LMFDB - Embedded newform 4200.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.a (trivial)

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:	yes
Analytic conductor:	$33.5371688489$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +3.00000 q^{13} +5.00000 q^{17} +4.00000 q^{19} -1.00000 q^{21} +5.00000 q^{23} -1.00000 q^{27} +9.00000 q^{29} -5.00000 q^{31} -2.00000 q^{33} -10.0000 q^{37} -3.00000 q^{39} +1.00000 q^{41} -3.00000 q^{43} -12.0000 q^{47} +1.00000 q^{49} -5.00000 q^{51} -11.0000 q^{53} -4.00000 q^{57} +5.00000 q^{59} +9.00000 q^{61} +1.00000 q^{63} +4.00000 q^{67} -5.00000 q^{69} -4.00000 q^{71} +6.00000 q^{73} +2.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -5.00000 q^{83} -9.00000 q^{87} -6.00000 q^{89} +3.00000 q^{91} +5.00000 q^{93} +18.0000 q^{97} +2.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	9.00000	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$61$	9.00000	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	20.0000	1.52057	0.760286	−	0.649589i	$-0.225059\pi$
$173$	20.0000	1.52057	0.760286	+	0.649589i	$0.225059\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.00000	−0.375823
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−9.00000	−0.665299
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.0000	0.731272
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.00000	0.347524
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	15.0000	1.00901
$222$	0	0
$223$	−11.0000	−0.736614	−0.368307	−	0.929704i	$-0.620063\pi$
$223$	−11.0000	−0.736614	−0.368307	+	0.929704i	$0.620063\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.0000	−0.909398
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	5.00000	0.316862
$250$	0	0
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−5.00000	−0.299342
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−18.0000	−1.05518
$292$	0	0
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	15.0000	0.867472
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.00000	0.0561656	0.0280828	−	0.999606i	$-0.491060\pi$
$317$	1.00000	0.0561656	0.0280828	+	0.999606i	$0.491060\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.0000	−1.10600
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	−11.0000	−0.571092
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.0000	1.39057
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.00000	−0.152499
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	25.0000	1.26430
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	0	0
$396$	0	0
$397$	33.0000	1.65622	0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000	1.65622	0.828111	+	0.560564i	$0.189416\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	5.00000	0.246034
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	31.0000	1.51445	0.757225	−	0.653155i	$-0.226555\pi$
$419$	31.0000	1.51445	0.757225	+	0.653155i	$0.226555\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.00000	0.435541
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−1.00000	−0.0481683	−0.0240842	−	0.999710i	$-0.507667\pi$
$431$	−1.00000	−0.0481683	−0.0240842	+	0.999710i	$0.507667\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.0000	0.956730
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.0000	0.520282
$448$	0	0
$449$	−32.0000	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	10.0000	0.469841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.00000	0.323921	0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000	0.323921	0.161961	+	0.986797i	$0.448218\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.0000	−0.503655
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−30.0000	−1.36788
$482$	0	0
$483$	−5.00000	−0.227508
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	45.0000	2.02670
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	25.0000	1.11915	0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000	1.11915	0.559577	+	0.828778i	$0.310964\pi$
$500$	0	0
$501$	14.0000	0.625474
$502$	0	0
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−20.0000	−0.877903
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.0000	−1.08902
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	5.00000	0.216982
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	0	0
$549$	9.00000	0.384111
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	15.0000	0.632175	0.316087	−	0.948730i	$-0.397631\pi$
$563$	15.0000	0.632175	0.316087	+	0.948730i	$0.397631\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−5.00000	−0.207435
$582$	0	0
$583$	−22.0000	−0.911147
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	3.00000	0.123404
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	7.00000	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$599$	7.00000	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	−36.0000	−1.45640
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−50.0000	−1.99363
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.0000	0.393141	0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000	0.393141	0.196570	+	0.980490i	$0.437020\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	5.00000	0.195965
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	−15.0000	−0.582552
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.0000	1.74241
$668$	0	0
$669$	11.0000	0.425285
$670$	0	0
$671$	18.0000	0.694882
$672$	0	0
$673$	−27.0000	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$673$	−27.0000	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	18.0000	0.690777
$680$	0	0
$681$	7.00000	0.268241
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	−33.0000	−1.25720
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.00000	0.189389
$698$	0	0
$699$	−4.00000	−0.151294
$700$	0	0
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.0000	1.35201	0.676004	−	0.736898i	$-0.263710\pi$
$709$	36.0000	1.35201	0.676004	+	0.736898i	$0.263710\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	−25.0000	−0.936257
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	0	0
$726$	0	0
$727$	49.0000	1.81731	0.908655	−	0.417548i	$-0.137111\pi$
$727$	49.0000	1.81731	0.908655	+	0.417548i	$0.137111\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.0000	−0.554795
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	31.0000	1.14035	0.570177	−	0.821522i	$-0.306875\pi$
$739$	31.0000	1.14035	0.570177	+	0.821522i	$0.306875\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.00000	−0.182940
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−36.0000	−1.31366	−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000	−1.31366	−0.656829	+	0.754039i	$0.728103\pi$
$752$	0	0
$753$	−17.0000	−0.619514
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	0	0
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.0000	0.541619
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	0	0
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	0	0
$789$	23.0000	0.818822
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	27.0000	0.958798
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−60.0000	−2.12265
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	3.00000	0.104828
$820$	0	0
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	−46.0000	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.0000	−1.59958	−0.799788	−	0.600282i	$-0.795055\pi$
$827$	−46.0000	−1.59958	−0.799788	+	0.600282i	$0.795055\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	5.00000	0.173240
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	−24.0000	−0.826604
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−50.0000	−1.71398
$852$	0	0
$853$	25.0000	0.855984	0.427992	−	0.903783i	$-0.359221\pi$
$853$	25.0000	0.855984	0.427992	+	0.903783i	$0.359221\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	−1.00000	−0.0340799
$862$	0	0
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	28.0000	0.949835
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	18.0000	0.609208
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	41.0000	1.38133	0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000	1.38133	0.690663	+	0.723177i	$0.257319\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−48.0000	−1.60626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−15.0000	−0.500835
$898$	0	0
$899$	−45.0000	−1.50083
$900$	0	0
$901$	−55.0000	−1.83232
$902$	0	0
$903$	3.00000	0.0998337
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47.0000	1.56061	0.780305	−	0.625400i	$-0.215064\pi$
$907$	47.0000	1.56061	0.780305	+	0.625400i	$0.215064\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.00000	−0.264183
$918$	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
$921$	−6.00000	−0.197707
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.00000	0.0328443
$928$	0	0
$929$	5.00000	0.164045	0.0820223	−	0.996630i	$-0.473862\pi$
$929$	5.00000	0.164045	0.0820223	+	0.996630i	$0.473862\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−2.00000	−0.0654771
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	5.00000	0.162822
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	0	0
$951$	−1.00000	−0.0324272
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.0000	−0.581857
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	−20.0000	−0.642493
$970$	0	0
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	0	0
$976$	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
$979$	−12.0000	−0.383522
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−15.0000	−0.476972
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	0	0
$993$	15.0000	0.476011
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.l.1.1	✓	1
4.3	odd	2		8400.2.a.bu.1.1		1
5.2	odd	4		4200.2.t.o.1849.2		2
5.3	odd	4		4200.2.t.o.1849.1		2
5.4	even	2		4200.2.a.v.1.1	yes	1
20.19	odd	2		8400.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4200.2.a.l.1.1	✓	1	1.1	even	1	trivial
4200.2.a.v.1.1	yes	1	5.4	even	2
4200.2.t.o.1849.1		2	5.3	odd	4
4200.2.t.o.1849.2		2	5.2	odd	4
8400.2.a.s.1.1		1	20.19	odd	2
8400.2.a.bu.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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