LMFDB - Newform orbit 930.2.bn.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(19,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 15, 4]))

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

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

chi := DirichletCharacter("930.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.bn (of order $30$, 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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$112$
Relative dimension:	$14$ over $\Q(\zeta_{30})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-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) = $ $ 112 q + 28 q^{4} - 2 q^{5} + 56 q^{6} - 14 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 112 q + 28 q^{4} - 2 q^{5} + 56 q^{6} - 14 q^{9} - 4 q^{10} + 18 q^{11} + 8 q^{14} + 8 q^{15} - 28 q^{16} + 16 q^{19} + 2 q^{20} + 28 q^{21} + 14 q^{24} + 14 q^{25} + 12 q^{26} + 16 q^{29} - 4 q^{30} + 10 q^{34} - 38 q^{35} - 56 q^{36} + 16 q^{39} - 6 q^{40} + 20 q^{41} + 2 q^{44} + 2 q^{45} - 2 q^{46} + 38 q^{49} + 8 q^{50} - 10 q^{51} - 28 q^{54} - 46 q^{55} + 12 q^{56} + 60 q^{59} - 8 q^{60} + 88 q^{61} + 28 q^{64} - 28 q^{65} + 6 q^{66} + 46 q^{69} + 26 q^{70} + 116 q^{71} - 34 q^{74} + 8 q^{75} + 24 q^{76} - 40 q^{79} - 12 q^{80} + 14 q^{81} - 8 q^{84} + 18 q^{85} - 38 q^{86} - 60 q^{89} + 4 q^{90} - 92 q^{91} + 132 q^{94} + 132 q^{95} - 14 q^{96} + 2 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} $
19.1	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	−2.15286	−	0.604297i	0.500000	+	0.866025i	2.63601	+	2.37347i	0.951057	+	0.309017i	−0.669131	−	0.743145i	1.75431	−	1.38651i
19.2	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	−1.44692	−	1.70482i	0.500000	+	0.866025i	−3.41731	−	3.07696i	0.951057	+	0.309017i	−0.669131	−	0.743145i	2.22971	−	0.168512i
19.3	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	−0.509764	+	2.17719i	0.500000	+	0.866025i	0.159624	+	0.143726i	0.951057	+	0.309017i	−0.669131	−	0.743145i	−1.46175	−	1.69213i
19.4	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	−0.181057	+	2.22873i	0.500000	+	0.866025i	3.89646	+	3.50839i	0.951057	+	0.309017i	−0.669131	−	0.743145i	−1.69665	−	1.45649i
19.5	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	0.230445	−	2.22416i	0.500000	+	0.866025i	0.872484	+	0.785588i	0.951057	+	0.309017i	−0.669131	−	0.743145i	1.66393	+	1.49376i
19.6	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	1.47095	+	1.68413i	0.500000	+	0.866025i	−2.07600	−	1.86924i	0.951057	+	0.309017i	−0.669131	−	0.743145i	−2.22710	+	0.200114i
19.7	−0.587785	+	0.809017i	0.406737	−	0.913545i	−0.309017	−	0.951057i	2.21090	+	0.334560i	0.500000	+	0.866025i	−0.274259	−	0.246944i	0.951057	+	0.309017i	−0.669131	−	0.743145i	−1.57020	+	1.59200i
19.8	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	−2.19398	−	0.431813i	0.500000	+	0.866025i	2.07600	+	1.86924i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	−1.63893	+	1.52115i
19.9	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	−1.83960	+	1.27116i	0.500000	+	0.866025i	−3.89646	−	3.50839i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	−0.0529000	+	2.23544i
19.10	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	−1.63062	+	1.53006i	0.500000	+	0.866025i	−0.159624	−	0.143726i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	0.279393	+	2.21854i
19.11	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	−1.39519	−	1.74741i	0.500000	+	0.866025i	0.274259	+	0.246944i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	−2.23376	+	0.101626i
19.12	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	1.59977	+	1.56229i	0.500000	+	0.866025i	−2.63601	−	2.37347i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	2.20424	−	0.375951i
19.13	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	1.81096	−	1.31165i	0.500000	+	0.866025i	−0.872484	−	0.785588i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	0.00330584	−	2.23607i
19.14	0.587785	−	0.809017i	−0.406737	+	0.913545i	−0.309017	−	0.951057i	2.19988	+	0.400657i	0.500000	+	0.866025i	3.41731	+	3.07696i	−0.951057	−	0.309017i	−0.669131	−	0.743145i	1.61720	−	1.54424i
49.1	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	−2.15286	+	0.604297i	0.500000	−	0.866025i	2.63601	−	2.37347i	0.951057	−	0.309017i	−0.669131	+	0.743145i	1.75431	+	1.38651i
49.2	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	−1.44692	+	1.70482i	0.500000	−	0.866025i	−3.41731	+	3.07696i	0.951057	−	0.309017i	−0.669131	+	0.743145i	2.22971	+	0.168512i
49.3	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	−0.509764	−	2.17719i	0.500000	−	0.866025i	0.159624	−	0.143726i	0.951057	−	0.309017i	−0.669131	+	0.743145i	−1.46175	+	1.69213i
49.4	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	−0.181057	−	2.22873i	0.500000	−	0.866025i	3.89646	−	3.50839i	0.951057	−	0.309017i	−0.669131	+	0.743145i	−1.69665	+	1.45649i
49.5	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	0.230445	+	2.22416i	0.500000	−	0.866025i	0.872484	−	0.785588i	0.951057	−	0.309017i	−0.669131	+	0.743145i	1.66393	−	1.49376i
49.6	−0.587785	−	0.809017i	0.406737	+	0.913545i	−0.309017	+	0.951057i	1.47095	−	1.68413i	0.500000	−	0.866025i	−2.07600	+	1.86924i	0.951057	−	0.309017i	−0.669131	+	0.743145i	−2.22710	−	0.200114i

See next 80 embeddings (of 112 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
31.g	even	15	1	inner
155.u	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bn.a	✓	112
5.b	even	2	1	inner	930.2.bn.a	✓	112
31.g	even	15	1	inner	930.2.bn.a	✓	112
155.u	even	30	1	inner	930.2.bn.a	✓	112

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bn.a	✓	112	1.a	even	1	1	trivial
930.2.bn.a	✓	112	5.b	even	2	1	inner
930.2.bn.a	✓	112	31.g	even	15	1	inner
930.2.bn.a	✓	112	155.u	even	30	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{112} + 30 T_{7}^{110} + 915 T_{7}^{108} + 20732 T_{7}^{106} + 14405 T_{7}^{104} + \cdots + 81\!\cdots\!36 $ T7^112 + 30*T7^110 + 915*T7^108 + 20732*T7^106 + 14405*T7^104 - 9407268*T7^102 - 247887195*T7^100 - 4087655394*T7^98 - 13980516399*T7^96 + 1410534718530*T7^94 + 36821204889428*T7^92 + 365612063487588*T7^90 - 1119267384587579*T7^88 - 92532262517505806*T7^86 - 1519490075152295271*T7^84 - 11351588866824097768*T7^82 + 52723188294010346442*T7^80 + 2527039698785196664964*T7^78 + 32202831566514359207251*T7^76 + 199548657886328421709092*T7^74 - 69575886423464363250243*T7^72 - 14514830556434620679380596*T7^70 - 153447814134257783506778259*T7^68 - 754483841699593793504608222*T7^66 + 750654182493339579255247131*T7^64 + 43836575645891607430498843152*T7^62 + 353132577396730373250090565238*T7^60 + 1269729894910553456939963401924*T7^58 - 1146772415220836273650687182615*T7^56 - 36925131837364573238859889518722*T7^54 - 180668866046903958065237207945504*T7^52 - 273074961566316484366653754086896*T7^50 + 1423771253142721239539208717918187*T7^48 + 9852819772387423828411695640197446*T7^46 + 26300290050872343361316337502272552*T7^44 + 19335793473255532174436226088635600*T7^42 - 98570793160546958377798818502706131*T7^40 - 374847783994907768787521598345434886*T7^38 - 559345361414256457664387818452323368*T7^36 - 32035981240043592290581616266479546*T7^34 + 1699911765394282301535421945242153368*T7^32 + 4353670721322725851320209506505403470*T7^30 + 6861190668644413627405263503134858954*T7^28 + 7575271909450238868424377463038510840*T7^26 + 5415496148986903987668265450459445141*T7^24 + 2038568112202465994914779366121513422*T7^22 + 358886218671875207812948804931545679*T7^20 + 30171091842106955871083592743155870*T7^18 + 6368563891684136178607312959196273*T7^16 - 5908092103792257695108353206496*T7^14 + 10755603554136094056853110715648*T7^12 - 224729449444223718488028647424*T7^10 + 6219726831885389109102772224*T7^8 - 636628789554000830110105600*T7^6 + 18458225112971925144993792*T7^4 - 206707017906624393117696*T7^2 + 819909086042795278336 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bn (of order \(30\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.14
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more