LMFDB - Embedded newform 888.2.a.e.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.73205 q^{5} -2.00000 q^{7} +1.00000 q^{9} -5.46410 q^{11} -3.46410 q^{13} -2.73205 q^{15} -2.73205 q^{17} +1.46410 q^{19} +2.00000 q^{21} -4.73205 q^{23} +2.46410 q^{25} -1.00000 q^{27} +4.19615 q^{29} -4.00000 q^{31} +5.46410 q^{33} -5.46410 q^{35} -1.00000 q^{37} +3.46410 q^{39} -2.00000 q^{41} +2.92820 q^{43} +2.73205 q^{45} +6.92820 q^{47} -3.00000 q^{49} +2.73205 q^{51} -10.3923 q^{53} -14.9282 q^{55} -1.46410 q^{57} +4.73205 q^{59} -8.92820 q^{61} -2.00000 q^{63} -9.46410 q^{65} -6.00000 q^{67} +4.73205 q^{69} +4.00000 q^{71} +1.46410 q^{73} -2.46410 q^{75} +10.9282 q^{77} -9.46410 q^{79} +1.00000 q^{81} -9.46410 q^{83} -7.46410 q^{85} -4.19615 q^{87} +8.19615 q^{89} +6.92820 q^{91} +4.00000 q^{93} +4.00000 q^{95} -18.3923 q^{97} -5.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	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
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	−2.73205	−0.705412
$16$	0	0
$17$	−2.73205	−0.662620	−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205	−0.662620	−0.331310	+	0.943522i	$0.607491\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.19615	0.779206	0.389603	−	0.920983i	$-0.372612\pi$
$29$	4.19615	0.779206	0.389603	+	0.920983i	$0.372612\pi$
$30$	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
$33$	5.46410	0.951178
$34$	0	0
$35$	−5.46410	−0.923602
$36$	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
$43$	2.92820	0.446547	0.223273	−	0.974756i	$-0.428326\pi$
$43$	2.92820	0.446547	0.223273	+	0.974756i	$0.428326\pi$
$44$	0	0
$45$	2.73205	0.407270
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.73205	0.382564
$52$	0	0
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	−14.9282	−2.01292
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	4.73205	0.616061	0.308030	−	0.951377i	$-0.400330\pi$
$59$	4.73205	0.616061	0.308030	+	0.951377i	$0.400330\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−9.46410	−1.17388
$66$	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
$69$	4.73205	0.569672
$70$	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
$73$	1.46410	0.171360	0.0856801	−	0.996323i	$-0.472694\pi$
$73$	1.46410	0.171360	0.0856801	+	0.996323i	$0.472694\pi$
$74$	0	0
$75$	−2.46410	−0.284530
$76$	0	0
$77$	10.9282	1.24538
$78$	0	0
$79$	−9.46410	−1.06479	−0.532397	−	0.846495i	$-0.678709\pi$
$79$	−9.46410	−1.06479	−0.532397	+	0.846495i	$0.678709\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.46410	−1.03882	−0.519410	−	0.854525i	$-0.673848\pi$
$83$	−9.46410	−1.03882	−0.519410	+	0.854525i	$0.673848\pi$
$84$	0	0
$85$	−7.46410	−0.809595
$86$	0	0
$87$	−4.19615	−0.449875
$88$	0	0
$89$	8.19615	0.868790	0.434395	−	0.900722i	$-0.356962\pi$
$89$	8.19615	0.868790	0.434395	+	0.900722i	$0.356962\pi$
$90$	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
$97$	−18.3923	−1.86746	−0.933728	−	0.357984i	$-0.883464\pi$
$97$	−18.3923	−1.86746	−0.933728	+	0.357984i	$0.883464\pi$
$98$	0	0
$99$	−5.46410	−0.549163

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.73205 q^{5} -2.00000 q^{7} +1.00000 q^{9} -5.46410 q^{11} -3.46410 q^{13} -2.73205 q^{15} -2.73205 q^{17} +1.46410 q^{19} +2.00000 q^{21} -4.73205 q^{23} +2.46410 q^{25} -1.00000 q^{27} +4.19615 q^{29} -4.00000 q^{31} +5.46410 q^{33} -5.46410 q^{35} -1.00000 q^{37} +3.46410 q^{39} -2.00000 q^{41} +2.92820 q^{43} +2.73205 q^{45} +6.92820 q^{47} -3.00000 q^{49} +2.73205 q^{51} -10.3923 q^{53} -14.9282 q^{55} -1.46410 q^{57} +4.73205 q^{59} -8.92820 q^{61} -2.00000 q^{63} -9.46410 q^{65} -6.00000 q^{67} +4.73205 q^{69} +4.00000 q^{71} +1.46410 q^{73} -2.46410 q^{75} +10.9282 q^{77} -9.46410 q^{79} +1.00000 q^{81} -9.46410 q^{83} -7.46410 q^{85} -4.19615 q^{87} +8.19615 q^{89} +6.92820 q^{91} +4.00000 q^{93} +4.00000 q^{95} -18.3923 q^{97} -5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} + 4 q^{21} - 6 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{35} - 2 q^{37} - 4 q^{41} - 8 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 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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