LMFDB - Embedded newform 1344.4.p.c.223.6

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			223.6
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.5

$q$-expansion

$f(q)$	$=$	$q+3.00000i q^{3} +20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} -39.2814 q^{11} -36.2449 q^{13} +62.1010i q^{15} +92.2048i q^{17} +87.5184i q^{19} +(-42.7820 - 35.4499i) q^{21} -100.990i q^{23} +303.504 q^{25} -27.0000i q^{27} +36.2034i q^{29} -264.306 q^{31} -117.844i q^{33} +(-244.609 + 295.201i) q^{35} -229.534i q^{37} -108.735i q^{39} -337.096i q^{41} -368.528 q^{43} -186.303 q^{45} +314.672 q^{47} +(-63.7338 - 337.027i) q^{49} -276.614 q^{51} -108.563i q^{53} -813.139 q^{55} -262.555 q^{57} -185.373i q^{59} -163.504 q^{61} +(106.350 - 128.346i) q^{63} -750.282 q^{65} -458.260 q^{67} +302.969 q^{69} -584.527i q^{71} +1099.24i q^{73} +910.512i q^{75} +(464.175 - 560.180i) q^{77} +438.627i q^{79} +81.0000 q^{81} -722.114i q^{83} +1908.67i q^{85} -108.610 q^{87} -452.334i q^{89} +(428.293 - 516.877i) q^{91} -792.917i q^{93} +1811.66i q^{95} -598.000i q^{97} +353.533 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	20.7003			1.85149			0.925747	−	0.378143i	$-0.123437\pi$
$5$	20.7003			1.85149			0.925747	+	0.378143i	$0.123437\pi$
$6$		0			0
$7$	−11.8166	+	14.2607i	−0.638039	+	0.770004i
$7$	−11.8166	+	14.2607i	−0.638039	+	0.770004i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−39.2814			−1.07671			−0.538354	−	0.842719i	$-0.680954\pi$
$11$	−39.2814			−1.07671			−0.538354	+	0.842719i	$0.680954\pi$
$12$		0			0
$13$	−36.2449			−0.773271			−0.386636	−	0.922233i	$-0.626363\pi$
$13$	−36.2449			−0.773271			−0.386636	+	0.922233i	$0.626363\pi$
$14$		0			0
$15$			62.1010i			1.06896i
$16$		0			0
$17$			92.2048i			1.31547i	0.753251	+	0.657733i	$0.228485\pi$
$17$			92.2048i			1.31547i	−0.753251	+	0.657733i	$0.771515\pi$
$18$		0			0
$19$			87.5184i			1.05674i	0.849013	+	0.528371i	$0.177197\pi$
$19$			87.5184i			1.05674i	−0.849013	+	0.528371i	$0.822803\pi$
$20$		0			0
$21$	−42.7820	−	35.4499i	−0.444562	−	0.368372i
$22$		0			0
$23$		−	100.990i		−	0.915558i	−0.889066	−	0.457779i	$-0.848645\pi$
$23$		−	100.990i		−	0.915558i	0.889066	−	0.457779i	$-0.151355\pi$
$24$		0			0
$25$	303.504			2.42803
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$			36.2034i			0.231821i	0.993260	+	0.115911i	$0.0369786\pi$
$29$			36.2034i			0.231821i	−0.993260	+	0.115911i	$0.963021\pi$
$30$		0			0
$31$	−264.306			−1.53131			−0.765656	−	0.643250i	$-0.777586\pi$
$31$	−264.306			−1.53131			−0.765656	+	0.643250i	$0.777586\pi$
$32$		0			0
$33$		−	117.844i		−	0.621638i
$34$		0			0
$35$	−244.609	+	295.201i	−1.18133	+	1.42566i
$36$		0			0
$37$		−	229.534i		−	1.01987i	−0.860213	−	0.509936i	$-0.829669\pi$
$37$		−	229.534i		−	1.01987i	0.860213	−	0.509936i	$-0.170331\pi$
$38$		0			0
$39$		−	108.735i		−	0.446448i
$40$		0			0
$41$		−	337.096i		−	1.28404i	−0.766688	−	0.642019i	$-0.778097\pi$
$41$		−	337.096i		−	1.28404i	0.766688	−	0.642019i	$-0.221903\pi$
$42$		0			0
$43$	−368.528			−1.30697			−0.653487	−	0.756937i	$-0.726695\pi$
$43$	−368.528			−1.30697			−0.653487	+	0.756937i	$0.726695\pi$
$44$		0			0
$45$	−186.303			−0.617165
$46$		0			0
$47$	314.672			0.976588			0.488294	−	0.872679i	$-0.337619\pi$
$47$	314.672			0.976588			0.488294	+	0.872679i	$0.337619\pi$
$48$		0			0
$49$	−63.7338	−	337.027i	−0.185813	−	0.982585i
$50$		0			0
$51$	−276.614			−0.759485
$52$		0			0
$53$		−	108.563i		−	0.281362i	−0.990055	−	0.140681i	$-0.955071\pi$
$53$		−	108.563i		−	0.281362i	0.990055	−	0.140681i	$-0.0449293\pi$
$54$		0			0
$55$	−813.139			−1.99352
$56$		0			0
$57$	−262.555			−0.610110
$58$		0			0
$59$		−	185.373i		−	0.409043i	−0.978862	−	0.204521i	$-0.934436\pi$
$59$		−	185.373i		−	0.409043i	0.978862	−	0.204521i	$-0.0655638\pi$
$60$		0			0
$61$	−163.504			−0.343189			−0.171594	−	0.985168i	$-0.554892\pi$
$61$	−163.504			−0.343189			−0.171594	+	0.985168i	$0.554892\pi$
$62$		0			0
$63$	106.350	−	128.346i	0.212680	−	0.256668i
$64$		0			0
$65$	−750.282			−1.43171
$66$		0			0
$67$	−458.260			−0.835603			−0.417802	−	0.908538i	$-0.637199\pi$
$67$	−458.260			−0.835603			−0.417802	+	0.908538i	$0.637199\pi$
$68$		0			0
$69$	302.969			0.528597
$70$		0			0
$71$		−	584.527i		−	0.977051i	−0.872550	−	0.488525i	$-0.837535\pi$
$71$		−	584.527i		−	0.977051i	0.872550	−	0.488525i	$-0.162465\pi$
$72$		0			0
$73$			1099.24i			1.76241i	0.472736	+	0.881204i	$0.343266\pi$
$73$			1099.24i			1.76241i	−0.472736	+	0.881204i	$0.656734\pi$
$74$		0			0
$75$			910.512i			1.40183i
$76$		0			0
$77$	464.175	−	560.180i	0.686982	−	0.829070i
$78$		0			0
$79$			438.627i			0.624676i	0.949971	+	0.312338i	$0.101112\pi$
$79$			438.627i			0.624676i	−0.949971	+	0.312338i	$0.898888\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	722.114i		−	0.954967i	−0.878641	−	0.477484i	$-0.841549\pi$
$83$		−	722.114i		−	0.954967i	0.878641	−	0.477484i	$-0.158451\pi$
$84$		0			0
$85$			1908.67i			2.43558i
$86$		0			0
$87$	−108.610			−0.133842
$88$		0			0
$89$		−	452.334i		−	0.538734i	−0.963038	−	0.269367i	$-0.913186\pi$
$89$		−	452.334i		−	0.538734i	0.963038	−	0.269367i	$-0.0868144\pi$
$90$		0			0
$91$	428.293	−	516.877i	0.493377	−	0.595422i
$92$		0			0
$93$		−	792.917i		−	0.884103i
$94$		0			0
$95$			1811.66i			1.95655i
$96$		0			0
$97$		−	598.000i		−	0.625956i	−0.949760	−	0.312978i	$-0.898673\pi$
$97$		−	598.000i		−	0.625956i	0.949760	−	0.312978i	$-0.101327\pi$
$98$		0			0
$99$	353.533			0.358903
$100$		0			0
$101$	1415.04			1.39407			0.697037	−	0.717035i	$-0.254501\pi$
$101$	1415.04			1.39407			0.697037	+	0.717035i	$0.254501\pi$
$102$		0			0
$103$	−1138.86			−1.08947			−0.544733	−	0.838609i	$-0.683369\pi$
$103$	−1138.86			−1.08947			−0.544733	+	0.838609i	$0.683369\pi$
$104$		0			0
$105$	−885.602	−	733.826i	−0.823104	−	0.682039i
$106$		0			0
$107$	1503.38			1.35829			0.679147	−	0.734002i	$-0.262350\pi$
$107$	1503.38			1.35829			0.679147	+	0.734002i	$0.262350\pi$
$108$		0			0
$109$		−	359.283i		−	0.315716i	−0.987462	−	0.157858i	$-0.949541\pi$
$109$		−	359.283i		−	0.315716i	0.987462	−	0.157858i	$-0.0504589\pi$
$110$		0			0
$111$	688.603			0.588823
$112$		0			0
$113$	−2363.20			−1.96736			−0.983679	−	0.179932i	$-0.942412\pi$
$113$	−2363.20			−1.96736			−0.983679	+	0.179932i	$0.942412\pi$
$114$		0			0
$115$		−	2090.52i		−	1.69515i
$116$		0			0
$117$	326.204			0.257757
$118$		0			0
$119$	−1314.90	−	1089.55i	−1.01291	−	0.839319i
$120$		0			0
$121$	212.030			0.159301
$122$		0			0
$123$	1011.29			0.741340
$124$		0			0
$125$	3695.09			2.64399
$126$		0			0
$127$			1620.69i			1.13238i	0.824274	+	0.566192i	$0.191584\pi$
$127$			1620.69i			1.13238i	−0.824274	+	0.566192i	$0.808416\pi$
$128$		0			0
$129$		−	1105.58i		−	0.754582i
$130$		0			0
$131$		−	503.229i		−	0.335629i	−0.985819	−	0.167814i	$-0.946329\pi$
$131$		−	503.229i		−	0.335629i	0.985819	−	0.167814i	$-0.0536709\pi$
$132$		0			0
$133$	−1248.07	−	1034.17i	−0.813696	−	0.674243i
$134$		0			0
$135$		−	558.909i		−	0.356320i
$136$		0			0
$137$	−577.769			−0.360308			−0.180154	−	0.983638i	$-0.557660\pi$
$137$	−577.769			−0.360308			−0.180154	+	0.983638i	$0.557660\pi$
$138$		0			0
$139$			412.449i			0.251680i	0.992051	+	0.125840i	$0.0401625\pi$
$139$			412.449i			0.251680i	−0.992051	+	0.125840i	$0.959837\pi$
$140$		0			0
$141$			944.017i			0.563834i
$142$		0			0
$143$	1423.75			0.832588
$144$		0			0
$145$			749.423i			0.429215i
$146$		0			0
$147$	1011.08	−	191.201i	0.567296	−	0.107279i
$148$		0			0
$149$			2985.68i			1.64158i	0.571227	+	0.820792i	$0.306468\pi$
$149$			2985.68i			1.64158i	−0.571227	+	0.820792i	$0.693532\pi$
$150$		0			0
$151$			2788.11i			1.50261i	0.659957	+	0.751303i	$0.270574\pi$
$151$			2788.11i			1.50261i	−0.659957	+	0.751303i	$0.729426\pi$
$152$		0			0
$153$		−	829.843i		−	0.438489i
$154$		0			0
$155$	−5471.21			−2.83522
$156$		0			0
$157$	1616.65			0.821801			0.410900	−	0.911680i	$-0.365214\pi$
$157$	1616.65			0.821801			0.410900	+	0.911680i	$0.365214\pi$
$158$		0			0
$159$	325.688			0.162445
$160$		0			0
$161$	1440.18	+	1193.36i	0.704983	+	0.584161i
$162$		0			0
$163$	−1934.76			−0.929708			−0.464854	−	0.885387i	$-0.653893\pi$
$163$	−1934.76			−0.929708			−0.464854	+	0.885387i	$0.653893\pi$
$164$		0			0
$165$		−	2439.42i		−	1.15096i
$166$		0			0
$167$	−816.453			−0.378318			−0.189159	−	0.981947i	$-0.560576\pi$
$167$	−816.453			−0.378318			−0.189159	+	0.981947i	$0.560576\pi$
$168$		0			0
$169$	−883.307			−0.402051
$170$		0			0
$171$		−	787.666i		−	0.352247i
$172$		0			0
$173$	−3644.18			−1.60151			−0.800757	−	0.598989i	$-0.795569\pi$
$173$	−3644.18			−1.60151			−0.800757	+	0.598989i	$0.795569\pi$
$174$		0			0
$175$	−3586.40	+	4328.17i	−1.54918	+	1.86959i
$176$		0			0
$177$	556.119			0.236161
$178$		0			0
$179$	3415.80			1.42631			0.713154	−	0.701008i	$-0.247266\pi$
$179$	3415.80			1.42631			0.713154	+	0.701008i	$0.247266\pi$
$180$		0			0
$181$	−3513.95			−1.44304			−0.721519	−	0.692395i	$-0.756556\pi$
$181$	−3513.95			−1.44304			−0.721519	+	0.692395i	$0.756556\pi$
$182$		0			0
$183$		−	490.512i		−	0.198140i
$184$		0			0
$185$		−	4751.44i		−	1.88829i
$186$		0			0
$187$		−	3621.93i		−	1.41637i
$188$		0			0
$189$	385.038	+	319.049i	0.148187	+	0.122791i
$190$		0			0
$191$			3058.15i			1.15854i	0.815137	+	0.579268i	$0.196661\pi$
$191$			3058.15i			1.15854i	−0.815137	+	0.579268i	$0.803339\pi$
$192$		0			0
$193$	−1894.89			−0.706721			−0.353361	−	0.935487i	$-0.614961\pi$
$193$	−1894.89			−0.706721			−0.353361	+	0.935487i	$0.614961\pi$
$194$		0			0
$195$		−	2250.85i		−	0.826597i
$196$		0			0
$197$		−	109.557i		−	0.0396224i	−0.999804	−	0.0198112i	$-0.993693\pi$
$197$		−	109.557i		−	0.0396224i	0.999804	−	0.0198112i	$-0.00630651\pi$
$198$		0			0
$199$	−1525.71			−0.543492			−0.271746	−	0.962369i	$-0.587601\pi$
$199$	−1525.71			−0.543492			−0.271746	+	0.962369i	$0.587601\pi$
$200$		0			0
$201$		−	1374.78i		−	0.482436i
$202$		0			0
$203$	−516.286	−	427.803i	−0.178503	−	0.147911i
$204$		0			0
$205$		−	6978.01i		−	2.37739i
$206$		0			0
$207$			908.908i			0.305186i
$208$		0			0
$209$		−	3437.85i		−	1.13780i
$210$		0			0
$211$	−5626.27			−1.83568			−0.917841	−	0.396949i	$-0.870069\pi$
$211$	−5626.27			−1.83568			−0.917841	+	0.396949i	$0.870069\pi$
$212$		0			0
$213$	1753.58			0.564101
$214$		0			0
$215$	−7628.64			−2.41986
$216$		0			0
$217$	3123.21	−	3769.18i	0.977037	−	1.17912i
$218$		0			0
$219$	−3297.71			−1.01753
$220$		0			0
$221$		−	3341.95i		−	1.01721i
$222$		0			0
$223$	1409.96			0.423398			0.211699	−	0.977335i	$-0.432100\pi$
$223$	1409.96			0.423398			0.211699	+	0.977335i	$0.432100\pi$
$224$		0			0
$225$	−2731.54			−0.809344
$226$		0			0
$227$			848.622i			0.248128i	0.992274	+	0.124064i	$0.0395928\pi$
$227$			848.622i			0.248128i	−0.992274	+	0.124064i	$0.960407\pi$
$228$		0			0
$229$	5880.63			1.69696			0.848479	−	0.529230i	$-0.177519\pi$
$229$	5880.63			1.69696			0.848479	+	0.529230i	$0.177519\pi$
$230$		0			0
$231$	1680.54	+	1392.52i	0.478664	+	0.396629i
$232$		0			0
$233$	−3101.62			−0.872077			−0.436038	−	0.899928i	$-0.643619\pi$
$233$	−3101.62			−0.872077			−0.436038	+	0.899928i	$0.643619\pi$
$234$		0			0
$235$	6513.82			1.80815
$236$		0			0
$237$	−1315.88			−0.360657
$238$		0			0
$239$		−	6225.18i		−	1.68483i	−0.538831	−	0.842414i	$-0.681134\pi$
$239$		−	6225.18i		−	1.68483i	0.538831	−	0.842414i	$-0.318866\pi$
$240$		0			0
$241$		−	201.092i		−	0.0537489i	−0.999639	−	0.0268744i	$-0.991445\pi$
$241$		−	201.092i		−	0.0537489i	0.999639	−	0.0268744i	$-0.00855543\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−1319.31	−	6976.57i	−0.344031	−	1.81925i
$246$		0			0
$247$		−	3172.10i		−	0.817149i
$248$		0			0
$249$	2166.34			0.551351
$250$		0			0
$251$			4561.95i			1.14720i	0.819135	+	0.573601i	$0.194454\pi$
$251$			4561.95i			1.14720i	−0.819135	+	0.573601i	$0.805546\pi$
$252$		0			0
$253$			3967.02i			0.985789i
$254$		0			0
$255$	−5726.01			−1.40618
$256$		0			0
$257$		−	1939.24i		−	0.470687i	−0.971912	−	0.235343i	$-0.924379\pi$
$257$		−	1939.24i		−	0.470687i	0.971912	−	0.235343i	$-0.0756215\pi$
$258$		0			0
$259$	3273.32	+	2712.33i	0.785305	+	0.650718i
$260$		0			0
$261$		−	325.831i		−	0.0772737i
$262$		0			0
$263$			1244.55i			0.291796i	0.989300	+	0.145898i	$0.0466072\pi$
$263$			1244.55i			0.291796i	−0.989300	+	0.145898i	$0.953393\pi$
$264$		0			0
$265$		−	2247.28i		−	0.520941i
$266$		0			0
$267$	1357.00			0.311038
$268$		0			0
$269$	188.083			0.0426305			0.0213153	−	0.999773i	$-0.493215\pi$
$269$	188.083			0.0426305			0.0213153	+	0.999773i	$0.493215\pi$
$270$		0			0
$271$	−504.590			−0.113106			−0.0565529	−	0.998400i	$-0.518011\pi$
$271$	−504.590			−0.113106			−0.0565529	+	0.998400i	$0.518011\pi$
$272$		0			0
$273$	1550.63	+	1284.88i	0.343767	+	0.284851i
$274$		0			0
$275$	−11922.1			−2.61428
$276$		0			0
$277$		−	4147.02i		−	0.899532i	−0.893146	−	0.449766i	$-0.851507\pi$
$277$		−	4147.02i		−	0.899532i	0.893146	−	0.449766i	$-0.148493\pi$
$278$		0			0
$279$	2378.75			0.510437
$280$		0			0
$281$	−2018.75			−0.428572			−0.214286	−	0.976771i	$-0.568742\pi$
$281$	−2018.75			−0.428572			−0.214286	+	0.976771i	$0.568742\pi$
$282$		0			0
$283$			50.2649i			0.0105581i	0.999986	+	0.00527905i	$0.00168038\pi$
$283$			50.2649i			0.0105581i	−0.999986	+	0.00527905i	$0.998320\pi$
$284$		0			0
$285$	−5434.98			−1.12962
$286$		0			0
$287$	4807.22	+	3983.35i	0.988715	+	0.819267i
$288$		0			0
$289$	−3588.72			−0.730453
$290$		0			0
$291$	1794.00			0.361396
$292$		0			0
$293$	2247.98			0.448221			0.224110	−	0.974564i	$-0.428052\pi$
$293$	2247.98			0.448221			0.224110	+	0.974564i	$0.428052\pi$
$294$		0			0
$295$		−	3837.29i		−	0.757340i
$296$		0			0
$297$			1060.60i			0.207213i
$298$		0			0
$299$			3660.36i			0.707974i
$300$		0			0
$301$	4354.76	−	5255.45i	0.833901	−	1.00638i
$302$		0			0
$303$			4245.11i			0.804869i
$304$		0			0
$305$	−3384.59			−0.635413
$306$		0			0
$307$		−	1759.91i		−	0.327177i	−0.986529	−	0.163588i	$-0.947693\pi$
$307$		−	1759.91i		−	0.327177i	0.986529	−	0.163588i	$-0.0523069\pi$
$308$		0			0
$309$		−	3416.57i		−	0.629004i
$310$		0			0
$311$	−8517.10			−1.55293			−0.776464	−	0.630162i	$-0.782988\pi$
$311$	−8517.10			−1.55293			−0.776464	+	0.630162i	$0.782988\pi$
$312$		0			0
$313$		−	10113.5i		−	1.82636i	−0.407556	−	0.913180i	$-0.633619\pi$
$313$		−	10113.5i		−	1.82636i	0.407556	−	0.913180i	$-0.366381\pi$
$314$		0			0
$315$	2201.48	−	2656.81i	0.393775	−	0.475220i
$316$		0			0
$317$			6511.01i			1.15361i	0.816882	+	0.576805i	$0.195701\pi$
$317$			6511.01i			1.15361i	−0.816882	+	0.576805i	$0.804299\pi$
$318$		0			0
$319$		−	1422.12i		−	0.249604i
$320$		0			0
$321$			4510.15i			0.784212i
$322$		0			0
$323$	−8069.62			−1.39011
$324$		0			0
$325$	−11000.5			−1.87753
$326$		0			0
$327$	1077.85			0.182279
$328$		0			0
$329$	−3718.37	+	4487.44i	−0.623101	+	0.751977i
$330$		0			0
$331$	2031.11			0.337280			0.168640	−	0.985678i	$-0.446062\pi$
$331$	2031.11			0.337280			0.168640	+	0.985678i	$0.446062\pi$
$332$		0			0
$333$			2065.81i			0.339957i
$334$		0			0
$335$	−9486.15			−1.54712
$336$		0			0
$337$	−9702.01			−1.56826			−0.784128	−	0.620599i	$-0.786889\pi$
$337$	−9702.01			−1.56826			−0.784128	+	0.620599i	$0.786889\pi$
$338$		0			0
$339$		−	7089.61i		−	1.13585i
$340$		0			0
$341$	10382.3			1.64878
$342$		0			0
$343$	5559.35	+	3073.64i	0.875150	+	0.483851i
$344$		0			0
$345$	6271.57			0.978695
$346$		0			0
$347$	4652.17			0.719716			0.359858	−	0.933007i	$-0.382825\pi$
$347$	4652.17			0.719716			0.359858	+	0.933007i	$0.382825\pi$
$348$		0			0
$349$	5507.67			0.844753			0.422376	−	0.906421i	$-0.361196\pi$
$349$	5507.67			0.844753			0.422376	+	0.906421i	$0.361196\pi$
$350$		0			0
$351$			978.612i			0.148816i
$352$		0			0
$353$			8066.25i			1.21621i	0.793856	+	0.608106i	$0.208071\pi$
$353$			8066.25i			1.21621i	−0.793856	+	0.608106i	$0.791929\pi$
$354$		0			0
$355$		−	12099.9i		−	1.80900i
$356$		0			0
$357$	3268.65	−	3944.71i	0.484581	−	0.584807i
$358$		0			0
$359$		−	2334.87i		−	0.343258i	−0.985162	−	0.171629i	$-0.945097\pi$
$359$		−	2334.87i		−	0.343258i	0.985162	−	0.171629i	$-0.0549030\pi$
$360$		0			0
$361$	−800.476			−0.116704
$362$		0			0
$363$			636.090i			0.0919726i
$364$		0			0
$365$			22754.6i			3.26309i
$366$		0			0
$367$	8870.84			1.26173			0.630864	−	0.775894i	$-0.282701\pi$
$367$	8870.84			1.26173			0.630864	+	0.775894i	$0.282701\pi$
$368$		0			0
$369$			3033.87i			0.428013i
$370$		0			0
$371$	1548.17	+	1282.84i	0.216650	+	0.179520i
$372$		0			0
$373$			8302.86i			1.15256i	0.817251	+	0.576281i	$0.195497\pi$
$373$			8302.86i			1.15256i	−0.817251	+	0.576281i	$0.804503\pi$
$374$		0			0
$375$			11085.3i			1.52651i
$376$		0			0
$377$		−	1312.19i		−	0.179261i
$378$		0			0
$379$	530.766			0.0719357			0.0359679	−	0.999353i	$-0.488549\pi$
$379$	530.766			0.0719357			0.0359679	+	0.999353i	$0.488549\pi$
$380$		0			0
$381$	−4862.06			−0.653782
$382$		0			0
$383$	−534.478			−0.0713069			−0.0356535	−	0.999364i	$-0.511351\pi$
$383$	−534.478			−0.0713069			−0.0356535	+	0.999364i	$0.511351\pi$
$384$		0			0
$385$	9608.57	−	11595.9i	1.27194	−	1.53502i
$386$		0			0
$387$	3316.75			0.435658
$388$		0			0
$389$			11253.1i			1.46672i	0.679842	+	0.733358i	$0.262048\pi$
$389$			11253.1i			1.46672i	−0.679842	+	0.733358i	$0.737952\pi$
$390$		0			0
$391$	9311.74			1.20439
$392$		0			0
$393$	1509.69			0.193775
$394$		0			0
$395$			9079.73i			1.15658i
$396$		0			0
$397$	8896.66			1.12471			0.562356	−	0.826895i	$-0.309895\pi$
$397$	8896.66			1.12471			0.562356	+	0.826895i	$0.309895\pi$
$398$		0			0
$399$	3102.52	−	3744.22i	0.389274	−	0.469788i
$400$		0			0
$401$	6210.10			0.773360			0.386680	−	0.922214i	$-0.373622\pi$
$401$	6210.10			0.773360			0.386680	+	0.922214i	$0.373622\pi$
$402$		0			0
$403$	9579.73			1.18412
$404$		0			0
$405$	1676.73			0.205722
$406$		0			0
$407$			9016.44i			1.09810i
$408$		0			0
$409$			3899.47i			0.471434i	0.971822	+	0.235717i	$0.0757439\pi$
$409$			3899.47i			0.471434i	−0.971822	+	0.235717i	$0.924256\pi$
$410$		0			0
$411$		−	1733.31i		−	0.208024i
$412$		0			0
$413$	2643.54	+	2190.49i	0.314965	+	0.260985i
$414$		0			0
$415$		−	14948.0i		−	1.76812i
$416$		0			0
$417$	−1237.35			−0.145307
$418$		0			0
$419$		−	12136.9i		−	1.41510i	−0.706663	−	0.707550i	$-0.749800\pi$
$419$		−	12136.9i		−	1.41510i	0.706663	−	0.707550i	$-0.250200\pi$
$420$		0			0
$421$			5388.65i			0.623817i	0.950112	+	0.311908i	$0.100968\pi$
$421$			5388.65i			0.623817i	−0.950112	+	0.311908i	$0.899032\pi$
$422$		0			0
$423$	−2832.05			−0.325529
$424$		0			0
$425$			27984.5i			3.19400i
$426$		0			0
$427$	1932.07	−	2331.68i	0.218968	−	0.264257i
$428$		0			0
$429$			4271.25i			0.480695i
$430$		0			0
$431$			2284.44i			0.255308i	0.991819	+	0.127654i	$0.0407447\pi$
$431$			2284.44i			0.255308i	−0.991819	+	0.127654i	$0.959255\pi$
$432$		0			0
$433$			1711.79i			0.189985i	0.995478	+	0.0949925i	$0.0302827\pi$
$433$			1711.79i			0.189985i	−0.995478	+	0.0949925i	$0.969717\pi$
$434$		0			0
$435$	−2248.27			−0.247808
$436$		0			0
$437$	8838.47			0.967509
$438$		0			0
$439$	−2820.52			−0.306643			−0.153322	−	0.988176i	$-0.548997\pi$
$439$	−2820.52			−0.306643			−0.153322	+	0.988176i	$0.548997\pi$
$440$		0			0
$441$	573.604	+	3033.24i	0.0619376	+	0.327528i
$442$		0			0
$443$	−13392.2			−1.43630			−0.718150	−	0.695888i	$-0.755011\pi$
$443$	−13392.2			−1.43630			−0.718150	+	0.695888i	$0.755011\pi$
$444$		0			0
$445$		−	9363.47i		−	0.997463i
$446$		0			0
$447$	−8957.03			−0.947769
$448$		0			0
$449$	−15186.9			−1.59624			−0.798120	−	0.602498i	$-0.794172\pi$
$449$	−15186.9			−1.59624			−0.798120	+	0.602498i	$0.794172\pi$
$450$		0			0
$451$			13241.6i			1.38254i
$452$		0			0
$453$	−8364.34			−0.867530
$454$		0			0
$455$	8865.81	−	10699.5i	0.913485	−	1.10242i
$456$		0			0
$457$	−19132.7			−1.95840			−0.979202	−	0.202888i	$-0.934967\pi$
$457$	−19132.7			−1.95840			−0.979202	+	0.202888i	$0.934967\pi$
$458$		0			0
$459$	2489.53			0.253162
$460$		0			0
$461$	237.892			0.0240341			0.0120170	−	0.999928i	$-0.496175\pi$
$461$	237.892			0.0240341			0.0120170	+	0.999928i	$0.496175\pi$
$462$		0			0
$463$			6527.48i			0.655200i	0.944816	+	0.327600i	$0.106240\pi$
$463$			6527.48i			0.655200i	−0.944816	+	0.327600i	$0.893760\pi$
$464$		0			0
$465$		−	16413.6i		−	1.63691i
$466$		0			0
$467$			488.599i			0.0484147i	0.999707	+	0.0242073i	$0.00770619\pi$
$467$			488.599i			0.0484147i	−0.999707	+	0.0242073i	$0.992294\pi$
$468$		0			0
$469$	5415.10	−	6535.10i	0.533147	−	0.643418i
$470$		0			0
$471$			4849.95i			0.474467i
$472$		0			0
$473$	14476.3			1.40723
$474$		0			0
$475$			26562.2i			2.56580i
$476$		0			0
$477$			977.063i			0.0937875i
$478$		0			0
$479$	−5872.67			−0.560186			−0.280093	−	0.959973i	$-0.590365\pi$
$479$	−5872.67			−0.560186			−0.280093	+	0.959973i	$0.590365\pi$
$480$		0			0
$481$			8319.45i			0.788637i
$482$		0			0
$483$	−3580.08	+	4320.55i	−0.337266	+	0.407022i
$484$		0			0
$485$		−	12378.8i		−	1.15895i
$486$		0			0
$487$			14307.7i			1.33130i	0.746263	+	0.665652i	$0.231846\pi$
$487$			14307.7i			1.33130i	−0.746263	+	0.665652i	$0.768154\pi$
$488$		0			0
$489$		−	5804.29i		−	0.536767i
$490$		0			0
$491$	10665.1			0.980264			0.490132	−	0.871648i	$-0.336948\pi$
$491$	10665.1			0.980264			0.490132	+	0.871648i	$0.336948\pi$
$492$		0			0
$493$	−3338.13			−0.304953
$494$		0			0
$495$	7318.25			0.664507
$496$		0			0
$497$	8335.75	+	6907.15i	0.752333	+	0.623397i
$498$		0			0
$499$	11905.8			1.06809			0.534046	−	0.845455i	$-0.320671\pi$
$499$	11905.8			1.06809			0.534046	+	0.845455i	$0.320671\pi$
$500$		0			0
$501$		−	2449.36i		−	0.218422i
$502$		0			0
$503$	11100.0			0.983941			0.491971	−	0.870612i	$-0.336277\pi$
$503$	11100.0			0.983941			0.491971	+	0.870612i	$0.336277\pi$
$504$		0			0
$505$	29291.8			2.58112
$506$		0			0
$507$		−	2649.92i		−	0.232125i
$508$		0			0
$509$	−8580.71			−0.747216			−0.373608	−	0.927587i	$-0.621880\pi$
$509$	−8580.71			−0.747216			−0.373608	+	0.927587i	$0.621880\pi$
$510$		0			0
$511$	−15675.8	−	12989.3i	−1.35706	−	1.12448i
$512$		0			0
$513$	2363.00			0.203370
$514$		0			0
$515$	−23574.7			−2.01714
$516$		0			0
$517$	−12360.8			−1.05150
$518$		0			0
$519$		−	10932.5i		−	0.924635i
$520$		0			0
$521$		−	4351.09i		−	0.365882i	−0.983124	−	0.182941i	$-0.941438\pi$
$521$		−	4351.09i		−	0.365882i	0.983124	−	0.182941i	$-0.0585618\pi$
$522$		0			0
$523$		−	3685.85i		−	0.308166i	−0.988058	−	0.154083i	$-0.950758\pi$
$523$		−	3685.85i		−	0.308166i	0.988058	−	0.154083i	$-0.0492424\pi$
$524$		0			0
$525$	−12984.5	−	10759.2i	−1.07941	−	0.894419i
$526$		0			0
$527$		−	24370.2i		−	2.01439i
$528$		0			0
$529$	1968.06			0.161754
$530$		0			0
$531$			1668.36i			0.136348i
$532$		0			0
$533$			12218.0i			0.992910i
$534$		0			0
$535$	31120.5			2.51487
$536$		0			0
$537$			10247.4i			0.823479i
$538$		0			0
$539$	2503.55	+	13238.9i	0.200066	+	1.05796i
$540$		0			0
$541$		−	21536.0i		−	1.71147i	−0.517413	−	0.855736i	$-0.673105\pi$
$541$		−	21536.0i		−	1.71147i	0.517413	−	0.855736i	$-0.326895\pi$
$542$		0			0
$543$		−	10541.8i		−	0.833138i
$544$		0			0
$545$		−	7437.28i		−	0.584547i
$546$		0			0
$547$	−5550.29			−0.433845			−0.216923	−	0.976189i	$-0.569602\pi$
$547$	−5550.29			−0.433845			−0.216923	+	0.976189i	$0.569602\pi$
$548$		0			0
$549$	1471.54			0.114396
$550$		0			0
$551$	−3168.47			−0.244975
$552$		0			0
$553$	−6255.12	−	5183.10i	−0.481003	−	0.398568i
$554$		0			0
$555$	14254.3			1.09020
$556$		0			0
$557$			14848.3i			1.12952i	0.825256	+	0.564758i	$0.191031\pi$
$557$			14848.3i			1.12952i	−0.825256	+	0.564758i	$0.808969\pi$
$558$		0			0
$559$	13357.2			1.01065
$560$		0			0
$561$	10865.8			0.817744
$562$		0			0
$563$			1189.55i			0.0890469i	0.999008	+	0.0445235i	$0.0141769\pi$
$563$			1189.55i			0.0890469i	−0.999008	+	0.0445235i	$0.985823\pi$
$564$		0			0
$565$	−48919.1			−3.64255
$566$		0			0
$567$	−957.148	+	1155.11i	−0.0708932	+	0.0855560i
$568$		0			0
$569$	9572.43			0.705267			0.352634	−	0.935761i	$-0.385286\pi$
$569$	9572.43			0.705267			0.352634	+	0.935761i	$0.385286\pi$
$570$		0			0
$571$	19487.5			1.42825			0.714123	−	0.700021i	$-0.246826\pi$
$571$	19487.5			1.42825			0.714123	+	0.700021i	$0.246826\pi$
$572$		0			0
$573$	−9174.46			−0.668881
$574$		0			0
$575$		−	30650.8i		−	2.22300i
$576$		0			0
$577$			24751.1i			1.78579i	0.450263	+	0.892896i	$0.351330\pi$
$577$			24751.1i			1.78579i	−0.450263	+	0.892896i	$0.648670\pi$
$578$		0			0
$579$		−	5684.67i		−	0.408026i
$580$		0			0
$581$	10297.8	+	8532.96i	0.735329	+	0.609306i
$582$		0			0
$583$			4264.49i			0.302945i
$584$		0			0
$585$	6752.54			0.477236
$586$		0			0
$587$			15015.1i			1.05577i	0.849315	+	0.527886i	$0.177015\pi$
$587$			15015.1i			1.05577i	−0.849315	+	0.527886i	$0.822985\pi$
$588$		0			0
$589$		−	23131.6i		−	1.61820i
$590$		0			0
$591$	328.671			0.0228760
$592$		0			0
$593$			18080.4i			1.25206i	0.779799	+	0.626030i	$0.215321\pi$
$593$			18080.4i			1.25206i	−0.779799	+	0.626030i	$0.784679\pi$
$594$		0			0
$595$	−27218.9	−	22554.1i	−1.87541	−	1.55399i
$596$		0			0
$597$		−	4577.13i		−	0.313785i
$598$		0			0
$599$		−	18896.2i		−	1.28894i	−0.764628	−	0.644472i	$-0.777077\pi$
$599$		−	18896.2i		−	1.28894i	0.764628	−	0.644472i	$-0.222923\pi$
$600$		0			0
$601$		−	316.737i		−	0.0214974i	−0.999942	−	0.0107487i	$-0.996579\pi$
$601$		−	316.737i		−	0.0214974i	0.999942	−	0.0107487i	$-0.00342149\pi$
$602$		0			0
$603$	4124.34			0.278534
$604$		0			0
$605$	4389.09			0.294945
$606$		0			0
$607$	−24428.2			−1.63346			−0.816730	−	0.577020i	$-0.804216\pi$
$607$	−24428.2			−1.63346			−0.816730	+	0.577020i	$0.804216\pi$
$608$		0			0
$609$	1283.41	−	1548.86i	0.0853964	−	0.103059i
$610$		0			0
$611$	−11405.3			−0.755168
$612$		0			0
$613$			24938.5i			1.64316i	0.570092	+	0.821581i	$0.306908\pi$
$613$			24938.5i			1.64316i	−0.570092	+	0.821581i	$0.693092\pi$
$614$		0			0
$615$	20934.0			1.37259
$616$		0			0
$617$	−10603.6			−0.691873			−0.345936	−	0.938258i	$-0.612439\pi$
$617$	−10603.6			−0.691873			−0.345936	+	0.938258i	$0.612439\pi$
$618$		0			0
$619$			17013.3i			1.10472i	0.833604	+	0.552362i	$0.186273\pi$
$619$			17013.3i			1.10472i	−0.833604	+	0.552362i	$0.813727\pi$
$620$		0			0
$621$	−2726.72			−0.176199
$622$		0			0
$623$	6450.59	+	5345.07i	0.414827	+	0.343733i
$624$		0			0
$625$	38551.7			2.46731
$626$		0			0
$627$	10313.5			0.656911
$628$		0			0
$629$	21164.2			1.34161
$630$		0			0
$631$			2157.13i			0.136092i	0.997682	+	0.0680460i	$0.0216765\pi$
$631$			2157.13i			0.136092i	−0.997682	+	0.0680460i	$0.978324\pi$
$632$		0			0
$633$		−	16878.8i		−	1.05983i
$634$		0			0
$635$			33548.8i			2.09660i
$636$		0			0
$637$	2310.02	+	12215.5i	0.143684	+	0.759805i
$638$		0			0
$639$			5260.75i			0.325684i
$640$		0			0
$641$	−22030.9			−1.35752			−0.678758	−	0.734362i	$-0.737481\pi$
$641$	−22030.9			−1.35752			−0.678758	+	0.734362i	$0.737481\pi$
$642$		0			0
$643$		−	30067.6i		−	1.84409i	−0.387081	−	0.922046i	$-0.626517\pi$
$643$		−	30067.6i		−	1.84409i	0.387081	−	0.922046i	$-0.373483\pi$
$644$		0			0
$645$		−	22885.9i		−	1.39711i
$646$		0			0
$647$	−6954.93			−0.422607			−0.211303	−	0.977421i	$-0.567771\pi$
$647$	−6954.93			−0.422607			−0.211303	+	0.977421i	$0.567771\pi$
$648$		0			0
$649$			7281.72i			0.440420i
$650$		0			0
$651$	11307.5	+	9369.62i	0.680763	+	0.564092i
$652$		0			0
$653$			1199.25i			0.0718686i	0.999354	+	0.0359343i	$0.0114407\pi$
$653$			1199.25i			0.0718686i	−0.999354	+	0.0359343i	$0.988559\pi$
$654$		0			0
$655$		−	10417.0i		−	0.621415i
$656$		0			0
$657$		−	9893.12i		−	0.587469i
$658$		0			0
$659$	13611.9			0.804619			0.402310	−	0.915504i	$-0.368208\pi$
$659$	13611.9			0.804619			0.402310	+	0.915504i	$0.368208\pi$
$660$		0			0
$661$	−30065.6			−1.76916			−0.884582	−	0.466385i	$-0.845556\pi$
$661$	−30065.6			−1.76916			−0.884582	+	0.466385i	$0.845556\pi$
$662$		0			0
$663$	10025.9			0.587288
$664$		0			0
$665$	−25835.5	−	21407.8i	−1.50655	−	1.24836i
$666$		0			0
$667$	3656.18			0.212246
$668$		0			0
$669$			4229.88i			0.244449i
$670$		0			0
$671$	6422.67			0.369515
$672$		0			0
$673$	10262.1			0.587777			0.293888	−	0.955840i	$-0.405051\pi$
$673$	10262.1			0.587777			0.293888	+	0.955840i	$0.405051\pi$
$674$		0			0
$675$		−	8194.61i		−	0.467275i
$676$		0			0
$677$	−426.182			−0.0241943			−0.0120971	−	0.999927i	$-0.503851\pi$
$677$	−426.182			−0.0241943			−0.0120971	+	0.999927i	$0.503851\pi$
$678$		0			0
$679$	8527.88	+	7066.35i	0.481988	+	0.399384i
$680$		0			0
$681$	−2545.87			−0.143257
$682$		0			0
$683$	2385.48			0.133643			0.0668213	−	0.997765i	$-0.478714\pi$
$683$	2385.48			0.133643			0.0668213	+	0.997765i	$0.478714\pi$
$684$		0			0
$685$	−11960.0			−0.667108
$686$		0			0
$687$			17641.9i			0.979739i
$688$		0			0
$689$			3934.84i			0.217570i
$690$		0			0
$691$		−	9485.21i		−	0.522192i	−0.965313	−	0.261096i	$-0.915916\pi$
$691$		−	9485.21i		−	0.522192i	0.965313	−	0.261096i	$-0.0840839\pi$
$692$		0			0
$693$	−4177.57	+	5041.62i	−0.228994	+	0.276357i
$694$		0			0
$695$			8537.84i			0.465984i
$696$		0			0
$697$	31081.9			1.68911
$698$		0			0
$699$		−	9304.86i		−	0.503494i
$700$		0			0
$701$		−	13463.5i		−	0.725404i	−0.931905	−	0.362702i	$-0.881854\pi$
$701$		−	13463.5i		−	0.725404i	0.931905	−	0.362702i	$-0.118146\pi$
$702$		0			0
$703$	20088.5			1.07774
$704$		0			0
$705$			19541.5i			1.04393i
$706$		0			0
$707$	−16721.0	+	20179.4i	−0.889473	+	1.07344i
$708$		0			0
$709$			1482.11i			0.0785075i	0.999229	+	0.0392538i	$0.0124981\pi$
$709$			1482.11i			0.0785075i	−0.999229	+	0.0392538i	$0.987502\pi$
$710$		0			0
$711$		−	3947.65i		−	0.208225i
$712$		0			0
$713$			26692.2i			1.40200i
$714$		0			0
$715$	29472.1			1.54153
$716$		0			0
$717$	18675.6			0.972735
$718$		0			0
$719$	33887.8			1.75772			0.878862	−	0.477077i	$-0.158304\pi$
$719$	33887.8			1.75772			0.878862	+	0.477077i	$0.158304\pi$
$720$		0			0
$721$	13457.5	−	16240.9i	0.695122	−	0.838893i
$722$		0			0
$723$	603.276			0.0310319
$724$		0			0
$725$			10987.9i			0.562869i
$726$		0			0
$727$	9840.41			0.502009			0.251004	−	0.967986i	$-0.419239\pi$
$727$	9840.41			0.502009			0.251004	+	0.967986i	$0.419239\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$		−	33980.0i		−	1.71928i
$732$		0			0
$733$	13241.0			0.667212			0.333606	−	0.942713i	$-0.391735\pi$
$733$	13241.0			0.667212			0.333606	+	0.942713i	$0.391735\pi$
$734$		0			0
$735$	20929.7	−	3957.93i	1.05035	−	0.198627i
$736$		0			0
$737$	18001.1			0.899701
$738$		0			0
$739$	6185.80			0.307914			0.153957	−	0.988078i	$-0.450798\pi$
$739$	6185.80			0.307914			0.153957	+	0.988078i	$0.450798\pi$
$740$		0			0
$741$	9516.29			0.471781
$742$		0			0
$743$			11911.6i			0.588148i	0.955783	+	0.294074i	$0.0950112\pi$
$743$			11911.6i			0.588148i	−0.955783	+	0.294074i	$0.904989\pi$
$744$		0			0
$745$			61804.5i			3.03939i
$746$		0			0
$747$			6499.02i			0.318322i
$748$		0			0
$749$	−17764.9	+	21439.3i	−0.866645	+	1.04589i
$750$		0			0
$751$			28902.1i			1.40433i	0.712015	+	0.702165i	$0.247783\pi$
$751$			28902.1i			1.40433i	−0.712015	+	0.702165i	$0.752217\pi$
$752$		0			0
$753$	−13685.8			−0.662337
$754$		0			0
$755$			57714.9i			2.78207i
$756$		0			0
$757$			7827.05i			0.375798i	0.982188	+	0.187899i	$0.0601678\pi$
$757$			7827.05i			0.375798i	−0.982188	+	0.187899i	$0.939832\pi$
$758$		0			0
$759$	−11901.1			−0.569145
$760$		0			0
$761$			4882.67i			0.232584i	0.993215	+	0.116292i	$0.0371008\pi$
$761$			4882.67i			0.232584i	−0.993215	+	0.116292i	$0.962899\pi$
$762$		0			0
$763$	5123.62	+	4245.52i	0.243103	+	0.201439i
$764$		0			0
$765$		−	17178.0i		−	0.811860i
$766$		0			0
$767$			6718.83i			0.316301i
$768$		0			0
$769$			20071.4i			0.941215i	0.882343	+	0.470607i	$0.155965\pi$
$769$			20071.4i			0.941215i	−0.882343	+	0.470607i	$0.844035\pi$
$770$		0			0
$771$	5817.72			0.271751
$772$		0			0
$773$	4965.70			0.231053			0.115526	−	0.993304i	$-0.463145\pi$
$773$	4965.70			0.231053			0.115526	+	0.993304i	$0.463145\pi$
$774$		0			0
$775$	−80217.8			−3.71808
$776$		0			0
$777$	−8136.98	+	9819.95i	−0.375692	+	0.453396i
$778$		0			0
$779$	29502.1			1.35690
$780$		0			0
$781$			22961.1i			1.05200i
$782$		0			0
$783$	977.493			0.0446140
$784$		0			0
$785$	33465.2			1.52156
$786$		0			0
$787$		−	40092.6i		−	1.81594i	−0.419035	−	0.907970i	$-0.637631\pi$
$787$		−	40092.6i		−	1.81594i	0.419035	−	0.907970i	$-0.362369\pi$
$788$		0			0
$789$	−3733.66			−0.168469
$790$		0			0
$791$	27925.1	−	33700.9i	1.25525	−	1.51487i
$792$		0			0
$793$	5926.18			0.265378
$794$		0			0
$795$	6741.84			0.300765
$796$		0			0
$797$	26919.4			1.19640			0.598202	−	0.801345i	$-0.295882\pi$
$797$	26919.4			1.19640			0.598202	+	0.801345i	$0.295882\pi$
$798$		0			0
$799$			29014.3i			1.28467i
$800$		0			0
$801$			4071.01i			0.179578i
$802$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} -39.2814 q^{11} -36.2449 q^{13} +62.1010i q^{15} +92.2048i q^{17} +87.5184i q^{19} +(-42.7820 - 35.4499i) q^{21} -100.990i q^{23} +303.504 q^{25} -27.0000i q^{27} +36.2034i q^{29} -264.306 q^{31} -117.844i q^{33} +(-244.609 + 295.201i) q^{35} -229.534i q^{37} -108.735i q^{39} -337.096i q^{41} -368.528 q^{43} -186.303 q^{45} +314.672 q^{47} +(-63.7338 - 337.027i) q^{49} -276.614 q^{51} -108.563i q^{53} -813.139 q^{55} -262.555 q^{57} -185.373i q^{59} -163.504 q^{61} +(106.350 - 128.346i) q^{63} -750.282 q^{65} -458.260 q^{67} +302.969 q^{69} -584.527i q^{71} +1099.24i q^{73} +910.512i q^{75} +(464.175 - 560.180i) q^{77} +438.627i q^{79} +81.0000 q^{81} -722.114i q^{83} +1908.67i q^{85} -108.610 q^{87} -452.334i q^{89} +(428.293 - 516.877i) q^{91} -792.917i q^{93} +1811.66i q^{95} -598.000i q^{97} +353.533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32q - 288q^{9} + O(q^{10}) \)	\( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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