LMFDB - Embedded newform 5000.2.a.n.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5000,2,Mod(1,5000)] 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("5000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5000 = 2^{3} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,3,0,0,0,2,0,-3,0,5] 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:	$39.9252010106$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
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			1.5
Root			$-2.14505$ of defining polynomial
Character	$\chi$	$=$	5000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} -0.0889565 q^{11} -1.98618 q^{13} +3.64217 q^{17} +2.32477 q^{19} -0.577346 q^{21} -1.90531 q^{23} -5.61844 q^{27} -4.96781 q^{29} -3.82913 q^{31} -0.117241 q^{33} -3.20858 q^{37} -2.61770 q^{39} +4.17513 q^{41} -1.93550 q^{43} +0.291012 q^{47} -6.80810 q^{49} +4.80023 q^{51} -8.63482 q^{53} +3.06395 q^{57} -3.70737 q^{59} +6.71522 q^{61} +0.553262 q^{63} +15.1109 q^{67} -2.51112 q^{69} -10.5420 q^{71} -4.60998 q^{73} +0.0389683 q^{77} -13.4735 q^{79} -3.61593 q^{81} -7.36387 q^{83} -6.54738 q^{87} +11.2130 q^{89} +0.870066 q^{91} -5.04664 q^{93} +3.13492 q^{97} +0.112350 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 7 q^{17} - 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} - 27 q^{31} + 3 q^{33} - 10 q^{37} - 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 + 5 * q^11 - 2 * q^13 - 7 * q^17 - 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 - 27 * q^31 + 3 * q^33 - 10 * q^37 - 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 - 10 * q^51 - 2 * q^53 - 2 * q^57 - 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 - 13 * q^71 - 12 * q^73 - 41 * q^77 - 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 - 26 * q^91 - 40 * q^93 + 8 * q^97 + 6 * 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
$3$	1.31796	0.760924	0.380462	−	0.924796i	$-0.375765\pi$
$3$	1.31796	0.760924	0.380462	+	0.924796i	$0.375765\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.438060	−0.165571	−0.0827856	−	0.996567i	$-0.526382\pi$
$7$	−0.438060	−0.165571	−0.0827856	+	0.996567i	$0.526382\pi$
$8$	0	0
$9$	−1.26298	−0.420994
$10$	0	0
$11$	−0.0889565	−0.0268214	−0.0134107	−	0.999910i	$-0.504269\pi$
$11$	−0.0889565	−0.0268214	−0.0134107	+	0.999910i	$0.504269\pi$
$12$	0	0
$13$	−1.98618	−0.550867	−0.275434	−	0.961320i	$-0.588821\pi$
$13$	−1.98618	−0.550867	−0.275434	+	0.961320i	$0.588821\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.64217	0.883355	0.441678	−	0.897174i	$-0.354383\pi$
$17$	3.64217	0.883355	0.441678	+	0.897174i	$0.354383\pi$
$18$	0	0
$19$	2.32477	0.533339	0.266669	−	0.963788i	$-0.414077\pi$
$19$	2.32477	0.533339	0.266669	+	0.963788i	$0.414077\pi$
$20$	0	0
$21$	−0.577346	−0.125987
$22$	0	0
$23$	−1.90531	−0.397284	−0.198642	−	0.980072i	$-0.563653\pi$
$23$	−1.90531	−0.397284	−0.198642	+	0.980072i	$0.563653\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.61844	−1.08127
$28$	0	0
$29$	−4.96781	−0.922500	−0.461250	−	0.887270i	$-0.652599\pi$
$29$	−4.96781	−0.922500	−0.461250	+	0.887270i	$0.652599\pi$
$30$	0	0
$31$	−3.82913	−0.687732	−0.343866	−	0.939019i	$-0.611737\pi$
$31$	−3.82913	−0.687732	−0.343866	+	0.939019i	$0.611737\pi$
$32$	0	0
$33$	−0.117241	−0.0204091
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.20858	−0.527487	−0.263743	−	0.964593i	$-0.584957\pi$
$37$	−3.20858	−0.527487	−0.263743	+	0.964593i	$0.584957\pi$
$38$	0	0
$39$	−2.61770	−0.419168
$40$	0	0
$41$	4.17513	0.652046	0.326023	−	0.945362i	$-0.394291\pi$
$41$	4.17513	0.652046	0.326023	+	0.945362i	$0.394291\pi$
$42$	0	0
$43$	−1.93550	−0.295161	−0.147581	−	0.989050i	$-0.547149\pi$
$43$	−1.93550	−0.295161	−0.147581	+	0.989050i	$0.547149\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.291012	0.0424484	0.0212242	−	0.999775i	$-0.493244\pi$
$47$	0.291012	0.0424484	0.0212242	+	0.999775i	$0.493244\pi$
$48$	0	0
$49$	−6.80810	−0.972586
$50$	0	0
$51$	4.80023	0.672167
$52$	0	0
$53$	−8.63482	−1.18608	−0.593042	−	0.805172i	$-0.702073\pi$
$53$	−8.63482	−1.18608	−0.593042	+	0.805172i	$0.702073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.06395	0.405831
$58$	0	0
$59$	−3.70737	−0.482658	−0.241329	−	0.970443i	$-0.577583\pi$
$59$	−3.70737	−0.482658	−0.241329	+	0.970443i	$0.577583\pi$
$60$	0	0
$61$	6.71522	0.859796	0.429898	−	0.902877i	$-0.358550\pi$
$61$	6.71522	0.859796	0.429898	+	0.902877i	$0.358550\pi$
$62$	0	0
$63$	0.553262	0.0697045
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.1109	1.84609	0.923047	−	0.384688i	$-0.125691\pi$
$67$	15.1109	1.84609	0.923047	+	0.384688i	$0.125691\pi$
$68$	0	0
$69$	−2.51112	−0.302303
$70$	0	0
$71$	−10.5420	−1.25110	−0.625552	−	0.780183i	$-0.715126\pi$
$71$	−10.5420	−1.25110	−0.625552	+	0.780183i	$0.715126\pi$
$72$	0	0
$73$	−4.60998	−0.539558	−0.269779	−	0.962922i	$-0.586951\pi$
$73$	−4.60998	−0.539558	−0.269779	+	0.962922i	$0.586951\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.0389683	0.00444085
$78$	0	0
$79$	−13.4735	−1.51589	−0.757945	−	0.652318i	$-0.773796\pi$
$79$	−13.4735	−1.51589	−0.757945	+	0.652318i	$0.773796\pi$
$80$	0	0
$81$	−3.61593	−0.401770
$82$	0	0
$83$	−7.36387	−0.808290	−0.404145	−	0.914695i	$-0.632431\pi$
$83$	−7.36387	−0.808290	−0.404145	+	0.914695i	$0.632431\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.54738	−0.701953
$88$	0	0
$89$	11.2130	1.18858	0.594288	−	0.804252i	$-0.297434\pi$
$89$	11.2130	1.18858	0.594288	+	0.804252i	$0.297434\pi$
$90$	0	0
$91$	0.870066	0.0912077
$92$	0	0
$93$	−5.04664	−0.523312
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.13492	0.318303	0.159152	−	0.987254i	$-0.449124\pi$
$97$	3.13492	0.318303	0.159152	+	0.987254i	$0.449124\pi$
$98$	0	0
$99$	0.112350	0.0112916

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	5000.2.a.n.1.5	yes	8
4.3	odd	2		10000.2.a.bg.1.4		8
5.4	even	2		5000.2.a.k.1.4	✓	8
20.19	odd	2		10000.2.a.bl.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.4	✓	8	5.4	even	2
5000.2.a.n.1.5	yes	8	1.1	even	1	trivial
10000.2.a.bg.1.4		8	4.3	odd	2
10000.2.a.bl.1.5		8	20.19	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:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} -0.0889565 q^{11} -1.98618 q^{13} +3.64217 q^{17} +2.32477 q^{19} -0.577346 q^{21} -1.90531 q^{23} -5.61844 q^{27} -4.96781 q^{29} -3.82913 q^{31} -0.117241 q^{33} -3.20858 q^{37} -2.61770 q^{39} +4.17513 q^{41} -1.93550 q^{43} +0.291012 q^{47} -6.80810 q^{49} +4.80023 q^{51} -8.63482 q^{53} +3.06395 q^{57} -3.70737 q^{59} +6.71522 q^{61} +0.553262 q^{63} +15.1109 q^{67} -2.51112 q^{69} -10.5420 q^{71} -4.60998 q^{73} +0.0389683 q^{77} -13.4735 q^{79} -3.61593 q^{81} -7.36387 q^{83} -6.54738 q^{87} +11.2130 q^{89} +0.870066 q^{91} -5.04664 q^{93} +3.13492 q^{97} +0.112350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 7 q^{17} - 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} - 27 q^{31} + 3 q^{33} - 10 q^{37} - 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 + 5 * q^11 - 2 * q^13 - 7 * q^17 - 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 - 27 * q^31 + 3 * q^33 - 10 * q^37 - 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 - 10 * q^51 - 2 * q^53 - 2 * q^57 - 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 - 13 * q^71 - 12 * q^73 - 41 * q^77 - 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 - 26 * q^91 - 40 * q^93 + 8 * q^97 + 6 * q^99

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