LMFDB - Embedded newform 448.2.ba.c.81.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(81,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 8]))

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

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

chi := DirichletCharacter("448.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.ba (of order $12$, degree $4$, 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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			81.10
Character	$\chi$	$=$	448.81
Dual form			448.2.ba.c.177.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.20008 - 0.589510i) q^{3} +(2.32205 + 0.622192i) q^{5} +(-2.41058 + 1.09047i) q^{7} +(1.89476 - 1.09394i) q^{9} +O(q^{10})$	\(q+(2.20008 - 0.589510i) q^{3} +(2.32205 + 0.622192i) q^{5} +(-2.41058 + 1.09047i) q^{7} +(1.89476 - 1.09394i) q^{9} +(0.00762177 + 0.0284448i) q^{11} +(4.38886 + 4.38886i) q^{13} +5.47550 q^{15} +(1.36101 - 2.35734i) q^{17} +(1.53778 - 5.73909i) q^{19} +(-4.66062 + 3.82019i) q^{21} +(-3.33422 + 1.92501i) q^{23} +(0.674681 + 0.389527i) q^{25} +(-1.30797 + 1.30797i) q^{27} +(-4.93772 - 4.93772i) q^{29} +(1.29224 - 2.23823i) q^{31} +(0.0335370 + 0.0580879i) q^{33} +(-6.27597 + 1.03229i) q^{35} +(-7.83686 - 2.09988i) q^{37} +(12.2431 + 7.06857i) q^{39} -0.207557i q^{41} +(0.278444 - 0.278444i) q^{43} +(5.08039 - 1.36129i) q^{45} +(1.91470 + 3.31636i) q^{47} +(4.62174 - 5.25733i) q^{49} +(1.60466 - 5.98867i) q^{51} +(-0.328098 - 1.22448i) q^{53} +0.0707926i q^{55} -13.5330i q^{57} +(-0.0558664 - 0.208496i) q^{59} +(-1.59004 + 5.93409i) q^{61} +(-3.37456 + 4.70322i) q^{63} +(7.46045 + 12.9219i) q^{65} +(-11.9741 + 3.20844i) q^{67} +(-6.20075 + 6.20075i) q^{69} -7.37112i q^{71} +(3.67756 + 2.12324i) q^{73} +(1.71398 + 0.459261i) q^{75} +(-0.0493911 - 0.0602571i) q^{77} +(4.51126 + 7.81374i) q^{79} +(-5.38841 + 9.33299i) q^{81} +(7.55784 + 7.55784i) q^{83} +(4.62706 - 4.62706i) q^{85} +(-13.7742 - 7.95255i) q^{87} +(-3.03921 + 1.75469i) q^{89} +(-15.3656 - 5.79375i) q^{91} +(1.52358 - 5.68608i) q^{93} +(7.14164 - 12.3697i) q^{95} -11.1522 q^{97} +(0.0455585 + 0.0455585i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{5}+O(q^{10}) $ 48 * q + 4 * q^5	$ 48 q + 4 q^{5} + 4 q^{11} - 24 q^{13} + 40 q^{15} + 8 q^{17} + 4 q^{19} - 8 q^{21} + 24 q^{27} + 24 q^{29} - 28 q^{31} + 16 q^{33} - 28 q^{35} - 24 q^{37} + 40 q^{43} - 28 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} + 20 q^{59} + 8 q^{61} + 16 q^{63} + 8 q^{65} - 48 q^{67} - 40 q^{69} + 4 q^{75} - 20 q^{77} + 36 q^{79} + 8 q^{83} - 64 q^{91} + 8 q^{93} + 4 q^{95} - 48 q^{97} + 24 q^{99}+O(q^{100}) $ 48 * q + 4 * q^5 + 4 * q^11 - 24 * q^13 + 40 * q^15 + 8 * q^17 + 4 * q^19 - 8 * q^21 + 24 * q^27 + 24 * q^29 - 28 * q^31 + 16 * q^33 - 28 * q^35 - 24 * q^37 + 40 * q^43 - 28 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 + 20 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 - 48 * q^67 - 40 * q^69 + 4 * q^75 - 20 * q^77 + 36 * q^79 + 8 * q^83 - 64 * q^91 + 8 * q^93 + 4 * q^95 - 48 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.20008	−	0.589510i	1.27022	−	0.340354i	0.440104	−	0.897947i	$-0.354942\pi$
$3$	2.20008	−	0.589510i	1.27022	−	0.340354i	0.830115	+	0.557593i	$0.188275\pi$
$4$		0			0
$5$	2.32205	+	0.622192i	1.03845	+	0.278253i	0.737472	−	0.675377i	$-0.236019\pi$
$5$	2.32205	+	0.622192i	1.03845	+	0.278253i	0.300981	+	0.953630i	$0.402686\pi$
$6$		0			0
$7$	−2.41058	+	1.09047i	−0.911112	+	0.412159i
$7$	−2.41058	+	1.09047i	−0.911112	+	0.412159i
$8$		0			0
$9$	1.89476	−	1.09394i	0.631588	−	0.364648i
$10$		0			0
$11$	0.00762177	+	0.0284448i	0.00229805	+	0.00857644i	0.967065	−	0.254528i	$-0.0819202\pi$
$11$	0.00762177	+	0.0284448i	0.00229805	+	0.00857644i	−0.964767	+	0.263105i	$0.915254\pi$
$12$		0			0
$13$	4.38886	+	4.38886i	1.21725	+	1.21725i	0.968591	+	0.248660i	$0.0799900\pi$
$13$	4.38886	+	4.38886i	1.21725	+	1.21725i	0.248660	+	0.968591i	$0.420010\pi$
$14$		0			0
$15$	5.47550			1.41377
$16$		0			0
$17$	1.36101	−	2.35734i	0.330094	−	0.571739i	−0.652436	−	0.757844i	$-0.726253\pi$
$17$	1.36101	−	2.35734i	0.330094	−	0.571739i	0.982530	+	0.186105i	$0.0595863\pi$
$18$		0			0
$19$	1.53778	−	5.73909i	0.352792	−	1.31664i	−0.530448	−	0.847717i	$-0.677976\pi$
$19$	1.53778	−	5.73909i	0.352792	−	1.31664i	0.883240	−	0.468921i	$-0.155357\pi$
$20$		0			0
$21$	−4.66062	+	3.82019i	−1.01703	+	0.833633i
$22$		0			0
$23$	−3.33422	+	1.92501i	−0.695233	+	0.401393i	−0.805570	−	0.592501i	$-0.798141\pi$
$23$	−3.33422	+	1.92501i	−0.695233	+	0.401393i	0.110336	+	0.993894i	$0.464807\pi$
$24$		0			0
$25$	0.674681	+	0.389527i	0.134936	+	0.0779054i
$26$		0			0
$27$	−1.30797	+	1.30797i	−0.251719	+	0.251719i
$28$		0			0
$29$	−4.93772	−	4.93772i	−0.916911	−	0.916911i	0.0798926	−	0.996803i	$-0.474542\pi$
$29$	−4.93772	−	4.93772i	−0.916911	−	0.916911i	−0.996803	+	0.0798926i	$0.974542\pi$
$30$		0			0
$31$	1.29224	−	2.23823i	0.232094	−	0.401998i	−0.726330	−	0.687346i	$-0.758776\pi$
$31$	1.29224	−	2.23823i	0.232094	−	0.401998i	0.958424	+	0.285348i	$0.0921091\pi$
$32$		0			0
$33$	0.0335370	+	0.0580879i	0.00583805	+	0.0101118i
$34$		0			0
$35$	−6.27597	+	1.03229i	−1.06083	+	0.174489i
$36$		0			0
$37$	−7.83686	−	2.09988i	−1.28837	−	0.345218i	−0.451330	−	0.892357i	$-0.649050\pi$
$37$	−7.83686	−	2.09988i	−1.28837	−	0.345218i	−0.837042	+	0.547139i	$0.815717\pi$
$38$		0			0
$39$	12.2431	+	7.06857i	1.96047	+	1.13188i
$40$		0			0
$41$		−	0.207557i		−	0.0324150i	−0.999869	−	0.0162075i	$-0.994841\pi$
$41$		−	0.207557i		−	0.0324150i	0.999869	−	0.0162075i	$-0.00515924\pi$
$42$		0			0
$43$	0.278444	−	0.278444i	0.0424623	−	0.0424623i	−0.685557	−	0.728019i	$-0.740441\pi$
$43$	0.278444	−	0.278444i	0.0424623	−	0.0424623i	0.728019	+	0.685557i	$0.240441\pi$
$44$		0			0
$45$	5.08039	−	1.36129i	0.757339	−	0.202928i
$46$		0			0
$47$	1.91470	+	3.31636i	0.279288	+	0.483741i	0.971208	−	0.238233i	$-0.0765683\pi$
$47$	1.91470	+	3.31636i	0.279288	+	0.483741i	−0.691920	+	0.721974i	$0.743235\pi$
$48$		0			0
$49$	4.62174	−	5.25733i	0.660249	−	0.751047i
$50$		0			0
$51$	1.60466	−	5.98867i	0.224697	−	0.838582i
$52$		0			0
$53$	−0.328098	−	1.22448i	−0.0450677	−	0.168195i	0.939724	−	0.341934i	$-0.111082\pi$
$53$	−0.328098	−	1.22448i	−0.0450677	−	0.168195i	−0.984792	+	0.173739i	$0.944415\pi$
$54$		0			0
$55$			0.0707926i			0.00954567i
$56$		0			0
$57$		−	13.5330i		−	1.79249i
$58$		0			0
$59$	−0.0558664	−	0.208496i	−0.00727319	−	0.0271439i	0.962194	−	0.272365i	$-0.0878060\pi$
$59$	−0.0558664	−	0.208496i	−0.00727319	−	0.0271439i	−0.969467	+	0.245222i	$0.921139\pi$
$60$		0			0
$61$	−1.59004	+	5.93409i	−0.203583	+	0.759783i	0.786294	+	0.617853i	$0.211997\pi$
$61$	−1.59004	+	5.93409i	−0.203583	+	0.759783i	−0.989877	+	0.141930i	$0.954669\pi$
$62$		0			0
$63$	−3.37456	+	4.70322i	−0.425154	+	0.592550i
$64$		0			0
$65$	7.46045	+	12.9219i	0.925355	+	1.60276i
$66$		0			0
$67$	−11.9741	+	3.20844i	−1.46287	+	0.391974i	−0.900479	−	0.434901i	$-0.856784\pi$
$67$	−11.9741	+	3.20844i	−1.46287	+	0.391974i	−0.562387	+	0.826874i	$0.690117\pi$
$68$		0			0
$69$	−6.20075	+	6.20075i	−0.746482	+	0.746482i
$70$		0			0
$71$		−	7.37112i		−	0.874791i	−0.899269	−	0.437395i	$-0.855901\pi$
$71$		−	7.37112i		−	0.874791i	0.899269	−	0.437395i	$-0.144099\pi$
$72$		0			0
$73$	3.67756	+	2.12324i	0.430426	+	0.248507i	0.699528	−	0.714605i	$-0.253394\pi$
$73$	3.67756	+	2.12324i	0.430426	+	0.248507i	−0.269102	+	0.963112i	$0.586727\pi$
$74$		0			0
$75$	1.71398	+	0.459261i	0.197914	+	0.0530308i
$76$		0			0
$77$	−0.0493911	−	0.0602571i	−0.00562864	−	0.00686693i
$78$		0			0
$79$	4.51126	+	7.81374i	0.507557	+	0.879114i	0.999962	+	0.00874800i	$0.00278461\pi$
$79$	4.51126	+	7.81374i	0.507557	+	0.879114i	−0.492405	+	0.870366i	$0.663882\pi$
$80$		0			0
$81$	−5.38841	+	9.33299i	−0.598712	+	1.03700i
$82$		0			0
$83$	7.55784	+	7.55784i	0.829580	+	0.829580i	0.987459	−	0.157878i	$-0.0504653\pi$
$83$	7.55784	+	7.55784i	0.829580	+	0.829580i	−0.157878	+	0.987459i	$0.550465\pi$
$84$		0			0
$85$	4.62706	−	4.62706i	0.501875	−	0.501875i
$86$		0			0
$87$	−13.7742	−	7.95255i	−1.47675	−	0.852603i
$88$		0			0
$89$	−3.03921	+	1.75469i	−0.322155	+	0.185996i	−0.652353	−	0.757915i	$-0.726218\pi$
$89$	−3.03921	+	1.75469i	−0.322155	+	0.185996i	0.330198	+	0.943912i	$0.392885\pi$
$90$		0			0
$91$	−15.3656	−	5.79375i	−1.61075	−	0.607350i
$92$		0			0
$93$	1.52358	−	5.68608i	0.157988	−	0.589619i
$94$		0			0
$95$	7.14164	−	12.3697i	0.732716	−	1.26910i
$96$		0			0
$97$	−11.1522			−1.13233			−0.566167	−	0.824290i	$-0.691574\pi$
$97$	−11.1522			−1.13233			−0.566167	+	0.824290i	$0.691574\pi$
$98$		0			0
$99$	0.0455585	+	0.0455585i	0.00457880	+	0.00457880i
$100$		0			0
$101$	−3.70400	−	13.8235i	−0.368561	−	1.37549i	−0.862528	−	0.506009i	$-0.831120\pi$
$101$	−3.70400	−	13.8235i	−0.368561	−	1.37549i	0.493967	−	0.869481i	$-0.335546\pi$
$102$		0			0
$103$	3.78718	−	2.18653i	0.373162	−	0.215445i	−0.301677	−	0.953410i	$-0.597546\pi$
$103$	3.78718	−	2.18653i	0.373162	−	0.215445i	0.674839	+	0.737965i	$0.264213\pi$
$104$		0			0
$105$	−13.1991	+	5.97087i	−1.28810	+	0.582698i
$106$		0			0
$107$	17.4266	+	4.66945i	1.68470	+	0.451413i	0.969013	−	0.247011i	$-0.0794483\pi$
$107$	17.4266	+	4.66945i	1.68470	+	0.451413i	0.715684	+	0.698424i	$0.246115\pi$
$108$		0			0
$109$	−11.3254	+	3.03462i	−1.08477	+	0.290664i	−0.756550	−	0.653936i	$-0.773116\pi$
$109$	−11.3254	+	3.03462i	−1.08477	+	0.290664i	−0.328223	+	0.944600i	$0.606450\pi$
$110$		0			0
$111$	−18.4796			−1.75401
$112$		0			0
$113$	−5.25802			−0.494633			−0.247316	−	0.968935i	$-0.579549\pi$
$113$	−5.25802			−0.494633			−0.247316	+	0.968935i	$0.579549\pi$
$114$		0			0
$115$	−8.93997	+	2.39546i	−0.833657	+	0.223378i
$116$		0			0
$117$	13.1170	+	3.51469i	1.21267	+	0.324934i
$118$		0			0
$119$	−0.710207	+	7.16669i	−0.0651046	+	0.656969i
$120$		0			0
$121$	9.52553	−	5.49957i	0.865957	−	0.499961i
$122$		0			0
$123$	−0.122357	−	0.456643i	−0.0110326	−	0.0411742i
$124$		0			0
$125$	−7.17502	−	7.17502i	−0.641753	−	0.641753i
$126$		0			0
$127$	−7.05793			−0.626290			−0.313145	−	0.949705i	$-0.601383\pi$
$127$	−7.05793			−0.626290			−0.313145	+	0.949705i	$0.601383\pi$
$128$		0			0
$129$	0.448454	−	0.776745i	0.0394842	−	0.0683886i
$130$		0			0
$131$	2.28296	−	8.52012i	0.199463	−	0.744407i	−0.791603	−	0.611036i	$-0.790753\pi$
$131$	2.28296	−	8.52012i	0.199463	−	0.744407i	0.991066	−	0.133371i	$-0.0425802\pi$
$132$		0			0
$133$	2.55137	+	15.5114i	0.221232	+	1.34501i
$134$		0			0
$135$	−3.85098	+	2.22336i	−0.331440	+	0.191357i
$136$		0			0
$137$	8.17549	+	4.72012i	0.698479	+	0.403267i	0.806781	−	0.590851i	$-0.201208\pi$
$137$	8.17549	+	4.72012i	0.698479	+	0.403267i	−0.108302	+	0.994118i	$0.534541\pi$
$138$		0			0
$139$	−9.26594	+	9.26594i	−0.785927	+	0.785927i	−0.980824	−	0.194897i	$-0.937563\pi$
$139$	−9.26594	+	9.26594i	−0.785927	+	0.785927i	0.194897	+	0.980824i	$0.437563\pi$
$140$		0			0
$141$	6.16753	+	6.16753i	0.519400	+	0.519400i
$142$		0			0
$143$	−0.0913895	+	0.158291i	−0.00764237	+	0.0132370i
$144$		0			0
$145$	−8.39343	−	14.5378i	−0.697036	−	1.20730i
$146$		0			0
$147$	7.06897	−	14.2911i	0.583039	−	1.17871i
$148$		0			0
$149$	15.3266	+	4.10675i	1.25560	+	0.336438i	0.824498	−	0.565864i	$-0.191457\pi$
$149$	15.3266	+	4.10675i	1.25560	+	0.336438i	0.431104	+	0.902302i	$0.358124\pi$
$150$		0			0
$151$	−1.97991	−	1.14310i	−0.161122	−	0.0930241i	0.417271	−	0.908782i	$-0.362987\pi$
$151$	−1.97991	−	1.14310i	−0.161122	−	0.0930241i	−0.578393	+	0.815758i	$0.696320\pi$
$152$		0			0
$153$		−	5.95547i		−	0.481472i
$154$		0			0
$155$	4.39326	−	4.39326i	0.352875	−	0.352875i
$156$		0			0
$157$	−5.66229	+	1.51721i	−0.451900	+	0.121086i	−0.477588	−	0.878584i	$-0.658489\pi$
$157$	−5.66229	+	1.51721i	−0.451900	+	0.121086i	0.0256882	+	0.999670i	$0.491822\pi$
$158$		0			0
$159$	−1.44368	−	2.50053i	−0.114492	−	0.198305i
$160$		0			0
$161$	5.93822	−	8.27627i	0.467997	−	0.652261i
$162$		0			0
$163$	−3.81383	+	14.2334i	−0.298722	+	1.11485i	0.639494	+	0.768796i	$0.279144\pi$
$163$	−3.81383	+	14.2334i	−0.298722	+	1.11485i	−0.938216	+	0.346050i	$0.887523\pi$
$164$		0			0
$165$	0.0417330	+	0.155750i	0.00324891	+	0.0121251i
$166$		0			0
$167$		−	15.6157i		−	1.20838i	−0.796841	−	0.604190i	$-0.793497\pi$
$167$		−	15.6157i		−	1.20838i	0.796841	−	0.604190i	$-0.206503\pi$
$168$		0			0
$169$			25.5242i			1.96340i
$170$		0			0
$171$	−3.36450	−	12.5565i	−0.257290	−	0.960218i
$172$		0			0
$173$	−0.316668	+	1.18182i	−0.0240758	+	0.0898522i	−0.976918	−	0.213613i	$-0.931477\pi$
$173$	−0.316668	+	1.18182i	−0.0240758	+	0.0898522i	0.952843	+	0.303465i	$0.0981436\pi$
$174$		0			0
$175$	−2.05114	−	0.203264i	−0.155051	−	0.0153653i
$176$		0			0
$177$	−0.245822	−	0.425775i	−0.0184771	−	0.0320032i
$178$		0			0
$179$	1.14594	−	0.307054i	0.0856516	−	0.0229503i	−0.215739	−	0.976451i	$-0.569216\pi$
$179$	1.14594	−	0.307054i	0.0856516	−	0.0229503i	0.301390	+	0.953501i	$0.402549\pi$
$180$		0			0
$181$	−1.91395	+	1.91395i	−0.142263	+	0.142263i	−0.774651	−	0.632389i	$-0.782075\pi$
$181$	−1.91395	+	1.91395i	−0.142263	+	0.142263i	0.632389	+	0.774651i	$0.282075\pi$
$182$		0			0
$183$			13.9928i			1.03438i
$184$		0			0
$185$	−16.8911	−	9.75206i	−1.24186	−	0.716986i
$186$		0			0
$187$	0.0774275	+	0.0207466i	0.00566206	+	0.00151714i
$188$		0			0
$189$	1.72665	−	4.57926i	0.125596	−	0.333092i
$190$		0			0
$191$	−3.36453	−	5.82754i	−0.243449	−	0.421666i	0.718246	−	0.695790i	$-0.244945\pi$
$191$	−3.36453	−	5.82754i	−0.243449	−	0.421666i	−0.961694	+	0.274124i	$0.911612\pi$
$192$		0			0
$193$	9.93003	−	17.1993i	0.714779	−	1.23803i	−0.248265	−	0.968692i	$-0.579860\pi$
$193$	9.93003	−	17.1993i	0.714779	−	1.23803i	0.963045	−	0.269342i	$-0.0868063\pi$
$194$		0			0
$195$	24.0312	+	24.0312i	1.72091	+	1.72091i
$196$		0			0
$197$	9.36967	−	9.36967i	0.667561	−	0.667561i	−0.289590	−	0.957151i	$-0.593519\pi$
$197$	9.36967	−	9.36967i	0.667561	−	0.667561i	0.957151	+	0.289590i	$0.0935188\pi$
$198$		0			0
$199$	4.53338	+	2.61735i	0.321363	+	0.185539i	0.652000	−	0.758219i	$-0.273930\pi$
$199$	4.53338	+	2.61735i	0.321363	+	0.185539i	−0.330637	+	0.943758i	$0.607264\pi$
$200$		0			0
$201$	−24.4525	+	14.1177i	−1.72475	+	0.995784i
$202$		0			0
$203$	17.2872	+	6.51830i	1.21332	+	0.457495i
$204$		0			0
$205$	0.129141	−	0.481959i	0.00901957	−	0.0336615i
$206$		0			0
$207$	−4.21171	+	7.29490i	−0.292734	+	0.507030i
$208$		0			0
$209$	0.174968			0.0121028
$210$		0			0
$211$	3.18196	+	3.18196i	0.219055	+	0.219055i	0.808100	−	0.589045i	$-0.200496\pi$
$211$	3.18196	+	3.18196i	0.219055	+	0.219055i	−0.589045	+	0.808100i	$0.700496\pi$
$212$		0			0
$213$	−4.34535	−	16.2171i	−0.297738	−	1.11118i
$214$		0			0
$215$	0.819807	−	0.473316i	0.0559104	−	0.0322799i
$216$		0			0
$217$	−0.674322	+	6.80457i	−0.0457759	+	0.461924i
$218$		0			0
$219$	9.34261	+	2.50335i	0.631315	+	0.169160i
$220$		0			0
$221$	16.3193	−	4.37275i	1.09776	−	0.294143i
$222$		0			0
$223$	−7.91238			−0.529852			−0.264926	−	0.964269i	$-0.585348\pi$
$223$	−7.91238			−0.529852			−0.264926	+	0.964269i	$0.585348\pi$
$224$		0			0
$225$	1.70448			0.113632
$226$		0			0
$227$	7.36819	−	1.97430i	0.489044	−	0.131039i	−0.00586685	−	0.999983i	$-0.501867\pi$
$227$	7.36819	−	1.97430i	0.489044	−	0.131039i	0.494911	+	0.868944i	$0.335201\pi$
$228$		0			0
$229$	4.13322	+	1.10749i	0.273131	+	0.0731852i	0.392785	−	0.919630i	$-0.371512\pi$
$229$	4.13322	+	1.10749i	0.273131	+	0.0731852i	−0.119654	+	0.992816i	$0.538178\pi$
$230$		0			0
$231$	−0.144187	−	0.103454i	−0.00948679	−	0.00680677i
$232$		0			0
$233$	18.3239	−	10.5793i	1.20044	−	0.693074i	0.239787	−	0.970826i	$-0.422922\pi$
$233$	18.3239	−	10.5793i	1.20044	−	0.693074i	0.960653	+	0.277751i	$0.0895891\pi$
$234$		0			0
$235$	2.38262	+	8.89208i	0.155425	+	0.580055i
$236$		0			0
$237$	14.5314	+	14.5314i	0.943918	+	0.943918i
$238$		0			0
$239$	23.1747			1.49905			0.749523	−	0.661978i	$-0.230283\pi$
$239$	23.1747			1.49905			0.749523	+	0.661978i	$0.230283\pi$
$240$		0			0
$241$	−3.03376	+	5.25462i	−0.195421	+	0.338480i	−0.947039	−	0.321120i	$-0.895941\pi$
$241$	−3.03376	+	5.25462i	−0.195421	+	0.338480i	0.751617	+	0.659600i	$0.229274\pi$
$242$		0			0
$243$	−4.91679	+	18.3497i	−0.315412	+	1.17714i
$244$		0			0
$245$	14.0030	−	9.33218i	0.894619	−	0.596211i
$246$		0			0
$247$	31.9372	−	18.4389i	2.03211	−	1.17324i
$248$		0			0
$249$	21.0833	+	12.1724i	1.33610	+	0.771397i
$250$		0			0
$251$	6.15437	−	6.15437i	0.388461	−	0.388461i	−0.485677	−	0.874138i	$-0.661427\pi$
$251$	6.15437	−	6.15437i	0.388461	−	0.388461i	0.874138	+	0.485677i	$0.161427\pi$
$252$		0			0
$253$	−0.0801694	−	0.0801694i	−0.00504020	−	0.00504020i
$254$		0			0
$255$	7.45221	−	12.9076i	0.466676	−	0.808306i
$256$		0			0
$257$	11.0543	+	19.1467i	0.689550	+	1.19434i	0.971983	+	0.235049i	$0.0755252\pi$
$257$	11.0543	+	19.1467i	0.689550	+	1.19434i	−0.282433	+	0.959287i	$0.591142\pi$
$258$		0			0
$259$	21.1812	−	3.48395i	1.31614	−	0.216482i
$260$		0			0
$261$	−14.7574	−	3.95423i	−0.913459	−	0.244761i
$262$		0			0
$263$	3.81595	+	2.20314i	0.235302	+	0.135852i	0.613016	−	0.790071i	$-0.289956\pi$
$263$	3.81595	+	2.20314i	0.235302	+	0.135852i	−0.377714	+	0.925922i	$0.623290\pi$
$264$		0			0
$265$		−	3.04744i		−	0.187203i
$266$		0			0
$267$	−5.65210	+	5.65210i	−0.345903	+	0.345903i
$268$		0			0
$269$	0.818934	−	0.219433i	0.0499313	−	0.0133791i	−0.233767	−	0.972293i	$-0.575105\pi$
$269$	0.818934	−	0.219433i	0.0499313	−	0.0133791i	0.283698	+	0.958914i	$0.408439\pi$
$270$		0			0
$271$	5.86644	+	10.1610i	0.356361	+	0.617235i	0.987350	−	0.158557i	$-0.0506840\pi$
$271$	5.86644	+	10.1610i	0.356361	+	0.617235i	−0.630989	+	0.775792i	$0.717351\pi$
$272$		0			0
$273$	−37.2211	−	3.68855i	−2.25272	−	0.223241i
$274$		0			0
$275$	−0.00593777	+	0.0221601i	−0.000358061	+	0.00133630i
$276$		0			0
$277$	−4.14686	−	15.4763i	−0.249161	−	0.929880i	−0.971246	−	0.238076i	$-0.923483\pi$
$277$	−4.14686	−	15.4763i	−0.249161	−	0.929880i	0.722086	−	0.691803i	$-0.243184\pi$
$278$		0			0
$279$		−	5.65456i		−	0.338529i
$280$		0			0
$281$			21.3810i			1.27548i	0.770250	+	0.637742i	$0.220131\pi$
$281$			21.3810i			1.27548i	−0.770250	+	0.637742i	$0.779869\pi$
$282$		0			0
$283$	−4.99404	−	18.6380i	−0.296865	−	1.10791i	−0.939725	−	0.341931i	$-0.888919\pi$
$283$	−4.99404	−	18.6380i	−0.296865	−	1.10791i	0.642860	−	0.765983i	$-0.277748\pi$
$284$		0			0
$285$	8.42014	−	31.4244i	0.498766	−	1.86142i
$286$		0			0
$287$	0.226335	+	0.500333i	0.0133602	+	0.0295337i
$288$		0			0
$289$	4.79530	+	8.30570i	0.282076	+	0.488571i
$290$		0			0
$291$	−24.5358	+	6.57434i	−1.43831	+	0.385395i
$292$		0			0
$293$	−19.0966	+	19.0966i	−1.11563	+	1.11563i	−0.123260	+	0.992374i	$0.539335\pi$
$293$	−19.0966	+	19.0966i	−1.11563	+	1.11563i	−0.992374	+	0.123260i	$0.960665\pi$
$294$		0			0
$295$		−	0.518899i		−	0.0302115i
$296$		0			0
$297$	−0.0471740	−	0.0272359i	−0.00273731	−	0.00158039i
$298$		0			0
$299$	−23.0820	−	6.18482i	−1.33487	−	0.357677i
$300$		0			0
$301$	−0.367575	+	0.974845i	−0.0211867	+	0.0561891i
$302$		0			0
$303$	−16.2982	−	28.2293i	−0.936307	−	1.62173i
$304$		0			0
$305$	−7.38429	+	12.7900i	−0.422823	+	0.732352i
$306$		0			0
$307$	−2.51869	−	2.51869i	−0.143749	−	0.143749i	0.631570	−	0.775319i	$-0.282411\pi$
$307$	−2.51869	−	2.51869i	−0.143749	−	0.143749i	−0.775319	+	0.631570i	$0.782411\pi$
$308$		0			0
$309$	7.04312	−	7.04312i	0.400669	−	0.400669i
$310$		0			0
$311$	2.62390	+	1.51491i	0.148788	+	0.0859027i	0.572546	−	0.819873i	$-0.305956\pi$
$311$	2.62390	+	1.51491i	0.148788	+	0.0859027i	−0.423758	+	0.905776i	$0.639289\pi$
$312$		0			0
$313$	23.7963	−	13.7388i	1.34504	−	0.776562i	0.357501	−	0.933913i	$-0.383629\pi$
$313$	23.7963	−	13.7388i	1.34504	−	0.776562i	0.987543	+	0.157351i	$0.0502954\pi$
$314$		0			0
$315$	−10.7622	+	8.82150i	−0.606382	+	0.497035i
$316$		0			0
$317$	−2.19301	+	8.18443i	−0.123172	+	0.459683i	−0.999768	−	0.0215435i	$-0.993142\pi$
$317$	−2.19301	+	8.18443i	−0.123172	+	0.459683i	0.876596	+	0.481227i	$0.159809\pi$
$318$		0			0
$319$	0.102818	−	0.178087i	0.00575672	−	0.00997094i
$320$		0			0
$321$	41.0927			2.29357
$322$		0			0
$323$	−11.4361	−	11.4361i	−0.636319	−	0.636319i
$324$		0			0
$325$	1.25150	+	4.67066i	0.0694207	+	0.259082i
$326$		0			0
$327$	−23.1278	+	13.3528i	−1.27897	+	0.738413i
$328$		0			0
$329$	−8.23193	−	5.90641i	−0.453841	−	0.325631i
$330$		0			0
$331$	−22.7094	−	6.08497i	−1.24822	−	0.334460i	−0.426573	−	0.904453i	$-0.640279\pi$
$331$	−22.7094	−	6.08497i	−1.24822	−	0.334460i	−0.821650	+	0.569993i	$0.806946\pi$
$332$		0			0
$333$	−17.1461	+	4.59430i	−0.939603	+	0.251766i
$334$		0			0
$335$	−29.8007			−1.62819
$336$		0			0
$337$	−8.50787			−0.463453			−0.231726	−	0.972781i	$-0.574437\pi$
$337$	−8.50787			−0.463453			−0.231726	+	0.972781i	$0.574437\pi$
$338$		0			0
$339$	−11.5681	+	3.09966i	−0.628291	+	0.168350i
$340$		0			0
$341$	0.0735152	+	0.0196983i	0.00398107	+	0.00106672i
$342$		0			0
$343$	−5.40810	+	17.7131i	−0.292010	+	0.956415i
$344$		0			0
$345$	−18.2565	+	10.5404i	−0.982898	+	0.567477i
$346$		0			0
$347$	3.28024	+	12.2420i	0.176093	+	0.657187i	0.996363	+	0.0852109i	$0.0271564\pi$
$347$	3.28024	+	12.2420i	0.176093	+	0.657187i	−0.820270	+	0.571976i	$0.806177\pi$
$348$		0			0
$349$	−10.5665	−	10.5665i	−0.565613	−	0.565613i	0.365283	−	0.930896i	$-0.380972\pi$
$349$	−10.5665	−	10.5665i	−0.565613	−	0.565613i	−0.930896	+	0.365283i	$0.880972\pi$
$350$		0			0
$351$	−11.4810			−0.612809
$352$		0			0
$353$	1.14985	−	1.99161i	0.0612006	−	0.106003i	−0.833802	−	0.552064i	$-0.813840\pi$
$353$	1.14985	−	1.99161i	0.0612006	−	0.106003i	0.895002	+	0.446061i	$0.147174\pi$
$354$		0			0
$355$	4.58625	−	17.1161i	0.243413	−	0.908430i
$356$		0			0
$357$	2.66232	+	16.1860i	0.140905	+	0.856653i
$358$		0			0
$359$	6.94555	−	4.01001i	0.366572	−	0.211640i	−0.305388	−	0.952228i	$-0.598786\pi$
$359$	6.94555	−	4.01001i	0.366572	−	0.211640i	0.671960	+	0.740588i	$0.265453\pi$
$360$		0			0
$361$	−14.1179	−	8.15098i	−0.743048	−	0.428999i
$362$		0			0
$363$	17.7149	−	17.7149i	0.929791	−	0.929791i
$364$		0			0
$365$	7.21843	+	7.21843i	0.377830	+	0.377830i
$366$		0			0
$367$	7.61255	−	13.1853i	0.397372	−	0.688269i	−0.596029	−	0.802963i	$-0.703256\pi$
$367$	7.61255	−	13.1853i	0.397372	−	0.688269i	0.993401	+	0.114694i	$0.0365889\pi$
$368$		0			0
$369$	−0.227056	−	0.393272i	−0.0118201	−	0.0204729i
$370$		0			0
$371$	2.12616	+	2.59391i	0.110385	+	0.134669i
$372$		0			0
$373$	10.8743	+	2.91375i	0.563048	+	0.150868i	0.529106	−	0.848556i	$-0.322527\pi$
$373$	10.8743	+	2.91375i	0.563048	+	0.150868i	0.0339416	+	0.999424i	$0.489194\pi$
$374$		0			0
$375$	−20.0154	−	11.5559i	−1.03359	−	0.596744i
$376$		0			0
$377$		−	43.3419i		−	2.23222i
$378$		0			0
$379$	−2.15234	+	2.15234i	−0.110558	+	0.110558i	−0.760222	−	0.649663i	$-0.774910\pi$
$379$	−2.15234	+	2.15234i	−0.110558	+	0.110558i	0.649663	+	0.760222i	$0.274910\pi$
$380$		0			0
$381$	−15.5280	+	4.16072i	−0.795525	+	0.213160i
$382$		0			0
$383$	15.2044	+	26.3347i	0.776906	+	1.34564i	0.933717	+	0.358012i	$0.116545\pi$
$383$	15.2044	+	26.3347i	0.776906	+	1.34564i	−0.156811	+	0.987629i	$0.550121\pi$
$384$		0			0
$385$	−0.0771973	−	0.170651i	−0.00393434	−	0.00869717i
$386$		0			0
$387$	0.222984	−	0.832187i	0.0113349	−	0.0423025i
$388$		0			0
$389$	−0.190562	−	0.711187i	−0.00966187	−	0.0360586i	0.960927	−	0.276803i	$-0.0892750\pi$
$389$	−0.190562	−	0.711187i	−0.00966187	−	0.0360586i	−0.970588	+	0.240745i	$0.922608\pi$
$390$		0			0
$391$			10.4799i			0.529989i
$392$		0			0
$393$		−	20.0908i		−	1.01345i
$394$		0			0
$395$	5.61375	+	20.9508i	0.282458	+	1.05415i
$396$		0			0
$397$	−4.39295	+	16.3947i	−0.220476	+	0.822827i	0.763691	+	0.645582i	$0.223385\pi$
$397$	−4.39295	+	16.3947i	−0.220476	+	0.822827i	−0.984167	+	0.177245i	$0.943281\pi$
$398$		0			0
$399$	14.7574	+	32.6223i	0.738792	+	1.63316i
$400$		0			0
$401$	−14.7845	−	25.6075i	−0.738302	−	1.27878i	−0.953260	−	0.302153i	$-0.902295\pi$
$401$	−14.7845	−	25.6075i	−0.738302	−	1.27878i	0.214958	−	0.976623i	$-0.431039\pi$
$402$		0			0
$403$	15.4947	−	4.15180i	0.771848	−	0.206816i
$404$		0			0
$405$	−18.3191	+	18.3191i	−0.910283	+	0.910283i
$406$		0			0
$407$		−	0.238923i		−	0.0118430i
$408$		0			0
$409$	14.1848	+	8.18961i	0.701394	+	0.404950i	0.807866	−	0.589366i	$-0.200622\pi$
$409$	14.1848	+	8.18961i	0.701394	+	0.404950i	−0.106472	+	0.994316i	$0.533956\pi$
$410$		0			0
$411$	20.7693	+	5.56512i	1.02447	+	0.274507i
$412$		0			0
$413$	0.362030	+	0.441675i	0.0178143	+	0.0217334i
$414$		0			0
$415$	12.8473	+	22.2521i	0.630648	+	1.09231i
$416$		0			0
$417$	−14.9235	+	25.8482i	−0.730805	+	1.26579i
$418$		0			0
$419$	−25.7529	−	25.7529i	−1.25811	−	1.25811i	−0.951995	−	0.306115i	$-0.900971\pi$
$419$	−25.7529	−	25.7529i	−1.25811	−	1.25811i	−0.306115	−	0.951995i	$-0.599029\pi$
$420$		0			0
$421$	−21.6928	+	21.6928i	−1.05724	+	1.05724i	−0.0589820	+	0.998259i	$0.518785\pi$
$421$	−21.6928	+	21.6928i	−1.05724	+	1.05724i	−0.998259	+	0.0589820i	$0.981215\pi$
$422$		0			0
$423$	7.25582	+	4.18915i	0.352790	+	0.203683i
$424$		0			0
$425$	1.83650	−	1.06030i	0.0890832	−	0.0514322i
$426$		0			0
$427$	−2.63806	−	16.0385i	−0.127665	−	0.776156i
$428$		0			0
$429$	−0.107750	+	0.402129i	−0.00520222	+	0.0194150i
$430$		0			0
$431$	13.2778	−	22.9977i	0.639567	−	1.10776i	−0.345961	−	0.938249i	$-0.612447\pi$
$431$	13.2778	−	22.9977i	0.639567	−	1.10776i	0.985528	−	0.169513i	$-0.0542196\pi$
$432$		0			0
$433$	29.2189			1.40417			0.702085	−	0.712093i	$-0.252253\pi$
$433$	29.2189			1.40417			0.702085	+	0.712093i	$0.252253\pi$
$434$		0			0
$435$	−27.0365	−	27.0365i	−1.29630	−	1.29630i
$436$		0			0
$437$	5.92052	+	22.0957i	0.283217	+	1.05698i
$438$		0			0
$439$	3.28383	−	1.89592i	0.156729	−	0.0904874i	−0.419584	−	0.907716i	$-0.637824\pi$
$439$	3.28383	−	1.89592i	0.156729	−	0.0904874i	0.576313	+	0.817229i	$0.304491\pi$
$440$		0			0
$441$	3.00590	−	15.0173i	0.143138	−	0.715110i
$442$		0			0
$443$	−3.60402	−	0.965694i	−0.171232	−	0.0458815i	0.172184	−	0.985065i	$-0.444918\pi$
$443$	−3.60402	−	0.965694i	−0.171232	−	0.0458815i	−0.343416	+	0.939183i	$0.611584\pi$
$444$		0			0
$445$	−8.14895	+	2.18351i	−0.386298	+	0.103508i
$446$		0			0
$447$	36.1407			1.70940
$448$		0			0
$449$	18.7626			0.885460			0.442730	−	0.896655i	$-0.354010\pi$
$449$	18.7626			0.885460			0.442730	+	0.896655i	$0.354010\pi$
$450$		0			0
$451$	0.00590393	−	0.00158195i	0.000278005	−	7.44913e-5i
$452$		0			0
$453$	−5.02983	−	1.34774i	−0.236322	−	0.0633222i
$454$		0			0
$455$	−32.0749	−	23.0138i	−1.50370	−	1.07890i
$456$		0			0
$457$	−15.8043	+	9.12461i	−0.739294	+	0.426831i	−0.821812	−	0.569758i	$-0.807037\pi$
$457$	−15.8043	+	9.12461i	−0.739294	+	0.426831i	0.0825187	+	0.996590i	$0.473704\pi$
$458$		0			0
$459$	1.30317	+	4.86349i	0.0608266	+	0.227008i
$460$		0			0
$461$	7.85090	+	7.85090i	0.365653	+	0.365653i	0.865889	−	0.500236i	$-0.166753\pi$
$461$	7.85090	+	7.85090i	0.365653	+	0.365653i	−0.500236	+	0.865889i	$0.666753\pi$
$462$		0			0
$463$	−2.40372			−0.111710			−0.0558551	−	0.998439i	$-0.517788\pi$
$463$	−2.40372			−0.111710			−0.0558551	+	0.998439i	$0.517788\pi$
$464$		0			0
$465$	7.07567	−	12.2554i	0.328126	−	0.568331i
$466$		0			0
$467$	−1.96510	+	7.33384i	−0.0909338	+	0.339369i	−0.996372	−	0.0851094i	$-0.972876\pi$
$467$	−1.96510	+	7.33384i	−0.0909338	+	0.339369i	0.905438	+	0.424479i	$0.139543\pi$
$468$		0			0
$469$	25.3657	−	20.7916i	1.17128	−	0.960066i
$470$		0			0
$471$	−11.5631	+	6.67596i	−0.532800	+	0.307612i
$472$		0			0
$473$	0.0100425	+	0.00579805i	0.000461756	+	0.000266595i
$474$		0			0
$475$	3.27305	−	3.27305i	0.150178	−	0.150178i
$476$		0			0
$477$	−1.96117	−	1.96117i	−0.0897960	−	0.0897960i
$478$		0			0
$479$	−17.2984	+	29.9617i	−0.790385	+	1.36899i	0.135344	+	0.990799i	$0.456786\pi$
$479$	−17.2984	+	29.9617i	−0.790385	+	1.36899i	−0.925729	+	0.378188i	$0.876547\pi$
$480$		0			0
$481$	−25.1788	−	43.6109i	−1.14805	−	1.98849i
$482$		0			0
$483$	8.18563	−	21.7091i	0.372459	−	0.987799i
$484$		0			0
$485$	−25.8960	−	6.93881i	−1.17588	−	0.315075i
$486$		0			0
$487$	4.11458	+	2.37555i	0.186449	+	0.107647i	0.590319	−	0.807170i	$-0.299002\pi$
$487$	4.11458	+	2.37555i	0.186449	+	0.107647i	−0.403870	+	0.914816i	$0.632335\pi$
$488$		0			0
$489$			33.5630i			1.51777i
$490$		0			0
$491$	16.3309	−	16.3309i	0.737003	−	0.737003i	−0.234994	−	0.971997i	$-0.575507\pi$
$491$	16.3309	−	16.3309i	0.737003	−	0.737003i	0.971997	+	0.234994i	$0.0755071\pi$
$492$		0			0
$493$	−18.3602	+	4.91959i	−0.826900	+	0.221567i
$494$		0			0
$495$	0.0774431	+	0.134135i	0.00348081	+	0.00602893i
$496$		0			0
$497$	8.03799	+	17.7686i	0.360553	+	0.797032i
$498$		0			0
$499$	−5.22895	+	19.5147i	−0.234080	+	0.873599i	0.744481	+	0.667643i	$0.232697\pi$
$499$	−5.22895	+	19.5147i	−0.234080	+	0.873599i	−0.978561	+	0.205955i	$0.933970\pi$
$500$		0			0
$501$	−9.20562	−	34.3558i	−0.411277	−	1.53491i
$502$		0			0
$503$			7.84369i			0.349733i	0.984592	+	0.174866i	$0.0559493\pi$
$503$			7.84369i			0.349733i	−0.984592	+	0.174866i	$0.944051\pi$
$504$		0			0
$505$		−	34.4035i		−	1.53094i
$506$		0			0
$507$	15.0468	+	56.1553i	0.668250	+	2.49394i
$508$		0			0
$509$	5.62933	−	21.0090i	0.249516	−	0.931206i	−0.721544	−	0.692369i	$-0.756567\pi$
$509$	5.62933	−	21.0090i	0.249516	−	0.931206i	0.971060	−	0.238837i	$-0.0767661\pi$
$510$		0			0
$511$	−11.1804	−	1.10796i	−0.494591	−	0.0490131i
$512$		0			0
$513$	5.49518	+	9.51793i	0.242618	+	0.420227i
$514$		0			0
$515$	10.1545	−	2.72088i	0.447459	−	0.119896i
$516$		0			0
$517$	−0.0797399	+	0.0797399i	−0.00350696	+	0.00350696i
$518$		0			0
$519$			2.78678i			0.122326i
$520$		0			0
$521$	6.01150	+	3.47074i	0.263369	+	0.152056i	0.625870	−	0.779927i	$-0.284744\pi$
$521$	6.01150	+	3.47074i	0.263369	+	0.152056i	−0.362502	+	0.931983i	$0.618077\pi$
$522$		0			0
$523$	6.91185	+	1.85202i	0.302234	+	0.0809833i	0.406749	−	0.913540i	$-0.366662\pi$
$523$	6.91185	+	1.85202i	0.302234	+	0.0809833i	−0.104515	+	0.994523i	$0.533329\pi$
$524$		0			0
$525$	−4.63250	+	0.761968i	−0.202179	+	0.0332550i
$526$		0			0
$527$	−3.51751	−	6.09251i	−0.153225	−	0.265394i
$528$		0			0
$529$	−4.08864	+	7.08173i	−0.177767	+	0.307902i
$530$		0			0
$531$	−0.333937	−	0.333937i	−0.0144916	−	0.0144916i
$532$		0			0
$533$	0.910940	−	0.910940i	0.0394572	−	0.0394572i
$534$		0			0
$535$	37.5603	+	21.6854i	1.62387	+	0.937543i
$536$		0			0
$537$	2.34015	−	1.35109i	0.100985	−	0.0583037i
$538$		0			0
$539$	0.184770	+	0.0913946i	0.00795859	+	0.00393664i
$540$		0			0
$541$	−11.4554	+	42.7523i	−0.492508	+	1.83806i	0.0510566	+	0.998696i	$0.483741\pi$
$541$	−11.4554	+	42.7523i	−0.492508	+	1.83806i	−0.543564	+	0.839368i	$0.682926\pi$
$542$		0			0
$543$	−3.08255	+	5.33914i	−0.132285	+	0.229124i
$544$		0			0
$545$	−28.1862			−1.20736
$546$		0			0
$547$	−1.52168	−	1.52168i	−0.0650623	−	0.0650623i	0.673827	−	0.738889i	$-0.264649\pi$
$547$	−1.52168	−	1.52168i	−0.0650623	−	0.0650623i	−0.738889	+	0.673827i	$0.764649\pi$
$548$		0			0
$549$	3.47882	+	12.9831i	0.148472	+	0.554106i
$550$		0			0
$551$	−35.9311	+	20.7449i	−1.53072	+	0.883761i
$552$		0			0
$553$	−19.3954	−	13.9162i	−0.824776	−	0.591777i
$554$		0			0
$555$	−42.9107	−	11.4979i	−1.82146	−	0.488058i
$556$		0			0
$557$	44.6337	−	11.9596i	1.89119	−	0.506743i	0.892773	−	0.450508i	$-0.148757\pi$
$557$	44.6337	−	11.9596i	1.89119	−	0.506743i	0.998418	−	0.0562353i	$-0.0179097\pi$
$558$		0			0
$559$	2.44410			0.103375
$560$		0			0
$561$	0.182577			0.00770841
$562$		0			0
$563$	1.63604	−	0.438376i	0.0689510	−	0.0184754i	−0.224179	−	0.974548i	$-0.571970\pi$
$563$	1.63604	−	0.438376i	0.0689510	−	0.0184754i	0.293130	+	0.956073i	$0.405303\pi$
$564$		0			0
$565$	−12.2094	−	3.27150i	−0.513653	−	0.137633i
$566$		0			0
$567$	2.81179	−	28.3738i	0.118084	−	1.19159i
$568$		0			0
$569$	4.81619	−	2.78063i	0.201905	−	0.116570i	−0.395639	−	0.918406i	$-0.629477\pi$
$569$	4.81619	−	2.78063i	0.201905	−	0.116570i	0.597544	+	0.801836i	$0.296143\pi$
$570$		0			0
$571$	−3.19199	−	11.9127i	−0.133581	−	0.498529i	0.866419	−	0.499317i	$-0.166416\pi$
$571$	−3.19199	−	11.9127i	−0.133581	−	0.498529i	−1.00000		0.000788088i	$0.999749\pi$
$572$		0			0
$573$	−10.8376	−	10.8376i	−0.452749	−	0.452749i
$574$		0			0
$575$	−2.99938			−0.125083
$576$		0			0
$577$	−14.2426	+	24.6689i	−0.592927	+	1.02698i	0.400909	+	0.916118i	$0.368695\pi$
$577$	−14.2426	+	24.6689i	−0.592927	+	1.02698i	−0.993836	+	0.110862i	$0.964639\pi$
$578$		0			0
$579$	11.7077	−	43.6938i	0.486556	−	1.81585i
$580$		0			0
$581$	−26.4603	−	9.97713i	−1.09776	−	0.413921i
$582$		0			0
$583$	0.0323293	−	0.0186654i	0.00133894	−	0.000773040i
$584$		0			0
$585$	28.2716	+	16.3226i	1.16889	+	0.674857i
$586$		0			0
$587$	−15.1948	+	15.1948i	−0.627157	+	0.627157i	−0.947352	−	0.320195i	$-0.896252\pi$
$587$	−15.1948	+	15.1948i	−0.627157	+	0.627157i	0.320195	+	0.947352i	$0.396252\pi$
$588$		0			0
$589$	−10.8582	−	10.8582i	−0.447405	−	0.447405i
$590$		0			0
$591$	15.0905	−	26.1376i	0.620741	−	1.07516i
$592$		0			0
$593$	15.3004	+	26.5011i	0.628313	+	1.08827i	0.987890	+	0.155155i	$0.0495877\pi$
$593$	15.3004	+	26.5011i	0.628313	+	1.08827i	−0.359577	+	0.933115i	$0.617079\pi$
$594$		0			0
$595$	−6.10820	+	16.1995i	−0.250412	+	0.664117i
$596$		0			0
$597$	11.5168	+	3.08591i	0.471350	+	0.126298i
$598$		0			0
$599$	11.2663	+	6.50461i	0.460329	+	0.265771i	0.712183	−	0.701994i	$-0.247707\pi$
$599$	11.2663	+	6.50461i	0.460329	+	0.265771i	−0.251853	+	0.967765i	$0.581040\pi$
$600$		0			0
$601$		−	39.6831i		−	1.61871i	−0.587322	−	0.809353i	$-0.699818\pi$
$601$		−	39.6831i		−	1.61871i	0.587322	−	0.809353i	$-0.300182\pi$
$602$		0			0
$603$	−19.1782	+	19.1782i	−0.780996	+	0.780996i
$604$		0			0
$605$	25.5406	−	6.84358i	1.03837	−	0.278231i
$606$		0			0
$607$	−0.982401	−	1.70157i	−0.0398744	−	0.0690645i	0.845399	−	0.534135i	$-0.179362\pi$
$607$	−0.982401	−	1.70157i	−0.0398744	−	0.0690645i	−0.885274	+	0.465070i	$0.846029\pi$
$608$		0			0
$609$	41.8758	+	4.14982i	1.69689	+	0.168159i
$610$		0			0
$611$	−6.15168	+	22.9584i	−0.248870	+	0.928797i
$612$		0			0
$613$	−1.30222	−	4.85993i	−0.0525960	−	0.196291i	0.934629	−	0.355625i	$-0.115732\pi$
$613$	−1.30222	−	4.85993i	−0.0525960	−	0.196291i	−0.987225	+	0.159334i	$0.949065\pi$
$614$		0			0
$615$		−	1.13648i		−	0.0458273i
$616$		0			0
$617$			21.7544i			0.875798i	0.899024	+	0.437899i	$0.144277\pi$
$617$			21.7544i			0.875798i	−0.899024	+	0.437899i	$0.855723\pi$
$618$		0			0
$619$	−11.1362	−	41.5607i	−0.447600	−	1.67047i	−0.708979	−	0.705230i	$-0.750844\pi$
$619$	−11.1362	−	41.5607i	−0.447600	−	1.67047i	0.261378	−	0.965236i	$-0.415823\pi$
$620$		0			0
$621$	1.84320	−	6.87892i	0.0739651	−	0.276041i
$622$		0			0
$623$	5.41280	−	7.54397i	0.216859	−	0.302243i
$624$		0			0
$625$	−14.1442	−	24.4984i	−0.565767	−	0.979937i
$626$		0			0
$627$	0.384944	−	0.103146i	0.0153732	−	0.00411923i
$628$		0			0
$629$	−15.6162	+	15.6162i	−0.622658	+	0.622658i
$630$		0			0
$631$		−	15.4311i		−	0.614303i	−0.951661	−	0.307152i	$-0.900624\pi$
$631$		−	15.4311i		−	0.614303i	0.951661	−	0.307152i	$-0.0993758\pi$
$632$		0			0
$633$	8.87636	+	5.12477i	0.352804	+	0.203691i
$634$		0			0
$635$	−16.3889	−	4.39139i	−0.650373	−	0.174267i
$636$		0			0
$637$	43.3578	−	2.78948i	1.71790	−	0.110523i
$638$		0			0
$639$	−8.06358	−	13.9665i	−0.318990	−	0.552507i
$640$		0			0
$641$	7.89923	−	13.6819i	0.312001	−	0.540402i	−0.666794	−	0.745242i	$-0.732334\pi$
$641$	7.89923	−	13.6819i	0.312001	−	0.540402i	0.978795	+	0.204840i	$0.0656674\pi$
$642$		0			0
$643$	15.9560	+	15.9560i	0.629244	+	0.629244i	0.947878	−	0.318634i	$-0.103224\pi$
$643$	15.9560	+	15.9560i	0.629244	+	0.629244i	−0.318634	+	0.947878i	$0.603224\pi$
$644$		0			0
$645$	1.52462	−	1.52462i	0.0600318	−	0.0600318i
$646$		0			0
$647$	12.7953	+	7.38738i	0.503036	+	0.290428i	0.729966	−	0.683483i	$-0.239536\pi$
$647$	12.7953	+	7.38738i	0.503036	+	0.290428i	−0.226931	+	0.973911i	$0.572869\pi$
$648$		0			0
$649$	0.00550484	−	0.00317822i	0.000216084	−	0.000124756i
$650$		0			0
$651$	2.52780	+	15.3681i	0.0990724	+	0.602325i
$652$		0			0
$653$	1.57289	−	5.87011i	0.0615520	−	0.229715i	−0.928297	−	0.371840i	$-0.878727\pi$
$653$	1.57289	−	5.87011i	0.0615520	−	0.229715i	0.989849	+	0.142125i	$0.0453936\pi$
$654$		0			0
$655$	10.6023	−	18.3637i	0.414267	−	0.717531i
$656$		0			0
$657$	9.29082			0.362469
$658$		0			0
$659$	16.6838	+	16.6838i	0.649907	+	0.649907i	0.952970	−	0.303063i	$-0.0980094\pi$
$659$	16.6838	+	16.6838i	0.649907	+	0.649907i	−0.303063	+	0.952970i	$0.598009\pi$
$660$		0			0
$661$	10.6101	+	39.5975i	0.412686	+	1.54016i	0.789427	+	0.613845i	$0.210378\pi$
$661$	10.6101	+	39.5975i	0.412686	+	1.54016i	−0.376741	+	0.926319i	$0.622955\pi$
$662$		0			0
$663$	33.3261	−	19.2408i	1.29428	−	0.747251i
$664$		0			0
$665$	−3.72667	+	37.6058i	−0.144514	+	1.45829i
$666$		0			0
$667$	25.9686	+	6.95827i	1.00551	+	0.269425i
$668$		0			0
$669$	−17.4079	+	4.66443i	−0.673028	+	0.180337i
$670$		0			0
$671$	−0.180913			−0.00698407
$672$		0			0
$673$	−1.26298			−0.0486841			−0.0243421	−	0.999704i	$-0.507749\pi$
$673$	−1.26298			−0.0486841			−0.0243421	+	0.999704i	$0.507749\pi$
$674$		0			0
$675$	−1.39195	+	0.372972i	−0.0535762	+	0.0143557i
$676$		0			0
$677$	20.3687	+	5.45778i	0.782833	+	0.209759i	0.628033	−	0.778186i	$-0.283860\pi$
$677$	20.3687	+	5.45778i	0.782833	+	0.209759i	0.154800	+	0.987946i	$0.450527\pi$
$678$		0			0
$679$	26.8832	−	12.1612i	1.03168	−	0.466702i
$680$		0			0
$681$	15.0467	−	8.68724i	0.576593	−	0.332896i
$682$		0			0
$683$	−10.1519	−	37.8876i	−0.388453	−	1.44973i	−0.832651	−	0.553798i	$-0.813178\pi$
$683$	−10.1519	−	37.8876i	−0.388453	−	1.44973i	0.444197	−	0.895929i	$-0.353489\pi$
$684$		0			0
$685$	16.0471	+	16.0471i	0.613128	+	0.613128i
$686$		0			0
$687$	9.74631			0.371845
$688$		0			0
$689$	3.93408	−	6.81403i	0.149877	−	0.259594i
$690$		0			0
$691$	3.46195	−	12.9202i	0.131699	−	0.491507i	−0.868291	−	0.496056i	$-0.834781\pi$
$691$	3.46195	−	12.9202i	0.131699	−	0.491507i	0.999990	+	0.00454863i	$0.00144788\pi$
$692$		0			0
$693$	−0.159502	−	0.0601419i	−0.00605899	−	0.00228460i
$694$		0			0
$695$	−27.2812	+	15.7508i	−1.03483	+	0.597462i
$696$		0			0
$697$	−0.489284	−	0.282488i	−0.0185329	−	0.0107000i
$698$		0			0
$699$	34.0775	−	34.0775i	1.28893	−	1.28893i
$700$		0			0
$701$	−26.5854	−	26.5854i	−1.00412	−	1.00412i	−0.999991	−	0.00412374i	$-0.998687\pi$
$701$	−26.5854	−	26.5854i	−1.00412	−	1.00412i	−0.00412374	−	0.999991i	$-0.501313\pi$
$702$		0			0
$703$	−24.1028	+	41.7473i	−0.909054	+	1.57453i
$704$		0			0
$705$	10.4839	+	18.1587i	0.394848	+	0.683897i
$706$		0			0
$707$	24.0029	+	29.2835i	0.902722	+	1.10132i
$708$		0			0
$709$	−8.02123	−	2.14928i	−0.301244	−	0.0807180i	0.105031	−	0.994469i	$-0.466506\pi$
$709$	−8.02123	−	2.14928i	−0.301244	−	0.0807180i	−0.406275	+	0.913751i	$0.633172\pi$
$710$		0			0
$711$	17.0956	+	9.87013i	0.641134	+	0.370159i
$712$		0			0
$713$			9.95034i			0.372643i
$714$		0			0
$715$	−0.310699	+	0.310699i	−0.0116195	+	0.0116195i
$716$		0			0
$717$	50.9862	−	13.6617i	1.90412	−	0.510206i
$718$		0			0
$719$	−16.9795	−	29.4094i	−0.633230	−	1.09679i	−0.986887	−	0.161411i	$-0.948395\pi$
$719$	−16.9795	−	29.4094i	−0.633230	−	1.09679i	0.353657	−	0.935375i	$-0.384938\pi$
$720$		0			0
$721$	−6.74493	+	9.40060i	−0.251194	+	0.350097i
$722$		0			0
$723$	−3.57686	+	13.3490i	−0.133025	+	0.496456i
$724$		0			0
$725$	−1.40801	−	5.25476i	−0.0522921	−	0.195157i
$726$		0			0
$727$			8.94504i			0.331753i	0.986147	+	0.165877i	$0.0530454\pi$
$727$			8.94504i			0.331753i	−0.986147	+	0.165877i	$0.946955\pi$
$728$		0			0
$729$			10.9390i			0.405147i
$730$		0			0
$731$	−0.277422	−	1.03535i	−0.0102608	−	0.0382939i
$732$		0			0
$733$	−8.59937	+	32.0933i	−0.317625	+	1.18539i	0.603896	+	0.797063i	$0.293614\pi$
$733$	−8.59937	+	32.0933i	−0.317625	+	1.18539i	−0.921521	+	0.388329i	$0.873052\pi$
$734$		0			0
$735$	25.3063	−	28.7865i	0.933439	−	1.06181i
$736$		0			0
$737$	−0.182527	−	0.316146i	−0.00672348	−	0.0116454i
$738$		0			0
$739$	−51.2188	+	13.7240i	−1.88411	+	0.504847i	−0.884875	+	0.465828i	$0.845757\pi$
$739$	−51.2188	+	13.7240i	−1.88411	+	0.504847i	−0.999238	+	0.0390191i	$0.987577\pi$
$740$		0			0
$741$	59.3945	−	59.3945i	2.18191	−	2.18191i
$742$		0			0
$743$			36.7472i			1.34812i	0.738675	+	0.674062i	$0.235452\pi$
$743$			36.7472i			1.34812i	−0.738675	+	0.674062i	$0.764548\pi$
$744$		0			0
$745$	33.0340	+	19.0722i	1.21027	+	0.698750i
$746$		0			0
$747$	22.5882	+	6.05248i	0.826458	+	0.221449i
$748$		0			0
$749$	−47.1001	+	7.74718i	−1.72100	+	0.283076i
$750$		0			0
$751$	0.273033	+	0.472907i	0.00996312	+	0.0172566i	0.870964	−	0.491347i	$-0.163495\pi$
$751$	0.273033	+	0.472907i	0.00996312	+	0.0172566i	−0.861001	+	0.508603i	$0.830162\pi$
$752$		0			0
$753$	9.91206	−	17.1682i	0.361216	−	0.625644i
$754$		0			0
$755$	−3.88622	−	3.88622i	−0.141434	−	0.141434i
$756$		0			0
$757$	−1.43871	+	1.43871i	−0.0522908	+	0.0522908i	−0.732769	−	0.680478i	$-0.761772\pi$
$757$	−1.43871	+	1.43871i	−0.0522908	+	0.0522908i	0.680478	+	0.732769i	$0.261772\pi$
$758$		0			0
$759$	−0.223640	−	0.129119i	−0.00811761	−	0.00468671i
$760$		0			0
$761$	−44.8946	+	25.9199i	−1.62743	+	0.939596i	−0.642571	+	0.766226i	$0.722132\pi$
$761$	−44.8946	+	25.9199i	−1.62743	+	0.939596i	−0.984857	+	0.173369i	$0.944535\pi$
$762$		0			0
$763$	23.9915	−	19.6652i	0.868549	−	0.711927i
$764$		0			0
$765$	3.70545	−	13.8289i	0.133971	−	0.499986i
$766$		0			0
$767$	0.669871	−	1.16025i	0.0241877	−	0.0418942i
$768$		0			0
$769$	14.1856			0.511545			0.255773	−	0.966737i	$-0.417670\pi$
$769$	14.1856			0.511545			0.255773	+	0.966737i	$0.417670\pi$
$770$		0			0
$771$	35.6076	+	35.6076i	1.28238	+	1.28238i
$772$		0			0
$773$	−1.26443	−	4.71890i	−0.0454782	−	0.169727i	0.939452	−	0.342682i	$-0.111335\pi$
$773$	−1.26443	−	4.71890i	−0.0454782	−	0.169727i	−0.984930	+	0.172955i	$0.944669\pi$
$774$		0			0
$775$	1.74370	−	1.00673i	0.0626356	−	0.0361627i
$776$		0			0
$777$	44.5465	−	20.1515i	1.59810	−	0.722931i
$778$		0			0
$779$	−1.19119	−	0.319179i	−0.0426788	−	0.0114358i
$780$		0			0
$781$	0.209670	−	0.0561809i	0.00750259	−	0.00201031i
$782$		0			0
$783$	12.9168			0.461607
$784$		0			0
$785$	−14.0921			−0.502970
$786$		0			0
$787$	26.9638	−	7.22492i	0.961154	−	0.257541i	0.256065	−	0.966659i	$-0.417574\pi$
$787$	26.9638	−	7.22492i	0.961154	−	0.257541i	0.705089	+	0.709119i	$0.250907\pi$
$788$		0			0
$789$	9.69419	+	2.59755i	0.345122	+	0.0924752i
$790$		0			0
$791$	12.6748	−	5.73372i	0.450666	−	0.203868i
$792$		0			0
$793$	−33.0223	+	19.0655i	−1.17266	+	0.677034i
$794$		0			0
$795$	−1.79650	−	6.70462i	−0.0637152	−	0.237788i
$796$		0			0
$797$	−17.1613	−	17.1613i	−0.607884	−	0.607884i	0.334509	−	0.942393i	$-0.391430\pi$
$797$	−17.1613	−	17.1613i	−0.607884	−	0.607884i	−0.942393	+	0.334509i	$0.891430\pi$
$798$		0			0
$799$	10.4237			0.368765
$800$		0			0
$801$	−3.83906	+	6.64944i	−0.135646	+	0.234946i
$802$		0			0
$803$	−0.0323657	+	0.120790i	−0.00114216	+	0.00426260i
$804$		0			0
$805$	18.9383	−	15.5232i	0.667487	−	0.547121i
$806$		0			0
$807$	1.67237	−	0.965541i	0.0588701	−	0.0339886i
$808$		0			0
$809$	−44.9216	−	25.9355i	−1.57936	−	0.911844i	−0.994948	−	0.100388i	$-0.967992\pi$
$809$	−44.9216	−	25.9355i	−1.57936	−	0.911844i	−0.584413	−	0.811457i	$-0.698675\pi$
$810$		0			0
$811$	−28.9865	+	28.9865i	−1.01785	+	1.01785i	−0.0180155	+	0.999838i	$0.505735\pi$
$811$	−28.9865	+	28.9865i	−1.01785	+	1.01785i	−0.999838	+	0.0180155i	$0.994265\pi$
$812$		0			0
$813$	18.8967	+	18.8967i	0.662734	+	0.662734i
$814$		0			0
$815$	−17.7118	+	30.6778i	−0.620418	+	1.07460i
$816$		0			0
$817$	−1.16983	−	2.02620i	−0.0409271	−	0.0708878i
$818$		0			0
$819$	−35.4522	+	5.83130i	−1.23880	+	0.203762i
$820$		0			0
$821$	−6.16671	−	1.65236i	−0.215220	−	0.0576679i	0.149598	−	0.988747i	$-0.452202\pi$
$821$	−6.16671	−	1.65236i	−0.215220	−	0.0576679i	−0.364818	+	0.931079i	$0.618869\pi$
$822$		0			0
$823$	−24.1611	−	13.9494i	−0.842202	−	0.486245i	0.0158104	−	0.999875i	$-0.494967\pi$
$823$	−24.1611	−	13.9494i	−0.842202	−	0.486245i	−0.858012	+	0.513630i	$0.828301\pi$
$824$		0			0
$825$			0.0522544i			0.00181926i
$826$		0			0
$827$	−33.3036	+	33.3036i	−1.15808	+	1.15808i	−0.173190	+	0.984888i	$0.555408\pi$
$827$	−33.3036	+	33.3036i	−1.15808	+	1.15808i	−0.984888	+	0.173190i	$0.944592\pi$
$828$		0			0
$829$	−10.0105	+	2.68229i	−0.347678	+	0.0931599i	−0.428432	−	0.903574i	$-0.640934\pi$
$829$	−10.0105	+	2.68229i	−0.347678	+	0.0931599i	0.0807543	+	0.996734i	$0.474267\pi$
$830$		0			0
$831$	−18.2469	−	31.6045i	−0.632976	−	1.09635i
$832$		0			0
$833$	−6.10306	−	18.0503i	−0.211459	−	0.625406i
$834$		0			0
$835$	9.71597	−	36.2605i	0.336235	−	1.25485i
$836$		0			0
$837$	1.23732	+	4.61775i	0.0427681	+	0.159613i
$838$		0			0
$839$		−	26.9095i		−	0.929018i	−0.885568	−	0.464509i	$-0.846231\pi$
$839$		−	26.9095i		−	0.929018i	0.885568	−	0.464509i	$-0.153769\pi$
$840$		0			0
$841$			19.7621i			0.681451i
$842$		0			0
$843$	12.6043	+	47.0399i	0.434116	+	1.62014i
$844$		0			0
$845$	−15.8809	+	59.2685i	−0.546321	+	2.03890i
$846$		0			0
$847$	−16.9649	+	23.6444i	−0.582920	+	0.812432i
$848$		0			0
$849$	−21.9746	−	38.0611i	−0.754166	−	1.30625i
$850$		0			0
$851$	30.1721	−	8.08459i	1.03429	−	0.277136i
$852$		0			0
$853$	16.4841	−	16.4841i	0.564406	−	0.564406i	−0.366150	−	0.930556i	$-0.619324\pi$
$853$	16.4841	−	16.4841i	0.564406	−	0.564406i	0.930556	+	0.366150i	$0.119324\pi$
$854$		0			0
$855$		−	31.2502i		−	1.06873i
$856$		0			0
$857$	−7.09763	−	4.09782i	−0.242450	−	0.139979i	0.373852	−	0.927488i	$-0.378037\pi$
$857$	−7.09763	−	4.09782i	−0.242450	−	0.139979i	−0.616302	+	0.787510i	$0.711370\pi$
$858$		0			0
$859$	−15.8468	−	4.24615i	−0.540687	−	0.144877i	−0.0218682	−	0.999761i	$-0.506961\pi$
$859$	−15.8468	−	4.24615i	−0.540687	−	0.144877i	−0.518819	+	0.854884i	$0.673628\pi$
$860$		0			0
$861$	0.792908	+	0.967346i	0.0270222	+	0.0329671i
$862$		0			0
$863$	−11.8882	−	20.5910i	−0.404680	−	0.700927i	0.589604	−	0.807693i	$-0.299284\pi$
$863$	−11.8882	−	20.5910i	−0.404680	−	0.700927i	−0.994284	+	0.106766i	$0.965951\pi$
$864$		0			0
$865$	−1.47064	+	2.54722i	−0.0500033	+	0.0866082i
$866$		0			0
$867$	15.4463	+	15.4463i	0.524585	+	0.524585i
$868$		0			0
$869$	−0.187877	+	0.187877i	−0.00637328	+	0.00637328i
$870$		0			0
$871$	−66.6339	−	38.4711i	−2.25780	−	1.30354i
$872$		0			0
$873$	−21.1308	+	12.1999i	−0.715169	+	0.412903i
$874$		0			0
$875$	25.1201	+	9.47177i	0.849214	+	0.320204i
$876$		0			0
$877$	13.4774	−	50.2984i	0.455100	−	1.69846i	−0.232693	−	0.972550i	$-0.574754\pi$
$877$	13.4774	−	50.2984i	0.455100	−	1.69846i	0.687793	−	0.725907i	$-0.258580\pi$
$878$		0			0
$879$	−30.7564	+	53.2717i	−1.03739	+	1.79681i
$880$		0			0
$881$	−41.6166			−1.40210			−0.701050	−	0.713112i	$-0.747285\pi$
$881$	−41.6166			−1.40210			−0.701050	+	0.713112i	$0.747285\pi$
$882$		0			0
$883$	7.97461	+	7.97461i	0.268367	+	0.268367i	0.828442	−	0.560075i	$-0.189228\pi$
$883$	7.97461	+	7.97461i	0.268367	+	0.268367i	−0.560075	+	0.828442i	$0.689228\pi$
$884$		0			0
$885$	−0.305897	−	1.14162i	−0.0102826	−	0.0383752i
$886$		0			0
$887$	24.1323	−	13.9328i	0.810284	−	0.467818i	−0.0367704	−	0.999324i	$-0.511707\pi$
$887$	24.1323	−	13.9328i	0.810284	−	0.467818i	0.847055	+	0.531506i	$0.178374\pi$
$888$		0			0
$889$	17.0137	−	7.69647i	0.570620	−	0.258131i
$890$		0			0
$891$	−0.306545	−	0.0821384i	−0.0102696	−	0.00275174i
$892$		0			0
$893$	21.9773	−	5.88880i	0.735442	−	0.197061i
$894$		0			0
$895$	2.85198			0.0953312
$896$		0			0
$897$	−54.4284			−1.81731
$898$		0			0
$899$	−17.4325	+	4.67101i	−0.581405	+	0.155787i
$900$		0			0
$901$	−3.33305	−	0.893089i	−0.111040	−	0.0297531i
$902$		0			0
$903$	−0.234014	+	2.36143i	−0.00778749	+	0.0785834i
$904$		0			0
$905$	−5.63514	+	3.25345i	−0.187318	+	0.108148i
$906$		0			0
$907$	12.9779	+	48.4340i	0.430923	+	1.60823i	0.750639	+	0.660712i	$0.229746\pi$
$907$	12.9779	+	48.4340i	0.430923	+	1.60823i	−0.319717	+	0.947513i	$0.603588\pi$
$908$		0			0
$909$	−22.1403	−	22.1403i	−0.734348	−	0.734348i
$910$		0			0
$911$	−53.4186			−1.76984			−0.884919	−	0.465746i	$-0.845786\pi$
$911$	−53.4186			−1.76984			−0.884919	+	0.465746i	$0.845786\pi$
$912$		0			0
$913$	−0.157377	+	0.272585i	−0.00520843	+	0.00902126i
$914$		0			0
$915$	−8.70624	+	32.4921i	−0.287819	+	1.07416i
$916$		0			0
$917$	3.78771	+	23.0279i	0.125081	+	0.760448i
$918$		0			0
$919$	−38.4269	+	22.1858i	−1.26759	+	0.731841i	−0.974531	−	0.224254i	$-0.928005\pi$
$919$	−38.4269	+	22.1858i	−1.26759	+	0.731841i	−0.293055	+	0.956096i	$0.594672\pi$
$920$		0			0
$921$	−7.02612	−	4.05653i	−0.231519	−	0.133667i
$922$		0			0
$923$	32.3508	−	32.3508i	1.06484	−	1.06484i
$924$		0			0
$925$	−4.46942	−	4.46942i	−0.146954	−	0.146954i
$926$		0			0
$927$	4.78387	−	8.28591i	0.157123	−	0.272145i
$928$		0			0
$929$	−19.1351	−	33.1430i	−0.627804	−	1.08739i	−0.987992	−	0.154508i	$-0.950621\pi$
$929$	−19.1351	−	33.1430i	−0.627804	−	1.08739i	0.360188	−	0.932880i	$-0.382713\pi$
$930$		0			0
$931$	−23.0650	−	34.6092i	−0.755926	−	1.13427i
$932$		0			0
$933$	6.66586	+	1.78611i	0.218230	+	0.0584746i
$934$		0			0
$935$	0.166882	+	0.0963495i	0.00545763	+	0.00315097i
$936$		0			0
$937$			28.5675i			0.933259i	0.884453	+	0.466629i	$0.154532\pi$
$937$			28.5675i			0.933259i	−0.884453	+	0.466629i	$0.845468\pi$
$938$		0			0
$939$	44.2546	−	44.2546i	1.44419	−	1.44419i
$940$		0			0
$941$	−22.7200	+	6.08780i	−0.740650	+	0.198457i	−0.609367	−	0.792888i	$-0.708576\pi$
$941$	−22.7200	+	6.08780i	−0.740650	+	0.198457i	−0.131283	+	0.991345i	$0.541910\pi$
$942$		0			0
$943$	0.399551	+	0.692043i	0.0130112	+	0.0225360i
$944$		0			0
$945$	6.85856	−	9.55897i	0.223109	−	0.310953i
$946$		0			0
$947$	10.4404	−	38.9640i	0.339267	−	1.26616i	−0.559902	−	0.828559i	$-0.689161\pi$
$947$	10.4404	−	38.9640i	0.339267	−	1.26616i	0.899169	−	0.437602i	$-0.144172\pi$
$948$		0			0
$949$	6.82169	+	25.4589i	0.221442	+	0.826431i
$950$		0			0
$951$			19.2992i			0.625820i
$952$		0			0
$953$			13.3303i			0.431812i	0.976414	+	0.215906i	$0.0692704\pi$
$953$			13.3303i			0.431812i	−0.976414	+	0.215906i	$0.930730\pi$
$954$		0			0
$955$	−4.18677	−	15.6252i	−0.135481	−	0.505621i
$956$		0			0
$957$	0.121225	−	0.452418i	0.00391865	−	0.0146246i
$958$		0			0
$959$	−24.8548	−	2.46307i	−0.802603	−	0.0795366i
$960$		0			0
$961$	12.1602	+	21.0621i	0.392265	+	0.679423i
$962$		0			0
$963$	38.1275	−	10.2162i	1.22864	−	0.329213i
$964$		0			0
$965$	33.7594	−	33.7594i	1.08675	−	1.08675i
$966$		0			0
$967$		−	16.7779i		−	0.539540i	−0.962925	−	0.269770i	$-0.913052\pi$
$967$		−	16.7779i		−	0.539540i	0.962925	−	0.269770i	$-0.0869476\pi$
$968$		0			0
$969$	−31.9019	−	18.4186i	−1.02484	−	0.591690i
$970$		0			0
$971$	−0.305865	−	0.0819563i	−0.00981568	−	0.00263010i	0.253908	−	0.967228i	$-0.418284\pi$
$971$	−0.305865	−	0.0819563i	−0.00981568	−	0.00263010i	−0.263724	+	0.964598i	$0.584951\pi$
$972$		0			0
$973$	12.2320	−	32.4405i	0.392140	−	1.03999i
$974$		0			0
$975$	5.50680	+	9.53806i	0.176359	+	0.305463i
$976$		0			0
$977$	−3.58053	+	6.20166i	−0.114551	+	0.198409i	−0.917600	−	0.397504i	$-0.869876\pi$
$977$	−3.58053	+	6.20166i	−0.114551	+	0.198409i	0.803049	+	0.595913i	$0.203210\pi$
$978$		0			0
$979$	−0.0730759	−	0.0730759i	−0.00233552	−	0.00233552i
$980$		0			0
$981$	−18.1392	+	18.1392i	−0.579140	+	0.579140i
$982$		0			0
$983$	−12.8516	−	7.41988i	−0.409903	−	0.236657i	0.280845	−	0.959753i	$-0.409385\pi$
$983$	−12.8516	−	7.41988i	−0.409903	−	0.236657i	−0.690748	+	0.723096i	$0.742719\pi$
$984$		0			0
$985$	27.5866	−	15.9271i	0.878982	−	0.507481i
$986$		0			0
$987$	−21.5928	−	8.14178i	−0.687307	−	0.259156i
$988$		0			0
$989$	−0.392385	+	1.46440i	−0.0124771	+	0.0465653i
$990$		0			0
$991$	0.108889	−	0.188601i	0.00345898	−	0.00599112i	−0.864291	−	0.502993i	$-0.832232\pi$
$991$	0.108889	−	0.188601i	0.00345898	−	0.00599112i	0.867750	+	0.497001i	$0.165566\pi$
$992$		0			0
$993$	−53.5497			−1.69935
$994$		0			0
$995$	8.89826	+	8.89826i	0.282094	+	0.282094i
$996$		0			0
$997$	−5.92816	−	22.1242i	−0.187747	−	0.700680i	−0.994026	−	0.109145i	$-0.965189\pi$
$997$	−5.92816	−	22.1242i	−0.187747	−	0.700680i	0.806279	−	0.591535i	$-0.201478\pi$
$998$		0			0
$999$	12.9969	−	7.50379i	0.411205	−	0.237409i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.ba.c.81.10		48
4.3	odd	2		112.2.w.c.109.4	yes	48
7.2	even	3	inner	448.2.ba.c.401.3		48
8.3	odd	2		896.2.ba.f.417.10		48
8.5	even	2		896.2.ba.e.417.3		48
16.3	odd	4		896.2.ba.f.865.3		48
16.5	even	4	inner	448.2.ba.c.305.3		48
16.11	odd	4		112.2.w.c.53.5	yes	48
16.13	even	4		896.2.ba.e.865.10		48
28.3	even	6		784.2.m.k.589.12		24
28.11	odd	6		784.2.m.j.589.12		24
28.19	even	6		784.2.x.o.765.5		48
28.23	odd	6		112.2.w.c.93.5	yes	48
28.27	even	2		784.2.x.o.557.4		48
56.37	even	6		896.2.ba.e.289.10		48
56.51	odd	6		896.2.ba.f.289.3		48
112.11	odd	12		784.2.m.j.197.12		24
112.27	even	4		784.2.x.o.165.5		48
112.37	even	12	inner	448.2.ba.c.177.10		48
112.51	odd	12		896.2.ba.f.737.10		48
112.59	even	12		784.2.m.k.197.12		24
112.75	even	12		784.2.x.o.373.4		48
112.93	even	12		896.2.ba.e.737.3		48
112.107	odd	12		112.2.w.c.37.4	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.w.c.37.4	✓	48	112.107	odd	12
112.2.w.c.53.5	yes	48	16.11	odd	4
112.2.w.c.93.5	yes	48	28.23	odd	6
112.2.w.c.109.4	yes	48	4.3	odd	2
448.2.ba.c.81.10		48	1.1	even	1	trivial
448.2.ba.c.177.10		48	112.37	even	12	inner
448.2.ba.c.305.3		48	16.5	even	4	inner
448.2.ba.c.401.3		48	7.2	even	3	inner
784.2.m.j.197.12		24	112.11	odd	12
784.2.m.j.589.12		24	28.11	odd	6
784.2.m.k.197.12		24	112.59	even	12
784.2.m.k.589.12		24	28.3	even	6
784.2.x.o.165.5		48	112.27	even	4
784.2.x.o.373.4		48	112.75	even	12
784.2.x.o.557.4		48	28.27	even	2
784.2.x.o.765.5		48	28.19	even	6
896.2.ba.e.289.10		48	56.37	even	6
896.2.ba.e.417.3		48	8.5	even	2
896.2.ba.e.737.3		48	112.93	even	12
896.2.ba.e.865.10		48	16.13	even	4
896.2.ba.f.289.3		48	56.51	odd	6
896.2.ba.f.417.10		48	8.3	odd	2
896.2.ba.f.737.10		48	112.51	odd	12
896.2.ba.f.865.3		48	16.3	odd	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.ba (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.20008 - 0.589510i) q^{3} +(2.32205 + 0.622192i) q^{5} +(-2.41058 + 1.09047i) q^{7} +(1.89476 - 1.09394i) q^{9} +O(q^{10})\)	\(q+(2.20008 - 0.589510i) q^{3} +(2.32205 + 0.622192i) q^{5} +(-2.41058 + 1.09047i) q^{7} +(1.89476 - 1.09394i) q^{9} +(0.00762177 + 0.0284448i) q^{11} +(4.38886 + 4.38886i) q^{13} +5.47550 q^{15} +(1.36101 - 2.35734i) q^{17} +(1.53778 - 5.73909i) q^{19} +(-4.66062 + 3.82019i) q^{21} +(-3.33422 + 1.92501i) q^{23} +(0.674681 + 0.389527i) q^{25} +(-1.30797 + 1.30797i) q^{27} +(-4.93772 - 4.93772i) q^{29} +(1.29224 - 2.23823i) q^{31} +(0.0335370 + 0.0580879i) q^{33} +(-6.27597 + 1.03229i) q^{35} +(-7.83686 - 2.09988i) q^{37} +(12.2431 + 7.06857i) q^{39} -0.207557i q^{41} +(0.278444 - 0.278444i) q^{43} +(5.08039 - 1.36129i) q^{45} +(1.91470 + 3.31636i) q^{47} +(4.62174 - 5.25733i) q^{49} +(1.60466 - 5.98867i) q^{51} +(-0.328098 - 1.22448i) q^{53} +0.0707926i q^{55} -13.5330i q^{57} +(-0.0558664 - 0.208496i) q^{59} +(-1.59004 + 5.93409i) q^{61} +(-3.37456 + 4.70322i) q^{63} +(7.46045 + 12.9219i) q^{65} +(-11.9741 + 3.20844i) q^{67} +(-6.20075 + 6.20075i) q^{69} -7.37112i q^{71} +(3.67756 + 2.12324i) q^{73} +(1.71398 + 0.459261i) q^{75} +(-0.0493911 - 0.0602571i) q^{77} +(4.51126 + 7.81374i) q^{79} +(-5.38841 + 9.33299i) q^{81} +(7.55784 + 7.55784i) q^{83} +(4.62706 - 4.62706i) q^{85} +(-13.7742 - 7.95255i) q^{87} +(-3.03921 + 1.75469i) q^{89} +(-15.3656 - 5.79375i) q^{91} +(1.52358 - 5.68608i) q^{93} +(7.14164 - 12.3697i) q^{95} -11.1522 q^{97} +(0.0455585 + 0.0455585i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{5}+O(q^{10}) \) 48 * q + 4 * q^5	\( 48 q + 4 q^{5} + 4 q^{11} - 24 q^{13} + 40 q^{15} + 8 q^{17} + 4 q^{19} - 8 q^{21} + 24 q^{27} + 24 q^{29} - 28 q^{31} + 16 q^{33} - 28 q^{35} - 24 q^{37} + 40 q^{43} - 28 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} + 20 q^{59} + 8 q^{61} + 16 q^{63} + 8 q^{65} - 48 q^{67} - 40 q^{69} + 4 q^{75} - 20 q^{77} + 36 q^{79} + 8 q^{83} - 64 q^{91} + 8 q^{93} + 4 q^{95} - 48 q^{97} + 24 q^{99}+O(q^{100}) \) 48 * q + 4 * q^5 + 4 * q^11 - 24 * q^13 + 40 * q^15 + 8 * q^17 + 4 * q^19 - 8 * q^21 + 24 * q^27 + 24 * q^29 - 28 * q^31 + 16 * q^33 - 28 * q^35 - 24 * q^37 + 40 * q^43 - 28 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 + 20 * q^59 + 8 * q^61 + 16 * q^63 + 8 * q^65 - 48 * q^67 - 40 * q^69 + 4 * q^75 - 20 * q^77 + 36 * q^79 + 8 * q^83 - 64 * q^91 + 8 * q^93 + 4 * q^95 - 48 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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