LMFDB - Embedded newform 219.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [219,2,Mod(1,219)] 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("219.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(219, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 219 = 3 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	219.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$1.74872380427$
Analytic rank:	$1$
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	$\chi$	$=$	219.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.00000 q^{12} -4.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} +6.00000 q^{20} -4.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +8.00000 q^{28} -6.00000 q^{29} -10.0000 q^{31} +12.0000 q^{35} -2.00000 q^{36} -7.00000 q^{37} -4.00000 q^{39} +2.00000 q^{43} -3.00000 q^{45} -3.00000 q^{47} +4.00000 q^{48} +9.00000 q^{49} +3.00000 q^{51} +8.00000 q^{52} +9.00000 q^{53} -1.00000 q^{57} -9.00000 q^{59} +6.00000 q^{60} -1.00000 q^{61} -4.00000 q^{63} -8.00000 q^{64} +12.0000 q^{65} -13.0000 q^{67} -6.00000 q^{68} +6.00000 q^{69} +12.0000 q^{71} +1.00000 q^{73} +4.00000 q^{75} +2.00000 q^{76} +11.0000 q^{79} -12.0000 q^{80} +1.00000 q^{81} +15.0000 q^{83} +8.00000 q^{84} -9.00000 q^{85} -6.00000 q^{87} -18.0000 q^{89} +16.0000 q^{91} -12.0000 q^{92} -10.0000 q^{93} +3.00000 q^{95} +5.00000 q^{97} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	4.00000	1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	6.00000	1.34164
$21$	−4.00000	−0.872872
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	8.00000	1.51186
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.0000	2.02837
$36$	−2.00000	−0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	4.00000	0.577350
$49$	9.00000	1.28571
$50$	0	0
$51$	3.00000	0.420084
$52$	8.00000	1.10940
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	6.00000	0.774597
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	−8.00000	−1.00000
$65$	12.0000	1.48842
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−6.00000	−0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	0	0
$75$	4.00000	0.461880
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	8.00000	0.872872
$85$	−9.00000	−0.976187
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	−12.0000	−1.25109
$93$	−10.0000	−1.03695
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	−2.00000	−0.192450
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	−16.0000	−1.51186
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	12.0000	1.11417
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	20.0000	1.79605
$125$	3.00000	0.268328
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−24.0000	−2.02837
$141$	−3.00000	−0.252646
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	18.0000	1.49482
$146$	0	0
$147$	9.00000	0.742307
$148$	14.0000	1.15079
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	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
$153$	3.00000	0.242536
$154$	0	0
$155$	30.0000	2.40966
$156$	8.00000	0.640513
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	6.00000	0.447214
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	21.0000	1.54395
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−8.00000	−0.577350
$193$	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
$195$	12.0000	0.859338
$196$	−18.0000	−1.28571
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	0	0
$203$	24.0000	1.68447
$204$	−6.00000	−0.420084
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	−16.0000	−1.10940
$209$	0	0
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−18.0000	−1.23625
$213$	12.0000	0.822226
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	40.0000	2.71538
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	30.0000	1.99117	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000	1.99117	0.995585	+	0.0938647i	$0.0299221\pi$
$228$	2.00000	0.132453
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	18.0000	1.17170
$237$	11.0000	0.714527
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	−12.0000	−0.774597
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−27.0000	−1.72497
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	15.0000	0.950586
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	8.00000	0.503953
$253$	0	0
$254$	0	0
$255$	−9.00000	−0.563602
$256$	16.0000	1.00000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	28.0000	1.73984
$260$	−24.0000	−1.48842
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−27.0000	−1.65860
$266$	0	0
$267$	−18.0000	−1.10158
$268$	26.0000	1.58820
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	12.0000	0.727607
$273$	16.0000	0.968364
$274$	0	0
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−24.0000	−1.42414
$285$	3.00000	0.177705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	5.00000	0.293105
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	27.0000	1.57200
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.0000	−1.38796
$300$	−8.00000	−0.461880
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−15.0000	−0.861727
$304$	−4.00000	−0.229416
$305$	3.00000	0.171780
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	−22.0000	−1.23760
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	0	0
$320$	24.0000	1.34164
$321$	−15.0000	−0.837218
$322$	0	0
$323$	−3.00000	−0.166924
$324$	−2.00000	−0.111111
$325$	−16.0000	−0.887520
$326$	0	0
$327$	5.00000	0.276501
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−30.0000	−1.64646
$333$	−7.00000	−0.383598
$334$	0	0
$335$	39.0000	2.13080
$336$	−16.0000	−0.872872
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	18.0000	0.976187
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−18.0000	−0.969087
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	12.0000	0.643268
$349$	−31.0000	−1.65939	−0.829696	−	0.558216i	$-0.811486\pi$
$349$	−31.0000	−1.65939	−0.829696	+	0.558216i	$0.811486\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	36.0000	1.90800
$357$	−12.0000	−0.635107
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−11.0000	−0.577350
$364$	−32.0000	−1.67726
$365$	−3.00000	−0.157027
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	24.0000	1.25109
$369$	0	0
$370$	0	0
$371$	−36.0000	−1.86903
$372$	20.0000	1.03695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−6.00000	−0.307794
$381$	11.0000	0.563547
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.00000	0.101666
$388$	−10.0000	−0.507673
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	−33.0000	−1.66041
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	16.0000	0.800000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	40.0000	1.99254
$404$	30.0000	1.49256
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	8.00000	0.394132
$413$	36.0000	1.77144
$414$	0	0
$415$	−45.0000	−2.20896
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−24.0000	−1.17108
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	4.00000	0.193574
$428$	30.0000	1.45010
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	4.00000	0.192450
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	−10.0000	−0.478913
$437$	−6.00000	−0.287019
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	14.0000	0.664411
$445$	54.0000	2.55985
$446$	0	0
$447$	0	0
$448$	32.0000	1.51186
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	−10.0000	−0.469841
$454$	0	0
$455$	−48.0000	−2.25027
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	36.0000	1.67851
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	−24.0000	−1.11417
$465$	30.0000	1.39122
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	8.00000	0.369800
$469$	52.0000	2.40114
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	24.0000	1.10004
$477$	9.00000	0.412082
$478$	0	0
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	0	0
$481$	28.0000	1.27669
$482$	0	0
$483$	−24.0000	−1.09204
$484$	22.0000	1.00000
$485$	−15.0000	−0.681115
$486$	0	0
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	0	0
$495$	0	0
$496$	−40.0000	−1.79605
$497$	−48.0000	−2.15309
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	−6.00000	−0.268328
$501$	24.0000	1.07224
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	45.0000	2.00247
$506$	0	0
$507$	3.00000	0.133235
$508$	−22.0000	−0.976092
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	12.0000	0.528783
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	−6.00000	−0.262111
$525$	−16.0000	−0.698297
$526$	0	0
$527$	−30.0000	−1.30682
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−9.00000	−0.390567
$532$	−8.00000	−0.346844
$533$	0	0
$534$	0	0
$535$	45.0000	1.94552
$536$	0	0
$537$	9.00000	0.388379
$538$	0	0
$539$	0	0
$540$	6.00000	0.258199
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−15.0000	−0.642529
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	36.0000	1.53784
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−44.0000	−1.87107
$554$	0	0
$555$	21.0000	0.891400
$556$	−16.0000	−0.678551
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	48.0000	2.02837
$561$	0	0
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	6.00000	0.252646
$565$	18.0000	0.757266
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	−8.00000	−0.333333
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	−36.0000	−1.49482
$581$	−60.0000	−2.48922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	−18.0000	−0.742307
$589$	10.0000	0.412043
$590$	0	0
$591$	6.00000	0.246807
$592$	−28.0000	−1.15079
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	36.0000	1.47586
$596$	0	0
$597$	−10.0000	−0.409273
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	0	0
$603$	−13.0000	−0.529401
$604$	20.0000	0.813788
$605$	33.0000	1.34164
$606$	0	0
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	12.0000	0.485468
$612$	−6.00000	−0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−60.0000	−2.40966
$621$	6.00000	0.240772
$622$	0	0
$623$	72.0000	2.88462
$624$	−16.0000	−0.640513
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−21.0000	−0.837325
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	−1.00000	−0.0397464
$634$	0	0
$635$	−33.0000	−1.30957
$636$	−18.0000	−0.713746
$637$	−36.0000	−1.42637
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	48.0000	1.89146
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	40.0000	1.56772
$652$	−16.0000	−0.626608
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	0	0
$657$	1.00000	0.0390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−36.0000	−1.39393
$668$	−48.0000	−1.85718
$669$	20.0000	0.773245
$670$	0	0
$671$	0	0
$672$	0	0
$673$	47.0000	1.81172	0.905858	−	0.423581i	$-0.139227\pi$
$673$	47.0000	1.81172	0.905858	+	0.423581i	$0.139227\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	−6.00000	−0.230769
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	2.00000	0.0764719
$685$	54.0000	2.06323
$686$	0	0
$687$	−16.0000	−0.610438
$688$	8.00000	0.304997
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	36.0000	1.36851
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	0	0
$698$	0	0
$699$	3.00000	0.113470
$700$	32.0000	1.20949
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	7.00000	0.264010
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	60.0000	2.25653
$708$	18.0000	0.676481
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−60.0000	−2.24702
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	3.00000	0.112037
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−12.0000	−0.447214
$721$	16.0000	0.595871
$722$	0	0
$723$	−28.0000	−1.04133
$724$	20.0000	0.743294
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	2.00000	0.0739221
$733$	−25.0000	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$734$	0	0
$735$	−27.0000	−0.995910
$736$	0	0
$737$	0	0
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−42.0000	−1.54395
$741$	4.00000	0.146944
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.0000	0.548821
$748$	0	0
$749$	60.0000	2.19235
$750$	0	0
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	−12.0000	−0.437595
$753$	−12.0000	−0.437304
$754$	0	0
$755$	30.0000	1.09181
$756$	8.00000	0.290957
$757$	11.0000	0.399802	0.199901	−	0.979816i	$-0.435938\pi$
$757$	11.0000	0.399802	0.199901	+	0.979816i	$0.435938\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	−9.00000	−0.325396
$766$	0	0
$767$	36.0000	1.29988
$768$	16.0000	0.577350
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−28.0000	−1.00774
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	0	0
$777$	28.0000	1.00449
$778$	0	0
$779$	0	0
$780$	−24.0000	−0.859338
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	36.0000	1.28571
$785$	−6.00000	−0.214149
$786$	0	0
$787$	23.0000	0.819861	0.409931	−	0.912117i	$-0.365553\pi$
$787$	23.0000	0.819861	0.409931	+	0.912117i	$0.365553\pi$
$788$	−12.0000	−0.427482
$789$	−9.00000	−0.320408
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	−27.0000	−0.957591
$796$	20.0000	0.708881
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	0	0
$804$	26.0000	0.916949
$805$	72.0000	2.53767
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	−48.0000	−1.68447
$813$	20.0000	0.701431
$814$	0	0
$815$	−24.0000	−0.840683
$816$	12.0000	0.420084
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	16.0000	0.559085
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	−12.0000	−0.417029
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	32.0000	1.10940
$833$	27.0000	0.935495
$834$	0	0
$835$	−72.0000	−2.49166
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	15.0000	0.516627
$844$	2.00000	0.0688428
$845$	−9.00000	−0.309609
$846$	0	0
$847$	44.0000	1.51186
$848$	36.0000	1.23625
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−42.0000	−1.43974
$852$	−24.0000	−0.822226
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	0	0
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	0	0
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	0	0
$867$	−8.00000	−0.271694
$868$	−80.0000	−2.71538
$869$	0	0
$870$	0	0
$871$	52.0000	1.76195
$872$	0	0
$873$	5.00000	0.169224
$874$	0	0
$875$	−12.0000	−0.405674
$876$	−2.00000	−0.0675737
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	39.0000	1.31394	0.656972	−	0.753915i	$-0.271837\pi$
$881$	39.0000	1.31394	0.656972	+	0.753915i	$0.271837\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	24.0000	0.807207
$885$	27.0000	0.907595
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	−44.0000	−1.47571
$890$	0	0
$891$	0	0
$892$	−40.0000	−1.33930
$893$	3.00000	0.100391
$894$	0	0
$895$	−27.0000	−0.902510
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	60.0000	2.00111
$900$	−8.00000	−0.266667
$901$	27.0000	0.899500
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	30.0000	0.997234
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	−60.0000	−1.99117
$909$	−15.0000	−0.497519
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	0	0
$915$	3.00000	0.0991769
$916$	32.0000	1.05731
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	−6.00000	−0.196537
$933$	30.0000	0.982156
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	−18.0000	−0.587095
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	−36.0000	−1.17170
$945$	12.0000	0.390360
$946$	0	0
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	−22.0000	−0.714527
$949$	−4.00000	−0.129845
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	0	0
$959$	72.0000	2.32500
$960$	24.0000	0.774597
$961$	69.0000	2.22581
$962$	0	0
$963$	−15.0000	−0.483368
$964$	56.0000	1.80364
$965$	−42.0000	−1.35203
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	−3.00000	−0.0963739
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	−2.00000	−0.0641500
$973$	−32.0000	−1.02587
$974$	0	0
$975$	−16.0000	−0.512410
$976$	−4.00000	−0.128037
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	54.0000	1.72497
$981$	5.00000	0.159638
$982$	0	0
$983$	−15.0000	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	12.0000	0.381964
$988$	−8.00000	−0.254514
$989$	12.0000	0.381578
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	30.0000	0.951064
$996$	−30.0000	−0.950586
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	0	0
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	219.2.a.b.1.1	✓	1
3.2	odd	2		657.2.a.c.1.1		1
4.3	odd	2		3504.2.a.c.1.1		1
5.4	even	2		5475.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
219.2.a.b.1.1	✓	1	1.1	even	1	trivial
657.2.a.c.1.1		1	3.2	odd	2
3504.2.a.c.1.1		1	4.3	odd	2
5475.2.a.e.1.1		1	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 219 = 3 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	219.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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