LMFDB - Newform orbit 666.2.w.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(7,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([12, 16]))

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

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

chi := DirichletCharacter("666.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.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:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$114$
Relative dimension:	$19$ 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) = $ $ 114 q + 6 q^{3} - 3 q^{7} + 57 q^{8} + 6 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 114 q + 6 q^{3} - 3 q^{7} + 57 q^{8} + 6 q^{9} + 3 q^{10} - 6 q^{11} + 6 q^{12} - 3 q^{13} - 6 q^{15} + 3 q^{18} + 12 q^{21} + 18 q^{23} + 3 q^{24} - 18 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{30} + 42 q^{31} + 51 q^{33} + 9 q^{35} - 3 q^{37} - 21 q^{38} + 30 q^{41} - 21 q^{42} + 42 q^{43} - 69 q^{45} + 6 q^{46} + 21 q^{49} - 3 q^{52} + 3 q^{53} - 45 q^{54} + 3 q^{56} - 54 q^{57} - 9 q^{58} + 3 q^{59} + 18 q^{60} - 48 q^{61} + 21 q^{62} + 39 q^{63} - 57 q^{64} + 18 q^{65} - 9 q^{66} - 12 q^{67} - 27 q^{69} + 18 q^{70} + 27 q^{71} + 3 q^{72} + 72 q^{73} + 12 q^{74} + 15 q^{75} + 12 q^{77} + 63 q^{78} + 48 q^{79} + 6 q^{80} + 102 q^{81} + 24 q^{82} + 9 q^{83} + 3 q^{86} - 57 q^{87} - 12 q^{88} - 30 q^{89} - 24 q^{90} - 51 q^{91} - 6 q^{92} + 54 q^{93} + 6 q^{94} - 105 q^{95} + 3 q^{96} - 72 q^{97} - 21 q^{98} - 3 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	−0.173648	+	0.984808i	−1.68648	+	0.394709i	−0.939693	−	0.342020i	−0.372325	+	0.312417i	−0.0958588	−	1.72940i	0.702678	+	0.255754i	0.500000	−	0.866025i	2.68841	−	1.33134i	−0.243018	−	0.420919i
7.2	−0.173648	+	0.984808i	−1.66402	−	0.480649i	−0.939693	−	0.342020i	0.737582	−	0.618905i	0.762301	−	1.55528i	−4.50888	−	1.64110i	0.500000	−	0.866025i	2.53795	+	1.59962i	0.481423	+	0.833848i
7.3	−0.173648	+	0.984808i	−1.64212	+	0.550843i	−0.939693	−	0.342020i	2.94025	−	2.46716i	−0.257323	−	1.71283i	1.44013	+	0.524163i	0.500000	−	0.866025i	2.39314	−	1.80911i	1.91911	+	3.32400i
7.4	−0.173648	+	0.984808i	−1.58144	−	0.706424i	−0.939693	−	0.342020i	−1.51681	+	1.27276i	0.970307	−	1.43475i	0.981241	+	0.357143i	0.500000	−	0.866025i	2.00193	+	2.23434i	−0.990030	−	1.71478i
7.5	−0.173648	+	0.984808i	−1.29296	+	1.15249i	−0.939693	−	0.342020i	−3.01823	+	2.53259i	−0.910465	−	1.47345i	−1.70169	−	0.619366i	0.500000	−	0.866025i	0.343513	−	2.98027i	−1.97001	−	3.41215i
7.6	−0.173648	+	0.984808i	−0.999066	+	1.41487i	−0.939693	−	0.342020i	0.204526	−	0.171618i	−1.21989	−	1.22958i	4.22180	+	1.53661i	0.500000	−	0.866025i	−1.00373	−	2.82710i	0.133495	+	0.231220i
7.7	−0.173648	+	0.984808i	−0.743439	−	1.56438i	−0.939693	−	0.342020i	−2.45674	+	2.06145i	1.66971	−	0.460492i	−0.0178929	−	0.00651249i	0.500000	−	0.866025i	−1.89460	+	2.32605i	−1.60352	−	2.77738i
7.8	−0.173648	+	0.984808i	−0.738294	−	1.56682i	−0.939693	−	0.342020i	3.22693	−	2.70772i	1.67122	−	0.455002i	1.24827	+	0.454331i	0.500000	−	0.866025i	−1.90985	+	2.31355i	2.10623	+	3.64810i
7.9	−0.173648	+	0.984808i	−0.341647	+	1.69802i	−0.939693	−	0.342020i	2.72591	−	2.28731i	−1.61290	−	0.631315i	−3.28664	−	1.19624i	0.500000	−	0.866025i	−2.76655	−	1.16025i	1.77921	+	3.08168i
7.10	−0.173648	+	0.984808i	0.305057	−	1.70498i	−0.939693	−	0.342020i	−1.01624	+	0.852724i	1.62610	+	0.596488i	−3.99613	−	1.45447i	0.500000	−	0.866025i	−2.81388	−	1.04023i	−0.663302	−	1.14887i
7.11	−0.173648	+	0.984808i	0.451353	−	1.67221i	−0.939693	−	0.342020i	1.54753	−	1.29853i	1.56843	+	0.734872i	−1.48626	−	0.540954i	0.500000	−	0.866025i	−2.59256	−	1.50951i	1.01008	+	1.74951i
7.12	−0.173648	+	0.984808i	0.500064	+	1.65829i	−0.939693	−	0.342020i	0.700905	−	0.588129i	−1.71994	+	0.204507i	1.06392	+	0.387234i	0.500000	−	0.866025i	−2.49987	+	1.65850i	0.457483	+	0.792385i
7.13	−0.173648	+	0.984808i	0.528375	+	1.64949i	−0.939693	−	0.342020i	−0.363365	+	0.304899i	−1.71618	+	0.233917i	−0.815742	−	0.296906i	0.500000	−	0.866025i	−2.44164	+	1.74310i	−0.237170	−	0.410790i
7.14	−0.173648	+	0.984808i	0.904730	−	1.47698i	−0.939693	−	0.342020i	0.614270	−	0.515434i	1.29743	+	1.14746i	4.27118	+	1.55458i	0.500000	−	0.866025i	−1.36293	−	2.67253i	0.400936	+	0.694442i
7.15	−0.173648	+	0.984808i	1.47887	+	0.901635i	−0.939693	−	0.342020i	−3.07190	+	2.57763i	−1.14474	+	1.29983i	4.75356	+	1.73015i	0.500000	−	0.866025i	1.37411	+	2.66680i	−2.00504	−	3.47283i
7.16	−0.173648	+	0.984808i	1.52498	+	0.821234i	−0.939693	−	0.342020i	0.131855	−	0.110639i	−1.07357	+	1.35921i	−0.775971	−	0.282431i	0.500000	−	0.866025i	1.65115	+	2.50474i	0.0860619	+	0.149064i
7.17	−0.173648	+	0.984808i	1.61046	−	0.637513i	−0.939693	−	0.342020i	−0.916401	+	0.768952i	0.348175	+	1.69670i	0.00657077	+	0.00239156i	0.500000	−	0.866025i	2.18715	−	2.05338i	−0.598138	−	1.03601i
7.18	−0.173648	+	0.984808i	1.71849	−	0.216331i	−0.939693	−	0.342020i	−1.49707	+	1.25619i	−0.0853680	+	1.72995i	−4.10029	−	1.49238i	0.500000	−	0.866025i	2.90640	−	0.743524i	−0.977141	−	1.69246i
7.19	−0.173648	+	0.984808i	1.72741	+	0.126708i	−0.939693	−	0.342020i	2.16536	−	1.81695i	−0.424744	+	1.67916i	1.67381	+	0.609219i	0.500000	−	0.866025i	2.96789	+	0.437752i	1.41334	+	2.44798i
49.1	0.939693	+	0.342020i	−1.72677	−	0.135090i	0.766044	+	0.642788i	−0.119543	+	0.677960i	−1.57643	−	0.717535i	−0.540678	−	0.453683i	0.500000	+	0.866025i	2.96350	+	0.466540i	−0.344209	+	0.596188i

See next 80 embeddings (of 114 total)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.w.b	✓	114
9.c	even	3	1		666.2.y.b	yes	114
37.f	even	9	1		666.2.y.b	yes	114
333.w	even	9	1	inner	666.2.w.b	✓	114

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.w.b	✓	114	1.a	even	1	1	trivial
666.2.w.b	✓	114	333.w	even	9	1	inner
666.2.y.b	yes	114	9.c	even	3	1
666.2.y.b	yes	114	37.f	even	9	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{114} + 25 T_{5}^{111} + 84 T_{5}^{110} + 339 T_{5}^{109} + 9708 T_{5}^{108} + \cdots + 41\!\cdots\!49 $ T5^114 + 25*T5^111 + 84*T5^110 + 339*T5^109 + 9708*T5^108 - 6771*T5^107 + 17178*T5^106 + 235153*T5^105 + 778992*T5^104 + 783357*T5^103 + 63130687*T5^102 - 41634495*T5^101 + 179480460*T5^100 + 1373930058*T5^99 + 3533031165*T5^98 + 2679816612*T5^97 + 218692563266*T5^96 - 210628537638*T5^95 + 610851425205*T5^94 + 3265036903217*T5^93 + 6717106539414*T5^92 - 17905355402322*T5^91 + 499565671031262*T5^90 - 622462353825387*T5^89 + 1288322975853132*T5^88 + 4907597625519635*T5^87 + 8067392828835252*T5^86 - 26572310617350486*T5^85 + 604371830007233252*T5^84 - 879376441705904427*T5^83 + 2212903109260981404*T5^82 - 310964914611258222*T5^81 + 7311006304421397129*T5^80 - 50098598292635600382*T5^79 + 431099594007162504828*T5^78 - 904404386570088516462*T5^77 + 2159827808883540207642*T5^76 - 2195609992292858369457*T5^75 + 2484396044134314058440*T5^74 - 16691541824195416494603*T5^73 + 158818410154499093624871*T5^72 - 432874920443098740231474*T5^71 + 1410503389755033903339987*T5^70 - 3009829700994620683268118*T5^69 + 5185076192176590746110890*T5^68 - 12134591018828557372922535*T5^67 + 30910797274332397501497804*T5^66 - 61801358161913538148374948*T5^65 + 174040711121473812205530939*T5^64 - 270448941372351024716900064*T5^63 + 377253591679331916825598974*T5^62 - 699232196108372334545603790*T5^61 + 1195086148576981916435218022*T5^60 - 2402862541592210271146310084*T5^59 + 8242909498330409084356244574*T5^58 - 10728394878436381863867541528*T5^57 + 15190488687602592220190595846*T5^56 - 27756349907832735459942781905*T5^55 + 26494961072731132723766851260*T5^54 - 70976495736313406959712863047*T5^53 + 289923933220192327409434363080*T5^52 - 216613936563973686378418261297*T5^51 + 416969938107992937697047907581*T5^50 - 436427794928298960838690703367*T5^49 - 788775447022858616750973520138*T5^48 - 784840891437221625399756039687*T5^47 + 3030218164843427640997187683089*T5^46 + 390523340961709144294351720677*T5^45 + 8618803261072834809901681262904*T5^44 + 1729522838296955583806097563007*T5^43 - 13772419155737919333249543679028*T5^42 - 18382874724555436799475282453657*T5^41 + 11813293856820906809642469618528*T5^40 - 1308472138703532010040798834363*T5^39 + 59655154216311027889080355076355*T5^38 + 69393095865333657794606179610676*T5^37 - 49708703345591143060985945600373*T5^36 - 131659773263832092987154223482387*T5^35 + 51972437968932290608619102365545*T5^34 + 131850678657024132880916028710317*T5^33 + 92019596101247598667162999572594*T5^32 - 36979942063546897110380614511328*T5^31 - 94401577574374990856964184908311*T5^30 - 20093927983582883590560273502710*T5^29 + 184812648318045511232547637622691*T5^28 + 97150631949881160165902207636787*T5^27 + 29075564425103288885262906838926*T5^26 - 10661523845091732838532034047763*T5^25 + 20312378640334814747925855269979*T5^24 + 78229967501836667248557663508389*T5^23 + 76283897536331243680877398648620*T5^22 + 29281821302495681713570892626125*T5^21 + 11826957989902976586487293432210*T5^20 + 9639794888805854971878527979270*T5^19 + 15413955156013321908705754933497*T5^18 + 13138674298383019434597575194212*T5^17 + 6686490902827929564997368150723*T5^16 + 1411761914697710232211827688083*T5^15 + 83105160747738647036735273607*T5^14 + 23667359192727408789062274009*T5^13 + 43390298521630400376182894478*T5^12 + 7458029462129682109010543406*T5^11 + 1047088545100800272599418073*T5^10 + 117082917552425216950419129*T5^9 + 3820770208960112888843661*T5^8 + 4966700646434448428408313*T5^7 + 1336828562226644385211074*T5^6 + 59954536083893089954077*T5^5 + 2444172623200291469121*T5^4 - 667452682694520811332*T5^3 - 74121617565858472398*T5^2 + 2931090201752072949*T5 + 413530398804003849 acting on $S_{2}^{\mathrm{new}}(666, [\chi])$.

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:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.w (of order \(9\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.19
Significant digits:
Format:

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