LMFDB - Embedded newform 3720.2.a.t.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:	4.4.92692.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 8x + 9 $ x^4 - 2x^3 - 8x^2 + 8*x + 9
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			$1.58074$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.30703 q^{7} +1.00000 q^{9} +3.72629 q^{11} +1.41926 q^{13} -1.00000 q^{15} +7.66274 q^{17} +6.88777 q^{19} -4.30703 q^{21} +1.22503 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.92052 q^{29} -1.00000 q^{31} -3.72629 q^{33} +4.30703 q^{35} -0.580739 q^{37} -1.41926 q^{39} -6.16400 q^{41} -10.0492 q^{43} +1.00000 q^{45} +6.82422 q^{47} +11.5505 q^{49} -7.66274 q^{51} -2.93898 q^{53} +3.72629 q^{55} -6.88777 q^{57} -5.08200 q^{59} +4.32296 q^{61} +4.30703 q^{63} +1.41926 q^{65} -14.3563 q^{67} -1.22503 q^{69} -6.25582 q^{71} -4.75709 q^{73} -1.00000 q^{75} +16.0492 q^{77} -6.67761 q^{79} +1.00000 q^{81} +3.21268 q^{83} +7.66274 q^{85} -1.92052 q^{87} +3.09435 q^{89} +6.11280 q^{91} +1.00000 q^{93} +6.88777 q^{95} -3.71395 q^{97} +3.72629 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - q^7 + 4 * q^9 + q^11 + 10 * q^13 - 4 * q^15 + 12 * q^17 + 5 * q^19 + q^21 + q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 - 4 * q^31 - q^33 - q^35 + 2 * q^37 - 10 * q^39 + 4 * q^41 - 9 * q^43 + 4 * q^45 + 5 * q^49 - 12 * q^51 + 13 * q^53 + q^55 - 5 * q^57 - 6 * q^59 - q^63 + 10 * q^65 - 8 * q^67 - q^69 - 9 * q^71 + 7 * q^73 - 4 * q^75 + 33 * q^77 + 5 * q^79 + 4 * q^81 + 2 * q^83 + 12 * q^85 - 2 * q^87 + 5 * q^89 - 2 * q^91 + 4 * q^93 + 5 * q^95 + 6 * 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
$7$	4.30703	1.62790	0.813952	−	0.580932i	$-0.197312\pi$
$7$	4.30703	1.62790	0.813952	+	0.580932i	$0.197312\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.72629	1.12352	0.561760	−	0.827301i	$-0.310125\pi$
$11$	3.72629	1.12352	0.561760	+	0.827301i	$0.310125\pi$
$12$	0	0
$13$	1.41926	0.393632	0.196816	−	0.980440i	$-0.436940\pi$
$13$	1.41926	0.393632	0.196816	+	0.980440i	$0.436940\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.66274	1.85849	0.929244	−	0.369467i	$-0.120460\pi$
$17$	7.66274	1.85849	0.929244	+	0.369467i	$0.120460\pi$
$18$	0	0
$19$	6.88777	1.58016	0.790081	−	0.613002i	$-0.210038\pi$
$19$	6.88777	1.58016	0.790081	+	0.613002i	$0.210038\pi$
$20$	0	0
$21$	−4.30703	−0.939871
$22$	0	0
$23$	1.22503	0.255436	0.127718	−	0.991811i	$-0.459235\pi$
$23$	1.22503	0.255436	0.127718	+	0.991811i	$0.459235\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.92052	0.356632	0.178316	−	0.983973i	$-0.442935\pi$
$29$	1.92052	0.356632	0.178316	+	0.983973i	$0.442935\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−3.72629	−0.648664
$34$	0	0
$35$	4.30703	0.728021
$36$	0	0
$37$	−0.580739	−0.0954730	−0.0477365	−	0.998860i	$-0.515201\pi$
$37$	−0.580739	−0.0954730	−0.0477365	+	0.998860i	$0.515201\pi$
$38$	0	0
$39$	−1.41926	−0.227264
$40$	0	0
$41$	−6.16400	−0.962656	−0.481328	−	0.876541i	$-0.659845\pi$
$41$	−6.16400	−0.962656	−0.481328	+	0.876541i	$0.659845\pi$
$42$	0	0
$43$	−10.0492	−1.53250	−0.766248	−	0.642545i	$-0.777879\pi$
$43$	−10.0492	−1.53250	−0.766248	+	0.642545i	$0.777879\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.82422	0.995415	0.497707	−	0.867345i	$-0.334175\pi$
$47$	6.82422	0.995415	0.497707	+	0.867345i	$0.334175\pi$
$48$	0	0
$49$	11.5505	1.65007
$50$	0	0
$51$	−7.66274	−1.07300
$52$	0	0
$53$	−2.93898	−0.403699	−0.201850	−	0.979417i	$-0.564695\pi$
$53$	−2.93898	−0.403699	−0.201850	+	0.979417i	$0.564695\pi$
$54$	0	0
$55$	3.72629	0.502453
$56$	0	0
$57$	−6.88777	−0.912307
$58$	0	0
$59$	−5.08200	−0.661620	−0.330810	−	0.943697i	$-0.607322\pi$
$59$	−5.08200	−0.661620	−0.330810	+	0.943697i	$0.607322\pi$
$60$	0	0
$61$	4.32296	0.553498	0.276749	−	0.960942i	$-0.410743\pi$
$61$	4.32296	0.553498	0.276749	+	0.960942i	$0.410743\pi$
$62$	0	0
$63$	4.30703	0.542635
$64$	0	0
$65$	1.41926	0.176038
$66$	0	0
$67$	−14.3563	−1.75390	−0.876949	−	0.480583i	$-0.840425\pi$
$67$	−14.3563	−1.75390	−0.876949	+	0.480583i	$0.840425\pi$
$68$	0	0
$69$	−1.22503	−0.147476
$70$	0	0
$71$	−6.25582	−0.742430	−0.371215	−	0.928547i	$-0.621059\pi$
$71$	−6.25582	−0.742430	−0.371215	+	0.928547i	$0.621059\pi$
$72$	0	0
$73$	−4.75709	−0.556775	−0.278387	−	0.960469i	$-0.589800\pi$
$73$	−4.75709	−0.556775	−0.278387	+	0.960469i	$0.589800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	16.0492	1.82898
$78$	0	0
$79$	−6.67761	−0.751290	−0.375645	−	0.926764i	$-0.622579\pi$
$79$	−6.67761	−0.751290	−0.375645	+	0.926764i	$0.622579\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.21268	0.352638	0.176319	−	0.984333i	$-0.443581\pi$
$83$	3.21268	0.352638	0.176319	+	0.984333i	$0.443581\pi$
$84$	0	0
$85$	7.66274	0.831141
$86$	0	0
$87$	−1.92052	−0.205902
$88$	0	0
$89$	3.09435	0.328000	0.164000	−	0.986460i	$-0.447560\pi$
$89$	3.09435	0.328000	0.164000	+	0.986460i	$0.447560\pi$
$90$	0	0
$91$	6.11280	0.640795
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	6.88777	0.706670
$96$	0	0
$97$	−3.71395	−0.377094	−0.188547	−	0.982064i	$-0.560378\pi$
$97$	−3.71395	−0.377094	−0.188547	+	0.982064i	$0.560378\pi$
$98$	0	0
$99$	3.72629	0.374506

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.t.1.4	✓	4
4.3	odd	2		7440.2.a.ca.1.1		4

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.30703 q^{7} +1.00000 q^{9} +3.72629 q^{11} +1.41926 q^{13} -1.00000 q^{15} +7.66274 q^{17} +6.88777 q^{19} -4.30703 q^{21} +1.22503 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.92052 q^{29} -1.00000 q^{31} -3.72629 q^{33} +4.30703 q^{35} -0.580739 q^{37} -1.41926 q^{39} -6.16400 q^{41} -10.0492 q^{43} +1.00000 q^{45} +6.82422 q^{47} +11.5505 q^{49} -7.66274 q^{51} -2.93898 q^{53} +3.72629 q^{55} -6.88777 q^{57} -5.08200 q^{59} +4.32296 q^{61} +4.30703 q^{63} +1.41926 q^{65} -14.3563 q^{67} -1.22503 q^{69} -6.25582 q^{71} -4.75709 q^{73} -1.00000 q^{75} +16.0492 q^{77} -6.67761 q^{79} +1.00000 q^{81} +3.21268 q^{83} +7.66274 q^{85} -1.92052 q^{87} +3.09435 q^{89} +6.11280 q^{91} +1.00000 q^{93} +6.88777 q^{95} -3.71395 q^{97} +3.72629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - q^7 + 4 * q^9 + q^11 + 10 * q^13 - 4 * q^15 + 12 * q^17 + 5 * q^19 + q^21 + q^23 + 4 * q^25 - 4 * q^27 + 2 * q^29 - 4 * q^31 - q^33 - q^35 + 2 * q^37 - 10 * q^39 + 4 * q^41 - 9 * q^43 + 4 * q^45 + 5 * q^49 - 12 * q^51 + 13 * q^53 + q^55 - 5 * q^57 - 6 * q^59 - q^63 + 10 * q^65 - 8 * q^67 - q^69 - 9 * q^71 + 7 * q^73 - 4 * q^75 + 33 * q^77 + 5 * q^79 + 4 * q^81 + 2 * q^83 + 12 * q^85 - 2 * q^87 + 5 * q^89 - 2 * q^91 + 4 * q^93 + 5 * q^95 + 6 * q^97 + q^99

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