LMFDB - Newform orbit 273.2.bt.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [273,2,Mod(136,273)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 2, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("273.136"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$273 = 3 \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	273.bt (of order $12$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [40] 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:	$2.17991597518$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$10$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$ -expansion

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

\operatorname{Tr}(f)(q) =

40 q - 2 q^{7} + 20 q^{9} - 8 q^{11} + 24 q^{12} - 18 q^{14} - 64 q^{16} + 8 q^{17} - 14 q^{19} - 14 q^{20} - 8 q^{21} + 4 q^{22} + 18 q^{24} - 24 q^{25} - 10 q^{26} - 2 q^{28} + 8 q^{29} - 8 q^{31} + 10 q^{32}+ \cdots - 4 q^{99}+O(q^{100})

40 * q - 2 * q^7 + 20 * q^9 - 8 * q^11 + 24 * q^12 - 18 * q^14 - 64 * q^16 + 8 * q^17 - 14 * q^19 - 14 * q^20 - 8 * q^21 + 4 * q^22 + 18 * q^24 - 24 * q^25 - 10 * q^26 - 2 * q^28 + 8 * q^29 - 8 * q^31 + 10 * q^32 - 8 * q^33 + 24 * q^34 - 22 * q^35 + 12 * q^37 + 8 * q^38 - 24 * q^39 - 30 * q^40 + 2 * q^41 + 6 * q^42 - 66 * q^43 + 28 * q^44 + 40 * q^46 + 10 * q^47 - 24 * q^48 + 38 * q^49 - 20 * q^50 + 40 * q^52 - 8 * q^53 + 42 * q^55 + 20 * q^56 - 14 * q^57 - 48 * q^58 - 26 * q^59 + 14 * q^60 - 12 * q^61 - 24 * q^62 - 4 * q^63 - 44 * q^65 - 18 * q^66 + 46 * q^67 - 4 * q^69 + 32 * q^70 - 6 * q^71 - 18 * q^72 + 10 * q^73 + 40 * q^74 + 48 * q^75 + 64 * q^76 - 24 * q^77 + 8 * q^78 + 34 * q^80 - 20 * q^81 + 24 * q^82 + 12 * q^83 + 20 * q^84 + 2 * q^85 + 12 * q^86 - 84 * q^88 - 16 * q^89 + 26 * q^91 + 236 * q^92 + 22 * q^93 + 30 * q^94 + 26 * q^96 + 62 * q^97 - 14 * q^98 - 4 * 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	$a_{2}$			$a_{3}$					$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
136.1	−1.75053	+	1.75053i	−0.866025	+	0.500000i	−	4.12868i	−0.286571	+	1.06950i	0.640737	−	2.39126i	1.02891	−	2.43749i	3.72630	+	3.72630i	0.500000	−	0.866025i	−1.37053	−	2.37383i
136.2	−1.59737	+	1.59737i	−0.866025	+	0.500000i	−	3.10321i	0.609466	−	2.27456i	0.584679	−	2.18205i	1.40839	+	2.23974i	1.76223	+	1.76223i	0.500000	−	0.866025i	2.65977	+	4.60686i
136.3	−1.03264	+	1.03264i	−0.866025	+	0.500000i	−	0.132693i	0.456824	−	1.70489i	0.377973	−	1.41061i	−2.58707	+	0.554121i	−1.92826	−	1.92826i	0.500000	−	0.866025i	1.28880	+	2.23227i
136.4	−0.783932	+	0.783932i	−0.866025	+	0.500000i		0.770902i	−0.994915	+	3.71307i	0.286939	−	1.07087i	0.0389254	+	2.64546i	−2.17220	−	2.17220i	0.500000	−	0.866025i	−2.13085	−	3.69074i
136.5	−0.326747	+	0.326747i	−0.866025	+	0.500000i		1.78647i	−0.562238	+	2.09830i	0.119598	−	0.446344i	2.25993	−	1.37576i	−1.23722	−	1.23722i	0.500000	−	0.866025i	−0.501904	−	0.869322i
136.6	−0.184060	+	0.184060i	−0.866025	+	0.500000i		1.93224i	0.900201	−	3.35960i	0.0673706	−	0.251431i	0.939961	−	2.47315i	−0.723769	−	0.723769i	0.500000	−	0.866025i	0.452676	+	0.784058i
136.7	1.07562	−	1.07562i	−0.866025	+	0.500000i	−	0.313900i	−0.962166	+	3.59085i	−0.393703	+	1.46932i	−2.21486	−	1.44721i	1.81360	+	1.81360i	0.500000	−	0.866025i	2.82746	+	4.89730i
136.8	1.09526	−	1.09526i	−0.866025	+	0.500000i	−	0.399185i	0.128983	−	0.481371i	−0.400893	+	1.49615i	2.51963	+	0.807124i	1.75331	+	1.75331i	0.500000	−	0.866025i	−0.385956	−	0.668496i
136.9	1.57115	−	1.57115i	−0.866025	+	0.500000i	−	2.93701i	0.922649	−	3.44337i	−0.575080	+	2.14623i	−2.64488	+	0.0678683i	−1.47218	−	1.47218i	0.500000	−	0.866025i	−3.96043	−	6.85967i
136.10	1.93326	−	1.93326i	−0.866025	+	0.500000i	−	5.47495i	−0.212233	+	0.792066i	−0.707621	+	2.64088i	2.21517	−	1.44673i	−6.71797	−	6.71797i	0.500000	−	0.866025i	1.12096	+	1.94157i
145.1	−1.96855	+	1.96855i	0.866025	+	0.500000i	−	5.75037i	−1.75415	+	0.470023i	−2.68909	+	0.720539i	2.27503	+	1.35065i	7.38279	+	7.38279i	0.500000	+	0.866025i	2.52787	−	4.37840i
145.2	−1.65256	+	1.65256i	0.866025	+	0.500000i	−	3.46192i	2.69306	−	0.721604i	−2.25744	+	0.604879i	−2.54381	−	0.727358i	2.41591	+	2.41591i	0.500000	+	0.866025i	−3.25795	+	5.64294i
145.3	−1.22934	+	1.22934i	0.866025	+	0.500000i	−	1.02253i	−1.95691	+	0.524353i	−1.67930	+	0.449968i	−1.82011	−	1.92021i	−1.20163	−	1.20163i	0.500000	+	0.866025i	1.76110	−	3.05031i
145.4	−0.507981	+	0.507981i	0.866025	+	0.500000i		1.48391i	1.10096	−	0.295002i	−0.693915	+	0.185934i	0.718657	+	2.54628i	−1.76976	−	1.76976i	0.500000	+	0.866025i	−0.409414	+	0.709125i
145.5	−0.240784	+	0.240784i	0.866025	+	0.500000i		1.88405i	2.06336	−	0.552877i	−0.328916	+	0.0881329i	1.35855	−	2.27032i	−0.935214	−	0.935214i	0.500000	+	0.866025i	−0.363700	+	0.629948i
145.6	0.465913	−	0.465913i	0.866025	+	0.500000i		1.56585i	−3.81958	+	1.02345i	0.636449	−	0.170536i	−2.61148	+	0.424440i	1.66138	+	1.66138i	0.500000	+	0.866025i	−1.30275	+	2.25643i
145.7	0.837153	−	0.837153i	0.866025	+	0.500000i		0.598351i	−1.58368	+	0.424345i	1.14357	−	0.306419i	2.46703	+	0.955901i	2.17522	+	2.17522i	0.500000	+	0.866025i	−0.970539	+	1.68102i
145.8	0.884731	−	0.884731i	0.866025	+	0.500000i		0.434503i	3.68041	−	0.986163i	1.20856	−	0.323834i	−2.53212	−	0.767050i	2.15388	+	2.15388i	0.500000	+	0.866025i	2.38368	−	4.12866i
145.9	1.55234	−	1.55234i	0.866025	+	0.500000i	−	2.81950i	−0.926472	+	0.248247i	2.12053	−	0.568195i	0.619609	−	2.57218i	−1.27214	−	1.27214i	0.500000	+	0.866025i	−1.05283	+	1.82356i
145.10	1.85908	−	1.85908i	0.866025	+	0.500000i	−	4.91234i	0.502992	−	0.134776i	2.53955	−	0.680470i	−1.89546	+	1.84587i	−5.41427	−	5.41427i	0.500000	+	0.866025i	0.684542	−	1.18566i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.ba	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	273.2.bt.b	✓	40
3.b	odd	2	1		819.2.et.d		40
7.d	odd	6	1		273.2.cg.b	yes	40
13.f	odd	12	1		273.2.cg.b	yes	40
21.g	even	6	1		819.2.gh.d		40
39.k	even	12	1		819.2.gh.d		40
91.ba	even	12	1	inner	273.2.bt.b	✓	40
273.bs	odd	12	1		819.2.et.d		40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
273.2.bt.b	✓	40	1.a	even	1	1	trivial
273.2.bt.b	✓	40	91.ba	even	12	1	inner
273.2.cg.b	yes	40	7.d	odd	6	1
273.2.cg.b	yes	40	13.f	odd	12	1
819.2.et.d		40	3.b	odd	2	1
819.2.et.d		40	273.bs	odd	12	1
819.2.gh.d		40	21.g	even	6	1
819.2.gh.d		40	39.k	even	12	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{40} + 168 T_{2}^{36} - 2 T_{2}^{35} + 8 T_{2}^{33} + 10786 T_{2}^{32} - 308 T_{2}^{31} + \cdots + 59049$ T2^40 + 168*T2^36 - 2*T2^35 + 8*T2^33 + 10786*T2^32 - 308*T2^31 + 2*T2^30 + 1192*T2^29 + 335772*T2^28 - 18394*T2^27 + 312*T2^26 + 78396*T2^25 + 5319379*T2^24 - 434822*T2^23 + 19768*T2^22 + 2099310*T2^21 + 41488543*T2^20 - 3569838*T2^19 + 514796*T2^18 + 22154080*T2^17 + 152167654*T2^16 - 2900766*T2^15 + 5373984*T2^14 + 84950940*T2^13 + 242842212*T2^12 + 41355270*T2^11 + 20420874*T2^10 + 99595710*T2^9 + 155258613*T2^8 + 69922008*T2^7 + 23363478*T2^6 + 20070342*T2^5 + 21521619*T2^4 + 11795220*T2^3 + 3792258*T2^2 + 669222*T2 + 59049 acting on $S_{2}^{\mathrm{new}}(273, [\chi])$ .

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$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 136.10
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