LMFDB - Embedded newform 688.2.bg.c.225.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(17,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 38]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("688.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.bg (of order $21$, degree $12$, not 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:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			225.2
Character	$\chi$	$=$	688.225
Dual form			688.2.bg.c.529.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.240184 + 0.222858i) q^{3} +(0.0260188 - 0.00392170i) q^{5} +(-1.56464 - 2.71003i) q^{7} +(-0.216168 - 2.88456i) q^{9} +O(q^{10})$	\(q+(0.240184 + 0.222858i) q^{3} +(0.0260188 - 0.00392170i) q^{5} +(-1.56464 - 2.71003i) q^{7} +(-0.216168 - 2.88456i) q^{9} +(-3.36665 - 1.62129i) q^{11} +(-1.78860 + 4.55728i) q^{13} +(0.00712328 + 0.00485657i) q^{15} +(-5.84766 - 0.881392i) q^{17} +(-0.296638 + 3.95836i) q^{19} +(0.228152 - 0.999599i) q^{21} +(-0.284113 + 0.193705i) q^{23} +(-4.77720 + 1.47357i) q^{25} +(1.20379 - 1.50950i) q^{27} +(0.971806 - 0.901704i) q^{29} +(-2.06955 - 0.638371i) q^{31} +(-0.447297 - 1.13969i) q^{33} +(-0.0513379 - 0.0643757i) q^{35} +(5.01814 - 8.69167i) q^{37} +(-1.44522 + 0.695983i) q^{39} +(-0.431814 - 1.89190i) q^{41} +(-6.54856 - 0.341120i) q^{43} +(-0.0169368 - 0.0742049i) q^{45} +(10.0138 - 4.82240i) q^{47} +(-1.39618 + 2.41825i) q^{49} +(-1.20809 - 1.51490i) q^{51} +(-2.12587 - 5.41662i) q^{53} +(-0.0939542 - 0.0289810i) q^{55} +(-0.953402 + 0.884627i) q^{57} +(-0.107460 + 0.134751i) q^{59} +(5.31524 - 1.63953i) q^{61} +(-7.47901 + 5.09911i) q^{63} +(-0.0286650 + 0.125589i) q^{65} +(0.0822742 - 1.09787i) q^{67} +(-0.111408 - 0.0167921i) q^{69} +(4.37443 + 2.98244i) q^{71} +(-2.70214 + 6.88494i) q^{73} +(-1.47581 - 0.710711i) q^{75} +(0.873829 + 11.6604i) q^{77} +(-2.70765 - 4.68979i) q^{79} +(-7.95548 + 1.19910i) q^{81} +(1.49132 + 1.38374i) q^{83} -0.155605 q^{85} +0.434365 q^{87} +(2.93360 + 2.72198i) q^{89} +(15.1489 - 2.28333i) q^{91} +(-0.354806 - 0.614542i) q^{93} +(0.00780535 + 0.104155i) q^{95} +(-8.11049 - 3.90581i) q^{97} +(-3.94895 + 10.0618i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{5}{21}\right)$	$1$

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
$3$	0.240184	+	0.222858i	0.138670	+	0.128667i	0.746459	−	0.665432i	$-0.231753\pi$
$3$	0.240184	+	0.222858i	0.138670	+	0.128667i	−0.607788	+	0.794099i	$0.707943\pi$
$4$		0			0
$5$	0.0260188	−	0.00392170i	0.0116360	−	0.00175384i	−0.143222	−	0.989691i	$-0.545746\pi$
$5$	0.0260188	−	0.00392170i	0.0116360	−	0.00175384i	0.154858	+	0.987937i	$0.450508\pi$
$6$		0			0
$7$	−1.56464	−	2.71003i	−0.591377	−	1.02430i	−0.994047	−	0.108950i	$-0.965251\pi$
$7$	−1.56464	−	2.71003i	−0.591377	−	1.02430i	0.402670	−	0.915345i	$-0.368082\pi$
$8$		0			0
$9$	−0.216168	−	2.88456i	−0.0720559	−	0.961519i
$10$		0			0
$11$	−3.36665	−	1.62129i	−1.01508	−	0.488838i	−0.149052	−	0.988829i	$-0.547622\pi$
$11$	−3.36665	−	1.62129i	−1.01508	−	0.488838i	−0.866030	+	0.499992i	$0.833336\pi$
$12$		0			0
$13$	−1.78860	+	4.55728i	−0.496069	+	1.26396i	0.435431	+	0.900222i	$0.356596\pi$
$13$	−1.78860	+	4.55728i	−0.496069	+	1.26396i	−0.931500	+	0.363741i	$0.881499\pi$
$14$		0			0
$15$	0.00712328	+	0.00485657i	0.00183922	+	0.00125396i
$16$		0			0
$17$	−5.84766	−	0.881392i	−1.41827	−	0.213769i	−0.605261	−	0.796027i	$-0.706931\pi$
$17$	−5.84766	−	0.881392i	−1.41827	−	0.213769i	−0.813004	+	0.582258i	$0.802169\pi$
$18$		0			0
$19$	−0.296638	+	3.95836i	−0.0680535	+	0.908110i	0.853807	+	0.520589i	$0.174288\pi$
$19$	−0.296638	+	3.95836i	−0.0680535	+	0.908110i	−0.921861	+	0.387521i	$0.873331\pi$
$20$		0			0
$21$	0.228152	−	0.999599i	0.0497868	−	0.218130i
$22$		0			0
$23$	−0.284113	+	0.193705i	−0.0592417	+	0.0403903i	−0.592579	−	0.805512i	$-0.701890\pi$
$23$	−0.284113	+	0.193705i	−0.0592417	+	0.0403903i	0.533337	+	0.845903i	$0.320938\pi$
$24$		0			0
$25$	−4.77720	+	1.47357i	−0.955440	+	0.294714i
$26$		0			0
$27$	1.20379	−	1.50950i	0.231669	−	0.290503i
$28$		0			0
$29$	0.971806	−	0.901704i	0.180460	−	0.167442i	−0.584771	−	0.811199i	$-0.698815\pi$
$29$	0.971806	−	0.901704i	0.180460	−	0.167442i	0.765230	+	0.643756i	$0.222625\pi$
$30$		0			0
$31$	−2.06955	−	0.638371i	−0.371702	−	0.114655i	0.103276	−	0.994653i	$-0.467068\pi$
$31$	−2.06955	−	0.638371i	−0.371702	−	0.114655i	−0.474977	+	0.879998i	$0.657544\pi$
$32$		0			0
$33$	−0.447297	−	1.13969i	−0.0778643	−	0.198395i
$34$		0			0
$35$	−0.0513379	−	0.0643757i	−0.00867769	−	0.0108815i
$36$		0			0
$37$	5.01814	−	8.69167i	0.824977	−	1.42890i	−0.0769599	−	0.997034i	$-0.524521\pi$
$37$	5.01814	−	8.69167i	0.824977	−	1.42890i	0.901937	−	0.431868i	$-0.142145\pi$
$38$		0			0
$39$	−1.44522	+	0.695983i	−0.231421	+	0.111446i
$40$		0			0
$41$	−0.431814	−	1.89190i	−0.0674381	−	0.295465i	0.929951	−	0.367684i	$-0.119849\pi$
$41$	−0.431814	−	1.89190i	−0.0674381	−	0.295465i	−0.997389	+	0.0722186i	$0.976992\pi$
$42$		0			0
$43$	−6.54856	−	0.341120i	−0.998646	−	0.0520203i
$43$	−6.54856	−	0.341120i	−0.998646	−	0.0520203i
$44$		0			0
$45$	−0.0169368	−	0.0742049i	−0.00252479	−	0.0110618i
$46$		0			0
$47$	10.0138	−	4.82240i	1.46066	−	0.703419i	0.476255	−	0.879307i	$-0.341994\pi$
$47$	10.0138	−	4.82240i	1.46066	−	0.703419i	0.984410	+	0.175888i	$0.0562798\pi$
$48$		0			0
$49$	−1.39618	+	2.41825i	−0.199454	+	0.345464i
$50$		0			0
$51$	−1.20809	−	1.51490i	−0.169166	−	0.212128i
$52$		0			0
$53$	−2.12587	−	5.41662i	−0.292010	−	0.744030i	−0.999311	−	0.0371173i	$-0.988182\pi$
$53$	−2.12587	−	5.41662i	−0.292010	−	0.744030i	0.707301	−	0.706913i	$-0.249913\pi$
$54$		0			0
$55$	−0.0939542	−	0.0289810i	−0.0126688	−	0.00390780i
$56$		0			0
$57$	−0.953402	+	0.884627i	−0.126281	+	0.117172i
$58$		0			0
$59$	−0.107460	+	0.134751i	−0.0139901	+	0.0175431i	−0.788777	−	0.614679i	$-0.789286\pi$
$59$	−0.107460	+	0.134751i	−0.0139901	+	0.0175431i	0.774787	+	0.632222i	$0.217857\pi$
$60$		0			0
$61$	5.31524	−	1.63953i	0.680547	−	0.209921i	0.0648393	−	0.997896i	$-0.479347\pi$
$61$	5.31524	−	1.63953i	0.680547	−	0.209921i	0.615707	+	0.787975i	$0.288870\pi$
$62$		0			0
$63$	−7.47901	+	5.09911i	−0.942267	+	0.642427i
$64$		0			0
$65$	−0.0286650	+	0.125589i	−0.00355545	+	0.0155774i
$66$		0			0
$67$	0.0822742	−	1.09787i	0.0100514	−	0.134126i	−0.989921	−	0.141624i	$-0.954768\pi$
$67$	0.0822742	−	1.09787i	0.0100514	−	0.134126i	0.999972	+	0.00749766i	$0.00238660\pi$
$68$		0			0
$69$	−0.111408	−	0.0167921i	−0.0134120	−	0.00202153i
$70$		0			0
$71$	4.37443	+	2.98244i	0.519150	+	0.353950i	0.794373	−	0.607430i	$-0.207799\pi$
$71$	4.37443	+	2.98244i	0.519150	+	0.353950i	−0.275223	+	0.961380i	$0.588752\pi$
$72$		0			0
$73$	−2.70214	+	6.88494i	−0.316261	+	0.805821i	0.681029	+	0.732256i	$0.261533\pi$
$73$	−2.70214	+	6.88494i	−0.316261	+	0.805821i	−0.997290	+	0.0735646i	$0.976562\pi$
$74$		0			0
$75$	−1.47581	−	0.710711i	−0.170411	−	0.0820658i
$76$		0			0
$77$	0.873829	+	11.6604i	0.0995821	+	1.32883i
$78$		0			0
$79$	−2.70765	−	4.68979i	−0.304635	−	0.527643i	0.672545	−	0.740056i	$-0.265201\pi$
$79$	−2.70765	−	4.68979i	−0.304635	−	0.527643i	−0.977180	+	0.212413i	$0.931868\pi$
$80$		0			0
$81$	−7.95548	+	1.19910i	−0.883942	+	0.133233i
$82$		0			0
$83$	1.49132	+	1.38374i	0.163694	+	0.151885i	0.757767	−	0.652526i	$-0.226291\pi$
$83$	1.49132	+	1.38374i	0.163694	+	0.151885i	−0.594073	+	0.804411i	$0.702481\pi$
$84$		0			0
$85$	−0.155605			−0.0168778
$86$		0			0
$87$	0.434365			0.0465688
$88$		0			0
$89$	2.93360	+	2.72198i	0.310961	+	0.288530i	0.820178	−	0.572109i	$-0.193874\pi$
$89$	2.93360	+	2.72198i	0.310961	+	0.288530i	−0.509217	+	0.860638i	$0.670065\pi$
$90$		0			0
$91$	15.1489	−	2.28333i	1.58804	−	0.239358i
$92$		0			0
$93$	−0.354806	−	0.614542i	−0.0367917	−	0.0637251i
$94$		0			0
$95$	0.00780535	+	0.104155i	0.000800812	+	0.0106861i
$96$		0			0
$97$	−8.11049	−	3.90581i	−0.823495	−	0.396574i	−0.0258237	−	0.999667i	$-0.508221\pi$
$97$	−8.11049	−	3.90581i	−0.823495	−	0.396574i	−0.797672	+	0.603092i	$0.793935\pi$
$98$		0			0
$99$	−3.94895	+	10.0618i	−0.396884	+	1.01124i
$100$		0			0
$101$	12.6400	+	8.61780i	1.25773	+	0.857503i	0.994242	−	0.107161i	$-0.0341759\pi$
$101$	12.6400	+	8.61780i	1.25773	+	0.857503i	0.263484	+	0.964664i	$0.415128\pi$
$102$		0			0
$103$	−15.8309	−	2.38613i	−1.55987	−	0.235112i	−0.688315	−	0.725412i	$-0.741649\pi$
$103$	−15.8309	−	2.38613i	−1.55987	−	0.235112i	−0.871554	+	0.490300i	$0.836887\pi$
$104$		0			0
$105$	0.00201611	−	0.0269031i	0.000196752	−	0.00262547i
$106$		0			0
$107$	3.68337	−	16.1379i	0.356085	−	1.56011i	−0.406765	−	0.913533i	$-0.633343\pi$
$107$	3.68337	−	16.1379i	0.356085	−	1.56011i	0.762850	−	0.646576i	$-0.223800\pi$
$108$		0			0
$109$	1.69999	−	1.15903i	0.162829	−	0.111015i	−0.479164	−	0.877725i	$-0.659060\pi$
$109$	1.69999	−	1.15903i	0.162829	−	0.111015i	0.641993	+	0.766710i	$0.278108\pi$
$110$		0			0
$111$	3.14229	−	0.969268i	0.298253	−	0.0919988i
$112$		0			0
$113$	7.27343	−	9.12059i	0.684226	−	0.857993i	−0.311509	−	0.950243i	$-0.600835\pi$
$113$	7.27343	−	9.12059i	0.684226	−	0.857993i	0.995736	+	0.0922502i	$0.0294060\pi$
$114$		0			0
$115$	−0.00663263	+	0.00615418i	−0.000618496	+	0.000573880i
$116$		0			0
$117$	13.5324	+	4.17419i	1.25107	+	0.385904i
$118$		0			0
$119$	6.76086	+	17.2264i	0.619767	+	1.57914i
$120$		0			0
$121$	1.84733	+	2.31648i	0.167939	+	0.210589i
$122$		0			0
$123$	0.317911	−	0.550638i	0.0286651	−	0.0496494i
$124$		0			0
$125$	−0.237053	+	0.114159i	−0.0212026	+	0.0102106i
$126$		0			0
$127$	1.67817	+	7.35255i	0.148914	+	0.652433i	0.993188	+	0.116522i	$0.0371746\pi$
$127$	1.67817	+	7.35255i	0.148914	+	0.652433i	−0.844275	+	0.535911i	$0.819968\pi$
$128$		0			0
$129$	−1.49684	−	1.54133i	−0.131789	−	0.135707i
$130$		0			0
$131$	−1.97245	−	8.64186i	−0.172334	−	0.755043i	−0.985034	−	0.172361i	$-0.944861\pi$
$131$	−1.97245	−	8.64186i	−0.172334	−	0.755043i	0.812700	−	0.582682i	$-0.197997\pi$
$132$		0			0
$133$	11.1914	−	5.38950i	0.970418	−	0.467329i
$134$		0			0
$135$	0.0254012	−	0.0439962i	0.00218619	−	0.00378659i
$136$		0			0
$137$	−10.9540	−	13.7359i	−0.935863	−	1.17354i	−0.984617	−	0.174725i	$-0.944096\pi$
$137$	−10.9540	−	13.7359i	−0.935863	−	1.17354i	0.0487538	−	0.998811i	$-0.484475\pi$
$138$		0			0
$139$	5.64316	+	14.3785i	0.478646	+	1.21957i	0.942663	+	0.333745i	$0.108312\pi$
$139$	5.64316	+	14.3785i	0.478646	+	1.21957i	−0.464017	+	0.885826i	$0.653592\pi$
$140$		0			0
$141$	3.47987	+	1.07340i	0.293058	+	0.0903964i
$142$		0			0
$143$	13.4103	−	12.4429i	1.12142	−	1.04053i
$144$		0			0
$145$	0.0217490	−	0.0272724i	0.00180616	−	0.00226485i
$146$		0			0
$147$	−0.874266	+	0.269675i	−0.0721083	+	0.0222425i
$148$		0			0
$149$	−5.99816	+	4.08947i	−0.491388	+	0.335023i	−0.783532	−	0.621352i	$-0.786584\pi$
$149$	−5.99816	+	4.08947i	−0.491388	+	0.335023i	0.292144	+	0.956374i	$0.405631\pi$
$150$		0			0
$151$	−2.23243	+	9.78091i	−0.181673	+	0.795960i	0.799162	+	0.601116i	$0.205277\pi$
$151$	−2.23243	+	9.78091i	−0.181673	+	0.795960i	−0.980834	+	0.194844i	$0.937580\pi$
$152$		0			0
$153$	−1.27835	+	17.0584i	−0.103349	+	1.37909i
$154$		0			0
$155$	−0.0563506	−	0.00849349i	−0.00452619	−	0.000682213i
$156$		0			0
$157$	7.98170	+	5.44183i	0.637009	+	0.434305i	0.838287	−	0.545229i	$-0.183557\pi$
$157$	7.98170	+	5.44183i	0.637009	+	0.434305i	−0.201279	+	0.979534i	$0.564510\pi$
$158$		0			0
$159$	0.696540	−	1.77475i	0.0552392	−	0.140747i
$160$		0			0
$161$	0.969481	+	0.466877i	0.0764058	+	0.0367951i
$162$		0			0
$163$	0.507211	+	6.76827i	0.0397279	+	0.530131i	0.981194	+	0.193024i	$0.0618295\pi$
$163$	0.507211	+	6.76827i	0.0397279	+	0.530131i	−0.941466	+	0.337107i	$0.890551\pi$
$164$		0			0
$165$	−0.0161077	−	0.0278993i	−0.00125398	−	0.00217196i
$166$		0			0
$167$	−13.8043	+	2.08066i	−1.06821	+	0.161006i	−0.659534	−	0.751675i	$-0.729246\pi$
$167$	−13.8043	+	2.08066i	−1.06821	+	0.161006i	−0.408672	+	0.912681i	$0.634008\pi$
$168$		0			0
$169$	−8.04007	−	7.46009i	−0.618467	−	0.573853i
$170$		0			0
$171$	11.4822			0.878069
$172$		0			0
$173$	11.5110			0.875168			0.437584	−	0.899178i	$-0.355834\pi$
$173$	11.5110			0.875168			0.437584	+	0.899178i	$0.355834\pi$
$174$		0			0
$175$	11.4680	+	10.6408i	0.866900	+	0.804366i
$176$		0			0
$177$	−0.0558405	+	0.00841661i	−0.00419723	+	0.000632631i
$178$		0			0
$179$	0.378359	+	0.655338i	0.0282799	+	0.0489823i	0.879819	−	0.475309i	$-0.157664\pi$
$179$	0.378359	+	0.655338i	0.0282799	+	0.0489823i	−0.851539	+	0.524291i	$0.824330\pi$
$180$		0			0
$181$	−0.0777475	−	1.03747i	−0.00577892	−	0.0771144i	0.993569	−	0.113231i	$-0.0361201\pi$
$181$	−0.0777475	−	1.03747i	−0.00577892	−	0.0771144i	−0.999348	+	0.0361170i	$0.988501\pi$
$182$		0			0
$183$	1.64202	+	0.790755i	0.121382	+	0.0584543i
$184$		0			0
$185$	0.0964797	−	0.245826i	0.00709333	−	0.0180735i
$186$		0			0
$187$	18.2580	+	12.4481i	1.33516	+	0.910294i
$188$		0			0
$189$	−5.97428	−	0.900478i	−0.434565	−	0.0655001i
$190$		0			0
$191$	1.03588	−	13.8228i	0.0749534	−	1.00018i	−0.825265	−	0.564745i	$-0.808974\pi$
$191$	1.03588	−	13.8228i	0.0749534	−	1.00018i	0.900219	−	0.435438i	$-0.143407\pi$
$192$		0			0
$193$	−3.28671	+	14.4000i	−0.236583	+	1.03654i	0.707470	+	0.706743i	$0.249836\pi$
$193$	−3.28671	+	14.4000i	−0.236583	+	1.03654i	−0.944053	+	0.329794i	$0.893021\pi$
$194$		0			0
$195$	−0.0348735	+	0.0237764i	−0.00249734	+	0.00170266i
$196$		0			0
$197$	−7.25577	+	2.23811i	−0.516952	+	0.159459i	−0.542243	−	0.840221i	$-0.682425\pi$
$197$	−7.25577	+	2.23811i	−0.516952	+	0.159459i	0.0252911	+	0.999680i	$0.491949\pi$
$198$		0			0
$199$	15.6429	−	19.6156i	1.10890	−	1.39051i	0.196844	−	0.980435i	$-0.436931\pi$
$199$	15.6429	−	19.6156i	1.10890	−	1.39051i	0.912051	−	0.410076i	$-0.134498\pi$
$200$		0			0
$201$	0.264431	−	0.245356i	0.0186515	−	0.0173061i
$202$		0			0
$203$	−3.96417	−	1.22278i	−0.278230	−	0.0858226i
$204$		0			0
$205$	−0.0186548	−	0.0475315i	−0.00130290	−	0.00331975i
$206$		0			0
$207$	0.620169	+	0.777668i	0.0431048	+	0.0540517i
$208$		0			0
$209$	7.41633	−	12.8455i	0.512998	−	0.888539i
$210$		0			0
$211$	−14.4636	+	6.96528i	−0.995712	+	0.479510i	−0.859481	−	0.511167i	$-0.829213\pi$
$211$	−14.4636	+	6.96528i	−0.995712	+	0.479510i	−0.136231	+	0.990677i	$0.543499\pi$
$212$		0			0
$213$	0.386008	+	1.69121i	0.0264489	+	0.115880i
$214$		0			0
$215$	−0.171723	+	0.0168060i	−0.0117114	+	0.00114616i
$216$		0			0
$217$	1.50809	+	6.60736i	0.102376	+	0.448537i
$218$		0			0
$219$	−2.18338	+	1.05146i	−0.147539	+	0.0710510i
$220$		0			0
$221$	14.4759	−	25.0730i	0.973754	−	1.68659i
$222$		0			0
$223$	9.41103	+	11.8011i	0.630209	+	0.790257i	0.989741	−	0.142876i	$-0.0456349\pi$
$223$	9.41103	+	11.8011i	0.630209	+	0.790257i	−0.359532	+	0.933133i	$0.617063\pi$
$224$		0			0
$225$	5.28328	+	13.4616i	0.352219	+	0.897438i
$226$		0			0
$227$	−19.2669	−	5.94305i	−1.27879	−	0.394454i	−0.420346	−	0.907364i	$-0.638091\pi$
$227$	−19.2669	−	5.94305i	−1.27879	−	0.394454i	−0.858442	+	0.512910i	$0.828567\pi$
$228$		0			0
$229$	−9.80918	+	9.10159i	−0.648209	+	0.601450i	−0.934186	−	0.356786i	$-0.883872\pi$
$229$	−9.80918	+	9.10159i	−0.648209	+	0.601450i	0.285977	+	0.958236i	$0.407682\pi$
$230$		0			0
$231$	−2.38875	+	2.99539i	−0.157168	+	0.197082i
$232$		0			0
$233$	8.84939	−	2.72968i	0.579743	−	0.178827i	0.00900522	−	0.999959i	$-0.497134\pi$
$233$	8.84939	−	2.72968i	0.579743	−	0.178827i	0.570738	+	0.821132i	$0.306657\pi$
$234$		0			0
$235$	0.241635	−	0.164744i	0.0157625	−	0.0107467i
$236$		0			0
$237$	0.394824	−	1.72984i	0.0256466	−	0.112365i
$238$		0			0
$239$	1.43677	−	19.1723i	0.0929366	−	1.24015i	−0.734506	−	0.678602i	$-0.762586\pi$
$239$	1.43677	−	19.1723i	0.0929366	−	1.24015i	0.827442	−	0.561550i	$-0.189795\pi$
$240$		0			0
$241$	−5.72297	−	0.862599i	−0.368649	−	0.0555649i	−0.0378946	−	0.999282i	$-0.512065\pi$
$241$	−5.72297	−	0.862599i	−0.368649	−	0.0555649i	−0.330755	+	0.943717i	$0.607303\pi$
$242$		0			0
$243$	−6.96372	−	4.74778i	−0.446723	−	0.304571i
$244$		0			0
$245$	−0.0268432	+	0.0683953i	−0.00171495	+	0.00436961i
$246$		0			0
$247$	−17.5088	−	8.43180i	−1.11406	−	0.536503i
$248$		0			0
$249$	0.0498128	+	0.664706i	0.00315676	+	0.0421240i
$250$		0			0
$251$	3.71016	+	6.42619i	0.234184	+	0.405618i	0.959035	−	0.283287i	$-0.0914250\pi$
$251$	3.71016	+	6.42619i	0.234184	+	0.405618i	−0.724852	+	0.688905i	$0.758092\pi$
$252$		0			0
$253$	1.27056	−	0.191506i	0.0798795	−	0.0120399i
$254$		0			0
$255$	−0.0373740	−	0.0346780i	−0.00234045	−	0.00217162i
$256$		0			0
$257$	−22.5274			−1.40522			−0.702609	−	0.711576i	$-0.747982\pi$
$257$	−22.5274			−1.40522			−0.702609	+	0.711576i	$0.747982\pi$
$258$		0			0
$259$	−31.4063			−1.95149
$260$		0			0
$261$	−2.81109	−	2.60831i	−0.174002	−	0.161450i
$262$		0			0
$263$	−5.37090	+	0.809533i	−0.331184	+	0.0499180i	−0.312530	−	0.949908i	$-0.601176\pi$
$263$	−5.37090	+	0.809533i	−0.331184	+	0.0499180i	−0.0186545	+	0.999826i	$0.505938\pi$
$264$		0			0
$265$	−0.0765549	−	0.132597i	−0.00470273	−	0.00814536i
$266$		0			0
$267$	0.0979876	+	1.30755i	0.00599675	+	0.0800210i
$268$		0			0
$269$	2.16753	+	1.04383i	0.132156	+	0.0636432i	0.498793	−	0.866721i	$-0.333777\pi$
$269$	2.16753	+	1.04383i	0.132156	+	0.0636432i	−0.366637	+	0.930364i	$0.619491\pi$
$270$		0			0
$271$	1.20518	−	3.07074i	0.0732093	−	0.186534i	−0.889585	−	0.456770i	$-0.849006\pi$
$271$	1.20518	−	3.07074i	0.0732093	−	0.186534i	0.962794	+	0.270236i	$0.0871016\pi$
$272$		0			0
$273$	4.14738	+	2.82764i	0.251011	+	0.171136i
$274$		0			0
$275$	18.4722	+	2.78424i	1.11392	+	0.167896i
$276$		0			0
$277$	−1.03223	+	13.7742i	−0.0620207	+	0.827609i	0.876283	+	0.481797i	$0.160016\pi$
$277$	−1.03223	+	13.7742i	−0.0620207	+	0.827609i	−0.938304	+	0.345812i	$0.887603\pi$
$278$		0			0
$279$	−1.39405	+	6.10772i	−0.0834595	+	0.365660i
$280$		0			0
$281$	14.8789	−	10.1443i	0.887603	−	0.605157i	−0.0313571	−	0.999508i	$-0.509983\pi$
$281$	14.8789	−	10.1443i	0.887603	−	0.605157i	0.918960	+	0.394351i	$0.129031\pi$
$282$		0			0
$283$	12.7020	−	3.91805i	0.755057	−	0.232904i	0.106752	−	0.994286i	$-0.465955\pi$
$283$	12.7020	−	3.91805i	0.755057	−	0.232904i	0.648304	+	0.761382i	$0.275479\pi$
$284$		0			0
$285$	−0.0213371	+	0.0267559i	−0.00126390	+	0.00158488i
$286$		0			0
$287$	−4.45148	+	4.13037i	−0.262763	+	0.243808i
$288$		0			0
$289$	17.1735	+	5.29732i	1.01021	+	0.311607i
$290$		0			0
$291$	−1.07757	−	2.74560i	−0.0631682	−	0.160950i
$292$		0			0
$293$	−2.31732	−	2.90583i	−0.135379	−	0.169760i	0.709521	−	0.704685i	$-0.248912\pi$
$293$	−2.31732	−	2.90583i	−0.135379	−	0.169760i	−0.844900	+	0.534924i	$0.820340\pi$
$294$		0			0
$295$	−0.00226753	+	0.00392748i	−0.000132021	+	0.000228667i
$296$		0			0
$297$	−6.50006	+	3.13026i	−0.377172	+	0.181636i
$298$		0			0
$299$	−0.374604	−	1.64125i	−0.0216639	−	0.0949157i
$300$		0			0
$301$	9.32167	+	18.2805i	0.537292	+	1.05367i
$302$		0			0
$303$	1.11538	+	4.88679i	0.0640767	+	0.280738i
$304$		0			0
$305$	0.131866	−	0.0635035i	0.00755064	−	0.00363620i
$306$		0			0
$307$	4.87728	−	8.44770i	0.278361	−	0.482136i	−0.692617	−	0.721306i	$-0.743542\pi$
$307$	4.87728	−	8.44770i	0.278361	−	0.482136i	0.970978	+	0.239170i	$0.0768755\pi$
$308$		0			0
$309$	−3.27057	−	4.10117i	−0.186056	−	0.233307i
$310$		0			0
$311$	1.43950	+	3.66777i	0.0816263	+	0.207980i	0.965860	−	0.259064i	$-0.0834141\pi$
$311$	1.43950	+	3.66777i	0.0816263	+	0.207980i	−0.884234	+	0.467044i	$0.845319\pi$
$312$		0			0
$313$	−21.8743	−	6.74733i	−1.23641	−	0.381382i	−0.393444	−	0.919349i	$-0.628716\pi$
$313$	−21.8743	−	6.74733i	−1.23641	−	0.381382i	−0.842966	+	0.537967i	$0.819193\pi$
$314$		0			0
$315$	−0.174598	+	0.162003i	−0.00983747	+	0.00912783i
$316$		0			0
$317$	6.42280	−	8.05394i	0.360740	−	0.452354i	−0.568031	−	0.823007i	$-0.692295\pi$
$317$	6.42280	−	8.05394i	0.360740	−	0.452354i	0.928771	+	0.370653i	$0.120866\pi$
$318$		0			0
$319$	−4.73365	+	1.46014i	−0.265034	+	0.0817520i
$320$		0			0
$321$	4.48115	−	3.05520i	0.250113	−	0.170524i
$322$		0			0
$323$	5.22351	−	22.8857i	0.290644	−	1.27339i
$324$		0			0
$325$	1.82903	−	24.4067i	0.101456	−	1.35384i
$326$		0			0
$327$	0.666611	+	0.100475i	0.0368637	+	0.00555630i
$328$		0			0
$329$	−28.7368	−	19.5924i	−1.58431	−	1.08017i
$330$		0			0
$331$	4.35233	−	11.0896i	0.239226	−	0.609537i	−0.759956	−	0.649974i	$-0.774780\pi$
$331$	4.35233	−	11.0896i	0.239226	−	0.609537i	0.999182	+	0.0404371i	$0.0128750\pi$
$332$		0			0
$333$	−26.1564	−	12.5963i	−1.43336	−	0.690270i
$334$		0			0
$335$	−0.00216485	−	0.0288880i	−0.000118279	−	0.00157832i
$336$		0			0
$337$	1.63994	+	2.84045i	0.0893330	+	0.154729i	0.907229	−	0.420636i	$-0.138193\pi$
$337$	1.63994	+	2.84045i	0.0893330	+	0.154729i	−0.817896	+	0.575366i	$0.804860\pi$
$338$		0			0
$339$	3.77956	−	0.569677i	0.205278	−	0.0309406i
$340$		0			0
$341$	5.93245	+	5.50451i	0.321260	+	0.298086i
$342$		0			0
$343$	−13.1669			−0.710945
$344$		0			0
$345$	−0.00296456			−0.000159607
$346$		0			0
$347$	−9.18271	−	8.52031i	−0.492954	−	0.457394i	0.394095	−	0.919070i	$-0.371058\pi$
$347$	−9.18271	−	8.52031i	−0.492954	−	0.457394i	−0.887049	+	0.461675i	$0.847248\pi$
$348$		0			0
$349$	−31.6460	+	4.76987i	−1.69397	+	0.255325i	−0.923900	−	0.382633i	$-0.875017\pi$
$349$	−31.6460	+	4.76987i	−1.69397	+	0.255325i	−0.770071	+	0.637958i	$0.779779\pi$
$350$		0			0
$351$	4.72613	+	8.18589i	0.252262	+	0.436930i
$352$		0			0
$353$	−0.179426	−	2.39427i	−0.00954988	−	0.127434i	0.990391	−	0.138295i	$-0.0441620\pi$
$353$	−0.179426	−	2.39427i	−0.00954988	−	0.127434i	−0.999941	+	0.0108603i	$0.996543\pi$
$354$		0			0
$355$	0.125514	+	0.0604442i	0.00666157	+	0.00320805i
$356$		0			0
$357$	−2.21519	+	5.64422i	−0.117240	+	0.298724i
$358$		0			0
$359$	−6.81331	−	4.64523i	−0.359593	−	0.245166i	0.370024	−	0.929022i	$-0.379349\pi$
$359$	−6.81331	−	4.64523i	−0.359593	−	0.245166i	−0.729617	+	0.683856i	$0.760302\pi$
$360$		0			0
$361$	3.20715	+	0.483400i	0.168797	+	0.0254421i
$362$		0			0
$363$	−0.0725471	+	0.968074i	−0.00380773	+	0.0508107i
$364$		0			0
$365$	−0.0433057	+	0.189735i	−0.00226672	+	0.00993117i
$366$		0			0
$367$	−8.55721	+	5.83421i	−0.446683	+	0.304543i	−0.765691	−	0.643209i	$-0.777603\pi$
$367$	−8.55721	+	5.83421i	−0.446683	+	0.304543i	0.319008	+	0.947752i	$0.396650\pi$
$368$		0			0
$369$	−5.36396	+	1.65456i	−0.279236	+	0.0861330i
$370$		0			0
$371$	−11.3530	+	14.2362i	−0.589418	+	0.739107i
$372$		0			0
$373$	−8.47754	+	7.86601i	−0.438950	+	0.407286i	−0.868447	−	0.495782i	$-0.834882\pi$
$373$	−8.47754	+	7.86601i	−0.438950	+	0.407286i	0.429497	+	0.903068i	$0.358691\pi$
$374$		0			0
$375$	−0.0823774	−	0.0254101i	−0.00425395	−	0.00131217i
$376$		0			0
$377$	2.37115	+	6.04159i	0.122120	+	0.311158i
$378$		0			0
$379$	−7.24547	−	9.08554i	−0.372175	−	0.466693i	0.560110	−	0.828419i	$-0.310759\pi$
$379$	−7.24547	−	9.08554i	−0.372175	−	0.466693i	−0.932284	+	0.361726i	$0.882188\pi$
$380$		0			0
$381$	−1.23551	+	2.13996i	−0.0632969	+	0.109633i
$382$		0			0
$383$	−15.8309	+	7.62376i	−0.808921	+	0.389556i	−0.792167	−	0.610304i	$-0.791047\pi$
$383$	−15.8309	+	7.62376i	−0.808921	+	0.389556i	−0.0167535	+	0.999860i	$0.505333\pi$
$384$		0			0
$385$	0.0684647	+	0.299964i	0.00348929	+	0.0152876i
$386$		0			0
$387$	0.431607	+	18.9634i	0.0219398	+	0.963966i
$388$		0			0
$389$	−1.45910	−	6.39272i	−0.0739792	−	0.324124i	0.924374	−	0.381487i	$-0.124588\pi$
$389$	−1.45910	−	6.39272i	−0.0739792	−	0.324124i	−0.998353	+	0.0573631i	$0.981731\pi$
$390$		0			0
$391$	1.83213	−	0.882306i	0.0926546	−	0.0446201i
$392$		0			0
$393$	1.45216	−	2.51521i	0.0732517	−	0.126876i
$394$		0			0
$395$	−0.0888418	−	0.111404i	−0.00447012	−	0.00560535i
$396$		0			0
$397$	−10.7843	−	27.4779i	−0.541248	−	1.37908i	−0.895973	−	0.444108i	$-0.853521\pi$
$397$	−10.7843	−	27.4779i	−0.541248	−	1.37908i	0.354725	−	0.934971i	$-0.384575\pi$
$398$		0			0
$399$	3.88909	+	1.19963i	0.194698	+	0.0600565i
$400$		0			0
$401$	−18.2470	+	16.9307i	−0.911210	+	0.845479i	−0.988452	−	0.151533i	$-0.951579\pi$
$401$	−18.2470	+	16.9307i	−0.911210	+	0.845479i	0.0772425	+	0.997012i	$0.475388\pi$
$402$		0			0
$403$	6.61084	−	8.28973i	0.329309	−	0.412941i
$404$		0			0
$405$	−0.202289	+	0.0623980i	−0.0100518	+	0.00310058i
$406$		0			0
$407$	−30.9860	+	21.1259i	−1.53592	+	1.04717i
$408$		0			0
$409$	2.84836	−	12.4795i	0.140843	−	0.617072i	−0.854398	−	0.519620i	$-0.826074\pi$
$409$	2.84836	−	12.4795i	0.140843	−	0.617072i	0.995240	−	0.0974519i	$-0.0310692\pi$
$410$		0			0
$411$	0.430178	−	5.74033i	0.0212191	−	0.283150i
$412$		0			0
$413$	0.533315	+	0.0803843i	0.0262427	+	0.00395545i
$414$		0			0
$415$	0.0442289	+	0.0301548i	0.00217111	+	0.00148024i
$416$		0			0
$417$	−1.84898	+	4.71112i	−0.0905449	+	0.230705i
$418$		0			0
$419$	25.2233	+	12.1469i	1.23224	+	0.593415i	0.932695	−	0.360667i	$-0.117451\pi$
$419$	25.2233	+	12.1469i	1.23224	+	0.593415i	0.299545	+	0.954082i	$0.403165\pi$
$420$		0			0
$421$	1.03389	+	13.7963i	0.0503885	+	0.672388i	0.964017	+	0.265839i	$0.0856490\pi$
$421$	1.03389	+	13.7963i	0.0503885	+	0.672388i	−0.913629	+	0.406549i	$0.866732\pi$
$422$		0			0
$423$	−16.0751	−	27.8430i	−0.781600	−	1.35377i
$424$		0			0
$425$	29.2342	−	4.40635i	1.41807	−	0.213739i
$426$		0			0
$427$	−12.7596	−	11.8392i	−0.617481	−	0.572938i
$428$		0			0
$429$	5.99394			0.289390
$430$		0			0
$431$	32.0319			1.54292			0.771462	−	0.636276i	$-0.219526\pi$
$431$	32.0319			1.54292			0.771462	+	0.636276i	$0.219526\pi$
$432$		0			0
$433$	−0.278233	−	0.258163i	−0.0133710	−	0.0124065i	0.673459	−	0.739225i	$-0.264808\pi$
$433$	−0.278233	−	0.258163i	−0.0133710	−	0.0124065i	−0.686830	+	0.726818i	$0.740998\pi$
$434$		0			0
$435$	0.0113016	−	0.00170345i	0.000541872	−	8.16741e-5i
$436$		0			0
$437$	−0.682476	−	1.18208i	−0.0326473	−	0.0565467i
$438$		0			0
$439$	−0.364516	−	4.86413i	−0.0173974	−	0.232152i	−0.999124	−	0.0418540i	$-0.986674\pi$
$439$	−0.364516	−	4.86413i	−0.0173974	−	0.232152i	0.981726	−	0.190298i	$-0.0609455\pi$
$440$		0			0
$441$	7.27738	+	3.50460i	0.346542	+	0.166886i
$442$		0			0
$443$	−2.16832	+	5.52479i	−0.103020	+	0.262491i	−0.973085	−	0.230446i	$-0.925981\pi$
$443$	−2.16832	+	5.52479i	−0.103020	+	0.262491i	0.870065	+	0.492937i	$0.164077\pi$
$444$		0			0
$445$	0.0870035	+	0.0593180i	0.00412436	+	0.00281194i
$446$		0			0
$447$	−2.35203	−	0.354512i	−0.111247	−	0.0167679i
$448$		0			0
$449$	2.44448	−	32.6193i	0.115362	−	1.53940i	−0.576336	−	0.817213i	$-0.695518\pi$
$449$	2.44448	−	32.6193i	0.115362	−	1.53940i	0.691698	−	0.722187i	$-0.256863\pi$
$450$		0			0
$451$	−1.61356	+	7.06946i	−0.0759795	+	0.332888i
$452$		0			0
$453$	−2.71595	+	1.85170i	−0.127607	+	0.0870007i
$454$		0			0
$455$	0.385201	−	0.118819i	0.0180585	−	0.00557031i
$456$		0			0
$457$	18.1197	−	22.7213i	0.847602	−	1.06286i	−0.149647	−	0.988739i	$-0.547814\pi$
$457$	18.1197	−	22.7213i	0.847602	−	1.06286i	0.997249	−	0.0741197i	$-0.0236147\pi$
$458$		0			0
$459$	−8.36979	+	7.76603i	−0.390668	+	0.362487i
$460$		0			0
$461$	−13.8948	−	4.28598i	−0.647146	−	0.199618i	−0.0462305	−	0.998931i	$-0.514721\pi$
$461$	−13.8948	−	4.28598i	−0.647146	−	0.199618i	−0.600915	+	0.799313i	$0.705197\pi$
$462$		0			0
$463$	2.25869	+	5.75504i	0.104970	+	0.267459i	0.973703	−	0.227822i	$-0.0731604\pi$
$463$	2.25869	+	5.75504i	0.104970	+	0.267459i	−0.868733	+	0.495281i	$0.835065\pi$
$464$		0			0
$465$	−0.0116417	−	0.0145982i	−0.000539870	−	0.000676975i
$466$		0			0
$467$	−7.90363	+	13.6895i	−0.365736	+	0.633474i	−0.988894	−	0.148622i	$-0.952516\pi$
$467$	−7.90363	+	13.6895i	−0.365736	+	0.633474i	0.623158	+	0.782096i	$0.285850\pi$
$468$		0			0
$469$	−3.10400	+	1.49481i	−0.143329	+	0.0690237i
$470$		0			0
$471$	0.704320	+	3.08583i	0.0324534	+	0.142187i
$472$		0			0
$473$	21.4936	+	11.7656i	0.988278	+	0.540981i
$474$		0			0
$475$	−4.41583	−	19.3470i	−0.202612	−	0.887702i
$476$		0			0
$477$	−15.1650	+	7.30309i	−0.694358	+	0.334385i
$478$		0			0
$479$	19.2621	−	33.3629i	0.880107	−	1.52439i	0.0288864	−	0.999583i	$-0.490804\pi$
$479$	19.2621	−	33.3629i	0.880107	−	1.52439i	0.851221	−	0.524808i	$-0.175863\pi$
$480$		0			0
$481$	30.6350	+	38.4150i	1.39683	+	1.75157i
$482$		0			0
$483$	0.128806	+	0.328193i	0.00586089	+	0.0149333i
$484$		0			0
$485$	−0.226342	−	0.0698174i	−0.0102777	−	0.00317024i
$486$		0			0
$487$	−9.12024	+	8.46235i	−0.413277	+	0.383465i	−0.859229	−	0.511592i	$-0.829056\pi$
$487$	−9.12024	+	8.46235i	−0.413277	+	0.383465i	0.445951	+	0.895057i	$0.352866\pi$
$488$		0			0
$489$	−1.38654	+	1.73867i	−0.0627015	+	0.0786252i
$490$		0			0
$491$	36.8246	−	11.3589i	1.66187	−	0.512619i	0.685381	−	0.728185i	$-0.259636\pi$
$491$	36.8246	−	11.3589i	1.66187	−	0.512619i	0.976489	+	0.215566i	$0.0691597\pi$
$492$		0			0
$493$	−6.47754	+	4.41631i	−0.291734	+	0.198901i
$494$		0			0
$495$	−0.0632876	+	0.277281i	−0.00284457	+	0.0124629i
$496$		0			0
$497$	1.23810	−	16.5213i	0.0555363	−	0.741081i
$498$		0			0
$499$	−41.3979	−	6.23973i	−1.85322	−	0.279329i	−0.874578	−	0.484884i	$-0.838862\pi$
$499$	−41.3979	−	6.23973i	−1.85322	−	0.279329i	−0.978645	+	0.205556i	$0.934100\pi$
$500$		0			0
$501$	−3.77926	−	2.57665i	−0.168845	−	0.115116i
$502$		0			0
$503$	5.36535	−	13.6707i	0.239229	−	0.609546i	−0.759953	−	0.649978i	$-0.774778\pi$
$503$	5.36535	−	13.6707i	0.239229	−	0.609546i	0.999182	+	0.0404321i	$0.0128734\pi$
$504$		0			0
$505$	0.362674	+	0.174654i	0.0161388	+	0.00777202i
$506$		0			0
$507$	−0.268553	−	3.58359i	−0.0119269	−	0.159153i
$508$		0			0
$509$	11.0632	+	19.1620i	0.490368	+	0.849342i	0.999939	−	0.0110868i	$-0.00352910\pi$
$509$	11.0632	+	19.1620i	0.490368	+	0.849342i	−0.509571	+	0.860429i	$0.670196\pi$
$510$		0			0
$511$	22.8863	−	3.44955i	1.01243	−	0.152599i
$512$		0			0
$513$	5.61806	+	5.21280i	0.248043	+	0.230151i
$514$		0			0
$515$	−0.421259			−0.0185629
$516$		0			0
$517$	−41.5315			−1.82655
$518$		0			0
$519$	2.76477	+	2.56533i	0.121360	+	0.112606i
$520$		0			0
$521$	25.2437	−	3.80487i	1.10594	−	0.166694i	0.429413	−	0.903108i	$-0.358720\pi$
$521$	25.2437	−	3.80487i	1.10594	−	0.166694i	0.676531	+	0.736414i	$0.263482\pi$
$522$		0			0
$523$	14.2828	+	24.7386i	0.624544	+	1.08174i	0.988629	+	0.150376i	$0.0480484\pi$
$523$	14.2828	+	24.7386i	0.624544	+	1.08174i	−0.364085	+	0.931366i	$0.618618\pi$
$524$		0			0
$525$	0.383053	+	5.11148i	0.0167178	+	0.223083i
$526$		0			0
$527$	11.5393	+	5.55706i	0.502662	+	0.242069i
$528$		0			0
$529$	−8.35964	+	21.3000i	−0.363463	+	0.926088i
$530$		0			0
$531$	0.411926	+	0.280846i	0.0178761	+	0.0121877i
$532$		0			0
$533$	9.39428	+	1.41596i	0.406911	+	0.0613320i
$534$		0			0
$535$	0.0325488	−	0.434333i	0.00140721	−	0.0187779i
$536$		0			0
$537$	−0.0551715	+	0.241722i	−0.00238083	+	0.0104311i
$538$		0			0
$539$	8.62112	−	5.87778i	0.371338	−	0.253174i
$540$		0			0
$541$	−1.66509	+	0.513613i	−0.0715879	+	0.0220819i	−0.330342	−	0.943861i	$-0.607164\pi$
$541$	−1.66509	+	0.513613i	−0.0715879	+	0.0220819i	0.258754	+	0.965943i	$0.416688\pi$
$542$		0			0
$543$	0.212535	−	0.266510i	0.00912073	−	0.0114370i
$544$		0			0
$545$	0.0396863	−	0.0368235i	0.00169997	−	0.00157735i
$546$		0			0
$547$	−10.3354	−	3.18805i	−0.441909	−	0.136311i	0.0658126	−	0.997832i	$-0.479036\pi$
$547$	−10.3354	−	3.18805i	−0.441909	−	0.136311i	−0.507722	+	0.861521i	$0.669512\pi$
$548$		0			0
$549$	−5.87831	−	14.9777i	−0.250880	−	0.639233i
$550$		0			0
$551$	3.28100	+	4.11424i	0.139775	+	0.175273i
$552$		0			0
$553$	−8.47299	+	14.6756i	−0.360308	+	0.624072i
$554$		0			0
$555$	0.0779574	−	0.0375423i	0.00330911	−	0.00159358i
$556$		0			0
$557$	−0.0339125	−	0.148580i	−0.00143692	−	0.00629554i	0.974204	−	0.225668i	$-0.0724563\pi$
$557$	−0.0339125	−	0.148580i	−0.00143692	−	0.00629554i	−0.975641	+	0.219372i	$0.929599\pi$
$558$		0			0
$559$	13.2674	−	29.2335i	0.561149	−	1.23645i
$560$		0			0
$561$	1.61112	+	7.05878i	0.0680215	+	0.298022i
$562$		0			0
$563$	−21.4601	+	10.3347i	−0.904437	+	0.435554i	−0.827489	−	0.561481i	$-0.810232\pi$
$563$	−21.4601	+	10.3347i	−0.904437	+	0.435554i	−0.0769474	+	0.997035i	$0.524517\pi$
$564$		0			0
$565$	0.153478	−	0.265831i	0.00645685	−	0.0111836i
$566$		0			0
$567$	15.6970	+	19.6834i	0.659213	+	0.826627i
$568$		0			0
$569$	11.4363	+	29.1391i	0.479433	+	1.22158i	0.942187	+	0.335088i	$0.108766\pi$
$569$	11.4363	+	29.1391i	0.479433	+	1.22158i	−0.462754	+	0.886487i	$0.653139\pi$
$570$		0			0
$571$	4.21255	+	1.29940i	0.176290	+	0.0543782i	0.381643	−	0.924310i	$-0.375358\pi$
$571$	4.21255	+	1.29940i	0.176290	+	0.0543782i	−0.205354	+	0.978688i	$0.565834\pi$
$572$		0			0
$573$	3.32933	−	3.08917i	0.139085	−	0.129052i
$574$		0			0
$575$	1.07183	−	1.34403i	0.0446983	−	0.0560499i
$576$		0			0
$577$	24.4740	−	7.54924i	1.01887	−	0.314279i	0.260059	−	0.965593i	$-0.416258\pi$
$577$	24.4740	−	7.54924i	1.01887	−	0.314279i	0.758809	+	0.651314i	$0.225782\pi$
$578$		0			0
$579$	−3.99858	+	2.72619i	−0.166175	+	0.113297i
$580$		0			0
$581$	1.41661	−	6.20657i	0.0587709	−	0.257492i
$582$		0			0
$583$	−1.62488	+	21.6825i	−0.0672956	+	0.897997i
$584$		0			0
$585$	0.368466	+	0.0555373i	0.0152342	+	0.00229619i
$586$		0			0
$587$	−21.8609	−	14.9045i	−0.902298	−	0.615176i	0.0207824	−	0.999784i	$-0.493384\pi$
$587$	−21.8609	−	14.9045i	−0.902298	−	0.615176i	−0.923080	+	0.384608i	$0.874337\pi$
$588$		0			0
$589$	3.14081	−	8.00265i	0.129415	−	0.329744i
$590$		0			0
$591$	−2.24150	−	1.07945i	−0.0922031	−	0.0444027i
$592$		0			0
$593$	−2.82909	−	37.7516i	−0.116177	−	1.55027i	−0.685448	−	0.728121i	$-0.740394\pi$
$593$	−2.82909	−	37.7516i	−0.116177	−	1.55027i	0.569272	−	0.822149i	$-0.307225\pi$
$594$		0			0
$595$	0.243466	+	0.421696i	0.00998114	+	0.0172878i
$596$		0			0
$597$	8.12867	−	1.22520i	0.332684	−	0.0501441i
$598$		0			0
$599$	−5.30017	−	4.91784i	−0.216559	−	0.200938i	0.564432	−	0.825480i	$-0.309095\pi$
$599$	−5.30017	−	4.91784i	−0.216559	−	0.200938i	−0.780991	+	0.624542i	$0.785286\pi$
$600$		0			0
$601$	1.42190			0.0580006			0.0290003	−	0.999579i	$-0.490768\pi$
$601$	1.42190			0.0580006			0.0290003	+	0.999579i	$0.490768\pi$
$602$		0			0
$603$	−3.18466			−0.129689
$604$		0			0
$605$	0.0571498	+	0.0530272i	0.00232347	+	0.00215586i
$606$		0			0
$607$	11.6641	−	1.75808i	0.473432	−	0.0713583i	0.0920071	−	0.995758i	$-0.470672\pi$
$607$	11.6641	−	1.75808i	0.473432	−	0.0713583i	0.381424	+	0.924400i	$0.375434\pi$
$608$		0			0
$609$	−0.679623	−	1.17714i	−0.0275397	−	0.0477002i
$610$		0			0
$611$	4.06631	+	54.2611i	0.164505	+	2.19517i
$612$		0			0
$613$	21.6428	+	10.4226i	0.874145	+	0.420966i	0.816483	−	0.577370i	$-0.195921\pi$
$613$	21.6428	+	10.4226i	0.874145	+	0.420966i	0.0576626	+	0.998336i	$0.481635\pi$
$614$		0			0
$615$	0.00611222	−	0.0155737i	0.000246469	−	0.000627992i
$616$		0			0
$617$	1.32647	+	0.904369i	0.0534015	+	0.0364085i	0.589727	−	0.807603i	$-0.299236\pi$
$617$	1.32647	+	0.904369i	0.0534015	+	0.0364085i	−0.536325	+	0.844011i	$0.680188\pi$
$618$		0			0
$619$	48.0499	+	7.24235i	1.93129	+	0.291095i	0.997731	−	0.0673331i	$-0.0214490\pi$
$619$	48.0499	+	7.24235i	1.93129	+	0.291095i	0.933557	+	0.358428i	$0.116687\pi$
$620$		0			0
$621$	−0.0496137	+	0.662048i	−0.00199093	+	0.0265671i
$622$		0			0
$623$	2.78664	−	12.2091i	0.111644	−	0.489146i
$624$		0			0
$625$	20.6474	−	14.0772i	0.825896	−	0.563086i
$626$		0			0
$627$	4.64400	−	1.43249i	0.185464	−	0.0572080i
$628$		0			0
$629$	−37.0051	+	46.4030i	−1.47549	+	1.85021i
$630$		0			0
$631$	−14.5938	+	13.5411i	−0.580971	+	0.539062i	−0.914975	−	0.403509i	$-0.867790\pi$
$631$	−14.5938	+	13.5411i	−0.580971	+	0.539062i	0.334005	+	0.942571i	$0.391600\pi$
$632$		0			0
$633$	−5.02619	−	1.55037i	−0.199773	−	0.0616218i
$634$		0			0
$635$	0.0724985	+	0.184723i	0.00287701	+	0.00733051i
$636$		0			0
$637$	−8.52344	−	10.6881i	−0.337711	−	0.423476i
$638$		0			0
$639$	7.65740	−	13.2630i	0.302922	−	0.524677i
$640$		0			0
$641$	−19.5056	+	9.39338i	−0.770423	+	0.371016i	−0.777438	−	0.628959i	$-0.783481\pi$
$641$	−19.5056	+	9.39338i	−0.770423	+	0.371016i	0.00701500	+	0.999975i	$0.497767\pi$
$642$		0			0
$643$	0.473254	+	2.07346i	0.0186633	+	0.0817694i	0.983402	−	0.181440i	$-0.0580760\pi$
$643$	0.473254	+	2.07346i	0.0186633	+	0.0817694i	−0.964739	+	0.263210i	$0.915219\pi$
$644$		0			0
$645$	−0.0449906	−	0.0342334i	−0.00177150	−	0.00134794i
$646$		0			0
$647$	−0.593026	−	2.59822i	−0.0233143	−	0.102146i	0.961932	−	0.273289i	$-0.0881114\pi$
$647$	−0.593026	−	2.59822i	−0.0233143	−	0.102146i	−0.985246	+	0.171142i	$0.945254\pi$
$648$		0			0
$649$	0.580250	−	0.279434i	0.0227768	−	0.0109687i
$650$		0			0
$651$	−1.11029	+	1.92307i	−0.0435155	+	0.0753711i
$652$		0			0
$653$	8.03489	+	10.0754i	0.314429	+	0.394282i	0.913783	−	0.406202i	$-0.133147\pi$
$653$	8.03489	+	10.0754i	0.314429	+	0.394282i	−0.599354	+	0.800484i	$0.704576\pi$
$654$		0			0
$655$	−0.0852115	−	0.217115i	−0.00332949	−	0.00848340i
$656$		0			0
$657$	20.4441	+	6.30617i	0.797601	+	0.246027i
$658$		0			0
$659$	−28.5593	+	26.4992i	−1.11251	+	1.03226i	−0.113261	+	0.993565i	$0.536130\pi$
$659$	−28.5593	+	26.4992i	−1.11251	+	1.03226i	−0.999251	+	0.0386951i	$0.987680\pi$
$660$		0			0
$661$	24.8200	−	31.1233i	0.965387	−	1.21056i	−0.0121787	−	0.999926i	$-0.503877\pi$
$661$	24.8200	−	31.1233i	0.965387	−	1.21056i	0.977566	−	0.210631i	$-0.0675519\pi$
$662$		0			0
$663$	9.06460	−	2.79606i	0.352040	−	0.108590i
$664$		0			0
$665$	0.270051	−	0.184118i	0.0104721	−	0.00713977i
$666$		0			0
$667$	−0.101438	+	0.444430i	−0.00392770	+	0.0172084i
$668$		0			0
$669$	−0.369584	+	4.93175i	−0.0142889	+	0.190673i
$670$		0			0
$671$	−20.5527	−	3.09782i	−0.793428	−	0.119590i
$672$		0			0
$673$	−30.9218	−	21.0822i	−1.19195	−	0.812657i	−0.205813	−	0.978591i	$-0.565984\pi$
$673$	−30.9218	−	21.0822i	−1.19195	−	0.812657i	−0.986137	+	0.165934i	$0.946936\pi$
$674$		0			0
$675$	−3.52637	+	8.98505i	−0.135730	+	0.345835i
$676$		0			0
$677$	4.02286	+	1.93731i	0.154611	+	0.0744567i	0.509589	−	0.860418i	$-0.329797\pi$
$677$	4.02286	+	1.93731i	0.154611	+	0.0744567i	−0.354978	+	0.934875i	$0.615512\pi$
$678$		0			0
$679$	2.10512	+	28.0908i	0.0807870	+	1.07803i
$680$		0			0
$681$	−3.30314	−	5.72121i	−0.126577	−	0.219237i
$682$		0			0
$683$	37.5870	−	5.66533i	1.43823	−	0.216778i	0.616812	−	0.787110i	$-0.288424\pi$
$683$	37.5870	−	5.66533i	1.43823	−	0.216778i	0.821414	+	0.570333i	$0.193186\pi$
$684$		0			0
$685$	−0.338878	−	0.314433i	−0.0129479	−	0.0120139i
$686$		0			0
$687$	−4.38437			−0.167274
$688$		0			0
$689$	28.4874			1.08528
$690$		0			0
$691$	−4.15300	−	3.85342i	−0.157988	−	0.146591i	0.597218	−	0.802079i	$-0.296273\pi$
$691$	−4.15300	−	3.85342i	−0.157988	−	0.146591i	−0.755206	+	0.655488i	$0.772463\pi$
$692$		0			0
$693$	33.4463	−	5.04122i	1.27052	−	0.191500i
$694$		0			0
$695$	0.203216	+	0.351981i	0.00770844	+	0.0133514i
$696$		0			0
$697$	0.857594	+	11.4438i	0.0324837	+	0.433464i
$698$		0			0
$699$	2.73381	+	1.31654i	0.103402	+	0.0497959i
$700$		0			0
$701$	−9.33779	+	23.7923i	−0.352683	+	0.898623i	0.638916	+	0.769276i	$0.279383\pi$
$701$	−9.33779	+	23.7923i	−0.352683	+	0.898623i	−0.991600	+	0.129346i	$0.958712\pi$
$702$		0			0
$703$	32.9162	+	22.4419i	1.24146	+	0.846412i
$704$		0			0
$705$	0.0947515	+	0.0142815i	0.00356855	+	0.000537872i
$706$		0			0
$707$	3.57750	−	47.7385i	0.134546	−	1.79539i
$708$		0			0
$709$	7.82205	−	34.2706i	0.293763	−	1.28706i	−0.585480	−	0.810687i	$-0.699094\pi$
$709$	7.82205	−	34.2706i	0.293763	−	1.28706i	0.879243	−	0.476373i	$-0.158049\pi$
$710$		0			0
$711$	−12.9427	+	8.82416i	−0.485388	+	0.330932i
$712$		0			0
$713$	0.711641	−	0.219512i	0.0266512	−	0.00822080i
$714$		0			0
$715$	0.300122	−	0.376341i	0.0112239	−	0.0140743i
$716$		0			0
$717$	4.61779	−	4.28469i	0.172455	−	0.160015i
$718$		0			0
$719$	−4.82456	−	1.48818i	−0.179926	−	0.0554998i	0.203483	−	0.979078i	$-0.434774\pi$
$719$	−4.82456	−	1.48818i	−0.179926	−	0.0554998i	−0.383409	+	0.923579i	$0.625250\pi$
$720$		0			0
$721$	18.3032	+	46.6357i	0.681646	+	1.73681i
$722$		0			0
$723$	−1.18233	−	1.48259i	−0.0439713	−	0.0551383i
$724$		0			0
$725$	−3.31379	+	5.73965i	−0.123071	+	0.213165i
$726$		0			0
$727$	−14.8389	+	7.14602i	−0.550343	+	0.265031i	−0.688324	−	0.725404i	$-0.741653\pi$
$727$	−14.8389	+	7.14602i	−0.550343	+	0.265031i	0.137981	+	0.990435i	$0.455939\pi$
$728$		0			0
$729$	4.75628	+	20.8386i	0.176158	+	0.771800i
$730$		0			0
$731$	37.9931	+	7.76660i	1.40522	+	0.287258i
$732$		0			0
$733$	−2.52808	−	11.0763i	−0.0933769	−	0.409111i	0.906538	−	0.422123i	$-0.138715\pi$
$733$	−2.52808	−	11.0763i	−0.0933769	−	0.409111i	−0.999915	+	0.0130123i	$0.995858\pi$
$734$		0			0
$735$	−0.0216898	+	0.0104452i	−0.000800039	+	0.000385278i
$736$		0			0
$737$	−2.05696	+	3.56276i	−0.0757691	+	0.131236i
$738$		0			0
$739$	18.4052	+	23.0794i	0.677047	+	0.848990i	0.995078	−	0.0990901i	$-0.0315932\pi$
$739$	18.4052	+	23.0794i	0.677047	+	0.848990i	−0.318032	+	0.948080i	$0.603022\pi$
$740$		0			0
$741$	−2.32624	−	5.92717i	−0.0854566	−	0.217740i
$742$		0			0
$743$	16.0390	+	4.94738i	0.588414	+	0.181502i	0.574644	−	0.818403i	$-0.305140\pi$
$743$	16.0390	+	4.94738i	0.588414	+	0.181502i	0.0137702	+	0.999905i	$0.495617\pi$
$744$		0			0
$745$	−0.140027	+	0.129926i	−0.00513019	+	0.00476012i
$746$		0			0
$747$	3.66911	−	4.60092i	0.134246	−	0.168339i
$748$		0			0
$749$	−49.4973	+	15.2679i	−1.80859	+	0.557877i
$750$		0			0
$751$	44.5875	−	30.3993i	1.62702	−	1.10928i	0.715015	−	0.699110i	$-0.246420\pi$
$751$	44.5875	−	30.3993i	1.62702	−	1.10928i	0.912007	−	0.410175i	$-0.134532\pi$
$752$		0			0
$753$	−0.541008	+	2.37031i	−0.0197154	+	0.0863789i
$754$		0			0
$755$	−0.0197273	+	0.263242i	−0.000717950	+	0.00958037i
$756$		0			0
$757$	−19.4613	−	2.93332i	−0.707332	−	0.106613i	−0.214483	−	0.976728i	$-0.568807\pi$
$757$	−19.4613	−	2.93332i	−0.707332	−	0.106613i	−0.492850	+	0.870115i	$0.664045\pi$
$758$		0			0
$759$	0.347847	+	0.237158i	0.0126261	+	0.00860830i
$760$		0			0
$761$	13.7137	−	34.9420i	0.497122	−	1.26665i	−0.433662	−	0.901076i	$-0.642779\pi$
$761$	13.7137	−	34.9420i	0.497122	−	1.26665i	0.930784	−	0.365571i	$-0.119126\pi$
$762$		0			0
$763$	−5.80088	−	2.79356i	−0.210006	−	0.101134i
$764$		0			0
$765$	0.0336369	+	0.448853i	0.00121614	+	0.0162283i
$766$		0			0
$767$	−0.421894	−	0.730742i	−0.0152337	−	0.0263856i
$768$		0			0
$769$	30.4201	−	4.58509i	1.09698	−	0.165342i	0.424475	−	0.905439i	$-0.360459\pi$
$769$	30.4201	−	4.58509i	1.09698	−	0.165342i	0.672500	+	0.740097i	$0.265220\pi$
$770$		0			0
$771$	−5.41072	−	5.02041i	−0.194862	−	0.180806i
$772$		0			0
$773$	−41.3365			−1.48677			−0.743386	−	0.668863i	$-0.766781\pi$
$773$	−41.3365			−1.48677			−0.743386	+	0.668863i	$0.766781\pi$
$774$		0			0
$775$	10.8273			0.388929
$776$		0			0
$777$	−7.54328	−	6.99915i	−0.270614	−	0.251093i
$778$		0			0
$779$	7.61692	−	1.14807i	0.272905	−	0.0411338i
$780$		0			0
$781$	−9.89177	−	17.1330i	−0.353955	−	0.613068i
$782$		0			0
$783$	−0.191276	−	2.55240i	−0.00683564	−	0.0912153i
$784$		0			0
$785$	0.229015	+	0.110288i	0.00817391	+	0.00393635i
$786$		0			0
$787$	−3.59809	+	9.16779i	−0.128258	+	0.326796i	−0.980550	−	0.196270i	$-0.937117\pi$
$787$	−3.59809	+	9.16779i	−0.128258	+	0.326796i	0.852292	+	0.523067i	$0.175212\pi$
$788$		0			0
$789$	−1.47042	−	1.00251i	−0.0523482	−	0.0356904i
$790$		0			0
$791$	−36.0973	−	5.44080i	−1.28347	−	0.193453i
$792$		0			0
$793$	−2.03503	+	27.1555i	−0.0722659	+	0.964321i
$794$		0			0
$795$	0.0111631	−	0.0489086i	0.000395913	−	0.00173461i
$796$		0			0
$797$	−12.9391	+	8.82176i	−0.458328	+	0.312483i	−0.770383	−	0.637581i	$-0.779935\pi$
$797$	−12.9391	+	8.82176i	−0.458328	+	0.312483i	0.312055	+	0.950064i	$0.398983\pi$
$798$		0			0
$799$	−62.8078	+	19.3736i	−2.22198	+	0.685390i
$800$		0			0
$801$	7.21757	−	9.05054i	0.255020	−	0.319785i
$802$		0			0
$803$	20.2596	−	18.7982i	0.714947	−	0.663374i
$804$		0			0
$805$	0.0270557	+	0.00834557i	0.000953587	+	0.000294143i
$806$		0			0
$807$	0.287980	+	0.733762i	0.0101374	+	0.0258296i
$808$		0			0
$809$	32.3483	+	40.5635i	1.13731	+	1.42614i	0.889268	+	0.457387i	$0.151215\pi$
$809$	32.3483	+	40.5635i	1.13731	+	1.42614i	0.248039	+	0.968750i	$0.420214\pi$
$810$		0			0
$811$	5.67356	−	9.82690i	0.199226	−	0.345069i	−0.749052	−	0.662511i	$-0.769491\pi$
$811$	5.67356	−	9.82690i	0.199226	−	0.345069i	0.948278	+	0.317442i	$0.102824\pi$
$812$		0			0
$813$	0.973804	−	0.468959i	0.0341528	−	0.0164471i
$814$		0			0
$815$	0.0397401	+	0.174113i	0.00139204	+	0.00609891i
$816$		0			0
$817$	3.29283	−	25.8204i	0.115202	−	0.903341i
$818$		0			0
$819$	−9.86109	−	43.2043i	−0.344574	−	1.50968i
$820$		0			0
$821$	12.6582	−	6.09588i	0.441775	−	0.212748i	−0.199755	−	0.979846i	$-0.564014\pi$
$821$	12.6582	−	6.09588i	0.441775	−	0.212748i	0.641529	+	0.767098i	$0.278300\pi$
$822$		0			0
$823$	−4.75616	+	8.23791i	−0.165789	+	0.287156i	−0.936935	−	0.349503i	$-0.886351\pi$
$823$	−4.75616	+	8.23791i	−0.165789	+	0.287156i	0.771146	+	0.636658i	$0.219684\pi$
$824$		0			0
$825$	3.81625	+	4.78542i	0.132865	+	0.166607i
$826$		0			0
$827$	−12.3551	−	31.4803i	−0.429629	−	1.09468i	−0.967773	−	0.251823i	$-0.918970\pi$
$827$	−12.3551	−	31.4803i	−0.429629	−	1.09468i	0.538145	−	0.842852i	$-0.319125\pi$
$828$		0			0
$829$	29.4594	+	9.08703i	1.02317	+	0.315606i	0.760540	−	0.649291i	$-0.224934\pi$
$829$	29.4594	+	9.08703i	1.02317	+	0.315606i	0.262628	+	0.964897i	$0.415411\pi$
$830$		0			0
$831$	−3.31761	+	3.07829i	−0.115087	+	0.106785i
$832$		0			0
$833$	10.2958	−	12.9105i	0.356728	−	0.447323i
$834$		0			0
$835$	−0.351011	+	0.108272i	−0.0121472	+	0.00374692i
$836$		0			0
$837$	−3.45491	+	2.35552i	−0.119419	+	0.0814187i
$838$		0			0
$839$	−4.39393	+	19.2511i	−0.151695	+	0.664621i	0.840697	+	0.541506i	$0.182146\pi$
$839$	−4.39393	+	19.2511i	−0.151695	+	0.664621i	−0.992392	+	0.123115i	$0.960712\pi$
$840$		0			0
$841$	−2.03584	+	27.1663i	−0.0702013	+	0.936771i
$842$		0			0
$843$	5.83442	+	0.879398i	0.200948	+	0.0302881i
$844$		0			0
$845$	−0.238449	−	0.162572i	−0.00820290	−	0.00559264i
$846$		0			0
$847$	3.38732	−	8.63076i	0.116390	−	0.296557i
$848$		0			0
$849$	3.92399	+	1.88970i	0.134671	+	0.0648542i
$850$		0			0
$851$	0.257902	+	3.44146i	0.00884075	+	0.117972i
$852$		0			0
$853$	−26.3040	−	45.5599i	−0.900632	−	1.55994i	−0.826676	−	0.562678i	$-0.809771\pi$
$853$	−26.3040	−	45.5599i	−0.900632	−	1.55994i	−0.0739557	−	0.997262i	$-0.523562\pi$
$854$		0			0
$855$	0.298754	−	0.0450299i	0.0102172	−	0.00153999i
$856$		0			0
$857$	−29.8684	−	27.7139i	−1.02029	−	0.946687i	−0.0216959	−	0.999765i	$-0.506907\pi$
$857$	−29.8684	−	27.7139i	−1.02029	−	0.946687i	−0.998590	+	0.0530773i	$0.983097\pi$
$858$		0			0
$859$	−36.3447			−1.24006			−0.620032	−	0.784577i	$-0.712880\pi$
$859$	−36.3447			−1.24006			−0.620032	+	0.784577i	$0.712880\pi$
$860$		0			0
$861$	−1.98966			−0.0678075
$862$		0			0
$863$	−21.1458	−	19.6204i	−0.719811	−	0.667887i	0.232646	−	0.972562i	$-0.425262\pi$
$863$	−21.1458	−	19.6204i	−0.719811	−	0.667887i	−0.952456	+	0.304675i	$0.901452\pi$
$864$		0			0
$865$	0.299503	−	0.0451429i	0.0101834	−	0.00153490i
$866$		0			0
$867$	2.94425	+	5.09959i	0.0999920	+	0.173191i
$868$		0			0
$869$	1.51219	+	20.1788i	0.0512975	+	0.684518i
$870$		0			0
$871$	4.85616	+	2.33860i	0.164545	+	0.0792406i
$872$		0			0
$873$	−9.51329	+	24.2395i	−0.321976	+	0.820382i
$874$		0			0
$875$	0.680274	+	0.463803i	0.0229975	+	0.0156794i
$876$		0			0
$877$	4.72066	+	0.711525i	0.159405	+	0.0240265i	0.228260	−	0.973600i	$-0.426696\pi$
$877$	4.72066	+	0.711525i	0.159405	+	0.0240265i	−0.0688546	+	0.997627i	$0.521934\pi$
$878$		0			0
$879$	0.0910043	−	1.21437i	0.00306950	−	0.0409596i
$880$		0			0
$881$	6.83266	−	29.9359i	0.230198	−	1.00856i	−0.719277	−	0.694723i	$-0.755527\pi$
$881$	6.83266	−	29.9359i	0.230198	−	1.00856i	0.949475	−	0.313841i	$-0.101616\pi$
$882$		0			0
$883$	17.0934	−	11.6541i	0.575238	−	0.392191i	−0.240468	−	0.970657i	$-0.577301\pi$
$883$	17.0934	−	11.6541i	0.575238	−	0.392191i	0.815706	+	0.578466i	$0.196349\pi$
$884$		0			0
$885$	−0.00141990		0.000437980i	−4.77293e−5		1.47225e-5i
$886$		0			0
$887$	1.55633	−	1.95158i	0.0522565	−	0.0655276i	−0.755016	−	0.655706i	$-0.772371\pi$
$887$	1.55633	−	1.95158i	0.0522565	−	0.0655276i	0.807273	+	0.590178i	$0.200943\pi$
$888$		0			0
$889$	17.2999	−	16.0520i	0.580220	−	0.538365i
$890$		0			0
$891$	28.7274	+	8.86121i	0.962403	+	0.296862i
$892$		0			0
$893$	16.1183	+	41.0688i	0.539379	+	1.37432i
$894$		0			0
$895$	0.0124145	+	0.0155673i	0.000414971	+	0.000520357i
$896$		0			0
$897$	0.275791	−	0.477685i	0.00920841	−	0.0159494i
$898$		0			0
$899$	−2.58682	+	1.24575i	−0.0862753	+	0.0415480i
$900$		0			0
$901$	7.65717	+	33.5483i	0.255097	+	1.11765i
$902$		0			0
$903$	−1.83505	+	6.46810i	−0.0610666	+	0.215245i
$904$		0			0
$905$	−0.00609153	−	0.0266887i	−0.000202489	−	0.000887164i
$906$		0			0
$907$	−20.8618	+	10.0465i	−0.692703	+	0.333588i	−0.746895	−	0.664942i	$-0.768456\pi$
$907$	−20.8618	+	10.0465i	−0.692703	+	0.333588i	0.0541915	+	0.998531i	$0.482742\pi$
$908$		0			0
$909$	22.1262	−	38.3237i	0.733879	−	1.27112i
$910$		0			0
$911$	−10.8608	−	13.6190i	−0.359834	−	0.451218i	0.568655	−	0.822576i	$-0.307464\pi$
$911$	−10.8608	−	13.6190i	−0.359834	−	0.451218i	−0.928490	+	0.371358i	$0.878892\pi$
$912$		0			0
$913$	−2.77729	−	7.07643i	−0.0919150	−	0.234196i
$914$		0			0
$915$	0.0458245	+	0.0141350i	0.00151491	+	0.000467288i
$916$		0			0
$917$	−20.3335	+	18.8668i	−0.671472	+	0.623035i
$918$		0			0
$919$	−8.79901	+	11.0336i	−0.290252	+	0.363965i	−0.905483	−	0.424382i	$-0.860491\pi$
$919$	−8.79901	+	11.0336i	−0.290252	+	0.363965i	0.615231	+	0.788347i	$0.289063\pi$
$920$		0			0
$921$	3.05408	−	0.942060i	0.100636	−	0.0310419i
$922$		0			0
$923$	−21.4159	+	14.6011i	−0.704914	+	0.480602i
$924$		0			0
$925$	−11.1649	+	48.9165i	−0.367098	+	1.60836i
$926$		0			0
$927$	−3.46079	+	46.1810i	−0.113667	+	1.51678i
$928$		0			0
$929$	−22.8960	−	3.45101i	−0.751192	−	0.113224i	−0.237724	−	0.971333i	$-0.576401\pi$
$929$	−22.8960	−	3.45101i	−0.751192	−	0.113224i	−0.513468	+	0.858109i	$0.671640\pi$
$930$		0			0
$931$	−9.15814	−	6.24392i	−0.300146	−	0.204636i
$932$		0			0
$933$	−0.471650	+	1.20174i	−0.0154411	+	0.0393434i
$934$		0			0
$935$	0.523868	+	0.252282i	0.0171323	+	0.00825050i
$936$		0			0
$937$	2.99630	+	39.9829i	0.0978850	+	1.30618i	0.802525	+	0.596619i	$0.203490\pi$
$937$	2.99630	+	39.9829i	0.0978850	+	1.30618i	−0.704640	+	0.709565i	$0.748891\pi$
$938$		0			0
$939$	−3.75016	−	6.49548i	−0.122382	−	0.211972i
$940$		0			0
$941$	33.4023	−	5.03458i	1.08888	−	0.164123i	0.420025	−	0.907513i	$-0.362021\pi$
$941$	33.4023	−	5.03458i	1.08888	−	0.164123i	0.668858	+	0.743390i	$0.266783\pi$
$942$		0			0
$943$	0.489155	+	0.453870i	0.0159291	+	0.0147800i
$944$		0			0
$945$	−0.158975			−0.00517145
$946$		0			0
$947$	6.10756			0.198469			0.0992345	−	0.995064i	$-0.468361\pi$
$947$	6.10756			0.198469			0.0992345	+	0.995064i	$0.468361\pi$
$948$		0			0
$949$	−26.5436	−	24.6288i	−0.861641	−	0.799486i
$950$		0			0
$951$	3.33754	−	0.503053i	0.108227	−	0.0163126i
$952$		0			0
$953$	−22.1509	−	38.3665i	−0.717538	−	1.24281i	−0.961973	−	0.273146i	$-0.911936\pi$
$953$	−22.1509	−	38.3665i	−0.717538	−	1.24281i	0.244435	−	0.969666i	$-0.421397\pi$
$954$		0			0
$955$	−0.0272567	−	0.363715i	−0.000882006	−	0.0117695i
$956$		0			0
$957$	−1.46235	−	0.704231i	−0.0472711	−	0.0227646i
$958$		0			0
$959$	−20.0856	+	51.1773i	−0.648599	+	1.65260i
$960$		0			0
$961$	−21.7379	−	14.8206i	−0.701222	−	0.478085i
$962$		0			0
$963$	−47.3469	−	7.13640i	−1.52573	−	0.229967i
$964$		0			0
$965$	−0.0290437	+	0.387561i	−0.000934949	+	0.0124760i
$966$		0			0
$967$	11.5379	−	50.5507i	0.371033	−	1.62560i	−0.352851	−	0.935680i	$-0.614788\pi$
$967$	11.5379	−	50.5507i	0.371033	−	1.62560i	0.723884	−	0.689922i	$-0.242355\pi$
$968$		0			0
$969$	6.35487	−	4.33268i	0.204148	−	0.139186i
$970$		0			0
$971$	−39.1975	+	12.0908i	−1.25791	+	0.388013i	−0.850859	−	0.525394i	$-0.823918\pi$
$971$	−39.1975	+	12.0908i	−1.25791	+	0.388013i	−0.407048	+	0.913407i	$0.633442\pi$
$972$		0			0
$973$	30.1368	−	37.7903i	0.966140	−	1.21150i
$974$		0			0
$975$	5.87854	−	5.45449i	0.188264	−	0.174683i
$976$		0			0
$977$	−38.3948	−	11.8432i	−1.22836	−	0.378899i	−0.388371	−	0.921503i	$-0.626962\pi$
$977$	−38.3948	−	11.8432i	−1.22836	−	0.378899i	−0.839988	+	0.542605i	$0.817438\pi$
$978$		0			0
$979$	−5.46326	−	13.9202i	−0.174607	−	0.444891i
$980$		0			0
$981$	−3.71078	−	4.65317i	−0.118476	−	0.148564i
$982$		0			0
$983$	21.7405	−	37.6556i	0.693414	−	1.20103i	−0.277298	−	0.960784i	$-0.589439\pi$
$983$	21.7405	−	37.6556i	0.693414	−	1.20103i	0.970712	−	0.240245i	$-0.0772276\pi$
$984$		0			0
$985$	−0.180009	+	0.0866878i	−0.00573557	+	0.00276210i
$986$		0			0
$987$	−2.53579	−	11.1100i	−0.0807152	−	0.353636i
$988$		0			0
$989$	1.92661	−	1.17157i	0.0612626	−	0.0372538i
$990$		0			0
$991$	5.52848	+	24.2218i	0.175618	+	0.769432i	0.983620	+	0.180253i	$0.0576917\pi$
$991$	5.52848	+	24.2218i	0.175618	+	0.769432i	−0.808002	+	0.589179i	$0.799451\pi$
$992$		0			0
$993$	3.51676	−	1.69358i	0.111601	−	0.0537442i
$994$		0			0
$995$	0.330083	−	0.571720i	0.0104643	−	0.0181247i
$996$		0			0
$997$	17.7893	+	22.3071i	0.563394	+	0.706474i	0.979181	−	0.202988i	$-0.0650652\pi$
$997$	17.7893	+	22.3071i	0.563394	+	0.706474i	−0.415787	+	0.909462i	$0.636494\pi$
$998$		0			0
$999$	−7.07931	−	18.0378i	−0.223980	−	0.570690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.225.2		36
4.3	odd	2		43.2.g.a.10.1	✓	36
12.11	even	2		387.2.y.c.10.3		36
43.13	even	21	inner	688.2.bg.c.529.2		36
172.23	odd	42		1849.2.a.n.1.16		18
172.63	even	42		1849.2.a.o.1.3		18
172.99	odd	42		43.2.g.a.13.1	yes	36
516.443	even	42		387.2.y.c.271.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.10.1	✓	36	4.3	odd	2
43.2.g.a.13.1	yes	36	172.99	odd	42
387.2.y.c.10.3		36	12.11	even	2
387.2.y.c.271.3		36	516.443	even	42
688.2.bg.c.225.2		36	1.1	even	1	trivial
688.2.bg.c.529.2		36	43.13	even	21	inner
1849.2.a.n.1.16		18	172.23	odd	42
1849.2.a.o.1.3		18	172.63	even	42

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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.bg (of order \(21\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.240184 + 0.222858i) q^{3} +(0.0260188 - 0.00392170i) q^{5} +(-1.56464 - 2.71003i) q^{7} +(-0.216168 - 2.88456i) q^{9} +O(q^{10})\)	\(q+(0.240184 + 0.222858i) q^{3} +(0.0260188 - 0.00392170i) q^{5} +(-1.56464 - 2.71003i) q^{7} +(-0.216168 - 2.88456i) q^{9} +(-3.36665 - 1.62129i) q^{11} +(-1.78860 + 4.55728i) q^{13} +(0.00712328 + 0.00485657i) q^{15} +(-5.84766 - 0.881392i) q^{17} +(-0.296638 + 3.95836i) q^{19} +(0.228152 - 0.999599i) q^{21} +(-0.284113 + 0.193705i) q^{23} +(-4.77720 + 1.47357i) q^{25} +(1.20379 - 1.50950i) q^{27} +(0.971806 - 0.901704i) q^{29} +(-2.06955 - 0.638371i) q^{31} +(-0.447297 - 1.13969i) q^{33} +(-0.0513379 - 0.0643757i) q^{35} +(5.01814 - 8.69167i) q^{37} +(-1.44522 + 0.695983i) q^{39} +(-0.431814 - 1.89190i) q^{41} +(-6.54856 - 0.341120i) q^{43} +(-0.0169368 - 0.0742049i) q^{45} +(10.0138 - 4.82240i) q^{47} +(-1.39618 + 2.41825i) q^{49} +(-1.20809 - 1.51490i) q^{51} +(-2.12587 - 5.41662i) q^{53} +(-0.0939542 - 0.0289810i) q^{55} +(-0.953402 + 0.884627i) q^{57} +(-0.107460 + 0.134751i) q^{59} +(5.31524 - 1.63953i) q^{61} +(-7.47901 + 5.09911i) q^{63} +(-0.0286650 + 0.125589i) q^{65} +(0.0822742 - 1.09787i) q^{67} +(-0.111408 - 0.0167921i) q^{69} +(4.37443 + 2.98244i) q^{71} +(-2.70214 + 6.88494i) q^{73} +(-1.47581 - 0.710711i) q^{75} +(0.873829 + 11.6604i) q^{77} +(-2.70765 - 4.68979i) q^{79} +(-7.95548 + 1.19910i) q^{81} +(1.49132 + 1.38374i) q^{83} -0.155605 q^{85} +0.434365 q^{87} +(2.93360 + 2.72198i) q^{89} +(15.1489 - 2.28333i) q^{91} +(-0.354806 - 0.614542i) q^{93} +(0.00780535 + 0.104155i) q^{95} +(-8.11049 - 3.90581i) q^{97} +(-3.94895 + 10.0618i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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