LMFDB - Embedded newform 378.4.g.f.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} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 $ x^8 - x^7 + 18x^6 + 9x^5 + 283x^4 - 48x^3 + 186x^2 + 40x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.3
Root			$2.24123 - 3.88192i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.f.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.59791 - 9.69587i) q^{5} +(-16.7643 + 7.87127i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.59791 - 9.69587i) q^{5} +(-16.7643 + 7.87127i) q^{7} -8.00000 q^{8} +(-11.1958 - 19.3917i) q^{10} +(-9.69889 - 16.7990i) q^{11} -39.5348 q^{13} +(-3.13090 + 36.9080i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(17.9246 + 31.0464i) q^{17} +(-28.0911 + 48.6552i) q^{19} -44.7833 q^{20} -38.7956 q^{22} +(-20.4926 + 35.4943i) q^{23} +(-0.173272 - 0.300115i) q^{25} +(-39.5348 + 68.4763i) q^{26} +(60.7956 + 42.3308i) q^{28} -179.308 q^{29} +(-111.582 - 193.266i) q^{31} +(16.0000 + 27.7128i) q^{32} +71.6986 q^{34} +(-17.5265 + 206.608i) q^{35} +(-13.6130 + 23.5785i) q^{37} +(56.1821 + 97.3103i) q^{38} +(-44.7833 + 77.5670i) q^{40} +100.096 q^{41} -61.7538 q^{43} +(-38.7956 + 67.1959i) q^{44} +(40.9853 + 70.9886i) q^{46} +(-231.260 + 400.554i) q^{47} +(219.086 - 263.913i) q^{49} -0.693086 q^{50} +(79.0696 + 136.953i) q^{52} +(174.811 + 302.782i) q^{53} -217.174 q^{55} +(134.115 - 62.9702i) q^{56} +(-179.308 + 310.571i) q^{58} +(121.964 + 211.248i) q^{59} +(-32.5638 + 56.4022i) q^{61} -446.328 q^{62} +64.0000 q^{64} +(-221.312 + 383.324i) q^{65} +(-85.4760 - 148.049i) q^{67} +(71.6986 - 124.186i) q^{68} +(340.328 + 236.964i) q^{70} -1009.91 q^{71} +(-573.557 - 993.429i) q^{73} +(27.2261 + 47.1570i) q^{74} +224.729 q^{76} +(294.825 + 205.281i) q^{77} +(-340.780 + 590.248i) q^{79} +(89.5666 + 155.134i) q^{80} +(100.096 - 173.371i) q^{82} -908.609 q^{83} +401.362 q^{85} +(-61.7538 + 106.961i) q^{86} +(77.5911 + 134.392i) q^{88} +(-315.344 + 546.192i) q^{89} +(662.775 - 311.189i) q^{91} +163.941 q^{92} +(462.519 + 801.107i) q^{94} +(314.503 + 544.735i) q^{95} +1330.32 q^{97} +(-238.025 - 643.382i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - 2 * q^5 + 6 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8} + 4 q^{10} + 32 q^{11} - 4 q^{13} + 36 q^{14} - 64 q^{16} - 58 q^{17} + 70 q^{19} + 16 q^{20} + 128 q^{22} + 86 q^{23} - 156 q^{25} - 4 q^{26} + 48 q^{28} - 212 q^{29} - 64 q^{31} + 128 q^{32} - 232 q^{34} + 8 q^{35} - 146 q^{37} - 140 q^{38} + 16 q^{40} - 780 q^{41} + 880 q^{43} + 128 q^{44} - 172 q^{46} - 306 q^{47} + 50 q^{49} - 624 q^{50} + 8 q^{52} - 90 q^{53} - 64 q^{55} - 48 q^{56} - 212 q^{58} + 148 q^{59} - 364 q^{61} - 256 q^{62} + 512 q^{64} + 1296 q^{65} - 954 q^{67} - 232 q^{68} + 20 q^{70} - 1360 q^{71} - 54 q^{73} + 292 q^{74} - 560 q^{76} + 2224 q^{77} - 226 q^{79} - 32 q^{80} - 780 q^{82} - 3136 q^{83} + 3920 q^{85} + 880 q^{86} - 256 q^{88} - 1458 q^{89} + 3836 q^{91} - 688 q^{92} + 612 q^{94} + 1310 q^{95} - 4344 q^{97} + 776 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - 2 * q^5 + 6 * q^7 - 64 * q^8 + 4 * q^10 + 32 * q^11 - 4 * q^13 + 36 * q^14 - 64 * q^16 - 58 * q^17 + 70 * q^19 + 16 * q^20 + 128 * q^22 + 86 * q^23 - 156 * q^25 - 4 * q^26 + 48 * q^28 - 212 * q^29 - 64 * q^31 + 128 * q^32 - 232 * q^34 + 8 * q^35 - 146 * q^37 - 140 * q^38 + 16 * q^40 - 780 * q^41 + 880 * q^43 + 128 * q^44 - 172 * q^46 - 306 * q^47 + 50 * q^49 - 624 * q^50 + 8 * q^52 - 90 * q^53 - 64 * q^55 - 48 * q^56 - 212 * q^58 + 148 * q^59 - 364 * q^61 - 256 * q^62 + 512 * q^64 + 1296 * q^65 - 954 * q^67 - 232 * q^68 + 20 * q^70 - 1360 * q^71 - 54 * q^73 + 292 * q^74 - 560 * q^76 + 2224 * q^77 - 226 * q^79 - 32 * q^80 - 780 * q^82 - 3136 * q^83 + 3920 * q^85 + 880 * q^86 - 256 * q^88 - 1458 * q^89 + 3836 * q^91 - 688 * q^92 + 612 * q^94 + 1310 * q^95 - 4344 * q^97 + 776 * 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
$5$	5.59791	−	9.69587i	0.500693	−	0.867225i	−0.499307	−	0.866425i	$-0.666412\pi$
$5$	5.59791	−	9.69587i	0.500693	−	0.867225i	1.00000		0.000799938i	$-0.000254628\pi$
$6$		0			0
$7$	−16.7643	+	7.87127i	−0.905189	+	0.425009i
$7$	−16.7643	+	7.87127i	−0.905189	+	0.425009i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−11.1958	−	19.3917i	−0.354043	−	0.613221i
$11$	−9.69889	−	16.7990i	−0.265848	−	0.460462i	0.701938	−	0.712238i	$-0.252319\pi$
$11$	−9.69889	−	16.7990i	−0.265848	−	0.460462i	−0.967785	+	0.251777i	$0.918985\pi$
$12$		0			0
$13$	−39.5348			−0.843460			−0.421730	−	0.906721i	$-0.638577\pi$
$13$	−39.5348			−0.843460			−0.421730	+	0.906721i	$0.638577\pi$
$14$	−3.13090	+	36.9080i	−0.0597692	+	0.704576i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	17.9246	+	31.0464i	0.255727	+	0.442933i	0.965093	−	0.261908i	$-0.0843517\pi$
$17$	17.9246	+	31.0464i	0.255727	+	0.442933i	−0.709366	+	0.704841i	$0.751018\pi$
$18$		0			0
$19$	−28.0911	+	48.6552i	−0.339186	+	0.587487i	−0.984280	−	0.176616i	$-0.943485\pi$
$19$	−28.0911	+	48.6552i	−0.339186	+	0.587487i	0.645094	+	0.764103i	$0.276818\pi$
$20$	−44.7833			−0.500693
$21$		0			0
$22$	−38.7956			−0.375966
$23$	−20.4926	+	35.4943i	−0.185783	+	0.321786i	−0.943840	−	0.330403i	$-0.892815\pi$
$23$	−20.4926	+	35.4943i	−0.185783	+	0.321786i	0.758057	+	0.652188i	$0.226149\pi$
$24$		0			0
$25$	−0.173272	−	0.300115i	−0.00138617	−	0.00240092i
$26$	−39.5348	+	68.4763i	−0.298208	+	0.516512i
$27$		0			0
$28$	60.7956	+	42.3308i	0.410331	+	0.285706i
$29$	−179.308			−1.14816			−0.574082	−	0.818798i	$-0.694641\pi$
$29$	−179.308			−1.14816			−0.574082	+	0.818798i	$0.694641\pi$
$30$		0			0
$31$	−111.582	−	193.266i	−0.646474	−	1.11973i	−0.983959	−	0.178395i	$-0.942909\pi$
$31$	−111.582	−	193.266i	−0.646474	−	1.11973i	0.337484	−	0.941331i	$-0.390424\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	71.6986			0.361653
$35$	−17.5265	+	206.608i	−0.0846435	+	0.997801i
$36$		0			0
$37$	−13.6130	+	23.5785i	−0.0604857	+	0.104764i	−0.894683	−	0.446702i	$-0.852598\pi$
$37$	−13.6130	+	23.5785i	−0.0604857	+	0.104764i	0.834197	+	0.551467i	$0.185932\pi$
$38$	56.1821	+	97.3103i	0.239841	+	0.415416i
$39$		0			0
$40$	−44.7833	+	77.5670i	−0.177022	+	0.306610i
$41$	100.096			0.381277			0.190639	−	0.981660i	$-0.438944\pi$
$41$	100.096			0.381277			0.190639	+	0.981660i	$0.438944\pi$
$42$		0			0
$43$	−61.7538			−0.219009			−0.109504	−	0.993986i	$-0.534926\pi$
$43$	−61.7538			−0.219009			−0.109504	+	0.993986i	$0.534926\pi$
$44$	−38.7956	+	67.1959i	−0.132924	+	0.230231i
$45$		0			0
$46$	40.9853	+	70.9886i	0.131368	+	0.227537i
$47$	−231.260	+	400.554i	−0.717717	+	1.24312i	0.244185	+	0.969729i	$0.421479\pi$
$47$	−231.260	+	400.554i	−0.717717	+	1.24312i	−0.961902	+	0.273394i	$0.911854\pi$
$48$		0			0
$49$	219.086	−	263.913i	0.638735	−	0.769427i
$50$	−0.693086			−0.00196034
$51$		0			0
$52$	79.0696	+	136.953i	0.210865	+	0.365229i
$53$	174.811	+	302.782i	0.453060	+	0.784723i	0.998574	−	0.0533783i	$-0.0169989\pi$
$53$	174.811	+	302.782i	0.453060	+	0.784723i	−0.545514	+	0.838102i	$0.683666\pi$
$54$		0			0
$55$	−217.174			−0.532432
$56$	134.115	−	62.9702i	0.320033	−	0.150263i
$57$		0			0
$58$	−179.308	+	310.571i	−0.405937	+	0.703104i
$59$	121.964	+	211.248i	0.269125	+	0.466139i	0.968636	−	0.248483i	$-0.0799320\pi$
$59$	121.964	+	211.248i	0.269125	+	0.466139i	−0.699511	+	0.714622i	$0.746599\pi$
$60$		0			0
$61$	−32.5638	+	56.4022i	−0.0683503	+	0.118386i	−0.898175	−	0.439638i	$-0.855107\pi$
$61$	−32.5638	+	56.4022i	−0.0683503	+	0.118386i	0.829825	+	0.558024i	$0.188440\pi$
$62$	−446.328			−0.914253
$63$		0			0
$64$	64.0000			0.125000
$65$	−221.312	+	383.324i	−0.422314	+	0.731470i
$66$		0			0
$67$	−85.4760	−	148.049i	−0.155859	−	0.269956i	0.777513	−	0.628867i	$-0.216481\pi$
$67$	−85.4760	−	148.049i	−0.155859	−	0.269956i	−0.933371	+	0.358912i	$0.883148\pi$
$68$	71.6986	−	124.186i	0.127864	−	0.221466i
$69$		0			0
$70$	340.328	+	236.964i	0.581100	+	0.404609i
$71$	−1009.91			−1.68809			−0.844045	−	0.536272i	$-0.819832\pi$
$71$	−1009.91			−1.68809			−0.844045	+	0.536272i	$0.819832\pi$
$72$		0			0
$73$	−573.557	−	993.429i	−0.919585	−	1.59277i	−0.800046	−	0.599939i	$-0.795192\pi$
$73$	−573.557	−	993.429i	−0.919585	−	1.59277i	−0.119539	−	0.992829i	$-0.538142\pi$
$74$	27.2261	+	47.1570i	0.0427699	+	0.0740796i
$75$		0			0
$76$	224.729			0.339186
$77$	294.825	+	205.281i	0.436343	+	0.303818i
$78$		0			0
$79$	−340.780	+	590.248i	−0.485325	+	0.840608i	−0.999858	−	0.0168627i	$-0.994632\pi$
$79$	−340.780	+	590.248i	−0.485325	+	0.840608i	0.514532	+	0.857471i	$0.327966\pi$
$80$	89.5666	+	155.134i	0.125173	+	0.216806i
$81$		0			0
$82$	100.096	−	173.371i	0.134802	−	0.233484i
$83$	−908.609			−1.20160			−0.600800	−	0.799400i	$-0.705151\pi$
$83$	−908.609			−1.20160			−0.600800	+	0.799400i	$0.705151\pi$
$84$		0			0
$85$	401.362			0.512163
$86$	−61.7538	+	106.961i	−0.0774312	+	0.134115i
$87$		0			0
$88$	77.5911	+	134.392i	0.0939914	+	0.162798i
$89$	−315.344	+	546.192i	−0.375578	+	0.650520i	−0.990413	−	0.138136i	$-0.955889\pi$
$89$	−315.344	+	546.192i	−0.375578	+	0.650520i	0.614836	+	0.788655i	$0.289222\pi$
$90$		0			0
$91$	662.775	−	311.189i	0.763491	−	0.358478i
$92$	163.941			0.185783
$93$		0			0
$94$	462.519	+	801.107i	0.507502	+	0.879020i
$95$	314.503	+	544.735i	0.339656	+	0.588301i
$96$		0			0
$97$	1330.32			1.39252			0.696258	−	0.717792i	$-0.254847\pi$
$97$	1330.32			1.39252			0.696258	+	0.717792i	$0.254847\pi$
$98$	−238.025	−	643.382i	−0.245349	−	0.663177i
$99$		0			0
$100$	−0.693086	+	1.20046i	−0.000693086	+	0.00120046i
$101$	−252.164	−	436.762i	−0.248429	−	0.430291i	0.714661	−	0.699471i	$-0.246581\pi$
$101$	−252.164	−	436.762i	−0.248429	−	0.430291i	−0.963090	+	0.269180i	$0.913247\pi$
$102$		0			0
$103$	−80.1333	+	138.795i	−0.0766580	+	0.132775i	−0.901806	−	0.432141i	$-0.857758\pi$
$103$	−80.1333	+	138.795i	−0.0766580	+	0.132775i	0.825148	+	0.564916i	$0.191092\pi$
$104$	316.278			0.298208
$105$		0			0
$106$	699.246			0.640724
$107$	984.295	−	1704.85i	0.889303	−	1.54032i	0.0486022	−	0.998818i	$-0.484523\pi$
$107$	984.295	−	1704.85i	0.889303	−	1.54032i	0.840701	−	0.541500i	$-0.182143\pi$
$108$		0			0
$109$	−577.764	−	1000.72i	−0.507704	−	0.879369i	−0.999960	−	0.00891861i	$-0.997161\pi$
$109$	−577.764	−	1000.72i	−0.507704	−	0.879369i	0.492256	−	0.870450i	$-0.336172\pi$
$110$	−217.174	+	376.157i	−0.188243	+	0.326047i
$111$		0			0
$112$	25.0472	−	295.264i	0.0211316	−	0.249105i
$113$	1441.21			1.19980			0.599901	−	0.800074i	$-0.295206\pi$
$113$	1441.21			1.19980			0.599901	+	0.800074i	$0.295206\pi$
$114$		0			0
$115$	229.432	+	397.388i	0.186040	+	0.322231i
$116$	358.617	+	621.143i	0.287041	+	0.497169i
$117$		0			0
$118$	487.857			0.380601
$119$	−544.869	−	379.383i	−0.419732	−	0.292252i
$120$		0			0
$121$	477.363	−	826.817i	0.358650	−	0.621200i
$122$	65.1276	+	112.804i	0.0483310	+	0.0837117i
$123$		0			0
$124$	−446.328	+	773.062i	−0.323237	+	0.559863i
$125$	1395.60			0.998609
$126$		0			0
$127$	−989.467			−0.691346			−0.345673	−	0.938355i	$-0.612349\pi$
$127$	−989.467			−0.691346			−0.345673	+	0.938355i	$0.612349\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	442.625	+	766.649i	0.298621	+	0.517227i
$131$	77.8773	−	134.887i	0.0519402	−	0.0899631i	−0.838886	−	0.544307i	$-0.816793\pi$
$131$	77.8773	−	134.887i	0.0519402	−	0.0899631i	0.890827	+	0.454344i	$0.150126\pi$
$132$		0			0
$133$	87.9504	−	1036.78i	0.0573403	−	0.675944i
$134$	−341.904			−0.220418
$135$		0			0
$136$	−143.397	−	248.371i	−0.0904133	−	0.156600i
$137$	−491.335	−	851.017i	−0.306406	−	0.530710i	0.671168	−	0.741306i	$-0.265793\pi$
$137$	−491.335	−	851.017i	−0.306406	−	0.530710i	−0.977573	+	0.210596i	$0.932460\pi$
$138$		0			0
$139$	−2232.74			−1.36243			−0.681216	−	0.732082i	$-0.738549\pi$
$139$	−2232.74			−1.36243			−0.681216	+	0.732082i	$0.738549\pi$
$140$	750.763	−	352.501i	0.453222	−	0.212799i
$141$		0			0
$142$	−1009.91	+	1749.22i	−0.596830	+	1.03374i
$143$	383.444	+	664.144i	0.224232	+	0.388381i
$144$		0			0
$145$	−1003.75	+	1738.55i	−0.574877	+	0.995716i
$146$	−2294.23			−1.30049
$147$		0			0
$148$	108.904			0.0604857
$149$	1535.08	−	2658.84i	0.844019	−	1.46188i	−0.0424511	−	0.999099i	$-0.513517\pi$
$149$	1535.08	−	2658.84i	0.844019	−	1.46188i	0.886470	−	0.462786i	$-0.153150\pi$
$150$		0			0
$151$	−1362.59	−	2360.07i	−0.734343	−	1.27192i	−0.955011	−	0.296571i	$-0.904157\pi$
$151$	−1362.59	−	2360.07i	−0.734343	−	1.27192i	0.220668	−	0.975349i	$-0.429176\pi$
$152$	224.729	−	389.241i	0.119920	−	0.207708i
$153$		0			0
$154$	650.382	−	305.370i	0.340320	−	0.159789i
$155$	−2498.50			−1.29474
$156$		0			0
$157$	−636.491	−	1102.43i	−0.323551	−	0.560407i	0.657667	−	0.753309i	$-0.271543\pi$
$157$	−636.491	−	1102.43i	−0.323551	−	0.560407i	−0.981218	+	0.192902i	$0.938210\pi$
$158$	681.559	+	1180.50i	0.343177	+	0.594400i
$159$		0			0
$160$	358.266			0.177022
$161$	64.1604	−	756.341i	0.0314071	−	0.370236i
$162$		0			0
$163$	141.011	−	244.238i	0.0677597	−	0.117363i	−0.830155	−	0.557532i	$-0.811748\pi$
$163$	141.011	−	244.238i	0.0677597	−	0.117363i	0.897915	+	0.440169i	$0.145082\pi$
$164$	−200.192	−	346.743i	−0.0953193	−	0.165098i
$165$		0			0
$166$	−908.609	+	1573.76i	−0.424830	+	0.735827i
$167$	4159.02			1.92715			0.963576	−	0.267435i	$-0.0861761\pi$
$167$	4159.02			1.92715			0.963576	+	0.267435i	$0.0861761\pi$
$168$		0			0
$169$	−633.999			−0.288575
$170$	401.362	−	695.180i	0.181077	−	0.313635i
$171$		0			0
$172$	123.508	+	213.922i	0.0547522	+	0.0948335i
$173$	−118.262	+	204.835i	−0.0519727	+	0.0900194i	−0.890841	−	0.454315i	$-0.849884\pi$
$173$	−118.262	+	204.835i	−0.0519727	+	0.0900194i	0.838869	+	0.544334i	$0.183218\pi$
$174$		0			0
$175$	5.26707	+	3.66737i	0.00227516	+	0.00158415i
$176$	310.364			0.132924
$177$		0			0
$178$	630.688	+	1092.38i	0.265574	+	0.459987i
$179$	−1580.82	−	2738.06i	−0.660088	−	1.14331i	−0.980592	−	0.196059i	$-0.937185\pi$
$179$	−1580.82	−	2738.06i	−0.660088	−	1.14331i	0.320504	−	0.947247i	$-0.396148\pi$
$180$		0			0
$181$	−3304.47			−1.35701			−0.678506	−	0.734595i	$-0.737372\pi$
$181$	−3304.47			−1.35701			−0.678506	+	0.734595i	$0.737372\pi$
$182$	123.780	−	1459.15i	0.0504129	−	0.594282i
$183$		0			0
$184$	163.941	−	283.954i	0.0656842	−	0.113768i
$185$	152.409	+	263.981i	0.0605695	+	0.104909i
$186$		0			0
$187$	347.698	−	602.231i	0.135969	−	0.235505i
$188$	1850.08			0.717717
$189$		0			0
$190$	1258.01			0.480346
$191$	1038.38	−	1798.53i	0.393374	−	0.681344i	−0.599518	−	0.800361i	$-0.704641\pi$
$191$	1038.38	−	1798.53i	0.393374	−	0.681344i	0.992892	+	0.119017i	$0.0379744\pi$
$192$		0			0
$193$	−1306.12	−	2262.27i	−0.487133	−	0.843739i	0.512758	−	0.858533i	$-0.328624\pi$
$193$	−1306.12	−	2262.27i	−0.487133	−	0.843739i	−0.999891	+	0.0147947i	$0.995291\pi$
$194$	1330.32	−	2304.19i	0.492329	−	0.852738i
$195$		0			0
$196$	−1352.39	−	231.110i	−0.492855	−	0.0842239i
$197$	−3418.62			−1.23638			−0.618189	−	0.786029i	$-0.712134\pi$
$197$	−3418.62			−1.23638			−0.618189	+	0.786029i	$0.712134\pi$
$198$		0			0
$199$	1812.08	+	3138.61i	0.645501	+	1.11804i	0.984185	+	0.177141i	$0.0566849\pi$
$199$	1812.08	+	3138.61i	0.645501	+	1.11804i	−0.338684	+	0.940900i	$0.609982\pi$
$200$	1.38617	+	2.40092i	0.000490086	+	0.000848854i
$201$		0			0
$202$	−1008.66			−0.351331
$203$	3005.99	−	1411.39i	1.03931	−	0.487979i
$204$		0			0
$205$	560.329	−	970.517i	0.190903	−	0.330653i
$206$	160.267	+	277.590i	0.0542054	+	0.0938864i
$207$		0			0
$208$	316.278	−	547.810i	0.105433	−	0.182614i
$209$	1089.81			0.360687
$210$		0			0
$211$	4087.38			1.33359			0.666794	−	0.745242i	$-0.267666\pi$
$211$	4087.38			1.33359			0.666794	+	0.745242i	$0.267666\pi$
$212$	699.246	−	1211.13i	0.226530	−	0.392362i
$213$		0			0
$214$	−1968.59	−	3409.70i	−0.628832	−	1.08917i
$215$	−345.693	+	598.757i	−0.109656	+	0.189930i
$216$		0			0
$217$	3391.84	+	2361.68i	1.06108	+	0.738807i
$218$	−2311.05			−0.718002
$219$		0			0
$220$	434.348	+	752.313i	0.133108	+	0.230550i
$221$	−708.647	−	1227.41i	−0.215696	−	0.373596i
$222$		0			0
$223$	−3393.50			−1.01904			−0.509520	−	0.860459i	$-0.670177\pi$
$223$	−3393.50			−1.01904			−0.509520	+	0.860459i	$0.670177\pi$
$224$	−486.364	−	338.647i	−0.145074	−	0.101012i
$225$		0			0
$226$	1441.21	−	2496.25i	0.424194	−	0.734726i
$227$	1276.99	+	2211.80i	0.373377	+	0.646707i	0.990083	−	0.140486i	$-0.0448666\pi$
$227$	1276.99	+	2211.80i	0.373377	+	0.646707i	−0.616706	+	0.787194i	$0.711533\pi$
$228$		0			0
$229$	351.347	−	608.550i	0.101387	−	0.175608i	−0.810869	−	0.585227i	$-0.801005\pi$
$229$	351.347	−	608.550i	0.101387	−	0.175608i	0.912256	+	0.409620i	$0.134339\pi$
$230$	917.728			0.263101
$231$		0			0
$232$	1434.47			0.405937
$233$	−205.573	+	356.063i	−0.0578006	+	0.100114i	−0.893478	−	0.449107i	$-0.851742\pi$
$233$	−205.573	+	356.063i	−0.0578006	+	0.100114i	0.835677	+	0.549221i	$0.185075\pi$
$234$		0			0
$235$	2589.14	+	4484.53i	0.718711	+	1.24484i
$236$	487.857	−	844.994i	0.134563	−	0.233070i
$237$		0			0
$238$	−1201.98	+	564.359i	−0.327364	+	0.153706i
$239$	−178.567			−0.0483286			−0.0241643	−	0.999708i	$-0.507692\pi$
$239$	−178.567			−0.0483286			−0.0241643	+	0.999708i	$0.507692\pi$
$240$		0			0
$241$	2030.95	+	3517.71i	0.542843	+	0.940232i	0.998739	+	0.0501987i	$0.0159854\pi$
$241$	2030.95	+	3517.71i	0.542843	+	0.940232i	−0.455896	+	0.890033i	$0.650681\pi$
$242$	−954.726	−	1653.63i	−0.253604	−	0.439255i
$243$		0			0
$244$	260.510			0.0683503
$245$	−1332.44	−	3601.60i	−0.347456	−	0.939173i
$246$		0			0
$247$	1110.57	−	1923.57i	0.286090	−	0.495522i
$248$	892.655	+	1546.12i	0.228563	+	0.395883i
$249$		0			0
$250$	1395.60	−	2417.25i	0.353062	−	0.611521i
$251$	−2040.70			−0.513178			−0.256589	−	0.966521i	$-0.582599\pi$
$251$	−2040.70			−0.513178			−0.256589	+	0.966521i	$0.582599\pi$
$252$		0			0
$253$	795.023			0.197560
$254$	−989.467	+	1713.81i	−0.244428	+	0.423361i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−2283.44	+	3955.04i	−0.554231	+	0.959956i	0.443732	+	0.896159i	$0.353654\pi$
$257$	−2283.44	+	3955.04i	−0.554231	+	0.959956i	−0.997963	+	0.0637964i	$0.979679\pi$
$258$		0			0
$259$	42.6211	−	502.430i	0.0102253	−	0.120539i
$260$	1770.50			0.422314
$261$		0			0
$262$	−155.755	−	269.775i	−0.0367273	−	0.0636135i
$263$	2668.42	+	4621.83i	0.625634	+	1.08363i	0.988418	+	0.151756i	$0.0484928\pi$
$263$	2668.42	+	4621.83i	0.625634	+	1.08363i	−0.362784	+	0.931873i	$0.618174\pi$
$264$		0			0
$265$	3914.32			0.907376
$266$	−1707.81	−	1189.12i	−0.393657	−	0.274096i
$267$		0			0
$268$	−341.904	+	592.195i	−0.0779295	+	0.134978i
$269$	3209.62	+	5559.22i	0.727486	+	1.26004i	0.957942	+	0.286960i	$0.0926448\pi$
$269$	3209.62	+	5559.22i	0.727486	+	1.26004i	−0.230456	+	0.973083i	$0.574022\pi$
$270$		0			0
$271$	−3540.34	+	6132.05i	−0.793580	+	1.37452i	0.130157	+	0.991493i	$0.458452\pi$
$271$	−3540.34	+	6132.05i	−0.793580	+	1.37452i	−0.923737	+	0.383028i	$0.874881\pi$
$272$	−573.589			−0.127864
$273$		0			0
$274$	−1965.34			−0.433323
$275$	−3.36108	+	5.82157i	−0.000737022	+	0.00127656i
$276$		0			0
$277$	−2333.66	−	4042.02i	−0.506195	−	0.876756i	−0.999974	−	0.00716871i	$-0.997718\pi$
$277$	−2333.66	−	4042.02i	−0.506195	−	0.876756i	0.493779	−	0.869588i	$-0.335615\pi$
$278$	−2232.74	+	3867.21i	−0.481693	+	0.834316i
$279$		0			0
$280$	140.212	−	1652.86i	0.0299260	−	0.352776i
$281$	−8234.13			−1.74807			−0.874034	−	0.485865i	$-0.838505\pi$
$281$	−8234.13			−1.74807			−0.874034	+	0.485865i	$0.838505\pi$
$282$		0			0
$283$	−101.417	−	175.660i	−0.0213026	−	0.0368971i	0.855178	−	0.518335i	$-0.173448\pi$
$283$	−101.417	−	175.660i	−0.0213026	−	0.0368971i	−0.876480	+	0.481438i	$0.840115\pi$
$284$	2019.82	+	3498.43i	0.422023	+	0.730964i
$285$		0			0
$286$	1533.77			0.317112
$287$	−1678.04	+	787.882i	−0.345128	+	0.162046i
$288$		0			0
$289$	1813.91	−	3141.79i	0.369207	−	0.639485i
$290$	2007.51	+	3477.10i	0.406499	+	0.704078i
$291$		0			0
$292$	−2294.23	+	3973.72i	−0.459793	+	0.796384i
$293$	3612.05			0.720199			0.360100	−	0.932914i	$-0.382743\pi$
$293$	3612.05			0.720199			0.360100	+	0.932914i	$0.382743\pi$
$294$		0			0
$295$	2730.98			0.538997
$296$	108.904	−	188.628i	0.0213849	−	0.0370398i
$297$		0			0
$298$	−3070.16	−	5317.68i	−0.596812	−	1.03371i
$299$	810.172	−	1403.26i	0.156701	−	0.271413i
$300$		0			0
$301$	1035.26	−	486.081i	0.198244	−	0.0930806i
$302$	−5450.35			−1.03852
$303$		0			0
$304$	−449.457	−	778.483i	−0.0847965	−	0.146872i
$305$	364.579	+	631.469i	0.0684450	+	0.118550i
$306$		0			0
$307$	−4684.69			−0.870910			−0.435455	−	0.900210i	$-0.643413\pi$
$307$	−4684.69			−0.870910			−0.435455	+	0.900210i	$0.643413\pi$
$308$	121.465	−	1431.86i	0.0224712	−	0.264896i
$309$		0			0
$310$	−2498.50	+	4327.54i	−0.457760	+	0.792863i
$311$	3810.20	+	6599.46i	0.694716	+	1.20328i	0.970276	+	0.242000i	$0.0778033\pi$
$311$	3810.20	+	6599.46i	0.694716	+	1.20328i	−0.275560	+	0.961284i	$0.588863\pi$
$312$		0			0
$313$	2901.80	−	5026.07i	0.524024	−	0.907636i	−0.475585	−	0.879670i	$-0.657764\pi$
$313$	2901.80	−	5026.07i	0.524024	−	0.907636i	0.999609	−	0.0279665i	$-0.00890316\pi$
$314$	−2545.96			−0.457570
$315$		0			0
$316$	2726.24			0.485325
$317$	−2131.98	+	3692.69i	−0.377740	+	0.654266i	−0.990733	−	0.135823i	$-0.956632\pi$
$317$	−2131.98	+	3692.69i	−0.377740	+	0.654266i	0.612993	+	0.790089i	$0.289966\pi$
$318$		0			0
$319$	1739.09	+	3012.20i	0.305237	+	0.528685i
$320$	358.266	−	620.536i	0.0625866	−	0.108403i
$321$		0			0
$322$	−1245.86	−	867.470i	−0.215618	−	0.150131i
$323$	−2014.09			−0.346956
$324$		0			0
$325$	6.85026	+	11.8650i	0.00116918	+	0.00202508i
$326$	−282.022	−	488.476i	−0.0479133	−	0.0829883i
$327$		0			0
$328$	−800.768			−0.134802
$329$	724.051	−	8535.32i	0.121332	−	1.43030i
$330$		0			0
$331$	−4174.62	+	7230.66i	−0.693226	+	1.20070i	0.277549	+	0.960712i	$0.410478\pi$
$331$	−4174.62	+	7230.66i	−0.693226	+	1.20070i	−0.970775	+	0.239992i	$0.922855\pi$
$332$	1817.22	+	3147.51i	0.300400	+	0.520308i
$333$		0			0
$334$	4159.02	−	7203.63i	0.681351	−	1.18013i
$335$	−1913.95			−0.312150
$336$		0			0
$337$	9133.23			1.47632			0.738158	−	0.674628i	$-0.235696\pi$
$337$	9133.23			1.47632			0.738158	+	0.674628i	$0.235696\pi$
$338$	−633.999	+	1098.12i	−0.102027	+	0.176715i
$339$		0			0
$340$	−802.725	−	1390.36i	−0.128041	−	0.221773i
$341$	−2164.44	+	3748.92i	−0.343728	+	0.595354i
$342$		0			0
$343$	−1595.50	+	6148.82i	−0.251164	+	0.967945i
$344$	494.031			0.0774312
$345$		0			0
$346$	236.524	+	409.671i	0.0367502	+	0.0636533i
$347$	−425.840	−	737.577i	−0.0658798	−	0.114107i	0.831204	−	0.555967i	$-0.187652\pi$
$347$	−425.840	−	737.577i	−0.0658798	−	0.114107i	−0.897084	+	0.441860i	$0.854319\pi$
$348$		0			0
$349$	2348.88			0.360266			0.180133	−	0.983642i	$-0.442347\pi$
$349$	2348.88			0.360266			0.180133	+	0.983642i	$0.442347\pi$
$350$	11.6191	−	5.45547i	0.00177448	−	0.000833163i
$351$		0			0
$352$	310.364	−	537.567i	0.0469957	−	0.0813989i
$353$	393.157	+	680.969i	0.0592795	+	0.102675i	0.894142	−	0.447783i	$-0.147786\pi$
$353$	393.157	+	680.969i	0.0592795	+	0.102675i	−0.834863	+	0.550458i	$0.814453\pi$
$354$		0			0
$355$	−5653.40	+	9791.97i	−0.845214	+	1.46395i
$356$	2522.75			0.375578
$357$		0			0
$358$	−6323.27			−0.933506
$359$	3690.81	−	6392.67i	0.542601	−	0.939812i	−0.456153	−	0.889901i	$-0.650773\pi$
$359$	3690.81	−	6392.67i	0.542601	−	0.939812i	0.998754	−	0.0499105i	$-0.0158936\pi$
$360$		0			0
$361$	1851.28	+	3206.52i	0.269906	+	0.467491i
$362$	−3304.47	+	5723.50i	−0.479776	+	0.830996i
$363$		0			0
$364$	−2403.54	−	1673.54i	−0.346098	−	0.240982i
$365$	−12842.9			−1.84172
$366$		0			0
$367$	1220.46	+	2113.90i	0.173590	+	0.300667i	0.939672	−	0.342075i	$-0.111130\pi$
$367$	1220.46	+	2113.90i	0.173590	+	0.300667i	−0.766082	+	0.642742i	$0.777797\pi$
$368$	−327.882	−	567.908i	−0.0464457	−	0.0804464i
$369$		0			0
$370$	609.637			0.0856582
$371$	−5313.88	−	3699.96i	−0.743619	−	0.517769i
$372$		0			0
$373$	5938.04	−	10285.0i	0.824290	−	1.42771i	−0.0781707	−	0.996940i	$-0.524908\pi$
$373$	5938.04	−	10285.0i	0.824290	−	1.42771i	0.902461	−	0.430772i	$-0.141759\pi$
$374$	−695.397	−	1204.46i	−0.0961446	−	0.166527i
$375$		0			0
$376$	1850.08	−	3204.43i	0.253751	−	0.439510i
$377$	7088.92			0.968430
$378$		0			0
$379$	−4202.37			−0.569554			−0.284777	−	0.958594i	$-0.591920\pi$
$379$	−4202.37			−0.569554			−0.284777	+	0.958594i	$0.591920\pi$
$380$	1258.01	−	2178.94i	0.169828	−	0.294151i
$381$		0			0
$382$	−2076.76	−	3597.05i	−0.278158	−	0.481783i
$383$	−7048.40	+	12208.2i	−0.940357	+	1.62875i	−0.175565	+	0.984468i	$0.556175\pi$
$383$	−7048.40	+	12208.2i	−0.940357	+	1.62875i	−0.764791	+	0.644278i	$0.777158\pi$
$384$		0			0
$385$	3640.78	−	1709.44i	0.481952	−	0.226288i
$386$	−5224.48			−0.688910
$387$		0			0
$388$	−2660.65	−	4608.38i	−0.348129	−	0.602977i
$389$	5668.66	+	9818.40i	0.738849	+	1.27972i	0.953014	+	0.302927i	$0.0979637\pi$
$389$	5668.66	+	9818.40i	0.738849	+	1.27972i	−0.214165	+	0.976798i	$0.568703\pi$
$390$		0			0
$391$	−1469.29			−0.190039
$392$	−1752.69	+	2111.31i	−0.225827	+	0.272033i
$393$		0			0
$394$	−3418.62	+	5921.22i	−0.437126	+	0.757124i
$395$	3815.31	+	6608.31i	0.485998	+	0.841773i
$396$		0			0
$397$	−493.685	+	855.088i	−0.0624115	+	0.108100i	−0.895543	−	0.444975i	$-0.853212\pi$
$397$	−493.685	+	855.088i	−0.0624115	+	0.108100i	0.833131	+	0.553075i	$0.186546\pi$
$398$	7248.31			0.912877
$399$		0			0
$400$	5.54469			0.000693086
$401$	−4246.79	+	7355.66i	−0.528864	+	0.916020i	0.470569	+	0.882363i	$0.344049\pi$
$401$	−4246.79	+	7355.66i	−0.528864	+	0.916020i	−0.999433	+	0.0336569i	$0.989285\pi$
$402$		0			0
$403$	4411.37	+	7640.72i	0.545275	+	0.944445i
$404$	−1008.66	+	1747.05i	−0.124214	+	0.215146i
$405$		0			0
$406$	561.397	−	6617.91i	0.0686248	−	0.808969i
$407$	528.126			0.0643200
$408$		0			0
$409$	−4632.80	−	8024.25i	−0.560091	−	0.970107i	−0.997488	−	0.0708381i	$-0.977433\pi$
$409$	−4632.80	−	8024.25i	−0.560091	−	0.970107i	0.437396	−	0.899269i	$-0.355901\pi$
$410$	−1120.66	−	1941.03i	−0.134989	−	0.233807i
$411$		0			0
$412$	641.066			0.0766580
$413$	−3707.44	−	2581.43i	−0.441723	−	0.307563i
$414$		0			0
$415$	−5086.31	+	8809.75i	−0.601632	+	1.04206i
$416$	−632.557	−	1095.62i	−0.0745521	−	0.129128i
$417$		0			0
$418$	1089.81	−	1887.60i	0.127522	−	0.220875i
$419$	−4398.11			−0.512797			−0.256398	−	0.966571i	$-0.582536\pi$
$419$	−4398.11			−0.512797			−0.256398	+	0.966571i	$0.582536\pi$
$420$		0			0
$421$	−9335.15			−1.08068			−0.540341	−	0.841446i	$-0.681705\pi$
$421$	−9335.15			−1.08068			−0.540341	+	0.841446i	$0.681705\pi$
$422$	4087.38	−	7079.56i	0.471495	−	0.816653i
$423$		0			0
$424$	−1398.49	−	2422.26i	−0.160181	−	0.277442i
$425$	6.21166	−	10.7589i	0.000708964	−	0.00122796i
$426$		0			0
$427$	101.954	−	1201.86i	0.0115548	−	0.136211i
$428$	−7874.36			−0.889303
$429$		0			0
$430$	691.385	+	1197.51i	0.0775385	+	0.134301i
$431$	138.876	+	240.540i	0.0155207	+	0.0268827i	0.873681	−	0.486498i	$-0.161726\pi$
$431$	138.876	+	240.540i	0.0155207	+	0.0268827i	−0.858161	+	0.513381i	$0.828393\pi$
$432$		0			0
$433$	706.915			0.0784577			0.0392289	−	0.999230i	$-0.487510\pi$
$433$	706.915			0.0784577			0.0392289	+	0.999230i	$0.487510\pi$
$434$	7482.39	−	3513.17i	0.827572	−	0.388565i
$435$		0			0
$436$	−2311.05	+	4002.86i	−0.253852	+	0.439684i
$437$	−1151.32	−	1994.14i	−0.126030	−	0.218290i
$438$		0			0
$439$	−710.440	+	1230.52i	−0.0772379	+	0.133780i	−0.902057	−	0.431616i	$-0.857943\pi$
$439$	−710.440	+	1230.52i	−0.0772379	+	0.133780i	0.824819	+	0.565396i	$0.191277\pi$
$440$	1737.39			0.188243
$441$		0			0
$442$	−2834.59			−0.305040
$443$	6029.64	−	10443.6i	0.646674	−	1.12007i	−0.337238	−	0.941420i	$-0.609493\pi$
$443$	6029.64	−	10443.6i	0.646674	−	1.12007i	0.983912	−	0.178653i	$-0.0571741\pi$
$444$		0			0
$445$	3530.54	+	6115.07i	0.376098	+	0.651421i
$446$	−3393.50	+	5877.72i	−0.360285	+	0.624032i
$447$		0			0
$448$	−1072.92	+	503.761i	−0.113149	+	0.0531261i
$449$	1519.39			0.159698			0.0798491	−	0.996807i	$-0.474556\pi$
$449$	1519.39			0.159698			0.0798491	+	0.996807i	$0.474556\pi$
$450$		0			0
$451$	−970.820	−	1681.51i	−0.101362	−	0.175564i
$452$	−2882.42	−	4992.50i	−0.299951	−	0.519530i
$453$		0			0
$454$	5107.94			0.528034
$455$	692.907	−	8168.19i	0.0713934	−	0.841606i
$456$		0			0
$457$	2724.80	−	4719.49i	0.278908	−	0.483082i	−0.692206	−	0.721700i	$-0.743361\pi$
$457$	2724.80	−	4719.49i	0.278908	−	0.483082i	0.971114	+	0.238618i	$0.0766944\pi$
$458$	−702.694	−	1217.10i	−0.0716915	−	0.124173i
$459$		0			0
$460$	917.728	−	1589.55i	0.0930202	−	0.161116i
$461$	9435.17			0.953232			0.476616	−	0.879112i	$-0.341863\pi$
$461$	9435.17			0.953232			0.476616	+	0.879112i	$0.341863\pi$
$462$		0			0
$463$	−17319.2			−1.73843			−0.869214	−	0.494436i	$-0.835375\pi$
$463$	−17319.2			−1.73843			−0.869214	+	0.494436i	$0.835375\pi$
$464$	1434.47	−	2484.57i	0.143520	−	0.248585i
$465$		0			0
$466$	411.146	+	712.126i	0.0408712	+	0.0707910i
$467$	3336.47	−	5778.93i	0.330606	−	0.572627i	−0.652024	−	0.758198i	$-0.726080\pi$
$467$	3336.47	−	5778.93i	0.330606	−	0.572627i	0.982631	+	0.185571i	$0.0594134\pi$
$468$		0			0
$469$	2598.28	+	1809.13i	0.255815	+	0.178120i
$470$	10356.6			1.01641
$471$		0			0
$472$	−975.715	−	1689.99i	−0.0951502	−	0.164805i
$473$	598.944	+	1037.40i	0.0582230	+	0.100845i
$474$		0			0
$475$	19.4695			0.00188068
$476$	−224.481	+	2646.25i	−0.0216157	+	0.254812i
$477$		0			0
$478$	−178.567	+	309.287i	−0.0170868	+	0.0295951i
$479$	7748.69	+	13421.1i	0.739138	+	1.28022i	0.952884	+	0.303335i	$0.0981000\pi$
$479$	7748.69	+	13421.1i	0.739138	+	1.28022i	−0.213747	+	0.976889i	$0.568567\pi$
$480$		0			0
$481$	538.189	−	932.171i	0.0510173	−	0.0883645i
$482$	8123.81			0.767696
$483$		0			0
$484$	−3818.90			−0.358650
$485$	7447.04	−	12898.7i	0.697222	−	1.20762i
$486$		0			0
$487$	1862.88	+	3226.60i	0.173337	+	0.300229i	0.939585	−	0.342317i	$-0.111212\pi$
$487$	1862.88	+	3226.60i	0.173337	+	0.300229i	−0.766247	+	0.642546i	$0.777878\pi$
$488$	260.510	−	451.217i	0.0241655	−	0.0418558i
$489$		0			0
$490$	−7570.59	−	1293.74i	−0.697968	−	0.119276i
$491$	−18496.7			−1.70009			−0.850045	−	0.526711i	$-0.823425\pi$
$491$	−18496.7			−1.70009			−0.850045	+	0.526711i	$0.823425\pi$
$492$		0			0
$493$	−3214.04	−	5566.88i	−0.293617	−	0.508559i
$494$	−2221.15	−	3847.14i	−0.202296	−	0.350387i
$495$		0			0
$496$	3570.62			0.323237
$497$	16930.5	−	7949.28i	1.52804	−	0.717453i
$498$		0			0
$499$	1851.59	−	3207.04i	0.166109	−	0.287709i	−0.770940	−	0.636908i	$-0.780213\pi$
$499$	1851.59	−	3207.04i	0.166109	−	0.287709i	0.937049	+	0.349199i	$0.113546\pi$
$500$	−2791.20	−	4834.50i	−0.249652	−	0.432410i
$501$		0			0
$502$	−2040.70	+	3534.59i	−0.181436	+	0.314256i
$503$	−6872.91			−0.609240			−0.304620	−	0.952474i	$-0.598529\pi$
$503$	−6872.91			−0.609240			−0.304620	+	0.952474i	$0.598529\pi$
$504$		0			0
$505$	−5646.38			−0.497546
$506$	795.023	−	1377.02i	0.0698480	−	0.120980i
$507$		0			0
$508$	1978.93	+	3427.61i	0.172837	+	0.299362i
$509$	4973.42	−	8614.22i	0.433091	−	0.750135i	−0.564047	−	0.825743i	$-0.690756\pi$
$509$	4973.42	−	8614.22i	0.433091	−	0.750135i	0.997138	+	0.0756076i	$0.0240896\pi$
$510$		0			0
$511$	17434.8	+	12139.6i	1.50934	+	1.05093i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	4566.89	+	7910.08i	0.391900	+	0.678791i
$515$	897.159	+	1553.92i	0.0767641	+	0.132959i
$516$		0			0
$517$	8971.85			0.763214
$518$	−827.613	−	576.252i	−0.0701993	−	0.0488785i
$519$		0			0
$520$	1770.50	−	3066.59i	0.149311	−	0.258614i
$521$	4227.85	+	7322.84i	0.355519	+	0.615777i	0.987207	−	0.159446i	$-0.0509709\pi$
$521$	4227.85	+	7322.84i	0.355519	+	0.615777i	−0.631688	+	0.775223i	$0.717638\pi$
$522$		0			0
$523$	3554.65	−	6156.84i	0.297197	−	0.514761i	−0.678296	−	0.734788i	$-0.737281\pi$
$523$	3554.65	−	6156.84i	0.297197	−	0.514761i	0.975494	+	0.220028i	$0.0706148\pi$
$524$	−623.018			−0.0519402
$525$		0			0
$526$	10673.7			0.884780
$527$	4000.13	−	6928.43i	0.330642	−	0.572689i
$528$		0			0
$529$	5243.60	+	9082.19i	0.430969	+	0.746461i
$530$	3914.32	−	6779.80i	0.320806	−	0.555652i
$531$		0			0
$532$	−3767.43	+	1768.90i	−0.307028	+	0.144157i
$533$	−3957.27			−0.321592
$534$		0			0
$535$	−11020.0	−	19087.2i	−0.890535	−	1.54245i
$536$	683.808	+	1184.39i	0.0551045	+	0.0954437i
$537$		0			0
$538$	12838.5			1.02882
$539$	−6558.36	−	1120.76i	−0.524098	−	0.0895629i
$540$		0			0
$541$	6264.28	−	10850.1i	0.497823	−	0.862256i	−0.502173	−	0.864767i	$-0.667466\pi$
$541$	6264.28	−	10850.1i	0.497823	−	0.862256i	0.999997	+	0.00251141i	$0.000799407\pi$
$542$	7080.68	+	12264.1i	0.561146	+	0.971933i
$543$		0			0
$544$	−573.589	+	993.484i	−0.0452066	+	0.0783002i
$545$	−12937.1			−1.01681
$546$		0			0
$547$	4698.99			0.367302			0.183651	−	0.982991i	$-0.441208\pi$
$547$	4698.99			0.367302			0.183651	+	0.982991i	$0.441208\pi$
$548$	−1965.34	+	3404.07i	−0.153203	+	0.265355i
$549$		0			0
$550$	6.72217	+	11.6431i	0.000521153	+	0.000902664i
$551$	5036.97	−	8724.28i	0.389441	−	0.674532i
$552$		0			0
$553$	1066.95	−	12577.5i	0.0820456	−	0.967177i
$554$	−9334.65			−0.715868
$555$		0			0
$556$	4465.47	+	7734.42i	0.340608	+	0.589951i
$557$	73.3775	+	127.094i	0.00558188	+	0.00966809i	0.868803	−	0.495158i	$-0.164890\pi$
$557$	73.3775	+	127.094i	0.00558188	+	0.00966809i	−0.863221	+	0.504826i	$0.831557\pi$
$558$		0			0
$559$	2441.43			0.184725
$560$	−2722.63	−	1895.72i	−0.205450	−	0.143051i
$561$		0			0
$562$	−8234.13	+	14261.9i	−0.618035	+	1.07047i
$563$	−7340.75	−	12714.5i	−0.549513	−	0.951784i	−0.998308	−	0.0581491i	$-0.981480\pi$
$563$	−7340.75	−	12714.5i	−0.549513	−	0.951784i	0.448795	−	0.893635i	$-0.351853\pi$
$564$		0			0
$565$	8067.77	−	13973.8i	0.600732	−	1.04050i
$566$	−405.669			−0.0301264
$567$		0			0
$568$	8079.29			0.596830
$569$	10220.1	−	17701.8i	0.752987	−	1.30421i	−0.193382	−	0.981123i	$-0.561946\pi$
$569$	10220.1	−	17701.8i	0.752987	−	1.30421i	0.946369	−	0.323088i	$-0.104721\pi$
$570$		0			0
$571$	9365.03	+	16220.7i	0.686364	+	1.18882i	0.973006	+	0.230780i	$0.0741278\pi$
$571$	9365.03	+	16220.7i	0.686364	+	1.18882i	−0.286641	+	0.958038i	$0.592539\pi$
$572$	1533.77	−	2656.58i	0.112116	−	0.194191i
$573$		0			0
$574$	−313.391	+	3694.34i	−0.0227886	+	0.268639i
$575$	14.2032			0.00103011
$576$		0			0
$577$	−3087.66	−	5347.98i	−0.222774	−	0.385856i	0.732875	−	0.680363i	$-0.238178\pi$
$577$	−3087.66	−	5347.98i	−0.222774	−	0.385856i	−0.955649	+	0.294507i	$0.904845\pi$
$578$	−3627.83	−	6283.58i	−0.261069	−	0.452184i
$579$		0			0
$580$	8030.03			0.574877
$581$	15232.2	−	7151.90i	1.08768	−	0.510690i
$582$		0			0
$583$	3390.95	−	5873.30i	0.240890	−	0.417234i
$584$	4588.45	+	7947.43i	0.325122	+	0.563129i
$585$		0			0
$586$	3612.05	−	6256.26i	0.254629	−	0.441030i
$587$	6964.63			0.489712			0.244856	−	0.969559i	$-0.421259\pi$
$587$	6964.63			0.489712			0.244856	+	0.969559i	$0.421259\pi$
$588$		0			0
$589$	12537.8			0.877100
$590$	2730.98	−	4730.20i	0.190564	−	0.330067i
$591$		0			0
$592$	−217.809	−	377.256i	−0.0151214	−	0.0261911i
$593$	2472.15	−	4281.89i	0.171196	−	0.296520i	−0.767642	−	0.640878i	$-0.778570\pi$
$593$	2472.15	−	4281.89i	0.171196	−	0.296520i	0.938838	+	0.344359i	$0.111904\pi$
$594$		0			0
$595$	−6728.58	+	3159.23i	−0.463605	+	0.217674i
$596$	−12280.7			−0.844019
$597$		0			0
$598$	−1620.34	−	2806.52i	−0.110804	−	0.191918i
$599$	−3158.77	−	5471.16i	−0.215466	−	0.373198i	0.737951	−	0.674855i	$-0.235794\pi$
$599$	−3158.77	−	5471.16i	−0.215466	−	0.373198i	−0.953417	+	0.301657i	$0.902460\pi$
$600$		0			0
$601$	−22912.5			−1.55511			−0.777555	−	0.628816i	$-0.783540\pi$
$601$	−22912.5			−1.55511			−0.777555	+	0.628816i	$0.783540\pi$
$602$	193.345	−	2279.21i	0.0130900	−	0.154308i
$603$		0			0
$604$	−5450.35	+	9440.29i	−0.367172	+	0.635960i
$605$	−5344.47	−	9256.90i	−0.359147	−	0.622060i
$606$		0			0
$607$	−1850.95	+	3205.93i	−0.123769	+	0.214374i	−0.921251	−	0.388969i	$-0.872831\pi$
$607$	−1850.95	+	3205.93i	−0.123769	+	0.214374i	0.797482	+	0.603342i	$0.206165\pi$
$608$	−1797.83			−0.119920
$609$		0			0
$610$	1458.32			0.0967958
$611$	9142.81	−	15835.8i	0.605366	−	1.04852i
$612$		0			0
$613$	−3816.69	−	6610.70i	−0.251476	−	0.435569i	0.712457	−	0.701716i	$-0.247582\pi$
$613$	−3816.69	−	6610.70i	−0.251476	−	0.435569i	−0.963932	+	0.266148i	$0.914249\pi$
$614$	−4684.69	+	8114.12i	−0.307913	+	0.533322i
$615$		0			0
$616$	−2358.60	−	1642.25i	−0.154270	−	0.107416i
$617$	4657.33			0.303885			0.151942	−	0.988389i	$-0.451447\pi$
$617$	4657.33			0.303885			0.151942	+	0.988389i	$0.451447\pi$
$618$		0			0
$619$	3276.14	+	5674.44i	0.212729	+	0.368457i	0.952568	−	0.304327i	$-0.0984316\pi$
$619$	3276.14	+	5674.44i	0.212729	+	0.368457i	−0.739839	+	0.672784i	$0.765098\pi$
$620$	4997.01	+	8655.07i	0.323685	+	0.560639i
$621$		0			0
$622$	15240.8			0.982477
$623$	987.312	−	11638.7i	0.0634925	−	0.748467i
$624$		0			0
$625$	7834.10	−	13569.1i	0.501382	−	0.868420i
$626$	−5803.60	−	10052.1i	−0.370541	−	0.641796i
$627$		0			0
$628$	−2545.96	+	4409.74i	−0.161775	+	0.280203i
$629$	−976.036			−0.0618714
$630$		0			0
$631$	−17505.8			−1.10443			−0.552215	−	0.833702i	$-0.686217\pi$
$631$	−17505.8			−1.10443			−0.552215	+	0.833702i	$0.686217\pi$
$632$	2726.24	−	4721.98i	0.171588	−	0.297200i
$633$		0			0
$634$	4263.95	+	7385.38i	0.267103	+	0.462636i
$635$	−5538.95	+	9593.74i	−0.346152	+	0.599553i
$636$		0			0
$637$	−8661.53	+	10433.8i	−0.538748	+	0.648981i
$638$	6956.37			0.431670
$639$		0			0
$640$	−716.533	−	1241.07i	−0.0442554	−	0.0766526i
$641$	5989.15	+	10373.5i	0.369044	+	0.639203i	0.989416	−	0.145105i	$-0.0463519\pi$
$641$	5989.15	+	10373.5i	0.369044	+	0.639203i	−0.620373	+	0.784307i	$0.713019\pi$
$642$		0			0
$643$	−14582.6			−0.894370			−0.447185	−	0.894441i	$-0.647573\pi$
$643$	−14582.6			−0.894370			−0.447185	+	0.894441i	$0.647573\pi$
$644$	−2748.36	+	1290.42i	−0.168169	+	0.0789594i
$645$		0			0
$646$	−2014.09	+	3488.51i	−0.122668	+	0.212467i
$647$	13631.5	+	23610.5i	0.828300	+	1.43466i	0.899371	+	0.437187i	$0.144025\pi$
$647$	13631.5	+	23610.5i	0.828300	+	1.43466i	−0.0710702	+	0.997471i	$0.522641\pi$
$648$		0			0
$649$	2365.84	−	4097.75i	0.143093	−	0.247844i
$650$	27.4010			0.00165347
$651$		0			0
$652$	−1128.09			−0.0677597
$653$	−13800.8	+	23903.6i	−0.827053	+	1.43250i	0.0732872	+	0.997311i	$0.476651\pi$
$653$	−13800.8	+	23903.6i	−0.827053	+	1.43250i	−0.900340	+	0.435187i	$0.856682\pi$
$654$		0			0
$655$	−871.900	−	1510.18i	−0.0520122	−	0.0900877i
$656$	−800.768	+	1386.97i	−0.0476596	+	0.0825489i
$657$		0			0
$658$	−14059.6	−	9789.42i	−0.832977	−	0.579987i
$659$	10780.5			0.637254			0.318627	−	0.947880i	$-0.396778\pi$
$659$	10780.5			0.637254			0.318627	+	0.947880i	$0.396778\pi$
$660$		0			0
$661$	5517.83	+	9557.17i	0.324688	+	0.562376i	0.981449	−	0.191722i	$-0.0614073\pi$
$661$	5517.83	+	9557.17i	0.324688	+	0.562376i	−0.656761	+	0.754099i	$0.728074\pi$
$662$	8349.24	+	14461.3i	0.490185	+	0.849025i
$663$		0			0
$664$	7268.87			0.424830
$665$	−9560.19	−	6656.58i	−0.557486	−	0.388167i
$666$		0			0
$667$	3674.50	−	6364.42i	0.213309	−	0.369462i
$668$	−8318.03	−	14407.3i	−0.481788	−	0.834481i
$669$		0			0
$670$	−1913.95	+	3315.06i	−0.110362	+	0.191152i
$671$	1263.33			0.0726831
$672$		0			0
$673$	−28007.8			−1.60419			−0.802097	−	0.597193i	$-0.796283\pi$
$673$	−28007.8			−1.60419			−0.802097	+	0.597193i	$0.796283\pi$
$674$	9133.23	−	15819.2i	0.521957	−	0.904055i
$675$		0			0
$676$	1268.00	+	2196.24i	0.0721437	+	0.124957i
$677$	2.53680	−	4.39387i	0.000144014	−	0.000249439i	−0.865953	−	0.500125i	$-0.833287\pi$
$677$	2.53680	−	4.39387i	0.000144014	−	0.000249439i	0.866097	+	0.499875i	$0.166621\pi$
$678$		0			0
$679$	−22302.0	+	10471.3i	−1.26049	+	0.591831i
$680$	−3210.90			−0.181077
$681$		0			0
$682$	4328.88	+	7497.85i	0.243052	+	0.420979i
$683$	4976.40	+	8619.37i	0.278794	+	0.482886i	0.971085	−	0.238732i	$-0.0767319\pi$
$683$	4976.40	+	8619.37i	0.278794	+	0.482886i	−0.692291	+	0.721618i	$0.743399\pi$
$684$		0			0
$685$	−11001.8			−0.613660
$686$	9054.56	+	8912.31i	0.503943	+	0.496026i
$687$		0			0
$688$	494.031	−	855.686i	0.0273761	−	0.0474168i
$689$	−6911.14	−	11970.4i	−0.382138	−	0.661883i
$690$		0			0
$691$	6159.06	−	10667.8i	0.339076	−	0.587298i	−0.645183	−	0.764028i	$-0.723219\pi$
$691$	6159.06	−	10667.8i	0.339076	−	0.587298i	0.984259	+	0.176731i	$0.0565521\pi$
$692$	946.094			0.0519727
$693$		0			0
$694$	−1703.36			−0.0931681
$695$	−12498.7	+	21648.3i	−0.682160	+	1.18154i
$696$		0			0
$697$	1794.18	+	3107.62i	0.0975030	+	0.168880i
$698$	2348.88	−	4068.38i	0.127373	−	0.220617i
$699$		0			0
$700$	2.16999	−	25.5804i	0.000117168	−	0.00138121i
$701$	−15992.8			−0.861685			−0.430843	−	0.902427i	$-0.641784\pi$
$701$	−15992.8			−0.861685			−0.430843	+	0.902427i	$0.641784\pi$
$702$		0			0
$703$	−764.810	−	1324.69i	−0.0410318	−	0.0710692i
$704$	−620.729	−	1075.13i	−0.0332310	−	0.0575577i
$705$		0			0
$706$	1572.63			0.0838338
$707$	7665.24	+	5337.17i	0.407752	+	0.283911i
$708$		0			0
$709$	9594.98	−	16619.0i	0.508247	−	0.880310i	−0.491707	−	0.870761i	$-0.663627\pi$
$709$	9594.98	−	16619.0i	0.508247	−	0.880310i	0.999954	−	0.00954922i	$-0.00303966\pi$
$710$	11306.8	+	19583.9i	0.597657	+	1.03517i
$711$		0			0
$712$	2522.75	−	4369.54i	0.132787	−	0.229993i
$713$	9146.43			0.480416
$714$		0			0
$715$	8585.94			0.449085
$716$	−6323.27	+	10952.2i	−0.330044	+	0.571653i
$717$		0			0
$718$	−7381.62	−	12785.3i	−0.383677	−	0.664547i
$719$	−5469.47	+	9473.40i	−0.283695	+	0.491375i	−0.972292	−	0.233770i	$-0.924894\pi$
$719$	−5469.47	+	9473.40i	−0.283695	+	0.491375i	0.688597	+	0.725145i	$0.258227\pi$
$720$		0			0
$721$	250.889	−	2957.56i	0.0129592	−	0.152767i
$722$	7405.13			0.381704
$723$		0			0
$724$	6608.93	+	11447.0i	0.339253	+	0.587603i
$725$	31.0691	+	53.8132i	0.00159155	+	0.00275665i
$726$		0			0
$727$	−24352.3			−1.24233			−0.621166	−	0.783679i	$-0.713341\pi$
$727$	−24352.3			−1.24233			−0.621166	+	0.783679i	$0.713341\pi$
$728$	−5302.20	+	2489.51i	−0.269935	+	0.126741i
$729$		0			0
$730$	−12842.9	+	22244.5i	−0.651146	+	1.12782i
$731$	−1106.92	−	1917.23i	−0.0560065	−	0.0970061i
$732$		0			0
$733$	993.651	−	1721.05i	0.0500700	−	0.0867239i	−0.839904	−	0.542735i	$-0.817389\pi$
$733$	993.651	−	1721.05i	0.0500700	−	0.0867239i	0.889974	+	0.456011i	$0.150722\pi$
$734$	4881.85			0.245494
$735$		0			0
$736$	−1311.53			−0.0656842
$737$	−1658.04	+	2871.82i	−0.0828695	+	0.143534i
$738$		0			0
$739$	−6181.23	−	10706.2i	−0.307687	−	0.532929i	0.670169	−	0.742208i	$-0.266221\pi$
$739$	−6181.23	−	10706.2i	−0.307687	−	0.532929i	−0.977856	+	0.209280i	$0.932888\pi$
$740$	609.637	−	1055.92i	0.0302848	−	0.0524547i
$741$		0			0
$742$	−11722.4	+	5503.95i	−0.579976	+	0.272313i
$743$	−7775.46			−0.383922			−0.191961	−	0.981403i	$-0.561485\pi$
$743$	−7775.46			−0.383922			−0.191961	+	0.981403i	$0.561485\pi$
$744$		0			0
$745$	−17186.5	−	29767.9i	−0.845188	−	1.46391i
$746$	−11876.1	−	20570.0i	−0.582861	−	1.00954i
$747$		0			0
$748$	−2781.59			−0.135969
$749$	−3081.73	+	36328.3i	−0.150339	+	1.77224i
$750$		0			0
$751$	−2550.98	+	4418.42i	−0.123950	+	0.214688i	−0.921322	−	0.388800i	$-0.872890\pi$
$751$	−2550.98	+	4418.42i	−0.123950	+	0.214688i	0.797372	+	0.603488i	$0.206223\pi$
$752$	−3700.16	−	6408.86i	−0.179429	−	0.310781i
$753$		0			0
$754$	7088.92	−	12278.4i	0.342392	−	0.593040i
$755$	−30510.6			−1.47072
$756$		0			0
$757$	10003.6			0.480302			0.240151	−	0.970736i	$-0.422803\pi$
$757$	10003.6			0.480302			0.240151	+	0.970736i	$0.422803\pi$
$758$	−4202.37	+	7278.71i	−0.201368	+	0.348779i
$759$		0			0
$760$	−2516.02	−	4357.88i	−0.120086	−	0.207996i
$761$	−18874.3	+	32691.2i	−0.899070	+	1.55724i	−0.0703850	+	0.997520i	$0.522423\pi$
$761$	−18874.3	+	32691.2i	−0.899070	+	1.55724i	−0.828685	+	0.559715i	$0.810911\pi$
$762$		0			0
$763$	17562.7	+	12228.6i	0.833307	+	0.580217i
$764$	−8307.03			−0.393374
$765$		0			0
$766$	14096.8	+	24416.4i	0.664933	+	1.15170i
$767$	−4821.84	−	8351.67i	−0.226997	−	0.393170i
$768$		0			0
$769$	4279.68			0.200688			0.100344	−	0.994953i	$-0.468006\pi$
$769$	4279.68			0.200688			0.100344	+	0.994953i	$0.468006\pi$
$770$	679.951	−	8015.46i	0.0318230	−	0.375139i
$771$		0			0
$772$	−5224.48	+	9049.07i	−0.243566	+	0.421869i
$773$	4215.77	+	7301.93i	0.196159	+	0.339757i	0.947280	−	0.320408i	$-0.103820\pi$
$773$	4215.77	+	7301.93i	0.196159	+	0.339757i	−0.751121	+	0.660165i	$0.770487\pi$
$774$		0			0
$775$	−38.6680	+	66.9749i	−0.00179225	+	0.00310427i
$776$	−10642.6			−0.492329
$777$		0			0
$778$	22674.6			1.04489
$779$	−2811.80	+	4870.18i	−0.129324	+	0.223995i
$780$		0			0
$781$	9795.02	+	16965.5i	0.448775	+	0.777301i
$782$	−1469.29	+	2544.89i	−0.0671890	+	0.116375i
$783$		0			0
$784$	1904.20	+	5147.05i	0.0867438	+	0.234469i
$785$	−14252.1			−0.647998
$786$		0			0
$787$	−5126.16	−	8878.78i	−0.232183	−	0.402153i	0.726267	−	0.687412i	$-0.241253\pi$
$787$	−5126.16	−	8878.78i	−0.232183	−	0.402153i	−0.958450	+	0.285260i	$0.907920\pi$
$788$	6837.24	+	11842.4i	0.309095	+	0.535368i
$789$		0			0
$790$	15261.2			0.687304
$791$	−24160.9	+	11344.2i	−1.08605	+	0.509927i
$792$		0			0
$793$	1287.40	−	2229.85i	0.0576508	−	0.0998540i
$794$	987.371	+	1710.18i	0.0441316	+	0.0764381i
$795$		0			0
$796$	7248.31	−	12554.4i	0.322751	−	0.559021i
$797$	28639.0			1.27283			0.636415	−	0.771347i	$-0.280417\pi$
$797$	28639.0			1.27283			0.636415	+	0.771347i	$0.280417\pi$
$798$		0			0
$799$	−16581.0			−0.734159
$800$	5.54469	−	9.60369i	0.000245043	−	0.000424427i
$801$		0			0
$802$	8493.58	+	14711.3i	0.373964	+	0.647724i
$803$	−11125.7	+	19270.3i	−0.488939	+	0.846868i
$804$		0			0
$805$	−6974.22	−	4856.02i	−0.305353	−	0.212612i
$806$	17645.5			0.771136
$807$		0			0
$808$	2017.32	+	3494.09i	0.0878328	+	0.152131i
$809$	−8194.17	−	14192.7i	−0.356108	−	0.616797i	0.631199	−	0.775621i	$-0.282563\pi$
$809$	−8194.17	−	14192.7i	−0.356108	−	0.616797i	−0.987307	+	0.158824i	$0.949230\pi$
$810$		0			0
$811$	31757.9			1.37506			0.687529	−	0.726157i	$-0.258695\pi$
$811$	31757.9			1.37506			0.687529	+	0.726157i	$0.258695\pi$
$812$	−10901.2	−	7590.28i	−0.471128	−	0.328038i
$813$		0			0
$814$	528.126	−	914.741i	0.0227405	−	0.0393878i
$815$	−1578.73	−	2734.45i	−0.0678535	−	0.117526i
$816$		0			0
$817$	1734.73	−	3004.64i	0.0742846	−	0.128665i
$818$	−18531.2			−0.792089
$819$		0			0
$820$	−4482.63			−0.190903
$821$	−6783.08	+	11748.6i	−0.288345	+	0.499428i	−0.973415	−	0.229049i	$-0.926438\pi$
$821$	−6783.08	+	11748.6i	−0.288345	+	0.499428i	0.685070	+	0.728477i	$0.259772\pi$
$822$		0			0
$823$	−10783.6	−	18677.7i	−0.456734	−	0.791086i	0.542052	−	0.840345i	$-0.317647\pi$
$823$	−10783.6	−	18677.7i	−0.456734	−	0.791086i	−0.998786	+	0.0492588i	$0.984314\pi$
$824$	641.066	−	1110.36i	0.0271027	−	0.0469432i
$825$		0			0
$826$	−8178.61	+	3840.06i	−0.344516	+	0.161759i
$827$	45594.6			1.91715			0.958574	−	0.284845i	$-0.0919421\pi$
$827$	45594.6			1.91715			0.958574	+	0.284845i	$0.0919421\pi$
$828$		0			0
$829$	7067.31	+	12240.9i	0.296089	+	0.512841i	0.975238	−	0.221160i	$-0.0709842\pi$
$829$	7067.31	+	12240.9i	0.296089	+	0.512841i	−0.679149	+	0.734001i	$0.737651\pi$
$830$	10172.6	+	17619.5i	0.425418	+	0.736846i
$831$		0			0
$832$	−2530.23			−0.105433
$833$	12120.6	+	2071.29i	0.504146	+	0.0861534i
$834$		0			0
$835$	23281.8	−	40325.3i	0.964911	−	1.67127i
$836$	−2179.62	−	3775.21i	−0.0901718	−	0.156182i
$837$		0			0
$838$	−4398.11	+	7617.75i	−0.181301	+	0.314023i
$839$	21256.2			0.874666			0.437333	−	0.899300i	$-0.355923\pi$
$839$	21256.2			0.874666			0.437333	+	0.899300i	$0.355923\pi$
$840$		0			0
$841$	7762.52			0.318280
$842$	−9335.15	+	16169.0i	−0.382079	+	0.661780i
$843$		0			0
$844$	−8174.77	−	14159.1i	−0.333397	−	0.577461i
$845$	−3549.07	+	6147.17i	−0.144487	+	0.250259i
$846$		0			0
$847$	−1494.58	+	17618.5i	−0.0606308	+	0.714733i
$848$	−5593.97			−0.226530
$849$		0			0
$850$	−12.4233	−	21.5178i	−0.000501314	−	0.000868301i
$851$	−557.934	−	966.371i	−0.0224744	−	0.0389269i
$852$		0			0
$853$	−42560.6			−1.70838			−0.854189	−	0.519962i	$-0.825946\pi$
$853$	−42560.6			−1.70838			−0.854189	+	0.519962i	$0.825946\pi$
$854$	−1979.74	−	1378.45i	−0.0793269	−	0.0552338i
$855$		0			0
$856$	−7874.36	+	13638.8i	−0.314416	+	0.544585i
$857$	−10487.2	−	18164.4i	−0.418013	−	0.724019i	0.577727	−	0.816230i	$-0.303940\pi$
$857$	−10487.2	−	18164.4i	−0.418013	−	0.724019i	−0.995740	+	0.0922108i	$0.970607\pi$
$858$		0			0
$859$	−21779.9	+	37723.9i	−0.865099	+	1.49840i	0.00184898	+	0.999998i	$0.499411\pi$
$859$	−21779.9	+	37723.9i	−0.865099	+	1.49840i	−0.866948	+	0.498398i	$0.833922\pi$
$860$	2765.54			0.109656
$861$		0			0
$862$	555.504			0.0219496
$863$	18441.1	−	31941.0i	0.727397	−	1.25989i	−0.230582	−	0.973053i	$-0.574063\pi$
$863$	18441.1	−	31941.0i	0.727397	−	1.25989i	0.957980	−	0.286836i	$-0.0926036\pi$
$864$		0			0
$865$	1324.04	+	2293.30i	0.0520447	+	0.0901440i
$866$	706.915	−	1224.41i	0.0277390	−	0.0480453i
$867$		0			0
$868$	1397.41	−	16473.0i	0.0546441	−	0.644161i
$869$	13220.7			0.516091
$870$		0			0
$871$	3379.28	+	5853.08i	0.131461	+	0.227697i
$872$	4622.11	+	8005.73i	0.179500	+	0.310904i
$873$		0			0
$874$	−4605.28			−0.178233
$875$	−23396.3	+	10985.1i	−0.903930	+	0.424417i
$876$		0			0
$877$	13395.6	−	23201.9i	0.515778	−	0.893354i	−0.484054	−	0.875038i	$-0.660836\pi$
$877$	13395.6	−	23201.9i	0.515778	−	0.893354i	0.999832	−	0.0183162i	$-0.00583056\pi$
$878$	1420.88	+	2461.04i	0.0546154	+	0.0945967i
$879$		0			0
$880$	1737.39	−	3009.25i	0.0665540	−	0.115275i
$881$	−39675.6			−1.51726			−0.758630	−	0.651522i	$-0.774131\pi$
$881$	−39675.6			−1.51726			−0.758630	+	0.651522i	$0.774131\pi$
$882$		0			0
$883$	33838.4			1.28964			0.644820	−	0.764335i	$-0.276932\pi$
$883$	33838.4			1.28964			0.644820	+	0.764335i	$0.276932\pi$
$884$	−2834.59	+	4909.65i	−0.107848	+	0.186798i
$885$		0			0
$886$	−12059.3	−	20887.3i	−0.457268	−	0.792011i
$887$	19071.6	−	33033.0i	0.721942	−	1.25044i	−0.238278	−	0.971197i	$-0.576583\pi$
$887$	19071.6	−	33033.0i	0.721942	−	1.25044i	0.960220	−	0.279243i	$-0.0900836\pi$
$888$		0			0
$889$	16587.8	−	7788.36i	0.625799	−	0.293828i
$890$	14122.2			0.531883
$891$		0			0
$892$	6787.01	+	11755.4i	0.254760	+	0.441257i
$893$	−12992.7	−	22504.0i	−0.486879	−	0.843299i
$894$		0			0
$895$	−35397.1			−1.32201
$896$	−200.378	+	2362.11i	−0.00747115	+	0.0880720i
$897$		0			0
$898$	1519.39	−	2631.66i	0.0564618	−	0.0977947i
$899$	20007.6	+	34654.2i	0.742258	+	1.28563i
$900$		0			0
$901$	−6266.86	+	10854.5i	−0.231720	+	0.401350i
$902$	−3883.28			−0.143347
$903$		0			0
$904$	−11529.7			−0.424194
$905$	−18498.1	+	32039.7i	−0.679446	+	1.17683i
$906$		0			0
$907$	13008.0	+	22530.5i	0.476212	+	0.824823i	0.999629	−	0.0272538i	$-0.00867624\pi$
$907$	13008.0	+	22530.5i	0.476212	+	0.824823i	−0.523417	+	0.852077i	$0.675343\pi$
$908$	5107.94	−	8847.21i	0.186688	−	0.323354i
$909$		0			0
$910$	−13454.8	−	9368.34i	−0.490135	−	0.341272i
$911$	−37680.4			−1.37037			−0.685185	−	0.728369i	$-0.740279\pi$
$911$	−37680.4			−1.37037			−0.685185	+	0.728369i	$0.740279\pi$
$912$		0			0
$913$	8812.50	+	15263.7i	0.319443	+	0.553291i
$914$	−5449.60	−	9438.98i	−0.197217	−	0.341591i
$915$		0			0
$916$	−2810.77			−0.101387
$917$	−243.826	+	2874.29i	−0.00878064	+	0.103509i
$918$		0			0
$919$	5999.92	−	10392.2i	0.215364	−	0.373021i	−0.738021	−	0.674777i	$-0.764240\pi$
$919$	5999.92	−	10392.2i	0.215364	−	0.373021i	0.953385	+	0.301757i	$0.0975730\pi$
$920$	−1835.46	−	3179.10i	−0.0657752	−	0.113926i
$921$		0			0
$922$	9435.17	−	16342.2i	0.337018	−	0.583733i
$923$	39926.6			1.42384
$924$		0			0
$925$	9.43502			0.000335375
$926$	−17319.2	+	29997.8i	−0.614627	+	1.06457i
$927$		0			0
$928$	−2868.94	−	4969.14i	−0.101484	−	0.175776i
$929$	−5321.14	+	9216.49i	−0.187924	+	0.325493i	−0.944558	−	0.328345i	$-0.893509\pi$
$929$	−5321.14	+	9216.49i	−0.187924	+	0.325493i	0.756634	+	0.653838i	$0.226842\pi$
$930$		0			0
$931$	6686.38	+	18073.3i	0.235378	+	0.636228i
$932$	1644.59			0.0578006
$933$		0			0
$934$	−6672.93	−	11557.9i	−0.233774	−	0.404909i
$935$	−3892.77	−	6742.47i	−0.136157	−	0.235832i
$936$		0			0
$937$	29565.1			1.03079			0.515395	−	0.856952i	$-0.327645\pi$
$937$	29565.1			1.03079			0.515395	+	0.856952i	$0.327645\pi$
$938$	5731.79	−	2691.22i	0.199520	−	0.0936795i
$939$		0			0
$940$	10356.6	−	17938.1i	0.359356	−	0.622422i
$941$	−20553.3	−	35599.3i	−0.712028	−	1.23327i	−0.964095	−	0.265558i	$-0.914444\pi$
$941$	−20553.3	−	35599.3i	−0.712028	−	1.23327i	0.252067	−	0.967710i	$-0.418890\pi$
$942$		0			0
$943$	−2051.23	+	3552.83i	−0.0708348	+	0.122689i
$944$	−3902.86			−0.134563
$945$		0			0
$946$	2395.77			0.0823397
$947$	−10606.7	+	18371.4i	−0.363962	+	0.630401i	−0.988609	−	0.150507i	$-0.951910\pi$
$947$	−10606.7	+	18371.4i	−0.363962	+	0.630401i	0.624647	+	0.780907i	$0.285243\pi$
$948$		0			0
$949$	22675.4	+	39275.0i	0.775634	+	1.34344i
$950$	19.4695	−	33.7222i	0.000664921	−	0.00115168i
$951$		0			0
$952$	4358.95	+	3035.06i	0.148398	+	0.103327i
$953$	36533.1			1.24179			0.620894	−	0.783894i	$-0.286770\pi$
$953$	36533.1			1.24179			0.620894	+	0.783894i	$0.286770\pi$
$954$		0			0
$955$	−11625.5	−	20136.0i	−0.393919	−	0.682288i
$956$	357.134	+	618.574i	0.0120822	+	0.0209269i
$957$		0			0
$958$	30994.8			1.04530
$959$	14935.5	+	10399.3i	0.502911	+	0.350168i
$960$		0			0
$961$	−10005.6	+	17330.1i	−0.335858	+	0.581724i
$962$	−1076.38	−	1864.34i	−0.0360747	−	0.0624832i
$963$		0			0
$964$	8123.81	−	14070.9i	0.271421	−	0.470116i
$965$	−29246.2			−0.975615
$966$		0			0
$967$	−2657.34			−0.0883706			−0.0441853	−	0.999023i	$-0.514069\pi$
$967$	−2657.34			−0.0883706			−0.0441853	+	0.999023i	$0.514069\pi$
$968$	−3818.90	+	6614.54i	−0.126802	+	0.219627i
$969$		0			0
$970$	−14894.1	−	25797.3i	−0.493011	−	0.853919i
$971$	17188.4	−	29771.2i	0.568076	−	0.983937i	−0.428680	−	0.903456i	$-0.641021\pi$
$971$	17188.4	−	29771.2i	0.568076	−	0.983937i	0.996756	−	0.0804803i	$-0.0256454\pi$
$972$		0			0
$973$	37430.3	−	17574.5i	1.23326	−	0.579046i
$974$	7451.52			0.245136
$975$		0			0
$976$	−521.021	−	902.435i	−0.0170876	−	0.0295966i
$977$	26246.5	+	45460.2i	0.859466	+	1.48864i	0.872439	+	0.488723i	$0.162537\pi$
$977$	26246.5	+	45460.2i	0.859466	+	1.48864i	−0.0129729	+	0.999916i	$0.504130\pi$
$978$		0			0
$979$	12234.0			0.399386
$980$	−9811.41	+	11818.9i	−0.319810	+	0.385246i
$981$		0			0
$982$	−18496.7	+	32037.2i	−0.601072	+	1.04109i
$983$	−14154.9	−	24517.0i	−0.459279	−	0.795495i	0.539644	−	0.841893i	$-0.318559\pi$
$983$	−14154.9	−	24517.0i	−0.459279	−	0.795495i	−0.998923	+	0.0463988i	$0.985226\pi$
$984$		0			0
$985$	−19137.1	+	33146.5i	−0.619046	+	1.07222i
$986$	−12856.2			−0.415237
$987$		0			0
$988$	−8884.60			−0.286090
$989$	1265.50	−	2191.91i	0.0406881	−	0.0704738i
$990$		0			0
$991$	8153.36	+	14122.0i	0.261352	+	0.452675i	0.966601	−	0.256284i	$-0.0824983\pi$
$991$	8153.36	+	14122.0i	0.261352	+	0.452675i	−0.705249	+	0.708959i	$0.749165\pi$
$992$	3570.62	−	6184.50i	0.114282	−	0.197942i
$993$		0			0
$994$	3161.93	−	37273.8i	0.100896	−	1.18939i
$995$	40575.4			1.29279
$996$		0			0
$997$	20310.9	+	35179.6i	0.645190	+	1.11750i	0.984258	+	0.176739i	$0.0565550\pi$
$997$	20310.9	+	35179.6i	0.645190	+	1.11750i	−0.339068	+	0.940762i	$0.610112\pi$
$998$	−3703.17	−	6414.08i	−0.117457	−	0.203441i
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.59791 - 9.69587i) q^{5} +(-16.7643 + 7.87127i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.59791 - 9.69587i) q^{5} +(-16.7643 + 7.87127i) q^{7} -8.00000 q^{8} +(-11.1958 - 19.3917i) q^{10} +(-9.69889 - 16.7990i) q^{11} -39.5348 q^{13} +(-3.13090 + 36.9080i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(17.9246 + 31.0464i) q^{17} +(-28.0911 + 48.6552i) q^{19} -44.7833 q^{20} -38.7956 q^{22} +(-20.4926 + 35.4943i) q^{23} +(-0.173272 - 0.300115i) q^{25} +(-39.5348 + 68.4763i) q^{26} +(60.7956 + 42.3308i) q^{28} -179.308 q^{29} +(-111.582 - 193.266i) q^{31} +(16.0000 + 27.7128i) q^{32} +71.6986 q^{34} +(-17.5265 + 206.608i) q^{35} +(-13.6130 + 23.5785i) q^{37} +(56.1821 + 97.3103i) q^{38} +(-44.7833 + 77.5670i) q^{40} +100.096 q^{41} -61.7538 q^{43} +(-38.7956 + 67.1959i) q^{44} +(40.9853 + 70.9886i) q^{46} +(-231.260 + 400.554i) q^{47} +(219.086 - 263.913i) q^{49} -0.693086 q^{50} +(79.0696 + 136.953i) q^{52} +(174.811 + 302.782i) q^{53} -217.174 q^{55} +(134.115 - 62.9702i) q^{56} +(-179.308 + 310.571i) q^{58} +(121.964 + 211.248i) q^{59} +(-32.5638 + 56.4022i) q^{61} -446.328 q^{62} +64.0000 q^{64} +(-221.312 + 383.324i) q^{65} +(-85.4760 - 148.049i) q^{67} +(71.6986 - 124.186i) q^{68} +(340.328 + 236.964i) q^{70} -1009.91 q^{71} +(-573.557 - 993.429i) q^{73} +(27.2261 + 47.1570i) q^{74} +224.729 q^{76} +(294.825 + 205.281i) q^{77} +(-340.780 + 590.248i) q^{79} +(89.5666 + 155.134i) q^{80} +(100.096 - 173.371i) q^{82} -908.609 q^{83} +401.362 q^{85} +(-61.7538 + 106.961i) q^{86} +(77.5911 + 134.392i) q^{88} +(-315.344 + 546.192i) q^{89} +(662.775 - 311.189i) q^{91} +163.941 q^{92} +(462.519 + 801.107i) q^{94} +(314.503 + 544.735i) q^{95} +1330.32 q^{97} +(-238.025 - 643.382i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - 2 * q^5 + 6 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8} + 4 q^{10} + 32 q^{11} - 4 q^{13} + 36 q^{14} - 64 q^{16} - 58 q^{17} + 70 q^{19} + 16 q^{20} + 128 q^{22} + 86 q^{23} - 156 q^{25} - 4 q^{26} + 48 q^{28} - 212 q^{29} - 64 q^{31} + 128 q^{32} - 232 q^{34} + 8 q^{35} - 146 q^{37} - 140 q^{38} + 16 q^{40} - 780 q^{41} + 880 q^{43} + 128 q^{44} - 172 q^{46} - 306 q^{47} + 50 q^{49} - 624 q^{50} + 8 q^{52} - 90 q^{53} - 64 q^{55} - 48 q^{56} - 212 q^{58} + 148 q^{59} - 364 q^{61} - 256 q^{62} + 512 q^{64} + 1296 q^{65} - 954 q^{67} - 232 q^{68} + 20 q^{70} - 1360 q^{71} - 54 q^{73} + 292 q^{74} - 560 q^{76} + 2224 q^{77} - 226 q^{79} - 32 q^{80} - 780 q^{82} - 3136 q^{83} + 3920 q^{85} + 880 q^{86} - 256 q^{88} - 1458 q^{89} + 3836 q^{91} - 688 q^{92} + 612 q^{94} + 1310 q^{95} - 4344 q^{97} + 776 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - 2 * q^5 + 6 * q^7 - 64 * q^8 + 4 * q^10 + 32 * q^11 - 4 * q^13 + 36 * q^14 - 64 * q^16 - 58 * q^17 + 70 * q^19 + 16 * q^20 + 128 * q^22 + 86 * q^23 - 156 * q^25 - 4 * q^26 + 48 * q^28 - 212 * q^29 - 64 * q^31 + 128 * q^32 - 232 * q^34 + 8 * q^35 - 146 * q^37 - 140 * q^38 + 16 * q^40 - 780 * q^41 + 880 * q^43 + 128 * q^44 - 172 * q^46 - 306 * q^47 + 50 * q^49 - 624 * q^50 + 8 * q^52 - 90 * q^53 - 64 * q^55 - 48 * q^56 - 212 * q^58 + 148 * q^59 - 364 * q^61 - 256 * q^62 + 512 * q^64 + 1296 * q^65 - 954 * q^67 - 232 * q^68 + 20 * q^70 - 1360 * q^71 - 54 * q^73 + 292 * q^74 - 560 * q^76 + 2224 * q^77 - 226 * q^79 - 32 * q^80 - 780 * q^82 - 3136 * q^83 + 3920 * q^85 + 880 * q^86 - 256 * q^88 - 1458 * q^89 + 3836 * q^91 - 688 * q^92 + 612 * q^94 + 1310 * q^95 - 4344 * q^97 + 776 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more