LMFDB - Newform orbit 714.2.y.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [714,2,Mod(47,714)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(714, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 10, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("714.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 714 = 2 \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	714.y (of order $12$, degree $4$, 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:	$5.70131870432$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$24$ over $\Q(\zeta_{12})$
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) = $ $ 96 q - 48 q^{2} - 48 q^{4} + 96 q^{8} + 4 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 96 q - 48 q^{2} - 48 q^{4} + 96 q^{8} + 4 q^{9} + 4 q^{11} - 48 q^{16} - 12 q^{17} + 4 q^{18} - 8 q^{21} - 8 q^{22} + 16 q^{29} - 48 q^{32} + 12 q^{33} - 8 q^{36} + 10 q^{39} + 28 q^{42} + 4 q^{44} + 36 q^{45} - 44 q^{51} + 54 q^{54} + 28 q^{57} - 8 q^{58} - 50 q^{63} + 96 q^{64} - 8 q^{65} - 12 q^{66} + 8 q^{67} + 12 q^{68} + 48 q^{71} + 4 q^{72} + 72 q^{73} + 18 q^{75} - 24 q^{77} - 20 q^{78} - 32 q^{79} + 12 q^{81} + 24 q^{82} - 20 q^{84} - 24 q^{85} + 4 q^{88} - 36 q^{91} - 36 q^{93} + 12 q^{95} - 4 q^{99}+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_{3} $			$ a_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $	$ a_{9} $			$ a_{10} $
47.1	−0.500000	+	0.866025i	−1.73205		0.000276224i	−0.500000	−	0.866025i	−2.73499	+	0.732838i	0.866265	−	1.49986i	−0.735897	+	2.54135i	1.00000	3.00000		0.000956869i	0.732838	−	2.73499i
47.2	−0.500000	+	0.866025i	−1.71843	−	0.216829i	−0.500000	−	0.866025i	3.06133	−	0.820281i	1.04699	−	1.37979i	2.56182	+	0.661102i	1.00000	2.90597	+	0.745208i	−0.820281	+	3.06133i
47.3	−0.500000	+	0.866025i	−1.69451	−	0.358643i	−0.500000	−	0.866025i	3.83391	−	1.02729i	1.15785	−	1.28817i	−2.15635	−	1.53302i	1.00000	2.74275	+	1.21545i	−1.02729	+	3.83391i
47.4	−0.500000	+	0.866025i	−1.62546	−	0.598226i	−0.500000	−	0.866025i	−0.682953	+	0.182997i	1.33081	−	1.10858i	−1.35359	−	2.27328i	1.00000	2.28425	+	1.94479i	0.182997	−	0.682953i
47.5	−0.500000	+	0.866025i	−1.55356	+	0.765798i	−0.500000	−	0.866025i	0.0456622	−	0.0122351i	0.113581	−	1.72832i	0.766453	+	2.53230i	1.00000	1.82711	−	2.37943i	−0.0122351	+	0.0456622i
47.6	−0.500000	+	0.866025i	−1.24578	−	1.20334i	−0.500000	−	0.866025i	−1.93129	+	0.517488i	1.66502	−	0.477203i	1.74678	−	1.98715i	1.00000	0.103923	+	2.99820i	0.517488	−	1.93129i
47.7	−0.500000	+	0.866025i	−1.13473	−	1.30858i	−0.500000	−	0.866025i	−1.27501	+	0.341638i	1.70063	−	0.328420i	−1.98354	+	1.75088i	1.00000	−0.424755	+	2.96978i	0.341638	−	1.27501i
47.8	−0.500000	+	0.866025i	−0.915075	+	1.47059i	−0.500000	−	0.866025i	2.07309	−	0.555482i	−0.816032	−	1.52777i	−2.14321	+	1.55133i	1.00000	−1.32528	−	2.69140i	−0.555482	+	2.07309i
47.9	−0.500000	+	0.866025i	−0.894606	+	1.48313i	−0.500000	−	0.866025i	−1.87565	+	0.502578i	−0.837126	−	1.51632i	2.64388	−	0.0994322i	1.00000	−1.39936	−	2.65364i	0.502578	−	1.87565i
47.10	−0.500000	+	0.866025i	−0.718731	+	1.57589i	−0.500000	−	0.866025i	−3.96209	+	1.06164i	−1.00539	−	1.41038i	−2.51330	−	0.826632i	1.00000	−1.96685	−	2.26528i	1.06164	−	3.96209i
47.11	−0.500000	+	0.866025i	−0.698364	−	1.58502i	−0.500000	−	0.866025i	2.27220	−	0.608834i	1.72185	+	0.187709i	2.41627	+	1.07780i	1.00000	−2.02458	+	2.21384i	−0.608834	+	2.27220i
47.12	−0.500000	+	0.866025i	−0.0314383	+	1.73177i	−0.500000	−	0.866025i	1.98744	−	0.532534i	−1.48403	−	0.893109i	−0.126655	−	2.64272i	1.00000	−2.99802	−	0.108887i	−0.532534	+	1.98744i
47.13	−0.500000	+	0.866025i	0.327055	−	1.70089i	−0.500000	−	0.866025i	−2.51579	+	0.674105i	1.30949	+	1.13368i	2.20221	+	1.46638i	1.00000	−2.78607	−	1.11257i	0.674105	−	2.51579i
47.14	−0.500000	+	0.866025i	0.421508	−	1.67998i	−0.500000	−	0.866025i	1.17352	−	0.314445i	1.24415	+	1.20503i	1.24347	−	2.33533i	1.00000	−2.64466	−	1.41625i	−0.314445	+	1.17352i
47.15	−0.500000	+	0.866025i	0.759585	+	1.55661i	−0.500000	−	0.866025i	−2.00853	+	0.538183i	−1.72786	−	0.120484i	2.64324	+	0.115266i	1.00000	−1.84606	+	2.36475i	0.538183	−	2.00853i
47.16	−0.500000	+	0.866025i	0.801658	+	1.53536i	−0.500000	−	0.866025i	1.24398	−	0.333324i	−1.73049	−	0.0734257i	0.197789	+	2.63835i	1.00000	−1.71469	+	2.46168i	−0.333324	+	1.24398i
47.17	−0.500000	+	0.866025i	1.11533	−	1.32515i	−0.500000	−	0.866025i	1.06290	−	0.284804i	0.589948	+	1.62848i	−2.50834	+	0.841556i	1.00000	−0.512057	−	2.95598i	−0.284804	+	1.06290i
47.18	−0.500000	+	0.866025i	1.15926	+	1.28690i	−0.500000	−	0.866025i	−0.763126	+	0.204479i	−1.69412	+	0.360494i	−1.25407	−	2.32966i	1.00000	−0.312245	+	2.98371i	0.204479	−	0.763126i
47.19	−0.500000	+	0.866025i	1.32134	−	1.11985i	−0.500000	−	0.866025i	−2.81813	+	0.755117i	0.309151	+	1.70424i	−1.04856	+	2.42910i	1.00000	0.491864	−	2.95940i	0.755117	−	2.81813i
47.20	−0.500000	+	0.866025i	1.47129	+	0.913957i	−0.500000	−	0.866025i	3.64888	−	0.977714i	−1.52715	+	0.817192i	−2.60089	+	0.485166i	1.00000	1.32936	+	2.68939i	−0.977714	+	3.64888i

See all 96 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
51.f	odd	4	1	inner
357.y	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	714.2.y.a	✓	96
3.b	odd	2	1		714.2.y.b	yes	96
7.d	odd	6	1	inner	714.2.y.a	✓	96
17.c	even	4	1		714.2.y.b	yes	96
21.g	even	6	1		714.2.y.b	yes	96
51.f	odd	4	1	inner	714.2.y.a	✓	96
119.m	odd	12	1		714.2.y.b	yes	96
357.y	even	12	1	inner	714.2.y.a	✓	96

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.y.a	✓	96	1.a	even	1	1	trivial
714.2.y.a	✓	96	7.d	odd	6	1	inner
714.2.y.a	✓	96	51.f	odd	4	1	inner
714.2.y.a	✓	96	357.y	even	12	1	inner
714.2.y.b	yes	96	3.b	odd	2	1
714.2.y.b	yes	96	17.c	even	4	1
714.2.y.b	yes	96	21.g	even	6	1
714.2.y.b	yes	96	119.m	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{96} - 770 T_{5}^{92} - 492 T_{5}^{91} + 6024 T_{5}^{89} + 366823 T_{5}^{88} + \cdots + 10\!\cdots\!56 $ T5^96 - 770*T5^92 - 492*T5^91 + 6024*T5^89 + 366823*T5^88 + 378840*T5^87 + 121032*T5^86 - 3784860*T5^85 - 111748066*T5^84 - 170452632*T5^83 - 75050352*T5^82 + 1532605104*T5^81 + 25280128911*T5^80 + 47815726008*T5^79 + 30636478728*T5^78 - 349113035256*T5^77 - 4261702575108*T5^76 - 9254436786336*T5^75 - 7059240730152*T5^74 + 56100020044176*T5^73 + 551872926702634*T5^72 + 1285003909595088*T5^71 + 1170773089670592*T5^70 - 5789257725415728*T5^69 - 52452285275211000*T5^68 - 126445879917072144*T5^67 - 127120071188759688*T5^66 + 426408596261761200*T5^65 + 3728580929358892645*T5^64 + 9064117201118087520*T5^63 + 10043413686998653176*T5^62 - 19330320116680721760*T5^61 - 185750538065907145226*T5^60 - 449289328961269312620*T5^59 - 518505672459715119072*T5^58 + 616519032292041156504*T5^57 + 6903310460217118344889*T5^56 + 16521570839482640297400*T5^55 + 19602720514957026089376*T5^54 - 12137873696214783305436*T5^53 - 189427669458545378015766*T5^52 - 446082774084384798661152*T5^51 - 530429438747043135136872*T5^50 + 131974550563085946655728*T5^49 + 3968261071587699949232661*T5^48 + 9163241105408077019212776*T5^47 + 10818362132134161068532600*T5^46 + 787581700707446248700832*T5^45 - 62232478377326607332850688*T5^44 - 139810182280792638700981536*T5^43 - 159834140633251371392878752*T5^42 - 52091380719728824886499984*T5^41 + 740645686030512052791401778*T5^40 + 1608401952428527580896416288*T5^39 + 1751881103251058355480958968*T5^38 + 1032936712800988917950281440*T5^37 - 6423910648619838944293349780*T5^36 - 13256938576278365696227746888*T5^35 - 12997583267639364938342440728*T5^34 - 11227072797748026928056945792*T5^33 + 41315033788367601436967163607*T5^32 + 80575587595646915529421379184*T5^31 + 68632279516667282543948352816*T5^30 + 83566067146661669093158950648*T5^29 - 187419685227307299781906978542*T5^28 - 339603232111734539060705016492*T5^27 - 214385896056548448237422536920*T5^26 - 380399692233316454246527353336*T5^25 + 587709793906535403857647020175*T5^24 + 1048521496016191005056547593640*T5^23 + 570187511908769320929180992160*T5^22 + 1067435444087183118024009854388*T5^21 - 1237340773064940981713357009038*T5^20 - 2227741375538271190261420324176*T5^19 - 1033669093568239103091705283392*T5^18 - 1945644969113849330416530060576*T5^17 + 1659403640761332461951722215249*T5^16 + 3323670754621502311044466147080*T5^15 + 2062696064908306900332066753312*T5^14 + 1730220158648852578105866477888*T5^13 - 1108387137550461925966849059336*T5^12 - 2781217166046230329715158931232*T5^11 - 2238226784897845426127616190464*T5^10 - 1296087178125821205520145045568*T5^9 + 655892774873943944507713691568*T5^8 + 1561959470375138920512088677504*T5^7 + 1183572087562420470136244367360*T5^6 + 618916456372373382651071759616*T5^5 + 134704007674599137697590253440*T5^4 - 20069540876351862430737850368*T5^3 + 1247960588145021454169475072*T5^2 - 51733640849365393176609792*T5 + 1072297322910375213465856 acting on $S_{2}^{\mathrm{new}}(714, [\chi])$.

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:	\( N \)	\(=\)	\( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	714.y (of order \(12\), degree \(4\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 47.24
Significant digits:
Format:

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