LMFDB - Newform orbit 2883.1.z.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2883,1,Mod(2,2883)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2883, base_ring=CyclotomicField(310))

chi = DirichletCharacter(H, H._module([155, 208]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2883.2");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2883 = 3 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2883.z (of order $310$, degree $120$, 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:	$1.43880443142$
Analytic rank:	$0$
Dimension:	$120$
Coefficient field:	$\Q(\zeta_{155})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{120} - x^{119} + x^{115} - x^{114} + x^{110} - x^{109} + x^{105} - x^{104} + x^{100} - x^{99} + \cdots + 1 $ x^120 - x^119 + x^115 - x^114 + x^110 - x^109 + x^105 - x^104 + x^100 - x^99 + x^95 - x^94 + x^90 - x^88 + x^85 - x^83 + x^80 - x^78 + x^75 - x^73 + x^70 - x^68 + x^65 - x^63 + x^60 - x^57 + x^55 - x^52 + x^50 - x^47 + x^45 - x^42 + x^40 - x^37 + x^35 - x^32 + x^30 - x^26 + x^25 - x^21 + x^20 - x^16 + x^15 - x^11 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{155}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{155} - \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{310}^{107} q^{3} - \zeta_{310}^{81} q^{4} + (\zeta_{310}^{138} - \zeta_{310}^{89}) q^{7} - \zeta_{310}^{59} q^{9} - \zeta_{310}^{33} q^{12} + (\zeta_{310}^{128} + \zeta_{310}^{126}) q^{13} - \zeta_{310}^{7} q^{16} + \cdots + ( - \zeta_{310}^{95} + \zeta_{310}^{22}) q^{97} +O(q^{100}) $ q - z^107 * q^3 - z^81 * q^4 + (z^138 - z^89) * q^7 - z^59 * q^9 - z^33 * q^12 + (z^128 + z^126) * q^13 - z^7 * q^16 + (z^142 + z^84) * q^19 + (z^90 - z^41) * q^21 - z^75 * q^25 - z^11 * q^27 + (z^64 - z^15) * q^28 - z^127 * q^31 + z^140 * q^36 + (-z^87 - z^43) * q^37 + (z^80 + z^78) * q^39 + (-z^151 - z^65) * q^43 + z^114 * q^48 + (-z^121 + z^72 - z^23) * q^49 + (z^54 + z^52) * q^52 + (z^94 + z^36) * q^57 + (z^86 + z^34) * q^61 + (z^148 + z^42) * q^63 + z^88 * q^64 + (-z^119 + z^56) * q^67 + (-z^93 - z^29) * q^73 - z^27 * q^75 + (z^68 + z^10) * q^76 + (-z^105 - z^83) * q^79 + z^118 * q^81 + (z^122 + z^16) * q^84 + (-z^111 - z^109 + z^62 + z^60) * q^91 - z^79 * q^93 + (-z^95 + z^22) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 120 q + q^{3} + q^{4} + 2 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 2 q^{19} - 3 q^{21} - 4 q^{25} + q^{27} - 3 q^{28} + q^{31} - 4 q^{36} + 2 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + 3 q^{49}+ \cdots - 3 q^{97}+O(q^{100}) $ 120 * q + q^3 + q^4 + 2 * q^7 + q^9 + q^12 + 2 * q^13 + q^16 + 2 * q^19 - 3 * q^21 - 4 * q^25 + q^27 - 3 * q^28 + q^31 - 4 * q^36 + 2 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + 3 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 + 2 * q^67 - 29 * q^73 + q^75 - 3 * q^76 - 3 * q^79 + q^81 + 2 * q^84 - 32 * q^91 + q^93 - 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times$.

$n$	$962$	$964$
$\chi(n)$	$-1$	$-\zeta_{310}^{59}$

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

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

2.1

0.843240	−	0.537537i
−0.131363	−	0.991334i
0.270221	+	0.962798i
−0.0910811	−	0.995843i
0.111245	+	0.993793i
−0.366239	+	0.930521i
−0.674136	+	0.738607i
0.986883	+	0.161437i
−0.941224	−	0.337784i
0.771804	−	0.635861i
−0.983408	−	0.181405i
−0.403648	+	0.914914i
0.717774	−	0.696277i
−0.983408	+	0.181405i
−0.366239	−	0.930521i
−0.784532	−	0.620088i
0.947876	+	0.318639i
0.595954	−	0.803019i
0.796938	−	0.604061i
−0.989952	−	0.141403i

−0.511656

0.859190i

−0.403648

−

0.914914i

−0.00468152

−

0.461938i

−0.476416

−

0.879220i

8.1

0.999179

0.0405256i

0.947876

0.318639i

−0.0391311

−

1.28670i

0.996715

0.0809846i

35.1

0.843240

0.537537i

−0.171430

−

0.985196i

−1.70167

0.771448i

0.422108

0.906546i

47.1

−0.328229

−

0.944598i

−0.893295

−

0.449470i

−1.96528

0.280717i

−0.784532

0.620088i

95.1

0.595954

−

0.803019i

0.385023

−

0.922907i

0.471460

−

0.560581i

−0.289679

−

0.957124i

101.1

0.659028

0.752118i

0.864331

0.502923i

−1.01784

−

1.37149i

−0.131363

0.991334i

128.1

−0.579556

0.814932i

0.230981

−

0.972958i

−0.141497

−

1.99128i

−0.328229

−

0.944598i

140.1

0.0708797

−

0.997485i

0.843240

0.537537i

−1.21644

0.581474i

−0.989952

−

0.141403i

188.1

−0.674136

0.738607i

0.934184

−

0.356791i

−1.64251

0.268687i

−0.0910811

−

0.995843i

194.1

−0.0910811

0.995843i

0.745401

0.666616i

0.729908

0.245367i

−0.983408

−

0.181405i

221.1

−0.784532

−

0.620088i

0.595954

−

0.803019i

1.86351

0.543448i

0.230981

0.972958i

233.1

0.458499

−

0.888695i

−0.784532

−

0.620088i

−0.0444865

0.0414375i

−0.579556

−

0.814932i

281.1

0.745401

−

0.666616i

0.902221

0.431273i

1.70757

1.04046i

0.111245

−

0.993793i

287.1

−0.784532

0.620088i

0.595954

0.803019i

1.86351

−

0.543448i

0.230981

−

0.972958i

314.1

0.659028

−

0.752118i

0.864331

−

0.502923i

−1.01784

1.37149i

−0.131363

−

0.991334i

326.1

0.771804

−

0.635861i

0.717774

0.696277i

0.620644

−

1.09196i

0.191362

−

0.981520i

380.1

−0.989952

−

0.141403i

0.422108

0.906546i

−0.114399

0.141760i

0.960010

0.279964i

407.1

0.717774

0.696277i

0.992615

−

0.121311i

−0.719731

−

1.10420i

0.0303978

0.999538i

419.1

0.960010

−

0.279964i

−0.643650

−

0.765320i

0.415421

−

1.92244i

0.843240

−

0.537537i

467.1

0.864331

−

0.502923i

−0.476416

0.879220i

−0.252751

0.605849i

0.494138

−

0.869384i

See next 80 embeddings (of 120 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
961.l	even	155	1	inner
2883.z	odd	310	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2883.1.z.a	✓	120
3.b	odd	2	1	CM	2883.1.z.a	✓	120
961.l	even	155	1	inner	2883.1.z.a	✓	120
2883.z	odd	310	1	inner	2883.1.z.a	✓	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2883.1.z.a	✓	120	1.a	even	1	1	trivial
2883.1.z.a	✓	120	3.b	odd	2	1	CM
2883.1.z.a	✓	120	961.l	even	155	1	inner
2883.1.z.a	✓	120	2883.z	odd	310	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2883, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{120} $ T^120
$3$	$ T^{120} - T^{119} + \cdots + 1 $ T^120 - T^119 + T^115 - T^114 + T^110 - T^109 + T^105 - T^104 + T^100 - T^99 + T^95 - T^94 + T^90 - T^88 + T^85 - T^83 + T^80 - T^78 + T^75 - T^73 + T^70 - T^68 + T^65 - T^63 + T^60 - T^57 + T^55 - T^52 + T^50 - T^47 + T^45 - T^42 + T^40 - T^37 + T^35 - T^32 + T^30 - T^26 + T^25 - T^21 + T^20 - T^16 + T^15 - T^11 + T^10 - T^6 + T^5 - T + 1
$5$	$ T^{120} $ T^120
$7$	$ T^{120} - 2 T^{119} + \cdots + 1 $ T^120 - 2T^119 + 5T^117 - 5T^116 - 8T^115 + 21T^114 + 93T^113 - 241T^112 + 179T^111 + 306T^110 - 698T^109 - 124T^108 + 1943T^107 + 4939T^106 - 16580T^105 + 41396T^104 - 33232T^103 - 41361T^102 + 108073T^101 + 7629T^100 + 130063T^99 - 573500T^98 - 3921247T^97 + 9738676T^96 + 4284916T^95 - 35181312T^94 + 29242176T^93 + 85700678T^92 - 187103290T^91 - 378211438T^90 + 1218491828T^89 - 74114463T^88 - 3403286759T^87 + 3505830243T^86 + 6383921091T^85 - 16373326880T^84 - 15682588377T^83 + 74202187519T^82 - 113128243015T^81 - 7062681069T^80 + 353911646327T^79 - 393507201825T^78 - 523236031470T^77 + 1198750523930T^76 + 524909357111T^75 - 2132128767705T^74 - 906894175553T^73 + 13258970840235T^72 - 16504860696029T^71 - 10696067060965T^70 + 56264745022482T^69 - 34583320219756T^68 - 16575795715192T^67 + 13453977626127T^66 - 71328209069294T^65 + 183517819183764T^64 - 309527624435486T^63 + 226955072448796T^62 + 215325955433253T^61 + 253021556445925T^60 - 1765682819499578T^59 + 4708564910709178T^58 - 3277561579465693T^57 + 2174871409080591T^56 - 2410638781581414T^55 + 4016800290998873T^54 - 9498032304826429T^53 + 8556387278392003T^52 - 11978623061885290T^51 + 14547127446855196T^50 - 33824402333801236T^49 + 42542288056755963T^48 - 2928269756966052T^47 - 64326678770065924T^46 + 62536515893218920T^45 - 1436453377326914T^44 - 47795033667055629T^43 + 90226017859084101T^42 - 57307576187902919T^41 - 5926006917053551T^40 + 76245989232392559T^39 - 39632791561920179T^38 - 59918756198838773T^37 + 43083587242999021T^36 + 24087121837803023T^35 - 10399096225388914T^34 - 296325976485825T^33 - 35327988292660177T^32 + 6986565017270068T^31 + 64163252515759644T^30 - 53979560045387687T^29 + 2990623083172617T^28 + 10569730936386790T^27 - 3232598121988266T^26 + 4087426101523941T^25 - 4952649019846760T^24 + 553450359928607T^23 + 1702031906780769T^22 - 803560110332506T^21 + 113412119294544T^20 + 173554919431751T^19 - 135033425236262T^18 + 35132206562349T^17 + 11153922949631T^16 - 13443506556900T^15 + 4877747561709T^14 - 823573255097T^13 + 113993593008T^12 - 21237383316T^11 - 517826646T^10 - 791924016T^9 - 57655467T^8 + 5119302T^7 + 14607724T^6 + 6862483T^5 + 1273325T^4 + 94545T^3 + 3787T^2 + 59T + 1
$11$	$ T^{120} $ T^120
$13$	$ T^{120} - 2 T^{119} + \cdots + 1 $ T^120 - 2T^119 + 5T^117 - 5T^116 - 8T^115 + 21T^114 - 55T^112 + 55T^111 + 89T^110 - 233T^109 + 610T^107 - 610T^106 - 987T^105 + 2615T^104 - 5022T^103 + 17911T^102 + 59620T^101 + 81905T^100 + 69675T^99 - 3467381T^98 + 7011585T^97 + 2475T^96 - 17499218T^95 + 17499592T^94 + 27898078T^93 - 73295718T^92 + 191989077T^90 - 191989076T^89 - 310681477T^88 + 813236707T^87 + 8083937T^86 - 2194876371T^85 + 2156440202T^84 + 5119201658T^83 + 152189702T^82 - 101945557401T^81 + 305697257208T^80 - 93100370336T^79 - 886178610739T^78 + 658557576449T^77 + 3710085626025T^76 - 8100501003789T^75 + 32528527158T^74 + 20743783911032T^73 - 20730587017298T^72 - 33797596470833T^71 + 88325018671810T^70 - 4719351874T^69 - 231127953656921T^68 + 230932913629441T^67 + 373399407786651T^66 - 956982600862275T^65 - 158108569199113T^64 + 3154136326422398T^63 - 3656213032403690T^62 - 5077517386375539T^61 + 22791563482533094T^60 - 35338235598799099T^59 + 23646745047455122T^58 + 16296710882908016T^57 - 63774764900188497T^56 + 87840507988375951T^55 - 73589223916387188T^54 + 32864933718357397T^53 + 8526633965863655T^52 - 31352651518529447T^51 + 32872422881990330T^50 - 22400701078966534T^49 + 12931050004112464T^48 - 21733556530857025T^47 + 90678236007870915T^46 - 340292972786109413T^45 + 1080821811847108468T^44 - 2957716300908310107T^43 + 6989468305313933805T^42 - 14321353741494111674T^41 + 25722748503038918447T^40 - 41022934664718391762T^39 + 58725332580895294446T^38 - 76070994518404711790T^37 + 89724373388801236337T^36 - 96840153572207223181T^35 + 96000856953273911239T^34 - 87621711243544482761T^33 + 73707732348038286555T^32 - 57113965901763023122T^31 + 40656389625552099794T^30 - 26430642257136465346T^29 + 15522253586248298412T^28 - 8080432653226307644T^27 + 3605267592036331060T^26 - 1288989337771141921T^25 + 306830924763904887T^24 - 4132980527240566T^23 - 34338490325158170T^22 + 11713253689052076T^21 + 4432279130899188T^20 - 7225035533822594T^19 + 4566701134370238T^18 - 1899421301861553T^17 + 546356593704527T^16 - 92864011580505T^15 - 2869810799024T^14 + 8015576223894T^13 - 2914324300031T^12 + 577545321776T^11 - 46516568736T^10 - 7672491227T^9 + 3006344760T^8 - 533411179T^7 + 85433145T^6 - 12140486T^5 + 1064075T^4 - 29517T^3 - 274T^2 + 59*T + 1
$17$	$ T^{120} $ T^120
$19$	$ T^{120} - 2 T^{119} + \cdots + 1 $ T^120 - 2T^119 - 119T^117 + 243T^116 - 8T^115 + 6779T^114 - 14136T^113 + 937T^112 - 246147T^111 + 524175T^110 - 52437T^109 + 6399733T^108 - 13916096T^107 + 1866396T^106 - 126895922T^105 + 281674815T^104 - 47449003T^103 + 1995757394T^102 - 4519847299T^101 + 917335139T^100 - 25566946709T^99 + 59028337365T^98 - 14020991049T^97 + 271889712089T^96 - 639213525368T^95 + 173902340488T^94 - 2434103091422T^93 + 5818347684907T^92 - 1782786779884T^91 + 18540996557558T^90 - 44971785958877T^89 + 15310036033561T^88 - 121150869018964T^87 + 297435955223532T^86 - 111234858762635T^85 + 683390571026781T^84 - 1692998703656080T^83 + 688802962783601T^82 - 3344108668160718T^81 + 8328611024341618T^80 - 3655082799602401T^79 + 14248330084104153T^78 - 35518053358425286T^77 + 16685364415559771T^76 - 53000230934009192T^75 + 131564834129448877T^74 - 65696698798557214T^73 + 172415166616260377T^72 - 423737811167611603T^71 + 223447742912377793T^70 - 490935150960304588T^69 + 1186877625368090347T^68 - 656840468177002583T^67 + 1223501816803238238T^66 - 2889339670478022495T^65 + 1668102438138940485T^64 - 2666249360973358371T^63 + 6104606501668569322T^62 - 3655139374487978816T^61 + 5070877702842949179T^60 - 11168764693507286309T^59 + 6895284330377512501T^58 - 8392531594512986645T^57 + 17639313455786479360T^56 - 11163710684504164941T^55 + 12040096368834584195T^54 - 23950658081781337001T^53 + 15448175625374815496T^52 - 14898523532564513104T^51 + 27816089826321092086T^50 - 18175266716723673254T^49 + 15805851550452597890T^48 - 27460663577413915076T^47 + 18063207208551904437T^46 - 14273613108766200998T^45 + 22872103834943022431T^44 - 15043253361541054545T^43 + 10877918894424961716T^42 - 15928399700296499520T^41 + 10394610102447389706T^40 - 6921311261405645361T^39 + 9173329190787569295T^38 - 5884608748065474083T^37 + 3624760016840296104T^36 - 4307898184629765428T^35 + 2684743370745125241T^34 - 1531246208834208952T^33 + 1617998306510157731T^32 - 965166355540146394T^31 + 506825743398074775T^30 - 472707581806091750T^29 + 264888490855914898T^28 - 125302054753472457T^27 + 102100025203927637T^26 - 52347294201158992T^25 + 21498635241058180T^24 - 15271233367869822T^23 + 6880188049702343T^22 - 2078514166245415T^21 + 1235162298889332T^20 - 427095806584873T^19 + 58547528015373T^18 - 53765855714827T^17 + 24779596631889T^16 - 3367390538920T^15 + 2537635943380T^14 - 771213206336T^13 - 109866452203T^12 + 11485340066T^11 + 20775411659T^10 - 1589154581T^9 + 89950103T^8 - 69594015T^7 - 4285319T^6 + 2545702T^5 + 746480T^4 + 35490T^3 + 3105T^2 + 90T + 1
$23$	$ T^{120} $ T^120
$29$	$ T^{120} $ T^120
$31$	$ T^{120} - T^{119} + \cdots + 1 $ T^120 - T^119 + T^115 - T^114 + T^110 - T^109 + T^105 - T^104 + T^100 - T^99 + T^95 - T^94 + T^90 - T^88 + T^85 - T^83 + T^80 - T^78 + T^75 - T^73 + T^70 - T^68 + T^65 - T^63 + T^60 - T^57 + T^55 - T^52 + T^50 - T^47 + T^45 - T^42 + T^40 - T^37 + T^35 - T^32 + T^30 - T^26 + T^25 - T^21 + T^20 - T^16 + T^15 - T^11 + T^10 - T^6 + T^5 - T + 1
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$41$	$ T^{120} $ T^120
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$47$	$ T^{120} $ T^120
$53$	$ T^{120} $ T^120
$59$	$ T^{120} $ T^120
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$71$	$ T^{120} $ T^120
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$83$	$ T^{120} $ T^120
$89$	$ T^{120} $ T^120
$97$	$ T^{120} + 3 T^{119} + \cdots + 1 $ T^120 + 3T^119 + 5T^118 + 5T^117 - 13T^115 - 34T^114 - 55T^113 - 55T^112 + 144T^110 + 532T^109 - 7109T^108 - 20098T^107 - 30132T^106 - 24723T^105 + 13272T^104 + 98542T^103 + 223224T^102 + 326771T^101 + 271229T^100 - 145460T^99 - 1068658T^98 - 3346538T^97 + 17504336T^96 + 45363910T^95 + 64901072T^94 + 56371671T^93 - 16428729T^92 - 190411207T^91 - 457675505T^90 - 700610819T^89 - 635514512T^88 + 173964984T^87 + 2036874825T^86 + 6567871004T^85 - 16928332466T^84 - 38488315189T^83 - 64181831288T^82 - 69832374802T^81 + 6568308633T^80 + 204904181028T^79 + 513545369683T^78 + 803954293373T^77 + 761338535351T^76 - 113750674368T^75 - 2241246299640T^74 - 6462197731993T^73 + 2725522347334T^72 + 9062862001454T^71 + 31758123335797T^70 + 36158102260369T^69 + 42677312754384T^68 + 23476686262935T^67 - 55661942612202T^66 - 212775595604634T^65 - 410718532688697T^64 - 520208795126175T^63 - 259711542173806T^62 + 654206176378508T^61 + 739597711236985T^60 + 1203666372598270T^59 + 2011831643785389T^58 - 574301024580691T^57 - 1856403705679888T^56 - 753742446288978T^55 + 1296302986773836T^54 + 4222226528362618T^53 + 7762923760911406T^52 + 9385713855290287T^51 + 3540620452190204T^50 - 7173045327118882T^49 + 19200542750368101T^48 + 80198909132330909T^47 + 92877270067613362T^46 + 24752366028452900T^45 - 18550963057779249T^44 - 5807769098316639T^43 - 7799584791759291T^42 - 70055311132998226T^41 - 161479952221268800T^40 - 209584233140562831T^39 - 133695681170028252T^38 + 96660653302714546T^37 + 352819201987001510T^36 + 439119281983774244T^35 + 289187489658911892T^34 + 37173337010875990T^33 - 107563105281171514T^32 - 90140719305092171T^31 - 4191816150817977T^30 + 60279338950017091T^29 + 90839095908854677T^28 + 113451819309056653T^27 + 135220416072117455T^26 + 140032765705140490T^25 + 120140082450462754T^24 + 85924528701036460T^23 + 52207708496536523T^22 + 27137336466784591T^21 + 12186737962751882T^20 + 4903944190992516T^19 + 1894443451944776T^18 + 730341532091062T^17 + 264406009631630T^16 + 79084377698398T^15 + 15854589194606T^14 + 650578259301T^13 - 803656471021T^12 - 301053559169T^11 - 44725937460T^10 + 2436965900T^9 + 2355746892T^8 + 523474556T^7 + 83639571T^6 + 11003551T^5 + 1034868T^4 + 30344T^3 - 279T^2 - 60*T + 1
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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:	\( N \)	\(=\)	\( 2883 = 3 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2883.z (of order \(310\), degree \(120\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{310}^{107} q^{3} - \zeta_{310}^{81} q^{4} + (\zeta_{310}^{138} - \zeta_{310}^{89}) q^{7} - \zeta_{310}^{59} q^{9} - \zeta_{310}^{33} q^{12} + (\zeta_{310}^{128} + \zeta_{310}^{126}) q^{13} - \zeta_{310}^{7} q^{16} + \cdots + ( - \zeta_{310}^{95} + \zeta_{310}^{22}) q^{97} +O(q^{100}) \) q - z^107 * q^3 - z^81 * q^4 + (z^138 - z^89) * q^7 - z^59 * q^9 - z^33 * q^12 + (z^128 + z^126) * q^13 - z^7 * q^16 + (z^142 + z^84) * q^19 + (z^90 - z^41) * q^21 - z^75 * q^25 - z^11 * q^27 + (z^64 - z^15) * q^28 - z^127 * q^31 + z^140 * q^36 + (-z^87 - z^43) * q^37 + (z^80 + z^78) * q^39 + (-z^151 - z^65) * q^43 + z^114 * q^48 + (-z^121 + z^72 - z^23) * q^49 + (z^54 + z^52) * q^52 + (z^94 + z^36) * q^57 + (z^86 + z^34) * q^61 + (z^148 + z^42) * q^63 + z^88 * q^64 + (-z^119 + z^56) * q^67 + (-z^93 - z^29) * q^73 - z^27 * q^75 + (z^68 + z^10) * q^76 + (-z^105 - z^83) * q^79 + z^118 * q^81 + (z^122 + z^16) * q^84 + (-z^111 - z^109 + z^62 + z^60) * q^91 - z^79 * q^93 + (-z^95 + z^22) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + q^{3} + q^{4} + 2 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 2 q^{19} - 3 q^{21} - 4 q^{25} + q^{27} - 3 q^{28} + q^{31} - 4 q^{36} + 2 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + 3 q^{49}+ \cdots - 3 q^{97}+O(q^{100}) \) 120 * q + q^3 + q^4 + 2 * q^7 + q^9 + q^12 + 2 * q^13 + q^16 + 2 * q^19 - 3 * q^21 - 4 * q^25 + q^27 - 3 * q^28 + q^31 - 4 * q^36 + 2 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + 3 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 + 2 * q^67 - 29 * q^73 + q^75 - 3 * q^76 - 3 * q^79 + q^81 + 2 * q^84 - 32 * q^91 + q^93 - 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2883.1.z.a

Level	$2883$
Weight	$1$
Character orbit	2883.z
Analytic conductor	$1.439$
Analytic rank	$0$
Dimension	$120$
Projective image	$D_{155}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.43880443142\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Coefficient field:	\(\Q(\zeta_{155})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{120} - x^{119} + x^{115} - x^{114} + x^{110} - x^{109} + x^{105} - x^{104} + x^{100} - x^{99} + \cdots + 1 \) x^120 - x^119 + x^115 - x^114 + x^110 - x^109 + x^105 - x^104 + x^100 - x^99 + x^95 - x^94 + x^90 - x^88 + x^85 - x^83 + x^80 - x^78 + x^75 - x^73 + x^70 - x^68 + x^65 - x^63 + x^60 - x^57 + x^55 - x^52 + x^50 - x^47 + x^45 - x^42 + x^40 - x^37 + x^35 - x^32 + x^30 - x^26 + x^25 - x^21 + x^20 - x^16 + x^15 - x^11 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{155}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{155} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
961.l	even	155	1	inner
2883.z	odd	310	1	inner

\(n\)	\(962\)	\(964\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{310}^{59}\)