LMFDB - Embedded newform 1860.2.a.h.1.1

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			1.1
Root			$4.47467$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -4.47467 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.47467 q^{13} +1.00000 q^{15} +1.07331 q^{17} +4.00000 q^{19} -4.47467 q^{21} -1.07331 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.54798 q^{29} -1.00000 q^{31} +2.00000 q^{33} -4.47467 q^{35} +2.47467 q^{37} +4.47467 q^{39} +4.00000 q^{41} +2.00000 q^{43} +1.00000 q^{45} -8.02265 q^{47} +13.0226 q^{49} +1.07331 q^{51} +10.0226 q^{53} +2.00000 q^{55} +4.00000 q^{57} +9.54798 q^{59} -0.146623 q^{61} -4.47467 q^{63} +4.47467 q^{65} -4.32804 q^{67} -1.07331 q^{69} +12.4973 q^{71} +15.4240 q^{73} +1.00000 q^{75} -8.94933 q^{77} +9.87602 q^{79} +1.00000 q^{81} -2.92669 q^{83} +1.07331 q^{85} -3.54798 q^{87} +15.4014 q^{89} -20.0226 q^{91} -1.00000 q^{93} +4.00000 q^{95} +1.85338 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.47467	−1.69127	−0.845633	−	0.533765i	$-0.820777\pi$
$7$	−4.47467	−1.69127	−0.845633	+	0.533765i	$0.820777\pi$
$8$	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
$13$	4.47467	1.24105	0.620525	−	0.784187i	$-0.286920\pi$
$13$	4.47467	1.24105	0.620525	+	0.784187i	$0.286920\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.07331	0.260316	0.130158	−	0.991493i	$-0.458451\pi$
$17$	1.07331	0.260316	0.130158	+	0.991493i	$0.458451\pi$
$18$	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
$21$	−4.47467	−0.976452
$22$	0	0
$23$	−1.07331	−0.223801	−0.111900	−	0.993719i	$-0.535694\pi$
$23$	−1.07331	−0.223801	−0.111900	+	0.993719i	$0.535694\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.54798	−0.658843	−0.329422	−	0.944183i	$-0.606854\pi$
$29$	−3.54798	−0.658843	−0.329422	+	0.944183i	$0.606854\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−4.47467	−0.756357
$36$	0	0
$37$	2.47467	0.406833	0.203416	−	0.979092i	$-0.434795\pi$
$37$	2.47467	0.406833	0.203416	+	0.979092i	$0.434795\pi$
$38$	0	0
$39$	4.47467	0.716520
$40$	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
$47$	−8.02265	−1.17022	−0.585112	−	0.810953i	$-0.698949\pi$
$47$	−8.02265	−1.17022	−0.585112	+	0.810953i	$0.698949\pi$
$48$	0	0
$49$	13.0226	1.86038
$50$	0	0
$51$	1.07331	0.150294
$52$	0	0
$53$	10.0226	1.37672	0.688358	−	0.725371i	$-0.258332\pi$
$53$	10.0226	1.37672	0.688358	+	0.725371i	$0.258332\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	9.54798	1.24304	0.621520	−	0.783398i	$-0.286515\pi$
$59$	9.54798	1.24304	0.621520	+	0.783398i	$0.286515\pi$
$60$	0	0
$61$	−0.146623	−0.0187731	−0.00938657	−	0.999956i	$-0.502988\pi$
$61$	−0.146623	−0.0187731	−0.00938657	+	0.999956i	$0.502988\pi$
$62$	0	0
$63$	−4.47467	−0.563755
$64$	0	0
$65$	4.47467	0.555014
$66$	0	0
$67$	−4.32804	−0.528755	−0.264377	−	0.964419i	$-0.585166\pi$
$67$	−4.32804	−0.528755	−0.264377	+	0.964419i	$0.585166\pi$
$68$	0	0
$69$	−1.07331	−0.129212
$70$	0	0
$71$	12.4973	1.48316	0.741579	−	0.670865i	$-0.234077\pi$
$71$	12.4973	1.48316	0.741579	+	0.670865i	$0.234077\pi$
$72$	0	0
$73$	15.4240	1.80524	0.902621	−	0.430435i	$-0.141640\pi$
$73$	15.4240	1.80524	0.902621	+	0.430435i	$0.141640\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−8.94933	−1.01987
$78$	0	0
$79$	9.87602	1.11114	0.555570	−	0.831470i	$-0.312500\pi$
$79$	9.87602	1.11114	0.555570	+	0.831470i	$0.312500\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.92669	−0.321246	−0.160623	−	0.987016i	$-0.551350\pi$
$83$	−2.92669	−0.321246	−0.160623	+	0.987016i	$0.551350\pi$
$84$	0	0
$85$	1.07331	0.116417
$86$	0	0
$87$	−3.54798	−0.380383
$88$	0	0
$89$	15.4014	1.63254	0.816270	−	0.577670i	$-0.196038\pi$
$89$	15.4014	1.63254	0.816270	+	0.577670i	$0.196038\pi$
$90$	0	0
$91$	−20.0226	−2.09894
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	1.85338	0.188182	0.0940910	−	0.995564i	$-0.470006\pi$
$97$	1.85338	0.188182	0.0940910	+	0.995564i	$0.470006\pi$
$98$	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.1.1	✓	3
3.2	odd	2		5580.2.a.i.1.1		3
4.3	odd	2		7440.2.a.bq.1.3		3
5.2	odd	4		9300.2.g.r.3349.1		6
5.3	odd	4		9300.2.g.r.3349.6		6
5.4	even	2		9300.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.h.1.1	✓	3	1.1	even	1	trivial
5580.2.a.i.1.1		3	3.2	odd	2
7440.2.a.bq.1.3		3	4.3	odd	2
9300.2.a.t.1.3		3	5.4	even	2
9300.2.g.r.3349.1		6	5.2	odd	4
9300.2.g.r.3349.6		6	5.3	odd	4

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Elliptic curves over $\Q$ over number fields
Genus 2 curves
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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.47467 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.47467 q^{13} +1.00000 q^{15} +1.07331 q^{17} +4.00000 q^{19} -4.47467 q^{21} -1.07331 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.54798 q^{29} -1.00000 q^{31} +2.00000 q^{33} -4.47467 q^{35} +2.47467 q^{37} +4.47467 q^{39} +4.00000 q^{41} +2.00000 q^{43} +1.00000 q^{45} -8.02265 q^{47} +13.0226 q^{49} +1.07331 q^{51} +10.0226 q^{53} +2.00000 q^{55} +4.00000 q^{57} +9.54798 q^{59} -0.146623 q^{61} -4.47467 q^{63} +4.47467 q^{65} -4.32804 q^{67} -1.07331 q^{69} +12.4973 q^{71} +15.4240 q^{73} +1.00000 q^{75} -8.94933 q^{77} +9.87602 q^{79} +1.00000 q^{81} -2.92669 q^{83} +1.07331 q^{85} -3.54798 q^{87} +15.4014 q^{89} -20.0226 q^{91} -1.00000 q^{93} +4.00000 q^{95} +1.85338 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 6 q^{11} + 3 q^{15} + 2 q^{17} + 12 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} + 6 q^{33} - 6 q^{37} + 12 q^{41} + 6 q^{43} + 3 q^{45} + 4 q^{47} + 11 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 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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