LMFDB - Embedded newform 384.2.d.a.193.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,2,Mod(193,384)] 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("384.193"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-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:	no
Analytic conductor:	$3.06625543762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.2.d.a.193.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} -8.00000i q^{29} +4.00000 q^{31} -4.00000 q^{33} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} +8.00000i q^{53} -4.00000 q^{57} -12.0000i q^{59} +12.0000i q^{61} +4.00000 q^{63} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +6.00000 q^{73} +5.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +8.00000 q^{87} +6.00000 q^{89} -16.0000i q^{91} +4.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 2 q^{9} - 4 q^{17} - 16 q^{23} + 10 q^{25} + 8 q^{31} - 8 q^{33} - 8 q^{39} - 12 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 16 q^{71} + 12 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 2 * q^9 - 4 * q^17 - 16 * q^23 + 10 * q^25 + 8 * q^31 - 8 * q^33 - 8 * q^39 - 12 * q^41 + 16 * q^47 + 18 * q^49 - 8 * q^57 + 8 * q^63 + 16 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 12 * q^89 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 8.00000i	− 1.48556i	−0.669534	−	0.742781i	$-0.733506\pi$
$29$	− 8.00000i	− 1.48556i	0.669534	−	0.742781i	$-0.266494\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	8.00000i	1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$	8.00000i	1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	5.00000i	0.577350i
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 16.0000i	− 1.67726i
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i

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	384.2.d.a.193.2	yes	2
3.2	odd	2		1152.2.d.a.577.1		2
4.3	odd	2		384.2.d.b.193.1	yes	2
8.3	odd	2		384.2.d.b.193.2	yes	2
8.5	even	2	inner	384.2.d.a.193.1	✓	2
12.11	even	2		1152.2.d.f.577.2		2
16.3	odd	4		768.2.a.f.1.1		1
16.5	even	4		768.2.a.g.1.1		1
16.11	odd	4		768.2.a.b.1.1		1
16.13	even	4		768.2.a.c.1.1		1
24.5	odd	2		1152.2.d.a.577.2		2
24.11	even	2		1152.2.d.f.577.1		2
48.5	odd	4		2304.2.a.k.1.1		1
48.11	even	4		2304.2.a.f.1.1		1
48.29	odd	4		2304.2.a.j.1.1		1
48.35	even	4		2304.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.d.a.193.1	✓	2	8.5	even	2	inner
384.2.d.a.193.2	yes	2	1.1	even	1	trivial
384.2.d.b.193.1	yes	2	4.3	odd	2
384.2.d.b.193.2	yes	2	8.3	odd	2
768.2.a.b.1.1		1	16.11	odd	4
768.2.a.c.1.1		1	16.13	even	4
768.2.a.f.1.1		1	16.3	odd	4
768.2.a.g.1.1		1	16.5	even	4
1152.2.d.a.577.1		2	3.2	odd	2
1152.2.d.a.577.2		2	24.5	odd	2
1152.2.d.f.577.1		2	24.11	even	2
1152.2.d.f.577.2		2	12.11	even	2
2304.2.a.f.1.1		1	48.11	even	4
2304.2.a.g.1.1		1	48.35	even	4
2304.2.a.j.1.1		1	48.29	odd	4
2304.2.a.k.1.1		1	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} -8.00000i q^{29} +4.00000 q^{31} -4.00000 q^{33} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} +8.00000i q^{53} -4.00000 q^{57} -12.0000i q^{59} +12.0000i q^{61} +4.00000 q^{63} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +6.00000 q^{73} +5.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +8.00000 q^{87} +6.00000 q^{89} -16.0000i q^{91} +4.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 2 q^{9} - 4 q^{17} - 16 q^{23} + 10 q^{25} + 8 q^{31} - 8 q^{33} - 8 q^{39} - 12 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 16 q^{71} + 12 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 2 * q^9 - 4 * q^17 - 16 * q^23 + 10 * q^25 + 8 * q^31 - 8 * q^33 - 8 * q^39 - 12 * q^41 + 16 * q^47 + 18 * q^49 - 8 * q^57 + 8 * q^63 + 16 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 12 * q^89 - 4 * q^97

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