LMFDB - Embedded newform 936.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,4,Mod(1,936)] 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("936.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-12,0,44] 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:	$55.2257877654$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.55744$ of defining polynomial
Character	$\chi$	$=$	936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-19.1149 q^{5} +35.1149 q^{7} -26.0000 q^{11} -13.0000 q^{13} +36.2298 q^{17} -95.5744 q^{19} +161.379 q^{23} +240.379 q^{25} +91.3785 q^{29} -266.723 q^{31} -671.217 q^{35} -149.608 q^{37} +77.8041 q^{41} +183.608 q^{43} +60.6893 q^{47} +890.055 q^{49} -281.540 q^{53} +496.987 q^{55} +542.527 q^{59} +65.0810 q^{61} +248.493 q^{65} -1033.94 q^{67} -1041.81 q^{71} +483.311 q^{73} -912.987 q^{77} -1337.05 q^{79} -812.825 q^{83} -692.527 q^{85} -936.885 q^{89} -456.493 q^{91} +1826.89 q^{95} -954.068 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} + 44 q^{7} - 52 q^{11} - 26 q^{13} + 20 q^{17} - 60 q^{19} + 8 q^{23} + 166 q^{25} - 132 q^{29} - 140 q^{31} - 608 q^{35} + 68 q^{37} - 28 q^{41} - 36 q^{47} + 626 q^{49} - 668 q^{53} + 312 q^{55}+ \cdots - 1436 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 + 44 * q^7 - 52 * q^11 - 26 * q^13 + 20 * q^17 - 60 * q^19 + 8 * q^23 + 166 * q^25 - 132 * q^29 - 140 * q^31 - 608 * q^35 + 68 * q^37 - 28 * q^41 - 36 * q^47 + 626 * q^49 - 668 * q^53 + 312 * q^55 + 508 * q^59 + 340 * q^61 + 156 * q^65 - 940 * q^67 - 300 * q^71 + 1124 * q^73 - 1144 * q^77 - 1520 * q^79 - 524 * q^83 - 808 * q^85 - 1900 * q^89 - 572 * q^91 + 2080 * q^95 - 1436 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−19.1149	−1.70969	−0.854843	−	0.518886i	$-0.826347\pi$
$5$	−19.1149	−1.70969	−0.854843	+	0.518886i	$0.826347\pi$
$6$	0	0
$7$	35.1149	1.89603	0.948013	−	0.318233i	$-0.103089\pi$
$7$	35.1149	1.89603	0.948013	+	0.318233i	$0.103089\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	36.2298	0.516883	0.258441	−	0.966027i	$-0.416791\pi$
$17$	36.2298	0.516883	0.258441	+	0.966027i	$0.416791\pi$
$18$	0	0
$19$	−95.5744	−1.15401	−0.577007	−	0.816739i	$-0.695779\pi$
$19$	−95.5744	−1.15401	−0.577007	+	0.816739i	$0.695779\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	161.379	1.46303	0.731516	−	0.681824i	$-0.238813\pi$
$23$	161.379	1.46303	0.731516	+	0.681824i	$0.238813\pi$
$24$	0	0
$25$	240.379	1.92303
$26$	0	0
$27$	0	0
$28$	0	0
$29$	91.3785	0.585123	0.292561	−	0.956247i	$-0.405492\pi$
$29$	91.3785	0.585123	0.292561	+	0.956247i	$0.405492\pi$
$30$	0	0
$31$	−266.723	−1.54532	−0.772660	−	0.634821i	$-0.781074\pi$
$31$	−266.723	−1.54532	−0.772660	+	0.634821i	$0.781074\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−671.217	−3.24161
$36$	0	0
$37$	−149.608	−0.664742	−0.332371	−	0.943149i	$-0.607849\pi$
$37$	−149.608	−0.664742	−0.332371	+	0.943149i	$0.607849\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	77.8041	0.296365	0.148183	−	0.988960i	$-0.452658\pi$
$41$	77.8041	0.296365	0.148183	+	0.988960i	$0.452658\pi$
$42$	0	0
$43$	183.608	0.651163	0.325581	−	0.945514i	$-0.394440\pi$
$43$	183.608	0.651163	0.325581	+	0.945514i	$0.394440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	60.6893	0.188350	0.0941749	−	0.995556i	$-0.469979\pi$
$47$	60.6893	0.188350	0.0941749	+	0.995556i	$0.469979\pi$
$48$	0	0
$49$	890.055	2.59491
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−281.540	−0.729671	−0.364835	−	0.931072i	$-0.618875\pi$
$53$	−281.540	−0.729671	−0.364835	+	0.931072i	$0.618875\pi$
$54$	0	0
$55$	496.987	1.21843
$56$	0	0
$57$	0	0
$58$	0	0
$59$	542.527	1.19714	0.598568	−	0.801072i	$-0.295737\pi$
$59$	542.527	1.19714	0.598568	+	0.801072i	$0.295737\pi$
$60$	0	0
$61$	65.0810	0.136603	0.0683014	−	0.997665i	$-0.478242\pi$
$61$	65.0810	0.136603	0.0683014	+	0.997665i	$0.478242\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	248.493	0.474182
$66$	0	0
$67$	−1033.94	−1.88531	−0.942656	−	0.333767i	$-0.891680\pi$
$67$	−1033.94	−1.88531	−0.942656	+	0.333767i	$0.891680\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1041.81	−1.74141	−0.870706	−	0.491803i	$-0.836338\pi$
$71$	−1041.81	−1.74141	−0.870706	+	0.491803i	$0.836338\pi$
$72$	0	0
$73$	483.311	0.774894	0.387447	−	0.921892i	$-0.373357\pi$
$73$	483.311	0.774894	0.387447	+	0.921892i	$0.373357\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−912.987	−1.35123
$78$	0	0
$79$	−1337.05	−1.90418	−0.952091	−	0.305815i	$-0.901071\pi$
$79$	−1337.05	−1.90418	−0.952091	+	0.305815i	$0.901071\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−812.825	−1.07493	−0.537465	−	0.843286i	$-0.680618\pi$
$83$	−812.825	−1.07493	−0.537465	+	0.843286i	$0.680618\pi$
$84$	0	0
$85$	−692.527	−0.883707
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−936.885	−1.11584	−0.557919	−	0.829895i	$-0.688400\pi$
$89$	−936.885	−1.11584	−0.557919	+	0.829895i	$0.688400\pi$
$90$	0	0
$91$	−456.493	−0.525863
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1826.89	1.97300
$96$	0	0
$97$	−954.068	−0.998669	−0.499335	−	0.866409i	$-0.666422\pi$
$97$	−954.068	−0.998669	−0.499335	+	0.866409i	$0.666422\pi$
$98$	0	0
$99$	0	0

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	936.4.a.c.1.1		2
3.2	odd	2		312.4.a.f.1.2	✓	2
4.3	odd	2		1872.4.a.v.1.1		2
12.11	even	2		624.4.a.l.1.2		2
24.5	odd	2		2496.4.a.u.1.1		2
24.11	even	2		2496.4.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.f.1.2	✓	2	3.2	odd	2
624.4.a.l.1.2		2	12.11	even	2
936.4.a.c.1.1		2	1.1	even	1	trivial
1872.4.a.v.1.1		2	4.3	odd	2
2496.4.a.u.1.1		2	24.5	odd	2
2496.4.a.bd.1.1		2	24.11	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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)

\(f(q)\)	\(=\)	\(q-19.1149 q^{5} +35.1149 q^{7} -26.0000 q^{11} -13.0000 q^{13} +36.2298 q^{17} -95.5744 q^{19} +161.379 q^{23} +240.379 q^{25} +91.3785 q^{29} -266.723 q^{31} -671.217 q^{35} -149.608 q^{37} +77.8041 q^{41} +183.608 q^{43} +60.6893 q^{47} +890.055 q^{49} -281.540 q^{53} +496.987 q^{55} +542.527 q^{59} +65.0810 q^{61} +248.493 q^{65} -1033.94 q^{67} -1041.81 q^{71} +483.311 q^{73} -912.987 q^{77} -1337.05 q^{79} -812.825 q^{83} -692.527 q^{85} -936.885 q^{89} -456.493 q^{91} +1826.89 q^{95} -954.068 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} + 44 q^{7} - 52 q^{11} - 26 q^{13} + 20 q^{17} - 60 q^{19} + 8 q^{23} + 166 q^{25} - 132 q^{29} - 140 q^{31} - 608 q^{35} + 68 q^{37} - 28 q^{41} - 36 q^{47} + 626 q^{49} - 668 q^{53} + 312 q^{55}+ \cdots - 1436 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 + 44 * q^7 - 52 * q^11 - 26 * q^13 + 20 * q^17 - 60 * q^19 + 8 * q^23 + 166 * q^25 - 132 * q^29 - 140 * q^31 - 608 * q^35 + 68 * q^37 - 28 * q^41 - 36 * q^47 + 626 * q^49 - 668 * q^53 + 312 * q^55 + 508 * q^59 + 340 * q^61 + 156 * q^65 - 940 * q^67 - 300 * q^71 + 1124 * q^73 - 1144 * q^77 - 1520 * q^79 - 524 * q^83 - 808 * q^85 - 1900 * q^89 - 572 * q^91 + 2080 * q^95 - 1436 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more