LMFDB - Embedded newform 688.2.bg.c.225.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("688.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			225.1
Character	$\chi$	$=$	688.225
Dual form			688.2.bg.c.529.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0129300 + 0.0119973i) q^{3} +(-3.39819 + 0.512194i) q^{5} +(0.134521 + 0.232998i) q^{7} +(-0.224167 - 2.99130i) q^{9} +O(q^{10})$	\(q+(0.0129300 + 0.0119973i) q^{3} +(-3.39819 + 0.512194i) q^{5} +(0.134521 + 0.232998i) q^{7} +(-0.224167 - 2.99130i) q^{9} +(2.96782 + 1.42923i) q^{11} +(-0.736485 + 1.87653i) q^{13} +(-0.0500834 - 0.0341463i) q^{15} +(6.37398 + 0.960723i) q^{17} +(0.449267 - 5.99506i) q^{19} +(-0.00105598 + 0.00462655i) q^{21} +(1.83546 - 1.25139i) q^{23} +(6.50747 - 2.00729i) q^{25} +(0.0659813 - 0.0827379i) q^{27} +(1.75989 - 1.63294i) q^{29} +(4.93996 + 1.52378i) q^{31} +(0.0212271 + 0.0540856i) q^{33} +(-0.576470 - 0.722870i) q^{35} +(2.63757 - 4.56841i) q^{37} +(-0.0320360 + 0.0154277i) q^{39} +(-0.643386 - 2.81886i) q^{41} +(4.01778 + 5.18242i) q^{43} +(2.29389 + 10.0502i) q^{45} +(5.31835 - 2.56118i) q^{47} +(3.46381 - 5.99949i) q^{49} +(0.0708893 + 0.0888924i) q^{51} +(2.34453 + 5.97377i) q^{53} +(-10.8173 - 3.33669i) q^{55} +(0.0777333 - 0.0721259i) q^{57} +(-8.36244 + 10.4862i) q^{59} +(-9.50158 + 2.93085i) q^{61} +(0.666812 - 0.454625i) q^{63} +(1.54156 - 6.75403i) q^{65} +(0.950106 - 12.6783i) q^{67} +(0.0387457 + 0.00583998i) q^{69} +(4.98725 + 3.40025i) q^{71} +(0.609746 - 1.55361i) q^{73} +(0.108223 + 0.0521176i) q^{75} +(0.0662286 + 0.883759i) q^{77} +(-6.97172 - 12.0754i) q^{79} +(-8.89671 + 1.34096i) q^{81} +(5.60384 + 5.19960i) q^{83} -22.1521 q^{85} +0.0423461 q^{87} +(-1.32654 - 1.23085i) q^{89} +(-0.536302 + 0.0808345i) q^{91} +(0.0455924 + 0.0789684i) q^{93} +(1.54394 + 20.6024i) q^{95} +(2.36508 + 1.13896i) q^{97} +(3.60997 - 9.19804i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.0129300	+	0.0119973i	0.00746512	+	0.00692662i	0.683896	−	0.729579i	$-0.260284\pi$
$3$	0.0129300	+	0.0119973i	0.00746512	+	0.00692662i	−0.676431	+	0.736506i	$0.736474\pi$
$4$		0			0
$5$	−3.39819	+	0.512194i	−1.51972	+	0.229060i	−0.855233	−	0.518243i	$-0.826586\pi$
$5$	−3.39819	+	0.512194i	−1.51972	+	0.229060i	−0.664483	+	0.747304i	$0.731348\pi$
$6$		0			0
$7$	0.134521	+	0.232998i	0.0508443	+	0.0880650i	0.890327	−	0.455321i	$-0.150475\pi$
$7$	0.134521	+	0.232998i	0.0508443	+	0.0880650i	−0.839483	+	0.543386i	$0.817142\pi$
$8$		0			0
$9$	−0.224167	−	2.99130i	−0.0747223	−	0.997100i
$10$		0			0
$11$	2.96782	+	1.42923i	0.894833	+	0.430929i	0.824020	−	0.566561i	$-0.191726\pi$
$11$	2.96782	+	1.42923i	0.894833	+	0.430929i	0.0708129	+	0.997490i	$0.477441\pi$
$12$		0			0
$13$	−0.736485	+	1.87653i	−0.204264	+	0.520457i	−0.995892	−	0.0905467i	$-0.971139\pi$
$13$	−0.736485	+	1.87653i	−0.204264	+	0.520457i	0.791628	+	0.611003i	$0.209234\pi$
$14$		0			0
$15$	−0.0500834	−	0.0341463i	−0.0129315	−	0.00881653i
$16$		0			0
$17$	6.37398	+	0.960723i	1.54592	+	0.233010i	0.865911	−	0.500197i	$-0.166739\pi$
$17$	6.37398	+	0.960723i	1.54592	+	0.233010i	0.680006	+	0.733207i	$0.261977\pi$
$18$		0			0
$19$	0.449267	−	5.99506i	0.103069	−	1.37536i	−0.670623	−	0.741798i	$-0.733973\pi$
$19$	0.449267	−	5.99506i	0.103069	−	1.37536i	0.773692	−	0.633562i	$-0.218408\pi$
$20$		0			0
$21$	−0.00105598	+	0.00462655i	−0.000230434	+	0.00100960i
$22$		0			0
$23$	1.83546	−	1.25139i	0.382719	−	0.260934i	−0.356644	−	0.934240i	$-0.616079\pi$
$23$	1.83546	−	1.25139i	0.382719	−	0.260934i	0.739363	+	0.673307i	$0.235127\pi$
$24$		0			0
$25$	6.50747	−	2.00729i	1.30149	−	0.401458i
$26$		0			0
$27$	0.0659813	−	0.0827379i	0.0126981	−	0.0159229i
$28$		0			0
$29$	1.75989	−	1.63294i	0.326803	−	0.303229i	−0.499658	−	0.866223i	$-0.666541\pi$
$29$	1.75989	−	1.63294i	0.326803	−	0.303229i	0.826461	+	0.562994i	$0.190351\pi$
$30$		0			0
$31$	4.93996	+	1.52378i	0.887243	+	0.273678i	0.704688	−	0.709517i	$-0.251087\pi$
$31$	4.93996	+	1.52378i	0.887243	+	0.273678i	0.182555	+	0.983196i	$0.441563\pi$
$32$		0			0
$33$	0.0212271	+	0.0540856i	0.00369516	+	0.00941510i
$34$		0			0
$35$	−0.576470	−	0.722870i	−0.0974411	−	0.122187i
$36$		0			0
$37$	2.63757	−	4.56841i	0.433615	−	0.751042i	−0.563567	−	0.826070i	$-0.690571\pi$
$37$	2.63757	−	4.56841i	0.433615	−	0.751042i	0.997181	+	0.0750281i	$0.0239046\pi$
$38$		0			0
$39$	−0.0320360	+	0.0154277i	−0.00512986	+	0.00247041i
$40$		0			0
$41$	−0.643386	−	2.81886i	−0.100480	−	0.440232i	−0.999994	−	0.00334069i	$-0.998937\pi$
$41$	−0.643386	−	2.81886i	−0.100480	−	0.440232i	0.899514	−	0.436891i	$-0.143921\pi$
$42$		0			0
$43$	4.01778	+	5.18242i	0.612705	+	0.790311i
$43$	4.01778	+	5.18242i	0.612705	+	0.790311i
$44$		0			0
$45$	2.29389	+	10.0502i	0.341953	+	1.49819i
$46$		0			0
$47$	5.31835	−	2.56118i	0.775761	−	0.373587i	−0.00373559	−	0.999993i	$-0.501189\pi$
$47$	5.31835	−	2.56118i	0.775761	−	0.373587i	0.779497	+	0.626406i	$0.215475\pi$
$48$		0			0
$49$	3.46381	−	5.99949i	0.494830	−	0.857070i
$50$		0			0
$51$	0.0708893	+	0.0888924i	0.00992649	+	0.0124474i
$52$		0			0
$53$	2.34453	+	5.97377i	0.322046	+	0.820561i	0.996604	+	0.0823389i	$0.0262390\pi$
$53$	2.34453	+	5.97377i	0.322046	+	0.820561i	−0.674558	+	0.738222i	$0.735666\pi$
$54$		0			0
$55$	−10.8173	−	3.33669i	−1.45860	−	0.449919i
$56$		0			0
$57$	0.0777333	−	0.0721259i	0.0102960	−	0.00955331i
$58$		0			0
$59$	−8.36244	+	10.4862i	−1.08870	+	1.36518i	−0.163128	+	0.986605i	$0.552158\pi$
$59$	−8.36244	+	10.4862i	−1.08870	+	1.36518i	−0.925569	+	0.378578i	$0.876413\pi$
$60$		0			0
$61$	−9.50158	+	2.93085i	−1.21655	+	0.375257i	−0.835600	−	0.549338i	$-0.814880\pi$
$61$	−9.50158	+	2.93085i	−1.21655	+	0.375257i	−0.380952	+	0.924595i	$0.624404\pi$
$62$		0			0
$63$	0.666812	−	0.454625i	0.0840104	−	0.0572773i
$64$		0			0
$65$	1.54156	−	6.75403i	0.191208	−	0.837735i
$66$		0			0
$67$	0.950106	−	12.6783i	0.116074	−	1.54890i	−0.570169	−	0.821527i	$-0.693122\pi$
$67$	0.950106	−	12.6783i	0.116074	−	1.54890i	0.686243	−	0.727372i	$-0.259259\pi$
$68$		0			0
$69$	0.0387457	+	0.00583998i	0.00466443	+	0.000703050i
$70$		0			0
$71$	4.98725	+	3.40025i	0.591877	+	0.403535i	0.821885	−	0.569654i	$-0.192923\pi$
$71$	4.98725	+	3.40025i	0.591877	+	0.403535i	−0.230007	+	0.973189i	$0.573875\pi$
$72$		0			0
$73$	0.609746	−	1.55361i	0.0713654	−	0.181836i	−0.890740	−	0.454512i	$-0.849814\pi$
$73$	0.609746	−	1.55361i	0.0713654	−	0.181836i	0.962106	+	0.272676i	$0.0879088\pi$
$74$		0			0
$75$	0.108223	+	0.0521176i	0.0124966	+	0.00601803i
$76$		0			0
$77$	0.0662286	+	0.883759i	0.00754745	+	0.100714i
$78$		0			0
$79$	−6.97172	−	12.0754i	−0.784380	−	1.35859i	−0.929369	−	0.369153i	$-0.879648\pi$
$79$	−6.97172	−	12.0754i	−0.784380	−	1.35859i	0.144989	−	0.989433i	$-0.453685\pi$
$80$		0			0
$81$	−8.89671	+	1.34096i	−0.988523	+	0.148996i
$82$		0			0
$83$	5.60384	+	5.19960i	0.615101	+	0.570731i	0.924966	−	0.380050i	$-0.124093\pi$
$83$	5.60384	+	5.19960i	0.615101	+	0.570731i	−0.309864	+	0.950781i	$0.600284\pi$
$84$		0			0
$85$	−22.1521			−2.40273
$86$		0			0
$87$	0.0423461			0.00453998
$88$		0			0
$89$	−1.32654	−	1.23085i	−0.140613	−	0.130470i	0.606730	−	0.794908i	$-0.292481\pi$
$89$	−1.32654	−	1.23085i	−0.140613	−	0.130470i	−0.747344	+	0.664437i	$0.768671\pi$
$90$		0			0
$91$	−0.536302	+	0.0808345i	−0.0562197	+	0.00847375i
$92$		0			0
$93$	0.0455924	+	0.0789684i	0.00472771	+	0.00818864i
$94$		0			0
$95$	1.54394	+	20.6024i	0.158405	+	2.11377i
$96$		0			0
$97$	2.36508	+	1.13896i	0.240138	+	0.115644i	0.550082	−	0.835110i	$-0.314596\pi$
$97$	2.36508	+	1.13896i	0.240138	+	0.115644i	−0.309945	+	0.950755i	$0.600311\pi$
$98$		0			0
$99$	3.60997	−	9.19804i	0.362815	−	0.924438i
$100$		0			0
$101$	7.32150	+	4.99172i	0.728517	+	0.496694i	0.869880	−	0.493264i	$-0.164196\pi$
$101$	7.32150	+	4.99172i	0.728517	+	0.496694i	−0.141363	+	0.989958i	$0.545149\pi$
$102$		0			0
$103$	0.748654	+	0.112841i	0.0737671	+	0.0111186i	0.185822	−	0.982583i	$-0.440505\pi$
$103$	0.748654	+	0.112841i	0.0737671	+	0.0111186i	−0.112055	+	0.993702i	$0.535743\pi$
$104$		0			0
$105$	0.00121872	−	0.0162627i	0.000118935	−	0.00158708i
$106$		0			0
$107$	−0.737592	+	3.23160i	−0.0713057	+	0.312411i	−0.997986	−	0.0634408i	$-0.979793\pi$
$107$	−0.737592	+	3.23160i	−0.0713057	+	0.312411i	0.926680	+	0.375852i	$0.122650\pi$
$108$		0			0
$109$	8.34007	−	5.68616i	0.798833	−	0.544635i	−0.0937085	−	0.995600i	$-0.529872\pi$
$109$	8.34007	−	5.68616i	0.798833	−	0.544635i	0.892542	+	0.450964i	$0.148920\pi$
$110$		0			0
$111$	0.0889122	−	0.0274258i	0.00843917	−	0.00260314i
$112$		0			0
$113$	−3.49529	+	4.38295i	−0.328809	+	0.412313i	−0.918566	−	0.395267i	$-0.870652\pi$
$113$	−3.49529	+	4.38295i	−0.328809	+	0.412313i	0.589757	+	0.807581i	$0.299223\pi$
$114$		0			0
$115$	−5.59627	+	5.19258i	−0.521855	+	0.484211i
$116$		0			0
$117$	5.77837	+	1.78239i	0.534211	+	0.164782i
$118$		0			0
$119$	0.633591	+	1.61436i	0.0580812	+	0.147988i
$120$		0			0
$121$	−0.0931005	−	0.116744i	−0.00846368	−	0.0106131i
$122$		0			0
$123$	0.0254996	−	0.0441666i	0.00229922	−	0.00398237i
$124$		0			0
$125$	−5.60427	+	2.69887i	−0.501261	+	0.241395i
$126$		0			0
$127$	−2.06730	−	9.05743i	−0.183443	−	0.803717i	−0.979975	−	0.199121i	$-0.936191\pi$
$127$	−2.06730	−	9.05743i	−0.183443	−	0.803717i	0.796532	−	0.604597i	$-0.206666\pi$
$128$		0			0
$129$	−0.0102251	+	0.115211i	−0.000900267	+	0.0101437i
$130$		0			0
$131$	−1.20701	−	5.28827i	−0.105457	−	0.462039i	−0.999890	−	0.0148359i	$-0.995277\pi$
$131$	−1.20701	−	5.28827i	−0.105457	−	0.462039i	0.894433	−	0.447203i	$-0.147580\pi$
$132$		0			0
$133$	1.45727	−	0.701786i	0.126362	−	0.0608525i
$134$		0			0
$135$	−0.181839	+	0.314954i	−0.0156502	+	0.0271069i
$136$		0			0
$137$	6.88665	+	8.63559i	0.588366	+	0.737788i	0.983514	−	0.180829i	$-0.0578782\pi$
$137$	6.88665	+	8.63559i	0.588366	+	0.737788i	−0.395148	+	0.918617i	$0.629307\pi$
$138$		0			0
$139$	2.80876	+	7.15661i	0.238236	+	0.607016i	0.999123	−	0.0418687i	$-0.0133311\pi$
$139$	2.80876	+	7.15661i	0.238236	+	0.607016i	−0.760887	+	0.648884i	$0.775236\pi$
$140$		0			0
$141$	0.0994933	+	0.0306896i	0.00837885	+	0.00258453i
$142$		0			0
$143$	−4.86775	+	4.51662i	−0.407062	+	0.377698i
$144$		0			0
$145$	−5.14405	+	6.45043i	−0.427190	+	0.535679i
$146$		0			0
$147$	0.116764	−	0.0360170i	0.00963056	−	0.00297064i
$148$		0			0
$149$	−0.776822	+	0.529628i	−0.0636397	+	0.0433888i	−0.594722	−	0.803931i	$-0.702738\pi$
$149$	−0.776822	+	0.529628i	−0.0636397	+	0.0433888i	0.531082	+	0.847320i	$0.321785\pi$
$150$		0			0
$151$	−2.44733	+	10.7224i	−0.199161	+	0.872579i	0.772277	+	0.635285i	$0.219118\pi$
$151$	−2.44733	+	10.7224i	−0.199161	+	0.872579i	−0.971438	+	0.237294i	$0.923740\pi$
$152$		0			0
$153$	1.44498	−	19.2819i	0.116819	−	1.55885i
$154$		0			0
$155$	−17.5674	−	2.64786i	−1.41105	−	0.212681i
$156$		0			0
$157$	−15.8917	−	10.8348i	−1.26829	−	0.864708i	−0.273038	−	0.962003i	$-0.588029\pi$
$157$	−15.8917	−	10.8348i	−1.26829	−	0.864708i	−0.995256	+	0.0972949i	$0.968981\pi$
$158$		0			0
$159$	−0.0413542	+	0.105369i	−0.00327960	+	0.00835628i
$160$		0			0
$161$	0.538481	+	0.259319i	0.0424382	+	0.0204372i
$162$		0			0
$163$	−1.43404	−	19.1360i	−0.112323	−	1.49884i	−0.714023	−	0.700122i	$-0.753129\pi$
$163$	−1.43404	−	19.1360i	−0.112323	−	1.49884i	0.601700	−	0.798722i	$-0.294490\pi$
$164$		0			0
$165$	−0.0998359	−	0.172921i	−0.00777221	−	0.0134619i
$166$		0			0
$167$	−6.00399	+	0.904956i	−0.464603	+	0.0700276i	−0.377172	−	0.926143i	$-0.623104\pi$
$167$	−6.00399	+	0.904956i	−0.464603	+	0.0700276i	−0.0874302	+	0.996171i	$0.527865\pi$
$168$		0			0
$169$	6.55071	+	6.07817i	0.503901	+	0.467551i
$170$		0			0
$171$	−18.0337			−1.37907
$172$		0			0
$173$	−11.0260			−0.838294			−0.419147	−	0.907918i	$-0.637671\pi$
$173$	−11.0260			−0.838294			−0.419147	+	0.907918i	$0.637671\pi$
$174$		0			0
$175$	1.34309	+	1.24620i	0.101528	+	0.0942042i
$176$		0			0
$177$	−0.233931	+	0.0352595i	−0.0175834	+	0.00265027i
$178$		0			0
$179$	4.06679	+	7.04389i	0.303966	+	0.526485i	0.977031	−	0.213099i	$-0.0683557\pi$
$179$	4.06679	+	7.04389i	0.303966	+	0.526485i	−0.673064	+	0.739584i	$0.735022\pi$
$180$		0			0
$181$	0.451827	+	6.02922i	0.0335841	+	0.448148i	0.988495	+	0.151255i	$0.0483315\pi$
$181$	0.451827	+	6.02922i	0.0335841	+	0.448148i	−0.954911	+	0.296893i	$0.904049\pi$
$182$		0			0
$183$	−0.158017	−	0.0760971i	−0.0116810	−	0.00562526i
$184$		0			0
$185$	−6.62306	+	16.8753i	−0.486937	+	1.24069i
$186$		0			0
$187$	17.5438	+	11.9611i	1.28293	+	0.874685i
$188$		0			0
$189$	0.0281537	+	0.00424348i	0.00204788	+	0.000308668i
$190$		0			0
$191$	1.15304	−	15.3862i	0.0834307	−	1.11331i	−0.785662	−	0.618656i	$-0.787677\pi$
$191$	1.15304	−	15.3862i	0.0834307	−	1.11331i	0.869093	−	0.494649i	$-0.164703\pi$
$192$		0			0
$193$	0.542270	−	2.37584i	0.0390335	−	0.171017i	−0.951654	−	0.307172i	$-0.900617\pi$
$193$	0.542270	−	2.37584i	0.0390335	−	0.171017i	0.990688	+	0.136155i	$0.0434745\pi$
$194$		0			0
$195$	0.100962	−	0.0688349i	0.00723006	−	0.00492937i
$196$		0			0
$197$	−2.38077	+	0.734371i	−0.169623	+	0.0523218i	−0.378403	−	0.925641i	$-0.623527\pi$
$197$	−2.38077	+	0.734371i	−0.169623	+	0.0523218i	0.208780	+	0.977963i	$0.433051\pi$
$198$		0			0
$199$	−5.15076	+	6.45884i	−0.365127	+	0.457855i	−0.930128	−	0.367235i	$-0.880304\pi$
$199$	−5.15076	+	6.45884i	−0.365127	+	0.457855i	0.565001	+	0.825090i	$0.308876\pi$
$200$		0			0
$201$	0.164390	−	0.152531i	0.0115951	−	0.0107587i
$202$		0			0
$203$	0.617214	+	0.190385i	0.0433199	+	0.0133624i
$204$		0			0
$205$	3.63015	+	9.24947i	0.253541	+	0.646011i
$206$		0			0
$207$	−4.15474	−	5.20988i	−0.288775	−	0.362112i
$208$		0			0
$209$	9.90166	−	17.1502i	0.684912	−	1.18630i
$210$		0			0
$211$	−12.8995	+	6.21209i	−0.888041	+	0.427658i	−0.821555	−	0.570129i	$-0.806893\pi$
$211$	−12.8995	+	6.21209i	−0.888041	+	0.427658i	−0.0664862	+	0.997787i	$0.521179\pi$
$212$		0			0
$213$	0.0236913	+	0.103798i	0.00162330	+	0.00711215i
$214$		0			0
$215$	−16.3076	−	15.5529i	−1.11217	−	1.06070i
$216$		0			0
$217$	0.309494	+	1.35598i	0.0210098	+	0.0920501i
$218$		0			0
$219$	0.0265230	−	0.0127728i	0.00179226	−	0.000863107i
$220$		0			0
$221$	−6.49717	+	11.2534i	−0.437047	+	0.756987i
$222$		0			0
$223$	4.25565	+	5.33642i	0.284980	+	0.357353i	0.903631	−	0.428313i	$-0.140892\pi$
$223$	4.25565	+	5.33642i	0.284980	+	0.357353i	−0.618651	+	0.785666i	$0.712320\pi$
$224$		0			0
$225$	−7.46317	−	19.0158i	−0.497544	−	1.26772i
$226$		0			0
$227$	22.4926	+	6.93804i	1.49288	+	0.460494i	0.930472	−	0.366364i	$-0.119397\pi$
$227$	22.4926	+	6.93804i	1.49288	+	0.460494i	0.562412	+	0.826857i	$0.309874\pi$
$228$		0			0
$229$	−11.5699	+	10.7353i	−0.764562	+	0.709410i	−0.962642	−	0.270776i	$-0.912720\pi$
$229$	−11.5699	+	10.7353i	−0.764562	+	0.709410i	0.198080	+	0.980186i	$0.436529\pi$
$230$		0			0
$231$	−0.00974636	+	0.0122215i	−0.000641263	+	0.000804119i
$232$		0			0
$233$	−2.67367	+	0.824717i	−0.175158	+	0.0540290i	−0.381093	−	0.924537i	$-0.624452\pi$
$233$	−2.67367	+	0.824717i	−0.175158	+	0.0540290i	0.205936	+	0.978566i	$0.433976\pi$
$234$		0			0
$235$	−16.7609	+	11.4274i	−1.09336	+	0.745442i
$236$		0			0
$237$	0.0547273	−	0.239776i	0.00355492	−	0.0155751i
$238$		0			0
$239$	−0.382464	+	5.10363i	−0.0247396	+	0.330127i	0.971078	+	0.238763i	$0.0767418\pi$
$239$	−0.382464	+	5.10363i	−0.0247396	+	0.330127i	−0.995818	+	0.0913641i	$0.970877\pi$
$240$		0			0
$241$	5.77630	+	0.870636i	0.372084	+	0.0560826i	0.332423	−	0.943131i	$-0.392134\pi$
$241$	5.77630	+	0.870636i	0.372084	+	0.0560826i	0.0396614	+	0.999213i	$0.487372\pi$
$242$		0			0
$243$	−0.393434	−	0.268239i	−0.0252388	−	0.0172075i
$244$		0			0
$245$	−8.69776	+	22.1615i	−0.555680	+	1.41585i
$246$		0			0
$247$	10.9190	+	5.25833i	0.694762	+	0.334580i
$248$		0			0
$249$	0.0100765	+	0.134461i	0.000638572	+	0.00852115i
$250$		0			0
$251$	−4.39608	−	7.61423i	−0.277478	−	0.480606i	0.693279	−	0.720669i	$-0.256165\pi$
$251$	−4.39608	−	7.61423i	−0.277478	−	0.480606i	−0.970757	+	0.240063i	$0.922832\pi$
$252$		0			0
$253$	7.23584	−	1.09063i	0.454914	−	0.0685672i
$254$		0			0
$255$	−0.286425	−	0.265764i	−0.0179367	−	0.0166428i
$256$		0			0
$257$	−6.02274			−0.375688			−0.187844	−	0.982199i	$-0.560150\pi$
$257$	−6.02274			−0.375688			−0.187844	+	0.982199i	$0.560150\pi$
$258$		0			0
$259$	1.41924			0.0881874
$260$		0			0
$261$	−5.27912	−	4.89830i	−0.326769	−	0.303197i
$262$		0			0
$263$	23.8955	−	3.60167i	1.47346	−	0.222089i	0.637319	−	0.770600i	$-0.280044\pi$
$263$	23.8955	−	3.60167i	1.47346	−	0.222089i	0.836143	+	0.548512i	$0.184805\pi$
$264$		0			0
$265$	−11.0269	−	19.0991i	−0.677377	−	1.17325i
$266$		0			0
$267$	−0.00238531	−	0.0318298i	−0.000145979	−	0.00194795i
$268$		0			0
$269$	3.94932	+	1.90189i	0.240794	+	0.115960i	0.550389	−	0.834908i	$-0.314479\pi$
$269$	3.94932	+	1.90189i	0.240794	+	0.115960i	−0.309595	+	0.950868i	$0.600194\pi$
$270$		0			0
$271$	−10.3289	+	26.3177i	−0.627437	+	1.59868i	0.163333	+	0.986571i	$0.447775\pi$
$271$	−10.3289	+	26.3177i	−0.627437	+	1.59868i	−0.790770	+	0.612113i	$0.790320\pi$
$272$		0			0
$273$	−0.00790416	−	0.00538896i	−0.000478381	−	0.000326155i
$274$		0			0
$275$	22.1819	+	3.34339i	1.33762	+	0.201614i
$276$		0			0
$277$	1.72237	−	22.9834i	0.103487	−	1.38094i	−0.667748	−	0.744387i	$-0.732742\pi$
$277$	1.72237	−	22.9834i	0.103487	−	1.38094i	0.771235	−	0.636550i	$-0.219639\pi$
$278$		0			0
$279$	3.45070	−	15.1185i	0.206588	−	0.905121i
$280$		0			0
$281$	14.7612	−	10.0640i	0.880582	−	0.600371i	−0.0363832	−	0.999338i	$-0.511584\pi$
$281$	14.7612	−	10.0640i	0.880582	−	0.600371i	0.916965	+	0.398967i	$0.130631\pi$
$282$		0			0
$283$	13.4802	−	4.15809i	0.801315	−	0.247173i	0.133063	−	0.991108i	$-0.457519\pi$
$283$	13.4802	−	4.15809i	0.801315	−	0.247173i	0.668252	+	0.743935i	$0.267043\pi$
$284$		0			0
$285$	−0.227210	+	0.284912i	−0.0134587	+	0.0168767i
$286$		0			0
$287$	0.570239	−	0.529105i	0.0336602	−	0.0312321i
$288$		0			0
$289$	23.4599	+	7.23642i	1.37999	+	0.425672i
$290$		0			0
$291$	0.0169160	+	0.0431012i	0.000991633	+	0.00252664i
$292$		0			0
$293$	3.88985	+	4.87772i	0.227247	+	0.284959i	0.882363	−	0.470569i	$-0.155951\pi$
$293$	3.88985	+	4.87772i	0.227247	+	0.284959i	−0.655115	+	0.755529i	$0.727380\pi$
$294$		0			0
$295$	23.0462	−	39.9172i	1.34180	−	2.32407i
$296$		0			0
$297$	0.314072	−	0.151249i	0.0182243	−	0.00877637i
$298$		0			0
$299$	0.996495	+	4.36593i	0.0576288	+	0.252488i
$300$		0			0
$301$	−0.667016	+	1.63328i	−0.0384462	+	0.0941407i
$302$		0			0
$303$	0.0347799	+	0.152381i	0.00199805	+	0.00875404i
$304$		0			0
$305$	30.7870	−	14.8262i	1.76286	−	0.848947i
$306$		0			0
$307$	2.49936	−	4.32902i	0.142646	−	0.247070i	−0.785846	−	0.618422i	$-0.787772\pi$
$307$	2.49936	−	4.32902i	0.142646	−	0.247070i	0.928492	+	0.371352i	$0.121106\pi$
$308$		0			0
$309$	0.00832628	+	0.0104408i	0.000473666	+	0.000593958i
$310$		0			0
$311$	−1.80281	−	4.59348i	−0.102228	−	0.260473i	0.870604	−	0.491984i	$-0.163728\pi$
$311$	−1.80281	−	4.59348i	−0.102228	−	0.260473i	−0.972832	+	0.231511i	$0.925633\pi$
$312$		0			0
$313$	8.00980	+	2.47070i	0.452741	+	0.139652i	0.512735	−	0.858547i	$-0.328632\pi$
$313$	8.00980	+	2.47070i	0.452741	+	0.139652i	−0.0599948	+	0.998199i	$0.519108\pi$
$314$		0			0
$315$	−2.03310	+	1.88644i	−0.114552	+	0.106289i
$316$		0			0
$317$	11.2630	−	14.1233i	0.632593	−	0.793246i	−0.357462	−	0.933928i	$-0.616358\pi$
$317$	11.2630	−	14.1233i	0.632593	−	0.793246i	0.990055	+	0.140681i	$0.0449293\pi$
$318$		0			0
$319$	7.55688	−	2.33099i	0.423104	−	0.130510i
$320$		0			0
$321$	−0.0483074	+	0.0329354i	−0.00269626	+	0.00183828i
$322$		0			0
$323$	8.62321	−	37.7808i	0.479808	−	2.10218i
$324$		0			0
$325$	−1.02591	+	13.6898i	−0.0569073	+	0.759375i
$326$		0			0
$327$	0.176055	+	0.0265360i	0.00973587	+	0.00146745i
$328$		0			0
$329$	1.31218	+	0.894631i	0.0723430	+	0.0493226i
$330$		0			0
$331$	−0.691785	+	1.76264i	−0.0380239	+	0.0968834i	−0.948624	−	0.316404i	$-0.897524\pi$
$331$	−0.691785	+	1.76264i	−0.0380239	+	0.0968834i	0.910600	+	0.413288i	$0.135620\pi$
$332$		0			0
$333$	−14.2568	−	6.86569i	−0.781265	−	0.376238i
$334$		0			0
$335$	3.26511	+	43.5698i	0.178392	+	2.38047i
$336$		0			0
$337$	−0.158818	−	0.275080i	−0.00865135	−	0.0149846i	0.861667	−	0.507474i	$-0.169421\pi$
$337$	−0.158818	−	0.275080i	−0.00865135	−	0.0149846i	−0.870319	+	0.492489i	$0.836087\pi$
$338$		0			0
$339$	−0.0977774	+	0.0147376i	−0.00531054	+	0.000800435i
$340$		0			0
$341$	12.4831	+	11.5826i	0.675999	+	0.627235i
$342$		0			0
$343$	3.74713			0.202326
$344$		0			0
$345$	−0.134656			−0.00724966
$346$		0			0
$347$	−20.3150	−	18.8496i	−1.09057	−	1.01190i	−0.999872	−	0.0160182i	$-0.994901\pi$
$347$	−20.3150	−	18.8496i	−1.09057	−	1.01190i	−0.0906939	−	0.995879i	$-0.528908\pi$
$348$		0			0
$349$	26.8636	−	4.04904i	1.43798	−	0.216740i	0.616668	−	0.787224i	$-0.288482\pi$
$349$	26.8636	−	4.04904i	1.43798	−	0.216740i	0.821309	+	0.570484i	$0.193244\pi$
$350$		0			0
$351$	0.106666	+	0.184751i	0.00569342	+	0.00986130i
$352$		0			0
$353$	1.46955	+	19.6097i	0.0782160	+	1.04372i	0.888859	+	0.458180i	$0.151499\pi$
$353$	1.46955	+	19.6097i	0.0782160	+	1.04372i	−0.810643	+	0.585540i	$0.800882\pi$
$354$		0			0
$355$	−18.6892	−	9.00024i	−0.991919	−	0.477683i
$356$		0			0
$357$	−0.0111756	+	0.0284750i	−0.000591476	+	0.00150706i
$358$		0			0
$359$	6.52863	+	4.45114i	0.344568	+	0.234922i	0.723222	−	0.690616i	$-0.242660\pi$
$359$	6.52863	+	4.45114i	0.344568	+	0.234922i	−0.378654	+	0.925538i	$0.623613\pi$
$360$		0			0
$361$	−16.9511	−	2.55496i	−0.892162	−	0.134472i
$362$		0			0
$363$	0.000196825	−	0.00262645i	1.03307e−5	−	0.000137853i
$364$		0			0
$365$	−1.27628	+	5.59176i	−0.0668037	+	0.292686i
$366$		0			0
$367$	−16.6615	+	11.3596i	−0.869723	+	0.592967i	−0.913847	−	0.406059i	$-0.866903\pi$
$367$	−16.6615	+	11.3596i	−0.869723	+	0.592967i	0.0441239	+	0.999026i	$0.485950\pi$
$368$		0			0
$369$	−8.28783	+	2.55646i	−0.431447	+	0.133084i
$370$		0			0
$371$	−1.07649	+	1.34987i	−0.0558884	+	0.0700819i
$372$		0			0
$373$	5.79496	−	5.37693i	0.300051	−	0.278407i	−0.515750	−	0.856739i	$-0.672487\pi$
$373$	5.79496	−	5.37693i	0.300051	−	0.278407i	0.815801	+	0.578332i	$0.196296\pi$
$374$		0			0
$375$	−0.104842	−	0.0323395i	−0.00541402	−	0.00167000i
$376$		0			0
$377$	1.76813	+	4.50512i	0.0910633	+	0.232026i
$378$		0			0
$379$	15.2679	+	19.1453i	0.784258	+	0.983428i	0.999976	+	0.00697248i	$0.00221943\pi$
$379$	15.2679	+	19.1453i	0.784258	+	0.983428i	−0.215718	+	0.976456i	$0.569209\pi$
$380$		0			0
$381$	0.0819343	−	0.141914i	0.00419762	−	0.00727049i
$382$		0			0
$383$	−24.8833	+	11.9832i	−1.27148	+	0.612311i	−0.943186	−	0.332266i	$-0.892187\pi$
$383$	−24.8833	+	11.9832i	−1.27148	+	0.612311i	−0.328291	+	0.944577i	$0.606473\pi$
$384$		0			0
$385$	−0.677714	−	2.96926i	−0.0345395	−	0.151327i
$386$		0			0
$387$	14.6015	−	13.1801i	0.742237	−	0.669983i
$388$		0			0
$389$	−6.15713	−	26.9761i	−0.312179	−	1.36774i	−0.850930	−	0.525279i	$-0.823961\pi$
$389$	−6.15713	−	26.9761i	−0.312179	−	1.36774i	0.538751	−	0.842465i	$-0.318896\pi$
$390$		0			0
$391$	12.9014	−	6.21299i	0.652452	−	0.314205i
$392$		0			0
$393$	0.0478381	−	0.0828581i	0.00241311	−	0.00417964i
$394$		0			0
$395$	29.8762	+	37.4635i	1.50323	+	1.88499i
$396$		0			0
$397$	3.68917	+	9.39985i	0.185154	+	0.471765i	0.993069	−	0.117534i	$-0.0374990\pi$
$397$	3.68917	+	9.39985i	0.185154	+	0.471765i	−0.807915	+	0.589300i	$0.799404\pi$
$398$		0			0
$399$	0.0272620	+	0.00840921i	0.00136481	+	0.000420987i
$400$		0			0
$401$	−24.5225	+	22.7536i	−1.22460	+	1.13626i	−0.238314	+	0.971188i	$0.576595\pi$
$401$	−24.5225	+	22.7536i	−1.22460	+	1.13626i	−0.986282	+	0.165070i	$0.947215\pi$
$402$		0			0
$403$	−6.49762	+	8.14776i	−0.323670	+	0.405869i
$404$		0			0
$405$	29.5459	−	9.11369i	1.46815	−	0.452863i
$406$		0			0
$407$	14.3572	−	9.78855i	0.711658	−	0.485200i
$408$		0			0
$409$	−5.20581	+	22.8081i	−0.257411	+	1.12779i	0.666598	+	0.745418i	$0.267750\pi$
$409$	−5.20581	+	22.8081i	−0.257411	+	1.12779i	−0.924008	+	0.382372i	$0.875107\pi$
$410$		0			0
$411$	−0.0145592	+	0.194279i	−0.000718152	+	0.00958307i
$412$		0			0
$413$	−3.56818	−	0.537817i	−0.175579	−	0.0264643i
$414$		0			0
$415$	−21.7061	−	14.7990i	−1.06551	−	0.726453i
$416$		0			0
$417$	−0.0495425	+	0.126232i	−0.00242611	+	0.00618162i
$418$		0			0
$419$	−16.3740	−	7.88529i	−0.799921	−	0.385221i	−0.0111722	−	0.999938i	$-0.503556\pi$
$419$	−16.3740	−	7.88529i	−0.799921	−	0.385221i	−0.788748	+	0.614716i	$0.789271\pi$
$420$		0			0
$421$	−2.18943	−	29.2159i	−0.106706	−	1.42390i	−0.751598	−	0.659621i	$-0.770717\pi$
$421$	−2.18943	−	29.2159i	−0.106706	−	1.42390i	0.644892	−	0.764274i	$-0.276902\pi$
$422$		0			0
$423$	−8.85347	−	15.3347i	−0.430470	−	0.745597i
$424$		0			0
$425$	43.4069	−	6.54254i	2.10555	−	0.317360i
$426$		0			0
$427$	−1.96105	−	1.81959i	−0.0949018	−	0.0880560i
$428$		0			0
$429$	−0.117127			−0.00565494
$430$		0			0
$431$	−2.95198			−0.142192			−0.0710959	−	0.997469i	$-0.522650\pi$
$431$	−2.95198			−0.142192			−0.0710959	+	0.997469i	$0.522650\pi$
$432$		0			0
$433$	−24.6385	−	22.8611i	−1.18405	−	1.09864i	−0.993128	−	0.117031i	$-0.962662\pi$
$433$	−24.6385	−	22.8611i	−1.18405	−	1.09864i	−0.190920	−	0.981606i	$-0.561147\pi$
$434$		0			0
$435$	−0.143900	+	0.0216894i	−0.00689947	+	0.00103993i
$436$		0			0
$437$	−6.67757	−	11.5659i	−0.319431	−	0.553271i
$438$		0			0
$439$	−1.61347	−	21.5302i	−0.0770066	−	1.02758i	−0.893158	−	0.449743i	$-0.851516\pi$
$439$	−1.61347	−	21.5302i	−0.0770066	−	1.02758i	0.816151	−	0.577838i	$-0.196104\pi$
$440$		0			0
$441$	−18.7228	−	9.01640i	−0.891560	−	0.429353i
$442$		0			0
$443$	−1.29471	+	3.29887i	−0.0615135	+	0.156734i	−0.958326	−	0.285677i	$-0.907782\pi$
$443$	−1.29471	+	3.29887i	−0.0615135	+	0.156734i	0.896812	+	0.442411i	$0.145877\pi$
$444$		0			0
$445$	5.13828	+	3.50322i	0.243578	+	0.166069i
$446$		0			0
$447$	−0.0163984	−	0.00247166i	−0.000775616	−	0.000116905i
$448$		0			0
$449$	−1.72113	+	22.9668i	−0.0812249	+	1.08387i	0.796478	+	0.604667i	$0.206694\pi$
$449$	−1.72113	+	22.9668i	−0.0812249	+	1.08387i	−0.877703	+	0.479204i	$0.840925\pi$
$450$		0			0
$451$	2.11934	−	9.28543i	0.0997957	−	0.437234i
$452$		0			0
$453$	−0.160284	+	0.109280i	−0.00753078	+	0.00513440i
$454$		0			0
$455$	1.78105	−	0.549381i	0.0834969	−	0.0257554i
$456$		0			0
$457$	−5.04145	+	6.32177i	−0.235829	+	0.295720i	−0.885637	−	0.464378i	$-0.846278\pi$
$457$	−5.04145	+	6.32177i	−0.235829	+	0.295720i	0.649808	+	0.760098i	$0.274849\pi$
$458$		0			0
$459$	0.500052	−	0.463980i	0.0233404	−	0.0216567i
$460$		0			0
$461$	9.39710	+	2.89862i	0.437667	+	0.135002i	0.505755	−	0.862677i	$-0.331214\pi$
$461$	9.39710	+	2.89862i	0.437667	+	0.135002i	−0.0680886	+	0.997679i	$0.521690\pi$
$462$		0			0
$463$	11.0144	+	28.0644i	0.511885	+	1.30426i	0.920210	+	0.391425i	$0.128018\pi$
$463$	11.0144	+	28.0644i	0.511885	+	1.30426i	−0.408325	+	0.912837i	$0.633887\pi$
$464$		0			0
$465$	−0.195379	−	0.244997i	−0.00906047	−	0.0113615i
$466$		0			0
$467$	−12.1322	+	21.0135i	−0.561409	+	0.972389i	0.435965	+	0.899964i	$0.356407\pi$
$467$	−12.1322	+	21.0135i	−0.561409	+	0.972389i	−0.997374	+	0.0724256i	$0.976926\pi$
$468$		0			0
$469$	3.08183	−	1.48413i	0.142305	−	0.0685307i
$470$		0			0
$471$	−0.0754915	−	0.330750i	−0.00347846	−	0.0152401i
$472$		0			0
$473$	4.51720	+	21.1228i	0.207701	+	0.971229i
$474$		0			0
$475$	−9.11022	−	39.9145i	−0.418005	−	1.83140i
$476$		0			0
$477$	17.3438	−	8.35233i	0.794117	−	0.382427i
$478$		0			0
$479$	−0.0305712	+	0.0529509i	−0.00139683	+	0.00241939i	−0.866723	−	0.498790i	$-0.833778\pi$
$479$	−0.0305712	+	0.0529509i	−0.00139683	+	0.00241939i	0.865326	+	0.501209i	$0.167111\pi$
$480$		0			0
$481$	6.63024	+	8.31406i	0.302313	+	0.379089i
$482$		0			0
$483$	0.00385143	+	0.00981328i	0.000175246	+	0.000446520i
$484$		0			0
$485$	−8.62036	−	2.65903i	−0.391430	−	0.120740i
$486$		0			0
$487$	17.0165	−	15.7890i	0.771093	−	0.715470i	−0.192952	−	0.981208i	$-0.561806\pi$
$487$	17.0165	−	15.7890i	0.771093	−	0.715470i	0.964045	+	0.265738i	$0.0856157\pi$
$488$		0			0
$489$	0.211037	−	0.264632i	0.00954342	−	0.0119671i
$490$		0			0
$491$	9.90993	−	3.05681i	0.447229	−	0.137952i	−0.0629566	−	0.998016i	$-0.520053\pi$
$491$	9.90993	−	3.05681i	0.447229	−	0.137952i	0.510185	+	0.860064i	$0.329577\pi$
$492$		0			0
$493$	12.7863	−	8.71755i	0.575866	−	0.392619i
$494$		0			0
$495$	−7.55616	+	33.1057i	−0.339624	+	1.48799i
$496$		0			0
$497$	−0.121359	+	1.61943i	−0.00544370	+	0.0726412i
$498$		0			0
$499$	11.9560	+	1.80208i	0.535224	+	0.0806720i	0.411092	−	0.911594i	$-0.365147\pi$
$499$	11.9560	+	1.80208i	0.535224	+	0.0806720i	0.124132	+	0.992266i	$0.460385\pi$
$500$		0			0
$501$	−0.0884884	−	0.0603304i	−0.00395337	−	0.00269536i
$502$		0			0
$503$	6.09669	−	15.5341i	0.271838	−	0.692632i	−0.728136	−	0.685433i	$-0.759613\pi$
$503$	6.09669	−	15.5341i	0.271838	−	0.692632i	0.999974	−	0.00719925i	$-0.00229161\pi$
$504$		0			0
$505$	−27.4366	−	13.2128i	−1.22091	−	0.587960i
$506$		0			0
$507$	0.0117791	+	0.157181i	0.000523128	+	0.00698066i
$508$		0			0
$509$	−16.1598	−	27.9896i	−0.716271	−	1.24062i	−0.962467	−	0.271397i	$-0.912514\pi$
$509$	−16.1598	−	27.9896i	−0.716271	−	1.24062i	0.246197	−	0.969220i	$-0.420819\pi$
$510$		0			0
$511$	0.444012	−	0.0669240i	0.0196419	−	0.00296054i
$512$		0			0
$513$	−0.466375	−	0.432733i	−0.0205910	−	0.0191056i
$514$		0			0
$515$	−2.60186			−0.114652
$516$		0			0
$517$	19.4445			0.855166
$518$		0			0
$519$	−0.142566	−	0.132282i	−0.00625796	−	0.00580654i
$520$		0			0
$521$	−36.0446	+	5.43284i	−1.57914	+	0.238017i	−0.879295	−	0.476277i	$-0.841986\pi$
$521$	−36.0446	+	5.43284i	−1.57914	+	0.238017i	−0.699845	+	0.714294i	$0.746748\pi$
$522$		0			0
$523$	13.2662	+	22.9777i	0.580088	+	1.00474i	0.995468	+	0.0950946i	$0.0303153\pi$
$523$	13.2662	+	22.9777i	0.580088	+	1.00474i	−0.415380	+	0.909648i	$0.636351\pi$
$524$		0			0
$525$	0.00241506	+	0.0322268i	0.000105402	+	0.00140649i
$526$		0			0
$527$	30.0233	+	14.4585i	1.30784	+	0.629820i
$528$		0			0
$529$	−6.59993	+	16.8163i	−0.286953	+	0.731145i
$530$		0			0
$531$	33.2419	+	22.6639i	1.44257	+	0.983531i
$532$		0			0
$533$	5.76353	+	0.868712i	0.249646	+	0.0376281i
$534$		0			0
$535$	0.851268	−	11.3594i	0.0368035	−	0.491109i
$536$		0			0
$537$	−0.0319239	+	0.139868i	−0.00137762	+	0.00603573i
$538$		0			0
$539$	18.8546	−	12.8549i	0.812126	−	0.553698i
$540$		0			0
$541$	−15.5995	+	4.81180i	−0.670674	+	0.206875i	−0.611346	−	0.791363i	$-0.709372\pi$
$541$	−15.5995	+	4.81180i	−0.670674	+	0.206875i	−0.0593272	+	0.998239i	$0.518896\pi$
$542$		0			0
$543$	−0.0664920	+	0.0833783i	−0.00285344	+	0.00357810i
$544$		0			0
$545$	−25.4287	+	23.5944i	−1.08925	+	1.01067i
$546$		0			0
$547$	−25.6168	−	7.90174i	−1.09530	−	0.337854i	−0.306103	−	0.951999i	$-0.599025\pi$
$547$	−25.6168	−	7.90174i	−1.09530	−	0.337854i	−0.789194	+	0.614144i	$0.789501\pi$
$548$		0			0
$549$	10.8970	+	27.7651i	0.465072	+	1.18498i
$550$		0			0
$551$	−8.99889	−	11.2843i	−0.383366	−	0.480725i
$552$		0			0
$553$	1.87569	−	3.24879i	0.0797626	−	0.138153i
$554$		0			0
$555$	−0.288093	+	0.138738i	−0.0122289	+	0.00588911i
$556$		0			0
$557$	−0.445481	−	1.95178i	−0.0188756	−	0.0826996i	0.964613	−	0.263670i	$-0.0849332\pi$
$557$	−0.445481	−	1.95178i	−0.0188756	−	0.0826996i	−0.983488	+	0.180971i	$0.942076\pi$
$558$		0			0
$559$	−12.6840	+	3.72272i	−0.536477	+	0.157454i
$560$		0			0
$561$	0.0833395	+	0.365134i	0.00351860	+	0.0154160i
$562$		0			0
$563$	−27.9871	+	13.4779i	−1.17952	+	0.568025i	−0.917770	−	0.397112i	$-0.870012\pi$
$563$	−27.9871	+	13.4779i	−1.17952	+	0.568025i	−0.261745	+	0.965137i	$0.584298\pi$
$564$		0			0
$565$	9.63272	−	16.6844i	0.405252	−	0.701916i
$566$		0			0
$567$	−1.50924	−	1.89253i	−0.0633821	−	0.0794787i
$568$		0			0
$569$	0.695255	+	1.77148i	0.0291466	+	0.0742643i	0.944706	−	0.327918i	$-0.106347\pi$
$569$	0.695255	+	1.77148i	0.0291466	+	0.0742643i	−0.915560	+	0.402182i	$0.868252\pi$
$570$		0			0
$571$	1.82295	+	0.562306i	0.0762882	+	0.0235318i	0.332664	−	0.943045i	$-0.392052\pi$
$571$	1.82295	+	0.562306i	0.0762882	+	0.0235318i	−0.256376	+	0.966577i	$0.582529\pi$
$572$		0			0
$573$	0.199501	−	0.185110i	0.00833426	−	0.00773307i
$574$		0			0
$575$	9.43228	−	11.8277i	0.393353	−	0.493249i
$576$		0			0
$577$	11.4249	−	3.52412i	0.475626	−	0.146711i	−0.0476684	−	0.998863i	$-0.515179\pi$
$577$	11.4249	−	3.52412i	0.475626	−	0.146711i	0.523294	+	0.852152i	$0.324703\pi$
$578$		0			0
$579$	0.0355151	−	0.0242138i	0.00147596	−	0.00100629i
$580$		0			0
$581$	−0.457661	+	2.00514i	−0.0189870	+	0.0831873i
$582$		0			0
$583$	−1.57973	+	21.0800i	−0.0654256	+	0.873044i
$584$		0			0
$585$	−20.5489	−	3.09725i	−0.849593	−	0.128056i
$586$		0			0
$587$	−10.5277	−	7.17769i	−0.434526	−	0.296255i	0.326233	−	0.945289i	$-0.394221\pi$
$587$	−10.5277	−	7.17769i	−0.434526	−	0.296255i	−0.760759	+	0.649034i	$0.775173\pi$
$588$		0			0
$589$	11.3545	−	28.9308i	0.467854	−	1.19207i
$590$		0			0
$591$	−0.0395938	−	0.0190674i	−0.00162867	−	0.000784326i
$592$		0			0
$593$	1.69336	+	22.5963i	0.0695380	+	0.927920i	0.917468	+	0.397811i	$0.130230\pi$
$593$	1.69336	+	22.5963i	0.0695380	+	0.927920i	−0.847930	+	0.530109i	$0.822151\pi$
$594$		0			0
$595$	−2.97993	−	5.16139i	−0.122165	−	0.211596i
$596$		0			0
$597$	−0.144088	+	0.0217177i	−0.00589711	+	0.000888846i
$598$		0			0
$599$	−19.4871	−	18.0814i	−0.796221	−	0.738785i	0.173015	−	0.984919i	$-0.444649\pi$
$599$	−19.4871	−	18.0814i	−0.796221	−	0.738785i	−0.969236	+	0.246134i	$0.920840\pi$
$600$		0			0
$601$	5.51022			0.224766			0.112383	−	0.993665i	$-0.464152\pi$
$601$	5.51022			0.224766			0.112383	+	0.993665i	$0.464152\pi$
$602$		0			0
$603$	−38.1376			−1.55308
$604$		0			0
$605$	0.376169	+	0.349034i	0.0152934	+	0.0141902i
$606$		0			0
$607$	−2.78541	+	0.419833i	−0.113056	+	0.0170405i	−0.205327	−	0.978693i	$-0.565826\pi$
$607$	−2.78541	+	0.419833i	−0.113056	+	0.0170405i	0.0922704	+	0.995734i	$0.470588\pi$
$608$		0			0
$609$	0.00569646	+	0.00986655i	0.000230832	+	0.000399813i
$610$		0			0
$611$	0.889259	+	11.8663i	0.0359756	+	0.480061i
$612$		0			0
$613$	3.46419	+	1.66826i	0.139917	+	0.0673805i	0.502532	−	0.864558i	$-0.332402\pi$
$613$	3.46419	+	1.66826i	0.139917	+	0.0673805i	−0.362615	+	0.931939i	$0.618116\pi$
$614$		0			0
$615$	−0.0640306	+	0.163147i	−0.00258196	+	0.00657873i
$616$		0			0
$617$	33.6620	+	22.9504i	1.35518	+	0.923947i	0.999941	−	0.0108275i	$-0.00344656\pi$
$617$	33.6620	+	22.9504i	1.35518	+	0.923947i	0.355241	+	0.934775i	$0.384399\pi$
$618$		0			0
$619$	−32.9850	−	4.97168i	−1.32578	−	0.199829i	−0.552296	−	0.833648i	$-0.686248\pi$
$619$	−32.9850	−	4.97168i	−1.32578	−	0.199829i	−0.773481	+	0.633819i	$0.781486\pi$
$620$		0			0
$621$	0.0175681	−	0.234431i	0.000704985	−	0.00940737i
$622$		0			0
$623$	0.108338	−	0.474659i	0.00434046	−	0.0190168i
$624$		0			0
$625$	−10.4715	+	7.13936i	−0.418861	+	0.285574i
$626$		0			0
$627$	0.333783	−	0.102958i	0.0133300	−	0.00411177i
$628$		0			0
$629$	21.2008	−	26.5850i	0.845332	−	1.06001i
$630$		0			0
$631$	−19.6700	+	18.2511i	−0.783049	+	0.726563i	−0.966556	−	0.256455i	$-0.917445\pi$
$631$	−19.6700	+	18.2511i	−0.783049	+	0.726563i	0.183507	+	0.983018i	$0.441255\pi$
$632$		0			0
$633$	−0.241319	−	0.0744370i	−0.00959156	−	0.00295860i
$634$		0			0
$635$	11.6642	+	29.7200i	0.462881	+	1.17940i
$636$		0			0
$637$	8.70720	+	10.9185i	0.344992	+	0.432606i
$638$		0			0
$639$	9.05319	−	15.6806i	0.358139	−	0.620314i
$640$		0			0
$641$	−35.6933	+	17.1890i	−1.40980	+	0.678923i	−0.975123	−	0.221663i	$-0.928851\pi$
$641$	−35.6933	+	17.1890i	−1.40980	+	0.678923i	−0.434676	+	0.900587i	$0.643137\pi$
$642$		0			0
$643$	−2.12257	−	9.29957i	−0.0837059	−	0.366739i	0.915675	−	0.401919i	$-0.131657\pi$
$643$	−2.12257	−	9.29957i	−0.0837059	−	0.366739i	−0.999381	+	0.0351799i	$0.988800\pi$
$644$		0			0
$645$	−0.0242636	−	0.396745i	−0.000955380	−	0.0156218i
$646$		0			0
$647$	5.25960	+	23.0438i	0.206776	+	0.905946i	0.966696	+	0.255929i	$0.0823813\pi$
$647$	5.25960	+	23.0438i	0.206776	+	0.905946i	−0.759919	+	0.650017i	$0.774762\pi$
$648$		0			0
$649$	−39.8054	+	19.1693i	−1.56250	+	0.752460i
$650$		0			0
$651$	−0.0122663	+	0.0212459i	−0.000480755	+	0.000832692i
$652$		0			0
$653$	0.844715	+	1.05924i	0.0330562	+	0.0414512i	0.798085	−	0.602545i	$-0.205847\pi$
$653$	0.844715	+	1.05924i	0.0330562	+	0.0414512i	−0.765029	+	0.643996i	$0.777275\pi$
$654$		0			0
$655$	6.81028	+	17.3523i	0.266100	+	0.678011i
$656$		0			0
$657$	−4.78399	−	1.47567i	−0.186641	−	0.0575712i
$658$		0			0
$659$	−11.3008	+	10.4856i	−0.440216	+	0.408461i	−0.868895	−	0.494996i	$-0.835169\pi$
$659$	−11.3008	+	10.4856i	−0.440216	+	0.408461i	0.428679	+	0.903457i	$0.358979\pi$
$660$		0			0
$661$	−9.58249	+	12.0161i	−0.372716	+	0.467371i	−0.932449	−	0.361302i	$-0.882332\pi$
$661$	−9.58249	+	12.0161i	−0.372716	+	0.467371i	0.559733	+	0.828673i	$0.310904\pi$
$662$		0			0
$663$	−0.219018	+	0.0675582i	−0.00850597	+	0.00262374i
$664$		0			0
$665$	−4.59264	+	3.13121i	−0.178095	+	0.121423i
$666$		0			0
$667$	1.18675	−	5.19950i	0.0459512	−	0.201325i
$668$		0			0
$669$	−0.00899695	+	0.120056i	−0.000347842	+	0.00464163i
$670$		0			0
$671$	−32.3879	−	4.88169i	−1.25032	−	0.188455i
$672$		0			0
$673$	−37.4014	−	25.4998i	−1.44172	−	0.982946i	−0.996147	−	0.0876984i	$-0.972049\pi$
$673$	−37.4014	−	25.4998i	−1.44172	−	0.982946i	−0.445570	−	0.895247i	$-0.646999\pi$
$674$		0			0
$675$	0.263293	−	0.670858i	0.0101341	−	0.0258213i
$676$		0			0
$677$	−5.42739	−	2.61370i	−0.208592	−	0.100452i	0.326671	−	0.945138i	$-0.394073\pi$
$677$	−5.42739	−	2.61370i	−0.208592	−	0.100452i	−0.535263	+	0.844686i	$0.679787\pi$
$678$		0			0
$679$	0.0527781	+	0.704274i	0.00202544	+	0.0270276i
$680$		0			0
$681$	0.207591	+	0.359558i	0.00795489	+	0.0137783i
$682$		0			0
$683$	2.80582	−	0.422910i	0.107362	−	0.0161822i	−0.0951416	−	0.995464i	$-0.530330\pi$
$683$	2.80582	−	0.422910i	0.107362	−	0.0161822i	0.202503	+	0.979282i	$0.435092\pi$
$684$		0			0
$685$	−27.8252	−	25.8180i	−1.06315	−	0.986457i
$686$		0			0
$687$	−0.278393			−0.0106214
$688$		0			0
$689$	−12.9367			−0.492849
$690$		0			0
$691$	−18.2969	−	16.9771i	−0.696048	−	0.645839i	0.250606	−	0.968089i	$-0.419370\pi$
$691$	−18.2969	−	16.9771i	−0.696048	−	0.645839i	−0.946654	+	0.322251i	$0.895561\pi$
$692$		0			0
$693$	2.62874	−	0.396219i	0.0998577	−	0.0150511i
$694$		0			0
$695$	−13.2103	−	22.8809i	−0.501094	−	0.867921i
$696$		0			0
$697$	−1.39279	−	18.5855i	−0.0527556	−	0.703975i
$698$		0			0
$699$	−0.0444648	−	0.0214131i	−0.00168181	−	0.000809918i
$700$		0			0
$701$	3.98473	−	10.1529i	0.150501	−	0.383471i	−0.835663	−	0.549243i	$-0.814916\pi$
$701$	3.98473	−	10.1529i	0.150501	−	0.383471i	0.986164	+	0.165772i	$0.0530115\pi$
$702$		0			0
$703$	−26.2029	−	17.8648i	−0.988262	−	0.673785i
$704$		0			0
$705$	−0.353816	−	0.0533292i	−0.0133255	−	0.00200849i
$706$		0			0
$707$	−0.178161	+	2.37739i	−0.00670042	+	0.0894109i
$708$		0			0
$709$	1.03191	−	4.52108i	0.0387541	−	0.169793i	−0.951847	−	0.306572i	$-0.900818\pi$
$709$	1.03191	−	4.52108i	0.0387541	−	0.169793i	0.990602	+	0.136779i	$0.0436751\pi$
$710$		0			0
$711$	−34.5583	+	23.5614i	−1.29604	+	0.883622i
$712$		0			0
$713$	10.9739	−	3.38501i	0.410977	−	0.126770i
$714$		0			0
$715$	14.2282	−	17.8415i	0.532103	−	0.667236i
$716$		0			0
$717$	−0.0661749	+	0.0614013i	−0.00247135	+	0.00229307i
$718$		0			0
$719$	40.4359	+	12.4728i	1.50801	+	0.465158i	0.935076	−	0.354447i	$-0.115331\pi$
$719$	40.4359	+	12.4728i	1.50801	+	0.465158i	0.572929	+	0.819605i	$0.305807\pi$
$720$		0			0
$721$	0.0744182	+	0.189614i	0.00277148	+	0.00706161i
$722$		0			0
$723$	0.0642421	+	0.0805570i	0.00238919	+	0.00299595i
$724$		0			0
$725$	8.17464	−	14.1589i	0.303599	−	0.525848i
$726$		0			0
$727$	−2.50295	+	1.20536i	−0.0928293	+	0.0447042i	−0.479722	−	0.877420i	$-0.659263\pi$
$727$	−2.50295	+	1.20536i	−0.0928293	+	0.0447042i	0.386893	+	0.922125i	$0.373548\pi$
$728$		0			0
$729$	6.00433	+	26.3067i	0.222382	+	0.974321i
$730$		0			0
$731$	20.6304	+	36.8926i	0.763042	+	1.36452i
$732$		0			0
$733$	9.70495	+	42.5201i	0.358460	+	1.57052i	0.757031	+	0.653380i	$0.226649\pi$
$733$	9.70495	+	42.5201i	0.358460	+	1.57052i	−0.398570	+	0.917138i	$0.630493\pi$
$734$		0			0
$735$	−0.378340	+	0.182199i	−0.0139553	+	0.00672050i
$736$		0			0
$737$	20.9399	−	36.2690i	0.771332	−	1.33599i
$738$		0			0
$739$	1.70413	+	2.13692i	0.0626876	+	0.0786077i	0.812187	−	0.583398i	$-0.198277\pi$
$739$	1.70413	+	2.13692i	0.0626876	+	0.0786077i	−0.749499	+	0.662005i	$0.769706\pi$
$740$		0			0
$741$	0.0780973	+	0.198989i	0.00286898	+	0.00731003i
$742$		0			0
$743$	−38.0996	−	11.7522i	−1.39774	−	0.431145i	−0.497849	−	0.867264i	$-0.665877\pi$
$743$	−38.0996	−	11.7522i	−1.39774	−	0.431145i	−0.899889	+	0.436119i	$0.856353\pi$
$744$		0			0
$745$	2.36851	−	2.19766i	0.0867756	−	0.0805160i
$746$		0			0
$747$	14.2974	−	17.9284i	0.523114	−	0.655964i
$748$		0			0
$749$	−0.852179	+	0.262862i	−0.0311379	+	0.00960478i
$750$		0			0
$751$	−37.1303	+	25.3150i	−1.35491	+	0.923759i	−0.999938	−	0.0111198i	$-0.996460\pi$
$751$	−37.1303	+	25.3150i	−1.35491	+	0.923759i	−0.354967	+	0.934879i	$0.615508\pi$
$752$		0			0
$753$	0.0345087	−	0.151193i	0.00125757	−	0.00550977i
$754$		0			0
$755$	2.82450	−	37.6903i	0.102794	−	1.37169i
$756$		0			0
$757$	−12.7269	−	1.91827i	−0.462566	−	0.0697207i	−0.0863751	−	0.996263i	$-0.527528\pi$
$757$	−12.7269	−	1.91827i	−0.462566	−	0.0697207i	−0.376191	+	0.926542i	$0.622766\pi$
$758$		0			0
$759$	0.106644	+	0.0727085i	0.00387092	+	0.00263915i
$760$		0			0
$761$	−3.08333	+	7.85619i	−0.111771	+	0.284787i	−0.975804	−	0.218648i	$-0.929835\pi$
$761$	−3.08333	+	7.85619i	−0.111771	+	0.284787i	0.864033	+	0.503435i	$0.167931\pi$
$762$		0			0
$763$	2.44678	+	1.17831i	0.0885795	+	0.0426576i
$764$		0			0
$765$	4.96576	+	66.2635i	0.179537	+	2.39576i
$766$		0			0
$767$	−13.5188	−	23.4153i	−0.488137	−	0.845478i
$768$		0			0
$769$	−29.1469	+	4.39319i	−1.05106	+	0.158422i	−0.651783	−	0.758406i	$-0.725979\pi$
$769$	−29.1469	+	4.39319i	−1.05106	+	0.158422i	−0.399282	+	0.916828i	$0.630740\pi$
$770$		0			0
$771$	−0.0778739	−	0.0722564i	−0.00280456	−	0.00260225i
$772$		0			0
$773$	−32.7084			−1.17644			−0.588220	−	0.808701i	$-0.700171\pi$
$773$	−32.7084			−1.17644			−0.588220	+	0.808701i	$0.700171\pi$
$774$		0			0
$775$	35.2053			1.26461
$776$		0			0
$777$	0.0183507	+	0.0170270i	0.000658329	+	0.000610840i
$778$		0			0
$779$	−17.1883	+	2.59072i	−0.615834	+	0.0928220i
$780$		0			0
$781$	9.94154	+	17.2193i	0.355736	+	0.616154i
$782$		0			0
$783$	−0.0189862	−	0.253353i	−0.000678510	−	0.00905409i
$784$		0			0
$785$	59.5524	+	28.6789i	2.12552	+	1.02359i
$786$		0			0
$787$	−1.27793	+	3.25611i	−0.0455532	+	0.116068i	−0.951840	−	0.306596i	$-0.900810\pi$
$787$	−1.27793	+	3.25611i	−0.0455532	+	0.116068i	0.906286	+	0.422664i	$0.138905\pi$
$788$		0			0
$789$	0.352179	+	0.240111i	0.0125379	+	0.00854819i
$790$		0			0
$791$	−1.49141	−	0.224794i	−0.0530285	−	0.00799275i
$792$		0			0
$793$	1.49794	−	19.9886i	0.0531932	−	0.709814i
$794$		0			0
$795$	0.0865599	−	0.379244i	0.00306996	−	0.0134504i
$796$		0			0
$797$	8.19917	−	5.59010i	0.290429	−	0.198011i	−0.409337	−	0.912383i	$-0.634240\pi$
$797$	8.19917	−	5.59010i	0.290429	−	0.198011i	0.699766	+	0.714372i	$0.253287\pi$
$798$		0			0
$799$	36.3597	−	11.2155i	1.28631	−	0.396775i
$800$		0			0
$801$	−3.38449	+	4.24401i	−0.119585	+	0.149955i
$802$		0			0
$803$	4.03008	−	3.73937i	0.142218	−	0.131959i
$804$		0			0
$805$	−1.96268	−	0.605407i	−0.0691754	−	0.0213378i
$806$		0			0
$807$	0.0282471	+	0.0719724i	0.000994345	+	0.00253355i
$808$		0			0
$809$	−4.95941	−	6.21890i	−0.174363	−	0.218645i	0.686969	−	0.726687i	$-0.258941\pi$
$809$	−4.95941	−	6.21890i	−0.174363	−	0.218645i	−0.861332	+	0.508042i	$0.830369\pi$
$810$		0			0
$811$	−23.5508	+	40.7913i	−0.826982	+	1.43238i	0.0734128	+	0.997302i	$0.476611\pi$
$811$	−23.5508	+	40.7913i	−0.826982	+	1.43238i	−0.900395	+	0.435073i	$0.856722\pi$
$812$		0			0
$813$	−0.449292	+	0.216368i	−0.0157574	+	0.00758835i
$814$		0			0
$815$	14.6745	+	64.2931i	0.514024	+	2.25209i
$816$		0			0
$817$	32.8740	−	21.7585i	1.15011	−	0.761234i
$818$		0			0
$819$	0.362021	+	1.58612i	0.0126500	+	0.0554235i
$820$		0			0
$821$	−39.4286	+	18.9878i	−1.37607	+	0.662679i	−0.968157	−	0.250344i	$-0.919456\pi$
$821$	−39.4286	+	18.9878i	−1.37607	+	0.662679i	−0.407909	+	0.913022i	$0.633742\pi$
$822$		0			0
$823$	27.9019	−	48.3275i	0.972599	−	1.68459i	0.284956	−	0.958541i	$-0.408021\pi$
$823$	27.9019	−	48.3275i	0.972599	−	1.68459i	0.687642	−	0.726050i	$-0.258646\pi$
$824$		0			0
$825$	0.246700	+	0.309352i	0.00858899	+	0.0107703i
$826$		0			0
$827$	1.87676	+	4.78192i	0.0652615	+	0.166284i	0.959784	−	0.280739i	$-0.0905796\pi$
$827$	1.87676	+	4.78192i	0.0652615	+	0.166284i	−0.894523	+	0.447023i	$0.852484\pi$
$828$		0			0
$829$	−3.41761	−	1.05419i	−0.118698	−	0.0366136i	0.234837	−	0.972035i	$-0.424544\pi$
$829$	−3.41761	−	1.05419i	−0.118698	−	0.0366136i	−0.353535	+	0.935421i	$0.615021\pi$
$830$		0			0
$831$	0.298008	−	0.276511i	0.0103378	−	0.00959205i
$832$		0			0
$833$	27.8421	−	34.9129i	0.964671	−	1.20966i
$834$		0			0
$835$	19.9392	−	6.15042i	0.690023	−	0.212844i
$836$		0			0
$837$	0.452019	−	0.308182i	0.0156241	−	0.0106523i
$838$		0			0
$839$	−9.15090	+	40.0927i	−0.315924	+	1.38415i	0.528706	+	0.848805i	$0.322677\pi$
$839$	−9.15090	+	40.0927i	−0.315924	+	1.38415i	−0.844631	+	0.535350i	$0.820180\pi$
$840$		0			0
$841$	−1.73645	+	23.1713i	−0.0598776	+	0.799012i
$842$		0			0
$843$	0.311603	+	0.0469667i	0.0107322	+	0.00161762i
$844$		0			0
$845$	−25.3737	−	17.2995i	−0.872883	−	0.595122i
$846$		0			0
$847$	0.0146772	−	0.0373969i	0.000504314	−	0.00128497i
$848$		0			0
$849$	0.224184	+	0.107961i	0.00769399	+	0.00370523i
$850$		0			0
$851$	−0.875727	−	11.6858i	−0.0300195	−	0.400583i
$852$		0			0
$853$	10.5523	+	18.2771i	0.361304	+	0.625797i	0.988176	−	0.153325i	$-0.0489982\pi$
$853$	10.5523	+	18.2771i	0.361304	+	0.625797i	−0.626871	+	0.779123i	$0.715665\pi$
$854$		0			0
$855$	61.2820	−	9.23678i	2.09580	−	0.315891i
$856$		0			0
$857$	−23.2674	−	21.5890i	−0.794799	−	0.737466i	0.174152	−	0.984719i	$-0.444282\pi$
$857$	−23.2674	−	21.5890i	−0.794799	−	0.737466i	−0.968951	+	0.247253i	$0.920472\pi$
$858$		0			0
$859$	29.8703			1.01916			0.509581	−	0.860423i	$-0.329800\pi$
$859$	29.8703			1.01916			0.509581	+	0.860423i	$0.329800\pi$
$860$		0			0
$861$	0.0137210			0.000467610
$862$		0			0
$863$	21.8250	+	20.2507i	0.742933	+	0.689341i	0.957843	−	0.287293i	$-0.0927553\pi$
$863$	21.8250	+	20.2507i	0.742933	+	0.689341i	−0.214910	+	0.976634i	$0.568946\pi$
$864$		0			0
$865$	37.4685	−	5.64747i	1.27397	−	0.192020i
$866$		0			0
$867$	0.216519	+	0.375021i	0.00735336	+	0.0127364i
$868$		0			0
$869$	−3.43237	−	45.8018i	−0.116435	−	1.55372i
$870$		0			0
$871$	23.0915	+	11.1203i	0.782425	+	0.376796i
$872$		0			0
$873$	2.87681	−	7.32999i	0.0973652	−	0.248082i
$874$		0			0
$875$	−1.38273	−	0.942727i	−0.0467447	−	0.0318700i
$876$		0			0
$877$	−10.5286	−	1.58692i	−0.355524	−	0.0535866i	−0.0311478	−	0.999515i	$-0.509916\pi$
$877$	−10.5286	−	1.58692i	−0.355524	−	0.0535866i	−0.324376	+	0.945928i	$0.605154\pi$
$878$		0			0
$879$	−0.00822360	+	0.109736i	−0.000277375	+	0.00370131i
$880$		0			0
$881$	1.95405	−	8.56125i	0.0658336	−	0.288436i	−0.931285	−	0.364291i	$-0.881311\pi$
$881$	1.95405	−	8.56125i	0.0658336	−	0.288436i	0.997119	+	0.0758550i	$0.0241686\pi$
$882$		0			0
$883$	37.3482	−	25.4636i	1.25687	−	0.856918i	0.262712	−	0.964874i	$-0.415383\pi$
$883$	37.3482	−	25.4636i	1.25687	−	0.856918i	0.994156	+	0.107956i	$0.0344306\pi$
$884$		0			0
$885$	0.776883	−	0.239637i	0.0261146	−	0.00805530i
$886$		0			0
$887$	18.9884	−	23.8107i	0.637569	−	0.799486i	−0.353128	−	0.935575i	$-0.614882\pi$
$887$	18.9884	−	23.8107i	0.637569	−	0.799486i	0.990697	+	0.136089i	$0.0434532\pi$
$888$		0			0
$889$	1.83227	−	1.70010i	0.0614523	−	0.0570194i
$890$		0			0
$891$	−28.3204	−	8.73569i	−0.948770	−	0.292657i
$892$		0			0
$893$	−12.9651	−	33.0345i	−0.433860	−	1.10546i
$894$		0			0
$895$	−17.4276	−	21.8535i	−0.582539	−	0.730481i
$896$		0			0
$897$	−0.0394945	+	0.0684065i	−0.00131868	+	0.00228403i
$898$		0			0
$899$	11.1820	−	5.38497i	0.372941	−	0.179599i
$900$		0			0
$901$	9.20487	+	40.3291i	0.306659	+	1.34356i
$902$		0			0
$903$	−0.0282194	+	0.0131159i	−0.000939083	+	0.000436470i
$904$		0			0
$905$	−4.62352	−	20.2570i	−0.153691	−	0.673365i
$906$		0			0
$907$	33.3055	−	16.0391i	1.10589	−	0.532570i	0.210386	−	0.977618i	$-0.432528\pi$
$907$	33.3055	−	16.0391i	1.10589	−	0.532570i	0.895506	+	0.445049i	$0.146814\pi$
$908$		0			0
$909$	13.2905	−	23.0198i	0.440818	−	0.763518i
$910$		0			0
$911$	−2.30448	−	2.88972i	−0.0763507	−	0.0957407i	0.742189	−	0.670190i	$-0.233788\pi$
$911$	−2.30448	−	2.88972i	−0.0763507	−	0.0957407i	−0.818540	+	0.574450i	$0.805216\pi$
$912$		0			0
$913$	9.19979	+	23.4407i	0.304469	+	0.775773i
$914$		0			0
$915$	0.575949	+	0.177657i	0.0190403	+	0.00587315i
$916$		0			0
$917$	1.06979	−	0.992618i	0.0353275	−	0.0327791i
$918$		0			0
$919$	9.80585	−	12.2961i	0.323465	−	0.405612i	−0.593337	−	0.804954i	$-0.702190\pi$
$919$	9.80585	−	12.2961i	0.323465	−	0.405612i	0.916802	+	0.399342i	$0.130761\pi$
$920$		0			0
$921$	0.0842531	−	0.0259886i	0.00277623	−	0.000856355i
$922$		0			0
$923$	−10.0537	+	6.85450i	−0.330922	+	0.225619i
$924$		0			0
$925$	7.99381	−	35.0232i	0.262835	−	1.15156i
$926$		0			0
$927$	0.169719	−	2.26474i	0.00557431	−	0.0743840i
$928$		0			0
$929$	18.5921	+	2.80230i	0.609986	+	0.0919406i	0.446769	−	0.894649i	$-0.352575\pi$
$929$	18.5921	+	2.80230i	0.609986	+	0.0919406i	0.163217	+	0.986590i	$0.447813\pi$
$930$		0			0
$931$	−34.4111	−	23.4611i	−1.12778	−	0.768907i
$932$		0			0
$933$	0.0317989	−	0.0810224i	0.00104105	−	0.00265255i
$934$		0			0
$935$	−65.7434	−	31.6604i	−2.15004	−	1.03540i
$936$		0			0
$937$	0.851277	+	11.3595i	0.0278100	+	0.371099i	0.993694	+	0.112124i	$0.0357655\pi$
$937$	0.851277	+	11.3595i	0.0278100	+	0.371099i	−0.965884	+	0.258974i	$0.916615\pi$
$938$		0			0
$939$	0.0739249	+	0.128042i	0.00241245	+	0.00417848i
$940$		0			0
$941$	38.2402	−	5.76378i	1.24660	−	0.187894i	0.507593	−	0.861597i	$-0.330535\pi$
$941$	38.2402	−	5.76378i	1.24660	−	0.187894i	0.739002	+	0.673703i	$0.235297\pi$
$942$		0			0
$943$	−4.70841	−	4.36877i	−0.153327	−	0.142267i
$944$		0			0
$945$	−0.0978450			−0.00318290
$946$		0			0
$947$	13.9735			0.454077			0.227039	−	0.973886i	$-0.427096\pi$
$947$	13.9735			0.454077			0.227039	+	0.973886i	$0.427096\pi$
$948$		0			0
$949$	2.46633	+	2.28842i	0.0800604	+	0.0742852i
$950$		0			0
$951$	0.315072	−	0.0474894i	0.0102169	−	0.00153995i
$952$		0			0
$953$	29.1009	+	50.4042i	0.942670	+	1.63275i	0.760350	+	0.649514i	$0.225028\pi$
$953$	29.1009	+	50.4042i	0.942670	+	1.63275i	0.182321	+	0.983239i	$0.441639\pi$
$954$		0			0
$955$	3.96249	+	52.8757i	0.128223	+	1.71102i
$956$		0			0
$957$	0.125676	+	0.0605222i	0.00406252	+	0.00195641i
$958$		0			0
$959$	−1.08567	+	2.76625i	−0.0350582	+	0.0893268i
$960$		0			0
$961$	−3.53207	−	2.40813i	−0.113938	−	0.0776815i
$962$		0			0
$963$	9.83204	+	1.48194i	0.316833	+	0.0477549i
$964$		0			0
$965$	−0.625843	+	8.35130i	−0.0201466	+	0.268838i
$966$		0			0
$967$	−4.46525	+	19.5635i	−0.143593	+	0.629121i	0.850991	+	0.525181i	$0.176002\pi$
$967$	−4.46525	+	19.5635i	−0.143593	+	0.629121i	−0.994584	+	0.103940i	$0.966855\pi$
$968$		0			0
$969$	0.564763	−	0.385049i	0.0181428	−	0.0123696i
$970$		0			0
$971$	−56.6062	+	17.4607i	−1.81658	+	0.560340i	−0.999861	−	0.0166975i	$-0.994685\pi$
$971$	−56.6062	+	17.4607i	−1.81658	+	0.560340i	−0.816718	+	0.577038i	$0.804209\pi$
$972$		0			0
$973$	−1.28964	+	1.61715i	−0.0413439	+	0.0518436i
$974$		0			0
$975$	−0.177505	+	0.164701i	−0.00568472	+	0.00527465i
$976$		0			0
$977$	−41.6110	−	12.8353i	−1.33125	−	0.410637i	−0.454144	−	0.890928i	$-0.650055\pi$
$977$	−41.6110	−	12.8353i	−1.33125	−	0.410637i	−0.877109	+	0.480291i	$0.840531\pi$
$978$		0			0
$979$	−2.17778	−	5.54889i	−0.0696022	−	0.177343i
$980$		0			0
$981$	−18.8786	−	23.6730i	−0.602747	−	0.755821i
$982$		0			0
$983$	1.58270	−	2.74132i	0.0504804	−	0.0874346i	−0.839681	−	0.543080i	$-0.817258\pi$
$983$	1.58270	−	2.74132i	0.0504804	−	0.0874346i	0.890161	+	0.455645i	$0.150591\pi$
$984$		0			0
$985$	7.71417	−	3.71495i	0.245794	−	0.118368i
$986$		0			0
$987$	0.00623337	+	0.0273102i	0.000198410	+	0.000869292i
$988$		0			0
$989$	13.8597	+	4.48429i	0.440713	+	0.142592i
$990$		0			0
$991$	0.746829	+	3.27207i	0.0237238	+	0.103941i	0.985404	−	0.170234i	$-0.0544524\pi$
$991$	0.746829	+	3.27207i	0.0237238	+	0.103941i	−0.961680	+	0.274175i	$0.911595\pi$
$992$		0			0
$993$	−0.0300916	+	0.0144914i	−0.000954928	+	0.000459869i
$994$		0			0
$995$	14.1951	−	24.5866i	0.450013	−	0.779446i
$996$		0			0
$997$	33.5063	+	42.0156i	1.06116	+	1.33065i	0.941193	+	0.337869i	$0.109706\pi$
$997$	33.5063	+	42.0156i	1.06116	+	1.33065i	0.119963	+	0.992778i	$0.461722\pi$
$998$		0			0
$999$	−0.203950	−	0.519657i	−0.00645271	−	0.0164412i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.225.1		36
4.3	odd	2		43.2.g.a.10.2	✓	36
12.11	even	2		387.2.y.c.10.2		36
43.13	even	21	inner	688.2.bg.c.529.1		36
172.23	odd	42		1849.2.a.n.1.9		18
172.63	even	42		1849.2.a.o.1.10		18
172.99	odd	42		43.2.g.a.13.2	yes	36
516.443	even	42		387.2.y.c.271.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.10.2	✓	36	4.3	odd	2
43.2.g.a.13.2	yes	36	172.99	odd	42
387.2.y.c.10.2		36	12.11	even	2
387.2.y.c.271.2		36	516.443	even	42
688.2.bg.c.225.1		36	1.1	even	1	trivial
688.2.bg.c.529.1		36	43.13	even	21	inner
1849.2.a.n.1.9		18	172.23	odd	42
1849.2.a.o.1.10		18	172.63	even	42

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.0129300 + 0.0119973i) q^{3} +(-3.39819 + 0.512194i) q^{5} +(0.134521 + 0.232998i) q^{7} +(-0.224167 - 2.99130i) q^{9} +O(q^{10})\)	\(q+(0.0129300 + 0.0119973i) q^{3} +(-3.39819 + 0.512194i) q^{5} +(0.134521 + 0.232998i) q^{7} +(-0.224167 - 2.99130i) q^{9} +(2.96782 + 1.42923i) q^{11} +(-0.736485 + 1.87653i) q^{13} +(-0.0500834 - 0.0341463i) q^{15} +(6.37398 + 0.960723i) q^{17} +(0.449267 - 5.99506i) q^{19} +(-0.00105598 + 0.00462655i) q^{21} +(1.83546 - 1.25139i) q^{23} +(6.50747 - 2.00729i) q^{25} +(0.0659813 - 0.0827379i) q^{27} +(1.75989 - 1.63294i) q^{29} +(4.93996 + 1.52378i) q^{31} +(0.0212271 + 0.0540856i) q^{33} +(-0.576470 - 0.722870i) q^{35} +(2.63757 - 4.56841i) q^{37} +(-0.0320360 + 0.0154277i) q^{39} +(-0.643386 - 2.81886i) q^{41} +(4.01778 + 5.18242i) q^{43} +(2.29389 + 10.0502i) q^{45} +(5.31835 - 2.56118i) q^{47} +(3.46381 - 5.99949i) q^{49} +(0.0708893 + 0.0888924i) q^{51} +(2.34453 + 5.97377i) q^{53} +(-10.8173 - 3.33669i) q^{55} +(0.0777333 - 0.0721259i) q^{57} +(-8.36244 + 10.4862i) q^{59} +(-9.50158 + 2.93085i) q^{61} +(0.666812 - 0.454625i) q^{63} +(1.54156 - 6.75403i) q^{65} +(0.950106 - 12.6783i) q^{67} +(0.0387457 + 0.00583998i) q^{69} +(4.98725 + 3.40025i) q^{71} +(0.609746 - 1.55361i) q^{73} +(0.108223 + 0.0521176i) q^{75} +(0.0662286 + 0.883759i) q^{77} +(-6.97172 - 12.0754i) q^{79} +(-8.89671 + 1.34096i) q^{81} +(5.60384 + 5.19960i) q^{83} -22.1521 q^{85} +0.0423461 q^{87} +(-1.32654 - 1.23085i) q^{89} +(-0.536302 + 0.0808345i) q^{91} +(0.0455924 + 0.0789684i) q^{93} +(1.54394 + 20.6024i) q^{95} +(2.36508 + 1.13896i) q^{97} +(3.60997 - 9.19804i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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