LMFDB - Embedded newform 68.2.h.a.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(9,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("68.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.h (of order $8$, degree $4$, 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:	$0.542982733745$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			25.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	68.25
Dual form			68.2.h.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.70711i) q^{3} +(1.70711 + 0.707107i) q^{5} +(-0.707107 + 0.292893i) q^{7} +(-0.292893 - 0.292893i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 1.70711i) q^{3} +(1.70711 + 0.707107i) q^{5} +(-0.707107 + 0.292893i) q^{7} +(-0.292893 - 0.292893i) q^{9} +(-1.29289 - 3.12132i) q^{11} -6.82843i q^{13} +(-2.41421 + 2.41421i) q^{15} +(-3.00000 + 2.82843i) q^{17} +(1.82843 - 1.82843i) q^{19} -1.41421i q^{21} +(2.70711 + 6.53553i) q^{23} +(-1.12132 - 1.12132i) q^{25} +(-4.41421 + 1.82843i) q^{27} +(3.70711 + 1.53553i) q^{29} +(2.46447 - 5.94975i) q^{31} +6.24264 q^{33} -1.41421 q^{35} +(-2.53553 + 6.12132i) q^{37} +(11.6569 + 4.82843i) q^{39} +(-1.12132 + 0.464466i) q^{41} +(-3.00000 - 3.00000i) q^{43} +(-0.292893 - 0.707107i) q^{45} +3.65685i q^{47} +(-4.53553 + 4.53553i) q^{49} +(-2.70711 - 7.12132i) q^{51} +(-4.17157 + 4.17157i) q^{53} -6.24264i q^{55} +(1.82843 + 4.41421i) q^{57} +(-9.82843 - 9.82843i) q^{59} +(1.70711 - 0.707107i) q^{61} +(0.292893 + 0.121320i) q^{63} +(4.82843 - 11.6569i) q^{65} +11.3137 q^{67} -13.0711 q^{69} +(0.464466 - 1.12132i) q^{71} +(-7.94975 - 3.29289i) q^{73} +(2.70711 - 1.12132i) q^{75} +(1.82843 + 1.82843i) q^{77} +(4.36396 + 10.5355i) q^{79} -10.0711i q^{81} +(0.171573 - 0.171573i) q^{83} +(-7.12132 + 2.70711i) q^{85} +(-5.24264 + 5.24264i) q^{87} +10.8284i q^{89} +(2.00000 + 4.82843i) q^{91} +(8.41421 + 8.41421i) q^{93} +(4.41421 - 1.82843i) q^{95} +(0.878680 + 0.363961i) q^{97} +(-0.535534 + 1.29289i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^9	$ 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 12 * q^27 + 12 * q^29 + 24 * q^31 + 8 * q^33 + 4 * q^37 + 24 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 - 4 * q^49 - 8 * q^51 - 28 * q^53 - 4 * q^57 - 28 * q^59 + 4 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 16 * q^71 - 12 * q^73 + 8 * q^75 - 4 * q^77 - 8 * q^79 + 12 * q^83 - 20 * q^85 - 4 * q^87 + 8 * q^91 + 28 * q^93 + 12 * q^95 + 12 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$e\left(\frac{5}{8}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	+	1.70711i	−0.408248	+	0.985599i	0.577350	+	0.816497i	$0.304087\pi$
$3$	−0.707107	+	1.70711i	−0.408248	+	0.985599i	−0.985599	+	0.169102i	$0.945913\pi$
$4$		0			0
$5$	1.70711	+	0.707107i	0.763441	+	0.316228i	0.730213	−	0.683220i	$-0.239421\pi$
$5$	1.70711	+	0.707107i	0.763441	+	0.316228i	0.0332288	+	0.999448i	$0.489421\pi$
$6$		0			0
$7$	−0.707107	+	0.292893i	−0.267261	+	0.110703i	−0.512290	−	0.858813i	$-0.671203\pi$
$7$	−0.707107	+	0.292893i	−0.267261	+	0.110703i	0.245029	+	0.969516i	$0.421203\pi$
$8$		0			0
$9$	−0.292893	−	0.292893i	−0.0976311	−	0.0976311i
$10$		0			0
$11$	−1.29289	−	3.12132i	−0.389822	−	0.941113i	−0.989977	−	0.141230i	$-0.954894\pi$
$11$	−1.29289	−	3.12132i	−0.389822	−	0.941113i	0.600155	−	0.799884i	$-0.295106\pi$
$12$		0			0
$13$		−	6.82843i		−	1.89386i	−0.321433	−	0.946932i	$-0.604164\pi$
$13$		−	6.82843i		−	1.89386i	0.321433	−	0.946932i	$-0.395836\pi$
$14$		0			0
$15$	−2.41421	+	2.41421i	−0.623347	+	0.623347i
$16$		0			0
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$18$		0			0
$19$	1.82843	−	1.82843i	0.419470	−	0.419470i	−0.465551	−	0.885021i	$-0.654144\pi$
$19$	1.82843	−	1.82843i	0.419470	−	0.419470i	0.885021	+	0.465551i	$0.154144\pi$
$20$		0			0
$21$		−	1.41421i		−	0.308607i
$22$		0			0
$23$	2.70711	+	6.53553i	0.564471	+	1.36275i	0.906158	+	0.422939i	$0.139002\pi$
$23$	2.70711	+	6.53553i	0.564471	+	1.36275i	−0.341687	+	0.939814i	$0.610998\pi$
$24$		0			0
$25$	−1.12132	−	1.12132i	−0.224264	−	0.224264i
$26$		0			0
$27$	−4.41421	+	1.82843i	−0.849516	+	0.351881i
$28$		0			0
$29$	3.70711	+	1.53553i	0.688392	+	0.285141i	0.699330	−	0.714799i	$-0.253482\pi$
$29$	3.70711	+	1.53553i	0.688392	+	0.285141i	−0.0109378	+	0.999940i	$0.503482\pi$
$30$		0			0
$31$	2.46447	−	5.94975i	0.442631	−	1.06861i	−0.532391	−	0.846499i	$-0.678706\pi$
$31$	2.46447	−	5.94975i	0.442631	−	1.06861i	0.975022	−	0.222108i	$-0.0712936\pi$
$32$		0			0
$33$	6.24264			1.08670
$34$		0			0
$35$	−1.41421			−0.239046
$36$		0			0
$37$	−2.53553	+	6.12132i	−0.416839	+	1.00634i	0.566418	+	0.824118i	$0.308329\pi$
$37$	−2.53553	+	6.12132i	−0.416839	+	1.00634i	−0.983258	+	0.182221i	$0.941671\pi$
$38$		0			0
$39$	11.6569	+	4.82843i	1.86659	+	0.773167i
$40$		0			0
$41$	−1.12132	+	0.464466i	−0.175121	+	0.0725374i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−1.12132	+	0.464466i	−0.175121	+	0.0725374i	0.293400	+	0.955990i	$0.405213\pi$
$42$		0			0
$43$	−3.00000	−	3.00000i	−0.457496	−	0.457496i	0.440337	−	0.897833i	$-0.354859\pi$
$43$	−3.00000	−	3.00000i	−0.457496	−	0.457496i	−0.897833	+	0.440337i	$0.854859\pi$
$44$		0			0
$45$	−0.292893	−	0.707107i	−0.0436619	−	0.105409i
$46$		0			0
$47$			3.65685i			0.533407i	0.963779	+	0.266704i	$0.0859344\pi$
$47$			3.65685i			0.533407i	−0.963779	+	0.266704i	$0.914066\pi$
$48$		0			0
$49$	−4.53553	+	4.53553i	−0.647933	+	0.647933i
$50$		0			0
$51$	−2.70711	−	7.12132i	−0.379071	−	0.997184i
$52$		0			0
$53$	−4.17157	+	4.17157i	−0.573010	+	0.573010i	−0.932968	−	0.359959i	$-0.882791\pi$
$53$	−4.17157	+	4.17157i	−0.573010	+	0.573010i	0.359959	+	0.932968i	$0.382791\pi$
$54$		0			0
$55$		−	6.24264i		−	0.841757i
$56$		0			0
$57$	1.82843	+	4.41421i	0.242181	+	0.584677i
$58$		0			0
$59$	−9.82843	−	9.82843i	−1.27955	−	1.27955i	−0.940918	−	0.338634i	$-0.890035\pi$
$59$	−9.82843	−	9.82843i	−1.27955	−	1.27955i	−0.338634	−	0.940918i	$-0.609965\pi$
$60$		0			0
$61$	1.70711	−	0.707107i	0.218573	−	0.0905357i	−0.270710	−	0.962661i	$-0.587259\pi$
$61$	1.70711	−	0.707107i	0.218573	−	0.0905357i	0.489283	+	0.872125i	$0.337259\pi$
$62$		0			0
$63$	0.292893	+	0.121320i	0.0369011	+	0.0152849i
$64$		0			0
$65$	4.82843	−	11.6569i	0.598893	−	1.44585i
$66$		0			0
$67$	11.3137			1.38219			0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137			1.38219			0.691095	+	0.722764i	$0.257129\pi$
$68$		0			0
$69$	−13.0711			−1.57357
$70$		0			0
$71$	0.464466	−	1.12132i	0.0551220	−	0.133076i	−0.893919	−	0.448228i	$-0.852055\pi$
$71$	0.464466	−	1.12132i	0.0551220	−	0.133076i	0.949041	+	0.315152i	$0.102055\pi$
$72$		0			0
$73$	−7.94975	−	3.29289i	−0.930448	−	0.385404i	−0.134599	−	0.990900i	$-0.542975\pi$
$73$	−7.94975	−	3.29289i	−0.930448	−	0.385404i	−0.795848	+	0.605496i	$0.792975\pi$
$74$		0			0
$75$	2.70711	−	1.12132i	0.312590	−	0.129479i
$76$		0			0
$77$	1.82843	+	1.82843i	0.208369	+	0.208369i
$78$		0			0
$79$	4.36396	+	10.5355i	0.490984	+	1.18534i	0.954220	+	0.299105i	$0.0966882\pi$
$79$	4.36396	+	10.5355i	0.490984	+	1.18534i	−0.463236	+	0.886235i	$0.653312\pi$
$80$		0			0
$81$		−	10.0711i		−	1.11901i
$82$		0			0
$83$	0.171573	−	0.171573i	0.0188326	−	0.0188326i	−0.697628	−	0.716460i	$-0.745761\pi$
$83$	0.171573	−	0.171573i	0.0188326	−	0.0188326i	0.716460	+	0.697628i	$0.245761\pi$
$84$		0			0
$85$	−7.12132	+	2.70711i	−0.772416	+	0.293627i
$86$		0			0
$87$	−5.24264	+	5.24264i	−0.562070	+	0.562070i
$88$		0			0
$89$			10.8284i			1.14781i	0.818922	+	0.573905i	$0.194572\pi$
$89$			10.8284i			1.14781i	−0.818922	+	0.573905i	$0.805428\pi$
$90$		0			0
$91$	2.00000	+	4.82843i	0.209657	+	0.506157i
$92$		0			0
$93$	8.41421	+	8.41421i	0.872513	+	0.872513i
$94$		0			0
$95$	4.41421	−	1.82843i	0.452889	−	0.187593i
$96$		0			0
$97$	0.878680	+	0.363961i	0.0892164	+	0.0369546i	0.426845	−	0.904325i	$-0.359625\pi$
$97$	0.878680	+	0.363961i	0.0892164	+	0.0369546i	−0.337629	+	0.941279i	$0.609625\pi$
$98$		0			0
$99$	−0.535534	+	1.29289i	−0.0538232	+	0.129941i
$100$		0			0
$101$	12.8284			1.27648			0.638238	−	0.769839i	$-0.279664\pi$
$101$	12.8284			1.27648			0.638238	+	0.769839i	$0.279664\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	1.00000	−	2.41421i	0.0975900	−	0.235603i
$106$		0			0
$107$	3.53553	+	1.46447i	0.341793	+	0.141575i	0.546976	−	0.837148i	$-0.315779\pi$
$107$	3.53553	+	1.46447i	0.341793	+	0.141575i	−0.205183	+	0.978724i	$0.565779\pi$
$108$		0			0
$109$	4.53553	−	1.87868i	0.434425	−	0.179945i	−0.154744	−	0.987955i	$-0.549455\pi$
$109$	4.53553	−	1.87868i	0.434425	−	0.179945i	0.589169	+	0.808010i	$0.299455\pi$
$110$		0			0
$111$	−8.65685	−	8.65685i	−0.821672	−	0.821672i
$112$		0			0
$113$	−0.535534	−	1.29289i	−0.0503788	−	0.121625i	0.896687	−	0.442666i	$-0.145967\pi$
$113$	−0.535534	−	1.29289i	−0.0503788	−	0.121625i	−0.947065	+	0.321041i	$0.895967\pi$
$114$		0			0
$115$			13.0711i			1.21888i
$116$		0			0
$117$	−2.00000	+	2.00000i	−0.184900	+	0.184900i
$118$		0			0
$119$	1.29289	−	2.87868i	0.118519	−	0.263888i
$120$		0			0
$121$	−0.292893	+	0.292893i	−0.0266267	+	0.0266267i
$122$		0			0
$123$		−	2.24264i		−	0.202212i
$124$		0			0
$125$	−4.65685	−	11.2426i	−0.416522	−	1.00557i
$126$		0			0
$127$	−0.171573	−	0.171573i	−0.0152246	−	0.0152246i	0.699453	−	0.714678i	$-0.253427\pi$
$127$	−0.171573	−	0.171573i	−0.0152246	−	0.0152246i	−0.714678	+	0.699453i	$0.753427\pi$
$128$		0			0
$129$	7.24264	−	3.00000i	0.637679	−	0.264135i
$130$		0			0
$131$	−10.1213	−	4.19239i	−0.884304	−	0.366291i	−0.106139	−	0.994351i	$-0.533849\pi$
$131$	−10.1213	−	4.19239i	−0.884304	−	0.366291i	−0.778164	+	0.628061i	$0.783849\pi$
$132$		0			0
$133$	−0.757359	+	1.82843i	−0.0656714	+	0.158545i
$134$		0			0
$135$	−8.82843			−0.759830
$136$		0			0
$137$	12.1421			1.03737			0.518686	−	0.854965i	$-0.326421\pi$
$137$	12.1421			1.03737			0.518686	+	0.854965i	$0.326421\pi$
$138$		0			0
$139$	−6.70711	+	16.1924i	−0.568889	+	1.37342i	0.333603	+	0.942714i	$0.391736\pi$
$139$	−6.70711	+	16.1924i	−0.568889	+	1.37342i	−0.902492	+	0.430707i	$0.858264\pi$
$140$		0			0
$141$	−6.24264	−	2.58579i	−0.525725	−	0.217763i
$142$		0			0
$143$	−21.3137	+	8.82843i	−1.78234	+	0.738270i
$144$		0			0
$145$	5.24264	+	5.24264i	0.435378	+	0.435378i
$146$		0			0
$147$	−4.53553	−	10.9497i	−0.374085	−	0.903120i
$148$		0			0
$149$		−	9.65685i		−	0.791120i	−0.918440	−	0.395560i	$-0.870550\pi$
$149$		−	9.65685i		−	0.791120i	0.918440	−	0.395560i	$-0.129450\pi$
$150$		0			0
$151$	10.3137	−	10.3137i	0.839318	−	0.839318i	−0.149451	−	0.988769i	$-0.547751\pi$
$151$	10.3137	−	10.3137i	0.839318	−	0.839318i	0.988769	+	0.149451i	$0.0477507\pi$
$152$		0			0
$153$	1.70711	+	0.0502525i	0.138011	+	0.00406268i
$154$		0			0
$155$	8.41421	−	8.41421i	0.675846	−	0.675846i
$156$		0			0
$157$		−	2.34315i		−	0.187003i	−0.995619	−	0.0935017i	$-0.970194\pi$
$157$		−	2.34315i		−	0.187003i	0.995619	−	0.0935017i	$-0.0298061\pi$
$158$		0			0
$159$	−4.17157	−	10.0711i	−0.330827	−	0.798688i
$160$		0			0
$161$	−3.82843	−	3.82843i	−0.301722	−	0.301722i
$162$		0			0
$163$	−6.70711	+	2.77817i	−0.525341	+	0.217603i	−0.629561	−	0.776951i	$-0.716765\pi$
$163$	−6.70711	+	2.77817i	−0.525341	+	0.217603i	0.104220	+	0.994554i	$0.466765\pi$
$164$		0			0
$165$	10.6569	+	4.41421i	0.829635	+	0.343646i
$166$		0			0
$167$	−4.70711	+	11.3640i	−0.364247	+	0.879370i	0.630422	+	0.776252i	$0.282882\pi$
$167$	−4.70711	+	11.3640i	−0.364247	+	0.879370i	−0.994669	+	0.103117i	$0.967118\pi$
$168$		0			0
$169$	−33.6274			−2.58672
$170$		0			0
$171$	−1.07107			−0.0819066
$172$		0			0
$173$	9.60660	−	23.1924i	0.730376	−	1.76328i	0.0890357	−	0.996028i	$-0.471621\pi$
$173$	9.60660	−	23.1924i	0.730376	−	1.76328i	0.641341	−	0.767256i	$-0.278379\pi$
$174$		0			0
$175$	1.12132	+	0.464466i	0.0847639	+	0.0351103i
$176$		0			0
$177$	23.7279	−	9.82843i	1.78350	−	0.738750i
$178$		0			0
$179$	13.0000	+	13.0000i	0.971666	+	0.971666i	0.999609	−	0.0279439i	$-0.00889597\pi$
$179$	13.0000	+	13.0000i	0.971666	+	0.971666i	−0.0279439	+	0.999609i	$0.508896\pi$
$180$		0			0
$181$	1.46447	+	3.53553i	0.108853	+	0.262794i	0.968914	−	0.247396i	$-0.0795749\pi$
$181$	1.46447	+	3.53553i	0.108853	+	0.262794i	−0.860061	+	0.510190i	$0.829575\pi$
$182$		0			0
$183$			3.41421i			0.252386i
$184$		0			0
$185$	−8.65685	+	8.65685i	−0.636465	+	0.636465i
$186$		0			0
$187$	12.7071	+	5.70711i	0.929236	+	0.417345i
$188$		0			0
$189$	2.58579	−	2.58579i	0.188088	−	0.188088i
$190$		0			0
$191$		−	0.343146i		−	0.0248292i	−0.999923	−	0.0124146i	$-0.996048\pi$
$191$		−	0.343146i		−	0.0248292i	0.999923	−	0.0124146i	$-0.00395178\pi$
$192$		0			0
$193$	5.94975	+	14.3640i	0.428272	+	1.03394i	0.979835	+	0.199807i	$0.0640316\pi$
$193$	5.94975	+	14.3640i	0.428272	+	1.03394i	−0.551563	+	0.834133i	$0.685968\pi$
$194$		0			0
$195$	16.4853	+	16.4853i	1.18054	+	1.18054i
$196$		0			0
$197$	−11.6066	+	4.80761i	−0.826936	+	0.342528i	−0.755689	−	0.654931i	$-0.772698\pi$
$197$	−11.6066	+	4.80761i	−0.826936	+	0.342528i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	1.53553	+	0.636039i	0.108851	+	0.0450876i	0.436444	−	0.899731i	$-0.356238\pi$
$199$	1.53553	+	0.636039i	0.108851	+	0.0450876i	−0.327593	+	0.944819i	$0.606238\pi$
$200$		0			0
$201$	−8.00000	+	19.3137i	−0.564276	+	1.36228i
$202$		0			0
$203$	−3.07107			−0.215547
$204$		0			0
$205$	−2.24264			−0.156633
$206$		0			0
$207$	1.12132	−	2.70711i	0.0779372	−	0.188157i
$208$		0			0
$209$	−8.07107	−	3.34315i	−0.558287	−	0.231250i
$210$		0			0
$211$	−1.87868	+	0.778175i	−0.129334	+	0.0535717i	−0.446412	−	0.894828i	$-0.647298\pi$
$211$	−1.87868	+	0.778175i	−0.129334	+	0.0535717i	0.317078	+	0.948399i	$0.397298\pi$
$212$		0			0
$213$	1.58579	+	1.58579i	0.108656	+	0.108656i
$214$		0			0
$215$	−3.00000	−	7.24264i	−0.204598	−	0.493944i
$216$		0			0
$217$			4.92893i			0.334598i
$218$		0			0
$219$	11.2426	−	11.2426i	0.759707	−	0.759707i
$220$		0			0
$221$	19.3137	+	20.4853i	1.29918	+	1.37799i
$222$		0			0
$223$	12.1716	−	12.1716i	0.815069	−	0.815069i	−0.170320	−	0.985389i	$-0.554480\pi$
$223$	12.1716	−	12.1716i	0.815069	−	0.815069i	0.985389	+	0.170320i	$0.0544801\pi$
$224$		0			0
$225$			0.656854i			0.0437903i
$226$		0			0
$227$	−7.43503	−	17.9497i	−0.493480	−	1.19137i	−0.952938	−	0.303167i	$-0.901956\pi$
$227$	−7.43503	−	17.9497i	−0.493480	−	1.19137i	0.459457	−	0.888200i	$-0.348044\pi$
$228$		0			0
$229$	−2.65685	−	2.65685i	−0.175570	−	0.175570i	0.613852	−	0.789421i	$-0.289619\pi$
$229$	−2.65685	−	2.65685i	−0.175570	−	0.175570i	−0.789421	+	0.613852i	$0.789619\pi$
$230$		0			0
$231$	−4.41421	+	1.82843i	−0.290434	+	0.120302i
$232$		0			0
$233$	−8.29289	−	3.43503i	−0.543285	−	0.225036i	0.0941253	−	0.995560i	$-0.469995\pi$
$233$	−8.29289	−	3.43503i	−0.543285	−	0.225036i	−0.637411	+	0.770524i	$0.719995\pi$
$234$		0			0
$235$	−2.58579	+	6.24264i	−0.168678	+	0.407225i
$236$		0			0
$237$	−21.0711			−1.36871
$238$		0			0
$239$	22.6274			1.46365			0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274			1.46365			0.731823	+	0.681495i	$0.238670\pi$
$240$		0			0
$241$	7.60660	−	18.3640i	0.489984	−	1.18293i	−0.464743	−	0.885445i	$-0.653853\pi$
$241$	7.60660	−	18.3640i	0.489984	−	1.18293i	0.954728	−	0.297481i	$-0.0961466\pi$
$242$		0			0
$243$	3.94975	+	1.63604i	0.253376	+	0.104952i
$244$		0			0
$245$	−10.9497	+	4.53553i	−0.699554	+	0.289765i
$246$		0			0
$247$	−12.4853	−	12.4853i	−0.794419	−	0.794419i
$248$		0			0
$249$	0.171573	+	0.414214i	0.0108730	+	0.0262497i
$250$		0			0
$251$			22.9706i			1.44989i	0.688807	+	0.724945i	$0.258135\pi$
$251$			22.9706i			1.44989i	−0.688807	+	0.724945i	$0.741865\pi$
$252$		0			0
$253$	16.8995	−	16.8995i	1.06246	−	1.06246i
$254$		0			0
$255$	0.414214	−	14.0711i	0.0259391	−	0.881164i
$256$		0			0
$257$	−2.65685	+	2.65685i	−0.165730	+	0.165730i	−0.785100	−	0.619370i	$-0.787388\pi$
$257$	−2.65685	+	2.65685i	−0.165730	+	0.165730i	0.619370	+	0.785100i	$0.287388\pi$
$258$		0			0
$259$		−	5.07107i		−	0.315101i
$260$		0			0
$261$	−0.636039	−	1.53553i	−0.0393698	−	0.0950472i
$262$		0			0
$263$	−9.34315	−	9.34315i	−0.576123	−	0.576123i	0.357710	−	0.933833i	$-0.383558\pi$
$263$	−9.34315	−	9.34315i	−0.576123	−	0.576123i	−0.933833	+	0.357710i	$0.883558\pi$
$264$		0			0
$265$	−10.0711	+	4.17157i	−0.618661	+	0.256258i
$266$		0			0
$267$	−18.4853	−	7.65685i	−1.13128	−	0.468592i
$268$		0			0
$269$	−3.22183	+	7.77817i	−0.196438	+	0.474244i	−0.991151	−	0.132742i	$-0.957622\pi$
$269$	−3.22183	+	7.77817i	−0.196438	+	0.474244i	0.794712	+	0.606986i	$0.207622\pi$
$270$		0			0
$271$	−11.3137			−0.687259			−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137			−0.687259			−0.343629	+	0.939105i	$0.611656\pi$
$272$		0			0
$273$	−9.65685			−0.584459
$274$		0			0
$275$	−2.05025	+	4.94975i	−0.123635	+	0.298481i
$276$		0			0
$277$	−13.9497	−	5.77817i	−0.838159	−	0.347177i	−0.0780317	−	0.996951i	$-0.524864\pi$
$277$	−13.9497	−	5.77817i	−0.838159	−	0.347177i	−0.760127	+	0.649774i	$0.774864\pi$
$278$		0			0
$279$	−2.46447	+	1.02082i	−0.147544	+	0.0611146i
$280$		0			0
$281$	10.6569	+	10.6569i	0.635735	+	0.635735i	0.949500	−	0.313766i	$-0.101591\pi$
$281$	10.6569	+	10.6569i	0.635735	+	0.635735i	−0.313766	+	0.949500i	$0.601591\pi$
$282$		0			0
$283$	2.70711	+	6.53553i	0.160921	+	0.388497i	0.983688	−	0.179881i	$-0.0575712\pi$
$283$	2.70711	+	6.53553i	0.160921	+	0.388497i	−0.822768	+	0.568378i	$0.807571\pi$
$284$		0			0
$285$			8.82843i			0.522951i
$286$		0			0
$287$	0.656854	−	0.656854i	0.0387729	−	0.0387729i
$288$		0			0
$289$	1.00000	−	16.9706i	0.0588235	−	0.998268i
$290$		0			0
$291$	−1.24264	+	1.24264i	−0.0728449	+	0.0728449i
$292$		0			0
$293$			8.00000i			0.467365i	0.972313	+	0.233682i	$0.0750776\pi$
$293$			8.00000i			0.467365i	−0.972313	+	0.233682i	$0.924922\pi$
$294$		0			0
$295$	−9.82843	−	23.7279i	−0.572233	−	1.38149i
$296$		0			0
$297$	11.4142	+	11.4142i	0.662320	+	0.662320i
$298$		0			0
$299$	44.6274	−	18.4853i	2.58087	−	1.06903i
$300$		0			0
$301$	3.00000	+	1.24264i	0.172917	+	0.0716246i
$302$		0			0
$303$	−9.07107	+	21.8995i	−0.521119	+	1.25809i
$304$		0			0
$305$	3.41421			0.195497
$306$		0			0
$307$	6.34315			0.362022			0.181011	−	0.983481i	$-0.442063\pi$
$307$	6.34315			0.362022			0.181011	+	0.983481i	$0.442063\pi$
$308$		0			0
$309$	−2.82843	+	6.82843i	−0.160904	+	0.388456i
$310$		0			0
$311$	−11.7782	−	4.87868i	−0.667879	−	0.276645i	0.0228708	−	0.999738i	$-0.492719\pi$
$311$	−11.7782	−	4.87868i	−0.667879	−	0.276645i	−0.690750	+	0.723094i	$0.742719\pi$
$312$		0			0
$313$	21.0208	−	8.70711i	1.18817	−	0.492155i	0.301009	−	0.953621i	$-0.402677\pi$
$313$	21.0208	−	8.70711i	1.18817	−	0.492155i	0.887158	+	0.461466i	$0.152677\pi$
$314$		0			0
$315$	0.414214	+	0.414214i	0.0233383	+	0.0233383i
$316$		0			0
$317$	0.150758	+	0.363961i	0.00846739	+	0.0204421i	0.928057	−	0.372439i	$-0.121478\pi$
$317$	0.150758	+	0.363961i	0.00846739	+	0.0204421i	−0.919589	+	0.392881i	$0.871478\pi$
$318$		0			0
$319$		−	13.5563i		−	0.759010i
$320$		0			0
$321$	−5.00000	+	5.00000i	−0.279073	+	0.279073i
$322$		0			0
$323$	−0.313708	+	10.6569i	−0.0174552	+	0.592963i
$324$		0			0
$325$	−7.65685	+	7.65685i	−0.424726	+	0.424726i
$326$		0			0
$327$			9.07107i			0.501631i
$328$		0			0
$329$	−1.07107	−	2.58579i	−0.0590499	−	0.142559i
$330$		0			0
$331$	20.3137	+	20.3137i	1.11654	+	1.11654i	0.992245	+	0.124297i	$0.0396677\pi$
$331$	20.3137	+	20.3137i	1.11654	+	1.11654i	0.124297	+	0.992245i	$0.460332\pi$
$332$		0			0
$333$	2.53553	−	1.05025i	0.138946	−	0.0575535i
$334$		0			0
$335$	19.3137	+	8.00000i	1.05522	+	0.437087i
$336$		0			0
$337$	9.94975	−	24.0208i	0.541997	−	1.30850i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	9.94975	−	24.0208i	0.541997	−	1.30850i	0.923312	−	0.384052i	$-0.125472\pi$
$338$		0			0
$339$	2.58579			0.140441
$340$		0			0
$341$	−21.7574			−1.17823
$342$		0			0
$343$	3.92893	−	9.48528i	0.212142	−	0.512157i
$344$		0			0
$345$	−22.3137	−	9.24264i	−1.20133	−	0.497607i
$346$		0			0
$347$	−7.19239	+	2.97918i	−0.386108	+	0.159931i	−0.567289	−	0.823519i	$-0.692008\pi$
$347$	−7.19239	+	2.97918i	−0.386108	+	0.159931i	0.181182	+	0.983450i	$0.442008\pi$
$348$		0			0
$349$	19.8284	+	19.8284i	1.06139	+	1.06139i	0.997988	+	0.0634034i	$0.0201955\pi$
$349$	19.8284	+	19.8284i	1.06139	+	1.06139i	0.0634034	+	0.997988i	$0.479805\pi$
$350$		0			0
$351$	12.4853	+	30.1421i	0.666415	+	1.60887i
$352$		0			0
$353$			15.3137i			0.815066i	0.913190	+	0.407533i	$0.133611\pi$
$353$			15.3137i			0.815066i	−0.913190	+	0.407533i	$0.866389\pi$
$354$		0			0
$355$	1.58579	−	1.58579i	0.0841648	−	0.0841648i
$356$		0			0
$357$	4.00000	+	4.24264i	0.211702	+	0.224544i
$358$		0			0
$359$	−21.4853	+	21.4853i	−1.13395	+	1.13395i	−0.144436	+	0.989514i	$0.546137\pi$
$359$	−21.4853	+	21.4853i	−1.13395	+	1.13395i	−0.989514	+	0.144436i	$0.953863\pi$
$360$		0			0
$361$			12.3137i			0.648090i
$362$		0			0
$363$	−0.292893	−	0.707107i	−0.0153729	−	0.0371135i
$364$		0			0
$365$	−11.2426	−	11.2426i	−0.588467	−	0.588467i
$366$		0			0
$367$	−3.53553	+	1.46447i	−0.184553	+	0.0764445i	−0.473047	−	0.881037i	$-0.656846\pi$
$367$	−3.53553	+	1.46447i	−0.184553	+	0.0764445i	0.288493	+	0.957482i	$0.406846\pi$
$368$		0			0
$369$	0.464466	+	0.192388i	0.0241791	+	0.0100153i
$370$		0			0
$371$	1.72792	−	4.17157i	0.0897092	−	0.216577i
$372$		0			0
$373$	18.4853			0.957132			0.478566	−	0.878052i	$-0.341157\pi$
$373$	18.4853			0.957132			0.478566	+	0.878052i	$0.341157\pi$
$374$		0			0
$375$	22.4853			1.16113
$376$		0			0
$377$	10.4853	−	25.3137i	0.540019	−	1.30372i
$378$		0			0
$379$	−10.1213	−	4.19239i	−0.519897	−	0.215349i	0.107275	−	0.994229i	$-0.465788\pi$
$379$	−10.1213	−	4.19239i	−0.519897	−	0.215349i	−0.627172	+	0.778881i	$0.715788\pi$
$380$		0			0
$381$	0.414214	−	0.171573i	0.0212208	−	0.00878994i
$382$		0			0
$383$	−18.3137	−	18.3137i	−0.935787	−	0.935787i	0.0622724	−	0.998059i	$-0.480165\pi$
$383$	−18.3137	−	18.3137i	−0.935787	−	0.935787i	−0.998059	+	0.0622724i	$0.980165\pi$
$384$		0			0
$385$	1.82843	+	4.41421i	0.0931853	+	0.224969i
$386$		0			0
$387$			1.75736i			0.0893316i
$388$		0			0
$389$	−1.48528	+	1.48528i	−0.0753068	+	0.0753068i	−0.743757	−	0.668450i	$-0.766958\pi$
$389$	−1.48528	+	1.48528i	−0.0753068	+	0.0753068i	0.668450	+	0.743757i	$0.266958\pi$
$390$		0			0
$391$	−26.6066	−	11.9497i	−1.34555	−	0.604325i
$392$		0			0
$393$	14.3137	−	14.3137i	0.722031	−	0.722031i
$394$		0			0
$395$			21.0711i			1.06020i
$396$		0			0
$397$	−7.22183	−	17.4350i	−0.362453	−	0.875039i	−0.994940	−	0.100468i	$-0.967966\pi$
$397$	−7.22183	−	17.4350i	−0.362453	−	0.875039i	0.632487	−	0.774571i	$-0.282034\pi$
$398$		0			0
$399$	−2.58579	−	2.58579i	−0.129451	−	0.129451i
$400$		0			0
$401$	8.53553	−	3.53553i	0.426244	−	0.176556i	−0.159240	−	0.987240i	$-0.550904\pi$
$401$	8.53553	−	3.53553i	0.426244	−	0.176556i	0.585484	+	0.810684i	$0.300904\pi$
$402$		0			0
$403$	−40.6274	−	16.8284i	−2.02380	−	0.838284i
$404$		0			0
$405$	7.12132	−	17.1924i	0.353861	−	0.854297i
$406$		0			0
$407$	22.3848			1.10957
$408$		0			0
$409$	−11.6569			−0.576394			−0.288197	−	0.957571i	$-0.593056\pi$
$409$	−11.6569			−0.576394			−0.288197	+	0.957571i	$0.593056\pi$
$410$		0			0
$411$	−8.58579	+	20.7279i	−0.423506	+	1.02243i
$412$		0			0
$413$	9.82843	+	4.07107i	0.483625	+	0.200324i
$414$		0			0
$415$	0.414214	−	0.171573i	0.0203329	−	0.00842218i
$416$		0			0
$417$	−22.8995	−	22.8995i	−1.12139	−	1.12139i
$418$		0			0
$419$	−13.6360	−	32.9203i	−0.666164	−	1.60826i	−0.787974	−	0.615709i	$-0.788870\pi$
$419$	−13.6360	−	32.9203i	−0.666164	−	1.60826i	0.121810	−	0.992553i	$-0.461130\pi$
$420$		0			0
$421$		−	1.17157i		−	0.0570990i	−0.999592	−	0.0285495i	$-0.990911\pi$
$421$		−	1.17157i		−	0.0570990i	0.999592	−	0.0285495i	$-0.00908882\pi$
$422$		0			0
$423$	1.07107	−	1.07107i	0.0520771	−	0.0520771i
$424$		0			0
$425$	6.53553	+	0.192388i	0.317020	+	0.00933220i
$426$		0			0
$427$	−1.00000	+	1.00000i	−0.0483934	+	0.0483934i
$428$		0			0
$429$		−	42.6274i		−	2.05807i
$430$		0			0
$431$	7.87868	+	19.0208i	0.379503	+	0.916200i	0.992059	+	0.125774i	$0.0401413\pi$
$431$	7.87868	+	19.0208i	0.379503	+	0.916200i	−0.612556	+	0.790427i	$0.709859\pi$
$432$		0			0
$433$	5.82843	+	5.82843i	0.280096	+	0.280096i	0.833147	−	0.553051i	$-0.186537\pi$
$433$	5.82843	+	5.82843i	0.280096	+	0.280096i	−0.553051	+	0.833147i	$0.686537\pi$
$434$		0			0
$435$	−12.6569	+	5.24264i	−0.606850	+	0.251365i
$436$		0			0
$437$	16.8995	+	7.00000i	0.808412	+	0.334855i
$438$		0			0
$439$	−5.67767	+	13.7071i	−0.270980	+	0.654205i	−0.999526	−	0.0307930i	$-0.990197\pi$
$439$	−5.67767	+	13.7071i	−0.270980	+	0.654205i	0.728545	+	0.684998i	$0.240197\pi$
$440$		0			0
$441$	2.65685			0.126517
$442$		0			0
$443$	−21.6569			−1.02895			−0.514474	−	0.857506i	$-0.672013\pi$
$443$	−21.6569			−1.02895			−0.514474	+	0.857506i	$0.672013\pi$
$444$		0			0
$445$	−7.65685	+	18.4853i	−0.362970	+	0.876286i
$446$		0			0
$447$	16.4853	+	6.82843i	0.779727	+	0.322974i
$448$		0			0
$449$	−19.6066	+	8.12132i	−0.925293	+	0.383269i	−0.793891	−	0.608060i	$-0.791948\pi$
$449$	−19.6066	+	8.12132i	−0.925293	+	0.383269i	−0.131402	+	0.991329i	$0.541948\pi$
$450$		0			0
$451$	2.89949	+	2.89949i	0.136532	+	0.136532i
$452$		0			0
$453$	10.3137	+	24.8995i	0.484580	+	1.16988i
$454$		0			0
$455$			9.65685i			0.452720i
$456$		0			0
$457$	3.00000	−	3.00000i	0.140334	−	0.140334i	−0.633450	−	0.773784i	$-0.718362\pi$
$457$	3.00000	−	3.00000i	0.140334	−	0.140334i	0.773784	+	0.633450i	$0.218362\pi$
$458$		0			0
$459$	8.07107	−	17.9706i	0.376725	−	0.838794i
$460$		0			0
$461$	5.48528	−	5.48528i	0.255475	−	0.255475i	−0.567736	−	0.823211i	$-0.692180\pi$
$461$	5.48528	−	5.48528i	0.255475	−	0.255475i	0.823211	+	0.567736i	$0.192180\pi$
$462$		0			0
$463$		−	7.65685i		−	0.355844i	−0.984045	−	0.177922i	$-0.943062\pi$
$463$		−	7.65685i		−	0.355844i	0.984045	−	0.177922i	$-0.0569375\pi$
$464$		0			0
$465$	8.41421	+	20.3137i	0.390200	+	0.942026i
$466$		0			0
$467$	−19.9706	−	19.9706i	−0.924127	−	0.924127i	0.0731905	−	0.997318i	$-0.476682\pi$
$467$	−19.9706	−	19.9706i	−0.924127	−	0.924127i	−0.997318	+	0.0731905i	$0.976682\pi$
$468$		0			0
$469$	−8.00000	+	3.31371i	−0.369406	+	0.153013i
$470$		0			0
$471$	4.00000	+	1.65685i	0.184310	+	0.0763438i
$472$		0			0
$473$	−5.48528	+	13.2426i	−0.252214	+	0.608897i
$474$		0			0
$475$	−4.10051			−0.188144
$476$		0			0
$477$	2.44365			0.111887
$478$		0			0
$479$	3.15076	−	7.60660i	0.143962	−	0.347555i	−0.835408	−	0.549630i	$-0.814769\pi$
$479$	3.15076	−	7.60660i	0.143962	−	0.347555i	0.979370	+	0.202076i	$0.0647687\pi$
$480$		0			0
$481$	41.7990	+	17.3137i	1.90587	+	0.789437i
$482$		0			0
$483$	9.24264	−	3.82843i	0.420555	−	0.174199i
$484$		0			0
$485$	1.24264	+	1.24264i	0.0564254	+	0.0564254i
$486$		0			0
$487$	1.39340	+	3.36396i	0.0631409	+	0.152436i	0.952301	−	0.305161i	$-0.0987103\pi$
$487$	1.39340	+	3.36396i	0.0631409	+	0.152436i	−0.889160	+	0.457597i	$0.848710\pi$
$488$		0			0
$489$		−	13.4142i		−	0.606612i
$490$		0			0
$491$	−16.7990	+	16.7990i	−0.758128	+	0.758128i	−0.975981	−	0.217854i	$-0.930094\pi$
$491$	−16.7990	+	16.7990i	−0.758128	+	0.758128i	0.217854	+	0.975981i	$0.430094\pi$
$492$		0			0
$493$	−15.4645	+	5.87868i	−0.696485	+	0.264762i
$494$		0			0
$495$	−1.82843	+	1.82843i	−0.0821817	+	0.0821817i
$496$		0			0
$497$			0.928932i			0.0416683i
$498$		0			0
$499$	14.3640	+	34.6777i	0.643019	+	1.55239i	0.822587	+	0.568640i	$0.192530\pi$
$499$	14.3640	+	34.6777i	0.643019	+	1.55239i	−0.179567	+	0.983746i	$0.557470\pi$
$500$		0			0
$501$	−16.0711	−	16.0711i	−0.718002	−	0.718002i
$502$		0			0
$503$	7.29289	−	3.02082i	0.325174	−	0.134691i	−0.214124	−	0.976806i	$-0.568690\pi$
$503$	7.29289	−	3.02082i	0.325174	−	0.134691i	0.539298	+	0.842115i	$0.318690\pi$
$504$		0			0
$505$	21.8995	+	9.07107i	0.974515	+	0.403657i
$506$		0			0
$507$	23.7782	−	57.4056i	1.05603	−	2.54947i
$508$		0			0
$509$	6.68629			0.296365			0.148182	−	0.988960i	$-0.452658\pi$
$509$	6.68629			0.296365			0.148182	+	0.988960i	$0.452658\pi$
$510$		0			0
$511$	6.58579			0.291338
$512$		0			0
$513$	−4.72792	+	11.4142i	−0.208743	+	0.503950i
$514$		0			0
$515$	6.82843	+	2.82843i	0.300896	+	0.124635i
$516$		0			0
$517$	11.4142	−	4.72792i	0.501997	−	0.207934i
$518$		0			0
$519$	32.7990	+	32.7990i	1.43972	+	1.43972i
$520$		0			0
$521$	−10.5355	−	25.4350i	−0.461570	−	1.11433i	−0.967753	−	0.251902i	$-0.918944\pi$
$521$	−10.5355	−	25.4350i	−0.461570	−	1.11433i	0.506183	−	0.862426i	$-0.331056\pi$
$522$		0			0
$523$		−	34.0000i		−	1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$		−	34.0000i		−	1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$		0			0
$525$	−1.58579	+	1.58579i	−0.0692094	+	0.0692094i
$526$		0			0
$527$	9.43503	+	24.8198i	0.410996	+	1.08117i
$528$		0			0
$529$	−19.1213	+	19.1213i	−0.831362	+	0.831362i
$530$		0			0
$531$			5.75736i			0.249848i
$532$		0			0
$533$	3.17157	+	7.65685i	0.137376	+	0.331655i
$534$		0			0
$535$	5.00000	+	5.00000i	0.216169	+	0.216169i
$536$		0			0
$537$	−31.3848	+	13.0000i	−1.35435	+	0.560991i
$538$		0			0
$539$	20.0208	+	8.29289i	0.862358	+	0.357200i
$540$		0			0
$541$	−3.02082	+	7.29289i	−0.129875	+	0.313546i	−0.975418	−	0.220361i	$-0.929277\pi$
$541$	−3.02082	+	7.29289i	−0.129875	+	0.313546i	0.845543	+	0.533907i	$0.179277\pi$
$542$		0			0
$543$	−7.07107			−0.303449
$544$		0			0
$545$	9.07107			0.388562
$546$		0			0
$547$	−9.39340	+	22.6777i	−0.401633	+	0.969627i	0.585637	+	0.810573i	$0.300844\pi$
$547$	−9.39340	+	22.6777i	−0.401633	+	0.969627i	−0.987270	+	0.159054i	$0.949156\pi$
$548$		0			0
$549$	−0.707107	−	0.292893i	−0.0301786	−	0.0125004i
$550$		0			0
$551$	9.58579	−	3.97056i	0.408368	−	0.169152i
$552$		0			0
$553$	−6.17157	−	6.17157i	−0.262442	−	0.262442i
$554$		0			0
$555$	−8.65685	−	20.8995i	−0.367463	−	0.887134i
$556$		0			0
$557$		−	22.8284i		−	0.967272i	−0.875269	−	0.483636i	$-0.839316\pi$
$557$		−	22.8284i		−	0.967272i	0.875269	−	0.483636i	$-0.160684\pi$
$558$		0			0
$559$	−20.4853	+	20.4853i	−0.866435	+	0.866435i
$560$		0			0
$561$	−18.7279	+	17.6569i	−0.790693	+	0.745473i
$562$		0			0
$563$	27.2843	−	27.2843i	1.14989	−	1.14989i	0.163322	−	0.986573i	$-0.447779\pi$
$563$	27.2843	−	27.2843i	1.14989	−	1.14989i	0.986573	−	0.163322i	$-0.0522210\pi$
$564$		0			0
$565$		−	2.58579i		−	0.108785i
$566$		0			0
$567$	2.94975	+	7.12132i	0.123878	+	0.299067i
$568$		0			0
$569$	−19.4853	−	19.4853i	−0.816865	−	0.816865i	0.168787	−	0.985653i	$-0.446015\pi$
$569$	−19.4853	−	19.4853i	−0.816865	−	0.816865i	−0.985653	+	0.168787i	$0.946015\pi$
$570$		0			0
$571$	−23.5355	+	9.74874i	−0.984931	+	0.407972i	−0.816250	−	0.577699i	$-0.803951\pi$
$571$	−23.5355	+	9.74874i	−0.984931	+	0.407972i	−0.168681	+	0.985671i	$0.553951\pi$
$572$		0			0
$573$	0.585786	+	0.242641i	0.0244716	+	0.0101365i
$574$		0			0
$575$	4.29289	−	10.3640i	0.179026	−	0.432207i
$576$		0			0
$577$	1.79899			0.0748929			0.0374465	−	0.999299i	$-0.488078\pi$
$577$	1.79899			0.0748929			0.0374465	+	0.999299i	$0.488078\pi$
$578$		0			0
$579$	−28.7279			−1.19389
$580$		0			0
$581$	−0.0710678	+	0.171573i	−0.00294839	+	0.00711804i
$582$		0			0
$583$	18.4142	+	7.62742i	0.762639	+	0.315895i
$584$		0			0
$585$	−4.82843	+	2.00000i	−0.199631	+	0.0826898i
$586$		0			0
$587$	−20.6569	−	20.6569i	−0.852600	−	0.852600i	0.137853	−	0.990453i	$-0.455980\pi$
$587$	−20.6569	−	20.6569i	−0.852600	−	0.852600i	−0.990453	+	0.137853i	$0.955980\pi$
$588$		0			0
$589$	−6.37258	−	15.3848i	−0.262578	−	0.633919i
$590$		0			0
$591$		−	23.2132i		−	0.954864i
$592$		0			0
$593$	−23.4853	+	23.4853i	−0.964425	+	0.964425i	−0.999389	−	0.0349637i	$-0.988868\pi$
$593$	−23.4853	+	23.4853i	−0.964425	+	0.964425i	0.0349637	+	0.999389i	$0.488868\pi$
$594$		0			0
$595$	4.24264	−	4.00000i	0.173931	−	0.163984i
$596$		0			0
$597$	−2.17157	+	2.17157i	−0.0888766	+	0.0888766i
$598$		0			0
$599$		−	37.3137i		−	1.52460i	−0.647226	−	0.762298i	$-0.724071\pi$
$599$		−	37.3137i		−	1.52460i	0.647226	−	0.762298i	$-0.275929\pi$
$600$		0			0
$601$	1.94975	+	4.70711i	0.0795319	+	0.192007i	0.958644	−	0.284608i	$-0.0918635\pi$
$601$	1.94975	+	4.70711i	0.0795319	+	0.192007i	−0.879112	+	0.476615i	$0.841864\pi$
$602$		0			0
$603$	−3.31371	−	3.31371i	−0.134945	−	0.134945i
$604$		0			0
$605$	−0.707107	+	0.292893i	−0.0287480	+	0.0119078i
$606$		0			0
$607$	14.7071	+	6.09188i	0.596943	+	0.247262i	0.660634	−	0.750708i	$-0.270287\pi$
$607$	14.7071	+	6.09188i	0.596943	+	0.247262i	−0.0636914	+	0.997970i	$0.520287\pi$
$608$		0			0
$609$	2.17157	−	5.24264i	0.0879966	−	0.212443i
$610$		0			0
$611$	24.9706			1.01020
$612$		0			0
$613$	0.343146			0.0138595			0.00692976	−	0.999976i	$-0.497794\pi$
$613$	0.343146			0.0138595			0.00692976	+	0.999976i	$0.497794\pi$
$614$		0			0
$615$	1.58579	−	3.82843i	0.0639451	−	0.154377i
$616$		0			0
$617$	31.5061	+	13.0503i	1.26839	+	0.525383i	0.912474	−	0.409135i	$-0.134170\pi$
$617$	31.5061	+	13.0503i	1.26839	+	0.525383i	0.355914	+	0.934519i	$0.384170\pi$
$618$		0			0
$619$	−12.3640	+	5.12132i	−0.496950	+	0.205843i	−0.617058	−	0.786918i	$-0.711676\pi$
$619$	−12.3640	+	5.12132i	−0.496950	+	0.205843i	0.120109	+	0.992761i	$0.461676\pi$
$620$		0			0
$621$	−23.8995	−	23.8995i	−0.959054	−	0.959054i
$622$		0			0
$623$	−3.17157	−	7.65685i	−0.127066	−	0.306765i
$624$		0			0
$625$		−	14.5563i		−	0.582254i
$626$		0			0
$627$	11.4142	−	11.4142i	0.455840	−	0.455840i
$628$		0			0
$629$	−9.70711	−	25.5355i	−0.387048	−	1.01817i
$630$		0			0
$631$	10.3137	−	10.3137i	0.410582	−	0.410582i	−0.471359	−	0.881941i	$-0.656236\pi$
$631$	10.3137	−	10.3137i	0.410582	−	0.410582i	0.881941	+	0.471359i	$0.156236\pi$
$632$		0			0
$633$		−	3.75736i		−	0.149342i
$634$		0			0
$635$	−0.171573	−	0.414214i	−0.00680866	−	0.0164376i
$636$		0			0
$637$	30.9706	+	30.9706i	1.22710	+	1.22710i
$638$		0			0
$639$	−0.464466	+	0.192388i	−0.0183740	+	0.00761076i
$640$		0			0
$641$	−42.9203	−	17.7782i	−1.69525	−	0.702196i	−0.695385	−	0.718637i	$-0.744766\pi$
$641$	−42.9203	−	17.7782i	−1.69525	−	0.702196i	−0.999865	+	0.0164412i	$0.994766\pi$
$642$		0			0
$643$	2.12132	−	5.12132i	0.0836567	−	0.201965i	−0.876516	−	0.481373i	$-0.840138\pi$
$643$	2.12132	−	5.12132i	0.0836567	−	0.201965i	0.960172	+	0.279408i	$0.0901382\pi$
$644$		0			0
$645$	14.4853			0.570357
$646$		0			0
$647$	32.9706			1.29621			0.648103	−	0.761552i	$-0.275563\pi$
$647$	32.9706			1.29621			0.648103	+	0.761552i	$0.275563\pi$
$648$		0			0
$649$	−17.9706	+	43.3848i	−0.705406	+	1.70300i
$650$		0			0
$651$	−8.41421	−	3.48528i	−0.329779	−	0.136599i
$652$		0			0
$653$	−41.7487	+	17.2929i	−1.63375	+	0.676723i	−0.995645	−	0.0932228i	$-0.970283\pi$
$653$	−41.7487	+	17.2929i	−1.63375	+	0.676723i	−0.638109	+	0.769946i	$0.720283\pi$
$654$		0			0
$655$	−14.3137	−	14.3137i	−0.559283	−	0.559283i
$656$		0			0
$657$	1.36396	+	3.29289i	0.0532132	+	0.128468i
$658$		0			0
$659$		−	2.68629i		−	0.104643i	−0.998630	−	0.0523215i	$-0.983338\pi$
$659$		−	2.68629i		−	0.104643i	0.998630	−	0.0523215i	$-0.0166621\pi$
$660$		0			0
$661$	9.00000	−	9.00000i	0.350059	−	0.350059i	−0.510072	−	0.860132i	$-0.670381\pi$
$661$	9.00000	−	9.00000i	0.350059	−	0.350059i	0.860132	+	0.510072i	$0.170381\pi$
$662$		0			0
$663$	−48.6274	+	18.4853i	−1.88853	+	0.717909i
$664$		0			0
$665$	−2.58579	+	2.58579i	−0.100272	+	0.100272i
$666$		0			0
$667$			28.3848i			1.09906i
$668$		0			0
$669$	12.1716	+	29.3848i	0.470580	+	1.13608i
$670$		0			0
$671$	−4.41421	−	4.41421i	−0.170409	−	0.170409i
$672$		0			0
$673$	28.6777	−	11.8787i	1.10544	−	0.457889i	0.246077	−	0.969250i	$-0.420858\pi$
$673$	28.6777	−	11.8787i	1.10544	−	0.457889i	0.859366	+	0.511361i	$0.170858\pi$
$674$		0			0
$675$	7.00000	+	2.89949i	0.269430	+	0.111602i
$676$		0			0
$677$	0.636039	−	1.53553i	0.0244450	−	0.0590154i	−0.911186	−	0.411996i	$-0.864832\pi$
$677$	0.636039	−	1.53553i	0.0244450	−	0.0590154i	0.935631	+	0.352981i	$0.114832\pi$
$678$		0			0
$679$	−0.727922			−0.0279351
$680$		0			0
$681$	35.8995			1.37567
$682$		0			0
$683$	1.63604	−	3.94975i	0.0626013	−	0.151133i	−0.889483	−	0.456968i	$-0.848936\pi$
$683$	1.63604	−	3.94975i	0.0626013	−	0.151133i	0.952085	+	0.305835i	$0.0989355\pi$
$684$		0			0
$685$	20.7279	+	8.58579i	0.791973	+	0.328046i
$686$		0			0
$687$	6.41421	−	2.65685i	0.244718	−	0.101365i
$688$		0			0
$689$	28.4853	+	28.4853i	1.08520	+	1.08520i
$690$		0			0
$691$	2.70711	+	6.53553i	0.102983	+	0.248623i	0.966970	−	0.254892i	$-0.0820398\pi$
$691$	2.70711	+	6.53553i	0.102983	+	0.248623i	−0.863986	+	0.503515i	$0.832040\pi$
$692$		0			0
$693$		−	1.07107i		−	0.0406865i
$694$		0			0
$695$	−22.8995	+	22.8995i	−0.868627	+	0.868627i
$696$		0			0
$697$	2.05025	−	4.56497i	0.0776589	−	0.172911i
$698$		0			0
$699$	11.7279	−	11.7279i	0.443591	−	0.443591i
$700$		0			0
$701$			19.5147i			0.737061i	0.929616	+	0.368530i	$0.120139\pi$
$701$			19.5147i			0.737061i	−0.929616	+	0.368530i	$0.879861\pi$
$702$		0			0
$703$	6.55635	+	15.8284i	0.247277	+	0.596980i
$704$		0			0
$705$	−8.82843	−	8.82843i	−0.332498	−	0.332498i
$706$		0			0
$707$	−9.07107	+	3.75736i	−0.341153	+	0.141310i
$708$		0			0
$709$	−3.12132	−	1.29289i	−0.117224	−	0.0485556i	0.323301	−	0.946296i	$-0.395207\pi$
$709$	−3.12132	−	1.29289i	−0.117224	−	0.0485556i	−0.440524	+	0.897741i	$0.645207\pi$
$710$		0			0
$711$	1.80761	−	4.36396i	0.0677907	−	0.163661i
$712$		0			0
$713$	45.5563			1.70610
$714$		0			0
$715$	−42.6274			−1.59418
$716$		0			0
$717$	−16.0000	+	38.6274i	−0.597531	+	1.44257i
$718$		0			0
$719$	13.8787	+	5.74874i	0.517587	+	0.214392i	0.626157	−	0.779697i	$-0.284627\pi$
$719$	13.8787	+	5.74874i	0.517587	+	0.214392i	−0.108569	+	0.994089i	$0.534627\pi$
$720$		0			0
$721$	−2.82843	+	1.17157i	−0.105336	+	0.0436317i
$722$		0			0
$723$	25.9706	+	25.9706i	0.965856	+	0.965856i
$724$		0			0
$725$	−2.43503	−	5.87868i	−0.0904347	−	0.218329i
$726$		0			0
$727$			19.6569i			0.729032i	0.931197	+	0.364516i	$0.118766\pi$
$727$			19.6569i			0.729032i	−0.931197	+	0.364516i	$0.881234\pi$
$728$		0			0
$729$	15.7782	−	15.7782i	0.584377	−	0.584377i
$730$		0			0
$731$	17.4853	+	0.514719i	0.646716	+	0.0190376i
$732$		0			0
$733$	3.34315	−	3.34315i	0.123482	−	0.123482i	−0.642665	−	0.766147i	$-0.722171\pi$
$733$	3.34315	−	3.34315i	0.123482	−	0.123482i	0.766147	+	0.642665i	$0.222171\pi$
$734$		0			0
$735$		−	21.8995i		−	0.807775i
$736$		0			0
$737$	−14.6274	−	35.3137i	−0.538808	−	1.30080i
$738$		0			0
$739$	−11.9706	−	11.9706i	−0.440344	−	0.440344i	0.451783	−	0.892128i	$-0.350788\pi$
$739$	−11.9706	−	11.9706i	−0.440344	−	0.440344i	−0.892128	+	0.451783i	$0.850788\pi$
$740$		0			0
$741$	30.1421	−	12.4853i	1.10730	−	0.458658i
$742$		0			0
$743$	26.0208	+	10.7782i	0.954611	+	0.395413i	0.804962	−	0.593326i	$-0.202186\pi$
$743$	26.0208	+	10.7782i	0.954611	+	0.395413i	0.149649	+	0.988739i	$0.452186\pi$
$744$		0			0
$745$	6.82843	−	16.4853i	0.250174	−	0.603974i
$746$		0			0
$747$	−0.100505			−0.00367729
$748$		0			0
$749$	−2.92893			−0.107021
$750$		0			0
$751$	17.6360	−	42.5772i	0.643548	−	1.55366i	−0.178312	−	0.983974i	$-0.557064\pi$
$751$	17.6360	−	42.5772i	0.643548	−	1.55366i	0.821860	−	0.569689i	$-0.192936\pi$
$752$		0			0
$753$	−39.2132	−	16.2426i	−1.42901	−	0.591915i
$754$		0			0
$755$	24.8995	−	10.3137i	0.906185	−	0.375354i
$756$		0			0
$757$	−33.0000	−	33.0000i	−1.19941	−	1.19941i	−0.974345	−	0.225061i	$-0.927742\pi$
$757$	−33.0000	−	33.0000i	−1.19941	−	1.19941i	−0.225061	−	0.974345i	$-0.572258\pi$
$758$		0			0
$759$	16.8995	+	40.7990i	0.613413	+	1.48091i
$760$		0			0
$761$		−	17.8579i		−	0.647347i	−0.946169	−	0.323674i	$-0.895082\pi$
$761$		−	17.8579i		−	0.647347i	0.946169	−	0.323674i	$-0.104918\pi$
$762$		0			0
$763$	−2.65685	+	2.65685i	−0.0961846	+	0.0961846i
$764$		0			0
$765$	2.87868	+	1.29289i	0.104079	+	0.0467447i
$766$		0			0
$767$	−67.1127	+	67.1127i	−2.42330	+	2.42330i
$768$		0			0
$769$		−	3.79899i		−	0.136995i	−0.997651	−	0.0684975i	$-0.978179\pi$
$769$		−	3.79899i		−	0.136995i	0.997651	−	0.0684975i	$-0.0218205\pi$
$770$		0			0
$771$	−2.65685	−	6.41421i	−0.0956843	−	0.231002i
$772$		0			0
$773$	14.7990	+	14.7990i	0.532283	+	0.532283i	0.921251	−	0.388968i	$-0.127168\pi$
$773$	14.7990	+	14.7990i	0.532283	+	0.532283i	−0.388968	+	0.921251i	$0.627168\pi$
$774$		0			0
$775$	−9.43503	+	3.90812i	−0.338916	+	0.140384i
$776$		0			0
$777$	8.65685	+	3.58579i	0.310563	+	0.128639i
$778$		0			0
$779$	−1.20101	+	2.89949i	−0.0430307	+	0.103885i
$780$		0			0
$781$	−4.10051			−0.146728
$782$		0			0
$783$	−19.1716			−0.685136
$784$		0			0
$785$	1.65685	−	4.00000i	0.0591357	−	0.142766i
$786$		0			0
$787$	−32.6066	−	13.5061i	−1.16230	−	0.481440i	−0.283660	−	0.958925i	$-0.591549\pi$
$787$	−32.6066	−	13.5061i	−1.16230	−	0.481440i	−0.878640	+	0.477485i	$0.841549\pi$
$788$		0			0
$789$	22.5563	−	9.34315i	0.803027	−	0.332625i
$790$		0			0
$791$	0.757359	+	0.757359i	0.0269286	+	0.0269286i
$792$		0			0
$793$	−4.82843	−	11.6569i	−0.171462	−	0.413947i
$794$		0			0
$795$		−	20.1421i		−	0.714368i
$796$		0			0
$797$	−34.3137	+	34.3137i	−1.21545	+	1.21545i	−0.246247	+	0.969207i	$0.579197\pi$
$797$	−34.3137	+	34.3137i	−1.21545	+	1.21545i	−0.969207	+	0.246247i	$0.920803\pi$
$798$		0			0
$799$	−10.3431	−	10.9706i	−0.365914	−	0.388111i
$800$		0			0
$801$	3.17157	−	3.17157i	0.112062	−	0.112062i
$802$		0			0
$803$			29.0711i			1.02590i
$804$		0			0
$805$	−3.82843	−	9.24264i	−0.134934	−	0.325760i
$806$		0			0
$807$	−11.0000	−	11.0000i	−0.387218	−	0.387218i
$808$		0			0
$809$	27.5061	−	11.3934i	0.967063	−	0.400571i	0.157445	−	0.987528i	$-0.449674\pi$
$809$	27.5061	−	11.3934i	0.967063	−	0.400571i	0.809618	+	0.586957i	$0.199674\pi$
$810$		0			0
$811$	22.0208	+	9.12132i	0.773255	+	0.320293i	0.734190	−	0.678944i	$-0.237562\pi$
$811$	22.0208	+	9.12132i	0.773255	+	0.320293i	0.0390652	+	0.999237i	$0.487562\pi$
$812$		0			0
$813$	8.00000	−	19.3137i	0.280572	−	0.677361i
$814$		0			0
$815$	−13.4142			−0.469879
$816$		0			0
$817$	−10.9706			−0.383811
$818$		0			0
$819$	0.828427	−	2.00000i	0.0289476	−	0.0698857i
$820$		0			0
$821$	−5.46447	−	2.26346i	−0.190711	−	0.0789952i	0.285284	−	0.958443i	$-0.407912\pi$
$821$	−5.46447	−	2.26346i	−0.190711	−	0.0789952i	−0.475995	+	0.879448i	$0.657912\pi$
$822$		0			0
$823$	27.9203	−	11.5650i	0.973241	−	0.403130i	0.161323	−	0.986902i	$-0.448424\pi$
$823$	27.9203	−	11.5650i	0.973241	−	0.403130i	0.811918	+	0.583772i	$0.198424\pi$
$824$		0			0
$825$	−7.00000	−	7.00000i	−0.243709	−	0.243709i
$826$		0			0
$827$	1.39340	+	3.36396i	0.0484532	+	0.116976i	0.946253	−	0.323428i	$-0.104835\pi$
$827$	1.39340	+	3.36396i	0.0484532	+	0.116976i	−0.897800	+	0.440404i	$0.854835\pi$
$828$		0			0
$829$			29.6569i			1.03003i	0.857183	+	0.515013i	$0.172213\pi$
$829$			29.6569i			1.03003i	−0.857183	+	0.515013i	$0.827787\pi$
$830$		0			0
$831$	19.7279	−	19.7279i	0.684354	−	0.684354i
$832$		0			0
$833$	0.778175	−	26.4350i	0.0269622	−	0.915919i
$834$		0			0
$835$	−16.0711	+	16.0711i	−0.556162	+	0.556162i
$836$		0			0
$837$			30.7696i			1.06355i
$838$		0			0
$839$	−2.32233	−	5.60660i	−0.0801758	−	0.193561i	0.878709	−	0.477359i	$-0.158406\pi$
$839$	−2.32233	−	5.60660i	−0.0801758	−	0.193561i	−0.958884	+	0.283797i	$0.908406\pi$
$840$		0			0
$841$	−9.12132	−	9.12132i	−0.314528	−	0.314528i
$842$		0			0
$843$	−25.7279	+	10.6569i	−0.886117	+	0.367042i
$844$		0			0
$845$	−57.4056	−	23.7782i	−1.97481	−	0.817994i
$846$		0			0
$847$	0.121320	−	0.292893i	0.00416862	−	0.0100639i
$848$		0			0
$849$	−13.0711			−0.448598
$850$		0			0
$851$	−46.8701			−1.60668
$852$		0			0
$853$	8.29289	−	20.0208i	0.283943	−	0.685500i	−0.715977	−	0.698124i	$-0.754018\pi$
$853$	8.29289	−	20.0208i	0.283943	−	0.685500i	0.999920	+	0.0126242i	$0.00401850\pi$
$854$		0			0
$855$	−1.82843	−	0.757359i	−0.0625309	−	0.0259011i
$856$		0			0
$857$	−11.9497	+	4.94975i	−0.408196	+	0.169080i	−0.577326	−	0.816513i	$-0.695904\pi$
$857$	−11.9497	+	4.94975i	−0.408196	+	0.169080i	0.169131	+	0.985594i	$0.445904\pi$
$858$		0			0
$859$	11.8284	+	11.8284i	0.403581	+	0.403581i	0.879493	−	0.475912i	$-0.157882\pi$
$859$	11.8284	+	11.8284i	0.403581	+	0.403581i	−0.475912	+	0.879493i	$0.657882\pi$
$860$		0			0
$861$	0.656854	+	1.58579i	0.0223855	+	0.0540435i
$862$		0			0
$863$			14.0000i			0.476566i	0.971196	+	0.238283i	$0.0765845\pi$
$863$			14.0000i			0.476566i	−0.971196	+	0.238283i	$0.923415\pi$
$864$		0			0
$865$	32.7990	−	32.7990i	1.11520	−	1.11520i
$866$		0			0
$867$	28.2635	+	13.7071i	0.959877	+	0.465518i
$868$		0			0
$869$	27.2426	−	27.2426i	0.924143	−	0.924143i
$870$		0			0
$871$		−	77.2548i		−	2.61768i
$872$		0			0
$873$	−0.150758	−	0.363961i	−0.00510237	−	0.0123182i
$874$		0			0
$875$	6.58579	+	6.58579i	0.222640	+	0.222640i
$876$		0			0
$877$	−7.94975	+	3.29289i	−0.268444	+	0.111193i	−0.512845	−	0.858481i	$-0.671409\pi$
$877$	−7.94975	+	3.29289i	−0.268444	+	0.111193i	0.244401	+	0.969674i	$0.421409\pi$
$878$		0			0
$879$	−13.6569	−	5.65685i	−0.460634	−	0.190801i
$880$		0			0
$881$	−3.70711	+	8.94975i	−0.124896	+	0.301525i	−0.973943	−	0.226791i	$-0.927176\pi$
$881$	−3.70711	+	8.94975i	−0.124896	+	0.301525i	0.849048	+	0.528316i	$0.177176\pi$
$882$		0			0
$883$	14.3431			0.482685			0.241343	−	0.970440i	$-0.422412\pi$
$883$	14.3431			0.482685			0.241343	+	0.970440i	$0.422412\pi$
$884$		0			0
$885$	47.4558			1.59521
$886$		0			0
$887$	13.5772	−	32.7782i	0.455877	−	1.10058i	−0.514175	−	0.857685i	$-0.671902\pi$
$887$	13.5772	−	32.7782i	0.455877	−	1.10058i	0.970052	−	0.242898i	$-0.0780981\pi$
$888$		0			0
$889$	0.171573	+	0.0710678i	0.00575437	+	0.00238354i
$890$		0			0
$891$	−31.4350	+	13.0208i	−1.05311	+	0.436214i
$892$		0			0
$893$	6.68629	+	6.68629i	0.223748	+	0.223748i
$894$		0			0
$895$	13.0000	+	31.3848i	0.434542	+	1.04908i
$896$		0			0
$897$			89.2548i			2.98013i
$898$		0			0
$899$	18.2721	−	18.2721i	0.609408	−	0.609408i
$900$		0			0
$901$	0.715729	−	24.3137i	0.0238444	−	0.810007i
$902$		0			0
$903$	−4.24264	+	4.24264i	−0.141186	+	0.141186i
$904$		0			0
$905$			7.07107i			0.235050i
$906$		0			0
$907$	2.90812	+	7.02082i	0.0965624	+	0.233122i	0.964778	−	0.263064i	$-0.0847332\pi$
$907$	2.90812	+	7.02082i	0.0965624	+	0.233122i	−0.868216	+	0.496187i	$0.834733\pi$
$908$		0			0
$909$	−3.75736	−	3.75736i	−0.124624	−	0.124624i
$910$		0			0
$911$	2.80761	−	1.16295i	0.0930203	−	0.0385303i	−0.335688	−	0.941973i	$-0.608969\pi$
$911$	2.80761	−	1.16295i	0.0930203	−	0.0385303i	0.428708	+	0.903443i	$0.358969\pi$
$912$		0			0
$913$	−0.757359	−	0.313708i	−0.0250649	−	0.0103822i
$914$		0			0
$915$	−2.41421	+	5.82843i	−0.0798114	+	0.192682i
$916$		0			0
$917$	8.38478			0.276890
$918$		0			0
$919$	−14.3431			−0.473137			−0.236568	−	0.971615i	$-0.576023\pi$
$919$	−14.3431			−0.473137			−0.236568	+	0.971615i	$0.576023\pi$
$920$		0			0
$921$	−4.48528	+	10.8284i	−0.147795	+	0.356809i
$922$		0			0
$923$	−7.65685	−	3.17157i	−0.252028	−	0.104394i
$924$		0			0
$925$	9.70711	−	4.02082i	0.319168	−	0.132204i
$926$		0			0
$927$	−1.17157	−	1.17157i	−0.0384795	−	0.0384795i
$928$		0			0
$929$	18.9203	+	45.6777i	0.620755	+	1.49864i	0.850818	+	0.525460i	$0.176107\pi$
$929$	18.9203	+	45.6777i	0.620755	+	1.49864i	−0.230063	+	0.973176i	$0.573893\pi$
$930$		0			0
$931$			16.5858i			0.543577i
$932$		0			0
$933$	16.6569	−	16.6569i	0.545321	−	0.545321i
$934$		0			0
$935$	17.6569	+	18.7279i	0.577441	+	0.612469i
$936$		0			0
$937$	16.5147	−	16.5147i	0.539512	−	0.539512i	−0.383873	−	0.923386i	$-0.625410\pi$
$937$	16.5147	−	16.5147i	0.539512	−	0.539512i	0.923386	+	0.383873i	$0.125410\pi$
$938$		0			0
$939$			42.0416i			1.37198i
$940$		0			0
$941$	0.636039	+	1.53553i	0.0207343	+	0.0500570i	0.933908	−	0.357514i	$-0.116376\pi$
$941$	0.636039	+	1.53553i	0.0207343	+	0.0500570i	−0.913173	+	0.407571i	$0.866376\pi$
$942$		0			0
$943$	−6.07107	−	6.07107i	−0.197701	−	0.197701i
$944$		0			0
$945$	6.24264	−	2.58579i	0.203073	−	0.0841156i
$946$		0			0
$947$	8.36396	+	3.46447i	0.271792	+	0.112580i	0.514417	−	0.857540i	$-0.328008\pi$
$947$	8.36396	+	3.46447i	0.271792	+	0.112580i	−0.242625	+	0.970120i	$0.578008\pi$
$948$		0			0
$949$	−22.4853	+	54.2843i	−0.729903	+	1.76214i
$950$		0			0
$951$	−0.727922			−0.0236045
$952$		0			0
$953$	12.1421			0.393322			0.196661	−	0.980472i	$-0.436990\pi$
$953$	12.1421			0.393322			0.196661	+	0.980472i	$0.436990\pi$
$954$		0			0
$955$	0.242641	−	0.585786i	0.00785167	−	0.0189556i
$956$		0			0
$957$	23.1421	+	9.58579i	0.748079	+	0.309864i
$958$		0			0
$959$	−8.58579	+	3.55635i	−0.277250	+	0.114841i
$960$		0			0
$961$	−7.40559	−	7.40559i	−0.238890	−	0.238890i
$962$		0			0
$963$	−0.606602	−	1.46447i	−0.0195475	−	0.0471918i
$964$		0			0
$965$			28.7279i			0.924785i
$966$		0			0
$967$	−23.3431	+	23.3431i	−0.750665	+	0.750665i	−0.974603	−	0.223938i	$-0.928109\pi$
$967$	−23.3431	+	23.3431i	−0.750665	+	0.750665i	0.223938	+	0.974603i	$0.428109\pi$
$968$		0			0
$969$	−17.9706	−	8.07107i	−0.577298	−	0.259280i
$970$		0			0
$971$	26.3137	−	26.3137i	0.844447	−	0.844447i	−0.144987	−	0.989434i	$-0.546314\pi$
$971$	26.3137	−	26.3137i	0.844447	−	0.844447i	0.989434	+	0.144987i	$0.0463139\pi$
$972$		0			0
$973$		−	13.4142i		−	0.430040i
$974$		0			0
$975$	−7.65685	−	18.4853i	−0.245216	−	0.592003i
$976$		0			0
$977$	27.1421	+	27.1421i	0.868354	+	0.868354i	0.992290	−	0.123936i	$-0.0395519\pi$
$977$	27.1421	+	27.1421i	0.868354	+	0.868354i	−0.123936	+	0.992290i	$0.539552\pi$
$978$		0			0
$979$	33.7990	−	14.0000i	1.08022	−	0.447442i
$980$		0			0
$981$	−1.87868	−	0.778175i	−0.0599816	−	0.0248452i
$982$		0			0
$983$	−15.0503	+	36.3345i	−0.480029	+	1.15889i	0.479566	+	0.877506i	$0.340794\pi$
$983$	−15.0503	+	36.3345i	−0.480029	+	1.15889i	−0.959595	+	0.281386i	$0.909206\pi$
$984$		0			0
$985$	−23.2132			−0.739634
$986$		0			0
$987$	5.17157			0.164613
$988$		0			0
$989$	11.4853	−	27.7279i	0.365211	−	0.881697i
$990$		0			0
$991$	−41.9203	−	17.3640i	−1.33164	−	0.551584i	−0.400519	−	0.916288i	$-0.631170\pi$
$991$	−41.9203	−	17.3640i	−1.33164	−	0.551584i	−0.931123	+	0.364704i	$0.881170\pi$
$992$		0			0
$993$	−49.0416	+	20.3137i	−1.55629	+	0.644636i
$994$		0			0
$995$	2.17157	+	2.17157i	0.0688435	+	0.0688435i
$996$		0			0
$997$	−11.8492	−	28.6066i	−0.375269	−	0.905980i	−0.992839	−	0.119464i	$-0.961882\pi$
$997$	−11.8492	−	28.6066i	−0.375269	−	0.905980i	0.617569	−	0.786516i	$-0.288118\pi$
$998$		0			0
$999$		−	31.6569i		−	1.00158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.h.a.25.1	✓	4
3.2	odd	2		612.2.w.a.433.1		4
4.3	odd	2		272.2.v.c.161.1		4
17.2	even	8		1156.2.h.b.1001.1		4
17.3	odd	16		1156.2.e.f.905.1		8
17.4	even	4		1156.2.h.c.757.1		4
17.5	odd	16		1156.2.e.f.829.1		8
17.6	odd	16		1156.2.b.d.577.4		4
17.7	odd	16		1156.2.a.g.1.4		4
17.8	even	8		1156.2.h.c.733.1		4
17.9	even	8		1156.2.h.a.733.1		4
17.10	odd	16		1156.2.a.g.1.1		4
17.11	odd	16		1156.2.b.d.577.1		4
17.12	odd	16		1156.2.e.f.829.4		8
17.13	even	4		1156.2.h.a.757.1		4
17.14	odd	16		1156.2.e.f.905.4		8
17.15	even	8	inner	68.2.h.a.49.1	yes	4
17.16	even	2		1156.2.h.b.977.1		4
51.32	odd	8		612.2.w.a.253.1		4
68.7	even	16		4624.2.a.bl.1.1		4
68.15	odd	8		272.2.v.c.49.1		4
68.27	even	16		4624.2.a.bl.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.h.a.25.1	✓	4	1.1	even	1	trivial
68.2.h.a.49.1	yes	4	17.15	even	8	inner
272.2.v.c.49.1		4	68.15	odd	8
272.2.v.c.161.1		4	4.3	odd	2
612.2.w.a.253.1		4	51.32	odd	8
612.2.w.a.433.1		4	3.2	odd	2
1156.2.a.g.1.1		4	17.10	odd	16
1156.2.a.g.1.4		4	17.7	odd	16
1156.2.b.d.577.1		4	17.11	odd	16
1156.2.b.d.577.4		4	17.6	odd	16
1156.2.e.f.829.1		8	17.5	odd	16
1156.2.e.f.829.4		8	17.12	odd	16
1156.2.e.f.905.1		8	17.3	odd	16
1156.2.e.f.905.4		8	17.14	odd	16
1156.2.h.a.733.1		4	17.9	even	8
1156.2.h.a.757.1		4	17.13	even	4
1156.2.h.b.977.1		4	17.16	even	2
1156.2.h.b.1001.1		4	17.2	even	8
1156.2.h.c.733.1		4	17.8	even	8
1156.2.h.c.757.1		4	17.4	even	4
4624.2.a.bl.1.1		4	68.7	even	16
4624.2.a.bl.1.4		4	68.27	even	16

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:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.70711i) q^{3} +(1.70711 + 0.707107i) q^{5} +(-0.707107 + 0.292893i) q^{7} +(-0.292893 - 0.292893i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 1.70711i) q^{3} +(1.70711 + 0.707107i) q^{5} +(-0.707107 + 0.292893i) q^{7} +(-0.292893 - 0.292893i) q^{9} +(-1.29289 - 3.12132i) q^{11} -6.82843i q^{13} +(-2.41421 + 2.41421i) q^{15} +(-3.00000 + 2.82843i) q^{17} +(1.82843 - 1.82843i) q^{19} -1.41421i q^{21} +(2.70711 + 6.53553i) q^{23} +(-1.12132 - 1.12132i) q^{25} +(-4.41421 + 1.82843i) q^{27} +(3.70711 + 1.53553i) q^{29} +(2.46447 - 5.94975i) q^{31} +6.24264 q^{33} -1.41421 q^{35} +(-2.53553 + 6.12132i) q^{37} +(11.6569 + 4.82843i) q^{39} +(-1.12132 + 0.464466i) q^{41} +(-3.00000 - 3.00000i) q^{43} +(-0.292893 - 0.707107i) q^{45} +3.65685i q^{47} +(-4.53553 + 4.53553i) q^{49} +(-2.70711 - 7.12132i) q^{51} +(-4.17157 + 4.17157i) q^{53} -6.24264i q^{55} +(1.82843 + 4.41421i) q^{57} +(-9.82843 - 9.82843i) q^{59} +(1.70711 - 0.707107i) q^{61} +(0.292893 + 0.121320i) q^{63} +(4.82843 - 11.6569i) q^{65} +11.3137 q^{67} -13.0711 q^{69} +(0.464466 - 1.12132i) q^{71} +(-7.94975 - 3.29289i) q^{73} +(2.70711 - 1.12132i) q^{75} +(1.82843 + 1.82843i) q^{77} +(4.36396 + 10.5355i) q^{79} -10.0711i q^{81} +(0.171573 - 0.171573i) q^{83} +(-7.12132 + 2.70711i) q^{85} +(-5.24264 + 5.24264i) q^{87} +10.8284i q^{89} +(2.00000 + 4.82843i) q^{91} +(8.41421 + 8.41421i) q^{93} +(4.41421 - 1.82843i) q^{95} +(0.878680 + 0.363961i) q^{97} +(-0.535534 + 1.29289i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^9	\( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 12 * q^27 + 12 * q^29 + 24 * q^31 + 8 * q^33 + 4 * q^37 + 24 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 - 4 * q^49 - 8 * q^51 - 28 * q^53 - 4 * q^57 - 28 * q^59 + 4 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 16 * q^71 - 12 * q^73 + 8 * q^75 - 4 * q^77 - 8 * q^79 + 12 * q^83 - 20 * q^85 - 4 * q^87 + 8 * q^91 + 28 * q^93 + 12 * q^95 + 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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