LMFDB - Embedded newform 416.4.i.d.321.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,4,Mod(289,416)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("416.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	416.i (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:	$24.5447945624$
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} + 74x^{6} + 5367x^{4} + 8066x^{2} + 11881 $ x^8 + 74x^6 + 5367x^4 + 8066*x^2 + 11881
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			321.3
Root			$0.613091 + 1.06190i$ of defining polynomial
Character	$\chi$	$=$	416.321
Dual form			416.4.i.d.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.613091 - 1.06190i) q^{3} +13.8322 q^{5} +(-12.7720 - 22.1218i) q^{7} +(12.7482 + 22.0806i) q^{9} +(-31.0618 + 53.8007i) q^{11} +(-13.0000 + 45.0333i) q^{13} +(8.48037 - 14.6884i) q^{15} +(28.4161 + 49.2181i) q^{17} +(13.9982 + 24.2456i) q^{19} -31.3216 q^{21} +(-28.7124 + 49.7313i) q^{23} +66.3286 q^{25} +64.3702 q^{27} +(12.3322 - 21.3599i) q^{29} +297.130 q^{31} +(38.0874 + 65.9694i) q^{33} +(-176.664 - 305.992i) q^{35} +(-19.4161 + 33.6296i) q^{37} +(39.8509 + 41.4143i) q^{39} +(58.4020 - 101.155i) q^{41} +(40.6655 + 70.4347i) q^{43} +(176.336 + 305.422i) q^{45} -462.450 q^{47} +(-154.748 + 268.032i) q^{49} +69.6865 q^{51} +412.182 q^{53} +(-429.652 + 744.179i) q^{55} +34.3286 q^{57} +(4.18873 + 7.25510i) q^{59} +(403.800 + 699.403i) q^{61} +(325.641 - 564.027i) q^{63} +(-179.818 + 622.908i) q^{65} +(0.921795 - 1.59660i) q^{67} +(35.2066 + 60.9796i) q^{69} +(-329.212 - 570.213i) q^{71} -487.090 q^{73} +(40.6655 - 70.4347i) q^{75} +1586.89 q^{77} +1240.62 q^{79} +(-304.738 + 527.821i) q^{81} +438.544 q^{83} +(393.056 + 680.793i) q^{85} +(-15.1215 - 26.1911i) q^{87} +(-423.011 + 732.676i) q^{89} +(1162.25 - 287.583i) q^{91} +(182.168 - 315.524i) q^{93} +(193.625 + 335.369i) q^{95} +(359.129 + 622.030i) q^{97} -1583.93 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 40 q^{9} - 104 q^{13} + 180 q^{17} + 696 q^{21} + 152 q^{25} + 4 q^{29} + 636 q^{33} - 108 q^{37} - 716 q^{41} + 1600 q^{45} - 1096 q^{49} + 4528 q^{53} - 104 q^{57} + 580 q^{61} - 208 q^{65}+ \cdots - 156 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 40 * q^9 - 104 * q^13 + 180 * q^17 + 696 * q^21 + 152 * q^25 + 4 * q^29 + 636 * q^33 - 108 * q^37 - 716 * q^41 + 1600 * q^45 - 1096 * q^49 + 4528 * q^53 - 104 * q^57 + 580 * q^61 - 208 * q^65 + 3500 * q^69 + 2256 * q^73 + 3608 * q^77 - 1444 * q^81 + 920 * q^85 - 4236 * q^89 + 1552 * q^93 - 156 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.613091	−	1.06190i	0.117989	−	0.204364i	−0.800981	−	0.598689i	$-0.795688\pi$
$3$	0.613091	−	1.06190i	0.117989	−	0.204364i	0.918971	+	0.394326i	$0.129022\pi$
$4$		0			0
$5$	13.8322			1.23719			0.618593	−	0.785712i	$-0.287703\pi$
$5$	13.8322			1.23719			0.618593	+	0.785712i	$0.287703\pi$
$6$		0			0
$7$	−12.7720	−	22.1218i	−0.689623	−	1.19446i	−0.971960	−	0.235147i	$-0.924443\pi$
$7$	−12.7720	−	22.1218i	−0.689623	−	1.19446i	0.282336	−	0.959316i	$-0.408891\pi$
$8$		0			0
$9$	12.7482	+	22.0806i	0.472157	+	0.817800i
$10$		0			0
$11$	−31.0618	+	53.8007i	−0.851408	+	1.47468i	0.0285288	+	0.999593i	$0.490918\pi$
$11$	−31.0618	+	53.8007i	−0.851408	+	1.47468i	−0.879937	+	0.475090i	$0.842416\pi$
$12$		0			0
$13$	−13.0000	+	45.0333i	−0.277350	+	0.960769i
$13$	−13.0000	+	45.0333i	−0.277350	+	0.960769i
$14$		0			0
$15$	8.48037	−	14.6884i	0.145975	−	0.252836i
$16$		0			0
$17$	28.4161	+	49.2181i	0.405407	+	0.702185i	0.994369	−	0.105976i	$-0.0337966\pi$
$17$	28.4161	+	49.2181i	0.405407	+	0.702185i	−0.588962	+	0.808161i	$0.700463\pi$
$18$		0			0
$19$	13.9982	+	24.2456i	0.169021	+	0.292753i	0.938076	−	0.346429i	$-0.112606\pi$
$19$	13.9982	+	24.2456i	0.169021	+	0.292753i	−0.769055	+	0.639183i	$0.779273\pi$
$20$		0			0
$21$	−31.3216			−0.325473
$22$		0			0
$23$	−28.7124	+	49.7313i	−0.260302	+	0.450856i	−0.966322	−	0.257336i	$-0.917155\pi$
$23$	−28.7124	+	49.7313i	−0.260302	+	0.450856i	0.706020	+	0.708192i	$0.250489\pi$
$24$		0			0
$25$	66.3286			0.530629
$26$		0			0
$27$	64.3702			0.458817
$28$		0			0
$29$	12.3322	−	21.3599i	0.0789664	−	0.136774i	−0.823838	−	0.566826i	$-0.808171\pi$
$29$	12.3322	−	21.3599i	0.0789664	−	0.136774i	0.902804	+	0.430052i	$0.141505\pi$
$30$		0			0
$31$	297.130			1.72149			0.860745	−	0.509037i	$-0.169998\pi$
$31$	297.130			1.72149			0.860745	+	0.509037i	$0.169998\pi$
$32$		0			0
$33$	38.0874	+	65.9694i	0.200914	+	0.347994i
$34$		0			0
$35$	−176.664	−	305.992i	−0.853192	−	1.47777i
$36$		0			0
$37$	−19.4161	+	33.6296i	−0.0862698	+	0.149424i	−0.905932	−	0.423424i	$-0.860828\pi$
$37$	−19.4161	+	33.6296i	−0.0862698	+	0.149424i	0.819662	+	0.572848i	$0.194161\pi$
$38$		0			0
$39$	39.8509	+	41.4143i	0.163622	+	0.170041i
$40$		0			0
$41$	58.4020	−	101.155i	0.222460	−	0.385312i	−0.733094	−	0.680127i	$-0.761925\pi$
$41$	58.4020	−	101.155i	0.222460	−	0.385312i	0.955554	+	0.294815i	$0.0952580\pi$
$42$		0			0
$43$	40.6655	+	70.4347i	0.144219	+	0.249795i	0.929081	−	0.369875i	$-0.120600\pi$
$43$	40.6655	+	70.4347i	0.144219	+	0.249795i	−0.784862	+	0.619670i	$0.787266\pi$
$44$		0			0
$45$	176.336	+	305.422i	0.584146	+	1.01177i
$46$		0			0
$47$	−462.450			−1.43522			−0.717610	−	0.696445i	$-0.754764\pi$
$47$	−462.450			−1.43522			−0.717610	+	0.696445i	$0.754764\pi$
$48$		0			0
$49$	−154.748	+	268.032i	−0.451161	+	0.781434i
$50$		0			0
$51$	69.6865			0.191335
$52$		0			0
$53$	412.182			1.06826			0.534128	−	0.845404i	$-0.320640\pi$
$53$	412.182			1.06826			0.534128	+	0.845404i	$0.320640\pi$
$54$		0			0
$55$	−429.652	+	744.179i	−1.05335	+	1.82446i
$56$		0			0
$57$	34.3286			0.0797709
$58$		0			0
$59$	4.18873	+	7.25510i	0.00924283	+	0.0160090i	0.870610	−	0.491974i	$-0.163725\pi$
$59$	4.18873	+	7.25510i	0.00924283	+	0.0160090i	−0.861367	+	0.507983i	$0.830391\pi$
$60$		0			0
$61$	403.800	+	699.403i	0.847563	+	1.46802i	0.883377	+	0.468664i	$0.155264\pi$
$61$	403.800	+	699.403i	0.847563	+	1.46802i	−0.0358134	+	0.999358i	$0.511402\pi$
$62$		0			0
$63$	325.641	−	564.027i	0.651221	−	1.12795i
$64$		0			0
$65$	−179.818	+	622.908i	−0.343134	+	1.18865i
$66$		0			0
$67$	0.921795	−	1.59660i	0.00168082	−	0.00291127i	−0.865184	−	0.501455i	$-0.832798\pi$
$67$	0.921795	−	1.59660i	0.00168082	−	0.00291127i	0.866865	+	0.498544i	$0.166132\pi$
$68$		0			0
$69$	35.2066	+	60.9796i	0.0614257	+	0.106392i
$70$		0			0
$71$	−329.212	−	570.213i	−0.550286	−	0.953124i	−0.998254	−	0.0590740i	$-0.981185\pi$
$71$	−329.212	−	570.213i	−0.550286	−	0.953124i	0.447967	−	0.894050i	$-0.352148\pi$
$72$		0			0
$73$	−487.090			−0.780954			−0.390477	−	0.920613i	$-0.627690\pi$
$73$	−487.090			−0.780954			−0.390477	+	0.920613i	$0.627690\pi$
$74$		0			0
$75$	40.6655	−	70.4347i	0.0626086	−	0.108441i
$76$		0			0
$77$	1586.89			2.34861
$78$		0			0
$79$	1240.62			1.76684			0.883422	−	0.468577i	$-0.155233\pi$
$79$	1240.62			1.76684			0.883422	+	0.468577i	$0.155233\pi$
$80$		0			0
$81$	−304.738	+	527.821i	−0.418022	+	0.724034i
$82$		0			0
$83$	438.544			0.579957			0.289979	−	0.957033i	$-0.406352\pi$
$83$	438.544			0.579957			0.289979	+	0.957033i	$0.406352\pi$
$84$		0			0
$85$	393.056	+	680.793i	0.501563	+	0.868733i
$86$		0			0
$87$	−15.1215	−	26.1911i	−0.0186344	−	0.0322757i
$88$		0			0
$89$	−423.011	+	732.676i	−0.503809	+	0.872623i	0.496181	+	0.868219i	$0.334735\pi$
$89$	−423.011	+	732.676i	−0.503809	+	0.872623i	−0.999990	+	0.00440423i	$0.998598\pi$
$90$		0			0
$91$	1162.25	−	287.583i	1.33887	−	0.331284i
$92$		0			0
$93$	182.168	−	315.524i	0.203117	−	0.351810i
$94$		0			0
$95$	193.625	+	335.369i	0.209111	+	0.362190i
$96$		0			0
$97$	359.129	+	622.030i	0.375918	+	0.651109i	0.990464	−	0.137772i	$-0.0439941\pi$
$97$	359.129	+	622.030i	0.375918	+	0.651109i	−0.614546	+	0.788881i	$0.710661\pi$
$98$		0			0
$99$	−1583.93			−1.60799
$100$		0			0
$101$	612.192	−	1060.35i	0.603122	−	1.04464i	−0.389223	−	0.921144i	$-0.627256\pi$
$101$	612.192	−	1060.35i	0.603122	−	1.04464i	0.992345	−	0.123495i	$-0.0394103\pi$
$102$		0			0
$103$	−838.674			−0.802301			−0.401150	−	0.916012i	$-0.631390\pi$
$103$	−838.674			−0.802301			−0.401150	+	0.916012i	$0.631390\pi$
$104$		0			0
$105$	−433.245			−0.402671
$106$		0			0
$107$	−840.307	+	1455.45i	−0.759211	+	1.31499i	0.184043	+	0.982918i	$0.441082\pi$
$107$	−840.307	+	1455.45i	−0.759211	+	1.31499i	−0.943254	+	0.332073i	$0.892252\pi$
$108$		0			0
$109$	923.874			0.811845			0.405923	−	0.913907i	$-0.366950\pi$
$109$	923.874			0.811845			0.405923	+	0.913907i	$0.366950\pi$
$110$		0			0
$111$	23.8076	+	41.2360i	0.0203578	+	0.0352608i
$112$		0			0
$113$	−573.024	−	992.507i	−0.477040	−	0.826258i	0.522613	−	0.852570i	$-0.324957\pi$
$113$	−573.024	−	992.507i	−0.477040	−	0.826258i	−0.999654	+	0.0263115i	$0.991624\pi$
$114$		0			0
$115$	−397.154	+	687.891i	−0.322042	+	0.557793i
$116$		0			0
$117$	−1160.09	+	287.048i	−0.916670	+	0.226817i
$118$		0			0
$119$	725.861	−	1257.23i	0.559156	−	0.968486i
$120$		0			0
$121$	−1264.17	−	2189.61i	−0.949793	−	1.64509i
$122$		0			0
$123$	−71.6115	−	124.035i	−0.0524958	−	0.0909254i
$124$		0			0
$125$	−811.552			−0.580699
$126$		0			0
$127$	−1246.97	+	2159.82i	−0.871266	+	1.50908i	−0.0105775	+	0.999944i	$0.503367\pi$
$127$	−1246.97	+	2159.82i	−0.871266	+	1.50908i	−0.860688	+	0.509132i	$0.829966\pi$
$128$		0			0
$129$	99.7265			0.0680654
$130$		0			0
$131$	396.837			0.264670			0.132335	−	0.991205i	$-0.457753\pi$
$131$	396.837			0.264670			0.132335	+	0.991205i	$0.457753\pi$
$132$		0			0
$133$	357.570	−	619.329i	0.233122	−	0.403779i
$134$		0			0
$135$	890.379			0.567642
$136$		0			0
$137$	−796.661	−	1379.86i	−0.496813	−	0.860505i	0.503181	−	0.864181i	$-0.332163\pi$
$137$	−796.661	−	1379.86i	−0.496813	−	0.860505i	−0.999993	+	0.00367660i	$0.998830\pi$
$138$		0			0
$139$	−1418.11	−	2456.24i	−0.865344	−	1.49882i	−0.866706	−	0.498820i	$-0.833767\pi$
$139$	−1418.11	−	2456.24i	−0.865344	−	1.49882i	0.00136200	−	0.999999i	$-0.499566\pi$
$140$		0			0
$141$	−283.524	+	491.078i	−0.169341	+	0.293307i
$142$		0			0
$143$	−2019.02	−	2098.23i	−1.18069	−	1.22701i
$144$		0			0
$145$	170.580	−	295.454i	0.0976961	−	0.169215i
$146$		0			0
$147$	189.749	+	328.656i	0.106464	+	0.184402i
$148$		0			0
$149$	−9.66784	−	16.7452i	−0.00531557	−	0.00920684i	0.863355	−	0.504596i	$-0.168359\pi$
$149$	−9.66784	−	16.7452i	−0.00531557	−	0.00920684i	−0.868671	+	0.495390i	$0.835025\pi$
$150$		0			0
$151$	1268.20			0.683473			0.341736	−	0.939796i	$-0.388985\pi$
$151$	1268.20			0.683473			0.341736	+	0.939796i	$0.388985\pi$
$152$		0			0
$153$	−724.510	+	1254.89i	−0.382831	+	0.663083i
$154$		0			0
$155$	4109.95			2.12980
$156$		0			0
$157$	−290.839			−0.147844			−0.0739220	−	0.997264i	$-0.523552\pi$
$157$	−290.839			−0.147844			−0.0739220	+	0.997264i	$0.523552\pi$
$158$		0			0
$159$	252.705	−	437.698i	0.126043	−	0.218313i
$160$		0			0
$161$	1466.86			0.718041
$162$		0			0
$163$	1157.26	+	2004.44i	0.556097	+	0.963188i	0.997817	+	0.0660351i	$0.0210349\pi$
$163$	1157.26	+	2004.44i	0.556097	+	0.963188i	−0.441721	+	0.897153i	$0.645632\pi$
$164$		0			0
$165$	526.832	+	912.499i	0.248568	+	0.430533i
$166$		0			0
$167$	−1101.77	+	1908.32i	−0.510523	+	0.884251i	0.489403	+	0.872058i	$0.337215\pi$
$167$	−1101.77	+	1908.32i	−0.510523	+	0.884251i	−0.999926	+	0.0121933i	$0.996119\pi$
$168$		0			0
$169$	−1859.00	−	1170.87i	−0.846154	−	0.532939i
$170$		0			0
$171$	−356.904	+	618.177i	−0.159609	+	0.276451i
$172$		0			0
$173$	−1826.74	−	3164.01i	−0.802802	−	1.39049i	−0.917765	−	0.397124i	$-0.870008\pi$
$173$	−1826.74	−	3164.01i	−0.802802	−	1.39049i	0.114963	−	0.993370i	$-0.463325\pi$
$174$		0			0
$175$	−847.150	−	1467.31i	−0.365934	−	0.633817i
$176$		0			0
$177$	10.2723			0.00436222
$178$		0			0
$179$	1148.47	−	1989.21i	0.479558	−	0.830619i	−0.520167	−	0.854065i	$-0.674130\pi$
$179$	1148.47	−	1989.21i	0.479558	−	0.830619i	0.999725	+	0.0234455i	$0.00746363\pi$
$180$		0			0
$181$	3694.28			1.51709			0.758546	−	0.651620i	$-0.225910\pi$
$181$	3694.28			1.51709			0.758546	+	0.651620i	$0.225910\pi$
$182$		0			0
$183$	990.265			0.400014
$184$		0			0
$185$	−268.566	+	465.171i	−0.106732	+	0.184865i
$186$		0			0
$187$	−3530.62			−1.38067
$188$		0			0
$189$	−822.137	−	1423.98i	−0.316411	−	0.548040i
$190$		0			0
$191$	2326.87	+	4030.25i	0.881499	+	1.52680i	0.849675	+	0.527307i	$0.176798\pi$
$191$	2326.87	+	4030.25i	0.881499	+	1.52680i	0.0318235	+	0.999494i	$0.489869\pi$
$192$		0			0
$193$	−1168.60	+	2024.08i	−0.435844	+	0.754904i	−0.997364	−	0.0725588i	$-0.976883\pi$
$193$	−1168.60	+	2024.08i	−0.435844	+	0.754904i	0.561520	+	0.827463i	$0.310217\pi$
$194$		0			0
$195$	551.224	+	572.849i	0.202431	+	0.210372i
$196$		0			0
$197$	1759.00	−	3046.68i	0.636161	−	1.10186i	−0.350107	−	0.936710i	$-0.613855\pi$
$197$	1759.00	−	3046.68i	0.636161	−	1.10186i	0.986268	−	0.165153i	$-0.0528119\pi$
$198$		0			0
$199$	−543.974	−	942.191i	−0.193775	−	0.335629i	0.752723	−	0.658337i	$-0.228740\pi$
$199$	−543.974	−	942.191i	−0.193775	−	0.335629i	−0.946498	+	0.322709i	$0.895407\pi$
$200$		0			0
$201$	−1.13029	−	1.95772i	−0.000396639	−	0.000686998i
$202$		0			0
$203$	−630.026			−0.217828
$204$		0			0
$205$	807.826	−	1399.20i	0.275224	−	0.476703i
$206$		0			0
$207$	−1464.13			−0.491613
$208$		0			0
$209$	−1739.24			−0.575625
$210$		0			0
$211$	2037.62	−	3529.26i	0.664812	−	1.15149i	−0.314524	−	0.949249i	$-0.601845\pi$
$211$	2037.62	−	3529.26i	0.664812	−	1.15149i	0.979336	−	0.202239i	$-0.0648218\pi$
$212$		0			0
$213$	−807.349			−0.259712
$214$		0			0
$215$	562.491	+	974.264i	0.178426	+	0.309043i
$216$		0			0
$217$	−3794.95	−	6573.05i	−1.18718	−	2.05626i
$218$		0			0
$219$	−298.631	+	517.243i	−0.0921442	+	0.159598i
$220$		0			0
$221$	−2585.86	+	639.835i	−0.787077	+	0.194751i
$222$		0			0
$223$	1012.07	−	1752.95i	0.303915	−	0.526396i	−0.673104	−	0.739548i	$-0.735040\pi$
$223$	1012.07	−	1752.95i	0.303915	−	0.526396i	0.977019	+	0.213152i	$0.0683728\pi$
$224$		0			0
$225$	845.573	+	1464.58i	0.250540	+	0.433948i
$226$		0			0
$227$	−2264.35	−	3921.97i	−0.662071	−	1.14674i	−0.980071	−	0.198650i	$-0.936344\pi$
$227$	−2264.35	−	3921.97i	−0.662071	−	1.14674i	0.317999	−	0.948091i	$-0.396989\pi$
$228$		0			0
$229$	−24.4613			−0.00705872			−0.00352936	−	0.999994i	$-0.501123\pi$
$229$	−24.4613			−0.00705872			−0.00352936	+	0.999994i	$0.501123\pi$
$230$		0			0
$231$	972.906	−	1685.12i	0.277110	−	0.479969i
$232$		0			0
$233$	−2991.73			−0.841180			−0.420590	−	0.907251i	$-0.638177\pi$
$233$	−2991.73			−0.841180			−0.420590	+	0.907251i	$0.638177\pi$
$234$		0			0
$235$	−6396.69			−1.77563
$236$		0			0
$237$	760.613	−	1317.42i	0.208469	−	0.361079i
$238$		0			0
$239$	−2113.39			−0.571982			−0.285991	−	0.958232i	$-0.592323\pi$
$239$	−2113.39			−0.571982			−0.285991	+	0.958232i	$0.592323\pi$
$240$		0			0
$241$	−3485.44	−	6036.96i	−0.931606	−	1.61359i	−0.780577	−	0.625060i	$-0.785075\pi$
$241$	−3485.44	−	6036.96i	−0.931606	−	1.61359i	−0.151029	−	0.988529i	$-0.548259\pi$
$242$		0			0
$243$	1242.66	+	2152.35i	0.328053	+	0.568204i
$244$		0			0
$245$	−2140.50	+	3707.46i	−0.558170	+	0.966779i
$246$		0			0
$247$	−1273.84	+	315.192i	−0.328146	+	0.0811952i
$248$		0			0
$249$	268.867	−	465.692i	0.0684288	−	0.118522i
$250$		0			0
$251$	2853.18	+	4941.86i	0.717495	+	1.24274i	0.961989	+	0.273087i	$0.0880447\pi$
$251$	2853.18	+	4941.86i	0.717495	+	1.24274i	−0.244494	+	0.969651i	$0.578622\pi$
$252$		0			0
$253$	−1783.72	−	3089.49i	−0.443246	−	0.767725i
$254$		0			0
$255$	963.915			0.236717
$256$		0			0
$257$	−1002.07	+	1735.64i	−0.243220	+	0.421269i	−0.961630	−	0.274351i	$-0.911537\pi$
$257$	−1002.07	+	1735.64i	−0.243220	+	0.421269i	0.718410	+	0.695620i	$0.244870\pi$
$258$		0			0
$259$	991.929			0.237975
$260$		0			0
$261$	628.853			0.149138
$262$		0			0
$263$	270.030	−	467.705i	0.0633108	−	0.109658i	−0.832633	−	0.553826i	$-0.813167\pi$
$263$	270.030	−	467.705i	0.0633108	−	0.109658i	0.895943	+	0.444168i	$0.146501\pi$
$264$		0			0
$265$	5701.37			1.32163
$266$		0			0
$267$	518.688	+	898.394i	0.118888	+	0.205921i
$268$		0			0
$269$	−79.2764	−	137.311i	−0.0179687	−	0.0311226i	0.856901	−	0.515481i	$-0.172387\pi$
$269$	−79.2764	−	137.311i	−0.0179687	−	0.0311226i	−0.874870	+	0.484358i	$0.839053\pi$
$270$		0			0
$271$	3466.13	−	6003.51i	0.776945	−	1.34571i	−0.156749	−	0.987638i	$-0.550101\pi$
$271$	3466.13	−	6003.51i	0.776945	−	1.34571i	0.933695	−	0.358070i	$-0.116565\pi$
$272$		0			0
$273$	407.181	−	1410.52i	0.0902700	−	0.312704i
$274$		0			0
$275$	−2060.29	+	3568.52i	−0.451782	+	0.782510i
$276$		0			0
$277$	2241.25	+	3881.97i	0.486151	+	0.842039i	0.999873	−	0.0159180i	$-0.00506706\pi$
$277$	2241.25	+	3881.97i	0.486151	+	0.842039i	−0.513722	+	0.857957i	$0.671734\pi$
$278$		0			0
$279$	3787.89	+	6560.81i	0.812813	+	1.40783i
$280$		0			0
$281$	7631.00			1.62003			0.810013	−	0.586411i	$-0.199460\pi$
$281$	7631.00			1.62003			0.810013	+	0.586411i	$0.199460\pi$
$282$		0			0
$283$	−1201.42	+	2080.92i	−0.252357	+	0.437096i	−0.964174	−	0.265269i	$-0.914539\pi$
$283$	−1201.42	+	2080.92i	−0.252357	+	0.437096i	0.711817	+	0.702365i	$0.247873\pi$
$284$		0			0
$285$	474.839			0.0986914
$286$		0			0
$287$	−2983.64			−0.613654
$288$		0			0
$289$	841.553	−	1457.61i	0.171291	−	0.296685i
$290$		0			0
$291$	880.715			0.177417
$292$		0			0
$293$	1416.14	+	2452.82i	0.282360	+	0.489062i	0.971966	−	0.235123i	$-0.0755493\pi$
$293$	1416.14	+	2452.82i	0.282360	+	0.489062i	−0.689605	+	0.724185i	$0.742216\pi$
$294$		0			0
$295$	57.9392	+	100.354i	0.0114351	+	0.0198062i
$296$		0			0
$297$	−1999.46	+	3463.16i	−0.390641	+	0.676609i
$298$		0			0
$299$	−1866.30	−	1939.52i	−0.360974	−	0.375135i
$300$		0			0
$301$	1038.76	−	1799.18i	0.198914	−	0.344529i
$302$		0			0
$303$	−750.658	−	1300.18i	−0.142324	−	0.246513i
$304$		0			0
$305$	5585.43	+	9674.25i	1.04859	+	1.81622i
$306$		0			0
$307$	7397.22			1.37518			0.687592	−	0.726097i	$-0.258668\pi$
$307$	7397.22			1.37518			0.687592	+	0.726097i	$0.258668\pi$
$308$		0			0
$309$	−514.183	+	890.591i	−0.0946630	+	0.163961i
$310$		0			0
$311$	−5160.32			−0.940884			−0.470442	−	0.882431i	$-0.655905\pi$
$311$	−5160.32			−0.940884			−0.470442	+	0.882431i	$0.655905\pi$
$312$		0			0
$313$	3804.66			0.687067			0.343533	−	0.939140i	$-0.388376\pi$
$313$	3804.66			0.687067			0.343533	+	0.939140i	$0.388376\pi$
$314$		0			0
$315$	4504.32	−	7801.71i	0.805682	−	1.39548i
$316$		0			0
$317$	−7727.00			−1.36906			−0.684530	−	0.728985i	$-0.739992\pi$
$317$	−7727.00			−1.36906			−0.684530	+	0.728985i	$0.739992\pi$
$318$		0			0
$319$	766.119	+	1326.96i	0.134465	+	0.232901i
$320$		0			0
$321$	1030.37	+	1784.65i	0.179158	+	0.310310i
$322$		0			0
$323$	−795.547	+	1377.93i	−0.137045	+	0.237368i
$324$		0			0
$325$	−862.272	+	2987.00i	−0.147170	+	0.509812i
$326$		0			0
$327$	566.419	−	981.066i	0.0957891	−	0.165912i
$328$		0			0
$329$	5906.42	+	10230.2i	0.989761	+	1.71432i
$330$		0			0
$331$	3767.67	+	6525.80i	0.625650	+	1.08366i	0.988415	+	0.151776i	$0.0484994\pi$
$331$	3767.67	+	6525.80i	0.625650	+	1.08366i	−0.362765	+	0.931881i	$0.618167\pi$
$332$		0			0
$333$	−990.083			−0.162932
$334$		0			0
$335$	12.7504	−	22.0844i	0.00207949	−	0.00360179i
$336$		0			0
$337$	4662.39			0.753640			0.376820	−	0.926286i	$-0.377017\pi$
$337$	4662.39			0.753640			0.376820	+	0.926286i	$0.377017\pi$
$338$		0			0
$339$	−1405.26			−0.225143
$340$		0			0
$341$	−9229.41	+	15985.8i	−1.46569	+	2.53865i
$342$		0			0
$343$	−855.814			−0.134722
$344$		0			0
$345$	486.983	+	843.479i	0.0759950	+	0.131627i
$346$		0			0
$347$	−3601.76	−	6238.43i	−0.557212	−	0.965120i	−0.997728	−	0.0673751i	$-0.978538\pi$
$347$	−3601.76	−	6238.43i	−0.557212	−	0.965120i	0.440515	−	0.897745i	$-0.354796\pi$
$348$		0			0
$349$	3523.18	−	6102.33i	0.540377	−	0.935961i	−0.458505	−	0.888692i	$-0.651615\pi$
$349$	3523.18	−	6102.33i	0.540377	−	0.935961i	0.998882	−	0.0472689i	$-0.0150518\pi$
$350$		0			0
$351$	−836.813	+	2898.80i	−0.127253	+	0.440817i
$352$		0			0
$353$	−1851.90	+	3207.59i	−0.279226	+	0.483634i	−0.971193	−	0.238295i	$-0.923411\pi$
$353$	−1851.90	+	3207.59i	−0.279226	+	0.483634i	0.691966	+	0.721930i	$0.256745\pi$
$354$		0			0
$355$	−4553.72	−	7887.27i	−0.680807	−	1.17919i
$356$		0			0
$357$	−890.037	−	1541.59i	−0.131949	−	0.228542i
$358$		0			0
$359$	−1444.39			−0.212345			−0.106173	−	0.994348i	$-0.533860\pi$
$359$	−1444.39			−0.212345			−0.106173	+	0.994348i	$0.533860\pi$
$360$		0			0
$361$	3037.60	−	5261.28i	0.442864	−	0.767062i
$362$		0			0
$363$	−3100.21			−0.448262
$364$		0			0
$365$	−6737.51			−0.966185
$366$		0			0
$367$	−1549.30	+	2683.46i	−0.220361	+	0.381677i	−0.954918	−	0.296871i	$-0.904057\pi$
$367$	−1549.30	+	2683.46i	−0.220361	+	0.381677i	0.734556	+	0.678548i	$0.237390\pi$
$368$		0			0
$369$	2978.09			0.420144
$370$		0			0
$371$	−5264.39	−	9118.19i	−0.736694	−	1.27599i
$372$		0			0
$373$	2289.51	+	3965.54i	0.317818	+	0.550477i	0.980033	−	0.198837i	$-0.0637165\pi$
$373$	2289.51	+	3965.54i	0.317818	+	0.550477i	−0.662214	+	0.749315i	$0.730383\pi$
$374$		0			0
$375$	−497.555	+	861.790i	−0.0685163	+	0.118674i
$376$		0			0
$377$	801.590	+	833.037i	0.109507	+	0.113803i
$378$		0			0
$379$	−3386.92	+	5866.32i	−0.459036	+	0.795073i	−0.998910	−	0.0466724i	$-0.985138\pi$
$379$	−3386.92	+	5866.32i	−0.459036	+	0.795073i	0.539875	+	0.841745i	$0.318472\pi$
$380$		0			0
$381$	1529.01	+	2648.33i	0.205600	+	0.356110i
$382$		0			0
$383$	4722.07	+	8178.87i	0.629991	+	1.09118i	0.987553	+	0.157287i	$0.0502748\pi$
$383$	4722.07	+	8178.87i	0.629991	+	1.09118i	−0.357562	+	0.933889i	$0.616392\pi$
$384$		0			0
$385$	21950.1			2.90566
$386$		0			0
$387$	−1036.83	+	1795.84i	−0.136188	+	0.235885i
$388$		0			0
$389$	5899.20			0.768898			0.384449	−	0.923146i	$-0.374391\pi$
$389$	5899.20			0.768898			0.384449	+	0.923146i	$0.374391\pi$
$390$		0			0
$391$	−3263.57			−0.422112
$392$		0			0
$393$	243.297	−	421.403i	0.0312283	−	0.0540889i
$394$		0			0
$395$	17160.5			2.18592
$396$		0			0
$397$	−5915.29	−	10245.6i	−0.747808	−	1.29524i	−0.948871	−	0.315664i	$-0.897773\pi$
$397$	−5915.29	−	10245.6i	−0.747808	−	1.29524i	0.201063	−	0.979578i	$-0.435560\pi$
$398$		0			0
$399$	−438.446	−	759.410i	−0.0550119	−	0.0952833i
$400$		0			0
$401$	−700.942	+	1214.07i	−0.0872902	+	0.151191i	−0.906365	−	0.422496i	$-0.861154\pi$
$401$	−700.942	+	1214.07i	−0.0872902	+	0.151191i	0.819075	+	0.573687i	$0.194487\pi$
$402$		0			0
$403$	−3862.69	+	13380.8i	−0.477455	+	1.65395i
$404$		0			0
$405$	−4215.18	+	7300.91i	−0.517170	+	0.895765i
$406$		0			0
$407$	−1206.20	−	2089.20i	−0.146902	−	0.254441i
$408$		0			0
$409$	−6640.14	−	11501.1i	−0.802772	−	1.39044i	−0.917785	−	0.397078i	$-0.870024\pi$
$409$	−6640.14	−	11501.1i	−0.802772	−	1.39044i	0.115013	−	0.993364i	$-0.463309\pi$
$410$		0			0
$411$	−1953.70			−0.234474
$412$		0			0
$413$	106.997	−	185.324i	0.0127481	−	0.0220804i
$414$		0			0
$415$	6066.01			0.717515
$416$		0			0
$417$	−3477.73			−0.408405
$418$		0			0
$419$	−6089.01	+	10546.5i	−0.709947	+	1.22966i	0.254929	+	0.966960i	$0.417948\pi$
$419$	−6089.01	+	10546.5i	−0.709947	+	1.22966i	−0.964876	+	0.262705i	$0.915386\pi$
$420$		0			0
$421$	6141.75			0.710998			0.355499	−	0.934677i	$-0.384311\pi$
$421$	6141.75			0.710998			0.355499	+	0.934677i	$0.384311\pi$
$422$		0			0
$423$	−5895.43	−	10211.2i	−0.677649	−	1.17372i
$424$		0			0
$425$	1884.80	+	3264.57i	0.215121	+	0.372600i
$426$		0			0
$427$	10314.7	−	17865.6i	1.16900	−	2.02477i
$428$		0			0
$429$	−3465.96	+	857.602i	−0.390065	+	0.0965161i
$430$		0			0
$431$	4269.15	−	7394.38i	0.477117	−	0.826391i	−0.522539	−	0.852616i	$-0.675015\pi$
$431$	4269.15	−	7394.38i	0.477117	−	0.826391i	0.999656	+	0.0262241i	$0.00834834\pi$
$432$		0			0
$433$	7470.26	+	12938.9i	0.829094	+	1.43603i	0.898750	+	0.438462i	$0.144477\pi$
$433$	7470.26	+	12938.9i	0.829094	+	1.43603i	−0.0696557	+	0.997571i	$0.522190\pi$
$434$		0			0
$435$	−209.163	−	362.280i	−0.0230542	−	0.0399310i
$436$		0			0
$437$	−1607.68			−0.175986
$438$		0			0
$439$	2266.51	−	3925.71i	0.246411	−	0.426797i	−0.716116	−	0.697981i	$-0.754082\pi$
$439$	2266.51	−	3925.71i	0.246411	−	0.426797i	0.962528	+	0.271184i	$0.0874152\pi$
$440$		0			0
$441$	−7891.07			−0.852075
$442$		0			0
$443$	4626.67			0.496207			0.248103	−	0.968734i	$-0.420193\pi$
$443$	4626.67			0.496207			0.248103	+	0.968734i	$0.420193\pi$
$444$		0			0
$445$	−5851.15	+	10134.5i	−0.623306	+	1.07960i
$446$		0			0
$447$	−23.7091			−0.00250872
$448$		0			0
$449$	836.152	+	1448.26i	0.0878852	+	0.152222i	0.906617	−	0.421954i	$-0.138656\pi$
$449$	836.152	+	1448.26i	0.0878852	+	0.152222i	−0.818732	+	0.574176i	$0.805322\pi$
$450$		0			0
$451$	3628.15	+	6284.13i	0.378809	+	0.656116i
$452$		0			0
$453$	777.520	−	1346.70i	0.0806425	−	0.139677i
$454$		0			0
$455$	16076.5	−	3977.89i	1.65643	−	0.409860i
$456$		0			0
$457$	2887.88	−	5001.96i	0.295601	−	0.511996i	−0.679524	−	0.733654i	$-0.737814\pi$
$457$	2887.88	−	5001.96i	0.295601	−	0.511996i	0.975124	+	0.221658i	$0.0711469\pi$
$458$		0			0
$459$	1829.15	+	3168.18i	0.186007	+	0.322174i
$460$		0			0
$461$	−8876.18	−	15374.0i	−0.896757	−	1.55323i	−0.831615	−	0.555353i	$-0.812583\pi$
$461$	−8876.18	−	15374.0i	−0.896757	−	1.55323i	−0.0651426	−	0.997876i	$-0.520750\pi$
$462$		0			0
$463$	−3060.93			−0.307243			−0.153621	−	0.988130i	$-0.549094\pi$
$463$	−3060.93			−0.307243			−0.153621	+	0.988130i	$0.549094\pi$
$464$		0			0
$465$	2519.77	−	4364.38i	0.251294	−	0.435254i
$466$		0			0
$467$	1019.98			0.101068			0.0505342	−	0.998722i	$-0.483908\pi$
$467$	1019.98			0.101068			0.0505342	+	0.998722i	$0.483908\pi$
$468$		0			0
$469$	−47.0927			−0.00463654
$470$		0			0
$471$	−178.311	+	308.843i	−0.0174440	+	0.0302139i
$472$		0			0
$473$	−5052.58			−0.491158
$474$		0			0
$475$	928.481	+	1608.18i	0.0896876	+	0.155343i
$476$		0			0
$477$	5254.59	+	9101.22i	0.504384	+	0.873619i
$478$		0			0
$479$	2854.97	−	4944.96i	0.272332	−	0.471693i	−0.697126	−	0.716948i	$-0.745538\pi$
$479$	2854.97	−	4944.96i	0.272332	−	0.471693i	0.969459	+	0.245255i	$0.0788717\pi$
$480$		0			0
$481$	−1262.05	−	1311.56i	−0.119635	−	0.124328i
$482$		0			0
$483$	899.317	−	1557.66i	0.0847212	−	0.146741i
$484$		0			0
$485$	4967.53	+	8604.02i	0.465080	+	0.805543i
$486$		0			0
$487$	−313.040	−	542.202i	−0.0291278	−	0.0504507i	0.851094	−	0.525013i	$-0.175940\pi$
$487$	−313.040	−	542.202i	−0.0291278	−	0.0504507i	−0.880222	+	0.474562i	$0.842606\pi$
$488$		0			0
$489$	2838.03			0.262454
$490$		0			0
$491$	−8004.57	+	13864.3i	−0.735725	+	1.27431i	0.218679	+	0.975797i	$0.429825\pi$
$491$	−8004.57	+	13864.3i	−0.735725	+	1.27431i	−0.954404	+	0.298516i	$0.903508\pi$
$492$		0			0
$493$	1401.73			0.128054
$494$		0			0
$495$	−21909.2			−1.98939
$496$		0			0
$497$	−8409.41	+	14565.5i	−0.758981	+	1.31459i
$498$		0			0
$499$	−8716.04			−0.781931			−0.390965	−	0.920405i	$-0.627859\pi$
$499$	−8716.04			−0.781931			−0.390965	+	0.920405i	$0.627859\pi$
$500$		0			0
$501$	1350.97	+	2339.94i	0.120472	+	0.208664i
$502$		0			0
$503$	−6079.76	−	10530.5i	−0.538933	−	0.933459i	−0.998962	−	0.0455551i	$-0.985494\pi$
$503$	−6079.76	−	10530.5i	−0.538933	−	0.933459i	0.460029	−	0.887904i	$-0.347839\pi$
$504$		0			0
$505$	8467.94	−	14666.9i	0.746175	−	1.29241i
$506$		0			0
$507$	−2383.08	+	1256.23i	−0.208750	+	0.110042i
$508$		0			0
$509$	1267.48	−	2195.34i	0.110373	−	0.191172i	−0.805547	−	0.592531i	$-0.798129\pi$
$509$	1267.48	−	2195.34i	0.110373	−	0.191172i	0.915921	+	0.401359i	$0.131462\pi$
$510$		0			0
$511$	6221.12	+	10775.3i	0.538564	+	0.932820i
$512$		0			0
$513$	901.066	+	1560.69i	0.0775498	+	0.134320i
$514$		0			0
$515$	−11600.7			−0.992595
$516$		0			0
$517$	14364.6	−	24880.1i	1.22196	−	2.11649i
$518$		0			0
$519$	−4479.84			−0.378889
$520$		0			0
$521$	−19884.2			−1.67206			−0.836030	−	0.548684i	$-0.815129\pi$
$521$	−19884.2			−1.67206			−0.836030	+	0.548684i	$0.815129\pi$
$522$		0			0
$523$	−179.507	+	310.915i	−0.0150082	+	0.0259950i	−0.873432	−	0.486946i	$-0.838111\pi$
$523$	−179.507	+	310.915i	−0.0150082	+	0.0259950i	0.858424	+	0.512941i	$0.171444\pi$
$524$		0			0
$525$	−2077.52			−0.172705
$526$		0			0
$527$	8443.28	+	14624.2i	0.697903	+	1.20880i
$528$		0			0
$529$	4434.70	+	7681.13i	0.364486	+	0.631308i
$530$		0			0
$531$	−106.798	+	184.980i	−0.00872813	+	0.0151176i
$532$		0			0
$533$	3796.13	+	3945.05i	0.308496	+	0.320599i
$534$		0			0
$535$	−11623.3	+	20132.1i	−0.939285	+	1.62689i
$536$		0			0
$537$	−1408.24	−	2439.14i	−0.113166	−	0.196008i
$538$		0			0
$539$	−9613.53	−	16651.1i	−0.768245	−	1.33064i
$540$		0			0
$541$	−6607.14			−0.525070			−0.262535	−	0.964922i	$-0.584559\pi$
$541$	−6607.14			−0.525070			−0.262535	+	0.964922i	$0.584559\pi$
$542$		0			0
$543$	2264.93	−	3922.97i	0.179001	−	0.310038i
$544$		0			0
$545$	12779.2			1.00440
$546$		0			0
$547$	23607.2			1.84529			0.922644	−	0.385654i	$-0.126024\pi$
$547$	23607.2			1.84529			0.922644	+	0.385654i	$0.126024\pi$
$548$		0			0
$549$	−10295.5	+	17832.3i	−0.800366	+	1.38627i
$550$		0			0
$551$	690.511			0.0533880
$552$		0			0
$553$	−15845.2	−	27444.7i	−1.21846	−	2.11043i
$554$		0			0
$555$	329.311	+	570.384i	0.0251864	+	0.0436242i
$556$		0			0
$557$	6312.86	−	10934.2i	0.480223	−	0.831771i	−0.519519	−	0.854459i	$-0.673889\pi$
$557$	6312.86	−	10934.2i	0.480223	−	0.831771i	0.999743	+	0.0226876i	$0.00722230\pi$
$558$		0			0
$559$	−3700.56	+	915.651i	−0.279995	+	0.0692807i
$560$		0			0
$561$	−2164.59	+	3749.18i	−0.162904	+	0.282158i
$562$		0			0
$563$	−4243.47	−	7349.91i	−0.317657	−	0.550199i	0.662341	−	0.749202i	$-0.269563\pi$
$563$	−4243.47	−	7349.91i	−0.317657	−	0.550199i	−0.979999	+	0.199003i	$0.936230\pi$
$564$		0			0
$565$	−7926.16	−	13728.5i	−0.590188	−	1.02224i
$566$		0			0
$567$	15568.4			1.15311
$568$		0			0
$569$	2504.88	−	4338.57i	0.184552	−	0.319653i	−0.758874	−	0.651238i	$-0.774250\pi$
$569$	2504.88	−	4338.57i	0.184552	−	0.319653i	0.943425	+	0.331585i	$0.107583\pi$
$570$		0			0
$571$	−16275.6			−1.19284			−0.596419	−	0.802673i	$-0.703410\pi$
$571$	−16275.6			−1.19284			−0.596419	+	0.802673i	$0.703410\pi$
$572$		0			0
$573$	5706.32			0.416030
$574$		0			0
$575$	−1904.45	+	3298.61i	−0.138124	+	0.239237i
$576$		0			0
$577$	−20850.8			−1.50438			−0.752191	−	0.658945i	$-0.771003\pi$
$577$	−20850.8			−1.50438			−0.752191	+	0.658945i	$0.771003\pi$
$578$		0			0
$579$	1432.92	+	2481.89i	0.102850	+	0.178141i
$580$		0			0
$581$	−5601.09	−	9701.37i	−0.399952	−	0.692738i
$582$		0			0
$583$	−12803.1	+	22175.7i	−0.909522	+	1.57534i
$584$		0			0
$585$	−16046.5	+	3970.49i	−1.13409	+	0.280615i
$586$		0			0
$587$	8392.84	−	14536.8i	0.590135	−	1.02214i	−0.404078	−	0.914724i	$-0.632408\pi$
$587$	8392.84	−	14536.8i	0.590135	−	1.02214i	0.994214	−	0.107420i	$-0.0342590\pi$
$588$		0			0
$589$	4159.29	+	7204.09i	0.290968	+	0.503972i
$590$		0			0
$591$	−2156.86	−	3735.78i	−0.150120	−	0.260016i
$592$		0			0
$593$	22582.3			1.56382			0.781909	−	0.623393i	$-0.214246\pi$
$593$	22582.3			1.56382			0.781909	+	0.623393i	$0.214246\pi$
$594$		0			0
$595$	10040.2	−	17390.2i	0.691780	−	1.19820i
$596$		0			0
$597$	−1334.02			−0.0914538
$598$		0			0
$599$	27736.8			1.89198			0.945990	−	0.324195i	$-0.105093\pi$
$599$	27736.8			1.89198			0.945990	+	0.324195i	$0.105093\pi$
$600$		0			0
$601$	3514.04	−	6086.49i	0.238504	−	0.413100i	−0.721782	−	0.692121i	$-0.756676\pi$
$601$	3514.04	−	6086.49i	0.238504	−	0.413100i	0.960285	+	0.279021i	$0.0900098\pi$
$602$		0			0
$603$	47.0051			0.00317445
$604$		0			0
$605$	−17486.3	−	30287.1i	−1.17507	−	2.03528i
$606$		0			0
$607$	−7180.51	−	12437.0i	−0.480145	−	0.831635i	0.519596	−	0.854412i	$-0.326083\pi$
$607$	−7180.51	−	12437.0i	−0.480145	−	0.831635i	−0.999741	+	0.0227768i	$0.992749\pi$
$608$		0			0
$609$	−386.263	+	669.027i	−0.0257014	+	0.0445162i
$610$		0			0
$611$	6011.85	−	20825.7i	0.398058	−	1.37891i
$612$		0			0
$613$	12038.9	−	20852.0i	0.793224	−	1.37390i	−0.130737	−	0.991417i	$-0.541734\pi$
$613$	12038.9	−	20852.0i	0.793224	−	1.37390i	0.923961	−	0.382487i	$-0.124932\pi$
$614$		0			0
$615$	−990.541	−	1715.67i	−0.0649471	−	0.112492i
$616$		0			0
$617$	−13044.7	−	22594.1i	−0.851152	−	1.47424i	−0.880169	−	0.474660i	$-0.842571\pi$
$617$	−13044.7	−	22594.1i	−0.851152	−	1.47424i	0.0290175	−	0.999579i	$-0.490762\pi$
$618$		0			0
$619$	20791.1			1.35003			0.675013	−	0.737806i	$-0.264138\pi$
$619$	20791.1			1.35003			0.675013	+	0.737806i	$0.264138\pi$
$620$		0			0
$621$	−1848.22	+	3201.21i	−0.119431	+	0.206860i
$622$		0			0
$623$	21610.8			1.38975
$624$		0			0
$625$	−19516.6			−1.24906
$626$		0			0
$627$	−1066.31	+	1846.90i	−0.0679176	+	0.117637i
$628$		0			0
$629$	−2206.92			−0.139897
$630$		0			0
$631$	−5930.20	−	10271.4i	−0.374133	−	0.648017i	0.616064	−	0.787696i	$-0.288726\pi$
$631$	−5930.20	−	10271.4i	−0.374133	−	0.648017i	−0.990197	+	0.139679i	$0.955393\pi$
$632$		0			0
$633$	−2498.49	−	4327.51i	−0.156882	−	0.271727i
$634$		0			0
$635$	−17248.3	+	29874.9i	−1.07792	+	1.86701i
$636$		0			0
$637$	−10058.6	−	10453.2i	−0.625648	−	0.650192i
$638$		0			0
$639$	8393.76	−	14538.4i	0.519643	−	0.900048i
$640$		0			0
$641$	2421.50	+	4194.16i	0.149210	+	0.258439i	0.930936	−	0.365183i	$-0.118994\pi$
$641$	2421.50	+	4194.16i	0.149210	+	0.258439i	−0.781726	+	0.623622i	$0.785660\pi$
$642$		0			0
$643$	4804.75	+	8322.07i	0.294682	+	0.510405i	0.974911	−	0.222595i	$-0.0714528\pi$
$643$	4804.75	+	8322.07i	0.294682	+	0.510405i	−0.680229	+	0.733000i	$0.738119\pi$
$644$		0			0
$645$	1379.43			0.0842095
$646$		0			0
$647$	−8923.09	+	15455.2i	−0.542199	+	0.939116i	0.456578	+	0.889683i	$0.349075\pi$
$647$	−8923.09	+	15455.2i	−0.542199	+	0.939116i	−0.998777	+	0.0494331i	$0.984259\pi$
$648$		0			0
$649$	−520.439			−0.0314777
$650$		0			0
$651$	−9306.59			−0.560298
$652$		0			0
$653$	−9391.52	+	16266.6i	−0.562816	+	0.974826i	0.434433	+	0.900704i	$0.356949\pi$
$653$	−9391.52	+	16266.6i	−0.562816	+	0.974826i	−0.997249	+	0.0741216i	$0.976385\pi$
$654$		0			0
$655$	5489.11			0.327446
$656$		0			0
$657$	−6209.54	−	10755.2i	−0.368733	−	0.638664i
$658$		0			0
$659$	10374.1	+	17968.5i	0.613229	+	1.06214i	0.990692	+	0.136120i	$0.0434632\pi$
$659$	10374.1	+	17968.5i	0.613229	+	1.06214i	−0.377463	+	0.926025i	$0.623204\pi$
$660$		0			0
$661$	−2460.41	+	4261.55i	−0.144779	+	0.250764i	−0.929290	−	0.369350i	$-0.879580\pi$
$661$	−2460.41	+	4261.55i	−0.144779	+	0.250764i	0.784512	+	0.620114i	$0.212914\pi$
$662$		0			0
$663$	−905.925	+	3138.22i	−0.0530667	+	0.183828i
$664$		0			0
$665$	4945.96	−	8566.66i	0.288415	−	0.499550i
$666$		0			0
$667$	708.171	+	1226.59i	0.0411102	+	0.0712049i
$668$		0			0
$669$	−1240.98	−	2149.44i	−0.0717174	−	0.124218i
$670$		0			0
$671$	−50171.1			−2.88649
$672$		0			0
$673$	9886.94	−	17124.7i	0.566291	−	0.980844i	−0.430638	−	0.902525i	$-0.641711\pi$
$673$	9886.94	−	17124.7i	0.566291	−	0.980844i	0.996928	−	0.0783195i	$-0.0249554\pi$
$674$		0			0
$675$	4269.59			0.243462
$676$		0			0
$677$	−5214.00			−0.295997			−0.147999	−	0.988988i	$-0.547283\pi$
$677$	−5214.00			−0.295997			−0.147999	+	0.988988i	$0.547283\pi$
$678$		0			0
$679$	9173.60	−	15889.1i	0.518484	−	0.898040i
$680$		0			0
$681$	−5553.01			−0.312469
$682$		0			0
$683$	14970.7	+	25930.0i	0.838707	+	1.45268i	0.890977	+	0.454049i	$0.150021\pi$
$683$	14970.7	+	25930.0i	0.838707	+	1.45268i	−0.0522700	+	0.998633i	$0.516646\pi$
$684$		0			0
$685$	−11019.5	−	19086.4i	−0.614650	−	1.06460i
$686$		0			0
$687$	−14.9970	+	25.9755i	−0.000832854	+	0.00144254i
$688$		0			0
$689$	−5358.37	+	18561.9i	−0.296281	+	1.02635i
$690$		0			0
$691$	610.634	−	1057.65i	0.0336174	−	0.0582270i	−0.848727	−	0.528831i	$-0.822631\pi$
$691$	610.634	−	1057.65i	0.0336174	−	0.0582270i	0.882345	+	0.470604i	$0.155964\pi$
$692$		0			0
$693$	20230.0	+	35039.4i	1.10891	+	1.92069i
$694$		0			0
$695$	−19615.6	−	33975.2i	−1.07059	−	1.85432i
$696$		0			0
$697$	6638.22			0.360747
$698$		0			0
$699$	−1834.20	+	3176.93i	−0.0992502	+	0.171906i
$700$		0			0
$701$	22838.2			1.23051			0.615253	−	0.788329i	$-0.289054\pi$
$701$	22838.2			1.23051			0.615253	+	0.788329i	$0.289054\pi$
$702$		0			0
$703$	−1087.16			−0.0583258
$704$		0			0
$705$	−3921.75	+	6792.67i	−0.209506	+	0.362875i
$706$		0			0
$707$	−31275.7			−1.66371
$708$		0			0
$709$	4386.16	+	7597.05i	0.232335	+	0.402417i	0.958495	−	0.285110i	$-0.0920300\pi$
$709$	4386.16	+	7597.05i	0.232335	+	0.402417i	−0.726160	+	0.687526i	$0.758697\pi$
$710$		0			0
$711$	15815.7	+	27393.7i	0.834228	+	1.44493i
$712$		0			0
$713$	−8531.31	+	14776.7i	−0.448107	+	0.776144i
$714$		0			0
$715$	−27927.4	−	29023.0i	−1.46073	−	1.51804i
$716$		0			0
$717$	−1295.70	+	2244.22i	−0.0674878	+	0.116892i
$718$		0			0
$719$	2370.08	+	4105.10i	0.122933	+	0.212927i	0.920923	−	0.389744i	$-0.127436\pi$
$719$	2370.08	+	4105.10i	0.122933	+	0.212927i	−0.797990	+	0.602671i	$0.794103\pi$
$720$		0			0
$721$	10711.5	+	18552.9i	0.553285	+	0.958318i
$722$		0			0
$723$	−8547.57			−0.439678
$724$		0			0
$725$	817.975	−	1416.77i	0.0419019	−	0.0725761i
$726$		0			0
$727$	16666.9			0.850265			0.425132	−	0.905131i	$-0.360228\pi$
$727$	16666.9			0.850265			0.425132	+	0.905131i	$0.360228\pi$
$728$		0			0
$729$	−13408.4			−0.681216
$730$		0			0
$731$	−2311.11	+	4002.95i	−0.116935	+	0.202537i
$732$		0			0
$733$	31724.7			1.59861			0.799304	−	0.600927i	$-0.205202\pi$
$733$	31724.7			1.59861			0.799304	+	0.600927i	$0.205202\pi$
$734$		0			0
$735$	2624.64	+	4546.02i	0.131716	+	0.228139i
$736$		0			0
$737$	57.2653	+	99.1864i	0.00286214	+	0.00495736i
$738$		0			0
$739$	2672.71	−	4629.27i	0.133041	−	0.230434i	−0.791806	−	0.610772i	$-0.790859\pi$
$739$	2672.71	−	4629.27i	0.133041	−	0.230434i	0.924847	+	0.380338i	$0.124193\pi$
$740$		0			0
$741$	−446.272	+	1545.93i	−0.0221245	+	0.0766414i
$742$		0			0
$743$	7782.47	−	13479.6i	0.384268	−	0.665572i	−0.607399	−	0.794397i	$-0.707787\pi$
$743$	7782.47	−	13479.6i	0.384268	−	0.665572i	0.991667	+	0.128825i	$0.0411205\pi$
$744$		0			0
$745$	−133.727	−	231.622i	−0.00657635	−	0.0113906i
$746$		0			0
$747$	5590.67	+	9683.32i	0.273831	+	0.474289i
$748$		0			0
$749$	42929.6			2.09428
$750$		0			0
$751$	9995.88	−	17313.4i	0.485692	−	0.841243i	−0.514173	−	0.857687i	$-0.671901\pi$
$751$	9995.88	−	17313.4i	0.485692	−	0.841243i	0.999865	+	0.0164434i	$0.00523433\pi$
$752$		0			0
$753$	6997.04			0.338627
$754$		0			0
$755$	17541.9			0.845583
$756$		0			0
$757$	−5269.15	+	9126.44i	−0.252986	+	0.438185i	−0.964347	−	0.264642i	$-0.914746\pi$
$757$	−5269.15	+	9126.44i	−0.252986	+	0.438185i	0.711360	+	0.702827i	$0.248079\pi$
$758$		0			0
$759$	−4374.32			−0.209193
$760$		0			0
$761$	−4418.07	−	7652.33i	−0.210453	−	0.364516i	0.741403	−	0.671060i	$-0.234161\pi$
$761$	−4418.07	−	7652.33i	−0.210453	−	0.364516i	−0.951857	+	0.306544i	$0.900827\pi$
$762$		0			0
$763$	−11799.7	−	20437.7i	−0.559867	−	0.969719i
$764$		0			0
$765$	−10021.5	+	17357.8i	−0.473633	+	0.820357i
$766$		0			0
$767$	−381.175	+	94.3163i	−0.0179445	+	0.00444011i
$768$		0			0
$769$	−3299.23	+	5714.43i	−0.154711	+	0.267968i	−0.932954	−	0.359996i	$-0.882778\pi$
$769$	−3299.23	+	5714.43i	−0.154711	+	0.267968i	0.778242	+	0.627964i	$0.216111\pi$
$770$		0			0
$771$	1228.72	+	2128.21i	0.0573948	+	0.0994106i
$772$		0			0
$773$	3278.11	+	5677.84i	0.152529	+	0.264189i	0.932157	−	0.362055i	$-0.117925\pi$
$773$	3278.11	+	5677.84i	0.152529	+	0.264189i	−0.779627	+	0.626244i	$0.784591\pi$
$774$		0			0
$775$	19708.2			0.913472
$776$		0			0
$777$	608.143	−	1053.33i	0.0280785	−	0.0486334i
$778$		0			0
$779$	3270.09			0.150402
$780$		0			0
$781$	40903.8			1.87407
$782$		0			0
$783$	793.824	−	1374.94i	0.0362311	−	0.0627541i
$784$		0			0
$785$	−4022.93			−0.182910
$786$		0			0
$787$	5240.67	+	9077.10i	0.237369	+	0.411136i	0.959959	−	0.280142i	$-0.0903815\pi$
$787$	5240.67	+	9077.10i	0.237369	+	0.411136i	−0.722589	+	0.691278i	$0.757048\pi$
$788$		0			0
$789$	−331.106	−	573.492i	−0.0149400	−	0.0258769i
$790$		0			0
$791$	−14637.3	+	25352.6i	−0.657957	+	1.13961i
$792$		0			0
$793$	−36745.8	+	9092.24i	−1.64550	+	0.407156i
$794$		0			0
$795$	3495.46	−	6054.31i	0.155938	−	0.270093i
$796$		0			0
$797$	1047.41	+	1814.16i	0.0465508	+	0.0806284i	0.888362	−	0.459144i	$-0.151844\pi$
$797$	1047.41	+	1814.16i	0.0465508	+	0.0806284i	−0.841811	+	0.539772i	$0.818510\pi$
$798$		0			0
$799$	−13141.0	−	22760.9i	−0.581847	−	1.00779i
$800$		0			0
$801$	−21570.6			−0.951508
$802$		0			0
$803$	15129.9	−	26205.8i	0.664911	−	1.15166i
$804$		0			0
$805$	20289.8			0.888350
$806$		0			0
$807$	−194.415			−0.00848044
$808$		0			0
$809$	−14326.1	+	24813.5i	−0.622594	+	1.07837i	0.366406	+	0.930455i	$0.380588\pi$
$809$	−14326.1	+	24813.5i	−0.622594	+	1.07837i	−0.989001	+	0.147910i	$0.952745\pi$
$810$		0			0
$811$	−13299.6			−0.575845			−0.287923	−	0.957654i	$-0.592965\pi$
$811$	−13299.6			−0.575845			−0.287923	+	0.957654i	$0.592965\pi$
$812$		0			0
$813$	−4250.10	−	7361.39i	−0.183343	−	0.317559i
$814$		0			0
$815$	16007.4	+	27725.7i	0.687995	+	1.19164i
$816$		0			0
$817$	−1138.49	+	1971.92i	−0.0487522	+	0.0844414i
$818$		0			0
$819$	21166.7	+	21997.1i	0.903081	+	0.938510i
$820$		0			0
$821$	16726.6	−	28971.3i	0.711038	−	1.23155i	−0.253430	−	0.967354i	$-0.581559\pi$
$821$	16726.6	−	28971.3i	0.711038	−	1.23155i	0.964468	−	0.264200i	$-0.0851078\pi$
$822$		0			0
$823$	4275.00	+	7404.52i	0.181066	+	0.313615i	0.942244	−	0.334928i	$-0.108712\pi$
$823$	4275.00	+	7404.52i	0.181066	+	0.313615i	−0.761178	+	0.648543i	$0.775379\pi$
$824$		0			0
$825$	2526.29	+	4375.66i	0.106611	+	0.184656i
$826$		0			0
$827$	11423.5			0.480333			0.240167	−	0.970732i	$-0.422798\pi$
$827$	11423.5			0.480333			0.240167	+	0.970732i	$0.422798\pi$
$828$		0			0
$829$	−7274.46	+	12599.7i	−0.304768	+	0.527873i	−0.977210	−	0.212277i	$-0.931912\pi$
$829$	−7274.46	+	12599.7i	−0.304768	+	0.527873i	0.672442	+	0.740150i	$0.265245\pi$
$830$		0			0
$831$	5496.37			0.229443
$832$		0			0
$833$	−17589.4			−0.731615
$834$		0			0
$835$	−15239.8	+	26396.1i	−0.631611	+	1.09398i
$836$		0			0
$837$	19126.3			0.789848
$838$		0			0
$839$	−9473.23	−	16408.1i	−0.389812	−	0.675174i	0.602612	−	0.798034i	$-0.294127\pi$
$839$	−9473.23	−	16408.1i	−0.389812	−	0.675174i	−0.992424	+	0.122860i	$0.960793\pi$
$840$		0			0
$841$	11890.3	+	20594.7i	0.487529	+	0.844424i
$842$		0			0
$843$	4678.50	−	8103.40i	0.191146	−	0.331074i
$844$		0			0
$845$	−25714.0	−	16195.6i	−1.04685	−	0.659344i
$846$		0			0
$847$	−32292.1	+	55931.5i	−1.31000	+	2.26898i
$848$		0			0
$849$	1473.16	+	2551.59i	0.0595510	+	0.103145i
$850$		0			0
$851$	−1114.96	−	1931.17i	−0.0449124	−	0.0777905i
$852$		0			0
$853$	44528.2			1.78736			0.893679	−	0.448707i	$-0.148115\pi$
$853$	44528.2			1.78736			0.893679	+	0.448707i	$0.148115\pi$
$854$		0			0
$855$	−4936.76	+	8550.72i	−0.197466	+	0.342022i
$856$		0			0
$857$	3613.56			0.144034			0.0720169	−	0.997403i	$-0.477056\pi$
$857$	3613.56			0.144034			0.0720169	+	0.997403i	$0.477056\pi$
$858$		0			0
$859$	13773.8			0.547095			0.273548	−	0.961858i	$-0.411803\pi$
$859$	13773.8			0.547095			0.273548	+	0.961858i	$0.411803\pi$
$860$		0			0
$861$	−1829.24	+	3168.34i	−0.0724047	+	0.125409i
$862$		0			0
$863$	−1110.05			−0.0437849			−0.0218925	−	0.999760i	$-0.506969\pi$
$863$	−1110.05			−0.0437849			−0.0218925	+	0.999760i	$0.506969\pi$
$864$		0			0
$865$	−25267.8	−	43765.1i	−0.993216	−	1.72030i
$866$		0			0
$867$	−1031.90	−	1787.30i	−0.0404210	−	0.0700113i
$868$		0			0
$869$	−38536.0	+	66746.2i	−1.50431	+	2.60554i
$870$		0			0
$871$	59.9167	+	62.2672i	0.00233088	+	0.00242232i
$872$		0			0
$873$	−9156.53	+	15859.6i	−0.354985	+	0.614851i
$874$		0			0
$875$	10365.1	+	17953.0i	0.400464	+	0.693623i
$876$		0			0
$877$	−8606.62	−	14907.1i	−0.331385	−	0.573976i	0.651398	−	0.758736i	$-0.274183\pi$
$877$	−8606.62	−	14907.1i	−0.331385	−	0.573976i	−0.982784	+	0.184760i	$0.940849\pi$
$878$		0			0
$879$	3472.88			0.133262
$880$		0			0
$881$	−7567.74	+	13107.7i	−0.289403	+	0.501260i	−0.973667	−	0.227974i	$-0.926790\pi$
$881$	−7567.74	+	13107.7i	−0.289403	+	0.501260i	0.684265	+	0.729234i	$0.260123\pi$
$882$		0			0
$883$	−23884.1			−0.910266			−0.455133	−	0.890424i	$-0.650408\pi$
$883$	−23884.1			−0.910266			−0.455133	+	0.890424i	$0.650408\pi$
$884$		0			0
$885$	142.088			0.00539688
$886$		0			0
$887$	4591.78	−	7953.19i	0.173818	−	0.301062i	−0.765933	−	0.642920i	$-0.777723\pi$
$887$	4591.78	−	7953.19i	0.173818	−	0.301062i	0.939752	+	0.341858i	$0.111056\pi$
$888$		0			0
$889$	63705.3			2.40338
$890$		0			0
$891$	−18931.4	−	32790.2i	−0.711814	−	1.23290i
$892$		0			0
$893$	−6473.47	−	11212.4i	−0.242583	−	0.420165i
$894$		0			0
$895$	15885.9	−	27515.1i	0.593303	−	1.02763i
$896$		0			0
$897$	−3203.80	+	792.734i	−0.119255	+	0.0295080i
$898$		0			0
$899$	3664.26	−	6346.68i	0.135940	−	0.235455i
$900$		0			0
$901$	11712.6	+	20286.8i	0.433078	+	0.750113i
$902$		0			0
$903$	−1273.71	−	2206.13i	−0.0469395	−	0.0813015i
$904$		0			0
$905$	51099.8			1.87692
$906$		0			0
$907$	−6376.01	+	11043.6i	−0.233420	+	0.404295i	−0.958812	−	0.284040i	$-0.908325\pi$
$907$	−6376.01	+	11043.6i	−0.233420	+	0.404295i	0.725392	+	0.688336i	$0.241658\pi$
$908$		0			0
$909$	31217.5			1.13907
$910$		0			0
$911$	15783.7			0.574027			0.287014	−	0.957926i	$-0.407337\pi$
$911$	15783.7			0.574027			0.287014	+	0.957926i	$0.407337\pi$
$912$		0			0
$913$	−13622.0	+	23594.0i	−0.493781	+	0.855253i
$914$		0			0
$915$	13697.5			0.494891
$916$		0			0
$917$	−5068.40	−	8778.73i	−0.182523	−	0.316139i
$918$		0			0
$919$	−26544.7	−	45976.8i	−0.952806	−	1.65031i	−0.739311	−	0.673364i	$-0.764849\pi$
$919$	−26544.7	−	45976.8i	−0.952806	−	1.65031i	−0.213495	−	0.976944i	$-0.568485\pi$
$920$		0			0
$921$	4535.17	−	7855.14i	0.162257	−	0.281038i
$922$		0			0
$923$	29958.3	−	7412.77i	1.06835	−	0.264349i
$924$		0			0
$925$	−1287.84	+	2230.61i	−0.0457773	+	0.0792886i
$926$		0			0
$927$	−10691.6	−	18518.4i	−0.378812	−	0.656121i
$928$		0			0
$929$	24382.3	+	42231.4i	0.861096	+	1.49146i	0.870872	+	0.491510i	$0.163555\pi$
$929$	24382.3	+	42231.4i	0.861096	+	1.49146i	−0.00977577	+	0.999952i	$0.503112\pi$
$930$		0			0
$931$	−8664.78			−0.305023
$932$		0			0
$933$	−3163.74	+	5479.77i	−0.111014	+	0.192282i
$934$		0			0
$935$	−48836.1			−1.70814
$936$		0			0
$937$	−30351.0			−1.05819			−0.529095	−	0.848562i	$-0.677469\pi$
$937$	−30351.0			−1.05819			−0.529095	+	0.848562i	$0.677469\pi$
$938$		0			0
$939$	2332.60	−	4040.18i	0.0810666	−	0.140411i
$940$		0			0
$941$	9796.43			0.339378			0.169689	−	0.985498i	$-0.445724\pi$
$941$	9796.43			0.339378			0.169689	+	0.985498i	$0.445724\pi$
$942$		0			0
$943$	3353.72	+	5808.81i	0.115813	+	0.200595i
$944$		0			0
$945$	−11371.9	−	19696.8i	−0.391459	−	0.678027i
$946$		0			0
$947$	986.513	−	1708.69i	0.0338515	−	0.0586325i	−0.848603	−	0.529030i	$-0.822556\pi$
$947$	986.513	−	1708.69i	0.0338515	−	0.0586325i	0.882455	+	0.470397i	$0.155889\pi$
$948$		0			0
$949$	6332.17	−	21935.3i	0.216598	−	0.750316i
$950$		0			0
$951$	−4737.36	+	8205.34i	−0.161534	+	0.279786i
$952$		0			0
$953$	2765.13	+	4789.35i	0.0939888	+	0.162793i	0.909186	−	0.416390i	$-0.136705\pi$
$953$	2765.13	+	4789.35i	0.0939888	+	0.162793i	−0.815197	+	0.579183i	$0.803372\pi$
$954$		0			0
$955$	32185.6	+	55747.1i	1.09058	+	1.88894i
$956$		0			0
$957$	1878.80			0.0634619
$958$		0			0
$959$	−20349.9	+	35247.1i	−0.685227	+	1.18685i
$960$		0			0
$961$	58495.4			1.96353
$962$		0			0
$963$	−42849.7			−1.43387
$964$		0			0
$965$	−16164.3	+	27997.4i	−0.539220	+	0.933957i
$966$		0			0
$967$	30916.7			1.02814			0.514072	−	0.857747i	$-0.328136\pi$
$967$	30916.7			1.02814			0.514072	+	0.857747i	$0.328136\pi$
$968$		0			0
$969$	975.485	+	1689.59i	0.0323396	+	0.0560139i
$970$		0			0
$971$	−9462.74	−	16389.9i	−0.312743	−	0.541687i	0.666212	−	0.745762i	$-0.267915\pi$
$971$	−9462.74	−	16389.9i	−0.312743	−	0.541687i	−0.978955	+	0.204075i	$0.934581\pi$
$972$		0			0
$973$	−36224.3	+	62742.3i	−1.19352	+	2.06724i
$974$		0			0
$975$	2643.26	+	2746.95i	0.0868225	+	0.0902286i
$976$		0			0
$977$	12526.0	−	21695.6i	0.410176	−	0.710445i	−0.584733	−	0.811226i	$-0.698801\pi$
$977$	12526.0	−	21695.6i	0.410176	−	0.710445i	0.994909	+	0.100781i	$0.0321340\pi$
$978$		0			0
$979$	−26279.0	−	45516.5i	−0.857895	−	1.48592i
$980$		0			0
$981$	11777.8	+	20399.7i	0.383318	+	0.663927i
$982$		0			0
$983$	22680.6			0.735909			0.367954	−	0.929844i	$-0.380058\pi$
$983$	22680.6			0.735909			0.367954	+	0.929844i	$0.380058\pi$
$984$		0			0
$985$	24330.8	−	42142.2i	0.787050	−	1.36321i
$986$		0			0
$987$	14484.7			0.467125
$988$		0			0
$989$	−4670.41			−0.150162
$990$		0			0
$991$	10294.5	−	17830.7i	0.329986	−	0.571553i	−0.652522	−	0.757770i	$-0.726289\pi$
$991$	10294.5	−	17830.7i	0.329986	−	0.571553i	0.982509	+	0.186216i	$0.0596225\pi$
$992$		0			0
$993$	9239.70			0.295280
$994$		0			0
$995$	−7524.34	−	13032.5i	−0.239736	−	0.415235i
$996$		0			0
$997$	−15218.4	−	26359.0i	−0.483420	−	0.837309i	0.516398	−	0.856349i	$-0.327272\pi$
$997$	−15218.4	−	26359.0i	−0.483420	−	0.837309i	−0.999819	+	0.0190397i	$0.993939\pi$
$998$		0			0
$999$	−1249.82	+	2164.75i	−0.0395820	+	0.0685581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.4.i.d.321.3	yes	8
4.3	odd	2	inner	416.4.i.d.321.2	yes	8
13.3	even	3	inner	416.4.i.d.289.3	yes	8
52.3	odd	6	inner	416.4.i.d.289.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.4.i.d.289.2	✓	8	52.3	odd	6	inner
416.4.i.d.289.3	yes	8	13.3	even	3	inner
416.4.i.d.321.2	yes	8	4.3	odd	2	inner
416.4.i.d.321.3	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	416.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.613091 - 1.06190i) q^{3} +13.8322 q^{5} +(-12.7720 - 22.1218i) q^{7} +(12.7482 + 22.0806i) q^{9} +(-31.0618 + 53.8007i) q^{11} +(-13.0000 + 45.0333i) q^{13} +(8.48037 - 14.6884i) q^{15} +(28.4161 + 49.2181i) q^{17} +(13.9982 + 24.2456i) q^{19} -31.3216 q^{21} +(-28.7124 + 49.7313i) q^{23} +66.3286 q^{25} +64.3702 q^{27} +(12.3322 - 21.3599i) q^{29} +297.130 q^{31} +(38.0874 + 65.9694i) q^{33} +(-176.664 - 305.992i) q^{35} +(-19.4161 + 33.6296i) q^{37} +(39.8509 + 41.4143i) q^{39} +(58.4020 - 101.155i) q^{41} +(40.6655 + 70.4347i) q^{43} +(176.336 + 305.422i) q^{45} -462.450 q^{47} +(-154.748 + 268.032i) q^{49} +69.6865 q^{51} +412.182 q^{53} +(-429.652 + 744.179i) q^{55} +34.3286 q^{57} +(4.18873 + 7.25510i) q^{59} +(403.800 + 699.403i) q^{61} +(325.641 - 564.027i) q^{63} +(-179.818 + 622.908i) q^{65} +(0.921795 - 1.59660i) q^{67} +(35.2066 + 60.9796i) q^{69} +(-329.212 - 570.213i) q^{71} -487.090 q^{73} +(40.6655 - 70.4347i) q^{75} +1586.89 q^{77} +1240.62 q^{79} +(-304.738 + 527.821i) q^{81} +438.544 q^{83} +(393.056 + 680.793i) q^{85} +(-15.1215 - 26.1911i) q^{87} +(-423.011 + 732.676i) q^{89} +(1162.25 - 287.583i) q^{91} +(182.168 - 315.524i) q^{93} +(193.625 + 335.369i) q^{95} +(359.129 + 622.030i) q^{97} -1583.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 40 q^{9} - 104 q^{13} + 180 q^{17} + 696 q^{21} + 152 q^{25} + 4 q^{29} + 636 q^{33} - 108 q^{37} - 716 q^{41} + 1600 q^{45} - 1096 q^{49} + 4528 q^{53} - 104 q^{57} + 580 q^{61} - 208 q^{65}+ \cdots - 156 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 40 * q^9 - 104 * q^13 + 180 * q^17 + 696 * q^21 + 152 * q^25 + 4 * q^29 + 636 * q^33 - 108 * q^37 - 716 * q^41 + 1600 * q^45 - 1096 * q^49 + 4528 * q^53 - 104 * q^57 + 580 * q^61 - 208 * q^65 + 3500 * q^69 + 2256 * q^73 + 3608 * q^77 - 1444 * q^81 + 920 * q^85 - 4236 * q^89 + 1552 * q^93 - 156 * q^97

Introduction

L-functions

Modular forms

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Database

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