LMFDB - Embedded newform 4232.2.a.r.1.3

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,-2,0,0,0,0] 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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{10 +2 \sqrt{17}})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.662153$ 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)

$f(q)$	$=$	$q+1.56155 q^{3} -1.32431 q^{5} -4.71659 q^{7} -0.561553 q^{9} -1.32431 q^{11} +0.438447 q^{13} -2.06798 q^{15} +4.71659 q^{17} -6.04090 q^{19} -7.36520 q^{21} -3.24621 q^{25} -5.56155 q^{27} -0.438447 q^{29} +5.56155 q^{31} -2.06798 q^{33} +6.24621 q^{35} +8.10887 q^{37} +0.684658 q^{39} +10.6847 q^{41} -8.68951 q^{43} +0.743668 q^{45} +11.8078 q^{47} +15.2462 q^{49} +7.36520 q^{51} +6.04090 q^{53} +1.75379 q^{55} -9.43318 q^{57} +4.00000 q^{59} +3.39228 q^{61} +2.64861 q^{63} -0.580639 q^{65} +1.32431 q^{67} -2.43845 q^{71} -4.43845 q^{73} -5.06913 q^{75} +6.24621 q^{77} +2.64861 q^{79} -7.00000 q^{81} -13.4061 q^{83} -6.24621 q^{85} -0.684658 q^{87} +14.7304 q^{89} -2.06798 q^{91} +8.68466 q^{93} +8.00000 q^{95} -14.1498 q^{97} +0.743668 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 6 q^{9} + 10 q^{13} + 20 q^{25} - 14 q^{27} - 10 q^{29} + 14 q^{31} - 8 q^{35} - 22 q^{39} + 18 q^{41} + 6 q^{47} + 28 q^{49} + 40 q^{55} + 16 q^{59} - 18 q^{71} - 26 q^{73} - 78 q^{75}+ \cdots + 32 q^{95}+O(q^{100}) $ 4 * q - 2 * q^3 + 6 * q^9 + 10 * q^13 + 20 * q^25 - 14 * q^27 - 10 * q^29 + 14 * q^31 - 8 * q^35 - 22 * q^39 + 18 * q^41 + 6 * q^47 + 28 * q^49 + 40 * q^55 + 16 * q^59 - 18 * q^71 - 26 * q^73 - 78 * q^75 - 8 * q^77 - 28 * q^81 + 8 * q^85 + 22 * q^87 + 10 * q^93 + 32 * q^95

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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.32431	−0.592248	−0.296124	−	0.955149i	$-0.595694\pi$
$5$	−1.32431	−0.592248	−0.296124	+	0.955149i	$0.595694\pi$
$6$	0	0
$7$	−4.71659	−1.78270	−0.891352	−	0.453313i	$-0.850242\pi$
$7$	−4.71659	−1.78270	−0.891352	+	0.453313i	$0.850242\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−1.32431	−0.399294	−0.199647	−	0.979868i	$-0.563979\pi$
$11$	−1.32431	−0.399294	−0.199647	+	0.979868i	$0.563979\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	−2.06798	−0.533949
$16$	0	0
$17$	4.71659	1.14394	0.571970	−	0.820274i	$-0.306179\pi$
$17$	4.71659	1.14394	0.571970	+	0.820274i	$0.306179\pi$
$18$	0	0
$19$	−6.04090	−1.38588	−0.692938	−	0.720997i	$-0.743684\pi$
$19$	−6.04090	−1.38588	−0.692938	+	0.720997i	$0.743684\pi$
$20$	0	0
$21$	−7.36520	−1.60722
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.24621	−0.649242
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−0.438447	−0.0814176	−0.0407088	−	0.999171i	$-0.512962\pi$
$29$	−0.438447	−0.0814176	−0.0407088	+	0.999171i	$0.512962\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	−2.06798	−0.359988
$34$	0	0
$35$	6.24621	1.05580
$36$	0	0
$37$	8.10887	1.33309	0.666545	−	0.745465i	$-0.267772\pi$
$37$	8.10887	1.33309	0.666545	+	0.745465i	$0.267772\pi$
$38$	0	0
$39$	0.684658	0.109633
$40$	0	0
$41$	10.6847	1.66866	0.834332	−	0.551263i	$-0.185854\pi$
$41$	10.6847	1.66866	0.834332	+	0.551263i	$0.185854\pi$
$42$	0	0
$43$	−8.68951	−1.32514	−0.662569	−	0.749001i	$-0.730534\pi$
$43$	−8.68951	−1.32514	−0.662569	+	0.749001i	$0.730534\pi$
$44$	0	0
$45$	0.743668	0.110860
$46$	0	0
$47$	11.8078	1.72234	0.861170	−	0.508318i	$-0.169732\pi$
$47$	11.8078	1.72234	0.861170	+	0.508318i	$0.169732\pi$
$48$	0	0
$49$	15.2462	2.17803
$50$	0	0
$51$	7.36520	1.03133
$52$	0	0
$53$	6.04090	0.829781	0.414890	−	0.909871i	$-0.363820\pi$
$53$	6.04090	0.829781	0.414890	+	0.909871i	$0.363820\pi$
$54$	0	0
$55$	1.75379	0.236481
$56$	0	0
$57$	−9.43318	−1.24945
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	3.39228	0.434337	0.217169	−	0.976134i	$-0.430318\pi$
$61$	3.39228	0.434337	0.217169	+	0.976134i	$0.430318\pi$
$62$	0	0
$63$	2.64861	0.333694
$64$	0	0
$65$	−0.580639	−0.0720194
$66$	0	0
$67$	1.32431	0.161790	0.0808949	−	0.996723i	$-0.474222\pi$
$67$	1.32431	0.161790	0.0808949	+	0.996723i	$0.474222\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.43845	−0.289390	−0.144695	−	0.989476i	$-0.546220\pi$
$71$	−2.43845	−0.289390	−0.144695	+	0.989476i	$0.546220\pi$
$72$	0	0
$73$	−4.43845	−0.519481	−0.259740	−	0.965678i	$-0.583637\pi$
$73$	−4.43845	−0.519481	−0.259740	+	0.965678i	$0.583637\pi$
$74$	0	0
$75$	−5.06913	−0.585333
$76$	0	0
$77$	6.24621	0.711822
$78$	0	0
$79$	2.64861	0.297992	0.148996	−	0.988838i	$-0.452396\pi$
$79$	2.64861	0.297992	0.148996	+	0.988838i	$0.452396\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−13.4061	−1.47151	−0.735755	−	0.677248i	$-0.763173\pi$
$83$	−13.4061	−1.47151	−0.735755	+	0.677248i	$0.763173\pi$
$84$	0	0
$85$	−6.24621	−0.677497
$86$	0	0
$87$	−0.684658	−0.0734031
$88$	0	0
$89$	14.7304	1.56142	0.780710	−	0.624894i	$-0.214858\pi$
$89$	14.7304	1.56142	0.780710	+	0.624894i	$0.214858\pi$
$90$	0	0
$91$	−2.06798	−0.216783
$92$	0	0
$93$	8.68466	0.900557
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−14.1498	−1.43669	−0.718346	−	0.695686i	$-0.755100\pi$
$97$	−14.1498	−1.43669	−0.718346	+	0.695686i	$0.755100\pi$
$98$	0	0
$99$	0.743668	0.0747415

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.r.1.3	✓	4
4.3	odd	2		8464.2.a.bo.1.1		4
23.22	odd	2	inner	4232.2.a.r.1.4	yes	4
92.91	even	2		8464.2.a.bo.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.r.1.3	✓	4	1.1	even	1	trivial
4232.2.a.r.1.4	yes	4	23.22	odd	2	inner
8464.2.a.bo.1.1		4	4.3	odd	2
8464.2.a.bo.1.2		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)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.32431 q^{5} -4.71659 q^{7} -0.561553 q^{9} -1.32431 q^{11} +0.438447 q^{13} -2.06798 q^{15} +4.71659 q^{17} -6.04090 q^{19} -7.36520 q^{21} -3.24621 q^{25} -5.56155 q^{27} -0.438447 q^{29} +5.56155 q^{31} -2.06798 q^{33} +6.24621 q^{35} +8.10887 q^{37} +0.684658 q^{39} +10.6847 q^{41} -8.68951 q^{43} +0.743668 q^{45} +11.8078 q^{47} +15.2462 q^{49} +7.36520 q^{51} +6.04090 q^{53} +1.75379 q^{55} -9.43318 q^{57} +4.00000 q^{59} +3.39228 q^{61} +2.64861 q^{63} -0.580639 q^{65} +1.32431 q^{67} -2.43845 q^{71} -4.43845 q^{73} -5.06913 q^{75} +6.24621 q^{77} +2.64861 q^{79} -7.00000 q^{81} -13.4061 q^{83} -6.24621 q^{85} -0.684658 q^{87} +14.7304 q^{89} -2.06798 q^{91} +8.68466 q^{93} +8.00000 q^{95} -14.1498 q^{97} +0.743668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 6 q^{9} + 10 q^{13} + 20 q^{25} - 14 q^{27} - 10 q^{29} + 14 q^{31} - 8 q^{35} - 22 q^{39} + 18 q^{41} + 6 q^{47} + 28 q^{49} + 40 q^{55} + 16 q^{59} - 18 q^{71} - 26 q^{73} - 78 q^{75}+ \cdots + 32 q^{95}+O(q^{100}) \) 4 * q - 2 * q^3 + 6 * q^9 + 10 * q^13 + 20 * q^25 - 14 * q^27 - 10 * q^29 + 14 * q^31 - 8 * q^35 - 22 * q^39 + 18 * q^41 + 6 * q^47 + 28 * q^49 + 40 * q^55 + 16 * q^59 - 18 * q^71 - 26 * q^73 - 78 * q^75 - 8 * q^77 - 28 * q^81 + 8 * q^85 + 22 * q^87 + 10 * q^93 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more