LMFDB - Embedded newform 336.4.bc.e.257.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(17,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.17"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.bc (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 $ x^16 - 2x^15 - x^14 - 2x^13 + 9x^12 - 24x^11 + 714x^10 - 1940x^9 - 2834x^8 - 17460x^7 + 57834x^6 - 17496x^5 + 59049x^4 - 118098x^3 - 531441x^2 - 9565938x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{11} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			257.5
Root			$-0.0204843 - 2.99993i$ of defining polynomial
Character	$\chi$	$=$	336.257
Dual form			336.4.bc.e.17.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0354799 + 5.19603i) q^{3} +(5.27257 - 9.13236i) q^{5} +(-17.7029 - 5.44135i) q^{7} +(-26.9975 + 0.368709i) q^{9} +(26.6918 - 15.4105i) q^{11} +19.8400i q^{13} +(47.6391 + 27.0724i) q^{15} +(46.3453 + 80.2724i) q^{17} +(118.901 + 68.6474i) q^{19} +(27.6454 - 92.1777i) q^{21} +(37.6697 + 21.7486i) q^{23} +(6.89995 + 11.9511i) q^{25} +(-2.87369 - 140.267i) q^{27} +134.318i q^{29} +(-144.963 + 83.6945i) q^{31} +(81.0206 + 138.145i) q^{33} +(-143.032 + 132.979i) q^{35} +(191.747 - 332.115i) q^{37} +(-103.089 + 0.703923i) q^{39} +107.887 q^{41} +285.480 q^{43} +(-138.979 + 248.495i) q^{45} +(-120.906 + 209.416i) q^{47} +(283.783 + 192.655i) q^{49} +(-415.453 + 243.660i) q^{51} +(-432.694 + 249.816i) q^{53} -325.013i q^{55} +(-352.476 + 620.248i) q^{57} +(366.212 + 634.299i) q^{59} +(-265.207 - 153.117i) q^{61} +(479.939 + 140.376i) q^{63} +(181.187 + 104.608i) q^{65} +(280.049 + 485.060i) q^{67} +(-111.670 + 196.505i) q^{69} +74.2161i q^{71} +(141.409 - 81.6426i) q^{73} +(-61.8533 + 36.2764i) q^{75} +(-556.376 + 127.571i) q^{77} +(437.160 - 757.183i) q^{79} +(728.728 - 19.9084i) q^{81} -406.600 q^{83} +977.435 q^{85} +(-697.922 + 4.76560i) q^{87} +(526.091 - 911.216i) q^{89} +(107.957 - 351.226i) q^{91} +(-440.023 - 750.264i) q^{93} +(1253.83 - 723.897i) q^{95} -243.235i q^{97} +(-714.930 + 425.887i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 80 q^{7} + 18 q^{9} + 342 q^{19} - 450 q^{21} - 194 q^{25} - 804 q^{31} + 1332 q^{33} - 962 q^{37} - 594 q^{39} - 1732 q^{43} - 2394 q^{45} + 820 q^{49} - 1638 q^{51} - 2664 q^{57} - 4620 q^{61}+ \cdots + 4284 q^{99}+O(q^{100}) $ 16 * q - 80 * q^7 + 18 * q^9 + 342 * q^19 - 450 * q^21 - 194 * q^25 - 804 * q^31 + 1332 * q^33 - 962 * q^37 - 594 * q^39 - 1732 * q^43 - 2394 * q^45 + 820 * q^49 - 1638 * q^51 - 2664 * q^57 - 4620 * q^61 + 2016 * q^63 + 706 * q^67 + 3294 * q^73 - 6174 * q^75 + 2656 * q^79 + 126 * q^81 + 5232 * q^85 - 1026 * q^87 - 4098 * q^91 + 2016 * q^93 + 4284 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	0.0354799	+	5.19603i	0.00682811	+	0.999977i
$3$	0.0354799	+	5.19603i	0.00682811	+	0.999977i
$4$		0			0
$5$	5.27257	−	9.13236i	0.471593	−	0.816823i	−0.527879	−	0.849320i	$-0.677012\pi$
$5$	5.27257	−	9.13236i	0.471593	−	0.816823i	0.999472	+	0.0324964i	$0.0103457\pi$
$6$		0			0
$7$	−17.7029	−	5.44135i	−0.955865	−	0.293806i
$7$	−17.7029	−	5.44135i	−0.955865	−	0.293806i
$8$		0			0
$9$	−26.9975	+	0.368709i	−0.999907	+	0.0136559i
$10$		0			0
$11$	26.6918	−	15.4105i	0.731626	−	0.422405i	−0.0873906	−	0.996174i	$-0.527853\pi$
$11$	26.6918	−	15.4105i	0.731626	−	0.422405i	0.819017	+	0.573770i	$0.194519\pi$
$12$		0			0
$13$			19.8400i			0.423280i	0.977348	+	0.211640i	$0.0678804\pi$
$13$			19.8400i			0.423280i	−0.977348	+	0.211640i	$0.932120\pi$
$14$		0			0
$15$	47.6391	+	27.0724i	0.820025	+	0.466005i
$16$		0			0
$17$	46.3453	+	80.2724i	0.661199	+	1.14523i	0.980301	+	0.197510i	$0.0632854\pi$
$17$	46.3453	+	80.2724i	0.661199	+	1.14523i	−0.319102	+	0.947720i	$0.603381\pi$
$18$		0			0
$19$	118.901	+	68.6474i	1.43567	+	0.828884i	0.997545	−	0.0700296i	$-0.0223094\pi$
$19$	118.901	+	68.6474i	1.43567	+	0.828884i	0.438125	+	0.898914i	$0.355643\pi$
$20$		0			0
$21$	27.6454	−	92.1777i	0.287272	−	0.957849i
$22$		0			0
$23$	37.6697	+	21.7486i	0.341508	+	0.197170i	0.660939	−	0.750440i	$-0.270158\pi$
$23$	37.6697	+	21.7486i	0.341508	+	0.197170i	−0.319431	+	0.947610i	$0.603492\pi$
$24$		0			0
$25$	6.89995	+	11.9511i	0.0551996	+	0.0956085i
$26$		0			0
$27$	−2.87369	−	140.267i	−0.0204831	−	0.999790i
$28$		0			0
$29$			134.318i			0.860079i	0.902810	+	0.430039i	$0.141500\pi$
$29$			134.318i			0.860079i	−0.902810	+	0.430039i	$0.858500\pi$
$30$		0			0
$31$	−144.963	+	83.6945i	−0.839876	+	0.484903i	−0.857222	−	0.514947i	$-0.827812\pi$
$31$	−144.963	+	83.6945i	−0.839876	+	0.484903i	0.0173460	+	0.999850i	$0.494478\pi$
$32$		0			0
$33$	81.0206	+	138.145i	0.427390	+	0.728725i
$34$		0			0
$35$	−143.032	+	132.979i	−0.690767	+	0.642216i
$36$		0			0
$37$	191.747	−	332.115i	0.851972	−	1.47566i	−0.0274532	−	0.999623i	$-0.508740\pi$
$37$	191.747	−	332.115i	0.851972	−	1.47566i	0.879426	−	0.476036i	$-0.157927\pi$
$38$		0			0
$39$	−103.089	+	0.703923i	−0.423270	+	0.00289020i
$40$		0			0
$41$	107.887			0.410954			0.205477	−	0.978662i	$-0.434125\pi$
$41$	107.887			0.410954			0.205477	+	0.978662i	$0.434125\pi$
$42$		0			0
$43$	285.480			1.01245			0.506225	−	0.862402i	$-0.331041\pi$
$43$	285.480			1.01245			0.506225	+	0.862402i	$0.331041\pi$
$44$		0			0
$45$	−138.979	+	248.495i	−0.460395	+	0.823187i
$46$		0			0
$47$	−120.906	+	209.416i	−0.375234	+	0.649924i	−0.990362	−	0.138503i	$-0.955771\pi$
$47$	−120.906	+	209.416i	−0.375234	+	0.649924i	0.615128	+	0.788427i	$0.289104\pi$
$48$		0			0
$49$	283.783	+	192.655i	0.827357	+	0.561677i
$50$		0			0
$51$	−415.453	+	243.660i	−1.14069	+	0.669003i
$52$		0			0
$53$	−432.694	+	249.816i	−1.12142	+	0.647451i	−0.941762	−	0.336279i	$-0.890831\pi$
$53$	−432.694	+	249.816i	−1.12142	+	0.647451i	−0.179655	+	0.983730i	$0.557498\pi$
$54$		0			0
$55$		−	325.013i		−	0.796813i
$56$		0			0
$57$	−352.476	+	620.248i	−0.819062	+	1.44130i
$58$		0			0
$59$	366.212	+	634.299i	0.808081	+	1.39964i	0.914191	+	0.405284i	$0.132827\pi$
$59$	366.212	+	634.299i	0.808081	+	1.39964i	−0.106110	+	0.994354i	$0.533839\pi$
$60$		0			0
$61$	−265.207	−	153.117i	−0.556661	−	0.321388i	0.195143	−	0.980775i	$-0.437483\pi$
$61$	−265.207	−	153.117i	−0.556661	−	0.321388i	−0.751804	+	0.659387i	$0.770816\pi$
$62$		0			0
$63$	479.939	+	140.376i	0.959788	+	0.280725i
$64$		0			0
$65$	181.187	+	104.608i	0.345745	+	0.199616i
$66$		0			0
$67$	280.049	+	485.060i	0.510649	+	0.884470i	0.999924	+	0.0123404i	$0.00392817\pi$
$67$	280.049	+	485.060i	0.510649	+	0.884470i	−0.489275	+	0.872130i	$0.662739\pi$
$68$		0			0
$69$	−111.670	+	196.505i	−0.194833	+	0.342846i
$70$		0			0
$71$			74.2161i			0.124054i	0.998074	+	0.0620270i	$0.0197565\pi$
$71$			74.2161i			0.124054i	−0.998074	+	0.0620270i	$0.980244\pi$
$72$		0			0
$73$	141.409	−	81.6426i	0.226722	−	0.130898i	−0.382337	−	0.924023i	$-0.624881\pi$
$73$	141.409	−	81.6426i	0.226722	−	0.130898i	0.609059	+	0.793125i	$0.291547\pi$
$74$		0			0
$75$	−61.8533	+	36.2764i	−0.0952294	+	0.0558512i
$76$		0			0
$77$	−556.376	+	127.571i	−0.823441	+	0.188806i
$78$		0			0
$79$	437.160	−	757.183i	0.622586	−	1.07835i	−0.366416	−	0.930451i	$-0.619415\pi$
$79$	437.160	−	757.183i	0.622586	−	1.07835i	0.989002	−	0.147900i	$-0.0472515\pi$
$80$		0			0
$81$	728.728	−	19.9084i	0.999627	−	0.0273093i
$82$		0			0
$83$	−406.600			−0.537712			−0.268856	−	0.963180i	$-0.586646\pi$
$83$	−406.600			−0.537712			−0.268856	+	0.963180i	$0.586646\pi$
$84$		0			0
$85$	977.435			1.24727
$86$		0			0
$87$	−697.922	+	4.76560i	−0.860059	+	0.00587271i
$88$		0			0
$89$	526.091	−	911.216i	0.626579	−	1.08527i	−0.361655	−	0.932312i	$-0.617788\pi$
$89$	526.091	−	911.216i	0.626579	−	1.08527i	0.988233	−	0.152954i	$-0.0488786\pi$
$90$		0			0
$91$	107.957	−	351.226i	0.124362	−	0.404598i
$92$		0			0
$93$	−440.023	−	750.264i	−0.490626	−	0.836546i
$94$		0			0
$95$	1253.83	−	723.897i	1.35410	−	0.781793i
$96$		0			0
$97$		−	243.235i		−	0.254606i	−0.991864	−	0.127303i	$-0.959368\pi$
$97$		−	243.235i		−	0.254606i	0.991864	−	0.127303i	$-0.0406320\pi$
$98$		0			0
$99$	−714.930	+	425.887i	−0.725790	+	0.432356i
$100$		0			0
$101$	106.008	+	183.612i	0.104438	+	0.180892i	0.913508	−	0.406820i	$-0.133362\pi$
$101$	106.008	+	183.612i	0.104438	+	0.180892i	−0.809070	+	0.587712i	$0.800029\pi$
$102$		0			0
$103$	742.977	+	428.958i	0.710754	+	0.410354i	0.811340	−	0.584574i	$-0.198738\pi$
$103$	742.977	+	428.958i	0.710754	+	0.410354i	−0.100586	+	0.994928i	$0.532072\pi$
$104$		0			0
$105$	−696.038	−	738.481i	−0.646918	−	0.686366i
$106$		0			0
$107$	−1860.36	−	1074.08i	−1.68082	−	0.970424i	−0.961115	−	0.276148i	$-0.910942\pi$
$107$	−1860.36	−	1074.08i	−1.68082	−	0.970424i	−0.719708	−	0.694276i	$-0.755725\pi$
$108$		0			0
$109$	621.510	+	1076.49i	0.546145	+	0.945952i	0.998534	+	0.0541307i	$0.0172388\pi$
$109$	621.510	+	1076.49i	0.546145	+	0.945952i	−0.452388	+	0.891821i	$0.649428\pi$
$110$		0			0
$111$	1732.48	+	984.539i	1.48144	+	0.841877i
$112$		0			0
$113$		−	193.701i		−	0.161256i	−0.996744	−	0.0806278i	$-0.974307\pi$
$113$		−	193.701i		−	0.161256i	0.996744	−	0.0806278i	$-0.0256925\pi$
$114$		0			0
$115$	397.233	−	229.342i	0.322106	−	0.185968i
$116$		0			0
$117$	−7.31521	−	535.631i	−0.00578027	−	0.423240i
$118$		0			0
$119$	−383.654	−	1673.23i	−0.295542	−	1.28895i
$120$		0			0
$121$	−190.531	+	330.009i	−0.143149	+	0.247941i
$122$		0			0
$123$	3.82782	+	560.584i	0.00280604	+	0.410944i
$124$		0			0
$125$	1463.67			1.04731
$126$		0			0
$127$	−1010.51			−0.706052			−0.353026	−	0.935614i	$-0.614847\pi$
$127$	−1010.51			−0.706052			−0.353026	+	0.935614i	$0.614847\pi$
$128$		0			0
$129$	10.1288	+	1483.36i	0.00691312	+	1.01243i
$130$		0			0
$131$	179.314	−	310.581i	0.119593	−	0.207142i	−0.800013	−	0.599982i	$-0.795174\pi$
$131$	179.314	−	310.581i	0.119593	−	0.207142i	0.919607	+	0.392841i	$0.128508\pi$
$132$		0			0
$133$	−1731.35	−	1862.24i	−1.12878	−	1.21411i
$134$		0			0
$135$	−1296.12	−	713.323i	−0.826312	−	0.454763i
$136$		0			0
$137$	−1194.04	+	689.378i	−0.744625	+	0.429909i	−0.823748	−	0.566956i	$-0.808121\pi$
$137$	−1194.04	+	689.378i	−0.744625	+	0.429909i	0.0791238	+	0.996865i	$0.474788\pi$
$138$		0			0
$139$		−	474.488i		−	0.289536i	−0.989466	−	0.144768i	$-0.953756\pi$
$139$		−	474.488i		−	0.289536i	0.989466	−	0.144768i	$-0.0462436\pi$
$140$		0			0
$141$	−1092.42	−	620.803i	−0.652471	−	0.370787i
$142$		0			0
$143$	305.746	+	529.567i	0.178795	+	0.309683i
$144$		0			0
$145$	1226.64	+	708.203i	0.702533	+	0.405607i
$146$		0			0
$147$	−990.974	+	1481.38i	−0.556015	+	0.831173i
$148$		0			0
$149$	−519.125	−	299.717i	−0.285426	−	0.164791i	0.350451	−	0.936581i	$-0.386028\pi$
$149$	−519.125	−	299.717i	−0.285426	−	0.164791i	−0.635877	+	0.771790i	$0.719361\pi$
$150$		0			0
$151$	−997.588	−	1727.87i	−0.537633	−	0.931208i	−0.999031	−	0.0440145i	$-0.985985\pi$
$151$	−997.588	−	1727.87i	−0.537633	−	0.931208i	0.461398	−	0.887193i	$-0.347348\pi$
$152$		0			0
$153$	−1280.80	−	2150.06i	−0.676776	−	1.13609i
$154$		0			0
$155$			1765.14i			0.914707i
$156$		0			0
$157$	−765.966	+	442.231i	−0.389368	+	0.224802i	−0.681886	−	0.731458i	$-0.738840\pi$
$157$	−765.966	+	442.231i	−0.389368	+	0.224802i	0.292518	+	0.956260i	$0.405507\pi$
$158$		0			0
$159$	−1313.40	−	2239.43i	−0.655093	−	1.11697i
$160$		0			0
$161$	−548.520	−	589.988i	−0.268506	−	0.288805i
$162$		0			0
$163$	−338.492	+	586.286i	−0.162655	+	0.281727i	−0.935820	−	0.352478i	$-0.885339\pi$
$163$	−338.492	+	586.286i	−0.162655	+	0.281727i	0.773165	+	0.634205i	$0.218672\pi$
$164$		0			0
$165$	1688.78	−	11.5314i	0.796794	−	0.00544072i
$166$		0			0
$167$	3718.74			1.72314			0.861571	−	0.507637i	$-0.169481\pi$
$167$	3718.74			1.72314			0.861571	+	0.507637i	$0.169481\pi$
$168$		0			0
$169$	1803.37			0.820834
$170$		0			0
$171$	−3235.34	−	1809.47i	−1.44686	−	0.809202i
$172$		0			0
$173$	1318.97	−	2284.52i	0.579650	−	1.00398i	−0.415869	−	0.909425i	$-0.636523\pi$
$173$	1318.97	−	2284.52i	0.579650	−	1.00398i	0.995519	−	0.0945591i	$-0.0301441\pi$
$174$		0			0
$175$	−57.1190	−	249.113i	−0.0246731	−	0.107607i
$176$		0			0
$177$	−3282.84	+	1925.36i	−1.39409	+	0.817619i
$178$		0			0
$179$	−2861.56	+	1652.12i	−1.19488	+	0.689862i	−0.959409	−	0.282020i	$-0.908996\pi$
$179$	−2861.56	+	1652.12i	−1.19488	+	0.689862i	−0.235468	+	0.971882i	$0.575662\pi$
$180$		0			0
$181$		−	417.941i		−	0.171631i	−0.996311	−	0.0858157i	$-0.972650\pi$
$181$		−	417.941i		−	0.171631i	0.996311	−	0.0858157i	$-0.0273496\pi$
$182$		0			0
$183$	786.193	−	1383.46i	0.317580	−	0.558842i
$184$		0			0
$185$	−2022.00	−	3502.20i	−0.803569	−	1.39182i
$186$		0			0
$187$	2474.08	+	1428.41i	0.967501	+	0.558587i
$188$		0			0
$189$	−712.368	+	2498.76i	−0.274165	+	0.961683i
$190$		0			0
$191$	920.494	+	531.447i	0.348715	+	0.201331i	0.664119	−	0.747627i	$-0.268807\pi$
$191$	920.494	+	531.447i	0.348715	+	0.201331i	−0.315404	+	0.948957i	$0.602140\pi$
$192$		0			0
$193$	−1945.03	−	3368.89i	−0.725420	−	1.25646i	−0.958801	−	0.284079i	$-0.908312\pi$
$193$	−1945.03	−	3368.89i	−0.725420	−	1.25646i	0.233380	−	0.972386i	$-0.425021\pi$
$194$		0			0
$195$	−537.118	+	945.162i	−0.197250	+	0.347100i
$196$		0			0
$197$			3227.57i			1.16728i	0.812011	+	0.583642i	$0.198373\pi$
$197$			3227.57i			1.16728i	−0.812011	+	0.583642i	$0.801627\pi$
$198$		0			0
$199$	2398.78	−	1384.94i	0.854497	−	0.493344i	−0.00766855	−	0.999971i	$-0.502441\pi$
$199$	2398.78	−	1384.94i	0.854497	−	0.493344i	0.862166	+	0.506626i	$0.169108\pi$
$200$		0			0
$201$	−2510.45	+	1472.36i	−0.880963	+	0.516676i
$202$		0			0
$203$	730.874	−	2377.82i	0.252696	−	0.822119i
$204$		0			0
$205$	568.842	−	985.263i	0.193803	−	0.335677i
$206$		0			0
$207$	−1025.01	−	573.269i	−0.344169	−	0.192488i
$208$		0			0
$209$	4231.58			1.40050
$210$		0			0
$211$	1978.76			0.645610			0.322805	−	0.946465i	$-0.395374\pi$
$211$	1978.76			0.645610			0.322805	+	0.946465i	$0.395374\pi$
$212$		0			0
$213$	−385.629	+	2.63318i	−0.124051	+	0.000847053i
$214$		0			0
$215$	1505.22	−	2607.11i	0.477464	−	0.826993i
$216$		0			0
$217$	3021.68	−	692.838i	0.945275	−	0.216741i
$218$		0			0
$219$	429.235	+	731.869i	0.132443	+	0.225823i
$220$		0			0
$221$	−1592.61	+	919.492i	−0.484753	+	0.279872i
$222$		0			0
$223$		−	2434.70i		−	0.731120i	−0.930788	−	0.365560i	$-0.880878\pi$
$223$		−	2434.70i		−	0.731120i	0.930788	−	0.365560i	$-0.119122\pi$
$224$		0			0
$225$	−190.688	−	320.105i	−0.0565001	−	0.0948458i
$226$		0			0
$227$	−165.862	−	287.281i	−0.0484961	−	0.0839977i	0.840758	−	0.541411i	$-0.182110\pi$
$227$	−165.862	−	287.281i	−0.0484961	−	0.0839977i	−0.889254	+	0.457413i	$0.848776\pi$
$228$		0			0
$229$	1842.29	+	1063.65i	0.531624	+	0.306933i	0.741677	−	0.670757i	$-0.234031\pi$
$229$	1842.29	+	1063.65i	0.531624	+	0.306933i	−0.210054	+	0.977690i	$0.567364\pi$
$230$		0			0
$231$	−682.603	−	2886.42i	−0.194424	−	0.822132i
$232$		0			0
$233$	−809.394	−	467.304i	−0.227576	−	0.131391i	0.381877	−	0.924213i	$-0.375278\pi$
$233$	−809.394	−	467.304i	−0.227576	−	0.131391i	−0.609453	+	0.792822i	$0.708611\pi$
$234$		0			0
$235$	1274.97	+	2208.32i	0.353915	+	0.613000i
$236$		0			0
$237$	3949.86	+	2244.63i	1.08258	+	0.615209i
$238$		0			0
$239$			1923.73i			0.520651i	0.965521	+	0.260325i	$0.0838298\pi$
$239$			1923.73i			0.520651i	−0.965521	+	0.260325i	$0.916170\pi$
$240$		0			0
$241$	2846.13	−	1643.22i	0.760729	−	0.439207i	−0.0688286	−	0.997629i	$-0.521926\pi$
$241$	2846.13	−	1643.22i	0.760729	−	0.439207i	0.829557	+	0.558422i	$0.188593\pi$
$242$		0			0
$243$	129.300	+	3785.79i	0.0341342	+	0.999417i
$244$		0			0
$245$	3255.67	−	1575.82i	0.848967	−	0.410921i
$246$		0			0
$247$	−1361.97	+	2359.00i	−0.350850	+	0.607690i
$248$		0			0
$249$	−14.4261	−	2112.70i	−0.00367156	−	0.537700i
$250$		0			0
$251$	−4303.94			−1.08232			−0.541160	−	0.840919i	$-0.682015\pi$
$251$	−4303.94			−1.08232			−0.541160	+	0.840919i	$0.682015\pi$
$252$		0			0
$253$	1340.63			0.333142
$254$		0			0
$255$	34.6793	+	5078.78i	0.00851648	+	1.24724i
$256$		0			0
$257$	−2345.95	+	4063.31i	−0.569403	+	0.986235i	0.427222	+	0.904147i	$0.359492\pi$
$257$	−2345.95	+	4063.31i	−0.569403	+	0.986235i	−0.996625	+	0.0820884i	$0.973841\pi$
$258$		0			0
$259$	−5201.63	+	4836.03i	−1.24793	+	1.16022i
$260$		0			0
$261$	−49.5244	−	3626.26i	−0.0117451	−	0.859999i
$262$		0			0
$263$	−911.918	+	526.496i	−0.213807	+	0.123442i	−0.603079	−	0.797681i	$-0.706060\pi$
$263$	−911.918	+	526.496i	−0.213807	+	0.123442i	0.389272	+	0.921123i	$0.372726\pi$
$264$		0			0
$265$			5268.69i			1.22133i
$266$		0			0
$267$	4753.37	+	2701.25i	1.08952	+	0.619154i
$268$		0			0
$269$	1327.49	+	2299.28i	0.300887	+	0.521151i	0.976337	−	0.216255i	$-0.0693841\pi$
$269$	1327.49	+	2299.28i	0.300887	+	0.521151i	−0.675450	+	0.737405i	$0.736051\pi$
$270$		0			0
$271$	−2347.45	−	1355.30i	−0.526189	−	0.303795i	0.213274	−	0.976992i	$-0.431587\pi$
$271$	−2347.45	−	1355.30i	−0.526189	−	0.303795i	−0.739463	+	0.673197i	$0.764921\pi$
$272$		0			0
$273$	1828.81	+	548.485i	0.405438	+	0.121596i
$274$		0			0
$275$	368.345	+	212.664i	0.0807710	+	0.0466331i
$276$		0			0
$277$	−960.218	−	1663.15i	−0.208281	−	0.360754i	0.742892	−	0.669411i	$-0.233454\pi$
$277$	−960.218	−	1663.15i	−0.208281	−	0.360754i	−0.951173	+	0.308658i	$0.900120\pi$
$278$		0			0
$279$	3882.78	−	2312.99i	0.833176	−	0.496327i
$280$		0			0
$281$			5730.99i			1.21666i	0.793683	+	0.608331i	$0.208161\pi$
$281$			5730.99i			1.21666i	−0.793683	+	0.608331i	$0.791839\pi$
$282$		0			0
$283$	1588.75	−	917.265i	0.333715	−	0.192670i	−0.323774	−	0.946134i	$-0.604952\pi$
$283$	1588.75	−	917.265i	0.333715	−	0.192670i	0.657489	+	0.753464i	$0.271619\pi$
$284$		0			0
$285$	3805.88	+	6489.24i	0.791020	+	1.34873i
$286$		0			0
$287$	−1909.91	−	587.051i	−0.392817	−	0.120741i
$288$		0			0
$289$	−1839.27	+	3185.71i	−0.374368	+	0.648424i
$290$		0			0
$291$	1263.86	−	8.62995i	0.254600	−	0.00173848i
$292$		0			0
$293$	−1206.83			−0.240627			−0.120314	−	0.992736i	$-0.538390\pi$
$293$	−1206.83			−0.240627			−0.120314	+	0.992736i	$0.538390\pi$
$294$		0			0
$295$	7723.53			1.52434
$296$		0			0
$297$	−2238.29	−	3699.69i	−0.437302	−	0.722821i
$298$		0			0
$299$	−431.494	+	747.369i	−0.0834580	+	0.144553i
$300$		0			0
$301$	−5053.82	−	1553.40i	−0.967765	−	0.297463i
$302$		0			0
$303$	−950.293	+	557.338i	−0.180175	+	0.105671i
$304$		0			0
$305$	−2796.65	+	1614.65i	−0.525035	+	0.303129i
$306$		0			0
$307$		−	5508.71i		−	1.02410i	−0.858955	−	0.512050i	$-0.828886\pi$
$307$		−	5508.71i		−	1.02410i	0.858955	−	0.512050i	$-0.171114\pi$
$308$		0			0
$309$	−2202.52	+	3875.75i	−0.405491	+	0.713539i
$310$		0			0
$311$	−2114.51	−	3662.44i	−0.385540	−	0.667775i	0.606304	−	0.795233i	$-0.292652\pi$
$311$	−2114.51	−	3662.44i	−0.385540	−	0.667775i	−0.991844	+	0.127458i	$0.959318\pi$
$312$		0			0
$313$	−4672.64	−	2697.75i	−0.843813	−	0.487176i	0.0147455	−	0.999891i	$-0.495306\pi$
$313$	−4672.64	−	2697.75i	−0.843813	−	0.487176i	−0.858558	+	0.512716i	$0.828640\pi$
$314$		0			0
$315$	3812.48	−	3642.84i	0.681932	−	0.651590i
$316$		0			0
$317$	−6169.57	−	3562.00i	−1.09312	−	0.631111i	−0.158711	−	0.987325i	$-0.550734\pi$
$317$	−6169.57	−	3562.00i	−1.09312	−	0.631111i	−0.934404	+	0.356214i	$0.884067\pi$
$318$		0			0
$319$	2069.92	+	3585.20i	0.363301	+	0.629256i
$320$		0			0
$321$	5514.96	−	9704.62i	0.958925	−	1.68741i
$322$		0			0
$323$			12725.9i			2.19223i
$324$		0			0
$325$	−237.110	+	136.895i	−0.0404692	+	0.0233649i
$326$		0			0
$327$	−5571.41	+	3267.58i	−0.942200	+	0.552592i
$328$		0			0
$329$	3279.89	−	3049.37i	0.549624	−	0.510994i
$330$		0			0
$331$	886.224	−	1534.99i	0.147164	−	0.254896i	−0.783014	−	0.622004i	$-0.786319\pi$
$331$	886.224	−	1534.99i	0.147164	−	0.254896i	0.930178	+	0.367108i	$0.119652\pi$
$332$		0			0
$333$	−5054.23	+	9036.97i	−0.831741	+	1.48716i
$334$		0			0
$335$	5906.32			0.963275
$336$		0			0
$337$	7800.94			1.26096			0.630481	−	0.776205i	$-0.282858\pi$
$337$	7800.94			1.26096			0.630481	+	0.776205i	$0.282858\pi$
$338$		0			0
$339$	1006.48	−	6.87250i	0.161252	−	0.00110107i
$340$		0			0
$341$	−2579.56	+	4467.92i	−0.409650	+	0.709535i
$342$		0			0
$343$	−3975.47	−	4954.72i	−0.625818	−	0.779969i
$344$		0			0
$345$	1205.76	+	2055.90i	0.188163	+	0.320828i
$346$		0			0
$347$	−6209.38	+	3584.99i	−0.960625	+	0.554617i	−0.896365	−	0.443316i	$-0.853802\pi$
$347$	−6209.38	+	3584.99i	−0.960625	+	0.554617i	−0.0642596	+	0.997933i	$0.520469\pi$
$348$		0			0
$349$		−	5543.44i		−	0.850240i	−0.905137	−	0.425120i	$-0.860232\pi$
$349$		−	5543.44i		−	0.850240i	0.905137	−	0.425120i	$-0.139768\pi$
$350$		0			0
$351$	2782.90	−	57.0142i	0.423191	−	0.00867006i
$352$		0			0
$353$	−15.4669	−	26.7895i	−0.00233207	−	0.00403926i	0.864857	−	0.502018i	$-0.167409\pi$
$353$	−15.4669	−	26.7895i	−0.00233207	−	0.00403926i	−0.867189	+	0.497979i	$0.834076\pi$
$354$		0			0
$355$	677.768	+	391.310i	0.101330	+	0.0585030i
$356$		0			0
$357$	8680.56	−	2052.84i	1.28690	−	0.304336i
$358$		0			0
$359$	5694.34	+	3287.63i	0.837147	+	0.483327i	0.856294	−	0.516489i	$-0.172761\pi$
$359$	5694.34	+	3287.63i	0.837147	+	0.483327i	−0.0191462	+	0.999817i	$0.506095\pi$
$360$		0			0
$361$	5995.44	+	10384.4i	0.874099	+	1.51398i
$362$		0			0
$363$	−1721.50	−	978.296i	−0.248912	−	0.141452i
$364$		0			0
$365$		−	1721.87i		−	0.246922i
$366$		0			0
$367$	−2095.30	+	1209.72i	−0.298021	+	0.172062i	−0.641553	−	0.767078i	$-0.721710\pi$
$367$	−2095.30	+	1209.72i	−0.298021	+	0.172062i	0.343533	+	0.939141i	$0.388376\pi$
$368$		0			0
$369$	−2912.68	+	39.7789i	−0.410916	+	0.00561195i
$370$		0			0
$371$	9019.27	−	2068.02i	1.26215	−	0.289397i
$372$		0			0
$373$	−4895.66	+	8479.53i	−0.679591	+	1.17709i	0.295513	+	0.955339i	$0.404510\pi$
$373$	−4895.66	+	8479.53i	−0.679591	+	1.17709i	−0.975104	+	0.221748i	$0.928824\pi$
$374$		0			0
$375$	51.9307	+	7605.25i	0.00715117	+	1.04729i
$376$		0			0
$377$	−2664.88			−0.364054
$378$		0			0
$379$	−12660.9			−1.71595			−0.857977	−	0.513689i	$-0.828279\pi$
$379$	−12660.9			−1.71595			−0.857977	+	0.513689i	$0.828279\pi$
$380$		0			0
$381$	−35.8529	−	5250.66i	−0.00482100	−	0.706035i
$382$		0			0
$383$	−4549.42	+	7879.82i	−0.606957	+	1.05128i	0.384782	+	0.923007i	$0.374276\pi$
$383$	−4549.42	+	7879.82i	−0.606957	+	1.05128i	−0.991739	+	0.128272i	$0.959057\pi$
$384$		0			0
$385$	−1768.51	+	5753.66i	−0.234108	+	0.761645i
$386$		0			0
$387$	−7707.25	+	105.259i	−1.01236	+	0.0138259i
$388$		0			0
$389$	8450.00	−	4878.61i	1.10137	−	0.635875i	0.164787	−	0.986329i	$-0.447306\pi$
$389$	8450.00	−	4878.61i	1.10137	−	0.635875i	0.936580	+	0.350454i	$0.113973\pi$
$390$		0			0
$391$			4031.78i			0.521473i
$392$		0			0
$393$	1620.15	+	920.701i	0.207954	+	0.118176i
$394$		0			0
$395$	−4609.91	−	7984.61i	−0.587215	−	1.01709i
$396$		0			0
$397$	−7069.17	−	4081.39i	−0.893681	−	0.515967i	−0.0185365	−	0.999828i	$-0.505901\pi$
$397$	−7069.17	−	4081.39i	−0.893681	−	0.515967i	−0.875145	+	0.483861i	$0.839234\pi$
$398$		0			0
$399$	9614.82	−	9062.23i	1.20637	−	1.13704i
$400$		0			0
$401$	8885.22	+	5129.88i	1.10650	+	0.638838i	0.937921	−	0.346850i	$-0.112749\pi$
$401$	8885.22	+	5129.88i	1.10650	+	0.638838i	0.168579	+	0.985688i	$0.446082\pi$
$402$		0			0
$403$	−1660.50	−	2876.08i	−0.205250	−	0.355503i
$404$		0			0
$405$	3660.46	−	6759.98i	0.449111	−	0.829398i
$406$		0			0
$407$		−	11819.7i		−	1.43951i
$408$		0			0
$409$	−1795.47	+	1036.61i	−0.217067	+	0.125323i	−0.604591	−	0.796536i	$-0.706664\pi$
$409$	−1795.47	+	1036.61i	−0.217067	+	0.125323i	0.387525	+	0.921859i	$0.373330\pi$
$410$		0			0
$411$	−3624.40	−	6179.80i	−0.434984	−	0.741672i
$412$		0			0
$413$	−3031.57	−	13221.6i	−0.361196	−	1.57528i
$414$		0			0
$415$	−2143.83	+	3713.22i	−0.253581	+	0.439216i
$416$		0			0
$417$	2465.45	−	16.8348i	0.289530	−	0.00197699i
$418$		0			0
$419$	11576.2			1.34973			0.674863	−	0.737943i	$-0.264203\pi$
$419$	11576.2			1.34973			0.674863	+	0.737943i	$0.264203\pi$
$420$		0			0
$421$	−1493.04			−0.172842			−0.0864208	−	0.996259i	$-0.527543\pi$
$421$	−1493.04			−0.172842			−0.0864208	+	0.996259i	$0.527543\pi$
$422$		0			0
$423$	3186.95	−	5698.28i	0.366324	−	0.654988i
$424$		0			0
$425$	−639.560	+	1107.75i	−0.0729958	+	0.126432i
$426$		0			0
$427$	3861.76	+	4153.70i	0.437667	+	0.470754i
$428$		0			0
$429$	−2740.80	+	1607.45i	−0.308455	+	0.180906i
$430$		0			0
$431$	−3354.19	+	1936.54i	−0.374862	+	0.216427i	−0.675580	−	0.737286i	$-0.736107\pi$
$431$	−3354.19	+	1936.54i	−0.374862	+	0.216427i	0.300719	+	0.953713i	$0.402774\pi$
$432$		0			0
$433$		−	5450.01i		−	0.604875i	−0.953169	−	0.302437i	$-0.902200\pi$
$433$		−	5450.01i		−	0.604875i	0.953169	−	0.302437i	$-0.0978003\pi$
$434$		0			0
$435$	−3636.32	+	6398.81i	−0.400801	+	0.705286i
$436$		0			0
$437$	2985.98	+	5171.86i	0.326862	+	0.566141i
$438$		0			0
$439$	2350.20	+	1356.89i	0.255510	+	0.147519i	0.622284	−	0.782791i	$-0.286205\pi$
$439$	2350.20	+	1356.89i	0.255510	+	0.147519i	−0.366775	+	0.930310i	$0.619538\pi$
$440$		0			0
$441$	−7732.47	−	5096.57i	−0.834950	−	0.550326i
$442$		0			0
$443$	−9583.65	−	5533.13i	−1.02784	−	0.593424i	−0.111475	−	0.993767i	$-0.535557\pi$
$443$	−9583.65	−	5533.13i	−1.02784	−	0.593424i	−0.916365	+	0.400344i	$0.868891\pi$
$444$		0			0
$445$	−5547.70	−	9608.90i	−0.590980	−	1.02361i
$446$		0			0
$447$	1538.92	−	2708.03i	0.162838	−	0.286544i
$448$		0			0
$449$			3599.14i			0.378294i	0.981949	+	0.189147i	$0.0605722\pi$
$449$			3599.14i			0.378294i	−0.981949	+	0.189147i	$0.939428\pi$
$450$		0			0
$451$	2879.70	−	1662.60i	0.300665	−	0.173589i
$452$		0			0
$453$	8942.69	−	5244.81i	0.927515	−	0.543979i
$454$		0			0
$455$	−2638.31	−	2837.76i	−0.271837	−	0.292388i
$456$		0			0
$457$	−4157.54	+	7201.08i	−0.425562	+	0.737094i	−0.996473	−	0.0839176i	$-0.973257\pi$
$457$	−4157.54	+	7201.08i	−0.425562	+	0.737094i	0.570911	+	0.821012i	$0.306590\pi$
$458$		0			0
$459$	11126.4	−	6731.38i	1.13145	−	0.684518i
$460$		0			0
$461$	−5672.08			−0.573048			−0.286524	−	0.958073i	$-0.592500\pi$
$461$	−5672.08			−0.573048			−0.286524	+	0.958073i	$0.592500\pi$
$462$		0			0
$463$	−6332.06			−0.635585			−0.317792	−	0.948160i	$-0.602942\pi$
$463$	−6332.06			−0.635585			−0.317792	+	0.948160i	$0.602942\pi$
$464$		0			0
$465$	−9171.73	+	62.6271i	−0.914686	+	0.00624572i
$466$		0			0
$467$	6469.64	−	11205.7i	0.641069	−	1.11036i	−0.344126	−	0.938924i	$-0.611825\pi$
$467$	6469.64	−	11205.7i	0.641069	−	1.11036i	0.985195	−	0.171440i	$-0.0548420\pi$
$468$		0			0
$469$	−2318.30	−	10110.8i	−0.228249	−	0.995466i
$470$		0			0
$471$	−2325.02	−	3964.29i	−0.227455	−	0.387824i
$472$		0			0
$473$	7619.99	−	4399.40i	0.740735	−	0.427663i
$474$		0			0
$475$			1894.66i			0.183016i
$476$		0			0
$477$	11589.5	−	6903.94i	1.11247	−	0.662704i
$478$		0			0
$479$	1051.99	+	1822.10i	0.100348	+	0.173808i	0.911828	−	0.410572i	$-0.134671\pi$
$479$	1051.99	+	1822.10i	0.100348	+	0.173808i	−0.811480	+	0.584380i	$0.801338\pi$
$480$		0			0
$481$	6589.18	+	3804.26i	0.624617	+	0.360623i
$482$		0			0
$483$	3046.13	−	2871.06i	0.286964	−	0.270472i
$484$		0			0
$485$	−2221.31	−	1282.47i	−0.207968	−	0.120070i
$486$		0			0
$487$	−6295.98	−	10905.0i	−0.585828	−	1.01468i	−0.994772	−	0.102124i	$-0.967436\pi$
$487$	−6295.98	−	10905.0i	−0.585828	−	1.01468i	0.408944	−	0.912559i	$-0.365897\pi$
$488$		0			0
$489$	−3058.37	−	1738.01i	−0.282831	−	0.160727i
$490$		0			0
$491$			1372.36i			0.126138i	0.998009	+	0.0630689i	$0.0200888\pi$
$491$			1372.36i			0.126138i	−0.998009	+	0.0630689i	$0.979911\pi$
$492$		0			0
$493$	−10782.0	+	6225.02i	−0.984988	+	0.568683i
$494$		0			0
$495$	119.835	+	8774.52i	0.0108812	+	0.796738i
$496$		0			0
$497$	403.836	−	1313.84i	0.0364477	−	0.118579i
$498$		0			0
$499$	−4588.75	+	7947.95i	−0.411665	+	0.713025i	−0.995072	−	0.0991554i	$-0.968386\pi$
$499$	−4588.75	+	7947.95i	−0.411665	+	0.713025i	0.583407	+	0.812180i	$0.301719\pi$
$500$		0			0
$501$	131.941	+	19322.7i	0.0117658	+	1.72310i
$502$		0			0
$503$	1025.01			0.0908605			0.0454302	−	0.998968i	$-0.485534\pi$
$503$	1025.01			0.0908605			0.0454302	+	0.998968i	$0.485534\pi$
$504$		0			0
$505$	2235.75			0.197009
$506$		0			0
$507$	63.9835	+	9370.38i	0.00560474	+	0.820815i
$508$		0			0
$509$	2308.88	−	3999.09i	0.201059	−	0.348245i	−0.747811	−	0.663912i	$-0.768895\pi$
$509$	2308.88	−	3999.09i	0.201059	−	0.348245i	0.948870	+	0.315667i	$0.102228\pi$
$510$		0			0
$511$	−2947.59	+	675.851i	−0.255174	+	0.0585086i
$512$		0			0
$513$	9287.27	−	16875.1i	0.799304	−	1.45235i
$514$		0			0
$515$	7834.80	−	4523.42i	0.670374	−	0.387040i
$516$		0			0
$517$			7452.92i			0.634002i
$518$		0			0
$519$	11917.3	+	6772.36i	1.00792	+	0.572781i
$520$		0			0
$521$	−6012.38	−	10413.7i	−0.505580	−	0.875690i	−0.999979	−	0.00645499i	$-0.997945\pi$
$521$	−6012.38	−	10413.7i	−0.505580	−	0.875690i	0.494399	−	0.869235i	$-0.335388\pi$
$522$		0			0
$523$	−2045.16	−	1180.78i	−0.170992	−	0.0987222i	0.412062	−	0.911156i	$-0.364809\pi$
$523$	−2045.16	−	1180.78i	−0.170992	−	0.0987222i	−0.583053	+	0.812434i	$0.698142\pi$
$524$		0			0
$525$	1292.37	−	305.630i	0.107436	−	0.0254073i
$526$		0			0
$527$	−13436.7	−	7757.69i	−1.11065	−	0.641234i
$528$		0			0
$529$	−5137.49	−	8898.40i	−0.422248	−	0.731355i
$530$		0			0
$531$	−10120.7	−	16989.4i	−0.827119	−	1.38847i
$532$		0			0
$533$			2140.48i			0.173949i
$534$		0			0
$535$	−19617.8	+	11326.3i	−1.58533	+	0.915291i
$536$		0			0
$537$	−8686.00	−	14810.1i	−0.698005	−	1.19014i
$538$		0			0
$539$	10543.6	+	769.067i	0.842571	+	0.0614583i
$540$		0			0
$541$	1479.47	−	2562.52i	0.117574	−	0.203644i	−0.801232	−	0.598354i	$-0.795822\pi$
$541$	1479.47	−	2562.52i	0.117574	−	0.203644i	0.918806	+	0.394710i	$0.129155\pi$
$542$		0			0
$543$	2171.63	−	14.8285i	0.171627	−	0.00117192i
$544$		0			0
$545$	13107.8			1.03023
$546$		0			0
$547$	8615.33			0.673427			0.336714	−	0.941607i	$-0.390685\pi$
$547$	8615.33			0.673427			0.336714	+	0.941607i	$0.390685\pi$
$548$		0			0
$549$	7216.38	+	4036.00i	0.560997	+	0.313756i
$550$		0			0
$551$	−9220.61	+	15970.6i	−0.712906	+	1.23479i
$552$		0			0
$553$	−11859.1	+	11025.6i	−0.911934	+	0.847839i
$554$		0			0
$555$	18125.8	−	10630.6i	1.38630	−	0.813054i
$556$		0			0
$557$	−20665.9	+	11931.5i	−1.57207	+	0.907635i	−0.576155	+	0.817340i	$0.695448\pi$
$557$	−20665.9	+	11931.5i	−1.57207	+	0.907635i	−0.995915	+	0.0902950i	$0.971219\pi$
$558$		0			0
$559$			5663.94i			0.428550i
$560$		0			0
$561$	−7334.29	+	12906.1i	−0.551968	+	0.971292i
$562$		0			0
$563$	−5552.10	−	9616.52i	−0.415618	−	0.719872i	0.579875	−	0.814705i	$-0.303101\pi$
$563$	−5552.10	−	9616.52i	−0.415618	−	0.719872i	−0.995493	+	0.0948337i	$0.969768\pi$
$564$		0			0
$565$	−1768.95	−	1021.30i	−0.131717	−	0.0760470i
$566$		0			0
$567$	−13008.9	−	3612.83i	−0.963532	−	0.267592i
$568$		0			0
$569$	−20144.7	−	11630.5i	−1.48420	−	0.856903i	−0.484361	−	0.874868i	$-0.660948\pi$
$569$	−20144.7	−	11630.5i	−1.48420	−	0.856903i	−0.999839	+	0.0179647i	$0.994281\pi$
$570$		0			0
$571$	−1521.55	−	2635.40i	−0.111514	−	0.193149i	0.804867	−	0.593456i	$-0.202237\pi$
$571$	−1521.55	−	2635.40i	−0.111514	−	0.193149i	−0.916381	+	0.400307i	$0.868903\pi$
$572$		0			0
$573$	−2728.76	+	4801.77i	−0.198945	+	0.350082i
$574$		0			0
$575$			600.258i			0.0435348i
$576$		0			0
$577$	8570.32	−	4948.08i	0.618349	−	0.357004i	−0.157877	−	0.987459i	$-0.550465\pi$
$577$	8570.32	−	4948.08i	0.618349	−	0.357004i	0.776226	+	0.630455i	$0.217132\pi$
$578$		0			0
$579$	17435.8	−	10226.0i	1.25148	−	0.733983i
$580$		0			0
$581$	7197.98	+	2212.45i	0.513980	+	0.157983i
$582$		0			0
$583$	−7699.60	+	13336.1i	−0.546972	+	0.947384i
$584$		0			0
$585$	−4930.15	−	2757.35i	−0.348439	−	0.194876i
$586$		0			0
$587$	14853.9			1.04444			0.522219	−	0.852812i	$-0.325104\pi$
$587$	14853.9			1.04444			0.522219	+	0.852812i	$0.325104\pi$
$588$		0			0
$589$	−22981.7			−1.60771
$590$		0			0
$591$	−16770.6	+	114.514i	−1.16726	+	0.00797034i
$592$		0			0
$593$	2072.42	−	3589.54i	0.143515	−	0.248575i	−0.785303	−	0.619111i	$-0.787493\pi$
$593$	2072.42	−	3589.54i	0.143515	−	0.248575i	0.928818	+	0.370537i	$0.120826\pi$
$594$		0			0
$595$	−17303.4	−	5318.57i	−1.19222	−	0.366454i
$596$		0			0
$597$	7281.28	+	12415.0i	0.499167	+	0.851109i
$598$		0			0
$599$	−4311.66	+	2489.34i	−0.294107	+	0.169802i	−0.639792	−	0.768548i	$-0.720980\pi$
$599$	−4311.66	+	2489.34i	−0.294107	+	0.169802i	0.345686	+	0.938350i	$0.387646\pi$
$600$		0			0
$601$			6641.45i			0.450766i	0.974270	+	0.225383i	$0.0723633\pi$
$601$			6641.45i			0.450766i	−0.974270	+	0.225383i	$0.927637\pi$
$602$		0			0
$603$	−7739.48	−	12992.1i	−0.522680	−	0.877414i
$604$		0			0
$605$	2009.18	+	3480.00i	0.135016	+	0.233854i
$606$		0			0
$607$	6658.36	+	3844.21i	0.445230	+	0.257054i	0.705814	−	0.708398i	$-0.250582\pi$
$607$	6658.36	+	3844.21i	0.445230	+	0.257054i	−0.260583	+	0.965451i	$0.583915\pi$
$608$		0			0
$609$	12381.2	+	3713.28i	0.823826	+	0.247077i
$610$		0			0
$611$	−4154.82	−	2398.79i	−0.275100	−	0.158829i
$612$		0			0
$613$	−4708.37	−	8155.14i	−0.310227	−	0.537330i	0.668184	−	0.743996i	$-0.267072\pi$
$613$	−4708.37	−	8155.14i	−0.310227	−	0.537330i	−0.978411	+	0.206666i	$0.933739\pi$
$614$		0			0
$615$	5139.64	+	2920.76i	0.336992	+	0.191507i
$616$		0			0
$617$			9209.91i			0.600935i	0.953792	+	0.300468i	$0.0971428\pi$
$617$			9209.91i			0.600935i	−0.953792	+	0.300468i	$0.902857\pi$
$618$		0			0
$619$	8867.28	−	5119.53i	0.575777	−	0.332425i	−0.183676	−	0.982987i	$-0.558800\pi$
$619$	8867.28	−	5119.53i	0.575777	−	0.332425i	0.759454	+	0.650562i	$0.225466\pi$
$620$		0			0
$621$	2942.36	−	5346.31i	0.190133	−	0.345475i
$622$		0			0
$623$	−14271.6	+	13268.5i	−0.917782	+	0.853276i
$624$		0			0
$625$	6854.79	−	11872.8i	0.438706	−	0.759862i
$626$		0			0
$627$	150.136	+	21987.4i	0.00956276	+	1.40047i
$628$		0			0
$629$	35546.2			2.25329
$630$		0			0
$631$	25041.7			1.57987			0.789934	−	0.613192i	$-0.210115\pi$
$631$	25041.7			1.57987			0.789934	+	0.613192i	$0.210115\pi$
$632$		0			0
$633$	70.2063	+	10281.7i	0.00440830	+	0.645595i
$634$		0			0
$635$	−5328.00	+	9228.37i	−0.332969	+	0.576719i
$636$		0			0
$637$	−3822.29	+	5630.27i	−0.237747	+	0.350203i
$638$		0			0
$639$	−27.3642	−	2003.65i	−0.00169407	−	0.124042i
$640$		0			0
$641$	545.089	−	314.707i	0.0335877	−	0.0193919i	−0.483112	−	0.875558i	$-0.660494\pi$
$641$	545.089	−	314.707i	0.0335877	−	0.0193919i	0.516700	+	0.856167i	$0.327160\pi$
$642$		0			0
$643$		−	11568.9i		−	0.709538i	−0.934954	−	0.354769i	$-0.884559\pi$
$643$		−	11568.9i		−	0.709538i	0.934954	−	0.354769i	$-0.115441\pi$
$644$		0			0
$645$	13600.0	+	7728.65i	0.830234	+	0.471807i
$646$		0			0
$647$	9784.59	+	16947.4i	0.594547	+	1.02979i	0.993611	+	0.112862i	$0.0360018\pi$
$647$	9784.59	+	16947.4i	0.594547	+	1.02979i	−0.399064	+	0.916923i	$0.630665\pi$
$648$		0			0
$649$	19549.8	+	11287.1i	1.18243	+	0.682675i
$650$		0			0
$651$	3707.21	+	15676.1i	0.223191	+	0.943773i
$652$		0			0
$653$	−5834.29	−	3368.43i	−0.349638	−	0.201863i	0.314888	−	0.949129i	$-0.398033\pi$
$653$	−5834.29	−	3368.43i	−0.349638	−	0.201863i	−0.664526	+	0.747265i	$0.731366\pi$
$654$		0			0
$655$	−1890.89	−	3275.12i	−0.112799	−	0.195373i
$656$		0			0
$657$	−3787.59	+	2256.28i	−0.224913	+	0.133982i
$658$		0			0
$659$			3912.25i			0.231259i	0.993292	+	0.115629i	$0.0368885\pi$
$659$			3912.25i			0.231259i	−0.993292	+	0.115629i	$0.963112\pi$
$660$		0			0
$661$	24001.8	−	13857.5i	1.41235	−	0.815420i	0.416740	−	0.909026i	$-0.363173\pi$
$661$	24001.8	−	13857.5i	1.41235	−	0.815420i	0.995609	+	0.0936057i	$0.0298393\pi$
$662$		0			0
$663$	−4834.22	−	8242.61i	−0.283176	−	0.482830i
$664$		0			0
$665$	−26135.3	+	5992.54i	−1.52404	+	0.349445i
$666$		0			0
$667$	−2921.24	+	5059.73i	−0.169581	+	0.293724i
$668$		0			0
$669$	12650.8	−	86.3830i	0.731103	−	0.00499217i
$670$		0			0
$671$	−9438.48			−0.543023
$672$		0			0
$673$	27462.5			1.57296			0.786479	−	0.617617i	$-0.211902\pi$
$673$	27462.5			1.57296			0.786479	+	0.617617i	$0.211902\pi$
$674$		0			0
$675$	1656.51	−	1002.18i	0.0944578	−	0.0571464i
$676$		0			0
$677$	−4757.18	+	8239.68i	−0.270064	+	0.467765i	−0.968878	−	0.247539i	$-0.920378\pi$
$677$	−4757.18	+	8239.68i	−0.270064	+	0.467765i	0.698814	+	0.715303i	$0.253712\pi$
$678$		0			0
$679$	−1323.53	+	4305.96i	−0.0748046	+	0.243369i
$680$		0			0
$681$	1486.83	−	872.014i	0.0836646	−	0.0490685i
$682$		0			0
$683$	22760.4	−	13140.7i	1.27511	−	0.736186i	0.299166	−	0.954201i	$-0.403292\pi$
$683$	22760.4	−	13140.7i	1.27511	−	0.736186i	0.975945	+	0.218015i	$0.0699583\pi$
$684$		0			0
$685$			14539.2i			0.810969i
$686$		0			0
$687$	−5461.37	+	9610.33i	−0.303296	+	0.533707i
$688$		0			0
$689$	−4956.36	−	8584.67i	−0.274053	−	0.474673i
$690$		0			0
$691$	6698.34	+	3867.29i	0.368765	+	0.212907i	0.672919	−	0.739716i	$-0.265040\pi$
$691$	6698.34	+	3867.29i	0.368765	+	0.212907i	−0.304154	+	0.952623i	$0.598374\pi$
$692$		0			0
$693$	14973.7	−	3649.24i	0.820786	−	0.200033i
$694$		0			0
$695$	−4333.20	−	2501.77i	−0.236500	−	0.136543i
$696$		0			0
$697$	5000.05	+	8660.34i	0.271722	+	0.470637i
$698$		0			0
$699$	2399.41	−	4222.22i	0.129834	−	0.228468i
$700$		0			0
$701$		−	16405.7i		−	0.883929i	−0.897033	−	0.441964i	$-0.854282\pi$
$701$		−	16405.7i		−	0.883929i	0.897033	−	0.441964i	$-0.145718\pi$
$702$		0			0
$703$	45597.7	−	26325.9i	2.44630	−	1.41237i
$704$		0			0
$705$	−11429.3	+	6703.16i	−0.610569	+	0.358093i
$706$		0			0
$707$	−877.556	−	3827.29i	−0.0466816	−	0.203593i
$708$		0			0
$709$	5304.71	−	9188.03i	0.280991	−	0.486691i	−0.690638	−	0.723201i	$-0.742670\pi$
$709$	5304.71	−	9188.03i	0.280991	−	0.486691i	0.971629	+	0.236510i	$0.0760035\pi$
$710$		0			0
$711$	−11523.0	+	20603.2i	−0.607803	+	1.08675i
$712$		0			0
$713$	−7280.97			−0.382432
$714$		0			0
$715$	6448.26			0.337275
$716$		0			0
$717$	−9995.74	+	68.2536i	−0.520638	+	0.00355506i
$718$		0			0
$719$	10075.2	−	17450.8i	0.522591	−	0.905153i	−0.477064	−	0.878869i	$-0.658299\pi$
$719$	10075.2	−	17450.8i	0.522591	−	0.905153i	0.999654	−	0.0262848i	$-0.00836768\pi$
$720$		0			0
$721$	−10818.7	−	11636.6i	−0.558821	−	0.601067i
$722$		0			0
$723$	8639.18	+	14730.3i	0.444391	+	0.757712i
$724$		0			0
$725$	−1605.25	+	926.790i	−0.0822309	+	0.0474760i
$726$		0			0
$727$			26758.5i			1.36509i	0.730846	+	0.682543i	$0.239126\pi$
$727$			26758.5i			1.36509i	−0.730846	+	0.682543i	$0.760874\pi$
$728$		0			0
$729$	−19666.5	+	806.167i	−0.999161	+	0.0409575i
$730$		0			0
$731$	13230.7	+	22916.2i	0.669431	+	1.15949i
$732$		0			0
$733$	7106.39	+	4102.88i	0.358091	+	0.206744i	0.668243	−	0.743943i	$-0.267047\pi$
$733$	7106.39	+	4102.88i	0.358091	+	0.206744i	−0.310152	+	0.950687i	$0.600380\pi$
$734$		0			0
$735$	8303.54	+	16860.6i	0.416709	+	0.846141i
$736$		0			0
$737$	14950.1	+	8631.42i	0.747208	+	0.431401i
$738$		0			0
$739$	−10324.9	−	17883.2i	−0.513946	−	0.890180i	−0.999869	−	0.0161788i	$-0.994850\pi$
$739$	−10324.9	−	17883.2i	−0.513946	−	0.890180i	0.485923	−	0.874001i	$-0.338483\pi$
$740$		0			0
$741$	−12305.8	−	6993.13i	−0.610072	−	0.346693i
$742$		0			0
$743$		−	8375.29i		−	0.413539i	−0.978390	−	0.206769i	$-0.933705\pi$
$743$		−	8375.29i		−	0.413539i	0.978390	−	0.206769i	$-0.0662950\pi$
$744$		0			0
$745$	−5474.25	+	3160.56i	−0.269210	+	0.155428i
$746$		0			0
$747$	10977.2	−	149.917i	0.537662	−	0.00734294i
$748$		0			0
$749$	27089.3	+	29137.2i	1.32152	+	1.42143i
$750$		0			0
$751$	13298.8	−	23034.3i	0.646181	−	1.11922i	−0.337847	−	0.941201i	$-0.609699\pi$
$751$	13298.8	−	23034.3i	0.646181	−	1.11922i	0.984028	−	0.178016i	$-0.0569680\pi$
$752$		0			0
$753$	−152.703	−	22363.4i	−0.00739020	−	1.08230i
$754$		0			0
$755$	−21039.4			−1.01418
$756$		0			0
$757$	−26633.6			−1.27875			−0.639376	−	0.768894i	$-0.720807\pi$
$757$	−26633.6			−1.27875			−0.639376	+	0.768894i	$0.720807\pi$
$758$		0			0
$759$	47.5655	+	6965.97i	0.00227473	+	0.333134i
$760$		0			0
$761$	−4243.87	+	7350.60i	−0.202155	+	0.350143i	−0.949223	−	0.314605i	$-0.898128\pi$
$761$	−4243.87	+	7350.60i	−0.202155	+	0.350143i	0.747067	+	0.664748i	$0.231461\pi$
$762$		0			0
$763$	−5144.96	−	22438.8i	−0.244116	−	1.06466i
$764$		0			0
$765$	−26388.3	+	360.389i	−1.24715	+	0.0170326i
$766$		0			0
$767$	−12584.5	+	7265.67i	−0.592439	+	0.342045i
$768$		0			0
$769$			22673.5i			1.06324i	0.846984	+	0.531618i	$0.178416\pi$
$769$			22673.5i			1.06324i	−0.846984	+	0.531618i	$0.821584\pi$
$770$		0			0
$771$	−21196.3	−	12045.5i	−0.990100	−	0.562656i
$772$		0			0
$773$	18141.2	+	31421.4i	0.844103	+	1.46203i	0.886398	+	0.462925i	$0.153200\pi$
$773$	18141.2	+	31421.4i	0.844103	+	1.46203i	−0.0422941	+	0.999105i	$0.513467\pi$
$774$		0			0
$775$	−2000.48	−	1154.98i	−0.0927217	−	0.0535329i
$776$		0			0
$777$	−25312.7	−	26856.2i	−1.16871	−	1.23998i
$778$		0			0
$779$	12827.9	+	7406.17i	0.589994	+	0.340633i
$780$		0			0
$781$	1143.71	+	1980.96i	0.0524009	+	0.0907611i
$782$		0			0
$783$	18840.4	−	385.990i	0.859898	−	0.0176170i
$784$		0			0
$785$			9326.77i			0.424060i
$786$		0			0
$787$	1911.44	−	1103.57i	0.0865763	−	0.0499849i	−0.456087	−	0.889935i	$-0.650749\pi$
$787$	1911.44	−	1103.57i	0.0865763	−	0.0499849i	0.542663	+	0.839950i	$0.317416\pi$
$788$		0			0
$789$	−2768.04	−	4719.67i	−0.124899	−	0.212959i
$790$		0			0
$791$	−1054.00	+	3429.07i	−0.0473778	+	0.154139i
$792$		0			0
$793$	3037.86	−	5261.72i	0.136037	−	0.235623i
$794$		0			0
$795$	−27376.3	+	186.933i	−1.22130	+	0.00833940i
$796$		0			0
$797$	−43496.3			−1.93315			−0.966574	−	0.256388i	$-0.917467\pi$
$797$	−43496.3			−1.93315			−0.966574	+	0.256388i	$0.917467\pi$
$798$		0			0
$799$	−22413.7			−0.992417
$800$		0			0
$801$	−13867.1	+	24794.5i	−0.611700	+	1.09372i
$802$		0			0
$803$	2516.31	−	4358.38i	0.110584	−	0.191537i
$804$		0			0
$805$	−8280.09	+	1898.54i	−0.362528	+	0.0831237i
$806$		0			0
$807$	−11900.0	+	6979.26i	−0.519084	+	0.304438i
$808$		0			0
$809$	23156.7	−	13369.5i	1.00636	−	0.581024i	0.0962378	−	0.995358i	$-0.469319\pi$
$809$	23156.7	−	13369.5i	1.00636	−	0.581024i	0.910125	+	0.414335i	$0.135986\pi$
$810$		0			0
$811$			43614.5i			1.88842i	0.329338	+	0.944212i	$0.393174\pi$
$811$			43614.5i			1.88842i	−0.329338	+	0.944212i	$0.606826\pi$
$812$		0			0
$813$	6958.89	−	12245.5i	0.300195	−	0.528251i
$814$		0			0
$815$	3569.45	+	6182.47i	0.153414	+	0.265721i
$816$		0			0
$817$	33943.9	+	19597.5i	1.45354	+	0.839204i
$818$		0			0
$819$	−2785.06	+	9522.02i	−0.118825	+	0.406259i
$820$		0			0
$821$	3199.70	+	1847.35i	0.136017	+	0.0785297i	0.566465	−	0.824086i	$-0.308311\pi$
$821$	3199.70	+	1847.35i	0.136017	+	0.0785297i	−0.430447	+	0.902616i	$0.641644\pi$
$822$		0			0
$823$	8733.70	+	15127.2i	0.369912	+	0.640706i	0.989551	−	0.144180i	$-0.0460545\pi$
$823$	8733.70	+	15127.2i	0.369912	+	0.640706i	−0.619640	+	0.784887i	$0.712721\pi$
$824$		0			0
$825$	−1091.94	+	1921.48i	−0.0460805	+	0.0810875i
$826$		0			0
$827$			37321.1i			1.56926i	0.619962	+	0.784632i	$0.287148\pi$
$827$			37321.1i			1.56926i	−0.619962	+	0.784632i	$0.712852\pi$
$828$		0			0
$829$	39701.3	−	22921.6i	1.66331	−	0.960313i	0.692193	−	0.721712i	$-0.256645\pi$
$829$	39701.3	−	22921.6i	1.66331	−	0.960313i	0.971118	−	0.238601i	$-0.0766887\pi$
$830$		0			0
$831$	8607.69	−	5048.33i	0.359323	−	0.210740i
$832$		0			0
$833$	−2312.87	+	31708.6i	−0.0962020	+	1.31889i
$834$		0			0
$835$	19607.3	−	33960.9i	0.812622	−	1.40750i
$836$		0			0
$837$	12156.1	+	20093.0i	0.502004	+	0.829768i
$838$		0			0
$839$	−11522.6			−0.474141			−0.237071	−	0.971492i	$-0.576187\pi$
$839$	−11522.6			−0.474141			−0.237071	+	0.971492i	$0.576187\pi$
$840$		0			0
$841$	6347.59			0.260264
$842$		0			0
$843$	−29778.4	+	203.335i	−1.21663	+	0.00830750i
$844$		0			0
$845$	9508.41	−	16469.1i	0.387100	−	0.670477i
$846$		0			0
$847$	5168.64	−	4805.36i	0.209677	−	0.194940i
$848$		0			0
$849$	4822.51	+	8222.65i	0.194945	+	0.332392i
$850$		0			0
$851$	14446.1	−	8340.46i	0.581911	−	0.335966i
$852$		0			0
$853$		−	44074.7i		−	1.76915i	−0.466395	−	0.884577i	$-0.654447\pi$
$853$		−	44074.7i		−	1.76915i	0.466395	−	0.884577i	$-0.345553\pi$
$854$		0			0
$855$	−33583.3	+	20005.7i	−1.34330	+	0.800211i
$856$		0			0
$857$	−8037.61	−	13921.5i	−0.320373	−	0.554902i	0.660192	−	0.751097i	$-0.270475\pi$
$857$	−8037.61	−	13921.5i	−0.320373	−	0.554902i	−0.980565	+	0.196195i	$0.937142\pi$
$858$		0			0
$859$	−16666.7	−	9622.52i	−0.662002	−	0.382207i	0.131037	−	0.991377i	$-0.458169\pi$
$859$	−16666.7	−	9622.52i	−0.662002	−	0.382207i	−0.793040	+	0.609170i	$0.791503\pi$
$860$		0			0
$861$	2982.57	−	9944.78i	0.118056	−	0.393632i
$862$		0			0
$863$	10098.5	+	5830.35i	0.398326	+	0.229974i	0.685762	−	0.727826i	$-0.259469\pi$
$863$	10098.5	+	5830.35i	0.398326	+	0.229974i	−0.287435	+	0.957800i	$0.592803\pi$
$864$		0			0
$865$	−13908.7	−	24090.6i	−0.546718	−	0.946944i
$866$		0			0
$867$	−16618.3	−	9443.87i	−0.650965	−	0.369931i
$868$		0			0
$869$		−	26947.5i		−	1.05193i
$870$		0			0
$871$	−9623.61	+	5556.19i	−0.374378	+	0.216147i
$872$		0			0
$873$	89.6830	+	6566.73i	0.00347687	+	0.254582i
$874$		0			0
$875$	−25911.1	−	7964.32i	−1.00109	−	0.307707i
$876$		0			0
$877$	−18913.9	+	32759.9i	−0.728254	+	1.26137i	0.229367	+	0.973340i	$0.426334\pi$
$877$	−18913.9	+	32759.9i	−0.728254	+	1.26137i	−0.957621	+	0.288033i	$0.906999\pi$
$878$		0			0
$879$	−42.8182	−	6270.73i	−0.00164303	−	0.240622i
$880$		0			0
$881$	26234.8			1.00326			0.501631	−	0.865082i	$-0.332734\pi$
$881$	26234.8			1.00326			0.501631	+	0.865082i	$0.332734\pi$
$882$		0			0
$883$	6803.07			0.259277			0.129639	−	0.991561i	$-0.458618\pi$
$883$	6803.07			0.259277			0.129639	+	0.991561i	$0.458618\pi$
$884$		0			0
$885$	274.030	+	40131.7i	0.0104084	+	1.52431i
$886$		0			0
$887$	−6115.02	+	10591.5i	−0.231480	+	0.400934i	−0.958244	−	0.285953i	$-0.907690\pi$
$887$	−6115.02	+	10591.5i	−0.231480	+	0.400934i	0.726764	+	0.686887i	$0.241023\pi$
$888$		0			0
$889$	17889.0	+	5498.56i	0.674890	+	0.207442i
$890$		0			0
$891$	19144.3	−	11761.5i	0.719818	−	0.442227i
$892$		0			0
$893$	−28751.7	+	16599.8i	−1.07742	+	0.622051i
$894$		0			0
$895$			34843.7i			1.30134i
$896$		0			0
$897$	−3898.66	−	2215.54i	−0.145120	−	0.0824690i
$898$		0			0
$899$	−11241.7	−	19471.2i	−0.417055	−	0.722360i
$900$		0			0
$901$	−40106.7	−	23155.6i	−1.48296	−	0.856187i
$902$		0			0
$903$	7892.20	−	26314.9i	0.290848	−	0.969774i
$904$		0			0
$905$	−3816.79	−	2203.62i	−0.140193	−	0.0809402i
$906$		0			0
$907$	−9920.70	−	17183.2i	−0.363188	−	0.629060i	0.625296	−	0.780388i	$-0.284978\pi$
$907$	−9920.70	−	17183.2i	−0.363188	−	0.629060i	−0.988484	+	0.151328i	$0.951645\pi$
$908$		0			0
$909$	−2929.66	−	4917.98i	−0.106898	−	0.179449i
$910$		0			0
$911$		−	48153.2i		−	1.75125i	−0.482995	−	0.875623i	$-0.660451\pi$
$911$		−	48153.2i		−	1.75125i	0.482995	−	0.875623i	$-0.339549\pi$
$912$		0			0
$913$	−10852.9	+	6265.92i	−0.393404	+	0.227132i
$914$		0			0
$915$	−8488.97	−	14474.2i	−0.306707	−	0.522953i
$916$		0			0
$917$	−4864.35	+	4522.46i	−0.175175	+	0.162862i
$918$		0			0
$919$	11559.8	−	20022.1i	0.414932	−	0.718683i	−0.580490	−	0.814268i	$-0.697139\pi$
$919$	11559.8	−	20022.1i	0.414932	−	0.718683i	0.995421	+	0.0955849i	$0.0304721\pi$
$920$		0			0
$921$	28623.5	−	195.449i	1.02408	−	0.00699267i
$922$		0			0
$923$	−1472.45			−0.0525095
$924$		0			0
$925$	5292.17			0.188114
$926$		0			0
$927$	−20216.7	−	11306.8i	−0.716291	−	0.400610i
$928$		0			0
$929$	16001.3	−	27715.1i	0.565109	−	0.978797i	−0.431931	−	0.901907i	$-0.642168\pi$
$929$	16001.3	−	27715.1i	0.565109	−	0.978797i	0.997040	−	0.0768903i	$-0.0244991\pi$
$930$		0			0
$931$	20516.8	+	42387.9i	0.722246	+	1.49217i
$932$		0			0
$933$	18955.2	−	11117.0i	0.665127	−	0.390091i
$934$		0			0
$935$	26089.5	−	15062.8i	0.912534	−	0.526852i
$936$		0			0
$937$		−	22855.6i		−	0.796863i	−0.917198	−	0.398431i	$-0.869555\pi$
$937$		−	22855.6i		−	0.796863i	0.917198	−	0.398431i	$-0.130445\pi$
$938$		0			0
$939$	13851.8	−	24374.9i	0.481403	−	0.847120i
$940$		0			0
$941$	−27703.7	−	47984.2i	−0.959739	−	1.66232i	−0.723129	−	0.690713i	$-0.757297\pi$
$941$	−27703.7	−	47984.2i	−0.959739	−	1.66232i	−0.236610	−	0.971605i	$-0.576036\pi$
$942$		0			0
$943$	4064.07	+	2346.39i	0.140344	+	0.0810277i
$944$		0			0
$945$	19063.6	+	19680.5i	0.656231	+	0.677467i
$946$		0			0
$947$	−5965.73	−	3444.31i	−0.204710	−	0.118189i	0.394141	−	0.919050i	$-0.371042\pi$
$947$	−5965.73	−	3444.31i	−0.204710	−	0.118189i	−0.598850	+	0.800861i	$0.704376\pi$
$948$		0			0
$949$	1619.79	+	2805.56i	0.0554064	+	0.0959667i
$950$		0			0
$951$	18289.4	−	32183.7i	0.623632	−	1.09740i
$952$		0			0
$953$		−	50638.7i		−	1.72125i	−0.509242	−	0.860623i	$-0.670074\pi$
$953$		−	50638.7i		−	1.72125i	0.509242	−	0.860623i	$-0.329926\pi$
$954$		0			0
$955$	9706.74	−	5604.19i	0.328903	−	0.189892i
$956$		0			0
$957$	−18555.4	+	10882.6i	−0.626761	+	0.367589i
$958$		0			0
$959$	24889.1	−	5706.79i	0.838070	−	0.192160i
$960$		0			0
$961$	−885.947	+	1534.51i	−0.0297387	+	0.0515090i
$962$		0			0
$963$	50621.2	+	28311.6i	1.69392	+	0.947380i
$964$		0			0
$965$	−41021.2			−1.36841
$966$		0			0
$967$	15112.3			0.502562			0.251281	−	0.967914i	$-0.419148\pi$
$967$	15112.3			0.502562			0.251281	+	0.967914i	$0.419148\pi$
$968$		0			0
$969$	−66124.4	+	451.515i	−2.19218	+	0.0149688i
$970$		0			0
$971$	14046.2	−	24328.7i	0.464226	−	0.804063i	−0.534941	−	0.844890i	$-0.679666\pi$
$971$	14046.2	−	24328.7i	0.464226	−	0.804063i	0.999166	+	0.0408272i	$0.0129993\pi$
$972$		0			0
$973$	−2581.86	+	8399.80i	−0.0850674	+	0.276758i
$974$		0			0
$975$	−719.725	−	1227.17i	−0.0236407	−	0.0403087i
$976$		0			0
$977$	34769.9	−	20074.4i	1.13857	−	0.657356i	0.192497	−	0.981298i	$-0.438341\pi$
$977$	34769.9	−	20074.4i	1.13857	−	0.657356i	0.946077	+	0.323941i	$0.105008\pi$
$978$		0			0
$979$		−	32429.3i		−	1.05868i
$980$		0			0
$981$	−17176.1	−	28833.3i	−0.559012	−	0.938405i
$982$		0			0
$983$	7276.53	+	12603.3i	0.236099	+	0.408935i	0.959591	−	0.281397i	$-0.0907977\pi$
$983$	7276.53	+	12603.3i	0.236099	+	0.408935i	−0.723493	+	0.690332i	$0.757464\pi$
$984$		0			0
$985$	29475.4	+	17017.6i	0.953465	+	0.550483i
$986$		0			0
$987$	15961.0	+	16934.2i	0.514735	+	0.546122i
$988$		0			0
$989$	10754.0	+	6208.81i	0.345760	+	0.199624i
$990$		0			0
$991$	−16550.0	−	28665.4i	−0.530503	−	0.918857i	−0.999367	−	0.0355870i	$-0.988670\pi$
$991$	−16550.0	−	28665.4i	−0.530503	−	0.918857i	0.468864	−	0.883270i	$-0.344663\pi$
$992$		0			0
$993$	8007.28	+	4550.39i	0.255894	+	0.145420i
$994$		0			0
$995$		−	29208.7i		−	0.930631i
$996$		0			0
$997$	43647.5	−	25199.9i	1.38649	−	0.800491i	0.393573	−	0.919293i	$-0.371239\pi$
$997$	43647.5	−	25199.9i	1.38649	−	0.800491i	0.992918	+	0.118803i	$0.0379055\pi$
$998$		0			0
$999$	−47135.7	−	25941.3i	−1.49280	−	0.821568i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.bc.e.257.5		16
3.2	odd	2	inner	336.4.bc.e.257.2		16
4.3	odd	2		42.4.f.a.5.6	yes	16
7.3	odd	6	inner	336.4.bc.e.17.2		16
12.11	even	2		42.4.f.a.5.4	✓	16
21.17	even	6	inner	336.4.bc.e.17.5		16
28.3	even	6		42.4.f.a.17.4	yes	16
28.11	odd	6		294.4.f.a.227.1		16
28.19	even	6		294.4.d.a.293.2		16
28.23	odd	6		294.4.d.a.293.7		16
28.27	even	2		294.4.f.a.215.7		16
84.11	even	6		294.4.f.a.227.7		16
84.23	even	6		294.4.d.a.293.10		16
84.47	odd	6		294.4.d.a.293.15		16
84.59	odd	6		42.4.f.a.17.6	yes	16
84.83	odd	2		294.4.f.a.215.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.f.a.5.4	✓	16	12.11	even	2
42.4.f.a.5.6	yes	16	4.3	odd	2
42.4.f.a.17.4	yes	16	28.3	even	6
42.4.f.a.17.6	yes	16	84.59	odd	6
294.4.d.a.293.2		16	28.19	even	6
294.4.d.a.293.7		16	28.23	odd	6
294.4.d.a.293.10		16	84.23	even	6
294.4.d.a.293.15		16	84.47	odd	6
294.4.f.a.215.1		16	84.83	odd	2
294.4.f.a.215.7		16	28.27	even	2
294.4.f.a.227.1		16	28.11	odd	6
294.4.f.a.227.7		16	84.11	even	6
336.4.bc.e.17.2		16	7.3	odd	6	inner
336.4.bc.e.17.5		16	21.17	even	6	inner
336.4.bc.e.257.2		16	3.2	odd	2	inner
336.4.bc.e.257.5		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.bc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0354799 + 5.19603i) q^{3} +(5.27257 - 9.13236i) q^{5} +(-17.7029 - 5.44135i) q^{7} +(-26.9975 + 0.368709i) q^{9} +(26.6918 - 15.4105i) q^{11} +19.8400i q^{13} +(47.6391 + 27.0724i) q^{15} +(46.3453 + 80.2724i) q^{17} +(118.901 + 68.6474i) q^{19} +(27.6454 - 92.1777i) q^{21} +(37.6697 + 21.7486i) q^{23} +(6.89995 + 11.9511i) q^{25} +(-2.87369 - 140.267i) q^{27} +134.318i q^{29} +(-144.963 + 83.6945i) q^{31} +(81.0206 + 138.145i) q^{33} +(-143.032 + 132.979i) q^{35} +(191.747 - 332.115i) q^{37} +(-103.089 + 0.703923i) q^{39} +107.887 q^{41} +285.480 q^{43} +(-138.979 + 248.495i) q^{45} +(-120.906 + 209.416i) q^{47} +(283.783 + 192.655i) q^{49} +(-415.453 + 243.660i) q^{51} +(-432.694 + 249.816i) q^{53} -325.013i q^{55} +(-352.476 + 620.248i) q^{57} +(366.212 + 634.299i) q^{59} +(-265.207 - 153.117i) q^{61} +(479.939 + 140.376i) q^{63} +(181.187 + 104.608i) q^{65} +(280.049 + 485.060i) q^{67} +(-111.670 + 196.505i) q^{69} +74.2161i q^{71} +(141.409 - 81.6426i) q^{73} +(-61.8533 + 36.2764i) q^{75} +(-556.376 + 127.571i) q^{77} +(437.160 - 757.183i) q^{79} +(728.728 - 19.9084i) q^{81} -406.600 q^{83} +977.435 q^{85} +(-697.922 + 4.76560i) q^{87} +(526.091 - 911.216i) q^{89} +(107.957 - 351.226i) q^{91} +(-440.023 - 750.264i) q^{93} +(1253.83 - 723.897i) q^{95} -243.235i q^{97} +(-714.930 + 425.887i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{7} + 18 q^{9} + 342 q^{19} - 450 q^{21} - 194 q^{25} - 804 q^{31} + 1332 q^{33} - 962 q^{37} - 594 q^{39} - 1732 q^{43} - 2394 q^{45} + 820 q^{49} - 1638 q^{51} - 2664 q^{57} - 4620 q^{61}+ \cdots + 4284 q^{99}+O(q^{100}) \) 16 * q - 80 * q^7 + 18 * q^9 + 342 * q^19 - 450 * q^21 - 194 * q^25 - 804 * q^31 + 1332 * q^33 - 962 * q^37 - 594 * q^39 - 1732 * q^43 - 2394 * q^45 + 820 * q^49 - 1638 * q^51 - 2664 * q^57 - 4620 * q^61 + 2016 * q^63 + 706 * q^67 + 3294 * q^73 - 6174 * q^75 + 2656 * q^79 + 126 * q^81 + 5232 * q^85 - 1026 * q^87 - 4098 * q^91 + 2016 * q^93 + 4284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more