LMFDB - Embedded newform 1860.2.a.f.1.3

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,-3,0,4] 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:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +4.43807 q^{7} +1.00000 q^{9} +4.17226 q^{11} +4.43807 q^{13} +1.00000 q^{15} -1.35194 q^{17} +2.70388 q^{19} -4.43807 q^{21} -1.35194 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.25839 q^{29} -1.00000 q^{31} -4.17226 q^{33} -4.43807 q^{35} -10.7826 q^{37} -4.43807 q^{39} +2.17226 q^{41} +9.46838 q^{43} -1.00000 q^{45} -4.11644 q^{47} +12.6965 q^{49} +1.35194 q^{51} +12.8203 q^{53} -4.17226 q^{55} -2.70388 q^{57} +5.25839 q^{59} +6.00000 q^{61} +4.43807 q^{63} -4.43807 q^{65} -1.14195 q^{67} +1.35194 q^{69} -10.7268 q^{71} +10.4381 q^{73} -1.00000 q^{75} +18.5168 q^{77} -2.05582 q^{79} +1.00000 q^{81} -6.99258 q^{83} +1.35194 q^{85} +7.25839 q^{87} -2.55451 q^{89} +19.6965 q^{91} +1.00000 q^{93} -2.70388 q^{95} -15.9245 q^{97} +4.17226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} - 3 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 - 3 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 - 4 * q^39 - 8 * q^41 + 18 * q^43 - 3 * q^45 - 4 * q^47 + 7 * q^49 + 2 * q^51 + 26 * q^53 + 2 * q^55 - 4 * q^57 - 6 * q^59 + 18 * q^61 + 4 * q^63 - 4 * q^65 + 10 * q^67 + 2 * q^69 + 22 * q^73 - 3 * q^75 + 12 * q^77 + 3 * q^81 + 6 * q^83 + 2 * q^85 + 10 * q^89 + 28 * q^91 + 3 * q^93 - 4 * q^95 + 4 * q^97 - 2 * 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
$7$	4.43807	1.67743	0.838716	−	0.544569i	$-0.183307\pi$
$7$	4.43807	1.67743	0.838716	+	0.544569i	$0.183307\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.17226	1.25798	0.628992	−	0.777412i	$-0.283468\pi$
$11$	4.17226	1.25798	0.628992	+	0.777412i	$0.283468\pi$
$12$	0	0
$13$	4.43807	1.23090	0.615449	−	0.788176i	$-0.288975\pi$
$13$	4.43807	1.23090	0.615449	+	0.788176i	$0.288975\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.35194	−0.327893	−0.163947	−	0.986469i	$-0.552423\pi$
$17$	−1.35194	−0.327893	−0.163947	+	0.986469i	$0.552423\pi$
$18$	0	0
$19$	2.70388	0.620312	0.310156	−	0.950686i	$-0.399619\pi$
$19$	2.70388	0.620312	0.310156	+	0.950686i	$0.399619\pi$
$20$	0	0
$21$	−4.43807	−0.968466
$22$	0	0
$23$	−1.35194	−0.281899	−0.140949	−	0.990017i	$-0.545015\pi$
$23$	−1.35194	−0.281899	−0.140949	+	0.990017i	$0.545015\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.25839	−1.34785	−0.673925	−	0.738800i	$-0.735393\pi$
$29$	−7.25839	−1.34785	−0.673925	+	0.738800i	$0.735393\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−4.17226	−0.726297
$34$	0	0
$35$	−4.43807	−0.750171
$36$	0	0
$37$	−10.7826	−1.77265	−0.886323	−	0.463067i	$-0.846749\pi$
$37$	−10.7826	−1.77265	−0.886323	+	0.463067i	$0.846749\pi$
$38$	0	0
$39$	−4.43807	−0.710660
$40$	0	0
$41$	2.17226	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$41$	2.17226	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$42$	0	0
$43$	9.46838	1.44391	0.721957	−	0.691938i	$-0.243243\pi$
$43$	9.46838	1.44391	0.721957	+	0.691938i	$0.243243\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.11644	−0.600445	−0.300222	−	0.953869i	$-0.597061\pi$
$47$	−4.11644	−0.600445	−0.300222	+	0.953869i	$0.597061\pi$
$48$	0	0
$49$	12.6965	1.81378
$50$	0	0
$51$	1.35194	0.189309
$52$	0	0
$53$	12.8203	1.76101	0.880503	−	0.474040i	$-0.157205\pi$
$53$	12.8203	1.76101	0.880503	+	0.474040i	$0.157205\pi$
$54$	0	0
$55$	−4.17226	−0.562587
$56$	0	0
$57$	−2.70388	−0.358137
$58$	0	0
$59$	5.25839	0.684584	0.342292	−	0.939594i	$-0.388797\pi$
$59$	5.25839	0.684584	0.342292	+	0.939594i	$0.388797\pi$
$60$	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
$63$	4.43807	0.559144
$64$	0	0
$65$	−4.43807	−0.550475
$66$	0	0
$67$	−1.14195	−0.139511	−0.0697556	−	0.997564i	$-0.522222\pi$
$67$	−1.14195	−0.139511	−0.0697556	+	0.997564i	$0.522222\pi$
$68$	0	0
$69$	1.35194	0.162754
$70$	0	0
$71$	−10.7268	−1.27303	−0.636517	−	0.771263i	$-0.719625\pi$
$71$	−10.7268	−1.27303	−0.636517	+	0.771263i	$0.719625\pi$
$72$	0	0
$73$	10.4381	1.22168	0.610842	−	0.791753i	$-0.290831\pi$
$73$	10.4381	1.22168	0.610842	+	0.791753i	$0.290831\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	18.5168	2.11018
$78$	0	0
$79$	−2.05582	−0.231298	−0.115649	−	0.993290i	$-0.536895\pi$
$79$	−2.05582	−0.231298	−0.115649	+	0.993290i	$0.536895\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.99258	−0.767536	−0.383768	−	0.923430i	$-0.625374\pi$
$83$	−6.99258	−0.767536	−0.383768	+	0.923430i	$0.625374\pi$
$84$	0	0
$85$	1.35194	0.146638
$86$	0	0
$87$	7.25839	0.778181
$88$	0	0
$89$	−2.55451	−0.270778	−0.135389	−	0.990793i	$-0.543228\pi$
$89$	−2.55451	−0.270778	−0.135389	+	0.990793i	$0.543228\pi$
$90$	0	0
$91$	19.6965	2.06475
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−2.70388	−0.277412
$96$	0	0
$97$	−15.9245	−1.61689	−0.808446	−	0.588570i	$-0.799691\pi$
$97$	−15.9245	−1.61689	−0.808446	+	0.588570i	$0.799691\pi$
$98$	0	0
$99$	4.17226	0.419328

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.f.1.3	✓	3
3.2	odd	2		5580.2.a.l.1.3		3
4.3	odd	2		7440.2.a.bt.1.1		3
5.2	odd	4		9300.2.g.p.3349.6		6
5.3	odd	4		9300.2.g.p.3349.1		6
5.4	even	2		9300.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.f.1.3	✓	3	1.1	even	1	trivial
5580.2.a.l.1.3		3	3.2	odd	2
7440.2.a.bt.1.1		3	4.3	odd	2
9300.2.a.v.1.1		3	5.4	even	2
9300.2.g.p.3349.1		6	5.3	odd	4
9300.2.g.p.3349.6		6	5.2	odd	4

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.43807 q^{7} +1.00000 q^{9} +4.17226 q^{11} +4.43807 q^{13} +1.00000 q^{15} -1.35194 q^{17} +2.70388 q^{19} -4.43807 q^{21} -1.35194 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.25839 q^{29} -1.00000 q^{31} -4.17226 q^{33} -4.43807 q^{35} -10.7826 q^{37} -4.43807 q^{39} +2.17226 q^{41} +9.46838 q^{43} -1.00000 q^{45} -4.11644 q^{47} +12.6965 q^{49} +1.35194 q^{51} +12.8203 q^{53} -4.17226 q^{55} -2.70388 q^{57} +5.25839 q^{59} +6.00000 q^{61} +4.43807 q^{63} -4.43807 q^{65} -1.14195 q^{67} +1.35194 q^{69} -10.7268 q^{71} +10.4381 q^{73} -1.00000 q^{75} +18.5168 q^{77} -2.05582 q^{79} +1.00000 q^{81} -6.99258 q^{83} +1.35194 q^{85} +7.25839 q^{87} -2.55451 q^{89} +19.6965 q^{91} +1.00000 q^{93} -2.70388 q^{95} -15.9245 q^{97} +4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} - 3 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 - 3 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 - 4 * q^39 - 8 * q^41 + 18 * q^43 - 3 * q^45 - 4 * q^47 + 7 * q^49 + 2 * q^51 + 26 * q^53 + 2 * q^55 - 4 * q^57 - 6 * q^59 + 18 * q^61 + 4 * q^63 - 4 * q^65 + 10 * q^67 + 2 * q^69 + 22 * q^73 - 3 * q^75 + 12 * q^77 + 3 * q^81 + 6 * q^83 + 2 * q^85 + 10 * q^89 + 28 * q^91 + 3 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

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