LMFDB - Newform orbit 378.2.v.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [378,2,Mod(67,378)] 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("378.67"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8, 12])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [72,0,0,0,0,3,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

$\operatorname{Tr}(f)(q) = $ $ 72 q + 3 q^{6} + 6 q^{7} - 36 q^{8} + 24 q^{9} - 6 q^{10} + 12 q^{11} - 12 q^{13} - 3 q^{14} - 6 q^{15} - 12 q^{17} - 18 q^{19} - 30 q^{21} - 6 q^{22} + 30 q^{23} + 6 q^{25} - 18 q^{26} - 6 q^{27} + 3 q^{29}+ \cdots - 54 q^{99}+O(q^{100}) $

72 * q + 3 * q^6 + 6 * q^7 - 36 * q^8 + 24 * q^9 - 6 * q^10 + 12 * q^11 - 12 * q^13 - 3 * q^14 - 6 * q^15 - 12 * q^17 - 18 * q^19 - 30 * q^21 - 6 * q^22 + 30 * q^23 + 6 * q^25 - 18 * q^26 - 6 * q^27 + 3 * q^29 + 3 * q^30 + 9 * q^31 - 18 * q^33 + 9 * q^34 - 18 * q^35 + 9 * q^36 - 24 * q^39 - 6 * q^41 + 12 * q^42 + 24 * q^43 + 51 * q^45 + 45 * q^47 - 6 * q^48 - 12 * q^49 + 6 * q^50 - 51 * q^51 + 6 * q^52 - 15 * q^53 + 27 * q^54 + 72 * q^55 - 3 * q^56 - 63 * q^57 + 3 * q^58 - 30 * q^59 - 3 * q^60 + 9 * q^61 - 24 * q^62 - 39 * q^63 - 36 * q^64 + 9 * q^65 - 6 * q^67 + 9 * q^68 - 21 * q^69 - 33 * q^70 + 12 * q^71 - 12 * q^72 + 132 * q^73 - 18 * q^74 - 30 * q^75 - 33 * q^77 + 30 * q^78 - 9 * q^79 - 6 * q^80 + 36 * q^81 - 33 * q^82 - 18 * q^83 + 15 * q^84 + 21 * q^85 - 12 * q^86 - 39 * q^87 - 6 * q^88 - 36 * q^89 + 69 * q^90 + 39 * q^91 + 30 * q^92 - 66 * q^93 - 9 * q^94 - 33 * q^95 - 48 * q^97 - 12 * q^98 - 54 * 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} $
67.1	0.766044	−	0.642788i	−1.73077	+	0.0664993i	0.173648	−	0.984808i	−0.294369	+	1.66945i	−1.28310	+	1.16346i	−2.64575	−	0.00486080i	−0.500000	−	0.866025i	2.99116	−	0.230191i	0.847602	+	1.46809i
67.2	0.766044	−	0.642788i	−1.69832	−	0.340165i	0.173648	−	0.984808i	−0.294026	+	1.66751i	−1.51964	+	0.831077i	1.80729	−	1.93228i	−0.500000	−	0.866025i	2.76858	+	1.15542i	0.846615	+	1.46638i
67.3	0.766044	−	0.642788i	−1.42742	+	0.981051i	0.173648	−	0.984808i	0.343656	−	1.94897i	−0.462863	+	1.66906i	1.84248	+	1.89876i	−0.500000	−	0.866025i	1.07508	−	2.80075i	−0.989519	−	1.71390i
67.4	0.766044	−	0.642788i	−1.07856	−	1.35526i	0.173648	−	0.984808i	0.620533	−	3.51922i	−1.69736	−	0.344903i	−0.670349	−	2.55942i	−0.500000	−	0.866025i	−0.673434	+	2.92344i	−1.78675	−	3.09475i
67.5	0.766044	−	0.642788i	−0.402772	+	1.68457i	0.173648	−	0.984808i	−0.366960	+	2.08113i	0.774279	+	1.54935i	1.84638	+	1.89496i	−0.500000	−	0.866025i	−2.67555	−	1.35699i	1.05662	+	1.83012i
67.6	0.766044	−	0.642788i	−0.389111	−	1.68778i	0.173648	−	0.984808i	−0.0926928	+	0.525687i	−1.38296	−	1.04280i	−2.63702	+	0.214772i	−0.500000	−	0.866025i	−2.69718	+	1.31347i	0.266898	+	0.462282i
67.7	0.766044	−	0.642788i	0.586306	+	1.62980i	0.173648	−	0.984808i	0.483736	−	2.74340i	1.49675	+	0.871628i	0.286196	−	2.63023i	−0.500000	−	0.866025i	−2.31249	+	1.91112i	−1.39286	−	2.41251i
67.8	0.766044	−	0.642788i	0.654437	−	1.60366i	0.173648	−	0.984808i	0.190866	−	1.08246i	−0.529482	−	1.64914i	2.53672	+	0.751691i	−0.500000	−	0.866025i	−2.14342	−	2.09898i	−0.549577	−	0.951896i
67.9	0.766044	−	0.642788i	1.40163	+	1.01757i	0.173648	−	0.984808i	−0.289065	+	1.63937i	1.72779	−	0.121449i	1.93330	−	1.80620i	−0.500000	−	0.866025i	0.929121	+	2.85250i	0.832330	+	1.44164i
67.10	0.766044	−	0.642788i	1.58806	−	0.691418i	0.173648	−	0.984808i	−0.0337396	+	0.191347i	0.772092	−	1.55044i	−1.90403	−	1.83703i	−0.500000	−	0.866025i	2.04388	−	2.19603i	0.0971494	+	0.168268i
67.11	0.766044	−	0.642788i	1.71536	−	0.239862i	0.173648	−	0.984808i	−0.604303	+	3.42717i	1.15986	−	1.28636i	−0.0661038	+	2.64493i	−0.500000	−	0.866025i	2.88493	−	0.822900i	1.74002	+	3.01380i
67.12	0.766044	−	0.642788i	1.72085	+	0.196628i	0.173648	−	0.984808i	0.683660	−	3.87723i	1.44464	−	0.955517i	−1.15546	+	2.38011i	−0.500000	−	0.866025i	2.92267	+	0.676737i	−1.96852	−	3.40958i
79.1	0.766044	+	0.642788i	−1.73077	−	0.0664993i	0.173648	+	0.984808i	−0.294369	−	1.66945i	−1.28310	−	1.16346i	−2.64575	+	0.00486080i	−0.500000	+	0.866025i	2.99116	+	0.230191i	0.847602	−	1.46809i
79.2	0.766044	+	0.642788i	−1.69832	+	0.340165i	0.173648	+	0.984808i	−0.294026	−	1.66751i	−1.51964	−	0.831077i	1.80729	+	1.93228i	−0.500000	+	0.866025i	2.76858	−	1.15542i	0.846615	−	1.46638i
79.3	0.766044	+	0.642788i	−1.42742	−	0.981051i	0.173648	+	0.984808i	0.343656	+	1.94897i	−0.462863	−	1.66906i	1.84248	−	1.89876i	−0.500000	+	0.866025i	1.07508	+	2.80075i	−0.989519	+	1.71390i
79.4	0.766044	+	0.642788i	−1.07856	+	1.35526i	0.173648	+	0.984808i	0.620533	+	3.51922i	−1.69736	+	0.344903i	−0.670349	+	2.55942i	−0.500000	+	0.866025i	−0.673434	−	2.92344i	−1.78675	+	3.09475i
79.5	0.766044	+	0.642788i	−0.402772	−	1.68457i	0.173648	+	0.984808i	−0.366960	−	2.08113i	0.774279	−	1.54935i	1.84638	−	1.89496i	−0.500000	+	0.866025i	−2.67555	+	1.35699i	1.05662	−	1.83012i
79.6	0.766044	+	0.642788i	−0.389111	+	1.68778i	0.173648	+	0.984808i	−0.0926928	−	0.525687i	−1.38296	+	1.04280i	−2.63702	−	0.214772i	−0.500000	+	0.866025i	−2.69718	−	1.31347i	0.266898	−	0.462282i
79.7	0.766044	+	0.642788i	0.586306	−	1.62980i	0.173648	+	0.984808i	0.483736	+	2.74340i	1.49675	−	0.871628i	0.286196	+	2.63023i	−0.500000	+	0.866025i	−2.31249	−	1.91112i	−1.39286	+	2.41251i
79.8	0.766044	+	0.642788i	0.654437	+	1.60366i	0.173648	+	0.984808i	0.190866	+	1.08246i	−0.529482	+	1.64914i	2.53672	−	0.751691i	−0.500000	+	0.866025i	−2.14342	+	2.09898i	−0.549577	+	0.951896i

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
189.u	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.v.b	✓	72
7.c	even	3	1		378.2.w.a	yes	72
27.e	even	9	1		378.2.w.a	yes	72
189.u	even	9	1	inner	378.2.v.b	✓	72

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{72} - 3 T_{5}^{70} + 41 T_{5}^{69} + 105 T_{5}^{68} - 57 T_{5}^{67} + 5130 T_{5}^{66} + \cdots + 18\!\cdots\!89 $ T5^72 - 3*T5^70 + 41*T5^69 + 105*T5^68 - 57*T5^67 + 5130*T5^66 + 9936*T5^65 - 18243*T5^64 + 77040*T5^63 + 684360*T5^62 + 1479843*T5^61 + 15612168*T5^60 + 46857969*T5^59 + 48338505*T5^58 + 322650720*T5^57 + 1823500746*T5^56 + 4068399051*T5^55 + 23282298336*T5^54 + 71469798156*T5^53 + 15744317805*T5^52 - 53789537577*T5^51 + 1182475597032*T5^50 + 2296807167924*T5^49 + 5533800870708*T5^48 + 27577180014084*T5^47 + 30306589055244*T5^46 - 71658350816986*T5^45 + 249294252863079*T5^44 + 1109507766999348*T5^43 + 2008336889738956*T5^42 + 4131237297460089*T5^41 + 3239235387231465*T5^40 - 12018861886728870*T5^39 + 53783671329467874*T5^38 + 148878581567989266*T5^37 + 158934871912785432*T5^36 + 232878233408586516*T5^35 + 234809796156450426*T5^34 - 1478806906452196239*T5^33 + 4864471434612914184*T5^32 + 9329617112627489139*T5^31 + 9228536356235390199*T5^30 + 18995268316706593533*T5^29 + 3173884168110064812*T5^28 - 92748848927600278665*T5^27 + 348307523984541460887*T5^26 + 14906745777775980933*T5^25 + 462788024714640905058*T5^24 + 194590847871205097787*T5^23 - 1704968316355059282690*T5^22 - 292392160038065713311*T5^21 + 2663403843262000390164*T5^20 - 3679109001387446019975*T5^19 + 12361380473614112197821*T5^18 - 12468833609291124027471*T5^17 + 7220134855909346259132*T5^16 - 5586134727077598084705*T5^15 + 4971968520534394780068*T5^14 - 5486180487521730743061*T5^13 + 10990196962677287148591*T5^12 - 5024069518134931813341*T5^11 - 3013271376445732826871*T5^10 - 218339119162461893184*T5^9 + 3294152992846268845461*T5^8 + 592342883383696992810*T5^7 - 65484280268956311066*T5^6 - 395727475629131956494*T5^5 - 171730289677778725248*T5^4 + 62471912431530521313*T5^3 + 50169797917044942591*T5^2 + 6609195619994576349*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 $\F_{q}$
Belyi maps

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 67.12
Significant digits:
Format:

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