LMFDB - Newform orbit 378.2.v.a

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} + 6 q^{10} + 12 q^{11} + 12 q^{13} - 3 q^{14} - 6 q^{15} - 12 q^{17} + 18 q^{19} + 6 q^{22} + 12 q^{23} - 6 q^{25} - 18 q^{26} + 6 q^{27} + 3 q^{29} + 15 q^{30} - 9 q^{31}+ \cdots + 90 q^{99}+O(q^{100}) $

72 * q + 3 * q^6 - 6 * q^7 + 36 * q^8 + 6 * q^10 + 12 * q^11 + 12 * q^13 - 3 * q^14 - 6 * q^15 - 12 * q^17 + 18 * q^19 + 6 * q^22 + 12 * q^23 - 6 * q^25 - 18 * q^26 + 6 * q^27 + 3 * q^29 + 15 * q^30 - 9 * q^31 - 18 * q^33 - 9 * q^34 - 18 * q^35 - 3 * q^36 - 48 * q^39 - 6 * q^41 + 36 * q^42 - 24 * q^43 - 45 * q^45 - 27 * q^47 + 6 * q^48 + 48 * q^49 + 6 * q^50 + 15 * q^51 - 6 * q^52 - 15 * q^53 - 63 * q^54 - 72 * q^55 - 3 * q^56 + 57 * q^57 - 3 * q^58 - 30 * q^59 + 21 * q^60 + 9 * q^61 - 24 * q^62 - 21 * q^63 - 36 * q^64 + 45 * q^65 + 6 * q^67 + 9 * q^68 - 39 * q^69 + 15 * q^70 + 12 * q^71 - 60 * q^73 - 18 * q^74 + 42 * q^75 + 3 * q^77 + 6 * q^78 - 27 * q^79 - 6 * q^80 + 12 * q^81 + 33 * q^82 + 18 * q^83 - 21 * q^84 + 51 * q^85 - 12 * q^86 - 117 * q^87 + 6 * q^88 - 36 * q^89 - 69 * q^90 - 3 * q^91 + 12 * q^92 - 30 * q^93 - 27 * q^94 + 3 * q^95 + 48 * q^97 - 12 * q^98 + 90 * 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	−1.71753	−	0.223844i	0.173648	−	0.984808i	0.0598945	−	0.339678i	1.45959	−	0.932530i	−1.66417	+	2.05683i	0.500000	+	0.866025i	2.89979	+	0.768916i	0.172459	+	0.298708i
67.2	−0.766044	+	0.642788i	−1.42403	−	0.985977i	0.173648	−	0.984808i	−0.327106	+	1.85511i	1.72464	−	0.160044i	−1.01788	−	2.44211i	0.500000	+	0.866025i	1.05570	+	2.80811i	−0.941864	−	1.63136i
67.3	−0.766044	+	0.642788i	−1.38102	+	1.04536i	0.173648	−	0.984808i	−0.716530	+	4.06364i	0.385982	−	1.68850i	2.64332	+	0.113332i	0.500000	+	0.866025i	0.814451	−	2.88733i	−2.06317	−	3.57351i
67.4	−0.766044	+	0.642788i	−0.921674	−	1.46646i	0.173648	−	0.984808i	0.622926	−	3.53279i	1.64867	+	0.530936i	2.63270	+	0.262455i	0.500000	+	0.866025i	−1.30103	+	2.70320i	1.79365	+	3.10668i
67.5	−0.766044	+	0.642788i	−0.829129	+	1.52071i	0.173648	−	0.984808i	0.684323	−	3.88099i	−0.342341	−	1.69788i	−1.85424	+	1.88727i	0.500000	+	0.866025i	−1.62509	−	2.52172i	1.97043	+	3.41288i
67.6	−0.766044	+	0.642788i	−0.402311	+	1.68468i	0.173648	−	0.984808i	0.256458	−	1.45445i	−0.774703	−	1.54914i	2.04878	−	1.67407i	0.500000	+	0.866025i	−2.67629	−	1.35553i	0.738441	+	1.27902i
67.7	−0.766044	+	0.642788i	0.0412776	−	1.73156i	0.173648	−	0.984808i	−0.194594	+	1.10360i	1.08140	+	1.35298i	−0.0465876	+	2.64534i	0.500000	+	0.866025i	−2.99659	−	0.142949i	−0.560311	−	0.970488i
67.8	−0.766044	+	0.642788i	0.309985	+	1.70409i	0.173648	−	0.984808i	−0.504047	+	2.85859i	−1.33283	−	1.10615i	−2.58705	+	0.554250i	0.500000	+	0.866025i	−2.80782	+	1.05648i	−1.45134	−	2.51380i
67.9	−0.766044	+	0.642788i	0.600510	−	1.62462i	0.173648	−	0.984808i	0.421605	−	2.39104i	0.584268	+	1.63053i	−1.84906	−	1.89235i	0.500000	+	0.866025i	−2.27878	−	1.95120i	1.21396	+	2.10265i
67.10	−0.766044	+	0.642788i	1.54135	−	0.790081i	0.173648	−	0.984808i	−0.272791	+	1.54708i	−0.672891	+	1.59600i	2.24230	−	1.40432i	0.500000	+	0.866025i	1.75154	−	2.43559i	−0.785472	−	1.36048i
67.11	−0.766044	+	0.642788i	1.59651	+	0.671670i	0.173648	−	0.984808i	−0.153204	+	0.868862i	−1.65474	+	0.511691i	0.258184	+	2.63312i	0.500000	+	0.866025i	2.09772	+	2.14466i	−0.441133	−	0.764065i
67.12	−0.766044	+	0.642788i	1.64636	+	0.538065i	0.173648	−	0.984808i	0.470361	−	2.66755i	−1.60704	+	0.646075i	−1.97994	−	1.75495i	0.500000	+	0.866025i	2.42097	+	1.77169i	1.35435	+	2.34580i
79.1	−0.766044	−	0.642788i	−1.71753	+	0.223844i	0.173648	+	0.984808i	0.0598945	+	0.339678i	1.45959	+	0.932530i	−1.66417	−	2.05683i	0.500000	−	0.866025i	2.89979	−	0.768916i	0.172459	−	0.298708i
79.2	−0.766044	−	0.642788i	−1.42403	+	0.985977i	0.173648	+	0.984808i	−0.327106	−	1.85511i	1.72464	+	0.160044i	−1.01788	+	2.44211i	0.500000	−	0.866025i	1.05570	−	2.80811i	−0.941864	+	1.63136i
79.3	−0.766044	−	0.642788i	−1.38102	−	1.04536i	0.173648	+	0.984808i	−0.716530	−	4.06364i	0.385982	+	1.68850i	2.64332	−	0.113332i	0.500000	−	0.866025i	0.814451	+	2.88733i	−2.06317	+	3.57351i
79.4	−0.766044	−	0.642788i	−0.921674	+	1.46646i	0.173648	+	0.984808i	0.622926	+	3.53279i	1.64867	−	0.530936i	2.63270	−	0.262455i	0.500000	−	0.866025i	−1.30103	−	2.70320i	1.79365	−	3.10668i
79.5	−0.766044	−	0.642788i	−0.829129	−	1.52071i	0.173648	+	0.984808i	0.684323	+	3.88099i	−0.342341	+	1.69788i	−1.85424	−	1.88727i	0.500000	−	0.866025i	−1.62509	+	2.52172i	1.97043	−	3.41288i
79.6	−0.766044	−	0.642788i	−0.402311	−	1.68468i	0.173648	+	0.984808i	0.256458	+	1.45445i	−0.774703	+	1.54914i	2.04878	+	1.67407i	0.500000	−	0.866025i	−2.67629	+	1.35553i	0.738441	−	1.27902i
79.7	−0.766044	−	0.642788i	0.0412776	+	1.73156i	0.173648	+	0.984808i	−0.194594	−	1.10360i	1.08140	−	1.35298i	−0.0465876	−	2.64534i	0.500000	−	0.866025i	−2.99659	+	0.142949i	−0.560311	+	0.970488i
79.8	−0.766044	−	0.642788i	0.309985	−	1.70409i	0.173648	+	0.984808i	−0.504047	−	2.85859i	−1.33283	+	1.10615i	−2.58705	−	0.554250i	0.500000	−	0.866025i	−2.80782	−	1.05648i	−1.45134	+	2.51380i

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.a	✓	72
7.c	even	3	1		378.2.w.b	yes	72
27.e	even	9	1		378.2.w.b	yes	72
189.u	even	9	1	inner	378.2.v.a	✓	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.2.v.a	✓	72	1.a	even	1	1	trivial
378.2.v.a	✓	72	189.u	even	9	1	inner
378.2.w.b	yes	72	7.c	even	3	1
378.2.w.b	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} + 207 T_{5}^{67} + 6532 T_{5}^{66} + \cdots + 2678994081 $ T5^72 + 3*T5^70 + 41*T5^69 - 105*T5^68 + 207*T5^67 + 6532*T5^66 - 7476*T5^65 + 57867*T5^64 + 137570*T5^63 - 299070*T5^62 + 1828671*T5^61 + 26671486*T5^60 - 9793713*T5^59 + 209289699*T5^58 + 595899194*T5^57 - 627929310*T5^56 + 2608930119*T5^55 + 40891516774*T5^54 - 56380480758*T5^53 + 368006322243*T5^52 - 184503499891*T5^51 - 156317954676*T5^50 + 2511997936596*T5^49 + 11157616203154*T5^48 + 3993114932922*T5^47 + 105245881717650*T5^46 + 656306263560*T5^45 - 88460025930909*T5^44 + 513819558753660*T5^43 + 4005157529343414*T5^42 + 6309184326332661*T5^41 + 22649634787486365*T5^40 + 2954624543576262*T5^39 - 29419846024018392*T5^38 - 9171402915775050*T5^37 + 433813460584249188*T5^36 + 1888560291864149136*T5^35 + 4592772347896944576*T5^34 + 5926239747186010425*T5^33 + 3122056898493430104*T5^32 + 1353450509390639061*T5^31 + 24048111525966518859*T5^30 + 114270846032905389927*T5^29 + 309167715803977626480*T5^28 + 612236220846820278471*T5^27 + 1081184393082391948539*T5^26 + 2052674465051479464891*T5^25 + 4401747329240689770708*T5^24 + 9513074825609700866643*T5^23 + 18534325472099184858048*T5^22 + 31012000912299486284973*T5^21 + 43991661064989790741014*T5^20 + 52803874018586272387383*T5^19 + 53610725867069264324571*T5^18 + 45982536098859394345575*T5^17 + 33255678535528628244312*T5^16 + 20256847519427857768059*T5^15 + 10441432432426329974088*T5^14 + 4657265190148416049215*T5^13 + 1897192304230850953947*T5^12 + 755897807446795075113*T5^11 + 299448224543806652835*T5^10 + 111973243623721757082*T5^9 + 37508709350223612423*T5^8 + 10891741526108503326*T5^7 + 2615698995761976948*T5^6 + 470855522641846176*T5^5 + 54877934339340318*T5^4 + 3098164136998545*T5^3 + 80420091965631*T5^2 + 780945640767*T5 + 2678994081 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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