LMFDB - Embedded newform 888.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [888,2,Mod(1,888)] 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("888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 888 = 2^{3} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,-4] 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:	$7.09071569949$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.732051 q^{5} -2.00000 q^{7} +1.00000 q^{9} +1.46410 q^{11} +3.46410 q^{13} +0.732051 q^{15} +0.732051 q^{17} -5.46410 q^{19} +2.00000 q^{21} -1.26795 q^{23} -4.46410 q^{25} -1.00000 q^{27} -6.19615 q^{29} -4.00000 q^{31} -1.46410 q^{33} +1.46410 q^{35} -1.00000 q^{37} -3.46410 q^{39} -2.00000 q^{41} -10.9282 q^{43} -0.732051 q^{45} -6.92820 q^{47} -3.00000 q^{49} -0.732051 q^{51} +10.3923 q^{53} -1.07180 q^{55} +5.46410 q^{57} +1.26795 q^{59} +4.92820 q^{61} -2.00000 q^{63} -2.53590 q^{65} -6.00000 q^{67} +1.26795 q^{69} +4.00000 q^{71} -5.46410 q^{73} +4.46410 q^{75} -2.92820 q^{77} -2.53590 q^{79} +1.00000 q^{81} -2.53590 q^{83} -0.535898 q^{85} +6.19615 q^{87} -2.19615 q^{89} -6.92820 q^{91} +4.00000 q^{93} +4.00000 q^{95} +2.39230 q^{97} +1.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} + 4 q^{21} - 6 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{35} - 2 q^{37} - 4 q^{41} - 8 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 + 4 * q^21 - 6 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^31 + 4 * q^33 - 4 * q^35 - 2 * q^37 - 4 * q^41 - 8 * q^43 + 2 * q^45 - 6 * q^49 + 2 * q^51 - 16 * q^55 + 4 * q^57 + 6 * q^59 - 4 * q^61 - 4 * q^63 - 12 * q^65 - 12 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^77 - 12 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 + 6 * q^89 + 8 * q^93 + 8 * q^95 - 16 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.732051	−0.327383	−0.163692	−	0.986512i	$-0.552340\pi$
$5$	−0.732051	−0.327383	−0.163692	+	0.986512i	$0.552340\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	0.732051	0.177548	0.0887742	−	0.996052i	$-0.471705\pi$
$17$	0.732051	0.177548	0.0887742	+	0.996052i	$0.471705\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.19615	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$29$	−6.19615	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	1.46410	0.247478
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−3.46410	−0.554700
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−10.9282	−1.66654	−0.833268	−	0.552870i	$-0.813533\pi$
$43$	−10.9282	−1.66654	−0.833268	+	0.552870i	$0.813533\pi$
$44$	0	0
$45$	−0.732051	−0.109128
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−0.732051	−0.102508
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	−1.07180	−0.144521
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	1.26795	0.165073	0.0825365	−	0.996588i	$-0.473698\pi$
$59$	1.26795	0.165073	0.0825365	+	0.996588i	$0.473698\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−2.53590	−0.314539
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	1.26795	0.152643
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−5.46410	−0.639525	−0.319762	−	0.947498i	$-0.603603\pi$
$73$	−5.46410	−0.639525	−0.319762	+	0.947498i	$0.603603\pi$
$74$	0	0
$75$	4.46410	0.515470
$76$	0	0
$77$	−2.92820	−0.333700
$78$	0	0
$79$	−2.53590	−0.285311	−0.142655	−	0.989772i	$-0.545564\pi$
$79$	−2.53590	−0.285311	−0.142655	+	0.989772i	$0.545564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.53590	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$83$	−2.53590	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$84$	0	0
$85$	−0.535898	−0.0581263
$86$	0	0
$87$	6.19615	0.664297
$88$	0	0
$89$	−2.19615	−0.232792	−0.116396	−	0.993203i	$-0.537134\pi$
$89$	−2.19615	−0.232792	−0.116396	+	0.993203i	$0.537134\pi$
$90$	0	0
$91$	−6.92820	−0.726273
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.39230	0.242902	0.121451	−	0.992597i	$-0.461245\pi$
$97$	2.39230	0.242902	0.121451	+	0.992597i	$0.461245\pi$
$98$	0	0
$99$	1.46410	0.147148

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	888.2.a.e.1.1	✓	2
3.2	odd	2		2664.2.a.i.1.2		2
4.3	odd	2		1776.2.a.q.1.1		2
8.3	odd	2		7104.2.a.bd.1.2		2
8.5	even	2		7104.2.a.bi.1.2		2
12.11	even	2		5328.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
888.2.a.e.1.1	✓	2	1.1	even	1	trivial
1776.2.a.q.1.1		2	4.3	odd	2
2664.2.a.i.1.2		2	3.2	odd	2
5328.2.a.bb.1.2		2	12.11	even	2
7104.2.a.bd.1.2		2	8.3	odd	2
7104.2.a.bi.1.2		2	8.5	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	888.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.732051 q^{5} -2.00000 q^{7} +1.00000 q^{9} +1.46410 q^{11} +3.46410 q^{13} +0.732051 q^{15} +0.732051 q^{17} -5.46410 q^{19} +2.00000 q^{21} -1.26795 q^{23} -4.46410 q^{25} -1.00000 q^{27} -6.19615 q^{29} -4.00000 q^{31} -1.46410 q^{33} +1.46410 q^{35} -1.00000 q^{37} -3.46410 q^{39} -2.00000 q^{41} -10.9282 q^{43} -0.732051 q^{45} -6.92820 q^{47} -3.00000 q^{49} -0.732051 q^{51} +10.3923 q^{53} -1.07180 q^{55} +5.46410 q^{57} +1.26795 q^{59} +4.92820 q^{61} -2.00000 q^{63} -2.53590 q^{65} -6.00000 q^{67} +1.26795 q^{69} +4.00000 q^{71} -5.46410 q^{73} +4.46410 q^{75} -2.92820 q^{77} -2.53590 q^{79} +1.00000 q^{81} -2.53590 q^{83} -0.535898 q^{85} +6.19615 q^{87} -2.19615 q^{89} -6.92820 q^{91} +4.00000 q^{93} +4.00000 q^{95} +2.39230 q^{97} +1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} + 4 q^{21} - 6 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{35} - 2 q^{37} - 4 q^{41} - 8 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 + 4 * q^21 - 6 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^31 + 4 * q^33 - 4 * q^35 - 2 * q^37 - 4 * q^41 - 8 * q^43 + 2 * q^45 - 6 * q^49 + 2 * q^51 - 16 * q^55 + 4 * q^57 + 6 * q^59 - 4 * q^61 - 4 * q^63 - 12 * q^65 - 12 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^77 - 12 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 + 6 * q^89 + 8 * q^93 + 8 * q^95 - 16 * q^97 - 4 * q^99

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