LMFDB - Embedded newform 3720.2.a.t.1.2

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			1.2
Root			$-0.727241$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.58859 q^{7} +1.00000 q^{9} -0.861351 q^{11} +3.72724 q^{13} -1.00000 q^{15} +5.01664 q^{17} -2.31583 q^{19} +2.58859 q^{21} -5.33247 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.19836 q^{29} -1.00000 q^{31} +0.861351 q^{33} -2.58859 q^{35} +1.72724 q^{37} -3.72724 q^{39} -5.48776 q^{41} +3.77031 q^{43} +1.00000 q^{45} -0.437842 q^{47} -0.299193 q^{49} -5.01664 q^{51} -8.82023 q^{53} -0.861351 q^{55} +2.31583 q^{57} -4.74388 q^{59} -4.90896 q^{61} -2.58859 q^{63} +3.72724 q^{65} +6.35891 q^{67} +5.33247 q^{69} +15.7247 q^{71} +15.2535 q^{73} -1.00000 q^{75} +2.22969 q^{77} +9.05517 q^{79} +1.00000 q^{81} +13.6816 q^{83} +5.01664 q^{85} -6.19836 q^{87} -14.2702 q^{89} -9.64830 q^{91} +1.00000 q^{93} -2.31583 q^{95} -16.1527 q^{97} -0.861351 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.58859	−0.978396	−0.489198	−	0.872173i	$-0.662710\pi$
$7$	−2.58859	−0.978396	−0.489198	+	0.872173i	$0.662710\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.861351	−0.259707	−0.129854	−	0.991533i	$-0.541451\pi$
$11$	−0.861351	−0.259707	−0.129854	+	0.991533i	$0.541451\pi$
$12$	0	0
$13$	3.72724	1.03375	0.516875	−	0.856061i	$-0.327095\pi$
$13$	3.72724	1.03375	0.516875	+	0.856061i	$0.327095\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.01664	1.21671	0.608357	−	0.793664i	$-0.291829\pi$
$17$	5.01664	1.21671	0.608357	+	0.793664i	$0.291829\pi$
$18$	0	0
$19$	−2.31583	−0.531288	−0.265644	−	0.964071i	$-0.585585\pi$
$19$	−2.31583	−0.531288	−0.265644	+	0.964071i	$0.585585\pi$
$20$	0	0
$21$	2.58859	0.564877
$22$	0	0
$23$	−5.33247	−1.11190	−0.555949	−	0.831217i	$-0.687645\pi$
$23$	−5.33247	−1.11190	−0.555949	+	0.831217i	$0.687645\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.19836	1.15101	0.575503	−	0.817799i	$-0.304806\pi$
$29$	6.19836	1.15101	0.575503	+	0.817799i	$0.304806\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0.861351	0.149942
$34$	0	0
$35$	−2.58859	−0.437552
$36$	0	0
$37$	1.72724	0.283957	0.141978	−	0.989870i	$-0.454654\pi$
$37$	1.72724	0.283957	0.141978	+	0.989870i	$0.454654\pi$
$38$	0	0
$39$	−3.72724	−0.596836
$40$	0	0
$41$	−5.48776	−0.857044	−0.428522	−	0.903531i	$-0.640966\pi$
$41$	−5.48776	−0.857044	−0.428522	+	0.903531i	$0.640966\pi$
$42$	0	0
$43$	3.77031	0.574967	0.287484	−	0.957786i	$-0.407181\pi$
$43$	3.77031	0.574967	0.287484	+	0.957786i	$0.407181\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.437842	−0.0638658	−0.0319329	−	0.999490i	$-0.510166\pi$
$47$	−0.437842	−0.0638658	−0.0319329	+	0.999490i	$0.510166\pi$
$48$	0	0
$49$	−0.299193	−0.0427418
$50$	0	0
$51$	−5.01664	−0.702470
$52$	0	0
$53$	−8.82023	−1.21155	−0.605776	−	0.795635i	$-0.707137\pi$
$53$	−8.82023	−1.21155	−0.605776	+	0.795635i	$0.707137\pi$
$54$	0	0
$55$	−0.861351	−0.116145
$56$	0	0
$57$	2.31583	0.306739
$58$	0	0
$59$	−4.74388	−0.617601	−0.308800	−	0.951127i	$-0.599927\pi$
$59$	−4.74388	−0.617601	−0.308800	+	0.951127i	$0.599927\pi$
$60$	0	0
$61$	−4.90896	−0.628528	−0.314264	−	0.949336i	$-0.601758\pi$
$61$	−4.90896	−0.628528	−0.314264	+	0.949336i	$0.601758\pi$
$62$	0	0
$63$	−2.58859	−0.326132
$64$	0	0
$65$	3.72724	0.462307
$66$	0	0
$67$	6.35891	0.776864	0.388432	−	0.921477i	$-0.373017\pi$
$67$	6.35891	0.776864	0.388432	+	0.921477i	$0.373017\pi$
$68$	0	0
$69$	5.33247	0.641954
$70$	0	0
$71$	15.7247	1.86617	0.933087	−	0.359651i	$-0.117104\pi$
$71$	15.7247	1.86617	0.933087	+	0.359651i	$0.117104\pi$
$72$	0	0
$73$	15.2535	1.78529	0.892646	−	0.450759i	$-0.148847\pi$
$73$	15.2535	1.78529	0.892646	+	0.450759i	$0.148847\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.22969	0.254096
$78$	0	0
$79$	9.05517	1.01879	0.509393	−	0.860534i	$-0.329870\pi$
$79$	9.05517	1.01879	0.509393	+	0.860534i	$0.329870\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.6816	1.50175	0.750874	−	0.660445i	$-0.229632\pi$
$83$	13.6816	1.50175	0.750874	+	0.660445i	$0.229632\pi$
$84$	0	0
$85$	5.01664	0.544131
$86$	0	0
$87$	−6.19836	−0.664534
$88$	0	0
$89$	−14.2702	−1.51264	−0.756318	−	0.654204i	$-0.773004\pi$
$89$	−14.2702	−1.51264	−0.756318	+	0.654204i	$0.773004\pi$
$90$	0	0
$91$	−9.64830	−1.01142
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−2.31583	−0.237599
$96$	0	0
$97$	−16.1527	−1.64006	−0.820029	−	0.572321i	$-0.806043\pi$
$97$	−16.1527	−1.64006	−0.820029	+	0.572321i	$0.806043\pi$
$98$	0	0
$99$	−0.861351	−0.0865690

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.1.2	✓	4
4.3	odd	2		7440.2.a.ca.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.t.1.2	✓	4	1.1	even	1	trivial
7440.2.a.ca.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.58859 q^{7} +1.00000 q^{9} -0.861351 q^{11} +3.72724 q^{13} -1.00000 q^{15} +5.01664 q^{17} -2.31583 q^{19} +2.58859 q^{21} -5.33247 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.19836 q^{29} -1.00000 q^{31} +0.861351 q^{33} -2.58859 q^{35} +1.72724 q^{37} -3.72724 q^{39} -5.48776 q^{41} +3.77031 q^{43} +1.00000 q^{45} -0.437842 q^{47} -0.299193 q^{49} -5.01664 q^{51} -8.82023 q^{53} -0.861351 q^{55} +2.31583 q^{57} -4.74388 q^{59} -4.90896 q^{61} -2.58859 q^{63} +3.72724 q^{65} +6.35891 q^{67} +5.33247 q^{69} +15.7247 q^{71} +15.2535 q^{73} -1.00000 q^{75} +2.22969 q^{77} +9.05517 q^{79} +1.00000 q^{81} +13.6816 q^{83} +5.01664 q^{85} -6.19836 q^{87} -14.2702 q^{89} -9.64830 q^{91} +1.00000 q^{93} -2.31583 q^{95} -16.1527 q^{97} -0.861351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - q^{7} + 4 q^{9} + q^{11} + 10 q^{13} - 4 q^{15} + 12 q^{17} + 5 q^{19} + q^{21} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} - q^{35} + 2 q^{37} - 10 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 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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