LMFDB - Embedded newform 1860.2.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,2,Mod(1,1860)] 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("1860.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,3,0,0] 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:	$14.8521747760$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7636.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 16x - 18 $ x^3 - 16*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24586$ of defining polynomial
Character	$\chi$	$=$	1860.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.24586 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.24586 q^{13} +1.00000 q^{15} -5.95610 q^{17} +4.00000 q^{19} +1.24586 q^{21} +5.95610 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.20197 q^{29} -1.00000 q^{31} +2.00000 q^{33} +1.24586 q^{35} -3.24586 q^{37} -1.24586 q^{39} +4.00000 q^{41} +2.00000 q^{43} +1.00000 q^{45} +10.4478 q^{47} -5.44783 q^{49} -5.95610 q^{51} -8.44783 q^{53} +2.00000 q^{55} +4.00000 q^{57} -3.20197 q^{59} +13.9122 q^{61} +1.24586 q^{63} -1.24586 q^{65} -12.6663 q^{67} +5.95610 q^{69} -11.6937 q^{71} -1.73759 q^{73} +1.00000 q^{75} +2.49172 q^{77} +5.46438 q^{79} +1.00000 q^{81} -9.95610 q^{83} -5.95610 q^{85} +9.20197 q^{87} +16.7102 q^{89} -1.55217 q^{91} -1.00000 q^{93} +4.00000 q^{95} +15.9122 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 6 q^{11} + 3 q^{15} + 2 q^{17} + 12 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} + 6 q^{33} - 6 q^{37} + 12 q^{41} + 6 q^{43} + 3 q^{45} + 4 q^{47} + 11 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 6 * q^11 + 3 * q^15 + 2 * q^17 + 12 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 + 6 * q^33 - 6 * q^37 + 12 * q^41 + 6 * q^43 + 3 * q^45 + 4 * q^47 + 11 * q^49 + 2 * q^51 + 2 * q^53 + 6 * q^55 + 12 * q^57 + 14 * q^59 + 2 * q^61 - 2 * q^67 - 2 * q^69 - 4 * q^71 + 6 * q^73 + 3 * q^75 + 4 * q^79 + 3 * q^81 - 10 * q^83 + 2 * q^85 + 4 * q^87 + 34 * q^89 - 32 * q^91 - 3 * q^93 + 12 * q^95 + 8 * q^97 + 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.24586	0.470892	0.235446	−	0.971887i	$-0.424345\pi$
$7$	1.24586	0.470892	0.235446	+	0.971887i	$0.424345\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.24586	−0.345540	−0.172770	−	0.984962i	$-0.555272\pi$
$13$	−1.24586	−0.345540	−0.172770	+	0.984962i	$0.555272\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.95610	−1.44457	−0.722284	−	0.691597i	$-0.756907\pi$
$17$	−5.95610	−1.44457	−0.722284	+	0.691597i	$0.756907\pi$
$18$	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
$21$	1.24586	0.271869
$22$	0	0
$23$	5.95610	1.24193	0.620967	−	0.783837i	$-0.286740\pi$
$23$	5.95610	1.24193	0.620967	+	0.783837i	$0.286740\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.20197	1.70876	0.854381	−	0.519647i	$-0.173937\pi$
$29$	9.20197	1.70876	0.854381	+	0.519647i	$0.173937\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.24586	0.210589
$36$	0	0
$37$	−3.24586	−0.533616	−0.266808	−	0.963750i	$-0.585969\pi$
$37$	−3.24586	−0.533616	−0.266808	+	0.963750i	$0.585969\pi$
$38$	0	0
$39$	−1.24586	−0.199498
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	10.4478	1.52397	0.761986	−	0.647593i	$-0.224224\pi$
$47$	10.4478	1.52397	0.761986	+	0.647593i	$0.224224\pi$
$48$	0	0
$49$	−5.44783	−0.778261
$50$	0	0
$51$	−5.95610	−0.834021
$52$	0	0
$53$	−8.44783	−1.16040	−0.580199	−	0.814475i	$-0.697025\pi$
$53$	−8.44783	−1.16040	−0.580199	+	0.814475i	$0.697025\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−3.20197	−0.416860	−0.208430	−	0.978037i	$-0.566835\pi$
$59$	−3.20197	−0.416860	−0.208430	+	0.978037i	$0.566835\pi$
$60$	0	0
$61$	13.9122	1.78128	0.890638	−	0.454713i	$-0.150258\pi$
$61$	13.9122	1.78128	0.890638	+	0.454713i	$0.150258\pi$
$62$	0	0
$63$	1.24586	0.156964
$64$	0	0
$65$	−1.24586	−0.154530
$66$	0	0
$67$	−12.6663	−1.54744	−0.773720	−	0.633527i	$-0.781606\pi$
$67$	−12.6663	−1.54744	−0.773720	+	0.633527i	$0.781606\pi$
$68$	0	0
$69$	5.95610	0.717031
$70$	0	0
$71$	−11.6937	−1.38779	−0.693893	−	0.720078i	$-0.744106\pi$
$71$	−11.6937	−1.38779	−0.693893	+	0.720078i	$0.744106\pi$
$72$	0	0
$73$	−1.73759	−0.203369	−0.101685	−	0.994817i	$-0.532423\pi$
$73$	−1.73759	−0.203369	−0.101685	+	0.994817i	$0.532423\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.49172	0.283958
$78$	0	0
$79$	5.46438	0.614791	0.307395	−	0.951582i	$-0.400543\pi$
$79$	5.46438	0.614791	0.307395	+	0.951582i	$0.400543\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.95610	−1.09282	−0.546412	−	0.837516i	$-0.684007\pi$
$83$	−9.95610	−1.09282	−0.546412	+	0.837516i	$0.684007\pi$
$84$	0	0
$85$	−5.95610	−0.646030
$86$	0	0
$87$	9.20197	0.986554
$88$	0	0
$89$	16.7102	1.77128	0.885641	−	0.464370i	$-0.153719\pi$
$89$	16.7102	1.77128	0.885641	+	0.464370i	$0.153719\pi$
$90$	0	0
$91$	−1.55217	−0.162712
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	15.9122	1.61564	0.807820	−	0.589429i	$-0.200647\pi$
$97$	15.9122	1.61564	0.807820	+	0.589429i	$0.200647\pi$
$98$	0	0
$99$	2.00000	0.201008

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	1860.2.a.h.1.2	✓	3
3.2	odd	2		5580.2.a.i.1.2		3
4.3	odd	2		7440.2.a.bq.1.2		3
5.2	odd	4		9300.2.g.r.3349.2		6
5.3	odd	4		9300.2.g.r.3349.5		6
5.4	even	2		9300.2.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.h.1.2	✓	3	1.1	even	1	trivial
5580.2.a.i.1.2		3	3.2	odd	2
7440.2.a.bq.1.2		3	4.3	odd	2
9300.2.a.t.1.2		3	5.4	even	2
9300.2.g.r.3349.2		6	5.2	odd	4
9300.2.g.r.3349.5		6	5.3	odd	4

Overview	Random
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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:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.24586 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.24586 q^{13} +1.00000 q^{15} -5.95610 q^{17} +4.00000 q^{19} +1.24586 q^{21} +5.95610 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.20197 q^{29} -1.00000 q^{31} +2.00000 q^{33} +1.24586 q^{35} -3.24586 q^{37} -1.24586 q^{39} +4.00000 q^{41} +2.00000 q^{43} +1.00000 q^{45} +10.4478 q^{47} -5.44783 q^{49} -5.95610 q^{51} -8.44783 q^{53} +2.00000 q^{55} +4.00000 q^{57} -3.20197 q^{59} +13.9122 q^{61} +1.24586 q^{63} -1.24586 q^{65} -12.6663 q^{67} +5.95610 q^{69} -11.6937 q^{71} -1.73759 q^{73} +1.00000 q^{75} +2.49172 q^{77} +5.46438 q^{79} +1.00000 q^{81} -9.95610 q^{83} -5.95610 q^{85} +9.20197 q^{87} +16.7102 q^{89} -1.55217 q^{91} -1.00000 q^{93} +4.00000 q^{95} +15.9122 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 6 q^{11} + 3 q^{15} + 2 q^{17} + 12 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} + 6 q^{33} - 6 q^{37} + 12 q^{41} + 6 q^{43} + 3 q^{45} + 4 q^{47} + 11 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 6 * q^11 + 3 * q^15 + 2 * q^17 + 12 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 + 6 * q^33 - 6 * q^37 + 12 * q^41 + 6 * q^43 + 3 * q^45 + 4 * q^47 + 11 * q^49 + 2 * q^51 + 2 * q^53 + 6 * q^55 + 12 * q^57 + 14 * q^59 + 2 * q^61 - 2 * q^67 - 2 * q^69 - 4 * q^71 + 6 * q^73 + 3 * q^75 + 4 * q^79 + 3 * q^81 - 10 * q^83 + 2 * q^85 + 4 * q^87 + 34 * q^89 - 32 * q^91 - 3 * q^93 + 12 * q^95 + 8 * q^97 + 6 * q^99

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