LMFDB - Newform orbit 378.2.w.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(25,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([10, 12]))

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

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

chi := DirichletCharacter("378.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.w (of order $9$, degree $6$, 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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-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 + 3 q^{6} - 3 q^{7} - 36 q^{8} - 12 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 72 q + 3 q^{6} - 3 q^{7} - 36 q^{8} - 12 q^{9} + 12 q^{10} - 6 q^{11} - 12 q^{13} - 3 q^{14} - 6 q^{15} + 24 q^{17} + 36 q^{19} - 12 q^{21} - 6 q^{22} + 12 q^{23} - 30 q^{25} - 18 q^{26} - 6 q^{27} + 3 q^{29} + 3 q^{30} + 9 q^{31} - 18 q^{33} + 9 q^{34} + 9 q^{35} + 9 q^{36} + 3 q^{39} - 6 q^{41} + 3 q^{42} + 24 q^{43} - 30 q^{45} - 9 q^{47} - 6 q^{48} + 51 q^{49} + 6 q^{50} + 12 q^{51} + 6 q^{52} - 15 q^{53} - 27 q^{54} + 72 q^{55} + 6 q^{56} - 63 q^{57} + 3 q^{58} + 15 q^{59} - 3 q^{60} - 18 q^{61} - 24 q^{62} - 48 q^{63} - 36 q^{64} - 18 q^{65} - 36 q^{66} + 66 q^{67} - 18 q^{68} - 21 q^{69} - 6 q^{70} + 12 q^{71} - 12 q^{72} - 66 q^{73} + 9 q^{74} + 15 q^{75} - 15 q^{77} + 30 q^{78} + 9 q^{79} - 6 q^{80} - 33 q^{82} - 18 q^{83} - 12 q^{84} + 21 q^{85} - 12 q^{86} - 48 q^{87} + 12 q^{88} + 72 q^{89} + 69 q^{90} + 12 q^{91} + 30 q^{92} + 60 q^{93} - 36 q^{94} + 93 q^{95} - 48 q^{97} + 6 q^{98} - 54 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} $
25.1	−0.939693	+	0.342020i	−1.70460	+	0.307144i	0.766044	−	0.642788i	−2.61773	−	0.952774i	1.49675	−	0.871628i	−1.47144	−	2.19883i	−0.500000	+	0.866025i	2.81133	−	1.04711i	2.78573
25.2	−0.939693	+	0.342020i	−1.58205	−	0.705062i	0.766044	−	0.642788i	1.56427	+	0.569347i	1.72779	+	0.121449i	0.319985	−	2.62633i	−0.500000	+	0.866025i	2.00577	+	2.23089i	−1.66466
25.3	−0.939693	+	0.342020i	−1.25749	+	1.19110i	0.766044	−	0.642788i	1.98579	+	0.722770i	0.774279	−	1.54935i	2.63247	+	0.264792i	−0.500000	+	0.866025i	0.162584	−	2.99559i	−2.11324
25.4	−0.939693	+	0.342020i	−1.03071	−	1.39199i	0.766044	−	0.642788i	−3.69961	−	1.34655i	1.44464	+	0.955517i	0.644767	+	2.56598i	−0.500000	+	0.866025i	−0.875266	+	2.86948i	3.93704
25.5	−0.939693	+	0.342020i	−0.649954	−	1.60548i	0.766044	−	0.642788i	3.27017	+	1.19024i	1.15986	+	1.28636i	1.64949	+	2.06862i	−0.500000	+	0.866025i	−2.15512	+	2.08697i	−3.48004
25.6	−0.939693	+	0.342020i	−0.195246	−	1.72101i	0.766044	−	0.642788i	0.182581	+	0.0664541i	0.772092	+	1.55044i	−2.63939	−	0.183356i	−0.500000	+	0.866025i	−2.92376	+	0.672041i	−0.194299
25.7	−0.939693	+	0.342020i	−0.135903	+	1.72671i	0.766044	−	0.642788i	−1.85969	−	0.676871i	−0.462863	−	1.66906i	2.63192	+	0.270209i	−0.500000	+	0.866025i	−2.96306	−	0.469329i	1.97904
25.8	−0.939693	+	0.342020i	0.807797	+	1.53214i	0.766044	−	0.642788i	1.59297	+	0.579794i	−1.28310	−	1.16346i	−2.02988	+	1.69693i	−0.500000	+	0.866025i	−1.69493	+	2.47532i	−1.69520
25.9	−0.939693	+	0.342020i	1.06159	−	1.36859i	0.766044	−	0.642788i	−1.03287	−	0.375933i	−0.529482	+	1.64914i	2.42642	−	1.05474i	−0.500000	+	0.866025i	−0.746062	−	2.90575i	1.09915
25.10	−0.939693	+	0.342020i	1.14375	+	1.30071i	0.766044	−	0.642788i	1.59112	+	0.579119i	−1.51964	−	0.831077i	0.142417	−	2.64192i	−0.500000	+	0.866025i	−0.383668	+	2.97537i	−1.69323
25.11	−0.939693	+	0.342020i	1.65621	−	0.506908i	0.766044	−	0.642788i	0.501605	+	0.182569i	−1.38296	+	1.04280i	−1.88202	+	1.85957i	−0.500000	+	0.866025i	2.48609	−	1.67910i	−0.533797
25.12	−0.939693	+	0.342020i	1.71296	+	0.256429i	0.766044	−	0.642788i	−3.35800	−	1.22221i	−1.69736	+	0.344903i	−2.15868	−	1.52974i	−0.500000	+	0.866025i	2.86849	+	0.878508i	3.57351
121.1	−0.939693	−	0.342020i	−1.70460	−	0.307144i	0.766044	+	0.642788i	−2.61773	+	0.952774i	1.49675	+	0.871628i	−1.47144	+	2.19883i	−0.500000	−	0.866025i	2.81133	+	1.04711i	2.78573
121.2	−0.939693	−	0.342020i	−1.58205	+	0.705062i	0.766044	+	0.642788i	1.56427	−	0.569347i	1.72779	−	0.121449i	0.319985	+	2.62633i	−0.500000	−	0.866025i	2.00577	−	2.23089i	−1.66466
121.3	−0.939693	−	0.342020i	−1.25749	−	1.19110i	0.766044	+	0.642788i	1.98579	−	0.722770i	0.774279	+	1.54935i	2.63247	−	0.264792i	−0.500000	−	0.866025i	0.162584	+	2.99559i	−2.11324
121.4	−0.939693	−	0.342020i	−1.03071	+	1.39199i	0.766044	+	0.642788i	−3.69961	+	1.34655i	1.44464	−	0.955517i	0.644767	−	2.56598i	−0.500000	−	0.866025i	−0.875266	−	2.86948i	3.93704
121.5	−0.939693	−	0.342020i	−0.649954	+	1.60548i	0.766044	+	0.642788i	3.27017	−	1.19024i	1.15986	−	1.28636i	1.64949	−	2.06862i	−0.500000	−	0.866025i	−2.15512	−	2.08697i	−3.48004
121.6	−0.939693	−	0.342020i	−0.195246	+	1.72101i	0.766044	+	0.642788i	0.182581	−	0.0664541i	0.772092	−	1.55044i	−2.63939	+	0.183356i	−0.500000	−	0.866025i	−2.92376	−	0.672041i	−0.194299
121.7	−0.939693	−	0.342020i	−0.135903	−	1.72671i	0.766044	+	0.642788i	−1.85969	+	0.676871i	−0.462863	+	1.66906i	2.63192	−	0.270209i	−0.500000	−	0.866025i	−2.96306	+	0.469329i	1.97904
121.8	−0.939693	−	0.342020i	0.807797	−	1.53214i	0.766044	+	0.642788i	1.59297	−	0.579794i	−1.28310	+	1.16346i	−2.02988	−	1.69693i	−0.500000	−	0.866025i	−1.69493	−	2.47532i	−1.69520

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.w	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.2.w.a	yes	72
7.c	even	3	1		378.2.v.b	✓	72
27.e	even	9	1		378.2.v.b	✓	72
189.w	even	9	1	inner	378.2.w.a	yes	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.2.v.b	✓	72	7.c	even	3	1
378.2.v.b	✓	72	27.e	even	9	1
378.2.w.a	yes	72	1.a	even	1	1	trivial
378.2.w.a	yes	72	189.w	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{72} + 15 T_{5}^{70} + 41 T_{5}^{69} + 24 T_{5}^{68} + 87 T_{5}^{67} + 4671 T_{5}^{66} + \cdots + 18\!\cdots\!89 $ T5^72 + 15*T5^70 + 41*T5^69 + 24*T5^68 + 87*T5^67 + 4671*T5^66 - 4338*T5^65 + 53712*T5^64 + 95076*T5^63 - 17937*T5^62 - 229095*T5^61 + 14772387*T5^60 - 38131605*T5^59 + 194081535*T5^58 + 31699557*T5^57 - 462471291*T5^56 + 3051462573*T5^55 + 8931344493*T5^54 - 50473466892*T5^53 + 325492785603*T5^52 - 735019875192*T5^51 + 1354390480413*T5^50 - 885105679995*T5^49 + 3041258088258*T5^48 - 15833429254260*T5^47 + 113156767368387*T5^46 - 349554257554465*T5^45 + 741740488007652*T5^44 - 1185379488762117*T5^43 + 2426516223188953*T5^42 - 8906187929892207*T5^41 + 42703142782251981*T5^40 - 132137905919745834*T5^39 + 285011006749677171*T5^38 - 448631866518319269*T5^37 + 519898172172120753*T5^36 - 684339023190811446*T5^35 + 3032688723466235454*T5^34 - 10288768681735501500*T5^33 + 21323554006215968532*T5^32 - 28088169301516333293*T5^31 + 14328649008559522656*T5^30 + 69890343187127220*T5^29 + 157773824673148110528*T5^28 - 645764799057749859177*T5^27 + 1235293920761325919164*T5^26 - 1382326174351988005932*T5^25 + 663769771104347215155*T5^24 + 44093073904159507299*T5^23 + 2373001529017842555759*T5^22 - 6548859782218944699345*T5^21 + 4522211829776468186622*T5^20 + 925847595865329548022*T5^19 + 1243694961772491583710*T5^18 - 2721205665118893215502*T5^17 + 2585315019839919744087*T5^16 - 8971763548171099531311*T5^15 + 6637307176653855288048*T5^14 - 1027516924188626806449*T5^13 + 8606523634683733881063*T5^12 - 1787692755758981392332*T5^11 + 5464524292226310363570*T5^10 - 6851737115949100756590*T5^9 + 3339023103813328850889*T5^8 - 2075597708733278738514*T5^7 + 1496833694993140737804*T5^6 - 499505223939374010717*T5^5 + 238489701205308687045*T5^4 - 214390539934863832251*T5^3 + 110581006778874001944*T5^2 - 23745707845827812874*T5 + 1868500722364663689 acting on $S_{2}^{\mathrm{new}}(378, [\chi])$.

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.w (of order \(9\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 25.12
Significant digits:
Format:

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