LMFDB - Embedded newform 1386.2.g.a.881.6

Newspace parameters

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.g (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 4 x^{14} + 8 x^{12} + 80 x^{10} + 1189 x^{8} - 2028 x^{6} + 1800 x^{4} + 1080 x^{2} + 324$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.6
Root			$0.221383 + 0.534465i$ of defining polynomial
Character	$\chi$	$=$	1386.881
Dual form			1386.2.g.a.881.14

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.06893 q^{5} +(0.884677 + 2.49346i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.06893 q^{5} +(0.884677 + 2.49346i) q^{7} +1.00000i q^{8} -1.06893i q^{10} +1.00000i q^{11} +(2.49346 - 0.884677i) q^{14} +1.00000 q^{16} -3.91799 q^{17} +7.49534i q^{19} -1.06893 q^{20} +1.00000 q^{22} +5.33067i q^{23} -3.85739 q^{25} +(-0.884677 - 2.49346i) q^{28} -8.01200i q^{29} +3.65862i q^{31} -1.00000i q^{32} +3.91799i q^{34} +(0.945659 + 2.66534i) q^{35} -7.33067 q^{37} +7.49534 q^{38} +1.06893i q^{40} +2.50842 q^{41} +0.230645 q^{43} -1.00000i q^{44} +5.33067 q^{46} +0.370555 q^{47} +(-5.43469 + 4.41182i) q^{49} +3.85739i q^{50} +7.56132i q^{53} +1.06893i q^{55} +(-2.49346 + 0.884677i) q^{56} -8.01200 q^{58} +2.13786 q^{59} +14.2496i q^{61} +3.65862 q^{62} -1.00000 q^{64} +2.47329 q^{67} +3.91799 q^{68} +(2.66534 - 0.945659i) q^{70} -3.15461i q^{71} +0.370555i q^{73} +7.33067i q^{74} -7.49534i q^{76} +(-2.49346 + 0.884677i) q^{77} -4.45068 q^{79} +1.06893 q^{80} -2.50842i q^{82} +12.2229 q^{83} -4.18806 q^{85} -0.230645i q^{86} -1.00000 q^{88} +10.0038 q^{89} -5.33067i q^{92} -0.370555i q^{94} +8.01200i q^{95} +12.8528i q^{97} +(4.41182 + 5.43469i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 16q^{4} - 8q^{7} + O(q^{10}) $	$ 16q - 16q^{4} - 8q^{7} + 16q^{16} + 16q^{22} + 16q^{25} + 8q^{28} - 16q^{37} + 48q^{43} - 16q^{46} + 8q^{49} - 16q^{58} - 16q^{64} + 16q^{67} - 8q^{70} - 16q^{79} + 112q^{85} - 16q^{88} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.06893			0.478040			0.239020	−	0.971015i	$-0.423174\pi$
$5$	1.06893			0.478040			0.239020	+	0.971015i	$0.423174\pi$
$6$		0			0
$7$	0.884677	+	2.49346i	0.334377	+	0.942440i
$7$	0.884677	+	2.49346i	0.334377	+	0.942440i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	1.06893i		−	0.338026i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	2.49346	−	0.884677i	0.666405	−	0.236440i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.91799			−0.950252			−0.475126	−	0.879918i	$-0.657598\pi$
$17$	−3.91799			−0.950252			−0.475126	+	0.879918i	$0.657598\pi$
$18$		0			0
$19$			7.49534i			1.71955i	0.510674	+	0.859774i	$0.329396\pi$
$19$			7.49534i			1.71955i	−0.510674	+	0.859774i	$0.670604\pi$
$20$	−1.06893			−0.239020
$21$		0			0
$22$	1.00000			0.213201
$23$			5.33067i			1.11152i	0.831342	+	0.555761i	$0.187573\pi$
$23$			5.33067i			1.11152i	−0.831342	+	0.555761i	$0.812427\pi$
$24$		0			0
$25$	−3.85739			−0.771477
$26$		0			0
$27$		0			0
$28$	−0.884677	−	2.49346i	−0.167188	−	0.471220i
$29$		−	8.01200i		−	1.48779i	−0.668296	−	0.743895i	$-0.732976\pi$
$29$		−	8.01200i		−	1.48779i	0.668296	−	0.743895i	$-0.267024\pi$
$30$		0			0
$31$			3.65862i			0.657108i	0.944485	+	0.328554i	$0.106561\pi$
$31$			3.65862i			0.657108i	−0.944485	+	0.328554i	$0.893439\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			3.91799i			0.671930i
$35$	0.945659	+	2.66534i	0.159846	+	0.450524i
$36$		0			0
$37$	−7.33067			−1.20516			−0.602578	−	0.798060i	$-0.705860\pi$
$37$	−7.33067			−1.20516			−0.602578	+	0.798060i	$0.705860\pi$
$38$	7.49534			1.21590
$39$		0			0
$40$			1.06893i			0.169013i
$41$	2.50842			0.391749			0.195874	−	0.980629i	$-0.437246\pi$
$41$	2.50842			0.391749			0.195874	+	0.980629i	$0.437246\pi$
$42$		0			0
$43$	0.230645			0.0351730			0.0175865	−	0.999845i	$-0.494402\pi$
$43$	0.230645			0.0351730			0.0175865	+	0.999845i	$0.494402\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$	5.33067			0.785965
$47$	0.370555			0.0540510			0.0270255	−	0.999635i	$-0.491396\pi$
$47$	0.370555			0.0540510			0.0270255	+	0.999635i	$0.491396\pi$
$48$		0			0
$49$	−5.43469	+	4.41182i	−0.776384	+	0.630260i
$50$			3.85739i			0.545517i
$51$		0			0
$52$		0			0
$53$			7.56132i			1.03863i	0.854584	+	0.519313i	$0.173812\pi$
$53$			7.56132i			1.03863i	−0.854584	+	0.519313i	$0.826188\pi$
$54$		0			0
$55$			1.06893i			0.144135i
$56$	−2.49346	+	0.884677i	−0.333203	+	0.118220i
$57$		0			0
$58$	−8.01200			−1.05203
$59$	2.13786			0.278326			0.139163	−	0.990269i	$-0.455559\pi$
$59$	2.13786			0.278326			0.139163	+	0.990269i	$0.455559\pi$
$60$		0			0
$61$			14.2496i			1.82447i	0.409667	+	0.912235i	$0.365645\pi$
$61$			14.2496i			1.82447i	−0.409667	+	0.912235i	$0.634355\pi$
$62$	3.65862			0.464646
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	2.47329			0.302160			0.151080	−	0.988522i	$-0.451725\pi$
$67$	2.47329			0.302160			0.151080	+	0.988522i	$0.451725\pi$
$68$	3.91799			0.475126
$69$		0			0
$70$	2.66534	−	0.945659i	0.318569	−	0.113028i
$71$		−	3.15461i		−	0.374383i	−0.982323	−	0.187192i	$-0.940061\pi$
$71$		−	3.15461i		−	0.374383i	0.982323	−	0.187192i	$-0.0599385\pi$
$72$		0			0
$73$			0.370555i			0.0433701i	0.999765	+	0.0216851i	$0.00690311\pi$
$73$			0.370555i			0.0433701i	−0.999765	+	0.0216851i	$0.993097\pi$
$74$			7.33067i			0.852173i
$75$		0			0
$76$		−	7.49534i		−	0.859774i
$77$	−2.49346	+	0.884677i	−0.284156	+	0.100818i
$78$		0			0
$79$	−4.45068			−0.500740			−0.250370	−	0.968150i	$-0.580552\pi$
$79$	−4.45068			−0.500740			−0.250370	+	0.968150i	$0.580552\pi$
$80$	1.06893			0.119510
$81$		0			0
$82$		−	2.50842i		−	0.277008i
$83$	12.2229			1.34164			0.670818	−	0.741622i	$-0.265943\pi$
$83$	12.2229			1.34164			0.670818	+	0.741622i	$0.265943\pi$
$84$		0			0
$85$	−4.18806			−0.454259
$86$		−	0.230645i		−	0.0248711i
$87$		0			0
$88$	−1.00000			−0.106600
$89$	10.0038			1.06040			0.530198	−	0.847874i	$-0.322118\pi$
$89$	10.0038			1.06040			0.530198	+	0.847874i	$0.322118\pi$
$90$		0			0
$91$		0			0
$92$		−	5.33067i		−	0.555761i
$93$		0			0
$94$		−	0.370555i		−	0.0382198i
$95$			8.01200i			0.822013i
$96$		0			0
$97$			12.8528i			1.30501i	0.757786	+	0.652503i	$0.226281\pi$
$97$			12.8528i			1.30501i	−0.757786	+	0.652503i	$0.773719\pi$
$98$	4.41182	+	5.43469i	0.445661	+	0.548987i
$99$		0			0
$100$	3.85739			0.385739
$101$	14.0030			1.39335			0.696676	−	0.717386i	$-0.254661\pi$
$101$	14.0030			1.39335			0.696676	+	0.717386i	$0.254661\pi$
$102$		0			0
$103$		−	12.2229i		−	1.20436i	−0.798361	−	0.602179i	$-0.794299\pi$
$103$		−	12.2229i		−	1.20436i	0.798361	−	0.602179i	$-0.205701\pi$
$104$		0			0
$105$		0			0
$106$	7.56132			0.734420
$107$		−	20.4081i		−	1.97292i	−0.163988	−	0.986462i	$-0.552436\pi$
$107$		−	20.4081i		−	1.97292i	0.163988	−	0.986462i	$-0.447564\pi$
$108$		0			0
$109$	12.8814			1.23381			0.616906	−	0.787037i	$-0.288386\pi$
$109$	12.8814			1.23381			0.616906	+	0.787037i	$0.288386\pi$
$110$	1.06893			0.101919
$111$		0			0
$112$	0.884677	+	2.49346i	0.0835942	+	0.235610i
$113$		−	6.92396i		−	0.651352i	−0.945481	−	0.325676i	$-0.894408\pi$
$113$		−	6.92396i		−	0.651352i	0.945481	−	0.325676i	$-0.105592\pi$
$114$		0			0
$115$			5.69812i			0.531352i
$116$			8.01200i			0.743895i
$117$		0			0
$118$		−	2.13786i		−	0.196806i
$119$	−3.46616	−	9.76935i	−0.317742	−	0.895555i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	14.2496			1.29009
$123$		0			0
$124$		−	3.65862i		−	0.328554i
$125$	−9.46793			−0.846838
$126$		0			0
$127$	10.6267			0.942971			0.471485	−	0.881874i	$-0.343718\pi$
$127$	10.6267			0.942971			0.471485	+	0.881874i	$0.343718\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	−3.23670			−0.282792			−0.141396	−	0.989953i	$-0.545159\pi$
$131$	−3.23670			−0.282792			−0.141396	+	0.989953i	$0.545159\pi$
$132$		0			0
$133$	−18.6893	+	6.63096i	−1.62057	+	0.574977i
$134$		−	2.47329i		−	0.213659i
$135$		0			0
$136$		−	3.91799i		−	0.335965i
$137$			10.7159i			0.915523i	0.889075	+	0.457762i	$0.151349\pi$
$137$			10.7159i			0.915523i	−0.889075	+	0.457762i	$0.848651\pi$
$138$		0			0
$139$			6.24832i			0.529975i	0.964252	+	0.264988i	$0.0853679\pi$
$139$			6.24832i			0.529975i	−0.964252	+	0.264988i	$0.914632\pi$
$140$	−0.945659	−	2.66534i	−0.0799228	−	0.225262i
$141$		0			0
$142$	−3.15461			−0.264729
$143$		0			0
$144$		0			0
$145$		−	8.56427i		−	0.711224i
$146$	0.370555			0.0306673
$147$		0			0
$148$	7.33067			0.602578
$149$			12.8694i			1.05430i	0.849772	+	0.527150i	$0.176740\pi$
$149$			12.8694i			1.05430i	−0.849772	+	0.527150i	$0.823260\pi$
$150$		0			0
$151$	−8.61475			−0.701058			−0.350529	−	0.936552i	$-0.613998\pi$
$151$	−8.61475			−0.701058			−0.350529	+	0.936552i	$0.613998\pi$
$152$	−7.49534			−0.607952
$153$		0			0
$154$	0.884677	+	2.49346i	0.0712893	+	0.200929i
$155$			3.91082i			0.314124i
$156$		0			0
$157$			4.39973i			0.351137i	0.984467	+	0.175568i	$0.0561763\pi$
$157$			4.39973i			0.351137i	−0.984467	+	0.175568i	$0.943824\pi$
$158$			4.45068i			0.354077i
$159$		0			0
$160$		−	1.06893i		−	0.0845064i
$161$	−13.2918	+	4.71593i	−1.04754	+	0.371667i
$162$		0			0
$163$	12.6933			0.994217			0.497109	−	0.867688i	$-0.334395\pi$
$163$	12.6933			0.994217			0.497109	+	0.867688i	$0.334395\pi$
$164$	−2.50842			−0.195874
$165$		0			0
$166$		−	12.2229i		−	0.948680i
$167$	3.80680			0.294579			0.147290	−	0.989093i	$-0.452945\pi$
$167$	3.80680			0.294579			0.147290	+	0.989093i	$0.452945\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$			4.18806i			0.321210i
$171$		0			0
$172$	−0.230645			−0.0175865
$173$	−20.9353			−1.59168			−0.795842	−	0.605504i	$-0.792971\pi$
$173$	−20.9353			−1.59168			−0.795842	+	0.605504i	$0.792971\pi$
$174$		0			0
$175$	−3.41254	−	9.61824i	−0.257964	−	0.727071i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$		−	10.0038i		−	0.749813i
$179$			6.47329i			0.483836i	0.970297	+	0.241918i	$0.0777765\pi$
$179$			6.47329i			0.483836i	−0.970297	+	0.241918i	$0.922223\pi$
$180$		0			0
$181$		−	18.1804i		−	1.35134i	−0.737206	−	0.675668i	$-0.763855\pi$
$181$		−	18.1804i		−	1.35134i	0.737206	−	0.675668i	$-0.236145\pi$
$182$		0			0
$183$		0			0
$184$	−5.33067			−0.392982
$185$	−7.83598			−0.576113
$186$		0			0
$187$		−	3.91799i		−	0.286512i
$188$	−0.370555			−0.0270255
$189$		0			0
$190$	8.01200			0.581251
$191$			18.2320i			1.31922i	0.751607	+	0.659612i	$0.229279\pi$
$191$			18.2320i			1.31922i	−0.751607	+	0.659612i	$0.770721\pi$
$192$		0			0
$193$	−1.05343			−0.0758275			−0.0379137	−	0.999281i	$-0.512071\pi$
$193$	−1.05343			−0.0758275			−0.0379137	+	0.999281i	$0.512071\pi$
$194$	12.8528			0.922778
$195$		0			0
$196$	5.43469	−	4.41182i	0.388192	−	0.315130i
$197$		−	20.8494i		−	1.48546i	−0.669592	−	0.742729i	$-0.733531\pi$
$197$		−	20.8494i		−	1.48546i	0.669592	−	0.742729i	$-0.266469\pi$
$198$		0			0
$199$		−	14.3736i		−	1.01892i	−0.860496	−	0.509458i	$-0.829846\pi$
$199$		−	14.3736i		−	1.01892i	0.860496	−	0.509458i	$-0.170154\pi$
$200$		−	3.85739i		−	0.272758i
$201$		0			0
$202$		−	14.0030i		−	0.985249i
$203$	19.9776	−	7.08803i	1.40215	−	0.497482i
$204$		0			0
$205$	2.68132			0.187272
$206$	−12.2229			−0.851609
$207$		0			0
$208$		0			0
$209$	−7.49534			−0.518463
$210$		0			0
$211$	9.94542			0.684671			0.342335	−	0.939578i	$-0.388782\pi$
$211$	9.94542			0.684671			0.342335	+	0.939578i	$0.388782\pi$
$212$		−	7.56132i		−	0.519313i
$213$		0			0
$214$	−20.4081			−1.39507
$215$	0.246544			0.0168141
$216$		0			0
$217$	−9.12264	+	3.23670i	−0.619285	+	0.219722i
$218$		−	12.8814i		−	0.872437i
$219$		0			0
$220$		−	1.06893i		−	0.0720673i
$221$		0			0
$222$		0			0
$223$			20.5648i			1.37712i	0.725180	+	0.688560i	$0.241757\pi$
$223$			20.5648i			1.37712i	−0.725180	+	0.688560i	$0.758243\pi$
$224$	2.49346	−	0.884677i	0.166601	−	0.0591100i
$225$		0			0
$226$	−6.92396			−0.460575
$227$	−22.9250			−1.52159			−0.760794	−	0.648994i	$-0.775190\pi$
$227$	−22.9250			−1.52159			−0.760794	+	0.648994i	$0.775190\pi$
$228$		0			0
$229$			27.2136i			1.79832i	0.437617	+	0.899162i	$0.355823\pi$
$229$			27.2136i			1.79832i	−0.437617	+	0.899162i	$0.644177\pi$
$230$	5.69812			0.375723
$231$		0			0
$232$	8.01200			0.526013
$233$		−	14.4853i		−	0.948962i	−0.880266	−	0.474481i	$-0.842636\pi$
$233$		−	14.4853i		−	0.948962i	0.880266	−	0.474481i	$-0.157364\pi$
$234$		0			0
$235$	0.396097			0.0258385
$236$	−2.13786			−0.139163
$237$		0			0
$238$	−9.76935	+	3.46616i	−0.633253	+	0.224678i
$239$		−	7.81595i		−	0.505572i	−0.967522	−	0.252786i	$-0.918653\pi$
$239$		−	7.81595i		−	0.505572i	0.967522	−	0.252786i	$-0.0813469\pi$
$240$		0			0
$241$		−	23.1844i		−	1.49344i	−0.665140	−	0.746719i	$-0.731628\pi$
$241$		−	23.1844i		−	1.49344i	0.665140	−	0.746719i	$-0.268372\pi$
$242$			1.00000i			0.0642824i
$243$		0			0
$244$		−	14.2496i		−	0.912235i
$245$	−5.80931	+	4.71593i	−0.371143	+	0.301289i
$246$		0			0
$247$		0			0
$248$	−3.65862			−0.232323
$249$		0			0
$250$			9.46793i			0.598805i
$251$	−12.1117			−0.764484			−0.382242	−	0.924062i	$-0.624848\pi$
$251$	−12.1117			−0.764484			−0.382242	+	0.924062i	$0.624848\pi$
$252$		0			0
$253$	−5.33067			−0.335137
$254$		−	10.6267i		−	0.666781i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−10.0038			−0.624017			−0.312008	−	0.950079i	$-0.601002\pi$
$257$	−10.0038			−0.624017			−0.312008	+	0.950079i	$0.601002\pi$
$258$		0			0
$259$	−6.48528	−	18.2787i	−0.402976	−	1.13579i
$260$		0			0
$261$		0			0
$262$			3.23670i			0.199964i
$263$			7.37464i			0.454740i	0.973808	+	0.227370i	$0.0730127\pi$
$263$			7.37464i			0.454740i	−0.973808	+	0.227370i	$0.926987\pi$
$264$		0			0
$265$			8.08252i			0.496506i
$266$	6.63096	+	18.6893i	0.406570	+	1.14592i
$267$		0			0
$268$	−2.47329			−0.151080
$269$	−9.41082			−0.573788			−0.286894	−	0.957962i	$-0.592623\pi$
$269$	−9.41082			−0.573788			−0.286894	+	0.957962i	$0.592623\pi$
$270$		0			0
$271$		−	27.0725i		−	1.64454i	−0.569101	−	0.822268i	$-0.692709\pi$
$271$		−	27.0725i		−	1.64454i	0.569101	−	0.822268i	$-0.307291\pi$
$272$	−3.91799			−0.237563
$273$		0			0
$274$	10.7159			0.647373
$275$		−	3.85739i		−	0.232609i
$276$		0			0
$277$	17.8279			1.07118			0.535589	−	0.844479i	$-0.320090\pi$
$277$	17.8279			1.07118			0.535589	+	0.844479i	$0.320090\pi$
$278$	6.24832			0.374749
$279$		0			0
$280$	−2.66534	+	0.945659i	−0.159284	+	0.0565139i
$281$		−	22.9706i		−	1.37031i	−0.728398	−	0.685154i	$-0.759735\pi$
$281$		−	22.9706i		−	1.37031i	0.728398	−	0.685154i	$-0.240265\pi$
$282$		0			0
$283$			10.3615i			0.615926i	0.951398	+	0.307963i	$0.0996474\pi$
$283$			10.3615i			0.615926i	−0.951398	+	0.307963i	$0.900353\pi$
$284$			3.15461i			0.187192i
$285$		0			0
$286$		0			0
$287$	2.21914	+	6.25464i	0.130992	+	0.369200i
$288$		0			0
$289$	−1.64935			−0.0970206
$290$	−8.56427			−0.502911
$291$		0			0
$292$		−	0.370555i		−	0.0216851i
$293$	−10.9615			−0.640377			−0.320189	−	0.947354i	$-0.603746\pi$
$293$	−10.9615			−0.640377			−0.320189	+	0.947354i	$0.603746\pi$
$294$		0			0
$295$	2.28523			0.133051
$296$		−	7.33067i		−	0.426087i
$297$		0			0
$298$	12.8694			0.745503
$299$		0			0
$300$		0			0
$301$	0.204046	+	0.575104i	0.0117610	+	0.0331485i
$302$			8.61475i			0.495723i
$303$		0			0
$304$			7.49534i			0.429887i
$305$			15.2318i			0.872170i
$306$		0			0
$307$		−	30.4845i		−	1.73985i	−0.493188	−	0.869923i	$-0.664169\pi$
$307$		−	30.4845i		−	1.73985i	0.493188	−	0.869923i	$-0.335831\pi$
$308$	2.49346	−	0.884677i	0.142078	−	0.0504092i
$309$		0			0
$310$	3.91082			0.222119
$311$	8.19371			0.464623			0.232311	−	0.972641i	$-0.425371\pi$
$311$	8.19371			0.464623			0.232311	+	0.972641i	$0.425371\pi$
$312$		0			0
$313$			33.2694i			1.88050i	0.340487	+	0.940249i	$0.389408\pi$
$313$			33.2694i			1.88050i	−0.340487	+	0.940249i	$0.610592\pi$
$314$	4.39973			0.248291
$315$		0			0
$316$	4.45068			0.250370
$317$		−	14.6627i		−	0.823541i	−0.911288	−	0.411770i	$-0.864911\pi$
$317$		−	14.6627i		−	0.823541i	0.911288	−	0.411770i	$-0.135089\pi$
$318$		0			0
$319$	8.01200			0.448586
$320$	−1.06893			−0.0597550
$321$		0			0
$322$	4.71593	+	13.2918i	0.262808	+	0.740724i
$323$		−	29.3667i		−	1.63400i
$324$		0			0
$325$		0			0
$326$		−	12.6933i		−	0.703018i
$327$		0			0
$328$			2.50842i			0.138504i
$329$	0.327821	+	0.923963i	0.0180734	+	0.0509398i
$330$		0			0
$331$	−6.88936			−0.378673			−0.189337	−	0.981912i	$-0.560634\pi$
$331$	−6.88936			−0.378673			−0.189337	+	0.981912i	$0.560634\pi$
$332$	−12.2229			−0.670818
$333$		0			0
$334$		−	3.80680i		−	0.208299i
$335$	2.64377			0.144445
$336$		0			0
$337$	−5.12264			−0.279048			−0.139524	−	0.990219i	$-0.544557\pi$
$337$	−5.12264			−0.279048			−0.139524	+	0.990219i	$0.544557\pi$
$338$		−	13.0000i		−	0.707107i
$339$		0			0
$340$	4.18806			0.227129
$341$	−3.65862			−0.198126
$342$		0			0
$343$	−15.8086	−	9.64815i	−0.853586	−	0.520951i
$344$			0.230645i			0.0124355i
$345$		0			0
$346$			20.9353i			1.12549i
$347$			11.6399i			0.624862i	0.949940	+	0.312431i	$0.101143\pi$
$347$			11.6399i			0.624862i	−0.949940	+	0.312431i	$0.898857\pi$
$348$		0			0
$349$			27.3248i			1.46266i	0.682023	+	0.731330i	$0.261100\pi$
$349$			27.3248i			1.46266i	−0.682023	+	0.731330i	$0.738900\pi$
$350$	−9.61824	+	3.41254i	−0.514117	+	0.182408i
$351$		0			0
$352$	1.00000			0.0533002
$353$	1.45231			0.0772985			0.0386493	−	0.999253i	$-0.487694\pi$
$353$	1.45231			0.0772985			0.0386493	+	0.999253i	$0.487694\pi$
$354$		0			0
$355$		−	3.37206i		−	0.178970i
$356$	−10.0038			−0.530198
$357$		0			0
$358$	6.47329			0.342124
$359$			0.196041i			0.0103467i	0.999987	+	0.00517333i	$0.00164673\pi$
$359$			0.196041i			0.0103467i	−0.999987	+	0.00517333i	$0.998353\pi$
$360$		0			0
$361$	−37.1801			−1.95685
$362$	−18.1804			−0.955539
$363$		0			0
$364$		0			0
$365$			0.396097i			0.0207327i
$366$		0			0
$367$			2.02667i			0.105791i	0.998600	+	0.0528957i	$0.0168451\pi$
$367$			2.02667i			0.105791i	−0.998600	+	0.0528957i	$0.983155\pi$
$368$			5.33067i			0.277881i
$369$		0			0
$370$			7.83598i			0.407373i
$371$	−18.8538	+	6.68933i	−0.978843	+	0.347293i
$372$		0			0
$373$	26.2001			1.35659			0.678294	−	0.734791i	$-0.262720\pi$
$373$	26.2001			1.35659			0.678294	+	0.734791i	$0.262720\pi$
$374$	−3.91799			−0.202594
$375$		0			0
$376$			0.370555i			0.0191099i
$377$		0			0
$378$		0			0
$379$	8.69332			0.446546			0.223273	−	0.974756i	$-0.428326\pi$
$379$	8.69332			0.446546			0.223273	+	0.974756i	$0.428326\pi$
$380$		−	8.01200i		−	0.411007i
$381$		0			0
$382$	18.2320			0.932832
$383$	5.89330			0.301133			0.150567	−	0.988600i	$-0.451890\pi$
$383$	5.89330			0.301133			0.150567	+	0.988600i	$0.451890\pi$
$384$		0			0
$385$	−2.66534	+	0.945659i	−0.135838	+	0.0481952i
$386$			1.05343i			0.0536181i
$387$		0			0
$388$		−	12.8528i		−	0.652503i
$389$			11.5853i			0.587398i	0.955898	+	0.293699i	$0.0948864\pi$
$389$			11.5853i			0.587398i	−0.955898	+	0.293699i	$0.905114\pi$
$390$		0			0
$391$		−	20.8855i		−	1.05623i
$392$	−4.41182	−	5.43469i	−0.222830	−	0.274493i
$393$		0			0
$394$	−20.8494			−1.05038
$395$	−4.75747			−0.239374
$396$		0			0
$397$		−	28.1414i		−	1.41238i	−0.708024	−	0.706188i	$-0.750413\pi$
$397$		−	28.1414i		−	1.41238i	0.708024	−	0.706188i	$-0.249587\pi$
$398$	−14.3736			−0.720482
$399$		0			0
$400$	−3.85739			−0.192869
$401$		−	24.4547i		−	1.22121i	−0.791936	−	0.610605i	$-0.790927\pi$
$401$		−	24.4547i		−	1.22121i	0.791936	−	0.610605i	$-0.209073\pi$
$402$		0			0
$403$		0			0
$404$	−14.0030			−0.696676
$405$		0			0
$406$	−7.08803	−	19.9776i	−0.351773	−	0.991471i
$407$		−	7.33067i		−	0.363368i
$408$		0			0
$409$			10.8503i			0.536513i	0.963348	+	0.268257i	$0.0864475\pi$
$409$			10.8503i			0.536513i	−0.963348	+	0.268257i	$0.913553\pi$
$410$		−	2.68132i		−	0.132421i
$411$		0			0
$412$			12.2229i			0.602179i
$413$	1.89132	+	5.33067i	0.0930657	+	0.262305i
$414$		0			0
$415$	13.0654			0.641356
$416$		0			0
$417$		0			0
$418$			7.49534i			0.366609i
$419$	8.57709			0.419018			0.209509	−	0.977807i	$-0.432813\pi$
$419$	8.57709			0.419018			0.209509	+	0.977807i	$0.432813\pi$
$420$		0			0
$421$	2.36011			0.115025			0.0575124	−	0.998345i	$-0.481683\pi$
$421$	2.36011			0.115025			0.0575124	+	0.998345i	$0.481683\pi$
$422$		−	9.94542i		−	0.484135i
$423$		0			0
$424$	−7.56132			−0.367210
$425$	15.1132			0.733098
$426$		0			0
$427$	−35.5307	+	12.6063i	−1.71945	+	0.610060i
$428$			20.4081i			0.986462i
$429$		0			0
$430$		−	0.246544i		−	0.0118894i
$431$			39.7508i			1.91473i	0.288886	+	0.957363i	$0.406715\pi$
$431$			39.7508i			1.91473i	−0.288886	+	0.957363i	$0.593285\pi$
$432$		0			0
$433$		−	0.765276i		−	0.0367768i	−0.999831	−	0.0183884i	$-0.994146\pi$
$433$		−	0.765276i		−	0.0367768i	0.999831	−	0.0183884i	$-0.00585354\pi$
$434$	3.23670	+	9.12264i	0.155367	+	0.437901i
$435$		0			0
$436$	−12.8814			−0.616906
$437$	−39.9552			−1.91132
$438$		0			0
$439$		−	30.9149i		−	1.47549i	−0.675080	−	0.737744i	$-0.735891\pi$
$439$		−	30.9149i		−	1.47549i	0.675080	−	0.737744i	$-0.264109\pi$
$440$	−1.06893			−0.0509593
$441$		0			0
$442$		0			0
$443$			18.3641i			0.872506i	0.899824	+	0.436253i	$0.143695\pi$
$443$			18.3641i			0.872506i	−0.899824	+	0.436253i	$0.856305\pi$
$444$		0			0
$445$	10.6933			0.506912
$446$	20.5648			0.973771
$447$		0			0
$448$	−0.884677	−	2.49346i	−0.0417971	−	0.117805i
$449$		−	3.80133i		−	0.179396i	−0.995969	−	0.0896979i	$-0.971410\pi$
$449$		−	3.80133i		−	0.179396i	0.995969	−	0.0896979i	$-0.0285901\pi$
$450$		0			0
$451$			2.50842i			0.118117i
$452$			6.92396i			0.325676i
$453$		0			0
$454$			22.9250i			1.07592i
$455$		0			0
$456$		0			0
$457$	6.48528			0.303369			0.151684	−	0.988429i	$-0.451530\pi$
$457$	6.48528			0.303369			0.151684	+	0.988429i	$0.451530\pi$
$458$	27.2136			1.27161
$459$		0			0
$460$		−	5.69812i		−	0.265676i
$461$	−11.1839			−0.520885			−0.260442	−	0.965489i	$-0.583868\pi$
$461$	−11.1839			−0.520885			−0.260442	+	0.965489i	$0.583868\pi$
$462$		0			0
$463$	33.7840			1.57007			0.785037	−	0.619448i	$-0.212644\pi$
$463$	33.7840			1.57007			0.785037	+	0.619448i	$0.212644\pi$
$464$		−	8.01200i		−	0.371948i
$465$		0			0
$466$	−14.4853			−0.671018
$467$	−31.8968			−1.47601			−0.738005	−	0.674796i	$-0.764232\pi$
$467$	−31.8968			−1.47601			−0.738005	+	0.674796i	$0.764232\pi$
$468$		0			0
$469$	2.18806	+	6.16704i	0.101035	+	0.284767i
$470$		−	0.396097i		−	0.0182706i
$471$		0			0
$472$			2.13786i			0.0984030i
$473$			0.230645i			0.0106051i
$474$		0			0
$475$		−	28.9124i		−	1.32659i
$476$	3.46616	+	9.76935i	0.158871	+	0.447778i
$477$		0			0
$478$	−7.81595			−0.357493
$479$	40.1419			1.83413			0.917065	−	0.398737i	$-0.130551\pi$
$479$	40.1419			1.83413			0.917065	+	0.398737i	$0.130551\pi$
$480$		0			0
$481$		0			0
$482$	−23.1844			−1.05602
$483$		0			0
$484$	1.00000			0.0454545
$485$			13.7388i			0.623845i
$486$		0			0
$487$	−5.82394			−0.263908			−0.131954	−	0.991256i	$-0.542125\pi$
$487$	−5.82394			−0.263908			−0.131954	+	0.991256i	$0.542125\pi$
$488$	−14.2496			−0.645047
$489$		0			0
$490$	4.71593	+	5.80931i	0.213044	+	0.262438i
$491$		−	15.9920i		−	0.721710i	−0.932622	−	0.360855i	$-0.882485\pi$
$491$		−	15.9920i		−	0.721710i	0.932622	−	0.360855i	$-0.117515\pi$
$492$		0			0
$493$			31.3909i			1.41378i
$494$		0			0
$495$		0			0
$496$			3.65862i			0.164277i
$497$	7.86589	−	2.79081i	0.352833	−	0.125185i
$498$		0			0
$499$	17.6199			0.788776			0.394388	−	0.918944i	$-0.370957\pi$
$499$	17.6199			0.788776			0.394388	+	0.918944i	$0.370957\pi$
$500$	9.46793			0.423419
$501$		0			0
$502$			12.1117i			0.540572i
$503$	0.468921			0.0209081			0.0104541	−	0.999945i	$-0.496672\pi$
$503$	0.468921			0.0209081			0.0104541	+	0.999945i	$0.496672\pi$
$504$		0			0
$505$	14.9683			0.666079
$506$			5.33067i			0.236977i
$507$		0			0
$508$	−10.6267			−0.471485
$509$	−13.6865			−0.606645			−0.303323	−	0.952888i	$-0.598096\pi$
$509$	−13.6865			−0.606645			−0.303323	+	0.952888i	$0.598096\pi$
$510$		0			0
$511$	−0.923963	+	0.327821i	−0.0408737	+	0.0145020i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			10.0038i			0.441247i
$515$		−	13.0654i		−	0.575731i
$516$		0			0
$517$			0.370555i			0.0162970i
$518$	−18.2787	+	6.48528i	−0.803122	+	0.284947i
$519$		0			0
$520$		0			0
$521$	−5.70239			−0.249826			−0.124913	−	0.992168i	$-0.539865\pi$
$521$	−5.70239			−0.249826			−0.124913	+	0.992168i	$0.539865\pi$
$522$		0			0
$523$		−	3.94790i		−	0.172630i	−0.996268	−	0.0863148i	$-0.972491\pi$
$523$		−	3.94790i		−	0.172630i	0.996268	−	0.0863148i	$-0.0275091\pi$
$524$	3.23670			0.141396
$525$		0			0
$526$	7.37464			0.321550
$527$		−	14.3345i		−	0.624419i
$528$		0			0
$529$	−5.41607			−0.235481
$530$	8.08252			0.351082
$531$		0			0
$532$	18.6893	−	6.63096i	0.810285	−	0.287488i
$533$		0			0
$534$		0			0
$535$		−	21.8148i		−	0.943138i
$536$			2.47329i			0.106830i
$537$		0			0
$538$			9.41082i			0.405729i
$539$	−4.41182	−	5.43469i	−0.190030	−	0.234089i
$540$		0			0
$541$	30.6613			1.31823			0.659117	−	0.752040i	$-0.270930\pi$
$541$	30.6613			1.31823			0.659117	+	0.752040i	$0.270930\pi$
$542$	−27.0725			−1.16286
$543$		0			0
$544$			3.91799i			0.167982i
$545$	13.7693			0.589812
$546$		0			0
$547$	2.05458			0.0878475			0.0439238	−	0.999035i	$-0.486014\pi$
$547$	2.05458			0.0878475			0.0439238	+	0.999035i	$0.486014\pi$
$548$		−	10.7159i		−	0.457762i
$549$		0			0
$550$	−3.85739			−0.164480
$551$	60.0526			2.55833
$552$		0			0
$553$	−3.93741	−	11.0976i	−0.167436	−	0.471918i
$554$		−	17.8279i		−	0.757437i
$555$		0			0
$556$		−	6.24832i		−	0.264988i
$557$		−	20.8494i		−	0.883418i	−0.897158	−	0.441709i	$-0.854372\pi$
$557$		−	20.8494i		−	0.883418i	0.897158	−	0.441709i	$-0.145628\pi$
$558$		0			0
$559$		0			0
$560$	0.945659	+	2.66534i	0.0399614	+	0.112631i
$561$		0			0
$562$	−22.9706			−0.968955
$563$	41.7225			1.75839			0.879197	−	0.476459i	$-0.158080\pi$
$563$	41.7225			1.75839			0.879197	+	0.476459i	$0.158080\pi$
$564$		0			0
$565$		−	7.40124i		−	0.311372i
$566$	10.3615			0.435526
$567$		0			0
$568$	3.15461			0.132364
$569$		−	2.48528i		−	0.104188i	−0.998642	−	0.0520942i	$-0.983410\pi$
$569$		−	2.48528i		−	0.104188i	0.998642	−	0.0520942i	$-0.0165896\pi$
$570$		0			0
$571$	−19.4412			−0.813590			−0.406795	−	0.913520i	$-0.633354\pi$
$571$	−19.4412			−0.813590			−0.406795	+	0.913520i	$0.633354\pi$
$572$		0			0
$573$		0			0
$574$	6.25464	−	2.21914i	0.261063	−	0.0926251i
$575$		−	20.5625i		−	0.857514i
$576$		0			0
$577$			4.49810i			0.187258i	0.995607	+	0.0936292i	$0.0298468\pi$
$577$			4.49810i			0.187258i	−0.995607	+	0.0936292i	$0.970153\pi$
$578$			1.64935i			0.0686039i
$579$		0			0
$580$			8.56427i			0.355612i
$581$	10.8133	+	30.4773i	0.448612	+	1.26441i
$582$		0			0
$583$	−7.56132			−0.313158
$584$	−0.370555			−0.0153337
$585$		0			0
$586$			10.9615i			0.452815i
$587$	37.2388			1.53701			0.768504	−	0.639845i	$-0.221001\pi$
$587$	37.2388			1.53701			0.768504	+	0.639845i	$0.221001\pi$
$588$		0			0
$589$	−27.4226			−1.12993
$590$		−	2.28523i		−	0.0940812i
$591$		0			0
$592$	−7.33067			−0.301289
$593$	25.1127			1.03125			0.515627	−	0.856813i	$-0.327559\pi$
$593$	25.1127			1.03125			0.515627	+	0.856813i	$0.327559\pi$
$594$		0			0
$595$	−3.70508	−	10.4428i	−0.151894	−	0.428112i
$596$		−	12.8694i		−	0.527150i
$597$		0			0
$598$		0			0
$599$			3.99202i			0.163109i	0.996669	+	0.0815547i	$0.0259885\pi$
$599$			3.99202i			0.163109i	−0.996669	+	0.0815547i	$0.974011\pi$
$600$		0			0
$601$		−	24.0624i		−	0.981526i	−0.871293	−	0.490763i	$-0.836718\pi$
$601$		−	24.0624i		−	0.981526i	0.871293	−	0.490763i	$-0.163282\pi$
$602$	0.575104	−	0.204046i	0.0234395	−	0.00831631i
$603$		0			0
$604$	8.61475			0.350529
$605$	−1.06893			−0.0434582
$606$		0			0
$607$		−	15.9243i		−	0.646345i	−0.946340	−	0.323173i	$-0.895251\pi$
$607$		−	15.9243i		−	0.646345i	0.946340	−	0.323173i	$-0.104749\pi$
$608$	7.49534			0.303976
$609$		0			0
$610$	15.2318			0.616717
$611$		0			0
$612$		0			0
$613$	18.6105			0.751669			0.375835	−	0.926687i	$-0.377356\pi$
$613$	18.6105			0.751669			0.375835	+	0.926687i	$0.377356\pi$
$614$	−30.4845			−1.23026
$615$		0			0
$616$	−0.884677	−	2.49346i	−0.0356447	−	0.100464i
$617$		−	15.8053i		−	0.636299i	−0.948041	−	0.318150i	$-0.896939\pi$
$617$		−	15.8053i		−	0.636299i	0.948041	−	0.318150i	$-0.103061\pi$
$618$		0			0
$619$			7.09487i			0.285167i	0.989783	+	0.142583i	$0.0455409\pi$
$619$			7.09487i			0.285167i	−0.989783	+	0.142583i	$0.954459\pi$
$620$		−	3.91082i		−	0.157062i
$621$		0			0
$622$		−	8.19371i		−	0.328538i
$623$	8.85010	+	24.9440i	0.354572	+	0.999359i
$624$		0			0
$625$	9.16637			0.366655
$626$	33.2694			1.32971
$627$		0			0
$628$		−	4.39973i		−	0.175568i
$629$	28.7215			1.14520
$630$		0			0
$631$	12.3521			0.491730			0.245865	−	0.969304i	$-0.420928\pi$
$631$	12.3521			0.491730			0.245865	+	0.969304i	$0.420928\pi$
$632$		−	4.45068i		−	0.177038i
$633$		0			0
$634$	−14.6627			−0.582331
$635$	11.3593			0.450778
$636$		0			0
$637$		0			0
$638$		−	8.01200i		−	0.317198i
$639$		0			0
$640$			1.06893i			0.0422532i
$641$			7.32006i			0.289125i	0.989496	+	0.144563i	$0.0461775\pi$
$641$			7.32006i			0.289125i	−0.989496	+	0.144563i	$0.953823\pi$
$642$		0			0
$643$			37.8173i			1.49137i	0.666299	+	0.745685i	$0.267878\pi$
$643$			37.8173i			1.49137i	−0.666299	+	0.745685i	$0.732122\pi$
$644$	13.2918	−	4.71593i	0.523771	−	0.185834i
$645$		0			0
$646$	−29.3667			−1.15542
$647$	−1.94268			−0.0763748			−0.0381874	−	0.999271i	$-0.512158\pi$
$647$	−1.94268			−0.0763748			−0.0381874	+	0.999271i	$0.512158\pi$
$648$		0			0
$649$			2.13786i			0.0839184i
$650$		0			0
$651$		0			0
$652$	−12.6933			−0.497109
$653$			23.6305i			0.924734i	0.886689	+	0.462367i	$0.153000\pi$
$653$			23.6305i			0.924734i	−0.886689	+	0.462367i	$0.847000\pi$
$654$		0			0
$655$	−3.45981			−0.135186
$656$	2.50842			0.0979372
$657$		0			0
$658$	0.923963	−	0.327821i	0.0360198	−	0.0127798i
$659$			35.4535i			1.38107i	0.723297	+	0.690537i	$0.242626\pi$
$659$			35.4535i			1.38107i	−0.723297	+	0.690537i	$0.757374\pi$
$660$		0			0
$661$			4.84582i			0.188481i	0.995549	+	0.0942403i	$0.0300422\pi$
$661$			4.84582i			0.188481i	−0.995549	+	0.0942403i	$0.969958\pi$
$662$			6.88936i			0.267763i
$663$		0			0
$664$			12.2229i			0.474340i
$665$	−19.9776	+	7.08803i	−0.774698	+	0.274862i
$666$		0			0
$667$	42.7093			1.65371
$668$	−3.80680			−0.147290
$669$		0			0
$670$		−	2.64377i		−	0.102138i
$671$	−14.2496			−0.550098
$672$		0			0
$673$	−51.0185			−1.96662			−0.983310	−	0.181937i	$-0.941763\pi$
$673$	−51.0185			−1.96662			−0.983310	+	0.181937i	$0.941763\pi$
$674$			5.12264i			0.197317i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	24.3330			0.935195			0.467597	−	0.883942i	$-0.345120\pi$
$677$	24.3330			0.935195			0.467597	+	0.883942i	$0.345120\pi$
$678$		0			0
$679$	−32.0480	+	11.3706i	−1.22989	+	0.436363i
$680$		−	4.18806i		−	0.160605i
$681$		0			0
$682$			3.65862i			0.140096i
$683$			25.9028i			0.991144i	0.868567	+	0.495572i	$0.165042\pi$
$683$			25.9028i			0.991144i	−0.868567	+	0.495572i	$0.834958\pi$
$684$		0			0
$685$			11.4546i			0.437657i
$686$	−9.64815	+	15.8086i	−0.368368	+	0.603577i
$687$		0			0
$688$	0.230645			0.00879326
$689$		0			0
$690$		0			0
$691$			37.7917i			1.43766i	0.695184	+	0.718832i	$0.255323\pi$
$691$			37.7917i			1.43766i	−0.695184	+	0.718832i	$0.744677\pi$
$692$	20.9353			0.795842
$693$		0			0
$694$	11.6399			0.441844
$695$			6.67902i			0.253350i
$696$		0			0
$697$	−9.82795			−0.372260
$698$	27.3248			1.03426
$699$		0			0
$700$	3.41254	+	9.61824i	0.128982	+	0.363535i
$701$		−	31.8479i		−	1.20288i	−0.798918	−	0.601440i	$-0.794594\pi$
$701$		−	31.8479i		−	1.20288i	0.798918	−	0.601440i	$-0.205406\pi$
$702$		0			0
$703$		−	54.9459i		−	2.07232i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	1.45231i		−	0.0546583i
$707$	12.3882	+	34.9160i	0.465905	+	1.31315i
$708$		0			0
$709$	−10.7682			−0.404408			−0.202204	−	0.979343i	$-0.564810\pi$
$709$	−10.7682			−0.404408			−0.202204	+	0.979343i	$0.564810\pi$
$710$	−3.37206			−0.126551
$711$		0			0
$712$			10.0038i			0.374906i
$713$	−19.5029			−0.730390
$714$		0			0
$715$		0			0
$716$		−	6.47329i		−	0.241918i
$717$		0			0
$718$	0.196041			0.00731620
$719$	22.1726			0.826897			0.413449	−	0.910527i	$-0.364324\pi$
$719$	22.1726			0.826897			0.413449	+	0.910527i	$0.364324\pi$
$720$		0			0
$721$	30.4773	−	10.8133i	1.13503	−	0.402709i
$722$			37.1801i			1.38370i
$723$		0			0
$724$			18.1804i			0.675668i
$725$			30.9054i			1.14780i
$726$		0			0
$727$		−	28.6830i		−	1.06379i	−0.846810	−	0.531896i	$-0.821480\pi$
$727$		−	28.6830i		−	1.06379i	0.846810	−	0.531896i	$-0.178520\pi$
$728$		0			0
$729$		0			0
$730$	0.396097			0.0146602
$731$	−0.903665			−0.0334233
$732$		0			0
$733$			24.2234i			0.894712i	0.894356	+	0.447356i	$0.147634\pi$
$733$			24.2234i			0.894712i	−0.894356	+	0.447356i	$0.852366\pi$
$734$	2.02667			0.0748059
$735$		0			0
$736$	5.33067			0.196491
$737$			2.47329i			0.0911046i
$738$		0			0
$739$	5.17722			0.190447			0.0952235	−	0.995456i	$-0.469643\pi$
$739$	5.17722			0.190447			0.0952235	+	0.995456i	$0.469643\pi$
$740$	7.83598			0.288056
$741$		0			0
$742$	6.68933	+	18.8538i	0.245573	+	0.692147i
$743$		−	26.7625i		−	0.981822i	−0.871210	−	0.490911i	$-0.836664\pi$
$743$		−	26.7625i		−	0.981822i	0.871210	−	0.490911i	$-0.163336\pi$
$744$		0			0
$745$			13.7565i			0.503998i
$746$		−	26.2001i		−	0.959252i
$747$		0			0
$748$			3.91799i			0.143256i
$749$	50.8868	−	18.0546i	1.85936	−	0.659700i
$750$		0			0
$751$	54.2933			1.98119			0.990594	−	0.136830i	$-0.0436915\pi$
$751$	54.2933			1.98119			0.990594	+	0.136830i	$0.0436915\pi$
$752$	0.370555			0.0135127
$753$		0			0
$754$		0			0
$755$	−9.20857			−0.335134
$756$		0			0
$757$	−12.6294			−0.459022			−0.229511	−	0.973306i	$-0.573713\pi$
$757$	−12.6294			−0.459022			−0.229511	+	0.973306i	$0.573713\pi$
$758$		−	8.69332i		−	0.315756i
$759$		0			0
$760$	−8.01200			−0.290626
$761$	11.0599			0.400920			0.200460	−	0.979702i	$-0.435756\pi$
$761$	11.0599			0.400920			0.200460	+	0.979702i	$0.435756\pi$
$762$		0			0
$763$	11.3959	+	32.1192i	0.412558	+	1.16279i
$764$		−	18.2320i		−	0.659612i
$765$		0			0
$766$		−	5.89330i		−	0.212933i
$767$		0			0
$768$		0			0
$769$		−	3.96499i		−	0.142981i	−0.997441	−	0.0714906i	$-0.977224\pi$
$769$		−	3.96499i		−	0.142981i	0.997441	−	0.0714906i	$-0.0227756\pi$
$770$	0.945659	+	2.66534i	0.0340792	+	0.0960521i
$771$		0			0
$772$	1.05343			0.0379137
$773$	10.5369			0.378985			0.189492	−	0.981882i	$-0.439316\pi$
$773$	10.5369			0.378985			0.189492	+	0.981882i	$0.439316\pi$
$774$		0			0
$775$		−	14.1127i		−	0.506944i
$776$	−12.8528			−0.461389
$777$		0			0
$778$	11.5853			0.415353
$779$			18.8014i			0.673631i
$780$		0			0
$781$	3.15461			0.112881
$782$	−20.8855			−0.746865
$783$		0			0
$784$	−5.43469	+	4.41182i	−0.194096	+	0.157565i
$785$			4.70301i			0.167858i
$786$		0			0
$787$		−	14.4277i		−	0.514290i	−0.966373	−	0.257145i	$-0.917218\pi$
$787$		−	14.4277i		−	0.514290i	0.966373	−	0.257145i	$-0.0827818\pi$
$788$			20.8494i			0.742729i
$789$		0			0
$790$			4.75747i			0.169263i
$791$	17.2646	−	6.12547i	0.613860	−	0.217797i
$792$		0			0
$793$		0			0
$794$	−28.1414			−0.998701
$795$		0			0
$796$			14.3736i			0.509458i
$797$	−23.3640			−0.827596			−0.413798	−	0.910369i	$-0.635798\pi$
$797$	−23.3640			−0.827596			−0.413798	+	0.910369i	$0.635798\pi$
$798$		0			0
$799$	−1.45183			−0.0513620
$800$			3.85739i			0.136379i
$801$		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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.06893 q^{5} +(0.884677 + 2.49346i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.06893 q^{5} +(0.884677 + 2.49346i) q^{7} +1.00000i q^{8} -1.06893i q^{10} +1.00000i q^{11} +(2.49346 - 0.884677i) q^{14} +1.00000 q^{16} -3.91799 q^{17} +7.49534i q^{19} -1.06893 q^{20} +1.00000 q^{22} +5.33067i q^{23} -3.85739 q^{25} +(-0.884677 - 2.49346i) q^{28} -8.01200i q^{29} +3.65862i q^{31} -1.00000i q^{32} +3.91799i q^{34} +(0.945659 + 2.66534i) q^{35} -7.33067 q^{37} +7.49534 q^{38} +1.06893i q^{40} +2.50842 q^{41} +0.230645 q^{43} -1.00000i q^{44} +5.33067 q^{46} +0.370555 q^{47} +(-5.43469 + 4.41182i) q^{49} +3.85739i q^{50} +7.56132i q^{53} +1.06893i q^{55} +(-2.49346 + 0.884677i) q^{56} -8.01200 q^{58} +2.13786 q^{59} +14.2496i q^{61} +3.65862 q^{62} -1.00000 q^{64} +2.47329 q^{67} +3.91799 q^{68} +(2.66534 - 0.945659i) q^{70} -3.15461i q^{71} +0.370555i q^{73} +7.33067i q^{74} -7.49534i q^{76} +(-2.49346 + 0.884677i) q^{77} -4.45068 q^{79} +1.06893 q^{80} -2.50842i q^{82} +12.2229 q^{83} -4.18806 q^{85} -0.230645i q^{86} -1.00000 q^{88} +10.0038 q^{89} -5.33067i q^{92} -0.370555i q^{94} +8.01200i q^{95} +12.8528i q^{97} +(4.41182 + 5.43469i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 16q^{4} - 8q^{7} + O(q^{10}) \)	\( 16q - 16q^{4} - 8q^{7} + 16q^{16} + 16q^{22} + 16q^{25} + 8q^{28} - 16q^{37} + 48q^{43} - 16q^{46} + 8q^{49} - 16q^{58} - 16q^{64} + 16q^{67} - 8q^{70} - 16q^{79} + 112q^{85} - 16q^{88} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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