LMFDB - Newform orbit 510.2.bi.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [510,2,Mod(37,510)] 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("510.37"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(510, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.bi (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [80] 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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$10$ over $\Q(\zeta_{16})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 80 q + 48 q^{25} + 16 q^{28} + 32 q^{29} + 16 q^{30} + 16 q^{31} - 16 q^{33} - 32 q^{34} - 32 q^{35} + 48 q^{37} + 32 q^{41} + 32 q^{43} + 16 q^{44} - 64 q^{49} + 48 q^{50} - 32 q^{51} + 16 q^{52} - 32 q^{53}+ \cdots - 144 q^{98}+O(q^{100}) $

80 * q + 48 * q^25 + 16 * q^28 + 32 * q^29 + 16 * q^30 + 16 * q^31 - 16 * q^33 - 32 * q^34 - 32 * q^35 + 48 * q^37 + 32 * q^41 + 32 * q^43 + 16 * q^44 - 64 * q^49 + 48 * q^50 - 32 * q^51 + 16 * q^52 - 32 * q^53 + 16 * q^55 - 16 * q^56 + 32 * q^57 + 32 * q^58 - 16 * q^60 + 48 * q^61 + 32 * q^62 - 32 * q^65 - 16 * q^67 - 16 * q^70 - 80 * q^72 - 80 * q^73 - 32 * q^74 - 48 * q^77 - 32 * q^78 - 64 * q^79 - 16 * q^80 - 32 * q^82 - 32 * q^83 - 32 * q^85 - 48 * q^87 - 16 * q^88 - 32 * q^89 - 32 * q^92 - 48 * q^93 - 16 * q^94 - 96 * q^95 + 80 * q^97 - 144 * q^98

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} $
37.1	−0.923880	−	0.382683i	−0.831470	+	0.555570i	0.707107	+	0.707107i	−1.99873	−	1.00254i	0.980785	−	0.195090i	0.843659	−	0.167814i	−0.382683	−	0.923880i	0.382683	−	0.923880i	1.46293	+	1.69111i
37.2	−0.923880	−	0.382683i	−0.831470	+	0.555570i	0.707107	+	0.707107i	−0.0618065	+	2.23521i	0.980785	−	0.195090i	2.25037	−	0.447626i	−0.382683	−	0.923880i	0.382683	−	0.923880i	0.912481	−	2.04142i
37.3	−0.923880	−	0.382683i	−0.831470	+	0.555570i	0.707107	+	0.707107i	0.375191	+	2.20437i	0.980785	−	0.195090i	−2.75876	+	0.548752i	−0.382683	−	0.923880i	0.382683	−	0.923880i	0.496944	−	2.18015i
37.4	−0.923880	−	0.382683i	−0.831470	+	0.555570i	0.707107	+	0.707107i	1.78276	−	1.34973i	0.980785	−	0.195090i	−1.91059	+	0.380040i	−0.382683	−	0.923880i	0.382683	−	0.923880i	−2.16357	+	0.564761i
37.5	−0.923880	−	0.382683i	−0.831470	+	0.555570i	0.707107	+	0.707107i	2.23247	+	0.126850i	0.980785	−	0.195090i	4.29610	−	0.854548i	−0.382683	−	0.923880i	0.382683	−	0.923880i	−2.01399	−	0.971522i
37.6	−0.923880	−	0.382683i	0.831470	−	0.555570i	0.707107	+	0.707107i	−2.17616	+	0.514138i	−0.980785	+	0.195090i	2.04622	−	0.407018i	−0.382683	−	0.923880i	0.382683	−	0.923880i	2.20726	+	0.357778i
37.7	−0.923880	−	0.382683i	0.831470	−	0.555570i	0.707107	+	0.707107i	−0.995069	−	2.00246i	−0.980785	+	0.195090i	1.13886	−	0.226533i	−0.382683	−	0.923880i	0.382683	−	0.923880i	0.153017	+	2.23083i
37.8	−0.923880	−	0.382683i	0.831470	−	0.555570i	0.707107	+	0.707107i	−0.921394	+	2.03741i	−0.980785	+	0.195090i	−2.36711	+	0.470848i	−0.382683	−	0.923880i	0.382683	−	0.923880i	1.63094	−	1.52972i
37.9	−0.923880	−	0.382683i	0.831470	−	0.555570i	0.707107	+	0.707107i	1.80447	+	1.32056i	−0.980785	+	0.195090i	−2.13851	+	0.425376i	−0.382683	−	0.923880i	0.382683	−	0.923880i	−1.16176	−	1.91058i
37.10	−0.923880	−	0.382683i	0.831470	−	0.555570i	0.707107	+	0.707107i	1.80603	−	1.31843i	−0.980785	+	0.195090i	4.04132	−	0.803869i	−0.382683	−	0.923880i	0.382683	−	0.923880i	−2.17309	+	0.526937i
97.1	−0.382683	+	0.923880i	−0.195090	+	0.980785i	−0.707107	−	0.707107i	−2.18813	+	0.460513i	−0.831470	−	0.555570i	−2.55505	−	1.70723i	0.923880	−	0.382683i	−0.923880	−	0.382683i	0.411904	−	2.19780i
97.2	−0.382683	+	0.923880i	−0.195090	+	0.980785i	−0.707107	−	0.707107i	−1.85360	−	1.25066i	−0.831470	−	0.555570i	3.56871	+	2.38454i	0.923880	−	0.382683i	−0.923880	−	0.382683i	1.86480	−	1.23390i
97.3	−0.382683	+	0.923880i	−0.195090	+	0.980785i	−0.707107	−	0.707107i	−1.29477	+	1.82307i	−0.831470	−	0.555570i	0.403469	+	0.269590i	0.923880	−	0.382683i	−0.923880	−	0.382683i	−1.18881	−	1.89387i
97.4	−0.382683	+	0.923880i	−0.195090	+	0.980785i	−0.707107	−	0.707107i	1.15835	−	1.91265i	−0.831470	−	0.555570i	−0.539781	−	0.360670i	0.923880	−	0.382683i	−0.923880	−	0.382683i	1.32378	+	1.80212i
97.5	−0.382683	+	0.923880i	−0.195090	+	0.980785i	−0.707107	−	0.707107i	1.91712	+	1.15094i	−0.831470	−	0.555570i	−2.83276	−	1.89279i	0.923880	−	0.382683i	−0.923880	−	0.382683i	−1.79698	+	1.33074i
97.6	−0.382683	+	0.923880i	0.195090	−	0.980785i	−0.707107	−	0.707107i	−1.46240	−	1.69157i	0.831470	+	0.555570i	−1.38078	−	0.922607i	0.923880	−	0.382683i	−0.923880	−	0.382683i	2.12244	−	0.703742i
97.7	−0.382683	+	0.923880i	0.195090	−	0.980785i	−0.707107	−	0.707107i	−0.107286	+	2.23349i	0.831470	+	0.555570i	−2.05486	−	1.37301i	0.923880	−	0.382683i	−0.923880	−	0.382683i	−2.02242	−	0.953840i
97.8	−0.382683	+	0.923880i	0.195090	−	0.980785i	−0.707107	−	0.707107i	1.01463	−	1.99262i	0.831470	+	0.555570i	3.37479	+	2.25496i	0.923880	−	0.382683i	−0.923880	−	0.382683i	1.45266	+	1.69993i
97.9	−0.382683	+	0.923880i	0.195090	−	0.980785i	−0.707107	−	0.707107i	1.38580	+	1.75487i	0.831470	+	0.555570i	1.76005	+	1.17603i	0.923880	−	0.382683i	−0.923880	−	0.382683i	−2.15161	+	0.608751i
97.10	−0.382683	+	0.923880i	0.195090	−	0.980785i	−0.707107	−	0.707107i	2.19567	−	0.423144i	0.831470	+	0.555570i	−3.65460	−	2.44193i	0.923880	−	0.382683i	−0.923880	−	0.382683i	−0.449311	+	2.19046i

See all 80 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.r	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	510.2.bi.b	yes	80
5.c	odd	4	1		510.2.bd.b	✓	80
17.e	odd	16	1		510.2.bd.b	✓	80
85.r	even	16	1	inner	510.2.bi.b	yes	80

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.bd.b	✓	80	5.c	odd	4	1
510.2.bd.b	✓	80	17.e	odd	16	1
510.2.bi.b	yes	80	1.a	even	1	1	trivial
510.2.bi.b	yes	80	85.r	even	16	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{80} + 32 T_{7}^{78} - 160 T_{7}^{77} + 384 T_{7}^{76} - 4768 T_{7}^{75} + 18256 T_{7}^{74} + \cdots + 43\!\cdots\!96 $ T7^80 + 32*T7^78 - 160*T7^77 + 384*T7^76 - 4768*T7^75 + 18256*T7^74 - 52896*T7^73 + 470528*T7^72 - 1484000*T7^71 + 3997088*T7^70 - 36864384*T7^69 + 93322288*T7^68 - 103276288*T7^67 + 1891064384*T7^66 - 1094044928*T7^65 + 5039039040*T7^64 - 20832740736*T7^63 + 162614943936*T7^62 - 96936454144*T7^61 - 1117109335168*T7^60 - 2232534103552*T7^59 + 13321913743616*T7^58 + 55146791281664*T7^57 - 63596322903936*T7^56 - 546773930703360*T7^55 - 3058865024816128*T7^54 - 6232571864450048*T7^53 + 39531337925690368*T7^52 + 373203016704899072*T7^51 + 2251346995182051328*T7^50 + 7861439693690074112*T7^49 + 16925715013164986368*T7^48 + 53858663294369762304*T7^47 + 234412362366769222656*T7^46 + 210458408214307328000*T7^45 - 1769713410228366908928*T7^44 - 3986789010962558668800*T7^43 + 4623757570107252901888*T7^42 - 4274867692097715945472*T7^41 - 106098051861146801881088*T7^40 + 54711250369732232613888*T7^39 + 865748456635178394814464*T7^38 - 236220085174758653337600*T7^37 - 2099258788723434863702016*T7^36 + 9768779748081400115552256*T7^35 + 4951189816373902893006848*T7^34 - 81977344294715398165168128*T7^33 - 15830671770924271881157632*T7^32 + 422726641797072871339212800*T7^31 + 122404392599034900057751552*T7^30 - 1378306998819223993947357184*T7^29 - 80920920561698191626518528*T7^28 + 3058171466906935469009010688*T7^27 - 5876882619890613151056658432*T7^26 - 9590098371777526736962453504*T7^25 + 32422699816367081833321594880*T7^24 + 27958164162222423888955179008*T7^23 - 75171694067482139488371867648*T7^22 + 24100458379536005437932961792*T7^21 + 290096399139939793297316511744*T7^20 - 70249844783769638938458718208*T7^19 - 759599876945713165592362483712*T7^18 + 23400535317426515708504702976*T7^17 + 1070188601040505171673872596992*T7^16 - 457724546280067694191662071808*T7^15 - 905193082639670379035018395648*T7^14 + 1794401315188174545352730345472*T7^13 - 121695676833429778605939884032*T7^12 - 2758854387055031828186927202304*T7^11 + 3802002879219899842652165636096*T7^10 - 2740522735849951847432625061888*T7^9 + 945568270972826866494537203712*T7^8 - 55396259042164095008343326720*T7^7 + 140610741983044718266805125120*T7^6 - 362801858213034674494048829440*T7^5 + 350985584361133473953178714112*T7^4 - 153928918294089918830413348864*T7^3 + 30543772201840572630233513984*T7^2 - 682656155150569459884752896*T7 + 432876024591433999681847296 acting on $S_{2}^{\mathrm{new}}(510, [\chi])$.

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:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.bi (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 37.10
Significant digits:
Format:

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