LMFDB - Embedded newform 888.2.a.h.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 = [2,0,2,0,0,0,3] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.70156 q^{7} +1.00000 q^{9} -1.70156 q^{11} +3.70156 q^{13} +5.70156 q^{17} +1.70156 q^{19} -1.70156 q^{21} +7.70156 q^{23} -5.00000 q^{25} +1.00000 q^{27} +8.00000 q^{29} +3.40312 q^{31} -1.70156 q^{33} +1.00000 q^{37} +3.70156 q^{39} +2.00000 q^{41} +4.00000 q^{43} -11.4031 q^{47} -4.10469 q^{49} +5.70156 q^{51} +0.298438 q^{53} +1.70156 q^{57} -9.40312 q^{59} +6.00000 q^{61} -1.70156 q^{63} -15.4031 q^{67} +7.70156 q^{69} +14.8062 q^{71} +3.10469 q^{73} -5.00000 q^{75} +2.89531 q^{77} -7.40312 q^{79} +1.00000 q^{81} -9.70156 q^{83} +8.00000 q^{87} +16.5078 q^{89} -6.29844 q^{91} +3.40312 q^{93} -16.8062 q^{97} -1.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 3 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} + 5 q^{17} - 3 q^{19} + 3 q^{21} + 9 q^{23} - 10 q^{25} + 2 q^{27} + 16 q^{29} - 6 q^{31} + 3 q^{33} + 2 q^{37} + q^{39} + 4 q^{41} + 8 q^{43} - 10 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 3 * q^7 + 2 * q^9 + 3 * q^11 + q^13 + 5 * q^17 - 3 * q^19 + 3 * q^21 + 9 * q^23 - 10 * q^25 + 2 * q^27 + 16 * q^29 - 6 * q^31 + 3 * q^33 + 2 * q^37 + q^39 + 4 * q^41 + 8 * q^43 - 10 * q^47 + 11 * q^49 + 5 * q^51 + 7 * q^53 - 3 * q^57 - 6 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^67 + 9 * q^69 + 4 * q^71 - 13 * q^73 - 10 * q^75 + 25 * q^77 - 2 * q^79 + 2 * q^81 - 13 * q^83 + 16 * q^87 + q^89 - 19 * q^91 - 6 * q^93 - 8 * q^97 + 3 * 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	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.70156	−0.643130	−0.321565	−	0.946888i	$-0.604209\pi$
$7$	−1.70156	−0.643130	−0.321565	+	0.946888i	$0.604209\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.70156	−0.513040	−0.256520	−	0.966539i	$-0.582576\pi$
$11$	−1.70156	−0.513040	−0.256520	+	0.966539i	$0.582576\pi$
$12$	0	0
$13$	3.70156	1.02663	0.513314	−	0.858201i	$-0.328418\pi$
$13$	3.70156	1.02663	0.513314	+	0.858201i	$0.328418\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.70156	1.38283	0.691416	−	0.722457i	$-0.256987\pi$
$17$	5.70156	1.38283	0.691416	+	0.722457i	$0.256987\pi$
$18$	0	0
$19$	1.70156	0.390365	0.195183	−	0.980767i	$-0.437470\pi$
$19$	1.70156	0.390365	0.195183	+	0.980767i	$0.437470\pi$
$20$	0	0
$21$	−1.70156	−0.371311
$22$	0	0
$23$	7.70156	1.60589	0.802943	−	0.596055i	$-0.203266\pi$
$23$	7.70156	1.60589	0.802943	+	0.596055i	$0.203266\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	3.40312	0.611219	0.305610	−	0.952157i	$-0.401140\pi$
$31$	3.40312	0.611219	0.305610	+	0.952157i	$0.401140\pi$
$32$	0	0
$33$	−1.70156	−0.296204
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	3.70156	0.592724
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.4031	−1.66332	−0.831658	−	0.555288i	$-0.812608\pi$
$47$	−11.4031	−1.66332	−0.831658	+	0.555288i	$0.812608\pi$
$48$	0	0
$49$	−4.10469	−0.586384
$50$	0	0
$51$	5.70156	0.798378
$52$	0	0
$53$	0.298438	0.0409936	0.0204968	−	0.999790i	$-0.493475\pi$
$53$	0.298438	0.0409936	0.0204968	+	0.999790i	$0.493475\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.70156	0.225377
$58$	0	0
$59$	−9.40312	−1.22418	−0.612091	−	0.790787i	$-0.709671\pi$
$59$	−9.40312	−1.22418	−0.612091	+	0.790787i	$0.709671\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−1.70156	−0.214377
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.4031	−1.88179	−0.940896	−	0.338697i	$-0.890014\pi$
$67$	−15.4031	−1.88179	−0.940896	+	0.338697i	$0.890014\pi$
$68$	0	0
$69$	7.70156	0.927159
$70$	0	0
$71$	14.8062	1.75718	0.878589	−	0.477578i	$-0.158485\pi$
$71$	14.8062	1.75718	0.878589	+	0.477578i	$0.158485\pi$
$72$	0	0
$73$	3.10469	0.363376	0.181688	−	0.983356i	$-0.441844\pi$
$73$	3.10469	0.363376	0.181688	+	0.983356i	$0.441844\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	2.89531	0.329952
$78$	0	0
$79$	−7.40312	−0.832917	−0.416458	−	0.909155i	$-0.636729\pi$
$79$	−7.40312	−0.832917	−0.416458	+	0.909155i	$0.636729\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.70156	−1.06488	−0.532442	−	0.846466i	$-0.678726\pi$
$83$	−9.70156	−1.06488	−0.532442	+	0.846466i	$0.678726\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	16.5078	1.74982	0.874912	−	0.484282i	$-0.160919\pi$
$89$	16.5078	1.74982	0.874912	+	0.484282i	$0.160919\pi$
$90$	0	0
$91$	−6.29844	−0.660256
$92$	0	0
$93$	3.40312	0.352888
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.8062	−1.70642	−0.853208	−	0.521571i	$-0.825346\pi$
$97$	−16.8062	−1.70642	−0.853208	+	0.521571i	$0.825346\pi$
$98$	0	0
$99$	−1.70156	−0.171013

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.h.1.1	✓	2
3.2	odd	2		2664.2.a.k.1.1		2
4.3	odd	2		1776.2.a.n.1.2		2
8.3	odd	2		7104.2.a.bl.1.2		2
8.5	even	2		7104.2.a.be.1.1		2
12.11	even	2		5328.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
888.2.a.h.1.1	✓	2	1.1	even	1	trivial
1776.2.a.n.1.2		2	4.3	odd	2
2664.2.a.k.1.1		2	3.2	odd	2
5328.2.a.be.1.2		2	12.11	even	2
7104.2.a.be.1.1		2	8.5	even	2
7104.2.a.bl.1.2		2	8.3	odd	2

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.70156 q^{7} +1.00000 q^{9} -1.70156 q^{11} +3.70156 q^{13} +5.70156 q^{17} +1.70156 q^{19} -1.70156 q^{21} +7.70156 q^{23} -5.00000 q^{25} +1.00000 q^{27} +8.00000 q^{29} +3.40312 q^{31} -1.70156 q^{33} +1.00000 q^{37} +3.70156 q^{39} +2.00000 q^{41} +4.00000 q^{43} -11.4031 q^{47} -4.10469 q^{49} +5.70156 q^{51} +0.298438 q^{53} +1.70156 q^{57} -9.40312 q^{59} +6.00000 q^{61} -1.70156 q^{63} -15.4031 q^{67} +7.70156 q^{69} +14.8062 q^{71} +3.10469 q^{73} -5.00000 q^{75} +2.89531 q^{77} -7.40312 q^{79} +1.00000 q^{81} -9.70156 q^{83} +8.00000 q^{87} +16.5078 q^{89} -6.29844 q^{91} +3.40312 q^{93} -16.8062 q^{97} -1.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 3 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} + 5 q^{17} - 3 q^{19} + 3 q^{21} + 9 q^{23} - 10 q^{25} + 2 q^{27} + 16 q^{29} - 6 q^{31} + 3 q^{33} + 2 q^{37} + q^{39} + 4 q^{41} + 8 q^{43} - 10 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 3 * q^7 + 2 * q^9 + 3 * q^11 + q^13 + 5 * q^17 - 3 * q^19 + 3 * q^21 + 9 * q^23 - 10 * q^25 + 2 * q^27 + 16 * q^29 - 6 * q^31 + 3 * q^33 + 2 * q^37 + q^39 + 4 * q^41 + 8 * q^43 - 10 * q^47 + 11 * q^49 + 5 * q^51 + 7 * q^53 - 3 * q^57 - 6 * q^59 + 12 * q^61 + 3 * q^63 - 18 * q^67 + 9 * q^69 + 4 * q^71 - 13 * q^73 - 10 * q^75 + 25 * q^77 - 2 * q^79 + 2 * q^81 - 13 * q^83 + 16 * q^87 + q^89 - 19 * q^91 - 6 * q^93 - 8 * q^97 + 3 * q^99

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