LMFDB - Embedded newform 5800.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5800,2,Mod(1,5800)] 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("5800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5800 = 2^{3} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,1,0,0,0,-7,0,12,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$46.3132331723$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.3145252.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 11x^{3} + 9x^{2} + 22x - 11 $ x^5 - 2x^4 - 11x^3 + 9x^2 + 22x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.66804$ of defining polynomial
Character	$\chi$	$=$	5800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.31966 q^{3} -4.86927 q^{7} +8.02012 q^{9} -4.15085 q^{11} -2.66804 q^{13} +2.86927 q^{17} -0.651620 q^{19} -16.1643 q^{21} +4.65573 q^{23} +16.6651 q^{27} +1.00000 q^{29} -4.17251 q^{31} -13.7794 q^{33} -11.3238 q^{37} -8.85697 q^{39} +3.83684 q^{41} -12.0094 q^{43} -7.48692 q^{47} +16.7098 q^{49} +9.52500 q^{51} -1.61919 q^{53} -2.16316 q^{57} -3.33196 q^{59} -11.0459 q^{61} -39.0522 q^{63} +6.18368 q^{67} +15.4554 q^{69} +0.903244 q^{71} -5.03243 q^{73} +20.2116 q^{77} -11.9918 q^{79} +31.2620 q^{81} -6.22453 q^{83} +3.31966 q^{87} +4.63931 q^{89} +12.9914 q^{91} -13.8513 q^{93} -11.5565 q^{97} -33.2903 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 7 q^{7} + 12 q^{9} - 10 q^{11} - 3 q^{13} - 3 q^{17} + 2 q^{19} + 4 q^{21} - 13 q^{23} + 4 q^{27} + 5 q^{29} + 7 q^{31} - 12 q^{33} - 10 q^{37} - q^{39} + 4 q^{41} - 17 q^{43} - 6 q^{47}+ \cdots - 96 q^{99}+O(q^{100}) $ 5 * q + q^3 - 7 * q^7 + 12 * q^9 - 10 * q^11 - 3 * q^13 - 3 * q^17 + 2 * q^19 + 4 * q^21 - 13 * q^23 + 4 * q^27 + 5 * q^29 + 7 * q^31 - 12 * q^33 - 10 * q^37 - q^39 + 4 * q^41 - 17 * q^43 - 6 * q^47 + 28 * q^49 - 6 * q^51 - 5 * q^53 - 26 * q^57 - 27 * q^59 + 21 * q^61 - 54 * q^63 + 10 * q^67 + 25 * q^69 - 4 * q^71 - 23 * q^73 - 2 * q^77 - 3 * q^79 + 57 * q^81 + 12 * q^83 + q^87 - 8 * q^89 + 16 * q^91 - 46 * q^93 - 25 * q^97 - 96 * 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
$3$	3.31966	1.91660	0.958302	−	0.285756i	$-0.0922446\pi$
$3$	3.31966	1.91660	0.958302	+	0.285756i	$0.0922446\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.86927	−1.84041	−0.920206	−	0.391434i	$-0.871979\pi$
$7$	−4.86927	−1.84041	−0.920206	+	0.391434i	$0.871979\pi$
$8$	0	0
$9$	8.02012	2.67337
$10$	0	0
$11$	−4.15085	−1.25153	−0.625764	−	0.780012i	$-0.715213\pi$
$11$	−4.15085	−1.25153	−0.625764	+	0.780012i	$0.715213\pi$
$12$	0	0
$13$	−2.66804	−0.739980	−0.369990	−	0.929036i	$-0.620639\pi$
$13$	−2.66804	−0.739980	−0.369990	+	0.929036i	$0.620639\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.86927	0.695901	0.347951	−	0.937513i	$-0.386878\pi$
$17$	2.86927	0.695901	0.347951	+	0.937513i	$0.386878\pi$
$18$	0	0
$19$	−0.651620	−0.149492	−0.0747460	−	0.997203i	$-0.523815\pi$
$19$	−0.651620	−0.149492	−0.0747460	+	0.997203i	$0.523815\pi$
$20$	0	0
$21$	−16.1643	−3.52734
$22$	0	0
$23$	4.65573	0.970787	0.485393	−	0.874296i	$-0.338676\pi$
$23$	4.65573	0.970787	0.485393	+	0.874296i	$0.338676\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.6651	3.20720
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.17251	−0.749406	−0.374703	−	0.927145i	$-0.622255\pi$
$31$	−4.17251	−0.749406	−0.374703	+	0.927145i	$0.622255\pi$
$32$	0	0
$33$	−13.7794	−2.39869
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.3238	−1.86162	−0.930808	−	0.365509i	$-0.880895\pi$
$37$	−11.3238	−1.86162	−0.930808	+	0.365509i	$0.880895\pi$
$38$	0	0
$39$	−8.85697	−1.41825
$40$	0	0
$41$	3.83684	0.599214	0.299607	−	0.954063i	$-0.403144\pi$
$41$	3.83684	0.599214	0.299607	+	0.954063i	$0.403144\pi$
$42$	0	0
$43$	−12.0094	−1.83141	−0.915705	−	0.401851i	$-0.868367\pi$
$43$	−12.0094	−1.83141	−0.915705	+	0.401851i	$0.868367\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.48692	−1.09208	−0.546040	−	0.837759i	$-0.683865\pi$
$47$	−7.48692	−1.09208	−0.546040	+	0.837759i	$0.683865\pi$
$48$	0	0
$49$	16.7098	2.38712
$50$	0	0
$51$	9.52500	1.33377
$52$	0	0
$53$	−1.61919	−0.222413	−0.111206	−	0.993797i	$-0.535471\pi$
$53$	−1.61919	−0.222413	−0.111206	+	0.993797i	$0.535471\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.16316	−0.286517
$58$	0	0
$59$	−3.33196	−0.433785	−0.216892	−	0.976196i	$-0.569592\pi$
$59$	−3.33196	−0.433785	−0.216892	+	0.976196i	$0.569592\pi$
$60$	0	0
$61$	−11.0459	−1.41428	−0.707141	−	0.707072i	$-0.750015\pi$
$61$	−11.0459	−1.41428	−0.707141	+	0.707072i	$0.750015\pi$
$62$	0	0
$63$	−39.0522	−4.92011
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.18368	0.755457	0.377729	−	0.925916i	$-0.376705\pi$
$67$	6.18368	0.755457	0.377729	+	0.925916i	$0.376705\pi$
$68$	0	0
$69$	15.4554	1.86062
$70$	0	0
$71$	0.903244	0.107195	0.0535977	−	0.998563i	$-0.482931\pi$
$71$	0.903244	0.107195	0.0535977	+	0.998563i	$0.482931\pi$
$72$	0	0
$73$	−5.03243	−0.589001	−0.294501	−	0.955651i	$-0.595153\pi$
$73$	−5.03243	−0.589001	−0.294501	+	0.955651i	$0.595153\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.2116	2.30333
$78$	0	0
$79$	−11.9918	−1.34918	−0.674592	−	0.738191i	$-0.735680\pi$
$79$	−11.9918	−1.34918	−0.674592	+	0.738191i	$0.735680\pi$
$80$	0	0
$81$	31.2620	3.47356
$82$	0	0
$83$	−6.22453	−0.683231	−0.341616	−	0.939840i	$-0.610974\pi$
$83$	−6.22453	−0.683231	−0.341616	+	0.939840i	$0.610974\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.31966	0.355905
$88$	0	0
$89$	4.63931	0.491766	0.245883	−	0.969299i	$-0.420922\pi$
$89$	4.63931	0.491766	0.245883	+	0.969299i	$0.420922\pi$
$90$	0	0
$91$	12.9914	1.36187
$92$	0	0
$93$	−13.8513	−1.43631
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.5565	−1.17338	−0.586692	−	0.809810i	$-0.699570\pi$
$97$	−11.5565	−1.17338	−0.586692	+	0.809810i	$0.699570\pi$
$98$	0	0
$99$	−33.2903	−3.34580

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	5800.2.a.u.1.5		5
5.4	even	2		1160.2.a.h.1.1	✓	5
20.19	odd	2		2320.2.a.v.1.5		5
40.19	odd	2		9280.2.a.ci.1.1		5
40.29	even	2		9280.2.a.ck.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.h.1.1	✓	5	5.4	even	2
2320.2.a.v.1.5		5	20.19	odd	2
5800.2.a.u.1.5		5	1.1	even	1	trivial
9280.2.a.ci.1.1		5	40.19	odd	2
9280.2.a.ck.1.5		5	40.29	even	2

Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5800.a (trivial)

\(f(q)\)	\(=\)	\(q+3.31966 q^{3} -4.86927 q^{7} +8.02012 q^{9} -4.15085 q^{11} -2.66804 q^{13} +2.86927 q^{17} -0.651620 q^{19} -16.1643 q^{21} +4.65573 q^{23} +16.6651 q^{27} +1.00000 q^{29} -4.17251 q^{31} -13.7794 q^{33} -11.3238 q^{37} -8.85697 q^{39} +3.83684 q^{41} -12.0094 q^{43} -7.48692 q^{47} +16.7098 q^{49} +9.52500 q^{51} -1.61919 q^{53} -2.16316 q^{57} -3.33196 q^{59} -11.0459 q^{61} -39.0522 q^{63} +6.18368 q^{67} +15.4554 q^{69} +0.903244 q^{71} -5.03243 q^{73} +20.2116 q^{77} -11.9918 q^{79} +31.2620 q^{81} -6.22453 q^{83} +3.31966 q^{87} +4.63931 q^{89} +12.9914 q^{91} -13.8513 q^{93} -11.5565 q^{97} -33.2903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 7 q^{7} + 12 q^{9} - 10 q^{11} - 3 q^{13} - 3 q^{17} + 2 q^{19} + 4 q^{21} - 13 q^{23} + 4 q^{27} + 5 q^{29} + 7 q^{31} - 12 q^{33} - 10 q^{37} - q^{39} + 4 q^{41} - 17 q^{43} - 6 q^{47}+ \cdots - 96 q^{99}+O(q^{100}) \) 5 * q + q^3 - 7 * q^7 + 12 * q^9 - 10 * q^11 - 3 * q^13 - 3 * q^17 + 2 * q^19 + 4 * q^21 - 13 * q^23 + 4 * q^27 + 5 * q^29 + 7 * q^31 - 12 * q^33 - 10 * q^37 - q^39 + 4 * q^41 - 17 * q^43 - 6 * q^47 + 28 * q^49 - 6 * q^51 - 5 * q^53 - 26 * q^57 - 27 * q^59 + 21 * q^61 - 54 * q^63 + 10 * q^67 + 25 * q^69 - 4 * q^71 - 23 * q^73 - 2 * q^77 - 3 * q^79 + 57 * q^81 + 12 * q^83 + q^87 - 8 * q^89 + 16 * q^91 - 46 * q^93 - 25 * q^97 - 96 * q^99

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