LMFDB - Embedded newform 336.4.bc.d.17.6

Newspace parameters

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$ x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 $ x^12 - x^11 - 29x^9 + 6x^8 - 49x^7 + 1564x^6 - 441x^5 + 486x^4 - 21141x^3 - 59049x + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.6
Root			$2.70662 - 1.29391i$ of defining polynomial
Character	$\chi$	$=$	336.17
Dual form			336.4.bc.d.257.6

$q$-expansion

$f(q)$	$=$	$q+(5.18049 - 0.403134i) q^{3} +(5.80193 + 10.0492i) q^{5} +(18.4018 - 2.09174i) q^{7} +(26.6750 - 4.17686i) q^{9} +O(q^{10})$	\(q+(5.18049 - 0.403134i) q^{3} +(5.80193 + 10.0492i) q^{5} +(18.4018 - 2.09174i) q^{7} +(26.6750 - 4.17686i) q^{9} +(15.5157 + 8.95800i) q^{11} -62.4185i q^{13} +(34.1081 + 49.7211i) q^{15} +(-10.7082 + 18.5472i) q^{17} +(-9.50747 + 5.48914i) q^{19} +(94.4869 - 18.2546i) q^{21} +(-59.8367 + 34.5467i) q^{23} +(-4.82490 + 8.35697i) q^{25} +(136.506 - 32.3918i) q^{27} +265.583i q^{29} +(-8.85795 - 5.11414i) q^{31} +(83.9902 + 40.1519i) q^{33} +(127.786 + 172.788i) q^{35} +(-20.8257 - 36.0712i) q^{37} +(-25.1630 - 323.358i) q^{39} +31.0035 q^{41} +224.550 q^{43} +(196.741 + 243.829i) q^{45} +(-81.8595 - 141.785i) q^{47} +(334.249 - 76.9836i) q^{49} +(-47.9968 + 100.400i) q^{51} +(-456.586 - 263.610i) q^{53} +207.895i q^{55} +(-47.0405 + 32.2692i) q^{57} +(-205.978 + 356.765i) q^{59} +(223.807 - 129.215i) q^{61} +(482.129 - 132.659i) q^{63} +(627.258 - 362.148i) q^{65} +(161.737 - 280.137i) q^{67} +(-296.056 + 203.091i) q^{69} +45.4199i q^{71} +(-486.879 - 281.100i) q^{73} +(-21.6264 + 45.2383i) q^{75} +(304.254 + 132.388i) q^{77} +(144.610 + 250.473i) q^{79} +(694.108 - 222.835i) q^{81} +448.767 q^{83} -248.513 q^{85} +(107.066 + 1375.85i) q^{87} +(280.814 + 486.384i) q^{89} +(-130.563 - 1148.61i) q^{91} +(-47.9502 - 22.9228i) q^{93} +(-110.323 - 63.6953i) q^{95} +214.364i q^{97} +(451.297 + 174.147i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9	$ 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9 - 6 * q^15 - 300 * q^19 + 357 * q^21 - 42 * q^25 + 930 * q^31 - 855 * q^33 + 764 * q^37 + 426 * q^39 + 1012 * q^43 + 2367 * q^45 - 336 * q^49 + 1341 * q^51 + 270 * q^57 + 2358 * q^61 - 1071 * q^63 - 792 * q^67 - 2904 * q^73 + 2418 * q^75 - 1674 * q^79 + 837 * q^81 + 348 * q^85 - 1638 * q^87 + 1218 * q^91 - 1479 * q^93 + 3354 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	5.18049	−	0.403134i	0.996986	−	0.0775831i
$3$	5.18049	−	0.403134i	0.996986	−	0.0775831i
$4$		0			0
$5$	5.80193	+	10.0492i	0.518941	+	0.898832i	0.999758	+	0.0220109i	$0.00700684\pi$
$5$	5.80193	+	10.0492i	0.518941	+	0.898832i	−0.480817	+	0.876821i	$0.659660\pi$
$6$		0			0
$7$	18.4018	−	2.09174i	0.993601	−	0.112944i
$7$	18.4018	−	2.09174i	0.993601	−	0.112944i
$8$		0			0
$9$	26.6750	−	4.17686i	0.987962	−	0.154699i
$10$		0			0
$11$	15.5157	+	8.95800i	0.425287	+	0.245540i	0.697337	−	0.716743i	$-0.254368\pi$
$11$	15.5157	+	8.95800i	0.425287	+	0.245540i	−0.272050	+	0.962283i	$0.587701\pi$
$12$		0			0
$13$		−	62.4185i		−	1.33167i	−0.746097	−	0.665837i	$-0.768075\pi$
$13$		−	62.4185i		−	1.33167i	0.746097	−	0.665837i	$-0.231925\pi$
$14$		0			0
$15$	34.1081	+	49.7211i	0.587111	+	0.855862i
$16$		0			0
$17$	−10.7082	+	18.5472i	−0.152772	+	0.264609i	−0.932245	−	0.361826i	$-0.882153\pi$
$17$	−10.7082	+	18.5472i	−0.152772	+	0.264609i	0.779474	+	0.626435i	$0.215487\pi$
$18$		0			0
$19$	−9.50747	+	5.48914i	−0.114798	+	0.0662787i	−0.556300	−	0.830982i	$-0.687779\pi$
$19$	−9.50747	+	5.48914i	−0.114798	+	0.0662787i	0.441502	+	0.897261i	$0.354446\pi$
$20$		0			0
$21$	94.4869	−	18.2546i	0.981844	−	0.189690i
$22$		0			0
$23$	−59.8367	+	34.5467i	−0.542470	+	0.313195i	−0.746079	−	0.665857i	$-0.768066\pi$
$23$	−59.8367	+	34.5467i	−0.542470	+	0.313195i	0.203609	+	0.979052i	$0.434733\pi$
$24$		0			0
$25$	−4.82490	+	8.35697i	−0.0385992	+	0.0668557i
$26$		0			0
$27$	136.506	−	32.3918i	0.972982	−	0.230881i
$28$		0			0
$29$			265.583i			1.70061i	0.526294	+	0.850303i	$0.323581\pi$
$29$			265.583i			1.70061i	−0.526294	+	0.850303i	$0.676419\pi$
$30$		0			0
$31$	−8.85795	−	5.11414i	−0.0513205	−	0.0296299i	0.474120	−	0.880460i	$-0.342766\pi$
$31$	−8.85795	−	5.11414i	−0.0513205	−	0.0296299i	−0.525441	+	0.850830i	$0.676100\pi$
$32$		0			0
$33$	83.9902	+	40.1519i	0.443055	+	0.211805i
$34$		0			0
$35$	127.786	+	172.788i	0.617138	+	0.834470i
$36$		0			0
$37$	−20.8257	−	36.0712i	−0.0925331	−	0.160272i	0.816043	−	0.577991i	$-0.196163\pi$
$37$	−20.8257	−	36.0712i	−0.0925331	−	0.160272i	−0.908576	+	0.417719i	$0.862830\pi$
$38$		0			0
$39$	−25.1630	−	323.358i	−0.103315	−	1.32766i
$40$		0			0
$41$	31.0035			0.118096			0.0590480	−	0.998255i	$-0.481193\pi$
$41$	31.0035			0.118096			0.0590480	+	0.998255i	$0.481193\pi$
$42$		0			0
$43$	224.550			0.796363			0.398181	−	0.917307i	$-0.369641\pi$
$43$	224.550			0.796363			0.398181	+	0.917307i	$0.369641\pi$
$44$		0			0
$45$	196.741	+	243.829i	0.651742	+	0.807732i
$46$		0			0
$47$	−81.8595	−	141.785i	−0.254052	−	0.440031i	0.710586	−	0.703611i	$-0.248430\pi$
$47$	−81.8595	−	141.785i	−0.254052	−	0.440031i	−0.964638	+	0.263580i	$0.915097\pi$
$48$		0			0
$49$	334.249	−	76.9836i	0.974487	−	0.224442i
$50$		0			0
$51$	−47.9968	+	100.400i	−0.131782	+	0.275664i
$52$		0			0
$53$	−456.586	−	263.610i	−1.18334	−	0.683200i	−0.226553	−	0.973999i	$-0.572746\pi$
$53$	−456.586	−	263.610i	−1.18334	−	0.683200i	−0.956784	+	0.290799i	$0.906079\pi$
$54$		0			0
$55$			207.895i			0.509682i
$56$		0			0
$57$	−47.0405	+	32.2692i	−0.109310	+	0.0749853i
$58$		0			0
$59$	−205.978	+	356.765i	−0.454510	+	0.787234i	−0.998660	−	0.0517537i	$-0.983519\pi$
$59$	−205.978	+	356.765i	−0.454510	+	0.787234i	0.544150	+	0.838988i	$0.316852\pi$
$60$		0			0
$61$	223.807	−	129.215i	0.469764	−	0.271218i	−0.246377	−	0.969174i	$-0.579240\pi$
$61$	223.807	−	129.215i	0.469764	−	0.271218i	0.716141	+	0.697956i	$0.245907\pi$
$62$		0			0
$63$	482.129	−	132.659i	0.964168	−	0.265293i
$64$		0			0
$65$	627.258	−	362.148i	1.19695	−	0.691060i
$66$		0			0
$67$	161.737	−	280.137i	0.294915	−	0.510808i	−0.680050	−	0.733166i	$-0.738042\pi$
$67$	161.737	−	280.137i	0.294915	−	0.510808i	0.974965	+	0.222357i	$0.0713752\pi$
$68$		0			0
$69$	−296.056	+	203.091i	−0.516536	+	0.354338i
$70$		0			0
$71$			45.4199i			0.0759205i	0.999279	+	0.0379603i	$0.0120860\pi$
$71$			45.4199i			0.0759205i	−0.999279	+	0.0379603i	$0.987914\pi$
$72$		0			0
$73$	−486.879	−	281.100i	−0.780615	−	0.450688i	0.0560334	−	0.998429i	$-0.482155\pi$
$73$	−486.879	−	281.100i	−0.780615	−	0.450688i	−0.836648	+	0.547741i	$0.815488\pi$
$74$		0			0
$75$	−21.6264	+	45.2383i	−0.0332960	+	0.0696489i
$76$		0			0
$77$	304.254	+	132.388i	0.450298	+	0.195935i
$78$		0			0
$79$	144.610	+	250.473i	0.205949	+	0.356714i	0.950435	−	0.310925i	$-0.100639\pi$
$79$	144.610	+	250.473i	0.205949	+	0.356714i	−0.744486	+	0.667638i	$0.767305\pi$
$80$		0			0
$81$	694.108	−	222.835i	0.952137	−	0.305673i
$82$		0			0
$83$	448.767			0.593477			0.296738	−	0.954959i	$-0.404101\pi$
$83$	448.767			0.593477			0.296738	+	0.954959i	$0.404101\pi$
$84$		0			0
$85$	−248.513			−0.317118
$86$		0			0
$87$	107.066	+	1375.85i	0.131938	+	1.69548i
$88$		0			0
$89$	280.814	+	486.384i	0.334452	+	0.579288i	0.983379	−	0.181562i	$-0.0581155\pi$
$89$	280.814	+	486.384i	0.334452	+	0.579288i	−0.648927	+	0.760850i	$0.724782\pi$
$90$		0			0
$91$	−130.563	−	1148.61i	−0.150404	−	1.32315i
$92$		0			0
$93$	−47.9502	−	22.9228i	−0.0534645	−	0.0255590i
$94$		0			0
$95$	−110.323	−	63.6953i	−0.119147	−	0.0687895i
$96$		0			0
$97$			214.364i			0.224385i	0.993686	+	0.112192i	$0.0357873\pi$
$97$			214.364i			0.224385i	−0.993686	+	0.112192i	$0.964213\pi$
$98$		0			0
$99$	451.297	+	174.147i	0.458152	+	0.176793i
$100$		0			0
$101$	−858.845	+	1487.56i	−0.846122	+	1.46553i	0.0385219	+	0.999258i	$0.487735\pi$
$101$	−858.845	+	1487.56i	−0.846122	+	1.46553i	−0.884644	+	0.466268i	$0.845598\pi$
$102$		0			0
$103$	−1002.61	+	578.855i	−0.959123	+	0.553750i	−0.895903	−	0.444250i	$-0.853470\pi$
$103$	−1002.61	+	578.855i	−0.959123	+	0.553750i	−0.0632200	+	0.998000i	$0.520137\pi$
$104$		0			0
$105$	731.652	+	843.610i	0.680018	+	0.784075i
$106$		0			0
$107$	1054.64	−	608.897i	0.952859	−	0.550134i	0.0588912	−	0.998264i	$-0.481243\pi$
$107$	1054.64	−	608.897i	0.952859	−	0.550134i	0.893968	+	0.448131i	$0.147910\pi$
$108$		0			0
$109$	−649.132	+	1124.33i	−0.570418	+	0.987992i	0.426105	+	0.904674i	$0.359885\pi$
$109$	−649.132	+	1124.33i	−0.570418	+	0.987992i	−0.996523	+	0.0833189i	$0.973448\pi$
$110$		0			0
$111$	−122.429	−	178.471i	−0.104689	−	0.152610i
$112$		0			0
$113$		−	1437.86i		−	1.19701i	−0.801118	−	0.598506i	$-0.795761\pi$
$113$		−	1437.86i		−	1.19701i	0.801118	−	0.598506i	$-0.204239\pi$
$114$		0			0
$115$	−694.337	−	400.876i	−0.563019	−	0.325059i
$116$		0			0
$117$	−260.713	−	1665.01i	−0.206008	−	1.31564i
$118$		0			0
$119$	−158.254	+	363.699i	−0.121908	+	0.280170i
$120$		0			0
$121$	−505.009	−	874.701i	−0.379420	−	0.657175i
$122$		0			0
$123$	160.613	−	12.4986i	0.117740	−	0.00916226i
$124$		0			0
$125$	1338.51			0.957759
$126$		0			0
$127$	−2686.32			−1.87695			−0.938475	−	0.345347i	$-0.887761\pi$
$127$	−2686.32			−1.87695			−0.938475	+	0.345347i	$0.887761\pi$
$128$		0			0
$129$	1163.28	−	90.5238i	0.793962	−	0.0617843i
$130$		0			0
$131$	−801.637	−	1388.48i	−0.534651	−	0.926043i	−0.999180	−	0.0404852i	$-0.987110\pi$
$131$	−801.637	−	1388.48i	−0.534651	−	0.926043i	0.464529	−	0.885558i	$-0.346224\pi$
$132$		0			0
$133$	−163.472	+	120.897i	−0.106578	+	0.0788203i
$134$		0			0
$135$	1117.51	+	1183.84i	0.712444	+	0.754733i
$136$		0			0
$137$	−2007.90	−	1159.26i	−1.25216	−	0.722938i	−0.280626	−	0.959817i	$-0.590542\pi$
$137$	−2007.90	−	1159.26i	−1.25216	−	0.722938i	−0.971539	+	0.236879i	$0.923875\pi$
$138$		0			0
$139$		−	1841.57i		−	1.12374i	−0.827225	−	0.561871i	$-0.810082\pi$
$139$		−	1841.57i		−	1.12374i	0.827225	−	0.561871i	$-0.189918\pi$
$140$		0			0
$141$	−481.230	−	701.514i	−0.287425	−	0.418994i
$142$		0			0
$143$	559.144	−	968.466i	0.326979	−	0.566344i
$144$		0			0
$145$	−2668.91	+	1540.90i	−1.52856	+	0.882514i
$146$		0			0
$147$	1700.54	−	533.560i	0.954137	−	0.299369i
$148$		0			0
$149$	1126.68	−	650.488i	0.619470	−	0.357651i	−0.157193	−	0.987568i	$-0.550244\pi$
$149$	1126.68	−	650.488i	0.619470	−	0.357651i	0.776663	+	0.629917i	$0.216911\pi$
$150$		0			0
$151$	−1308.24	+	2265.94i	−0.705055	+	1.22119i	0.261616	+	0.965172i	$0.415745\pi$
$151$	−1308.24	+	2265.94i	−0.705055	+	1.22119i	−0.966672	+	0.256020i	$0.917589\pi$
$152$		0			0
$153$	−208.172	+	539.472i	−0.109998	+	0.285057i
$154$		0			0
$155$		−	118.688i		−	0.0615046i
$156$		0			0
$157$	−809.876	−	467.582i	−0.411689	−	0.237689i	0.279826	−	0.960051i	$-0.409723\pi$
$157$	−809.876	−	467.582i	−0.411689	−	0.237689i	−0.691515	+	0.722362i	$0.743056\pi$
$158$		0			0
$159$	−2471.61	−	1181.56i	−1.23278	−	0.589334i
$160$		0			0
$161$	−1028.84	+	760.883i	−0.503625	+	0.372460i
$162$		0			0
$163$	−259.079	−	448.738i	−0.124495	−	0.215631i	0.797041	−	0.603926i	$-0.206398\pi$
$163$	−259.079	−	448.738i	−0.124495	−	0.215631i	−0.921535	+	0.388295i	$0.873064\pi$
$164$		0			0
$165$	83.8094	+	1077.00i	0.0395428	+	0.508146i
$166$		0			0
$167$	−3767.97			−1.74595			−0.872977	−	0.487761i	$-0.837814\pi$
$167$	−3767.97			−1.74595			−0.872977	+	0.487761i	$0.837814\pi$
$168$		0			0
$169$	−1699.06			−0.773356
$170$		0			0
$171$	−230.684	+	186.134i	−0.103163	+	0.0832399i
$172$		0			0
$173$	−1196.53	−	2072.45i	−0.525841	−	0.910783i	−0.999547	−	0.0301000i	$-0.990417\pi$
$173$	−1196.53	−	2072.45i	−0.525841	−	0.910783i	0.473706	−	0.880683i	$-0.342916\pi$
$174$		0			0
$175$	−71.3059	+	163.875i	−0.0308013	+	0.0707875i
$176$		0			0
$177$	−923.244	+	1931.25i	−0.392064	+	0.820124i
$178$		0			0
$179$	554.381	+	320.072i	0.231488	+	0.133650i	0.611258	−	0.791431i	$-0.290664\pi$
$179$	554.381	+	320.072i	0.231488	+	0.133650i	−0.379770	+	0.925081i	$0.623997\pi$
$180$		0			0
$181$		−	4204.05i		−	1.72643i	−0.504833	−	0.863217i	$-0.668446\pi$
$181$		−	4204.05i		−	1.72643i	0.504833	−	0.863217i	$-0.331554\pi$
$182$		0			0
$183$	1107.34	−	759.623i	0.447306	−	0.306847i
$184$		0			0
$185$	241.659	−	418.565i	0.0960384	−	0.166343i
$186$		0			0
$187$	−332.291	+	191.848i	−0.129944	+	0.0750231i
$188$		0			0
$189$	2444.19	−	881.600i	0.940680	−	0.339296i
$190$		0			0
$191$	−1261.85	+	728.530i	−0.478033	+	0.275993i	−0.719597	−	0.694392i	$-0.755673\pi$
$191$	−1261.85	+	728.530i	−0.478033	+	0.275993i	0.241563	+	0.970385i	$0.422340\pi$
$192$		0			0
$193$	−914.633	+	1584.19i	−0.341123	+	0.590842i	−0.984642	−	0.174588i	$-0.944141\pi$
$193$	−914.633	+	1584.19i	−0.341123	+	0.590842i	0.643519	+	0.765430i	$0.277474\pi$
$194$		0			0
$195$	3103.51	−	2128.97i	1.13973	−	0.781840i
$196$		0			0
$197$			661.168i			0.239118i	0.992827	+	0.119559i	$0.0381481\pi$
$197$			661.168i			0.239118i	−0.992827	+	0.119559i	$0.961852\pi$
$198$		0			0
$199$	1687.99	+	974.564i	0.601300	+	0.347161i	0.769553	−	0.638583i	$-0.220479\pi$
$199$	1687.99	+	974.564i	0.601300	+	0.347161i	−0.168253	+	0.985744i	$0.553812\pi$
$200$		0			0
$201$	724.945	−	1516.45i	0.254396	−	0.532149i
$202$		0			0
$203$	555.532	+	4887.20i	0.192073	+	1.68972i
$204$		0			0
$205$	179.880	+	311.562i	0.0612849	+	0.106148i
$206$		0			0
$207$	−1451.84	+	1171.46i	−0.487489	+	0.393344i
$208$		0			0
$209$	−196.687			−0.0650963
$210$		0			0
$211$	−3341.96			−1.09038			−0.545189	−	0.838313i	$-0.683542\pi$
$211$	−3341.96			−1.09038			−0.545189	+	0.838313i	$0.683542\pi$
$212$		0			0
$213$	18.3103	+	235.298i	0.00589015	+	0.0756917i
$214$		0			0
$215$	1302.83	+	2256.56i	0.413265	+	0.715796i
$216$		0			0
$217$	−173.699	−	75.5805i	−0.0543386	−	0.0236440i
$218$		0			0
$219$	−2635.59	−	1259.96i	−0.813228	−	0.388767i
$220$		0			0
$221$	1157.68	+	668.390i	0.352372	+	0.203442i
$222$		0			0
$223$			2143.28i			0.643608i	0.946806	+	0.321804i	$0.104289\pi$
$223$			2143.28i			0.643608i	−0.946806	+	0.321804i	$0.895711\pi$
$224$		0			0
$225$	−93.7981	+	243.075i	−0.0277920	+	0.0720221i
$226$		0			0
$227$	1284.55	−	2224.91i	0.375589	−	0.650540i	−0.614826	−	0.788663i	$-0.710774\pi$
$227$	1284.55	−	2224.91i	0.375589	−	0.650540i	0.990415	+	0.138123i	$0.0441070\pi$
$228$		0			0
$229$	91.0827	−	52.5866i	0.0262835	−	0.0151748i	−0.486801	−	0.873513i	$-0.661836\pi$
$229$	91.0827	−	52.5866i	0.0262835	−	0.0151748i	0.513084	+	0.858338i	$0.328503\pi$
$230$		0			0
$231$	1629.56	+	563.180i	0.464142	+	0.160409i
$232$		0			0
$233$	−2273.94	+	1312.86i	−0.639360	+	0.369135i	−0.784368	−	0.620296i	$-0.787013\pi$
$233$	−2273.94	+	1312.86i	−0.639360	+	0.369135i	0.145008	+	0.989431i	$0.453679\pi$
$234$		0			0
$235$	949.887	−	1645.25i	0.263676	−	0.456700i
$236$		0			0
$237$	850.127	+	1239.27i	0.233003	+	0.339660i
$238$		0			0
$239$		−	6080.85i		−	1.64576i	−0.568212	−	0.822882i	$-0.692365\pi$
$239$		−	6080.85i		−	1.64576i	0.568212	−	0.822882i	$-0.307635\pi$
$240$		0			0
$241$	4008.74	+	2314.45i	1.07147	+	0.618616i	0.928584	−	0.371122i	$-0.121027\pi$
$241$	4008.74	+	2314.45i	1.07147	+	0.618616i	0.142891	+	0.989738i	$0.454360\pi$
$242$		0			0
$243$	3505.99	−	1434.21i	0.925552	−	0.378621i
$244$		0			0
$245$	2712.92	+	2912.30i	0.707437	+	0.759428i
$246$		0			0
$247$	342.624	+	593.442i	0.0882617	+	0.152874i
$248$		0			0
$249$	2324.83	−	180.913i	0.591688	−	0.0460438i
$250$		0			0
$251$	5967.85			1.50075			0.750373	−	0.661015i	$-0.229874\pi$
$251$	5967.85			1.50075			0.750373	+	0.661015i	$0.229874\pi$
$252$		0			0
$253$	−1237.88			−0.307607
$254$		0			0
$255$	−1287.42	+	100.184i	−0.316162	+	0.0246030i
$256$		0			0
$257$	2819.70	+	4883.86i	0.684389	+	1.18540i	0.973628	+	0.228140i	$0.0732643\pi$
$257$	2819.70	+	4883.86i	0.684389	+	1.18540i	−0.289239	+	0.957257i	$0.593402\pi$
$258$		0			0
$259$	−458.681	−	620.211i	−0.110043	−	0.148796i
$260$		0			0
$261$	1109.30	+	7084.42i	0.263081	+	1.68013i
$262$		0			0
$263$	−3018.63	−	1742.81i	−0.707745	−	0.408617i	0.102480	−	0.994735i	$-0.467322\pi$
$263$	−3018.63	−	1742.81i	−0.707745	−	0.408617i	−0.810226	+	0.586118i	$0.800655\pi$
$264$		0			0
$265$		−	6117.79i		−	1.41816i
$266$		0			0
$267$	1650.83	+	2406.50i	0.378387	+	0.551594i
$268$		0			0
$269$	1897.28	−	3286.18i	0.430033	−	0.744839i	−0.566842	−	0.823826i	$-0.691835\pi$
$269$	1897.28	−	3286.18i	0.430033	−	0.744839i	0.996876	+	0.0789869i	$0.0251685\pi$
$270$		0			0
$271$	6458.49	−	3728.81i	1.44769	−	0.835827i	0.449351	−	0.893356i	$-0.351655\pi$
$271$	6458.49	−	3728.81i	1.44769	−	0.835827i	0.998344	+	0.0575288i	$0.0183221\pi$
$272$		0			0
$273$	−1139.43	−	5897.72i	−0.252605	−	1.30750i
$274$		0			0
$275$	−149.723	+	86.4428i	−0.0328315	+	0.0189553i
$276$		0			0
$277$	1707.75	−	2957.90i	0.370428	−	0.641600i	−0.619203	−	0.785231i	$-0.712544\pi$
$277$	1707.75	−	2957.90i	0.370428	−	0.641600i	0.989631	+	0.143631i	$0.0458777\pi$
$278$		0			0
$279$	−257.646	−	99.4210i	−0.0552863	−	0.0213340i
$280$		0			0
$281$		−	2762.14i		−	0.586390i	−0.956053	−	0.293195i	$-0.905281\pi$
$281$		−	2762.14i		−	0.586390i	0.956053	−	0.293195i	$-0.0947185\pi$
$282$		0			0
$283$	4767.64	+	2752.60i	1.00144	+	0.578181i	0.908674	−	0.417507i	$-0.137096\pi$
$283$	4767.64	+	2752.60i	1.00144	+	0.578181i	0.0927647	+	0.995688i	$0.470430\pi$
$284$		0			0
$285$	−597.208	−	285.498i	−0.124125	−	0.0593383i
$286$		0			0
$287$	570.519	−	64.8515i	0.117340	−	0.0133382i
$288$		0			0
$289$	2227.17	+	3857.57i	0.453322	+	0.785176i
$290$		0			0
$291$	86.4172	+	1110.51i	0.0174085	+	0.223709i
$292$		0			0
$293$	−4101.08			−0.817705			−0.408853	−	0.912600i	$-0.634071\pi$
$293$	−4101.08			−0.817705			−0.408853	+	0.912600i	$0.634071\pi$
$294$		0			0
$295$	−4780.29			−0.943455
$296$		0			0
$297$	2408.15	+	720.235i	0.470488	+	0.140715i
$298$		0			0
$299$	2156.35	+	3734.91i	0.417074	+	0.722393i
$300$		0			0
$301$	4132.12	−	469.702i	0.791267	−	0.0899441i
$302$		0			0
$303$	−3849.55	+	8052.54i	−0.729871	+	1.52675i
$304$		0			0
$305$	2597.03	+	1499.40i	0.487560	+	0.281493i
$306$		0			0
$307$		−	8281.42i		−	1.53956i	−0.638308	−	0.769781i	$-0.720365\pi$
$307$		−	8281.42i		−	1.53956i	0.638308	−	0.769781i	$-0.279635\pi$
$308$		0			0
$309$	−4960.63	+	3402.94i	−0.913270	+	0.626493i
$310$		0			0
$311$	−3435.52	+	5950.50i	−0.626401	+	1.08496i	0.361868	+	0.932230i	$0.382139\pi$
$311$	−3435.52	+	5950.50i	−0.626401	+	1.08496i	−0.988268	+	0.152728i	$0.951194\pi$
$312$		0			0
$313$	−2922.40	+	1687.25i	−0.527743	+	0.304693i	−0.740097	−	0.672500i	$-0.765220\pi$
$313$	−2922.40	+	1687.25i	−0.527743	+	0.304693i	0.212354	+	0.977193i	$0.431887\pi$
$314$		0			0
$315$	4130.40	+	4075.36i	0.738800	+	0.728954i
$316$		0			0
$317$	−2052.28	+	1184.88i	−0.363620	+	0.209936i	−0.670667	−	0.741758i	$-0.733992\pi$
$317$	−2052.28	+	1184.88i	−0.363620	+	0.209936i	0.307048	+	0.951694i	$0.400659\pi$
$318$		0			0
$319$	−2379.09	+	4120.71i	−0.417566	+	0.723246i
$320$		0			0
$321$	5218.09	−	3579.55i	0.907306	−	0.622401i
$322$		0			0
$323$		−	235.116i		−	0.0405021i
$324$		0			0
$325$	521.629	+	301.163i	0.0890300	+	0.0514015i
$326$		0			0
$327$	−2909.57	+	6086.26i	−0.492047	+	1.02927i
$328$		0			0
$329$	−1802.94	−	2437.86i	−0.302125	−	0.408521i
$330$		0			0
$331$	1901.80	+	3294.02i	0.315808	+	0.546996i	0.979609	−	0.200913i	$-0.0643910\pi$
$331$	1901.80	+	3294.02i	0.315808	+	0.546996i	−0.663801	+	0.747910i	$0.731058\pi$
$332$		0			0
$333$	−706.189	−	875.212i	−0.116213	−	0.144028i
$334$		0			0
$335$	3753.55			0.612175
$336$		0			0
$337$	−592.955			−0.0958466			−0.0479233	−	0.998851i	$-0.515260\pi$
$337$	−592.955			−0.0958466			−0.0479233	+	0.998851i	$0.515260\pi$
$338$		0			0
$339$	−579.649	−	7448.81i	−0.0928679	−	1.19340i
$340$		0			0
$341$	−91.6249	−	158.699i	−0.0145506	−	0.0252024i
$342$		0			0
$343$	5989.74	−	2115.80i	0.942903	−	0.333068i
$344$		0			0
$345$	−3758.61	−	1796.82i	−0.586542	−	0.280399i
$346$		0			0
$347$	−4137.14	−	2388.58i	−0.640039	−	0.369527i	0.144590	−	0.989492i	$-0.453814\pi$
$347$	−4137.14	−	2388.58i	−0.640039	−	0.369527i	−0.784630	+	0.619965i	$0.787147\pi$
$348$		0			0
$349$		−	7358.26i		−	1.12859i	−0.825573	−	0.564296i	$-0.809148\pi$
$349$		−	7358.26i		−	1.12859i	0.825573	−	0.564296i	$-0.190852\pi$
$350$		0			0
$351$	−2021.84	−	8520.47i	−0.307459	−	1.29569i
$352$		0			0
$353$	−1652.78	+	2862.69i	−0.249202	+	0.431631i	−0.963305	−	0.268410i	$-0.913502\pi$
$353$	−1652.78	+	2862.69i	−0.249202	+	0.431631i	0.714102	+	0.700041i	$0.246835\pi$
$354$		0			0
$355$	−456.436	+	263.524i	−0.0682398	+	0.0393982i
$356$		0			0
$357$	−673.213	+	1947.94i	−0.0998046	+	0.288784i
$358$		0			0
$359$	−359.154	+	207.358i	−0.0528006	+	0.0304845i	−0.526168	−	0.850381i	$-0.676372\pi$
$359$	−359.154	+	207.358i	−0.0528006	+	0.0304845i	0.473367	+	0.880865i	$0.343038\pi$
$360$		0			0
$361$	−3369.24	+	5835.69i	−0.491214	+	0.850808i
$362$		0			0
$363$	−2968.81	−	4327.79i	−0.429263	−	0.625758i
$364$		0			0
$365$		−	6523.69i		−	0.935522i
$366$		0			0
$367$	65.9242	+	38.0613i	0.00937661	+	0.00541359i	0.504681	−	0.863306i	$-0.331610\pi$
$367$	65.9242	+	38.0613i	0.00937661	+	0.00541359i	−0.495304	+	0.868720i	$0.664943\pi$
$368$		0			0
$369$	827.018	−	129.497i	0.116674	−	0.0182693i
$370$		0			0
$371$	−8953.38	−	3895.82i	−1.25293	−	0.545178i
$372$		0			0
$373$	6150.49	+	10653.0i	0.853781	+	1.47879i	0.877771	+	0.479080i	$0.159030\pi$
$373$	6150.49	+	10653.0i	0.853781	+	1.47879i	−0.0239900	+	0.999712i	$0.507637\pi$
$374$		0			0
$375$	6934.13	−	539.598i	0.954872	−	0.0743059i
$376$		0			0
$377$	16577.3			2.26465
$378$		0			0
$379$	1429.02			0.193678			0.0968389	−	0.995300i	$-0.469127\pi$
$379$	1429.02			0.193678			0.0968389	+	0.995300i	$0.469127\pi$
$380$		0			0
$381$	−13916.5	+	1082.95i	−1.87129	+	0.145620i
$382$		0			0
$383$	2555.45	+	4426.17i	0.340933	+	0.590513i	0.984606	−	0.174787i	$-0.0559237\pi$
$383$	2555.45	+	4426.17i	0.340933	+	0.590513i	−0.643673	+	0.765300i	$0.722590\pi$
$384$		0			0
$385$	434.863	+	3825.63i	0.0575654	+	0.506421i
$386$		0			0
$387$	5989.87	−	937.915i	0.786776	−	0.123196i
$388$		0			0
$389$	−6339.84	−	3660.31i	−0.826331	−	0.477083i	0.0262636	−	0.999655i	$-0.491639\pi$
$389$	−6339.84	−	3660.31i	−0.826331	−	0.477083i	−0.852595	+	0.522572i	$0.824972\pi$
$390$		0			0
$391$		−	1479.73i		−	0.191390i
$392$		0			0
$393$	−4712.61	−	6869.82i	−0.604885	−	0.881772i
$394$		0			0
$395$	−1678.04	+	2906.45i	−0.213750	+	0.370227i
$396$		0			0
$397$	7516.61	−	4339.72i	0.950247	−	0.548625i	0.0570893	−	0.998369i	$-0.481818\pi$
$397$	7516.61	−	4339.72i	0.950247	−	0.548625i	0.893158	+	0.449744i	$0.148485\pi$
$398$		0			0
$399$	−798.129	+	692.207i	−0.100141	+	0.0868514i
$400$		0			0
$401$	−8447.68	+	4877.27i	−1.05201	+	0.607379i	−0.923212	−	0.384291i	$-0.874446\pi$
$401$	−8447.68	+	4877.27i	−1.05201	+	0.607379i	−0.128800	+	0.991671i	$0.541113\pi$
$402$		0			0
$403$	−319.217	+	552.899i	−0.0394573	+	0.0683421i
$404$		0			0
$405$	6266.49	+	5682.38i	0.768851	+	0.697185i
$406$		0			0
$407$		−	746.226i		−	0.0908822i
$408$		0			0
$409$	2935.32	+	1694.71i	0.354871	+	0.204885i	0.666828	−	0.745211i	$-0.267651\pi$
$409$	2935.32	+	1694.71i	0.354871	+	0.204885i	−0.311958	+	0.950096i	$0.600985\pi$
$410$		0			0
$411$	−10869.3	−	5196.10i	−1.30448	−	0.623612i
$412$		0			0
$413$	−3044.10	+	6995.95i	−0.362689	+	0.833531i
$414$		0			0
$415$	2603.72	+	4509.77i	0.307979	+	0.533436i
$416$		0			0
$417$	−742.400	−	9540.25i	−0.0871834	−	1.12036i
$418$		0			0
$419$	12777.4			1.48977			0.744887	−	0.667191i	$-0.232504\pi$
$419$	12777.4			1.48977			0.744887	+	0.667191i	$0.232504\pi$
$420$		0			0
$421$	11005.4			1.27404			0.637020	−	0.770848i	$-0.280167\pi$
$421$	11005.4			1.27404			0.637020	+	0.770848i	$0.280167\pi$
$422$		0			0
$423$	−2775.81	−	3440.19i	−0.319065	−	0.395432i
$424$		0			0
$425$	−103.332	−	178.976i	−0.0117937	−	0.0204273i
$426$		0			0
$427$	3848.17	−	2845.94i	0.436126	−	0.322540i
$428$		0			0
$429$	2506.22	−	5242.54i	0.282055	−	0.590005i
$430$		0			0
$431$	3279.13	+	1893.21i	0.366474	+	0.211584i	0.671917	−	0.740627i	$-0.265471\pi$
$431$	3279.13	+	1893.21i	0.366474	+	0.211584i	−0.305443	+	0.952210i	$0.598805\pi$
$432$		0			0
$433$			3191.67i			0.354230i	0.984190	+	0.177115i	$0.0566765\pi$
$433$			3191.67i			0.354230i	−0.984190	+	0.177115i	$0.943323\pi$
$434$		0			0
$435$	−13205.1	+	9058.53i	−1.45548	+	0.998444i
$436$		0			0
$437$	379.264	−	656.904i	0.0415163	−	0.0719084i
$438$		0			0
$439$	2872.90	−	1658.67i	0.312337	−	0.180328i	−0.335635	−	0.941992i	$-0.608951\pi$
$439$	2872.90	−	1658.67i	0.312337	−	0.180328i	0.647972	+	0.761664i	$0.275617\pi$
$440$		0			0
$441$	8594.54	−	3449.65i	0.928035	−	0.372492i
$442$		0			0
$443$	−9266.95	+	5350.27i	−0.993873	+	0.573813i	−0.906430	−	0.422356i	$-0.861203\pi$
$443$	−9266.95	+	5350.27i	−0.993873	+	0.573813i	−0.0874435	+	0.996169i	$0.527870\pi$
$444$		0			0
$445$	−3258.53	+	5643.94i	−0.347122	+	0.601232i
$446$		0			0
$447$	5574.51	−	3824.05i	0.589855	−	0.404634i
$448$		0			0
$449$		−	4017.92i		−	0.422310i	−0.977453	−	0.211155i	$-0.932278\pi$
$449$		−	4017.92i		−	0.422310i	0.977453	−	0.211155i	$-0.0677225\pi$
$450$		0			0
$451$	481.042	+	277.729i	0.0502248	+	0.0289973i
$452$		0			0
$453$	−5863.87	+	12266.1i	−0.608186	+	1.27221i
$454$		0			0
$455$	10785.1	−	7976.22i	1.11124	−	0.821826i
$456$		0			0
$457$	−4584.83	−	7941.15i	−0.469298	−	0.812848i	0.530086	−	0.847944i	$-0.322160\pi$
$457$	−4584.83	−	7941.15i	−0.469298	−	0.812848i	−0.999384	+	0.0350961i	$0.988826\pi$
$458$		0			0
$459$	−860.955	+	2878.65i	−0.0875510	+	0.292732i
$460$		0			0
$461$	1289.80			0.130308			0.0651542	−	0.997875i	$-0.479246\pi$
$461$	1289.80			0.130308			0.0651542	+	0.997875i	$0.479246\pi$
$462$		0			0
$463$	−6976.52			−0.700273			−0.350137	−	0.936699i	$-0.613865\pi$
$463$	−6976.52			−0.700273			−0.350137	+	0.936699i	$0.613865\pi$
$464$		0			0
$465$	−47.8470	−	614.860i	−0.00477172	−	0.0613192i
$466$		0			0
$467$	−3931.83	−	6810.13i	−0.389600	−	0.674808i	0.602795	−	0.797896i	$-0.294054\pi$
$467$	−3931.83	−	6810.13i	−0.389600	−	0.674808i	−0.992396	+	0.123088i	$0.960720\pi$
$468$		0			0
$469$	2390.27	−	5493.32i	0.235336	−	0.540849i
$470$		0			0
$471$	−4384.05	−	2095.82i	−0.428889	−	0.205032i
$472$		0			0
$473$	3484.06	+	2011.52i	0.338683	+	0.195539i
$474$		0			0
$475$		−	105.938i		−	0.0102332i
$476$		0			0
$477$	−13280.5	−	5124.69i	−1.27478	−	0.491915i
$478$		0			0
$479$	1847.13	−	3199.33i	0.176196	−	0.305180i	−0.764379	−	0.644767i	$-0.776954\pi$
$479$	1847.13	−	3199.33i	0.176196	−	0.305180i	0.940574	+	0.339588i	$0.110288\pi$
$480$		0			0
$481$	−2251.51	+	1299.91i	−0.213430	+	0.123224i
$482$		0			0
$483$	−5023.14	+	4356.51i	−0.473211	+	0.410410i
$484$		0			0
$485$	−2154.19	+	1243.72i	−0.201684	+	0.116442i
$486$		0			0
$487$	−526.359	+	911.681i	−0.0489766	+	0.0848300i	−0.889474	−	0.456985i	$-0.848929\pi$
$487$	−526.359	+	911.681i	−0.0489766	+	0.0848300i	0.840498	+	0.541815i	$0.182263\pi$
$488$		0			0
$489$	−1523.06	−	2220.24i	−0.140849	−	0.205322i
$490$		0			0
$491$		−	2378.40i		−	0.218607i	−0.994008	−	0.109303i	$-0.965138\pi$
$491$		−	2378.40i		−	0.218607i	0.994008	−	0.109303i	$-0.0348620\pi$
$492$		0			0
$493$	−4925.81	−	2843.92i	−0.449995	−	0.259805i
$494$		0			0
$495$	868.348	+	5545.59i	0.0788471	+	0.503547i
$496$		0			0
$497$	95.0069	+	835.807i	0.00857474	+	0.0754347i
$498$		0			0
$499$	9385.74	+	16256.6i	0.842011	+	1.45840i	0.888192	+	0.459472i	$0.151961\pi$
$499$	9385.74	+	16256.6i	0.842011	+	1.45840i	−0.0461818	+	0.998933i	$0.514705\pi$
$500$		0			0
$501$	−19519.9	+	1519.00i	−1.74069	+	0.135457i
$502$		0			0
$503$	−16095.2			−1.42674			−0.713370	−	0.700788i	$-0.752832\pi$
$503$	−16095.2			−1.42674			−0.713370	+	0.700788i	$0.752832\pi$
$504$		0			0
$505$	−19931.9			−1.75635
$506$		0			0
$507$	−8801.98	+	684.950i	−0.771025	+	0.0599994i
$508$		0			0
$509$	−1575.31	−	2728.52i	−0.137180	−	0.237603i	0.789248	−	0.614074i	$-0.210471\pi$
$509$	−1575.31	−	2728.52i	−0.137180	−	0.237603i	−0.926428	+	0.376472i	$0.877137\pi$
$510$		0			0
$511$	−9547.42	−	4154.30i	−0.826522	−	0.359639i
$512$		0			0
$513$	−1120.02	+	1057.26i	−0.0963940	+	0.0909928i
$514$		0			0
$515$	−11634.1	−	6716.95i	−0.995456	−	0.574727i
$516$		0			0
$517$		−	2933.19i		−	0.249519i
$518$		0			0
$519$	−7034.08	−	10253.9i	−0.594917	−	0.867241i
$520$		0			0
$521$	2489.60	−	4312.12i	0.209350	−	0.362605i	−0.742160	−	0.670223i	$-0.766198\pi$
$521$	2489.60	−	4312.12i	0.209350	−	0.362605i	0.951510	+	0.307618i	$0.0995318\pi$
$522$		0			0
$523$	−11977.0	+	6914.92i	−1.00137	+	0.578142i	−0.908654	−	0.417550i	$-0.862889\pi$
$523$	−11977.0	+	6914.92i	−1.00137	+	0.578142i	−0.0927181	+	0.995692i	$0.529556\pi$
$524$		0			0
$525$	−303.336	+	877.700i	−0.0252165	+	0.0729638i
$526$		0			0
$527$	189.705	−	109.526i	0.0156806	−	0.00905322i
$528$		0			0
$529$	−3696.55	+	6402.61i	−0.303818	+	0.526228i
$530$		0			0
$531$	−4004.31	+	10377.0i	−0.327254	+	0.848069i
$532$		0			0
$533$		−	1935.19i		−	0.157265i
$534$		0			0
$535$	12237.9	+	7065.56i	0.988955	+	0.570973i
$536$		0			0
$537$	3001.00	+	1434.64i	0.241159	+	0.115287i
$538$		0			0
$539$	5875.73	+	1799.75i	0.469547	+	0.143823i
$540$		0			0
$541$	−2658.97	−	4605.48i	−0.211309	−	0.365998i	0.740815	−	0.671709i	$-0.234439\pi$
$541$	−2658.97	−	4605.48i	−0.211309	−	0.365998i	−0.952124	+	0.305711i	$0.901106\pi$
$542$		0			0
$543$	−1694.79	−	21779.0i	−0.133942	−	1.72123i
$544$		0			0
$545$	−15064.9			−1.18405
$546$		0			0
$547$	9266.96			0.724363			0.362181	−	0.932108i	$-0.382032\pi$
$547$	9266.96			0.724363			0.362181	+	0.932108i	$0.382032\pi$
$548$		0			0
$549$	5430.34	−	4381.63i	0.422152	−	0.340625i
$550$		0			0
$551$	−1457.82	−	2525.03i	−0.112714	−	0.195226i
$552$		0			0
$553$	3185.01	+	4306.65i	0.244920	+	0.331171i
$554$		0			0
$555$	1083.17	−	2265.79i	0.0828435	−	0.173293i
$556$		0			0
$557$	125.920	+	72.6999i	0.00957881	+	0.00553033i	0.504782	−	0.863247i	$-0.331573\pi$
$557$	125.920	+	72.6999i	0.00957881	+	0.00553033i	−0.495203	+	0.868777i	$0.664906\pi$
$558$		0			0
$559$		−	14016.1i		−	1.06050i
$560$		0			0
$561$	−1644.09	+	1127.83i	−0.123732	+	0.0848785i
$562$		0			0
$563$	1958.12	−	3391.56i	0.146581	−	0.253885i	−0.783381	−	0.621542i	$-0.786507\pi$
$563$	1958.12	−	3391.56i	0.146581	−	0.253885i	0.929962	+	0.367657i	$0.119840\pi$
$564$		0			0
$565$	14449.4	−	8342.36i	1.07591	−	0.621178i
$566$		0			0
$567$	12306.7	−	5552.46i	0.911521	−	0.411254i
$568$		0			0
$569$	7404.97	−	4275.26i	0.545576	−	0.314988i	−0.201760	−	0.979435i	$-0.564666\pi$
$569$	7404.97	−	4275.26i	0.545576	−	0.314988i	0.747336	+	0.664447i	$0.231333\pi$
$570$		0			0
$571$	−11956.8	+	20709.8i	−0.876318	+	1.51783i	−0.0209659	+	0.999780i	$0.506674\pi$
$571$	−11956.8	+	20709.8i	−0.876318	+	1.51783i	−0.855352	+	0.518047i	$0.826659\pi$
$572$		0			0
$573$	−6243.32	+	4282.84i	−0.455180	+	0.312248i
$574$		0			0
$575$		−	666.737i		−	0.0483563i
$576$		0			0
$577$	11347.8	+	6551.66i	0.818745	+	0.472703i	0.849983	−	0.526809i	$-0.176612\pi$
$577$	11347.8	+	6551.66i	0.818745	+	0.472703i	−0.0312386	+	0.999512i	$0.509945\pi$
$578$		0			0
$579$	−4099.61	+	8575.60i	−0.294255	+	0.615527i
$580$		0			0
$581$	8258.10	−	938.706i	0.589679	−	0.0670294i
$582$		0			0
$583$	−4722.83	−	8180.18i	−0.335506	−	0.581113i
$584$		0			0
$585$	15219.5	−	12280.3i	1.07564	−	0.867908i
$586$		0			0
$587$	18034.7			1.26809			0.634047	−	0.773294i	$-0.281392\pi$
$587$	18034.7			1.26809			0.634047	+	0.773294i	$0.281392\pi$
$588$		0			0
$589$	112.289			0.00785532
$590$		0			0
$591$	266.539	+	3425.18i	0.0185515	+	0.238398i
$592$		0			0
$593$	−11358.1	−	19672.8i	−0.786547	−	1.36234i	−0.928071	−	0.372404i	$-0.878534\pi$
$593$	−11358.1	−	19672.8i	−0.786547	−	1.36234i	0.141524	−	0.989935i	$-0.454800\pi$
$594$		0			0
$595$	−4573.08	+	519.826i	−0.315089	+	0.0358165i
$596$		0			0
$597$	9137.52	+	4368.23i	0.626422	+	0.299464i
$598$		0			0
$599$	11720.4	+	6766.78i	0.799471	+	0.461575i	0.843286	−	0.537465i	$-0.180618\pi$
$599$	11720.4	+	6766.78i	0.799471	+	0.461575i	−0.0438153	+	0.999040i	$0.513951\pi$
$600$		0			0
$601$		−	3667.98i		−	0.248952i	−0.992223	−	0.124476i	$-0.960275\pi$
$601$		−	3667.98i		−	0.248952i	0.992223	−	0.124476i	$-0.0397250\pi$
$602$		0			0
$603$	3144.24	−	8148.20i	0.212344	−	0.550282i
$604$		0			0
$605$	5860.05	−	10149.9i	0.393794	−	0.682070i
$606$		0			0
$607$	6942.92	−	4008.50i	0.464258	−	0.268039i	−0.249575	−	0.968355i	$-0.580291\pi$
$607$	6942.92	−	4008.50i	0.464258	−	0.268039i	0.713833	+	0.700316i	$0.246958\pi$
$608$		0			0
$609$	4848.12	+	25094.1i	0.322588	+	1.66973i
$610$		0			0
$611$	−8849.99	+	5109.54i	−0.585977	+	0.338314i
$612$		0			0
$613$	−4698.26	+	8137.63i	−0.309561	+	0.536176i	−0.978266	−	0.207352i	$-0.933515\pi$
$613$	−4698.26	+	8137.63i	−0.309561	+	0.536176i	0.668705	+	0.743528i	$0.266849\pi$
$614$		0			0
$615$	1057.47	+	1541.53i	0.0693355	+	0.101074i
$616$		0			0
$617$			8906.76i			0.581155i	0.956851	+	0.290578i	$0.0938474\pi$
$617$			8906.76i			0.581155i	−0.956851	+	0.290578i	$0.906153\pi$
$618$		0			0
$619$	6091.73	+	3517.06i	0.395553	+	0.228373i	0.684563	−	0.728953i	$-0.259993\pi$
$619$	6091.73	+	3517.06i	0.395553	+	0.228373i	−0.289010	+	0.957326i	$0.593326\pi$
$620$		0			0
$621$	−7049.01	+	6654.03i	−0.455502	+	0.429979i
$622$		0			0
$623$	6184.86	+	8362.93i	0.397739	+	0.537807i
$624$		0			0
$625$	8369.05	+	14495.6i	0.535619	+	0.927720i
$626$		0			0
$627$	−1018.93	+	79.2911i	−0.0649000	+	0.00505037i
$628$		0			0
$629$	892.024			0.0565458
$630$		0			0
$631$	12628.6			0.796728			0.398364	−	0.917227i	$-0.369578\pi$
$631$	12628.6			0.796728			0.398364	+	0.917227i	$0.369578\pi$
$632$		0			0
$633$	−17313.0	+	1347.26i	−1.08709	+	0.0845950i
$634$		0			0
$635$	−15585.9	−	26995.5i	−0.974026	−	1.68706i
$636$		0			0
$637$	−4805.19	−	20863.3i	−0.298883	−	1.29770i
$638$		0			0
$639$	189.713	+	1211.58i	0.0117448	+	0.0750066i
$640$		0			0
$641$	8593.58	+	4961.51i	0.529526	+	0.305722i	0.740823	−	0.671700i	$-0.234436\pi$
$641$	8593.58	+	4961.51i	0.529526	+	0.305722i	−0.211297	+	0.977422i	$0.567769\pi$
$642$		0			0
$643$		−	294.191i		−	0.0180432i	−0.999959	−	0.00902160i	$-0.997128\pi$
$643$		−	294.191i		−	0.0180432i	0.999959	−	0.00902160i	$-0.00287170\pi$
$644$		0			0
$645$	7658.97	+	11164.9i	0.467553	+	0.681576i
$646$		0			0
$647$	−3859.39	+	6684.67i	−0.234511	+	0.406184i	−0.959130	−	0.282965i	$-0.908682\pi$
$647$	−3859.39	+	6684.67i	−0.234511	+	0.406184i	0.724620	+	0.689149i	$0.242015\pi$
$648$		0			0
$649$	−6391.80	+	3690.30i	−0.386595	+	0.223201i
$650$		0			0
$651$	−930.316	−	321.520i	−0.0560092	−	0.0193569i
$652$		0			0
$653$	9846.89	−	5685.11i	0.590105	−	0.340697i	−0.175034	−	0.984562i	$-0.556004\pi$
$653$	9846.89	−	5685.11i	0.590105	−	0.340697i	0.765139	+	0.643865i	$0.222670\pi$
$654$		0			0
$655$	9302.09	−	16111.7i	0.554905	−	0.961123i
$656$		0			0
$657$	−14161.6	−	5464.70i	−0.840938	−	0.324503i
$658$		0			0
$659$			19795.1i			1.17012i	0.810990	+	0.585060i	$0.198929\pi$
$659$			19795.1i			1.17012i	−0.810990	+	0.585060i	$0.801071\pi$
$660$		0			0
$661$	26896.6	+	15528.7i	1.58268	+	0.913763i	0.994466	+	0.105062i	$0.0335041\pi$
$661$	26896.6	+	15528.7i	1.58268	+	0.913763i	0.588219	+	0.808702i	$0.299829\pi$
$662$		0			0
$663$	6266.83	+	2995.88i	0.367094	+	0.175491i
$664$		0			0
$665$	−2163.38	−	941.337i	−0.126154	−	0.0548924i
$666$		0			0
$667$	−9175.03	−	15891.6i	−0.532621	−	0.922527i
$668$		0			0
$669$	864.027	+	11103.2i	0.0499331	+	0.641668i
$670$		0			0
$671$	4630.04			0.266380
$672$		0			0
$673$	5340.26			0.305872			0.152936	−	0.988236i	$-0.451127\pi$
$673$	5340.26			0.305872			0.152936	+	0.988236i	$0.451127\pi$
$674$		0			0
$675$	−387.928	+	1297.06i	−0.0221205	+	0.0739612i
$676$		0			0
$677$	12478.3	+	21613.0i	0.708389	+	1.22697i	0.965455	+	0.260571i	$0.0839110\pi$
$677$	12478.3	+	21613.0i	0.708389	+	1.22697i	−0.257066	+	0.966394i	$0.582756\pi$
$678$		0			0
$679$	448.394	+	3944.67i	0.0253428	+	0.222949i
$680$		0			0
$681$	5757.68	−	12044.0i	0.323986	−	0.677718i
$682$		0			0
$683$	9706.83	+	5604.24i	0.543809	+	0.313968i	0.746621	−	0.665249i	$-0.231675\pi$
$683$	9706.83	+	5604.24i	0.543809	+	0.313968i	−0.202812	+	0.979218i	$0.565008\pi$
$684$		0			0
$685$		−	26903.9i		−	1.50065i
$686$		0			0
$687$	450.654	−	309.143i	0.0250269	−	0.0171682i
$688$		0			0
$689$	−16454.1	+	28499.4i	−0.909800	+	1.57582i
$690$		0			0
$691$	9472.37	−	5468.88i	0.521485	−	0.301079i	−0.216057	−	0.976381i	$-0.569320\pi$
$691$	9472.37	−	5468.88i	0.521485	−	0.301079i	0.737542	+	0.675301i	$0.235986\pi$
$692$		0			0
$693$	8668.93	+	2260.62i	0.475188	+	0.123916i
$694$		0			0
$695$	18506.4	−	10684.7i	1.01006	−	0.583156i
$696$		0			0
$697$	−331.992	+	575.027i	−0.0180418	+	0.0312492i
$698$		0			0
$699$	−11250.9	+	7717.97i	−0.608795	+	0.417626i
$700$		0			0
$701$			27949.3i			1.50589i	0.658082	+	0.752947i	$0.271368\pi$
$701$			27949.3i			1.50589i	−0.658082	+	0.752947i	$0.728632\pi$
$702$		0			0
$703$	396.000	+	228.631i	0.0212453	+	0.0122660i
$704$		0			0
$705$	4257.62	−	8906.14i	0.227449	−	0.475780i
$706$		0			0
$707$	−12692.7	+	29170.3i	−0.675186	+	1.55171i
$708$		0			0
$709$	−10472.7	−	18139.3i	−0.554741	−	0.960840i	−0.997924	−	0.0644082i	$-0.979484\pi$
$709$	−10472.7	−	18139.3i	−0.554741	−	0.960840i	0.443183	−	0.896431i	$-0.353849\pi$
$710$		0			0
$711$	4903.67	+	6077.33i	0.258653	+	0.320560i
$712$		0			0
$713$	706.707			0.0371197
$714$		0			0
$715$	12976.5			0.678731
$716$		0			0
$717$	−2451.40	−	31501.8i	−0.127684	−	1.64080i
$718$		0			0
$719$	4150.10	+	7188.19i	0.215261	+	0.372843i	0.953353	−	0.301857i	$-0.0976064\pi$
$719$	4150.10	+	7188.19i	0.215261	+	0.372843i	−0.738092	+	0.674700i	$0.764273\pi$
$720$		0			0
$721$	−17238.9	+	12749.1i	−0.890443	+	0.658533i
$722$		0			0
$723$	21700.3	+	10373.9i	1.11624	+	0.533623i
$724$		0			0
$725$	−2219.47	−	1281.41i	−0.113695	−	0.0656420i
$726$		0			0
$727$		−	20951.5i		−	1.06884i	−0.845218	−	0.534421i	$-0.820530\pi$
$727$		−	20951.5i		−	1.06884i	0.845218	−	0.534421i	$-0.179470\pi$
$728$		0			0
$729$	17584.5	−	8843.31i	0.893388	−	0.449287i
$730$		0			0
$731$	−2404.53	+	4164.77i	−0.121662	+	0.210724i
$732$		0			0
$733$	4885.73	−	2820.78i	0.246192	−	0.142139i	−0.371827	−	0.928302i	$-0.621269\pi$
$733$	4885.73	−	2820.78i	0.246192	−	0.142139i	0.618019	+	0.786163i	$0.287935\pi$
$734$		0			0
$735$	15228.3	+	13993.5i	0.764223	+	0.702254i
$736$		0			0
$737$	5018.93	−	2897.68i	0.250848	−	0.144827i
$738$		0			0
$739$	−5691.08	+	9857.24i	−0.283288	+	0.490669i	−0.972193	−	0.234183i	$-0.924758\pi$
$739$	−5691.08	+	9857.24i	−0.283288	+	0.490669i	0.688905	+	0.724852i	$0.258092\pi$
$740$		0			0
$741$	2014.20	+	2936.20i	0.0998560	+	0.145565i
$742$		0			0
$743$			4665.46i			0.230362i	0.993345	+	0.115181i	$0.0367449\pi$
$743$			4665.46i			0.230362i	−0.993345	+	0.115181i	$0.963255\pi$
$744$		0			0
$745$	13073.8	+	7548.17i	0.642936	+	0.371200i
$746$		0			0
$747$	11970.8	−	1874.44i	0.586332	−	0.0918100i
$748$		0			0
$749$	18133.6	−	13410.8i	0.884628	−	0.654233i
$750$		0			0
$751$	−4780.43	−	8279.95i	−0.232277	−	0.402316i	0.726200	−	0.687483i	$-0.241284\pi$
$751$	−4780.43	−	8279.95i	−0.232277	−	0.402316i	−0.958478	+	0.285167i	$0.907951\pi$
$752$		0			0
$753$	30916.4	−	2405.84i	1.49622	−	0.116433i
$754$		0			0
$755$	−30361.4			−1.46353
$756$		0			0
$757$	31574.1			1.51596			0.757979	−	0.652279i	$-0.226187\pi$
$757$	31574.1			1.51596			0.757979	+	0.652279i	$0.226187\pi$
$758$		0			0
$759$	−6412.81	+	499.030i	−0.306680	+	0.0238651i
$760$		0			0
$761$	−11215.0	−	19424.9i	−0.534221	−	0.925298i	−0.999201	−	0.0399765i	$-0.987272\pi$
$761$	−11215.0	−	19424.9i	−0.534221	−	0.925298i	0.464980	−	0.885321i	$-0.346062\pi$
$762$		0			0
$763$	−9593.35	+	22047.4i	−0.455180	+	1.04610i
$764$		0			0
$765$	−6629.08	+	1038.01i	−0.313301	+	0.0490577i
$766$		0			0
$767$	22268.7	+	12856.8i	1.04834	+	0.605259i
$768$		0			0
$769$			26887.4i			1.26084i	0.776254	+	0.630420i	$0.217117\pi$
$769$			26887.4i			1.26084i	−0.776254	+	0.630420i	$0.782883\pi$
$770$		0			0
$771$	16576.3	+	24164.1i	0.774293	+	1.12873i
$772$		0			0
$773$	−11104.6	+	19233.7i	−0.516693	+	0.894938i	0.483119	+	0.875555i	$0.339504\pi$
$773$	−11104.6	+	19233.7i	−0.516693	+	0.894938i	−0.999812	+	0.0193838i	$0.993830\pi$
$774$		0			0
$775$	85.4774	−	49.3504i	0.00396185	−	0.00228738i
$776$		0			0
$777$	−2626.22	−	3028.09i	−0.121255	−	0.139810i
$778$		0			0
$779$	−294.765	+	170.183i	−0.0135572	+	0.00782725i
$780$		0			0
$781$	−406.872	+	704.722i	−0.0186415	+	0.0322880i
$782$		0			0
$783$	8602.71	+	36253.6i	0.392638	+	1.65466i
$784$		0			0
$785$		−	10851.5i		−	0.493385i
$786$		0			0
$787$	17403.2	+	10047.8i	0.788257	+	0.455100i	0.839348	−	0.543594i	$-0.182937\pi$
$787$	17403.2	+	10047.8i	0.788257	+	0.455100i	−0.0510919	+	0.998694i	$0.516270\pi$
$788$		0			0
$789$	−16340.6	−	7811.69i	−0.737314	−	0.352476i

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

\(f(q)\)	\(=\)	\(q+(5.18049 - 0.403134i) q^{3} +(5.80193 + 10.0492i) q^{5} +(18.4018 - 2.09174i) q^{7} +(26.6750 - 4.17686i) q^{9} +O(q^{10})\)	\(q+(5.18049 - 0.403134i) q^{3} +(5.80193 + 10.0492i) q^{5} +(18.4018 - 2.09174i) q^{7} +(26.6750 - 4.17686i) q^{9} +(15.5157 + 8.95800i) q^{11} -62.4185i q^{13} +(34.1081 + 49.7211i) q^{15} +(-10.7082 + 18.5472i) q^{17} +(-9.50747 + 5.48914i) q^{19} +(94.4869 - 18.2546i) q^{21} +(-59.8367 + 34.5467i) q^{23} +(-4.82490 + 8.35697i) q^{25} +(136.506 - 32.3918i) q^{27} +265.583i q^{29} +(-8.85795 - 5.11414i) q^{31} +(83.9902 + 40.1519i) q^{33} +(127.786 + 172.788i) q^{35} +(-20.8257 - 36.0712i) q^{37} +(-25.1630 - 323.358i) q^{39} +31.0035 q^{41} +224.550 q^{43} +(196.741 + 243.829i) q^{45} +(-81.8595 - 141.785i) q^{47} +(334.249 - 76.9836i) q^{49} +(-47.9968 + 100.400i) q^{51} +(-456.586 - 263.610i) q^{53} +207.895i q^{55} +(-47.0405 + 32.2692i) q^{57} +(-205.978 + 356.765i) q^{59} +(223.807 - 129.215i) q^{61} +(482.129 - 132.659i) q^{63} +(627.258 - 362.148i) q^{65} +(161.737 - 280.137i) q^{67} +(-296.056 + 203.091i) q^{69} +45.4199i q^{71} +(-486.879 - 281.100i) q^{73} +(-21.6264 + 45.2383i) q^{75} +(304.254 + 132.388i) q^{77} +(144.610 + 250.473i) q^{79} +(694.108 - 222.835i) q^{81} +448.767 q^{83} -248.513 q^{85} +(107.066 + 1375.85i) q^{87} +(280.814 + 486.384i) q^{89} +(-130.563 - 1148.61i) q^{91} +(-47.9502 - 22.9228i) q^{93} +(-110.323 - 63.6953i) q^{95} +214.364i q^{97} +(451.297 + 174.147i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9	\( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 56 * q^7 - 3 * q^9 - 6 * q^15 - 300 * q^19 + 357 * q^21 - 42 * q^25 + 930 * q^31 - 855 * q^33 + 764 * q^37 + 426 * q^39 + 1012 * q^43 + 2367 * q^45 - 336 * q^49 + 1341 * q^51 + 270 * q^57 + 2358 * q^61 - 1071 * q^63 - 792 * q^67 - 2904 * q^73 + 2418 * q^75 - 1674 * q^79 + 837 * q^81 + 348 * q^85 - 1638 * q^87 + 1218 * q^91 - 1479 * q^93 + 3354 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more