LMFDB - Embedded newform 3720.2.a.v.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 = [5,0,5,0,-5,0,3] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.2294036.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 8x^{2} + 6x - 4 $ x^5 - x^4 - 8x^3 + 8x^2 + 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.36670$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +2.13213 q^{7} +1.00000 q^{9} -3.64738 q^{11} +4.85276 q^{13} -1.00000 q^{15} +1.80665 q^{17} +4.72063 q^{19} +2.13213 q^{21} +6.38079 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.65940 q^{29} +1.00000 q^{31} -3.64738 q^{33} -2.13213 q^{35} -0.588501 q^{37} +4.85276 q^{39} -8.90806 q^{41} -6.18743 q^{43} -1.00000 q^{45} -2.73340 q^{47} -2.45403 q^{49} +1.80665 q^{51} +10.9208 q^{53} +3.64738 q^{55} +4.72063 q^{57} -5.58616 q^{59} +4.92676 q^{61} +2.13213 q^{63} -4.85276 q^{65} +13.9088 q^{67} +6.38079 q^{69} -14.5710 q^{71} -2.67218 q^{73} +1.00000 q^{75} -7.77669 q^{77} +14.4845 q^{79} +1.00000 q^{81} +10.0282 q^{83} -1.80665 q^{85} -4.65940 q^{87} +0.747764 q^{89} +10.3467 q^{91} +1.00000 q^{93} -4.72063 q^{95} +17.1723 q^{97} -3.64738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 5 * 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
$7$	2.13213	0.805869	0.402934	−	0.915229i	$-0.367990\pi$
$7$	2.13213	0.805869	0.402934	+	0.915229i	$0.367990\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.64738	−1.09973	−0.549864	−	0.835254i	$-0.685320\pi$
$11$	−3.64738	−1.09973	−0.549864	+	0.835254i	$0.685320\pi$
$12$	0	0
$13$	4.85276	1.34591	0.672956	−	0.739682i	$-0.265024\pi$
$13$	4.85276	1.34591	0.672956	+	0.739682i	$0.265024\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.80665	0.438176	0.219088	−	0.975705i	$-0.429692\pi$
$17$	1.80665	0.438176	0.219088	+	0.975705i	$0.429692\pi$
$18$	0	0
$19$	4.72063	1.08299	0.541493	−	0.840705i	$-0.317859\pi$
$19$	4.72063	1.08299	0.541493	+	0.840705i	$0.317859\pi$
$20$	0	0
$21$	2.13213	0.465268
$22$	0	0
$23$	6.38079	1.33049	0.665243	−	0.746627i	$-0.268328\pi$
$23$	6.38079	1.33049	0.665243	+	0.746627i	$0.268328\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.65940	−0.865230	−0.432615	−	0.901579i	$-0.642409\pi$
$29$	−4.65940	−0.865230	−0.432615	+	0.901579i	$0.642409\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.64738	−0.634928
$34$	0	0
$35$	−2.13213	−0.360395
$36$	0	0
$37$	−0.588501	−0.0967490	−0.0483745	−	0.998829i	$-0.515404\pi$
$37$	−0.588501	−0.0967490	−0.0483745	+	0.998829i	$0.515404\pi$
$38$	0	0
$39$	4.85276	0.777063
$40$	0	0
$41$	−8.90806	−1.39121	−0.695603	−	0.718427i	$-0.744863\pi$
$41$	−8.90806	−1.39121	−0.695603	+	0.718427i	$0.744863\pi$
$42$	0	0
$43$	−6.18743	−0.943575	−0.471787	−	0.881712i	$-0.656391\pi$
$43$	−6.18743	−0.943575	−0.471787	+	0.881712i	$0.656391\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.73340	−0.398708	−0.199354	−	0.979928i	$-0.563884\pi$
$47$	−2.73340	−0.398708	−0.199354	+	0.979928i	$0.563884\pi$
$48$	0	0
$49$	−2.45403	−0.350576
$50$	0	0
$51$	1.80665	0.252981
$52$	0	0
$53$	10.9208	1.50009	0.750046	−	0.661386i	$-0.230031\pi$
$53$	10.9208	1.50009	0.750046	+	0.661386i	$0.230031\pi$
$54$	0	0
$55$	3.64738	0.491813
$56$	0	0
$57$	4.72063	0.625263
$58$	0	0
$59$	−5.58616	−0.727256	−0.363628	−	0.931544i	$-0.618462\pi$
$59$	−5.58616	−0.727256	−0.363628	+	0.931544i	$0.618462\pi$
$60$	0	0
$61$	4.92676	0.630806	0.315403	−	0.948958i	$-0.397860\pi$
$61$	4.92676	0.630806	0.315403	+	0.948958i	$0.397860\pi$
$62$	0	0
$63$	2.13213	0.268623
$64$	0	0
$65$	−4.85276	−0.601910
$66$	0	0
$67$	13.9088	1.69923	0.849616	−	0.527401i	$-0.176834\pi$
$67$	13.9088	1.69923	0.849616	+	0.527401i	$0.176834\pi$
$68$	0	0
$69$	6.38079	0.768156
$70$	0	0
$71$	−14.5710	−1.72926	−0.864632	−	0.502405i	$-0.832449\pi$
$71$	−14.5710	−1.72926	−0.864632	+	0.502405i	$0.832449\pi$
$72$	0	0
$73$	−2.67218	−0.312755	−0.156377	−	0.987697i	$-0.549982\pi$
$73$	−2.67218	−0.312755	−0.156377	+	0.987697i	$0.549982\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−7.77669	−0.886236
$78$	0	0
$79$	14.4845	1.62964	0.814819	−	0.579715i	$-0.196836\pi$
$79$	14.4845	1.62964	0.814819	+	0.579715i	$0.196836\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0282	1.10073	0.550367	−	0.834923i	$-0.314488\pi$
$83$	10.0282	1.10073	0.550367	+	0.834923i	$0.314488\pi$
$84$	0	0
$85$	−1.80665	−0.195958
$86$	0	0
$87$	−4.65940	−0.499541
$88$	0	0
$89$	0.747764	0.0792629	0.0396314	−	0.999214i	$-0.487382\pi$
$89$	0.747764	0.0792629	0.0396314	+	0.999214i	$0.487382\pi$
$90$	0	0
$91$	10.3467	1.08463
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−4.72063	−0.484326
$96$	0	0
$97$	17.1723	1.74358	0.871792	−	0.489876i	$-0.162958\pi$
$97$	17.1723	1.74358	0.871792	+	0.489876i	$0.162958\pi$
$98$	0	0
$99$	−3.64738	−0.366576

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.v.1.3	✓	5
4.3	odd	2		7440.2.a.cd.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.v.1.3	✓	5	1.1	even	1	trivial
7440.2.a.cd.1.3		5	4.3	odd	2

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.13213 q^{7} +1.00000 q^{9} -3.64738 q^{11} +4.85276 q^{13} -1.00000 q^{15} +1.80665 q^{17} +4.72063 q^{19} +2.13213 q^{21} +6.38079 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.65940 q^{29} +1.00000 q^{31} -3.64738 q^{33} -2.13213 q^{35} -0.588501 q^{37} +4.85276 q^{39} -8.90806 q^{41} -6.18743 q^{43} -1.00000 q^{45} -2.73340 q^{47} -2.45403 q^{49} +1.80665 q^{51} +10.9208 q^{53} +3.64738 q^{55} +4.72063 q^{57} -5.58616 q^{59} +4.92676 q^{61} +2.13213 q^{63} -4.85276 q^{65} +13.9088 q^{67} +6.38079 q^{69} -14.5710 q^{71} -2.67218 q^{73} +1.00000 q^{75} -7.77669 q^{77} +14.4845 q^{79} +1.00000 q^{81} +10.0282 q^{83} -1.80665 q^{85} -4.65940 q^{87} +0.747764 q^{89} +10.3467 q^{91} +1.00000 q^{93} -4.72063 q^{95} +17.1723 q^{97} -3.64738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 5 * q^99

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