LMFDB - Newform orbit 240.4.bc.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,4,Mod(43,240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0, 3]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("240.43");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	240.bc (of order $4$, degree $2$, 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:	$14.1604584014$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$36$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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) = $ $ 72 q - 2 q^{2} - 10 q^{4} - 80 q^{8} - 648 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 72 q - 2 q^{2} - 10 q^{4} - 80 q^{8} - 648 q^{9} + 12 q^{12} - 110 q^{14} - 22 q^{16} - 124 q^{17} + 18 q^{18} + 12 q^{19} + 100 q^{20} - 238 q^{22} - 88 q^{23} + 126 q^{24} - 184 q^{25} + 12 q^{26} + 102 q^{28} - 282 q^{30} + 468 q^{32} - 186 q^{34} + 228 q^{35} + 90 q^{36} + 238 q^{38} - 882 q^{40} + 18 q^{42} - 432 q^{43} + 1770 q^{44} + 1106 q^{46} + 896 q^{47} - 432 q^{48} + 274 q^{50} - 372 q^{51} - 1272 q^{52} + 688 q^{55} + 622 q^{56} + 36 q^{57} - 246 q^{58} - 688 q^{59} - 570 q^{60} + 2552 q^{61} + 836 q^{62} - 634 q^{64} + 340 q^{65} + 750 q^{66} - 1848 q^{67} + 478 q^{68} + 264 q^{69} + 2458 q^{70} + 224 q^{71} + 720 q^{72} + 296 q^{73} + 1040 q^{74} - 552 q^{75} - 3766 q^{76} - 516 q^{78} + 928 q^{79} + 872 q^{80} + 5832 q^{81} - 472 q^{82} + 270 q^{84} - 3424 q^{85} - 1844 q^{86} + 2334 q^{88} - 1968 q^{89} - 848 q^{91} + 818 q^{92} - 792 q^{93} - 310 q^{94} - 1240 q^{95} - 510 q^{96} - 1176 q^{97} + 2812 q^{98}+O(q^{100}) $

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_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $	$ a_{10} $
43.1	−2.82692	−	0.0922271i	3.00000i	7.98299	+	0.521438i	−8.95290	+	6.69669i	0.276681	−	8.48077i	10.4217	−	10.4217i	−22.5192	−	2.21031i	−9.00000	25.9268	−	18.1053i
43.2	−2.78101	−	0.515716i	3.00000i	7.46807	+	2.86842i	11.1109	+	1.24445i	1.54715	−	8.34304i	−9.00178	+	9.00178i	−19.2895	−	11.8285i	−9.00000	−30.2577	−	9.19087i
43.3	−2.76388	−	0.600811i	3.00000i	7.27805	+	3.32114i	3.16032	−	10.7244i	1.80243	−	8.29164i	16.9592	−	16.9592i	−18.1203	−	13.5519i	−9.00000	−15.1781	+	27.7421i
43.4	−2.72050	+	0.773864i	3.00000i	6.80227	−	4.21060i	−7.14805	−	8.59682i	−2.32159	−	8.16151i	3.15918	−	3.15918i	−15.2472	+	16.7190i	−9.00000	26.0991	+	17.8561i
43.5	−2.62097	+	1.06326i	3.00000i	5.73896	−	5.57354i	2.42765	+	10.9136i	−3.18978	−	7.86291i	−14.0460	+	14.0460i	−9.11553	+	20.7101i	−9.00000	−17.9668	−	26.0230i
43.6	−2.51616	+	1.29187i	3.00000i	4.66216	−	6.50110i	8.76435	−	6.94162i	−3.87560	−	7.54849i	−6.47277	+	6.47277i	−3.33220	+	22.3807i	−9.00000	−13.0849	+	28.7886i
43.7	−2.48314	−	1.35426i	3.00000i	4.33195	+	6.72564i	−11.1716	+	0.442714i	4.06279	−	7.44941i	−20.1262	+	20.1262i	−1.64853	−	22.5673i	−9.00000	28.3401	+	14.0299i
43.8	−2.21529	−	1.75855i	3.00000i	1.81500	+	7.79139i	−6.29836	−	9.23746i	5.27565	−	6.64586i	4.86481	−	4.86481i	9.68082	−	20.4519i	−9.00000	−2.29188	+	31.5396i
43.9	−2.09083	+	1.90484i	3.00000i	0.743142	−	7.96541i	−4.16734	+	10.3746i	−5.71453	−	6.27249i	22.3735	−	22.3735i	13.6191	+	18.0699i	−9.00000	−11.0489	−	29.6298i
43.10	−2.02696	−	1.97267i	3.00000i	0.217117	+	7.99705i	7.64845	+	8.15483i	5.91802	−	6.08087i	19.8243	−	19.8243i	15.3355	−	16.6380i	−9.00000	0.583738	−	31.6174i
43.11	−1.83847	−	2.14943i	3.00000i	−1.24007	+	7.90330i	10.7962	−	2.90553i	6.44828	−	5.51540i	−14.7876	+	14.7876i	19.2674	−	11.8645i	−9.00000	−26.0937	−	17.8639i
43.12	−1.42312	+	2.44433i	3.00000i	−3.94945	−	6.95714i	1.10167	−	11.1259i	−7.33298	−	4.26937i	2.79711	−	2.79711i	22.6261	+	0.247124i	−9.00000	25.6276	+	18.5264i
43.13	−1.27014	+	2.52720i	3.00000i	−4.77348	−	6.41981i	9.81262	+	5.35841i	−7.58160	−	3.81043i	4.30891	−	4.30891i	22.2871	−	3.90946i	−9.00000	−26.0052	+	17.9925i
43.14	−1.25059	+	2.53693i	3.00000i	−4.87203	−	6.34534i	−10.2861	+	4.38137i	−7.61079	−	3.75178i	−18.0427	+	18.0427i	22.1906	−	4.42456i	−9.00000	1.74848	−	31.5744i
43.15	−1.03418	−	2.63258i	3.00000i	−5.86096	+	5.44511i	−4.01543	+	10.4344i	7.89774	−	3.10253i	−6.76247	+	6.76247i	20.3959	+	9.79823i	−9.00000	31.6220	−	0.220050i
43.16	−0.471133	−	2.78891i	3.00000i	−7.55607	+	2.62790i	−10.9299	−	2.35302i	8.36674	−	1.41340i	20.3270	−	20.3270i	10.8889	+	19.8351i	−9.00000	−1.41292	+	31.5912i
43.17	−0.0374011	+	2.82818i	3.00000i	−7.99720	−	0.211554i	−11.0865	−	1.44584i	−8.48454	−	0.112203i	7.76015	−	7.76015i	0.897418	−	22.6096i	−9.00000	4.50374	−	31.3004i
43.18	0.133644	−	2.82527i	3.00000i	−7.96428	−	0.755158i	8.78941	−	6.90987i	8.47580	+	0.400931i	9.84227	−	9.84227i	−3.19790	+	22.4003i	−9.00000	−18.3476	−	25.7559i
43.19	0.252562	−	2.81713i	3.00000i	−7.87242	−	1.42300i	−8.62322	−	7.11619i	8.45139	+	0.757686i	−16.3978	+	16.3978i	−5.99705	+	21.8182i	−9.00000	−22.2251	+	22.4954i
43.20	0.265755	+	2.81591i	3.00000i	−7.85875	+	1.49669i	10.7738	+	2.98740i	−8.44774	+	0.797266i	6.90177	−	6.90177i	−6.30305	−	21.7318i	−9.00000	−5.54906	+	31.1321i

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.j	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.4.bc.a	yes	72
5.c	odd	4	1		240.4.y.b	✓	72
16.f	odd	4	1		240.4.y.b	✓	72
80.j	even	4	1	inner	240.4.bc.a	yes	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.4.y.b	✓	72	5.c	odd	4	1
240.4.y.b	✓	72	16.f	odd	4	1
240.4.bc.a	yes	72	1.a	even	1	1	trivial
240.4.bc.a	yes	72	80.j	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{72} - 296 T_{7}^{69} + 4866416 T_{7}^{68} + 3147680 T_{7}^{67} + 43808 T_{7}^{66} + \cdots + 67\!\cdots\!00 $ T7^72 - 296*T7^69 + 4866416*T7^68 + 3147680*T7^67 + 43808*T7^66 - 5045566912*T7^65 + 9990912183296*T7^64 + 11961177868608*T7^63 + 6234244545024*T7^62 - 17531750054894080*T7^61 + 11368554756064707712*T7^60 + 17703863561737536512*T7^59 + 24792121818699939840*T7^58 - 27179991948157522015744*T7^57 + 7920396454063474090655744*T7^56 + 12919433810962694875561984*T7^55 + 40331279143580218848782336*T7^54 - 22945333370879861503952556032*T7^53 + 3531437419818302501270213599232*T7^52 + 4619053402463147186775687770112*T7^51 + 34850699901415861577982495784960*T7^50 - 11391835810034460678570089919070208*T7^49 + 1029743785867571944273014993502859264*T7^48 + 489423570153411805237237432698404864*T7^47 + 17514104076096582421760817609014509568*T7^46 - 3431226658909919874960688725491228672000*T7^45 + 199336633213682829629118666806935791730688*T7^44 - 182196321150935439475513483716705982349312*T7^43 + 5301562158607136992759789717104830693244928*T7^42 - 636580988732871173175657708806856161234518016*T7^41 + 25995105417733879322581770985120056848633823232*T7^40 - 71341721082317982726520368590978869145623855104*T7^39 + 981741939292031199387745099612551461697128235008*T7^38 - 73770396621554601498525827680317042903947231100928*T7^37 + 2302214232617682922827826434912607296325504335872000*T7^36 - 10922774362512795670774409021257393321492614209339392*T7^35 + 112531212578719878114399436521968740503150772829028352*T7^34 - 5394701182490489715412635876322687872786883773529587712*T7^33 + 137825883016874396057460792666984227287629149101036142592*T7^32 - 907887692899332994032656858516125723670200582896525246464*T7^31 + 8026464968890745479694135178330489554674893401580315344896*T7^30 - 250349437074156121767870566715658972119917472389775101001728*T7^29 + 5474672469491484280526837325188355601673761220147305122889728*T7^28 - 44067254720921005207052282719145950959978070146220366403469312*T7^27 + 355382206497991211078758788140074386106072565907468208578756608*T7^26 - 7355933001544156761322484298500547142696912575815369877624979456*T7^25 + 139941420751042608677340779236945524476967094187177109771142561792*T7^24 - 1266146131928184158626235356061549940086518432809373086794507091968*T7^23 + 9621459722674802919675279644629993321203564689388787570170423934976*T7^22 - 135337344047663911389383511663149240481063323077487421600408195301376*T7^21 + 2208514640842064317918361827131508487583829803768788420553758446977024*T7^20 - 21155513451815226546073738390500313986850851639990800755591059647496192*T7^19 + 154298606734354083060785098995258745040753852088416138269517499594702848*T7^18 - 1526513431767171257581594731173555790442022579782806072576572755288260608*T7^17 + 20375004575162872592773282345390746310044426128232397011055783980814041088*T7^16 - 196379673560085281141916264541339081924441839971583266854677311998876712960*T7^15 + 1388128323919099953268393350845048076076298101796556508627115251130754400256*T7^14 - 10163716880872931118334734622537996362209221817421155224879765790904461295616*T7^13 + 102449364096253792076729482075175036438115236129055920290484517554705240424448*T7^12 - 936447934388481822938176490037112484846531996473693178143947791895238996393984*T7^11 + 6409541256827440224714886870002411345200045964874500045073557030566145736835072*T7^10 - 36524043281834174302468205847912236975357125917298134432358279447714037904703488*T7^9 + 242193549260043248457071438168710922662986331962693505484778466374557180532621312*T7^8 - 1838860820669972022911932785898272976602259685814647674311729357526218514599247872*T7^7 + 11837001757577406973511574324531020015901867119549319323039750846851006669478952960*T7^6 - 54526138322878454974112040737284382497326073976824423809764950410304295513956024320*T7^5 + 170458116458331396036233863418428679322670925462868490415948997140597966004201455616*T7^4 - 333250911910746735291702210475794685555477470053538187345603555973557874073041633280*T7^3 + 347261637472300742110559574192339591233989293044524194026016385064862567680389939200*T7^2 - 68699484763088180441276563614622251808052696797016940382538140537552977205080883200*T7 + 6795480262472477381666453873268370597505428198867271755768764689903972603500953600 acting on $S_{4}^{\mathrm{new}}(240, [\chi])$.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.bc (of order \(4\), degree \(2\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 43.36
Significant digits:
Format:

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