LMFDB - Embedded newform 810.2.i.c.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,2,Mod(109,810)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("810.109"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,-2,0,0,0,0,8,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.46788256372$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	810.109
Dual form			810.2.i.c.379.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +(-2.50000 - 4.33013i) q^{11} +(-2.59808 - 1.50000i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 - 0.866025i) q^{16} +1.00000i q^{17} +6.00000 q^{19} +(2.23205 - 0.133975i) q^{20} +(-4.33013 - 2.50000i) q^{22} +(0.866025 + 0.500000i) q^{23} +(-1.96410 + 4.59808i) q^{25} -3.00000 q^{26} -4.00000i q^{28} +(4.50000 + 7.79423i) q^{29} +(2.50000 - 4.33013i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(0.500000 + 0.866025i) q^{34} +(8.00000 + 4.00000i) q^{35} -2.00000i q^{37} +(5.19615 - 3.00000i) q^{38} +(1.86603 - 1.23205i) q^{40} +(-1.00000 + 1.73205i) q^{41} +(-0.866025 + 0.500000i) q^{43} -5.00000 q^{44} +1.00000 q^{46} +(-11.2583 + 6.50000i) q^{47} +(4.50000 - 7.79423i) q^{49} +(0.598076 + 4.96410i) q^{50} +(-2.59808 + 1.50000i) q^{52} +(5.00000 - 10.0000i) q^{55} +(-2.00000 - 3.46410i) q^{56} +(7.79423 + 4.50000i) q^{58} +(2.00000 - 3.46410i) q^{59} +(-4.00000 - 6.92820i) q^{61} -5.00000i q^{62} -1.00000 q^{64} +(-0.401924 - 6.69615i) q^{65} +(3.46410 + 2.00000i) q^{67} +(0.866025 + 0.500000i) q^{68} +(8.92820 - 0.535898i) q^{70} +6.00000 q^{71} +2.00000i q^{73} +(-1.00000 - 1.73205i) q^{74} +(3.00000 - 5.19615i) q^{76} +(-17.3205 - 10.0000i) q^{77} +(4.50000 + 7.79423i) q^{79} +(1.00000 - 2.00000i) q^{80} +2.00000i q^{82} +(-3.46410 + 2.00000i) q^{83} +(-1.86603 + 1.23205i) q^{85} +(-0.500000 + 0.866025i) q^{86} +(-4.33013 + 2.50000i) q^{88} -14.0000 q^{89} -12.0000 q^{91} +(0.866025 - 0.500000i) q^{92} +(-6.50000 + 11.2583i) q^{94} +(7.39230 + 11.1962i) q^{95} +(-8.66025 + 5.00000i) q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} + 8 q^{10} - 10 q^{11} + 8 q^{14} - 2 q^{16} + 24 q^{19} + 2 q^{20} + 6 q^{25} - 12 q^{26} + 18 q^{29} + 10 q^{31} + 2 q^{34} + 32 q^{35} + 4 q^{40} - 4 q^{41} - 20 q^{44} + 4 q^{46}+ \cdots - 12 q^{95}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 + 8 * q^10 - 10 * q^11 + 8 * q^14 - 2 * q^16 + 24 * q^19 + 2 * q^20 + 6 * q^25 - 12 * q^26 + 18 * q^29 + 10 * q^31 + 2 * q^34 + 32 * q^35 + 4 * q^40 - 4 * q^41 - 20 * q^44 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 20 * q^55 - 8 * q^56 + 8 * q^59 - 16 * q^61 - 4 * q^64 - 12 * q^65 + 8 * q^70 + 24 * q^71 - 4 * q^74 + 12 * q^76 + 18 * q^79 + 4 * q^80 - 4 * q^85 - 2 * q^86 - 56 * q^89 - 48 * q^91 - 26 * q^94 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	1.23205	+	1.86603i	0.550990	+	0.834512i
$5$	1.23205	+	1.86603i	0.550990	+	0.834512i
$6$		0			0
$7$	3.46410	−	2.00000i	1.30931	−	0.755929i	0.327327	−	0.944911i	$-0.393852\pi$
$7$	3.46410	−	2.00000i	1.30931	−	0.755929i	0.981981	+	0.188982i	$0.0605189\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	−2.50000	−	4.33013i	−0.753778	−	1.30558i	−0.945979	−	0.324227i	$-0.894896\pi$
$11$	−2.50000	−	4.33013i	−0.753778	−	1.30558i	0.192201	−	0.981356i	$-0.438437\pi$
$12$		0			0
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	0.0943882	−	0.995535i	$-0.469911\pi$
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	−0.814965	+	0.579510i	$0.803244\pi$
$14$	2.00000	−	3.46410i	0.534522	−	0.925820i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	2.23205	−	0.133975i	0.499102	−	0.0299576i
$21$		0			0
$22$	−4.33013	−	2.50000i	−0.923186	−	0.533002i
$23$	0.866025	+	0.500000i	0.180579	+	0.104257i	0.587565	−	0.809177i	$-0.300087\pi$
$23$	0.866025	+	0.500000i	0.180579	+	0.104257i	−0.406986	+	0.913434i	$0.633420\pi$
$24$		0			0
$25$	−1.96410	+	4.59808i	−0.392820	+	0.919615i
$26$	−3.00000			−0.588348
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	$0.148230\pi$
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$	0.500000	+	0.866025i	0.0857493	+	0.148522i
$35$	8.00000	+	4.00000i	1.35225	+	0.676123i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	5.19615	−	3.00000i	0.842927	−	0.486664i
$39$		0			0
$40$	1.86603	−	1.23205i	0.295045	−	0.194804i
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	−0.933486	−	0.358614i	$-0.883249\pi$
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	0.777312	+	0.629115i	$0.216583\pi$
$42$		0			0
$43$	−0.866025	+	0.500000i	−0.132068	+	0.0762493i	−0.564578	−	0.825380i	$-0.690961\pi$
$43$	−0.866025	+	0.500000i	−0.132068	+	0.0762493i	0.432511	+	0.901629i	$0.357628\pi$
$44$	−5.00000			−0.753778
$45$		0			0
$46$	1.00000			0.147442
$47$	−11.2583	+	6.50000i	−1.64220	+	0.948122i	−0.662145	+	0.749375i	$0.730354\pi$
$47$	−11.2583	+	6.50000i	−1.64220	+	0.948122i	−0.980051	+	0.198747i	$0.936313\pi$
$48$		0			0
$49$	4.50000	−	7.79423i	0.642857	−	1.11346i
$50$	0.598076	+	4.96410i	0.0845807	+	0.702030i
$51$		0			0
$52$	−2.59808	+	1.50000i	−0.360288	+	0.208013i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	5.00000	−	10.0000i	0.674200	−	1.34840i
$56$	−2.00000	−	3.46410i	−0.267261	−	0.462910i
$57$		0			0
$58$	7.79423	+	4.50000i	1.02343	+	0.590879i
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	−0.999901	−	0.0140840i	$-0.995517\pi$
$61$	−4.00000	−	6.92820i	−0.512148	−	0.887066i	0.487753	−	0.872982i	$-0.337817\pi$
$62$		−	5.00000i		−	0.635001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−0.401924	−	6.69615i	−0.0498525	−	0.830555i
$66$		0			0
$67$	3.46410	+	2.00000i	0.423207	+	0.244339i	0.696449	−	0.717607i	$-0.254762\pi$
$67$	3.46410	+	2.00000i	0.423207	+	0.244339i	−0.273241	+	0.961946i	$0.588096\pi$
$68$	0.866025	+	0.500000i	0.105021	+	0.0606339i
$69$		0			0
$70$	8.92820	−	0.535898i	1.06712	−	0.0640521i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−1.00000	−	1.73205i	−0.116248	−	0.201347i
$75$		0			0
$76$	3.00000	−	5.19615i	0.344124	−	0.596040i
$77$	−17.3205	−	10.0000i	−1.97386	−	1.13961i
$78$		0			0
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	0.999974	+	0.00727784i	$0.00231663\pi$
$79$	4.50000	+	7.79423i	0.506290	+	0.876919i	−0.493684	+	0.869641i	$0.664350\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$			2.00000i			0.220863i
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	−0.677920	−	0.735135i	$-0.737119\pi$
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	0.297686	+	0.954664i	$0.403785\pi$
$84$		0			0
$85$	−1.86603	+	1.23205i	−0.202399	+	0.133635i
$86$	−0.500000	+	0.866025i	−0.0539164	+	0.0933859i
$87$		0			0
$88$	−4.33013	+	2.50000i	−0.461593	+	0.266501i
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$	0.866025	−	0.500000i	0.0902894	−	0.0521286i
$93$		0			0
$94$	−6.50000	+	11.2583i	−0.670424	+	1.16121i
$95$	7.39230	+	11.1962i	0.758434	+	1.14870i
$96$		0			0
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.870433	−	0.492287i	$-0.836161\pi$
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.00888289	+	0.999961i	$0.502828\pi$
$98$		−	9.00000i		−	0.909137i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	0.884941	+	0.465704i	$0.154199\pi$
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	−0.0391591	+	0.999233i	$0.512468\pi$
$102$		0			0
$103$	−8.66025	−	5.00000i	−0.853320	−	0.492665i	0.00844953	−	0.999964i	$-0.497310\pi$
$103$	−8.66025	−	5.00000i	−0.853320	−	0.492665i	−0.861770	+	0.507300i	$0.830644\pi$
$104$	−1.50000	+	2.59808i	−0.147087	+	0.254762i
$105$		0			0
$106$		0			0
$107$			6.00000i			0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$			6.00000i			0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$	−0.669873	−	11.1603i	−0.0638699	−	1.06409i
$111$		0			0
$112$	−3.46410	−	2.00000i	−0.327327	−	0.188982i
$113$	−2.59808	−	1.50000i	−0.244406	−	0.141108i	0.372794	−	0.927914i	$-0.378400\pi$
$113$	−2.59808	−	1.50000i	−0.244406	−	0.141108i	−0.617200	+	0.786806i	$0.711733\pi$
$114$		0			0
$115$	0.133975	+	2.23205i	0.0124932	+	0.208140i
$116$	9.00000			0.835629
$117$		0			0
$118$		−	4.00000i		−	0.368230i
$119$	2.00000	+	3.46410i	0.183340	+	0.317554i
$120$		0			0
$121$	−7.00000	+	12.1244i	−0.636364	+	1.10221i
$122$	−6.92820	−	4.00000i	−0.627250	−	0.362143i
$123$		0			0
$124$	−2.50000	−	4.33013i	−0.224507	−	0.388857i
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−3.69615	−	5.59808i	−0.324174	−	0.490984i
$131$	−1.50000	+	2.59808i	−0.131056	+	0.226995i	−0.924084	−	0.382190i	$-0.875170\pi$
$131$	−1.50000	+	2.59808i	−0.131056	+	0.226995i	0.793028	+	0.609185i	$0.208503\pi$
$132$		0			0
$133$	20.7846	−	12.0000i	1.80225	−	1.04053i
$134$	4.00000			0.345547
$135$		0			0
$136$	1.00000			0.0857493
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.985577	−	0.169226i	$-0.945873\pi$
$137$	−15.5885	+	9.00000i	−1.33181	+	0.768922i	−0.346235	+	0.938148i	$0.612540\pi$
$138$		0			0
$139$	−4.00000	+	6.92820i	−0.339276	+	0.587643i	−0.984297	−	0.176522i	$-0.943515\pi$
$139$	−4.00000	+	6.92820i	−0.339276	+	0.587643i	0.645021	+	0.764165i	$0.276849\pi$
$140$	7.46410	−	4.92820i	0.630832	−	0.416509i
$141$		0			0
$142$	5.19615	−	3.00000i	0.436051	−	0.251754i
$143$			15.0000i			1.25436i
$144$		0			0
$145$	−9.00000	+	18.0000i	−0.747409	+	1.49482i
$146$	1.00000	+	1.73205i	0.0827606	+	0.143346i
$147$		0			0
$148$	−1.73205	−	1.00000i	−0.142374	−	0.0821995i
$149$	6.50000	−	11.2583i	0.532501	−	0.922318i	−0.466779	−	0.884374i	$-0.654586\pi$
$149$	6.50000	−	11.2583i	0.532501	−	0.922318i	0.999280	−	0.0379444i	$-0.0120810\pi$
$150$		0			0
$151$	−6.50000	−	11.2583i	−0.528962	−	0.916190i	−0.999430	−	0.0337724i	$-0.989248\pi$
$151$	−6.50000	−	11.2583i	−0.528962	−	0.916190i	0.470467	−	0.882418i	$-0.344085\pi$
$152$		−	6.00000i		−	0.486664i
$153$		0			0
$154$	−20.0000			−1.61165
$155$	11.1603	−	0.669873i	0.896413	−	0.0538055i
$156$		0			0
$157$	11.2583	+	6.50000i	0.898513	+	0.518756i	0.876717	−	0.481006i	$-0.159728\pi$
$157$	11.2583	+	6.50000i	0.898513	+	0.518756i	0.0217953	+	0.999762i	$0.493062\pi$
$158$	7.79423	+	4.50000i	0.620076	+	0.358001i
$159$		0			0
$160$	−0.133975	−	2.23205i	−0.0105916	−	0.176459i
$161$	4.00000			0.315244
$162$		0			0
$163$			19.0000i			1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$			19.0000i			1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	1.00000	+	1.73205i	0.0780869	+	0.135250i
$165$		0			0
$166$	−2.00000	+	3.46410i	−0.155230	+	0.268866i
$167$	−10.3923	−	6.00000i	−0.804181	−	0.464294i	0.0407502	−	0.999169i	$-0.487025\pi$
$167$	−10.3923	−	6.00000i	−0.804181	−	0.464294i	−0.844931	+	0.534875i	$0.820359\pi$
$168$		0			0
$169$	−2.00000	−	3.46410i	−0.153846	−	0.266469i
$170$	−1.00000	+	2.00000i	−0.0766965	+	0.153393i
$171$		0			0
$172$			1.00000i			0.0762493i
$173$	13.8564	−	8.00000i	1.05348	−	0.608229i	0.129861	−	0.991532i	$-0.458547\pi$
$173$	13.8564	−	8.00000i	1.05348	−	0.608229i	0.923622	+	0.383304i	$0.125214\pi$
$174$		0			0
$175$	2.39230	+	19.8564i	0.180841	+	1.50100i
$176$	−2.50000	+	4.33013i	−0.188445	+	0.326396i
$177$		0			0
$178$	−12.1244	+	7.00000i	−0.908759	+	0.524672i
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	−10.3923	+	6.00000i	−0.770329	+	0.444750i
$183$		0			0
$184$	0.500000	−	0.866025i	0.0368605	−	0.0638442i
$185$	3.73205	−	2.46410i	0.274386	−	0.181164i
$186$		0			0
$187$	4.33013	−	2.50000i	0.316650	−	0.182818i
$188$			13.0000i			0.948122i
$189$		0			0
$190$	12.0000	+	6.00000i	0.870572	+	0.435286i
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	0.736839	−	0.676068i	$-0.236317\pi$
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	−0.953912	+	0.300088i	$0.902984\pi$
$192$		0			0
$193$	3.46410	+	2.00000i	0.249351	+	0.143963i	0.619467	−	0.785022i	$-0.287349\pi$
$193$	3.46410	+	2.00000i	0.249351	+	0.143963i	−0.370116	+	0.928986i	$0.620682\pi$
$194$	−5.00000	+	8.66025i	−0.358979	+	0.621770i
$195$		0			0
$196$	−4.50000	−	7.79423i	−0.321429	−	0.556731i
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$	−3.00000			−0.212664			−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000			−0.212664			−0.106332	+	0.994331i	$0.533911\pi$
$200$	4.59808	+	1.96410i	0.325133	+	0.138883i
$201$		0			0
$202$	14.7224	+	8.50000i	1.03587	+	0.598058i
$203$	31.1769	+	18.0000i	2.18819	+	1.26335i
$204$		0			0
$205$	−4.46410	+	0.267949i	−0.311786	+	0.0187144i
$206$	−10.0000			−0.696733
$207$		0			0
$208$			3.00000i			0.208013i
$209$	−15.0000	−	25.9808i	−1.03757	−	1.79713i
$210$		0			0
$211$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$211$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$	3.00000	+	5.19615i	0.205076	+	0.355202i
$215$	−2.00000	−	1.00000i	−0.136399	−	0.0681994i
$216$		0			0
$217$		−	20.0000i		−	1.35769i
$218$	6.92820	−	4.00000i	0.469237	−	0.270914i
$219$		0			0
$220$	−6.16025	−	9.33013i	−0.415324	−	0.629037i
$221$	1.50000	−	2.59808i	0.100901	−	0.174766i
$222$		0			0
$223$	−19.0526	+	11.0000i	−1.27585	+	0.736614i	−0.976083	−	0.217397i	$-0.930243\pi$
$223$	−19.0526	+	11.0000i	−1.27585	+	0.736614i	−0.299770	+	0.954011i	$0.596910\pi$
$224$	−4.00000			−0.267261
$225$		0			0
$226$	−3.00000			−0.199557
$227$	−1.73205	+	1.00000i	−0.114960	+	0.0663723i	−0.556378	−	0.830930i	$-0.687809\pi$
$227$	−1.73205	+	1.00000i	−0.114960	+	0.0663723i	0.441417	+	0.897302i	$0.354476\pi$
$228$		0			0
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	−0.831092	−	0.556136i	$-0.812283\pi$
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	0.897173	+	0.441679i	$0.145617\pi$
$230$	1.23205	+	1.86603i	0.0812390	+	0.123042i
$231$		0			0
$232$	7.79423	−	4.50000i	0.511716	−	0.295439i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	−26.0000	−	13.0000i	−1.69605	−	0.848026i
$236$	−2.00000	−	3.46410i	−0.130189	−	0.225494i
$237$		0			0
$238$	3.46410	+	2.00000i	0.224544	+	0.129641i
$239$	−8.00000	+	13.8564i	−0.517477	+	0.896296i	0.482317	+	0.875997i	$0.339795\pi$
$239$	−8.00000	+	13.8564i	−0.517477	+	0.896296i	−0.999794	+	0.0202996i	$0.993538\pi$
$240$		0			0
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	0.952141	+	0.305661i	$0.0988773\pi$
$241$	11.5000	+	19.9186i	0.740780	+	1.28307i	−0.211360	+	0.977408i	$0.567789\pi$
$242$			14.0000i			0.899954i
$243$		0			0
$244$	−8.00000			−0.512148
$245$	20.0885	−	1.20577i	1.28340	−	0.0770339i
$246$		0			0
$247$	−15.5885	−	9.00000i	−0.991870	−	0.572656i
$248$	−4.33013	−	2.50000i	−0.274963	−	0.158750i
$249$		0			0
$250$	−8.52628	+	7.23205i	−0.539249	+	0.457395i
$251$	−15.0000			−0.946792			−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000			−0.946792			−0.473396	+	0.880850i	$0.656972\pi$
$252$		0			0
$253$		−	5.00000i		−	0.314347i
$254$	2.00000	+	3.46410i	0.125491	+	0.217357i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−18.1865	−	10.5000i	−1.13444	−	0.654972i	−0.189396	−	0.981901i	$-0.560653\pi$
$257$	−18.1865	−	10.5000i	−1.13444	−	0.654972i	−0.945049	+	0.326929i	$0.893986\pi$
$258$		0			0
$259$	−4.00000	−	6.92820i	−0.248548	−	0.430498i
$260$	−6.00000	−	3.00000i	−0.372104	−	0.186052i
$261$		0			0
$262$			3.00000i			0.185341i
$263$	6.92820	−	4.00000i	0.427211	−	0.246651i	−0.270947	−	0.962594i	$-0.587337\pi$
$263$	6.92820	−	4.00000i	0.427211	−	0.246651i	0.698158	+	0.715944i	$0.254003\pi$
$264$		0			0
$265$		0			0
$266$	12.0000	−	20.7846i	0.735767	−	1.27439i
$267$		0			0
$268$	3.46410	−	2.00000i	0.211604	−	0.122169i
$269$	9.00000			0.548740			0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000			0.548740			0.274370	+	0.961624i	$0.411531\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$	0.866025	−	0.500000i	0.0525105	−	0.0303170i
$273$		0			0
$274$	−9.00000	+	15.5885i	−0.543710	+	0.941733i
$275$	24.8205	−	2.99038i	1.49673	−	0.180327i
$276$		0			0
$277$	−22.5167	+	13.0000i	−1.35290	+	0.781094i	−0.988654	−	0.150210i	$-0.952005\pi$
$277$	−22.5167	+	13.0000i	−1.35290	+	0.781094i	−0.364241	+	0.931305i	$0.618672\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$	4.00000	−	8.00000i	0.239046	−	0.478091i
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	17.3205	+	10.0000i	1.02960	+	0.594438i	0.916869	−	0.399188i	$-0.130708\pi$
$283$	17.3205	+	10.0000i	1.02960	+	0.594438i	0.112728	+	0.993626i	$0.464041\pi$
$284$	3.00000	−	5.19615i	0.178017	−	0.308335i
$285$		0			0
$286$	7.50000	+	12.9904i	0.443484	+	0.768137i
$287$			8.00000i			0.472225i
$288$		0			0
$289$	16.0000			0.941176
$290$	1.20577	+	20.0885i	0.0708053	+	1.17963i
$291$		0			0
$292$	1.73205	+	1.00000i	0.101361	+	0.0585206i
$293$	−5.19615	−	3.00000i	−0.303562	−	0.175262i	0.340480	−	0.940252i	$-0.389411\pi$
$293$	−5.19615	−	3.00000i	−0.303562	−	0.175262i	−0.644042	+	0.764990i	$0.722744\pi$
$294$		0			0
$295$	8.92820	−	0.535898i	0.519820	−	0.0312012i
$296$	−2.00000			−0.116248
$297$		0			0
$298$		−	13.0000i		−	0.753070i
$299$	−1.50000	−	2.59808i	−0.0867472	−	0.150251i
$300$		0			0
$301$	−2.00000	+	3.46410i	−0.115278	+	0.199667i
$302$	−11.2583	−	6.50000i	−0.647844	−	0.374033i
$303$		0			0
$304$	−3.00000	−	5.19615i	−0.172062	−	0.298020i
$305$	8.00000	−	16.0000i	0.458079	−	0.916157i
$306$		0			0
$307$			21.0000i			1.19853i	0.800549	+	0.599267i	$0.204541\pi$
$307$			21.0000i			1.19853i	−0.800549	+	0.599267i	$0.795459\pi$
$308$	−17.3205	+	10.0000i	−0.986928	+	0.569803i
$309$		0			0
$310$	9.33013	−	6.16025i	0.529916	−	0.349879i
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	12.1244	−	7.00000i	0.685309	−	0.395663i	−0.116543	−	0.993186i	$-0.537181\pi$
$313$	12.1244	−	7.00000i	0.685309	−	0.395663i	0.801852	+	0.597522i	$0.203848\pi$
$314$	13.0000			0.733632
$315$		0			0
$316$	9.00000			0.506290
$317$	3.46410	−	2.00000i	0.194563	−	0.112331i	−0.399554	−	0.916710i	$-0.630835\pi$
$317$	3.46410	−	2.00000i	0.194563	−	0.112331i	0.594117	+	0.804379i	$0.297502\pi$
$318$		0			0
$319$	22.5000	−	38.9711i	1.25976	−	2.18197i
$320$	−1.23205	−	1.86603i	−0.0688737	−	0.104314i
$321$		0			0
$322$	3.46410	−	2.00000i	0.193047	−	0.111456i
$323$			6.00000i			0.333849i
$324$		0			0
$325$	12.0000	−	9.00000i	0.665640	−	0.499230i
$326$	9.50000	+	16.4545i	0.526156	+	0.911330i
$327$		0			0
$328$	1.73205	+	1.00000i	0.0956365	+	0.0552158i
$329$	−26.0000	+	45.0333i	−1.43343	+	2.48277i
$330$		0			0
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	0.695266	−	0.718752i	$-0.255287\pi$
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	−0.970091	+	0.242742i	$0.921953\pi$
$332$			4.00000i			0.219529i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	0.535898	+	8.92820i	0.0292793	+	0.487800i
$336$		0			0
$337$	−13.8564	−	8.00000i	−0.754807	−	0.435788i	0.0726214	−	0.997360i	$-0.476864\pi$
$337$	−13.8564	−	8.00000i	−0.754807	−	0.435788i	−0.827428	+	0.561572i	$0.810197\pi$
$338$	−3.46410	−	2.00000i	−0.188422	−	0.108786i
$339$		0			0
$340$	0.133975	+	2.23205i	0.00726579	+	0.121050i
$341$	−25.0000			−1.35383
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$	0.500000	+	0.866025i	0.0269582	+	0.0466930i
$345$		0			0
$346$	8.00000	−	13.8564i	0.430083	−	0.744925i
$347$	5.19615	+	3.00000i	0.278944	+	0.161048i	0.632945	−	0.774197i	$-0.281846\pi$
$347$	5.19615	+	3.00000i	0.278944	+	0.161048i	−0.354001	+	0.935245i	$0.615179\pi$
$348$		0			0
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	0.996716	−	0.0809766i	$-0.0258039\pi$
$349$	8.00000	+	13.8564i	0.428230	+	0.741716i	−0.568486	+	0.822693i	$0.692471\pi$
$350$	12.0000	+	16.0000i	0.641427	+	0.855236i
$351$		0			0
$352$			5.00000i			0.266501i
$353$	−9.52628	+	5.50000i	−0.507033	+	0.292735i	−0.731613	−	0.681720i	$-0.761232\pi$
$353$	−9.52628	+	5.50000i	−0.507033	+	0.292735i	0.224580	+	0.974456i	$0.427899\pi$
$354$		0			0
$355$	7.39230	+	11.1962i	0.392343	+	0.594230i
$356$	−7.00000	+	12.1244i	−0.370999	+	0.642590i
$357$		0			0
$358$	17.3205	−	10.0000i	0.915417	−	0.528516i
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	−19.0526	+	11.0000i	−1.00138	+	0.578147i
$363$		0			0
$364$	−6.00000	+	10.3923i	−0.314485	+	0.544705i
$365$	−3.73205	+	2.46410i	−0.195344	+	0.128977i
$366$		0			0
$367$	19.0526	−	11.0000i	0.994535	−	0.574195i	0.0879086	−	0.996129i	$-0.471982\pi$
$367$	19.0526	−	11.0000i	0.994535	−	0.574195i	0.906627	+	0.421933i	$0.138648\pi$
$368$		−	1.00000i		−	0.0521286i
$369$		0			0
$370$	2.00000	−	4.00000i	0.103975	−	0.207950i
$371$		0			0
$372$		0			0
$373$	−18.1865	−	10.5000i	−0.941663	−	0.543669i	−0.0511818	−	0.998689i	$-0.516299\pi$
$373$	−18.1865	−	10.5000i	−0.941663	−	0.543669i	−0.890481	+	0.455020i	$0.849632\pi$
$374$	2.50000	−	4.33013i	0.129272	−	0.223906i
$375$		0			0
$376$	6.50000	+	11.2583i	0.335212	+	0.580604i
$377$		−	27.0000i		−	1.39057i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	13.3923	−	0.803848i	0.687011	−	0.0412365i
$381$		0			0
$382$	−5.19615	−	3.00000i	−0.265858	−	0.153493i
$383$	−16.4545	−	9.50000i	−0.840785	−	0.485427i	0.0167461	−	0.999860i	$-0.494669\pi$
$383$	−16.4545	−	9.50000i	−0.840785	−	0.485427i	−0.857531	+	0.514432i	$0.828003\pi$
$384$		0			0
$385$	−2.67949	−	44.6410i	−0.136560	−	2.27512i
$386$	4.00000			0.203595
$387$		0			0
$388$			10.0000i			0.507673i
$389$	2.50000	+	4.33013i	0.126755	+	0.219546i	0.922418	−	0.386194i	$-0.126210\pi$
$389$	2.50000	+	4.33013i	0.126755	+	0.219546i	−0.795663	+	0.605740i	$0.792877\pi$
$390$		0			0
$391$	−0.500000	+	0.866025i	−0.0252861	+	0.0437968i
$392$	−7.79423	−	4.50000i	−0.393668	−	0.227284i
$393$		0			0
$394$	−11.0000	−	19.0526i	−0.554172	−	0.959854i
$395$	−9.00000	+	18.0000i	−0.452839	+	0.905678i
$396$		0			0
$397$			13.0000i			0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$			13.0000i			0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$	−2.59808	+	1.50000i	−0.130230	+	0.0751882i
$399$		0			0
$400$	4.96410	−	0.598076i	0.248205	−	0.0299038i
$401$	−8.00000	+	13.8564i	−0.399501	+	0.691956i	−0.993664	−	0.112388i	$-0.964150\pi$
$401$	−8.00000	+	13.8564i	−0.399501	+	0.691956i	0.594163	+	0.804344i	$0.297483\pi$
$402$		0			0
$403$	−12.9904	+	7.50000i	−0.647097	+	0.373602i
$404$	17.0000			0.845782
$405$		0			0
$406$	36.0000			1.78665
$407$	−8.66025	+	5.00000i	−0.429273	+	0.247841i
$408$		0			0
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	−0.697406	−	0.716677i	$-0.745662\pi$
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	0.969363	+	0.245633i	$0.0789957\pi$
$410$	−3.73205	+	2.46410i	−0.184313	+	0.121693i
$411$		0			0
$412$	−8.66025	+	5.00000i	−0.426660	+	0.246332i
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$	1.50000	+	2.59808i	0.0735436	+	0.127381i
$417$		0			0
$418$	−25.9808	−	15.0000i	−1.27076	−	0.733674i
$419$	11.5000	−	19.9186i	0.561812	−	0.973087i	−0.435527	−	0.900176i	$-0.643438\pi$
$419$	11.5000	−	19.9186i	0.561812	−	0.973087i	0.997338	−	0.0729107i	$-0.0232288\pi$
$420$		0			0
$421$	13.0000	+	22.5167i	0.633581	+	1.09739i	0.986814	+	0.161859i	$0.0517491\pi$
$421$	13.0000	+	22.5167i	0.633581	+	1.09739i	−0.353233	+	0.935536i	$0.614918\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.59808	−	1.96410i	−0.223039	−	0.0952729i
$426$		0			0
$427$	−27.7128	−	16.0000i	−1.34112	−	0.774294i
$428$	5.19615	+	3.00000i	0.251166	+	0.145010i
$429$		0			0
$430$	−2.23205	+	0.133975i	−0.107639	+	0.00646083i
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		−	22.0000i		−	1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$		−	22.0000i		−	1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	−10.0000	−	17.3205i	−0.480015	−	0.831411i
$435$		0			0
$436$	4.00000	−	6.92820i	0.191565	−	0.331801i
$437$	5.19615	+	3.00000i	0.248566	+	0.143509i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$	−10.0000	−	5.00000i	−0.476731	−	0.238366i
$441$		0			0
$442$		−	3.00000i		−	0.142695i
$443$	25.9808	−	15.0000i	1.23438	−	0.712672i	0.266443	−	0.963851i	$-0.414152\pi$
$443$	25.9808	−	15.0000i	1.23438	−	0.712672i	0.967941	+	0.251179i	$0.0808184\pi$
$444$		0			0
$445$	−17.2487	−	26.1244i	−0.817667	−	1.23841i
$446$	−11.0000	+	19.0526i	−0.520865	+	0.902165i
$447$		0			0
$448$	−3.46410	+	2.00000i	−0.163663	+	0.0944911i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$	−2.59808	+	1.50000i	−0.122203	+	0.0705541i
$453$		0			0
$454$	−1.00000	+	1.73205i	−0.0469323	+	0.0812892i
$455$	−14.7846	−	22.3923i	−0.693113	−	1.04977i
$456$		0			0
$457$	13.8564	−	8.00000i	0.648175	−	0.374224i	−0.139581	−	0.990211i	$-0.544576\pi$
$457$	13.8564	−	8.00000i	0.648175	−	0.374224i	0.787757	+	0.615986i	$0.211242\pi$
$458$		−	2.00000i		−	0.0934539i
$459$		0			0
$460$	2.00000	+	1.00000i	0.0932505	+	0.0466252i
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	0.787668	−	0.616100i	$-0.211288\pi$
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	−0.927392	+	0.374091i	$0.877955\pi$
$462$		0			0
$463$	−5.19615	−	3.00000i	−0.241486	−	0.139422i	0.374374	−	0.927278i	$-0.377858\pi$
$463$	−5.19615	−	3.00000i	−0.241486	−	0.139422i	−0.615859	+	0.787856i	$0.711191\pi$
$464$	4.50000	−	7.79423i	0.208907	−	0.361838i
$465$		0			0
$466$	−7.00000	−	12.1244i	−0.324269	−	0.561650i
$467$			6.00000i			0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$			6.00000i			0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	−29.0167	+	1.74167i	−1.33844	+	0.0803372i
$471$		0			0
$472$	−3.46410	−	2.00000i	−0.159448	−	0.0920575i
$473$	4.33013	+	2.50000i	0.199099	+	0.114950i
$474$		0			0
$475$	−11.7846	+	27.5885i	−0.540715	+	1.26585i
$476$	4.00000			0.183340
$477$		0			0
$478$			16.0000i			0.731823i
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	0.887967	−	0.459908i	$-0.152118\pi$
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	−0.842275	+	0.539048i	$0.818784\pi$
$480$		0			0
$481$	−3.00000	+	5.19615i	−0.136788	+	0.236924i
$482$	19.9186	+	11.5000i	0.907267	+	0.523811i
$483$		0			0
$484$	7.00000	+	12.1244i	0.318182	+	0.551107i
$485$	−20.0000	−	10.0000i	−0.908153	−	0.454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	−6.92820	+	4.00000i	−0.313625	+	0.181071i
$489$		0			0
$490$	16.7942	−	11.0885i	0.758686	−	0.500925i
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	−0.457263	−	0.889332i	$-0.651170\pi$
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	0.998815	−	0.0486647i	$-0.0154966\pi$
$492$		0			0
$493$	−7.79423	+	4.50000i	−0.351034	+	0.202670i
$494$	−18.0000			−0.809858
$495$		0			0
$496$	−5.00000			−0.224507
$497$	20.7846	−	12.0000i	0.932317	−	0.538274i
$498$		0			0
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	−0.968500	−	0.249015i	$-0.919893\pi$
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	0.699903	+	0.714238i	$0.253227\pi$
$500$	−3.76795	+	10.5263i	−0.168508	+	0.470750i
$501$		0			0
$502$	−12.9904	+	7.50000i	−0.579789	+	0.334741i
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$	−17.0000	+	34.0000i	−0.756490	+	1.51298i
$506$	−2.50000	−	4.33013i	−0.111139	−	0.192498i
$507$		0			0
$508$	3.46410	+	2.00000i	0.153695	+	0.0887357i
$509$	10.5000	−	18.1865i	0.465404	−	0.806104i	−0.533815	−	0.845601i	$-0.679242\pi$
$509$	10.5000	−	18.1865i	0.465404	−	0.806104i	0.999220	+	0.0394971i	$0.0125756\pi$
$510$		0			0
$511$	4.00000	+	6.92820i	0.176950	+	0.306486i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−21.0000			−0.926270
$515$	−1.33975	−	22.3205i	−0.0590363	−	0.983559i
$516$		0			0
$517$	56.2917	+	32.5000i	2.47570	+	1.42935i
$518$	−6.92820	−	4.00000i	−0.304408	−	0.175750i
$519$		0			0
$520$	−6.69615	+	0.401924i	−0.293646	+	0.0176255i
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$		−	23.0000i		−	1.00572i	−0.864368	−	0.502860i	$-0.832281\pi$
$523$		−	23.0000i		−	1.00572i	0.864368	−	0.502860i	$-0.167719\pi$
$524$	1.50000	+	2.59808i	0.0655278	+	0.113497i
$525$		0			0
$526$	4.00000	−	6.92820i	0.174408	−	0.302084i
$527$	4.33013	+	2.50000i	0.188623	+	0.108902i
$528$		0			0
$529$	−11.0000	−	19.0526i	−0.478261	−	0.828372i
$530$		0			0
$531$		0			0
$532$		−	24.0000i		−	1.04053i
$533$	5.19615	−	3.00000i	0.225070	−	0.129944i
$534$		0			0
$535$	−11.1962	+	7.39230i	−0.484052	+	0.319597i
$536$	2.00000	−	3.46410i	0.0863868	−	0.149626i
$537$		0			0
$538$	7.79423	−	4.50000i	0.336033	−	0.194009i
$539$	−45.0000			−1.93829
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$	13.8564	−	8.00000i	0.595184	−	0.343629i
$543$		0			0
$544$	0.500000	−	0.866025i	0.0214373	−	0.0371305i
$545$	9.85641	+	14.9282i	0.422202	+	0.639454i
$546$		0			0
$547$	21.6506	−	12.5000i	0.925714	−	0.534461i	0.0402607	−	0.999189i	$-0.487181\pi$
$547$	21.6506	−	12.5000i	0.925714	−	0.534461i	0.885454	+	0.464728i	$0.153848\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	20.0000	−	15.0000i	0.852803	−	0.639602i
$551$	27.0000	+	46.7654i	1.15024	+	1.99227i
$552$		0			0
$553$	31.1769	+	18.0000i	1.32578	+	0.765438i
$554$	−13.0000	+	22.5167i	−0.552317	+	0.956641i
$555$		0			0
$556$	4.00000	+	6.92820i	0.169638	+	0.293821i
$557$			42.0000i			1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$			42.0000i			1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$	−0.535898	−	8.92820i	−0.0226458	−	0.377285i
$561$		0			0
$562$	−15.5885	−	9.00000i	−0.657559	−	0.379642i
$563$	−3.46410	−	2.00000i	−0.145994	−	0.0842900i	0.425223	−	0.905088i	$-0.360196\pi$
$563$	−3.46410	−	2.00000i	−0.145994	−	0.0842900i	−0.571218	+	0.820798i	$0.693529\pi$
$564$		0			0
$565$	−0.401924	−	6.69615i	−0.0169091	−	0.281709i
$566$	20.0000			0.840663
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	0.995861	−	0.0908852i	$-0.0289696\pi$
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	−0.576640	+	0.816999i	$0.695636\pi$
$570$		0			0
$571$	16.0000	−	27.7128i	0.669579	−	1.15975i	−0.308443	−	0.951243i	$-0.599808\pi$
$571$	16.0000	−	27.7128i	0.669579	−	1.15975i	0.978022	−	0.208502i	$-0.0668588\pi$
$572$	12.9904	+	7.50000i	0.543155	+	0.313591i
$573$		0			0
$574$	4.00000	+	6.92820i	0.166957	+	0.289178i
$575$	−4.00000	+	3.00000i	−0.166812	+	0.125109i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	13.8564	−	8.00000i	0.576351	−	0.332756i
$579$		0			0
$580$	11.0885	+	16.7942i	0.460423	+	0.697342i
$581$	−8.00000	+	13.8564i	−0.331896	+	0.574861i
$582$		0			0
$583$		0			0
$584$	2.00000			0.0827606
$585$		0			0
$586$	−6.00000			−0.247858
$587$	31.1769	−	18.0000i	1.28681	−	0.742940i	0.308725	−	0.951151i	$-0.400098\pi$
$587$	31.1769	−	18.0000i	1.28681	−	0.742940i	0.978084	+	0.208212i	$0.0667643\pi$
$588$		0			0
$589$	15.0000	−	25.9808i	0.618064	−	1.07052i
$590$	7.46410	−	4.92820i	0.307292	−	0.202891i
$591$		0			0
$592$	−1.73205	+	1.00000i	−0.0711868	+	0.0410997i
$593$		−	21.0000i		−	0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$		−	21.0000i		−	0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$	−6.50000	−	11.2583i	−0.266250	−	0.461159i
$597$		0			0
$598$	−2.59808	−	1.50000i	−0.106243	−	0.0613396i
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	−0.0157622	−	0.999876i	$-0.505017\pi$
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	0.873799	−	0.486287i	$-0.161649\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$			4.00000i			0.163028i
$603$		0			0
$604$	−13.0000			−0.528962
$605$	−31.2487	+	1.87564i	−1.27044	+	0.0762558i
$606$		0			0
$607$	8.66025	+	5.00000i	0.351509	+	0.202944i	0.665350	−	0.746532i	$-0.268282\pi$
$607$	8.66025	+	5.00000i	0.351509	+	0.202944i	−0.313841	+	0.949476i	$0.601616\pi$
$608$	−5.19615	−	3.00000i	−0.210732	−	0.121666i
$609$		0			0
$610$	−1.07180	−	17.8564i	−0.0433958	−	0.722985i
$611$	39.0000			1.57777
$612$		0			0
$613$			23.0000i			0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$613$			23.0000i			0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$614$	10.5000	+	18.1865i	0.423746	+	0.733949i
$615$		0			0
$616$	−10.0000	+	17.3205i	−0.402911	+	0.697863i
$617$	25.1147	+	14.5000i	1.01108	+	0.583748i	0.911508	−	0.411282i	$-0.134919\pi$
$617$	25.1147	+	14.5000i	1.01108	+	0.583748i	0.0995732	+	0.995030i	$0.468252\pi$
$618$		0			0
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	0.999657	+	0.0261952i	$0.00833914\pi$
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	−0.477143	+	0.878826i	$0.658328\pi$
$620$	5.00000	−	10.0000i	0.200805	−	0.401610i
$621$		0			0
$622$		−	18.0000i		−	0.721734i
$623$	−48.4974	+	28.0000i	−1.94301	+	1.12180i
$624$		0			0
$625$	−17.2846	−	18.0622i	−0.691384	−	0.722487i
$626$	7.00000	−	12.1244i	0.279776	−	0.484587i
$627$		0			0
$628$	11.2583	−	6.50000i	0.449256	−	0.259378i
$629$	2.00000			0.0797452
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$	7.79423	−	4.50000i	0.310038	−	0.179000i
$633$		0			0
$634$	2.00000	−	3.46410i	0.0794301	−	0.137577i
$635$	−7.46410	+	4.92820i	−0.296204	+	0.195570i
$636$		0			0
$637$	−23.3827	+	13.5000i	−0.926456	+	0.534889i
$638$		−	45.0000i		−	1.78157i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	−10.0000	−	17.3205i	−0.394976	−	0.684119i	0.598122	−	0.801405i	$-0.295914\pi$
$641$	−10.0000	−	17.3205i	−0.394976	−	0.684119i	−0.993098	+	0.117286i	$0.962581\pi$
$642$		0			0
$643$	18.1865	+	10.5000i	0.717207	+	0.414080i	0.813724	−	0.581252i	$-0.197437\pi$
$643$	18.1865	+	10.5000i	0.717207	+	0.414080i	−0.0965169	+	0.995331i	$0.530770\pi$
$644$	2.00000	−	3.46410i	0.0788110	−	0.136505i
$645$		0			0
$646$	3.00000	+	5.19615i	0.118033	+	0.204440i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$	5.89230	−	13.7942i	0.231115	−	0.541054i
$651$		0			0
$652$	16.4545	+	9.50000i	0.644407	+	0.372049i
$653$	−22.5167	−	13.0000i	−0.881145	−	0.508729i	−0.0101092	−	0.999949i	$-0.503218\pi$
$653$	−22.5167	−	13.0000i	−0.881145	−	0.508729i	−0.871036	+	0.491220i	$0.836551\pi$
$654$		0			0
$655$	−6.69615	+	0.401924i	−0.261640	+	0.0157045i
$656$	2.00000			0.0780869
$657$		0			0
$658$			52.0000i			2.02717i
$659$	−10.0000	−	17.3205i	−0.389545	−	0.674711i	0.602844	−	0.797859i	$-0.294034\pi$
$659$	−10.0000	−	17.3205i	−0.389545	−	0.674711i	−0.992388	+	0.123148i	$0.960701\pi$
$660$		0			0
$661$	−22.0000	+	38.1051i	−0.855701	+	1.48212i	0.0202925	+	0.999794i	$0.493540\pi$
$661$	−22.0000	+	38.1051i	−0.855701	+	1.48212i	−0.875993	+	0.482323i	$0.839793\pi$
$662$	−8.66025	−	5.00000i	−0.336590	−	0.194331i
$663$		0			0
$664$	2.00000	+	3.46410i	0.0776151	+	0.134433i
$665$	48.0000	+	24.0000i	1.86136	+	0.930680i
$666$		0			0
$667$			9.00000i			0.348481i
$668$	−10.3923	+	6.00000i	−0.402090	+	0.232147i
$669$		0			0
$670$	4.92820	+	7.46410i	0.190393	+	0.288363i
$671$	−20.0000	+	34.6410i	−0.772091	+	1.33730i
$672$		0			0
$673$	−41.5692	+	24.0000i	−1.60238	+	0.925132i	−0.611365	+	0.791349i	$0.709379\pi$
$673$	−41.5692	+	24.0000i	−1.60238	+	0.925132i	−0.991011	+	0.133783i	$0.957287\pi$
$674$	−16.0000			−0.616297
$675$		0			0
$676$	−4.00000			−0.153846
$677$	3.46410	−	2.00000i	0.133136	−	0.0768662i	−0.431953	−	0.901896i	$-0.642175\pi$
$677$	3.46410	−	2.00000i	0.133136	−	0.0768662i	0.565089	+	0.825030i	$0.308842\pi$
$678$		0			0
$679$	−20.0000	+	34.6410i	−0.767530	+	1.32940i
$680$	1.23205	+	1.86603i	0.0472470	+	0.0715588i
$681$		0			0
$682$	−21.6506	+	12.5000i	−0.829046	+	0.478650i
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$	−36.0000	−	18.0000i	−1.37549	−	0.687745i
$686$	−4.00000	−	6.92820i	−0.152721	−	0.264520i
$687$		0			0
$688$	0.866025	+	0.500000i	0.0330169	+	0.0190623i
$689$		0			0
$690$		0			0
$691$	−11.0000	−	19.0526i	−0.418460	−	0.724793i	0.577325	−	0.816514i	$-0.304097\pi$
$691$	−11.0000	−	19.0526i	−0.418460	−	0.724793i	−0.995785	+	0.0917209i	$0.970763\pi$
$692$		−	16.0000i		−	0.608229i
$693$		0			0
$694$	6.00000			0.227757
$695$	−17.8564	+	1.07180i	−0.677332	+	0.0406556i
$696$		0			0
$697$	−1.73205	−	1.00000i	−0.0656061	−	0.0378777i
$698$	13.8564	+	8.00000i	0.524473	+	0.302804i
$699$		0			0
$700$	18.3923	+	7.85641i	0.695164	+	0.296944i
$701$	−31.0000			−1.17085			−0.585427	−	0.810725i	$-0.699073\pi$
$701$	−31.0000			−1.17085			−0.585427	+	0.810725i	$0.699073\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$	2.50000	+	4.33013i	0.0942223	+	0.163198i
$705$		0			0
$706$	−5.50000	+	9.52628i	−0.206995	+	0.358526i
$707$	58.8897	+	34.0000i	2.21478	+	1.27870i
$708$		0			0
$709$	−12.0000	−	20.7846i	−0.450669	−	0.780582i	0.547758	−	0.836637i	$-0.315481\pi$
$709$	−12.0000	−	20.7846i	−0.450669	−	0.780582i	−0.998428	+	0.0560542i	$0.982148\pi$
$710$	12.0000	+	6.00000i	0.450352	+	0.225176i
$711$		0			0
$712$			14.0000i			0.524672i
$713$	4.33013	−	2.50000i	0.162165	−	0.0936257i
$714$		0			0
$715$	−27.9904	+	18.4808i	−1.04678	+	0.691141i
$716$	10.0000	−	17.3205i	0.373718	−	0.647298i
$717$		0			0
$718$	5.19615	−	3.00000i	0.193919	−	0.111959i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	−40.0000			−1.48968
$722$	14.7224	−	8.50000i	0.547912	−	0.316337i
$723$		0			0
$724$	−11.0000	+	19.0526i	−0.408812	+	0.708083i
$725$	−44.6769	+	5.38269i	−1.65926	+	0.199908i
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$			12.0000i			0.444750i
$729$		0			0
$730$	−2.00000	+	4.00000i	−0.0740233	+	0.148047i
$731$	−0.500000	−	0.866025i	−0.0184932	−	0.0320311i
$732$		0			0
$733$	−39.8372	−	23.0000i	−1.47142	−	0.849524i	−0.471935	−	0.881633i	$-0.656444\pi$
$733$	−39.8372	−	23.0000i	−1.47142	−	0.849524i	−0.999484	+	0.0321090i	$0.989778\pi$
$734$	11.0000	−	19.0526i	0.406017	−	0.703243i
$735$		0			0
$736$	−0.500000	−	0.866025i	−0.0184302	−	0.0319221i
$737$		−	20.0000i		−	0.736709i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$	−0.267949	−	4.46410i	−0.00985001	−	0.164104i
$741$		0			0
$742$		0			0
$743$	−7.79423	−	4.50000i	−0.285943	−	0.165089i	0.350168	−	0.936687i	$-0.386124\pi$
$743$	−7.79423	−	4.50000i	−0.285943	−	0.165089i	−0.636111	+	0.771598i	$0.719458\pi$
$744$		0			0
$745$	29.0167	−	1.74167i	1.06309	−	0.0638098i
$746$	−21.0000			−0.768865
$747$		0			0
$748$		−	5.00000i		−	0.182818i
$749$	12.0000	+	20.7846i	0.438470	+	0.759453i
$750$		0			0
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	−0.795075	−	0.606511i	$-0.792568\pi$
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	0.922792	+	0.385299i	$0.125902\pi$
$752$	11.2583	+	6.50000i	0.410549	+	0.237031i
$753$		0			0
$754$	−13.5000	−	23.3827i	−0.491641	−	0.851547i
$755$	13.0000	−	26.0000i	0.473118	−	0.946237i
$756$		0			0
$757$			43.0000i			1.56286i	0.623992	+	0.781431i	$0.285510\pi$
$757$			43.0000i			1.56286i	−0.623992	+	0.781431i	$0.714490\pi$
$758$	24.2487	−	14.0000i	0.880753	−	0.508503i
$759$		0			0
$760$	11.1962	−	7.39230i	0.406127	−	0.268147i
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	−0.736543	−	0.676391i	$-0.763543\pi$
$761$	6.00000	−	10.3923i	0.217500	−	0.376721i	0.954043	+	0.299670i	$0.0968765\pi$
$762$		0			0
$763$	27.7128	−	16.0000i	1.00327	−	0.579239i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	−19.0000			−0.686498
$767$	−10.3923	+	6.00000i	−0.375244	+	0.216647i
$768$		0			0
$769$	1.50000	−	2.59808i	0.0540914	−	0.0936890i	−0.837712	−	0.546113i	$-0.816107\pi$
$769$	1.50000	−	2.59808i	0.0540914	−	0.0936890i	0.891803	+	0.452423i	$0.149440\pi$
$770$	−24.6410	−	37.3205i	−0.888001	−	1.34494i
$771$		0			0
$772$	3.46410	−	2.00000i	0.124676	−	0.0719816i
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	15.0000	+	20.0000i	0.538816	+	0.718421i
$776$	5.00000	+	8.66025i	0.179490	+	0.310885i
$777$		0			0
$778$	4.33013	+	2.50000i	0.155243	+	0.0896293i
$779$	−6.00000	+	10.3923i	−0.214972	+	0.372343i
$780$		0			0
$781$	−15.0000	−	25.9808i	−0.536742	−	0.929665i
$782$			1.00000i			0.0357599i
$783$		0			0
$784$	−9.00000			−0.321429
$785$	1.74167	+	29.0167i	0.0621629	+	1.03565i
$786$		0			0
$787$	−11.2583	−	6.50000i	−0.401316	−	0.231700i	0.285736	−	0.958308i	$-0.407762\pi$
$787$	−11.2583	−	6.50000i	−0.401316	−	0.231700i	−0.687052	+	0.726609i	$0.741095\pi$
$788$	−19.0526	−	11.0000i	−0.678719	−	0.391859i
$789$		0			0
$790$	1.20577	+	20.0885i	0.0428994	+	0.714715i
$791$	−12.0000			−0.426671
$792$		0			0
$793$			24.0000i			0.852265i
$794$	6.50000	+	11.2583i	0.230676	+	0.399543i
$795$		0			0
$796$	−1.50000	+	2.59808i	−0.0531661	+	0.0920864i
$797$	12.1244	+	7.00000i	0.429467	+	0.247953i	0.699119	−	0.715005i	$-0.253576\pi$
$797$	12.1244	+	7.00000i	0.429467	+	0.247953i	−0.269653	+	0.962958i	$0.586909\pi$
$798$		0			0
$799$	−6.50000	−	11.2583i	−0.229953	−	0.398291i
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$			16.0000i			0.564980i
$803$	8.66025	−	5.00000i	0.305614	−	0.176446i
$804$		0			0
$805$	4.92820	+	7.46410i	0.173696	+	0.263075i
$806$	−7.50000	+	12.9904i	−0.264176	+	0.457567i
$807$		0			0
$808$	14.7224	−	8.50000i	0.517933	−	0.299029i
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$	−26.0000			−0.912983			−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000			−0.912983			−0.456492	+	0.889728i	$0.650894\pi$
$812$	31.1769	−	18.0000i	1.09410	−	0.631676i
$813$		0			0
$814$	−5.00000	+	8.66025i	−0.175250	+	0.303542i
$815$	−35.4545	+	23.4090i	−1.24192	+	0.819980i
$816$		0			0
$817$	−5.19615	+	3.00000i	−0.181790	+	0.104957i
$818$		−	11.0000i		−	0.384606i
$819$		0			0
$820$	−2.00000	+	4.00000i	−0.0698430	+	0.139686i
$821$	−23.0000	−	39.8372i	−0.802706	−	1.39033i	−0.917829	−	0.396976i	$-0.870060\pi$
$821$	−23.0000	−	39.8372i	−0.802706	−	1.39033i	0.115124	−	0.993351i	$-0.463274\pi$
$822$		0			0
$823$	20.7846	+	12.0000i	0.724506	+	0.418294i	0.816409	−	0.577474i	$-0.195962\pi$
$823$	20.7846	+	12.0000i	0.724506	+	0.418294i	−0.0919029	+	0.995768i	$0.529295\pi$
$824$	−5.00000	+	8.66025i	−0.174183	+	0.301694i
$825$		0			0
$826$	−8.00000	−	13.8564i	−0.278356	−	0.482126i
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	−8.92820	+	0.535898i	−0.309902	+	0.0186013i
$831$		0			0
$832$	2.59808	+	1.50000i	0.0900721	+	0.0520031i
$833$	7.79423	+	4.50000i	0.270054	+	0.155916i
$834$		0			0
$835$	−1.60770	−	26.7846i	−0.0556366	−	0.926920i
$836$	−30.0000			−1.03757
$837$		0			0
$838$		−	23.0000i		−	0.794522i
$839$	17.0000	+	29.4449i	0.586905	+	1.01655i	0.994635	+	0.103447i	$0.0329872\pi$
$839$	17.0000	+	29.4449i	0.586905	+	1.01655i	−0.407730	+	0.913103i	$0.633679\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$	22.5167	+	13.0000i	0.775975	+	0.448010i
$843$		0			0
$844$		0			0
$845$	4.00000	−	8.00000i	0.137604	−	0.275208i
$846$		0			0
$847$			56.0000i			1.92418i
$848$		0			0
$849$		0			0
$850$	−4.96410	+	0.598076i	−0.170267	+	0.0205138i
$851$	1.00000	−	1.73205i	0.0342796	−	0.0593739i
$852$		0			0
$853$	16.4545	−	9.50000i	0.563391	−	0.325274i	−0.191115	−	0.981568i	$-0.561210\pi$
$853$	16.4545	−	9.50000i	0.563391	−	0.325274i	0.754505	+	0.656294i	$0.227877\pi$
$854$	−32.0000			−1.09502
$855$		0			0
$856$	6.00000			0.205076
$857$	−5.19615	+	3.00000i	−0.177497	+	0.102478i	−0.586116	−	0.810227i	$-0.699344\pi$
$857$	−5.19615	+	3.00000i	−0.177497	+	0.102478i	0.408619	+	0.912705i	$0.366010\pi$
$858$		0			0
$859$	1.00000	−	1.73205i	0.0341196	−	0.0590968i	−0.848461	−	0.529257i	$-0.822471\pi$
$859$	1.00000	−	1.73205i	0.0341196	−	0.0590968i	0.882581	+	0.470160i	$0.155804\pi$
$860$	−1.86603	+	1.23205i	−0.0636309	+	0.0420126i
$861$		0			0
$862$	6.92820	−	4.00000i	0.235976	−	0.136241i
$863$		−	39.0000i		−	1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$		−	39.0000i		−	1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$		0			0
$865$	32.0000	+	16.0000i	1.08803	+	0.544016i
$866$	−11.0000	−	19.0526i	−0.373795	−	0.647432i
$867$		0			0
$868$	−17.3205	−	10.0000i	−0.587896	−	0.339422i
$869$	22.5000	−	38.9711i	0.763260	−	1.32201i
$870$		0			0
$871$	−6.00000	−	10.3923i	−0.203302	−	0.352130i
$872$		−	8.00000i		−	0.270914i
$873$		0			0
$874$	6.00000			0.202953
$875$	−34.1051	+	28.9282i	−1.15296	+	0.977952i
$876$		0			0
$877$	0.866025	+	0.500000i	0.0292436	+	0.0168838i	0.514551	−	0.857460i	$-0.327959\pi$
$877$	0.866025	+	0.500000i	0.0292436	+	0.0168838i	−0.485307	+	0.874344i	$0.661292\pi$
$878$		0			0
$879$		0			0
$880$	−11.1603	+	0.669873i	−0.376212	+	0.0225814i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	−1.50000	−	2.59808i	−0.0504505	−	0.0873828i
$885$		0			0
$886$	15.0000	−	25.9808i	0.503935	−	0.872841i
$887$	−44.1673	−	25.5000i	−1.48299	−	0.856206i	−0.483179	−	0.875521i	$-0.660518\pi$
$887$	−44.1673	−	25.5000i	−1.48299	−	0.856206i	−0.999813	+	0.0193153i	$0.993851\pi$
$888$		0			0
$889$	8.00000	+	13.8564i	0.268311	+	0.464729i
$890$	−28.0000	−	14.0000i	−0.938562	−	0.469281i
$891$		0			0
$892$			22.0000i			0.736614i
$893$	−67.5500	+	39.0000i	−2.26047	+	1.30509i
$894$		0			0
$895$	24.6410	+	37.3205i	0.823658	+	1.24749i
$896$	−2.00000	+	3.46410i	−0.0668153	+	0.115728i
$897$		0			0
$898$		0			0
$899$	45.0000			1.50083
$900$		0			0
$901$		0			0
$902$	8.66025	−	5.00000i	0.288355	−	0.166482i
$903$		0			0
$904$	−1.50000	+	2.59808i	−0.0498893	+	0.0864107i
$905$	−27.1051	−	41.0526i	−0.901005	−	1.36463i
$906$		0			0
$907$	−2.59808	+	1.50000i	−0.0862677	+	0.0498067i	−0.542513	−	0.840047i	$-0.682527\pi$
$907$	−2.59808	+	1.50000i	−0.0862677	+	0.0498067i	0.456246	+	0.889854i	$0.349194\pi$
$908$			2.00000i			0.0663723i
$909$		0			0
$910$	−24.0000	−	12.0000i	−0.795592	−	0.397796i
$911$	−18.0000	−	31.1769i	−0.596367	−	1.03294i	−0.993352	−	0.115113i	$-0.963277\pi$
$911$	−18.0000	−	31.1769i	−0.596367	−	1.03294i	0.396986	−	0.917825i	$-0.370056\pi$
$912$		0			0
$913$	17.3205	+	10.0000i	0.573225	+	0.330952i
$914$	8.00000	−	13.8564i	0.264616	−	0.458329i
$915$		0			0
$916$	−1.00000	−	1.73205i	−0.0330409	−	0.0572286i
$917$			12.0000i			0.396275i
$918$		0			0
$919$	−33.0000			−1.08857			−0.544285	−	0.838901i	$-0.683199\pi$
$919$	−33.0000			−1.08857			−0.544285	+	0.838901i	$0.683199\pi$
$920$	2.23205	−	0.133975i	0.0735885	−	0.00441701i
$921$		0			0
$922$	−5.19615	−	3.00000i	−0.171126	−	0.0987997i
$923$	−15.5885	−	9.00000i	−0.513100	−	0.296239i
$924$		0			0
$925$	9.19615	+	3.92820i	0.302368	+	0.129159i
$926$	−6.00000			−0.197172
$927$		0			0
$928$		−	9.00000i		−	0.295439i
$929$	3.00000	+	5.19615i	0.0984268	+	0.170480i	0.911034	−	0.412332i	$-0.135286\pi$
$929$	3.00000	+	5.19615i	0.0984268	+	0.170480i	−0.812607	+	0.582812i	$0.801952\pi$
$930$		0			0
$931$	27.0000	−	46.7654i	0.884889	−	1.53267i
$932$	−12.1244	−	7.00000i	−0.397146	−	0.229293i
$933$		0			0
$934$	3.00000	+	5.19615i	0.0981630	+	0.170023i
$935$	10.0000	+	5.00000i	0.327035	+	0.163517i
$936$		0			0
$937$		−	40.0000i		−	1.30674i	−0.757037	−	0.653372i	$-0.773354\pi$
$937$		−	40.0000i		−	1.30674i	0.757037	−	0.653372i	$-0.226646\pi$
$938$	13.8564	−	8.00000i	0.452428	−	0.261209i
$939$		0			0
$940$	−24.2583	+	16.0167i	−0.791219	+	0.522406i
$941$	15.5000	−	26.8468i	0.505286	−	0.875180i	−0.494696	−	0.869066i	$-0.664720\pi$
$941$	15.5000	−	26.8468i	0.505286	−	0.875180i	0.999981	−	0.00611403i	$-0.00194617\pi$
$942$		0			0
$943$	−1.73205	+	1.00000i	−0.0564033	+	0.0325645i
$944$	−4.00000			−0.130189
$945$		0			0
$946$	5.00000			0.162564
$947$	41.5692	−	24.0000i	1.35082	−	0.779895i	0.362454	−	0.932002i	$-0.381939\pi$
$947$	41.5692	−	24.0000i	1.35082	−	0.779895i	0.988364	+	0.152106i	$0.0486057\pi$
$948$		0			0
$949$	3.00000	−	5.19615i	0.0973841	−	0.168674i
$950$	3.58846	+	29.7846i	0.116425	+	0.966340i
$951$		0			0
$952$	3.46410	−	2.00000i	0.112272	−	0.0648204i
$953$			61.0000i			1.97598i	0.154506	+	0.987992i	$0.450622\pi$
$953$			61.0000i			1.97598i	−0.154506	+	0.987992i	$0.549378\pi$
$954$		0			0
$955$	6.00000	−	12.0000i	0.194155	−	0.388311i
$956$	8.00000	+	13.8564i	0.258738	+	0.448148i
$957$		0			0
$958$	1.73205	+	1.00000i	0.0559600	+	0.0323085i
$959$	−36.0000	+	62.3538i	−1.16250	+	2.01351i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$			6.00000i			0.193448i
$963$		0			0
$964$	23.0000			0.740780
$965$	0.535898	+	8.92820i	0.0172512	+	0.287409i
$966$		0			0
$967$	−1.73205	−	1.00000i	−0.0556990	−	0.0321578i	0.471892	−	0.881656i	$-0.343571\pi$
$967$	−1.73205	−	1.00000i	−0.0556990	−	0.0321578i	−0.527591	+	0.849499i	$0.676905\pi$
$968$	12.1244	+	7.00000i	0.389692	+	0.224989i
$969$		0			0
$970$	−22.3205	+	1.33975i	−0.716668	+	0.0430167i
$971$	9.00000			0.288824			0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000			0.288824			0.144412	+	0.989518i	$0.453871\pi$
$972$		0			0
$973$			32.0000i			1.02587i
$974$	−6.00000	−	10.3923i	−0.192252	−	0.332991i
$975$		0			0
$976$	−4.00000	+	6.92820i	−0.128037	+	0.221766i
$977$	33.7750	+	19.5000i	1.08056	+	0.623860i	0.931047	−	0.364900i	$-0.118897\pi$
$977$	33.7750	+	19.5000i	1.08056	+	0.623860i	0.149511	+	0.988760i	$0.452230\pi$
$978$		0			0
$979$	35.0000	+	60.6218i	1.11860	+	1.93748i
$980$	9.00000	−	18.0000i	0.287494	−	0.574989i
$981$		0			0
$982$		−	24.0000i		−	0.765871i
$983$	−25.1147	+	14.5000i	−0.801036	+	0.462478i	−0.843833	−	0.536606i	$-0.819706\pi$
$983$	−25.1147	+	14.5000i	−0.801036	+	0.462478i	0.0427975	+	0.999084i	$0.486373\pi$
$984$		0			0
$985$	41.0526	−	27.1051i	1.30804	−	0.863641i
$986$	−4.50000	+	7.79423i	−0.143309	+	0.248219i
$987$		0			0
$988$	−15.5885	+	9.00000i	−0.495935	+	0.286328i
$989$	−1.00000			−0.0317982
$990$		0			0
$991$	−17.0000			−0.540023			−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000			−0.540023			−0.270011	+	0.962857i	$0.587027\pi$
$992$	−4.33013	+	2.50000i	−0.137482	+	0.0793751i
$993$		0			0
$994$	12.0000	−	20.7846i	0.380617	−	0.659248i
$995$	−3.69615	−	5.59808i	−0.117176	−	0.177471i
$996$		0			0
$997$	12.9904	−	7.50000i	0.411409	−	0.237527i	−0.279986	−	0.960004i	$-0.590330\pi$
$997$	12.9904	−	7.50000i	0.411409	−	0.237527i	0.691395	+	0.722477i	$0.256996\pi$
$998$			12.0000i			0.379853i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.i.c.109.2		4
3.2	odd	2		810.2.i.d.109.1		4
5.4	even	2	inner	810.2.i.c.109.1		4
9.2	odd	6		810.2.i.d.379.2		4
9.4	even	3		270.2.c.b.109.2	yes	2
9.5	odd	6		270.2.c.a.109.1	✓	2
9.7	even	3	inner	810.2.i.c.379.1		4
15.14	odd	2		810.2.i.d.109.2		4
36.23	even	6		2160.2.f.d.1729.2		2
36.31	odd	6		2160.2.f.e.1729.1		2
45.4	even	6		270.2.c.b.109.1	yes	2
45.13	odd	12		1350.2.a.v.1.1		1
45.14	odd	6		270.2.c.a.109.2	yes	2
45.22	odd	12		1350.2.a.b.1.1		1
45.23	even	12		1350.2.a.j.1.1		1
45.29	odd	6		810.2.i.d.379.1		4
45.32	even	12		1350.2.a.l.1.1		1
45.34	even	6	inner	810.2.i.c.379.2		4
180.59	even	6		2160.2.f.d.1729.1		2
180.139	odd	6		2160.2.f.e.1729.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.a.109.1	✓	2	9.5	odd	6
270.2.c.a.109.2	yes	2	45.14	odd	6
270.2.c.b.109.1	yes	2	45.4	even	6
270.2.c.b.109.2	yes	2	9.4	even	3
810.2.i.c.109.1		4	5.4	even	2	inner
810.2.i.c.109.2		4	1.1	even	1	trivial
810.2.i.c.379.1		4	9.7	even	3	inner
810.2.i.c.379.2		4	45.34	even	6	inner
810.2.i.d.109.1		4	3.2	odd	2
810.2.i.d.109.2		4	15.14	odd	2
810.2.i.d.379.1		4	45.29	odd	6
810.2.i.d.379.2		4	9.2	odd	6
1350.2.a.b.1.1		1	45.22	odd	12
1350.2.a.j.1.1		1	45.23	even	12
1350.2.a.l.1.1		1	45.32	even	12
1350.2.a.v.1.1		1	45.13	odd	12
2160.2.f.d.1729.1		2	180.59	even	6
2160.2.f.d.1729.2		2	36.23	even	6
2160.2.f.e.1729.1		2	36.31	odd	6
2160.2.f.e.1729.2		2	180.139	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +(-2.50000 - 4.33013i) q^{11} +(-2.59808 - 1.50000i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 - 0.866025i) q^{16} +1.00000i q^{17} +6.00000 q^{19} +(2.23205 - 0.133975i) q^{20} +(-4.33013 - 2.50000i) q^{22} +(0.866025 + 0.500000i) q^{23} +(-1.96410 + 4.59808i) q^{25} -3.00000 q^{26} -4.00000i q^{28} +(4.50000 + 7.79423i) q^{29} +(2.50000 - 4.33013i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(0.500000 + 0.866025i) q^{34} +(8.00000 + 4.00000i) q^{35} -2.00000i q^{37} +(5.19615 - 3.00000i) q^{38} +(1.86603 - 1.23205i) q^{40} +(-1.00000 + 1.73205i) q^{41} +(-0.866025 + 0.500000i) q^{43} -5.00000 q^{44} +1.00000 q^{46} +(-11.2583 + 6.50000i) q^{47} +(4.50000 - 7.79423i) q^{49} +(0.598076 + 4.96410i) q^{50} +(-2.59808 + 1.50000i) q^{52} +(5.00000 - 10.0000i) q^{55} +(-2.00000 - 3.46410i) q^{56} +(7.79423 + 4.50000i) q^{58} +(2.00000 - 3.46410i) q^{59} +(-4.00000 - 6.92820i) q^{61} -5.00000i q^{62} -1.00000 q^{64} +(-0.401924 - 6.69615i) q^{65} +(3.46410 + 2.00000i) q^{67} +(0.866025 + 0.500000i) q^{68} +(8.92820 - 0.535898i) q^{70} +6.00000 q^{71} +2.00000i q^{73} +(-1.00000 - 1.73205i) q^{74} +(3.00000 - 5.19615i) q^{76} +(-17.3205 - 10.0000i) q^{77} +(4.50000 + 7.79423i) q^{79} +(1.00000 - 2.00000i) q^{80} +2.00000i q^{82} +(-3.46410 + 2.00000i) q^{83} +(-1.86603 + 1.23205i) q^{85} +(-0.500000 + 0.866025i) q^{86} +(-4.33013 + 2.50000i) q^{88} -14.0000 q^{89} -12.0000 q^{91} +(0.866025 - 0.500000i) q^{92} +(-6.50000 + 11.2583i) q^{94} +(7.39230 + 11.1962i) q^{95} +(-8.66025 + 5.00000i) q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} + 8 q^{10} - 10 q^{11} + 8 q^{14} - 2 q^{16} + 24 q^{19} + 2 q^{20} + 6 q^{25} - 12 q^{26} + 18 q^{29} + 10 q^{31} + 2 q^{34} + 32 q^{35} + 4 q^{40} - 4 q^{41} - 20 q^{44} + 4 q^{46}+ \cdots - 12 q^{95}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 + 8 * q^10 - 10 * q^11 + 8 * q^14 - 2 * q^16 + 24 * q^19 + 2 * q^20 + 6 * q^25 - 12 * q^26 + 18 * q^29 + 10 * q^31 + 2 * q^34 + 32 * q^35 + 4 * q^40 - 4 * q^41 - 20 * q^44 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 20 * q^55 - 8 * q^56 + 8 * q^59 - 16 * q^61 - 4 * q^64 - 12 * q^65 + 8 * q^70 + 24 * q^71 - 4 * q^74 + 12 * q^76 + 18 * q^79 + 4 * q^80 - 4 * q^85 - 2 * q^86 - 56 * q^89 - 48 * q^91 - 26 * q^94 - 12 * q^95

Introduction

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Database

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