LMFDB - Embedded newform 888.2.a.f.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,-2,0,2,0,8] 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:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.23607 q^{5} +4.00000 q^{7} +1.00000 q^{9} +6.47214 q^{11} -4.47214 q^{13} -3.23607 q^{15} -0.763932 q^{17} -4.00000 q^{21} -1.23607 q^{23} +5.47214 q^{25} -1.00000 q^{27} -9.70820 q^{29} +1.52786 q^{31} -6.47214 q^{33} +12.9443 q^{35} +1.00000 q^{37} +4.47214 q^{39} -10.9443 q^{41} -2.47214 q^{43} +3.23607 q^{45} +8.00000 q^{47} +9.00000 q^{49} +0.763932 q^{51} +8.47214 q^{53} +20.9443 q^{55} -10.1803 q^{59} +2.94427 q^{61} +4.00000 q^{63} -14.4721 q^{65} +12.9443 q^{67} +1.23607 q^{69} -4.94427 q^{71} +8.47214 q^{73} -5.47214 q^{75} +25.8885 q^{77} -8.94427 q^{79} +1.00000 q^{81} -14.4721 q^{83} -2.47214 q^{85} +9.70820 q^{87} +17.7082 q^{89} -17.8885 q^{91} -1.52786 q^{93} -13.4164 q^{97} +6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 8 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} - 6 q^{17} - 8 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 4 q^{33} + 8 q^{35} + 2 q^{37} - 4 q^{41} + 4 q^{43} + 2 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 8 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 - 6 * q^17 - 8 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 12 * q^31 - 4 * q^33 + 8 * q^35 + 2 * q^37 - 4 * q^41 + 4 * q^43 + 2 * q^45 + 16 * q^47 + 18 * q^49 + 6 * q^51 + 8 * q^53 + 24 * q^55 + 2 * q^59 - 12 * q^61 + 8 * q^63 - 20 * q^65 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 8 * q^73 - 2 * q^75 + 16 * q^77 + 2 * q^81 - 20 * q^83 + 4 * q^85 + 6 * q^87 + 22 * q^89 - 12 * q^93 + 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
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.47214	1.95142	0.975711	−	0.219061i	$-0.0702993\pi$
$11$	6.47214	1.95142	0.975711	+	0.219061i	$0.0702993\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−3.23607	−0.835549
$16$	0	0
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.70820	−1.80277	−0.901384	−	0.433020i	$-0.857448\pi$
$29$	−9.70820	−1.80277	−0.901384	+	0.433020i	$0.857448\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	0	0
$33$	−6.47214	−1.12665
$34$	0	0
$35$	12.9443	2.18798
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−10.9443	−1.70921	−0.854604	−	0.519280i	$-0.826200\pi$
$41$	−10.9443	−1.70921	−0.854604	+	0.519280i	$0.826200\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0.763932	0.106972
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	20.9443	2.82413
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.1803	−1.32537	−0.662684	−	0.748899i	$-0.730583\pi$
$59$	−10.1803	−1.32537	−0.662684	+	0.748899i	$0.730583\pi$
$60$	0	0
$61$	2.94427	0.376975	0.188488	−	0.982076i	$-0.439641\pi$
$61$	2.94427	0.376975	0.188488	+	0.982076i	$0.439641\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	1.23607	0.148805
$70$	0	0
$71$	−4.94427	−0.586777	−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427	−0.586777	−0.293389	+	0.955993i	$0.594783\pi$
$72$	0	0
$73$	8.47214	0.991589	0.495794	−	0.868440i	$-0.334877\pi$
$73$	8.47214	0.991589	0.495794	+	0.868440i	$0.334877\pi$
$74$	0	0
$75$	−5.47214	−0.631868
$76$	0	0
$77$	25.8885	2.95027
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.4721	−1.58852	−0.794262	−	0.607576i	$-0.792142\pi$
$83$	−14.4721	−1.58852	−0.794262	+	0.607576i	$0.792142\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	0	0
$87$	9.70820	1.04083
$88$	0	0
$89$	17.7082	1.87707	0.938533	−	0.345190i	$-0.112185\pi$
$89$	17.7082	1.87707	0.938533	+	0.345190i	$0.112185\pi$
$90$	0	0
$91$	−17.8885	−1.87523
$92$	0	0
$93$	−1.52786	−0.158432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	0	0
$99$	6.47214	0.650474

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.f.1.2	✓	2
3.2	odd	2		2664.2.a.j.1.1		2
4.3	odd	2		1776.2.a.p.1.2		2
8.3	odd	2		7104.2.a.bc.1.1		2
8.5	even	2		7104.2.a.bj.1.1		2
12.11	even	2		5328.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
888.2.a.f.1.2	✓	2	1.1	even	1	trivial
1776.2.a.p.1.2		2	4.3	odd	2
2664.2.a.j.1.1		2	3.2	odd	2
5328.2.a.ba.1.1		2	12.11	even	2
7104.2.a.bc.1.1		2	8.3	odd	2
7104.2.a.bj.1.1		2	8.5	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.23607 q^{5} +4.00000 q^{7} +1.00000 q^{9} +6.47214 q^{11} -4.47214 q^{13} -3.23607 q^{15} -0.763932 q^{17} -4.00000 q^{21} -1.23607 q^{23} +5.47214 q^{25} -1.00000 q^{27} -9.70820 q^{29} +1.52786 q^{31} -6.47214 q^{33} +12.9443 q^{35} +1.00000 q^{37} +4.47214 q^{39} -10.9443 q^{41} -2.47214 q^{43} +3.23607 q^{45} +8.00000 q^{47} +9.00000 q^{49} +0.763932 q^{51} +8.47214 q^{53} +20.9443 q^{55} -10.1803 q^{59} +2.94427 q^{61} +4.00000 q^{63} -14.4721 q^{65} +12.9443 q^{67} +1.23607 q^{69} -4.94427 q^{71} +8.47214 q^{73} -5.47214 q^{75} +25.8885 q^{77} -8.94427 q^{79} +1.00000 q^{81} -14.4721 q^{83} -2.47214 q^{85} +9.70820 q^{87} +17.7082 q^{89} -17.8885 q^{91} -1.52786 q^{93} -13.4164 q^{97} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 8 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} - 6 q^{17} - 8 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 4 q^{33} + 8 q^{35} + 2 q^{37} - 4 q^{41} + 4 q^{43} + 2 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 8 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 - 6 * q^17 - 8 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 12 * q^31 - 4 * q^33 + 8 * q^35 + 2 * q^37 - 4 * q^41 + 4 * q^43 + 2 * q^45 + 16 * q^47 + 18 * q^49 + 6 * q^51 + 8 * q^53 + 24 * q^55 + 2 * q^59 - 12 * q^61 + 8 * q^63 - 20 * q^65 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 8 * q^73 - 2 * q^75 + 16 * q^77 + 2 * q^81 - 20 * q^83 + 4 * q^85 + 6 * q^87 + 22 * q^89 - 12 * q^93 + 4 * q^99

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