LMFDB - Embedded newform 1344.4.p.d.223.15

Newspace parameters

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.15
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.d.223.16

$q$-expansion

$f(q)$	$=$	$q-3.00000i q^{3} +12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} -25.0429 q^{11} -64.1604 q^{13} -38.4675i q^{15} -77.4251i q^{17} -49.9995i q^{19} +(52.2158 - 18.9872i) q^{21} +80.5687i q^{23} +39.4169 q^{25} +27.0000i q^{27} +159.100i q^{29} +96.5596 q^{31} +75.1287i q^{33} +(81.1547 + 223.179i) q^{35} +274.101i q^{37} +192.481i q^{39} -299.796i q^{41} +385.824 q^{43} -115.403 q^{45} +418.362 q^{47} +(-262.886 + 220.318i) q^{49} -232.275 q^{51} +665.385i q^{53} -321.113 q^{55} -149.999 q^{57} +445.655i q^{59} -599.783 q^{61} +(-56.9617 - 156.647i) q^{63} -822.697 q^{65} -675.504 q^{67} +241.706 q^{69} +877.093i q^{71} +696.060i q^{73} -118.251i q^{75} +(-158.498 - 435.878i) q^{77} -1248.56i q^{79} +81.0000 q^{81} -238.303i q^{83} -992.784i q^{85} +477.299 q^{87} +743.503i q^{89} +(-406.076 - 1116.73i) q^{91} -289.679i q^{93} -641.120i q^{95} +1552.16i q^{97} +225.386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32q - 288q^{9} + O(q^{10}) $	$ 32q - 288q^{9} + 224q^{13} + 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} + 544q^{61} + 1536q^{65} + 144q^{69} + 1632q^{77} + 2592q^{81} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	12.8225			1.14688			0.573440	−	0.819247i	$-0.305608\pi$
$5$	12.8225			1.14688			0.573440	+	0.819247i	$0.305608\pi$
$6$		0			0
$7$	6.32908	+	17.4053i	0.341738	+	0.939795i
$7$	6.32908	+	17.4053i	0.341738	+	0.939795i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−25.0429			−0.686429			−0.343214	−	0.939257i	$-0.611516\pi$
$11$	−25.0429			−0.686429			−0.343214	+	0.939257i	$0.611516\pi$
$12$		0			0
$13$	−64.1604			−1.36884			−0.684419	−	0.729089i	$-0.739944\pi$
$13$	−64.1604			−1.36884			−0.684419	+	0.729089i	$0.739944\pi$
$14$		0			0
$15$		−	38.4675i		−	0.662152i
$16$		0			0
$17$		−	77.4251i		−	1.10461i	−0.833643	−	0.552304i	$-0.813749\pi$
$17$		−	77.4251i		−	1.10461i	0.833643	−	0.552304i	$-0.186251\pi$
$18$		0			0
$19$		−	49.9995i		−	0.603720i	−0.953352	−	0.301860i	$-0.902393\pi$
$19$		−	49.9995i		−	0.603720i	0.953352	−	0.301860i	$-0.0976075\pi$
$20$		0			0
$21$	52.2158	−	18.9872i	0.542591	−	0.197303i
$22$		0			0
$23$			80.5687i			0.730423i	0.930925	+	0.365212i	$0.119003\pi$
$23$			80.5687i			0.730423i	−0.930925	+	0.365212i	$0.880997\pi$
$24$		0			0
$25$	39.4169			0.315335
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$			159.100i			1.01876i	0.860541	+	0.509381i	$0.170126\pi$
$29$			159.100i			1.01876i	−0.860541	+	0.509381i	$0.829874\pi$
$30$		0			0
$31$	96.5596			0.559440			0.279720	−	0.960082i	$-0.409758\pi$
$31$	96.5596			0.559440			0.279720	+	0.960082i	$0.409758\pi$
$32$		0			0
$33$			75.1287i			0.396310i
$34$		0			0
$35$	81.1547	+	223.179i	0.391933	+	1.07783i
$36$		0			0
$37$			274.101i			1.21789i	0.793212	+	0.608946i	$0.208407\pi$
$37$			274.101i			1.21789i	−0.793212	+	0.608946i	$0.791593\pi$
$38$		0			0
$39$			192.481i			0.790299i
$40$		0			0
$41$		−	299.796i		−	1.14196i	−0.820964	−	0.570980i	$-0.806564\pi$
$41$		−	299.796i		−	1.14196i	0.820964	−	0.570980i	$-0.193436\pi$
$42$		0			0
$43$	385.824			1.36832			0.684158	−	0.729334i	$-0.260170\pi$
$43$	385.824			1.36832			0.684158	+	0.729334i	$0.260170\pi$
$44$		0			0
$45$	−115.403			−0.382294
$46$		0			0
$47$	418.362			1.29839			0.649196	−	0.760622i	$-0.275106\pi$
$47$	418.362			1.29839			0.649196	+	0.760622i	$0.275106\pi$
$48$		0			0
$49$	−262.886	+	220.318i	−0.766430	+	0.642327i
$50$		0			0
$51$	−232.275			−0.637746
$52$		0			0
$53$			665.385i			1.72448i	0.506497	+	0.862242i	$0.330940\pi$
$53$			665.385i			1.72448i	−0.506497	+	0.862242i	$0.669060\pi$
$54$		0			0
$55$	−321.113			−0.787252
$56$		0			0
$57$	−149.999			−0.348558
$58$		0			0
$59$			445.655i			0.983380i	0.870770	+	0.491690i	$0.163621\pi$
$59$			445.655i			0.983380i	−0.870770	+	0.491690i	$0.836379\pi$
$60$		0			0
$61$	−599.783			−1.25892			−0.629461	−	0.777032i	$-0.716724\pi$
$61$	−599.783			−1.25892			−0.629461	+	0.777032i	$0.716724\pi$
$62$		0			0
$63$	−56.9617	−	156.647i	−0.113913	−	0.313265i
$64$		0			0
$65$	−822.697			−1.56989
$66$		0			0
$67$	−675.504			−1.23173			−0.615866	−	0.787851i	$-0.711194\pi$
$67$	−675.504			−1.23173			−0.615866	+	0.787851i	$0.711194\pi$
$68$		0			0
$69$	241.706			0.421710
$70$		0			0
$71$			877.093i			1.46608i	0.680185	+	0.733041i	$0.261900\pi$
$71$			877.093i			1.46608i	−0.680185	+	0.733041i	$0.738100\pi$
$72$		0			0
$73$			696.060i			1.11600i	0.829842	+	0.557998i	$0.188430\pi$
$73$			696.060i			1.11600i	−0.829842	+	0.557998i	$0.811570\pi$
$74$		0			0
$75$		−	118.251i		−	0.182059i
$76$		0			0
$77$	−158.498	−	435.878i	−0.234579	−	0.645102i
$78$		0			0
$79$		−	1248.56i		−	1.77815i	−0.457760	−	0.889076i	$-0.651348\pi$
$79$		−	1248.56i		−	1.77815i	0.457760	−	0.889076i	$-0.348652\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	238.303i		−	0.315146i	−0.987507	−	0.157573i	$-0.949633\pi$
$83$		−	238.303i		−	0.315146i	0.987507	−	0.157573i	$-0.0503670\pi$
$84$		0			0
$85$		−	992.784i		−	1.26685i
$86$		0			0
$87$	477.299			0.588182
$88$		0			0
$89$			743.503i			0.885519i	0.896640	+	0.442759i	$0.146000\pi$
$89$			743.503i			0.885519i	−0.896640	+	0.442759i	$0.854000\pi$
$90$		0			0
$91$	−406.076	−	1116.73i	−0.467784	−	1.28643i
$92$		0			0
$93$		−	289.679i		−	0.322993i
$94$		0			0
$95$		−	641.120i		−	0.692395i
$96$		0			0
$97$			1552.16i			1.62472i	0.583153	+	0.812362i	$0.301819\pi$
$97$			1552.16i			1.62472i	−0.583153	+	0.812362i	$0.698181\pi$
$98$		0			0
$99$	225.386			0.228810
$100$		0			0
$101$	−1202.46			−1.18464			−0.592322	−	0.805701i	$-0.701789\pi$
$101$	−1202.46			−1.18464			−0.592322	+	0.805701i	$0.701789\pi$
$102$		0			0
$103$	404.562			0.387016			0.193508	−	0.981099i	$-0.438013\pi$
$103$	404.562			0.387016			0.193508	+	0.981099i	$0.438013\pi$
$104$		0			0
$105$	669.537	−	243.464i	0.622287	−	0.226282i
$106$		0			0
$107$	−1065.72			−0.962871			−0.481436	−	0.876481i	$-0.659884\pi$
$107$	−1065.72			−0.962871			−0.481436	+	0.876481i	$0.659884\pi$
$108$		0			0
$109$		−	359.515i		−	0.315920i	−0.987445	−	0.157960i	$-0.949508\pi$
$109$		−	359.515i		−	0.315920i	0.987445	−	0.157960i	$-0.0504918\pi$
$110$		0			0
$111$	822.304			0.703150
$112$		0			0
$113$	828.461			0.689691			0.344845	−	0.938660i	$-0.387931\pi$
$113$	828.461			0.689691			0.344845	+	0.938660i	$0.387931\pi$
$114$		0			0
$115$			1033.09i			0.837708i
$116$		0			0
$117$	577.443			0.456279
$118$		0			0
$119$	1347.60	−	490.029i	1.03811	−	0.377486i
$120$		0			0
$121$	−703.854			−0.528816
$122$		0			0
$123$	−899.389			−0.659310
$124$		0			0
$125$	−1097.39			−0.785229
$126$		0			0
$127$			876.572i			0.612466i	0.951957	+	0.306233i	$0.0990687\pi$
$127$			876.572i			0.612466i	−0.951957	+	0.306233i	$0.900931\pi$
$128$		0			0
$129$		−	1157.47i		−	0.789998i
$130$		0			0
$131$			1083.04i			0.722335i	0.932501	+	0.361167i	$0.117622\pi$
$131$			1083.04i			0.722335i	−0.932501	+	0.361167i	$0.882378\pi$
$132$		0			0
$133$	870.254	−	316.451i	0.567373	−	0.206314i
$134$		0			0
$135$			346.208i			0.220717i
$136$		0			0
$137$	−751.836			−0.468859			−0.234430	−	0.972133i	$-0.575322\pi$
$137$	−751.836			−0.468859			−0.234430	+	0.972133i	$0.575322\pi$
$138$		0			0
$139$			543.555i			0.331681i	0.986153	+	0.165841i	$0.0530337\pi$
$139$			543.555i			0.331681i	−0.986153	+	0.165841i	$0.946966\pi$
$140$		0			0
$141$		−	1255.09i		−	0.749626i
$142$		0			0
$143$	1606.76			0.939609
$144$		0			0
$145$			2040.06i			1.16840i
$146$		0			0
$147$	660.955	+	788.657i	0.370848	+	0.442499i
$148$		0			0
$149$			1056.44i			0.580855i	0.956897	+	0.290427i	$0.0937974\pi$
$149$			1056.44i			0.580855i	−0.956897	+	0.290427i	$0.906203\pi$
$150$		0			0
$151$			2092.13i			1.12752i	0.825939	+	0.563759i	$0.190645\pi$
$151$			2092.13i			1.12752i	−0.825939	+	0.563759i	$0.809355\pi$
$152$		0			0
$153$			696.826i			0.368203i
$154$		0			0
$155$	1238.14			0.641610
$156$		0			0
$157$	−1185.70			−0.602735			−0.301367	−	0.953508i	$-0.597443\pi$
$157$	−1185.70			−0.602735			−0.301367	+	0.953508i	$0.597443\pi$
$158$		0			0
$159$	1996.15			0.995631
$160$		0			0
$161$	−1402.32	+	509.925i	−0.686448	+	0.249613i
$162$		0			0
$163$	−903.635			−0.434222			−0.217111	−	0.976147i	$-0.569663\pi$
$163$	−903.635			−0.434222			−0.217111	+	0.976147i	$0.569663\pi$
$164$		0			0
$165$			963.338i			0.454520i
$166$		0			0
$167$	−474.632			−0.219929			−0.109964	−	0.993936i	$-0.535074\pi$
$167$	−474.632			−0.219929			−0.109964	+	0.993936i	$0.535074\pi$
$168$		0			0
$169$	1919.55			0.873715
$170$		0			0
$171$			449.996i			0.201240i
$172$		0			0
$173$	1641.33			0.721317			0.360658	−	0.932698i	$-0.382552\pi$
$173$	1641.33			0.721317			0.360658	+	0.932698i	$0.382552\pi$
$174$		0			0
$175$	249.473	+	686.061i	0.107762	+	0.296351i
$176$		0			0
$177$	1336.97			0.567755
$178$		0			0
$179$	−1802.11			−0.752491			−0.376246	−	0.926520i	$-0.622785\pi$
$179$	−1802.11			−0.752491			−0.376246	+	0.926520i	$0.622785\pi$
$180$		0			0
$181$	−2068.45			−0.849429			−0.424715	−	0.905327i	$-0.639626\pi$
$181$	−2068.45			−0.849429			−0.424715	+	0.905327i	$0.639626\pi$
$182$		0			0
$183$			1799.35i			0.726840i
$184$		0			0
$185$			3514.67i			1.39678i
$186$		0			0
$187$			1938.95i			0.758234i
$188$		0			0
$189$	−469.942	+	170.885i	−0.180864	+	0.0657675i
$190$		0			0
$191$			2179.61i			0.825711i	0.910797	+	0.412855i	$0.135469\pi$
$191$			2179.61i			0.825711i	−0.910797	+	0.412855i	$0.864531\pi$
$192$		0			0
$193$	−4489.81			−1.67453			−0.837263	−	0.546801i	$-0.815845\pi$
$193$	−4489.81			−1.67453			−0.837263	+	0.546801i	$0.815845\pi$
$194$		0			0
$195$			2468.09i			0.906378i
$196$		0			0
$197$		−	4519.08i		−	1.63437i	−0.576376	−	0.817185i	$-0.695533\pi$
$197$		−	4519.08i		−	1.63437i	0.576376	−	0.817185i	$-0.304467\pi$
$198$		0			0
$199$	826.598			0.294452			0.147226	−	0.989103i	$-0.452966\pi$
$199$	826.598			0.294452			0.147226	+	0.989103i	$0.452966\pi$
$200$		0			0
$201$			2026.51i			0.711140i
$202$		0			0
$203$	−2769.17	+	1006.95i	−0.957428	+	0.348150i
$204$		0			0
$205$		−	3844.14i		−	1.30969i
$206$		0			0
$207$		−	725.118i		−	0.243474i
$208$		0			0
$209$			1252.13i			0.414411i
$210$		0			0
$211$	4709.17			1.53646			0.768229	−	0.640175i	$-0.221138\pi$
$211$	4709.17			1.53646			0.768229	+	0.640175i	$0.221138\pi$
$212$		0			0
$213$	2631.28			0.846443
$214$		0			0
$215$	4947.23			1.56930
$216$		0			0
$217$	611.133	+	1680.65i	0.191182	+	0.525759i
$218$		0			0
$219$	2088.18			0.644321
$220$		0			0
$221$			4967.62i			1.51203i
$222$		0			0
$223$	5775.09			1.73421			0.867104	−	0.498127i	$-0.165979\pi$
$223$	5775.09			1.73421			0.867104	+	0.498127i	$0.165979\pi$
$224$		0			0
$225$	−354.752			−0.105112
$226$		0			0
$227$			2280.11i			0.666678i	0.942807	+	0.333339i	$0.108175\pi$
$227$			2280.11i			0.666678i	−0.942807	+	0.333339i	$0.891825\pi$
$228$		0			0
$229$	−3692.99			−1.06567			−0.532837	−	0.846218i	$-0.678874\pi$
$229$	−3692.99			−1.06567			−0.532837	+	0.846218i	$0.678874\pi$
$230$		0			0
$231$	−1307.63	+	475.495i	−0.372450	+	0.135434i
$232$		0			0
$233$	3140.07			0.882887			0.441443	−	0.897289i	$-0.354467\pi$
$233$	3140.07			0.882887			0.441443	+	0.897289i	$0.354467\pi$
$234$		0			0
$235$	5364.45			1.48910
$236$		0			0
$237$	−3745.68			−1.02662
$238$		0			0
$239$		−	310.784i		−	0.0841127i	−0.999115	−	0.0420563i	$-0.986609\pi$
$239$		−	310.784i		−	0.0841127i	0.999115	−	0.0420563i	$-0.0133909\pi$
$240$		0			0
$241$		−	457.833i		−	0.122372i	−0.998126	−	0.0611859i	$-0.980512\pi$
$241$		−	457.833i		−	0.122372i	0.998126	−	0.0611859i	$-0.0194883\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−3370.85	+	2825.04i	−0.879004	+	0.736673i
$246$		0			0
$247$			3207.99i			0.826394i
$248$		0			0
$249$	−714.909			−0.181950
$250$		0			0
$251$		−	807.916i		−	0.203168i	−0.994827	−	0.101584i	$-0.967609\pi$
$251$		−	807.916i		−	0.203168i	0.994827	−	0.101584i	$-0.0323911\pi$
$252$		0			0
$253$		−	2017.67i		−	0.501383i
$254$		0			0
$255$	−2978.35			−0.731418
$256$		0			0
$257$		−	1749.36i		−	0.424599i	−0.977205	−	0.212299i	$-0.931905\pi$
$257$		−	1749.36i		−	0.424599i	0.977205	−	0.212299i	$-0.0680952\pi$
$258$		0			0
$259$	−4770.80	+	1734.81i	−1.14457	+	0.416200i
$260$		0			0
$261$		−	1431.90i		−	0.339587i
$262$		0			0
$263$		−	1360.35i		−	0.318945i	−0.987202	−	0.159473i	$-0.949021\pi$
$263$		−	1360.35i		−	0.318945i	0.987202	−	0.159473i	$-0.0509793\pi$
$264$		0			0
$265$			8531.91i			1.97778i
$266$		0			0
$267$	2230.51			0.511255
$268$		0			0
$269$	−2224.06			−0.504102			−0.252051	−	0.967714i	$-0.581105\pi$
$269$	−2224.06			−0.504102			−0.252051	+	0.967714i	$0.581105\pi$
$270$		0			0
$271$	−2918.59			−0.654212			−0.327106	−	0.944988i	$-0.606073\pi$
$271$	−2918.59			−0.654212			−0.327106	+	0.944988i	$0.606073\pi$
$272$		0			0
$273$	−3350.18	+	1218.23i	−0.742719	+	0.270075i
$274$		0			0
$275$	−987.113			−0.216455
$276$		0			0
$277$			1632.84i			0.354180i	0.984195	+	0.177090i	$0.0566684\pi$
$277$			1632.84i			0.354180i	−0.984195	+	0.177090i	$0.943332\pi$
$278$		0			0
$279$	−869.037			−0.186480
$280$		0			0
$281$	−7615.42			−1.61672			−0.808360	−	0.588689i	$-0.799644\pi$
$281$	−7615.42			−1.61672			−0.808360	+	0.588689i	$0.799644\pi$
$282$		0			0
$283$			2867.07i			0.602226i	0.953589	+	0.301113i	$0.0973581\pi$
$283$			2867.07i			0.602226i	−0.953589	+	0.301113i	$0.902642\pi$
$284$		0			0
$285$	−1923.36			−0.399754
$286$		0			0
$287$	5218.03	−	1897.43i	1.07321	−	0.390251i
$288$		0			0
$289$	−1081.64			−0.220159
$290$		0			0
$291$	4656.49			0.938035
$292$		0			0
$293$	1133.07			0.225920			0.112960	−	0.993600i	$-0.463967\pi$
$293$	1133.07			0.225920			0.112960	+	0.993600i	$0.463967\pi$
$294$		0			0
$295$			5714.42i			1.12782i
$296$		0			0
$297$		−	676.158i		−	0.132103i
$298$		0			0
$299$		−	5169.32i		−	0.999831i
$300$		0			0
$301$	2441.91	+	6715.36i	0.467606	+	1.28594i
$302$		0			0
$303$			3607.37i			0.683954i
$304$		0			0
$305$	−7690.72			−1.44383
$306$		0			0
$307$		−	206.149i		−	0.0383242i	−0.999816	−	0.0191621i	$-0.993900\pi$
$307$		−	206.149i		−	0.0383242i	0.999816	−	0.0191621i	$-0.00609985\pi$
$308$		0			0
$309$		−	1213.69i		−	0.223444i
$310$		0			0
$311$	6296.00			1.14795			0.573977	−	0.818871i	$-0.305400\pi$
$311$	6296.00			1.14795			0.573977	+	0.818871i	$0.305400\pi$
$312$		0			0
$313$		−	7997.75i		−	1.44428i	−0.691747	−	0.722140i	$-0.743158\pi$
$313$		−	7997.75i		−	1.44428i	0.691747	−	0.722140i	$-0.256842\pi$
$314$		0			0
$315$	−730.392	−	2008.61i	−0.130644	−	0.359278i
$316$		0			0
$317$		−	5830.42i		−	1.03303i	−0.856279	−	0.516513i	$-0.827230\pi$
$317$		−	5830.42i		−	1.03303i	0.856279	−	0.516513i	$-0.172770\pi$
$318$		0			0
$319$		−	3984.32i		−	0.699307i
$320$		0			0
$321$			3197.17i			0.555914i
$322$		0			0
$323$	−3871.22			−0.666874
$324$		0			0
$325$	−2529.00			−0.431643
$326$		0			0
$327$	−1078.55			−0.182397
$328$		0			0
$329$	2647.85	+	7281.70i	0.443710	+	1.22022i
$330$		0			0
$331$	5022.44			0.834013			0.417007	−	0.908903i	$-0.363079\pi$
$331$	5022.44			0.834013			0.417007	+	0.908903i	$0.363079\pi$
$332$		0			0
$333$		−	2466.91i		−	0.405964i
$334$		0			0
$335$	−8661.66			−1.41265
$336$		0			0
$337$	8480.49			1.37081			0.685403	−	0.728164i	$-0.259626\pi$
$337$	8480.49			1.37081			0.685403	+	0.728164i	$0.259626\pi$
$338$		0			0
$339$		−	2485.38i		−	0.398193i
$340$		0			0
$341$	−2418.13			−0.384015
$342$		0			0
$343$	−5498.52	−	3181.18i	−0.865575	−	0.500780i
$344$		0			0
$345$	3099.28			0.483651
$346$		0			0
$347$	−9890.27			−1.53008			−0.765040	−	0.643983i	$-0.777281\pi$
$347$	−9890.27			−1.53008			−0.765040	+	0.643983i	$0.777281\pi$
$348$		0			0
$349$	6913.69			1.06041			0.530203	−	0.847871i	$-0.322116\pi$
$349$	6913.69			1.06041			0.530203	+	0.847871i	$0.322116\pi$
$350$		0			0
$351$		−	1732.33i		−	0.263433i
$352$		0			0
$353$			5062.99i			0.763388i	0.924289	+	0.381694i	$0.124659\pi$
$353$			5062.99i			0.763388i	−0.924289	+	0.381694i	$0.875341\pi$
$354$		0			0
$355$			11246.5i			1.68142i
$356$		0			0
$357$	−1470.09	−	4042.81i	−0.217942	−	0.599350i
$358$		0			0
$359$		−	7551.38i		−	1.11016i	−0.831798	−	0.555079i	$-0.812688\pi$
$359$		−	7551.38i		−	1.11016i	0.831798	−	0.555079i	$-0.187312\pi$
$360$		0			0
$361$	4359.05			0.635522
$362$		0			0
$363$			2111.56i			0.305312i
$364$		0			0
$365$			8925.25i			1.27991i
$366$		0			0
$367$	12597.1			1.79173			0.895864	−	0.444329i	$-0.146558\pi$
$367$	12597.1			1.79173			0.895864	+	0.444329i	$0.146558\pi$
$368$		0			0
$369$			2698.17i			0.380653i
$370$		0			0
$371$	−11581.2	+	4211.27i	−1.62066	+	0.589322i
$372$		0			0
$373$		−	9549.28i		−	1.32558i	−0.748803	−	0.662792i	$-0.769371\pi$
$373$		−	9549.28i		−	1.32558i	0.748803	−	0.662792i	$-0.230629\pi$
$374$		0			0
$375$			3292.17i			0.453352i
$376$		0			0
$377$		−	10207.9i		−	1.39452i
$378$		0			0
$379$	−1603.13			−0.217276			−0.108638	−	0.994081i	$-0.534649\pi$
$379$	−1603.13			−0.217276			−0.108638	+	0.994081i	$0.534649\pi$
$380$		0			0
$381$	2629.72			0.353608
$382$		0			0
$383$	402.889			0.0537511			0.0268756	−	0.999639i	$-0.491444\pi$
$383$	402.889			0.0537511			0.0268756	+	0.999639i	$0.491444\pi$
$384$		0			0
$385$	−2032.35	−	5589.05i	−0.269034	−	0.739855i
$386$		0			0
$387$	−3472.42			−0.456105
$388$		0			0
$389$			8540.21i			1.11313i	0.830806	+	0.556563i	$0.187880\pi$
$389$			8540.21i			1.11313i	−0.830806	+	0.556563i	$0.812120\pi$
$390$		0			0
$391$	6238.04			0.806831
$392$		0			0
$393$	3249.13			0.417040
$394$		0			0
$395$		−	16009.7i		−	2.03933i
$396$		0			0
$397$	11977.7			1.51421			0.757106	−	0.653292i	$-0.226613\pi$
$397$	11977.7			1.51421			0.757106	+	0.653292i	$0.226613\pi$
$398$		0			0
$399$	−949.352	−	2610.76i	−0.119115	−	0.327573i
$400$		0			0
$401$	58.8089			0.00732364			0.00366182	−	0.999993i	$-0.498834\pi$
$401$	58.8089			0.00732364			0.00366182	+	0.999993i	$0.498834\pi$
$402$		0			0
$403$	−6195.30			−0.765782
$404$		0			0
$405$	1038.62			0.127431
$406$		0			0
$407$		−	6864.29i		−	0.835996i
$408$		0			0
$409$			805.417i			0.0973723i	0.998814	+	0.0486862i	$0.0155034\pi$
$409$			805.417i			0.0973723i	−0.998814	+	0.0486862i	$0.984497\pi$
$410$		0			0
$411$			2255.51i			0.270696i
$412$		0			0
$413$	−7756.75	+	2820.59i	−0.924176	+	0.336058i
$414$		0			0
$415$		−	3055.64i		−	0.361435i
$416$		0			0
$417$	1630.66			0.191496
$418$		0			0
$419$		−	7784.37i		−	0.907616i	−0.891099	−	0.453808i	$-0.850065\pi$
$419$		−	7784.37i		−	0.907616i	0.891099	−	0.453808i	$-0.149935\pi$
$420$		0			0
$421$		−	4998.41i		−	0.578640i	−0.957232	−	0.289320i	$-0.906571\pi$
$421$		−	4998.41i		−	0.578640i	0.957232	−	0.289320i	$-0.0934292\pi$
$422$		0			0
$423$	−3765.26			−0.432797
$424$		0			0
$425$		−	3051.86i		−	0.348322i
$426$		0			0
$427$	−3796.07	−	10439.4i	−0.430222	−	1.18313i
$428$		0			0
$429$		−	4820.28i		−	0.542484i
$430$		0			0
$431$			5983.17i			0.668675i	0.942453	+	0.334338i	$0.108513\pi$
$431$			5983.17i			0.668675i	−0.942453	+	0.334338i	$0.891487\pi$
$432$		0			0
$433$		−	3739.75i		−	0.415060i	−0.978229	−	0.207530i	$-0.933458\pi$
$433$		−	3739.75i		−	0.415060i	0.978229	−	0.207530i	$-0.0665425\pi$
$434$		0			0
$435$	6120.18			0.674575
$436$		0			0
$437$	4028.40			0.440971
$438$		0			0
$439$	8032.79			0.873313			0.436656	−	0.899628i	$-0.356163\pi$
$439$	8032.79			0.873313			0.436656	+	0.899628i	$0.356163\pi$
$440$		0			0
$441$	2365.97	−	1982.86i	0.255477	−	0.214109i
$442$		0			0
$443$	−10025.1			−1.07519			−0.537595	−	0.843203i	$-0.680667\pi$
$443$	−10025.1			−1.07519			−0.537595	+	0.843203i	$0.680667\pi$
$444$		0			0
$445$			9533.58i			1.01558i
$446$		0			0
$447$	3169.33			0.335357
$448$		0			0
$449$	4911.85			0.516268			0.258134	−	0.966109i	$-0.416892\pi$
$449$	4911.85			0.516268			0.258134	+	0.966109i	$0.416892\pi$
$450$		0			0
$451$			7507.77i			0.783873i
$452$		0			0
$453$	6276.40			0.650973
$454$		0			0
$455$	−5206.91	−	14319.3i	−0.536492	−	1.47538i
$456$		0			0
$457$	10828.7			1.10842			0.554208	−	0.832378i	$-0.313021\pi$
$457$	10828.7			1.10842			0.554208	+	0.832378i	$0.313021\pi$
$458$		0			0
$459$	2090.48			0.212582
$460$		0			0
$461$	−7105.06			−0.717821			−0.358911	−	0.933372i	$-0.616852\pi$
$461$	−7105.06			−0.717821			−0.358911	+	0.933372i	$0.616852\pi$
$462$		0			0
$463$			14241.0i			1.42945i	0.699403	+	0.714727i	$0.253449\pi$
$463$			14241.0i			1.42945i	−0.699403	+	0.714727i	$0.746551\pi$
$464$		0			0
$465$		−	3714.41i		−	0.370434i
$466$		0			0
$467$			5568.41i			0.551767i	0.961191	+	0.275884i	$0.0889705\pi$
$467$			5568.41i			0.551767i	−0.961191	+	0.275884i	$0.911030\pi$
$468$		0			0
$469$	−4275.32	−	11757.3i	−0.420929	−	1.15757i
$470$		0			0
$471$			3557.11i			0.347989i
$472$		0			0
$473$	−9662.15			−0.939252
$474$		0			0
$475$		−	1970.83i		−	0.190374i
$476$		0			0
$477$		−	5988.46i		−	0.574828i
$478$		0			0
$479$	−14755.8			−1.40754			−0.703768	−	0.710430i	$-0.748501\pi$
$479$	−14755.8			−1.40754			−0.703768	+	0.710430i	$0.748501\pi$
$480$		0			0
$481$		−	17586.4i		−	1.66710i
$482$		0			0
$483$	1529.78	+	4206.96i	0.144114	+	0.396321i
$484$		0			0
$485$			19902.6i			1.86336i
$486$		0			0
$487$			2870.19i			0.267065i	0.991044	+	0.133533i	$0.0426321\pi$
$487$			2870.19i			0.267065i	−0.991044	+	0.133533i	$0.957368\pi$
$488$		0			0
$489$			2710.91i			0.250698i
$490$		0			0
$491$	13050.2			1.19949			0.599745	−	0.800191i	$-0.295269\pi$
$491$	13050.2			1.19949			0.599745	+	0.800191i	$0.295269\pi$
$492$		0			0
$493$	12318.3			1.12533
$494$		0			0
$495$	2890.02			0.262417
$496$		0			0
$497$	−15266.0	+	5551.19i	−1.37782	+	0.501016i
$498$		0			0
$499$	10021.7			0.899063			0.449531	−	0.893265i	$-0.351591\pi$
$499$	10021.7			0.899063			0.449531	+	0.893265i	$0.351591\pi$
$500$		0			0
$501$			1423.90i			0.126976i
$502$		0			0
$503$	−13625.2			−1.20779			−0.603894	−	0.797065i	$-0.706385\pi$
$503$	−13625.2			−1.20779			−0.603894	+	0.797065i	$0.706385\pi$
$504$		0			0
$505$	−15418.5			−1.35865
$506$		0			0
$507$		−	5758.66i		−	0.504440i
$508$		0			0
$509$	−6931.67			−0.603617			−0.301808	−	0.953369i	$-0.597590\pi$
$509$	−6931.67			−0.603617			−0.301808	+	0.953369i	$0.597590\pi$
$510$		0			0
$511$	−12115.1	+	4405.42i	−1.04881	+	0.381378i
$512$		0			0
$513$	1349.99			0.116186
$514$		0			0
$515$	5187.50			0.443861
$516$		0			0
$517$	−10477.0			−0.891253
$518$		0			0
$519$		−	4923.98i		−	0.416452i
$520$		0			0
$521$		−	20497.7i		−	1.72365i	−0.507207	−	0.861824i	$-0.669322\pi$
$521$		−	20497.7i		−	1.72365i	0.507207	−	0.861824i	$-0.330678\pi$
$522$		0			0
$523$		−	17220.5i		−	1.43977i	−0.694094	−	0.719885i	$-0.744195\pi$
$523$		−	17220.5i		−	1.43977i	0.694094	−	0.719885i	$-0.255805\pi$
$524$		0			0
$525$	2058.18	−	748.418i	0.171098	−	0.0622165i
$526$		0			0
$527$		−	7476.14i		−	0.617961i
$528$		0			0
$529$	5675.68			0.466482
$530$		0			0
$531$		−	4010.90i		−	0.327793i
$532$		0			0
$533$			19235.0i			1.56316i
$534$		0			0
$535$	−13665.2			−1.10430
$536$		0			0
$537$			5406.33i			0.434451i
$538$		0			0
$539$	6583.41	−	5517.41i	0.526100	−	0.440912i
$540$		0			0
$541$		−	6118.24i		−	0.486218i	−0.969999	−	0.243109i	$-0.921833\pi$
$541$		−	6118.24i		−	0.486218i	0.969999	−	0.243109i	$-0.0781672\pi$
$542$		0			0
$543$			6205.35i			0.490418i
$544$		0			0
$545$		−	4609.89i		−	0.362323i
$546$		0			0
$547$	391.181			0.0305772			0.0152886	−	0.999883i	$-0.495133\pi$
$547$	391.181			0.0305772			0.0152886	+	0.999883i	$0.495133\pi$
$548$		0			0
$549$	5398.04			0.419641
$550$		0			0
$551$	7954.91			0.615047
$552$		0			0
$553$	21731.5	−	7902.23i	1.67110	−	0.607662i
$554$		0			0
$555$	10544.0			0.806429
$556$		0			0
$557$			14903.1i			1.13369i	0.823824	+	0.566846i	$0.191837\pi$
$557$			14903.1i			1.13369i	−0.823824	+	0.566846i	$0.808163\pi$
$558$		0			0
$559$	−24754.6			−1.87300
$560$		0			0
$561$	5816.84			0.437767
$562$		0			0
$563$			18651.1i			1.39618i	0.716010	+	0.698090i	$0.245966\pi$
$563$			18651.1i			1.39618i	−0.716010	+	0.698090i	$0.754034\pi$
$564$		0			0
$565$	10623.0			0.790993
$566$		0			0
$567$	512.655	+	1409.83i	0.0379709	+	0.104422i
$568$		0			0
$569$	−14380.9			−1.05954			−0.529770	−	0.848141i	$-0.677722\pi$
$569$	−14380.9			−1.05954			−0.529770	+	0.848141i	$0.677722\pi$
$570$		0			0
$571$	26145.9			1.91623			0.958117	−	0.286376i	$-0.0924506\pi$
$571$	26145.9			1.91623			0.958117	+	0.286376i	$0.0924506\pi$
$572$		0			0
$573$	6538.82			0.476724
$574$		0			0
$575$			3175.77i			0.230328i
$576$		0			0
$577$			18408.8i			1.32820i	0.747645	+	0.664099i	$0.231185\pi$
$577$			18408.8i			1.32820i	−0.747645	+	0.664099i	$0.768815\pi$
$578$		0			0
$579$			13469.4i			0.966788i
$580$		0			0
$581$	4147.72	−	1508.24i	0.296173	−	0.107697i
$582$		0			0
$583$		−	16663.2i		−	1.18374i
$584$		0			0
$585$	7404.28			0.523298
$586$		0			0
$587$		−	25942.5i		−	1.82412i	−0.410053	−	0.912062i	$-0.634490\pi$
$587$		−	25942.5i		−	1.82412i	0.410053	−	0.912062i	$-0.365510\pi$
$588$		0			0
$589$		−	4827.94i		−	0.337745i
$590$		0			0
$591$	−13557.2			−0.943604
$592$		0			0
$593$		−	19221.9i		−	1.33111i	−0.746347	−	0.665557i	$-0.768194\pi$
$593$		−	19221.9i		−	1.33111i	0.746347	−	0.665557i	$-0.231806\pi$
$594$		0			0
$595$	17279.7	−	6283.41i	1.19058	−	0.432932i
$596$		0			0
$597$		−	2479.79i		−	0.170002i
$598$		0			0
$599$		−	14280.0i		−	0.974068i	−0.873383	−	0.487034i	$-0.838079\pi$
$599$		−	14280.0i		−	0.974068i	0.873383	−	0.487034i	$-0.161921\pi$
$600$		0			0
$601$			5677.17i			0.385319i	0.981266	+	0.192659i	$0.0617113\pi$
$601$			5677.17i			0.385319i	−0.981266	+	0.192659i	$0.938289\pi$
$602$		0			0
$603$	6079.54			0.410577
$604$		0			0
$605$	−9025.18			−0.606489
$606$		0			0
$607$	−16394.3			−1.09625			−0.548124	−	0.836397i	$-0.684658\pi$
$607$	−16394.3			−1.09625			−0.548124	+	0.836397i	$0.684658\pi$
$608$		0			0
$609$	3020.86	+	8307.52i	0.201004	+	0.552771i
$610$		0			0
$611$	−26842.3			−1.77729
$612$		0			0
$613$			2171.28i			0.143063i	0.997438	+	0.0715313i	$0.0227886\pi$
$613$			2171.28i			0.143063i	−0.997438	+	0.0715313i	$0.977211\pi$
$614$		0			0
$615$	−11532.4			−0.756150
$616$		0			0
$617$	−6024.75			−0.393108			−0.196554	−	0.980493i	$-0.562975\pi$
$617$	−6024.75			−0.393108			−0.196554	+	0.980493i	$0.562975\pi$
$618$		0			0
$619$		−	13135.5i		−	0.852926i	−0.904505	−	0.426463i	$-0.859760\pi$
$619$		−	13135.5i		−	0.852926i	0.904505	−	0.426463i	$-0.140240\pi$
$620$		0			0
$621$	−2175.36			−0.140570
$622$		0			0
$623$	−12940.9	+	4705.69i	−0.832206	+	0.302615i
$624$		0			0
$625$	−18998.4			−1.21590
$626$		0			0
$627$	3756.40			0.239260
$628$		0			0
$629$	21222.3			1.34529
$630$		0			0
$631$		−	13957.4i		−	0.880562i	−0.897860	−	0.440281i	$-0.854879\pi$
$631$		−	13957.4i		−	0.880562i	0.897860	−	0.440281i	$-0.145121\pi$
$632$		0			0
$633$		−	14127.5i		−	0.887074i
$634$		0			0
$635$			11239.9i			0.702426i
$636$		0			0
$637$	16866.8	−	14135.7i	1.04912	−	0.879242i
$638$		0			0
$639$		−	7893.84i		−	0.488694i
$640$		0			0
$641$	24561.9			1.51348			0.756738	−	0.653718i	$-0.226792\pi$
$641$	24561.9			1.51348			0.756738	+	0.653718i	$0.226792\pi$
$642$		0			0
$643$			15778.3i			0.967708i	0.875149	+	0.483854i	$0.160763\pi$
$643$			15778.3i			0.967708i	−0.875149	+	0.483854i	$0.839237\pi$
$644$		0			0
$645$		−	14841.7i		−	0.906033i
$646$		0			0
$647$	11167.7			0.678588			0.339294	−	0.940680i	$-0.389812\pi$
$647$	11167.7			0.678588			0.339294	+	0.940680i	$0.389812\pi$
$648$		0			0
$649$		−	11160.5i		−	0.675020i
$650$		0			0
$651$	5041.94	−	1833.40i	0.303547	−	0.110379i
$652$		0			0
$653$			15738.6i			0.943185i	0.881817	+	0.471592i	$0.156321\pi$
$653$			15738.6i			0.943185i	−0.881817	+	0.471592i	$0.843679\pi$
$654$		0			0
$655$			13887.3i			0.828432i
$656$		0			0
$657$		−	6264.54i		−	0.371999i
$658$		0			0
$659$	−21821.3			−1.28989			−0.644944	−	0.764230i	$-0.723119\pi$
$659$	−21821.3			−1.28989			−0.644944	+	0.764230i	$0.723119\pi$
$660$		0			0
$661$	−5703.63			−0.335621			−0.167810	−	0.985819i	$-0.553670\pi$
$661$	−5703.63			−0.335621			−0.167810	+	0.985819i	$0.553670\pi$
$662$		0			0
$663$	14902.9			0.872970
$664$		0			0
$665$	11158.8	−	4057.69i	0.650709	−	0.236618i
$666$		0			0
$667$	−12818.5			−0.744127
$668$		0			0
$669$		−	17325.3i		−	1.00125i
$670$		0			0
$671$	15020.3			0.864161
$672$		0			0
$673$	30185.3			1.72891			0.864456	−	0.502708i	$-0.167663\pi$
$673$	30185.3			1.72891			0.864456	+	0.502708i	$0.167663\pi$
$674$		0			0
$675$			1064.26i			0.0606863i
$676$		0			0
$677$	12990.9			0.737488			0.368744	−	0.929531i	$-0.379788\pi$
$677$	12990.9			0.737488			0.368744	+	0.929531i	$0.379788\pi$
$678$		0			0
$679$	−27015.8	+	9823.76i	−1.52691	+	0.555230i
$680$		0			0
$681$	6840.32			0.384907
$682$		0			0
$683$	3636.79			0.203745			0.101872	−	0.994797i	$-0.467517\pi$
$683$	3636.79			0.203745			0.101872	+	0.994797i	$0.467517\pi$
$684$		0			0
$685$	−9640.43			−0.537725
$686$		0			0
$687$			11079.0i			0.615267i
$688$		0			0
$689$		−	42691.3i		−	2.36054i
$690$		0			0
$691$			869.636i			0.0478763i	0.999713	+	0.0239381i	$0.00762048\pi$
$691$			869.636i			0.0478763i	−0.999713	+	0.0239381i	$0.992380\pi$
$692$		0			0
$693$	1426.49	+	3922.90i	0.0781929	+	0.215034i
$694$		0			0
$695$			6969.74i			0.380399i
$696$		0			0
$697$	−23211.7			−1.26142
$698$		0			0
$699$		−	9420.20i		−	0.509735i
$700$		0			0
$701$			8128.07i			0.437936i	0.975732	+	0.218968i	$0.0702690\pi$
$701$			8128.07i			0.437936i	−0.975732	+	0.218968i	$0.929731\pi$
$702$		0			0
$703$	13704.9			0.735265
$704$		0			0
$705$		−	16093.4i		−	0.859732i
$706$		0			0
$707$	−7610.45	−	20929.1i	−0.404838	−	1.11332i
$708$		0			0
$709$		−	6488.87i		−	0.343716i	−0.985122	−	0.171858i	$-0.945023\pi$
$709$		−	6488.87i		−	0.343716i	0.985122	−	0.171858i	$-0.0549770\pi$
$710$		0			0
$711$			11237.0i			0.592717i
$712$		0			0
$713$			7779.69i			0.408628i
$714$		0			0
$715$	20602.7			1.07762
$716$		0			0
$717$	−932.351			−0.0485625
$718$		0			0
$719$	19419.9			1.00729			0.503645	−	0.863911i	$-0.331992\pi$
$719$	19419.9			1.00729			0.503645	+	0.863911i	$0.331992\pi$
$720$		0			0
$721$	2560.50	+	7041.50i	0.132258	+	0.363716i
$722$		0			0
$723$	−1373.50			−0.0706514
$724$		0			0
$725$			6271.22i			0.321252i
$726$		0			0
$727$	−29343.3			−1.49695			−0.748475	−	0.663163i	$-0.769214\pi$
$727$	−29343.3			−1.49695			−0.748475	+	0.663163i	$0.769214\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$		−	29872.4i		−	1.51145i
$732$		0			0
$733$	27788.9			1.40028			0.700140	−	0.714006i	$-0.253121\pi$
$733$	27788.9			1.40028			0.700140	+	0.714006i	$0.253121\pi$
$734$		0			0
$735$	8475.11	+	10112.6i	0.425318	+	0.507493i
$736$		0			0
$737$	16916.6			0.845495
$738$		0			0
$739$	12588.2			0.626607			0.313304	−	0.949653i	$-0.398564\pi$
$739$	12588.2			0.626607			0.313304	+	0.949653i	$0.398564\pi$
$740$		0			0
$741$	9623.96			0.477119
$742$		0			0
$743$		−	8137.53i		−	0.401799i	−0.979612	−	0.200900i	$-0.935613\pi$
$743$		−	8137.53i		−	0.401799i	0.979612	−	0.200900i	$-0.0643865\pi$
$744$		0			0
$745$			13546.3i			0.666171i
$746$		0			0
$747$			2144.73i			0.105049i
$748$		0			0
$749$	−6745.03	−	18549.2i	−0.329050	−	0.904902i
$750$		0			0
$751$			2723.45i			0.132331i	0.997809	+	0.0661653i	$0.0210765\pi$
$751$			2723.45i			0.132331i	−0.997809	+	0.0661653i	$0.978924\pi$
$752$		0			0
$753$	−2423.75			−0.117299
$754$		0			0
$755$			26826.4i			1.29313i
$756$		0			0
$757$			8168.46i			0.392190i	0.980585	+	0.196095i	$0.0628261\pi$
$757$			8168.46i			0.392190i	−0.980585	+	0.196095i	$0.937174\pi$
$758$		0			0
$759$	−6053.02			−0.289474
$760$		0			0
$761$		−	24255.5i		−	1.15540i	−0.816248	−	0.577702i	$-0.803950\pi$
$761$		−	24255.5i		−	1.15540i	0.816248	−	0.577702i	$-0.196050\pi$
$762$		0			0
$763$	6257.46	−	2275.40i	0.296901	−	0.107962i
$764$		0			0
$765$			8935.06i			0.422285i
$766$		0			0
$767$		−	28593.4i		−	1.34609i
$768$		0			0
$769$			13226.7i			0.620241i	0.950697	+	0.310121i	$0.100369\pi$
$769$			13226.7i			0.620241i	−0.950697	+	0.310121i	$0.899631\pi$
$770$		0			0
$771$	−5248.07			−0.245142
$772$		0			0
$773$	24347.2			1.13287			0.566434	−	0.824107i	$-0.308323\pi$
$773$	24347.2			1.13287			0.566434	+	0.824107i	$0.308323\pi$
$774$		0			0
$775$	3806.08			0.176411
$776$		0			0
$777$	5204.42	+	14312.4i	0.240293	+	0.660817i
$778$		0			0
$779$	−14989.7			−0.689423
$780$		0			0
$781$		−	21964.9i		−	1.00636i
$782$		0			0
$783$	−4295.69			−0.196061
$784$		0			0
$785$	−15203.7			−0.691265
$786$		0			0
$787$			31493.4i			1.42645i	0.700933	+	0.713227i	$0.252767\pi$
$787$			31493.4i			1.42645i	−0.700933	+	0.713227i	$0.747233\pi$
$788$		0			0
$789$	−4081.04			−0.184143
$790$		0			0
$791$	5243.39	+	14419.6i	0.235694	+	0.648168i
$792$		0			0
$793$	38482.3			1.72326
$794$		0			0
$795$	25595.7			1.14187
$796$		0			0
$797$	−27985.5			−1.24378			−0.621892	−	0.783103i	$-0.713636\pi$
$797$	−27985.5			−1.24378			−0.621892	+	0.783103i	$0.713636\pi$
$798$		0			0
$799$		−	32391.7i		−	1.43421i
$800$		0			0
$801$		−	6691.53i		−	0.295173i
$802$

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +12.8225 q^{5} +(6.32908 + 17.4053i) q^{7} -9.00000 q^{9} -25.0429 q^{11} -64.1604 q^{13} -38.4675i q^{15} -77.4251i q^{17} -49.9995i q^{19} +(52.2158 - 18.9872i) q^{21} +80.5687i q^{23} +39.4169 q^{25} +27.0000i q^{27} +159.100i q^{29} +96.5596 q^{31} +75.1287i q^{33} +(81.1547 + 223.179i) q^{35} +274.101i q^{37} +192.481i q^{39} -299.796i q^{41} +385.824 q^{43} -115.403 q^{45} +418.362 q^{47} +(-262.886 + 220.318i) q^{49} -232.275 q^{51} +665.385i q^{53} -321.113 q^{55} -149.999 q^{57} +445.655i q^{59} -599.783 q^{61} +(-56.9617 - 156.647i) q^{63} -822.697 q^{65} -675.504 q^{67} +241.706 q^{69} +877.093i q^{71} +696.060i q^{73} -118.251i q^{75} +(-158.498 - 435.878i) q^{77} -1248.56i q^{79} +81.0000 q^{81} -238.303i q^{83} -992.784i q^{85} +477.299 q^{87} +743.503i q^{89} +(-406.076 - 1116.73i) q^{91} -289.679i q^{93} -641.120i q^{95} +1552.16i q^{97} +225.386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32q - 288q^{9} + O(q^{10}) \)	\( 32q - 288q^{9} + 224q^{13} + 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} + 544q^{61} + 1536q^{65} + 144q^{69} + 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Introduction

L-functions

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