LMFDB - Embedded newform 688.2.bg.c.273.1

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			273.1
Character	$\chi$	$=$	688.273
Dual form			688.2.bg.c.625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.922165 + 2.34964i) q^{3} +(-0.0373507 - 0.498411i) q^{5} +(1.65334 - 2.86367i) q^{7} +(-2.47126 - 2.29299i) q^{9} +O(q^{10})$	\(q+(-0.922165 + 2.34964i) q^{3} +(-0.0373507 - 0.498411i) q^{5} +(1.65334 - 2.86367i) q^{7} +(-2.47126 - 2.29299i) q^{9} +(-0.828392 + 3.62942i) q^{11} +(-3.87254 + 2.64025i) q^{13} +(1.20553 + 0.371857i) q^{15} +(-0.343032 + 4.57744i) q^{17} +(-4.44115 + 4.12079i) q^{19} +(5.20394 + 6.52553i) q^{21} +(0.371666 - 0.114644i) q^{23} +(4.69714 - 0.707979i) q^{25} +(0.844152 - 0.406522i) q^{27} +(-1.29147 - 3.29062i) q^{29} +(-0.861113 - 0.129792i) q^{31} +(-7.76391 - 5.29335i) q^{33} +(-1.48904 - 0.717083i) q^{35} +(-1.96656 - 3.40619i) q^{37} +(-2.63252 - 11.5338i) q^{39} +(-6.72350 + 8.43100i) q^{41} +(-6.26851 + 1.92504i) q^{43} +(-1.05055 + 1.31735i) q^{45} +(0.776794 + 3.40336i) q^{47} +(-1.96707 - 3.40706i) q^{49} +(-10.4390 - 5.02716i) q^{51} +(3.94414 + 2.68907i) q^{53} +(1.83988 + 0.277318i) q^{55} +(-5.58689 - 14.2352i) q^{57} +(6.05684 - 2.91682i) q^{59} +(-5.18836 + 0.782019i) q^{61} +(-10.6522 + 3.28577i) q^{63} +(1.46057 + 1.83150i) q^{65} +(7.73611 - 7.17806i) q^{67} +(-0.0733661 + 0.979002i) q^{69} +(-0.859578 - 0.265145i) q^{71} +(-0.798611 + 0.544484i) q^{73} +(-2.66804 + 11.6894i) q^{75} +(9.02385 + 8.37291i) q^{77} +(-0.500983 + 0.867728i) q^{79} +(-0.579056 - 7.72697i) q^{81} +(1.24798 - 3.17979i) q^{83} +2.29426 q^{85} +8.92272 q^{87} +(-0.672465 + 1.71341i) q^{89} +(1.15819 + 15.4549i) q^{91} +(1.09905 - 1.90361i) q^{93} +(2.21973 + 2.05961i) q^{95} +(-0.213753 + 0.936511i) q^{97} +(10.3694 - 7.06974i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{13}{21}\right)$	$1$

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
$3$	−0.922165	+	2.34964i	−0.532412	+	1.35656i	0.371335	+	0.928499i	$0.378900\pi$
$3$	−0.922165	+	2.34964i	−0.532412	+	1.35656i	−0.903748	+	0.428066i	$0.859195\pi$
$4$		0			0
$5$	−0.0373507	−	0.498411i	−0.0167038	−	0.222896i	−0.999308	−	0.0371862i	$-0.988161\pi$
$5$	−0.0373507	−	0.498411i	−0.0167038	−	0.222896i	0.982605	−	0.185710i	$-0.0594585\pi$
$6$		0			0
$7$	1.65334	−	2.86367i	0.624904	−	1.08237i	−0.363655	−	0.931534i	$-0.618471\pi$
$7$	1.65334	−	2.86367i	0.624904	−	1.08237i	0.988559	−	0.150832i	$-0.0481952\pi$
$8$		0			0
$9$	−2.47126	−	2.29299i	−0.823753	−	0.764331i
$10$		0			0
$11$	−0.828392	+	3.62942i	−0.249769	+	1.09431i	0.682026	+	0.731328i	$0.261099\pi$
$11$	−0.828392	+	3.62942i	−0.249769	+	1.09431i	−0.931796	+	0.362984i	$0.881758\pi$
$12$		0			0
$13$	−3.87254	+	2.64025i	−1.07405	+	0.732274i	−0.965129	−	0.261776i	$-0.915692\pi$
$13$	−3.87254	+	2.64025i	−1.07405	+	0.732274i	−0.108921	+	0.994050i	$0.534739\pi$
$14$		0			0
$15$	1.20553	+	0.371857i	0.311266	+	0.0960130i
$16$		0			0
$17$	−0.343032	+	4.57744i	−0.0831975	+	1.11019i	0.786823	+	0.617178i	$0.211724\pi$
$17$	−0.343032	+	4.57744i	−0.0831975	+	1.11019i	−0.870021	+	0.493015i	$0.835895\pi$
$18$		0			0
$19$	−4.44115	+	4.12079i	−1.01887	+	0.945374i	−0.998518	−	0.0544191i	$-0.982669\pi$
$19$	−4.44115	+	4.12079i	−1.01887	+	0.945374i	−0.0203524	+	0.999793i	$0.506479\pi$
$20$		0			0
$21$	5.20394	+	6.52553i	1.13559	+	1.42399i
$22$		0			0
$23$	0.371666	−	0.114644i	0.0774978	−	0.0239049i	−0.255764	−	0.966739i	$-0.582327\pi$
$23$	0.371666	−	0.114644i	0.0774978	−	0.0239049i	0.333262	+	0.942834i	$0.391851\pi$
$24$		0			0
$25$	4.69714	−	0.707979i	0.939427	−	0.141596i
$26$		0			0
$27$	0.844152	−	0.406522i	0.162457	−	0.0782352i
$28$		0			0
$29$	−1.29147	−	3.29062i	−0.239820	−	0.611053i	0.759396	−	0.650629i	$-0.225495\pi$
$29$	−1.29147	−	3.29062i	−0.239820	−	0.611053i	−0.999217	+	0.0395760i	$0.987399\pi$
$30$		0			0
$31$	−0.861113	−	0.129792i	−0.154660	−	0.0233113i	0.0712556	−	0.997458i	$-0.477299\pi$
$31$	−0.861113	−	0.129792i	−0.154660	−	0.0233113i	−0.225916	+	0.974147i	$0.572537\pi$
$32$		0			0
$33$	−7.76391	−	5.29335i	−1.35152	−	0.921454i
$34$		0			0
$35$	−1.48904	−	0.717083i	−0.251693	−	0.121209i
$36$		0			0
$37$	−1.96656	−	3.40619i	−0.323301	−	0.559974i	0.657866	−	0.753135i	$-0.271459\pi$
$37$	−1.96656	−	3.40619i	−0.323301	−	0.559974i	−0.981167	+	0.193161i	$0.938126\pi$
$38$		0			0
$39$	−2.63252	−	11.5338i	−0.421540	−	1.84689i
$40$		0			0
$41$	−6.72350	+	8.43100i	−1.05003	+	1.31670i	−0.103313	+	0.994649i	$0.532944\pi$
$41$	−6.72350	+	8.43100i	−1.05003	+	1.31670i	−0.946722	+	0.322053i	$0.895627\pi$
$42$		0			0
$43$	−6.26851	+	1.92504i	−0.955939	+	0.293566i
$43$	−6.26851	+	1.92504i	−0.955939	+	0.293566i
$44$		0			0
$45$	−1.05055	+	1.31735i	−0.156607	+	0.196379i
$46$		0			0
$47$	0.776794	+	3.40336i	0.113307	+	0.496430i	0.999454	+	0.0330270i	$0.0105147\pi$
$47$	0.776794	+	3.40336i	0.113307	+	0.496430i	−0.886147	+	0.463403i	$0.846628\pi$
$48$		0			0
$49$	−1.96707	−	3.40706i	−0.281010	−	0.486724i
$50$		0			0
$51$	−10.4390	−	5.02716i	−1.46175	−	0.703943i
$52$		0			0
$53$	3.94414	+	2.68907i	0.541770	+	0.369372i	0.803069	−	0.595885i	$-0.203199\pi$
$53$	3.94414	+	2.68907i	0.541770	+	0.369372i	−0.261300	+	0.965258i	$0.584151\pi$
$54$		0			0
$55$	1.83988	+	0.277318i	0.248090	+	0.0373935i
$56$		0			0
$57$	−5.58689	−	14.2352i	−0.740001	−	1.88549i
$58$		0			0
$59$	6.05684	−	2.91682i	0.788534	−	0.379738i	0.00413260	−	0.999991i	$-0.498685\pi$
$59$	6.05684	−	2.91682i	0.788534	−	0.379738i	0.784401	+	0.620253i	$0.212970\pi$
$60$		0			0
$61$	−5.18836	+	0.782019i	−0.664301	+	0.100127i	−0.472533	−	0.881313i	$-0.656660\pi$
$61$	−5.18836	+	0.782019i	−0.664301	+	0.100127i	−0.191768	+	0.981440i	$0.561422\pi$
$62$		0			0
$63$	−10.6522	+	3.28577i	−1.34205	+	0.413968i
$64$		0			0
$65$	1.46057	+	1.83150i	0.181162	+	0.227170i
$66$		0			0
$67$	7.73611	−	7.17806i	0.945116	−	0.876940i	−0.0474219	−	0.998875i	$-0.515101\pi$
$67$	7.73611	−	7.17806i	0.945116	−	0.876940i	0.992538	+	0.121935i	$0.0389101\pi$
$68$		0			0
$69$	−0.0733661	+	0.979002i	−0.00883224	+	0.117858i
$70$		0			0
$71$	−0.859578	−	0.265145i	−0.102013	−	0.0314669i	0.243328	−	0.969944i	$-0.421761\pi$
$71$	−0.859578	−	0.265145i	−0.102013	−	0.0314669i	−0.345342	+	0.938477i	$0.612237\pi$
$72$		0			0
$73$	−0.798611	+	0.544484i	−0.0934704	+	0.0637270i	−0.609153	−	0.793052i	$-0.708491\pi$
$73$	−0.798611	+	0.544484i	−0.0934704	+	0.0637270i	0.515683	+	0.856779i	$0.327538\pi$
$74$		0			0
$75$	−2.66804	+	11.6894i	−0.308079	+	1.34978i
$76$		0			0
$77$	9.02385	+	8.37291i	1.02836	+	0.954181i
$78$		0			0
$79$	−0.500983	+	0.867728i	−0.0563650	+	0.0976270i	−0.892831	−	0.450392i	$-0.851284\pi$
$79$	−0.500983	+	0.867728i	−0.0563650	+	0.0976270i	0.836466	+	0.548019i	$0.184618\pi$
$80$		0			0
$81$	−0.579056	−	7.72697i	−0.0643396	−	0.858552i
$82$		0			0
$83$	1.24798	−	3.17979i	0.136983	−	0.349027i	−0.845878	−	0.533377i	$-0.820923\pi$
$83$	1.24798	−	3.17979i	0.136983	−	0.349027i	0.982861	+	0.184350i	$0.0590179\pi$
$84$		0			0
$85$	2.29426			0.248847
$86$		0			0
$87$	8.92272			0.956616
$88$		0			0
$89$	−0.672465	+	1.71341i	−0.0712811	+	0.181621i	−0.962074	−	0.272788i	$-0.912054\pi$
$89$	−0.672465	+	1.71341i	−0.0712811	+	0.181621i	0.890793	+	0.454409i	$0.150150\pi$
$90$		0			0
$91$	1.15819	+	15.4549i	0.121411	+	1.62012i
$92$		0			0
$93$	1.09905	−	1.90361i	0.113966	−	0.197396i
$94$		0			0
$95$	2.21973	+	2.05961i	0.227739	+	0.211311i
$96$		0			0
$97$	−0.213753	+	0.936511i	−0.0217033	+	0.0950883i	−0.984620	−	0.174712i	$-0.944101\pi$
$97$	−0.213753	+	0.936511i	−0.0217033	+	0.0950883i	0.962916	+	0.269800i	$0.0869577\pi$
$98$		0			0
$99$	10.3694	−	7.06974i	1.04216	−	0.710536i
$100$		0			0
$101$	15.7085	+	4.84543i	1.56305	+	0.482138i	0.951133	−	0.308781i	$-0.0999210\pi$
$101$	15.7085	+	4.84543i	1.56305	+	0.482138i	0.611921	+	0.790919i	$0.290397\pi$
$102$		0			0
$103$	0.0427552	−	0.570528i	0.00421279	−	0.0562158i	−0.994701	−	0.102808i	$-0.967217\pi$
$103$	0.0427552	−	0.570528i	0.00421279	−	0.0562158i	0.998914	+	0.0465925i	$0.0148362\pi$
$104$		0			0
$105$	3.05803	−	2.83743i	0.298433	−	0.276905i
$106$		0			0
$107$	10.0832	+	12.6440i	0.974785	+	1.22234i	0.974969	+	0.222341i	$0.0713699\pi$
$107$	10.0832	+	12.6440i	0.974785	+	1.22234i	−0.000184398		1.00000i	$0.500059\pi$
$108$		0			0
$109$	−6.23712	+	1.92390i	−0.597407	+	0.184276i	−0.578688	−	0.815549i	$-0.696435\pi$
$109$	−6.23712	+	1.92390i	−0.597407	+	0.184276i	−0.0187194	+	0.999825i	$0.505959\pi$
$110$		0			0
$111$	9.81680	−	1.47965i	0.931770	−	0.140442i
$112$		0			0
$113$	1.53214	−	0.737838i	0.144131	−	0.0694100i	−0.360428	−	0.932787i	$-0.617369\pi$
$113$	1.53214	−	0.737838i	0.144131	−	0.0694100i	0.504559	+	0.863377i	$0.331655\pi$
$114$		0			0
$115$	−0.0710218	−	0.180961i	−0.00662281	−	0.0168747i
$116$		0			0
$117$	15.6241	+	2.35496i	1.44445	+	0.217716i
$118$		0			0
$119$	12.5411	+	8.55040i	1.14964	+	0.783814i
$120$		0			0
$121$	−2.57580	−	1.24044i	−0.234164	−	0.112767i
$122$		0			0
$123$	−13.6096	−	23.5726i	−1.22714	−	2.12547i
$124$		0			0
$125$	−1.08440	−	4.75105i	−0.0969913	−	0.424947i
$126$		0			0
$127$	−9.78615	+	12.2714i	−0.868380	+	1.08891i	0.126904	+	0.991915i	$0.459496\pi$
$127$	−9.78615	+	12.2714i	−0.868380	+	1.08891i	−0.995284	+	0.0969996i	$0.969075\pi$
$128$		0			0
$129$	1.25745	−	16.5039i	0.110712	−	1.45309i
$130$		0			0
$131$	6.68767	−	8.38607i	0.584304	−	0.732694i	−0.398536	−	0.917153i	$-0.630482\pi$
$131$	6.68767	−	8.38607i	0.584304	−	0.732694i	0.982840	+	0.184458i	$0.0590531\pi$
$132$		0			0
$133$	4.45784	+	19.5311i	0.386544	+	1.69356i
$134$		0			0
$135$	−0.234145	−	0.405551i	−0.0201520	−	0.0349042i
$136$		0			0
$137$	7.28496	+	3.50825i	0.622396	+	0.299730i	0.718378	−	0.695653i	$-0.244885\pi$
$137$	7.28496	+	3.50825i	0.622396	+	0.299730i	−0.0959821	+	0.995383i	$0.530599\pi$
$138$		0			0
$139$	−10.6300	−	7.24742i	−0.901625	−	0.614718i	0.0212677	−	0.999774i	$-0.493230\pi$
$139$	−10.6300	−	7.24742i	−0.901625	−	0.614718i	−0.922893	+	0.385056i	$0.874182\pi$
$140$		0			0
$141$	−8.71299	−	1.31327i	−0.733766	−	0.110597i
$142$		0			0
$143$	−6.37461	−	16.2422i	−0.533072	−	1.35824i
$144$		0			0
$145$	−1.59184	+	0.766591i	−0.132195	+	0.0636620i
$146$		0			0
$147$	9.81934	−	1.48003i	0.809885	−	0.122071i
$148$		0			0
$149$	−7.10954	+	2.19300i	−0.582436	+	0.179658i	−0.571952	−	0.820287i	$-0.693813\pi$
$149$	−7.10954	+	2.19300i	−0.582436	+	0.179658i	−0.0104843	+	0.999945i	$0.503337\pi$
$150$		0			0
$151$	2.97754	+	3.73371i	0.242309	+	0.303845i	0.888083	−	0.459683i	$-0.152037\pi$
$151$	2.97754	+	3.73371i	0.242309	+	0.303845i	−0.645775	+	0.763528i	$0.723465\pi$
$152$		0			0
$153$	11.3438	−	10.5255i	0.917089	−	0.850935i
$154$		0			0
$155$	−0.0325265	+	0.434036i	−0.00261259	+	0.0348626i
$156$		0			0
$157$	12.8438	+	3.96180i	1.02505	+	0.316186i	0.761297	−	0.648404i	$-0.224563\pi$
$157$	12.8438	+	3.96180i	1.02505	+	0.316186i	0.263754	+	0.964590i	$0.415039\pi$
$158$		0			0
$159$	−9.95550	+	6.78754i	−0.789522	+	0.538287i
$160$		0			0
$161$	0.286189	−	1.25387i	0.0225548	−	0.0988192i
$162$		0			0
$163$	13.7295	+	12.7392i	1.07538	+	0.997808i	1.00000		0.000653135i	$0.000207899\pi$
$163$	13.7295	+	12.7392i	1.07538	+	0.997808i	0.0753814	+	0.997155i	$0.475983\pi$
$164$		0			0
$165$	−2.34827	+	4.06733i	−0.182813	+	0.316641i
$166$		0			0
$167$	1.03540	+	13.8165i	0.0801219	+	1.06915i	0.881871	+	0.471491i	$0.156284\pi$
$167$	1.03540	+	13.8165i	0.0801219	+	1.06915i	−0.801749	+	0.597661i	$0.796097\pi$
$168$		0			0
$169$	3.27620	−	8.34761i	0.252015	−	0.642124i
$170$		0			0
$171$	20.4242			1.56188
$172$		0			0
$173$	−13.8705			−1.05456			−0.527278	−	0.849693i	$-0.676787\pi$
$173$	−13.8705			−1.05456			−0.527278	+	0.849693i	$0.676787\pi$
$174$		0			0
$175$	5.73855	−	14.6216i	0.433793	−	1.10529i
$176$		0			0
$177$	1.26807	+	16.9212i	0.0953138	+	1.27187i
$178$		0			0
$179$	3.09219	−	5.35583i	0.231121	−	0.400313i	−0.727017	−	0.686619i	$-0.759094\pi$
$179$	3.09219	−	5.35583i	0.231121	−	0.400313i	0.958138	+	0.286306i	$0.0924273\pi$
$180$		0			0
$181$	−2.71431	−	2.51851i	−0.201753	−	0.187200i	0.572825	−	0.819678i	$-0.305847\pi$
$181$	−2.71431	−	2.51851i	−0.201753	−	0.187200i	−0.774578	+	0.632478i	$0.782038\pi$
$182$		0			0
$183$	2.94706	−	12.9119i	0.217853	−	0.954477i
$184$		0			0
$185$	−1.62423	+	1.10738i	−0.119416	+	0.0814162i
$186$		0			0
$187$	−16.3293	−	5.03692i	−1.19412	−	0.368336i
$188$		0			0
$189$	0.231525	−	3.08949i	0.0168410	−	0.224727i
$190$		0			0
$191$	8.36809	−	7.76446i	0.605494	−	0.561816i	−0.316709	−	0.948523i	$-0.602578\pi$
$191$	8.36809	−	7.76446i	0.605494	−	0.561816i	0.922203	+	0.386706i	$0.126387\pi$
$192$		0			0
$193$	−5.26831	−	6.60625i	−0.379221	−	0.475528i	0.555191	−	0.831723i	$-0.312645\pi$
$193$	−5.26831	−	6.60625i	−0.379221	−	0.475528i	−0.934412	+	0.356195i	$0.884074\pi$
$194$		0			0
$195$	−5.65026	+	1.74287i	−0.404623	+	0.124810i
$196$		0			0
$197$	21.4229	−	3.22898i	1.52632	−	0.230056i	0.668385	−	0.743816i	$-0.266986\pi$
$197$	21.4229	−	3.22898i	1.52632	−	0.230056i	0.857934	+	0.513760i	$0.171748\pi$
$198$		0			0
$199$	−8.90021	+	4.28611i	−0.630919	+	0.303835i	−0.721878	−	0.692020i	$-0.756721\pi$
$199$	−8.90021	+	4.28611i	−0.630919	+	0.303835i	0.0909593	+	0.995855i	$0.471007\pi$
$200$		0			0
$201$	9.73188	+	24.7964i	0.686434	+	1.74900i
$202$		0			0
$203$	−11.5585	−	1.74216i	−0.811247	−	0.122276i
$204$		0			0
$205$	4.45323	+	3.03616i	0.311027	+	0.212055i
$206$		0			0
$207$	−1.18136	−	0.568914i	−0.0821103	−	0.0395422i
$208$		0			0
$209$	−11.2771	−	19.5324i	−0.780051	−	1.35109i
$210$		0			0
$211$	−3.40455	−	14.9163i	−0.234379	−	1.02688i	−0.945962	−	0.324278i	$-0.894879\pi$
$211$	−3.40455	−	14.9163i	−0.234379	−	1.02688i	0.711583	−	0.702602i	$-0.247978\pi$
$212$		0			0
$213$	1.41567	−	1.77519i	0.0969999	−	0.121634i
$214$		0			0
$215$	1.19360	+	3.05239i	0.0814025	+	0.208171i
$216$		0			0
$217$	−1.79539	+	2.25135i	−0.121879	+	0.152832i
$218$		0			0
$219$	−0.542889	−	2.37855i	−0.0366850	−	0.160728i
$220$		0			0
$221$	−10.7572	−	18.6320i	−0.723608	−	1.25333i
$222$		0			0
$223$	10.5564	+	5.08368i	0.706907	+	0.340428i	0.752555	−	0.658529i	$-0.228821\pi$
$223$	10.5564	+	5.08368i	0.706907	+	0.340428i	−0.0456482	+	0.998958i	$0.514535\pi$
$224$		0			0
$225$	−13.2312	−	9.02090i	−0.882082	−	0.601394i
$226$		0			0
$227$	8.93878	+	1.34730i	0.593288	+	0.0894238i	0.438821	−	0.898574i	$-0.355396\pi$
$227$	8.93878	+	1.34730i	0.593288	+	0.0894238i	0.154466	+	0.987998i	$0.450634\pi$
$228$		0			0
$229$	−5.69852	−	14.5196i	−0.376569	−	0.959482i	−0.986015	−	0.166655i	$-0.946704\pi$
$229$	−5.69852	−	14.5196i	−0.376569	−	0.959482i	0.609446	−	0.792827i	$-0.291392\pi$
$230$		0			0
$231$	−27.9948	+	13.4816i	−1.84192	+	0.887023i
$232$		0			0
$233$	4.64471	−	0.700078i	0.304285	−	0.0458636i	0.00487511	−	0.999988i	$-0.498448\pi$
$233$	4.64471	−	0.700078i	0.304285	−	0.0458636i	0.299410	+	0.954124i	$0.403210\pi$
$234$		0			0
$235$	1.66726	−	0.514280i	0.108760	−	0.0335480i
$236$		0			0
$237$	−1.57686	−	1.97732i	−0.102428	−	0.128441i
$238$		0			0
$239$	−3.01626	+	2.79868i	−0.195106	+	0.181032i	−0.771676	−	0.636016i	$-0.780581\pi$
$239$	−3.01626	+	2.79868i	−0.195106	+	0.181032i	0.576570	+	0.817048i	$0.304391\pi$
$240$		0			0
$241$	0.243946	−	3.25523i	0.0157139	−	0.209688i	−0.983820	−	0.179160i	$-0.942662\pi$
$241$	0.243946	−	3.25523i	0.0157139	−	0.209688i	0.999534	−	0.0305279i	$-0.00971886\pi$
$242$		0			0
$243$	21.3755	+	6.59347i	1.37124	+	0.422971i
$244$		0			0
$245$	−1.62465	+	1.10767i	−0.103795	+	0.0707662i
$246$		0			0
$247$	6.31862	−	27.6837i	0.402044	−	1.76147i
$248$		0			0
$249$	6.32052	+	5.86458i	0.400547	+	0.371653i
$250$		0			0
$251$	2.73625	−	4.73933i	0.172711	−	0.299144i	−0.766656	−	0.642058i	$-0.778081\pi$
$251$	2.73625	−	4.73933i	0.172711	−	0.299144i	0.939367	+	0.342914i	$0.111414\pi$
$252$		0			0
$253$	0.108206	+	1.44390i	0.00680283	+	0.0907774i
$254$		0			0
$255$	−2.11569	+	5.39068i	−0.132490	+	0.337578i
$256$		0			0
$257$	−8.41973			−0.525208			−0.262604	−	0.964904i	$-0.584581\pi$
$257$	−8.41973			−0.525208			−0.262604	+	0.964904i	$0.584581\pi$
$258$		0			0
$259$	−13.0056			−0.808128
$260$		0			0
$261$	−4.35380	+	11.0933i	−0.269494	+	0.686659i
$262$		0			0
$263$	−0.457346	−	6.10286i	−0.0282012	−	0.376319i	−0.993392	−	0.114772i	$-0.963386\pi$
$263$	−0.457346	−	6.10286i	−0.0282012	−	0.376319i	0.965191	−	0.261547i	$-0.0842327\pi$
$264$		0			0
$265$	1.19295	−	2.06624i	0.0732821	−	0.126928i
$266$		0			0
$267$	−3.40578	−	3.16010i	−0.208430	−	0.193395i
$268$		0			0
$269$	−0.510581	+	2.23700i	−0.0311307	+	0.136392i	−0.988105	−	0.153780i	$-0.950855\pi$
$269$	−0.510581	+	2.23700i	−0.0311307	+	0.136392i	0.956974	+	0.290172i	$0.0937126\pi$
$270$		0			0
$271$	−22.6364	+	15.4333i	−1.37506	+	0.937503i	−0.375120	+	0.926976i	$0.622399\pi$
$271$	−22.6364	+	15.4333i	−1.37506	+	0.937503i	−0.999945	+	0.0105270i	$0.996649\pi$
$272$		0			0
$273$	−37.3815	−	11.5307i	−2.26243	−	0.697868i
$274$		0			0
$275$	−1.32151	+	17.6344i	−0.0796902	+	1.06339i
$276$		0			0
$277$	19.4981	−	18.0916i	1.17153	−	1.08702i	0.176825	−	0.984242i	$-0.443417\pi$
$277$	19.4981	−	18.0916i	1.17153	−	1.08702i	0.994704	−	0.102778i	$-0.0327731\pi$
$278$		0			0
$279$	1.83042	+	2.29528i	0.109584	+	0.137415i
$280$		0			0
$281$	15.9359	−	4.91556i	0.950653	−	0.293238i	0.219625	−	0.975584i	$-0.429517\pi$
$281$	15.9359	−	4.91556i	0.950653	−	0.293238i	0.731029	+	0.682347i	$0.239041\pi$
$282$		0			0
$283$	16.9165	−	2.54976i	1.00558	−	0.151567i	0.374455	−	0.927245i	$-0.377830\pi$
$283$	16.9165	−	2.54976i	1.00558	−	0.151567i	0.631129	+	0.775678i	$0.282592\pi$
$284$		0			0
$285$	−6.88629	+	3.31626i	−0.407908	+	0.196438i
$286$		0			0
$287$	13.0274	+	33.1932i	0.768982	+	1.95933i
$288$		0			0
$289$	−4.02518	−	0.606699i	−0.236776	−	0.0356882i
$290$		0			0
$291$	−2.00335	−	1.36586i	−0.117438	−	0.0800681i
$292$		0			0
$293$	26.9499	+	12.9784i	1.57443	+	0.758206i	0.998251	−	0.0591160i	$-0.0188282\pi$
$293$	26.9499	+	12.9784i	1.57443	+	0.758206i	0.576181	+	0.817322i	$0.304542\pi$
$294$		0			0
$295$	−1.68000	−	2.90985i	−0.0978136	−	0.169418i
$296$		0			0
$297$	0.776151	+	3.40054i	0.0450369	+	0.197319i
$298$		0			0
$299$	−1.13660	+	1.42526i	−0.0657315	+	0.0824247i
$300$		0			0
$301$	−4.85130	+	21.1337i	−0.279624	+	1.21813i
$302$		0			0
$303$	−25.8708	+	32.4410i	−1.48624	+	1.86369i
$304$		0			0
$305$	0.583556	+	2.55673i	0.0334143	+	0.146398i
$306$		0			0
$307$	−6.72148	−	11.6419i	−0.383615	−	0.664441i	0.607961	−	0.793967i	$-0.291988\pi$
$307$	−6.72148	−	11.6419i	−0.383615	−	0.664441i	−0.991576	+	0.129526i	$0.958654\pi$
$308$		0			0
$309$	1.30111	+	0.626581i	0.0740175	+	0.0356449i
$310$		0			0
$311$	−12.5506	−	8.55684i	−0.711678	−	0.485214i	0.152562	−	0.988294i	$-0.451248\pi$
$311$	−12.5506	−	8.55684i	−0.711678	−	0.485214i	−0.864240	+	0.503080i	$0.832200\pi$
$312$		0			0
$313$	−11.2647	−	1.69789i	−0.636721	−	0.0959702i	−0.177254	−	0.984165i	$-0.556721\pi$
$313$	−11.2647	−	1.69789i	−0.636721	−	0.0959702i	−0.459467	+	0.888195i	$0.651960\pi$
$314$		0			0
$315$	2.03553	+	5.18645i	0.114689	+	0.292224i
$316$		0			0
$317$	20.5592	−	9.90081i	1.15472	−	0.556085i	0.244273	−	0.969707i	$-0.421451\pi$
$317$	20.5592	−	9.90081i	1.15472	−	0.556085i	0.910449	+	0.413622i	$0.135736\pi$
$318$		0			0
$319$	13.0129	−	1.96138i	0.728582	−	0.109816i
$320$		0			0
$321$	−39.0073	+	12.0321i	−2.17717	+	0.671569i
$322$		0			0
$323$	−17.3392	−	21.7427i	−0.964780	−	1.20980i
$324$		0			0
$325$	−16.3206	+	15.1433i	−0.905304	+	0.839999i
$326$		0			0
$327$	1.23119	−	16.4291i	0.0680851	−	0.908533i
$328$		0			0
$329$	11.0304	+	3.40242i	0.608125	+	0.187582i
$330$		0			0
$331$	−12.8486	+	8.76005i	−0.706225	+	0.481496i	−0.862396	−	0.506234i	$-0.831037\pi$
$331$	−12.8486	+	8.76005i	−0.706225	+	0.481496i	0.156171	+	0.987730i	$0.450085\pi$
$332$		0			0
$333$	−2.95048	+	12.9269i	−0.161685	+	0.708389i
$334$		0			0
$335$	−3.86657	−	3.58766i	−0.211254	−	0.196015i
$336$		0			0
$337$	−7.45198	+	12.9072i	−0.405935	+	0.703101i	−0.994430	−	0.105401i	$-0.966388\pi$
$337$	−7.45198	+	12.9072i	−0.405935	+	0.703101i	0.588494	+	0.808501i	$0.299721\pi$
$338$		0			0
$339$	0.320770	+	4.28038i	0.0174218	+	0.232478i
$340$		0			0
$341$	1.18441	−	3.01782i	0.0641393	−	0.163424i
$342$		0			0
$343$	10.1378			0.547391
$344$		0			0
$345$	0.490686			0.0264176
$346$		0			0
$347$	−9.92490	+	25.2882i	−0.532797	+	1.35754i	0.370622	+	0.928784i	$0.379145\pi$
$347$	−9.92490	+	25.2882i	−0.532797	+	1.35754i	−0.903419	+	0.428760i	$0.858951\pi$
$348$		0			0
$349$	2.11902	+	28.2763i	0.113428	+	1.51360i	0.706077	+	0.708135i	$0.250463\pi$
$349$	2.11902	+	28.2763i	0.113428	+	1.51360i	−0.592649	+	0.805461i	$0.701918\pi$
$350$		0			0
$351$	−2.19569	+	3.80305i	−0.117197	+	0.202992i
$352$		0			0
$353$	14.4582	+	13.4153i	0.769532	+	0.714022i	0.963712	−	0.266944i	$-0.0860139\pi$
$353$	14.4582	+	13.4153i	0.769532	+	0.714022i	−0.194179	+	0.980966i	$0.562204\pi$
$354$		0			0
$355$	−0.100045	+	0.438326i	−0.00530984	+	0.0232639i
$356$		0			0
$357$	−31.6554	+	21.5823i	−1.67538	+	1.14225i
$358$		0			0
$359$	26.9263	+	8.30565i	1.42111	+	0.438355i	0.907640	−	0.419750i	$-0.137882\pi$
$359$	26.9263	+	8.30565i	1.42111	+	0.438355i	0.513474	+	0.858105i	$0.328358\pi$
$360$		0			0
$361$	1.32308	−	17.6552i	0.0696357	−	0.929223i
$362$		0			0
$363$	5.28991	−	4.90832i	0.277648	−	0.257620i
$364$		0			0
$365$	0.301206	+	0.377700i	0.0157658	+	0.0197697i
$366$		0			0
$367$	28.2270	−	8.70689i	1.47344	−	0.454496i	0.548999	−	0.835823i	$-0.315009\pi$
$367$	28.2270	−	8.70689i	1.47344	−	0.454496i	0.924440	+	0.381327i	$0.124533\pi$
$368$		0			0
$369$	35.9478	−	5.41825i	1.87137	−	0.282063i
$370$		0			0
$371$	14.2216	−	6.84877i	0.738350	−	0.355570i
$372$		0			0
$373$	−11.7449	−	29.9254i	−0.608126	−	1.54948i	−0.820109	−	0.572207i	$-0.806087\pi$
$373$	−11.7449	−	29.9254i	−0.608126	−	1.54948i	0.211983	−	0.977273i	$-0.432008\pi$
$374$		0			0
$375$	12.1632	+	1.83331i	0.628107	+	0.0946719i
$376$		0			0
$377$	13.6893	+	9.33324i	0.705037	+	0.480686i
$378$		0			0
$379$	−16.2426	−	7.82203i	−0.834327	−	0.401791i	−0.0325909	−	0.999469i	$-0.510376\pi$
$379$	−16.2426	−	7.82203i	−0.834327	−	0.401791i	−0.801736	+	0.597678i	$0.796090\pi$
$380$		0			0
$381$	−19.8090	−	34.3102i	−1.01485	−	1.75777i
$382$		0			0
$383$	−5.11016	−	22.3891i	−0.261117	−	1.14403i	−0.920042	−	0.391819i	$-0.871846\pi$
$383$	−5.11016	−	22.3891i	−0.261117	−	1.14403i	0.658925	−	0.752208i	$-0.271011\pi$
$384$		0			0
$385$	3.83610	−	4.81032i	0.195506	−	0.245157i
$386$		0			0
$387$	19.9052	+	9.61638i	1.01184	+	0.488828i
$388$		0			0
$389$	−14.2264	+	17.8394i	−0.721309	+	0.904493i	−0.998411	−	0.0563472i	$-0.982055\pi$
$389$	−14.2264	+	17.8394i	−0.721309	+	0.904493i	0.277102	+	0.960841i	$0.410626\pi$
$390$		0			0
$391$	0.397282	+	1.74061i	0.0200914	+	0.0880263i
$392$		0			0
$393$	13.5371	+	23.4470i	0.682857	+	1.18274i
$394$		0			0
$395$	0.451197	+	0.217285i	0.0227022	+	0.0109328i
$396$		0			0
$397$	−7.13200	−	4.86251i	−0.357945	−	0.244043i	0.370973	−	0.928644i	$-0.379024\pi$
$397$	−7.13200	−	4.86251i	−0.357945	−	0.244043i	−0.728918	+	0.684601i	$0.759976\pi$
$398$		0			0
$399$	−50.0018	−	7.53656i	−2.50322	−	0.377300i
$400$		0			0
$401$	12.3578	+	31.4871i	0.617118	+	1.57239i	0.806979	+	0.590580i	$0.201101\pi$
$401$	12.3578	+	31.4871i	0.617118	+	1.57239i	−0.189862	+	0.981811i	$0.560804\pi$
$402$		0			0
$403$	3.67738	−	1.77093i	0.183183	−	0.0882164i
$404$		0			0
$405$	−3.82958	+	0.577216i	−0.190293	+	0.0286821i
$406$		0			0
$407$	13.9916	−	4.31583i	0.693536	−	0.213928i
$408$		0			0
$409$	−15.9968	−	20.0594i	−0.790992	−	0.991873i	−0.999903	−	0.0139379i	$-0.995563\pi$
$409$	−15.9968	−	20.0594i	−0.790992	−	0.991873i	0.208911	−	0.977935i	$-0.433008\pi$
$410$		0			0
$411$	−14.9611	+	13.8818i	−0.737975	+	0.684740i
$412$		0			0
$413$	1.66121	−	22.1673i	0.0817428	−	1.09078i
$414$		0			0
$415$	−1.63146	−	0.503237i	−0.0800850	−	0.0247029i
$416$		0			0
$417$	26.8314	−	18.2934i	1.31394	−	0.895830i
$418$		0			0
$419$	−5.52572	+	24.2097i	−0.269949	+	1.18272i	0.640123	+	0.768273i	$0.278883\pi$
$419$	−5.52572	+	24.2097i	−0.269949	+	1.18272i	−0.910072	+	0.414451i	$0.863974\pi$
$420$		0			0
$421$	−14.1439	−	13.1236i	−0.689330	−	0.639605i	0.255636	−	0.966773i	$-0.417715\pi$
$421$	−14.1439	−	13.1236i	−0.689330	−	0.639605i	−0.944966	+	0.327168i	$0.893906\pi$
$422$		0			0
$423$	5.88421	−	10.1918i	0.286100	−	0.495540i
$424$		0			0
$425$	1.62947	+	21.7437i	0.0790408	+	1.05473i
$426$		0			0
$427$	−6.33868	+	16.1507i	−0.306750	+	0.781587i
$428$		0			0
$429$	44.0418			2.12636
$430$		0			0
$431$	−32.4273			−1.56197			−0.780983	−	0.624552i	$-0.785282\pi$
$431$	−32.4273			−1.56197			−0.780983	+	0.624552i	$0.785282\pi$
$432$		0			0
$433$	−6.27055	+	15.9771i	−0.301343	+	0.767810i	0.697351	+	0.716730i	$0.254362\pi$
$433$	−6.27055	+	15.9771i	−0.301343	+	0.767810i	−0.998695	+	0.0510807i	$0.983733\pi$
$434$		0			0
$435$	−0.333270	−	4.44718i	−0.0159791	−	0.213226i
$436$		0			0
$437$	−1.17820	+	2.04071i	−0.0563611	+	0.0976204i
$438$		0			0
$439$	10.5781	+	9.81508i	0.504867	+	0.468448i	0.891005	−	0.453993i	$-0.150001\pi$
$439$	10.5781	+	9.81508i	0.504867	+	0.468448i	−0.386138	+	0.922441i	$0.626191\pi$
$440$		0			0
$441$	−2.95124	+	12.9302i	−0.140535	+	0.615725i
$442$		0			0
$443$	−15.7039	+	10.7067i	−0.746114	+	0.508692i	−0.875688	−	0.482878i	$-0.839591\pi$
$443$	−15.7039	+	10.7067i	−0.746114	+	0.508692i	0.129574	+	0.991570i	$0.458639\pi$
$444$		0			0
$445$	0.879101	+	0.271167i	0.0416734	+	0.0128545i
$446$		0			0
$447$	1.40341	−	18.7272i	0.0663788	−	0.885764i
$448$		0			0
$449$	−3.90802	+	3.62611i	−0.184431	+	0.171127i	−0.766985	−	0.641665i	$-0.778244\pi$
$449$	−3.90802	+	3.62611i	−0.184431	+	0.171127i	0.582554	+	0.812792i	$0.302053\pi$
$450$		0			0
$451$	−25.0300	−	31.3866i	−1.17862	−	1.47794i
$452$		0			0
$453$	−11.5187	+	3.55304i	−0.541194	+	0.166936i
$454$		0			0
$455$	7.65964	−	1.15451i	0.359089	−	0.0541240i
$456$		0			0
$457$	−7.16485	+	3.45041i	−0.335157	+	0.161403i	−0.593891	−	0.804545i	$-0.702409\pi$
$457$	−7.16485	+	3.45041i	−0.335157	+	0.161403i	0.258734	+	0.965949i	$0.416695\pi$
$458$		0			0
$459$	1.57126	+	4.00351i	0.0733402	+	0.186868i
$460$		0			0
$461$	−42.1232	−	6.34904i	−1.96187	−	0.295704i	−0.999732	−	0.0231454i	$-0.992632\pi$
$461$	−42.1232	−	6.34904i	−1.96187	−	0.295704i	−0.962139	−	0.272559i	$-0.912130\pi$
$462$		0			0
$463$	4.70372	+	3.20694i	0.218600	+	0.149039i	0.667670	−	0.744458i	$-0.267292\pi$
$463$	4.70372	+	3.20694i	0.218600	+	0.149039i	−0.449069	+	0.893497i	$0.648244\pi$
$464$		0			0
$465$	−0.989833	−	0.476678i	−0.0459024	−	0.0221054i
$466$		0			0
$467$	−3.69374	−	6.39775i	−0.170926	−	0.296053i	0.767818	−	0.640668i	$-0.221343\pi$
$467$	−3.69374	−	6.39775i	−0.170926	−	0.296053i	−0.938744	+	0.344616i	$0.888009\pi$
$468$		0			0
$469$	−7.76517	−	34.0214i	−0.358562	−	1.57096i
$470$		0			0
$471$	−21.1530	+	26.5250i	−0.974677	+	1.22221i
$472$		0			0
$473$	−1.79400	−	24.3457i	−0.0824883	−	1.11942i
$474$		0			0
$475$	−17.9433	+	22.5001i	−0.823294	+	1.03238i
$476$		0			0
$477$	−3.58098	−	15.6893i	−0.163962	−	0.718363i
$478$		0			0
$479$	10.3541	+	17.9337i	0.473089	+	0.819414i	0.999526	−	0.0308006i	$-0.00980570\pi$
$479$	10.3541	+	17.9337i	0.473089	+	0.819414i	−0.526437	+	0.850214i	$0.676472\pi$
$480$		0			0
$481$	16.6088	+	7.99837i	0.757295	+	0.364694i
$482$		0			0
$483$	2.68224	+	1.82872i	0.122046	+	0.0832096i
$484$		0			0
$485$	0.474751	+	0.0715573i	0.0215574	+	0.00324925i
$486$		0			0
$487$	−13.0064	−	33.1397i	−0.589376	−	1.50170i	−0.844912	−	0.534905i	$-0.820347\pi$
$487$	−13.0064	−	33.1397i	−0.589376	−	1.50170i	0.255537	−	0.966799i	$-0.417748\pi$
$488$		0			0
$489$	−42.5933	+	20.5119i	−1.92614	+	0.927579i
$490$		0			0
$491$	0.658494	−	0.0992520i	0.0297174	−	0.00447918i	−0.134167	−	0.990959i	$-0.542836\pi$
$491$	0.658494	−	0.0992520i	0.0297174	−	0.00447918i	0.163884	+	0.986480i	$0.447598\pi$
$492$		0			0
$493$	15.5056	−	4.78286i	0.698339	−	0.215409i
$494$		0			0
$495$	−3.91094	−	4.90417i	−0.175784	−	0.220426i
$496$		0			0
$497$	−2.18046	+	2.02317i	−0.0978070	+	0.0907517i
$498$		0			0
$499$	0.537738	−	7.17561i	0.0240724	−	0.321224i	−0.972148	−	0.234367i	$-0.924698\pi$
$499$	0.537738	−	7.17561i	0.0240724	−	0.321224i	0.996221	−	0.0868578i	$-0.0276826\pi$
$500$		0			0
$501$	−33.4186	−	10.3083i	−1.49303	−	0.460539i
$502$		0			0
$503$	34.3276	−	23.4041i	1.53059	−	1.04354i	0.552960	−	0.833208i	$-0.313498\pi$
$503$	34.3276	−	23.4041i	1.53059	−	1.04354i	0.977630	−	0.210331i	$-0.0674543\pi$
$504$		0			0
$505$	1.82829	−	8.01027i	0.0813579	−	0.356452i
$506$		0			0
$507$	16.5927	+	15.3958i	0.736907	+	0.683750i
$508$		0			0
$509$	15.3793	−	26.6378i	0.681677	−	1.18070i	−0.292792	−	0.956176i	$-0.594584\pi$
$509$	15.3793	−	26.6378i	0.681677	−	1.18070i	0.974469	−	0.224523i	$-0.0720824\pi$
$510$		0			0
$511$	0.238846	+	3.18718i	0.0105659	+	0.140992i
$512$		0			0
$513$	−2.07382	+	5.28400i	−0.0915612	+	0.233294i
$514$		0			0
$515$	−0.285955			−0.0126007
$516$		0			0
$517$	−12.9957			−0.571550
$518$		0			0
$519$	12.7909	−	32.5907i	0.561459	−	1.43057i
$520$		0			0
$521$	−1.14354	−	15.2594i	−0.0500992	−	0.668528i	−0.964562	−	0.263857i	$-0.915005\pi$
$521$	−1.14354	−	15.2594i	−0.0500992	−	0.668528i	0.914463	−	0.404671i	$-0.132614\pi$
$522$		0			0
$523$	0.672029	−	1.16399i	0.0293858	−	0.0508977i	−0.850959	−	0.525233i	$-0.823978\pi$
$523$	0.672029	−	1.16399i	0.0293858	−	0.0508977i	0.880344	+	0.474335i	$0.157312\pi$
$524$		0			0
$525$	29.0635	+	26.9670i	1.26844	+	1.17694i
$526$		0			0
$527$	0.889504	−	3.89717i	0.0387474	−	0.169763i
$528$		0			0
$529$	−18.8785	+	12.8711i	−0.820804	+	0.559615i
$530$		0			0
$531$	−21.6563	−	6.68008i	−0.939803	−	0.289891i
$532$		0			0
$533$	3.77704	−	50.4011i	0.163602	−	2.18312i
$534$		0			0
$535$	5.92529	−	5.49787i	0.256173	−	0.237693i
$536$		0			0
$537$	9.73276	+	12.2045i	0.419999	+	0.526663i
$538$		0			0
$539$	13.9952	−	4.31694i	0.602815	−	0.185944i
$540$		0			0
$541$	0.196272	−	0.0295833i	0.00843841	−	0.00127189i	−0.144822	−	0.989458i	$-0.546261\pi$
$541$	0.196272	−	0.0295833i	0.00843841	−	0.00127189i	0.153260	+	0.988186i	$0.451023\pi$
$542$		0			0
$543$	8.42065	−	4.05517i	0.361364	−	0.174024i
$544$		0			0
$545$	1.19185	+	3.03679i	0.0510533	+	0.130082i
$546$		0			0
$547$	7.85074	+	1.18331i	0.335673	+	0.0505946i	0.314716	−	0.949186i	$-0.398091\pi$
$547$	7.85074	+	1.18331i	0.335673	+	0.0505946i	0.0209569	+	0.999780i	$0.493329\pi$
$548$		0			0
$549$	14.6149	+	9.96430i	0.623751	+	0.425266i
$550$		0			0
$551$	19.2956	+	9.29226i	0.822019	+	0.395864i
$552$		0			0
$553$	1.65659	+	2.86930i	0.0704454	+	0.122015i
$554$		0			0
$555$	−1.10414	−	4.83754i	−0.0468680	−	0.205342i
$556$		0			0
$557$	−10.9192	+	13.6922i	−0.462659	+	0.580157i	−0.957357	−	0.288909i	$-0.906708\pi$
$557$	−10.9192	+	13.6922i	−0.462659	+	0.580157i	0.494697	+	0.869065i	$0.335279\pi$
$558$		0			0
$559$	19.1925	−	24.0053i	0.811755	−	1.01531i
$560$		0			0
$561$	26.8933	−	33.7231i	1.13543	−	1.42379i
$562$		0			0
$563$	5.52035	+	24.1862i	0.232655	+	1.01933i	0.947427	+	0.319971i	$0.103673\pi$
$563$	5.52035	+	24.1862i	0.232655	+	1.01933i	−0.714773	+	0.699357i	$0.753470\pi$
$564$		0			0
$565$	−0.424973	−	0.736075i	−0.0178788	−	0.0309669i
$566$		0			0
$567$	−23.0849	−	11.1171i	−0.969473	−	0.466874i
$568$		0			0
$569$	−23.8352	−	16.2505i	−0.999222	−	0.681258i	−0.0509448	−	0.998701i	$-0.516223\pi$
$569$	−23.8352	−	16.2505i	−0.999222	−	0.681258i	−0.948277	+	0.317443i	$0.897176\pi$
$570$		0			0
$571$	32.2418	+	4.85967i	1.34928	+	0.203371i	0.783605	−	0.621259i	$-0.213379\pi$
$571$	32.2418	+	4.85967i	1.34928	+	0.203371i	0.565672	+	0.824630i	$0.308617\pi$
$572$		0			0
$573$	10.5269	+	26.8221i	0.439768	+	1.12051i
$574$		0			0
$575$	1.66460	−	0.801630i	0.0694187	−	0.0334303i
$576$		0			0
$577$	33.2474	−	5.01124i	1.38411	−	0.208621i	0.585597	−	0.810602i	$-0.300860\pi$
$577$	33.2474	−	5.01124i	1.38411	−	0.208621i	0.798510	+	0.601982i	$0.205622\pi$
$578$		0			0
$579$	20.3806	−	6.28657i	0.846987	−	0.261261i
$580$		0			0
$581$	−7.04254	−	8.83106i	−0.292174	−	0.366374i
$582$		0			0
$583$	−13.0271	+	12.0873i	−0.539526	+	0.500607i
$584$		0			0
$585$	0.590165	−	7.87520i	0.0244003	−	0.325600i
$586$		0			0
$587$	−15.4987	−	4.78070i	−0.639698	−	0.197321i	−0.0420960	−	0.999114i	$-0.513404\pi$
$587$	−15.4987	−	4.78070i	−0.639698	−	0.197321i	−0.597602	+	0.801793i	$0.703880\pi$
$588$		0			0
$589$	4.35918	−	2.97204i	0.179617	−	0.122461i
$590$		0			0
$591$	−12.1685	+	53.3138i	−0.500546	+	2.19303i
$592$		0			0
$593$	32.4141	+	30.0759i	1.33109	+	1.23507i	0.950651	+	0.310263i	$0.100417\pi$
$593$	32.4141	+	30.0759i	1.33109	+	1.23507i	0.380438	+	0.924807i	$0.375773\pi$
$594$		0			0
$595$	3.79319	−	6.57000i	0.155506	−	0.269344i
$596$		0			0
$597$	−1.86336	−	24.8648i	−0.0762622	−	1.01765i
$598$		0			0
$599$	−1.51367	+	3.85676i	−0.0618468	+	0.157583i	−0.958457	−	0.285238i	$-0.907927\pi$
$599$	−1.51367	+	3.85676i	−0.0618468	+	0.157583i	0.896610	+	0.442821i	$0.146022\pi$
$600$		0			0
$601$	19.9921			0.815496			0.407748	−	0.913094i	$-0.366314\pi$
$601$	19.9921			0.815496			0.407748	+	0.913094i	$0.366314\pi$
$602$		0			0
$603$	−35.5772			−1.44881
$604$		0			0
$605$	−0.522042	+	1.33014i	−0.0212240	+	0.0540779i
$606$		0			0
$607$	1.35174	+	18.0378i	0.0548656	+	0.732130i	0.955017	+	0.296552i	$0.0958369\pi$
$607$	1.35174	+	18.0378i	0.0548656	+	0.732130i	−0.900151	+	0.435578i	$0.856544\pi$
$608$		0			0
$609$	14.7523	−	25.5517i	0.597793	−	1.03541i
$610$		0			0
$611$	−11.9939	−	11.1287i	−0.485221	−	0.450219i
$612$		0			0
$613$	−8.34759	+	36.5732i	−0.337156	+	1.47718i	0.467796	+	0.883836i	$0.345048\pi$
$613$	−8.34759	+	36.5732i	−0.337156	+	1.47718i	−0.804952	+	0.593340i	$0.797809\pi$
$614$		0			0
$615$	−11.2405	+	7.66365i	−0.453261	+	0.309028i
$616$		0			0
$617$	−8.65578	−	2.66996i	−0.348469	−	0.107488i	0.115580	−	0.993298i	$-0.463127\pi$
$617$	−8.65578	−	2.66996i	−0.348469	−	0.107488i	−0.464048	+	0.885810i	$0.653604\pi$
$618$		0			0
$619$	−0.370823	+	4.94829i	−0.0149046	+	0.198889i	0.984781	+	0.173802i	$0.0556052\pi$
$619$	−0.370823	+	4.94829i	−0.0149046	+	0.198889i	−0.999685	+	0.0250867i	$0.992014\pi$
$620$		0			0
$621$	0.267137	−	0.247867i	0.0107199	−	0.00994657i
$622$		0			0
$623$	3.79483	+	4.75857i	0.152037	+	0.190648i
$624$		0			0
$625$	20.3683	−	6.28279i	0.814732	−	0.251311i
$626$		0			0
$627$	56.2935	−	8.48488i	2.24815	−	0.338854i
$628$		0			0
$629$	16.2662	−	7.83340i	0.648576	−	0.312338i
$630$		0			0
$631$	−12.7382	−	32.4564i	−0.507100	−	1.29207i	−0.923748	−	0.383002i	$-0.874890\pi$
$631$	−12.7382	−	32.4564i	−0.507100	−	1.29207i	0.416647	−	0.909068i	$-0.363205\pi$
$632$		0			0
$633$	38.1875	+	5.75583i	1.51782	+	0.228774i
$634$		0			0
$635$	6.48174	+	4.41918i	0.257220	+	0.175370i
$636$		0			0
$637$	16.6131	+	8.00043i	0.658234	+	0.316989i
$638$		0			0
$639$	1.51626	+	2.62625i	0.0599825	+	0.103893i
$640$		0			0
$641$	9.77409	+	42.8231i	0.386053	+	1.69141i	0.678070	+	0.734997i	$0.262816\pi$
$641$	9.77409	+	42.8231i	0.386053	+	1.69141i	−0.292017	+	0.956413i	$0.594326\pi$
$642$		0			0
$643$	−14.3909	+	18.0457i	−0.567523	+	0.711651i	−0.979929	−	0.199349i	$-0.936117\pi$
$643$	−14.3909	+	18.0457i	−0.567523	+	0.711651i	0.412406	+	0.911000i	$0.364689\pi$
$644$		0			0
$645$	−8.27271	−	0.0102932i	−0.325738	−	0.000405295i
$646$		0			0
$647$	27.5277	−	34.5186i	1.08222	−	1.35707i	0.152714	−	0.988270i	$-0.451199\pi$
$647$	27.5277	−	34.5186i	1.08222	−	1.35707i	0.929511	−	0.368796i	$-0.120230\pi$
$648$		0			0
$649$	5.56894	+	24.3991i	0.218600	+	0.957749i
$650$		0			0
$651$	−3.63422	−	6.29465i	−0.142436	−	0.246707i
$652$		0			0
$653$	−5.99963	−	2.88927i	−0.234783	−	0.113066i	0.312793	−	0.949821i	$-0.398735\pi$
$653$	−5.99963	−	2.88927i	−0.234783	−	0.113066i	−0.547577	+	0.836755i	$0.684450\pi$
$654$		0			0
$655$	−4.42950	−	3.01998i	−0.173075	−	0.118000i
$656$		0			0
$657$	3.22207	+	0.485650i	0.125705	+	0.0189470i
$658$		0			0
$659$	3.46012	+	8.81623i	0.134787	+	0.343432i	0.982292	−	0.187355i	$-0.0599916\pi$
$659$	3.46012	+	8.81623i	0.134787	+	0.343432i	−0.847505	+	0.530787i	$0.821896\pi$
$660$		0			0
$661$	−15.1724	+	7.30666i	−0.590140	+	0.284196i	−0.705025	−	0.709183i	$-0.749064\pi$
$661$	−15.1724	+	7.30666i	−0.590140	+	0.284196i	0.114885	+	0.993379i	$0.463350\pi$
$662$		0			0
$663$	53.6984	−	8.09374i	2.08547	−	0.314335i
$664$		0			0
$665$	9.56800	−	2.95134i	0.371031	−	0.114448i
$666$		0			0
$667$	−0.857246	−	1.07495i	−0.0331927	−	0.0416223i
$668$		0			0
$669$	−21.6795	+	20.1157i	−0.838179	+	0.777717i
$670$		0			0
$671$	1.45972	−	19.4786i	0.0563517	−	0.751961i
$672$		0			0
$673$	17.2152	+	5.31018i	0.663597	+	0.204692i	0.608214	−	0.793773i	$-0.291886\pi$
$673$	17.2152	+	5.31018i	0.663597	+	0.204692i	0.0553823	+	0.998465i	$0.482362\pi$
$674$		0			0
$675$	3.67729	−	2.50713i	0.141539	−	0.0964995i
$676$		0			0
$677$	−1.20314	+	5.27130i	−0.0462404	+	0.202592i	−0.992771	−	0.120020i	$-0.961704\pi$
$677$	−1.20314	+	5.27130i	−0.0462404	+	0.202592i	0.946531	+	0.322613i	$0.104561\pi$
$678$		0			0
$679$	2.32845	+	2.16049i	0.0893578	+	0.0829120i
$680$		0			0
$681$	−11.4087	+	19.7605i	−0.437183	+	0.757223i
$682$		0			0
$683$	2.75414	+	36.7514i	0.105384	+	1.40626i	0.759818	+	0.650136i	$0.225288\pi$
$683$	2.75414	+	36.7514i	0.105384	+	1.40626i	−0.654434	+	0.756119i	$0.727093\pi$
$684$		0			0
$685$	1.47645	−	3.76194i	0.0564123	−	0.143736i
$686$		0			0
$687$	39.3708			1.50209
$688$		0			0
$689$	−22.3737			−0.852369
$690$		0			0
$691$	−3.64459	+	9.28626i	−0.138647	+	0.353266i	−0.983285	−	0.182071i	$-0.941720\pi$
$691$	−3.64459	+	9.28626i	−0.138647	+	0.353266i	0.844639	+	0.535337i	$0.179815\pi$
$692$		0			0
$693$	−3.10125	−	41.3833i	−0.117807	−	1.57202i
$694$		0			0
$695$	−3.21515	+	5.56881i	−0.121958	+	0.211237i
$696$		0			0
$697$	−36.2861	−	33.6685i	−1.37443	−	1.27529i
$698$		0			0
$699$	−2.63826	+	11.5590i	−0.0997883	+	0.437201i
$700$		0			0
$701$	9.62158	−	6.55988i	0.363402	−	0.247763i	−0.367829	−	0.929894i	$-0.619899\pi$
$701$	9.62158	−	6.55988i	0.363402	−	0.247763i	0.731231	+	0.682130i	$0.238946\pi$
$702$		0			0
$703$	22.7700	+	7.02361i	0.858786	+	0.264900i
$704$		0			0
$705$	−0.329113	+	4.39170i	−0.0123951	+	0.165401i
$706$		0			0
$707$	39.8472	−	36.9728i	1.49861	−	1.39051i
$708$		0			0
$709$	10.4090	+	13.0524i	0.390916	+	0.490194i	0.937879	−	0.346964i	$-0.112787\pi$
$709$	10.4090	+	13.0524i	0.390916	+	0.490194i	−0.546962	+	0.837157i	$0.684216\pi$
$710$		0			0
$711$	3.22775	−	0.995630i	0.121050	−	0.0373390i
$712$		0			0
$713$	−0.334926	+	0.0504820i	−0.0125431	+	0.00189057i
$714$		0			0
$715$	−7.85722	+	3.78384i	−0.293843	+	0.141507i
$716$		0			0
$717$	−3.79440	−	9.66798i	−0.141705	−	0.361057i
$718$		0			0
$719$	−31.8762	−	4.80456i	−1.18878	−	0.179180i	−0.475280	−	0.879834i	$-0.657653\pi$
$719$	−31.8762	−	4.80456i	−1.18878	−	0.179180i	−0.713501	+	0.700654i	$0.752891\pi$
$720$		0			0
$721$	−1.56312	−	1.06571i	−0.0582135	−	0.0396893i
$722$		0			0
$723$	7.42365	+	3.57504i	0.276089	+	0.132957i
$724$		0			0
$725$	−8.39591	−	14.5421i	−0.311816	−	0.540082i
$726$		0			0
$727$	4.68039	+	20.5061i	0.173586	+	0.760529i	0.984503	+	0.175368i	$0.0561116\pi$
$727$	4.68039	+	20.5061i	0.173586	+	0.760529i	−0.810917	+	0.585161i	$0.801031\pi$
$728$		0			0
$729$	−20.7105	+	25.9701i	−0.767054	+	0.961855i
$730$		0			0
$731$	−6.66147	−	29.3541i	−0.246383	−	1.08570i
$732$		0			0
$733$	2.81862	−	3.53443i	0.104108	−	0.130547i	−0.727048	−	0.686587i	$-0.759108\pi$
$733$	2.81862	−	3.53443i	0.104108	−	0.130547i	0.831156	+	0.556039i	$0.187680\pi$
$734$		0			0
$735$	−1.10442	−	4.83879i	−0.0407372	−	0.178481i
$736$		0			0
$737$	19.6437	+	34.0238i	0.723584	+	1.25328i
$738$		0			0
$739$	−6.30660	−	3.03710i	−0.231992	−	0.111722i	0.314277	−	0.949331i	$-0.398238\pi$
$739$	−6.30660	−	3.03710i	−0.231992	−	0.111722i	−0.546269	+	0.837610i	$0.683952\pi$
$740$		0			0
$741$	59.2199	+	40.3754i	2.17550	+	1.48323i
$742$		0			0
$743$	−22.7632	−	3.43099i	−0.835099	−	0.125871i	−0.282447	−	0.959283i	$-0.591146\pi$
$743$	−22.7632	−	3.43099i	−0.835099	−	0.125871i	−0.552652	+	0.833412i	$0.686384\pi$
$744$		0			0
$745$	1.35856	+	3.46156i	0.0497739	+	0.126822i
$746$		0			0
$747$	−10.3753	+	4.99649i	−0.379613	+	0.182812i
$748$		0			0
$749$	52.8793	−	7.97027i	1.93217	−	0.291227i
$750$		0			0
$751$	−13.6946	+	4.22423i	−0.499723	+	0.154144i	−0.534362	−	0.845256i	$-0.679448\pi$
$751$	−13.6946	+	4.22423i	−0.499723	+	0.154144i	0.0346389	+	0.999400i	$0.488972\pi$
$752$		0			0
$753$	8.61244	+	10.7997i	0.313855	+	0.393561i
$754$		0			0
$755$	1.74971	−	1.62349i	0.0636785	−	0.0590850i
$756$		0			0
$757$	2.72389	−	36.3478i	0.0990014	−	1.32108i	−0.697561	−	0.716526i	$-0.745731\pi$
$757$	2.72389	−	36.3478i	0.0990014	−	1.32108i	0.796562	−	0.604556i	$-0.206650\pi$
$758$		0			0
$759$	−3.49243	−	1.07727i	−0.126767	−	0.0391025i
$760$		0			0
$761$	−36.7441	+	25.0517i	−1.33197	+	0.908124i	−0.999389	−	0.0349488i	$-0.988873\pi$
$761$	−36.7441	+	25.0517i	−1.33197	+	0.908124i	−0.332585	+	0.943073i	$0.607921\pi$
$762$		0			0
$763$	−4.80268	+	21.0419i	−0.173869	+	0.761768i
$764$		0			0
$765$	−5.66971	−	5.26073i	−0.204989	−	0.190202i
$766$		0			0
$767$	−15.7542	+	27.2871i	−0.568852	+	0.985281i
$768$		0			0
$769$	0.188022	+	2.50898i	0.00678024	+	0.0904761i	0.999567	−	0.0294151i	$-0.00936447\pi$
$769$	0.188022	+	2.50898i	0.00678024	+	0.0904761i	−0.992787	+	0.119891i	$0.961745\pi$
$770$		0			0
$771$	7.76438	−	19.7833i	0.279627	−	0.712479i
$772$		0			0
$773$	−44.8085			−1.61165			−0.805824	−	0.592155i	$-0.798277\pi$
$773$	−44.8085			−1.61165			−0.805824	+	0.592155i	$0.798277\pi$
$774$		0			0
$775$	−4.13665			−0.148593
$776$		0			0
$777$	11.9933	−	30.5584i	0.430257	−	1.09628i
$778$		0			0
$779$	−4.88228	−	65.1495i	−0.174926	−	2.33422i
$780$		0			0
$781$	1.67439	−	2.90013i	0.0599143	−	0.103775i
$782$		0			0
$783$	−2.42791	−	2.25277i	−0.0867664	−	0.0805074i
$784$		0			0
$785$	1.49488	−	6.54949i	0.0533545	−	0.233761i
$786$		0			0
$787$	−2.14540	+	1.46271i	−0.0764753	+	0.0521400i	−0.600954	−	0.799284i	$-0.705213\pi$
$787$	−2.14540	+	1.46271i	−0.0764753	+	0.0521400i	0.524479	+	0.851423i	$0.324260\pi$
$788$		0			0
$789$	14.7613	+	4.55325i	0.525515	+	0.162100i
$790$		0			0
$791$	0.420219	−	5.60743i	0.0149413	−	0.199377i
$792$		0			0
$793$	18.0274	−	16.7270i	0.640172	−	0.593992i
$794$		0			0
$795$	3.75483	+	4.70841i	0.133170	+	0.166990i
$796$		0			0
$797$	22.5905	−	6.96825i	0.800197	−	0.246828i	0.132424	−	0.991193i	$-0.457724\pi$
$797$	22.5905	−	6.96825i	0.800197	−	0.246828i	0.667773	+	0.744365i	$0.267248\pi$
$798$		0			0
$799$	−15.8451	+	2.38827i	−0.560560	+	0.0844909i
$800$		0			0
$801$	5.59068	−	2.69233i	0.197537	−	0.0951288i
$802$		0			0
$803$	−1.31460	−	3.34954i	−0.0463912	−	0.118203i
$804$		0			0
$805$	−0.635634	−	0.0958065i	−0.0224032	−	0.00337673i
$806$		0			0
$807$	−4.78531	−	3.26257i	−0.168451	−	0.114848i
$808$		0			0
$809$	−36.0711	−	17.3709i	−1.26819	−	0.610729i	−0.325862	−	0.945417i	$-0.605655\pi$
$809$	−36.0711	−	17.3709i	−1.26819	−	0.610729i	−0.942329	+	0.334689i	$0.891369\pi$
$810$		0			0
$811$	−4.66195	−	8.07473i	−0.163703	−	0.283542i	0.772491	−	0.635026i	$-0.219011\pi$
$811$	−4.66195	−	8.07473i	−0.163703	−	0.283542i	−0.936194	+	0.351484i	$0.885677\pi$
$812$		0			0
$813$	−15.3880	−	67.4194i	−0.539682	−	2.36450i
$814$		0			0
$815$	5.83653	−	7.31878i	0.204445	−	0.256366i
$816$		0			0
$817$	19.9067	−	34.3806i	0.696448	−	1.20283i
$818$		0			0
$819$	32.5758	−	40.8488i	1.13829	−	1.42737i
$820$		0			0
$821$	0.416311	+	1.82398i	0.0145294	+	0.0636573i	0.981673	−	0.190574i	$-0.0610348\pi$
$821$	0.416311	+	1.82398i	0.0145294	+	0.0636573i	−0.967144	+	0.254231i	$0.918178\pi$
$822$		0			0
$823$	3.86127	+	6.68791i	0.134595	+	0.233126i	0.925443	−	0.378887i	$-0.123693\pi$
$823$	3.86127	+	6.68791i	0.134595	+	0.233126i	−0.790847	+	0.612013i	$0.790360\pi$
$824$		0			0
$825$	−40.2157	−	19.3669i	−1.40013	−	0.674268i
$826$		0			0
$827$	−8.94564	−	6.09903i	−0.311070	−	0.212084i	0.397714	−	0.917509i	$-0.369804\pi$
$827$	−8.94564	−	6.09903i	−0.311070	−	0.212084i	−0.708784	+	0.705425i	$0.750756\pi$
$828$		0			0
$829$	16.5194	+	2.48990i	0.573742	+	0.0864777i	0.429503	−	0.903065i	$-0.358689\pi$
$829$	16.5194	+	2.48990i	0.573742	+	0.0864777i	0.144239	+	0.989543i	$0.453927\pi$
$830$		0			0
$831$	24.5283	+	62.4970i	0.850877	+	2.16800i
$832$		0			0
$833$	16.2704	−	7.83542i	0.563736	−	0.271481i
$834$		0			0
$835$	6.84762	−	1.03211i	0.236972	−	0.0357177i
$836$		0			0
$837$	−0.779673	+	0.240497i	−0.0269494	+	0.00831280i
$838$		0			0
$839$	−1.42431	−	1.78602i	−0.0491725	−	0.0616604i	0.756637	−	0.653835i	$-0.226841\pi$
$839$	−1.42431	−	1.78602i	−0.0491725	−	0.0616604i	−0.805810	+	0.592175i	$0.798270\pi$
$840$		0			0
$841$	12.0982	−	11.2255i	0.417180	−	0.387087i
$842$		0			0
$843$	−3.14570	+	41.9765i	−0.108344	+	1.44575i
$844$		0			0
$845$	−4.28291	−	1.32110i	−0.147337	−	0.0454473i
$846$		0			0
$847$	−7.81089	+	5.32538i	−0.268385	+	0.182982i
$848$		0			0
$849$	−9.60884	+	42.0991i	−0.329774	+	1.44484i
$850$		0			0
$851$	−1.12140	−	1.04051i	−0.0384412	−	0.0356682i
$852$		0			0
$853$	−7.02133	+	12.1613i	−0.240406	+	0.416395i	−0.960830	−	0.277139i	$-0.910614\pi$
$853$	−7.02133	+	12.1613i	−0.240406	+	0.416395i	0.720424	+	0.693534i	$0.243947\pi$
$854$		0			0
$855$	−0.762859	−	10.1796i	−0.0260892	−	0.348136i
$856$		0			0
$857$	−14.6979	+	37.4496i	−0.502071	+	1.27926i	0.425280	+	0.905062i	$0.360176\pi$
$857$	−14.6979	+	37.4496i	−0.502071	+	1.27926i	−0.927351	+	0.374194i	$0.877920\pi$
$858$		0			0
$859$	34.4229			1.17449			0.587247	−	0.809407i	$-0.300212\pi$
$859$	34.4229			1.17449			0.587247	+	0.809407i	$0.300212\pi$
$860$		0			0
$861$	−90.0055			−3.06738
$862$		0			0
$863$	−7.40091	+	18.8572i	−0.251930	+	0.641907i	−0.999758	−	0.0219850i	$-0.993001\pi$
$863$	−7.40091	+	18.8572i	−0.251930	+	0.641907i	0.747828	+	0.663892i	$0.231097\pi$
$864$		0			0
$865$	0.518074	+	6.91322i	0.0176151	+	0.235057i
$866$		0			0
$867$	5.13741	−	8.89825i	0.174476	−	0.302201i
$868$		0			0
$869$	−2.73434	−	2.53710i	−0.0927561	−	0.0860651i
$870$		0			0
$871$	−11.0065	+	48.2226i	−0.372941	+	1.63396i
$872$		0			0
$873$	2.67565	−	1.82423i	0.0905571	−	0.0617408i
$874$		0			0
$875$	−15.3983	−	4.74975i	−0.520558	−	0.160571i
$876$		0			0
$877$	−3.27144	+	43.6544i	−0.110469	+	1.47410i	0.616457	+	0.787388i	$0.288567\pi$
$877$	−3.27144	+	43.6544i	−0.110469	+	1.47410i	−0.726926	+	0.686716i	$0.759052\pi$
$878$		0			0
$879$	−55.3469	+	51.3544i	−1.86680	+	1.73214i
$880$		0			0
$881$	2.02505	+	2.53934i	0.0682258	+	0.0855524i	0.814773	−	0.579780i	$-0.196861\pi$
$881$	2.02505	+	2.53934i	0.0682258	+	0.0855524i	−0.746548	+	0.665332i	$0.768290\pi$
$882$		0			0
$883$	14.6272	−	4.51190i	0.492245	−	0.151837i	−0.0386879	−	0.999251i	$-0.512318\pi$
$883$	14.6272	−	4.51190i	0.492245	−	0.151837i	0.530933	+	0.847414i	$0.321842\pi$
$884$		0			0
$885$	8.38635	−	1.26404i	0.281904	−	0.0424902i
$886$		0			0
$887$	−9.28365	+	4.47077i	−0.311714	+	0.150114i	−0.583200	−	0.812329i	$-0.698199\pi$
$887$	−9.28365	+	4.47077i	−0.311714	+	0.150114i	0.271485	+	0.962443i	$0.412485\pi$
$888$		0			0
$889$	18.9615	+	48.3132i	0.635949	+	1.62037i
$890$		0			0
$891$	28.5241	+	4.29932i	0.955593	+	0.144033i
$892$		0			0
$893$	−17.4744	−	11.9138i	−0.584757	−	0.398681i
$894$		0			0
$895$	−2.78490	−	1.34114i	−0.0930889	−	0.0448293i
$896$		0			0
$897$	−2.30070	−	3.98493i	−0.0768181	−	0.133053i
$898$		0			0
$899$	0.685008	+	3.00122i	0.0228463	+	0.100096i
$900$		0			0
$901$	−13.6620	+	17.1316i	−0.455148	+	0.570738i
$902$		0			0
$903$	−45.1828	−	30.8876i	−1.50359	−	1.02787i
$904$		0			0
$905$	−1.15387	+	1.44691i	−0.0383561	+	0.0480970i
$906$		0			0
$907$	−9.91878	−	43.4570i	−0.329348	−	1.44297i	−0.820377	−	0.571823i	$-0.806236\pi$
$907$	−9.91878	−	43.4570i	−0.329348	−	1.44297i	0.491029	−	0.871143i	$-0.336621\pi$
$908$		0			0
$909$	−27.7092	−	47.9938i	−0.919057	−	1.59185i
$910$		0			0
$911$	8.21706	+	3.95713i	0.272243	+	0.131105i	0.565025	−	0.825074i	$-0.308867\pi$
$911$	8.21706	+	3.95713i	0.272243	+	0.131105i	−0.292782	+	0.956179i	$0.594581\pi$
$912$		0			0
$913$	10.5070	+	7.16354i	0.347730	+	0.237079i
$914$		0			0
$915$	−6.54552	−	0.986578i	−0.216388	−	0.0326153i
$916$		0			0
$917$	−12.9579	−	33.0163i	−0.427909	−	1.09029i
$918$		0			0
$919$	−24.6919	+	11.8910i	−0.814511	+	0.392248i	−0.794283	−	0.607547i	$-0.792153\pi$
$919$	−24.6919	+	11.8910i	−0.814511	+	0.392248i	−0.0202279	+	0.999795i	$0.506439\pi$
$920$		0			0
$921$	33.5527	−	5.05725i	1.10560	−	0.166642i
$922$		0			0
$923$	4.02880	−	1.24272i	0.132609	−	0.0409046i
$924$		0			0
$925$	−11.6487	−	14.6070i	−0.383008	−	0.480276i
$926$		0			0
$927$	−1.41388	+	1.31189i	−0.0464378	+	0.0430880i
$928$		0			0
$929$	0.604172	−	8.06211i	0.0198222	−	0.264509i	−0.978484	−	0.206325i	$-0.933850\pi$
$929$	0.604172	−	8.06211i	0.0198222	−	0.264509i	0.998306	−	0.0581847i	$-0.0185312\pi$
$930$		0			0
$931$	22.7759	+	7.02542i	0.746448	+	0.230249i
$932$		0			0
$933$	31.6792	−	21.5985i	1.03713	−	0.707104i
$934$		0			0
$935$	−1.90055	+	8.32684i	−0.0621545	+	0.272317i
$936$		0			0
$937$	−27.3135	−	25.3432i	−0.892293	−	0.827927i	0.0935468	−	0.995615i	$-0.470180\pi$
$937$	−27.3135	−	25.3432i	−0.892293	−	0.827927i	−0.985840	+	0.167688i	$0.946370\pi$
$938$		0			0
$939$	14.3774	−	24.9023i	0.469188	−	0.812657i
$940$		0			0
$941$	2.71314	+	36.2043i	0.0884457	+	1.18023i	0.848077	+	0.529873i	$0.177760\pi$
$941$	2.71314	+	36.2043i	0.0884457	+	1.18023i	−0.759632	+	0.650354i	$0.774621\pi$
$942$		0			0
$943$	−1.53234	+	3.90433i	−0.0498997	+	0.127142i
$944$		0			0
$945$	−1.54848			−0.0503722
$946$		0			0
$947$	−46.5566			−1.51288			−0.756442	−	0.654060i	$-0.773064\pi$
$947$	−46.5566			−1.51288			−0.756442	+	0.654060i	$0.773064\pi$
$948$		0			0
$949$	1.65508	−	4.21707i	0.0537261	−	0.136892i
$950$		0			0
$951$	4.30431	+	57.4370i	0.139577	+	1.86252i
$952$		0			0
$953$	−15.7468	+	27.2743i	−0.510089	+	0.883500i	0.489843	+	0.871811i	$0.337054\pi$
$953$	−15.7468	+	27.2743i	−0.510089	+	0.883500i	−0.999932	+	0.0116891i	$0.996279\pi$
$954$		0			0
$955$	−4.18245	−	3.88074i	−0.135341	−	0.125578i
$956$		0			0
$957$	−7.39151	+	32.3843i	−0.238933	+	1.04684i
$958$		0			0
$959$	22.0910	−	15.0614i	0.713355	−	0.486357i
$960$		0			0
$961$	−28.8981	−	8.91388i	−0.932196	−	0.287545i
$962$		0			0
$963$	4.07428	−	54.3674i	0.131292	−	1.75197i
$964$		0			0
$965$	−3.09585	+	2.87253i	−0.0996590	+	0.0924701i
$966$		0			0
$967$	0.735828	+	0.922699i	0.0236626	+	0.0296720i	0.793522	−	0.608542i	$-0.208245\pi$
$967$	0.735828	+	0.922699i	0.0236626	+	0.0296720i	−0.769859	+	0.638214i	$0.779674\pi$
$968$		0			0
$969$	67.0771	−	20.6905i	2.15483	−	0.664676i
$970$		0			0
$971$	−39.4527	+	5.94653i	−1.26610	+	0.190833i	−0.747544	−	0.664213i	$-0.768767\pi$
$971$	−39.4527	+	5.94653i	−1.26610	+	0.190833i	−0.518552	+	0.855046i	$0.673529\pi$
$972$		0			0
$973$	−38.3292	+	18.4584i	−1.22878	+	0.591749i
$974$		0			0
$975$	−20.5310	−	52.3122i	−0.657518	−	1.67533i
$976$		0			0
$977$	31.2557	+	4.71104i	0.999959	+	0.150720i	0.628566	−	0.777756i	$-0.283642\pi$
$977$	31.2557	+	4.71104i	0.999959	+	0.150720i	0.371393	+	0.928476i	$0.378880\pi$
$978$		0			0
$979$	−5.66163	−	3.86003i	−0.180946	−	0.123367i
$980$		0			0
$981$	19.8250	+	9.54722i	0.632964	+	0.304819i
$982$		0			0
$983$	−12.1582	−	21.0586i	−0.387787	−	0.671666i	0.604365	−	0.796708i	$-0.293427\pi$
$983$	−12.1582	−	21.0586i	−0.387787	−	0.671666i	−0.992152	+	0.125042i	$0.960094\pi$
$984$		0			0
$985$	−2.40952	−	10.5568i	−0.0767738	−	0.336368i
$986$		0			0
$987$	−18.1663	+	22.7798i	−0.578240	+	0.725090i
$988$		0			0
$989$	−2.10910	+	1.43412i	−0.0670655	+	0.0456023i
$990$		0			0
$991$	2.01386	−	2.52530i	0.0639724	−	0.0802189i	−0.748816	−	0.662778i	$-0.769377\pi$
$991$	2.01386	−	2.52530i	0.0639724	−	0.0802189i	0.812788	+	0.582559i	$0.197949\pi$
$992$		0			0
$993$	−8.73439	−	38.2679i	−0.277178	−	1.21439i
$994$		0			0
$995$	2.46868	+	4.27587i	0.0782623	+	0.135554i
$996$		0			0
$997$	13.2681	+	6.38960i	0.420206	+	0.202361i	0.632024	−	0.774948i	$-0.282224\pi$
$997$	13.2681	+	6.38960i	0.420206	+	0.202361i	−0.211818	+	0.977309i	$0.567939\pi$
$998$		0			0
$999$	−3.04477	−	2.07589i	−0.0963322	−	0.0656782i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.273.1		36
4.3	odd	2		43.2.g.a.15.2	✓	36
12.11	even	2		387.2.y.c.316.2		36
43.23	even	21	inner	688.2.bg.c.625.1		36
172.23	odd	42		43.2.g.a.23.2	yes	36
172.111	odd	42		1849.2.a.n.1.8		18
172.147	even	42		1849.2.a.o.1.11		18
516.23	even	42		387.2.y.c.109.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.15.2	✓	36	4.3	odd	2
43.2.g.a.23.2	yes	36	172.23	odd	42
387.2.y.c.109.2		36	516.23	even	42
387.2.y.c.316.2		36	12.11	even	2
688.2.bg.c.273.1		36	1.1	even	1	trivial
688.2.bg.c.625.1		36	43.23	even	21	inner
1849.2.a.n.1.8		18	172.111	odd	42
1849.2.a.o.1.11		18	172.147	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)

\(f(q)\)	\(=\)	\(q+(-0.922165 + 2.34964i) q^{3} +(-0.0373507 - 0.498411i) q^{5} +(1.65334 - 2.86367i) q^{7} +(-2.47126 - 2.29299i) q^{9} +O(q^{10})\)	\(q+(-0.922165 + 2.34964i) q^{3} +(-0.0373507 - 0.498411i) q^{5} +(1.65334 - 2.86367i) q^{7} +(-2.47126 - 2.29299i) q^{9} +(-0.828392 + 3.62942i) q^{11} +(-3.87254 + 2.64025i) q^{13} +(1.20553 + 0.371857i) q^{15} +(-0.343032 + 4.57744i) q^{17} +(-4.44115 + 4.12079i) q^{19} +(5.20394 + 6.52553i) q^{21} +(0.371666 - 0.114644i) q^{23} +(4.69714 - 0.707979i) q^{25} +(0.844152 - 0.406522i) q^{27} +(-1.29147 - 3.29062i) q^{29} +(-0.861113 - 0.129792i) q^{31} +(-7.76391 - 5.29335i) q^{33} +(-1.48904 - 0.717083i) q^{35} +(-1.96656 - 3.40619i) q^{37} +(-2.63252 - 11.5338i) q^{39} +(-6.72350 + 8.43100i) q^{41} +(-6.26851 + 1.92504i) q^{43} +(-1.05055 + 1.31735i) q^{45} +(0.776794 + 3.40336i) q^{47} +(-1.96707 - 3.40706i) q^{49} +(-10.4390 - 5.02716i) q^{51} +(3.94414 + 2.68907i) q^{53} +(1.83988 + 0.277318i) q^{55} +(-5.58689 - 14.2352i) q^{57} +(6.05684 - 2.91682i) q^{59} +(-5.18836 + 0.782019i) q^{61} +(-10.6522 + 3.28577i) q^{63} +(1.46057 + 1.83150i) q^{65} +(7.73611 - 7.17806i) q^{67} +(-0.0733661 + 0.979002i) q^{69} +(-0.859578 - 0.265145i) q^{71} +(-0.798611 + 0.544484i) q^{73} +(-2.66804 + 11.6894i) q^{75} +(9.02385 + 8.37291i) q^{77} +(-0.500983 + 0.867728i) q^{79} +(-0.579056 - 7.72697i) q^{81} +(1.24798 - 3.17979i) q^{83} +2.29426 q^{85} +8.92272 q^{87} +(-0.672465 + 1.71341i) q^{89} +(1.15819 + 15.4549i) q^{91} +(1.09905 - 1.90361i) q^{93} +(2.21973 + 2.05961i) q^{95} +(-0.213753 + 0.936511i) q^{97} +(10.3694 - 7.06974i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more