LMFDB - Newform orbit 804.2.s.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(5,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 5]))

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

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

chi := DirichletCharacter("804.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.s (of order $22$, degree $10$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$200$
Relative dimension:	$20$ over $\Q(\zeta_{22})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-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) = $ $ 200 q - 10 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 200 q - 10 q^{9} + 2 q^{15} + 6 q^{19} - 10 q^{21} - 20 q^{25} - 44 q^{31} - 5 q^{33} + 78 q^{39} - 22 q^{43} - 22 q^{45} - 16 q^{49} + 36 q^{55} + 66 q^{57} + 176 q^{61} + 132 q^{63} + 46 q^{67} - 26 q^{73} - 165 q^{75} - 44 q^{79} + 42 q^{81} - 66 q^{87} - 20 q^{91} + 84 q^{93} - 55 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_{3} $				$ a_{5} $				$ a_{7} $				$ a_{9} $
5.1	0	−1.72660	−	0.137338i	0	2.48625	−	0.730028i	0	3.64531	+	3.15868i	0	2.96228	+	0.474256i	0
5.2	0	−1.58909	+	0.689041i	0	3.06595	−	0.900243i	0	0.591666	+	0.512682i	0	2.05045	−	2.18990i	0
5.3	0	−1.57919	−	0.711443i	0	−0.583807	+	0.171421i	0	−0.967947	−	0.838730i	0	1.98770	+	2.24701i	0
5.4	0	−1.56138	+	0.749732i	0	−3.06595	+	0.900243i	0	0.591666	+	0.512682i	0	1.87580	−	2.34123i	0
5.5	0	−1.53236	−	0.807386i	0	−3.45254	+	1.01376i	0	−0.403229	−	0.349400i	0	1.69626	+	2.47441i	0
5.6	0	−1.17979	−	1.26810i	0	2.83063	−	0.831147i	0	−2.86249	−	2.48036i	0	−0.216176	+	2.99220i	0
5.7	0	−1.02689	+	1.39481i	0	−2.48625	+	0.730028i	0	3.64531	+	3.15868i	0	−0.891005	−	2.86463i	0
5.8	0	−0.786808	−	1.54303i	0	0.966140	−	0.283684i	0	1.54930	+	1.34247i	0	−1.76187	+	2.42813i	0
5.9	0	−0.496478	+	1.65937i	0	0.583807	−	0.171421i	0	−0.967947	−	0.838730i	0	−2.50702	−	1.64768i	0
5.10	0	−0.393301	+	1.68681i	0	3.45254	−	1.01376i	0	−0.403229	−	0.349400i	0	−2.69063	−	1.32685i	0
5.11	0	0.0434100	−	1.73151i	0	−1.09364	+	0.321121i	0	2.09055	+	1.81147i	0	−2.99623	−	0.150329i	0
5.12	0	0.185769	+	1.72206i	0	−2.83063	+	0.831147i	0	−2.86249	−	2.48036i	0	−2.93098	+	0.639810i	0
5.13	0	0.429842	−	1.67787i	0	−3.03558	+	0.891328i	0	−1.25477	−	1.08727i	0	−2.63047	−	1.44244i	0
5.14	0	0.575290	−	1.63372i	0	2.05430	−	0.603197i	0	−2.91035	−	2.52183i	0	−2.33808	−	1.87973i	0
5.15	0	0.650892	+	1.60510i	0	−0.966140	+	0.283684i	0	1.54930	+	1.34247i	0	−2.15268	+	2.08949i	0
5.16	0	1.33701	+	1.10109i	0	1.09364	−	0.321121i	0	2.09055	+	1.81147i	0	0.575207	+	2.94434i	0
5.17	0	1.48256	−	0.895546i	0	−2.22357	+	0.652900i	0	2.01808	+	1.74867i	0	1.39599	−	2.65541i	0
5.18	0	1.54953	+	0.773916i	0	3.03558	−	0.891328i	0	−1.25477	−	1.08727i	0	1.80211	+	2.39842i	0
5.19	0	1.61142	+	0.635084i	0	−2.05430	+	0.603197i	0	−2.91035	−	2.52183i	0	2.19334	+	2.04677i	0
5.20	0	1.64768	−	0.533989i	0	2.22357	−	0.652900i	0	2.01808	+	1.74867i	0	2.42971	−	1.75969i	0

See next 80 embeddings (of 200 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
67.f	odd	22	1	inner
201.j	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	804.2.s.b	✓	200
3.b	odd	2	1	inner	804.2.s.b	✓	200
67.f	odd	22	1	inner	804.2.s.b	✓	200
201.j	even	22	1	inner	804.2.s.b	✓	200

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
804.2.s.b	✓	200	1.a	even	1	1	trivial
804.2.s.b	✓	200	3.b	odd	2	1	inner
804.2.s.b	✓	200	67.f	odd	22	1	inner
804.2.s.b	✓	200	201.j	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{200} + 60 T_{5}^{198} + 2469 T_{5}^{196} + 73443 T_{5}^{194} + 1818260 T_{5}^{192} + \cdots + 61\!\cdots\!81 $ T5^200 + 60*T5^198 + 2469*T5^196 + 73443*T5^194 + 1818260*T5^192 + 38428465*T5^190 + 735112005*T5^188 + 12986620935*T5^186 + 213844495923*T5^184 + 3333906909033*T5^182 + 50301735542716*T5^180 + 733193063906030*T5^178 + 10551356133459831*T5^176 + 145296081782217675*T5^174 + 1948934290010496267*T5^172 + 24807617339389461755*T5^170 + 315054904978698309801*T5^168 + 3821591568651120937214*T5^166 + 44477632772571838194687*T5^164 + 512863017757477926103752*T5^162 + 5876547952673929278456395*T5^160 + 65060016120386213925254900*T5^158 + 689572446682668599805488365*T5^156 + 7323353458082340276127625217*T5^154 + 76172553109205690508206528773*T5^152 + 741073801046260642003943856509*T5^150 + 7485466298978631823832817932938*T5^148 + 71113930335029164390769909058630*T5^146 + 657104781257966919012856193258365*T5^144 + 6333497003503012034104725514276840*T5^142 + 55842625756964064829333668080171798*T5^140 + 482144814395297163776476183091310363*T5^138 + 4272519265373858584795555056403873681*T5^136 + 35081457046201397104990520713910099457*T5^134 + 288196481586946963329872820359497833372*T5^132 + 2311143421783469988794046037159575584213*T5^130 + 18113881946821748278508164632040100539974*T5^128 + 140185100836449570533137148609239856381818*T5^126 + 1047023016660200904451217590905038049363426*T5^124 + 7531111909649252542786368497985190964402032*T5^122 + 52215605474933745854096202860686911767695095*T5^120 + 356518501333951551345391012788664602913501251*T5^118 + 2403677622070830840367253200209168982849867275*T5^116 + 15726270801020095339098931153658979465445600706*T5^114 + 101455884937566325209423961182101007044798405588*T5^112 + 626915216134130114355470387267199223685509264833*T5^110 + 3753497390672833248381896608407269948626780370995*T5^108 + 22013508184849090004479269184190377469502751496278*T5^106 + 124302479143602700918285523220275990174998589429758*T5^104 + 687721748927832830465734329243817286571704991384677*T5^102 + 3727797123766433354131169005151776150490614284160246*T5^100 + 19583275580636535277635413925834701474398544167770588*T5^98 + 100955999033030463062660312621335126146996268743359907*T5^96 + 504389953907615033344742387325064202994446876372755225*T5^94 + 2433999222968504041867253648610127657857554446830848580*T5^92 + 11409195866715578461662913179732062060974873208168735295*T5^90 + 51842426336459553521847622009287764037256186712049052141*T5^88 + 228159753579020088910469837413271234646456919893735258852*T5^86 + 975081435595590942416139045633728855802743083289305998879*T5^84 + 4052482877832402559640416737040604782687456936252623726249*T5^82 + 16327089845555390359261926778649429975821130956890488317227*T5^80 + 63458850383156526274425463597837337397065532659202717485091*T5^78 + 236790845610888041803858551881638889763007719985615041757898*T5^76 + 837890694929014363048950193105795929903555786037447291978263*T5^74 + 2796262606355836567850629803595414603781480474631201830366397*T5^72 + 8749866193518251017780547932045222987278537466736150779888264*T5^70 + 25260966822016714660572523900037891194769891786045848232826114*T5^68 + 66589956292120157236217641802981312047957255068089010714687581*T5^66 + 160631687559144349513660793929716544003296724612009197372918281*T5^64 + 354668354128499095617273348893575379767845654117722357416810776*T5^62 + 710604478690128797965560363076107311970678258273603149619887693*T5^60 + 1267568618615958775920025341302796410839629535520778332641313812*T5^58 + 2055706271524843478684028559541663152686570033841975716165009369*T5^56 + 3096582670169913125218907220576334966983144760030006550369827963*T5^54 + 4546451035615443829294103257238230595763293395127294373627141935*T5^52 + 6199574154413359689763891132082373942782243189229133885319684611*T5^50 + 8318403587249203514139018742346733520168132813929025257644315271*T5^48 + 10624503542622849488005381935832714451678713481575462185941268354*T5^46 + 14341482993873972073966295508011153884409131479743980829694032230*T5^44 + 17049400556273288776637093616878590578305126398145485374541358547*T5^42 + 20313502652962630424475255366767216097804558327564148587695306698*T5^40 + 21183023635697326466886714459500857629425031502452147257672695454*T5^38 + 23071218125851273043235080726019435409190131630207252143571537299*T5^36 + 20988520860421469257710656600982361091787983993254688942355287576*T5^34 + 19628468086316793084655854315454692978268109023680604259395094652*T5^32 + 17838046733067261574780257100862786860433242322931113376698439494*T5^30 + 15319010092866635594532236884863154393418491498160780052648859619*T5^28 + 16188929909559256702216639824488831677555907624398832383665910705*T5^26 + 14138235949735884984364547313075608534811707500681153317454476011*T5^24 + 7840416863500737895824265902501825524062401509011860619368998603*T5^22 + 4718502237165728876404662249310288496367007659496863852123347791*T5^20 + 818383192778719639398533572402584974987484497709985807329447184*T5^18 + 260472510600291915755066917585670882921002845390533082212970787*T5^16 + 55631127812686782724751491584171936509709577467626725919847167*T5^14 + 49966582437028460700102947735436504553881308580739197253726317*T5^12 + 26845634868916993752091160373699869203450947839796147020800618*T5^10 + 8982385335843219679451875099586802376094398641048156822193229*T5^8 + 1656517647691111803516349357279359787124594365877293848379201*T5^6 + 148659671780342279502983148043652984903060452375866660520289*T5^4 + 5129697249604066101286824378800458775270148920689860163953*T5^2 + 61415334133078640794462651616837084164720496293639256281 acting on $S_{2}^{\mathrm{new}}(804, [\chi])$.

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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.s (of order \(22\), degree \(10\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.20
Significant digits:
Format:

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