LMFDB - Embedded newform 378.4.g.e.109.3

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			109.3
Root			$-1.44566 + 2.50395i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.e.163.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(1.18685 + 2.05569i) q^{5} +(9.15909 - 16.0969i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(1.18685 + 2.05569i) q^{5} +(9.15909 - 16.0969i) q^{7} -8.00000 q^{8} +(-2.37371 + 4.11138i) q^{10} +(19.5527 - 33.8662i) q^{11} -24.4393 q^{13} +(37.0398 - 0.232922i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-23.8709 + 41.3456i) q^{17} +(-59.8352 - 103.638i) q^{19} -9.49483 q^{20} +78.2108 q^{22} +(-76.6464 - 132.755i) q^{23} +(59.6828 - 103.374i) q^{25} +(-24.4393 - 42.3301i) q^{26} +(37.4432 + 63.9219i) q^{28} +215.634 q^{29} +(29.6485 - 51.3527i) q^{31} +(16.0000 - 27.7128i) q^{32} -95.4835 q^{34} +(43.9608 - 0.276444i) q^{35} +(104.999 + 181.863i) q^{37} +(119.670 - 207.275i) q^{38} +(-9.49483 - 16.4455i) q^{40} +415.622 q^{41} -452.341 q^{43} +(78.2108 + 135.465i) q^{44} +(153.293 - 265.511i) q^{46} +(114.366 + 198.087i) q^{47} +(-175.222 - 294.866i) q^{49} +238.731 q^{50} +(48.8786 - 84.6602i) q^{52} +(220.936 - 382.673i) q^{53} +92.8247 q^{55} +(-73.2727 + 128.775i) q^{56} +(215.634 + 373.488i) q^{58} +(362.528 - 627.916i) q^{59} +(170.880 + 295.973i) q^{61} +118.594 q^{62} +64.0000 q^{64} +(-29.0059 - 50.2396i) q^{65} +(-125.678 + 217.681i) q^{67} +(-95.4835 - 165.382i) q^{68} +(44.4396 + 75.8659i) q^{70} +209.119 q^{71} +(-60.9267 + 105.528i) q^{73} +(-209.997 + 363.726i) q^{74} +478.682 q^{76} +(-366.058 - 624.922i) q^{77} +(-399.598 - 692.123i) q^{79} +(18.9897 - 32.8910i) q^{80} +(415.622 + 719.879i) q^{82} +116.801 q^{83} -113.325 q^{85} +(-452.341 - 783.477i) q^{86} +(-156.422 + 270.930i) q^{88} +(183.244 + 317.387i) q^{89} +(-223.842 + 393.398i) q^{91} +613.171 q^{92} +(-228.731 + 396.174i) q^{94} +(142.031 - 246.005i) q^{95} -1045.65 q^{97} +(335.501 - 598.360i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.18685	+	2.05569i	0.106155	+	0.183866i	0.914210	−	0.405242i	$-0.132813\pi$
$5$	1.18685	+	2.05569i	0.106155	+	0.183866i	−0.808054	+	0.589108i	$0.799479\pi$
$6$		0			0
$7$	9.15909	−	16.0969i	0.494544	−	0.869152i
$7$	9.15909	−	16.0969i	0.494544	−	0.869152i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−2.37371	+	4.11138i	−0.0750632	+	0.130013i
$11$	19.5527	−	33.8662i	0.535942	−	0.928278i	−0.463176	−	0.886267i	$-0.653290\pi$
$11$	19.5527	−	33.8662i	0.535942	−	0.928278i	0.999117	−	0.0420115i	$-0.0133766\pi$
$12$		0			0
$13$	−24.4393			−0.521403			−0.260702	−	0.965419i	$-0.583954\pi$
$13$	−24.4393			−0.521403			−0.260702	+	0.965419i	$0.583954\pi$
$14$	37.0398	−	0.232922i	0.707093	−	0.00444650i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−23.8709	+	41.3456i	−0.340561	+	0.589869i	−0.984537	−	0.175177i	$-0.943950\pi$
$17$	−23.8709	+	41.3456i	−0.340561	+	0.589869i	0.643976	+	0.765046i	$0.277284\pi$
$18$		0			0
$19$	−59.8352	−	103.638i	−0.722481	−	1.25137i	−0.960002	−	0.279992i	$-0.909668\pi$
$19$	−59.8352	−	103.638i	−0.722481	−	1.25137i	0.237521	−	0.971382i	$-0.423665\pi$
$20$	−9.49483			−0.106155
$21$		0			0
$22$	78.2108			0.757936
$23$	−76.6464	−	132.755i	−0.694864	−	1.20354i	−0.970226	−	0.242200i	$-0.922131\pi$
$23$	−76.6464	−	132.755i	−0.694864	−	1.20354i	0.275362	−	0.961341i	$-0.411202\pi$
$24$		0			0
$25$	59.6828	−	103.374i	0.477462	−	0.826989i
$26$	−24.4393	−	42.3301i	−0.184344	−	0.319293i
$27$		0			0
$28$	37.4432	+	63.9219i	0.252718	+	0.431432i
$29$	215.634			1.38076			0.690382	−	0.723445i	$-0.257443\pi$
$29$	215.634			1.38076			0.690382	+	0.723445i	$0.257443\pi$
$30$		0			0
$31$	29.6485	−	51.3527i	0.171775	−	0.297523i	−0.767265	−	0.641330i	$-0.778383\pi$
$31$	29.6485	−	51.3527i	0.171775	−	0.297523i	0.939041	+	0.343806i	$0.111716\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−95.4835			−0.481626
$35$	43.9608	−	0.276444i	0.212307	−	0.00133507i
$36$		0			0
$37$	104.999	+	181.863i	0.466532	+	0.808057i	0.999269	−	0.0382238i	$-0.0121700\pi$
$37$	104.999	+	181.863i	0.466532	+	0.808057i	−0.532737	+	0.846281i	$0.678837\pi$
$38$	119.670	−	207.275i	0.510871	−	0.884855i
$39$		0			0
$40$	−9.49483	−	16.4455i	−0.0375316	−	0.0650066i
$41$	415.622			1.58315			0.791577	−	0.611070i	$-0.209261\pi$
$41$	415.622			1.58315			0.791577	+	0.611070i	$0.209261\pi$
$42$		0			0
$43$	−452.341			−1.60422			−0.802109	−	0.597178i	$-0.796289\pi$
$43$	−452.341			−1.60422			−0.802109	+	0.597178i	$0.796289\pi$
$44$	78.2108	+	135.465i	0.267971	+	0.464139i
$45$		0			0
$46$	153.293	−	265.511i	0.491343	−	0.851032i
$47$	114.366	+	198.087i	0.354935	+	0.614766i	0.987107	−	0.160063i	$-0.0511697\pi$
$47$	114.366	+	198.087i	0.354935	+	0.614766i	−0.632172	+	0.774828i	$0.717836\pi$
$48$		0			0
$49$	−175.222	−	294.866i	−0.510852	−	0.859669i
$50$	238.731			0.675233
$51$		0			0
$52$	48.8786	−	84.6602i	0.130351	−	0.225774i
$53$	220.936	−	382.673i	0.572603	−	0.991777i	−0.423695	−	0.905805i	$-0.639267\pi$
$53$	220.936	−	382.673i	0.572603	−	0.991777i	0.996298	−	0.0859723i	$-0.0273996\pi$
$54$		0			0
$55$	92.8247			0.227572
$56$	−73.2727	+	128.775i	−0.174848	+	0.307292i
$57$		0			0
$58$	215.634	+	373.488i	0.488174	+	0.845542i
$59$	362.528	−	627.916i	0.799951	−	1.38556i	−0.119697	−	0.992810i	$-0.538192\pi$
$59$	362.528	−	627.916i	0.799951	−	1.38556i	0.919648	−	0.392745i	$-0.128474\pi$
$60$		0			0
$61$	170.880	+	295.973i	0.358671	+	0.621237i	0.987739	−	0.156113i	$-0.0498965\pi$
$61$	170.880	+	295.973i	0.358671	+	0.621237i	−0.629068	+	0.777351i	$0.716563\pi$
$62$	118.594			0.242927
$63$		0			0
$64$	64.0000			0.125000
$65$	−29.0059	−	50.2396i	−0.0553497	−	0.0958686i
$66$		0			0
$67$	−125.678	+	217.681i	−0.229165	+	0.396925i	−0.957561	−	0.288231i	$-0.906933\pi$
$67$	−125.678	+	217.681i	−0.229165	+	0.396925i	0.728396	+	0.685156i	$0.240266\pi$
$68$	−95.4835	−	165.382i	−0.170280	−	0.294934i
$69$		0			0
$70$	44.4396	+	75.8659i	0.0758793	+	0.129539i
$71$	209.119			0.349547			0.174773	−	0.984609i	$-0.444081\pi$
$71$	209.119			0.349547			0.174773	+	0.984609i	$0.444081\pi$
$72$		0			0
$73$	−60.9267	+	105.528i	−0.0976841	+	0.169194i	−0.910726	−	0.413012i	$-0.864477\pi$
$73$	−60.9267	+	105.528i	−0.0976841	+	0.169194i	0.813042	+	0.582206i	$0.197810\pi$
$74$	−209.997	+	363.726i	−0.329888	+	0.571382i
$75$		0			0
$76$	478.682			0.722481
$77$	−366.058	−	624.922i	−0.541768	−	0.924889i
$78$		0			0
$79$	−399.598	−	692.123i	−0.569092	−	0.985696i	−0.996656	−	0.0817107i	$-0.973962\pi$
$79$	−399.598	−	692.123i	−0.569092	−	0.985696i	0.427565	−	0.903985i	$-0.359372\pi$
$80$	18.9897	−	32.8910i	0.0265388	−	0.0459666i
$81$		0			0
$82$	415.622	+	719.879i	0.559729	+	0.969479i
$83$	116.801			0.154465			0.0772324	−	0.997013i	$-0.475392\pi$
$83$	116.801			0.154465			0.0772324	+	0.997013i	$0.475392\pi$
$84$		0			0
$85$	−113.325			−0.144609
$86$	−452.341	−	783.477i	−0.567176	−	0.982378i
$87$		0			0
$88$	−156.422	+	270.930i	−0.189484	+	0.328196i
$89$	183.244	+	317.387i	0.218245	+	0.378011i	0.954271	−	0.298942i	$-0.0966336\pi$
$89$	183.244	+	317.387i	0.218245	+	0.378011i	−0.736027	+	0.676952i	$0.763300\pi$
$90$		0			0
$91$	−223.842	+	393.398i	−0.257857	+	0.453179i
$92$	613.171			0.694864
$93$		0			0
$94$	−228.731	+	396.174i	−0.250977	+	0.434705i
$95$	142.031	−	246.005i	0.153391	−	0.265680i
$96$		0			0
$97$	−1045.65			−1.09454			−0.547268	−	0.836957i	$-0.684332\pi$
$97$	−1045.65			−1.09454			−0.547268	+	0.836957i	$0.684332\pi$
$98$	335.501	−	598.360i	0.345824	−	0.616770i
$99$		0			0
$100$	238.731	+	413.494i	0.238731	+	0.413494i
$101$	−581.936	+	1007.94i	−0.573314	+	0.993010i	0.422908	+	0.906173i	$0.361009\pi$
$101$	−581.936	+	1007.94i	−0.573314	+	0.993010i	−0.996223	+	0.0868370i	$0.972324\pi$
$102$		0			0
$103$	−79.1611	−	137.111i	−0.0757279	−	0.131165i	0.825675	−	0.564147i	$-0.190795\pi$
$103$	−79.1611	−	137.111i	−0.0757279	−	0.131165i	−0.901403	+	0.432982i	$0.857461\pi$
$104$	195.514			0.184344
$105$		0			0
$106$	883.746			0.809783
$107$	−79.6283	−	137.920i	−0.0719436	−	0.124610i	0.827809	−	0.561009i	$-0.189587\pi$
$107$	−79.6283	−	137.920i	−0.0719436	−	0.124610i	−0.899753	+	0.436399i	$0.856253\pi$
$108$		0			0
$109$	18.0923	−	31.3368i	0.0158984	−	0.0275369i	−0.857967	−	0.513705i	$-0.828272\pi$
$109$	18.0923	−	31.3368i	0.0158984	−	0.0275369i	0.873865	+	0.486168i	$0.161606\pi$
$110$	92.8247	+	160.777i	0.0804590	+	0.139359i
$111$		0			0
$112$	−296.318	+	1.86337i	−0.249995	+	0.00157207i
$113$	211.787			0.176312			0.0881561	−	0.996107i	$-0.471903\pi$
$113$	211.787			0.176312			0.0881561	+	0.996107i	$0.471903\pi$
$114$		0			0
$115$	181.936	−	315.123i	0.147527	−	0.255525i
$116$	−431.267	+	746.977i	−0.345191	+	0.597888i
$117$		0			0
$118$	1450.11			1.13130
$119$	446.901	+	762.935i	0.344263	+	0.587716i
$120$		0			0
$121$	−99.1152	−	171.673i	−0.0744667	−	0.128980i
$122$	−341.760	+	591.946i	−0.253619	+	0.439281i
$123$		0			0
$124$	118.594	+	205.411i	0.0858876	+	0.148762i
$125$	580.052			0.415051
$126$		0			0
$127$	−818.436			−0.571846			−0.285923	−	0.958253i	$-0.592300\pi$
$127$	−818.436			−0.571846			−0.285923	+	0.958253i	$0.592300\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	58.0117	−	100.479i	0.0391382	−	0.0677893i
$131$	−174.364	−	302.007i	−0.116292	−	0.201424i	0.802004	−	0.597319i	$-0.203767\pi$
$131$	−174.364	−	302.007i	−0.116292	−	0.201424i	−0.918295	+	0.395896i	$0.870434\pi$
$132$		0			0
$133$	−2216.28	+	13.9369i	−1.44493	+	0.00908635i
$134$	−502.713			−0.324088
$135$		0			0
$136$	190.967	−	330.764i	0.120406	−	0.208550i
$137$	−904.288	+	1566.27i	−0.563931	+	0.976757i	0.433217	+	0.901290i	$0.357378\pi$
$137$	−904.288	+	1566.27i	−0.563931	+	0.976757i	−0.997148	+	0.0754677i	$0.975955\pi$
$138$		0			0
$139$	1673.29			1.02106			0.510528	−	0.859861i	$-0.329450\pi$
$139$	1673.29			1.02106			0.510528	+	0.859861i	$0.329450\pi$
$140$	−86.9639	+	152.838i	−0.0524985	+	0.0922652i
$141$		0			0
$142$	209.119	+	362.204i	0.123583	+	0.214053i
$143$	−477.854	+	827.667i	−0.279442	+	0.484007i
$144$		0			0
$145$	255.925	+	443.276i	0.146575	+	0.253876i
$146$	−243.707			−0.138146
$147$		0			0
$148$	−839.989			−0.466532
$149$	−1605.66	−	2781.09i	−0.882824	−	1.52910i	−0.848187	−	0.529697i	$-0.822306\pi$
$149$	−1605.66	−	2781.09i	−0.882824	−	1.52910i	−0.0346375	−	0.999400i	$-0.511028\pi$
$150$		0			0
$151$	1423.80	−	2466.09i	0.767331	−	1.32906i	−0.171675	−	0.985154i	$-0.554918\pi$
$151$	1423.80	−	2466.09i	0.767331	−	1.32906i	0.939005	−	0.343902i	$-0.111749\pi$
$152$	478.682	+	829.101i	0.255436	+	0.442428i
$153$		0			0
$154$	716.339	−	1258.95i	0.374833	−	0.658762i
$155$	140.754			0.0729394
$156$		0			0
$157$	−206.423	+	357.536i	−0.104932	+	0.181748i	−0.913711	−	0.406366i	$-0.866796\pi$
$157$	−206.423	+	357.536i	−0.104932	+	0.181748i	0.808778	+	0.588114i	$0.200129\pi$
$158$	799.195	−	1384.25i	0.402409	−	0.696992i
$159$		0			0
$160$	75.9586			0.0375316
$161$	−2838.97	+	17.8526i	−1.38970	+	0.00873903i
$162$		0			0
$163$	−1027.57	−	1779.80i	−0.493774	−	0.855241i	0.506200	−	0.862416i	$-0.331050\pi$
$163$	−1027.57	−	1779.80i	−0.493774	−	0.855241i	−0.999974	+	0.00717453i	$0.997716\pi$
$164$	−831.244	+	1439.76i	−0.395788	+	0.685525i
$165$		0			0
$166$	116.801	+	202.305i	0.0546116	+	0.0945900i
$167$	−3780.59			−1.75180			−0.875900	−	0.482493i	$-0.839732\pi$
$167$	−3780.59			−1.75180			−0.875900	+	0.482493i	$0.839732\pi$
$168$		0			0
$169$	−1599.72			−0.728139
$170$	−113.325	−	196.284i	−0.0511272	−	0.0885549i
$171$		0			0
$172$	904.682	−	1566.95i	0.401054	−	0.694646i
$173$	1775.75	+	3075.68i	0.780390	+	1.35168i	0.931715	+	0.363191i	$0.118313\pi$
$173$	1775.75	+	3075.68i	0.780390	+	1.35168i	−0.151325	+	0.988484i	$0.548354\pi$
$174$		0			0
$175$	−1117.36	−	1907.52i	−0.482653	−	0.823970i
$176$	−625.686			−0.267971
$177$		0			0
$178$	−366.487	+	634.774i	−0.154322	+	0.267294i
$179$	−1974.70	+	3420.28i	−0.824558	+	1.42818i	0.0776984	+	0.996977i	$0.475243\pi$
$179$	−1974.70	+	3420.28i	−0.824558	+	1.42818i	−0.902257	+	0.431200i	$0.858090\pi$
$180$		0			0
$181$	−2448.62			−1.00555			−0.502774	−	0.864418i	$-0.667687\pi$
$181$	−2448.62			−1.00555			−0.502774	+	0.864418i	$0.667687\pi$
$182$	−905.226	+	5.69245i	−0.368680	+	0.00231842i
$183$		0			0
$184$	613.171	+	1062.04i	0.245672	+	0.425516i
$185$	−249.236	+	431.689i	−0.0990497	+	0.171559i
$186$		0			0
$187$	933.479	+	1616.83i	0.365041	+	0.632270i
$188$	−914.925			−0.354935
$189$		0			0
$190$	568.125			0.216927
$191$	381.726	+	661.169i	0.144611	+	0.250474i	0.929228	−	0.369507i	$-0.120474\pi$
$191$	381.726	+	661.169i	0.144611	+	0.250474i	−0.784617	+	0.619981i	$0.787140\pi$
$192$		0			0
$193$	1365.76	−	2365.57i	0.509376	−	0.882266i	−0.490565	−	0.871405i	$-0.663209\pi$
$193$	1365.76	−	2365.57i	0.509376	−	0.882266i	0.999941	−	0.0108609i	$-0.00345719\pi$
$194$	−1045.65	−	1811.12i	−0.386977	−	0.670264i
$195$		0			0
$196$	1371.89	−	17.2547i	0.499960	−	0.00628817i
$197$	3476.58			1.25734			0.628671	−	0.777671i	$-0.283599\pi$
$197$	3476.58			1.25734			0.628671	+	0.777671i	$0.283599\pi$
$198$		0			0
$199$	−1627.72	+	2819.30i	−0.579831	+	1.00430i	0.415668	+	0.909517i	$0.363548\pi$
$199$	−1627.72	+	2819.30i	−0.579831	+	1.00430i	−0.995498	+	0.0947796i	$0.969785\pi$
$200$	−477.462	+	826.989i	−0.168808	+	0.292385i
$201$		0			0
$202$	−2327.74			−0.810789
$203$	1975.01	−	3471.04i	0.682849	−	1.20009i
$204$		0			0
$205$	493.282	+	854.390i	0.168060	+	0.291089i
$206$	158.322	−	274.222i	0.0535477	−	0.0927473i
$207$		0			0
$208$	195.514	+	338.641i	0.0651754	+	0.112887i
$209$	−4679.76			−1.54883
$210$		0			0
$211$	1596.29			0.520822			0.260411	−	0.965498i	$-0.416142\pi$
$211$	1596.29			0.520822			0.260411	+	0.965498i	$0.416142\pi$
$212$	883.746	+	1530.69i	0.286301	+	0.495889i
$213$		0			0
$214$	159.257	−	275.841i	0.0508718	−	0.0881125i
$215$	−536.862	−	929.873i	−0.170296	−	0.294962i
$216$		0			0
$217$	−555.068	−	947.594i	−0.173643	−	0.296437i
$218$	72.3693			0.0224838
$219$		0			0
$220$	−185.649	+	321.554i	−0.0568931	+	0.0985417i
$221$	583.387	−	1010.46i	0.177570	−	0.307559i
$222$		0			0
$223$	5869.21			1.76247			0.881236	−	0.472677i	$-0.156712\pi$
$223$	5869.21			1.76247			0.881236	+	0.472677i	$0.156712\pi$
$224$	−299.546	−	511.375i	−0.0893493	−	0.152534i
$225$		0			0
$226$	211.787	+	366.827i	0.0623358	+	0.107969i
$227$	−2979.05	+	5159.86i	−0.871041	+	1.50869i	−0.0101189	+	0.999949i	$0.503221\pi$
$227$	−2979.05	+	5159.86i	−0.871041	+	1.50869i	−0.860922	+	0.508738i	$0.830112\pi$
$228$		0			0
$229$	−319.983	−	554.227i	−0.0923366	−	0.159932i	0.816157	−	0.577830i	$-0.196100\pi$
$229$	−319.983	−	554.227i	−0.0923366	−	0.159932i	−0.908494	+	0.417898i	$0.862767\pi$
$230$	727.744			0.208635
$231$		0			0
$232$	−1725.07			−0.488174
$233$	−20.5054	−	35.5163i	−0.00576545	−	0.00998606i	0.863128	−	0.504985i	$-0.168502\pi$
$233$	−20.5054	−	35.5163i	−0.00576545	−	0.00998606i	−0.868894	+	0.494999i	$0.835169\pi$
$234$		0			0
$235$	−271.470	+	470.201i	−0.0753565	+	0.130521i
$236$	1450.11	+	2511.67i	0.399975	+	0.692778i
$237$		0			0
$238$	−874.541	+	1536.99i	−0.238185	+	0.418606i
$239$	2995.72			0.810783			0.405391	−	0.914143i	$-0.367135\pi$
$239$	2995.72			0.810783			0.405391	+	0.914143i	$0.367135\pi$
$240$		0			0
$241$	332.680	−	576.218i	0.0889203	−	0.154014i	−0.818135	−	0.575027i	$-0.804992\pi$
$241$	332.680	−	576.218i	0.0889203	−	0.154014i	0.907055	+	0.421012i	$0.138325\pi$
$242$	198.230	−	343.345i	0.0526559	−	0.0912027i
$243$		0			0
$244$	−1367.04			−0.358671
$245$	398.191	−	710.166i	0.103835	−	0.185187i
$246$		0			0
$247$	1462.33	+	2532.83i	0.376704	+	0.652470i
$248$	−237.188	+	410.822i	−0.0607317	+	0.105190i
$249$		0			0
$250$	580.052	+	1004.68i	0.146743	+	0.254166i
$251$	−611.679			−0.153820			−0.0769100	−	0.997038i	$-0.524505\pi$
$251$	−611.679			−0.153820			−0.0769100	+	0.997038i	$0.524505\pi$
$252$		0			0
$253$	−5994.57			−1.48963
$254$	−818.436	−	1417.57i	−0.202178	−	0.350183i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−1049.17	−	1817.21i	−0.254651	−	0.441069i	0.710149	−	0.704051i	$-0.248627\pi$
$257$	−1049.17	−	1817.21i	−0.254651	−	0.441069i	−0.964801	+	0.262982i	$0.915294\pi$
$258$		0			0
$259$	3889.13	−	24.4565i	0.933045	−	0.00586738i
$260$	232.047			0.0553497
$261$		0			0
$262$	348.728	−	604.014i	0.0822308	−	0.142428i
$263$	2070.93	−	3586.95i	0.485547	−	0.840992i	−0.514315	−	0.857601i	$-0.671954\pi$
$263$	2070.93	−	3586.95i	0.485547	−	0.840992i	0.999862	+	0.0166096i	$0.00528725\pi$
$264$		0			0
$265$	1048.88			0.243140
$266$	−2240.42	−	3824.78i	−0.516425	−	0.881625i
$267$		0			0
$268$	−502.713	−	870.725i	−0.114582	−	0.198463i
$269$	−1941.13	+	3362.14i	−0.439974	+	0.762058i	−0.997687	−	0.0679765i	$-0.978346\pi$
$269$	−1941.13	+	3362.14i	−0.439974	+	0.762058i	0.557713	+	0.830034i	$0.311679\pi$
$270$		0			0
$271$	698.842	+	1210.43i	0.156648	+	0.271322i	0.933658	−	0.358166i	$-0.116598\pi$
$271$	698.842	+	1210.43i	0.156648	+	0.271322i	−0.777010	+	0.629488i	$0.783264\pi$
$272$	763.868			0.170280
$273$		0			0
$274$	−3617.15			−0.797519
$275$	−2333.92	−	4042.46i	−0.511784	−	0.886435i
$276$		0			0
$277$	−3305.04	+	5724.50i	−0.716898	+	1.24170i	0.245325	+	0.969441i	$0.421105\pi$
$277$	−3305.04	+	5724.50i	−0.716898	+	1.24170i	−0.962223	+	0.272262i	$0.912228\pi$
$278$	1673.29	+	2898.23i	0.360998	+	0.625267i
$279$		0			0
$280$	−351.686	+	2.21155i	−0.0750617	+	0.000472020i
$281$	7076.82			1.50238			0.751188	−	0.660088i	$-0.229481\pi$
$281$	7076.82			1.50238			0.751188	+	0.660088i	$0.229481\pi$
$282$		0			0
$283$	937.326	−	1623.50i	0.196884	−	0.341013i	−0.750632	−	0.660720i	$-0.770251\pi$
$283$	937.326	−	1623.50i	0.196884	−	0.341013i	0.947517	+	0.319707i	$0.103584\pi$
$284$	−418.238	+	724.409i	−0.0873867	+	0.151358i
$285$		0			0
$286$	−1911.42			−0.395190
$287$	3806.72	−	6690.24i	0.782939	−	1.37600i
$288$		0			0
$289$	1316.86	+	2280.87i	0.268037	+	0.464253i
$290$	−511.851	+	886.551i	−0.103645	+	0.179518i
$291$		0			0
$292$	−243.707	−	422.113i	−0.0488420	−	0.0845969i
$293$	6964.76			1.38869			0.694344	−	0.719643i	$-0.255694\pi$
$293$	6964.76			1.38869			0.694344	+	0.719643i	$0.255694\pi$
$294$		0			0
$295$	1721.07			0.339676
$296$	−839.989	−	1454.90i	−0.164944	−	0.285691i
$297$		0			0
$298$	3211.32	−	5562.17i	0.624251	−	1.08123i
$299$	1873.18	+	3244.45i	0.362305	+	0.627530i
$300$		0			0
$301$	−4143.03	+	7281.30i	−0.793356	+	1.39431i
$302$	5695.19			1.08517
$303$		0			0
$304$	−957.364	+	1658.20i	−0.180620	+	0.312844i
$305$	−405.619	+	702.553i	−0.0761498	+	0.131895i
$306$		0			0
$307$	5750.23			1.06900			0.534500	−	0.845169i	$-0.320500\pi$
$307$	5750.23			1.06900			0.534500	+	0.845169i	$0.320500\pi$
$308$	2896.91	−	18.2170i	0.535931	−	0.00337016i
$309$		0			0
$310$	140.754	+	243.793i	0.0257880	+	0.0446661i
$311$	−2392.17	+	4143.35i	−0.436165	+	0.755460i	−0.997390	−	0.0722036i	$-0.976997\pi$
$311$	−2392.17	+	4143.35i	−0.436165	+	0.755460i	0.561225	+	0.827663i	$0.310330\pi$
$312$		0			0
$313$	4449.28	+	7706.39i	0.803477	+	1.39166i	0.917314	+	0.398164i	$0.130353\pi$
$313$	4449.28	+	7706.39i	0.803477	+	1.39166i	−0.113837	+	0.993499i	$0.536314\pi$
$314$	−825.693			−0.148397
$315$		0			0
$316$	3196.78			0.569092
$317$	−2036.68	−	3527.63i	−0.360856	−	0.625021i	0.627246	−	0.778821i	$-0.284182\pi$
$317$	−2036.68	−	3527.63i	−0.360856	−	0.625021i	−0.988102	+	0.153800i	$0.950849\pi$
$318$		0			0
$319$	4216.22	−	7302.70i	0.740009	−	1.28173i
$320$	75.9586	+	131.564i	0.0132694	+	0.0229833i
$321$		0			0
$322$	−2869.89	−	4899.38i	−0.496685	−	0.847925i
$323$	5713.27			0.984195
$324$		0			0
$325$	−1458.60	+	2526.38i	−0.248950	+	0.431194i
$326$	2055.13	−	3559.59i	0.349151	−	0.604747i
$327$		0			0
$328$	−3324.98			−0.559729
$329$	4236.08	−	26.6383i	0.709856	−	0.00446387i
$330$		0			0
$331$	−1613.95	−	2795.44i	−0.268008	−	0.464203i	0.700340	−	0.713810i	$-0.253032\pi$
$331$	−1613.95	−	2795.44i	−0.268008	−	0.464203i	−0.968347	+	0.249607i	$0.919699\pi$
$332$	−233.602	+	404.611i	−0.0386162	+	0.0668853i
$333$		0			0
$334$	−3780.59	−	6548.17i	−0.619355	−	1.07275i
$335$	−596.647			−0.0973083
$336$		0			0
$337$	9639.76			1.55819			0.779097	−	0.626904i	$-0.215678\pi$
$337$	9639.76			1.55819			0.779097	+	0.626904i	$0.215678\pi$
$338$	−1599.72	−	2770.80i	−0.257436	−	0.445892i
$339$		0			0
$340$	226.650	−	392.569i	0.0361524	−	0.0626177i
$341$	−1159.42	−	2008.17i	−0.184123	−	0.318910i
$342$		0			0
$343$	−6351.32	+	119.832i	−0.999822	+	0.0188639i
$344$	3618.73			0.567176
$345$		0			0
$346$	−3551.49	+	6151.37i	−0.551819	+	0.955779i
$347$	832.775	−	1442.41i	0.128835	−	0.223149i	−0.794391	−	0.607407i	$-0.792210\pi$
$347$	832.775	−	1442.41i	0.128835	−	0.223149i	0.923225	+	0.384259i	$0.125543\pi$
$348$		0			0
$349$	4441.18			0.681177			0.340589	−	0.940212i	$-0.389374\pi$
$349$	4441.18			0.681177			0.340589	+	0.940212i	$0.389374\pi$
$350$	2186.56	−	3842.84i	0.333933	−	0.586881i
$351$		0			0
$352$	−625.686	−	1083.72i	−0.0947420	−	0.164098i
$353$	4399.42	−	7620.02i	0.663336	−	1.14893i	−0.316397	−	0.948627i	$-0.602473\pi$
$353$	4399.42	−	7620.02i	0.663336	−	1.14893i	0.979734	−	0.200305i	$-0.0641933\pi$
$354$		0			0
$355$	248.193	+	429.883i	0.0371063	+	0.0642700i
$356$	−1465.95			−0.218245
$357$		0			0
$358$	−7898.79			−1.16610
$359$	2769.49	+	4796.89i	0.407153	+	0.705210i	0.994569	−	0.104075i	$-0.0331883\pi$
$359$	2769.49	+	4796.89i	0.407153	+	0.705210i	−0.587417	+	0.809285i	$0.699855\pi$
$360$		0			0
$361$	−3731.01	+	6462.30i	−0.543958	+	0.942163i
$362$	−2448.62	−	4241.13i	−0.355515	−	0.615770i
$363$		0			0
$364$	−915.086	−	1562.21i	−0.131768	−	0.224950i
$365$	−289.244			−0.0414788
$366$		0			0
$367$	5721.93	−	9910.68i	0.813848	−	1.40963i	−0.0963034	−	0.995352i	$-0.530702\pi$
$367$	5721.93	−	9910.68i	0.813848	−	1.40963i	0.910152	−	0.414275i	$-0.135965\pi$
$368$	−1226.34	+	2124.09i	−0.173716	+	0.300885i
$369$		0			0
$370$	−996.944			−0.140077
$371$	−4136.29	−	7061.34i	−0.578828	−	0.988157i
$372$		0			0
$373$	−6187.63	−	10717.3i	−0.858936	−	1.48772i	−0.872945	−	0.487819i	$-0.837792\pi$
$373$	−6187.63	−	10717.3i	−0.858936	−	1.48772i	0.0140082	−	0.999902i	$-0.495541\pi$
$374$	−1866.96	+	3233.67i	−0.258123	+	0.447083i
$375$		0			0
$376$	−914.925	−	1584.70i	−0.125488	−	0.217352i
$377$	−5269.93			−0.719935
$378$		0			0
$379$	621.352			0.0842129			0.0421064	−	0.999113i	$-0.486593\pi$
$379$	621.352			0.0842129			0.0421064	+	0.999113i	$0.486593\pi$
$380$	568.125	+	984.021i	0.0766953	+	0.132840i
$381$		0			0
$382$	−763.452	+	1322.34i	−0.102256	+	0.177112i
$383$	4673.97	+	8095.56i	0.623574	+	1.08006i	0.988815	+	0.149149i	$0.0476533\pi$
$383$	4673.97	+	8095.56i	0.623574	+	1.08006i	−0.365241	+	0.930913i	$0.619013\pi$
$384$		0			0
$385$	850.189	−	1494.19i	0.112545	−	0.197795i
$386$	5463.04			0.720367
$387$		0			0
$388$	2091.31	−	3622.25i	0.273634	−	0.473948i
$389$	575.571	−	996.918i	0.0750195	−	0.129938i	−0.826075	−	0.563560i	$-0.809431\pi$
$389$	575.571	−	996.918i	0.0750195	−	0.129938i	0.901095	+	0.433622i	$0.142765\pi$
$390$		0			0
$391$	7318.46			0.946575
$392$	1401.78	+	2358.93i	0.180613	+	0.303939i
$393$		0			0
$394$	3476.58	+	6021.62i	0.444538	+	0.769962i
$395$	948.527	−	1642.90i	0.120824	−	0.209274i
$396$		0			0
$397$	2714.27	+	4701.26i	0.343137	+	0.594331i	0.985014	−	0.172477i	$-0.0551771\pi$
$397$	2714.27	+	4701.26i	0.343137	+	0.594331i	−0.641876	+	0.766808i	$0.721844\pi$
$398$	−6510.90			−0.820004
$399$		0			0
$400$	−1909.85			−0.238731
$401$	−2182.88	−	3780.87i	−0.271841	−	0.470842i	0.697493	−	0.716592i	$-0.254299\pi$
$401$	−2182.88	−	3780.87i	−0.271841	−	0.470842i	−0.969333	+	0.245750i	$0.920966\pi$
$402$		0			0
$403$	−724.589	+	1255.02i	−0.0895641	+	0.155130i
$404$	−2327.74	−	4031.77i	−0.286657	−	0.496505i
$405$		0			0
$406$	7987.02	−	50.2258i	0.976328	−	0.00613956i
$407$	8212.03			1.00014
$408$		0			0
$409$	−1654.33	+	2865.38i	−0.200003	+	0.346415i	−0.948529	−	0.316690i	$-0.897428\pi$
$409$	−1654.33	+	2865.38i	−0.200003	+	0.346415i	0.748526	+	0.663105i	$0.230762\pi$
$410$	−986.565	+	1708.78i	−0.118837	+	0.205831i
$411$		0			0
$412$	633.288			0.0757279
$413$	−6787.10	−	11586.7i	−0.808648	−	1.38050i
$414$		0			0
$415$	138.626	+	240.107i	0.0163973	+	0.0284009i
$416$	−391.029	+	677.282i	−0.0460860	+	0.0798232i
$417$		0			0
$418$	−4679.76	−	8105.58i	−0.547594	−	0.948461i
$419$	−3249.72			−0.378900			−0.189450	−	0.981890i	$-0.560670\pi$
$419$	−3249.72			−0.378900			−0.189450	+	0.981890i	$0.560670\pi$
$420$		0			0
$421$	−2932.21			−0.339447			−0.169724	−	0.985492i	$-0.554287\pi$
$421$	−2932.21			−0.339447			−0.169724	+	0.985492i	$0.554287\pi$
$422$	1596.29	+	2764.86i	0.184138	+	0.318937i
$423$		0			0
$424$	−1767.49	+	3061.39i	−0.202446	+	0.350646i
$425$	2849.36	+	4935.23i	0.325210	+	0.563280i
$426$		0			0
$427$	6329.36	−	39.8017i	0.717329	−	0.00451087i
$428$	637.027			0.0719436
$429$		0			0
$430$	1073.72	−	1859.75i	0.120418	−	0.208569i
$431$	3981.70	−	6896.50i	0.444992	−	0.770749i	−0.553059	−	0.833142i	$-0.686540\pi$
$431$	3981.70	−	6896.50i	0.444992	−	0.770749i	0.998052	+	0.0623926i	$0.0198731\pi$
$432$		0			0
$433$	−7441.66			−0.825920			−0.412960	−	0.910749i	$-0.635505\pi$
$433$	−7441.66			−0.825920			−0.412960	+	0.910749i	$0.635505\pi$
$434$	1086.21	−	1909.00i	0.120138	−	0.211140i
$435$		0			0
$436$	72.3693	+	125.347i	0.00794922	+	0.0137685i
$437$	−9172.31	+	15886.9i	−1.00405	+	1.73907i
$438$		0			0
$439$	−2822.84	−	4889.30i	−0.306895	−	0.531557i	0.670787	−	0.741650i	$-0.265957\pi$
$439$	−2822.84	−	4889.30i	−0.306895	−	0.531557i	−0.977681	+	0.210093i	$0.932623\pi$
$440$	−742.597			−0.0804590
$441$		0			0
$442$	2333.55			0.251121
$443$	−6952.30	−	12041.7i	−0.745630	−	1.29147i	−0.949900	−	0.312554i	$-0.898816\pi$
$443$	−6952.30	−	12041.7i	−0.745630	−	1.29147i	0.204270	−	0.978915i	$-0.434518\pi$
$444$		0			0
$445$	−434.966	+	753.384i	−0.0463357	+	0.0802558i
$446$	5869.21	+	10165.8i	0.623128	+	1.07929i
$447$		0			0
$448$	586.182	−	1030.20i	0.0618180	−	0.108644i
$449$	15582.7			1.63784			0.818921	−	0.573906i	$-0.194573\pi$
$449$	15582.7			1.63784			0.818921	+	0.573906i	$0.194573\pi$
$450$		0			0
$451$	8126.53	−	14075.6i	0.848477	−	1.46961i
$452$	−423.575	+	733.653i	−0.0440781	+	0.0763454i
$453$		0			0
$454$	−11916.2			−1.23184
$455$	−1074.37	+	6.75610i	−0.110697	+	0.000696111i
$456$		0			0
$457$	−4679.71	−	8105.49i	−0.479010	−	0.829670i	0.520700	−	0.853740i	$-0.325671\pi$
$457$	−4679.71	−	8105.49i	−0.479010	−	0.829670i	−0.999710	+	0.0240700i	$0.992338\pi$
$458$	639.966	−	1108.45i	0.0652918	−	0.113089i
$459$		0			0
$460$	727.744	+	1260.49i	0.0737636	+	0.127762i
$461$	−1558.11			−0.157415			−0.0787074	−	0.996898i	$-0.525079\pi$
$461$	−1558.11			−0.157415			−0.0787074	+	0.996898i	$0.525079\pi$
$462$		0			0
$463$	12753.6			1.28015			0.640074	−	0.768313i	$-0.278904\pi$
$463$	12753.6			1.28015			0.640074	+	0.768313i	$0.278904\pi$
$464$	−1725.07	−	2987.91i	−0.172595	−	0.298944i
$465$		0			0
$466$	41.0107	−	71.0326i	0.00407679	−	0.00706121i
$467$	5556.71	+	9624.51i	0.550608	+	0.953681i	0.998231	+	0.0594586i	$0.0189374\pi$
$467$	5556.71	+	9624.51i	0.550608	+	0.953681i	−0.447623	+	0.894223i	$0.647729\pi$
$468$		0			0
$469$	2352.90	+	4016.80i	0.231656	+	0.395476i
$470$	−1085.88			−0.106570
$471$		0			0
$472$	−2900.22	+	5023.33i	−0.282825	+	0.489868i
$473$	−8844.48	+	15319.1i	−0.859767	+	1.48916i
$474$		0			0
$475$	−14284.5			−1.37983
$476$	−3536.69	+	22.2402i	−0.340554	+	0.00214155i
$477$		0			0
$478$	2995.72	+	5188.74i	0.286655	+	0.496501i
$479$	−9495.17	+	16446.1i	−0.905731	+	1.56877i	−0.0857987	+	0.996312i	$0.527344\pi$
$479$	−9495.17	+	16446.1i	−0.905731	+	1.56877i	−0.819933	+	0.572460i	$0.805989\pi$
$480$		0			0
$481$	−2566.09	−	4444.61i	−0.243251	−	0.421323i
$482$	1330.72			0.125752
$483$		0			0
$484$	792.922			0.0744667
$485$	−1241.04	−	2149.54i	−0.116191	−	0.201249i
$486$		0			0
$487$	−2474.13	+	4285.31i	−0.230212	+	0.398739i	−0.957870	−	0.287201i	$-0.907275\pi$
$487$	−2474.13	+	4285.31i	−0.230212	+	0.398739i	0.727658	+	0.685940i	$0.240609\pi$
$488$	−1367.04	−	2367.78i	−0.126810	−	0.219641i
$489$		0			0
$490$	1628.23	−	20.4789i	0.150114	−	0.00188804i
$491$	9178.86			0.843658			0.421829	−	0.906675i	$-0.361388\pi$
$491$	9178.86			0.843658			0.421829	+	0.906675i	$0.361388\pi$
$492$		0			0
$493$	−5147.36	+	8915.49i	−0.470234	+	0.814469i
$494$	−2924.66	+	5065.66i	−0.266370	+	0.461366i
$495$		0			0
$496$	−948.752			−0.0858876
$497$	1915.34	−	3366.17i	0.172866	−	0.303810i
$498$		0			0
$499$	8374.31	+	14504.7i	0.751274	+	1.30125i	0.947205	+	0.320627i	$0.103894\pi$
$499$	8374.31	+	14504.7i	0.751274	+	1.30125i	−0.195931	+	0.980618i	$0.562773\pi$
$500$	−1160.10	+	2009.36i	−0.103763	+	0.179723i
$501$		0			0
$502$	−611.679	−	1059.46i	−0.0543836	−	0.0941952i
$503$	−10162.0			−0.900794			−0.450397	−	0.892828i	$-0.648718\pi$
$503$	−10162.0			−0.900794			−0.450397	+	0.892828i	$0.648718\pi$
$504$		0			0
$505$	−2762.69			−0.243442
$506$	−5994.57	−	10382.9i	−0.526663	−	0.912206i
$507$		0			0
$508$	1636.87	−	2835.14i	0.142961	−	0.247617i
$509$	2831.68	+	4904.62i	0.246586	+	0.427099i	0.962576	−	0.271011i	$-0.0873580\pi$
$509$	2831.68	+	4904.62i	0.246586	+	0.427099i	−0.715991	+	0.698110i	$0.754025\pi$
$510$		0			0
$511$	1140.65	+	1947.28i	0.0987461	+	0.168576i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	2098.34	−	3634.43i	0.180066	−	0.311883i
$515$	187.905	−	325.461i	0.0160778	−	0.0278476i
$516$		0			0
$517$	8944.62			0.760898
$518$	3931.49	+	6711.71i	0.333474	+	0.569297i
$519$		0			0
$520$	232.047	+	401.917i	0.0195691	+	0.0338947i
$521$	839.858	−	1454.68i	0.0706235	−	0.122324i	−0.828551	−	0.559913i	$-0.810834\pi$
$521$	839.858	−	1454.68i	0.0706235	−	0.122324i	0.899175	+	0.437590i	$0.144168\pi$
$522$		0			0
$523$	3522.27	+	6100.76i	0.294490	+	0.510072i	0.974866	−	0.222791i	$-0.0715169\pi$
$523$	3522.27	+	6100.76i	0.294490	+	0.510072i	−0.680376	+	0.732863i	$0.738184\pi$
$524$	1394.91			0.116292
$525$		0			0
$526$	8283.70			0.686667
$527$	1415.47	+	2451.67i	0.117000	+	0.202650i
$528$		0			0
$529$	−5665.85	+	9813.53i	−0.465673	+	0.806570i
$530$	1048.88	+	1816.71i	0.0859628	+	0.148892i
$531$		0			0
$532$	4384.29	−	7705.31i	0.357299	−	0.627946i
$533$	−10157.5			−0.825461
$534$		0			0
$535$	189.014	−	327.382i	0.0152744	−	0.0264560i
$536$	1005.43	−	1741.45i	0.0810220	−	0.140334i
$537$		0			0
$538$	−7764.54			−0.622217
$539$	−13412.1	+	168.688i	−1.07180	+	0.0134804i
$540$		0			0
$541$	−884.613	−	1532.20i	−0.0703004	−	0.121764i	0.828733	−	0.559645i	$-0.189062\pi$
$541$	−884.613	−	1532.20i	−0.0703004	−	0.121764i	−0.899033	+	0.437881i	$0.855729\pi$
$542$	−1397.68	+	2420.86i	−0.110767	+	0.191854i
$543$		0			0
$544$	763.868	+	1323.06i	0.0602032	+	0.104275i
$545$	85.8917			0.00675082
$546$		0			0
$547$	5279.46			0.412676			0.206338	−	0.978481i	$-0.433845\pi$
$547$	5279.46			0.412676			0.206338	+	0.978481i	$0.433845\pi$
$548$	−3617.15	−	6265.09i	−0.281966	−	0.488379i
$549$		0			0
$550$	4667.83	−	8084.92i	0.361886	−	0.626804i
$551$	−12902.5	−	22347.8i	−0.997576	−	1.72785i
$552$		0			0
$553$	−14801.0	+	93.0750i	−1.13816	+	0.00715724i
$554$	−13220.2			−1.01385
$555$		0			0
$556$	−3346.59	+	5796.46i	−0.255264	+	0.442131i
$557$	−10662.5	+	18468.1i	−0.811107	+	1.40488i	0.100984	+	0.994888i	$0.467801\pi$
$557$	−10662.5	+	18468.1i	−0.811107	+	1.40488i	−0.912090	+	0.409990i	$0.865532\pi$
$558$		0			0
$559$	11054.9			0.836444
$560$	−355.517	−	606.927i	−0.0268274	−	0.0457988i
$561$		0			0
$562$	7076.82	+	12257.4i	0.531170	+	0.920014i
$563$	−12490.1	+	21633.4i	−0.934979	+	1.61943i	−0.160308	+	0.987067i	$0.551249\pi$
$563$	−12490.1	+	21633.4i	−0.934979	+	1.61943i	−0.774671	+	0.632364i	$0.782085\pi$
$564$		0			0
$565$	251.361	+	435.369i	0.0187165	+	0.0324179i
$566$	3749.30			0.278436
$567$		0			0
$568$	−1672.95			−0.123583
$569$	−8527.32	−	14769.7i	−0.628267	−	1.08819i	−0.987899	−	0.155096i	$-0.950431\pi$
$569$	−8527.32	−	14769.7i	−0.628267	−	1.08819i	0.359633	−	0.933094i	$-0.382902\pi$
$570$		0			0
$571$	−7455.68	+	12913.6i	−0.546429	+	0.946442i	0.452087	+	0.891974i	$0.350680\pi$
$571$	−7455.68	+	12913.6i	−0.546429	+	0.946442i	−0.998516	+	0.0544681i	$0.982654\pi$
$572$	−1911.42	−	3310.67i	−0.139721	−	0.242004i
$573$		0			0
$574$	15394.6	−	96.8075i	1.11944	−	0.00703949i
$575$	−18297.9			−1.32709
$576$		0			0
$577$	−11224.7	+	19441.7i	−0.809860	+	1.40272i	0.103100	+	0.994671i	$0.467124\pi$
$577$	−11224.7	+	19441.7i	−0.809860	+	1.40272i	−0.912960	+	0.408048i	$0.866210\pi$
$578$	−2633.73	+	4561.75i	−0.189530	+	0.328276i
$579$		0			0
$580$	−2047.40			−0.146575
$581$	1069.79	−	1880.14i	0.0763897	−	0.134254i
$582$		0			0
$583$	−8639.80	−	14964.6i	−0.613763	−	1.06307i
$584$	487.414	−	844.226i	0.0345365	−	0.0598190i
$585$		0			0
$586$	6964.76	+	12063.3i	0.490976	+	0.850395i
$587$	7613.92			0.535367			0.267683	−	0.963507i	$-0.413742\pi$
$587$	7613.92			0.535367			0.267683	+	0.963507i	$0.413742\pi$
$588$		0			0
$589$	−7096.10			−0.496417
$590$	1721.07	+	2980.98i	0.120094	+	0.208008i
$591$		0			0
$592$	1679.98	−	2909.81i	0.116633	−	0.202014i
$593$	6920.65	+	11986.9i	0.479253	+	0.830090i	0.999717	−	0.0237934i	$-0.00757438\pi$
$593$	6920.65	+	11986.9i	0.479253	+	0.830090i	−0.520464	+	0.853884i	$0.674241\pi$
$594$		0			0
$595$	−1037.95	+	1824.18i	−0.0715158	+	0.125688i
$596$	12845.3			0.882824
$597$		0			0
$598$	−3746.37	+	6488.90i	−0.256188	+	0.443731i
$599$	−3038.99	+	5263.69i	−0.207295	+	0.359046i	−0.950862	−	0.309616i	$-0.899799\pi$
$599$	−3038.99	+	5263.69i	−0.207295	+	0.359046i	0.743566	+	0.668662i	$0.233133\pi$
$600$		0			0
$601$	−5719.48			−0.388190			−0.194095	−	0.980983i	$-0.562177\pi$
$601$	−5719.48			−0.388190			−0.194095	+	0.980983i	$0.562177\pi$
$602$	−16754.6	+	105.360i	−1.13433	+	0.00713315i
$603$		0			0
$604$	5695.19	+	9864.35i	0.383665	+	0.664528i
$605$	235.270	−	407.500i	0.0158101	−	0.0273839i
$606$		0			0
$607$	−14768.8	−	25580.3i	−0.987558	−	1.71050i	−0.629966	−	0.776623i	$-0.716931\pi$
$607$	−14768.8	−	25580.3i	−0.987558	−	1.71050i	−0.357592	−	0.933878i	$-0.616402\pi$
$608$	−3829.45			−0.255436
$609$		0			0
$610$	−1622.48			−0.107692
$611$	−2795.02	−	4841.11i	−0.185064	−	0.320541i
$612$		0			0
$613$	−10569.8	+	18307.5i	−0.696429	+	1.20625i	0.273267	+	0.961938i	$0.411896\pi$
$613$	−10569.8	+	18307.5i	−0.696429	+	1.20625i	−0.969697	+	0.244312i	$0.921438\pi$
$614$	5750.23	+	9959.69i	0.377949	+	0.654626i
$615$		0			0
$616$	2928.46	+	4999.38i	0.191544	+	0.326998i
$617$	6754.03			0.440692			0.220346	−	0.975422i	$-0.429281\pi$
$617$	6754.03			0.440692			0.220346	+	0.975422i	$0.429281\pi$
$618$		0			0
$619$	13387.6	−	23188.0i	0.869292	−	1.50566i	0.00657132	−	0.999978i	$-0.497908\pi$
$619$	13387.6	−	23188.0i	0.869292	−	1.50566i	0.862721	−	0.505680i	$-0.168758\pi$
$620$	−281.507	+	487.585i	−0.0182349	+	0.0315837i
$621$		0			0
$622$	−9568.66			−0.616830
$623$	6787.30	−	42.6814i	0.436481	−	0.00274478i
$624$		0			0
$625$	−6771.91	−	11729.3i	−0.433402	−	0.750675i
$626$	−8898.57	+	15412.8i	−0.568144	+	0.984055i
$627$		0			0
$628$	−825.693	−	1430.14i	−0.0524662	−	0.0908741i
$629$	−10025.6			−0.635530
$630$		0			0
$631$	17990.3			1.13500			0.567499	−	0.823374i	$-0.307911\pi$
$631$	17990.3			1.13500			0.567499	+	0.823374i	$0.307911\pi$
$632$	3196.78	+	5536.99i	0.201204	+	0.348496i
$633$		0			0
$634$	4073.36	−	7055.27i	0.255164	−	0.441957i
$635$	−971.363	−	1682.45i	−0.0607045	−	0.105143i
$636$		0			0
$637$	4282.31	+	7206.33i	0.266360	+	0.448234i
$638$	16864.9			1.04653
$639$		0			0
$640$	−151.917	+	263.128i	−0.00938290	+	0.0162517i
$641$	4678.17	−	8102.83i	0.288263	−	0.499287i	−0.685132	−	0.728419i	$-0.740256\pi$
$641$	4678.17	−	8102.83i	0.288263	−	0.499287i	0.973395	+	0.229132i	$0.0735888\pi$
$642$		0			0
$643$	14794.8			0.907388			0.453694	−	0.891158i	$-0.350106\pi$
$643$	14794.8			0.907388			0.453694	+	0.891158i	$0.350106\pi$
$644$	5616.09	−	9870.17i	0.343641	−	0.603943i
$645$		0			0
$646$	5713.27	+	9895.68i	0.347966	+	0.602694i
$647$	13672.9	−	23682.2i	0.830816	−	1.43902i	−0.0665758	−	0.997781i	$-0.521207\pi$
$647$	13672.9	−	23682.2i	0.830816	−	1.43902i	0.897392	−	0.441234i	$-0.145459\pi$
$648$		0			0
$649$	−14176.8	−	24554.9i	−0.857454	−	1.48515i
$650$	−5834.42			−0.352069
$651$		0			0
$652$	8220.52			0.493774
$653$	3351.78	+	5805.46i	0.200866	+	0.347910i	0.948808	−	0.315854i	$-0.102291\pi$
$653$	3351.78	+	5805.46i	0.200866	+	0.347910i	−0.747942	+	0.663764i	$0.768958\pi$
$654$		0			0
$655$	413.889	−	716.876i	0.0246900	−	0.0427644i
$656$	−3324.98	−	5759.03i	−0.197894	−	0.342763i
$657$		0			0
$658$	4282.22	+	7310.47i	0.253706	+	0.433118i
$659$	13321.2			0.787433			0.393717	−	0.919232i	$-0.371189\pi$
$659$	13321.2			0.787433			0.393717	+	0.919232i	$0.371189\pi$
$660$		0			0
$661$	9509.03	−	16470.1i	0.559544	−	0.969158i	−0.437991	−	0.898980i	$-0.644310\pi$
$661$	9509.03	−	16470.1i	0.559544	−	0.969158i	0.997534	−	0.0701787i	$-0.0223570\pi$
$662$	3227.89	−	5590.88i	0.189510	−	0.328241i
$663$		0			0
$664$	−934.409			−0.0546116
$665$	−2659.05	−	4539.45i	−0.155058	−	0.264710i
$666$		0			0
$667$	−16527.5	−	28626.5i	−0.959444	−	1.66181i
$668$	7561.17	−	13096.3i	0.437950	−	0.758552i
$669$		0			0
$670$	−596.647	−	1033.42i	−0.0344037	−	0.0595890i
$671$	13364.7			0.768908
$672$		0			0
$673$	−12786.8			−0.732386			−0.366193	−	0.930539i	$-0.619339\pi$
$673$	−12786.8			−0.732386			−0.366193	+	0.930539i	$0.619339\pi$
$674$	9639.76	+	16696.6i	0.550905	+	0.954195i
$675$		0			0
$676$	3199.44	−	5541.60i	0.182035	−	0.315293i
$677$	4555.10	+	7889.67i	0.258592	+	0.447895i	0.965865	−	0.259046i	$-0.0834081\pi$
$677$	4555.10	+	7889.67i	0.258592	+	0.447895i	−0.707273	+	0.706941i	$0.750075\pi$
$678$		0			0
$679$	−9577.23	+	16831.8i	−0.541297	+	0.951319i
$680$	906.599			0.0511272
$681$		0			0
$682$	2318.83	−	4016.34i	0.130195	−	0.225504i
$683$	−10067.1	+	17436.7i	−0.563993	+	0.976864i	0.433150	+	0.901322i	$0.357402\pi$
$683$	−10067.1	+	17436.7i	−0.563993	+	0.976864i	−0.997143	+	0.0755422i	$0.975931\pi$
$684$		0			0
$685$	−4293.03			−0.239457
$686$	−6558.87	−	10881.0i	−0.365042	−	0.605594i
$687$		0			0
$688$	3618.73	+	6267.82i	0.200527	+	0.347323i
$689$	−5399.53	+	9352.26i	−0.298557	+	0.517116i
$690$		0			0
$691$	15688.3	+	27173.0i	0.863693	+	1.49596i	0.868339	+	0.495972i	$0.165188\pi$
$691$	15688.3	+	27173.0i	0.863693	+	1.49596i	−0.00464517	+	0.999989i	$0.501479\pi$
$692$	−14206.0			−0.780390
$693$		0			0
$694$	3331.10			0.182200
$695$	1985.95	+	3439.77i	0.108391	+	0.187738i
$696$		0			0
$697$	−9921.26	+	17184.1i	−0.539160	+	0.933852i
$698$	4441.18	+	7692.35i	0.240833	+	0.417134i
$699$		0			0
$700$	8842.55	−	55.6057i	0.477453	−	0.00300242i
$701$	22364.5			1.20498			0.602492	−	0.798125i	$-0.294174\pi$
$701$	22364.5			1.20498			0.602492	+	0.798125i	$0.294174\pi$
$702$		0			0
$703$	12565.2	−	21763.6i	0.674121	−	1.16761i
$704$	1251.37	−	2167.44i	0.0669927	−	0.116035i
$705$		0			0
$706$	17597.7			0.938099
$707$	10894.8	+	18599.2i	0.579547	+	0.989385i
$708$		0			0
$709$	−5772.08	−	9997.54i	−0.305748	−	0.529571i	0.671680	−	0.740842i	$-0.265573\pi$
$709$	−5772.08	−	9997.54i	−0.305748	−	0.529571i	−0.977428	+	0.211271i	$0.932240\pi$
$710$	−496.387	+	859.767i	−0.0262381	+	0.0454457i
$711$		0			0
$712$	−1465.95	−	2539.10i	−0.0771611	−	0.133647i
$713$	−9089.81			−0.477442
$714$		0			0
$715$	−2268.57			−0.118657
$716$	−7898.79	−	13681.1i	−0.412279	−	0.714088i
$717$		0			0
$718$	−5538.97	+	9593.78i	−0.287901	+	0.498658i
$719$	940.338	+	1628.71i	0.0487742	+	0.0844795i	0.889382	−	0.457165i	$-0.151135\pi$
$719$	940.338	+	1628.71i	0.0487742	+	0.0844795i	−0.840608	+	0.541645i	$0.817802\pi$
$720$		0			0
$721$	−2932.11	+	18.4383i	−0.151453	+	0.000952399i
$722$	−14924.0			−0.769273
$723$		0			0
$724$	4897.23	−	8482.26i	0.251387	−	0.435415i
$725$	12869.6	−	22290.8i	0.659262	−	1.14188i
$726$		0			0
$727$	−2783.27			−0.141988			−0.0709942	−	0.997477i	$-0.522617\pi$
$727$	−2783.27			−0.141988			−0.0709942	+	0.997477i	$0.522617\pi$
$728$	1790.73	−	3147.18i	0.0911662	−	0.160223i
$729$		0			0
$730$	−289.244	−	500.986i	−0.0146650	−	0.0254004i
$731$	10797.8	−	18702.3i	0.546334	−	0.946278i
$732$		0			0
$733$	−14210.9	−	24614.0i	−0.716088	−	1.24030i	−0.962539	−	0.271145i	$-0.912598\pi$
$733$	−14210.9	−	24614.0i	−0.716088	−	1.24030i	0.246451	−	0.969155i	$-0.420736\pi$
$734$	22887.7			1.15096
$735$		0			0
$736$	−4905.37			−0.245672
$737$	4914.70	+	8512.51i	0.245638	+	0.425458i
$738$		0			0
$739$	−3402.66	+	5893.58i	−0.169376	+	0.293368i	−0.938201	−	0.346092i	$-0.887509\pi$
$739$	−3402.66	+	5893.58i	−0.169376	+	0.293368i	0.768825	+	0.639460i	$0.220842\pi$
$740$	−996.944	−	1726.76i	−0.0495249	−	0.0857796i
$741$		0			0
$742$	8094.31	−	14225.6i	0.400473	−	0.703825i
$743$	−4302.19			−0.212426			−0.106213	−	0.994343i	$-0.533872\pi$
$743$	−4302.19			−0.212426			−0.106213	+	0.994343i	$0.533872\pi$
$744$		0			0
$745$	3811.37	−	6601.48i	0.187433	−	0.324644i
$746$	12375.3	−	21434.6i	0.607360	−	1.05198i
$747$		0			0
$748$	−7467.83			−0.365041
$749$	−2949.42	+	18.5472i	−0.143884	+	0.000904805i
$750$		0			0
$751$	−9151.88	−	15851.5i	−0.444683	−	0.770213i	0.553347	−	0.832951i	$-0.313350\pi$
$751$	−9151.88	−	15851.5i	−0.444683	−	0.770213i	−0.998030	+	0.0627375i	$0.980017\pi$
$752$	1829.85	−	3169.39i	0.0887338	−	0.153691i
$753$		0			0
$754$	−5269.93	−	9127.79i	−0.254535	−	0.440868i
$755$	6759.35			0.325825
$756$		0			0
$757$	4475.34			0.214873			0.107437	−	0.994212i	$-0.465736\pi$
$757$	4475.34			0.214873			0.107437	+	0.994212i	$0.465736\pi$
$758$	621.352	+	1076.21i	0.0297738	+	0.0515696i
$759$		0			0
$760$	−1136.25	+	1968.04i	−0.0542317	+	0.0939321i
$761$	−12248.0	−	21214.1i	−0.583428	−	1.01053i	−0.995069	−	0.0991813i	$-0.968378\pi$
$761$	−12248.0	−	21214.1i	−0.583428	−	1.01053i	0.411641	−	0.911346i	$-0.364956\pi$
$762$		0			0
$763$	−338.717	−	578.247i	−0.0160713	−	0.0274364i
$764$	−3053.81			−0.144611
$765$		0			0
$766$	−9347.94	+	16191.1i	−0.440933	+	0.763719i
$767$	−8859.92	+	15345.8i	−0.417097	+	0.722433i
$768$		0			0
$769$	−11671.0			−0.547290			−0.273645	−	0.961831i	$-0.588229\pi$
$769$	−11671.0			−0.547290			−0.273645	+	0.961831i	$0.588229\pi$
$770$	3438.21	−	21.6209i	0.160915	−	0.00101190i
$771$		0			0
$772$	5463.04	+	9462.27i	0.254688	+	0.441133i
$773$	17808.5	−	30845.3i	0.828627	−	1.43522i	−0.0704885	−	0.997513i	$-0.522456\pi$
$773$	17808.5	−	30845.3i	0.828627	−	1.43522i	0.899115	−	0.437711i	$-0.144211\pi$
$774$		0			0
$775$	−3539.01	−	6129.75i	−0.164032	−	0.284112i
$776$	8365.23			0.386977
$777$		0			0
$778$	2302.28			0.106094
$779$	−24868.8	−	43074.1i	−1.14380	−	1.98112i
$780$		0			0
$781$	4088.83	−	7082.07i	0.187337	−	0.324477i
$782$	7318.46	+	12676.0i	0.334665	+	0.579656i
$783$		0			0
$784$	−2684.01	+	4786.88i	−0.122267	+	0.218061i
$785$	−979.977			−0.0445565
$786$		0			0
$787$	−21576.7	+	37372.0i	−0.977291	+	1.69272i	−0.305134	+	0.952310i	$0.598701\pi$
$787$	−21576.7	+	37372.0i	−0.977291	+	1.69272i	−0.672158	+	0.740408i	$0.734632\pi$
$788$	−6953.17	+	12043.2i	−0.314336	+	0.544445i
$789$		0			0
$790$	3794.11			0.170871
$791$	1939.78	−	3409.13i	0.0871942	−	0.153242i
$792$		0			0
$793$	−4176.19	−	7233.37i	−0.187012	−	0.323915i
$794$	−5428.54	+	9402.51i	−0.242635	+	0.420255i
$795$		0			0
$796$	−6510.90	−	11277.2i	−0.289915	−	0.502148i
$797$	15094.7			0.670866			0.335433	−	0.942064i	$-0.391117\pi$
$797$	15094.7			0.670866			0.335433	+	0.942064i	$0.391117\pi$
$798$		0			0
$799$	−10920.0			−0.483508
$800$	−1909.85	−	3307.95i	−0.0844042	−	0.146192i
$801$		0			0
$802$	4365.77	−	7561.73i	0.192220	−	0.332935i
$803$	2382.56	+	4126.72i	0.104706	+	0.181356i
$804$		0			0
$805$	−3406.14	−	5814.85i	−0.149131	−	0.254592i
$806$	−2898.35			−0.126663
$807$		0			0
$808$	4655.48	−	8063.54i	0.202697	−	0.351082i
$809$	16161.4	−	27992.3i	0.702354	−	1.21651i	−0.265284	−	0.964170i	$-0.585466\pi$
$809$	16161.4	−	27992.3i	0.702354	−	1.21651i	0.967638	−	0.252342i	$-0.0812008\pi$
$810$		0			0
$811$	24077.3			1.04250			0.521251	−	0.853403i	$-0.325465\pi$
$811$	24077.3			1.04250			0.521251	+	0.853403i	$0.325465\pi$
$812$	8074.01	+	13783.7i	0.348944	+	0.595706i
$813$		0			0
$814$	8212.03	+	14223.6i	0.353601	+	0.612455i
$815$	2439.14	−	4224.71i	0.104833	−	0.181577i
$816$		0			0
$817$	27065.9	+	46879.6i	1.15902	+	2.00748i
$818$	−6617.30			−0.282847
$819$		0			0
$820$	−3946.26			−0.168060
$821$	10389.9	+	17995.9i	0.441669	+	0.764994i	0.997814	−	0.0660921i	$-0.0210531\pi$
$821$	10389.9	+	17995.9i	0.441669	+	0.764994i	−0.556144	+	0.831086i	$0.687720\pi$
$822$		0			0
$823$	7628.70	−	13213.3i	0.323110	−	0.559644i	−0.658018	−	0.753002i	$-0.728605\pi$
$823$	7628.70	−	13213.3i	0.323110	−	0.559644i	0.981128	+	0.193359i	$0.0619382\pi$
$824$	633.288	+	1096.89i	0.0267738	+	0.0463737i
$825$		0			0
$826$	13281.7	−	23342.3i	0.559479	−	0.983273i
$827$	27962.7			1.17577			0.587884	−	0.808945i	$-0.299961\pi$
$827$	27962.7			1.17577			0.587884	+	0.808945i	$0.299961\pi$
$828$		0			0
$829$	7240.55	−	12541.0i	0.303347	−	0.525413i	−0.673545	−	0.739146i	$-0.735229\pi$
$829$	7240.55	−	12541.0i	0.303347	−	0.525413i	0.976892	+	0.213734i	$0.0685625\pi$
$830$	−277.251	+	480.214i	−0.0115946	+	0.0200825i
$831$		0			0
$832$	−1564.11			−0.0651754
$833$	16374.1	−	205.943i	0.681068	−	0.00856602i
$834$		0			0
$835$	−4487.00	−	7771.71i	−0.185963	−	0.322097i
$836$	9359.52	−	16211.2i	0.387208	−	0.670663i
$837$		0			0
$838$	−3249.72	−	5628.67i	−0.133961	−	0.232028i
$839$	33893.5			1.39468			0.697339	−	0.716742i	$-0.254367\pi$
$839$	33893.5			1.39468			0.697339	+	0.716742i	$0.254367\pi$
$840$		0			0
$841$	22108.8			0.906508
$842$	−2932.21	−	5078.74i	−0.120013	−	0.207868i
$843$		0			0
$844$	−3192.59	+	5529.73i	−0.130206	+	0.225523i
$845$	−1898.63	−	3288.53i	−0.0772958	−	0.133880i
$846$		0			0
$847$	−3671.21	+	23.0861i	−0.148930	+	0.000936538i
$848$	−7069.97			−0.286301
$849$		0			0
$850$	−5698.72	+	9870.47i	−0.229958	+	0.398299i
$851$	16095.5	−	27878.3i	0.648353	−	1.12298i
$852$		0			0
$853$	5250.20			0.210743			0.105371	−	0.994433i	$-0.466397\pi$
$853$	5250.20			0.210743			0.105371	+	0.994433i	$0.466397\pi$
$854$	6398.30	+	10923.0i	0.256376	+	0.437678i
$855$		0			0
$856$	637.027	+	1103.36i	0.0254359	+	0.0440563i
$857$	−23042.4	+	39910.5i	−0.918450	+	1.59080i	−0.116681	+	0.993169i	$0.537225\pi$
$857$	−23042.4	+	39910.5i	−0.918450	+	1.59080i	−0.801770	+	0.597633i	$0.796108\pi$
$858$		0			0
$859$	18043.8	+	31252.8i	0.716703	+	1.24137i	0.962299	+	0.271993i	$0.0876828\pi$
$859$	18043.8	+	31252.8i	0.716703	+	1.24137i	−0.245597	+	0.969372i	$0.578984\pi$
$860$	4294.90			0.170296
$861$		0			0
$862$	15926.8			0.629314
$863$	22120.3	+	38313.6i	0.872520	+	1.51125i	0.859381	+	0.511336i	$0.170849\pi$
$863$	22120.3	+	38313.6i	0.872520	+	1.51125i	0.0131396	+	0.999914i	$0.495817\pi$
$864$		0			0
$865$	−4215.10	+	7300.77i	−0.165685	+	0.286975i
$866$	−7441.66	−	12889.3i	−0.292007	−	0.505771i
$867$		0			0
$868$	4392.70	−	27.6231i	0.171772	−	0.00108017i
$869$	−31252.8			−1.22000
$870$		0			0
$871$	3071.49	−	5319.98i	0.119487	−	0.206958i
$872$	−144.739	+	250.695i	−0.00562095	+	0.00973577i
$873$		0			0
$874$	−36689.2			−1.41995
$875$	5312.75	−	9337.05i	0.205261	−	0.360743i
$876$		0			0
$877$	600.202	+	1039.58i	0.0231099	+	0.0400275i	0.877349	−	0.479853i	$-0.159310\pi$
$877$	600.202	+	1039.58i	0.0231099	+	0.0400275i	−0.854239	+	0.519880i	$0.825977\pi$
$878$	5645.68	−	9778.60i	0.217007	−	0.375868i
$879$		0			0
$880$	−742.597	−	1286.22i	−0.0284465	−	0.0492708i
$881$	−38060.7			−1.45550			−0.727751	−	0.685842i	$-0.759434\pi$
$881$	−38060.7			−1.45550			−0.727751	+	0.685842i	$0.759434\pi$
$882$		0			0
$883$	−47249.0			−1.80074			−0.900372	−	0.435121i	$-0.856706\pi$
$883$	−47249.0			−1.80074			−0.900372	+	0.435121i	$0.856706\pi$
$884$	2333.55	+	4041.82i	0.0887848	+	0.153780i
$885$		0			0
$886$	13904.6	−	24083.5i	0.527240	−	0.913206i
$887$	−1056.33	−	1829.62i	−0.0399867	−	0.0692590i	0.845339	−	0.534230i	$-0.179398\pi$
$887$	−1056.33	−	1829.62i	−0.0399867	−	0.0692590i	−0.885326	+	0.464971i	$0.846065\pi$
$888$		0			0
$889$	−7496.12	+	13174.3i	−0.282803	+	0.497021i
$890$	−1739.87			−0.0655286
$891$		0			0
$892$	−11738.4	+	20331.5i	−0.440618	+	0.763173i
$893$	13686.2	−	23705.2i	0.512868	−	0.888313i
$894$		0			0
$895$	−9374.71			−0.350125
$896$	2370.55	−	14.9070i	0.0883866	−	0.000555812i
$897$		0			0
$898$	15582.7	+	26989.9i	0.579065	+	1.00297i
$899$	6393.21	−	11073.4i	0.237181	−	0.410809i
$900$		0			0
$901$	10547.9	+	18269.5i	0.390012	+	0.675521i
$902$	32506.1			1.19993
$903$		0			0
$904$	−1694.30			−0.0623358
$905$	−2906.15	−	5033.60i	−0.106744	−	0.184887i
$906$		0			0
$907$	434.958	−	753.369i	0.0159234	−	0.0275802i	−0.857954	−	0.513727i	$-0.828265\pi$
$907$	434.958	−	753.369i	0.0159234	−	0.0275802i	0.873877	+	0.486147i	$0.161598\pi$
$908$	−11916.2	−	20639.4i	−0.435520	−	0.754343i
$909$		0			0
$910$	−1086.07	−	1854.11i	−0.0395637	−	0.0675419i
$911$	−36371.4			−1.32276			−0.661381	−	0.750050i	$-0.730029\pi$
$911$	−36371.4			−1.32276			−0.661381	+	0.750050i	$0.730029\pi$
$912$		0			0
$913$	2283.78	−	3955.61i	0.0827841	−	0.143386i
$914$	9359.42	−	16211.0i	0.338711	−	0.586665i
$915$		0			0
$916$	2559.86			0.0923366
$917$	−6458.40	+	40.6132i	−0.232579	+	0.00146256i
$918$		0			0
$919$	327.756	+	567.690i	0.0117646	+	0.0203769i	0.871848	−	0.489777i	$-0.162922\pi$
$919$	327.756	+	567.690i	0.0117646	+	0.0203769i	−0.860083	+	0.510154i	$0.829588\pi$
$920$	−1455.49	+	2520.98i	−0.0521587	+	0.0903416i
$921$		0			0
$922$	−1558.11	−	2698.72i	−0.0556545	−	0.0963965i
$923$	−5110.72			−0.182255
$924$		0			0
$925$	25066.4			0.891005
$926$	12753.6	+	22089.8i	0.452601	+	0.783927i
$927$		0			0
$928$	3450.14	−	5975.81i	0.122043	−	0.211385i
$929$	3797.48	+	6577.42i	0.134113	+	0.232291i	0.925258	−	0.379338i	$-0.123848\pi$
$929$	3797.48	+	6577.42i	0.134113	+	0.232291i	−0.791145	+	0.611628i	$0.790515\pi$
$930$		0			0
$931$	−20074.8	+	35803.0i	−0.706686	+	1.26036i
$932$	164.043			0.00576545
$933$		0			0
$934$	−11113.4	+	19249.0i	−0.389339	+	0.674354i
$935$	−2215.81	+	3837.89i	−0.0775022	+	0.134238i
$936$		0			0
$937$	37468.6			1.30634			0.653172	−	0.757210i	$-0.273438\pi$
$937$	37468.6			1.30634			0.653172	+	0.757210i	$0.273438\pi$
$938$	−4604.40	+	8092.14i	−0.160276	+	0.281682i
$939$		0			0
$940$	−1085.88	−	1880.80i	−0.0376783	−	0.0652607i
$941$	11501.0	−	19920.4i	0.398430	−	0.690102i	−0.595102	−	0.803650i	$-0.702888\pi$
$941$	11501.0	−	19920.4i	0.398430	−	0.690102i	0.993532	+	0.113549i	$0.0362217\pi$
$942$		0			0
$943$	−31855.9	−	55176.1i	−1.10008	−	1.90539i
$944$	−11600.9			−0.399975
$945$		0			0
$946$	−35377.9			−1.21589
$947$	17442.3	+	30211.0i	0.598520	+	1.03667i	0.993040	+	0.117780i	$0.0375777\pi$
$947$	17442.3	+	30211.0i	0.598520	+	1.03667i	−0.394519	+	0.918888i	$0.629089\pi$
$948$		0			0
$949$	1489.01	−	2579.04i	0.0509328	−	0.0882182i
$950$	−14284.5	−	24741.5i	−0.487843	−	0.844970i
$951$		0			0
$952$	−3575.21	−	6103.48i	−0.121715	−	0.207789i
$953$	−33867.3			−1.15118			−0.575588	−	0.817740i	$-0.695227\pi$
$953$	−33867.3			−1.15118			−0.575588	+	0.817740i	$0.695227\pi$
$954$		0			0
$955$	−906.106	+	1569.42i	−0.0307025	+	0.0531783i
$956$	−5991.44	+	10377.5i	−0.202696	+	0.351079i
$957$		0			0
$958$	−37980.7			−1.28090
$959$	16929.7	+	28901.9i	0.570062	+	0.973192i
$960$		0			0
$961$	13137.4	+	22754.7i	0.440987	+	0.763811i
$962$	5132.19	−	8889.21i	0.172005	−	0.297921i
$963$		0			0
$964$	1330.72	+	2304.87i	0.0444601	+	0.0770072i
$965$	6483.83			0.216292
$966$		0			0
$967$	5998.34			0.199476			0.0997382	−	0.995014i	$-0.468199\pi$
$967$	5998.34			0.199476			0.0997382	+	0.995014i	$0.468199\pi$
$968$	792.922	+	1373.38i	0.0263280	+	0.0456014i
$969$		0			0
$970$	2482.07	−	4299.08i	0.0821594	−	0.142304i
$971$	−1682.43	−	2914.05i	−0.0556042	−	0.0963093i	0.836883	−	0.547381i	$-0.184375\pi$
$971$	−1682.43	−	2914.05i	−0.0556042	−	0.0963093i	−0.892488	+	0.451072i	$0.851042\pi$
$972$		0			0
$973$	15325.8	−	26934.9i	0.504958	−	0.887454i
$974$	−9896.50			−0.325569
$975$		0			0
$976$	2734.08	−	4735.57i	0.0896679	−	0.155309i
$977$	3294.28	−	5705.85i	0.107874	−	0.186844i	−0.807035	−	0.590504i	$-0.798929\pi$
$977$	3294.28	−	5705.85i	0.107874	−	0.186844i	0.914909	+	0.403660i	$0.132262\pi$
$978$		0			0
$979$	14331.6			0.467866
$980$	1663.70	+	2799.70i	0.0542297	+	0.0912585i
$981$		0			0
$982$	9178.86	+	15898.2i	0.298278	+	0.516633i
$983$	6015.47	−	10419.1i	0.195182	−	0.338065i	−0.751778	−	0.659416i	$-0.770804\pi$
$983$	6015.47	−	10419.1i	0.195182	−	0.338065i	0.946960	+	0.321351i	$0.104137\pi$
$984$		0			0
$985$	4126.20	+	7146.78i	0.133474	+	0.231183i
$986$	−20589.4			−0.665011
$987$		0			0
$988$	−11698.6			−0.376704
$989$	34670.3	+	60050.7i	1.11471	+	1.93074i
$990$		0			0
$991$	4081.69	−	7069.69i	0.130837	−	0.226616i	−0.793163	−	0.609010i	$-0.791567\pi$
$991$	4081.69	−	7069.69i	0.130837	−	0.226616i	0.923999	+	0.382394i	$0.124900\pi$
$992$	−948.752	−	1643.29i	−0.0303658	−	0.0525952i
$993$		0			0
$994$	7745.71	−	48.7083i	0.247162	−	0.00155426i
$995$	−7727.48			−0.246209
$996$		0			0
$997$	−17358.0	+	30064.9i	−0.551387	+	0.955031i	0.446787	+	0.894640i	$0.352568\pi$
$997$	−17358.0	+	30064.9i	−0.551387	+	0.955031i	−0.998175	+	0.0603908i	$0.980765\pi$
$998$	−16748.6	+	29009.5i	−0.531231	+	0.920119i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.e.109.3	yes	8
3.2	odd	2		378.4.g.d.109.2	✓	8
7.2	even	3	inner	378.4.g.e.163.3	yes	8
21.2	odd	6		378.4.g.d.163.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.d.109.2	✓	8	3.2	odd	2
378.4.g.d.163.2	yes	8	21.2	odd	6
378.4.g.e.109.3	yes	8	1.1	even	1	trivial
378.4.g.e.163.3	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)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(1.18685 + 2.05569i) q^{5} +(9.15909 - 16.0969i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(1.18685 + 2.05569i) q^{5} +(9.15909 - 16.0969i) q^{7} -8.00000 q^{8} +(-2.37371 + 4.11138i) q^{10} +(19.5527 - 33.8662i) q^{11} -24.4393 q^{13} +(37.0398 - 0.232922i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-23.8709 + 41.3456i) q^{17} +(-59.8352 - 103.638i) q^{19} -9.49483 q^{20} +78.2108 q^{22} +(-76.6464 - 132.755i) q^{23} +(59.6828 - 103.374i) q^{25} +(-24.4393 - 42.3301i) q^{26} +(37.4432 + 63.9219i) q^{28} +215.634 q^{29} +(29.6485 - 51.3527i) q^{31} +(16.0000 - 27.7128i) q^{32} -95.4835 q^{34} +(43.9608 - 0.276444i) q^{35} +(104.999 + 181.863i) q^{37} +(119.670 - 207.275i) q^{38} +(-9.49483 - 16.4455i) q^{40} +415.622 q^{41} -452.341 q^{43} +(78.2108 + 135.465i) q^{44} +(153.293 - 265.511i) q^{46} +(114.366 + 198.087i) q^{47} +(-175.222 - 294.866i) q^{49} +238.731 q^{50} +(48.8786 - 84.6602i) q^{52} +(220.936 - 382.673i) q^{53} +92.8247 q^{55} +(-73.2727 + 128.775i) q^{56} +(215.634 + 373.488i) q^{58} +(362.528 - 627.916i) q^{59} +(170.880 + 295.973i) q^{61} +118.594 q^{62} +64.0000 q^{64} +(-29.0059 - 50.2396i) q^{65} +(-125.678 + 217.681i) q^{67} +(-95.4835 - 165.382i) q^{68} +(44.4396 + 75.8659i) q^{70} +209.119 q^{71} +(-60.9267 + 105.528i) q^{73} +(-209.997 + 363.726i) q^{74} +478.682 q^{76} +(-366.058 - 624.922i) q^{77} +(-399.598 - 692.123i) q^{79} +(18.9897 - 32.8910i) q^{80} +(415.622 + 719.879i) q^{82} +116.801 q^{83} -113.325 q^{85} +(-452.341 - 783.477i) q^{86} +(-156.422 + 270.930i) q^{88} +(183.244 + 317.387i) q^{89} +(-223.842 + 393.398i) q^{91} +613.171 q^{92} +(-228.731 + 396.174i) q^{94} +(142.031 - 246.005i) q^{95} -1045.65 q^{97} +(335.501 - 598.360i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

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