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Properties

Label 336.2.bs.a
Level $336$
Weight $2$
Character orbit 336.bs
Analytic conductor $2.683$
Analytic rank $0$
Dimension $128$
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(19,336)]
 
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([6, 9, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bs (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: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{4} - 24 q^{8} + 36 q^{10} - 8 q^{11} - 32 q^{14} - 4 q^{16} + 4 q^{18} + 16 q^{22} + 16 q^{23} - 32 q^{28} + 32 q^{29} - 24 q^{35} - 16 q^{37} + 60 q^{40} - 20 q^{42} - 16 q^{43} - 12 q^{44} - 20 q^{46} - 32 q^{50} - 144 q^{52} - 16 q^{53} - 56 q^{56} - 40 q^{58} - 96 q^{59} - 24 q^{60} - 32 q^{64} - 72 q^{66} + 16 q^{67} - 60 q^{68} + 28 q^{70} + 128 q^{71} - 4 q^{72} - 72 q^{74} + 72 q^{78} - 36 q^{80} + 64 q^{81} - 60 q^{82} + 24 q^{84} - 44 q^{86} - 48 q^{88} - 8 q^{91} + 56 q^{92} + 36 q^{94} + 60 q^{96} + 148 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41385 + 0.0322336i −0.258819 + 0.965926i 1.99792 − 0.0911467i −3.59645 + 0.963666i 0.334795 − 1.37401i −1.51089 + 2.17191i −2.82182 + 0.193268i −0.866025 − 0.500000i 5.05377 − 1.47840i
19.2 −1.41082 − 0.0979362i 0.258819 − 0.965926i 1.98082 + 0.276340i −0.896422 + 0.240196i −0.459746 + 1.33740i −0.720444 + 2.54577i −2.76751 − 0.583860i −0.866025 − 0.500000i 1.28821 − 0.251080i
19.3 −1.35023 − 0.420569i −0.258819 + 0.965926i 1.64624 + 1.13573i 2.34159 − 0.627426i 0.755704 − 1.19537i 2.44702 − 1.00602i −1.74516 − 2.22586i −0.866025 − 0.500000i −3.42556 − 0.137628i
19.4 −1.34503 + 0.436906i −0.258819 + 0.965926i 1.61823 − 1.17531i −1.46978 + 0.393827i −0.0738984 − 1.41228i 0.798557 − 2.52236i −1.66307 + 2.28784i −0.866025 − 0.500000i 1.80484 − 1.17187i
19.5 −1.29731 + 0.563017i 0.258819 − 0.965926i 1.36602 − 1.46081i 3.94718 − 1.05764i 0.208064 + 1.39882i 2.38481 − 1.14572i −0.949691 + 2.66422i −0.866025 − 0.500000i −4.52524 + 3.59442i
19.6 −1.27158 − 0.618944i 0.258819 − 0.965926i 1.23382 + 1.57407i 2.19308 − 0.587633i −0.926962 + 1.06806i −2.21461 − 1.44758i −0.594635 − 2.76521i −0.866025 − 0.500000i −3.15238 − 0.610169i
19.7 −1.19140 − 0.761944i 0.258819 − 0.965926i 0.838882 + 1.81557i −3.72912 + 0.999214i −1.04434 + 0.953601i 1.89040 − 1.85105i 0.383912 − 2.80225i −0.866025 − 0.500000i 5.20423 + 1.65091i
19.8 −1.09426 + 0.895880i 0.258819 − 0.965926i 0.394798 − 1.96065i −2.90593 + 0.778643i 0.582139 + 1.28884i 2.58569 + 0.560554i 1.32449 + 2.49914i −0.866025 − 0.500000i 2.48227 − 3.45540i
19.9 −1.02338 − 0.976055i −0.258819 + 0.965926i 0.0946319 + 1.99776i 0.892283 − 0.239086i 1.20767 − 0.735892i −1.55431 + 2.14106i 1.85308 − 2.13684i −0.866025 − 0.500000i −1.14651 − 0.626240i
19.10 −0.931793 + 1.06384i −0.258819 + 0.965926i −0.263523 − 1.98256i −0.506628 + 0.135751i −0.786428 − 1.17539i 1.23345 + 2.34064i 2.35468 + 1.56699i −0.866025 − 0.500000i 0.327655 − 0.665464i
19.11 −0.904988 − 1.08674i −0.258819 + 0.965926i −0.361993 + 1.96697i −2.68161 + 0.718534i 1.28394 − 0.592883i −0.164584 − 2.64063i 2.46517 − 1.38669i −0.866025 − 0.500000i 3.20768 + 2.26394i
19.12 −0.867553 + 1.11685i −0.258819 + 0.965926i −0.494702 − 1.93785i 1.83632 − 0.492041i −0.854254 − 1.12705i −1.58583 − 2.11781i 2.59347 + 1.12868i −0.866025 − 0.500000i −1.04357 + 2.47776i
19.13 −0.668833 − 1.24606i 0.258819 − 0.965926i −1.10533 + 1.66681i 0.513920 − 0.137704i −1.37671 + 0.323539i 1.08086 + 2.41490i 2.81622 + 0.262485i −0.866025 − 0.500000i −0.515314 − 0.548273i
19.14 −0.375468 + 1.36346i 0.258819 − 0.965926i −1.71805 − 1.02387i 2.98722 − 0.800424i 1.21982 + 0.715563i −1.52023 + 2.16539i 2.04108 − 1.95806i −0.866025 − 0.500000i −0.0302596 + 4.37349i
19.15 −0.341170 − 1.37244i 0.258819 − 0.965926i −1.76721 + 0.936473i −1.77524 + 0.475675i −1.41398 − 0.0256701i −2.64184 − 0.143862i 1.88817 + 2.10590i −0.866025 − 0.500000i 1.25850 + 2.27414i
19.16 0.0555893 + 1.41312i 0.258819 − 0.965926i −1.99382 + 0.157109i −2.15146 + 0.576482i 1.37936 + 0.312047i 0.0219879 − 2.64566i −0.332849 − 2.80877i −0.866025 − 0.500000i −0.934236 − 3.00822i
19.17 0.0897801 − 1.41136i −0.258819 + 0.965926i −1.98388 − 0.253424i 3.09513 − 0.829339i 1.34003 + 0.452008i 1.80424 + 1.93513i −0.535786 + 2.77722i −0.866025 − 0.500000i −0.892615 − 4.44281i
19.18 0.265903 + 1.38899i −0.258819 + 0.965926i −1.85859 + 0.738673i −1.15145 + 0.308529i −1.41048 − 0.102655i −2.57151 − 0.622366i −1.52021 − 2.38515i −0.866025 − 0.500000i −0.734716 − 1.51731i
19.19 0.278250 − 1.38657i −0.258819 + 0.965926i −1.84515 − 0.771627i −1.22597 + 0.328499i 1.26731 + 0.627640i −2.24886 − 1.39379i −1.58333 + 2.34373i −0.866025 − 0.500000i 0.114359 + 1.79130i
19.20 0.311784 − 1.37942i 0.258819 − 0.965926i −1.80558 − 0.860160i 1.69002 − 0.452840i −1.25172 − 0.658179i 1.34626 − 2.27763i −1.74947 + 2.22247i −0.866025 − 0.500000i −0.0977339 − 2.47244i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v 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.bs.a ✓ 128
7.d odd 6 1 inner 336.2.bs.a ✓ 128
16.f odd 4 1 inner 336.2.bs.a ✓ 128
112.v even 12 1 inner 336.2.bs.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bs.a ✓ 128 1.a even 1 1 trivial
336.2.bs.a ✓ 128 7.d odd 6 1 inner
336.2.bs.a ✓ 128 16.f odd 4 1 inner
336.2.bs.a ✓ 128 112.v even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(336, [\chi])\).