LMFDB - Embedded newform 888.2.a.h.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 = [2,0,2,0,0,0,3] 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			1.2
Root			$3.70156$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.70156 q^{7} +1.00000 q^{9} +4.70156 q^{11} -2.70156 q^{13} -0.701562 q^{17} -4.70156 q^{19} +4.70156 q^{21} +1.29844 q^{23} -5.00000 q^{25} +1.00000 q^{27} +8.00000 q^{29} -9.40312 q^{31} +4.70156 q^{33} +1.00000 q^{37} -2.70156 q^{39} +2.00000 q^{41} +4.00000 q^{43} +1.40312 q^{47} +15.1047 q^{49} -0.701562 q^{51} +6.70156 q^{53} -4.70156 q^{57} +3.40312 q^{59} +6.00000 q^{61} +4.70156 q^{63} -2.59688 q^{67} +1.29844 q^{69} -10.8062 q^{71} -16.1047 q^{73} -5.00000 q^{75} +22.1047 q^{77} +5.40312 q^{79} +1.00000 q^{81} -3.29844 q^{83} +8.00000 q^{87} -15.5078 q^{89} -12.7016 q^{91} -9.40312 q^{93} +8.80625 q^{97} +4.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.70156	1.77702	0.888512	−	0.458854i	$-0.151740\pi$
$7$	4.70156	1.77702	0.888512	+	0.458854i	$0.151740\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.70156	1.41757	0.708787	−	0.705422i	$-0.249243\pi$
$11$	4.70156	1.41757	0.708787	+	0.705422i	$0.249243\pi$
$12$	0	0
$13$	−2.70156	−0.749279	−0.374639	−	0.927171i	$-0.622233\pi$
$13$	−2.70156	−0.749279	−0.374639	+	0.927171i	$0.622233\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.701562	−0.170154	−0.0850769	−	0.996374i	$-0.527114\pi$
$17$	−0.701562	−0.170154	−0.0850769	+	0.996374i	$0.527114\pi$
$18$	0	0
$19$	−4.70156	−1.07861	−0.539306	−	0.842110i	$-0.681313\pi$
$19$	−4.70156	−1.07861	−0.539306	+	0.842110i	$0.681313\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	1.29844	0.270743	0.135372	−	0.990795i	$-0.456777\pi$
$23$	1.29844	0.270743	0.135372	+	0.990795i	$0.456777\pi$
$24$	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
$31$	−9.40312	−1.68885	−0.844425	−	0.535673i	$-0.820058\pi$
$31$	−9.40312	−1.68885	−0.844425	+	0.535673i	$0.820058\pi$
$32$	0	0
$33$	4.70156	0.818437
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.70156	−0.432596
$40$	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
$47$	1.40312	0.204667	0.102333	−	0.994750i	$-0.467369\pi$
$47$	1.40312	0.204667	0.102333	+	0.994750i	$0.467369\pi$
$48$	0	0
$49$	15.1047	2.15781
$50$	0	0
$51$	−0.701562	−0.0982383
$52$	0	0
$53$	6.70156	0.920530	0.460265	−	0.887781i	$-0.347754\pi$
$53$	6.70156	0.920530	0.460265	+	0.887781i	$0.347754\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.70156	−0.622737
$58$	0	0
$59$	3.40312	0.443049	0.221525	−	0.975155i	$-0.428897\pi$
$59$	3.40312	0.443049	0.221525	+	0.975155i	$0.428897\pi$
$60$	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
$63$	4.70156	0.592341
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.59688	−0.317259	−0.158629	−	0.987338i	$-0.550708\pi$
$67$	−2.59688	−0.317259	−0.158629	+	0.987338i	$0.550708\pi$
$68$	0	0
$69$	1.29844	0.156314
$70$	0	0
$71$	−10.8062	−1.28247	−0.641233	−	0.767346i	$-0.721577\pi$
$71$	−10.8062	−1.28247	−0.641233	+	0.767346i	$0.721577\pi$
$72$	0	0
$73$	−16.1047	−1.88491	−0.942455	−	0.334332i	$-0.891489\pi$
$73$	−16.1047	−1.88491	−0.942455	+	0.334332i	$0.891489\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	22.1047	2.51906
$78$	0	0
$79$	5.40312	0.607899	0.303949	−	0.952688i	$-0.401695\pi$
$79$	5.40312	0.607899	0.303949	+	0.952688i	$0.401695\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.29844	−0.362051	−0.181025	−	0.983478i	$-0.557942\pi$
$83$	−3.29844	−0.362051	−0.181025	+	0.983478i	$0.557942\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−15.5078	−1.64382	−0.821912	−	0.569614i	$-0.807093\pi$
$89$	−15.5078	−1.64382	−0.821912	+	0.569614i	$0.807093\pi$
$90$	0	0
$91$	−12.7016	−1.33149
$92$	0	0
$93$	−9.40312	−0.975059
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.80625	0.894139	0.447070	−	0.894499i	$-0.352468\pi$
$97$	8.80625	0.894139	0.447070	+	0.894499i	$0.352468\pi$
$98$	0	0
$99$	4.70156	0.472525

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.70156 q^{7} +1.00000 q^{9} +4.70156 q^{11} -2.70156 q^{13} -0.701562 q^{17} -4.70156 q^{19} +4.70156 q^{21} +1.29844 q^{23} -5.00000 q^{25} +1.00000 q^{27} +8.00000 q^{29} -9.40312 q^{31} +4.70156 q^{33} +1.00000 q^{37} -2.70156 q^{39} +2.00000 q^{41} +4.00000 q^{43} +1.40312 q^{47} +15.1047 q^{49} -0.701562 q^{51} +6.70156 q^{53} -4.70156 q^{57} +3.40312 q^{59} +6.00000 q^{61} +4.70156 q^{63} -2.59688 q^{67} +1.29844 q^{69} -10.8062 q^{71} -16.1047 q^{73} -5.00000 q^{75} +22.1047 q^{77} +5.40312 q^{79} +1.00000 q^{81} -3.29844 q^{83} +8.00000 q^{87} -15.5078 q^{89} -12.7016 q^{91} -9.40312 q^{93} +8.80625 q^{97} +4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 3 q^{7} + 2 q^{9} + 3 q^{11} + q^{13} + 5 q^{17} - 3 q^{19} + 3 q^{21} + 9 q^{23} - 10 q^{25} + 2 q^{27} + 16 q^{29} - 6 q^{31} + 3 q^{33} + 2 q^{37} + q^{39} + 4 q^{41} + 8 q^{43} - 10 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 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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