LMFDB - Newform orbit 399.2.ck.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,2,Mod(10,399)] 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("399.10"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 3, 17])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.ck (of order $18$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [84] 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:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$84$
Relative dimension:	$14$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 84 q - 6 q^{7} - 12 q^{10} + 6 q^{11} - 96 q^{12} + 27 q^{13} + 9 q^{14} - 39 q^{19} + 6 q^{21} - 12 q^{22} - 6 q^{23} + 9 q^{24} + 18 q^{25} - 42 q^{27} + 45 q^{28} - 6 q^{29} + 18 q^{31} - 51 q^{32} + 12 q^{33}+ \cdots - 6 q^{99}+O(q^{100}) $

84 * q - 6 * q^7 - 12 * q^10 + 6 * q^11 - 96 * q^12 + 27 * q^13 + 9 * q^14 - 39 * q^19 + 6 * q^21 - 12 * q^22 - 6 * q^23 + 9 * q^24 + 18 * q^25 - 42 * q^27 + 45 * q^28 - 6 * q^29 + 18 * q^31 - 51 * q^32 + 12 * q^33 + 6 * q^34 + 15 * q^35 + 18 * q^37 + 3 * q^38 + 18 * q^40 - 45 * q^41 + 15 * q^42 + 15 * q^43 - 36 * q^44 + 9 * q^45 - 24 * q^47 + 18 * q^49 - 27 * q^50 + 18 * q^51 - 84 * q^52 + 9 * q^53 - 30 * q^55 + 75 * q^56 - 6 * q^57 - 18 * q^58 + 24 * q^59 + 21 * q^61 - 45 * q^62 - 12 * q^63 + 84 * q^64 - 9 * q^65 + 12 * q^66 - 45 * q^67 + 12 * q^69 + 42 * q^70 + 42 * q^71 - 9 * q^72 - 72 * q^73 + 18 * q^74 + 57 * q^75 - 96 * q^76 - 51 * q^77 - 15 * q^78 - 42 * q^79 - 78 * q^80 + 24 * q^82 + 9 * q^84 - 48 * q^85 - 9 * q^86 + 108 * q^88 - 48 * q^89 - 21 * q^90 + 27 * q^91 + 6 * q^92 + 21 * q^93 + 30 * q^94 + 60 * q^95 - 21 * q^97 + 51 * q^98 - 6 * 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} $
10.1	−1.70778	−	2.03525i	0.173648	+	0.984808i	−0.878441	+	4.98189i	−0.754725	+	0.133078i	1.70778	−	2.03525i	2.63125	−	0.276653i	7.03780	−	4.06327i	−0.939693	+	0.342020i	1.55975	+	1.30878i
10.2	−1.59239	−	1.89774i	0.173648	+	0.984808i	−0.718403	+	4.07426i	−0.187693	+	0.0330953i	1.59239	−	1.89774i	−2.52570	−	0.787947i	4.58502	−	2.64716i	−0.939693	+	0.342020i	0.361686	+	0.303491i
10.3	−1.18779	−	1.41555i	0.173648	+	0.984808i	−0.245648	+	1.39314i	3.44602	−	0.607626i	1.18779	−	1.41555i	1.07748	−	2.41641i	−0.936773	+	0.540846i	−0.939693	+	0.342020i	−4.95326	−	4.15628i
10.4	−1.01309	−	1.20736i	0.173648	+	0.984808i	−0.0840576	+	0.476714i	0.743295	−	0.131063i	1.01309	−	1.20736i	0.344620	+	2.62321i	−2.06915	+	1.19462i	−0.939693	+	0.342020i	−0.911267	−	0.764644i
10.5	−0.998594	−	1.19008i	0.173648	+	0.984808i	−0.0717992	+	0.407193i	−4.35353	+	0.767645i	0.998594	−	1.19008i	2.61034	−	0.431444i	−2.13451	+	1.23236i	−0.939693	+	0.342020i	5.26097	+	4.41447i
10.6	−0.254619	−	0.303443i	0.173648	+	0.984808i	0.320049	−	1.81509i	−1.66318	+	0.293263i	0.254619	−	0.303443i	0.0963050	−	2.64400i	−1.31836	+	0.761157i	−0.939693	+	0.342020i	0.512465	+	0.430009i
10.7	−0.115373	−	0.137496i	0.173648	+	0.984808i	0.341702	−	1.93789i	−2.49094	+	0.439220i	0.115373	−	0.137496i	−1.76224	+	1.97345i	−0.616759	+	0.356086i	−0.939693	+	0.342020i	0.347778	+	0.291821i
10.8	0.0227955	+	0.0271666i	0.173648	+	0.984808i	0.347078	−	1.96838i	3.34814	−	0.590368i	−0.0227955	+	0.0271666i	2.07064	+	1.64695i	0.122811	−	0.0709048i	−0.939693	+	0.342020i	0.0923610	+	0.0775001i
10.9	0.265093	+	0.315925i	0.173648	+	0.984808i	0.317762	−	1.80212i	0.341437	−	0.0602045i	−0.265093	+	0.315925i	−2.36410	−	1.18787i	1.36789	−	0.789750i	−0.939693	+	0.342020i	0.109533	+	0.0919087i
10.10	0.849174	+	1.01201i	0.173648	+	0.984808i	0.0442365	−	0.250878i	0.224052	−	0.0395064i	−0.849174	+	1.01201i	1.93362	−	1.80586i	2.57963	−	1.48935i	−0.939693	+	0.342020i	0.230240	+	0.193194i
10.11	1.14592	+	1.36566i	0.173648	+	0.984808i	−0.204584	+	1.16026i	1.97843	−	0.348851i	−1.14592	+	1.36566i	−0.992518	+	2.45253i	1.26884	−	0.732568i	−0.939693	+	0.342020i	2.74354	+	2.30211i
10.12	1.27666	+	1.52146i	0.173648	+	0.984808i	−0.337696	+	1.91517i	−4.09501	+	0.722060i	−1.27666	+	1.52146i	−2.58773	−	0.551064i	0.0950963	−	0.0549038i	−0.939693	+	0.342020i	−6.32652	−	5.30858i
10.13	1.62266	+	1.93381i	0.173648	+	0.984808i	−0.759303	+	4.30622i	−1.95594	+	0.344886i	−1.62266	+	1.93381i	2.62508	+	0.330076i	−5.18711	+	2.99478i	−0.939693	+	0.342020i	−3.84078	−	3.22280i
10.14	1.68733	+	2.01088i	0.173648	+	0.984808i	−0.849267	+	4.81643i	3.71390	−	0.654861i	−1.68733	+	2.01088i	−0.919226	−	2.48093i	−6.57161	+	3.79412i	−0.939693	+	0.342020i	7.58342	+	6.36325i
40.1	−1.70778	+	2.03525i	0.173648	−	0.984808i	−0.878441	−	4.98189i	−0.754725	−	0.133078i	1.70778	+	2.03525i	2.63125	+	0.276653i	7.03780	+	4.06327i	−0.939693	−	0.342020i	1.55975	−	1.30878i
40.2	−1.59239	+	1.89774i	0.173648	−	0.984808i	−0.718403	−	4.07426i	−0.187693	−	0.0330953i	1.59239	+	1.89774i	−2.52570	+	0.787947i	4.58502	+	2.64716i	−0.939693	−	0.342020i	0.361686	−	0.303491i
40.3	−1.18779	+	1.41555i	0.173648	−	0.984808i	−0.245648	−	1.39314i	3.44602	+	0.607626i	1.18779	+	1.41555i	1.07748	+	2.41641i	−0.936773	−	0.540846i	−0.939693	−	0.342020i	−4.95326	+	4.15628i
40.4	−1.01309	+	1.20736i	0.173648	−	0.984808i	−0.0840576	−	0.476714i	0.743295	+	0.131063i	1.01309	+	1.20736i	0.344620	−	2.62321i	−2.06915	−	1.19462i	−0.939693	−	0.342020i	−0.911267	+	0.764644i
40.5	−0.998594	+	1.19008i	0.173648	−	0.984808i	−0.0717992	−	0.407193i	−4.35353	−	0.767645i	0.998594	+	1.19008i	2.61034	+	0.431444i	−2.13451	−	1.23236i	−0.939693	−	0.342020i	5.26097	−	4.41447i
40.6	−0.254619	+	0.303443i	0.173648	−	0.984808i	0.320049	+	1.81509i	−1.66318	−	0.293263i	0.254619	+	0.303443i	0.0963050	+	2.64400i	−1.31836	−	0.761157i	−0.939693	−	0.342020i	0.512465	−	0.430009i

See all 84 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.bf	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.2.ck.b	yes	84
7.d	odd	6	1		399.2.cd.b	✓	84
19.f	odd	18	1		399.2.cd.b	✓	84
133.bf	even	18	1	inner	399.2.ck.b	yes	84

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.cd.b	✓	84	7.d	odd	6	1
399.2.cd.b	✓	84	19.f	odd	18	1
399.2.ck.b	yes	84	1.a	even	1	1	trivial
399.2.ck.b	yes	84	133.bf	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{84} + 39 T_{2}^{79} - 682 T_{2}^{78} - 69 T_{2}^{77} + 90 T_{2}^{76} + 39 T_{2}^{74} + \cdots + 263169 $ T2^84 + 39*T2^79 - 682*T2^78 - 69*T2^77 + 90*T2^76 + 39*T2^74 - 25560*T2^73 + 330569*T2^72 - 23385*T2^71 + 42216*T2^70 - 216255*T2^69 + 541710*T2^68 + 8257785*T2^67 - 75063867*T2^66 - 931767*T2^65 - 7192263*T2^64 + 40841919*T2^63 - 105105210*T2^62 - 1216981935*T2^61 + 12112949646*T2^60 - 1997069772*T2^59 - 434812776*T2^58 + 4779980709*T2^57 + 10308853416*T2^56 + 102524591226*T2^55 - 873249587668*T2^54 + 76303958154*T2^53 + 265181627880*T2^52 - 483520124169*T2^51 - 1524231045429*T2^50 - 3515805566643*T2^49 + 44234949101962*T2^48 - 9913938166566*T2^47 - 5114896265967*T2^46 + 21500887584270*T2^45 + 52823477166228*T2^44 + 97521166558785*T2^43 - 932658138996158*T2^42 + 260264689353027*T2^41 - 443257685961714*T2^40 + 22546536540819*T2^39 - 21142267074132*T2^38 - 1242489987272784*T2^37 + 13333171474952912*T2^36 - 8119913771331411*T2^35 + 11647454604647697*T2^34 - 11633093079205908*T2^33 + 17522444518822329*T2^32 - 12962042349893154*T2^31 - 47723571417428443*T2^30 + 34132442970642471*T2^29 - 41731307386050987*T2^28 + 29905254314850690*T2^27 - 22178264009168895*T2^26 + 11273594983320015*T2^25 + 175629004455695617*T2^24 - 94507239515716161*T2^23 - 19366513034453238*T2^22 + 47601587218920168*T2^21 - 73319238841508706*T2^20 - 49009819527532833*T2^19 + 1706588347999417*T2^18 + 30857736035337627*T2^17 + 26288056838116632*T2^16 + 1352069198226075*T2^15 - 3041941826197521*T2^14 + 332185864296264*T2^13 + 621220589590528*T2^12 - 4327823617533*T2^11 - 114977395321416*T2^10 - 18418899137004*T2^9 + 4371363878166*T2^8 + 825401864718*T2^7 + 403438246938*T2^6 + 61572486474*T2^5 + 17752070727*T2^4 + 1019669553*T2^3 + 150257106*T2^2 - 7188669*T2 + 263169 acting on $S_{2}^{\mathrm{new}}(399, [\chi])$.

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:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.ck (of order \(18\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 10.14
Significant digits:
Format:

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