LMFDB - Newform orbit 714.2.y.b

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} - 6 q^{39} - 28 q^{42} - 4 q^{44} - 72 q^{45} + 28 q^{51} - 18 q^{54} - 36 q^{57} - 8 q^{58} - 30 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} - 12 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} + 28 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	−1.70680	−	0.294652i	−0.500000	−	0.866025i	0.682953	−	0.182997i	−1.10858	+	1.33081i	−1.35359	−	2.27328i	−1.00000	2.82636	+	1.00583i	0.182997	−	0.682953i
47.2	0.500000	−	0.866025i	−1.68055	+	0.419238i	−0.500000	−	0.866025i	1.93129	−	0.517488i	−0.477203	+	1.66502i	1.74678	−	1.98715i	−1.00000	2.64848	−	1.40910i	0.517488	−	1.93129i
47.3	0.500000	−	0.866025i	−1.64681	−	0.536663i	−0.500000	−	0.866025i	−3.83391	+	1.02729i	−1.28817	+	1.15785i	−2.15635	−	1.53302i	−1.00000	2.42399	+	1.76757i	−1.02729	+	3.83391i
47.4	0.500000	−	0.866025i	−1.63700	+	0.565895i	−0.500000	−	0.866025i	1.27501	−	0.341638i	−0.328420	+	1.70063i	−1.98354	+	1.75088i	−1.00000	2.35953	−	1.85274i	0.341638	−	1.27501i
47.5	0.500000	−	0.866025i	−1.59661	−	0.671433i	−0.500000	−	0.866025i	−3.06133	+	0.820281i	−1.37979	+	1.04699i	2.56182	+	0.661102i	−1.00000	2.09835	+	2.14404i	−0.820281	+	3.06133i
47.6	0.500000	−	0.866025i	−1.50014	−	0.865786i	−0.500000	−	0.866025i	2.73499	−	0.732838i	−1.49986	+	0.866265i	−0.735897	+	2.54135i	−1.00000	1.50083	+	2.59760i	0.732838	−	2.73499i
47.7	0.500000	−	0.866025i	−1.39731	+	1.02349i	−0.500000	−	0.866025i	−2.27220	+	0.608834i	0.187709	+	1.72185i	2.41627	+	1.07780i	−1.00000	0.904955	−	2.86025i	−0.608834	+	2.27220i
47.8	0.500000	−	0.866025i	−0.962525	−	1.43998i	−0.500000	−	0.866025i	−0.0456622	+	0.0122351i	−1.72832	+	0.113581i	0.766453	+	2.53230i	−1.00000	−1.14709	+	2.77204i	−0.0122351	+	0.0456622i
47.9	0.500000	−	0.866025i	−0.567208	+	1.63654i	−0.500000	−	0.866025i	2.51579	−	0.674105i	1.13368	+	1.30949i	2.20221	+	1.46638i	−1.00000	−2.35655	−	1.85652i	0.674105	−	2.51579i
47.10	0.500000	−	0.866025i	−0.474953	+	1.66566i	−0.500000	−	0.866025i	−1.17352	+	0.314445i	1.20503	+	1.24415i	1.24347	−	2.33533i	−1.00000	−2.54884	−	1.58222i	−0.314445	+	1.17352i
47.11	0.500000	−	0.866025i	−0.0571822	−	1.73111i	−0.500000	−	0.866025i	−2.07309	+	0.555482i	−1.52777	−	0.816032i	−2.14321	+	1.55133i	−1.00000	−2.99346	+	0.197977i	−0.555482	+	2.07309i
47.12	0.500000	−	0.866025i	−0.0331860	−	1.73173i	−0.500000	−	0.866025i	1.87565	−	0.502578i	−1.51632	−	0.837126i	2.64388	−	0.0994322i	−1.00000	−2.99780	+	0.114939i	0.502578	−	1.87565i
47.13	0.500000	−	0.866025i	0.165506	−	1.72413i	−0.500000	−	0.866025i	3.96209	−	1.06164i	−1.41038	−	1.00539i	−2.51330	−	0.826632i	−1.00000	−2.94522	−	0.570705i	1.06164	−	3.96209i
47.14	0.500000	−	0.866025i	0.303332	+	1.70528i	−0.500000	−	0.866025i	−1.06290	+	0.284804i	1.62848	+	0.589948i	−2.50834	+	0.841556i	−1.00000	−2.81598	+	1.03453i	−0.284804	+	1.06290i
47.15	0.500000	−	0.866025i	0.584386	+	1.63049i	−0.500000	−	0.866025i	2.81813	−	0.755117i	1.70424	+	0.309151i	−1.04856	+	2.42910i	−1.00000	−2.31699	+	1.90567i	0.755117	−	2.81813i
47.16	0.500000	−	0.866025i	0.838656	−	1.51547i	−0.500000	−	0.866025i	−1.98744	+	0.532534i	−0.893109	−	1.48403i	−0.126655	−	2.64272i	−1.00000	−1.59331	−	2.54192i	−0.532534	+	1.98744i
47.17	0.500000	−	0.866025i	0.951679	+	1.44717i	−0.500000	−	0.866025i	−3.81940	+	1.02341i	1.72913	−	0.100592i	0.816505	−	2.51661i	−1.00000	−1.18861	+	2.75449i	−1.02341	+	3.81940i
47.18	0.500000	−	0.866025i	1.04383	+	1.38218i	−0.500000	−	0.866025i	3.17743	−	0.851391i	1.71892	−	0.212892i	1.70783	−	2.02072i	−1.00000	−0.820844	+	2.88552i	0.851391	−	3.17743i
47.19	0.500000	−	0.866025i	1.43612	−	0.968270i	−0.500000	−	0.866025i	2.00853	−	0.538183i	−0.120484	−	1.72786i	2.64324	+	0.115266i	−1.00000	1.12491	−	2.78111i	0.538183	−	2.00853i
47.20	0.500000	−	0.866025i	1.46194	−	0.928835i	−0.500000	−	0.866025i	−1.24398	+	0.333324i	−0.0734257	−	1.73049i	0.197789	+	2.63835i	−1.00000	1.27453	−	2.71580i	−0.333324	+	1.24398i

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.b	yes	96
3.b	odd	2	1		714.2.y.a	✓	96
7.d	odd	6	1	inner	714.2.y.b	yes	96
17.c	even	4	1		714.2.y.a	✓	96
21.g	even	6	1		714.2.y.a	✓	96
51.f	odd	4	1	inner	714.2.y.b	yes	96
119.m	odd	12	1		714.2.y.a	✓	96
357.y	even	12	1	inner	714.2.y.b	yes	96

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

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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