LMFDB - Embedded newform 363.4.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.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:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 $ x^6 - x^5 - 30x^4 + 3x^3 + 211x^2 + 208x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-4.12458$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.12458 q^{2} -3.00000 q^{3} +18.2613 q^{4} +14.9367 q^{5} -15.3737 q^{6} -12.3403 q^{7} +52.5849 q^{8} +9.00000 q^{9} +76.5442 q^{10} -54.7839 q^{12} +30.5222 q^{13} -63.2387 q^{14} -44.8100 q^{15} +123.385 q^{16} +132.065 q^{17} +46.1212 q^{18} -22.3068 q^{19} +272.763 q^{20} +37.0208 q^{21} -158.801 q^{23} -157.755 q^{24} +98.1044 q^{25} +156.413 q^{26} -27.0000 q^{27} -225.349 q^{28} -47.7165 q^{29} -229.633 q^{30} +68.0343 q^{31} +211.617 q^{32} +676.778 q^{34} -184.323 q^{35} +164.352 q^{36} +158.409 q^{37} -114.313 q^{38} -91.5666 q^{39} +785.444 q^{40} +286.159 q^{41} +189.716 q^{42} +103.549 q^{43} +134.430 q^{45} -813.790 q^{46} -464.238 q^{47} -370.155 q^{48} -190.718 q^{49} +502.744 q^{50} -396.195 q^{51} +557.375 q^{52} -6.04835 q^{53} -138.364 q^{54} -648.912 q^{56} +66.9205 q^{57} -244.527 q^{58} -221.619 q^{59} -818.290 q^{60} -274.338 q^{61} +348.647 q^{62} -111.062 q^{63} +97.3683 q^{64} +455.900 q^{65} +187.178 q^{67} +2411.68 q^{68} +476.404 q^{69} -944.576 q^{70} -457.714 q^{71} +473.264 q^{72} -270.332 q^{73} +811.780 q^{74} -294.313 q^{75} -407.352 q^{76} -469.240 q^{78} +493.647 q^{79} +1842.96 q^{80} +81.0000 q^{81} +1466.45 q^{82} -271.033 q^{83} +676.048 q^{84} +1972.61 q^{85} +530.646 q^{86} +143.150 q^{87} -77.4891 q^{89} +688.898 q^{90} -376.652 q^{91} -2899.92 q^{92} -204.103 q^{93} -2379.02 q^{94} -333.190 q^{95} -634.851 q^{96} -1269.39 q^{97} -977.348 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * 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 $

$2$	5.12458	1.81181	0.905906	−	0.423478i	$-0.139191\pi$
$2$	5.12458	1.81181	0.905906	+	0.423478i	$0.139191\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	18.2613	2.28266
$5$	14.9367	1.33598	0.667989	−	0.744172i	$-0.267155\pi$
$5$	14.9367	1.33598	0.667989	+	0.744172i	$0.267155\pi$
$6$	−15.3737	−1.04605
$7$	−12.3403	−0.666312	−0.333156	−	0.942872i	$-0.608114\pi$
$7$	−12.3403	−0.666312	−0.333156	+	0.942872i	$0.608114\pi$
$8$	52.5849	2.32395
$9$	9.00000	0.333333
$10$	76.5442	2.42054
$11$	0	0
$11$	0	0
$12$	−54.7839	−1.31790
$13$	30.5222	0.651179	0.325590	−	0.945511i	$-0.394437\pi$
$13$	30.5222	0.651179	0.325590	+	0.945511i	$0.394437\pi$
$14$	−63.2387	−1.20723
$15$	−44.8100	−0.771327
$16$	123.385	1.92789
$17$	132.065	1.88415	0.942073	−	0.335409i	$-0.108874\pi$
$17$	132.065	1.88415	0.942073	+	0.335409i	$0.108874\pi$
$18$	46.1212	0.603937
$19$	−22.3068	−0.269344	−0.134672	−	0.990890i	$-0.542998\pi$
$19$	−22.3068	−0.269344	−0.134672	+	0.990890i	$0.542998\pi$
$20$	272.763	3.04959
$21$	37.0208	0.384695
$22$	0	0
$23$	−158.801	−1.43967	−0.719834	−	0.694147i	$-0.755782\pi$
$23$	−158.801	−1.43967	−0.719834	+	0.694147i	$0.755782\pi$
$24$	−157.755	−1.34173
$25$	98.1044	0.784835
$26$	156.413	1.17982
$27$	−27.0000	−0.192450
$28$	−225.349	−1.52097
$29$	−47.7165	−0.305543	−0.152771	−	0.988262i	$-0.548820\pi$
$29$	−47.7165	−0.305543	−0.152771	+	0.988262i	$0.548820\pi$
$30$	−229.633	−1.39750
$31$	68.0343	0.394172	0.197086	−	0.980386i	$-0.436852\pi$
$31$	68.0343	0.394172	0.197086	+	0.980386i	$0.436852\pi$
$32$	211.617	1.16903
$33$	0	0
$34$	676.778	3.41372
$35$	−184.323	−0.890177
$36$	164.352	0.760888
$37$	158.409	0.703846	0.351923	−	0.936029i	$-0.385528\pi$
$37$	158.409	0.703846	0.351923	+	0.936029i	$0.385528\pi$
$38$	−114.313	−0.488001
$39$	−91.5666	−0.375959
$40$	785.444	3.10474
$41$	286.159	1.09001	0.545007	−	0.838432i	$-0.316527\pi$
$41$	286.159	1.09001	0.545007	+	0.838432i	$0.316527\pi$
$42$	189.716	0.696996
$43$	103.549	0.367235	0.183617	−	0.982998i	$-0.441219\pi$
$43$	103.549	0.367235	0.183617	+	0.982998i	$0.441219\pi$
$44$	0	0
$45$	134.430	0.445326
$46$	−813.790	−2.60841
$47$	−464.238	−1.44077	−0.720383	−	0.693576i	$-0.756034\pi$
$47$	−464.238	−1.44077	−0.720383	+	0.693576i	$0.756034\pi$
$48$	−370.155	−1.11307
$49$	−190.718	−0.556029
$50$	502.744	1.42197
$51$	−396.195	−1.08781
$52$	557.375	1.48642
$53$	−6.04835	−0.0156756	−0.00783778	−	0.999969i	$-0.502495\pi$
$53$	−6.04835	−0.0156756	−0.00783778	+	0.999969i	$0.502495\pi$
$54$	−138.364	−0.348683
$55$	0	0
$56$	−648.912	−1.54847
$57$	66.9205	0.155506
$58$	−244.527	−0.553586
$59$	−221.619	−0.489023	−0.244511	−	0.969646i	$-0.578628\pi$
$59$	−221.619	−0.489023	−0.244511	+	0.969646i	$0.578628\pi$
$60$	−818.290	−1.76068
$61$	−274.338	−0.575825	−0.287913	−	0.957657i	$-0.592961\pi$
$61$	−274.338	−0.575825	−0.287913	+	0.957657i	$0.592961\pi$
$62$	348.647	0.714165
$63$	−111.062	−0.222104
$64$	97.3683	0.190172
$65$	455.900	0.869961
$66$	0	0
$67$	187.178	0.341304	0.170652	−	0.985331i	$-0.445413\pi$
$67$	187.178	0.341304	0.170652	+	0.985331i	$0.445413\pi$
$68$	2411.68	4.30087
$69$	476.404	0.831192
$70$	−944.576	−1.61283
$71$	−457.714	−0.765080	−0.382540	−	0.923939i	$-0.624951\pi$
$71$	−457.714	−0.765080	−0.382540	+	0.923939i	$0.624951\pi$
$72$	473.264	0.774649
$73$	−270.332	−0.433424	−0.216712	−	0.976236i	$-0.569533\pi$
$73$	−270.332	−0.433424	−0.216712	+	0.976236i	$0.569533\pi$
$74$	811.780	1.27524
$75$	−294.313	−0.453125
$76$	−407.352	−0.614822
$77$	0	0
$78$	−469.240	−0.681167
$79$	493.647	0.703033	0.351517	−	0.936182i	$-0.385666\pi$
$79$	493.647	0.703033	0.351517	+	0.936182i	$0.385666\pi$
$80$	1842.96	2.57562
$81$	81.0000	0.111111
$82$	1466.45	1.97490
$83$	−271.033	−0.358431	−0.179216	−	0.983810i	$-0.557356\pi$
$83$	−271.033	−0.358431	−0.179216	+	0.983810i	$0.557356\pi$
$84$	676.048	0.878130
$85$	1972.61	2.51718
$86$	530.646	0.665361
$87$	143.150	0.176405
$88$	0	0
$89$	−77.4891	−0.0922902	−0.0461451	−	0.998935i	$-0.514694\pi$
$89$	−77.4891	−0.0922902	−0.0461451	+	0.998935i	$0.514694\pi$
$90$	688.898	0.806847
$91$	−376.652	−0.433889
$92$	−2899.92	−3.28628
$93$	−204.103	−0.227575
$94$	−2379.02	−2.61040
$95$	−333.190	−0.359837
$96$	−634.851	−0.674940
$97$	−1269.39	−1.32873	−0.664364	−	0.747409i	$-0.731298\pi$
$97$	−1269.39	−1.32873	−0.664364	+	0.747409i	$0.731298\pi$
$98$	−977.348	−1.00742
$99$	0	0
$100$	1791.51	1.79151
$101$	−1238.06	−1.21972	−0.609859	−	0.792510i	$-0.708774\pi$
$101$	−1238.06	−1.21972	−0.609859	+	0.792510i	$0.708774\pi$
$102$	−2030.33	−1.97091
$103$	911.965	0.872413	0.436206	−	0.899847i	$-0.356322\pi$
$103$	911.965	0.872413	0.436206	+	0.899847i	$0.356322\pi$
$104$	1605.01	1.51331
$105$	552.968	0.513944
$106$	−30.9953	−0.0284012
$107$	−1583.34	−1.43054	−0.715269	−	0.698849i	$-0.753696\pi$
$107$	−1583.34	−1.43054	−0.715269	+	0.698849i	$0.753696\pi$
$108$	−493.055	−0.439299
$109$	296.898	0.260896	0.130448	−	0.991455i	$-0.458358\pi$
$109$	296.898	0.260896	0.130448	+	0.991455i	$0.458358\pi$
$110$	0	0
$111$	−475.228	−0.406366
$112$	−1522.60	−1.28458
$113$	−109.404	−0.0910787	−0.0455393	−	0.998963i	$-0.514501\pi$
$113$	−109.404	−0.0910787	−0.0455393	+	0.998963i	$0.514501\pi$
$114$	342.939	0.281747
$115$	−2371.96	−1.92336
$116$	−871.367	−0.697451
$117$	274.700	0.217060
$118$	−1135.70	−0.886017
$119$	−1629.72	−1.25543
$120$	−2356.33	−1.79252
$121$	0	0
$122$	−1405.86	−1.04329
$123$	−858.478	−0.629320
$124$	1242.40	0.899761
$125$	−401.731	−0.287455
$126$	−569.148	−0.402411
$127$	1387.37	0.969366	0.484683	−	0.874690i	$-0.338935\pi$
$127$	1387.37	0.969366	0.484683	+	0.874690i	$0.338935\pi$
$128$	−1193.97	−0.824474
$129$	−310.647	−0.212023
$130$	2336.30	1.57621
$131$	1515.23	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$131$	1515.23	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$132$	0	0
$133$	275.272	0.179467
$134$	959.207	0.618379
$135$	−403.290	−0.257109
$136$	6944.63	4.37865
$137$	−2970.09	−1.85220	−0.926101	−	0.377275i	$-0.876861\pi$
$137$	−2970.09	−1.85220	−0.926101	+	0.377275i	$0.876861\pi$
$138$	2441.37	1.50596
$139$	−2152.08	−1.31322	−0.656608	−	0.754232i	$-0.728010\pi$
$139$	−2152.08	−1.31322	−0.656608	+	0.754232i	$0.728010\pi$
$140$	−3365.97	−2.03198
$141$	1392.71	0.831827
$142$	−2345.59	−1.38618
$143$	0	0
$144$	1110.47	0.642630
$145$	−712.727	−0.408198
$146$	−1385.34	−0.785283
$147$	572.153	0.321023
$148$	2892.76	1.60664
$149$	−132.876	−0.0730579	−0.0365290	−	0.999333i	$-0.511630\pi$
$149$	−132.876	−0.0730579	−0.0365290	+	0.999333i	$0.511630\pi$
$150$	−1508.23	−0.820977
$151$	212.982	0.114783	0.0573915	−	0.998352i	$-0.481722\pi$
$151$	212.982	0.114783	0.0573915	+	0.998352i	$0.481722\pi$
$152$	−1173.00	−0.625941
$153$	1188.59	0.628049
$154$	0	0
$155$	1016.21	0.526604
$156$	−1672.13	−0.858187
$157$	−2925.09	−1.48693	−0.743464	−	0.668776i	$-0.766818\pi$
$157$	−2925.09	−1.48693	−0.743464	+	0.668776i	$0.766818\pi$
$158$	2529.73	1.27376
$159$	18.1451	0.00905029
$160$	3160.86	1.56180
$161$	1959.65	0.959267
$162$	415.091	0.201312
$163$	−633.267	−0.304302	−0.152151	−	0.988357i	$-0.548620\pi$
$163$	−633.267	−0.304302	−0.152151	+	0.988357i	$0.548620\pi$
$164$	5225.64	2.48813
$165$	0	0
$166$	−1388.93	−0.649410
$167$	−1533.33	−0.710496	−0.355248	−	0.934772i	$-0.615604\pi$
$167$	−1533.33	−0.710496	−0.355248	+	0.934772i	$0.615604\pi$
$168$	1946.74	0.894011
$169$	−1265.40	−0.575965
$170$	10108.8	4.56065
$171$	−200.761	−0.0897813
$172$	1890.94	0.838274
$173$	3326.68	1.46198	0.730991	−	0.682388i	$-0.239058\pi$
$173$	3326.68	1.46198	0.730991	+	0.682388i	$0.239058\pi$
$174$	733.582	0.319613
$175$	−1210.63	−0.522945
$176$	0	0
$177$	664.857	0.282337
$178$	−397.099	−0.167212
$179$	4261.40	1.77940	0.889698	−	0.456549i	$-0.150915\pi$
$179$	4261.40	1.77940	0.889698	+	0.456549i	$0.150915\pi$
$180$	2454.87	1.01653
$181$	741.930	0.304681	0.152340	−	0.988328i	$-0.451319\pi$
$181$	741.930	0.304681	0.152340	+	0.988328i	$0.451319\pi$
$182$	−1930.18	−0.786125
$183$	823.013	0.332453
$184$	−8350.55	−3.34571
$185$	2366.11	0.940323
$186$	−1045.94	−0.412323
$187$	0	0
$188$	−8477.59	−3.28879
$189$	333.187	0.128232
$190$	−1707.46	−0.651958
$191$	4132.16	1.56541	0.782704	−	0.622394i	$-0.213840\pi$
$191$	4132.16	1.56541	0.782704	+	0.622394i	$0.213840\pi$
$192$	−292.105	−0.109796
$193$	3081.78	1.14938	0.574692	−	0.818370i	$-0.305122\pi$
$193$	3081.78	1.14938	0.574692	+	0.818370i	$0.305122\pi$
$194$	−6505.07	−2.40741
$195$	−1367.70	−0.502272
$196$	−3482.76	−1.26923
$197$	3059.84	1.10662	0.553310	−	0.832975i	$-0.313364\pi$
$197$	3059.84	1.10662	0.553310	+	0.832975i	$0.313364\pi$
$198$	0	0
$199$	−971.928	−0.346222	−0.173111	−	0.984902i	$-0.555382\pi$
$199$	−971.928	−0.346222	−0.173111	+	0.984902i	$0.555382\pi$
$200$	5158.81	1.82391
$201$	−561.533	−0.197052
$202$	−6344.54	−2.20990
$203$	588.835	0.203587
$204$	−7235.04	−2.48311
$205$	4274.27	1.45623
$206$	4673.43	1.58065
$207$	−1429.21	−0.479889
$208$	3765.98	1.25540
$209$	0	0
$210$	2833.73	0.931170
$211$	3327.16	1.08555	0.542775	−	0.839878i	$-0.317374\pi$
$211$	3327.16	1.08555	0.542775	+	0.839878i	$0.317374\pi$
$212$	−110.451	−0.0357821
$213$	1373.14	0.441719
$214$	−8113.97	−2.59187
$215$	1546.68	0.490617
$216$	−1419.79	−0.447244
$217$	−839.562	−0.262641
$218$	1521.48	0.472694
$219$	810.996	0.250237
$220$	0	0
$221$	4030.91	1.22692
$222$	−2435.34	−0.736259
$223$	4915.02	1.47594	0.737969	−	0.674834i	$-0.235785\pi$
$223$	4915.02	1.47594	0.737969	+	0.674834i	$0.235785\pi$
$224$	−2611.41	−0.778939
$225$	882.939	0.261612
$226$	−560.651	−0.165017
$227$	−1062.64	−0.310706	−0.155353	−	0.987859i	$-0.549651\pi$
$227$	−1062.64	−0.310706	−0.155353	+	0.987859i	$0.549651\pi$
$228$	1222.06	0.354968
$229$	−4312.88	−1.24456	−0.622278	−	0.782796i	$-0.713793\pi$
$229$	−4312.88	−1.24456	−0.622278	+	0.782796i	$0.713793\pi$
$230$	−12155.3	−3.48477
$231$	0	0
$232$	−2509.17	−0.710065
$233$	5154.61	1.44931	0.724655	−	0.689111i	$-0.241999\pi$
$233$	5154.61	1.44931	0.724655	+	0.689111i	$0.241999\pi$
$234$	1407.72	0.393272
$235$	−6934.17	−1.92483
$236$	−4047.05	−1.11627
$237$	−1480.94	−0.405896
$238$	−8351.62	−2.27460
$239$	−124.293	−0.0336395	−0.0168197	−	0.999859i	$-0.505354\pi$
$239$	−124.293	−0.0336395	−0.0168197	+	0.999859i	$0.505354\pi$
$240$	−5528.89	−1.48703
$241$	−445.539	−0.119086	−0.0595429	−	0.998226i	$-0.518964\pi$
$241$	−445.539	−0.119086	−0.0595429	+	0.998226i	$0.518964\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−5009.76	−1.31442
$245$	−2848.69	−0.742841
$246$	−4399.34	−1.14021
$247$	−680.853	−0.175391
$248$	3577.58	0.916034
$249$	813.100	0.206940
$250$	−2058.70	−0.520815
$251$	1471.10	0.369940	0.184970	−	0.982744i	$-0.440781\pi$
$251$	1471.10	0.369940	0.184970	+	0.982744i	$0.440781\pi$
$252$	−2028.15	−0.506989
$253$	0	0
$254$	7109.70	1.75631
$255$	−5917.84	−1.45329
$256$	−6897.52	−1.68396
$257$	1463.01	0.355097	0.177549	−	0.984112i	$-0.443183\pi$
$257$	1463.01	0.355097	0.177549	+	0.984112i	$0.443183\pi$
$258$	−1591.94	−0.384146
$259$	−1954.81	−0.468981
$260$	8325.34	1.98583
$261$	−429.449	−0.101848
$262$	7764.91	1.83098
$263$	−734.717	−0.172261	−0.0861304	−	0.996284i	$-0.527450\pi$
$263$	−734.717	−0.172261	−0.0861304	+	0.996284i	$0.527450\pi$
$264$	0	0
$265$	−90.3423	−0.0209422
$266$	1410.65	0.325161
$267$	232.467	0.0532838
$268$	3418.11	0.779083
$269$	3427.29	0.776823	0.388411	−	0.921486i	$-0.373024\pi$
$269$	3427.29	0.776823	0.388411	+	0.921486i	$0.373024\pi$
$270$	−2066.69	−0.465833
$271$	1719.34	0.385398	0.192699	−	0.981258i	$-0.438276\pi$
$271$	1719.34	0.385398	0.192699	+	0.981258i	$0.438276\pi$
$272$	16294.8	3.63243
$273$	1129.96	0.250506
$274$	−15220.5	−3.35584
$275$	0	0
$276$	8699.76	1.89733
$277$	5156.07	1.11841	0.559203	−	0.829031i	$-0.311107\pi$
$277$	5156.07	1.11841	0.559203	+	0.829031i	$0.311107\pi$
$278$	−11028.5	−2.37930
$279$	612.309	0.131391
$280$	−9692.59	−2.06872
$281$	5020.69	1.06587	0.532934	−	0.846157i	$-0.321089\pi$
$281$	5020.69	1.06587	0.532934	+	0.846157i	$0.321089\pi$
$282$	7137.07	1.50711
$283$	−1878.02	−0.394477	−0.197238	−	0.980356i	$-0.563197\pi$
$283$	−1878.02	−0.394477	−0.197238	+	0.980356i	$0.563197\pi$
$284$	−8358.46	−1.74642
$285$	999.569	0.207752
$286$	0	0
$287$	−3531.28	−0.726289
$288$	1904.55	0.389677
$289$	12528.2	2.55000
$290$	−3652.42	−0.739578
$291$	3808.16	0.767142
$292$	−4936.61	−0.989361
$293$	4811.63	0.959379	0.479690	−	0.877438i	$-0.340749\pi$
$293$	4811.63	0.959379	0.479690	+	0.877438i	$0.340749\pi$
$294$	2932.05	0.581634
$295$	−3310.25	−0.653323
$296$	8329.93	1.63570
$297$	0	0
$298$	−680.934	−0.132367
$299$	−4846.96	−0.937482
$300$	−5374.54	−1.03433
$301$	−1277.82	−0.244693
$302$	1091.44	0.207965
$303$	3714.18	0.704205
$304$	−2752.33	−0.519266
$305$	−4097.69	−0.769289
$306$	6091.00	1.13791
$307$	−598.905	−0.111340	−0.0556699	−	0.998449i	$-0.517729\pi$
$307$	−598.905	−0.111340	−0.0556699	+	0.998449i	$0.517729\pi$
$308$	0	0
$309$	−2735.89	−0.503688
$310$	5207.63	0.954108
$311$	5131.15	0.935566	0.467783	−	0.883843i	$-0.345053\pi$
$311$	5131.15	0.935566	0.467783	+	0.883843i	$0.345053\pi$
$312$	−4815.02	−0.873708
$313$	−9541.45	−1.72305	−0.861525	−	0.507716i	$-0.830490\pi$
$313$	−9541.45	−1.72305	−0.861525	+	0.507716i	$0.830490\pi$
$314$	−14989.9	−2.69403
$315$	−1658.90	−0.296726
$316$	9014.64	1.60479
$317$	6603.42	1.16998	0.584992	−	0.811039i	$-0.301098\pi$
$317$	6603.42	1.16998	0.584992	+	0.811039i	$0.301098\pi$
$318$	92.9858	0.0163974
$319$	0	0
$320$	1454.36	0.254066
$321$	4750.03	0.825922
$322$	10042.4	1.73801
$323$	−2945.95	−0.507483
$324$	1479.17	0.253629
$325$	2994.36	0.511068
$326$	−3245.23	−0.551339
$327$	−890.693	−0.150628
$328$	15047.7	2.53313
$329$	5728.82	0.960000
$330$	0	0
$331$	−6549.79	−1.08764	−0.543820	−	0.839202i	$-0.683023\pi$
$331$	−6549.79	−1.08764	−0.543820	+	0.839202i	$0.683023\pi$
$332$	−4949.43	−0.818178
$333$	1425.68	0.234615
$334$	−7857.69	−1.28729
$335$	2795.81	0.455975
$336$	4567.81	0.741651
$337$	−2690.38	−0.434879	−0.217439	−	0.976074i	$-0.569770\pi$
$337$	−2690.38	−0.434879	−0.217439	+	0.976074i	$0.569770\pi$
$338$	−6484.62	−1.04354
$339$	328.213	0.0525843
$340$	36022.5	5.74587
$341$	0	0
$342$	−1028.82	−0.162667
$343$	6586.22	1.03680
$344$	5445.12	0.853434
$345$	7115.89	1.11045
$346$	17047.8	2.64884
$347$	−778.334	−0.120413	−0.0602063	−	0.998186i	$-0.519176\pi$
$347$	−778.334	−0.120413	−0.0602063	+	0.998186i	$0.519176\pi$
$348$	2614.10	0.402674
$349$	−5416.06	−0.830702	−0.415351	−	0.909661i	$-0.636341\pi$
$349$	−5416.06	−0.830702	−0.415351	+	0.909661i	$0.636341\pi$
$350$	−6203.99	−0.947478
$351$	−824.099	−0.125320
$352$	0	0
$353$	−4504.51	−0.679181	−0.339591	−	0.940573i	$-0.610289\pi$
$353$	−4504.51	−0.679181	−0.339591	+	0.940573i	$0.610289\pi$
$354$	3407.11	0.511542
$355$	−6836.73	−1.02213
$356$	−1415.05	−0.210667
$357$	4889.15	0.724822
$358$	21837.9	3.22393
$359$	8588.42	1.26262	0.631308	−	0.775532i	$-0.282518\pi$
$359$	8588.42	1.26262	0.631308	+	0.775532i	$0.282518\pi$
$360$	7068.99	1.03491
$361$	−6361.41	−0.927454
$362$	3802.08	0.552025
$363$	0	0
$364$	−6878.16	−0.990422
$365$	−4037.86	−0.579045
$366$	4217.59	0.602342
$367$	2178.08	0.309795	0.154897	−	0.987931i	$-0.450495\pi$
$367$	2178.08	0.309795	0.154897	+	0.987931i	$0.450495\pi$
$368$	−19593.7	−2.77552
$369$	2575.43	0.363338
$370$	12125.3	1.70369
$371$	74.6383	0.0104448
$372$	−3727.19	−0.519478
$373$	−12814.7	−1.77888	−0.889440	−	0.457052i	$-0.848905\pi$
$373$	−12814.7	−1.77888	−0.889440	+	0.457052i	$0.848905\pi$
$374$	0	0
$375$	1205.19	0.165963
$376$	−24411.9	−3.34826
$377$	−1456.41	−0.198963
$378$	1707.44	0.232332
$379$	7302.82	0.989764	0.494882	−	0.868960i	$-0.335211\pi$
$379$	7302.82	0.989764	0.494882	+	0.868960i	$0.335211\pi$
$380$	−6084.48	−0.821388
$381$	−4162.12	−0.559663
$382$	21175.6	2.83622
$383$	11314.0	1.50945	0.754726	−	0.656040i	$-0.227770\pi$
$383$	11314.0	1.50945	0.754726	+	0.656040i	$0.227770\pi$
$384$	3581.90	0.476010
$385$	0	0
$386$	15792.8	2.08247
$387$	931.942	0.122412
$388$	−23180.7	−3.03304
$389$	−2955.81	−0.385259	−0.192629	−	0.981272i	$-0.561701\pi$
$389$	−2955.81	−0.385259	−0.192629	+	0.981272i	$0.561701\pi$
$390$	−7008.89	−0.910023
$391$	−20972.1	−2.71254
$392$	−10028.9	−1.29218
$393$	−4545.69	−0.583460
$394$	15680.4	2.00499
$395$	7373.45	0.939236
$396$	0	0
$397$	12391.9	1.56658	0.783288	−	0.621659i	$-0.213541\pi$
$397$	12391.9	1.56658	0.783288	+	0.621659i	$0.213541\pi$
$398$	−4980.72	−0.627289
$399$	−825.816	−0.103615
$400$	12104.6	1.51308
$401$	7592.65	0.945533	0.472766	−	0.881188i	$-0.343255\pi$
$401$	7592.65	0.945533	0.472766	+	0.881188i	$0.343255\pi$
$402$	−2877.62	−0.357022
$403$	2076.56	0.256677
$404$	−22608.6	−2.78421
$405$	1209.87	0.148442
$406$	3017.53	0.368861
$407$	0	0
$408$	−20833.9	−2.52802
$409$	−7275.77	−0.879618	−0.439809	−	0.898091i	$-0.644954\pi$
$409$	−7275.77	−0.879618	−0.439809	+	0.898091i	$0.644954\pi$
$410$	21903.8	2.63842
$411$	8910.26	1.06937
$412$	16653.7	1.99143
$413$	2734.84	0.325842
$414$	−7324.11	−0.869469
$415$	−4048.34	−0.478856
$416$	6459.02	0.761248
$417$	6456.24	0.758186
$418$	0	0
$419$	11095.7	1.29370	0.646851	−	0.762616i	$-0.276085\pi$
$419$	11095.7	1.29370	0.646851	+	0.762616i	$0.276085\pi$
$420$	10097.9	1.17316
$421$	−5103.54	−0.590811	−0.295406	−	0.955372i	$-0.595455\pi$
$421$	−5103.54	−0.590811	−0.295406	+	0.955372i	$0.595455\pi$
$422$	17050.3	1.96681
$423$	−4178.14	−0.480255
$424$	−318.052	−0.0364292
$425$	12956.2	1.47874
$426$	7036.78	0.800312
$427$	3385.40	0.383679
$428$	−28913.9	−3.26544
$429$	0	0
$430$	7926.09	0.888906
$431$	−14437.7	−1.61355	−0.806774	−	0.590861i	$-0.798788\pi$
$431$	−14437.7	−1.61355	−0.806774	+	0.590861i	$0.798788\pi$
$432$	−3331.40	−0.371023
$433$	4236.83	0.470229	0.235114	−	0.971968i	$-0.424454\pi$
$433$	4236.83	0.470229	0.235114	+	0.971968i	$0.424454\pi$
$434$	−4302.40	−0.475857
$435$	2138.18	0.235673
$436$	5421.74	0.595538
$437$	3542.35	0.387766
$438$	4156.01	0.453383
$439$	1809.83	0.196761	0.0983807	−	0.995149i	$-0.468634\pi$
$439$	1809.83	0.196761	0.0983807	+	0.995149i	$0.468634\pi$
$440$	0	0
$441$	−1716.46	−0.185343
$442$	20656.7	2.22294
$443$	4797.37	0.514514	0.257257	−	0.966343i	$-0.417181\pi$
$443$	4797.37	0.514514	0.257257	+	0.966343i	$0.417181\pi$
$444$	−8678.28	−0.927597
$445$	−1157.43	−0.123298
$446$	25187.4	2.67412
$447$	398.628	0.0421800
$448$	−1201.55	−0.126714
$449$	−9177.20	−0.964585	−0.482293	−	0.876010i	$-0.660196\pi$
$449$	−9177.20	−0.964585	−0.482293	+	0.876010i	$0.660196\pi$
$450$	4524.69	0.473991
$451$	0	0
$452$	−1997.87	−0.207902
$453$	−638.946	−0.0662700
$454$	−5445.61	−0.562940
$455$	−5625.93	−0.579665
$456$	3519.01	0.361387
$457$	6179.47	0.632524	0.316262	−	0.948672i	$-0.397572\pi$
$457$	6179.47	0.632524	0.316262	+	0.948672i	$0.397572\pi$
$458$	−22101.7	−2.25490
$459$	−3565.76	−0.362604
$460$	−43315.2	−4.39039
$461$	−16827.2	−1.70004	−0.850022	−	0.526747i	$-0.823411\pi$
$461$	−16827.2	−1.70004	−0.850022	+	0.526747i	$0.823411\pi$
$462$	0	0
$463$	−818.694	−0.0821770	−0.0410885	−	0.999156i	$-0.513083\pi$
$463$	−818.694	−0.0821770	−0.0410885	+	0.999156i	$0.513083\pi$
$464$	−5887.51	−0.589053
$465$	−3048.62	−0.304035
$466$	26415.2	2.62588
$467$	−16789.7	−1.66367	−0.831836	−	0.555021i	$-0.812710\pi$
$467$	−16789.7	−1.66367	−0.831836	+	0.555021i	$0.812710\pi$
$468$	5016.38	0.495475
$469$	−2309.82	−0.227415
$470$	−35534.7	−3.48743
$471$	8775.27	0.858478
$472$	−11653.8	−1.13646
$473$	0	0
$474$	−7589.20	−0.735408
$475$	−2188.40	−0.211391
$476$	−29760.8	−2.86572
$477$	−54.4352	−0.00522519
$478$	−636.949	−0.0609485
$479$	−4656.56	−0.444183	−0.222092	−	0.975026i	$-0.571288\pi$
$479$	−4656.56	−0.444183	−0.222092	+	0.975026i	$0.571288\pi$
$480$	−9482.57	−0.901704
$481$	4835.00	0.458330
$482$	−2283.20	−0.215761
$483$	−5878.95	−0.553833
$484$	0	0
$485$	−18960.4	−1.77515
$486$	−1245.27	−0.116228
$487$	17627.2	1.64018	0.820089	−	0.572236i	$-0.193924\pi$
$487$	17627.2	1.64018	0.820089	+	0.572236i	$0.193924\pi$
$488$	−14426.0	−1.33819
$489$	1899.80	0.175689
$490$	−14598.3	−1.34589
$491$	−14954.8	−1.37454	−0.687269	−	0.726403i	$-0.741191\pi$
$491$	−14954.8	−1.37454	−0.687269	+	0.726403i	$0.741191\pi$
$492$	−15676.9	−1.43653
$493$	−6301.69	−0.575687
$494$	−3489.09	−0.317776
$495$	0	0
$496$	8394.41	0.759920
$497$	5648.32	0.509782
$498$	4166.80	0.374937
$499$	−21474.7	−1.92653	−0.963265	−	0.268554i	$-0.913454\pi$
$499$	−21474.7	−1.92653	−0.963265	+	0.268554i	$0.913454\pi$
$500$	−7336.14	−0.656164
$501$	4600.00	0.410205
$502$	7538.77	0.670263
$503$	11832.4	1.04887	0.524435	−	0.851450i	$-0.324277\pi$
$503$	11832.4	1.04887	0.524435	+	0.851450i	$0.324277\pi$
$504$	−5840.21	−0.516158
$505$	−18492.5	−1.62952
$506$	0	0
$507$	3796.19	0.332534
$508$	25335.2	2.21274
$509$	−13571.6	−1.18183	−0.590914	−	0.806735i	$-0.701233\pi$
$509$	−13571.6	−1.18183	−0.590914	+	0.806735i	$0.701233\pi$
$510$	−30326.4	−2.63309
$511$	3335.97	0.288796
$512$	−25795.1	−2.22655
$513$	602.284	0.0518353
$514$	7497.31	0.643370
$515$	13621.7	1.16552
$516$	−5672.83	−0.483977
$517$	0	0
$518$	−10017.6	−0.849706
$519$	−9980.04	−0.844075
$520$	23973.5	2.02174
$521$	−2124.44	−0.178644	−0.0893221	−	0.996003i	$-0.528470\pi$
$521$	−2124.44	−0.178644	−0.0893221	+	0.996003i	$0.528470\pi$
$522$	−2200.75	−0.184529
$523$	−5558.58	−0.464741	−0.232371	−	0.972627i	$-0.574648\pi$
$523$	−5558.58	−0.464741	−0.232371	+	0.972627i	$0.574648\pi$
$524$	27670.1	2.30682
$525$	3631.90	0.301922
$526$	−3765.11	−0.312104
$527$	8984.95	0.742677
$528$	0	0
$529$	13050.8	1.07264
$530$	−462.966	−0.0379433
$531$	−1994.57	−0.163008
$532$	5026.83	0.409663
$533$	8734.21	0.709794
$534$	1191.30	0.0965402
$535$	−23649.9	−1.91117
$536$	9842.72	0.793173
$537$	−12784.2	−1.02733
$538$	17563.4	1.40746
$539$	0	0
$540$	−7364.61	−0.586893
$541$	13186.0	1.04789	0.523946	−	0.851751i	$-0.324459\pi$
$541$	13186.0	1.04789	0.523946	+	0.851751i	$0.324459\pi$
$542$	8810.92	0.698268
$543$	−2225.79	−0.175908
$544$	27947.2	2.20262
$545$	4434.67	0.348551
$546$	5790.55	0.453869
$547$	20380.7	1.59308	0.796539	−	0.604587i	$-0.206662\pi$
$547$	20380.7	1.59308	0.796539	+	0.604587i	$0.206662\pi$
$548$	−54237.7	−4.22796
$549$	−2469.04	−0.191942
$550$	0	0
$551$	1064.40	0.0822961
$552$	25051.6	1.93165
$553$	−6091.74	−0.468439
$554$	26422.7	2.02634
$555$	−7098.32	−0.542895
$556$	−39299.8	−2.99763
$557$	−8523.55	−0.648392	−0.324196	−	0.945990i	$-0.605094\pi$
$557$	−8523.55	−0.648392	−0.324196	+	0.945990i	$0.605094\pi$
$558$	3137.82	0.238055
$559$	3160.55	0.239136
$560$	−22742.7	−1.71616
$561$	0	0
$562$	25728.9	1.93115
$563$	24660.1	1.84600	0.923000	−	0.384801i	$-0.125730\pi$
$563$	24660.1	1.84600	0.923000	+	0.384801i	$0.125730\pi$
$564$	25432.8	1.89878
$565$	−1634.14	−0.121679
$566$	−9624.08	−0.714718
$567$	−999.562	−0.0740346
$568$	−24068.9	−1.77801
$569$	6276.91	0.462464	0.231232	−	0.972899i	$-0.425724\pi$
$569$	6276.91	0.462464	0.231232	+	0.972899i	$0.425724\pi$
$570$	5122.37	0.376408
$571$	13032.1	0.955127	0.477563	−	0.878597i	$-0.341520\pi$
$571$	13032.1	0.955127	0.477563	+	0.878597i	$0.341520\pi$
$572$	0	0
$573$	−12396.5	−0.903789
$574$	−18096.3	−1.31590
$575$	−15579.1	−1.12990
$576$	876.314	0.0633908
$577$	7724.04	0.557289	0.278645	−	0.960394i	$-0.410115\pi$
$577$	7724.04	0.557289	0.278645	+	0.960394i	$0.410115\pi$
$578$	64201.6	4.62013
$579$	−9245.33	−0.663597
$580$	−13015.3	−0.931779
$581$	3344.63	0.238827
$582$	19515.2	1.38992
$583$	0	0
$584$	−14215.4	−1.00725
$585$	4103.10	0.289987
$586$	24657.6	1.73822
$587$	3858.98	0.271341	0.135670	−	0.990754i	$-0.456681\pi$
$587$	3858.98	0.271341	0.135670	+	0.990754i	$0.456681\pi$
$588$	10448.3	0.732788
$589$	−1517.63	−0.106168
$590$	−16963.7	−1.18370
$591$	−9179.51	−0.638908
$592$	19545.3	1.35694
$593$	26982.6	1.86854	0.934270	−	0.356567i	$-0.116053\pi$
$593$	26982.6	1.86854	0.934270	+	0.356567i	$0.116053\pi$
$594$	0	0
$595$	−24342.6	−1.67722
$596$	−2426.49	−0.166767
$597$	2915.78	0.199891
$598$	−24838.6	−1.69854
$599$	−22912.4	−1.56289	−0.781447	−	0.623971i	$-0.785518\pi$
$599$	−22912.4	−1.56289	−0.781447	+	0.623971i	$0.785518\pi$
$600$	−15476.4	−1.05304
$601$	3091.89	0.209851	0.104926	−	0.994480i	$-0.466540\pi$
$601$	3091.89	0.209851	0.104926	+	0.994480i	$0.466540\pi$
$602$	−6548.31	−0.443338
$603$	1684.60	0.113768
$604$	3889.33	0.262011
$605$	0	0
$606$	19033.6	1.27589
$607$	−13059.0	−0.873225	−0.436612	−	0.899650i	$-0.643822\pi$
$607$	−13059.0	−0.873225	−0.436612	+	0.899650i	$0.643822\pi$
$608$	−4720.50	−0.314871
$609$	−1766.50	−0.117541
$610$	−20999.0	−1.39381
$611$	−14169.5	−0.938197
$612$	21705.1	1.43362
$613$	−27025.2	−1.78065	−0.890325	−	0.455326i	$-0.849523\pi$
$613$	−27025.2	−1.78065	−0.890325	+	0.455326i	$0.849523\pi$
$614$	−3069.13	−0.201727
$615$	−12822.8	−0.840757
$616$	0	0
$617$	23885.0	1.55847	0.779235	−	0.626732i	$-0.215608\pi$
$617$	23885.0	1.55847	0.779235	+	0.626732i	$0.215608\pi$
$618$	−14020.3	−0.912588
$619$	−14347.6	−0.931631	−0.465815	−	0.884882i	$-0.654239\pi$
$619$	−14347.6	−0.931631	−0.465815	+	0.884882i	$0.654239\pi$
$620$	18557.3	1.20206
$621$	4287.63	0.277064
$622$	26295.0	1.69507
$623$	956.236	0.0614940
$624$	−11297.9	−0.724807
$625$	−18263.6	−1.16887
$626$	−48895.9	−3.12184
$627$	0	0
$628$	−53416.0	−3.39415
$629$	20920.3	1.32615
$630$	−8501.18	−0.537611
$631$	4672.42	0.294780	0.147390	−	0.989078i	$-0.452913\pi$
$631$	4672.42	0.294780	0.147390	+	0.989078i	$0.452913\pi$
$632$	25958.4	1.63381
$633$	−9981.48	−0.626743
$634$	33839.7	2.11979
$635$	20722.7	1.29505
$636$	331.353	0.0206588
$637$	−5821.13	−0.362074
$638$	0	0
$639$	−4119.43	−0.255027
$640$	−17833.9	−1.10148
$641$	1007.49	0.0620802	0.0310401	−	0.999518i	$-0.490118\pi$
$641$	1007.49	0.0620802	0.0310401	+	0.999518i	$0.490118\pi$
$642$	24341.9	1.49642
$643$	−238.620	−0.0146349	−0.00731747	−	0.999973i	$-0.502329\pi$
$643$	−238.620	−0.0146349	−0.00731747	+	0.999973i	$0.502329\pi$
$644$	35785.8	2.18969
$645$	−4640.04	−0.283258
$646$	−15096.8	−0.919464
$647$	8030.33	0.487952	0.243976	−	0.969781i	$-0.421548\pi$
$647$	8030.33	0.487952	0.243976	+	0.969781i	$0.421548\pi$
$648$	4259.38	0.258216
$649$	0	0
$650$	15344.8	0.925960
$651$	2518.68	0.151636
$652$	−11564.3	−0.694620
$653$	292.050	0.0175020	0.00875101	−	0.999962i	$-0.497214\pi$
$653$	292.050	0.0175020	0.00875101	+	0.999962i	$0.497214\pi$
$654$	−4564.43	−0.272910
$655$	22632.5	1.35011
$656$	35307.8	2.10143
$657$	−2432.99	−0.144475
$658$	29357.8	1.73934
$659$	−9034.21	−0.534025	−0.267013	−	0.963693i	$-0.586037\pi$
$659$	−9034.21	−0.534025	−0.267013	+	0.963693i	$0.586037\pi$
$660$	0	0
$661$	11884.6	0.699332	0.349666	−	0.936874i	$-0.386295\pi$
$661$	11884.6	0.699332	0.349666	+	0.936874i	$0.386295\pi$
$662$	−33564.9	−1.97060
$663$	−12092.7	−0.708361
$664$	−14252.3	−0.832975
$665$	4111.65	0.239764
$666$	7306.02	0.425079
$667$	7577.45	0.439880
$668$	−28000.7	−1.62182
$669$	−14745.1	−0.852133
$670$	14327.4	0.826141
$671$	0	0
$672$	7834.23	0.449720
$673$	−7483.49	−0.428629	−0.214314	−	0.976765i	$-0.568752\pi$
$673$	−7483.49	−0.428629	−0.214314	+	0.976765i	$0.568752\pi$
$674$	−13787.0	−0.787919
$675$	−2648.82	−0.151042
$676$	−23107.8	−1.31474
$677$	−31698.6	−1.79952	−0.899761	−	0.436384i	$-0.856259\pi$
$677$	−31698.6	−1.79952	−0.899761	+	0.436384i	$0.856259\pi$
$678$	1681.95	0.0952729
$679$	15664.6	0.885347
$680$	103730.	5.84978
$681$	3187.93	0.179386
$682$	0	0
$683$	−7905.89	−0.442914	−0.221457	−	0.975170i	$-0.571081\pi$
$683$	−7905.89	−0.442914	−0.221457	+	0.975170i	$0.571081\pi$
$684$	−3666.17	−0.204941
$685$	−44363.3	−2.47450
$686$	33751.6	1.87849
$687$	12938.6	0.718545
$688$	12776.4	0.707989
$689$	−184.609	−0.0102076
$690$	36465.9	2.01193
$691$	−6903.03	−0.380034	−0.190017	−	0.981781i	$-0.560854\pi$
$691$	−6903.03	−0.380034	−0.190017	+	0.981781i	$0.560854\pi$
$692$	60749.5	3.33721
$693$	0	0
$694$	−3988.64	−0.218165
$695$	−32144.9	−1.75443
$696$	7527.51	0.409956
$697$	37791.6	2.05374
$698$	−27755.0	−1.50508
$699$	−15463.8	−0.836760
$700$	−22107.8	−1.19371
$701$	20702.1	1.11542	0.557709	−	0.830036i	$-0.311680\pi$
$701$	20702.1	1.11542	0.557709	+	0.830036i	$0.311680\pi$
$702$	−4223.16	−0.227056
$703$	−3533.61	−0.189577
$704$	0	0
$705$	20802.5	1.11130
$706$	−23083.7	−1.23055
$707$	15278.0	0.812713
$708$	12141.2	0.644481
$709$	−6302.17	−0.333826	−0.166913	−	0.985972i	$-0.553380\pi$
$709$	−6302.17	−0.333826	−0.166913	+	0.985972i	$0.553380\pi$
$710$	−35035.4	−1.85191
$711$	4442.82	0.234344
$712$	−4074.76	−0.214477
$713$	−10803.9	−0.567476
$714$	25054.9	1.31324
$715$	0	0
$716$	77818.7	4.06176
$717$	372.879	0.0194218
$718$	44012.0	2.28762
$719$	1385.72	0.0718756	0.0359378	−	0.999354i	$-0.488558\pi$
$719$	1385.72	0.0718756	0.0359378	+	0.999354i	$0.488558\pi$
$720$	16586.7	0.858539
$721$	−11253.9	−0.581299
$722$	−32599.5	−1.68037
$723$	1336.62	0.0687542
$724$	13548.6	0.695484
$725$	−4681.20	−0.239801
$726$	0	0
$727$	−11818.1	−0.602900	−0.301450	−	0.953482i	$-0.597471\pi$
$727$	−11818.1	−0.602900	−0.301450	+	0.953482i	$0.597471\pi$
$728$	−19806.2	−1.00833
$729$	729.000	0.0370370
$730$	−20692.3	−1.04912
$731$	13675.2	0.691924
$732$	15029.3	0.758878
$733$	−9537.65	−0.480602	−0.240301	−	0.970698i	$-0.577246\pi$
$733$	−9537.65	−0.480602	−0.240301	+	0.970698i	$0.577246\pi$
$734$	11161.7	0.561290
$735$	8546.07	0.428880
$736$	−33605.1	−1.68301
$737$	0	0
$738$	13198.0	0.658300
$739$	−3411.51	−0.169817	−0.0849084	−	0.996389i	$-0.527060\pi$
$739$	−3411.51	−0.169817	−0.0849084	+	0.996389i	$0.527060\pi$
$740$	43208.2	2.14644
$741$	2042.56	0.101262
$742$	382.490	0.0189240
$743$	−2452.91	−0.121115	−0.0605576	−	0.998165i	$-0.519288\pi$
$743$	−2452.91	−0.121115	−0.0605576	+	0.998165i	$0.519288\pi$
$744$	−10732.7	−0.528872
$745$	−1984.73	−0.0976037
$746$	−65670.2	−3.22300
$747$	−2439.30	−0.119477
$748$	0	0
$749$	19538.9	0.953185
$750$	6176.11	0.300693
$751$	−11556.1	−0.561500	−0.280750	−	0.959781i	$-0.590583\pi$
$751$	−11556.1	−0.561500	−0.280750	+	0.959781i	$0.590583\pi$
$752$	−57280.0	−2.77764
$753$	−4413.30	−0.213585
$754$	−7463.51	−0.360484
$755$	3181.24	0.153347
$756$	6084.44	0.292710
$757$	25642.8	1.23118	0.615589	−	0.788067i	$-0.288918\pi$
$757$	25642.8	1.23118	0.615589	+	0.788067i	$0.288918\pi$
$758$	37423.9	1.79327
$759$	0	0
$760$	−17520.8	−0.836243
$761$	−6054.02	−0.288381	−0.144191	−	0.989550i	$-0.546058\pi$
$761$	−6054.02	−0.288381	−0.144191	+	0.989550i	$0.546058\pi$
$762$	−21329.1	−1.01401
$763$	−3663.80	−0.173838
$764$	75458.7	3.57330
$765$	17753.5	0.839058
$766$	57979.7	2.73484
$767$	−6764.30	−0.318442
$768$	20692.5	0.972237
$769$	23509.7	1.10245	0.551224	−	0.834357i	$-0.314161\pi$
$769$	23509.7	1.10245	0.551224	+	0.834357i	$0.314161\pi$
$770$	0	0
$771$	−4389.03	−0.205015
$772$	56277.3	2.62366
$773$	4985.36	0.231967	0.115984	−	0.993251i	$-0.462998\pi$
$773$	4985.36	0.231967	0.115984	+	0.993251i	$0.462998\pi$
$774$	4775.81	0.221787
$775$	6674.46	0.309360
$776$	−66750.6	−3.08789
$777$	5864.44	0.270766
$778$	−15147.3	−0.698016
$779$	−6383.30	−0.293589
$780$	−24976.0	−1.14652
$781$	0	0
$782$	−107473.	−4.91462
$783$	1288.35	0.0588017
$784$	−23531.7	−1.07196
$785$	−43691.1	−1.98650
$786$	−23294.7	−1.05712
$787$	23093.6	1.04599	0.522996	−	0.852335i	$-0.324814\pi$
$787$	23093.6	1.04599	0.522996	+	0.852335i	$0.324814\pi$
$788$	55876.6	2.52604
$789$	2204.15	0.0994548
$790$	37785.8	1.70172
$791$	1350.08	0.0606868
$792$	0	0
$793$	−8373.39	−0.374965
$794$	63503.2	2.83834
$795$	271.027	0.0120910
$796$	−17748.7	−0.790308
$797$	17376.5	0.772281	0.386141	−	0.922440i	$-0.373808\pi$
$797$	17376.5	0.772281	0.386141	+	0.922440i	$0.373808\pi$
$798$	−4231.96	−0.187732
$799$	−61309.6	−2.71461
$800$	20760.6	0.917496
$801$	−697.402	−0.0307634
$802$	38909.1	1.71313
$803$	0	0
$804$	−10254.3	−0.449804
$805$	29270.7	1.28156
$806$	10641.5	0.465050
$807$	−10281.9	−0.448499
$808$	−65103.3	−2.83456
$809$	3335.36	0.144951	0.0724753	−	0.997370i	$-0.476910\pi$
$809$	3335.36	0.144951	0.0724753	+	0.997370i	$0.476910\pi$
$810$	6200.08	0.268949
$811$	4873.73	0.211023	0.105512	−	0.994418i	$-0.466352\pi$
$811$	4873.73	0.211023	0.105512	+	0.994418i	$0.466352\pi$
$812$	10752.9	0.464720
$813$	−5158.03	−0.222509
$814$	0	0
$815$	−9458.91	−0.406541
$816$	−48884.5	−2.09718
$817$	−2309.85	−0.0989125
$818$	−37285.3	−1.59370
$819$	−3389.87	−0.144630
$820$	78053.7	3.32409
$821$	18640.4	0.792392	0.396196	−	0.918166i	$-0.370330\pi$
$821$	18640.4	0.792392	0.396196	+	0.918166i	$0.370330\pi$
$822$	45661.4	1.93750
$823$	18819.8	0.797106	0.398553	−	0.917145i	$-0.369512\pi$
$823$	18819.8	0.797106	0.398553	+	0.917145i	$0.369512\pi$
$824$	47955.6	2.02744
$825$	0	0
$826$	14014.9	0.590364
$827$	−34989.3	−1.47122	−0.735610	−	0.677406i	$-0.763104\pi$
$827$	−34989.3	−1.47122	−0.735610	+	0.677406i	$0.763104\pi$
$828$	−26099.3	−1.09543
$829$	14231.8	0.596250	0.298125	−	0.954527i	$-0.403639\pi$
$829$	14231.8	0.596250	0.298125	+	0.954527i	$0.403639\pi$
$830$	−20746.0	−0.867597
$831$	−15468.2	−0.645712
$832$	2971.89	0.123836
$833$	−25187.2	−1.04764
$834$	33085.5	1.37369
$835$	−22902.9	−0.949207
$836$	0	0
$837$	−1836.93	−0.0758584
$838$	56860.9	2.34395
$839$	−25354.5	−1.04331	−0.521654	−	0.853157i	$-0.674685\pi$
$839$	−25354.5	−1.04331	−0.521654	+	0.853157i	$0.674685\pi$
$840$	29077.8	1.19438
$841$	−22112.1	−0.906644
$842$	−26153.5	−1.07044
$843$	−15062.1	−0.615380
$844$	60758.3	2.47795
$845$	−18900.8	−0.769476
$846$	−21411.2	−0.870133
$847$	0	0
$848$	−746.276	−0.0302208
$849$	5634.07	0.227751
$850$	66394.9	2.67921
$851$	−25155.6	−1.01330
$852$	25075.4	1.00830
$853$	−9664.40	−0.387928	−0.193964	−	0.981009i	$-0.562135\pi$
$853$	−9664.40	−0.387928	−0.193964	+	0.981009i	$0.562135\pi$
$854$	17348.7	0.695154
$855$	−2998.71	−0.119946
$856$	−83260.0	−3.32449
$857$	−38619.7	−1.53935	−0.769676	−	0.638435i	$-0.779582\pi$
$857$	−38619.7	−1.53935	−0.769676	+	0.638435i	$0.779582\pi$
$858$	0	0
$859$	−30501.9	−1.21154	−0.605769	−	0.795641i	$-0.707134\pi$
$859$	−30501.9	−1.21154	−0.605769	+	0.795641i	$0.707134\pi$
$860$	28244.4	1.11991
$861$	10593.8	0.419323
$862$	−73987.1	−2.92344
$863$	−29563.6	−1.16611	−0.583056	−	0.812432i	$-0.698143\pi$
$863$	−29563.6	−1.16611	−0.583056	+	0.812432i	$0.698143\pi$
$864$	−5713.66	−0.224980
$865$	49689.5	1.95317
$866$	21712.0	0.851966
$867$	−37584.5	−1.47225
$868$	−15331.5	−0.599522
$869$	0	0
$870$	10957.3	0.426996
$871$	5713.07	0.222250
$872$	15612.3	0.606308
$873$	−11424.5	−0.442909
$874$	18153.1	0.702559
$875$	4957.47	0.191535
$876$	14809.8	0.571208
$877$	−23007.5	−0.885872	−0.442936	−	0.896553i	$-0.646063\pi$
$877$	−23007.5	−0.885872	−0.442936	+	0.896553i	$0.646063\pi$
$878$	9274.60	0.356495
$879$	−14434.9	−0.553898
$880$	0	0
$881$	34057.4	1.30241	0.651205	−	0.758902i	$-0.274264\pi$
$881$	34057.4	1.30241	0.651205	+	0.758902i	$0.274264\pi$
$882$	−8796.14	−0.335806
$883$	34653.7	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$883$	34653.7	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$884$	73609.8	2.80064
$885$	9930.76	0.377196
$886$	24584.5	0.932203
$887$	−5147.89	−0.194870	−0.0974348	−	0.995242i	$-0.531064\pi$
$887$	−5147.89	−0.194870	−0.0974348	+	0.995242i	$0.531064\pi$
$888$	−24989.8	−0.944372
$889$	−17120.6	−0.645900
$890$	−5931.34	−0.223392
$891$	0	0
$892$	89754.8	3.36907
$893$	10355.7	0.388062
$894$	2042.80	0.0764223
$895$	63651.2	2.37723
$896$	14733.8	0.549356
$897$	14540.9	0.541255
$898$	−47029.3	−1.74765
$899$	−3246.36	−0.120436
$900$	16123.6	0.597172
$901$	−798.776	−0.0295351
$902$	0	0
$903$	3833.47	0.141273
$904$	−5753.02	−0.211662
$905$	11082.0	0.407047
$906$	−3274.33	−0.120069
$907$	40684.2	1.48941	0.744706	−	0.667392i	$-0.232589\pi$
$907$	40684.2	1.48941	0.744706	+	0.667392i	$0.232589\pi$
$908$	−19405.3	−0.709237
$909$	−11142.5	−0.406573
$910$	−28830.5	−1.05024
$911$	25223.0	0.917315	0.458657	−	0.888613i	$-0.348331\pi$
$911$	25223.0	0.917315	0.458657	+	0.888613i	$0.348331\pi$
$912$	8256.98	0.299798
$913$	0	0
$914$	31667.2	1.14601
$915$	12293.1	0.444149
$916$	−78758.9	−2.84090
$917$	−18698.3	−0.673363
$918$	−18273.0	−0.656970
$919$	−9198.23	−0.330165	−0.165083	−	0.986280i	$-0.552789\pi$
$919$	−9198.23	−0.330165	−0.165083	+	0.986280i	$0.552789\pi$
$920$	−124729.	−4.46979
$921$	1796.71	0.0642820
$922$	−86232.2	−3.08016
$923$	−13970.4	−0.498205
$924$	0	0
$925$	15540.6	0.552403
$926$	−4195.46	−0.148889
$927$	8207.68	0.290804
$928$	−10097.6	−0.357189
$929$	−33459.9	−1.18168	−0.590841	−	0.806788i	$-0.701204\pi$
$929$	−33459.9	−1.18168	−0.590841	+	0.806788i	$0.701204\pi$
$930$	−15622.9	−0.550855
$931$	4254.31	0.149763
$932$	94129.9	3.30829
$933$	−15393.5	−0.540149
$934$	−86040.2	−3.01426
$935$	0	0
$936$	14445.1	0.504435
$937$	−31048.7	−1.08252	−0.541258	−	0.840857i	$-0.682052\pi$
$937$	−31048.7	−1.08252	−0.541258	+	0.840857i	$0.682052\pi$
$938$	−11836.9	−0.412034
$939$	28624.3	0.994803
$940$	−126627.	−4.39374
$941$	21640.9	0.749704	0.374852	−	0.927085i	$-0.377693\pi$
$941$	21640.9	0.749704	0.374852	+	0.927085i	$0.377693\pi$
$942$	44969.6	1.55540
$943$	−45442.4	−1.56926
$944$	−27344.5	−0.942783
$945$	4976.71	0.171315
$946$	0	0
$947$	−31725.1	−1.08862	−0.544311	−	0.838883i	$-0.683209\pi$
$947$	−31725.1	−1.08862	−0.544311	+	0.838883i	$0.683209\pi$
$948$	−27043.9	−0.926525
$949$	−8251.12	−0.282237
$950$	−11214.6	−0.383000
$951$	−19810.3	−0.675490
$952$	−85698.6	−2.91755
$953$	52072.7	1.76999	0.884995	−	0.465601i	$-0.154162\pi$
$953$	52072.7	1.76999	0.884995	+	0.465601i	$0.154162\pi$
$954$	−278.957	−0.00946706
$955$	61720.8	2.09135
$956$	−2269.75	−0.0767877
$957$	0	0
$958$	−23862.9	−0.804777
$959$	36651.7	1.23414
$960$	−4363.08	−0.146685
$961$	−25162.3	−0.844629
$962$	24777.3	0.830408
$963$	−14250.1	−0.476846
$964$	−8136.13	−0.271833
$965$	46031.5	1.53555
$966$	−30127.1	−1.00344
$967$	−4519.31	−0.150291	−0.0751454	−	0.997173i	$-0.523942\pi$
$967$	−4519.31	−0.150291	−0.0751454	+	0.997173i	$0.523942\pi$
$968$	0	0
$969$	8837.85	0.292996
$970$	−97164.2	−3.21624
$971$	24955.7	0.824785	0.412392	−	0.911006i	$-0.364693\pi$
$971$	24955.7	0.824785	0.412392	+	0.911006i	$0.364693\pi$
$972$	−4437.50	−0.146433
$973$	26557.3	0.875012
$974$	90332.2	2.97169
$975$	−8983.08	−0.295066
$976$	−33849.2	−1.11013
$977$	56789.9	1.85964	0.929820	−	0.368014i	$-0.119962\pi$
$977$	56789.9	1.85964	0.929820	+	0.368014i	$0.119962\pi$
$978$	9735.68	0.318316
$979$	0	0
$980$	−52020.8	−1.69566
$981$	2672.08	0.0869653
$982$	−76636.8	−2.49041
$983$	−31484.6	−1.02157	−0.510784	−	0.859709i	$-0.670645\pi$
$983$	−31484.6	−1.02157	−0.510784	+	0.859709i	$0.670645\pi$
$984$	−45143.0	−1.46251
$985$	45703.8	1.47842
$986$	−32293.5	−1.04304
$987$	−17186.4	−0.554256
$988$	−12433.3	−0.400359
$989$	−16443.7	−0.528696
$990$	0	0
$991$	9490.27	0.304206	0.152103	−	0.988365i	$-0.451395\pi$
$991$	9490.27	0.304206	0.152103	+	0.988365i	$0.451395\pi$
$992$	14397.2	0.460799
$993$	19649.4	0.627950
$994$	28945.2	0.923629
$995$	−14517.4	−0.462545
$996$	14848.3	0.472375
$997$	40887.6	1.29882	0.649409	−	0.760439i	$-0.275016\pi$
$997$	40887.6	1.29882	0.649409	+	0.760439i	$0.275016\pi$
$998$	−110049.	−3.49051
$999$	−4277.05	−0.135455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.6		6
3.2	odd	2		1089.4.a.bi.1.1		6
11.3	even	5		33.4.e.c.31.3	yes	12
11.4	even	5		33.4.e.c.16.3	✓	12
11.10	odd	2		363.4.a.u.1.1		6
33.14	odd	10		99.4.f.d.64.1		12
33.26	odd	10		99.4.f.d.82.1		12
33.32	even	2		1089.4.a.bk.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.16.3	✓	12	11.4	even	5
33.4.e.c.31.3	yes	12	11.3	even	5
99.4.f.d.64.1		12	33.14	odd	10
99.4.f.d.82.1		12	33.26	odd	10
363.4.a.u.1.1		6	11.10	odd	2
363.4.a.v.1.6		6	1.1	even	1	trivial
1089.4.a.bi.1.1		6	3.2	odd	2
1089.4.a.bk.1.6		6	33.32	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+5.12458 q^{2} -3.00000 q^{3} +18.2613 q^{4} +14.9367 q^{5} -15.3737 q^{6} -12.3403 q^{7} +52.5849 q^{8} +9.00000 q^{9} +76.5442 q^{10} -54.7839 q^{12} +30.5222 q^{13} -63.2387 q^{14} -44.8100 q^{15} +123.385 q^{16} +132.065 q^{17} +46.1212 q^{18} -22.3068 q^{19} +272.763 q^{20} +37.0208 q^{21} -158.801 q^{23} -157.755 q^{24} +98.1044 q^{25} +156.413 q^{26} -27.0000 q^{27} -225.349 q^{28} -47.7165 q^{29} -229.633 q^{30} +68.0343 q^{31} +211.617 q^{32} +676.778 q^{34} -184.323 q^{35} +164.352 q^{36} +158.409 q^{37} -114.313 q^{38} -91.5666 q^{39} +785.444 q^{40} +286.159 q^{41} +189.716 q^{42} +103.549 q^{43} +134.430 q^{45} -813.790 q^{46} -464.238 q^{47} -370.155 q^{48} -190.718 q^{49} +502.744 q^{50} -396.195 q^{51} +557.375 q^{52} -6.04835 q^{53} -138.364 q^{54} -648.912 q^{56} +66.9205 q^{57} -244.527 q^{58} -221.619 q^{59} -818.290 q^{60} -274.338 q^{61} +348.647 q^{62} -111.062 q^{63} +97.3683 q^{64} +455.900 q^{65} +187.178 q^{67} +2411.68 q^{68} +476.404 q^{69} -944.576 q^{70} -457.714 q^{71} +473.264 q^{72} -270.332 q^{73} +811.780 q^{74} -294.313 q^{75} -407.352 q^{76} -469.240 q^{78} +493.647 q^{79} +1842.96 q^{80} +81.0000 q^{81} +1466.45 q^{82} -271.033 q^{83} +676.048 q^{84} +1972.61 q^{85} +530.646 q^{86} +143.150 q^{87} -77.4891 q^{89} +688.898 q^{90} -376.652 q^{91} -2899.92 q^{92} -204.103 q^{93} -2379.02 q^{94} -333.190 q^{95} -634.851 q^{96} -1269.39 q^{97} -977.348 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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