LMFDB - Embedded newform 3720.2.a.i.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 = [2,0,-2,0,-2,0,2] 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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -2.47214 q^{19} -3.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} -1.00000 q^{31} -3.23607 q^{35} +9.23607 q^{37} -1.23607 q^{39} -6.00000 q^{41} -2.47214 q^{43} -1.00000 q^{45} -8.94427 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} +2.47214 q^{57} +11.2361 q^{59} -3.52786 q^{61} +3.23607 q^{63} -1.23607 q^{65} -9.70820 q^{67} +4.00000 q^{69} -4.76393 q^{71} -10.1803 q^{73} -1.00000 q^{75} +8.94427 q^{79} +1.00000 q^{81} +12.9443 q^{83} +4.47214 q^{85} +2.76393 q^{87} -2.76393 q^{89} +4.00000 q^{91} +1.00000 q^{93} +2.47214 q^{95} +0.472136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

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
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	9.23607	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$37$	9.23607	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$38$	0	0
$39$	−1.23607	−0.197929
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.47214	0.327442
$58$	0	0
$59$	11.2361	1.46281	0.731406	−	0.681943i	$-0.238865\pi$
$59$	11.2361	1.46281	0.731406	+	0.681943i	$0.238865\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−9.70820	−1.18605	−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820	−1.18605	−0.593023	+	0.805186i	$0.702066\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−4.76393	−0.565375	−0.282687	−	0.959212i	$-0.591226\pi$
$71$	−4.76393	−0.565375	−0.282687	+	0.959212i	$0.591226\pi$
$72$	0	0
$73$	−10.1803	−1.19152	−0.595759	−	0.803163i	$-0.703149\pi$
$73$	−10.1803	−1.19152	−0.595759	+	0.803163i	$0.703149\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	2.76393	0.296325
$88$	0	0
$89$	−2.76393	−0.292976	−0.146488	−	0.989212i	$-0.546797\pi$
$89$	−2.76393	−0.292976	−0.146488	+	0.989212i	$0.546797\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.47214	0.253636
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	0	0
$99$	0	0

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.i.1.2	✓	2
4.3	odd	2		7440.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.i.1.2	✓	2	1.1	even	1	trivial
7440.2.a.bi.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.23607 q^{7} +1.00000 q^{9} +1.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -2.47214 q^{19} -3.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.76393 q^{29} -1.00000 q^{31} -3.23607 q^{35} +9.23607 q^{37} -1.23607 q^{39} -6.00000 q^{41} -2.47214 q^{43} -1.00000 q^{45} -8.94427 q^{47} +3.47214 q^{49} +4.47214 q^{51} -0.472136 q^{53} +2.47214 q^{57} +11.2361 q^{59} -3.52786 q^{61} +3.23607 q^{63} -1.23607 q^{65} -9.70820 q^{67} +4.00000 q^{69} -4.76393 q^{71} -10.1803 q^{73} -1.00000 q^{75} +8.94427 q^{79} +1.00000 q^{81} +12.9443 q^{83} +4.47214 q^{85} +2.76393 q^{87} -2.76393 q^{89} +4.00000 q^{91} +1.00000 q^{93} +2.47214 q^{95} +0.472136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

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