LMFDB - Embedded newform 3840.2.k.bc.1921.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3840,2,Mod(1921,3840)] 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("3840.1921"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,-4,0,0,0,0,0,4,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1921.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	3840.1921
Dual form			3840.2.k.bc.1921.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -5.12311 q^{7} -1.00000 q^{9} +2.00000i q^{11} -5.12311i q^{13} +1.00000 q^{15} -1.12311 q^{17} -5.12311i q^{19} +5.12311i q^{21} +5.12311 q^{23} -1.00000 q^{25} +1.00000i q^{27} +8.24621i q^{29} -7.12311 q^{31} +2.00000 q^{33} -5.12311i q^{35} -5.12311i q^{37} -5.12311 q^{39} +2.00000 q^{41} -6.24621i q^{43} -1.00000i q^{45} -13.1231 q^{47} +19.2462 q^{49} +1.12311i q^{51} +10.0000i q^{53} -2.00000 q^{55} -5.12311 q^{57} +6.00000i q^{59} +2.00000i q^{61} +5.12311 q^{63} +5.12311 q^{65} +6.24621i q^{67} -5.12311i q^{69} +8.00000 q^{71} +4.24621 q^{73} +1.00000i q^{75} -10.2462i q^{77} -4.87689 q^{79} +1.00000 q^{81} +4.00000i q^{83} -1.12311i q^{85} +8.24621 q^{87} +10.0000 q^{89} +26.2462i q^{91} +7.12311i q^{93} +5.12311 q^{95} +10.0000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{9} + 4 q^{15} + 12 q^{17} + 4 q^{23} - 4 q^{25} - 12 q^{31} + 8 q^{33} - 4 q^{39} + 8 q^{41} - 36 q^{47} + 44 q^{49} - 8 q^{55} - 4 q^{57} + 4 q^{63} + 4 q^{65} + 32 q^{71} - 16 q^{73}+ \cdots + 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^9 + 4 * q^15 + 12 * q^17 + 4 * q^23 - 4 * q^25 - 12 * q^31 + 8 * q^33 - 4 * q^39 + 8 * q^41 - 36 * q^47 + 44 * q^49 - 8 * q^55 - 4 * q^57 + 4 * q^63 + 4 * q^65 + 32 * q^71 - 16 * q^73 - 36 * q^79 + 4 * q^81 + 40 * q^89 + 4 * q^95 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	− 5.12311i	− 1.42089i	−0.703751	−	0.710447i	$-0.748493\pi$
$13$	− 5.12311i	− 1.42089i	0.703751	−	0.710447i	$-0.251507\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.12311	−0.272393	−0.136197	−	0.990682i	$-0.543488\pi$
$17$	−1.12311	−0.272393	−0.136197	+	0.990682i	$0.543488\pi$
$18$	0	0
$19$	− 5.12311i	− 1.17532i	−0.809108	−	0.587661i	$-0.800049\pi$
$19$	− 5.12311i	− 1.17532i	0.809108	−	0.587661i	$-0.199951\pi$
$20$	0	0
$21$	5.12311i	1.11795i
$22$	0	0
$23$	5.12311	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	8.24621i	1.53128i	0.643268	+	0.765641i	$0.277578\pi$
$29$	8.24621i	1.53128i	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	−7.12311	−1.27935	−0.639674	−	0.768647i	$-0.720931\pi$
$31$	−7.12311	−1.27935	−0.639674	+	0.768647i	$0.720931\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	− 5.12311i	− 0.865963i
$36$	0	0
$37$	− 5.12311i	− 0.842233i	−0.907006	−	0.421117i	$-0.861638\pi$
$37$	− 5.12311i	− 0.842233i	0.907006	−	0.421117i	$-0.138362\pi$
$38$	0	0
$39$	−5.12311	−0.820353
$40$	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
$43$	− 6.24621i	− 0.952538i	−0.879300	−	0.476269i	$-0.841989\pi$
$43$	− 6.24621i	− 0.952538i	0.879300	−	0.476269i	$-0.158011\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−13.1231	−1.91420	−0.957101	−	0.289755i	$-0.906426\pi$
$47$	−13.1231	−1.91420	−0.957101	+	0.289755i	$0.906426\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	1.12311i	0.157266i
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−5.12311	−0.678572
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	2.00000i	0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$	2.00000i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	0	0
$63$	5.12311	0.645451
$64$	0	0
$65$	5.12311	0.635443
$66$	0	0
$67$	6.24621i	0.763096i	0.924349	+	0.381548i	$0.124609\pi$
$67$	6.24621i	0.763096i	−0.924349	+	0.381548i	$0.875391\pi$
$68$	0	0
$69$	− 5.12311i	− 0.616749i
$70$	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
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 10.2462i	− 1.16766i
$78$	0	0
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	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
$85$	− 1.12311i	− 0.121818i
$86$	0	0
$87$	8.24621	0.884087
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	26.2462i	2.75135i
$92$	0	0
$93$	7.12311i	0.738632i
$94$	0	0
$95$	5.12311	0.525620
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	14.0000i	1.39305i	0.717532	+	0.696526i	$0.245272\pi$
$101$	14.0000i	1.39305i	−0.717532	+	0.696526i	$0.754728\pi$
$102$	0	0
$103$	11.3693	1.12025	0.560126	−	0.828407i	$-0.310753\pi$
$103$	11.3693	1.12025	0.560126	+	0.828407i	$0.310753\pi$
$104$	0	0
$105$	−5.12311	−0.499964
$106$	0	0
$107$	2.24621i	0.217149i	0.994088	+	0.108575i	$0.0346287\pi$
$107$	2.24621i	0.217149i	−0.994088	+	0.108575i	$0.965371\pi$
$108$	0	0
$109$	− 4.24621i	− 0.406713i	−0.979105	−	0.203357i	$-0.934815\pi$
$109$	− 4.24621i	− 0.406713i	0.979105	−	0.203357i	$-0.0651851\pi$
$110$	0	0
$111$	−5.12311	−0.486264
$112$	0	0
$113$	6.87689	0.646924	0.323462	−	0.946241i	$-0.395153\pi$
$113$	6.87689	0.646924	0.323462	+	0.946241i	$0.395153\pi$
$114$	0	0
$115$	5.12311i	0.477732i
$116$	0	0
$117$	5.12311i	0.473631i
$118$	0	0
$119$	5.75379	0.527449
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−6.87689	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$127$	−6.87689	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	8.24621i	0.720475i	0.932861	+	0.360237i	$0.117304\pi$
$131$	8.24621i	0.720475i	−0.932861	+	0.360237i	$0.882696\pi$
$132$	0	0
$133$	26.2462i	2.27584i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.87689	0.245790	0.122895	−	0.992420i	$-0.460782\pi$
$137$	2.87689	0.245790	0.122895	+	0.992420i	$0.460782\pi$
$138$	0	0
$139$	15.3693i	1.30361i	0.758387	+	0.651804i	$0.225988\pi$
$139$	15.3693i	1.30361i	−0.758387	+	0.651804i	$0.774012\pi$
$140$	0	0
$141$	13.1231i	1.10516i
$142$	0	0
$143$	10.2462	0.856831
$144$	0	0
$145$	−8.24621	−0.684811
$146$	0	0
$147$	− 19.2462i	− 1.58740i
$148$	0	0
$149$	20.2462i	1.65863i	0.558778	+	0.829317i	$0.311270\pi$
$149$	20.2462i	1.65863i	−0.558778	+	0.829317i	$0.688730\pi$
$150$	0	0
$151$	−17.3693	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$151$	−17.3693	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	− 7.12311i	− 0.572142i
$156$	0	0
$157$	19.3693i	1.54584i	0.634504	+	0.772920i	$0.281205\pi$
$157$	19.3693i	1.54584i	−0.634504	+	0.772920i	$0.718795\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−26.2462	−2.06849
$162$	0	0
$163$	− 8.49242i	− 0.665178i	−0.943072	−	0.332589i	$-0.892078\pi$
$163$	− 8.49242i	− 0.665178i	0.943072	−	0.332589i	$-0.107922\pi$
$164$	0	0
$165$	2.00000i	0.155700i
$166$	0	0
$167$	10.8769	0.841679	0.420840	−	0.907135i	$-0.361735\pi$
$167$	10.8769	0.841679	0.420840	+	0.907135i	$0.361735\pi$
$168$	0	0
$169$	−13.2462	−1.01894
$170$	0	0
$171$	5.12311i	0.391774i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	5.12311	0.387270
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	12.2462i	0.915325i	0.889126	+	0.457662i	$0.151313\pi$
$179$	12.2462i	0.915325i	−0.889126	+	0.457662i	$0.848687\pi$
$180$	0	0
$181$	− 0.246211i	− 0.0183007i	−0.999958	−	0.00915037i	$-0.997087\pi$
$181$	− 0.246211i	− 0.0183007i	0.999958	−	0.00915037i	$-0.00291269\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	5.12311	0.376658
$186$	0	0
$187$	− 2.24621i	− 0.164259i
$188$	0	0
$189$	− 5.12311i	− 0.372651i
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	24.2462	1.74528	0.872640	−	0.488364i	$-0.162406\pi$
$193$	24.2462	1.74528	0.872640	+	0.488364i	$0.162406\pi$
$194$	0	0
$195$	− 5.12311i	− 0.366873i
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	19.6155	1.39051	0.695254	−	0.718764i	$-0.255292\pi$
$199$	19.6155	1.39051	0.695254	+	0.718764i	$0.255292\pi$
$200$	0	0
$201$	6.24621	0.440574
$202$	0	0
$203$	− 42.2462i	− 2.96510i
$204$	0	0
$205$	2.00000i	0.139686i
$206$	0	0
$207$	−5.12311	−0.356080
$208$	0	0
$209$	10.2462	0.708745
$210$	0	0
$211$	− 10.8769i	− 0.748796i	−0.927268	−	0.374398i	$-0.877849\pi$
$211$	− 10.8769i	− 0.748796i	0.927268	−	0.374398i	$-0.122151\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	6.24621	0.425988
$216$	0	0
$217$	36.4924	2.47727
$218$	0	0
$219$	− 4.24621i	− 0.286932i
$220$	0	0
$221$	5.75379i	0.387042i
$222$	0	0
$223$	14.8769	0.996231	0.498115	−	0.867111i	$-0.334026\pi$
$223$	14.8769	0.996231	0.498115	+	0.867111i	$0.334026\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 10.0000i	− 0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$	− 10.0000i	− 0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$	0	0
$231$	−10.2462	−0.674151
$232$	0	0
$233$	−25.6155	−1.67813	−0.839065	−	0.544032i	$-0.816897\pi$
$233$	−25.6155	−1.67813	−0.839065	+	0.544032i	$0.816897\pi$
$234$	0	0
$235$	− 13.1231i	− 0.856057i
$236$	0	0
$237$	4.87689i	0.316788i
$238$	0	0
$239$	−6.24621	−0.404034	−0.202017	−	0.979382i	$-0.564750\pi$
$239$	−6.24621	−0.404034	−0.202017	+	0.979382i	$0.564750\pi$
$240$	0	0
$241$	−7.75379	−0.499465	−0.249733	−	0.968315i	$-0.580343\pi$
$241$	−7.75379	−0.499465	−0.249733	+	0.968315i	$0.580343\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	19.2462i	1.22960i
$246$	0	0
$247$	−26.2462	−1.67001
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	− 4.24621i	− 0.268018i	−0.990980	−	0.134009i	$-0.957215\pi$
$251$	− 4.24621i	− 0.268018i	0.990980	−	0.134009i	$-0.0427852\pi$
$252$	0	0
$253$	10.2462i	0.644174i
$254$	0	0
$255$	−1.12311	−0.0703316
$256$	0	0
$257$	31.8617	1.98748	0.993740	−	0.111714i	$-0.0356342\pi$
$257$	31.8617	1.98748	0.993740	+	0.111714i	$0.0356342\pi$
$258$	0	0
$259$	26.2462i	1.63086i
$260$	0	0
$261$	− 8.24621i	− 0.510428i
$262$	0	0
$263$	−15.3693	−0.947713	−0.473856	−	0.880602i	$-0.657138\pi$
$263$	−15.3693	−0.947713	−0.473856	+	0.880602i	$0.657138\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	8.24621i	0.502780i	0.967886	+	0.251390i	$0.0808877\pi$
$269$	8.24621i	0.502780i	−0.967886	+	0.251390i	$0.919112\pi$
$270$	0	0
$271$	−7.12311	−0.432698	−0.216349	−	0.976316i	$-0.569415\pi$
$271$	−7.12311	−0.432698	−0.216349	+	0.976316i	$0.569415\pi$
$272$	0	0
$273$	26.2462	1.58849
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	− 14.8769i	− 0.893866i	−0.894567	−	0.446933i	$-0.852516\pi$
$277$	− 14.8769i	− 0.893866i	0.894567	−	0.446933i	$-0.147484\pi$
$278$	0	0
$279$	7.12311	0.426449
$280$	0	0
$281$	−0.246211	−0.0146877	−0.00734387	−	0.999973i	$-0.502338\pi$
$281$	−0.246211	−0.0146877	−0.00734387	+	0.999973i	$0.502338\pi$
$282$	0	0
$283$	30.2462i	1.79795i	0.437999	+	0.898975i	$0.355687\pi$
$283$	30.2462i	1.79795i	−0.437999	+	0.898975i	$0.644313\pi$
$284$	0	0
$285$	− 5.12311i	− 0.303467i
$286$	0	0
$287$	−10.2462	−0.604815
$288$	0	0
$289$	−15.7386	−0.925802
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	− 28.7386i	− 1.67893i	−0.543415	−	0.839464i	$-0.682869\pi$
$293$	− 28.7386i	− 1.67893i	0.543415	−	0.839464i	$-0.317131\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	− 26.2462i	− 1.51786i
$300$	0	0
$301$	32.0000i	1.84445i
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	− 14.2462i	− 0.813074i	−0.913634	−	0.406537i	$-0.866736\pi$
$307$	− 14.2462i	− 0.813074i	0.913634	−	0.406537i	$-0.133264\pi$
$308$	0	0
$309$	− 11.3693i	− 0.646778i
$310$	0	0
$311$	30.2462	1.71511	0.857553	−	0.514396i	$-0.171984\pi$
$311$	30.2462	1.71511	0.857553	+	0.514396i	$0.171984\pi$
$312$	0	0
$313$	−20.7386	−1.17222	−0.586108	−	0.810233i	$-0.699341\pi$
$313$	−20.7386	−1.17222	−0.586108	+	0.810233i	$0.699341\pi$
$314$	0	0
$315$	5.12311i	0.288654i
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	−16.4924	−0.923398
$320$	0	0
$321$	2.24621	0.125371
$322$	0	0
$323$	5.75379i	0.320149i
$324$	0	0
$325$	5.12311i	0.284179i
$326$	0	0
$327$	−4.24621	−0.234816
$328$	0	0
$329$	67.2311	3.70657
$330$	0	0
$331$	− 1.12311i	− 0.0617315i	−0.999524	−	0.0308657i	$-0.990174\pi$
$331$	− 1.12311i	− 0.0617315i	0.999524	−	0.0308657i	$-0.00982643\pi$
$332$	0	0
$333$	5.12311i	0.280744i
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	−24.7386	−1.34760	−0.673800	−	0.738914i	$-0.735339\pi$
$337$	−24.7386	−1.34760	−0.673800	+	0.738914i	$0.735339\pi$
$338$	0	0
$339$	− 6.87689i	− 0.373502i
$340$	0	0
$341$	− 14.2462i	− 0.771476i
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	5.12311	0.275819
$346$	0	0
$347$	10.2462i	0.550045i	0.961438	+	0.275023i	$0.0886854\pi$
$347$	10.2462i	0.550045i	−0.961438	+	0.275023i	$0.911315\pi$
$348$	0	0
$349$	− 10.4924i	− 0.561646i	−0.959760	−	0.280823i	$-0.909393\pi$
$349$	− 10.4924i	− 0.561646i	0.959760	−	0.280823i	$-0.0906075\pi$
$350$	0	0
$351$	5.12311	0.273451
$352$	0	0
$353$	−27.8617	−1.48293	−0.741465	−	0.670991i	$-0.765869\pi$
$353$	−27.8617	−1.48293	−0.741465	+	0.670991i	$0.765869\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	− 5.75379i	− 0.304523i
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−7.24621	−0.381380
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	4.24621i	0.222257i
$366$	0	0
$367$	7.36932	0.384675	0.192338	−	0.981329i	$-0.438393\pi$
$367$	7.36932	0.384675	0.192338	+	0.981329i	$0.438393\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	− 51.2311i	− 2.65978i
$372$	0	0
$373$	17.1231i	0.886601i	0.896373	+	0.443300i	$0.146193\pi$
$373$	17.1231i	0.886601i	−0.896373	+	0.443300i	$0.853807\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	42.2462	2.17579
$378$	0	0
$379$	− 3.36932i	− 0.173070i	−0.996249	−	0.0865351i	$-0.972421\pi$
$379$	− 3.36932i	− 0.173070i	0.996249	−	0.0865351i	$-0.0275795\pi$
$380$	0	0
$381$	6.87689i	0.352314i
$382$	0	0
$383$	5.12311	0.261778	0.130889	−	0.991397i	$-0.458217\pi$
$383$	5.12311	0.261778	0.130889	+	0.991397i	$0.458217\pi$
$384$	0	0
$385$	10.2462	0.522195
$386$	0	0
$387$	6.24621i	0.317513i
$388$	0	0
$389$	10.4924i	0.531987i	0.963975	+	0.265993i	$0.0857000\pi$
$389$	10.4924i	0.531987i	−0.963975	+	0.265993i	$0.914300\pi$
$390$	0	0
$391$	−5.75379	−0.290982
$392$	0	0
$393$	8.24621	0.415966
$394$	0	0
$395$	− 4.87689i	− 0.245383i
$396$	0	0
$397$	15.3693i	0.771364i	0.922632	+	0.385682i	$0.126034\pi$
$397$	15.3693i	0.771364i	−0.922632	+	0.385682i	$0.873966\pi$
$398$	0	0
$399$	26.2462	1.31395
$400$	0	0
$401$	−11.7538	−0.586956	−0.293478	−	0.955966i	$-0.594813\pi$
$401$	−11.7538	−0.586956	−0.293478	+	0.955966i	$0.594813\pi$
$402$	0	0
$403$	36.4924i	1.81782i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	10.2462	0.507886
$408$	0	0
$409$	−18.4924	−0.914391	−0.457196	−	0.889366i	$-0.651146\pi$
$409$	−18.4924	−0.914391	−0.457196	+	0.889366i	$0.651146\pi$
$410$	0	0
$411$	− 2.87689i	− 0.141907i
$412$	0	0
$413$	− 30.7386i	− 1.51255i
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	15.3693	0.752639
$418$	0	0
$419$	− 0.246211i	− 0.0120282i	−0.999982	−	0.00601410i	$-0.998086\pi$
$419$	− 0.246211i	− 0.0120282i	0.999982	−	0.00601410i	$-0.00191436\pi$
$420$	0	0
$421$	30.9848i	1.51011i	0.655662	+	0.755054i	$0.272390\pi$
$421$	30.9848i	1.51011i	−0.655662	+	0.755054i	$0.727610\pi$
$422$	0	0
$423$	13.1231	0.638067
$424$	0	0
$425$	1.12311	0.0544786
$426$	0	0
$427$	− 10.2462i	− 0.495849i
$428$	0	0
$429$	− 10.2462i	− 0.494692i
$430$	0	0
$431$	16.4924	0.794412	0.397206	−	0.917729i	$-0.369980\pi$
$431$	16.4924	0.794412	0.397206	+	0.917729i	$0.369980\pi$
$432$	0	0
$433$	0.246211	0.0118322	0.00591608	−	0.999982i	$-0.498117\pi$
$433$	0.246211	0.0118322	0.00591608	+	0.999982i	$0.498117\pi$
$434$	0	0
$435$	8.24621i	0.395376i
$436$	0	0
$437$	− 26.2462i	− 1.25553i
$438$	0	0
$439$	2.63068	0.125556	0.0627778	−	0.998028i	$-0.480004\pi$
$439$	2.63068	0.125556	0.0627778	+	0.998028i	$0.480004\pi$
$440$	0	0
$441$	−19.2462	−0.916486
$442$	0	0
$443$	− 3.50758i	− 0.166650i	−0.996522	−	0.0833250i	$-0.973446\pi$
$443$	− 3.50758i	− 0.166650i	0.996522	−	0.0833250i	$-0.0265540\pi$
$444$	0	0
$445$	10.0000i	0.474045i
$446$	0	0
$447$	20.2462	0.957613
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	4.00000i	0.188353i
$452$	0	0
$453$	17.3693i	0.816082i
$454$	0	0
$455$	−26.2462	−1.23044
$456$	0	0
$457$	−6.49242	−0.303703	−0.151851	−	0.988403i	$-0.548524\pi$
$457$	−6.49242	−0.303703	−0.151851	+	0.988403i	$0.548524\pi$
$458$	0	0
$459$	− 1.12311i	− 0.0524221i
$460$	0	0
$461$	40.2462i	1.87445i	0.348721	+	0.937226i	$0.386616\pi$
$461$	40.2462i	1.87445i	−0.348721	+	0.937226i	$0.613384\pi$
$462$	0	0
$463$	37.1231	1.72526	0.862629	−	0.505838i	$-0.168817\pi$
$463$	37.1231	1.72526	0.862629	+	0.505838i	$0.168817\pi$
$464$	0	0
$465$	−7.12311	−0.330326
$466$	0	0
$467$	18.7386i	0.867121i	0.901125	+	0.433560i	$0.142743\pi$
$467$	18.7386i	0.867121i	−0.901125	+	0.433560i	$0.857257\pi$
$468$	0	0
$469$	− 32.0000i	− 1.47762i
$470$	0	0
$471$	19.3693	0.892491
$472$	0	0
$473$	12.4924	0.574402
$474$	0	0
$475$	5.12311i	0.235064i
$476$	0	0
$477$	− 10.0000i	− 0.457869i
$478$	0	0
$479$	36.4924	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$479$	36.4924	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$480$	0	0
$481$	−26.2462	−1.19672
$482$	0	0
$483$	26.2462i	1.19424i
$484$	0	0
$485$	10.0000i	0.454077i
$486$	0	0
$487$	13.6155	0.616978	0.308489	−	0.951228i	$-0.400177\pi$
$487$	13.6155	0.616978	0.308489	+	0.951228i	$0.400177\pi$
$488$	0	0
$489$	−8.49242	−0.384041
$490$	0	0
$491$	− 1.50758i	− 0.0680360i	−0.999421	−	0.0340180i	$-0.989170\pi$
$491$	− 1.50758i	− 0.0680360i	0.999421	−	0.0340180i	$-0.0108304\pi$
$492$	0	0
$493$	− 9.26137i	− 0.417111i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−40.9848	−1.83842
$498$	0	0
$499$	4.63068i	0.207298i	0.994614	+	0.103649i	$0.0330518\pi$
$499$	4.63068i	0.207298i	−0.994614	+	0.103649i	$0.966948\pi$
$500$	0	0
$501$	− 10.8769i	− 0.485944i
$502$	0	0
$503$	1.61553	0.0720328	0.0360164	−	0.999351i	$-0.488533\pi$
$503$	1.61553	0.0720328	0.0360164	+	0.999351i	$0.488533\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	13.2462i	0.588285i
$508$	0	0
$509$	− 14.0000i	− 0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$	− 14.0000i	− 0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$	0	0
$511$	−21.7538	−0.962331
$512$	0	0
$513$	5.12311	0.226191
$514$	0	0
$515$	11.3693i	0.500992i
$516$	0	0
$517$	− 26.2462i	− 1.15431i
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	18.7386i	0.819383i	0.912224	+	0.409692i	$0.134364\pi$
$523$	18.7386i	0.819383i	−0.912224	+	0.409692i	$0.865636\pi$
$524$	0	0
$525$	− 5.12311i	− 0.223591i
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 10.2462i	− 0.443813i
$534$	0	0
$535$	−2.24621	−0.0971122
$536$	0	0
$537$	12.2462	0.528463
$538$	0	0
$539$	38.4924i	1.65799i
$540$	0	0
$541$	8.24621i	0.354532i	0.984163	+	0.177266i	$0.0567253\pi$
$541$	8.24621i	0.354532i	−0.984163	+	0.177266i	$0.943275\pi$
$542$	0	0
$543$	−0.246211	−0.0105659
$544$	0	0
$545$	4.24621	0.181888
$546$	0	0
$547$	9.75379i	0.417042i	0.978018	+	0.208521i	$0.0668649\pi$
$547$	9.75379i	0.417042i	−0.978018	+	0.208521i	$0.933135\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	42.2462	1.79975
$552$	0	0
$553$	24.9848	1.06246
$554$	0	0
$555$	− 5.12311i	− 0.217464i
$556$	0	0
$557$	− 1.50758i	− 0.0638781i	−0.999490	−	0.0319391i	$-0.989832\pi$
$557$	− 1.50758i	− 0.0638781i	0.999490	−	0.0319391i	$-0.0101682\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	−2.24621	−0.0948351
$562$	0	0
$563$	22.7386i	0.958319i	0.877728	+	0.479160i	$0.159058\pi$
$563$	22.7386i	0.958319i	−0.877728	+	0.479160i	$0.840942\pi$
$564$	0	0
$565$	6.87689i	0.289313i
$566$	0	0
$567$	−5.12311	−0.215150
$568$	0	0
$569$	−24.7386	−1.03710	−0.518549	−	0.855048i	$-0.673528\pi$
$569$	−24.7386	−1.03710	−0.518549	+	0.855048i	$0.673528\pi$
$570$	0	0
$571$	2.38447i	0.0997870i	0.998755	+	0.0498935i	$0.0158882\pi$
$571$	2.38447i	0.0997870i	−0.998755	+	0.0498935i	$0.984112\pi$
$572$	0	0
$573$	4.00000i	0.167102i
$574$	0	0
$575$	−5.12311	−0.213648
$576$	0	0
$577$	42.9848	1.78948	0.894741	−	0.446585i	$-0.147360\pi$
$577$	42.9848	1.78948	0.894741	+	0.446585i	$0.147360\pi$
$578$	0	0
$579$	− 24.2462i	− 1.00764i
$580$	0	0
$581$	− 20.4924i	− 0.850169i
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	−5.12311	−0.211814
$586$	0	0
$587$	− 46.7386i	− 1.92911i	−0.263882	−	0.964555i	$-0.585003\pi$
$587$	− 46.7386i	− 1.92911i	0.263882	−	0.964555i	$-0.414997\pi$
$588$	0	0
$589$	36.4924i	1.50364i
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	39.8617	1.63693	0.818463	−	0.574560i	$-0.194827\pi$
$593$	39.8617	1.63693	0.818463	+	0.574560i	$0.194827\pi$
$594$	0	0
$595$	5.75379i	0.235882i
$596$	0	0
$597$	− 19.6155i	− 0.802810i
$598$	0	0
$599$	24.4924	1.00073	0.500367	−	0.865814i	$-0.333199\pi$
$599$	24.4924	1.00073	0.500367	+	0.865814i	$0.333199\pi$
$600$	0	0
$601$	30.9848	1.26390	0.631949	−	0.775010i	$-0.282255\pi$
$601$	30.9848	1.26390	0.631949	+	0.775010i	$0.282255\pi$
$602$	0	0
$603$	− 6.24621i	− 0.254365i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	−13.1231	−0.532650	−0.266325	−	0.963883i	$-0.585810\pi$
$607$	−13.1231	−0.532650	−0.266325	+	0.963883i	$0.585810\pi$
$608$	0	0
$609$	−42.2462	−1.71190
$610$	0	0
$611$	67.2311i	2.71988i
$612$	0	0
$613$	− 7.36932i	− 0.297644i	−0.988864	−	0.148822i	$-0.952452\pi$
$613$	− 7.36932i	− 0.297644i	0.988864	−	0.148822i	$-0.0475481\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−9.61553	−0.387107	−0.193553	−	0.981090i	$-0.562001\pi$
$617$	−9.61553	−0.387107	−0.193553	+	0.981090i	$0.562001\pi$
$618$	0	0
$619$	− 21.1231i	− 0.849009i	−0.905426	−	0.424505i	$-0.860448\pi$
$619$	− 21.1231i	− 0.849009i	0.905426	−	0.424505i	$-0.139552\pi$
$620$	0	0
$621$	5.12311i	0.205583i
$622$	0	0
$623$	−51.2311	−2.05253
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 10.2462i	− 0.409194i
$628$	0	0
$629$	5.75379i	0.229419i
$630$	0	0
$631$	13.8617	0.551827	0.275914	−	0.961182i	$-0.411020\pi$
$631$	13.8617	0.551827	0.275914	+	0.961182i	$0.411020\pi$
$632$	0	0
$633$	−10.8769	−0.432318
$634$	0	0
$635$	− 6.87689i	− 0.272901i
$636$	0	0
$637$	− 98.6004i	− 3.90669i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−36.2462	−1.43164	−0.715820	−	0.698285i	$-0.753947\pi$
$641$	−36.2462	−1.43164	−0.715820	+	0.698285i	$0.753947\pi$
$642$	0	0
$643$	32.4924i	1.28138i	0.767801	+	0.640688i	$0.221351\pi$
$643$	32.4924i	1.28138i	−0.767801	+	0.640688i	$0.778649\pi$
$644$	0	0
$645$	− 6.24621i	− 0.245944i
$646$	0	0
$647$	−13.1231	−0.515923	−0.257961	−	0.966155i	$-0.583051\pi$
$647$	−13.1231	−0.515923	−0.257961	+	0.966155i	$0.583051\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	− 36.4924i	− 1.43025i
$652$	0	0
$653$	− 44.7386i	− 1.75076i	−0.483437	−	0.875379i	$-0.660612\pi$
$653$	− 44.7386i	− 1.75076i	0.483437	−	0.875379i	$-0.339388\pi$
$654$	0	0
$655$	−8.24621	−0.322206
$656$	0	0
$657$	−4.24621	−0.165660
$658$	0	0
$659$	− 26.0000i	− 1.01282i	−0.862294	−	0.506408i	$-0.830973\pi$
$659$	− 26.0000i	− 1.01282i	0.862294	−	0.506408i	$-0.169027\pi$
$660$	0	0
$661$	− 2.00000i	− 0.0777910i	−0.999243	−	0.0388955i	$-0.987616\pi$
$661$	− 2.00000i	− 0.0777910i	0.999243	−	0.0388955i	$-0.0123839\pi$
$662$	0	0
$663$	5.75379	0.223459
$664$	0	0
$665$	−26.2462	−1.01778
$666$	0	0
$667$	42.2462i	1.63578i
$668$	0	0
$669$	− 14.8769i	− 0.575174i
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−20.2462	−0.780434	−0.390217	−	0.920723i	$-0.627600\pi$
$673$	−20.2462	−0.780434	−0.390217	+	0.920723i	$0.627600\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	4.73863i	0.182120i	0.995845	+	0.0910602i	$0.0290256\pi$
$677$	4.73863i	0.182120i	−0.995845	+	0.0910602i	$0.970974\pi$
$678$	0	0
$679$	−51.2311	−1.96607
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 9.75379i	− 0.373218i	−0.982434	−	0.186609i	$-0.940250\pi$
$683$	− 9.75379i	− 0.373218i	0.982434	−	0.186609i	$-0.0597498\pi$
$684$	0	0
$685$	2.87689i	0.109920i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	51.2311	1.95175
$690$	0	0
$691$	13.1231i	0.499226i	0.968346	+	0.249613i	$0.0803035\pi$
$691$	13.1231i	0.499226i	−0.968346	+	0.249613i	$0.919697\pi$
$692$	0	0
$693$	10.2462i	0.389221i
$694$	0	0
$695$	−15.3693	−0.582991
$696$	0	0
$697$	−2.24621	−0.0850813
$698$	0	0
$699$	25.6155i	0.968868i
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	−26.2462	−0.989895
$704$	0	0
$705$	−13.1231	−0.494245
$706$	0	0
$707$	− 71.7235i	− 2.69744i
$708$	0	0
$709$	− 50.9848i	− 1.91478i	−0.288805	−	0.957388i	$-0.593258\pi$
$709$	− 50.9848i	− 1.91478i	0.288805	−	0.957388i	$-0.406742\pi$
$710$	0	0
$711$	4.87689	0.182898
$712$	0	0
$713$	−36.4924	−1.36665
$714$	0	0
$715$	10.2462i	0.383187i
$716$	0	0
$717$	6.24621i	0.233269i
$718$	0	0
$719$	22.2462	0.829644	0.414822	−	0.909903i	$-0.363844\pi$
$719$	22.2462	0.829644	0.414822	+	0.909903i	$0.363844\pi$
$720$	0	0
$721$	−58.2462	−2.16920
$722$	0	0
$723$	7.75379i	0.288367i
$724$	0	0
$725$	− 8.24621i	− 0.306257i
$726$	0	0
$727$	29.6155	1.09838	0.549190	−	0.835698i	$-0.314936\pi$
$727$	29.6155	1.09838	0.549190	+	0.835698i	$0.314936\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	7.01515i	0.259465i
$732$	0	0
$733$	− 37.6155i	− 1.38936i	−0.719318	−	0.694681i	$-0.755546\pi$
$733$	− 37.6155i	− 1.38936i	0.719318	−	0.694681i	$-0.244454\pi$
$734$	0	0
$735$	19.2462	0.709907
$736$	0	0
$737$	−12.4924	−0.460164
$738$	0	0
$739$	33.1231i	1.21845i	0.792996	+	0.609227i	$0.208520\pi$
$739$	33.1231i	1.21845i	−0.792996	+	0.609227i	$0.791480\pi$
$740$	0	0
$741$	26.2462i	0.964179i
$742$	0	0
$743$	−19.8617	−0.728657	−0.364328	−	0.931271i	$-0.618701\pi$
$743$	−19.8617	−0.728657	−0.364328	+	0.931271i	$0.618701\pi$
$744$	0	0
$745$	−20.2462	−0.741764
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	− 11.5076i	− 0.420478i
$750$	0	0
$751$	−2.63068	−0.0959950	−0.0479975	−	0.998847i	$-0.515284\pi$
$751$	−2.63068	−0.0959950	−0.0479975	+	0.998847i	$0.515284\pi$
$752$	0	0
$753$	−4.24621	−0.154741
$754$	0	0
$755$	− 17.3693i	− 0.632134i
$756$	0	0
$757$	13.6155i	0.494865i	0.968905	+	0.247432i	$0.0795868\pi$
$757$	13.6155i	0.494865i	−0.968905	+	0.247432i	$0.920413\pi$
$758$	0	0
$759$	10.2462	0.371914
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	21.7538i	0.787540i
$764$	0	0
$765$	1.12311i	0.0406060i
$766$	0	0
$767$	30.7386	1.10991
$768$	0	0
$769$	31.7538	1.14507	0.572535	−	0.819880i	$-0.305960\pi$
$769$	31.7538	1.14507	0.572535	+	0.819880i	$0.305960\pi$
$770$	0	0
$771$	− 31.8617i	− 1.14747i
$772$	0	0
$773$	47.4773i	1.70764i	0.520569	+	0.853819i	$0.325720\pi$
$773$	47.4773i	1.70764i	−0.520569	+	0.853819i	$0.674280\pi$
$774$	0	0
$775$	7.12311	0.255870
$776$	0	0
$777$	26.2462	0.941578
$778$	0	0
$779$	− 10.2462i	− 0.367109i
$780$	0	0
$781$	16.0000i	0.572525i
$782$	0	0
$783$	−8.24621	−0.294696
$784$	0	0
$785$	−19.3693	−0.691321
$786$	0	0
$787$	− 38.2462i	− 1.36333i	−0.731664	−	0.681665i	$-0.761256\pi$
$787$	− 38.2462i	− 1.36333i	0.731664	−	0.681665i	$-0.238744\pi$
$788$	0	0
$789$	15.3693i	0.547162i
$790$	0	0
$791$	−35.2311	−1.25267
$792$	0	0
$793$	10.2462	0.363854
$794$	0	0
$795$	10.0000i	0.354663i
$796$	0	0
$797$	− 11.7538i	− 0.416341i	−0.978093	−	0.208170i	$-0.933249\pi$
$797$	− 11.7538i	− 0.416341i	0.978093	−	0.208170i	$-0.0667508\pi$
$798$	0	0
$799$	14.7386	0.521415
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	8.49242i	0.299691i
$804$	0	0
$805$	− 26.2462i	− 0.925057i
$806$	0	0
$807$	8.24621	0.290280
$808$	0	0
$809$	12.2462	0.430554	0.215277	−	0.976553i	$-0.430935\pi$
$809$	12.2462	0.430554	0.215277	+	0.976553i	$0.430935\pi$
$810$	0	0
$811$	− 17.1231i	− 0.601274i	−0.953739	−	0.300637i	$-0.902801\pi$
$811$	− 17.1231i	− 0.601274i	0.953739	−	0.300637i	$-0.0971992\pi$
$812$	0	0
$813$	7.12311i	0.249818i
$814$	0	0
$815$	8.49242	0.297477
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	− 26.2462i	− 0.917117i
$820$	0	0
$821$	24.7386i	0.863384i	0.902021	+	0.431692i	$0.142083\pi$
$821$	24.7386i	0.863384i	−0.902021	+	0.431692i	$0.857917\pi$
$822$	0	0
$823$	−20.6307	−0.719140	−0.359570	−	0.933118i	$-0.617077\pi$
$823$	−20.6307	−0.719140	−0.359570	+	0.933118i	$0.617077\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	23.5076i	0.817439i	0.912660	+	0.408719i	$0.134024\pi$
$827$	23.5076i	0.817439i	−0.912660	+	0.408719i	$0.865976\pi$
$828$	0	0
$829$	20.7386i	0.720283i	0.932898	+	0.360141i	$0.117272\pi$
$829$	20.7386i	0.720283i	−0.932898	+	0.360141i	$0.882728\pi$
$830$	0	0
$831$	−14.8769	−0.516074
$832$	0	0
$833$	−21.6155	−0.748934
$834$	0	0
$835$	10.8769i	0.376410i
$836$	0	0
$837$	− 7.12311i	− 0.246211i
$838$	0	0
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	0	0
$843$	0.246211i	0.00847997i
$844$	0	0
$845$	− 13.2462i	− 0.455684i
$846$	0	0
$847$	−35.8617	−1.23222
$848$	0	0
$849$	30.2462	1.03805
$850$	0	0
$851$	− 26.2462i	− 0.899709i
$852$	0	0
$853$	21.1231i	0.723241i	0.932325	+	0.361621i	$0.117776\pi$
$853$	21.1231i	0.723241i	−0.932325	+	0.361621i	$0.882224\pi$
$854$	0	0
$855$	−5.12311	−0.175207
$856$	0	0
$857$	−30.8769	−1.05473	−0.527367	−	0.849637i	$-0.676821\pi$
$857$	−30.8769	−1.05473	−0.527367	+	0.849637i	$0.676821\pi$
$858$	0	0
$859$	25.1231i	0.857189i	0.903497	+	0.428595i	$0.140991\pi$
$859$	25.1231i	0.857189i	−0.903497	+	0.428595i	$0.859009\pi$
$860$	0	0
$861$	10.2462i	0.349190i
$862$	0	0
$863$	−47.3693	−1.61247	−0.806235	−	0.591595i	$-0.798498\pi$
$863$	−47.3693	−1.61247	−0.806235	+	0.591595i	$0.798498\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	15.7386i	0.534512i
$868$	0	0
$869$	− 9.75379i	− 0.330875i
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	5.12311i	0.173193i
$876$	0	0
$877$	− 53.6155i	− 1.81047i	−0.424914	−	0.905234i	$-0.639696\pi$
$877$	− 53.6155i	− 1.81047i	0.424914	−	0.905234i	$-0.360304\pi$
$878$	0	0
$879$	−28.7386	−0.969330
$880$	0	0
$881$	−14.4924	−0.488262	−0.244131	−	0.969742i	$-0.578503\pi$
$881$	−14.4924	−0.488262	−0.244131	+	0.969742i	$0.578503\pi$
$882$	0	0
$883$	0.492423i	0.0165713i	0.999966	+	0.00828567i	$0.00263744\pi$
$883$	0.492423i	0.0165713i	−0.999966	+	0.00828567i	$0.997363\pi$
$884$	0	0
$885$	6.00000i	0.201688i
$886$	0	0
$887$	6.38447	0.214370	0.107185	−	0.994239i	$-0.465816\pi$
$887$	6.38447	0.214370	0.107185	+	0.994239i	$0.465816\pi$
$888$	0	0
$889$	35.2311	1.18161
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	67.2311i	2.24980i
$894$	0	0
$895$	−12.2462	−0.409346
$896$	0	0
$897$	−26.2462	−0.876335
$898$	0	0
$899$	− 58.7386i	− 1.95904i
$900$	0	0
$901$	− 11.2311i	− 0.374161i
$902$	0	0
$903$	32.0000	1.06489
$904$	0	0
$905$	0.246211	0.00818434
$906$	0	0
$907$	28.9848i	0.962426i	0.876604	+	0.481213i	$0.159804\pi$
$907$	28.9848i	0.962426i	−0.876604	+	0.481213i	$0.840196\pi$
$908$	0	0
$909$	− 14.0000i	− 0.464351i
$910$	0	0
$911$	9.26137	0.306843	0.153421	−	0.988161i	$-0.450971\pi$
$911$	9.26137	0.306843	0.153421	+	0.988161i	$0.450971\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	2.00000i	0.0661180i
$916$	0	0
$917$	− 42.2462i	− 1.39509i
$918$	0	0
$919$	39.1231	1.29055	0.645276	−	0.763949i	$-0.276742\pi$
$919$	39.1231	1.29055	0.645276	+	0.763949i	$0.276742\pi$
$920$	0	0
$921$	−14.2462	−0.469429
$922$	0	0
$923$	− 40.9848i	− 1.34903i
$924$	0	0
$925$	5.12311i	0.168447i
$926$	0	0
$927$	−11.3693	−0.373417
$928$	0	0
$929$	14.4924	0.475481	0.237740	−	0.971329i	$-0.423593\pi$
$929$	14.4924	0.475481	0.237740	+	0.971329i	$0.423593\pi$
$930$	0	0
$931$	− 98.6004i	− 3.23150i
$932$	0	0
$933$	− 30.2462i	− 0.990217i
$934$	0	0
$935$	2.24621	0.0734590
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	20.7386i	0.676780i
$940$	0	0
$941$	− 43.4773i	− 1.41732i	−0.705551	−	0.708659i	$-0.749300\pi$
$941$	− 43.4773i	− 1.41732i	0.705551	−	0.708659i	$-0.250700\pi$
$942$	0	0
$943$	10.2462	0.333663
$944$	0	0
$945$	5.12311	0.166655
$946$	0	0
$947$	26.7386i	0.868889i	0.900699	+	0.434444i	$0.143055\pi$
$947$	26.7386i	0.868889i	−0.900699	+	0.434444i	$0.856945\pi$
$948$	0	0
$949$	− 21.7538i	− 0.706158i
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−45.1231	−1.46168	−0.730840	−	0.682548i	$-0.760872\pi$
$953$	−45.1231	−1.46168	−0.730840	+	0.682548i	$0.760872\pi$
$954$	0	0
$955$	− 4.00000i	− 0.129437i
$956$	0	0
$957$	16.4924i	0.533124i
$958$	0	0
$959$	−14.7386	−0.475935
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	− 2.24621i	− 0.0723831i
$964$	0	0
$965$	24.2462i	0.780513i
$966$	0	0
$967$	−51.3693	−1.65193	−0.825963	−	0.563724i	$-0.809368\pi$
$967$	−51.3693	−1.65193	−0.825963	+	0.563724i	$0.809368\pi$
$968$	0	0
$969$	5.75379	0.184838
$970$	0	0
$971$	32.2462i	1.03483i	0.855735	+	0.517415i	$0.173106\pi$
$971$	32.2462i	1.03483i	−0.855735	+	0.517415i	$0.826894\pi$
$972$	0	0
$973$	− 78.7386i	− 2.52424i
$974$	0	0
$975$	5.12311	0.164071
$976$	0	0
$977$	30.8769	0.987839	0.493920	−	0.869508i	$-0.335564\pi$
$977$	30.8769	0.987839	0.493920	+	0.869508i	$0.335564\pi$
$978$	0	0
$979$	20.0000i	0.639203i
$980$	0	0
$981$	4.24621i	0.135571i
$982$	0	0
$983$	−47.3693	−1.51085	−0.755423	−	0.655237i	$-0.772569\pi$
$983$	−47.3693	−1.51085	−0.755423	+	0.655237i	$0.772569\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	− 67.2311i	− 2.13999i
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	15.1231	0.480401	0.240201	−	0.970723i	$-0.422787\pi$
$991$	15.1231	0.480401	0.240201	+	0.970723i	$0.422787\pi$
$992$	0	0
$993$	−1.12311	−0.0356407
$994$	0	0
$995$	19.6155i	0.621854i
$996$	0	0
$997$	− 5.61553i	− 0.177846i	−0.996039	−	0.0889228i	$-0.971658\pi$
$997$	− 5.61553i	− 0.177846i	0.996039	−	0.0889228i	$-0.0283424\pi$
$998$	0	0
$999$	5.12311	0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.bc.1921.1		4
4.3	odd	2		3840.2.k.bd.1921.4		4
8.3	odd	2		3840.2.k.bd.1921.2		4
8.5	even	2	inner	3840.2.k.bc.1921.3		4
16.3	odd	4		1920.2.a.z.1.1	yes	2
16.5	even	4		1920.2.a.y.1.2	✓	2
16.11	odd	4		1920.2.a.ba.1.1	yes	2
16.13	even	4		1920.2.a.bb.1.2	yes	2
48.5	odd	4		5760.2.a.cj.1.2		2
48.11	even	4		5760.2.a.cg.1.1		2
48.29	odd	4		5760.2.a.cc.1.2		2
48.35	even	4		5760.2.a.bx.1.1		2
80.19	odd	4		9600.2.a.dg.1.2		2
80.29	even	4		9600.2.a.cl.1.1		2
80.59	odd	4		9600.2.a.ct.1.2		2
80.69	even	4		9600.2.a.cw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.y.1.2	✓	2	16.5	even	4
1920.2.a.z.1.1	yes	2	16.3	odd	4
1920.2.a.ba.1.1	yes	2	16.11	odd	4
1920.2.a.bb.1.2	yes	2	16.13	even	4
3840.2.k.bc.1921.1		4	1.1	even	1	trivial
3840.2.k.bc.1921.3		4	8.5	even	2	inner
3840.2.k.bd.1921.2		4	8.3	odd	2
3840.2.k.bd.1921.4		4	4.3	odd	2
5760.2.a.bx.1.1		2	48.35	even	4
5760.2.a.cc.1.2		2	48.29	odd	4
5760.2.a.cg.1.1		2	48.11	even	4
5760.2.a.cj.1.2		2	48.5	odd	4
9600.2.a.cl.1.1		2	80.29	even	4
9600.2.a.ct.1.2		2	80.59	odd	4
9600.2.a.cw.1.1		2	80.69	even	4
9600.2.a.dg.1.2		2	80.19	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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -5.12311 q^{7} -1.00000 q^{9} +2.00000i q^{11} -5.12311i q^{13} +1.00000 q^{15} -1.12311 q^{17} -5.12311i q^{19} +5.12311i q^{21} +5.12311 q^{23} -1.00000 q^{25} +1.00000i q^{27} +8.24621i q^{29} -7.12311 q^{31} +2.00000 q^{33} -5.12311i q^{35} -5.12311i q^{37} -5.12311 q^{39} +2.00000 q^{41} -6.24621i q^{43} -1.00000i q^{45} -13.1231 q^{47} +19.2462 q^{49} +1.12311i q^{51} +10.0000i q^{53} -2.00000 q^{55} -5.12311 q^{57} +6.00000i q^{59} +2.00000i q^{61} +5.12311 q^{63} +5.12311 q^{65} +6.24621i q^{67} -5.12311i q^{69} +8.00000 q^{71} +4.24621 q^{73} +1.00000i q^{75} -10.2462i q^{77} -4.87689 q^{79} +1.00000 q^{81} +4.00000i q^{83} -1.12311i q^{85} +8.24621 q^{87} +10.0000 q^{89} +26.2462i q^{91} +7.12311i q^{93} +5.12311 q^{95} +10.0000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{9} + 4 q^{15} + 12 q^{17} + 4 q^{23} - 4 q^{25} - 12 q^{31} + 8 q^{33} - 4 q^{39} + 8 q^{41} - 36 q^{47} + 44 q^{49} - 8 q^{55} - 4 q^{57} + 4 q^{63} + 4 q^{65} + 32 q^{71} - 16 q^{73}+ \cdots + 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^9 + 4 * q^15 + 12 * q^17 + 4 * q^23 - 4 * q^25 - 12 * q^31 + 8 * q^33 - 4 * q^39 + 8 * q^41 - 36 * q^47 + 44 * q^49 - 8 * q^55 - 4 * q^57 + 4 * q^63 + 4 * q^65 + 32 * q^71 - 16 * q^73 - 36 * q^79 + 4 * q^81 + 40 * q^89 + 4 * q^95 + 40 * q^97

Introduction

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