LMFDB - Embedded newform 378.4.g.e.163.1

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			163.1
Root			$1.39378 + 2.41410i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.e.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-8.97703 + 15.5487i) q^{5} +(13.2278 - 12.9624i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-8.97703 + 15.5487i) q^{5} +(13.2278 - 12.9624i) q^{7} -8.00000 q^{8} +(17.9541 + 31.0973i) q^{10} +(-19.8381 - 34.3607i) q^{11} +28.4065 q^{13} +(-9.22380 - 35.8737i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(7.38252 + 12.7869i) q^{17} +(-33.9153 + 58.7430i) q^{19} +71.8162 q^{20} -79.3526 q^{22} +(-88.8684 + 153.925i) q^{23} +(-98.6740 - 170.908i) q^{25} +(28.4065 - 49.2016i) q^{26} +(-71.3589 - 19.8976i) q^{28} -209.301 q^{29} +(-143.931 - 249.296i) q^{31} +(16.0000 + 27.7128i) q^{32} +29.5301 q^{34} +(82.8023 + 322.039i) q^{35} +(-183.917 + 318.554i) q^{37} +(67.8305 + 117.486i) q^{38} +(71.8162 - 124.389i) q^{40} -294.638 q^{41} -348.661 q^{43} +(-79.3526 + 137.443i) q^{44} +(177.737 + 307.849i) q^{46} +(-119.010 + 206.132i) q^{47} +(6.95004 - 342.930i) q^{49} -394.696 q^{50} +(-56.8131 - 98.4031i) q^{52} +(-183.607 - 318.017i) q^{53} +712.350 q^{55} +(-105.822 + 103.700i) q^{56} +(-209.301 + 362.520i) q^{58} +(127.872 + 221.481i) q^{59} +(149.170 - 258.371i) q^{61} -575.724 q^{62} +64.0000 q^{64} +(-255.006 + 441.684i) q^{65} +(499.195 + 864.631i) q^{67} +(29.5301 - 51.1476i) q^{68} +(640.590 + 178.621i) q^{70} +659.220 q^{71} +(-319.532 - 553.446i) q^{73} +(367.835 + 637.109i) q^{74} +271.322 q^{76} +(-707.814 - 197.366i) q^{77} +(338.790 - 586.802i) q^{79} +(-143.632 - 248.779i) q^{80} +(-294.638 + 510.328i) q^{82} +830.441 q^{83} -265.092 q^{85} +(-348.661 + 603.899i) q^{86} +(158.705 + 274.885i) q^{88} +(-231.740 + 401.385i) q^{89} +(375.756 - 368.218i) q^{91} +710.948 q^{92} +(238.021 + 412.264i) q^{94} +(-608.916 - 1054.67i) q^{95} +373.154 q^{97} +(-587.021 - 354.967i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.97703	+	15.5487i	−0.802930	+	1.39072i	0.114750	+	0.993394i	$0.463393\pi$
$5$	−8.97703	+	15.5487i	−0.802930	+	1.39072i	−0.917680	+	0.397321i	$0.869940\pi$
$6$		0			0
$7$	13.2278	−	12.9624i	0.714235	−	0.699906i
$7$	13.2278	−	12.9624i	0.714235	−	0.699906i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	17.9541	+	31.0973i	0.567757	+	0.983384i
$11$	−19.8381	−	34.3607i	−0.543766	−	0.941830i	−0.998683	−	0.0512966i	$-0.983665\pi$
$11$	−19.8381	−	34.3607i	−0.543766	−	0.941830i	0.454918	−	0.890534i	$-0.349669\pi$
$12$		0			0
$13$	28.4065			0.606043			0.303021	−	0.952984i	$-0.402005\pi$
$13$	28.4065			0.606043			0.303021	+	0.952984i	$0.402005\pi$
$14$	−9.22380	−	35.8737i	−0.176083	−	0.684832i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	7.38252	+	12.7869i	0.105325	+	0.182428i	0.913871	−	0.406005i	$-0.133078\pi$
$17$	7.38252	+	12.7869i	0.105325	+	0.182428i	−0.808546	+	0.588433i	$0.799745\pi$
$18$		0			0
$19$	−33.9153	+	58.7430i	−0.409510	+	0.709293i	−0.994835	−	0.101506i	$-0.967634\pi$
$19$	−33.9153	+	58.7430i	−0.409510	+	0.709293i	0.585325	+	0.810799i	$0.300967\pi$
$20$	71.8162			0.802930
$21$		0			0
$22$	−79.3526			−0.769001
$23$	−88.8684	+	153.925i	−0.805667	+	1.39546i	0.110172	+	0.993913i	$0.464860\pi$
$23$	−88.8684	+	153.925i	−0.805667	+	1.39546i	−0.915840	+	0.401544i	$0.868474\pi$
$24$		0			0
$25$	−98.6740	−	170.908i	−0.789392	−	1.36727i
$26$	28.4065	−	49.2016i	0.214268	−	0.371124i
$27$		0			0
$28$	−71.3589	−	19.8976i	−0.481627	−	0.134296i
$29$	−209.301			−1.34021			−0.670107	−	0.742264i	$-0.733752\pi$
$29$	−209.301			−1.34021			−0.670107	+	0.742264i	$0.733752\pi$
$30$		0			0
$31$	−143.931	−	249.296i	−0.833895	−	1.44435i	−0.894927	−	0.446213i	$-0.852772\pi$
$31$	−143.931	−	249.296i	−0.833895	−	1.44435i	0.0610314	−	0.998136i	$-0.480561\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	29.5301			0.148952
$35$	82.8023	+	322.039i	0.399890	+	1.55527i
$36$		0			0
$37$	−183.917	+	318.554i	−0.817185	+	1.41541i	0.0905633	+	0.995891i	$0.471133\pi$
$37$	−183.917	+	318.554i	−0.817185	+	1.41541i	−0.907748	+	0.419515i	$0.862200\pi$
$38$	67.8305	+	117.486i	0.289567	+	0.501546i
$39$		0			0
$40$	71.8162	−	124.389i	0.283879	−	0.491692i
$41$	−294.638			−1.12231			−0.561156	−	0.827710i	$-0.689643\pi$
$41$	−294.638			−1.12231			−0.561156	+	0.827710i	$0.689643\pi$
$42$		0			0
$43$	−348.661			−1.23652			−0.618259	−	0.785974i	$-0.712162\pi$
$43$	−348.661			−1.23652			−0.618259	+	0.785974i	$0.712162\pi$
$44$	−79.3526	+	137.443i	−0.271883	+	0.470915i
$45$		0			0
$46$	177.737	+	307.849i	0.569693	+	0.986737i
$47$	−119.010	+	206.132i	−0.369350	+	0.639733i	−0.989464	−	0.144779i	$-0.953753\pi$
$47$	−119.010	+	206.132i	−0.369350	+	0.639733i	0.620114	+	0.784512i	$0.287086\pi$
$48$		0			0
$49$	6.95004	−	342.930i	0.0202625	−	0.999795i
$50$	−394.696			−1.11637
$51$		0			0
$52$	−56.8131	−	98.4031i	−0.151511	−	0.262424i
$53$	−183.607	−	318.017i	−0.475856	−	0.824206i	0.523762	−	0.851865i	$-0.324528\pi$
$53$	−183.607	−	318.017i	−0.475856	−	0.824206i	−0.999617	+	0.0276585i	$0.991195\pi$
$54$		0			0
$55$	712.350			1.74642
$56$	−105.822	+	103.700i	−0.252520	+	0.247454i
$57$		0			0
$58$	−209.301	+	362.520i	−0.473837	+	0.820711i
$59$	127.872	+	221.481i	0.282162	+	0.488719i	0.971917	−	0.235324i	$-0.0756151\pi$
$59$	127.872	+	221.481i	0.282162	+	0.488719i	−0.689755	+	0.724043i	$0.742282\pi$
$60$		0			0
$61$	149.170	−	258.371i	0.313104	−	0.542311i	−0.665929	−	0.746015i	$-0.731965\pi$
$61$	149.170	−	258.371i	0.313104	−	0.542311i	0.979033	+	0.203704i	$0.0652980\pi$
$62$	−575.724			−1.17931
$63$		0			0
$64$	64.0000			0.125000
$65$	−255.006	+	441.684i	−0.486610	+	0.842833i
$66$		0			0
$67$	499.195	+	864.631i	0.910245	+	1.57659i	0.813718	+	0.581260i	$0.197440\pi$
$67$	499.195	+	864.631i	0.910245	+	1.57659i	0.0965267	+	0.995330i	$0.469227\pi$
$68$	29.5301	−	51.1476i	0.0526625	−	0.0912140i
$69$		0			0
$70$	640.590	+	178.621i	1.09379	+	0.304990i
$71$	659.220			1.10190			0.550951	−	0.834538i	$-0.314265\pi$
$71$	659.220			1.10190			0.550951	+	0.834538i	$0.314265\pi$
$72$		0			0
$73$	−319.532	−	553.446i	−0.512307	−	0.887341i	−0.999898	−	0.0142693i	$-0.995458\pi$
$73$	−319.532	−	553.446i	−0.512307	−	0.887341i	0.487591	−	0.873072i	$-0.337876\pi$
$74$	367.835	+	637.109i	0.577837	+	1.00084i
$75$		0			0
$76$	271.322			0.409510
$77$	−707.814	−	197.366i	−1.04757	−	0.292103i
$78$		0			0
$79$	338.790	−	586.802i	0.482492	−	0.835700i	−0.517306	−	0.855800i	$-0.673065\pi$
$79$	338.790	−	586.802i	0.482492	−	0.835700i	0.999798	+	0.0201001i	$0.00639848\pi$
$80$	−143.632	−	248.779i	−0.200732	−	0.347679i
$81$		0			0
$82$	−294.638	+	510.328i	−0.396797	+	0.687273i
$83$	830.441			1.09823			0.549113	−	0.835748i	$-0.314966\pi$
$83$	830.441			1.09823			0.549113	+	0.835748i	$0.314966\pi$
$84$		0			0
$85$	−265.092			−0.338274
$86$	−348.661	+	603.899i	−0.437175	+	0.757210i
$87$		0			0
$88$	158.705	+	274.885i	0.192250	+	0.332987i
$89$	−231.740	+	401.385i	−0.276004	+	0.478053i	−0.970388	−	0.241552i	$-0.922344\pi$
$89$	−231.740	+	401.385i	−0.276004	+	0.478053i	0.694384	+	0.719605i	$0.255677\pi$
$90$		0			0
$91$	375.756	−	368.218i	0.432857	−	0.424173i
$92$	710.948			0.805667
$93$		0			0
$94$	238.021	+	412.264i	0.261170	+	0.452360i
$95$	−608.916	−	1054.67i	−0.657616	−	1.13902i
$96$		0			0
$97$	373.154			0.390598			0.195299	−	0.980744i	$-0.437432\pi$
$97$	373.154			0.390598			0.195299	+	0.980744i	$0.437432\pi$
$98$	−587.021	−	354.967i	−0.605083	−	0.365889i
$99$		0			0
$100$	−394.696	+	683.634i	−0.394696	+	0.683634i
$101$	−138.026	−	239.068i	−0.135981	−	0.235526i	0.789991	−	0.613119i	$-0.210085\pi$
$101$	−138.026	−	239.068i	−0.135981	−	0.235526i	−0.925972	+	0.377593i	$0.876752\pi$
$102$		0			0
$103$	44.3595	−	76.8329i	0.0424356	−	0.0735007i	−0.844027	−	0.536300i	$-0.819822\pi$
$103$	44.3595	−	76.8329i	0.0424356	−	0.0735007i	0.886463	+	0.462799i	$0.153155\pi$
$104$	−227.252			−0.214268
$105$		0			0
$106$	−734.428			−0.672962
$107$	−344.439	+	596.585i	−0.311197	+	0.539010i	−0.978622	−	0.205668i	$-0.934063\pi$
$107$	−344.439	+	596.585i	−0.311197	+	0.539010i	0.667424	+	0.744678i	$0.267397\pi$
$108$		0			0
$109$	−292.449	−	506.537i	−0.256987	−	0.445114i	0.708447	−	0.705765i	$-0.249396\pi$
$109$	−292.449	−	506.537i	−0.256987	−	0.445114i	−0.965433	+	0.260650i	$0.916063\pi$
$110$	712.350	−	1233.83i	0.617454	−	1.06946i
$111$		0			0
$112$	73.7904	+	286.990i	0.0622548	+	0.242125i
$113$	−1397.04			−1.16303			−0.581517	−	0.813534i	$-0.697541\pi$
$113$	−1397.04			−1.16303			−0.581517	+	0.813534i	$0.697541\pi$
$114$		0			0
$115$	−1595.55	−	2763.57i	−1.29379	−	2.24091i
$116$	418.602	+	725.040i	0.335054	+	0.580330i
$117$		0			0
$118$	511.489			0.399037
$119$	263.404	+	73.4472i	0.202909	+	0.0565789i
$120$		0			0
$121$	−121.604	+	210.624i	−0.0913627	+	0.158245i
$122$	−298.341	−	516.742i	−0.221398	−	0.383472i
$123$		0			0
$124$	−575.724	+	997.183i	−0.416948	+	0.722175i
$125$	1298.94			0.929446
$126$		0			0
$127$	−988.221			−0.690476			−0.345238	−	0.938515i	$-0.612202\pi$
$127$	−988.221			−0.690476			−0.345238	+	0.938515i	$0.612202\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	510.012	+	883.368i	0.344085	+	0.595973i
$131$	−565.462	+	979.409i	−0.377135	+	0.653217i	−0.990644	−	0.136471i	$-0.956424\pi$
$131$	−565.462	+	979.409i	−0.377135	+	0.653217i	0.613509	+	0.789688i	$0.289757\pi$
$132$		0			0
$133$	312.828	+	1216.67i	0.203952	+	0.793220i
$134$	1996.78			1.28728
$135$		0			0
$136$	−59.0601	−	102.295i	−0.0372380	−	0.0644981i
$137$	451.777	+	782.501i	0.281737	+	0.487983i	0.971813	−	0.235755i	$-0.0757561\pi$
$137$	451.777	+	782.501i	0.281737	+	0.487983i	−0.690076	+	0.723737i	$0.742423\pi$
$138$		0			0
$139$	−1220.74			−0.744903			−0.372451	−	0.928052i	$-0.621483\pi$
$139$	−1220.74			−0.744903			−0.372451	+	0.928052i	$0.621483\pi$
$140$	949.971	−	930.914i	0.573480	−	0.561976i
$141$		0			0
$142$	659.220	−	1141.80i	0.389581	−	0.674775i
$143$	−563.533	−	976.068i	−0.329545	−	0.570789i
$144$		0			0
$145$	1878.90	−	3254.35i	1.07610	−	1.86386i
$146$	−1278.13			−0.724511
$147$		0			0
$148$	1471.34			0.817185
$149$	1003.09	−	1737.41i	0.551520	−	0.955261i	−0.446645	−	0.894711i	$-0.647381\pi$
$149$	1003.09	−	1737.41i	0.551520	−	0.955261i	0.998165	−	0.0605494i	$-0.0192853\pi$
$150$		0			0
$151$	1138.85	+	1972.55i	0.613765	+	1.06307i	0.990600	+	0.136791i	$0.0436790\pi$
$151$	1138.85	+	1972.55i	0.613765	+	1.06307i	−0.376835	+	0.926280i	$0.622988\pi$
$152$	271.322	−	469.944i	0.144784	−	0.250773i
$153$		0			0
$154$	−1049.66	+	1028.60i	−0.549247	+	0.538229i
$155$	5168.29			2.67824
$156$		0			0
$157$	150.782	+	261.163i	0.0766481	+	0.132758i	0.901802	−	0.432150i	$-0.142245\pi$
$157$	150.782	+	261.163i	0.0766481	+	0.132758i	−0.825154	+	0.564908i	$0.808912\pi$
$158$	−677.580	−	1173.60i	−0.341173	−	0.590929i
$159$		0			0
$160$	−574.530			−0.283879
$161$	819.705	+	3188.04i	0.401253	+	1.56058i
$162$		0			0
$163$	−1052.33	+	1822.70i	−0.505676	+	0.875857i	0.494302	+	0.869290i	$0.335424\pi$
$163$	−1052.33	+	1822.70i	−0.505676	+	0.875857i	−0.999978	+	0.00656657i	$0.997910\pi$
$164$	589.277	+	1020.66i	0.280578	+	0.485975i
$165$		0			0
$166$	830.441	−	1438.37i	0.388281	−	0.672523i
$167$	1229.42			0.569674			0.284837	−	0.958576i	$-0.408061\pi$
$167$	1229.42			0.569674			0.284837	+	0.958576i	$0.408061\pi$
$168$		0			0
$169$	−1390.07			−0.632712
$170$	−265.092	+	459.153i	−0.119598	+	0.207150i
$171$		0			0
$172$	697.322	+	1207.80i	0.309130	+	0.535428i
$173$	1110.79	−	1923.95i	0.488162	−	0.845521i	−0.511746	−	0.859137i	$-0.671001\pi$
$173$	1110.79	−	1923.95i	0.488162	−	0.845521i	0.999907	+	0.0136163i	$0.00433433\pi$
$174$		0			0
$175$	−3520.63	−	981.688i	−1.52077	−	0.424049i
$176$	634.821			0.271883
$177$		0			0
$178$	463.479	+	802.770i	0.195164	+	0.338034i
$179$	−2152.54	−	3728.31i	−0.898818	−	1.55680i	−0.829007	−	0.559239i	$-0.811093\pi$
$179$	−2152.54	−	3728.31i	−0.898818	−	1.55680i	−0.0698117	−	0.997560i	$-0.522240\pi$
$180$		0			0
$181$	3415.77			1.40272			0.701360	−	0.712808i	$-0.252577\pi$
$181$	3415.77			1.40272			0.701360	+	0.712808i	$0.252577\pi$
$182$	−262.016	−	1019.05i	−0.106714	−	0.415037i
$183$		0			0
$184$	710.948	−	1231.40i	0.284846	−	0.493369i
$185$	−3302.06	−	5719.34i	−1.31228	−	2.27294i
$186$		0			0
$187$	292.911	−	507.336i	0.114544	−	0.198396i
$188$	952.083			0.369350
$189$		0			0
$190$	−2435.67			−0.930009
$191$	468.636	−	811.700i	0.177535	−	0.307500i	−0.763500	−	0.645807i	$-0.776521\pi$
$191$	468.636	−	811.700i	0.177535	−	0.307500i	0.941036	+	0.338307i	$0.109854\pi$
$192$		0			0
$193$	−1371.32	−	2375.20i	−0.511450	−	0.885858i	−0.999912	−	0.0132725i	$-0.995775\pi$
$193$	−1371.32	−	2375.20i	−0.511450	−	0.885858i	0.488462	−	0.872585i	$-0.337558\pi$
$194$	373.154	−	646.321i	0.138097	−	0.239191i
$195$		0			0
$196$	−1201.84	+	661.784i	−0.437989	+	0.241175i
$197$	1790.49			0.647548			0.323774	−	0.946134i	$-0.395048\pi$
$197$	1790.49			0.647548			0.323774	+	0.946134i	$0.395048\pi$
$198$		0			0
$199$	1910.81	+	3309.62i	0.680672	+	1.17896i	0.974776	+	0.223185i	$0.0716454\pi$
$199$	1910.81	+	3309.62i	0.680672	+	1.17896i	−0.294104	+	0.955773i	$0.595021\pi$
$200$	789.392	+	1367.27i	0.279092	+	0.483402i
$201$		0			0
$202$	−552.103			−0.192306
$203$	−2768.59	+	2713.05i	−0.957228	+	0.938025i
$204$		0			0
$205$	2644.98	−	4581.23i	0.901137	−	1.56082i
$206$	−88.7190	−	153.666i	−0.0300065	−	0.0519728i
$207$		0			0
$208$	−227.252	+	393.613i	−0.0757553	+	0.131212i
$209$	2691.26			0.890711
$210$		0			0
$211$	−66.8406			−0.0218080			−0.0109040	−	0.999941i	$-0.503471\pi$
$211$	−66.8406			−0.0218080			−0.0109040	+	0.999941i	$0.503471\pi$
$212$	−734.428	+	1272.07i	−0.237928	+	0.412103i
$213$		0			0
$214$	688.877	+	1193.17i	0.220050	+	0.381138i
$215$	3129.94	−	5421.21i	0.992838	−	1.71965i
$216$		0			0
$217$	−5135.37	−	1431.94i	−1.60651	−	0.447956i
$218$	−1169.80			−0.363434
$219$		0			0
$220$	−1424.70	−	2467.65i	−0.436606	−	0.756223i
$221$	209.712	+	363.231i	0.0638314	+	0.110559i
$222$		0			0
$223$	−5993.39			−1.79976			−0.899882	−	0.436134i	$-0.856348\pi$
$223$	−5993.39			−1.79976			−0.899882	+	0.436134i	$0.856348\pi$
$224$	570.871	+	159.181i	0.170281	+	0.0474809i
$225$		0			0
$226$	−1397.04	+	2419.75i	−0.411195	+	0.712210i
$227$	1535.08	+	2658.83i	0.448840	+	0.777414i	0.998311	−	0.0580988i	$-0.0185038\pi$
$227$	1535.08	+	2658.83i	0.448840	+	0.777414i	−0.549470	+	0.835513i	$0.685171\pi$
$228$		0			0
$229$	−65.7688	+	113.915i	−0.0189787	+	0.0328721i	−0.875359	−	0.483474i	$-0.839375\pi$
$229$	−65.7688	+	113.915i	−0.0189787	+	0.0328721i	0.856380	+	0.516346i	$0.172708\pi$
$230$	−6382.20			−1.82969
$231$		0			0
$232$	1674.41			0.473837
$233$	−248.293	+	430.056i	−0.0698121	+	0.120918i	−0.898818	−	0.438321i	$-0.855573\pi$
$233$	−248.293	+	430.056i	−0.0698121	+	0.120918i	0.829006	+	0.559239i	$0.188907\pi$
$234$		0			0
$235$	−2136.72	−	3700.91i	−0.593124	−	1.02732i
$236$	511.489	−	885.925i	0.141081	−	0.244359i
$237$		0			0
$238$	390.618	−	382.782i	0.106387	−	0.104252i
$239$	−1827.43			−0.494588			−0.247294	−	0.968941i	$-0.579541\pi$
$239$	−1827.43			−0.494588			−0.247294	+	0.968941i	$0.579541\pi$
$240$		0			0
$241$	3097.90	+	5365.72i	0.828022	+	1.43418i	0.899588	+	0.436740i	$0.143867\pi$
$241$	3097.90	+	5365.72i	0.828022	+	1.43418i	−0.0715663	+	0.997436i	$0.522800\pi$
$242$	243.207	+	421.248i	0.0646032	+	0.111896i
$243$		0			0
$244$	−1193.36			−0.313104
$245$	5269.71	+	3186.55i	1.37416	+	0.830944i
$246$		0			0
$247$	−963.415	+	1668.68i	−0.248181	+	0.429862i
$248$	1151.45	+	1994.37i	0.294827	+	0.510654i
$249$		0			0
$250$	1298.94	−	2249.83i	0.328609	−	0.569167i
$251$	−2249.36			−0.565652			−0.282826	−	0.959171i	$-0.591272\pi$
$251$	−2249.36			−0.565652			−0.282826	+	0.959171i	$0.591272\pi$
$252$		0			0
$253$	7051.94			1.75238
$254$	−988.221	+	1711.65i	−0.244120	+	0.422828i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1183.64	−	2050.13i	0.287291	−	0.497602i	−0.685871	−	0.727723i	$-0.740579\pi$
$257$	1183.64	−	2050.13i	0.287291	−	0.497602i	0.973162	+	0.230121i	$0.0739121\pi$
$258$		0			0
$259$	1696.42	+	6597.80i	0.406990	+	1.58289i
$260$	2040.05			0.486610
$261$		0			0
$262$	1130.92	+	1958.82i	0.266675	+	0.461894i
$263$	−802.307	−	1389.64i	−0.188108	−	0.325813i	0.756511	−	0.653980i	$-0.226902\pi$
$263$	−802.307	−	1389.64i	−0.188108	−	0.325813i	−0.944619	+	0.328168i	$0.893569\pi$
$264$		0			0
$265$	6592.98			1.52831
$266$	2420.15	+	674.832i	0.557854	+	0.155551i
$267$		0			0
$268$	1996.78	−	3458.53i	0.455122	−	0.788295i
$269$	−197.727	−	342.474i	−0.0448166	−	0.0776246i	0.842747	−	0.538310i	$-0.180937\pi$
$269$	−197.727	−	342.474i	−0.0448166	−	0.0776246i	−0.887564	+	0.460685i	$0.847604\pi$
$270$		0			0
$271$	1722.52	−	2983.49i	0.386109	−	0.668761i	−0.605813	−	0.795607i	$-0.707152\pi$
$271$	1722.52	−	2983.49i	0.386109	−	0.668761i	0.991922	+	0.126846i	$0.0404854\pi$
$272$	−236.241			−0.0526625
$273$		0			0
$274$	1807.11			0.398436
$275$	−3915.02	+	6781.01i	−0.858489	+	1.48695i
$276$		0			0
$277$	3826.62	+	6627.90i	0.830034	+	1.43766i	0.898011	+	0.439974i	$0.145012\pi$
$277$	3826.62	+	6627.90i	0.830034	+	1.43766i	−0.0679770	+	0.997687i	$0.521654\pi$
$278$	−1220.74	+	2114.38i	−0.263363	+	0.456158i
$279$		0			0
$280$	−662.418	−	2576.31i	−0.141382	−	0.549872i
$281$	3291.80			0.698835			0.349417	−	0.936967i	$-0.386380\pi$
$281$	3291.80			0.698835			0.349417	+	0.936967i	$0.386380\pi$
$282$		0			0
$283$	2728.08	+	4725.16i	0.573029	+	0.992516i	0.996253	+	0.0864903i	$0.0275651\pi$
$283$	2728.08	+	4725.16i	0.573029	+	0.992516i	−0.423224	+	0.906025i	$0.639102\pi$
$284$	−1318.44	−	2283.61i	−0.275476	−	0.477138i
$285$		0			0
$286$	−2254.13			−0.466048
$287$	−3897.42	+	3819.23i	−0.801594	+	0.785513i
$288$		0			0
$289$	2347.50	−	4065.98i	0.477813	−	0.827597i
$290$	−3757.80	−	6508.70i	−0.760916	−	1.31795i
$291$		0			0
$292$	−1278.13	+	2213.78i	−0.256153	+	0.443671i
$293$	−4614.48			−0.920072			−0.460036	−	0.887900i	$-0.652163\pi$
$293$	−4614.48			−0.920072			−0.460036	+	0.887900i	$0.652163\pi$
$294$		0			0
$295$	−4591.65			−0.906225
$296$	1471.34	−	2548.44i	0.288919	−	0.500422i
$297$		0			0
$298$	−2006.18	−	3474.81i	−0.389983	−	0.675471i
$299$	−2524.44	+	4372.47i	−0.488269	+	0.845707i
$300$		0			0
$301$	−4612.02	+	4519.50i	−0.883165	+	0.865447i
$302$	4555.41			0.867994
$303$		0			0
$304$	−542.644	−	939.887i	−0.102378	−	0.177323i
$305$	2678.21	+	4638.80i	0.502800	+	0.870876i
$306$		0			0
$307$	7520.35			1.39808			0.699038	−	0.715085i	$-0.253612\pi$
$307$	7520.35			1.39808			0.699038	+	0.715085i	$0.253612\pi$
$308$	731.932	+	2846.67i	0.135408	+	0.526636i
$309$		0			0
$310$	5168.29	−	8951.73i	0.946900	−	1.64008i
$311$	−4683.36	−	8111.81i	−0.853919	−	1.47903i	−0.877644	−	0.479312i	$-0.840886\pi$
$311$	−4683.36	−	8111.81i	−0.853919	−	1.47903i	0.0237254	−	0.999719i	$-0.492447\pi$
$312$		0			0
$313$	1402.09	−	2428.49i	0.253197	−	0.438550i	−0.711207	−	0.702982i	$-0.751851\pi$
$313$	1402.09	−	2428.49i	0.253197	−	0.438550i	0.964404	+	0.264432i	$0.0851846\pi$
$314$	603.130			0.108397
$315$		0			0
$316$	−2710.32			−0.482492
$317$	−4383.08	+	7591.72i	−0.776588	+	1.34509i	0.157310	+	0.987549i	$0.449718\pi$
$317$	−4383.08	+	7591.72i	−0.776588	+	1.34509i	−0.933898	+	0.357540i	$0.883615\pi$
$318$		0			0
$319$	4152.14	+	7191.72i	0.728763	+	1.26225i
$320$	−574.530	+	995.115i	−0.100366	+	0.173839i
$321$		0			0
$322$	6341.55	+	1768.27i	1.09752	+	0.306030i
$323$	−1001.52			−0.172527
$324$		0			0
$325$	−2802.99	−	4854.92i	−0.478405	−	0.828622i
$326$	2104.67	+	3645.39i	0.357567	+	0.619324i
$327$		0			0
$328$	2357.11			0.396797
$329$	1097.73	+	4269.34i	0.183951	+	0.715430i
$330$		0			0
$331$	−3337.31	+	5780.40i	−0.554185	+	0.959877i	0.443781	+	0.896135i	$0.353637\pi$
$331$	−3337.31	+	5780.40i	−0.554185	+	0.959877i	−0.997966	+	0.0637421i	$0.979697\pi$
$332$	−1660.88	−	2876.73i	−0.274556	−	0.475546i
$333$		0			0
$334$	1229.42	−	2129.42i	0.201410	−	0.348853i
$335$	−17925.2			−2.92345
$336$		0			0
$337$	−3025.35			−0.489024			−0.244512	−	0.969646i	$-0.578628\pi$
$337$	−3025.35			−0.489024			−0.244512	+	0.969646i	$0.578628\pi$
$338$	−1390.07	+	2407.67i	−0.223698	+	0.387456i
$339$		0			0
$340$	530.184	+	918.306i	0.0845685	+	0.146477i
$341$	−5710.64	+	9891.12i	−0.906888	+	1.57078i
$342$		0			0
$343$	−4353.27	−	4626.30i	−0.685290	−	0.728270i
$344$	2789.29			0.437175
$345$		0			0
$346$	−2221.58	−	3847.90i	−0.345182	−	0.597873i
$347$	4438.95	+	7688.48i	0.686730	+	1.18945i	0.972890	+	0.231269i	$0.0742877\pi$
$347$	4438.95	+	7688.48i	0.686730	+	1.18945i	−0.286160	+	0.958182i	$0.592379\pi$
$348$		0			0
$349$	−7994.22			−1.22613			−0.613067	−	0.790031i	$-0.710064\pi$
$349$	−7994.22			−1.22613			−0.613067	+	0.790031i	$0.710064\pi$
$350$	−5220.97	+	5116.23i	−0.797350	+	0.781354i
$351$		0			0
$352$	634.821	−	1099.54i	0.0961251	−	0.166494i
$353$	−5726.13	−	9917.96i	−0.863375	−	1.49541i	−0.868652	−	0.495423i	$-0.835013\pi$
$353$	−5726.13	−	9917.96i	−0.863375	−	1.49541i	0.00527653	−	0.999986i	$-0.498320\pi$
$354$		0			0
$355$	−5917.84	+	10250.0i	−0.884750	+	1.53243i
$356$	1853.92			0.276004
$357$		0			0
$358$	−8610.16			−1.27112
$359$	3639.69	−	6304.12i	0.535084	−	0.926793i	−0.464075	−	0.885796i	$-0.653613\pi$
$359$	3639.69	−	6304.12i	0.535084	−	0.926793i	0.999159	−	0.0409973i	$-0.0130535\pi$
$360$		0			0
$361$	1129.01	+	1955.50i	0.164603	+	0.285100i
$362$	3415.77	−	5916.29i	0.495936	−	0.858987i
$363$		0			0
$364$	−2027.06	−	565.222i	−0.291887	−	0.0813892i
$365$	11473.8			1.64539
$366$		0			0
$367$	1050.66	+	1819.80i	0.149439	+	0.258837i	0.931020	−	0.364967i	$-0.118920\pi$
$367$	1050.66	+	1819.80i	0.149439	+	0.258837i	−0.781581	+	0.623804i	$0.785586\pi$
$368$	−1421.90	−	2462.79i	−0.201417	−	0.348864i
$369$		0			0
$370$	−13208.3			−1.85585
$371$	−6550.99	−	1826.67i	−0.916740	−	0.255622i
$372$		0			0
$373$	−1979.37	+	3428.37i	−0.274767	+	0.475910i	−0.970076	−	0.242801i	$-0.921934\pi$
$373$	−1979.37	+	3428.37i	−0.274767	+	0.475910i	0.695310	+	0.718710i	$0.255267\pi$
$374$	−585.822	−	1014.67i	−0.0809950	−	0.140287i
$375$		0			0
$376$	952.083	−	1649.06i	0.130585	−	0.226180i
$377$	−5945.52			−0.812227
$378$		0			0
$379$	−3876.03			−0.525325			−0.262662	−	0.964888i	$-0.584601\pi$
$379$	−3876.03			−0.525325			−0.262662	+	0.964888i	$0.584601\pi$
$380$	−2435.67	+	4218.70i	−0.328808	+	0.569512i
$381$		0			0
$382$	−937.271	−	1623.40i	−0.125537	−	0.217436i
$383$	3281.16	−	5683.14i	0.437753	−	0.758211i	−0.559763	−	0.828653i	$-0.689108\pi$
$383$	3281.16	−	5683.14i	0.437753	−	0.758211i	0.997516	+	0.0704423i	$0.0224411\pi$
$384$		0			0
$385$	9422.83	−	9233.80i	1.24736	−	1.22233i
$386$	−5485.29			−0.723300
$387$		0			0
$388$	−746.307	−	1292.64i	−0.0976495	−	0.169134i
$389$	−4275.65	−	7405.64i	−0.557286	−	0.965247i	−0.997722	−	0.0674631i	$-0.978510\pi$
$389$	−4275.65	−	7405.64i	−0.557286	−	0.965247i	0.440436	−	0.897784i	$-0.354824\pi$
$390$		0			0
$391$	−2624.29			−0.339427
$392$	−55.6003	+	2743.44i	−0.00716388	+	0.353481i
$393$		0			0
$394$	1790.49	−	3101.22i	0.228943	−	0.396541i
$395$	6082.65	+	10535.5i	0.774814	+	1.34202i
$396$		0			0
$397$	1246.36	−	2158.76i	0.157564	−	0.272910i	−0.776425	−	0.630209i	$-0.782969\pi$
$397$	1246.36	−	2158.76i	0.157564	−	0.272910i	0.933990	+	0.357300i	$0.116302\pi$
$398$	7643.24			0.962616
$399$		0			0
$400$	3157.57			0.394696
$401$	−5375.99	+	9311.49i	−0.669487	+	1.15959i	0.308561	+	0.951205i	$0.400153\pi$
$401$	−5375.99	+	9311.49i	−0.669487	+	1.15959i	−0.978048	+	0.208381i	$0.933181\pi$
$402$		0			0
$403$	−4088.58	−	7081.63i	−0.505376	−	0.875337i
$404$	−552.103	+	956.271i	−0.0679905	+	0.117763i
$405$		0			0
$406$	1930.55	+	7508.40i	0.235989	+	0.917822i
$407$	14594.3			1.77743
$408$		0			0
$409$	−6071.96	−	10516.9i	−0.734081	−	1.27147i	−0.955125	−	0.296202i	$-0.904280\pi$
$409$	−6071.96	−	10516.9i	−0.734081	−	1.27147i	0.221044	−	0.975264i	$-0.429054\pi$
$410$	−5289.95	−	9162.46i	−0.637200	−	1.10366i
$411$		0			0
$412$	−354.876			−0.0424356
$413$	4562.41	+	1272.18i	0.543587	+	0.151573i
$414$		0			0
$415$	−7454.89	+	12912.2i	−0.881798	+	1.52732i
$416$	454.505	+	787.225i	0.0535671	+	0.0927810i
$417$		0			0
$418$	2691.26	−	4661.40i	0.314914	−	0.545447i
$419$	14733.1			1.71780			0.858902	−	0.512140i	$-0.171147\pi$
$419$	14733.1			1.71780			0.858902	+	0.512140i	$0.171147\pi$
$420$		0			0
$421$	8152.89			0.943818			0.471909	−	0.881647i	$-0.343565\pi$
$421$	8152.89			0.943818			0.471909	+	0.881647i	$0.343565\pi$
$422$	−66.8406	+	115.771i	−0.00771030	+	0.0133546i
$423$		0			0
$424$	1468.86	+	2544.13i	0.168240	+	0.291401i
$425$	1456.93	−	2523.47i	0.166285	−	0.288015i
$426$		0			0
$427$	−1375.92	−	5351.29i	−0.155938	−	0.606481i
$428$	2755.51			0.311197
$429$		0			0
$430$	−6259.88	−	10842.4i	−0.702042	−	1.21597i
$431$	−6721.50	−	11642.0i	−0.751191	−	1.30110i	−0.947246	−	0.320508i	$-0.896146\pi$
$431$	−6721.50	−	11642.0i	−0.751191	−	1.30110i	0.196055	−	0.980593i	$-0.437187\pi$
$432$		0			0
$433$	10303.9			1.14359			0.571796	−	0.820396i	$-0.306247\pi$
$433$	10303.9			1.14359			0.571796	+	0.820396i	$0.306247\pi$
$434$	−7615.56	+	7462.79i	−0.842301	+	0.825404i
$435$		0			0
$436$	−1169.80	+	2026.15i	−0.128493	+	0.222557i
$437$	−6027.99	−	10440.8i	−0.659858	−	1.14291i
$438$		0			0
$439$	−1049.70	+	1818.14i	−0.114122	+	0.197665i	−0.917428	−	0.397901i	$-0.869739\pi$
$439$	−1049.70	+	1818.14i	−0.114122	+	0.197665i	0.803306	+	0.595566i	$0.203072\pi$
$440$	−5698.80			−0.617454
$441$		0			0
$442$	838.847			0.0902712
$443$	−3171.89	+	5493.87i	−0.340182	+	0.589213i	−0.984466	−	0.175573i	$-0.943822\pi$
$443$	−3171.89	+	5493.87i	−0.340182	+	0.589213i	0.644284	+	0.764786i	$0.277156\pi$
$444$		0			0
$445$	−4160.67	−	7206.48i	−0.443224	−	0.767686i
$446$	−5993.39	+	10380.9i	−0.636313	+	1.10213i
$447$		0			0
$448$	846.580	−	829.596i	0.0892793	−	0.0874883i
$449$	12278.8			1.29058			0.645290	−	0.763938i	$-0.276737\pi$
$449$	12278.8			1.29058			0.645290	+	0.763938i	$0.276737\pi$
$450$		0			0
$451$	5845.08	+	10124.0i	0.610275	+	1.05703i
$452$	2794.09	+	4839.51i	0.290759	+	0.503609i
$453$		0			0
$454$	6140.32			0.634756
$455$	2352.13	+	9148.01i	0.242350	+	0.942562i
$456$		0			0
$457$	−5848.66	+	10130.2i	−0.598663	+	1.03691i	0.394356	+	0.918958i	$0.370968\pi$
$457$	−5848.66	+	10130.2i	−0.598663	+	1.03691i	−0.993019	+	0.117957i	$0.962366\pi$
$458$	131.538	+	227.830i	0.0134200	+	0.0232441i
$459$		0			0
$460$	−6382.20	+	11054.3i	−0.646894	+	1.12045i
$461$	−13465.2			−1.36039			−0.680194	−	0.733033i	$-0.738104\pi$
$461$	−13465.2			−1.36039			−0.680194	+	0.733033i	$0.738104\pi$
$462$		0			0
$463$	7792.30			0.782157			0.391079	−	0.920357i	$-0.372102\pi$
$463$	7792.30			0.782157			0.391079	+	0.920357i	$0.372102\pi$
$464$	1674.41	−	2900.16i	0.167527	−	0.290165i
$465$		0			0
$466$	496.586	+	860.112i	0.0493646	+	0.0855020i
$467$	7086.38	−	12274.0i	0.702181	−	1.21621i	−0.265518	−	0.964106i	$-0.585543\pi$
$467$	7086.38	−	12274.0i	0.702181	−	1.21621i	0.967699	−	0.252108i	$-0.0811238\pi$
$468$		0			0
$469$	17811.0	+	4966.39i	1.75359	+	0.488969i
$470$	−8546.88			−0.838804
$471$		0			0
$472$	−1022.98	−	1771.85i	−0.0997593	−	0.172788i
$473$	6916.79	+	11980.2i	0.672377	+	1.16459i
$474$		0			0
$475$	13386.2			1.29306
$476$	−272.379	−	1059.35i	−0.0262279	−	0.102007i
$477$		0			0
$478$	−1827.43	+	3165.20i	−0.174863	+	0.302872i
$479$	−2984.18	−	5168.75i	−0.284657	−	0.493040i	0.687869	−	0.725835i	$-0.258546\pi$
$479$	−2984.18	−	5168.75i	−0.284657	−	0.493040i	−0.972526	+	0.232795i	$0.925213\pi$
$480$		0			0
$481$	−5224.46	+	9049.03i	−0.495249	+	0.857797i
$482$	12391.6			1.17100
$483$		0			0
$484$	972.830			0.0913627
$485$	−3349.81	+	5802.04i	−0.313623	+	0.543211i
$486$		0			0
$487$	−1845.34	−	3196.22i	−0.171705	−	0.297401i	0.767311	−	0.641275i	$-0.221594\pi$
$487$	−1845.34	−	3196.22i	−0.171705	−	0.297401i	−0.939016	+	0.343874i	$0.888261\pi$
$488$	−1193.36	+	2066.97i	−0.110699	+	0.191736i
$489$		0			0
$490$	10789.0	−	5940.85i	0.994686	−	0.547715i
$491$	11918.0			1.09542			0.547708	−	0.836669i	$-0.315500\pi$
$491$	11918.0			1.09542			0.547708	+	0.836669i	$0.315500\pi$
$492$		0			0
$493$	−1545.17	−	2676.31i	−0.141158	−	0.244493i
$494$	1926.83	+	3337.37i	0.175490	+	0.303958i
$495$		0			0
$496$	4605.79			0.416948
$497$	8720.04	−	8545.11i	0.787017	−	0.771228i
$498$		0			0
$499$	−6982.72	+	12094.4i	−0.626432	+	1.08501i	0.361830	+	0.932244i	$0.382152\pi$
$499$	−6982.72	+	12094.4i	−0.626432	+	1.08501i	−0.988262	+	0.152768i	$0.951181\pi$
$500$	−2597.88	−	4499.66i	−0.232362	−	0.402462i
$501$		0			0
$502$	−2249.36	+	3896.01i	−0.199988	+	0.346390i
$503$	−8268.76			−0.732974			−0.366487	−	0.930423i	$-0.619440\pi$
$503$	−8268.76			−0.732974			−0.366487	+	0.930423i	$0.619440\pi$
$504$		0			0
$505$	4956.25			0.436733
$506$	7051.94	−	12214.3i	0.619559	−	1.07311i
$507$		0			0
$508$	1976.44	+	3423.30i	0.172619	+	0.298985i
$509$	10568.7	−	18305.6i	0.920336	−	1.59407i	0.121441	−	0.992599i	$-0.461248\pi$
$509$	10568.7	−	18305.6i	0.920336	−	1.59407i	0.798895	−	0.601470i	$-0.205418\pi$
$510$		0			0
$511$	−11400.7	−	3178.96i	−0.986963	−	0.275203i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−2367.29	−	4100.27i	−0.203145	−	0.351858i
$515$	796.433	+	1379.46i	0.0681457	+	0.118032i
$516$		0			0
$517$	9443.78			0.803360
$518$	13124.1	+	3659.52i	1.11321	+	0.310405i
$519$		0			0
$520$	2040.05	−	3533.47i	0.172043	−	0.297986i
$521$	−8634.33	−	14955.1i	−0.726060	−	1.25757i	−0.958536	−	0.284970i	$-0.908016\pi$
$521$	−8634.33	−	14955.1i	−0.726060	−	1.25757i	0.232477	−	0.972602i	$-0.425317\pi$
$522$		0			0
$523$	1000.71	−	1733.28i	0.0836674	−	0.144916i	−0.821155	−	0.570705i	$-0.806670\pi$
$523$	1000.71	−	1733.28i	0.0836674	−	0.144916i	0.904823	+	0.425789i	$0.140003\pi$
$524$	4523.70			0.377135
$525$		0			0
$526$	−3209.23			−0.266025
$527$	2125.14	−	3680.86i	0.175660	−	0.304252i
$528$		0			0
$529$	−9711.70	−	16821.2i	−0.798200	−	1.38252i
$530$	6592.98	−	11419.4i	0.540341	−	0.935898i
$531$		0			0
$532$	3589.00	−	3517.00i	0.292486	−	0.286619i
$533$	−8369.65			−0.680169
$534$		0			0
$535$	−6184.07	−	10711.1i	−0.499739	−	0.865574i
$536$	−3993.56	−	6917.05i	−0.321820	−	0.557409i
$537$		0			0
$538$	−790.910			−0.0633802
$539$	−11921.2	+	6564.28i	−0.952655	+	0.524570i
$540$		0			0
$541$	−206.979	+	358.499i	−0.0164487	+	0.0284899i	−0.874133	−	0.485687i	$-0.838569\pi$
$541$	−206.979	+	358.499i	−0.0164487	+	0.0284899i	0.857684	+	0.514177i	$0.171903\pi$
$542$	−3445.04	−	5966.99i	−0.273021	−	0.472886i
$543$		0			0
$544$	−236.241	+	409.181i	−0.0186190	+	0.0322490i
$545$	10501.3			0.825369
$546$		0			0
$547$	−17009.5			−1.32957			−0.664786	−	0.747034i	$-0.731477\pi$
$547$	−17009.5			−1.32957			−0.664786	+	0.747034i	$0.731477\pi$
$548$	1807.11	−	3130.01i	0.140868	−	0.243991i
$549$		0			0
$550$	7830.04	+	13562.0i	0.607043	+	1.05143i
$551$	7098.50	−	12295.0i	0.548832	−	0.950604i
$552$		0			0
$553$	−3124.93	−	12153.6i	−0.240299	−	0.934585i
$554$	15306.5			1.17384
$555$		0			0
$556$	2441.47	+	4228.75i	0.186226	+	0.322552i
$557$	−4146.02	−	7181.13i	−0.315391	−	0.546273i	0.664130	−	0.747617i	$-0.268802\pi$
$557$	−4146.02	−	7181.13i	−0.315391	−	0.546273i	−0.979521	+	0.201344i	$0.935469\pi$
$558$		0			0
$559$	−9904.25			−0.749383
$560$	−5124.72	−	1428.97i	−0.386713	−	0.107830i
$561$		0			0
$562$	3291.80	−	5701.57i	0.247075	−	0.427947i
$563$	7875.43	+	13640.6i	0.589538	+	1.02111i	0.994293	+	0.106685i	$0.0340235\pi$
$563$	7875.43	+	13640.6i	0.589538	+	1.02111i	−0.404755	+	0.914425i	$0.632643\pi$
$564$		0			0
$565$	12541.3	−	21722.2i	0.933835	−	1.61745i
$566$	10912.3			0.810386
$567$		0			0
$568$	−5273.76			−0.389581
$569$	−8339.46	+	14444.4i	−0.614426	+	1.06422i	0.376059	+	0.926596i	$0.377279\pi$
$569$	−8339.46	+	14444.4i	−0.614426	+	1.06422i	−0.990485	+	0.137622i	$0.956054\pi$
$570$		0			0
$571$	2338.21	+	4049.89i	0.171368	+	0.296817i	0.938898	−	0.344195i	$-0.111848\pi$
$571$	2338.21	+	4049.89i	0.171368	+	0.296817i	−0.767531	+	0.641012i	$0.778515\pi$
$572$	−2254.13	+	3904.27i	−0.164773	+	0.285395i
$573$		0			0
$574$	2717.68	+	10569.8i	0.197620	+	0.768595i
$575$	35076.0			2.54395
$576$		0			0
$577$	4491.29	+	7779.13i	0.324046	+	0.561265i	0.981319	−	0.192388i	$-0.0616233\pi$
$577$	4491.29	+	7779.13i	0.324046	+	0.561265i	−0.657273	+	0.753653i	$0.728290\pi$
$578$	−4694.99	−	8131.97i	−0.337865	−	0.585199i
$579$		0			0
$580$	−15031.2			−1.07610
$581$	10984.9	−	10764.5i	0.784391	−	0.768655i
$582$		0			0
$583$	−7284.84	+	12617.7i	−0.517508	+	0.896351i
$584$	2556.26	+	4427.56i	0.181128	+	0.313723i
$585$		0			0
$586$	−4614.48	+	7992.52i	−0.325294	+	0.563427i
$587$	−3547.35			−0.249429			−0.124714	−	0.992193i	$-0.539801\pi$
$587$	−3547.35			−0.249429			−0.124714	+	0.992193i	$0.539801\pi$
$588$		0			0
$589$	19525.8			1.36595
$590$	−4591.65	+	7952.98i	−0.320399	+	0.554947i
$591$		0			0
$592$	−2942.68	−	5096.87i	−0.204296	−	0.353851i
$593$	3489.19	−	6043.45i	0.241625	−	0.418507i	−0.719552	−	0.694438i	$-0.755653\pi$
$593$	3489.19	−	6043.45i	0.241625	−	0.418507i	0.961177	+	0.275931i	$0.0889862\pi$
$594$		0			0
$595$	−3506.59	+	3436.24i	−0.241607	+	0.236760i
$596$	−8024.73			−0.551520
$597$		0			0
$598$	5048.89	+	8744.93i	0.345258	+	0.598005i
$599$	3375.23	+	5846.06i	0.230230	+	0.398771i	0.957876	−	0.287183i	$-0.0927187\pi$
$599$	3375.23	+	5846.06i	0.230230	+	0.398771i	−0.727645	+	0.685953i	$0.759385\pi$
$600$		0			0
$601$	27765.5			1.88449			0.942243	−	0.334929i	$-0.108712\pi$
$601$	27765.5			1.88449			0.942243	+	0.334929i	$0.108712\pi$
$602$	3215.98	+	12507.8i	0.217730	+	0.846808i
$603$		0			0
$604$	4555.41	−	7890.20i	0.306882	−	0.531536i
$605$	−2183.28	−	3781.55i	−0.146716	−	0.254119i
$606$		0			0
$607$	−7448.52	+	12901.2i	−0.498066	+	0.862675i	−0.999998	−	0.00223201i	$-0.999290\pi$
$607$	−7448.52	+	12901.2i	−0.498066	+	0.862675i	0.501932	+	0.864907i	$0.332623\pi$
$608$	−2170.58			−0.144784
$609$		0			0
$610$	10712.9			0.711067
$611$	−3380.67	+	5855.50i	−0.223842	+	0.387706i
$612$		0			0
$613$	−9490.34	−	16437.7i	−0.625304	−	1.08306i	−0.988482	−	0.151338i	$-0.951642\pi$
$613$	−9490.34	−	16437.7i	−0.625304	−	1.08306i	0.363179	−	0.931720i	$-0.381691\pi$
$614$	7520.35	−	13025.6i	0.494294	−	0.856143i
$615$		0			0
$616$	5662.51	+	1578.93i	0.370372	+	0.103274i
$617$	−18594.0			−1.21324			−0.606618	−	0.794993i	$-0.707474\pi$
$617$	−18594.0			−1.21324			−0.606618	+	0.794993i	$0.707474\pi$
$618$		0			0
$619$	8274.89	+	14332.5i	0.537311	+	0.930651i	0.999048	+	0.0436334i	$0.0138933\pi$
$619$	8274.89	+	14332.5i	0.537311	+	0.930651i	−0.461736	+	0.887017i	$0.652773\pi$
$620$	−10336.6	−	17903.5i	−0.669559	−	1.15971i
$621$		0			0
$622$	−18733.4			−1.20762
$623$	2137.52	+	8313.35i	0.137461	+	0.534619i
$624$		0			0
$625$	673.628	−	1166.76i	0.0431122	−	0.0746725i
$626$	−2804.17	−	4856.97i	−0.179037	−	0.310102i
$627$		0			0
$628$	603.130	−	1044.65i	0.0383240	−	0.0663792i
$629$	−5431.10			−0.344280
$630$		0			0
$631$	−4468.32			−0.281904			−0.140952	−	0.990016i	$-0.545016\pi$
$631$	−4468.32			−0.281904			−0.140952	+	0.990016i	$0.545016\pi$
$632$	−2710.32	+	4694.41i	−0.170587	+	0.295465i
$633$		0			0
$634$	8766.16	+	15183.4i	0.549131	+	0.951122i
$635$	8871.29	−	15365.5i	0.554404	−	0.960255i
$636$		0			0
$637$	197.426	−	9741.44i	0.0122799	−	0.605918i
$638$	16608.6			1.03063
$639$		0			0
$640$	1149.06	+	1990.23i	0.0709696	+	0.122923i
$641$	−9245.29	−	16013.3i	−0.569683	−	0.986721i	−0.996597	−	0.0824282i	$-0.973732\pi$
$641$	−9245.29	−	16013.3i	−0.569683	−	0.986721i	0.426914	−	0.904292i	$-0.359601\pi$
$642$		0			0
$643$	−8584.51			−0.526501			−0.263250	−	0.964728i	$-0.584795\pi$
$643$	−8584.51			−0.526501			−0.263250	+	0.964728i	$0.584795\pi$
$644$	9404.28	−	9215.62i	0.575436	−	0.563892i
$645$		0			0
$646$	−1001.52	+	1734.68i	−0.0609973	+	0.105650i
$647$	9936.10	+	17209.8i	0.603753	+	1.04573i	0.992247	+	0.124280i	$0.0396621\pi$
$647$	9936.10	+	17209.8i	0.603753	+	1.04573i	−0.388494	+	0.921451i	$0.627005\pi$
$648$		0			0
$649$	5073.50	−	8787.56i	0.306860	−	0.531497i
$650$	−11211.9			−0.676567
$651$		0			0
$652$	8418.68			0.505676
$653$	−9867.12	+	17090.4i	−0.591318	+	1.02419i	0.402738	+	0.915315i	$0.368059\pi$
$653$	−9867.12	+	17090.4i	−0.591318	+	1.02419i	−0.994055	+	0.108877i	$0.965275\pi$
$654$		0			0
$655$	−10152.3	−	17584.4i	−0.605626	−	1.04897i
$656$	2357.11	−	4082.63i	0.140289	−	0.242988i
$657$		0			0
$658$	8492.45	+	2368.02i	0.503146	+	0.140296i
$659$	1670.68			0.0987564			0.0493782	−	0.998780i	$-0.484276\pi$
$659$	1670.68			0.0987564			0.0493782	+	0.998780i	$0.484276\pi$
$660$		0			0
$661$	2828.16	+	4898.51i	0.166418	+	0.288245i	0.937158	−	0.348905i	$-0.113446\pi$
$661$	2828.16	+	4898.51i	0.166418	+	0.288245i	−0.770740	+	0.637150i	$0.780113\pi$
$662$	6674.63	+	11560.8i	0.391868	+	0.678736i
$663$		0			0
$664$	−6643.53			−0.388281
$665$	−21725.8	−	6057.99i	−1.26690	−	0.353261i
$666$		0			0
$667$	18600.3	−	32216.6i	1.07977	−	1.87021i
$668$	−2458.84	−	4258.84i	−0.142418	−	0.246676i
$669$		0			0
$670$	−17925.2	+	31047.3i	−1.03360	+	1.79024i
$671$	−11837.1			−0.681020
$672$		0			0
$673$	342.738			0.0196309			0.00981544	−	0.999952i	$-0.496876\pi$
$673$	342.738			0.0196309			0.00981544	+	0.999952i	$0.496876\pi$
$674$	−3025.35	+	5240.06i	−0.172896	+	0.299465i
$675$		0			0
$676$	2780.14	+	4815.34i	0.158178	+	0.273972i
$677$	−9873.37	+	17101.2i	−0.560509	+	0.970830i	0.436943	+	0.899489i	$0.356061\pi$
$677$	−9873.37	+	17101.2i	−0.560509	+	0.970830i	−0.997452	+	0.0713405i	$0.977272\pi$
$678$		0			0
$679$	4936.01	−	4836.98i	0.278979	−	0.273382i
$680$	2120.74			0.119598
$681$		0			0
$682$	11421.3	+	19782.2i	0.641266	+	1.11071i
$683$	−1739.93	−	3013.65i	−0.0974766	−	0.168834i	0.813163	−	0.582036i	$-0.197744\pi$
$683$	−1739.93	−	3013.65i	−0.0974766	−	0.168834i	−0.910640	+	0.413202i	$0.864410\pi$
$684$		0			0
$685$	−16222.5			−0.904860
$686$	−12366.3	+	2913.79i	−0.688259	+	0.162171i
$687$		0			0
$688$	2789.29	−	4831.19i	0.154565	−	0.267714i
$689$	−5215.64	−	9033.75i	−0.288389	−	0.499504i
$690$		0			0
$691$	1399.18	−	2423.45i	0.0770295	−	0.133419i	−0.824938	−	0.565224i	$-0.808790\pi$
$691$	1399.18	−	2423.45i	0.0770295	−	0.133419i	0.901967	+	0.431805i	$0.142123\pi$
$692$	−8886.34			−0.488162
$693$		0			0
$694$	17755.8			0.971182
$695$	10958.6	−	18980.8i	0.598105	−	1.03595i
$696$		0			0
$697$	−2175.17	−	3767.51i	−0.118207	−	0.204741i
$698$	−7994.22	+	13846.4i	−0.433504	+	0.750850i
$699$		0			0
$700$	3640.60	+	14159.2i	0.196574	+	0.764525i
$701$	−4777.03			−0.257384			−0.128692	−	0.991685i	$-0.541078\pi$
$701$	−4777.03			−0.257384			−0.128692	+	0.991685i	$0.541078\pi$
$702$		0			0
$703$	−12475.2	−	21607.7i	−0.669291	−	1.15925i
$704$	−1269.64	−	2199.08i	−0.0679707	−	0.117729i
$705$		0			0
$706$	−22904.5			−1.22100
$707$	−4924.68	−	1373.19i	−0.261969	−	0.0730469i
$708$		0			0
$709$	−9952.32	+	17237.9i	−0.527175	+	0.913094i	0.472323	+	0.881425i	$0.343416\pi$
$709$	−9952.32	+	17237.9i	−0.527175	+	0.913094i	−0.999498	+	0.0316689i	$0.989918\pi$
$710$	11835.7	+	20500.0i	0.625613	+	1.08359i
$711$		0			0
$712$	1853.92	−	3211.08i	0.0975821	−	0.169017i
$713$	51163.7			2.68737
$714$		0			0
$715$	20235.4			1.05841
$716$	−8610.16	+	14913.2i	−0.449409	+	0.778399i
$717$		0			0
$718$	−7279.37	−	12608.2i	−0.378362	−	0.655342i
$719$	−9113.53	+	15785.1i	−0.472708	+	0.818755i	−0.999512	−	0.0312322i	$-0.990057\pi$
$719$	−9113.53	+	15785.1i	−0.472708	+	0.818755i	0.526804	+	0.849987i	$0.323390\pi$
$720$		0			0
$721$	−409.163	−	1591.34i	−0.0211346	−	0.0821977i
$722$	4516.04			0.232783
$723$		0			0
$724$	−6831.54	−	11832.6i	−0.350680	−	0.607395i
$725$	20652.6	+	35771.3i	1.05796	+	1.83243i
$726$		0			0
$727$	−23706.6			−1.20940			−0.604698	−	0.796455i	$-0.706706\pi$
$727$	−23706.6			−1.20940			−0.604698	+	0.796455i	$0.706706\pi$
$728$	−3006.05	+	2945.75i	−0.153038	+	0.149968i
$729$		0			0
$730$	11473.8	−	19873.2i	0.581731	−	1.00759i
$731$	−2574.00	−	4458.29i	−0.130236	−	0.225576i
$732$		0			0
$733$	7052.15	−	12214.7i	0.355358	−	0.615498i	−0.631821	−	0.775114i	$-0.717692\pi$
$733$	7052.15	−	12214.7i	0.355358	−	0.615498i	0.987179	+	0.159616i	$0.0510257\pi$
$734$	4202.66			0.211339
$735$		0			0
$736$	−5687.58			−0.284846
$737$	19806.2	−	34305.4i	0.989920	−	1.71459i
$738$		0			0
$739$	−12281.0	−	21271.3i	−0.611317	−	1.05883i	−0.991019	−	0.133724i	$-0.957307\pi$
$739$	−12281.0	−	21271.3i	−0.611317	−	1.05883i	0.379701	−	0.925109i	$-0.376027\pi$
$740$	−13208.3	+	22877.4i	−0.656142	+	1.13647i
$741$		0			0
$742$	−9714.87	+	9519.98i	−0.480653	+	0.471010i
$743$	−24154.7			−1.19267			−0.596333	−	0.802738i	$-0.703376\pi$
$743$	−24154.7			−1.19267			−0.596333	+	0.802738i	$0.703376\pi$
$744$		0			0
$745$	18009.6	+	31193.5i	0.885663	+	1.53401i
$746$	3958.74	+	6856.74i	0.194289	+	0.336519i
$747$		0			0
$748$	−2343.29			−0.114544
$749$	3177.03	+	12356.3i	0.154988	+	0.602789i
$750$		0			0
$751$	11784.7	−	20411.7i	0.572608	−	0.991787i	−0.423688	−	0.905808i	$-0.639265\pi$
$751$	11784.7	−	20411.7i	0.572608	−	0.991787i	0.996297	−	0.0859790i	$-0.0274018\pi$
$752$	−1904.17	−	3298.11i	−0.0923375	−	0.159933i
$753$		0			0
$754$	−5945.52	+	10297.9i	−0.287166	+	0.497386i
$755$	−40894.0			−1.97124
$756$		0			0
$757$	−13175.7			−0.632603			−0.316302	−	0.948659i	$-0.602441\pi$
$757$	−13175.7			−0.632603			−0.316302	+	0.948659i	$0.602441\pi$
$758$	−3876.03	+	6713.47i	−0.185730	+	0.321694i
$759$		0			0
$760$	4871.33	+	8437.39i	0.232502	+	0.402706i
$761$	−16037.4	+	27777.6i	−0.763936	+	1.32318i	0.176872	+	0.984234i	$0.443402\pi$
$761$	−16037.4	+	27777.6i	−0.763936	+	1.32318i	−0.940807	+	0.338942i	$0.889931\pi$
$762$		0			0
$763$	−10434.4	−	2909.52i	−0.495087	−	0.138049i
$764$	−3749.08			−0.177535
$765$		0			0
$766$	−6562.32	−	11366.3i	−0.309538	−	0.536136i
$767$	3632.41	+	6291.52i	0.171002	+	0.296185i
$768$		0			0
$769$	−25596.8			−1.20032			−0.600159	−	0.799881i	$-0.704896\pi$
$769$	−25596.8			−1.20032			−0.600159	+	0.799881i	$0.704896\pi$
$770$	−6570.58	−	25554.6i	−0.307516	−	1.19601i
$771$		0			0
$772$	−5485.29	+	9500.80i	−0.255725	+	0.442929i
$773$	1382.15	+	2393.96i	0.0643112	+	0.111390i	0.896388	−	0.443270i	$-0.146182\pi$
$773$	1382.15	+	2393.96i	0.0643112	+	0.111390i	−0.832077	+	0.554660i	$0.812848\pi$
$774$		0			0
$775$	−28404.5	+	49198.0i	−1.31654	+	2.28032i
$776$	−2985.23			−0.138097
$777$		0			0
$778$	−17102.6			−0.788121
$779$	9992.73	−	17307.9i	0.459598	−	0.796047i
$780$		0			0
$781$	−13077.7	−	22651.3i	−0.599177	−	1.03780i
$782$	−2624.29	+	4545.41i	−0.120006	+	0.207856i
$783$		0			0
$784$	4696.17	+	2839.74i	0.213929	+	0.129361i
$785$	−5414.31			−0.246172
$786$		0			0
$787$	15484.6	+	26820.2i	0.701357	+	1.21479i	0.967990	+	0.250988i	$0.0807555\pi$
$787$	15484.6	+	26820.2i	0.701357	+	1.21479i	−0.266633	+	0.963798i	$0.585911\pi$
$788$	−3580.98	−	6202.43i	−0.161887	−	0.280397i
$789$		0			0
$790$	24330.6			1.09575
$791$	−18479.8	+	18109.1i	−0.830680	+	0.814015i
$792$		0			0
$793$	4237.42	−	7339.42i	0.189754	−	0.328664i
$794$	−2492.72	−	4317.52i	−0.111415	−	0.192976i
$795$		0			0
$796$	7643.24	−	13238.5i	0.340336	−	0.589479i
$797$	43554.4			1.93573			0.967865	−	0.251472i	$-0.0809146\pi$
$797$	43554.4			1.93573			0.967865	+	0.251472i	$0.0809146\pi$
$798$		0			0
$799$	−3514.39			−0.155607
$800$	3157.57	−	5469.07i	0.139546	−	0.241701i
$801$		0			0
$802$	10752.0	+	18623.0i	0.473399	+	0.819951i
$803$	−12677.8	+	21958.7i	−0.557150	+	0.965012i
$804$		0			0
$805$	−56928.3	−	15873.8i	−2.49249	−	0.695003i
$806$	−16354.3			−0.714710
$807$		0			0
$808$	1104.21	+	1912.54i	0.0480766	+	0.0832710i
$809$	8215.93	+	14230.4i	0.357054	+	0.618436i	0.987467	−	0.157824i	$-0.0504479\pi$
$809$	8215.93	+	14230.4i	0.357054	+	0.618436i	−0.630413	+	0.776260i	$0.717115\pi$
$810$		0			0
$811$	4634.20			0.200652			0.100326	−	0.994955i	$-0.468011\pi$
$811$	4634.20			0.200652			0.100326	+	0.994955i	$0.468011\pi$
$812$	14935.5	+	4164.59i	0.645484	+	0.179986i
$813$		0			0
$814$	14594.3	−	25278.1i	0.628416	−	1.08845i
$815$	−18893.7	−	32724.8i	−0.812045	−	1.40650i
$816$		0			0
$817$	11824.9	−	20481.4i	0.506367	−	0.877054i
$818$	−24287.8			−1.03815
$819$		0			0
$820$	−21159.8			−0.901137
$821$	−20651.6	+	35769.5i	−0.877886	+	1.52054i	−0.0242290	+	0.999706i	$0.507713\pi$
$821$	−20651.6	+	35769.5i	−0.877886	+	1.52054i	−0.853657	+	0.520836i	$0.825620\pi$
$822$		0			0
$823$	−11976.2	−	20743.3i	−0.507245	−	0.878574i	−0.999965	−	0.00838602i	$-0.997331\pi$
$823$	−11976.2	−	20743.3i	−0.507245	−	0.878574i	0.492720	−	0.870188i	$-0.336003\pi$
$824$	−354.876	+	614.663i	−0.0150033	+	0.0259864i
$825$		0			0
$826$	6765.88	−	6630.15i	0.285006	−	0.279289i
$827$	−1967.92			−0.0827465			−0.0413732	−	0.999144i	$-0.513173\pi$
$827$	−1967.92			−0.0827465			−0.0413732	+	0.999144i	$0.513173\pi$
$828$		0			0
$829$	2589.87	+	4485.78i	0.108504	+	0.187934i	0.915164	−	0.403081i	$-0.132061\pi$
$829$	2589.87	+	4485.78i	0.108504	+	0.187934i	−0.806660	+	0.591015i	$0.798727\pi$
$830$	14909.8	+	25824.5i	0.623525	+	1.07998i
$831$		0			0
$832$	1818.02			0.0757553
$833$	4436.31	−	2442.81i	0.184525	−	0.101607i
$834$		0			0
$835$	−11036.6	+	19115.9i	−0.457408	+	0.792254i
$836$	−5382.53	−	9322.81i	−0.222678	−	0.385689i
$837$		0			0
$838$	14733.1	−	25518.5i	0.607335	−	1.05194i
$839$	−29027.4			−1.19444			−0.597222	−	0.802076i	$-0.703729\pi$
$839$	−29027.4			−1.19444			−0.597222	+	0.802076i	$0.703729\pi$
$840$		0			0
$841$	19417.9			0.796176
$842$	8152.89	−	14121.2i	0.333690	−	0.577968i
$843$		0			0
$844$	133.681	+	231.543i	0.00545201	+	0.00944315i
$845$	12478.7	−	21613.7i	0.508023	−	0.879922i
$846$		0			0
$847$	1121.65	+	4362.37i	0.0455021	+	0.176969i
$848$	5875.42			0.237928
$849$		0			0
$850$	−2913.85	−	5046.94i	−0.117581	−	0.203657i
$851$	−32688.9	−	56618.9i	−1.31676	−	2.28069i
$852$		0			0
$853$	12483.2			0.501076			0.250538	−	0.968107i	$-0.419393\pi$
$853$	12483.2			0.501076			0.250538	+	0.968107i	$0.419393\pi$
$854$	−10644.6	−	2968.13i	−0.426524	−	0.118931i
$855$		0			0
$856$	2755.51	−	4772.68i	0.110025	−	0.190569i
$857$	5458.03	+	9453.59i	0.217553	+	0.376813i	0.954059	−	0.299618i	$-0.0968592\pi$
$857$	5458.03	+	9453.59i	0.217553	+	0.376813i	−0.736506	+	0.676431i	$0.763526\pi$
$858$		0			0
$859$	14683.5	−	25432.6i	0.583231	−	1.01019i	−0.411863	−	0.911246i	$-0.635122\pi$
$859$	14683.5	−	25432.6i	0.583231	−	1.01019i	0.995093	−	0.0989394i	$-0.0315450\pi$
$860$	−25039.5			−0.992838
$861$		0			0
$862$	−26886.0			−1.06234
$863$	−1384.63	+	2398.25i	−0.0546157	+	0.0945971i	−0.892041	−	0.451955i	$-0.850727\pi$
$863$	−1384.63	+	2398.25i	−0.0546157	+	0.0945971i	0.837425	+	0.546552i	$0.184060\pi$
$864$		0			0
$865$	19943.2	+	34542.7i	0.783919	+	1.35779i
$866$	10303.9	−	17846.9i	0.404321	−	0.700304i
$867$		0			0
$868$	5310.36	+	20653.3i	0.207656	+	0.807626i
$869$	−26883.9			−1.04945
$870$		0			0
$871$	14180.4	+	24561.2i	0.551647	+	0.955481i
$872$	2339.59	+	4052.30i	0.0908585	+	0.157372i
$873$		0			0
$874$	−24112.0			−0.933180
$875$	17182.1	−	16837.4i	0.663843	−	0.650525i
$876$		0			0
$877$	2048.40	−	3547.93i	0.0788705	−	0.136608i	−0.823892	−	0.566746i	$-0.808202\pi$
$877$	2048.40	−	3547.93i	0.0788705	−	0.136608i	0.902763	+	0.430139i	$0.141535\pi$
$878$	2099.41	+	3636.28i	0.0806965	+	0.139770i
$879$		0			0
$880$	−5698.80	+	9870.61i	−0.218303	+	0.378112i
$881$	−15164.0			−0.579896			−0.289948	−	0.957042i	$-0.593638\pi$
$881$	−15164.0			−0.579896			−0.289948	+	0.957042i	$0.593638\pi$
$882$		0			0
$883$	20459.2			0.779736			0.389868	−	0.920871i	$-0.372521\pi$
$883$	20459.2			0.779736			0.389868	+	0.920871i	$0.372521\pi$
$884$	838.847	−	1452.93i	0.0319157	−	0.0552796i
$885$		0			0
$886$	6343.77	+	10987.7i	0.240545	+	0.416637i
$887$	−24017.5	+	41599.5i	−0.909164	+	1.57472i	−0.0939363	+	0.995578i	$0.529945\pi$
$887$	−24017.5	+	41599.5i	−0.909164	+	1.57472i	−0.815228	+	0.579140i	$0.803388\pi$
$888$		0			0
$889$	−13072.0	+	12809.8i	−0.493162	+	0.483268i
$890$	−16642.7			−0.626813
$891$		0			0
$892$	11986.8	+	20761.7i	0.449941	+	0.779321i
$893$	−8072.54	−	13982.0i	−0.302505	−	0.523955i
$894$		0			0
$895$	77293.7			2.88675
$896$	−590.323	−	2295.92i	−0.0220104	−	0.0856040i
$897$		0			0
$898$	12278.8	−	21267.4i	0.456289	−	0.790315i
$899$	30124.9	+	52177.8i	1.11760	+	1.93574i
$900$		0			0
$901$	2710.96	−	4695.53i	0.100239	−	0.173619i
$902$	23380.3			0.863059
$903$		0			0
$904$	11176.4			0.411195
$905$	−30663.5	+	53110.7i	−1.12628	+	1.95078i
$906$		0			0
$907$	10652.5	+	18450.6i	0.389977	+	0.675460i	0.992446	−	0.122683i	$-0.0391497\pi$
$907$	10652.5	+	18450.6i	0.389977	+	0.675460i	−0.602469	+	0.798142i	$0.705816\pi$
$908$	6140.32	−	10635.3i	0.224420	−	0.388707i
$909$		0			0
$910$	18197.0	+	5074.01i	0.662883	+	0.184837i
$911$	−16526.4			−0.601036			−0.300518	−	0.953776i	$-0.597160\pi$
$911$	−16526.4			−0.601036			−0.300518	+	0.953776i	$0.597160\pi$
$912$		0			0
$913$	−16474.4	−	28534.5i	−0.597177	−	1.03434i
$914$	11697.3	+	20260.4i	0.423319	+	0.733209i
$915$		0			0
$916$	526.150			0.0189787
$917$	5215.71	+	20285.2i	0.187828	+	0.730509i
$918$		0			0
$919$	−24244.3	+	41992.4i	−0.870235	+	1.50729i	−0.00848156	+	0.999964i	$0.502700\pi$
$919$	−24244.3	+	41992.4i	−0.870235	+	1.50729i	−0.861753	+	0.507327i	$0.830634\pi$
$920$	12764.4	+	22108.6i	0.457423	+	0.792281i
$921$		0			0
$922$	−13465.2	+	23322.5i	−0.480969	+	0.833064i
$923$	18726.2			0.667800
$924$		0			0
$925$	72591.5			2.58032
$926$	7792.30	−	13496.7i	0.276534	−	0.478972i
$927$		0			0
$928$	−3348.82	−	5800.32i	−0.118459	−	0.205178i
$929$	−6715.15	+	11631.0i	−0.237155	+	0.410764i	−0.959897	−	0.280354i	$-0.909548\pi$
$929$	−6715.15	+	11631.0i	−0.237155	+	0.410764i	0.722742	+	0.691118i	$0.242882\pi$
$930$		0			0
$931$	19909.0	+	12038.8i	0.700849	+	0.423798i
$932$	1986.34			0.0698121
$933$		0			0
$934$	−14172.8	−	24548.0i	−0.496517	−	0.859993i
$935$	5258.94	+	9108.75i	0.183942	+	0.318597i
$936$		0			0
$937$	13716.5			0.478227			0.239114	−	0.970992i	$-0.423143\pi$
$937$	13716.5			0.478227			0.239114	+	0.970992i	$0.423143\pi$
$938$	26413.0	−	25883.2i	0.919420	−	0.900976i
$939$		0			0
$940$	−8546.88	+	14803.6i	−0.296562	+	0.513661i
$941$	−19160.9	−	33187.7i	−0.663791	−	1.14972i	−0.979612	−	0.200901i	$-0.935613\pi$
$941$	−19160.9	−	33187.7i	−0.663791	−	1.14972i	0.315820	−	0.948819i	$-0.397720\pi$
$942$		0			0
$943$	26184.0	−	45352.1i	0.904210	−	1.56614i
$944$	−4091.91			−0.141081
$945$		0			0
$946$	27667.1			0.950884
$947$	27580.9	−	47771.6i	0.946420	−	1.63925i	0.193537	−	0.981093i	$-0.438004\pi$
$947$	27580.9	−	47771.6i	0.946420	−	1.63925i	0.752883	−	0.658155i	$-0.228663\pi$
$948$		0			0
$949$	−9076.80	−	15721.5i	−0.310480	−	0.537767i
$950$	13386.2	−	23185.6i	0.457165	−	0.791832i
$951$		0			0
$952$	−2107.23	−	587.577i	−0.0717393	−	0.0200037i
$953$	11111.7			0.377695			0.188848	−	0.982006i	$-0.439525\pi$
$953$	11111.7			0.377695			0.188848	+	0.982006i	$0.439525\pi$
$954$		0			0
$955$	8413.91	+	14573.3i	0.285097	+	0.493802i
$956$	3654.85	+	6330.39i	0.123647	+	0.214163i
$957$		0			0
$958$	−11936.7			−0.402565
$959$	16119.2	+	4494.64i	0.542768	+	0.151345i
$960$		0			0
$961$	−26536.7	+	45962.9i	−0.890763	+	1.54285i
$962$	10448.9	+	18098.1i	0.350194	+	0.606554i
$963$		0			0
$964$	12391.6	−	21462.9i	0.414011	−	0.717088i
$965$	49241.6			1.64263
$966$		0			0
$967$	−25253.4			−0.839809			−0.419905	−	0.907568i	$-0.637936\pi$
$967$	−25253.4			−0.839809			−0.419905	+	0.907568i	$0.637936\pi$
$968$	972.830	−	1684.99i	0.0323016	−	0.0559480i
$969$		0			0
$970$	6699.62	+	11604.1i	0.221765	+	0.384108i
$971$	−6336.04	+	10974.3i	−0.209406	+	0.362702i	−0.951528	−	0.307564i	$-0.900486\pi$
$971$	−6336.04	+	10974.3i	−0.209406	+	0.362702i	0.742122	+	0.670265i	$0.233820\pi$
$972$		0			0
$973$	−16147.7	+	15823.7i	−0.532035	+	0.521362i
$974$	−7381.35			−0.242827
$975$		0			0
$976$	2386.73	+	4133.93i	0.0782759	+	0.135578i
$977$	3115.03	+	5395.39i	0.102005	+	0.176677i	0.912511	−	0.409053i	$-0.134141\pi$
$977$	3115.03	+	5395.39i	0.102005	+	0.176677i	−0.810506	+	0.585731i	$0.800808\pi$
$978$		0			0
$979$	18389.1			0.600326
$980$	499.125	−	24627.9i	0.0162694	−	0.802765i
$981$		0			0
$982$	11918.0	−	20642.5i	0.387288	−	0.670803i
$983$	13293.4	+	23024.8i	0.431325	+	0.747078i	0.996988	−	0.0775594i	$-0.0247127\pi$
$983$	13293.4	+	23024.8i	0.431325	+	0.747078i	−0.565662	+	0.824637i	$0.691379\pi$
$984$		0			0
$985$	−16073.3	+	27839.7i	−0.519936	+	0.900555i
$986$	−6180.67			−0.199628
$987$		0			0
$988$	7707.32			0.248181
$989$	30985.0	−	53667.5i	0.996223	−	1.72551i
$990$		0			0
$991$	−7430.78	−	12870.5i	−0.238190	−	0.412558i	0.722005	−	0.691888i	$-0.243221\pi$
$991$	−7430.78	−	12870.5i	−0.238190	−	0.412558i	−0.960195	+	0.279330i	$0.909888\pi$
$992$	4605.79	−	7977.46i	0.147413	−	0.255327i
$993$		0			0
$994$	−6080.52	−	23648.7i	−0.194026	−	0.754618i
$995$	−68613.6			−2.18613
$996$		0			0
$997$	−3129.05	−	5419.68i	−0.0993962	−	0.172159i	0.812039	−	0.583604i	$-0.198358\pi$
$997$	−3129.05	−	5419.68i	−0.0993962	−	0.172159i	−0.911435	+	0.411444i	$0.865024\pi$
$998$	13965.4	+	24188.9i	0.442954	+	0.767219i
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-8.97703 + 15.5487i) q^{5} +(13.2278 - 12.9624i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-8.97703 + 15.5487i) q^{5} +(13.2278 - 12.9624i) q^{7} -8.00000 q^{8} +(17.9541 + 31.0973i) q^{10} +(-19.8381 - 34.3607i) q^{11} +28.4065 q^{13} +(-9.22380 - 35.8737i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(7.38252 + 12.7869i) q^{17} +(-33.9153 + 58.7430i) q^{19} +71.8162 q^{20} -79.3526 q^{22} +(-88.8684 + 153.925i) q^{23} +(-98.6740 - 170.908i) q^{25} +(28.4065 - 49.2016i) q^{26} +(-71.3589 - 19.8976i) q^{28} -209.301 q^{29} +(-143.931 - 249.296i) q^{31} +(16.0000 + 27.7128i) q^{32} +29.5301 q^{34} +(82.8023 + 322.039i) q^{35} +(-183.917 + 318.554i) q^{37} +(67.8305 + 117.486i) q^{38} +(71.8162 - 124.389i) q^{40} -294.638 q^{41} -348.661 q^{43} +(-79.3526 + 137.443i) q^{44} +(177.737 + 307.849i) q^{46} +(-119.010 + 206.132i) q^{47} +(6.95004 - 342.930i) q^{49} -394.696 q^{50} +(-56.8131 - 98.4031i) q^{52} +(-183.607 - 318.017i) q^{53} +712.350 q^{55} +(-105.822 + 103.700i) q^{56} +(-209.301 + 362.520i) q^{58} +(127.872 + 221.481i) q^{59} +(149.170 - 258.371i) q^{61} -575.724 q^{62} +64.0000 q^{64} +(-255.006 + 441.684i) q^{65} +(499.195 + 864.631i) q^{67} +(29.5301 - 51.1476i) q^{68} +(640.590 + 178.621i) q^{70} +659.220 q^{71} +(-319.532 - 553.446i) q^{73} +(367.835 + 637.109i) q^{74} +271.322 q^{76} +(-707.814 - 197.366i) q^{77} +(338.790 - 586.802i) q^{79} +(-143.632 - 248.779i) q^{80} +(-294.638 + 510.328i) q^{82} +830.441 q^{83} -265.092 q^{85} +(-348.661 + 603.899i) q^{86} +(158.705 + 274.885i) q^{88} +(-231.740 + 401.385i) q^{89} +(375.756 - 368.218i) q^{91} +710.948 q^{92} +(238.021 + 412.264i) q^{94} +(-608.916 - 1054.67i) q^{95} +373.154 q^{97} +(-587.021 - 354.967i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

Introduction

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