LMFDB - Embedded newform 3720.2.a.r.1.4

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 = [4,0,-4,0,-4,0,1] 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:	$4$
Coefficient field:	4.4.78292.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 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			1.4
Root			$2.68461$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.68461 q^{7} +1.00000 q^{9} +3.24072 q^{11} -4.97040 q^{13} +1.00000 q^{15} +0.921359 q^{17} +1.24072 q^{19} -2.68461 q^{21} +6.57637 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.89176 q^{29} +1.00000 q^{31} -3.24072 q^{33} -2.68461 q^{35} +2.97040 q^{37} +4.97040 q^{39} +4.00000 q^{41} +5.65501 q^{43} -1.00000 q^{45} -8.86216 q^{47} +0.207146 q^{49} -0.921359 q^{51} +0.792854 q^{53} -3.24072 q^{55} -1.24072 q^{57} -7.00397 q^{59} +10.8957 q^{61} +2.68461 q^{63} +4.97040 q^{65} -12.3396 q^{67} -6.57637 q^{69} +15.5468 q^{71} +7.09891 q^{73} -1.00000 q^{75} +8.70008 q^{77} -1.04986 q^{79} +1.00000 q^{81} -16.3887 q^{83} -0.921359 q^{85} +3.89176 q^{87} -3.13248 q^{89} -13.3436 q^{91} -1.00000 q^{93} -1.24072 q^{95} -4.57157 q^{97} +3.24072 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + 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
$7$	2.68461	1.01469	0.507344	−	0.861744i	$-0.330627\pi$
$7$	2.68461	1.01469	0.507344	+	0.861744i	$0.330627\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.24072	0.977114	0.488557	−	0.872532i	$-0.337523\pi$
$11$	3.24072	0.977114	0.488557	+	0.872532i	$0.337523\pi$
$12$	0	0
$13$	−4.97040	−1.37854	−0.689270	−	0.724504i	$-0.742069\pi$
$13$	−4.97040	−1.37854	−0.689270	+	0.724504i	$0.742069\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.921359	0.223462	0.111731	−	0.993738i	$-0.464360\pi$
$17$	0.921359	0.223462	0.111731	+	0.993738i	$0.464360\pi$
$18$	0	0
$19$	1.24072	0.284641	0.142320	−	0.989821i	$-0.454544\pi$
$19$	1.24072	0.284641	0.142320	+	0.989821i	$0.454544\pi$
$20$	0	0
$21$	−2.68461	−0.585831
$22$	0	0
$23$	6.57637	1.37127	0.685634	−	0.727946i	$-0.259525\pi$
$23$	6.57637	1.37127	0.685634	+	0.727946i	$0.259525\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.89176	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$29$	−3.89176	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.24072	−0.564137
$34$	0	0
$35$	−2.68461	−0.453782
$36$	0	0
$37$	2.97040	0.488331	0.244165	−	0.969734i	$-0.421486\pi$
$37$	2.97040	0.488331	0.244165	+	0.969734i	$0.421486\pi$
$38$	0	0
$39$	4.97040	0.795901
$40$	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
$43$	5.65501	0.862381	0.431191	−	0.902261i	$-0.358094\pi$
$43$	5.65501	0.862381	0.431191	+	0.902261i	$0.358094\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.86216	−1.29268	−0.646339	−	0.763050i	$-0.723701\pi$
$47$	−8.86216	−1.29268	−0.646339	+	0.763050i	$0.723701\pi$
$48$	0	0
$49$	0.207146	0.0295923
$50$	0	0
$51$	−0.921359	−0.129016
$52$	0	0
$53$	0.792854	0.108907	0.0544534	−	0.998516i	$-0.482658\pi$
$53$	0.792854	0.108907	0.0544534	+	0.998516i	$0.482658\pi$
$54$	0	0
$55$	−3.24072	−0.436979
$56$	0	0
$57$	−1.24072	−0.164337
$58$	0	0
$59$	−7.00397	−0.911840	−0.455920	−	0.890021i	$-0.650690\pi$
$59$	−7.00397	−0.911840	−0.455920	+	0.890021i	$0.650690\pi$
$60$	0	0
$61$	10.8957	1.39506	0.697528	−	0.716558i	$-0.254283\pi$
$61$	10.8957	1.39506	0.697528	+	0.716558i	$0.254283\pi$
$62$	0	0
$63$	2.68461	0.338229
$64$	0	0
$65$	4.97040	0.616502
$66$	0	0
$67$	−12.3396	−1.50753	−0.753763	−	0.657147i	$-0.771763\pi$
$67$	−12.3396	−1.50753	−0.753763	+	0.657147i	$0.771763\pi$
$68$	0	0
$69$	−6.57637	−0.791702
$70$	0	0
$71$	15.5468	1.84506	0.922531	−	0.385923i	$-0.126117\pi$
$71$	15.5468	1.84506	0.922531	+	0.385923i	$0.126117\pi$
$72$	0	0
$73$	7.09891	0.830864	0.415432	−	0.909624i	$-0.363630\pi$
$73$	7.09891	0.830864	0.415432	+	0.909624i	$0.363630\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.70008	0.991466
$78$	0	0
$79$	−1.04986	−0.118119	−0.0590595	−	0.998254i	$-0.518810\pi$
$79$	−1.04986	−0.118119	−0.0590595	+	0.998254i	$0.518810\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.3887	−1.79889	−0.899445	−	0.437034i	$-0.856029\pi$
$83$	−16.3887	−1.79889	−0.899445	+	0.437034i	$0.856029\pi$
$84$	0	0
$85$	−0.921359	−0.0999354
$86$	0	0
$87$	3.89176	0.417240
$88$	0	0
$89$	−3.13248	−0.332042	−0.166021	−	0.986122i	$-0.553092\pi$
$89$	−3.13248	−0.332042	−0.166021	+	0.986122i	$0.553092\pi$
$90$	0	0
$91$	−13.3436	−1.39879
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−1.24072	−0.127295
$96$	0	0
$97$	−4.57157	−0.464173	−0.232087	−	0.972695i	$-0.574555\pi$
$97$	−4.57157	−0.464173	−0.232087	+	0.972695i	$0.574555\pi$
$98$	0	0
$99$	3.24072	0.325705

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.r.1.4	✓	4
4.3	odd	2		7440.2.a.bx.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.r.1.4	✓	4	1.1	even	1	trivial
7440.2.a.bx.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.68461 q^{7} +1.00000 q^{9} +3.24072 q^{11} -4.97040 q^{13} +1.00000 q^{15} +0.921359 q^{17} +1.24072 q^{19} -2.68461 q^{21} +6.57637 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.89176 q^{29} +1.00000 q^{31} -3.24072 q^{33} -2.68461 q^{35} +2.97040 q^{37} +4.97040 q^{39} +4.00000 q^{41} +5.65501 q^{43} -1.00000 q^{45} -8.86216 q^{47} +0.207146 q^{49} -0.921359 q^{51} +0.792854 q^{53} -3.24072 q^{55} -1.24072 q^{57} -7.00397 q^{59} +10.8957 q^{61} +2.68461 q^{63} +4.97040 q^{65} -12.3396 q^{67} -6.57637 q^{69} +15.5468 q^{71} +7.09891 q^{73} -1.00000 q^{75} +8.70008 q^{77} -1.04986 q^{79} +1.00000 q^{81} -16.3887 q^{83} -0.921359 q^{85} +3.89176 q^{87} -3.13248 q^{89} -13.3436 q^{91} -1.00000 q^{93} -1.24072 q^{95} -4.57157 q^{97} +3.24072 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 + q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 - 7 * q^19 - q^21 - q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 + 2 * q^39 + 16 * q^41 - 5 * q^43 - 4 * q^45 - 7 * q^49 - 4 * q^51 + 11 * q^53 - q^55 + 7 * q^57 - 6 * q^59 + 4 * q^61 + q^63 + 2 * q^65 - 12 * q^67 + q^69 + 17 * q^71 + 3 * q^73 - 4 * q^75 + 11 * q^77 + 3 * q^79 + 4 * q^81 - 10 * q^83 - 4 * q^85 - 2 * q^87 + 17 * q^89 + 6 * q^91 - 4 * q^93 + 7 * q^95 - 2 * q^97 + q^99

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