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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Newspace parameters

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

Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{4} - 24q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{4} - 24q^{8} + 36q^{10} - 8q^{11} - 32q^{14} - 4q^{16} + 4q^{18} + 16q^{22} + 16q^{23} - 32q^{28} + 32q^{29} - 24q^{35} - 16q^{37} + 60q^{40} - 20q^{42} - 16q^{43} - 12q^{44} - 20q^{46} - 32q^{50} - 144q^{52} - 16q^{53} - 56q^{56} - 40q^{58} - 96q^{59} - 24q^{60} - 32q^{64} - 72q^{66} + 16q^{67} - 60q^{68} + 28q^{70} + 128q^{71} - 4q^{72} - 72q^{74} + 72q^{78} - 36q^{80} + 64q^{81} - 60q^{82} + 24q^{84} - 44q^{86} - 48q^{88} - 8q^{91} + 56q^{92} + 36q^{94} + 60q^{96} + 148q^{98} - 16q^{99} + O(q^{100}) \)

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.

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 283.32
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 \(S_{2}^{\mathrm{new}}(336, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database