LMFDB - Embedded newform 378.4.g.e.109.4

Newspace parameters

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$ x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 $ x^8 - x^7 + 66x^6 + 59x^5 + 3770x^4 + 721x^3 + 29779x^2 + 1374x + 209764
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.4
Root			$4.05517 - 7.02376i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.e.163.4

$q$-expansion

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(8.00592 + 13.8667i) q^{5} +(1.70902 + 18.4412i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(8.00592 + 13.8667i) q^{5} +(1.70902 + 18.4412i) q^{7} -8.00000 q^{8} +(-16.0118 + 27.7333i) q^{10} +(-7.89520 + 13.6749i) q^{11} -50.4536 q^{13} +(-30.2321 + 21.4013i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(18.4772 - 32.0034i) q^{17} +(44.6871 + 77.4002i) q^{19} -64.0474 q^{20} -31.5808 q^{22} +(-30.6840 - 53.1462i) q^{23} +(-65.6896 + 113.778i) q^{25} +(-50.4536 - 87.3882i) q^{26} +(-67.3004 - 30.9623i) q^{28} +126.366 q^{29} +(129.992 - 225.153i) q^{31} +(16.0000 - 27.7128i) q^{32} +73.9087 q^{34} +(-242.036 + 171.338i) q^{35} +(123.048 + 213.126i) q^{37} +(-89.3741 + 154.800i) q^{38} +(-64.0474 - 110.933i) q^{40} -516.432 q^{41} -34.2518 q^{43} +(-31.5808 - 54.6996i) q^{44} +(61.3680 - 106.292i) q^{46} +(-182.851 - 316.707i) q^{47} +(-337.159 + 63.0328i) q^{49} -262.759 q^{50} +(100.907 - 174.776i) q^{52} +(23.5816 - 40.8445i) q^{53} -252.834 q^{55} +(-13.6721 - 147.530i) q^{56} +(126.366 + 218.872i) q^{58} +(-358.163 + 620.357i) q^{59} +(348.209 + 603.115i) q^{61} +519.968 q^{62} +64.0000 q^{64} +(-403.928 - 699.623i) q^{65} +(-55.3121 + 95.8034i) q^{67} +(73.9087 + 128.014i) q^{68} +(-538.802 - 247.882i) q^{70} -607.199 q^{71} +(140.081 - 242.627i) q^{73} +(-246.096 + 426.251i) q^{74} -357.496 q^{76} +(-265.675 - 122.227i) q^{77} +(441.645 + 764.951i) q^{79} +(128.095 - 221.867i) q^{80} +(-516.432 - 894.487i) q^{82} -63.1517 q^{83} +591.707 q^{85} +(-34.2518 - 59.3258i) q^{86} +(63.1616 - 109.399i) q^{88} +(337.796 + 585.080i) q^{89} +(-86.2261 - 930.426i) q^{91} +245.472 q^{92} +(365.701 - 633.413i) q^{94} +(-715.522 + 1239.32i) q^{95} +976.191 q^{97} +(-446.335 - 520.943i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	8.00592	+	13.8667i	0.716072	+	1.24027i	0.962545	+	0.271123i	$0.0873949\pi$
$5$	8.00592	+	13.8667i	0.716072	+	1.24027i	−0.246473	+	0.969150i	$0.579272\pi$
$6$		0			0
$7$	1.70902	+	18.4412i	0.0922783	+	0.995733i
$7$	1.70902	+	18.4412i	0.0922783	+	0.995733i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−16.0118	+	27.7333i	−0.506339	+	0.877005i
$11$	−7.89520	+	13.6749i	−0.216409	+	0.374831i	−0.953707	−	0.300736i	$-0.902768\pi$
$11$	−7.89520	+	13.6749i	−0.216409	+	0.374831i	0.737299	+	0.675567i	$0.236101\pi$
$12$		0			0
$13$	−50.4536			−1.07641			−0.538204	−	0.842815i	$-0.680897\pi$
$13$	−50.4536			−1.07641			−0.538204	+	0.842815i	$0.680897\pi$
$14$	−30.2321	+	21.4013i	−0.577134	+	0.408554i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	18.4772	−	32.0034i	0.263610	−	0.456586i	−0.703588	−	0.710608i	$-0.748420\pi$
$17$	18.4772	−	32.0034i	0.263610	−	0.456586i	0.967199	+	0.254022i	$0.0817534\pi$
$18$		0			0
$19$	44.6871	+	77.4002i	0.539574	+	0.934570i	0.998927	+	0.0463161i	$0.0147482\pi$
$19$	44.6871	+	77.4002i	0.539574	+	0.934570i	−0.459352	+	0.888254i	$0.651919\pi$
$20$	−64.0474			−0.716072
$21$		0			0
$22$	−31.5808			−0.306048
$23$	−30.6840	−	53.1462i	−0.278176	−	0.481816i	0.692755	−	0.721173i	$-0.256397\pi$
$23$	−30.6840	−	53.1462i	−0.278176	−	0.481816i	−0.970932	+	0.239357i	$0.923063\pi$
$24$		0			0
$25$	−65.6896	+	113.778i	−0.525517	+	0.910222i
$26$	−50.4536	−	87.3882i	−0.380568	−	0.659163i
$27$		0			0
$28$	−67.3004	−	30.9623i	−0.454235	−	0.208976i
$29$	126.366			0.809156			0.404578	−	0.914504i	$-0.367418\pi$
$29$	126.366			0.809156			0.404578	+	0.914504i	$0.367418\pi$
$30$		0			0
$31$	129.992	−	225.153i	0.753137	−	1.30447i	−0.193159	−	0.981167i	$-0.561873\pi$
$31$	129.992	−	225.153i	0.753137	−	1.30447i	0.946295	−	0.323303i	$-0.104793\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	73.9087			0.372801
$35$	−242.036	+	171.338i	−1.16890	+	0.827467i
$36$		0			0
$37$	123.048	+	213.126i	0.546729	+	0.946963i	0.998496	+	0.0548266i	$0.0174606\pi$
$37$	123.048	+	213.126i	0.546729	+	0.946963i	−0.451767	+	0.892136i	$0.649206\pi$
$38$	−89.3741	+	154.800i	−0.381537	+	0.660841i
$39$		0			0
$40$	−64.0474	−	110.933i	−0.253170	−	0.438503i
$41$	−516.432			−1.96715			−0.983575	−	0.180500i	$-0.942228\pi$
$41$	−516.432			−1.96715			−0.983575	+	0.180500i	$0.942228\pi$
$42$		0			0
$43$	−34.2518			−0.121473			−0.0607366	−	0.998154i	$-0.519345\pi$
$43$	−34.2518			−0.121473			−0.0607366	+	0.998154i	$0.519345\pi$
$44$	−31.5808	−	54.6996i	−0.108204	−	0.187415i
$45$		0			0
$46$	61.3680	−	106.292i	0.196700	−	0.340695i
$47$	−182.851	−	316.707i	−0.567479	−	0.982902i	−0.996814	−	0.0797571i	$-0.974586\pi$
$47$	−182.851	−	316.707i	−0.567479	−	0.982902i	0.429335	−	0.903145i	$-0.358748\pi$
$48$		0			0
$49$	−337.159	+	63.0328i	−0.982969	+	0.183769i
$50$	−262.759			−0.743193
$51$		0			0
$52$	100.907	−	174.776i	0.269102	−	0.466098i
$53$	23.5816	−	40.8445i	0.0611166	−	0.105857i	−0.833848	−	0.551994i	$-0.813867\pi$
$53$	23.5816	−	40.8445i	0.0611166	−	0.105857i	0.894965	+	0.446137i	$0.147201\pi$
$54$		0			0
$55$	−252.834			−0.619856
$56$	−13.6721	−	147.530i	−0.0326253	−	0.352045i
$57$		0			0
$58$	126.366	+	218.872i	0.286080	+	0.495505i
$59$	−358.163	+	620.357i	−0.790320	+	1.36887i	0.135449	+	0.990784i	$0.456752\pi$
$59$	−358.163	+	620.357i	−0.790320	+	1.36887i	−0.925769	+	0.378090i	$0.876581\pi$
$60$		0			0
$61$	348.209	+	603.115i	0.730878	+	1.26592i	0.956508	+	0.291705i	$0.0942227\pi$
$61$	348.209	+	603.115i	0.730878	+	1.26592i	−0.225630	+	0.974213i	$0.572444\pi$
$62$	519.968			1.06510
$63$		0			0
$64$	64.0000			0.125000
$65$	−403.928	−	699.623i	−0.770785	−	1.33504i
$66$		0			0
$67$	−55.3121	+	95.8034i	−0.100857	+	0.174690i	−0.912038	−	0.410105i	$-0.865492\pi$
$67$	−55.3121	+	95.8034i	−0.100857	+	0.174690i	0.811181	+	0.584796i	$0.198825\pi$
$68$	73.9087	+	128.014i	0.131805	+	0.228293i
$69$		0			0
$70$	−538.802	−	247.882i	−0.919987	−	0.423250i
$71$	−607.199			−1.01495			−0.507474	−	0.861667i	$-0.669421\pi$
$71$	−607.199			−1.01495			−0.507474	+	0.861667i	$0.669421\pi$
$72$		0			0
$73$	140.081	−	242.627i	0.224592	−	0.389004i	−0.731605	−	0.681729i	$-0.761228\pi$
$73$	140.081	−	242.627i	0.224592	−	0.389004i	0.956197	+	0.292724i	$0.0945618\pi$
$74$	−246.096	+	426.251i	−0.386596	+	0.669604i
$75$		0			0
$76$	−357.496			−0.539574
$77$	−265.675	−	122.227i	−0.393201	−	0.180896i
$78$		0			0
$79$	441.645	+	764.951i	0.628974	+	1.08941i	0.987758	+	0.155994i	$0.0498580\pi$
$79$	441.645	+	764.951i	0.628974	+	1.08941i	−0.358784	+	0.933420i	$0.616809\pi$
$80$	128.095	−	221.867i	0.179018	−	0.310068i
$81$		0			0
$82$	−516.432	−	894.487i	−0.695492	−	1.20463i
$83$	−63.1517			−0.0835157			−0.0417578	−	0.999128i	$-0.513296\pi$
$83$	−63.1517			−0.0835157			−0.0417578	+	0.999128i	$0.513296\pi$
$84$		0			0
$85$	591.707			0.755055
$86$	−34.2518	−	59.3258i	−0.0429473	−	0.0743869i
$87$		0			0
$88$	63.1616	−	109.399i	0.0765120	−	0.132523i
$89$	337.796	+	585.080i	0.402318	+	0.696836i	0.994005	−	0.109332i	$-0.0348712\pi$
$89$	337.796	+	585.080i	0.402318	+	0.696836i	−0.591687	+	0.806168i	$0.701538\pi$
$90$		0			0
$91$	−86.2261	−	930.426i	−0.0993291	−	1.07182i
$92$	245.472			0.278176
$93$		0			0
$94$	365.701	−	633.413i	0.401268	−	0.695017i
$95$	−715.522	+	1239.32i	−0.772748	+	1.33844i
$96$		0			0
$97$	976.191			1.02183			0.510913	−	0.859632i	$-0.329307\pi$
$97$	976.191			1.02183			0.510913	+	0.859632i	$0.329307\pi$
$98$	−446.335	−	520.943i	−0.460067	−	0.536971i
$99$		0			0
$100$	−262.759	−	455.111i	−0.262759	−	0.455111i
$101$	613.782	−	1063.10i	0.604689	−	1.04735i	−0.387411	−	0.921907i	$-0.626631\pi$
$101$	613.782	−	1063.10i	0.604689	−	1.04735i	0.992100	−	0.125446i	$-0.0400361\pi$
$102$		0			0
$103$	419.207	+	726.088i	0.401026	+	0.694598i	0.993850	−	0.110734i	$-0.0353203\pi$
$103$	419.207	+	726.088i	0.401026	+	0.694598i	−0.592824	+	0.805332i	$0.701987\pi$
$104$	403.629			0.380568
$105$		0			0
$106$	94.3264			0.0864320
$107$	−215.777	−	373.737i	−0.194953	−	0.337669i	0.751932	−	0.659241i	$-0.229122\pi$
$107$	−215.777	−	373.737i	−0.194953	−	0.337669i	−0.946885	+	0.321572i	$0.895789\pi$
$108$		0			0
$109$	653.539	−	1131.96i	0.574291	−	0.994701i	−0.421827	−	0.906676i	$-0.638611\pi$
$109$	653.539	−	1131.96i	0.574291	−	0.994701i	0.996118	−	0.0880250i	$-0.0280555\pi$
$110$	−252.834	−	437.921i	−0.219152	−	0.379583i
$111$		0			0
$112$	241.857	−	171.211i	0.204048	−	0.144445i
$113$	1008.94			0.839943			0.419971	−	0.907537i	$-0.362040\pi$
$113$	1008.94			0.839943			0.419971	+	0.907537i	$0.362040\pi$
$114$		0			0
$115$	491.307	−	850.969i	0.398388	−	0.690029i
$116$	−252.731	+	437.744i	−0.202289	+	0.350375i
$117$		0			0
$118$	−1432.65			−1.11768
$119$	621.760	+	286.047i	0.478963	+	0.220352i
$120$		0			0
$121$	540.831	+	936.748i	0.406335	+	0.703792i
$122$	−696.418	+	1206.23i	−0.516809	+	0.895140i
$123$		0			0
$124$	519.968	+	900.610i	0.376568	+	0.652235i
$125$	−102.144			−0.0730883
$126$		0			0
$127$	−882.265			−0.616444			−0.308222	−	0.951315i	$-0.599734\pi$
$127$	−882.265			−0.616444			−0.308222	+	0.951315i	$0.599734\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	807.855	−	1399.25i	0.545028	−	0.944015i
$131$	1023.93	+	1773.50i	0.682912	+	1.18284i	0.974088	+	0.226168i	$0.0726200\pi$
$131$	1023.93	+	1773.50i	0.682912	+	1.18284i	−0.291176	+	0.956669i	$0.594047\pi$
$132$		0			0
$133$	−1350.99	+	956.363i	−0.880792	+	0.623513i
$134$	−221.248			−0.142634
$135$		0			0
$136$	−147.817	+	256.027i	−0.0932002	+	0.161428i
$137$	1250.79	−	2166.44i	0.780018	−	1.35103i	−0.151913	−	0.988394i	$-0.548543\pi$
$137$	1250.79	−	2166.44i	0.780018	−	1.35103i	0.931931	−	0.362637i	$-0.118123\pi$
$138$		0			0
$139$	−949.028			−0.579104			−0.289552	−	0.957162i	$-0.593506\pi$
$139$	−949.028			−0.579104			−0.289552	+	0.957162i	$0.593506\pi$
$140$	−109.458	−	1181.11i	−0.0660779	−	0.713016i
$141$		0			0
$142$	−607.199	−	1051.70i	−0.358838	−	0.621526i
$143$	398.341	−	689.947i	0.232944	−	0.403471i
$144$		0			0
$145$	1011.67	+	1752.27i	0.579414	+	1.00357i
$146$	560.323			0.317621
$147$		0			0
$148$	−984.385			−0.546729
$149$	−398.773	−	690.696i	−0.219254	−	0.379758i	0.735326	−	0.677713i	$-0.237029\pi$
$149$	−398.773	−	690.696i	−0.219254	−	0.379758i	−0.954580	+	0.297955i	$0.903696\pi$
$150$		0			0
$151$	−1354.87	+	2346.70i	−0.730181	+	1.26471i	0.226624	+	0.973982i	$0.427231\pi$
$151$	−1354.87	+	2346.70i	−0.730181	+	1.26471i	−0.956806	+	0.290729i	$0.906102\pi$
$152$	−357.496	−	619.202i	−0.190768	−	0.330420i
$153$		0			0
$154$	−53.9722	−	582.389i	−0.0282416	−	0.304742i
$155$	4162.82			2.15720
$156$		0			0
$157$	−1674.15	+	2899.70i	−0.851028	+	1.47402i	0.0292542	+	0.999572i	$0.490687\pi$
$157$	−1674.15	+	2899.70i	−0.851028	+	1.47402i	−0.880282	+	0.474451i	$0.842647\pi$
$158$	−883.290	+	1529.90i	−0.444752	+	0.770332i
$159$		0			0
$160$	512.379			0.253170
$161$	927.643	−	656.679i	0.454090	−	0.321451i
$162$		0			0
$163$	1351.28	+	2340.48i	0.649327	+	1.12467i	0.983284	+	0.182079i	$0.0582826\pi$
$163$	1351.28	+	2340.48i	0.649327	+	1.12467i	−0.333957	+	0.942588i	$0.608384\pi$
$164$	1032.86	−	1788.97i	0.491787	−	0.851801i
$165$		0			0
$166$	−63.1517	−	109.382i	−0.0295273	−	0.0511427i
$167$	−737.971			−0.341951			−0.170976	−	0.985275i	$-0.554692\pi$
$167$	−737.971			−0.341951			−0.170976	+	0.985275i	$0.554692\pi$
$168$		0			0
$169$	348.564			0.158654
$170$	591.707	+	1024.87i	0.266952	+	0.462375i
$171$		0			0
$172$	68.5036	−	118.652i	0.0303683	−	0.0525995i
$173$	1942.56	+	3364.61i	0.853699	+	1.47865i	0.877847	+	0.478942i	$0.158979\pi$
$173$	1942.56	+	3364.61i	0.853699	+	1.47865i	−0.0241476	+	0.999708i	$0.507687\pi$
$174$		0			0
$175$	−2210.47	−	1016.95i	−0.954832	−	0.439281i
$176$	252.647			0.108204
$177$		0			0
$178$	−675.593	+	1170.16i	−0.284482	+	0.492737i
$179$	1291.06	−	2236.18i	0.539096	−	0.933741i	−0.459857	−	0.887993i	$-0.652099\pi$
$179$	1291.06	−	2236.18i	0.539096	−	0.933741i	0.998953	−	0.0457482i	$-0.0145672\pi$
$180$		0			0
$181$	2978.56			1.22317			0.611587	−	0.791177i	$-0.290531\pi$
$181$	2978.56			1.22317			0.611587	+	0.791177i	$0.290531\pi$
$182$	1525.32	−	1079.77i	0.621232	−	0.439770i
$183$		0			0
$184$	245.472	+	425.170i	0.0983502	+	0.170348i
$185$	−1970.23	+	3412.53i	−0.782994	+	1.35619i
$186$		0			0
$187$	291.762	+	505.347i	0.114095	+	0.197618i
$188$	1462.81			0.567479
$189$		0			0
$190$	−2862.09			−1.09283
$191$	−2063.72	−	3574.47i	−0.781809	−	1.35413i	−0.930887	−	0.365308i	$-0.880964\pi$
$191$	−2063.72	−	3574.47i	−0.781809	−	1.35413i	0.149077	−	0.988826i	$-0.452370\pi$
$192$		0			0
$193$	1508.85	−	2613.41i	0.562743	−	0.974700i	−0.434512	−	0.900666i	$-0.643079\pi$
$193$	1508.85	−	2613.41i	0.562743	−	0.974700i	0.997256	−	0.0740342i	$-0.0235874\pi$
$194$	976.191	+	1690.81i	0.361270	+	0.625738i
$195$		0			0
$196$	455.965	−	1294.02i	0.166168	−	0.471581i
$197$	3959.32			1.43193			0.715964	−	0.698137i	$-0.245988\pi$
$197$	3959.32			1.43193			0.715964	+	0.698137i	$0.245988\pi$
$198$		0			0
$199$	−2346.11	+	4063.57i	−0.835734	+	1.44753i	0.0576981	+	0.998334i	$0.481624\pi$
$199$	−2346.11	+	4063.57i	−0.835734	+	1.44753i	−0.893432	+	0.449199i	$0.851709\pi$
$200$	525.517	−	910.222i	0.185798	−	0.321812i
$201$		0			0
$202$	2455.13			0.855160
$203$	215.961	+	2330.34i	0.0746675	+	0.805703i
$204$		0			0
$205$	−4134.52	−	7161.19i	−1.40862	−	2.43980i
$206$	−838.414	+	1452.18i	−0.283568	+	0.491155i
$207$		0			0
$208$	403.629	+	699.105i	0.134551	+	0.233049i
$209$	−1411.25			−0.467074
$210$		0			0
$211$	−1542.58			−0.503297			−0.251648	−	0.967819i	$-0.580973\pi$
$211$	−1542.58			−0.503297			−0.251648	+	0.967819i	$0.580973\pi$
$212$	94.3264	+	163.378i	0.0305583	+	0.0529286i
$213$		0			0
$214$	431.555	−	747.475i	0.137853	−	0.238768i
$215$	−274.217	−	474.958i	−0.0869835	−	0.150660i
$216$		0			0
$217$	4374.25	+	2012.42i	1.36840	+	0.629549i
$218$	2614.16			0.812170
$219$		0			0
$220$	505.667	−	875.841i	0.154964	−	0.268406i
$221$	−932.239	+	1614.69i	−0.283752	+	0.491473i
$222$		0			0
$223$	5343.30			1.60455			0.802273	−	0.596958i	$-0.203624\pi$
$223$	5343.30			1.60455			0.802273	+	0.596958i	$0.203624\pi$
$224$	538.403	+	247.698i	0.160596	+	0.0738840i
$225$		0			0
$226$	1008.94	+	1747.54i	0.296965	+	0.514358i
$227$	−1902.99	+	3296.08i	−0.556415	+	0.963738i	0.441377	+	0.897322i	$0.354490\pi$
$227$	−1902.99	+	3296.08i	−0.556415	+	0.963738i	−0.997792	+	0.0664167i	$0.978843\pi$
$228$		0			0
$229$	−599.426	−	1038.24i	−0.172975	−	0.299601i	0.766484	−	0.642264i	$-0.222005\pi$
$229$	−599.426	−	1038.24i	−0.172975	−	0.299601i	−0.939459	+	0.342663i	$0.888671\pi$
$230$	1965.23			0.563406
$231$		0			0
$232$	−1010.93			−0.286080
$233$	1305.87	+	2261.84i	0.367170	+	0.635956i	0.989122	−	0.147099i	$-0.0469936\pi$
$233$	1305.87	+	2261.84i	0.367170	+	0.635956i	−0.621952	+	0.783055i	$0.713660\pi$
$234$		0			0
$235$	2927.78	−	5071.06i	0.812711	−	1.40766i
$236$	−1432.65	−	2481.43i	−0.395160	−	0.684437i
$237$		0			0
$238$	126.311	+	1362.97i	0.0344014	+	0.371210i
$239$	3318.79			0.898220			0.449110	−	0.893477i	$-0.351741\pi$
$239$	3318.79			0.898220			0.449110	+	0.893477i	$0.351741\pi$
$240$		0			0
$241$	1388.03	−	2404.14i	0.370999	−	0.642590i	−0.618720	−	0.785611i	$-0.712349\pi$
$241$	1388.03	−	2404.14i	0.370999	−	0.642590i	0.989720	+	0.143022i	$0.0456819\pi$
$242$	−1081.66	+	1873.50i	−0.287322	+	0.497656i
$243$		0			0
$244$	−2785.67			−0.730878
$245$	−3573.32	−	4170.63i	−0.931800	−	1.08756i
$246$		0			0
$247$	−2254.62	−	3905.12i	−0.580802	−	1.00598i
$248$	−1039.94	+	1801.22i	−0.266274	+	0.461200i
$249$		0			0
$250$	−102.144	−	176.919i	−0.0258406	−	0.0447573i
$251$	−1339.17			−0.336764			−0.168382	−	0.985722i	$-0.553854\pi$
$251$	−1339.17			−0.336764			−0.168382	+	0.985722i	$0.553854\pi$
$252$		0			0
$253$	969.026			0.240799
$254$	−882.265	−	1528.13i	−0.217946	−	0.377493i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−1513.62	−	2621.66i	−0.367381	−	0.636322i	0.621774	−	0.783196i	$-0.286412\pi$
$257$	−1513.62	−	2621.66i	−0.367381	−	0.636322i	−0.989155	+	0.146874i	$0.953079\pi$
$258$		0			0
$259$	−3720.01	+	2633.39i	−0.892471	+	0.631781i
$260$	3231.42			0.770785
$261$		0			0
$262$	−2047.87	+	3547.01i	−0.482892	+	0.836393i
$263$	3106.65	−	5380.87i	0.728381	−	1.26159i	−0.229186	−	0.973383i	$-0.573606\pi$
$263$	3106.65	−	5380.87i	0.728381	−	1.26159i	0.957567	−	0.288210i	$-0.0930602\pi$
$264$		0			0
$265$	755.170			0.175056
$266$	−3007.45	−	1383.61i	−0.693229	−	0.318927i
$267$		0			0
$268$	−221.248	−	383.214i	−0.0504287	−	0.0873451i
$269$	−309.272	+	535.674i	−0.0700990	+	0.121415i	−0.898945	−	0.438063i	$-0.855665\pi$
$269$	−309.272	+	535.674i	−0.0700990	+	0.121415i	0.828846	+	0.559478i	$0.188998\pi$
$270$		0			0
$271$	−1368.21	−	2369.81i	−0.306690	−	0.531202i	0.670946	−	0.741506i	$-0.265888\pi$
$271$	−1368.21	−	2369.81i	−0.306690	−	0.531202i	−0.977636	+	0.210304i	$0.932555\pi$
$272$	−591.269			−0.131805
$273$		0			0
$274$	5003.17			1.10311
$275$	−1037.27	−	1796.60i	−0.227453	−	0.393960i
$276$		0			0
$277$	2065.70	−	3577.91i	0.448073	−	0.776085i	−0.550188	−	0.835041i	$-0.685444\pi$
$277$	2065.70	−	3577.91i	0.448073	−	0.776085i	0.998261	+	0.0589561i	$0.0187772\pi$
$278$	−949.028	−	1643.76i	−0.204744	−	0.354628i
$279$		0			0
$280$	1936.29	−	1370.70i	0.413269	−	0.292554i
$281$	−1741.24			−0.369658			−0.184829	−	0.982771i	$-0.559173\pi$
$281$	−1741.24			−0.369658			−0.184829	+	0.982771i	$0.559173\pi$
$282$		0			0
$283$	−1707.54	+	2957.54i	−0.358666	+	0.621227i	−0.987738	−	0.156119i	$-0.950102\pi$
$283$	−1707.54	+	2957.54i	−0.358666	+	0.621227i	0.629072	+	0.777347i	$0.283435\pi$
$284$	1214.40	−	2103.40i	0.253737	−	0.439485i
$285$		0			0
$286$	1593.37			0.329432
$287$	−882.592	−	9523.65i	−0.181525	−	1.95876i
$288$		0			0
$289$	1773.69	+	3072.12i	0.361019	+	0.625304i
$290$	−2023.35	+	3504.54i	−0.409707	+	0.709634i
$291$		0			0
$292$	560.323	+	970.507i	0.112296	+	0.194502i
$293$	−8652.18			−1.72514			−0.862569	−	0.505939i	$-0.831146\pi$
$293$	−8652.18			−1.72514			−0.862569	+	0.505939i	$0.831146\pi$
$294$		0			0
$295$	−11469.7			−2.26370
$296$	−984.385	−	1705.00i	−0.193298	−	0.334802i
$297$		0			0
$298$	797.547	−	1381.39i	0.155036	−	0.268530i
$299$	1548.12	+	2681.42i	0.299431	+	0.518630i
$300$		0			0
$301$	−58.5369	−	631.645i	−0.0112093	−	0.120955i
$302$	−5419.46			−1.03263
$303$		0			0
$304$	714.993	−	1238.40i	0.134894	−	0.233643i
$305$	−5575.47	+	9656.99i	−1.04672	+	1.81298i
$306$		0			0
$307$	−695.737			−0.129341			−0.0646707	−	0.997907i	$-0.520600\pi$
$307$	−695.737			−0.129341			−0.0646707	+	0.997907i	$0.520600\pi$
$308$	954.756	−	675.872i	0.176631	−	0.125037i
$309$		0			0
$310$	4162.82	+	7210.22i	0.762685	+	1.32101i
$311$	−3347.24	+	5797.60i	−0.610305	+	1.05708i	0.380884	+	0.924623i	$0.375620\pi$
$311$	−3347.24	+	5797.60i	−0.610305	+	1.05708i	−0.991189	+	0.132456i	$0.957714\pi$
$312$		0			0
$313$	5092.10	+	8819.78i	0.919561	+	1.59273i	0.800083	+	0.599890i	$0.204789\pi$
$313$	5092.10	+	8819.78i	0.919561	+	1.59273i	0.119478	+	0.992837i	$0.461878\pi$
$314$	−6696.58			−1.20353
$315$		0			0
$316$	−3533.16			−0.628974
$317$	1765.83	+	3058.50i	0.312867	+	0.541901i	0.978982	−	0.203948i	$-0.0653773\pi$
$317$	1765.83	+	3058.50i	0.312867	+	0.541901i	−0.666115	+	0.745849i	$0.732044\pi$
$318$		0			0
$319$	−997.683	+	1728.04i	−0.175108	+	0.303296i
$320$	512.379	+	887.467i	0.0895090	+	0.155034i
$321$		0			0
$322$	2065.04	+	950.046i	0.357393	+	0.164422i
$323$	3302.76			0.568949
$324$		0			0
$325$	3314.28	−	5740.50i	0.565671	−	0.979771i
$326$	−2702.56	+	4680.96i	−0.459143	+	0.795260i
$327$		0			0
$328$	4131.46			0.695492
$329$	5527.97	−	3913.25i	0.926342	−	0.655758i
$330$		0			0
$331$	−5436.54	−	9416.37i	−0.902778	−	1.56366i	−0.823873	−	0.566775i	$-0.808191\pi$
$331$	−5436.54	−	9416.37i	−0.902778	−	1.56366i	−0.0789050	−	0.996882i	$-0.525142\pi$
$332$	126.303	−	218.764i	0.0208789	−	0.0361634i
$333$		0			0
$334$	−737.971	−	1278.20i	−0.120898	−	0.209402i
$335$	−1771.30			−0.288885
$336$		0			0
$337$	−978.004			−0.158087			−0.0790434	−	0.996871i	$-0.525187\pi$
$337$	−978.004			−0.158087			−0.0790434	+	0.996871i	$0.525187\pi$
$338$	348.564	+	603.730i	0.0560928	+	0.0971556i
$339$		0			0
$340$	−1183.41	+	2049.73i	−0.188764	+	0.326948i
$341$	2052.63	+	3555.25i	0.325970	+	0.564597i
$342$		0			0
$343$	−1738.61	−	6109.90i	−0.273692	−	0.961817i
$344$	274.014			0.0429473
$345$		0			0
$346$	−3885.12	+	6729.22i	−0.603656	+	1.04556i
$347$	−2759.56	+	4779.70i	−0.426919	+	0.739446i	−0.996598	−	0.0824220i	$-0.973734\pi$
$347$	−2759.56	+	4779.70i	−0.426919	+	0.739446i	0.569678	+	0.821868i	$0.307068\pi$
$348$		0			0
$349$	−234.793			−0.0360120			−0.0180060	−	0.999838i	$-0.505732\pi$
$349$	−234.793			−0.0360120			−0.0180060	+	0.999838i	$0.505732\pi$
$350$	−449.059	−	4845.59i	−0.0685806	−	0.740022i
$351$		0			0
$352$	252.647	+	437.597i	0.0382560	+	0.0662613i
$353$	−1597.40	+	2766.78i	−0.240853	+	0.417170i	−0.960958	−	0.276696i	$-0.910761\pi$
$353$	−1597.40	+	2766.78i	−0.240853	+	0.417170i	0.720104	+	0.693866i	$0.244094\pi$
$354$		0			0
$355$	−4861.19	−	8419.83i	−0.726775	−	1.25881i
$356$	−2702.37			−0.402318
$357$		0			0
$358$	5164.23			0.762396
$359$	−5938.35	−	10285.5i	−0.873020	−	1.51211i	−0.858857	−	0.512215i	$-0.828825\pi$
$359$	−5938.35	−	10285.5i	−0.873020	−	1.51211i	−0.0141629	−	0.999900i	$-0.504508\pi$
$360$		0			0
$361$	−564.365	+	977.509i	−0.0822810	+	0.142515i
$362$	2978.56	+	5159.01i	0.432457	+	0.749038i
$363$		0			0
$364$	3395.54	+	1562.16i	0.488942	+	0.224943i
$365$	4485.90			0.643295
$366$		0			0
$367$	2974.87	−	5152.62i	0.423125	−	0.732874i	−0.573118	−	0.819473i	$-0.694267\pi$
$367$	2974.87	−	5152.62i	0.423125	−	0.732874i	0.996243	+	0.0865989i	$0.0275998\pi$
$368$	−490.944	+	850.340i	−0.0695441	+	0.120454i
$369$		0			0
$370$	−7880.91			−1.10732
$371$	793.525	+	365.070i	0.111045	+	0.0510875i
$372$		0			0
$373$	1311.66	+	2271.86i	0.182078	+	0.315368i	0.942588	−	0.333958i	$-0.108384\pi$
$373$	1311.66	+	2271.86i	0.182078	+	0.315368i	−0.760510	+	0.649326i	$0.775051\pi$
$374$	−583.524	+	1010.69i	−0.0806773	+	0.139737i
$375$		0			0
$376$	1462.81	+	2533.65i	0.200634	+	0.347508i
$377$	−6375.60			−0.870982
$378$		0			0
$379$	3206.76			0.434618			0.217309	−	0.976103i	$-0.430272\pi$
$379$	3206.76			0.434618			0.217309	+	0.976103i	$0.430272\pi$
$380$	−2862.09	−	4957.28i	−0.386374	−	0.669219i
$381$		0			0
$382$	4127.44	−	7148.94i	0.552823	−	0.957517i
$383$	−3223.28	−	5582.89i	−0.430031	−	0.744836i	0.566844	−	0.823825i	$-0.308164\pi$
$383$	−3223.28	−	5582.89i	−0.430031	−	0.744836i	−0.996875	+	0.0789891i	$0.974831\pi$
$384$		0			0
$385$	−432.097	−	4662.57i	−0.0571993	−	0.617211i
$386$	6035.40			0.795839
$387$		0			0
$388$	−1952.38	+	3381.62i	−0.255457	+	0.442464i
$389$	1476.74	−	2557.79i	0.192478	−	0.333381i	−0.753593	−	0.657341i	$-0.771681\pi$
$389$	1476.74	−	2557.79i	0.192478	−	0.333381i	0.946071	+	0.323960i	$0.105014\pi$
$390$		0			0
$391$	−2267.81			−0.293320
$392$	2697.27	−	504.263i	0.347532	−	0.0649722i
$393$		0			0
$394$	3959.32	+	6857.74i	0.506263	+	0.876873i
$395$	−7071.55	+	12248.3i	−0.900780	+	1.56020i
$396$		0			0
$397$	2303.53	+	3989.83i	0.291211	+	0.504393i	0.974096	−	0.226133i	$-0.0726084\pi$
$397$	2303.53	+	3989.83i	0.291211	+	0.504393i	−0.682885	+	0.730526i	$0.739275\pi$
$398$	−9384.42			−1.18191
$399$		0			0
$400$	2102.07			0.262759
$401$	−5282.15	−	9148.95i	−0.657800	−	1.13934i	−0.981184	−	0.193076i	$-0.938154\pi$
$401$	−5282.15	−	9148.95i	−0.657800	−	1.13934i	0.323384	−	0.946268i	$-0.395180\pi$
$402$		0			0
$403$	−6558.56	+	11359.8i	−0.810682	+	1.40414i
$404$	2455.13	+	4252.41i	0.302345	+	0.523676i
$405$		0			0
$406$	−3820.31	+	2704.40i	−0.466992	+	0.330583i
$407$	−3885.96			−0.473267
$408$		0			0
$409$	1360.31	−	2356.12i	0.164457	−	0.284848i	−0.772005	−	0.635616i	$-0.780746\pi$
$409$	1360.31	−	2356.12i	0.164457	−	0.284848i	0.936462	+	0.350768i	$0.114080\pi$
$410$	8269.03	−	14322.4i	0.996045	−	1.72520i
$411$		0			0
$412$	−3353.66			−0.401026
$413$	−12052.3	−	5544.77i	−1.43596	−	0.660630i
$414$		0			0
$415$	−505.588	−	875.704i	−0.0598032	−	0.103582i
$416$	−807.257	+	1398.21i	−0.0951419	+	0.164791i
$417$		0			0
$418$	−1411.25	−	2444.36i	−0.165136	−	0.286023i
$419$	13647.0			1.59116			0.795582	−	0.605846i	$-0.207165\pi$
$419$	13647.0			1.59116			0.795582	+	0.605846i	$0.207165\pi$
$420$		0			0
$421$	9396.95			1.08784			0.543918	−	0.839138i	$-0.316940\pi$
$421$	9396.95			1.08784			0.543918	+	0.839138i	$0.316940\pi$
$422$	−1542.58	−	2671.83i	−0.177942	−	0.308205i
$423$		0			0
$424$	−188.653	+	326.756i	−0.0216080	+	0.0374261i
$425$	2427.52	+	4204.58i	0.277063	+	0.479888i
$426$		0			0
$427$	−10527.1	+	7452.14i	−1.19307	+	0.844577i
$428$	1726.22			0.194953
$429$		0			0
$430$	548.435	−	949.916i	0.0615067	−	0.106533i
$431$	−3631.56	+	6290.04i	−0.405861	+	0.702971i	−0.994421	−	0.105482i	$-0.966362\pi$
$431$	−3631.56	+	6290.04i	−0.405861	+	0.702971i	0.588561	+	0.808453i	$0.299695\pi$
$432$		0			0
$433$	3288.50			0.364977			0.182489	−	0.983208i	$-0.441585\pi$
$433$	3288.50			0.364977			0.182489	+	0.983208i	$0.441585\pi$
$434$	888.634	+	9588.85i	0.0982853	+	1.06055i
$435$		0			0
$436$	2614.16	+	4527.85i	0.287146	+	0.497351i
$437$	2742.35	−	4749.90i	0.300194	−	0.519951i
$438$		0			0
$439$	−443.607	−	768.350i	−0.0482283	−	0.0835339i	0.840903	−	0.541185i	$-0.182024\pi$
$439$	−443.607	−	768.350i	−0.0482283	−	0.0835339i	−0.889132	+	0.457651i	$0.848691\pi$
$440$	2022.67			0.219152
$441$		0			0
$442$	−3728.96			−0.401286
$443$	−3940.57	−	6825.27i	−0.422624	−	0.732006i	0.573571	−	0.819156i	$-0.305557\pi$
$443$	−3940.57	−	6825.27i	−0.422624	−	0.732006i	−0.996195	+	0.0871497i	$0.972224\pi$
$444$		0			0
$445$	−5408.74	+	9368.22i	−0.576177	+	0.997969i
$446$	5343.30	+	9254.86i	0.567292	+	0.982579i
$447$		0			0
$448$	109.377	+	1180.24i	0.0115348	+	0.124467i
$449$	13593.0			1.42872			0.714359	−	0.699780i	$-0.246719\pi$
$449$	13593.0			1.42872			0.714359	+	0.699780i	$0.246719\pi$
$450$		0			0
$451$	4077.34	−	7062.16i	0.425708	−	0.737348i
$452$	−2017.89	+	3495.09i	−0.209986	+	0.363706i
$453$		0			0
$454$	−7611.97			−0.786889
$455$	12211.6	−	8644.59i	1.25822	−	0.890692i
$456$		0			0
$457$	4175.62	+	7232.39i	0.427412	+	0.740300i	0.996642	−	0.0818784i	$-0.0260919\pi$
$457$	4175.62	+	7232.39i	0.427412	+	0.740300i	−0.569230	+	0.822178i	$0.692759\pi$
$458$	1198.85	−	2076.47i	0.122312	−	0.211850i
$459$		0			0
$460$	1965.23	+	3403.88i	0.199194	+	0.345014i
$461$	−9295.22			−0.939093			−0.469546	−	0.882908i	$-0.655583\pi$
$461$	−9295.22			−0.939093			−0.469546	+	0.882908i	$0.655583\pi$
$462$		0			0
$463$	272.006			0.0273028			0.0136514	−	0.999907i	$-0.495654\pi$
$463$	272.006			0.0273028			0.0136514	+	0.999907i	$0.495654\pi$
$464$	−1010.93	−	1750.97i	−0.101144	−	0.175187i
$465$		0			0
$466$	−2611.74	+	4523.67i	−0.259628	+	0.449689i
$467$	4440.72	+	7691.56i	0.440026	+	0.762147i	0.997691	−	0.0679189i	$-0.0216359\pi$
$467$	4440.72	+	7691.56i	0.440026	+	0.762147i	−0.557665	+	0.830066i	$0.688303\pi$
$468$		0			0
$469$	−1861.26	−	856.294i	−0.183252	−	0.0843070i
$470$	11711.1			1.14935
$471$		0			0
$472$	2865.31	−	4962.85i	0.279420	−	0.483970i
$473$	270.425	−	468.390i	0.0262878	−	0.0455319i
$474$		0			0
$475$	−11741.9			−1.13422
$476$	−2234.42	+	1581.74i	−0.215156	+	0.152309i
$477$		0			0
$478$	3318.79	+	5748.31i	0.317569	+	0.550045i
$479$	−9482.73	+	16424.6i	−0.904545	+	1.56672i	−0.0830174	+	0.996548i	$0.526456\pi$
$479$	−9482.73	+	16424.6i	−0.904545	+	1.56672i	−0.821527	+	0.570169i	$0.806878\pi$
$480$		0			0
$481$	−6208.22	−	10752.9i	−0.588504	−	1.01932i
$482$	5552.12			0.524672
$483$		0			0
$484$	−4326.65			−0.406335
$485$	7815.31	+	13536.5i	0.731701	+	1.26734i
$486$		0			0
$487$	−8984.26	+	15561.2i	−0.835966	+	1.44794i	0.0572748	+	0.998358i	$0.481759\pi$
$487$	−8984.26	+	15561.2i	−0.835966	+	1.44794i	−0.893241	+	0.449578i	$0.851574\pi$
$488$	−2785.67	−	4824.92i	−0.258405	−	0.447570i
$489$		0			0
$490$	3650.42	−	10359.8i	0.336549	−	0.955119i
$491$	5379.38			0.494435			0.247218	−	0.968960i	$-0.420484\pi$
$491$	5379.38			0.494435			0.247218	+	0.968960i	$0.420484\pi$
$492$		0			0
$493$	2334.88	−	4044.13i	0.213302	−	0.369449i
$494$	4509.24	−	7810.24i	0.410689	−	0.711335i
$495$		0			0
$496$	−4159.74			−0.376568
$497$	−1037.71	−	11197.5i	−0.0936577	−	1.01062i
$498$		0			0
$499$	−5839.80	−	10114.8i	−0.523899	−	0.907419i	−0.999613	−	0.0278191i	$-0.991144\pi$
$499$	−5839.80	−	10114.8i	−0.523899	−	0.907419i	0.475714	−	0.879600i	$-0.342190\pi$
$500$	204.288	−	353.837i	0.0182721	−	0.0316482i
$501$		0			0
$502$	−1339.17	−	2319.52i	−0.119064	−	0.206225i
$503$	1522.74			0.134981			0.0674906	−	0.997720i	$-0.478501\pi$
$503$	1522.74			0.134981			0.0674906	+	0.997720i	$0.478501\pi$
$504$		0			0
$505$	19655.6			1.73200
$506$	969.026	+	1678.40i	0.0851353	+	0.147459i
$507$		0			0
$508$	1764.53	−	3056.25i	0.154111	−	0.266928i
$509$	−2821.79	−	4887.49i	−0.245725	−	0.425607i	0.716611	−	0.697474i	$-0.245693\pi$
$509$	−2821.79	−	4887.49i	−0.245725	−	0.425607i	−0.962335	+	0.271866i	$0.912359\pi$
$510$		0			0
$511$	4713.74	+	2168.61i	0.408070	+	0.187737i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	3027.23	−	5243.32i	0.259777	−	0.449948i
$515$	−6712.28	+	11626.0i	−0.574327	+	0.994764i
$516$		0			0
$517$	5774.57			0.491229
$518$	−8281.18	−	3809.85i	−0.702421	−	0.323156i
$519$		0			0
$520$	3231.42	+	5596.98i	0.272514	+	0.472008i
$521$	8363.93	−	14486.8i	0.703322	−	1.21819i	−0.263972	−	0.964530i	$-0.585033\pi$
$521$	8363.93	−	14486.8i	0.703322	−	1.21819i	0.967294	−	0.253659i	$-0.0816340\pi$
$522$		0			0
$523$	2319.53	+	4017.55i	0.193931	+	0.335899i	0.946550	−	0.322558i	$-0.104543\pi$
$523$	2319.53	+	4017.55i	0.193931	+	0.335899i	−0.752618	+	0.658457i	$0.771209\pi$
$524$	−8191.46			−0.682912
$525$		0			0
$526$	12426.6			1.03009
$527$	−4803.76	−	8320.36i	−0.397069	−	0.687743i
$528$		0			0
$529$	4200.48	−	7275.45i	0.345236	−	0.597966i
$530$	755.170	+	1307.99i	0.0618915	+	0.107199i
$531$		0			0
$532$	−610.968	−	6592.68i	−0.0497910	−	0.537272i
$533$	26055.8			2.11746
$534$		0			0
$535$	3455.00	−	5984.23i	0.279201	−	0.483590i
$536$	442.497	−	766.427i	0.0356585	−	0.0617623i
$537$		0			0
$538$	−1237.09			−0.0991349
$539$	1799.97	−	5108.26i	0.143841	−	0.408216i
$540$		0			0
$541$	−8647.63	−	14978.1i	−0.687229	−	1.19032i	−0.972731	−	0.231937i	$-0.925494\pi$
$541$	−8647.63	−	14978.1i	−0.687229	−	1.19032i	0.285502	−	0.958378i	$-0.407840\pi$
$542$	2736.42	−	4739.62i	0.216862	−	0.375617i
$543$		0			0
$544$	−591.269	−	1024.11i	−0.0466001	−	0.0807138i
$545$	20928.7			1.64493
$546$		0			0
$547$	2832.09			0.221373			0.110687	−	0.993855i	$-0.464695\pi$
$547$	2832.09			0.221373			0.110687	+	0.993855i	$0.464695\pi$
$548$	5003.17	+	8665.75i	0.390009	+	0.675515i
$549$		0			0
$550$	2074.53	−	3593.20i	0.160833	−	0.278572i
$551$	5646.91	+	9780.73i	0.436600	+	0.756213i
$552$		0			0
$553$	−13351.9	+	9451.79i	−1.02673	+	0.726819i
$554$	8262.82			0.633671
$555$		0			0
$556$	1898.06	−	3287.53i	0.144776	−	0.250760i
$557$	4779.35	−	8278.07i	0.363568	−	0.629718i	−0.624977	−	0.780643i	$-0.714892\pi$
$557$	4779.35	−	8278.07i	0.363568	−	0.629718i	0.988545	+	0.150925i	$0.0482251\pi$
$558$		0			0
$559$	1728.13			0.130755
$560$	4310.41	+	1983.05i	0.325265	+	0.149642i
$561$		0			0
$562$	−1741.24	−	3015.92i	−0.130694	−	0.226368i
$563$	−2642.28	+	4576.56i	−0.197795	+	0.342591i	−0.947813	−	0.318826i	$-0.896711\pi$
$563$	−2642.28	+	4576.56i	−0.197795	+	0.342591i	0.750018	+	0.661417i	$0.230045\pi$
$564$		0			0
$565$	8077.53	+	13990.7i	0.601459	+	1.04176i
$566$	−6830.14			−0.507230
$567$		0			0
$568$	4857.59			0.358838
$569$	2532.77	+	4386.89i	0.186607	+	0.323213i	0.944117	−	0.329611i	$-0.106918\pi$
$569$	2532.77	+	4386.89i	0.186607	+	0.323213i	−0.757510	+	0.652824i	$0.773584\pi$
$570$		0			0
$571$	6695.36	−	11596.7i	0.490704	−	0.849925i	−0.509239	−	0.860625i	$-0.670073\pi$
$571$	6695.36	−	11596.7i	0.490704	−	0.849925i	0.999943	+	0.0107008i	$0.00340623\pi$
$572$	1593.37	+	2759.79i	0.116472	+	0.201735i
$573$		0			0
$574$	15612.8	−	11052.3i	1.13531	−	0.803686i
$575$	8062.48			0.584746
$576$		0			0
$577$	12027.8	−	20832.7i	0.867804	−	1.50308i	0.00356838	−	0.999994i	$-0.498864\pi$
$577$	12027.8	−	20832.7i	0.867804	−	1.50308i	0.864236	−	0.503087i	$-0.167803\pi$
$578$	−3547.38	+	6144.24i	−0.255279	+	0.442157i
$579$		0			0
$580$	−8093.39			−0.579414
$581$	−107.927	−	1164.60i	−0.00770669	−	0.0831593i
$582$		0			0
$583$	372.363	+	644.952i	0.0264523	+	0.0458168i
$584$	−1120.65	+	1941.01i	−0.0794052	+	0.137534i
$585$		0			0
$586$	−8652.18	−	14986.0i	−0.609929	−	1.05643i
$587$	27108.0			1.90608			0.953039	−	0.302849i	$-0.0979378\pi$
$587$	27108.0			1.90608			0.953039	+	0.302849i	$0.0979378\pi$
$588$		0			0
$589$	23235.8			1.62549
$590$	−11469.7	−	19866.1i	−0.800340	−	1.38623i
$591$		0			0
$592$	1968.77	−	3410.01i	0.136682	−	0.236741i
$593$	−1276.43	−	2210.85i	−0.0883927	−	0.153101i	0.818439	−	0.574593i	$-0.194840\pi$
$593$	−1276.43	−	2210.85i	−0.0883927	−	0.153101i	−0.906832	+	0.421492i	$0.861506\pi$
$594$		0			0
$595$	1011.24	+	10911.8i	0.0696752	+	0.751833i
$596$	3190.19			0.219254
$597$		0			0
$598$	−3096.23	+	5362.84i	−0.211730	+	0.366727i
$599$	4297.99	−	7444.33i	0.293173	−	0.507791i	−0.681385	−	0.731925i	$-0.738622\pi$
$599$	4297.99	−	7444.33i	0.293173	−	0.507791i	0.974558	+	0.224134i	$0.0719553\pi$
$600$		0			0
$601$	6026.80			0.409049			0.204524	−	0.978861i	$-0.434435\pi$
$601$	6026.80			0.409049			0.204524	+	0.978861i	$0.434435\pi$
$602$	1035.51	−	733.034i	0.0701064	−	0.0496283i
$603$		0			0
$604$	−5419.46	−	9386.78i	−0.365091	−	0.632355i
$605$	−8659.71	+	14999.1i	−0.581929	+	1.00793i
$606$		0			0
$607$	−652.013	−	1129.32i	−0.0435986	−	0.0755151i	0.843403	−	0.537282i	$-0.180549\pi$
$607$	−652.013	−	1129.32i	−0.0435986	−	0.0755151i	−0.887001	+	0.461767i	$0.847216\pi$
$608$	2859.97			0.190768
$609$		0			0
$610$	−22301.9			−1.48029
$611$	9225.47	+	15979.0i	0.610839	+	1.05800i
$612$		0			0
$613$	−8953.92	+	15508.6i	−0.589960	+	1.02184i	0.404277	+	0.914637i	$0.367523\pi$
$613$	−8953.92	+	15508.6i	−0.589960	+	1.02184i	−0.994237	+	0.107204i	$0.965810\pi$
$614$	−695.737	−	1205.05i	−0.0457291	−	0.0792051i
$615$		0			0
$616$	2125.40	+	977.814i	0.139018	+	0.0639565i
$617$	26896.2			1.75494			0.877472	−	0.479628i	$-0.159228\pi$
$617$	26896.2			1.75494			0.877472	+	0.479628i	$0.159228\pi$
$618$		0			0
$619$	−5252.53	+	9097.65i	−0.341061	+	0.590735i	−0.984630	−	0.174653i	$-0.944120\pi$
$619$	−5252.53	+	9097.65i	−0.341061	+	0.590735i	0.643569	+	0.765388i	$0.277453\pi$
$620$	−8325.64	+	14420.4i	−0.539300	+	0.934094i
$621$		0			0
$622$	−13389.0			−0.863101
$623$	−10212.3	+	7229.29i	−0.656737	+	0.464905i
$624$		0			0
$625$	7393.45	+	12805.8i	0.473181	+	0.819573i
$626$	−10184.2	+	17639.6i	−0.650228	+	1.12623i
$627$		0			0
$628$	−6696.58	−	11598.8i	−0.425514	−	0.737012i
$629$	9094.32			0.576493
$630$		0			0
$631$	−23670.2			−1.49334			−0.746668	−	0.665197i	$-0.768348\pi$
$631$	−23670.2			−1.49334			−0.746668	+	0.665197i	$0.768348\pi$
$632$	−3533.16	−	6119.61i	−0.222376	−	0.385166i
$633$		0			0
$634$	−3531.66	+	6117.01i	−0.221230	+	0.383182i
$635$	−7063.35	−	12234.1i	−0.441418	−	0.764558i
$636$		0			0
$637$	17010.9	−	3180.23i	1.05808	−	0.197811i
$638$	−3990.73			−0.247640
$639$		0			0
$640$	−1024.76	+	1774.93i	−0.0632924	+	0.109626i
$641$	−11664.4	+	20203.4i	−0.718747	+	1.24491i	0.242749	+	0.970089i	$0.421951\pi$
$641$	−11664.4	+	20203.4i	−0.718747	+	1.24491i	−0.961496	+	0.274817i	$0.911383\pi$
$642$		0			0
$643$	13384.9			0.820914			0.410457	−	0.911880i	$-0.365369\pi$
$643$	13384.9			0.820914			0.410457	+	0.911880i	$0.365369\pi$
$644$	419.516	+	4526.81i	0.0256696	+	0.276989i
$645$		0			0
$646$	3302.76	+	5720.55i	0.201154	+	0.348409i
$647$	4779.80	−	8278.86i	0.290438	−	0.503053i	−0.683475	−	0.729974i	$-0.739532\pi$
$647$	4779.80	−	8278.86i	0.290438	−	0.503053i	0.973913	+	0.226920i	$0.0728657\pi$
$648$		0			0
$649$	−5655.54	−	9795.69i	−0.342064	−	0.592472i
$650$	13257.1			0.799979
$651$		0			0
$652$	−10810.2			−0.649327
$653$	−10103.9	−	17500.5i	−0.605507	−	1.04877i	−0.991971	−	0.126464i	$-0.959637\pi$
$653$	−10103.9	−	17500.5i	−0.605507	−	1.04877i	0.386464	−	0.922304i	$-0.373696\pi$
$654$		0			0
$655$	−16395.1	+	28397.1i	−0.978027	+	1.69399i
$656$	4131.46	+	7155.89i	0.245894	+	0.425900i
$657$		0			0
$658$	12305.9	+	5661.47i	0.729080	+	0.335421i
$659$	−22497.2			−1.32984			−0.664920	−	0.746914i	$-0.731534\pi$
$659$	−22497.2			−1.32984			−0.664920	+	0.746914i	$0.731534\pi$
$660$		0			0
$661$	−12408.4	+	21492.0i	−0.730152	+	1.26466i	0.226667	+	0.973972i	$0.427217\pi$
$661$	−12408.4	+	21492.0i	−0.730152	+	1.26466i	−0.956818	+	0.290687i	$0.906116\pi$
$662$	10873.1	−	18832.7i	0.638360	−	1.10567i
$663$		0			0
$664$	505.214			0.0295273
$665$	−24077.5	−	11077.1i	−1.40404	−	0.645942i
$666$		0			0
$667$	−3877.40	−	6715.86i	−0.225088	−	0.389864i
$668$	1475.94	−	2556.41i	0.0854879	−	0.148069i
$669$		0			0
$670$	−1771.30	−	3067.98i	−0.102136	−	0.176905i
$671$	−10996.7			−0.632673
$672$		0			0
$673$	2331.38			0.133534			0.0667669	−	0.997769i	$-0.478732\pi$
$673$	2331.38			0.133534			0.0667669	+	0.997769i	$0.478732\pi$
$674$	−978.004	−	1693.95i	−0.0558922	−	0.0968080i
$675$		0			0
$676$	−697.128	+	1207.46i	−0.0396636	+	0.0686994i
$677$	−837.086	−	1449.88i	−0.0475212	−	0.0823090i	0.841286	−	0.540590i	$-0.181799\pi$
$677$	−837.086	−	1449.88i	−0.0475212	−	0.0823090i	−0.888808	+	0.458281i	$0.848465\pi$
$678$		0			0
$679$	1668.33	+	18002.2i	0.0942924	+	1.01747i
$680$	−4733.66			−0.266952
$681$		0			0
$682$	−4105.25	+	7110.50i	−0.230496	+	0.399231i
$683$	1402.89	−	2429.87i	0.0785944	−	0.136129i	−0.824049	−	0.566518i	$-0.808290\pi$
$683$	1402.89	−	2429.87i	0.0785944	−	0.136129i	0.902644	+	0.430389i	$0.141623\pi$
$684$		0			0
$685$	40055.0			2.23419
$686$	8844.04	−	9121.26i	0.492226	−	0.507655i
$687$		0			0
$688$	274.014	+	474.607i	0.0151842	+	0.0262997i
$689$	−1189.78	+	2060.75i	−0.0657864	+	0.113945i
$690$		0			0
$691$	−2493.67	−	4319.16i	−0.137284	−	0.237784i	0.789183	−	0.614158i	$-0.210504\pi$
$691$	−2493.67	−	4319.16i	−0.137284	−	0.237784i	−0.926468	+	0.376374i	$0.877171\pi$
$692$	−15540.5			−0.853699
$693$		0			0
$694$	−11038.2			−0.603755
$695$	−7597.85	−	13159.9i	−0.414680	−	0.718247i
$696$		0			0
$697$	−9542.20	+	16527.6i	−0.518561	+	0.898173i
$698$	−234.793	−	406.674i	−0.0127322	−	0.0220528i
$699$		0			0
$700$	7943.75	−	5623.39i	0.428922	−	0.303634i
$701$	−15941.5			−0.858921			−0.429460	−	0.903086i	$-0.641296\pi$
$701$	−15941.5			−0.858921			−0.429460	+	0.903086i	$0.641296\pi$
$702$		0			0
$703$	−10997.3	+	19047.9i	−0.590002	+	1.02191i
$704$	−505.293	+	875.193i	−0.0270511	+	0.0468538i
$705$		0			0
$706$	−6389.61			−0.340618
$707$	20653.9	+	9502.04i	1.09868	+	0.505461i
$708$		0			0
$709$	−3071.65	−	5320.25i	−0.162706	−	0.281814i	0.773133	−	0.634244i	$-0.218689\pi$
$709$	−3071.65	−	5320.25i	−0.162706	−	0.281814i	−0.935838	+	0.352430i	$0.885355\pi$
$710$	9722.38	−	16839.7i	0.513908	−	0.890114i
$711$		0			0
$712$	−2702.37	−	4680.64i	−0.142241	−	0.246369i
$713$	−15954.7			−0.838019
$714$		0			0
$715$	12756.4			0.667218
$716$	5164.23	+	8944.70i	0.269548	+	0.466870i
$717$		0			0
$718$	11876.7	−	20571.1i	0.617318	−	1.06923i
$719$	2001.23	+	3466.24i	0.103802	+	0.179790i	0.913248	−	0.407404i	$-0.133566\pi$
$719$	2001.23	+	3466.24i	0.103802	+	0.179790i	−0.809446	+	0.587194i	$0.800233\pi$
$720$		0			0
$721$	−12673.5	+	8971.60i	−0.654628	+	0.463412i
$722$	−2257.46			−0.116363
$723$		0			0
$724$	−5957.12	+	10318.0i	−0.305794	+	0.529650i
$725$	−8300.92	+	14377.6i	−0.425225	+	0.736512i
$726$		0			0
$727$	−37145.8			−1.89500			−0.947498	−	0.319763i	$-0.896397\pi$
$727$	−37145.8			−1.89500			−0.947498	+	0.319763i	$0.896397\pi$
$728$	689.809	+	7443.41i	0.0351181	+	0.378944i
$729$		0			0
$730$	4485.90	+	7769.81i	0.227439	+	0.393936i
$731$	−632.876	+	1096.17i	−0.0320216	+	0.0554630i
$732$		0			0
$733$	97.6418	+	169.121i	0.00492017	+	0.00852198i	0.868475	−	0.495733i	$-0.165101\pi$
$733$	97.6418	+	169.121i	0.00492017	+	0.00852198i	−0.863555	+	0.504255i	$0.831767\pi$
$734$	11899.5			0.598389
$735$		0			0
$736$	−1963.78			−0.0983502
$737$	−873.401	−	1512.78i	−0.0436528	−	0.0756089i
$738$		0			0
$739$	2574.43	−	4459.05i	0.128149	−	0.221960i	−0.794811	−	0.606858i	$-0.792430\pi$
$739$	2574.43	−	4459.05i	0.128149	−	0.221960i	0.922959	+	0.384897i	$0.125763\pi$
$740$	−7880.91	−	13650.1i	−0.391497	−	0.678093i
$741$		0			0
$742$	161.206	+	1739.50i	0.00797580	+	0.0860632i
$743$	−21310.2			−1.05221			−0.526106	−	0.850419i	$-0.676349\pi$
$743$	−21310.2			−1.05221			−0.526106	+	0.850419i	$0.676349\pi$
$744$		0			0
$745$	6385.10	−	11059.3i	0.314003	−	0.543869i
$746$	−2623.31	+	4543.71i	−0.128748	+	0.222999i
$747$		0			0
$748$	−2334.10			−0.114095
$749$	6523.41	−	4617.93i	0.318238	−	0.225281i
$750$		0			0
$751$	−7507.47	−	13003.3i	−0.364782	−	0.631822i	0.623959	−	0.781457i	$-0.285523\pi$
$751$	−7507.47	−	13003.3i	−0.364782	−	0.631822i	−0.988741	+	0.149635i	$0.952190\pi$
$752$	−2925.61	+	5067.31i	−0.141870	+	0.245726i
$753$		0			0
$754$	−6375.60	−	11042.9i	−0.307939	−	0.533365i
$755$	−43387.8			−2.09145
$756$		0			0
$757$	3951.68			0.189731			0.0948654	−	0.995490i	$-0.469758\pi$
$757$	3951.68			0.189731			0.0948654	+	0.995490i	$0.469758\pi$
$758$	3206.76	+	5554.27i	0.153661	+	0.266148i
$759$		0			0
$760$	5724.18	−	9914.57i	0.273208	−	0.473209i
$761$	16117.9	+	27917.1i	0.767773	+	1.32982i	0.938768	+	0.344549i	$0.111968\pi$
$761$	16117.9	+	27917.1i	0.767773	+	1.32982i	−0.170996	+	0.985272i	$0.554698\pi$
$762$		0			0
$763$	21991.7	+	10117.5i	1.04345	+	0.480051i
$764$	16509.8			0.781809
$765$		0			0
$766$	6446.56	−	11165.8i	0.304078	−	0.526678i
$767$	18070.6	−	31299.2i	0.850707	−	1.47347i
$768$		0			0
$769$	9297.59			0.435994			0.217997	−	0.975949i	$-0.430048\pi$
$769$	9297.59			0.435994			0.217997	+	0.975949i	$0.430048\pi$
$770$	7643.70	−	5410.98i	0.357740	−	0.253244i
$771$		0			0
$772$	6035.40	+	10453.6i	0.281372	+	0.487350i
$773$	17411.2	−	30157.0i	0.810138	−	1.40320i	−0.102630	−	0.994720i	$-0.532726\pi$
$773$	17411.2	−	30157.0i	0.810138	−	1.40320i	0.912767	−	0.408480i	$-0.133941\pi$
$774$		0			0
$775$	17078.2	+	29580.4i	0.791572	+	1.37104i
$776$	−7809.53			−0.361270
$777$		0			0
$778$	5906.97			0.272205
$779$	−23077.8	−	39972.0i	−1.06142	−	1.83844i
$780$		0			0
$781$	4793.96	−	8303.39i	0.219643	−	0.380433i
$782$	−2267.81	−	3927.97i	−0.103704	−	0.179621i
$783$		0			0
$784$	3570.68	+	4167.54i	0.162658	+	0.189848i

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(8.00592 + 13.8667i) q^{5} +(1.70902 + 18.4412i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(8.00592 + 13.8667i) q^{5} +(1.70902 + 18.4412i) q^{7} -8.00000 q^{8} +(-16.0118 + 27.7333i) q^{10} +(-7.89520 + 13.6749i) q^{11} -50.4536 q^{13} +(-30.2321 + 21.4013i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(18.4772 - 32.0034i) q^{17} +(44.6871 + 77.4002i) q^{19} -64.0474 q^{20} -31.5808 q^{22} +(-30.6840 - 53.1462i) q^{23} +(-65.6896 + 113.778i) q^{25} +(-50.4536 - 87.3882i) q^{26} +(-67.3004 - 30.9623i) q^{28} +126.366 q^{29} +(129.992 - 225.153i) q^{31} +(16.0000 - 27.7128i) q^{32} +73.9087 q^{34} +(-242.036 + 171.338i) q^{35} +(123.048 + 213.126i) q^{37} +(-89.3741 + 154.800i) q^{38} +(-64.0474 - 110.933i) q^{40} -516.432 q^{41} -34.2518 q^{43} +(-31.5808 - 54.6996i) q^{44} +(61.3680 - 106.292i) q^{46} +(-182.851 - 316.707i) q^{47} +(-337.159 + 63.0328i) q^{49} -262.759 q^{50} +(100.907 - 174.776i) q^{52} +(23.5816 - 40.8445i) q^{53} -252.834 q^{55} +(-13.6721 - 147.530i) q^{56} +(126.366 + 218.872i) q^{58} +(-358.163 + 620.357i) q^{59} +(348.209 + 603.115i) q^{61} +519.968 q^{62} +64.0000 q^{64} +(-403.928 - 699.623i) q^{65} +(-55.3121 + 95.8034i) q^{67} +(73.9087 + 128.014i) q^{68} +(-538.802 - 247.882i) q^{70} -607.199 q^{71} +(140.081 - 242.627i) q^{73} +(-246.096 + 426.251i) q^{74} -357.496 q^{76} +(-265.675 - 122.227i) q^{77} +(441.645 + 764.951i) q^{79} +(128.095 - 221.867i) q^{80} +(-516.432 - 894.487i) q^{82} -63.1517 q^{83} +591.707 q^{85} +(-34.2518 - 59.3258i) q^{86} +(63.1616 - 109.399i) q^{88} +(337.796 + 585.080i) q^{89} +(-86.2261 - 930.426i) q^{91} +245.472 q^{92} +(365.701 - 633.413i) q^{94} +(-715.522 + 1239.32i) q^{95} +976.191 q^{97} +(-446.335 - 520.943i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more