# Properties

 Label 336.2.bu.a Level 336 Weight 2 Character orbit 336.bu Analytic conductor 2.683 Analytic rank 0 Dimension 240 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bu (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$60$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 2q^{3} - 4q^{4} - 8q^{6} - 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 2q^{3} - 4q^{4} - 8q^{6} - 16q^{7} - 4q^{10} - 2q^{12} - 16q^{13} - 20q^{16} + 16q^{18} - 4q^{19} + 2q^{21} - 40q^{22} - 22q^{24} - 8q^{27} - 4q^{28} - 26q^{30} - 4q^{33} + 16q^{36} - 4q^{37} - 4q^{39} + 8q^{40} - 18q^{42} - 16q^{43} + 18q^{45} - 20q^{46} - 88q^{48} - 16q^{49} + 6q^{51} + 8q^{52} + 14q^{54} - 32q^{55} - 36q^{58} - 42q^{60} - 4q^{61} - 64q^{64} - 30q^{66} - 36q^{67} - 20q^{69} + 116q^{70} - 46q^{72} - 24q^{75} - 112q^{76} - 92q^{78} - 4q^{81} - 32q^{82} + 44q^{84} - 56q^{85} - 4q^{87} - 20q^{88} + 28q^{90} - 40q^{91} - 14q^{93} + 72q^{94} + 36q^{96} - 32q^{97} + 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41143 0.0886154i −1.41967 + 0.992245i 1.98429 + 0.250150i 2.16856 + 0.581063i 2.09169 1.27469i 2.64341 0.111231i −2.77853 0.528909i 1.03090 2.81731i −3.00929 1.01230i
11.2 −1.38808 + 0.270599i 0.495329 + 1.65971i 1.85355 0.751228i −1.75704 0.470799i −1.13667 2.16979i 2.40884 + 1.09429i −2.36960 + 1.54434i −2.50930 + 1.64421i 2.56632 + 0.178054i
11.3 −1.37508 0.330401i 1.72844 0.111804i 1.78167 + 0.908653i −0.754514 0.202171i −2.41368 0.417338i −2.17010 + 1.51350i −2.14971 1.83813i 2.97500 0.386494i 0.970717 + 0.527293i
11.4 −1.37478 + 0.331648i −1.48697 0.888215i 1.78002 0.911883i −1.88832 0.505975i 2.33882 + 0.727948i 0.358137 + 2.62140i −2.14470 + 1.84397i 1.42215 + 2.64150i 2.76383 + 0.0693439i
11.5 −1.37118 + 0.346233i 1.54917 0.774639i 1.76025 0.949493i 2.52702 + 0.677113i −1.85598 + 1.59854i 0.885088 2.49331i −2.08486 + 1.91138i 1.79987 2.40010i −3.69943 0.0535028i
11.6 −1.35622 + 0.400842i 0.102444 1.72902i 1.67865 1.08726i −0.613008 0.164255i 0.554126 + 2.38599i −2.64440 0.0846229i −1.84080 + 2.14743i −2.97901 0.354255i 0.897213 0.0229536i
11.7 −1.33024 0.480058i −0.332434 1.69985i 1.53909 + 1.27719i 0.419196 + 0.112323i −0.373809 + 2.42080i 1.95186 1.78613i −1.43423 2.43782i −2.77897 + 1.13018i −0.503710 0.350655i
11.8 −1.32564 0.492619i −1.70566 + 0.301217i 1.51465 + 1.30607i −3.93617 1.05469i 2.40948 + 0.440934i −1.10769 2.40271i −1.36449 2.47753i 2.81854 1.02755i 4.69840 + 3.33718i
11.9 −1.30635 + 0.541709i 1.11887 + 1.32217i 1.41310 1.41532i 3.95276 + 1.05914i −2.17787 1.12111i −1.79142 + 1.94700i −1.07931 + 2.61440i −0.496255 + 2.95867i −5.73744 + 0.757640i
11.10 −1.25435 0.653142i −1.45082 0.946102i 1.14681 + 1.63854i 3.01819 + 0.808721i 1.20191 + 2.13434i −2.25133 + 1.38979i −0.368304 2.80435i 1.20978 + 2.74525i −3.25767 2.98573i
11.11 −1.14092 0.835638i −0.269460 + 1.71096i 0.603419 + 1.90680i −1.67179 0.447955i 1.73718 1.72691i −0.837482 + 2.50971i 0.904939 2.67975i −2.85478 0.922073i 1.53306 + 1.90810i
11.12 −1.12647 + 0.855017i −1.69811 0.341188i 0.537892 1.92631i 0.483584 + 0.129576i 2.20460 1.06758i 0.378986 2.61847i 1.04111 + 2.62985i 2.76718 + 1.15875i −0.655534 + 0.267508i
11.13 −1.11569 + 0.869039i −0.392190 + 1.68706i 0.489541 1.93916i −1.41326 0.378681i −1.02856 2.22307i −1.15905 2.37836i 1.13903 + 2.58894i −2.69237 1.32330i 1.90585 0.805684i
11.14 −1.10334 0.884670i 1.72780 + 0.121317i 0.434720 + 1.95218i 2.01953 + 0.541132i −1.79902 1.66238i 2.33730 + 1.23976i 1.24739 2.53851i 2.97056 + 0.419221i −1.74951 2.38367i
11.15 −1.02071 0.978849i 0.0677712 + 1.73072i 0.0837109 + 1.99825i 3.51084 + 0.940727i 1.62494 1.83291i −0.174511 2.63999i 1.87054 2.12158i −2.99081 + 0.234586i −2.66273 4.39679i
11.16 −0.950619 + 1.04705i 1.24919 1.19980i −0.192647 1.99070i 0.322006 + 0.0862812i 0.0687593 + 2.44852i 1.62394 + 2.08874i 2.26751 + 1.69069i 0.120939 2.99756i −0.396446 + 0.255137i
11.17 −0.934536 + 1.06143i 1.64436 + 0.544135i −0.253284 1.98390i −3.28878 0.881227i −2.11428 + 1.23686i −2.48766 + 0.900855i 2.34248 + 1.58518i 2.40783 + 1.78951i 4.00885 2.66729i
11.18 −0.863239 1.12019i 1.08000 1.35410i −0.509638 + 1.93398i −1.29824 0.347862i −2.44915 0.0408893i −2.01473 1.71490i 2.60636 1.09860i −0.667192 2.92487i 0.731019 + 1.75456i
11.19 −0.860221 + 1.12251i −1.55800 + 0.756720i −0.520041 1.93121i 1.89562 + 0.507929i 0.490803 2.39982i −0.588039 + 2.57958i 2.61514 + 1.07751i 1.85475 2.35795i −2.20080 + 1.69091i
11.20 −0.674285 1.24312i 1.47021 + 0.915682i −1.09068 + 1.67643i −3.56829 0.956121i 0.146958 2.44508i 2.17230 1.51033i 2.81943 + 0.225451i 1.32305 + 2.69250i 1.21747 + 5.08050i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.60 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
16.f odd 4 1 inner
21.h odd 6 1 inner
48.k even 4 1 inner
112.u odd 12 1 inner
336.bu 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.bu.a 240
3.b odd 2 1 inner 336.2.bu.a 240
7.c even 3 1 inner 336.2.bu.a 240
16.f odd 4 1 inner 336.2.bu.a 240
21.h odd 6 1 inner 336.2.bu.a 240
48.k even 4 1 inner 336.2.bu.a 240
112.u odd 12 1 inner 336.2.bu.a 240
336.bu even 12 1 inner 336.2.bu.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bu.a 240 1.a even 1 1 trivial
336.2.bu.a 240 3.b odd 2 1 inner
336.2.bu.a 240 7.c even 3 1 inner
336.2.bu.a 240 16.f odd 4 1 inner
336.2.bu.a 240 21.h odd 6 1 inner
336.2.bu.a 240 48.k even 4 1 inner
336.2.bu.a 240 112.u odd 12 1 inner
336.2.bu.a 240 336.bu 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