LMFDB - Embedded newform 390.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,4,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.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:	$23.0107449022$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -28.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -28.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -36.0000 q^{11} +12.0000 q^{12} +13.0000 q^{13} -56.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +42.0000 q^{17} +18.0000 q^{18} -112.000 q^{19} -20.0000 q^{20} -84.0000 q^{21} -72.0000 q^{22} -168.000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} +27.0000 q^{27} -112.000 q^{28} -210.000 q^{29} -30.0000 q^{30} -76.0000 q^{31} +32.0000 q^{32} -108.000 q^{33} +84.0000 q^{34} +140.000 q^{35} +36.0000 q^{36} +278.000 q^{37} -224.000 q^{38} +39.0000 q^{39} -40.0000 q^{40} +150.000 q^{41} -168.000 q^{42} -460.000 q^{43} -144.000 q^{44} -45.0000 q^{45} -336.000 q^{46} -264.000 q^{47} +48.0000 q^{48} +441.000 q^{49} +50.0000 q^{50} +126.000 q^{51} +52.0000 q^{52} +582.000 q^{53} +54.0000 q^{54} +180.000 q^{55} -224.000 q^{56} -336.000 q^{57} -420.000 q^{58} -204.000 q^{59} -60.0000 q^{60} +614.000 q^{61} -152.000 q^{62} -252.000 q^{63} +64.0000 q^{64} -65.0000 q^{65} -216.000 q^{66} -304.000 q^{67} +168.000 q^{68} -504.000 q^{69} +280.000 q^{70} +1080.00 q^{71} +72.0000 q^{72} -934.000 q^{73} +556.000 q^{74} +75.0000 q^{75} -448.000 q^{76} +1008.00 q^{77} +78.0000 q^{78} +128.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +300.000 q^{82} +348.000 q^{83} -336.000 q^{84} -210.000 q^{85} -920.000 q^{86} -630.000 q^{87} -288.000 q^{88} -834.000 q^{89} -90.0000 q^{90} -364.000 q^{91} -672.000 q^{92} -228.000 q^{93} -528.000 q^{94} +560.000 q^{95} +96.0000 q^{96} -1582.00 q^{97} +882.000 q^{98} -324.000 q^{99} +O(q^{100})\)

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.00000	0.408248
$7$	−28.0000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−28.0000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−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
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	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	$0.736363\pi$
$20$	−20.0000	−0.223607
$21$	−84.0000	−0.872872
$22$	−72.0000	−0.697748
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	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	$0.734713\pi$
$30$	−30.0000	−0.182574
$31$	−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
$35$	140.000	0.676123
$36$	36.0000	0.166667
$37$	278.000	1.23521	0.617607	−	0.786487i	$-0.288102\pi$
$37$	278.000	1.23521	0.617607	+	0.786487i	$0.288102\pi$
$38$	−224.000	−0.956253
$39$	39.0000	0.160128
$40$	−40.0000	−0.158114
$41$	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	$-0.803642\pi$
$43$	−460.000	−1.63138	−0.815690	+	0.578489i	$0.803642\pi$
$44$	−144.000	−0.493382
$45$	−45.0000	−0.149071
$46$	−336.000	−1.07697
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	48.0000	0.144338
$49$	441.000	1.28571
$50$	50.0000	0.141421
$51$	126.000	0.345952
$52$	52.0000	0.138675
$53$	582.000	1.50837	0.754187	−	0.656659i	$-0.228031\pi$
$53$	582.000	1.50837	0.754187	+	0.656659i	$0.228031\pi$
$54$	54.0000	0.136083
$55$	180.000	0.441294
$56$	−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	$0.572262\pi$
$60$	−60.0000	−0.129099
$61$	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
$65$	−65.0000	−0.124035
$66$	−216.000	−0.402845
$67$	−304.000	−0.554321	−0.277161	−	0.960824i	$-0.589393\pi$
$67$	−304.000	−0.554321	−0.277161	+	0.960824i	$0.589393\pi$
$68$	168.000	0.299603
$69$	−504.000	−0.879340
$70$	280.000	0.478091
$71$	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	$-0.769342\pi$
$73$	−934.000	−1.49749	−0.748743	+	0.662861i	$0.769342\pi$
$74$	556.000	0.873428
$75$	75.0000	0.115470
$76$	−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	$0.470947\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	300.000	0.404018
$83$	348.000	0.460216	0.230108	−	0.973165i	$-0.426092\pi$
$83$	348.000	0.460216	0.230108	+	0.973165i	$0.426092\pi$
$84$	−336.000	−0.436436
$85$	−210.000	−0.267973
$86$	−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	$0.665437\pi$
$90$	−90.0000	−0.105409
$91$	−364.000	−0.419314
$92$	−672.000	−0.761531
$93$	−228.000	−0.254220
$94$	−528.000	−0.579352
$95$	560.000	0.604787
$96$	96.0000	0.102062
$97$	−1582.00	−1.65596	−0.827978	−	0.560760i	$-0.810509\pi$
$97$	−1582.00	−1.65596	−0.827978	+	0.560760i	$0.810509\pi$
$98$	882.000	0.909137
$99$	−324.000	−0.328921
$100$	100.000	0.100000
$101$	−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	$-0.235526\pi$
$103$	1544.00	1.47704	0.738519	+	0.674233i	$0.235526\pi$
$104$	104.000	0.0980581
$105$	420.000	0.390360
$106$	1164.00	1.06658
$107$	60.0000	0.0542095	0.0271048	−	0.999633i	$-0.491371\pi$
$107$	60.0000	0.0542095	0.0271048	+	0.999633i	$0.491371\pi$
$108$	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	$0.328792\pi$
$110$	360.000	0.312042
$111$	834.000	0.713151
$112$	−448.000	−0.377964
$113$	1698.00	1.41358	0.706789	−	0.707424i	$-0.250143\pi$
$113$	1698.00	1.41358	0.706789	+	0.707424i	$0.250143\pi$
$114$	−672.000	−0.552093
$115$	840.000	0.681134
$116$	−840.000	−0.672345
$117$	117.000	0.0924500
$118$	−408.000	−0.318300
$119$	−1176.00	−0.905914
$120$	−120.000	−0.0912871
$121$	−35.0000	−0.0262960
$122$	1228.00	0.911294
$123$	450.000	0.329879
$124$	−304.000	−0.220161
$125$	−125.000	−0.0894427
$126$	−504.000	−0.356348
$127$	−520.000	−0.363327	−0.181664	−	0.983361i	$-0.558148\pi$
$127$	−520.000	−0.363327	−0.181664	+	0.983361i	$0.558148\pi$
$128$	128.000	0.0883883
$129$	−1380.00	−0.941878
$130$	−130.000	−0.0877058
$131$	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
$135$	−135.000	−0.0860663
$136$	336.000	0.211851
$137$	−618.000	−0.385396	−0.192698	−	0.981258i	$-0.561724\pi$
$137$	−618.000	−0.385396	−0.192698	+	0.981258i	$0.561724\pi$
$138$	−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	$0.448891\pi$
$140$	560.000	0.338062
$141$	−792.000	−0.473039
$142$	2160.00	1.27650
$143$	−468.000	−0.273679
$144$	144.000	0.0833333
$145$	1050.00	0.601364
$146$	−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	$0.563959\pi$
$150$	150.000	0.0816497
$151$	−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
$155$	380.000	0.196918
$156$	156.000	0.0800641
$157$	−2458.00	−1.24949	−0.624744	−	0.780829i	$-0.714797\pi$
$157$	−2458.00	−1.24949	−0.624744	+	0.780829i	$0.714797\pi$
$158$	256.000	0.128900
$159$	1746.00	0.870860
$160$	−160.000	−0.0790569
$161$	4704.00	2.30265
$162$	162.000	0.0785674
$163$	−592.000	−0.284473	−0.142236	−	0.989833i	$-0.545429\pi$
$163$	−592.000	−0.284473	−0.142236	+	0.989833i	$0.545429\pi$
$164$	600.000	0.285684
$165$	540.000	0.254781
$166$	696.000	0.325422
$167$	1968.00	0.911907	0.455953	−	0.890004i	$-0.349298\pi$
$167$	1968.00	0.911907	0.455953	+	0.890004i	$0.349298\pi$
$168$	−672.000	−0.308607
$169$	169.000	0.0769231
$170$	−420.000	−0.189485
$171$	−1008.00	−0.450782
$172$	−1840.00	−0.815690
$173$	2214.00	0.972990	0.486495	−	0.873683i	$-0.338275\pi$
$173$	2214.00	0.972990	0.486495	+	0.873683i	$0.338275\pi$
$174$	−1260.00	−0.548968
$175$	−700.000	−0.302372
$176$	−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	$0.277324\pi$
$180$	−180.000	−0.0745356
$181$	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
$185$	−1390.00	−0.552404
$186$	−456.000	−0.179761
$187$	−1512.00	−0.591275
$188$	−1056.00	−0.409663
$189$	−756.000	−0.290957
$190$	1120.00	0.427649
$191$	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	$-0.843336\pi$
$193$	−4726.00	−1.76262	−0.881308	+	0.472542i	$0.843336\pi$
$194$	−3164.00	−1.17094
$195$	−195.000	−0.0716115
$196$	1764.00	0.642857
$197$	258.000	0.0933083	0.0466542	−	0.998911i	$-0.485144\pi$
$197$	258.000	0.0933083	0.0466542	+	0.998911i	$0.485144\pi$
$198$	−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	$0.320026\pi$
$200$	200.000	0.0707107
$201$	−912.000	−0.320037
$202$	−2100.00	−0.731463
$203$	5880.00	2.03298
$204$	504.000	0.172976
$205$	−750.000	−0.255523
$206$	3088.00	1.04442
$207$	−1512.00	−0.507687
$208$	208.000	0.0693375
$209$	4032.00	1.33445
$210$	840.000	0.276026
$211$	−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
$215$	2300.00	0.729575
$216$	216.000	0.0680414
$217$	2128.00	0.665705
$218$	2332.00	0.724509
$219$	−2802.00	−0.864574
$220$	720.000	0.220647
$221$	546.000	0.166190
$222$	1668.00	0.504274
$223$	−4516.00	−1.35612	−0.678058	−	0.735009i	$-0.737178\pi$
$223$	−4516.00	−1.35612	−0.678058	+	0.735009i	$0.737178\pi$
$224$	−896.000	−0.267261
$225$	225.000	0.0666667
$226$	3396.00	0.999551
$227$	372.000	0.108769	0.0543844	−	0.998520i	$-0.482680\pi$
$227$	372.000	0.108769	0.0543844	+	0.998520i	$0.482680\pi$
$228$	−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	$0.703659\pi$
$230$	1680.00	0.481634
$231$	3024.00	0.861319
$232$	−1680.00	−0.475420
$233$	−1446.00	−0.406569	−0.203285	−	0.979120i	$-0.565162\pi$
$233$	−1446.00	−0.406569	−0.203285	+	0.979120i	$0.565162\pi$
$234$	234.000	0.0653720
$235$	1320.00	0.366414
$236$	−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	$0.654909\pi$
$240$	−240.000	−0.0645497
$241$	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
$245$	−2205.00	−0.574989
$246$	900.000	0.233260
$247$	−1456.00	−0.375073
$248$	−608.000	−0.155678
$249$	1044.00	0.265706
$250$	−250.000	−0.0632456
$251$	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
$255$	−630.000	−0.154714
$256$	256.000	0.0625000
$257$	−5694.00	−1.38203	−0.691015	−	0.722840i	$-0.742836\pi$
$257$	−5694.00	−1.38203	−0.691015	+	0.722840i	$0.742836\pi$
$258$	−2760.00	−0.666008
$259$	−7784.00	−1.86747
$260$	−260.000	−0.0620174
$261$	−1890.00	−0.448230
$262$	24.0000	0.00565926
$263$	−960.000	−0.225080	−0.112540	−	0.993647i	$-0.535899\pi$
$263$	−960.000	−0.225080	−0.112540	+	0.993647i	$0.535899\pi$
$264$	−864.000	−0.201422
$265$	−2910.00	−0.674566
$266$	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	$0.586558\pi$
$270$	−270.000	−0.0608581
$271$	−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
$275$	−900.000	−0.197353
$276$	−2016.00	−0.439670
$277$	−4498.00	−0.975663	−0.487831	−	0.872938i	$-0.662212\pi$
$277$	−4498.00	−0.975663	−0.487831	+	0.872938i	$0.662212\pi$
$278$	1048.00	0.226097
$279$	−684.000	−0.146774
$280$	1120.00	0.239046
$281$	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	$-0.639332\pi$
$283$	−4036.00	−0.847757	−0.423879	+	0.905719i	$0.639332\pi$
$284$	4320.00	0.902623
$285$	1680.00	0.349174
$286$	−936.000	−0.193520
$287$	−4200.00	−0.863826
$288$	288.000	0.0589256
$289$	−3149.00	−0.640953
$290$	2100.00	0.425228
$291$	−4746.00	−0.956067
$292$	−3736.00	−0.748743
$293$	1674.00	0.333775	0.166888	−	0.985976i	$-0.446628\pi$
$293$	1674.00	0.333775	0.166888	+	0.985976i	$0.446628\pi$
$294$	2646.00	0.524891
$295$	1020.00	0.201311
$296$	2224.00	0.436714
$297$	−972.000	−0.189903
$298$	−1452.00	−0.282255
$299$	−2184.00	−0.422421
$300$	300.000	0.0577350
$301$	12880.0	2.46641
$302$	−5048.00	−0.961854
$303$	−3150.00	−0.597237
$304$	−1792.00	−0.338086
$305$	−3070.00	−0.576353
$306$	756.000	0.141234
$307$	−8968.00	−1.66720	−0.833601	−	0.552368i	$-0.813724\pi$
$307$	−8968.00	−1.66720	−0.833601	+	0.552368i	$0.813724\pi$
$308$	4032.00	0.745924
$309$	4632.00	0.852768
$310$	760.000	0.139242
$311$	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	$-0.443778\pi$
$313$	1946.00	0.351420	0.175710	+	0.984442i	$0.443778\pi$
$314$	−4916.00	−0.883522
$315$	1260.00	0.225374
$316$	512.000	0.0911464
$317$	474.000	0.0839826	0.0419913	−	0.999118i	$-0.486630\pi$
$317$	474.000	0.0839826	0.0419913	+	0.999118i	$0.486630\pi$
$318$	3492.00	0.615791
$319$	7560.00	1.32689
$320$	−320.000	−0.0559017
$321$	180.000	0.0312979
$322$	9408.00	1.62822
$323$	−4704.00	−0.810333
$324$	324.000	0.0555556
$325$	325.000	0.0554700
$326$	−1184.00	−0.201152
$327$	3498.00	0.591559
$328$	1200.00	0.202009
$329$	7392.00	1.23871
$330$	1080.00	0.180158
$331$	−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
$335$	1520.00	0.247900
$336$	−1344.00	−0.218218
$337$	2882.00	0.465853	0.232927	−	0.972494i	$-0.425170\pi$
$337$	2882.00	0.465853	0.232927	+	0.972494i	$0.425170\pi$
$338$	338.000	0.0543928
$339$	5094.00	0.816130
$340$	−840.000	−0.133986
$341$	2736.00	0.434495
$342$	−2016.00	−0.318751
$343$	−2744.00	−0.431959
$344$	−3680.00	−0.576780
$345$	2520.00	0.393253
$346$	4428.00	0.688008
$347$	−7548.00	−1.16772	−0.583859	−	0.811855i	$-0.698458\pi$
$347$	−7548.00	−1.16772	−0.583859	+	0.811855i	$0.698458\pi$
$348$	−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	$0.629131\pi$
$350$	−1400.00	−0.213809
$351$	351.000	0.0533761
$352$	−1152.00	−0.174437
$353$	−9138.00	−1.37781	−0.688905	−	0.724852i	$-0.741908\pi$
$353$	−9138.00	−1.37781	−0.688905	+	0.724852i	$0.741908\pi$
$354$	−1224.00	−0.183771
$355$	−5400.00	−0.807330
$356$	−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	$0.602860\pi$
$360$	−360.000	−0.0527046
$361$	5685.00	0.828838
$362$	1372.00	0.199201
$363$	−105.000	−0.0151820
$364$	−1456.00	−0.209657
$365$	4670.00	0.669696
$366$	3684.00	0.526136
$367$	−3976.00	−0.565519	−0.282760	−	0.959191i	$-0.591250\pi$
$367$	−3976.00	−0.565519	−0.282760	+	0.959191i	$0.591250\pi$
$368$	−2688.00	−0.380765
$369$	1350.00	0.190456
$370$	−2780.00	−0.390609
$371$	−16296.0	−2.28045
$372$	−912.000	−0.127110
$373$	−11314.0	−1.57055	−0.785277	−	0.619144i	$-0.787480\pi$
$373$	−11314.0	−1.57055	−0.785277	+	0.619144i	$0.787480\pi$
$374$	−3024.00	−0.418094
$375$	−375.000	−0.0516398
$376$	−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	$0.495686\pi$
$380$	2240.00	0.302394
$381$	−1560.00	−0.209767
$382$	8880.00	1.18937
$383$	3960.00	0.528320	0.264160	−	0.964479i	$-0.414905\pi$
$383$	3960.00	0.528320	0.264160	+	0.964479i	$0.414905\pi$
$384$	384.000	0.0510310
$385$	−5040.00	−0.667175
$386$	−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	$0.801274\pi$
$390$	−390.000	−0.0506370
$391$	−7056.00	−0.912627
$392$	3528.00	0.454569
$393$	36.0000	0.00462076
$394$	516.000	0.0659789
$395$	−640.000	−0.0815238
$396$	−1296.00	−0.164461
$397$	−7570.00	−0.956996	−0.478498	−	0.878089i	$-0.658819\pi$
$397$	−7570.00	−0.956996	−0.478498	+	0.878089i	$0.658819\pi$
$398$	6016.00	0.757675
$399$	9408.00	1.18042
$400$	400.000	0.0500000
$401$	−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
$405$	−405.000	−0.0496904
$406$	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	$0.561296\pi$
$410$	−1500.00	−0.180682
$411$	−1854.00	−0.222509
$412$	6176.00	0.738519
$413$	5712.00	0.680555
$414$	−3024.00	−0.358989
$415$	−1740.00	−0.205815
$416$	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	$0.522063\pi$
$420$	1680.00	0.195180
$421$	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
$425$	1050.00	0.119841
$426$	6480.00	0.736988
$427$	−17192.0	−1.94843
$428$	240.000	0.0271048
$429$	−1404.00	−0.158009
$430$	4600.00	0.515888
$431$	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	$-0.303784\pi$
$433$	10418.0	1.15625	0.578126	+	0.815947i	$0.303784\pi$
$434$	4256.00	0.470725
$435$	3150.00	0.347198
$436$	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	$0.509830\pi$
$440$	1440.00	0.156021
$441$	3969.00	0.428571
$442$	1092.00	0.117514
$443$	−14580.0	−1.56369	−0.781847	−	0.623470i	$-0.785722\pi$
$443$	−14580.0	−1.56369	−0.781847	+	0.623470i	$0.785722\pi$
$444$	3336.00	0.356576
$445$	4170.00	0.444218
$446$	−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	$0.817268\pi$
$450$	450.000	0.0471405
$451$	−5400.00	−0.563805
$452$	6792.00	0.706789
$453$	−7572.00	−0.785350
$454$	744.000	0.0769111
$455$	1820.00	0.187523
$456$	−2688.00	−0.276046
$457$	−6982.00	−0.714670	−0.357335	−	0.933976i	$-0.616315\pi$
$457$	−6982.00	−0.714670	−0.357335	+	0.933976i	$0.616315\pi$
$458$	−8276.00	−0.844350
$459$	1134.00	0.115317
$460$	3360.00	0.340567
$461$	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	$-0.628518\pi$
$463$	−7828.00	−0.785741	−0.392870	+	0.919594i	$0.628518\pi$
$464$	−3360.00	−0.336173
$465$	1140.00	0.113691
$466$	−2892.00	−0.287488
$467$	2076.00	0.205708	0.102854	−	0.994696i	$-0.467202\pi$
$467$	2076.00	0.205708	0.102854	+	0.994696i	$0.467202\pi$
$468$	468.000	0.0462250
$469$	8512.00	0.838055
$470$	2640.00	0.259094
$471$	−7374.00	−0.721393
$472$	−1632.00	−0.159150
$473$	16560.0	1.60979
$474$	768.000	0.0744207
$475$	−2800.00	−0.270469
$476$	−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	$0.178376\pi$
$480$	−480.000	−0.0456435
$481$	3614.00	0.342587
$482$	10516.0	0.993757
$483$	14112.0	1.32944
$484$	−140.000	−0.0131480
$485$	7910.00	0.740566
$486$	486.000	0.0453609
$487$	3044.00	0.283238	0.141619	−	0.989921i	$-0.454769\pi$
$487$	3044.00	0.283238	0.141619	+	0.989921i	$0.454769\pi$
$488$	4912.00	0.455647
$489$	−1776.00	−0.164240
$490$	−4410.00	−0.406579
$491$	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
$495$	1620.00	0.147098
$496$	−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	$0.253134\pi$
$500$	−500.000	−0.0447214
$501$	5904.00	0.526490
$502$	14040.0	1.24828
$503$	−7152.00	−0.633980	−0.316990	−	0.948429i	$-0.602672\pi$
$503$	−7152.00	−0.633980	−0.316990	+	0.948429i	$0.602672\pi$
$504$	−2016.00	−0.178174
$505$	5250.00	0.462618
$506$	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	$0.120011\pi$
$510$	−1260.00	−0.109399
$511$	26152.0	2.26399
$512$	512.000	0.0441942
$513$	−3024.00	−0.260259
$514$	−11388.0	−0.977243
$515$	−7720.00	−0.660551
$516$	−5520.00	−0.470939
$517$	9504.00	0.808482
$518$	−15568.0	−1.32050
$519$	6642.00	0.561756
$520$	−520.000	−0.0438529
$521$	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	$-0.611543\pi$
$523$	−8212.00	−0.686588	−0.343294	+	0.939228i	$0.611543\pi$
$524$	48.0000	0.00400170
$525$	−2100.00	−0.174574
$526$	−1920.00	−0.159156
$527$	−3192.00	−0.263844
$528$	−1728.00	−0.142427
$529$	16057.0	1.31972
$530$	−5820.00	−0.476990
$531$	−1836.00	−0.150048
$532$	12544.0	1.02228
$533$	1950.00	0.158469
$534$	−5004.00	−0.405514
$535$	−300.000	−0.0242432
$536$	−2432.00	−0.195982
$537$	9252.00	0.743488
$538$	−4740.00	−0.379844
$539$	−15876.0	−1.26870
$540$	−540.000	−0.0430331
$541$	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
$545$	−5830.00	−0.458220
$546$	−2184.00	−0.171184
$547$	16292.0	1.27348	0.636742	−	0.771077i	$-0.280282\pi$
$547$	16292.0	1.27348	0.636742	+	0.771077i	$0.280282\pi$
$548$	−2472.00	−0.192698
$549$	5526.00	0.429588
$550$	−1800.00	−0.139550
$551$	23520.0	1.81849
$552$	−4032.00	−0.310894
$553$	−3584.00	−0.275601
$554$	−8996.00	−0.689898
$555$	−4170.00	−0.318931
$556$	2096.00	0.159874
$557$	−10686.0	−0.812891	−0.406446	−	0.913675i	$-0.633232\pi$
$557$	−10686.0	−0.812891	−0.406446	+	0.913675i	$0.633232\pi$
$558$	−1368.00	−0.103785
$559$	−5980.00	−0.452463
$560$	2240.00	0.169031
$561$	−4536.00	−0.341373
$562$	14124.0	1.06012
$563$	19572.0	1.46512	0.732559	−	0.680704i	$-0.238326\pi$
$563$	19572.0	1.46512	0.732559	+	0.680704i	$0.238326\pi$
$564$	−3168.00	−0.236519
$565$	−8490.00	−0.632172
$566$	−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	$0.557512\pi$
$570$	3360.00	0.246903
$571$	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
$575$	−4200.00	−0.304612
$576$	576.000	0.0416667
$577$	26474.0	1.91010	0.955049	−	0.296447i	$-0.0958017\pi$
$577$	26474.0	1.91010	0.955049	+	0.296447i	$0.0958017\pi$
$578$	−6298.00	−0.453222
$579$	−14178.0	−1.01765
$580$	4200.00	0.300682
$581$	−9744.00	−0.695782
$582$	−9492.00	−0.676041
$583$	−20952.0	−1.48841
$584$	−7472.00	−0.529441
$585$	−585.000	−0.0413449
$586$	3348.00	0.236015
$587$	4884.00	0.343414	0.171707	−	0.985148i	$-0.445072\pi$
$587$	4884.00	0.343414	0.171707	+	0.985148i	$0.445072\pi$
$588$	5292.00	0.371154
$589$	8512.00	0.595468
$590$	2040.00	0.142348
$591$	774.000	0.0538716
$592$	4448.00	0.308804
$593$	5862.00	0.405942	0.202971	−	0.979185i	$-0.434940\pi$
$593$	5862.00	0.405942	0.202971	+	0.979185i	$0.434940\pi$
$594$	−1944.00	−0.134282
$595$	5880.00	0.405137
$596$	−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	$0.564803\pi$
$600$	600.000	0.0408248
$601$	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
$605$	175.000	0.0117599
$606$	−6300.00	−0.422310
$607$	7784.00	0.520499	0.260249	−	0.965541i	$-0.416195\pi$
$607$	7784.00	0.520499	0.260249	+	0.965541i	$0.416195\pi$
$608$	−3584.00	−0.239063
$609$	17640.0	1.17374
$610$	−6140.00	−0.407543
$611$	−3432.00	−0.227240
$612$	1512.00	0.0998676
$613$	−21994.0	−1.44915	−0.724575	−	0.689196i	$-0.757964\pi$
$613$	−21994.0	−1.44915	−0.724575	+	0.689196i	$0.757964\pi$
$614$	−17936.0	−1.17889
$615$	−2250.00	−0.147526
$616$	8064.00	0.527448
$617$	−9546.00	−0.622865	−0.311432	−	0.950268i	$-0.600809\pi$
$617$	−9546.00	−0.622865	−0.311432	+	0.950268i	$0.600809\pi$
$618$	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	$0.705963\pi$
$620$	1520.00	0.0984591
$621$	−4536.00	−0.293113
$622$	1344.00	0.0866391
$623$	23352.0	1.50173
$624$	624.000	0.0400320
$625$	625.000	0.0400000
$626$	3892.00	0.248491
$627$	12096.0	0.770443
$628$	−9832.00	−0.624744
$629$	11676.0	0.740147
$630$	2520.00	0.159364
$631$	−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
$635$	2600.00	0.162485
$636$	6984.00	0.435430
$637$	5733.00	0.356593
$638$	15120.0	0.938255
$639$	9720.00	0.601748
$640$	−640.000	−0.0395285
$641$	−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	$-0.307270\pi$
$643$	18560.0	1.13831	0.569156	+	0.822229i	$0.307270\pi$
$644$	18816.0	1.15133
$645$	6900.00	0.421221
$646$	−9408.00	−0.572992
$647$	−27072.0	−1.64499	−0.822496	−	0.568771i	$-0.807419\pi$
$647$	−27072.0	−1.64499	−0.822496	+	0.568771i	$0.807419\pi$
$648$	648.000	0.0392837
$649$	7344.00	0.444187
$650$	650.000	0.0392232
$651$	6384.00	0.384345
$652$	−2368.00	−0.142236
$653$	24150.0	1.44726	0.723631	−	0.690187i	$-0.242472\pi$
$653$	24150.0	1.44726	0.723631	+	0.690187i	$0.242472\pi$
$654$	6996.00	0.418295
$655$	−60.0000	−0.00357923
$656$	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	$0.609271\pi$
$660$	2160.00	0.127391
$661$	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
$665$	−15680.0	−0.914352
$666$	5004.00	0.291143
$667$	35280.0	2.04805
$668$	7872.00	0.455953
$669$	−13548.0	−0.782954
$670$	3040.00	0.175292
$671$	−22104.0	−1.27171
$672$	−2688.00	−0.154303
$673$	−7918.00	−0.453516	−0.226758	−	0.973951i	$-0.572813\pi$
$673$	−7918.00	−0.453516	−0.226758	+	0.973951i	$0.572813\pi$
$674$	5764.00	0.329408
$675$	675.000	0.0384900
$676$	676.000	0.0384615
$677$	−1578.00	−0.0895827	−0.0447913	−	0.998996i	$-0.514262\pi$
$677$	−1578.00	−0.0895827	−0.0447913	+	0.998996i	$0.514262\pi$
$678$	10188.0	0.577091
$679$	44296.0	2.50357
$680$	−1680.00	−0.0947427
$681$	1116.00	0.0627977
$682$	5472.00	0.307234
$683$	−8580.00	−0.480680	−0.240340	−	0.970689i	$-0.577259\pi$
$683$	−8580.00	−0.480680	−0.240340	+	0.970689i	$0.577259\pi$
$684$	−4032.00	−0.225391
$685$	3090.00	0.172354
$686$	−5488.00	−0.305441
$687$	−12414.0	−0.689409
$688$	−7360.00	−0.407845
$689$	7566.00	0.418348
$690$	5040.00	0.278072
$691$	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
$695$	−2620.00	−0.142996
$696$	−5040.00	−0.274484
$697$	6300.00	0.342367
$698$	−10292.0	−0.558106
$699$	−4338.00	−0.234733
$700$	−2800.00	−0.151186
$701$	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
$705$	3960.00	0.211549
$706$	−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	$0.430332\pi$
$710$	−10800.0	−0.570869
$711$	1152.00	0.0607642
$712$	−6672.00	−0.351185
$713$	12768.0	0.670639
$714$	−7056.00	−0.369838
$715$	2340.00	0.122393
$716$	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	$0.595952\pi$
$720$	−720.000	−0.0372678
$721$	−43232.0	−2.23307
$722$	11370.0	0.586077
$723$	15774.0	0.811399
$724$	2744.00	0.140856
$725$	−5250.00	−0.268938
$726$	−210.000	−0.0107353
$727$	−10864.0	−0.554228	−0.277114	−	0.960837i	$-0.589378\pi$
$727$	−10864.0	−0.554228	−0.277114	+	0.960837i	$0.589378\pi$
$728$	−2912.00	−0.148250
$729$	729.000	0.0370370
$730$	9340.00	0.473546
$731$	−19320.0	−0.977532
$732$	7368.00	0.372034
$733$	31286.0	1.57650	0.788250	−	0.615355i	$-0.210987\pi$
$733$	31286.0	1.57650	0.788250	+	0.615355i	$0.210987\pi$
$734$	−7952.00	−0.399882
$735$	−6615.00	−0.331970
$736$	−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	$0.336775\pi$
$740$	−5560.00	−0.276202
$741$	−4368.00	−0.216549
$742$	−32592.0	−1.61252
$743$	−13608.0	−0.671910	−0.335955	−	0.941878i	$-0.609059\pi$
$743$	−13608.0	−0.671910	−0.335955	+	0.941878i	$0.609059\pi$
$744$	−1824.00	−0.0898805
$745$	3630.00	0.178514
$746$	−22628.0	−1.11055
$747$	3132.00	0.153405
$748$	−6048.00	−0.295637
$749$	−1680.00	−0.0819571
$750$	−750.000	−0.0365148
$751$	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
$755$	12620.0	0.608330
$756$	−3024.00	−0.145479
$757$	−27634.0	−1.32678	−0.663392	−	0.748272i	$-0.730884\pi$
$757$	−27634.0	−1.32678	−0.663392	+	0.748272i	$0.730884\pi$
$758$	400.000	0.0191671
$759$	18144.0	0.867702
$760$	4480.00	0.213825
$761$	−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
$765$	−1890.00	−0.0893243
$766$	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	$0.584813\pi$
$770$	−10080.0	−0.471764
$771$	−17082.0	−0.797916
$772$	−18904.0	−0.881308
$773$	−10374.0	−0.482700	−0.241350	−	0.970438i	$-0.577590\pi$
$773$	−10374.0	−0.482700	−0.241350	+	0.970438i	$0.577590\pi$
$774$	−8280.00	−0.384520
$775$	−1900.00	−0.0880645
$776$	−12656.0	−0.585469
$777$	−23352.0	−1.07818
$778$	−24900.0	−1.14744
$779$	−16800.0	−0.772686
$780$	−780.000	−0.0358057
$781$	−38880.0	−1.78135
$782$	−14112.0	−0.645325
$783$	−5670.00	−0.258786
$784$	7056.00	0.321429
$785$	12290.0	0.558788
$786$	72.0000	0.00326737
$787$	−39544.0	−1.79109	−0.895547	−	0.444966i	$-0.853216\pi$
$787$	−39544.0	−1.79109	−0.895547	+	0.444966i	$0.853216\pi$
$788$	1032.00	0.0466542
$789$	−2880.00	−0.129950
$790$	−1280.00	−0.0576460
$791$	−47544.0	−2.13713
$792$	−2592.00	−0.116291
$793$	7982.00	0.357439
$794$	−15140.0	−0.676698
$795$	−8730.00	−0.389461
$796$	12032.0	0.535757
$797$	−12210.0	−0.542660	−0.271330	−	0.962486i	$-0.587464\pi$
$797$	−12210.0	−0.542660	−0.271330	+	0.962486i	$0.587464\pi$
$798$	18816.0	0.834686
$799$	−11088.0	−0.490945
$800$	800.000	0.0353553
$801$	−7506.00	−0.331100
$802$	−16020.0	−0.705344
$803$	33624.0	1.47767
$804$	−3648.00	−0.160019
$805$	−23520.0	−1.02978
$806$	−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	$0.404493\pi$
$810$	−810.000	−0.0351364
$811$	−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
$815$	2960.00	0.127220
$816$	2016.00	0.0864879
$817$	51520.0	2.20619
$818$	−6332.00	−0.270652
$819$	−3276.00	−0.139771
$820$	−3000.00	−0.127762
$821$	−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	$-0.382425\pi$
$823$	17048.0	0.722061	0.361030	+	0.932554i	$0.382425\pi$
$824$	12352.0	0.522212
$825$	−2700.00	−0.113942
$826$	11424.0	0.481225
$827$	26652.0	1.12065	0.560327	−	0.828271i	$-0.310675\pi$
$827$	26652.0	1.12065	0.560327	+	0.828271i	$0.310675\pi$
$828$	−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	$0.558649\pi$
$830$	−3480.00	−0.145533
$831$	−13494.0	−0.563299
$832$	832.000	0.0346688
$833$	18522.0	0.770407
$834$	3144.00	0.130537
$835$	−9840.00	−0.407817
$836$	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	$0.535438\pi$
$840$	3360.00	0.138013
$841$	19711.0	0.808192
$842$	28108.0	1.15043
$843$	21186.0	0.865581
$844$	−18064.0	−0.736716
$845$	−845.000	−0.0344010
$846$	−4752.00	−0.193117
$847$	980.000	0.0397558
$848$	9312.00	0.377094
$849$	−12108.0	−0.489453
$850$	2100.00	0.0847405
$851$	−46704.0	−1.88131
$852$	12960.0	0.521129
$853$	−11338.0	−0.455106	−0.227553	−	0.973766i	$-0.573073\pi$
$853$	−11338.0	−0.455106	−0.227553	+	0.973766i	$0.573073\pi$
$854$	−34384.0	−1.37775
$855$	5040.00	0.201596
$856$	480.000	0.0191660
$857$	−47430.0	−1.89052	−0.945261	−	0.326314i	$-0.894193\pi$
$857$	−47430.0	−1.89052	−0.945261	+	0.326314i	$0.894193\pi$
$858$	−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	$0.434476\pi$
$860$	9200.00	0.364788
$861$	−12600.0	−0.498730
$862$	3408.00	0.134660
$863$	34392.0	1.35657	0.678283	−	0.734800i	$-0.262724\pi$
$863$	34392.0	1.35657	0.678283	+	0.734800i	$0.262724\pi$
$864$	864.000	0.0340207
$865$	−11070.0	−0.435134
$866$	20836.0	0.817594
$867$	−9447.00	−0.370054
$868$	8512.00	0.332853
$869$	−4608.00	−0.179880
$870$	6300.00	0.245506
$871$	−3952.00	−0.153741
$872$	9328.00	0.362255
$873$	−14238.0	−0.551985
$874$	37632.0	1.45643
$875$	3500.00	0.135225
$876$	−11208.0	−0.432287
$877$	2606.00	0.100340	0.0501701	−	0.998741i	$-0.484024\pi$
$877$	2606.00	0.100340	0.0501701	+	0.998741i	$0.484024\pi$
$878$	−1136.00	−0.0436653
$879$	5022.00	0.192705
$880$	2880.00	0.110324
$881$	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	$-0.379074\pi$
$883$	19460.0	0.741655	0.370827	+	0.928702i	$0.379074\pi$
$884$	2184.00	0.0830949
$885$	3060.00	0.116227
$886$	−29160.0	−1.10570
$887$	46728.0	1.76885	0.884427	−	0.466679i	$-0.154550\pi$
$887$	46728.0	1.76885	0.884427	+	0.466679i	$0.154550\pi$
$888$	6672.00	0.252137
$889$	14560.0	0.549299
$890$	8340.00	0.314109
$891$	−2916.00	−0.109640
$892$	−18064.0	−0.678058
$893$	29568.0	1.10801
$894$	−4356.00	−0.162960
$895$	−15420.0	−0.575904
$896$	−3584.00	−0.133631
$897$	−6552.00	−0.243885
$898$	−31956.0	−1.18751
$899$	15960.0	0.592098
$900$	900.000	0.0333333
$901$	24444.0	0.903827
$902$	−10800.0	−0.398670
$903$	38640.0	1.42399
$904$	13584.0	0.499776
$905$	−3430.00	−0.125986
$906$	−15144.0	−0.555326
$907$	32300.0	1.18247	0.591237	−	0.806498i	$-0.298640\pi$
$907$	32300.0	1.18247	0.591237	+	0.806498i	$0.298640\pi$
$908$	1488.00	0.0543844
$909$	−9450.00	−0.344815
$910$	3640.00	0.132599
$911$	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
$915$	−9210.00	−0.332758
$916$	−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	$0.723182\pi$
$920$	6720.00	0.240817
$921$	−26904.0	−0.962559
$922$	23892.0	0.853407
$923$	14040.0	0.500685
$924$	12096.0	0.430659
$925$	6950.00	0.247043
$926$	−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	$0.336837\pi$
$930$	2280.00	0.0803916
$931$	−49392.0	−1.73873
$932$	−5784.00	−0.203285
$933$	2016.00	0.0707405
$934$	4152.00	0.145458
$935$	7560.00	0.264426
$936$	936.000	0.0326860
$937$	16106.0	0.561537	0.280768	−	0.959776i	$-0.409411\pi$
$937$	16106.0	0.561537	0.280768	+	0.959776i	$0.409411\pi$
$938$	17024.0	0.592594
$939$	5838.00	0.202892
$940$	5280.00	0.183207
$941$	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
$945$	3780.00	0.130120
$946$	33120.0	1.13829
$947$	−21636.0	−0.742424	−0.371212	−	0.928548i	$-0.621058\pi$
$947$	−21636.0	−0.742424	−0.371212	+	0.928548i	$0.621058\pi$
$948$	1536.00	0.0526234
$949$	−12142.0	−0.415328
$950$	−5600.00	−0.191251
$951$	1422.00	0.0484874
$952$	−9408.00	−0.320289
$953$	−56742.0	−1.92870	−0.964351	−	0.264625i	$-0.914752\pi$
$953$	−56742.0	−1.92870	−0.964351	+	0.264625i	$0.914752\pi$
$954$	10476.0	0.355527
$955$	−22200.0	−0.752225
$956$	−13824.0	−0.467678
$957$	22680.0	0.766082
$958$	35520.0	1.19791
$959$	17304.0	0.582665
$960$	−960.000	−0.0322749
$961$	−24015.0	−0.806116
$962$	7228.00	0.242245
$963$	540.000	0.0180698
$964$	21032.0	0.702692
$965$	23630.0	0.788266
$966$	28224.0	0.940054
$967$	−13708.0	−0.455863	−0.227932	−	0.973677i	$-0.573196\pi$
$967$	−13708.0	−0.455863	−0.227932	+	0.973677i	$0.573196\pi$
$968$	−280.000	−0.00929705
$969$	−14112.0	−0.467846
$970$	15820.0	0.523659
$971$	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
$975$	975.000	0.0320256
$976$	9824.00	0.322191
$977$	−13914.0	−0.455628	−0.227814	−	0.973705i	$-0.573158\pi$
$977$	−13914.0	−0.455628	−0.227814	+	0.973705i	$0.573158\pi$
$978$	−3552.00	−0.116135
$979$	30024.0	0.980154
$980$	−8820.00	−0.287494
$981$	10494.0	0.341537
$982$	5400.00	0.175480
$983$	10056.0	0.326283	0.163142	−	0.986603i	$-0.447837\pi$
$983$	10056.0	0.326283	0.163142	+	0.986603i	$0.447837\pi$
$984$	3600.00	0.116630
$985$	−1290.00	−0.0417287
$986$	−17640.0	−0.569749
$987$	22176.0	0.715167
$988$	−5824.00	−0.187537
$989$	77280.0	2.48469
$990$	3240.00	0.104014
$991$	−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
$995$	−15040.0	−0.479196
$996$	4176.00	0.132853
$997$	−27250.0	−0.865613	−0.432806	−	0.901487i	$-0.642477\pi$
$997$	−27250.0	−0.865613	−0.432806	+	0.901487i	$0.642477\pi$
$998$	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	390.4.a.k.1.1	✓	1
3.2	odd	2		1170.4.a.d.1.1		1
5.4	even	2		1950.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.k.1.1	✓	1	1.1	even	1	trivial
1170.4.a.d.1.1		1	3.2	odd	2
1950.4.a.b.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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