LMFDB - Embedded newform 2320.2.d.c.929.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(929,2320)] 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("2320.929"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-3,0,0,0,-2,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.3
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.c.929.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287i q^{3} +(0.686141 - 2.12819i) q^{5} -5.04868i q^{7} +2.37228 q^{9} -0.627719 q^{11} -4.25639i q^{13} +(1.68614 + 0.543620i) q^{15} -1.58457i q^{17} +4.00000 q^{19} +4.00000 q^{21} +3.46410i q^{23} +(-4.05842 - 2.92048i) q^{25} +4.25639i q^{27} +1.00000 q^{29} +3.37228 q^{31} -0.497333i q^{33} +(-10.7446 - 3.46410i) q^{35} +3.16915i q^{37} +3.37228 q^{39} -4.74456 q^{41} -10.8896i q^{43} +(1.62772 - 5.04868i) q^{45} +10.8896i q^{47} -18.4891 q^{49} +1.25544 q^{51} -4.25639i q^{53} +(-0.430703 + 1.33591i) q^{55} +3.16915i q^{57} -10.7446 q^{59} +6.00000 q^{61} -11.9769i q^{63} +(-9.05842 - 2.92048i) q^{65} +1.87953i q^{67} -2.74456 q^{69} -6.74456 q^{71} +6.92820i q^{73} +(2.31386 - 3.21543i) q^{75} +3.16915i q^{77} +11.3723 q^{79} +3.74456 q^{81} -9.80240i q^{83} +(-3.37228 - 1.08724i) q^{85} +0.792287i q^{87} +0.744563 q^{89} -21.4891 q^{91} +2.67181i q^{93} +(2.74456 - 8.51278i) q^{95} -6.92820i q^{97} -1.48913 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 2 q^{9} - 14 q^{11} + q^{15} + 16 q^{19} + 16 q^{21} + q^{25} + 4 q^{29} + 2 q^{31} - 20 q^{35} + 2 q^{39} + 4 q^{41} + 18 q^{45} - 28 q^{49} + 28 q^{51} + 27 q^{55} - 20 q^{59} + 24 q^{61}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q - 3 * q^5 - 2 * q^9 - 14 * q^11 + q^15 + 16 * q^19 + 16 * q^21 + q^25 + 4 * q^29 + 2 * q^31 - 20 * q^35 + 2 * q^39 + 4 * q^41 + 18 * q^45 - 28 * q^49 + 28 * q^51 + 27 * q^55 - 20 * q^59 + 24 * q^61 - 19 * q^65 + 12 * q^69 - 4 * q^71 + 15 * q^75 + 34 * q^79 - 8 * q^81 - 2 * q^85 - 20 * q^89 - 40 * q^91 - 12 * q^95 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			0.792287i			0.457427i	0.973494	+	0.228714i	$0.0734519\pi$
$3$			0.792287i			0.457427i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	0.686141	−	2.12819i	0.306851	−	0.951757i
$5$	0.686141	−	2.12819i	0.306851	−	0.951757i
$6$		0			0
$7$		−	5.04868i		−	1.90822i	−0.299456	−	0.954110i	$-0.596805\pi$
$7$		−	5.04868i		−	1.90822i	0.299456	−	0.954110i	$-0.403195\pi$
$8$		0			0
$9$	2.37228			0.790760
$10$		0			0
$11$	−0.627719			−0.189264			−0.0946322	−	0.995512i	$-0.530167\pi$
$11$	−0.627719			−0.189264			−0.0946322	+	0.995512i	$0.530167\pi$
$12$		0			0
$13$		−	4.25639i		−	1.18051i	−0.807217	−	0.590255i	$-0.799027\pi$
$13$		−	4.25639i		−	1.18051i	0.807217	−	0.590255i	$-0.200973\pi$
$14$		0			0
$15$	1.68614	+	0.543620i	0.435360	+	0.140362i
$16$		0			0
$17$		−	1.58457i		−	0.384316i	−0.981364	−	0.192158i	$-0.938451\pi$
$17$		−	1.58457i		−	0.384316i	0.981364	−	0.192158i	$-0.0615486\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$		0			0
$25$	−4.05842	−	2.92048i	−0.811684	−	0.584096i
$26$		0			0
$27$			4.25639i			0.819142i
$28$		0			0
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	3.37228			0.605680			0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228			0.605680			0.302840	+	0.953041i	$0.402065\pi$
$32$		0			0
$33$		−	0.497333i		−	0.0865746i
$34$		0			0
$35$	−10.7446	−	3.46410i	−1.81616	−	0.585540i
$36$		0			0
$37$			3.16915i			0.521005i	0.965473	+	0.260502i	$0.0838882\pi$
$37$			3.16915i			0.521005i	−0.965473	+	0.260502i	$0.916112\pi$
$38$		0			0
$39$	3.37228			0.539997
$40$		0			0
$41$	−4.74456			−0.740976			−0.370488	−	0.928837i	$-0.620810\pi$
$41$	−4.74456			−0.740976			−0.370488	+	0.928837i	$0.620810\pi$
$42$		0			0
$43$		−	10.8896i		−	1.66065i	−0.557276	−	0.830327i	$-0.688154\pi$
$43$		−	10.8896i		−	1.66065i	0.557276	−	0.830327i	$-0.311846\pi$
$44$		0			0
$45$	1.62772	−	5.04868i	0.242646	−	0.752612i
$46$		0			0
$47$			10.8896i			1.58842i	0.607645	+	0.794208i	$0.292114\pi$
$47$			10.8896i			1.58842i	−0.607645	+	0.794208i	$0.707886\pi$
$48$		0			0
$49$	−18.4891			−2.64130
$50$		0			0
$51$	1.25544			0.175796
$52$		0			0
$53$		−	4.25639i		−	0.584660i	−0.956318	−	0.292330i	$-0.905569\pi$
$53$		−	4.25639i		−	0.584660i	0.956318	−	0.292330i	$-0.0944306\pi$
$54$		0			0
$55$	−0.430703	+	1.33591i	−0.0580760	+	0.180134i
$56$		0			0
$57$			3.16915i			0.419764i
$58$		0			0
$59$	−10.7446			−1.39882			−0.699411	−	0.714719i	$-0.746554\pi$
$59$	−10.7446			−1.39882			−0.699411	+	0.714719i	$0.746554\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		−	11.9769i		−	1.50894i
$64$		0			0
$65$	−9.05842	−	2.92048i	−1.12356	−	0.362241i
$66$		0			0
$67$			1.87953i			0.229621i	0.993387	+	0.114810i	$0.0366261\pi$
$67$			1.87953i			0.229621i	−0.993387	+	0.114810i	$0.963374\pi$
$68$		0			0
$69$	−2.74456			−0.330407
$70$		0			0
$71$	−6.74456			−0.800432			−0.400216	−	0.916421i	$-0.631065\pi$
$71$	−6.74456			−0.800432			−0.400216	+	0.916421i	$0.631065\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	2.31386	−	3.21543i	0.267181	−	0.371286i
$76$		0			0
$77$			3.16915i			0.361158i
$78$		0			0
$79$	11.3723			1.27948			0.639741	−	0.768591i	$-0.279042\pi$
$79$	11.3723			1.27948			0.639741	+	0.768591i	$0.279042\pi$
$80$		0			0
$81$	3.74456			0.416063
$82$		0			0
$83$		−	9.80240i		−	1.07595i	−0.842960	−	0.537976i	$-0.819189\pi$
$83$		−	9.80240i		−	1.07595i	0.842960	−	0.537976i	$-0.180811\pi$
$84$		0			0
$85$	−3.37228	−	1.08724i	−0.365775	−	0.117928i
$86$		0			0
$87$			0.792287i			0.0849421i
$88$		0			0
$89$	0.744563			0.0789235			0.0394617	−	0.999221i	$-0.487436\pi$
$89$	0.744563			0.0789235			0.0394617	+	0.999221i	$0.487436\pi$
$90$		0			0
$91$	−21.4891			−2.25267
$92$		0			0
$93$			2.67181i			0.277054i
$94$		0			0
$95$	2.74456	−	8.51278i	0.281586	−	0.873393i
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$	−1.48913			−0.149663

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	2320.2.d.c.929.3		4
4.3	odd	2		145.2.b.a.59.3	yes	4
5.4	even	2	inner	2320.2.d.c.929.2		4
12.11	even	2		1305.2.c.e.784.1		4
20.3	even	4		725.2.a.g.1.4		4
20.7	even	4		725.2.a.g.1.1		4
20.19	odd	2		145.2.b.a.59.2	✓	4
60.23	odd	4		6525.2.a.bk.1.2		4
60.47	odd	4		6525.2.a.bk.1.3		4
60.59	even	2		1305.2.c.e.784.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.a.59.2	✓	4	20.19	odd	2
145.2.b.a.59.3	yes	4	4.3	odd	2
725.2.a.g.1.1		4	20.7	even	4
725.2.a.g.1.4		4	20.3	even	4
1305.2.c.e.784.1		4	12.11	even	2
1305.2.c.e.784.3		4	60.59	even	2
2320.2.d.c.929.2		4	5.4	even	2	inner
2320.2.d.c.929.3		4	1.1	even	1	trivial
6525.2.a.bk.1.2		4	60.23	odd	4
6525.2.a.bk.1.3		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.792287i q^{3} +(0.686141 - 2.12819i) q^{5} -5.04868i q^{7} +2.37228 q^{9} -0.627719 q^{11} -4.25639i q^{13} +(1.68614 + 0.543620i) q^{15} -1.58457i q^{17} +4.00000 q^{19} +4.00000 q^{21} +3.46410i q^{23} +(-4.05842 - 2.92048i) q^{25} +4.25639i q^{27} +1.00000 q^{29} +3.37228 q^{31} -0.497333i q^{33} +(-10.7446 - 3.46410i) q^{35} +3.16915i q^{37} +3.37228 q^{39} -4.74456 q^{41} -10.8896i q^{43} +(1.62772 - 5.04868i) q^{45} +10.8896i q^{47} -18.4891 q^{49} +1.25544 q^{51} -4.25639i q^{53} +(-0.430703 + 1.33591i) q^{55} +3.16915i q^{57} -10.7446 q^{59} +6.00000 q^{61} -11.9769i q^{63} +(-9.05842 - 2.92048i) q^{65} +1.87953i q^{67} -2.74456 q^{69} -6.74456 q^{71} +6.92820i q^{73} +(2.31386 - 3.21543i) q^{75} +3.16915i q^{77} +11.3723 q^{79} +3.74456 q^{81} -9.80240i q^{83} +(-3.37228 - 1.08724i) q^{85} +0.792287i q^{87} +0.744563 q^{89} -21.4891 q^{91} +2.67181i q^{93} +(2.74456 - 8.51278i) q^{95} -6.92820i q^{97} -1.48913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 2 q^{9} - 14 q^{11} + q^{15} + 16 q^{19} + 16 q^{21} + q^{25} + 4 q^{29} + 2 q^{31} - 20 q^{35} + 2 q^{39} + 4 q^{41} + 18 q^{45} - 28 q^{49} + 28 q^{51} + 27 q^{55} - 20 q^{59} + 24 q^{61}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q - 3 * q^5 - 2 * q^9 - 14 * q^11 + q^15 + 16 * q^19 + 16 * q^21 + q^25 + 4 * q^29 + 2 * q^31 - 20 * q^35 + 2 * q^39 + 4 * q^41 + 18 * q^45 - 28 * q^49 + 28 * q^51 + 27 * q^55 - 20 * q^59 + 24 * q^61 - 19 * q^65 + 12 * q^69 - 4 * q^71 + 15 * q^75 + 34 * q^79 - 8 * q^81 - 2 * q^85 - 20 * q^89 - 40 * q^91 - 12 * q^95 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more