LMFDB - Embedded newform 378.4.g.d.163.3

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			163.3
Root			$-3.50329 - 6.06788i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.d.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.21575 - 3.83779i) q^{5} +(-11.5959 + 14.4407i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.21575 - 3.83779i) q^{5} +(-11.5959 + 14.4407i) q^{7} +8.00000 q^{8} +(4.43150 + 7.67558i) q^{10} +(19.8193 + 34.3281i) q^{11} +37.4863 q^{13} +(-13.4162 - 34.5255i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-57.0112 - 98.7463i) q^{17} +(67.5634 - 117.023i) q^{19} -17.7260 q^{20} -79.2774 q^{22} +(-96.1989 + 166.621i) q^{23} +(52.6809 + 91.2460i) q^{25} +(-37.4863 + 64.9282i) q^{26} +(73.2160 + 11.2880i) q^{28} -129.302 q^{29} +(122.291 + 211.813i) q^{31} +(-16.0000 - 27.7128i) q^{32} +228.045 q^{34} +(29.7269 + 76.4998i) q^{35} +(-136.629 + 236.649i) q^{37} +(135.127 + 234.047i) q^{38} +(17.7260 - 30.7023i) q^{40} -365.448 q^{41} +57.2537 q^{43} +(79.2774 - 137.312i) q^{44} +(-192.398 - 333.243i) q^{46} +(-172.495 + 298.771i) q^{47} +(-74.0693 - 334.907i) q^{49} -210.724 q^{50} +(-74.9727 - 129.856i) q^{52} +(300.911 + 521.193i) q^{53} +175.659 q^{55} +(-92.7674 + 115.526i) q^{56} +(129.302 - 223.957i) q^{58} +(-15.7631 - 27.3024i) q^{59} +(-431.259 + 746.963i) q^{61} -489.162 q^{62} +64.0000 q^{64} +(83.0604 - 143.865i) q^{65} +(341.295 + 591.141i) q^{67} +(-228.045 + 394.985i) q^{68} +(-162.228 - 25.0114i) q^{70} -664.860 q^{71} +(-471.122 - 816.007i) q^{73} +(-273.259 - 473.298i) q^{74} -540.507 q^{76} +(-725.546 - 111.860i) q^{77} +(1.66271 - 2.87990i) q^{79} +(35.4520 + 61.4047i) q^{80} +(365.448 - 632.975i) q^{82} +54.0900 q^{83} -505.290 q^{85} +(-57.2537 + 99.1664i) q^{86} +(158.555 + 274.625i) q^{88} +(-142.700 + 247.163i) q^{89} +(-434.689 + 541.330i) q^{91} +769.591 q^{92} +(-344.991 - 597.542i) q^{94} +(-299.407 - 518.589i) q^{95} +240.309 q^{97} +(654.145 + 206.615i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 16 q^{4} + 4 q^{5} + 25 q^{7} + 64 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 16 * q^4 + 4 * q^5 + 25 * q^7 + 64 * q^8	$ 8 q - 8 q^{2} - 16 q^{4} + 4 q^{5} + 25 q^{7} + 64 q^{8} + 8 q^{10} + 56 q^{11} - 18 q^{13} - 22 q^{14} - 64 q^{16} - 118 q^{17} + 37 q^{19} - 32 q^{20} - 224 q^{22} + 200 q^{23} - 104 q^{25} + 18 q^{26} - 56 q^{28} - 524 q^{29} + 276 q^{31} - 128 q^{32} + 472 q^{34} + 290 q^{35} - 185 q^{37} + 74 q^{38} + 32 q^{40} + 60 q^{41} - 1556 q^{43} + 224 q^{44} + 400 q^{46} + 30 q^{47} - 1159 q^{49} + 416 q^{50} + 36 q^{52} + 480 q^{53} + 1456 q^{55} + 200 q^{56} + 524 q^{58} - 296 q^{59} + 474 q^{61} - 1104 q^{62} + 512 q^{64} + 1542 q^{65} + 1319 q^{67} - 472 q^{68} - 32 q^{70} - 1852 q^{71} - 1423 q^{73} - 370 q^{74} - 296 q^{76} + 1228 q^{77} + 765 q^{79} + 64 q^{80} - 60 q^{82} - 1660 q^{83} - 584 q^{85} + 1556 q^{86} + 448 q^{88} - 864 q^{89} - 738 q^{91} - 1600 q^{92} + 60 q^{94} + 1766 q^{95} + 1088 q^{97} + 2704 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 16 * q^4 + 4 * q^5 + 25 * q^7 + 64 * q^8 + 8 * q^10 + 56 * q^11 - 18 * q^13 - 22 * q^14 - 64 * q^16 - 118 * q^17 + 37 * q^19 - 32 * q^20 - 224 * q^22 + 200 * q^23 - 104 * q^25 + 18 * q^26 - 56 * q^28 - 524 * q^29 + 276 * q^31 - 128 * q^32 + 472 * q^34 + 290 * q^35 - 185 * q^37 + 74 * q^38 + 32 * q^40 + 60 * q^41 - 1556 * q^43 + 224 * q^44 + 400 * q^46 + 30 * q^47 - 1159 * q^49 + 416 * q^50 + 36 * q^52 + 480 * q^53 + 1456 * q^55 + 200 * q^56 + 524 * q^58 - 296 * q^59 + 474 * q^61 - 1104 * q^62 + 512 * q^64 + 1542 * q^65 + 1319 * q^67 - 472 * q^68 - 32 * q^70 - 1852 * q^71 - 1423 * q^73 - 370 * q^74 - 296 * q^76 + 1228 * q^77 + 765 * q^79 + 64 * q^80 - 60 * q^82 - 1660 * q^83 - 584 * q^85 + 1556 * q^86 + 448 * q^88 - 864 * q^89 - 738 * q^91 - 1600 * q^92 + 60 * q^94 + 1766 * q^95 + 1088 * q^97 + 2704 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	2.21575	−	3.83779i	0.198183	−	0.343263i	−0.749756	−	0.661714i	$-0.769829\pi$
$5$	2.21575	−	3.83779i	0.198183	−	0.343263i	0.947939	+	0.318451i	$0.103163\pi$
$6$		0			0
$7$	−11.5959	+	14.4407i	−0.626121	+	0.779726i
$7$	−11.5959	+	14.4407i	−0.626121	+	0.779726i
$8$	8.00000			0.353553
$9$		0			0
$10$	4.43150	+	7.67558i	0.140136	+	0.242723i
$11$	19.8193	+	34.3281i	0.543251	+	0.940938i	0.998715	+	0.0506835i	$0.0161400\pi$
$11$	19.8193	+	34.3281i	0.543251	+	0.940938i	−0.455464	+	0.890254i	$0.650527\pi$
$12$		0			0
$13$	37.4863			0.799757			0.399878	−	0.916568i	$-0.369052\pi$
$13$	37.4863			0.799757			0.399878	+	0.916568i	$0.369052\pi$
$14$	−13.4162	−	34.5255i	−0.256116	−	0.659094i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−57.0112	−	98.7463i	−0.813367	−	1.40879i	−0.910494	−	0.413521i	$-0.864299\pi$
$17$	−57.0112	−	98.7463i	−0.813367	−	1.40879i	0.0971273	−	0.995272i	$-0.469035\pi$
$18$		0			0
$19$	67.5634	−	117.023i	0.815796	−	1.41300i	−0.0929597	−	0.995670i	$-0.529633\pi$
$19$	67.5634	−	117.023i	0.815796	−	1.41300i	0.908755	−	0.417329i	$-0.137034\pi$
$20$	−17.7260			−0.198183
$21$		0			0
$22$	−79.2774			−0.768272
$23$	−96.1989	+	166.621i	−0.872124	+	1.51056i	−0.0123284	+	0.999924i	$0.503924\pi$
$23$	−96.1989	+	166.621i	−0.872124	+	1.51056i	−0.859795	+	0.510639i	$0.829409\pi$
$24$		0			0
$25$	52.6809	+	91.2460i	0.421447	+	0.729968i
$26$	−37.4863	+	64.9282i	−0.282757	+	0.489749i
$27$		0			0
$28$	73.2160	+	11.2880i	0.494161	+	0.0761867i
$29$	−129.302			−0.827957			−0.413978	−	0.910287i	$-0.635861\pi$
$29$	−129.302			−0.827957			−0.413978	+	0.910287i	$0.635861\pi$
$30$		0			0
$31$	122.291	+	211.813i	0.708517	+	1.22719i	0.965407	+	0.260747i	$0.0839687\pi$
$31$	122.291	+	211.813i	0.708517	+	1.22719i	−0.256890	+	0.966441i	$0.582698\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	228.045			1.15027
$35$	29.7269	+	76.4998i	0.143564	+	0.369452i
$36$		0			0
$37$	−136.629	+	236.649i	−0.607073	+	1.05148i	0.384647	+	0.923064i	$0.374323\pi$
$37$	−136.629	+	236.649i	−0.607073	+	1.05148i	−0.991720	+	0.128418i	$0.959010\pi$
$38$	135.127	+	234.047i	0.576855	+	0.999141i
$39$		0			0
$40$	17.7260	−	30.7023i	0.0700682	−	0.121362i
$41$	−365.448			−1.39203			−0.696017	−	0.718025i	$-0.745046\pi$
$41$	−365.448			−1.39203			−0.696017	+	0.718025i	$0.745046\pi$
$42$		0			0
$43$	57.2537			0.203049			0.101525	−	0.994833i	$-0.467628\pi$
$43$	57.2537			0.203049			0.101525	+	0.994833i	$0.467628\pi$
$44$	79.2774	−	137.312i	0.271625	−	0.470469i
$45$		0			0
$46$	−192.398	−	333.243i	−0.616685	−	1.06813i
$47$	−172.495	+	298.771i	−0.535341	+	0.927238i	0.463805	+	0.885937i	$0.346484\pi$
$47$	−172.495	+	298.771i	−0.535341	+	0.927238i	−0.999147	+	0.0413012i	$0.986850\pi$
$48$		0			0
$49$	−74.0693	−	334.907i	−0.215946	−	0.976405i
$50$	−210.724			−0.596016
$51$		0			0
$52$	−74.9727	−	129.856i	−0.199939	−	0.346305i
$53$	300.911	+	521.193i	0.779874	+	1.35078i	0.932014	+	0.362422i	$0.118050\pi$
$53$	300.911	+	521.193i	0.779874	+	1.35078i	−0.152140	+	0.988359i	$0.548616\pi$
$54$		0			0
$55$	175.659			0.430652
$56$	−92.7674	+	115.526i	−0.221367	+	0.275675i
$57$		0			0
$58$	129.302	−	223.957i	0.292727	−	0.507018i
$59$	−15.7631	−	27.3024i	−0.0347827	−	0.0602454i	0.848110	−	0.529820i	$-0.177741\pi$
$59$	−15.7631	−	27.3024i	−0.0347827	−	0.0602454i	−0.882893	+	0.469575i	$0.844407\pi$
$60$		0			0
$61$	−431.259	+	746.963i	−0.905199	+	1.56785i	−0.0845483	+	0.996419i	$0.526945\pi$
$61$	−431.259	+	746.963i	−0.905199	+	1.56785i	−0.820650	+	0.571431i	$0.806389\pi$
$62$	−489.162			−1.00199
$63$		0			0
$64$	64.0000			0.125000
$65$	83.0604	−	143.865i	0.158498	−	0.274527i
$66$		0			0
$67$	341.295	+	591.141i	0.622326	+	1.07790i	0.989051	+	0.147571i	$0.0471456\pi$
$67$	341.295	+	591.141i	0.622326	+	1.07790i	−0.366725	+	0.930329i	$0.619521\pi$
$68$	−228.045	+	394.985i	−0.406684	+	0.704397i
$69$		0			0
$70$	−162.228	−	25.0114i	−0.277000	−	0.0427061i
$71$	−664.860			−1.11133			−0.555665	−	0.831406i	$-0.687536\pi$
$71$	−664.860			−1.11133			−0.555665	+	0.831406i	$0.687536\pi$
$72$		0			0
$73$	−471.122	−	816.007i	−0.755351	−	1.30831i	−0.945199	−	0.326494i	$-0.894133\pi$
$73$	−471.122	−	816.007i	−0.755351	−	1.30831i	0.189848	−	0.981813i	$-0.439200\pi$
$74$	−273.259	−	473.298i	−0.429266	−	0.743510i
$75$		0			0
$76$	−540.507			−0.815796
$77$	−725.546	−	111.860i	−1.07381	−	0.165554i
$78$		0			0
$79$	1.66271	−	2.87990i	0.00236797	−	0.00410145i	−0.864839	−	0.502049i	$-0.832580\pi$
$79$	1.66271	−	2.87990i	0.00236797	−	0.00410145i	0.867207	+	0.497948i	$0.165913\pi$
$80$	35.4520	+	61.4047i	0.0495457	+	0.0858156i
$81$		0			0
$82$	365.448	−	632.975i	0.492159	−	0.852444i
$83$	54.0900			0.0715319			0.0357660	−	0.999360i	$-0.488613\pi$
$83$	54.0900			0.0715319			0.0357660	+	0.999360i	$0.488613\pi$
$84$		0			0
$85$	−505.290			−0.644781
$86$	−57.2537	+	99.1664i	−0.0717887	+	0.124342i
$87$		0			0
$88$	158.555	+	274.625i	0.192068	+	0.332672i
$89$	−142.700	+	247.163i	−0.169957	+	0.294374i	−0.938404	−	0.345539i	$-0.887696\pi$
$89$	−142.700	+	247.163i	−0.169957	+	0.294374i	0.768448	+	0.639913i	$0.221029\pi$
$90$		0			0
$91$	−434.689	+	541.330i	−0.500744	+	0.623591i
$92$	769.591			0.872124
$93$		0			0
$94$	−344.991	−	597.542i	−0.378543	−	0.655656i
$95$	−299.407	−	518.589i	−0.323353	−	0.560064i
$96$		0			0
$97$	240.309			0.251543			0.125771	−	0.992059i	$-0.459859\pi$
$97$	240.309			0.251543			0.125771	+	0.992059i	$0.459859\pi$
$98$	654.145	+	206.615i	0.674272	+	0.212972i
$99$		0			0
$100$	210.724	−	364.984i	0.210724	−	0.364984i
$101$	222.821	+	385.937i	0.219520	+	0.380220i	0.954661	−	0.297694i	$-0.0962176\pi$
$101$	222.821	+	385.937i	0.219520	+	0.380220i	−0.735141	+	0.677914i	$0.762884\pi$
$102$		0			0
$103$	423.094	−	732.821i	0.404745	−	0.701039i	−0.589547	−	0.807734i	$-0.700694\pi$
$103$	423.094	−	732.821i	0.404745	−	0.701039i	0.994292	+	0.106695i	$0.0340270\pi$
$104$	299.891			0.282757
$105$		0			0
$106$	−1203.64			−1.10291
$107$	281.156	−	486.976i	0.254022	−	0.439979i	−0.710607	−	0.703589i	$-0.751580\pi$
$107$	281.156	−	486.976i	0.254022	−	0.439979i	0.964629	+	0.263610i	$0.0849131\pi$
$108$		0			0
$109$	557.817	+	966.168i	0.490176	+	0.849010i	0.999936	−	0.0113064i	$-0.00359902\pi$
$109$	557.817	+	966.168i	0.490176	+	0.849010i	−0.509760	+	0.860317i	$0.670266\pi$
$110$	−175.659	+	304.250i	−0.152258	+	0.263719i
$111$		0			0
$112$	−107.329	−	276.204i	−0.0905506	−	0.233025i
$113$	−1518.31			−1.26399			−0.631995	−	0.774972i	$-0.717764\pi$
$113$	−1518.31			−1.26399			−0.631995	+	0.774972i	$0.717764\pi$
$114$		0			0
$115$	426.305	+	738.382i	0.345680	+	0.598735i
$116$	258.604	+	447.915i	0.206989	+	0.358516i
$117$		0			0
$118$	63.0523			0.0491901
$119$	2087.06	+	321.771i	1.60774	+	0.247871i
$120$		0			0
$121$	−120.113	+	208.041i	−0.0902423	+	0.156304i
$122$	−862.519	−	1493.93i	−0.640072	−	1.10864i
$123$		0			0
$124$	489.162	−	847.253i	0.354258	−	0.613594i
$125$	1020.85			0.730460
$126$		0			0
$127$	−803.079			−0.561116			−0.280558	−	0.959837i	$-0.590519\pi$
$127$	−803.079			−0.561116			−0.280558	+	0.959837i	$0.590519\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	166.121	+	287.730i	0.112075	+	0.194120i
$131$	445.107	−	770.948i	0.296864	−	0.514183i	−0.678553	−	0.734552i	$-0.737393\pi$
$131$	445.107	−	770.948i	0.296864	−	0.514183i	0.975417	+	0.220368i	$0.0707259\pi$
$132$		0			0
$133$	906.442	+	2332.66i	0.590966	+	1.52081i
$134$	−1365.18			−0.880102
$135$		0			0
$136$	−456.089	−	789.970i	−0.287569	−	0.498084i
$137$	1022.28	+	1770.64i	0.637514	+	1.10421i	0.985976	+	0.166884i	$0.0533706\pi$
$137$	1022.28	+	1770.64i	0.637514	+	1.10421i	−0.348462	+	0.937323i	$0.613296\pi$
$138$		0			0
$139$	587.471			0.358479			0.179240	−	0.983805i	$-0.442636\pi$
$139$	587.471			0.358479			0.179240	+	0.983805i	$0.442636\pi$
$140$	205.549	−	255.976i	0.124086	−	0.154528i
$141$		0			0
$142$	664.860	−	1151.57i	0.392914	−	0.680547i
$143$	742.955	+	1286.84i	0.434468	+	0.752521i
$144$		0			0
$145$	−286.501	+	496.233i	−0.164087	+	0.284207i
$146$	1884.49			1.06823
$147$		0			0
$148$	1093.03			0.607073
$149$	1356.66	−	2349.80i	0.745918	−	1.29197i	−0.203847	−	0.979003i	$-0.565345\pi$
$149$	1356.66	−	2349.80i	0.745918	−	1.29197i	0.949765	−	0.312964i	$-0.101322\pi$
$150$		0			0
$151$	83.2159	+	144.134i	0.0448478	+	0.0776786i	0.887578	−	0.460658i	$-0.152386\pi$
$151$	83.2159	+	144.134i	0.0448478	+	0.0776786i	−0.842730	+	0.538336i	$0.819053\pi$
$152$	540.507	−	936.186i	0.288427	−	0.499571i
$153$		0			0
$154$	919.294	−	1144.82i	0.481031	−	0.599042i
$155$	1083.86			0.561663
$156$		0			0
$157$	267.786	+	463.819i	0.136125	+	0.235776i	0.926027	−	0.377458i	$-0.123202\pi$
$157$	267.786	+	463.819i	0.136125	+	0.235776i	−0.789901	+	0.613234i	$0.789868\pi$
$158$	3.32542	+	5.75980i	0.00167441	+	0.00290016i
$159$		0			0
$160$	−141.808			−0.0700682
$161$	−1290.62	−	3321.31i	−0.631770	−	1.62581i
$162$		0			0
$163$	−601.378	+	1041.62i	−0.288979	+	0.500526i	−0.973566	−	0.228404i	$-0.926649\pi$
$163$	−601.378	+	1041.62i	−0.288979	+	0.500526i	0.684587	+	0.728931i	$0.259982\pi$
$164$	730.896	+	1265.95i	0.348009	+	0.602769i
$165$		0			0
$166$	−54.0900	+	93.6866i	−0.0252903	+	0.0438042i
$167$	−3467.14			−1.60656			−0.803279	−	0.595604i	$-0.796913\pi$
$167$	−3467.14			−1.60656			−0.803279	+	0.595604i	$0.796913\pi$
$168$		0			0
$169$	−791.774			−0.360389
$170$	505.290	−	875.188i	0.227965	−	0.394846i
$171$		0			0
$172$	−114.507	−	198.333i	−0.0507623	−	0.0879229i
$173$	208.096	−	360.433i	0.0914524	−	0.158400i	−0.816670	−	0.577105i	$-0.804182\pi$
$173$	208.096	−	360.433i	0.0914524	−	0.158400i	0.908122	+	0.418705i	$0.137516\pi$
$174$		0			0
$175$	−1928.54	−	297.331i	−0.833052	−	0.128435i
$176$	−634.219			−0.271625
$177$		0			0
$178$	−285.400	−	494.327i	−0.120178	−	0.208154i
$179$	187.817	+	325.309i	0.0784251	+	0.135836i	0.902571	−	0.430542i	$-0.141678\pi$
$179$	187.817	+	325.309i	0.0784251	+	0.135836i	−0.824145	+	0.566378i	$0.808344\pi$
$180$		0			0
$181$	2201.29			0.903981			0.451990	−	0.892023i	$-0.350714\pi$
$181$	2201.29			0.903981			0.451990	+	0.892023i	$0.350714\pi$
$182$	−502.923	−	1294.23i	−0.204830	−	0.527115i
$183$		0			0
$184$	−769.591	+	1332.97i	−0.308342	+	0.534065i
$185$	605.473	+	1048.71i	0.240623	+	0.416771i
$186$		0			0
$187$	2259.85	−	3914.17i	0.883724	−	1.53066i
$188$	1379.96			0.535341
$189$		0			0
$190$	1197.63			0.457290
$191$	2176.64	−	3770.05i	0.824587	−	1.42823i	−0.0776467	−	0.996981i	$-0.524741\pi$
$191$	2176.64	−	3770.05i	0.824587	−	1.42823i	0.902234	−	0.431246i	$-0.141926\pi$
$192$		0			0
$193$	1113.21	+	1928.14i	0.415184	+	0.719121i	0.995448	−	0.0953083i	$-0.0303837\pi$
$193$	1113.21	+	1928.14i	0.415184	+	0.719121i	−0.580263	+	0.814429i	$0.697050\pi$
$194$	−240.309	+	416.227i	−0.0889339	+	0.154038i
$195$		0			0
$196$	−1012.01	+	926.398i	−0.368810	+	0.337609i
$197$	3746.39			1.35492			0.677460	−	0.735559i	$-0.263081\pi$
$197$	3746.39			1.35492			0.677460	+	0.735559i	$0.263081\pi$
$198$		0			0
$199$	1432.02	+	2480.33i	0.510116	+	0.883547i	0.999931	+	0.0117211i	$0.00373101\pi$
$199$	1432.02	+	2480.33i	0.510116	+	0.883547i	−0.489815	+	0.871826i	$0.662936\pi$
$200$	421.447	+	729.968i	0.149004	+	0.258083i
$201$		0			0
$202$	−891.284			−0.310448
$203$	1499.37	−	1867.21i	0.518401	−	0.645579i
$204$		0			0
$205$	−809.742	+	1402.51i	−0.275877	+	0.477833i
$206$	846.189	+	1465.64i	0.286198	+	0.495709i
$207$		0			0
$208$	−299.891	+	519.426i	−0.0999696	+	0.173152i
$209$	5356.25			1.77273
$210$		0			0
$211$	3729.13			1.21670			0.608350	−	0.793669i	$-0.291832\pi$
$211$	3729.13			1.21670			0.608350	+	0.793669i	$0.291832\pi$
$212$	1203.64	−	2084.77i	0.389937	−	0.675391i
$213$		0			0
$214$	562.311	+	973.952i	0.179621	+	0.311112i
$215$	126.860	−	219.728i	0.0402408	−	0.0696992i
$216$		0			0
$217$	−4476.81	−	690.207i	−1.40049	−	0.215918i
$218$	−2231.27			−0.693214
$219$		0			0
$220$	−351.318	−	608.500i	−0.107663	−	0.186478i
$221$	−2137.14	−	3701.64i	−0.650496	−	1.12669i
$222$		0			0
$223$	152.890			0.0459115			0.0229557	−	0.999736i	$-0.492692\pi$
$223$	152.890			0.0459115			0.0229557	+	0.999736i	$0.492692\pi$
$224$	585.728	+	90.3039i	0.174712	+	0.0269361i
$225$		0			0
$226$	1518.31	−	2629.80i	0.446888	−	0.774033i
$227$	1693.04	+	2932.43i	0.495027	+	0.857411i	0.999984	−	0.00573340i	$-0.00182501\pi$
$227$	1693.04	+	2932.43i	0.495027	+	0.857411i	−0.504957	+	0.863144i	$0.668492\pi$
$228$		0			0
$229$	386.178	−	668.880i	0.111438	−	0.193017i	−0.804912	−	0.593394i	$-0.797788\pi$
$229$	386.178	−	668.880i	0.111438	−	0.193017i	0.916350	+	0.400377i	$0.131121\pi$
$230$	−1705.22			−0.488865
$231$		0			0
$232$	−1034.41			−0.292727
$233$	−1781.93	+	3086.39i	−0.501021	+	0.867794i	0.498978	+	0.866614i	$0.333709\pi$
$233$	−1781.93	+	3086.39i	−0.501021	+	0.867794i	−0.999999	+	0.00117935i	$0.999625\pi$
$234$		0			0
$235$	764.413	+	1324.00i	0.212191	+	0.367525i
$236$	−63.0523	+	109.210i	−0.0173913	+	0.0301227i
$237$		0			0
$238$	−2644.39	+	3293.13i	−0.720211	+	0.896899i
$239$	−5832.92			−1.57866			−0.789331	−	0.613968i	$-0.789572\pi$
$239$	−5832.92			−1.57866			−0.789331	+	0.613968i	$0.789572\pi$
$240$		0			0
$241$	−1846.61	−	3198.42i	−0.493570	−	0.854889i	0.506402	−	0.862297i	$-0.330975\pi$
$241$	−1846.61	−	3198.42i	−0.493570	−	0.854889i	−0.999973	+	0.00740841i	$0.997642\pi$
$242$	−240.225	−	416.082i	−0.0638110	−	0.110524i
$243$		0			0
$244$	3450.08			0.905199
$245$	−1449.42	−	457.808i	−0.377960	−	0.119381i
$246$		0			0
$247$	2532.71	−	4386.78i	0.652438	−	1.13006i
$248$	978.324	+	1694.51i	0.250499	+	0.433876i
$249$		0			0
$250$	−1020.85	+	1768.16i	−0.258257	+	0.447313i
$251$	1571.78			0.395259			0.197630	−	0.980277i	$-0.436676\pi$
$251$	1571.78			0.395259			0.197630	+	0.980277i	$0.436676\pi$
$252$		0			0
$253$	−7626.39			−1.89513
$254$	803.079	−	1390.97i	0.198384	−	0.343612i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−1826.14	+	3162.97i	−0.443236	+	0.767707i	−0.997927	−	0.0643490i	$-0.979503\pi$
$257$	−1826.14	+	3162.97i	−0.443236	+	0.767707i	0.554692	+	0.832056i	$0.312836\pi$
$258$		0			0
$259$	−1833.04	−	4717.19i	−0.439767	−	1.13171i
$260$	−664.483			−0.158498
$261$		0			0
$262$	890.214	+	1541.90i	0.209914	+	0.363582i
$263$	355.268	+	615.342i	0.0832956	+	0.144272i	0.904664	−	0.426126i	$-0.140122\pi$
$263$	355.268	+	615.342i	0.0832956	+	0.144272i	−0.821368	+	0.570399i	$0.806789\pi$
$264$		0			0
$265$	2666.98			0.618230
$266$	−4946.72	−	762.655i	−1.14024	−	0.175795i
$267$		0			0
$268$	1365.18	−	2364.56i	0.311163	−	0.538950i
$269$	−4.13321	−	7.15894i	−0.000936827	−	0.00162263i	0.865557	−	0.500811i	$-0.166965\pi$
$269$	−4.13321	−	7.15894i	−0.000936827	−	0.00162263i	−0.866493	+	0.499188i	$0.833632\pi$
$270$		0			0
$271$	1361.85	−	2358.79i	0.305264	−	0.528732i	−0.672056	−	0.740500i	$-0.734589\pi$
$271$	1361.85	−	2358.79i	0.305264	−	0.528732i	0.977320	+	0.211768i	$0.0679221\pi$
$272$	1824.36			0.406684
$273$		0			0
$274$	−4089.13			−0.901581
$275$	−2088.20	+	3616.87i	−0.457903	+	0.793111i
$276$		0			0
$277$	−620.284	−	1074.36i	−0.134546	−	0.233041i	0.790878	−	0.611974i	$-0.209624\pi$
$277$	−620.284	−	1074.36i	−0.134546	−	0.233041i	−0.925424	+	0.378933i	$0.876291\pi$
$278$	−587.471	+	1017.53i	−0.126742	+	0.219523i
$279$		0			0
$280$	237.815	+	611.998i	0.0507577	+	0.130621i
$281$	−7346.62			−1.55965			−0.779826	−	0.625996i	$-0.784693\pi$
$281$	−7346.62			−1.55965			−0.779826	+	0.625996i	$0.784693\pi$
$282$		0			0
$283$	−2058.87	−	3566.06i	−0.432462	−	0.749047i	0.564622	−	0.825349i	$-0.309022\pi$
$283$	−2058.87	−	3566.06i	−0.432462	−	0.749047i	−0.997085	+	0.0763025i	$0.975689\pi$
$284$	1329.72	+	2303.14i	0.277832	+	0.481220i
$285$		0			0
$286$	−2971.82			−0.614431
$287$	4237.71	−	5277.34i	0.871582	−	1.08541i
$288$		0			0
$289$	−4044.05	+	7004.50i	−0.823132	+	1.42571i
$290$	−573.001	−	992.467i	−0.116027	−	0.200964i
$291$		0			0
$292$	−1884.49	+	3264.03i	−0.377676	+	0.654154i
$293$	2410.10			0.480545			0.240273	−	0.970705i	$-0.422763\pi$
$293$	2410.10			0.480545			0.240273	+	0.970705i	$0.422763\pi$
$294$		0			0
$295$	−139.708			−0.0275733
$296$	−1093.03	+	1893.19i	−0.214633	+	0.371755i
$297$		0			0
$298$	2713.32	+	4699.60i	0.527443	+	0.913559i
$299$	−3606.14	+	6246.02i	−0.697487	+	1.20808i
$300$		0			0
$301$	−663.910	+	826.786i	−0.127133	+	0.158323i
$302$	−332.864			−0.0634244
$303$		0			0
$304$	1081.01	+	1872.37i	0.203949	+	0.353250i
$305$	1911.13	+	3310.17i	0.358789	+	0.621442i
$306$		0			0
$307$	−5172.84			−0.961661			−0.480830	−	0.876814i	$-0.659665\pi$
$307$	−5172.84			−0.961661			−0.480830	+	0.876814i	$0.659665\pi$
$308$	1063.60	+	2737.09i	0.196767	+	0.506364i
$309$		0			0
$310$	−1083.86	+	1877.30i	−0.198578	+	0.343947i
$311$	1714.23	+	2969.14i	0.312557	+	0.541365i	0.978915	−	0.204267i	$-0.0654811\pi$
$311$	1714.23	+	2969.14i	0.312557	+	0.541365i	−0.666358	+	0.745632i	$0.732148\pi$
$312$		0			0
$313$	2292.03	−	3969.91i	0.413908	−	0.716909i	−0.581406	−	0.813614i	$-0.697497\pi$
$313$	2292.03	−	3969.91i	0.413908	−	0.716909i	0.995313	+	0.0967050i	$0.0308303\pi$
$314$	−1071.14			−0.192510
$315$		0			0
$316$	−13.3017			−0.00236797
$317$	74.0671	−	128.288i	0.0131231	−	0.0227299i	−0.859389	−	0.511322i	$-0.829156\pi$
$317$	74.0671	−	128.288i	0.0131231	−	0.0227299i	0.872512	+	0.488592i	$0.162489\pi$
$318$		0			0
$319$	−2562.68	−	4438.69i	−0.449788	−	0.779055i
$320$	141.808	−	245.619i	0.0247728	−	0.0429078i
$321$		0			0
$322$	7043.29	+	1085.89i	1.21897	+	0.187933i
$323$	−15407.5			−2.65417
$324$		0			0
$325$	1974.81	+	3420.48i	0.337055	+	0.583797i
$326$	−1202.76	−	2083.24i	−0.204339	−	0.353926i
$327$		0			0
$328$	−2923.59			−0.492159
$329$	−2314.23	−	5955.48i	−0.387804	−	0.997983i
$330$		0			0
$331$	3893.30	−	6743.40i	0.646512	−	1.11979i	−0.337438	−	0.941348i	$-0.609560\pi$
$331$	3893.30	−	6743.40i	0.646512	−	1.11979i	0.983950	−	0.178444i	$-0.0571062\pi$
$332$	−108.180	−	187.373i	−0.0178830	−	0.0309742i
$333$		0			0
$334$	3467.14	−	6005.26i	0.568004	−	0.983811i
$335$	3024.90			0.493337
$336$		0			0
$337$	906.591			0.146544			0.0732718	−	0.997312i	$-0.476656\pi$
$337$	906.591			0.146544			0.0732718	+	0.997312i	$0.476656\pi$
$338$	791.774	−	1371.39i	0.127417	−	0.220692i
$339$		0			0
$340$	1010.58	+	1750.38i	0.161195	+	0.279199i
$341$	−4847.43	+	8396.00i	−0.769804	+	1.33334i
$342$		0			0
$343$	5695.20	+	2813.94i	0.896537	+	0.442969i
$344$	458.030			0.0717887
$345$		0			0
$346$	416.192	+	720.866i	0.0646666	+	0.112006i
$347$	2098.16	+	3634.12i	0.324597	+	0.562219i	0.981431	−	0.191817i	$-0.0614379\pi$
$347$	2098.16	+	3634.12i	0.324597	+	0.562219i	−0.656834	+	0.754036i	$0.728105\pi$
$348$		0			0
$349$	−5970.17			−0.915690			−0.457845	−	0.889032i	$-0.651379\pi$
$349$	−5970.17			−0.915690			−0.457845	+	0.889032i	$0.651379\pi$
$350$	2443.53	−	3043.00i	0.373178	−	0.464729i
$351$		0			0
$352$	634.219	−	1098.50i	0.0960340	−	0.166336i
$353$	5523.89	+	9567.65i	0.832880	+	1.44259i	0.895745	+	0.444569i	$0.146643\pi$
$353$	5523.89	+	9567.65i	0.832880	+	1.44259i	−0.0628644	+	0.998022i	$0.520024\pi$
$354$		0			0
$355$	−1473.16	+	2551.60i	−0.220246	+	0.381478i
$356$	1141.60			0.169957
$357$		0			0
$358$	−751.268			−0.110910
$359$	3781.82	−	6550.31i	0.555980	−	0.962986i	−0.441846	−	0.897091i	$-0.645676\pi$
$359$	3781.82	−	6550.31i	0.555980	−	0.962986i	0.997827	−	0.0658953i	$-0.0209903\pi$
$360$		0			0
$361$	−5700.14	−	9872.93i	−0.831045	−	1.43941i
$362$	−2201.29	+	3812.74i	−0.319605	+	0.553573i
$363$		0			0
$364$	2744.60	+	423.145i	0.395209	+	0.0609309i
$365$	−4175.55			−0.598790
$366$		0			0
$367$	1716.04	+	2972.26i	0.244077	+	0.422754i	0.961872	−	0.273501i	$-0.0881816\pi$
$367$	1716.04	+	2972.26i	0.244077	+	0.422754i	−0.717794	+	0.696255i	$0.754848\pi$
$368$	−1539.18	−	2665.94i	−0.218031	−	0.377641i
$369$		0			0
$370$	−2421.89			−0.340292
$371$	−11015.8	−	1698.34i	−1.54153	−	0.237664i
$372$		0			0
$373$	2636.84	−	4567.15i	0.366034	−	0.633989i	−0.622908	−	0.782295i	$-0.714049\pi$
$373$	2636.84	−	4567.15i	0.366034	−	0.633989i	0.988941	+	0.148306i	$0.0473822\pi$
$374$	4519.70	+	7828.34i	0.624887	+	1.08234i
$375$		0			0
$376$	−1379.96	+	2390.17i	−0.189272	+	0.327828i
$377$	−4847.05			−0.662164
$378$		0			0
$379$	8856.92			1.20039			0.600197	−	0.799852i	$-0.295089\pi$
$379$	8856.92			1.20039			0.600197	+	0.799852i	$0.295089\pi$
$380$	−1197.63	+	2074.36i	−0.161677	+	0.280032i
$381$		0			0
$382$	4353.28	+	7540.10i	0.583071	+	1.00991i
$383$	520.852	−	902.142i	0.0694890	−	0.120358i	−0.829187	−	0.558971i	$-0.811196\pi$
$383$	520.852	−	902.142i	0.0694890	−	0.120358i	0.898676	+	0.438612i	$0.144530\pi$
$384$		0			0
$385$	−2036.93	+	2536.64i	−0.269640	+	0.335790i
$386$	−4452.84			−0.587160
$387$		0			0
$388$	−480.618	−	832.454i	−0.0628857	−	0.108921i
$389$	−4666.34	−	8082.33i	−0.608207	−	1.05345i	−0.991536	−	0.129834i	$-0.958556\pi$
$389$	−4666.34	−	8082.33i	−0.608207	−	1.05345i	0.383328	−	0.923612i	$-0.374778\pi$
$390$		0			0
$391$	21937.6			2.83743
$392$	−592.555	−	2679.26i	−0.0763483	−	0.345211i
$393$		0			0
$394$	−3746.39	+	6488.94i	−0.479037	+	0.829716i
$395$	−7.36831	−	12.7623i	−0.000938582	−	0.00162567i
$396$		0			0
$397$	−3411.16	+	5908.31i	−0.431238	+	0.746926i	−0.996980	−	0.0776562i	$-0.975256\pi$
$397$	−3411.16	+	5908.31i	−0.431238	+	0.746926i	0.565742	+	0.824582i	$0.308590\pi$
$398$	−5728.08			−0.721414
$399$		0			0
$400$	−1685.79			−0.210724
$401$	4054.97	−	7023.42i	0.504977	−	0.874646i	−0.495007	−	0.868889i	$-0.664834\pi$
$401$	4054.97	−	7023.42i	0.504977	−	0.874646i	0.999983	−	0.00575638i	$-0.00183232\pi$
$402$		0			0
$403$	4584.22	+	7940.11i	0.566641	+	0.981452i
$404$	891.284	−	1543.75i	0.109760	−	0.190110i
$405$		0			0
$406$	1734.73	+	4464.20i	0.212053	+	0.545701i
$407$	−10831.6			−1.31917
$408$		0			0
$409$	−2216.52	−	3839.13i	−0.267970	−	0.464138i	0.700367	−	0.713783i	$-0.253020\pi$
$409$	−2216.52	−	3839.13i	−0.267970	−	0.464138i	−0.968338	+	0.249644i	$0.919686\pi$
$410$	−1619.48	−	2805.03i	−0.195075	−	0.337879i
$411$		0			0
$412$	−3384.76			−0.404745
$413$	577.055	+	88.9667i	0.0687530	+	0.0105999i
$414$		0			0
$415$	119.850	−	207.586i	0.0141764	−	0.0245542i
$416$	−599.781	−	1038.85i	−0.0706892	−	0.122437i
$417$		0			0
$418$	−5356.25	+	9277.30i	−0.626753	+	1.08557i
$419$	6782.36			0.790787			0.395394	−	0.918512i	$-0.370608\pi$
$419$	6782.36			0.790787			0.395394	+	0.918512i	$0.370608\pi$
$420$		0			0
$421$	−4798.62			−0.555512			−0.277756	−	0.960652i	$-0.589591\pi$
$421$	−4798.62			−0.555512			−0.277756	+	0.960652i	$0.589591\pi$
$422$	−3729.13	+	6459.04i	−0.430168	+	0.745073i
$423$		0			0
$424$	2407.29	+	4169.55i	0.275727	+	0.477573i
$425$	6006.80	−	10404.1i	0.685583	−	1.18746i
$426$		0			0
$427$	−5785.85	−	14889.4i	−0.655730	−	1.68747i
$428$	−2249.25			−0.254022
$429$		0			0
$430$	253.720	+	439.456i	0.0284546	+	0.0492848i
$431$	6474.64	+	11214.4i	0.723602	+	1.25332i	0.959547	+	0.281549i	$0.0908482\pi$
$431$	6474.64	+	11214.4i	0.723602	+	1.25332i	−0.235945	+	0.971766i	$0.575818\pi$
$432$		0			0
$433$	937.240			0.104020			0.0520102	−	0.998647i	$-0.483437\pi$
$433$	937.240			0.104020			0.0520102	+	0.998647i	$0.483437\pi$
$434$	5672.28	−	7063.86i	0.627369	−	0.781281i
$435$		0			0
$436$	2231.27	−	3864.67i	0.245088	−	0.424505i
$437$	12999.1	+	22515.0i	1.42295	+	2.46462i
$438$		0			0
$439$	6800.15	−	11778.2i	0.739302	−	1.28051i	−0.213509	−	0.976941i	$-0.568489\pi$
$439$	6800.15	−	11778.2i	0.739302	−	1.28051i	0.952810	−	0.303567i	$-0.0981775\pi$
$440$	1405.27			0.152258
$441$		0			0
$442$	8548.56			0.919940
$443$	2316.24	−	4011.84i	0.248415	−	0.430267i	−0.714671	−	0.699460i	$-0.753424\pi$
$443$	2316.24	−	4011.84i	0.248415	−	0.430267i	0.963086	+	0.269193i	$0.0867571\pi$
$444$		0			0
$445$	632.374	+	1095.30i	0.0673650	+	0.116680i
$446$	−152.890	+	264.813i	−0.0162322	+	0.0281149i
$447$		0			0
$448$	−742.139	+	924.207i	−0.0782651	+	0.0974658i
$449$	254.426			0.0267419			0.0133709	−	0.999911i	$-0.495744\pi$
$449$	254.426			0.0267419			0.0133709	+	0.999911i	$0.495744\pi$
$450$		0			0
$451$	−7242.94	−	12545.1i	−0.756224	−	1.30982i
$452$	3036.63	+	5259.59i	0.315998	+	0.547324i
$453$		0			0
$454$	−6772.16			−0.700073
$455$	1114.35	+	2867.70i	0.114817	+	0.295472i
$456$		0			0
$457$	−8205.25	+	14211.9i	−0.839881	+	1.45472i	0.0501133	+	0.998744i	$0.484042\pi$
$457$	−8205.25	+	14211.9i	−0.839881	+	1.45472i	−0.889994	+	0.455972i	$0.849292\pi$
$458$	772.356	+	1337.76i	0.0787988	+	0.136483i
$459$		0			0
$460$	1705.22	−	2953.53i	0.172840	−	0.299367i
$461$	−5842.56			−0.590272			−0.295136	−	0.955455i	$-0.595365\pi$
$461$	−5842.56			−0.590272			−0.295136	+	0.955455i	$0.595365\pi$
$462$		0			0
$463$	6699.12			0.672429			0.336215	−	0.941785i	$-0.390853\pi$
$463$	6699.12			0.672429			0.336215	+	0.941785i	$0.390853\pi$
$464$	1034.41	−	1791.66i	0.103495	−	0.179258i
$465$		0			0
$466$	−3563.85	−	6172.77i	−0.354275	−	0.613623i
$467$	912.817	−	1581.05i	0.0904500	−	0.156664i	−0.817251	−	0.576282i	$-0.804503\pi$
$467$	912.817	−	1581.05i	0.0904500	−	0.156664i	0.907701	+	0.419619i	$0.137836\pi$
$468$		0			0
$469$	−12494.1	−	1926.27i	−1.23012	−	0.189652i
$470$	−3057.65			−0.300083
$471$		0			0
$472$	−126.105	−	218.420i	−0.0122975	−	0.0212999i
$473$	1134.73	+	1965.41i	0.110307	+	0.191057i
$474$		0			0
$475$	14237.2			1.37526
$476$	−3059.48	−	7873.35i	−0.294603	−	0.758139i
$477$		0			0
$478$	5832.92	−	10102.9i	0.558141	−	0.966729i
$479$	−5518.07	−	9557.58i	−0.526362	−	0.911685i	−0.999528	−	0.0307121i	$-0.990223\pi$
$479$	−5518.07	−	9557.58i	−0.526362	−	0.911685i	0.473167	−	0.880973i	$-0.343111\pi$
$480$		0			0
$481$	−5121.73	+	8871.10i	−0.485511	+	0.840930i
$482$	7386.43			0.698014
$483$		0			0
$484$	960.900			0.0902423
$485$	532.464	−	922.255i	0.0498515	−	0.0863453i
$486$		0			0
$487$	−4092.78	−	7088.90i	−0.380824	−	0.659607i	0.610356	−	0.792127i	$-0.291026\pi$
$487$	−4092.78	−	7088.90i	−0.380824	−	0.659607i	−0.991180	+	0.132520i	$0.957693\pi$
$488$	−3450.08	+	5975.71i	−0.320036	+	0.554319i
$489$		0			0
$490$	2242.37	−	2052.67i	0.206735	−	0.189245i
$491$	19006.2			1.74692			0.873459	−	0.486897i	$-0.161871\pi$
$491$	19006.2			1.74692			0.873459	+	0.486897i	$0.161871\pi$
$492$		0			0
$493$	7371.65	+	12768.1i	0.673433	+	1.16642i
$494$	5065.41	+	8773.55i	0.461343	+	0.799070i
$495$		0			0
$496$	−3913.30			−0.354258
$497$	7709.66	−	9601.07i	0.695826	−	0.866533i
$498$		0			0
$499$	−8246.79	+	14283.9i	−0.739834	+	1.28143i	0.212736	+	0.977110i	$0.431763\pi$
$499$	−8246.79	+	14283.9i	−0.739834	+	1.28143i	−0.952570	+	0.304320i	$0.901571\pi$
$500$	−2041.70	−	3536.32i	−0.182615	−	0.316298i
$501$		0			0
$502$	−1571.78	+	2722.41i	−0.139745	+	0.242046i
$503$	10368.0			0.919059			0.459530	−	0.888162i	$-0.348018\pi$
$503$	10368.0			0.919059			0.459530	+	0.888162i	$0.348018\pi$
$504$		0			0
$505$	1974.86			0.174020
$506$	7626.39	−	13209.3i	0.670029	−	1.16052i
$507$		0			0
$508$	1606.16	+	2781.95i	0.140279	+	0.242970i
$509$	−3254.37	+	5636.74i	−0.283394	+	0.490853i	−0.972218	−	0.234075i	$-0.924794\pi$
$509$	−3254.37	+	5636.74i	−0.283394	+	0.490853i	0.688825	+	0.724928i	$0.258127\pi$
$510$		0			0
$511$	17246.8	+	2659.01i	1.49306	+	0.230191i
$512$	512.000			0.0441942
$513$		0			0
$514$	−3652.28	−	6325.94i	−0.313415	−	0.542851i
$515$	−1874.94	−	3247.50i	−0.160427	−	0.277868i
$516$		0			0
$517$	−13675.0			−1.16330
$518$	10003.4	+	1542.27i	0.848506	+	0.130817i
$519$		0			0
$520$	664.483	−	1150.92i	0.0560375	−	0.0970598i
$521$	−6759.54	−	11707.9i	−0.568409	−	0.984513i	−0.996724	−	0.0808831i	$-0.974226\pi$
$521$	−6759.54	−	11707.9i	−0.568409	−	0.984513i	0.428315	−	0.903630i	$-0.359107\pi$
$522$		0			0
$523$	−929.020	+	1609.11i	−0.0776734	+	0.134534i	−0.902246	−	0.431222i	$-0.858082\pi$
$523$	−929.020	+	1609.11i	−0.0776734	+	0.134534i	0.824572	+	0.565757i	$0.191416\pi$
$524$	−3560.85			−0.296864
$525$		0			0
$526$	−1421.07			−0.117798
$527$	13943.9	−	24151.5i	1.15257	−	1.99631i
$528$		0			0
$529$	−12424.9	−	21520.6i	−1.02120	−	1.76877i
$530$	−2666.98	+	4619.34i	−0.218577	+	0.378587i
$531$		0			0
$532$	6267.68	−	7805.32i	0.510787	−	0.636097i
$533$	−13699.3			−1.11329
$534$		0			0
$535$	−1245.94	−	2158.03i	−0.100686	−	0.174392i
$536$	2730.36	+	4729.13i	0.220026	+	0.381095i
$537$		0			0
$538$	16.5329			0.00132487
$539$	10028.7	−	9180.30i	0.801424	−	0.733624i
$540$		0			0
$541$	−6846.27	+	11858.1i	−0.544075	+	0.942365i	0.454590	+	0.890701i	$0.349786\pi$
$541$	−6846.27	+	11858.1i	−0.544075	+	0.942365i	−0.998665	+	0.0516640i	$0.983548\pi$
$542$	2723.70	+	4717.58i	0.215854	+	0.373870i
$543$		0			0
$544$	−1824.36	+	3159.88i	−0.143784	+	0.249042i
$545$	4943.94			0.388578
$546$		0			0
$547$	−9054.00			−0.707717			−0.353858	−	0.935299i	$-0.615131\pi$
$547$	−9054.00			−0.707717			−0.353858	+	0.935299i	$0.615131\pi$
$548$	4089.13	−	7082.58i	0.318757	−	0.552104i
$549$		0			0
$550$	−4176.40	−	7233.74i	−0.323786	−	0.560814i
$551$	−8736.07	+	15131.3i	−0.675443	+	1.16990i
$552$		0			0
$553$	22.3072	+	57.4059i	0.00171537	+	0.00441437i
$554$	2481.14			0.190277
$555$		0			0
$556$	−1174.94	−	2035.06i	−0.0896198	−	0.155226i
$557$	−4717.22	−	8170.46i	−0.358842	−	0.621533i	0.628926	−	0.777465i	$-0.283495\pi$
$557$	−4717.22	−	8170.46i	−0.358842	−	0.621533i	−0.987768	+	0.155933i	$0.950162\pi$
$558$		0			0
$559$	2146.23			0.162390
$560$	−1297.83	−	200.091i	−0.0979343	−	0.0150989i
$561$		0			0
$562$	7346.62	−	12724.7i	0.551421	−	0.955088i
$563$	3205.10	+	5551.39i	0.239927	+	0.415565i	0.960693	−	0.277613i	$-0.0895433\pi$
$563$	3205.10	+	5551.39i	0.239927	+	0.415565i	−0.720766	+	0.693178i	$0.756210\pi$
$564$		0			0
$565$	−3364.20	+	5826.97i	−0.250501	+	0.433880i
$566$	8235.46			0.611594
$567$		0			0
$568$	−5318.88			−0.392914
$569$	2432.99	−	4214.06i	0.179255	−	0.310479i	−0.762370	−	0.647141i	$-0.775965\pi$
$569$	2432.99	−	4214.06i	0.179255	−	0.310479i	0.941626	+	0.336662i	$0.109298\pi$
$570$		0			0
$571$	−9349.38	−	16193.6i	−0.685218	−	1.18683i	−0.973368	−	0.229247i	$-0.926374\pi$
$571$	−9349.38	−	16193.6i	−0.685218	−	1.18683i	0.288150	−	0.957585i	$-0.406960\pi$
$572$	2971.82	−	5147.34i	0.217234	−	0.376261i
$573$		0			0
$574$	4902.91	+	12617.3i	0.356522	+	0.917482i
$575$	−20271.4			−1.47022
$576$		0			0
$577$	9174.11	+	15890.0i	0.661912	+	1.14647i	0.980113	+	0.198442i	$0.0635882\pi$
$577$	9174.11	+	15890.0i	0.661912	+	1.14647i	−0.318200	+	0.948023i	$0.603079\pi$
$578$	−8088.10	−	14009.0i	−0.582042	−	1.00813i
$579$		0			0
$580$	2292.00			0.164087
$581$	−627.223	+	781.099i	−0.0447876	+	0.0557753i
$582$		0			0
$583$	−11927.7	+	20659.4i	−0.847334	+	1.46763i
$584$	−3768.98	−	6528.06i	−0.267057	−	0.462556i
$585$		0			0
$586$	−2410.10	+	4174.42i	−0.169898	+	0.294273i
$587$	17246.6			1.21268			0.606340	−	0.795206i	$-0.292637\pi$
$587$	17246.6			1.21268			0.606340	+	0.795206i	$0.292637\pi$
$588$		0			0
$589$	33049.5			2.31202
$590$	139.708	−	241.982i	0.00974863	−	0.0168851i
$591$		0			0
$592$	−2186.07	−	3786.38i	−0.151768	−	0.262870i
$593$	8806.40	−	15253.1i	0.609841	−	1.05628i	−0.381426	−	0.924400i	$-0.624567\pi$
$593$	8806.40	−	15253.1i	0.609841	−	1.05628i	0.991266	−	0.131875i	$-0.0420999\pi$
$594$		0			0
$595$	5859.30	−	7296.76i	0.403711	−	0.502753i
$596$	−10853.3			−0.745918
$597$		0			0
$598$	−7212.29	−	12492.0i	−0.493198	−	0.854244i
$599$	6533.22	+	11315.9i	0.445643	+	0.771876i	0.998097	−	0.0616672i	$-0.0196418\pi$
$599$	6533.22	+	11315.9i	0.445643	+	0.771876i	−0.552454	+	0.833544i	$0.686308\pi$
$600$		0			0
$601$	−10232.8			−0.694516			−0.347258	−	0.937770i	$-0.612887\pi$
$601$	−10232.8			−0.694516			−0.347258	+	0.937770i	$0.612887\pi$
$602$	−768.125	−	1976.71i	−0.0520041	−	0.133828i
$603$		0			0
$604$	332.864	−	576.537i	0.0224239	−	0.0388393i
$605$	532.279	+	921.934i	0.0357689	+	0.0619536i
$606$		0			0
$607$	5116.84	−	8862.63i	0.342152	−	0.592625i	−0.642680	−	0.766135i	$-0.722178\pi$
$607$	5116.84	−	8862.63i	0.342152	−	0.592625i	0.984832	+	0.173510i	$0.0555109\pi$
$608$	−4324.06			−0.288427
$609$		0			0
$610$	−7644.51			−0.507405
$611$	−6466.22	+	11199.8i	−0.428143	+	0.741565i
$612$		0			0
$613$	1435.08	+	2485.63i	0.0945550	+	0.163774i	0.909423	−	0.415873i	$-0.136524\pi$
$613$	1435.08	+	2485.63i	0.0945550	+	0.163774i	−0.814868	+	0.579647i	$0.803191\pi$
$614$	5172.84	−	8959.63i	0.339998	−	0.588894i
$615$		0			0
$616$	−5804.37	−	894.882i	−0.379651	−	0.0585322i
$617$	−743.774			−0.0485304			−0.0242652	−	0.999706i	$-0.507725\pi$
$617$	−743.774			−0.0485304			−0.0242652	+	0.999706i	$0.507725\pi$
$618$		0			0
$619$	11435.6	+	19807.0i	0.742543	+	1.28612i	0.951334	+	0.308162i	$0.0997140\pi$
$619$	11435.6	+	19807.0i	0.742543	+	1.28612i	−0.208790	+	0.977960i	$0.566953\pi$
$620$	−2167.72	−	3754.60i	−0.140416	−	0.243207i
$621$		0			0
$622$	−6856.94			−0.442023
$623$	−1914.48	−	4926.78i	−0.123117	−	0.316833i
$624$		0			0
$625$	−4323.17	+	7487.95i	−0.276683	+	0.479228i
$626$	4584.06	+	7939.82i	0.292677	+	0.506931i
$627$		0			0
$628$	1071.14	−	1855.28i	0.0680626	−	0.117888i
$629$	31157.6			1.97509
$630$		0			0
$631$	−16861.8			−1.06380			−0.531901	−	0.846807i	$-0.678522\pi$
$631$	−16861.8			−1.06380			−0.531901	+	0.846807i	$0.678522\pi$
$632$	13.3017	−	23.0392i	0.000837204	−	0.00145008i
$633$		0			0
$634$	148.134	+	256.576i	0.00927943	+	0.0160724i
$635$	−1779.42	+	3082.05i	−0.111203	+	0.192610i
$636$		0			0
$637$	−2776.59	−	12554.4i	−0.172704	−	0.780887i
$638$	10250.7			0.636096
$639$		0			0
$640$	283.616	+	491.237i	0.0175170	+	0.0303404i
$641$	4259.46	+	7377.61i	0.262463	+	0.454599i	0.966896	−	0.255172i	$-0.0821320\pi$
$641$	4259.46	+	7377.61i	0.262463	+	0.454599i	−0.704433	+	0.709771i	$0.748799\pi$
$642$		0			0
$643$	−12045.2			−0.738748			−0.369374	−	0.929281i	$-0.620428\pi$
$643$	−12045.2			−0.738748			−0.369374	+	0.929281i	$0.620428\pi$
$644$	−8924.11	+	11113.5i	−0.546055	+	0.680018i
$645$		0			0
$646$	15407.5	−	26686.5i	0.938389	−	1.62534i
$647$	2078.82	+	3600.63i	0.126317	+	0.218787i	0.922247	−	0.386601i	$-0.126351\pi$
$647$	2078.82	+	3600.63i	0.126317	+	0.218787i	−0.795930	+	0.605389i	$0.793018\pi$
$648$		0			0
$649$	624.828	−	1082.23i	0.0377914	−	0.0654566i
$650$	−7899.26			−0.476668
$651$		0			0
$652$	4811.03			0.288979
$653$	15174.8	−	26283.5i	0.909394	−	1.57512i	0.0944872	−	0.995526i	$-0.469879\pi$
$653$	15174.8	−	26283.5i	0.909394	−	1.57512i	0.814907	−	0.579591i	$-0.196788\pi$
$654$		0			0
$655$	−1972.49	−	3416.45i	−0.117667	−	0.203804i
$656$	2923.59	−	5063.80i	0.174004	−	0.301384i
$657$		0			0
$658$	12629.4	+	1947.13i	0.748246	+	0.115360i
$659$	14482.7			0.856093			0.428046	−	0.903757i	$-0.359202\pi$
$659$	14482.7			0.856093			0.428046	+	0.903757i	$0.359202\pi$
$660$		0			0
$661$	3262.70	+	5651.16i	0.191988	+	0.332533i	0.945909	−	0.324432i	$-0.105173\pi$
$661$	3262.70	+	5651.16i	0.191988	+	0.332533i	−0.753921	+	0.656965i	$0.771840\pi$
$662$	7786.61	+	13486.8i	0.457153	+	0.791812i
$663$		0			0
$664$	432.720			0.0252903
$665$	10960.7	+	1689.85i	0.639155	+	0.0985409i
$666$		0			0
$667$	12438.7	−	21544.4i	0.722081	−	1.25068i
$668$	6934.27	+	12010.5i	0.401639	+	0.695660i
$669$		0			0
$670$	−3024.90	+	5239.28i	−0.174421	+	0.302106i
$671$	−34189.1			−1.96700
$672$		0			0
$673$	1453.70			0.0832629			0.0416314	−	0.999133i	$-0.486744\pi$
$673$	1453.70			0.0832629			0.0416314	+	0.999133i	$0.486744\pi$
$674$	−906.591	+	1570.26i	−0.0518110	+	0.0897392i
$675$		0			0
$676$	1583.55	+	2742.79i	0.0900972	+	0.156053i
$677$	−7385.35	+	12791.8i	−0.419265	+	0.726187i	−0.995866	−	0.0908383i	$-0.971045\pi$
$677$	−7385.35	+	12791.8i	−0.419265	+	0.726187i	0.576601	+	0.817026i	$0.304379\pi$
$678$		0			0
$679$	−2786.60	+	3470.23i	−0.157496	+	0.196135i
$680$	−4042.32			−0.227965
$681$		0			0
$682$	−9694.87	−	16792.0i	−0.544334	−	0.942814i
$683$	−4539.15	−	7862.04i	−0.254298	−	0.440457i	0.710406	−	0.703792i	$-0.248511\pi$
$683$	−4539.15	−	7862.04i	−0.254298	−	0.440457i	−0.964705	+	0.263334i	$0.915178\pi$
$684$		0			0
$685$	9060.49			0.505377
$686$	−10569.1	+	7050.44i	−0.588236	+	0.392401i
$687$		0			0
$688$	−458.030	+	793.331i	−0.0253811	+	0.0439614i
$689$	11280.1	+	19537.6i	0.623710	+	1.08030i
$690$		0			0
$691$	−2234.84	+	3870.86i	−0.123035	+	0.213104i	−0.920963	−	0.389649i	$-0.872596\pi$
$691$	−2234.84	+	3870.86i	−0.123035	+	0.213104i	0.797928	+	0.602753i	$0.205930\pi$
$692$	−1664.77			−0.0914524
$693$		0			0
$694$	−8392.65			−0.459050
$695$	1301.69	−	2254.59i	0.0710444	−	0.123053i
$696$		0			0
$697$	20834.6	+	36086.6i	1.13224	+	1.96109i
$698$	5970.17	−	10340.6i	0.323745	−	0.560743i
$699$		0			0
$700$	2827.10	+	7275.33i	0.152649	+	0.392831i
$701$	−29024.1			−1.56380			−0.781901	−	0.623403i	$-0.785750\pi$
$701$	−29024.1			−1.56380			−0.781901	+	0.623403i	$0.785750\pi$
$702$		0			0
$703$	18462.3	+	31977.6i	0.990496	+	1.71559i
$704$	1268.44	+	2197.00i	0.0679063	+	0.117617i
$705$		0			0
$706$	−22095.5			−1.17787
$707$	−8157.03	−	1257.60i	−0.433913	−	0.0668980i
$708$		0			0
$709$	33.0540	−	57.2511i	0.00175087	−	0.00303260i	−0.865149	−	0.501516i	$-0.832776\pi$
$709$	33.0540	−	57.2511i	0.00175087	−	0.00303260i	0.866900	+	0.498483i	$0.166109\pi$
$710$	−2946.33	−	5103.19i	−0.155738	−	0.269746i
$711$		0			0
$712$	−1141.60	+	1977.31i	−0.0600888	+	0.104077i
$713$	−47056.8			−2.47166
$714$		0			0
$715$	6584.81			0.344417
$716$	751.268	−	1301.23i	0.0392126	−	0.0679182i
$717$		0			0
$718$	7563.64	+	13100.6i	0.393137	+	0.680934i
$719$	−14833.0	+	25691.4i	−0.769368	+	1.33259i	0.168537	+	0.985695i	$0.446096\pi$
$719$	−14833.0	+	25691.4i	−0.769368	+	1.33259i	−0.937906	+	0.346890i	$0.887238\pi$
$720$		0			0
$721$	5676.30	+	14607.5i	0.293199	+	0.754525i
$722$	22800.5			1.17527
$723$		0			0
$724$	−4402.58	−	7625.49i	−0.225995	−	0.391435i
$725$	−6811.74	−	11798.3i	−0.348940	−	0.604382i
$726$		0			0
$727$	23619.7			1.20496			0.602481	−	0.798133i	$-0.294179\pi$
$727$	23619.7			1.20496			0.602481	+	0.798133i	$0.294179\pi$
$728$	−3477.51	+	4330.64i	−0.177040	+	0.220473i
$729$		0			0
$730$	4175.55	−	7232.27i	0.211704	−	0.366683i
$731$	−3264.10	−	5653.59i	−0.165154	−	0.286054i
$732$		0			0
$733$	−10767.9	+	18650.5i	−0.542593	+	0.939798i	0.456161	+	0.889897i	$0.349224\pi$
$733$	−10767.9	+	18650.5i	−0.542593	+	0.939798i	−0.998754	+	0.0499012i	$0.984109\pi$
$734$	−6864.15			−0.345177
$735$		0			0
$736$	6156.73			0.308342
$737$	−13528.5	+	23432.0i	−0.676158	+	1.17114i
$738$		0			0
$739$	−5373.78	−	9307.66i	−0.267494	−	0.463312i	0.700720	−	0.713436i	$-0.252862\pi$
$739$	−5373.78	−	9307.66i	−0.267494	−	0.463312i	−0.968214	+	0.250124i	$0.919529\pi$
$740$	2421.89	−	4194.84i	0.120311	−	0.208386i
$741$		0			0
$742$	13957.4	−	17381.5i	0.690554	−	0.859966i
$743$	19106.9			0.943426			0.471713	−	0.881752i	$-0.343636\pi$
$743$	19106.9			0.943426			0.471713	+	0.881752i	$0.343636\pi$
$744$		0			0
$745$	−6012.03	−	10413.1i	−0.295656	−	0.512091i
$746$	5273.69	+	9134.29i	0.258825	+	0.448298i
$747$		0			0
$748$	−18078.8			−0.883724
$749$	3772.03	+	9707.03i	0.184015	+	0.473547i
$750$		0			0
$751$	2371.18	−	4107.00i	0.115214	−	0.199556i	−0.802651	−	0.596448i	$-0.796578\pi$
$751$	2371.18	−	4107.00i	0.115214	−	0.199556i	0.917865	+	0.396892i	$0.129911\pi$
$752$	−2759.93	−	4780.33i	−0.133835	−	0.231810i
$753$		0			0
$754$	4847.05	−	8395.34i	0.234110	−	0.405491i
$755$	737.543			0.0355522
$756$		0			0
$757$	−3717.28			−0.178476			−0.0892382	−	0.996010i	$-0.528443\pi$
$757$	−3717.28			−0.178476			−0.0892382	+	0.996010i	$0.528443\pi$
$758$	−8856.92	+	15340.6i	−0.424403	+	0.735088i
$759$		0			0
$760$	−2395.26	−	4148.71i	−0.114323	−	0.198013i
$761$	−2047.43	+	3546.26i	−0.0975288	+	0.168925i	−0.910661	−	0.413154i	$-0.864427\pi$
$761$	−2047.43	+	3546.26i	−0.0975288	+	0.168925i	0.813132	+	0.582079i	$0.197760\pi$
$762$		0			0
$763$	−20420.6	−	3148.32i	−0.968905	−	0.149380i
$764$	−17413.1			−0.824587
$765$		0			0
$766$	1041.70	+	1804.28i	0.0491361	+	0.0851063i
$767$	−590.900	−	1023.47i	−0.0278177	−	0.0481816i
$768$		0			0
$769$	16871.2			0.791145			0.395573	−	0.918435i	$-0.370546\pi$
$769$	16871.2			0.791145			0.395573	+	0.918435i	$0.370546\pi$
$770$	−2356.67	−	6064.70i	−0.110297	−	0.283840i
$771$		0			0
$772$	4452.84	−	7712.54i	0.207592	−	0.359560i
$773$	−14015.1	−	24274.9i	−0.652121	−	1.12951i	−0.982607	−	0.185696i	$-0.940546\pi$
$773$	−14015.1	−	24274.9i	−0.652121	−	1.12951i	0.330487	−	0.943811i	$-0.392787\pi$
$774$		0			0
$775$	−12884.7	+	22317.0i	−0.597205	+	1.03439i
$776$	1922.47			0.0889339
$777$		0			0
$778$	18665.4			0.860135
$779$	−24690.9	+	42766.0i	−1.13562	+	1.96694i
$780$		0			0
$781$	−13177.1	−	22823.4i	−0.603730	−	1.04569i
$782$	−21937.6	+	37997.1i	−1.00318	+	1.73756i
$783$		0			0
$784$	5233.16	+	1652.92i	0.238391	+	0.0752971i
$785$	2373.39			0.107911
$786$		0			0
$787$	−2603.42	−	4509.26i	−0.117919	−	0.204241i	0.801024	−	0.598632i	$-0.204289\pi$
$787$	−2603.42	−	4509.26i	−0.117919	−	0.204241i	−0.918943	+	0.394391i	$0.870956\pi$
$788$	−7492.78	−	12977.9i	−0.338730	−	0.586698i
$789$		0			0
$790$	29.4732			0.00132736
$791$	17606.2	−	21925.5i	0.791410	−	0.985566i
$792$		0			0
$793$	−16166.3	+	28000.9i	−0.723939	+	1.25390i
$794$	−6822.33	−	11816.6i	−0.304931	−	0.528156i
$795$		0			0
$796$	5728.08	−	9921.32i	0.255058	−	0.441774i
$797$	4553.19			0.202362			0.101181	−	0.994868i	$-0.467738\pi$
$797$	4553.19			0.202362			0.101181	+	0.994868i	$0.467738\pi$
$798$		0			0
$799$	39336.7			1.74172
$800$	1685.79	−	2919.87i	0.0745020	−	0.129041i
$801$		0			0
$802$	8109.95	+	14046.8i	0.357073	+	0.618468i
$803$	18674.7	−	32345.4i	0.820690	−	1.42148i
$804$		0			0
$805$	−15606.2	−	2406.06i	−0.683287	−	0.105345i
$806$	−18336.9			−0.801352
$807$		0			0
$808$	1782.57	+	3087.50i	0.0776120	+	0.134428i
$809$	17523.7	+	30351.9i	0.761557	+	1.31906i	0.942048	+	0.335479i	$0.108898\pi$
$809$	17523.7	+	30351.9i	0.761557	+	1.31906i	−0.180490	+	0.983577i	$0.557768\pi$
$810$		0			0
$811$	36540.6			1.58214			0.791069	−	0.611727i	$-0.209525\pi$
$811$	36540.6			1.58214			0.791069	+	0.611727i	$0.209525\pi$
$812$	−9466.96	−	1459.56i	−0.409144	−	0.0630793i
$813$		0			0
$814$	10831.6	−	18760.9i	0.466398	−	0.807824i
$815$	2665.01	+	4615.93i	0.114541	+	0.198391i
$816$		0			0
$817$	3868.26	−	6700.02i	0.165647	−	0.286908i
$818$	8866.08			0.378967
$819$		0			0
$820$	6477.94			0.275877
$821$	4052.84	−	7019.73i	0.172284	−	0.298405i	−0.766934	−	0.641726i	$-0.778219\pi$
$821$	4052.84	−	7019.73i	0.172284	−	0.298405i	0.939218	+	0.343321i	$0.111552\pi$
$822$		0			0
$823$	−8980.10	−	15554.0i	−0.380348	−	0.658783i	0.610764	−	0.791813i	$-0.290863\pi$
$823$	−8980.10	−	15554.0i	−0.380348	−	0.658783i	−0.991112	+	0.133030i	$0.957529\pi$
$824$	3384.76	−	5862.57i	0.143099	−	0.247855i
$825$		0			0
$826$	−731.149	+	910.521i	−0.0307990	+	0.0383548i
$827$	527.061			0.0221617			0.0110808	−	0.999939i	$-0.496473\pi$
$827$	527.061			0.0221617			0.0110808	+	0.999939i	$0.496473\pi$
$828$		0			0
$829$	−14390.3	−	24924.7i	−0.602888	−	1.04423i	−0.992381	−	0.123204i	$-0.960683\pi$
$829$	−14390.3	−	24924.7i	−0.602888	−	1.04423i	0.389493	−	0.921029i	$-0.372650\pi$
$830$	239.700	+	415.172i	0.0100242	+	0.0173625i
$831$		0			0
$832$	2399.13			0.0999696
$833$	−28848.0	+	26407.5i	−1.19991	+	1.09840i
$834$		0			0
$835$	−7682.31	+	13306.1i	−0.318392	+	0.551471i
$836$	−10712.5	−	18554.6i	−0.443181	−	0.767613i
$837$		0			0
$838$	−6782.36	+	11747.4i	−0.279585	+	0.484256i
$839$	2811.89			0.115706			0.0578530	−	0.998325i	$-0.481575\pi$
$839$	2811.89			0.115706			0.0578530	+	0.998325i	$0.481575\pi$
$840$		0			0
$841$	−7670.04			−0.314488
$842$	4798.62	−	8311.46i	0.196403	−	0.340180i
$843$		0			0
$844$	−7458.25	−	12918.1i	−0.304175	−	0.526846i
$845$	−1754.37	+	3038.67i	−0.0714228	+	0.123708i
$846$		0			0
$847$	−1611.45	−	4146.94i	−0.0653719	−	0.168230i
$848$	−9629.15			−0.389937
$849$		0			0
$850$	12013.6	+	20808.2i	0.484780	+	0.839664i
$851$	−26287.2	−	45530.7i	−1.05889	−	1.83404i
$852$		0			0
$853$	−26873.5			−1.07870			−0.539351	−	0.842081i	$-0.681330\pi$
$853$	−26873.5			−1.07870			−0.539351	+	0.842081i	$0.681330\pi$
$854$	31575.1	+	4868.05i	1.26520	+	0.195060i
$855$		0			0
$856$	2249.25	−	3895.81i	0.0898103	−	0.155556i
$857$	10187.0	+	17644.4i	0.406045	+	0.703290i	0.994442	−	0.105282i	$-0.0335744\pi$
$857$	10187.0	+	17644.4i	0.406045	+	0.703290i	−0.588398	+	0.808572i	$0.700241\pi$
$858$		0			0
$859$	−19982.8	+	34611.3i	−0.793720	+	1.37476i	0.129929	+	0.991523i	$0.458525\pi$
$859$	−19982.8	+	34611.3i	−0.793720	+	1.37476i	−0.923649	+	0.383240i	$0.874808\pi$
$860$	−1014.88			−0.0402408
$861$		0			0
$862$	−25898.6			−1.02333
$863$	−21047.6	+	36455.6i	−0.830208	+	1.43796i	0.0676645	+	0.997708i	$0.478445\pi$
$863$	−21047.6	+	36455.6i	−0.830208	+	1.43796i	−0.897873	+	0.440255i	$0.854888\pi$
$864$		0			0
$865$	−922.179	−	1597.26i	−0.0362486	−	0.0627844i
$866$	−937.240	+	1623.35i	−0.0367768	+	0.0636993i
$867$		0			0
$868$	6562.67	+	16888.5i	0.256626	+	0.660408i
$869$	131.815			0.00514561
$870$		0			0
$871$	12793.9	+	22159.7i	0.497710	+	0.862058i
$872$	4462.54	+	7729.35i	0.173304	+	0.300171i
$873$		0			0
$874$	−51996.2			−2.01235
$875$	−11837.7	+	14741.8i	−0.457356	+	0.569559i
$876$		0			0
$877$	−6378.94	+	11048.6i	−0.245612	+	0.425412i	−0.962303	−	0.271978i	$-0.912322\pi$
$877$	−6378.94	+	11048.6i	−0.245612	+	0.425412i	0.716692	+	0.697390i	$0.245655\pi$
$878$	13600.3	+	23556.4i	0.522765	+	0.905456i
$879$		0			0
$880$	−1405.27	+	2434.00i	−0.0538314	+	0.0932388i
$881$	−21457.2			−0.820557			−0.410278	−	0.911960i	$-0.634568\pi$
$881$	−21457.2			−0.820557			−0.410278	+	0.911960i	$0.634568\pi$
$882$		0			0
$883$	20778.9			0.791920			0.395960	−	0.918268i	$-0.370412\pi$
$883$	20778.9			0.791920			0.395960	+	0.918268i	$0.370412\pi$
$884$	−8548.56	+	14806.5i	−0.325248	+	0.563346i
$885$		0			0
$886$	4632.47	+	8023.68i	0.175656	+	0.304245i
$887$	8337.09	−	14440.3i	0.315594	−	0.546625i	−0.663969	−	0.747760i	$-0.731130\pi$
$887$	8337.09	−	14440.3i	0.315594	−	0.546625i	0.979564	+	0.201135i	$0.0644629\pi$
$888$		0			0
$889$	9312.43	−	11597.0i	0.351326	−	0.437517i
$890$	−2529.50			−0.0952685
$891$		0			0
$892$	−305.780	−	529.626i	−0.0114779	−	0.0198803i
$893$	23308.8	+	40372.0i	0.873458	+	1.51287i
$894$		0			0
$895$	1664.62			0.0621700
$896$	−858.634	−	2209.63i	−0.0320145	−	0.0823867i
$897$		0			0
$898$	−254.426	+	440.679i	−0.00945468	+	0.0163760i
$899$	−15812.4	−	27387.8i	−0.586621	−	1.01606i
$900$		0			0
$901$	34310.6	−	59427.7i	1.26865	−	2.19736i
$902$	28971.8			1.06946
$903$		0			0
$904$	−12146.5			−0.446888
$905$	4877.51	−	8448.09i	0.179153	−	0.310303i
$906$		0			0
$907$	−9844.63	−	17051.4i	−0.360403	−	0.624237i	0.627624	−	0.778517i	$-0.284028\pi$
$907$	−9844.63	−	17051.4i	−0.360403	−	0.624237i	−0.988027	+	0.154280i	$0.950694\pi$
$908$	6772.16	−	11729.7i	0.247513	−	0.428706i
$909$		0			0
$910$	−6081.35	−	937.584i	−0.221533	−	0.0341545i
$911$	−110.121			−0.00400492			−0.00200246	−	0.999998i	$-0.500637\pi$
$911$	−110.121			−0.00400492			−0.00200246	+	0.999998i	$0.500637\pi$
$912$		0			0
$913$	1072.03	+	1856.81i	0.0388597	+	0.0673071i
$914$	−16410.5	−	28423.8i	−0.593885	−	1.02864i
$915$		0			0
$916$	−3089.42			−0.111438
$917$	5971.62	+	15367.5i	0.215049	+	0.553413i
$918$		0			0
$919$	−57.8619	+	100.220i	−0.00207692	+	0.00359733i	−0.867062	−	0.498200i	$-0.833994\pi$
$919$	−57.8619	+	100.220i	−0.00207692	+	0.00359733i	0.864985	+	0.501798i	$0.167328\pi$
$920$	3410.44	+	5907.06i	0.122216	+	0.211685i
$921$		0			0
$922$	5842.56	−	10119.6i	0.208693	−	0.361466i
$923$	−24923.2			−0.888793
$924$		0			0
$925$	−28791.0			−1.02340
$926$	−6699.12	+	11603.2i	−0.237740	+	0.411777i
$927$		0			0
$928$	2068.83	+	3583.32i	0.0731817	+	0.126754i
$929$	9408.07	−	16295.3i	0.332259	−	0.575490i	−0.650695	−	0.759339i	$-0.725522\pi$
$929$	9408.07	−	16295.3i	0.332259	−	0.575490i	0.982954	+	0.183849i	$0.0588558\pi$
$930$		0			0
$931$	−44196.3	−	13959.6i	−1.55583	−	0.491416i
$932$	14255.4			0.501021
$933$		0			0
$934$	1825.63	+	3162.09i	0.0639578	+	0.110778i
$935$	−10014.5	−	17345.7i	−0.350278	−	0.606699i
$936$		0			0
$937$	6981.55			0.243412			0.121706	−	0.992566i	$-0.461163\pi$
$937$	6981.55			0.243412			0.121706	+	0.992566i	$0.461163\pi$
$938$	15830.5	−	19714.2i	0.551050	−	0.686239i
$939$		0			0
$940$	3057.65	−	5296.01i	0.106095	−	0.183763i
$941$	−14876.6	−	25767.1i	−0.515372	−	0.892650i	−0.999841	−	0.0178417i	$-0.994321\pi$
$941$	−14876.6	−	25767.1i	−0.515372	−	0.892650i	0.484469	−	0.874808i	$-0.339013\pi$
$942$		0			0
$943$	35155.7	−	60891.5i	1.21403	−	2.10276i
$944$	504.418			0.0173913
$945$		0			0
$946$	−4538.93			−0.155997
$947$	−13793.6	+	23891.2i	−0.473317	+	0.819809i	−0.999533	−	0.0305416i	$-0.990277\pi$
$947$	−13793.6	+	23891.2i	−0.473317	+	0.819809i	0.526217	+	0.850351i	$0.323610\pi$
$948$		0			0
$949$	−17660.6	−	30589.1i	−0.604098	−	1.04633i
$950$	−14237.2	+	24659.6i	−0.486227	+	0.842171i
$951$		0			0
$952$	16696.5	+	2574.16i	0.568422	+	0.0876357i
$953$	53424.4			1.81593			0.907967	−	0.419042i	$-0.137634\pi$
$953$	53424.4			1.81593			0.907967	+	0.419042i	$0.137634\pi$
$954$		0			0
$955$	−9645.79	−	16707.0i	−0.326838	−	0.566100i
$956$	11665.8	+	20205.8i	0.394666	+	0.683581i
$957$		0			0
$958$	22072.3			0.744388
$959$	−37423.7	−	5769.75i	−1.26014	−	0.194281i
$960$		0			0
$961$	−15014.4	+	26005.8i	−0.503992	+	0.872940i
$962$	−10243.5	−	17742.2i	−0.343308	−	0.594627i
$963$		0			0
$964$	−7386.43	+	12793.7i	−0.246785	+	0.427445i
$965$	9866.38			0.329130
$966$		0			0
$967$	−55913.3			−1.85941			−0.929705	−	0.368305i	$-0.879938\pi$
$967$	−55913.3			−1.85941			−0.929705	+	0.368305i	$0.879938\pi$
$968$	−960.900	+	1664.33i	−0.0319055	+	0.0552619i
$969$		0			0
$970$	1064.93	+	1844.51i	0.0352503	+	0.0610553i
$971$	−1286.67	+	2228.58i	−0.0425244	+	0.0736544i	−0.886504	−	0.462720i	$-0.846873\pi$
$971$	−1286.67	+	2228.58i	−0.0425244	+	0.0736544i	0.843980	+	0.536375i	$0.180207\pi$
$972$		0			0
$973$	−6812.27	+	8483.51i	−0.224451	+	0.279516i
$974$	16371.1			0.538567
$975$		0			0
$976$	−6900.15	−	11951.4i	−0.226300	−	0.391963i
$977$	−13958.3	−	24176.5i	−0.457078	−	0.791683i	0.541727	−	0.840555i	$-0.317771\pi$
$977$	−13958.3	−	24176.5i	−0.457078	−	0.791683i	−0.998805	+	0.0488718i	$0.984437\pi$
$978$		0			0
$979$	−11312.9			−0.369316
$980$	1312.95	+	5936.56i	0.0427967	+	0.193507i
$981$		0			0
$982$	−19006.2	+	32919.7i	−0.617629	+	1.06976i
$983$	19945.8	+	34547.2i	0.647175	+	1.12094i	0.983794	+	0.179300i	$0.0573832\pi$
$983$	19945.8	+	34547.2i	0.647175	+	1.12094i	−0.336619	+	0.941641i	$0.609283\pi$
$984$		0			0
$985$	8301.07	−	14377.9i	0.268522	−	0.465094i
$986$	−29486.6			−0.952378
$987$		0			0
$988$	−20261.6			−0.652438
$989$	−5507.74	+	9539.69i	−0.177084	+	0.306719i
$990$		0			0
$991$	3506.45	+	6073.36i	0.112398	+	0.194679i	0.916737	−	0.399492i	$-0.130814\pi$
$991$	3506.45	+	6073.36i	0.112398	+	0.194679i	−0.804339	+	0.594171i	$0.797480\pi$
$992$	3913.30	−	6778.03i	0.125249	−	0.216938i
$993$		0			0
$994$	8919.87	+	22954.6i	0.284629	+	0.732470i
$995$	12692.0			0.404385
$996$		0			0
$997$	8436.08	+	14611.7i	0.267977	+	0.464150i	0.968339	−	0.249638i	$-0.0803115\pi$
$997$	8436.08	+	14611.7i	0.267977	+	0.464150i	−0.700362	+	0.713788i	$0.746978\pi$
$998$	−16493.6	−	28567.7i	−0.523142	−	0.906108i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.d.109.3	✓	8	7.4	even	3	inner
378.4.g.d.163.3	yes	8	1.1	even	1	trivial
378.4.g.e.109.2	yes	8	21.11	odd	6
378.4.g.e.163.2	yes	8	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.21575 - 3.83779i) q^{5} +(-11.5959 + 14.4407i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.21575 - 3.83779i) q^{5} +(-11.5959 + 14.4407i) q^{7} +8.00000 q^{8} +(4.43150 + 7.67558i) q^{10} +(19.8193 + 34.3281i) q^{11} +37.4863 q^{13} +(-13.4162 - 34.5255i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-57.0112 - 98.7463i) q^{17} +(67.5634 - 117.023i) q^{19} -17.7260 q^{20} -79.2774 q^{22} +(-96.1989 + 166.621i) q^{23} +(52.6809 + 91.2460i) q^{25} +(-37.4863 + 64.9282i) q^{26} +(73.2160 + 11.2880i) q^{28} -129.302 q^{29} +(122.291 + 211.813i) q^{31} +(-16.0000 - 27.7128i) q^{32} +228.045 q^{34} +(29.7269 + 76.4998i) q^{35} +(-136.629 + 236.649i) q^{37} +(135.127 + 234.047i) q^{38} +(17.7260 - 30.7023i) q^{40} -365.448 q^{41} +57.2537 q^{43} +(79.2774 - 137.312i) q^{44} +(-192.398 - 333.243i) q^{46} +(-172.495 + 298.771i) q^{47} +(-74.0693 - 334.907i) q^{49} -210.724 q^{50} +(-74.9727 - 129.856i) q^{52} +(300.911 + 521.193i) q^{53} +175.659 q^{55} +(-92.7674 + 115.526i) q^{56} +(129.302 - 223.957i) q^{58} +(-15.7631 - 27.3024i) q^{59} +(-431.259 + 746.963i) q^{61} -489.162 q^{62} +64.0000 q^{64} +(83.0604 - 143.865i) q^{65} +(341.295 + 591.141i) q^{67} +(-228.045 + 394.985i) q^{68} +(-162.228 - 25.0114i) q^{70} -664.860 q^{71} +(-471.122 - 816.007i) q^{73} +(-273.259 - 473.298i) q^{74} -540.507 q^{76} +(-725.546 - 111.860i) q^{77} +(1.66271 - 2.87990i) q^{79} +(35.4520 + 61.4047i) q^{80} +(365.448 - 632.975i) q^{82} +54.0900 q^{83} -505.290 q^{85} +(-57.2537 + 99.1664i) q^{86} +(158.555 + 274.625i) q^{88} +(-142.700 + 247.163i) q^{89} +(-434.689 + 541.330i) q^{91} +769.591 q^{92} +(-344.991 - 597.542i) q^{94} +(-299.407 - 518.589i) q^{95} +240.309 q^{97} +(654.145 + 206.615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 16 q^{4} + 4 q^{5} + 25 q^{7} + 64 q^{8}+O(q^{10}) \) 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