LMFDB - Embedded newform 588.4.i.l.373.3

Newspace parameters

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$x^{8} - 2 x^{7} + 27 x^{6} + 10 x^{5} + 446 x^{4} + 62 x^{3} + 3061 x^{2} + 2142 x + 14161$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			373.3
Root			$-1.65506 - 2.86665i$ of defining polynomial
Character	$\chi$	$=$	588.373
Dual form			588.4.i.l.361.3

$q$-expansion

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{3} +(4.08470 - 7.07491i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(1.50000 + 2.59808i) q^{3} +(4.08470 - 7.07491i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-18.9094 - 32.7521i) q^{11} -39.9319 q^{13} +24.5082 q^{15} +(4.96798 + 8.60480i) q^{17} +(-45.2229 + 78.3284i) q^{19} +(-59.2973 + 102.706i) q^{23} +(29.1304 + 50.4554i) q^{25} -27.0000 q^{27} -78.4061 q^{29} +(46.0055 + 79.6838i) q^{31} +(56.7283 - 98.2563i) q^{33} +(-166.218 + 287.897i) q^{37} +(-59.8979 - 103.746i) q^{39} -71.7451 q^{41} -115.947 q^{43} +(36.7623 + 63.6742i) q^{45} +(153.964 - 266.673i) q^{47} +(-14.9039 + 25.8144i) q^{51} +(201.575 + 349.138i) q^{53} -308.958 q^{55} -271.337 q^{57} +(296.855 + 514.168i) q^{59} +(166.585 - 288.534i) q^{61} +(-163.110 + 282.515i) q^{65} +(371.756 + 643.900i) q^{67} -355.784 q^{69} -728.272 q^{71} +(-400.874 - 694.335i) q^{73} +(-87.3912 + 151.366i) q^{75} +(-533.953 + 924.833i) q^{79} +(-40.5000 - 70.1481i) q^{81} -906.756 q^{83} +81.1709 q^{85} +(-117.609 - 203.705i) q^{87} +(556.634 - 964.119i) q^{89} +(-138.016 + 239.051i) q^{93} +(369.444 + 639.896i) q^{95} -1480.94 q^{97} +340.370 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q + 12q^{3} - 36q^{9} + O(q^{10}) $	$ 8q + 12q^{3} - 36q^{9} + 48q^{17} + 192q^{19} - 192q^{23} - 324q^{25} - 216q^{27} + 192q^{29} + 48q^{31} - 256q^{37} - 2016q^{41} - 224q^{43} + 864q^{47} - 144q^{51} + 648q^{53} - 4704q^{55} + 1152q^{57} + 336q^{59} + 960q^{61} + 360q^{65} - 720q^{67} - 1152q^{69} - 2688q^{71} + 672q^{73} + 972q^{75} + 1984q^{79} - 324q^{81} - 6240q^{83} + 1360q^{85} + 288q^{87} + 2160q^{89} - 144q^{93} + 3744q^{95} - 4032q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	1.50000	+	2.59808i	0.288675	+	0.500000i
$3$	1.50000	+	2.59808i	0.288675	+	0.500000i
$4$		0			0
$5$	4.08470	−	7.07491i	0.365347	−	0.632799i	−0.623485	−	0.781835i	$-0.714284\pi$
$5$	4.08470	−	7.07491i	0.365347	−	0.632799i	0.988832	+	0.149036i	$0.0476170\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	−18.9094	−	32.7521i	−0.518310	−	0.897739i	−0.999774	−	0.0212729i	$-0.993228\pi$
$11$	−18.9094	−	32.7521i	−0.518310	−	0.897739i	0.481464	−	0.876466i	$-0.340105\pi$
$12$		0			0
$13$	−39.9319			−0.851933			−0.425966	−	0.904739i	$-0.640066\pi$
$13$	−39.9319			−0.851933			−0.425966	+	0.904739i	$0.640066\pi$
$14$		0			0
$15$	24.5082			0.421866
$16$		0			0
$17$	4.96798	+	8.60480i	0.0708772	+	0.122763i	0.899286	−	0.437361i	$-0.144087\pi$
$17$	4.96798	+	8.60480i	0.0708772	+	0.122763i	−0.828409	+	0.560124i	$0.810753\pi$
$18$		0			0
$19$	−45.2229	+	78.3284i	−0.546045	+	0.945777i	0.452496	+	0.891767i	$0.350534\pi$
$19$	−45.2229	+	78.3284i	−0.546045	+	0.945777i	−0.998540	+	0.0540105i	$0.982800\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−59.2973	+	102.706i	−0.537580	+	0.931116i	0.461454	+	0.887164i	$0.347328\pi$
$23$	−59.2973	+	102.706i	−0.537580	+	0.931116i	−0.999034	+	0.0439513i	$0.986005\pi$
$24$		0			0
$25$	29.1304	+	50.4554i	0.233043	+	0.403643i
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−78.4061			−0.502057			−0.251028	−	0.967980i	$-0.580769\pi$
$29$	−78.4061			−0.502057			−0.251028	+	0.967980i	$0.580769\pi$
$30$		0			0
$31$	46.0055	+	79.6838i	0.266543	+	0.461666i	0.967967	−	0.251079i	$-0.0807853\pi$
$31$	46.0055	+	79.6838i	0.266543	+	0.461666i	−0.701424	+	0.712744i	$0.747452\pi$
$32$		0			0
$33$	56.7283	−	98.2563i	0.299246	−	0.518310i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−166.218	+	287.897i	−0.738541	+	1.27919i	0.214611	+	0.976700i	$0.431152\pi$
$37$	−166.218	+	287.897i	−0.738541	+	1.27919i	−0.953152	+	0.302491i	$0.902182\pi$
$38$		0			0
$39$	−59.8979	−	103.746i	−0.245932	−	0.425966i
$40$		0			0
$41$	−71.7451			−0.273285			−0.136643	−	0.990620i	$-0.543631\pi$
$41$	−71.7451			−0.273285			−0.136643	+	0.990620i	$0.543631\pi$
$42$		0			0
$43$	−115.947			−0.411202			−0.205601	−	0.978636i	$-0.565915\pi$
$43$	−115.947			−0.411202			−0.205601	+	0.978636i	$0.565915\pi$
$44$		0			0
$45$	36.7623	+	63.6742i	0.121782	+	0.210933i
$46$		0			0
$47$	153.964	−	266.673i	0.477828	−	0.827623i	−0.521849	−	0.853038i	$-0.674758\pi$
$47$	153.964	−	266.673i	0.477828	−	0.827623i	0.999677	+	0.0254154i	$0.00809085\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−14.9039	+	25.8144i	−0.0409210	+	0.0708772i
$52$		0			0
$53$	201.575	+	349.138i	0.522424	+	0.904865i	0.999660	+	0.0260895i	$0.00830548\pi$
$53$	201.575	+	349.138i	0.522424	+	0.904865i	−0.477236	+	0.878775i	$0.658361\pi$
$54$		0			0
$55$	−308.958			−0.757451
$56$		0			0
$57$	−271.337			−0.630518
$58$		0			0
$59$	296.855	+	514.168i	0.655038	+	1.13456i	0.981884	+	0.189481i	$0.0606806\pi$
$59$	296.855	+	514.168i	0.655038	+	1.13456i	−0.326847	+	0.945077i	$0.605986\pi$
$60$		0			0
$61$	166.585	−	288.534i	0.349657	−	0.605623i	−0.636532	−	0.771250i	$-0.719632\pi$
$61$	166.585	−	288.534i	0.349657	−	0.605623i	0.986188	+	0.165628i	$0.0529649\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−163.110	+	282.515i	−0.311251	+	0.539102i
$66$		0			0
$67$	371.756	+	643.900i	0.677869	+	1.17410i	0.975622	+	0.219460i	$0.0704295\pi$
$67$	371.756	+	643.900i	0.677869	+	1.17410i	−0.297753	+	0.954643i	$0.596237\pi$
$68$		0			0
$69$	−355.784			−0.620744
$70$		0			0
$71$	−728.272			−1.21732			−0.608662	−	0.793429i	$-0.708294\pi$
$71$	−728.272			−1.21732			−0.608662	+	0.793429i	$0.708294\pi$
$72$		0			0
$73$	−400.874	−	694.335i	−0.642723	−	1.11323i	−0.984822	−	0.173566i	$-0.944471\pi$
$73$	−400.874	−	694.335i	−0.642723	−	1.11323i	0.342099	−	0.939664i	$-0.388862\pi$
$74$		0			0
$75$	−87.3912	+	151.366i	−0.134548	+	0.233043i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−533.953	+	924.833i	−0.760435	+	1.31711i	0.182191	+	0.983263i	$0.441681\pi$
$79$	−533.953	+	924.833i	−0.760435	+	1.31711i	−0.942626	+	0.333849i	$0.891652\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−906.756			−1.19915			−0.599575	−	0.800319i	$-0.704664\pi$
$83$	−906.756			−1.19915			−0.599575	+	0.800319i	$0.704664\pi$
$84$		0			0
$85$	81.1709			0.103579
$86$		0			0
$87$	−117.609	−	203.705i	−0.144931	−	0.251028i
$88$		0			0
$89$	556.634	−	964.119i	0.662957	−	1.14827i	−0.316878	−	0.948466i	$-0.602635\pi$
$89$	556.634	−	964.119i	0.662957	−	1.14827i	0.979835	−	0.199808i	$-0.0640320\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−138.016	+	239.051i	−0.153889	+	0.266543i
$94$		0			0
$95$	369.444	+	639.896i	0.398991	+	0.691073i
$96$		0			0
$97$	−1480.94			−1.55017			−0.775084	−	0.631858i	$-0.782293\pi$
$97$	−1480.94			−1.55017			−0.775084	+	0.631858i	$0.782293\pi$
$98$		0			0
$99$	340.370			0.345540
$100$		0			0
$101$	−278.158	−	481.784i	−0.274037	−	0.474647i	0.695854	−	0.718183i	$-0.255026\pi$
$101$	−278.158	−	481.784i	−0.274037	−	0.474647i	−0.969892	+	0.243536i	$0.921693\pi$
$102$		0			0
$103$	276.217	−	478.423i	0.264238	−	0.457674i	−0.703126	−	0.711066i	$-0.748213\pi$
$103$	276.217	−	478.423i	0.264238	−	0.457674i	0.967364	+	0.253392i	$0.0815462\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−266.902	+	462.288i	−0.241144	+	0.417674i	−0.961040	−	0.276408i	$-0.910856\pi$
$107$	−266.902	+	462.288i	−0.241144	+	0.417674i	0.719896	+	0.694082i	$0.244189\pi$
$108$		0			0
$109$	−547.314	−	947.976i	−0.480947	−	0.833025i	0.518814	−	0.854887i	$-0.326374\pi$
$109$	−547.314	−	947.976i	−0.480947	−	0.833025i	−0.999761	+	0.0218626i	$0.993040\pi$
$110$		0			0
$111$	−997.306			−0.852794
$112$		0			0
$113$	−1425.18			−1.18646			−0.593228	−	0.805035i	$-0.702147\pi$
$113$	−1425.18			−1.18646			−0.593228	+	0.805035i	$0.702147\pi$
$114$		0			0
$115$	484.423	+	839.046i	0.392806	+	0.680360i
$116$		0			0
$117$	179.694	−	311.239i	0.141989	−	0.245932i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−49.6330	+	85.9668i	−0.0372900	+	0.0645882i
$122$		0			0
$123$	−107.618	−	186.399i	−0.0788907	−	0.136643i
$124$		0			0
$125$	1497.13			1.07126
$126$		0			0
$127$	−786.485			−0.549522			−0.274761	−	0.961513i	$-0.588599\pi$
$127$	−786.485			−0.549522			−0.274761	+	0.961513i	$0.588599\pi$
$128$		0			0
$129$	−173.920	−	301.238i	−0.118704	−	0.205601i
$130$		0			0
$131$	−13.8651	+	24.0151i	−0.00924735	+	0.0160169i	−0.870612	−	0.491970i	$-0.836277\pi$
$131$	−13.8651	+	24.0151i	−0.00924735	+	0.0160169i	0.861365	+	0.507987i	$0.169610\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−110.287	+	191.023i	−0.0703110	+	0.121782i
$136$		0			0
$137$	1181.49	+	2046.40i	0.736798	+	1.27617i	0.953930	+	0.300029i	$0.0969964\pi$
$137$	1181.49	+	2046.40i	0.736798	+	1.27617i	−0.217132	+	0.976142i	$0.569670\pi$
$138$		0			0
$139$	−2513.28			−1.53362			−0.766811	−	0.641873i	$-0.778158\pi$
$139$	−2513.28			−1.53362			−0.766811	+	0.641873i	$0.778158\pi$
$140$		0			0
$141$	923.782			0.551748
$142$		0			0
$143$	755.090	+	1307.85i	0.441565	+	0.764813i
$144$		0			0
$145$	−320.266	+	554.716i	−0.183425	+	0.317701i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1132.92	+	1962.27i	−0.622900	+	1.07889i	0.366043	+	0.930598i	$0.380712\pi$
$149$	−1132.92	+	1962.27i	−0.622900	+	1.07889i	−0.988943	+	0.148296i	$0.952621\pi$
$150$		0			0
$151$	−141.573	−	245.212i	−0.0762984	−	0.132153i	0.825352	−	0.564619i	$-0.190977\pi$
$151$	−141.573	−	245.212i	−0.0762984	−	0.132153i	−0.901650	+	0.432466i	$0.857644\pi$
$152$		0			0
$153$	−89.4237			−0.0472515
$154$		0			0
$155$	751.675			0.389522
$156$		0			0
$157$	96.2904	+	166.780i	0.0489478	+	0.0847801i	0.889461	−	0.457011i	$-0.151080\pi$
$157$	96.2904	+	166.780i	0.0489478	+	0.0847801i	−0.840513	+	0.541791i	$0.817747\pi$
$158$		0			0
$159$	−604.725	+	1047.41i	−0.301622	+	0.522424i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	421.417	−	729.915i	0.202502	−	0.350744i	−0.746832	−	0.665013i	$-0.768426\pi$
$163$	421.417	−	729.915i	0.202502	−	0.350744i	0.949334	+	0.314269i	$0.101759\pi$
$164$		0			0
$165$	−463.436	−	802.695i	−0.218657	−	0.378726i
$166$		0			0
$167$	3859.21			1.78823			0.894116	−	0.447836i	$-0.147805\pi$
$167$	3859.21			1.78823			0.894116	+	0.447836i	$0.147805\pi$
$168$		0			0
$169$	−602.441			−0.274211
$170$		0			0
$171$	−407.006	−	704.955i	−0.182015	−	0.315259i
$172$		0			0
$173$	755.329	−	1308.27i	0.331946	−	0.574947i	−0.650948	−	0.759123i	$-0.725628\pi$
$173$	755.329	−	1308.27i	0.331946	−	0.574947i	0.982893	+	0.184176i	$0.0589616\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−890.565	+	1542.50i	−0.378186	+	0.655038i
$178$		0			0
$179$	1232.31	+	2134.43i	0.514566	+	0.891255i	0.999857	+	0.0169022i	$0.00538039\pi$
$179$	1232.31	+	2134.43i	0.514566	+	0.891255i	−0.485291	+	0.874353i	$0.661286\pi$
$180$		0			0
$181$	3297.36			1.35409			0.677047	−	0.735940i	$-0.263259\pi$
$181$	3297.36			1.35409			0.677047	+	0.735940i	$0.263259\pi$
$182$		0			0
$183$	999.511			0.403749
$184$		0			0
$185$	1357.90	+	2351.95i	0.539647	+	0.934696i
$186$		0			0
$187$	187.883	−	325.424i	0.0734727	−	0.127258i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2364.85	−	4096.05i	0.895889	−	1.55173i	0.0631890	−	0.998002i	$-0.479873\pi$
$191$	2364.85	−	4096.05i	0.895889	−	1.55173i	0.832700	−	0.553724i	$-0.186794\pi$
$192$		0			0
$193$	−2276.59	−	3943.17i	−0.849080	−	1.47065i	−0.882030	−	0.471193i	$-0.843823\pi$
$193$	−2276.59	−	3943.17i	−0.849080	−	1.47065i	0.0329498	−	0.999457i	$-0.489510\pi$
$194$		0			0
$195$	−978.660			−0.359402
$196$		0			0
$197$	−3109.06			−1.12442			−0.562212	−	0.826993i	$-0.690050\pi$
$197$	−3109.06			−1.12442			−0.562212	+	0.826993i	$0.690050\pi$
$198$		0			0
$199$	−221.756	−	384.092i	−0.0789943	−	0.136822i	0.823822	−	0.566849i	$-0.191838\pi$
$199$	−221.756	−	384.092i	−0.0789943	−	0.136822i	−0.902816	+	0.430027i	$0.858504\pi$
$200$		0			0
$201$	−1115.27	+	1931.70i	−0.391368	+	0.677869i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−293.057	+	507.590i	−0.0998439	+	0.172935i
$206$		0			0
$207$	−533.675	−	924.353i	−0.179193	−	0.310372i
$208$		0			0
$209$	3420.56			1.13208
$210$		0			0
$211$	4653.28			1.51822			0.759112	−	0.650960i	$-0.225633\pi$
$211$	4653.28			1.51822			0.759112	+	0.650960i	$0.225633\pi$
$212$		0			0
$213$	−1092.41	−	1892.11i	−0.351411	−	0.608662i
$214$		0			0
$215$	−473.607	+	820.311i	−0.150231	+	0.260208i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1202.62	−	2083.00i	0.371077	−	0.642723i
$220$		0			0
$221$	−198.381	−	343.606i	−0.0603826	−	0.104586i
$222$		0			0
$223$	−2778.90			−0.834481			−0.417240	−	0.908796i	$-0.637003\pi$
$223$	−2778.90			−0.834481			−0.417240	+	0.908796i	$0.637003\pi$
$224$		0			0
$225$	−524.347			−0.155362
$226$		0			0
$227$	1608.61	+	2786.20i	0.470340	+	0.814653i	0.999425	−	0.0339159i	$-0.0107978\pi$
$227$	1608.61	+	2786.20i	0.470340	+	0.814653i	−0.529084	+	0.848569i	$0.677464\pi$
$228$		0			0
$229$	−1864.01	+	3228.56i	−0.537891	+	0.931655i	0.461126	+	0.887335i	$0.347446\pi$
$229$	−1864.01	+	3228.56i	−0.537891	+	0.931655i	−0.999017	+	0.0443203i	$0.985888\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1801.76	−	3120.73i	0.506596	−	0.877451i	−0.493374	−	0.869817i	$-0.664237\pi$
$233$	1801.76	−	3120.73i	0.506596	−	0.877451i	0.999971	−	0.00763377i	$-0.00242993\pi$
$234$		0			0
$235$	−1257.79	−	2178.56i	−0.349146	−	0.604739i
$236$		0			0
$237$	−3203.72			−0.878075
$238$		0			0
$239$	2348.76			0.635684			0.317842	−	0.948144i	$-0.397042\pi$
$239$	2348.76			0.635684			0.317842	+	0.948144i	$0.397042\pi$
$240$		0			0
$241$	−2546.70	−	4411.01i	−0.680693	−	1.17900i	−0.974770	−	0.223213i	$-0.928345\pi$
$241$	−2546.70	−	4411.01i	−0.680693	−	1.17900i	0.294076	−	0.955782i	$-0.404988\pi$
$242$		0			0
$243$	121.500	−	210.444i	0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1805.84	−	3127.80i	0.465193	−	0.805738i
$248$		0			0
$249$	−1360.13	−	2355.82i	−0.346165	−	0.599575i
$250$		0			0
$251$	5939.02			1.49350			0.746749	−	0.665106i	$-0.231614\pi$
$251$	5939.02			1.49350			0.746749	+	0.665106i	$0.231614\pi$
$252$		0			0
$253$	4485.11			1.11453
$254$		0			0
$255$	121.756	+	210.888i	0.0299007	+	0.0517895i
$256$		0			0
$257$	−757.944	+	1312.80i	−0.183966	+	0.318638i	−0.943228	−	0.332147i	$-0.892227\pi$
$257$	−757.944	+	1312.80i	−0.183966	+	0.318638i	0.759262	+	0.650785i	$0.225560\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	352.827	−	611.115i	0.0836761	−	0.144931i
$262$		0			0
$263$	3216.96	+	5571.94i	0.754244	+	1.30639i	0.945749	+	0.324898i	$0.105330\pi$
$263$	3216.96	+	5571.94i	0.754244	+	1.30639i	−0.191505	+	0.981492i	$0.561337\pi$
$264$		0			0
$265$	3293.50			0.763464
$266$		0			0
$267$	3339.81			0.765516
$268$		0			0
$269$	−3474.90	−	6018.69i	−0.787614	−	1.36419i	−0.927425	−	0.374009i	$-0.877983\pi$
$269$	−3474.90	−	6018.69i	−0.787614	−	1.36419i	0.139811	−	0.990178i	$-0.455350\pi$
$270$		0			0
$271$	−480.997	+	833.111i	−0.107817	+	0.186745i	−0.914886	−	0.403713i	$-0.867719\pi$
$271$	−480.997	+	833.111i	−0.107817	+	0.186745i	0.807068	+	0.590458i	$0.201053\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1101.68	−	1908.16i	0.241577	−	0.418424i
$276$		0			0
$277$	−380.005	−	658.188i	−0.0824271	−	0.142768i	0.821865	−	0.569682i	$-0.192934\pi$
$277$	−380.005	−	658.188i	−0.0824271	−	0.142768i	−0.904292	+	0.426915i	$0.859600\pi$
$278$		0			0
$279$	−828.099			−0.177695
$280$		0			0
$281$	−4412.07			−0.936662			−0.468331	−	0.883553i	$-0.655145\pi$
$281$	−4412.07			−0.936662			−0.468331	+	0.883553i	$0.655145\pi$
$282$		0			0
$283$	−1301.07	−	2253.53i	−0.273289	−	0.473351i	0.696413	−	0.717641i	$-0.254778\pi$
$283$	−1301.07	−	2253.53i	−0.273289	−	0.473351i	−0.969702	+	0.244291i	$0.921445\pi$
$284$		0			0
$285$	−1108.33	+	1919.69i	−0.230358	+	0.398991i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2407.14	−	4169.29i	0.489953	−	0.848623i
$290$		0			0
$291$	−2221.41	−	3847.59i	−0.447495	−	0.775084i
$292$		0			0
$293$	9332.18			1.86072			0.930361	−	0.366644i	$-0.119493\pi$
$293$	9332.18			1.86072			0.930361	+	0.366644i	$0.119493\pi$
$294$		0			0
$295$	4850.26			0.957264
$296$		0			0
$297$	510.555	+	884.306i	0.0997488	+	0.172770i
$298$		0			0
$299$	2367.85	−	4101.24i	0.457982	−	0.793248i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	834.474	−	1445.35i	0.158216	−	0.274037i
$304$		0			0
$305$	−1360.90	−	2357.15i	−0.255492	−	0.442525i
$306$		0			0
$307$	−4895.51			−0.910103			−0.455051	−	0.890465i	$-0.650379\pi$
$307$	−4895.51			−0.910103			−0.455051	+	0.890465i	$0.650379\pi$
$308$		0			0
$309$	1657.30			0.305116
$310$		0			0
$311$	3752.64	+	6499.77i	0.684221	+	1.18511i	0.973681	+	0.227916i	$0.0731910\pi$
$311$	3752.64	+	6499.77i	0.684221	+	1.18511i	−0.289460	+	0.957190i	$0.593476\pi$
$312$		0			0
$313$	−3174.64	+	5498.63i	−0.573294	+	0.992974i	0.422931	+	0.906162i	$0.361001\pi$
$313$	−3174.64	+	5498.63i	−0.573294	+	0.992974i	−0.996225	+	0.0868124i	$0.972332\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3999.80	−	6927.86i	0.708679	−	1.22747i	−0.256669	−	0.966499i	$-0.582625\pi$
$317$	3999.80	−	6927.86i	0.708679	−	1.22747i	0.965347	−	0.260968i	$-0.0840417\pi$
$318$		0			0
$319$	1482.61	+	2567.96i	0.260221	+	0.450716i
$320$		0			0
$321$	−1601.41			−0.278449
$322$		0			0
$323$	−898.666			−0.154808
$324$		0			0
$325$	−1163.23	−	2014.78i	−0.198537	−	0.343876i
$326$		0			0
$327$	1641.94	−	2843.93i	0.277675	−	0.480947i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1060.78	−	1837.32i	0.176150	−	0.305101i	−0.764409	−	0.644732i	$-0.776969\pi$
$331$	1060.78	−	1837.32i	0.176150	−	0.305101i	0.940559	+	0.339631i	$0.110302\pi$
$332$		0			0
$333$	−1495.96	−	2591.08i	−0.246180	−	0.426397i
$334$		0			0
$335$	6074.05			0.990629
$336$		0			0
$337$	−9114.19			−1.47324			−0.736620	−	0.676307i	$-0.763579\pi$
$337$	−9114.19			−1.47324			−0.736620	+	0.676307i	$0.763579\pi$
$338$		0			0
$339$	−2137.77	−	3702.72i	−0.342500	−	0.593228i
$340$		0			0
$341$	1739.87	−	3013.55i	0.276303	−	0.478572i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1453.27	+	2517.14i	−0.226787	+	0.392806i
$346$		0			0
$347$	836.534	+	1448.92i	0.129416	+	0.224156i	0.923451	−	0.383717i	$-0.125356\pi$
$347$	836.534	+	1448.92i	0.129416	+	0.224156i	−0.794034	+	0.607873i	$0.792023\pi$
$348$		0			0
$349$	3467.56			0.531845			0.265923	−	0.963994i	$-0.414323\pi$
$349$	3467.56			0.531845			0.265923	+	0.963994i	$0.414323\pi$
$350$		0			0
$351$	1078.16			0.163955
$352$		0			0
$353$	1992.12	+	3450.45i	0.300368	+	0.520252i	0.976219	−	0.216785i	$-0.0695572\pi$
$353$	1992.12	+	3450.45i	0.300368	+	0.520252i	−0.675851	+	0.737038i	$0.736224\pi$
$354$		0			0
$355$	−2974.78	+	5152.46i	−0.444746	+	0.770322i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1085.47	+	1880.08i	−0.159579	+	0.276398i	−0.934717	−	0.355394i	$-0.884347\pi$
$359$	−1085.47	+	1880.08i	−0.159579	+	0.276398i	0.775138	+	0.631792i	$0.217680\pi$
$360$		0			0
$361$	−660.724	−	1144.41i	−0.0963295	−	0.166848i
$362$		0			0
$363$	−297.798			−0.0430588
$364$		0			0
$365$	−6549.81			−0.939268
$366$		0			0
$367$	1796.06	+	3110.86i	0.255459	+	0.442468i	0.965020	−	0.262176i	$-0.0844400\pi$
$367$	1796.06	+	3110.86i	0.255459	+	0.442468i	−0.709561	+	0.704644i	$0.751107\pi$
$368$		0			0
$369$	322.853	−	559.197i	0.0455475	−	0.0788907i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3719.90	+	6443.05i	−0.516378	+	0.894393i	0.483441	+	0.875377i	$0.339387\pi$
$373$	−3719.90	+	6443.05i	−0.516378	+	0.894393i	−0.999819	+	0.0190163i	$0.993947\pi$
$374$		0			0
$375$	2245.70	+	3889.66i	0.309246	+	0.535630i
$376$		0			0
$377$	3130.91			0.427719
$378$		0			0
$379$	11243.9			1.52390			0.761951	−	0.647635i	$-0.224242\pi$
$379$	11243.9			1.52390			0.761951	+	0.647635i	$0.224242\pi$
$380$		0			0
$381$	−1179.73	−	2043.35i	−0.158633	−	0.274761i
$382$		0			0
$383$	1770.62	−	3066.81i	0.236226	−	0.409156i	−0.723402	−	0.690427i	$-0.757423\pi$
$383$	1770.62	−	3066.81i	0.236226	−	0.409156i	0.959628	+	0.281271i	$0.0907561\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	521.759	−	903.714i	0.0685336	−	0.118704i
$388$		0			0
$389$	5279.10	+	9143.66i	0.688074	+	1.19178i	0.972460	+	0.233070i	$0.0748770\pi$
$389$	5279.10	+	9143.66i	0.688074	+	1.19178i	−0.284386	+	0.958710i	$0.591790\pi$
$390$		0			0
$391$	−1178.35			−0.152409
$392$		0			0
$393$	−83.1908			−0.0106779
$394$		0			0
$395$	4362.08	+	7555.34i	0.555645	+	0.962406i
$396$		0			0
$397$	−270.287	+	468.151i	−0.0341696	+	0.0591834i	−0.882604	−	0.470116i	$-0.844212\pi$
$397$	−270.287	+	468.151i	−0.0341696	+	0.0591834i	0.848435	+	0.529300i	$0.177545\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1083.52	+	1876.70i	−0.134933	+	0.233711i	−0.925572	−	0.378572i	$-0.876415\pi$
$401$	−1083.52	+	1876.70i	−0.134933	+	0.233711i	0.790639	+	0.612283i	$0.209749\pi$
$402$		0			0
$403$	−1837.09	−	3181.93i	−0.227077	−	0.393308i
$404$		0			0
$405$	−661.722			−0.0811882
$406$		0			0
$407$	12572.3			1.53117
$408$		0			0
$409$	−478.845	−	829.384i	−0.0578908	−	0.100270i	0.835628	−	0.549296i	$-0.185104\pi$
$409$	−478.845	−	829.384i	−0.0578908	−	0.100270i	−0.893518	+	0.449027i	$0.851771\pi$
$410$		0			0
$411$	−3544.46	+	6139.19i	−0.425390	+	0.736798i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−3703.83	+	6415.22i	−0.438105	+	0.758821i
$416$		0			0
$417$	−3769.92	−	6529.69i	−0.442719	−	0.766811i
$418$		0			0
$419$	−6464.42			−0.753718			−0.376859	−	0.926271i	$-0.622996\pi$
$419$	−6464.42			−0.753718			−0.376859	+	0.926271i	$0.622996\pi$
$420$		0			0
$421$	−6201.23			−0.717885			−0.358943	−	0.933360i	$-0.616863\pi$
$421$	−6201.23			−0.717885			−0.358943	+	0.933360i	$0.616863\pi$
$422$		0			0
$423$	1385.67	+	2400.06i	0.159276	+	0.275874i
$424$		0			0
$425$	−289.439	+	501.322i	−0.0330349	+	0.0572181i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2265.27	+	3923.56i	−0.254938	+	0.441565i
$430$		0			0
$431$	−707.796	−	1225.94i	−0.0791029	−	0.137010i	0.823760	−	0.566938i	$-0.191872\pi$
$431$	−707.796	−	1225.94i	−0.0791029	−	0.137010i	−0.902863	+	0.429928i	$0.858539\pi$
$432$		0			0
$433$	−8905.73			−0.988411			−0.494206	−	0.869345i	$-0.664541\pi$
$433$	−8905.73			−0.988411			−0.494206	+	0.869345i	$0.664541\pi$
$434$		0			0
$435$	−1921.59			−0.211801
$436$		0			0
$437$	−5363.19	−	9289.32i	−0.587085	−	1.01686i
$438$		0			0
$439$	5438.25	−	9419.33i	0.591238	−	1.02405i	−0.402828	−	0.915276i	$-0.631973\pi$
$439$	5438.25	−	9419.33i	0.591238	−	1.02405i	0.994066	−	0.108779i	$-0.0346940\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5701.48	+	9875.25i	−0.611479	+	1.05911i	0.379512	+	0.925187i	$0.376092\pi$
$443$	−5701.48	+	9875.25i	−0.611479	+	1.05911i	−0.990991	+	0.133927i	$0.957241\pi$
$444$		0			0
$445$	−4547.37	−	7876.28i	−0.484418	−	0.839037i
$446$		0			0
$447$	−6797.49			−0.719262
$448$		0			0
$449$	−12689.0			−1.33370			−0.666852	−	0.745190i	$-0.732359\pi$
$449$	−12689.0			−1.33370			−0.666852	+	0.745190i	$0.732359\pi$
$450$		0			0
$451$	1356.66	+	2349.80i	0.141646	+	0.245339i
$452$		0			0
$453$	424.720	−	735.636i	0.0440509	−	0.0762984i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1135.02	+	1965.92i	−0.116180	+	0.201229i	−0.918251	−	0.395999i	$-0.870398\pi$
$457$	−1135.02	+	1965.92i	−0.116180	+	0.201229i	0.802071	+	0.597229i	$0.203732\pi$
$458$		0			0
$459$	−134.135	−	232.329i	−0.0136403	−	0.0236257i
$460$		0			0
$461$	−10731.2			−1.08417			−0.542085	−	0.840324i	$-0.682365\pi$
$461$	−10731.2			−1.08417			−0.542085	+	0.840324i	$0.682365\pi$
$462$		0			0
$463$	−3307.74			−0.332017			−0.166008	−	0.986124i	$-0.553088\pi$
$463$	−3307.74			−0.332017			−0.166008	+	0.986124i	$0.553088\pi$
$464$		0			0
$465$	1127.51	+	1952.91i	0.112445	+	0.194761i
$466$		0			0
$467$	−4623.35	+	8007.87i	−0.458122	+	0.793490i	−0.998862	−	0.0476996i	$-0.984811\pi$
$467$	−4623.35	+	8007.87i	−0.458122	+	0.793490i	0.540740	+	0.841190i	$0.318144\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−288.871	+	500.339i	−0.0282600	+	0.0489478i
$472$		0			0
$473$	2192.48	+	3797.49i	0.213130	+	0.369152i
$474$		0			0
$475$	−5269.45			−0.509008
$476$		0			0
$477$	−3628.35			−0.348283
$478$		0			0
$479$	7611.66	+	13183.8i	0.726066	+	1.25758i	0.958534	+	0.284978i	$0.0919865\pi$
$479$	7611.66	+	13183.8i	0.726066	+	1.25758i	−0.232468	+	0.972604i	$0.574680\pi$
$480$		0			0
$481$	6637.39	−	11496.3i	0.629187	−	1.08978i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6049.19	+	10477.5i	−0.566349	+	0.980946i
$486$		0			0
$487$	3879.36	+	6719.25i	0.360966	+	0.625212i	0.988120	−	0.153683i	$-0.0491136\pi$
$487$	3879.36	+	6719.25i	0.360966	+	0.625212i	−0.627154	+	0.778895i	$0.715780\pi$
$488$		0			0
$489$	2528.50			0.233830
$490$		0			0
$491$	−8342.30			−0.766767			−0.383384	−	0.923589i	$-0.625241\pi$
$491$	−8342.30			−0.766767			−0.383384	+	0.923589i	$0.625241\pi$
$492$		0			0
$493$	−389.520	−	674.668i	−0.0355844	−	0.0616340i
$494$		0			0
$495$	1390.31	−	2408.09i	0.126242	−	0.218657i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1293.47	−	2240.36i	0.116040	−	0.200987i	−0.802155	−	0.597116i	$-0.796313\pi$
$499$	1293.47	−	2240.36i	0.116040	−	0.200987i	0.918195	+	0.396129i	$0.129647\pi$
$500$		0			0
$501$	5788.82	+	10026.5i	0.516218	+	0.894116i
$502$		0			0
$503$	−409.682			−0.0363157			−0.0181578	−	0.999835i	$-0.505780\pi$
$503$	−409.682			−0.0363157			−0.0181578	+	0.999835i	$0.505780\pi$
$504$		0			0
$505$	−4544.77			−0.400475
$506$		0			0
$507$	−903.662	−	1565.19i	−0.0791578	−	0.137105i
$508$		0			0
$509$	2195.25	−	3802.29i	0.191165	−	0.331107i	−0.754472	−	0.656333i	$-0.772107\pi$
$509$	2195.25	−	3802.29i	0.191165	−	0.331107i	0.945637	+	0.325225i	$0.105440\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1221.02	−	2114.87i	0.105086	−	0.182015i
$514$		0			0
$515$	−2256.53	−	3908.43i	−0.193077	−	0.334419i
$516$		0			0
$517$	−11645.5			−0.990652
$518$		0			0
$519$	4531.98			0.383298
$520$		0			0
$521$	−9714.76	−	16826.5i	−0.816912	−	1.41493i	−0.907947	−	0.419086i	$-0.862351\pi$
$521$	−9714.76	−	16826.5i	−0.816912	−	1.41493i	0.0910341	−	0.995848i	$-0.470983\pi$
$522$		0			0
$523$	8974.35	−	15544.0i	0.750326	−	1.29960i	−0.197338	−	0.980336i	$-0.563230\pi$
$523$	8974.35	−	15544.0i	0.750326	−	1.29960i	0.947664	−	0.319268i	$-0.103437\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−457.109	+	791.735i	−0.0377836	+	0.0654431i
$528$		0			0
$529$	−948.833	−	1643.43i	−0.0779841	−	0.135072i
$530$		0			0
$531$	−5343.39			−0.436692
$532$		0			0
$533$	2864.92			0.232821
$534$		0			0
$535$	2180.43	+	3776.62i	0.176202	+	0.305192i
$536$		0			0
$537$	−3696.94	+	6403.28i	−0.297085	+	0.514566i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11547.0	+	20000.0i	−0.917641	+	1.58940i	−0.114652	+	0.993406i	$0.536575\pi$
$541$	−11547.0	+	20000.0i	−0.917641	+	1.58940i	−0.802989	+	0.595994i	$0.796758\pi$
$542$		0			0
$543$	4946.04	+	8566.80i	0.390893	+	0.677047i
$544$		0			0
$545$	−8942.47			−0.702850
$546$		0			0
$547$	−2266.68			−0.177178			−0.0885889	−	0.996068i	$-0.528236\pi$
$547$	−2266.68			−0.177178			−0.0885889	+	0.996068i	$0.528236\pi$
$548$		0			0
$549$	1499.27	+	2596.81i	0.116552	+	0.201874i
$550$		0			0
$551$	3545.75	−	6141.42i	0.274145	−	0.474834i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4073.70	+	7055.85i	−0.311565	+	0.539647i
$556$		0			0
$557$	5519.27	+	9559.65i	0.419854	+	0.727209i	0.995924	−	0.0901913i	$-0.0287478\pi$
$557$	5519.27	+	9559.65i	0.419854	+	0.727209i	−0.576070	+	0.817400i	$0.695415\pi$
$558$		0			0
$559$	4629.97			0.350316
$560$		0			0
$561$	1127.30			0.0848390
$562$		0			0
$563$	−3529.68	−	6113.58i	−0.264224	−	0.457650i	0.703136	−	0.711055i	$-0.251783\pi$
$563$	−3529.68	−	6113.58i	−0.264224	−	0.457650i	−0.967360	+	0.253406i	$0.918449\pi$
$564$		0			0
$565$	−5821.43	+	10083.0i	−0.433468	+	0.750788i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	676.403	−	1171.56i	0.0498353	−	0.0863172i	−0.840032	−	0.542537i	$-0.817464\pi$
$569$	676.403	−	1171.56i	0.0498353	−	0.0863172i	0.889867	+	0.456220i	$0.150797\pi$
$570$		0			0
$571$	10419.6	+	18047.3i	0.763655	+	1.32269i	0.940955	+	0.338532i	$0.109930\pi$
$571$	10419.6	+	18047.3i	0.763655	+	1.32269i	−0.177300	+	0.984157i	$0.556736\pi$
$572$		0			0
$573$	14189.1			1.03448
$574$		0			0
$575$	−6909.42			−0.501117
$576$		0			0
$577$	5007.32	+	8672.92i	0.361278	+	0.625751i	0.988171	−	0.153353i	$-0.0490073\pi$
$577$	5007.32	+	8672.92i	0.361278	+	0.625751i	−0.626894	+	0.779105i	$0.715674\pi$
$578$		0			0
$579$	6829.77	−	11829.5i	0.490217	−	0.849080i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7623.34	−	13204.0i	0.541555	−	0.938001i
$584$		0			0
$585$	−1467.99	−	2542.63i	−0.103750	−	0.179701i
$586$		0			0
$587$	−20864.8			−1.46709			−0.733546	−	0.679640i	$-0.762136\pi$
$587$	−20864.8			−1.46709			−0.733546	+	0.679640i	$0.762136\pi$
$588$		0			0
$589$	−8322.01			−0.582177
$590$		0			0
$591$	−4663.60	−	8077.59i	−0.324593	−	0.562212i
$592$		0			0
$593$	8773.84	−	15196.7i	0.607586	−	1.05237i	−0.384051	−	0.923312i	$-0.625471\pi$
$593$	8773.84	−	15196.7i	0.607586	−	1.05237i	0.991637	−	0.129058i	$-0.0411952\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	665.268	−	1152.28i	0.0456074	−	0.0789943i
$598$		0			0
$599$	65.3409	+	113.174i	0.00445703	+	0.00771980i	0.868245	−	0.496135i	$-0.165248\pi$
$599$	65.3409	+	113.174i	0.00445703	+	0.00771980i	−0.863788	+	0.503855i	$0.831915\pi$
$600$		0			0
$601$	5964.47			0.404818			0.202409	−	0.979301i	$-0.435123\pi$
$601$	5964.47			0.404818			0.202409	+	0.979301i	$0.435123\pi$
$602$		0			0
$603$	−6691.60			−0.451912
$604$		0			0
$605$	405.472	+	702.298i	0.0272476	+	0.0471942i
$606$		0			0
$607$	−4955.82	+	8583.73i	−0.331384	+	0.573975i	−0.982784	−	0.184761i	$-0.940849\pi$
$607$	−4955.82	+	8583.73i	−0.331384	+	0.573975i	0.651399	+	0.758735i	$0.274182\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6148.07	+	10648.8i	−0.407077	+	0.705079i
$612$		0			0
$613$	3241.26	+	5614.03i	0.213562	+	0.369900i	0.952827	−	0.303515i	$-0.0981602\pi$
$613$	3241.26	+	5614.03i	0.213562	+	0.369900i	−0.739265	+	0.673415i	$0.764827\pi$
$614$		0			0
$615$	−1758.34			−0.115290
$616$		0			0
$617$	7189.36			0.469097			0.234548	−	0.972104i	$-0.424639\pi$
$617$	7189.36			0.469097			0.234548	+	0.972104i	$0.424639\pi$
$618$		0			0
$619$	930.696	+	1612.01i	0.0604327	+	0.104672i	0.894659	−	0.446750i	$-0.147419\pi$
$619$	930.696	+	1612.01i	0.0604327	+	0.104672i	−0.834226	+	0.551422i	$0.814085\pi$
$620$		0			0
$621$	1601.03	−	2773.06i	0.103457	−	0.179193i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2474.04	−	4285.16i	0.158338	−	0.274250i
$626$		0			0
$627$	5130.84	+	8886.87i	0.326804	+	0.566041i
$628$		0			0
$629$	−3303.06			−0.209383
$630$		0			0
$631$	−29032.0			−1.83161			−0.915803	−	0.401627i	$-0.868445\pi$
$631$	−29032.0			−1.83161			−0.915803	+	0.401627i	$0.868445\pi$
$632$		0			0
$633$	6979.93	+	12089.6i	0.438274	+	0.759112i
$634$		0			0
$635$	−3212.56	+	5564.32i	−0.200766	+	0.347737i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3277.23	−	5676.32i	0.202887	−	0.351411i
$640$		0			0
$641$	−12228.9	−	21181.0i	−0.753527	−	1.30515i	−0.946103	−	0.323866i	$-0.895017\pi$
$641$	−12228.9	−	21181.0i	−0.753527	−	1.30515i	0.192576	−	0.981282i	$-0.438316\pi$
$642$		0			0
$643$	−10968.2			−0.672695			−0.336348	−	0.941738i	$-0.609192\pi$
$643$	−10968.2			−0.672695			−0.336348	+	0.941738i	$0.609192\pi$
$644$		0			0
$645$	−2841.64			−0.173472
$646$		0			0
$647$	2742.37	+	4749.92i	0.166636	+	0.288622i	0.937235	−	0.348698i	$-0.113376\pi$
$647$	2742.37	+	4749.92i	0.166636	+	0.288622i	−0.770599	+	0.637320i	$0.780043\pi$
$648$		0			0
$649$	11226.7	−	19445.2i	0.679025	−	1.17611i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5164.98	−	8946.01i	0.309527	−	0.536117i	−0.668732	−	0.743504i	$-0.733163\pi$
$653$	5164.98	−	8946.01i	0.309527	−	0.536117i	0.978259	+	0.207387i	$0.0664959\pi$
$654$		0			0
$655$	113.270	+	196.189i	0.00675698	+	0.0117034i
$656$		0			0
$657$	7215.74			0.428482
$658$		0			0
$659$	26822.2			1.58550			0.792751	−	0.609546i	$-0.208648\pi$
$659$	26822.2			1.58550			0.792751	+	0.609546i	$0.208648\pi$
$660$		0			0
$661$	1249.38	+	2163.99i	0.0735177	+	0.127336i	0.900441	−	0.434979i	$-0.143244\pi$
$661$	1249.38	+	2163.99i	0.0735177	+	0.127336i	−0.826923	+	0.562315i	$0.809911\pi$
$662$		0			0
$663$	595.143	−	1030.82i	0.0348619	−	0.0603826i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4649.27	−	8052.77i	0.269896	−	0.467473i
$668$		0			0
$669$	−4168.36	−	7219.81i	−0.240894	−	0.417240i
$670$		0			0
$671$	−12600.1			−0.724922
$672$		0			0
$673$	10092.1			0.578044			0.289022	−	0.957322i	$-0.406670\pi$
$673$	10092.1			0.578044			0.289022	+	0.957322i	$0.406670\pi$
$674$		0			0
$675$	−786.521	−	1362.29i	−0.0448492	−	0.0776811i
$676$		0			0
$677$	−13183.2	+	22833.9i	−0.748406	+	1.29628i	0.200181	+	0.979759i	$0.435847\pi$
$677$	−13183.2	+	22833.9i	−0.748406	+	1.29628i	−0.948586	+	0.316518i	$0.897486\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−4825.83	+	8358.59i	−0.271551	+	0.470340i
$682$		0			0
$683$	−11285.4	−	19546.9i	−0.632245	−	1.09508i	−0.987092	−	0.160155i	$-0.948800\pi$
$683$	−11285.4	−	19546.9i	−0.632245	−	1.09508i	0.354847	−	0.934924i	$-0.384533\pi$
$684$		0			0
$685$	19304.1			1.07675
$686$		0			0
$687$	−11184.0			−0.621103
$688$		0			0
$689$	−8049.28	−	13941.8i	−0.445070	−	0.770884i
$690$		0			0
$691$	−5965.75	+	10333.0i	−0.328434	+	0.568864i	−0.982201	−	0.187832i	$-0.939854\pi$
$691$	−5965.75	+	10333.0i	−0.328434	+	0.568864i	0.653768	+	0.756695i	$0.273187\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10266.0	+	17781.2i	−0.560304	+	0.970475i
$696$		0			0
$697$	−356.428	−	617.352i	−0.0193697	−	0.0335493i
$698$		0			0
$699$	10810.5			0.584967
$700$		0			0
$701$	16592.6			0.893997			0.446999	−	0.894535i	$-0.352493\pi$
$701$	16592.6			0.893997			0.446999	+	0.894535i	$0.352493\pi$
$702$		0			0
$703$	−15033.7	−	26039.1i	−0.806553	−	1.39699i
$704$		0			0
$705$	3773.38	−	6535.68i	0.201580	−	0.349146i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−9357.00	+	16206.8i	−0.495641	+	0.858476i	−0.999987	−	0.00502575i	$-0.998400\pi$
$709$	−9357.00	+	16206.8i	−0.495641	+	0.858476i	0.504346	+	0.863502i	$0.331734\pi$
$710$		0			0
$711$	−4805.57	−	8323.50i	−0.253478	−	0.439037i
$712$		0			0
$713$	−10912.0			−0.573152
$714$		0			0
$715$	12337.3			0.645298
$716$		0			0
$717$	3523.14	+	6102.25i	0.183506	+	0.317842i
$718$		0			0
$719$	−12427.4	+	21524.9i	−0.644594	+	1.11647i	0.339801	+	0.940497i	$0.389640\pi$
$719$	−12427.4	+	21524.9i	−0.644594	+	1.11647i	−0.984395	+	0.175972i	$0.943693\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	7640.09	−	13233.0i	0.392998	−	0.680693i
$724$		0			0
$725$	−2284.00	−	3956.01i	−0.117001	−	0.202652i
$726$		0			0
$727$	22506.0			1.14814			0.574071	−	0.818805i	$-0.305363\pi$
$727$	22506.0			1.14814			0.574071	+	0.818805i	$0.305363\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−576.020	−	997.696i	−0.0291448	−	0.0504803i
$732$		0			0
$733$	−11896.0	+	20604.5i	−0.599440	+	1.03826i	0.393464	+	0.919340i	$0.371277\pi$
$733$	−11896.0	+	20604.5i	−0.599440	+	1.03826i	−0.992904	+	0.118920i	$0.962057\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14059.4	−	24351.6i	0.702692	−	1.21710i
$738$		0			0
$739$	−2401.62	−	4159.72i	−0.119547	−	0.207061i	0.800041	−	0.599945i	$-0.204811\pi$
$739$	−2401.62	−	4159.72i	−0.119547	−	0.207061i	−0.919588	+	0.392884i	$0.871477\pi$
$740$		0			0
$741$	10835.0			0.537159
$742$		0			0
$743$	5076.10			0.250638			0.125319	−	0.992116i	$-0.460005\pi$
$743$	5076.10			0.250638			0.125319	+	0.992116i	$0.460005\pi$
$744$		0			0
$745$	9255.24	+	16030.6i	0.455149	+	0.788341i
$746$		0			0
$747$	4080.40	−	7067.46i	0.199858	−	0.346165i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4579.36	−	7931.68i	0.222508	−	0.385394i	−0.733061	−	0.680163i	$-0.761909\pi$
$751$	4579.36	−	7931.68i	0.222508	−	0.385394i	0.955569	+	0.294768i	$0.0952425\pi$
$752$		0			0
$753$	8908.54	+	15430.0i	0.431136	+	0.746749i
$754$		0			0
$755$	−2313.14			−0.111502
$756$		0			0
$757$	33682.2			1.61717			0.808587	−	0.588377i	$-0.200233\pi$
$757$	33682.2			1.61717			0.808587	+	0.588377i	$0.200233\pi$
$758$		0			0
$759$	6727.67	+	11652.7i	0.321738	+	0.557266i
$760$		0			0
$761$	6673.14	−	11558.2i	0.317873	−	0.550571i	−0.662171	−	0.749352i	$-0.730365\pi$
$761$	6673.14	−	11558.2i	0.317873	−	0.550571i	0.980044	+	0.198781i	$0.0636983\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−365.269	+	632.665i	−0.0172632	+	0.0299007i
$766$		0			0
$767$	−11854.0	−	20531.7i	−0.558048	−	0.966568i
$768$		0			0
$769$	15530.6			0.728281			0.364140	−	0.931344i	$-0.381363\pi$
$769$	15530.6			0.728281			0.364140	+	0.931344i	$0.381363\pi$
$770$		0			0
$771$	−4547.66			−0.212425
$772$		0			0
$773$	−5507.62	−	9539.47i	−0.256268	−	0.443869i	0.708971	−	0.705238i	$-0.249160\pi$
$773$	−5507.62	−	9539.47i	−0.256268	−	0.443869i	−0.965239	+	0.261368i	$0.915826\pi$
$774$		0			0
$775$	−2680.32	+	4642.45i	−0.124232	+	0.215176i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3244.52	−	5619.67i	0.149226	−	0.258467i
$780$		0			0
$781$	13771.2	+	23852.4i	0.630951	+	1.09284i
$782$		0			0
$783$	2116.96			0.0966209
$784$		0			0
$785$	1573.27			0.0715317
$786$		0			0
$787$	10596.3	+	18353.3i	0.479946	+	0.831291i	0.999735	−	0.0230036i	$-0.00732292\pi$
$787$	10596.3	+	18353.3i	0.479946	+	0.831291i	−0.519789	+	0.854294i	$0.673990\pi$
$788$		0			0
$789$	−9650.88	+	16715.8i	−0.435463	+	0.754244i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6652.07	+	11521.7i	−0.297884	+	0.515950i
$794$		0			0
$795$	4940.25	+	8556.76i	0.220393	+	0.381732i
$796$		0			0
$797$	−18189.9			−0.808432			−0.404216	−	0.914663i	$-0.632456\pi$
$797$	−18189.9			−0.808432			−0.404216	+	0.914663i	$0.632456\pi$

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{3} +(4.08470 - 7.07491i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 2.59808i) q^{3} +(4.08470 - 7.07491i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(-18.9094 - 32.7521i) q^{11} -39.9319 q^{13} +24.5082 q^{15} +(4.96798 + 8.60480i) q^{17} +(-45.2229 + 78.3284i) q^{19} +(-59.2973 + 102.706i) q^{23} +(29.1304 + 50.4554i) q^{25} -27.0000 q^{27} -78.4061 q^{29} +(46.0055 + 79.6838i) q^{31} +(56.7283 - 98.2563i) q^{33} +(-166.218 + 287.897i) q^{37} +(-59.8979 - 103.746i) q^{39} -71.7451 q^{41} -115.947 q^{43} +(36.7623 + 63.6742i) q^{45} +(153.964 - 266.673i) q^{47} +(-14.9039 + 25.8144i) q^{51} +(201.575 + 349.138i) q^{53} -308.958 q^{55} -271.337 q^{57} +(296.855 + 514.168i) q^{59} +(166.585 - 288.534i) q^{61} +(-163.110 + 282.515i) q^{65} +(371.756 + 643.900i) q^{67} -355.784 q^{69} -728.272 q^{71} +(-400.874 - 694.335i) q^{73} +(-87.3912 + 151.366i) q^{75} +(-533.953 + 924.833i) q^{79} +(-40.5000 - 70.1481i) q^{81} -906.756 q^{83} +81.1709 q^{85} +(-117.609 - 203.705i) q^{87} +(556.634 - 964.119i) q^{89} +(-138.016 + 239.051i) q^{93} +(369.444 + 639.896i) q^{95} -1480.94 q^{97} +340.370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q + 12q^{3} - 36q^{9} + O(q^{10}) \)	\( 8q + 12q^{3} - 36q^{9} + 48q^{17} + 192q^{19} - 192q^{23} - 324q^{25} - 216q^{27} + 192q^{29} + 48q^{31} - 256q^{37} - 2016q^{41} - 224q^{43} + 864q^{47} - 144q^{51} + 648q^{53} - 4704q^{55} + 1152q^{57} + 336q^{59} + 960q^{61} + 360q^{65} - 720q^{67} - 1152q^{69} - 2688q^{71} + 672q^{73} + 972q^{75} + 1984q^{79} - 324q^{81} - 6240q^{83} + 1360q^{85} + 288q^{87} + 2160q^{89} - 144q^{93} + 3744q^{95} - 4032q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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