LMFDB - Embedded newform 378.4.g.e.109.1

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			109.1
Root			$1.39378 - 2.41410i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.e.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-8.97703 - 15.5487i) q^{5} +(13.2278 + 12.9624i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-8.97703 - 15.5487i) q^{5} +(13.2278 + 12.9624i) q^{7} -8.00000 q^{8} +(17.9541 - 31.0973i) q^{10} +(-19.8381 + 34.3607i) q^{11} +28.4065 q^{13} +(-9.22380 + 35.8737i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(7.38252 - 12.7869i) q^{17} +(-33.9153 - 58.7430i) q^{19} +71.8162 q^{20} -79.3526 q^{22} +(-88.8684 - 153.925i) q^{23} +(-98.6740 + 170.908i) q^{25} +(28.4065 + 49.2016i) q^{26} +(-71.3589 + 19.8976i) q^{28} -209.301 q^{29} +(-143.931 + 249.296i) q^{31} +(16.0000 - 27.7128i) q^{32} +29.5301 q^{34} +(82.8023 - 322.039i) q^{35} +(-183.917 - 318.554i) q^{37} +(67.8305 - 117.486i) q^{38} +(71.8162 + 124.389i) q^{40} -294.638 q^{41} -348.661 q^{43} +(-79.3526 - 137.443i) q^{44} +(177.737 - 307.849i) q^{46} +(-119.010 - 206.132i) q^{47} +(6.95004 + 342.930i) q^{49} -394.696 q^{50} +(-56.8131 + 98.4031i) q^{52} +(-183.607 + 318.017i) q^{53} +712.350 q^{55} +(-105.822 - 103.700i) q^{56} +(-209.301 - 362.520i) q^{58} +(127.872 - 221.481i) q^{59} +(149.170 + 258.371i) q^{61} -575.724 q^{62} +64.0000 q^{64} +(-255.006 - 441.684i) q^{65} +(499.195 - 864.631i) q^{67} +(29.5301 + 51.1476i) q^{68} +(640.590 - 178.621i) q^{70} +659.220 q^{71} +(-319.532 + 553.446i) q^{73} +(367.835 - 637.109i) q^{74} +271.322 q^{76} +(-707.814 + 197.366i) q^{77} +(338.790 + 586.802i) q^{79} +(-143.632 + 248.779i) q^{80} +(-294.638 - 510.328i) q^{82} +830.441 q^{83} -265.092 q^{85} +(-348.661 - 603.899i) q^{86} +(158.705 - 274.885i) q^{88} +(-231.740 - 401.385i) q^{89} +(375.756 + 368.218i) q^{91} +710.948 q^{92} +(238.021 - 412.264i) q^{94} +(-608.916 + 1054.67i) q^{95} +373.154 q^{97} +(-587.021 + 354.967i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

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

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
$5$	−8.97703	−	15.5487i	−0.802930	−	1.39072i	−0.917680	−	0.397321i	$-0.869940\pi$
$5$	−8.97703	−	15.5487i	−0.802930	−	1.39072i	0.114750	−	0.993394i	$-0.463393\pi$
$6$		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	0.454918	+	0.890534i	$0.349669\pi$
$11$	−19.8381	+	34.3607i	−0.543766	+	0.941830i	−0.998683	+	0.0512966i	$0.983665\pi$
$12$		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	−0.808546	−	0.588433i	$-0.799745\pi$
$17$	7.38252	−	12.7869i	0.105325	−	0.182428i	0.913871	+	0.406005i	$0.133078\pi$
$18$		0			0
$19$	−33.9153	−	58.7430i	−0.409510	−	0.709293i	0.585325	−	0.810799i	$-0.300967\pi$
$19$	−33.9153	−	58.7430i	−0.409510	−	0.709293i	−0.994835	+	0.101506i	$0.967634\pi$
$20$	71.8162			0.802930
$21$		0			0
$22$	−79.3526			−0.769001
$23$	−88.8684	−	153.925i	−0.805667	−	1.39546i	−0.915840	−	0.401544i	$-0.868474\pi$
$23$	−88.8684	−	153.925i	−0.805667	−	1.39546i	0.110172	−	0.993913i	$-0.464860\pi$
$24$		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	0.0610314	+	0.998136i	$0.480561\pi$
$31$	−143.931	+	249.296i	−0.833895	+	1.44435i	−0.894927	+	0.446213i	$0.852772\pi$
$32$	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	−0.907748	−	0.419515i	$-0.862200\pi$
$37$	−183.917	−	318.554i	−0.817185	−	1.41541i	0.0905633	−	0.995891i	$-0.471133\pi$
$38$	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	0.620114	−	0.784512i	$-0.287086\pi$
$47$	−119.010	−	206.132i	−0.369350	−	0.639733i	−0.989464	+	0.144779i	$0.953753\pi$
$48$		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	−0.999617	−	0.0276585i	$-0.991195\pi$
$53$	−183.607	+	318.017i	−0.475856	+	0.824206i	0.523762	+	0.851865i	$0.324528\pi$
$54$		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	−0.689755	−	0.724043i	$-0.742282\pi$
$59$	127.872	−	221.481i	0.282162	−	0.488719i	0.971917	+	0.235324i	$0.0756151\pi$
$60$		0			0
$61$	149.170	+	258.371i	0.313104	+	0.542311i	0.979033	−	0.203704i	$-0.0652980\pi$
$61$	149.170	+	258.371i	0.313104	+	0.542311i	−0.665929	+	0.746015i	$0.731965\pi$
$62$	−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	0.0965267	−	0.995330i	$-0.469227\pi$
$67$	499.195	−	864.631i	0.910245	−	1.57659i	0.813718	−	0.581260i	$-0.197440\pi$
$68$	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	0.487591	+	0.873072i	$0.337876\pi$
$73$	−319.532	+	553.446i	−0.512307	+	0.887341i	−0.999898	+	0.0142693i	$0.995458\pi$
$74$	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	0.999798	−	0.0201001i	$-0.00639848\pi$
$79$	338.790	+	586.802i	0.482492	+	0.835700i	−0.517306	+	0.855800i	$0.673065\pi$
$80$	−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	0.694384	−	0.719605i	$-0.255677\pi$
$89$	−231.740	−	401.385i	−0.276004	−	0.478053i	−0.970388	+	0.241552i	$0.922344\pi$
$90$		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	−0.925972	−	0.377593i	$-0.876752\pi$
$101$	−138.026	+	239.068i	−0.135981	+	0.235526i	0.789991	+	0.613119i	$0.210085\pi$
$102$		0			0
$103$	44.3595	+	76.8329i	0.0424356	+	0.0735007i	0.886463	−	0.462799i	$-0.153155\pi$
$103$	44.3595	+	76.8329i	0.0424356	+	0.0735007i	−0.844027	+	0.536300i	$0.819822\pi$
$104$	−227.252			−0.214268
$105$		0			0
$106$	−734.428			−0.672962
$107$	−344.439	−	596.585i	−0.311197	−	0.539010i	0.667424	−	0.744678i	$-0.267397\pi$
$107$	−344.439	−	596.585i	−0.311197	−	0.539010i	−0.978622	+	0.205668i	$0.934063\pi$
$108$		0			0
$109$	−292.449	+	506.537i	−0.256987	+	0.445114i	−0.965433	−	0.260650i	$-0.916063\pi$
$109$	−292.449	+	506.537i	−0.256987	+	0.445114i	0.708447	+	0.705765i	$0.249396\pi$
$110$	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	0.613509	−	0.789688i	$-0.289757\pi$
$131$	−565.462	−	979.409i	−0.377135	−	0.653217i	−0.990644	+	0.136471i	$0.956424\pi$
$132$		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	−0.690076	−	0.723737i	$-0.742423\pi$
$137$	451.777	−	782.501i	0.281737	−	0.487983i	0.971813	+	0.235755i	$0.0757561\pi$
$138$		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	0.998165	+	0.0605494i	$0.0192853\pi$
$149$	1003.09	+	1737.41i	0.551520	+	0.955261i	−0.446645	+	0.894711i	$0.647381\pi$
$150$		0			0
$151$	1138.85	−	1972.55i	0.613765	−	1.06307i	−0.376835	−	0.926280i	$-0.622988\pi$
$151$	1138.85	−	1972.55i	0.613765	−	1.06307i	0.990600	−	0.136791i	$-0.0436790\pi$
$152$	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	−0.825154	−	0.564908i	$-0.808912\pi$
$157$	150.782	−	261.163i	0.0766481	−	0.132758i	0.901802	+	0.432150i	$0.142245\pi$
$158$	−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	−0.999978	−	0.00656657i	$-0.997910\pi$
$163$	−1052.33	−	1822.70i	−0.505676	−	0.875857i	0.494302	−	0.869290i	$-0.335424\pi$
$164$	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	0.999907	−	0.0136163i	$-0.00433433\pi$
$173$	1110.79	+	1923.95i	0.488162	+	0.845521i	−0.511746	+	0.859137i	$0.671001\pi$
$174$		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	−0.0698117	+	0.997560i	$0.522240\pi$
$179$	−2152.54	+	3728.31i	−0.898818	+	1.55680i	−0.829007	+	0.559239i	$0.811093\pi$
$180$		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	0.941036	−	0.338307i	$-0.109854\pi$
$191$	468.636	+	811.700i	0.177535	+	0.307500i	−0.763500	+	0.645807i	$0.776521\pi$
$192$		0			0
$193$	−1371.32	+	2375.20i	−0.511450	+	0.885858i	0.488462	+	0.872585i	$0.337558\pi$
$193$	−1371.32	+	2375.20i	−0.511450	+	0.885858i	−0.999912	+	0.0132725i	$0.995775\pi$
$194$	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	−0.294104	−	0.955773i	$-0.595021\pi$
$199$	1910.81	−	3309.62i	0.680672	−	1.17896i	0.974776	−	0.223185i	$-0.0716454\pi$
$200$	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	−0.549470	−	0.835513i	$-0.685171\pi$
$227$	1535.08	−	2658.83i	0.448840	−	0.777414i	0.998311	+	0.0580988i	$0.0185038\pi$
$228$		0			0
$229$	−65.7688	−	113.915i	−0.0189787	−	0.0328721i	0.856380	−	0.516346i	$-0.172708\pi$
$229$	−65.7688	−	113.915i	−0.0189787	−	0.0328721i	−0.875359	+	0.483474i	$0.839375\pi$
$230$	−6382.20			−1.82969
$231$		0			0
$232$	1674.41			0.473837
$233$	−248.293	−	430.056i	−0.0698121	−	0.120918i	0.829006	−	0.559239i	$-0.188907\pi$
$233$	−248.293	−	430.056i	−0.0698121	−	0.120918i	−0.898818	+	0.438321i	$0.855573\pi$
$234$		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	−0.0715663	−	0.997436i	$-0.522800\pi$
$241$	3097.90	−	5365.72i	0.828022	−	1.43418i	0.899588	−	0.436740i	$-0.143867\pi$
$242$	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	0.973162	−	0.230121i	$-0.0739121\pi$
$257$	1183.64	+	2050.13i	0.287291	+	0.497602i	−0.685871	+	0.727723i	$0.740579\pi$
$258$		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	−0.944619	−	0.328168i	$-0.893569\pi$
$263$	−802.307	+	1389.64i	−0.188108	+	0.325813i	0.756511	+	0.653980i	$0.226902\pi$
$264$		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	−0.887564	−	0.460685i	$-0.847604\pi$
$269$	−197.727	+	342.474i	−0.0448166	+	0.0776246i	0.842747	+	0.538310i	$0.180937\pi$
$270$		0			0
$271$	1722.52	+	2983.49i	0.386109	+	0.668761i	0.991922	−	0.126846i	$-0.0404854\pi$
$271$	1722.52	+	2983.49i	0.386109	+	0.668761i	−0.605813	+	0.795607i	$0.707152\pi$
$272$	−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	−0.0679770	−	0.997687i	$-0.521654\pi$
$277$	3826.62	−	6627.90i	0.830034	−	1.43766i	0.898011	−	0.439974i	$-0.145012\pi$
$278$	−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	−0.423224	−	0.906025i	$-0.639102\pi$
$283$	2728.08	−	4725.16i	0.573029	−	0.992516i	0.996253	−	0.0864903i	$-0.0275651\pi$
$284$	−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	0.0237254	+	0.999719i	$0.492447\pi$
$311$	−4683.36	+	8111.81i	−0.853919	+	1.47903i	−0.877644	+	0.479312i	$0.840886\pi$
$312$		0			0
$313$	1402.09	+	2428.49i	0.253197	+	0.438550i	0.964404	−	0.264432i	$-0.0851846\pi$
$313$	1402.09	+	2428.49i	0.253197	+	0.438550i	−0.711207	+	0.702982i	$0.751851\pi$
$314$	603.130			0.108397
$315$		0			0
$316$	−2710.32			−0.482492
$317$	−4383.08	−	7591.72i	−0.776588	−	1.34509i	−0.933898	−	0.357540i	$-0.883615\pi$
$317$	−4383.08	−	7591.72i	−0.776588	−	1.34509i	0.157310	−	0.987549i	$-0.449718\pi$
$318$		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	−0.997966	−	0.0637421i	$-0.979697\pi$
$331$	−3337.31	−	5780.40i	−0.554185	−	0.959877i	0.443781	−	0.896135i	$-0.353637\pi$
$332$	−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	−0.286160	−	0.958182i	$-0.592379\pi$
$347$	4438.95	−	7688.48i	0.686730	−	1.18945i	0.972890	−	0.231269i	$-0.0742877\pi$
$348$		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	0.00527653	+	0.999986i	$0.498320\pi$
$353$	−5726.13	+	9917.96i	−0.863375	+	1.49541i	−0.868652	+	0.495423i	$0.835013\pi$
$354$		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	0.999159	+	0.0409973i	$0.0130535\pi$
$359$	3639.69	+	6304.12i	0.535084	+	0.926793i	−0.464075	+	0.885796i	$0.653613\pi$
$360$		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	−0.781581	−	0.623804i	$-0.785586\pi$
$367$	1050.66	−	1819.80i	0.149439	−	0.258837i	0.931020	+	0.364967i	$0.118920\pi$
$368$	−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	0.695310	−	0.718710i	$-0.255267\pi$
$373$	−1979.37	−	3428.37i	−0.274767	−	0.475910i	−0.970076	+	0.242801i	$0.921934\pi$
$374$	−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	0.997516	−	0.0704423i	$-0.0224411\pi$
$383$	3281.16	+	5683.14i	0.437753	+	0.758211i	−0.559763	+	0.828653i	$0.689108\pi$
$384$		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	0.440436	+	0.897784i	$0.354824\pi$
$389$	−4275.65	+	7405.64i	−0.557286	+	0.965247i	−0.997722	+	0.0674631i	$0.978510\pi$
$390$		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	0.933990	−	0.357300i	$-0.116302\pi$
$397$	1246.36	+	2158.76i	0.157564	+	0.272910i	−0.776425	+	0.630209i	$0.782969\pi$
$398$	7643.24			0.962616
$399$		0			0
$400$	3157.57			0.394696
$401$	−5375.99	−	9311.49i	−0.669487	−	1.15959i	−0.978048	−	0.208381i	$-0.933181\pi$
$401$	−5375.99	−	9311.49i	−0.669487	−	1.15959i	0.308561	−	0.951205i	$-0.400153\pi$
$402$		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	0.221044	+	0.975264i	$0.429054\pi$
$409$	−6071.96	+	10516.9i	−0.734081	+	1.27147i	−0.955125	+	0.296202i	$0.904280\pi$
$410$	−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	0.196055	+	0.980593i	$0.437187\pi$
$431$	−6721.50	+	11642.0i	−0.751191	+	1.30110i	−0.947246	+	0.320508i	$0.896146\pi$
$432$		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	0.803306	−	0.595566i	$-0.203072\pi$
$439$	−1049.70	−	1818.14i	−0.114122	−	0.197665i	−0.917428	+	0.397901i	$0.869739\pi$
$440$	−5698.80			−0.617454
$441$		0			0
$442$	838.847			0.0902712
$443$	−3171.89	−	5493.87i	−0.340182	−	0.589213i	0.644284	−	0.764786i	$-0.277156\pi$
$443$	−3171.89	−	5493.87i	−0.340182	−	0.589213i	−0.984466	+	0.175573i	$0.943822\pi$
$444$		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	−0.993019	−	0.117957i	$-0.962366\pi$
$457$	−5848.66	−	10130.2i	−0.598663	−	1.03691i	0.394356	−	0.918958i	$-0.370968\pi$
$458$	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	0.967699	+	0.252108i	$0.0811238\pi$
$467$	7086.38	+	12274.0i	0.702181	+	1.21621i	−0.265518	+	0.964106i	$0.585543\pi$
$468$		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	−0.972526	−	0.232795i	$-0.925213\pi$
$479$	−2984.18	+	5168.75i	−0.284657	+	0.493040i	0.687869	+	0.725835i	$0.258546\pi$
$480$		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	−0.939016	−	0.343874i	$-0.888261\pi$
$487$	−1845.34	+	3196.22i	−0.171705	+	0.297401i	0.767311	+	0.641275i	$0.221594\pi$
$488$	−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	−0.988262	−	0.152768i	$-0.951181\pi$
$499$	−6982.72	−	12094.4i	−0.626432	−	1.08501i	0.361830	−	0.932244i	$-0.382152\pi$
$500$	−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	0.798895	+	0.601470i	$0.205418\pi$
$509$	10568.7	+	18305.6i	0.920336	+	1.59407i	0.121441	+	0.992599i	$0.461248\pi$
$510$		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	0.232477	+	0.972602i	$0.425317\pi$
$521$	−8634.33	+	14955.1i	−0.726060	+	1.25757i	−0.958536	+	0.284970i	$0.908016\pi$
$522$		0			0
$523$	1000.71	+	1733.28i	0.0836674	+	0.144916i	0.904823	−	0.425789i	$-0.140003\pi$
$523$	1000.71	+	1733.28i	0.0836674	+	0.144916i	−0.821155	+	0.570705i	$0.806670\pi$
$524$	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	0.857684	−	0.514177i	$-0.171903\pi$
$541$	−206.979	−	358.499i	−0.0164487	−	0.0284899i	−0.874133	+	0.485687i	$0.838569\pi$
$542$	−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	−0.979521	−	0.201344i	$-0.935469\pi$
$557$	−4146.02	+	7181.13i	−0.315391	+	0.546273i	0.664130	+	0.747617i	$0.268802\pi$
$558$		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	−0.404755	−	0.914425i	$-0.632643\pi$
$563$	7875.43	−	13640.6i	0.589538	−	1.02111i	0.994293	−	0.106685i	$-0.0340235\pi$
$564$		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	−0.990485	−	0.137622i	$-0.956054\pi$
$569$	−8339.46	−	14444.4i	−0.614426	−	1.06422i	0.376059	−	0.926596i	$-0.377279\pi$
$570$		0			0
$571$	2338.21	−	4049.89i	0.171368	−	0.296817i	−0.767531	−	0.641012i	$-0.778515\pi$
$571$	2338.21	−	4049.89i	0.171368	−	0.296817i	0.938898	+	0.344195i	$0.111848\pi$
$572$	−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	−0.657273	−	0.753653i	$-0.728290\pi$
$577$	4491.29	−	7779.13i	0.324046	−	0.561265i	0.981319	+	0.192388i	$0.0616233\pi$
$578$	−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	0.961177	−	0.275931i	$-0.0889862\pi$
$593$	3489.19	+	6043.45i	0.241625	+	0.418507i	−0.719552	+	0.694438i	$0.755653\pi$
$594$		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	−0.727645	−	0.685953i	$-0.759385\pi$
$599$	3375.23	−	5846.06i	0.230230	−	0.398771i	0.957876	+	0.287183i	$0.0927187\pi$
$600$		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	0.501932	−	0.864907i	$-0.332623\pi$
$607$	−7448.52	−	12901.2i	−0.498066	−	0.862675i	−0.999998	+	0.00223201i	$0.999290\pi$
$608$	−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	0.363179	+	0.931720i	$0.381691\pi$
$613$	−9490.34	+	16437.7i	−0.625304	+	1.08306i	−0.988482	+	0.151338i	$0.951642\pi$
$614$	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	−0.461736	−	0.887017i	$-0.652773\pi$
$619$	8274.89	−	14332.5i	0.537311	−	0.930651i	0.999048	−	0.0436334i	$-0.0138933\pi$
$620$	−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	0.426914	+	0.904292i	$0.359601\pi$
$641$	−9245.29	+	16013.3i	−0.569683	+	0.986721i	−0.996597	+	0.0824282i	$0.973732\pi$
$642$		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	−0.388494	−	0.921451i	$-0.627005\pi$
$647$	9936.10	−	17209.8i	0.603753	−	1.04573i	0.992247	−	0.124280i	$-0.0396621\pi$
$648$		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	−0.994055	−	0.108877i	$-0.965275\pi$
$653$	−9867.12	−	17090.4i	−0.591318	−	1.02419i	0.402738	−	0.915315i	$-0.368059\pi$
$654$		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	−0.770740	−	0.637150i	$-0.780113\pi$
$661$	2828.16	−	4898.51i	0.166418	−	0.288245i	0.937158	+	0.348905i	$0.113446\pi$
$662$	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	−0.997452	−	0.0713405i	$-0.977272\pi$
$677$	−9873.37	−	17101.2i	−0.560509	−	0.970830i	0.436943	−	0.899489i	$-0.356061\pi$
$678$		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	−0.910640	−	0.413202i	$-0.864410\pi$
$683$	−1739.93	+	3013.65i	−0.0974766	+	0.168834i	0.813163	+	0.582036i	$0.197744\pi$
$684$		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	0.901967	−	0.431805i	$-0.142123\pi$
$691$	1399.18	+	2423.45i	0.0770295	+	0.133419i	−0.824938	+	0.565224i	$0.808790\pi$
$692$	−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	−0.999498	−	0.0316689i	$-0.989918\pi$
$709$	−9952.32	−	17237.9i	−0.527175	−	0.913094i	0.472323	−	0.881425i	$-0.343416\pi$
$710$	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	0.526804	−	0.849987i	$-0.323390\pi$
$719$	−9113.53	−	15785.1i	−0.472708	−	0.818755i	−0.999512	+	0.0312322i	$0.990057\pi$
$720$		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	0.987179	−	0.159616i	$-0.0510257\pi$
$733$	7052.15	+	12214.7i	0.355358	+	0.615498i	−0.631821	+	0.775114i	$0.717692\pi$
$734$	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	0.379701	+	0.925109i	$0.376027\pi$
$739$	−12281.0	+	21271.3i	−0.611317	+	1.05883i	−0.991019	+	0.133724i	$0.957307\pi$
$740$	−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	0.996297	+	0.0859790i	$0.0274018\pi$
$751$	11784.7	+	20411.7i	0.572608	+	0.991787i	−0.423688	+	0.905808i	$0.639265\pi$
$752$	−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	−0.940807	−	0.338942i	$-0.889931\pi$
$761$	−16037.4	−	27777.6i	−0.763936	−	1.32318i	0.176872	−	0.984234i	$-0.443402\pi$
$762$		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	−0.832077	−	0.554660i	$-0.812848\pi$
$773$	1382.15	−	2393.96i	0.0643112	−	0.111390i	0.896388	+	0.443270i	$0.146182\pi$
$774$		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	−0.266633	−	0.963798i	$-0.585911\pi$
$787$	15484.6	−	26820.2i	0.701357	−	1.21479i	0.967990	−	0.250988i	$-0.0807555\pi$
$788$	−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	−0.630413	−	0.776260i	$-0.717115\pi$
$809$	8215.93	−	14230.4i	0.357054	−	0.618436i	0.987467	+	0.157824i	$0.0504479\pi$
$810$		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	−0.853657	−	0.520836i	$-0.825620\pi$
$821$	−20651.6	−	35769.5i	−0.877886	−	1.52054i	−0.0242290	−	0.999706i	$-0.507713\pi$
$822$		0			0
$823$	−11976.2	+	20743.3i	−0.507245	+	0.878574i	0.492720	+	0.870188i	$0.336003\pi$
$823$	−11976.2	+	20743.3i	−0.507245	+	0.878574i	−0.999965	+	0.00838602i	$0.997331\pi$
$824$	−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	−0.806660	−	0.591015i	$-0.798727\pi$
$829$	2589.87	−	4485.78i	0.108504	−	0.187934i	0.915164	+	0.403081i	$0.132061\pi$
$830$	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	−0.736506	−	0.676431i	$-0.763526\pi$
$857$	5458.03	−	9453.59i	0.217553	−	0.376813i	0.954059	+	0.299618i	$0.0968592\pi$
$858$		0			0
$859$	14683.5	+	25432.6i	0.583231	+	1.01019i	0.995093	+	0.0989394i	$0.0315450\pi$
$859$	14683.5	+	25432.6i	0.583231	+	1.01019i	−0.411863	+	0.911246i	$0.635122\pi$
$860$	−25039.5			−0.992838
$861$		0			0
$862$	−26886.0			−1.06234
$863$	−1384.63	−	2398.25i	−0.0546157	−	0.0945971i	0.837425	−	0.546552i	$-0.184060\pi$
$863$	−1384.63	−	2398.25i	−0.0546157	−	0.0945971i	−0.892041	+	0.451955i	$0.850727\pi$
$864$		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	0.902763	−	0.430139i	$-0.141535\pi$
$877$	2048.40	+	3547.93i	0.0788705	+	0.136608i	−0.823892	+	0.566746i	$0.808202\pi$
$878$	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	−0.815228	−	0.579140i	$-0.803388\pi$
$887$	−24017.5	−	41599.5i	−0.909164	−	1.57472i	−0.0939363	−	0.995578i	$-0.529945\pi$
$888$		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	−0.602469	−	0.798142i	$-0.705816\pi$
$907$	10652.5	−	18450.6i	0.389977	−	0.675460i	0.992446	+	0.122683i	$0.0391497\pi$
$908$	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	−0.861753	−	0.507327i	$-0.830634\pi$
$919$	−24244.3	−	41992.4i	−0.870235	−	1.50729i	−0.00848156	−	0.999964i	$-0.502700\pi$
$920$	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	0.722742	−	0.691118i	$-0.242882\pi$
$929$	−6715.15	−	11631.0i	−0.237155	−	0.410764i	−0.959897	+	0.280354i	$0.909548\pi$
$930$		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	0.315820	+	0.948819i	$0.397720\pi$
$941$	−19160.9	+	33187.7i	−0.663791	+	1.14972i	−0.979612	+	0.200901i	$0.935613\pi$
$942$		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	0.752883	+	0.658155i	$0.228663\pi$
$947$	27580.9	+	47771.6i	0.946420	+	1.63925i	0.193537	+	0.981093i	$0.438004\pi$
$948$		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	0.742122	−	0.670265i	$-0.233820\pi$
$971$	−6336.04	−	10974.3i	−0.209406	−	0.362702i	−0.951528	+	0.307564i	$0.900486\pi$
$972$		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	−0.810506	−	0.585731i	$-0.800808\pi$
$977$	3115.03	−	5395.39i	0.102005	−	0.176677i	0.912511	+	0.409053i	$0.134141\pi$
$978$		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	−0.565662	−	0.824637i	$-0.691379\pi$
$983$	13293.4	−	23024.8i	0.431325	−	0.747078i	0.996988	+	0.0775594i	$0.0247127\pi$
$984$		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	−0.960195	−	0.279330i	$-0.909888\pi$
$991$	−7430.78	+	12870.5i	−0.238190	+	0.412558i	0.722005	+	0.691888i	$0.243221\pi$
$992$	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	−0.911435	−	0.411444i	$-0.865024\pi$
$997$	−3129.05	+	5419.68i	−0.0993962	+	0.172159i	0.812039	+	0.583604i	$0.198358\pi$
$998$	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.109.1	yes	8
3.2	odd	2		378.4.g.d.109.4	✓	8
7.2	even	3	inner	378.4.g.e.163.1	yes	8
21.2	odd	6		378.4.g.d.163.4	yes	8

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

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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