LMFDB - Embedded newform 1160.2.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1160,2,Mod(1,1160)] 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("1160.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,-3,0,-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:	$9.26264663447$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.93543 q^{3} -1.00000 q^{5} -1.25410 q^{7} +5.61676 q^{9} +2.50820 q^{11} -2.93543 q^{13} -2.93543 q^{15} +7.12497 q^{17} +4.85446 q^{19} -3.68133 q^{21} +1.57277 q^{23} +1.00000 q^{25} +7.68133 q^{27} -1.00000 q^{29} -4.61676 q^{31} +7.36266 q^{33} +1.25410 q^{35} +9.87086 q^{37} -8.61676 q^{39} +0.508203 q^{41} +1.38324 q^{43} -5.61676 q^{45} -1.36266 q^{47} -5.42723 q^{49} +20.9149 q^{51} +2.23769 q^{53} -2.50820 q^{55} +14.2499 q^{57} -11.6608 q^{59} +3.41082 q^{61} -7.04399 q^{63} +2.93543 q^{65} -6.72532 q^{67} +4.61676 q^{69} +14.7253 q^{71} -12.3585 q^{73} +2.93543 q^{75} -3.14554 q^{77} +3.91903 q^{79} +5.69774 q^{81} -1.87086 q^{83} -7.12497 q^{85} -2.93543 q^{87} -11.8709 q^{89} +3.68133 q^{91} -13.5522 q^{93} -4.85446 q^{95} +5.31450 q^{97} +14.0880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} + 3 q^{25} + 16 q^{27} - 3 q^{29} + q^{31} + 8 q^{33} + 3 q^{35} + 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 + 6 * q^11 - q^13 - q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 + 3 * q^25 + 16 * q^27 - 3 * q^29 + q^31 + 8 * q^33 + 3 * q^35 + 14 * q^37 - 11 * q^39 + 19 * q^43 - 2 * q^45 + 10 * q^47 - 10 * q^49 + 26 * q^51 + 9 * q^53 - 6 * q^55 + 10 * q^57 + q^59 + 7 * q^61 + q^65 + 8 * q^67 - q^69 + 16 * q^71 + 9 * q^73 + q^75 - 22 * q^77 + 7 * q^79 + 7 * q^81 + 10 * q^83 - 5 * q^85 - q^87 - 20 * q^89 + 4 * q^91 - 18 * q^93 - 2 * q^95 - 9 * 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
$3$	2.93543	1.69477	0.847386	−	0.530977i	$-0.178175\pi$
$3$	2.93543	1.69477	0.847386	+	0.530977i	$0.178175\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.25410	−0.474006	−0.237003	−	0.971509i	$-0.576165\pi$
$7$	−1.25410	−0.474006	−0.237003	+	0.971509i	$0.576165\pi$
$8$	0	0
$9$	5.61676	1.87225
$10$	0	0
$11$	2.50820	0.756252	0.378126	−	0.925754i	$-0.376569\pi$
$11$	2.50820	0.756252	0.378126	+	0.925754i	$0.376569\pi$
$12$	0	0
$13$	−2.93543	−0.814142	−0.407071	−	0.913396i	$-0.633450\pi$
$13$	−2.93543	−0.814142	−0.407071	+	0.913396i	$0.633450\pi$
$14$	0	0
$15$	−2.93543	−0.757925
$16$	0	0
$17$	7.12497	1.72806	0.864029	−	0.503442i	$-0.167933\pi$
$17$	7.12497	1.72806	0.864029	+	0.503442i	$0.167933\pi$
$18$	0	0
$19$	4.85446	1.11369	0.556845	−	0.830617i	$-0.312012\pi$
$19$	4.85446	1.11369	0.556845	+	0.830617i	$0.312012\pi$
$20$	0	0
$21$	−3.68133	−0.803332
$22$	0	0
$23$	1.57277	0.327945	0.163973	−	0.986465i	$-0.447569\pi$
$23$	1.57277	0.327945	0.163973	+	0.986465i	$0.447569\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.68133	1.47827
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.61676	−0.829195	−0.414598	−	0.910005i	$-0.636078\pi$
$31$	−4.61676	−0.829195	−0.414598	+	0.910005i	$0.636078\pi$
$32$	0	0
$33$	7.36266	1.28167
$34$	0	0
$35$	1.25410	0.211982
$36$	0	0
$37$	9.87086	1.62276	0.811380	−	0.584519i	$-0.198717\pi$
$37$	9.87086	1.62276	0.811380	+	0.584519i	$0.198717\pi$
$38$	0	0
$39$	−8.61676	−1.37979
$40$	0	0
$41$	0.508203	0.0793680	0.0396840	−	0.999212i	$-0.487365\pi$
$41$	0.508203	0.0793680	0.0396840	+	0.999212i	$0.487365\pi$
$42$	0	0
$43$	1.38324	0.210942	0.105471	−	0.994422i	$-0.466365\pi$
$43$	1.38324	0.210942	0.105471	+	0.994422i	$0.466365\pi$
$44$	0	0
$45$	−5.61676	−0.837298
$46$	0	0
$47$	−1.36266	−0.198765	−0.0993823	−	0.995049i	$-0.531687\pi$
$47$	−1.36266	−0.198765	−0.0993823	+	0.995049i	$0.531687\pi$
$48$	0	0
$49$	−5.42723	−0.775318
$50$	0	0
$51$	20.9149	2.92867
$52$	0	0
$53$	2.23769	0.307371	0.153686	−	0.988120i	$-0.450886\pi$
$53$	2.23769	0.307371	0.153686	+	0.988120i	$0.450886\pi$
$54$	0	0
$55$	−2.50820	−0.338206
$56$	0	0
$57$	14.2499	1.88745
$58$	0	0
$59$	−11.6608	−1.51810	−0.759050	−	0.651032i	$-0.774336\pi$
$59$	−11.6608	−1.51810	−0.759050	+	0.651032i	$0.774336\pi$
$60$	0	0
$61$	3.41082	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$61$	3.41082	0.436711	0.218356	+	0.975869i	$0.429931\pi$
$62$	0	0
$63$	−7.04399	−0.887460
$64$	0	0
$65$	2.93543	0.364096
$66$	0	0
$67$	−6.72532	−0.821629	−0.410814	−	0.911719i	$-0.634756\pi$
$67$	−6.72532	−0.821629	−0.410814	+	0.911719i	$0.634756\pi$
$68$	0	0
$69$	4.61676	0.555793
$70$	0	0
$71$	14.7253	1.74757	0.873787	−	0.486309i	$-0.161657\pi$
$71$	14.7253	1.74757	0.873787	+	0.486309i	$0.161657\pi$
$72$	0	0
$73$	−12.3585	−1.44645	−0.723226	−	0.690611i	$-0.757342\pi$
$73$	−12.3585	−1.44645	−0.723226	+	0.690611i	$0.757342\pi$
$74$	0	0
$75$	2.93543	0.338955
$76$	0	0
$77$	−3.14554	−0.358468
$78$	0	0
$79$	3.91903	0.440925	0.220462	−	0.975395i	$-0.429243\pi$
$79$	3.91903	0.440925	0.220462	+	0.975395i	$0.429243\pi$
$80$	0	0
$81$	5.69774	0.633082
$82$	0	0
$83$	−1.87086	−0.205354	−0.102677	−	0.994715i	$-0.532741\pi$
$83$	−1.87086	−0.205354	−0.102677	+	0.994715i	$0.532741\pi$
$84$	0	0
$85$	−7.12497	−0.772811
$86$	0	0
$87$	−2.93543	−0.314711
$88$	0	0
$89$	−11.8709	−1.25831	−0.629155	−	0.777280i	$-0.716599\pi$
$89$	−11.8709	−1.25831	−0.629155	+	0.777280i	$0.716599\pi$
$90$	0	0
$91$	3.68133	0.385908
$92$	0	0
$93$	−13.5522	−1.40530
$94$	0	0
$95$	−4.85446	−0.498057
$96$	0	0
$97$	5.31450	0.539606	0.269803	−	0.962916i	$-0.413041\pi$
$97$	5.31450	0.539606	0.269803	+	0.962916i	$0.413041\pi$
$98$	0	0
$99$	14.0880	1.41590

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	1160.2.a.g.1.3	✓	3
4.3	odd	2		2320.2.a.p.1.1		3
5.4	even	2		5800.2.a.q.1.1		3
8.3	odd	2		9280.2.a.bq.1.3		3
8.5	even	2		9280.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.g.1.3	✓	3	1.1	even	1	trivial
2320.2.a.p.1.1		3	4.3	odd	2
5800.2.a.q.1.1		3	5.4	even	2
9280.2.a.bo.1.1		3	8.5	even	2
9280.2.a.bq.1.3		3	8.3	odd	2

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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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.93543 q^{3} -1.00000 q^{5} -1.25410 q^{7} +5.61676 q^{9} +2.50820 q^{11} -2.93543 q^{13} -2.93543 q^{15} +7.12497 q^{17} +4.85446 q^{19} -3.68133 q^{21} +1.57277 q^{23} +1.00000 q^{25} +7.68133 q^{27} -1.00000 q^{29} -4.61676 q^{31} +7.36266 q^{33} +1.25410 q^{35} +9.87086 q^{37} -8.61676 q^{39} +0.508203 q^{41} +1.38324 q^{43} -5.61676 q^{45} -1.36266 q^{47} -5.42723 q^{49} +20.9149 q^{51} +2.23769 q^{53} -2.50820 q^{55} +14.2499 q^{57} -11.6608 q^{59} +3.41082 q^{61} -7.04399 q^{63} +2.93543 q^{65} -6.72532 q^{67} +4.61676 q^{69} +14.7253 q^{71} -12.3585 q^{73} +2.93543 q^{75} -3.14554 q^{77} +3.91903 q^{79} +5.69774 q^{81} -1.87086 q^{83} -7.12497 q^{85} -2.93543 q^{87} -11.8709 q^{89} +3.68133 q^{91} -13.5522 q^{93} -4.85446 q^{95} +5.31450 q^{97} +14.0880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} + 3 q^{25} + 16 q^{27} - 3 q^{29} + q^{31} + 8 q^{33} + 3 q^{35} + 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 + 6 * q^11 - q^13 - q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 + 3 * q^25 + 16 * q^27 - 3 * q^29 + q^31 + 8 * q^33 + 3 * q^35 + 14 * q^37 - 11 * q^39 + 19 * q^43 - 2 * q^45 + 10 * q^47 - 10 * q^49 + 26 * q^51 + 9 * q^53 - 6 * q^55 + 10 * q^57 + q^59 + 7 * q^61 + q^65 + 8 * q^67 - q^69 + 16 * q^71 + 9 * q^73 + q^75 - 22 * q^77 + 7 * q^79 + 7 * q^81 + 10 * q^83 - 5 * q^85 - q^87 - 20 * q^89 + 4 * q^91 - 18 * q^93 - 2 * q^95 - 9 * q^97

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