LMFDB - Embedded newform 4232.2.a.s.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -0.0681483 q^{5} -2.73205 q^{7} -1.00000 q^{9} +2.73205 q^{11} +0.0352762 q^{13} -0.0963763 q^{15} +1.55051 q^{17} +2.09638 q^{19} -3.86370 q^{21} -4.99536 q^{25} -5.65685 q^{27} -1.93890 q^{29} -9.32780 q^{31} +3.86370 q^{33} +0.186185 q^{35} +9.08516 q^{37} +0.0498881 q^{39} -4.46410 q^{41} +2.02922 q^{43} +0.0681483 q^{45} -0.944060 q^{47} +0.464102 q^{49} +2.19275 q^{51} -12.4670 q^{53} -0.186185 q^{55} +2.96472 q^{57} -9.47067 q^{59} -6.81017 q^{61} +2.73205 q^{63} -0.00240401 q^{65} -3.19151 q^{67} +10.4130 q^{71} +12.9236 q^{73} -7.06450 q^{75} -7.46410 q^{77} -10.1928 q^{79} -5.00000 q^{81} +7.16228 q^{83} -0.105665 q^{85} -2.74202 q^{87} +12.2037 q^{89} -0.0963763 q^{91} -13.1915 q^{93} -0.142865 q^{95} +11.1732 q^{97} -2.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 16 q^{17} + 4 q^{19} + 4 q^{25} - 8 q^{29} - 8 q^{31} + 8 q^{35} + 24 q^{37} - 8 q^{39} - 4 q^{41} + 8 q^{45} - 8 q^{47} - 12 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 8 * q^5 - 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 16 * q^17 + 4 * q^19 + 4 * q^25 - 8 * q^29 - 8 * q^31 + 8 * q^35 + 24 * q^37 - 8 * q^39 - 4 * q^41 + 8 * q^45 - 8 * q^47 - 12 * q^49 - 8 * q^55 + 16 * q^57 - 8 * q^59 + 4 * q^63 + 16 * q^65 + 32 * q^67 - 16 * q^75 - 16 * q^77 - 32 * q^79 - 20 * q^81 - 8 * q^83 - 44 * q^85 + 24 * q^87 + 16 * q^89 + 4 * q^91 - 8 * q^93 - 16 * q^97 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−0.0681483	−0.0304769	−0.0152384	−	0.999884i	$-0.504851\pi$
$5$	−0.0681483	−0.0304769	−0.0152384	+	0.999884i	$0.504851\pi$
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	0.0352762	0.00978385	0.00489193	−	0.999988i	$-0.498443\pi$
$13$	0.0352762	0.00978385	0.00489193	+	0.999988i	$0.498443\pi$
$14$	0	0
$15$	−0.0963763	−0.0248843
$16$	0	0
$17$	1.55051	0.376054	0.188027	−	0.982164i	$-0.439791\pi$
$17$	1.55051	0.376054	0.188027	+	0.982164i	$0.439791\pi$
$18$	0	0
$19$	2.09638	0.480942	0.240471	−	0.970656i	$-0.422698\pi$
$19$	2.09638	0.480942	0.240471	+	0.970656i	$0.422698\pi$
$20$	0	0
$21$	−3.86370	−0.843129
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.99536	−0.999071
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−1.93890	−0.360045	−0.180022	−	0.983663i	$-0.557617\pi$
$29$	−1.93890	−0.360045	−0.180022	+	0.983663i	$0.557617\pi$
$30$	0	0
$31$	−9.32780	−1.67532	−0.837662	−	0.546190i	$-0.816078\pi$
$31$	−9.32780	−1.67532	−0.837662	+	0.546190i	$0.816078\pi$
$32$	0	0
$33$	3.86370	0.672584
$34$	0	0
$35$	0.186185	0.0314710
$36$	0	0
$37$	9.08516	1.49359	0.746796	−	0.665053i	$-0.231591\pi$
$37$	9.08516	1.49359	0.746796	+	0.665053i	$0.231591\pi$
$38$	0	0
$39$	0.0498881	0.00798848
$40$	0	0
$41$	−4.46410	−0.697176	−0.348588	−	0.937276i	$-0.613339\pi$
$41$	−4.46410	−0.697176	−0.348588	+	0.937276i	$0.613339\pi$
$42$	0	0
$43$	2.02922	0.309454	0.154727	−	0.987957i	$-0.450550\pi$
$43$	2.02922	0.309454	0.154727	+	0.987957i	$0.450550\pi$
$44$	0	0
$45$	0.0681483	0.0101590
$46$	0	0
$47$	−0.944060	−0.137705	−0.0688526	−	0.997627i	$-0.521934\pi$
$47$	−0.944060	−0.137705	−0.0688526	+	0.997627i	$0.521934\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.19275	0.307047
$52$	0	0
$53$	−12.4670	−1.71248	−0.856239	−	0.516581i	$-0.827205\pi$
$53$	−12.4670	−1.71248	−0.856239	+	0.516581i	$0.827205\pi$
$54$	0	0
$55$	−0.186185	−0.0251051
$56$	0	0
$57$	2.96472	0.392687
$58$	0	0
$59$	−9.47067	−1.23298	−0.616488	−	0.787364i	$-0.711445\pi$
$59$	−9.47067	−1.23298	−0.616488	+	0.787364i	$0.711445\pi$
$60$	0	0
$61$	−6.81017	−0.871953	−0.435976	−	0.899958i	$-0.643597\pi$
$61$	−6.81017	−0.871953	−0.435976	+	0.899958i	$0.643597\pi$
$62$	0	0
$63$	2.73205	0.344206
$64$	0	0
$65$	−0.00240401	−0.000298181 0
$66$	0	0
$67$	−3.19151	−0.389905	−0.194952	−	0.980813i	$-0.562455\pi$
$67$	−3.19151	−0.389905	−0.194952	+	0.980813i	$0.562455\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.4130	1.23579	0.617896	−	0.786260i	$-0.287985\pi$
$71$	10.4130	1.23579	0.617896	+	0.786260i	$0.287985\pi$
$72$	0	0
$73$	12.9236	1.51259	0.756294	−	0.654232i	$-0.227008\pi$
$73$	12.9236	1.51259	0.756294	+	0.654232i	$0.227008\pi$
$74$	0	0
$75$	−7.06450	−0.815738
$76$	0	0
$77$	−7.46410	−0.850613
$78$	0	0
$79$	−10.1928	−1.14677	−0.573387	−	0.819285i	$-0.694371\pi$
$79$	−10.1928	−1.14677	−0.573387	+	0.819285i	$0.694371\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	7.16228	0.786163	0.393081	−	0.919504i	$-0.371409\pi$
$83$	7.16228	0.786163	0.393081	+	0.919504i	$0.371409\pi$
$84$	0	0
$85$	−0.105665	−0.0114609
$86$	0	0
$87$	−2.74202	−0.293975
$88$	0	0
$89$	12.2037	1.29359	0.646796	−	0.762663i	$-0.276109\pi$
$89$	12.2037	1.29359	0.646796	+	0.762663i	$0.276109\pi$
$90$	0	0
$91$	−0.0963763	−0.0101030
$92$	0	0
$93$	−13.1915	−1.36790
$94$	0	0
$95$	−0.142865	−0.0146576
$96$	0	0
$97$	11.1732	1.13447	0.567236	−	0.823555i	$-0.308013\pi$
$97$	11.1732	1.13447	0.567236	+	0.823555i	$0.308013\pi$
$98$	0	0
$99$	−2.73205	−0.274581

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.s.1.4	✓	4
4.3	odd	2		8464.2.a.bl.1.2		4
23.22	odd	2		4232.2.a.u.1.3	yes	4
92.91	even	2		8464.2.a.bn.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.s.1.4	✓	4	1.1	even	1	trivial
4232.2.a.u.1.3	yes	4	23.22	odd	2
8464.2.a.bl.1.2		4	4.3	odd	2
8464.2.a.bn.1.1		4	92.91	even	2

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:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -0.0681483 q^{5} -2.73205 q^{7} -1.00000 q^{9} +2.73205 q^{11} +0.0352762 q^{13} -0.0963763 q^{15} +1.55051 q^{17} +2.09638 q^{19} -3.86370 q^{21} -4.99536 q^{25} -5.65685 q^{27} -1.93890 q^{29} -9.32780 q^{31} +3.86370 q^{33} +0.186185 q^{35} +9.08516 q^{37} +0.0498881 q^{39} -4.46410 q^{41} +2.02922 q^{43} +0.0681483 q^{45} -0.944060 q^{47} +0.464102 q^{49} +2.19275 q^{51} -12.4670 q^{53} -0.186185 q^{55} +2.96472 q^{57} -9.47067 q^{59} -6.81017 q^{61} +2.73205 q^{63} -0.00240401 q^{65} -3.19151 q^{67} +10.4130 q^{71} +12.9236 q^{73} -7.06450 q^{75} -7.46410 q^{77} -10.1928 q^{79} -5.00000 q^{81} +7.16228 q^{83} -0.105665 q^{85} -2.74202 q^{87} +12.2037 q^{89} -0.0963763 q^{91} -13.1915 q^{93} -0.142865 q^{95} +11.1732 q^{97} -2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 16 q^{17} + 4 q^{19} + 4 q^{25} - 8 q^{29} - 8 q^{31} + 8 q^{35} + 24 q^{37} - 8 q^{39} - 4 q^{41} + 8 q^{45} - 8 q^{47} - 12 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 8 * q^5 - 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 16 * q^17 + 4 * q^19 + 4 * q^25 - 8 * q^29 - 8 * q^31 + 8 * q^35 + 24 * q^37 - 8 * q^39 - 4 * q^41 + 8 * q^45 - 8 * q^47 - 12 * q^49 - 8 * q^55 + 16 * q^57 - 8 * q^59 + 4 * q^63 + 16 * q^65 + 32 * q^67 - 16 * q^75 - 16 * q^77 - 32 * q^79 - 20 * q^81 - 8 * q^83 - 44 * q^85 + 24 * q^87 + 16 * q^89 + 4 * q^91 - 8 * q^93 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more