LMFDB - Embedded newform 3720.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-3,0,-2] 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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -5.41855 q^{11} -1.17009 q^{13} -1.00000 q^{15} +7.12783 q^{17} -4.00000 q^{19} +1.17009 q^{21} -6.97107 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.21953 q^{29} -1.00000 q^{31} -5.41855 q^{33} -1.17009 q^{35} -2.09171 q^{37} -1.17009 q^{39} -5.26180 q^{41} -6.00000 q^{43} -1.00000 q^{45} -1.52586 q^{47} -5.63090 q^{49} +7.12783 q^{51} +0.474142 q^{53} +5.41855 q^{55} -4.00000 q^{57} +14.6381 q^{59} -11.9421 q^{61} +1.17009 q^{63} +1.17009 q^{65} -13.3268 q^{67} -6.97107 q^{69} -8.14116 q^{71} -8.74539 q^{73} +1.00000 q^{75} -6.34017 q^{77} +9.46800 q^{79} +1.00000 q^{81} -3.86603 q^{83} -7.12783 q^{85} +9.21953 q^{87} -2.04226 q^{89} -1.36910 q^{91} -1.00000 q^{93} +4.00000 q^{95} -2.73820 q^{97} -5.41855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * 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
$7$	1.17009	0.442251	0.221126	−	0.975245i	$-0.429027\pi$
$7$	1.17009	0.442251	0.221126	+	0.975245i	$0.429027\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.41855	−1.63375	−0.816877	−	0.576812i	$-0.804297\pi$
$11$	−5.41855	−1.63375	−0.816877	+	0.576812i	$0.804297\pi$
$12$	0	0
$13$	−1.17009	−0.324524	−0.162262	−	0.986748i	$-0.551879\pi$
$13$	−1.17009	−0.324524	−0.162262	+	0.986748i	$0.551879\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.12783	1.72875	0.864376	−	0.502846i	$-0.167714\pi$
$17$	7.12783	1.72875	0.864376	+	0.502846i	$0.167714\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.17009	0.255334
$22$	0	0
$23$	−6.97107	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$23$	−6.97107	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.21953	1.71202	0.856012	−	0.516955i	$-0.172935\pi$
$29$	9.21953	1.71202	0.856012	+	0.516955i	$0.172935\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−5.41855	−0.943249
$34$	0	0
$35$	−1.17009	−0.197781
$36$	0	0
$37$	−2.09171	−0.343875	−0.171937	−	0.985108i	$-0.555003\pi$
$37$	−2.09171	−0.343875	−0.171937	+	0.985108i	$0.555003\pi$
$38$	0	0
$39$	−1.17009	−0.187364
$40$	0	0
$41$	−5.26180	−0.821754	−0.410877	−	0.911691i	$-0.634777\pi$
$41$	−5.26180	−0.821754	−0.410877	+	0.911691i	$0.634777\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.52586	−0.222569	−0.111285	−	0.993789i	$-0.535497\pi$
$47$	−1.52586	−0.222569	−0.111285	+	0.993789i	$0.535497\pi$
$48$	0	0
$49$	−5.63090	−0.804414
$50$	0	0
$51$	7.12783	0.998095
$52$	0	0
$53$	0.474142	0.0651284	0.0325642	−	0.999470i	$-0.489633\pi$
$53$	0.474142	0.0651284	0.0325642	+	0.999470i	$0.489633\pi$
$54$	0	0
$55$	5.41855	0.730637
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	14.6381	1.90572	0.952858	−	0.303416i	$-0.0981272\pi$
$59$	14.6381	1.90572	0.952858	+	0.303416i	$0.0981272\pi$
$60$	0	0
$61$	−11.9421	−1.52903	−0.764517	−	0.644603i	$-0.777023\pi$
$61$	−11.9421	−1.52903	−0.764517	+	0.644603i	$0.777023\pi$
$62$	0	0
$63$	1.17009	0.147417
$64$	0	0
$65$	1.17009	0.145131
$66$	0	0
$67$	−13.3268	−1.62813	−0.814066	−	0.580772i	$-0.802751\pi$
$67$	−13.3268	−1.62813	−0.814066	+	0.580772i	$0.802751\pi$
$68$	0	0
$69$	−6.97107	−0.839218
$70$	0	0
$71$	−8.14116	−0.966178	−0.483089	−	0.875571i	$-0.660485\pi$
$71$	−8.14116	−0.966178	−0.483089	+	0.875571i	$0.660485\pi$
$72$	0	0
$73$	−8.74539	−1.02357	−0.511785	−	0.859113i	$-0.671016\pi$
$73$	−8.74539	−1.02357	−0.511785	+	0.859113i	$0.671016\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.34017	−0.722530
$78$	0	0
$79$	9.46800	1.06523	0.532617	−	0.846357i	$-0.321209\pi$
$79$	9.46800	1.06523	0.532617	+	0.846357i	$0.321209\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.86603	−0.424352	−0.212176	−	0.977231i	$-0.568055\pi$
$83$	−3.86603	−0.424352	−0.212176	+	0.977231i	$0.568055\pi$
$84$	0	0
$85$	−7.12783	−0.773121
$86$	0	0
$87$	9.21953	0.988438
$88$	0	0
$89$	−2.04226	−0.216479	−0.108240	−	0.994125i	$-0.534521\pi$
$89$	−2.04226	−0.216479	−0.108240	+	0.994125i	$0.534521\pi$
$90$	0	0
$91$	−1.36910	−0.143521
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−2.73820	−0.278023	−0.139011	−	0.990291i	$-0.544392\pi$
$97$	−2.73820	−0.278023	−0.139011	+	0.990291i	$0.544392\pi$
$98$	0	0
$99$	−5.41855	−0.544585

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	3720.2.a.o.1.3	✓	3
4.3	odd	2		7440.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.o.1.3	✓	3	1.1	even	1	trivial
7440.2.a.bo.1.1		3	4.3	odd	2

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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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -5.41855 q^{11} -1.17009 q^{13} -1.00000 q^{15} +7.12783 q^{17} -4.00000 q^{19} +1.17009 q^{21} -6.97107 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.21953 q^{29} -1.00000 q^{31} -5.41855 q^{33} -1.17009 q^{35} -2.09171 q^{37} -1.17009 q^{39} -5.26180 q^{41} -6.00000 q^{43} -1.00000 q^{45} -1.52586 q^{47} -5.63090 q^{49} +7.12783 q^{51} +0.474142 q^{53} +5.41855 q^{55} -4.00000 q^{57} +14.6381 q^{59} -11.9421 q^{61} +1.17009 q^{63} +1.17009 q^{65} -13.3268 q^{67} -6.97107 q^{69} -8.14116 q^{71} -8.74539 q^{73} +1.00000 q^{75} -6.34017 q^{77} +9.46800 q^{79} +1.00000 q^{81} -3.86603 q^{83} -7.12783 q^{85} +9.21953 q^{87} -2.04226 q^{89} -1.36910 q^{91} -1.00000 q^{93} +4.00000 q^{95} -2.73820 q^{97} -5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

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