LMFDB - Embedded newform 688.2.u.e.465.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(97,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 0, 10]))

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

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

chi := DirichletCharacter("688.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.u (of order $7$, degree $6$, 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:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} + 14 x^{10} - 22 x^{9} + 40 x^{8} + 42 x^{7} + 239 x^{6} + 518 x^{5} + 2208 x^{4} + \cdots + 19321 $ x^12 - 5x^11 + 14x^10 - 22x^9 + 40x^8 + 42x^7 + 239x^6 + 518x^5 + 2208x^4 + 3625x^3 + 13209x^2 + 17931*x + 19321
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 172)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			465.1
Root			$1.65762 + 2.07859i$ of defining polynomial
Character	$\chi$	$=$	688.465
Dual form			688.2.u.e.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0990311 + 0.433884i) q^{3} +(-0.256654 + 0.321834i) q^{5} -2.98873 q^{7} +(2.52446 - 1.21572i) q^{9} +O(q^{10})$	\(q+(0.0990311 + 0.433884i) q^{3} +(-0.256654 + 0.321834i) q^{5} -2.98873 q^{7} +(2.52446 - 1.21572i) q^{9} +(1.01994 - 0.491178i) q^{11} +(2.06702 - 2.59196i) q^{13} +(-0.165055 - 0.0794865i) q^{15} +(1.81161 + 2.27169i) q^{17} +(4.36557 + 2.10235i) q^{19} +(-0.295978 - 1.29676i) q^{21} +(6.41416 - 3.08890i) q^{23} +(1.07490 + 4.70944i) q^{25} +(1.60992 + 2.01877i) q^{27} +(-0.272729 + 1.19490i) q^{29} +(-0.367962 + 1.61215i) q^{31} +(0.314120 + 0.393894i) q^{33} +(0.767071 - 0.961877i) q^{35} -1.26184 q^{37} +(1.32931 + 0.640162i) q^{39} +(-0.806062 + 3.53159i) q^{41} +(6.39381 - 1.45574i) q^{43} +(-0.256654 + 1.12448i) q^{45} +(5.14013 + 2.47536i) q^{47} +1.93252 q^{49} +(-0.806243 + 1.01100i) q^{51} +(-6.86195 - 8.60461i) q^{53} +(-0.103694 + 0.454315i) q^{55} +(-0.479847 + 2.10235i) q^{57} +(-6.08948 - 7.63597i) q^{59} +(0.319226 + 1.39862i) q^{61} +(-7.54493 + 3.63345i) q^{63} +(0.303673 + 1.33048i) q^{65} +(-3.11712 - 1.50113i) q^{67} +(1.97542 + 2.47710i) q^{69} +(3.84310 + 1.85074i) q^{71} +(8.45481 - 10.6020i) q^{73} +(-1.93690 + 0.932762i) q^{75} +(-3.04833 + 1.46800i) q^{77} -3.59704 q^{79} +(4.52446 - 5.67349i) q^{81} +(-0.491730 - 2.15441i) q^{83} -1.19607 q^{85} -0.545458 q^{87} +(4.11589 + 18.0329i) q^{89} +(-6.17778 + 7.74669i) q^{91} -0.735924 q^{93} +(-1.79705 + 0.865414i) q^{95} +(8.57167 - 4.12790i) q^{97} +(1.97767 - 2.47991i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q + 10 * q^3 + 3 * q^5 + 12 * q^9	$ 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9} - q^{11} - 4 q^{13} - q^{15} + 17 q^{17} - 13 q^{19} - 7 q^{21} + 14 q^{23} + q^{25} + 22 q^{27} - 6 q^{29} + 23 q^{31} - 2 q^{33} + 16 q^{35} - 24 q^{37} - q^{39} + 8 q^{41} + 7 q^{43} + 3 q^{45} - 22 q^{47} + 24 q^{49} + 13 q^{51} + 27 q^{53} + 29 q^{55} - 19 q^{57} + 17 q^{59} + 8 q^{61} + 7 q^{63} + 29 q^{65} - 29 q^{67} + 17 q^{71} - 10 q^{73} - 5 q^{75} + 15 q^{77} - 14 q^{79} + 36 q^{81} + 6 q^{83} - 70 q^{85} - 12 q^{87} - 16 q^{89} - 46 q^{91} + 46 q^{93} - 67 q^{95} + 10 q^{97} - q^{99}+O(q^{100}) $ 12 * q + 10 * q^3 + 3 * q^5 + 12 * q^9 - q^11 - 4 * q^13 - q^15 + 17 * q^17 - 13 * q^19 - 7 * q^21 + 14 * q^23 + q^25 + 22 * q^27 - 6 * q^29 + 23 * q^31 - 2 * q^33 + 16 * q^35 - 24 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 3 * q^45 - 22 * q^47 + 24 * q^49 + 13 * q^51 + 27 * q^53 + 29 * q^55 - 19 * q^57 + 17 * q^59 + 8 * q^61 + 7 * q^63 + 29 * q^65 - 29 * q^67 + 17 * q^71 - 10 * q^73 - 5 * q^75 + 15 * q^77 - 14 * q^79 + 36 * q^81 + 6 * q^83 - 70 * q^85 - 12 * q^87 - 16 * q^89 - 46 * q^91 + 46 * q^93 - 67 * q^95 + 10 * q^97 - q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{3}{7}\right)$	$1$

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
$3$	0.0990311	+	0.433884i	0.0571757	+	0.250503i	0.995437	−	0.0954255i	$-0.0304212\pi$
$3$	0.0990311	+	0.433884i	0.0571757	+	0.250503i	−0.938261	+	0.345928i	$0.887564\pi$
$4$		0			0
$5$	−0.256654	+	0.321834i	−0.114779	+	0.143929i	−0.835902	−	0.548879i	$-0.815055\pi$
$5$	−0.256654	+	0.321834i	−0.114779	+	0.143929i	0.721123	+	0.692807i	$0.243626\pi$
$6$		0			0
$7$	−2.98873			−1.12963			−0.564817	−	0.825216i	$-0.691053\pi$
$7$	−2.98873			−1.12963			−0.564817	+	0.825216i	$0.691053\pi$
$8$		0			0
$9$	2.52446	−	1.21572i	0.841486	−	0.405238i
$10$		0			0
$11$	1.01994	−	0.491178i	0.307524	−	0.148096i	−0.273757	−	0.961799i	$-0.588266\pi$
$11$	1.01994	−	0.491178i	0.307524	−	0.148096i	0.581280	+	0.813703i	$0.302552\pi$
$12$		0			0
$13$	2.06702	−	2.59196i	0.573289	−	0.718881i	−0.407663	−	0.913132i	$-0.633656\pi$
$13$	2.06702	−	2.59196i	0.573289	−	0.718881i	0.980952	+	0.194251i	$0.0622276\pi$
$14$		0			0
$15$	−0.165055	−	0.0794865i	−0.0426171	−	0.0205233i
$16$		0			0
$17$	1.81161	+	2.27169i	0.439381	+	0.550966i	0.951380	−	0.308020i	$-0.0996664\pi$
$17$	1.81161	+	2.27169i	0.439381	+	0.550966i	−0.511999	+	0.858986i	$0.671095\pi$
$18$		0			0
$19$	4.36557	+	2.10235i	1.00153	+	0.482311i	0.861457	−	0.507830i	$-0.169552\pi$
$19$	4.36557	+	2.10235i	1.00153	+	0.482311i	0.140073	+	0.990141i	$0.455266\pi$
$20$		0			0
$21$	−0.295978	−	1.29676i	−0.0645876	−	0.282977i
$22$		0			0
$23$	6.41416	−	3.08890i	1.33744	−	0.644079i	0.377955	−	0.925824i	$-0.376627\pi$
$23$	6.41416	−	3.08890i	1.33744	−	0.644079i	0.959489	+	0.281744i	$0.0909130\pi$
$24$		0			0
$25$	1.07490	+	4.70944i	0.214980	+	0.941888i
$26$		0			0
$27$	1.60992	+	2.01877i	0.309829	+	0.388513i
$28$		0			0
$29$	−0.272729	+	1.19490i	−0.0506445	+	0.221888i	−0.993917	−	0.110131i	$-0.964873\pi$
$29$	−0.272729	+	1.19490i	−0.0506445	+	0.221888i	0.943272	+	0.332020i	$0.107730\pi$
$30$		0			0
$31$	−0.367962	+	1.61215i	−0.0660879	+	0.289550i	−0.997163	−	0.0752787i	$-0.976015\pi$
$31$	−0.367962	+	1.61215i	−0.0660879	+	0.289550i	0.931075	+	0.364829i	$0.118873\pi$
$32$		0			0
$33$	0.314120	+	0.393894i	0.0546812	+	0.0685681i
$34$		0			0
$35$	0.767071	−	0.961877i	0.129659	−	0.162587i
$36$		0			0
$37$	−1.26184			−0.207446			−0.103723	−	0.994606i	$-0.533076\pi$
$37$	−1.26184			−0.207446			−0.103723	+	0.994606i	$0.533076\pi$
$38$		0			0
$39$	1.32931	+	0.640162i	0.212860	+	0.102508i
$40$		0			0
$41$	−0.806062	+	3.53159i	−0.125886	+	0.551542i	0.872169	+	0.489204i	$0.162713\pi$
$41$	−0.806062	+	3.53159i	−0.125886	+	0.551542i	−0.998055	+	0.0623376i	$0.980144\pi$
$42$		0			0
$43$	6.39381	−	1.45574i	0.975047	−	0.221998i
$43$	6.39381	−	1.45574i	0.975047	−	0.221998i
$44$		0			0
$45$	−0.256654	+	1.12448i	−0.0382598	+	0.167627i
$46$		0			0
$47$	5.14013	+	2.47536i	0.749766	+	0.361068i	0.769424	−	0.638739i	$-0.220544\pi$
$47$	5.14013	+	2.47536i	0.749766	+	0.361068i	−0.0196582	+	0.999807i	$0.506258\pi$
$48$		0			0
$49$	1.93252			0.276074
$50$		0			0
$51$	−0.806243	+	1.01100i	−0.112897	+	0.141568i
$52$		0			0
$53$	−6.86195	−	8.60461i	−0.942562	−	1.18193i	−0.983158	−	0.182759i	$-0.941497\pi$
$53$	−6.86195	−	8.60461i	−0.942562	−	1.18193i	0.0405962	−	0.999176i	$-0.487074\pi$
$54$		0			0
$55$	−0.103694	+	0.454315i	−0.0139821	+	0.0612598i
$56$		0			0
$57$	−0.479847	+	2.10235i	−0.0635573	+	0.278463i
$58$		0			0
$59$	−6.08948	−	7.63597i	−0.792783	−	0.994118i	−0.999875	−	0.0157969i	$-0.994971\pi$
$59$	−6.08948	−	7.63597i	−0.792783	−	0.994118i	0.207092	−	0.978321i	$-0.433600\pi$
$60$		0			0
$61$	0.319226	+	1.39862i	0.0408727	+	0.179075i	0.991244	−	0.132043i	$-0.0421537\pi$
$61$	0.319226	+	1.39862i	0.0408727	+	0.179075i	−0.950371	+	0.311118i	$0.899297\pi$
$62$		0			0
$63$	−7.54493	+	3.63345i	−0.950572	+	0.457771i
$64$		0			0
$65$	0.303673	+	1.33048i	0.0376660	+	0.165025i
$66$		0			0
$67$	−3.11712	−	1.50113i	−0.380817	−	0.183392i	0.233675	−	0.972315i	$-0.424925\pi$
$67$	−3.11712	−	1.50113i	−0.380817	−	0.183392i	−0.614492	+	0.788923i	$0.710639\pi$
$68$		0			0
$69$	1.97542	+	2.47710i	0.237813	+	0.298208i
$70$		0			0
$71$	3.84310	+	1.85074i	0.456092	+	0.219642i	0.647801	−	0.761810i	$-0.275689\pi$
$71$	3.84310	+	1.85074i	0.456092	+	0.219642i	−0.191709	+	0.981452i	$0.561403\pi$
$72$		0			0
$73$	8.45481	−	10.6020i	0.989560	−	1.24087i	0.0190478	−	0.999819i	$-0.493937\pi$
$73$	8.45481	−	10.6020i	0.989560	−	1.24087i	0.970512	−	0.241051i	$-0.0774921\pi$
$74$		0			0
$75$	−1.93690	+	0.932762i	−0.223654	+	0.107706i
$76$		0			0
$77$	−3.04833	+	1.46800i	−0.347389	+	0.167294i
$78$		0			0
$79$	−3.59704			−0.404699			−0.202350	−	0.979313i	$-0.564858\pi$
$79$	−3.59704			−0.404699			−0.202350	+	0.979313i	$0.564858\pi$
$80$		0			0
$81$	4.52446	−	5.67349i	0.502718	−	0.630388i
$82$		0			0
$83$	−0.491730	−	2.15441i	−0.0539744	−	0.236477i	0.940744	−	0.339117i	$-0.110128\pi$
$83$	−0.491730	−	2.15441i	−0.0539744	−	0.236477i	−0.994719	+	0.102640i	$0.967271\pi$
$84$		0			0
$85$	−1.19607			−0.129732
$86$		0			0
$87$	−0.545458			−0.0584793
$88$		0			0
$89$	4.11589	+	18.0329i	0.436283	+	1.91148i	0.410610	+	0.911811i	$0.365316\pi$
$89$	4.11589	+	18.0329i	0.436283	+	1.91148i	0.0256732	+	0.999670i	$0.491827\pi$
$90$		0			0
$91$	−6.17778	+	7.74669i	−0.647607	+	0.812073i
$92$		0			0
$93$	−0.735924			−0.0763118
$94$		0			0
$95$	−1.79705	+	0.865414i	−0.184373	+	0.0887896i
$96$		0			0
$97$	8.57167	−	4.12790i	0.870321	−	0.419125i	0.0552414	−	0.998473i	$-0.482407\pi$
$97$	8.57167	−	4.12790i	0.870321	−	0.419125i	0.815080	+	0.579348i	$0.196693\pi$
$98$		0			0
$99$	1.97767	−	2.47991i	0.198763	−	0.249241i
$100$		0			0
$101$	−3.55656	−	1.71275i	−0.353891	−	0.170425i	0.248488	−	0.968635i	$-0.420066\pi$
$101$	−3.55656	−	1.71275i	−0.353891	−	0.170425i	−0.602379	+	0.798210i	$0.705781\pi$
$102$		0			0
$103$	1.47604	+	1.85089i	0.145438	+	0.182374i	0.849215	−	0.528048i	$-0.177076\pi$
$103$	1.47604	+	1.85089i	0.145438	+	0.182374i	−0.703777	+	0.710421i	$0.748504\pi$
$104$		0			0
$105$	0.493307	+	0.237564i	0.0481418	+	0.0231839i
$106$		0			0
$107$	−0.714908	−	3.13221i	−0.0691127	−	0.302803i	0.928544	−	0.371223i	$-0.121061\pi$
$107$	−0.714908	−	3.13221i	−0.0691127	−	0.302803i	−0.997657	+	0.0684200i	$0.978204\pi$
$108$		0			0
$109$	4.48584	−	2.16027i	0.429666	−	0.206916i	−0.206536	−	0.978439i	$-0.566219\pi$
$109$	4.48584	−	2.16027i	0.429666	−	0.206916i	0.636202	+	0.771523i	$0.280505\pi$
$110$		0			0
$111$	−0.124962	−	0.547494i	−0.0118609	−	0.0519658i
$112$		0			0
$113$	−7.13136	−	8.94244i	−0.670862	−	0.841234i	0.323615	−	0.946189i	$-0.395102\pi$
$113$	−7.13136	−	8.94244i	−0.670862	−	0.841234i	−0.994477	+	0.104955i	$0.966530\pi$
$114$		0			0
$115$	−0.652109	+	2.85708i	−0.0608095	+	0.266424i
$116$		0			0
$117$	2.06702	−	9.05622i	0.191096	−	0.837248i
$118$		0			0
$119$	−5.41442	−	6.78947i	−0.496339	−	0.622390i
$120$		0			0
$121$	−6.05936	+	7.59820i	−0.550851	+	0.690746i
$122$		0			0
$123$	−1.61212			−0.145360
$124$		0			0
$125$	−3.64592	−	1.75578i	−0.326101	−	0.157042i
$126$		0			0
$127$	2.12080	−	9.29184i	0.188191	−	0.824517i	−0.789379	−	0.613906i	$-0.789597\pi$
$127$	2.12080	−	9.29184i	0.188191	−	0.824517i	0.977570	−	0.210611i	$-0.0675455\pi$
$128$		0			0
$129$	1.26481	+	2.63001i	0.111360	+	0.231559i
$130$		0			0
$131$	−3.34132	+	14.6393i	−0.291933	+	1.27904i	0.589899	+	0.807477i	$0.299167\pi$
$131$	−3.34132	+	14.6393i	−0.291933	+	1.27904i	−0.881832	+	0.471564i	$0.843690\pi$
$132$		0			0
$133$	−13.0475	−	6.28335i	−1.13136	−	0.544836i
$134$		0			0
$135$	−1.06290			−0.0914800
$136$		0			0
$137$	−1.49710	+	1.87730i	−0.127906	+	0.160389i	−0.841661	−	0.540007i	$-0.818422\pi$
$137$	−1.49710	+	1.87730i	−0.127906	+	0.160389i	0.713755	+	0.700396i	$0.246993\pi$
$138$		0			0
$139$	−11.6367	−	14.5920i	−0.987016	−	1.23768i	−0.971314	−	0.237802i	$-0.923573\pi$
$139$	−11.6367	−	14.5920i	−0.987016	−	1.23768i	−0.0157019	−	0.999877i	$-0.504998\pi$
$140$		0			0
$141$	−0.564984	+	2.47536i	−0.0475803	+	0.208463i
$142$		0			0
$143$	0.835126	−	3.65892i	0.0698367	−	0.305975i
$144$		0			0
$145$	−0.314564	−	0.394451i	−0.0261231	−	0.0327574i
$146$		0			0
$147$	0.191380	+	0.838489i	0.0157847	+	0.0691574i
$148$		0			0
$149$	−19.5575	+	9.41842i	−1.60222	+	0.771587i	−0.999649	−	0.0264814i	$-0.991570\pi$
$149$	−19.5575	+	9.41842i	−1.60222	+	0.771587i	−0.602567	+	0.798068i	$0.705855\pi$
$150$		0			0
$151$	2.82255	+	12.3664i	0.229696	+	1.00636i	0.949889	+	0.312589i	$0.101196\pi$
$151$	2.82255	+	12.3664i	0.229696	+	1.00636i	−0.720193	+	0.693774i	$0.755947\pi$
$152$		0			0
$153$	7.33507	+	3.53238i	0.593005	+	0.285576i
$154$		0			0
$155$	−0.424405	−	0.532187i	−0.0340891	−	0.0427463i
$156$		0			0
$157$	−18.1652	−	8.74790i	−1.44974	−	0.698158i	−0.467190	−	0.884157i	$-0.654734\pi$
$157$	−18.1652	−	8.74790i	−1.44974	−	0.698158i	−0.982550	+	0.185999i	$0.940448\pi$
$158$		0			0
$159$	3.05386	−	3.82941i	0.242187	−	0.303692i
$160$		0			0
$161$	−19.1702	+	9.23188i	−1.51082	+	0.727574i
$162$		0			0
$163$	12.4599	−	6.00039i	0.975937	−	0.469987i	0.123231	−	0.992378i	$-0.460674\pi$
$163$	12.4599	−	6.00039i	0.975937	−	0.469987i	0.852706	+	0.522391i	$0.174960\pi$
$164$		0			0
$165$	−0.207389			−0.0161452
$166$		0			0
$167$	−7.44195	+	9.33191i	−0.575875	+	0.722125i	−0.981403	−	0.191958i	$-0.938516\pi$
$167$	−7.44195	+	9.33191i	−0.575875	+	0.722125i	0.405528	+	0.914083i	$0.367088\pi$
$168$		0			0
$169$	0.447076	+	1.95877i	0.0343905	+	0.150674i
$170$		0			0
$171$	13.5766			1.03822
$172$		0			0
$173$	−11.6203			−0.883474			−0.441737	−	0.897145i	$-0.645638\pi$
$173$	−11.6203			−0.883474			−0.441737	+	0.897145i	$0.645638\pi$
$174$		0			0
$175$	−3.21258	−	14.0753i	−0.242849	−	1.06399i
$176$		0			0
$177$	2.71007	−	3.39833i	0.203702	−	0.255434i
$178$		0			0
$179$	−20.5632			−1.53697			−0.768484	−	0.639869i	$-0.778989\pi$
$179$	−20.5632			−1.53697			−0.768484	+	0.639869i	$0.778989\pi$
$180$		0			0
$181$	17.8889	−	8.61483i	1.32967	−	0.640335i	0.372007	−	0.928230i	$-0.378670\pi$
$181$	17.8889	−	8.61483i	1.32967	−	0.640335i	0.957662	+	0.287895i	$0.0929553\pi$
$182$		0			0
$183$	−0.575225	+	0.277014i	−0.0425219	+	0.0204774i
$184$		0			0
$185$	0.323858	−	0.406105i	0.0238105	−	0.0298574i
$186$		0			0
$187$	2.96354	+	1.42717i	0.216716	+	0.104365i
$188$		0			0
$189$	−4.81161	−	6.03357i	−0.349993	−	0.438877i
$190$		0			0
$191$	−6.71493	−	3.23374i	−0.485875	−	0.233985i	0.174875	−	0.984591i	$-0.444048\pi$
$191$	−6.71493	−	3.23374i	−0.485875	−	0.233985i	−0.660751	+	0.750605i	$0.729762\pi$
$192$		0			0
$193$	3.71093	+	16.2587i	0.267119	+	1.17032i	0.913348	+	0.407180i	$0.133488\pi$
$193$	3.71093	+	16.2587i	0.267119	+	1.17032i	−0.646229	+	0.763143i	$0.723655\pi$
$194$		0			0
$195$	−0.547200	+	0.263517i	−0.0391858	+	0.0188709i
$196$		0			0
$197$	−2.01165	−	8.81363i	−0.143324	−	0.627945i	−0.994650	−	0.103307i	$-0.967058\pi$
$197$	−2.01165	−	8.81363i	−0.143324	−	0.627945i	0.851325	−	0.524639i	$-0.175800\pi$
$198$		0			0
$199$	2.72630	+	3.41868i	0.193263	+	0.242344i	0.869016	−	0.494784i	$-0.164753\pi$
$199$	2.72630	+	3.41868i	0.193263	+	0.242344i	−0.675753	+	0.737128i	$0.736182\pi$
$200$		0			0
$201$	0.342622	−	1.50113i	0.0241667	−	0.105881i
$202$		0			0
$203$	0.815115	−	3.57125i	0.0572098	−	0.250653i
$204$		0			0
$205$	−0.929708	−	1.16582i	−0.0649336	−	0.0814242i
$206$		0			0
$207$	12.4371	−	15.5956i	0.864436	−	1.08397i
$208$		0			0
$209$	5.48525			0.379422
$210$		0			0
$211$	12.4986	+	6.01902i	0.860441	+	0.414367i	0.811442	−	0.584432i	$-0.198683\pi$
$211$	12.4986	+	6.01902i	0.860441	+	0.414367i	0.0489986	+	0.998799i	$0.484397\pi$
$212$		0			0
$213$	−0.422419	+	1.85074i	−0.0289437	+	0.126810i
$214$		0			0
$215$	−1.17249	+	2.43137i	−0.0799634	+	0.165818i
$216$		0			0
$217$	1.09974	−	4.81828i	0.0746552	−	0.327086i
$218$		0			0
$219$	5.43732	+	2.61848i	0.367420	+	0.176940i
$220$		0			0
$221$	9.63278			0.647971
$222$		0			0
$223$	−14.9459	+	18.7415i	−1.00085	+	1.25502i	−0.0340674	+	0.999420i	$0.510846\pi$
$223$	−14.9459	+	18.7415i	−1.00085	+	1.25502i	−0.966781	+	0.255605i	$0.917725\pi$
$224$		0			0
$225$	8.43887	+	10.5820i	0.562592	+	0.705468i
$226$		0			0
$227$	−3.33881	+	14.6283i	−0.221605	+	0.970913i	0.734666	+	0.678429i	$0.237339\pi$
$227$	−3.33881	+	14.6283i	−0.221605	+	0.970913i	−0.956270	+	0.292484i	$0.905518\pi$
$228$		0			0
$229$	−3.81566	+	16.7175i	−0.252146	+	1.10472i	0.677283	+	0.735723i	$0.263157\pi$
$229$	−3.81566	+	16.7175i	−0.252146	+	1.10472i	−0.929429	+	0.369001i	$0.879700\pi$
$230$		0			0
$231$	−0.938820	−	1.17724i	−0.0617698	−	0.0774569i
$232$		0			0
$233$	−5.71901	−	25.0566i	−0.374665	−	1.64151i	−0.713490	−	0.700665i	$-0.752887\pi$
$233$	−5.71901	−	25.0566i	−0.374665	−	1.64151i	0.338825	−	0.940849i	$-0.389970\pi$
$234$		0			0
$235$	−2.11589	+	1.01896i	−0.138026	+	0.0664696i
$236$		0			0
$237$	−0.356219	−	1.56070i	−0.0231389	−	0.101378i
$238$		0			0
$239$	−19.0412	−	9.16975i	−1.23167	−	0.593142i	−0.299133	−	0.954212i	$-0.596697\pi$
$239$	−19.0412	−	9.16975i	−1.23167	−	0.593142i	−0.932539	+	0.361070i	$0.882412\pi$
$240$		0			0
$241$	12.0483	+	15.1081i	0.776098	+	0.973196i	0.999999	−	0.00141563i	$-0.000450609\pi$
$241$	12.0483	+	15.1081i	0.776098	+	0.973196i	−0.223901	+	0.974612i	$0.571879\pi$
$242$		0			0
$243$	9.88889	+	4.76224i	0.634372	+	0.305498i
$244$		0			0
$245$	−0.495990	+	0.621951i	−0.0316876	+	0.0397350i
$246$		0			0
$247$	14.4729	−	6.96980i	0.920891	−	0.443478i
$248$		0			0
$249$	0.886067	−	0.426707i	0.0561522	−	0.0270415i
$250$		0			0
$251$	−25.4283			−1.60502			−0.802510	−	0.596639i	$-0.796502\pi$
$251$	−25.4283			−1.60502			−0.802510	+	0.596639i	$0.796502\pi$
$252$		0			0
$253$	5.02487	−	6.30098i	0.315911	−	0.396139i
$254$		0			0
$255$	−0.118448	−	0.518954i	−0.00741749	−	0.0324981i
$256$		0			0
$257$	13.4695			0.840204			0.420102	−	0.907477i	$-0.361994\pi$
$257$	13.4695			0.840204			0.420102	+	0.907477i	$0.361994\pi$
$258$		0			0
$259$	3.77131			0.234338
$260$		0			0
$261$	0.764170	+	3.34805i	0.0473010	+	0.207239i
$262$		0			0
$263$	−19.1927	+	24.0669i	−1.18347	+	1.48403i	−0.345415	+	0.938450i	$0.612262\pi$
$263$	−19.1927	+	24.0669i	−1.18347	+	1.48403i	−0.838059	+	0.545580i	$0.816309\pi$
$264$		0			0
$265$	4.53041			0.278301
$266$		0			0
$267$	−7.41657	+	3.57163i	−0.453887	+	0.218580i
$268$		0			0
$269$	25.9522	−	12.4979i	1.58233	−	0.762011i	0.583586	−	0.812052i	$-0.301649\pi$
$269$	25.9522	−	12.4979i	1.58233	−	0.762011i	0.998747	+	0.0500404i	$0.0159350\pi$
$270$		0			0
$271$	12.8472	−	16.1098i	0.780409	−	0.978602i	−0.219586	−	0.975593i	$-0.570471\pi$
$271$	12.8472	−	16.1098i	0.780409	−	0.978602i	0.999995	−	0.00300915i	$-0.000957843\pi$
$272$		0			0
$273$	−3.97295	−	1.91327i	−0.240454	−	0.115797i
$274$		0			0
$275$	3.40950	+	4.27538i	0.205601	+	0.257815i
$276$		0			0
$277$	5.56067	+	2.67788i	0.334108	+	0.160898i	0.593414	−	0.804897i	$-0.297780\pi$
$277$	5.56067	+	2.67788i	0.334108	+	0.160898i	−0.259306	+	0.965795i	$0.583494\pi$
$278$		0			0
$279$	1.03101	+	4.51714i	0.0617248	+	0.270434i
$280$		0			0
$281$	−15.1870	+	7.31367i	−0.905980	+	0.436297i	−0.828045	−	0.560661i	$-0.810547\pi$
$281$	−15.1870	+	7.31367i	−0.905980	+	0.436297i	−0.0779352	+	0.996958i	$0.524833\pi$
$282$		0			0
$283$	−1.87178	−	8.20081i	−0.111266	−	0.487487i	−0.999600	−	0.0282895i	$-0.990994\pi$
$283$	−1.87178	−	8.20081i	−0.111266	−	0.487487i	0.888334	−	0.459198i	$-0.151863\pi$
$284$		0			0
$285$	−0.553453	−	0.694008i	−0.0327837	−	0.0411095i
$286$		0			0
$287$	2.40910	−	10.5550i	0.142205	−	0.623041i
$288$		0			0
$289$	1.90422	−	8.34293i	0.112013	−	0.490761i
$290$		0			0
$291$	2.63989	+	3.31032i	0.154753	+	0.194054i
$292$		0			0
$293$	−2.85780	+	3.58357i	−0.166955	+	0.209354i	−0.858270	−	0.513199i	$-0.828460\pi$
$293$	−2.85780	+	3.58357i	−0.166955	+	0.209354i	0.691315	+	0.722553i	$0.257032\pi$
$294$		0			0
$295$	4.02041			0.234077
$296$		0			0
$297$	2.63359	+	1.26827i	0.152817	+	0.0735926i
$298$		0			0
$299$	5.25190	−	23.0101i	0.303725	−	1.33071i
$300$		0			0
$301$	−19.1094	+	4.35080i	−1.10145	+	0.250776i
$302$		0			0
$303$	0.390924	−	1.71275i	0.0224580	−	0.0983949i
$304$		0			0
$305$	−0.532055	−	0.256224i	−0.0304654	−	0.0146713i
$306$		0			0
$307$	−1.81686			−0.103694			−0.0518468	−	0.998655i	$-0.516511\pi$
$307$	−1.81686			−0.103694			−0.0518468	+	0.998655i	$0.516511\pi$
$308$		0			0
$309$	−0.656897	+	0.823723i	−0.0373696	+	0.0468600i
$310$		0			0
$311$	11.2179	+	14.0668i	0.636111	+	0.797657i	0.990511	−	0.137436i	$-0.0438863\pi$
$311$	11.2179	+	14.0668i	0.636111	+	0.797657i	−0.354400	+	0.935094i	$0.615315\pi$
$312$		0			0
$313$	5.90236	−	25.8599i	0.333621	−	1.46169i	−0.478441	−	0.878120i	$-0.658798\pi$
$313$	5.90236	−	25.8599i	0.333621	−	1.46169i	0.812062	−	0.583571i	$-0.198345\pi$
$314$		0			0
$315$	0.767071	−	3.36076i	0.0432196	−	0.189357i
$316$		0			0
$317$	18.5780	+	23.2961i	1.04345	+	1.30844i	0.949808	+	0.312833i	$0.101278\pi$
$317$	18.5780	+	23.2961i	1.04345	+	1.30844i	0.0936375	+	0.995606i	$0.470151\pi$
$318$		0			0
$319$	0.308743	+	1.35269i	0.0172863	+	0.0757361i
$320$		0			0
$321$	1.28822	−	0.620374i	0.0719014	−	0.0346259i
$322$		0			0
$323$	3.13284	+	13.7259i	0.174316	+	0.763727i
$324$		0			0
$325$	14.4285	+	6.94842i	0.800351	+	0.385429i
$326$		0			0
$327$	1.38154	+	1.73240i	0.0763995	+	0.0958020i
$328$		0			0
$329$	−15.3625	−	7.39818i	−0.846961	−	0.407875i
$330$		0			0
$331$	−4.08549	+	5.12304i	−0.224559	+	0.281588i	−0.881329	−	0.472503i	$-0.843351\pi$
$331$	−4.08549	+	5.12304i	−0.224559	+	0.281588i	0.656770	+	0.754091i	$0.271922\pi$
$332$		0			0
$333$	−3.18547	+	1.53404i	−0.174563	+	0.0840650i
$334$		0			0
$335$	1.28314	−	0.617926i	0.0701053	−	0.0337609i
$336$		0			0
$337$	−9.06955			−0.494050			−0.247025	−	0.969009i	$-0.579453\pi$
$337$	−9.06955			−0.494050			−0.247025	+	0.969009i	$0.579453\pi$
$338$		0			0
$339$	3.17375	−	3.97976i	0.172375	−	0.216151i
$340$		0			0
$341$	0.416551	+	1.82503i	0.0225575	+	0.0988309i
$342$		0			0
$343$	15.1453			0.817772
$344$		0			0
$345$	−1.30422			−0.0702167
$346$		0			0
$347$	−3.99101	−	17.4857i	−0.214248	−	0.938683i	−0.961643	−	0.274303i	$-0.911553\pi$
$347$	−3.99101	−	17.4857i	−0.214248	−	0.938683i	0.747395	−	0.664380i	$-0.231304\pi$
$348$		0			0
$349$	4.66116	−	5.84491i	0.249506	−	0.312871i	−0.641268	−	0.767317i	$-0.721591\pi$
$349$	4.66116	−	5.84491i	0.249506	−	0.312871i	0.890774	+	0.454446i	$0.150163\pi$
$350$		0			0
$351$	8.56032			0.456916
$352$		0			0
$353$	5.83501	−	2.81000i	0.310567	−	0.149561i	−0.272108	−	0.962267i	$-0.587721\pi$
$353$	5.83501	−	2.81000i	0.310567	−	0.149561i	0.582674	+	0.812706i	$0.302006\pi$
$354$		0			0
$355$	−1.58198	+	0.761841i	−0.0839627	+	0.0404343i
$356$		0			0
$357$	2.40965	−	3.02160i	0.127532	−	0.159920i
$358$		0			0
$359$	21.9779	+	10.5840i	1.15995	+	0.558603i	0.912008	−	0.410172i	$-0.134531\pi$
$359$	21.9779	+	10.5840i	1.15995	+	0.558603i	0.247943	+	0.968775i	$0.420246\pi$
$360$		0			0
$361$	2.79202	+	3.50108i	0.146948	+	0.184268i
$362$		0			0
$363$	−3.89680	−	1.87660i	−0.204529	−	0.0984960i
$364$		0			0
$365$	1.24212	+	5.44210i	0.0650157	+	0.284852i
$366$		0			0
$367$	2.22706	−	1.07249i	0.116251	−	0.0559837i	−0.374854	−	0.927084i	$-0.622307\pi$
$367$	2.22706	−	1.07249i	0.116251	−	0.0559837i	0.491106	+	0.871100i	$0.336593\pi$
$368$		0			0
$369$	2.25854	+	9.89529i	0.117575	+	0.515128i
$370$		0			0
$371$	20.5085	+	25.7169i	1.06475	+	1.33515i
$372$		0			0
$373$	−4.41261	+	19.3329i	−0.228476	+	1.00102i	0.722406	+	0.691469i	$0.243036\pi$
$373$	−4.41261	+	19.3329i	−0.228476	+	1.00102i	−0.950883	+	0.309552i	$0.899821\pi$
$374$		0			0
$375$	0.400746	−	1.75578i	0.0206944	−	0.0906682i
$376$		0			0
$377$	2.53341	+	3.17680i	0.130477	+	0.163613i
$378$		0			0
$379$	23.1925	−	29.0825i	1.19132	−	1.49387i	0.364686	−	0.931130i	$-0.381176\pi$
$379$	23.1925	−	29.0825i	1.19132	−	1.49387i	0.826635	−	0.562739i	$-0.190252\pi$
$380$		0			0
$381$	4.24160			0.217304
$382$		0			0
$383$	18.7161	+	9.01319i	0.956348	+	0.460553i	0.845907	−	0.533330i	$-0.179060\pi$
$383$	18.7161	+	9.01319i	0.956348	+	0.460553i	0.110440	+	0.993883i	$0.464774\pi$
$384$		0			0
$385$	0.309915	−	1.35783i	0.0157947	−	0.0692012i
$386$		0			0
$387$	14.3712	−	11.4480i	0.730527	−	0.581934i
$388$		0			0
$389$	4.66942	−	20.4581i	0.236749	−	1.03727i	−0.707158	−	0.707055i	$-0.750023\pi$
$389$	4.66942	−	20.4581i	0.236749	−	1.03727i	0.943907	−	0.330210i	$-0.107120\pi$
$390$		0			0
$391$	18.6370	+	8.97510i	0.942513	+	0.453890i
$392$		0			0
$393$	−6.68264			−0.337095
$394$		0			0
$395$	0.923197	−	1.15765i	0.0464511	−	0.0582478i
$396$		0			0
$397$	−12.5037	−	15.6792i	−0.627544	−	0.786915i	0.361840	−	0.932240i	$-0.382149\pi$
$397$	−12.5037	−	15.6792i	−0.627544	−	0.786915i	−0.989384	+	0.145325i	$0.953577\pi$
$398$		0			0
$399$	1.43413	−	6.28335i	0.0717965	−	0.314561i
$400$		0			0
$401$	1.96981	−	8.63031i	0.0983677	−	0.430977i	−0.901631	−	0.432506i	$-0.857629\pi$
$401$	1.96981	−	8.63031i	0.0983677	−	0.430977i	0.999999	+	0.00152873i	$0.000486611\pi$
$402$		0			0
$403$	3.41804	+	4.28609i	0.170265	+	0.213505i
$404$		0			0
$405$	0.664703	+	2.91225i	0.0330293	+	0.144711i
$406$		0			0
$407$	−1.28701	+	0.619789i	−0.0637945	+	0.0307218i
$408$		0			0
$409$	5.73810	+	25.1403i	0.283731	+	1.24311i	0.892969	+	0.450119i	$0.148618\pi$
$409$	5.73810	+	25.1403i	0.283731	+	1.24311i	−0.609238	+	0.792988i	$0.708524\pi$
$410$		0			0
$411$	−0.962791	−	0.463656i	−0.0474910	−	0.0228704i
$412$		0			0
$413$	18.1998	+	22.8219i	0.895555	+	1.12299i
$414$		0			0
$415$	0.819568	+	0.394683i	0.0402310	+	0.0193742i
$416$		0			0
$417$	5.17884	−	6.49406i	0.253609	−	0.318015i
$418$		0			0
$419$	−9.09770	+	4.38122i	−0.444452	+	0.214037i	−0.642704	−	0.766114i	$-0.722188\pi$
$419$	−9.09770	+	4.38122i	−0.444452	+	0.214037i	0.198253	+	0.980151i	$0.436473\pi$
$420$		0			0
$421$	−2.49620	+	1.20210i	−0.121657	+	0.0585870i	−0.493722	−	0.869620i	$-0.664364\pi$
$421$	−2.49620	+	1.20210i	−0.121657	+	0.0585870i	0.372065	+	0.928207i	$0.378650\pi$
$422$		0			0
$423$	15.9854			0.777236
$424$		0			0
$425$	−8.75109	+	10.9735i	−0.424490	+	0.532294i
$426$		0			0
$427$	−0.954080	−	4.18010i	−0.0461712	−	0.202289i
$428$		0			0
$429$	1.67025			0.0806405
$430$		0			0
$431$	−37.1361			−1.78878			−0.894391	−	0.447286i	$-0.852391\pi$
$431$	−37.1361			−1.78878			−0.894391	+	0.447286i	$0.852391\pi$
$432$		0			0
$433$	2.32887	+	10.2034i	0.111918	+	0.490346i	0.999556	+	0.0298035i	$0.00948815\pi$
$433$	2.32887	+	10.2034i	0.111918	+	0.490346i	−0.887637	+	0.460543i	$0.847655\pi$
$434$		0			0
$435$	0.139994	−	0.175547i	0.00671221	−	0.00841685i
$436$		0			0
$437$	34.4954			1.65014
$438$		0			0
$439$	−15.2683	+	7.35285i	−0.728718	+	0.350932i	−0.761180	−	0.648541i	$-0.775380\pi$
$439$	−15.2683	+	7.35285i	−0.728718	+	0.350932i	0.0324617	+	0.999473i	$0.489665\pi$
$440$		0			0
$441$	4.87857	−	2.34939i	0.232313	−	0.111876i
$442$		0			0
$443$	−7.34612	+	9.21174i	−0.349025	+	0.437663i	−0.925094	−	0.379738i	$-0.876014\pi$
$443$	−7.34612	+	9.21174i	−0.349025	+	0.437663i	0.576070	+	0.817401i	$0.304586\pi$
$444$		0			0
$445$	−6.85996	−	3.30358i	−0.325193	−	0.156605i
$446$		0			0
$447$	−6.02330	−	7.55298i	−0.284892	−	0.357244i
$448$		0			0
$449$	−27.8623	−	13.4178i	−1.31490	−	0.633225i	−0.360785	−	0.932649i	$-0.617491\pi$
$449$	−27.8623	−	13.4178i	−1.31490	−	0.633225i	−0.954120	+	0.299424i	$0.903205\pi$
$450$		0			0
$451$	0.912502	+	3.99793i	0.0429680	+	0.188255i
$452$		0			0
$453$	−5.08605	+	2.44931i	−0.238964	+	0.115079i
$454$		0			0
$455$	−0.907597	−	3.97644i	−0.0425488	−	0.186418i
$456$		0			0
$457$	−11.7947	−	14.7901i	−0.551735	−	0.691854i	0.425271	−	0.905066i	$-0.360179\pi$
$457$	−11.7947	−	14.7901i	−0.551735	−	0.691854i	−0.977006	+	0.213212i	$0.931607\pi$
$458$		0			0
$459$	−1.66948	+	7.31446i	−0.0779246	+	0.341410i
$460$		0			0
$461$	4.63438	−	20.3045i	0.215844	−	0.945676i	−0.744667	−	0.667436i	$-0.767392\pi$
$461$	4.63438	−	20.3045i	0.215844	−	0.945676i	0.960512	−	0.278240i	$-0.0897511\pi$
$462$		0			0
$463$	−15.6149	−	19.5805i	−0.725686	−	0.909982i	0.272959	−	0.962026i	$-0.411998\pi$
$463$	−15.6149	−	19.5805i	−0.725686	−	0.909982i	−0.998645	+	0.0520442i	$0.983426\pi$
$464$		0			0
$465$	0.188878	−	0.236846i	0.00875901	−	0.0109835i
$466$		0			0
$467$	6.44116			0.298061			0.149031	−	0.988833i	$-0.452385\pi$
$467$	6.44116			0.298061			0.149031	+	0.988833i	$0.452385\pi$
$468$		0			0
$469$	9.31624	+	4.48647i	0.430184	+	0.207166i
$470$		0			0
$471$	1.99665	−	8.74790i	0.0920008	−	0.403082i
$472$		0			0
$473$	5.80628	−	4.62526i	0.266973	−	0.212670i
$474$		0			0
$475$	−5.20833	+	22.8192i	−0.238975	+	1.04702i
$476$		0			0
$477$	−27.7835	−	13.3798i	−1.27212	−	0.612620i
$478$		0			0
$479$	−16.9258			−0.773361			−0.386680	−	0.922214i	$-0.626378\pi$
$479$	−16.9258			−0.773361			−0.386680	+	0.922214i	$0.626378\pi$
$480$		0			0
$481$	−2.60826	+	3.27065i	−0.118926	+	0.149129i
$482$		0			0
$483$	−5.90401	−	7.40340i	−0.268642	−	0.336866i
$484$		0			0
$485$	−0.871457	+	3.81810i	−0.0395708	+	0.173371i
$486$		0			0
$487$	3.60922	−	15.8130i	0.163549	−	0.716556i	−0.824935	−	0.565228i	$-0.808788\pi$
$487$	3.60922	−	15.8130i	0.163549	−	0.716556i	0.988484	−	0.151327i	$-0.0483548\pi$
$488$		0			0
$489$	3.83739	+	4.81194i	0.173533	+	0.217603i
$490$		0			0
$491$	−3.06749	−	13.4395i	−0.138434	−	0.606518i	−0.995780	−	0.0917769i	$-0.970745\pi$
$491$	−3.06749	−	13.4395i	−0.138434	−	0.606518i	0.857346	−	0.514741i	$-0.172112\pi$
$492$		0			0
$493$	−3.20853	+	1.54515i	−0.144505	+	0.0695900i
$494$		0			0
$495$	0.290545	+	1.27296i	0.0130590	+	0.0572154i
$496$		0			0
$497$	−11.4860	−	5.53136i	−0.515217	−	0.248115i
$498$		0			0
$499$	3.14965	+	3.94954i	0.140998	+	0.176806i	0.847316	−	0.531088i	$-0.178217\pi$
$499$	3.14965	+	3.94954i	0.140998	+	0.176806i	−0.706319	+	0.707894i	$0.749645\pi$
$500$		0			0
$501$	−4.78595	−	2.30479i	−0.213820	−	0.102970i
$502$		0			0
$503$	−13.0168	+	16.3226i	−0.580392	+	0.727788i	−0.982180	−	0.187945i	$-0.939817\pi$
$503$	−13.0168	+	16.3226i	−0.580392	+	0.727788i	0.401788	+	0.915733i	$0.368389\pi$
$504$		0			0
$505$	1.46403	−	0.705039i	0.0651484	−	0.0313738i
$506$		0			0
$507$	−0.805603	+	0.387958i	−0.0357781	+	0.0172298i
$508$		0			0
$509$	24.6278			1.09161			0.545803	−	0.837913i	$-0.316225\pi$
$509$	24.6278			1.09161			0.545803	+	0.837913i	$0.316225\pi$
$510$		0			0
$511$	−25.2692	+	31.6865i	−1.11784	+	1.40173i
$512$		0			0
$513$	2.78404	+	12.1977i	0.122918	+	0.538541i
$514$		0			0
$515$	−0.974511			−0.0429421
$516$		0			0
$517$	6.45847			0.284043
$518$		0			0
$519$	−1.15077	−	5.04185i	−0.0505132	−	0.221313i
$520$		0			0
$521$	2.07325	−	2.59977i	0.0908308	−	0.113898i	−0.734339	−	0.678783i	$-0.762508\pi$
$521$	2.07325	−	2.59977i	0.0908308	−	0.113898i	0.825170	+	0.564884i	$0.191079\pi$
$522$		0			0
$523$	−4.41262			−0.192951			−0.0964753	−	0.995335i	$-0.530757\pi$
$523$	−4.41262			−0.192951			−0.0964753	+	0.995335i	$0.530757\pi$
$524$		0			0
$525$	5.78888	−	2.78778i	0.252647	−	0.121669i
$526$		0			0
$527$	−4.32890	+	2.08469i	−0.188570	+	0.0908105i
$528$		0			0
$529$	17.2599	−	21.6432i	0.750430	−	0.941010i
$530$		0			0
$531$	−24.6558	−	11.8736i	−1.06997	−	0.515271i
$532$		0			0
$533$	7.48760	+	9.38916i	0.324324	+	0.406690i
$534$		0			0
$535$	1.19154	+	0.573815i	0.0515147	+	0.0248082i
$536$		0			0
$537$	−2.03640	−	8.92206i	−0.0878772	−	0.385015i
$538$		0			0
$539$	1.97106	−	0.949210i	0.0848994	−	0.0408854i
$540$		0			0
$541$	6.05696	+	26.5373i	0.260409	+	1.14093i	0.920809	+	0.390013i	$0.127529\pi$
$541$	6.05696	+	26.5373i	0.260409	+	1.14093i	−0.660400	+	0.750914i	$0.729613\pi$
$542$		0			0
$543$	5.50939	+	6.90856i	0.236431	+	0.296474i
$544$		0			0
$545$	−0.456063	+	1.99814i	−0.0195356	+	0.0855909i
$546$		0			0
$547$	−6.28867	+	27.5525i	−0.268884	+	1.17806i	0.642429	+	0.766345i	$0.277927\pi$
$547$	−6.28867	+	27.5525i	−0.268884	+	1.17806i	−0.911313	+	0.411714i	$0.864930\pi$
$548$		0			0
$549$	2.50620	+	3.14267i	0.106962	+	0.134126i
$550$		0			0
$551$	−3.70272	+	4.64307i	−0.157741	+	0.197801i
$552$		0			0
$553$	10.7506			0.457162
$554$		0			0
$555$	0.208274	+	0.100300i	0.00884075	+	0.00425748i
$556$		0			0
$557$	−6.44394	+	28.2327i	−0.273038	+	1.19626i	0.633367	+	0.773851i	$0.281672\pi$
$557$	−6.44394	+	28.2327i	−0.273038	+	1.19626i	−0.906406	+	0.422408i	$0.861185\pi$
$558$		0			0
$559$	9.44294	−	19.5816i	0.399394	−	0.828212i
$560$		0			0
$561$	−0.325741	+	1.42717i	−0.0137528	+	0.0602550i
$562$		0			0
$563$	14.5512	+	7.00750i	0.613261	+	0.295331i	0.714613	−	0.699520i	$-0.246603\pi$
$563$	14.5512	+	7.00750i	0.613261	+	0.295331i	−0.101352	+	0.994851i	$0.532317\pi$
$564$		0			0
$565$	4.70828			0.198079
$566$		0			0
$567$	−13.5224	+	16.9565i	−0.567887	+	0.712108i
$568$		0			0
$569$	−16.1665	−	20.2722i	−0.677735	−	0.849853i	0.317408	−	0.948289i	$-0.397187\pi$
$569$	−16.1665	−	20.2722i	−0.677735	−	0.849853i	−0.995143	+	0.0984360i	$0.968616\pi$
$570$		0			0
$571$	8.31830	−	36.4449i	0.348110	−	1.52517i	−0.433357	−	0.901223i	$-0.642671\pi$
$571$	8.31830	−	36.4449i	0.348110	−	1.52517i	0.781467	−	0.623947i	$-0.214472\pi$
$572$		0			0
$573$	0.738080	−	3.23374i	0.0308337	−	0.135091i
$574$		0			0
$575$	21.4415	+	26.8868i	0.894174	+	1.12126i
$576$		0			0
$577$	−7.00642	−	30.6971i	−0.291681	−	1.27794i	−0.882185	−	0.470904i	$-0.843928\pi$
$577$	−7.00642	−	30.6971i	−0.291681	−	1.27794i	0.590504	−	0.807035i	$-0.298929\pi$
$578$		0			0
$579$	−6.68687	+	3.22023i	−0.277897	+	0.133828i
$580$		0			0
$581$	1.46965	+	6.43896i	0.0609713	+	0.267133i
$582$		0			0
$583$	−11.2252	−	5.40576i	−0.464899	−	0.223884i
$584$		0			0
$585$	2.38409	+	2.98956i	0.0985701	+	0.123603i
$586$		0			0
$587$	−25.8145	−	12.4316i	−1.06548	−	0.513108i	−0.182833	−	0.983144i	$-0.558527\pi$
$587$	−25.8145	−	12.4316i	−1.06548	−	0.513108i	−0.882647	+	0.470036i	$0.844241\pi$
$588$		0			0
$589$	−4.99566	+	6.26436i	−0.205842	+	0.258118i
$590$		0			0
$591$	3.62488	−	1.74565i	0.149107	−	0.0718064i
$592$		0			0
$593$	−31.8001	+	15.3141i	−1.30587	+	0.628875i	−0.951908	−	0.306384i	$-0.900881\pi$
$593$	−31.8001	+	15.3141i	−1.30587	+	0.628875i	−0.353964	+	0.935259i	$0.615167\pi$
$594$		0			0
$595$	3.57472			0.146549
$596$		0			0
$597$	−1.21332	+	1.52145i	−0.0496579	+	0.0622690i
$598$		0			0
$599$	−0.876100	−	3.83845i	−0.0357965	−	0.156835i	0.953871	−	0.300217i	$-0.0970592\pi$
$599$	−0.876100	−	3.83845i	−0.0357965	−	0.156835i	−0.989667	+	0.143383i	$0.954202\pi$
$600$		0			0
$601$	−13.5242			−0.551662			−0.275831	−	0.961206i	$-0.588953\pi$
$601$	−13.5242			−0.551662			−0.275831	+	0.961206i	$0.588953\pi$
$602$		0			0
$603$	−9.69399			−0.394770
$604$		0			0
$605$	−0.890201	−	3.90022i	−0.0361918	−	0.158567i
$606$		0			0
$607$	13.8272	−	17.3387i	0.561228	−	0.703758i	−0.417556	−	0.908651i	$-0.637113\pi$
$607$	13.8272	−	17.3387i	0.561228	−	0.703758i	0.978784	+	0.204893i	$0.0656848\pi$
$608$		0			0
$609$	1.63023			0.0660602
$610$		0			0
$611$	17.0408	−	8.20642i	0.689397	−	0.331996i
$612$		0			0
$613$	−3.45396	+	1.66334i	−0.139504	+	0.0671817i	−0.502334	−	0.864674i	$-0.667525\pi$
$613$	−3.45396	+	1.66334i	−0.139504	+	0.0671817i	0.362829	+	0.931856i	$0.381811\pi$
$614$		0			0
$615$	0.413759	−	0.518837i	0.0166844	−	0.0209215i
$616$		0			0
$617$	−15.3326	−	7.38379i	−0.617267	−	0.297260i	0.0989991	−	0.995088i	$-0.468436\pi$
$617$	−15.3326	−	7.38379i	−0.617267	−	0.297260i	−0.716266	+	0.697828i	$0.754150\pi$
$618$		0			0
$619$	15.3890	+	19.2972i	0.618538	+	0.775622i	0.988138	−	0.153569i	$-0.0490767\pi$
$619$	15.3890	+	19.2972i	0.618538	+	0.775622i	−0.369600	+	0.929191i	$0.620505\pi$
$620$		0			0
$621$	16.5620	+	7.97586i	0.664612	+	0.320060i
$622$		0			0
$623$	−12.3013	−	53.8954i	−0.492841	−	2.15928i
$624$		0			0
$625$	−20.2601	+	9.75674i	−0.810403	+	0.390269i
$626$		0			0
$627$	0.543210	+	2.37996i	0.0216937	+	0.0950464i
$628$		0			0
$629$	−2.28597	−	2.86652i	−0.0911477	−	0.114296i
$630$		0			0
$631$	−8.66377	+	37.9584i	−0.344899	+	1.51110i	0.443690	+	0.896181i	$0.353669\pi$
$631$	−8.66377	+	37.9584i	−0.344899	+	1.51110i	−0.788589	+	0.614921i	$0.789188\pi$
$632$		0			0
$633$	−1.37380	+	6.01902i	−0.0546037	+	0.239235i
$634$		0			0
$635$	2.44612	+	3.06734i	0.0970713	+	0.121724i
$636$		0			0
$637$	3.99456	−	5.00902i	0.158270	−	0.198465i
$638$		0			0
$639$	11.9517			0.472802
$640$		0			0
$641$	−42.0780	−	20.2637i	−1.66198	−	0.800369i	−0.998645	−	0.0520485i	$-0.983425\pi$
$641$	−42.0780	−	20.2637i	−1.66198	−	0.800369i	−0.663338	−	0.748320i	$-0.730861\pi$
$642$		0			0
$643$	−1.74012	+	7.62399i	−0.0686238	+	0.300661i	−0.997580	−	0.0695293i	$-0.977850\pi$
$643$	−1.74012	+	7.62399i	−0.0686238	+	0.300661i	0.928956	+	0.370190i	$0.120707\pi$
$644$		0			0
$645$	−1.17105	−	0.267945i	−0.0461099	−	0.0105503i
$646$		0			0
$647$	8.51060	−	37.2874i	0.334586	−	1.46592i	−0.475556	−	0.879685i	$-0.657753\pi$
$647$	8.51060	−	37.2874i	0.334586	−	1.46592i	0.810142	−	0.586233i	$-0.199390\pi$
$648$		0			0
$649$	−9.96152	−	4.79722i	−0.391024	−	0.188307i
$650$		0			0
$651$	2.19948			0.0862044
$652$		0			0
$653$	5.95232	−	7.46397i	0.232932	−	0.292088i	−0.651604	−	0.758559i	$-0.725904\pi$
$653$	5.95232	−	7.46397i	0.232932	−	0.292088i	0.884536	+	0.466472i	$0.154475\pi$
$654$		0			0
$655$	−3.85386	−	4.83259i	−0.150583	−	0.188825i
$656$		0			0
$657$	8.45481	−	37.0429i	0.329853	−	1.44518i
$658$		0			0
$659$	4.45062	−	19.4994i	0.173372	−	0.759591i	−0.811223	−	0.584737i	$-0.801198\pi$
$659$	4.45062	−	19.4994i	0.173372	−	0.759591i	0.984594	−	0.174853i	$-0.0559452\pi$
$660$		0			0
$661$	27.0897	+	33.9694i	1.05367	+	1.32126i	0.944961	+	0.327183i	$0.106099\pi$
$661$	27.0897	+	33.9694i	1.05367	+	1.32126i	0.108706	+	0.994074i	$0.465329\pi$
$662$		0			0
$663$	0.953945	+	4.17951i	0.0370482	+	0.162319i
$664$		0			0
$665$	5.37090	−	2.58649i	0.208275	−	0.100300i
$666$		0			0
$667$	1.94161	+	8.50674i	0.0751794	+	0.329382i
$668$		0			0
$669$	−9.61174	−	4.62877i	−0.371611	−	0.178959i
$670$		0			0
$671$	1.01256	+	1.26971i	0.0390895	+	0.0490167i
$672$		0			0
$673$	−9.56779	−	4.60761i	−0.368811	−	0.177610i	0.240293	−	0.970700i	$-0.422757\pi$
$673$	−9.56779	−	4.60761i	−0.368811	−	0.177610i	−0.609104	+	0.793090i	$0.708471\pi$
$674$		0			0
$675$	−7.77678	+	9.75178i	−0.299329	+	0.375346i
$676$		0			0
$677$	−19.0117	+	9.15554i	−0.730678	+	0.351876i	−0.761951	−	0.647635i	$-0.775758\pi$
$677$	−19.0117	+	9.15554i	−0.730678	+	0.351876i	0.0312727	+	0.999511i	$0.490044\pi$
$678$		0			0
$679$	−25.6184	+	12.3372i	−0.983145	+	0.473458i
$680$		0			0
$681$	−6.67762			−0.255887
$682$		0			0
$683$	−3.04142	+	3.81382i	−0.116377	+	0.145932i	−0.836608	−	0.547803i	$-0.815464\pi$
$683$	−3.04142	+	3.81382i	−0.116377	+	0.145932i	0.720231	+	0.693735i	$0.244036\pi$
$684$		0			0
$685$	−0.219944	−	0.963636i	−0.00840361	−	0.0368186i
$686$		0			0
$687$	−7.63132			−0.291153
$688$		0			0
$689$	−36.4867			−1.39003
$690$		0			0
$691$	−6.12181	−	26.8214i	−0.232885	−	1.02033i	−0.947233	−	0.320545i	$-0.896134\pi$
$691$	−6.12181	−	26.8214i	−0.232885	−	1.02033i	0.714349	−	0.699790i	$-0.246723\pi$
$692$		0			0
$693$	−5.91071	+	7.41180i	−0.224529	+	0.281551i
$694$		0			0
$695$	7.68283			0.291426
$696$		0			0
$697$	−9.48295	+	4.56675i	−0.359192	+	0.172978i
$698$		0			0
$699$	10.3053	−	4.96277i	0.389782	−	0.187709i
$700$		0			0
$701$	−26.0022	+	32.6057i	−0.982090	+	1.23150i	−0.00926540	+	0.999957i	$0.502949\pi$
$701$	−26.0022	+	32.6057i	−0.982090	+	1.23150i	−0.972824	+	0.231544i	$0.925622\pi$
$702$		0			0
$703$	−5.50867	−	2.65283i	−0.207763	−	0.100054i
$704$		0			0
$705$	−0.651650	−	0.817143i	−0.0245425	−	0.0307754i
$706$		0			0
$707$	10.6296	+	5.11895i	0.399768	+	0.192518i
$708$		0			0
$709$	−10.0194	−	43.8977i	−0.376285	−	1.64861i	−0.708723	−	0.705487i	$-0.750728\pi$
$709$	−10.0194	−	43.8977i	−0.376285	−	1.64861i	0.332438	−	0.943125i	$-0.392129\pi$
$710$		0			0
$711$	−9.08059	+	4.37298i	−0.340549	+	0.164000i
$712$		0			0
$713$	2.61959	+	11.4772i	0.0981044	+	0.429823i
$714$		0			0
$715$	0.963229	+	1.20785i	0.0360227	+	0.0451711i
$716$		0			0
$717$	2.09294	−	9.16975i	0.0781621	−	0.342451i
$718$		0			0
$719$	1.58420	−	6.94085i	0.0590808	−	0.258850i	−0.936759	−	0.349976i	$-0.886190\pi$
$719$	1.58420	−	6.94085i	0.0590808	−	0.258850i	0.995839	+	0.0911265i	$0.0290468\pi$
$720$		0			0
$721$	−4.41147	−	5.53181i	−0.164292	−	0.206016i
$722$		0			0
$723$	−5.36199	+	6.72372i	−0.199415	+	0.250058i
$724$		0			0
$725$	−5.92049			−0.219881
$726$		0			0
$727$	−15.6637	−	7.54325i	−0.580935	−	0.279764i	0.120249	−	0.992744i	$-0.461631\pi$
$727$	−15.6637	−	7.54325i	−0.580935	−	0.279764i	−0.701184	+	0.712980i	$0.747345\pi$
$728$		0			0
$729$	3.75733	−	16.4619i	0.139160	−	0.609702i
$730$		0			0
$731$	14.8901	+	11.8875i	0.550730	+	0.439676i
$732$		0			0
$733$	−2.34017	+	10.2530i	−0.0864363	+	0.378702i	−0.999581	−	0.0289325i	$-0.990789\pi$
$733$	−2.34017	+	10.2530i	−0.0864363	+	0.378702i	0.913145	+	0.407635i	$0.133646\pi$
$734$		0			0
$735$	−0.318973	−	0.153609i	−0.0117655	−	0.00566596i
$736$		0			0
$737$	−3.91660			−0.144270
$738$		0			0
$739$	−3.04018	+	3.81226i	−0.111835	+	0.140236i	−0.834598	−	0.550859i	$-0.814300\pi$
$739$	−3.04018	+	3.81226i	−0.111835	+	0.140236i	0.722763	+	0.691095i	$0.242872\pi$
$740$		0			0
$741$	4.45735	+	5.58934i	0.163745	+	0.205330i
$742$		0			0
$743$	8.62968	−	37.8091i	0.316592	−	1.38708i	−0.526894	−	0.849931i	$-0.676644\pi$
$743$	8.62968	−	37.8091i	0.316592	−	1.38708i	0.843486	−	0.537151i	$-0.180499\pi$
$744$		0			0
$745$	1.98836	−	8.71157i	0.0728478	−	0.319167i
$746$		0			0
$747$	−3.86050	−	4.84092i	−0.141248	−	0.177120i
$748$		0			0
$749$	2.13667	+	9.36135i	0.0780721	+	0.342056i
$750$		0			0
$751$	−0.302735	+	0.145789i	−0.0110469	+	0.00531993i	−0.439399	−	0.898292i	$-0.644809\pi$
$751$	−0.302735	+	0.145789i	−0.0110469	+	0.00531993i	0.428352	+	0.903612i	$0.359094\pi$
$752$		0			0
$753$	−2.51819	−	11.0329i	−0.0917680	−	0.402062i
$754$		0			0
$755$	−4.70435	−	2.26549i	−0.171209	−	0.0824498i
$756$		0			0
$757$	−5.73517	−	7.19168i	−0.208448	−	0.261386i	0.666606	−	0.745410i	$-0.267746\pi$
$757$	−5.73517	−	7.19168i	−0.208448	−	0.261386i	−0.875055	+	0.484024i	$0.839175\pi$
$758$		0			0
$759$	3.23151	+	1.55621i	0.117296	+	0.0564870i
$760$		0			0
$761$	−9.02659	+	11.3190i	−0.327214	+	0.410313i	−0.918041	−	0.396485i	$-0.870230\pi$
$761$	−9.02659	+	11.3190i	−0.327214	+	0.410313i	0.590828	+	0.806798i	$0.298801\pi$
$762$		0			0
$763$	−13.4070	+	6.45646i	−0.485365	+	0.233740i
$764$		0			0
$765$	−3.01942	+	1.45408i	−0.109167	+	0.0525722i
$766$		0			0
$767$	−32.3792			−1.16915
$768$		0			0
$769$	−7.04703	+	8.83670i	−0.254122	+	0.318659i	−0.892486	−	0.451076i	$-0.851040\pi$
$769$	−7.04703	+	8.83670i	−0.254122	+	0.318659i	0.638363	+	0.769735i	$0.279612\pi$
$770$		0			0
$771$	1.33390	+	5.84419i	0.0480392	+	0.210473i
$772$		0			0
$773$	18.9181			0.680438			0.340219	−	0.940346i	$-0.389499\pi$
$773$	18.9181			0.680438			0.340219	+	0.940346i	$0.389499\pi$
$774$		0			0
$775$	−7.98783			−0.286931
$776$		0			0
$777$	0.373478	+	1.63631i	0.0133984	+	0.0587024i
$778$		0			0
$779$	−10.9435	+	13.7228i	−0.392093	+	0.491669i
$780$		0			0
$781$	4.82877			0.172787
$782$		0			0
$783$	−2.85131	+	1.37312i	−0.101898	+	0.0490713i
$784$		0			0
$785$	7.47755	−	3.60100i	0.266885	−	0.128525i
$786$		0			0
$787$	12.5083	−	15.6850i	0.445874	−	0.559109i	−0.507206	−	0.861825i	$-0.669322\pi$
$787$	12.5083	−	15.6850i	0.445874	−	0.559109i	0.953081	+	0.302716i	$0.0978933\pi$
$788$		0			0
$789$	−12.3429	−	5.94404i	−0.439420	−	0.211613i
$790$		0			0
$791$	21.3137	+	26.7266i	0.757828	+	0.950287i
$792$		0			0
$793$	4.28502	+	2.06356i	0.152165	+	0.0732790i
$794$		0			0
$795$	0.448652	+	1.96567i	0.0159120	+	0.0697152i
$796$		0			0
$797$	18.1833	−	8.75663i	0.644087	−	0.310176i	−0.0831762	−	0.996535i	$-0.526506\pi$
$797$	18.1833	−	8.75663i	0.644087	−	0.310176i	0.727263	+	0.686359i	$0.240792\pi$
$798$		0			0
$799$	3.68868	+	16.1612i	0.130496	+	0.571741i
$800$		0			0
$801$	32.3132	+	40.5195i	1.14173	+	1.43169i
$802$		0			0
$803$	3.41594	−	14.9662i	0.120546	−	0.528146i
$804$		0			0
$805$	1.94898	−	8.53903i	0.0686925	−	0.300961i
$806$		0			0
$807$	7.99272	+	10.0225i	0.281357	+	0.352810i
$808$		0			0
$809$	1.60995	−	2.01881i	0.0566027	−	0.0709775i	−0.752724	−	0.658336i	$-0.771261\pi$
$809$	1.60995	−	2.01881i	0.0566027	−	0.0709775i	0.809327	+	0.587358i	$0.199832\pi$
$810$		0			0
$811$	−33.7281			−1.18435			−0.592176	−	0.805809i	$-0.701731\pi$
$811$	−33.7281			−1.18435			−0.592176	+	0.805809i	$0.701731\pi$
$812$		0			0
$813$	8.26206	+	3.97880i	0.289763	+	0.139543i
$814$		0			0
$815$	−1.26676	+	5.55006i	−0.0443728	+	0.194410i
$816$		0			0
$817$	30.9731	+	7.08690i	1.08361	+	0.247939i
$818$		0			0
$819$	−6.17778	+	27.0666i	−0.215869	+	0.945784i
$820$		0			0
$821$	33.7492	+	16.2528i	1.17785	+	0.567225i	0.917286	−	0.398230i	$-0.130375\pi$
$821$	33.7492	+	16.2528i	1.17785	+	0.567225i	0.260569	+	0.965455i	$0.416090\pi$
$822$		0			0
$823$	39.2010			1.36646			0.683231	−	0.730203i	$-0.260574\pi$
$823$	39.2010			1.36646			0.683231	+	0.730203i	$0.260574\pi$
$824$		0			0
$825$	−1.51737	+	1.90272i	−0.0528281	+	0.0662444i
$826$		0			0
$827$	−21.2263	−	26.6169i	−0.738110	−	0.925561i	0.261099	−	0.965312i	$-0.415915\pi$
$827$	−21.2263	−	26.6169i	−0.738110	−	0.925561i	−0.999210	+	0.0397510i	$0.987344\pi$
$828$		0			0
$829$	10.5589	−	46.2617i	0.366727	−	1.60673i	−0.368982	−	0.929436i	$-0.620294\pi$
$829$	10.5589	−	46.2617i	0.366727	−	1.60673i	0.735709	−	0.677298i	$-0.236849\pi$
$830$		0			0
$831$	−0.611208	+	2.67788i	−0.0212026	+	0.0928945i
$832$		0			0
$833$	3.50098	+	4.39009i	0.121302	+	0.152107i
$834$		0			0
$835$	−1.09332	−	4.79015i	−0.0378359	−	0.165770i
$836$		0			0
$837$	−3.84694	+	1.85259i	−0.132970	+	0.0640349i
$838$		0			0
$839$	5.66352	+	24.8135i	0.195526	+	0.856657i	0.973559	+	0.228434i	$0.0733605\pi$
$839$	5.66352	+	24.8135i	0.195526	+	0.856657i	−0.778033	+	0.628223i	$0.783782\pi$
$840$		0			0
$841$	24.7747	+	11.9309i	0.854299	+	0.411409i
$842$		0			0
$843$	−4.67727	−	5.86511i	−0.161094	−	0.202005i
$844$		0			0
$845$	−0.745143	−	0.358842i	−0.0256337	−	0.0123445i
$846$		0			0
$847$	18.1098	−	22.7090i	0.622261	−	0.780290i
$848$		0			0
$849$	3.37283	−	1.62427i	0.115755	−	0.0557448i
$850$		0			0
$851$	−8.09367	+	3.89771i	−0.277447	+	0.133612i
$852$		0			0
$853$	−20.8798			−0.714911			−0.357455	−	0.933930i	$-0.616356\pi$
$853$	−20.8798			−0.714911			−0.357455	+	0.933930i	$0.616356\pi$
$854$		0			0
$855$	−3.48448	+	4.36940i	−0.119167	+	0.149430i
$856$		0			0
$857$	−2.55116	−	11.1774i	−0.0871459	−	0.381811i	0.912481	−	0.409119i	$-0.134164\pi$
$857$	−2.55116	−	11.1774i	−0.0871459	−	0.381811i	−0.999627	+	0.0273075i	$0.991307\pi$
$858$		0			0
$859$	−2.00539			−0.0684230			−0.0342115	−	0.999415i	$-0.510892\pi$
$859$	−2.00539			−0.0684230			−0.0342115	+	0.999415i	$0.510892\pi$
$860$		0			0
$861$	4.81821			0.164204
$862$		0			0
$863$	0.739352	+	3.23931i	0.0251678	+	0.110268i	0.985952	−	0.167031i	$-0.0534180\pi$
$863$	0.739352	+	3.23931i	0.0251678	+	0.110268i	−0.960784	+	0.277299i	$0.910561\pi$
$864$		0			0
$865$	2.98240	−	3.73981i	0.101405	−	0.127157i
$866$		0			0
$867$	3.80844			0.129341
$868$		0			0
$869$	−3.66877	+	1.76679i	−0.124455	+	0.0599342i
$870$		0			0
$871$	−10.3340	+	4.97661i	−0.350155	+	0.168626i
$872$		0			0
$873$	16.6205	−	20.8414i	0.562518	−	0.705375i
$874$		0			0
$875$	10.8967	+	5.24756i	0.368375	+	0.177400i
$876$		0			0
$877$	20.6831	+	25.9358i	0.698420	+	0.875791i	0.996904	−	0.0786233i	$-0.0250524\pi$
$877$	20.6831	+	25.9358i	0.698420	+	0.875791i	−0.298484	+	0.954415i	$0.596481\pi$
$878$		0			0
$879$	−1.83786	−	0.885069i	−0.0619896	−	0.0298526i
$880$		0			0
$881$	−1.42119	−	6.22663i	−0.0478810	−	0.209781i	0.945329	−	0.326120i	$-0.105741\pi$
$881$	−1.42119	−	6.22663i	−0.0478810	−	0.209781i	−0.993210	+	0.116339i	$0.962884\pi$
$882$		0			0
$883$	−37.5787	+	18.0969i	−1.26462	+	0.609011i	−0.941394	−	0.337310i	$-0.890483\pi$
$883$	−37.5787	+	18.0969i	−1.26462	+	0.609011i	−0.323230	+	0.946321i	$0.604769\pi$
$884$		0			0
$885$	0.398146	+	1.74439i	0.0133835	+	0.0586370i
$886$		0			0
$887$	17.5493	+	22.0061i	0.589248	+	0.738893i	0.983659	−	0.180041i	$-0.0576229\pi$
$887$	17.5493	+	22.0061i	0.589248	+	0.738893i	−0.394411	+	0.918934i	$0.629051\pi$
$888$		0			0
$889$	−6.33851	+	27.7708i	−0.212587	+	0.931403i
$890$		0			0
$891$	1.82799	−	8.00894i	0.0612399	−	0.268309i
$892$		0			0
$893$	17.2355	+	21.6127i	0.576766	+	0.723241i
$894$		0			0
$895$	5.27765	−	6.61796i	0.176412	−	0.221214i
$896$		0			0
$897$	10.5038			0.350712
$898$		0			0
$899$	−1.82601	−	0.879359i	−0.0609008	−	0.0293283i
$900$		0			0
$901$	7.11582	−	31.1765i	0.237062	−	1.03864i
$902$		0			0
$903$	−3.78017	−	7.86039i	−0.125796	−	0.261577i
$904$		0			0
$905$	−1.81871	+	7.96829i	−0.0604560	+	0.264875i
$906$		0			0
$907$	−40.5603	−	19.5328i	−1.34678	−	0.648577i	−0.385134	−	0.922861i	$-0.625845\pi$
$907$	−40.5603	−	19.5328i	−1.34678	−	0.648577i	−0.961649	+	0.274284i	$0.911559\pi$
$908$		0			0
$909$	−11.0606			−0.366857
$910$		0			0
$911$	14.7285	−	18.4690i	0.487977	−	0.611904i	−0.475493	−	0.879719i	$-0.657730\pi$
$911$	14.7285	−	18.4690i	0.487977	−	0.611904i	0.963470	+	0.267815i	$0.0863018\pi$
$912$		0			0
$913$	−1.55973	−	1.95584i	−0.0516196	−	0.0647290i
$914$		0			0
$915$	0.0584815	−	0.256224i	0.00193334	−	0.00847050i
$916$		0			0
$917$	9.98632	−	43.7529i	0.329777	−	1.44485i
$918$		0			0
$919$	18.5309	+	23.2370i	0.611278	+	0.766519i	0.987088	−	0.160178i	$-0.0512069\pi$
$919$	18.5309	+	23.2370i	0.611278	+	0.766519i	−0.375810	+	0.926697i	$0.622635\pi$
$920$		0			0
$921$	−0.179925	−	0.788305i	−0.00592874	−	0.0259755i
$922$		0			0
$923$	12.7408	−	6.13565i	0.419369	−	0.201957i
$924$		0			0
$925$	−1.35635	−	5.94258i	−0.0445967	−	0.195391i
$926$		0			0
$927$	5.97634	+	2.87806i	0.196289	+	0.0945278i
$928$		0			0
$929$	−2.06382	−	2.58795i	−0.0677119	−	0.0849080i	0.746822	−	0.665024i	$-0.231579\pi$
$929$	−2.06382	−	2.58795i	−0.0677119	−	0.0849080i	−0.814534	+	0.580116i	$0.803007\pi$
$930$		0			0
$931$	8.43655	+	4.06283i	0.276497	+	0.133154i
$932$		0			0
$933$	−4.99245	+	6.26033i	−0.163445	+	0.204954i
$934$		0			0
$935$	−1.21992	+	0.587481i	−0.0398955	+	0.0192127i
$936$		0			0
$937$	−1.03891	+	0.500311i	−0.0339396	+	0.0163444i	−0.450777	−	0.892637i	$-0.648853\pi$
$937$	−1.03891	+	0.500311i	−0.0339396	+	0.0163444i	0.416837	+	0.908981i	$0.363139\pi$
$938$		0			0
$939$	11.8047			0.385233
$940$		0			0
$941$	−28.5224	+	35.7660i	−0.929804	+	1.16594i	0.0560669	+	0.998427i	$0.482144\pi$
$941$	−28.5224	+	35.7660i	−0.929804	+	1.16594i	−0.985870	+	0.167510i	$0.946427\pi$
$942$		0			0
$943$	5.73850	+	25.1420i	0.186871	+	0.818737i
$944$		0			0
$945$	3.17673			0.103339
$946$		0			0
$947$	38.1369			1.23928			0.619642	−	0.784885i	$-0.287278\pi$
$947$	38.1369			1.23928			0.619642	+	0.784885i	$0.287278\pi$
$948$		0			0
$949$	−10.0037	−	43.8291i	−0.324734	−	1.42275i
$950$		0			0
$951$	−8.26800	+	10.3677i	−0.268108	+	0.336197i
$952$		0			0
$953$	21.7434			0.704337			0.352168	−	0.935937i	$-0.385444\pi$
$953$	21.7434			0.704337			0.352168	+	0.935937i	$0.385444\pi$
$954$		0			0
$955$	2.76414	−	1.33114i	0.0894456	−	0.0430747i
$956$		0			0
$957$	−0.556335	+	0.267917i	−0.0179838	+	0.00866052i
$958$		0			0
$959$	4.47443	−	5.61076i	0.144487	−	0.181181i
$960$		0			0
$961$	25.4664	+	12.2640i	0.821497	+	0.395612i
$962$		0			0
$963$	−5.61264	−	7.03802i	−0.180865	−	0.226797i
$964$		0			0
$965$	−6.18502	−	2.97855i	−0.199103	−	0.0958829i
$966$		0			0
$967$	−4.70961	−	20.6342i	−0.151451	−	0.663550i	−0.992464	−	0.122535i	$-0.960897\pi$
$967$	−4.70961	−	20.6342i	−0.151451	−	0.663550i	0.841013	−	0.541015i	$-0.181960\pi$
$968$		0			0
$969$	−5.64518	+	2.71857i	−0.181349	+	0.0873332i
$970$		0			0
$971$	7.19043	+	31.5033i	0.230752	+	1.01099i	0.949018	+	0.315222i	$0.102079\pi$
$971$	7.19043	+	31.5033i	0.230752	+	1.01099i	−0.718266	+	0.695768i	$0.755064\pi$
$972$		0			0
$973$	34.7791	+	43.6116i	1.11497	+	1.39812i
$974$		0			0
$975$	−1.58593	+	6.94842i	−0.0507904	+	0.222527i
$976$		0			0
$977$	−6.66896	+	29.2186i	−0.213359	+	0.934786i	0.748907	+	0.662675i	$0.230579\pi$
$977$	−6.66896	+	29.2186i	−0.213359	+	0.934786i	−0.962266	+	0.272111i	$0.912278\pi$
$978$		0			0
$979$	13.0553	+	16.3708i	0.417249	+	0.523214i
$980$		0			0
$981$	8.69805	−	10.9070i	0.277707	−	0.348234i
$982$		0			0
$983$	8.59558			0.274156			0.137078	−	0.990560i	$-0.456229\pi$
$983$	8.59558			0.274156			0.137078	+	0.990560i	$0.456229\pi$
$984$		0			0
$985$	3.35283	+	1.61464i	0.106830	+	0.0514466i
$986$		0			0
$987$	1.68859	−	7.39818i	0.0537483	−	0.235487i
$988$		0			0
$989$	36.5143	−	29.0871i	1.16109	−	0.924917i
$990$		0			0
$991$	−5.25773	+	23.0356i	−0.167017	+	0.731750i	0.820162	+	0.572132i	$0.193884\pi$
$991$	−5.25773	+	23.0356i	−0.167017	+	0.731750i	−0.987179	+	0.159618i	$0.948974\pi$
$992$		0			0
$993$	−2.62740	−	1.26529i	−0.0833779	−	0.0401527i
$994$		0			0
$995$	−1.79997			−0.0570627
$996$		0			0
$997$	−21.0278	+	26.3681i	−0.665958	+	0.835085i	−0.993977	−	0.109586i	$-0.965047\pi$
$997$	−21.0278	+	26.3681i	−0.665958	+	0.835085i	0.328019	+	0.944671i	$0.393619\pi$
$998$		0			0
$999$	−2.03146	−	2.54737i	−0.0642727	−	0.0805954i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.u.e.465.1		12
4.3	odd	2		172.2.i.b.121.1	✓	12
43.16	even	7	inner	688.2.u.e.145.1		12
172.39	even	14		7396.2.a.i.1.4		6
172.47	odd	14		7396.2.a.j.1.3		6
172.59	odd	14		172.2.i.b.145.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
172.2.i.b.121.1	✓	12	4.3	odd	2
172.2.i.b.145.1	yes	12	172.59	odd	14
688.2.u.e.145.1		12	43.16	even	7	inner
688.2.u.e.465.1		12	1.1	even	1	trivial
7396.2.a.i.1.4		6	172.39	even	14
7396.2.a.j.1.3		6	172.47	odd	14

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.u (of order \(7\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0990311 + 0.433884i) q^{3} +(-0.256654 + 0.321834i) q^{5} -2.98873 q^{7} +(2.52446 - 1.21572i) q^{9} +O(q^{10})\)	\(q+(0.0990311 + 0.433884i) q^{3} +(-0.256654 + 0.321834i) q^{5} -2.98873 q^{7} +(2.52446 - 1.21572i) q^{9} +(1.01994 - 0.491178i) q^{11} +(2.06702 - 2.59196i) q^{13} +(-0.165055 - 0.0794865i) q^{15} +(1.81161 + 2.27169i) q^{17} +(4.36557 + 2.10235i) q^{19} +(-0.295978 - 1.29676i) q^{21} +(6.41416 - 3.08890i) q^{23} +(1.07490 + 4.70944i) q^{25} +(1.60992 + 2.01877i) q^{27} +(-0.272729 + 1.19490i) q^{29} +(-0.367962 + 1.61215i) q^{31} +(0.314120 + 0.393894i) q^{33} +(0.767071 - 0.961877i) q^{35} -1.26184 q^{37} +(1.32931 + 0.640162i) q^{39} +(-0.806062 + 3.53159i) q^{41} +(6.39381 - 1.45574i) q^{43} +(-0.256654 + 1.12448i) q^{45} +(5.14013 + 2.47536i) q^{47} +1.93252 q^{49} +(-0.806243 + 1.01100i) q^{51} +(-6.86195 - 8.60461i) q^{53} +(-0.103694 + 0.454315i) q^{55} +(-0.479847 + 2.10235i) q^{57} +(-6.08948 - 7.63597i) q^{59} +(0.319226 + 1.39862i) q^{61} +(-7.54493 + 3.63345i) q^{63} +(0.303673 + 1.33048i) q^{65} +(-3.11712 - 1.50113i) q^{67} +(1.97542 + 2.47710i) q^{69} +(3.84310 + 1.85074i) q^{71} +(8.45481 - 10.6020i) q^{73} +(-1.93690 + 0.932762i) q^{75} +(-3.04833 + 1.46800i) q^{77} -3.59704 q^{79} +(4.52446 - 5.67349i) q^{81} +(-0.491730 - 2.15441i) q^{83} -1.19607 q^{85} -0.545458 q^{87} +(4.11589 + 18.0329i) q^{89} +(-6.17778 + 7.74669i) q^{91} -0.735924 q^{93} +(-1.79705 + 0.865414i) q^{95} +(8.57167 - 4.12790i) q^{97} +(1.97767 - 2.47991i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q + 10 * q^3 + 3 * q^5 + 12 * q^9	\( 12 q + 10 q^{3} + 3 q^{5} + 12 q^{9} - q^{11} - 4 q^{13} - q^{15} + 17 q^{17} - 13 q^{19} - 7 q^{21} + 14 q^{23} + q^{25} + 22 q^{27} - 6 q^{29} + 23 q^{31} - 2 q^{33} + 16 q^{35} - 24 q^{37} - q^{39} + 8 q^{41} + 7 q^{43} + 3 q^{45} - 22 q^{47} + 24 q^{49} + 13 q^{51} + 27 q^{53} + 29 q^{55} - 19 q^{57} + 17 q^{59} + 8 q^{61} + 7 q^{63} + 29 q^{65} - 29 q^{67} + 17 q^{71} - 10 q^{73} - 5 q^{75} + 15 q^{77} - 14 q^{79} + 36 q^{81} + 6 q^{83} - 70 q^{85} - 12 q^{87} - 16 q^{89} - 46 q^{91} + 46 q^{93} - 67 q^{95} + 10 q^{97} - q^{99}+O(q^{100}) \) 12 * q + 10 * q^3 + 3 * q^5 + 12 * q^9 - q^11 - 4 * q^13 - q^15 + 17 * q^17 - 13 * q^19 - 7 * q^21 + 14 * q^23 + q^25 + 22 * q^27 - 6 * q^29 + 23 * q^31 - 2 * q^33 + 16 * q^35 - 24 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 3 * q^45 - 22 * q^47 + 24 * q^49 + 13 * q^51 + 27 * q^53 + 29 * q^55 - 19 * q^57 + 17 * q^59 + 8 * q^61 + 7 * q^63 + 29 * q^65 - 29 * q^67 + 17 * q^71 - 10 * q^73 - 5 * q^75 + 15 * q^77 - 14 * q^79 + 36 * q^81 + 6 * q^83 - 70 * q^85 - 12 * q^87 - 16 * q^89 - 46 * q^91 + 46 * q^93 - 67 * q^95 + 10 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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