LMFDB - Embedded newform 2299.4.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,4,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2299.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,1,21,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.645391103$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ of defining polynomial
Character	$\chi$	$=$	2299.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.96257 q^{2} +6.71610 q^{3} +7.70200 q^{4} +18.1342 q^{5} +26.6130 q^{6} +25.8362 q^{7} -1.18085 q^{8} +18.1060 q^{9} +71.8581 q^{10} +51.7274 q^{12} +4.56640 q^{13} +102.378 q^{14} +121.791 q^{15} -66.2952 q^{16} +62.5850 q^{17} +71.7464 q^{18} +19.0000 q^{19} +139.670 q^{20} +173.518 q^{21} -52.7502 q^{23} -7.93070 q^{24} +203.849 q^{25} +18.0947 q^{26} -59.7330 q^{27} +198.990 q^{28} -171.620 q^{29} +482.606 q^{30} +168.749 q^{31} -253.253 q^{32} +247.998 q^{34} +468.519 q^{35} +139.452 q^{36} -147.534 q^{37} +75.2889 q^{38} +30.6684 q^{39} -21.4138 q^{40} -308.774 q^{41} +687.580 q^{42} +448.950 q^{43} +328.338 q^{45} -209.027 q^{46} +113.335 q^{47} -445.245 q^{48} +324.509 q^{49} +807.768 q^{50} +420.327 q^{51} +35.1704 q^{52} +155.402 q^{53} -236.696 q^{54} -30.5086 q^{56} +127.606 q^{57} -680.059 q^{58} +182.347 q^{59} +938.035 q^{60} -404.080 q^{61} +668.681 q^{62} +467.790 q^{63} -473.172 q^{64} +82.8080 q^{65} -106.400 q^{67} +482.030 q^{68} -354.276 q^{69} +1856.54 q^{70} +472.079 q^{71} -21.3805 q^{72} -843.821 q^{73} -584.616 q^{74} +1369.07 q^{75} +146.338 q^{76} +121.526 q^{78} +591.036 q^{79} -1202.21 q^{80} -890.035 q^{81} -1223.54 q^{82} -290.388 q^{83} +1336.44 q^{84} +1134.93 q^{85} +1779.00 q^{86} -1152.62 q^{87} -964.896 q^{89} +1301.06 q^{90} +117.978 q^{91} -406.282 q^{92} +1133.34 q^{93} +449.099 q^{94} +344.550 q^{95} -1700.87 q^{96} -219.495 q^{97} +1285.89 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 21 q^{4} + 14 q^{5} + 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 88 q^{10} - 115 q^{12} - 65 q^{13} + 37 q^{14} + 140 q^{15} + 33 q^{16} - 29 q^{17} - 138 q^{18} + 57 q^{19} + 100 q^{20}+ \cdots + 2450 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 21 * q^4 + 14 * q^5 + 65 * q^6 + 35 * q^7 - 27 * q^8 + 48 * q^9 + 88 * q^10 - 115 * q^12 - 65 * q^13 + 37 * q^14 + 140 * q^15 + 33 * q^16 - 29 * q^17 - 138 * q^18 + 57 * q^19 + 100 * q^20 + 25 * q^21 - 101 * q^23 + 489 * q^24 - 37 * q^25 + 299 * q^26 - 377 * q^27 + 493 * q^28 - 377 * q^29 + 376 * q^30 - 140 * q^31 - 579 * q^32 + 329 * q^34 + 438 * q^35 + 1074 * q^36 - 290 * q^37 - 57 * q^38 + 371 * q^39 + 72 * q^40 - 956 * q^41 + 1385 * q^42 + 570 * q^43 + 230 * q^45 + 239 * q^46 + 66 * q^47 - 1411 * q^48 - 98 * q^49 + 1639 * q^50 - 187 * q^51 - 697 * q^52 + 817 * q^53 + 947 * q^54 - 915 * q^56 + 19 * q^57 + 647 * q^58 + 265 * q^59 + 1360 * q^60 - 988 * q^61 + 1436 * q^62 + 1304 * q^63 - 159 * q^64 + 240 * q^65 - 207 * q^67 + 1529 * q^68 + 807 * q^69 + 2040 * q^70 + 846 * q^71 - 2826 * q^72 - 627 * q^73 + 770 * q^74 + 2031 * q^75 + 399 * q^76 - 1817 * q^78 - 382 * q^79 - 1472 * q^80 - 501 * q^81 + 872 * q^82 + 766 * q^83 - 907 * q^84 + 1242 * q^85 + 2592 * q^86 + 1671 * q^87 - 172 * q^89 + 1888 * q^90 - 457 * q^91 - 2677 * q^92 + 892 * q^93 + 96 * q^94 + 266 * q^95 - 863 * q^96 - 2450 * q^97 + 2450 * q^98

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 $

$2$	3.96257	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$2$	3.96257	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$3$	6.71610	1.29251	0.646257	−	0.763120i	$-0.276333\pi$
$3$	6.71610	1.29251	0.646257	+	0.763120i	$0.276333\pi$
$4$	7.70200	0.962750
$5$	18.1342	1.62197	0.810986	−	0.585065i	$-0.198931\pi$
$5$	18.1342	1.62197	0.810986	+	0.585065i	$0.198931\pi$
$6$	26.6130	1.81079
$7$	25.8362	1.39502	0.697512	−	0.716573i	$-0.254290\pi$
$7$	25.8362	1.39502	0.697512	+	0.716573i	$0.254290\pi$
$8$	−1.18085	−0.0521866
$9$	18.1060	0.670593
$10$	71.8581	2.27235
$11$	0	0
$11$	0	0
$12$	51.7274	1.24437
$13$	4.56640	0.0974224	0.0487112	−	0.998813i	$-0.484489\pi$
$13$	4.56640	0.0974224	0.0487112	+	0.998813i	$0.484489\pi$
$14$	102.378	1.95440
$15$	121.791	2.09642
$16$	−66.2952	−1.03586
$17$	62.5850	0.892888	0.446444	−	0.894812i	$-0.352690\pi$
$17$	62.5850	0.892888	0.446444	+	0.894812i	$0.352690\pi$
$18$	71.7464	0.939488
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	139.670	1.56155
$21$	173.518	1.80309
$22$	0	0
$23$	−52.7502	−0.478225	−0.239113	−	0.970992i	$-0.576856\pi$
$23$	−52.7502	−0.478225	−0.239113	+	0.970992i	$0.576856\pi$
$24$	−7.93070	−0.0674520
$25$	203.849	1.63079
$26$	18.0947	0.136487
$27$	−59.7330	−0.425764
$28$	198.990	1.34306
$29$	−171.620	−1.09894	−0.549468	−	0.835515i	$-0.685169\pi$
$29$	−171.620	−1.09894	−0.549468	+	0.835515i	$0.685169\pi$
$30$	482.606	2.93705
$31$	168.749	0.977685	0.488842	−	0.872372i	$-0.337419\pi$
$31$	168.749	0.977685	0.488842	+	0.872372i	$0.337419\pi$
$32$	−253.253	−1.39904
$33$	0	0
$34$	247.998	1.25092
$35$	468.519	2.26269
$36$	139.452	0.645613
$37$	−147.534	−0.655528	−0.327764	−	0.944760i	$-0.606295\pi$
$37$	−147.534	−0.655528	−0.327764	+	0.944760i	$0.606295\pi$
$38$	75.2889	0.321407
$39$	30.6684	0.125920
$40$	−21.4138	−0.0846453
$41$	−308.774	−1.17616	−0.588078	−	0.808804i	$-0.700115\pi$
$41$	−308.774	−1.17616	−0.588078	+	0.808804i	$0.700115\pi$
$42$	687.580	2.52609
$43$	448.950	1.59219	0.796096	−	0.605170i	$-0.206895\pi$
$43$	448.950	1.59219	0.796096	+	0.605170i	$0.206895\pi$
$44$	0	0
$45$	328.338	1.08768
$46$	−209.027	−0.669985
$47$	113.335	0.351737	0.175868	−	0.984414i	$-0.443727\pi$
$47$	113.335	0.351737	0.175868	+	0.984414i	$0.443727\pi$
$48$	−445.245	−1.33887
$49$	324.509	0.946091
$50$	807.768	2.28471
$51$	420.327	1.15407
$52$	35.1704	0.0937934
$53$	155.402	0.402758	0.201379	−	0.979513i	$-0.435458\pi$
$53$	155.402	0.402758	0.201379	+	0.979513i	$0.435458\pi$
$54$	−236.696	−0.596487
$55$	0	0
$56$	−30.5086	−0.0728016
$57$	127.606	0.296523
$58$	−680.059	−1.53959
$59$	182.347	0.402365	0.201183	−	0.979554i	$-0.435522\pi$
$59$	182.347	0.402365	0.201183	+	0.979554i	$0.435522\pi$
$60$	938.035	2.01833
$61$	−404.080	−0.848149	−0.424075	−	0.905627i	$-0.639401\pi$
$61$	−404.080	−0.848149	−0.424075	+	0.905627i	$0.639401\pi$
$62$	668.681	1.36972
$63$	467.790	0.935492
$64$	−473.172	−0.924164
$65$	82.8080	0.158016
$66$	0	0
$67$	−106.400	−0.194013	−0.0970064	−	0.995284i	$-0.530927\pi$
$67$	−106.400	−0.194013	−0.0970064	+	0.995284i	$0.530927\pi$
$68$	482.030	0.859628
$69$	−354.276	−0.618113
$70$	1856.54	3.16999
$71$	472.079	0.789091	0.394546	−	0.918876i	$-0.370902\pi$
$71$	472.079	0.789091	0.394546	+	0.918876i	$0.370902\pi$
$72$	−21.3805	−0.0349960
$73$	−843.821	−1.35290	−0.676451	−	0.736488i	$-0.736483\pi$
$73$	−843.821	−1.35290	−0.676451	+	0.736488i	$0.736483\pi$
$74$	−584.616	−0.918382
$75$	1369.07	2.10782
$76$	146.338	0.220870
$77$	0	0
$78$	121.526	0.176411
$79$	591.036	0.841731	0.420866	−	0.907123i	$-0.361726\pi$
$79$	591.036	0.841731	0.420866	+	0.907123i	$0.361726\pi$
$80$	−1202.21	−1.68014
$81$	−890.035	−1.22090
$82$	−1223.54	−1.64777
$83$	−290.388	−0.384027	−0.192013	−	0.981392i	$-0.561502\pi$
$83$	−290.388	−0.384027	−0.192013	+	0.981392i	$0.561502\pi$
$84$	1336.44	1.73592
$85$	1134.93	1.44824
$86$	1779.00	2.23063
$87$	−1152.62	−1.42039
$88$	0	0
$89$	−964.896	−1.14920	−0.574600	−	0.818435i	$-0.694842\pi$
$89$	−964.896	−1.14920	−0.574600	+	0.818435i	$0.694842\pi$
$90$	1301.06	1.52382
$91$	117.978	0.135907
$92$	−406.282	−0.460411
$93$	1133.34	1.26367
$94$	449.099	0.492777
$95$	344.550	0.372106
$96$	−1700.87	−1.80828
$97$	−219.495	−0.229756	−0.114878	−	0.993380i	$-0.536648\pi$
$97$	−219.495	−0.229756	−0.114878	+	0.993380i	$0.536648\pi$
$98$	1285.89	1.32546
$99$	0	0
$100$	1570.05	1.57005
$101$	−1447.94	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$101$	−1447.94	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$102$	1665.58	1.61683
$103$	883.567	0.845247	0.422623	−	0.906305i	$-0.361109\pi$
$103$	883.567	0.845247	0.422623	+	0.906305i	$0.361109\pi$
$104$	−5.39223	−0.00508415
$105$	3146.62	2.92456
$106$	615.793	0.564256
$107$	1307.82	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$107$	1307.82	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$108$	−460.064	−0.409904
$109$	−870.507	−0.764949	−0.382475	−	0.923966i	$-0.624928\pi$
$109$	−870.507	−0.764949	−0.382475	+	0.923966i	$0.624928\pi$
$110$	0	0
$111$	−990.856	−0.847279
$112$	−1712.82	−1.44505
$113$	−1181.41	−0.983521	−0.491761	−	0.870730i	$-0.663646\pi$
$113$	−1181.41	−0.983521	−0.491761	+	0.870730i	$0.663646\pi$
$114$	505.648	0.415423
$115$	−956.583	−0.775668
$116$	−1321.82	−1.05800
$117$	82.6792	0.0653307
$118$	722.564	0.563707
$119$	1616.96	1.24560
$120$	−143.817	−0.109405
$121$	0	0
$122$	−1601.20	−1.18824
$123$	−2073.76	−1.52020
$124$	1299.71	0.941266
$125$	1429.87	1.02313
$126$	1853.65	1.31061
$127$	−887.509	−0.620108	−0.310054	−	0.950719i	$-0.600347\pi$
$127$	−887.509	−0.620108	−0.310054	+	0.950719i	$0.600347\pi$
$128$	151.044	0.104301
$129$	3015.19	2.05793
$130$	328.133	0.221378
$131$	2344.76	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$131$	2344.76	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$132$	0	0
$133$	490.888	0.320040
$134$	−421.619	−0.271809
$135$	−1083.21	−0.690577
$136$	−73.9034	−0.0465968
$137$	2244.82	1.39991	0.699956	−	0.714186i	$-0.253203\pi$
$137$	2244.82	1.39991	0.699956	+	0.714186i	$0.253203\pi$
$138$	−1403.84	−0.865964
$139$	296.146	0.180711	0.0903554	−	0.995910i	$-0.471200\pi$
$139$	296.146	0.180711	0.0903554	+	0.995910i	$0.471200\pi$
$140$	3608.53	2.17840
$141$	761.170	0.454625
$142$	1870.65	1.10550
$143$	0	0
$144$	−1200.34	−0.694642
$145$	−3112.20	−1.78244
$146$	−3343.70	−1.89539
$147$	2179.44	1.22284
$148$	−1136.31	−0.631109
$149$	−1791.09	−0.984780	−0.492390	−	0.870375i	$-0.663877\pi$
$149$	−1791.09	−0.984780	−0.492390	+	0.870375i	$0.663877\pi$
$150$	5425.05	2.95302
$151$	2352.65	1.26792	0.633960	−	0.773366i	$-0.281429\pi$
$151$	2352.65	1.26792	0.633960	+	0.773366i	$0.281429\pi$
$152$	−22.4361	−0.0119724
$153$	1133.16	0.598764
$154$	0	0
$155$	3060.13	1.58578
$156$	236.208	0.121229
$157$	−1438.26	−0.731118	−0.365559	−	0.930788i	$-0.619122\pi$
$157$	−1438.26	−0.731118	−0.365559	+	0.930788i	$0.619122\pi$
$158$	2342.02	1.17925
$159$	1043.70	0.520570
$160$	−4592.54	−2.26920
$161$	−1362.86	−0.667135
$162$	−3526.83	−1.71046
$163$	127.493	0.0612640	0.0306320	−	0.999531i	$-0.490248\pi$
$163$	127.493	0.0612640	0.0306320	+	0.999531i	$0.490248\pi$
$164$	−2378.18	−1.13234
$165$	0	0
$166$	−1150.68	−0.538014
$167$	−3419.05	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$167$	−3419.05	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$168$	−204.899	−0.0940971
$169$	−2176.15	−0.990509
$170$	4497.24	2.02896
$171$	344.014	0.153844
$172$	3457.81	1.53288
$173$	362.598	0.159352	0.0796758	−	0.996821i	$-0.474611\pi$
$173$	362.598	0.159352	0.0796758	+	0.996821i	$0.474611\pi$
$174$	−4567.34	−1.98994
$175$	5266.69	2.27500
$176$	0	0
$177$	1224.66	0.520063
$178$	−3823.47	−1.61001
$179$	2417.89	1.00962	0.504809	−	0.863231i	$-0.331563\pi$
$179$	2417.89	1.00962	0.504809	+	0.863231i	$0.331563\pi$
$180$	2528.86	1.04717
$181$	−2444.64	−1.00391	−0.501957	−	0.864892i	$-0.667387\pi$
$181$	−2444.64	−1.00391	−0.501957	+	0.864892i	$0.667387\pi$
$182$	467.498	0.190403
$183$	−2713.84	−1.09624
$184$	62.2900	0.0249570
$185$	−2675.42	−1.06325
$186$	4490.93	1.77038
$187$	0	0
$188$	872.908	0.338635
$189$	−1543.27	−0.593951
$190$	1365.30	0.521314
$191$	−1387.66	−0.525693	−0.262846	−	0.964838i	$-0.584661\pi$
$191$	−1387.66	−0.525693	−0.262846	+	0.964838i	$0.584661\pi$
$192$	−3177.87	−1.19449
$193$	3208.03	1.19647	0.598237	−	0.801319i	$-0.295868\pi$
$193$	3208.03	1.19647	0.598237	+	0.801319i	$0.295868\pi$
$194$	−869.764	−0.321884
$195$	556.147	0.204238
$196$	2499.37	0.910849
$197$	3445.36	1.24605	0.623025	−	0.782202i	$-0.285903\pi$
$197$	3445.36	1.24605	0.623025	+	0.782202i	$0.285903\pi$
$198$	0	0
$199$	2025.71	0.721602	0.360801	−	0.932643i	$-0.382503\pi$
$199$	2025.71	0.721602	0.360801	+	0.932643i	$0.382503\pi$
$200$	−240.715	−0.0851056
$201$	−714.595	−0.250764
$202$	−5737.59	−1.99849
$203$	−4434.02	−1.53304
$204$	3237.36	1.11108
$205$	−5599.37	−1.90769
$206$	3501.20	1.18418
$207$	−955.095	−0.320694
$208$	−302.730	−0.100916
$209$	0	0
$210$	12468.7	4.09725
$211$	−4309.54	−1.40607	−0.703036	−	0.711155i	$-0.748173\pi$
$211$	−4309.54	−1.40607	−0.703036	+	0.711155i	$0.748173\pi$
$212$	1196.91	0.387755
$213$	3170.53	1.01991
$214$	5182.33	1.65540
$215$	8141.35	2.58249
$216$	70.5357	0.0222192
$217$	4359.84	1.36389
$218$	−3449.45	−1.07168
$219$	−5667.19	−1.74864
$220$	0	0
$221$	285.788	0.0869873
$222$	−3926.34	−1.18702
$223$	825.648	0.247935	0.123968	−	0.992286i	$-0.460438\pi$
$223$	825.648	0.247935	0.123968	+	0.992286i	$0.460438\pi$
$224$	−6543.09	−1.95169
$225$	3690.89	1.09360
$226$	−4681.43	−1.37790
$227$	1501.19	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$227$	1501.19	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$228$	982.821	0.285478
$229$	−5250.40	−1.51509	−0.757547	−	0.652781i	$-0.773602\pi$
$229$	−5250.40	−1.51509	−0.757547	+	0.652781i	$0.773602\pi$
$230$	−3790.53	−1.08670
$231$	0	0
$232$	202.658	0.0573497
$233$	−2139.06	−0.601435	−0.300717	−	0.953713i	$-0.597226\pi$
$233$	−2139.06	−0.601435	−0.300717	+	0.953713i	$0.597226\pi$
$234$	327.623	0.0915272
$235$	2055.24	0.570508
$236$	1404.44	0.387377
$237$	3969.46	1.08795
$238$	6407.32	1.74506
$239$	−3772.70	−1.02107	−0.510534	−	0.859857i	$-0.670552\pi$
$239$	−3772.70	−1.02107	−0.510534	+	0.859857i	$0.670552\pi$
$240$	−8074.17	−2.17160
$241$	−6415.39	−1.71474	−0.857369	−	0.514702i	$-0.827903\pi$
$241$	−6415.39	−1.71474	−0.857369	+	0.514702i	$0.827903\pi$
$242$	0	0
$243$	−4364.77	−1.15226
$244$	−3112.22	−0.816555
$245$	5884.71	1.53453
$246$	−8217.41	−2.12977
$247$	86.7616	0.0223502
$248$	−199.267	−0.0510221
$249$	−1950.27	−0.496360
$250$	5665.95	1.43339
$251$	−6277.31	−1.57857	−0.789283	−	0.614029i	$-0.789548\pi$
$251$	−6277.31	−1.57857	−0.789283	+	0.614029i	$0.789548\pi$
$252$	3602.92	0.900645
$253$	0	0
$254$	−3516.82	−0.868760
$255$	7622.30	1.87187
$256$	4383.90	1.07029
$257$	−3183.98	−0.772807	−0.386404	−	0.922330i	$-0.626283\pi$
$257$	−3183.98	−0.772807	−0.386404	+	0.922330i	$0.626283\pi$
$258$	11947.9	2.88312
$259$	−3811.73	−0.914476
$260$	637.787	0.152130
$261$	−3107.36	−0.736938
$262$	9291.31	2.19091
$263$	2624.18	0.615261	0.307630	−	0.951506i	$-0.400464\pi$
$263$	2624.18	0.615261	0.307630	+	0.951506i	$0.400464\pi$
$264$	0	0
$265$	2818.10	0.653261
$266$	1945.18	0.448371
$267$	−6480.34	−1.48536
$268$	−819.495	−0.186786
$269$	7444.76	1.68742	0.843708	−	0.536803i	$-0.180368\pi$
$269$	7444.76	1.68742	0.843708	+	0.536803i	$0.180368\pi$
$270$	−4292.30	−0.967486
$271$	4004.49	0.897621	0.448810	−	0.893627i	$-0.351848\pi$
$271$	4004.49	0.897621	0.448810	+	0.893627i	$0.351848\pi$
$272$	−4149.08	−0.924909
$273$	792.355	0.175661
$274$	8895.27	1.96125
$275$	0	0
$276$	−2728.63	−0.595088
$277$	5830.66	1.26473	0.632365	−	0.774671i	$-0.282084\pi$
$277$	5830.66	1.26473	0.632365	+	0.774671i	$0.282084\pi$
$278$	1173.50	0.253172
$279$	3055.37	0.655628
$280$	−553.250	−0.118082
$281$	7504.37	1.59314	0.796572	−	0.604544i	$-0.206645\pi$
$281$	7504.37	1.59314	0.796572	+	0.604544i	$0.206645\pi$
$282$	3016.19	0.636921
$283$	−5910.87	−1.24157	−0.620785	−	0.783980i	$-0.713186\pi$
$283$	−5910.87	−1.24157	−0.620785	+	0.783980i	$0.713186\pi$
$284$	3635.95	0.759697
$285$	2314.03	0.480952
$286$	0	0
$287$	−7977.54	−1.64076
$288$	−4585.40	−0.938184
$289$	−996.118	−0.202752
$290$	−12332.3	−2.49717
$291$	−1474.15	−0.296962
$292$	−6499.11	−1.30251
$293$	−3245.59	−0.647131	−0.323566	−	0.946206i	$-0.604882\pi$
$293$	−3245.59	−0.647131	−0.323566	+	0.946206i	$0.604882\pi$
$294$	8636.18	1.71317
$295$	3306.72	0.652625
$296$	174.216	0.0342098
$297$	0	0
$298$	−7097.35	−1.37966
$299$	−240.878	−0.0465898
$300$	10544.6	2.02931
$301$	11599.2	2.22115
$302$	9322.54	1.77633
$303$	−9724.54	−1.84376
$304$	−1259.61	−0.237643
$305$	−7327.66	−1.37567
$306$	4490.25	0.838857
$307$	−7489.14	−1.39227	−0.696137	−	0.717909i	$-0.745099\pi$
$307$	−7489.14	−1.39227	−0.696137	+	0.717909i	$0.745099\pi$
$308$	0	0
$309$	5934.12	1.09249
$310$	12126.0	2.22165
$311$	2136.71	0.389588	0.194794	−	0.980844i	$-0.437596\pi$
$311$	2136.71	0.389588	0.194794	+	0.980844i	$0.437596\pi$
$312$	−36.2147	−0.00657133
$313$	2212.15	0.399483	0.199742	−	0.979849i	$-0.435990\pi$
$313$	2212.15	0.399483	0.199742	+	0.979849i	$0.435990\pi$
$314$	−5699.21	−1.02428
$315$	8483.00	1.51734
$316$	4552.16	0.810377
$317$	429.326	0.0760674	0.0380337	−	0.999276i	$-0.487891\pi$
$317$	429.326	0.0760674	0.0380337	+	0.999276i	$0.487891\pi$
$318$	4135.73	0.729309
$319$	0	0
$320$	−8580.60	−1.49897
$321$	8783.43	1.52724
$322$	−5400.45	−0.934644
$323$	1189.11	0.204842
$324$	−6855.05	−1.17542
$325$	930.857	0.158876
$326$	505.201	0.0858297
$327$	−5846.42	−0.988708
$328$	364.615	0.0613796
$329$	2928.15	0.490681
$330$	0	0
$331$	−765.454	−0.127109	−0.0635546	−	0.997978i	$-0.520244\pi$
$331$	−765.454	−0.127109	−0.0635546	+	0.997978i	$0.520244\pi$
$332$	−2236.57	−0.369722
$333$	−2671.26	−0.439592
$334$	−13548.3	−2.21954
$335$	−1929.48	−0.314684
$336$	−11503.4	−1.86775
$337$	3049.81	0.492978	0.246489	−	0.969146i	$-0.420723\pi$
$337$	3049.81	0.492978	0.246489	+	0.969146i	$0.420723\pi$
$338$	−8623.15	−1.38768
$339$	−7934.48	−1.27121
$340$	8741.22	1.39429
$341$	0	0
$342$	1363.18	0.215533
$343$	−477.732	−0.0752045
$344$	−530.142	−0.0830912
$345$	−6424.50	−1.00256
$346$	1436.82	0.223249
$347$	5907.00	0.913845	0.456922	−	0.889507i	$-0.348952\pi$
$347$	5907.00	0.913845	0.456922	+	0.889507i	$0.348952\pi$
$348$	−8877.48	−1.36748
$349$	12107.4	1.85700	0.928502	−	0.371327i	$-0.121097\pi$
$349$	12107.4	1.85700	0.928502	+	0.371327i	$0.121097\pi$
$350$	20869.6	3.18723
$351$	−272.765	−0.0414789
$352$	0	0
$353$	−2420.40	−0.364943	−0.182471	−	0.983211i	$-0.558410\pi$
$353$	−2420.40	−0.364943	−0.182471	+	0.983211i	$0.558410\pi$
$354$	4852.81	0.728599
$355$	8560.77	1.27988
$356$	−7431.63	−1.10639
$357$	10859.7	1.60995
$358$	9581.07	1.41446
$359$	1455.80	0.214023	0.107011	−	0.994258i	$-0.465872\pi$
$359$	1455.80	0.214023	0.107011	+	0.994258i	$0.465872\pi$
$360$	−387.717	−0.0567625
$361$	361.000	0.0526316
$362$	−9687.06	−1.40647
$363$	0	0
$364$	908.670	0.130844
$365$	−15302.0	−2.19437
$366$	−10753.8	−1.53582
$367$	8783.80	1.24935	0.624674	−	0.780886i	$-0.285232\pi$
$367$	8783.80	1.24935	0.624674	+	0.780886i	$0.285232\pi$
$368$	3497.08	0.495375
$369$	−5590.66	−0.788721
$370$	−10601.6	−1.48959
$371$	4015.00	0.561856
$372$	8728.95	1.21660
$373$	9199.84	1.27708	0.638538	−	0.769590i	$-0.279539\pi$
$373$	9199.84	1.27708	0.638538	+	0.769590i	$0.279539\pi$
$374$	0	0
$375$	9603.13	1.32241
$376$	−133.832	−0.0183560
$377$	−783.688	−0.107061
$378$	−6115.34	−0.832114
$379$	−6161.38	−0.835063	−0.417531	−	0.908662i	$-0.637105\pi$
$379$	−6161.38	−0.835063	−0.417531	+	0.908662i	$0.637105\pi$
$380$	2653.72	0.358245
$381$	−5960.60	−0.801498
$382$	−5498.70	−0.736486
$383$	2630.79	0.350985	0.175492	−	0.984481i	$-0.443848\pi$
$383$	2630.79	0.350985	0.175492	+	0.984481i	$0.443848\pi$
$384$	1014.43	0.134810
$385$	0	0
$386$	12712.1	1.67624
$387$	8128.69	1.06771
$388$	−1690.55	−0.221197
$389$	−5866.48	−0.764633	−0.382317	−	0.924031i	$-0.624874\pi$
$389$	−5866.48	−0.764633	−0.382317	+	0.924031i	$0.624874\pi$
$390$	2203.77	0.286134
$391$	−3301.37	−0.427001
$392$	−383.196	−0.0493733
$393$	15747.7	2.02129
$394$	13652.5	1.74569
$395$	10718.0	1.36526
$396$	0	0
$397$	−14254.0	−1.80199	−0.900993	−	0.433833i	$-0.857161\pi$
$397$	−14254.0	−1.80199	−0.900993	+	0.433833i	$0.857161\pi$
$398$	8027.03	1.01095
$399$	3296.85	0.413657
$400$	−13514.2	−1.68928
$401$	9909.27	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$401$	9909.27	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$402$	−2831.64	−0.351316
$403$	770.576	0.0952484
$404$	−11152.1	−1.37336
$405$	−16140.1	−1.98026
$406$	−17570.1	−2.14776
$407$	0	0
$408$	−496.343	−0.0602270
$409$	−5805.87	−0.701912	−0.350956	−	0.936392i	$-0.614143\pi$
$409$	−5805.87	−0.701912	−0.350956	+	0.936392i	$0.614143\pi$
$410$	−22187.9	−2.67264
$411$	15076.4	1.80941
$412$	6805.23	0.813761
$413$	4711.15	0.561309
$414$	−3784.64	−0.449287
$415$	−5265.95	−0.622881
$416$	−1156.45	−0.136298
$417$	1988.95	0.233571
$418$	0	0
$419$	12260.9	1.42955	0.714777	−	0.699353i	$-0.246528\pi$
$419$	12260.9	1.42955	0.714777	+	0.699353i	$0.246528\pi$
$420$	24235.3	2.81562
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	−17076.9	−1.96988
$423$	2052.05	0.235872
$424$	−183.507	−0.0210186
$425$	12757.9	1.45612
$426$	12563.5	1.42888
$427$	−10439.9	−1.18319
$428$	10072.8	1.13759
$429$	0	0
$430$	32260.7	3.61802
$431$	2770.16	0.309591	0.154796	−	0.987946i	$-0.450528\pi$
$431$	2770.16	0.309591	0.154796	+	0.987946i	$0.450528\pi$
$432$	3960.01	0.441033
$433$	5663.00	0.628513	0.314257	−	0.949338i	$-0.398245\pi$
$433$	5663.00	0.628513	0.314257	+	0.949338i	$0.398245\pi$
$434$	17276.2	1.91079
$435$	−20901.8	−2.30383
$436$	−6704.65	−0.736455
$437$	−1002.25	−0.109712
$438$	−22456.6	−2.44982
$439$	8399.20	0.913148	0.456574	−	0.889685i	$-0.349076\pi$
$439$	8399.20	0.913148	0.456574	+	0.889685i	$0.349076\pi$
$440$	0	0
$441$	5875.56	0.634442
$442$	1132.46	0.121868
$443$	6154.68	0.660085	0.330043	−	0.943966i	$-0.392937\pi$
$443$	6154.68	0.660085	0.330043	+	0.943966i	$0.392937\pi$
$444$	−7631.58	−0.815717
$445$	−17497.6	−1.86397
$446$	3271.69	0.347352
$447$	−12029.2	−1.27284
$448$	−12225.0	−1.28923
$449$	3445.03	0.362095	0.181048	−	0.983474i	$-0.442051\pi$
$449$	3445.03	0.362095	0.181048	+	0.983474i	$0.442051\pi$
$450$	14625.4	1.53211
$451$	0	0
$452$	−9099.24	−0.946885
$453$	15800.6	1.63880
$454$	5948.57	0.614935
$455$	2139.44	0.220437
$456$	−150.683	−0.0154745
$457$	502.346	0.0514196	0.0257098	−	0.999669i	$-0.491815\pi$
$457$	502.346	0.0514196	0.0257098	+	0.999669i	$0.491815\pi$
$458$	−20805.1	−2.12262
$459$	−3738.39	−0.380159
$460$	−7367.60	−0.746774
$461$	−546.259	−0.0551883	−0.0275942	−	0.999619i	$-0.508785\pi$
$461$	−546.259	−0.0551883	−0.0275942	+	0.999619i	$0.508785\pi$
$462$	0	0
$463$	18540.2	1.86098	0.930490	−	0.366316i	$-0.119381\pi$
$463$	18540.2	1.86098	0.930490	+	0.366316i	$0.119381\pi$
$464$	11377.6	1.13835
$465$	20552.1	2.04964
$466$	−8476.18	−0.842599
$467$	12475.1	1.23614	0.618070	−	0.786123i	$-0.287915\pi$
$467$	12475.1	1.23614	0.618070	+	0.786123i	$0.287915\pi$
$468$	636.795	0.0628972
$469$	−2748.98	−0.270653
$470$	8144.05	0.799271
$471$	−9659.49	−0.944981
$472$	−215.324	−0.0209981
$473$	0	0
$474$	15729.3	1.52420
$475$	3873.13	0.374130
$476$	12453.8	1.19920
$477$	2813.71	0.270086
$478$	−14949.6	−1.43050
$479$	10569.2	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$479$	10569.2	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$480$	−30843.9	−2.93297
$481$	−673.701	−0.0638631
$482$	−25421.5	−2.40232
$483$	−9153.13	−0.862282
$484$	0	0
$485$	−3980.36	−0.372657
$486$	−17295.7	−1.61430
$487$	−11227.9	−1.04473	−0.522366	−	0.852721i	$-0.674951\pi$
$487$	−11227.9	−1.04473	−0.522366	+	0.852721i	$0.674951\pi$
$488$	477.157	0.0442621
$489$	856.256	0.0791846
$490$	23318.6	2.14985
$491$	536.840	0.0493427	0.0246713	−	0.999696i	$-0.492146\pi$
$491$	536.840	0.0493427	0.0246713	+	0.999696i	$0.492146\pi$
$492$	−15972.1	−1.46357
$493$	−10740.9	−0.981226
$494$	343.799	0.0313123
$495$	0	0
$496$	−11187.3	−1.01275
$497$	12196.7	1.10080
$498$	−7728.11	−0.695391
$499$	1319.91	0.118412	0.0592058	−	0.998246i	$-0.481143\pi$
$499$	1319.91	0.118412	0.0592058	+	0.998246i	$0.481143\pi$
$500$	11012.8	0.985018
$501$	−22962.7	−2.04770
$502$	−24874.3	−2.21154
$503$	−1749.27	−0.155062	−0.0775310	−	0.996990i	$-0.524704\pi$
$503$	−1749.27	−0.155062	−0.0775310	+	0.996990i	$0.524704\pi$
$504$	−552.390	−0.0488202
$505$	−26257.3	−2.31373
$506$	0	0
$507$	−14615.2	−1.28025
$508$	−6835.59	−0.597009
$509$	1882.19	0.163903	0.0819516	−	0.996636i	$-0.473885\pi$
$509$	1882.19	0.163903	0.0819516	+	0.996636i	$0.473885\pi$
$510$	30203.9	2.62245
$511$	−21801.1	−1.88733
$512$	16163.2	1.39515
$513$	−1134.93	−0.0976769
$514$	−12616.8	−1.08269
$515$	16022.8	1.37097
$516$	23223.0	1.98127
$517$	0	0
$518$	−15104.3	−1.28116
$519$	2435.25	0.205964
$520$	−97.7837	−0.00824635
$521$	−3238.50	−0.272325	−0.136163	−	0.990686i	$-0.543477\pi$
$521$	−3238.50	−0.272325	−0.136163	+	0.990686i	$0.543477\pi$
$522$	−12313.1	−1.03244
$523$	−99.0144	−0.00827839	−0.00413919	−	0.999991i	$-0.501318\pi$
$523$	−99.0144	−0.00827839	−0.00413919	+	0.999991i	$0.501318\pi$
$524$	18059.4	1.50559
$525$	35371.6	2.94046
$526$	10398.5	0.861969
$527$	10561.2	0.872963
$528$	0	0
$529$	−9384.42	−0.771301
$530$	11166.9	0.915207
$531$	3301.57	0.269823
$532$	3780.82	0.308119
$533$	−1409.98	−0.114584
$534$	−25678.8	−2.08096
$535$	23716.2	1.91653
$536$	125.643	0.0101249
$537$	16238.8	1.30495
$538$	29500.4	2.36404
$539$	0	0
$540$	−8342.88	−0.664853
$541$	−17183.7	−1.36559	−0.682794	−	0.730611i	$-0.739235\pi$
$541$	−17183.7	−1.36559	−0.682794	+	0.730611i	$0.739235\pi$
$542$	15868.1	1.25755
$543$	−16418.4	−1.29757
$544$	−15849.8	−1.24918
$545$	−15786.0	−1.24073
$546$	3139.77	0.246098
$547$	1965.86	0.153664	0.0768319	−	0.997044i	$-0.475520\pi$
$547$	1965.86	0.153664	0.0768319	+	0.997044i	$0.475520\pi$
$548$	17289.6	1.34777
$549$	−7316.27	−0.568762
$550$	0	0
$551$	−3260.79	−0.252113
$552$	418.346	0.0322572
$553$	15270.1	1.17423
$554$	23104.4	1.77186
$555$	−17968.4	−1.37426
$556$	2280.92	0.173979
$557$	−6039.93	−0.459461	−0.229731	−	0.973254i	$-0.573785\pi$
$557$	−6039.93	−0.459461	−0.229731	+	0.973254i	$0.573785\pi$
$558$	12107.1	0.918523
$559$	2050.09	0.155115
$560$	−31060.5	−2.34384
$561$	0	0
$562$	29736.6	2.23197
$563$	−5260.06	−0.393757	−0.196878	−	0.980428i	$-0.563080\pi$
$563$	−5260.06	−0.393757	−0.196878	+	0.980428i	$0.563080\pi$
$564$	5862.53	0.437690
$565$	−21424.0	−1.59524
$566$	−23422.3	−1.73942
$567$	−22995.1	−1.70318
$568$	−557.454	−0.0411800
$569$	−20567.4	−1.51534	−0.757672	−	0.652635i	$-0.773663\pi$
$569$	−20567.4	−1.51534	−0.757672	+	0.652635i	$0.773663\pi$
$570$	9169.52	0.673805
$571$	−11462.4	−0.840080	−0.420040	−	0.907506i	$-0.637984\pi$
$571$	−11462.4	−0.840080	−0.420040	+	0.907506i	$0.637984\pi$
$572$	0	0
$573$	−9319.64	−0.679466
$574$	−31611.6	−2.29868
$575$	−10753.1	−0.779886
$576$	−8567.25	−0.619737
$577$	−27029.6	−1.95019	−0.975094	−	0.221790i	$-0.928810\pi$
$577$	−27029.6	−1.95019	−0.975094	+	0.221790i	$0.928810\pi$
$578$	−3947.19	−0.284051
$579$	21545.5	1.54646
$580$	−23970.2	−1.71605
$581$	−7502.52	−0.535726
$582$	−5841.42	−0.416039
$583$	0	0
$584$	996.425	0.0706034
$585$	1499.32	0.105965
$586$	−12860.9	−0.906619
$587$	15200.4	1.06881	0.534403	−	0.845230i	$-0.320536\pi$
$587$	15200.4	1.06881	0.534403	+	0.845230i	$0.320536\pi$
$588$	16786.0	1.17729
$589$	3206.23	0.224296
$590$	13103.1	0.914316
$591$	23139.4	1.61054
$592$	9780.83	0.679036
$593$	19026.7	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$593$	19026.7	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$594$	0	0
$595$	29322.2	2.02033
$596$	−13795.0	−0.948096
$597$	13604.9	0.932681
$598$	−954.499	−0.0652715
$599$	3927.31	0.267889	0.133945	−	0.990989i	$-0.457236\pi$
$599$	3927.31	0.267889	0.133945	+	0.990989i	$0.457236\pi$
$600$	−1616.67	−0.110000
$601$	−13718.1	−0.931069	−0.465534	−	0.885030i	$-0.654138\pi$
$601$	−13718.1	−0.931069	−0.465534	+	0.885030i	$0.654138\pi$
$602$	45962.6	3.11178
$603$	−1926.48	−0.130104
$604$	18120.1	1.22069
$605$	0	0
$606$	−38534.2	−2.58308
$607$	−26461.5	−1.76942	−0.884712	−	0.466138i	$-0.845645\pi$
$607$	−26461.5	−1.76942	−0.884712	+	0.466138i	$0.845645\pi$
$608$	−4811.81	−0.320961
$609$	−29779.3	−1.98148
$610$	−29036.4	−1.92729
$611$	517.534	0.0342671
$612$	8727.63	0.576460
$613$	233.384	0.0153773	0.00768865	−	0.999970i	$-0.497553\pi$
$613$	233.384	0.0153773	0.00768865	+	0.999970i	$0.497553\pi$
$614$	−29676.3	−1.95055
$615$	−37605.9	−2.46572
$616$	0	0
$617$	4202.77	0.274225	0.137113	−	0.990555i	$-0.456218\pi$
$617$	4202.77	0.274225	0.137113	+	0.990555i	$0.456218\pi$
$618$	23514.4	1.53056
$619$	−23009.4	−1.49407	−0.747033	−	0.664787i	$-0.768522\pi$
$619$	−23009.4	−1.49407	−0.747033	+	0.664787i	$0.768522\pi$
$620$	23569.1	1.52671
$621$	3150.93	0.203611
$622$	8466.89	0.545806
$623$	−24929.2	−1.60316
$624$	−2033.17	−0.130436
$625$	448.344	0.0286940
$626$	8765.82	0.559668
$627$	0	0
$628$	−11077.5	−0.703884
$629$	−9233.45	−0.585313
$630$	33614.5	2.12577
$631$	18819.3	1.18730	0.593650	−	0.804723i	$-0.297686\pi$
$631$	18819.3	1.18730	0.593650	+	0.804723i	$0.297686\pi$
$632$	−697.924	−0.0439271
$633$	−28943.3	−1.81737
$634$	1701.24	0.106569
$635$	−16094.3	−1.00580
$636$	8038.56	0.501178
$637$	1481.84	0.0921705
$638$	0	0
$639$	8547.46	0.529159
$640$	2739.06	0.169173
$641$	25253.7	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$641$	25253.7	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$642$	34805.0	2.13963
$643$	11712.1	0.718324	0.359162	−	0.933275i	$-0.383063\pi$
$643$	11712.1	0.718324	0.359162	+	0.933275i	$0.383063\pi$
$644$	−10496.8	−0.642284
$645$	54678.1	3.33791
$646$	4711.96	0.286981
$647$	−26533.3	−1.61226	−0.806131	−	0.591737i	$-0.798442\pi$
$647$	−26533.3	−1.61226	−0.806131	+	0.591737i	$0.798442\pi$
$648$	1051.00	0.0637146
$649$	0	0
$650$	3688.59	0.222582
$651$	29281.1	1.76285
$652$	981.952	0.0589819
$653$	27898.9	1.67193	0.835964	−	0.548785i	$-0.184909\pi$
$653$	27898.9	1.67193	0.835964	+	0.548785i	$0.184909\pi$
$654$	−23166.9	−1.38516
$655$	42520.4	2.53650
$656$	20470.2	1.21834
$657$	−15278.2	−0.907245
$658$	11603.0	0.687436
$659$	1274.66	0.0753468	0.0376734	−	0.999290i	$-0.488005\pi$
$659$	1274.66	0.0753468	0.0376734	+	0.999290i	$0.488005\pi$
$660$	0	0
$661$	−5049.52	−0.297131	−0.148565	−	0.988903i	$-0.547466\pi$
$661$	−5049.52	−0.297131	−0.148565	+	0.988903i	$0.547466\pi$
$662$	−3033.17	−0.178078
$663$	1919.38	0.112432
$664$	342.904	0.0200411
$665$	8901.86	0.519097
$666$	−10585.1	−0.615860
$667$	9053.01	0.525538
$668$	−26333.6	−1.52526
$669$	5545.14	0.320460
$670$	−7645.73	−0.440866
$671$	0	0
$672$	−43944.1	−2.52259
$673$	−8398.64	−0.481045	−0.240523	−	0.970644i	$-0.577319\pi$
$673$	−8398.64	−0.481045	−0.240523	+	0.970644i	$0.577319\pi$
$674$	12085.1	0.690653
$675$	−12176.5	−0.694333
$676$	−16760.7	−0.953612
$677$	−9875.31	−0.560619	−0.280309	−	0.959910i	$-0.590437\pi$
$677$	−9875.31	−0.560619	−0.280309	+	0.959910i	$0.590437\pi$
$678$	−31441.0	−1.78095
$679$	−5670.91	−0.320515
$680$	−1340.18	−0.0755787
$681$	10082.1	0.567325
$682$	0	0
$683$	−8653.78	−0.484814	−0.242407	−	0.970175i	$-0.577937\pi$
$683$	−8653.78	−0.484814	−0.242407	+	0.970175i	$0.577937\pi$
$684$	2649.60	0.148114
$685$	40708.0	2.27062
$686$	−1893.05	−0.105360
$687$	−35262.2	−1.95828
$688$	−29763.2	−1.64929
$689$	709.629	0.0392376
$690$	−25457.6	−1.40457
$691$	2916.50	0.160563	0.0802813	−	0.996772i	$-0.474418\pi$
$691$	2916.50	0.160563	0.0802813	+	0.996772i	$0.474418\pi$
$692$	2792.73	0.153416
$693$	0	0
$694$	23406.9	1.28028
$695$	5370.37	0.293108
$696$	1361.07	0.0741254
$697$	−19324.6	−1.05017
$698$	47976.5	2.60163
$699$	−14366.1	−0.777363
$700$	40564.0	2.19025
$701$	9070.78	0.488729	0.244364	−	0.969683i	$-0.421421\pi$
$701$	9070.78	0.488729	0.244364	+	0.969683i	$0.421421\pi$
$702$	−1080.85	−0.0581112
$703$	−2803.16	−0.150388
$704$	0	0
$705$	13803.2	0.737389
$706$	−9591.00	−0.511278
$707$	−37409.4	−1.98999
$708$	9432.34	0.500690
$709$	−5957.11	−0.315549	−0.157774	−	0.987475i	$-0.550432\pi$
$709$	−5957.11	−0.315549	−0.157774	+	0.987475i	$0.550432\pi$
$710$	33922.7	1.79309
$711$	10701.3	0.564459
$712$	1139.40	0.0599729
$713$	−8901.55	−0.467553
$714$	43032.2	2.25552
$715$	0	0
$716$	18622.6	0.972010
$717$	−25337.8	−1.31975
$718$	5768.71	0.299842
$719$	−31140.8	−1.61524	−0.807620	−	0.589703i	$-0.799245\pi$
$719$	−31140.8	−1.61524	−0.807620	+	0.589703i	$0.799245\pi$
$720$	−21767.2	−1.12669
$721$	22828.0	1.17914
$722$	1430.49	0.0737359
$723$	−43086.4	−2.21632
$724$	−18828.6	−0.966519
$725$	−34984.7	−1.79214
$726$	0	0
$727$	14969.7	0.763682	0.381841	−	0.924228i	$-0.375290\pi$
$727$	14969.7	0.763682	0.381841	+	0.924228i	$0.375290\pi$
$728$	−139.315	−0.00709251
$729$	−5283.30	−0.268420
$730$	−60635.4	−3.07427
$731$	28097.5	1.42165
$732$	−20902.0	−1.05541
$733$	12414.1	0.625545	0.312772	−	0.949828i	$-0.398742\pi$
$733$	12414.1	0.625545	0.312772	+	0.949828i	$0.398742\pi$
$734$	34806.5	1.75031
$735$	39522.3	1.98341
$736$	13359.1	0.669055
$737$	0	0
$738$	−22153.4	−1.10498
$739$	−1324.11	−0.0659111	−0.0329555	−	0.999457i	$-0.510492\pi$
$739$	−1324.11	−0.0659111	−0.0329555	+	0.999457i	$0.510492\pi$
$740$	−20606.1	−1.02364
$741$	582.700	0.0288880
$742$	15909.8	0.787150
$743$	4391.55	0.216838	0.108419	−	0.994105i	$-0.465421\pi$
$743$	4391.55	0.216838	0.108419	+	0.994105i	$0.465421\pi$
$744$	−1338.30	−0.0659468
$745$	−32480.1	−1.59729
$746$	36455.0	1.78916
$747$	−5257.76	−0.257525
$748$	0	0
$749$	33789.0	1.64836
$750$	38053.1	1.85267
$751$	−31947.5	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$751$	−31947.5	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$752$	−7513.58	−0.364351
$753$	−42159.0	−2.04032
$754$	−3105.42	−0.149990
$755$	42663.4	2.05653
$756$	−11886.3	−0.571826
$757$	18569.8	0.891585	0.445793	−	0.895136i	$-0.352922\pi$
$757$	18569.8	0.891585	0.445793	+	0.895136i	$0.352922\pi$
$758$	−24414.9	−1.16991
$759$	0	0
$760$	−406.861	−0.0194190
$761$	−5507.32	−0.262339	−0.131170	−	0.991360i	$-0.541873\pi$
$761$	−5507.32	−0.262339	−0.131170	+	0.991360i	$0.541873\pi$
$762$	−23619.3	−1.12288
$763$	−22490.6	−1.06712
$764$	−10687.7	−0.506111
$765$	20549.0	0.971178
$766$	10424.7	0.491723
$767$	832.669	0.0391994
$768$	29442.7	1.38336
$769$	14977.9	0.702362	0.351181	−	0.936308i	$-0.385780\pi$
$769$	14977.9	0.702362	0.351181	+	0.936308i	$0.385780\pi$
$770$	0	0
$771$	−21384.0	−0.998864
$772$	24708.3	1.15190
$773$	19545.6	0.909450	0.454725	−	0.890632i	$-0.349738\pi$
$773$	19545.6	0.909450	0.454725	+	0.890632i	$0.349738\pi$
$774$	32210.5	1.49585
$775$	34399.4	1.59440
$776$	259.190	0.0119902
$777$	−25600.0	−1.18197
$778$	−23246.4	−1.07124
$779$	−5866.70	−0.269829
$780$	4283.44	0.196631
$781$	0	0
$782$	−13081.9	−0.598221
$783$	10251.4	0.467887
$784$	−21513.4	−0.980020
$785$	−26081.7	−1.18585
$786$	62401.3	2.83178
$787$	4274.62	0.193613	0.0968067	−	0.995303i	$-0.469137\pi$
$787$	4274.62	0.193613	0.0968067	+	0.995303i	$0.469137\pi$
$788$	26536.2	1.19963
$789$	17624.2	0.795233
$790$	42470.7	1.91271
$791$	−30523.2	−1.37204
$792$	0	0
$793$	−1845.19	−0.0826287
$794$	−56482.6	−2.52455
$795$	18926.6	0.844350
$796$	15602.0	0.694722
$797$	−25450.6	−1.13112	−0.565562	−	0.824706i	$-0.691341\pi$
$797$	−25450.6	−1.13112	−0.565562	+	0.824706i	$0.691341\pi$
$798$	13064.0	0.579525
$799$	7093.08	0.314062
$800$	−51625.4	−2.28154
$801$	−17470.4	−0.770645
$802$	39266.2	1.72885
$803$	0	0
$804$	−5503.81	−0.241423
$805$	−24714.5	−1.08207
$806$	3053.46	0.133441
$807$	49999.7	2.18101
$808$	1709.80	0.0744439
$809$	−4002.04	−0.173924	−0.0869619	−	0.996212i	$-0.527716\pi$
$809$	−4002.04	−0.173924	−0.0869619	+	0.996212i	$0.527716\pi$
$810$	−63956.2	−2.77431
$811$	37915.1	1.64165	0.820826	−	0.571179i	$-0.193514\pi$
$811$	37915.1	1.64165	0.820826	+	0.571179i	$0.193514\pi$
$812$	−34150.8	−1.47593
$813$	26894.5	1.16019
$814$	0	0
$815$	2311.99	0.0993685
$816$	−27865.7	−1.19546
$817$	8530.05	0.365274
$818$	−23006.2	−0.983366
$819$	2136.12	0.0911379
$820$	−43126.3	−1.83663
$821$	−4739.43	−0.201470	−0.100735	−	0.994913i	$-0.532119\pi$
$821$	−4739.43	−0.201470	−0.100735	+	0.994913i	$0.532119\pi$
$822$	59741.5	2.53494
$823$	20752.2	0.878952	0.439476	−	0.898254i	$-0.355164\pi$
$823$	20752.2	0.878952	0.439476	+	0.898254i	$0.355164\pi$
$824$	−1043.36	−0.0441106
$825$	0	0
$826$	18668.3	0.786384
$827$	−34264.8	−1.44075	−0.720377	−	0.693583i	$-0.756031\pi$
$827$	−34264.8	−1.44075	−0.720377	+	0.693583i	$0.756031\pi$
$828$	−7356.14	−0.308748
$829$	−39707.5	−1.66357	−0.831784	−	0.555100i	$-0.812680\pi$
$829$	−39707.5	−1.66357	−0.831784	+	0.555100i	$0.812680\pi$
$830$	−20866.7	−0.872645
$831$	39159.3	1.63468
$832$	−2160.69	−0.0900343
$833$	20309.4	0.844753
$834$	7881.35	0.327229
$835$	−62001.8	−2.56965
$836$	0	0
$837$	−10079.9	−0.416263
$838$	48584.6	2.00278
$839$	−4524.04	−0.186159	−0.0930794	−	0.995659i	$-0.529671\pi$
$839$	−4524.04	−0.186159	−0.0930794	+	0.995659i	$0.529671\pi$
$840$	−3715.68	−0.152623
$841$	5064.59	0.207659
$842$	23132.9	0.946809
$843$	50400.1	2.05916
$844$	−33192.1	−1.35370
$845$	−39462.7	−1.60658
$846$	8131.39	0.330453
$847$	0	0
$848$	−10302.4	−0.417201
$849$	−39698.0	−1.60475
$850$	50554.1	2.03999
$851$	7782.47	0.313490
$852$	24419.4	0.981920
$853$	7595.54	0.304884	0.152442	−	0.988312i	$-0.451286\pi$
$853$	7595.54	0.304884	0.152442	+	0.988312i	$0.451286\pi$
$854$	−41368.8	−1.65762
$855$	6238.42	0.249531
$856$	−1544.34	−0.0616639
$857$	19528.9	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$857$	19528.9	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$858$	0	0
$859$	25980.8	1.03196	0.515979	−	0.856601i	$-0.327428\pi$
$859$	25980.8	1.03196	0.515979	+	0.856601i	$0.327428\pi$
$860$	62704.7	2.48629
$861$	−53578.0	−2.12071
$862$	10977.0	0.433732
$863$	−48294.6	−1.90494	−0.952472	−	0.304625i	$-0.901469\pi$
$863$	−48294.6	−1.90494	−0.952472	+	0.304625i	$0.901469\pi$
$864$	15127.6	0.595660
$865$	6575.43	0.258464
$866$	22440.1	0.880536
$867$	−6690.03	−0.262059
$868$	33579.4	1.31309
$869$	0	0
$870$	−82825.1	−3.22763
$871$	−485.866	−0.0189012
$872$	1027.94	0.0399201
$873$	−3974.17	−0.154072
$874$	−3971.51	−0.153705
$875$	36942.3	1.42729
$876$	−43648.7	−1.68351
$877$	−44377.5	−1.70869	−0.854346	−	0.519705i	$-0.826042\pi$
$877$	−44377.5	−1.70869	−0.854346	+	0.519705i	$0.826042\pi$
$878$	33282.5	1.27930
$879$	−21797.7	−0.836426
$880$	0	0
$881$	−12139.4	−0.464231	−0.232116	−	0.972688i	$-0.574565\pi$
$881$	−12139.4	−0.464231	−0.232116	+	0.972688i	$0.574565\pi$
$882$	23282.4	0.888841
$883$	5048.07	0.192391	0.0961954	−	0.995362i	$-0.469333\pi$
$883$	5048.07	0.192391	0.0961954	+	0.995362i	$0.469333\pi$
$884$	2201.14	0.0837470
$885$	22208.2	0.843527
$886$	24388.4	0.924767
$887$	−20373.4	−0.771221	−0.385610	−	0.922662i	$-0.626009\pi$
$887$	−20373.4	−0.771221	−0.385610	+	0.922662i	$0.626009\pi$
$888$	1170.05	0.0442166
$889$	−22929.9	−0.865065
$890$	−69335.6	−2.61139
$891$	0	0
$892$	6359.14	0.238699
$893$	2153.37	0.0806940
$894$	−47666.5	−1.78323
$895$	43846.5	1.63757
$896$	3902.40	0.145502
$897$	−1617.76	−0.0602180
$898$	13651.2	0.507289
$899$	−28960.8	−1.07441
$900$	28427.3	1.05286
$901$	9725.85	0.359617
$902$	0	0
$903$	77901.2	2.87086
$904$	1395.07	0.0513267
$905$	−44331.6	−1.62832
$906$	62611.1	2.29593
$907$	−7456.13	−0.272962	−0.136481	−	0.990643i	$-0.543579\pi$
$907$	−7456.13	−0.272962	−0.136481	+	0.990643i	$0.543579\pi$
$908$	11562.2	0.422581
$909$	−26216.5	−0.956596
$910$	8477.71	0.308828
$911$	−10653.2	−0.387440	−0.193720	−	0.981057i	$-0.562055\pi$
$911$	−10653.2	−0.387440	−0.193720	+	0.981057i	$0.562055\pi$
$912$	−8459.66	−0.307157
$913$	0	0
$914$	1990.59	0.0720380
$915$	−49213.3	−1.77808
$916$	−40438.6	−1.45866
$917$	60579.8	2.18159
$918$	−14813.6	−0.532596
$919$	12569.7	0.451183	0.225591	−	0.974222i	$-0.427569\pi$
$919$	12569.7	0.451183	0.225591	+	0.974222i	$0.427569\pi$
$920$	1129.58	0.0404795
$921$	−50297.8	−1.79953
$922$	−2164.59	−0.0773179
$923$	2155.70	0.0768752
$924$	0	0
$925$	−30074.8	−1.06903
$926$	73466.8	2.60720
$927$	15997.9	0.566816
$928$	43463.4	1.53745
$929$	4920.06	0.173759	0.0868795	−	0.996219i	$-0.472311\pi$
$929$	4920.06	0.173759	0.0868795	+	0.996219i	$0.472311\pi$
$930$	81439.4	2.87151
$931$	6165.67	0.217048
$932$	−16475.0	−0.579031
$933$	14350.4	0.503548
$934$	49433.4	1.73181
$935$	0	0
$936$	−97.6317	−0.00340939
$937$	−1991.87	−0.0694465	−0.0347233	−	0.999397i	$-0.511055\pi$
$937$	−1991.87	−0.0694465	−0.0347233	+	0.999397i	$0.511055\pi$
$938$	−10893.0	−0.379179
$939$	14857.0	0.516337
$940$	15829.5	0.549256
$941$	7640.33	0.264684	0.132342	−	0.991204i	$-0.457750\pi$
$941$	7640.33	0.264684	0.132342	+	0.991204i	$0.457750\pi$
$942$	−38276.5	−1.32390
$943$	16287.9	0.562467
$944$	−12088.7	−0.416795
$945$	−27986.0	−0.963371
$946$	0	0
$947$	−6521.15	−0.223769	−0.111884	−	0.993721i	$-0.535689\pi$
$947$	−6521.15	−0.223769	−0.111884	+	0.993721i	$0.535689\pi$
$948$	30572.8	1.04742
$949$	−3853.22	−0.131803
$950$	15347.6	0.524149
$951$	2883.40	0.0983182
$952$	−1909.38	−0.0650037
$953$	35757.9	1.21544	0.607719	−	0.794152i	$-0.292085\pi$
$953$	35757.9	1.21544	0.607719	+	0.794152i	$0.292085\pi$
$954$	11149.6	0.378386
$955$	−25164.1	−0.852659
$956$	−29057.3	−0.983034
$957$	0	0
$958$	41881.3	1.41244
$959$	57997.6	1.95291
$960$	−57628.1	−1.93744
$961$	−1314.75	−0.0441323
$962$	−2669.59	−0.0894710
$963$	23679.3	0.792374
$964$	−49411.4	−1.65086
$965$	58175.1	1.94065
$966$	−36270.0	−1.20804
$967$	6342.30	0.210915	0.105457	−	0.994424i	$-0.466369\pi$
$967$	6342.30	0.210915	0.105457	+	0.994424i	$0.466369\pi$
$968$	0	0
$969$	7986.21	0.264762
$970$	−15772.5	−0.522086
$971$	30351.2	1.00311	0.501553	−	0.865127i	$-0.332762\pi$
$971$	30351.2	1.00311	0.501553	+	0.865127i	$0.332762\pi$
$972$	−33617.5	−1.10934
$973$	7651.29	0.252096
$974$	−44491.4	−1.46365
$975$	6251.73	0.205349
$976$	26788.5	0.878566
$977$	39843.6	1.30472	0.652359	−	0.757910i	$-0.273780\pi$
$977$	39843.6	1.30472	0.652359	+	0.757910i	$0.273780\pi$
$978$	3392.98	0.110936
$979$	0	0
$980$	45324.1	1.47737
$981$	−15761.4	−0.512969
$982$	2127.27	0.0691282
$983$	24068.7	0.780949	0.390475	−	0.920614i	$-0.372311\pi$
$983$	24068.7	0.780949	0.390475	+	0.920614i	$0.372311\pi$
$984$	2448.79	0.0793340
$985$	62478.9	2.02106
$986$	−42561.5	−1.37468
$987$	19665.8	0.634213
$988$	668.238	0.0215177
$989$	−23682.2	−0.761426
$990$	0	0
$991$	3235.83	0.103723	0.0518615	−	0.998654i	$-0.483485\pi$
$991$	3235.83	0.103723	0.0518615	+	0.998654i	$0.483485\pi$
$992$	−42736.2	−1.36782
$993$	−5140.86	−0.164290
$994$	48330.4	1.54220
$995$	36734.6	1.17042
$996$	−15021.0	−0.477871
$997$	−19444.5	−0.617665	−0.308833	−	0.951116i	$-0.599938\pi$
$997$	−19444.5	−0.617665	−0.308833	+	0.951116i	$0.599938\pi$
$998$	5230.25	0.165892
$999$	8812.68	0.279100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.4.a.h.1.3		3
11.10	odd	2		19.4.a.b.1.1	✓	3
33.32	even	2		171.4.a.f.1.3		3
44.43	even	2		304.4.a.i.1.1		3
55.32	even	4		475.4.b.f.324.2		6
55.43	even	4		475.4.b.f.324.5		6
55.54	odd	2		475.4.a.f.1.3		3
77.76	even	2		931.4.a.c.1.1		3
88.21	odd	2		1216.4.a.s.1.1		3
88.43	even	2		1216.4.a.u.1.3		3
209.208	even	2		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	11.10	odd	2
171.4.a.f.1.3		3	33.32	even	2
304.4.a.i.1.1		3	44.43	even	2
361.4.a.i.1.3		3	209.208	even	2
475.4.a.f.1.3		3	55.54	odd	2
475.4.b.f.324.2		6	55.32	even	4
475.4.b.f.324.5		6	55.43	even	4
931.4.a.c.1.1		3	77.76	even	2
1216.4.a.s.1.1		3	88.21	odd	2
1216.4.a.u.1.3		3	88.43	even	2
2299.4.a.h.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(f(q)\)	\(=\)	\(q+3.96257 q^{2} +6.71610 q^{3} +7.70200 q^{4} +18.1342 q^{5} +26.6130 q^{6} +25.8362 q^{7} -1.18085 q^{8} +18.1060 q^{9} +71.8581 q^{10} +51.7274 q^{12} +4.56640 q^{13} +102.378 q^{14} +121.791 q^{15} -66.2952 q^{16} +62.5850 q^{17} +71.7464 q^{18} +19.0000 q^{19} +139.670 q^{20} +173.518 q^{21} -52.7502 q^{23} -7.93070 q^{24} +203.849 q^{25} +18.0947 q^{26} -59.7330 q^{27} +198.990 q^{28} -171.620 q^{29} +482.606 q^{30} +168.749 q^{31} -253.253 q^{32} +247.998 q^{34} +468.519 q^{35} +139.452 q^{36} -147.534 q^{37} +75.2889 q^{38} +30.6684 q^{39} -21.4138 q^{40} -308.774 q^{41} +687.580 q^{42} +448.950 q^{43} +328.338 q^{45} -209.027 q^{46} +113.335 q^{47} -445.245 q^{48} +324.509 q^{49} +807.768 q^{50} +420.327 q^{51} +35.1704 q^{52} +155.402 q^{53} -236.696 q^{54} -30.5086 q^{56} +127.606 q^{57} -680.059 q^{58} +182.347 q^{59} +938.035 q^{60} -404.080 q^{61} +668.681 q^{62} +467.790 q^{63} -473.172 q^{64} +82.8080 q^{65} -106.400 q^{67} +482.030 q^{68} -354.276 q^{69} +1856.54 q^{70} +472.079 q^{71} -21.3805 q^{72} -843.821 q^{73} -584.616 q^{74} +1369.07 q^{75} +146.338 q^{76} +121.526 q^{78} +591.036 q^{79} -1202.21 q^{80} -890.035 q^{81} -1223.54 q^{82} -290.388 q^{83} +1336.44 q^{84} +1134.93 q^{85} +1779.00 q^{86} -1152.62 q^{87} -964.896 q^{89} +1301.06 q^{90} +117.978 q^{91} -406.282 q^{92} +1133.34 q^{93} +449.099 q^{94} +344.550 q^{95} -1700.87 q^{96} -219.495 q^{97} +1285.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 21 q^{4} + 14 q^{5} + 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 88 q^{10} - 115 q^{12} - 65 q^{13} + 37 q^{14} + 140 q^{15} + 33 q^{16} - 29 q^{17} - 138 q^{18} + 57 q^{19} + 100 q^{20}+ \cdots + 2450 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 21 * q^4 + 14 * q^5 + 65 * q^6 + 35 * q^7 - 27 * q^8 + 48 * q^9 + 88 * q^10 - 115 * q^12 - 65 * q^13 + 37 * q^14 + 140 * q^15 + 33 * q^16 - 29 * q^17 - 138 * q^18 + 57 * q^19 + 100 * q^20 + 25 * q^21 - 101 * q^23 + 489 * q^24 - 37 * q^25 + 299 * q^26 - 377 * q^27 + 493 * q^28 - 377 * q^29 + 376 * q^30 - 140 * q^31 - 579 * q^32 + 329 * q^34 + 438 * q^35 + 1074 * q^36 - 290 * q^37 - 57 * q^38 + 371 * q^39 + 72 * q^40 - 956 * q^41 + 1385 * q^42 + 570 * q^43 + 230 * q^45 + 239 * q^46 + 66 * q^47 - 1411 * q^48 - 98 * q^49 + 1639 * q^50 - 187 * q^51 - 697 * q^52 + 817 * q^53 + 947 * q^54 - 915 * q^56 + 19 * q^57 + 647 * q^58 + 265 * q^59 + 1360 * q^60 - 988 * q^61 + 1436 * q^62 + 1304 * q^63 - 159 * q^64 + 240 * q^65 - 207 * q^67 + 1529 * q^68 + 807 * q^69 + 2040 * q^70 + 846 * q^71 - 2826 * q^72 - 627 * q^73 + 770 * q^74 + 2031 * q^75 + 399 * q^76 - 1817 * q^78 - 382 * q^79 - 1472 * q^80 - 501 * q^81 + 872 * q^82 + 766 * q^83 - 907 * q^84 + 1242 * q^85 + 2592 * q^86 + 1671 * q^87 - 172 * q^89 + 1888 * q^90 - 457 * q^91 - 2677 * q^92 + 892 * q^93 + 96 * q^94 + 266 * q^95 - 863 * q^96 - 2450 * q^97 + 2450 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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