LMFDB - Embedded newform 68.2.e.a.13.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(13,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(4))

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("68.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.e (of order $4$, degree $2$, 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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.2
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	68.13
Dual form			68.2.e.a.21.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30278 + 1.30278i) q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.30278 + 1.30278i) q^{7} +0.394449i q^{9} +O(q^{10})$	\(q+(1.30278 + 1.30278i) q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.30278 + 1.30278i) q^{7} +0.394449i q^{9} +(3.30278 - 3.30278i) q^{11} -4.60555 q^{13} -2.60555i q^{15} +(-3.60555 - 2.00000i) q^{17} +6.60555i q^{19} -3.39445 q^{21} +(-0.697224 + 0.697224i) q^{23} -3.00000i q^{25} +(3.39445 - 3.39445i) q^{27} +(5.60555 + 5.60555i) q^{29} +(0.697224 + 0.697224i) q^{31} +8.60555 q^{33} +2.60555 q^{35} +(3.00000 + 3.00000i) q^{37} +(-6.00000 - 6.00000i) q^{39} +(1.00000 - 1.00000i) q^{41} +10.6056i q^{43} +(0.394449 - 0.394449i) q^{45} -4.00000 q^{47} +3.60555i q^{49} +(-2.09167 - 7.30278i) q^{51} -9.21110i q^{53} -6.60555 q^{55} +(-8.60555 + 8.60555i) q^{57} +1.39445i q^{59} +(8.21110 - 8.21110i) q^{61} +(-0.513878 - 0.513878i) q^{63} +(4.60555 + 4.60555i) q^{65} -5.21110 q^{67} -1.81665 q^{69} +(-7.90833 - 7.90833i) q^{71} +(-7.00000 - 7.00000i) q^{73} +(3.90833 - 3.90833i) q^{75} +8.60555i q^{77} +(3.30278 - 3.30278i) q^{79} +10.0278 q^{81} +3.81665i q^{83} +(1.60555 + 5.60555i) q^{85} +14.6056i q^{87} -13.8167 q^{89} +(6.00000 - 6.00000i) q^{91} +1.81665i q^{93} +(6.60555 - 6.60555i) q^{95} +(-0.394449 - 0.394449i) q^{97} +(1.30278 + 1.30278i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7	$ 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 + 6 * q^11 - 4 * q^13 - 28 * q^21 - 10 * q^23 + 28 * q^27 + 8 * q^29 + 10 * q^31 + 20 * q^33 - 4 * q^35 + 12 * q^37 - 24 * q^39 + 4 * q^41 + 16 * q^45 - 16 * q^47 - 30 * q^51 - 12 * q^55 - 20 * q^57 + 4 * q^61 + 34 * q^63 + 4 * q^65 + 8 * q^67 + 36 * q^69 - 10 * q^71 - 28 * q^73 - 6 * q^75 + 6 * q^79 - 32 * q^81 - 8 * q^85 - 12 * q^89 + 24 * q^91 + 12 * q^95 - 16 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$35$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.30278	+	1.30278i	0.752158	+	0.752158i	0.974882	−	0.222724i	$-0.0714948\pi$
$3$	1.30278	+	1.30278i	0.752158	+	0.752158i	−0.222724	+	0.974882i	$0.571495\pi$
$4$		0			0
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	0.447214	−	0.894427i	$-0.352416\pi$
$5$	−1.00000	−	1.00000i	−0.447214	−	0.447214i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	−1.30278	+	1.30278i	−0.492403	+	0.492403i	−0.909063	−	0.416660i	$-0.863201\pi$
$7$	−1.30278	+	1.30278i	−0.492403	+	0.492403i	0.416660	+	0.909063i	$0.363201\pi$
$8$		0			0
$9$			0.394449i			0.131483i
$10$		0			0
$11$	3.30278	−	3.30278i	0.995824	−	0.995824i	−0.00416699	−	0.999991i	$-0.501326\pi$
$11$	3.30278	−	3.30278i	0.995824	−	0.995824i	0.999991	+	0.00416699i	$0.00132640\pi$
$12$		0			0
$13$	−4.60555			−1.27735			−0.638675	−	0.769477i	$-0.720517\pi$
$13$	−4.60555			−1.27735			−0.638675	+	0.769477i	$0.720517\pi$
$14$		0			0
$15$		−	2.60555i		−	0.672750i
$16$		0			0
$17$	−3.60555	−	2.00000i	−0.874475	−	0.485071i
$17$	−3.60555	−	2.00000i	−0.874475	−	0.485071i
$18$		0			0
$19$			6.60555i			1.51542i	0.652593	+	0.757709i	$0.273681\pi$
$19$			6.60555i			1.51542i	−0.652593	+	0.757709i	$0.726319\pi$
$20$		0			0
$21$	−3.39445			−0.740729
$22$		0			0
$23$	−0.697224	+	0.697224i	−0.145381	+	0.145381i	−0.776051	−	0.630670i	$-0.782780\pi$
$23$	−0.697224	+	0.697224i	−0.145381	+	0.145381i	0.630670	+	0.776051i	$0.282780\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$	3.39445	−	3.39445i	0.653262	−	0.653262i
$28$		0			0
$29$	5.60555	+	5.60555i	1.04092	+	1.04092i	0.999126	+	0.0417987i	$0.0133088\pi$
$29$	5.60555	+	5.60555i	1.04092	+	1.04092i	0.0417987	+	0.999126i	$0.486691\pi$
$30$		0			0
$31$	0.697224	+	0.697224i	0.125225	+	0.125225i	0.766942	−	0.641717i	$-0.221777\pi$
$31$	0.697224	+	0.697224i	0.125225	+	0.125225i	−0.641717	+	0.766942i	$0.721777\pi$
$32$		0			0
$33$	8.60555			1.49803
$34$		0			0
$35$	2.60555			0.440419
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		0			0
$39$	−6.00000	−	6.00000i	−0.960769	−	0.960769i
$40$		0			0
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$			10.6056i			1.61733i	0.588268	+	0.808666i	$0.299810\pi$
$43$			10.6056i			1.61733i	−0.588268	+	0.808666i	$0.700190\pi$
$44$		0			0
$45$	0.394449	−	0.394449i	0.0588009	−	0.0588009i
$46$		0			0
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$			3.60555i			0.515079i
$50$		0			0
$51$	−2.09167	−	7.30278i	−0.292893	−	1.02259i
$52$		0			0
$53$		−	9.21110i		−	1.26524i	−0.774461	−	0.632621i	$-0.781979\pi$
$53$		−	9.21110i		−	1.26524i	0.774461	−	0.632621i	$-0.218021\pi$
$54$		0			0
$55$	−6.60555			−0.890692
$56$		0			0
$57$	−8.60555	+	8.60555i	−1.13983	+	1.13983i
$58$		0			0
$59$			1.39445i			0.181542i	0.995872	+	0.0907709i	$0.0289331\pi$
$59$			1.39445i			0.181542i	−0.995872	+	0.0907709i	$0.971067\pi$
$60$		0			0
$61$	8.21110	−	8.21110i	1.05132	−	1.05132i	0.0527143	−	0.998610i	$-0.483213\pi$
$61$	8.21110	−	8.21110i	1.05132	−	1.05132i	0.998610	−	0.0527143i	$-0.0167873\pi$
$62$		0			0
$63$	−0.513878	−	0.513878i	−0.0647426	−	0.0647426i
$64$		0			0
$65$	4.60555	+	4.60555i	0.571248	+	0.571248i
$66$		0			0
$67$	−5.21110			−0.636638			−0.318319	−	0.947984i	$-0.603118\pi$
$67$	−5.21110			−0.636638			−0.318319	+	0.947984i	$0.603118\pi$
$68$		0			0
$69$	−1.81665			−0.218699
$70$		0			0
$71$	−7.90833	−	7.90833i	−0.938546	−	0.938546i	0.0596723	−	0.998218i	$-0.480994\pi$
$71$	−7.90833	−	7.90833i	−0.938546	−	0.938546i	−0.998218	+	0.0596723i	$0.980994\pi$
$72$		0			0
$73$	−7.00000	−	7.00000i	−0.819288	−	0.819288i	0.166717	−	0.986005i	$-0.446683\pi$
$73$	−7.00000	−	7.00000i	−0.819288	−	0.819288i	−0.986005	+	0.166717i	$0.946683\pi$
$74$		0			0
$75$	3.90833	−	3.90833i	0.451295	−	0.451295i
$76$		0			0
$77$			8.60555i			0.980694i
$78$		0			0
$79$	3.30278	−	3.30278i	0.371591	−	0.371591i	−0.496465	−	0.868057i	$-0.665369\pi$
$79$	3.30278	−	3.30278i	0.371591	−	0.371591i	0.868057	+	0.496465i	$0.165369\pi$
$80$		0			0
$81$	10.0278			1.11420
$82$		0			0
$83$			3.81665i			0.418932i	0.977816	+	0.209466i	$0.0671726\pi$
$83$			3.81665i			0.418932i	−0.977816	+	0.209466i	$0.932827\pi$
$84$		0			0
$85$	1.60555	+	5.60555i	0.174146	+	0.608007i
$86$		0			0
$87$			14.6056i			1.56588i
$88$		0			0
$89$	−13.8167			−1.46456			−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167			−1.46456			−0.732281	+	0.681002i	$0.761544\pi$
$90$		0			0
$91$	6.00000	−	6.00000i	0.628971	−	0.628971i
$92$		0			0
$93$			1.81665i			0.188378i
$94$		0			0
$95$	6.60555	−	6.60555i	0.677715	−	0.677715i
$96$		0			0
$97$	−0.394449	−	0.394449i	−0.0400502	−	0.0400502i	0.686798	−	0.726848i	$-0.259016\pi$
$97$	−0.394449	−	0.394449i	−0.0400502	−	0.0400502i	−0.726848	+	0.686798i	$0.759016\pi$
$98$		0			0
$99$	1.30278	+	1.30278i	0.130934	+	0.130934i
$100$		0			0
$101$	3.39445			0.337760			0.168880	−	0.985637i	$-0.445985\pi$
$101$	3.39445			0.337760			0.168880	+	0.985637i	$0.445985\pi$
$102$		0			0
$103$	1.21110			0.119333			0.0596667	−	0.998218i	$-0.480996\pi$
$103$	1.21110			0.119333			0.0596667	+	0.998218i	$0.480996\pi$
$104$		0			0
$105$	3.39445	+	3.39445i	0.331264	+	0.331264i
$106$		0			0
$107$	9.30278	+	9.30278i	0.899333	+	0.899333i	0.995377	−	0.0960438i	$-0.0306189\pi$
$107$	9.30278	+	9.30278i	0.899333	+	0.899333i	−0.0960438	+	0.995377i	$0.530619\pi$
$108$		0			0
$109$	8.21110	−	8.21110i	0.786481	−	0.786481i	−0.194435	−	0.980916i	$-0.562287\pi$
$109$	8.21110	−	8.21110i	0.786481	−	0.786481i	0.980916	+	0.194435i	$0.0622872\pi$
$110$		0			0
$111$			7.81665i			0.741924i
$112$		0			0
$113$	2.39445	−	2.39445i	0.225251	−	0.225251i	−0.585454	−	0.810705i	$-0.699084\pi$
$113$	2.39445	−	2.39445i	0.225251	−	0.225251i	0.810705	+	0.585454i	$0.199084\pi$
$114$		0			0
$115$	1.39445			0.130033
$116$		0			0
$117$		−	1.81665i		−	0.167950i
$118$		0			0
$119$	7.30278	−	2.09167i	0.669444	−	0.191743i
$120$		0			0
$121$		−	10.8167i		−	0.983332i
$122$		0			0
$123$	2.60555			0.234935
$124$		0			0
$125$	−8.00000	+	8.00000i	−0.715542	+	0.715542i
$126$		0			0
$127$		−	6.60555i		−	0.586148i	−0.956090	−	0.293074i	$-0.905322\pi$
$127$		−	6.60555i		−	0.586148i	0.956090	−	0.293074i	$-0.0946782\pi$
$128$		0			0
$129$	−13.8167	+	13.8167i	−1.21649	+	1.21649i
$130$		0			0
$131$	−2.69722	−	2.69722i	−0.235658	−	0.235658i	0.579392	−	0.815049i	$-0.303290\pi$
$131$	−2.69722	−	2.69722i	−0.235658	−	0.235658i	−0.815049	+	0.579392i	$0.803290\pi$
$132$		0			0
$133$	−8.60555	−	8.60555i	−0.746196	−	0.746196i
$134$		0			0
$135$	−6.78890			−0.584295
$136$		0			0
$137$	4.60555			0.393479			0.196739	−	0.980456i	$-0.436965\pi$
$137$	4.60555			0.393479			0.196739	+	0.980456i	$0.436965\pi$
$138$		0			0
$139$	−3.30278	−	3.30278i	−0.280138	−	0.280138i	0.553026	−	0.833164i	$-0.313473\pi$
$139$	−3.30278	−	3.30278i	−0.280138	−	0.280138i	−0.833164	+	0.553026i	$0.813473\pi$
$140$		0			0
$141$	−5.21110	−	5.21110i	−0.438854	−	0.438854i
$142$		0			0
$143$	−15.2111	+	15.2111i	−1.27202	+	1.27202i
$144$		0			0
$145$		−	11.2111i		−	0.931031i
$146$		0			0
$147$	−4.69722	+	4.69722i	−0.387421	+	0.387421i
$148$		0			0
$149$	−20.4222			−1.67305			−0.836526	−	0.547927i	$-0.815417\pi$
$149$	−20.4222			−1.67305			−0.836526	+	0.547927i	$0.815417\pi$
$150$		0			0
$151$			2.60555i			0.212037i	0.994364	+	0.106018i	$0.0338102\pi$
$151$			2.60555i			0.212037i	−0.994364	+	0.106018i	$0.966190\pi$
$152$		0			0
$153$	0.788897	−	1.42221i	0.0637786	−	0.114978i
$154$		0			0
$155$		−	1.39445i		−	0.112005i
$156$		0			0
$157$	20.4222			1.62987			0.814935	−	0.579553i	$-0.196773\pi$
$157$	20.4222			1.62987			0.814935	+	0.579553i	$0.196773\pi$
$158$		0			0
$159$	12.0000	−	12.0000i	0.951662	−	0.951662i
$160$		0			0
$161$		−	1.81665i		−	0.143172i
$162$		0			0
$163$	−0.697224	+	0.697224i	−0.0546108	+	0.0546108i	−0.733885	−	0.679274i	$-0.762295\pi$
$163$	−0.697224	+	0.697224i	−0.0546108	+	0.0546108i	0.679274	+	0.733885i	$0.262295\pi$
$164$		0			0
$165$	−8.60555	−	8.60555i	−0.669941	−	0.669941i
$166$		0			0
$167$	4.09167	+	4.09167i	0.316623	+	0.316623i	0.847469	−	0.530845i	$-0.178126\pi$
$167$	4.09167	+	4.09167i	0.316623	+	0.316623i	−0.530845	+	0.847469i	$0.678126\pi$
$168$		0			0
$169$	8.21110			0.631623
$170$		0			0
$171$	−2.60555			−0.199251
$172$		0			0
$173$	6.81665	+	6.81665i	0.518261	+	0.518261i	0.917045	−	0.398784i	$-0.130568\pi$
$173$	6.81665	+	6.81665i	0.518261	+	0.518261i	−0.398784	+	0.917045i	$0.630568\pi$
$174$		0			0
$175$	3.90833	+	3.90833i	0.295442	+	0.295442i
$176$		0			0
$177$	−1.81665	+	1.81665i	−0.136548	+	0.136548i
$178$		0			0
$179$		−	21.0278i		−	1.57169i	−0.618425	−	0.785844i	$-0.712229\pi$
$179$		−	21.0278i		−	1.57169i	0.618425	−	0.785844i	$-0.287771\pi$
$180$		0			0
$181$	−13.0000	+	13.0000i	−0.966282	+	0.966282i	−0.999450	−	0.0331674i	$-0.989441\pi$
$181$	−13.0000	+	13.0000i	−0.966282	+	0.966282i	0.0331674	+	0.999450i	$0.489441\pi$
$182$		0			0
$183$	21.3944			1.58152
$184$		0			0
$185$		−	6.00000i		−	0.441129i
$186$		0			0
$187$	−18.5139	+	5.30278i	−1.35387	+	0.387777i
$188$		0			0
$189$			8.84441i			0.643336i
$190$		0			0
$191$	21.2111			1.53478			0.767391	−	0.641180i	$-0.221555\pi$
$191$	21.2111			1.53478			0.767391	+	0.641180i	$0.221555\pi$
$192$		0			0
$193$	6.21110	−	6.21110i	0.447085	−	0.447085i	−0.447299	−	0.894384i	$-0.647614\pi$
$193$	6.21110	−	6.21110i	0.447085	−	0.447085i	0.894384	+	0.447299i	$0.147614\pi$
$194$		0			0
$195$			12.0000i			0.859338i
$196$		0			0
$197$	−10.3944	+	10.3944i	−0.740574	+	0.740574i	−0.972688	−	0.232115i	$-0.925436\pi$
$197$	−10.3944	+	10.3944i	−0.740574	+	0.740574i	0.232115	+	0.972688i	$0.425436\pi$
$198$		0			0
$199$	15.1194	+	15.1194i	1.07179	+	1.07179i	0.997216	+	0.0745723i	$0.0237592\pi$
$199$	15.1194	+	15.1194i	1.07179	+	1.07179i	0.0745723	+	0.997216i	$0.476241\pi$
$200$		0			0
$201$	−6.78890	−	6.78890i	−0.478852	−	0.478852i
$202$		0			0
$203$	−14.6056			−1.02511
$204$		0			0
$205$	−2.00000			−0.139686
$206$		0			0
$207$	−0.275019	−	0.275019i	−0.0191152	−	0.0191152i
$208$		0			0
$209$	21.8167	+	21.8167i	1.50909	+	1.50909i
$210$		0			0
$211$	−2.51388	+	2.51388i	−0.173063	+	0.173063i	−0.788323	−	0.615261i	$-0.789051\pi$
$211$	−2.51388	+	2.51388i	−0.173063	+	0.173063i	0.615261	+	0.788323i	$0.289051\pi$
$212$		0			0
$213$		−	20.6056i		−	1.41187i
$214$		0			0
$215$	10.6056	−	10.6056i	0.723293	−	0.723293i
$216$		0			0
$217$	−1.81665			−0.123322
$218$		0			0
$219$		−	18.2389i		−	1.23247i
$220$		0			0
$221$	16.6056	+	9.21110i	1.11701	+	0.619606i
$222$		0			0
$223$		−	3.81665i		−	0.255582i	−0.991801	−	0.127791i	$-0.959211\pi$
$223$		−	3.81665i		−	0.255582i	0.991801	−	0.127791i	$-0.0407887\pi$
$224$		0			0
$225$	1.18335			0.0788897
$226$		0			0
$227$	3.30278	−	3.30278i	0.219213	−	0.219213i	−0.588954	−	0.808167i	$-0.700460\pi$
$227$	3.30278	−	3.30278i	0.219213	−	0.219213i	0.808167	+	0.588954i	$0.200460\pi$
$228$		0			0
$229$			2.60555i			0.172180i	0.996287	+	0.0860898i	$0.0274372\pi$
$229$			2.60555i			0.172180i	−0.996287	+	0.0860898i	$0.972563\pi$
$230$		0			0
$231$	−11.2111	+	11.2111i	−0.737636	+	0.737636i
$232$		0			0
$233$	−1.60555	−	1.60555i	−0.105183	−	0.105183i	0.652557	−	0.757740i	$-0.273696\pi$
$233$	−1.60555	−	1.60555i	−0.105183	−	0.105183i	−0.757740	+	0.652557i	$0.773696\pi$
$234$		0			0
$235$	4.00000	+	4.00000i	0.260931	+	0.260931i
$236$		0			0
$237$	8.60555			0.558991
$238$		0			0
$239$	−29.2111			−1.88951			−0.944755	−	0.327779i	$-0.893700\pi$
$239$	−29.2111			−1.88951			−0.944755	+	0.327779i	$0.893700\pi$
$240$		0			0
$241$	9.00000	+	9.00000i	0.579741	+	0.579741i	0.934832	−	0.355091i	$-0.115550\pi$
$241$	9.00000	+	9.00000i	0.579741	+	0.579741i	−0.355091	+	0.934832i	$0.615550\pi$
$242$		0			0
$243$	2.88057	+	2.88057i	0.184789	+	0.184789i
$244$		0			0
$245$	3.60555	−	3.60555i	0.230350	−	0.230350i
$246$		0			0
$247$		−	30.4222i		−	1.93572i
$248$		0			0
$249$	−4.97224	+	4.97224i	−0.315103	+	0.315103i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			4.60555i			0.289549i
$254$		0			0
$255$	−5.21110	+	9.39445i	−0.326332	+	0.588303i
$256$		0			0
$257$		−	26.6056i		−	1.65961i	−0.558054	−	0.829804i	$-0.688452\pi$
$257$		−	26.6056i		−	1.65961i	0.558054	−	0.829804i	$-0.311548\pi$
$258$		0			0
$259$	−7.81665			−0.485703
$260$		0			0
$261$	−2.21110	+	2.21110i	−0.136864	+	0.136864i
$262$		0			0
$263$			13.0278i			0.803326i	0.915788	+	0.401663i	$0.131568\pi$
$263$			13.0278i			0.803326i	−0.915788	+	0.401663i	$0.868432\pi$
$264$		0			0
$265$	−9.21110	+	9.21110i	−0.565834	+	0.565834i
$266$		0			0
$267$	−18.0000	−	18.0000i	−1.10158	−	1.10158i
$268$		0			0
$269$	−11.4222	−	11.4222i	−0.696424	−	0.696424i	0.267213	−	0.963637i	$-0.413897\pi$
$269$	−11.4222	−	11.4222i	−0.696424	−	0.696424i	−0.963637	+	0.267213i	$0.913897\pi$
$270$		0			0
$271$	−13.2111			−0.802517			−0.401259	−	0.915965i	$-0.631427\pi$
$271$	−13.2111			−0.802517			−0.401259	+	0.915965i	$0.631427\pi$
$272$		0			0
$273$	15.6333			0.946171
$274$		0			0
$275$	−9.90833	−	9.90833i	−0.597495	−	0.597495i
$276$		0			0
$277$	−3.60555	−	3.60555i	−0.216637	−	0.216637i	0.590443	−	0.807079i	$-0.298953\pi$
$277$	−3.60555	−	3.60555i	−0.216637	−	0.216637i	−0.807079	+	0.590443i	$0.798953\pi$
$278$		0			0
$279$	−0.275019	+	0.275019i	−0.0164650	+	0.0164650i
$280$		0			0
$281$			10.4222i			0.621737i	0.950453	+	0.310868i	$0.100620\pi$
$281$			10.4222i			0.621737i	−0.950453	+	0.310868i	$0.899380\pi$
$282$		0			0
$283$	−23.1194	+	23.1194i	−1.37431	+	1.37431i	−0.520361	+	0.853946i	$0.674203\pi$
$283$	−23.1194	+	23.1194i	−1.37431	+	1.37431i	−0.853946	+	0.520361i	$0.825797\pi$
$284$		0			0
$285$	17.2111			1.01950
$286$		0			0
$287$			2.60555i			0.153801i
$288$		0			0
$289$	9.00000	+	14.4222i	0.529412	+	0.848365i
$290$		0			0
$291$		−	1.02776i		−	0.0602481i
$292$		0			0
$293$	21.6333			1.26383			0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333			1.26383			0.631916	+	0.775037i	$0.282269\pi$
$294$		0			0
$295$	1.39445	−	1.39445i	0.0811879	−	0.0811879i
$296$		0			0
$297$		−	22.4222i		−	1.30107i
$298$		0			0
$299$	3.21110	−	3.21110i	0.185703	−	0.185703i
$300$		0			0
$301$	−13.8167	−	13.8167i	−0.796379	−	0.796379i
$302$		0			0
$303$	4.42221	+	4.42221i	0.254049	+	0.254049i
$304$		0			0
$305$	−16.4222			−0.940333
$306$		0			0
$307$	6.78890			0.387463			0.193731	−	0.981055i	$-0.437941\pi$
$307$	6.78890			0.387463			0.193731	+	0.981055i	$0.437941\pi$
$308$		0			0
$309$	1.57779	+	1.57779i	0.0897576	+	0.0897576i
$310$		0			0
$311$	8.09167	+	8.09167i	0.458837	+	0.458837i	0.898274	−	0.439437i	$-0.144822\pi$
$311$	8.09167	+	8.09167i	0.458837	+	0.458837i	−0.439437	+	0.898274i	$0.644822\pi$
$312$		0			0
$313$	−8.21110	+	8.21110i	−0.464119	+	0.464119i	−0.900003	−	0.435884i	$-0.856436\pi$
$313$	−8.21110	+	8.21110i	−0.464119	+	0.464119i	0.435884	+	0.900003i	$0.356436\pi$
$314$		0			0
$315$			1.02776i			0.0579075i
$316$		0			0
$317$	12.3944	−	12.3944i	0.696142	−	0.696142i	−0.267434	−	0.963576i	$-0.586176\pi$
$317$	12.3944	−	12.3944i	0.696142	−	0.696142i	0.963576	+	0.267434i	$0.0861759\pi$
$318$		0			0
$319$	37.0278			2.07316
$320$		0			0
$321$			24.2389i			1.35288i
$322$		0			0
$323$	13.2111	−	23.8167i	0.735085	−	1.32519i
$324$		0			0
$325$			13.8167i			0.766410i
$326$		0			0
$327$	21.3944			1.18312
$328$		0			0
$329$	5.21110	−	5.21110i	0.287297	−	0.287297i
$330$		0			0
$331$			10.6056i			0.582934i	0.956581	+	0.291467i	$0.0941433\pi$
$331$			10.6056i			0.582934i	−0.956581	+	0.291467i	$0.905857\pi$
$332$		0			0
$333$	−1.18335	+	1.18335i	−0.0648470	+	0.0648470i
$334$		0			0
$335$	5.21110	+	5.21110i	0.284713	+	0.284713i
$336$		0			0
$337$	−1.78890	−	1.78890i	−0.0974475	−	0.0974475i	0.656702	−	0.754150i	$-0.271951\pi$
$337$	−1.78890	−	1.78890i	−0.0974475	−	0.0974475i	−0.754150	+	0.656702i	$0.771951\pi$
$338$		0			0
$339$	6.23886			0.338848
$340$		0			0
$341$	4.60555			0.249405
$342$		0			0
$343$	−13.8167	−	13.8167i	−0.746029	−	0.746029i
$344$		0			0
$345$	1.81665	+	1.81665i	0.0978054	+	0.0978054i
$346$		0			0
$347$	12.5139	−	12.5139i	0.671780	−	0.671780i	−0.286346	−	0.958126i	$-0.592441\pi$
$347$	12.5139	−	12.5139i	0.671780	−	0.671780i	0.958126	+	0.286346i	$0.0924407\pi$
$348$		0			0
$349$			19.6333i			1.05095i	0.850810	+	0.525473i	$0.176112\pi$
$349$			19.6333i			1.05095i	−0.850810	+	0.525473i	$0.823888\pi$
$350$		0			0
$351$	−15.6333	+	15.6333i	−0.834444	+	0.834444i
$352$		0			0
$353$	−16.7889			−0.893583			−0.446791	−	0.894638i	$-0.647433\pi$
$353$	−16.7889			−0.893583			−0.446791	+	0.894638i	$0.647433\pi$
$354$		0			0
$355$			15.8167i			0.839461i
$356$		0			0
$357$	12.2389	+	6.78890i	0.647749	+	0.359307i
$358$		0			0
$359$		−	9.39445i		−	0.495820i	−0.968783	−	0.247910i	$-0.920256\pi$
$359$		−	9.39445i		−	0.495820i	0.968783	−	0.247910i	$-0.0797437\pi$
$360$		0			0
$361$	−24.6333			−1.29649
$362$		0			0
$363$	14.0917	−	14.0917i	0.739621	−	0.739621i
$364$		0			0
$365$			14.0000i			0.732793i
$366$		0			0
$367$	3.90833	−	3.90833i	0.204013	−	0.204013i	−0.597704	−	0.801717i	$-0.703920\pi$
$367$	3.90833	−	3.90833i	0.204013	−	0.204013i	0.801717	+	0.597704i	$0.203920\pi$
$368$		0			0
$369$	0.394449	+	0.394449i	0.0205342	+	0.0205342i
$370$		0			0
$371$	12.0000	+	12.0000i	0.623009	+	0.623009i
$372$		0			0
$373$	−15.0278			−0.778108			−0.389054	−	0.921215i	$-0.627198\pi$
$373$	−15.0278			−0.778108			−0.389054	+	0.921215i	$0.627198\pi$
$374$		0			0
$375$	−20.8444			−1.07640
$376$		0			0
$377$	−25.8167	−	25.8167i	−1.32963	−	1.32963i
$378$		0			0
$379$	11.7250	+	11.7250i	0.602272	+	0.602272i	0.940915	−	0.338643i	$-0.109968\pi$
$379$	11.7250	+	11.7250i	0.602272	+	0.602272i	−0.338643	+	0.940915i	$0.609968\pi$
$380$		0			0
$381$	8.60555	−	8.60555i	0.440876	−	0.440876i
$382$		0			0
$383$			26.6056i			1.35948i	0.733453	+	0.679740i	$0.237907\pi$
$383$			26.6056i			1.35948i	−0.733453	+	0.679740i	$0.762093\pi$
$384$		0			0
$385$	8.60555	−	8.60555i	0.438580	−	0.438580i
$386$		0			0
$387$	−4.18335			−0.212651
$388$		0			0
$389$		−	31.8167i		−	1.61317i	−0.591119	−	0.806584i	$-0.701314\pi$
$389$		−	31.8167i		−	1.61317i	0.591119	−	0.806584i	$-0.298686\pi$
$390$		0			0
$391$	3.90833	−	1.11943i	0.197653	−	0.0566120i
$392$		0			0
$393$		−	7.02776i		−	0.354503i
$394$		0			0
$395$	−6.60555			−0.332361
$396$		0			0
$397$	−9.00000	+	9.00000i	−0.451697	+	0.451697i	−0.895918	−	0.444220i	$-0.853481\pi$
$397$	−9.00000	+	9.00000i	−0.451697	+	0.451697i	0.444220	+	0.895918i	$0.353481\pi$
$398$		0			0
$399$		−	22.4222i		−	1.12251i
$400$		0			0
$401$	−7.00000	+	7.00000i	−0.349563	+	0.349563i	−0.859947	−	0.510384i	$-0.829503\pi$
$401$	−7.00000	+	7.00000i	−0.349563	+	0.349563i	0.510384	+	0.859947i	$0.329503\pi$
$402$		0			0
$403$	−3.21110	−	3.21110i	−0.159956	−	0.159956i
$404$		0			0
$405$	−10.0278	−	10.0278i	−0.498283	−	0.498283i
$406$		0			0
$407$	19.8167			0.982275
$408$		0			0
$409$	8.78890			0.434583			0.217292	−	0.976107i	$-0.430278\pi$
$409$	8.78890			0.434583			0.217292	+	0.976107i	$0.430278\pi$
$410$		0			0
$411$	6.00000	+	6.00000i	0.295958	+	0.295958i
$412$		0			0
$413$	−1.81665	−	1.81665i	−0.0893917	−	0.0893917i
$414$		0			0
$415$	3.81665	−	3.81665i	0.187352	−	0.187352i
$416$		0			0
$417$		−	8.60555i		−	0.421416i
$418$		0			0
$419$	−1.30278	+	1.30278i	−0.0636448	+	0.0636448i	−0.738213	−	0.674568i	$-0.764330\pi$
$419$	−1.30278	+	1.30278i	−0.0636448	+	0.0636448i	0.674568	+	0.738213i	$0.264330\pi$
$420$		0			0
$421$	11.3944			0.555331			0.277666	−	0.960678i	$-0.410439\pi$
$421$	11.3944			0.555331			0.277666	+	0.960678i	$0.410439\pi$
$422$		0			0
$423$		−	1.57779i		−	0.0767150i
$424$		0			0
$425$	−6.00000	+	10.8167i	−0.291043	+	0.524685i
$426$		0			0
$427$			21.3944i			1.03535i
$428$		0			0
$429$	−39.6333			−1.91351
$430$		0			0
$431$	−27.1194	+	27.1194i	−1.30630	+	1.30630i	−0.382230	+	0.924067i	$0.624844\pi$
$431$	−27.1194	+	27.1194i	−1.30630	+	1.30630i	−0.924067	+	0.382230i	$0.875156\pi$
$432$		0			0
$433$		−	21.0278i		−	1.01053i	−0.862964	−	0.505265i	$-0.831395\pi$
$433$		−	21.0278i		−	1.01053i	0.862964	−	0.505265i	$-0.168605\pi$
$434$		0			0
$435$	14.6056	−	14.6056i	0.700283	−	0.700283i
$436$		0			0
$437$	−4.60555	−	4.60555i	−0.220313	−	0.220313i
$438$		0			0
$439$	−22.6972	−	22.6972i	−1.08328	−	1.08328i	−0.996202	−	0.0870779i	$-0.972247\pi$
$439$	−22.6972	−	22.6972i	−1.08328	−	1.08328i	−0.0870779	−	0.996202i	$-0.527753\pi$
$440$		0			0
$441$	−1.42221			−0.0677241
$442$		0			0
$443$	−2.78890			−0.132505			−0.0662523	−	0.997803i	$-0.521104\pi$
$443$	−2.78890			−0.132505			−0.0662523	+	0.997803i	$0.521104\pi$
$444$		0			0
$445$	13.8167	+	13.8167i	0.654972	+	0.654972i
$446$		0			0
$447$	−26.6056	−	26.6056i	−1.25840	−	1.25840i
$448$		0			0
$449$	4.81665	−	4.81665i	0.227312	−	0.227312i	−0.584257	−	0.811569i	$-0.698614\pi$
$449$	4.81665	−	4.81665i	0.227312	−	0.227312i	0.811569	+	0.584257i	$0.198614\pi$
$450$		0			0
$451$		−	6.60555i		−	0.311043i
$452$		0			0
$453$	−3.39445	+	3.39445i	−0.159485	+	0.159485i
$454$		0			0
$455$	−12.0000			−0.562569
$456$		0			0
$457$			29.3944i			1.37501i	0.726178	+	0.687507i	$0.241295\pi$
$457$			29.3944i			1.37501i	−0.726178	+	0.687507i	$0.758705\pi$
$458$		0			0
$459$	−19.0278	+	5.44996i	−0.888140	+	0.254382i
$460$		0			0
$461$		−	14.7889i		−	0.688788i	−0.938825	−	0.344394i	$-0.888084\pi$
$461$		−	14.7889i		−	0.688788i	0.938825	−	0.344394i	$-0.111916\pi$
$462$		0			0
$463$	3.63331			0.168854			0.0844271	−	0.996430i	$-0.473094\pi$
$463$	3.63331			0.168854			0.0844271	+	0.996430i	$0.473094\pi$
$464$		0			0
$465$	1.81665	−	1.81665i	0.0842453	−	0.0842453i
$466$		0			0
$467$			34.2389i			1.58439i	0.610271	+	0.792193i	$0.291061\pi$
$467$			34.2389i			1.58439i	−0.610271	+	0.792193i	$0.708939\pi$
$468$		0			0
$469$	6.78890	−	6.78890i	0.313482	−	0.313482i
$470$		0			0
$471$	26.6056	+	26.6056i	1.22592	+	1.22592i
$472$		0			0
$473$	35.0278	+	35.0278i	1.61058	+	1.61058i
$474$		0			0
$475$	19.8167			0.909250
$476$		0			0
$477$	3.63331			0.166358
$478$		0			0
$479$	8.69722	+	8.69722i	0.397386	+	0.397386i	0.877310	−	0.479924i	$-0.159336\pi$
$479$	8.69722	+	8.69722i	0.397386	+	0.397386i	−0.479924	+	0.877310i	$0.659336\pi$
$480$		0			0
$481$	−13.8167	−	13.8167i	−0.629985	−	0.629985i
$482$		0			0
$483$	2.36669	−	2.36669i	0.107688	−	0.107688i
$484$		0			0
$485$			0.788897i			0.0358220i
$486$		0			0
$487$	22.3305	−	22.3305i	1.01189	−	1.01189i	0.0119646	−	0.999928i	$-0.496191\pi$
$487$	22.3305	−	22.3305i	1.01189	−	1.01189i	0.999928	−	0.0119646i	$-0.00380853\pi$
$488$		0			0
$489$	−1.81665			−0.0821519
$490$		0			0
$491$		−	11.8167i		−	0.533278i	−0.963796	−	0.266639i	$-0.914087\pi$
$491$		−	11.8167i		−	0.533278i	0.963796	−	0.266639i	$-0.0859132\pi$
$492$		0			0
$493$	−9.00000	−	31.4222i	−0.405340	−	1.41518i
$494$		0			0
$495$		−	2.60555i		−	0.117111i
$496$		0			0
$497$	20.6056			0.924285
$498$		0			0
$499$	6.69722	−	6.69722i	0.299809	−	0.299809i	−0.541130	−	0.840939i	$-0.682003\pi$
$499$	6.69722	−	6.69722i	0.299809	−	0.299809i	0.840939	+	0.541130i	$0.182003\pi$
$500$		0			0
$501$			10.6611i			0.476301i
$502$		0			0
$503$	9.48612	−	9.48612i	0.422965	−	0.422965i	−0.463258	−	0.886223i	$-0.653320\pi$
$503$	9.48612	−	9.48612i	0.422965	−	0.422965i	0.886223	+	0.463258i	$0.153320\pi$
$504$		0			0
$505$	−3.39445	−	3.39445i	−0.151051	−	0.151051i
$506$		0			0
$507$	10.6972	+	10.6972i	0.475080	+	0.475080i
$508$		0			0
$509$	11.2111			0.496923			0.248462	−	0.968642i	$-0.420075\pi$
$509$	11.2111			0.496923			0.248462	+	0.968642i	$0.420075\pi$
$510$		0			0
$511$	18.2389			0.806840
$512$		0			0
$513$	22.4222	+	22.4222i	0.989965	+	0.989965i
$514$		0			0
$515$	−1.21110	−	1.21110i	−0.0533676	−	0.0533676i
$516$		0			0
$517$	−13.2111	+	13.2111i	−0.581024	+	0.581024i
$518$		0			0
$519$			17.7611i			0.779628i
$520$		0			0
$521$	13.0000	−	13.0000i	0.569540	−	0.569540i	−0.362459	−	0.932000i	$-0.618063\pi$
$521$	13.0000	−	13.0000i	0.569540	−	0.569540i	0.932000	+	0.362459i	$0.118063\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$		0			0
$525$			10.1833i			0.444438i
$526$		0			0
$527$	−1.11943	−	3.90833i	−0.0487631	−	0.170249i
$528$		0			0
$529$			22.0278i			0.957729i
$530$		0			0
$531$	−0.550039			−0.0238696
$532$		0			0
$533$	−4.60555	+	4.60555i	−0.199489	+	0.199489i
$534$		0			0
$535$		−	18.6056i		−	0.804388i
$536$		0			0
$537$	27.3944	−	27.3944i	1.18216	−	1.18216i
$538$		0			0
$539$	11.9083	+	11.9083i	0.512928	+	0.512928i
$540$		0			0
$541$	3.00000	+	3.00000i	0.128980	+	0.128980i	0.768650	−	0.639670i	$-0.220929\pi$
$541$	3.00000	+	3.00000i	0.128980	+	0.128980i	−0.639670	+	0.768650i	$0.720929\pi$
$542$		0			0
$543$	−33.8722			−1.45359
$544$		0			0
$545$	−16.4222			−0.703450
$546$		0			0
$547$	6.51388	+	6.51388i	0.278513	+	0.278513i	0.832515	−	0.554002i	$-0.186900\pi$
$547$	6.51388	+	6.51388i	0.278513	+	0.278513i	−0.554002	+	0.832515i	$0.686900\pi$
$548$		0			0
$549$	3.23886	+	3.23886i	0.138231	+	0.138231i
$550$		0			0
$551$	−37.0278	+	37.0278i	−1.57744	+	1.57744i
$552$		0			0
$553$			8.60555i			0.365945i
$554$		0			0
$555$	7.81665	−	7.81665i	0.331798	−	0.331798i
$556$		0			0
$557$	−28.6056			−1.21206			−0.606028	−	0.795443i	$-0.707238\pi$
$557$	−28.6056			−1.21206			−0.606028	+	0.795443i	$0.707238\pi$
$558$		0			0
$559$		−	48.8444i		−	2.06590i
$560$		0			0
$561$	−31.0278	−	17.2111i	−1.30999	−	0.726653i
$562$		0			0
$563$		−	13.3944i		−	0.564509i	−0.959340	−	0.282254i	$-0.908918\pi$
$563$		−	13.3944i		−	0.564509i	0.959340	−	0.282254i	$-0.0910822\pi$
$564$		0			0
$565$	−4.78890			−0.201470
$566$		0			0
$567$	−13.0639	+	13.0639i	−0.548633	+	0.548633i
$568$		0			0
$569$			1.57779i			0.0661446i	0.999453	+	0.0330723i	$0.0105292\pi$
$569$			1.57779i			0.0661446i	−0.999453	+	0.0330723i	$0.989471\pi$
$570$		0			0
$571$	−1.30278	+	1.30278i	−0.0545195	+	0.0545195i	−0.733841	−	0.679321i	$-0.762274\pi$
$571$	−1.30278	+	1.30278i	−0.0545195	+	0.0545195i	0.679321	+	0.733841i	$0.262274\pi$
$572$		0			0
$573$	27.6333	+	27.6333i	1.15440	+	1.15440i
$574$		0			0
$575$	2.09167	+	2.09167i	0.0872288	+	0.0872288i
$576$		0			0
$577$	33.4500			1.39254			0.696270	−	0.717780i	$-0.254842\pi$
$577$	33.4500			1.39254			0.696270	+	0.717780i	$0.254842\pi$
$578$		0			0
$579$	16.1833			0.672557
$580$		0			0
$581$	−4.97224	−	4.97224i	−0.206283	−	0.206283i
$582$		0			0
$583$	−30.4222	−	30.4222i	−1.25996	−	1.25996i
$584$		0			0
$585$	−1.81665	+	1.81665i	−0.0751094	+	0.0751094i
$586$		0			0
$587$		−	2.60555i		−	0.107543i	−0.998553	−	0.0537713i	$-0.982876\pi$
$587$		−	2.60555i		−	0.107543i	0.998553	−	0.0537713i	$-0.0171242\pi$
$588$		0			0
$589$	−4.60555	+	4.60555i	−0.189768	+	0.189768i
$590$		0			0
$591$	−27.0833			−1.11406
$592$		0			0
$593$		−	30.4222i		−	1.24929i	−0.780909	−	0.624645i	$-0.785244\pi$
$593$		−	30.4222i		−	1.24929i	0.780909	−	0.624645i	$-0.214756\pi$
$594$		0			0
$595$	−9.39445	−	5.21110i	−0.385135	−	0.213634i
$596$		0			0
$597$			39.3944i			1.61231i
$598$		0			0
$599$	−46.0555			−1.88178			−0.940889	−	0.338716i	$-0.890007\pi$
$599$	−46.0555			−1.88178			−0.940889	+	0.338716i	$0.890007\pi$
$600$		0			0
$601$	10.2111	−	10.2111i	0.416520	−	0.416520i	−0.467483	−	0.884002i	$-0.654839\pi$
$601$	10.2111	−	10.2111i	0.416520	−	0.416520i	0.884002	+	0.467483i	$0.154839\pi$
$602$		0			0
$603$		−	2.05551i		−	0.0837070i
$604$		0			0
$605$	−10.8167	+	10.8167i	−0.439760	+	0.439760i
$606$		0			0
$607$	−8.51388	−	8.51388i	−0.345568	−	0.345568i	0.512888	−	0.858456i	$-0.328576\pi$
$607$	−8.51388	−	8.51388i	−0.345568	−	0.345568i	−0.858456	+	0.512888i	$0.828576\pi$
$608$		0			0
$609$	−19.0278	−	19.0278i	−0.771044	−	0.771044i
$610$		0			0
$611$	18.4222			0.745283
$612$		0			0
$613$	11.5778			0.467623			0.233811	−	0.972282i	$-0.424880\pi$
$613$	11.5778			0.467623			0.233811	+	0.972282i	$0.424880\pi$
$614$		0			0
$615$	−2.60555	−	2.60555i	−0.105066	−	0.105066i
$616$		0			0
$617$	−5.60555	−	5.60555i	−0.225671	−	0.225671i	0.585210	−	0.810881i	$-0.301012\pi$
$617$	−5.60555	−	5.60555i	−0.225671	−	0.225671i	−0.810881	+	0.585210i	$0.801012\pi$
$618$		0			0
$619$	32.5139	−	32.5139i	1.30684	−	1.30684i	0.383162	−	0.923681i	$-0.374835\pi$
$619$	32.5139	−	32.5139i	1.30684	−	1.30684i	0.923681	−	0.383162i	$-0.125165\pi$
$620$		0			0
$621$			4.73338i			0.189944i
$622$		0			0
$623$	18.0000	−	18.0000i	0.721155	−	0.721155i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			56.8444i			2.27015i
$628$		0			0
$629$	−4.81665	−	16.8167i	−0.192053	−	0.670524i
$630$		0			0
$631$			26.2389i			1.04455i	0.852776	+	0.522276i	$0.174917\pi$
$631$			26.2389i			1.04455i	−0.852776	+	0.522276i	$0.825083\pi$
$632$		0			0
$633$	−6.55004			−0.260341
$634$		0			0
$635$	−6.60555	+	6.60555i	−0.262133	+	0.262133i
$636$		0			0
$637$		−	16.6056i		−	0.657936i
$638$		0			0
$639$	3.11943	−	3.11943i	0.123403	−	0.123403i
$640$		0			0
$641$	9.18335	+	9.18335i	0.362720	+	0.362720i	0.864814	−	0.502093i	$-0.167437\pi$
$641$	9.18335	+	9.18335i	0.362720	+	0.362720i	−0.502093	+	0.864814i	$0.667437\pi$
$642$		0			0
$643$	−21.1194	−	21.1194i	−0.832869	−	0.832869i	0.155039	−	0.987908i	$-0.450450\pi$
$643$	−21.1194	−	21.1194i	−0.832869	−	0.832869i	−0.987908	+	0.155039i	$0.950450\pi$
$644$		0			0
$645$	27.6333			1.08806
$646$		0			0
$647$	−2.78890			−0.109643			−0.0548214	−	0.998496i	$-0.517459\pi$
$647$	−2.78890			−0.109643			−0.0548214	+	0.998496i	$0.517459\pi$
$648$		0			0
$649$	4.60555	+	4.60555i	0.180784	+	0.180784i
$650$		0			0
$651$	−2.36669	−	2.36669i	−0.0927580	−	0.0927580i
$652$		0			0
$653$	3.18335	−	3.18335i	0.124574	−	0.124574i	−0.642071	−	0.766645i	$-0.721925\pi$
$653$	3.18335	−	3.18335i	0.124574	−	0.124574i	0.766645	+	0.642071i	$0.221925\pi$
$654$		0			0
$655$			5.39445i			0.210779i
$656$		0			0
$657$	2.76114	−	2.76114i	0.107722	−	0.107722i
$658$		0			0
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$		−	13.2111i		−	0.513852i	−0.966431	−	0.256926i	$-0.917290\pi$
$661$		−	13.2111i		−	0.513852i	0.966431	−	0.256926i	$-0.0827097\pi$
$662$		0			0
$663$	9.63331	+	33.6333i	0.374127	+	1.30621i
$664$		0			0
$665$			17.2111i			0.667418i
$666$		0			0
$667$	−7.81665			−0.302662
$668$		0			0
$669$	4.97224	−	4.97224i	0.192238	−	0.192238i
$670$		0			0
$671$		−	54.2389i		−	2.09387i
$672$		0			0
$673$	−1.60555	+	1.60555i	−0.0618895	+	0.0618895i	−0.737374	−	0.675485i	$-0.763934\pi$
$673$	−1.60555	+	1.60555i	−0.0618895	+	0.0618895i	0.675485	+	0.737374i	$0.263934\pi$
$674$		0			0
$675$	−10.1833	−	10.1833i	−0.391957	−	0.391957i
$676$		0			0
$677$	20.0278	+	20.0278i	0.769729	+	0.769729i	0.978059	−	0.208329i	$-0.0668026\pi$
$677$	20.0278	+	20.0278i	0.769729	+	0.769729i	−0.208329	+	0.978059i	$0.566803\pi$
$678$		0			0
$679$	1.02776			0.0394417
$680$		0			0
$681$	8.60555			0.329765
$682$		0			0
$683$	31.7250	+	31.7250i	1.21392	+	1.21392i	0.969726	+	0.244197i	$0.0785244\pi$
$683$	31.7250	+	31.7250i	1.21392	+	1.21392i	0.244197	+	0.969726i	$0.421476\pi$
$684$		0			0
$685$	−4.60555	−	4.60555i	−0.175969	−	0.175969i
$686$		0			0
$687$	−3.39445	+	3.39445i	−0.129506	+	0.129506i
$688$		0			0
$689$			42.4222i			1.61616i
$690$		0			0
$691$	12.5139	−	12.5139i	0.476050	−	0.476050i	−0.427816	−	0.903866i	$-0.640717\pi$
$691$	12.5139	−	12.5139i	0.476050	−	0.476050i	0.903866	+	0.427816i	$0.140717\pi$
$692$		0			0
$693$	−3.39445			−0.128944
$694$		0			0
$695$			6.60555i			0.250563i
$696$		0			0
$697$	−5.60555	+	1.60555i	−0.212325	+	0.0608146i
$698$		0			0
$699$		−	4.18335i		−	0.158229i
$700$		0			0
$701$	13.8167			0.521848			0.260924	−	0.965359i	$-0.415973\pi$
$701$	13.8167			0.521848			0.260924	+	0.965359i	$0.415973\pi$
$702$		0			0
$703$	−19.8167	+	19.8167i	−0.747399	+	0.747399i
$704$		0			0
$705$			10.4222i			0.392523i
$706$		0			0
$707$	−4.42221	+	4.42221i	−0.166314	+	0.166314i
$708$		0			0
$709$	21.6056	+	21.6056i	0.811414	+	0.811414i	0.984846	−	0.173432i	$-0.0554858\pi$
$709$	21.6056	+	21.6056i	0.811414	+	0.811414i	−0.173432	+	0.984846i	$0.555486\pi$
$710$		0			0
$711$	1.30278	+	1.30278i	0.0488579	+	0.0488579i
$712$		0			0
$713$	−0.972244			−0.0364108
$714$		0			0
$715$	30.4222			1.13773
$716$		0			0
$717$	−38.0555	−	38.0555i	−1.42121	−	1.42121i
$718$		0			0
$719$	27.7250	+	27.7250i	1.03397	+	1.03397i	0.999402	+	0.0345649i	$0.0110045\pi$
$719$	27.7250	+	27.7250i	1.03397	+	1.03397i	0.0345649	+	0.999402i	$0.488995\pi$
$720$		0			0
$721$	−1.57779	+	1.57779i	−0.0587602	+	0.0587602i
$722$		0			0
$723$			23.4500i			0.872113i
$724$		0			0
$725$	16.8167	−	16.8167i	0.624555	−	0.624555i
$726$		0			0
$727$	22.4222			0.831594			0.415797	−	0.909458i	$-0.363503\pi$
$727$	22.4222			0.831594			0.415797	+	0.909458i	$0.363503\pi$
$728$		0			0
$729$		−	22.5778i		−	0.836215i
$730$		0			0
$731$	21.2111	−	38.2389i	0.784521	−	1.41432i
$732$		0			0
$733$		−	26.0555i		−	0.962382i	−0.876616	−	0.481191i	$-0.840204\pi$
$733$		−	26.0555i		−	0.962382i	0.876616	−	0.481191i	$-0.159796\pi$
$734$		0			0
$735$	9.39445			0.346519
$736$		0			0
$737$	−17.2111	+	17.2111i	−0.633979	+	0.633979i
$738$		0			0
$739$			44.6611i			1.64288i	0.570292	+	0.821442i	$0.306830\pi$
$739$			44.6611i			1.64288i	−0.570292	+	0.821442i	$0.693170\pi$
$740$		0			0
$741$	39.6333	−	39.6333i	1.45597	−	1.45597i
$742$		0			0
$743$	−17.7250	−	17.7250i	−0.650266	−	0.650266i	0.302791	−	0.953057i	$-0.402082\pi$
$743$	−17.7250	−	17.7250i	−0.650266	−	0.650266i	−0.953057	+	0.302791i	$0.902082\pi$
$744$		0			0
$745$	20.4222	+	20.4222i	0.748212	+	0.748212i
$746$		0			0
$747$	−1.50547			−0.0550824
$748$		0			0
$749$	−24.2389			−0.885669
$750$		0			0
$751$	5.30278	+	5.30278i	0.193501	+	0.193501i	0.797207	−	0.603706i	$-0.206310\pi$
$751$	5.30278	+	5.30278i	0.193501	+	0.193501i	−0.603706	+	0.797207i	$0.706310\pi$
$752$		0			0
$753$	−15.6333	−	15.6333i	−0.569709	−	0.569709i
$754$		0			0
$755$	2.60555	−	2.60555i	0.0948257	−	0.0948257i
$756$		0			0
$757$		−	2.97224i		−	0.108028i	−0.998540	−	0.0540140i	$-0.982798\pi$
$757$		−	2.97224i		−	0.108028i	0.998540	−	0.0540140i	$-0.0172016\pi$
$758$		0			0
$759$	−6.00000	+	6.00000i	−0.217786	+	0.217786i
$760$		0			0
$761$	33.4500			1.21256			0.606280	−	0.795251i	$-0.292661\pi$
$761$	33.4500			1.21256			0.606280	+	0.795251i	$0.292661\pi$
$762$		0			0
$763$			21.3944i			0.774531i
$764$		0			0
$765$	−2.21110	+	0.633308i	−0.0799426	+	0.0228973i
$766$		0			0
$767$		−	6.42221i		−	0.231892i
$768$		0			0
$769$	−11.3944			−0.410894			−0.205447	−	0.978668i	$-0.565865\pi$
$769$	−11.3944			−0.410894			−0.205447	+	0.978668i	$0.565865\pi$
$770$		0			0
$771$	34.6611	−	34.6611i	1.24829	−	1.24829i
$772$		0			0
$773$			21.3944i			0.769505i	0.923020	+	0.384752i	$0.125713\pi$
$773$			21.3944i			0.769505i	−0.923020	+	0.384752i	$0.874287\pi$
$774$		0			0
$775$	2.09167	−	2.09167i	0.0751351	−	0.0751351i
$776$		0			0
$777$	−10.1833	−	10.1833i	−0.365326	−	0.365326i
$778$		0			0
$779$	6.60555	+	6.60555i	0.236668	+	0.236668i
$780$		0			0
$781$	−52.2389			−1.86925
$782$		0			0
$783$	38.0555			1.35999
$784$		0			0
$785$	−20.4222	−	20.4222i	−0.728900	−	0.728900i
$786$		0			0
$787$	−17.7250	−	17.7250i	−0.631827	−	0.631827i	0.316699	−	0.948526i	$-0.397426\pi$
$787$	−17.7250	−	17.7250i	−0.631827	−	0.631827i	−0.948526	+	0.316699i	$0.897426\pi$
$788$		0			0
$789$	−16.9722	+	16.9722i	−0.604228	+	0.604228i
$790$		0			0
$791$			6.23886i			0.221828i
$792$		0			0
$793$	−37.8167	+	37.8167i	−1.34291	+	1.34291i
$794$		0			0
$795$	−24.0000			−0.851192
$796$		0			0
$797$			39.6333i			1.40388i	0.712234	+	0.701942i	$0.247683\pi$
$797$			39.6333i			1.40388i	−0.712234	+	0.701942i	$0.752317\pi$
$798$		0			0
$799$	14.4222	+	8.00000i	0.510221	+	0.283020i
$800$		0			0
$801$		−	5.44996i		−	0.192565i
$802$		0			0
$803$	−46.2389			−1.63173
$804$		0			0
$805$	−1.81665	+	1.81665i	−0.0640286	+	0.0640286i
$806$		0			0
$807$		−	29.7611i		−	1.04764i
$808$		0			0
$809$	0.816654	−	0.816654i	0.0287120	−	0.0287120i	−0.692605	−	0.721317i	$-0.743537\pi$
$809$	0.816654	−	0.816654i	0.0287120	−	0.0287120i	0.721317	+	0.692605i	$0.243537\pi$
$810$		0			0
$811$	−21.7250	−	21.7250i	−0.762867	−	0.762867i	0.213972	−	0.976840i	$-0.431360\pi$
$811$	−21.7250	−	21.7250i	−0.762867	−	0.762867i	−0.976840	+	0.213972i	$0.931360\pi$
$812$		0			0
$813$	−17.2111	−	17.2111i	−0.603620	−	0.603620i
$814$		0			0
$815$	1.39445			0.0488454
$816$		0			0
$817$	−70.0555			−2.45093
$818$		0			0
$819$	2.36669	+	2.36669i	0.0826989	+	0.0826989i
$820$		0			0
$821$	−8.81665	−	8.81665i	−0.307703	−	0.307703i	0.536315	−	0.844018i	$-0.319816\pi$
$821$	−8.81665	−	8.81665i	−0.307703	−	0.307703i	−0.844018	+	0.536315i	$0.819816\pi$
$822$		0			0
$823$	2.09167	−	2.09167i	0.0729111	−	0.0729111i	−0.669711	−	0.742622i	$-0.733582\pi$
$823$	2.09167	−	2.09167i	0.0729111	−	0.0729111i	0.742622	+	0.669711i	$0.233582\pi$
$824$		0			0
$825$		−	25.8167i		−	0.898821i
$826$		0			0
$827$	2.69722	−	2.69722i	0.0937917	−	0.0937917i	−0.658654	−	0.752446i	$-0.728874\pi$
$827$	2.69722	−	2.69722i	0.0937917	−	0.0937917i	0.752446	+	0.658654i	$0.228874\pi$
$828$		0			0
$829$	38.8444			1.34912			0.674561	−	0.738219i	$-0.264333\pi$
$829$	38.8444			1.34912			0.674561	+	0.738219i	$0.264333\pi$
$830$		0			0
$831$		−	9.39445i		−	0.325890i
$832$		0			0
$833$	7.21110	−	13.0000i	0.249850	−	0.450423i
$834$		0			0
$835$		−	8.18335i		−	0.283196i
$836$		0			0
$837$	4.73338			0.163610
$838$		0			0
$839$	−4.09167	+	4.09167i	−0.141260	+	0.141260i	−0.774201	−	0.632940i	$-0.781848\pi$
$839$	−4.09167	+	4.09167i	−0.141260	+	0.141260i	0.632940	+	0.774201i	$0.281848\pi$
$840$		0			0
$841$			33.8444i			1.16705i
$842$		0			0
$843$	−13.5778	+	13.5778i	−0.467644	+	0.467644i
$844$		0			0
$845$	−8.21110	−	8.21110i	−0.282471	−	0.282471i
$846$		0			0
$847$	14.0917	+	14.0917i	0.484196	+	0.484196i
$848$		0			0
$849$	−60.2389			−2.06739
$850$		0			0
$851$	−4.18335			−0.143403
$852$		0			0
$853$	0.577795	+	0.577795i	0.0197833	+	0.0197833i	0.716929	−	0.697146i	$-0.245547\pi$
$853$	0.577795	+	0.577795i	0.0197833	+	0.0197833i	−0.697146	+	0.716929i	$0.745547\pi$
$854$		0			0
$855$	2.60555	+	2.60555i	0.0891080	+	0.0891080i
$856$		0			0
$857$	24.6333	−	24.6333i	0.841458	−	0.841458i	−0.147591	−	0.989049i	$-0.547152\pi$
$857$	24.6333	−	24.6333i	0.841458	−	0.841458i	0.989049	+	0.147591i	$0.0471519\pi$
$858$		0			0
$859$			19.8167i			0.676136i	0.941122	+	0.338068i	$0.109773\pi$
$859$			19.8167i			0.676136i	−0.941122	+	0.338068i	$0.890227\pi$
$860$		0			0
$861$	−3.39445	+	3.39445i	−0.115683	+	0.115683i
$862$		0			0
$863$	48.4777			1.65020			0.825100	−	0.564986i	$-0.191119\pi$
$863$	48.4777			1.65020			0.825100	+	0.564986i	$0.191119\pi$
$864$		0			0
$865$		−	13.6333i		−	0.463546i
$866$		0			0
$867$	−7.06392	+	30.5139i	−0.239903	+	1.03631i
$868$		0			0
$869$		−	21.8167i		−	0.740079i
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$	0.155590	−	0.155590i	0.00526592	−	0.00526592i
$874$		0			0
$875$		−	20.8444i		−	0.704670i
$876$		0			0
$877$	16.2111	−	16.2111i	0.547410	−	0.547410i	−0.378281	−	0.925691i	$-0.623485\pi$
$877$	16.2111	−	16.2111i	0.547410	−	0.547410i	0.925691	+	0.378281i	$0.123485\pi$
$878$		0			0
$879$	28.1833	+	28.1833i	0.950601	+	0.950601i
$880$		0			0
$881$	−9.78890	−	9.78890i	−0.329796	−	0.329796i	0.522713	−	0.852509i	$-0.324920\pi$
$881$	−9.78890	−	9.78890i	−0.329796	−	0.329796i	−0.852509	+	0.522713i	$0.824920\pi$
$882$		0			0
$883$	54.4222			1.83145			0.915727	−	0.401802i	$-0.131616\pi$
$883$	54.4222			1.83145			0.915727	+	0.401802i	$0.131616\pi$
$884$		0			0
$885$	3.63331			0.122132
$886$		0			0
$887$	−15.3028	−	15.3028i	−0.513817	−	0.513817i	0.401877	−	0.915694i	$-0.368358\pi$
$887$	−15.3028	−	15.3028i	−0.513817	−	0.513817i	−0.915694	+	0.401877i	$0.868358\pi$
$888$		0			0
$889$	8.60555	+	8.60555i	0.288621	+	0.288621i
$890$		0			0
$891$	33.1194	−	33.1194i	1.10954	−	1.10954i
$892$		0			0
$893$		−	26.4222i		−	0.884185i
$894$		0			0
$895$	−21.0278	+	21.0278i	−0.702880	+	0.702880i
$896$		0			0
$897$	8.36669			0.279356
$898$		0			0
$899$			7.81665i			0.260700i
$900$		0			0
$901$	−18.4222	+	33.2111i	−0.613733	+	1.10642i
$902$		0			0
$903$		−	36.0000i		−	1.19800i
$904$		0			0
$905$	26.0000			0.864269
$906$		0			0
$907$	−29.5416	+	29.5416i	−0.980914	+	0.980914i	−0.999821	−	0.0189074i	$-0.993981\pi$
$907$	−29.5416	+	29.5416i	−0.980914	+	0.980914i	0.0189074	+	0.999821i	$0.493981\pi$
$908$		0			0
$909$			1.33894i			0.0444097i
$910$		0			0
$911$	−27.7250	+	27.7250i	−0.918570	+	0.918570i	−0.996925	−	0.0783559i	$-0.975033\pi$
$911$	−27.7250	+	27.7250i	−0.918570	+	0.918570i	0.0783559	+	0.996925i	$0.475033\pi$
$912$		0			0
$913$	12.6056	+	12.6056i	0.417183	+	0.417183i
$914$		0			0
$915$	−21.3944	−	21.3944i	−0.707279	−	0.707279i
$916$		0			0
$917$	7.02776			0.232077
$918$		0			0
$919$	−16.8444			−0.555646			−0.277823	−	0.960632i	$-0.589613\pi$
$919$	−16.8444			−0.555646			−0.277823	+	0.960632i	$0.589613\pi$
$920$		0			0
$921$	8.84441	+	8.84441i	0.291433	+	0.291433i
$922$		0			0
$923$	36.4222	+	36.4222i	1.19885	+	1.19885i
$924$		0			0
$925$	9.00000	−	9.00000i	0.295918	−	0.295918i
$926$		0			0
$927$			0.477718i			0.0156903i
$928$		0			0
$929$	1.00000	−	1.00000i	0.0328089	−	0.0328089i	−0.690512	−	0.723321i	$-0.742615\pi$
$929$	1.00000	−	1.00000i	0.0328089	−	0.0328089i	0.723321	+	0.690512i	$0.242615\pi$
$930$		0			0
$931$	−23.8167			−0.780559
$932$		0			0
$933$			21.0833i			0.690235i
$934$		0			0
$935$	23.8167	+	13.2111i	0.778888	+	0.432049i
$936$		0			0
$937$			22.4222i			0.732502i	0.930516	+	0.366251i	$0.119359\pi$
$937$			22.4222i			0.732502i	−0.930516	+	0.366251i	$0.880641\pi$
$938$		0			0
$939$	−21.3944			−0.698181
$940$		0			0
$941$	20.0278	−	20.0278i	0.652886	−	0.652886i	−0.300801	−	0.953687i	$-0.597254\pi$
$941$	20.0278	−	20.0278i	0.652886	−	0.652886i	0.953687	+	0.300801i	$0.0972539\pi$
$942$		0			0
$943$			1.39445i			0.0454095i
$944$		0			0
$945$	8.84441	−	8.84441i	0.287709	−	0.287709i
$946$		0			0
$947$	−41.7250	−	41.7250i	−1.35588	−	1.35588i	−0.878931	−	0.476948i	$-0.841743\pi$
$947$	−41.7250	−	41.7250i	−1.35588	−	1.35588i	−0.476948	−	0.878931i	$-0.658257\pi$
$948$		0			0
$949$	32.2389	+	32.2389i	1.04652	+	1.04652i
$950$		0			0
$951$	32.2944			1.04722
$952$		0			0
$953$	−0.238859			−0.00773740			−0.00386870	−	0.999993i	$-0.501231\pi$
$953$	−0.238859			−0.00773740			−0.00386870	+	0.999993i	$0.501231\pi$
$954$		0			0
$955$	−21.2111	−	21.2111i	−0.686375	−	0.686375i
$956$		0			0
$957$	48.2389	+	48.2389i	1.55934	+	1.55934i
$958$		0			0
$959$	−6.00000	+	6.00000i	−0.193750	+	0.193750i
$960$		0			0
$961$		−	30.0278i		−	0.968637i
$962$		0			0
$963$	−3.66947	+	3.66947i	−0.118247	+	0.118247i
$964$		0			0
$965$	−12.4222			−0.399885
$966$		0			0
$967$		−	0.183346i		−	0.00589602i	−0.999996	−	0.00294801i	$-0.999062\pi$
$967$		−	0.183346i		−	0.00589602i	0.999996	−	0.00294801i	$-0.000938381\pi$
$968$		0			0
$969$	48.2389	−	13.8167i	1.54966	−	0.443855i
$970$		0			0
$971$		−	18.9722i		−	0.608848i	−0.952537	−	0.304424i	$-0.901536\pi$
$971$		−	18.9722i		−	0.608848i	0.952537	−	0.304424i	$-0.0984640\pi$
$972$		0			0
$973$	8.60555			0.275881
$974$		0			0
$975$	−18.0000	+	18.0000i	−0.576461	+	0.576461i
$976$		0			0
$977$			40.8444i			1.30673i	0.757044	+	0.653364i	$0.226643\pi$
$977$			40.8444i			1.30673i	−0.757044	+	0.653364i	$0.773357\pi$
$978$		0			0
$979$	−45.6333	+	45.6333i	−1.45845	+	1.45845i
$980$		0			0
$981$	3.23886	+	3.23886i	0.103409	+	0.103409i
$982$		0			0
$983$	−17.1194	−	17.1194i	−0.546025	−	0.546025i	0.379264	−	0.925289i	$-0.376177\pi$
$983$	−17.1194	−	17.1194i	−0.546025	−	0.546025i	−0.925289	+	0.379264i	$0.876177\pi$
$984$		0			0
$985$	20.7889			0.662389
$986$		0			0
$987$	13.5778			0.432186
$988$		0			0
$989$	−7.39445	−	7.39445i	−0.235130	−	0.235130i
$990$		0			0
$991$	−5.48612	−	5.48612i	−0.174272	−	0.174272i	0.614581	−	0.788854i	$-0.289325\pi$
$991$	−5.48612	−	5.48612i	−0.174272	−	0.174272i	−0.788854	+	0.614581i	$0.789325\pi$
$992$		0			0
$993$	−13.8167	+	13.8167i	−0.438458	+	0.438458i
$994$		0			0
$995$		−	30.2389i		−	0.958636i
$996$		0			0
$997$	−11.2389	+	11.2389i	−0.355938	+	0.355938i	−0.862313	−	0.506375i	$-0.830985\pi$
$997$	−11.2389	+	11.2389i	−0.355938	+	0.355938i	0.506375	+	0.862313i	$0.330985\pi$
$998$		0			0
$999$	20.3667			0.644374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.e.a.13.2	✓	4
3.2	odd	2		612.2.k.e.217.1		4
4.3	odd	2		272.2.o.g.81.1		4
5.2	odd	4		1700.2.m.a.149.2		4
5.3	odd	4		1700.2.m.b.149.1		4
5.4	even	2		1700.2.o.c.1101.1		4
8.3	odd	2		1088.2.o.s.897.2		4
8.5	even	2		1088.2.o.t.897.1		4
12.11	even	2		2448.2.be.u.1441.2		4
17.2	even	8		1156.2.a.h.1.3		4
17.3	odd	16		1156.2.h.e.1001.3		16
17.4	even	4	inner	68.2.e.a.21.2	yes	4
17.5	odd	16		1156.2.h.e.733.2		16
17.6	odd	16		1156.2.h.e.757.3		16
17.7	odd	16		1156.2.h.e.977.3		16
17.8	even	8		1156.2.b.a.577.2		4
17.9	even	8		1156.2.b.a.577.3		4
17.10	odd	16		1156.2.h.e.977.2		16
17.11	odd	16		1156.2.h.e.757.2		16
17.12	odd	16		1156.2.h.e.733.3		16
17.13	even	4		1156.2.e.c.905.1		4
17.14	odd	16		1156.2.h.e.1001.2		16
17.15	even	8		1156.2.a.h.1.2		4
17.16	even	2		1156.2.e.c.829.1		4
51.38	odd	4		612.2.k.e.361.1		4
68.15	odd	8		4624.2.a.bq.1.3		4
68.19	odd	8		4624.2.a.bq.1.2		4
68.55	odd	4		272.2.o.g.225.1		4
85.4	even	4		1700.2.o.c.701.1		4
85.38	odd	4		1700.2.m.a.1449.2		4
85.72	odd	4		1700.2.m.b.1449.1		4
136.21	even	4		1088.2.o.t.769.1		4
136.123	odd	4		1088.2.o.s.769.2		4
204.191	even	4		2448.2.be.u.1585.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.e.a.13.2	✓	4	1.1	even	1	trivial
68.2.e.a.21.2	yes	4	17.4	even	4	inner
272.2.o.g.81.1		4	4.3	odd	2
272.2.o.g.225.1		4	68.55	odd	4
612.2.k.e.217.1		4	3.2	odd	2
612.2.k.e.361.1		4	51.38	odd	4
1088.2.o.s.769.2		4	136.123	odd	4
1088.2.o.s.897.2		4	8.3	odd	2
1088.2.o.t.769.1		4	136.21	even	4
1088.2.o.t.897.1		4	8.5	even	2
1156.2.a.h.1.2		4	17.15	even	8
1156.2.a.h.1.3		4	17.2	even	8
1156.2.b.a.577.2		4	17.8	even	8
1156.2.b.a.577.3		4	17.9	even	8
1156.2.e.c.829.1		4	17.16	even	2
1156.2.e.c.905.1		4	17.13	even	4
1156.2.h.e.733.2		16	17.5	odd	16
1156.2.h.e.733.3		16	17.12	odd	16
1156.2.h.e.757.2		16	17.11	odd	16
1156.2.h.e.757.3		16	17.6	odd	16
1156.2.h.e.977.2		16	17.10	odd	16
1156.2.h.e.977.3		16	17.7	odd	16
1156.2.h.e.1001.2		16	17.14	odd	16
1156.2.h.e.1001.3		16	17.3	odd	16
1700.2.m.a.149.2		4	5.2	odd	4
1700.2.m.a.1449.2		4	85.38	odd	4
1700.2.m.b.149.1		4	5.3	odd	4
1700.2.m.b.1449.1		4	85.72	odd	4
1700.2.o.c.701.1		4	85.4	even	4
1700.2.o.c.1101.1		4	5.4	even	2
2448.2.be.u.1441.2		4	12.11	even	2
2448.2.be.u.1585.2		4	204.191	even	4
4624.2.a.bq.1.2		4	68.19	odd	8
4624.2.a.bq.1.3		4	68.15	odd	8

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.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.30278 + 1.30278i) q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.30278 + 1.30278i) q^{7} +0.394449i q^{9} +O(q^{10})\)	\(q+(1.30278 + 1.30278i) q^{3} +(-1.00000 - 1.00000i) q^{5} +(-1.30278 + 1.30278i) q^{7} +0.394449i q^{9} +(3.30278 - 3.30278i) q^{11} -4.60555 q^{13} -2.60555i q^{15} +(-3.60555 - 2.00000i) q^{17} +6.60555i q^{19} -3.39445 q^{21} +(-0.697224 + 0.697224i) q^{23} -3.00000i q^{25} +(3.39445 - 3.39445i) q^{27} +(5.60555 + 5.60555i) q^{29} +(0.697224 + 0.697224i) q^{31} +8.60555 q^{33} +2.60555 q^{35} +(3.00000 + 3.00000i) q^{37} +(-6.00000 - 6.00000i) q^{39} +(1.00000 - 1.00000i) q^{41} +10.6056i q^{43} +(0.394449 - 0.394449i) q^{45} -4.00000 q^{47} +3.60555i q^{49} +(-2.09167 - 7.30278i) q^{51} -9.21110i q^{53} -6.60555 q^{55} +(-8.60555 + 8.60555i) q^{57} +1.39445i q^{59} +(8.21110 - 8.21110i) q^{61} +(-0.513878 - 0.513878i) q^{63} +(4.60555 + 4.60555i) q^{65} -5.21110 q^{67} -1.81665 q^{69} +(-7.90833 - 7.90833i) q^{71} +(-7.00000 - 7.00000i) q^{73} +(3.90833 - 3.90833i) q^{75} +8.60555i q^{77} +(3.30278 - 3.30278i) q^{79} +10.0278 q^{81} +3.81665i q^{83} +(1.60555 + 5.60555i) q^{85} +14.6056i q^{87} -13.8167 q^{89} +(6.00000 - 6.00000i) q^{91} +1.81665i q^{93} +(6.60555 - 6.60555i) q^{95} +(-0.394449 - 0.394449i) q^{97} +(1.30278 + 1.30278i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7	\( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 + 6 * q^11 - 4 * q^13 - 28 * q^21 - 10 * q^23 + 28 * q^27 + 8 * q^29 + 10 * q^31 + 20 * q^33 - 4 * q^35 + 12 * q^37 - 24 * q^39 + 4 * q^41 + 16 * q^45 - 16 * q^47 - 30 * q^51 - 12 * q^55 - 20 * q^57 + 4 * q^61 + 34 * q^63 + 4 * q^65 + 8 * q^67 + 36 * q^69 - 10 * q^71 - 28 * q^73 - 6 * q^75 + 6 * q^79 - 32 * q^81 - 8 * q^85 - 12 * q^89 + 24 * q^91 + 12 * q^95 - 16 * q^97 - 2 * q^99

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