LMFDB - Embedded newform 648.4.i.r.217.1

Newspace parameters

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
Defining polynomial:	$x^{4} - x^{3} + 2 x^{2} + x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.r.433.1

$q$-expansion

$f(q)$	$=$	$q+(-5.70820 - 9.88690i) q^{5} +(14.9164 - 25.8360i) q^{7} +O(q^{10})$	$q+(-5.70820 - 9.88690i) q^{5} +(14.9164 - 25.8360i) q^{7} +(33.1246 - 57.3735i) q^{11} +(-19.9164 - 34.4962i) q^{13} -107.416 q^{17} +70.3313 q^{19} +(3.45898 + 5.99113i) q^{23} +(-2.66718 + 4.61970i) q^{25} +(18.3344 - 31.7561i) q^{29} +(115.666 + 200.339i) q^{31} -340.584 q^{35} +36.8359 q^{37} +(214.663 + 371.806i) q^{41} +(37.1672 - 64.3755i) q^{43} +(26.2918 - 45.5387i) q^{47} +(-273.498 - 473.713i) q^{49} -288.170 q^{53} -756.328 q^{55} +(-391.872 - 678.743i) q^{59} +(-219.579 + 380.322i) q^{61} +(-227.374 + 393.823i) q^{65} +(109.169 + 189.086i) q^{67} -790.492 q^{71} +1098.00 q^{73} +(-988.200 - 1711.61i) q^{77} +(-219.913 + 380.901i) q^{79} +(-25.4226 + 44.0333i) q^{83} +(613.155 + 1062.02i) q^{85} -719.745 q^{89} -1188.33 q^{91} +(-401.465 - 695.358i) q^{95} +(599.822 - 1038.92i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4q + 4q^{5} + 6q^{7} + O(q^{10}) $	$ 4q + 4q^{5} + 6q^{7} + 52q^{11} - 26q^{13} - 376q^{17} - 148q^{19} + 148q^{23} - 118q^{25} + 288q^{29} + 248q^{31} - 1416q^{35} + 684q^{37} + 256q^{43} + 132q^{47} - 772q^{49} - 1904q^{53} - 1952q^{55} - 1004q^{59} + 34q^{61} - 668q^{65} + 866q^{67} - 1552q^{71} + 3748q^{73} - 2316q^{77} - 182q^{79} - 1336q^{83} - 16q^{85} - 1752q^{89} - 3036q^{91} - 3028q^{95} + 38q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.70820	−	9.88690i	−0.510557	−	0.884311i	−0.999925	−	0.0122337i	$-0.996106\pi$
$5$	−5.70820	−	9.88690i	−0.510557	−	0.884311i	0.489368	−	0.872077i	$-0.337228\pi$
$6$		0			0
$7$	14.9164	−	25.8360i	0.805410	−	1.39501i	−0.110603	−	0.993865i	$-0.535278\pi$
$7$	14.9164	−	25.8360i	0.805410	−	1.39501i	0.916014	−	0.401147i	$-0.131388\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	33.1246	−	57.3735i	0.907950	−	1.57261i	0.0910416	−	0.995847i	$-0.470980\pi$
$11$	33.1246	−	57.3735i	0.907950	−	1.57261i	0.816908	−	0.576768i	$-0.195686\pi$
$12$		0			0
$13$	−19.9164	−	34.4962i	−0.424909	−	0.735964i	0.571503	−	0.820600i	$-0.306361\pi$
$13$	−19.9164	−	34.4962i	−0.424909	−	0.735964i	−0.996412	+	0.0846360i	$0.973027\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−107.416			−1.53249			−0.766244	−	0.642549i	$-0.777877\pi$
$17$	−107.416			−1.53249			−0.766244	+	0.642549i	$0.777877\pi$
$18$		0			0
$19$	70.3313			0.849216			0.424608	−	0.905377i	$-0.360412\pi$
$19$	70.3313			0.849216			0.424608	+	0.905377i	$0.360412\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.45898	+	5.99113i	0.0313586	+	0.0543146i	0.881279	−	0.472597i	$-0.156683\pi$
$23$	3.45898	+	5.99113i	0.0313586	+	0.0543146i	−0.849920	+	0.526911i	$0.823350\pi$
$24$		0			0
$25$	−2.66718	+	4.61970i	−0.0213375	+	0.0369576i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	18.3344	−	31.7561i	0.117400	−	0.203343i	−0.801336	−	0.598214i	$-0.795877\pi$
$29$	18.3344	−	31.7561i	0.117400	−	0.203343i	0.918737	+	0.394871i	$0.129211\pi$
$30$		0			0
$31$	115.666	+	200.339i	0.670134	+	1.16071i	0.977866	+	0.209233i	$0.0670968\pi$
$31$	115.666	+	200.339i	0.670134	+	1.16071i	−0.307732	+	0.951473i	$0.599570\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−340.584			−1.64483
$36$		0			0
$37$	36.8359			0.163670			0.0818350	−	0.996646i	$-0.473922\pi$
$37$	36.8359			0.163670			0.0818350	+	0.996646i	$0.473922\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	214.663	+	371.806i	0.817674	+	1.41625i	0.907391	+	0.420286i	$0.138070\pi$
$41$	214.663	+	371.806i	0.817674	+	1.41625i	−0.0897170	+	0.995967i	$0.528596\pi$
$42$		0			0
$43$	37.1672	−	64.3755i	0.131813	−	0.228306i	−0.792563	−	0.609790i	$-0.791254\pi$
$43$	37.1672	−	64.3755i	0.131813	−	0.228306i	0.924375	+	0.381484i	$0.124587\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	26.2918	−	45.5387i	0.0815969	−	0.141330i	−0.822339	−	0.568998i	$-0.807331\pi$
$47$	26.2918	−	45.5387i	0.0815969	−	0.141330i	0.903936	+	0.427668i	$0.140665\pi$
$48$		0			0
$49$	−273.498	−	473.713i	−0.797372	−	1.38109i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−288.170			−0.746853			−0.373427	−	0.927660i	$-0.621817\pi$
$53$	−288.170			−0.746853			−0.373427	+	0.927660i	$0.621817\pi$
$54$		0			0
$55$	−756.328			−1.85424
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−391.872	−	678.743i	−0.864702	−	1.49771i	−0.867343	−	0.497712i	$-0.834174\pi$
$59$	−391.872	−	678.743i	−0.864702	−	1.49771i	0.00264051	−	0.999997i	$-0.499159\pi$
$60$		0			0
$61$	−219.579	+	380.322i	−0.460889	+	0.798282i	−0.999005	−	0.0445878i	$-0.985803\pi$
$61$	−219.579	+	380.322i	−0.460889	+	0.798282i	0.538117	+	0.842870i	$0.319136\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−227.374	+	393.823i	−0.433881	+	0.751504i
$66$		0			0
$67$	109.169	+	189.086i	0.199061	+	0.344784i	0.948224	−	0.317602i	$-0.102877\pi$
$67$	109.169	+	189.086i	0.199061	+	0.344784i	−0.749163	+	0.662385i	$0.769544\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−790.492			−1.32133			−0.660663	−	0.750682i	$-0.729725\pi$
$71$	−790.492			−1.32133			−0.660663	+	0.750682i	$0.729725\pi$
$72$		0			0
$73$	1098.00			1.76042			0.880211	−	0.474582i	$-0.157401\pi$
$73$	1098.00			1.76042			0.880211	+	0.474582i	$0.157401\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−988.200	−	1711.61i	−1.46254	−	2.53320i
$78$		0			0
$79$	−219.913	+	380.901i	−0.313192	+	0.542465i	−0.979052	−	0.203613i	$-0.934732\pi$
$79$	−219.913	+	380.901i	−0.313192	+	0.542465i	0.665859	+	0.746077i	$0.268065\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−25.4226	+	44.0333i	−0.0336204	+	0.0582323i	−0.882346	−	0.470601i	$-0.844037\pi$
$83$	−25.4226	+	44.0333i	−0.0336204	+	0.0582323i	0.848726	+	0.528833i	$0.177370\pi$
$84$		0			0
$85$	613.155	+	1062.02i	0.782423	+	1.35520i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−719.745			−0.857222			−0.428611	−	0.903489i	$-0.640997\pi$
$89$	−719.745			−0.857222			−0.428611	+	0.903489i	$0.640997\pi$
$90$		0			0
$91$	−1188.33			−1.36890
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−401.465	−	695.358i	−0.433573	−	0.750971i
$96$		0			0
$97$	599.822	−	1038.92i	0.627863	−	1.08749i	−0.360117	−	0.932907i	$-0.617263\pi$
$97$	599.822	−	1038.92i	0.627863	−	1.08749i	0.987980	−	0.154583i	$-0.0494034\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	122.991	−	213.026i	0.121169	−	0.209870i	−0.799060	−	0.601251i	$-0.794669\pi$
$101$	122.991	−	213.026i	0.121169	−	0.209870i	0.920229	+	0.391381i	$0.128002\pi$
$102$		0			0
$103$	−287.910	−	498.675i	−0.275424	−	0.477048i	0.694818	−	0.719185i	$-0.255485\pi$
$103$	−287.910	−	498.675i	−0.275424	−	0.477048i	−0.970242	+	0.242138i	$0.922151\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1015.89			0.917849			0.458924	−	0.888475i	$-0.348235\pi$
$107$	1015.89			0.917849			0.458924	+	0.888475i	$0.348235\pi$
$108$		0			0
$109$	1471.64			1.29319			0.646595	−	0.762834i	$-0.276193\pi$
$109$	1471.64			1.29319			0.646595	+	0.762834i	$0.276193\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−951.787	−	1648.54i	−0.792359	−	1.37241i	−0.924503	−	0.381176i	$-0.875519\pi$
$113$	−951.787	−	1648.54i	−0.792359	−	1.37241i	0.132143	−	0.991231i	$-0.457814\pi$
$114$		0			0
$115$	39.4891	−	68.3972i	0.0320207	−	0.0554615i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1602.27	+	2775.21i	−1.23428	+	2.13784i
$120$		0			0
$121$	−1528.98	−	2648.27i	−1.14875	−	1.98968i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1366.15			−0.977539
$126$		0			0
$127$	230.310			0.160919			0.0804593	−	0.996758i	$-0.474361\pi$
$127$	230.310			0.160919			0.0804593	+	0.996758i	$0.474361\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	397.155	+	687.892i	0.264882	+	0.458789i	0.967533	−	0.252746i	$-0.0813336\pi$
$131$	397.155	+	687.892i	0.264882	+	0.458789i	−0.702651	+	0.711535i	$0.748000\pi$
$132$		0			0
$133$	1049.09	−	1817.08i	0.683967	−	1.18467i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−341.538	+	591.561i	−0.212989	+	0.368909i	−0.952649	−	0.304073i	$-0.901653\pi$
$137$	−341.538	+	591.561i	−0.212989	+	0.368909i	0.739659	+	0.672982i	$0.234987\pi$
$138$		0			0
$139$	756.320	+	1309.99i	0.461513	+	0.799363i	0.999037	−	0.0438851i	$-0.0139736\pi$
$139$	756.320	+	1309.99i	0.461513	+	0.799363i	−0.537524	+	0.843248i	$0.680640\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2638.89			−1.54318
$144$		0			0
$145$	−418.625			−0.239758
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1153.57	+	1998.04i	0.634255	+	1.09856i	0.986672	+	0.162719i	$0.0520263\pi$
$149$	1153.57	+	1998.04i	0.634255	+	1.09856i	−0.352418	+	0.935843i	$0.614640\pi$
$150$		0			0
$151$	−1261.59	+	2185.13i	−0.679910	+	1.17764i	0.295098	+	0.955467i	$0.404648\pi$
$151$	−1261.59	+	2185.13i	−0.679910	+	1.17764i	−0.975008	+	0.222171i	$0.928686\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1320.49	−	2287.15i	0.684284	−	1.18521i
$156$		0			0
$157$	1459.49	+	2527.91i	0.741912	+	1.28503i	0.951624	+	0.307266i	$0.0994141\pi$
$157$	1459.49	+	2527.91i	0.741912	+	1.28503i	−0.209712	+	0.977763i	$0.567253\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	206.382			0.101026
$162$		0			0
$163$	1096.99			0.527133			0.263567	−	0.964641i	$-0.415101\pi$
$163$	1096.99			0.527133			0.263567	+	0.964641i	$0.415101\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−366.365	−	634.562i	−0.169761	−	0.294035i	0.768575	−	0.639760i	$-0.220966\pi$
$167$	−366.365	−	634.562i	−0.169761	−	0.294035i	−0.938336	+	0.345725i	$0.887633\pi$
$168$		0			0
$169$	305.173	−	528.576i	0.138905	−	0.240590i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	361.337	−	625.855i	0.158798	−	0.275045i	−0.775638	−	0.631178i	$-0.782572\pi$
$173$	361.337	−	625.855i	0.158798	−	0.275045i	0.934435	+	0.356133i	$0.115905\pi$
$174$		0			0
$175$	79.5696	+	137.819i	0.0343708	+	0.0595320i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1010.16			−0.421806			−0.210903	−	0.977507i	$-0.567640\pi$
$179$	−1010.16			−0.421806			−0.210903	+	0.977507i	$0.567640\pi$
$180$		0			0
$181$	−4256.79			−1.74809			−0.874045	−	0.485844i	$-0.838512\pi$
$181$	−4256.79			−1.74809			−0.874045	+	0.485844i	$0.838512\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−210.267	−	364.193i	−0.0835629	−	0.144735i
$186$		0			0
$187$	−3558.13	+	6162.86i	−1.39142	+	2.41001i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1714.69	−	2969.93i	0.649585	−	1.12511i	−0.333637	−	0.942702i	$-0.608276\pi$
$191$	1714.69	−	2969.93i	0.649585	−	1.12511i	0.983222	−	0.182413i	$-0.0583908\pi$
$192$		0			0
$193$	−483.670	−	837.742i	−0.180390	−	0.312445i	0.761623	−	0.648020i	$-0.224403\pi$
$193$	−483.670	−	837.742i	−0.180390	−	0.312445i	−0.942014	+	0.335575i	$0.891069\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	468.413			0.169406			0.0847032	−	0.996406i	$-0.473006\pi$
$197$	468.413			0.169406			0.0847032	+	0.996406i	$0.473006\pi$
$198$		0			0
$199$	−2673.47			−0.952347			−0.476174	−	0.879351i	$-0.657977\pi$
$199$	−2673.47			−0.952347			−0.476174	+	0.879351i	$0.657977\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−546.966	−	947.373i	−0.189111	−	0.327549i
$204$		0			0
$205$	2450.67	−	4244.69i	0.834939	−	1.44616i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2329.70	−	4035.15i	0.771045	−	1.33549i
$210$		0			0
$211$	−1435.31	−	2486.03i	−0.468297	−	0.811114i	0.531046	−	0.847343i	$-0.321799\pi$
$211$	−1435.31	−	2486.03i	−0.468297	−	0.811114i	−0.999344	+	0.0362284i	$0.988466\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−848.631			−0.269192
$216$		0			0
$217$	6901.26			2.15893
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2139.35	+	3705.46i	0.651168	+	1.12786i
$222$		0			0
$223$	623.300	−	1079.59i	0.187172	−	0.324191i	−0.757135	−	0.653259i	$-0.773401\pi$
$223$	623.300	−	1079.59i	0.187172	−	0.324191i	0.944306	+	0.329068i	$0.106735\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	280.231	−	485.374i	0.0819364	−	0.141918i	−0.822145	−	0.569278i	$-0.807223\pi$
$227$	280.231	−	485.374i	0.0819364	−	0.141918i	0.904082	+	0.427360i	$0.140556\pi$
$228$		0			0
$229$	572.690	+	991.929i	0.165260	+	0.286238i	0.936747	−	0.350006i	$-0.113820\pi$
$229$	572.690	+	991.929i	0.165260	+	0.286238i	−0.771488	+	0.636244i	$0.780487\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	623.709			0.175367			0.0876836	−	0.996148i	$-0.472054\pi$
$233$	623.709			0.175367			0.0876836	+	0.996148i	$0.472054\pi$
$234$		0			0
$235$	−600.316			−0.166639
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3132.90	+	5426.34i	0.847910	+	1.46862i	0.883070	+	0.469241i	$0.155472\pi$
$239$	3132.90	+	5426.34i	0.847910	+	1.46862i	−0.0351605	+	0.999382i	$0.511194\pi$
$240$		0			0
$241$	320.327	−	554.822i	0.0856185	−	0.148296i	−0.820036	−	0.572312i	$-0.806047\pi$
$241$	320.327	−	554.822i	0.0856185	−	0.148296i	0.905655	+	0.424016i	$0.139380\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3122.37	+	5408.10i	−0.814208	+	1.41025i
$246$		0			0
$247$	−1400.75	−	2426.16i	−0.360839	−	0.624992i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7748.91			1.94863			0.974316	−	0.225183i	$-0.0722980\pi$
$251$	7748.91			1.94863			0.974316	+	0.225183i	$0.0722980\pi$
$252$		0			0
$253$	458.310			0.113888
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1595.88	+	2764.14i	0.387347	+	0.670904i	0.992092	−	0.125515i	$-0.0400583\pi$
$257$	1595.88	+	2764.14i	0.387347	+	0.670904i	−0.604745	+	0.796419i	$0.706725\pi$
$258$		0			0
$259$	549.460	−	951.692i	0.131821	−	0.228321i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1888.21	+	3270.47i	−0.442706	+	0.766790i	−0.997889	−	0.0649384i	$-0.979315\pi$
$263$	−1888.21	+	3270.47i	−0.442706	+	0.766790i	0.555183	+	0.831728i	$0.312648\pi$
$264$		0			0
$265$	1644.93	+	2849.11i	0.381311	+	0.660451i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4065.91			0.921572			0.460786	−	0.887511i	$-0.347568\pi$
$269$	4065.91			0.921572			0.460786	+	0.887511i	$0.347568\pi$
$270$		0			0
$271$	−5508.17			−1.23468			−0.617339	−	0.786697i	$-0.711789\pi$
$271$	−5508.17			−1.23468			−0.617339	+	0.786697i	$0.711789\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	176.699	+	306.051i	0.0387467	+	0.0671113i
$276$		0			0
$277$	4153.45	−	7194.00i	0.900928	−	1.56045i	0.0746351	−	0.997211i	$-0.476221\pi$
$277$	4153.45	−	7194.00i	0.900928	−	1.56045i	0.826292	−	0.563241i	$-0.190446\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1481.34	+	2565.75i	−0.314481	+	0.544697i	−0.979327	−	0.202283i	$-0.935164\pi$
$281$	−1481.34	+	2565.75i	−0.314481	+	0.544697i	0.664846	+	0.746980i	$0.268497\pi$
$282$		0			0
$283$	2254.83	+	3905.48i	0.473625	+	0.820343i	0.999544	−	0.0301919i	$-0.00961185\pi$
$283$	2254.83	+	3905.48i	0.473625	+	0.820343i	−0.525919	+	0.850535i	$0.676279\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12808.0			2.63425
$288$		0			0
$289$	6625.28			1.34852
$290$		0			0
$291$		0			0
$292$		0			0
$293$	907.228	+	1571.37i	0.180890	+	0.313311i	0.942184	−	0.335096i	$-0.108769\pi$
$293$	907.228	+	1571.37i	0.180890	+	0.313311i	−0.761294	+	0.648407i	$0.775435\pi$
$294$		0			0
$295$	−4473.77	+	7748.80i	−0.882960	+	1.52933i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	137.781	−	238.644i	0.0266491	−	0.0461576i
$300$		0			0
$301$	−1108.80	−	1920.50i	−0.212326	−	0.367760i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5013.61			0.941240
$306$		0			0
$307$	4395.62			0.817171			0.408585	−	0.912720i	$-0.366022\pi$
$307$	4395.62			0.817171			0.408585	+	0.912720i	$0.366022\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−688.614	−	1192.71i	−0.125555	−	0.217468i	0.796395	−	0.604777i	$-0.206738\pi$
$311$	−688.614	−	1192.71i	−0.125555	−	0.217468i	−0.921950	+	0.387309i	$0.873405\pi$
$312$		0			0
$313$	−3673.97	+	6363.51i	−0.663467	+	1.14916i	0.316231	+	0.948682i	$0.397582\pi$
$313$	−3673.97	+	6363.51i	−0.663467	+	1.14916i	−0.979698	+	0.200477i	$0.935751\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1303.89	−	2258.40i	0.231021	−	0.400141i	−0.727088	−	0.686545i	$-0.759127\pi$
$317$	1303.89	−	2258.40i	0.231021	−	0.400141i	0.958109	+	0.286404i	$0.0924600\pi$
$318$		0			0
$319$	−1214.64	−	2103.81i	−0.213187	−	0.369251i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7554.73			−1.30141
$324$		0			0
$325$	212.483			0.0362659
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−784.358	−	1358.55i	−0.131438	−	0.227657i
$330$		0			0
$331$	3270.67	−	5664.96i	0.543118	−	0.940708i	−0.455605	−	0.890182i	$-0.650577\pi$
$331$	3270.67	−	5664.96i	0.543118	−	0.940708i	0.998723	−	0.0505260i	$-0.0160898\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1246.31	−	2158.68i	0.203264	−	0.352064i
$336$		0			0
$337$	−2157.83	−	3737.47i	−0.348797	−	0.604134i	0.637239	−	0.770666i	$-0.280076\pi$
$337$	−2157.83	−	3737.47i	−0.348797	−	0.604134i	−0.986036	+	0.166532i	$0.946743\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15325.5			2.43379
$342$		0			0
$343$	−6085.80			−0.958025
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1947.88	+	3373.82i	0.301347	+	0.521949i	0.976441	−	0.215783i	$-0.0692303\pi$
$347$	1947.88	+	3373.82i	0.301347	+	0.521949i	−0.675094	+	0.737732i	$0.735897\pi$
$348$		0			0
$349$	2438.71	−	4223.97i	0.374044	−	0.647863i	−0.616140	−	0.787637i	$-0.711304\pi$
$349$	2438.71	−	4223.97i	0.374044	−	0.647863i	0.990183	+	0.139774i	$0.0446377\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	869.896	−	1506.70i	0.131161	−	0.227178i	−0.792963	−	0.609269i	$-0.791463\pi$
$353$	869.896	−	1506.70i	0.131161	−	0.227178i	0.924124	+	0.382092i	$0.124796\pi$
$354$		0			0
$355$	4512.29	+	7815.52i	0.674613	+	1.16846i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−188.828			−0.0277604			−0.0138802	−	0.999904i	$-0.504418\pi$
$359$	−188.828			−0.0277604			−0.0138802	+	0.999904i	$0.504418\pi$
$360$		0			0
$361$	−1912.51			−0.278833
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6267.59	−	10855.8i	−0.898796	−	1.55676i
$366$		0			0
$367$	3169.07	−	5489.00i	0.450747	−	0.780718i	−0.547685	−	0.836685i	$-0.684491\pi$
$367$	3169.07	−	5489.00i	0.450747	−	0.780718i	0.998433	+	0.0559670i	$0.0178242\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4298.47	+	7445.16i	−0.601523	+	1.04187i
$372$		0			0
$373$	−4320.70	−	7483.68i	−0.599779	−	1.03885i	−0.992853	−	0.119341i	$-0.961922\pi$
$373$	−4320.70	−	7483.68i	−0.599779	−	1.03885i	0.393075	−	0.919507i	$-0.371411\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1460.62			−0.199538
$378$		0			0
$379$	−4143.88			−0.561627			−0.280814	−	0.959762i	$-0.590604\pi$
$379$	−4143.88			−0.561627			−0.280814	+	0.959762i	$0.590604\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2096.97	−	3632.06i	−0.279766	−	0.484568i	0.691561	−	0.722318i	$-0.256923\pi$
$383$	−2096.97	−	3632.06i	−0.279766	−	0.484568i	−0.971326	+	0.237750i	$0.923590\pi$
$384$		0			0
$385$	−11281.7	+	19540.5i	−1.49343	+	2.58669i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5435.21	−	9414.07i	0.708422	−	1.22702i	−0.257020	−	0.966406i	$-0.582741\pi$
$389$	5435.21	−	9414.07i	0.708422	−	1.22702i	0.965442	−	0.260617i	$-0.0839261\pi$
$390$		0			0
$391$	−371.551	−	643.546i	−0.0480567	−	0.0832366i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5021.24			0.639610
$396$		0			0
$397$	−8890.87			−1.12398			−0.561990	−	0.827144i	$-0.689964\pi$
$397$	−8890.87			−1.12398			−0.561990	+	0.827144i	$0.689964\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7547.71	+	13073.0i	0.939936	+	1.62802i	0.765586	+	0.643334i	$0.222449\pi$
$401$	7547.71	+	13073.0i	0.939936	+	1.62802i	0.174351	+	0.984684i	$0.444217\pi$
$402$		0			0
$403$	4607.29	−	7980.06i	0.569492	−	0.986389i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1220.18	−	2113.41i	0.148604	−	0.257390i
$408$		0			0
$409$	−809.169	−	1401.52i	−0.0978260	−	0.169440i	0.812959	−	0.582322i	$-0.197856\pi$
$409$	−809.169	−	1401.52i	−0.0978260	−	0.169440i	−0.910785	+	0.412882i	$0.864522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23381.3			−2.78576
$414$		0			0
$415$	580.470			0.0686606
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5005.94	+	8670.54i	0.583666	+	1.01094i	0.995040	+	0.0994728i	$0.0317156\pi$
$419$	5005.94	+	8670.54i	0.583666	+	1.01094i	−0.411374	+	0.911467i	$0.634951\pi$
$420$		0			0
$421$	−3617.22	+	6265.20i	−0.418747	+	0.725291i	−0.995814	−	0.0914063i	$-0.970864\pi$
$421$	−3617.22	+	6265.20i	−0.418747	+	0.725291i	0.577067	+	0.816697i	$0.304197\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	286.499	−	496.231i	0.0326994	−	0.0566371i
$426$		0			0
$427$	6550.66	+	11346.1i	0.742409	+	1.28589i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5379.03			−0.601157			−0.300579	−	0.953757i	$-0.597180\pi$
$431$	−5379.03			−0.601157			−0.300579	+	0.953757i	$0.597180\pi$
$432$		0			0
$433$	−1602.96			−0.177906			−0.0889530	−	0.996036i	$-0.528352\pi$
$433$	−1602.96			−0.177906			−0.0889530	+	0.996036i	$0.528352\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	243.274	+	421.364i	0.0266302	+	0.0461249i
$438$		0			0
$439$	−3362.93	+	5824.77i	−0.365613	+	0.633260i	−0.988874	−	0.148753i	$-0.952474\pi$
$439$	−3362.93	+	5824.77i	−0.365613	+	0.633260i	0.623261	+	0.782014i	$0.285807\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	584.961	−	1013.18i	0.0627367	−	0.108663i	−0.832951	−	0.553347i	$-0.813350\pi$
$443$	584.961	−	1013.18i	0.0627367	−	0.108663i	0.895688	+	0.444683i	$0.146684\pi$
$444$		0			0
$445$	4108.45	+	7116.04i	0.437661	+	0.758051i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2996.56			0.314958			0.157479	−	0.987522i	$-0.449663\pi$
$449$	2996.56			0.314958			0.157479	+	0.987522i	$0.449663\pi$
$450$		0			0
$451$	28442.5			2.96963
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6783.20	+	11748.9i	0.698904	+	1.21054i
$456$		0			0
$457$	5576.04	−	9657.98i	0.570757	−	0.988580i	−0.425731	−	0.904850i	$-0.639983\pi$
$457$	5576.04	−	9657.98i	0.570757	−	0.988580i	0.996488	−	0.0837307i	$-0.0266835\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1473.59	+	2552.33i	−0.148876	+	0.257860i	−0.930812	−	0.365498i	$-0.880899\pi$
$461$	−1473.59	+	2552.33i	−0.148876	+	0.257860i	0.781936	+	0.623358i	$0.214232\pi$
$462$		0			0
$463$	−3281.69	−	5684.05i	−0.329402	−	0.570541i	0.652991	−	0.757365i	$-0.273514\pi$
$463$	−3281.69	−	5684.05i	−0.329402	−	0.570541i	−0.982393	+	0.186824i	$0.940180\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2727.33			0.270248			0.135124	−	0.990829i	$-0.456857\pi$
$467$	2727.33			0.270248			0.135124	+	0.990829i	$0.456857\pi$
$468$		0			0
$469$	6513.62			0.641303
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2462.30	−	4264.82i	−0.239358	−	0.414581i
$474$		0			0
$475$	−187.586	+	324.909i	−0.0181201	+	0.0313850i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7256.06	−	12567.9i	0.692146	−	1.19883i	−0.278988	−	0.960295i	$-0.589999\pi$
$479$	7256.06	−	12567.9i	0.692146	−	1.19883i	0.971133	−	0.238537i	$-0.0766678\pi$
$480$		0			0
$481$	−733.639	−	1270.70i	−0.0695448	−	0.120455i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13695.6			−1.28224
$486$		0			0
$487$	−14189.9			−1.32034			−0.660170	−	0.751116i	$-0.729516\pi$
$487$	−14189.9			−1.32034			−0.660170	+	0.751116i	$0.729516\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3607.96	−	6249.17i	−0.331619	−	0.574381i	0.651211	−	0.758897i	$-0.274261\pi$
$491$	−3607.96	−	6249.17i	−0.331619	−	0.574381i	−0.982829	+	0.184516i	$0.940928\pi$
$492$		0			0
$493$	−1969.41	+	3411.12i	−0.179915	+	0.311621i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11791.3	+	20423.1i	−1.06421	+	1.84327i
$498$		0			0
$499$	−6516.28	−	11286.5i	−0.584587	−	1.01253i	−0.994927	−	0.100601i	$-0.967923\pi$
$499$	−6516.28	−	11286.5i	−0.584587	−	1.01253i	0.410340	−	0.911932i	$-0.365410\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9502.84			−0.842367			−0.421184	−	0.906975i	$-0.638385\pi$
$503$	−9502.84			−0.842367			−0.421184	+	0.906975i	$0.638385\pi$
$504$		0			0
$505$	−2808.22			−0.247454
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1897.15	+	3285.96i	0.165206	+	0.286145i	0.936728	−	0.350057i	$-0.113838\pi$
$509$	1897.15	+	3285.96i	0.165206	+	0.286145i	−0.771523	+	0.636202i	$0.780505\pi$
$510$		0			0
$511$	16378.2	−	28367.8i	1.41786	−	2.45581i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3286.90	+	5693.08i	−0.281239	+	0.487120i
$516$		0			0
$517$	−1741.81	−	3016.91i	−0.148172	−	0.256641i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14811.2			−1.24547			−0.622735	−	0.782432i	$-0.713979\pi$
$521$	−14811.2			−1.24547			−0.622735	+	0.782432i	$0.713979\pi$
$522$		0			0
$523$	−345.532			−0.0288892			−0.0144446	−	0.999896i	$-0.504598\pi$
$523$	−345.532			−0.0288892			−0.0144446	+	0.999896i	$0.504598\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12424.4	−	21519.7i	−1.02697	−	1.77877i
$528$		0			0
$529$	6059.57	−	10495.5i	0.498033	−	0.862619i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8550.61	−	14810.1i	0.694875	−	1.20356i
$534$		0			0
$535$	−5798.91	−	10044.0i	−0.468614	−	0.811664i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−36238.1			−2.89589
$540$		0			0
$541$	−5474.45			−0.435056			−0.217528	−	0.976054i	$-0.569799\pi$
$541$	−5474.45			−0.435056			−0.217528	+	0.976054i	$0.569799\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8400.43	−	14550.0i	−0.660247	−	1.14358i
$546$		0			0
$547$	−488.639	+	846.348i	−0.0381951	+	0.0661558i	−0.884491	−	0.466557i	$-0.845494\pi$
$547$	−488.639	+	846.348i	−0.0381951	+	0.0661558i	0.846296	+	0.532713i	$0.178827\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1289.48	−	2233.44i	0.0996981	−	0.172682i
$552$		0			0
$553$	6560.63	+	11363.3i	0.504496	+	0.873813i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18833.3			1.43266			0.716330	−	0.697762i	$-0.245821\pi$
$557$	18833.3			1.43266			0.716330	+	0.697762i	$0.245821\pi$
$558$		0			0
$559$	−2960.95			−0.224033
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10483.5	−	18158.0i	−0.784772	−	1.35927i	−0.929135	−	0.369741i	$-0.879447\pi$
$563$	−10483.5	−	18158.0i	−0.784772	−	1.35927i	0.144363	−	0.989525i	$-0.453887\pi$
$564$		0			0
$565$	−10866.0	+	18820.4i	−0.809090	+	1.40138i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6744.33	−	11681.5i	0.496902	−	0.860659i	−0.503092	−	0.864233i	$-0.667804\pi$
$569$	6744.33	−	11681.5i	0.496902	−	0.860659i	0.999994	+	0.00357372i	$0.00113755\pi$
$570$		0			0
$571$	1124.45	+	1947.60i	0.0824110	+	0.142740i	0.904285	−	0.426929i	$-0.140405\pi$
$571$	1124.45	+	1947.60i	0.0824110	+	0.142740i	−0.821874	+	0.569669i	$0.807071\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−36.9030			−0.00267645
$576$		0			0
$577$	−1968.87			−0.142054			−0.0710270	−	0.997474i	$-0.522628\pi$
$577$	−1968.87			−0.142054			−0.0710270	+	0.997474i	$0.522628\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	758.428	+	1313.64i	0.0541565	+	0.0938018i
$582$		0			0
$583$	−9545.53	+	16533.3i	−0.678105	+	1.17451i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2053.83	−	3557.34i	0.144413	−	0.250131i	−0.784741	−	0.619824i	$-0.787204\pi$
$587$	2053.83	−	3557.34i	0.144413	−	0.250131i	0.929154	+	0.369693i	$0.120537\pi$
$588$		0			0
$589$	8134.91	+	14090.1i	0.569088	+	0.985690i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7497.21			0.519180			0.259590	−	0.965719i	$-0.416413\pi$
$593$	7497.21			0.519180			0.259590	+	0.965719i	$0.416413\pi$
$594$		0			0
$595$	36584.3			2.52069
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13553.6	−	23475.4i	−0.924513	−	1.60130i	−0.792343	−	0.610076i	$-0.791139\pi$
$599$	−13553.6	−	23475.4i	−0.924513	−	1.60130i	−0.132170	−	0.991227i	$-0.542195\pi$
$600$		0			0
$601$	−3401.08	+	5890.85i	−0.230837	+	0.399822i	−0.958055	−	0.286585i	$-0.907480\pi$
$601$	−3401.08	+	5890.85i	−0.230837	+	0.399822i	0.727218	+	0.686407i	$0.240813\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17455.5	+	30233.7i	−1.17300	+	2.03170i
$606$		0			0
$607$	−8496.22	−	14715.9i	−0.568123	−	0.984019i	−0.996752	−	0.0805376i	$-0.974336\pi$
$607$	−8496.22	−	14715.9i	−0.568123	−	0.984019i	0.428628	−	0.903481i	$-0.358997\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2094.55			−0.138685
$612$		0			0
$613$	−13766.6			−0.907062			−0.453531	−	0.891241i	$-0.649836\pi$
$613$	−13766.6			−0.907062			−0.453531	+	0.891241i	$0.649836\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8678.00	−	15030.7i	−0.566229	−	0.980737i	−0.996934	−	0.0782443i	$-0.975069\pi$
$617$	−8678.00	−	15030.7i	−0.566229	−	0.980737i	0.430706	−	0.902493i	$-0.358265\pi$
$618$		0			0
$619$	−1462.45	+	2533.03i	−0.0949608	+	0.164477i	−0.909592	−	0.415502i	$-0.863606\pi$
$619$	−1462.45	+	2533.03i	−0.0949608	+	0.164477i	0.814631	+	0.579979i	$0.196939\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10736.0	+	18595.3i	−0.690416	+	1.19583i
$624$		0			0
$625$	8131.67	+	14084.5i	0.520427	+	0.901406i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3956.78			−0.250822
$630$		0			0
$631$	6158.40			0.388530			0.194265	−	0.980949i	$-0.437768\pi$
$631$	6158.40			0.388530			0.194265	+	0.980949i	$0.437768\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1314.65	−	2277.05i	−0.0821582	−	0.142302i
$636$		0			0
$637$	−10894.2	+	18869.3i	−0.677621	+	1.17367i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5907.12	−	10231.4i	0.363990	−	0.630448i	−0.624624	−	0.780926i	$-0.714748\pi$
$641$	5907.12	−	10231.4i	0.363990	−	0.630448i	0.988613	+	0.150477i	$0.0480811\pi$
$642$		0			0
$643$	8457.37	+	14648.6i	0.518703	+	0.898420i	0.999764	+	0.0217330i	$0.00691836\pi$
$643$	8457.37	+	14648.6i	0.518703	+	0.898420i	−0.481061	+	0.876687i	$0.659748\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	28652.4			1.74103			0.870513	−	0.492145i	$-0.163787\pi$
$647$	28652.4			1.74103			0.870513	+	0.492145i	$0.163787\pi$
$648$		0			0
$649$	−51922.5			−3.14042
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2749.24	−	4761.82i	−0.164757	−	0.285367i	0.771812	−	0.635851i	$-0.219351\pi$
$653$	−2749.24	−	4761.82i	−0.164757	−	0.285367i	−0.936569	+	0.350484i	$0.886017\pi$
$654$		0			0
$655$	4534.08	−	7853.26i	0.270475	−	0.468477i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9392.57	+	16268.4i	−0.555209	+	0.961650i	0.442679	+	0.896680i	$0.354028\pi$
$659$	−9392.57	+	16268.4i	−0.555209	+	0.961650i	−0.997887	+	0.0649692i	$0.979305\pi$
$660$		0			0
$661$	8652.23	+	14986.1i	0.509127	+	0.881834i	0.999944	+	0.0105712i	$0.00336497\pi$
$661$	8652.23	+	14986.1i	0.509127	+	0.881834i	−0.490817	+	0.871263i	$0.663302\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−23953.7			−1.39682
$666$		0			0
$667$	253.673			0.0147260
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14546.9	+	25196.0i	0.836927	+	1.44960i
$672$		0			0
$673$	6855.73	−	11874.5i	0.392673	−	0.680129i	−0.600128	−	0.799904i	$-0.704884\pi$
$673$	6855.73	−	11874.5i	0.392673	−	0.680129i	0.992801	+	0.119774i	$0.0382172\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15283.4	+	26471.7i	−0.867637	+	1.50279i	−0.00323298	+	0.999995i	$0.501029\pi$
$677$	−15283.4	+	26471.7i	−0.867637	+	1.50279i	−0.864404	+	0.502797i	$0.832304\pi$
$678$		0			0
$679$	−17894.4	−	30994.0i	−1.01137	−	1.75175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4445.72			−0.249064			−0.124532	−	0.992216i	$-0.539743\pi$
$683$	−4445.72			−0.249064			−0.124532	+	0.992216i	$0.539743\pi$
$684$		0			0
$685$	7798.27			0.434973
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5739.32	+	9940.79i	0.317345	+	0.549657i
$690$		0			0
$691$	6099.36	−	10564.4i	0.335790	−	0.581605i	−0.647846	−	0.761771i	$-0.724330\pi$
$691$	6099.36	−	10564.4i	0.335790	−	0.581605i	0.983636	+	0.180166i	$0.0576635\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8634.46	−	14955.3i	0.471257	−	0.816242i
$696$		0			0
$697$	−23058.3	−	39938.1i	−1.25308	−	2.17039i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21613.4			−1.16452			−0.582258	−	0.813004i	$-0.697831\pi$
$701$	−21613.4			−1.16452			−0.582258	+	0.813004i	$0.697831\pi$
$702$		0			0
$703$	2590.72			0.138991
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3669.16	−	6355.17i	−0.195181	−	0.338063i
$708$		0			0
$709$	2462.43	−	4265.05i	0.130435	−	0.225920i	−0.793409	−	0.608688i	$-0.791696\pi$
$709$	2462.43	−	4265.05i	0.130435	−	0.225920i	0.923844	+	0.382768i	$0.125029\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−800.170	+	1385.94i	−0.0420289	+	0.0727962i
$714$		0			0
$715$	15063.3	+	26090.5i	0.787884	+	1.36465i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	205.354			0.0106515			0.00532573	−	0.999986i	$-0.498305\pi$
$719$	205.354			0.0106515			0.00532573	+	0.999986i	$0.498305\pi$
$720$		0			0
$721$	−17178.3			−0.887316
$722$		0			0
$723$		0			0
$724$		0			0
$725$	97.8023	+	169.399i	0.00501005	+	0.00867766i
$726$		0			0
$727$	−7898.02	+	13679.8i	−0.402918	+	0.697875i	−0.994077	−	0.108681i	$-0.965337\pi$
$727$	−7898.02	+	13679.8i	−0.402918	+	0.697875i	0.591159	+	0.806555i	$0.298671\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3992.37	+	6914.98i	−0.202001	+	0.349877i
$732$		0			0
$733$	−16062.7	−	27821.4i	−0.809398	−	1.40192i	−0.913282	−	0.407328i	$-0.866460\pi$
$733$	−16062.7	−	27821.4i	−0.809398	−	1.40192i	0.103884	−	0.994589i	$-0.466873\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14464.7			0.722949
$738$		0			0
$739$	28938.9			1.44051			0.720255	−	0.693710i	$-0.244025\pi$
$739$	28938.9			1.44051			0.720255	+	0.693710i	$0.244025\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18054.2	+	31270.8i	0.891447	+	1.54403i	0.838141	+	0.545454i	$0.183643\pi$
$743$	18054.2	+	31270.8i	0.891447	+	1.54403i	0.0533063	+	0.998578i	$0.483024\pi$
$744$		0			0
$745$	13169.6	−	22810.4i	0.647647	−	1.12176i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15153.4	−	26246.5i	0.739245	−	1.28041i
$750$		0			0
$751$	11615.8	+	20119.2i	0.564405	+	0.977578i	0.997105	+	0.0760399i	$0.0242277\pi$
$751$	11615.8	+	20119.2i	0.564405	+	0.977578i	−0.432700	+	0.901538i	$0.642439\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	28805.5			1.38853
$756$		0			0
$757$	22762.6			1.09289			0.546447	−	0.837494i	$-0.315980\pi$
$757$	22762.6			1.09289			0.546447	+	0.837494i	$0.315980\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20259.3	−	35090.1i	−0.965044	−	1.67150i	−0.709497	−	0.704708i	$-0.751078\pi$
$761$	−20259.3	−	35090.1i	−0.965044	−	1.67150i	−0.255547	−	0.966797i	$-0.582255\pi$
$762$		0			0
$763$	21951.6	−	38021.3i	1.04155	−	1.80401i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15609.4	+	27036.2i	−0.734840	+	1.27278i
$768$		0			0
$769$	13700.6	+	23730.1i	0.642464	+	1.11278i	0.984881	+	0.173232i	$0.0554212\pi$
$769$	13700.6	+	23730.1i	0.642464	+	1.11278i	−0.342417	+	0.939548i	$0.611246\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	35698.4			1.66104			0.830520	−	0.556989i	$-0.188043\pi$
$773$	35698.4			1.66104			0.830520	+	0.556989i	$0.188043\pi$
$774$		0			0
$775$	−1234.01			−0.0571959
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15097.5	+	26149.6i	0.694382	+	1.20270i
$780$		0			0
$781$	−26184.7	+	45353.3i	−1.19970	+	2.07794i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16662.2	−	28859.7i	0.757577	−	1.31216i
$786$		0			0
$787$	4783.18	+	8284.71i	0.216648	+	0.375245i	0.953781	−	0.300502i	$-0.0971543\pi$
$787$	4783.18	+	8284.71i	0.216648	+	0.375245i	−0.737133	+	0.675748i	$0.763821\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−56789.0			−2.55270
$792$		0			0
$793$	17492.9			0.783343
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2653.01	−	4595.14i	−0.117910	−	0.204226i	0.801029	−	0.598625i	$-0.204286\pi$
$797$	−2653.01	−	4595.14i	−0.117910	−	0.204226i	−0.918939	+	0.394399i	$0.870953\pi$
$798$		0			0
$799$	−2824.17	+	4891.61i	−0.125046	+	0.216586i
$800$		0			0
$801$

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.70820 - 9.88690i) q^{5} +(14.9164 - 25.8360i) q^{7} +O(q^{10})\)	\(q+(-5.70820 - 9.88690i) q^{5} +(14.9164 - 25.8360i) q^{7} +(33.1246 - 57.3735i) q^{11} +(-19.9164 - 34.4962i) q^{13} -107.416 q^{17} +70.3313 q^{19} +(3.45898 + 5.99113i) q^{23} +(-2.66718 + 4.61970i) q^{25} +(18.3344 - 31.7561i) q^{29} +(115.666 + 200.339i) q^{31} -340.584 q^{35} +36.8359 q^{37} +(214.663 + 371.806i) q^{41} +(37.1672 - 64.3755i) q^{43} +(26.2918 - 45.5387i) q^{47} +(-273.498 - 473.713i) q^{49} -288.170 q^{53} -756.328 q^{55} +(-391.872 - 678.743i) q^{59} +(-219.579 + 380.322i) q^{61} +(-227.374 + 393.823i) q^{65} +(109.169 + 189.086i) q^{67} -790.492 q^{71} +1098.00 q^{73} +(-988.200 - 1711.61i) q^{77} +(-219.913 + 380.901i) q^{79} +(-25.4226 + 44.0333i) q^{83} +(613.155 + 1062.02i) q^{85} -719.745 q^{89} -1188.33 q^{91} +(-401.465 - 695.358i) q^{95} +(599.822 - 1038.92i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q + 4q^{5} + 6q^{7} + O(q^{10}) \)	\( 4q + 4q^{5} + 6q^{7} + 52q^{11} - 26q^{13} - 376q^{17} - 148q^{19} + 148q^{23} - 118q^{25} + 288q^{29} + 248q^{31} - 1416q^{35} + 684q^{37} + 256q^{43} + 132q^{47} - 772q^{49} - 1904q^{53} - 1952q^{55} - 1004q^{59} + 34q^{61} - 668q^{65} + 866q^{67} - 1552q^{71} + 3748q^{73} - 2316q^{77} - 182q^{79} - 1336q^{83} - 16q^{85} - 1752q^{89} - 3036q^{91} - 3028q^{95} + 38q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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