LMFDB - Embedded newform 384.2.d.a.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(193,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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)

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

Display 100 coefficients 10 coefficients

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 $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See 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
$100$	0	0
$101$	8.00000i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$	8.00000i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	20.0000i	1.69638i	0.529694	+	0.848189i	$0.322307\pi$
$139$	20.0000i	1.69638i	−0.529694	+	0.848189i	$0.677693\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	32.0000	2.52195
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−20.0000	−1.51186
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	− 4.00000i	− 0.297318i	−0.988889	−	0.148659i	$-0.952504\pi$
$181$	− 4.00000i	− 0.297318i	0.988889	−	0.148659i	$-0.0474956\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	32.0000i	2.24596i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	8.00000i	0.548151i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	− 16.0000i	− 0.994192i
$260$	0	0
$261$	8.00000i	0.495188i
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	24.0000i	1.46331i	0.681677	+	0.731653i	$0.261251\pi$
$269$	24.0000i	1.46331i	−0.681677	+	0.731653i	$0.738749\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	16.0000	0.968364
$274$	0	0
$275$	20.0000i	1.20605i
$276$	0	0
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	− 32.0000i	− 1.85061i
$300$	0	0
$301$	− 16.0000i	− 0.922225i
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.00000i	0.449325i	0.974437	+	0.224662i	$0.0721279\pi$
$317$	8.00000i	0.449325i	−0.974437	+	0.224662i	$0.927872\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	20.0000i	1.10940i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	− 14.0000i	− 0.760376i
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	28.0000i	1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$	28.0000i	1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 32.0000i	− 1.66136i
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.0000	1.64808
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	− 4.00000i	− 0.204926i
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	− 32.0000i	− 1.62246i	−0.584724	−	0.811232i	$-0.698797\pi$
$389$	− 32.0000i	− 1.62246i	0.584724	−	0.811232i	$-0.301203\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	48.0000i	2.36193i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	20.0000i	0.977064i	0.872546	+	0.488532i	$0.162467\pi$
$419$	20.0000i	0.977064i	−0.872546	+	0.488532i	$0.837533\pi$
$420$	0	0
$421$	− 20.0000i	− 0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$	− 20.0000i	− 0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−10.0000	−0.485071
$426$	0	0
$427$	− 48.0000i	− 2.32288i
$428$	0	0
$429$	− 16.0000i	− 0.772487i
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	12.0000i	0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	48.0000i	2.21643i
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	20.0000i	0.917663i
$476$	0	0
$477$	− 8.00000i	− 0.366295i
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	32.0000i	1.45605i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	16.0000i	0.720604i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	− 24.0000i	− 1.06378i	−0.846813	−	0.531891i	$-0.821482\pi$
$509$	− 24.0000i	− 1.06378i	0.846813	−	0.531891i	$-0.178518\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	− 20.0000i	− 0.872872i
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	36.0000i	1.55063i
$540$	0	0
$541$	− 28.0000i	− 1.20381i	−0.798566	−	0.601907i	$-0.794408\pi$
$541$	− 28.0000i	− 1.20381i	0.798566	−	0.601907i	$-0.205592\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	0	0
$549$	− 12.0000i	− 0.512148i
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	−40.0000	−1.66812
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	− 18.0000i	− 0.748054i
$580$	0	0
$581$	− 16.0000i	− 0.663792i
$582$	0	0
$583$	−32.0000	−1.32530
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	0	0
$589$	16.0000i	0.659269i
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000i	0.163709i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	−32.0000	−1.29671
$610$	0	0
$611$	32.0000i	1.29458i
$612$	0	0
$613$	36.0000i	1.45403i	0.686624	+	0.727013i	$0.259092\pi$
$613$	36.0000i	1.45403i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	0	0
$621$	8.00000i	0.321029i
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	− 8.00000i	− 0.318981i
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.0000i	1.42637i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	− 16.0000i	− 0.627089i
$652$	0	0
$653$	16.0000i	0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$	16.0000i	0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	4.00000i	0.155582i	0.996970	+	0.0777910i	$0.0247867\pi$
$661$	4.00000i	0.155582i	−0.996970	+	0.0777910i	$0.975213\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	64.0000i	2.47809i
$668$	0	0
$669$	− 20.0000i	− 0.773245i
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	16.0000i	0.614930i	0.951559	+	0.307465i	$0.0994807\pi$
$677$	16.0000i	0.614930i	−0.951559	+	0.307465i	$0.900519\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	−32.0000	−1.21910
$690$	0	0
$691$	− 44.0000i	− 1.67384i	−0.547326	−	0.836919i	$-0.684354\pi$
$691$	− 44.0000i	− 1.67384i	0.547326	−	0.836919i	$-0.315646\pi$
$692$	0	0
$693$	16.0000i	0.607790i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	22.0000i	0.832116i
$700$	0	0
$701$	8.00000i	0.302156i	0.988522	+	0.151078i	$0.0482744\pi$
$701$	8.00000i	0.302156i	−0.988522	+	0.151078i	$0.951726\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 32.0000i	− 1.20348i
$708$	0	0
$709$	− 20.0000i	− 0.751116i	−0.926799	−	0.375558i	$-0.877451\pi$
$709$	− 20.0000i	− 0.751116i	0.926799	−	0.375558i	$-0.122549\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	− 40.0000i	− 1.48556i
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	− 28.0000i	− 1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$733$	− 28.0000i	− 1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	− 28.0000i	− 1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$	− 28.0000i	− 1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	− 16.0000i	− 0.584627i
$750$	0	0
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000i	0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$	12.0000i	0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	− 16.0000i	− 0.579239i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	2.00000i	0.0720282i
$772$	0	0
$773$	− 8.00000i	− 0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$	− 8.00000i	− 0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	32.0000i	1.14505i
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 12.0000i	− 0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$	− 12.0000i	− 0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	56.0000	1.99113
$792$	0	0
$793$	−48.0000	−1.70453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.0000i	− 0.566749i	−0.959009	−	0.283375i	$-0.908546\pi$
$797$	− 16.0000i	− 0.566749i	0.959009	−	0.283375i	$-0.0914540\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	− 44.0000i	− 1.54505i	−0.634985	−	0.772524i	$-0.718994\pi$
$811$	− 44.0000i	− 1.54505i	0.634985	−	0.772524i	$-0.281006\pi$
$812$	0	0
$813$	12.0000i	0.420858i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	16.0000i	0.559085i
$820$	0	0
$821$	24.0000i	0.837606i	0.908077	+	0.418803i	$0.137550\pi$
$821$	24.0000i	0.837606i	−0.908077	+	0.418803i	$0.862450\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	4.00000i	0.138926i	0.997585	+	0.0694629i	$0.0221285\pi$
$829$	4.00000i	0.138926i	−0.997585	+	0.0694629i	$0.977871\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.0000	0.687208
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 32.0000i	− 1.09695i
$852$	0	0
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	− 28.0000i	− 0.955348i	−0.878537	−	0.477674i	$-0.841480\pi$
$859$	− 28.0000i	− 0.955348i	0.878537	−	0.477674i	$-0.158520\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	16.0000i	0.542763i
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.0000i	− 0.675352i	−0.941262	−	0.337676i	$-0.890359\pi$
$877$	− 20.0000i	− 0.675352i	0.941262	−	0.337676i	$-0.109641\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	− 32.0000i	− 1.06726i
$900$	0	0
$901$	− 16.0000i	− 0.533037i
$902$	0	0
$903$	16.0000	0.532447
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	− 8.00000i	− 0.265343i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.0000i	1.58510i
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	32.0000i	1.05329i
$924$	0	0
$925$	20.0000i	0.657596i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	36.0000i	1.17985i
$932$	0	0
$933$	32.0000i	1.04763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	− 22.0000i	− 0.717943i
$940$	0	0
$941$	− 32.0000i	− 1.04317i	−0.853199	−	0.521585i	$-0.825341\pi$
$941$	− 32.0000i	− 1.04317i	0.853199	−	0.521585i	$-0.174659\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	24.0000i	0.779073i
$950$	0	0
$951$	−8.00000	−0.259418
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.0000i	1.03441i
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	20.0000i	0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$	20.0000i	0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$	0	0
$973$	− 80.0000i	− 2.56468i
$974$	0	0
$975$	−20.0000	−0.640513
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	24.0000i	0.767043i
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 32.0000i	− 1.01857i
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 44.0000i	− 1.39349i	−0.717317	−	0.696747i	$-0.754630\pi$
$997$	− 44.0000i	− 1.39349i	0.717317	−	0.696747i	$-0.245370\pi$
$998$	0	0
$999$	4.00000	0.126554

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 $\F_{q}$

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

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