LMFDB - Newform orbit 816.2.bh.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(35,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("816.35");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.bh (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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$124$
Relative dimension:	$62$ 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) = $ $ 124 q - 4 q^{2} - 4 q^{5} + 6 q^{6} + 12 q^{7} + 8 q^{8} - 12 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 124 q - 4 q^{2} - 4 q^{5} + 6 q^{6} + 12 q^{7} + 8 q^{8} - 12 q^{9} - 4 q^{11} - 12 q^{12} + 8 q^{13} + 12 q^{14} - 12 q^{15} + 16 q^{18} - 4 q^{19} - 52 q^{20} + 12 q^{21} - 16 q^{22} - 16 q^{24} + 28 q^{26} - 12 q^{27} + 8 q^{29} - 24 q^{30} + 16 q^{32} - 4 q^{34} + 24 q^{35} + 56 q^{36} - 8 q^{37} - 8 q^{38} - 36 q^{39} + 36 q^{40} - 24 q^{41} - 12 q^{42} - 20 q^{43} + 20 q^{44} - 52 q^{45} - 4 q^{46} - 32 q^{47} + 20 q^{48} + 108 q^{49} - 40 q^{50} - 8 q^{52} + 16 q^{53} + 50 q^{54} - 40 q^{55} - 24 q^{56} - 24 q^{57} - 36 q^{58} + 4 q^{59} - 12 q^{60} + 24 q^{61} - 28 q^{62} + 36 q^{63} - 48 q^{64} + 92 q^{66} + 4 q^{67} - 8 q^{68} - 72 q^{70} + 24 q^{72} + 92 q^{74} + 8 q^{75} - 56 q^{76} - 36 q^{77} - 32 q^{78} + 16 q^{80} - 36 q^{81} - 20 q^{82} + 36 q^{83} + 36 q^{84} - 4 q^{85} + 80 q^{87} + 60 q^{88} + 16 q^{89} - 44 q^{90} - 12 q^{91} + 104 q^{92} - 24 q^{93} - 32 q^{94} - 56 q^{95} + 20 q^{96} + 16 q^{97} - 92 q^{98} + 56 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} $
35.1	−1.41408	+	0.0193758i	−1.72466	−	0.159874i	1.99925	−	0.0547979i	1.58607	+	1.58607i	2.44190	+	0.192658i	1.50080	−2.82604	+	0.116226i	2.94888	+	0.551455i	−2.27356	−	2.21210i
35.2	−1.41120	+	0.0922243i	1.09232	+	1.34419i	1.98299	−	0.260294i	−0.312654	−	0.312654i	−1.66545	−	1.79618i	−2.39885	−2.77440	+	0.550208i	−0.613668	+	2.93656i	0.470053	+	0.412385i
35.3	−1.40896	−	0.121776i	−0.955174	−	1.44487i	1.97034	+	0.343157i	−1.16277	−	1.16277i	1.16985	+	2.15208i	−2.27492	−2.73434	−	0.723435i	−1.17528	+	2.76020i	1.49669	+	1.77989i
35.4	−1.38464	+	0.287702i	−0.954322	+	1.44543i	1.83445	−	0.796729i	−1.50951	−	1.50951i	0.905539	−	2.27596i	3.43490	−2.31084	+	1.63096i	−1.17854	−	2.75881i	2.52442	+	1.65584i
35.5	−1.37382	+	0.335584i	1.63225	−	0.579434i	1.77477	−	0.922064i	1.75683	+	1.75683i	−2.04798	+	1.34380i	1.77368	−2.12878	+	1.86233i	2.32851	−	1.89157i	−3.00313	−	1.82400i
35.6	−1.35099	+	0.418117i	−1.72576	+	0.147446i	1.65036	−	1.12975i	−2.11786	−	2.11786i	2.26984	−	0.920770i	−3.48796	−1.75725	+	2.21632i	2.95652	−	0.508915i	3.74673	+	1.97570i
35.7	−1.33501	−	0.466627i	−0.461119	−	1.66954i	1.56452	+	1.24591i	2.64828	+	2.64828i	−0.163453	+	2.44403i	4.61896	−1.50728	−	2.39335i	−2.57474	+	1.53972i	−2.29973	−	4.77124i
35.8	−1.31634	+	0.516966i	1.62317	+	0.604407i	1.46549	−	1.36100i	−2.41460	−	2.41460i	−2.44910	+	0.0435202i	2.57003	−1.22549	+	2.54915i	2.26938	+	1.96212i	4.42670	+	1.93017i
35.9	−1.26166	−	0.638910i	0.881475	−	1.49097i	1.18359	+	1.61218i	−0.0939607	−	0.0939607i	−2.06472	+	1.31792i	−0.105597	−0.463253	−	2.79023i	−1.44600	−	2.62851i	0.0585143	+	0.178579i
35.10	−1.21433	+	0.724839i	1.01832	−	1.40108i	0.949217	−	1.76039i	−1.21434	−	1.21434i	−0.221025	+	2.43950i	−3.59563	0.123336	+	2.82574i	−0.926045	−	2.85350i	2.35482	+	0.594414i
35.11	−1.20913	−	0.733493i	1.40161	+	1.01759i	0.923975	+	1.77377i	−2.32657	−	2.32657i	−0.948335	−	2.25846i	−2.46056	0.183848	−	2.82245i	0.929037	+	2.85252i	1.10659	+	4.51964i
35.12	−1.16677	−	0.799147i	−1.69103	+	0.374733i	0.722727	+	1.86485i	0.455424	+	0.455424i	2.27252	+	0.914152i	−4.72509	0.647031	−	2.75343i	2.71915	−	1.26737i	−0.167426	−	0.895328i
35.13	−1.15730	−	0.812808i	1.40099	−	1.01844i	0.678686	+	1.88133i	−1.62872	−	1.62872i	−2.44916	+	0.0398998i	2.91139	0.743713	−	2.72890i	0.925566	−	2.85365i	0.561082	+	3.20876i
35.14	−1.14105	+	0.835462i	−1.51128	−	0.846192i	0.604005	−	1.90661i	2.28493	+	2.28493i	2.43141	−	0.297066i	−3.55506	0.903702	+	2.68017i	1.56792	+	2.55766i	−4.51620	−	0.698254i
35.15	−1.08045	−	0.912489i	1.04619	+	1.38039i	0.334729	+	1.97179i	2.83480	+	2.83480i	0.129236	−	2.44608i	−1.14023	1.43758	−	2.43585i	−0.810962	+	2.88831i	−0.476127	−	5.64957i
35.16	−1.04000	+	0.958328i	−0.343694	+	1.69761i	0.163215	−	1.99333i	0.141788	+	0.141788i	−1.26942	−	2.09489i	−1.49884	1.74052	+	2.22948i	−2.76375	−	1.16692i	−0.283338	−	0.0115806i
35.17	−1.03888	−	0.959546i	−1.47856	+	0.902142i	0.158543	+	1.99371i	0.812410	+	0.812410i	2.40169	+	0.481530i	3.63748	1.74835	−	2.22335i	1.37228	−	2.66774i	−0.0644515	−	1.62354i
35.18	−0.969764	−	1.02935i	−0.112313	+	1.72841i	−0.119114	+	1.99645i	−2.07215	−	2.07215i	1.88805	−	1.56054i	3.25344	2.17055	−	1.81348i	−2.97477	−	0.388246i	−0.123466	+	4.14246i
35.19	−0.945277	+	1.05188i	0.208059	−	1.71951i	−0.212904	−	1.98864i	−2.52621	−	2.52621i	1.61204	+	1.84426i	1.58640	2.29306	+	1.65586i	−2.91342	−	0.715518i	5.04524	−	0.269303i
35.20	−0.745678	+	1.20165i	1.61122	+	0.635578i	−0.887929	−	1.79209i	1.74078	+	1.74078i	−1.96520	+	1.46219i	−1.17168	2.81557	+	0.269341i	2.19208	+	2.04812i	−3.38987	+	0.793748i

See next 80 embeddings (of 124 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
48.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	816.2.bh.c	✓	124
3.b	odd	2	1		816.2.bh.d	yes	124
16.f	odd	4	1		816.2.bh.d	yes	124
48.k	even	4	1	inner	816.2.bh.c	✓	124

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
816.2.bh.c	✓	124	1.a	even	1	1	trivial
816.2.bh.c	✓	124	48.k	even	4	1	inner
816.2.bh.d	yes	124	3.b	odd	2	1
816.2.bh.d	yes	124	16.f	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{124} + 4 T_{5}^{123} + 8 T_{5}^{122} - 16 T_{5}^{121} + 1904 T_{5}^{120} + 7504 T_{5}^{119} + \cdots + 14\!\cdots\!24 $ T5^124 + 4*T5^123 + 8*T5^122 - 16*T5^121 + 1904*T5^120 + 7504*T5^119 + 14912*T5^118 - 29120*T5^117 + 1644492*T5^116 + 6385920*T5^115 + 12616224*T5^114 - 24109792*T5^113 + 854295112*T5^112 + 3267381488*T5^111 + 6420897408*T5^110 - 12039464640*T5^109 + 298544724422*T5^108 + 1123576212776*T5^107 + 2197156628976*T5^106 - 4053507857856*T5^105 + 74410680425976*T5^104 + 275121339337744*T5^103 + 535460910122368*T5^102 - 974636764467456*T5^101 + 13693747642638908*T5^100 + 49615846794044640*T5^99 + 96099438375324960*T5^98 - 172994242982739872*T5^97 + 1901845480238617208*T5^96 + 6728929682396316112*T5^95 + 12963460301499592832*T5^94 - 23121665169621169600*T5^93 + 202200101914645520769*T5^92 + 695336543142086638004*T5^91 + 1331082248794209273672*T5^90 - 2354463744628615932848*T5^89 + 16608241873267917949208*T5^88 + 55190458860545175358432*T5^87 + 104814453674018295584576*T5^86 - 183816469590126747898816*T5^85 + 1059570641554880119579184*T5^84 + 3379703737856708518194496*T5^83 + 6354008892438418888350592*T5^82 - 11029807015084131015476224*T5^81 + 52614802400243526558919744*T5^80 + 159925655244666205285261824*T5^79 + 296873594259871211024197120*T5^78 - 508510564465191087716344064*T5^77 + 2031590251262989012697575168*T5^76 + 5842082324125132959216407552*T5^75 + 10678663381028750546649745408*T5^74 - 17970593755304017828714458112*T5^73 + 60764493353976812860361651200*T5^72 + 164204561133768642671299663872*T5^71 + 294844813097072416049632755712*T5^70 - 485008203558929565584556023808*T5^69 + 1399039665366458157979996599808*T5^68 + 3531366644257568300130102495232*T5^67 + 6219979733946133983245598584832*T5^66 - 9951732486928124720751164764160*T5^65 + 24600298327300257007977739759616*T5^64 + 57662220715557397337087466504192*T5^63 + 99636745845294296861923080077312*T5^62 - 154466174774040799192056803672064*T5^61 + 327561076650538571097540559577088*T5^60 + 708095597812667788890607057698816*T5^59 + 1202723796757648032544213785313280*T5^58 - 1803729306227426529861352731967488*T5^57 + 3276805877315008952738729653551104*T5^56 + 6468758767204535287473831928266752*T5^55 + 10842179785106259819575140835524608*T5^54 - 15746572114567427334628383861702656*T5^53 + 24468110638422685567874616127651840*T5^52 + 43447630639951136178940956031057920*T5^51 + 72241725369798593026364266956980224*T5^50 - 101976271924341227109007468288802816*T5^49 + 135676296496517434269814815273517056*T5^48 + 211851502716590988558565549127761920*T5^47 + 351590422931476266906675208555331584*T5^46 - 484925358245374553907842256681828352*T5^45 + 555793160595522609689555551405539328*T5^44 + 739296857607948123943137920512425984*T5^43 + 1232475716365655152085561309668048896*T5^42 - 1670169407858628788451435133351755776*T5^41 + 1668890429667495643372148194620735488*T5^40 + 1814088615893626166746753829080924160*T5^39 + 3058009166335137805976021239014096896*T5^38 - 4090326980073125495315828637649862656*T5^37 + 3622694304625225463223625544081866752*T5^36 + 3055280707493324741705227831391289344*T5^35 + 5249750217786073635535875251386712064*T5^34 - 6950254127160944320569592093987569664*T5^33 + 5556474376628685513511331917484523520*T5^32 + 3408048695358372090810848835362881536*T5^31 + 6049860237079644902481409779729170432*T5^30 - 7934394959529681913365712917002977280*T5^29 + 5824381834980213508175058518321135616*T5^28 + 2380322199130109880892740551758053376*T5^27 + 4498505028100601196654043541350121472*T5^26 - 5844520809462529631379714237358145536*T5^25 + 3999916472728675575188756846197342208*T5^24 + 941985343204056221957465907006537728*T5^23 + 2049679835669484687960439805838360576*T5^22 - 2644222539417560650034332004775886848*T5^21 + 1717857788742549221303667592613855232*T5^20 + 164194571232938666749044800113082368*T5^19 + 527528749299704467338195318125625344*T5^18 - 679907246878065573781074681947750400*T5^17 + 430374818694847275483141946315112448*T5^16 - 2691235289851558909102170969210880*T5^15 + 65446185567163921096535425654194176*T5^14 - 84702836420927813722203007102222336*T5^13 + 53863939352164756239771939318530048*T5^12 - 3608114472037717580617476812570624*T5^11 + 2540590345481942974470042012352512*T5^10 - 3278509211468033385531279503523840*T5^9 + 2214998956496303201744176078127104*T5^8 - 160335619876122117692396998230016*T5^7 + 30554756290558019025132874366976*T5^6 - 35491918765838267840657732141056*T5^5 + 25282257416629237016797458202624*T5^4 - 2014232228041205936864966475776*T5^3 + 25000657457430896988492136448*T5^2 + 26893814695947044551552663552*T5 + 14465164968790829194201268224 acting on $S_{2}^{\mathrm{new}}(816, [\chi])$.

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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.bh (of order \(4\), degree \(2\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 35.62
Significant digits:
Format:

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