LMFDB - Newform orbit 4752.2.d.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4752,2,Mod(3455,4752)] 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("4752.3455"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4752, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4752 = 2^{4} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4752.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-8,0,4,0,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$37.9449110405$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 49x^{4} + 48x^{2} + 9 $ x^8 + 14x^6 + 49x^4 + 48*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{5} - \beta_{5} q^{7} - q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{7} - \beta_{5} - \beta_1) q^{17} + ( - \beta_{6} + \beta_1) q^{19} + \beta_{2} q^{23} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{25}+ \cdots + (\beta_{3} - 2 \beta_{2} + 2) q^{97}+O(q^{100}) $ q - b1 * q^5 - b5 * q^7 - q^11 + (-b2 + 1) * q^13 + (b7 - b5 - b1) * q^17 + (-b6 + b1) * q^19 + b2 * q^23 + (-b4 - b3 + b2 - 2) * q^25 + (b7 - b6 + b5) * q^29 + (-b7 - b1) * q^31 + (b4 + b3 - 2b2 + 4) q^35 + (-b4 + b2 - 1) * q^37 + (b7 + b6 + b5) * q^41 + (b7 - b6 - b5 - b1) * q^43 + (-b3 - b2 - 1) * q^47 + (-b4 + b2 - 2) * q^49 - 2b5 q^53 + b1 * q^55 + (-b4 - b3 + 2b2 - 1) q^59 + (2b3 - b2 + 1) q^61 + (b7 + 2b5) q^65 + (b7 - 2b6 - 3b1) * q^67 + (-b4 - b3 + b2 - 1) * q^71 + (-2b2 + 4) q^73 + b5 * q^77 + (b7 + b6 + 3b5 + 3b1) * q^79 + (-b4 + b3 + 2b2 - 8) q^83 + (-2b4 + b2 - 3) q^85 + 2b6 q^89 + (-b7 + 2b6 + 3b1) * q^91 + (2b4 + 2b3 - 2b2 + 8) q^95 + (b3 - 2b2 + 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{11} + 4 q^{13} + 4 q^{23} - 16 q^{25} + 28 q^{35} - 4 q^{37} - 16 q^{47} - 12 q^{49} - 4 q^{59} + 12 q^{61} - 8 q^{71} + 24 q^{73} - 52 q^{83} - 20 q^{85} + 64 q^{95} + 12 q^{97}+O(q^{100}) $ 8 * q - 8 * q^11 + 4 * q^13 + 4 * q^23 - 16 * q^25 + 28 * q^35 - 4 * q^37 - 16 * q^47 - 12 * q^49 - 4 * q^59 + 12 * q^61 - 8 * q^71 + 24 * q^73 - 52 * q^83 - 20 * q^85 + 64 * q^95 + 12 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 14x^{6} + 49x^{4} + 48x^{2} + 9 $ x^8 + 14*x^6 + 49*x^4 + 48*x^2 + 9 :

$\beta_{1}$	$=$	$ ( -\nu^{7} - 8\nu^{5} + 26\nu^{3} + 147\nu ) / 27 $ (-v^7 - 8v^5 + 26v^3 + 147*v) / 27
$\beta_{2}$	$=$	$ ( \nu^{6} + 17\nu^{4} + 73\nu^{2} + 51 ) / 9 $ (v^6 + 17v^4 + 73v^2 + 51) / 9
$\beta_{3}$	$=$	$ ( -\nu^{6} - 8\nu^{4} + 26\nu^{2} + 84 ) / 9 $ (-v^6 - 8v^4 + 26v^2 + 84) / 9
$\beta_{4}$	$=$	$ ( -2\nu^{6} - 25\nu^{4} - 65\nu^{2} - 30 ) / 3 $ (-2v^6 - 25v^4 - 65*v^2 - 30) / 3
$\beta_{5}$	$=$	$ ( 5\nu^{7} + 67\nu^{5} + 194\nu^{3} + 48\nu ) / 27 $ (5v^7 + 67v^5 + 194v^3 + 48v) / 27
$\beta_{6}$	$=$	$ ( 7\nu^{7} + 83\nu^{5} + 169\nu^{3} + 24\nu ) / 27 $ (7v^7 + 83v^5 + 169v^3 + 24v) / 27
$\beta_{7}$	$=$	$ ( 20\nu^{7} + 268\nu^{5} + 830\nu^{3} + 570\nu ) / 27 $ (20v^7 + 268v^5 + 830v^3 + 570v) / 27

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{7} + 2\beta_{6} + 2\beta_{5} + 4\beta_1 ) / 6 $ (-b7 + 2b6 + 2b5 + 4*b1) / 6
$\nu^{2}$	$=$	$ ( -\beta_{4} + 3\beta_{3} - 3\beta_{2} - 21 ) / 6 $ (-b4 + 3b3 - 3b2 - 21) / 6
$\nu^{3}$	$=$	$ ( 5\beta_{7} - 7\beta_{6} - 13\beta_{5} - 14\beta_1 ) / 3 $ (5b7 - 7b6 - 13b5 - 14b1) / 3
$\nu^{4}$	$=$	$ ( 11\beta_{4} - 27\beta_{3} + 39\beta_{2} + 141 ) / 6 $ (11b4 - 27b3 + 39*b2 + 141) / 6
$\nu^{5}$	$=$	$ ( -91\beta_{7} + 110\beta_{6} + 260\beta_{5} + 250\beta_1 ) / 6 $ (-91b7 + 110b6 + 260b5 + 250b1) / 6
$\nu^{6}$	$=$	$ -19\beta_{4} + 40\beta_{3} - 65\beta_{2} - 195 $ -19b4 + 40b3 - 65*b2 - 195
$\nu^{7}$	$=$	$ ( 841\beta_{7} - 950\beta_{6} - 2462\beta_{5} - 2302\beta_1 ) / 6 $ (841b7 - 950b6 - 2462b5 - 2302b1) / 6

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4752\mathbb{Z}\right)^\times$.

$n$	$353$	$1189$	$1729$	$4159$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

3455.1

		1.09993i
		0.494171i
		3.04208i
		1.81430i
	−	1.81430i
	−	3.04208i
	−	0.494171i
	−	1.09993i

−

4.30215i

3.97172i

3455.2

−

2.56581i

−

0.0832230i

3455.3

−

1.55553i

−

3.14415i

3455.4

−

0.698863i

2.88666i

3455.5

0.698863i

−

2.88666i

3455.6

1.55553i

3.14415i

3455.7

2.56581i

0.0832230i

3455.8

4.30215i

−

3.97172i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4752.2.d.k	✓	8
3.b	odd	2	1		4752.2.d.l	yes	8
4.b	odd	2	1		4752.2.d.l	yes	8
12.b	even	2	1	inner	4752.2.d.k	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4752.2.d.k	✓	8	1.a	even	1	1	trivial
4752.2.d.k	✓	8	12.b	even	2	1	inner
4752.2.d.l	yes	8	3.b	odd	2	1
4752.2.d.l	yes	8	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4752, [\chi])$:

$ T_{5}^{8} + 28T_{5}^{6} + 196T_{5}^{4} + 384T_{5}^{2} + 144 $

T5^8 + 28*T5^6 + 196*T5^4 + 384*T5^2 + 144

$ T_{23}^{4} - 2T_{23}^{3} - 26T_{23}^{2} + 42T_{23} + 117 $

T23^4 - 2*T23^3 - 26*T23^2 + 42*T23 + 117

$ T_{47}^{4} + 8T_{47}^{3} - 110T_{47}^{2} - 504T_{47} + 81 $

T47^4 + 8*T47^3 - 110*T47^2 - 504*T47 + 81

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 28 T^{6} + \cdots + 144 $ T^8 + 28T^6 + 196T^4 + 384*T^2 + 144
$7$	$ T^{8} + 34 T^{6} + \cdots + 9 $ T^8 + 34T^6 + 370T^4 + 1302*T^2 + 9
$11$	$ (T + 1)^{8} $ (T + 1)^8
$13$	$ (T^{4} - 2 T^{3} + \cdots + 132)^{2} $ (T^4 - 2T^3 - 26T^2 + 12*T + 132)^2
$17$	$ T^{8} + 82 T^{6} + \cdots + 131769 $ T^8 + 82T^6 + 2386T^4 + 29526*T^2 + 131769
$19$	$ T^{8} + 112 T^{6} + \cdots + 11664 $ T^8 + 112T^6 + 3556T^4 + 22896*T^2 + 11664
$23$	$ (T^{4} - 2 T^{3} + \cdots + 117)^{2} $ (T^4 - 2T^3 - 26T^2 + 42*T + 117)^2
$29$	$ T^{8} + 150 T^{6} + \cdots + 6561 $ T^8 + 150T^6 + 6570T^4 + 75330*T^2 + 6561
$31$	$ T^{8} + 100 T^{6} + \cdots + 144 $ T^8 + 100T^6 + 772T^4 + 960*T^2 + 144
$37$	$ (T^{4} + 2 T^{3} - 62 T^{2} + \cdots - 39)^{2} $ (T^4 + 2T^3 - 62T^2 - 138*T - 39)^2
$41$	$ T^{8} + 214 T^{6} + \cdots + 4198401 $ T^8 + 214T^6 + 15418T^4 + 445074*T^2 + 4198401
$43$	$ T^{8} + 102 T^{6} + \cdots + 6561 $ T^8 + 102T^6 + 2682T^4 + 20898*T^2 + 6561
$47$	$ (T^{4} + 8 T^{3} - 110 T^{2} + \cdots + 81)^{2} $ (T^4 + 8T^3 - 110T^2 - 504*T + 81)^2
$53$	$ T^{8} + 136 T^{6} + \cdots + 2304 $ T^8 + 136T^6 + 5920T^4 + 83328*T^2 + 2304
$59$	$ (T^{4} + 2 T^{3} + \cdots - 351)^{2} $ (T^4 + 2T^3 - 134T^2 + 486*T - 351)^2
$61$	$ (T^{4} - 6 T^{3} + \cdots + 15172)^{2} $ (T^4 - 6T^3 - 278T^2 + 1212*T + 15172)^2
$67$	$ T^{8} + 468 T^{6} + \cdots + 156816 $ T^8 + 468T^6 + 65700T^4 + 2897856*T^2 + 156816
$71$	$ (T^{4} + 4 T^{3} + \cdots + 144)^{2} $ (T^4 + 4T^3 - 92T^2 + 192*T + 144)^2
$73$	$ (T^{4} - 12 T^{3} + \cdots + 1552)^{2} $ (T^4 - 12T^3 - 56T^2 + 432*T + 1552)^2
$79$	$ T^{8} + 486 T^{6} + \cdots + 13549761 $ T^8 + 486T^6 + 67050T^4 + 1945458*T^2 + 13549761
$83$	$ (T^{4} + 26 T^{3} + \cdots - 27612)^{2} $ (T^4 + 26T^3 - 26T^2 - 4620*T - 27612)^2
$89$	$ T^{8} + 304 T^{6} + \cdots + 36864 $ T^8 + 304T^6 + 14464T^4 + 181248*T^2 + 36864
$97$	$ (T^{4} - 6 T^{3} + \cdots - 1859)^{2} $ (T^4 - 6T^3 - 122T^2 + 1014*T - 1859)^2
show more
show less

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:	\( N \)	\(=\)	\( 4752 = 2^{4} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4752.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{5} - \beta_{5} q^{7} - q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{7} - \beta_{5} - \beta_1) q^{17} + ( - \beta_{6} + \beta_1) q^{19} + \beta_{2} q^{23} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{25}+ \cdots + (\beta_{3} - 2 \beta_{2} + 2) q^{97}+O(q^{100}) \) q - b1 * q^5 - b5 * q^7 - q^11 + (-b2 + 1) * q^13 + (b7 - b5 - b1) * q^17 + (-b6 + b1) * q^19 + b2 * q^23 + (-b4 - b3 + b2 - 2) * q^25 + (b7 - b6 + b5) * q^29 + (-b7 - b1) * q^31 + (b4 + b3 - 2b2 + 4) q^35 + (-b4 + b2 - 1) * q^37 + (b7 + b6 + b5) * q^41 + (b7 - b6 - b5 - b1) * q^43 + (-b3 - b2 - 1) * q^47 + (-b4 + b2 - 2) * q^49 - 2b5 q^53 + b1 * q^55 + (-b4 - b3 + 2b2 - 1) q^59 + (2b3 - b2 + 1) q^61 + (b7 + 2b5) q^65 + (b7 - 2b6 - 3b1) * q^67 + (-b4 - b3 + b2 - 1) * q^71 + (-2b2 + 4) q^73 + b5 * q^77 + (b7 + b6 + 3b5 + 3b1) * q^79 + (-b4 + b3 + 2b2 - 8) q^83 + (-2b4 + b2 - 3) q^85 + 2b6 q^89 + (-b7 + 2b6 + 3b1) * q^91 + (2b4 + 2b3 - 2b2 + 8) q^95 + (b3 - 2b2 + 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{11} + 4 q^{13} + 4 q^{23} - 16 q^{25} + 28 q^{35} - 4 q^{37} - 16 q^{47} - 12 q^{49} - 4 q^{59} + 12 q^{61} - 8 q^{71} + 24 q^{73} - 52 q^{83} - 20 q^{85} + 64 q^{95} + 12 q^{97}+O(q^{100}) \) 8 * q - 8 * q^11 + 4 * q^13 + 4 * q^23 - 16 * q^25 + 28 * q^35 - 4 * q^37 - 16 * q^47 - 12 * q^49 - 4 * q^59 + 12 * q^61 - 8 * q^71 + 24 * q^73 - 52 * q^83 - 20 * q^85 + 64 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more