LMFDB - Newform orbit 666.2.e.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [666,2,Mod(223,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("666.223"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [22,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 22 q + 11 q^{2} - 4 q^{3} - 11 q^{4} + q^{5} - 5 q^{6} - 5 q^{7} - 22 q^{8} - 4 q^{9} + 2 q^{10} - 2 q^{11} - q^{12} + q^{13} + 5 q^{14} + 9 q^{15} - 11 q^{16} - 24 q^{17} - 5 q^{18} + 20 q^{19} + q^{20}+ \cdots - 18 q^{99}+O(q^{100}) $

22 * q + 11 * q^2 - 4 * q^3 - 11 * q^4 + q^5 - 5 * q^6 - 5 * q^7 - 22 * q^8 - 4 * q^9 + 2 * q^10 - 2 * q^11 - q^12 + q^13 + 5 * q^14 + 9 * q^15 - 11 * q^16 - 24 * q^17 - 5 * q^18 + 20 * q^19 + q^20 + 4 * q^21 + 2 * q^22 + 4 * q^24 - 12 * q^25 + 2 * q^26 - 13 * q^27 + 10 * q^28 + 9 * q^29 - 12 * q^30 - 21 * q^31 + 11 * q^32 + 24 * q^33 - 12 * q^34 - 6 * q^35 - q^36 + 22 * q^37 + 10 * q^38 - 17 * q^39 - q^40 + 33 * q^41 + 8 * q^42 - 6 * q^43 + 4 * q^44 - 48 * q^45 + 4 * q^47 + 5 * q^48 - 44 * q^49 + 12 * q^50 + 27 * q^51 + q^52 - 2 * q^54 + 44 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 8 * q^59 - 21 * q^60 - 39 * q^61 - 42 * q^62 - 47 * q^63 + 22 * q^64 - 12 * q^65 - 6 * q^66 + 4 * q^67 + 12 * q^68 + 18 * q^69 - 3 * q^70 - 54 * q^71 + 4 * q^72 + 60 * q^73 + 11 * q^74 + 65 * q^75 - 10 * q^76 + 27 * q^77 + 8 * q^78 - 26 * q^79 - 2 * q^80 - 52 * q^81 + 66 * q^82 + 10 * q^83 + 4 * q^84 - 33 * q^85 + 6 * q^86 + 18 * q^87 + 2 * q^88 + 22 * q^89 - 9 * q^90 + 38 * q^91 + 7 * q^93 - 4 * q^94 + 38 * q^95 + q^96 - 8 * q^97 - 88 * q^98 - 18 * q^99

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} $
223.1	0.500000	−	0.866025i	−1.67730	−	0.432045i	−0.500000	−	0.866025i	−1.28024	−	2.21744i	−1.21281	+	1.23656i	−1.14205	+	1.97809i	−1.00000	2.62668	+	1.44934i	−2.56048
223.2	0.500000	−	0.866025i	−1.53708	+	0.798365i	−0.500000	−	0.866025i	1.54068	+	2.66854i	−0.0771356	+	1.73033i	−1.98771	+	3.44281i	−1.00000	1.72523	−	2.45430i	3.08137
223.3	0.500000	−	0.866025i	−1.37289	−	1.05602i	−0.500000	−	0.866025i	1.74283	+	3.01866i	−1.60098	+	0.660945i	1.12153	−	1.94254i	−1.00000	0.769638	+	2.89960i	3.48565
223.4	0.500000	−	0.866025i	−1.25796	+	1.19060i	−0.500000	−	0.866025i	−1.61947	−	2.80500i	0.402106	+	1.68473i	2.18497	−	3.78448i	−1.00000	0.164952	−	2.99546i	−3.23894
223.5	0.500000	−	0.866025i	−0.924201	−	1.46487i	−0.500000	−	0.866025i	1.28880	+	2.23226i	−1.73072	+	0.0679447i	−1.87984	+	3.25598i	−1.00000	−1.29171	+	2.70767i	2.57759
223.6	0.500000	−	0.866025i	−0.309813	+	1.70412i	−0.500000	−	0.866025i	−0.646008	−	1.11892i	1.32090	+	1.12036i	−0.189687	+	0.328547i	−1.00000	−2.80803	−	1.05592i	−1.29202
223.7	0.500000	−	0.866025i	0.546833	−	1.64346i	−0.500000	−	0.866025i	0.961530	+	1.66542i	−1.14986	−	1.29530i	2.14544	−	3.71602i	−1.00000	−2.40195	−	1.79740i	1.92306
223.8	0.500000	−	0.866025i	0.690057	−	1.58865i	−0.500000	−	0.866025i	0.294009	+	0.509238i	−1.03079	−	1.39193i	−0.187265	+	0.324353i	−1.00000	−2.04764	−	2.19252i	0.588018
223.9	0.500000	−	0.866025i	0.994972	+	1.41776i	−0.500000	−	0.866025i	0.132922	+	0.230227i	1.72530	−	0.152794i	1.42395	−	2.46635i	−1.00000	−1.02006	+	2.82125i	0.265844
223.10	0.500000	−	0.866025i	1.21435	−	1.23506i	−0.500000	−	0.866025i	−1.99108	−	3.44865i	−0.462416	−	1.66918i	−1.37418	+	2.38014i	−1.00000	−0.0507255	−	2.99957i	−3.98216
223.11	0.500000	−	0.866025i	1.63304	+	0.577226i	−0.500000	−	0.866025i	0.0760328	+	0.131693i	1.31641	−	1.12564i	−2.61516	+	4.52959i	−1.00000	2.33362	+	1.88526i	0.152066
445.1	0.500000	+	0.866025i	−1.67730	+	0.432045i	−0.500000	+	0.866025i	−1.28024	+	2.21744i	−1.21281	−	1.23656i	−1.14205	−	1.97809i	−1.00000	2.62668	−	1.44934i	−2.56048
445.2	0.500000	+	0.866025i	−1.53708	−	0.798365i	−0.500000	+	0.866025i	1.54068	−	2.66854i	−0.0771356	−	1.73033i	−1.98771	−	3.44281i	−1.00000	1.72523	+	2.45430i	3.08137
445.3	0.500000	+	0.866025i	−1.37289	+	1.05602i	−0.500000	+	0.866025i	1.74283	−	3.01866i	−1.60098	−	0.660945i	1.12153	+	1.94254i	−1.00000	0.769638	−	2.89960i	3.48565
445.4	0.500000	+	0.866025i	−1.25796	−	1.19060i	−0.500000	+	0.866025i	−1.61947	+	2.80500i	0.402106	−	1.68473i	2.18497	+	3.78448i	−1.00000	0.164952	+	2.99546i	−3.23894
445.5	0.500000	+	0.866025i	−0.924201	+	1.46487i	−0.500000	+	0.866025i	1.28880	−	2.23226i	−1.73072	−	0.0679447i	−1.87984	−	3.25598i	−1.00000	−1.29171	−	2.70767i	2.57759
445.6	0.500000	+	0.866025i	−0.309813	−	1.70412i	−0.500000	+	0.866025i	−0.646008	+	1.11892i	1.32090	−	1.12036i	−0.189687	−	0.328547i	−1.00000	−2.80803	+	1.05592i	−1.29202
445.7	0.500000	+	0.866025i	0.546833	+	1.64346i	−0.500000	+	0.866025i	0.961530	−	1.66542i	−1.14986	+	1.29530i	2.14544	+	3.71602i	−1.00000	−2.40195	+	1.79740i	1.92306
445.8	0.500000	+	0.866025i	0.690057	+	1.58865i	−0.500000	+	0.866025i	0.294009	−	0.509238i	−1.03079	+	1.39193i	−0.187265	−	0.324353i	−1.00000	−2.04764	+	2.19252i	0.588018
445.9	0.500000	+	0.866025i	0.994972	−	1.41776i	−0.500000	+	0.866025i	0.132922	−	0.230227i	1.72530	+	0.152794i	1.42395	+	2.46635i	−1.00000	−1.02006	−	2.82125i	0.265844

See all 22 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	666.2.e.f	✓	22
3.b	odd	2	1		1998.2.e.f		22
9.c	even	3	1	inner	666.2.e.f	✓	22
9.c	even	3	1		5994.2.a.bc		11
9.d	odd	6	1		1998.2.e.f		22
9.d	odd	6	1		5994.2.a.bd		11

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
666.2.e.f	✓	22	1.a	even	1	1	trivial
666.2.e.f	✓	22	9.c	even	3	1	inner
1998.2.e.f		22	3.b	odd	2	1
1998.2.e.f		22	9.d	odd	6	1
5994.2.a.bc		11	9.c	even	3	1
5994.2.a.bd		11	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{22} - T_{5}^{21} + 34 T_{5}^{20} - 47 T_{5}^{19} + 767 T_{5}^{18} - 1088 T_{5}^{17} + 10146 T_{5}^{16} + \cdots + 2916 $ T5^22 - T5^21 + 34*T5^20 - 47*T5^19 + 767*T5^18 - 1088*T5^17 + 10146*T5^16 - 14427*T5^15 + 97194*T5^14 - 128549*T5^13 + 593705*T5^12 - 689036*T5^11 + 2524789*T5^10 - 2507158*T5^9 + 5810017*T5^8 - 3676446*T5^7 + 7198950*T5^6 - 5419854*T5^5 + 3727224*T5^4 - 1206360*T5^3 + 294273*T5^2 - 34506*T5 + 2916 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 $\F_{q}$
Belyi maps

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 223.11
Significant digits:
Format:

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