LMFDB - Newform orbit 990.2.bg.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [990,2,Mod(31,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 0, 18])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.31"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	990.bg (of order $15$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [88,11,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.90518980011$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$11$ over $\Q(\zeta_{15})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 88 q + 11 q^{2} - 8 q^{3} + 11 q^{4} + 11 q^{5} + q^{6} - 2 q^{7} - 22 q^{8} + 6 q^{9} + 88 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 7 q^{15} + 11 q^{16} + 18 q^{17} - 6 q^{18} + 12 q^{19}+ \cdots - 128 q^{99}+O(q^{100}) $

88 * q + 11 * q^2 - 8 * q^3 + 11 * q^4 + 11 * q^5 + q^6 - 2 * q^7 - 22 * q^8 + 6 * q^9 + 88 * q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 7 * q^15 + 11 * q^16 + 18 * q^17 - 6 * q^18 + 12 * q^19 + 11 * q^20 + 36 * q^21 + 3 * q^22 + 46 * q^23 + 7 * q^24 + 11 * q^25 + 26 * q^26 + 16 * q^27 + 4 * q^28 - 9 * q^29 - 8 * q^30 - 20 * q^31 - 44 * q^32 + 35 * q^33 + 26 * q^34 + 4 * q^35 + 15 * q^36 + 4 * q^37 + 4 * q^38 - 33 * q^39 + 11 * q^40 + 22 * q^41 - 29 * q^42 - 2 * q^43 + 14 * q^44 + 8 * q^46 - 16 * q^47 + q^48 - 5 * q^49 + 11 * q^50 - 13 * q^51 - 13 * q^52 + 64 * q^53 - 4 * q^54 + 4 * q^55 + 8 * q^56 - 25 * q^57 - 9 * q^58 - 27 * q^59 + q^60 + 4 * q^61 + q^63 - 22 * q^64 + 22 * q^65 + 40 * q^66 - 6 * q^67 - 9 * q^68 - 54 * q^69 - 2 * q^70 + 52 * q^71 + 6 * q^72 - 22 * q^73 - 2 * q^74 + q^75 + 4 * q^76 + 12 * q^77 - 32 * q^78 + 2 * q^79 - 22 * q^80 - 30 * q^81 + 36 * q^82 + 33 * q^83 + 8 * q^84 - 4 * q^85 - 17 * q^86 - 192 * q^87 + 3 * q^88 - 76 * q^89 + 6 * q^90 - 24 * q^91 - 19 * q^92 - 28 * q^93 - q^94 - 6 * q^95 + 7 * q^96 - 38 * q^97 - 128 * q^99

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} $
31.1	−0.978148	+	0.207912i	−1.73205	−	0.00389470i	0.913545	−	0.406737i	−0.978148	−	0.207912i	1.69501	−	0.356303i	−0.310018	+	2.94962i	−0.809017	+	0.587785i	2.99997	+	0.0134916i	1.00000
31.2	−0.978148	+	0.207912i	−1.63543	−	0.570424i	0.913545	−	0.406737i	−0.978148	−	0.207912i	1.71829	+	0.217935i	0.378746	−	3.60353i	−0.809017	+	0.587785i	2.34923	+	1.86577i	1.00000
31.3	−0.978148	+	0.207912i	−1.49597	+	0.872961i	0.913545	−	0.406737i	−0.978148	−	0.207912i	1.28178	−	1.16492i	−0.0811866	+	0.772439i	−0.809017	+	0.587785i	1.47588	−	2.61185i	1.00000
31.4	−0.978148	+	0.207912i	−1.20451	+	1.24465i	0.913545	−	0.406737i	−0.978148	−	0.207912i	0.919407	−	1.46789i	0.420802	−	4.00367i	−0.809017	+	0.587785i	−0.0983267	−	2.99839i	1.00000
31.5	−0.978148	+	0.207912i	−0.860003	−	1.50346i	0.913545	−	0.406737i	−0.978148	−	0.207912i	1.15380	+	1.29180i	−0.324508	+	3.08749i	−0.809017	+	0.587785i	−1.52079	+	2.58596i	1.00000
31.6	−0.978148	+	0.207912i	−0.556540	+	1.64020i	0.913545	−	0.406737i	−0.978148	−	0.207912i	0.203361	−	1.72007i	−0.187148	+	1.78059i	−0.809017	+	0.587785i	−2.38053	−	1.82568i	1.00000
31.7	−0.978148	+	0.207912i	0.157407	−	1.72488i	0.913545	−	0.406737i	−0.978148	−	0.207912i	0.204657	+	1.71992i	0.125602	−	1.19502i	−0.809017	+	0.587785i	−2.95045	−	0.543016i	1.00000
31.8	−0.978148	+	0.207912i	1.16782	−	1.27914i	0.913545	−	0.406737i	−0.978148	−	0.207912i	−0.876347	+	1.49399i	−0.169813	+	1.61567i	−0.809017	+	0.587785i	−0.272413	−	2.98761i	1.00000
31.9	−0.978148	+	0.207912i	1.22140	+	1.22809i	0.913545	−	0.406737i	−0.978148	−	0.207912i	−1.45004	−	0.947306i	0.0524430	−	0.498962i	−0.809017	+	0.587785i	−0.0163876	+	2.99996i	1.00000
31.10	−0.978148	+	0.207912i	1.35544	+	1.07832i	0.913545	−	0.406737i	−0.978148	−	0.207912i	−1.55002	−	0.772946i	0.475009	−	4.51941i	−0.809017	+	0.587785i	0.674443	+	2.92320i	1.00000
31.11	−0.978148	+	0.207912i	1.70882	−	0.282739i	0.913545	−	0.406737i	−0.978148	−	0.207912i	−1.61269	+	0.631844i	−0.170872	+	1.62574i	−0.809017	+	0.587785i	2.84012	−	0.966301i	1.00000
301.1	−0.104528	+	0.994522i	−1.66547	−	0.475629i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	0.647112	−	1.60663i	0.131019	−	0.145512i	0.309017	−	0.951057i	2.54755	+	1.58429i	1.00000
301.2	−0.104528	+	0.994522i	−1.64649	+	0.537666i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	−0.362616	−	1.69367i	−0.184614	+	0.205034i	0.309017	−	0.951057i	2.42183	−	1.77052i	1.00000
301.3	−0.104528	+	0.994522i	−1.30271	+	1.14147i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	−0.999047	−	1.41489i	−1.41375	+	1.57013i	0.309017	−	0.951057i	0.394092	−	2.97400i	1.00000
301.4	−0.104528	+	0.994522i	−1.11905	−	1.32201i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	1.43175	−	0.974733i	−2.02500	+	2.24899i	0.309017	−	0.951057i	−0.495446	+	2.95881i	1.00000
301.5	−0.104528	+	0.994522i	−0.845166	−	1.51185i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	1.59191	−	0.682504i	3.24799	−	3.60726i	0.309017	−	0.951057i	−1.57139	+	2.55553i	1.00000
301.6	−0.104528	+	0.994522i	−0.0126374	+	1.73200i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	−1.72120	−	0.193612i	2.08037	−	2.31049i	0.309017	−	0.951057i	−2.99968	−	0.0437761i	1.00000
301.7	−0.104528	+	0.994522i	−0.00703919	+	1.73204i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	−1.72181	−	0.188048i	1.23119	−	1.36737i	0.309017	−	0.951057i	−2.99990	−	0.0243843i	1.00000
301.8	−0.104528	+	0.994522i	0.386303	−	1.68842i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	1.63879	+	0.560675i	−1.26492	+	1.40484i	0.309017	−	0.951057i	−2.70154	−	1.30449i	1.00000
301.9	−0.104528	+	0.994522i	1.36777	+	1.06264i	−0.978148	−	0.207912i	−0.104528	−	0.994522i	−1.19979	+	1.24920i	0.339825	−	0.377414i	0.309017	−	0.951057i	0.741590	+	2.90690i	1.00000

See all 88 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
11.c	even	5	1	inner
99.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	990.2.bg.a	✓	88
9.c	even	3	1	inner	990.2.bg.a	✓	88
11.c	even	5	1	inner	990.2.bg.a	✓	88
99.m	even	15	1	inner	990.2.bg.a	✓	88

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
990.2.bg.a	✓	88	1.a	even	1	1	trivial
990.2.bg.a	✓	88	9.c	even	3	1	inner
990.2.bg.a	✓	88	11.c	even	5	1	inner
990.2.bg.a	✓	88	99.m	even	15	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{88} + 2 T_{7}^{87} - 34 T_{7}^{86} + 4 T_{7}^{85} + 480 T_{7}^{84} - 2654 T_{7}^{83} + \cdots + 26\!\cdots\!61 $ T7^88 + 2*T7^87 - 34*T7^86 + 4*T7^85 + 480*T7^84 - 2654*T7^83 + 2046*T7^82 + 59701*T7^81 - 347783*T7^80 - 785000*T7^79 + 10200607*T7^78 + 8525791*T7^77 - 99912415*T7^76 + 238319304*T7^75 - 588920740*T7^74 - 11446375771*T7^73 + 37698545578*T7^72 + 134794873326*T7^71 - 495115783370*T7^70 + 300446327032*T7^69 + 1286385210465*T7^68 - 27252868332116*T7^67 + 72128602376947*T7^66 + 214474354452022*T7^65 - 769287963592539*T7^64 - 312240422950426*T7^63 + 1182308884987746*T7^62 - 12542862758411356*T7^61 + 66311532956379807*T7^60 + 126200848698386750*T7^59 - 566683373890330650*T7^58 - 1967586703598297672*T7^57 + 5369268825679487872*T7^56 + 3527726010412939491*T7^55 - 1188133417108033624*T7^54 + 84173281773643236930*T7^53 - 484894740779321095768*T7^52 - 173377841585632485737*T7^51 + 3369977998068340085086*T7^50 - 4060385308059027009112*T7^49 + 7644013058192814518744*T7^48 - 40375801458017028345017*T7^47 + 81348367351961674902639*T7^46 - 133345284033209861396188*T7^45 + 329189148680043503447970*T7^44 - 319665166294352587390648*T7^43 - 534177565035799462898737*T7^42 + 1173890135289948325117847*T7^41 - 1235555838428135783187248*T7^40 + 3642148327788948808875854*T7^39 - 4381718202882256675901395*T7^38 - 2122814160701775937425323*T7^37 + 6603258127476756902117874*T7^36 - 38368081849309854942005653*T7^35 + 156843871580820799496559087*T7^34 - 296884615923157363108320863*T7^33 + 377286236487732729496272598*T7^32 - 405873222583945859626281781*T7^31 + 176018321677774591269588763*T7^30 + 240532489261911205029700402*T7^29 - 120851278713429129896963686*T7^28 - 517810582380846849279955119*T7^27 + 1151335619578844017653566239*T7^26 - 1723627724101520852163820852*T7^25 + 1428569305814031553371286741*T7^24 - 329259943752134720245457331*T7^23 + 83921710668150528073148591*T7^22 - 253638071880656540177896537*T7^21 + 29210274223232313959297662*T7^20 - 6931143461416222722064802*T7^19 + 61341886311975723302003810*T7^18 + 75843477341341037145760262*T7^17 - 73988250005571912026884535*T7^16 - 18015506166995166809658498*T7^15 + 27472359734450891759925816*T7^14 - 6705282352769742483302931*T7^13 - 2253600255651920381695365*T7^12 + 3012998063065418335913487*T7^11 - 860608049892251158875762*T7^10 - 280747598305240787659434*T7^9 + 370061000273902912962165*T7^8 + 17741052683949435864714*T7^7 - 32535230242491851456241*T7^6 - 4840075917891514862676*T7^5 + 2146197082507641799542*T7^4 + 201754414532399626854*T7^3 - 93128571978371605764*T7^2 + 1921471987139275824*T7 + 2688029375043408561 acting on $S_{2}^{\mathrm{new}}(990, [\chi])$.

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	990.bg (of order \(15\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.11
Significant digits:
Format:

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