LMFDB - Embedded newform 378.4.g.d.109.2

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			109.2
Root			$-1.44566 + 2.50395i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.d.163.2

$q$-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	0.808054	−	0.589108i	$-0.200521\pi$
$5$	−1.18685	−	2.05569i	−0.106155	−	0.183866i	−0.914210	+	0.405242i	$0.867187\pi$
$6$		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	$0.346710\pi$
$11$	−19.5527	+	33.8662i	−0.535942	+	0.928278i	−0.999117	+	0.0420115i	$0.986623\pi$
$12$		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	−0.643976	−	0.765046i	$-0.722716\pi$
$17$	23.8709	−	41.3456i	0.340561	−	0.589869i	0.984537	+	0.175177i	$0.0560497\pi$
$18$		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	$0.0778690\pi$
$23$	76.6464	+	132.755i	0.694864	+	1.20354i	−0.275362	+	0.961341i	$0.588798\pi$
$24$		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	$-0.742557\pi$
$29$	−215.634			−1.38076			−0.690382	+	0.723445i	$0.742557\pi$
$30$		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	$-0.790739\pi$
$41$	−415.622			−1.58315			−0.791577	+	0.611070i	$0.790739\pi$
$42$		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	0.632172	−	0.774828i	$-0.282164\pi$
$47$	−114.366	−	198.087i	−0.354935	−	0.614766i	−0.987107	+	0.160063i	$0.948830\pi$
$48$		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	$0.360733\pi$
$53$	−220.936	+	382.673i	−0.572603	+	0.991777i	−0.996298	+	0.0859723i	$0.972600\pi$
$54$		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	$0.461808\pi$
$59$	−362.528	+	627.916i	−0.799951	+	1.38556i	−0.919648	+	0.392745i	$0.871526\pi$
$60$		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	$-0.555919\pi$
$71$	−209.119			−0.349547			−0.174773	+	0.984609i	$0.555919\pi$
$72$		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	$-0.524608\pi$
$83$	−116.801			−0.154465			−0.0772324	+	0.997013i	$0.524608\pi$
$84$		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	0.736027	−	0.676952i	$-0.236700\pi$
$89$	−183.244	−	317.387i	−0.218245	−	0.378011i	−0.954271	+	0.298942i	$0.903366\pi$
$90$		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	$-0.638991\pi$
$101$	581.936	−	1007.94i	0.573314	−	0.993010i	0.996223	−	0.0868370i	$-0.0276759\pi$
$102$		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	0.899753	−	0.436399i	$-0.143747\pi$
$107$	79.6283	+	137.920i	0.0719436	+	0.124610i	−0.827809	+	0.561009i	$0.810413\pi$
$108$		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	$-0.528097\pi$
$113$	−211.787			−0.176312			−0.0881561	+	0.996107i	$0.528097\pi$
$114$		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	0.918295	−	0.395896i	$-0.129566\pi$
$131$	174.364	+	302.007i	0.116292	+	0.201424i	−0.802004	+	0.597319i	$0.796233\pi$
$132$		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	$-0.642622\pi$
$137$	904.288	−	1566.27i	0.563931	−	0.976757i	0.997148	−	0.0754677i	$-0.0240450\pi$
$138$		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	$0.177694\pi$
$149$	1605.66	+	2781.09i	0.882824	+	1.52910i	0.0346375	+	0.999400i	$0.488972\pi$
$150$		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	$-0.160268\pi$
$167$	3780.59			1.75180			0.875900	+	0.482493i	$0.160268\pi$
$168$		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	$-0.881687\pi$
$173$	−1775.75	−	3075.68i	−0.780390	−	1.35168i	0.151325	−	0.988484i	$-0.451646\pi$
$174$		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	$-0.524757\pi$
$179$	1974.70	−	3420.28i	0.824558	−	1.42818i	0.902257	−	0.431200i	$-0.141910\pi$
$180$		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	0.784617	−	0.619981i	$-0.212860\pi$
$191$	−381.726	−	661.169i	−0.144611	−	0.250474i	−0.929228	+	0.369507i	$0.879526\pi$
$192$		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	$-0.716401\pi$
$197$	−3476.58			−1.25734			−0.628671	+	0.777671i	$0.716401\pi$
$198$		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	$-0.496779\pi$
$227$	2979.05	−	5159.86i	0.871041	−	1.50869i	0.860922	−	0.508738i	$-0.169888\pi$
$228$		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	0.868894	−	0.494999i	$-0.164831\pi$
$233$	20.5054	+	35.5163i	0.00576545	+	0.00998606i	−0.863128	+	0.504985i	$0.831498\pi$
$234$		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	$-0.632865\pi$
$239$	−2995.72			−0.810783			−0.405391	+	0.914143i	$0.632865\pi$
$240$		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	$-0.475495\pi$
$251$	611.679			0.153820			0.0769100	+	0.997038i	$0.475495\pi$
$252$		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	0.964801	−	0.262982i	$-0.0847060\pi$
$257$	1049.17	+	1817.21i	0.254651	+	0.441069i	−0.710149	+	0.704051i	$0.751373\pi$
$258$		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	−0.999862	−	0.0166096i	$-0.994713\pi$
$263$	−2070.93	+	3586.95i	−0.485547	+	0.840992i	0.514315	+	0.857601i	$0.328046\pi$
$264$		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	−0.557713	−	0.830034i	$-0.688321\pi$
$269$	1941.13	−	3362.14i	0.439974	−	0.762058i	0.997687	+	0.0679765i	$0.0216543\pi$
$270$		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	$-0.770519\pi$
$281$	−7076.82			−1.50238			−0.751188	+	0.660088i	$0.770519\pi$
$282$		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	$-0.744306\pi$
$293$	−6964.76			−1.38869			−0.694344	+	0.719643i	$0.744306\pi$
$294$		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	−0.561225	−	0.827663i	$-0.689670\pi$
$311$	2392.17	−	4143.35i	0.436165	−	0.755460i	0.997390	+	0.0722036i	$0.0230032\pi$
$312$		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	0.988102	−	0.153800i	$-0.0491512\pi$
$317$	2036.68	+	3527.63i	0.360856	+	0.625021i	−0.627246	+	0.778821i	$0.715818\pi$
$318$		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	−0.923225	−	0.384259i	$-0.874457\pi$
$347$	−832.775	+	1442.41i	−0.128835	+	0.223149i	0.794391	+	0.607407i	$0.207790\pi$
$348$		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	$0.397527\pi$
$353$	−4399.42	+	7620.02i	−0.663336	+	1.14893i	−0.979734	+	0.200305i	$0.935807\pi$
$354$		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	0.587417	−	0.809285i	$-0.300145\pi$
$359$	−2769.49	−	4796.89i	−0.407153	−	0.705210i	−0.994569	+	0.104075i	$0.966812\pi$
$360$		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	$-0.952347\pi$
$383$	−4673.97	−	8095.56i	−0.623574	−	1.08006i	0.365241	−	0.930913i	$-0.380987\pi$
$384$		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	−0.901095	−	0.433622i	$-0.857235\pi$
$389$	−575.571	+	996.918i	−0.0750195	+	0.129938i	0.826075	+	0.563560i	$0.190569\pi$
$390$		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	0.969333	−	0.245750i	$-0.0790343\pi$
$401$	2182.88	+	3780.87i	0.271841	+	0.470842i	−0.697493	+	0.716592i	$0.745701\pi$
$402$		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	$-0.439330\pi$
$419$	3249.72			0.378900			0.189450	+	0.981890i	$0.439330\pi$
$420$		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	−0.998052	−	0.0623926i	$-0.980127\pi$
$431$	−3981.70	+	6896.50i	−0.444992	+	0.770749i	0.553059	+	0.833142i	$0.313460\pi$
$432$		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	$0.101184\pi$
$443$	6952.30	+	12041.7i	0.745630	+	1.29147i	−0.204270	+	0.978915i	$0.565482\pi$
$444$		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	$-0.805427\pi$
$449$	−15582.7			−1.63784			−0.818921	+	0.573906i	$0.805427\pi$
$450$		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	$-0.474921\pi$
$461$	1558.11			0.157415			0.0787074	+	0.996898i	$0.474921\pi$
$462$		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	$-0.981063\pi$
$467$	−5556.71	−	9624.51i	−0.550608	−	0.953681i	0.447623	−	0.894223i	$-0.352271\pi$
$468$		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	$-0.472656\pi$
$479$	9495.17	−	16446.1i	0.905731	−	1.56877i	0.819933	−	0.572460i	$-0.194011\pi$
$480$		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	$-0.638612\pi$
$491$	−9178.86			−0.843658			−0.421829	+	0.906675i	$0.638612\pi$
$492$		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	$-0.351282\pi$
$503$	10162.0			0.900794			0.450397	+	0.892828i	$0.351282\pi$
$504$		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	0.715991	−	0.698110i	$-0.245975\pi$
$509$	−2831.68	−	4904.62i	−0.246586	−	0.427099i	−0.962576	+	0.271011i	$0.912642\pi$
$510$		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	−0.899175	−	0.437590i	$-0.855832\pi$
$521$	−839.858	+	1454.68i	−0.0706235	+	0.122324i	0.828551	+	0.559913i	$0.189166\pi$
$522$		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	$-0.532199\pi$
$557$	10662.5	−	18468.1i	0.811107	−	1.40488i	0.912090	−	0.409990i	$-0.134468\pi$
$558$		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	$-0.448751\pi$
$563$	12490.1	−	21633.4i	0.934979	−	1.61943i	0.774671	−	0.632364i	$-0.217915\pi$
$564$		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	$0.0495687\pi$
$569$	8527.32	+	14769.7i	0.628267	+	1.08819i	−0.359633	+	0.933094i	$0.617098\pi$
$570$		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	$-0.586258\pi$
$587$	−7613.92			−0.535367			−0.267683	+	0.963507i	$0.586258\pi$
$588$		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	0.520464	−	0.853884i	$-0.325759\pi$
$593$	−6920.65	−	11986.9i	−0.479253	−	0.830090i	−0.999717	+	0.0237934i	$0.992426\pi$
$594$		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	−0.743566	−	0.668662i	$-0.766867\pi$
$599$	3038.99	−	5263.69i	0.207295	−	0.359046i	0.950862	+	0.309616i	$0.100201\pi$
$600$		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	$-0.570719\pi$
$617$	−6754.03			−0.440692			−0.220346	+	0.975422i	$0.570719\pi$
$618$		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	−0.973395	−	0.229132i	$-0.926411\pi$
$641$	−4678.17	+	8102.83i	−0.288263	+	0.499287i	0.685132	+	0.728419i	$0.259744\pi$
$642$		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	$0.478793\pi$
$647$	−13672.9	+	23682.2i	−0.830816	+	1.43902i	−0.897392	+	0.441234i	$0.854541\pi$
$648$		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	0.747942	−	0.663764i	$-0.231042\pi$
$653$	−3351.78	−	5805.46i	−0.200866	−	0.347910i	−0.948808	+	0.315854i	$0.897709\pi$
$654$		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	$-0.628811\pi$
$659$	−13321.2			−0.787433			−0.393717	+	0.919232i	$0.628811\pi$
$660$		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	0.707273	−	0.706941i	$-0.249925\pi$
$677$	−4555.10	−	7889.67i	−0.258592	−	0.447895i	−0.965865	+	0.259046i	$0.916592\pi$
$678$		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	$-0.642598\pi$
$683$	10067.1	−	17436.7i	0.563993	−	0.976864i	0.997143	−	0.0755422i	$-0.0240688\pi$
$684$		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	$-0.705826\pi$
$701$	−22364.5			−1.20498			−0.602492	+	0.798125i	$0.705826\pi$
$702$		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	0.840608	−	0.541645i	$-0.182198\pi$
$719$	−940.338	−	1628.71i	−0.0487742	−	0.0844795i	−0.889382	+	0.457165i	$0.848865\pi$
$720$		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	$-0.466128\pi$
$743$	4302.19			0.212426			0.106213	+	0.994343i	$0.466128\pi$
$744$		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	$0.0316224\pi$
$761$	12248.0	+	21214.1i	0.583428	+	1.01053i	−0.411641	+	0.911346i	$0.635044\pi$
$762$		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	$0.477544\pi$
$773$	−17808.5	+	30845.3i	−0.828627	+	1.43522i	−0.899115	+	0.437711i	$0.855789\pi$
$774$		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	$-0.608883\pi$
$797$	−15094.7			−0.670866			−0.335433	+	0.942064i	$0.608883\pi$
$798$		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	$0.414534\pi$
$809$	−16161.4	+	27992.3i	−0.702354	+	1.21651i	−0.967638	+	0.252342i	$0.918799\pi$
$810$		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	0.556144	−	0.831086i	$-0.312280\pi$
$821$	−10389.9	−	17995.9i	−0.441669	−	0.764994i	−0.997814	+	0.0660921i	$0.978947\pi$
$822$		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	$-0.700039\pi$
$827$	−27962.7			−1.17577			−0.587884	+	0.808945i	$0.700039\pi$
$828$		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	$-0.745633\pi$
$839$	−33893.5			−1.39468			−0.697339	+	0.716742i	$0.745633\pi$
$840$		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	$-0.462775\pi$
$857$	23042.4	−	39910.5i	0.918450	−	1.59080i	0.801770	−	0.597633i	$-0.203892\pi$
$858$		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	$-0.829151\pi$
$863$	−22120.3	−	38313.6i	−0.872520	−	1.51125i	−0.0131396	−	0.999914i	$-0.504183\pi$
$864$		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	$-0.240566\pi$
$881$	38060.7			1.45550			0.727751	+	0.685842i	$0.240566\pi$
$882$		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	0.885326	−	0.464971i	$-0.153935\pi$
$887$	1056.33	+	1829.62i	0.0399867	+	0.0692590i	−0.845339	+	0.534230i	$0.820602\pi$
$888$		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	$-0.269971\pi$
$911$	36371.4			1.32276			0.661381	+	0.750050i	$0.269971\pi$
$912$		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	0.791145	−	0.611628i	$-0.209485\pi$
$929$	−3797.48	−	6577.42i	−0.134113	−	0.232291i	−0.925258	+	0.379338i	$0.876152\pi$
$930$		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	−0.993532	−	0.113549i	$-0.963778\pi$
$941$	−11501.0	+	19920.4i	−0.398430	+	0.690102i	0.595102	+	0.803650i	$0.297112\pi$
$942$		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	$-0.962422\pi$
$947$	−17442.3	−	30211.0i	−0.598520	−	1.03667i	0.394519	−	0.918888i	$-0.370911\pi$
$948$		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	$-0.304773\pi$
$953$	33867.3			1.15118			0.575588	+	0.817740i	$0.304773\pi$
$954$		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	0.892488	−	0.451072i	$-0.148958\pi$
$971$	1682.43	+	2914.05i	0.0556042	+	0.0963093i	−0.836883	+	0.547381i	$0.815625\pi$
$972$		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	−0.914909	−	0.403660i	$-0.867738\pi$
$977$	−3294.28	+	5705.85i	−0.107874	+	0.186844i	0.807035	+	0.590504i	$0.201071\pi$
$978$		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	−0.946960	−	0.321351i	$-0.895863\pi$
$983$	−6015.47	+	10419.1i	−0.195182	+	0.338065i	0.751778	+	0.659416i	$0.229196\pi$
$984$		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.d.109.2	✓	8
3.2	odd	2		378.4.g.e.109.3	yes	8
7.2	even	3	inner	378.4.g.d.163.2	yes	8
21.2	odd	6		378.4.g.e.163.3	yes	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more