# 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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120q - 4q^{4} - 8q^{5} - 8q^{6} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q - 4q^{4} - 8q^{5} - 8q^{6} + 24q^{8} - 8q^{10} + 4q^{11} + 8q^{13} + 32q^{14} + 8q^{15} + 12q^{16} - 16q^{17} - 4q^{18} - 8q^{19} - 64q^{20} + 12q^{21} + 24q^{22} - 8q^{24} - 8q^{26} - 8q^{28} - 64q^{29} - 52q^{31} - 4q^{33} + 32q^{34} - 8q^{35} + 24q^{37} - 40q^{38} - 60q^{40} + 24q^{42} - 32q^{43} - 12q^{44} - 8q^{45} + 4q^{46} - 8q^{47} + 32q^{48} + 20q^{49} - 16q^{50} - 8q^{51} - 16q^{52} + 4q^{54} - 104q^{56} + 4q^{58} + 52q^{59} - 16q^{60} + 4q^{61} - 144q^{62} + 8q^{63} - 64q^{64} + 24q^{65} + 24q^{66} + 12q^{68} + 24q^{69} - 48q^{70} + 4q^{72} + 16q^{74} + 16q^{75} - 8q^{76} + 8q^{77} - 56q^{78} + 28q^{79} + 68q^{80} + 60q^{81} + 28q^{82} + 8q^{83} + 24q^{84} - 80q^{85} + 36q^{86} + 40q^{88} + 8q^{90} + 36q^{91} - 184q^{92} + 4q^{93} + 60q^{94} - 16q^{95} + 36q^{96} + 24q^{97} + 124q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 277.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{5}^{120} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database