LMFDB - Embedded newform 1950.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,4,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1950.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1950.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:	$115.053724511$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -4.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} -4.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +2.00000 q^{11} -12.0000 q^{12} +13.0000 q^{13} +8.00000 q^{14} +16.0000 q^{16} +6.00000 q^{17} -18.0000 q^{18} -36.0000 q^{19} +12.0000 q^{21} -4.00000 q^{22} +20.0000 q^{23} +24.0000 q^{24} -26.0000 q^{26} -27.0000 q^{27} -16.0000 q^{28} -14.0000 q^{29} -152.000 q^{31} -32.0000 q^{32} -6.00000 q^{33} -12.0000 q^{34} +36.0000 q^{36} +258.000 q^{37} +72.0000 q^{38} -39.0000 q^{39} +84.0000 q^{41} -24.0000 q^{42} +188.000 q^{43} +8.00000 q^{44} -40.0000 q^{46} -254.000 q^{47} -48.0000 q^{48} -327.000 q^{49} -18.0000 q^{51} +52.0000 q^{52} -366.000 q^{53} +54.0000 q^{54} +32.0000 q^{56} +108.000 q^{57} +28.0000 q^{58} +550.000 q^{59} -14.0000 q^{61} +304.000 q^{62} -36.0000 q^{63} +64.0000 q^{64} +12.0000 q^{66} -448.000 q^{67} +24.0000 q^{68} -60.0000 q^{69} +926.000 q^{71} -72.0000 q^{72} -254.000 q^{73} -516.000 q^{74} -144.000 q^{76} -8.00000 q^{77} +78.0000 q^{78} +1328.00 q^{79} +81.0000 q^{81} -168.000 q^{82} -186.000 q^{83} +48.0000 q^{84} -376.000 q^{86} +42.0000 q^{87} -16.0000 q^{88} -336.000 q^{89} -52.0000 q^{91} +80.0000 q^{92} +456.000 q^{93} +508.000 q^{94} +96.0000 q^{96} -614.000 q^{97} +654.000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	6.00000	0.408248
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	2.00000	0.0548202	0.0274101	−	0.999624i	$-0.491274\pi$
$11$	2.00000	0.0548202	0.0274101	+	0.999624i	$0.491274\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	8.00000	0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	6.00000	0.0856008	0.0428004	−	0.999084i	$-0.486372\pi$
$17$	6.00000	0.0856008	0.0428004	+	0.999084i	$0.486372\pi$
$18$	−18.0000	−0.235702
$19$	−36.0000	−0.434682	−0.217341	−	0.976096i	$-0.569738\pi$
$19$	−36.0000	−0.434682	−0.217341	+	0.976096i	$0.569738\pi$
$20$	0	0
$21$	12.0000	0.124696
$22$	−4.00000	−0.0387638
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−26.0000	−0.196116
$27$	−27.0000	−0.192450
$28$	−16.0000	−0.107990
$29$	−14.0000	−0.0896460	−0.0448230	−	0.998995i	$-0.514272\pi$
$29$	−14.0000	−0.0896460	−0.0448230	+	0.998995i	$0.514272\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	−32.0000	−0.176777
$33$	−6.00000	−0.0316505
$34$	−12.0000	−0.0605289
$35$	0	0
$36$	36.0000	0.166667
$37$	258.000	1.14635	0.573175	−	0.819433i	$-0.305712\pi$
$37$	258.000	1.14635	0.573175	+	0.819433i	$0.305712\pi$
$38$	72.0000	0.307367
$39$	−39.0000	−0.160128
$40$	0	0
$41$	84.0000	0.319966	0.159983	−	0.987120i	$-0.448856\pi$
$41$	84.0000	0.319966	0.159983	+	0.987120i	$0.448856\pi$
$42$	−24.0000	−0.0881733
$43$	188.000	0.666738	0.333369	−	0.942796i	$-0.391815\pi$
$43$	188.000	0.666738	0.333369	+	0.942796i	$0.391815\pi$
$44$	8.00000	0.0274101
$45$	0	0
$46$	−40.0000	−0.128210
$47$	−254.000	−0.788292	−0.394146	−	0.919048i	$-0.628960\pi$
$47$	−254.000	−0.788292	−0.394146	+	0.919048i	$0.628960\pi$
$48$	−48.0000	−0.144338
$49$	−327.000	−0.953353
$50$	0	0
$51$	−18.0000	−0.0494217
$52$	52.0000	0.138675
$53$	−366.000	−0.948565	−0.474283	−	0.880373i	$-0.657293\pi$
$53$	−366.000	−0.948565	−0.474283	+	0.880373i	$0.657293\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	32.0000	0.0763604
$57$	108.000	0.250964
$58$	28.0000	0.0633893
$59$	550.000	1.21363	0.606813	−	0.794845i	$-0.292448\pi$
$59$	550.000	1.21363	0.606813	+	0.794845i	$0.292448\pi$
$60$	0	0
$61$	−14.0000	−0.0293855	−0.0146928	−	0.999892i	$-0.504677\pi$
$61$	−14.0000	−0.0293855	−0.0146928	+	0.999892i	$0.504677\pi$
$62$	304.000	0.622710
$63$	−36.0000	−0.0719932
$64$	64.0000	0.125000
$65$	0	0
$66$	12.0000	0.0223803
$67$	−448.000	−0.816894	−0.408447	−	0.912782i	$-0.633930\pi$
$67$	−448.000	−0.816894	−0.408447	+	0.912782i	$0.633930\pi$
$68$	24.0000	0.0428004
$69$	−60.0000	−0.104683
$70$	0	0
$71$	926.000	1.54783	0.773915	−	0.633289i	$-0.218296\pi$
$71$	926.000	1.54783	0.773915	+	0.633289i	$0.218296\pi$
$72$	−72.0000	−0.117851
$73$	−254.000	−0.407239	−0.203620	−	0.979050i	$-0.565271\pi$
$73$	−254.000	−0.407239	−0.203620	+	0.979050i	$0.565271\pi$
$74$	−516.000	−0.810592
$75$	0	0
$76$	−144.000	−0.217341
$77$	−8.00000	−0.0118401
$78$	78.0000	0.113228
$79$	1328.00	1.89129	0.945644	−	0.325205i	$-0.105433\pi$
$79$	1328.00	1.89129	0.945644	+	0.325205i	$0.105433\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−168.000	−0.226250
$83$	−186.000	−0.245978	−0.122989	−	0.992408i	$-0.539248\pi$
$83$	−186.000	−0.245978	−0.122989	+	0.992408i	$0.539248\pi$
$84$	48.0000	0.0623480
$85$	0	0
$86$	−376.000	−0.471455
$87$	42.0000	0.0517572
$88$	−16.0000	−0.0193819
$89$	−336.000	−0.400179	−0.200089	−	0.979778i	$-0.564123\pi$
$89$	−336.000	−0.400179	−0.200089	+	0.979778i	$0.564123\pi$
$90$	0	0
$91$	−52.0000	−0.0599020
$92$	80.0000	0.0906584
$93$	456.000	0.508441
$94$	508.000	0.557406
$95$	0	0
$96$	96.0000	0.102062
$97$	−614.000	−0.642704	−0.321352	−	0.946960i	$-0.604137\pi$
$97$	−614.000	−0.642704	−0.321352	+	0.946960i	$0.604137\pi$
$98$	654.000	0.674122
$99$	18.0000	0.0182734
$100$	0	0
$101$	−1606.00	−1.58221	−0.791104	−	0.611682i	$-0.790493\pi$
$101$	−1606.00	−1.58221	−0.791104	+	0.611682i	$0.790493\pi$
$102$	36.0000	0.0349464
$103$	−208.000	−0.198979	−0.0994896	−	0.995039i	$-0.531721\pi$
$103$	−208.000	−0.198979	−0.0994896	+	0.995039i	$0.531721\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	732.000	0.670737
$107$	248.000	0.224066	0.112033	−	0.993704i	$-0.464264\pi$
$107$	248.000	0.224066	0.112033	+	0.993704i	$0.464264\pi$
$108$	−108.000	−0.0962250
$109$	−542.000	−0.476277	−0.238138	−	0.971231i	$-0.576537\pi$
$109$	−542.000	−0.476277	−0.238138	+	0.971231i	$0.576537\pi$
$110$	0	0
$111$	−774.000	−0.661845
$112$	−64.0000	−0.0539949
$113$	2042.00	1.69996	0.849979	−	0.526817i	$-0.176615\pi$
$113$	2042.00	1.69996	0.849979	+	0.526817i	$0.176615\pi$
$114$	−216.000	−0.177458
$115$	0	0
$116$	−56.0000	−0.0448230
$117$	117.000	0.0924500
$118$	−1100.00	−0.858163
$119$	−24.0000	−0.0184880
$120$	0	0
$121$	−1327.00	−0.996995
$122$	28.0000	0.0207787
$123$	−252.000	−0.184732
$124$	−608.000	−0.440323
$125$	0	0
$126$	72.0000	0.0509069
$127$	488.000	0.340968	0.170484	−	0.985360i	$-0.445467\pi$
$127$	488.000	0.340968	0.170484	+	0.985360i	$0.445467\pi$
$128$	−128.000	−0.0883883
$129$	−564.000	−0.384941
$130$	0	0
$131$	1744.00	1.16316	0.581580	−	0.813489i	$-0.302435\pi$
$131$	1744.00	1.16316	0.581580	+	0.813489i	$0.302435\pi$
$132$	−24.0000	−0.0158252
$133$	144.000	0.0938826
$134$	896.000	0.577631
$135$	0	0
$136$	−48.0000	−0.0302645
$137$	828.000	0.516356	0.258178	−	0.966097i	$-0.416878\pi$
$137$	828.000	0.516356	0.258178	+	0.966097i	$0.416878\pi$
$138$	120.000	0.0740223
$139$	−404.000	−0.246524	−0.123262	−	0.992374i	$-0.539336\pi$
$139$	−404.000	−0.246524	−0.123262	+	0.992374i	$0.539336\pi$
$140$	0	0
$141$	762.000	0.455120
$142$	−1852.00	−1.09448
$143$	26.0000	0.0152044
$144$	144.000	0.0833333
$145$	0	0
$146$	508.000	0.287962
$147$	981.000	0.550418
$148$	1032.00	0.573175
$149$	2928.00	1.60987	0.804937	−	0.593361i	$-0.202199\pi$
$149$	2928.00	1.60987	0.804937	+	0.593361i	$0.202199\pi$
$150$	0	0
$151$	1944.00	1.04769	0.523843	−	0.851815i	$-0.324498\pi$
$151$	1944.00	1.04769	0.523843	+	0.851815i	$0.324498\pi$
$152$	288.000	0.153683
$153$	54.0000	0.0285336
$154$	16.0000	0.00837219
$155$	0	0
$156$	−156.000	−0.0800641
$157$	−3590.00	−1.82492	−0.912462	−	0.409161i	$-0.865822\pi$
$157$	−3590.00	−1.82492	−0.912462	+	0.409161i	$0.865822\pi$
$158$	−2656.00	−1.33734
$159$	1098.00	0.547654
$160$	0	0
$161$	−80.0000	−0.0391608
$162$	−162.000	−0.0785674
$163$	2284.00	1.09753	0.548763	−	0.835978i	$-0.315099\pi$
$163$	2284.00	1.09753	0.548763	+	0.835978i	$0.315099\pi$
$164$	336.000	0.159983
$165$	0	0
$166$	372.000	0.173933
$167$	−3174.00	−1.47073	−0.735364	−	0.677673i	$-0.762989\pi$
$167$	−3174.00	−1.47073	−0.735364	+	0.677673i	$0.762989\pi$
$168$	−96.0000	−0.0440867
$169$	169.000	0.0769231
$170$	0	0
$171$	−324.000	−0.144894
$172$	752.000	0.333369
$173$	1358.00	0.596802	0.298401	−	0.954441i	$-0.403547\pi$
$173$	1358.00	0.596802	0.298401	+	0.954441i	$0.403547\pi$
$174$	−84.0000	−0.0365978
$175$	0	0
$176$	32.0000	0.0137051
$177$	−1650.00	−0.700687
$178$	672.000	0.282969
$179$	708.000	0.295634	0.147817	−	0.989015i	$-0.452775\pi$
$179$	708.000	0.295634	0.147817	+	0.989015i	$0.452775\pi$
$180$	0	0
$181$	−546.000	−0.224220	−0.112110	−	0.993696i	$-0.535761\pi$
$181$	−546.000	−0.224220	−0.112110	+	0.993696i	$0.535761\pi$
$182$	104.000	0.0423571
$183$	42.0000	0.0169657
$184$	−160.000	−0.0641052
$185$	0	0
$186$	−912.000	−0.359522
$187$	12.0000	0.00469266
$188$	−1016.00	−0.394146
$189$	108.000	0.0415653
$190$	0	0
$191$	−3472.00	−1.31531	−0.657657	−	0.753317i	$-0.728453\pi$
$191$	−3472.00	−1.31531	−0.657657	+	0.753317i	$0.728453\pi$
$192$	−192.000	−0.0721688
$193$	310.000	0.115618	0.0578090	−	0.998328i	$-0.481589\pi$
$193$	310.000	0.115618	0.0578090	+	0.998328i	$0.481589\pi$
$194$	1228.00	0.454460
$195$	0	0
$196$	−1308.00	−0.476676
$197$	−1020.00	−0.368893	−0.184447	−	0.982843i	$-0.559049\pi$
$197$	−1020.00	−0.368893	−0.184447	+	0.982843i	$0.559049\pi$
$198$	−36.0000	−0.0129213
$199$	−3256.00	−1.15986	−0.579929	−	0.814667i	$-0.696920\pi$
$199$	−3256.00	−1.15986	−0.579929	+	0.814667i	$0.696920\pi$
$200$	0	0
$201$	1344.00	0.471634
$202$	3212.00	1.11879
$203$	56.0000	0.0193617
$204$	−72.0000	−0.0247108
$205$	0	0
$206$	416.000	0.140699
$207$	180.000	0.0604390
$208$	208.000	0.0693375
$209$	−72.0000	−0.0238294
$210$	0	0
$211$	−4564.00	−1.48909	−0.744547	−	0.667570i	$-0.767334\pi$
$211$	−4564.00	−1.48909	−0.744547	+	0.667570i	$0.767334\pi$
$212$	−1464.00	−0.474283
$213$	−2778.00	−0.893640
$214$	−496.000	−0.158439
$215$	0	0
$216$	216.000	0.0680414
$217$	608.000	0.190202
$218$	1084.00	0.336779
$219$	762.000	0.235120
$220$	0	0
$221$	78.0000	0.0237414
$222$	1548.00	0.467995
$223$	72.0000	0.0216210	0.0108105	−	0.999942i	$-0.496559\pi$
$223$	72.0000	0.0216210	0.0108105	+	0.999942i	$0.496559\pi$
$224$	128.000	0.0381802
$225$	0	0
$226$	−4084.00	−1.20205
$227$	−2694.00	−0.787696	−0.393848	−	0.919176i	$-0.628856\pi$
$227$	−2694.00	−0.787696	−0.393848	+	0.919176i	$0.628856\pi$
$228$	432.000	0.125482
$229$	5922.00	1.70889	0.854447	−	0.519538i	$-0.173896\pi$
$229$	5922.00	1.70889	0.854447	+	0.519538i	$0.173896\pi$
$230$	0	0
$231$	24.0000	0.00683586
$232$	112.000	0.0316947
$233$	5122.00	1.44014	0.720072	−	0.693900i	$-0.244109\pi$
$233$	5122.00	1.44014	0.720072	+	0.693900i	$0.244109\pi$
$234$	−234.000	−0.0653720
$235$	0	0
$236$	2200.00	0.606813
$237$	−3984.00	−1.09194
$238$	48.0000	0.0130730
$239$	5022.00	1.35919	0.679595	−	0.733588i	$-0.262156\pi$
$239$	5022.00	1.35919	0.679595	+	0.733588i	$0.262156\pi$
$240$	0	0
$241$	−1218.00	−0.325553	−0.162777	−	0.986663i	$-0.552045\pi$
$241$	−1218.00	−0.325553	−0.162777	+	0.986663i	$0.552045\pi$
$242$	2654.00	0.704982
$243$	−243.000	−0.0641500
$244$	−56.0000	−0.0146928
$245$	0	0
$246$	504.000	0.130625
$247$	−468.000	−0.120559
$248$	1216.00	0.311355
$249$	558.000	0.142015
$250$	0	0
$251$	−2112.00	−0.531109	−0.265554	−	0.964096i	$-0.585555\pi$
$251$	−2112.00	−0.531109	−0.265554	+	0.964096i	$0.585555\pi$
$252$	−144.000	−0.0359966
$253$	40.0000	0.00993984
$254$	−976.000	−0.241101
$255$	0	0
$256$	256.000	0.0625000
$257$	−2814.00	−0.683006	−0.341503	−	0.939881i	$-0.610936\pi$
$257$	−2814.00	−0.683006	−0.341503	+	0.939881i	$0.610936\pi$
$258$	1128.00	0.272195
$259$	−1032.00	−0.247588
$260$	0	0
$261$	−126.000	−0.0298820
$262$	−3488.00	−0.822478
$263$	4044.00	0.948151	0.474076	−	0.880484i	$-0.342782\pi$
$263$	4044.00	0.948151	0.474076	+	0.880484i	$0.342782\pi$
$264$	48.0000	0.0111901
$265$	0	0
$266$	−288.000	−0.0663850
$267$	1008.00	0.231043
$268$	−1792.00	−0.408447
$269$	−1470.00	−0.333188	−0.166594	−	0.986026i	$-0.553277\pi$
$269$	−1470.00	−0.333188	−0.166594	+	0.986026i	$0.553277\pi$
$270$	0	0
$271$	−1844.00	−0.413340	−0.206670	−	0.978411i	$-0.566263\pi$
$271$	−1844.00	−0.413340	−0.206670	+	0.978411i	$0.566263\pi$
$272$	96.0000	0.0214002
$273$	156.000	0.0345844
$274$	−1656.00	−0.365119
$275$	0	0
$276$	−240.000	−0.0523417
$277$	−5766.00	−1.25071	−0.625353	−	0.780342i	$-0.715045\pi$
$277$	−5766.00	−1.25071	−0.625353	+	0.780342i	$0.715045\pi$
$278$	808.000	0.174319
$279$	−1368.00	−0.293548
$280$	0	0
$281$	−7468.00	−1.58542	−0.792711	−	0.609598i	$-0.791331\pi$
$281$	−7468.00	−1.58542	−0.792711	+	0.609598i	$0.791331\pi$
$282$	−1524.00	−0.321819
$283$	−1228.00	−0.257940	−0.128970	−	0.991648i	$-0.541167\pi$
$283$	−1228.00	−0.257940	−0.128970	+	0.991648i	$0.541167\pi$
$284$	3704.00	0.773915
$285$	0	0
$286$	−52.0000	−0.0107511
$287$	−336.000	−0.0691061
$288$	−288.000	−0.0589256
$289$	−4877.00	−0.992673
$290$	0	0
$291$	1842.00	0.371065
$292$	−1016.00	−0.203620
$293$	−6608.00	−1.31755	−0.658777	−	0.752338i	$-0.728926\pi$
$293$	−6608.00	−1.31755	−0.658777	+	0.752338i	$0.728926\pi$
$294$	−1962.00	−0.389205
$295$	0	0
$296$	−2064.00	−0.405296
$297$	−54.0000	−0.0105502
$298$	−5856.00	−1.13835
$299$	260.000	0.0502883
$300$	0	0
$301$	−752.000	−0.144002
$302$	−3888.00	−0.740825
$303$	4818.00	0.913488
$304$	−576.000	−0.108671
$305$	0	0
$306$	−108.000	−0.0201763
$307$	−7664.00	−1.42478	−0.712390	−	0.701784i	$-0.752387\pi$
$307$	−7664.00	−1.42478	−0.712390	+	0.701784i	$0.752387\pi$
$308$	−32.0000	−0.00592003
$309$	624.000	0.114881
$310$	0	0
$311$	−2340.00	−0.426653	−0.213327	−	0.976981i	$-0.568430\pi$
$311$	−2340.00	−0.426653	−0.213327	+	0.976981i	$0.568430\pi$
$312$	312.000	0.0566139
$313$	−6710.00	−1.21173	−0.605865	−	0.795567i	$-0.707173\pi$
$313$	−6710.00	−1.21173	−0.605865	+	0.795567i	$0.707173\pi$
$314$	7180.00	1.29042
$315$	0	0
$316$	5312.00	0.945644
$317$	−4164.00	−0.737771	−0.368886	−	0.929475i	$-0.620261\pi$
$317$	−4164.00	−0.737771	−0.368886	+	0.929475i	$0.620261\pi$
$318$	−2196.00	−0.387250
$319$	−28.0000	−0.00491442
$320$	0	0
$321$	−744.000	−0.129365
$322$	160.000	0.0276908
$323$	−216.000	−0.0372092
$324$	324.000	0.0555556
$325$	0	0
$326$	−4568.00	−0.776068
$327$	1626.00	0.274979
$328$	−672.000	−0.113125
$329$	1016.00	0.170255
$330$	0	0
$331$	−10072.0	−1.67253	−0.836265	−	0.548326i	$-0.815265\pi$
$331$	−10072.0	−1.67253	−0.836265	+	0.548326i	$0.815265\pi$
$332$	−744.000	−0.122989
$333$	2322.00	0.382117
$334$	6348.00	1.03996
$335$	0	0
$336$	192.000	0.0311740
$337$	−2990.00	−0.483311	−0.241655	−	0.970362i	$-0.577690\pi$
$337$	−2990.00	−0.483311	−0.241655	+	0.970362i	$0.577690\pi$
$338$	−338.000	−0.0543928
$339$	−6126.00	−0.981471
$340$	0	0
$341$	−304.000	−0.0482772
$342$	648.000	0.102456
$343$	2680.00	0.421885
$344$	−1504.00	−0.235727
$345$	0	0
$346$	−2716.00	−0.422003
$347$	−6564.00	−1.01549	−0.507743	−	0.861508i	$-0.669520\pi$
$347$	−6564.00	−1.01549	−0.507743	+	0.861508i	$0.669520\pi$
$348$	168.000	0.0258786
$349$	−674.000	−0.103376	−0.0516882	−	0.998663i	$-0.516460\pi$
$349$	−674.000	−0.103376	−0.0516882	+	0.998663i	$0.516460\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	−64.0000	−0.00969094
$353$	10732.0	1.61815	0.809075	−	0.587706i	$-0.199969\pi$
$353$	10732.0	1.61815	0.809075	+	0.587706i	$0.199969\pi$
$354$	3300.00	0.495461
$355$	0	0
$356$	−1344.00	−0.200089
$357$	72.0000	0.0106741
$358$	−1416.00	−0.209044
$359$	−4842.00	−0.711841	−0.355921	−	0.934516i	$-0.615833\pi$
$359$	−4842.00	−0.711841	−0.355921	+	0.934516i	$0.615833\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	1092.00	0.158548
$363$	3981.00	0.575615
$364$	−208.000	−0.0299510
$365$	0	0
$366$	−84.0000	−0.0119966
$367$	6280.00	0.893224	0.446612	−	0.894728i	$-0.352630\pi$
$367$	6280.00	0.893224	0.446612	+	0.894728i	$0.352630\pi$
$368$	320.000	0.0453292
$369$	756.000	0.106655
$370$	0	0
$371$	1464.00	0.204871
$372$	1824.00	0.254220
$373$	−6434.00	−0.893136	−0.446568	−	0.894750i	$-0.647354\pi$
$373$	−6434.00	−0.893136	−0.446568	+	0.894750i	$0.647354\pi$
$374$	−24.0000	−0.00331821
$375$	0	0
$376$	2032.00	0.278703
$377$	−182.000	−0.0248633
$378$	−216.000	−0.0293911
$379$	−9068.00	−1.22900	−0.614501	−	0.788916i	$-0.710643\pi$
$379$	−9068.00	−1.22900	−0.614501	+	0.788916i	$0.710643\pi$
$380$	0	0
$381$	−1464.00	−0.196858
$382$	6944.00	0.930068
$383$	−3162.00	−0.421855	−0.210928	−	0.977502i	$-0.567648\pi$
$383$	−3162.00	−0.421855	−0.210928	+	0.977502i	$0.567648\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−620.000	−0.0817543
$387$	1692.00	0.222246
$388$	−2456.00	−0.321352
$389$	−3666.00	−0.477824	−0.238912	−	0.971041i	$-0.576791\pi$
$389$	−3666.00	−0.477824	−0.238912	+	0.971041i	$0.576791\pi$
$390$	0	0
$391$	120.000	0.0155209
$392$	2616.00	0.337061
$393$	−5232.00	−0.671551
$394$	2040.00	0.260847
$395$	0	0
$396$	72.0000	0.00913671
$397$	−11054.0	−1.39744	−0.698721	−	0.715394i	$-0.746247\pi$
$397$	−11054.0	−1.39744	−0.698721	+	0.715394i	$0.746247\pi$
$398$	6512.00	0.820143
$399$	−432.000	−0.0542031
$400$	0	0
$401$	−5328.00	−0.663510	−0.331755	−	0.943366i	$-0.607641\pi$
$401$	−5328.00	−0.663510	−0.331755	+	0.943366i	$0.607641\pi$
$402$	−2688.00	−0.333496
$403$	−1976.00	−0.244247
$404$	−6424.00	−0.791104
$405$	0	0
$406$	−112.000	−0.0136908
$407$	516.000	0.0628432
$408$	144.000	0.0174732
$409$	−12074.0	−1.45971	−0.729854	−	0.683603i	$-0.760412\pi$
$409$	−12074.0	−1.45971	−0.729854	+	0.683603i	$0.760412\pi$
$410$	0	0
$411$	−2484.00	−0.298118
$412$	−832.000	−0.0994896
$413$	−2200.00	−0.262118
$414$	−360.000	−0.0427368
$415$	0	0
$416$	−416.000	−0.0490290
$417$	1212.00	0.142331
$418$	144.000	0.0168499
$419$	13584.0	1.58382	0.791911	−	0.610636i	$-0.209086\pi$
$419$	13584.0	1.58382	0.791911	+	0.610636i	$0.209086\pi$
$420$	0	0
$421$	−7406.00	−0.857355	−0.428677	−	0.903458i	$-0.641020\pi$
$421$	−7406.00	−0.857355	−0.428677	+	0.903458i	$0.641020\pi$
$422$	9128.00	1.05295
$423$	−2286.00	−0.262764
$424$	2928.00	0.335369
$425$	0	0
$426$	5556.00	0.631899
$427$	56.0000	0.00634667
$428$	992.000	0.112033
$429$	−78.0000	−0.00877826
$430$	0	0
$431$	−10134.0	−1.13257	−0.566285	−	0.824210i	$-0.691620\pi$
$431$	−10134.0	−1.13257	−0.566285	+	0.824210i	$0.691620\pi$
$432$	−432.000	−0.0481125
$433$	−9406.00	−1.04393	−0.521967	−	0.852966i	$-0.674802\pi$
$433$	−9406.00	−1.04393	−0.521967	+	0.852966i	$0.674802\pi$
$434$	−1216.00	−0.134493
$435$	0	0
$436$	−2168.00	−0.238138
$437$	−720.000	−0.0788153
$438$	−1524.00	−0.166255
$439$	4088.00	0.444441	0.222220	−	0.974996i	$-0.428670\pi$
$439$	4088.00	0.444441	0.222220	+	0.974996i	$0.428670\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	−156.000	−0.0167877
$443$	5328.00	0.571424	0.285712	−	0.958315i	$-0.407770\pi$
$443$	5328.00	0.571424	0.285712	+	0.958315i	$0.407770\pi$
$444$	−3096.00	−0.330923
$445$	0	0
$446$	−144.000	−0.0152883
$447$	−8784.00	−0.929461
$448$	−256.000	−0.0269975
$449$	13160.0	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$449$	13160.0	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$450$	0	0
$451$	168.000	0.0175406
$452$	8168.00	0.849979
$453$	−5832.00	−0.604881
$454$	5388.00	0.556985
$455$	0	0
$456$	−864.000	−0.0887292
$457$	9146.00	0.936175	0.468087	−	0.883682i	$-0.344943\pi$
$457$	9146.00	0.936175	0.468087	+	0.883682i	$0.344943\pi$
$458$	−11844.0	−1.20837
$459$	−162.000	−0.0164739
$460$	0	0
$461$	5580.00	0.563745	0.281873	−	0.959452i	$-0.409044\pi$
$461$	5580.00	0.563745	0.281873	+	0.959452i	$0.409044\pi$
$462$	−48.0000	−0.00483368
$463$	−14788.0	−1.48436	−0.742178	−	0.670203i	$-0.766207\pi$
$463$	−14788.0	−1.48436	−0.742178	+	0.670203i	$0.766207\pi$
$464$	−224.000	−0.0224115
$465$	0	0
$466$	−10244.0	−1.01834
$467$	−12376.0	−1.22632	−0.613162	−	0.789957i	$-0.710103\pi$
$467$	−12376.0	−1.22632	−0.613162	+	0.789957i	$0.710103\pi$
$468$	468.000	0.0462250
$469$	1792.00	0.176433
$470$	0	0
$471$	10770.0	1.05362
$472$	−4400.00	−0.429081
$473$	376.000	0.0365507
$474$	7968.00	0.772115
$475$	0	0
$476$	−96.0000	−0.00924402
$477$	−3294.00	−0.316188
$478$	−10044.0	−0.961092
$479$	834.000	0.0795541	0.0397771	−	0.999209i	$-0.487335\pi$
$479$	834.000	0.0795541	0.0397771	+	0.999209i	$0.487335\pi$
$480$	0	0
$481$	3354.00	0.317940
$482$	2436.00	0.230201
$483$	240.000	0.0226095
$484$	−5308.00	−0.498497
$485$	0	0
$486$	486.000	0.0453609
$487$	13192.0	1.22749	0.613744	−	0.789505i	$-0.289663\pi$
$487$	13192.0	1.22749	0.613744	+	0.789505i	$0.289663\pi$
$488$	112.000	0.0103893
$489$	−6852.00	−0.633657
$490$	0	0
$491$	16568.0	1.52282	0.761409	−	0.648272i	$-0.224508\pi$
$491$	16568.0	1.52282	0.761409	+	0.648272i	$0.224508\pi$
$492$	−1008.00	−0.0923662
$493$	−84.0000	−0.00767377
$494$	936.000	0.0852482
$495$	0	0
$496$	−2432.00	−0.220161
$497$	−3704.00	−0.334300
$498$	−1116.00	−0.100420
$499$	−10136.0	−0.909318	−0.454659	−	0.890666i	$-0.650239\pi$
$499$	−10136.0	−0.909318	−0.454659	+	0.890666i	$0.650239\pi$
$500$	0	0
$501$	9522.00	0.849125
$502$	4224.00	0.375550
$503$	−10412.0	−0.922959	−0.461479	−	0.887151i	$-0.652681\pi$
$503$	−10412.0	−0.922959	−0.461479	+	0.887151i	$0.652681\pi$
$504$	288.000	0.0254535
$505$	0	0
$506$	−80.0000	−0.00702853
$507$	−507.000	−0.0444116
$508$	1952.00	0.170484
$509$	−4180.00	−0.363999	−0.181999	−	0.983299i	$-0.558257\pi$
$509$	−4180.00	−0.363999	−0.181999	+	0.983299i	$0.558257\pi$
$510$	0	0
$511$	1016.00	0.0879554
$512$	−512.000	−0.0441942
$513$	972.000	0.0836547
$514$	5628.00	0.482958
$515$	0	0
$516$	−2256.00	−0.192471
$517$	−508.000	−0.0432143
$518$	2064.00	0.175071
$519$	−4074.00	−0.344564
$520$	0	0
$521$	−14610.0	−1.22855	−0.614276	−	0.789091i	$-0.710552\pi$
$521$	−14610.0	−1.22855	−0.614276	+	0.789091i	$0.710552\pi$
$522$	252.000	0.0211298
$523$	2172.00	0.181596	0.0907982	−	0.995869i	$-0.471058\pi$
$523$	2172.00	0.181596	0.0907982	+	0.995869i	$0.471058\pi$
$524$	6976.00	0.581580
$525$	0	0
$526$	−8088.00	−0.670444
$527$	−912.000	−0.0753840
$528$	−96.0000	−0.00791262
$529$	−11767.0	−0.967124
$530$	0	0
$531$	4950.00	0.404542
$532$	576.000	0.0469413
$533$	1092.00	0.0887425
$534$	−2016.00	−0.163372
$535$	0	0
$536$	3584.00	0.288816
$537$	−2124.00	−0.170684
$538$	2940.00	0.235599
$539$	−654.000	−0.0522630
$540$	0	0
$541$	−11758.0	−0.934410	−0.467205	−	0.884149i	$-0.654739\pi$
$541$	−11758.0	−0.934410	−0.467205	+	0.884149i	$0.654739\pi$
$542$	3688.00	0.292275
$543$	1638.00	0.129454
$544$	−192.000	−0.0151322
$545$	0	0
$546$	−312.000	−0.0244549
$547$	−340.000	−0.0265765	−0.0132883	−	0.999912i	$-0.504230\pi$
$547$	−340.000	−0.0265765	−0.0132883	+	0.999912i	$0.504230\pi$
$548$	3312.00	0.258178
$549$	−126.000	−0.00979517
$550$	0	0
$551$	504.000	0.0389676
$552$	480.000	0.0370112
$553$	−5312.00	−0.408480
$554$	11532.0	0.884382
$555$	0	0
$556$	−1616.00	−0.123262
$557$	3768.00	0.286634	0.143317	−	0.989677i	$-0.454223\pi$
$557$	3768.00	0.286634	0.143317	+	0.989677i	$0.454223\pi$
$558$	2736.00	0.207570
$559$	2444.00	0.184920
$560$	0	0
$561$	−36.0000	−0.00270931
$562$	14936.0	1.12106
$563$	10172.0	0.761454	0.380727	−	0.924687i	$-0.375674\pi$
$563$	10172.0	0.761454	0.380727	+	0.924687i	$0.375674\pi$
$564$	3048.00	0.227560
$565$	0	0
$566$	2456.00	0.182391
$567$	−324.000	−0.0239977
$568$	−7408.00	−0.547241
$569$	−5506.00	−0.405665	−0.202833	−	0.979213i	$-0.565015\pi$
$569$	−5506.00	−0.405665	−0.202833	+	0.979213i	$0.565015\pi$
$570$	0	0
$571$	2340.00	0.171499	0.0857495	−	0.996317i	$-0.472672\pi$
$571$	2340.00	0.171499	0.0857495	+	0.996317i	$0.472672\pi$
$572$	104.000	0.00760220
$573$	10416.0	0.759397
$574$	672.000	0.0488654
$575$	0	0
$576$	576.000	0.0416667
$577$	20094.0	1.44978	0.724891	−	0.688864i	$-0.241890\pi$
$577$	20094.0	1.44978	0.724891	+	0.688864i	$0.241890\pi$
$578$	9754.00	0.701925
$579$	−930.000	−0.0667521
$580$	0	0
$581$	744.000	0.0531262
$582$	−3684.00	−0.262383
$583$	−732.000	−0.0520006
$584$	2032.00	0.143981
$585$	0	0
$586$	13216.0	0.931652
$587$	7118.00	0.500496	0.250248	−	0.968182i	$-0.419488\pi$
$587$	7118.00	0.500496	0.250248	+	0.968182i	$0.419488\pi$
$588$	3924.00	0.275209
$589$	5472.00	0.382801
$590$	0	0
$591$	3060.00	0.212981
$592$	4128.00	0.286587
$593$	10328.0	0.715211	0.357606	−	0.933873i	$-0.383593\pi$
$593$	10328.0	0.715211	0.357606	+	0.933873i	$0.383593\pi$
$594$	108.000	0.00746009
$595$	0	0
$596$	11712.0	0.804937
$597$	9768.00	0.669644
$598$	−520.000	−0.0355592
$599$	−19732.0	−1.34596	−0.672978	−	0.739662i	$-0.734985\pi$
$599$	−19732.0	−1.34596	−0.672978	+	0.739662i	$0.734985\pi$
$600$	0	0
$601$	−12026.0	−0.816224	−0.408112	−	0.912932i	$-0.633813\pi$
$601$	−12026.0	−0.816224	−0.408112	+	0.912932i	$0.633813\pi$
$602$	1504.00	0.101825
$603$	−4032.00	−0.272298
$604$	7776.00	0.523843
$605$	0	0
$606$	−9636.00	−0.645934
$607$	−17016.0	−1.13782	−0.568911	−	0.822399i	$-0.692635\pi$
$607$	−17016.0	−1.13782	−0.568911	+	0.822399i	$0.692635\pi$
$608$	1152.00	0.0768417
$609$	−168.000	−0.0111785
$610$	0	0
$611$	−3302.00	−0.218633
$612$	216.000	0.0142668
$613$	−11654.0	−0.767864	−0.383932	−	0.923361i	$-0.625430\pi$
$613$	−11654.0	−0.767864	−0.383932	+	0.923361i	$0.625430\pi$
$614$	15328.0	1.00747
$615$	0	0
$616$	64.0000	0.00418609
$617$	−11612.0	−0.757669	−0.378834	−	0.925465i	$-0.623675\pi$
$617$	−11612.0	−0.757669	−0.378834	+	0.925465i	$0.623675\pi$
$618$	−1248.00	−0.0812329
$619$	4024.00	0.261290	0.130645	−	0.991429i	$-0.458295\pi$
$619$	4024.00	0.261290	0.130645	+	0.991429i	$0.458295\pi$
$620$	0	0
$621$	−540.000	−0.0348945
$622$	4680.00	0.301690
$623$	1344.00	0.0864305
$624$	−624.000	−0.0400320
$625$	0	0
$626$	13420.0	0.856823
$627$	216.000	0.0137579
$628$	−14360.0	−0.912462
$629$	1548.00	0.0981285
$630$	0	0
$631$	−1088.00	−0.0686412	−0.0343206	−	0.999411i	$-0.510927\pi$
$631$	−1088.00	−0.0686412	−0.0343206	+	0.999411i	$0.510927\pi$
$632$	−10624.0	−0.668671
$633$	13692.0	0.859729
$634$	8328.00	0.521683
$635$	0	0
$636$	4392.00	0.273827
$637$	−4251.00	−0.264412
$638$	56.0000	0.00347502
$639$	8334.00	0.515944
$640$	0	0
$641$	−7078.00	−0.436138	−0.218069	−	0.975933i	$-0.569976\pi$
$641$	−7078.00	−0.436138	−0.218069	+	0.975933i	$0.569976\pi$
$642$	1488.00	0.0914746
$643$	−8336.00	−0.511259	−0.255630	−	0.966775i	$-0.582283\pi$
$643$	−8336.00	−0.511259	−0.255630	+	0.966775i	$0.582283\pi$
$644$	−320.000	−0.0195804
$645$	0	0
$646$	432.000	0.0263109
$647$	−32.0000	−0.00194444	−0.000972218	−	1.00000i	$-0.500309\pi$
$647$	−32.0000	−0.00194444	−0.000972218		1.00000i	$0.500309\pi$
$648$	−648.000	−0.0392837
$649$	1100.00	0.0665312
$650$	0	0
$651$	−1824.00	−0.109813
$652$	9136.00	0.548763
$653$	15822.0	0.948182	0.474091	−	0.880476i	$-0.342777\pi$
$653$	15822.0	0.948182	0.474091	+	0.880476i	$0.342777\pi$
$654$	−3252.00	−0.194439
$655$	0	0
$656$	1344.00	0.0799914
$657$	−2286.00	−0.135746
$658$	−2032.00	−0.120388
$659$	21540.0	1.27326	0.636631	−	0.771169i	$-0.280328\pi$
$659$	21540.0	1.27326	0.636631	+	0.771169i	$0.280328\pi$
$660$	0	0
$661$	8270.00	0.486635	0.243317	−	0.969947i	$-0.421764\pi$
$661$	8270.00	0.486635	0.243317	+	0.969947i	$0.421764\pi$
$662$	20144.0	1.18266
$663$	−234.000	−0.0137071
$664$	1488.00	0.0869663
$665$	0	0
$666$	−4644.00	−0.270197
$667$	−280.000	−0.0162543
$668$	−12696.0	−0.735364
$669$	−216.000	−0.0124829
$670$	0	0
$671$	−28.0000	−0.00161092
$672$	−384.000	−0.0220433
$673$	−8482.00	−0.485820	−0.242910	−	0.970049i	$-0.578102\pi$
$673$	−8482.00	−0.485820	−0.242910	+	0.970049i	$0.578102\pi$
$674$	5980.00	0.341752
$675$	0	0
$676$	676.000	0.0384615
$677$	−2550.00	−0.144763	−0.0723814	−	0.997377i	$-0.523060\pi$
$677$	−2550.00	−0.144763	−0.0723814	+	0.997377i	$0.523060\pi$
$678$	12252.0	0.694005
$679$	2456.00	0.138811
$680$	0	0
$681$	8082.00	0.454777
$682$	608.000	0.0341371
$683$	31534.0	1.76664	0.883320	−	0.468771i	$-0.155303\pi$
$683$	31534.0	1.76664	0.883320	+	0.468771i	$0.155303\pi$
$684$	−1296.00	−0.0724471
$685$	0	0
$686$	−5360.00	−0.298317
$687$	−17766.0	−0.986631
$688$	3008.00	0.166684
$689$	−4758.00	−0.263085
$690$	0	0
$691$	33832.0	1.86256	0.931281	−	0.364302i	$-0.118693\pi$
$691$	33832.0	1.86256	0.931281	+	0.364302i	$0.118693\pi$
$692$	5432.00	0.298401
$693$	−72.0000	−0.00394669
$694$	13128.0	0.718058
$695$	0	0
$696$	−336.000	−0.0182989
$697$	504.000	0.0273893
$698$	1348.00	0.0730982
$699$	−15366.0	−0.831467
$700$	0	0
$701$	19422.0	1.04645	0.523223	−	0.852196i	$-0.324729\pi$
$701$	19422.0	1.04645	0.523223	+	0.852196i	$0.324729\pi$
$702$	702.000	0.0377426
$703$	−9288.00	−0.498298
$704$	128.000	0.00685253
$705$	0	0
$706$	−21464.0	−1.14420
$707$	6424.00	0.341725
$708$	−6600.00	−0.350343
$709$	−1894.00	−0.100325	−0.0501627	−	0.998741i	$-0.515974\pi$
$709$	−1894.00	−0.100325	−0.0501627	+	0.998741i	$0.515974\pi$
$710$	0	0
$711$	11952.0	0.630429
$712$	2688.00	0.141485
$713$	−3040.00	−0.159676
$714$	−144.000	−0.00754771
$715$	0	0
$716$	2832.00	0.147817
$717$	−15066.0	−0.784728
$718$	9684.00	0.503348
$719$	−20156.0	−1.04547	−0.522734	−	0.852496i	$-0.675088\pi$
$719$	−20156.0	−1.04547	−0.522734	+	0.852496i	$0.675088\pi$
$720$	0	0
$721$	832.000	0.0429754
$722$	11126.0	0.573500
$723$	3654.00	0.187958
$724$	−2184.00	−0.112110
$725$	0	0
$726$	−7962.00	−0.407021
$727$	−11128.0	−0.567696	−0.283848	−	0.958869i	$-0.591611\pi$
$727$	−11128.0	−0.567696	−0.283848	+	0.958869i	$0.591611\pi$
$728$	416.000	0.0211786
$729$	729.000	0.0370370
$730$	0	0
$731$	1128.00	0.0570733
$732$	168.000	0.00848287
$733$	−16202.0	−0.816418	−0.408209	−	0.912888i	$-0.633847\pi$
$733$	−16202.0	−0.816418	−0.408209	+	0.912888i	$0.633847\pi$
$734$	−12560.0	−0.631605
$735$	0	0
$736$	−640.000	−0.0320526
$737$	−896.000	−0.0447823
$738$	−1512.00	−0.0754167
$739$	−5328.00	−0.265215	−0.132607	−	0.991169i	$-0.542335\pi$
$739$	−5328.00	−0.265215	−0.132607	+	0.991169i	$0.542335\pi$
$740$	0	0
$741$	1404.00	0.0696049
$742$	−2928.00	−0.144866
$743$	20482.0	1.01132	0.505661	−	0.862732i	$-0.331249\pi$
$743$	20482.0	1.01132	0.505661	+	0.862732i	$0.331249\pi$
$744$	−3648.00	−0.179761
$745$	0	0
$746$	12868.0	0.631543
$747$	−1674.00	−0.0819926
$748$	48.0000	0.00234633
$749$	−992.000	−0.0483937
$750$	0	0
$751$	8040.00	0.390657	0.195329	−	0.980738i	$-0.437423\pi$
$751$	8040.00	0.390657	0.195329	+	0.980738i	$0.437423\pi$
$752$	−4064.00	−0.197073
$753$	6336.00	0.306636
$754$	364.000	0.0175810
$755$	0	0
$756$	432.000	0.0207827
$757$	15822.0	0.759657	0.379829	−	0.925057i	$-0.375983\pi$
$757$	15822.0	0.759657	0.379829	+	0.925057i	$0.375983\pi$
$758$	18136.0	0.869036
$759$	−120.000	−0.00573877
$760$	0	0
$761$	−1452.00	−0.0691655	−0.0345828	−	0.999402i	$-0.511010\pi$
$761$	−1452.00	−0.0691655	−0.0345828	+	0.999402i	$0.511010\pi$
$762$	2928.00	0.139200
$763$	2168.00	0.102866
$764$	−13888.0	−0.657657
$765$	0	0
$766$	6324.00	0.298297
$767$	7150.00	0.336599
$768$	−768.000	−0.0360844
$769$	32298.0	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	32298.0	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	8442.00	0.394334
$772$	1240.00	0.0578090
$773$	−18736.0	−0.871781	−0.435891	−	0.900000i	$-0.643567\pi$
$773$	−18736.0	−0.871781	−0.435891	+	0.900000i	$0.643567\pi$
$774$	−3384.00	−0.157152
$775$	0	0
$776$	4912.00	0.227230
$777$	3096.00	0.142945
$778$	7332.00	0.337873
$779$	−3024.00	−0.139083
$780$	0	0
$781$	1852.00	0.0848525
$782$	−240.000	−0.0109749
$783$	378.000	0.0172524
$784$	−5232.00	−0.238338
$785$	0	0
$786$	10464.0	0.474858
$787$	40816.0	1.84871	0.924354	−	0.381536i	$-0.124605\pi$
$787$	40816.0	1.84871	0.924354	+	0.381536i	$0.124605\pi$
$788$	−4080.00	−0.184447
$789$	−12132.0	−0.547415
$790$	0	0
$791$	−8168.00	−0.367156
$792$	−144.000	−0.00646063
$793$	−182.000	−0.00815008
$794$	22108.0	0.988141
$795$	0	0
$796$	−13024.0	−0.579929
$797$	−4518.00	−0.200798	−0.100399	−	0.994947i	$-0.532012\pi$
$797$	−4518.00	−0.200798	−0.100399	+	0.994947i	$0.532012\pi$
$798$	864.000	0.0383274
$799$	−1524.00	−0.0674784
$800$	0	0
$801$	−3024.00	−0.133393
$802$	10656.0	0.469173
$803$	−508.000	−0.0223249
$804$	5376.00	0.235817
$805$	0	0
$806$	3952.00	0.172709
$807$	4410.00	0.192366
$808$	12848.0	0.559395
$809$	−5058.00	−0.219814	−0.109907	−	0.993942i	$-0.535055\pi$
$809$	−5058.00	−0.219814	−0.109907	+	0.993942i	$0.535055\pi$
$810$	0	0
$811$	−22564.0	−0.976978	−0.488489	−	0.872570i	$-0.662452\pi$
$811$	−22564.0	−0.976978	−0.488489	+	0.872570i	$0.662452\pi$
$812$	224.000	0.00968086
$813$	5532.00	0.238642
$814$	−1032.00	−0.0444368
$815$	0	0
$816$	−288.000	−0.0123554
$817$	−6768.00	−0.289819
$818$	24148.0	1.03217
$819$	−468.000	−0.0199673
$820$	0	0
$821$	32584.0	1.38513	0.692564	−	0.721357i	$-0.256481\pi$
$821$	32584.0	1.38513	0.692564	+	0.721357i	$0.256481\pi$
$822$	4968.00	0.210802
$823$	9288.00	0.393389	0.196695	−	0.980465i	$-0.436979\pi$
$823$	9288.00	0.393389	0.196695	+	0.980465i	$0.436979\pi$
$824$	1664.00	0.0703497
$825$	0	0
$826$	4400.00	0.185346
$827$	−20586.0	−0.865593	−0.432796	−	0.901492i	$-0.642473\pi$
$827$	−20586.0	−0.865593	−0.432796	+	0.901492i	$0.642473\pi$
$828$	720.000	0.0302195
$829$	−46118.0	−1.93214	−0.966070	−	0.258280i	$-0.916844\pi$
$829$	−46118.0	−1.93214	−0.966070	+	0.258280i	$0.916844\pi$
$830$	0	0
$831$	17298.0	0.722095
$832$	832.000	0.0346688
$833$	−1962.00	−0.0816078
$834$	−2424.00	−0.100643
$835$	0	0
$836$	−288.000	−0.0119147
$837$	4104.00	0.169480
$838$	−27168.0	−1.11993
$839$	−39230.0	−1.61427	−0.807133	−	0.590369i	$-0.798982\pi$
$839$	−39230.0	−1.61427	−0.807133	+	0.590369i	$0.798982\pi$
$840$	0	0
$841$	−24193.0	−0.991964
$842$	14812.0	0.606241
$843$	22404.0	0.915344
$844$	−18256.0	−0.744547
$845$	0	0
$846$	4572.00	0.185802
$847$	5308.00	0.215331
$848$	−5856.00	−0.237141
$849$	3684.00	0.148922
$850$	0	0
$851$	5160.00	0.207853
$852$	−11112.0	−0.446820
$853$	18674.0	0.749573	0.374786	−	0.927111i	$-0.377716\pi$
$853$	18674.0	0.749573	0.374786	+	0.927111i	$0.377716\pi$
$854$	−112.000	−0.00448778
$855$	0	0
$856$	−1984.00	−0.0792193
$857$	−41678.0	−1.66125	−0.830626	−	0.556830i	$-0.812017\pi$
$857$	−41678.0	−1.66125	−0.830626	+	0.556830i	$0.812017\pi$
$858$	156.000	0.00620717
$859$	−14740.0	−0.585474	−0.292737	−	0.956193i	$-0.594566\pi$
$859$	−14740.0	−0.585474	−0.292737	+	0.956193i	$0.594566\pi$
$860$	0	0
$861$	1008.00	0.0398984
$862$	20268.0	0.800848
$863$	24982.0	0.985396	0.492698	−	0.870200i	$-0.336011\pi$
$863$	24982.0	0.985396	0.492698	+	0.870200i	$0.336011\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	18812.0	0.738173
$867$	14631.0	0.573120
$868$	2432.00	0.0951008
$869$	2656.00	0.103681
$870$	0	0
$871$	−5824.00	−0.226566
$872$	4336.00	0.168389
$873$	−5526.00	−0.214235
$874$	1440.00	0.0557308
$875$	0	0
$876$	3048.00	0.117560
$877$	−1134.00	−0.0436630	−0.0218315	−	0.999762i	$-0.506950\pi$
$877$	−1134.00	−0.0436630	−0.0218315	+	0.999762i	$0.506950\pi$
$878$	−8176.00	−0.314267
$879$	19824.0	0.760690
$880$	0	0
$881$	34950.0	1.33654	0.668272	−	0.743917i	$-0.267034\pi$
$881$	34950.0	1.33654	0.668272	+	0.743917i	$0.267034\pi$
$882$	5886.00	0.224707
$883$	3068.00	0.116927	0.0584634	−	0.998290i	$-0.481380\pi$
$883$	3068.00	0.116927	0.0584634	+	0.998290i	$0.481380\pi$
$884$	312.000	0.0118707
$885$	0	0
$886$	−10656.0	−0.404058
$887$	14080.0	0.532988	0.266494	−	0.963837i	$-0.414135\pi$
$887$	14080.0	0.532988	0.266494	+	0.963837i	$0.414135\pi$
$888$	6192.00	0.233998
$889$	−1952.00	−0.0736423
$890$	0	0
$891$	162.000	0.00609114
$892$	288.000	0.0108105
$893$	9144.00	0.342657
$894$	17568.0	0.657228
$895$	0	0
$896$	512.000	0.0190901
$897$	−780.000	−0.0290339
$898$	−26320.0	−0.978073
$899$	2128.00	0.0789464
$900$	0	0
$901$	−2196.00	−0.0811980
$902$	−336.000	−0.0124031
$903$	2256.00	0.0831395
$904$	−16336.0	−0.601026
$905$	0	0
$906$	11664.0	0.427716
$907$	24876.0	0.910688	0.455344	−	0.890316i	$-0.349516\pi$
$907$	24876.0	0.910688	0.455344	+	0.890316i	$0.349516\pi$
$908$	−10776.0	−0.393848
$909$	−14454.0	−0.527403
$910$	0	0
$911$	51456.0	1.87136	0.935682	−	0.352843i	$-0.114785\pi$
$911$	51456.0	1.87136	0.935682	+	0.352843i	$0.114785\pi$
$912$	1728.00	0.0627410
$913$	−372.000	−0.0134846
$914$	−18292.0	−0.661975
$915$	0	0
$916$	23688.0	0.854447
$917$	−6976.00	−0.251219
$918$	324.000	0.0116488
$919$	−31032.0	−1.11388	−0.556938	−	0.830554i	$-0.688024\pi$
$919$	−31032.0	−1.11388	−0.556938	+	0.830554i	$0.688024\pi$
$920$	0	0
$921$	22992.0	0.822597
$922$	−11160.0	−0.398628
$923$	12038.0	0.429291
$924$	96.0000	0.00341793
$925$	0	0
$926$	29576.0	1.04960
$927$	−1872.00	−0.0663264
$928$	448.000	0.0158473
$929$	50820.0	1.79478	0.897390	−	0.441239i	$-0.145461\pi$
$929$	50820.0	1.79478	0.897390	+	0.441239i	$0.145461\pi$
$930$	0	0
$931$	11772.0	0.414406
$932$	20488.0	0.720072
$933$	7020.00	0.246328
$934$	24752.0	0.867142
$935$	0	0
$936$	−936.000	−0.0326860
$937$	−5982.00	−0.208563	−0.104281	−	0.994548i	$-0.533254\pi$
$937$	−5982.00	−0.208563	−0.104281	+	0.994548i	$0.533254\pi$
$938$	−3584.00	−0.124757
$939$	20130.0	0.699593
$940$	0	0
$941$	−20224.0	−0.700620	−0.350310	−	0.936634i	$-0.613924\pi$
$941$	−20224.0	−0.700620	−0.350310	+	0.936634i	$0.613924\pi$
$942$	−21540.0	−0.745022
$943$	1680.00	0.0580152
$944$	8800.00	0.303406
$945$	0	0
$946$	−752.000	−0.0258453
$947$	−8478.00	−0.290917	−0.145458	−	0.989364i	$-0.546466\pi$
$947$	−8478.00	−0.290917	−0.145458	+	0.989364i	$0.546466\pi$
$948$	−15936.0	−0.545968
$949$	−3302.00	−0.112948
$950$	0	0
$951$	12492.0	0.425953
$952$	192.000	0.00653651
$953$	−40918.0	−1.39083	−0.695417	−	0.718607i	$-0.744780\pi$
$953$	−40918.0	−1.39083	−0.695417	+	0.718607i	$0.744780\pi$
$954$	6588.00	0.223579
$955$	0	0
$956$	20088.0	0.679595
$957$	84.0000	0.00283734
$958$	−1668.00	−0.0562533
$959$	−3312.00	−0.111522
$960$	0	0
$961$	−6687.00	−0.224464
$962$	−6708.00	−0.224818
$963$	2232.00	0.0746887
$964$	−4872.00	−0.162777
$965$	0	0
$966$	−480.000	−0.0159873
$967$	4624.00	0.153772	0.0768862	−	0.997040i	$-0.475502\pi$
$967$	4624.00	0.153772	0.0768862	+	0.997040i	$0.475502\pi$
$968$	10616.0	0.352491
$969$	648.000	0.0214827
$970$	0	0
$971$	15300.0	0.505665	0.252832	−	0.967510i	$-0.418638\pi$
$971$	15300.0	0.505665	0.252832	+	0.967510i	$0.418638\pi$
$972$	−972.000	−0.0320750
$973$	1616.00	0.0532442
$974$	−26384.0	−0.867965
$975$	0	0
$976$	−224.000	−0.00734638
$977$	−19584.0	−0.641298	−0.320649	−	0.947198i	$-0.603901\pi$
$977$	−19584.0	−0.641298	−0.320649	+	0.947198i	$0.603901\pi$
$978$	13704.0	0.448063
$979$	−672.000	−0.0219379
$980$	0	0
$981$	−4878.00	−0.158759
$982$	−33136.0	−1.07679
$983$	17582.0	0.570477	0.285238	−	0.958457i	$-0.407927\pi$
$983$	17582.0	0.570477	0.285238	+	0.958457i	$0.407927\pi$
$984$	2016.00	0.0653127
$985$	0	0
$986$	168.000	0.00542618
$987$	−3048.00	−0.0982968
$988$	−1872.00	−0.0602796
$989$	3760.00	0.120891
$990$	0	0
$991$	47904.0	1.53554	0.767770	−	0.640725i	$-0.221366\pi$
$991$	47904.0	1.53554	0.767770	+	0.640725i	$0.221366\pi$
$992$	4864.00	0.155678
$993$	30216.0	0.965635
$994$	7408.00	0.236386
$995$	0	0
$996$	2232.00	0.0710077
$997$	44578.0	1.41605	0.708024	−	0.706189i	$-0.249587\pi$
$997$	44578.0	1.41605	0.708024	+	0.706189i	$0.249587\pi$
$998$	20272.0	0.642985
$999$	−6966.00	−0.220615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.4.a.a.1.1		1
5.4	even	2		78.4.a.f.1.1	✓	1
15.14	odd	2		234.4.a.c.1.1		1
20.19	odd	2		624.4.a.c.1.1		1
40.19	odd	2		2496.4.a.l.1.1		1
40.29	even	2		2496.4.a.c.1.1		1
60.59	even	2		1872.4.a.f.1.1		1
65.34	odd	4		1014.4.b.g.337.2		2
65.44	odd	4		1014.4.b.g.337.1		2
65.64	even	2		1014.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.f.1.1	✓	1	5.4	even	2
234.4.a.c.1.1		1	15.14	odd	2
624.4.a.c.1.1		1	20.19	odd	2
1014.4.a.e.1.1		1	65.64	even	2
1014.4.b.g.337.1		2	65.44	odd	4
1014.4.b.g.337.2		2	65.34	odd	4
1872.4.a.f.1.1		1	60.59	even	2
1950.4.a.a.1.1		1	1.1	even	1	trivial
2496.4.a.c.1.1		1	40.29	even	2
2496.4.a.l.1.1		1	40.19	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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