LMFDB - Embedded newform 1950.4.a.b.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
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)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -36.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -56.0000 q^{14} +16.0000 q^{16} -42.0000 q^{17} -18.0000 q^{18} -112.000 q^{19} -84.0000 q^{21} +72.0000 q^{22} +168.000 q^{23} +24.0000 q^{24} +26.0000 q^{26} -27.0000 q^{27} +112.000 q^{28} -210.000 q^{29} -76.0000 q^{31} -32.0000 q^{32} +108.000 q^{33} +84.0000 q^{34} +36.0000 q^{36} -278.000 q^{37} +224.000 q^{38} +39.0000 q^{39} +150.000 q^{41} +168.000 q^{42} +460.000 q^{43} -144.000 q^{44} -336.000 q^{46} +264.000 q^{47} -48.0000 q^{48} +441.000 q^{49} +126.000 q^{51} -52.0000 q^{52} -582.000 q^{53} +54.0000 q^{54} -224.000 q^{56} +336.000 q^{57} +420.000 q^{58} -204.000 q^{59} +614.000 q^{61} +152.000 q^{62} +252.000 q^{63} +64.0000 q^{64} -216.000 q^{66} +304.000 q^{67} -168.000 q^{68} -504.000 q^{69} +1080.00 q^{71} -72.0000 q^{72} +934.000 q^{73} +556.000 q^{74} -448.000 q^{76} -1008.00 q^{77} -78.0000 q^{78} +128.000 q^{79} +81.0000 q^{81} -300.000 q^{82} -348.000 q^{83} -336.000 q^{84} -920.000 q^{86} +630.000 q^{87} +288.000 q^{88} -834.000 q^{89} -364.000 q^{91} +672.000 q^{92} +228.000 q^{93} -528.000 q^{94} +96.0000 q^{96} +1582.00 q^{97} -882.000 q^{98} -324.000 q^{99} +O(q^{100})\)

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
$7$	28.0000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	28.0000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−56.0000	−1.06904
$15$	0	0
$16$	16.0000	0.250000
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	−18.0000	−0.235702
$19$	−112.000	−1.35235	−0.676173	−	0.736743i	$-0.736363\pi$
$19$	−112.000	−1.35235	−0.676173	+	0.736743i	$0.736363\pi$
$20$	0	0
$21$	−84.0000	−0.872872
$22$	72.0000	0.697748
$23$	168.000	1.52306	0.761531	−	0.648129i	$-0.224448\pi$
$23$	168.000	1.52306	0.761531	+	0.648129i	$0.224448\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	112.000	0.755929
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−76.0000	−0.440323	−0.220161	−	0.975463i	$-0.570658\pi$
$31$	−76.0000	−0.440323	−0.220161	+	0.975463i	$0.570658\pi$
$32$	−32.0000	−0.176777
$33$	108.000	0.569709
$34$	84.0000	0.423702
$35$	0	0
$36$	36.0000	0.166667
$37$	−278.000	−1.23521	−0.617607	−	0.786487i	$-0.711898\pi$
$37$	−278.000	−1.23521	−0.617607	+	0.786487i	$0.711898\pi$
$38$	224.000	0.956253
$39$	39.0000	0.160128
$40$	0	0
$41$	150.000	0.571367	0.285684	−	0.958324i	$-0.407779\pi$
$41$	150.000	0.571367	0.285684	+	0.958324i	$0.407779\pi$
$42$	168.000	0.617213
$43$	460.000	1.63138	0.815690	−	0.578489i	$-0.196358\pi$
$43$	460.000	1.63138	0.815690	+	0.578489i	$0.196358\pi$
$44$	−144.000	−0.493382
$45$	0	0
$46$	−336.000	−1.07697
$47$	264.000	0.819327	0.409663	−	0.912237i	$-0.365646\pi$
$47$	264.000	0.819327	0.409663	+	0.912237i	$0.365646\pi$
$48$	−48.0000	−0.144338
$49$	441.000	1.28571
$50$	0	0
$51$	126.000	0.345952
$52$	−52.0000	−0.138675
$53$	−582.000	−1.50837	−0.754187	−	0.656659i	$-0.771969\pi$
$53$	−582.000	−1.50837	−0.754187	+	0.656659i	$0.771969\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−224.000	−0.534522
$57$	336.000	0.780777
$58$	420.000	0.950840
$59$	−204.000	−0.450145	−0.225072	−	0.974342i	$-0.572262\pi$
$59$	−204.000	−0.450145	−0.225072	+	0.974342i	$0.572262\pi$
$60$	0	0
$61$	614.000	1.28876	0.644382	−	0.764703i	$-0.277115\pi$
$61$	614.000	1.28876	0.644382	+	0.764703i	$0.277115\pi$
$62$	152.000	0.311355
$63$	252.000	0.503953
$64$	64.0000	0.125000
$65$	0	0
$66$	−216.000	−0.402845
$67$	304.000	0.554321	0.277161	−	0.960824i	$-0.410607\pi$
$67$	304.000	0.554321	0.277161	+	0.960824i	$0.410607\pi$
$68$	−168.000	−0.299603
$69$	−504.000	−0.879340
$70$	0	0
$71$	1080.00	1.80525	0.902623	−	0.430433i	$-0.141639\pi$
$71$	1080.00	1.80525	0.902623	+	0.430433i	$0.141639\pi$
$72$	−72.0000	−0.117851
$73$	934.000	1.49749	0.748743	−	0.662861i	$-0.230658\pi$
$73$	934.000	1.49749	0.748743	+	0.662861i	$0.230658\pi$
$74$	556.000	0.873428
$75$	0	0
$76$	−448.000	−0.676173
$77$	−1008.00	−1.49185
$78$	−78.0000	−0.113228
$79$	128.000	0.182293	0.0911464	−	0.995838i	$-0.470947\pi$
$79$	128.000	0.182293	0.0911464	+	0.995838i	$0.470947\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−300.000	−0.404018
$83$	−348.000	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$83$	−348.000	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$84$	−336.000	−0.436436
$85$	0	0
$86$	−920.000	−1.15356
$87$	630.000	0.776357
$88$	288.000	0.348874
$89$	−834.000	−0.993301	−0.496651	−	0.867951i	$-0.665437\pi$
$89$	−834.000	−0.993301	−0.496651	+	0.867951i	$0.665437\pi$
$90$	0	0
$91$	−364.000	−0.419314
$92$	672.000	0.761531
$93$	228.000	0.254220
$94$	−528.000	−0.579352
$95$	0	0
$96$	96.0000	0.102062
$97$	1582.00	1.65596	0.827978	−	0.560760i	$-0.189491\pi$
$97$	1582.00	1.65596	0.827978	+	0.560760i	$0.189491\pi$
$98$	−882.000	−0.909137
$99$	−324.000	−0.328921
$100$	0	0
$101$	−1050.00	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$101$	−1050.00	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$102$	−252.000	−0.244625
$103$	−1544.00	−1.47704	−0.738519	−	0.674233i	$-0.764474\pi$
$103$	−1544.00	−1.47704	−0.738519	+	0.674233i	$0.764474\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	1164.00	1.06658
$107$	−60.0000	−0.0542095	−0.0271048	−	0.999633i	$-0.508629\pi$
$107$	−60.0000	−0.0542095	−0.0271048	+	0.999633i	$0.508629\pi$
$108$	−108.000	−0.0962250
$109$	1166.00	1.02461	0.512305	−	0.858803i	$-0.328792\pi$
$109$	1166.00	1.02461	0.512305	+	0.858803i	$0.328792\pi$
$110$	0	0
$111$	834.000	0.713151
$112$	448.000	0.377964
$113$	−1698.00	−1.41358	−0.706789	−	0.707424i	$-0.749857\pi$
$113$	−1698.00	−1.41358	−0.706789	+	0.707424i	$0.749857\pi$
$114$	−672.000	−0.552093
$115$	0	0
$116$	−840.000	−0.672345
$117$	−117.000	−0.0924500
$118$	408.000	0.318300
$119$	−1176.00	−0.905914
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	−1228.00	−0.911294
$123$	−450.000	−0.329879
$124$	−304.000	−0.220161
$125$	0	0
$126$	−504.000	−0.356348
$127$	520.000	0.363327	0.181664	−	0.983361i	$-0.441852\pi$
$127$	520.000	0.363327	0.181664	+	0.983361i	$0.441852\pi$
$128$	−128.000	−0.0883883
$129$	−1380.00	−0.941878
$130$	0	0
$131$	12.0000	0.00800340	0.00400170	−	0.999992i	$-0.498726\pi$
$131$	12.0000	0.00800340	0.00400170	+	0.999992i	$0.498726\pi$
$132$	432.000	0.284854
$133$	−3136.00	−2.04455
$134$	−608.000	−0.391964
$135$	0	0
$136$	336.000	0.211851
$137$	618.000	0.385396	0.192698	−	0.981258i	$-0.438276\pi$
$137$	618.000	0.385396	0.192698	+	0.981258i	$0.438276\pi$
$138$	1008.00	0.621787
$139$	524.000	0.319749	0.159874	−	0.987137i	$-0.448891\pi$
$139$	524.000	0.319749	0.159874	+	0.987137i	$0.448891\pi$
$140$	0	0
$141$	−792.000	−0.473039
$142$	−2160.00	−1.27650
$143$	468.000	0.273679
$144$	144.000	0.0833333
$145$	0	0
$146$	−1868.00	−1.05888
$147$	−1323.00	−0.742307
$148$	−1112.00	−0.617607
$149$	−726.000	−0.399169	−0.199585	−	0.979881i	$-0.563959\pi$
$149$	−726.000	−0.399169	−0.199585	+	0.979881i	$0.563959\pi$
$150$	0	0
$151$	−2524.00	−1.36027	−0.680133	−	0.733089i	$-0.738078\pi$
$151$	−2524.00	−1.36027	−0.680133	+	0.733089i	$0.738078\pi$
$152$	896.000	0.478126
$153$	−378.000	−0.199735
$154$	2016.00	1.05490
$155$	0	0
$156$	156.000	0.0800641
$157$	2458.00	1.24949	0.624744	−	0.780829i	$-0.285203\pi$
$157$	2458.00	1.24949	0.624744	+	0.780829i	$0.285203\pi$
$158$	−256.000	−0.128900
$159$	1746.00	0.870860
$160$	0	0
$161$	4704.00	2.30265
$162$	−162.000	−0.0785674
$163$	592.000	0.284473	0.142236	−	0.989833i	$-0.454571\pi$
$163$	592.000	0.284473	0.142236	+	0.989833i	$0.454571\pi$
$164$	600.000	0.285684
$165$	0	0
$166$	696.000	0.325422
$167$	−1968.00	−0.911907	−0.455953	−	0.890004i	$-0.650702\pi$
$167$	−1968.00	−0.911907	−0.455953	+	0.890004i	$0.650702\pi$
$168$	672.000	0.308607
$169$	169.000	0.0769231
$170$	0	0
$171$	−1008.00	−0.450782
$172$	1840.00	0.815690
$173$	−2214.00	−0.972990	−0.486495	−	0.873683i	$-0.661725\pi$
$173$	−2214.00	−0.972990	−0.486495	+	0.873683i	$0.661725\pi$
$174$	−1260.00	−0.548968
$175$	0	0
$176$	−576.000	−0.246691
$177$	612.000	0.259891
$178$	1668.00	0.702370
$179$	3084.00	1.28776	0.643880	−	0.765127i	$-0.277324\pi$
$179$	3084.00	1.28776	0.643880	+	0.765127i	$0.277324\pi$
$180$	0	0
$181$	686.000	0.281713	0.140856	−	0.990030i	$-0.455014\pi$
$181$	686.000	0.281713	0.140856	+	0.990030i	$0.455014\pi$
$182$	728.000	0.296500
$183$	−1842.00	−0.744069
$184$	−1344.00	−0.538484
$185$	0	0
$186$	−456.000	−0.179761
$187$	1512.00	0.591275
$188$	1056.00	0.409663
$189$	−756.000	−0.290957
$190$	0	0
$191$	4440.00	1.68203	0.841013	−	0.541014i	$-0.181960\pi$
$191$	4440.00	1.68203	0.841013	+	0.541014i	$0.181960\pi$
$192$	−192.000	−0.0721688
$193$	4726.00	1.76262	0.881308	−	0.472542i	$-0.156664\pi$
$193$	4726.00	1.76262	0.881308	+	0.472542i	$0.156664\pi$
$194$	−3164.00	−1.17094
$195$	0	0
$196$	1764.00	0.642857
$197$	−258.000	−0.0933083	−0.0466542	−	0.998911i	$-0.514856\pi$
$197$	−258.000	−0.0933083	−0.0466542	+	0.998911i	$0.514856\pi$
$198$	648.000	0.232583
$199$	3008.00	1.07151	0.535757	−	0.844372i	$-0.320026\pi$
$199$	3008.00	1.07151	0.535757	+	0.844372i	$0.320026\pi$
$200$	0	0
$201$	−912.000	−0.320037
$202$	2100.00	0.731463
$203$	−5880.00	−2.03298
$204$	504.000	0.172976
$205$	0	0
$206$	3088.00	1.04442
$207$	1512.00	0.507687
$208$	−208.000	−0.0693375
$209$	4032.00	1.33445
$210$	0	0
$211$	−4516.00	−1.47343	−0.736716	−	0.676202i	$-0.763625\pi$
$211$	−4516.00	−1.47343	−0.736716	+	0.676202i	$0.763625\pi$
$212$	−2328.00	−0.754187
$213$	−3240.00	−1.04226
$214$	120.000	0.0383319
$215$	0	0
$216$	216.000	0.0680414
$217$	−2128.00	−0.665705
$218$	−2332.00	−0.724509
$219$	−2802.00	−0.864574
$220$	0	0
$221$	546.000	0.166190
$222$	−1668.00	−0.504274
$223$	4516.00	1.35612	0.678058	−	0.735009i	$-0.262822\pi$
$223$	4516.00	1.35612	0.678058	+	0.735009i	$0.262822\pi$
$224$	−896.000	−0.267261
$225$	0	0
$226$	3396.00	0.999551
$227$	−372.000	−0.108769	−0.0543844	−	0.998520i	$-0.517320\pi$
$227$	−372.000	−0.108769	−0.0543844	+	0.998520i	$0.517320\pi$
$228$	1344.00	0.390388
$229$	−4138.00	−1.19409	−0.597045	−	0.802208i	$-0.703659\pi$
$229$	−4138.00	−1.19409	−0.597045	+	0.802208i	$0.703659\pi$
$230$	0	0
$231$	3024.00	0.861319
$232$	1680.00	0.475420
$233$	1446.00	0.406569	0.203285	−	0.979120i	$-0.434838\pi$
$233$	1446.00	0.406569	0.203285	+	0.979120i	$0.434838\pi$
$234$	234.000	0.0653720
$235$	0	0
$236$	−816.000	−0.225072
$237$	−384.000	−0.105247
$238$	2352.00	0.640578
$239$	−3456.00	−0.935356	−0.467678	−	0.883899i	$-0.654909\pi$
$239$	−3456.00	−0.935356	−0.467678	+	0.883899i	$0.654909\pi$
$240$	0	0
$241$	5258.00	1.40538	0.702692	−	0.711494i	$-0.251981\pi$
$241$	5258.00	1.40538	0.702692	+	0.711494i	$0.251981\pi$
$242$	70.0000	0.0185941
$243$	−243.000	−0.0641500
$244$	2456.00	0.644382
$245$	0	0
$246$	900.000	0.233260
$247$	1456.00	0.375073
$248$	608.000	0.155678
$249$	1044.00	0.265706
$250$	0	0
$251$	7020.00	1.76533	0.882666	−	0.470000i	$-0.155746\pi$
$251$	7020.00	1.76533	0.882666	+	0.470000i	$0.155746\pi$
$252$	1008.00	0.251976
$253$	−6048.00	−1.50290
$254$	−1040.00	−0.256911
$255$	0	0
$256$	256.000	0.0625000
$257$	5694.00	1.38203	0.691015	−	0.722840i	$-0.257164\pi$
$257$	5694.00	1.38203	0.691015	+	0.722840i	$0.257164\pi$
$258$	2760.00	0.666008
$259$	−7784.00	−1.86747
$260$	0	0
$261$	−1890.00	−0.448230
$262$	−24.0000	−0.00565926
$263$	960.000	0.225080	0.112540	−	0.993647i	$-0.464101\pi$
$263$	960.000	0.225080	0.112540	+	0.993647i	$0.464101\pi$
$264$	−864.000	−0.201422
$265$	0	0
$266$	6272.00	1.44572
$267$	2502.00	0.573483
$268$	1216.00	0.277161
$269$	−2370.00	−0.537180	−0.268590	−	0.963255i	$-0.586558\pi$
$269$	−2370.00	−0.537180	−0.268590	+	0.963255i	$0.586558\pi$
$270$	0	0
$271$	−508.000	−0.113870	−0.0569351	−	0.998378i	$-0.518133\pi$
$271$	−508.000	−0.113870	−0.0569351	+	0.998378i	$0.518133\pi$
$272$	−672.000	−0.149801
$273$	1092.00	0.242091
$274$	−1236.00	−0.272516
$275$	0	0
$276$	−2016.00	−0.439670
$277$	4498.00	0.975663	0.487831	−	0.872938i	$-0.337788\pi$
$277$	4498.00	0.975663	0.487831	+	0.872938i	$0.337788\pi$
$278$	−1048.00	−0.226097
$279$	−684.000	−0.146774
$280$	0	0
$281$	7062.00	1.49923	0.749615	−	0.661874i	$-0.230239\pi$
$281$	7062.00	1.49923	0.749615	+	0.661874i	$0.230239\pi$
$282$	1584.00	0.334489
$283$	4036.00	0.847757	0.423879	−	0.905719i	$-0.360668\pi$
$283$	4036.00	0.847757	0.423879	+	0.905719i	$0.360668\pi$
$284$	4320.00	0.902623
$285$	0	0
$286$	−936.000	−0.193520
$287$	4200.00	0.863826
$288$	−288.000	−0.0589256
$289$	−3149.00	−0.640953
$290$	0	0
$291$	−4746.00	−0.956067
$292$	3736.00	0.748743
$293$	−1674.00	−0.333775	−0.166888	−	0.985976i	$-0.553372\pi$
$293$	−1674.00	−0.333775	−0.166888	+	0.985976i	$0.553372\pi$
$294$	2646.00	0.524891
$295$	0	0
$296$	2224.00	0.436714
$297$	972.000	0.189903
$298$	1452.00	0.282255
$299$	−2184.00	−0.422421
$300$	0	0
$301$	12880.0	2.46641
$302$	5048.00	0.961854
$303$	3150.00	0.597237
$304$	−1792.00	−0.338086
$305$	0	0
$306$	756.000	0.141234
$307$	8968.00	1.66720	0.833601	−	0.552368i	$-0.186276\pi$
$307$	8968.00	1.66720	0.833601	+	0.552368i	$0.186276\pi$
$308$	−4032.00	−0.745924
$309$	4632.00	0.852768
$310$	0	0
$311$	672.000	0.122526	0.0612631	−	0.998122i	$-0.480487\pi$
$311$	672.000	0.122526	0.0612631	+	0.998122i	$0.480487\pi$
$312$	−312.000	−0.0566139
$313$	−1946.00	−0.351420	−0.175710	−	0.984442i	$-0.556222\pi$
$313$	−1946.00	−0.351420	−0.175710	+	0.984442i	$0.556222\pi$
$314$	−4916.00	−0.883522
$315$	0	0
$316$	512.000	0.0911464
$317$	−474.000	−0.0839826	−0.0419913	−	0.999118i	$-0.513370\pi$
$317$	−474.000	−0.0839826	−0.0419913	+	0.999118i	$0.513370\pi$
$318$	−3492.00	−0.615791
$319$	7560.00	1.32689
$320$	0	0
$321$	180.000	0.0312979
$322$	−9408.00	−1.62822
$323$	4704.00	0.810333
$324$	324.000	0.0555556
$325$	0	0
$326$	−1184.00	−0.201152
$327$	−3498.00	−0.591559
$328$	−1200.00	−0.202009
$329$	7392.00	1.23871
$330$	0	0
$331$	−6928.00	−1.15045	−0.575223	−	0.817997i	$-0.695085\pi$
$331$	−6928.00	−1.15045	−0.575223	+	0.817997i	$0.695085\pi$
$332$	−1392.00	−0.230108
$333$	−2502.00	−0.411738
$334$	3936.00	0.644815
$335$	0	0
$336$	−1344.00	−0.218218
$337$	−2882.00	−0.465853	−0.232927	−	0.972494i	$-0.574830\pi$
$337$	−2882.00	−0.465853	−0.232927	+	0.972494i	$0.574830\pi$
$338$	−338.000	−0.0543928
$339$	5094.00	0.816130
$340$	0	0
$341$	2736.00	0.434495
$342$	2016.00	0.318751
$343$	2744.00	0.431959
$344$	−3680.00	−0.576780
$345$	0	0
$346$	4428.00	0.688008
$347$	7548.00	1.16772	0.583859	−	0.811855i	$-0.301542\pi$
$347$	7548.00	1.16772	0.583859	+	0.811855i	$0.301542\pi$
$348$	2520.00	0.388179
$349$	−5146.00	−0.789281	−0.394640	−	0.918836i	$-0.629131\pi$
$349$	−5146.00	−0.789281	−0.394640	+	0.918836i	$0.629131\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	1152.00	0.174437
$353$	9138.00	1.37781	0.688905	−	0.724852i	$-0.258092\pi$
$353$	9138.00	1.37781	0.688905	+	0.724852i	$0.258092\pi$
$354$	−1224.00	−0.183771
$355$	0	0
$356$	−3336.00	−0.496651
$357$	3528.00	0.523030
$358$	−6168.00	−0.910584
$359$	−4320.00	−0.635100	−0.317550	−	0.948242i	$-0.602860\pi$
$359$	−4320.00	−0.635100	−0.317550	+	0.948242i	$0.602860\pi$
$360$	0	0
$361$	5685.00	0.828838
$362$	−1372.00	−0.199201
$363$	105.000	0.0151820
$364$	−1456.00	−0.209657
$365$	0	0
$366$	3684.00	0.526136
$367$	3976.00	0.565519	0.282760	−	0.959191i	$-0.408750\pi$
$367$	3976.00	0.565519	0.282760	+	0.959191i	$0.408750\pi$
$368$	2688.00	0.380765
$369$	1350.00	0.190456
$370$	0	0
$371$	−16296.0	−2.28045
$372$	912.000	0.127110
$373$	11314.0	1.57055	0.785277	−	0.619144i	$-0.212520\pi$
$373$	11314.0	1.57055	0.785277	+	0.619144i	$0.212520\pi$
$374$	−3024.00	−0.418094
$375$	0	0
$376$	−2112.00	−0.289676
$377$	2730.00	0.372950
$378$	1512.00	0.205738
$379$	200.000	0.0271064	0.0135532	−	0.999908i	$-0.495686\pi$
$379$	200.000	0.0271064	0.0135532	+	0.999908i	$0.495686\pi$
$380$	0	0
$381$	−1560.00	−0.209767
$382$	−8880.00	−1.18937
$383$	−3960.00	−0.528320	−0.264160	−	0.964479i	$-0.585095\pi$
$383$	−3960.00	−0.528320	−0.264160	+	0.964479i	$0.585095\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−9452.00	−1.24636
$387$	4140.00	0.543793
$388$	6328.00	0.827978
$389$	−12450.0	−1.62273	−0.811363	−	0.584543i	$-0.801274\pi$
$389$	−12450.0	−1.62273	−0.811363	+	0.584543i	$0.801274\pi$
$390$	0	0
$391$	−7056.00	−0.912627
$392$	−3528.00	−0.454569
$393$	−36.0000	−0.00462076
$394$	516.000	0.0659789
$395$	0	0
$396$	−1296.00	−0.164461
$397$	7570.00	0.956996	0.478498	−	0.878089i	$-0.341181\pi$
$397$	7570.00	0.956996	0.478498	+	0.878089i	$0.341181\pi$
$398$	−6016.00	−0.757675
$399$	9408.00	1.18042
$400$	0	0
$401$	−8010.00	−0.997507	−0.498754	−	0.866744i	$-0.666209\pi$
$401$	−8010.00	−0.997507	−0.498754	+	0.866744i	$0.666209\pi$
$402$	1824.00	0.226301
$403$	988.000	0.122124
$404$	−4200.00	−0.517222
$405$	0	0
$406$	11760.0	1.43753
$407$	10008.0	1.21887
$408$	−1008.00	−0.122312
$409$	−3166.00	−0.382759	−0.191380	−	0.981516i	$-0.561296\pi$
$409$	−3166.00	−0.382759	−0.191380	+	0.981516i	$0.561296\pi$
$410$	0	0
$411$	−1854.00	−0.222509
$412$	−6176.00	−0.738519
$413$	−5712.00	−0.680555
$414$	−3024.00	−0.358989
$415$	0	0
$416$	416.000	0.0490290
$417$	−1572.00	−0.184607
$418$	−8064.00	−0.943596
$419$	−1188.00	−0.138515	−0.0692573	−	0.997599i	$-0.522063\pi$
$419$	−1188.00	−0.138515	−0.0692573	+	0.997599i	$0.522063\pi$
$420$	0	0
$421$	14054.0	1.62696	0.813480	−	0.581593i	$-0.197570\pi$
$421$	14054.0	1.62696	0.813480	+	0.581593i	$0.197570\pi$
$422$	9032.00	1.04187
$423$	2376.00	0.273109
$424$	4656.00	0.533291
$425$	0	0
$426$	6480.00	0.736988
$427$	17192.0	1.94843
$428$	−240.000	−0.0271048
$429$	−1404.00	−0.158009
$430$	0	0
$431$	1704.00	0.190438	0.0952190	−	0.995456i	$-0.469645\pi$
$431$	1704.00	0.190438	0.0952190	+	0.995456i	$0.469645\pi$
$432$	−432.000	−0.0481125
$433$	−10418.0	−1.15625	−0.578126	−	0.815947i	$-0.696216\pi$
$433$	−10418.0	−1.15625	−0.578126	+	0.815947i	$0.696216\pi$
$434$	4256.00	0.470725
$435$	0	0
$436$	4664.00	0.512305
$437$	−18816.0	−2.05971
$438$	5604.00	0.611346
$439$	−568.000	−0.0617521	−0.0308760	−	0.999523i	$-0.509830\pi$
$439$	−568.000	−0.0617521	−0.0308760	+	0.999523i	$0.509830\pi$
$440$	0	0
$441$	3969.00	0.428571
$442$	−1092.00	−0.117514
$443$	14580.0	1.56369	0.781847	−	0.623470i	$-0.214278\pi$
$443$	14580.0	1.56369	0.781847	+	0.623470i	$0.214278\pi$
$444$	3336.00	0.356576
$445$	0	0
$446$	−9032.00	−0.958918
$447$	2178.00	0.230461
$448$	1792.00	0.188982
$449$	−15978.0	−1.67940	−0.839698	−	0.543054i	$-0.817268\pi$
$449$	−15978.0	−1.67940	−0.839698	+	0.543054i	$0.817268\pi$
$450$	0	0
$451$	−5400.00	−0.563805
$452$	−6792.00	−0.706789
$453$	7572.00	0.785350
$454$	744.000	0.0769111
$455$	0	0
$456$	−2688.00	−0.276046
$457$	6982.00	0.714670	0.357335	−	0.933976i	$-0.383685\pi$
$457$	6982.00	0.714670	0.357335	+	0.933976i	$0.383685\pi$
$458$	8276.00	0.844350
$459$	1134.00	0.115317
$460$	0	0
$461$	11946.0	1.20690	0.603450	−	0.797401i	$-0.293792\pi$
$461$	11946.0	1.20690	0.603450	+	0.797401i	$0.293792\pi$
$462$	−6048.00	−0.609044
$463$	7828.00	0.785741	0.392870	−	0.919594i	$-0.371482\pi$
$463$	7828.00	0.785741	0.392870	+	0.919594i	$0.371482\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	−2892.00	−0.287488
$467$	−2076.00	−0.205708	−0.102854	−	0.994696i	$-0.532798\pi$
$467$	−2076.00	−0.205708	−0.102854	+	0.994696i	$0.532798\pi$
$468$	−468.000	−0.0462250
$469$	8512.00	0.838055
$470$	0	0
$471$	−7374.00	−0.721393
$472$	1632.00	0.159150
$473$	−16560.0	−1.60979
$474$	768.000	0.0744207
$475$	0	0
$476$	−4704.00	−0.452957
$477$	−5238.00	−0.502791
$478$	6912.00	0.661396
$479$	17760.0	1.69410	0.847051	−	0.531511i	$-0.178376\pi$
$479$	17760.0	1.69410	0.847051	+	0.531511i	$0.178376\pi$
$480$	0	0
$481$	3614.00	0.342587
$482$	−10516.0	−0.993757
$483$	−14112.0	−1.32944
$484$	−140.000	−0.0131480
$485$	0	0
$486$	486.000	0.0453609
$487$	−3044.00	−0.283238	−0.141619	−	0.989921i	$-0.545231\pi$
$487$	−3044.00	−0.283238	−0.141619	+	0.989921i	$0.545231\pi$
$488$	−4912.00	−0.455647
$489$	−1776.00	−0.164240
$490$	0	0
$491$	2700.00	0.248166	0.124083	−	0.992272i	$-0.460401\pi$
$491$	2700.00	0.248166	0.124083	+	0.992272i	$0.460401\pi$
$492$	−1800.00	−0.164940
$493$	8820.00	0.805746
$494$	−2912.00	−0.265217
$495$	0	0
$496$	−1216.00	−0.110081
$497$	30240.0	2.72927
$498$	−2088.00	−0.187883
$499$	15608.0	1.40022	0.700110	−	0.714035i	$-0.253134\pi$
$499$	15608.0	1.40022	0.700110	+	0.714035i	$0.253134\pi$
$500$	0	0
$501$	5904.00	0.526490
$502$	−14040.0	−1.24828
$503$	7152.00	0.633980	0.316990	−	0.948429i	$-0.397328\pi$
$503$	7152.00	0.633980	0.316990	+	0.948429i	$0.397328\pi$
$504$	−2016.00	−0.178174
$505$	0	0
$506$	12096.0	1.06271
$507$	−507.000	−0.0444116
$508$	2080.00	0.181664
$509$	21354.0	1.85953	0.929764	−	0.368157i	$-0.120011\pi$
$509$	21354.0	1.85953	0.929764	+	0.368157i	$0.120011\pi$
$510$	0	0
$511$	26152.0	2.26399
$512$	−512.000	−0.0441942
$513$	3024.00	0.260259
$514$	−11388.0	−0.977243
$515$	0	0
$516$	−5520.00	−0.470939
$517$	−9504.00	−0.808482
$518$	15568.0	1.32050
$519$	6642.00	0.561756
$520$	0	0
$521$	7746.00	0.651360	0.325680	−	0.945480i	$-0.394407\pi$
$521$	7746.00	0.651360	0.325680	+	0.945480i	$0.394407\pi$
$522$	3780.00	0.316947
$523$	8212.00	0.686588	0.343294	−	0.939228i	$-0.388457\pi$
$523$	8212.00	0.686588	0.343294	+	0.939228i	$0.388457\pi$
$524$	48.0000	0.00400170
$525$	0	0
$526$	−1920.00	−0.159156
$527$	3192.00	0.263844
$528$	1728.00	0.142427
$529$	16057.0	1.31972
$530$	0	0
$531$	−1836.00	−0.150048
$532$	−12544.0	−1.02228
$533$	−1950.00	−0.158469
$534$	−5004.00	−0.405514
$535$	0	0
$536$	−2432.00	−0.195982
$537$	−9252.00	−0.743488
$538$	4740.00	0.379844
$539$	−15876.0	−1.26870
$540$	0	0
$541$	9758.00	0.775470	0.387735	−	0.921771i	$-0.373258\pi$
$541$	9758.00	0.775470	0.387735	+	0.921771i	$0.373258\pi$
$542$	1016.00	0.0805183
$543$	−2058.00	−0.162647
$544$	1344.00	0.105926
$545$	0	0
$546$	−2184.00	−0.171184
$547$	−16292.0	−1.27348	−0.636742	−	0.771077i	$-0.719718\pi$
$547$	−16292.0	−1.27348	−0.636742	+	0.771077i	$0.719718\pi$
$548$	2472.00	0.192698
$549$	5526.00	0.429588
$550$	0	0
$551$	23520.0	1.81849
$552$	4032.00	0.310894
$553$	3584.00	0.275601
$554$	−8996.00	−0.689898
$555$	0	0
$556$	2096.00	0.159874
$557$	10686.0	0.812891	0.406446	−	0.913675i	$-0.366768\pi$
$557$	10686.0	0.812891	0.406446	+	0.913675i	$0.366768\pi$
$558$	1368.00	0.103785
$559$	−5980.00	−0.452463
$560$	0	0
$561$	−4536.00	−0.341373
$562$	−14124.0	−1.06012
$563$	−19572.0	−1.46512	−0.732559	−	0.680704i	$-0.761674\pi$
$563$	−19572.0	−1.46512	−0.732559	+	0.680704i	$0.761674\pi$
$564$	−3168.00	−0.236519
$565$	0	0
$566$	−8072.00	−0.599455
$567$	2268.00	0.167984
$568$	−8640.00	−0.638251
$569$	−4878.00	−0.359396	−0.179698	−	0.983722i	$-0.557512\pi$
$569$	−4878.00	−0.359396	−0.179698	+	0.983722i	$0.557512\pi$
$570$	0	0
$571$	5612.00	0.411305	0.205652	−	0.978625i	$-0.434068\pi$
$571$	5612.00	0.411305	0.205652	+	0.978625i	$0.434068\pi$
$572$	1872.00	0.136840
$573$	−13320.0	−0.971119
$574$	−8400.00	−0.610817
$575$	0	0
$576$	576.000	0.0416667
$577$	−26474.0	−1.91010	−0.955049	−	0.296447i	$-0.904198\pi$
$577$	−26474.0	−1.91010	−0.955049	+	0.296447i	$0.904198\pi$
$578$	6298.00	0.453222
$579$	−14178.0	−1.01765
$580$	0	0
$581$	−9744.00	−0.695782
$582$	9492.00	0.676041
$583$	20952.0	1.48841
$584$	−7472.00	−0.529441
$585$	0	0
$586$	3348.00	0.236015
$587$	−4884.00	−0.343414	−0.171707	−	0.985148i	$-0.554928\pi$
$587$	−4884.00	−0.343414	−0.171707	+	0.985148i	$0.554928\pi$
$588$	−5292.00	−0.371154
$589$	8512.00	0.595468
$590$	0	0
$591$	774.000	0.0538716
$592$	−4448.00	−0.308804
$593$	−5862.00	−0.405942	−0.202971	−	0.979185i	$-0.565060\pi$
$593$	−5862.00	−0.405942	−0.202971	+	0.979185i	$0.565060\pi$
$594$	−1944.00	−0.134282
$595$	0	0
$596$	−2904.00	−0.199585
$597$	−9024.00	−0.618639
$598$	4368.00	0.298697
$599$	−5928.00	−0.404360	−0.202180	−	0.979348i	$-0.564803\pi$
$599$	−5928.00	−0.404360	−0.202180	+	0.979348i	$0.564803\pi$
$600$	0	0
$601$	14762.0	1.00192	0.500961	−	0.865470i	$-0.332980\pi$
$601$	14762.0	1.00192	0.500961	+	0.865470i	$0.332980\pi$
$602$	−25760.0	−1.74402
$603$	2736.00	0.184774
$604$	−10096.0	−0.680133
$605$	0	0
$606$	−6300.00	−0.422310
$607$	−7784.00	−0.520499	−0.260249	−	0.965541i	$-0.583805\pi$
$607$	−7784.00	−0.520499	−0.260249	+	0.965541i	$0.583805\pi$
$608$	3584.00	0.239063
$609$	17640.0	1.17374
$610$	0	0
$611$	−3432.00	−0.227240
$612$	−1512.00	−0.0998676
$613$	21994.0	1.44915	0.724575	−	0.689196i	$-0.242036\pi$
$613$	21994.0	1.44915	0.724575	+	0.689196i	$0.242036\pi$
$614$	−17936.0	−1.17889
$615$	0	0
$616$	8064.00	0.527448
$617$	9546.00	0.622865	0.311432	−	0.950268i	$-0.399191\pi$
$617$	9546.00	0.622865	0.311432	+	0.950268i	$0.399191\pi$
$618$	−9264.00	−0.602998
$619$	−18568.0	−1.20567	−0.602836	−	0.797865i	$-0.705963\pi$
$619$	−18568.0	−1.20567	−0.602836	+	0.797865i	$0.705963\pi$
$620$	0	0
$621$	−4536.00	−0.293113
$622$	−1344.00	−0.0866391
$623$	−23352.0	−1.50173
$624$	624.000	0.0400320
$625$	0	0
$626$	3892.00	0.248491
$627$	−12096.0	−0.770443
$628$	9832.00	0.624744
$629$	11676.0	0.740147
$630$	0	0
$631$	−3580.00	−0.225860	−0.112930	−	0.993603i	$-0.536024\pi$
$631$	−3580.00	−0.225860	−0.112930	+	0.993603i	$0.536024\pi$
$632$	−1024.00	−0.0644502
$633$	13548.0	0.850687
$634$	948.000	0.0593847
$635$	0	0
$636$	6984.00	0.435430
$637$	−5733.00	−0.356593
$638$	−15120.0	−0.938255
$639$	9720.00	0.601748
$640$	0	0
$641$	−20718.0	−1.27662	−0.638309	−	0.769780i	$-0.720366\pi$
$641$	−20718.0	−1.27662	−0.638309	+	0.769780i	$0.720366\pi$
$642$	−360.000	−0.0221309
$643$	−18560.0	−1.13831	−0.569156	−	0.822229i	$-0.692730\pi$
$643$	−18560.0	−1.13831	−0.569156	+	0.822229i	$0.692730\pi$
$644$	18816.0	1.15133
$645$	0	0
$646$	−9408.00	−0.572992
$647$	27072.0	1.64499	0.822496	−	0.568771i	$-0.192581\pi$
$647$	27072.0	1.64499	0.822496	+	0.568771i	$0.192581\pi$
$648$	−648.000	−0.0392837
$649$	7344.00	0.444187
$650$	0	0
$651$	6384.00	0.384345
$652$	2368.00	0.142236
$653$	−24150.0	−1.44726	−0.723631	−	0.690187i	$-0.757528\pi$
$653$	−24150.0	−1.44726	−0.723631	+	0.690187i	$0.757528\pi$
$654$	6996.00	0.418295
$655$	0	0
$656$	2400.00	0.142842
$657$	8406.00	0.499162
$658$	−14784.0	−0.875897
$659$	−11388.0	−0.673162	−0.336581	−	0.941655i	$-0.609271\pi$
$659$	−11388.0	−0.673162	−0.336581	+	0.941655i	$0.609271\pi$
$660$	0	0
$661$	1598.00	0.0940318	0.0470159	−	0.998894i	$-0.485029\pi$
$661$	1598.00	0.0940318	0.0470159	+	0.998894i	$0.485029\pi$
$662$	13856.0	0.813488
$663$	−1638.00	−0.0959497
$664$	2784.00	0.162711
$665$	0	0
$666$	5004.00	0.291143
$667$	−35280.0	−2.04805
$668$	−7872.00	−0.455953
$669$	−13548.0	−0.782954
$670$	0	0
$671$	−22104.0	−1.27171
$672$	2688.00	0.154303
$673$	7918.00	0.453516	0.226758	−	0.973951i	$-0.427187\pi$
$673$	7918.00	0.453516	0.226758	+	0.973951i	$0.427187\pi$
$674$	5764.00	0.329408
$675$	0	0
$676$	676.000	0.0384615
$677$	1578.00	0.0895827	0.0447913	−	0.998996i	$-0.485738\pi$
$677$	1578.00	0.0895827	0.0447913	+	0.998996i	$0.485738\pi$
$678$	−10188.0	−0.577091
$679$	44296.0	2.50357
$680$	0	0
$681$	1116.00	0.0627977
$682$	−5472.00	−0.307234
$683$	8580.00	0.480680	0.240340	−	0.970689i	$-0.422741\pi$
$683$	8580.00	0.480680	0.240340	+	0.970689i	$0.422741\pi$
$684$	−4032.00	−0.225391
$685$	0	0
$686$	−5488.00	−0.305441
$687$	12414.0	0.689409
$688$	7360.00	0.407845
$689$	7566.00	0.418348
$690$	0	0
$691$	14168.0	0.779994	0.389997	−	0.920816i	$-0.372476\pi$
$691$	14168.0	0.779994	0.389997	+	0.920816i	$0.372476\pi$
$692$	−8856.00	−0.486495
$693$	−9072.00	−0.497283
$694$	−15096.0	−0.825701
$695$	0	0
$696$	−5040.00	−0.274484
$697$	−6300.00	−0.342367
$698$	10292.0	0.558106
$699$	−4338.00	−0.234733
$700$	0	0
$701$	11814.0	0.636532	0.318266	−	0.948002i	$-0.396900\pi$
$701$	11814.0	0.636532	0.318266	+	0.948002i	$0.396900\pi$
$702$	−702.000	−0.0377426
$703$	31136.0	1.67044
$704$	−2304.00	−0.123346
$705$	0	0
$706$	−18276.0	−0.974258
$707$	−29400.0	−1.56393
$708$	2448.00	0.129946
$709$	8198.00	0.434249	0.217124	−	0.976144i	$-0.430332\pi$
$709$	8198.00	0.434249	0.217124	+	0.976144i	$0.430332\pi$
$710$	0	0
$711$	1152.00	0.0607642
$712$	6672.00	0.351185
$713$	−12768.0	−0.670639
$714$	−7056.00	−0.369838
$715$	0	0
$716$	12336.0	0.643880
$717$	10368.0	0.540028
$718$	8640.00	0.449083
$719$	−11448.0	−0.593795	−0.296897	−	0.954909i	$-0.595952\pi$
$719$	−11448.0	−0.593795	−0.296897	+	0.954909i	$0.595952\pi$
$720$	0	0
$721$	−43232.0	−2.23307
$722$	−11370.0	−0.586077
$723$	−15774.0	−0.811399
$724$	2744.00	0.140856
$725$	0	0
$726$	−210.000	−0.0107353
$727$	10864.0	0.554228	0.277114	−	0.960837i	$-0.410622\pi$
$727$	10864.0	0.554228	0.277114	+	0.960837i	$0.410622\pi$
$728$	2912.00	0.148250
$729$	729.000	0.0370370
$730$	0	0
$731$	−19320.0	−0.977532
$732$	−7368.00	−0.372034
$733$	−31286.0	−1.57650	−0.788250	−	0.615355i	$-0.789013\pi$
$733$	−31286.0	−1.57650	−0.788250	+	0.615355i	$0.789013\pi$
$734$	−7952.00	−0.399882
$735$	0	0
$736$	−5376.00	−0.269242
$737$	−10944.0	−0.546984
$738$	−2700.00	−0.134673
$739$	19712.0	0.981215	0.490607	−	0.871381i	$-0.336775\pi$
$739$	19712.0	0.981215	0.490607	+	0.871381i	$0.336775\pi$
$740$	0	0
$741$	−4368.00	−0.216549
$742$	32592.0	1.61252
$743$	13608.0	0.671910	0.335955	−	0.941878i	$-0.390941\pi$
$743$	13608.0	0.671910	0.335955	+	0.941878i	$0.390941\pi$
$744$	−1824.00	−0.0898805
$745$	0	0
$746$	−22628.0	−1.11055
$747$	−3132.00	−0.153405
$748$	6048.00	0.295637
$749$	−1680.00	−0.0819571
$750$	0	0
$751$	1952.00	0.0948462	0.0474231	−	0.998875i	$-0.484899\pi$
$751$	1952.00	0.0948462	0.0474231	+	0.998875i	$0.484899\pi$
$752$	4224.00	0.204832
$753$	−21060.0	−1.01922
$754$	−5460.00	−0.263715
$755$	0	0
$756$	−3024.00	−0.145479
$757$	27634.0	1.32678	0.663392	−	0.748272i	$-0.269116\pi$
$757$	27634.0	1.32678	0.663392	+	0.748272i	$0.269116\pi$
$758$	−400.000	−0.0191671
$759$	18144.0	0.867702
$760$	0	0
$761$	−5370.00	−0.255798	−0.127899	−	0.991787i	$-0.540823\pi$
$761$	−5370.00	−0.255798	−0.127899	+	0.991787i	$0.540823\pi$
$762$	3120.00	0.148328
$763$	32648.0	1.54907
$764$	17760.0	0.841013
$765$	0	0
$766$	7920.00	0.373579
$767$	2652.00	0.124848
$768$	−768.000	−0.0360844
$769$	−11230.0	−0.526611	−0.263306	−	0.964712i	$-0.584813\pi$
$769$	−11230.0	−0.526611	−0.263306	+	0.964712i	$0.584813\pi$
$770$	0	0
$771$	−17082.0	−0.797916
$772$	18904.0	0.881308
$773$	10374.0	0.482700	0.241350	−	0.970438i	$-0.422410\pi$
$773$	10374.0	0.482700	0.241350	+	0.970438i	$0.422410\pi$
$774$	−8280.00	−0.384520
$775$	0	0
$776$	−12656.0	−0.585469
$777$	23352.0	1.07818
$778$	24900.0	1.14744
$779$	−16800.0	−0.772686
$780$	0	0
$781$	−38880.0	−1.78135
$782$	14112.0	0.645325
$783$	5670.00	0.258786
$784$	7056.00	0.321429
$785$	0	0
$786$	72.0000	0.00326737
$787$	39544.0	1.79109	0.895547	−	0.444966i	$-0.146784\pi$
$787$	39544.0	1.79109	0.895547	+	0.444966i	$0.146784\pi$
$788$	−1032.00	−0.0466542
$789$	−2880.00	−0.129950
$790$	0	0
$791$	−47544.0	−2.13713
$792$	2592.00	0.116291
$793$	−7982.00	−0.357439
$794$	−15140.0	−0.676698
$795$	0	0
$796$	12032.0	0.535757
$797$	12210.0	0.542660	0.271330	−	0.962486i	$-0.412536\pi$
$797$	12210.0	0.542660	0.271330	+	0.962486i	$0.412536\pi$
$798$	−18816.0	−0.834686
$799$	−11088.0	−0.490945
$800$	0	0
$801$	−7506.00	−0.331100
$802$	16020.0	0.705344
$803$	−33624.0	−1.47767
$804$	−3648.00	−0.160019
$805$	0	0
$806$	−1976.00	−0.0863544
$807$	7110.00	0.310141
$808$	8400.00	0.365731
$809$	13602.0	0.591126	0.295563	−	0.955323i	$-0.404493\pi$
$809$	13602.0	0.591126	0.295563	+	0.955323i	$0.404493\pi$
$810$	0	0
$811$	−24040.0	−1.04089	−0.520443	−	0.853896i	$-0.674233\pi$
$811$	−24040.0	−1.04089	−0.520443	+	0.853896i	$0.674233\pi$
$812$	−23520.0	−1.01649
$813$	1524.00	0.0657429
$814$	−20016.0	−0.861868
$815$	0	0
$816$	2016.00	0.0864879
$817$	−51520.0	−2.20619
$818$	6332.00	0.270652
$819$	−3276.00	−0.139771
$820$	0	0
$821$	−1878.00	−0.0798327	−0.0399164	−	0.999203i	$-0.512709\pi$
$821$	−1878.00	−0.0798327	−0.0399164	+	0.999203i	$0.512709\pi$
$822$	3708.00	0.157337
$823$	−17048.0	−0.722061	−0.361030	−	0.932554i	$-0.617575\pi$
$823$	−17048.0	−0.722061	−0.361030	+	0.932554i	$0.617575\pi$
$824$	12352.0	0.522212
$825$	0	0
$826$	11424.0	0.481225
$827$	−26652.0	−1.12065	−0.560327	−	0.828271i	$-0.689325\pi$
$827$	−26652.0	−1.12065	−0.560327	+	0.828271i	$0.689325\pi$
$828$	6048.00	0.253844
$829$	−8746.00	−0.366419	−0.183209	−	0.983074i	$-0.558649\pi$
$829$	−8746.00	−0.366419	−0.183209	+	0.983074i	$0.558649\pi$
$830$	0	0
$831$	−13494.0	−0.563299
$832$	−832.000	−0.0346688
$833$	−18522.0	−0.770407
$834$	3144.00	0.130537
$835$	0	0
$836$	16128.0	0.667223
$837$	2052.00	0.0847401
$838$	2376.00	0.0979446
$839$	−5400.00	−0.222203	−0.111102	−	0.993809i	$-0.535438\pi$
$839$	−5400.00	−0.222203	−0.111102	+	0.993809i	$0.535438\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	−28108.0	−1.15043
$843$	−21186.0	−0.865581
$844$	−18064.0	−0.736716
$845$	0	0
$846$	−4752.00	−0.193117
$847$	−980.000	−0.0397558
$848$	−9312.00	−0.377094
$849$	−12108.0	−0.489453
$850$	0	0
$851$	−46704.0	−1.88131
$852$	−12960.0	−0.521129
$853$	11338.0	0.455106	0.227553	−	0.973766i	$-0.426927\pi$
$853$	11338.0	0.455106	0.227553	+	0.973766i	$0.426927\pi$
$854$	−34384.0	−1.37775
$855$	0	0
$856$	480.000	0.0191660
$857$	47430.0	1.89052	0.945261	−	0.326314i	$-0.105807\pi$
$857$	47430.0	1.89052	0.945261	+	0.326314i	$0.105807\pi$
$858$	2808.00	0.111729
$859$	10292.0	0.408799	0.204400	−	0.978888i	$-0.434476\pi$
$859$	10292.0	0.408799	0.204400	+	0.978888i	$0.434476\pi$
$860$	0	0
$861$	−12600.0	−0.498730
$862$	−3408.00	−0.134660
$863$	−34392.0	−1.35657	−0.678283	−	0.734800i	$-0.737276\pi$
$863$	−34392.0	−1.35657	−0.678283	+	0.734800i	$0.737276\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	20836.0	0.817594
$867$	9447.00	0.370054
$868$	−8512.00	−0.332853
$869$	−4608.00	−0.179880
$870$	0	0
$871$	−3952.00	−0.153741
$872$	−9328.00	−0.362255
$873$	14238.0	0.551985
$874$	37632.0	1.45643
$875$	0	0
$876$	−11208.0	−0.432287
$877$	−2606.00	−0.100340	−0.0501701	−	0.998741i	$-0.515976\pi$
$877$	−2606.00	−0.100340	−0.0501701	+	0.998741i	$0.515976\pi$
$878$	1136.00	0.0436653
$879$	5022.00	0.192705
$880$	0	0
$881$	1290.00	0.0493317	0.0246658	−	0.999696i	$-0.492148\pi$
$881$	1290.00	0.0493317	0.0246658	+	0.999696i	$0.492148\pi$
$882$	−7938.00	−0.303046
$883$	−19460.0	−0.741655	−0.370827	−	0.928702i	$-0.620926\pi$
$883$	−19460.0	−0.741655	−0.370827	+	0.928702i	$0.620926\pi$
$884$	2184.00	0.0830949
$885$	0	0
$886$	−29160.0	−1.10570
$887$	−46728.0	−1.76885	−0.884427	−	0.466679i	$-0.845450\pi$
$887$	−46728.0	−1.76885	−0.884427	+	0.466679i	$0.845450\pi$
$888$	−6672.00	−0.252137
$889$	14560.0	0.549299
$890$	0	0
$891$	−2916.00	−0.109640
$892$	18064.0	0.678058
$893$	−29568.0	−1.10801
$894$	−4356.00	−0.162960
$895$	0	0
$896$	−3584.00	−0.133631
$897$	6552.00	0.243885
$898$	31956.0	1.18751
$899$	15960.0	0.592098
$900$	0	0
$901$	24444.0	0.903827
$902$	10800.0	0.398670
$903$	−38640.0	−1.42399
$904$	13584.0	0.499776
$905$	0	0
$906$	−15144.0	−0.555326
$907$	−32300.0	−1.18247	−0.591237	−	0.806498i	$-0.701360\pi$
$907$	−32300.0	−1.18247	−0.591237	+	0.806498i	$0.701360\pi$
$908$	−1488.00	−0.0543844
$909$	−9450.00	−0.344815
$910$	0	0
$911$	21432.0	0.779444	0.389722	−	0.920932i	$-0.372571\pi$
$911$	21432.0	0.779444	0.389722	+	0.920932i	$0.372571\pi$
$912$	5376.00	0.195194
$913$	12528.0	0.454125
$914$	−13964.0	−0.505348
$915$	0	0
$916$	−16552.0	−0.597045
$917$	336.000	0.0121000
$918$	−2268.00	−0.0815416
$919$	−35944.0	−1.29019	−0.645094	−	0.764103i	$-0.723182\pi$
$919$	−35944.0	−1.29019	−0.645094	+	0.764103i	$0.723182\pi$
$920$	0	0
$921$	−26904.0	−0.962559
$922$	−23892.0	−0.853407
$923$	−14040.0	−0.500685
$924$	12096.0	0.430659
$925$	0	0
$926$	−15656.0	−0.555603
$927$	−13896.0	−0.492346
$928$	6720.00	0.237710
$929$	27774.0	0.980878	0.490439	−	0.871476i	$-0.336837\pi$
$929$	27774.0	0.980878	0.490439	+	0.871476i	$0.336837\pi$
$930$	0	0
$931$	−49392.0	−1.73873
$932$	5784.00	0.203285
$933$	−2016.00	−0.0707405
$934$	4152.00	0.145458
$935$	0	0
$936$	936.000	0.0326860
$937$	−16106.0	−0.561537	−0.280768	−	0.959776i	$-0.590589\pi$
$937$	−16106.0	−0.561537	−0.280768	+	0.959776i	$0.590589\pi$
$938$	−17024.0	−0.592594
$939$	5838.00	0.202892
$940$	0	0
$941$	38802.0	1.34422	0.672109	−	0.740452i	$-0.265389\pi$
$941$	38802.0	1.34422	0.672109	+	0.740452i	$0.265389\pi$
$942$	14748.0	0.510102
$943$	25200.0	0.870228
$944$	−3264.00	−0.112536
$945$	0	0
$946$	33120.0	1.13829
$947$	21636.0	0.742424	0.371212	−	0.928548i	$-0.378942\pi$
$947$	21636.0	0.742424	0.371212	+	0.928548i	$0.378942\pi$
$948$	−1536.00	−0.0526234
$949$	−12142.0	−0.415328
$950$	0	0
$951$	1422.00	0.0484874
$952$	9408.00	0.320289
$953$	56742.0	1.92870	0.964351	−	0.264625i	$-0.0852482\pi$
$953$	56742.0	1.92870	0.964351	+	0.264625i	$0.0852482\pi$
$954$	10476.0	0.355527
$955$	0	0
$956$	−13824.0	−0.467678
$957$	−22680.0	−0.766082
$958$	−35520.0	−1.19791
$959$	17304.0	0.582665
$960$	0	0
$961$	−24015.0	−0.806116
$962$	−7228.00	−0.242245
$963$	−540.000	−0.0180698
$964$	21032.0	0.702692
$965$	0	0
$966$	28224.0	0.940054
$967$	13708.0	0.455863	0.227932	−	0.973677i	$-0.426804\pi$
$967$	13708.0	0.455863	0.227932	+	0.973677i	$0.426804\pi$
$968$	280.000	0.00929705
$969$	−14112.0	−0.467846
$970$	0	0
$971$	6924.00	0.228838	0.114419	−	0.993433i	$-0.463499\pi$
$971$	6924.00	0.228838	0.114419	+	0.993433i	$0.463499\pi$
$972$	−972.000	−0.0320750
$973$	14672.0	0.483415
$974$	6088.00	0.200279
$975$	0	0
$976$	9824.00	0.322191
$977$	13914.0	0.455628	0.227814	−	0.973705i	$-0.426842\pi$
$977$	13914.0	0.455628	0.227814	+	0.973705i	$0.426842\pi$
$978$	3552.00	0.116135
$979$	30024.0	0.980154
$980$	0	0
$981$	10494.0	0.341537
$982$	−5400.00	−0.175480
$983$	−10056.0	−0.326283	−0.163142	−	0.986603i	$-0.552163\pi$
$983$	−10056.0	−0.326283	−0.163142	+	0.986603i	$0.552163\pi$
$984$	3600.00	0.116630
$985$	0	0
$986$	−17640.0	−0.569749
$987$	−22176.0	−0.715167
$988$	5824.00	0.187537
$989$	77280.0	2.48469
$990$	0	0
$991$	−4792.00	−0.153605	−0.0768027	−	0.997046i	$-0.524471\pi$
$991$	−4792.00	−0.153605	−0.0768027	+	0.997046i	$0.524471\pi$
$992$	2432.00	0.0778388
$993$	20784.0	0.664210
$994$	−60480.0	−1.92989
$995$	0	0
$996$	4176.00	0.132853
$997$	27250.0	0.865613	0.432806	−	0.901487i	$-0.357523\pi$
$997$	27250.0	0.865613	0.432806	+	0.901487i	$0.357523\pi$
$998$	−31216.0	−0.990105
$999$	7506.00	0.237717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.4.a.b.1.1		1
5.4	even	2		390.4.a.k.1.1	✓	1
15.14	odd	2		1170.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.k.1.1	✓	1	5.4	even	2
1170.4.a.d.1.1		1	15.14	odd	2
1950.4.a.b.1.1		1	1.1	even	1	trivial

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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