LMFDB - Embedded newform 1050.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.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:	$8.38429221223$
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	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{29} +3.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -3.00000 q^{41} +1.00000 q^{42} +1.00000 q^{43} +2.00000 q^{44} +7.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} -1.00000 q^{58} -3.00000 q^{59} +5.00000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +12.0000 q^{67} +1.00000 q^{68} +7.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000 q^{77} -1.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} -1.00000 q^{86} -1.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} -1.00000 q^{91} -7.00000 q^{92} -3.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	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	$0.402509\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	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
$22$	−2.00000	−0.426401
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	1.00000	0.154303
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	7.00000	1.03209
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	−1.00000	−0.131306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−3.00000	−0.381000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	1.00000	0.121268
$69$	7.00000	0.842701
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	2.00000	0.227921
$78$	−1.00000	−0.113228
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−1.00000	−0.107833
$87$	−1.00000	−0.107211
$88$	−2.00000	−0.213201
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−7.00000	−0.729800
$93$	−3.00000	−0.311086
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−1.00000	−0.101015
$99$	2.00000	0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	1.00000	0.0990148
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	11.0000	1.06841
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−1.00000	−0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	1.00000	0.0928477
$117$	−1.00000	−0.0924500
$118$	3.00000	0.276172
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.00000	−0.452679
$123$	3.00000	0.270501
$124$	3.00000	0.269408
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	−1.00000	−0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−2.00000	−0.174078
$133$	4.00000	0.346844
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	−7.00000	−0.595880
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	−4.00000	−0.335673
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	−1.00000	−0.0824786
$148$	6.00000	0.493197
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	−2.00000	−0.161165
$155$	0	0
$156$	1.00000	0.0800641
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	2.00000	0.159111
$159$	11.0000	0.872357
$160$	0	0
$161$	−7.00000	−0.551677
$162$	−1.00000	−0.0785674
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−3.00000	−0.232845
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	1.00000	0.0762493
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	2.00000	0.150756
$177$	3.00000	0.225494
$178$	−10.0000	−0.749532
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	1.00000	0.0741249
$183$	−5.00000	−0.369611
$184$	7.00000	0.516047
$185$	0	0
$186$	3.00000	0.219971
$187$	2.00000	0.146254
$188$	12.0000	0.875190
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	−2.00000	−0.142134
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	1.00000	0.0701862
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−17.0000	−1.18445
$207$	−7.00000	−0.486534
$208$	−1.00000	−0.0693375
$209$	8.00000	0.553372
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−11.0000	−0.755483
$213$	−4.00000	−0.274075
$214$	−18.0000	−1.23045
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.00000	0.203653
$218$	4.00000	0.270914
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	6.00000	0.402694
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	−4.00000	−0.264906
$229$	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
$232$	−1.00000	−0.0656532
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−3.00000	−0.195283
$237$	2.00000	0.129914
$238$	−1.00000	−0.0648204
$239$	−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
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	5.00000	0.320092
$245$	0	0
$246$	−3.00000	−0.191273
$247$	−4.00000	−0.254514
$248$	−3.00000	−0.190500
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	1.00000	0.0629941
$253$	−14.0000	−0.880172
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	1.00000	0.0622573
$259$	6.00000	0.372822
$260$	0	0
$261$	1.00000	0.0618984
$262$	−8.00000	−0.494242
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	1.00000	0.0606339
$273$	1.00000	0.0605228
$274$	4.00000	0.241649
$275$	0	0
$276$	7.00000	0.421350
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	4.00000	0.239904
$279$	3.00000	0.179605
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	12.0000	0.714590
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	2.00000	0.118262
$287$	−3.00000	−0.177084
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000	0.819288
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−6.00000	−0.348743
$297$	−2.00000	−0.116052
$298$	−5.00000	−0.289642
$299$	7.00000	0.404820
$300$	0	0
$301$	1.00000	0.0576390
$302$	−22.0000	−1.26596
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	2.00000	0.113961
$309$	−17.0000	−0.967096
$310$	0	0
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	−1.00000	−0.0566139
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−2.00000	−0.112509
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	−11.0000	−0.616849
$319$	2.00000	0.111979
$320$	0	0
$321$	−18.0000	−1.00466
$322$	7.00000	0.390095
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	−19.0000	−1.05231
$327$	4.00000	0.221201
$328$	3.00000	0.165647
$329$	12.0000	0.661581
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	3.00000	0.164646
$333$	6.00000	0.328798
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	12.0000	0.652714
$339$	6.00000	0.325875
$340$	0	0
$341$	6.00000	0.324918
$342$	−4.00000	−0.216295
$343$	1.00000	0.0539949
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	−1.00000	−0.0536056
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−2.00000	−0.106600
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−3.00000	−0.159448
$355$	0	0
$356$	10.0000	0.529999
$357$	−1.00000	−0.0529256
$358$	−10.0000	−0.528516
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	18.0000	0.946059
$363$	7.00000	0.367405
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	5.00000	0.261354
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	−7.00000	−0.364900
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−11.0000	−0.571092
$372$	−3.00000	−0.155543
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	−1.00000	−0.0515026
$378$	1.00000	0.0514344
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	14.0000	0.717242
$382$	13.0000	0.665138
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	1.00000	0.0508329
$388$	10.0000	0.507673
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	−1.00000	−0.0505076
$393$	−8.00000	−0.403547
$394$	27.0000	1.36024
$395$	0	0
$396$	2.00000	0.100504
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	24.0000	1.20301
$399$	−4.00000	−0.200250
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	12.0000	0.598506
$403$	−3.00000	−0.149441
$404$	0	0
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	12.0000	0.594818
$408$	1.00000	0.0495074
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	17.0000	0.837530
$413$	−3.00000	−0.147620
$414$	7.00000	0.344031
$415$	0	0
$416$	1.00000	0.0490290
$417$	4.00000	0.195881
$418$	−8.00000	−0.391293
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−15.0000	−0.730189
$423$	12.0000	0.583460
$424$	11.0000	0.534207
$425$	0	0
$426$	4.00000	0.193801
$427$	5.00000	0.241967
$428$	18.0000	0.870063
$429$	2.00000	0.0965609
$430$	0	0
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−28.0000	−1.33942
$438$	14.0000	0.668946
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	1.00000	0.0475651
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	−13.0000	−0.615568
$447$	−5.00000	−0.236492
$448$	1.00000	0.0472456
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−6.00000	−0.282216
$453$	−22.0000	−1.03365
$454$	7.00000	0.328526
$455$	0	0
$456$	4.00000	0.187317
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	−14.0000	−0.654177
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	2.00000	0.0930484
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	20.0000	0.926482
$467$	39.0000	1.80470	0.902352	−	0.430999i	$-0.141839\pi$
$467$	39.0000	1.80470	0.902352	+	0.430999i	$0.141839\pi$
$468$	−1.00000	−0.0462250
$469$	12.0000	0.554109
$470$	0	0
$471$	18.0000	0.829396
$472$	3.00000	0.138086
$473$	2.00000	0.0919601
$474$	−2.00000	−0.0918630
$475$	0	0
$476$	1.00000	0.0458349
$477$	−11.0000	−0.503655
$478$	24.0000	1.09773
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	26.0000	1.18427
$483$	7.00000	0.318511
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−5.00000	−0.226339
$489$	−19.0000	−0.859210
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	3.00000	0.135250
$493$	1.00000	0.0450377
$494$	4.00000	0.179969
$495$	0	0
$496$	3.00000	0.134704
$497$	4.00000	0.179425
$498$	3.00000	0.134433
$499$	−27.0000	−1.20869	−0.604343	−	0.796724i	$-0.706564\pi$
$499$	−27.0000	−1.20869	−0.604343	+	0.796724i	$0.706564\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	15.0000	0.669483
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	14.0000	0.622376
$507$	12.0000	0.532939
$508$	−14.0000	−0.621150
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−5.00000	−0.220541
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	24.0000	1.05552
$518$	−6.00000	−0.263625
$519$	−12.0000	−0.526742
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	−1.00000	−0.0437688
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	11.0000	0.479623
$527$	3.00000	0.130682
$528$	−2.00000	−0.0870388
$529$	26.0000	1.13043
$530$	0	0
$531$	−3.00000	−0.130189
$532$	4.00000	0.173422
$533$	3.00000	0.129944
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	−10.0000	−0.431532
$538$	−18.0000	−0.776035
$539$	2.00000	0.0861461
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	20.0000	0.859074
$543$	18.0000	0.772454
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	15.0000	0.641354	0.320677	−	0.947189i	$-0.396090\pi$
$547$	15.0000	0.641354	0.320677	+	0.947189i	$0.396090\pi$
$548$	−4.00000	−0.170872
$549$	5.00000	0.213395
$550$	0	0
$551$	4.00000	0.170406
$552$	−7.00000	−0.297940
$553$	−2.00000	−0.0850487
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−3.00000	−0.127000
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	−41.0000	−1.72794	−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000	−1.72794	−0.863972	+	0.503540i	$0.832031\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−20.0000	−0.840663
$567$	1.00000	0.0419961
$568$	−4.00000	−0.167836
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	19.0000	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$572$	−2.00000	−0.0836242
$573$	13.0000	0.543083
$574$	3.00000	0.125218
$575$	0	0
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	16.0000	0.665512
$579$	2.00000	0.0831172
$580$	0	0
$581$	3.00000	0.124461
$582$	10.0000	0.414513
$583$	−22.0000	−0.911147
$584$	−14.0000	−0.579324
$585$	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	$0.311874\pi$
$588$	−1.00000	−0.0412393
$589$	12.0000	0.494451
$590$	0	0
$591$	27.0000	1.11063
$592$	6.00000	0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	5.00000	0.204808
$597$	24.0000	0.982255
$598$	−7.00000	−0.286251
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	−1.00000	−0.0407570
$603$	12.0000	0.488678
$604$	22.0000	0.895167
$605$	0	0
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	−12.0000	−0.485468
$612$	1.00000	0.0404226
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	17.0000	0.683840
$619$	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
$621$	7.00000	0.280900
$622$	−34.0000	−1.36328
$623$	10.0000	0.400642
$624$	1.00000	0.0400320
$625$	0	0
$626$	18.0000	0.719425
$627$	−8.00000	−0.319489
$628$	−18.0000	−0.718278
$629$	6.00000	0.239236
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	2.00000	0.0795557
$633$	−15.0000	−0.596196
$634$	−17.0000	−0.675156
$635$	0	0
$636$	11.0000	0.436178
$637$	−1.00000	−0.0396214
$638$	−2.00000	−0.0791808
$639$	4.00000	0.158238
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	18.0000	0.710403
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	−7.00000	−0.275839
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−1.00000	−0.0392837
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−3.00000	−0.117579
$652$	19.0000	0.744097
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−3.00000	−0.117130
$657$	14.0000	0.546192
$658$	−12.0000	−0.467809
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	3.00000	0.116598
$663$	1.00000	0.0388368
$664$	−3.00000	−0.116423
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−7.00000	−0.271041
$668$	2.00000	0.0773823
$669$	−13.0000	−0.502609
$670$	0	0
$671$	10.0000	0.386046
$672$	1.00000	0.0385758
$673$	−27.0000	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$673$	−27.0000	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	−12.0000	−0.461538
$677$	20.0000	0.768662	0.384331	−	0.923195i	$-0.374432\pi$
$677$	20.0000	0.768662	0.384331	+	0.923195i	$0.374432\pi$
$678$	−6.00000	−0.230429
$679$	10.0000	0.383765
$680$	0	0
$681$	7.00000	0.268241
$682$	−6.00000	−0.229752
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−14.0000	−0.534133
$688$	1.00000	0.0381246
$689$	11.0000	0.419067
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	12.0000	0.456172
$693$	2.00000	0.0759737
$694$	26.0000	0.986947
$695$	0	0
$696$	1.00000	0.0379049
$697$	−3.00000	−0.113633
$698$	1.00000	0.0378506
$699$	20.0000	0.756469
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	−1.00000	−0.0377426
$703$	24.0000	0.905177
$704$	2.00000	0.0753778
$705$	0	0
$706$	10.0000	0.376355
$707$	0	0
$708$	3.00000	0.112747
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	−10.0000	−0.374766
$713$	−21.0000	−0.786456
$714$	1.00000	0.0374241
$715$	0	0
$716$	10.0000	0.373718
$717$	24.0000	0.896296
$718$	9.00000	0.335877
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	0	0
$721$	17.0000	0.633113
$722$	3.00000	0.111648
$723$	26.0000	0.966950
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−7.00000	−0.259794
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.00000	0.0369863
$732$	−5.00000	−0.184805
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	13.0000	0.479839
$735$	0	0
$736$	7.00000	0.258023
$737$	24.0000	0.884051
$738$	3.00000	0.110432
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	11.0000	0.403823
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	3.00000	0.109985
$745$	0	0
$746$	−4.00000	−0.146450
$747$	3.00000	0.109764
$748$	2.00000	0.0731272
$749$	18.0000	0.657706
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	12.0000	0.437595
$753$	15.0000	0.546630
$754$	1.00000	0.0364179
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	5.00000	0.181608
$759$	14.0000	0.508168
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	−14.0000	−0.507166
$763$	−4.00000	−0.144810
$764$	−13.0000	−0.470323
$765$	0	0
$766$	−20.0000	−0.722629
$767$	3.00000	0.108324
$768$	−1.00000	−0.0360844
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−5.00000	−0.180071
$772$	−2.00000	−0.0719816
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−10.0000	−0.358979
$777$	−6.00000	−0.215249
$778$	−22.0000	−0.788738
$779$	−12.0000	−0.429945
$780$	0	0
$781$	8.00000	0.286263
$782$	7.00000	0.250319
$783$	−1.00000	−0.0357371
$784$	1.00000	0.0357143
$785$	0	0
$786$	8.00000	0.285351
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−27.0000	−0.961835
$789$	11.0000	0.391610
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−2.00000	−0.0710669
$793$	−5.00000	−0.177555
$794$	3.00000	0.106466
$795$	0	0
$796$	−24.0000	−0.850657
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	4.00000	0.141598
$799$	12.0000	0.424529
$800$	0	0
$801$	10.0000	0.353333
$802$	−16.0000	−0.564980
$803$	28.0000	0.988099
$804$	−12.0000	−0.423207
$805$	0	0
$806$	3.00000	0.105670
$807$	−18.0000	−0.633630
$808$	0	0
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	1.00000	0.0350931
$813$	20.0000	0.701431
$814$	−12.0000	−0.420600
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	4.00000	0.139942
$818$	14.0000	0.489499
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−4.00000	−0.139516
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	3.00000	0.104383
$827$	50.0000	1.73867	0.869335	−	0.494223i	$-0.164547\pi$
$827$	50.0000	1.73867	0.869335	+	0.494223i	$0.164547\pi$
$828$	−7.00000	−0.243267
$829$	3.00000	0.104194	0.0520972	−	0.998642i	$-0.483409\pi$
$829$	3.00000	0.104194	0.0520972	+	0.998642i	$0.483409\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	−1.00000	−0.0346688
$833$	1.00000	0.0346479
$834$	−4.00000	−0.138509
$835$	0	0
$836$	8.00000	0.276686
$837$	−3.00000	−0.103695
$838$	25.0000	0.863611
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−34.0000	−1.17172
$843$	0	0
$844$	15.0000	0.516321
$845$	0	0
$846$	−12.0000	−0.412568
$847$	−7.00000	−0.240523
$848$	−11.0000	−0.377742
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−42.0000	−1.43974
$852$	−4.00000	−0.137038
$853$	−43.0000	−1.47229	−0.736146	−	0.676823i	$-0.763356\pi$
$853$	−43.0000	−1.47229	−0.736146	+	0.676823i	$0.763356\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−18.0000	−0.615227
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	−2.00000	−0.0682789
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	−27.0000	−0.919624
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	16.0000	0.543388
$868$	3.00000	0.101827
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−12.0000	−0.406604
$872$	4.00000	0.135457
$873$	10.0000	0.338449
$874$	28.0000	0.947114
$875$	0	0
$876$	−14.0000	−0.473016
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−7.00000	−0.236239
$879$	0	0
$880$	0	0
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	−1.00000	−0.0336718
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−50.0000	−1.67884	−0.839418	−	0.543487i	$-0.817104\pi$
$887$	−50.0000	−1.67884	−0.839418	+	0.543487i	$0.817104\pi$
$888$	6.00000	0.201347
$889$	−14.0000	−0.469545
$890$	0	0
$891$	2.00000	0.0670025
$892$	13.0000	0.435272
$893$	48.0000	1.60626
$894$	5.00000	0.167225
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−7.00000	−0.233723
$898$	16.0000	0.533927
$899$	3.00000	0.100056
$900$	0	0
$901$	−11.0000	−0.366463
$902$	6.00000	0.199778
$903$	−1.00000	−0.0332779
$904$	6.00000	0.199557
$905$	0	0
$906$	22.0000	0.730901
$907$	11.0000	0.365249	0.182625	−	0.983183i	$-0.441541\pi$
$907$	11.0000	0.365249	0.182625	+	0.983183i	$0.441541\pi$
$908$	−7.00000	−0.232303
$909$	0	0
$910$	0	0
$911$	−19.0000	−0.629498	−0.314749	−	0.949175i	$-0.601920\pi$
$911$	−19.0000	−0.629498	−0.314749	+	0.949175i	$0.601920\pi$
$912$	−4.00000	−0.132453
$913$	6.00000	0.198571
$914$	−17.0000	−0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	8.00000	0.264183
$918$	1.00000	0.0330049
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	−32.0000	−1.05386
$923$	−4.00000	−0.131662
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	36.0000	1.18303
$927$	17.0000	0.558353
$928$	−1.00000	−0.0328266
$929$	25.0000	0.820223	0.410112	−	0.912035i	$-0.365490\pi$
$929$	25.0000	0.820223	0.410112	+	0.912035i	$0.365490\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−20.0000	−0.655122
$933$	−34.0000	−1.11311
$934$	−39.0000	−1.27612
$935$	0	0
$936$	1.00000	0.0326860
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−12.0000	−0.391814
$939$	18.0000	0.587408
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−18.0000	−0.586472
$943$	21.0000	0.683854
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	2.00000	0.0649570
$949$	−14.0000	−0.454459
$950$	0	0
$951$	−17.0000	−0.551263
$952$	−1.00000	−0.0324102
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	11.0000	0.356138
$955$	0	0
$956$	−24.0000	−0.776215
$957$	−2.00000	−0.0646508
$958$	−8.00000	−0.258468
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−22.0000	−0.709677
$962$	6.00000	0.193448
$963$	18.0000	0.580042
$964$	−26.0000	−0.837404
$965$	0	0
$966$	−7.00000	−0.225221
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	7.00000	0.224989
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	22.0000	0.704925
$975$	0	0
$976$	5.00000	0.160046
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	19.0000	0.607553
$979$	20.0000	0.639203
$980$	0	0
$981$	−4.00000	−0.127710
$982$	−30.0000	−0.957338
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−3.00000	−0.0956365
$985$	0	0
$986$	−1.00000	−0.0318465
$987$	−12.0000	−0.381964
$988$	−4.00000	−0.127257
$989$	−7.00000	−0.222587
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−3.00000	−0.0952501
$993$	3.00000	0.0952021
$994$	−4.00000	−0.126872
$995$	0	0
$996$	−3.00000	−0.0950586
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	27.0000	0.854670
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.a.e.1.1	✓	1
3.2	odd	2		3150.2.a.bl.1.1		1
4.3	odd	2		8400.2.a.bt.1.1		1
5.2	odd	4		1050.2.g.j.799.1		2
5.3	odd	4		1050.2.g.j.799.2		2
5.4	even	2		1050.2.a.o.1.1	yes	1
7.6	odd	2		7350.2.a.bj.1.1		1
15.2	even	4		3150.2.g.g.2899.2		2
15.8	even	4		3150.2.g.g.2899.1		2
15.14	odd	2		3150.2.a.c.1.1		1
20.19	odd	2		8400.2.a.t.1.1		1
35.34	odd	2		7350.2.a.ca.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.e.1.1	✓	1	1.1	even	1	trivial
1050.2.a.o.1.1	yes	1	5.4	even	2
1050.2.g.j.799.1		2	5.2	odd	4
1050.2.g.j.799.2		2	5.3	odd	4
3150.2.a.c.1.1		1	15.14	odd	2
3150.2.a.bl.1.1		1	3.2	odd	2
3150.2.g.g.2899.1		2	15.8	even	4
3150.2.g.g.2899.2		2	15.2	even	4
7350.2.a.bj.1.1		1	7.6	odd	2
7350.2.a.ca.1.1		1	35.34	odd	2
8400.2.a.t.1.1		1	20.19	odd	2
8400.2.a.bt.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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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