LMFDB - Newform orbit 455.2.l.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [455,2,Mod(16,455)] 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("455.16"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(455, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 455 = 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	455.l (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [38,4] 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:	$3.63319329197$
Analytic rank:	$0$
Dimension:	$38$
Relative dimension:	$19$ 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) = $ $ 38 q + 4 q^{2} - 2 q^{3} + 44 q^{4} - 19 q^{5} - q^{7} + 12 q^{8} - 19 q^{9} - 2 q^{10} - 9 q^{11} - 4 q^{12} + 2 q^{13} - 3 q^{14} - 2 q^{15} + 56 q^{16} + 6 q^{17} - 23 q^{18} + 2 q^{19} - 22 q^{20}+ \cdots + 52 q^{99}+O(q^{100}) $

38 * q + 4 * q^2 - 2 * q^3 + 44 * q^4 - 19 * q^5 - q^7 + 12 * q^8 - 19 * q^9 - 2 * q^10 - 9 * q^11 - 4 * q^12 + 2 * q^13 - 3 * q^14 - 2 * q^15 + 56 * q^16 + 6 * q^17 - 23 * q^18 + 2 * q^19 - 22 * q^20 - 17 * q^21 - 5 * q^22 - 20 * q^23 + 2 * q^24 - 19 * q^25 + 12 * q^26 + 34 * q^27 - 2 * q^28 + 7 * q^29 - 9 * q^31 + 28 * q^32 + 7 * q^33 - 8 * q^34 - 4 * q^35 - 20 * q^36 + 26 * q^37 + 5 * q^38 + 3 * q^39 - 6 * q^40 + 14 * q^41 - 66 * q^42 + 4 * q^43 - 21 * q^44 + 38 * q^45 + 38 * q^46 - 4 * q^47 - 37 * q^48 - 5 * q^49 - 2 * q^50 - 2 * q^51 - 53 * q^52 - q^53 + 32 * q^54 - 9 * q^55 + 8 * q^56 + 10 * q^57 - 28 * q^58 - 4 * q^60 + 15 * q^61 - 6 * q^62 + 68 * q^63 + 68 * q^64 + 8 * q^65 + 20 * q^66 - 47 * q^67 + 20 * q^68 + 2 * q^69 + 3 * q^70 - 23 * q^71 - 103 * q^72 + 15 * q^73 - 70 * q^74 + 4 * q^75 + 5 * q^76 - 65 * q^77 + 24 * q^78 - 3 * q^79 - 28 * q^80 - 23 * q^81 - 47 * q^82 - 8 * q^83 - 45 * q^84 - 3 * q^85 + 5 * q^86 + 128 * q^87 - 9 * q^88 - 32 * q^89 + 46 * q^90 + 60 * q^91 - 106 * q^92 + 62 * q^93 + 40 * q^94 - 4 * q^95 - 56 * q^96 + 37 * q^97 + 70 * q^98 + 52 * 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} $
16.1	−2.74039	−0.182108	+	0.315421i	5.50972	−0.500000	+	0.866025i	0.499047	−	0.864375i	−0.413858	−	2.61318i	−9.61800	1.43367	+	2.48319i	1.37019	−	2.37325i
16.2	−2.41578	0.261635	−	0.453165i	3.83599	−0.500000	+	0.866025i	−0.632052	+	1.09475i	1.65525	+	2.06401i	−4.43534	1.36309	+	2.36095i	1.20789	−	2.09213i
16.3	−2.21654	−1.22503	+	2.12182i	2.91305	−0.500000	+	0.866025i	2.71533	−	4.70309i	−2.59082	+	0.536353i	−2.02382	−1.50140	−	2.60051i	1.10827	−	1.91958i
16.4	−1.82106	0.929351	−	1.60968i	1.31627	−0.500000	+	0.866025i	−1.69241	+	2.93134i	2.13637	−	1.56074i	1.24511	−0.227386	−	0.393844i	0.910532	−	1.57709i
16.5	−1.56699	1.38100	−	2.39196i	0.455451	−0.500000	+	0.866025i	−2.16401	+	3.74817i	−2.08092	+	1.63394i	2.42029	−2.31431	−	4.00850i	0.783494	−	1.35705i
16.6	−1.36671	−1.19212	+	2.06481i	−0.132098	−0.500000	+	0.866025i	1.62928	−	2.82200i	0.361080	+	2.62100i	2.91396	−1.34229	−	2.32491i	0.683356	−	1.18361i
16.7	−1.13200	−0.884561	+	1.53211i	−0.718583	−0.500000	+	0.866025i	1.00132	−	1.73434i	2.00877	−	1.72188i	3.07743	−0.0648974	−	0.112406i	0.565998	−	0.980338i
16.8	−0.460333	0.706974	−	1.22451i	−1.78809	−0.500000	+	0.866025i	−0.325444	+	0.563685i	−1.42578	+	2.22871i	1.74378	0.500376	+	0.866676i	0.230167	−	0.398660i
16.9	−0.307288	0.823211	−	1.42584i	−1.90557	−0.500000	+	0.866025i	−0.252963	+	0.438145i	2.33734	+	1.23970i	1.20014	0.144647	+	0.250535i	0.153644	−	0.266119i
16.10	0.414405	−1.18735	+	2.05655i	−1.82827	−0.500000	+	0.866025i	−0.492043	+	0.852244i	0.223348	−	2.63631i	−1.58645	−1.31960	−	2.28561i	−0.207202	+	0.358885i
16.11	0.616455	−1.45860	+	2.52637i	−1.61998	−0.500000	+	0.866025i	−0.899162	+	1.55739i	−1.60065	+	2.10664i	−2.23156	−2.75503	−	4.77185i	−0.308228	+	0.533866i
16.12	0.752383	0.162148	−	0.280848i	−1.43392	−0.500000	+	0.866025i	0.121997	−	0.211305i	−2.32268	−	1.26695i	−2.58362	1.44742	+	2.50700i	−0.376192	+	0.651583i
16.13	0.970955	1.13697	−	1.96930i	−1.05725	−0.500000	+	0.866025i	1.10395	−	1.91210i	−0.574936	−	2.58253i	−2.96845	−1.08542	−	1.88001i	−0.485478	+	0.840872i
16.14	1.49905	−0.443947	+	0.768938i	0.247155	−0.500000	+	0.866025i	−0.665499	+	1.15268i	0.0520434	+	2.64524i	−2.62760	1.10582	+	1.91534i	−0.749526	+	1.29822i
16.15	1.79890	−0.217038	+	0.375921i	1.23603	−0.500000	+	0.866025i	−0.390429	+	0.676242i	2.63010	−	0.287369i	−1.37431	1.40579	+	2.43490i	−0.899448	+	1.55789i
16.16	2.34023	0.780077	−	1.35113i	3.47668	−0.500000	+	0.866025i	1.82556	−	3.16196i	1.33454	−	2.28452i	3.45577	0.282960	+	0.490101i	−1.17012	+	2.02670i
16.17	2.35621	1.60078	−	2.77263i	3.55174	−0.500000	+	0.866025i	3.77177	−	6.53290i	−2.14038	+	1.55524i	3.65623	−3.62498	−	6.27865i	−1.17811	+	2.04054i
16.18	2.56715	−1.69976	+	2.94407i	4.59025	−0.500000	+	0.866025i	−4.36354	+	7.55787i	2.54857	+	0.710483i	6.64957	−4.27837	−	7.41035i	−1.28357	+	2.22322i
16.19	2.71135	−0.291634	+	0.505124i	5.35142	−0.500000	+	0.866025i	−0.790721	+	1.36957i	−2.63739	+	0.210237i	9.08688	1.32990	+	2.30345i	−1.35568	+	2.34810i
256.1	−2.74039	−0.182108	−	0.315421i	5.50972	−0.500000	−	0.866025i	0.499047	+	0.864375i	−0.413858	+	2.61318i	−9.61800	1.43367	−	2.48319i	1.37019	+	2.37325i

See all 38 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	455.2.l.b	yes	38
7.c	even	3	1		455.2.k.a	✓	38
13.c	even	3	1		455.2.k.a	✓	38
91.h	even	3	1	inner	455.2.l.b	yes	38

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
455.2.k.a	✓	38	7.c	even	3	1
455.2.k.a	✓	38	13.c	even	3	1
455.2.l.b	yes	38	1.a	even	1	1	trivial
455.2.l.b	yes	38	91.h	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{19} - 2 T_{2}^{18} - 28 T_{2}^{17} + 54 T_{2}^{16} + 322 T_{2}^{15} - 594 T_{2}^{14} - 1969 T_{2}^{13} + \cdots + 177 $ T2^19 - 2*T2^18 - 28*T2^17 + 54*T2^16 + 322*T2^15 - 594*T2^14 - 1969*T2^13 + 3448*T2^12 + 6912*T2^11 - 11445*T2^10 - 13965*T2^9 + 22025*T2^8 + 15232*T2^7 - 23717*T2^6 - 7432*T2^5 + 12859*T2^4 + 787*T2^3 - 2697*T2^2 + 41*T2 + 177 acting on $S_{2}^{\mathrm{new}}(455, [\chi])$.

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Level:	\( N \)	\(=\)	\( 455 = 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	455.l (of order \(3\), degree \(2\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 16.19
Significant digits:
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