LMFDB - Embedded newform 336.4.q.e.289.1

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.4.q.e.193.1

$q$-expansion

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{3} +(1.50000 - 2.59808i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(1.50000 + 2.59808i) q^{3} +(1.50000 - 2.59808i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-7.50000 - 12.9904i) q^{11} -64.0000 q^{13} +9.00000 q^{15} +(-42.0000 - 72.7461i) q^{17} +(-8.00000 + 13.8564i) q^{19} +(52.5000 - 18.1865i) q^{21} +(-42.0000 + 72.7461i) q^{23} +(58.0000 + 100.459i) q^{25} -27.0000 q^{27} -297.000 q^{29} +(-126.500 - 219.104i) q^{31} +(22.5000 - 38.9711i) q^{33} +(-42.0000 - 36.3731i) q^{35} +(158.000 - 273.664i) q^{37} +(-96.0000 - 166.277i) q^{39} +360.000 q^{41} -26.0000 q^{43} +(13.5000 + 23.3827i) q^{45} +(-15.0000 + 25.9808i) q^{47} +(-318.500 - 127.306i) q^{49} +(126.000 - 218.238i) q^{51} +(-181.500 - 314.367i) q^{53} -45.0000 q^{55} -48.0000 q^{57} +(-7.50000 - 12.9904i) q^{59} +(59.0000 - 102.191i) q^{61} +(126.000 + 109.119i) q^{63} +(-96.0000 + 166.277i) q^{65} +(-185.000 - 320.429i) q^{67} -252.000 q^{69} +342.000 q^{71} +(-181.000 - 313.501i) q^{73} +(-174.000 + 301.377i) q^{75} +(-262.500 + 90.9327i) q^{77} +(233.500 - 404.434i) q^{79} +(-40.5000 - 70.1481i) q^{81} -477.000 q^{83} -252.000 q^{85} +(-445.500 - 771.629i) q^{87} +(-453.000 + 784.619i) q^{89} +(-224.000 + 1163.94i) q^{91} +(379.500 - 657.313i) q^{93} +(24.0000 + 41.5692i) q^{95} +503.000 q^{97} +135.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} + O(q^{10}) $	$ 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} - 15q^{11} - 128q^{13} + 18q^{15} - 84q^{17} - 16q^{19} + 105q^{21} - 84q^{23} + 116q^{25} - 54q^{27} - 594q^{29} - 253q^{31} + 45q^{33} - 84q^{35} + 316q^{37} - 192q^{39} + 720q^{41} - 52q^{43} + 27q^{45} - 30q^{47} - 637q^{49} + 252q^{51} - 363q^{53} - 90q^{55} - 96q^{57} - 15q^{59} + 118q^{61} + 252q^{63} - 192q^{65} - 370q^{67} - 504q^{69} + 684q^{71} - 362q^{73} - 348q^{75} - 525q^{77} + 467q^{79} - 81q^{81} - 954q^{83} - 504q^{85} - 891q^{87} - 906q^{89} - 448q^{91} + 759q^{93} + 48q^{95} + 1006q^{97} + 270q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	1.50000	−	2.59808i	0.134164	−	0.232379i	−0.791114	−	0.611669i	$-0.790498\pi$
$5$	1.50000	−	2.59808i	0.134164	−	0.232379i	0.925278	+	0.379290i	$0.123832\pi$
$6$		0			0
$7$	3.50000	−	18.1865i	0.188982	−	0.981981i
$7$	3.50000	−	18.1865i	0.188982	−	0.981981i
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	0.744740	−	0.667355i	$-0.232573\pi$
$11$	−7.50000	−	12.9904i	−0.205576	−	0.356068i	−0.950316	+	0.311287i	$0.899240\pi$
$12$		0			0
$13$	−64.0000			−1.36542			−0.682708	−	0.730691i	$-0.739198\pi$
$13$	−64.0000			−1.36542			−0.682708	+	0.730691i	$0.739198\pi$
$14$		0			0
$15$	9.00000			0.154919
$16$		0			0
$17$	−42.0000	−	72.7461i	−0.599206	−	1.03785i	−0.992939	−	0.118630i	$-0.962150\pi$
$17$	−42.0000	−	72.7461i	−0.599206	−	1.03785i	0.393733	−	0.919225i	$-0.371183\pi$
$18$		0			0
$19$	−8.00000	+	13.8564i	−0.0965961	+	0.167309i	−0.910274	−	0.414007i	$-0.864129\pi$
$19$	−8.00000	+	13.8564i	−0.0965961	+	0.167309i	0.813678	+	0.581317i	$0.197462\pi$
$20$		0			0
$21$	52.5000	−	18.1865i	0.545545	−	0.188982i
$22$		0			0
$23$	−42.0000	+	72.7461i	−0.380765	+	0.659505i	−0.991172	−	0.132583i	$-0.957673\pi$
$23$	−42.0000	+	72.7461i	−0.380765	+	0.659505i	0.610406	+	0.792088i	$0.291006\pi$
$24$		0			0
$25$	58.0000	+	100.459i	0.464000	+	0.803672i
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−297.000			−1.90178			−0.950888	−	0.309535i	$-0.899827\pi$
$29$	−297.000			−1.90178			−0.950888	+	0.309535i	$0.899827\pi$
$30$		0			0
$31$	−126.500	−	219.104i	−0.732906	−	1.26943i	−0.955636	−	0.294550i	$-0.904830\pi$
$31$	−126.500	−	219.104i	−0.732906	−	1.26943i	0.222731	−	0.974880i	$-0.428503\pi$
$32$		0			0
$33$	22.5000	−	38.9711i	0.118689	−	0.205576i
$34$		0			0
$35$	−42.0000	−	36.3731i	−0.202837	−	0.175662i
$36$		0			0
$37$	158.000	−	273.664i	0.702028	−	1.21595i	−0.265725	−	0.964049i	$-0.585611\pi$
$37$	158.000	−	273.664i	0.702028	−	1.21595i	0.967753	−	0.251900i	$-0.0810553\pi$
$38$		0			0
$39$	−96.0000	−	166.277i	−0.394162	−	0.682708i
$40$		0			0
$41$	360.000			1.37128			0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000			1.37128			0.685641	+	0.727940i	$0.259522\pi$
$42$		0			0
$43$	−26.0000			−0.0922084			−0.0461042	−	0.998937i	$-0.514681\pi$
$43$	−26.0000			−0.0922084			−0.0461042	+	0.998937i	$0.514681\pi$
$44$		0			0
$45$	13.5000	+	23.3827i	0.0447214	+	0.0774597i
$46$		0			0
$47$	−15.0000	+	25.9808i	−0.0465527	+	0.0806316i	−0.888363	−	0.459142i	$-0.848157\pi$
$47$	−15.0000	+	25.9808i	−0.0465527	+	0.0806316i	0.841810	+	0.539774i	$0.181490\pi$
$48$		0			0
$49$	−318.500	−	127.306i	−0.928571	−	0.371154i
$50$		0			0
$51$	126.000	−	218.238i	0.345952	−	0.599206i
$52$		0			0
$53$	−181.500	−	314.367i	−0.470395	−	0.814748i	0.529032	−	0.848602i	$-0.322555\pi$
$53$	−181.500	−	314.367i	−0.470395	−	0.814748i	−0.999427	+	0.0338538i	$0.989222\pi$
$54$		0			0
$55$	−45.0000			−0.110324
$56$		0			0
$57$	−48.0000			−0.111540
$58$		0			0
$59$	−7.50000	−	12.9904i	−0.0165494	−	0.0286645i	0.857632	−	0.514264i	$-0.171935\pi$
$59$	−7.50000	−	12.9904i	−0.0165494	−	0.0286645i	−0.874182	+	0.485599i	$0.838601\pi$
$60$		0			0
$61$	59.0000	−	102.191i	0.123839	−	0.214495i	−0.797440	−	0.603399i	$-0.793813\pi$
$61$	59.0000	−	102.191i	0.123839	−	0.214495i	0.921279	+	0.388903i	$0.127146\pi$
$62$		0			0
$63$	126.000	+	109.119i	0.251976	+	0.218218i
$64$		0			0
$65$	−96.0000	+	166.277i	−0.183190	+	0.317294i
$66$		0			0
$67$	−185.000	−	320.429i	−0.337334	−	0.584279i	0.646597	−	0.762832i	$-0.276192\pi$
$67$	−185.000	−	320.429i	−0.337334	−	0.584279i	−0.983930	+	0.178553i	$0.942858\pi$
$68$		0			0
$69$	−252.000			−0.439670
$70$		0			0
$71$	342.000			0.571661			0.285831	−	0.958280i	$-0.407731\pi$
$71$	342.000			0.571661			0.285831	+	0.958280i	$0.407731\pi$
$72$		0			0
$73$	−181.000	−	313.501i	−0.290198	−	0.502638i	0.683658	−	0.729802i	$-0.260388\pi$
$73$	−181.000	−	313.501i	−0.290198	−	0.502638i	−0.973856	+	0.227165i	$0.927054\pi$
$74$		0			0
$75$	−174.000	+	301.377i	−0.267891	+	0.464000i
$76$		0			0
$77$	−262.500	+	90.9327i	−0.388502	+	0.134581i
$78$		0			0
$79$	233.500	−	404.434i	0.332542	−	0.575979i	−0.650468	−	0.759534i	$-0.725427\pi$
$79$	233.500	−	404.434i	0.332542	−	0.575979i	0.983010	+	0.183555i	$0.0587604\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−477.000			−0.630814			−0.315407	−	0.948957i	$-0.602141\pi$
$83$	−477.000			−0.630814			−0.315407	+	0.948957i	$0.602141\pi$
$84$		0			0
$85$	−252.000			−0.321568
$86$		0			0
$87$	−445.500	−	771.629i	−0.548996	−	0.950888i
$88$		0			0
$89$	−453.000	+	784.619i	−0.539527	+	0.934488i	0.459402	+	0.888228i	$0.348064\pi$
$89$	−453.000	+	784.619i	−0.539527	+	0.934488i	−0.998929	+	0.0462600i	$0.985270\pi$
$90$		0			0
$91$	−224.000	+	1163.94i	−0.258039	+	1.34081i
$92$		0			0
$93$	379.500	−	657.313i	0.423143	−	0.732906i
$94$		0			0
$95$	24.0000	+	41.5692i	0.0259195	+	0.0448938i
$96$		0			0
$97$	503.000			0.526515			0.263257	−	0.964726i	$-0.415203\pi$
$97$	503.000			0.526515			0.263257	+	0.964726i	$0.415203\pi$
$98$		0			0
$99$	135.000			0.137051
$100$		0			0
$101$	543.000	+	940.504i	0.534956	+	0.926570i	0.999165	+	0.0408451i	$0.0130050\pi$
$101$	543.000	+	940.504i	0.534956	+	0.926570i	−0.464210	+	0.885725i	$0.653662\pi$
$102$		0			0
$103$	868.000	−	1503.42i	0.830355	−	1.43822i	−0.0674017	−	0.997726i	$-0.521471\pi$
$103$	868.000	−	1503.42i	0.830355	−	1.43822i	0.897757	−	0.440491i	$-0.145196\pi$
$104$		0			0
$105$	31.5000	−	163.679i	0.0292770	−	0.152128i
$106$		0			0
$107$	−676.500	+	1171.73i	−0.611212	+	1.05865i	0.379824	+	0.925059i	$0.375985\pi$
$107$	−676.500	+	1171.73i	−0.611212	+	1.05865i	−0.991036	+	0.133592i	$0.957349\pi$
$108$		0			0
$109$	185.000	+	320.429i	0.162567	+	0.281574i	0.935789	−	0.352562i	$-0.114689\pi$
$109$	185.000	+	320.429i	0.162567	+	0.281574i	−0.773222	+	0.634136i	$0.781356\pi$
$110$		0			0
$111$	948.000			0.810632
$112$		0			0
$113$	−648.000			−0.539458			−0.269729	−	0.962936i	$-0.586934\pi$
$113$	−648.000			−0.539458			−0.269729	+	0.962936i	$0.586934\pi$
$114$		0			0
$115$	126.000	+	218.238i	0.102170	+	0.176964i
$116$		0			0
$117$	288.000	−	498.831i	0.227569	−	0.394162i
$118$		0			0
$119$	−1470.00	+	509.223i	−1.13239	+	0.392272i
$120$		0			0
$121$	553.000	−	957.824i	0.415477	−	0.719627i
$122$		0			0
$123$	540.000	+	935.307i	0.395855	+	0.685641i
$124$		0			0
$125$	723.000			0.517337
$126$		0			0
$127$	−377.000			−0.263412			−0.131706	−	0.991289i	$-0.542046\pi$
$127$	−377.000			−0.263412			−0.131706	+	0.991289i	$0.542046\pi$
$128$		0			0
$129$	−39.0000	−	67.5500i	−0.0266183	−	0.0461042i
$130$		0			0
$131$	−325.500	+	563.783i	−0.217092	+	0.376015i	−0.953918	−	0.300068i	$-0.902991\pi$
$131$	−325.500	+	563.783i	−0.217092	+	0.376015i	0.736826	+	0.676083i	$0.236324\pi$
$132$		0			0
$133$	224.000	+	193.990i	0.146040	+	0.126474i
$134$		0			0
$135$	−40.5000	+	70.1481i	−0.0258199	+	0.0447214i
$136$		0			0
$137$	885.000	+	1532.86i	0.551903	+	0.955923i	0.998137	+	0.0610074i	$0.0194313\pi$
$137$	885.000	+	1532.86i	0.551903	+	0.955923i	−0.446235	+	0.894916i	$0.647235\pi$
$138$		0			0
$139$	1558.00			0.950704			0.475352	−	0.879796i	$-0.342321\pi$
$139$	1558.00			0.950704			0.475352	+	0.879796i	$0.342321\pi$
$140$		0			0
$141$	−90.0000			−0.0537544
$142$		0			0
$143$	480.000	+	831.384i	0.280697	+	0.486181i
$144$		0			0
$145$	−445.500	+	771.629i	−0.255150	+	0.441933i
$146$		0			0
$147$	−147.000	−	1018.45i	−0.0824786	−	0.571429i
$148$		0			0
$149$	−1227.00	+	2125.23i	−0.674629	+	1.16849i	0.301948	+	0.953324i	$0.402363\pi$
$149$	−1227.00	+	2125.23i	−0.674629	+	1.16849i	−0.976577	+	0.215168i	$0.930970\pi$
$150$		0			0
$151$	629.500	+	1090.33i	0.339258	+	0.587612i	0.984293	−	0.176540i	$-0.0564906\pi$
$151$	629.500	+	1090.33i	0.339258	+	0.587612i	−0.645035	+	0.764153i	$0.723157\pi$
$152$		0			0
$153$	756.000			0.399470
$154$		0			0
$155$	−759.000			−0.393318
$156$		0			0
$157$	98.0000	+	169.741i	0.0498169	+	0.0862854i	0.889859	−	0.456236i	$-0.150803\pi$
$157$	98.0000	+	169.741i	0.0498169	+	0.0862854i	−0.840042	+	0.542522i	$0.817470\pi$
$158$		0			0
$159$	544.500	−	943.102i	0.271583	−	0.470395i
$160$		0			0
$161$	1176.00	+	1018.45i	0.575663	+	0.498539i
$162$		0			0
$163$	−626.000	+	1084.26i	−0.300810	+	0.521019i	−0.976320	−	0.216332i	$-0.930590\pi$
$163$	−626.000	+	1084.26i	−0.300810	+	0.521019i	0.675509	+	0.737351i	$0.263924\pi$
$164$		0			0
$165$	−67.5000	−	116.913i	−0.0318477	−	0.0551618i
$166$		0			0
$167$	2646.00			1.22607			0.613035	−	0.790056i	$-0.289949\pi$
$167$	2646.00			1.22607			0.613035	+	0.790056i	$0.289949\pi$
$168$		0			0
$169$	1899.00			0.864360
$170$		0			0
$171$	−72.0000	−	124.708i	−0.0321987	−	0.0557698i
$172$		0			0
$173$	393.000	−	680.696i	0.172712	−	0.299147i	−0.766655	−	0.642059i	$-0.778080\pi$
$173$	393.000	−	680.696i	0.172712	−	0.299147i	0.939367	+	0.342913i	$0.111414\pi$
$174$		0			0
$175$	2030.00	−	703.213i	0.876878	−	0.303759i
$176$		0			0
$177$	22.5000	−	38.9711i	0.00955482	−	0.0165494i
$178$		0			0
$179$	1446.00	+	2504.55i	0.603794	+	1.04580i	0.992241	+	0.124331i	$0.0396784\pi$
$179$	1446.00	+	2504.55i	0.603794	+	1.04580i	−0.388447	+	0.921471i	$0.626988\pi$
$180$		0			0
$181$	1352.00			0.555212			0.277606	−	0.960695i	$-0.410459\pi$
$181$	1352.00			0.555212			0.277606	+	0.960695i	$0.410459\pi$
$182$		0			0
$183$	354.000			0.142997
$184$		0			0
$185$	−474.000	−	820.992i	−0.188374	−	0.326273i
$186$		0			0
$187$	−630.000	+	1091.19i	−0.246365	+	0.426716i
$188$		0			0
$189$	−94.5000	+	491.036i	−0.0363696	+	0.188982i
$190$		0			0
$191$	1956.00	−	3387.89i	0.741001	−	1.28345i	−0.211039	−	0.977478i	$-0.567685\pi$
$191$	1956.00	−	3387.89i	0.741001	−	1.28345i	0.952040	−	0.305974i	$-0.0989820\pi$
$192$		0			0
$193$	−746.500	−	1292.98i	−0.278416	−	0.482230i	0.692575	−	0.721345i	$-0.256476\pi$
$193$	−746.500	−	1292.98i	−0.278416	−	0.482230i	−0.970991	+	0.239115i	$0.923143\pi$
$194$		0			0
$195$	−576.000			−0.211529
$196$		0			0
$197$	−4086.00			−1.47774			−0.738872	−	0.673846i	$-0.764641\pi$
$197$	−4086.00			−1.47774			−0.738872	+	0.673846i	$0.764641\pi$
$198$		0			0
$199$	−1778.00	−	3079.59i	−0.633362	−	1.09702i	−0.986860	−	0.161580i	$-0.948341\pi$
$199$	−1778.00	−	3079.59i	−0.633362	−	1.09702i	0.353497	−	0.935436i	$-0.384992\pi$
$200$		0			0
$201$	555.000	−	961.288i	0.194760	−	0.337334i
$202$		0			0
$203$	−1039.50	+	5401.40i	−0.359402	+	1.86751i
$204$		0			0
$205$	540.000	−	935.307i	0.183977	−	0.318657i
$206$		0			0
$207$	−378.000	−	654.715i	−0.126922	−	0.219835i
$208$		0			0
$209$	240.000			0.0794313
$210$		0			0
$211$	−1250.00			−0.407837			−0.203918	−	0.978988i	$-0.565368\pi$
$211$	−1250.00			−0.407837			−0.203918	+	0.978988i	$0.565368\pi$
$212$		0			0
$213$	513.000	+	888.542i	0.165024	+	0.285831i
$214$		0			0
$215$	−39.0000	+	67.5500i	−0.0123711	+	0.0214273i
$216$		0			0
$217$	−4427.50	+	1533.73i	−1.38506	+	0.479799i
$218$		0			0
$219$	543.000	−	940.504i	0.167546	−	0.290198i
$220$		0			0
$221$	2688.00	+	4655.75i	0.818165	+	1.41710i
$222$		0			0
$223$	−425.000			−0.127624			−0.0638119	−	0.997962i	$-0.520326\pi$
$223$	−425.000			−0.127624			−0.0638119	+	0.997962i	$0.520326\pi$
$224$		0			0
$225$	−1044.00			−0.309333
$226$		0			0
$227$	1927.50	+	3338.53i	0.563580	+	0.976149i	0.997180	+	0.0750439i	$0.0239097\pi$
$227$	1927.50	+	3338.53i	0.563580	+	0.976149i	−0.433600	+	0.901105i	$0.642757\pi$
$228$		0			0
$229$	1094.00	−	1894.86i	0.315692	−	0.546795i	−0.663892	−	0.747828i	$-0.731097\pi$
$229$	1094.00	−	1894.86i	0.315692	−	0.546795i	0.979584	+	0.201033i	$0.0644299\pi$
$230$		0			0
$231$	−630.000	−	545.596i	−0.179441	−	0.155401i
$232$		0			0
$233$	−426.000	+	737.854i	−0.119778	+	0.207461i	−0.919679	−	0.392670i	$-0.871551\pi$
$233$	−426.000	+	737.854i	−0.119778	+	0.207461i	0.799902	+	0.600131i	$0.204885\pi$
$234$		0			0
$235$	45.0000	+	77.9423i	0.0124914	+	0.0216357i
$236$		0			0
$237$	1401.00			0.383986
$238$		0			0
$239$	−5508.00			−1.49072			−0.745362	−	0.666660i	$-0.767723\pi$
$239$	−5508.00			−1.49072			−0.745362	+	0.666660i	$0.767723\pi$
$240$		0			0
$241$	−395.500	−	685.026i	−0.105711	−	0.183097i	0.808317	−	0.588747i	$-0.200379\pi$
$241$	−395.500	−	685.026i	−0.105711	−	0.183097i	−0.914029	+	0.405650i	$0.867045\pi$
$242$		0			0
$243$	121.500	−	210.444i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−808.500	+	636.529i	−0.210829	+	0.165985i
$246$		0			0
$247$	512.000	−	886.810i	0.131894	−	0.228447i
$248$		0			0
$249$	−715.500	−	1239.28i	−0.182100	−	0.315407i
$250$		0			0
$251$	−5265.00			−1.32400			−0.662000	−	0.749504i	$-0.730292\pi$
$251$	−5265.00			−1.32400			−0.662000	+	0.749504i	$0.730292\pi$
$252$		0			0
$253$	1260.00			0.313105
$254$		0			0
$255$	−378.000	−	654.715i	−0.0928285	−	0.160784i
$256$		0			0
$257$	3435.00	−	5949.59i	0.833733	−	1.44407i	−0.0613246	−	0.998118i	$-0.519532\pi$
$257$	3435.00	−	5949.59i	0.833733	−	1.44407i	0.895058	−	0.445950i	$-0.147134\pi$
$258$		0			0
$259$	−4424.00	−	3831.30i	−1.06137	−	0.919171i
$260$		0			0
$261$	1336.50	−	2314.89i	0.316963	−	0.548996i
$262$		0			0
$263$	−111.000	−	192.258i	−0.0260249	−	0.0450765i	0.852720	−	0.522369i	$-0.174952\pi$
$263$	−111.000	−	192.258i	−0.0260249	−	0.0450765i	−0.878745	+	0.477292i	$0.841618\pi$
$264$		0			0
$265$	−1089.00			−0.252441
$266$		0			0
$267$	−2718.00			−0.622992
$268$		0			0
$269$	−3925.50	−	6799.17i	−0.889747	−	1.54109i	−0.840174	−	0.542317i	$-0.817547\pi$
$269$	−3925.50	−	6799.17i	−0.889747	−	1.54109i	−0.0495729	−	0.998771i	$-0.515786\pi$
$270$		0			0
$271$	2591.50	−	4488.61i	0.580895	−	1.00614i	−0.414479	−	0.910059i	$-0.636036\pi$
$271$	2591.50	−	4488.61i	0.580895	−	1.00614i	0.995374	−	0.0960800i	$-0.0306305\pi$
$272$		0			0
$273$	−3360.00	+	1163.94i	−0.744895	+	0.258039i
$274$		0			0
$275$	870.000	−	1506.88i	0.190774	−	0.330431i
$276$		0			0
$277$	2480.00	+	4295.49i	0.537938	+	0.931736i	0.999015	+	0.0443755i	$0.0141298\pi$
$277$	2480.00	+	4295.49i	0.537938	+	0.931736i	−0.461077	+	0.887360i	$0.652537\pi$
$278$		0			0
$279$	2277.00			0.488604
$280$		0			0
$281$	−774.000			−0.164317			−0.0821583	−	0.996619i	$-0.526181\pi$
$281$	−774.000			−0.164317			−0.0821583	+	0.996619i	$0.526181\pi$
$282$		0			0
$283$	1849.00	+	3202.56i	0.388380	+	0.672695i	0.992232	−	0.124402i	$-0.0397013\pi$
$283$	1849.00	+	3202.56i	0.388380	+	0.672695i	−0.603852	+	0.797097i	$0.706368\pi$
$284$		0			0
$285$	−72.0000	+	124.708i	−0.0149646	+	0.0259195i
$286$		0			0
$287$	1260.00	−	6547.15i	0.259148	−	1.34657i
$288$		0			0
$289$	−1071.50	+	1855.89i	−0.218095	+	0.377751i
$290$		0			0
$291$	754.500	+	1306.83i	0.151992	+	0.263257i
$292$		0			0
$293$	−6273.00			−1.25076			−0.625380	−	0.780321i	$-0.715056\pi$
$293$	−6273.00			−1.25076			−0.625380	+	0.780321i	$0.715056\pi$
$294$		0			0
$295$	−45.0000			−0.00888136
$296$		0			0
$297$	202.500	+	350.740i	0.0395631	+	0.0685253i
$298$		0			0
$299$	2688.00	−	4655.75i	0.519903	−	0.900499i
$300$		0			0
$301$	−91.0000	+	472.850i	−0.0174258	+	0.0905469i
$302$		0			0
$303$	−1629.00	+	2821.51i	−0.308857	+	0.534956i
$304$		0			0
$305$	−177.000	−	306.573i	−0.0332295	−	0.0575551i
$306$		0			0
$307$	1684.00			0.313065			0.156533	−	0.987673i	$-0.449968\pi$
$307$	1684.00			0.313065			0.156533	+	0.987673i	$0.449968\pi$
$308$		0			0
$309$	5208.00			0.958812
$310$		0			0
$311$	−660.000	−	1143.15i	−0.120338	−	0.208432i	0.799563	−	0.600582i	$-0.205065\pi$
$311$	−660.000	−	1143.15i	−0.120338	−	0.208432i	−0.919901	+	0.392151i	$0.871731\pi$
$312$		0			0
$313$	4251.50	−	7363.81i	0.767760	−	1.32980i	−0.171014	−	0.985269i	$-0.554704\pi$
$313$	4251.50	−	7363.81i	0.767760	−	1.32980i	0.938775	−	0.344531i	$-0.111962\pi$
$314$		0			0
$315$	472.500	−	163.679i	0.0845154	−	0.0292770i
$316$		0			0
$317$	1288.50	−	2231.75i	0.228295	−	0.395418i	−0.729008	−	0.684505i	$-0.760018\pi$
$317$	1288.50	−	2231.75i	0.228295	−	0.395418i	0.957303	+	0.289087i	$0.0933517\pi$
$318$		0			0
$319$	2227.50	+	3858.14i	0.390959	+	0.677162i
$320$		0			0
$321$	−4059.00			−0.705767
$322$		0			0
$323$	1344.00			0.231524
$324$		0			0
$325$	−3712.00	−	6429.37i	−0.633553	−	1.09735i
$326$		0			0
$327$	−555.000	+	961.288i	−0.0938580	+	0.162567i
$328$		0			0
$329$	420.000	+	363.731i	0.0703810	+	0.0609517i
$330$		0			0
$331$	−242.000	+	419.156i	−0.0401859	+	0.0696040i	−0.885419	−	0.464794i	$-0.846128\pi$
$331$	−242.000	+	419.156i	−0.0401859	+	0.0696040i	0.845233	+	0.534398i	$0.179462\pi$
$332$		0			0
$333$	1422.00	+	2462.98i	0.234009	+	0.405316i
$334$		0			0
$335$	−1110.00			−0.181032
$336$		0			0
$337$	−8359.00			−1.35117			−0.675584	−	0.737283i	$-0.736109\pi$
$337$	−8359.00			−1.35117			−0.675584	+	0.737283i	$0.736109\pi$
$338$		0			0
$339$	−972.000	−	1683.55i	−0.155728	−	0.269729i
$340$		0			0
$341$	−1897.50	+	3286.57i	−0.301335	+	0.521928i
$342$		0			0
$343$	−3430.00	+	5346.84i	−0.539949	+	0.841698i
$344$		0			0
$345$	−378.000	+	654.715i	−0.0589879	+	0.102170i
$346$		0			0
$347$	−930.000	−	1610.81i	−0.143876	−	0.249201i	0.785077	−	0.619398i	$-0.212623\pi$
$347$	−930.000	−	1610.81i	−0.143876	−	0.249201i	−0.928953	+	0.370197i	$0.879290\pi$
$348$		0			0
$349$	−1918.00			−0.294178			−0.147089	−	0.989123i	$-0.546990\pi$
$349$	−1918.00			−0.294178			−0.147089	+	0.989123i	$0.546990\pi$
$350$		0			0
$351$	1728.00			0.262774
$352$		0			0
$353$	1524.00	+	2639.65i	0.229786	+	0.398000i	0.957744	−	0.287620i	$-0.0928642\pi$
$353$	1524.00	+	2639.65i	0.229786	+	0.398000i	−0.727959	+	0.685621i	$0.759531\pi$
$354$		0			0
$355$	513.000	−	888.542i	0.0766964	−	0.132842i
$356$		0			0
$357$	−3528.00	−	3055.34i	−0.523030	−	0.452957i
$358$		0			0
$359$	−15.0000	+	25.9808i	−0.00220521	+	0.00381953i	−0.867126	−	0.498089i	$-0.834035\pi$
$359$	−15.0000	+	25.9808i	−0.00220521	+	0.00381953i	0.864921	+	0.501909i	$0.167369\pi$
$360$		0			0
$361$	3301.50	+	5718.37i	0.481338	+	0.833703i
$362$		0			0
$363$	3318.00			0.479752
$364$		0			0
$365$	−1086.00			−0.155737
$366$		0			0
$367$	−5655.50	−	9795.61i	−0.804400	−	1.39326i	−0.916696	−	0.399586i	$-0.869154\pi$
$367$	−5655.50	−	9795.61i	−0.804400	−	1.39326i	0.112296	−	0.993675i	$-0.464180\pi$
$368$		0			0
$369$	−1620.00	+	2805.92i	−0.228547	+	0.395855i
$370$		0			0
$371$	−6352.50	+	2200.57i	−0.888963	+	0.307946i
$372$		0			0
$373$	−604.000	+	1046.16i	−0.0838443	+	0.145223i	−0.904898	−	0.425628i	$-0.860053\pi$
$373$	−604.000	+	1046.16i	−0.0838443	+	0.145223i	0.821054	+	0.570851i	$0.193387\pi$
$374$		0			0
$375$	1084.50	+	1878.41i	0.149342	+	0.258668i
$376$		0			0
$377$	19008.0			2.59672
$378$		0			0
$379$	−7640.00			−1.03546			−0.517731	−	0.855543i	$-0.673223\pi$
$379$	−7640.00			−1.03546			−0.517731	+	0.855543i	$0.673223\pi$
$380$		0			0
$381$	−565.500	−	979.475i	−0.0760405	−	0.131706i
$382$		0			0
$383$	6375.00	−	11041.8i	0.850515	−	1.47314i	−0.0302291	−	0.999543i	$-0.509624\pi$
$383$	6375.00	−	11041.8i	0.850515	−	1.47314i	0.880744	−	0.473592i	$-0.157043\pi$
$384$		0			0
$385$	−157.500	+	818.394i	−0.0208492	+	0.108336i
$386$		0			0
$387$	117.000	−	202.650i	0.0153681	−	0.0266183i
$388$		0			0
$389$	−1563.00	−	2707.20i	−0.203720	−	0.352854i	0.746004	−	0.665942i	$-0.231970\pi$
$389$	−1563.00	−	2707.20i	−0.203720	−	0.352854i	−0.949724	+	0.313087i	$0.898637\pi$
$390$		0			0
$391$	7056.00			0.912627
$392$		0			0
$393$	−1953.00			−0.250676
$394$		0			0
$395$	−700.500	−	1213.30i	−0.0892303	−	0.154551i
$396$		0			0
$397$	2966.00	−	5137.26i	0.374960	−	0.649450i	−0.615361	−	0.788246i	$-0.710990\pi$
$397$	2966.00	−	5137.26i	0.374960	−	0.649450i	0.990321	+	0.138795i	$0.0443230\pi$
$398$		0			0
$399$	−168.000	+	872.954i	−0.0210790	+	0.109530i
$400$		0			0
$401$	−804.000	+	1392.57i	−0.100124	+	0.173420i	−0.911736	−	0.410777i	$-0.865257\pi$
$401$	−804.000	+	1392.57i	−0.100124	+	0.173420i	0.811611	+	0.584198i	$0.198591\pi$
$402$		0			0
$403$	8096.00	+	14022.7i	1.00072	+	1.73330i
$404$		0			0
$405$	−243.000			−0.0298142
$406$		0			0
$407$	−4740.00			−0.577280
$408$		0			0
$409$	2232.50	+	3866.80i	0.269902	+	0.467484i	0.968836	−	0.247702i	$-0.0796753\pi$
$409$	2232.50	+	3866.80i	0.269902	+	0.467484i	−0.698934	+	0.715186i	$0.746342\pi$
$410$		0			0
$411$	−2655.00	+	4598.59i	−0.318641	+	0.551903i
$412$		0			0
$413$	−262.500	+	90.9327i	−0.0312755	+	0.0108342i
$414$		0			0
$415$	−715.500	+	1239.28i	−0.0846326	+	0.146588i
$416$		0			0
$417$	2337.00	+	4047.80i	0.274445	+	0.475352i
$418$		0			0
$419$	1584.00			0.184686			0.0923430	−	0.995727i	$-0.470564\pi$
$419$	1584.00			0.184686			0.0923430	+	0.995727i	$0.470564\pi$
$420$		0			0
$421$	−1330.00			−0.153967			−0.0769837	−	0.997032i	$-0.524529\pi$
$421$	−1330.00			−0.153967			−0.0769837	+	0.997032i	$0.524529\pi$
$422$		0			0
$423$	−135.000	−	233.827i	−0.0155176	−	0.0268772i
$424$		0			0
$425$	4872.00	−	8438.55i	0.556063	−	0.963129i
$426$		0			0
$427$	−1652.00	−	1430.67i	−0.187227	−	0.162143i
$428$		0			0
$429$	−1440.00	+	2494.15i	−0.162060	+	0.280697i
$430$		0			0
$431$	4794.00	+	8303.45i	0.535775	+	0.927989i	0.999125	+	0.0418139i	$0.0133137\pi$
$431$	4794.00	+	8303.45i	0.535775	+	0.927989i	−0.463351	+	0.886175i	$0.653353\pi$
$432$		0			0
$433$	494.000			0.0548271			0.0274135	−	0.999624i	$-0.491273\pi$
$433$	494.000			0.0548271			0.0274135	+	0.999624i	$0.491273\pi$
$434$		0			0
$435$	−2673.00			−0.294622
$436$		0			0
$437$	−672.000	−	1163.94i	−0.0735609	−	0.127411i
$438$		0			0
$439$	−8004.50	+	13864.2i	−0.870237	+	1.50729i	−0.00848508	+	0.999964i	$0.502701\pi$
$439$	−8004.50	+	13864.2i	−0.870237	+	1.50729i	−0.861752	+	0.507330i	$0.830632\pi$
$440$		0			0
$441$	2425.50	−	1909.59i	0.261905	−	0.206197i
$442$		0			0
$443$	3886.50	−	6731.62i	0.416824	−	0.721961i	−0.578794	−	0.815474i	$-0.696476\pi$
$443$	3886.50	−	6731.62i	0.416824	−	0.721961i	0.995618	+	0.0935130i	$0.0298097\pi$
$444$		0			0
$445$	1359.00	+	2353.86i	0.144770	+	0.250749i
$446$		0			0
$447$	−7362.00			−0.778995
$448$		0			0
$449$	864.000			0.0908122			0.0454061	−	0.998969i	$-0.485542\pi$
$449$	864.000			0.0908122			0.0454061	+	0.998969i	$0.485542\pi$
$450$		0			0
$451$	−2700.00	−	4676.54i	−0.281903	−	0.488269i
$452$		0			0
$453$	−1888.50	+	3270.98i	−0.195871	+	0.339258i
$454$		0			0
$455$	2688.00	+	2327.88i	0.276957	+	0.239852i
$456$		0			0
$457$	−1259.50	+	2181.52i	−0.128921	+	0.223298i	−0.923259	−	0.384179i	$-0.874485\pi$
$457$	−1259.50	+	2181.52i	−0.128921	+	0.223298i	0.794338	+	0.607476i	$0.207818\pi$
$458$		0			0
$459$	1134.00	+	1964.15i	0.115317	+	0.199735i
$460$		0			0
$461$	−342.000			−0.0345521			−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000			−0.0345521			−0.0172761	+	0.999851i	$0.505499\pi$
$462$		0			0
$463$	4336.00			0.435229			0.217614	−	0.976035i	$-0.430172\pi$
$463$	4336.00			0.435229			0.217614	+	0.976035i	$0.430172\pi$
$464$		0			0
$465$	−1138.50	−	1971.94i	−0.113541	−	0.196659i
$466$		0			0
$467$	9318.00	−	16139.2i	0.923310	−	1.59922i	0.129052	−	0.991638i	$-0.458806\pi$
$467$	9318.00	−	16139.2i	0.923310	−	1.59922i	0.794257	−	0.607581i	$-0.207860\pi$
$468$		0			0
$469$	−6475.00	+	2243.01i	−0.637500	+	0.220837i
$470$		0			0
$471$	−294.000	+	509.223i	−0.0287618	+	0.0498169i
$472$		0			0
$473$	195.000	+	337.750i	0.0189558	+	0.0328325i
$474$		0			0
$475$	−1856.00			−0.179282
$476$		0			0
$477$	3267.00			0.313597
$478$		0			0
$479$	7539.00	+	13057.9i	0.719135	+	1.24558i	0.961343	+	0.275354i	$0.0887951\pi$
$479$	7539.00	+	13057.9i	0.719135	+	1.24558i	−0.242208	+	0.970224i	$0.577872\pi$
$480$		0			0
$481$	−10112.0	+	17514.5i	−0.958560	+	1.66028i
$482$		0			0
$483$	−882.000	+	4583.01i	−0.0830898	+	0.431747i
$484$		0			0
$485$	754.500	−	1306.83i	0.0706393	−	0.122351i
$486$		0			0
$487$	3110.50	+	5387.54i	0.289425	+	0.501300i	0.973673	−	0.227950i	$-0.0732024\pi$
$487$	3110.50	+	5387.54i	0.289425	+	0.501300i	−0.684247	+	0.729250i	$0.739869\pi$
$488$		0			0
$489$	−3756.00			−0.347346
$490$		0			0
$491$	7371.00			0.677492			0.338746	−	0.940878i	$-0.389997\pi$
$491$	7371.00			0.677492			0.338746	+	0.940878i	$0.389997\pi$
$492$		0			0
$493$	12474.0	+	21605.6i	1.13956	+	1.97377i
$494$		0			0
$495$	202.500	−	350.740i	0.0183873	−	0.0318477i
$496$		0			0
$497$	1197.00	−	6219.79i	0.108034	−	0.561360i
$498$		0			0
$499$	2137.00	−	3701.39i	0.191714	−	0.332058i	−0.754104	−	0.656755i	$-0.771929\pi$
$499$	2137.00	−	3701.39i	0.191714	−	0.332058i	0.945818	+	0.324696i	$0.105262\pi$
$500$		0			0
$501$	3969.00	+	6874.51i	0.353936	+	0.613035i
$502$		0			0
$503$	2520.00			0.223382			0.111691	−	0.993743i	$-0.464373\pi$
$503$	2520.00			0.223382			0.111691	+	0.993743i	$0.464373\pi$
$504$		0			0
$505$	3258.00			0.287087
$506$		0			0
$507$	2848.50	+	4933.75i	0.249519	+	0.432180i
$508$		0			0
$509$	7138.50	−	12364.2i	0.621628	−	1.07669i	−0.367555	−	0.930002i	$-0.619805\pi$
$509$	7138.50	−	12364.2i	0.621628	−	1.07669i	0.989183	−	0.146689i	$-0.0468616\pi$
$510$		0			0
$511$	−6335.00	+	2194.51i	−0.548423	+	0.189979i
$512$		0			0
$513$	216.000	−	374.123i	0.0185899	−	0.0321987i
$514$		0			0
$515$	−2604.00	−	4510.26i	−0.222808	−	0.385914i
$516$		0			0
$517$	450.000			0.0382804
$518$		0			0
$519$	2358.00			0.199431
$520$		0			0
$521$	3153.00	+	5461.16i	0.265135	+	0.459228i	0.967599	−	0.252492i	$-0.0812501\pi$
$521$	3153.00	+	5461.16i	0.265135	+	0.459228i	−0.702464	+	0.711719i	$0.747917\pi$
$522$		0			0
$523$	4036.00	−	6990.56i	0.337442	−	0.584466i	−0.646509	−	0.762906i	$-0.723772\pi$
$523$	4036.00	−	6990.56i	0.337442	−	0.584466i	0.983951	+	0.178440i	$0.0571051\pi$
$524$		0			0
$525$	4872.00	+	4219.28i	0.405012	+	0.350751i
$526$		0			0
$527$	−10626.0	+	18404.8i	−0.878322	+	1.52130i
$528$		0			0
$529$	2555.50	+	4426.26i	0.210035	+	0.363792i
$530$		0			0
$531$	135.000			0.0110330
$532$		0			0
$533$	−23040.0			−1.87237
$534$		0			0
$535$	2029.50	+	3515.20i	0.164005	+	0.284066i
$536$		0			0
$537$	−4338.00	+	7513.64i	−0.348601	+	0.603794i
$538$		0			0
$539$	735.000	+	5092.23i	0.0587360	+	0.406935i
$540$		0			0
$541$	11429.0	−	19795.6i	0.908264	−	1.57316i	0.0917903	−	0.995778i	$-0.470741\pi$
$541$	11429.0	−	19795.6i	0.908264	−	1.57316i	0.816474	−	0.577382i	$-0.195926\pi$
$542$		0			0
$543$	2028.00	+	3512.60i	0.160276	+	0.277606i
$544$		0			0
$545$	1110.00			0.0872425
$546$		0			0
$547$	24724.0			1.93258			0.966291	−	0.257454i	$-0.0828835\pi$
$547$	24724.0			1.93258			0.966291	+	0.257454i	$0.0828835\pi$
$548$		0			0
$549$	531.000	+	919.719i	0.0412796	+	0.0714985i
$550$		0			0
$551$	2376.00	−	4115.35i	0.183704	−	0.318185i
$552$		0			0
$553$	−6538.00	−	5662.07i	−0.502756	−	0.435399i
$554$		0			0
$555$	1422.00	−	2462.98i	0.108758	−	0.188374i
$556$		0			0
$557$	4921.50	+	8524.29i	0.374382	+	0.648448i	0.990234	−	0.139413i	$-0.0445216\pi$
$557$	4921.50	+	8524.29i	0.374382	+	0.648448i	−0.615853	+	0.787861i	$0.711188\pi$
$558$		0			0
$559$	1664.00			0.125903
$560$		0			0
$561$	−3780.00			−0.284477
$562$		0			0
$563$	−6685.50	−	11579.6i	−0.500462	−	0.866826i	−1.00000		0.000533812i	$-0.999830\pi$
$563$	−6685.50	−	11579.6i	−0.500462	−	0.866826i	0.499538	−	0.866292i	$-0.333503\pi$
$564$		0			0
$565$	−972.000	+	1683.55i	−0.0723758	+	0.125359i
$566$		0			0
$567$	−1417.50	+	491.036i	−0.104990	+	0.0363696i
$568$		0			0
$569$	2616.00	−	4531.04i	0.192739	−	0.333834i	−0.753418	−	0.657542i	$-0.771596\pi$
$569$	2616.00	−	4531.04i	0.192739	−	0.333834i	0.946157	+	0.323708i	$0.104930\pi$
$570$		0			0
$571$	−7199.00	−	12469.0i	−0.527616	−	0.913858i	−0.999482	−	0.0321874i	$-0.989753\pi$
$571$	−7199.00	−	12469.0i	−0.527616	−	0.913858i	0.471866	−	0.881670i	$-0.343581\pi$
$572$		0			0
$573$	11736.0			0.855634
$574$		0			0
$575$	−9744.00			−0.706701
$576$		0			0
$577$	−9935.50	−	17208.8i	−0.716846	−	1.24161i	−0.962243	−	0.272191i	$-0.912252\pi$
$577$	−9935.50	−	17208.8i	−0.716846	−	1.24161i	0.245397	−	0.969423i	$-0.421082\pi$
$578$		0			0
$579$	2239.50	−	3878.93i	0.160743	−	0.278416i
$580$		0			0
$581$	−1669.50	+	8674.98i	−0.119213	+	0.619447i
$582$		0			0
$583$	−2722.50	+	4715.51i	−0.193404	+	0.334985i
$584$		0			0
$585$	−864.000	−	1496.49i	−0.0610633	−	0.105765i
$586$		0			0
$587$	16137.0			1.13466			0.567330	−	0.823491i	$-0.307976\pi$
$587$	16137.0			1.13466			0.567330	+	0.823491i	$0.307976\pi$
$588$		0			0
$589$	4048.00			0.283183
$590$		0			0
$591$	−6129.00	−	10615.7i	−0.426588	−	0.738872i
$592$		0			0
$593$	10662.0	−	18467.1i	0.738340	−	1.27884i	−0.214902	−	0.976636i	$-0.568943\pi$
$593$	10662.0	−	18467.1i	0.738340	−	1.27884i	0.953242	−	0.302207i	$-0.0977235\pi$
$594$		0			0
$595$	−882.000	+	4583.01i	−0.0607705	+	0.315773i
$596$		0			0
$597$	5334.00	−	9238.76i	0.365672	−	0.633362i
$598$		0			0
$599$	−4323.00	−	7487.66i	−0.294880	−	0.510747i	0.680077	−	0.733141i	$-0.261946\pi$
$599$	−4323.00	−	7487.66i	−0.294880	−	0.510747i	−0.974957	+	0.222394i	$0.928613\pi$
$600$		0			0
$601$	11195.0			0.759823			0.379911	−	0.925023i	$-0.375954\pi$
$601$	11195.0			0.759823			0.379911	+	0.925023i	$0.375954\pi$
$602$		0			0
$603$	3330.00			0.224889
$604$		0			0
$605$	−1659.00	−	2873.47i	−0.111484	−	0.193096i
$606$		0			0
$607$	−4485.50	+	7769.11i	−0.299935	+	0.519503i	−0.976121	−	0.217228i	$-0.930298\pi$
$607$	−4485.50	+	7769.11i	−0.299935	+	0.519503i	0.676185	+	0.736731i	$0.263632\pi$
$608$		0			0
$609$	−15592.5	+	5401.40i	−1.03750	+	0.359402i
$610$		0			0
$611$	960.000	−	1662.77i	0.0635637	−	0.110096i
$612$		0			0
$613$	6386.00	+	11060.9i	0.420764	+	0.728784i	0.996014	−	0.0891932i	$-0.0284288\pi$
$613$	6386.00	+	11060.9i	0.420764	+	0.728784i	−0.575251	+	0.817977i	$0.695096\pi$
$614$		0			0
$615$	3240.00			0.212438
$616$		0			0
$617$	12762.0			0.832705			0.416352	−	0.909203i	$-0.363308\pi$
$617$	12762.0			0.832705			0.416352	+	0.909203i	$0.363308\pi$
$618$		0			0
$619$	6421.00	+	11121.5i	0.416933	+	0.722150i	0.995629	−	0.0933936i	$-0.0297715\pi$
$619$	6421.00	+	11121.5i	0.416933	+	0.722150i	−0.578696	+	0.815543i	$0.696438\pi$
$620$		0			0
$621$	1134.00	−	1964.15i	0.0732783	−	0.126922i
$622$		0			0
$623$	12684.0	+	10984.7i	0.815688	+	0.706407i
$624$		0			0
$625$	−6165.50	+	10679.0i	−0.394592	+	0.683453i
$626$		0			0
$627$	360.000	+	623.538i	0.0229298	+	0.0397157i
$628$		0			0
$629$	−26544.0			−1.68264
$630$		0			0
$631$	−21365.0			−1.34790			−0.673952	−	0.738775i	$-0.735404\pi$
$631$	−21365.0			−1.34790			−0.673952	+	0.738775i	$0.735404\pi$
$632$		0			0
$633$	−1875.00	−	3247.60i	−0.117732	−	0.203918i
$634$		0			0
$635$	−565.500	+	979.475i	−0.0353404	+	0.0612114i
$636$		0			0
$637$	20384.0	+	8147.57i	1.26789	+	0.506779i
$638$		0			0
$639$	−1539.00	+	2665.63i	−0.0952768	+	0.165024i
$640$		0			0
$641$	−4137.00	−	7165.49i	−0.254917	−	0.441529i	0.709956	−	0.704246i	$-0.248715\pi$
$641$	−4137.00	−	7165.49i	−0.254917	−	0.441529i	−0.964873	+	0.262717i	$0.915381\pi$
$642$		0			0
$643$	−27998.0			−1.71716			−0.858580	−	0.512680i	$-0.828653\pi$
$643$	−27998.0			−1.71716			−0.858580	+	0.512680i	$0.828653\pi$
$644$		0			0
$645$	−234.000			−0.0142849
$646$		0			0
$647$	−8733.00	−	15126.0i	−0.530649	−	0.919110i	−0.999360	−	0.0357592i	$-0.988615\pi$
$647$	−8733.00	−	15126.0i	−0.530649	−	0.919110i	0.468712	−	0.883351i	$-0.344718\pi$
$648$		0			0
$649$	−112.500	+	194.856i	−0.00680433	+	0.0117854i
$650$		0			0
$651$	−10626.0	−	9202.39i	−0.639732	−	0.554024i
$652$		0			0
$653$	−1078.50	+	1868.02i	−0.0646324	+	0.111947i	−0.896531	−	0.442981i	$-0.853921\pi$
$653$	−1078.50	+	1868.02i	−0.0646324	+	0.111947i	0.831898	+	0.554928i	$0.187254\pi$
$654$		0			0
$655$	976.500	+	1691.35i	0.0582519	+	0.100895i
$656$		0			0
$657$	3258.00			0.193465
$658$		0			0
$659$	−19944.0			−1.17892			−0.589460	−	0.807798i	$-0.700659\pi$
$659$	−19944.0			−1.17892			−0.589460	+	0.807798i	$0.700659\pi$
$660$		0			0
$661$	−13753.0	−	23820.9i	−0.809273	−	1.40170i	−0.913368	−	0.407135i	$-0.866528\pi$
$661$	−13753.0	−	23820.9i	−0.809273	−	1.40170i	0.104095	−	0.994567i	$-0.466806\pi$
$662$		0			0
$663$	−8064.00	+	13967.3i	−0.472368	+	0.818165i
$664$		0			0
$665$	840.000	−	290.985i	0.0489832	−	0.0169683i
$666$		0			0
$667$	12474.0	−	21605.6i	0.724131	−	1.25423i
$668$		0			0
$669$	−637.500	−	1104.18i	−0.0368418	−	0.0638119i
$670$		0			0
$671$	−1770.00			−0.101833
$672$		0			0
$673$	−19123.0			−1.09530			−0.547650	−	0.836707i	$-0.684478\pi$
$673$	−19123.0			−1.09530			−0.547650	+	0.836707i	$0.684478\pi$
$674$		0			0
$675$	−1566.00	−	2712.39i	−0.0892968	−	0.154667i
$676$		0			0
$677$	−6928.50	+	12000.5i	−0.393329	+	0.681266i	−0.992886	−	0.119066i	$-0.962010\pi$
$677$	−6928.50	+	12000.5i	−0.393329	+	0.681266i	0.599557	+	0.800332i	$0.295343\pi$
$678$		0			0
$679$	1760.50	−	9147.83i	0.0995019	−	0.517027i
$680$		0			0
$681$	−5782.50	+	10015.6i	−0.325383	+	0.563580i
$682$		0			0
$683$	−11122.5	−	19264.7i	−0.623120	−	1.07927i	−0.988901	−	0.148574i	$-0.952532\pi$
$683$	−11122.5	−	19264.7i	−0.623120	−	1.07927i	0.365782	−	0.930701i	$-0.380802\pi$
$684$		0			0
$685$	5310.00			0.296182
$686$		0			0
$687$	6564.00			0.364530
$688$		0			0
$689$	11616.0	+	20119.5i	0.642285	+	1.11247i
$690$		0			0
$691$	−320.000	+	554.256i	−0.0176170	+	0.0305136i	−0.874700	−	0.484666i	$-0.838941\pi$
$691$	−320.000	+	554.256i	−0.0176170	+	0.0305136i	0.857082	+	0.515179i	$0.172275\pi$
$692$		0			0
$693$	472.500	−	2455.18i	0.0259001	−	0.134581i
$694$		0			0
$695$	2337.00	−	4047.80i	0.127550	−	0.220924i
$696$		0			0
$697$	−15120.0	−	26188.6i	−0.821680	−	1.42319i
$698$		0			0
$699$	−2556.00			−0.138307
$700$		0			0
$701$	−15561.0			−0.838418			−0.419209	−	0.907890i	$-0.637693\pi$
$701$	−15561.0			−0.838418			−0.419209	+	0.907890i	$0.637693\pi$
$702$		0			0
$703$	2528.00	+	4378.62i	0.135626	+	0.234912i
$704$		0			0
$705$	−135.000	+	233.827i	−0.00721191	+	0.0124914i
$706$		0			0
$707$	19005.0	−	6583.53i	1.01097	−	0.350211i
$708$		0			0
$709$	−2767.00	+	4792.58i	−0.146568	+	0.253864i	−0.929957	−	0.367668i	$-0.880156\pi$
$709$	−2767.00	+	4792.58i	−0.146568	+	0.253864i	0.783389	+	0.621532i	$0.213489\pi$
$710$		0			0
$711$	2101.50	+	3639.90i	0.110847	+	0.191993i
$712$		0			0
$713$	21252.0			1.11626
$714$		0			0
$715$	2880.00			0.150638
$716$		0			0
$717$	−8262.00	−	14310.2i	−0.430335	−	0.745362i
$718$		0			0
$719$	−10923.0	+	18919.2i	−0.566564	+	0.981317i	0.430339	+	0.902667i	$0.358394\pi$
$719$	−10923.0	+	18919.2i	−0.566564	+	0.981317i	−0.996902	+	0.0786494i	$0.974939\pi$
$720$		0			0
$721$	−24304.0	−	21047.9i	−1.25538	−	1.08719i
$722$		0			0
$723$	1186.50	−	2055.08i	0.0610324	−	0.105711i
$724$		0			0
$725$	−17226.0	−	29836.3i	−0.882424	−	1.52840i
$726$		0			0
$727$	11089.0			0.565706			0.282853	−	0.959163i	$-0.408719\pi$
$727$	11089.0			0.565706			0.282853	+	0.959163i	$0.408719\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	1092.00	+	1891.40i	0.0552518	+	0.0956990i
$732$		0			0
$733$	−5881.00	+	10186.2i	−0.296343	+	0.513282i	−0.975296	−	0.220900i	$-0.929101\pi$
$733$	−5881.00	+	10186.2i	−0.296343	+	0.513282i	0.678953	+	0.734182i	$0.262434\pi$
$734$		0			0
$735$	−2866.50	−	1145.75i	−0.143854	−	0.0574989i
$736$		0			0
$737$	−2775.00	+	4806.44i	−0.138695	+	0.240227i
$738$		0			0
$739$	−11363.0	−	19681.3i	−0.565622	−	0.979686i	−0.996992	−	0.0775108i	$-0.975303\pi$
$739$	−11363.0	−	19681.3i	−0.565622	−	0.979686i	0.431369	−	0.902175i	$-0.358031\pi$
$740$		0			0
$741$	3072.00			0.152298
$742$		0			0
$743$	−6678.00			−0.329734			−0.164867	−	0.986316i	$-0.552719\pi$
$743$	−6678.00			−0.329734			−0.164867	+	0.986316i	$0.552719\pi$
$744$		0			0
$745$	3681.00	+	6375.68i	0.181022	+	0.313539i
$746$		0			0
$747$	2146.50	−	3717.85i	0.105136	−	0.182100i
$748$		0			0
$749$	18942.0	+	16404.3i	0.924066	+	0.800265i
$750$		0			0
$751$	−9993.50	+	17309.2i	−0.485577	+	0.841043i	−0.999863	−	0.0165754i	$-0.994724\pi$
$751$	−9993.50	+	17309.2i	−0.485577	+	0.841043i	0.514286	+	0.857619i	$0.328057\pi$
$752$		0			0
$753$	−7897.50	−	13678.9i	−0.382206	−	0.662000i
$754$		0			0
$755$	3777.00			0.182065
$756$		0			0
$757$	314.000			0.0150760			0.00753799	−	0.999972i	$-0.497601\pi$
$757$	314.000			0.0150760			0.00753799	+	0.999972i	$0.497601\pi$
$758$		0			0
$759$	1890.00	+	3273.58i	0.0903856	+	0.156552i
$760$		0			0
$761$	5748.00	−	9955.83i	0.273804	−	0.474242i	−0.696029	−	0.718014i	$-0.745051\pi$
$761$	5748.00	−	9955.83i	0.273804	−	0.474242i	0.969833	+	0.243772i	$0.0783847\pi$
$762$		0			0
$763$	6475.00	−	2243.01i	0.307222	−	0.106425i
$764$		0			0
$765$	1134.00	−	1964.15i	0.0535946	−	0.0928285i
$766$		0			0
$767$	480.000	+	831.384i	0.0225969	+	0.0391389i
$768$		0			0
$769$	2765.00			0.129660			0.0648299	−	0.997896i	$-0.479350\pi$
$769$	2765.00			0.129660			0.0648299	+	0.997896i	$0.479350\pi$
$770$		0			0
$771$	20610.0			0.962712
$772$		0			0
$773$	7023.00	+	12164.2i	0.326778	+	0.565997i	0.981871	−	0.189552i	$-0.0607036\pi$
$773$	7023.00	+	12164.2i	0.326778	+	0.565997i	−0.655092	+	0.755549i	$0.727370\pi$
$774$		0			0
$775$	14674.0	−	25416.1i	0.680136	−	1.17803i
$776$		0			0
$777$	3318.00	−	17240.8i	0.153195	−	0.796025i
$778$		0			0
$779$	−2880.00	+	4988.31i	−0.132460	+	0.229428i
$780$		0			0
$781$	−2565.00	−	4442.71i	−0.117520	−	0.203550i
$782$		0			0
$783$	8019.00			0.365997
$784$		0			0
$785$	588.000			0.0267345
$786$		0			0
$787$	−9257.00	−	16033.6i	−0.419284	−	0.726221i	0.576584	−	0.817038i	$-0.304385\pi$
$787$	−9257.00	−	16033.6i	−0.419284	−	0.726221i	−0.995868	+	0.0908171i	$0.971052\pi$
$788$		0			0
$789$	333.000	−	576.773i	0.0150255	−	0.0260249i
$790$		0			0
$791$	−2268.00	+	11784.9i	−0.101948	+	0.529737i
$792$		0			0
$793$	−3776.00	+	6540.22i	−0.169092

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.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{3} +(1.50000 - 2.59808i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 2.59808i) q^{3} +(1.50000 - 2.59808i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(-7.50000 - 12.9904i) q^{11} -64.0000 q^{13} +9.00000 q^{15} +(-42.0000 - 72.7461i) q^{17} +(-8.00000 + 13.8564i) q^{19} +(52.5000 - 18.1865i) q^{21} +(-42.0000 + 72.7461i) q^{23} +(58.0000 + 100.459i) q^{25} -27.0000 q^{27} -297.000 q^{29} +(-126.500 - 219.104i) q^{31} +(22.5000 - 38.9711i) q^{33} +(-42.0000 - 36.3731i) q^{35} +(158.000 - 273.664i) q^{37} +(-96.0000 - 166.277i) q^{39} +360.000 q^{41} -26.0000 q^{43} +(13.5000 + 23.3827i) q^{45} +(-15.0000 + 25.9808i) q^{47} +(-318.500 - 127.306i) q^{49} +(126.000 - 218.238i) q^{51} +(-181.500 - 314.367i) q^{53} -45.0000 q^{55} -48.0000 q^{57} +(-7.50000 - 12.9904i) q^{59} +(59.0000 - 102.191i) q^{61} +(126.000 + 109.119i) q^{63} +(-96.0000 + 166.277i) q^{65} +(-185.000 - 320.429i) q^{67} -252.000 q^{69} +342.000 q^{71} +(-181.000 - 313.501i) q^{73} +(-174.000 + 301.377i) q^{75} +(-262.500 + 90.9327i) q^{77} +(233.500 - 404.434i) q^{79} +(-40.5000 - 70.1481i) q^{81} -477.000 q^{83} -252.000 q^{85} +(-445.500 - 771.629i) q^{87} +(-453.000 + 784.619i) q^{89} +(-224.000 + 1163.94i) q^{91} +(379.500 - 657.313i) q^{93} +(24.0000 + 41.5692i) q^{95} +503.000 q^{97} +135.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} + O(q^{10}) \)	\( 2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} - 15q^{11} - 128q^{13} + 18q^{15} - 84q^{17} - 16q^{19} + 105q^{21} - 84q^{23} + 116q^{25} - 54q^{27} - 594q^{29} - 253q^{31} + 45q^{33} - 84q^{35} + 316q^{37} - 192q^{39} + 720q^{41} - 52q^{43} + 27q^{45} - 30q^{47} - 637q^{49} + 252q^{51} - 363q^{53} - 90q^{55} - 96q^{57} - 15q^{59} + 118q^{61} + 252q^{63} - 192q^{65} - 370q^{67} - 504q^{69} + 684q^{71} - 362q^{73} - 348q^{75} - 525q^{77} + 467q^{79} - 81q^{81} - 954q^{83} - 504q^{85} - 891q^{87} - 906q^{89} - 448q^{91} + 759q^{93} + 48q^{95} + 1006q^{97} + 270q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about