LMFDB - Embedded newform 5800.2.a.v.1.3

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 = [5,0,1,0,0,0,1,0,12,0,-4] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.6083172.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 $ x^5 - x^4 - 13x^3 + 10x^2 + 40*x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$0.689426$ 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)

$f(q)$	$=$	$q+0.689426 q^{3} -4.55109 q^{7} -2.52469 q^{9} -0.973602 q^{11} -2.28418 q^{13} -5.82707 q^{17} -0.302375 q^{19} -3.13764 q^{21} +1.41345 q^{23} -3.80887 q^{27} -1.00000 q^{29} -2.55109 q^{31} -0.671227 q^{33} -1.02640 q^{37} -1.57477 q^{39} +4.65483 q^{41} +3.92994 q^{43} -1.07648 q^{47} +13.7124 q^{49} -4.01733 q^{51} +4.85075 q^{53} -0.208465 q^{57} +1.66303 q^{59} +12.1969 q^{61} +11.4901 q^{63} -0.352455 q^{67} +0.974469 q^{69} -6.94651 q^{71} -16.9793 q^{73} +4.43095 q^{77} +9.96540 q^{79} +4.94814 q^{81} +13.9201 q^{83} -0.689426 q^{87} +4.82690 q^{89} +10.3955 q^{91} -1.75879 q^{93} -13.7388 q^{97} +2.45805 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + q^{7} + 12 q^{9} - 4 q^{11} - 13 q^{13} - 3 q^{17} + 8 q^{21} + 7 q^{23} + 4 q^{27} - 5 q^{29} + 11 q^{31} - 4 q^{33} - 6 q^{37} + 21 q^{39} + 16 q^{41} - 9 q^{43} - 2 q^{47} + 16 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 5 * q + q^3 + q^7 + 12 * q^9 - 4 * q^11 - 13 * q^13 - 3 * q^17 + 8 * q^21 + 7 * q^23 + 4 * q^27 - 5 * q^29 + 11 * q^31 - 4 * q^33 - 6 * q^37 + 21 * q^39 + 16 * q^41 - 9 * q^43 - 2 * q^47 + 16 * q^49 + 2 * q^51 - 7 * q^53 - 2 * q^57 + 5 * q^59 + 7 * q^61 + 28 * q^63 + 4 * q^67 + 23 * q^69 + 12 * q^71 - 7 * q^73 - 26 * q^77 + 45 * q^79 - 7 * q^81 + 22 * q^83 - q^87 + 24 * q^89 - 20 * q^91 + 10 * q^93 - 17 * q^97 + 10 * 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
$3$	0.689426	0.398040	0.199020	−	0.979995i	$-0.436224\pi$
$3$	0.689426	0.398040	0.199020	+	0.979995i	$0.436224\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.55109	−1.72015	−0.860075	−	0.510168i	$-0.829583\pi$
$7$	−4.55109	−1.72015	−0.860075	+	0.510168i	$0.829583\pi$
$8$	0	0
$9$	−2.52469	−0.841564
$10$	0	0
$11$	−0.973602	−0.293552	−0.146776	−	0.989170i	$-0.546890\pi$
$11$	−0.973602	−0.293552	−0.146776	+	0.989170i	$0.546890\pi$
$12$	0	0
$13$	−2.28418	−0.633516	−0.316758	−	0.948506i	$-0.602594\pi$
$13$	−2.28418	−0.633516	−0.316758	+	0.948506i	$0.602594\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.82707	−1.41327	−0.706636	−	0.707578i	$-0.749788\pi$
$17$	−5.82707	−1.41327	−0.706636	+	0.707578i	$0.749788\pi$
$18$	0	0
$19$	−0.302375	−0.0693697	−0.0346848	−	0.999398i	$-0.511043\pi$
$19$	−0.302375	−0.0693697	−0.0346848	+	0.999398i	$0.511043\pi$
$20$	0	0
$21$	−3.13764	−0.684689
$22$	0	0
$23$	1.41345	0.294724	0.147362	−	0.989083i	$-0.452922\pi$
$23$	1.41345	0.294724	0.147362	+	0.989083i	$0.452922\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.80887	−0.733017
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.55109	−0.458189	−0.229095	−	0.973404i	$-0.573577\pi$
$31$	−2.55109	−0.458189	−0.229095	+	0.973404i	$0.573577\pi$
$32$	0	0
$33$	−0.671227	−0.116846
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.02640	−0.168739	−0.0843694	−	0.996435i	$-0.526888\pi$
$37$	−1.02640	−0.168739	−0.0843694	+	0.996435i	$0.526888\pi$
$38$	0	0
$39$	−1.57477	−0.252165
$40$	0	0
$41$	4.65483	0.726962	0.363481	−	0.931602i	$-0.381588\pi$
$41$	4.65483	0.726962	0.363481	+	0.931602i	$0.381588\pi$
$42$	0	0
$43$	3.92994	0.599310	0.299655	−	0.954048i	$-0.403128\pi$
$43$	3.92994	0.599310	0.299655	+	0.954048i	$0.403128\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.07648	−0.157020	−0.0785102	−	0.996913i	$-0.525016\pi$
$47$	−1.07648	−0.157020	−0.0785102	+	0.996913i	$0.525016\pi$
$48$	0	0
$49$	13.7124	1.95892
$50$	0	0
$51$	−4.01733	−0.562539
$52$	0	0
$53$	4.85075	0.666302	0.333151	−	0.942874i	$-0.391888\pi$
$53$	4.85075	0.666302	0.333151	+	0.942874i	$0.391888\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.208465	−0.0276119
$58$	0	0
$59$	1.66303	0.216508	0.108254	−	0.994123i	$-0.465474\pi$
$59$	1.66303	0.216508	0.108254	+	0.994123i	$0.465474\pi$
$60$	0	0
$61$	12.1969	1.56165	0.780824	−	0.624752i	$-0.214800\pi$
$61$	12.1969	1.56165	0.780824	+	0.624752i	$0.214800\pi$
$62$	0	0
$63$	11.4901	1.44762
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.352455	−0.0430592	−0.0215296	−	0.999768i	$-0.506854\pi$
$67$	−0.352455	−0.0430592	−0.0215296	+	0.999768i	$0.506854\pi$
$68$	0	0
$69$	0.974469	0.117312
$70$	0	0
$71$	−6.94651	−0.824399	−0.412199	−	0.911094i	$-0.635239\pi$
$71$	−6.94651	−0.824399	−0.412199	+	0.911094i	$0.635239\pi$
$72$	0	0
$73$	−16.9793	−1.98728	−0.993640	−	0.112605i	$-0.964080\pi$
$73$	−16.9793	−1.98728	−0.993640	+	0.112605i	$0.964080\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.43095	0.504954
$78$	0	0
$79$	9.96540	1.12120	0.560598	−	0.828088i	$-0.310571\pi$
$79$	9.96540	1.12120	0.560598	+	0.828088i	$0.310571\pi$
$80$	0	0
$81$	4.94814	0.549793
$82$	0	0
$83$	13.9201	1.52793	0.763965	−	0.645257i	$-0.223250\pi$
$83$	13.9201	1.52793	0.763965	+	0.645257i	$0.223250\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.689426	−0.0739143
$88$	0	0
$89$	4.82690	0.511650	0.255825	−	0.966723i	$-0.417653\pi$
$89$	4.82690	0.511650	0.255825	+	0.966723i	$0.417653\pi$
$90$	0	0
$91$	10.3955	1.08974
$92$	0	0
$93$	−1.75879	−0.182378
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.7388	−1.39496	−0.697482	−	0.716602i	$-0.745696\pi$
$97$	−13.7388	−1.39496	−0.697482	+	0.716602i	$0.745696\pi$
$98$	0	0
$99$	2.45805	0.247043

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.v.1.3		5
5.4	even	2		1160.2.a.i.1.3	✓	5
20.19	odd	2		2320.2.a.w.1.3		5
40.19	odd	2		9280.2.a.ch.1.3		5
40.29	even	2		9280.2.a.cj.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.i.1.3	✓	5	5.4	even	2
2320.2.a.w.1.3		5	20.19	odd	2
5800.2.a.v.1.3		5	1.1	even	1	trivial
9280.2.a.ch.1.3		5	40.19	odd	2
9280.2.a.cj.1.3		5	40.29	even	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:	\( N \)	\(=\)	\( 5800 = 2^{3} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.689426 q^{3} -4.55109 q^{7} -2.52469 q^{9} -0.973602 q^{11} -2.28418 q^{13} -5.82707 q^{17} -0.302375 q^{19} -3.13764 q^{21} +1.41345 q^{23} -3.80887 q^{27} -1.00000 q^{29} -2.55109 q^{31} -0.671227 q^{33} -1.02640 q^{37} -1.57477 q^{39} +4.65483 q^{41} +3.92994 q^{43} -1.07648 q^{47} +13.7124 q^{49} -4.01733 q^{51} +4.85075 q^{53} -0.208465 q^{57} +1.66303 q^{59} +12.1969 q^{61} +11.4901 q^{63} -0.352455 q^{67} +0.974469 q^{69} -6.94651 q^{71} -16.9793 q^{73} +4.43095 q^{77} +9.96540 q^{79} +4.94814 q^{81} +13.9201 q^{83} -0.689426 q^{87} +4.82690 q^{89} +10.3955 q^{91} -1.75879 q^{93} -13.7388 q^{97} +2.45805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + q^{7} + 12 q^{9} - 4 q^{11} - 13 q^{13} - 3 q^{17} + 8 q^{21} + 7 q^{23} + 4 q^{27} - 5 q^{29} + 11 q^{31} - 4 q^{33} - 6 q^{37} + 21 q^{39} + 16 q^{41} - 9 q^{43} - 2 q^{47} + 16 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 5 * q + q^3 + q^7 + 12 * q^9 - 4 * q^11 - 13 * q^13 - 3 * q^17 + 8 * q^21 + 7 * q^23 + 4 * q^27 - 5 * q^29 + 11 * q^31 - 4 * q^33 - 6 * q^37 + 21 * q^39 + 16 * q^41 - 9 * q^43 - 2 * q^47 + 16 * q^49 + 2 * q^51 - 7 * q^53 - 2 * q^57 + 5 * q^59 + 7 * q^61 + 28 * q^63 + 4 * q^67 + 23 * q^69 + 12 * q^71 - 7 * q^73 - 26 * q^77 + 45 * q^79 - 7 * q^81 + 22 * q^83 - q^87 + 24 * q^89 - 20 * q^91 + 10 * q^93 - 17 * q^97 + 10 * q^99

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