LMFDB - Embedded newform 588.4.i.l.373.1

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			373.1
Root			$1.44795 + 2.50793i$ of defining polynomial
Character	$\chi$	$=$	588.373
Dual form			588.4.i.l.361.1

$q$-expansion

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{3} +(-8.32734 + 14.4234i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(1.50000 + 2.59808i) q^{3} +(-8.32734 + 14.4234i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(35.8800 + 62.1460i) q^{11} +65.3878 q^{13} -49.9641 q^{15} +(45.2158 + 78.3160i) q^{17} +(81.9092 - 141.871i) q^{19} +(-39.6144 + 68.6142i) q^{23} +(-76.1893 - 131.964i) q^{25} -27.0000 q^{27} -43.2995 q^{29} +(67.8179 + 117.464i) q^{31} +(-107.640 + 186.438i) q^{33} +(-135.370 + 234.468i) q^{37} +(98.0817 + 169.882i) q^{39} -152.241 q^{41} -177.641 q^{43} +(-74.9461 - 129.810i) q^{45} +(-22.8166 + 39.5194i) q^{47} +(-135.647 + 234.948i) q^{51} +(79.2188 + 137.211i) q^{53} -1195.14 q^{55} +491.455 q^{57} +(-195.884 - 339.282i) q^{59} +(275.647 - 477.435i) q^{61} +(-544.506 + 943.113i) q^{65} +(-229.315 - 397.185i) q^{67} -237.687 q^{69} -486.786 q^{71} +(287.446 + 497.871i) q^{73} +(228.568 - 395.891i) q^{75} +(334.160 - 578.782i) q^{79} +(-40.5000 - 70.1481i) q^{81} -76.2450 q^{83} -1506.11 q^{85} +(-64.9493 - 112.495i) q^{87} +(683.401 - 1183.69i) q^{89} +(-203.454 + 352.392i) q^{93} +(1364.17 + 2362.82i) q^{95} -242.655 q^{97} -645.840 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.32734	+	14.4234i	−0.744820	+	1.29007i	0.205459	+	0.978666i	$0.434131\pi$
$5$	−8.32734	+	14.4234i	−0.744820	+	1.29007i	−0.950279	+	0.311401i	$0.899202\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	35.8800	+	62.1460i	0.983475	+	1.70343i	0.648526	+	0.761192i	$0.275386\pi$
$11$	35.8800	+	62.1460i	0.983475	+	1.70343i	0.334949	+	0.942236i	$0.391281\pi$
$12$		0			0
$13$	65.3878			1.39502			0.697512	−	0.716573i	$-0.254291\pi$
$13$	65.3878			1.39502			0.697512	+	0.716573i	$0.254291\pi$
$14$		0			0
$15$	−49.9641			−0.860044
$16$		0			0
$17$	45.2158	+	78.3160i	0.645085	+	1.11732i	0.984282	+	0.176604i	$0.0565112\pi$
$17$	45.2158	+	78.3160i	0.645085	+	1.11732i	−0.339197	+	0.940715i	$0.610155\pi$
$18$		0			0
$19$	81.9092	−	141.871i	0.989014	−	1.71302i	0.366488	−	0.930423i	$-0.380560\pi$
$19$	81.9092	−	141.871i	0.989014	−	1.71302i	0.622526	−	0.782599i	$-0.286107\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−39.6144	+	68.6142i	−0.359138	+	0.622045i	−0.987817	−	0.155619i	$-0.950263\pi$
$23$	−39.6144	+	68.6142i	−0.359138	+	0.622045i	0.628679	+	0.777665i	$0.283596\pi$
$24$		0			0
$25$	−76.1893	−	131.964i	−0.609514	−	1.05571i
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−43.2995			−0.277259			−0.138630	−	0.990344i	$-0.544270\pi$
$29$	−43.2995			−0.277259			−0.138630	+	0.990344i	$0.544270\pi$
$30$		0			0
$31$	67.8179	+	117.464i	0.392918	+	0.680554i	0.992833	−	0.119510i	$-0.0381323\pi$
$31$	67.8179	+	117.464i	0.392918	+	0.680554i	−0.599915	+	0.800064i	$0.704799\pi$
$32$		0			0
$33$	−107.640	+	186.438i	−0.567810	+	0.983475i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−135.370	+	234.468i	−0.601479	+	1.04179i	0.391118	+	0.920340i	$0.372088\pi$
$37$	−135.370	+	234.468i	−0.601479	+	1.04179i	−0.992597	+	0.121452i	$0.961245\pi$
$38$		0			0
$39$	98.0817	+	169.882i	0.402709	+	0.697512i
$40$		0			0
$41$	−152.241			−0.579902			−0.289951	−	0.957041i	$-0.593639\pi$
$41$	−152.241			−0.579902			−0.289951	+	0.957041i	$0.593639\pi$
$42$		0			0
$43$	−177.641			−0.630001			−0.315001	−	0.949091i	$-0.602005\pi$
$43$	−177.641			−0.630001			−0.315001	+	0.949091i	$0.602005\pi$
$44$		0			0
$45$	−74.9461	−	129.810i	−0.248273	−	0.430022i
$46$		0			0
$47$	−22.8166	+	39.5194i	−0.0708114	+	0.122649i	−0.899257	−	0.437420i	$-0.855892\pi$
$47$	−22.8166	+	39.5194i	−0.0708114	+	0.122649i	0.828446	+	0.560069i	$0.189226\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−135.647	+	234.948i	−0.372440	+	0.645085i
$52$		0			0
$53$	79.2188	+	137.211i	0.205312	+	0.355611i	0.950232	−	0.311543i	$-0.100846\pi$
$53$	79.2188	+	137.211i	0.205312	+	0.355611i	−0.744920	+	0.667154i	$0.767512\pi$
$54$		0			0
$55$	−1195.14			−2.93005
$56$		0			0
$57$	491.455			1.14201
$58$		0			0
$59$	−195.884	−	339.282i	−0.432237	−	0.748656i	0.564829	−	0.825208i	$-0.308942\pi$
$59$	−195.884	−	339.282i	−0.432237	−	0.748656i	−0.997066	+	0.0765519i	$0.975609\pi$
$60$		0			0
$61$	275.647	−	477.435i	0.578574	−	1.00212i	−0.417069	−	0.908875i	$-0.636943\pi$
$61$	275.647	−	477.435i	0.578574	−	1.00212i	0.995643	−	0.0932450i	$-0.0297240\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−544.506	+	943.113i	−1.03904	+	1.79967i
$66$		0			0
$67$	−229.315	−	397.185i	−0.418139	−	0.724237i	0.577614	−	0.816310i	$-0.303984\pi$
$67$	−229.315	−	397.185i	−0.418139	−	0.724237i	−0.995752	+	0.0920729i	$0.970651\pi$
$68$		0			0
$69$	−237.687			−0.414697
$70$		0			0
$71$	−486.786			−0.813674			−0.406837	−	0.913501i	$-0.633368\pi$
$71$	−486.786			−0.813674			−0.406837	+	0.913501i	$0.633368\pi$
$72$		0			0
$73$	287.446	+	497.871i	0.460863	+	0.798238i	0.999004	−	0.0446164i	$-0.0142066\pi$
$73$	287.446	+	497.871i	0.460863	+	0.798238i	−0.538141	+	0.842855i	$0.680873\pi$
$74$		0			0
$75$	228.568	−	395.891i	0.351903	−	0.609514i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	334.160	−	578.782i	0.475897	−	0.824279i	−0.523721	−	0.851890i	$-0.675457\pi$
$79$	334.160	−	578.782i	0.475897	−	0.824279i	0.999619	+	0.0276111i	$0.00878999\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−76.2450			−0.100831			−0.0504155	−	0.998728i	$-0.516055\pi$
$83$	−76.2450			−0.100831			−0.0504155	+	0.998728i	$0.516055\pi$
$84$		0			0
$85$	−1506.11			−1.92189
$86$		0			0
$87$	−64.9493	−	112.495i	−0.0800379	−	0.138630i
$88$		0			0
$89$	683.401	−	1183.69i	0.813937	−	1.40978i	−0.0961518	−	0.995367i	$-0.530653\pi$
$89$	683.401	−	1183.69i	0.813937	−	1.40978i	0.910089	−	0.414413i	$-0.136013\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−203.454	+	352.392i	−0.226851	+	0.392918i
$94$		0			0
$95$	1364.17	+	2362.82i	1.47327	+	2.55179i
$96$		0			0
$97$	−242.655			−0.253999			−0.127000	−	0.991903i	$-0.540535\pi$
$97$	−242.655			−0.253999			−0.127000	+	0.991903i	$0.540535\pi$
$98$		0			0
$99$	−645.840			−0.655650
$100$		0			0
$101$	347.371	+	601.665i	0.342225	+	0.592751i	0.984846	−	0.173433i	$-0.0554861\pi$
$101$	347.371	+	601.665i	0.342225	+	0.592751i	−0.642620	+	0.766185i	$0.722153\pi$
$102$		0			0
$103$	−87.1089	+	150.877i	−0.0833310	+	0.144334i	−0.904679	−	0.426094i	$-0.859889\pi$
$103$	−87.1089	+	150.877i	−0.0833310	+	0.144334i	0.821348	+	0.570428i	$0.193223\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−23.8025	+	41.2272i	−0.0215054	+	0.0372485i	−0.876578	−	0.481260i	$-0.840179\pi$
$107$	−23.8025	+	41.2272i	−0.0215054	+	0.0372485i	0.855072	+	0.518509i	$0.173513\pi$
$108$		0			0
$109$	816.549	+	1414.30i	0.717534	+	1.24281i	0.961974	+	0.273141i	$0.0880626\pi$
$109$	816.549	+	1414.30i	0.717534	+	1.24281i	−0.244440	+	0.969664i	$0.578604\pi$
$110$		0			0
$111$	−812.221			−0.694528
$112$		0			0
$113$	−1127.29			−0.938464			−0.469232	−	0.883075i	$-0.655469\pi$
$113$	−1127.29			−0.938464			−0.469232	+	0.883075i	$0.655469\pi$
$114$		0			0
$115$	−659.766	−	1142.75i	−0.534987	−	0.926624i
$116$		0			0
$117$	−294.245	+	509.647i	−0.232504	+	0.402709i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1909.25	+	3306.91i	−1.43445	+	2.48453i
$122$		0			0
$123$	−228.361	−	395.533i	−0.167403	−	0.289951i
$124$		0			0
$125$	455.982			0.326274
$126$		0			0
$127$	2398.66			1.67596			0.837979	−	0.545702i	$-0.183737\pi$
$127$	2398.66			1.67596			0.837979	+	0.545702i	$0.183737\pi$
$128$		0			0
$129$	−266.462	−	461.526i	−0.181866	−	0.315001i
$130$		0			0
$131$	774.482	−	1341.44i	0.516541	−	0.894675i	−0.483275	−	0.875469i	$-0.660553\pi$
$131$	774.482	−	1341.44i	0.516541	−	0.894675i	0.999816	−	0.0192062i	$-0.00611389\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	224.838	−	389.431i	0.143341	−	0.248273i
$136$		0			0
$137$	−761.634	−	1319.19i	−0.474969	−	0.822671i	0.524620	−	0.851337i	$-0.324207\pi$
$137$	−761.634	−	1319.19i	−0.474969	−	0.822671i	−0.999589	+	0.0286657i	$0.990874\pi$
$138$		0			0
$139$	−551.445			−0.336496			−0.168248	−	0.985745i	$-0.553811\pi$
$139$	−551.445			−0.336496			−0.168248	+	0.985745i	$0.553811\pi$
$140$		0			0
$141$	−136.899			−0.0817660
$142$		0			0
$143$	2346.11	+	4063.59i	1.37197	+	2.37632i
$144$		0			0
$145$	360.570	−	624.526i	0.206508	−	0.357683i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1200.12	−	2078.67i	0.659851	−	1.14290i	−0.320803	−	0.947146i	$-0.603953\pi$
$149$	1200.12	−	2078.67i	0.659851	−	1.14290i	0.980654	−	0.195749i	$-0.0627139\pi$
$150$		0			0
$151$	−310.780	−	538.287i	−0.167490	−	0.290100i	0.770047	−	0.637987i	$-0.220233\pi$
$151$	−310.780	−	538.287i	−0.167490	−	0.290100i	−0.937537	+	0.347887i	$0.886899\pi$
$152$		0			0
$153$	−813.884			−0.430056
$154$		0			0
$155$	−2258.97			−1.17061
$156$		0			0
$157$	−1345.79	−	2330.97i	−0.684112	−	1.18492i	−0.973715	−	0.227770i	$-0.926857\pi$
$157$	−1345.79	−	2330.97i	−0.684112	−	1.18492i	0.289603	−	0.957147i	$-0.406477\pi$
$158$		0			0
$159$	−237.657	+	411.633i	−0.118537	+	0.205312i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−528.653	+	915.654i	−0.254032	+	0.439997i	−0.964632	−	0.263599i	$-0.915090\pi$
$163$	−528.653	+	915.654i	−0.254032	+	0.439997i	0.710600	+	0.703596i	$0.248424\pi$
$164$		0			0
$165$	−1792.71	−	3105.06i	−0.845832	−	1.46502i
$166$		0			0
$167$	243.840			0.112987			0.0564937	−	0.998403i	$-0.482008\pi$
$167$	243.840			0.112987			0.0564937	+	0.998403i	$0.482008\pi$
$168$		0			0
$169$	2078.56			0.946090
$170$		0			0
$171$	737.183	+	1276.84i	0.329671	+	0.571007i
$172$		0			0
$173$	−843.632	+	1461.21i	−0.370753	+	0.642162i	−0.989682	−	0.143285i	$-0.954234\pi$
$173$	−843.632	+	1461.21i	−0.370753	+	0.642162i	0.618929	+	0.785447i	$0.287567\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	587.653	−	1017.85i	0.249552	−	0.432237i
$178$		0			0
$179$	179.570	+	311.024i	0.0749814	+	0.129872i	0.901078	−	0.433657i	$-0.142777\pi$
$179$	179.570	+	311.024i	0.0749814	+	0.129872i	−0.826097	+	0.563528i	$0.809444\pi$
$180$		0			0
$181$	−2982.58			−1.22483			−0.612413	−	0.790538i	$-0.709801\pi$
$181$	−2982.58			−1.22483			−0.612413	+	0.790538i	$0.709801\pi$
$182$		0			0
$183$	1653.88			0.668080
$184$		0			0
$185$	−2254.55	−	3904.99i	−0.895988	−	1.55190i
$186$		0			0
$187$	−3244.68	+	5619.96i	−1.26885	+	2.19771i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1745.39	+	3023.10i	−0.661213	+	1.14525i	0.319084	+	0.947726i	$0.396625\pi$
$191$	−1745.39	+	3023.10i	−0.661213	+	1.14525i	−0.980297	+	0.197528i	$0.936709\pi$
$192$		0			0
$193$	−802.114	−	1389.30i	−0.299158	−	0.518156i	0.676786	−	0.736180i	$-0.263372\pi$
$193$	−802.114	−	1389.30i	−0.299158	−	0.518156i	−0.975943	+	0.218024i	$0.930039\pi$
$194$		0			0
$195$	−3267.04			−1.19978
$196$		0			0
$197$	−455.935			−0.164893			−0.0824467	−	0.996595i	$-0.526273\pi$
$197$	−455.935			−0.164893			−0.0824467	+	0.996595i	$0.526273\pi$
$198$		0			0
$199$	1042.08	+	1804.94i	0.371212	+	0.642958i	0.989752	−	0.142795i	$-0.0456091\pi$
$199$	1042.08	+	1804.94i	0.371212	+	0.642958i	−0.618541	+	0.785753i	$0.712276\pi$
$200$		0			0
$201$	687.945	−	1191.56i	0.241412	−	0.418139i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1267.76	−	2195.83i	0.431923	−	0.748113i
$206$		0			0
$207$	−356.530	−	617.528i	−0.119713	−	0.207348i
$208$		0			0
$209$	11755.6			3.89068
$210$		0			0
$211$	2568.06			0.837880			0.418940	−	0.908014i	$-0.362402\pi$
$211$	2568.06			0.837880			0.418940	+	0.908014i	$0.362402\pi$
$212$		0			0
$213$	−730.178	−	1264.71i	−0.234887	−	0.406837i
$214$		0			0
$215$	1479.28	−	2562.19i	0.469238	−	0.812743i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−862.338	+	1493.61i	−0.266079	+	0.460863i
$220$		0			0
$221$	2956.56	+	5120.91i	0.899908	+	1.55869i
$222$		0			0
$223$	4659.55			1.39922			0.699612	−	0.714523i	$-0.253356\pi$
$223$	4659.55			1.39922			0.699612	+	0.714523i	$0.253356\pi$
$224$		0			0
$225$	1371.41			0.406343
$226$		0			0
$227$	2414.39	+	4181.84i	0.705940	+	1.22272i	0.966351	+	0.257227i	$0.0828087\pi$
$227$	2414.39	+	4181.84i	0.705940	+	1.22272i	−0.260411	+	0.965498i	$0.583858\pi$
$228$		0			0
$229$	−1611.29	+	2790.84i	−0.464966	+	0.805345i	−0.999200	−	0.0399917i	$-0.987267\pi$
$229$	−1611.29	+	2790.84i	−0.464966	+	0.805345i	0.534234	+	0.845337i	$0.320600\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2851.21	−	4938.44i	0.801670	−	1.38853i	−0.116847	−	0.993150i	$-0.537279\pi$
$233$	2851.21	−	4938.44i	0.801670	−	1.38853i	0.918517	−	0.395383i	$-0.129388\pi$
$234$		0			0
$235$	−380.003	−	658.184i	−0.105484	−	0.182703i
$236$		0			0
$237$	2004.96			0.549519
$238$		0			0
$239$	1994.47			0.539796			0.269898	−	0.962889i	$-0.413010\pi$
$239$	1994.47			0.539796			0.269898	+	0.962889i	$0.413010\pi$
$240$		0			0
$241$	−710.443	−	1230.52i	−0.189891	−	0.328900i	0.755323	−	0.655353i	$-0.227480\pi$
$241$	−710.443	−	1230.52i	−0.189891	−	0.328900i	−0.945214	+	0.326452i	$0.894147\pi$
$242$		0			0
$243$	121.500	−	210.444i	0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5355.86	−	9276.62i	1.37970	−	2.38971i
$248$		0			0
$249$	−114.367	−	198.090i	−0.0291074	−	0.0504155i
$250$		0			0
$251$	3855.44			0.969534			0.484767	−	0.874643i	$-0.338904\pi$
$251$	3855.44			0.969534			0.484767	+	0.874643i	$0.338904\pi$
$252$		0			0
$253$	−5685.46			−1.41281
$254$		0			0
$255$	−2259.16	−	3912.99i	−0.554801	−	0.960944i
$256$		0			0
$257$	2026.64	−	3510.24i	0.491900	−	0.851996i	−0.508056	−	0.861324i	$-0.669636\pi$
$257$	2026.64	−	3510.24i	0.491900	−	0.851996i	0.999956	+	0.00932793i	$0.00296922\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	194.848	−	337.486i	0.0462099	−	0.0800379i
$262$		0			0
$263$	250.393	+	433.694i	0.0587069	+	0.101683i	0.893885	−	0.448296i	$-0.147969\pi$
$263$	250.393	+	433.694i	0.0587069	+	0.101683i	−0.835178	+	0.549979i	$0.814636\pi$
$264$		0			0
$265$	−2638.73			−0.611683
$266$		0			0
$267$	4100.41			0.939853
$268$		0			0
$269$	1640.87	+	2842.07i	0.371917	+	0.644179i	0.989860	−	0.142043i	$-0.0453672\pi$
$269$	1640.87	+	2842.07i	0.371917	+	0.644179i	−0.617943	+	0.786223i	$0.712034\pi$
$270$		0			0
$271$	1662.18	−	2878.98i	0.372584	−	0.645334i	−0.617378	−	0.786666i	$-0.711805\pi$
$271$	1662.18	−	2878.98i	0.372584	−	0.645334i	0.989962	+	0.141332i	$0.0451386\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5467.34	−	9469.71i	1.19888	−	2.07653i
$276$		0			0
$277$	−4309.87	−	7464.92i	−0.934856	−	1.61922i	−0.774890	−	0.632096i	$-0.782195\pi$
$277$	−4309.87	−	7464.92i	−0.934856	−	1.61922i	−0.159966	−	0.987122i	$-0.551139\pi$
$278$		0			0
$279$	−1220.72			−0.261945
$280$		0			0
$281$	−7710.80			−1.63697			−0.818484	−	0.574529i	$-0.805185\pi$
$281$	−7710.80			−1.63697			−0.818484	+	0.574529i	$0.805185\pi$
$282$		0			0
$283$	−69.6354	−	120.612i	−0.0146268	−	0.0253344i	0.858619	−	0.512614i	$-0.171323\pi$
$283$	−69.6354	−	120.612i	−0.0146268	−	0.0253344i	−0.873246	+	0.487279i	$0.837989\pi$
$284$		0			0
$285$	−4092.52	+	7088.45i	−0.850596	+	1.47327i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1632.43	+	2827.46i	−0.332268	+	0.575506i
$290$		0			0
$291$	−363.983	−	630.437i	−0.0733232	−	0.127000i
$292$		0			0
$293$	4437.09			0.884702			0.442351	−	0.896842i	$-0.354145\pi$
$293$	4437.09			0.884702			0.442351	+	0.896842i	$0.354145\pi$
$294$		0			0
$295$	6524.79			1.28776
$296$		0			0
$297$	−968.760	−	1677.94i	−0.189270	−	0.327825i
$298$		0			0
$299$	−2590.30	+	4486.53i	−0.501006	+	0.867768i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1042.11	+	1804.99i	−0.197584	+	0.342225i
$304$		0			0
$305$	4590.82	+	7951.53i	0.861867	+	1.49280i
$306$		0			0
$307$	299.480			0.0556749			0.0278375	−	0.999612i	$-0.491138\pi$
$307$	299.480			0.0556749			0.0278375	+	0.999612i	$0.491138\pi$
$308$		0			0
$309$	−522.654			−0.0962224
$310$		0			0
$311$	456.972	+	791.498i	0.0833199	+	0.144314i	0.904674	−	0.426104i	$-0.140114\pi$
$311$	456.972	+	791.498i	0.0833199	+	0.144314i	−0.821354	+	0.570419i	$0.806781\pi$
$312$		0			0
$313$	1943.69	−	3366.57i	0.351002	−	0.607954i	−0.635423	−	0.772164i	$-0.719174\pi$
$313$	1943.69	−	3366.57i	0.351002	−	0.607954i	0.986425	+	0.164210i	$0.0525075\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1439.65	+	2493.55i	−0.255075	+	0.441803i	−0.964916	−	0.262559i	$-0.915434\pi$
$317$	−1439.65	+	2493.55i	−0.255075	+	0.441803i	0.709841	+	0.704362i	$0.248767\pi$
$318$		0			0
$319$	−1553.59	−	2690.89i	−0.272678	−	0.472291i
$320$		0			0
$321$	−142.815			−0.0248323
$322$		0			0
$323$	14814.4			2.55199
$324$		0			0
$325$	−4981.85	−	8628.81i	−0.850287	−	1.47274i
$326$		0			0
$327$	−2449.65	+	4242.91i	−0.414269	+	0.717534i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1405.39	−	2434.21i	0.233376	−	0.404219i	−0.725424	−	0.688303i	$-0.758356\pi$
$331$	1405.39	−	2434.21i	0.233376	−	0.404219i	0.958799	+	0.284084i	$0.0916894\pi$
$332$		0			0
$333$	−1218.33	−	2110.21i	−0.200493	−	0.347264i
$334$		0			0
$335$	7638.34			1.24575
$336$		0			0
$337$	2960.90			0.478607			0.239303	−	0.970945i	$-0.423081\pi$
$337$	2960.90			0.478607			0.239303	+	0.970945i	$0.423081\pi$
$338$		0			0
$339$	−1690.93	−	2928.78i	−0.270911	−	0.469232i
$340$		0			0
$341$	−4866.61	+	8429.22i	−0.772850	+	1.33862i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1979.30	−	3428.24i	0.308875	−	0.534987i
$346$		0			0
$347$	2142.26	+	3710.50i	0.331419	+	0.574034i	0.982790	−	0.184725i	$-0.0591393\pi$
$347$	2142.26	+	3710.50i	0.331419	+	0.574034i	−0.651371	+	0.758759i	$0.725806\pi$
$348$		0			0
$349$	−3295.36			−0.505434			−0.252717	−	0.967540i	$-0.581324\pi$
$349$	−3295.36			−0.505434			−0.252717	+	0.967540i	$0.581324\pi$
$350$		0			0
$351$	−1765.47			−0.268472
$352$		0			0
$353$	−5314.83	−	9205.56i	−0.801360	−	1.38800i	−0.918721	−	0.394906i	$-0.870777\pi$
$353$	−5314.83	−	9205.56i	−0.801360	−	1.38800i	0.117362	−	0.993089i	$-0.462556\pi$
$354$		0			0
$355$	4053.63	−	7021.09i	0.606040	−	1.04969i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1592.51	+	2758.30i	−0.234121	+	0.405509i	−0.959017	−	0.283349i	$-0.908554\pi$
$359$	−1592.51	+	2758.30i	−0.234121	+	0.405509i	0.724896	+	0.688858i	$0.241888\pi$
$360$		0			0
$361$	−9988.74	−	17301.0i	−1.45630	−	2.52238i
$362$		0			0
$363$	−11455.5			−1.65636
$364$		0			0
$365$	−9574.65			−1.37304
$366$		0			0
$367$	3801.52	+	6584.42i	0.540702	+	0.936523i	0.998864	+	0.0476544i	$0.0151746\pi$
$367$	3801.52	+	6584.42i	0.540702	+	0.936523i	−0.458162	+	0.888869i	$0.651492\pi$
$368$		0			0
$369$	685.083	−	1186.60i	0.0966504	−	0.167403i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−755.978	+	1309.39i	−0.104941	+	0.181763i	−0.913714	−	0.406358i	$-0.866799\pi$
$373$	−755.978	+	1309.39i	−0.104941	+	0.181763i	0.808773	+	0.588121i	$0.200132\pi$
$374$		0			0
$375$	683.972	+	1184.67i	0.0941871	+	0.163137i
$376$		0			0
$377$	−2831.26			−0.386783
$378$		0			0
$379$	8848.36			1.19923			0.599617	−	0.800287i	$-0.295320\pi$
$379$	8848.36			1.19923			0.599617	+	0.800287i	$0.295320\pi$
$380$		0			0
$381$	3597.99	+	6231.90i	0.483807	+	0.837979i
$382$		0			0
$383$	2067.05	−	3580.24i	0.275774	−	0.477654i	−0.694556	−	0.719438i	$-0.744399\pi$
$383$	2067.05	−	3580.24i	0.275774	−	0.477654i	0.970330	+	0.241784i	$0.0777326\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	799.386	−	1384.58i	0.105000	−	0.181866i
$388$		0			0
$389$	−3815.30	−	6608.30i	−0.497284	−	0.861322i	0.502711	−	0.864455i	$-0.332336\pi$
$389$	−3815.30	−	6608.30i	−0.497284	−	0.861322i	−0.999995	+	0.00313295i	$0.999003\pi$
$390$		0			0
$391$	−7164.79			−0.926698
$392$		0			0
$393$	4646.89			0.596450
$394$		0			0
$395$	5565.32	+	9639.43i	0.708916	+	1.22788i
$396$		0			0
$397$	−5809.25	+	10061.9i	−0.734403	+	1.27202i	0.220582	+	0.975368i	$0.429204\pi$
$397$	−5809.25	+	10061.9i	−0.734403	+	1.27202i	−0.954985	+	0.296654i	$0.904129\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1523.51	+	2638.79i	−0.189726	+	0.328616i	−0.945159	−	0.326611i	$-0.894093\pi$
$401$	−1523.51	+	2638.79i	−0.189726	+	0.328616i	0.755433	+	0.655226i	$0.227427\pi$
$402$		0			0
$403$	4434.46	+	7680.71i	0.548130	+	0.949388i
$404$		0			0
$405$	1349.03			0.165516
$406$		0			0
$407$	−19428.3			−2.36616
$408$		0			0
$409$	5791.42	+	10031.0i	0.700164	+	1.21272i	0.968408	+	0.249369i	$0.0802233\pi$
$409$	5791.42	+	10031.0i	0.700164	+	1.21272i	−0.268244	+	0.963351i	$0.586443\pi$
$410$		0			0
$411$	2284.90	−	3957.57i	0.274224	−	0.474969i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	634.918	−	1099.71i	0.0751010	−	0.130079i
$416$		0			0
$417$	−827.167	−	1432.70i	−0.0971380	−	0.168248i
$418$		0			0
$419$	1318.83			0.153768			0.0768841	−	0.997040i	$-0.475503\pi$
$419$	1318.83			0.153768			0.0768841	+	0.997040i	$0.475503\pi$
$420$		0			0
$421$	−9733.56			−1.12680			−0.563402	−	0.826183i	$-0.690508\pi$
$421$	−9733.56			−1.12680			−0.563402	+	0.826183i	$0.690508\pi$
$422$		0			0
$423$	−205.349	−	355.675i	−0.0236038	−	0.0408830i
$424$		0			0
$425$	6889.92	−	11933.7i	0.786377	−	1.36204i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−7038.34	+	12190.8i	−0.792108	+	1.37197i
$430$		0			0
$431$	7214.47	+	12495.8i	0.806284	+	1.39653i	0.915421	+	0.402499i	$0.131858\pi$
$431$	7214.47	+	12495.8i	0.806284	+	1.39653i	−0.109136	+	0.994027i	$0.534808\pi$
$432$		0			0
$433$	16620.3			1.84463			0.922313	−	0.386444i	$-0.126297\pi$
$433$	16620.3			1.84463			0.922313	+	0.386444i	$0.126297\pi$
$434$		0			0
$435$	2163.42			0.238455
$436$		0			0
$437$	6489.57	+	11240.3i	0.710385	+	1.23042i
$438$		0			0
$439$	−257.783	+	446.493i	−0.0280257	+	0.0485420i	−0.879698	−	0.475533i	$-0.842255\pi$
$439$	−257.783	+	446.493i	−0.0280257	+	0.0485420i	0.851672	+	0.524075i	$0.175589\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4488.05	+	7773.52i	−0.481340	+	0.833705i	−0.999771	−	0.0214146i	$-0.993183\pi$
$443$	−4488.05	+	7773.52i	−0.481340	+	0.833705i	0.518431	+	0.855120i	$0.326516\pi$
$444$		0			0
$445$	11381.8	+	19713.9i	1.21247	+	2.10007i
$446$		0			0
$447$	7200.73			0.761930
$448$		0			0
$449$	2106.83			0.221442			0.110721	−	0.993852i	$-0.464684\pi$
$449$	2106.83			0.221442			0.110721	+	0.993852i	$0.464684\pi$
$450$		0			0
$451$	−5462.39	−	9461.14i	−0.570319	−	0.987822i
$452$		0			0
$453$	932.340	−	1614.86i	0.0967001	−	0.167490i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7863.08	+	13619.3i	−0.804857	+	1.39405i	0.111531	+	0.993761i	$0.464425\pi$
$457$	−7863.08	+	13619.3i	−0.804857	+	1.39405i	−0.916388	+	0.400292i	$0.868909\pi$
$458$		0			0
$459$	−1220.83	−	2114.53i	−0.124147	−	0.215028i
$460$		0			0
$461$	7921.16			0.800272			0.400136	−	0.916456i	$-0.368963\pi$
$461$	7921.16			0.800272			0.400136	+	0.916456i	$0.368963\pi$
$462$		0			0
$463$	−5821.20			−0.584306			−0.292153	−	0.956372i	$-0.594372\pi$
$463$	−5821.20			−0.584306			−0.292153	+	0.956372i	$0.594372\pi$
$464$		0			0
$465$	−3388.46	−	5868.98i	−0.337927	−	0.585306i
$466$		0			0
$467$	5893.94	−	10208.6i	0.584024	−	1.01156i	−0.410973	−	0.911648i	$-0.634811\pi$
$467$	5893.94	−	10208.6i	0.584024	−	1.01156i	0.994996	−	0.0999112i	$-0.0318559\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4037.36	−	6992.92i	0.394972	−	0.684112i
$472$		0			0
$473$	−6373.77	−	11039.7i	−0.619590	−	1.07316i
$474$		0			0
$475$	−24962.4			−2.41127
$476$		0			0
$477$	−1425.94			−0.136875
$478$		0			0
$479$	−2657.21	−	4602.43i	−0.253468	−	0.439020i	0.711010	−	0.703182i	$-0.248238\pi$
$479$	−2657.21	−	4602.43i	−0.253468	−	0.439020i	−0.964478	+	0.264162i	$0.914905\pi$
$480$		0			0
$481$	−8851.56	+	15331.3i	−0.839078	+	1.45332i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2020.67	−	3499.91i	0.189184	−	0.327676i
$486$		0			0
$487$	9331.34	+	16162.4i	0.868261	+	1.50387i	0.863772	+	0.503883i	$0.168096\pi$
$487$	9331.34	+	16162.4i	0.868261	+	1.50387i	0.00448933	+	0.999990i	$0.498571\pi$
$488$		0			0
$489$	−3171.92			−0.293331
$490$		0			0
$491$	13519.1			1.24259			0.621293	−	0.783578i	$-0.286608\pi$
$491$	13519.1			1.24259			0.621293	+	0.783578i	$0.286608\pi$
$492$		0			0
$493$	−1957.82	−	3391.05i	−0.178856	−	0.309787i
$494$		0			0
$495$	5378.13	−	9315.19i	0.488341	−	0.845832i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6916.17	+	11979.2i	−0.620462	+	1.07467i	0.368938	+	0.929454i	$0.379721\pi$
$499$	−6916.17	+	11979.2i	−0.620462	+	1.07467i	−0.989400	+	0.145217i	$0.953612\pi$
$500$		0			0
$501$	365.760	+	633.515i	0.0326167	+	0.0564937i
$502$		0			0
$503$	−13469.0			−1.19394			−0.596970	−	0.802264i	$-0.703629\pi$
$503$	−13469.0			−1.19394			−0.596970	+	0.802264i	$0.703629\pi$
$504$		0			0
$505$	−11570.7			−1.01958
$506$		0			0
$507$	3117.84	+	5400.26i	0.273113	+	0.473045i
$508$		0			0
$509$	4387.62	−	7599.59i	0.382079	−	0.661779i	−0.609281	−	0.792955i	$-0.708542\pi$
$509$	4387.62	−	7599.59i	0.382079	−	0.661779i	0.991359	+	0.131175i	$0.0418750\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−2211.55	+	3830.51i	−0.190336	+	0.329671i
$514$		0			0
$515$	−1450.77	−	2512.81i	−0.124133	−	0.215005i
$516$		0			0
$517$	−3274.63			−0.278565
$518$		0			0
$519$	−5061.79			−0.428108
$520$		0			0
$521$	−830.382	−	1438.26i	−0.0698267	−	0.120943i	0.828998	−	0.559251i	$-0.188911\pi$
$521$	−830.382	−	1438.26i	−0.0698267	−	0.120943i	−0.898825	+	0.438308i	$0.855578\pi$
$522$		0			0
$523$	3162.41	−	5477.45i	0.264402	−	0.457958i	−0.703005	−	0.711185i	$-0.748159\pi$
$523$	3162.41	−	5477.45i	0.264402	−	0.457958i	0.967407	+	0.253227i	$0.0814920\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6132.88	+	10622.5i	−0.506931	+	0.878030i
$528$		0			0
$529$	2944.90	+	5100.71i	0.242040	+	0.419225i
$530$		0			0
$531$	3525.92			0.288158
$532$		0			0
$533$	−9954.68			−0.808977
$534$		0			0
$535$	−396.424	−	686.626i	−0.0320353	−	0.0554868i
$536$		0			0
$537$	−538.709	+	933.071i	−0.0432905	+	0.0749814i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2233.94	+	3869.30i	−0.177532	+	0.307494i	−0.941034	−	0.338311i	$-0.890145\pi$
$541$	−2233.94	+	3869.30i	−0.177532	+	0.307494i	0.763503	+	0.645804i	$0.223478\pi$
$542$		0			0
$543$	−4473.87	−	7748.97i	−0.353577	−	0.612413i
$544$		0			0
$545$	−27198.7			−2.13774
$546$		0			0
$547$	−10939.0			−0.855063			−0.427531	−	0.904000i	$-0.640617\pi$
$547$	−10939.0			−0.855063			−0.427531	+	0.904000i	$0.640617\pi$
$548$		0			0
$549$	2480.83	+	4296.92i	0.192858	+	0.334040i
$550$		0			0
$551$	−3546.63	+	6142.94i	−0.274213	+	0.474951i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6763.65	−	11715.0i	0.517299	−	0.895988i
$556$		0			0
$557$	7162.57	+	12405.9i	0.544862	+	0.943728i	0.998616	+	0.0526011i	$0.0167512\pi$
$557$	7162.57	+	12405.9i	0.544862	+	0.943728i	−0.453754	+	0.891127i	$0.649915\pi$
$558$		0			0
$559$	−11615.6			−0.878867
$560$		0			0
$561$	−19468.1			−1.46514
$562$		0			0
$563$	10201.4	+	17669.4i	0.763656	+	1.32269i	0.940954	+	0.338533i	$0.109931\pi$
$563$	10201.4	+	17669.4i	0.763656	+	1.32269i	−0.177299	+	0.984157i	$0.556736\pi$
$564$		0			0
$565$	9387.32	−	16259.3i	0.698987	−	1.21068i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3625.21	−	6279.05i	0.267095	−	0.462622i	−0.701016	−	0.713146i	$-0.747270\pi$
$569$	3625.21	−	6279.05i	0.267095	−	0.462622i	0.968110	+	0.250524i	$0.0806030\pi$
$570$		0			0
$571$	−12447.4	−	21559.5i	−0.912272	−	1.58010i	−0.810847	−	0.585258i	$-0.800993\pi$
$571$	−12447.4	−	21559.5i	−0.912272	−	1.58010i	−0.101424	−	0.994843i	$-0.532340\pi$
$572$		0			0
$573$	−10472.3			−0.763503
$574$		0			0
$575$	12072.8			0.875599
$576$		0			0
$577$	681.723	+	1180.78i	0.0491863	+	0.0851932i	0.889570	−	0.456798i	$-0.151004\pi$
$577$	681.723	+	1180.78i	0.0491863	+	0.0851932i	−0.840384	+	0.541991i	$0.817671\pi$
$578$		0			0
$579$	2406.34	−	4167.90i	0.172719	−	0.299158i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5684.74	+	9846.26i	−0.403839	+	0.699469i
$584$		0			0
$585$	−4900.56	−	8488.01i	−0.346347	−	0.599891i
$586$		0			0
$587$	−23854.8			−1.67733			−0.838666	−	0.544647i	$-0.816664\pi$
$587$	−23854.8			−1.67733			−0.838666	+	0.544647i	$0.816664\pi$
$588$		0			0
$589$	22219.6			1.55440
$590$		0			0
$591$	−683.902	−	1184.55i	−0.0476006	−	0.0824467i
$592$		0			0
$593$	1718.60	−	2976.71i	0.119013	−	0.206136i	−0.800364	−	0.599514i	$-0.795360\pi$
$593$	1718.60	−	2976.71i	0.119013	−	0.206136i	0.919377	+	0.393378i	$0.128694\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3126.24	+	5414.81i	−0.214319	+	0.371212i
$598$		0			0
$599$	−8255.87	−	14299.6i	−0.563148	−	0.975401i	−0.997219	−	0.0745226i	$-0.976257\pi$
$599$	−8255.87	−	14299.6i	−0.563148	−	0.975401i	0.434071	−	0.900879i	$-0.357077\pi$
$600$		0			0
$601$	11148.3			0.756656			0.378328	−	0.925672i	$-0.376499\pi$
$601$	11148.3			0.756656			0.378328	+	0.925672i	$0.376499\pi$
$602$		0			0
$603$	4127.67			0.278759
$604$		0			0
$605$	−31797.9	−	55075.6i	−2.13681	−	3.70106i
$606$		0			0
$607$	−195.791	+	339.121i	−0.0130921	+	0.0226763i	−0.872497	−	0.488619i	$-0.837501\pi$
$607$	−195.791	+	339.121i	−0.0130921	+	0.0226763i	0.859405	+	0.511295i	$0.170834\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1491.92	+	2584.09i	−0.0987836	+	0.171098i
$612$		0			0
$613$	5870.25	+	10167.6i	0.386782	+	0.669925i	0.992015	−	0.126123i	$-0.0402534\pi$
$613$	5870.25	+	10167.6i	0.386782	+	0.669925i	−0.605233	+	0.796048i	$0.706920\pi$
$614$		0			0
$615$	7606.56			0.498742
$616$		0			0
$617$	−8818.95			−0.575426			−0.287713	−	0.957717i	$-0.592895\pi$
$617$	−8818.95			−0.575426			−0.287713	+	0.957717i	$0.592895\pi$
$618$		0			0
$619$	−7962.35	−	13791.2i	−0.517018	−	0.895501i	−0.999805	−	0.0197631i	$-0.993709\pi$
$619$	−7962.35	−	13791.2i	−0.517018	−	0.895501i	0.482787	−	0.875738i	$-0.339625\pi$
$620$		0			0
$621$	1069.59	−	1852.58i	0.0691162	−	0.119713i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5726.55	−	9918.67i	0.366499	−	0.634795i
$626$		0			0
$627$	17633.4	+	30542.0i	1.12314	+	1.94534i
$628$		0			0
$629$	−24483.5			−1.55202
$630$		0			0
$631$	29206.4			1.84261			0.921307	−	0.388836i	$-0.127123\pi$
$631$	29206.4			1.84261			0.921307	+	0.388836i	$0.127123\pi$
$632$		0			0
$633$	3852.09	+	6672.02i	0.241875	+	0.418940i
$634$		0			0
$635$	−19974.5	+	34596.8i	−1.24829	+	2.16210i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2190.54	−	3794.12i	0.135612	−	0.234887i
$640$		0			0
$641$	−2806.51	−	4861.01i	−0.172933	−	0.299530i	0.766511	−	0.642232i	$-0.221991\pi$
$641$	−2806.51	−	4861.01i	−0.172933	−	0.299530i	−0.939444	+	0.342702i	$0.888658\pi$
$642$		0			0
$643$	18995.0			1.16499			0.582496	−	0.812834i	$-0.302076\pi$
$643$	18995.0			1.16499			0.582496	+	0.812834i	$0.302076\pi$
$644$		0			0
$645$	8875.68			0.541829
$646$		0			0
$647$	11332.5	+	19628.5i	0.688604	+	1.19270i	0.972290	+	0.233779i	$0.0751093\pi$
$647$	11332.5	+	19628.5i	0.688604	+	1.19270i	−0.283686	+	0.958917i	$0.591557\pi$
$648$		0			0
$649$	14056.7	−	24346.9i	0.850188	−	1.47257i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13263.9	+	22973.7i	−0.794879	+	1.37677i	0.128037	+	0.991769i	$0.459132\pi$
$653$	−13263.9	+	22973.7i	−0.794879	+	1.37677i	−0.922916	+	0.385001i	$0.874201\pi$
$654$		0			0
$655$	12898.8	+	22341.3i	0.769460	+	1.33274i
$656$		0			0
$657$	−5174.03			−0.307242
$658$		0			0
$659$	−17114.8			−1.01168			−0.505839	−	0.862628i	$-0.668817\pi$
$659$	−17114.8			−1.01168			−0.505839	+	0.862628i	$0.668817\pi$
$660$		0			0
$661$	−595.908	−	1032.14i	−0.0350653	−	0.0607348i	0.847960	−	0.530060i	$-0.177831\pi$
$661$	−595.908	−	1032.14i	−0.0350653	−	0.0607348i	−0.883025	+	0.469325i	$0.844497\pi$
$662$		0			0
$663$	−8869.68	+	15362.7i	−0.519562	+	0.899908i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1715.29	−	2970.96i	0.0995744	−	0.172468i
$668$		0			0
$669$	6989.33	+	12105.9i	0.403921	+	0.699612i
$670$		0			0
$671$	39560.9			2.27605
$672$		0			0
$673$	−9415.20			−0.539271			−0.269636	−	0.962962i	$-0.586903\pi$
$673$	−9415.20			−0.539271			−0.269636	+	0.962962i	$0.586903\pi$
$674$		0			0
$675$	2057.11	+	3563.02i	0.117301	+	0.203171i
$676$		0			0
$677$	3958.56	−	6856.43i	0.224727	−	0.389238i	−0.731511	−	0.681830i	$-0.761184\pi$
$677$	3958.56	−	6856.43i	0.224727	−	0.389238i	0.956237	+	0.292592i	$0.0945177\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7243.16	+	12545.5i	−0.407575	+	0.705940i
$682$		0			0
$683$	−14617.2	−	25317.7i	−0.818904	−	1.41838i	−0.906490	−	0.422227i	$-0.861249\pi$
$683$	−14617.2	−	25317.7i	−0.818904	−	1.41838i	0.0875859	−	0.996157i	$-0.472085\pi$
$684$		0			0
$685$	25369.6			1.41507
$686$		0			0
$687$	−9667.76			−0.536897
$688$		0			0
$689$	5179.94	+	8971.93i	0.286415	+	0.496086i
$690$		0			0
$691$	−611.228	+	1058.68i	−0.0336501	+	0.0582836i	−0.882360	−	0.470575i	$-0.844047\pi$
$691$	−611.228	+	1058.68i	−0.0336501	+	0.0582836i	0.848710	+	0.528859i	$0.177380\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4592.07	−	7953.70i	0.250629	−	0.434102i
$696$		0			0
$697$	−6883.68	−	11922.9i	−0.374086	−	0.647936i
$698$		0			0
$699$	17107.3			0.925688
$700$		0			0
$701$	13054.6			0.703372			0.351686	−	0.936118i	$-0.385608\pi$
$701$	13054.6			0.703372			0.351686	+	0.936118i	$0.385608\pi$
$702$		0			0
$703$	22176.1	+	38410.2i	1.18974	+	2.06069i
$704$		0			0
$705$	1140.01	−	1974.55i	0.0609010	−	0.105484i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9987.87	−	17299.5i	0.529058	−	0.916356i	−0.470367	−	0.882471i	$-0.655879\pi$
$709$	9987.87	−	17299.5i	0.529058	−	0.916356i	0.999426	−	0.0338854i	$-0.0107881\pi$
$710$		0			0
$711$	3007.44	+	5209.03i	0.158632	+	0.274760i
$712$		0			0
$713$	−10746.3			−0.564447
$714$		0			0
$715$	−78147.5			−4.08749
$716$		0			0
$717$	2991.70	+	5181.77i	0.155826	+	0.269898i
$718$		0			0
$719$	3256.13	−	5639.78i	0.168892	−	0.292529i	−0.769139	−	0.639082i	$-0.779315\pi$
$719$	3256.13	−	5639.78i	0.168892	−	0.292529i	0.938030	+	0.346553i	$0.112648\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	2131.33	−	3691.57i	0.109633	−	0.189891i
$724$		0			0
$725$	3298.96	+	5713.97i	0.168994	+	0.292705i
$726$		0			0
$727$	−11437.1			−0.583467			−0.291733	−	0.956500i	$-0.594232\pi$
$727$	−11437.1			−0.583467			−0.291733	+	0.956500i	$0.594232\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−8032.19	−	13912.2i	−0.406404	−	0.703913i
$732$		0			0
$733$	−6138.02	+	10631.4i	−0.309295	+	0.535714i	−0.978208	−	0.207626i	$-0.933426\pi$
$733$	−6138.02	+	10631.4i	−0.309295	+	0.535714i	0.668914	+	0.743340i	$0.266760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16455.6	−	28502.0i	0.822458	−	1.42454i
$738$		0			0
$739$	−6046.91	−	10473.5i	−0.301000	−	0.521347i	0.675363	−	0.737486i	$-0.263987\pi$
$739$	−6046.91	−	10473.5i	−0.301000	−	0.521347i	−0.976363	+	0.216138i	$0.930654\pi$
$740$		0			0
$741$	32135.2			1.59314
$742$		0			0
$743$	34331.6			1.69516			0.847580	−	0.530668i	$-0.178059\pi$
$743$	34331.6			1.69516			0.847580	+	0.530668i	$0.178059\pi$
$744$		0			0
$745$	19987.6	+	34619.6i	0.982940	+	1.70250i
$746$		0			0
$747$	343.102	−	594.271i	0.0168052	−	0.0291074i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	7278.96	−	12607.5i	0.353679	−	0.612590i	−0.633212	−	0.773979i	$-0.718264\pi$
$751$	7278.96	−	12607.5i	0.353679	−	0.612590i	0.986891	+	0.161388i	$0.0515971\pi$
$752$		0			0
$753$	5783.16	+	10016.7i	0.279880	+	0.484767i
$754$		0			0
$755$	10351.9			0.498998
$756$		0			0
$757$	27115.5			1.30189			0.650944	−	0.759126i	$-0.274373\pi$
$757$	27115.5			1.30189			0.650944	+	0.759126i	$0.274373\pi$
$758$		0			0
$759$	−8528.19	−	14771.3i	−0.407844	−	0.706407i
$760$		0			0
$761$	−5481.98	+	9495.06i	−0.261132	+	0.452294i	−0.966543	−	0.256504i	$-0.917429\pi$
$761$	−5481.98	+	9495.06i	−0.261132	+	0.452294i	0.705411	+	0.708799i	$0.250762\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6777.49	−	11739.0i	0.320315	−	0.554801i
$766$		0			0
$767$	−12808.4	−	22184.9i	−0.602981	−	1.04439i
$768$		0			0
$769$	7835.53			0.367434			0.183717	−	0.982979i	$-0.441187\pi$
$769$	7835.53			0.367434			0.183717	+	0.982979i	$0.441187\pi$
$770$		0			0
$771$	12159.8			0.567997
$772$		0			0
$773$	−14383.9	−	24913.6i	−0.669278	−	1.15922i	−0.978106	−	0.208106i	$-0.933270\pi$
$773$	−14383.9	−	24913.6i	−0.669278	−	1.15922i	0.308828	−	0.951118i	$-0.400063\pi$
$774$		0			0
$775$	10334.0	−	17899.0i	0.478978	−	0.829614i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12469.9	+	21598.5i	−0.573531	+	0.993386i
$780$		0			0
$781$	−17465.9	−	30251.8i	−0.800228	−	1.38603i
$782$		0			0
$783$	1169.09			0.0533586
$784$		0			0
$785$	44827.3			2.03816
$786$		0			0
$787$	−1347.91	−	2334.65i	−0.0610520	−	0.105745i	0.833884	−	0.551940i	$-0.186112\pi$
$787$	−1347.91	−	2334.65i	−0.0610520	−	0.105745i	−0.894936	+	0.446195i	$0.852779\pi$
$788$		0			0
$789$	−751.180	+	1301.08i	−0.0338944	+	0.0587069i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18024.0	−	31218.4i	0.807125	−	1.39798i
$794$		0			0
$795$	−3958.09	−	6855.62i	−0.176578	−	0.305841i
$796$		0			0
$797$	−12592.1			−0.559643			−0.279822	−	0.960052i	$-0.590275\pi$
$797$	−12592.1			−0.559643			−0.279822	+	0.960052i	$0.590275\pi$
$798$		0

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)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{3} +(-8.32734 + 14.4234i) q^{5} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 2.59808i) q^{3} +(-8.32734 + 14.4234i) q^{5} +(-4.50000 + 7.79423i) q^{9} +(35.8800 + 62.1460i) q^{11} +65.3878 q^{13} -49.9641 q^{15} +(45.2158 + 78.3160i) q^{17} +(81.9092 - 141.871i) q^{19} +(-39.6144 + 68.6142i) q^{23} +(-76.1893 - 131.964i) q^{25} -27.0000 q^{27} -43.2995 q^{29} +(67.8179 + 117.464i) q^{31} +(-107.640 + 186.438i) q^{33} +(-135.370 + 234.468i) q^{37} +(98.0817 + 169.882i) q^{39} -152.241 q^{41} -177.641 q^{43} +(-74.9461 - 129.810i) q^{45} +(-22.8166 + 39.5194i) q^{47} +(-135.647 + 234.948i) q^{51} +(79.2188 + 137.211i) q^{53} -1195.14 q^{55} +491.455 q^{57} +(-195.884 - 339.282i) q^{59} +(275.647 - 477.435i) q^{61} +(-544.506 + 943.113i) q^{65} +(-229.315 - 397.185i) q^{67} -237.687 q^{69} -486.786 q^{71} +(287.446 + 497.871i) q^{73} +(228.568 - 395.891i) q^{75} +(334.160 - 578.782i) q^{79} +(-40.5000 - 70.1481i) q^{81} -76.2450 q^{83} -1506.11 q^{85} +(-64.9493 - 112.495i) q^{87} +(683.401 - 1183.69i) q^{89} +(-203.454 + 352.392i) q^{93} +(1364.17 + 2362.82i) q^{95} -242.655 q^{97} -645.840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q + 12q^{3} - 36q^{9} + O(q^{10}) \)	\( 8q + 12q^{3} - 36q^{9} + 48q^{17} + 192q^{19} - 192q^{23} - 324q^{25} - 216q^{27} + 192q^{29} + 48q^{31} - 256q^{37} - 2016q^{41} - 224q^{43} + 864q^{47} - 144q^{51} + 648q^{53} - 4704q^{55} + 1152q^{57} + 336q^{59} + 960q^{61} + 360q^{65} - 720q^{67} - 1152q^{69} - 2688q^{71} + 672q^{73} + 972q^{75} + 1984q^{79} - 324q^{81} - 6240q^{83} + 1360q^{85} + 288q^{87} + 2160q^{89} - 144q^{93} + 3744q^{95} - 4032q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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