LMFDB - Newform orbit 242.2.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [242,2,Mod(23,242)] 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("242.23"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([14])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	242.e (of order $11$, degree $10$, minimal)

Newform invariants

comment:select newform

sage:traces = [50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.93237972891$
Analytic rank:	$0$
Dimension:	$50$
Relative dimension:	$5$ over $\Q(\zeta_{11})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 50 q + 5 q^{2} - 10 q^{3} - 5 q^{4} + q^{5} - q^{6} + 5 q^{8} + 40 q^{9} - q^{10} + 11 q^{11} + q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} - 5 q^{16} - 2 q^{17} + 4 q^{18} - q^{19} + q^{20} - 2 q^{21}+ \cdots - 22 q^{99}+O(q^{100}) $

50 * q + 5 * q^2 - 10 * q^3 - 5 * q^4 + q^5 - q^6 + 5 * q^8 + 40 * q^9 - q^10 + 11 * q^11 + q^12 + 32 * q^13 + 11 * q^14 + 24 * q^15 - 5 * q^16 - 2 * q^17 + 4 * q^18 - q^19 + q^20 - 2 * q^21 + 22 * q^22 - 31 * q^23 - q^24 - 2 * q^25 + q^26 - 106 * q^27 - 3 * q^29 - 13 * q^30 - 18 * q^31 + 5 * q^32 + 2 * q^34 - 4 * q^35 - 4 * q^36 - 17 * q^37 + 34 * q^38 - 103 * q^39 + 10 * q^40 - 6 * q^41 - 42 * q^42 - 5 * q^43 - 11 * q^44 - 39 * q^45 - 46 * q^46 - 7 * q^47 + q^48 + 43 * q^49 + 2 * q^50 + 25 * q^51 - q^52 + 25 * q^53 - 4 * q^54 + 22 * q^55 - 11 * q^56 - 41 * q^57 - 8 * q^58 - 30 * q^59 - 31 * q^60 - 40 * q^61 - 4 * q^62 + 45 * q^63 - 5 * q^64 - 43 * q^65 - 55 * q^66 + 64 * q^67 - 2 * q^68 + 8 * q^69 + 15 * q^70 - 5 * q^71 + 4 * q^72 - 34 * q^73 - 5 * q^74 + 11 * q^75 + 10 * q^76 - 22 * q^77 + 4 * q^78 + 47 * q^79 + q^80 + 18 * q^81 + 28 * q^82 - 9 * q^83 - 46 * q^84 + 100 * q^85 + 5 * q^86 - 87 * q^87 + 11 * q^88 - 3 * q^89 + 28 * q^90 + q^91 + 46 * q^92 + 47 * q^93 + 18 * q^94 - 129 * q^95 - q^96 + 3 * q^97 + 45 * q^98 - 22 * 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} $
23.1	−0.415415	+	0.909632i	−3.35440	−0.654861	−	0.755750i	−2.65093	−	0.778384i	1.39347	−	3.05127i	0.220513	+	1.53370i	0.959493	−	0.281733i	8.25199	1.80928	−	2.08802i
23.2	−0.415415	+	0.909632i	−2.04457	−0.654861	−	0.755750i	2.69702	+	0.791917i	0.849346	−	1.85981i	−0.325139	−	2.26139i	0.959493	−	0.281733i	1.18028	−1.84074	+	2.12432i
23.3	−0.415415	+	0.909632i	0.647458	−0.654861	−	0.755750i	−2.94126	−	0.863632i	−0.268964	+	0.588948i	−0.524581	−	3.64854i	0.959493	−	0.281733i	−2.58080	2.00743	−	2.31670i
23.4	−0.415415	+	0.909632i	1.72431	−0.654861	−	0.755750i	0.126082	+	0.0370210i	−0.716304	+	1.56849i	0.366718	+	2.55058i	0.959493	−	0.281733i	−0.0267559	−0.0860519	+	0.0993091i
23.5	−0.415415	+	0.909632i	2.02720	−0.654861	−	0.755750i	3.72858	+	1.09481i	−0.842131	+	1.84401i	−0.174034	−	1.21043i	0.959493	−	0.281733i	1.10956	−2.54478	+	2.93684i
45.1	0.654861	+	0.755750i	−2.99061	−0.142315	+	0.989821i	−2.11935	−	1.36203i	−1.95843	−	2.26015i	4.22943	−	1.24187i	−0.841254	+	0.540641i	5.94373	−0.358531	−	2.49364i
45.2	0.654861	+	0.755750i	−1.95439	−0.142315	+	0.989821i	0.0295064	+	0.0189626i	−1.27985	−	1.47703i	−3.22164	+	0.945958i	−0.841254	+	0.540641i	0.819623	0.00499160	+	0.0347174i
45.3	0.654861	+	0.755750i	−0.774435	−0.142315	+	0.989821i	1.60144	+	1.02918i	−0.507147	−	0.585279i	2.49752	−	0.733338i	−0.841254	+	0.540641i	−2.40025	0.270915	+	1.88425i
45.4	0.654861	+	0.755750i	2.35545	−0.142315	+	0.989821i	−0.952611	−	0.612206i	1.54249	+	1.78013i	1.97851	−	0.580944i	−0.841254	+	0.540641i	2.54813	−0.161153	−	1.12084i
45.5	0.654861	+	0.755750i	2.36398	−0.142315	+	0.989821i	0.599770	+	0.385449i	1.54808	+	1.78658i	−3.73462	+	1.09658i	−0.841254	+	0.540641i	2.58841	0.101463	+	0.705691i
67.1	0.959493	−	0.281733i	−3.22135	0.841254	−	0.540641i	0.201689	+	0.232762i	−3.09087	+	0.907560i	1.49651	+	3.27690i	0.654861	−	0.755750i	7.37712	0.259096	+	0.166511i
67.2	0.959493	−	0.281733i	−0.585904	0.841254	−	0.540641i	2.26003	+	2.60821i	−0.562171	+	0.165068i	0.767980	+	1.68164i	0.654861	−	0.755750i	−2.65672	2.90330	+	1.86584i
67.3	0.959493	−	0.281733i	−0.494695	0.841254	−	0.540641i	−1.78488	−	2.05986i	−0.474657	+	0.139372i	−1.73849	−	3.80675i	0.654861	−	0.755750i	−2.75528	−2.29291	−	1.47356i
67.4	0.959493	−	0.281733i	1.23115	0.841254	−	0.540641i	0.845962	+	0.976293i	1.18128	−	0.346856i	−0.388652	−	0.851030i	0.654861	−	0.755750i	−1.48426	1.08675	+	0.698411i
67.5	0.959493	−	0.281733i	2.07080	0.841254	−	0.540641i	−0.867939	−	1.00166i	1.98692	−	0.583412i	0.558619	+	1.22321i	0.654861	−	0.755750i	1.28821	−1.11498	−	0.716555i
89.1	0.142315	−	0.989821i	−2.14993	−0.959493	−	0.281733i	1.09234	+	2.39190i	−0.305966	+	2.12804i	1.56763	−	1.00746i	−0.415415	+	0.909632i	1.62218	2.52301	−	0.740822i
89.2	0.142315	−	0.989821i	−1.81479	−0.959493	−	0.281733i	−0.307404	−	0.673122i	−0.258272	+	1.79632i	−1.40327	+	0.901829i	−0.415415	+	0.909632i	0.293474	−0.710018	+	0.208480i
89.3	0.142315	−	0.989821i	−0.123120	−0.959493	−	0.281733i	−1.36045	−	2.97896i	−0.0175218	+	0.121867i	−2.84598	+	1.82900i	−0.415415	+	0.909632i	−2.98484	−3.14225	+	0.922649i
89.4	0.142315	−	0.989821i	1.14558	−0.959493	−	0.281733i	−0.915212	−	2.00403i	0.163033	−	1.13392i	2.37710	−	1.52767i	−0.415415	+	0.909632i	−1.68765	−2.11388	+	0.620692i
89.5	0.142315	−	0.989821i	1.94226	−0.959493	−	0.281733i	1.07530	+	2.35459i	0.276413	−	1.92249i	2.15045	−	1.38201i	−0.415415	+	0.909632i	0.772376	2.48365	−	0.729267i

See all 50 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
121.e	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	242.2.e.a	✓	50
121.e	even	11	1	inner	242.2.e.a	✓	50

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
242.2.e.a	✓	50	1.a	even	1	1	trivial
242.2.e.a	✓	50	121.e	even	11	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{25} + 5 T_{3}^{24} - 35 T_{3}^{23} - 189 T_{3}^{22} + 539 T_{3}^{21} + 3114 T_{3}^{20} + \cdots + 131 $ T3^25 + 5*T3^24 - 35*T3^23 - 189*T3^22 + 539*T3^21 + 3114*T3^20 - 4848*T3^19 - 29285*T3^18 + 28489*T3^17 + 172898*T3^16 - 115246*T3^15 - 664279*T3^14 + 325175*T3^13 + 1667088*T3^12 - 623793*T3^11 - 2676794*T3^10 + 757141*T3^9 + 2630748*T3^8 - 505002*T3^7 - 1474602*T3^6 + 136885*T3^5 + 418407*T3^4 + 935*T3^3 - 46275*T3^2 - 3937*T3 + 131 acting on $S_{2}^{\mathrm{new}}(242, [\chi])$.

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Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.e (of order \(11\), degree \(10\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 23.5
Significant digits:
Format:

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