LMFDB - Embedded newform 1368.2.g.b.685.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(685,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1368.685");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.g (of order $2$, degree $1$, 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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.6
Root			$1.12629 - 0.855255i$ of defining polynomial
Character	$\chi$	$=$	1368.685
Dual form			1368.2.g.b.685.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.855255 + 1.12629i) q^{2} +(-0.537077 - 1.92654i) q^{4} -1.51356i q^{5} +0.580162 q^{7} +(2.62919 + 1.04278i) q^{8} +O(q^{10})$	\(q+(-0.855255 + 1.12629i) q^{2} +(-0.537077 - 1.92654i) q^{4} -1.51356i q^{5} +0.580162 q^{7} +(2.62919 + 1.04278i) q^{8} +(1.70471 + 1.29448i) q^{10} +1.31799i q^{11} +3.89230i q^{13} +(-0.496186 + 0.653433i) q^{14} +(-3.42310 + 2.06940i) q^{16} +1.20142 q^{17} +1.00000i q^{19} +(-2.91593 + 0.812898i) q^{20} +(-1.48444 - 1.12722i) q^{22} -5.85527 q^{23} +2.70913 q^{25} +(-4.38387 - 3.32891i) q^{26} +(-0.311591 - 1.11770i) q^{28} -1.29188i q^{29} +2.96413 q^{31} +(0.596871 - 5.62528i) q^{32} +(-1.02752 + 1.35316i) q^{34} -0.878110i q^{35} -1.18418i q^{37} +(-1.12629 - 0.855255i) q^{38} +(1.57830 - 3.97943i) q^{40} +9.04577 q^{41} +8.38816i q^{43} +(2.53915 - 0.707861i) q^{44} +(5.00775 - 6.59475i) q^{46} +12.8560 q^{47} -6.66341 q^{49} +(-2.31700 + 3.05128i) q^{50} +(7.49866 - 2.09046i) q^{52} +3.07183i q^{53} +1.99485 q^{55} +(1.52535 + 0.604978i) q^{56} +(1.45503 + 1.10488i) q^{58} -0.258163i q^{59} +14.7200i q^{61} +(-2.53509 + 3.33848i) q^{62} +(5.82524 + 5.48330i) q^{64} +5.89123 q^{65} -9.54884i q^{67} +(-0.645257 - 2.31459i) q^{68} +(0.989010 + 0.751008i) q^{70} +6.93697 q^{71} +15.2934 q^{73} +(1.33373 + 1.01277i) q^{74} +(1.92654 - 0.537077i) q^{76} +0.764646i q^{77} -13.1332 q^{79} +(3.13216 + 5.18106i) q^{80} +(-7.73645 + 10.1882i) q^{82} +3.70615i q^{83} -1.81843i q^{85} +(-9.44754 - 7.17402i) q^{86} +(-1.37436 + 3.46523i) q^{88} +8.36653 q^{89} +2.25816i q^{91} +(3.14473 + 11.2804i) q^{92} +(-10.9952 + 14.4796i) q^{94} +1.51356 q^{95} +17.0442 q^{97} +(5.69892 - 7.50496i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) $ 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8	$ 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44} - 48 q^{46} - 24 q^{47} + 24 q^{49} - 12 q^{50} + 8 q^{52} + 16 q^{55} + 48 q^{56} + 38 q^{58} + 16 q^{62} + 14 q^{64} - 16 q^{65} + 26 q^{68} - 32 q^{70} - 48 q^{71} + 20 q^{74} - 4 q^{76} - 48 q^{79} - 4 q^{80} - 12 q^{82} - 48 q^{86} + 40 q^{88} + 16 q^{89} - 62 q^{92} - 36 q^{94} - 16 q^{95} + 32 q^{97} + 48 q^{98}+O(q^{100}) $ 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	−0.855255	+	1.12629i	−0.604757	+	0.796410i
$2$	−0.855255	+	1.12629i	−0.604757	+	0.796410i
$3$		0			0
$3$		0			0
$4$	−0.537077	−	1.92654i	−0.268538	−	0.963269i
$5$		−	1.51356i		−	0.676885i	−0.940987	−	0.338442i	$-0.890100\pi$
$5$		−	1.51356i		−	0.676885i	0.940987	−	0.338442i	$-0.109900\pi$
$6$		0			0
$7$	0.580162			0.219281			0.109640	−	0.993971i	$-0.465030\pi$
$7$	0.580162			0.219281			0.109640	+	0.993971i	$0.465030\pi$
$8$	2.62919	+	1.04278i	0.929558	+	0.368677i
$9$		0			0
$10$	1.70471	+	1.29448i	0.539078	+	0.409351i
$11$			1.31799i			0.397388i	0.980062	+	0.198694i	$0.0636700\pi$
$11$			1.31799i			0.397388i	−0.980062	+	0.198694i	$0.936330\pi$
$12$		0			0
$13$			3.89230i			1.07953i	0.841816	+	0.539765i	$0.181487\pi$
$13$			3.89230i			1.07953i	−0.841816	+	0.539765i	$0.818513\pi$
$14$	−0.496186	+	0.653433i	−0.132611	+	0.174637i
$15$		0			0
$16$	−3.42310	+	2.06940i	−0.855774	+	0.517349i
$17$	1.20142			0.291388			0.145694	−	0.989330i	$-0.453459\pi$
$17$	1.20142			0.291388			0.145694	+	0.989330i	$0.453459\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$	−2.91593	+	0.812898i	−0.652022	+	0.181770i
$21$		0			0
$22$	−1.48444	−	1.12722i	−0.316484	−	0.240323i
$23$	−5.85527			−1.22091			−0.610454	−	0.792052i	$-0.709013\pi$
$23$	−5.85527			−1.22091			−0.610454	+	0.792052i	$0.709013\pi$
$24$		0			0
$25$	2.70913			0.541827
$26$	−4.38387	−	3.32891i	−0.859748	−	0.652853i
$27$		0			0
$28$	−0.311591	−	1.11770i	−0.0588853	−	0.211226i
$29$		−	1.29188i		−	0.239896i	−0.992780	−	0.119948i	$-0.961727\pi$
$29$		−	1.29188i		−	0.239896i	0.992780	−	0.119948i	$-0.0382727\pi$
$30$		0			0
$31$	2.96413			0.532374			0.266187	−	0.963921i	$-0.414236\pi$
$31$	2.96413			0.532374			0.266187	+	0.963921i	$0.414236\pi$
$32$	0.596871	−	5.62528i	0.105513	−	0.994418i
$33$		0			0
$34$	−1.02752	+	1.35316i	−0.176219	+	0.232064i
$35$		−	0.878110i		−	0.148428i
$36$		0			0
$37$		−	1.18418i		−	0.194677i	−0.995251	−	0.0973387i	$-0.968967\pi$
$37$		−	1.18418i		−	0.194677i	0.995251	−	0.0973387i	$-0.0310330\pi$
$38$	−1.12629	−	0.855255i	−0.182709	−	0.138741i
$39$		0			0
$40$	1.57830	−	3.97943i	0.249552	−	0.629204i
$41$	9.04577			1.41271			0.706356	−	0.707856i	$-0.250338\pi$
$41$	9.04577			1.41271			0.706356	+	0.707856i	$0.250338\pi$
$42$		0			0
$43$			8.38816i			1.27918i	0.768715	+	0.639592i	$0.220896\pi$
$43$			8.38816i			1.27918i	−0.768715	+	0.639592i	$0.779104\pi$
$44$	2.53915	−	0.707861i	0.382792	−	0.106714i
$45$		0			0
$46$	5.00775	−	6.59475i	0.738352	−	0.972343i
$47$	12.8560			1.87524			0.937620	−	0.347661i	$-0.113024\pi$
$47$	12.8560			1.87524			0.937620	+	0.347661i	$0.113024\pi$
$48$		0			0
$49$	−6.66341			−0.951916
$50$	−2.31700	+	3.05128i	−0.327673	+	0.431516i
$51$		0			0
$52$	7.49866	−	2.09046i	1.03988	−	0.289895i
$53$			3.07183i			0.421949i	0.977492	+	0.210974i	$0.0676636\pi$
$53$			3.07183i			0.421949i	−0.977492	+	0.210974i	$0.932336\pi$
$54$		0			0
$55$	1.99485			0.268986
$56$	1.52535	+	0.604978i	0.203834	+	0.0808436i
$57$		0			0
$58$	1.45503	+	1.10488i	0.191055	+	0.145079i
$59$		−	0.258163i		−	0.0336100i	−0.999859	−	0.0168050i	$-0.994651\pi$
$59$		−	0.258163i		−	0.0336100i	0.999859	−	0.0168050i	$-0.00534945\pi$
$60$		0			0
$61$			14.7200i			1.88470i	0.334623	+	0.942352i	$0.391391\pi$
$61$			14.7200i			1.88470i	−0.334623	+	0.942352i	$0.608609\pi$
$62$	−2.53509	+	3.33848i	−0.321957	+	0.423988i
$63$		0			0
$64$	5.82524	+	5.48330i	0.728155	+	0.685413i
$65$	5.89123			0.730717
$66$		0			0
$67$		−	9.54884i		−	1.16658i	−0.812265	−	0.583288i	$-0.801766\pi$
$67$		−	9.54884i		−	1.16658i	0.812265	−	0.583288i	$-0.198234\pi$
$68$	−0.645257	−	2.31459i	−0.0782489	−	0.280685i
$69$		0			0
$70$	0.989010	+	0.751008i	0.118209	+	0.0897627i
$71$	6.93697			0.823267			0.411634	−	0.911349i	$-0.364958\pi$
$71$	6.93697			0.823267			0.411634	+	0.911349i	$0.364958\pi$
$72$		0			0
$73$	15.2934			1.78996			0.894978	−	0.446110i	$-0.147191\pi$
$73$	15.2934			1.78996			0.894978	+	0.446110i	$0.147191\pi$
$74$	1.33373	+	1.01277i	0.155043	+	0.117732i
$75$		0			0
$76$	1.92654	−	0.537077i	0.220989	−	0.0616069i
$77$			0.764646i			0.0871395i
$78$		0			0
$79$	−13.1332			−1.47760			−0.738798	−	0.673927i	$-0.764606\pi$
$79$	−13.1332			−1.47760			−0.738798	+	0.673927i	$0.764606\pi$
$80$	3.13216	+	5.18106i	0.350186	+	0.579261i
$81$		0			0
$82$	−7.73645	+	10.1882i	−0.854348	+	1.12510i
$83$			3.70615i			0.406803i	0.979095	+	0.203402i	$0.0651997\pi$
$83$			3.70615i			0.406803i	−0.979095	+	0.203402i	$0.934800\pi$
$84$		0			0
$85$		−	1.81843i		−	0.197236i
$86$	−9.44754	−	7.17402i	−1.01875	−	0.773595i
$87$		0			0
$88$	−1.37436	+	3.46523i	−0.146508	+	0.369395i
$89$	8.36653			0.886850			0.443425	−	0.896311i	$-0.353763\pi$
$89$	8.36653			0.886850			0.443425	+	0.896311i	$0.353763\pi$
$90$		0			0
$91$			2.25816i			0.236720i
$92$	3.14473	+	11.2804i	0.327861	+	1.17606i
$93$		0			0
$94$	−10.9952	+	14.4796i	−1.13406	+	1.49346i
$95$	1.51356			0.155288
$96$		0			0
$97$	17.0442			1.73057			0.865286	−	0.501278i	$-0.167137\pi$
$97$	17.0442			1.73057			0.865286	+	0.501278i	$0.167137\pi$
$98$	5.69892	−	7.50496i	0.575678	−	0.758116i
$99$		0			0
$100$	−1.45501	−	5.21925i	−0.145501	−	0.521925i
$101$			1.97721i			0.196739i	0.995150	+	0.0983697i	$0.0313628\pi$
$101$			1.97721i			0.196739i	−0.995150	+	0.0983697i	$0.968637\pi$
$102$		0			0
$103$	−2.30853			−0.227466			−0.113733	−	0.993511i	$-0.536281\pi$
$103$	−2.30853			−0.227466			−0.113733	+	0.993511i	$0.536281\pi$
$104$	−4.05879	+	10.2336i	−0.397997	+	1.00348i
$105$		0			0
$106$	−3.45979	−	2.62720i	−0.336044	−	0.255176i
$107$			15.0396i			1.45393i	0.686675	+	0.726965i	$0.259070\pi$
$107$			15.0396i			1.45393i	−0.686675	+	0.726965i	$0.740930\pi$
$108$		0			0
$109$		−	13.4002i		−	1.28351i	−0.766911	−	0.641753i	$-0.778207\pi$
$109$		−	13.4002i		−	1.28351i	0.766911	−	0.641753i	$-0.221793\pi$
$110$	−1.70611	+	2.24679i	−0.162671	+	0.214223i
$111$		0			0
$112$	−1.98595	+	1.20059i	−0.187655	+	0.113445i
$113$	−4.89717			−0.460687			−0.230343	−	0.973109i	$-0.573985\pi$
$113$	−4.89717			−0.460687			−0.230343	+	0.973109i	$0.573985\pi$
$114$		0			0
$115$			8.86230i			0.826414i
$116$	−2.48885	+	0.693837i	−0.231084	+	0.0644212i
$117$		0			0
$118$	0.290768	+	0.220795i	0.0267673	+	0.0203259i
$119$	0.697020			0.0638957
$120$		0			0
$121$	9.26291			0.842083
$122$	−16.5791	−	12.5894i	−1.50100	−	1.13979i
$123$		0			0
$124$	−1.59197	−	5.71051i	−0.142963	−	0.512819i
$125$		−	11.6682i		−	1.04364i
$126$		0			0
$127$	−2.61625			−0.232155			−0.116078	−	0.993240i	$-0.537032\pi$
$127$	−2.61625			−0.232155			−0.116078	+	0.993240i	$0.537032\pi$
$128$	−11.1579	+	1.87131i	−0.986226	+	0.165402i
$129$		0			0
$130$	−5.03851	+	6.63526i	−0.441906	+	0.581951i
$131$		−	2.64498i		−	0.231093i	−0.993302	−	0.115547i	$-0.963138\pi$
$131$		−	2.64498i		−	0.231093i	0.993302	−	0.115547i	$-0.0368620\pi$
$132$		0			0
$133$			0.580162i			0.0503064i
$134$	10.7548	+	8.16670i	0.929073	+	0.705495i
$135$		0			0
$136$	3.15877	+	1.25281i	0.270862	+	0.107428i
$137$	−4.88035			−0.416957			−0.208478	−	0.978027i	$-0.566851\pi$
$137$	−4.88035			−0.416957			−0.208478	+	0.978027i	$0.566851\pi$
$138$		0			0
$139$			10.3334i			0.876468i	0.898861	+	0.438234i	$0.144396\pi$
$139$			10.3334i			0.876468i	−0.898861	+	0.438234i	$0.855604\pi$
$140$	−1.69171	+	0.471613i	−0.142976	+	0.0398585i
$141$		0			0
$142$	−5.93288	+	7.81307i	−0.497877	+	0.655659i
$143$	−5.13000			−0.428992
$144$		0			0
$145$	−1.95533			−0.162382
$146$	−13.0798	+	17.2249i	−1.08249	+	1.42554i
$147$		0			0
$148$	−2.28136	+	0.635994i	−0.187527	+	0.0522784i
$149$			8.35324i			0.684324i	0.939641	+	0.342162i	$0.111159\pi$
$149$			8.35324i			0.684324i	−0.939641	+	0.342162i	$0.888841\pi$
$150$		0			0
$151$	−5.84941			−0.476019			−0.238009	−	0.971263i	$-0.576495\pi$
$151$	−5.84941			−0.476019			−0.238009	+	0.971263i	$0.576495\pi$
$152$	−1.04278	+	2.62919i	−0.0845802	+	0.213255i
$153$		0			0
$154$	−0.861216	−	0.653967i	−0.0693988	−	0.0526982i
$155$		−	4.48639i		−	0.360356i
$156$		0			0
$157$		−	0.361645i		−	0.0288624i	−0.999896	−	0.0144312i	$-0.995406\pi$
$157$		−	0.361645i		−	0.0288624i	0.999896	−	0.0144312i	$-0.00459375\pi$
$158$	11.2322	−	14.7918i	0.893587	−	1.17677i
$159$		0			0
$160$	−8.51420	−	0.903401i	−0.673107	−	0.0714201i
$161$	−3.39700			−0.267721
$162$		0			0
$163$			0.146029i			0.0114379i	0.999984	+	0.00571894i	$0.00182041\pi$
$163$			0.146029i			0.0114379i	−0.999984	+	0.00571894i	$0.998180\pi$
$164$	−4.85828	−	17.4270i	−0.379368	−	1.36082i
$165$		0			0
$166$	−4.17422	−	3.16971i	−0.323982	−	0.246017i
$167$	−0.406603			−0.0314639			−0.0157319	−	0.999876i	$-0.505008\pi$
$167$	−0.406603			−0.0314639			−0.0157319	+	0.999876i	$0.505008\pi$
$168$		0			0
$169$	−2.14999			−0.165384
$170$	2.04808	+	1.55522i	0.157081	+	0.119280i
$171$		0			0
$172$	16.1601	−	4.50509i	1.23220	−	0.343510i
$173$		−	12.6466i		−	0.961507i	−0.876856	−	0.480753i	$-0.840363\pi$
$173$		−	12.6466i		−	0.961507i	0.876856	−	0.480753i	$-0.159637\pi$
$174$		0			0
$175$	1.57174			0.118812
$176$	−2.72744	−	4.51160i	−0.205589	−	0.340075i
$177$		0			0
$178$	−7.15552	+	9.42317i	−0.536329	+	0.706297i
$179$			5.55598i			0.415274i	0.978206	+	0.207637i	$0.0665772\pi$
$179$			5.55598i			0.415274i	−0.978206	+	0.207637i	$0.933423\pi$
$180$		0			0
$181$		−	9.13401i		−	0.678926i	−0.940619	−	0.339463i	$-0.889755\pi$
$181$		−	9.13401i		−	0.678926i	0.940619	−	0.339463i	$-0.110245\pi$
$182$	−2.54336	−	1.93131i	−0.188526	−	0.143158i
$183$		0			0
$184$	−15.3946	−	6.10573i	−1.13490	−	0.450120i
$185$	−1.79232			−0.131774
$186$		0			0
$187$			1.58346i			0.115794i
$188$	−6.90466	−	24.7676i	−0.503574	−	1.80636i
$189$		0			0
$190$	−1.29448	+	1.70471i	−0.0939115	+	0.123673i
$191$	−5.47532			−0.396180			−0.198090	−	0.980184i	$-0.563474\pi$
$191$	−5.47532			−0.396180			−0.198090	+	0.980184i	$0.563474\pi$
$192$		0			0
$193$	16.9697			1.22151			0.610753	−	0.791821i	$-0.290867\pi$
$193$	16.9697			1.22151			0.610753	+	0.791821i	$0.290867\pi$
$194$	−14.5771	+	19.1967i	−1.04658	+	1.37825i
$195$		0			0
$196$	3.57876	+	12.8373i	0.255626	+	0.916951i
$197$			23.7727i			1.69374i	0.531803	+	0.846868i	$0.321515\pi$
$197$			23.7727i			1.69374i	−0.531803	+	0.846868i	$0.678485\pi$
$198$		0			0
$199$	−17.5747			−1.24584			−0.622918	−	0.782287i	$-0.714053\pi$
$199$	−17.5747			−1.24584			−0.622918	+	0.782287i	$0.714053\pi$
$200$	7.12282	+	2.82502i	0.503659	+	0.199759i
$201$		0			0
$202$	−2.22692	−	1.69102i	−0.156685	−	0.118979i
$203$		−	0.749498i		−	0.0526044i
$204$		0			0
$205$		−	13.6913i		−	0.956244i
$206$	1.97438	−	2.60008i	0.137562	−	0.181156i
$207$		0			0
$208$	−8.05472	−	13.3237i	−0.558494	−	0.923834i
$209$	−1.31799			−0.0911671
$210$		0			0
$211$			17.2804i			1.18963i	0.803863	+	0.594815i	$0.202775\pi$
$211$			17.2804i			1.18963i	−0.803863	+	0.594815i	$0.797225\pi$
$212$	5.91800	−	1.64981i	0.406450	−	0.113309i
$213$		0			0
$214$	−16.9390	−	12.8627i	−1.15792	−	0.879274i
$215$	12.6960			0.865860
$216$		0			0
$217$	1.71968			0.116739
$218$	15.0926	+	11.4606i	1.02220	+	0.776209i
$219$		0			0
$220$	−1.07139	−	3.84316i	−0.0722331	−	0.259106i
$221$			4.67630i			0.314562i
$222$		0			0
$223$	11.6027			0.776973			0.388487	−	0.921454i	$-0.372998\pi$
$223$	11.6027			0.776973			0.388487	+	0.921454i	$0.372998\pi$
$224$	0.346282	−	3.26357i	0.0231369	−	0.218057i
$225$		0			0
$226$	4.18833	−	5.51565i	0.278604	−	0.366896i
$227$			27.4484i			1.82182i	0.412609	+	0.910908i	$0.364618\pi$
$227$			27.4484i			1.82182i	−0.412609	+	0.910908i	$0.635382\pi$
$228$		0			0
$229$		−	21.8680i		−	1.44508i	−0.691329	−	0.722540i	$-0.742975\pi$
$229$		−	21.8680i		−	1.44508i	0.691329	−	0.722540i	$-0.257025\pi$
$230$	−9.98156	−	7.57953i	−0.658165	−	0.499780i
$231$		0			0
$232$	1.34714	−	3.39659i	0.0884439	−	0.222997i
$233$	22.1975			1.45421			0.727103	−	0.686528i	$-0.240866\pi$
$233$	22.1975			1.45421			0.727103	+	0.686528i	$0.240866\pi$
$234$		0			0
$235$		−	19.4583i		−	1.26932i
$236$	−0.497361	+	0.138654i	−0.0323755	+	0.00902557i
$237$		0			0
$238$	−0.596130	+	0.785049i	−0.0386414	+	0.0508872i
$239$	−21.1350			−1.36711			−0.683555	−	0.729899i	$-0.739567\pi$
$239$	−21.1350			−1.36711			−0.683555	+	0.729899i	$0.739567\pi$
$240$		0			0
$241$	−17.2437			−1.11076			−0.555382	−	0.831595i	$-0.687428\pi$
$241$	−17.2437			−1.11076			−0.555382	+	0.831595i	$0.687428\pi$
$242$	−7.92215	+	10.4328i	−0.509255	+	0.670643i
$243$		0			0
$244$	28.3587	−	7.90578i	1.81548	−	0.506116i
$245$			10.0855i			0.644338i
$246$		0			0
$247$	−3.89230			−0.247661
$248$	7.79325	+	3.09092i	0.494872	+	0.196274i
$249$		0			0
$250$	13.1419	+	9.97933i	0.831165	+	0.631148i
$251$			6.08851i			0.384303i	0.981365	+	0.192152i	$0.0615465\pi$
$251$			6.08851i			0.384303i	−0.981365	+	0.192152i	$0.938453\pi$
$252$		0			0
$253$		−	7.71717i		−	0.485174i
$254$	2.23757	−	2.94667i	0.140397	−	0.184891i
$255$		0			0
$256$	7.43518	−	14.1675i	0.464699	−	0.885469i
$257$	−10.1741			−0.634643			−0.317322	−	0.948318i	$-0.602783\pi$
$257$	−10.1741			−0.634643			−0.317322	+	0.948318i	$0.602783\pi$
$258$		0			0
$259$		−	0.687014i		−	0.0426890i
$260$	−3.16404	−	11.3497i	−0.196226	−	0.703877i
$261$		0			0
$262$	2.97903	+	2.26213i	0.184045	+	0.139755i
$263$	15.8414			0.976822			0.488411	−	0.872614i	$-0.337577\pi$
$263$	15.8414			0.976822			0.488411	+	0.872614i	$0.337577\pi$
$264$		0			0
$265$	4.64940			0.285611
$266$	−0.653433	−	0.496186i	−0.0400645	−	0.0304231i
$267$		0			0
$268$	−18.3962	+	5.12846i	−1.12373	+	0.313271i
$269$		−	22.8533i		−	1.39339i	−0.717367	−	0.696695i	$-0.754653\pi$
$269$		−	22.8533i		−	1.39339i	0.717367	−	0.696695i	$-0.245347\pi$
$270$		0			0
$271$	12.2830			0.746142			0.373071	−	0.927803i	$-0.378305\pi$
$271$	12.2830			0.746142			0.373071	+	0.927803i	$0.378305\pi$
$272$	−4.11259	+	2.48622i	−0.249362	+	0.150749i
$273$		0			0
$274$	4.17395	−	5.49671i	0.252157	−	0.332068i
$275$			3.57060i			0.215316i
$276$		0			0
$277$			0.643776i			0.0386808i	0.999813	+	0.0193404i	$0.00615662\pi$
$277$			0.643776i			0.0386808i	−0.999813	+	0.0193404i	$0.993843\pi$
$278$	−11.6385	−	8.83770i	−0.698028	−	0.530050i
$279$		0			0
$280$	0.915671	−	2.30871i	0.0547218	−	0.137972i
$281$	8.93027			0.532735			0.266368	−	0.963872i	$-0.414177\pi$
$281$	8.93027			0.532735			0.266368	+	0.963872i	$0.414177\pi$
$282$		0			0
$283$			14.7397i			0.876187i	0.898929	+	0.438094i	$0.144346\pi$
$283$			14.7397i			0.876187i	−0.898929	+	0.438094i	$0.855654\pi$
$284$	−3.72569	−	13.3643i	−0.221079	−	0.793028i
$285$		0			0
$286$	4.38746	−	5.77789i	0.259436	−	0.341654i
$287$	5.24801			0.309780
$288$		0			0
$289$	−15.5566			−0.915093
$290$	1.67231	−	2.20228i	0.0982015	−	0.129322i
$291$		0			0
$292$	−8.21373	−	29.4633i	−0.480672	−	1.72421i
$293$		−	23.9816i		−	1.40102i	−0.713643	−	0.700509i	$-0.752956\pi$
$293$		−	23.9816i		−	1.40102i	0.713643	−	0.700509i	$-0.247044\pi$
$294$		0			0
$295$	−0.390746			−0.0227501
$296$	1.23483	−	3.11342i	0.0717730	−	0.180964i
$297$		0			0
$298$	−9.40820	−	7.14415i	−0.545002	−	0.413850i
$299$		−	22.7905i		−	1.31801i
$300$		0			0
$301$			4.86649i			0.280500i
$302$	5.00274	−	6.58816i	0.287875	−	0.379106i
$303$		0			0
$304$	−2.06940	−	3.42310i	−0.118688	−	0.196328i
$305$	22.2796			1.27573
$306$		0			0
$307$		−	1.82132i		−	0.103948i	−0.998648	−	0.0519740i	$-0.983449\pi$
$307$		−	1.82132i		−	0.103948i	0.998648	−	0.0519740i	$-0.0165513\pi$
$308$	1.47312	−	0.410674i	0.0839388	−	0.0234003i
$309$		0			0
$310$	5.05300	+	3.83701i	0.286991	+	0.217928i
$311$	4.53302			0.257044			0.128522	−	0.991707i	$-0.458977\pi$
$311$	4.53302			0.257044			0.128522	+	0.991707i	$0.458977\pi$
$312$		0			0
$313$	12.4149			0.701730			0.350865	−	0.936426i	$-0.385888\pi$
$313$	12.4149			0.701730			0.350865	+	0.936426i	$0.385888\pi$
$314$	0.407319	+	0.309299i	0.0229863	+	0.0174547i
$315$		0			0
$316$	7.05352	+	25.3015i	0.396792	+	1.42332i
$317$		−	23.0948i		−	1.29713i	−0.761157	−	0.648567i	$-0.775368\pi$
$317$		−	23.0948i		−	1.29713i	0.761157	−	0.648567i	$-0.224632\pi$
$318$		0			0
$319$	1.70268			0.0953317
$320$	8.29931	−	8.81685i	0.463945	−	0.492877i
$321$		0			0
$322$	2.90530	−	3.82602i	0.161906	−	0.213216i
$323$			1.20142i			0.0668490i
$324$		0			0
$325$			10.5448i			0.584918i
$326$	−0.164472	−	0.124892i	−0.00910924	−	0.00691714i
$327$		0			0
$328$	23.7830	+	9.43271i	1.31320	+	0.520834i
$329$	7.45856			0.411204
$330$		0			0
$331$		−	26.3743i		−	1.44966i	−0.688927	−	0.724831i	$-0.741918\pi$
$331$		−	26.3743i		−	1.44966i	0.688927	−	0.724831i	$-0.258082\pi$
$332$	7.14004	−	1.99049i	0.391861	−	0.109242i
$333$		0			0
$334$	0.347749	−	0.457954i	0.0190280	−	0.0250581i
$335$	−14.4527			−0.789638
$336$		0			0
$337$	3.82102			0.208144			0.104072	−	0.994570i	$-0.466813\pi$
$337$	3.82102			0.208144			0.104072	+	0.994570i	$0.466813\pi$
$338$	1.83879	−	2.42152i	0.100017	−	0.131714i
$339$		0			0
$340$	−3.50327	+	0.976635i	−0.189991	+	0.0529655i
$341$			3.90669i			0.211559i
$342$		0			0
$343$	−7.92699			−0.428017
$344$	−8.74697	+	22.0540i	−0.471605	+	1.18907i
$345$		0			0
$346$	14.2438	+	10.8161i	0.765754	+	0.581478i
$347$		−	29.8808i		−	1.60408i	−0.597268	−	0.802042i	$-0.703747\pi$
$347$		−	29.8808i		−	1.60408i	0.597268	−	0.802042i	$-0.296253\pi$
$348$		0			0
$349$			12.7016i			0.679902i	0.940443	+	0.339951i	$0.110411\pi$
$349$			12.7016i			0.679902i	−0.940443	+	0.339951i	$0.889589\pi$
$350$	−1.34424	+	1.77024i	−0.0718524	+	0.0946232i
$351$		0			0
$352$	7.41404	+	0.786668i	0.395170	+	0.0419296i
$353$	−22.8398			−1.21564			−0.607820	−	0.794074i	$-0.707956\pi$
$353$	−22.8398			−1.21564			−0.607820	+	0.794074i	$0.707956\pi$
$354$		0			0
$355$		−	10.4995i		−	0.557257i
$356$	−4.49347	−	16.1184i	−0.238153	−	0.854275i
$357$		0			0
$358$	−6.25767	−	4.75178i	−0.330728	−	0.251140i
$359$	−35.4003			−1.86835			−0.934177	−	0.356810i	$-0.883864\pi$
$359$	−35.4003			−1.86835			−0.934177	+	0.356810i	$0.883864\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	10.2876	+	7.81191i	0.540703	+	0.410585i
$363$		0			0
$364$	4.35044	−	1.21281i	0.228025	−	0.0635684i
$365$		−	23.1475i		−	1.21159i
$366$		0			0
$367$	−17.2148			−0.898608			−0.449304	−	0.893379i	$-0.648328\pi$
$367$	−17.2148			−0.898608			−0.449304	+	0.893379i	$0.648328\pi$
$368$	20.0432	−	12.1169i	1.04482	−	0.631636i
$369$		0			0
$370$	1.53289	−	2.01868i	0.0796914	−	0.104946i
$371$			1.78216i			0.0925251i
$372$		0			0
$373$		−	20.3506i		−	1.05372i	−0.849953	−	0.526858i	$-0.823370\pi$
$373$		−	20.3506i		−	1.05372i	0.849953	−	0.526858i	$-0.176630\pi$
$374$	−1.78344	−	1.35426i	−0.0922196	−	0.0700273i
$375$		0			0
$376$	33.8008	+	13.4059i	1.74314	+	0.691358i
$377$	5.02837			0.258974
$378$		0			0
$379$			4.22765i			0.217160i	0.994088	+	0.108580i	$0.0346303\pi$
$379$			4.22765i			0.217160i	−0.994088	+	0.108580i	$0.965370\pi$
$380$	−0.812898	−	2.91593i	−0.0417008	−	0.149584i
$381$		0			0
$382$	4.68280	−	6.16682i	0.239593	−	0.315522i
$383$	−9.71281			−0.496301			−0.248151	−	0.968721i	$-0.579823\pi$
$383$	−9.71281			−0.496301			−0.248151	+	0.968721i	$0.579823\pi$
$384$		0			0
$385$	1.15734			0.0589834
$386$	−14.5134	+	19.1129i	−0.738714	+	0.972819i
$387$		0			0
$388$	−9.15403	−	32.8362i	−0.464725	−	1.66701i
$389$			22.3291i			1.13213i	0.824361	+	0.566064i	$0.191534\pi$
$389$			22.3291i			1.13213i	−0.824361	+	0.566064i	$0.808466\pi$
$390$		0			0
$391$	−7.03466			−0.355758
$392$	−17.5194	−	6.94844i	−0.884861	−	0.350949i
$393$		0			0
$394$	−26.7751	−	20.3317i	−1.34891	−	1.02430i
$395$			19.8778i			1.00016i
$396$		0			0
$397$		−	23.9210i		−	1.20056i	−0.799789	−	0.600281i	$-0.795055\pi$
$397$		−	23.9210i		−	1.20056i	0.799789	−	0.600281i	$-0.204945\pi$
$398$	15.0308	−	19.7943i	0.753428	−	0.992197i
$399$		0			0
$400$	−9.27363	+	5.60628i	−0.463681	+	0.280314i
$401$	16.5237			0.825154			0.412577	−	0.910923i	$-0.364629\pi$
$401$	16.5237			0.825154			0.412577	+	0.910923i	$0.364629\pi$
$402$		0			0
$403$			11.5373i			0.574713i
$404$	3.80916	−	1.06191i	0.189513	−	0.0528321i
$405$		0			0
$406$	0.844155	+	0.641012i	0.0418947	+	0.0318129i
$407$	1.56073			0.0773625
$408$		0			0
$409$	16.0228			0.792278			0.396139	−	0.918191i	$-0.370350\pi$
$409$	16.0228			0.792278			0.396139	+	0.918191i	$0.370350\pi$
$410$	15.4205	+	11.7096i	0.761562	+	0.578295i
$411$		0			0
$412$	1.23986	+	4.44747i	0.0610834	+	0.219111i
$413$		−	0.149776i		−	0.00737002i
$414$		0			0
$415$	5.60949			0.275359
$416$	21.8953	+	2.32320i	1.07350	+	0.113904i
$417$		0			0
$418$	1.12722	−	1.48444i	0.0551339	−	0.0726064i
$419$		−	12.7496i		−	0.622858i	−0.950269	−	0.311429i	$-0.899192\pi$
$419$		−	12.7496i		−	0.622858i	0.950269	−	0.311429i	$-0.100808\pi$
$420$		0			0
$421$			1.89293i			0.0922560i	0.998936	+	0.0461280i	$0.0146882\pi$
$421$			1.89293i			0.0922560i	−0.998936	+	0.0461280i	$0.985312\pi$
$422$	−19.4628	−	14.7791i	−0.947434	−	0.719437i
$423$		0			0
$424$	−3.20323	+	8.07642i	−0.155563	+	0.392225i
$425$	3.25482			0.157882
$426$		0			0
$427$			8.53999i			0.413279i
$428$	28.9743	−	8.07740i	1.40052	−	0.390436i
$429$		0			0
$430$	−10.8583	+	14.2994i	−0.523635	+	0.689580i
$431$	−37.2166			−1.79266			−0.896331	−	0.443386i	$-0.853777\pi$
$431$	−37.2166			−1.79266			−0.896331	+	0.443386i	$0.853777\pi$
$432$		0			0
$433$	−36.0314			−1.73156			−0.865780	−	0.500425i	$-0.833177\pi$
$433$	−36.0314			−1.73156			−0.865780	+	0.500425i	$0.833177\pi$
$434$	−1.47076	+	1.93686i	−0.0705988	+	0.0929723i
$435$		0			0
$436$	−25.8160	+	7.19693i	−1.23636	+	0.344671i
$437$		−	5.85527i		−	0.280095i
$438$		0			0
$439$	−18.6188			−0.888629			−0.444315	−	0.895871i	$-0.646553\pi$
$439$	−18.6188			−0.888629			−0.444315	+	0.895871i	$0.646553\pi$
$440$	5.24484	+	2.08018i	0.250038	+	0.0991689i
$441$		0			0
$442$	−5.26689	−	3.99943i	−0.250520	−	0.190233i
$443$		−	2.26887i		−	0.107797i	−0.998546	−	0.0538987i	$-0.982835\pi$
$443$		−	2.26887i		−	0.107797i	0.998546	−	0.0538987i	$-0.0171648\pi$
$444$		0			0
$445$		−	12.6633i		−	0.600296i
$446$	−9.92326	+	13.0680i	−0.469880	+	0.618789i
$447$		0			0
$448$	3.37958	+	3.18120i	0.159670	+	0.150298i
$449$	−0.248909			−0.0117467			−0.00587337	−	0.999983i	$-0.501870\pi$
$449$	−0.248909			−0.0117467			−0.00587337	+	0.999983i	$0.501870\pi$
$450$		0			0
$451$			11.9222i			0.561395i
$452$	2.63016	+	9.43458i	0.123712	+	0.443765i
$453$		0			0
$454$	−30.9150	−	23.4754i	−1.45091	−	1.10176i
$455$	3.41787			0.160232
$456$		0			0
$457$	−6.98767			−0.326869			−0.163435	−	0.986554i	$-0.552257\pi$
$457$	−6.98767			−0.326869			−0.163435	+	0.986554i	$0.552257\pi$
$458$	24.6298	+	18.7027i	1.15088	+	0.873922i
$459$		0			0
$460$	17.0736	−	4.75974i	0.796059	−	0.221924i
$461$			34.1624i			1.59110i	0.605888	+	0.795550i	$0.292818\pi$
$461$			34.1624i			1.59110i	−0.605888	+	0.795550i	$0.707182\pi$
$462$		0			0
$463$	34.5311			1.60480			0.802399	−	0.596788i	$-0.203556\pi$
$463$	34.5311			1.60480			0.802399	+	0.596788i	$0.203556\pi$
$464$	2.67341	+	4.42222i	0.124110	+	0.205296i
$465$		0			0
$466$	−18.9845	+	25.0009i	−0.879441	+	1.15815i
$467$		−	39.6496i		−	1.83476i	−0.398008	−	0.917382i	$-0.630298\pi$
$467$		−	39.6496i		−	1.83476i	0.398008	−	0.917382i	$-0.369702\pi$
$468$		0			0
$469$		−	5.53987i		−	0.255808i
$470$	21.9158	+	16.6418i	1.01090	+	0.767631i
$471$		0			0
$472$	0.269206	−	0.678759i	0.0123912	−	0.0312424i
$473$	−11.0555			−0.508332
$474$		0			0
$475$			2.70913i			0.124304i
$476$	−0.374353	−	1.34284i	−0.0171585	−	0.0615488i
$477$		0			0
$478$	18.0758	−	23.8042i	0.826770	−	1.08878i
$479$	24.8669			1.13620			0.568098	−	0.822961i	$-0.307679\pi$
$479$	24.8669			1.13620			0.568098	+	0.822961i	$0.307679\pi$
$480$		0			0
$481$	4.60917			0.210160
$482$	14.7478	−	19.4215i	0.671742	−	0.884624i
$483$		0			0
$484$	−4.97489	−	17.8453i	−0.226132	−	0.811152i
$485$		−	25.7974i		−	1.17140i
$486$		0			0
$487$	−39.7320			−1.80043			−0.900214	−	0.435448i	$-0.856590\pi$
$487$	−39.7320			−1.80043			−0.900214	+	0.435448i	$0.856590\pi$
$488$	−15.3497	+	38.7016i	−0.694847	+	1.75194i
$489$		0			0
$490$	−11.3592	−	8.62566i	−0.513157	−	0.389668i
$491$			3.45561i			0.155949i	0.996955	+	0.0779747i	$0.0248453\pi$
$491$			3.45561i			0.155949i	−0.996955	+	0.0779747i	$0.975155\pi$
$492$		0			0
$493$		−	1.55209i		−	0.0699027i
$494$	3.32891	−	4.38387i	0.149775	−	0.197240i
$495$		0			0
$496$	−10.1465	+	6.13397i	−0.455592	+	0.275423i
$497$	4.02457			0.180527
$498$		0			0
$499$			17.5837i			0.787156i	0.919291	+	0.393578i	$0.128763\pi$
$499$			17.5837i			0.787156i	−0.919291	+	0.393578i	$0.871237\pi$
$500$	−22.4793	+	6.26674i	−1.00531	+	0.280257i
$501$		0			0
$502$	−6.85745	−	5.20723i	−0.306063	−	0.232410i
$503$	−2.72712			−0.121596			−0.0607981	−	0.998150i	$-0.519365\pi$
$503$	−2.72712			−0.121596			−0.0607981	+	0.998150i	$0.519365\pi$
$504$		0			0
$505$	2.99262			0.133170
$506$	8.69180	+	6.60015i	0.386398	+	0.293412i
$507$		0			0
$508$	1.40513	+	5.04031i	0.0623425	+	0.223628i
$509$			6.73123i			0.298356i	0.988810	+	0.149178i	$0.0476628\pi$
$509$			6.73123i			0.298356i	−0.988810	+	0.149178i	$0.952337\pi$
$510$		0			0
$511$	8.87264			0.392503
$512$	9.59779	+	20.4910i	0.424166	+	0.905584i
$513$		0			0
$514$	8.70146	−	11.4590i	0.383805	−	0.505436i
$515$			3.49410i			0.153968i
$516$		0			0
$517$			16.9441i			0.745198i
$518$	0.773780	+	0.587572i	0.0339979	+	0.0258164i
$519$		0			0
$520$	15.4891	+	6.14323i	0.679244	+	0.269398i
$521$	−38.2190			−1.67440			−0.837202	−	0.546894i	$-0.815810\pi$
$521$	−38.2190			−1.67440			−0.837202	+	0.546894i	$0.815810\pi$
$522$		0			0
$523$		−	5.37182i		−	0.234893i	−0.993079	−	0.117447i	$-0.962529\pi$
$523$		−	5.37182i		−	0.234893i	0.993079	−	0.117447i	$-0.0374709\pi$
$524$	−5.09566	+	1.42056i	−0.222605	+	0.0620574i
$525$		0			0
$526$	−13.5484	+	17.8421i	−0.590739	+	0.777951i
$527$	3.56118			0.155127
$528$		0			0
$529$	11.2842			0.490616
$530$	−3.97643	+	5.23660i	−0.172725	+	0.227463i
$531$		0			0
$532$	1.11770	−	0.311591i	0.0484586	−	0.0135092i
$533$			35.2089i			1.52506i
$534$		0			0
$535$	22.7633			0.984143
$536$	9.95729	−	25.1057i	0.430090	−	1.08440i
$537$		0			0
$538$	25.7395	+	19.5454i	1.10971	+	0.842662i
$539$		−	8.78229i		−	0.378280i
$540$		0			0
$541$			34.3497i			1.47681i	0.674359	+	0.738404i	$0.264420\pi$
$541$			34.3497i			1.47681i	−0.674359	+	0.738404i	$0.735580\pi$
$542$	−10.5051	+	13.8343i	−0.451234	+	0.594235i
$543$		0			0
$544$	0.717095	−	6.75834i	0.0307452	−	0.289761i
$545$	−20.2820			−0.868786
$546$		0			0
$547$		−	2.58358i		−	0.110466i	−0.998473	−	0.0552330i	$-0.982410\pi$
$547$		−	2.58358i		−	0.110466i	0.998473	−	0.0552330i	$-0.0175902\pi$
$548$	2.62112	+	9.40218i	0.111969	+	0.401641i
$549$		0			0
$550$	−4.02155	−	3.05378i	−0.171480	−	0.130214i
$551$	1.29188			0.0550358
$552$		0			0
$553$	−7.61936			−0.324008
$554$	−0.725081	−	0.550593i	−0.0308057	−	0.0233924i
$555$		0			0
$556$	19.9077	−	5.54983i	0.844275	−	0.235365i
$557$			0.498941i			0.0211408i	0.999944	+	0.0105704i	$0.00336472\pi$
$557$			0.498941i			0.0211408i	−0.999944	+	0.0105704i	$0.996635\pi$
$558$		0			0
$559$	−32.6492			−1.38092
$560$	1.81716	+	3.00586i	0.0767890	+	0.127021i
$561$		0			0
$562$	−7.63766	+	10.0581i	−0.322175	+	0.424276i
$563$		−	32.5553i		−	1.37204i	−0.727581	−	0.686021i	$-0.759356\pi$
$563$		−	32.5553i		−	1.37204i	0.727581	−	0.686021i	$-0.240644\pi$
$564$		0			0
$565$			7.41216i			0.311832i
$566$	−16.6013	−	12.6062i	−0.697804	−	0.529880i
$567$		0			0
$568$	18.2386	+	7.23371i	0.765275	+	0.303520i
$569$	−13.6834			−0.573636			−0.286818	−	0.957985i	$-0.592598\pi$
$569$	−13.6834			−0.573636			−0.286818	+	0.957985i	$0.592598\pi$
$570$		0			0
$571$			12.2520i			0.512732i	0.966580	+	0.256366i	$0.0825253\pi$
$571$			12.2520i			0.512732i	−0.966580	+	0.256366i	$0.917475\pi$
$572$	2.75520	+	9.88314i	0.115201	+	0.413235i
$573$		0			0
$574$	−4.48839	+	5.91081i	−0.187342	+	0.246712i
$575$	−15.8627			−0.661521
$576$		0			0
$577$	14.6672			0.610605			0.305302	−	0.952255i	$-0.401242\pi$
$577$	14.6672			0.610605			0.305302	+	0.952255i	$0.401242\pi$
$578$	13.3048	−	17.5213i	0.553409	−	0.728789i
$579$		0			0
$580$	1.05017	+	3.76703i	0.0436057	+	0.156417i
$581$			2.15017i			0.0892040i
$582$		0			0
$583$	−4.04864			−0.167677
$584$	40.2092	+	15.9476i	1.66387	+	0.659915i
$585$		0			0
$586$	27.0103	+	20.5104i	1.11579	+	0.847275i
$587$			5.53206i			0.228333i	0.993462	+	0.114166i	$0.0364197\pi$
$587$			5.53206i			0.228333i	−0.993462	+	0.114166i	$0.963580\pi$
$588$		0			0
$589$			2.96413i			0.122135i
$590$	0.334187	−	0.440095i	0.0137583	−	0.0181184i
$591$		0			0
$592$	2.45053	+	4.05355i	0.100716	+	0.166600i
$593$	−21.5870			−0.886470			−0.443235	−	0.896405i	$-0.646169\pi$
$593$	−21.5870			−0.886470			−0.443235	+	0.896405i	$0.646169\pi$
$594$		0			0
$595$		−	1.05498i		−	0.0432500i
$596$	16.0928	−	4.48633i	0.659188	−	0.183767i
$597$		0			0
$598$	25.6688	+	19.4917i	1.04967	+	0.797073i
$599$	28.3513			1.15840			0.579202	−	0.815184i	$-0.303364\pi$
$599$	28.3513			1.15840			0.579202	+	0.815184i	$0.303364\pi$
$600$		0			0
$601$	−21.9758			−0.896410			−0.448205	−	0.893931i	$-0.647936\pi$
$601$	−21.9758			−0.896410			−0.448205	+	0.893931i	$0.647936\pi$
$602$	−5.48110	−	4.16209i	−0.223393	−	0.169634i
$603$		0			0
$604$	3.14158	+	11.2691i	0.127829	+	0.458534i
$605$		−	14.0200i		−	0.569993i
$606$		0			0
$607$	−7.89484			−0.320441			−0.160221	−	0.987081i	$-0.551221\pi$
$607$	−7.89484			−0.320441			−0.160221	+	0.987081i	$0.551221\pi$
$608$	5.62528	+	0.596871i	0.228135	+	0.0242063i
$609$		0			0
$610$	−19.0548	+	25.0934i	−0.771505	+	1.01600i
$611$			50.0394i			2.02438i
$612$		0			0
$613$		−	29.5174i		−	1.19220i	−0.802912	−	0.596098i	$-0.796717\pi$
$613$		−	29.5174i		−	1.19220i	0.802912	−	0.596098i	$-0.203283\pi$
$614$	2.05134	+	1.55769i	0.0827853	+	0.0628633i
$615$		0			0
$616$	−0.797354	+	2.01040i	−0.0321263	+	0.0810012i
$617$	−3.08379			−0.124149			−0.0620744	−	0.998072i	$-0.519772\pi$
$617$	−3.08379			−0.124149			−0.0620744	+	0.998072i	$0.519772\pi$
$618$		0			0
$619$			10.9487i			0.440066i	0.975492	+	0.220033i	$0.0706164\pi$
$619$			10.9487i			0.440066i	−0.975492	+	0.220033i	$0.929384\pi$
$620$	−8.64320	+	2.40954i	−0.347119	+	0.0967693i
$621$		0			0
$622$	−3.87689	+	5.10551i	−0.155449	+	0.204712i
$623$	4.85394			0.194469
$624$		0			0
$625$	−4.11492			−0.164597
$626$	−10.6179	+	13.9828i	−0.424376	+	0.558865i
$627$		0			0
$628$	−0.696723	+	0.194231i	−0.0278023	+	0.00775067i
$629$		−	1.42270i		−	0.0567267i
$630$		0			0
$631$	−20.7432			−0.825773			−0.412886	−	0.910783i	$-0.635479\pi$
$631$	−20.7432			−0.825773			−0.412886	+	0.910783i	$0.635479\pi$
$632$	−34.5295	−	13.6949i	−1.37351	−	0.544756i
$633$		0			0
$634$	26.0116	+	19.7520i	1.03305	+	0.784451i
$635$			3.95986i			0.157142i
$636$		0			0
$637$		−	25.9360i		−	1.02762i
$638$	−1.45622	+	1.91772i	−0.0576525	+	0.0759231i
$639$		0			0
$640$	2.83234	+	16.8881i	0.111958	+	0.667562i
$641$	34.5310			1.36389			0.681946	−	0.731403i	$-0.261134\pi$
$641$	34.5310			1.36389			0.681946	+	0.731403i	$0.261134\pi$
$642$		0			0
$643$		−	21.5300i		−	0.849062i	−0.905413	−	0.424531i	$-0.860439\pi$
$643$		−	21.5300i		−	0.849062i	0.905413	−	0.424531i	$-0.139561\pi$
$644$	1.82445	+	6.54446i	0.0718935	+	0.257888i
$645$		0			0
$646$	−1.35316	−	1.02752i	−0.0532392	−	0.0404274i
$647$	−22.7527			−0.894502			−0.447251	−	0.894409i	$-0.647597\pi$
$647$	−22.7527			−0.894502			−0.447251	+	0.894409i	$0.647597\pi$
$648$		0			0
$649$	0.340256			0.0133562
$650$	−11.8765	−	9.01846i	−0.465835	−	0.353733i
$651$		0			0
$652$	0.281331	−	0.0784288i	0.0110178	−	0.00307151i
$653$		−	35.6805i		−	1.39629i	−0.715958	−	0.698143i	$-0.754010\pi$
$653$		−	35.6805i		−	1.39629i	0.715958	−	0.698143i	$-0.245990\pi$
$654$		0			0
$655$	−4.00334			−0.156423
$656$	−30.9646	+	18.7193i	−1.20896	+	0.730866i
$657$		0			0
$658$	−6.37897	+	8.40053i	−0.248678	+	0.327487i
$659$		−	35.8589i		−	1.39686i	−0.715676	−	0.698432i	$-0.753881\pi$
$659$		−	35.8589i		−	1.39686i	0.715676	−	0.698432i	$-0.246119\pi$
$660$		0			0
$661$			25.4409i			0.989537i	0.869025	+	0.494769i	$0.164747\pi$
$661$			25.4409i			0.989537i	−0.869025	+	0.494769i	$0.835253\pi$
$662$	29.7052	+	22.5567i	1.15453	+	0.876693i
$663$		0			0
$664$	−3.86468	+	9.74416i	−0.149979	+	0.378147i
$665$	0.878110			0.0340516
$666$		0			0
$667$			7.56429i			0.292890i
$668$	0.218377	+	0.783335i	0.00844925	+	0.0303082i
$669$		0			0
$670$	12.3608	−	16.2780i	0.477539	−	0.628876i
$671$	−19.4008			−0.748959
$672$		0			0
$673$	34.2142			1.31886			0.659431	−	0.751765i	$-0.270797\pi$
$673$	34.2142			1.31886			0.659431	+	0.751765i	$0.270797\pi$
$674$	−3.26795	+	4.30359i	−0.125877	+	0.165768i
$675$		0			0
$676$	1.15471	+	4.14204i	0.0444120	+	0.159309i
$677$			37.9103i			1.45701i	0.685040	+	0.728505i	$0.259785\pi$
$677$			37.9103i			1.45701i	−0.685040	+	0.728505i	$0.740215\pi$
$678$		0			0
$679$	9.88837			0.379481
$680$	1.89621	−	4.78098i	0.0727164	−	0.183342i
$681$		0			0
$682$	−4.40008	−	3.34121i	−0.168488	−	0.127942i
$683$			30.3542i			1.16147i	0.814092	+	0.580736i	$0.197235\pi$
$683$			30.3542i			1.16147i	−0.814092	+	0.580736i	$0.802765\pi$
$684$		0			0
$685$			7.38671i			0.282232i
$686$	6.77960	−	8.92812i	0.258846	−	0.340877i
$687$		0			0
$688$	−17.3584	−	28.7135i	−0.661785	−	1.09469i
$689$	−11.9565			−0.455506
$690$		0			0
$691$		−	3.50694i		−	0.133410i	−0.997773	−	0.0667052i	$-0.978751\pi$
$691$		−	3.50694i		−	0.133410i	0.997773	−	0.0667052i	$-0.0212487\pi$
$692$	−24.3642	+	6.79222i	−0.926189	+	0.258201i
$693$		0			0
$694$	33.6545	+	25.5557i	1.27751	+	0.970081i
$695$	15.6402			0.593268
$696$		0			0
$697$	10.8678			0.411647
$698$	−14.3058	−	10.8631i	−0.541481	−	0.411175i
$699$		0			0
$700$	−0.844143	−	3.02801i	−0.0319056	−	0.114448i
$701$			13.2168i			0.499190i	0.968350	+	0.249595i	$0.0802975\pi$
$701$			13.2168i			0.499190i	−0.968350	+	0.249595i	$0.919703\pi$
$702$		0			0
$703$	1.18418			0.0446621
$704$	−7.22692	+	7.67759i	−0.272375	+	0.289360i
$705$		0			0
$706$	19.5339	−	25.7244i	0.735167	−	0.968149i
$707$			1.14710i			0.0431411i
$708$		0			0
$709$			23.2882i			0.874605i	0.899314	+	0.437302i	$0.144066\pi$
$709$			23.2882i			0.874605i	−0.899314	+	0.437302i	$0.855934\pi$
$710$	11.8256	+	8.97978i	0.443805	+	0.337005i
$711$		0			0
$712$	21.9972	+	8.72441i	0.824379	+	0.326961i
$713$	−17.3558			−0.649979
$714$		0			0
$715$			7.76457i			0.290378i
$716$	10.7038	−	2.98399i	0.400020	−	0.111517i
$717$		0			0
$718$	30.2763	−	39.8711i	1.12990	−	1.48798i
$719$	−0.192843			−0.00719184			−0.00359592	−	0.999994i	$-0.501145\pi$
$719$	−0.192843			−0.00719184			−0.00359592	+	0.999994i	$0.501145\pi$
$720$		0			0
$721$	−1.33932			−0.0498789
$722$	0.855255	−	1.12629i	0.0318293	−	0.0419163i
$723$		0			0
$724$	−17.5970	+	4.90567i	−0.653988	+	0.182318i
$725$		−	3.49987i		−	0.129982i
$726$		0			0
$727$	14.5604			0.540014			0.270007	−	0.962858i	$-0.412974\pi$
$727$	14.5604			0.540014			0.270007	+	0.962858i	$0.412974\pi$
$728$	−2.35476	+	5.93713i	−0.0872731	+	0.220045i
$729$		0			0
$730$	26.0709	+	19.7970i	0.964926	+	0.732720i
$731$			10.0777i			0.372739i
$732$		0			0
$733$		−	29.0856i		−	1.07430i	−0.843487	−	0.537150i	$-0.819501\pi$
$733$		−	29.0856i		−	1.07430i	0.843487	−	0.537150i	$-0.180499\pi$
$734$	14.7231	−	19.3890i	0.543439	−	0.715660i
$735$		0			0
$736$	−3.49484	+	32.9375i	−0.128822	+	1.21409i
$737$	12.5853			0.463584
$738$		0			0
$739$			28.8090i			1.05975i	0.848074	+	0.529877i	$0.177762\pi$
$739$			28.8090i			1.05975i	−0.848074	+	0.529877i	$0.822238\pi$
$740$	0.962615	+	3.45298i	0.0353864	+	0.126934i
$741$		0			0
$742$	−2.00724	−	1.52420i	−0.0736879	−	0.0559552i
$743$	−42.2025			−1.54826			−0.774129	−	0.633027i	$-0.781812\pi$
$743$	−42.2025			−1.54826			−0.774129	+	0.633027i	$0.781812\pi$
$744$		0			0
$745$	12.6431			0.463208
$746$	22.9208	+	17.4050i	0.839191	+	0.637242i
$747$		0			0
$748$	3.05060	−	0.850440i	0.111541	−	0.0310952i
$749$			8.72538i			0.318818i
$750$		0			0
$751$	2.64036			0.0963483			0.0481741	−	0.998839i	$-0.484660\pi$
$751$	2.64036			0.0963483			0.0481741	+	0.998839i	$0.484660\pi$
$752$	−44.0073	+	26.6042i	−1.60478	+	0.970155i
$753$		0			0
$754$	−4.30054	+	5.66343i	−0.156617	+	0.206250i
$755$			8.85344i			0.322210i
$756$		0			0
$757$			4.51764i			0.164197i	0.996624	+	0.0820983i	$0.0261621\pi$
$757$			4.51764i			0.164197i	−0.996624	+	0.0820983i	$0.973838\pi$
$758$	−4.76158	−	3.61572i	−0.172948	−	0.131329i
$759$		0			0
$760$	3.97943	+	1.57830i	0.144349	+	0.0572511i
$761$	23.5124			0.852323			0.426162	−	0.904647i	$-0.359866\pi$
$761$	23.5124			0.852323			0.426162	+	0.904647i	$0.359866\pi$
$762$		0			0
$763$		−	7.77428i		−	0.281448i
$764$	2.94067	+	10.5484i	0.106390	+	0.381628i
$765$		0			0
$766$	8.30693	−	10.9395i	0.300142	−	0.395259i
$767$	1.00485			0.0362830
$768$		0			0
$769$	−29.6727			−1.07002			−0.535012	−	0.844845i	$-0.679693\pi$
$769$	−29.6727			−1.07002			−0.535012	+	0.844845i	$0.679693\pi$
$770$	−0.989819	+	1.30350i	−0.0356706	+	0.0469750i
$771$		0			0
$772$	−9.11403	−	32.6928i	−0.328021	−	1.17664i
$773$			10.1647i			0.365598i	0.983150	+	0.182799i	$0.0585158\pi$
$773$			10.1647i			0.365598i	−0.983150	+	0.182799i	$0.941484\pi$
$774$		0			0
$775$	8.03023			0.288454
$776$	44.8123	+	17.7732i	1.60867	+	0.638022i
$777$		0			0
$778$	−25.1491	−	19.0970i	−0.901638	−	0.684662i
$779$			9.04577i			0.324098i
$780$		0			0
$781$			9.14285i			0.327157i
$782$	6.01643	−	7.92309i	0.215147	−	0.283329i
$783$		0			0
$784$	22.8095	−	13.7893i	0.814625	−	0.492473i
$785$	−0.547372			−0.0195365
$786$		0			0
$787$			18.8481i			0.671863i	0.941886	+	0.335931i	$0.109051\pi$
$787$			18.8481i			0.671863i	−0.941886	+	0.335931i	$0.890949\pi$
$788$	45.7991	−	12.7678i	1.63152	−	0.454833i
$789$		0			0
$790$	−22.3883	−	17.0006i	−0.796540	−	0.604855i
$791$	−2.84115			−0.101020
$792$		0			0
$793$	−57.2947			−2.03459
$794$	26.9421	+	20.4586i	0.956140	+	0.726049i
$795$		0			0
$796$	9.43896	+	33.8583i	0.334555	+	1.20008i
$797$			13.9098i			0.492709i	0.969180	+	0.246355i	$0.0792328\pi$
$797$			13.9098i			0.492709i	−0.969180	+	0.246355i	$0.920767\pi$
$798$		0			0
$799$	15.4455			0.546423
$800$	1.61700	−	15.2396i	0.0571697	−	0.538802i
$801$		0			0
$802$	−14.1320	+	18.6105i	−0.499018	+	0.657161i
$803$			20.1565i			0.711308i
$804$		0			0
$805$			5.14157i			0.181217i
$806$	−12.9944	−	9.86732i	−0.457707	−	0.347562i
$807$		0			0
$808$	−2.06178	+	5.19844i	−0.0725332	+	0.182881i
$809$	−19.7630			−0.694831			−0.347415	−	0.937711i	$-0.612941\pi$
$809$	−19.7630			−0.694831			−0.347415	+	0.937711i	$0.612941\pi$
$810$		0			0
$811$		−	30.1099i		−	1.05730i	−0.848839	−	0.528651i	$-0.822698\pi$
$811$		−	30.1099i		−	1.05730i	0.848839	−	0.528651i	$-0.177302\pi$
$812$	−1.44394	+	0.402538i	−0.0506722	+	0.0141263i
$813$		0			0
$814$	−1.33482	+	1.75784i	−0.0467855	+	0.0616123i
$815$	0.221024			0.00774213
$816$		0			0
$817$	−8.38816			−0.293465
$818$	−13.7036	+	18.0464i	−0.479135	+	0.630978i
$819$		0			0
$820$	−26.3769	+	7.35330i	−0.921120	+	0.256788i
$821$			26.1788i			0.913645i	0.889558	+	0.456823i	$0.151013\pi$
$821$			26.1788i			0.913645i	−0.889558	+	0.456823i	$0.848987\pi$
$822$		0			0
$823$	41.9817			1.46339			0.731696	−	0.681631i	$-0.238729\pi$
$823$	41.9817			1.46339			0.731696	+	0.681631i	$0.238729\pi$
$824$	−6.06955	−	2.40728i	−0.211443	−	0.0838615i
$825$		0			0
$826$	0.168692	+	0.128097i	0.00586956	+	0.00445707i
$827$		−	20.6394i		−	0.717703i	−0.933395	−	0.358851i	$-0.883168\pi$
$827$		−	20.6394i		−	0.717703i	0.933395	−	0.358851i	$-0.116832\pi$
$828$		0			0
$829$			36.6292i			1.27218i	0.771613	+	0.636092i	$0.219450\pi$
$829$			36.6292i			1.27218i	−0.771613	+	0.636092i	$0.780550\pi$
$830$	−4.79754	+	6.31793i	−0.166525	+	0.219299i
$831$		0			0
$832$	−21.3426	+	22.6736i	−0.739923	+	0.786065i
$833$	−8.00558			−0.277377
$834$		0			0
$835$			0.615418i			0.0212974i
$836$	0.707861	+	2.53915i	0.0244819	+	0.0878184i
$837$		0			0
$838$	14.3598	+	10.9042i	0.496051	+	0.376678i
$839$	−3.71855			−0.128379			−0.0641893	−	0.997938i	$-0.520446\pi$
$839$	−3.71855			−0.128379			−0.0641893	+	0.997938i	$0.520446\pi$
$840$		0			0
$841$	27.3311			0.942450
$842$	−2.13200	−	1.61894i	−0.0734736	−	0.0557924i
$843$		0			0
$844$	33.2913	−	9.28089i	1.14593	−	0.319461i
$845$			3.25414i			0.111946i
$846$		0			0
$847$	5.37399			0.184652
$848$	−6.35684	−	10.5152i	−0.218295	−	0.361093i
$849$		0			0
$850$	−2.78370	+	3.66588i	−0.0954801	+	0.125739i
$851$			6.93367i			0.237683i
$852$		0			0
$853$			49.7965i			1.70500i	0.522726	+	0.852501i	$0.324915\pi$
$853$			49.7965i			1.70500i	−0.522726	+	0.852501i	$0.675085\pi$
$854$	−9.61854	−	7.30387i	−0.329140	−	0.249933i
$855$		0			0
$856$	−15.6829	+	39.5418i	−0.536030	+	1.35151i
$857$	−35.6409			−1.21747			−0.608735	−	0.793374i	$-0.708323\pi$
$857$	−35.6409			−1.21747			−0.608735	+	0.793374i	$0.708323\pi$
$858$		0			0
$859$		−	43.5752i		−	1.48677i	−0.668866	−	0.743383i	$-0.733220\pi$
$859$		−	43.5752i		−	1.48677i	0.668866	−	0.743383i	$-0.266780\pi$
$860$	−6.81873	−	24.4593i	−0.232517	−	0.834056i
$861$		0			0
$862$	31.8297	−	41.9169i	1.08412	−	1.42769i
$863$	43.6579			1.48613			0.743067	−	0.669217i	$-0.233370\pi$
$863$	43.6579			1.48613			0.743067	+	0.669217i	$0.233370\pi$
$864$		0			0
$865$	−19.1415			−0.650829
$866$	30.8161	−	40.5820i	1.04717	−	1.37903i
$867$		0			0
$868$	−0.923598	−	3.31302i	−0.0313490	−	0.112451i
$869$		−	17.3093i		−	0.587179i
$870$		0			0
$871$	37.1669			1.25935
$872$	13.9734	−	35.2316i	0.473199	−	1.19309i
$873$		0			0
$874$	6.59475	+	5.00775i	0.223071	+	0.169390i
$875$		−	6.76947i		−	0.228850i
$876$		0			0
$877$		−	5.23078i		−	0.176631i	−0.996093	−	0.0883154i	$-0.971852\pi$
$877$		−	5.23078i		−	0.176631i	0.996093	−	0.0883154i	$-0.0281483\pi$
$878$	15.9239	−	20.9703i	0.537405	−	0.707713i
$879$		0			0
$880$	−6.82858	+	4.12815i	−0.230191	+	0.139160i
$881$	22.0358			0.742405			0.371203	−	0.928552i	$-0.378946\pi$
$881$	22.0358			0.742405			0.371203	+	0.928552i	$0.378946\pi$
$882$		0			0
$883$		−	56.9658i		−	1.91705i	−0.285008	−	0.958525i	$-0.591996\pi$
$883$		−	56.9658i		−	1.91705i	0.285008	−	0.958525i	$-0.408004\pi$
$884$	9.00907	−	2.51153i	0.303008	−	0.0844720i
$885$		0			0
$886$	2.55542	+	1.94047i	0.0858510	+	0.0651912i
$887$	−3.28694			−0.110365			−0.0551823	−	0.998476i	$-0.517574\pi$
$887$	−3.28694			−0.110365			−0.0551823	+	0.998476i	$0.517574\pi$
$888$		0			0
$889$	−1.51785			−0.0509071
$890$	14.2625	+	10.8303i	0.478082	+	0.363033i
$891$		0			0
$892$	−6.23153	−	22.3530i	−0.208647	−	0.748434i
$893$			12.8560i			0.430210i
$894$		0			0
$895$	8.40932			0.281092
$896$	−6.47337	+	1.08566i	−0.216260	+	0.0362695i
$897$		0			0
$898$	0.212881	−	0.280345i	0.00710392	−	0.00935522i
$899$		−	3.82929i		−	0.127714i
$900$		0			0
$901$			3.69057i			0.122951i
$902$	−13.4279	−	10.1965i	−0.447101	−	0.339508i
$903$		0			0
$904$	−12.8756	−	5.10665i	−0.428235	−	0.169845i
$905$	−13.8249			−0.459555
$906$		0			0
$907$			0.0491571i			0.00163223i	1.00000		0.000816117i	$0.000259778\pi$
$907$			0.0491571i			0.00163223i	−1.00000		0.000816117i	$0.999740\pi$
$908$	52.8804	−	14.7419i	1.75490	−	0.489228i
$909$		0			0
$910$	−2.92315	+	3.84952i	−0.0969014	+	0.127610i
$911$	−32.4990			−1.07674			−0.538370	−	0.842708i	$-0.680960\pi$
$911$	−32.4990			−1.07674			−0.538370	+	0.842708i	$0.680960\pi$
$912$		0			0
$913$	−4.88466			−0.161659
$914$	5.97624	−	7.87017i	0.197676	−	0.260322i
$915$		0			0
$916$	−42.1296	+	11.7448i	−1.39200	+	0.388059i
$917$		−	1.53452i		−	0.0506742i
$918$		0			0
$919$	−19.8145			−0.653619			−0.326810	−	0.945090i	$-0.605974\pi$
$919$	−19.8145			−0.653619			−0.326810	+	0.945090i	$0.605974\pi$
$920$	−9.24139	+	23.3006i	−0.304680	+	0.768200i
$921$		0			0
$922$	−38.4769	−	29.2175i	−1.26717	−	0.962229i
$923$			27.0008i			0.888742i
$924$		0			0
$925$		−	3.20809i		−	0.105481i
$926$	−29.5329	+	38.8922i	−0.970513	+	1.27808i
$927$		0			0
$928$	−7.26717	−	0.771084i	−0.238557	−	0.0253121i
$929$	−43.9293			−1.44127			−0.720637	−	0.693313i	$-0.756150\pi$
$929$	−43.9293			−1.44127			−0.720637	+	0.693313i	$0.756150\pi$
$930$		0			0
$931$		−	6.66341i		−	0.218385i
$932$	−11.9218	−	42.7643i	−0.390510	−	1.40079i
$933$		0			0
$934$	44.6571	+	33.9105i	1.46122	+	1.10959i
$935$	2.39666			0.0783793
$936$		0			0
$937$	−23.9026			−0.780863			−0.390431	−	0.920632i	$-0.627674\pi$
$937$	−23.9026			−0.780863			−0.390431	+	0.920632i	$0.627674\pi$
$938$	6.23953	+	4.73801i	0.203728	+	0.154701i
$939$		0			0
$940$	−37.4872	+	10.4506i	−1.22270	+	0.340862i
$941$		−	20.4109i		−	0.665375i	−0.943037	−	0.332688i	$-0.892045\pi$
$941$		−	20.4109i		−	0.665375i	0.943037	−	0.332688i	$-0.107955\pi$
$942$		0			0
$943$	−52.9654			−1.72479
$944$	0.534242	+	0.883718i	0.0173881	+	0.0287626i
$945$		0			0
$946$	9.45527	−	12.4517i	0.307417	−	0.404841i
$947$		−	2.72706i		−	0.0886176i	−0.999018	−	0.0443088i	$-0.985891\pi$
$947$		−	2.72706i		−	0.0886176i	0.999018	−	0.0443088i	$-0.0141085\pi$
$948$		0			0
$949$			59.5265i			1.93231i
$950$	−3.05128	−	2.31700i	−0.0989967	−	0.0751734i
$951$		0			0
$952$	1.83260	+	0.726835i	0.0593947	+	0.0235569i
$953$	32.7679			1.06146			0.530729	−	0.847542i	$-0.321918\pi$
$953$	32.7679			1.06146			0.530729	+	0.847542i	$0.321918\pi$
$954$		0			0
$955$			8.28723i			0.268168i
$956$	11.3511	+	40.7174i	0.367122	+	1.31690i
$957$		0			0
$958$	−21.2675	+	28.0074i	−0.687123	+	0.904879i
$959$	−2.83139			−0.0914305
$960$		0			0
$961$	−22.2139			−0.716578
$962$	−3.94202	+	5.19128i	−0.127096	+	0.167374i
$963$		0			0
$964$	9.26119	+	33.2206i	0.298283	+	1.06996i
$965$		−	25.6847i		−	0.826818i
$966$		0			0
$967$	36.4163			1.17107			0.585535	−	0.810647i	$-0.300885\pi$
$967$	36.4163			1.17107			0.585535	+	0.810647i	$0.300885\pi$
$968$	24.3539	+	9.65913i	0.782764	+	0.310456i
$969$		0			0
$970$	29.0554	+	22.0633i	0.932914	+	0.708411i
$971$		−	6.40329i		−	0.205491i	−0.994708	−	0.102746i	$-0.967237\pi$
$971$		−	6.40329i		−	0.205491i	0.994708	−	0.102746i	$-0.0327628\pi$
$972$		0			0
$973$			5.99505i			0.192192i
$974$	33.9810	−	44.7499i	1.08882	−	1.43388i
$975$		0			0
$976$	−30.4616	−	50.3880i	−0.975051	−	1.61288i
$977$	−11.6473			−0.372629			−0.186315	−	0.982490i	$-0.559654\pi$
$977$	−11.6473			−0.372629			−0.186315	+	0.982490i	$0.559654\pi$
$978$		0			0
$979$			11.0270i			0.352424i
$980$	19.4301	−	5.41668i	0.620670	−	0.173029i
$981$		0			0
$982$	−3.89203	−	2.95543i	−0.124200	−	0.0943115i
$983$	34.7061			1.10695			0.553477	−	0.832864i	$-0.313301\pi$
$983$	34.7061			1.10695			0.553477	+	0.832864i	$0.313301\pi$
$984$		0			0
$985$	35.9815			1.14646
$986$	1.74811	+	1.32743i	0.0556712	+	0.0422741i
$987$		0			0
$988$	2.09046	+	7.49866i	0.0665065	+	0.238564i
$989$		−	49.1150i		−	1.56176i
$990$		0			0
$991$	3.30503			0.104988			0.0524939	−	0.998621i	$-0.483283\pi$
$991$	3.30503			0.104988			0.0524939	+	0.998621i	$0.483283\pi$
$992$	1.76920	−	16.6741i	0.0561723	−	0.529402i
$993$		0			0
$994$	−3.44203	+	4.53285i	−0.109175	+	0.143773i
$995$			26.6003i			0.843288i
$996$		0			0
$997$			15.9009i			0.503586i	0.967781	+	0.251793i	$0.0810203\pi$
$997$			15.9009i			0.503586i	−0.967781	+	0.251793i	$0.918980\pi$
$998$	−19.8045	−	15.0386i	−0.626899	−	0.476038i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.g.b.685.6		16
3.2	odd	2		152.2.c.b.77.11	✓	16
4.3	odd	2		5472.2.g.b.2737.7		16
8.3	odd	2		5472.2.g.b.2737.10		16
8.5	even	2	inner	1368.2.g.b.685.5		16
12.11	even	2		608.2.c.b.305.12		16
24.5	odd	2		152.2.c.b.77.12	yes	16
24.11	even	2		608.2.c.b.305.5		16
48.5	odd	4		4864.2.a.bq.1.3		8
48.11	even	4		4864.2.a.bp.1.6		8
48.29	odd	4		4864.2.a.bo.1.6		8
48.35	even	4		4864.2.a.bn.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.11	✓	16	3.2	odd	2
152.2.c.b.77.12	yes	16	24.5	odd	2
608.2.c.b.305.5		16	24.11	even	2
608.2.c.b.305.12		16	12.11	even	2
1368.2.g.b.685.5		16	8.5	even	2	inner
1368.2.g.b.685.6		16	1.1	even	1	trivial
4864.2.a.bn.1.3		8	48.35	even	4
4864.2.a.bo.1.6		8	48.29	odd	4
4864.2.a.bp.1.6		8	48.11	even	4
4864.2.a.bq.1.3		8	48.5	odd	4
5472.2.g.b.2737.7		16	4.3	odd	2
5472.2.g.b.2737.10		16	8.3	odd	2

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.855255 + 1.12629i) q^{2} +(-0.537077 - 1.92654i) q^{4} -1.51356i q^{5} +0.580162 q^{7} +(2.62919 + 1.04278i) q^{8} +O(q^{10})\)	\(q+(-0.855255 + 1.12629i) q^{2} +(-0.537077 - 1.92654i) q^{4} -1.51356i q^{5} +0.580162 q^{7} +(2.62919 + 1.04278i) q^{8} +(1.70471 + 1.29448i) q^{10} +1.31799i q^{11} +3.89230i q^{13} +(-0.496186 + 0.653433i) q^{14} +(-3.42310 + 2.06940i) q^{16} +1.20142 q^{17} +1.00000i q^{19} +(-2.91593 + 0.812898i) q^{20} +(-1.48444 - 1.12722i) q^{22} -5.85527 q^{23} +2.70913 q^{25} +(-4.38387 - 3.32891i) q^{26} +(-0.311591 - 1.11770i) q^{28} -1.29188i q^{29} +2.96413 q^{31} +(0.596871 - 5.62528i) q^{32} +(-1.02752 + 1.35316i) q^{34} -0.878110i q^{35} -1.18418i q^{37} +(-1.12629 - 0.855255i) q^{38} +(1.57830 - 3.97943i) q^{40} +9.04577 q^{41} +8.38816i q^{43} +(2.53915 - 0.707861i) q^{44} +(5.00775 - 6.59475i) q^{46} +12.8560 q^{47} -6.66341 q^{49} +(-2.31700 + 3.05128i) q^{50} +(7.49866 - 2.09046i) q^{52} +3.07183i q^{53} +1.99485 q^{55} +(1.52535 + 0.604978i) q^{56} +(1.45503 + 1.10488i) q^{58} -0.258163i q^{59} +14.7200i q^{61} +(-2.53509 + 3.33848i) q^{62} +(5.82524 + 5.48330i) q^{64} +5.89123 q^{65} -9.54884i q^{67} +(-0.645257 - 2.31459i) q^{68} +(0.989010 + 0.751008i) q^{70} +6.93697 q^{71} +15.2934 q^{73} +(1.33373 + 1.01277i) q^{74} +(1.92654 - 0.537077i) q^{76} +0.764646i q^{77} -13.1332 q^{79} +(3.13216 + 5.18106i) q^{80} +(-7.73645 + 10.1882i) q^{82} +3.70615i q^{83} -1.81843i q^{85} +(-9.44754 - 7.17402i) q^{86} +(-1.37436 + 3.46523i) q^{88} +8.36653 q^{89} +2.25816i q^{91} +(3.14473 + 11.2804i) q^{92} +(-10.9952 + 14.4796i) q^{94} +1.51356 q^{95} +17.0442 q^{97} +(5.69892 - 7.50496i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8	\( 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44} - 48 q^{46} - 24 q^{47} + 24 q^{49} - 12 q^{50} + 8 q^{52} + 16 q^{55} + 48 q^{56} + 38 q^{58} + 16 q^{62} + 14 q^{64} - 16 q^{65} + 26 q^{68} - 32 q^{70} - 48 q^{71} + 20 q^{74} - 4 q^{76} - 48 q^{79} - 4 q^{80} - 12 q^{82} - 48 q^{86} + 40 q^{88} + 16 q^{89} - 62 q^{92} - 36 q^{94} - 16 q^{95} + 32 q^{97} + 48 q^{98}+O(q^{100}) \) 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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