LMFDB - Embedded newform 200.2.c.a.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,2,Mod(49,200)] 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("200.49"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$1.59700804043$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.2.c.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -4.00000i q^{13} +5.00000i q^{17} -1.00000 q^{19} -6.00000 q^{21} +2.00000i q^{23} -9.00000i q^{27} +8.00000 q^{29} +10.0000 q^{31} +3.00000i q^{33} -6.00000i q^{37} +12.0000 q^{39} -3.00000 q^{41} -4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -15.0000 q^{51} -6.00000i q^{53} -3.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} -12.0000i q^{63} -1.00000i q^{67} -6.00000 q^{69} -12.0000 q^{71} -3.00000i q^{73} +2.00000i q^{77} -6.00000 q^{79} +9.00000 q^{81} +13.0000i q^{83} +24.0000i q^{87} +9.00000 q^{89} +8.00000 q^{91} +30.0000i q^{93} -14.0000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9} + 2 q^{11} - 2 q^{19} - 12 q^{21} + 16 q^{29} + 20 q^{31} + 24 q^{39} - 6 q^{41} + 6 q^{49} - 30 q^{51} - 16 q^{59} + 20 q^{61} - 12 q^{69} - 24 q^{71} - 12 q^{79} + 18 q^{81} + 18 q^{89}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 + 2 * q^11 - 2 * q^19 - 12 * q^21 + 16 * q^29 + 20 * q^31 + 24 * q^39 - 6 * q^41 + 6 * q^49 - 30 * q^51 - 16 * q^59 + 20 * q^61 - 12 * q^69 - 24 * q^71 - 12 * q^79 + 18 * q^81 + 18 * q^89 + 16 * q^91 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000i	1.21268i	0.795206	+	0.606339i	$0.207363\pi$
$17$	5.00000i	1.21268i	−0.795206	+	0.606339i	$0.792637\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	3.00000i	0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	12.0000	1.92154
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−15.0000	−2.10042
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.00000i	− 0.397360i
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	− 12.0000i	− 1.51186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.00000i	− 0.122169i	−0.998133	−	0.0610847i	$-0.980544\pi$
$67$	− 1.00000i	− 0.122169i	0.998133	−	0.0610847i	$-0.0194560\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	− 3.00000i	− 0.351123i	−0.984468	−	0.175562i	$-0.943826\pi$
$73$	− 3.00000i	− 0.351123i	0.984468	−	0.175562i	$-0.0561742\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	13.0000i	1.42694i	0.700688	+	0.713468i	$0.252876\pi$
$83$	13.0000i	1.42694i	−0.700688	+	0.713468i	$0.747124\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	24.0000i	2.57307i
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	30.0000i	3.11086i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	18.0000	1.70848
$112$	0	0
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	24.0000i	2.21880i
$118$	0	0
$119$	−10.0000	−0.916698
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 9.00000i	− 0.811503i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.0000i	1.11066i	0.831628	+	0.555332i	$0.187409\pi$
$137$	13.0000i	1.11066i	−0.831628	+	0.555332i	$0.812591\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	− 30.0000i	− 2.42536i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	− 7.00000i	− 0.548282i	−0.961689	−	0.274141i	$-0.911606\pi$
$163$	− 7.00000i	− 0.548282i	0.961689	−	0.274141i	$-0.0883936\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	− 8.00000i	− 0.608229i	−0.952636	−	0.304114i	$-0.901639\pi$
$173$	− 8.00000i	− 0.608229i	0.952636	−	0.304114i	$-0.0983605\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 24.0000i	− 1.80395i
$178$	0	0
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	30.0000i	2.21766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.00000i	0.365636i
$188$	0	0
$189$	18.0000	1.30931
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	0	0
$203$	16.0000i	1.12298i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.0000i	− 0.834058i
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	− 36.0000i	− 2.46668i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.0000i	1.35769i
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 18.0000i	− 1.16923i
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	−39.0000	−2.47152
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	−48.0000	−2.97113
$262$	0	0
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.0000i	1.65237i
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	24.0000i	1.45255i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	−60.0000	−3.59211
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	1.00000i	0.0594438i	0.999558	+	0.0297219i	$0.00946217\pi$
$283$	1.00000i	0.0594438i	−0.999558	+	0.0297219i	$0.990538\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.00000i	− 0.354169i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	42.0000	2.46208
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.00000i	− 0.522233i
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	18.0000i	1.03407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.0000i	0.741949i	0.928643	+	0.370975i	$0.120976\pi$
$307$	13.0000i	0.741949i	−0.928643	+	0.370975i	$0.879024\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	45.0000	2.51166
$322$	0	0
$323$	− 5.00000i	− 0.278207i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 42.0000i	− 2.32261i
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	0	0
$333$	36.0000i	1.97279i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.0000i	1.03500i	0.855684	+	0.517498i	$0.173136\pi$
$337$	19.0000i	1.03500i	−0.855684	+	0.517498i	$0.826864\pi$
$338$	0	0
$339$	−27.0000	−1.46644
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.0000i	0.912608i	0.889824	+	0.456304i	$0.150827\pi$
$347$	17.0000i	0.912608i	−0.889824	+	0.456304i	$0.849173\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−36.0000	−1.92154
$352$	0	0
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 30.0000i	− 1.58777i
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 30.0000i	− 1.57459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0000i	1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$	20.0000i	1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 32.0000i	− 1.64808i
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.0000i	1.21999i
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	0	0
$393$	− 36.0000i	− 1.81596i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.00000i	0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$	4.00000i	0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	0	0
$403$	− 40.0000i	− 1.99254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	−39.0000	−1.92373
$412$	0	0
$413$	− 16.0000i	− 0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 27.0000i	− 1.32220i
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	0	0
$423$	− 24.0000i	− 1.16692i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.0000i	0.967868i
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	− 5.00000i	− 0.240285i	−0.992757	−	0.120142i	$-0.961665\pi$
$433$	− 5.00000i	− 0.240285i	0.992757	−	0.120142i	$-0.0383351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	21.0000i	0.997740i	0.866677	+	0.498870i	$0.166252\pi$
$443$	21.0000i	0.997740i	−0.866677	+	0.498870i	$0.833748\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 24.0000i	− 1.13516i
$448$	0	0
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	0	0
$453$	6.00000i	0.281905i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000i	0.0467780i	0.999726	+	0.0233890i	$0.00744563\pi$
$457$	1.00000i	0.0467780i	−0.999726	+	0.0233890i	$0.992554\pi$
$458$	0	0
$459$	45.0000	2.10042
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	42.0000	1.93526
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	36.0000i	1.64833i
$478$	0	0
$479$	−22.0000	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$479$	−22.0000	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	− 12.0000i	− 0.546019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	0	0
$489$	21.0000	0.949653
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	40.0000i	1.80151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	36.0000	1.60836
$502$	0	0
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	9.00000i	0.397360i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	0	0
$523$	35.0000i	1.53044i	0.643767	+	0.765222i	$0.277371\pi$
$523$	35.0000i	1.53044i	−0.643767	+	0.765222i	$0.722629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	50.0000i	2.17803i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	48.0000	2.08302
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	57.0000i	2.45973i
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	0	0
$543$	6.00000i	0.257485i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 19.0000i	− 0.812381i	−0.913788	−	0.406191i	$-0.866857\pi$
$547$	− 19.0000i	− 0.812381i	0.913788	−	0.406191i	$-0.133143\pi$
$548$	0	0
$549$	−60.0000	−2.56074
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.0000i	− 1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	− 28.0000i	− 1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	32.0000i	1.34864i	0.738440	+	0.674320i	$0.235563\pi$
$563$	32.0000i	1.34864i	−0.738440	+	0.674320i	$0.764437\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.0000i	0.755929i
$568$	0	0
$569$	−17.0000	−0.712677	−0.356339	−	0.934357i	$-0.615975\pi$
$569$	−17.0000	−0.712677	−0.356339	+	0.934357i	$0.615975\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	− 30.0000i	− 1.25327i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 13.0000i	− 0.541197i	−0.962692	−	0.270599i	$-0.912778\pi$
$577$	− 13.0000i	− 0.541197i	0.962692	−	0.270599i	$-0.0872216\pi$
$578$	0	0
$579$	−33.0000	−1.37143
$580$	0	0
$581$	−26.0000	−1.07866
$582$	0	0
$583$	− 6.00000i	− 0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.00000i	0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$	3.00000i	0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	− 39.0000i	− 1.60154i	−0.598973	−	0.800769i	$-0.704424\pi$
$593$	− 39.0000i	− 1.60154i	0.598973	−	0.800769i	$-0.295576\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	84.0000i	3.43789i
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	43.0000	1.75401	0.877003	−	0.480484i	$-0.159539\pi$
$601$	43.0000	1.75401	0.877003	+	0.480484i	$0.159539\pi$
$602$	0	0
$603$	6.00000i	0.244339i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	− 30.0000i	− 1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$	− 30.0000i	− 1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	18.0000i	0.721155i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.00000i	− 0.119808i
$628$	0	0
$629$	30.0000	1.19618
$630$	0	0
$631$	10.0000	0.398094	0.199047	−	0.979990i	$-0.436215\pi$
$631$	10.0000	0.398094	0.199047	+	0.979990i	$0.436215\pi$
$632$	0	0
$633$	− 3.00000i	− 0.119239i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	72.0000	2.84828
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	−60.0000	−2.35159
$652$	0	0
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.0000i	0.702247i
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	60.0000i	2.33021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	−12.0000	−0.463947
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32.0000i	− 1.22986i	−0.788582	−	0.614930i	$-0.789184\pi$
$677$	− 32.0000i	− 1.22986i	0.788582	−	0.614930i	$-0.210816\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	0	0
$681$	36.0000	1.37952
$682$	0	0
$683$	3.00000i	0.114792i	0.998351	+	0.0573959i	$0.0182797\pi$
$683$	3.00000i	0.114792i	−0.998351	+	0.0573959i	$0.981720\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.0000i	− 0.457829i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	0	0
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 15.0000i	− 0.568166i
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−28.0000	−1.05755	−0.528773	−	0.848763i	$-0.677348\pi$
$701$	−28.0000	−1.05755	−0.528773	+	0.848763i	$0.677348\pi$
$702$	0	0
$703$	6.00000i	0.226294i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 72.0000i	− 2.68889i
$718$	0	0
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 21.0000i	− 0.780998i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.00000i	− 0.0368355i
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	− 30.0000i	− 1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$	− 30.0000i	− 1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 78.0000i	− 2.85387i
$748$	0	0
$749$	30.0000	1.09618
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	45.0000i	1.63989i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.0000i	1.15545i
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	−90.0000	−3.24127
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	36.0000i	1.29149i
$778$	0	0
$779$	3.00000	0.107486
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	− 72.0000i	− 2.57307i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	0	0
$789$	90.0000	3.20408
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	−20.0000	−0.707549
$800$	0	0
$801$	−54.0000	−1.90800
$802$	0	0
$803$	− 3.00000i	− 0.105868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.0000i	1.68968i
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	− 18.0000i	− 0.631288i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	0	0
$819$	−48.0000	−1.67726
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−24.0000	−0.832551
$832$	0	0
$833$	15.0000i	0.519719i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 90.0000i	− 3.11086i
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 20.0000i	− 0.687208i
$848$	0	0
$849$	−3.00000	−0.102960
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 24.0000i	− 0.815083i
$868$	0	0
$869$	−6.00000	−0.203536
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	84.0000i	2.84297i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.00000i	0.135070i	0.997717	+	0.0675352i	$0.0215135\pi$
$877$	4.00000i	0.135070i	−0.997717	+	0.0675352i	$0.978487\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	− 5.00000i	− 0.168263i	−0.996455	−	0.0841317i	$-0.973188\pi$
$883$	− 5.00000i	− 0.168263i	0.996455	−	0.0841317i	$-0.0268116\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	− 4.00000i	− 0.133855i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.0000i	0.801337i
$898$	0	0
$899$	80.0000	2.66815
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	24.0000i	0.798670i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	−36.0000	−1.19404
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	13.0000i	0.430237i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 24.0000i	− 0.792550i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	−39.0000	−1.28509
$922$	0	0
$923$	48.0000i	1.57994i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	24.0000i	0.788263i
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	42.0000i	1.37502i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.00000i	− 0.163343i	−0.996659	−	0.0816714i	$-0.973974\pi$
$937$	− 5.00000i	− 0.163343i	0.996659	−	0.0816714i	$-0.0260258\pi$
$938$	0	0
$939$	−66.0000	−2.15383
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	0	0
$943$	− 6.00000i	− 0.195387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.0000i	1.29983i	0.760009	+	0.649913i	$0.225195\pi$
$947$	40.0000i	1.29983i	−0.760009	+	0.649913i	$0.774805\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−36.0000	−1.16738
$952$	0	0
$953$	39.0000i	1.26333i	0.775240	+	0.631667i	$0.217629\pi$
$953$	39.0000i	1.26333i	−0.775240	+	0.631667i	$0.782371\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	−26.0000	−0.839584
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	90.0000i	2.90021i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	−31.0000	−0.994837	−0.497419	−	0.867511i	$-0.665719\pi$
$971$	−31.0000	−0.994837	−0.497419	+	0.867511i	$0.665719\pi$
$972$	0	0
$973$	− 18.0000i	− 0.577054i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.0000i	0.543878i	0.962314	+	0.271939i	$0.0876649\pi$
$977$	17.0000i	0.543878i	−0.962314	+	0.271939i	$0.912335\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	84.0000	2.68191
$982$	0	0
$983$	− 6.00000i	− 0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$	− 6.00000i	− 0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	− 81.0000i	− 2.57046i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 12.0000i	− 0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	− 12.0000i	− 0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$	0	0
$999$	−54.0000	−1.70848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.c.a.49.2		2
3.2	odd	2		1800.2.f.f.649.2		2
4.3	odd	2		400.2.c.a.49.1		2
5.2	odd	4		200.2.a.e.1.1	yes	1
5.3	odd	4		200.2.a.a.1.1	✓	1
5.4	even	2	inner	200.2.c.a.49.1		2
8.3	odd	2		1600.2.c.b.449.2		2
8.5	even	2		1600.2.c.a.449.1		2
12.11	even	2		3600.2.f.n.2449.1		2
15.2	even	4		1800.2.a.h.1.1		1
15.8	even	4		1800.2.a.r.1.1		1
15.14	odd	2		1800.2.f.f.649.1		2
20.3	even	4		400.2.a.h.1.1		1
20.7	even	4		400.2.a.a.1.1		1
20.19	odd	2		400.2.c.a.49.2		2
35.13	even	4		9800.2.a.bq.1.1		1
35.27	even	4		9800.2.a.c.1.1		1
40.3	even	4		1600.2.a.b.1.1		1
40.13	odd	4		1600.2.a.x.1.1		1
40.19	odd	2		1600.2.c.b.449.1		2
40.27	even	4		1600.2.a.y.1.1		1
40.29	even	2		1600.2.c.a.449.2		2
40.37	odd	4		1600.2.a.a.1.1		1
60.23	odd	4		3600.2.a.m.1.1		1
60.47	odd	4		3600.2.a.bf.1.1		1
60.59	even	2		3600.2.f.n.2449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.a.a.1.1	✓	1	5.3	odd	4
200.2.a.e.1.1	yes	1	5.2	odd	4
200.2.c.a.49.1		2	5.4	even	2	inner
200.2.c.a.49.2		2	1.1	even	1	trivial
400.2.a.a.1.1		1	20.7	even	4
400.2.a.h.1.1		1	20.3	even	4
400.2.c.a.49.1		2	4.3	odd	2
400.2.c.a.49.2		2	20.19	odd	2
1600.2.a.a.1.1		1	40.37	odd	4
1600.2.a.b.1.1		1	40.3	even	4
1600.2.a.x.1.1		1	40.13	odd	4
1600.2.a.y.1.1		1	40.27	even	4
1600.2.c.a.449.1		2	8.5	even	2
1600.2.c.a.449.2		2	40.29	even	2
1600.2.c.b.449.1		2	40.19	odd	2
1600.2.c.b.449.2		2	8.3	odd	2
1800.2.a.h.1.1		1	15.2	even	4
1800.2.a.r.1.1		1	15.8	even	4
1800.2.f.f.649.1		2	15.14	odd	2
1800.2.f.f.649.2		2	3.2	odd	2
3600.2.a.m.1.1		1	60.23	odd	4
3600.2.a.bf.1.1		1	60.47	odd	4
3600.2.f.n.2449.1		2	12.11	even	2
3600.2.f.n.2449.2		2	60.59	even	2
9800.2.a.c.1.1		1	35.27	even	4
9800.2.a.bq.1.1		1	35.13	even	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -4.00000i q^{13} +5.00000i q^{17} -1.00000 q^{19} -6.00000 q^{21} +2.00000i q^{23} -9.00000i q^{27} +8.00000 q^{29} +10.0000 q^{31} +3.00000i q^{33} -6.00000i q^{37} +12.0000 q^{39} -3.00000 q^{41} -4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -15.0000 q^{51} -6.00000i q^{53} -3.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} -12.0000i q^{63} -1.00000i q^{67} -6.00000 q^{69} -12.0000 q^{71} -3.00000i q^{73} +2.00000i q^{77} -6.00000 q^{79} +9.00000 q^{81} +13.0000i q^{83} +24.0000i q^{87} +9.00000 q^{89} +8.00000 q^{91} +30.0000i q^{93} -14.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9} + 2 q^{11} - 2 q^{19} - 12 q^{21} + 16 q^{29} + 20 q^{31} + 24 q^{39} - 6 q^{41} + 6 q^{49} - 30 q^{51} - 16 q^{59} + 20 q^{61} - 12 q^{69} - 24 q^{71} - 12 q^{79} + 18 q^{81} + 18 q^{89}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 + 2 * q^11 - 2 * q^19 - 12 * q^21 + 16 * q^29 + 20 * q^31 + 24 * q^39 - 6 * q^41 + 6 * q^49 - 30 * q^51 - 16 * q^59 + 20 * q^61 - 12 * q^69 - 24 * q^71 - 12 * q^79 + 18 * q^81 + 18 * q^89 + 16 * q^91 - 12 * q^99

Introduction

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