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Properties

Label 273.2.cc.a
Level $273$
Weight $2$
Character orbit 273.cc
Analytic conductor $2.180$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(50,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cc (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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 12 q^{6} - 48 q^{10} - 32 q^{13} + 4 q^{15} + 40 q^{16} - 16 q^{18} - 8 q^{19} - 4 q^{21} - 16 q^{22} - 88 q^{24} + 24 q^{27} - 72 q^{30} + 16 q^{31} + 48 q^{34} + 12 q^{36} + 48 q^{37} + 56 q^{39} + 32 q^{40} - 28 q^{45} + 72 q^{46} + 24 q^{48} - 144 q^{52} - 108 q^{54} - 28 q^{57} - 120 q^{58} - 116 q^{60} - 48 q^{61} + 16 q^{63} + 40 q^{66} - 16 q^{67} + 72 q^{69} + 48 q^{70} + 52 q^{72} - 16 q^{73} + 60 q^{75} + 16 q^{76} - 4 q^{78} + 16 q^{79} - 20 q^{81} + 120 q^{82} + 72 q^{84} - 40 q^{85} - 24 q^{87} + 72 q^{88} + 16 q^{91} + 92 q^{93} - 96 q^{94} + 28 q^{96} + 96 q^{97} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −2.59589 − 0.695566i −1.18735 − 1.26103i 4.52278 + 2.61123i 0.855518 − 0.855518i 2.20509 + 4.09938i 0.258819 + 0.965926i −6.12370 − 6.12370i −0.180409 + 2.99457i −2.81590 + 1.62576i
50.2 −2.46820 − 0.661352i 1.55028 − 0.772424i 3.92257 + 2.26470i 2.12589 − 2.12589i −4.33724 + 0.881217i −0.258819 − 0.965926i −4.57023 − 4.57023i 1.80672 − 2.39494i −6.65309 + 3.84116i
50.3 −2.24002 − 0.600211i 1.07699 − 1.35650i 2.92537 + 1.68896i −2.36227 + 2.36227i −3.22666 + 2.39216i 0.258819 + 0.965926i −2.25953 − 2.25953i −0.680177 − 2.92188i 6.70937 − 3.87366i
50.4 −2.14995 − 0.576078i 1.44893 + 0.949002i 2.55837 + 1.47708i −1.66934 + 1.66934i −2.56843 − 2.87500i −0.258819 − 0.965926i −1.50172 − 1.50172i 1.19879 + 2.75007i 4.55066 − 2.62732i
50.5 −2.03579 − 0.545489i −0.604204 + 1.62325i 2.11484 + 1.22100i 2.41109 − 2.41109i 2.11550 − 2.97501i −0.258819 − 0.965926i −0.658724 − 0.658724i −2.26987 − 1.96155i −6.22370 + 3.59326i
50.6 −1.84038 − 0.493129i −1.11254 − 1.32750i 1.41178 + 0.815092i −1.68431 + 1.68431i 1.39288 + 2.99173i −0.258819 − 0.965926i 0.498237 + 0.498237i −0.524491 + 2.95380i 3.93035 − 2.26919i
50.7 −1.78824 − 0.479158i −1.54238 + 0.788082i 1.23617 + 0.713702i 0.0341444 − 0.0341444i 3.13576 − 0.670240i 0.258819 + 0.965926i 0.749576 + 0.749576i 1.75785 − 2.43104i −0.0774191 + 0.0446979i
50.8 −1.74208 − 0.466790i 0.804463 + 1.53390i 1.08492 + 0.626376i 0.450139 − 0.450139i −0.685436 − 3.04769i 0.258819 + 0.965926i 0.952961 + 0.952961i −1.70568 + 2.46793i −0.994301 + 0.574060i
50.9 −1.06694 − 0.285886i 1.72116 − 0.193962i −0.675418 − 0.389953i 1.84315 − 1.84315i −1.89182 − 0.285109i 0.258819 + 0.965926i 2.17126 + 2.17126i 2.92476 − 0.667676i −2.49346 + 1.43960i
50.10 −0.790296 − 0.211759i −0.427528 + 1.67846i −1.15232 − 0.665295i −2.01842 + 2.01842i 0.693303 − 1.23595i −0.258819 − 0.965926i 1.92687 + 1.92687i −2.63444 − 1.43518i 2.02257 − 1.16773i
50.11 −0.751420 − 0.201342i −1.18266 − 1.26543i −1.20796 − 0.697414i 2.48091 − 2.48091i 0.633888 + 1.18899i −0.258819 − 0.965926i 1.86742 + 1.86742i −0.202642 + 2.99315i −2.36372 + 1.36469i
50.12 −0.609155 − 0.163223i −1.67612 − 0.436610i −1.38762 − 0.801144i −1.27442 + 1.27442i 0.949752 + 0.539544i 0.258819 + 0.965926i 1.60638 + 1.60638i 2.61874 + 1.46362i 0.984333 − 0.568305i
50.13 −0.397883 − 0.106613i −1.64214 + 0.550805i −1.58511 − 0.915161i 0.406846 − 0.406846i 0.712102 − 0.0440836i −0.258819 − 0.965926i 1.11566 + 1.11566i 2.39323 − 1.80899i −0.205252 + 0.118502i
50.14 −0.307973 − 0.0825212i 1.72958 + 0.0924370i −1.64401 − 0.949171i −2.80070 + 2.80070i −0.525037 − 0.171195i 0.258819 + 0.965926i 0.878890 + 0.878890i 2.98291 + 0.319755i 1.09366 − 0.631424i
50.15 0.307973 + 0.0825212i −0.784738 + 1.54408i −1.64401 − 0.949171i 2.80070 − 2.80070i −0.369098 + 0.410778i 0.258819 + 0.965926i −0.878890 − 0.878890i −1.76837 − 2.42340i 1.09366 − 0.631424i
50.16 0.397883 + 0.106613i 1.29808 − 1.14673i −1.58511 − 0.915161i −0.406846 + 0.406846i 0.638740 − 0.317873i −0.258819 − 0.965926i −1.11566 − 1.11566i 0.370020 − 2.97709i −0.205252 + 0.118502i
50.17 0.609155 + 0.163223i 0.459944 − 1.66987i −1.38762 − 0.801144i 1.27442 − 1.27442i 0.552737 − 0.942134i 0.258819 + 0.965926i −1.60638 − 1.60638i −2.57690 − 1.53609i 0.984333 − 0.568305i
50.18 0.751420 + 0.201342i −0.504568 − 1.65693i −1.20796 − 0.697414i −2.48091 + 2.48091i −0.0455330 − 1.34664i −0.258819 − 0.965926i −1.86742 − 1.86742i −2.49082 + 1.67207i −2.36372 + 1.36469i
50.19 0.790296 + 0.211759i 1.66735 + 0.468978i −1.15232 − 0.665295i 2.01842 − 2.01842i 1.21839 + 0.723709i −0.258819 − 0.965926i −1.92687 − 1.92687i 2.56012 + 1.56390i 2.02257 − 1.16773i
50.20 1.06694 + 0.285886i −1.02855 + 1.39358i −0.675418 − 0.389953i −1.84315 + 1.84315i −1.49581 + 1.19282i 0.258819 + 0.965926i −2.17126 − 2.17126i −0.884154 − 2.86675i −2.49346 + 1.43960i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cc.a ✓ 112
3.b odd 2 1 inner 273.2.cc.a ✓ 112
13.f odd 12 1 inner 273.2.cc.a ✓ 112
39.k even 12 1 inner 273.2.cc.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cc.a ✓ 112 1.a even 1 1 trivial
273.2.cc.a ✓ 112 3.b odd 2 1 inner
273.2.cc.a ✓ 112 13.f odd 12 1 inner
273.2.cc.a ✓ 112 39.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(273, [\chi])\).