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Properties

Label 663.2.bk.a
Level $663$
Weight $2$
Character orbit 663.bk
Analytic conductor $5.294$
Analytic rank $0$
Dimension $160$
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] = [663,2,Mod(4,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.bk (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: \(5.29408165401\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 72 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 72 q^{4} - 12 q^{11} - 16 q^{13} - 16 q^{14} - 56 q^{16} + 20 q^{17} - 72 q^{20} - 20 q^{22} - 4 q^{23} - 96 q^{28} + 12 q^{29} + 16 q^{30} - 12 q^{33} + 12 q^{37} + 80 q^{38} + 8 q^{39} + 12 q^{41} + 12 q^{45} + 12 q^{46} - 16 q^{48} - 40 q^{52} - 104 q^{55} + 20 q^{56} - 12 q^{61} - 12 q^{62} - 48 q^{64} + 68 q^{65} - 96 q^{67} + 88 q^{68} - 36 q^{69} - 56 q^{74} + 16 q^{75} + 12 q^{78} + 144 q^{80} + 80 q^{81} - 60 q^{82} + 12 q^{85} - 100 q^{88} - 32 q^{91} - 16 q^{92} - 64 q^{95} + 48 q^{97} + 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.40321 + 2.43042i 0.258819 + 0.965926i −2.93797 − 5.08872i −1.10816 − 1.10816i −2.71078 − 0.726353i 2.06892 + 0.554366i 10.8775 −0.866025 + 0.500000i 4.24828 − 1.13832i
4.2 −1.31893 + 2.28445i −0.258819 − 0.965926i −2.47915 − 4.29401i −1.83196 − 1.83196i 2.54798 + 0.682728i −0.551277 − 0.147714i 7.80358 −0.866025 + 0.500000i 6.60125 − 1.76880i
4.3 −1.24096 + 2.14941i −0.258819 − 0.965926i −2.07998 − 3.60263i 1.53700 + 1.53700i 2.39736 + 0.642370i 4.32503 + 1.15889i 5.36085 −0.866025 + 0.500000i −5.21099 + 1.39628i
4.4 −1.12117 + 1.94193i 0.258819 + 0.965926i −1.51405 − 2.62242i 2.94385 + 2.94385i −2.16594 − 0.580361i −0.231880 − 0.0621320i 2.30537 −0.866025 + 0.500000i −9.01730 + 2.41618i
4.5 −1.07043 + 1.85404i 0.258819 + 0.965926i −1.29164 − 2.23719i −0.907333 − 0.907333i −2.06791 − 0.554096i 0.185748 + 0.0497710i 1.24874 −0.866025 + 0.500000i 2.65347 − 0.710995i
4.6 −1.06635 + 1.84698i −0.258819 − 0.965926i −1.27422 − 2.20702i −1.78870 − 1.78870i 2.06004 + 0.551986i −2.81705 − 0.754826i 1.16967 −0.866025 + 0.500000i 5.21109 − 1.39631i
4.7 −0.975846 + 1.69022i 0.258819 + 0.965926i −0.904551 − 1.56673i −2.86604 − 2.86604i −1.88519 − 0.505135i 2.96314 + 0.793970i −0.372573 −0.866025 + 0.500000i 7.64104 − 2.04741i
4.8 −0.962735 + 1.66751i −0.258819 − 0.965926i −0.853717 − 1.47868i 1.37658 + 1.37658i 1.85986 + 0.498348i 0.768177 + 0.205832i −0.563326 −0.866025 + 0.500000i −3.62075 + 0.970176i
4.9 −0.956419 + 1.65657i 0.258819 + 0.965926i −0.829474 − 1.43669i −0.00173979 − 0.00173979i −1.84766 − 0.495079i −2.32467 − 0.622894i −0.652377 −0.866025 + 0.500000i 0.00454606 − 0.00121811i
4.10 −0.830936 + 1.43922i −0.258819 − 0.965926i −0.380909 − 0.659753i −1.53696 − 1.53696i 1.60524 + 0.430124i 2.34758 + 0.629033i −2.05770 −0.866025 + 0.500000i 3.48915 − 0.934914i
4.11 −0.819994 + 1.42027i −0.258819 − 0.965926i −0.344780 − 0.597177i 1.16160 + 1.16160i 1.58411 + 0.424460i −5.06174 − 1.35629i −2.14911 −0.866025 + 0.500000i −2.60229 + 0.697282i
4.12 −0.805114 + 1.39450i 0.258819 + 0.965926i −0.296418 − 0.513411i 0.638177 + 0.638177i −1.55536 − 0.416758i −2.23352 − 0.598470i −2.26586 −0.866025 + 0.500000i −1.40374 + 0.376132i
4.13 −0.516293 + 0.894246i −0.258819 − 0.965926i 0.466883 + 0.808664i −0.976675 − 0.976675i 0.997402 + 0.267253i 1.63464 + 0.438000i −3.02937 −0.866025 + 0.500000i 1.37764 − 0.369137i
4.14 −0.506318 + 0.876969i −0.258819 − 0.965926i 0.487284 + 0.844001i 1.97244 + 1.97244i 0.978131 + 0.262089i 1.16641 + 0.312538i −3.01216 −0.866025 + 0.500000i −2.72845 + 0.731086i
4.15 −0.392121 + 0.679173i 0.258819 + 0.965926i 0.692483 + 1.19942i 0.395877 + 0.395877i −0.757519 − 0.202977i 3.97438 + 1.06493i −2.65463 −0.866025 + 0.500000i −0.424101 + 0.113637i
4.16 −0.357917 + 0.619931i 0.258819 + 0.965926i 0.743790 + 1.28828i 1.51606 + 1.51606i −0.691443 − 0.185272i 0.909274 + 0.243639i −2.49653 −0.866025 + 0.500000i −1.48248 + 0.397228i
4.17 −0.270724 + 0.468907i 0.258819 + 0.965926i 0.853417 + 1.47816i −2.51716 − 2.51716i −0.522998 − 0.140137i 1.95366 + 0.523482i −2.00706 −0.866025 + 0.500000i 1.86177 − 0.498860i
4.18 −0.168070 + 0.291106i −0.258819 − 0.965926i 0.943505 + 1.63420i −0.342697 − 0.342697i 0.324686 + 0.0869994i −3.51667 − 0.942288i −1.30658 −0.866025 + 0.500000i 0.157358 − 0.0421640i
4.19 −0.121476 + 0.210403i −0.258819 − 0.965926i 0.970487 + 1.68093i −1.56852 − 1.56852i 0.234674 + 0.0628806i 1.42160 + 0.380918i −0.957468 −0.866025 + 0.500000i 0.520558 − 0.139483i
4.20 −0.107662 + 0.186476i 0.258819 + 0.965926i 0.976818 + 1.69190i 2.08922 + 2.08922i −0.207987 − 0.0557299i −5.04278 − 1.35121i −0.851311 −0.866025 + 0.500000i −0.614519 + 0.164660i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner
17.c even 4 1 inner
221.s even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.bk.a ✓ 160
13.e even 6 1 inner 663.2.bk.a ✓ 160
17.c even 4 1 inner 663.2.bk.a ✓ 160
221.s even 12 1 inner 663.2.bk.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.bk.a ✓ 160 1.a even 1 1 trivial
663.2.bk.a ✓ 160 13.e even 6 1 inner
663.2.bk.a ✓ 160 17.c even 4 1 inner
663.2.bk.a ✓ 160 221.s even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).