LMFDB - Newform orbit 510.2.bd.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [510,2,Mod(7,510)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(510, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 11])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("510.7"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [64] 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:	$64$
Relative dimension:	$8$ 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) = $ $ 64 q - 16 q^{15} + 16 q^{20} + 16 q^{25} - 16 q^{28} + 32 q^{29} + 16 q^{31} + 16 q^{33} + 32 q^{35} - 16 q^{37} - 16 q^{40} + 48 q^{41} + 16 q^{44} + 32 q^{47} - 64 q^{49} - 16 q^{50} + 32 q^{51} + 16 q^{52}+ \cdots - 80 q^{97}+O(q^{100}) $

64 * q - 16 * q^15 + 16 * q^20 + 16 * q^25 - 16 * q^28 + 32 * q^29 + 16 * q^31 + 16 * q^33 + 32 * q^35 - 16 * q^37 - 16 * q^40 + 48 * q^41 + 16 * q^44 + 32 * q^47 - 64 * q^49 - 16 * q^50 + 32 * q^51 + 16 * q^52 + 80 * q^53 - 16 * q^55 + 16 * q^56 - 32 * q^57 + 64 * q^59 - 48 * q^61 + 32 * q^65 + 16 * q^67 - 48 * q^68 - 64 * q^71 + 16 * q^73 + 48 * q^74 - 32 * q^75 - 80 * q^77 - 32 * q^78 - 64 * q^79 - 16 * q^80 - 48 * q^82 - 32 * q^83 - 16 * q^85 - 64 * q^86 - 32 * q^87 - 16 * q^88 - 32 * q^89 - 16 * q^90 - 16 * q^92 - 16 * q^93 - 16 * q^94 - 112 * q^95 - 80 * q^97

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} $
7.1	0.923880	−	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	−2.08794	+	0.800320i	−0.555570	−	0.831470i	2.90113	−	1.93847i	0.382683	−	0.923880i	−0.923880	+	0.382683i	−1.62273	+	1.53842i
7.2	0.923880	−	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	−1.53450	−	1.62644i	−0.555570	−	0.831470i	−4.11436	+	2.74913i	0.382683	−	0.923880i	−0.923880	+	0.382683i	−2.04010	−	0.915410i
7.3	0.923880	−	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	0.602255	−	2.15344i	−0.555570	−	0.831470i	3.47469	−	2.32171i	0.382683	−	0.923880i	−0.923880	+	0.382683i	−0.267673	−	2.21999i
7.4	0.923880	−	0.382683i	−0.195090	−	0.980785i	0.707107	−	0.707107i	2.15266	−	0.605039i	−0.555570	−	0.831470i	−0.954897	+	0.638042i	0.382683	−	0.923880i	−0.923880	+	0.382683i	1.75726	−	1.38277i
7.5	0.923880	−	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	−1.54906	+	1.61258i	0.555570	+	0.831470i	−2.38187	+	1.59151i	0.382683	−	0.923880i	−0.923880	+	0.382683i	−0.814041	+	2.08263i
7.6	0.923880	−	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	0.282288	−	2.21818i	0.555570	+	0.831470i	−1.13986	+	0.761632i	0.382683	−	0.923880i	−0.923880	+	0.382683i	−0.588060	−	2.15736i
7.7	0.923880	−	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	0.583501	+	2.15859i	0.555570	+	0.831470i	2.99482	−	2.00107i	0.382683	−	0.923880i	−0.923880	+	0.382683i	1.36514	+	1.77098i
7.8	0.923880	−	0.382683i	0.195090	+	0.980785i	0.707107	−	0.707107i	2.19965	−	0.401940i	0.555570	+	0.831470i	1.83348	−	1.22509i	0.382683	−	0.923880i	−0.923880	+	0.382683i	1.87839	−	1.21311i
73.1	0.923880	+	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	−2.08794	−	0.800320i	−0.555570	+	0.831470i	2.90113	+	1.93847i	0.382683	+	0.923880i	−0.923880	−	0.382683i	−1.62273	−	1.53842i
73.2	0.923880	+	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	−1.53450	+	1.62644i	−0.555570	+	0.831470i	−4.11436	−	2.74913i	0.382683	+	0.923880i	−0.923880	−	0.382683i	−2.04010	+	0.915410i
73.3	0.923880	+	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	0.602255	+	2.15344i	−0.555570	+	0.831470i	3.47469	+	2.32171i	0.382683	+	0.923880i	−0.923880	−	0.382683i	−0.267673	+	2.21999i
73.4	0.923880	+	0.382683i	−0.195090	+	0.980785i	0.707107	+	0.707107i	2.15266	+	0.605039i	−0.555570	+	0.831470i	−0.954897	−	0.638042i	0.382683	+	0.923880i	−0.923880	−	0.382683i	1.75726	+	1.38277i
73.5	0.923880	+	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	−1.54906	−	1.61258i	0.555570	−	0.831470i	−2.38187	−	1.59151i	0.382683	+	0.923880i	−0.923880	−	0.382683i	−0.814041	−	2.08263i
73.6	0.923880	+	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	0.282288	+	2.21818i	0.555570	−	0.831470i	−1.13986	−	0.761632i	0.382683	+	0.923880i	−0.923880	−	0.382683i	−0.588060	+	2.15736i
73.7	0.923880	+	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	0.583501	−	2.15859i	0.555570	−	0.831470i	2.99482	+	2.00107i	0.382683	+	0.923880i	−0.923880	−	0.382683i	1.36514	−	1.77098i
73.8	0.923880	+	0.382683i	0.195090	−	0.980785i	0.707107	+	0.707107i	2.19965	+	0.401940i	0.555570	−	0.831470i	1.83348	+	1.22509i	0.382683	+	0.923880i	−0.923880	−	0.382683i	1.87839	+	1.21311i
133.1	−0.382683	+	0.923880i	−0.831470	+	0.555570i	−0.707107	−	0.707107i	−2.18876	+	0.457512i	−0.195090	−	0.980785i	2.02243	−	0.402287i	0.923880	−	0.382683i	0.382683	−	0.923880i	0.414918	−	2.19724i
133.2	−0.382683	+	0.923880i	−0.831470	+	0.555570i	−0.707107	−	0.707107i	−1.35939	−	1.77540i	−0.195090	−	0.980785i	−0.304082	+	0.0604858i	0.923880	−	0.382683i	0.382683	−	0.923880i	2.16047	−	0.576500i
133.3	−0.382683	+	0.923880i	−0.831470	+	0.555570i	−0.707107	−	0.707107i	1.40077	+	1.74294i	−0.195090	−	0.980785i	−1.26601	+	0.251826i	0.923880	−	0.382683i	0.382683	−	0.923880i	−2.14632	+	0.627144i
133.4	−0.382683	+	0.923880i	−0.831470	+	0.555570i	−0.707107	−	0.707107i	2.13357	+	0.669244i	−0.195090	−	0.980785i	0.0888617	−	0.0176757i	0.923880	−	0.382683i	0.382683	−	0.923880i	−1.43478	+	1.71505i

See all 64 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
85.o	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.bd.a	✓	64
5.c	odd	4	1		510.2.bi.a	yes	64
17.e	odd	16	1		510.2.bi.a	yes	64
85.o	even	16	1	inner	510.2.bd.a	✓	64

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.bd.a	✓	64	1.a	even	1	1	trivial
510.2.bd.a	✓	64	85.o	even	16	1	inner
510.2.bi.a	yes	64	5.c	odd	4	1
510.2.bi.a	yes	64	17.e	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{64} + 32 T_{7}^{62} - 128 T_{7}^{61} + 896 T_{7}^{60} - 3328 T_{7}^{59} + 29296 T_{7}^{58} + \cdots + 4228120576 $ T7^64 + 32*T7^62 - 128*T7^61 + 896*T7^60 - 3328*T7^59 + 29296*T7^58 - 88096*T7^57 + 585216*T7^56 - 2408992*T7^55 + 11938528*T7^54 - 42560896*T7^53 + 182155312*T7^52 - 495597184*T7^51 + 1490922432*T7^50 - 2934045184*T7^49 + 6662615104*T7^48 - 12975531904*T7^47 + 54986103616*T7^46 - 150859837696*T7^45 + 348657093504*T7^44 - 296388596992*T7^43 - 1014195031296*T7^42 + 10037083793408*T7^41 - 14838386747264*T7^40 - 23822227662336*T7^39 + 106533965433856*T7^38 + 413698660444160*T7^37 + 1117496065027072*T7^36 + 1431801352055808*T7^35 + 9787502161504256*T7^34 + 13523260089011200*T7^33 + 19357242331107328*T7^32 + 38184190541138944*T7^31 + 62786390336109568*T7^30 - 42495538580885504*T7^29 - 102012769628970496*T7^28 + 575487903692152832*T7^27 + 1754895313931712512*T7^26 + 191301749541019648*T7^25 - 8205261478707445760*T7^24 - 21572886156055089152*T7^23 - 20596593125437523968*T7^22 + 45682475513140035584*T7^21 + 250668629024557649920*T7^20 + 630834474132818837504*T7^19 + 1108350980867514220544*T7^18 + 1480570794538785538048*T7^17 + 1542996168649223635968*T7^16 + 1259021141383186038784*T7^15 + 800527689456236429312*T7^14 + 393272572510701977600*T7^13 + 150050706438498041856*T7^12 + 45380924283357495296*T7^11 + 11577697603587276800*T7^10 + 2485252277194719232*T7^9 + 382197132074024960*T7^8 + 47319901377658880*T7^7 + 9624102566100992*T7^6 + 1132521617620992*T7^5 - 149065299656704*T7^4 - 31818644455424*T7^3 + 386208366592*T7^2 + 155008892928*T7 + 4228120576 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.bd (of order \(16\), degree \(8\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.8
Significant digits:
Format:

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