LMFDB - Newform orbit 374.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [374,2,Mod(65,374)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(374, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 9]))

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

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

chi := DirichletCharacter("374.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 374 = 2 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	374.m (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:	$2.98640503560$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$9$ 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) = $ $ 72 q+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 72 q - 24 q^{11} + 8 q^{12} - 16 q^{14} + 16 q^{19} + 8 q^{22} + 16 q^{23} + 8 q^{24} - 16 q^{25} - 24 q^{27} + 32 q^{31} - 16 q^{33} - 16 q^{37} - 8 q^{38} + 32 q^{41} + 56 q^{43} + 24 q^{44} - 16 q^{46} - 64 q^{49} - 104 q^{50} + 32 q^{51} - 8 q^{54} + 48 q^{55} - 40 q^{57} - 16 q^{59} - 32 q^{60} + 32 q^{62} - 48 q^{63} + 48 q^{65} + 32 q^{70} - 32 q^{72} + 32 q^{74} - 64 q^{75} - 16 q^{80} + 88 q^{81} + 72 q^{83} - 32 q^{84} - 64 q^{85} - 32 q^{86} - 144 q^{87} + 8 q^{88} + 80 q^{90} - 128 q^{91} + 16 q^{92} - 32 q^{93} - 32 q^{95} - 64 q^{97} - 16 q^{98} + 32 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} $
65.1	−0.382683	−	0.923880i	−0.594235	+	2.98742i	−0.707107	+	0.707107i	3.49407	−	2.33467i	2.98742	−	0.594235i	1.76187	−	2.63683i	0.923880	+	0.382683i	−5.79995	−	2.40242i	−3.49407	−	2.33467i
65.2	−0.382683	−	0.923880i	−0.467613	+	2.35085i	−0.707107	+	0.707107i	−1.32362	+	0.884417i	2.35085	−	0.467613i	−0.427024	+	0.639087i	0.923880	+	0.382683i	−2.53619	−	1.05052i	1.32362	+	0.884417i
65.3	−0.382683	−	0.923880i	−0.359176	+	1.80570i	−0.707107	+	0.707107i	−0.982686	+	0.656610i	1.80570	−	0.359176i	2.42854	−	3.63456i	0.923880	+	0.382683i	−0.359898	−	0.149075i	0.982686	+	0.656610i
65.4	−0.382683	−	0.923880i	−0.142967	+	0.718746i	−0.707107	+	0.707107i	1.68394	−	1.12517i	0.718746	−	0.142967i	−2.75324	+	4.12052i	0.923880	+	0.382683i	2.27548	+	0.942536i	−1.68394	−	1.12517i
65.5	−0.382683	−	0.923880i	0.0793166	−	0.398751i	−0.707107	+	0.707107i	−0.844776	+	0.564461i	−0.398751	+	0.0793166i	0.673267	−	1.00761i	0.923880	+	0.382683i	2.61893	+	1.08480i	0.844776	+	0.564461i
65.6	−0.382683	−	0.923880i	0.202383	−	1.01745i	−0.707107	+	0.707107i	2.05599	−	1.37377i	−1.01745	+	0.202383i	1.07222	−	1.60470i	0.923880	+	0.382683i	1.77739	+	0.736221i	−2.05599	−	1.37377i
65.7	−0.382683	−	0.923880i	0.340491	−	1.71176i	−0.707107	+	0.707107i	−2.30866	+	1.54259i	−1.71176	+	0.340491i	−1.99595	+	2.98716i	0.923880	+	0.382683i	−0.0425580	−	0.0176281i	2.30866	+	1.54259i
65.8	−0.382683	−	0.923880i	0.559476	−	2.81267i	−0.707107	+	0.707107i	2.76761	−	1.84926i	−2.81267	+	0.559476i	−1.27163	+	1.90314i	0.923880	+	0.382683i	−4.82648	−	1.99919i	−2.76761	−	1.84926i
65.9	−0.382683	−	0.923880i	0.599098	−	3.01187i	−0.707107	+	0.707107i	−2.69411	+	1.80015i	−3.01187	+	0.599098i	2.69154	−	4.02817i	0.923880	+	0.382683i	−5.94080	−	2.46076i	2.69411	+	1.80015i
109.1	0.923880	+	0.382683i	−1.33452	+	1.99725i	0.707107	+	0.707107i	0.212507	+	1.06835i	−1.99725	+	1.33452i	3.77224	+	0.750346i	0.382683	+	0.923880i	−1.06001	−	2.55910i	−0.212507	+	1.06835i
109.2	0.923880	+	0.382683i	−1.20660	+	1.80581i	0.707107	+	0.707107i	0.705255	+	3.54556i	−1.80581	+	1.20660i	−2.74517	−	0.546049i	0.382683	+	0.923880i	−0.657010	−	1.58616i	−0.705255	+	3.54556i
109.3	0.923880	+	0.382683i	−0.599837	+	0.897720i	0.707107	+	0.707107i	−0.383244	−	1.92670i	−0.897720	+	0.599837i	1.02760	+	0.204401i	0.382683	+	0.923880i	0.701954	+	1.69467i	0.383244	−	1.92670i
109.4	0.923880	+	0.382683i	−0.466714	+	0.698488i	0.707107	+	0.707107i	0.197102	+	0.990901i	−0.698488	+	0.466714i	−3.92075	−	0.779885i	0.382683	+	0.923880i	0.877988	+	2.11965i	−0.197102	+	0.990901i
109.5	0.923880	+	0.382683i	0.0929082	−	0.139047i	0.707107	+	0.707107i	0.00269154	+	0.0135313i	0.139047	−	0.0929082i	1.55404	+	0.309118i	0.382683	+	0.923880i	1.13735	+	2.74580i	−0.00269154	+	0.0135313i
109.6	0.923880	+	0.382683i	0.659488	−	0.986994i	0.707107	+	0.707107i	0.544847	+	2.73913i	0.986994	−	0.659488i	1.36742	+	0.271996i	0.382683	+	0.923880i	0.608818	+	1.46982i	−0.544847	+	2.73913i
109.7	0.923880	+	0.382683i	0.744966	−	1.11492i	0.707107	+	0.707107i	−0.769484	−	3.86846i	1.11492	−	0.744966i	−3.98082	−	0.791834i	0.382683	+	0.923880i	0.459977	+	1.11048i	0.769484	−	3.86846i
109.8	0.923880	+	0.382683i	1.56358	−	2.34006i	0.707107	+	0.707107i	−0.363537	−	1.82762i	2.34006	−	1.56358i	0.482872	+	0.0960492i	0.382683	+	0.923880i	−1.88305	−	4.54608i	0.363537	−	1.82762i
109.9	0.923880	+	0.382683i	1.63653	−	2.44924i	0.707107	+	0.707107i	0.619229	+	3.11308i	2.44924	−	1.63653i	−0.819398	−	0.162988i	0.382683	+	0.923880i	−2.17249	−	5.24484i	−0.619229	+	3.11308i
131.1	−0.923880	+	0.382683i	−1.95575	+	1.30679i	0.707107	−	0.707107i	−0.243709	−	0.0484767i	1.30679	−	1.95575i	−0.266359	−	1.33908i	−0.382683	+	0.923880i	0.969205	−	2.33987i	0.243709	−	0.0484767i
131.2	−0.923880	+	0.382683i	−1.86936	+	1.24907i	0.707107	−	0.707107i	−0.532510	−	0.105923i	1.24907	−	1.86936i	0.675335	+	3.39514i	−0.382683	+	0.923880i	0.786299	−	1.89829i	0.532510	−	0.105923i

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
187.m	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	374.2.m.a	✓	72
11.b	odd	2	1		374.2.m.b	yes	72
17.e	odd	16	1		374.2.m.b	yes	72
187.m	even	16	1	inner	374.2.m.a	✓	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
374.2.m.a	✓	72	1.a	even	1	1	trivial
374.2.m.a	✓	72	187.m	even	16	1	inner
374.2.m.b	yes	72	11.b	odd	2	1
374.2.m.b	yes	72	17.e	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{72} + 32 T_{7}^{70} + 128 T_{7}^{69} + 584 T_{7}^{68} + 3712 T_{7}^{67} + 7096 T_{7}^{66} + \cdots + 31\!\cdots\!12 $ T7^72 + 32*T7^70 + 128*T7^69 + 584*T7^68 + 3712*T7^67 + 7096*T7^66 + 82656*T7^65 + 12576*T7^64 + 182896*T7^63 + 4029064*T7^62 - 12331968*T7^61 + 45169296*T7^60 - 57675632*T7^59 - 328380024*T7^58 + 3470190112*T7^57 - 4498671143*T7^56 + 11001647216*T7^55 + 21736773752*T7^54 + 410960020192*T7^53 + 2175418062356*T7^52 + 1297657371264*T7^51 + 44297597412896*T7^50 + 98620014439408*T7^49 + 128624695329408*T7^48 + 946188802157056*T7^47 + 3942171108145248*T7^46 + 1824510332725728*T7^45 + 15367699525014964*T7^44 + 45983232080519072*T7^43 + 78249662402750200*T7^42 + 431071136709855904*T7^41 + 212836473018384890*T7^40 + 380703752178942192*T7^39 + 1941309840473477976*T7^38 - 826426292280241120*T7^37 + 17422429526997388016*T7^36 - 40081226636272388016*T7^35 + 123002258778033441464*T7^34 - 185807474528847598544*T7^33 + 172926507017162969288*T7^32 - 766165257354894569840*T7^31 + 1887098744778809947432*T7^30 - 4997626950678879310192*T7^29 + 6472764916225568875156*T7^28 - 6704729307550439546560*T7^27 + 13673117292553447571984*T7^26 - 14179318619225096741568*T7^25 + 48640599818519663716641*T7^24 - 64042530799492504685504*T7^23 + 54748371000996590245208*T7^22 - 143321228789526720912496*T7^21 + 61233090867113365982328*T7^20 - 33616593371584245013760*T7^19 + 250964709936582939481392*T7^18 - 228777533160211497998304*T7^17 + 206673275593981845688480*T7^16 - 148440172819545867013760*T7^15 + 247452974843154497192992*T7^14 + 5881958030635339435328*T7^13 - 368893058235253754225120*T7^12 - 370247368065591477495168*T7^11 + 307427542918945861550528*T7^10 + 423411432375276920476032*T7^9 + 126316789889633703575568*T7^8 - 250215065975130705802496*T7^7 - 353442288899464595516928*T7^6 + 76877897192374120696832*T7^5 + 258932642376752447386112*T7^4 + 71439433387999105769472*T7^3 - 52750482818564783263744*T7^2 - 75822919106858605002752*T7 + 31723996066834187558912 acting on $S_{2}^{\mathrm{new}}(374, [\chi])$.

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:	\( N \)	\(=\)	\( 374 = 2 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	374.m (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 65.9
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more