LMFDB - Embedded newform 363.4.a.v.1.1

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			1.1
Root			$4.80390$ 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)

$f(q)$	$=$	\(q-3.80390 q^{2} -3.00000 q^{3} +6.46966 q^{4} -2.57942 q^{5} +11.4117 q^{6} +1.46925 q^{7} +5.82127 q^{8} +9.00000 q^{9} +9.81184 q^{10} -19.4090 q^{12} -25.3227 q^{13} -5.58890 q^{14} +7.73825 q^{15} -73.9008 q^{16} +71.0533 q^{17} -34.2351 q^{18} +24.4559 q^{19} -16.6879 q^{20} -4.40776 q^{21} -191.800 q^{23} -17.4638 q^{24} -118.347 q^{25} +96.3249 q^{26} -27.0000 q^{27} +9.50557 q^{28} +71.9978 q^{29} -29.4355 q^{30} -197.641 q^{31} +234.541 q^{32} -270.280 q^{34} -3.78982 q^{35} +58.2269 q^{36} +239.066 q^{37} -93.0279 q^{38} +75.9680 q^{39} -15.0155 q^{40} +250.936 q^{41} +16.7667 q^{42} -435.218 q^{43} -23.2147 q^{45} +729.590 q^{46} +496.677 q^{47} +221.702 q^{48} -340.841 q^{49} +450.179 q^{50} -213.160 q^{51} -163.829 q^{52} -15.6662 q^{53} +102.705 q^{54} +8.55293 q^{56} -73.3677 q^{57} -273.872 q^{58} +60.3380 q^{59} +50.0638 q^{60} -128.445 q^{61} +751.807 q^{62} +13.2233 q^{63} -300.965 q^{64} +65.3177 q^{65} +889.699 q^{67} +459.690 q^{68} +575.401 q^{69} +14.4161 q^{70} +106.009 q^{71} +52.3914 q^{72} +770.733 q^{73} -909.382 q^{74} +355.040 q^{75} +158.221 q^{76} -288.975 q^{78} +1110.45 q^{79} +190.621 q^{80} +81.0000 q^{81} -954.537 q^{82} +1175.97 q^{83} -28.5167 q^{84} -183.276 q^{85} +1655.53 q^{86} -215.993 q^{87} +1335.58 q^{89} +88.3066 q^{90} -37.2054 q^{91} -1240.88 q^{92} +592.923 q^{93} -1889.31 q^{94} -63.0820 q^{95} -703.623 q^{96} -665.725 q^{97} +1296.53 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.80390	−1.34488	−0.672441	−	0.740151i	$-0.734754\pi$
$2$	−3.80390	−1.34488	−0.672441	+	0.740151i	$0.734754\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	6.46966	0.808707
$5$	−2.57942	−0.230710	−0.115355	−	0.993324i	$-0.536801\pi$
$5$	−2.57942	−0.230710	−0.115355	+	0.993324i	$0.536801\pi$
$6$	11.4117	0.776468
$7$	1.46925	0.0793323	0.0396661	−	0.999213i	$-0.487371\pi$
$7$	1.46925	0.0793323	0.0396661	+	0.999213i	$0.487371\pi$
$8$	5.82127	0.257266
$9$	9.00000	0.333333
$10$	9.81184	0.310278
$11$	0	0
$11$	0	0
$12$	−19.4090	−0.466907
$13$	−25.3227	−0.540249	−0.270125	−	0.962825i	$-0.587065\pi$
$13$	−25.3227	−0.540249	−0.270125	+	0.962825i	$0.587065\pi$
$14$	−5.58890	−0.106693
$15$	7.73825	0.133200
$16$	−73.9008	−1.15470
$17$	71.0533	1.01370	0.506852	−	0.862033i	$-0.330809\pi$
$17$	71.0533	1.01370	0.506852	+	0.862033i	$0.330809\pi$
$18$	−34.2351	−0.448294
$19$	24.4559	0.295293	0.147647	−	0.989040i	$-0.452830\pi$
$19$	24.4559	0.295293	0.147647	+	0.989040i	$0.452830\pi$
$20$	−16.6879	−0.186577
$21$	−4.40776	−0.0458025
$22$	0	0
$23$	−191.800	−1.73883	−0.869416	−	0.494080i	$-0.835505\pi$
$23$	−191.800	−1.73883	−0.869416	+	0.494080i	$0.835505\pi$
$24$	−17.4638	−0.148533
$25$	−118.347	−0.946773
$26$	96.3249	0.726572
$27$	−27.0000	−0.192450
$28$	9.50557	0.0641566
$29$	71.9978	0.461022	0.230511	−	0.973070i	$-0.425960\pi$
$29$	71.9978	0.461022	0.230511	+	0.973070i	$0.425960\pi$
$30$	−29.4355	−0.179139
$31$	−197.641	−1.14508	−0.572539	−	0.819878i	$-0.694041\pi$
$31$	−197.641	−1.14508	−0.572539	+	0.819878i	$0.694041\pi$
$32$	234.541	1.29567
$33$	0	0
$34$	−270.280	−1.36331
$35$	−3.78982	−0.0183027
$36$	58.2269	0.269569
$37$	239.066	1.06222	0.531110	−	0.847303i	$-0.321775\pi$
$37$	239.066	1.06222	0.531110	+	0.847303i	$0.321775\pi$
$38$	−93.0279	−0.397135
$39$	75.9680	0.311913
$40$	−15.0155	−0.0593539
$41$	250.936	0.955846	0.477923	−	0.878402i	$-0.341390\pi$
$41$	250.936	0.955846	0.477923	+	0.878402i	$0.341390\pi$
$42$	16.7667	0.0615990
$43$	−435.218	−1.54349	−0.771746	−	0.635930i	$-0.780617\pi$
$43$	−435.218	−1.54349	−0.771746	+	0.635930i	$0.780617\pi$
$44$	0	0
$45$	−23.2147	−0.0769033
$46$	729.590	2.33852
$47$	496.677	1.54144	0.770721	−	0.637173i	$-0.219896\pi$
$47$	496.677	1.54144	0.770721	+	0.637173i	$0.219896\pi$
$48$	221.702	0.666666
$49$	−340.841	−0.993706
$50$	450.179	1.27330
$51$	−213.160	−0.585262
$52$	−163.829	−0.436904
$53$	−15.6662	−0.0406023	−0.0203011	−	0.999794i	$-0.506462\pi$
$53$	−15.6662	−0.0406023	−0.0203011	+	0.999794i	$0.506462\pi$
$54$	102.705	0.258823
$55$	0	0
$56$	8.55293	0.0204095
$57$	−73.3677	−0.170488
$58$	−273.872	−0.620021
$59$	60.3380	0.133141	0.0665707	−	0.997782i	$-0.478794\pi$
$59$	60.3380	0.133141	0.0665707	+	0.997782i	$0.478794\pi$
$60$	50.0638	0.107720
$61$	−128.445	−0.269602	−0.134801	−	0.990873i	$-0.543039\pi$
$61$	−128.445	−0.269602	−0.134801	+	0.990873i	$0.543039\pi$
$62$	751.807	1.53999
$63$	13.2233	0.0264441
$64$	−300.965	−0.587822
$65$	65.3177	0.124641
$66$	0	0
$67$	889.699	1.62230	0.811149	−	0.584839i	$-0.198842\pi$
$67$	889.699	1.62230	0.811149	+	0.584839i	$0.198842\pi$
$68$	459.690	0.819789
$69$	575.401	1.00392
$70$	14.4161	0.0246150
$71$	106.009	0.177196	0.0885982	−	0.996067i	$-0.471761\pi$
$71$	106.009	0.177196	0.0885982	+	0.996067i	$0.471761\pi$
$72$	52.3914	0.0857554
$73$	770.733	1.23572	0.617860	−	0.786288i	$-0.288000\pi$
$73$	770.733	1.23572	0.617860	+	0.786288i	$0.288000\pi$
$74$	−909.382	−1.42856
$75$	355.040	0.546620
$76$	158.221	0.238806
$77$	0	0
$78$	−288.975	−0.419486
$79$	1110.45	1.58146	0.790730	−	0.612165i	$-0.209701\pi$
$79$	1110.45	1.58146	0.790730	+	0.612165i	$0.209701\pi$
$80$	190.621	0.266401
$81$	81.0000	0.111111
$82$	−954.537	−1.28550
$83$	1175.97	1.55518	0.777590	−	0.628772i	$-0.216442\pi$
$83$	1175.97	1.55518	0.777590	+	0.628772i	$0.216442\pi$
$84$	−28.5167	−0.0370408
$85$	−183.276	−0.233871
$86$	1655.53	2.07582
$87$	−215.993	−0.266171
$88$	0	0
$89$	1335.58	1.59069	0.795343	−	0.606159i	$-0.207291\pi$
$89$	1335.58	1.59069	0.795343	+	0.606159i	$0.207291\pi$
$90$	88.3066	0.103426
$91$	−37.2054	−0.0428592
$92$	−1240.88	−1.40621
$93$	592.923	0.661111
$94$	−1889.31	−2.07306
$95$	−63.0820	−0.0681271
$96$	−703.623	−0.748055
$97$	−665.725	−0.696847	−0.348423	−	0.937337i	$-0.613283\pi$
$97$	−665.725	−0.696847	−0.348423	+	0.937337i	$0.613283\pi$
$98$	1296.53	1.33642
$99$	0	0
$100$	−765.662	−0.765662
$101$	−442.098	−0.435549	−0.217774	−	0.975999i	$-0.569880\pi$
$101$	−442.098	−0.435549	−0.217774	+	0.975999i	$0.569880\pi$
$102$	810.839	0.787108
$103$	421.967	0.403666	0.201833	−	0.979420i	$-0.435310\pi$
$103$	421.967	0.403666	0.201833	+	0.979420i	$0.435310\pi$
$104$	−147.410	−0.138988
$105$	11.3695	0.0105671
$106$	59.5927	0.0546052
$107$	171.229	0.154704	0.0773518	−	0.997004i	$-0.475354\pi$
$107$	171.229	0.154704	0.0773518	+	0.997004i	$0.475354\pi$
$108$	−174.681	−0.155636
$109$	−1058.07	−0.929772	−0.464886	−	0.885371i	$-0.653905\pi$
$109$	−1058.07	−0.929772	−0.464886	+	0.885371i	$0.653905\pi$
$110$	0	0
$111$	−717.197	−0.613274
$112$	−108.579	−0.0916050
$113$	155.617	0.129551	0.0647753	−	0.997900i	$-0.479367\pi$
$113$	155.617	0.129551	0.0647753	+	0.997900i	$0.479367\pi$
$114$	279.084	0.229286
$115$	494.733	0.401166
$116$	465.801	0.372832
$117$	−227.904	−0.180083
$118$	−229.520	−0.179059
$119$	104.395	0.0804194
$120$	45.0464	0.0342680
$121$	0	0
$122$	488.592	0.362582
$123$	−752.809	−0.551858
$124$	−1278.67	−0.926032
$125$	627.692	0.449140
$126$	−50.3001	−0.0355642
$127$	−875.437	−0.611673	−0.305836	−	0.952084i	$-0.598936\pi$
$127$	−875.437	−0.611673	−0.305836	+	0.952084i	$0.598936\pi$
$128$	−731.489	−0.505118
$129$	1305.66	0.891136
$130$	−248.462	−0.167627
$131$	−1240.16	−0.827127	−0.413564	−	0.910475i	$-0.635716\pi$
$131$	−1240.16	−0.827127	−0.413564	+	0.910475i	$0.635716\pi$
$132$	0	0
$133$	35.9320	0.0234263
$134$	−3384.33	−2.18180
$135$	69.6442	0.0444001
$136$	413.620	0.260792
$137$	298.904	0.186402	0.0932012	−	0.995647i	$-0.470290\pi$
$137$	298.904	0.186402	0.0932012	+	0.995647i	$0.470290\pi$
$138$	−2188.77	−1.35015
$139$	−1463.28	−0.892905	−0.446452	−	0.894807i	$-0.647313\pi$
$139$	−1463.28	−0.892905	−0.446452	+	0.894807i	$0.647313\pi$
$140$	−24.5188	−0.0148016
$141$	−1490.03	−0.889952
$142$	−403.247	−0.238308
$143$	0	0
$144$	−665.107	−0.384900
$145$	−185.712	−0.106362
$146$	−2931.79	−1.66190
$147$	1022.52	0.573717
$148$	1546.67	0.859026
$149$	2088.32	1.14820	0.574100	−	0.818785i	$-0.305352\pi$
$149$	2088.32	1.14820	0.574100	+	0.818785i	$0.305352\pi$
$150$	−1350.54	−0.735139
$151$	2818.20	1.51882	0.759410	−	0.650613i	$-0.225488\pi$
$151$	2818.20	1.51882	0.759410	+	0.650613i	$0.225488\pi$
$152$	142.364	0.0759690
$153$	639.480	0.337901
$154$	0	0
$155$	509.799	0.264181
$156$	491.487	0.252246
$157$	−30.0537	−0.0152774	−0.00763869	−	0.999971i	$-0.502431\pi$
$157$	−30.0537	−0.0152774	−0.00763869	+	0.999971i	$0.502431\pi$
$158$	−4224.04	−2.12688
$159$	46.9986	0.0234417
$160$	−604.979	−0.298924
$161$	−281.804	−0.137946
$162$	−308.116	−0.149431
$163$	−576.514	−0.277031	−0.138516	−	0.990360i	$-0.544233\pi$
$163$	−576.514	−0.277031	−0.138516	+	0.990360i	$0.544233\pi$
$164$	1623.47	0.773000
$165$	0	0
$166$	−4473.29	−2.09153
$167$	−720.139	−0.333689	−0.166844	−	0.985983i	$-0.553358\pi$
$167$	−720.139	−0.333689	−0.166844	+	0.985983i	$0.553358\pi$
$168$	−25.6588	−0.0117834
$169$	−1555.76	−0.708131
$170$	697.164	0.314529
$171$	220.103	0.0984311
$172$	−2815.71	−1.24823
$173$	4232.96	1.86026	0.930132	−	0.367226i	$-0.119692\pi$
$173$	4232.96	1.86026	0.930132	+	0.367226i	$0.119692\pi$
$174$	821.617	0.357969
$175$	−173.881	−0.0751097
$176$	0	0
$177$	−181.014	−0.0768692
$178$	−5080.41	−2.13929
$179$	−3089.40	−1.29001	−0.645007	−	0.764177i	$-0.723146\pi$
$179$	−3089.40	−1.29001	−0.645007	+	0.764177i	$0.723146\pi$
$180$	−150.191	−0.0621923
$181$	2389.59	0.981310	0.490655	−	0.871354i	$-0.336758\pi$
$181$	2389.59	0.981310	0.490655	+	0.871354i	$0.336758\pi$
$182$	141.526	0.0576406
$183$	385.335	0.155655
$184$	−1116.52	−0.447343
$185$	−616.650	−0.245065
$186$	−2255.42	−0.889116
$187$	0	0
$188$	3213.33	1.24657
$189$	−39.6699	−0.0152675
$190$	239.958	0.0916229
$191$	−1804.91	−0.683764	−0.341882	−	0.939743i	$-0.611064\pi$
$191$	−1804.91	−0.683764	−0.341882	+	0.939743i	$0.611064\pi$
$192$	902.894	0.339379
$193$	1029.17	0.383841	0.191920	−	0.981411i	$-0.438528\pi$
$193$	1029.17	0.383841	0.191920	+	0.981411i	$0.438528\pi$
$194$	2532.35	0.937177
$195$	−195.953	−0.0719615
$196$	−2205.13	−0.803618
$197$	2483.31	0.898116	0.449058	−	0.893503i	$-0.351760\pi$
$197$	2483.31	0.898116	0.449058	+	0.893503i	$0.351760\pi$
$198$	0	0
$199$	3936.02	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$199$	3936.02	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$200$	−688.927	−0.243573
$201$	−2669.10	−0.936635
$202$	1681.70	0.585762
$203$	105.783	0.0365740
$204$	−1379.07	−0.473305
$205$	−647.270	−0.220523
$206$	−1605.12	−0.542883
$207$	−1726.20	−0.579611
$208$	1871.36	0.623826
$209$	0	0
$210$	−43.2483	−0.0142115
$211$	−924.114	−0.301510	−0.150755	−	0.988571i	$-0.548170\pi$
$211$	−924.114	−0.301510	−0.150755	+	0.988571i	$0.548170\pi$
$212$	−101.355	−0.0328353
$213$	−318.027	−0.102304
$214$	−651.336	−0.208058
$215$	1122.61	0.356099
$216$	−157.174	−0.0495109
$217$	−290.385	−0.0908416
$218$	4024.81	1.25043
$219$	−2312.20	−0.713443
$220$	0	0
$221$	−1799.26	−0.547653
$222$	2728.15	0.824780
$223$	6068.10	1.82220	0.911099	−	0.412188i	$-0.135235\pi$
$223$	6068.10	1.82220	0.911099	+	0.412188i	$0.135235\pi$
$224$	344.601	0.102788
$225$	−1065.12	−0.315591
$226$	−591.951	−0.174230
$227$	4143.05	1.21138	0.605692	−	0.795699i	$-0.292896\pi$
$227$	4143.05	1.21138	0.605692	+	0.795699i	$0.292896\pi$
$228$	−474.664	−0.137875
$229$	5521.15	1.59322	0.796610	−	0.604493i	$-0.206624\pi$
$229$	5521.15	1.59322	0.796610	+	0.604493i	$0.206624\pi$
$230$	−1881.92	−0.539521
$231$	0	0
$232$	419.118	0.118605
$233$	4823.91	1.35633	0.678164	−	0.734910i	$-0.262776\pi$
$233$	4823.91	1.35633	0.678164	+	0.734910i	$0.262776\pi$
$234$	866.924	0.242191
$235$	−1281.14	−0.355626
$236$	390.366	0.107672
$237$	−3331.35	−0.913056
$238$	−397.110	−0.108155
$239$	−281.091	−0.0760764	−0.0380382	−	0.999276i	$-0.512111\pi$
$239$	−281.091	−0.0760764	−0.0380382	+	0.999276i	$0.512111\pi$
$240$	−571.863	−0.153807
$241$	−70.9520	−0.0189644	−0.00948220	−	0.999955i	$-0.503018\pi$
$241$	−70.9520	−0.0189644	−0.00948220	+	0.999955i	$0.503018\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−830.995	−0.218029
$245$	879.171	0.229258
$246$	2863.61	0.742184
$247$	−619.289	−0.159532
$248$	−1150.52	−0.294590
$249$	−3527.92	−0.897883
$250$	−2387.68	−0.604040
$251$	−3991.19	−1.00367	−0.501836	−	0.864963i	$-0.667342\pi$
$251$	−3991.19	−1.00367	−0.501836	+	0.864963i	$0.667342\pi$
$252$	85.5502	0.0213855
$253$	0	0
$254$	3330.07	0.822628
$255$	549.828	0.135026
$256$	5190.23	1.26715
$257$	−491.529	−0.119303	−0.0596513	−	0.998219i	$-0.518999\pi$
$257$	−491.529	−0.119303	−0.0596513	+	0.998219i	$0.518999\pi$
$258$	−4966.58	−1.19847
$259$	351.249	0.0842684
$260$	422.583	0.100798
$261$	647.980	0.153674
$262$	4717.46	1.11239
$263$	251.552	0.0589786	0.0294893	−	0.999565i	$-0.490612\pi$
$263$	251.552	0.0589786	0.0294893	+	0.999565i	$0.490612\pi$
$264$	0	0
$265$	40.4097	0.00936734
$266$	−136.682	−0.0315056
$267$	−4006.74	−0.918384
$268$	5756.05	1.31196
$269$	17.7838	0.00403084	0.00201542	−	0.999998i	$-0.499358\pi$
$269$	17.7838	0.00403084	0.00201542	+	0.999998i	$0.499358\pi$
$270$	−264.920	−0.0597130
$271$	−20.6908	−0.00463791	−0.00231896	−	0.999997i	$-0.500738\pi$
$271$	−20.6908	−0.00463791	−0.00231896	+	0.999997i	$0.500738\pi$
$272$	−5250.89	−1.17052
$273$	111.616	0.0247448
$274$	−1137.00	−0.250689
$275$	0	0
$276$	3722.65	0.811874
$277$	8024.17	1.74052	0.870262	−	0.492588i	$-0.163949\pi$
$277$	8024.17	1.74052	0.870262	+	0.492588i	$0.163949\pi$
$278$	5566.17	1.20085
$279$	−1778.77	−0.381692
$280$	−22.0615	−0.00470868
$281$	−2925.95	−0.621165	−0.310583	−	0.950546i	$-0.600524\pi$
$281$	−2925.95	−0.621165	−0.310583	+	0.950546i	$0.600524\pi$
$282$	5667.93	1.19688
$283$	−8667.94	−1.82069	−0.910346	−	0.413848i	$-0.864184\pi$
$283$	−8667.94	−1.82069	−0.910346	+	0.413848i	$0.864184\pi$
$284$	685.842	0.143300
$285$	189.246	0.0393332
$286$	0	0
$287$	368.690	0.0758295
$288$	2110.87	0.431890
$289$	135.570	0.0275942
$290$	706.430	0.143045
$291$	1997.18	0.402325
$292$	4986.38	0.999335
$293$	5072.03	1.01130	0.505651	−	0.862738i	$-0.331252\pi$
$293$	5072.03	1.01130	0.505651	+	0.862738i	$0.331252\pi$
$294$	−3889.58	−0.771581
$295$	−155.637	−0.0307170
$296$	1391.67	0.273274
$297$	0	0
$298$	−7943.76	−1.54419
$299$	4856.90	0.939404
$300$	2296.99	0.442055
$301$	−639.447	−0.122449
$302$	−10720.1	−2.04263
$303$	1326.30	0.251464
$304$	−1807.31	−0.340975
$305$	331.313	0.0621998
$306$	−2432.52	−0.454437
$307$	3010.24	0.559620	0.279810	−	0.960055i	$-0.409729\pi$
$307$	3010.24	0.559620	0.279810	+	0.960055i	$0.409729\pi$
$308$	0	0
$309$	−1265.90	−0.233057
$310$	−1939.22	−0.355292
$311$	−2888.96	−0.526745	−0.263373	−	0.964694i	$-0.584835\pi$
$311$	−2888.96	−0.526745	−0.263373	+	0.964694i	$0.584835\pi$
$312$	442.230	0.0802447
$313$	8776.88	1.58498	0.792490	−	0.609885i	$-0.208785\pi$
$313$	8776.88	1.58498	0.792490	+	0.609885i	$0.208785\pi$
$314$	114.321	0.0205463
$315$	−34.1084	−0.00610092
$316$	7184.23	1.27894
$317$	−2738.39	−0.485185	−0.242592	−	0.970128i	$-0.577998\pi$
$317$	−2738.39	−0.485185	−0.242592	+	0.970128i	$0.577998\pi$
$318$	−178.778	−0.0315263
$319$	0	0
$320$	776.313	0.135616
$321$	−513.686	−0.0893182
$322$	1071.95	0.185521
$323$	1737.67	0.299340
$324$	524.042	0.0898564
$325$	2996.85	0.511494
$326$	2193.00	0.372574
$327$	3174.22	0.536804
$328$	1460.77	0.245907
$329$	729.744	0.122286
$330$	0	0
$331$	3715.53	0.616990	0.308495	−	0.951226i	$-0.400175\pi$
$331$	3715.53	0.616990	0.308495	+	0.951226i	$0.400175\pi$
$332$	7608.15	1.25768
$333$	2151.59	0.354074
$334$	2739.34	0.448772
$335$	−2294.90	−0.374280
$336$	325.737	0.0528882
$337$	6704.10	1.08367	0.541833	−	0.840486i	$-0.317730\pi$
$337$	6704.10	1.08367	0.541833	+	0.840486i	$0.317730\pi$
$338$	5917.97	0.952352
$339$	−466.851	−0.0747961
$340$	−1185.73	−0.189134
$341$	0	0
$342$	−837.251	−0.132378
$343$	−1004.74	−0.158165
$344$	−2533.52	−0.397088
$345$	−1484.20	−0.231613
$346$	−16101.7	−2.50183
$347$	8547.28	1.32231	0.661156	−	0.750249i	$-0.270066\pi$
$347$	8547.28	1.32231	0.661156	+	0.750249i	$0.270066\pi$
$348$	−1397.40	−0.215255
$349$	3589.58	0.550561	0.275280	−	0.961364i	$-0.411229\pi$
$349$	3589.58	0.550561	0.275280	+	0.961364i	$0.411229\pi$
$350$	661.427	0.101014
$351$	683.712	0.103971
$352$	0	0
$353$	−8820.68	−1.32996	−0.664982	−	0.746859i	$-0.731561\pi$
$353$	−8820.68	−1.32996	−0.664982	+	0.746859i	$0.731561\pi$
$354$	688.559	0.103380
$355$	−273.441	−0.0408810
$356$	8640.74	1.28640
$357$	−313.186	−0.0464302
$358$	11751.8	1.73492
$359$	−12990.2	−1.90974	−0.954872	−	0.297018i	$-0.904008\pi$
$359$	−12990.2	−1.90974	−0.954872	+	0.297018i	$0.904008\pi$
$360$	−135.139	−0.0197846
$361$	−6260.91	−0.912802
$362$	−9089.77	−1.31975
$363$	0	0
$364$	−240.706	−0.0346606
$365$	−1988.04	−0.285093
$366$	−1465.78	−0.209337
$367$	−7657.56	−1.08916	−0.544579	−	0.838709i	$-0.683311\pi$
$367$	−7657.56	−1.08916	−0.544579	+	0.838709i	$0.683311\pi$
$368$	14174.2	2.00783
$369$	2258.43	0.318615
$370$	2345.68	0.329583
$371$	−23.0177	−0.00322107
$372$	3836.01	0.534645
$373$	10252.1	1.42314	0.711570	−	0.702615i	$-0.247984\pi$
$373$	10252.1	1.42314	0.711570	+	0.702615i	$0.247984\pi$
$374$	0	0
$375$	−1883.08	−0.259311
$376$	2891.29	0.396561
$377$	−1823.17	−0.249067
$378$	150.900	0.0205330
$379$	−521.447	−0.0706726	−0.0353363	−	0.999375i	$-0.511250\pi$
$379$	−521.447	−0.0706726	−0.0353363	+	0.999375i	$0.511250\pi$
$380$	−408.119	−0.0550949
$381$	2626.31	0.353150
$382$	6865.71	0.919582
$383$	−2090.55	−0.278909	−0.139454	−	0.990228i	$-0.544535\pi$
$383$	−2090.55	−0.278909	−0.139454	+	0.990228i	$0.544535\pi$
$384$	2194.47	0.291630
$385$	0	0
$386$	−3914.86	−0.516220
$387$	−3916.97	−0.514498
$388$	−4307.01	−0.563545
$389$	2482.02	0.323505	0.161753	−	0.986831i	$-0.448285\pi$
$389$	2482.02	0.323505	0.161753	+	0.986831i	$0.448285\pi$
$390$	745.386	0.0967797
$391$	−13628.1	−1.76266
$392$	−1984.13	−0.255647
$393$	3720.49	0.477542
$394$	−9446.28	−1.20786
$395$	−2864.31	−0.364859
$396$	0	0
$397$	2074.35	0.262239	0.131119	−	0.991367i	$-0.458143\pi$
$397$	2074.35	0.262239	0.131119	+	0.991367i	$0.458143\pi$
$398$	−14972.2	−1.88565
$399$	−107.796	−0.0135252
$400$	8745.91	1.09324
$401$	−10370.6	−1.29147	−0.645737	−	0.763560i	$-0.723450\pi$
$401$	−10370.6	−1.29147	−0.645737	+	0.763560i	$0.723450\pi$
$402$	10153.0	1.25966
$403$	5004.80	0.618627
$404$	−2860.23	−0.352232
$405$	−208.933	−0.0256344
$406$	−402.388	−0.0491877
$407$	0	0
$408$	−1240.86	−0.150568
$409$	−3415.16	−0.412882	−0.206441	−	0.978459i	$-0.566188\pi$
$409$	−3415.16	−0.412882	−0.206441	+	0.978459i	$0.566188\pi$
$410$	2462.15	0.296578
$411$	−896.713	−0.107619
$412$	2729.98	0.326448
$413$	88.6519	0.0105624
$414$	6566.31	0.779508
$415$	−3033.33	−0.358795
$416$	−5939.21	−0.699984
$417$	4389.84	0.515519
$418$	0	0
$419$	−7647.52	−0.891661	−0.445830	−	0.895117i	$-0.647092\pi$
$419$	−7647.52	−0.891661	−0.445830	+	0.895117i	$0.647092\pi$
$420$	73.5565	0.00854569
$421$	−7614.05	−0.881440	−0.440720	−	0.897645i	$-0.645277\pi$
$421$	−7614.05	−0.881440	−0.440720	+	0.897645i	$0.645277\pi$
$422$	3515.24	0.405495
$423$	4470.09	0.513814
$424$	−91.1972	−0.0104456
$425$	−8408.92	−0.959747
$426$	1209.74	0.137587
$427$	−188.718	−0.0213881
$428$	1107.79	0.125110
$429$	0	0
$430$	−4270.29	−0.478911
$431$	6492.48	0.725595	0.362798	−	0.931868i	$-0.381822\pi$
$431$	6492.48	0.725595	0.362798	+	0.931868i	$0.381822\pi$
$432$	1995.32	0.222222
$433$	−1837.96	−0.203988	−0.101994	−	0.994785i	$-0.532522\pi$
$433$	−1837.96	−0.203988	−0.101994	+	0.994785i	$0.532522\pi$
$434$	1104.60	0.122171
$435$	557.136	0.0614084
$436$	−6845.38	−0.751913
$437$	−4690.66	−0.513466
$438$	8795.38	0.959497
$439$	−16807.1	−1.82725	−0.913623	−	0.406562i	$-0.866727\pi$
$439$	−16807.1	−1.82725	−0.913623	+	0.406562i	$0.866727\pi$
$440$	0	0
$441$	−3067.57	−0.331235
$442$	6844.20	0.736528
$443$	12104.8	1.29823	0.649114	−	0.760692i	$-0.275140\pi$
$443$	12104.8	1.29823	0.649114	+	0.760692i	$0.275140\pi$
$444$	−4640.02	−0.495959
$445$	−3445.01	−0.366987
$446$	−23082.5	−2.45064
$447$	−6264.96	−0.662914
$448$	−442.194	−0.0466332
$449$	6734.71	0.707863	0.353931	−	0.935271i	$-0.384845\pi$
$449$	6734.71	0.707863	0.353931	+	0.935271i	$0.384845\pi$
$450$	4051.61	0.424433
$451$	0	0
$452$	1006.79	0.104768
$453$	−8454.59	−0.876891
$454$	−15759.8	−1.62917
$455$	95.9683	0.00988805
$456$	−427.093	−0.0438607
$457$	−5950.32	−0.609068	−0.304534	−	0.952501i	$-0.598501\pi$
$457$	−5950.32	−0.609068	−0.304534	+	0.952501i	$0.598501\pi$
$458$	−21001.9	−2.14269
$459$	−1918.44	−0.195087
$460$	3200.75	0.324426
$461$	2917.31	0.294735	0.147367	−	0.989082i	$-0.452920\pi$
$461$	2917.31	0.294735	0.147367	+	0.989082i	$0.452920\pi$
$462$	0	0
$463$	−537.106	−0.0539123	−0.0269562	−	0.999637i	$-0.508581\pi$
$463$	−537.106	−0.0539123	−0.0269562	+	0.999637i	$0.508581\pi$
$464$	−5320.69	−0.532342
$465$	−1529.40	−0.152525
$466$	−18349.7	−1.82410
$467$	12124.7	1.20143	0.600713	−	0.799465i	$-0.294883\pi$
$467$	12124.7	1.20143	0.600713	+	0.799465i	$0.294883\pi$
$468$	−1474.46	−0.145635
$469$	1307.19	0.128701
$470$	4873.31	0.478275
$471$	90.1612	0.00882040
$472$	351.244	0.0342528
$473$	0	0
$474$	12672.1	1.22795
$475$	−2894.27	−0.279576
$476$	675.402	0.0650357
$477$	−140.996	−0.0135341
$478$	1069.24	0.102314
$479$	−3854.34	−0.367660	−0.183830	−	0.982958i	$-0.558850\pi$
$479$	−3854.34	−0.367660	−0.183830	+	0.982958i	$0.558850\pi$
$480$	1814.94	0.172584
$481$	−6053.78	−0.573864
$482$	269.894	0.0255049
$483$	845.411	0.0796429
$484$	0	0
$485$	1717.18	0.160770
$486$	924.348	0.0862742
$487$	−5700.80	−0.530448	−0.265224	−	0.964187i	$-0.585446\pi$
$487$	−5700.80	−0.530448	−0.265224	+	0.964187i	$0.585446\pi$
$488$	−747.713	−0.0693594
$489$	1729.54	0.159944
$490$	−3344.28	−0.308325
$491$	17006.5	1.56312	0.781562	−	0.623827i	$-0.214423\pi$
$491$	17006.5	1.56312	0.781562	+	0.623827i	$0.214423\pi$
$492$	−4870.42	−0.446292
$493$	5115.68	0.467340
$494$	2355.71	0.214552
$495$	0	0
$496$	14605.8	1.32222
$497$	155.754	0.0140574
$498$	13419.9	1.20755
$499$	−8473.24	−0.760149	−0.380074	−	0.924956i	$-0.624102\pi$
$499$	−8473.24	−0.760149	−0.380074	+	0.924956i	$0.624102\pi$
$500$	4060.95	0.363223
$501$	2160.42	0.192655
$502$	15182.1	1.34982
$503$	−10347.5	−0.917243	−0.458622	−	0.888632i	$-0.651657\pi$
$503$	−10347.5	−0.917243	−0.458622	+	0.888632i	$0.651657\pi$
$504$	76.9763	0.00680317
$505$	1140.36	0.100485
$506$	0	0
$507$	4667.29	0.408839
$508$	−5663.78	−0.494664
$509$	1904.10	0.165811	0.0829057	−	0.996557i	$-0.473580\pi$
$509$	1904.10	0.165811	0.0829057	+	0.996557i	$0.473580\pi$
$510$	−2091.49	−0.181594
$511$	1132.40	0.0980325
$512$	−13891.2	−1.19904
$513$	−660.310	−0.0568292
$514$	1869.73	0.160448
$515$	−1088.43	−0.0931298
$516$	8447.14	0.720668
$517$	0	0
$518$	−1336.11	−0.113331
$519$	−12698.9	−1.07402
$520$	380.232	0.0320659
$521$	19351.2	1.62724	0.813619	−	0.581399i	$-0.197494\pi$
$521$	19351.2	1.62724	0.813619	+	0.581399i	$0.197494\pi$
$522$	−2464.85	−0.206674
$523$	2872.65	0.240176	0.120088	−	0.992763i	$-0.461682\pi$
$523$	2872.65	0.240176	0.120088	+	0.992763i	$0.461682\pi$
$524$	−8023.44	−0.668904
$525$	521.644	0.0433646
$526$	−956.879	−0.0793193
$527$	−14043.1	−1.16077
$528$	0	0
$529$	24620.4	2.02354
$530$	−153.714	−0.0125980
$531$	543.042	0.0443805
$532$	232.468	0.0189450
$533$	−6354.38	−0.516395
$534$	15241.2	1.23512
$535$	−441.669	−0.0356917
$536$	5179.18	0.417363
$537$	9268.19	0.744790
$538$	−67.6478	−0.00542101
$539$	0	0
$540$	450.574	0.0359067
$541$	13187.9	1.04804	0.524021	−	0.851705i	$-0.324431\pi$
$541$	13187.9	1.04804	0.524021	+	0.851705i	$0.324431\pi$
$542$	78.7056	0.00623744
$543$	−7168.78	−0.566559
$544$	16664.9	1.31342
$545$	2729.21	0.214508
$546$	−424.577	−0.0332788
$547$	−3584.57	−0.280192	−0.140096	−	0.990138i	$-0.544741\pi$
$547$	−3584.57	−0.280192	−0.140096	+	0.990138i	$0.544741\pi$
$548$	1933.81	0.150745
$549$	−1156.01	−0.0898672
$550$	0	0
$551$	1760.77	0.136137
$552$	3349.57	0.258274
$553$	1631.53	0.125461
$554$	−30523.1	−2.34080
$555$	1849.95	0.141488
$556$	−9466.92	−0.722099
$557$	3922.45	0.298383	0.149192	−	0.988808i	$-0.452333\pi$
$557$	3922.45	0.298383	0.149192	+	0.988808i	$0.452333\pi$
$558$	6766.26	0.513331
$559$	11020.9	0.833871
$560$	280.071	0.0211342
$561$	0	0
$562$	11130.0	0.835394
$563$	−17585.3	−1.31640	−0.658200	−	0.752843i	$-0.728682\pi$
$563$	−17585.3	−1.31640	−0.658200	+	0.752843i	$0.728682\pi$
$564$	−9639.98	−0.719710
$565$	−401.401	−0.0298886
$566$	32972.0	2.44862
$567$	119.010	0.00881470
$568$	617.107	0.0455866
$569$	5757.89	0.424224	0.212112	−	0.977245i	$-0.431966\pi$
$569$	5757.89	0.424224	0.212112	+	0.977245i	$0.431966\pi$
$570$	−719.873	−0.0528985
$571$	−13983.0	−1.02482	−0.512408	−	0.858742i	$-0.671246\pi$
$571$	−13983.0	−1.02482	−0.512408	+	0.858742i	$0.671246\pi$
$572$	0	0
$573$	5414.74	0.394772
$574$	−1402.46	−0.101982
$575$	22698.9	1.64628
$576$	−2708.68	−0.195941
$577$	−19913.3	−1.43674	−0.718370	−	0.695661i	$-0.755112\pi$
$577$	−19913.3	−1.43674	−0.718370	+	0.695661i	$0.755112\pi$
$578$	−515.697	−0.0371110
$579$	−3087.51	−0.221611
$580$	−1201.49	−0.0860161
$581$	1727.80	0.123376
$582$	−7597.06	−0.541079
$583$	0	0
$584$	4486.65	0.317909
$585$	587.859	0.0415470
$586$	−19293.5	−1.36008
$587$	2361.42	0.166041	0.0830207	−	0.996548i	$-0.473543\pi$
$587$	2361.42	0.166041	0.0830207	+	0.996548i	$0.473543\pi$
$588$	6615.38	0.463969
$589$	−4833.49	−0.338134
$590$	592.027	0.0413108
$591$	−7449.94	−0.518527
$592$	−17667.2	−1.22655
$593$	24598.2	1.70341	0.851707	−	0.524018i	$-0.175567\pi$
$593$	24598.2	1.70341	0.851707	+	0.524018i	$0.175567\pi$
$594$	0	0
$595$	−269.279	−0.0185536
$596$	13510.7	0.928558
$597$	−11808.1	−0.809500
$598$	−18475.2	−1.26339
$599$	−14590.6	−0.995252	−0.497626	−	0.867392i	$-0.665795\pi$
$599$	−14590.6	−0.995252	−0.497626	+	0.867392i	$0.665795\pi$
$600$	2066.78	0.140627
$601$	−12232.5	−0.830243	−0.415121	−	0.909766i	$-0.636261\pi$
$601$	−12232.5	−0.830243	−0.415121	+	0.909766i	$0.636261\pi$
$602$	2432.39	0.164679
$603$	8007.29	0.540766
$604$	18232.8	1.22828
$605$	0	0
$606$	−5045.10	−0.338190
$607$	−8501.02	−0.568444	−0.284222	−	0.958758i	$-0.591735\pi$
$607$	−8501.02	−0.568444	−0.284222	+	0.958758i	$0.591735\pi$
$608$	5735.92	0.382602
$609$	−317.349	−0.0211160
$610$	−1260.28	−0.0836514
$611$	−12577.2	−0.832763
$612$	4137.21	0.273263
$613$	18556.9	1.22268	0.611341	−	0.791367i	$-0.290630\pi$
$613$	18556.9	1.22268	0.611341	+	0.791367i	$0.290630\pi$
$614$	−11450.6	−0.752623
$615$	1941.81	0.127319
$616$	0	0
$617$	−21751.6	−1.41926	−0.709632	−	0.704572i	$-0.751139\pi$
$617$	−21751.6	−1.41926	−0.709632	+	0.704572i	$0.751139\pi$
$618$	4815.36	0.313434
$619$	7389.61	0.479828	0.239914	−	0.970794i	$-0.422881\pi$
$619$	7389.61	0.479828	0.239914	+	0.970794i	$0.422881\pi$
$620$	3298.22	0.213645
$621$	5178.61	0.334639
$622$	10989.3	0.708410
$623$	1962.31	0.126193
$624$	−5614.09	−0.360166
$625$	13174.2	0.843152
$626$	−33386.4	−2.13161
$627$	0	0
$628$	−194.437	−0.0123549
$629$	16986.4	1.07678
$630$	129.745	0.00820501
$631$	12663.5	0.798931	0.399466	−	0.916748i	$-0.369196\pi$
$631$	12663.5	0.798931	0.399466	+	0.916748i	$0.369196\pi$
$632$	6464.23	0.406856
$633$	2772.34	0.174077
$634$	10416.6	0.652516
$635$	2258.12	0.141119
$636$	304.065	0.0189575
$637$	8631.01	0.536849
$638$	0	0
$639$	954.081	0.0590655
$640$	1886.81	0.116536
$641$	−10874.0	−0.670044	−0.335022	−	0.942210i	$-0.608744\pi$
$641$	−10874.0	−0.670044	−0.335022	+	0.942210i	$0.608744\pi$
$642$	1954.01	0.120122
$643$	8200.41	0.502944	0.251472	−	0.967865i	$-0.419085\pi$
$643$	8200.41	0.502944	0.251472	+	0.967865i	$0.419085\pi$
$644$	−1823.17	−0.111558
$645$	−3367.83	−0.205594
$646$	−6609.94	−0.402577
$647$	−22028.4	−1.33853	−0.669264	−	0.743025i	$-0.733390\pi$
$647$	−22028.4	−1.33853	−0.669264	+	0.743025i	$0.733390\pi$
$648$	471.523	0.0285851
$649$	0	0
$650$	−11399.7	−0.687898
$651$	871.155	0.0524474
$652$	−3729.85	−0.224037
$653$	20838.9	1.24884	0.624418	−	0.781091i	$-0.285336\pi$
$653$	20838.9	1.24884	0.624418	+	0.781091i	$0.285336\pi$
$654$	−12074.4	−0.721938
$655$	3198.90	0.190826
$656$	−18544.4	−1.10372
$657$	6936.60	0.411906
$658$	−2775.88	−0.164460
$659$	−8237.87	−0.486953	−0.243476	−	0.969907i	$-0.578288\pi$
$659$	−8237.87	−0.486953	−0.243476	+	0.969907i	$0.578288\pi$
$660$	0	0
$661$	−19161.2	−1.12751	−0.563756	−	0.825941i	$-0.690644\pi$
$661$	−19161.2	−1.12751	−0.563756	+	0.825941i	$0.690644\pi$
$662$	−14133.5	−0.829779
$663$	5397.78	0.316187
$664$	6845.66	0.400095
$665$	−92.6835	−0.00540468
$666$	−8184.44	−0.476187
$667$	−13809.2	−0.801641
$668$	−4659.06	−0.269857
$669$	−18204.3	−1.05205
$670$	8729.58	0.503363
$671$	0	0
$672$	−1033.80	−0.0593449
$673$	−33012.1	−1.89082	−0.945411	−	0.325879i	$-0.894340\pi$
$673$	−33012.1	−1.89082	−0.945411	+	0.325879i	$0.894340\pi$
$674$	−25501.7	−1.45740
$675$	3195.36	0.182207
$676$	−10065.3	−0.572670
$677$	6411.69	0.363990	0.181995	−	0.983299i	$-0.441745\pi$
$677$	6411.69	0.363990	0.181995	+	0.983299i	$0.441745\pi$
$678$	1775.85	0.100592
$679$	−978.120	−0.0552825
$680$	−1066.90	−0.0601672
$681$	−12429.2	−0.699393
$682$	0	0
$683$	7840.57	0.439255	0.219627	−	0.975584i	$-0.429516\pi$
$683$	7840.57	0.439255	0.219627	+	0.975584i	$0.429516\pi$
$684$	1423.99	0.0796019
$685$	−770.999	−0.0430049
$686$	3821.92	0.212714
$687$	−16563.4	−0.919847
$688$	32163.0	1.78227
$689$	396.710	0.0219353
$690$	5645.75	0.311493
$691$	−19507.9	−1.07397	−0.536986	−	0.843591i	$-0.680437\pi$
$691$	−19507.9	−1.07397	−0.536986	+	0.843591i	$0.680437\pi$
$692$	27385.8	1.50441
$693$	0	0
$694$	−32513.0	−1.77835
$695$	3774.41	0.206002
$696$	−1257.35	−0.0684769
$697$	17829.9	0.968944
$698$	−13654.4	−0.740439
$699$	−14471.7	−0.783077
$700$	−1124.95	−0.0607417
$701$	−8579.85	−0.462277	−0.231139	−	0.972921i	$-0.574245\pi$
$701$	−8579.85	−0.462277	−0.231139	+	0.972921i	$0.574245\pi$
$702$	−2600.77	−0.139829
$703$	5846.57	0.313667
$704$	0	0
$705$	3843.41	0.205321
$706$	33553.0	1.78865
$707$	−649.555	−0.0345531
$708$	−1171.10	−0.0621647
$709$	3458.48	0.183196	0.0915979	−	0.995796i	$-0.470803\pi$
$709$	3458.48	0.183196	0.0915979	+	0.995796i	$0.470803\pi$
$710$	1040.14	0.0549801
$711$	9994.05	0.527153
$712$	7774.77	0.409230
$713$	37907.7	1.99110
$714$	1191.33	0.0624431
$715$	0	0
$716$	−19987.3	−1.04324
$717$	843.272	0.0439227
$718$	49413.5	2.56838
$719$	29934.6	1.55267	0.776336	−	0.630319i	$-0.217076\pi$
$719$	29934.6	1.55267	0.776336	+	0.630319i	$0.217076\pi$
$720$	1715.59	0.0888002
$721$	619.976	0.0320237
$722$	23815.9	1.22761
$723$	212.856	0.0109491
$724$	15459.8	0.793592
$725$	−8520.69	−0.436483
$726$	0	0
$727$	−23967.3	−1.22269	−0.611347	−	0.791363i	$-0.709372\pi$
$727$	−23967.3	−1.22269	−0.611347	+	0.791363i	$0.709372\pi$
$728$	−216.583	−0.0110262
$729$	729.000	0.0370370
$730$	7562.31	0.383416
$731$	−30923.7	−1.56464
$732$	2492.99	0.125879
$733$	−22000.9	−1.10862	−0.554312	−	0.832309i	$-0.687019\pi$
$733$	−22000.9	−1.10862	−0.554312	+	0.832309i	$0.687019\pi$
$734$	29128.6	1.46479
$735$	−2637.51	−0.132362
$736$	−44985.1	−2.25295
$737$	0	0
$738$	−8590.84	−0.428500
$739$	−24018.7	−1.19559	−0.597795	−	0.801649i	$-0.703956\pi$
$739$	−24018.7	−1.19559	−0.597795	+	0.801649i	$0.703956\pi$
$740$	−3989.51	−0.198186
$741$	1857.87	0.0921059
$742$	87.5569	0.00433196
$743$	21187.3	1.04614	0.523072	−	0.852288i	$-0.324786\pi$
$743$	21187.3	1.04614	0.523072	+	0.852288i	$0.324786\pi$
$744$	3451.57	0.170081
$745$	−5386.65	−0.264901
$746$	−38997.8	−1.91396
$747$	10583.8	0.518393
$748$	0	0
$749$	251.578	0.0122730
$750$	7163.03	0.348743
$751$	6625.32	0.321919	0.160960	−	0.986961i	$-0.448541\pi$
$751$	6625.32	0.321919	0.160960	+	0.986961i	$0.448541\pi$
$752$	−36704.8	−1.77990
$753$	11973.6	0.579470
$754$	6935.18	0.334966
$755$	−7269.30	−0.350407
$756$	−256.651	−0.0123469
$757$	31645.0	1.51936	0.759682	−	0.650295i	$-0.225355\pi$
$757$	31645.0	1.51936	0.759682	+	0.650295i	$0.225355\pi$
$758$	1983.53	0.0950463
$759$	0	0
$760$	−367.217	−0.0175268
$761$	28599.5	1.36233	0.681164	−	0.732131i	$-0.261474\pi$
$761$	28599.5	1.36233	0.681164	+	0.732131i	$0.261474\pi$
$762$	−9990.22	−0.474944
$763$	−1554.58	−0.0737609
$764$	−11677.2	−0.552965
$765$	−1649.48	−0.0779571
$766$	7952.24	0.375100
$767$	−1527.92	−0.0719296
$768$	−15570.7	−0.731587
$769$	26098.1	1.22382	0.611912	−	0.790926i	$-0.290400\pi$
$769$	26098.1	1.22382	0.611912	+	0.790926i	$0.290400\pi$
$770$	0	0
$771$	1474.59	0.0688794
$772$	6658.38	0.310415
$773$	−6276.05	−0.292023	−0.146011	−	0.989283i	$-0.546644\pi$
$773$	−6276.05	−0.292023	−0.146011	+	0.989283i	$0.546644\pi$
$774$	14899.7	0.691939
$775$	23390.2	1.08413
$776$	−3875.37	−0.179275
$777$	−1053.75	−0.0486524
$778$	−9441.36	−0.435076
$779$	6136.88	0.282255
$780$	−1267.75	−0.0581958
$781$	0	0
$782$	51839.8	2.37057
$783$	−1943.94	−0.0887238
$784$	25188.4	1.14743
$785$	77.5211	0.00352464
$786$	−14152.4	−0.642238
$787$	22287.2	1.00947	0.504736	−	0.863274i	$-0.331590\pi$
$787$	22287.2	1.00947	0.504736	+	0.863274i	$0.331590\pi$
$788$	16066.2	0.726313
$789$	−754.657	−0.0340513
$790$	10895.6	0.490692
$791$	228.641	0.0102775
$792$	0	0
$793$	3252.57	0.145652
$794$	−7890.63	−0.352680
$795$	−121.229	−0.00540824
$796$	25464.7	1.13388
$797$	−39299.1	−1.74661	−0.873304	−	0.487175i	$-0.838027\pi$
$797$	−39299.1	−1.74661	−0.873304	+	0.487175i	$0.838027\pi$
$798$	410.045	0.0181898
$799$	35290.5	1.56256
$800$	−27757.1	−1.22670
$801$	12020.2	0.530229
$802$	39448.6	1.73688
$803$	0	0
$804$	−17268.1	−0.757463
$805$	726.889	0.0318254
$806$	−19037.8	−0.831981
$807$	−53.3514	−0.00232721
$808$	−2573.57	−0.112052
$809$	15023.1	0.652885	0.326443	−	0.945217i	$-0.394150\pi$
$809$	15023.1	0.652885	0.326443	+	0.945217i	$0.394150\pi$
$810$	794.759	0.0344753
$811$	−20100.0	−0.870290	−0.435145	−	0.900360i	$-0.643303\pi$
$811$	−20100.0	−0.870290	−0.435145	+	0.900360i	$0.643303\pi$
$812$	684.380	0.0295776
$813$	62.0723	0.00267770
$814$	0	0
$815$	1487.07	0.0639139
$816$	15752.7	0.675802
$817$	−10643.7	−0.455783
$818$	12990.9	0.555277
$819$	−334.849	−0.0142864
$820$	−4187.61	−0.178339
$821$	6017.68	0.255808	0.127904	−	0.991787i	$-0.459175\pi$
$821$	6017.68	0.255808	0.127904	+	0.991787i	$0.459175\pi$
$822$	3411.01	0.144735
$823$	−20919.9	−0.886052	−0.443026	−	0.896509i	$-0.646095\pi$
$823$	−20919.9	−0.886052	−0.443026	+	0.896509i	$0.646095\pi$
$824$	2456.38	0.103850
$825$	0	0
$826$	−337.223	−0.0142052
$827$	17107.5	0.719331	0.359665	−	0.933081i	$-0.382891\pi$
$827$	17107.5	0.719331	0.359665	+	0.933081i	$0.382891\pi$
$828$	−11167.9	−0.468736
$829$	−1185.27	−0.0496576	−0.0248288	−	0.999692i	$-0.507904\pi$
$829$	−1185.27	−0.0496576	−0.0248288	+	0.999692i	$0.507904\pi$
$830$	11538.5	0.482537
$831$	−24072.5	−1.00489
$832$	7621.23	0.317570
$833$	−24217.9	−1.00732
$834$	−16698.5	−0.693312
$835$	1857.54	0.0769854
$836$	0	0
$837$	5336.31	0.220370
$838$	29090.4	1.19918
$839$	10636.8	0.437692	0.218846	−	0.975759i	$-0.429771\pi$
$839$	10636.8	0.437692	0.218846	+	0.975759i	$0.429771\pi$
$840$	66.1846	0.00271856
$841$	−19205.3	−0.787458
$842$	28963.1	1.18543
$843$	8777.85	0.358630
$844$	−5978.70	−0.243833
$845$	4012.96	0.163373
$846$	−17003.8	−0.691019
$847$	0	0
$848$	1157.75	0.0468834
$849$	26003.8	1.05118
$850$	31986.7	1.29075
$851$	−45852.9	−1.84702
$852$	−2057.52	−0.0827343
$853$	8100.14	0.325139	0.162569	−	0.986697i	$-0.448022\pi$
$853$	8100.14	0.325139	0.162569	+	0.986697i	$0.448022\pi$
$854$	717.866	0.0287645
$855$	−567.738	−0.0227090
$856$	996.767	0.0398000
$857$	−17919.7	−0.714266	−0.357133	−	0.934054i	$-0.616246\pi$
$857$	−17919.7	−0.714266	−0.357133	+	0.934054i	$0.616246\pi$
$858$	0	0
$859$	8475.92	0.336665	0.168332	−	0.985730i	$-0.446162\pi$
$859$	8475.92	0.336665	0.168332	+	0.985730i	$0.446162\pi$
$860$	7262.90	0.287980
$861$	−1106.07	−0.0437802
$862$	−24696.7	−0.975840
$863$	38487.1	1.51810	0.759048	−	0.651035i	$-0.225665\pi$
$863$	38487.1	1.51810	0.759048	+	0.651035i	$0.225665\pi$
$864$	−6332.61	−0.249352
$865$	−10918.6	−0.429181
$866$	6991.42	0.274340
$867$	−406.711	−0.0159315
$868$	−1878.69	−0.0734643
$869$	0	0
$870$	−2119.29	−0.0825870
$871$	−22529.5	−0.876446
$872$	−6159.33	−0.239199
$873$	−5991.53	−0.232282
$874$	17842.8	0.690551
$875$	922.239	0.0356313
$876$	−14959.1	−0.576966
$877$	35473.2	1.36584	0.682922	−	0.730492i	$-0.260709\pi$
$877$	35473.2	1.36584	0.682922	+	0.730492i	$0.260709\pi$
$878$	63932.7	2.45743
$879$	−15216.1	−0.583875
$880$	0	0
$881$	26126.3	0.999112	0.499556	−	0.866282i	$-0.333496\pi$
$881$	26126.3	0.999112	0.499556	+	0.866282i	$0.333496\pi$
$882$	11668.7	0.445473
$883$	19009.3	0.724477	0.362239	−	0.932085i	$-0.382013\pi$
$883$	19009.3	0.724477	0.362239	+	0.932085i	$0.382013\pi$
$884$	−11640.6	−0.442891
$885$	466.910	0.0177345
$886$	−46045.3	−1.74596
$887$	40183.1	1.52110	0.760551	−	0.649279i	$-0.224929\pi$
$887$	40183.1	1.52110	0.760551	+	0.649279i	$0.224929\pi$
$888$	−4175.00	−0.157775
$889$	−1286.24	−0.0485254
$890$	13104.5	0.493555
$891$	0	0
$892$	39258.5	1.47362
$893$	12146.7	0.455177
$894$	23831.3	0.891541
$895$	7968.84	0.297619
$896$	−1074.74	−0.0400722
$897$	−14570.7	−0.542365
$898$	−25618.1	−0.951992
$899$	−14229.7	−0.527906
$900$	−6890.96	−0.255221
$901$	−1113.14	−0.0411586
$902$	0	0
$903$	1918.34	0.0706959
$904$	905.888	0.0333290
$905$	−6163.75	−0.226398
$906$	32160.4	1.17931
$907$	5208.17	0.190666	0.0953332	−	0.995445i	$-0.469608\pi$
$907$	5208.17	0.190666	0.0953332	+	0.995445i	$0.469608\pi$
$908$	26804.1	0.979655
$909$	−3978.89	−0.145183
$910$	−365.054	−0.0132983
$911$	−13167.6	−0.478881	−0.239441	−	0.970911i	$-0.576964\pi$
$911$	−13167.6	−0.478881	−0.239441	+	0.970911i	$0.576964\pi$
$912$	5421.93	0.196862
$913$	0	0
$914$	22634.4	0.819125
$915$	−993.939	−0.0359111
$916$	35719.9	1.28845
$917$	−1822.12	−0.0656179
$918$	7297.55	0.262369
$919$	−27692.5	−0.994004	−0.497002	−	0.867749i	$-0.665566\pi$
$919$	−27692.5	−0.994004	−0.497002	+	0.867749i	$0.665566\pi$
$920$	2879.97	0.103206
$921$	−9030.71	−0.323097
$922$	−11097.2	−0.396384
$923$	−2684.43	−0.0957303
$924$	0	0
$925$	−28292.6	−1.00568
$926$	2043.10	0.0725057
$927$	3797.70	0.134555
$928$	16886.4	0.597332
$929$	−8856.73	−0.312788	−0.156394	−	0.987695i	$-0.549987\pi$
$929$	−8856.73	−0.312788	−0.156394	+	0.987695i	$0.549987\pi$
$930$	5817.67	0.205128
$931$	−8335.59	−0.293435
$932$	31209.0	1.09687
$933$	8666.88	0.304117
$934$	−46121.3	−1.61578
$935$	0	0
$936$	−1326.69	−0.0463293
$937$	7266.73	0.253355	0.126677	−	0.991944i	$-0.459569\pi$
$937$	7266.73	0.253355	0.126677	+	0.991944i	$0.459569\pi$
$938$	−4972.44	−0.173087
$939$	−26330.6	−0.915088
$940$	−8288.51	−0.287597
$941$	−31760.8	−1.10029	−0.550144	−	0.835070i	$-0.685427\pi$
$941$	−31760.8	−1.10029	−0.550144	+	0.835070i	$0.685427\pi$
$942$	−342.964	−0.0118624
$943$	−48129.7	−1.66206
$944$	−4459.03	−0.153738
$945$	102.325	0.00352237
$946$	0	0
$947$	22691.1	0.778630	0.389315	−	0.921105i	$-0.372712\pi$
$947$	22691.1	0.778630	0.389315	+	0.921105i	$0.372712\pi$
$948$	−21552.7	−0.738395
$949$	−19517.0	−0.667597
$950$	11009.5	0.375996
$951$	8215.18	0.280122
$952$	607.714	0.0206892
$953$	−8492.60	−0.288670	−0.144335	−	0.989529i	$-0.546104\pi$
$953$	−8492.60	−0.288670	−0.144335	+	0.989529i	$0.546104\pi$
$954$	536.334	0.0182017
$955$	4655.62	0.157751
$956$	−1818.56	−0.0615235
$957$	0	0
$958$	14661.5	0.494459
$959$	439.167	0.0147877
$960$	−2328.94	−0.0782981
$961$	9271.02	0.311202
$962$	23028.0	0.771780
$963$	1541.06	0.0515679
$964$	−459.035	−0.0153367
$965$	−2654.66	−0.0885559
$966$	−3215.86	−0.107110
$967$	−27202.5	−0.904627	−0.452313	−	0.891859i	$-0.649401\pi$
$967$	−27202.5	−0.904627	−0.452313	+	0.891859i	$0.649401\pi$
$968$	0	0
$969$	−5213.02	−0.172824
$970$	−6531.99	−0.216216
$971$	31503.2	1.04118	0.520589	−	0.853807i	$-0.325712\pi$
$971$	31503.2	1.04118	0.520589	+	0.853807i	$0.325712\pi$
$972$	−1572.13	−0.0518786
$973$	−2149.93	−0.0708362
$974$	21685.3	0.713389
$975$	−8990.55	−0.295311
$976$	9492.19	0.311309
$977$	1930.23	0.0632073	0.0316036	−	0.999500i	$-0.489939\pi$
$977$	1930.23	0.0632073	0.0316036	+	0.999500i	$0.489939\pi$
$978$	−6579.01	−0.215106
$979$	0	0
$980$	5687.94	0.185403
$981$	−9522.67	−0.309924
$982$	−64691.1	−2.10222
$983$	−27211.8	−0.882930	−0.441465	−	0.897278i	$-0.645541\pi$
$983$	−27211.8	−0.882930	−0.441465	+	0.897278i	$0.645541\pi$
$984$	−4382.31	−0.141974
$985$	−6405.50	−0.207204
$986$	−19459.5	−0.628517
$987$	−2189.23	−0.0706019
$988$	−4006.59	−0.129015
$989$	83475.1	2.68388
$990$	0	0
$991$	−37496.9	−1.20195	−0.600974	−	0.799269i	$-0.705220\pi$
$991$	−37496.9	−1.20195	−0.600974	+	0.799269i	$0.705220\pi$
$992$	−46355.0	−1.48364
$993$	−11146.6	−0.356220
$994$	−592.473	−0.0189055
$995$	−10152.6	−0.323477
$996$	−22824.4	−0.726125
$997$	−13984.8	−0.444235	−0.222118	−	0.975020i	$-0.571297\pi$
$997$	−13984.8	−0.444235	−0.222118	+	0.975020i	$0.571297\pi$
$998$	32231.4	1.02231
$999$	−6454.78	−0.204425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.1		6
3.2	odd	2		1089.4.a.bi.1.6		6
11.5	even	5		33.4.e.c.25.3	yes	12
11.9	even	5		33.4.e.c.4.3	✓	12
11.10	odd	2		363.4.a.u.1.6		6
33.5	odd	10		99.4.f.d.91.1		12
33.20	odd	10		99.4.f.d.37.1		12
33.32	even	2		1089.4.a.bk.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.4.3	✓	12	11.9	even	5
33.4.e.c.25.3	yes	12	11.5	even	5
99.4.f.d.37.1		12	33.20	odd	10
99.4.f.d.91.1		12	33.5	odd	10
363.4.a.u.1.6		6	11.10	odd	2
363.4.a.v.1.1		6	1.1	even	1	trivial
1089.4.a.bi.1.6		6	3.2	odd	2
1089.4.a.bk.1.1		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)

\(f(q)\)	\(=\)	\(q-3.80390 q^{2} -3.00000 q^{3} +6.46966 q^{4} -2.57942 q^{5} +11.4117 q^{6} +1.46925 q^{7} +5.82127 q^{8} +9.00000 q^{9} +9.81184 q^{10} -19.4090 q^{12} -25.3227 q^{13} -5.58890 q^{14} +7.73825 q^{15} -73.9008 q^{16} +71.0533 q^{17} -34.2351 q^{18} +24.4559 q^{19} -16.6879 q^{20} -4.40776 q^{21} -191.800 q^{23} -17.4638 q^{24} -118.347 q^{25} +96.3249 q^{26} -27.0000 q^{27} +9.50557 q^{28} +71.9978 q^{29} -29.4355 q^{30} -197.641 q^{31} +234.541 q^{32} -270.280 q^{34} -3.78982 q^{35} +58.2269 q^{36} +239.066 q^{37} -93.0279 q^{38} +75.9680 q^{39} -15.0155 q^{40} +250.936 q^{41} +16.7667 q^{42} -435.218 q^{43} -23.2147 q^{45} +729.590 q^{46} +496.677 q^{47} +221.702 q^{48} -340.841 q^{49} +450.179 q^{50} -213.160 q^{51} -163.829 q^{52} -15.6662 q^{53} +102.705 q^{54} +8.55293 q^{56} -73.3677 q^{57} -273.872 q^{58} +60.3380 q^{59} +50.0638 q^{60} -128.445 q^{61} +751.807 q^{62} +13.2233 q^{63} -300.965 q^{64} +65.3177 q^{65} +889.699 q^{67} +459.690 q^{68} +575.401 q^{69} +14.4161 q^{70} +106.009 q^{71} +52.3914 q^{72} +770.733 q^{73} -909.382 q^{74} +355.040 q^{75} +158.221 q^{76} -288.975 q^{78} +1110.45 q^{79} +190.621 q^{80} +81.0000 q^{81} -954.537 q^{82} +1175.97 q^{83} -28.5167 q^{84} -183.276 q^{85} +1655.53 q^{86} -215.993 q^{87} +1335.58 q^{89} +88.3066 q^{90} -37.2054 q^{91} -1240.88 q^{92} +592.923 q^{93} -1889.31 q^{94} -63.0820 q^{95} -703.623 q^{96} -665.725 q^{97} +1296.53 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 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

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