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Properties

Label 336.2.bq.b
Level $336$
Weight $2$
Character orbit 336.bq
Analytic conductor $2.683$
Analytic rank $0$
Dimension $120$
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] = [336,2,Mod(37,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bq (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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8} - 8 q^{10} + 4 q^{11} + 8 q^{13} + 32 q^{14} + 8 q^{15} + 12 q^{16} - 16 q^{17} - 4 q^{18} - 8 q^{19} - 64 q^{20} + 12 q^{21} + 24 q^{22} - 8 q^{24} - 8 q^{26} - 8 q^{28} - 64 q^{29} - 52 q^{31} - 4 q^{33} + 32 q^{34} - 8 q^{35} + 24 q^{37} - 40 q^{38} - 60 q^{40} + 24 q^{42} - 32 q^{43} - 12 q^{44} - 8 q^{45} + 4 q^{46} - 8 q^{47} + 32 q^{48} + 20 q^{49} - 16 q^{50} - 8 q^{51} - 16 q^{52} + 4 q^{54} - 104 q^{56} + 4 q^{58} + 52 q^{59} - 16 q^{60} + 4 q^{61} - 144 q^{62} + 8 q^{63} - 64 q^{64} + 24 q^{65} + 24 q^{66} + 12 q^{68} + 24 q^{69} - 48 q^{70} + 4 q^{72} + 16 q^{74} + 16 q^{75} - 8 q^{76} + 8 q^{77} - 56 q^{78} + 28 q^{79} + 68 q^{80} + 60 q^{81} + 28 q^{82} + 8 q^{83} + 24 q^{84} - 80 q^{85} + 36 q^{86} + 40 q^{88} + 8 q^{90} + 36 q^{91} - 184 q^{92} + 4 q^{93} + 60 q^{94} - 16 q^{95} + 36 q^{96} + 24 q^{97} + 124 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.39055 − 0.257600i 0.965926 + 0.258819i 1.86728 + 0.716415i −0.588887 + 0.157792i −1.27650 − 0.608725i 2.58295 − 0.573024i −2.41201 − 1.47723i 0.866025 + 0.500000i 0.859526 − 0.0677205i
37.2 −1.37310 + 0.338535i −0.965926 − 0.258819i 1.77079 − 0.929682i 0.552230 − 0.147970i 1.41393 + 0.0283840i −2.64020 + 0.171366i −2.11673 + 1.87602i 0.866025 + 0.500000i −0.708172 + 0.390126i
37.3 −1.33496 + 0.466767i 0.965926 + 0.258819i 1.56426 − 1.24623i 3.48274 − 0.933198i −1.41028 + 0.105348i 0.681804 + 2.55639i −1.50653 + 2.39382i 0.866025 + 0.500000i −4.21375 + 2.87141i
37.4 −1.30822 − 0.537189i −0.965926 − 0.258819i 1.42286 + 1.40552i −1.52423 + 0.408415i 1.12460 + 0.857476i 1.09198 − 2.40989i −1.10637 − 2.60306i 0.866025 + 0.500000i 2.21341 + 0.284503i
37.5 −1.30731 + 0.539398i 0.965926 + 0.258819i 1.41810 − 1.41032i −3.17326 + 0.850273i −1.40237 + 0.182663i −2.50019 − 0.865491i −1.09317 + 2.60864i 0.866025 + 0.500000i 3.68979 − 2.82322i
37.6 −1.17868 − 0.781485i 0.965926 + 0.258819i 0.778563 + 1.84224i −1.63016 + 0.436800i −0.936252 − 1.05992i −2.22530 + 1.43110i 0.522007 − 2.77984i 0.866025 + 0.500000i 2.26279 + 0.759099i
37.7 −1.02134 + 0.978190i −0.965926 − 0.258819i 0.0862884 − 1.99814i −3.48258 + 0.933154i 1.23972 − 0.680516i 1.89801 − 1.84325i 1.86643 + 2.12519i 0.866025 + 0.500000i 2.64411 − 4.35970i
37.8 −0.978024 + 1.02150i −0.965926 − 0.258819i −0.0869382 − 1.99811i 0.298815 − 0.0800672i 1.20908 − 0.733565i −2.15167 + 1.53959i 2.12610 + 1.86539i 0.866025 + 0.500000i −0.210459 + 0.383548i
37.9 −0.975065 − 1.02433i 0.965926 + 0.258819i −0.0984971 + 1.99757i 3.52312 − 0.944018i −0.676725 − 1.24179i −0.745861 − 2.53844i 2.14221 − 1.84687i 0.866025 + 0.500000i −4.40226 − 2.68836i
37.10 −0.948203 − 1.04924i −0.965926 − 0.258819i −0.201821 + 1.98979i −2.71136 + 0.726506i 0.644330 + 1.25890i −0.393486 + 2.61633i 2.27914 − 1.67497i 0.866025 + 0.500000i 3.33320 + 2.15600i
37.11 −0.687104 + 1.23608i 0.965926 + 0.258819i −1.05578 − 1.69863i 0.137710 − 0.0368993i −0.983612 + 1.01612i −2.08565 + 1.62790i 2.82506 − 0.137888i 0.866025 + 0.500000i −0.0490107 + 0.195574i
37.12 −0.485523 − 1.32826i −0.965926 − 0.258819i −1.52854 + 1.28980i −0.879995 + 0.235794i 0.125201 + 1.40866i −1.69105 − 2.03479i 2.45532 + 1.40406i 0.866025 + 0.500000i 0.740453 + 1.05438i
37.13 −0.453022 + 1.33969i 0.965926 + 0.258819i −1.58954 − 1.21382i 1.78314 − 0.477791i −0.784323 + 1.17679i 0.786362 − 2.52619i 2.34624 − 1.57961i 0.866025 + 0.500000i −0.167710 + 2.60531i
37.14 −0.244495 − 1.39292i 0.965926 + 0.258819i −1.88044 + 0.681122i −3.28948 + 0.881413i 0.124350 − 1.40874i 2.12312 + 1.57873i 1.40851 + 2.45278i 0.866025 + 0.500000i 2.03200 + 4.36648i
37.15 −0.218928 − 1.39717i −0.965926 − 0.258819i −1.90414 + 0.611757i 2.14612 − 0.575051i −0.150145 + 1.40622i 2.53649 − 0.752486i 1.27160 + 2.52647i 0.866025 + 0.500000i −1.27329 − 2.87259i
37.16 −0.212205 − 1.39820i 0.965926 + 0.258819i −1.90994 + 0.593412i 2.46123 − 0.659483i 0.156907 − 1.40548i −1.68235 + 2.04198i 1.23501 + 2.54455i 0.866025 + 0.500000i −1.44438 − 3.30134i
37.17 −0.120841 + 1.40904i −0.965926 − 0.258819i −1.97080 − 0.340539i 0.433260 − 0.116092i 0.481410 − 1.32975i 2.04258 + 1.68163i 0.717986 − 2.73578i 0.866025 + 0.500000i 0.111222 + 0.624510i
37.18 0.0367692 + 1.41374i −0.965926 − 0.258819i −1.99730 + 0.103964i −0.377111 + 0.101047i 0.330385 − 1.37508i −1.14433 − 2.38548i −0.220416 − 2.81983i 0.866025 + 0.500000i −0.156719 − 0.529419i
37.19 0.426746 + 1.34829i 0.965926 + 0.258819i −1.63577 + 1.15076i −3.88430 + 1.04080i 0.0632421 + 1.41280i −0.638884 − 2.56746i −2.24962 − 1.71442i 0.866025 + 0.500000i −3.06091 − 4.79301i
37.20 0.608093 − 1.27680i 0.965926 + 0.258819i −1.26045 − 1.55283i 2.60505 − 0.698021i 0.917833 − 1.07591i 2.64133 − 0.152886i −2.74912 + 0.665077i 0.866025 + 0.500000i 0.692877 − 3.75059i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bq.b ✓ 120
7.c even 3 1 inner 336.2.bq.b ✓ 120
16.e even 4 1 inner 336.2.bq.b ✓ 120
112.w even 12 1 inner 336.2.bq.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bq.b ✓ 120 1.a even 1 1 trivial
336.2.bq.b ✓ 120 7.c even 3 1 inner
336.2.bq.b ✓ 120 16.e even 4 1 inner
336.2.bq.b ✓ 120 112.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{120} + 8 T_{5}^{119} + 32 T_{5}^{118} + 176 T_{5}^{117} - 47 T_{5}^{116} - 4896 T_{5}^{115} + \cdots + 13\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display