LMFDB - Newform orbit 162.2.g.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,2,Mod(7,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("162.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	162.g (of order $27$, degree $18$, 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:	$1.29357651274$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$5$ over $\Q(\zeta_{27})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

$q$-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) = $ $ 90 q - 9 q^{6}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 90 q - 9 q^{6} - 18 q^{13} - 9 q^{18} - 9 q^{20} - 54 q^{21} + 27 q^{23} - 18 q^{25} - 27 q^{26} - 27 q^{27} - 18 q^{28} - 27 q^{29} + 9 q^{30} + 54 q^{31} - 63 q^{33} - 27 q^{35} - 9 q^{36} - 18 q^{38} - 9 q^{41} - 9 q^{42} - 36 q^{43} + 63 q^{45} + 18 q^{46} - 27 q^{47} - 9 q^{48} - 36 q^{51} + 36 q^{52} - 27 q^{53} - 54 q^{55} - 81 q^{57} - 9 q^{58} - 45 q^{59} - 63 q^{63} + 9 q^{65} + 36 q^{66} + 81 q^{67} + 36 q^{68} + 18 q^{69} - 72 q^{70} + 72 q^{71} + 18 q^{72} - 36 q^{73} + 45 q^{74} + 216 q^{75} - 18 q^{76} + 144 q^{77} + 54 q^{78} - 99 q^{79} + 18 q^{80} + 144 q^{81} + 72 q^{82} + 45 q^{83} + 18 q^{84} - 117 q^{85} + 72 q^{86} + 81 q^{87} - 18 q^{88} + 45 q^{89} + 162 q^{90} - 63 q^{91} + 36 q^{92} + 45 q^{93} - 72 q^{94} + 45 q^{95} + 18 q^{96} + 117 q^{97} + 36 q^{98} - 81 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} $
7.1	0.597159	+	0.802123i	−1.58049	+	0.708545i	−0.286803	+	0.957990i	−0.763144	+	0.501928i	−1.51215	−	0.844638i	−2.48772	+	2.63683i	−0.939693	+	0.342020i	1.99593	−	2.23970i	−0.858326	−	0.312405i
7.2	0.597159	+	0.802123i	0.156275	+	1.72499i	−0.286803	+	0.957990i	3.59342	−	2.36343i	−1.29033	+	1.15544i	−0.0885835	+	0.0938930i	−0.939693	+	0.342020i	−2.95116	+	0.539145i	4.04160	+	1.47102i
7.3	0.597159	+	0.802123i	0.427379	−	1.67850i	−0.286803	+	0.957990i	1.23282	−	0.810835i	1.60157	−	0.659517i	0.694984	−	0.736640i	−0.939693	+	0.342020i	−2.63469	−	1.43471i	1.38658	+	0.504672i
7.4	0.597159	+	0.802123i	0.497048	+	1.65920i	−0.286803	+	0.957990i	−3.09160	+	2.03338i	−1.03407	+	1.38950i	2.90797	−	3.08226i	−0.939693	+	0.342020i	−2.50589	+	1.64940i	−3.47720	−	1.26560i
7.5	0.597159	+	0.802123i	1.73136	−	0.0489119i	−0.286803	+	0.957990i	−0.660279	+	0.434273i	1.07313	+	1.35956i	−2.31933	+	2.45835i	−0.939693	+	0.342020i	2.99522	−	0.169368i	−0.742632	−	0.270296i
13.1	0.893633	+	0.448799i	−1.57770	−	0.714739i	0.597159	+	0.802123i	0.891212	+	2.97686i	−1.08911	−	1.34679i	0.275210	+	0.638009i	0.173648	+	0.984808i	1.97830	+	2.25529i	−0.539594	+	3.06019i
13.2	0.893633	+	0.448799i	−0.977769	−	1.42967i	0.597159	+	0.802123i	−1.15296	−	3.85116i	−0.232129	−	1.71643i	−0.663579	−	1.53835i	0.173648	+	0.984808i	−1.08794	+	2.79578i	0.698073	−	3.95897i
13.3	0.893633	+	0.448799i	−0.530234	+	1.64889i	0.597159	+	0.802123i	−0.536865	−	1.79325i	−1.21386	+	1.23554i	1.99514	+	4.62524i	0.173648	+	0.984808i	−2.43770	−	1.74860i	0.325051	−	1.84346i
13.4	0.893633	+	0.448799i	1.02523	−	1.39603i	0.597159	+	0.802123i	0.664151	+	2.21842i	1.54272	−	0.787412i	−0.590051	−	1.36789i	0.173648	+	0.984808i	−0.897788	−	2.86251i	−0.402118	+	2.28052i
13.5	0.893633	+	0.448799i	1.51179	+	0.845280i	0.597159	+	0.802123i	−0.379535	−	1.26773i	0.971622	+	1.43386i	−0.768169	−	1.78082i	0.173648	+	0.984808i	1.57100	+	2.55577i	0.229793	−	1.30322i
25.1	0.893633	−	0.448799i	−1.57770	+	0.714739i	0.597159	−	0.802123i	0.891212	−	2.97686i	−1.08911	+	1.34679i	0.275210	−	0.638009i	0.173648	−	0.984808i	1.97830	−	2.25529i	−0.539594	−	3.06019i
25.2	0.893633	−	0.448799i	−0.977769	+	1.42967i	0.597159	−	0.802123i	−1.15296	+	3.85116i	−0.232129	+	1.71643i	−0.663579	+	1.53835i	0.173648	−	0.984808i	−1.08794	−	2.79578i	0.698073	+	3.95897i
25.3	0.893633	−	0.448799i	−0.530234	−	1.64889i	0.597159	−	0.802123i	−0.536865	+	1.79325i	−1.21386	−	1.23554i	1.99514	−	4.62524i	0.173648	−	0.984808i	−2.43770	+	1.74860i	0.325051	+	1.84346i
25.4	0.893633	−	0.448799i	1.02523	+	1.39603i	0.597159	−	0.802123i	0.664151	−	2.21842i	1.54272	+	0.787412i	−0.590051	+	1.36789i	0.173648	−	0.984808i	−0.897788	+	2.86251i	−0.402118	−	2.28052i
25.5	0.893633	−	0.448799i	1.51179	−	0.845280i	0.597159	−	0.802123i	−0.379535	+	1.26773i	0.971622	−	1.43386i	−0.768169	+	1.78082i	0.173648	−	0.984808i	1.57100	−	2.55577i	0.229793	+	1.30322i
31.1	0.396080	+	0.918216i	−1.62304	+	0.604777i	−0.686242	+	0.727374i	−0.102641	+	1.76229i	−1.19817	−	1.25076i	−1.24262	+	0.294506i	−0.939693	−	0.342020i	2.26849	−	1.96315i	−1.65881	+	0.603759i
31.2	0.396080	+	0.918216i	−1.57515	−	0.720348i	−0.686242	+	0.727374i	0.193287	−	3.31862i	0.0375499	−	1.73164i	2.86227	−	0.678371i	−0.939693	−	0.342020i	1.96220	+	2.26931i	3.12376	−	1.13696i
31.3	0.396080	+	0.918216i	0.178274	−	1.72285i	−0.686242	+	0.727374i	−0.253242	+	4.34800i	1.65256	−	0.518693i	3.96081	−	0.938728i	−0.939693	−	0.342020i	−2.93644	−	0.614278i	−4.09271	+	1.48962i
31.4	0.396080	+	0.918216i	0.360138	+	1.69420i	−0.686242	+	0.727374i	−0.0437737	+	0.751566i	−1.41299	+	1.00172i	−2.40536	+	0.570080i	−0.939693	−	0.342020i	−2.74060	+	1.22029i	−0.707437	+	0.257486i
31.5	0.396080	+	0.918216i	1.72468	+	0.159609i	−0.686242	+	0.727374i	0.0350699	−	0.602127i	0.536555	+	1.64685i	0.510080	−	0.120891i	−0.939693	−	0.342020i	2.94905	+	0.550550i	0.566774	−	0.206289i

See all 90 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
81.g	even	27	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.2.g.b	✓	90
3.b	odd	2	1		486.2.g.b		90
81.g	even	27	1	inner	162.2.g.b	✓	90
81.h	odd	54	1		486.2.g.b		90

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.2.g.b	✓	90	1.a	even	1	1	trivial
162.2.g.b	✓	90	81.g	even	27	1	inner
486.2.g.b		90	3.b	odd	2	1
486.2.g.b		90	81.h	odd	54	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{90} + 9 T_{5}^{88} + 30 T_{5}^{87} + 45 T_{5}^{86} + 1377 T_{5}^{85} + 2673 T_{5}^{84} + 10341 T_{5}^{83} + 18657 T_{5}^{82} + 199554 T_{5}^{81} + 401679 T_{5}^{80} + 1119420 T_{5}^{79} + \cdots + 21\!\cdots\!84 $ T5^90 + 9*T5^88 + 30*T5^87 + 45*T5^86 + 1377*T5^85 + 2673*T5^84 + 10341*T5^83 + 18657*T5^82 + 199554*T5^81 + 401679*T5^80 + 1119420*T5^79 + 1084239*T5^78 - 24612417*T5^77 + 98528886*T5^76 - 65730204*T5^75 - 929247633*T5^74 + 1639840059*T5^73 + 24185855529*T5^72 - 42151663548*T5^71 + 256550208768*T5^70 + 505217165517*T5^69 - 310113318528*T5^68 + 14626993751289*T5^67 - 46862802103995*T5^66 + 98648407198593*T5^65 + 241283604609126*T5^64 - 2179837803028608*T5^63 + 2431845517854726*T5^62 + 17722648468233045*T5^61 - 83512317182777046*T5^60 + 82185776764317441*T5^59 + 366205736133129213*T5^58 - 1094104737479163285*T5^57 + 2008298410228848933*T5^56 - 16143925626605563065*T5^55 + 99310226536061712972*T5^54 - 314876374223690518785*T5^53 + 434559364641225455751*T5^52 - 60548283948033569529*T5^51 - 707149525081102433379*T5^50 + 2133896119490138973651*T5^49 - 3220125759018869838327*T5^48 + 5794485434190770233518*T5^47 + 19929529166113712619936*T5^46 - 147405157172736804808227*T5^45 + 163498481799491804973759*T5^44 - 31668841512366898040721*T5^43 + 153273797907869100335454*T5^42 + 836861199459883872543153*T5^41 - 467112100191177372223113*T5^40 + 151377945412765544333901*T5^39 - 4483674900541729648436337*T5^38 - 37558115046638207961898428*T5^37 + 39543162387433124901825798*T5^36 + 35909817816133618000250553*T5^35 + 285463096760160390000918627*T5^34 - 108181032051909813359804010*T5^33 - 744000885593744946674536440*T5^32 - 615473213179954777712304876*T5^31 - 213902949342769725630796398*T5^30 + 3746152965213370716483385773*T5^29 + 2848456034563110747036788709*T5^28 - 6513357017984872248246957444*T5^27 - 2641351231633686074707117176*T5^26 - 8220157629446437962791465475*T5^25 + 19987311860779359930380418600*T5^24 + 186072182037808652969492085*T5^23 + 10025537701520304834165724716*T5^22 - 19274893745098595880695693103*T5^21 + 22468899189394876965923342214*T5^20 - 9324570534844648648583017521*T5^19 + 13466359432484510396981260743*T5^18 - 13052959761107015672418190566*T5^17 + 18846158645335012763819787936*T5^16 - 7219160523294617290624580304*T5^15 + 14143023309965392116437798016*T5^14 - 3524514765645467317889355840*T5^13 + 7598161538874375068591376192*T5^12 - 4769047955389781332202675328*T5^11 + 3485221352150551854315641856*T5^10 - 3276843512126123653663371264*T5^9 + 675639319029391357274492928*T5^8 - 1071059594812861120105494528*T5^7 + 860783773094835233074532352*T5^6 - 293935089809926304086597632*T5^5 + 304525784995279173771411456*T5^4 - 57505636468225888844120064*T5^3 + 3839935105319228540780544*T5^2 - 9862674662200699668529152*T5 + 2163784203674077249142784 acting on $S_{2}^{\mathrm{new}}(162, [\chi])$.

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.g (of order \(27\), degree \(18\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 157.5
Significant digits:
Format:

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