LMFDB - Embedded newform 690.3.b.a.599.2

Newspace parameters

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Character	$\chi$	$=$	690.599
Dual form			690.3.b.a.599.1

$q$-expansion

$f(q)$	$=$	$q-1.41421 q^{2} +(-2.99999 + 0.00596006i) q^{3} +2.00000 q^{4} +(1.21044 - 4.85127i) q^{5} +(4.24263 - 0.00842880i) q^{6} +6.90074i q^{7} -2.82843 q^{8} +(8.99993 - 0.0357603i) q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +(-2.99999 + 0.00596006i) q^{3} +2.00000 q^{4} +(1.21044 - 4.85127i) q^{5} +(4.24263 - 0.00842880i) q^{6} +6.90074i q^{7} -2.82843 q^{8} +(8.99993 - 0.0357603i) q^{9} +(-1.71182 + 6.86073i) q^{10} -1.93119i q^{11} +(-5.99999 + 0.0119201i) q^{12} +4.38117i q^{13} -9.75913i q^{14} +(-3.60241 + 14.5610i) q^{15} +4.00000 q^{16} +6.60993 q^{17} +(-12.7278 + 0.0505727i) q^{18} -16.9559 q^{19} +(2.42089 - 9.70254i) q^{20} +(-0.0411289 - 20.7022i) q^{21} +2.73112i q^{22} -4.79583 q^{23} +(8.48526 - 0.0168576i) q^{24} +(-22.0697 - 11.7444i) q^{25} -6.19592i q^{26} +(-26.9995 + 0.160921i) q^{27} +13.8015i q^{28} +5.45358i q^{29} +(5.09457 - 20.5924i) q^{30} +19.1195 q^{31} -5.65685 q^{32} +(0.0115100 + 5.79356i) q^{33} -9.34785 q^{34} +(33.4774 + 8.35296i) q^{35} +(17.9999 - 0.0715206i) q^{36} -4.29657i q^{37} +23.9793 q^{38} +(-0.0261121 - 13.1435i) q^{39} +(-3.42365 + 13.7215i) q^{40} -32.7285i q^{41} +(0.0581650 + 29.2773i) q^{42} +29.3732i q^{43} -3.86238i q^{44} +(10.7204 - 43.7044i) q^{45} +6.78233 q^{46} -23.8924 q^{47} +(-12.0000 + 0.0238402i) q^{48} +1.37973 q^{49} +(31.2112 + 16.6091i) q^{50} +(-19.8297 + 0.0393956i) q^{51} +8.76235i q^{52} -28.4218 q^{53} +(38.1831 - 0.227576i) q^{54} +(-9.36873 - 2.33760i) q^{55} -19.5183i q^{56} +(50.8677 - 0.101058i) q^{57} -7.71253i q^{58} +82.3938i q^{59} +(-7.20482 + 29.1220i) q^{60} +64.8349 q^{61} -27.0391 q^{62} +(0.246773 + 62.1062i) q^{63} +8.00000 q^{64} +(21.2543 + 5.30316i) q^{65} +(-0.0162776 - 8.19333i) q^{66} +62.3462i q^{67} +13.2199 q^{68} +(14.3875 - 0.0285834i) q^{69} +(-47.3442 - 11.8129i) q^{70} +130.312i q^{71} +(-25.4556 + 0.101145i) q^{72} +10.5750i q^{73} +6.07626i q^{74} +(66.2788 + 35.1015i) q^{75} -33.9119 q^{76} +13.3267 q^{77} +(0.0369280 + 18.5877i) q^{78} +72.0807 q^{79} +(4.84177 - 19.4051i) q^{80} +(80.9974 - 0.643680i) q^{81} +46.2851i q^{82} -93.0145 q^{83} +(-0.0822577 - 41.4044i) q^{84} +(8.00094 - 32.0665i) q^{85} -41.5400i q^{86} +(-0.0325037 - 16.3607i) q^{87} +5.46223i q^{88} +164.282i q^{89} +(-15.1610 + 61.8073i) q^{90} -30.2334 q^{91} -9.59166 q^{92} +(-57.3585 + 0.113954i) q^{93} +33.7889 q^{94} +(-20.5242 + 82.2578i) q^{95} +(16.9705 - 0.0337152i) q^{96} +69.5724i q^{97} -1.95123 q^{98} +(-0.0690599 - 17.3806i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88q + 176q^{4} + 8q^{9} + O(q^{10}) $	$ 88q + 176q^{4} + 8q^{9} + 32q^{10} - 12q^{15} + 352q^{16} - 16q^{19} - 176q^{21} + 72q^{25} - 72q^{30} + 32q^{31} + 160q^{34} + 16q^{36} + 144q^{39} + 64q^{40} + 92q^{45} - 360q^{49} + 48q^{51} - 144q^{54} + 16q^{55} - 24q^{60} + 208q^{61} + 704q^{64} + 512q^{66} + 304q^{70} + 536q^{75} - 32q^{76} + 448q^{79} - 24q^{81} - 352q^{84} - 96q^{85} + 32q^{90} - 64q^{91} + 160q^{94} + 296q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$	−2.99999	+	0.00596006i	−0.999998	+	0.00198669i
$3$	−2.99999	+	0.00596006i	−0.999998	+	0.00198669i
$4$	2.00000			0.500000
$5$	1.21044	−	4.85127i	0.242089	−	0.970254i
$5$	1.21044	−	4.85127i	0.242089	−	0.970254i
$6$	4.24263	−	0.00842880i	0.707105	−	0.00140480i
$7$			6.90074i			0.985821i	0.870080	+	0.492910i	$0.164067\pi$
$7$			6.90074i			0.985821i	−0.870080	+	0.492910i	$0.835933\pi$
$8$	−2.82843			−0.353553
$9$	8.99993	−	0.0357603i	0.999992	−	0.00397337i
$10$	−1.71182	+	6.86073i	−0.171182	+	0.686073i
$11$		−	1.93119i		−	0.175563i	−0.996140	−	0.0877814i	$-0.972022\pi$
$11$		−	1.93119i		−	0.175563i	0.996140	−	0.0877814i	$-0.0279777\pi$
$12$	−5.99999	+	0.0119201i	−0.499999	+	0.000993343i
$13$			4.38117i			0.337013i	0.985701	+	0.168507i	$0.0538945\pi$
$13$			4.38117i			0.337013i	−0.985701	+	0.168507i	$0.946106\pi$
$14$		−	9.75913i		−	0.697080i
$15$	−3.60241	+	14.5610i	−0.240161	+	0.970733i
$16$	4.00000			0.250000
$17$	6.60993			0.388819			0.194410	−	0.980920i	$-0.437721\pi$
$17$	6.60993			0.388819			0.194410	+	0.980920i	$0.437721\pi$
$18$	−12.7278	+	0.0505727i	−0.707101	+	0.00280959i
$19$	−16.9559			−0.892417			−0.446209	−	0.894929i	$-0.647226\pi$
$19$	−16.9559			−0.892417			−0.446209	+	0.894929i	$0.647226\pi$
$20$	2.42089	−	9.70254i	0.121044	−	0.485127i
$21$	−0.0411289	−	20.7022i	−0.00195852	−	0.985819i
$22$			2.73112i			0.124142i
$23$	−4.79583			−0.208514
$23$	−4.79583			−0.208514
$24$	8.48526	−	0.0168576i	0.353553	−	0.000702400i
$25$	−22.0697	−	11.7444i	−0.882786	−	0.469775i
$26$		−	6.19592i		−	0.238304i
$27$	−26.9995	+	0.160921i	−0.999982	+	0.00596003i
$28$			13.8015i			0.492910i
$29$			5.45358i			0.188055i	0.995570	+	0.0940273i	$0.0299741\pi$
$29$			5.45358i			0.188055i	−0.995570	+	0.0940273i	$0.970026\pi$
$30$	5.09457	−	20.5924i	0.169819	−	0.686412i
$31$	19.1195			0.616759			0.308380	−	0.951263i	$-0.400213\pi$
$31$	19.1195			0.616759			0.308380	+	0.951263i	$0.400213\pi$
$32$	−5.65685			−0.176777
$33$	0.0115100	+	5.79356i	0.000348788	+	0.175562i
$34$	−9.34785			−0.274937
$35$	33.4774	+	8.35296i	0.956497	+	0.238656i
$36$	17.9999	−	0.0715206i	0.499996	−	0.00198668i
$37$		−	4.29657i		−	0.116123i	−0.998313	−	0.0580617i	$-0.981508\pi$
$37$		−	4.29657i		−	0.116123i	0.998313	−	0.0580617i	$-0.0184920\pi$
$38$	23.9793			0.631034
$39$	−0.0261121	−	13.1435i	−0.000669540	−	0.337013i
$40$	−3.42365	+	13.7215i	−0.0855912	+	0.343037i
$41$		−	32.7285i		−	0.798256i	−0.916895	−	0.399128i	$-0.869313\pi$
$41$		−	32.7285i		−	0.798256i	0.916895	−	0.399128i	$-0.130687\pi$
$42$	0.0581650	+	29.2773i	0.00138488	+	0.697079i
$43$			29.3732i			0.683098i	0.939864	+	0.341549i	$0.110951\pi$
$43$			29.3732i			0.683098i	−0.939864	+	0.341549i	$0.889049\pi$
$44$		−	3.86238i		−	0.0877814i
$45$	10.7204	−	43.7044i	0.238231	−	0.971208i
$46$	6.78233			0.147442
$47$	−23.8924			−0.508348			−0.254174	−	0.967158i	$-0.581804\pi$
$47$	−23.8924			−0.508348			−0.254174	+	0.967158i	$0.581804\pi$
$48$	−12.0000	+	0.0238402i	−0.250000	+	0.000496672i
$49$	1.37973			0.0281577
$50$	31.2112	+	16.6091i	0.624224	+	0.332181i
$51$	−19.8297	+	0.0393956i	−0.388818	+	0.000772462i
$52$			8.76235i			0.168507i
$53$	−28.4218			−0.536260			−0.268130	−	0.963383i	$-0.586406\pi$
$53$	−28.4218			−0.536260			−0.268130	+	0.963383i	$0.586406\pi$
$54$	38.1831	−	0.227576i	0.707094	−	0.00421438i
$55$	−9.36873	−	2.33760i	−0.170340	−	0.0425017i
$56$		−	19.5183i		−	0.348540i
$57$	50.8677	−	0.101058i	0.892416	−	0.00177295i
$58$		−	7.71253i		−	0.132975i
$59$			82.3938i			1.39650i	0.715852	+	0.698252i	$0.246039\pi$
$59$			82.3938i			1.39650i	−0.715852	+	0.698252i	$0.753961\pi$
$60$	−7.20482	+	29.1220i	−0.120080	+	0.485367i
$61$	64.8349			1.06287			0.531433	−	0.847100i	$-0.321654\pi$
$61$	64.8349			1.06287			0.531433	+	0.847100i	$0.321654\pi$
$62$	−27.0391			−0.436115
$63$	0.246773	+	62.1062i	0.00391703	+	0.985813i
$64$	8.00000			0.125000
$65$	21.2543	+	5.30316i	0.326989	+	0.0815871i
$66$	−0.0162776	−	8.19333i	−0.000246631	−	0.124141i
$67$			62.3462i			0.930540i	0.885169	+	0.465270i	$0.154043\pi$
$67$			62.3462i			0.930540i	−0.885169	+	0.465270i	$0.845957\pi$
$68$	13.2199			0.194410
$69$	14.3875	−	0.0285834i	0.208514	−	0.000414253i
$70$	−47.3442	−	11.8129i	−0.676345	−	0.168755i
$71$			130.312i			1.83538i	0.397300	+	0.917689i	$0.369947\pi$
$71$			130.312i			1.83538i	−0.397300	+	0.917689i	$0.630053\pi$
$72$	−25.4556	+	0.101145i	−0.353551	+	0.00140480i
$73$			10.5750i			0.144863i	0.997373	+	0.0724314i	$0.0230758\pi$
$73$			10.5750i			0.144863i	−0.997373	+	0.0724314i	$0.976924\pi$
$74$			6.07626i			0.0821117i
$75$	66.2788	+	35.1015i	0.883718	+	0.468020i
$76$	−33.9119			−0.446209
$77$	13.3267			0.173073
$78$	0.0369280	+	18.5877i	0.000473436	+	0.238304i
$79$	72.0807			0.912414			0.456207	−	0.889874i	$-0.349208\pi$
$79$	72.0807			0.912414			0.456207	+	0.889874i	$0.349208\pi$
$80$	4.84177	−	19.4051i	0.0605221	−	0.242564i
$81$	80.9974	−	0.643680i	0.999968	−	0.00794667i
$82$			46.2851i			0.564452i
$83$	−93.0145			−1.12066			−0.560328	−	0.828271i	$-0.689325\pi$
$83$	−93.0145			−1.12066			−0.560328	+	0.828271i	$0.689325\pi$
$84$	−0.0822577	−	41.4044i	−0.000979258	−	0.492909i
$85$	8.00094	−	32.0665i	0.0941287	−	0.377253i
$86$		−	41.5400i		−	0.483023i
$87$	−0.0325037	−	16.3607i	−0.000373606	−	0.188054i
$88$			5.46223i			0.0620708i
$89$			164.282i			1.84587i	0.384959	+	0.922934i	$0.374216\pi$
$89$			164.282i			1.84587i	−0.384959	+	0.922934i	$0.625784\pi$
$90$	−15.1610	+	61.8073i	−0.168455	+	0.686748i
$91$	−30.2334			−0.332235
$92$	−9.59166			−0.104257
$93$	−57.3585	+	0.113954i	−0.616758	+	0.00122531i
$94$	33.7889			0.359456
$95$	−20.5242	+	82.2578i	−0.216044	+	0.865872i
$96$	16.9705	−	0.0337152i	0.176776	−	0.000351200i
$97$			69.5724i			0.717242i	0.933483	+	0.358621i	$0.116753\pi$
$97$			69.5724i			0.717242i	−0.933483	+	0.358621i	$0.883247\pi$
$98$	−1.95123			−0.0199105
$99$	−0.0690599	−	17.3806i	−0.000697575	−	0.175561i
$100$	−44.1393	−	23.4887i	−0.441393	−	0.234887i
$101$			131.778i			1.30474i	0.757903	+	0.652368i	$0.226224\pi$
$101$			131.778i			1.30474i	−0.757903	+	0.652368i	$0.773776\pi$
$102$	28.0435	−	0.0557137i	0.274936	−	0.000546213i
$103$			36.5773i			0.355120i	0.984110	+	0.177560i	$0.0568204\pi$
$103$			36.5773i			0.355120i	−0.984110	+	0.177560i	$0.943180\pi$
$104$		−	12.3918i		−	0.119152i
$105$	−100.482	−	24.8593i	−0.956969	−	0.236755i
$106$	40.1944			0.379193
$107$	79.1711			0.739917			0.369958	−	0.929048i	$-0.379372\pi$
$107$	79.1711			0.739917			0.369958	+	0.929048i	$0.379372\pi$
$108$	−53.9990	+	0.321842i	−0.499991	+	0.00298001i
$109$	−87.8564			−0.806022			−0.403011	−	0.915195i	$-0.632036\pi$
$109$	−87.8564			−0.806022			−0.403011	+	0.915195i	$0.632036\pi$
$110$	13.2494	+	3.30586i	0.120449	+	0.0300533i
$111$	0.0256078	+	12.8897i	0.000230701	+	0.116123i
$112$			27.6030i			0.246455i
$113$	5.97436			0.0528704			0.0264352	−	0.999651i	$-0.491584\pi$
$113$	5.97436			0.0528704			0.0264352	+	0.999651i	$0.491584\pi$
$114$	−71.9378	+	0.142918i	−0.631033	+	0.00125367i
$115$	−5.80508	+	23.2659i	−0.0504790	+	0.202312i
$116$			10.9072i			0.0940273i
$117$	0.156672	+	39.4303i	0.00133908	+	0.337011i
$118$		−	116.522i		−	0.987478i
$119$			45.6134i			0.383306i
$120$	10.1891	−	41.1847i	0.0849096	−	0.343206i
$121$	117.271			0.969178
$122$	−91.6903			−0.751560
$123$	0.195064	+	98.1853i	0.00158589	+	0.798255i
$124$	38.2391			0.308380
$125$	−83.6892	+	92.8500i	−0.669514	+	0.742800i
$126$	−0.348989	−	87.8314i	−0.00276976	−	0.697075i
$127$			23.9163i			0.188317i	0.995557	+	0.0941586i	$0.0300161\pi$
$127$			23.9163i			0.188317i	−0.995557	+	0.0941586i	$0.969984\pi$
$128$	−11.3137			−0.0883883
$129$	−0.175066	−	88.1194i	−0.00135710	−	0.683096i
$130$	−30.0581	−	7.49980i	−0.231216	−	0.0576908i
$131$		−	4.82841i		−	0.0368581i	−0.999830	−	0.0184291i	$-0.994134\pi$
$131$		−	4.82841i		−	0.0368581i	0.999830	−	0.0184291i	$-0.00586648\pi$
$132$	0.0230200	+	11.5871i	0.000174394	+	0.0877812i
$133$		−	117.009i		−	0.879764i
$134$		−	88.1709i		−	0.657991i
$135$	−31.9007	+	131.177i	−0.236302	+	0.971680i
$136$	−18.6957			−0.137468
$137$	260.934			1.90463			0.952314	−	0.305119i	$-0.0986964\pi$
$137$	260.934			1.90463			0.952314	+	0.305119i	$0.0986964\pi$
$138$	−20.3469	+	0.0404231i	−0.147442	+	0.000292921i
$139$	−10.8481			−0.0780440			−0.0390220	−	0.999238i	$-0.512424\pi$
$139$	−10.8481			−0.0780440			−0.0390220	+	0.999238i	$0.512424\pi$
$140$	66.9548	+	16.7059i	0.478248	+	0.119328i
$141$	71.6769	−	0.142400i	0.508347	−	0.00100993i
$142$		−	184.289i		−	1.29781i
$143$	8.46088			0.0591670
$144$	35.9997	−	0.143041i	0.249998	−	0.000993342i
$145$	26.4568	+	6.60125i	0.182461	+	0.0455259i
$146$		−	14.9553i		−	0.102434i
$147$	−4.13917	+	0.00822326i	−0.0281576	+	5.59405e-5i
$148$		−	8.59314i		−	0.0580617i
$149$		−	88.4787i		−	0.593817i	−0.954906	−	0.296908i	$-0.904044\pi$
$149$		−	88.4787i		−	0.593817i	0.954906	−	0.296908i	$-0.0959556\pi$
$150$	−93.7324	−	49.6410i	−0.624883	−	0.330940i
$151$	−74.3683			−0.492505			−0.246253	−	0.969206i	$-0.579199\pi$
$151$	−74.3683			−0.492505			−0.246253	+	0.969206i	$0.579199\pi$
$152$	47.9586			0.315517
$153$	59.4889	−	0.236373i	0.388816	−	0.00154492i
$154$	−18.8467			−0.122381
$155$	23.1431	−	92.7541i	0.149310	−	0.598413i
$156$	−0.0522241	−	26.2870i	−0.000334770	−	0.168506i
$157$		−	160.285i		−	1.02093i	−0.859900	−	0.510463i	$-0.829474\pi$
$157$		−	160.285i		−	1.02093i	0.859900	−	0.510463i	$-0.170526\pi$
$158$	−101.937			−0.645174
$159$	85.2651	−	0.169395i	0.536259	−	0.00106538i
$160$	−6.84730	+	27.4429i	−0.0427956	+	0.171518i
$161$		−	33.0948i		−	0.205558i
$162$	−114.548	+	0.910301i	−0.707084	+	0.00561914i
$163$			147.791i			0.906694i	0.891334	+	0.453347i	$0.149770\pi$
$163$			147.791i			0.906694i	−0.891334	+	0.453347i	$0.850230\pi$
$164$		−	65.4570i		−	0.399128i
$165$	28.1201	+	6.95693i	0.170425	+	0.0421632i
$166$	131.542			0.792424
$167$	253.815			1.51985			0.759926	−	0.650010i	$-0.225235\pi$
$167$	253.815			1.51985			0.759926	+	0.650010i	$0.225235\pi$
$168$	0.116330	+	58.5546i	0.000692440	+	0.348540i
$169$	149.805			0.886422
$170$	−11.3150	+	45.3489i	−0.0665590	+	0.266758i
$171$	−152.602	+	0.606349i	−0.892410	+	0.00354590i
$172$			58.7464i			0.341549i
$173$	−141.013			−0.815104			−0.407552	−	0.913182i	$-0.633617\pi$
$173$	−141.013			−0.815104			−0.407552	+	0.913182i	$0.633617\pi$
$174$	0.0459672	+	23.1375i	0.000264179	+	0.132974i
$175$	81.0449	−	152.297i	0.463114	−	0.870269i
$176$		−	7.72476i		−	0.0438907i
$177$	−0.491072	−	247.181i	−0.00277442	−	1.39650i
$178$		−	232.330i		−	1.30523i
$179$		−	125.991i		−	0.703860i	−0.936026	−	0.351930i	$-0.885525\pi$
$179$		−	125.991i		−	0.703860i	0.936026	−	0.351930i	$-0.114475\pi$
$180$	21.4408	−	87.4088i	0.119116	−	0.485604i
$181$	−226.332			−1.25045			−0.625227	−	0.780443i	$-0.714994\pi$
$181$	−226.332			−1.25045			−0.625227	+	0.780443i	$0.714994\pi$
$182$	42.7564			0.234925
$183$	−194.504	+	0.386420i	−1.06286	+	0.00211158i
$184$	13.5647			0.0737210
$185$	−20.8438	−	5.20075i	−0.112669	−	0.0281122i
$186$	81.1172	−	0.161155i	0.436114	−	0.000866424i
$187$		−	12.7650i		−	0.0682622i
$188$	−47.7847			−0.254174
$189$	−1.11047	−	186.317i	−0.00587552	−	0.985803i
$190$	29.0256	−	116.330i	0.152766	−	0.612264i
$191$			161.079i			0.843345i	0.906748	+	0.421673i	$0.138557\pi$
$191$			161.079i			0.843345i	−0.906748	+	0.421673i	$0.861443\pi$
$192$	−24.0000	+	0.0476805i	−0.125000	+	0.000248336i
$193$		−	23.7745i		−	0.123184i	−0.998101	−	0.0615921i	$-0.980382\pi$
$193$		−	23.7745i		−	0.123184i	0.998101	−	0.0615921i	$-0.0196178\pi$
$194$		−	98.3903i		−	0.507166i
$195$	−63.7943	−	15.7828i	−0.327150	−	0.0809373i
$196$	2.75945			0.0140788
$197$	−132.922			−0.674730			−0.337365	−	0.941374i	$-0.609536\pi$
$197$	−132.922			−0.674730			−0.337365	+	0.941374i	$0.609536\pi$
$198$	0.0976655	+	24.5798i	0.000493260	+	0.124141i
$199$	148.011			0.743775			0.371887	−	0.928278i	$-0.378711\pi$
$199$	148.011			0.743775			0.371887	+	0.928278i	$0.378711\pi$
$200$	62.4224	+	33.2181i	0.312112	+	0.166091i
$201$	−0.371587	−	187.038i	−0.00184869	−	0.930539i
$202$		−	186.363i		−	0.922587i
$203$	−37.6338			−0.185388
$204$	−39.6595	+	0.0787911i	−0.194409	+	0.000386231i
$205$	−158.775	−	39.6160i	−0.774512	−	0.193249i
$206$		−	51.7282i		−	0.251108i
$207$	−43.1621	+	0.171500i	−0.208513	+	0.000828504i
$208$			17.5247i			0.0842533i
$209$			32.7451i			0.156675i
$210$	142.103	+	35.1564i	0.676679	+	0.167411i
$211$	−193.048			−0.914920			−0.457460	−	0.889230i	$-0.651241\pi$
$211$	−193.048			−0.914920			−0.457460	+	0.889230i	$0.651241\pi$
$212$	−56.8435			−0.268130
$213$	−0.776666	−	390.935i	−0.00364632	−	1.83537i
$214$	−111.965			−0.523200
$215$	142.497	+	35.5546i	0.662778	+	0.165370i
$216$	76.3662	−	0.455153i	0.353547	−	0.00210719i
$217$			131.939i			0.608014i
$218$	124.248			0.569944
$219$	−0.0630276	−	31.7249i	−0.000287797	−	0.144863i
$220$	−18.7375	−	4.67519i	−0.0851702	−	0.0212509i
$221$			28.9592i			0.131037i
$222$	−0.0362149	−	18.2288i	−0.000163130	−	0.0821115i
$223$		−	44.5616i		−	0.199828i	−0.994996	−	0.0999140i	$-0.968143\pi$
$223$		−	44.5616i		−	0.199828i	0.994996	−	0.0999140i	$-0.0318568\pi$
$224$		−	39.0365i		−	0.174270i
$225$	−199.045	−	104.909i	−0.884646	−	0.466264i
$226$	−8.44901			−0.0373850
$227$	186.388			0.821094			0.410547	−	0.911839i	$-0.365338\pi$
$227$	186.388			0.821094			0.410547	+	0.911839i	$0.365338\pi$
$228$	101.735	−	0.202117i	0.446208	−	0.000886477i
$229$	127.546			0.556970			0.278485	−	0.960441i	$-0.410168\pi$
$229$	127.546			0.556970			0.278485	+	0.960441i	$0.410168\pi$
$230$	8.20962	−	32.9029i	0.0356940	−	0.143056i
$231$	−39.9799	+	0.0794277i	−0.173073	+	0.000343843i
$232$		−	15.4251i		−	0.0664873i
$233$	−119.516			−0.512945			−0.256473	−	0.966551i	$-0.582560\pi$
$233$	−119.516			−0.512945			−0.256473	+	0.966551i	$0.582560\pi$
$234$	−0.221568	−	55.7628i	−0.000946871	−	0.238303i
$235$	−28.9203	+	115.908i	−0.123065	+	0.493227i
$236$			164.788i			0.698252i
$237$	−216.242	+	0.429605i	−0.912412	+	0.00181268i
$238$		−	64.5071i		−	0.271038i
$239$			404.639i			1.69305i	0.532348	+	0.846526i	$0.321310\pi$
$239$			404.639i			1.69305i	−0.532348	+	0.846526i	$0.678690\pi$
$240$	−14.4096	+	58.2440i	−0.0600401	+	0.242683i
$241$	−156.085			−0.647654			−0.323827	−	0.946116i	$-0.604970\pi$
$241$	−156.085			−0.647654			−0.323827	+	0.946116i	$0.604970\pi$
$242$	−165.846			−0.685312
$243$	−242.988	+	2.41379i	−0.999951	+	0.00993328i
$244$	129.670			0.531433
$245$	1.67008	−	6.69343i	0.00681666	−	0.0273201i
$246$	−0.275862	−	138.855i	−0.00112139	−	0.564451i
$247$		−	74.2869i		−	0.300757i
$248$	−54.0782			−0.218057
$249$	279.043	−	0.554372i	1.12065	−	0.00222639i
$250$	118.354	−	131.310i	0.473418	−	0.525239i
$251$			150.203i			0.598416i	0.954188	+	0.299208i	$0.0967225\pi$
$251$			150.203i			0.598416i	−0.954188	+	0.299208i	$0.903277\pi$
$252$	0.493545	+	124.212i	0.00195851	+	0.492906i
$253$			9.26166i			0.0366074i
$254$		−	33.8227i		−	0.133160i
$255$	−23.8117	+	96.2471i	−0.0933790	+	0.377440i
$256$	16.0000			0.0625000
$257$	−11.3432			−0.0441368			−0.0220684	−	0.999756i	$-0.507025\pi$
$257$	−11.3432			−0.0441368			−0.0220684	+	0.999756i	$0.507025\pi$
$258$	0.247581	+	124.620i	0.000959616	+	0.483022i
$259$	29.6495			0.114477
$260$	42.5085	+	10.6063i	0.163494	+	0.0407935i
$261$	0.195022	+	49.0819i	0.000747210	+	0.188053i
$262$			6.82841i			0.0260626i
$263$	−147.415			−0.560511			−0.280256	−	0.959925i	$-0.590419\pi$
$263$	−147.415			−0.560511			−0.280256	+	0.959925i	$0.590419\pi$
$264$	−0.0325552	−	16.3867i	−0.000123315	−	0.0620707i
$265$	−34.4029	+	137.882i	−0.129822	+	0.520308i
$266$			165.475i			0.622087i
$267$	−0.979132	−	492.846i	−0.00366716	−	1.84586i
$268$			124.692i			0.465270i
$269$			205.779i			0.764976i	0.923960	+	0.382488i	$0.124933\pi$
$269$			205.779i			0.764976i	−0.923960	+	0.382488i	$0.875067\pi$
$270$	45.1144	−	185.512i	0.167090	−	0.687081i
$271$	−239.892			−0.885209			−0.442605	−	0.896717i	$-0.645945\pi$
$271$	−239.892			−0.885209			−0.442605	+	0.896717i	$0.645945\pi$
$272$	26.4397			0.0972048
$273$	90.6999	−	0.180193i	0.332234	−	0.000660046i
$274$	−369.016			−1.34678
$275$	−22.6806	+	42.6207i	−0.0824750	+	0.154984i
$276$	28.7749	−	0.0571669i	0.104257	−	0.000207126i
$277$			165.615i			0.597890i	0.954270	+	0.298945i	$0.0966347\pi$
$277$			165.615i			0.597890i	−0.954270	+	0.298945i	$0.903365\pi$
$278$	15.3415			0.0551854
$279$	172.075	−	0.683721i	0.616755	−	0.00245061i
$280$	−94.6883	−	23.6257i	−0.338173	−	0.0843776i
$281$		−	215.556i		−	0.767104i	−0.923519	−	0.383552i	$-0.874701\pi$
$281$		−	215.556i		−	0.767104i	0.923519	−	0.383552i	$-0.125299\pi$
$282$	−101.366	+	0.201384i	−0.359456	+	0.000714127i
$283$		−	261.575i		−	0.924293i	−0.886804	−	0.462147i	$-0.847079\pi$
$283$		−	261.575i		−	0.924293i	0.886804	−	0.462147i	$-0.152921\pi$
$284$			260.624i			0.917689i
$285$	61.0822	−	246.895i	0.214323	−	0.866299i
$286$	−11.9655			−0.0418374
$287$	225.851			0.786938
$288$	−50.9113	+	0.202291i	−0.176775	+	0.000702399i
$289$	−245.309			−0.848820
$290$	−37.4156	−	9.33558i	−0.129019	−	0.0321917i
$291$	−0.414656	−	208.717i	−0.00142493	−	0.717240i
$292$			21.1500i			0.0724314i
$293$	−529.882			−1.80847			−0.904235	−	0.427036i	$-0.859558\pi$
$293$	−529.882			−1.80847			−0.904235	+	0.427036i	$0.859558\pi$
$294$	5.85367	−	0.0116294i	0.0199105	−	3.95559e-5i
$295$	399.715	+	99.7330i	1.35496	+	0.338078i
$296$			12.1525i			0.0410558i
$297$	0.310769	+	52.1412i	0.00104636	+	0.175560i
$298$			125.128i			0.419892i
$299$		−	21.0114i		−	0.0702721i
$300$	132.558	+	70.2030i	0.441859	+	0.234010i
$301$	−202.697			−0.673412
$302$	105.173			0.348254
$303$	−0.785407	−	395.334i	−0.00259210	−	1.30473i
$304$	−67.8237			−0.223104
$305$	78.4789	−	314.531i	0.257308	−	1.03125i
$306$	−84.1300	+	0.334282i	−0.274935	+	0.00109242i
$307$			122.092i			0.397693i	0.980031	+	0.198847i	$0.0637195\pi$
$307$			122.092i			0.397693i	−0.980031	+	0.198847i	$0.936280\pi$
$308$	26.6533			0.0865367
$309$	−0.218003	−	109.732i	−0.000705512	−	0.355119i
$310$	−32.7293	+	131.174i	−0.105578	+	0.423142i
$311$			50.4037i			0.162070i	0.996711	+	0.0810348i	$0.0258225\pi$
$311$			50.4037i			0.162070i	−0.996711	+	0.0810348i	$0.974178\pi$
$312$	0.0738561	+	37.1754i	0.000236718	+	0.119152i
$313$		−	491.711i		−	1.57096i	−0.618887	−	0.785480i	$-0.712416\pi$
$313$		−	491.711i		−	1.57096i	0.618887	−	0.785480i	$-0.287584\pi$
$314$			226.678i			0.721904i
$315$	301.593	+	73.9789i	0.957437	+	0.234854i
$316$	144.161			0.456207
$317$	534.614			1.68648			0.843240	−	0.537538i	$-0.180645\pi$
$317$	534.614			1.68648			0.843240	+	0.537538i	$0.180645\pi$
$318$	−120.583	+	0.239561i	−0.379192	+	0.000753337i
$319$	10.5319			0.0330154
$320$	9.68354	−	38.8102i	0.0302611	−	0.121282i
$321$	−237.513	+	0.471865i	−0.739915	+	0.00146998i
$322$			46.8031i			0.145351i
$323$	−112.077			−0.346989
$324$	161.995	−	1.28736i	0.499984	−	0.00397333i
$325$	51.4541	−	96.6910i	0.158320	−	0.297511i
$326$		−	209.008i		−	0.641129i
$327$	263.569	−	0.523630i	0.806021	−	0.00160131i
$328$			92.5702i			0.282226i
$329$		−	164.875i		−	0.501140i
$330$	−39.7678	−	9.83859i	−0.120508	−	0.0298139i
$331$	360.609			1.08945			0.544727	−	0.838614i	$-0.316633\pi$
$331$	360.609			1.08945			0.544727	+	0.838614i	$0.316633\pi$
$332$	−186.029			−0.560328
$333$	−0.153647	−	38.6688i	−0.000461401	−	0.116123i
$334$	−358.949			−1.07470
$335$	302.458	+	75.4665i	0.902861	+	0.225273i
$336$	−0.164515	−	82.8088i	−0.000489629	−	0.246455i
$337$			518.017i			1.53714i	0.639765	+	0.768571i	$0.279032\pi$
$337$			518.017i			1.53714i	−0.639765	+	0.768571i	$0.720968\pi$
$338$	−211.857			−0.626795
$339$	−17.9230	+	0.0356075i	−0.0528703	+	0.000105037i
$340$	16.0019	−	64.1331i	0.0470643	−	0.188627i
$341$		−	36.9235i		−	0.108280i
$342$	215.812	−	0.857507i	0.631029	−	0.00250733i
$343$			347.658i			1.01358i
$344$		−	83.0800i		−	0.241512i
$345$	17.2765	−	69.8321i	0.0500769	−	0.202412i
$346$	199.422			0.576365
$347$	341.360			0.983746			0.491873	−	0.870667i	$-0.336313\pi$
$347$	341.360			0.983746			0.491873	+	0.870667i	$0.336313\pi$
$348$	−0.0650074	−	32.7214i	−0.000186803	−	0.0940271i
$349$	−0.400677			−0.00114807			−0.000574036	−	1.00000i	$-0.500183\pi$
$349$	−0.400677			−0.00114807			−0.000574036		1.00000i	$0.500183\pi$
$350$	−114.615	+	215.381i	−0.327471	+	0.615373i
$351$	−0.705022	−	118.290i	−0.00200861	−	0.337007i
$352$			10.9245i			0.0310354i
$353$	−271.119			−0.768042			−0.384021	−	0.923324i	$-0.625461\pi$
$353$	−271.119			−0.768042			−0.384021	+	0.923324i	$0.625461\pi$
$354$	0.694481	+	349.567i	0.00196181	+	0.987476i
$355$	632.178	+	157.735i	1.78078	+	0.444324i
$356$			328.564i			0.922934i
$357$	−0.271859	−	136.840i	−0.000761509	−	0.383305i
$358$			178.178i			0.497704i
$359$			348.233i			0.970008i	0.874512	+	0.485004i	$0.161182\pi$
$359$			348.233i			0.970008i	−0.874512	+	0.485004i	$0.838818\pi$
$360$	−30.3219	+	123.615i	−0.0842276	+	0.343374i
$361$	−73.4964			−0.203591
$362$	320.082			0.884205
$363$	−351.811	+	0.698939i	−0.969176	+	0.00192545i
$364$	−60.4667			−0.166117
$365$	51.3021	+	12.8004i	0.140554	+	0.0350696i
$366$	275.070	−	0.546480i	0.751559	−	0.00149311i
$367$		−	546.201i		−	1.48828i	−0.668021	−	0.744142i	$-0.732858\pi$
$367$		−	546.201i		−	1.48828i	0.668021	−	0.744142i	$-0.267142\pi$
$368$	−19.1833			−0.0521286
$369$	−1.17038	−	294.554i	−0.00317176	−	0.798250i
$370$	29.4776	+	7.35497i	0.0796692	+	0.0198783i
$371$		−	196.131i		−	0.528656i
$372$	−114.717	+	0.227907i	−0.308379	+	0.000612654i
$373$			561.225i			1.50463i	0.658806	+	0.752313i	$0.271062\pi$
$373$			561.225i			1.50463i	−0.658806	+	0.752313i	$0.728938\pi$
$374$			18.0525i			0.0482686i
$375$	250.514	−	279.048i	0.668036	−	0.744129i
$376$	67.5778			0.179728
$377$	−23.8931			−0.0633769
$378$	1.57045	+	263.492i	0.00415462	+	0.697068i
$379$	−64.2274			−0.169465			−0.0847327	−	0.996404i	$-0.527004\pi$
$379$	−64.2274			−0.169465			−0.0847327	+	0.996404i	$0.527004\pi$
$380$	−41.0484	+	164.516i	−0.108022	+	0.432936i
$381$	−0.142542	−	71.7487i	−0.000374127	−	0.188317i
$382$		−	227.800i		−	0.596335i
$383$	54.9496			0.143471			0.0717357	−	0.997424i	$-0.477146\pi$
$383$	54.9496			0.143471			0.0717357	+	0.997424i	$0.477146\pi$
$384$	33.9411	−	0.0674304i	0.0883882	−	0.000175600i
$385$	16.1311	−	64.6512i	0.0418991	−	0.167925i
$386$			33.6223i			0.0871044i
$387$	1.05039	+	264.357i	0.00271420	+	0.683092i
$388$			139.145i			0.358621i
$389$		−	333.955i		−	0.858497i	−0.903186	−	0.429248i	$-0.858778\pi$
$389$		−	333.955i		−	0.858497i	0.903186	−	0.429248i	$-0.141222\pi$
$390$	90.2187	+	22.3202i	0.231330	+	0.0572313i
$391$	−31.7001			−0.0810744
$392$	−3.90246			−0.00995525
$393$	0.0287776	+	14.4852i	7.32255e−5	+	0.0368580i
$394$	187.980			0.477106
$395$	87.2495	−	349.683i	0.220885	−	0.885273i
$396$	−0.138120	−	34.7612i	−0.000348788	−	0.0877807i
$397$		−	188.644i		−	0.475175i	−0.971366	−	0.237587i	$-0.923643\pi$
$397$		−	188.644i		−	0.475175i	0.971366	−	0.237587i	$-0.0763566\pi$
$398$	−209.319			−0.525928
$399$	0.697378	+	351.025i	0.00174781	+	0.879762i
$400$	−88.2786	−	46.9775i	−0.220697	−	0.117444i
$401$			200.819i			0.500796i	0.968143	+	0.250398i	$0.0805616\pi$
$401$			200.819i			0.500796i	−0.968143	+	0.250398i	$0.919438\pi$
$402$	0.525504	+	264.512i	0.00130722	+	0.657990i
$403$			83.7660i			0.207856i
$404$			263.557i			0.652368i
$405$	94.9201	−	393.720i	0.234371	−	0.972147i
$406$	53.2222			0.131089
$407$	−8.29749			−0.0203870
$408$	56.0870	−	0.111427i	0.137468	−	0.000273107i
$409$	324.047			0.792292			0.396146	−	0.918188i	$-0.370347\pi$
$409$	324.047			0.792292			0.396146	+	0.918188i	$0.370347\pi$
$410$	224.542	+	56.0255i	0.547662	+	0.136647i
$411$	−782.801	+	1.55518i	−1.90462	+	0.00378390i
$412$			73.1547i			0.177560i
$413$	−568.579			−1.37670
$414$	61.0405	−	0.242538i	0.147441	−	0.000585841i
$415$	−112.589	+	451.238i	−0.271298	+	1.08732i
$416$		−	24.7837i		−	0.0595761i
$417$	32.5443	−	0.0646554i	0.0780438	−	0.000155049i
$418$		−	46.3086i		−	0.110786i
$419$		−	51.6049i		−	0.123162i	−0.998102	−	0.0615811i	$-0.980386\pi$
$419$		−	51.6049i		−	0.123162i	0.998102	−	0.0615811i	$-0.0196143\pi$
$420$	−200.963	−	49.7186i	−0.478484	−	0.118378i
$421$	−444.406			−1.05560			−0.527798	−	0.849370i	$-0.676982\pi$
$421$	−444.406			−1.05560			−0.527798	+	0.849370i	$0.676982\pi$
$422$	273.011			0.646946
$423$	−215.030	+	0.854398i	−0.508344	+	0.00201985i
$424$	80.3889			0.189596
$425$	−145.879	−	77.6294i	−0.343244	−	0.182658i
$426$	1.09837	+	552.865i	0.00257834	+	1.29781i
$427$			447.409i			1.04780i
$428$	158.342			0.369958
$429$	−25.3826	+	0.0504274i	−0.0591669	+	0.000117546i
$430$	−201.522	−	50.2818i	−0.468655	−	0.116934i
$431$		−	175.252i		−	0.406618i	−0.979115	−	0.203309i	$-0.934830\pi$
$431$		−	175.252i		−	0.406618i	0.979115	−	0.203309i	$-0.0651695\pi$
$432$	−107.998	+	0.643683i	−0.249996	+	0.00149001i
$433$			687.223i			1.58712i	0.608492	+	0.793560i	$0.291775\pi$
$433$			687.223i			1.58712i	−0.608492	+	0.793560i	$0.708225\pi$
$434$		−	186.590i		−	0.429931i
$435$	−79.4096	−	19.6460i	−0.182551	−	0.0451633i
$436$	−175.713			−0.403011
$437$	81.3178			0.186082
$438$	0.0891344	+	44.8658i	0.000203503	+	0.102433i
$439$	63.9919			0.145767			0.0728837	−	0.997340i	$-0.476780\pi$
$439$	63.9919			0.145767			0.0728837	+	0.997340i	$0.476780\pi$
$440$	26.4988	+	6.61172i	0.0602245	+	0.0150266i
$441$	12.4174	−	0.0493394i	0.0281575	−	0.000111881i
$442$		−	40.9545i		−	0.0926573i
$443$	−619.458			−1.39833			−0.699163	−	0.714962i	$-0.746444\pi$
$443$	−619.458			−1.39833			−0.699163	+	0.714962i	$0.746444\pi$
$444$	0.0512156	+	25.7794i	0.000115350	+	0.0580616i
$445$	796.977	+	198.854i	1.79096	+	0.446863i
$446$			63.0197i			0.141300i
$447$	0.527339	+	265.436i	0.00117973	+	0.593816i
$448$			55.2060i			0.123228i
$449$		−	321.746i		−	0.716583i	−0.933610	−	0.358291i	$-0.883359\pi$
$449$		−	321.746i		−	0.716583i	0.933610	−	0.358291i	$-0.116641\pi$
$450$	281.493	+	148.364i	0.625539	+	0.329698i
$451$	−63.2050			−0.140144
$452$	11.9487			0.0264352
$453$	223.104	−	0.443239i	0.492504	−	0.000978453i
$454$	−263.593			−0.580601
$455$	−36.5958	+	146.670i	−0.0804302	+	0.322352i
$456$	−143.876	+	0.285836i	−0.315517	+	0.000626834i
$457$			694.280i			1.51921i	0.650383	+	0.759607i	$0.274608\pi$
$457$			694.280i			1.51921i	−0.650383	+	0.759607i	$0.725392\pi$
$458$	−180.378			−0.393837
$459$	−178.465	+	1.06367i	−0.388812	+	0.00231737i
$460$	−11.6102	+	46.5318i	−0.0252395	+	0.101156i
$461$		−	878.547i		−	1.90574i	−0.303376	−	0.952871i	$-0.598114\pi$
$461$		−	878.547i		−	1.90574i	0.303376	−	0.952871i	$-0.401886\pi$
$462$	56.5401	−	0.112328i	0.122381	−	0.000243133i
$463$		−	266.191i		−	0.574926i	−0.957792	−	0.287463i	$-0.907188\pi$
$463$		−	266.191i		−	0.574926i	0.957792	−	0.287463i	$-0.0928119\pi$
$464$			21.8143i			0.0470136i
$465$	−68.8764	+	278.400i	−0.148121	+	0.598709i
$466$	169.022			0.362707
$467$	240.484			0.514956			0.257478	−	0.966284i	$-0.417109\pi$
$467$	240.484			0.514956			0.257478	+	0.966284i	$0.417109\pi$
$468$	0.313344	+	78.8605i	0.000669539	+	0.168505i
$469$	−430.235			−0.917346
$470$	40.8995	−	163.919i	0.0870203	−	0.348764i
$471$	0.955311	+	480.855i	0.00202826	+	1.02092i
$472$		−	233.045i		−	0.493739i
$473$	56.7252			0.119927
$474$	305.812	−	0.607554i	0.645173	−	0.00128176i
$475$	374.212	+	199.137i	0.787814	+	0.419235i
$476$			91.2268i			0.191653i
$477$	−255.794	+	1.01637i	−0.536255	+	0.00213076i
$478$		−	572.246i		−	1.19717i
$479$			477.380i			0.996617i	0.867000	+	0.498309i	$0.166045\pi$
$479$			477.380i			0.996617i	−0.867000	+	0.498309i	$0.833955\pi$
$480$	20.3783	−	82.3694i	0.0424548	−	0.171603i
$481$	18.8240			0.0391352
$482$	220.737			0.457961
$483$	0.197247	+	99.2842i	0.000408379	+	0.205557i
$484$	234.541			0.484589
$485$	337.515	+	84.2135i	0.695907	+	0.173636i
$486$	343.637	−	3.41361i	0.707072	−	0.00702389i
$487$		−	521.050i		−	1.06992i	−0.844878	−	0.534959i	$-0.820327\pi$
$487$		−	521.050i		−	1.06992i	0.844878	−	0.534959i	$-0.179673\pi$
$488$	−183.381			−0.375780
$489$	−0.880844	−	443.372i	−0.00180132	−	0.906692i
$490$	−2.36185	+	9.46594i	−0.00482010	+	0.0193182i
$491$			763.114i			1.55420i	0.629375	+	0.777102i	$0.283311\pi$
$491$			763.114i			1.55420i	−0.629375	+	0.777102i	$0.716689\pi$
$492$	0.390128	+	196.371i	0.000792943	+	0.399127i
$493$			36.0478i			0.0731192i
$494$			105.058i			0.212667i
$495$	−84.4015	−	20.7032i	−0.170508	−	0.0418246i
$496$	76.4782			0.154190
$497$	−899.249			−1.80935
$498$	−394.626	+	0.784000i	−0.792422	+	0.00157430i
$499$	−605.819			−1.21407			−0.607033	−	0.794677i	$-0.707640\pi$
$499$	−605.819			−1.21407			−0.607033	+	0.794677i	$0.707640\pi$
$500$	−167.378	+	185.700i	−0.334757	+	0.371400i
$501$	−761.444	+	1.51275i	−1.51985	+	0.00301947i
$502$		−	212.418i		−	0.423144i
$503$	−325.402			−0.646923			−0.323462	−	0.946241i	$-0.604847\pi$
$503$	−325.402			−0.646923			−0.323462	+	0.946241i	$0.604847\pi$
$504$	−0.697978	−	175.663i	−0.00138488	−	0.348537i
$505$	639.292	+	159.510i	1.26593	+	0.315862i
$506$		−	13.0980i		−	0.0258853i
$507$	−449.415	+	0.892849i	−0.886420	+	0.00176104i
$508$			47.8326i			0.0941586i
$509$		−	428.784i		−	0.842404i	−0.906967	−	0.421202i	$-0.861608\pi$
$509$		−	428.784i		−	0.842404i	0.906967	−	0.421202i	$-0.138392\pi$
$510$	33.6748	−	136.114i	0.0660289	−	0.266890i
$511$	−72.9753			−0.142809
$512$	−22.6274			−0.0441942
$513$	457.802	−	2.72856i	0.892402	−	0.00531883i
$514$	16.0416			0.0312094
$515$	177.447	+	44.2748i	0.344557	+	0.0859705i
$516$	−0.350132	−	176.239i	−0.000678551	−	0.341548i
$517$			46.1407i			0.0892470i
$518$	−41.9307			−0.0809474
$519$	423.038	−	0.840446i	0.815102	−	0.00161936i
$520$	−60.1161	−	14.9996i	−0.115608	−	0.0288454i
$521$		−	700.778i		−	1.34506i	−0.740069	−	0.672531i	$-0.765207\pi$
$521$		−	700.778i		−	1.34506i	0.740069	−	0.672531i	$-0.234793\pi$
$522$	−0.275802	−	69.4122i	−0.000528357	−	0.132974i
$523$			411.838i			0.787454i	0.919228	+	0.393727i	$0.128814\pi$
$523$			411.838i			0.787454i	−0.919228	+	0.393727i	$0.871186\pi$
$524$		−	9.65682i		−	0.0184291i
$525$	−242.227	+	457.373i	−0.461384	+	0.871187i
$526$	208.476			0.396341
$527$	126.379			0.239808
$528$	0.0460400	+	23.1742i	8.71971e−5	+	0.0438906i
$529$	23.0000			0.0434783
$530$	48.6531	−	194.994i	0.0917982	−	0.367913i
$531$	2.94643	+	741.538i	0.00554883	+	1.39649i
$532$		−	234.017i		−	0.439882i
$533$	143.389			0.269023
$534$	1.38470	+	696.989i	0.00259307	+	1.30522i
$535$	95.8321	−	384.080i	0.179125	−	0.717907i
$536$		−	176.342i		−	0.328996i
$537$	0.750914	+	377.972i	0.00139835	+	0.703859i
$538$		−	291.015i		−	0.540920i
$539$		−	2.66452i		−	0.00494344i
$540$	−63.8014	+	262.354i	−0.118151	+	0.485840i
$541$	598.358			1.10602			0.553011	−	0.833174i	$-0.313479\pi$
$541$	598.358			1.10602			0.553011	+	0.833174i	$0.313479\pi$
$542$	339.258			0.625937
$543$	678.995	−	1.34895i	1.25045	−	0.00248426i
$544$	−37.3914			−0.0687342
$545$	−106.345	+	426.215i	−0.195129	+	0.782047i
$546$	−128.269	+	0.254831i	−0.234925	+	0.000466723i
$547$		−	779.530i		−	1.42510i	−0.701621	−	0.712550i	$-0.747540\pi$
$547$		−	779.530i		−	1.42510i	0.701621	−	0.712550i	$-0.252460\pi$
$548$	521.868			0.952314
$549$	583.509	−	2.31851i	1.06286	−	0.00422316i
$550$	32.0752	−	60.2748i	0.0583186	−	0.109591i
$551$		−	92.4706i		−	0.167823i
$552$	−40.6939	+	0.0808462i	−0.0737208	+	0.000146461i
$553$			497.410i			0.899476i
$554$		−	234.216i		−	0.422772i
$555$	62.5623	+	15.4780i	0.112725	+	0.0278883i
$556$	−21.6962			−0.0390220
$557$	185.855			0.333672			0.166836	−	0.985985i	$-0.446645\pi$
$557$	185.855			0.333672			0.166836	+	0.985985i	$0.446645\pi$
$558$	−243.350	+	0.966927i	−0.436111	+	0.00173284i
$559$	−128.689			−0.230213
$560$	133.910	+	33.4118i	0.239124	+	0.0596640i
$561$	0.0760803	+	38.2950i	0.000135616	+	0.0682620i
$562$			304.843i			0.542425i
$563$	−252.783			−0.448994			−0.224497	−	0.974475i	$-0.572074\pi$
$563$	−252.783			−0.448994			−0.224497	+	0.974475i	$0.572074\pi$
$564$	143.354	−	0.284800i	0.254174	−	0.000504964i
$565$	7.23162	−	28.9832i	0.0127993	−	0.0512977i
$566$			369.923i			0.653574i
$567$	4.44187	+	558.943i	0.00783399	+	0.985789i
$568$		−	368.577i		−	0.648904i
$569$		−	155.819i		−	0.273848i	−0.990582	−	0.136924i	$-0.956278\pi$
$569$		−	155.819i		−	0.273848i	0.990582	−	0.136924i	$-0.0437216\pi$
$570$	−86.3832	+	349.163i	−0.151550	+	0.612566i
$571$	217.567			0.381029			0.190514	−	0.981684i	$-0.438984\pi$
$571$	217.567			0.381029			0.190514	+	0.981684i	$0.438984\pi$
$572$	16.9218			0.0295835
$573$	−0.960040	−	483.236i	−0.00167546	−	0.843344i
$574$	−319.402			−0.556449
$575$	105.842	+	56.3240i	0.184074	+	0.0979548i
$576$	71.9994	−	0.286082i	0.124999	−	0.000496671i
$577$			824.725i			1.42933i	0.699466	+	0.714666i	$0.253421\pi$
$577$			824.725i			1.42933i	−0.699466	+	0.714666i	$0.746579\pi$
$578$	346.919			0.600206
$579$	0.141698	+	71.3235i	0.000244728	+	0.123184i
$580$	52.9136	+	13.2025i	0.0912304	+	0.0227629i
$581$		−	641.869i		−	1.10477i
$582$	0.586412	+	295.170i	0.00100758	+	0.507165i
$583$			54.8878i			0.0941472i
$584$		−	29.9106i		−	0.0512168i
$585$	191.476	+	46.9680i	0.327310	+	0.0802872i
$586$	749.366			1.27878
$587$	455.443			0.775883			0.387942	−	0.921684i	$-0.373186\pi$
$587$	455.443			0.775883			0.387942	+	0.921684i	$0.373186\pi$
$588$	−8.27834	+	0.0164465i	−0.0140788	+	2.79703e-5i
$589$	−324.190			−0.550407
$590$	−565.282	−	141.044i	−0.958105	−	0.239057i
$591$	398.765	−	0.792223i	0.674729	−	0.00134048i
$592$		−	17.1863i		−	0.0290309i
$593$	264.393			0.445857			0.222928	−	0.974835i	$-0.428438\pi$
$593$	264.393			0.445857			0.222928	+	0.974835i	$0.428438\pi$
$594$	−0.439493	−	73.7388i	−0.000739888	−	0.124139i
$595$	221.283	+	55.2124i	0.371904	+	0.0927940i
$596$		−	176.957i		−	0.296908i
$597$	−444.033	+	0.882155i	−0.743773	+	0.00147765i
$598$			29.7146i			0.0496899i
$599$		−	1090.92i		−	1.82124i	−0.413249	−	0.910618i	$-0.635606\pi$
$599$		−	1090.92i		−	1.82124i	0.413249	−	0.910618i	$-0.364394\pi$
$600$	−187.465	−	99.2821i	−0.312441	−	0.165470i
$601$	−73.3335			−0.122019			−0.0610096	−	0.998137i	$-0.519432\pi$
$601$	−73.3335			−0.122019			−0.0610096	+	0.998137i	$0.519432\pi$
$602$	286.657			0.476174
$603$	2.22952	+	561.111i	0.00369738	+	0.930533i
$604$	−148.737			−0.246253
$605$	141.949	−	568.911i	0.234627	−	0.940349i
$606$	1.11073	+	559.087i	0.00183289	+	0.922585i
$607$			486.978i			0.802271i	0.916019	+	0.401135i	$0.131384\pi$
$607$			486.978i			0.802271i	−0.916019	+	0.401135i	$0.868616\pi$
$608$	95.9172			0.157759
$609$	112.901	−	0.224300i	0.185388	−	0.000368308i
$610$	−110.986	+	444.815i	−0.181944	+	0.729204i
$611$		−	104.677i		−	0.171320i
$612$	118.978	−	0.472746i	0.194408	−	0.000772461i
$613$			977.971i			1.59538i	0.603065	+	0.797692i	$0.293946\pi$
$613$			977.971i			1.59538i	−0.603065	+	0.797692i	$0.706054\pi$
$614$		−	172.664i		−	0.281211i
$615$	476.560	+	117.901i	0.774894	+	0.191710i
$616$	−37.6935			−0.0611907
$617$	119.147			0.193107			0.0965534	−	0.995328i	$-0.469218\pi$
$617$	119.147			0.193107			0.0965534	+	0.995328i	$0.469218\pi$
$618$	0.308303	+	155.184i	0.000498872	+	0.251107i
$619$	−661.995			−1.06946			−0.534730	−	0.845023i	$-0.679587\pi$
$619$	−661.995			−1.06946			−0.534730	+	0.845023i	$0.679587\pi$
$620$	46.2862	−	185.508i	0.0746552	−	0.299207i
$621$	129.485	−	0.771749i	0.208511	−	0.00124275i
$622$		−	71.2815i		−	0.114601i
$623$	−1133.67			−1.81969
$624$	−0.104448	−	52.5740i	−0.000167385	−	0.0842532i
$625$	349.139	+	518.389i	0.558623	+	0.829422i
$626$			695.384i			1.11084i
$627$	−0.195163	−	98.2352i	−0.000311265	−	0.156675i
$628$		−	320.571i		−	0.510463i
$629$		−	28.4000i		−	0.0451510i
$630$	−426.517	−	104.622i	−0.677010	−	0.166067i
$631$	−334.926			−0.530786			−0.265393	−	0.964140i	$-0.585502\pi$
$631$	−334.926			−0.530786			−0.265393	+	0.964140i	$0.585502\pi$
$632$	−203.875			−0.322587
$633$	579.143	−	1.15058i	0.914918	−	0.00181766i
$634$	−756.058			−1.19252
$635$	116.024	+	28.9493i	0.182715	+	0.0455894i
$636$	170.530	−	0.338791i	0.268129	−	0.000532690i
$637$			6.04482i			0.00948952i
$638$	−14.8944			−0.0233454
$639$	4.65999	+	1172.80i	0.00729263	+	1.83536i
$640$	−13.6946	+	54.8859i	−0.0213978	+	0.0857592i
$641$			21.9255i			0.0342052i	0.999854	+	0.0171026i	$0.00544419\pi$
$641$			21.9255i			0.0342052i	−0.999854	+	0.0171026i	$0.994556\pi$
$642$	335.894	−	0.667317i	0.523199	−	0.00103943i
$643$		−	624.566i		−	0.971332i	−0.874145	−	0.485666i	$-0.838577\pi$
$643$		−	624.566i		−	0.971332i	0.874145	−	0.485666i	$-0.161423\pi$
$644$		−	66.1896i		−	0.102779i
$645$	−427.703	−	105.814i	−0.663106	−	0.164053i
$646$	158.501			0.245358
$647$	−548.096			−0.847134			−0.423567	−	0.905865i	$-0.639222\pi$
$647$	−548.096			−0.847134			−0.423567	+	0.905865i	$0.639222\pi$
$648$	−229.095	+	1.82060i	−0.353542	+	0.00280957i
$649$	159.118			0.245174
$650$	−72.7671	+	136.742i	−0.111949	+	0.210372i
$651$	−0.786365	−	395.816i	−0.00120793	−	0.608013i
$652$			295.582i			0.453347i
$653$	1074.01			1.64473			0.822366	−	0.568959i	$-0.192654\pi$
$653$	1074.01			1.64473			0.822366	+	0.568959i	$0.192654\pi$
$654$	−372.743	+	0.740524i	−0.569943	+	0.00113230i
$655$	−23.4239	−	5.84452i	−0.0357617	−	0.00892293i
$656$		−	130.914i		−	0.199564i
$657$	0.378165	+	95.1741i	0.000575593	+	0.144862i
$658$			233.169i			0.354359i
$659$		−	624.157i		−	0.947127i	−0.880760	−	0.473564i	$-0.842967\pi$
$659$		−	624.157i		−	0.947127i	0.880760	−	0.473564i	$-0.157033\pi$
$660$	56.2401	+	13.9139i	0.0852123	+	0.0210816i
$661$	497.027			0.751931			0.375966	−	0.926634i	$-0.377311\pi$
$661$	497.027			0.751931			0.375966	+	0.926634i	$0.377311\pi$
$662$	−509.978			−0.770360
$663$	−0.172599	−	86.8775i	−0.000260330	−	0.131037i
$664$	263.085			0.396212
$665$	−567.640	−	141.632i	−0.853594	−	0.212981i
$666$	0.217289	+	54.6859i	0.000326260	+	0.0821110i
$667$		−	26.1545i		−	0.0392121i
$668$	507.631			0.759926
$669$	0.265590	+	133.685i	0.000396996	+	0.199828i
$670$	−427.741	−	106.726i	−0.638419	−	0.159292i
$671$		−	125.208i		−	0.186600i
$672$	0.232660	+	117.109i	0.000346220	+	0.174270i
$673$		−	599.940i		−	0.891442i	−0.895172	−	0.445721i	$-0.852947\pi$
$673$		−	599.940i		−	0.891442i	0.895172	−	0.445721i	$-0.147053\pi$
$674$		−	732.586i		−	1.08692i
$675$	597.760	+	313.541i	0.885570	+	0.464505i
$676$	299.611			0.443211
$677$	−302.950			−0.447488			−0.223744	−	0.974648i	$-0.571828\pi$
$677$	−302.950			−0.447488			−0.223744	+	0.974648i	$0.571828\pi$
$678$	25.3470	−	0.0503566i	0.0373849	−	7.42723e-5i
$679$	−480.102			−0.707072
$680$	−22.6301	+	90.6979i	−0.0332795	+	0.133379i
$681$	−559.164	+	1.11089i	−0.821093	+	0.00163126i
$682$			52.2177i			0.0765655i
$683$	287.077			0.420318			0.210159	−	0.977667i	$-0.432602\pi$
$683$	287.077			0.420318			0.210159	+	0.977667i	$0.432602\pi$
$684$	−305.204	+	1.21270i	−0.446205	+	0.00177295i
$685$	315.846	−	1265.86i	0.461089	−	1.84797i
$686$		−	491.662i		−	0.716709i
$687$	−382.638	+	0.760183i	−0.556969	+	0.00110653i
$688$			117.493i			0.170774i
$689$		−	124.521i		−	0.180727i
$690$	−24.4327	+	98.7575i	−0.0354097	+	0.143127i
$691$	−364.267			−0.527159			−0.263580	−	0.964638i	$-0.584903\pi$
$691$	−364.267			−0.527159			−0.263580	+	0.964638i	$0.584903\pi$
$692$	−282.026			−0.407552
$693$	119.939	−	0.476565i	0.173072	−	0.000687684i
$694$	−482.756			−0.695613
$695$	−13.1310	+	52.6271i	−0.0188936	+	0.0757225i
$696$	0.0919343	+	46.2751i	0.000132090	+	0.0664872i
$697$		−	216.333i		−	0.310377i
$698$	0.566643			0.000811810
$699$	358.548	−	0.712324i	0.512944	−	0.00101906i
$700$	162.090	−	304.594i	0.231557	−	0.435134i
$701$		−	281.045i		−	0.400921i	−0.979702	−	0.200460i	$-0.935756\pi$
$701$		−	281.045i		−	0.400921i	0.979702	−	0.200460i	$-0.0642438\pi$
$702$	0.997052	+	167.287i	0.00142030	+	0.238300i
$703$			72.8523i			0.103631i
$704$		−	15.4495i		−	0.0219453i
$705$	86.0700	−	347.897i	0.122085	−	0.493470i
$706$	383.420			0.543087
$707$	−909.368			−1.28624
$708$	−0.982144	−	494.362i	−0.00138721	−	0.698251i
$709$	−734.625			−1.03614			−0.518071	−	0.855337i	$-0.673350\pi$
$709$	−734.625			−1.03614			−0.518071	+	0.855337i	$0.673350\pi$
$710$	−894.035	−	223.071i	−1.25920	−	0.314185i
$711$	648.721	−	2.57763i	0.912407	−	0.00362535i
$712$		−	464.660i		−	0.652613i
$713$	−91.6941			−0.128603
$714$	0.384466	+	193.521i	0.000538468	+	0.271038i
$715$	10.2414	−	41.0460i	0.0143237	−	0.0574070i
$716$		−	251.982i		−	0.351930i
$717$	−2.41167	−	1213.92i	−0.00336356	−	1.69305i
$718$		−	492.476i		−	0.685899i
$719$		−	976.727i		−	1.35845i	−0.733929	−	0.679226i	$-0.762316\pi$
$719$		−	976.727i		−	1.35845i	0.733929	−	0.679226i	$-0.237684\pi$
$720$	42.8817	−	174.818i	0.0595579	−	0.242802i
$721$	−252.411			−0.350085
$722$	103.940			0.143961
$723$	468.253	−	0.930274i	0.647653	−	0.00128669i
$724$	−452.664			−0.625227
$725$	64.0489	−	120.359i	0.0883433	−	0.166012i
$726$	497.536	−	0.988449i	0.685311	−	0.00136150i
$727$		−	1318.22i		−	1.81323i	−0.421956	−	0.906616i	$-0.638656\pi$
$727$		−	1318.22i		−	1.81323i	0.421956	−	0.906616i	$-0.361344\pi$
$728$	85.5129			0.117463
$729$	728.948	−	8.68957i	0.999929	−	0.0119198i
$730$	−72.5522	−	18.1025i	−0.0993865	−	0.0247980i
$731$			194.155i			0.265602i
$732$	−389.008	+	0.772839i	−0.531432	+	0.00105579i
$733$		−	272.542i		−	0.371817i	−0.982567	−	0.185908i	$-0.940477\pi$
$733$		−	272.542i		−	0.371817i	0.982567	−	0.185908i	$-0.0595228\pi$
$734$			772.444i			1.05238i
$735$	−4.97034	+	20.0902i	−0.00676237	+	0.0273336i
$736$	27.1293			0.0368605
$737$	120.402			0.163368
$738$	1.65517	+	416.563i	0.00224278	+	0.564448i
$739$	−1298.32			−1.75685			−0.878427	−	0.477876i	$-0.841407\pi$
$739$	−1298.32			−1.75685			−0.878427	+	0.477876i	$0.841407\pi$
$740$	−41.6876	−	10.4015i	−0.0563346	−	0.0140561i
$741$	0.442754	+	222.860i	0.000597509	+	0.300756i
$742$			277.372i			0.373816i
$743$	302.960			0.407753			0.203876	−	0.978997i	$-0.434646\pi$
$743$	302.960			0.407753			0.203876	+	0.978997i	$0.434646\pi$
$744$	162.234	−	0.322310i	0.218057	−	0.000433212i
$745$	−429.234	−	107.098i	−0.576153	−	0.143756i
$746$		−	793.692i		−	1.06393i
$747$	−837.124	+	3.32622i	−1.12065	+	0.00445278i
$748$		−	25.5301i		−	0.0341311i
$749$			546.340i			0.729425i
$750$	−354.280	+	394.634i	−0.472373	+	0.526178i
$751$	708.772			0.943770			0.471885	−	0.881660i	$-0.343574\pi$
$751$	708.772			0.943770			0.471885	+	0.881660i	$0.343574\pi$
$752$	−95.5694			−0.127087
$753$	−0.895216	−	450.607i	−0.00118887	−	0.598415i
$754$	33.7899			0.0448142
$755$	−90.0185	+	360.781i	−0.119230	+	0.477855i
$756$	−2.22095	−	372.634i	−0.00293776	−	0.492902i
$757$		−	916.947i		−	1.21129i	−0.795735	−	0.605645i	$-0.792915\pi$
$757$		−	916.947i		−	1.21129i	0.795735	−	0.605645i	$-0.207085\pi$
$758$	90.8312			0.119830
$759$	−0.0552001	−	27.7849i	−7.27274e−5	−	0.0366073i
$760$	58.0512	−	232.660i	0.0763831	−	0.306132i
$761$			239.914i			0.315261i	0.987498	+	0.157631i	$0.0503855\pi$
$761$			239.914i			0.315261i	−0.987498	+	0.157631i	$0.949614\pi$
$762$	0.201585	+	101.468i	0.000264548	+	0.133160i
$763$		−	606.275i		−	0.794593i
$764$			322.158i			0.421673i
$765$	70.8612	−	288.883i	0.0926290	−	0.377624i
$766$	−77.7104			−0.101450
$767$	−360.982			−0.470641
$768$	−47.9999	+	0.0953610i	−0.0624999	+	0.000124168i
$769$	391.054			0.508522			0.254261	−	0.967136i	$-0.418168\pi$
$769$	391.054			0.508522			0.254261	+	0.967136i	$0.418168\pi$
$770$	−22.8129	+	91.4306i	−0.0296271	+	0.118741i
$771$	34.0294	−	0.0676059i	0.0441367	−	8.76860e-5i
$772$		−	47.5491i		−	0.0615921i
$773$	319.039			0.412729			0.206364	−	0.978475i	$-0.433837\pi$
$773$	319.039			0.412729			0.206364	+	0.978475i	$0.433837\pi$
$774$	−1.48548	−	373.857i	−0.00191923	−	0.483019i
$775$	−421.962	−	224.547i	−0.544467	−	0.289738i
$776$		−	196.781i		−	0.253583i
$777$	−88.9484	+	0.176713i	−0.114477	+	0.000227430i
$778$			472.284i			0.607049i
$779$			554.942i			0.712378i
$780$	−127.589	−	31.5655i	−0.163575	−	0.0404687i
$781$	251.657			0.322224
$782$	44.8307			0.0573283

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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +(-2.99999 + 0.00596006i) q^{3} +2.00000 q^{4} +(1.21044 - 4.85127i) q^{5} +(4.24263 - 0.00842880i) q^{6} +6.90074i q^{7} -2.82843 q^{8} +(8.99993 - 0.0357603i) q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +(-2.99999 + 0.00596006i) q^{3} +2.00000 q^{4} +(1.21044 - 4.85127i) q^{5} +(4.24263 - 0.00842880i) q^{6} +6.90074i q^{7} -2.82843 q^{8} +(8.99993 - 0.0357603i) q^{9} +(-1.71182 + 6.86073i) q^{10} -1.93119i q^{11} +(-5.99999 + 0.0119201i) q^{12} +4.38117i q^{13} -9.75913i q^{14} +(-3.60241 + 14.5610i) q^{15} +4.00000 q^{16} +6.60993 q^{17} +(-12.7278 + 0.0505727i) q^{18} -16.9559 q^{19} +(2.42089 - 9.70254i) q^{20} +(-0.0411289 - 20.7022i) q^{21} +2.73112i q^{22} -4.79583 q^{23} +(8.48526 - 0.0168576i) q^{24} +(-22.0697 - 11.7444i) q^{25} -6.19592i q^{26} +(-26.9995 + 0.160921i) q^{27} +13.8015i q^{28} +5.45358i q^{29} +(5.09457 - 20.5924i) q^{30} +19.1195 q^{31} -5.65685 q^{32} +(0.0115100 + 5.79356i) q^{33} -9.34785 q^{34} +(33.4774 + 8.35296i) q^{35} +(17.9999 - 0.0715206i) q^{36} -4.29657i q^{37} +23.9793 q^{38} +(-0.0261121 - 13.1435i) q^{39} +(-3.42365 + 13.7215i) q^{40} -32.7285i q^{41} +(0.0581650 + 29.2773i) q^{42} +29.3732i q^{43} -3.86238i q^{44} +(10.7204 - 43.7044i) q^{45} +6.78233 q^{46} -23.8924 q^{47} +(-12.0000 + 0.0238402i) q^{48} +1.37973 q^{49} +(31.2112 + 16.6091i) q^{50} +(-19.8297 + 0.0393956i) q^{51} +8.76235i q^{52} -28.4218 q^{53} +(38.1831 - 0.227576i) q^{54} +(-9.36873 - 2.33760i) q^{55} -19.5183i q^{56} +(50.8677 - 0.101058i) q^{57} -7.71253i q^{58} +82.3938i q^{59} +(-7.20482 + 29.1220i) q^{60} +64.8349 q^{61} -27.0391 q^{62} +(0.246773 + 62.1062i) q^{63} +8.00000 q^{64} +(21.2543 + 5.30316i) q^{65} +(-0.0162776 - 8.19333i) q^{66} +62.3462i q^{67} +13.2199 q^{68} +(14.3875 - 0.0285834i) q^{69} +(-47.3442 - 11.8129i) q^{70} +130.312i q^{71} +(-25.4556 + 0.101145i) q^{72} +10.5750i q^{73} +6.07626i q^{74} +(66.2788 + 35.1015i) q^{75} -33.9119 q^{76} +13.3267 q^{77} +(0.0369280 + 18.5877i) q^{78} +72.0807 q^{79} +(4.84177 - 19.4051i) q^{80} +(80.9974 - 0.643680i) q^{81} +46.2851i q^{82} -93.0145 q^{83} +(-0.0822577 - 41.4044i) q^{84} +(8.00094 - 32.0665i) q^{85} -41.5400i q^{86} +(-0.0325037 - 16.3607i) q^{87} +5.46223i q^{88} +164.282i q^{89} +(-15.1610 + 61.8073i) q^{90} -30.2334 q^{91} -9.59166 q^{92} +(-57.3585 + 0.113954i) q^{93} +33.7889 q^{94} +(-20.5242 + 82.2578i) q^{95} +(16.9705 - 0.0337152i) q^{96} +69.5724i q^{97} -1.95123 q^{98} +(-0.0690599 - 17.3806i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88q + 176q^{4} + 8q^{9} + O(q^{10}) \)	\( 88q + 176q^{4} + 8q^{9} + 32q^{10} - 12q^{15} + 352q^{16} - 16q^{19} - 176q^{21} + 72q^{25} - 72q^{30} + 32q^{31} + 160q^{34} + 16q^{36} + 144q^{39} + 64q^{40} + 92q^{45} - 360q^{49} + 48q^{51} - 144q^{54} + 16q^{55} - 24q^{60} + 208q^{61} + 704q^{64} + 512q^{66} + 304q^{70} + 536q^{75} - 32q^{76} + 448q^{79} - 24q^{81} - 352q^{84} - 96q^{85} + 32q^{90} - 64q^{91} + 160q^{94} + 296q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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Groups

Properties

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