LMFDB - Embedded newform 378.4.g.e.163.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
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			163.4
Root			$4.05517 + 7.02376i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.e.109.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.00592 - 13.8667i) q^{5} +(1.70902 - 18.4412i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(8.00592 - 13.8667i) q^{5} +(1.70902 - 18.4412i) q^{7} -8.00000 q^{8} +(-16.0118 - 27.7333i) q^{10} +(-7.89520 - 13.6749i) q^{11} -50.4536 q^{13} +(-30.2321 - 21.4013i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(18.4772 + 32.0034i) q^{17} +(44.6871 - 77.4002i) q^{19} -64.0474 q^{20} -31.5808 q^{22} +(-30.6840 + 53.1462i) q^{23} +(-65.6896 - 113.778i) q^{25} +(-50.4536 + 87.3882i) q^{26} +(-67.3004 + 30.9623i) q^{28} +126.366 q^{29} +(129.992 + 225.153i) q^{31} +(16.0000 + 27.7128i) q^{32} +73.9087 q^{34} +(-242.036 - 171.338i) q^{35} +(123.048 - 213.126i) q^{37} +(-89.3741 - 154.800i) q^{38} +(-64.0474 + 110.933i) q^{40} -516.432 q^{41} -34.2518 q^{43} +(-31.5808 + 54.6996i) q^{44} +(61.3680 + 106.292i) q^{46} +(-182.851 + 316.707i) q^{47} +(-337.159 - 63.0328i) q^{49} -262.759 q^{50} +(100.907 + 174.776i) q^{52} +(23.5816 + 40.8445i) q^{53} -252.834 q^{55} +(-13.6721 + 147.530i) q^{56} +(126.366 - 218.872i) q^{58} +(-358.163 - 620.357i) q^{59} +(348.209 - 603.115i) q^{61} +519.968 q^{62} +64.0000 q^{64} +(-403.928 + 699.623i) q^{65} +(-55.3121 - 95.8034i) q^{67} +(73.9087 - 128.014i) q^{68} +(-538.802 + 247.882i) q^{70} -607.199 q^{71} +(140.081 + 242.627i) q^{73} +(-246.096 - 426.251i) q^{74} -357.496 q^{76} +(-265.675 + 122.227i) q^{77} +(441.645 - 764.951i) q^{79} +(128.095 + 221.867i) q^{80} +(-516.432 + 894.487i) q^{82} -63.1517 q^{83} +591.707 q^{85} +(-34.2518 + 59.3258i) q^{86} +(63.1616 + 109.399i) q^{88} +(337.796 - 585.080i) q^{89} +(-86.2261 + 930.426i) q^{91} +245.472 q^{92} +(365.701 + 633.413i) q^{94} +(-715.522 - 1239.32i) q^{95} +976.191 q^{97} +(-446.335 + 520.943i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

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

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	−0.246473	−	0.969150i	$-0.579272\pi$
$5$	8.00592	−	13.8667i	0.716072	−	1.24027i	0.962545	−	0.271123i	$-0.0873949\pi$
$6$		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	0.737299	−	0.675567i	$-0.236101\pi$
$11$	−7.89520	−	13.6749i	−0.216409	−	0.374831i	−0.953707	+	0.300736i	$0.902768\pi$
$12$		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	0.967199	−	0.254022i	$-0.0817534\pi$
$17$	18.4772	+	32.0034i	0.263610	+	0.456586i	−0.703588	+	0.710608i	$0.748420\pi$
$18$		0			0
$19$	44.6871	−	77.4002i	0.539574	−	0.934570i	−0.459352	−	0.888254i	$-0.651919\pi$
$19$	44.6871	−	77.4002i	0.539574	−	0.934570i	0.998927	−	0.0463161i	$-0.0147482\pi$
$20$	−64.0474			−0.716072
$21$		0			0
$22$	−31.5808			−0.306048
$23$	−30.6840	+	53.1462i	−0.278176	+	0.481816i	−0.970932	−	0.239357i	$-0.923063\pi$
$23$	−30.6840	+	53.1462i	−0.278176	+	0.481816i	0.692755	+	0.721173i	$0.256397\pi$
$24$		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	0.946295	+	0.323303i	$0.104793\pi$
$31$	129.992	+	225.153i	0.753137	+	1.30447i	−0.193159	+	0.981167i	$0.561873\pi$
$32$	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	−0.451767	−	0.892136i	$-0.649206\pi$
$37$	123.048	−	213.126i	0.546729	−	0.946963i	0.998496	−	0.0548266i	$-0.0174606\pi$
$38$	−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	0.429335	+	0.903145i	$0.358748\pi$
$47$	−182.851	+	316.707i	−0.567479	+	0.982902i	−0.996814	+	0.0797571i	$0.974586\pi$
$48$		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	0.894965	−	0.446137i	$-0.147201\pi$
$53$	23.5816	+	40.8445i	0.0611166	+	0.105857i	−0.833848	+	0.551994i	$0.813867\pi$
$54$		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	−0.925769	−	0.378090i	$-0.876581\pi$
$59$	−358.163	−	620.357i	−0.790320	−	1.36887i	0.135449	−	0.990784i	$-0.456752\pi$
$60$		0			0
$61$	348.209	−	603.115i	0.730878	−	1.26592i	−0.225630	−	0.974213i	$-0.572444\pi$
$61$	348.209	−	603.115i	0.730878	−	1.26592i	0.956508	−	0.291705i	$-0.0942227\pi$
$62$	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	0.811181	−	0.584796i	$-0.198825\pi$
$67$	−55.3121	−	95.8034i	−0.100857	−	0.174690i	−0.912038	+	0.410105i	$0.865492\pi$
$68$	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	0.956197	−	0.292724i	$-0.0945618\pi$
$73$	140.081	+	242.627i	0.224592	+	0.389004i	−0.731605	+	0.681729i	$0.761228\pi$
$74$	−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	−0.358784	−	0.933420i	$-0.616809\pi$
$79$	441.645	−	764.951i	0.628974	−	1.08941i	0.987758	−	0.155994i	$-0.0498580\pi$
$80$	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	−0.591687	−	0.806168i	$-0.701538\pi$
$89$	337.796	−	585.080i	0.402318	−	0.696836i	0.994005	+	0.109332i	$0.0348712\pi$
$90$		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	0.992100	+	0.125446i	$0.0400361\pi$
$101$	613.782	+	1063.10i	0.604689	+	1.04735i	−0.387411	+	0.921907i	$0.626631\pi$
$102$		0			0
$103$	419.207	−	726.088i	0.401026	−	0.694598i	−0.592824	−	0.805332i	$-0.701987\pi$
$103$	419.207	−	726.088i	0.401026	−	0.694598i	0.993850	+	0.110734i	$0.0353203\pi$
$104$	403.629			0.380568
$105$		0			0
$106$	94.3264			0.0864320
$107$	−215.777	+	373.737i	−0.194953	+	0.337669i	−0.946885	−	0.321572i	$-0.895789\pi$
$107$	−215.777	+	373.737i	−0.194953	+	0.337669i	0.751932	+	0.659241i	$0.229122\pi$
$108$		0			0
$109$	653.539	+	1131.96i	0.574291	+	0.994701i	0.996118	+	0.0880250i	$0.0280555\pi$
$109$	653.539	+	1131.96i	0.574291	+	0.994701i	−0.421827	+	0.906676i	$0.638611\pi$
$110$	−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	−0.291176	−	0.956669i	$-0.594047\pi$
$131$	1023.93	−	1773.50i	0.682912	−	1.18284i	0.974088	−	0.226168i	$-0.0726200\pi$
$132$		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	0.931931	+	0.362637i	$0.118123\pi$
$137$	1250.79	+	2166.44i	0.780018	+	1.35103i	−0.151913	+	0.988394i	$0.548543\pi$
$138$		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	−0.954580	−	0.297955i	$-0.903696\pi$
$149$	−398.773	+	690.696i	−0.219254	+	0.379758i	0.735326	+	0.677713i	$0.237029\pi$
$150$		0			0
$151$	−1354.87	−	2346.70i	−0.730181	−	1.26471i	−0.956806	−	0.290729i	$-0.906102\pi$
$151$	−1354.87	−	2346.70i	−0.730181	−	1.26471i	0.226624	−	0.973982i	$-0.427231\pi$
$152$	−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	−0.880282	−	0.474451i	$-0.842647\pi$
$157$	−1674.15	−	2899.70i	−0.851028	−	1.47402i	0.0292542	−	0.999572i	$-0.490687\pi$
$158$	−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	−0.333957	−	0.942588i	$-0.608384\pi$
$163$	1351.28	−	2340.48i	0.649327	−	1.12467i	0.983284	−	0.182079i	$-0.0582826\pi$
$164$	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	−0.0241476	−	0.999708i	$-0.507687\pi$
$173$	1942.56	−	3364.61i	0.853699	−	1.47865i	0.877847	−	0.478942i	$-0.158979\pi$
$174$		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	0.998953	+	0.0457482i	$0.0145672\pi$
$179$	1291.06	+	2236.18i	0.539096	+	0.933741i	−0.459857	+	0.887993i	$0.652099\pi$
$180$		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	0.149077	+	0.988826i	$0.452370\pi$
$191$	−2063.72	+	3574.47i	−0.781809	+	1.35413i	−0.930887	+	0.365308i	$0.880964\pi$
$192$		0			0
$193$	1508.85	+	2613.41i	0.562743	+	0.974700i	0.997256	+	0.0740342i	$0.0235874\pi$
$193$	1508.85	+	2613.41i	0.562743	+	0.974700i	−0.434512	+	0.900666i	$0.643079\pi$
$194$	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	−0.893432	−	0.449199i	$-0.851709\pi$
$199$	−2346.11	−	4063.57i	−0.835734	−	1.44753i	0.0576981	−	0.998334i	$-0.481624\pi$
$200$	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	−0.997792	−	0.0664167i	$-0.978843\pi$
$227$	−1902.99	−	3296.08i	−0.556415	−	0.963738i	0.441377	−	0.897322i	$-0.354490\pi$
$228$		0			0
$229$	−599.426	+	1038.24i	−0.172975	+	0.299601i	−0.939459	−	0.342663i	$-0.888671\pi$
$229$	−599.426	+	1038.24i	−0.172975	+	0.299601i	0.766484	+	0.642264i	$0.222005\pi$
$230$	1965.23			0.563406
$231$		0			0
$232$	−1010.93			−0.286080
$233$	1305.87	−	2261.84i	0.367170	−	0.635956i	−0.621952	−	0.783055i	$-0.713660\pi$
$233$	1305.87	−	2261.84i	0.367170	−	0.635956i	0.989122	+	0.147099i	$0.0469936\pi$
$234$		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	0.989720	−	0.143022i	$-0.0456819\pi$
$241$	1388.03	+	2404.14i	0.370999	+	0.642590i	−0.618720	+	0.785611i	$0.712349\pi$
$242$	−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	−0.989155	−	0.146874i	$-0.953079\pi$
$257$	−1513.62	+	2621.66i	−0.367381	+	0.636322i	0.621774	+	0.783196i	$0.286412\pi$
$258$		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	0.957567	+	0.288210i	$0.0930602\pi$
$263$	3106.65	+	5380.87i	0.728381	+	1.26159i	−0.229186	+	0.973383i	$0.573606\pi$
$264$		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	0.828846	−	0.559478i	$-0.188998\pi$
$269$	−309.272	−	535.674i	−0.0700990	−	0.121415i	−0.898945	+	0.438063i	$0.855665\pi$
$270$		0			0
$271$	−1368.21	+	2369.81i	−0.306690	+	0.531202i	−0.977636	−	0.210304i	$-0.932555\pi$
$271$	−1368.21	+	2369.81i	−0.306690	+	0.531202i	0.670946	+	0.741506i	$0.265888\pi$
$272$	−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	0.998261	−	0.0589561i	$-0.0187772\pi$
$277$	2065.70	+	3577.91i	0.448073	+	0.776085i	−0.550188	+	0.835041i	$0.685444\pi$
$278$	−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	0.629072	−	0.777347i	$-0.283435\pi$
$283$	−1707.54	−	2957.54i	−0.358666	−	0.621227i	−0.987738	+	0.156119i	$0.950102\pi$
$284$	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	−0.991189	−	0.132456i	$-0.957714\pi$
$311$	−3347.24	−	5797.60i	−0.610305	−	1.05708i	0.380884	−	0.924623i	$-0.375620\pi$
$312$		0			0
$313$	5092.10	−	8819.78i	0.919561	−	1.59273i	0.119478	−	0.992837i	$-0.461878\pi$
$313$	5092.10	−	8819.78i	0.919561	−	1.59273i	0.800083	−	0.599890i	$-0.204789\pi$
$314$	−6696.58			−1.20353
$315$		0			0
$316$	−3533.16			−0.628974
$317$	1765.83	−	3058.50i	0.312867	−	0.541901i	−0.666115	−	0.745849i	$-0.732044\pi$
$317$	1765.83	−	3058.50i	0.312867	−	0.541901i	0.978982	+	0.203948i	$0.0653773\pi$
$318$		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	−0.0789050	+	0.996882i	$0.525142\pi$
$331$	−5436.54	+	9416.37i	−0.902778	+	1.56366i	−0.823873	+	0.566775i	$0.808191\pi$
$332$	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	0.569678	−	0.821868i	$-0.307068\pi$
$347$	−2759.56	−	4779.70i	−0.426919	−	0.739446i	−0.996598	+	0.0824220i	$0.973734\pi$
$348$		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	0.720104	−	0.693866i	$-0.244094\pi$
$353$	−1597.40	−	2766.78i	−0.240853	−	0.417170i	−0.960958	+	0.276696i	$0.910761\pi$
$354$		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	−0.0141629	+	0.999900i	$0.504508\pi$
$359$	−5938.35	+	10285.5i	−0.873020	+	1.51211i	−0.858857	+	0.512215i	$0.828825\pi$
$360$		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	0.996243	−	0.0865989i	$-0.0275998\pi$
$367$	2974.87	+	5152.62i	0.423125	+	0.732874i	−0.573118	+	0.819473i	$0.694267\pi$
$368$	−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	−0.760510	−	0.649326i	$-0.775051\pi$
$373$	1311.66	−	2271.86i	0.182078	−	0.315368i	0.942588	+	0.333958i	$0.108384\pi$
$374$	−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	−0.996875	−	0.0789891i	$-0.974831\pi$
$383$	−3223.28	+	5582.89i	−0.430031	+	0.744836i	0.566844	+	0.823825i	$0.308164\pi$
$384$		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	0.946071	−	0.323960i	$-0.105014\pi$
$389$	1476.74	+	2557.79i	0.192478	+	0.333381i	−0.753593	+	0.657341i	$0.771681\pi$
$390$		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	−0.682885	−	0.730526i	$-0.739275\pi$
$397$	2303.53	−	3989.83i	0.291211	−	0.504393i	0.974096	+	0.226133i	$0.0726084\pi$
$398$	−9384.42			−1.18191
$399$		0			0
$400$	2102.07			0.262759
$401$	−5282.15	+	9148.95i	−0.657800	+	1.13934i	0.323384	+	0.946268i	$0.395180\pi$
$401$	−5282.15	+	9148.95i	−0.657800	+	1.13934i	−0.981184	+	0.193076i	$0.938154\pi$
$402$		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	0.936462	−	0.350768i	$-0.114080\pi$
$409$	1360.31	+	2356.12i	0.164457	+	0.284848i	−0.772005	+	0.635616i	$0.780746\pi$
$410$	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	0.588561	−	0.808453i	$-0.299695\pi$
$431$	−3631.56	−	6290.04i	−0.405861	−	0.702971i	−0.994421	+	0.105482i	$0.966362\pi$
$432$		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	−0.889132	−	0.457651i	$-0.848691\pi$
$439$	−443.607	+	768.350i	−0.0482283	+	0.0835339i	0.840903	+	0.541185i	$0.182024\pi$
$440$	2022.67			0.219152
$441$		0			0
$442$	−3728.96			−0.401286
$443$	−3940.57	+	6825.27i	−0.422624	+	0.732006i	−0.996195	−	0.0871497i	$-0.972224\pi$
$443$	−3940.57	+	6825.27i	−0.422624	+	0.732006i	0.573571	+	0.819156i	$0.305557\pi$
$444$		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	−0.569230	−	0.822178i	$-0.692759\pi$
$457$	4175.62	−	7232.39i	0.427412	−	0.740300i	0.996642	+	0.0818784i	$0.0260919\pi$
$458$	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	−0.557665	−	0.830066i	$-0.688303\pi$
$467$	4440.72	−	7691.56i	0.440026	−	0.762147i	0.997691	+	0.0679189i	$0.0216359\pi$
$468$		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	−0.821527	−	0.570169i	$-0.806878\pi$
$479$	−9482.73	−	16424.6i	−0.904545	−	1.56672i	−0.0830174	−	0.996548i	$-0.526456\pi$
$480$		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	−0.893241	−	0.449578i	$-0.851574\pi$
$487$	−8984.26	−	15561.2i	−0.835966	−	1.44794i	0.0572748	−	0.998358i	$-0.481759\pi$
$488$	−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	0.475714	+	0.879600i	$0.342190\pi$
$499$	−5839.80	+	10114.8i	−0.523899	+	0.907419i	−0.999613	+	0.0278191i	$0.991144\pi$
$500$	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	−0.962335	−	0.271866i	$-0.912359\pi$
$509$	−2821.79	+	4887.49i	−0.245725	+	0.425607i	0.716611	+	0.697474i	$0.245693\pi$
$510$		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	0.967294	+	0.253659i	$0.0816340\pi$
$521$	8363.93	+	14486.8i	0.703322	+	1.21819i	−0.263972	+	0.964530i	$0.585033\pi$
$522$		0			0
$523$	2319.53	−	4017.55i	0.193931	−	0.335899i	−0.752618	−	0.658457i	$-0.771209\pi$
$523$	2319.53	−	4017.55i	0.193931	−	0.335899i	0.946550	+	0.322558i	$0.104543\pi$
$524$	−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	0.285502	+	0.958378i	$0.407840\pi$
$541$	−8647.63	+	14978.1i	−0.687229	+	1.19032i	−0.972731	+	0.231937i	$0.925494\pi$
$542$	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	0.988545	−	0.150925i	$-0.0482251\pi$
$557$	4779.35	+	8278.07i	0.363568	+	0.629718i	−0.624977	+	0.780643i	$0.714892\pi$
$558$		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	0.750018	−	0.661417i	$-0.230045\pi$
$563$	−2642.28	−	4576.56i	−0.197795	−	0.342591i	−0.947813	+	0.318826i	$0.896711\pi$
$564$		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	−0.757510	−	0.652824i	$-0.773584\pi$
$569$	2532.77	−	4386.89i	0.186607	−	0.323213i	0.944117	+	0.329611i	$0.106918\pi$
$570$		0			0
$571$	6695.36	+	11596.7i	0.490704	+	0.849925i	0.999943	−	0.0107008i	$-0.00340623\pi$
$571$	6695.36	+	11596.7i	0.490704	+	0.849925i	−0.509239	+	0.860625i	$0.670073\pi$
$572$	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	0.864236	+	0.503087i	$0.167803\pi$
$577$	12027.8	+	20832.7i	0.867804	+	1.50308i	0.00356838	+	0.999994i	$0.498864\pi$
$578$	−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	−0.906832	−	0.421492i	$-0.861506\pi$
$593$	−1276.43	+	2210.85i	−0.0883927	+	0.153101i	0.818439	+	0.574593i	$0.194840\pi$
$594$		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	0.974558	−	0.224134i	$-0.0719553\pi$
$599$	4297.99	+	7444.33i	0.293173	+	0.507791i	−0.681385	+	0.731925i	$0.738622\pi$
$600$		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	−0.887001	−	0.461767i	$-0.847216\pi$
$607$	−652.013	+	1129.32i	−0.0435986	+	0.0755151i	0.843403	+	0.537282i	$0.180549\pi$
$608$	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	−0.994237	−	0.107204i	$-0.965810\pi$
$613$	−8953.92	−	15508.6i	−0.589960	−	1.02184i	0.404277	−	0.914637i	$-0.367523\pi$
$614$	−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	0.643569	−	0.765388i	$-0.277453\pi$
$619$	−5252.53	−	9097.65i	−0.341061	−	0.590735i	−0.984630	+	0.174653i	$0.944120\pi$
$620$	−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	−0.961496	−	0.274817i	$-0.911383\pi$
$641$	−11664.4	−	20203.4i	−0.718747	−	1.24491i	0.242749	−	0.970089i	$-0.421951\pi$
$642$		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	0.973913	−	0.226920i	$-0.0728657\pi$
$647$	4779.80	+	8278.86i	0.290438	+	0.503053i	−0.683475	+	0.729974i	$0.739532\pi$
$648$		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	0.386464	+	0.922304i	$0.373696\pi$
$653$	−10103.9	+	17500.5i	−0.605507	+	1.04877i	−0.991971	+	0.126464i	$0.959637\pi$
$654$		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	−0.956818	−	0.290687i	$-0.906116\pi$
$661$	−12408.4	−	21492.0i	−0.730152	−	1.26466i	0.226667	−	0.973972i	$-0.427217\pi$
$662$	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	−0.888808	−	0.458281i	$-0.848465\pi$
$677$	−837.086	+	1449.88i	−0.0475212	+	0.0823090i	0.841286	+	0.540590i	$0.181799\pi$
$678$		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	0.902644	−	0.430389i	$-0.141623\pi$
$683$	1402.89	+	2429.87i	0.0785944	+	0.136129i	−0.824049	+	0.566518i	$0.808290\pi$
$684$		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	−0.926468	−	0.376374i	$-0.877171\pi$
$691$	−2493.67	+	4319.16i	−0.137284	+	0.237784i	0.789183	+	0.614158i	$0.210504\pi$
$692$	−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	−0.935838	−	0.352430i	$-0.885355\pi$
$709$	−3071.65	+	5320.25i	−0.162706	+	0.281814i	0.773133	+	0.634244i	$0.218689\pi$
$710$	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	−0.809446	−	0.587194i	$-0.800233\pi$
$719$	2001.23	−	3466.24i	0.103802	−	0.179790i	0.913248	+	0.407404i	$0.133566\pi$
$720$		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	−0.863555	−	0.504255i	$-0.831767\pi$
$733$	97.6418	−	169.121i	0.00492017	−	0.00852198i	0.868475	+	0.495733i	$0.165101\pi$
$734$	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	0.922959	−	0.384897i	$-0.125763\pi$
$739$	2574.43	+	4459.05i	0.128149	+	0.221960i	−0.794811	+	0.606858i	$0.792430\pi$
$740$	−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	−0.988741	−	0.149635i	$-0.952190\pi$
$751$	−7507.47	+	13003.3i	−0.364782	+	0.631822i	0.623959	+	0.781457i	$0.285523\pi$
$752$	−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	−0.170996	−	0.985272i	$-0.554698\pi$
$761$	16117.9	−	27917.1i	0.767773	−	1.32982i	0.938768	−	0.344549i	$-0.111968\pi$
$762$		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	0.912767	+	0.408480i	$0.133941\pi$
$773$	17411.2	+	30157.0i	0.810138	+	1.40320i	−0.102630	+	0.994720i	$0.532726\pi$
$774$		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
$785$	−53612.3			−2.43759
$786$		0			0
$787$	6385.52	+	11060.1i	0.289224	+	0.500951i	0.973625	−	0.228155i	$-0.0732694\pi$
$787$	6385.52	+	11060.1i	0.289224	+	0.500951i	−0.684401	+	0.729106i	$0.739936\pi$
$788$	−7918.64	−	13715.5i	−0.357982	−	0.620043i
$789$		0			0
$790$	−28286.2			−1.27390
$791$	1724.30	−	18606.2i	0.0775085	−	0.836359i
$792$		0			0
$793$	−17568.4	+	30429.3i	−0.786723	+	1.36264i
$794$	−4607.06	−	7979.66i	−0.205917	−	0.356660i
$795$		0			0
$796$	−9384.42	+	16254.3i	−0.417867	+	0.723767i
$797$	−32051.9			−1.42451			−0.712256	−	0.701920i	$-0.752326\pi$
$797$	−32051.9			−1.42451			−0.712256	+	0.701920i	$0.752326\pi$
$798$		0			0
$799$	−13514.2			−0.598373
$800$	2102.07	−	3640.89i	0.0928992	−	0.160906i
$801$		0			0
$802$	10564.3	+	18297.9i	0.465135	+	0.805637i
$803$	2211.93	−	3831.18i	0.0972072	−	0.168368i
$804$		0			0
$805$	16532.6	−	7606.00i	0.723847	−	0.333014i
$806$	−26234.2			−1.14648
$807$		0			0
$808$	−4910.26	−	8504.82i	−0.213790	−	0.370295i
$809$	−6858.63	−	11879.5i	−0.298068	−	0.516268i	0.677626	−	0.735407i	$-0.263009\pi$
$809$	−6858.63	−	11879.5i	−0.298068	−	0.516268i	−0.975694	+	0.219138i	$0.929675\pi$
$810$		0			0
$811$	28519.9			1.23486			0.617428	−	0.786627i	$-0.288175\pi$
$811$	28519.9			1.23486			0.617428	+	0.786627i	$0.288175\pi$
$812$	−8504.46	+	3912.57i	−0.367547	+	0.169094i
$813$		0			0
$814$	−3885.96	+	6730.68i	−0.167325	+	0.289816i
$815$	−21636.5	−	37475.4i	−0.929929	−	1.61068i
$816$		0			0
$817$	−1530.61	+	2651.10i	−0.0655439	+	0.113525i
$818$	5441.23			0.232577
$819$		0			0
$820$	33076.1			1.40862
$821$	12195.5	−	21123.2i	0.518423	−	0.897935i	−0.481348	−	0.876530i	$-0.659853\pi$
$821$	12195.5	−	21123.2i	0.518423	−	0.897935i	0.999771	−	0.0214051i	$-0.00681399\pi$
$822$		0			0
$823$	3620.56	+	6270.99i	0.153347	+	0.265605i	0.932456	−	0.361284i	$-0.117661\pi$
$823$	3620.56	+	6270.99i	0.153347	+	0.265605i	−0.779109	+	0.626889i	$0.784328\pi$
$824$	−3353.66	+	5808.70i	−0.141784	+	0.245577i
$825$		0			0
$826$	−2448.43	+	26419.9i	−0.103138	+	1.11291i
$827$	14936.2			0.628034			0.314017	−	0.949417i	$-0.398325\pi$
$827$	14936.2			0.628034			0.314017	+	0.949417i	$0.398325\pi$
$828$		0			0
$829$	−18501.7	−	32045.8i	−0.775138	−	1.34258i	−0.934717	−	0.355393i	$-0.884347\pi$
$829$	−18501.7	−	32045.8i	−0.775138	−	1.34258i	0.159579	−	0.987185i	$-0.448986\pi$
$830$	1011.18	+	1751.41i	0.0422873	+	0.0732437i
$831$		0			0
$832$	−3229.03			−0.134551
$833$	−4212.47	−	11954.9i	−0.175214	−	0.497254i
$834$		0			0
$835$	−5908.14	+	10233.2i	−0.244862	+	0.424113i
$836$	2822.51	+	4888.73i	0.116768	+	0.202249i
$837$		0			0
$838$	13647.0	−	23637.2i	0.562561	−	0.974385i
$839$	7871.79			0.323915			0.161957	−	0.986798i	$-0.448219\pi$
$839$	7871.79			0.323915			0.161957	+	0.986798i	$0.448219\pi$
$840$		0			0
$841$	−8420.71			−0.345267
$842$	9396.95	−	16276.0i	0.384608	−	0.666161i
$843$		0			0
$844$	3085.16	+	5343.66i	0.125824	+	0.217934i
$845$	2790.58	−	4833.42i	0.113608	−	0.196775i
$846$		0			0
$847$	−16350.5	−	11574.5i	−0.663294	−	0.469546i
$848$	−754.611			−0.0305583
$849$		0			0
$850$	−4855.03	−	8409.17i	−0.195913	−	0.339332i
$851$	7551.21	+	13079.1i	0.304174	+	0.526845i
$852$		0			0
$853$	24025.1			0.964365			0.482183	−	0.876071i	$-0.339844\pi$
$853$	24025.1			0.964365			0.482183	+	0.876071i	$0.339844\pi$
$854$	−23434.6	+	10781.3i	−0.939010	+	0.432002i
$855$		0			0
$856$	1726.22	−	2989.90i	0.0689264	−	0.119384i
$857$	15474.3	+	26802.3i	0.616793	+	1.06832i	0.990067	+	0.140596i	$0.0449019\pi$
$857$	15474.3	+	26802.3i	0.616793	+	1.06832i	−0.373274	+	0.927721i	$0.621765\pi$
$858$		0			0
$859$	−15004.5	+	25988.6i	−0.595981	+	1.03227i	0.397427	+	0.917634i	$0.369903\pi$
$859$	−15004.5	+	25988.6i	−0.595981	+	1.03227i	−0.993408	+	0.114635i	$0.963430\pi$
$860$	2193.74			0.0869835
$861$		0			0
$862$	−14526.2			−0.573974
$863$	18673.7	−	32343.7i	0.736568	−	1.27577i	−0.217463	−	0.976068i	$-0.569778\pi$
$863$	18673.7	−	32343.7i	0.736568	−	1.27577i	0.954032	−	0.299706i	$-0.0968884\pi$
$864$		0			0
$865$	−31103.9	−	53873.6i	−1.22262	−	2.11764i
$866$	3288.50	−	5695.84i	0.129039	−	0.223502i
$867$		0			0
$868$	−15719.7	−	11128.0i	−0.614703	−	0.435149i
$869$	−13947.5			−0.544461
$870$		0			0
$871$	2790.69	+	4833.62i	0.108564	+	0.188038i
$872$	−5228.32	−	9055.71i	−0.203043	−	0.351680i
$873$		0			0
$874$	10969.4			0.424538
$875$	−174.566	+	1883.66i	−0.00674447	+	0.0727765i
$876$		0			0
$877$	−10752.7	+	18624.2i	−0.414015	+	0.717096i	−0.995325	−	0.0965870i	$-0.969207\pi$
$877$	−10752.7	+	18624.2i	−0.414015	+	0.717096i	0.581309	+	0.813683i	$0.302541\pi$
$878$	887.215	+	1536.70i	0.0341026	+	0.0590674i
$879$		0			0
$880$	2022.67	−	3503.37i	0.0774820	−	0.134203i
$881$	−48536.5			−1.85611			−0.928057	−	0.372439i	$-0.878522\pi$
$881$	−48536.5			−1.85611			−0.928057	+	0.372439i	$0.878522\pi$
$882$		0			0
$883$	−11336.1			−0.432037			−0.216019	−	0.976389i	$-0.569307\pi$
$883$	−11336.1			−0.432037			−0.216019	+	0.976389i	$0.569307\pi$
$884$	−3728.96	+	6458.74i	−0.141876	+	0.245736i
$885$		0			0
$886$	7881.15	+	13650.5i	0.298840	+	0.517606i
$887$	−10674.1	+	18488.1i	−0.404059	+	0.699851i	−0.994211	−	0.107441i	$-0.965734\pi$
$887$	−10674.1	+	18488.1i	−0.404059	+	0.699851i	0.590152	+	0.807292i	$0.299068\pi$
$888$		0			0
$889$	−1507.81	+	16270.1i	−0.0568844	+	0.613813i
$890$	−21635.0			−0.814838
$891$		0			0
$892$	−10686.6	−	18509.7i	−0.401136	−	0.694788i
$893$	16342.1	+	28305.4i	0.612394	+	1.06070i
$894$		0			0
$895$	41344.4			1.54412
$896$	−1934.86	−	1369.69i	−0.0721418	−	0.0510692i
$897$		0			0
$898$	13593.0	−	23543.8i	0.505128	−	0.874907i
$899$	16426.5	+	28451.6i	0.609405	+	1.05552i
$900$		0			0
$901$	−871.442	+	1509.38i	−0.0322219	+	0.0558100i
$902$	16309.3			0.602042
$903$		0			0
$904$	−8071.56			−0.296965
$905$	23846.1	−	41302.7i	0.875880	−	1.51707i
$906$		0			0
$907$	24408.7	+	42277.1i	0.893581	+	1.54773i	0.835551	+	0.549414i	$0.185149\pi$
$907$	24408.7	+	42277.1i	0.893581	+	1.54773i	0.0580308	+	0.998315i	$0.481518\pi$
$908$	−7611.97	+	13184.3i	−0.278207	+	0.481869i
$909$		0			0
$910$	27184.5	−	12506.5i	0.990282	−	0.455590i
$911$	39929.6			1.45217			0.726085	−	0.687604i	$-0.241338\pi$
$911$	39929.6			1.45217			0.726085	+	0.687604i	$0.241338\pi$
$912$		0			0
$913$	498.596	+	863.593i	0.0180735	+	0.0313042i
$914$	−8351.25	−	14464.8i	−0.302226	−	0.523471i
$915$		0			0
$916$	4795.41			0.172975
$917$	−30955.7	−	21913.5i	−1.11477	−	0.789148i
$918$		0			0
$919$	−3435.58	+	5950.61i	−0.123318	+	0.213593i	−0.921074	−	0.389387i	$-0.872687\pi$
$919$	−3435.58	+	5950.61i	−0.123318	+	0.213593i	0.797756	+	0.602980i	$0.206020\pi$
$920$	−3930.46	−	6807.76i	−0.140852	−	0.243962i
$921$		0			0
$922$	−9295.22	+	16099.8i	−0.332019	+	0.575074i
$923$	30635.4			1.09250
$924$		0			0
$925$	−32331.9			−1.14926
$926$	272.006	−	471.129i	0.00965301	−	0.0167195i
$927$		0			0
$928$	2021.85	+	3501.95i	0.0715199	+	0.123876i
$929$	−12994.3	+	22506.7i	−0.458910	+	0.794856i	−0.998904	−	0.0468132i	$-0.985093\pi$
$929$	−12994.3	+	22506.7i	−0.458910	+	0.794856i	0.539993	+	0.841669i	$0.318427\pi$
$930$		0			0
$931$	−19945.4	+	23279.4i	−0.702130	+	0.819497i
$932$	−10447.0			−0.367170
$933$		0			0
$934$	−8881.44	−	15383.1i	−0.311145	−	0.538919i
$935$	−4671.65	−	8091.53i	−0.163400	−	0.283018i
$936$		0			0
$937$	−1527.60			−0.0532600			−0.0266300	−	0.999645i	$-0.508478\pi$
$937$	−1527.60			−0.0532600			−0.0266300	+	0.999645i	$0.508478\pi$
$938$	−378.118	+	4080.10i	−0.0131620	+	0.142025i
$939$		0			0
$940$	11711.1	−	20284.2i	0.406355	−	0.703828i
$941$	−27306.8	−	47296.7i	−0.945989	−	1.63850i	−0.753759	−	0.657151i	$-0.771761\pi$
$941$	−27306.8	−	47296.7i	−0.945989	−	1.63850i	−0.192230	−	0.981350i	$-0.561572\pi$
$942$		0			0
$943$	15846.2	−	27446.4i	0.547214	−	0.947803i
$944$	11461.2			0.395160
$945$		0			0
$946$	1081.70			0.0371766
$947$	−19593.8	+	33937.5i	−0.672348	+	1.16454i	0.304889	+	0.952388i	$0.401381\pi$
$947$	−19593.8	+	33937.5i	−0.672348	+	1.16454i	−0.977237	+	0.212153i	$0.931953\pi$
$948$		0			0
$949$	−7067.57	−	12241.4i	−0.241752	−	0.418728i
$950$	−11741.9	+	20337.6i	−0.401008	+	0.694566i
$951$		0			0
$952$	−4974.08	+	2288.38i	−0.169339	+	0.0779063i
$953$	20700.0			0.703608			0.351804	−	0.936074i	$-0.385568\pi$
$953$	20700.0			0.703608			0.351804	+	0.936074i	$0.385568\pi$
$954$		0			0
$955$	33044.0	+	57233.9i	1.11966	+	1.93931i
$956$	−6637.57	−	11496.6i	−0.224555	−	0.388940i
$957$		0			0
$958$	−37930.9			−1.27922
$959$	42089.4	−	19363.7i	1.41724	−	0.652019i
$960$		0			0
$961$	−18900.3	+	32736.2i	−0.634429	+	1.09886i
$962$	12416.4	+	21505.9i	0.416135	+	0.720767i
$963$		0			0
$964$	5552.12	−	9616.55i	0.185500	−	0.321295i
$965$	48319.0			1.61186
$966$		0			0
$967$	35879.3			1.19318			0.596588	−	0.802547i	$-0.296522\pi$
$967$	35879.3			1.19318			0.596588	+	0.802547i	$0.296522\pi$
$968$	−4326.65	+	7493.98i	−0.143661	+	0.248828i
$969$		0			0
$970$	−15630.6	−	27073.0i	−0.517391	−	0.896147i
$971$	−15267.2	+	26443.6i	−0.504581	+	0.873959i	0.495405	+	0.868662i	$0.335020\pi$
$971$	−15267.2	+	26443.6i	−0.504581	+	0.873959i	−0.999986	+	0.00529741i	$0.998314\pi$
$972$		0			0
$973$	−1621.91	+	17501.3i	−0.0534388	+	0.576633i
$974$	−35937.0			−1.18224
$975$		0			0
$976$	5571.34	+	9649.85i	0.182720	+	0.316480i
$977$	9083.40	+	15732.9i	0.297445	+	0.515190i	0.975551	−	0.219775i	$-0.0705322\pi$
$977$	9083.40	+	15732.9i	0.297445	+	0.515190i	−0.678106	+	0.734964i	$0.737199\pi$
$978$		0			0
$979$	−10667.9			−0.348260
$980$	21594.1	+	4037.09i	0.703877	+	0.131592i
$981$		0			0
$982$	5379.38	−	9317.35i	0.174809	−	0.302779i
$983$	8205.99	+	14213.2i	0.266257	+	0.461170i	0.967892	−	0.251366i	$-0.0808797\pi$
$983$	8205.99	+	14213.2i	0.266257	+	0.461170i	−0.701635	+	0.712536i	$0.747546\pi$
$984$		0			0
$985$	31698.0	−	54902.5i	1.02536	−	1.77598i
$986$	9339.52			0.301654
$987$		0			0
$988$	18037.0			0.580802
$989$	1050.98	−	1820.35i	0.0337910	−	0.0585277i
$990$		0			0
$991$	1544.14	+	2674.52i	0.0494966	+	0.0857306i	0.889712	−	0.456522i	$-0.150905\pi$
$991$	1544.14	+	2674.52i	0.0494966	+	0.0857306i	−0.840216	+	0.542252i	$0.817572\pi$
$992$	−4159.74	+	7204.88i	−0.133137	+	0.230600i
$993$		0			0
$994$	18356.9	+	12994.9i	0.585761	+	0.414661i
$995$	−75131.0			−2.39378
$996$		0			0
$997$	12206.0	+	21141.4i	0.387730	+	0.671568i	0.992144	−	0.125102i	$-0.0399259\pi$
$997$	12206.0	+	21141.4i	0.387730	+	0.671568i	−0.604414	+	0.796671i	$0.706593\pi$
$998$	11679.6	+	20229.7i	0.370452	+	0.641642i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.e.163.4	yes	8
3.2	odd	2		378.4.g.d.163.1	yes	8
7.4	even	3	inner	378.4.g.e.109.4	yes	8
21.11	odd	6		378.4.g.d.109.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.d.109.1	✓	8	21.11	odd	6
378.4.g.d.163.1	yes	8	3.2	odd	2
378.4.g.e.109.4	yes	8	7.4	even	3	inner
378.4.g.e.163.4	yes	8	1.1	even	1	trivial

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

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