LMFDB - Embedded newform 378.4.g.d.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [378,4,Mod(109,378)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,-8,0,-16,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			109.1
Root			$4.05517 - 7.02376i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.d.163.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$-0.912605\pi$
$5$	−8.00592	−	13.8667i	−0.716072	−	1.24027i	0.246473	−	0.969150i	$-0.420728\pi$
$6$		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	$-0.763899\pi$
$11$	7.89520	−	13.6749i	0.216409	−	0.374831i	0.953707	+	0.300736i	$0.0972324\pi$
$12$		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	$-0.918247\pi$
$17$	−18.4772	+	32.0034i	−0.263610	+	0.456586i	0.703588	+	0.710608i	$0.251580\pi$
$18$		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	0.970932	−	0.239357i	$-0.0769367\pi$
$23$	30.6840	+	53.1462i	0.278176	+	0.481816i	−0.692755	+	0.721173i	$0.743603\pi$
$24$		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	$-0.632582\pi$
$29$	−126.366			−0.809156			−0.404578	+	0.914504i	$0.632582\pi$
$30$		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	$-0.0577717\pi$
$41$	516.432			1.96715			0.983575	+	0.180500i	$0.0577717\pi$
$42$		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	$0.0254145\pi$
$47$	182.851	+	316.707i	0.567479	+	0.982902i	−0.429335	+	0.903145i	$0.641252\pi$
$48$		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	$-0.852799\pi$
$53$	−23.5816	+	40.8445i	−0.0611166	+	0.105857i	0.833848	+	0.551994i	$0.186133\pi$
$54$		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	$-0.543248\pi$
$59$	358.163	−	620.357i	0.790320	−	1.36887i	0.925769	−	0.378090i	$-0.123419\pi$
$60$		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	$-0.330579\pi$
$71$	607.199			1.01495			0.507474	+	0.861667i	$0.330579\pi$
$72$		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	$-0.486704\pi$
$83$	63.1517			0.0835157			0.0417578	+	0.999128i	$0.486704\pi$
$84$		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	$-0.298462\pi$
$89$	−337.796	−	585.080i	−0.402318	−	0.696836i	−0.994005	+	0.109332i	$0.965129\pi$
$90$		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	$0.373369\pi$
$101$	−613.782	+	1063.10i	−0.604689	+	1.04735i	−0.992100	+	0.125446i	$0.959964\pi$
$102$		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	0.946885	−	0.321572i	$-0.104211\pi$
$107$	215.777	+	373.737i	0.194953	+	0.337669i	−0.751932	+	0.659241i	$0.770878\pi$
$108$		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	$-0.637960\pi$
$113$	−1008.94			−0.839943			−0.419971	+	0.907537i	$0.637960\pi$
$114$		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	$-0.927380\pi$
$131$	−1023.93	−	1773.50i	−0.682912	−	1.18284i	0.291176	−	0.956669i	$-0.405953\pi$
$132$		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	$0.451457\pi$
$137$	−1250.79	+	2166.44i	−0.780018	+	1.35103i	−0.931931	+	0.362637i	$0.881877\pi$
$138$		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	$-0.0963044\pi$
$149$	398.773	+	690.696i	0.219254	+	0.379758i	−0.735326	+	0.677713i	$0.762971\pi$
$150$		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	$-0.445308\pi$
$167$	737.971			0.341951			0.170976	+	0.985275i	$0.445308\pi$
$168$		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	$-0.841021\pi$
$173$	−1942.56	−	3364.61i	−0.853699	−	1.47865i	0.0241476	−	0.999708i	$-0.492313\pi$
$174$		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	$0.347901\pi$
$179$	−1291.06	+	2236.18i	−0.539096	+	0.933741i	−0.998953	+	0.0457482i	$0.985433\pi$
$180$		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	$0.119036\pi$
$191$	2063.72	+	3574.47i	0.781809	+	1.35413i	−0.149077	+	0.988826i	$0.547630\pi$
$192$		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	$-0.754012\pi$
$197$	−3959.32			−1.43193			−0.715964	+	0.698137i	$0.754012\pi$
$198$		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	$-0.645510\pi$
$227$	1902.99	−	3296.08i	0.556415	−	0.963738i	0.997792	−	0.0664167i	$-0.0211567\pi$
$228$		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	0.621952	−	0.783055i	$-0.286340\pi$
$233$	−1305.87	−	2261.84i	−0.367170	−	0.635956i	−0.989122	+	0.147099i	$0.953006\pi$
$234$		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	$-0.648259\pi$
$239$	−3318.79			−0.898220			−0.449110	+	0.893477i	$0.648259\pi$
$240$		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	$-0.446146\pi$
$251$	1339.17			0.336764			0.168382	+	0.985722i	$0.446146\pi$
$252$		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	$-0.0469213\pi$
$257$	1513.62	+	2621.66i	0.367381	+	0.636322i	−0.621774	+	0.783196i	$0.713588\pi$
$258$		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	$0.426394\pi$
$263$	−3106.65	+	5380.87i	−0.728381	+	1.26159i	−0.957567	+	0.288210i	$0.906940\pi$
$264$		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	$-0.811002\pi$
$269$	309.272	−	535.674i	0.0700990	−	0.121415i	0.898945	+	0.438063i	$0.144335\pi$
$270$		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	$-0.440827\pi$
$281$	1741.24			0.369658			0.184829	+	0.982771i	$0.440827\pi$
$282$		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	$-0.168854\pi$
$293$	8652.18			1.72514			0.862569	+	0.505939i	$0.168854\pi$
$294$		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	$-0.624380\pi$
$311$	3347.24	−	5797.60i	0.610305	−	1.05708i	0.991189	−	0.132456i	$-0.0422864\pi$
$312$		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	0.666115	−	0.745849i	$-0.267956\pi$
$317$	−1765.83	−	3058.50i	−0.312867	−	0.541901i	−0.978982	+	0.203948i	$0.934623\pi$
$318$		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	−0.569678	−	0.821868i	$-0.692932\pi$
$347$	2759.56	−	4779.70i	0.426919	−	0.739446i	0.996598	+	0.0824220i	$0.0262655\pi$
$348$		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	$-0.755906\pi$
$353$	1597.40	−	2766.78i	0.240853	−	0.417170i	0.960958	+	0.276696i	$0.0892394\pi$
$354$		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	$0.171175\pi$
$359$	5938.35	+	10285.5i	0.873020	+	1.51211i	0.0141629	+	0.999900i	$0.495492\pi$
$360$		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	0.996875	−	0.0789891i	$-0.0251692\pi$
$383$	3223.28	+	5582.89i	0.430031	+	0.744836i	−0.566844	+	0.823825i	$0.691836\pi$
$384$		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	$-0.894986\pi$
$389$	−1476.74	+	2557.79i	−0.192478	+	0.333381i	0.753593	+	0.657341i	$0.228319\pi$
$390$		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	$0.0618462\pi$
$401$	5282.15	+	9148.95i	0.657800	+	1.13934i	−0.323384	+	0.946268i	$0.604820\pi$
$402$		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	$-0.792835\pi$
$419$	−13647.0			−1.59116			−0.795582	+	0.605846i	$0.792835\pi$
$420$		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	$-0.700305\pi$
$431$	3631.56	−	6290.04i	0.405861	−	0.702971i	0.994421	+	0.105482i	$0.0336385\pi$
$432$		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	0.996195	−	0.0871497i	$-0.0277758\pi$
$443$	3940.57	+	6825.27i	0.422624	+	0.732006i	−0.573571	+	0.819156i	$0.694443\pi$
$444$		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	$-0.753281\pi$
$449$	−13593.0			−1.42872			−0.714359	+	0.699780i	$0.753281\pi$
$450$		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	$-0.344417\pi$
$461$	9295.22			0.939093			0.469546	+	0.882908i	$0.344417\pi$
$462$		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	$-0.311697\pi$
$467$	−4440.72	−	7691.56i	−0.440026	−	0.762147i	−0.997691	+	0.0679189i	$0.978364\pi$
$468$		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	$-0.473544\pi$
$479$	9482.73	−	16424.6i	0.904545	−	1.56672i	0.821527	−	0.570169i	$-0.193122\pi$
$480$		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	$-0.579516\pi$
$491$	−5379.38			−0.494435			−0.247218	+	0.968960i	$0.579516\pi$
$492$		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	$-0.521499\pi$
$503$	−1522.74			−0.134981			−0.0674906	+	0.997720i	$0.521499\pi$
$504$		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	$-0.0876408\pi$
$509$	2821.79	+	4887.49i	0.245725	+	0.425607i	−0.716611	+	0.697474i	$0.754307\pi$
$510$		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	$0.414967\pi$
$521$	−8363.93	+	14486.8i	−0.703322	+	1.21819i	−0.967294	+	0.253659i	$0.918366\pi$
$522$		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	−0.988545	−	0.150925i	$-0.951775\pi$
$557$	−4779.35	+	8278.07i	−0.363568	+	0.629718i	0.624977	+	0.780643i	$0.285108\pi$
$558$		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	$-0.769955\pi$
$563$	2642.28	−	4576.56i	0.197795	−	0.342591i	0.947813	+	0.318826i	$0.103289\pi$
$564$		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	$-0.226416\pi$
$569$	−2532.77	−	4386.89i	−0.186607	−	0.323213i	−0.944117	+	0.329611i	$0.893082\pi$
$570$		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	$-0.902062\pi$
$587$	−27108.0			−1.90608			−0.953039	+	0.302849i	$0.902062\pi$
$588$		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	$-0.138494\pi$
$593$	1276.43	+	2210.85i	0.0883927	+	0.153101i	−0.818439	+	0.574593i	$0.805160\pi$
$594$		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	$-0.928045\pi$
$599$	−4297.99	+	7444.33i	−0.293173	+	0.507791i	0.681385	+	0.731925i	$0.261378\pi$
$600$		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	$-0.840772\pi$
$617$	−26896.2			−1.75494			−0.877472	+	0.479628i	$0.840772\pi$
$618$		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	$-0.578049\pi$
$641$	11664.4	−	20203.4i	0.718747	−	1.24491i	0.961496	−	0.274817i	$-0.0886174\pi$
$642$		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	$-0.927134\pi$
$647$	−4779.80	+	8278.86i	−0.290438	+	0.503053i	0.683475	+	0.729974i	$0.260468\pi$
$648$		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	$0.0403629\pi$
$653$	10103.9	+	17500.5i	0.605507	+	1.04877i	−0.386464	+	0.922304i	$0.626304\pi$
$654$		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	$-0.268466\pi$
$659$	22497.2			1.32984			0.664920	+	0.746914i	$0.268466\pi$
$660$		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	0.888808	−	0.458281i	$-0.151535\pi$
$677$	837.086	+	1449.88i	0.0475212	+	0.0823090i	−0.841286	+	0.540590i	$0.818201\pi$
$678$		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	$-0.858377\pi$
$683$	−1402.89	+	2429.87i	−0.0785944	+	0.136129i	0.824049	+	0.566518i	$0.191710\pi$
$684$		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	$-0.358704\pi$
$701$	15941.5			0.858921			0.429460	+	0.903086i	$0.358704\pi$
$702$		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	0.809446	−	0.587194i	$-0.199767\pi$
$719$	−2001.23	−	3466.24i	−0.103802	−	0.179790i	−0.913248	+	0.407404i	$0.866434\pi$
$720$		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	$-0.323651\pi$
$743$	21310.2			1.05221			0.526106	+	0.850419i	$0.323651\pi$
$744$		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	$-0.888032\pi$
$761$	−16117.9	−	27917.1i	−0.767773	−	1.32982i	0.170996	−	0.985272i	$-0.445302\pi$
$762$		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	$0.467274\pi$
$773$	−17411.2	+	30157.0i	−0.810138	+	1.40320i	−0.912767	+	0.408480i	$0.866059\pi$
$774$		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	−0.684401	−	0.729106i	$-0.739936\pi$
$787$	6385.52	−	11060.1i	0.289224	−	0.500951i	0.973625	+	0.228155i	$0.0732694\pi$
$788$	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	$-0.247674\pi$
$797$	32051.9			1.42451			0.712256	+	0.701920i	$0.247674\pi$
$798$		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	$-0.736991\pi$
$809$	6858.63	−	11879.5i	0.298068	−	0.516268i	0.975694	+	0.219138i	$0.0703246\pi$
$810$		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	−0.999771	−	0.0214051i	$-0.993186\pi$
$821$	−12195.5	−	21123.2i	−0.518423	−	0.897935i	0.481348	−	0.876530i	$-0.340147\pi$
$822$		0			0
$823$	3620.56	−	6270.99i	0.153347	−	0.265605i	−0.779109	−	0.626889i	$-0.784328\pi$
$823$	3620.56	−	6270.99i	0.153347	−	0.265605i	0.932456	+	0.361284i	$0.117661\pi$
$824$	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	$-0.601675\pi$
$827$	−14936.2			−0.628034			−0.314017	+	0.949417i	$0.601675\pi$
$828$		0			0
$829$	−18501.7	+	32045.8i	−0.775138	+	1.34258i	0.159579	+	0.987185i	$0.448986\pi$
$829$	−18501.7	+	32045.8i	−0.775138	+	1.34258i	−0.934717	+	0.355393i	$0.884347\pi$
$830$	−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	$-0.551781\pi$
$839$	−7871.79			−0.323915			−0.161957	+	0.986798i	$0.551781\pi$
$840$		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	0.373274	+	0.927721i	$0.378235\pi$
$857$	−15474.3	+	26802.3i	−0.616793	+	1.06832i	−0.990067	+	0.140596i	$0.955098\pi$
$858$		0			0
$859$	−15004.5	−	25988.6i	−0.595981	−	1.03227i	−0.993408	−	0.114635i	$-0.963430\pi$
$859$	−15004.5	−	25988.6i	−0.595981	−	1.03227i	0.397427	−	0.917634i	$-0.369903\pi$
$860$	−2193.74			−0.0869835
$861$		0			0
$862$	−14526.2			−0.573974
$863$	−18673.7	−	32343.7i	−0.736568	−	1.27577i	−0.954032	−	0.299706i	$-0.903112\pi$
$863$	−18673.7	−	32343.7i	−0.736568	−	1.27577i	0.217463	−	0.976068i	$-0.430222\pi$
$864$		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	0.581309	−	0.813683i	$-0.302541\pi$
$877$	−10752.7	−	18624.2i	−0.414015	−	0.717096i	−0.995325	+	0.0965870i	$0.969207\pi$
$878$	−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	$-0.121478\pi$
$881$	48536.5			1.85611			0.928057	+	0.372439i	$0.121478\pi$
$882$		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	$-0.0342656\pi$
$887$	10674.1	+	18488.1i	0.404059	+	0.699851i	−0.590152	+	0.807292i	$0.700932\pi$
$888$		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	0.0580308	−	0.998315i	$-0.481518\pi$
$907$	24408.7	−	42277.1i	0.893581	−	1.54773i	0.835551	−	0.549414i	$-0.185149\pi$
$908$	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	$-0.758662\pi$
$911$	−39929.6			−1.45217			−0.726085	+	0.687604i	$0.758662\pi$
$912$		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	0.797756	−	0.602980i	$-0.206020\pi$
$919$	−3435.58	−	5950.61i	−0.123318	−	0.213593i	−0.921074	+	0.389387i	$0.872687\pi$
$920$	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	$-0.0149065\pi$
$929$	12994.3	+	22506.7i	0.458910	+	0.794856i	−0.539993	+	0.841669i	$0.681573\pi$
$930$		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	0.192230	−	0.981350i	$-0.438428\pi$
$941$	27306.8	−	47296.7i	0.945989	−	1.63850i	0.753759	−	0.657151i	$-0.228239\pi$
$942$		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	0.977237	+	0.212153i	$0.0680474\pi$
$947$	19593.8	+	33937.5i	0.672348	+	1.16454i	−0.304889	+	0.952388i	$0.598619\pi$
$948$		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	$-0.614432\pi$
$953$	−20700.0			−0.703608			−0.351804	+	0.936074i	$0.614432\pi$
$954$		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	0.999986	+	0.00529741i	$0.00168623\pi$
$971$	15267.2	+	26443.6i	0.504581	+	0.873959i	−0.495405	+	0.868662i	$0.664980\pi$
$972$		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	$-0.929468\pi$
$977$	−9083.40	+	15732.9i	−0.297445	+	0.515190i	0.678106	+	0.734964i	$0.262801\pi$
$978$		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	$-0.919120\pi$
$983$	−8205.99	+	14213.2i	−0.266257	+	0.461170i	0.701635	+	0.712536i	$0.252454\pi$
$984$		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	−0.840216	−	0.542252i	$-0.817572\pi$
$991$	1544.14	−	2674.52i	0.0494966	−	0.0857306i	0.889712	+	0.456522i	$0.150905\pi$
$992$	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	−0.604414	−	0.796671i	$-0.706593\pi$
$997$	12206.0	−	21141.4i	0.387730	−	0.671568i	0.992144	+	0.125102i	$0.0399259\pi$
$998$	−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.d.109.1	✓	8
3.2	odd	2		378.4.g.e.109.4	yes	8
7.2	even	3	inner	378.4.g.d.163.1	yes	8
21.2	odd	6		378.4.g.e.163.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.d.109.1	✓	8	1.1	even	1	trivial
378.4.g.d.163.1	yes	8	7.2	even	3	inner
378.4.g.e.109.4	yes	8	3.2	odd	2
378.4.g.e.163.4	yes	8	21.2	odd	6

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 $\F_{q}$
Belyi maps

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

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

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