117.20 −0.209002 + 1.39868i −1.90798 − 0.511242i −1.91264 − 0.584656i 1.79531 + 1.33299i 1.11384 − 2.56181i 1.57290 + 2.12743i 1.21749 − 2.55298i 0.780947 + 0.450880i −2.23966 + 2.23247i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
35.k even 12 1 inner
40.i odd 4 1 inner
56.j odd 6 1 inner
280.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bv.e ✓ 160
5.c odd 4 1 inner 280.2.bv.e ✓ 160
7.d odd 6 1 inner 280.2.bv.e ✓ 160
8.b even 2 1 inner 280.2.bv.e ✓ 160
35.k even 12 1 inner 280.2.bv.e ✓ 160
40.i odd 4 1 inner 280.2.bv.e ✓ 160
56.j odd 6 1 inner 280.2.bv.e ✓ 160
280.bv even 12 1 inner 280.2.bv.e ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bv.e ✓ 160 1.a even 1 1 trivial
280.2.bv.e ✓ 160 5.c odd 4 1 inner
280.2.bv.e ✓ 160 7.d odd 6 1 inner
280.2.bv.e ✓ 160 8.b even 2 1 inner
280.2.bv.e ✓ 160 35.k even 12 1 inner
280.2.bv.e ✓ 160 40.i odd 4 1 inner
280.2.bv.e ✓ 160 56.j odd 6 1 inner
280.2.bv.e ✓ 160 280.bv even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} - 544 T_{3}^{156} + 166312 T_{3}^{152} - 34827424 T_{3}^{148} + 5517313412 T_{3}^{144} + \cdots + 25600000000 \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display