LMFDB - Newform orbit 97.3.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [97,3,Mod(5,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(96)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97.5"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 97 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	97.l (of order $96$, degree $32$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.64305856429$
Analytic rank:	$0$
Dimension:	$480$
Relative dimension:	$15$ over $\Q(\zeta_{96})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{96}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 480 q - 32 q^{2} - 32 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 32 q^{8} + 64 q^{9} - 32 q^{10} - 32 q^{11} - 224 q^{12} - 32 q^{13} - 32 q^{14} + 224 q^{15} - 32 q^{16} + 48 q^{17} - 32 q^{18}+ \cdots - 512 q^{99}+O(q^{100}) $

480 * q - 32 * q^2 - 32 * q^3 - 32 * q^4 - 32 * q^5 - 32 * q^6 - 32 * q^7 - 32 * q^8 + 64 * q^9 - 32 * q^10 - 32 * q^11 - 224 * q^12 - 32 * q^13 - 32 * q^14 + 224 * q^15 - 32 * q^16 + 48 * q^17 - 32 * q^18 - 32 * q^19 - 416 * q^20 - 32 * q^21 - 32 * q^22 - 144 * q^23 - 32 * q^24 + 288 * q^25 + 48 * q^26 - 32 * q^27 - 272 * q^28 + 256 * q^29 - 32 * q^30 - 32 * q^31 - 32 * q^32 + 144 * q^33 - 432 * q^34 - 16 * q^35 - 48 * q^36 - 272 * q^37 - 32 * q^38 - 384 * q^39 + 544 * q^40 - 32 * q^41 - 32 * q^42 - 32 * q^43 + 496 * q^44 - 240 * q^45 - 32 * q^46 - 800 * q^47 - 912 * q^48 - 32 * q^49 - 32 * q^50 + 160 * q^51 + 352 * q^52 - 32 * q^53 + 1376 * q^54 - 352 * q^55 - 32 * q^56 - 32 * q^57 - 32 * q^58 - 544 * q^59 + 1408 * q^60 - 16 * q^61 - 48 * q^62 - 288 * q^63 - 32 * q^64 + 544 * q^65 - 1696 * q^66 - 32 * q^67 + 1696 * q^68 - 32 * q^69 - 1728 * q^70 + 16 * q^71 - 96 * q^72 - 176 * q^73 - 752 * q^74 - 32 * q^75 + 2368 * q^76 - 32 * q^77 - 32 * q^78 + 1776 * q^79 + 1760 * q^80 - 160 * q^81 + 80 * q^82 + 384 * q^83 + 2272 * q^84 + 944 * q^85 + 1888 * q^86 + 1760 * q^87 + 2272 * q^88 + 640 * q^89 + 2240 * q^90 + 928 * q^91 - 624 * q^92 + 784 * q^93 + 224 * q^94 - 448 * q^95 - 192 * q^97 - 768 * q^98 - 512 * 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} $
5.1	−1.65450	−	3.35499i	−2.33421	+	2.66165i	−6.08354	+	7.92822i	6.93452	+	0.227013i	12.7918	+	3.42754i	−3.26636	−	5.25273i	21.9887	+	4.37383i	−0.461140	−	3.50270i	−10.7115	−	23.6408i
5.2	−1.40167	−	2.84231i	2.77755	−	3.16719i	−3.67900	+	4.79457i	−0.883234	−	0.0289141i	−12.8954	−	3.45530i	−2.96741	−	4.77199i	6.35145	+	1.26338i	−1.14157	−	8.67110i	1.15582	+	2.55096i
5.3	−1.37439	−	2.78699i	−0.762503	+	0.869468i	−3.44331	+	4.48741i	−5.14179	−	0.168325i	3.47117	+	0.930098i	3.28413	+	5.28130i	5.04784	+	1.00408i	1.00017	+	7.59706i	6.59770	+	14.5614i
5.4	−0.892161	−	1.80912i	1.10269	−	1.25737i	−0.0419330	+	0.0546482i	9.36675	+	0.306636i	−3.25852	−	0.873117i	2.51256	+	4.04053i	−7.77728	−	1.54700i	0.809667	+	6.15003i	−7.80191	−	17.2192i
5.5	−0.696622	−	1.41261i	−3.05902	+	3.48815i	0.924862	−	1.20530i	0.338209	+	0.0110718i	7.05838	+	1.89129i	−3.69079	−	5.93527i	−8.52600	−	1.69593i	−1.63482	−	12.4177i	−0.219963	−	0.485470i
5.6	−0.390169	−	0.791184i	−2.36070	+	2.69186i	1.96131	−	2.55602i	0.944383	+	0.0309159i	3.05082	+	0.817466i	5.11796	+	8.23036i	−6.24836	−	1.24288i	−0.498478	−	3.78632i	−0.344009	−	0.759243i
5.7	−0.324869	−	0.658770i	0.314308	−	0.358400i	2.10661	−	2.74539i	−9.68590	−	0.317084i	−0.338212	−	0.0906237i	−4.51165	−	7.25532i	−5.37457	−	1.06907i	1.14507	+	8.69771i	2.93777	+	6.48379i
5.8	−0.294635	−	0.597460i	2.48200	−	2.83018i	2.16490	−	2.82135i	−0.576421	−	0.0188701i	−2.42220	−	0.649027i	1.69403	+	2.72423i	−4.93693	−	0.982017i	−0.674853	−	5.12602i	0.158559	+	0.349948i
5.9	0.353009	+	0.715832i	−0.0724022	+	0.0825589i	2.04725	−	2.66802i	4.63816	+	0.151838i	−0.0846570	−	0.0226838i	−6.73702	−	10.8340i	5.76378	+	1.14649i	1.17316	+	8.91105i	1.52862	+	3.37374i
5.10	0.661299	+	1.34098i	−0.943730	+	1.07612i	1.07413	−	1.39983i	3.07311	+	0.100603i	−2.06714	−	0.553889i	2.69693	+	4.33701i	8.45325	+	1.68146i	0.907332	+	6.89187i	1.89734	+	4.18752i
5.11	0.846508	+	1.71655i	−2.95510	+	3.36964i	0.205082	−	0.267268i	−8.04921	−	0.263504i	−8.28567	−	2.22014i	−0.649793	−	1.04495i	8.14099	+	1.61934i	−1.44715	−	10.9922i	−6.36141	−	14.0399i
5.12	0.907123	+	1.83946i	2.64934	−	3.02099i	−0.125706	+	0.163823i	−3.20791	−	0.105016i	7.96027	+	2.13295i	1.56170	+	2.51141i	7.63088	+	1.51788i	−0.932656	−	7.08423i	−2.71680	−	5.99610i
5.13	1.42932	+	2.89837i	−3.72788	+	4.25083i	−3.92256	+	5.11198i	8.57096	+	0.280584i	−17.6488	−	4.72899i	−0.656059	−	1.05503i	−7.74484	−	1.54054i	−2.99775	−	22.7702i	11.4374	+	25.2429i
5.14	1.52675	+	3.09595i	−0.360809	+	0.411424i	−4.81886	+	6.28006i	−3.75239	−	0.122840i	−1.82461	−	0.488903i	0.812792	+	1.30708i	−13.2575	−	2.63708i	1.13565	+	8.62611i	−5.34866	−	11.8047i
5.15	1.53926	+	3.12131i	2.56943	−	2.92987i	−4.93820	+	6.43558i	6.28191	+	0.205648i	13.1000	+	3.51014i	−4.41226	−	7.09549i	−14.0352	−	2.79178i	−0.807446	−	6.13316i	9.02759	+	19.9243i
7.1	−3.72531	+	0.244169i	1.40357	−	4.13477i	9.85250	−	1.29711i	1.47402	+	2.37041i	−4.21913	+	15.7460i	8.21761	+	0.269017i	−21.7406	+	4.32448i	−7.98615	−	6.12799i	−6.06995	−	8.47060i
7.2	−3.50846	+	0.229957i	−0.807400	+	2.37852i	8.29065	−	1.09148i	−3.05573	−	4.91402i	2.28577	−	8.53062i	−4.50388	−	0.147442i	−15.0427	+	2.99218i	2.13470	+	1.63802i	11.8509	+	16.5380i
7.3	−2.91985	+	0.191377i	−1.27371	+	3.75224i	4.52310	−	0.595478i	4.36164	+	7.01409i	3.00096	−	11.1997i	0.0597225	+	0.00195511i	−1.61326	+	0.320898i	−5.31678	−	4.07971i	−14.0777	−	19.6453i
7.4	−2.57378	+	0.168694i	0.766859	−	2.25909i	2.63009	−	0.346257i	−1.40224	−	2.25499i	−1.59263	+	5.94377i	−6.95923	−	0.227822i	3.40811	−	0.677915i	2.62474	+	2.01404i	3.98946	+	5.56729i
7.5	−1.75983	+	0.115345i	0.716397	−	2.11044i	−0.882081	+	0.116128i	3.67912	+	5.91650i	−1.01731	+	3.79665i	−3.93366	−	0.128775i	8.45779	−	1.68236i	3.19945	+	2.45503i	−7.15707	−	9.98768i

See next 80 embeddings (of 480 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
97.l	odd	96	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	97.3.l.a	✓	480
97.l	odd	96	1	inner	97.3.l.a	✓	480

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
97.3.l.a	✓	480	1.a	even	1	1	trivial
97.3.l.a	✓	480	97.l	odd	96	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(97, [\chi])$.

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:	\( N \)	\(=\)	\( 97 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	97.l (of order \(96\), degree \(32\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.15
Significant digits:
Format:

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