LMFDB - Embedded newform 363.4.a.q.1.3

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 28x^{2} - 2x + 72 $ x^4 - x^3 - 28x^2 - 2x + 72
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81838$ 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)

$f(q)$	$=$	\(q+1.81838 q^{2} +3.00000 q^{3} -4.69349 q^{4} -19.2178 q^{5} +5.45514 q^{6} +14.1043 q^{7} -23.0816 q^{8} +9.00000 q^{9} -34.9453 q^{10} -14.0805 q^{12} +54.2664 q^{13} +25.6471 q^{14} -57.6534 q^{15} -4.42322 q^{16} +78.2846 q^{17} +16.3654 q^{18} +90.2036 q^{19} +90.1986 q^{20} +42.3130 q^{21} -122.719 q^{23} -69.2448 q^{24} +244.324 q^{25} +98.6770 q^{26} +27.0000 q^{27} -66.1986 q^{28} -123.798 q^{29} -104.836 q^{30} +212.442 q^{31} +176.610 q^{32} +142.351 q^{34} -271.054 q^{35} -42.2414 q^{36} +429.123 q^{37} +164.024 q^{38} +162.799 q^{39} +443.578 q^{40} +224.405 q^{41} +76.9412 q^{42} -408.062 q^{43} -172.960 q^{45} -223.150 q^{46} +224.749 q^{47} -13.2697 q^{48} -144.068 q^{49} +444.274 q^{50} +234.854 q^{51} -254.699 q^{52} +283.806 q^{53} +49.0963 q^{54} -325.551 q^{56} +270.611 q^{57} -225.112 q^{58} -306.981 q^{59} +270.596 q^{60} +111.455 q^{61} +386.301 q^{62} +126.939 q^{63} +356.529 q^{64} -1042.88 q^{65} -153.581 q^{67} -367.428 q^{68} -368.157 q^{69} -492.880 q^{70} -323.300 q^{71} -207.734 q^{72} -194.522 q^{73} +780.309 q^{74} +732.971 q^{75} -423.370 q^{76} +296.031 q^{78} -788.138 q^{79} +85.0045 q^{80} +81.0000 q^{81} +408.054 q^{82} +870.482 q^{83} -198.596 q^{84} -1504.46 q^{85} -742.013 q^{86} -371.394 q^{87} +788.215 q^{89} -314.508 q^{90} +765.392 q^{91} +575.981 q^{92} +637.327 q^{93} +408.679 q^{94} -1733.51 q^{95} +529.829 q^{96} -852.379 q^{97} -261.970 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20}+ \cdots - 615 q^{98}+O(q^{100}) $ 4 * q - q^2 + 12 * q^3 + 25 * q^4 + 14 * q^5 - 3 * q^6 + 20 * q^7 - 75 * q^8 + 36 * q^9 + 3 * q^10 + 75 * q^12 + 32 * q^13 + 62 * q^14 + 42 * q^15 + 289 * q^16 + 92 * q^17 - 9 * q^18 - 34 * q^19 + 391 * q^20 + 60 * q^21 - 26 * q^23 - 225 * q^24 + 334 * q^25 - 181 * q^26 + 108 * q^27 + 692 * q^28 + 174 * q^29 + 9 * q^30 + 422 * q^31 - 1271 * q^32 - 477 * q^34 - 82 * q^35 + 225 * q^36 + 518 * q^37 - 798 * q^38 + 96 * q^39 + 423 * q^40 + 428 * q^41 + 186 * q^42 - 550 * q^43 + 126 * q^45 - 2004 * q^46 + 556 * q^47 + 867 * q^48 + 282 * q^49 + 2074 * q^50 + 276 * q^51 + 467 * q^52 + 882 * q^53 - 27 * q^54 - 2112 * q^56 - 102 * q^57 - 225 * q^58 - 158 * q^59 + 1173 * q^60 + 290 * q^61 + 1142 * q^62 + 180 * q^63 + 3097 * q^64 - 1636 * q^65 + 992 * q^67 - 2033 * q^68 - 78 * q^69 + 948 * q^70 - 42 * q^71 - 675 * q^72 + 1274 * q^73 + 1509 * q^74 + 1002 * q^75 + 632 * q^76 - 543 * q^78 + 362 * q^79 + 1423 * q^80 + 324 * q^81 + 2403 * q^82 + 1500 * q^83 + 2076 * q^84 - 2388 * q^85 - 1272 * q^86 + 522 * q^87 + 1428 * q^89 + 27 * q^90 + 1750 * q^91 + 896 * q^92 + 1266 * q^93 - 1728 * q^94 - 4452 * q^95 - 3813 * q^96 + 1052 * q^97 - 615 * 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 $

$2$	1.81838	0.642895	0.321447	−	0.946927i	$-0.395831\pi$
$2$	1.81838	0.642895	0.321447	+	0.946927i	$0.395831\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.69349	−0.586686
$5$	−19.2178	−1.71889	−0.859446	−	0.511226i	$-0.829191\pi$
$5$	−19.2178	−1.71889	−0.859446	+	0.511226i	$0.829191\pi$
$6$	5.45514	0.371175
$7$	14.1043	0.761563	0.380781	−	0.924665i	$-0.375655\pi$
$7$	14.1043	0.761563	0.380781	+	0.924665i	$0.375655\pi$
$8$	−23.0816	−1.02007
$9$	9.00000	0.333333
$10$	−34.9453	−1.10507
$11$	0	0
$11$	0	0
$12$	−14.0805	−0.338724
$13$	54.2664	1.15775	0.578877	−	0.815415i	$-0.303491\pi$
$13$	54.2664	1.15775	0.578877	+	0.815415i	$0.303491\pi$
$14$	25.6471	0.489605
$15$	−57.6534	−0.992403
$16$	−4.42322	−0.0691128
$17$	78.2846	1.11687	0.558436	−	0.829548i	$-0.311402\pi$
$17$	78.2846	1.11687	0.558436	+	0.829548i	$0.311402\pi$
$18$	16.3654	0.214298
$19$	90.2036	1.08916	0.544582	−	0.838708i	$-0.316688\pi$
$19$	90.2036	1.08916	0.544582	+	0.838708i	$0.316688\pi$
$20$	90.1986	1.00845
$21$	42.3130	0.439689
$22$	0	0
$23$	−122.719	−1.11255	−0.556276	−	0.830998i	$-0.687770\pi$
$23$	−122.719	−1.11255	−0.556276	+	0.830998i	$0.687770\pi$
$24$	−69.2448	−0.588939
$25$	244.324	1.95459
$26$	98.6770	0.744314
$27$	27.0000	0.192450
$28$	−66.1986	−0.446799
$29$	−123.798	−0.792715	−0.396357	−	0.918096i	$-0.629726\pi$
$29$	−123.798	−0.792715	−0.396357	+	0.918096i	$0.629726\pi$
$30$	−104.836	−0.638011
$31$	212.442	1.23083	0.615416	−	0.788202i	$-0.288988\pi$
$31$	212.442	1.23083	0.615416	+	0.788202i	$0.288988\pi$
$32$	176.610	0.975640
$33$	0	0
$34$	142.351	0.718031
$35$	−271.054	−1.30904
$36$	−42.2414	−0.195562
$37$	429.123	1.90669	0.953343	−	0.301889i	$-0.0976171\pi$
$37$	429.123	1.90669	0.953343	+	0.301889i	$0.0976171\pi$
$38$	164.024	0.700218
$39$	162.799	0.668429
$40$	443.578	1.75339
$41$	224.405	0.854784	0.427392	−	0.904066i	$-0.359432\pi$
$41$	224.405	0.854784	0.427392	+	0.904066i	$0.359432\pi$
$42$	76.9412	0.282673
$43$	−408.062	−1.44718	−0.723592	−	0.690228i	$-0.757510\pi$
$43$	−408.062	−1.44718	−0.723592	+	0.690228i	$0.757510\pi$
$44$	0	0
$45$	−172.960	−0.572964
$46$	−223.150	−0.715253
$47$	224.749	0.697510	0.348755	−	0.937214i	$-0.386605\pi$
$47$	224.749	0.697510	0.348755	+	0.937214i	$0.386605\pi$
$48$	−13.2697	−0.0399023
$49$	−144.068	−0.420022
$50$	444.274	1.25660
$51$	234.854	0.644826
$52$	−254.699	−0.679238
$53$	283.806	0.735543	0.367771	−	0.929916i	$-0.380121\pi$
$53$	283.806	0.735543	0.367771	+	0.929916i	$0.380121\pi$
$54$	49.0963	0.123725
$55$	0	0
$56$	−325.551	−0.776849
$57$	270.611	0.628829
$58$	−225.112	−0.509632
$59$	−306.981	−0.677381	−0.338690	−	0.940898i	$-0.609984\pi$
$59$	−306.981	−0.677381	−0.338690	+	0.940898i	$0.609984\pi$
$60$	270.596	0.582229
$61$	111.455	0.233940	0.116970	−	0.993135i	$-0.462682\pi$
$61$	111.455	0.233940	0.116970	+	0.993135i	$0.462682\pi$
$62$	386.301	0.791295
$63$	126.939	0.253854
$64$	356.529	0.696347
$65$	−1042.88	−1.99005
$66$	0	0
$67$	−153.581	−0.280043	−0.140021	−	0.990148i	$-0.544717\pi$
$67$	−153.581	−0.280043	−0.140021	+	0.990148i	$0.544717\pi$
$68$	−367.428	−0.655253
$69$	−368.157	−0.642332
$70$	−492.880	−0.841578
$71$	−323.300	−0.540403	−0.270202	−	0.962804i	$-0.587090\pi$
$71$	−323.300	−0.540403	−0.270202	+	0.962804i	$0.587090\pi$
$72$	−207.734	−0.340024
$73$	−194.522	−0.311877	−0.155939	−	0.987767i	$-0.549840\pi$
$73$	−194.522	−0.311877	−0.155939	+	0.987767i	$0.549840\pi$
$74$	780.309	1.22580
$75$	732.971	1.12848
$76$	−423.370	−0.638998
$77$	0	0
$78$	296.031	0.429730
$79$	−788.138	−1.12244	−0.561218	−	0.827668i	$-0.689667\pi$
$79$	−788.138	−1.12244	−0.561218	+	0.827668i	$0.689667\pi$
$80$	85.0045	0.118797
$81$	81.0000	0.111111
$82$	408.054	0.549536
$83$	870.482	1.15118	0.575589	−	0.817739i	$-0.304773\pi$
$83$	870.482	1.15118	0.575589	+	0.817739i	$0.304773\pi$
$84$	−198.596	−0.257959
$85$	−1504.46	−1.91978
$86$	−742.013	−0.930387
$87$	−371.394	−0.457674
$88$	0	0
$89$	788.215	0.938771	0.469385	−	0.882993i	$-0.344475\pi$
$89$	788.215	0.938771	0.469385	+	0.882993i	$0.344475\pi$
$90$	−314.508	−0.368356
$91$	765.392	0.881702
$92$	575.981	0.652719
$93$	637.327	0.710621
$94$	408.679	0.448426
$95$	−1733.51	−1.87216
$96$	529.829	0.563286
$97$	−852.379	−0.892226	−0.446113	−	0.894977i	$-0.647192\pi$
$97$	−852.379	−0.892226	−0.446113	+	0.894977i	$0.647192\pi$
$98$	−261.970	−0.270030
$99$	0	0
$100$	−1146.73	−1.14673
$101$	1070.69	1.05483	0.527415	−	0.849608i	$-0.323161\pi$
$101$	1070.69	1.05483	0.527415	+	0.849608i	$0.323161\pi$
$102$	427.054	0.414555
$103$	1705.74	1.63176	0.815879	−	0.578222i	$-0.196253\pi$
$103$	1705.74	1.63176	0.815879	+	0.578222i	$0.196253\pi$
$104$	−1252.56	−1.18099
$105$	−813.163	−0.755777
$106$	516.068	0.472877
$107$	652.066	0.589136	0.294568	−	0.955630i	$-0.404824\pi$
$107$	652.066	0.589136	0.294568	+	0.955630i	$0.404824\pi$
$108$	−126.724	−0.112908
$109$	2066.09	1.81555	0.907777	−	0.419453i	$-0.137778\pi$
$109$	2066.09	1.81555	0.907777	+	0.419453i	$0.137778\pi$
$110$	0	0
$111$	1287.37	1.10083
$112$	−62.3866	−0.0526337
$113$	−181.559	−0.151147	−0.0755736	−	0.997140i	$-0.524079\pi$
$113$	−181.559	−0.151147	−0.0755736	+	0.997140i	$0.524079\pi$
$114$	492.073	0.404271
$115$	2358.39	1.91236
$116$	581.045	0.465075
$117$	488.398	0.385918
$118$	−558.208	−0.435484
$119$	1104.15	0.850568
$120$	1330.73	1.01232
$121$	0	0
$122$	202.668	0.150399
$123$	673.215	0.493510
$124$	−997.097	−0.722112
$125$	−2293.14	−1.64084
$126$	230.824	0.163202
$127$	−755.589	−0.527934	−0.263967	−	0.964532i	$-0.585031\pi$
$127$	−755.589	−0.527934	−0.263967	+	0.964532i	$0.585031\pi$
$128$	−764.571	−0.527962
$129$	−1224.19	−0.835532
$130$	−1896.36	−1.27939
$131$	252.052	0.168106	0.0840531	−	0.996461i	$-0.473213\pi$
$131$	252.052	0.168106	0.0840531	+	0.996461i	$0.473213\pi$
$132$	0	0
$133$	1272.26	0.829467
$134$	−279.268	−0.180038
$135$	−518.881	−0.330801
$136$	−1806.94	−1.13929
$137$	−2149.24	−1.34031	−0.670153	−	0.742223i	$-0.733771\pi$
$137$	−2149.24	−1.34031	−0.670153	+	0.742223i	$0.733771\pi$
$138$	−669.450	−0.412952
$139$	607.426	0.370656	0.185328	−	0.982677i	$-0.440665\pi$
$139$	607.426	0.370656	0.185328	+	0.982677i	$0.440665\pi$
$140$	1272.19	0.767998
$141$	674.246	0.402708
$142$	−587.882	−0.347422
$143$	0	0
$144$	−39.8090	−0.0230376
$145$	2379.13	1.36259
$146$	−353.714	−0.200504
$147$	−432.203	−0.242500
$148$	−2014.09	−1.11863
$149$	−2679.77	−1.47339	−0.736697	−	0.676224i	$-0.763615\pi$
$149$	−2679.77	−1.47339	−0.736697	+	0.676224i	$0.763615\pi$
$150$	1332.82	0.725496
$151$	−529.332	−0.285275	−0.142637	−	0.989775i	$-0.545558\pi$
$151$	−529.332	−0.285275	−0.142637	+	0.989775i	$0.545558\pi$
$152$	−2082.04	−1.11103
$153$	704.562	0.372291
$154$	0	0
$155$	−4082.68	−2.11567
$156$	−764.097	−0.392158
$157$	111.990	0.0569283	0.0284642	−	0.999595i	$-0.490938\pi$
$157$	111.990	0.0569283	0.0284642	+	0.999595i	$0.490938\pi$
$158$	−1433.13	−0.721608
$159$	851.418	0.424666
$160$	−3394.05	−1.67702
$161$	−1730.87	−0.847278
$162$	147.289	0.0714327
$163$	2318.99	1.11434	0.557171	−	0.830398i	$-0.311887\pi$
$163$	2318.99	1.11434	0.557171	+	0.830398i	$0.311887\pi$
$164$	−1053.24	−0.501490
$165$	0	0
$166$	1582.87	0.740087
$167$	3361.08	1.55741	0.778707	−	0.627387i	$-0.215876\pi$
$167$	3361.08	1.55741	0.778707	+	0.627387i	$0.215876\pi$
$168$	−976.652	−0.448514
$169$	747.844	0.340393
$170$	−2735.68	−1.23422
$171$	811.832	0.363055
$172$	1915.24	0.849043
$173$	−872.145	−0.383283	−0.191642	−	0.981465i	$-0.561381\pi$
$173$	−872.145	−0.383283	−0.191642	+	0.981465i	$0.561381\pi$
$174$	−675.336	−0.294236
$175$	3446.03	1.48854
$176$	0	0
$177$	−920.942	−0.391086
$178$	1433.27	0.603531
$179$	1774.29	0.740874	0.370437	−	0.928858i	$-0.379208\pi$
$179$	1774.29	0.740874	0.370437	+	0.928858i	$0.379208\pi$
$180$	811.787	0.336150
$181$	−969.085	−0.397964	−0.198982	−	0.980003i	$-0.563764\pi$
$181$	−969.085	−0.397964	−0.198982	+	0.980003i	$0.563764\pi$
$182$	1391.77	0.566842
$183$	334.365	0.135065
$184$	2832.55	1.13488
$185$	−8246.80	−3.27739
$186$	1158.90	0.456855
$187$	0	0
$188$	−1054.86	−0.409220
$189$	380.817	0.146563
$190$	−3152.19	−1.20360
$191$	796.117	0.301597	0.150798	−	0.988565i	$-0.451816\pi$
$191$	796.117	0.301597	0.150798	+	0.988565i	$0.451816\pi$
$192$	1069.59	0.402036
$193$	−3167.89	−1.18150	−0.590750	−	0.806855i	$-0.701168\pi$
$193$	−3167.89	−1.18150	−0.590750	+	0.806855i	$0.701168\pi$
$194$	−1549.95	−0.573608
$195$	−3128.64	−1.14896
$196$	676.180	0.246421
$197$	1309.07	0.473438	0.236719	−	0.971578i	$-0.423928\pi$
$197$	1309.07	0.473438	0.236719	+	0.971578i	$0.423928\pi$
$198$	0	0
$199$	2639.16	0.940125	0.470063	−	0.882633i	$-0.344231\pi$
$199$	2639.16	0.940125	0.470063	+	0.882633i	$0.344231\pi$
$200$	−5639.38	−1.99382
$201$	−460.742	−0.161683
$202$	1946.93	0.678145
$203$	−1746.09	−0.603702
$204$	−1102.28	−0.378311
$205$	−4312.57	−1.46928
$206$	3101.68	1.04905
$207$	−1104.47	−0.370850
$208$	−240.032	−0.0800156
$209$	0	0
$210$	−1478.64	−0.485885
$211$	−1067.34	−0.348239	−0.174120	−	0.984725i	$-0.555708\pi$
$211$	−1067.34	−0.348239	−0.174120	+	0.984725i	$0.555708\pi$
$212$	−1332.04	−0.431533
$213$	−969.899	−0.312002
$214$	1185.70	0.378753
$215$	7842.06	2.48755
$216$	−623.203	−0.196313
$217$	2996.36	0.937356
$218$	3756.94	1.16721
$219$	−583.565	−0.180062
$220$	0	0
$221$	4248.23	1.29306
$222$	2340.93	0.707715
$223$	2488.35	0.747229	0.373615	−	0.927584i	$-0.378118\pi$
$223$	2488.35	0.747229	0.373615	+	0.927584i	$0.378118\pi$
$224$	2490.96	0.743011
$225$	2198.91	0.651530
$226$	−330.143	−0.0971717
$227$	−4030.74	−1.17854	−0.589272	−	0.807935i	$-0.700585\pi$
$227$	−4030.74	−1.17854	−0.589272	+	0.807935i	$0.700585\pi$
$228$	−1270.11	−0.368926
$229$	2301.18	0.664046	0.332023	−	0.943271i	$-0.392269\pi$
$229$	2301.18	0.664046	0.332023	+	0.943271i	$0.392269\pi$
$230$	4288.45	1.22944
$231$	0	0
$232$	2857.46	0.808627
$233$	1153.47	0.324320	0.162160	−	0.986765i	$-0.448154\pi$
$233$	1153.47	0.324320	0.162160	+	0.986765i	$0.448154\pi$
$234$	888.093	0.248105
$235$	−4319.18	−1.19894
$236$	1440.81	0.397410
$237$	−2364.41	−0.648038
$238$	2007.77	0.546826
$239$	−633.939	−0.171574	−0.0857869	−	0.996314i	$-0.527340\pi$
$239$	−633.939	−0.171574	−0.0857869	+	0.996314i	$0.527340\pi$
$240$	255.014	0.0685877
$241$	−3185.15	−0.851343	−0.425671	−	0.904878i	$-0.639962\pi$
$241$	−3185.15	−0.851343	−0.425671	+	0.904878i	$0.639962\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−523.113	−0.137250
$245$	2768.66	0.721973
$246$	1224.16	0.317275
$247$	4895.02	1.26098
$248$	−4903.51	−1.25554
$249$	2611.45	0.664633
$250$	−4169.80	−1.05489
$251$	−2805.97	−0.705622	−0.352811	−	0.935695i	$-0.614774\pi$
$251$	−2805.97	−0.705622	−0.352811	+	0.935695i	$0.614774\pi$
$252$	−595.787	−0.148933
$253$	0	0
$254$	−1373.95	−0.339406
$255$	−4513.38	−1.10839
$256$	−4242.52	−1.03577
$257$	−3930.62	−0.954029	−0.477014	−	0.878895i	$-0.658281\pi$
$257$	−3930.62	−0.954029	−0.477014	+	0.878895i	$0.658281\pi$
$258$	−2226.04	−0.537159
$259$	6052.50	1.45206
$260$	4894.75	1.16754
$261$	−1114.18	−0.264238
$262$	458.327	0.108075
$263$	−7363.55	−1.72645	−0.863224	−	0.504821i	$-0.831559\pi$
$263$	−7363.55	−1.72645	−0.863224	+	0.504821i	$0.831559\pi$
$264$	0	0
$265$	−5454.13	−1.26432
$266$	2313.46	0.533260
$267$	2364.64	0.542000
$268$	720.829	0.164297
$269$	−5.56813	−0.00126206	−0.000631032	−	1.00000i	$-0.500201\pi$
$269$	−5.56813	−0.00126206	−0.000631032		1.00000i	$0.500201\pi$
$270$	−943.523	−0.212670
$271$	2378.32	0.533108	0.266554	−	0.963820i	$-0.414115\pi$
$271$	2378.32	0.533108	0.266554	+	0.963820i	$0.414115\pi$
$272$	−346.270	−0.0771901
$273$	2296.18	0.509051
$274$	−3908.14	−0.861676
$275$	0	0
$276$	1727.94	0.376847
$277$	5629.04	1.22100	0.610499	−	0.792017i	$-0.290969\pi$
$277$	5629.04	1.22100	0.610499	+	0.792017i	$0.290969\pi$
$278$	1104.53	0.238293
$279$	1911.98	0.410277
$280$	6256.37	1.33532
$281$	3642.04	0.773189	0.386594	−	0.922250i	$-0.373651\pi$
$281$	3642.04	0.773189	0.386594	+	0.922250i	$0.373651\pi$
$282$	1226.04	0.258899
$283$	−4526.44	−0.950773	−0.475386	−	0.879777i	$-0.657692\pi$
$283$	−4526.44	−0.950773	−0.475386	+	0.879777i	$0.657692\pi$
$284$	1517.40	0.317047
$285$	−5200.54	−1.08089
$286$	0	0
$287$	3165.08	0.650972
$288$	1589.49	0.325213
$289$	1215.49	0.247402
$290$	4326.16	0.876003
$291$	−2557.14	−0.515127
$292$	912.985	0.182974
$293$	−3472.04	−0.692282	−0.346141	−	0.938182i	$-0.612508\pi$
$293$	−3472.04	−0.692282	−0.346141	+	0.938182i	$0.612508\pi$
$294$	−785.909	−0.155902
$295$	5899.49	1.16434
$296$	−9904.85	−1.94496
$297$	0	0
$298$	−4872.85	−0.947237
$299$	−6659.52	−1.28806
$300$	−3440.19	−0.662066
$301$	−5755.45	−1.10212
$302$	−962.528	−0.183402
$303$	3212.08	0.609007
$304$	−398.990	−0.0752752
$305$	−2141.92	−0.402118
$306$	1281.16	0.239344
$307$	6657.30	1.23763	0.618815	−	0.785537i	$-0.287613\pi$
$307$	6657.30	1.23763	0.618815	+	0.785537i	$0.287613\pi$
$308$	0	0
$309$	5117.21	0.942096
$310$	−7423.86	−1.36015
$311$	−722.660	−0.131763	−0.0658815	−	0.997827i	$-0.520986\pi$
$311$	−722.660	−0.131763	−0.0658815	+	0.997827i	$0.520986\pi$
$312$	−3757.67	−0.681846
$313$	−4727.15	−0.853657	−0.426828	−	0.904333i	$-0.640369\pi$
$313$	−4727.15	−0.853657	−0.426828	+	0.904333i	$0.640369\pi$
$314$	203.640	0.0365989
$315$	−2439.49	−0.436348
$316$	3699.12	0.658518
$317$	11094.3	1.96567	0.982837	−	0.184474i	$-0.0590583\pi$
$317$	11094.3	1.96567	0.982837	+	0.184474i	$0.0590583\pi$
$318$	1548.20	0.273015
$319$	0	0
$320$	−6851.71	−1.19694
$321$	1956.20	0.340138
$322$	−3147.38	−0.544710
$323$	7061.56	1.21646
$324$	−380.173	−0.0651874
$325$	13258.6	2.26293
$326$	4216.82	0.716404
$327$	6198.26	1.04821
$328$	−5179.62	−0.871942
$329$	3169.93	0.531198
$330$	0	0
$331$	−3064.62	−0.508902	−0.254451	−	0.967086i	$-0.581895\pi$
$331$	−3064.62	−0.508902	−0.254451	+	0.967086i	$0.581895\pi$
$332$	−4085.60	−0.675381
$333$	3862.11	0.635562
$334$	6111.73	1.00125
$335$	2951.48	0.481363
$336$	−187.160	−0.0303881
$337$	7186.71	1.16168	0.580838	−	0.814019i	$-0.302725\pi$
$337$	7186.71	1.16168	0.580838	+	0.814019i	$0.302725\pi$
$338$	1359.86	0.218837
$339$	−544.677	−0.0872648
$340$	7061.16	1.12631
$341$	0	0
$342$	1476.22	0.233406
$343$	−6869.77	−1.08144
$344$	9418.73	1.47623
$345$	7075.17	1.10410
$346$	−1585.89	−0.246411
$347$	−10910.7	−1.68794	−0.843971	−	0.536389i	$-0.819788\pi$
$347$	−10910.7	−1.68794	−0.843971	+	0.536389i	$0.819788\pi$
$348$	1743.14	0.268511
$349$	7635.29	1.17108	0.585541	−	0.810643i	$-0.300882\pi$
$349$	7635.29	1.17108	0.585541	+	0.810643i	$0.300882\pi$
$350$	6266.19	0.956977
$351$	1465.19	0.222810
$352$	0	0
$353$	5344.39	0.805817	0.402908	−	0.915240i	$-0.367999\pi$
$353$	5344.39	0.805817	0.402908	+	0.915240i	$0.367999\pi$
$354$	−1674.62	−0.251427
$355$	6213.11	0.928895
$356$	−3699.48	−0.550764
$357$	3312.46	0.491076
$358$	3226.33	0.476304
$359$	3249.23	0.477682	0.238841	−	0.971059i	$-0.423233\pi$
$359$	3249.23	0.477682	0.238841	+	0.971059i	$0.423233\pi$
$360$	3992.20	0.584465
$361$	1277.69	0.186279
$362$	−1762.17	−0.255849
$363$	0	0
$364$	−3592.36	−0.517283
$365$	3738.28	0.536083
$366$	608.003	0.0868329
$367$	2898.60	0.412277	0.206138	−	0.978523i	$-0.433910\pi$
$367$	2898.60	0.412277	0.206138	+	0.978523i	$0.433910\pi$
$368$	542.813	0.0768915
$369$	2019.64	0.284928
$370$	−14995.8	−2.10702
$371$	4002.90	0.560162
$372$	−2991.29	−0.416912
$373$	−10883.1	−1.51074	−0.755372	−	0.655297i	$-0.772544\pi$
$373$	−10883.1	−1.51074	−0.755372	+	0.655297i	$0.772544\pi$
$374$	0	0
$375$	−6879.42	−0.947338
$376$	−5187.56	−0.711511
$377$	−6718.08	−0.917768
$378$	692.471	0.0942245
$379$	521.221	0.0706420	0.0353210	−	0.999376i	$-0.488755\pi$
$379$	521.221	0.0706420	0.0353210	+	0.999376i	$0.488755\pi$
$380$	8136.23	1.09837
$381$	−2266.77	−0.304803
$382$	1447.64	0.193895
$383$	−10668.3	−1.42330	−0.711649	−	0.702536i	$-0.752051\pi$
$383$	−10668.3	−1.42330	−0.711649	+	0.702536i	$0.752051\pi$
$384$	−2293.71	−0.304819
$385$	0	0
$386$	−5760.43	−0.759580
$387$	−3672.56	−0.482395
$388$	4000.63	0.523457
$389$	2186.58	0.284997	0.142499	−	0.989795i	$-0.454486\pi$
$389$	2186.58	0.284997	0.142499	+	0.989795i	$0.454486\pi$
$390$	−5689.07	−0.738659
$391$	−9607.01	−1.24258
$392$	3325.31	0.428453
$393$	756.157	0.0970562
$394$	2380.39	0.304371
$395$	15146.3	1.92935
$396$	0	0
$397$	2360.44	0.298406	0.149203	−	0.988807i	$-0.452329\pi$
$397$	2360.44	0.298406	0.149203	+	0.988807i	$0.452329\pi$
$398$	4798.99	0.604401
$399$	3816.79	0.478893
$400$	−1080.70	−0.135087
$401$	5041.37	0.627815	0.313908	−	0.949454i	$-0.398362\pi$
$401$	5041.37	0.627815	0.313908	+	0.949454i	$0.398362\pi$
$402$	−837.804	−0.103945
$403$	11528.5	1.42500
$404$	−5025.29	−0.618855
$405$	−1556.64	−0.190988
$406$	−3175.06	−0.388117
$407$	0	0
$408$	−5420.81	−0.657769
$409$	−574.833	−0.0694955	−0.0347477	−	0.999396i	$-0.511063\pi$
$409$	−574.833	−0.0694955	−0.0347477	+	0.999396i	$0.511063\pi$
$410$	−7841.89	−0.944594
$411$	−6447.72	−0.773826
$412$	−8005.85	−0.957330
$413$	−4329.76	−0.515868
$414$	−2008.35	−0.238418
$415$	−16728.7	−1.97875
$416$	9583.98	1.12955
$417$	1822.28	0.213998
$418$	0	0
$419$	690.078	0.0804594	0.0402297	−	0.999190i	$-0.487191\pi$
$419$	690.078	0.0804594	0.0402297	+	0.999190i	$0.487191\pi$
$420$	3816.57	0.443404
$421$	13982.6	1.61869	0.809345	−	0.587334i	$-0.199823\pi$
$421$	13982.6	1.61869	0.809345	+	0.587334i	$0.199823\pi$
$422$	−1940.82	−0.223881
$423$	2022.74	0.232503
$424$	−6550.70	−0.750307
$425$	19126.8	2.18303
$426$	−1763.65	−0.200584
$427$	1572.00	0.178160
$428$	−3060.47	−0.345638
$429$	0	0
$430$	14259.8	1.59923
$431$	11745.4	1.31266	0.656332	−	0.754472i	$-0.272107\pi$
$431$	11745.4	1.31266	0.656332	+	0.754472i	$0.272107\pi$
$432$	−119.427	−0.0133008
$433$	167.526	0.0185930	0.00929651	−	0.999957i	$-0.497041\pi$
$433$	167.526	0.0185930	0.00929651	+	0.999957i	$0.497041\pi$
$434$	5448.53	0.602621
$435$	7137.38	0.786693
$436$	−9697.17	−1.06516
$437$	−11069.7	−1.21175
$438$	−1061.14	−0.115761
$439$	−3975.68	−0.432229	−0.216115	−	0.976368i	$-0.569339\pi$
$439$	−3975.68	−0.432229	−0.216115	+	0.976368i	$0.569339\pi$
$440$	0	0
$441$	−1296.61	−0.140007
$442$	7724.90	0.831303
$443$	−6117.57	−0.656105	−0.328053	−	0.944659i	$-0.606392\pi$
$443$	−6117.57	−0.656105	−0.328053	+	0.944659i	$0.606392\pi$
$444$	−6042.26	−0.645839
$445$	−15147.8	−1.61365
$446$	4524.77	0.480390
$447$	−8039.32	−0.850664
$448$	5028.61	0.530312
$449$	12725.6	1.33755	0.668773	−	0.743467i	$-0.266820\pi$
$449$	12725.6	1.33755	0.668773	+	0.743467i	$0.266820\pi$
$450$	3998.46	0.418865
$451$	0	0
$452$	852.145	0.0886760
$453$	−1588.00	−0.164703
$454$	−7329.42	−0.757680
$455$	−14709.1	−1.51555
$456$	−6246.13	−0.641451
$457$	−13086.5	−1.33952	−0.669759	−	0.742579i	$-0.733603\pi$
$457$	−13086.5	−1.33952	−0.669759	+	0.742579i	$0.733603\pi$
$458$	4184.43	0.426911
$459$	2113.69	0.214942
$460$	−11069.1	−1.12195
$461$	7251.82	0.732649	0.366324	−	0.930487i	$-0.380616\pi$
$461$	7251.82	0.732649	0.366324	+	0.930487i	$0.380616\pi$
$462$	0	0
$463$	−1559.25	−0.156511	−0.0782553	−	0.996933i	$-0.524935\pi$
$463$	−1559.25	−0.156511	−0.0782553	+	0.996933i	$0.524935\pi$
$464$	547.586	0.0547867
$465$	−12248.0	−1.22148
$466$	2097.45	0.208503
$467$	9284.25	0.919966	0.459983	−	0.887928i	$-0.347856\pi$
$467$	9284.25	0.919966	0.459983	+	0.887928i	$0.347856\pi$
$468$	−2292.29	−0.226413
$469$	−2166.15	−0.213270
$470$	−7853.91	−0.770795
$471$	335.969	0.0328676
$472$	7085.60	0.690977
$473$	0	0
$474$	−4299.40	−0.416620
$475$	22038.9	2.12887
$476$	−5182.33	−0.499017
$477$	2554.25	0.245181
$478$	−1152.74	−0.110304
$479$	−10669.5	−1.01775	−0.508873	−	0.860842i	$-0.669938\pi$
$479$	−10669.5	−1.01775	−0.508873	+	0.860842i	$0.669938\pi$
$480$	−10182.2	−0.968228
$481$	23287.0	2.20747
$482$	−5791.82	−0.547324
$483$	−5192.61	−0.489176
$484$	0	0
$485$	16380.8	1.53364
$486$	441.867	0.0412417
$487$	−15539.4	−1.44591	−0.722955	−	0.690895i	$-0.757217\pi$
$487$	−15539.4	−1.44591	−0.722955	+	0.690895i	$0.757217\pi$
$488$	−2572.56	−0.238636
$489$	6956.98	0.643365
$490$	5034.48	0.464152
$491$	−14350.7	−1.31902	−0.659511	−	0.751695i	$-0.729237\pi$
$491$	−14350.7	−1.31902	−0.659511	+	0.751695i	$0.729237\pi$
$492$	−3159.73	−0.289536
$493$	−9691.49	−0.885361
$494$	8901.02	0.810680
$495$	0	0
$496$	−939.679	−0.0850662
$497$	−4559.93	−0.411551
$498$	4748.60	0.427289
$499$	9290.76	0.833490	0.416745	−	0.909023i	$-0.363171\pi$
$499$	9290.76	0.833490	0.416745	+	0.909023i	$0.363171\pi$
$500$	10762.8	0.962657
$501$	10083.2	0.899174
$502$	−5102.32	−0.453640
$503$	−11744.5	−1.04107	−0.520537	−	0.853839i	$-0.674268\pi$
$503$	−11744.5	−1.04107	−0.520537	+	0.853839i	$0.674268\pi$
$504$	−2929.96	−0.258950
$505$	−20576.4	−1.81314
$506$	0	0
$507$	2243.53	0.196526
$508$	3546.35	0.309732
$509$	2755.59	0.239959	0.119980	−	0.992776i	$-0.461717\pi$
$509$	2755.59	0.239959	0.119980	+	0.992776i	$0.461717\pi$
$510$	−8207.04	−0.712576
$511$	−2743.60	−0.237514
$512$	−1597.94	−0.137929
$513$	2435.50	0.209610
$514$	−7147.37	−0.613340
$515$	−32780.5	−2.80482
$516$	5745.71	0.490195
$517$	0	0
$518$	11005.7	0.933523
$519$	−2616.44	−0.221289
$520$	24071.4	2.03000
$521$	11366.3	0.955794	0.477897	−	0.878416i	$-0.341399\pi$
$521$	11366.3	0.955794	0.477897	+	0.878416i	$0.341399\pi$
$522$	−2026.01	−0.169877
$523$	1742.55	0.145691	0.0728455	−	0.997343i	$-0.476792\pi$
$523$	1742.55	0.145691	0.0728455	+	0.997343i	$0.476792\pi$
$524$	−1183.01	−0.0986257
$525$	10338.1	0.859411
$526$	−13389.7	−1.10992
$527$	16631.0	1.37468
$528$	0	0
$529$	2892.96	0.237771
$530$	−9917.68	−0.812824
$531$	−2762.82	−0.225794
$532$	−5971.35	−0.486637
$533$	12177.6	0.989629
$534$	4299.82	0.348449
$535$	−12531.3	−1.01266
$536$	3544.89	0.285664
$537$	5322.86	0.427744
$538$	−10.1250	−0.000811374 0
$539$	0	0
$540$	2435.36	0.194076
$541$	−16859.5	−1.33983	−0.669913	−	0.742439i	$-0.733669\pi$
$541$	−16859.5	−1.33983	−0.669913	+	0.742439i	$0.733669\pi$
$542$	4324.68	0.342733
$543$	−2907.25	−0.229765
$544$	13825.8	1.08966
$545$	−39705.7	−3.12074
$546$	4175.32	0.327266
$547$	−3129.06	−0.244587	−0.122293	−	0.992494i	$-0.539025\pi$
$547$	−3129.06	−0.244587	−0.122293	+	0.992494i	$0.539025\pi$
$548$	10087.4	0.786339
$549$	1003.10	0.0779801
$550$	0	0
$551$	−11167.0	−0.863397
$552$	8497.65	0.655225
$553$	−11116.2	−0.854805
$554$	10235.7	0.784973
$555$	−24740.4	−1.89220
$556$	−2850.95	−0.217459
$557$	−12117.1	−0.921754	−0.460877	−	0.887464i	$-0.652465\pi$
$557$	−12117.1	−0.921754	−0.460877	+	0.887464i	$0.652465\pi$
$558$	3476.71	0.263765
$559$	−22144.1	−1.67548
$560$	1198.93	0.0904717
$561$	0	0
$562$	6622.62	0.497079
$563$	−8619.51	−0.645238	−0.322619	−	0.946529i	$-0.604563\pi$
$563$	−8619.51	−0.645238	−0.322619	+	0.946529i	$0.604563\pi$
$564$	−3164.57	−0.236263
$565$	3489.16	0.259806
$566$	−8230.78	−0.611247
$567$	1142.45	0.0846181
$568$	7462.28	0.551250
$569$	17596.1	1.29642	0.648212	−	0.761460i	$-0.275517\pi$
$569$	17596.1	1.29642	0.648212	+	0.761460i	$0.275517\pi$
$570$	−9456.57	−0.694898
$571$	16207.2	1.18783	0.593916	−	0.804527i	$-0.297581\pi$
$571$	16207.2	1.18783	0.593916	+	0.804527i	$0.297581\pi$
$572$	0	0
$573$	2388.35	0.174127
$574$	5755.33	0.418506
$575$	−29983.2	−2.17458
$576$	3208.77	0.232116
$577$	−16520.8	−1.19197	−0.595987	−	0.802994i	$-0.703239\pi$
$577$	−16520.8	−1.19197	−0.595987	+	0.802994i	$0.703239\pi$
$578$	2210.22	0.159053
$579$	−9503.66	−0.682139
$580$	−11166.4	−0.799414
$581$	12277.6	0.876695
$582$	−4649.85	−0.331173
$583$	0	0
$584$	4489.87	0.318137
$585$	−9385.93	−0.663351
$586$	−6313.49	−0.445065
$587$	−14514.9	−1.02060	−0.510301	−	0.859996i	$-0.670466\pi$
$587$	−14514.9	−1.02060	−0.510301	+	0.859996i	$0.670466\pi$
$588$	2028.54	0.142271
$589$	19163.1	1.34058
$590$	10727.5	0.748551
$591$	3927.21	0.273340
$592$	−1898.10	−0.131776
$593$	−6639.31	−0.459771	−0.229885	−	0.973218i	$-0.573835\pi$
$593$	−6639.31	−0.459771	−0.229885	+	0.973218i	$0.573835\pi$
$594$	0	0
$595$	−21219.4	−1.46203
$596$	12577.5	0.864420
$597$	7917.47	0.542781
$598$	−12109.5	−0.828087
$599$	2835.72	0.193430	0.0967148	−	0.995312i	$-0.469167\pi$
$599$	2835.72	0.193430	0.0967148	+	0.995312i	$0.469167\pi$
$600$	−16918.2	−1.15113
$601$	−12841.2	−0.871555	−0.435778	−	0.900054i	$-0.643527\pi$
$601$	−12841.2	−0.871555	−0.435778	+	0.900054i	$0.643527\pi$
$602$	−10465.6	−0.708548
$603$	−1382.23	−0.0933475
$604$	2484.42	0.167367
$605$	0	0
$606$	5840.78	0.391527
$607$	−15853.7	−1.06010	−0.530050	−	0.847966i	$-0.677827\pi$
$607$	−15853.7	−1.06010	−0.530050	+	0.847966i	$0.677827\pi$
$608$	15930.8	1.06263
$609$	−5238.27	−0.348548
$610$	−3894.83	−0.258520
$611$	12196.3	0.807545
$612$	−3306.85	−0.218418
$613$	−10069.5	−0.663467	−0.331733	−	0.943373i	$-0.607633\pi$
$613$	−10069.5	−0.663467	−0.331733	+	0.943373i	$0.607633\pi$
$614$	12105.5	0.795665
$615$	−12937.7	−0.848290
$616$	0	0
$617$	−4947.89	−0.322843	−0.161422	−	0.986886i	$-0.551608\pi$
$617$	−4947.89	−0.322843	−0.161422	+	0.986886i	$0.551608\pi$
$618$	9305.03	0.605669
$619$	−14401.3	−0.935114	−0.467557	−	0.883963i	$-0.654866\pi$
$619$	−14401.3	−0.935114	−0.467557	+	0.883963i	$0.654866\pi$
$620$	19162.0	1.24123
$621$	−3313.41	−0.214111
$622$	−1314.07	−0.0847097
$623$	11117.3	0.714933
$624$	−720.097	−0.0461970
$625$	13528.6	0.865833
$626$	−8595.77	−0.548811
$627$	0	0
$628$	−525.622	−0.0333991
$629$	33593.7	2.12952
$630$	−4435.92	−0.280526
$631$	−20392.2	−1.28653	−0.643264	−	0.765645i	$-0.722420\pi$
$631$	−20392.2	−1.28653	−0.643264	+	0.765645i	$0.722420\pi$
$632$	18191.5	1.14497
$633$	−3202.01	−0.201056
$634$	20173.7	1.26372
$635$	14520.8	0.907462
$636$	−3996.12	−0.249146
$637$	−7818.03	−0.486282
$638$	0	0
$639$	−2909.70	−0.180134
$640$	14693.4	0.907511
$641$	−19353.8	−1.19256	−0.596278	−	0.802778i	$-0.703354\pi$
$641$	−19353.8	−1.19256	−0.596278	+	0.802778i	$0.703354\pi$
$642$	3557.11	0.218673
$643$	−4443.84	−0.272547	−0.136274	−	0.990671i	$-0.543513\pi$
$643$	−4443.84	−0.272547	−0.136274	+	0.990671i	$0.543513\pi$
$644$	8123.83	0.497086
$645$	23526.2	1.43619
$646$	12840.6	0.782054
$647$	−5631.62	−0.342198	−0.171099	−	0.985254i	$-0.554732\pi$
$647$	−5631.62	−0.342198	−0.171099	+	0.985254i	$0.554732\pi$
$648$	−1869.61	−0.113341
$649$	0	0
$650$	24109.1	1.45483
$651$	8989.08	0.541183
$652$	−10884.2	−0.653769
$653$	18345.4	1.09940	0.549702	−	0.835361i	$-0.314741\pi$
$653$	18345.4	1.09940	0.549702	+	0.835361i	$0.314741\pi$
$654$	11270.8	0.673889
$655$	−4843.89	−0.288957
$656$	−992.592	−0.0590765
$657$	−1750.69	−0.103959
$658$	5764.15	0.341504
$659$	8999.50	0.531974	0.265987	−	0.963977i	$-0.414302\pi$
$659$	8999.50	0.531974	0.265987	+	0.963977i	$0.414302\pi$
$660$	0	0
$661$	−22524.0	−1.32539	−0.662695	−	0.748890i	$-0.730587\pi$
$661$	−22524.0	−1.32539	−0.662695	+	0.748890i	$0.730587\pi$
$662$	−5572.64	−0.327171
$663$	12744.7	0.746550
$664$	−20092.1	−1.17429
$665$	−24450.1	−1.42576
$666$	7022.78	0.408600
$667$	15192.4	0.881936
$668$	−15775.2	−0.913714
$669$	7465.05	0.431413
$670$	5366.92	0.309466
$671$	0	0
$672$	7472.89	0.428978
$673$	−22164.9	−1.26953	−0.634766	−	0.772704i	$-0.718904\pi$
$673$	−22164.9	−1.26953	−0.634766	+	0.772704i	$0.718904\pi$
$674$	13068.2	0.746836
$675$	6596.74	0.376161
$676$	−3510.00	−0.199704
$677$	−21982.0	−1.24792	−0.623958	−	0.781458i	$-0.714476\pi$
$677$	−21982.0	−1.24792	−0.623958	+	0.781458i	$0.714476\pi$
$678$	−990.430	−0.0561021
$679$	−12022.2	−0.679486
$680$	34725.3	1.95832
$681$	−12092.2	−0.680433
$682$	0	0
$683$	−14360.0	−0.804495	−0.402247	−	0.915531i	$-0.631771\pi$
$683$	−14360.0	−0.804495	−0.402247	+	0.915531i	$0.631771\pi$
$684$	−3810.33	−0.212999
$685$	41303.6	2.30384
$686$	−12491.9	−0.695250
$687$	6903.55	0.383387
$688$	1804.95	0.100019
$689$	15401.1	0.851577
$690$	12865.4	0.709820
$691$	28580.0	1.57342	0.786711	−	0.617322i	$-0.211782\pi$
$691$	28580.0	1.57342	0.786711	+	0.617322i	$0.211782\pi$
$692$	4093.41	0.224867
$693$	0	0
$694$	−19839.8	−1.08517
$695$	−11673.4	−0.637118
$696$	8572.38	0.466861
$697$	17567.5	0.954684
$698$	13883.9	0.752882
$699$	3460.42	0.187246
$700$	−16173.9	−0.873308
$701$	−5713.11	−0.307819	−0.153910	−	0.988085i	$-0.549186\pi$
$701$	−5713.11	−0.307819	−0.153910	+	0.988085i	$0.549186\pi$
$702$	2664.28	0.143243
$703$	38708.4	2.07669
$704$	0	0
$705$	−12957.5	−0.692211
$706$	9718.14	0.518055
$707$	15101.4	0.803320
$708$	4322.43	0.229445
$709$	17566.5	0.930501	0.465250	−	0.885179i	$-0.345964\pi$
$709$	17566.5	0.930501	0.465250	+	0.885179i	$0.345964\pi$
$710$	11297.8	0.597182
$711$	−7093.24	−0.374145
$712$	−18193.3	−0.957614
$713$	−26070.7	−1.36936
$714$	6023.31	0.315710
$715$	0	0
$716$	−8327.60	−0.434661
$717$	−1901.82	−0.0990582
$718$	5908.34	0.307099
$719$	3747.65	0.194386	0.0971931	−	0.995266i	$-0.469014\pi$
$719$	3747.65	0.194386	0.0971931	+	0.995266i	$0.469014\pi$
$720$	765.041	0.0395991
$721$	24058.3	1.24269
$722$	2323.32	0.119758
$723$	−9555.45	−0.491523
$724$	4548.39	0.233480
$725$	−30246.8	−1.54943
$726$	0	0
$727$	19566.4	0.998183	0.499092	−	0.866549i	$-0.333667\pi$
$727$	19566.4	0.998183	0.499092	+	0.866549i	$0.333667\pi$
$728$	−17666.5	−0.899400
$729$	729.000	0.0370370
$730$	6797.61	0.344645
$731$	−31945.0	−1.61632
$732$	−1569.34	−0.0792411
$733$	−5341.99	−0.269183	−0.134591	−	0.990901i	$-0.542972\pi$
$733$	−5341.99	−0.269183	−0.134591	+	0.990901i	$0.542972\pi$
$734$	5270.75	0.265051
$735$	8305.98	0.416831
$736$	−21673.4	−1.08545
$737$	0	0
$738$	3672.48	0.183179
$739$	−30927.1	−1.53947	−0.769737	−	0.638361i	$-0.779613\pi$
$739$	−30927.1	−1.53947	−0.769737	+	0.638361i	$0.779613\pi$
$740$	38706.3	1.92280
$741$	14685.1	0.728029
$742$	7278.79	0.360125
$743$	26069.2	1.28720	0.643598	−	0.765364i	$-0.277441\pi$
$743$	26069.2	1.28720	0.643598	+	0.765364i	$0.277441\pi$
$744$	−14710.5	−0.724885
$745$	51499.3	2.53260
$746$	−19789.7	−0.971249
$747$	7834.34	0.383726
$748$	0	0
$749$	9196.96	0.448664
$750$	−12509.4	−0.609039
$751$	−5928.45	−0.288059	−0.144029	−	0.989573i	$-0.546006\pi$
$751$	−5928.45	−0.288059	−0.144029	+	0.989573i	$0.546006\pi$
$752$	−994.113	−0.0482069
$753$	−8417.90	−0.407391
$754$	−12216.0	−0.590029
$755$	10172.6	0.490356
$756$	−1787.36	−0.0859864
$757$	12040.3	0.578089	0.289044	−	0.957316i	$-0.406663\pi$
$757$	12040.3	0.578089	0.289044	+	0.957316i	$0.406663\pi$
$758$	947.779	0.0454154
$759$	0	0
$760$	40012.3	1.90973
$761$	−23222.8	−1.10621	−0.553106	−	0.833111i	$-0.686558\pi$
$761$	−23222.8	−1.10621	−0.553106	+	0.833111i	$0.686558\pi$
$762$	−4121.85	−0.195956
$763$	29140.8	1.38266
$764$	−3736.57	−0.176943
$765$	−13540.1	−0.639927
$766$	−19399.0	−0.915030
$767$	−16658.7	−0.784240
$768$	−12727.6	−0.598003
$769$	−4256.37	−0.199595	−0.0997975	−	0.995008i	$-0.531819\pi$
$769$	−4256.37	−0.199595	−0.0997975	+	0.995008i	$0.531819\pi$
$770$	0	0
$771$	−11791.9	−0.550809
$772$	14868.5	0.693170
$773$	17107.1	0.795987	0.397994	−	0.917388i	$-0.369707\pi$
$773$	17107.1	0.795987	0.397994	+	0.917388i	$0.369707\pi$
$774$	−6678.11	−0.310129
$775$	51904.7	2.40577
$776$	19674.3	0.910135
$777$	18157.5	0.838348
$778$	3976.04	0.183223
$779$	20242.1	0.931000
$780$	14684.3	0.674078
$781$	0	0
$782$	−17469.2	−0.798846
$783$	−3342.55	−0.152558
$784$	637.242	0.0290289
$785$	−2152.19	−0.0978536
$786$	1374.98	0.0623969
$787$	21847.6	0.989557	0.494779	−	0.869019i	$-0.335249\pi$
$787$	21847.6	0.989557	0.494779	+	0.869019i	$0.335249\pi$
$788$	−6144.10	−0.277760
$789$	−22090.7	−0.996766
$790$	27541.7	1.24037
$791$	−2560.77	−0.115108
$792$	0	0
$793$	6048.27	0.270845
$794$	4292.18	0.191843
$795$	−16362.4	−0.729955
$796$	−12386.9	−0.551559
$797$	31883.5	1.41703	0.708513	−	0.705698i	$-0.249366\pi$
$797$	31883.5	1.41703	0.708513	+	0.705698i	$0.249366\pi$
$798$	6940.37	0.307878
$799$	17594.4	0.779029
$800$	43150.0	1.90698
$801$	7093.93	0.312924
$802$	9167.13	0.403619
$803$	0	0
$804$	2162.49	0.0948570
$805$	33263.5	1.45638
$806$	20963.2	0.916125
$807$	−16.7044	−0.000728652 0
$808$	−24713.3	−1.07600
$809$	12262.7	0.532921	0.266460	−	0.963846i	$-0.414146\pi$
$809$	12262.7	0.532921	0.266460	+	0.963846i	$0.414146\pi$
$810$	−2830.57	−0.122785
$811$	21255.2	0.920307	0.460154	−	0.887839i	$-0.347794\pi$
$811$	21255.2	0.920307	0.460154	+	0.887839i	$0.347794\pi$
$812$	8195.26	0.354184
$813$	7134.95	0.307790
$814$	0	0
$815$	−44566.0	−1.91543
$816$	−1038.81	−0.0445657
$817$	−36808.7	−1.57622
$818$	−1045.26	−0.0446783
$819$	6888.53	0.293901
$820$	20241.0	0.862008
$821$	−34754.2	−1.47738	−0.738690	−	0.674046i	$-0.764555\pi$
$821$	−34754.2	−1.47738	−0.738690	+	0.674046i	$0.764555\pi$
$822$	−11724.4	−0.497489
$823$	25847.0	1.09474	0.547368	−	0.836892i	$-0.315630\pi$
$823$	25847.0	1.09474	0.547368	+	0.836892i	$0.315630\pi$
$824$	−39371.1	−1.66451
$825$	0	0
$826$	−7873.15	−0.331649
$827$	13383.8	0.562758	0.281379	−	0.959597i	$-0.409208\pi$
$827$	13383.8	0.562758	0.281379	+	0.959597i	$0.409208\pi$
$828$	5183.82	0.217573
$829$	28625.8	1.19929	0.599647	−	0.800265i	$-0.295308\pi$
$829$	28625.8	1.19929	0.599647	+	0.800265i	$0.295308\pi$
$830$	−30419.2	−1.27213
$831$	16887.1	0.704944
$832$	19347.6	0.806198
$833$	−11278.3	−0.469111
$834$	3313.59	0.137578
$835$	−64592.6	−2.67703
$836$	0	0
$837$	5735.95	0.236874
$838$	1254.82	0.0517269
$839$	−43963.0	−1.80903	−0.904513	−	0.426447i	$-0.859765\pi$
$839$	−43963.0	−1.80903	−0.904513	+	0.426447i	$0.859765\pi$
$840$	18769.1	0.770947
$841$	−9063.03	−0.371603
$842$	25425.6	1.04065
$843$	10926.1	0.446401
$844$	5009.53	0.204307
$845$	−14371.9	−0.585099
$846$	3678.11	0.149475
$847$	0	0
$848$	−1255.34	−0.0508354
$849$	−13579.3	−0.548929
$850$	34779.8	1.40346
$851$	−52661.6	−2.12129
$852$	4552.21	0.183047
$853$	−46858.1	−1.88088	−0.940440	−	0.339959i	$-0.889587\pi$
$853$	−46858.1	−1.88088	−0.940440	+	0.339959i	$0.889587\pi$
$854$	2858.50	0.114538
$855$	−15601.6	−0.624052
$856$	−15050.7	−0.600962
$857$	13108.4	0.522489	0.261244	−	0.965273i	$-0.415867\pi$
$857$	13108.4	0.522489	0.261244	+	0.965273i	$0.415867\pi$
$858$	0	0
$859$	−18510.5	−0.735240	−0.367620	−	0.929976i	$-0.619827\pi$
$859$	−18510.5	−0.735240	−0.367620	+	0.929976i	$0.619827\pi$
$860$	−36806.6	−1.45941
$861$	9495.25	0.375839
$862$	21357.7	0.843905
$863$	−27702.6	−1.09271	−0.546354	−	0.837555i	$-0.683984\pi$
$863$	−27702.6	−1.09271	−0.546354	+	0.837555i	$0.683984\pi$
$864$	4768.46	0.187762
$865$	16760.7	0.658822
$866$	304.626	0.0119534
$867$	3646.46	0.142838
$868$	−14063.4	−0.549934
$869$	0	0
$870$	12978.5	0.505761
$871$	−8334.27	−0.324220
$872$	−47688.6	−1.85200
$873$	−7671.41	−0.297409
$874$	−20128.9	−0.779029
$875$	−32343.2	−1.24960
$876$	2738.96	0.105640
$877$	15341.4	0.590698	0.295349	−	0.955389i	$-0.404564\pi$
$877$	15341.4	0.590698	0.295349	+	0.955389i	$0.404564\pi$
$878$	−7229.29	−0.277878
$879$	−10416.1	−0.399689
$880$	0	0
$881$	39390.7	1.50636	0.753182	−	0.657812i	$-0.228518\pi$
$881$	39390.7	1.50636	0.753182	+	0.657812i	$0.228518\pi$
$882$	−2357.73	−0.0900100
$883$	−16427.5	−0.626082	−0.313041	−	0.949740i	$-0.601348\pi$
$883$	−16427.5	−0.626082	−0.313041	+	0.949740i	$0.601348\pi$
$884$	−19939.0	−0.758622
$885$	17698.5	0.672235
$886$	−11124.1	−0.421807
$887$	21130.1	0.799865	0.399933	−	0.916545i	$-0.369034\pi$
$887$	21130.1	0.799865	0.399933	+	0.916545i	$0.369034\pi$
$888$	−29714.5	−1.12292
$889$	−10657.1	−0.402055
$890$	−27544.4	−1.03740
$891$	0	0
$892$	−11679.0	−0.438389
$893$	20273.1	0.759703
$894$	−14618.5	−0.546887
$895$	−34097.9	−1.27348
$896$	−10783.8	−0.402077
$897$	−19978.6	−0.743662
$898$	23140.0	0.859901
$899$	−26300.0	−0.975699
$900$	−10320.6	−0.382244
$901$	22217.7	0.821507
$902$	0	0
$903$	−17266.3	−0.636310
$904$	4190.67	0.154181
$905$	18623.7	0.684057
$906$	−2887.58	−0.105887
$907$	53222.0	1.94841	0.974205	−	0.225665i	$-0.0724556\pi$
$907$	53222.0	1.94841	0.974205	+	0.225665i	$0.0724556\pi$
$908$	18918.2	0.691436
$909$	9636.23	0.351610
$910$	−26746.8	−0.974340
$911$	−19976.1	−0.726494	−0.363247	−	0.931693i	$-0.618332\pi$
$911$	−19976.1	−0.726494	−0.363247	+	0.931693i	$0.618332\pi$
$912$	−1196.97	−0.0434601
$913$	0	0
$914$	−23796.2	−0.861169
$915$	−6425.76	−0.232163
$916$	−10800.6	−0.389587
$917$	3555.03	0.128024
$918$	3843.49	0.138185
$919$	−8732.05	−0.313432	−0.156716	−	0.987644i	$-0.550091\pi$
$919$	−8732.05	−0.313432	−0.156716	+	0.987644i	$0.550091\pi$
$920$	−54435.4	−1.95074
$921$	19971.9	0.714546
$922$	13186.6	0.471016
$923$	−17544.3	−0.625654
$924$	0	0
$925$	104845.	3.72679
$926$	−2835.31	−0.100620
$927$	15351.6	0.543919
$928$	−21864.0	−0.773404
$929$	−35095.6	−1.23945	−0.619725	−	0.784819i	$-0.712756\pi$
$929$	−35095.6	−1.23945	−0.619725	+	0.784819i	$0.712756\pi$
$930$	−22271.6	−0.785284
$931$	−12995.4	−0.457473
$932$	−5413.81	−0.190274
$933$	−2167.98	−0.0760734
$934$	16882.3	0.591441
$935$	0	0
$936$	−11273.0	−0.393664
$937$	−18700.7	−0.652002	−0.326001	−	0.945369i	$-0.605701\pi$
$937$	−18700.7	−0.652002	−0.326001	+	0.945369i	$0.605701\pi$
$938$	−3938.89	−0.137110
$939$	−14181.5	−0.492859
$940$	20272.0	0.703404
$941$	31357.5	1.08632	0.543160	−	0.839629i	$-0.317228\pi$
$941$	31357.5	1.08632	0.543160	+	0.839629i	$0.317228\pi$
$942$	610.919	0.0211304
$943$	−27538.7	−0.950991
$944$	1357.84	0.0468157
$945$	−7318.47	−0.251926
$946$	0	0
$947$	−17691.0	−0.607054	−0.303527	−	0.952823i	$-0.598164\pi$
$947$	−17691.0	−0.607054	−0.303527	+	0.952823i	$0.598164\pi$
$948$	11097.3	0.380195
$949$	−10556.0	−0.361077
$950$	40075.1	1.36864
$951$	33282.9	1.13488
$952$	−25485.6	−0.867641
$953$	−22457.5	−0.763347	−0.381673	−	0.924297i	$-0.624652\pi$
$953$	−22457.5	−0.763347	−0.381673	+	0.924297i	$0.624652\pi$
$954$	4644.61	0.157626
$955$	−15299.6	−0.518412
$956$	2975.39	0.100660
$957$	0	0
$958$	−19401.2	−0.654304
$959$	−30313.6	−1.02073
$960$	−20555.1	−0.691056
$961$	15340.8	0.514947
$962$	42344.6	1.41917
$963$	5868.59	0.196379
$964$	14949.5	0.499471
$965$	60879.8	2.03087
$966$	−9442.15	−0.314489
$967$	28238.1	0.939065	0.469532	−	0.882915i	$-0.344422\pi$
$967$	28238.1	0.939065	0.469532	+	0.882915i	$0.344422\pi$
$968$	0	0
$969$	21184.7	0.702322
$970$	29786.6	0.985970
$971$	20726.0	0.684995	0.342497	−	0.939519i	$-0.388727\pi$
$971$	20726.0	0.684995	0.342497	+	0.939519i	$0.388727\pi$
$972$	−1140.52	−0.0376359
$973$	8567.34	0.282278
$974$	−28256.6	−0.929568
$975$	39775.7	1.30651
$976$	−492.990	−0.0161683
$977$	−11209.8	−0.367074	−0.183537	−	0.983013i	$-0.558755\pi$
$977$	−11209.8	−0.367074	−0.183537	+	0.983013i	$0.558755\pi$
$978$	12650.4	0.413616
$979$	0	0
$980$	−12994.7	−0.423571
$981$	18594.8	0.605185
$982$	−26095.1	−0.847992
$983$	53573.8	1.73829	0.869145	−	0.494557i	$-0.164670\pi$
$983$	53573.8	1.73829	0.869145	+	0.494557i	$0.164670\pi$
$984$	−15538.9	−0.503416
$985$	−25157.4	−0.813789
$986$	−17622.8	−0.569194
$987$	9509.80	0.306687
$988$	−22974.8	−0.739802
$989$	50077.0	1.61007
$990$	0	0
$991$	−43260.9	−1.38671	−0.693354	−	0.720597i	$-0.743868\pi$
$991$	−43260.9	−1.38671	−0.693354	+	0.720597i	$0.743868\pi$
$992$	37519.4	1.20085
$993$	−9193.85	−0.293815
$994$	−8291.69	−0.264584
$995$	−50718.8	−1.61597
$996$	−12256.8	−0.389931
$997$	−7938.54	−0.252173	−0.126086	−	0.992019i	$-0.540242\pi$
$997$	−7938.54	−0.252173	−0.126086	+	0.992019i	$0.540242\pi$
$998$	16894.1	0.535846
$999$	11586.3	0.366942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.q.1.3	✓	4
3.2	odd	2		1089.4.a.bf.1.2		4
11.10	odd	2		363.4.a.s.1.2	yes	4
33.32	even	2		1089.4.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.a.q.1.3	✓	4	1.1	even	1	trivial
363.4.a.s.1.2	yes	4	11.10	odd	2
1089.4.a.ba.1.3		4	33.32	even	2
1089.4.a.bf.1.2		4	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+1.81838 q^{2} +3.00000 q^{3} -4.69349 q^{4} -19.2178 q^{5} +5.45514 q^{6} +14.1043 q^{7} -23.0816 q^{8} +9.00000 q^{9} -34.9453 q^{10} -14.0805 q^{12} +54.2664 q^{13} +25.6471 q^{14} -57.6534 q^{15} -4.42322 q^{16} +78.2846 q^{17} +16.3654 q^{18} +90.2036 q^{19} +90.1986 q^{20} +42.3130 q^{21} -122.719 q^{23} -69.2448 q^{24} +244.324 q^{25} +98.6770 q^{26} +27.0000 q^{27} -66.1986 q^{28} -123.798 q^{29} -104.836 q^{30} +212.442 q^{31} +176.610 q^{32} +142.351 q^{34} -271.054 q^{35} -42.2414 q^{36} +429.123 q^{37} +164.024 q^{38} +162.799 q^{39} +443.578 q^{40} +224.405 q^{41} +76.9412 q^{42} -408.062 q^{43} -172.960 q^{45} -223.150 q^{46} +224.749 q^{47} -13.2697 q^{48} -144.068 q^{49} +444.274 q^{50} +234.854 q^{51} -254.699 q^{52} +283.806 q^{53} +49.0963 q^{54} -325.551 q^{56} +270.611 q^{57} -225.112 q^{58} -306.981 q^{59} +270.596 q^{60} +111.455 q^{61} +386.301 q^{62} +126.939 q^{63} +356.529 q^{64} -1042.88 q^{65} -153.581 q^{67} -367.428 q^{68} -368.157 q^{69} -492.880 q^{70} -323.300 q^{71} -207.734 q^{72} -194.522 q^{73} +780.309 q^{74} +732.971 q^{75} -423.370 q^{76} +296.031 q^{78} -788.138 q^{79} +85.0045 q^{80} +81.0000 q^{81} +408.054 q^{82} +870.482 q^{83} -198.596 q^{84} -1504.46 q^{85} -742.013 q^{86} -371.394 q^{87} +788.215 q^{89} -314.508 q^{90} +765.392 q^{91} +575.981 q^{92} +637.327 q^{93} +408.679 q^{94} -1733.51 q^{95} +529.829 q^{96} -852.379 q^{97} -261.970 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20}+ \cdots - 615 q^{98}+O(q^{100}) \) 4 * q - q^2 + 12 * q^3 + 25 * q^4 + 14 * q^5 - 3 * q^6 + 20 * q^7 - 75 * q^8 + 36 * q^9 + 3 * q^10 + 75 * q^12 + 32 * q^13 + 62 * q^14 + 42 * q^15 + 289 * q^16 + 92 * q^17 - 9 * q^18 - 34 * q^19 + 391 * q^20 + 60 * q^21 - 26 * q^23 - 225 * q^24 + 334 * q^25 - 181 * q^26 + 108 * q^27 + 692 * q^28 + 174 * q^29 + 9 * q^30 + 422 * q^31 - 1271 * q^32 - 477 * q^34 - 82 * q^35 + 225 * q^36 + 518 * q^37 - 798 * q^38 + 96 * q^39 + 423 * q^40 + 428 * q^41 + 186 * q^42 - 550 * q^43 + 126 * q^45 - 2004 * q^46 + 556 * q^47 + 867 * q^48 + 282 * q^49 + 2074 * q^50 + 276 * q^51 + 467 * q^52 + 882 * q^53 - 27 * q^54 - 2112 * q^56 - 102 * q^57 - 225 * q^58 - 158 * q^59 + 1173 * q^60 + 290 * q^61 + 1142 * q^62 + 180 * q^63 + 3097 * q^64 - 1636 * q^65 + 992 * q^67 - 2033 * q^68 - 78 * q^69 + 948 * q^70 - 42 * q^71 - 675 * q^72 + 1274 * q^73 + 1509 * q^74 + 1002 * q^75 + 632 * q^76 - 543 * q^78 + 362 * q^79 + 1423 * q^80 + 324 * q^81 + 2403 * q^82 + 1500 * q^83 + 2076 * q^84 - 2388 * q^85 - 1272 * q^86 + 522 * q^87 + 1428 * q^89 + 27 * q^90 + 1750 * q^91 + 896 * q^92 + 1266 * q^93 - 1728 * q^94 - 4452 * q^95 - 3813 * q^96 + 1052 * q^97 - 615 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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