LMFDB - Newform orbit 510.2.bd.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(7,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 4, 11]))

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

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

chi := DirichletCharacter("510.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.bd (of order $16$, degree $8$, 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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$10$ over $\Q(\zeta_{16})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-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) = $ $ 80 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 80 q + 16 q^{15} + 16 q^{25} - 16 q^{28} - 32 q^{29} + 16 q^{31} + 16 q^{33} + 32 q^{34} - 32 q^{35} - 16 q^{37} + 16 q^{40} + 32 q^{41} - 16 q^{44} - 32 q^{47} + 64 q^{49} - 48 q^{50} - 32 q^{51} + 16 q^{52} - 16 q^{55} - 16 q^{56} - 32 q^{57} + 48 q^{61} - 32 q^{65} + 16 q^{67} + 32 q^{68} - 32 q^{70} + 16 q^{73} + 32 q^{74} - 32 q^{75} - 80 q^{77} - 32 q^{78} + 64 q^{79} - 16 q^{80} + 32 q^{82} - 32 q^{83} - 16 q^{85} - 64 q^{87} - 16 q^{88} + 32 q^{89} - 16 q^{92} - 16 q^{93} + 16 q^{94} - 144 q^{95} - 80 q^{97}+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	−0.923880	+	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	−2.23502	+	0.0686129i	0.555570	+	0.831470i	−2.71503	+	1.81413i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	2.03863	−	0.918693i
7.2	−0.923880	+	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	−1.83266	−	1.28115i	0.555570	+	0.831470i	2.98823	−	1.99667i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	2.18343	+	0.482302i
7.3	−0.923880	+	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	−0.692683	+	2.12607i	0.555570	+	0.831470i	−0.601686	+	0.402034i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	−0.173657	−	2.22931i
7.4	−0.923880	+	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	1.67272	−	1.48392i	0.555570	+	0.831470i	3.92689	−	2.62387i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	−0.977515	+	2.01109i
7.5	−0.923880	+	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	2.23397	−	0.0968072i	0.555570	+	0.831470i	−2.29184	+	1.53136i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	−2.02687	+	0.944342i
7.6	−0.923880	+	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	−2.02294	+	0.952736i	−0.555570	−	0.831470i	2.65908	−	1.77674i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	1.50436	−	1.65436i
7.7	−0.923880	+	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	−1.43467	−	1.71514i	−0.555570	−	0.831470i	1.36903	−	0.914757i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	1.98182	+	1.03556i
7.8	−0.923880	+	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	−0.435085	−	2.19333i	−0.555570	−	0.831470i	−2.98077	+	1.99169i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	1.24132	+	1.85987i
7.9	−0.923880	+	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	1.93341	−	1.12335i	−0.555570	−	0.831470i	1.67263	−	1.11762i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	−1.35635	+	1.77773i
7.10	−0.923880	+	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	2.04760	+	0.898528i	−0.555570	−	0.831470i	−1.41341	+	0.944408i	−0.382683	+	0.923880i	−0.923880	+	0.382683i	−2.23558	−	0.0465502i
73.1	−0.923880	−	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	−2.23502	−	0.0686129i	0.555570	−	0.831470i	−2.71503	−	1.81413i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	2.03863	+	0.918693i
73.2	−0.923880	−	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	−1.83266	+	1.28115i	0.555570	−	0.831470i	2.98823	+	1.99667i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	2.18343	−	0.482302i
73.3	−0.923880	−	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	−0.692683	−	2.12607i	0.555570	−	0.831470i	−0.601686	−	0.402034i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	−0.173657	+	2.22931i
73.4	−0.923880	−	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	1.67272	+	1.48392i	0.555570	−	0.831470i	3.92689	+	2.62387i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	−0.977515	−	2.01109i
73.5	−0.923880	−	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	2.23397	+	0.0968072i	0.555570	−	0.831470i	−2.29184	−	1.53136i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	−2.02687	−	0.944342i
73.6	−0.923880	−	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	−2.02294	−	0.952736i	−0.555570	+	0.831470i	2.65908	+	1.77674i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	1.50436	+	1.65436i
73.7	−0.923880	−	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	−1.43467	+	1.71514i	−0.555570	+	0.831470i	1.36903	+	0.914757i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	1.98182	−	1.03556i
73.8	−0.923880	−	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	−0.435085	+	2.19333i	−0.555570	+	0.831470i	−2.98077	−	1.99169i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	1.24132	−	1.85987i
73.9	−0.923880	−	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	1.93341	+	1.12335i	−0.555570	+	0.831470i	1.67263	+	1.11762i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	−1.35635	−	1.77773i
73.10	−0.923880	−	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	2.04760	−	0.898528i	−0.555570	+	0.831470i	−1.41341	−	0.944408i	−0.382683	−	0.923880i	−0.923880	−	0.382683i	−2.23558	+	0.0465502i

See all 80 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.o	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	510.2.bd.b	✓	80
5.c	odd	4	1		510.2.bi.b	yes	80
17.e	odd	16	1		510.2.bi.b	yes	80
85.o	even	16	1	inner	510.2.bd.b	✓	80

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.bd.b	✓	80	1.a	even	1	1	trivial
510.2.bd.b	✓	80	85.o	even	16	1	inner
510.2.bi.b	yes	80	5.c	odd	4	1
510.2.bi.b	yes	80	17.e	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{80} - 32 T_{7}^{78} + 384 T_{7}^{76} + 2176 T_{7}^{75} - 5456 T_{7}^{74} - 74016 T_{7}^{73} + \cdots + 43\!\cdots\!96 $ T7^80 - 32*T7^78 + 384*T7^76 + 2176*T7^75 - 5456*T7^74 - 74016*T7^73 + 117248*T7^72 + 1010016*T7^71 + 1980512*T7^70 - 23809152*T7^69 - 78685136*T7^68 + 541577600*T7^67 + 1669031360*T7^66 - 1758882048*T7^65 - 24743093696*T7^64 - 12519544704*T7^63 + 440701555520*T7^62 + 620345393408*T7^61 - 4261221749888*T7^60 - 19637544851712*T7^59 + 63579205830400*T7^58 + 183075164533760*T7^57 + 15264762079360*T7^56 - 2939557144209920*T7^55 - 509354511133696*T7^54 + 4390991936692224*T7^53 + 11355732639941632*T7^52 + 280186826877398016*T7^51 + 451352745978068992*T7^50 - 923747605708440576*T7^49 - 4380210400071618560*T7^48 - 9748155715735192576*T7^47 + 111564260431788370944*T7^46 - 216328817713077719040*T7^45 + 1756067621486777590272*T7^44 - 6157186529681849532416*T7^43 + 22310118535259051816960*T7^42 - 98952928544502771531776*T7^41 + 320343119672664511725568*T7^40 - 1090634816475261077426176*T7^39 + 3516537131994745542080512*T7^38 - 10339742840851579299561472*T7^37 + 30842376376458312173899776*T7^36 - 86297007187947738259439616*T7^35 + 237222070776525015069999104*T7^34 - 623827224654746371410165760*T7^33 + 1586990899654683410226217984*T7^32 - 3870887979215652841024733184*T7^31 + 9068396360154772255525765120*T7^30 - 20146566091599916331307008000*T7^29 + 42628878930722814545016864768*T7^28 - 85064672128485996926374117376*T7^27 + 161526449831747327859238240256*T7^26 - 294482515636284339678319214592*T7^25 + 511679223088430847151325708288*T7^24 - 865279021992525936850522341376*T7^23 + 1416772970496333365522819448832*T7^22 - 2200597034878156164798490345472*T7^21 + 3317857049408106826713698402304*T7^20 - 4726860778817165251778736816128*T7^19 + 6334113326405258736009510649856*T7^18 - 8098296920598166391154653790208*T7^17 + 9458187696572899545887365398528*T7^16 - 9271088663988283962423020355584*T7^15 + 8997297396958710253317351538688*T7^14 - 7738339363628137883854954299392*T7^13 + 4979062565930226112492262653952*T7^12 - 4096972542821374709689382600704*T7^11 + 3316891527746199577303900487680*T7^10 - 1708879615855802381689344229376*T7^9 + 1454412368102903992907921883136*T7^8 - 1135202921709695236410057949184*T7^7 + 423543100198643484203281285120*T7^6 - 236750365302998420306339561472*T7^5 + 165424456918975704866182135808*T7^4 - 64379074559343820738209513472*T7^3 + 10847467262151315669924184064*T7^2 + 5947144773587138243838607360*T7 + 432876024591433999681847296 acting on $S_{2}^{\mathrm{new}}(510, [\chi])$.

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:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.bd (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.10
Significant digits:
Format:

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