LMFDB - Embedded newform 1638.2.y.c.827.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(827,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.827"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 8 x^{15} + 20 x^{14} - 12 x^{13} + 40 x^{12} + 40 x^{11} + 82 x^{10} + 104 x^{9} + 537 x^{8} + \cdots + 16 $ x^16 - 8x^15 + 20x^14 - 12x^13 + 40x^12 + 40x^11 + 82x^10 + 104x^9 + 537x^8 - 2528x^7 + 5494x^6 - 7284x^5 + 6002x^4 - 2984x^3 + 1000x^2 - 192*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			827.1
Root			$-0.777611 + 1.87732i$ of defining polynomial
Character	$\chi$	$=$	1638.827
Dual form			1638.2.y.c.1331.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.707107 + 0.707107i) q^{2} -1.00000i q^{4} +(-3.00337 + 3.00337i) q^{5} +(0.707107 - 0.707107i) q^{7} +(0.707107 + 0.707107i) q^{8} -4.24741i q^{10} +(2.53473 + 2.53473i) q^{11} +(2.84530 + 2.21455i) q^{13} +1.00000i q^{14} -1.00000 q^{16} -7.16415 q^{17} +(4.75928 + 4.75928i) q^{19} +(3.00337 + 3.00337i) q^{20} -3.58464 q^{22} +5.70954 q^{23} -13.0405i q^{25} +(-3.57786 + 0.446008i) q^{26} +(-0.707107 - 0.707107i) q^{28} +3.76120i q^{29} +(0.0835445 + 0.0835445i) q^{31} +(0.707107 - 0.707107i) q^{32} +(5.06582 - 5.06582i) q^{34} +4.24741i q^{35} +(-1.86236 + 1.86236i) q^{37} -6.73064 q^{38} -4.24741 q^{40} +(3.47354 - 3.47354i) q^{41} +7.91560i q^{43} +(2.53473 - 2.53473i) q^{44} +(-4.03725 + 4.03725i) q^{46} +(1.55421 + 1.55421i) q^{47} -1.00000i q^{49} +(9.22102 + 9.22102i) q^{50} +(2.21455 - 2.84530i) q^{52} +5.22095i q^{53} -15.2255 q^{55} +1.00000 q^{56} +(-2.65957 - 2.65957i) q^{58} +(-4.33095 - 4.33095i) q^{59} -12.2542 q^{61} -0.118150 q^{62} +1.00000i q^{64} +(-15.1966 + 1.89438i) q^{65} +(0.568664 + 0.568664i) q^{67} +7.16415i q^{68} +(-3.00337 - 3.00337i) q^{70} +(8.61297 - 8.61297i) q^{71} +(-3.94683 + 3.94683i) q^{73} -2.63378i q^{74} +(4.75928 - 4.75928i) q^{76} +3.58464 q^{77} -9.08222 q^{79} +(3.00337 - 3.00337i) q^{80} +4.91233i q^{82} +(-8.23786 + 8.23786i) q^{83} +(21.5166 - 21.5166i) q^{85} +(-5.59717 - 5.59717i) q^{86} +3.58464i q^{88} +(-8.69652 - 8.69652i) q^{89} +(3.57786 - 0.446008i) q^{91} -5.70954i q^{92} -2.19799 q^{94} -28.5878 q^{95} +(-9.42542 - 9.42542i) q^{97} +(0.707107 + 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{5} + 8 q^{11} + 4 q^{13} - 16 q^{16} + 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} - 24 q^{23} - 8 q^{26} - 8 q^{31} + 4 q^{34} - 4 q^{37} - 16 q^{38} + 16 q^{41} + 8 q^{44} - 4 q^{47} + 16 q^{50}+ \cdots - 28 q^{97}+O(q^{100}) $ 16 * q - 4 * q^5 + 8 * q^11 + 4 * q^13 - 16 * q^16 + 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 - 24 * q^23 - 8 * q^26 - 8 * q^31 + 4 * q^34 - 4 * q^37 - 16 * q^38 + 16 * q^41 + 8 * q^44 - 4 * q^47 + 16 * q^50 - 16 * q^55 + 16 * q^56 + 4 * q^58 + 8 * q^59 - 40 * q^61 - 24 * q^62 - 56 * q^65 - 4 * q^67 - 4 * q^70 + 24 * q^71 - 12 * q^73 + 16 * q^76 + 8 * q^77 + 16 * q^79 + 4 * q^80 + 8 * q^85 - 24 * q^86 - 16 * q^89 + 8 * q^91 - 8 * q^94 + 40 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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 $

$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.500000i
$5$	−3.00337	+	3.00337i	−1.34315	+	1.34315i	−0.450243	+	0.892906i	$0.648663\pi$
$5$	−3.00337	+	3.00337i	−1.34315	+	1.34315i	−0.892906	+	0.450243i	$0.851337\pi$
$6$		0			0
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$		0			0
$10$		−	4.24741i		−	1.34315i
$11$	2.53473	+	2.53473i	0.764249	+	0.764249i	0.977087	−	0.212839i	$-0.0682708\pi$
$11$	2.53473	+	2.53473i	0.764249	+	0.764249i	−0.212839	+	0.977087i	$0.568271\pi$
$12$		0			0
$13$	2.84530	+	2.21455i	0.789145	+	0.614207i
$13$	2.84530	+	2.21455i	0.789145	+	0.614207i
$14$			1.00000i			0.267261i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−7.16415			−1.73756			−0.868781	−	0.495196i	$-0.835096\pi$
$17$	−7.16415			−1.73756			−0.868781	+	0.495196i	$0.835096\pi$
$18$		0			0
$19$	4.75928	+	4.75928i	1.09185	+	1.09185i	0.995331	+	0.0965227i	$0.0307720\pi$
$19$	4.75928	+	4.75928i	1.09185	+	1.09185i	0.0965227	+	0.995331i	$0.469228\pi$
$20$	3.00337	+	3.00337i	0.671575	+	0.671575i
$21$		0			0
$22$	−3.58464			−0.764249
$23$	5.70954			1.19052			0.595261	−	0.803533i	$-0.297049\pi$
$23$	5.70954			1.19052			0.595261	+	0.803533i	$0.297049\pi$
$24$		0			0
$25$		−	13.0405i		−	2.60810i
$26$	−3.57786	+	0.446008i	−0.701676	+	0.0874694i
$27$		0			0
$28$	−0.707107	−	0.707107i	−0.133631	−	0.133631i
$29$			3.76120i			0.698438i	0.937041	+	0.349219i	$0.113553\pi$
$29$			3.76120i			0.698438i	−0.937041	+	0.349219i	$0.886447\pi$
$30$		0			0
$31$	0.0835445	+	0.0835445i	0.0150050	+	0.0150050i	0.714569	−	0.699564i	$-0.246623\pi$
$31$	0.0835445	+	0.0835445i	0.0150050	+	0.0150050i	−0.699564	+	0.714569i	$0.746623\pi$
$32$	0.707107	−	0.707107i	0.125000	−	0.125000i
$33$		0			0
$34$	5.06582	−	5.06582i	0.868781	−	0.868781i
$35$			4.24741i			0.717943i
$36$		0			0
$37$	−1.86236	+	1.86236i	−0.306171	+	0.306171i	−0.843422	−	0.537251i	$-0.819463\pi$
$37$	−1.86236	+	1.86236i	−0.306171	+	0.306171i	0.537251	+	0.843422i	$0.319463\pi$
$38$	−6.73064			−1.09185
$39$		0			0
$40$	−4.24741			−0.671575
$41$	3.47354	−	3.47354i	0.542476	−	0.542476i	−0.381778	−	0.924254i	$-0.624688\pi$
$41$	3.47354	−	3.47354i	0.542476	−	0.542476i	0.924254	+	0.381778i	$0.124688\pi$
$42$		0			0
$43$			7.91560i			1.20712i	0.797318	+	0.603559i	$0.206251\pi$
$43$			7.91560i			1.20712i	−0.797318	+	0.603559i	$0.793749\pi$
$44$	2.53473	−	2.53473i	0.382124	−	0.382124i
$45$		0			0
$46$	−4.03725	+	4.03725i	−0.595261	+	0.595261i
$47$	1.55421	+	1.55421i	0.226705	+	0.226705i	0.811315	−	0.584610i	$-0.198752\pi$
$47$	1.55421	+	1.55421i	0.226705	+	0.226705i	−0.584610	+	0.811315i	$0.698752\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$	9.22102	+	9.22102i	1.30405	+	1.30405i
$51$		0			0
$52$	2.21455	−	2.84530i	0.307103	−	0.394573i
$53$			5.22095i			0.717153i	0.933500	+	0.358576i	$0.116738\pi$
$53$			5.22095i			0.717153i	−0.933500	+	0.358576i	$0.883262\pi$
$54$		0			0
$55$	−15.2255			−2.05300
$56$	1.00000			0.133631
$57$		0			0
$58$	−2.65957	−	2.65957i	−0.349219	−	0.349219i
$59$	−4.33095	−	4.33095i	−0.563842	−	0.563842i	0.366554	−	0.930397i	$-0.380537\pi$
$59$	−4.33095	−	4.33095i	−0.563842	−	0.563842i	−0.930397	+	0.366554i	$0.880537\pi$
$60$		0			0
$61$	−12.2542			−1.56898			−0.784492	−	0.620139i	$-0.787076\pi$
$61$	−12.2542			−1.56898			−0.784492	+	0.620139i	$0.787076\pi$
$62$	−0.118150			−0.0150050
$63$		0			0
$64$			1.00000i			0.125000i
$65$	−15.1966	+	1.89438i	−1.88491	+	0.234969i
$66$		0			0
$67$	0.568664	+	0.568664i	0.0694734	+	0.0694734i	0.740990	−	0.671516i	$-0.234357\pi$
$67$	0.568664	+	0.568664i	0.0694734	+	0.0694734i	−0.671516	+	0.740990i	$0.734357\pi$
$68$			7.16415i			0.868781i
$69$		0			0
$70$	−3.00337	−	3.00337i	−0.358972	−	0.358972i
$71$	8.61297	−	8.61297i	1.02217	−	1.02217i	0.0224230	−	0.999749i	$-0.492862\pi$
$71$	8.61297	−	8.61297i	1.02217	−	1.02217i	0.999749	−	0.0224230i	$-0.00713805\pi$
$72$		0			0
$73$	−3.94683	+	3.94683i	−0.461942	+	0.461942i	−0.899292	−	0.437350i	$-0.855917\pi$
$73$	−3.94683	+	3.94683i	−0.461942	+	0.461942i	0.437350	+	0.899292i	$0.355917\pi$
$74$		−	2.63378i		−	0.306171i
$75$		0			0
$76$	4.75928	−	4.75928i	0.545927	−	0.545927i
$77$	3.58464			0.408508
$78$		0			0
$79$	−9.08222			−1.02183			−0.510915	−	0.859631i	$-0.670693\pi$
$79$	−9.08222			−1.02183			−0.510915	+	0.859631i	$0.670693\pi$
$80$	3.00337	−	3.00337i	0.335787	−	0.335787i
$81$		0			0
$82$			4.91233i			0.542476i
$83$	−8.23786	+	8.23786i	−0.904223	+	0.904223i	−0.995798	−	0.0915751i	$-0.970810\pi$
$83$	−8.23786	+	8.23786i	−0.904223	+	0.904223i	0.0915751	+	0.995798i	$0.470810\pi$
$84$		0			0
$85$	21.5166	−	21.5166i	2.33380	−	2.33380i
$86$	−5.59717	−	5.59717i	−0.603559	−	0.603559i
$87$		0			0
$88$			3.58464i			0.382124i
$89$	−8.69652	−	8.69652i	−0.921829	−	0.921829i	0.0753299	−	0.997159i	$-0.475999\pi$
$89$	−8.69652	−	8.69652i	−0.921829	−	0.921829i	−0.997159	+	0.0753299i	$0.975999\pi$
$90$		0			0
$91$	3.57786	−	0.446008i	0.375062	−	0.0467544i
$92$		−	5.70954i		−	0.595261i
$93$		0			0
$94$	−2.19799			−0.226705
$95$	−28.5878			−2.93304
$96$		0			0
$97$	−9.42542	−	9.42542i	−0.957006	−	0.957006i	0.0421070	−	0.999113i	$-0.486593\pi$
$97$	−9.42542	−	9.42542i	−0.957006	−	0.957006i	−0.999113	+	0.0421070i	$0.986593\pi$
$98$	0.707107	+	0.707107i	0.0714286	+	0.0714286i
$99$		0			0
$100$	−13.0405			−1.30405
$101$	−8.94096			−0.889658			−0.444829	−	0.895615i	$-0.646736\pi$
$101$	−8.94096			−0.889658			−0.444829	+	0.895615i	$0.646736\pi$
$102$		0			0
$103$			17.4234i			1.71678i	0.512995	+	0.858392i	$0.328536\pi$
$103$			17.4234i			1.71678i	−0.512995	+	0.858392i	$0.671464\pi$
$104$	0.446008	+	3.57786i	0.0437347	+	0.350838i
$105$		0			0
$106$	−3.69177	−	3.69177i	−0.358576	−	0.358576i
$107$		−	6.36508i		−	0.615335i	−0.951494	−	0.307668i	$-0.900452\pi$
$107$		−	6.36508i		−	0.615335i	0.951494	−	0.307668i	$-0.0995485\pi$
$108$		0			0
$109$	2.46859	+	2.46859i	0.236448	+	0.236448i	0.815378	−	0.578930i	$-0.196529\pi$
$109$	2.46859	+	2.46859i	0.236448	+	0.236448i	−0.578930	+	0.815378i	$0.696529\pi$
$110$	10.7660	−	10.7660i	1.02650	−	1.02650i
$111$		0			0
$112$	−0.707107	+	0.707107i	−0.0668153	+	0.0668153i
$113$		−	4.11099i		−	0.386729i	−0.981127	−	0.193365i	$-0.938060\pi$
$113$		−	4.11099i		−	0.386729i	0.981127	−	0.193365i	$-0.0619401\pi$
$114$		0			0
$115$	−17.1479	+	17.1479i	−1.59905	+	1.59905i
$116$	3.76120			0.349219
$117$		0			0
$118$	6.12490			0.563842
$119$	−5.06582	+	5.06582i	−0.464383	+	0.464383i
$120$		0			0
$121$			1.84968i			0.168153i
$122$	8.66500	−	8.66500i	0.784492	−	0.784492i
$123$		0			0
$124$	0.0835445	−	0.0835445i	0.00750252	−	0.00750252i
$125$	24.1486	+	24.1486i	2.15992	+	2.15992i
$126$		0			0
$127$		−	19.7825i		−	1.75541i	−0.479201	−	0.877705i	$-0.659073\pi$
$127$		−	19.7825i		−	1.75541i	0.479201	−	0.877705i	$-0.340927\pi$
$128$	−0.707107	−	0.707107i	−0.0625000	−	0.0625000i
$129$		0			0
$130$	9.40612	−	12.0852i	0.824971	−	1.05994i
$131$			12.9454i			1.13104i	0.824733	+	0.565522i	$0.191325\pi$
$131$			12.9454i			1.13104i	−0.824733	+	0.565522i	$0.808675\pi$
$132$		0			0
$133$	6.73064			0.583620
$134$	−0.804213			−0.0694734
$135$		0			0
$136$	−5.06582	−	5.06582i	−0.434391	−	0.434391i
$137$	6.35706	+	6.35706i	0.543120	+	0.543120i	0.924442	−	0.381322i	$-0.124531\pi$
$137$	6.35706	+	6.35706i	0.543120	+	0.543120i	−0.381322	+	0.924442i	$0.624531\pi$
$138$		0			0
$139$	−1.45054			−0.123033			−0.0615165	−	0.998106i	$-0.519594\pi$
$139$	−1.45054			−0.123033			−0.0615165	+	0.998106i	$0.519594\pi$
$140$	4.24741			0.358972
$141$		0			0
$142$			12.1806i			1.02217i
$143$	1.59878	+	12.8254i	0.133697	+	1.07251i
$144$		0			0
$145$	−11.2963	−	11.2963i	−0.938106	−	0.938106i
$146$		−	5.58166i		−	0.461942i
$147$		0			0
$148$	1.86236	+	1.86236i	0.153085	+	0.153085i
$149$	−5.20091	+	5.20091i	−0.426075	+	0.426075i	−0.887289	−	0.461214i	$-0.847414\pi$
$149$	−5.20091	+	5.20091i	−0.426075	+	0.426075i	0.461214	+	0.887289i	$0.347414\pi$
$150$		0			0
$151$	−7.76020	+	7.76020i	−0.631516	+	0.631516i	−0.948448	−	0.316932i	$-0.897347\pi$
$151$	−7.76020	+	7.76020i	−0.631516	+	0.631516i	0.316932	+	0.948448i	$0.397347\pi$
$152$			6.73064i			0.545927i
$153$		0			0
$154$	−2.53473	+	2.53473i	−0.204254	+	0.204254i
$155$	−0.501831			−0.0403080
$156$		0			0
$157$	11.7086			0.934446			0.467223	−	0.884139i	$-0.345254\pi$
$157$	11.7086			0.934446			0.467223	+	0.884139i	$0.345254\pi$
$158$	6.42210	−	6.42210i	0.510915	−	0.510915i
$159$		0			0
$160$			4.24741i			0.335787i
$161$	4.03725	−	4.03725i	0.318180	−	0.318180i
$162$		0			0
$163$	2.18476	−	2.18476i	0.171124	−	0.171124i	−0.616349	−	0.787473i	$-0.711389\pi$
$163$	2.18476	−	2.18476i	0.171124	−	0.171124i	0.787473	+	0.616349i	$0.211389\pi$
$164$	−3.47354	−	3.47354i	−0.271238	−	0.271238i
$165$		0			0
$166$		−	11.6501i		−	0.904223i
$167$	2.11815	+	2.11815i	0.163908	+	0.163908i	0.784295	−	0.620388i	$-0.213025\pi$
$167$	2.11815	+	2.11815i	0.163908	+	0.163908i	−0.620388	+	0.784295i	$0.713025\pi$
$168$		0			0
$169$	3.19151	+	12.6022i	0.245501	+	0.969396i
$170$			30.4291i			2.33380i
$171$		0			0
$172$	7.91560			0.603559
$173$	−5.06663			−0.385209			−0.192604	−	0.981276i	$-0.561693\pi$
$173$	−5.06663			−0.385209			−0.192604	+	0.981276i	$0.561693\pi$
$174$		0			0
$175$	−9.22102	−	9.22102i	−0.697044	−	0.697044i
$176$	−2.53473	−	2.53473i	−0.191062	−	0.191062i
$177$		0			0
$178$	12.2987			0.921829
$179$	−0.961445			−0.0718618			−0.0359309	−	0.999354i	$-0.511440\pi$
$179$	−0.961445			−0.0718618			−0.0359309	+	0.999354i	$0.511440\pi$
$180$		0			0
$181$			14.9714i			1.11282i	0.830909	+	0.556408i	$0.187821\pi$
$181$			14.9714i			1.11282i	−0.830909	+	0.556408i	$0.812179\pi$
$182$	−2.21455	+	2.84530i	−0.164154	+	0.210908i
$183$		0			0
$184$	4.03725	+	4.03725i	0.297630	+	0.297630i
$185$		−	11.1868i		−	0.822466i
$186$		0			0
$187$	−18.1592	−	18.1592i	−1.32793	−	1.32793i
$188$	1.55421	−	1.55421i	0.113353	−	0.113353i
$189$		0			0
$190$	20.2146	−	20.2146i	1.46652	−	1.46652i
$191$		−	19.5793i		−	1.41671i	−0.705856	−	0.708356i	$-0.749437\pi$
$191$		−	19.5793i		−	1.41671i	0.705856	−	0.708356i	$-0.250563\pi$
$192$		0			0
$193$	−4.06219	+	4.06219i	−0.292403	+	0.292403i	−0.838029	−	0.545626i	$-0.816292\pi$
$193$	−4.06219	+	4.06219i	−0.292403	+	0.292403i	0.545626	+	0.838029i	$0.316292\pi$
$194$	13.3296			0.957006
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	4.34520	−	4.34520i	0.309583	−	0.309583i	−0.535165	−	0.844748i	$-0.679751\pi$
$197$	4.34520	−	4.34520i	0.309583	−	0.309583i	0.844748	+	0.535165i	$0.179751\pi$
$198$		0			0
$199$			0.564211i			0.0399959i	0.999800	+	0.0199979i	$0.00636596\pi$
$199$			0.564211i			0.0399959i	−0.999800	+	0.0199979i	$0.993634\pi$
$200$	9.22102	−	9.22102i	0.652025	−	0.652025i
$201$		0			0
$202$	6.32221	−	6.32221i	0.444829	−	0.444829i
$203$	2.65957	+	2.65957i	0.186665	+	0.186665i
$204$		0			0
$205$			20.8647i			1.45725i
$206$	−12.3202	−	12.3202i	−0.858392	−	0.858392i
$207$		0			0
$208$	−2.84530	−	2.21455i	−0.197286	−	0.153552i
$209$			24.1269i			1.66890i
$210$		0			0
$211$	13.3542			0.919342			0.459671	−	0.888089i	$-0.347967\pi$
$211$	13.3542			0.919342			0.459671	+	0.888089i	$0.347967\pi$
$212$	5.22095			0.358576
$213$		0			0
$214$	4.50079	+	4.50079i	0.307668	+	0.307668i
$215$	−23.7735	−	23.7735i	−1.62134	−	1.62134i
$216$		0			0
$217$	0.118150			0.00802053
$218$	−3.49111			−0.236448
$219$		0			0
$220$			15.2255i			1.02650i
$221$	−20.3842	−	15.8654i	−1.37119	−	1.06722i
$222$		0			0
$223$	3.80927	+	3.80927i	0.255087	+	0.255087i	0.823053	−	0.567965i	$-0.192269\pi$
$223$	3.80927	+	3.80927i	0.255087	+	0.255087i	−0.567965	+	0.823053i	$0.692269\pi$
$224$		−	1.00000i		−	0.0668153i
$225$		0			0
$226$	2.90691	+	2.90691i	0.193365	+	0.193365i
$227$	−9.92499	+	9.92499i	−0.658744	+	0.658744i	−0.955083	−	0.296339i	$-0.904234\pi$
$227$	−9.92499	+	9.92499i	−0.658744	+	0.658744i	0.296339	+	0.955083i	$0.404234\pi$
$228$		0			0
$229$	6.50911	−	6.50911i	0.430134	−	0.430134i	−0.458540	−	0.888674i	$-0.651627\pi$
$229$	6.50911	−	6.50911i	0.430134	−	0.430134i	0.888674	+	0.458540i	$0.151627\pi$
$230$		−	24.2508i		−	1.59905i
$231$		0			0
$232$	−2.65957	+	2.65957i	−0.174609	+	0.174609i
$233$	−17.0947			−1.11991			−0.559955	−	0.828523i	$-0.689181\pi$
$233$	−17.0947			−1.11991			−0.559955	+	0.828523i	$0.689181\pi$
$234$		0			0
$235$	−9.33576			−0.608998
$236$	−4.33095	+	4.33095i	−0.281921	+	0.281921i
$237$		0			0
$238$		−	7.16415i		−	0.464383i
$239$	5.78149	−	5.78149i	0.373974	−	0.373974i	−0.494949	−	0.868922i	$-0.664813\pi$
$239$	5.78149	−	5.78149i	0.373974	−	0.373974i	0.868922	+	0.494949i	$0.164813\pi$
$240$		0			0
$241$	14.4471	−	14.4471i	0.930622	−	0.930622i	−0.0671225	−	0.997745i	$-0.521382\pi$
$241$	14.4471	−	14.4471i	0.930622	−	0.930622i	0.997745	+	0.0671225i	$0.0213818\pi$
$242$	−1.30792	−	1.30792i	−0.0840763	−	0.0840763i
$243$		0			0
$244$			12.2542i			0.784492i
$245$	3.00337	+	3.00337i	0.191878	+	0.191878i
$246$		0			0
$247$	3.00192	+	24.0813i	0.191008	+	1.53225i
$248$			0.118150i			0.00750252i
$249$		0			0
$250$	−34.1513			−2.15992
$251$	17.0939			1.07896			0.539479	−	0.841999i	$-0.318621\pi$
$251$	17.0939			1.07896			0.539479	+	0.841999i	$0.318621\pi$
$252$		0			0
$253$	14.4721	+	14.4721i	0.909855	+	0.909855i
$254$	13.9883	+	13.9883i	0.877705	+	0.877705i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−25.8713			−1.61381			−0.806903	−	0.590684i	$-0.798858\pi$
$257$	−25.8713			−1.61381			−0.806903	+	0.590684i	$0.798858\pi$
$258$		0			0
$259$			2.63378i			0.163655i
$260$	1.89438	+	15.1966i	0.117484	+	0.942455i
$261$		0			0
$262$	−9.15378	−	9.15378i	−0.565522	−	0.565522i
$263$			4.09575i			0.252555i	0.991995	+	0.126277i	$0.0403029\pi$
$263$			4.09575i			0.252555i	−0.991995	+	0.126277i	$0.959697\pi$
$264$		0			0
$265$	−15.6805	−	15.6805i	−0.963243	−	0.963243i
$266$	−4.75928	+	4.75928i	−0.291810	+	0.291810i
$267$		0			0
$268$	0.568664	−	0.568664i	0.0347367	−	0.0347367i
$269$		−	22.4500i		−	1.36880i	−0.729106	−	0.684400i	$-0.760064\pi$
$269$		−	22.4500i		−	1.36880i	0.729106	−	0.684400i	$-0.239936\pi$
$270$		0			0
$271$	15.1392	−	15.1392i	0.919643	−	0.919643i	−0.0773603	−	0.997003i	$-0.524649\pi$
$271$	15.1392	−	15.1392i	0.919643	−	0.919643i	0.997003	+	0.0773603i	$0.0246492\pi$
$272$	7.16415			0.434391
$273$		0			0
$274$	−8.99023			−0.543120
$275$	33.0541	−	33.0541i	1.99324	−	1.99324i
$276$		0			0
$277$			24.5278i			1.47373i	0.676037	+	0.736867i	$0.263696\pi$
$277$			24.5278i			1.47373i	−0.676037	+	0.736867i	$0.736304\pi$
$278$	1.02569	−	1.02569i	0.0615165	−	0.0615165i
$279$		0			0
$280$	−3.00337	+	3.00337i	−0.179486	+	0.179486i
$281$	5.13979	+	5.13979i	0.306614	+	0.306614i	0.843595	−	0.536981i	$-0.180435\pi$
$281$	5.13979	+	5.13979i	0.306614	+	0.306614i	−0.536981	+	0.843595i	$0.680435\pi$
$282$		0			0
$283$		−	2.58173i		−	0.153468i	−0.997052	−	0.0767340i	$-0.975551\pi$
$283$		−	2.58173i		−	0.153468i	0.997052	−	0.0767340i	$-0.0244492\pi$
$284$	−8.61297	−	8.61297i	−0.511086	−	0.511086i
$285$		0			0
$286$	−10.1994	−	7.93839i	−0.603103	−	0.469407i
$287$		−	4.91233i		−	0.289966i
$288$		0			0
$289$	34.3251			2.01912
$290$	15.9754			0.938106
$291$		0			0
$292$	3.94683	+	3.94683i	0.230971	+	0.230971i
$293$	−3.60392	−	3.60392i	−0.210543	−	0.210543i	0.593955	−	0.804498i	$-0.297566\pi$
$293$	−3.60392	−	3.60392i	−0.210543	−	0.210543i	−0.804498	+	0.593955i	$0.797566\pi$
$294$		0			0
$295$	26.0149			1.51465
$296$	−2.63378			−0.153085
$297$		0			0
$298$		−	7.35520i		−	0.426075i
$299$	16.2454	+	12.6441i	0.939494	+	0.731226i
$300$		0			0
$301$	5.59717	+	5.59717i	0.322616	+	0.322616i
$302$		−	10.9746i		−	0.631516i
$303$		0			0
$304$	−4.75928	−	4.75928i	−0.272963	−	0.272963i
$305$	36.8038	−	36.8038i	2.10738	−	2.10738i
$306$		0			0
$307$	−14.0322	+	14.0322i	−0.800858	+	0.800858i	−0.983230	−	0.182371i	$-0.941623\pi$
$307$	−14.0322	+	14.0322i	−0.800858	+	0.800858i	0.182371	+	0.983230i	$0.441623\pi$
$308$		−	3.58464i		−	0.204254i
$309$		0			0
$310$	0.354848	−	0.354848i	0.0201540	−	0.0201540i
$311$	22.4477			1.27289			0.636445	−	0.771322i	$-0.280404\pi$
$311$	22.4477			1.27289			0.636445	+	0.771322i	$0.280404\pi$
$312$		0			0
$313$	23.3587			1.32031			0.660156	−	0.751128i	$-0.270490\pi$
$313$	23.3587			1.32031			0.660156	+	0.751128i	$0.270490\pi$
$314$	−8.27922	+	8.27922i	−0.467223	+	0.467223i
$315$		0			0
$316$			9.08222i			0.510915i
$317$	−1.86545	+	1.86545i	−0.104774	+	0.104774i	−0.757551	−	0.652777i	$-0.773604\pi$
$317$	−1.86545	+	1.86545i	−0.104774	+	0.104774i	0.652777	+	0.757551i	$0.273604\pi$
$318$		0			0
$319$	−9.53362	+	9.53362i	−0.533780	+	0.533780i
$320$	−3.00337	−	3.00337i	−0.167894	−	0.167894i
$321$		0			0
$322$			5.70954i			0.318180i
$323$	−34.0962	−	34.0962i	−1.89716	−	1.89716i
$324$		0			0
$325$	28.8789	−	37.1042i	1.60191	−	2.05817i
$326$			3.08972i			0.171124i
$327$		0			0
$328$	4.91233			0.271238
$329$	2.19799			0.121179
$330$		0			0
$331$	−15.6984	−	15.6984i	−0.862862	−	0.862862i	0.128808	−	0.991670i	$-0.458885\pi$
$331$	−15.6984	−	15.6984i	−0.862862	−	0.862862i	−0.991670	+	0.128808i	$0.958885\pi$
$332$	8.23786	+	8.23786i	0.452112	+	0.452112i
$333$		0			0
$334$	−2.99552			−0.163908
$335$	−3.41582			−0.186626
$336$		0			0
$337$			5.07534i			0.276471i	0.990399	+	0.138236i	$0.0441431\pi$
$337$			5.07534i			0.276471i	−0.990399	+	0.138236i	$0.955857\pi$
$338$	−11.1678	−	6.65433i	−0.607449	−	0.361948i
$339$		0			0
$340$	−21.5166	−	21.5166i	−1.16690	−	1.16690i
$341$			0.423525i			0.0229352i
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.0381802	−	0.0381802i
$344$	−5.59717	+	5.59717i	−0.301779	+	0.301779i
$345$		0			0
$346$	3.58265	−	3.58265i	0.192604	−	0.192604i
$347$			14.2741i			0.766276i	0.923691	+	0.383138i	$0.125157\pi$
$347$			14.2741i			0.766276i	−0.923691	+	0.383138i	$0.874843\pi$
$348$		0			0
$349$	−6.17873	+	6.17873i	−0.330740	+	0.330740i	−0.852867	−	0.522127i	$-0.825139\pi$
$349$	−6.17873	+	6.17873i	−0.330740	+	0.330740i	0.522127	+	0.852867i	$0.325139\pi$
$350$	13.0405			0.697044
$351$		0			0
$352$	3.58464			0.191062
$353$	−18.3126	+	18.3126i	−0.974679	+	0.974679i	−0.999687	−	0.0250078i	$-0.992039\pi$
$353$	−18.3126	+	18.3126i	−0.974679	+	0.974679i	0.0250078	+	0.999687i	$0.492039\pi$
$354$		0			0
$355$			51.7359i			2.74586i
$356$	−8.69652	+	8.69652i	−0.460914	+	0.460914i
$357$		0			0
$358$	0.679845	−	0.679845i	0.0359309	−	0.0359309i
$359$	−0.831801	−	0.831801i	−0.0439008	−	0.0439008i	0.684816	−	0.728716i	$-0.259883\pi$
$359$	−0.831801	−	0.831801i	−0.0439008	−	0.0439008i	−0.728716	+	0.684816i	$0.759883\pi$
$360$		0			0
$361$			26.3015i			1.38429i
$362$	−10.5864	−	10.5864i	−0.556408	−	0.556408i
$363$		0			0
$364$	−0.446008	−	3.57786i	−0.0233772	−	0.187531i
$365$		−	23.7076i		−	1.24091i
$366$		0			0
$367$	33.1794			1.73195			0.865975	−	0.500088i	$-0.166699\pi$
$367$	33.1794			1.73195			0.865975	+	0.500088i	$0.166699\pi$
$368$	−5.70954			−0.297630
$369$		0			0
$370$	7.91023	+	7.91023i	0.411233	+	0.411233i
$371$	3.69177	+	3.69177i	0.191667	+	0.191667i
$372$		0			0
$373$	−30.4259			−1.57539			−0.787697	−	0.616063i	$-0.788727\pi$
$373$	−30.4259			−1.57539			−0.787697	+	0.616063i	$0.788727\pi$
$374$	25.6809			1.32793
$375$		0			0
$376$			2.19799i			0.113353i
$377$	−8.32938	+	10.7018i	−0.428985	+	0.551169i
$378$		0			0
$379$	12.1657	+	12.1657i	0.624911	+	0.624911i	0.946783	−	0.321873i	$-0.104312\pi$
$379$	12.1657	+	12.1657i	0.624911	+	0.624911i	−0.321873	+	0.946783i	$0.604312\pi$
$380$			28.5878i			1.46652i
$381$		0			0
$382$	13.8447	+	13.8447i	0.708356	+	0.708356i
$383$	26.5124	−	26.5124i	1.35472	−	1.35472i	0.474425	−	0.880296i	$-0.342656\pi$
$383$	26.5124	−	26.5124i	1.35472	−	1.35472i	0.880296	−	0.474425i	$-0.157344\pi$
$384$		0			0
$385$	−10.7660	+	10.7660i	−0.548687	+	0.548687i
$386$		−	5.74480i		−	0.292403i
$387$		0			0
$388$	−9.42542	+	9.42542i	−0.478503	+	0.478503i
$389$	3.73757			0.189502			0.0947512	−	0.995501i	$-0.469794\pi$
$389$	3.73757			0.189502			0.0947512	+	0.995501i	$0.469794\pi$
$390$		0			0
$391$	−40.9040			−2.06860
$392$	0.707107	−	0.707107i	0.0357143	−	0.0357143i
$393$		0			0
$394$			6.14504i			0.309583i
$395$	27.2773	−	27.2773i	1.37247	−	1.37247i
$396$		0			0
$397$	−22.9598	+	22.9598i	−1.15232	+	1.15232i	−0.166231	+	0.986087i	$0.553160\pi$
$397$	−22.9598	+	22.9598i	−1.15232	+	1.15232i	−0.986087	+	0.166231i	$0.946840\pi$
$398$	−0.398958	−	0.398958i	−0.0199979	−	0.0199979i
$399$		0			0
$400$			13.0405i			0.652025i
$401$	15.6263	+	15.6263i	0.780341	+	0.780341i	0.979888	−	0.199547i	$-0.0639472\pi$
$401$	15.6263	+	15.6263i	0.780341	+	0.780341i	−0.199547	+	0.979888i	$0.563947\pi$
$402$		0			0
$403$	0.0526958	+	0.422723i	0.00262496	+	0.0210573i
$404$			8.94096i			0.444829i
$405$		0			0
$406$	−3.76120			−0.186665
$407$	−9.44117			−0.467982
$408$		0			0
$409$	19.4055	+	19.4055i	0.959539	+	0.959539i	0.999213	−	0.0396740i	$-0.0126319\pi$
$409$	19.4055	+	19.4055i	0.959539	+	0.959539i	−0.0396740	+	0.999213i	$0.512632\pi$
$410$	−14.7536	−	14.7536i	−0.728627	−	0.728627i
$411$		0			0
$412$	17.4234			0.858392
$413$	−6.12490			−0.301386
$414$		0			0
$415$		−	49.4828i		−	2.42901i
$416$	3.57786	−	0.446008i	0.175419	−	0.0218673i
$417$		0			0
$418$	−17.0603	−	17.0603i	−0.834448	−	0.834448i
$419$		−	8.84076i		−	0.431899i	−0.976404	−	0.215950i	$-0.930715\pi$
$419$		−	8.84076i		−	0.431899i	0.976404	−	0.215950i	$-0.0692847\pi$
$420$		0			0
$421$	7.59479	+	7.59479i	0.370147	+	0.370147i	0.867531	−	0.497383i	$-0.165706\pi$
$421$	7.59479	+	7.59479i	0.370147	+	0.370147i	−0.497383	+	0.867531i	$0.665706\pi$
$422$	−9.44285	+	9.44285i	−0.459671	+	0.459671i
$423$		0			0
$424$	−3.69177	+	3.69177i	−0.179288	+	0.179288i
$425$			93.4241i			4.53173i
$426$		0			0
$427$	−8.66500	+	8.66500i	−0.419329	+	0.419329i
$428$	−6.36508			−0.307668
$429$		0			0
$430$	33.6208			1.62134
$431$	−2.99857	+	2.99857i	−0.144436	+	0.144436i	−0.775627	−	0.631191i	$-0.782566\pi$
$431$	−2.99857	+	2.99857i	−0.144436	+	0.144436i	0.631191	+	0.775627i	$0.282566\pi$
$432$		0			0
$433$			10.4330i			0.501380i	0.968067	+	0.250690i	$0.0806574\pi$
$433$			10.4330i			0.501380i	−0.968067	+	0.250690i	$0.919343\pi$
$434$	−0.0835445	+	0.0835445i	−0.00401026	+	0.00401026i
$435$		0			0
$436$	2.46859	−	2.46859i	0.118224	−	0.118224i
$437$	27.1733	+	27.1733i	1.29987	+	1.29987i
$438$		0			0
$439$			11.5804i			0.552703i	0.961057	+	0.276352i	$0.0891255\pi$
$439$			11.5804i			0.552703i	−0.961057	+	0.276352i	$0.910875\pi$
$440$	−10.7660	−	10.7660i	−0.513250	−	0.513250i
$441$		0			0
$442$	25.6323	−	3.19527i	1.21921	−	0.151983i
$443$			11.7229i			0.556971i	0.960440	+	0.278486i	$0.0898325\pi$
$443$			11.7229i			0.556971i	−0.960440	+	0.278486i	$0.910168\pi$
$444$		0			0
$445$	52.2378			2.47631
$446$	−5.38712			−0.255087
$447$		0			0
$448$	0.707107	+	0.707107i	0.0334077	+	0.0334077i
$449$	19.0283	+	19.0283i	0.898000	+	0.898000i	0.995259	−	0.0972590i	$-0.0310075\pi$
$449$	19.0283	+	19.0283i	0.898000	+	0.898000i	−0.0972590	+	0.995259i	$0.531008\pi$
$450$		0			0
$451$	17.6090			0.829174
$452$	−4.11099			−0.193365
$453$		0			0
$454$		−	14.0360i		−	0.658744i
$455$	−9.40612	+	12.0852i	−0.440966	+	0.566562i
$456$		0			0
$457$	−25.1280	−	25.1280i	−1.17544	−	1.17544i	−0.980894	−	0.194544i	$-0.937677\pi$
$457$	−25.1280	−	25.1280i	−1.17544	−	1.17544i	−0.194544	−	0.980894i	$-0.562323\pi$
$458$			9.20528i			0.430134i
$459$		0			0
$460$	17.1479	+	17.1479i	0.799524	+	0.799524i
$461$	21.5174	−	21.5174i	1.00216	−	1.00216i	0.00216692	−	0.999998i	$-0.499310\pi$
$461$	21.5174	−	21.5174i	1.00216	−	1.00216i	0.999998	−	0.00216692i	$-0.000689754\pi$
$462$		0			0
$463$	12.6794	−	12.6794i	0.589263	−	0.589263i	−0.348169	−	0.937432i	$-0.613196\pi$
$463$	12.6794	−	12.6794i	0.589263	−	0.589263i	0.937432	+	0.348169i	$0.113196\pi$
$464$		−	3.76120i		−	0.174609i
$465$		0			0
$466$	12.0878	−	12.0878i	0.559955	−	0.559955i
$467$	12.6572			0.585706			0.292853	−	0.956157i	$-0.405395\pi$
$467$	12.6572			0.585706			0.292853	+	0.956157i	$0.405395\pi$
$468$		0			0
$469$	0.804213			0.0371351
$470$	6.60138	−	6.60138i	0.304499	−	0.304499i
$471$		0			0
$472$		−	6.12490i		−	0.281921i
$473$	−20.0639	+	20.0639i	−0.922538	+	0.922538i
$474$		0			0
$475$	62.0634	−	62.0634i	2.84766	−	2.84766i
$476$	5.06582	+	5.06582i	0.232191	+	0.232191i
$477$		0			0
$478$			8.17627i			0.373974i
$479$	6.81265	+	6.81265i	0.311278	+	0.311278i	0.845405	−	0.534127i	$-0.179359\pi$
$479$	6.81265	+	6.81265i	0.311278	+	0.311278i	−0.534127	+	0.845405i	$0.679359\pi$
$480$		0			0
$481$	−9.42330	+	1.17469i	−0.429665	+	0.0535612i
$482$			20.4313i			0.930622i
$483$		0			0
$484$	1.84968			0.0840763
$485$	56.6161			2.57080
$486$		0			0
$487$	20.4577	+	20.4577i	0.927026	+	0.927026i	0.997513	−	0.0704871i	$-0.0224554\pi$
$487$	20.4577	+	20.4577i	0.927026	+	0.927026i	−0.0704871	+	0.997513i	$0.522455\pi$
$488$	−8.66500	−	8.66500i	−0.392246	−	0.392246i
$489$		0			0
$490$	−4.24741			−0.191878
$491$	−40.9998			−1.85030			−0.925148	−	0.379606i	$-0.876059\pi$
$491$	−40.9998			−1.85030			−0.925148	+	0.379606i	$0.876059\pi$
$492$		0			0
$493$		−	26.9458i		−	1.21358i
$494$	−19.1507	−	14.9054i	−0.861631	−	0.670624i
$495$		0			0
$496$	−0.0835445	−	0.0835445i	−0.00375126	−	0.00375126i
$497$		−	12.1806i		−	0.546374i
$498$		0			0
$499$	3.12135	+	3.12135i	0.139731	+	0.139731i	0.773512	−	0.633781i	$-0.218498\pi$
$499$	3.12135	+	3.12135i	0.139731	+	0.139731i	−0.633781	+	0.773512i	$0.718498\pi$
$500$	24.1486	−	24.1486i	1.07996	−	1.07996i
$501$		0			0
$502$	−12.0872	+	12.0872i	−0.539479	+	0.539479i
$503$		−	0.981307i		−	0.0437543i	−0.999761	−	0.0218771i	$-0.993036\pi$
$503$		−	0.981307i		−	0.0437543i	0.999761	−	0.0218771i	$-0.00696427\pi$
$504$		0			0
$505$	26.8530	−	26.8530i	1.19494	−	1.19494i
$506$	−20.4667			−0.909855
$507$		0			0
$508$	−19.7825			−0.877705
$509$	16.9483	−	16.9483i	0.751219	−	0.751219i	−0.223488	−	0.974707i	$-0.571744\pi$
$509$	16.9483	−	16.9483i	0.751219	−	0.751219i	0.974707	+	0.223488i	$0.0717443\pi$
$510$		0			0
$511$			5.58166i			0.246918i
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$		0			0
$514$	18.2938	−	18.2938i	0.806903	−	0.806903i
$515$	−52.3291	−	52.3291i	−2.30590	−	2.30590i
$516$		0			0
$517$			7.87901i			0.346518i
$518$	−1.86236	−	1.86236i	−0.0818276	−	0.0818276i
$519$		0			0
$520$	−12.0852	−	9.40612i	−0.529970	−	0.412485i
$521$			44.4621i			1.94792i	0.226723	+	0.973959i	$0.427199\pi$
$521$			44.4621i			1.94792i	−0.226723	+	0.973959i	$0.572801\pi$
$522$		0			0
$523$	1.28397			0.0561441			0.0280720	−	0.999606i	$-0.491063\pi$
$523$	1.28397			0.0561441			0.0280720	+	0.999606i	$0.491063\pi$
$524$	12.9454			0.565522
$525$		0			0
$526$	−2.89613	−	2.89613i	−0.126277	−	0.126277i
$527$	−0.598526	−	0.598526i	−0.0260722	−	0.0260722i
$528$		0			0
$529$	9.59885			0.417341
$530$	22.1755			0.963243
$531$		0			0
$532$		−	6.73064i		−	0.291810i
$533$	17.5756	−	2.19094i	0.761285	−	0.0949002i
$534$		0			0
$535$	19.1167	+	19.1167i	0.826487	+	0.826487i
$536$			0.804213i			0.0347367i
$537$		0			0
$538$	15.8745	+	15.8745i	0.684400	+	0.684400i
$539$	2.53473	−	2.53473i	0.109178	−	0.109178i
$540$		0			0
$541$	24.2007	−	24.2007i	1.04047	−	1.04047i	0.0413221	−	0.999146i	$-0.486843\pi$
$541$	24.2007	−	24.2007i	1.04047	−	1.04047i	0.999146	−	0.0413221i	$-0.0131570\pi$
$542$			21.4101i			0.919643i
$543$		0			0
$544$	−5.06582	+	5.06582i	−0.217195	+	0.217195i
$545$	−14.8282			−0.635170
$546$		0			0
$547$	−23.8487			−1.01970			−0.509849	−	0.860264i	$-0.670299\pi$
$547$	−23.8487			−1.01970			−0.509849	+	0.860264i	$0.670299\pi$
$548$	6.35706	−	6.35706i	0.271560	−	0.271560i
$549$		0			0
$550$			46.7455i			1.99324i
$551$	−17.9006	+	17.9006i	−0.762592	+	0.762592i
$552$		0			0
$553$	−6.42210	+	6.42210i	−0.273095	+	0.273095i
$554$	−17.3438	−	17.3438i	−0.736867	−	0.736867i
$555$		0			0
$556$			1.45054i			0.0615165i
$557$	26.4867	+	26.4867i	1.12228	+	1.12228i	0.991398	+	0.130881i	$0.0417805\pi$
$557$	26.4867	+	26.4867i	1.12228	+	1.12228i	0.130881	+	0.991398i	$0.458220\pi$
$558$		0			0
$559$	−17.5295	+	22.5223i	−0.741420	+	0.952591i
$560$		−	4.24741i		−	0.179486i
$561$		0			0
$562$	−7.26876			−0.306614
$563$	19.0628			0.803401			0.401701	−	0.915771i	$-0.368419\pi$
$563$	19.0628			0.803401			0.401701	+	0.915771i	$0.368419\pi$
$564$		0			0
$565$	12.3468	+	12.3468i	0.519435	+	0.519435i
$566$	1.82556	+	1.82556i	0.0767340	+	0.0767340i
$567$		0			0
$568$	12.1806			0.511086
$569$	−6.65003			−0.278784			−0.139392	−	0.990237i	$-0.544515\pi$
$569$	−6.65003			−0.278784			−0.139392	+	0.990237i	$0.544515\pi$
$570$		0			0
$571$		−	29.8722i		−	1.25011i	−0.780579	−	0.625057i	$-0.785076\pi$
$571$		−	29.8722i		−	1.25011i	0.780579	−	0.625057i	$-0.214924\pi$
$572$	12.8254	−	1.59878i	0.536255	−	0.0668484i
$573$		0			0
$574$	3.47354	+	3.47354i	0.144983	+	0.144983i
$575$		−	74.4552i		−	3.10500i
$576$		0			0
$577$	−8.74494	−	8.74494i	−0.364057	−	0.364057i	0.501247	−	0.865304i	$-0.332875\pi$
$577$	−8.74494	−	8.74494i	−0.364057	−	0.364057i	−0.865304	+	0.501247i	$0.832875\pi$
$578$	−24.2715	+	24.2715i	−1.00956	+	1.00956i
$579$		0			0
$580$	−11.2963	+	11.2963i	−0.469053	+	0.469053i
$581$			11.6501i			0.483328i
$582$		0			0
$583$	−13.2337	+	13.2337i	−0.548083	+	0.548083i
$584$	−5.58166			−0.230971
$585$		0			0
$586$	5.09672			0.210543
$587$	−24.7147	+	24.7147i	−1.02009	+	1.02009i	−0.0202912	+	0.999794i	$0.506459\pi$
$587$	−24.7147	+	24.7147i	−1.02009	+	1.02009i	−0.999794	+	0.0202912i	$0.993541\pi$
$588$		0			0
$589$			0.795223i			0.0327666i
$590$	−18.3953	+	18.3953i	−0.757324	+	0.757324i
$591$		0			0
$592$	1.86236	−	1.86236i	0.0765427	−	0.0765427i
$593$	17.5245	+	17.5245i	0.719645	+	0.719645i	0.968532	−	0.248888i	$-0.0800650\pi$
$593$	17.5245	+	17.5245i	0.719645	+	0.719645i	−0.248888	+	0.968532i	$0.580065\pi$
$594$		0			0
$595$		−	30.4291i		−	1.24747i
$596$	5.20091	+	5.20091i	0.213038	+	0.213038i
$597$		0			0
$598$	−20.4279	+	2.54650i	−0.835360	+	0.104134i
$599$		−	12.1383i		−	0.495958i	−0.968765	−	0.247979i	$-0.920234\pi$
$599$		−	12.1383i		−	0.495958i	0.968765	−	0.247979i	$-0.0797664\pi$
$600$		0			0
$601$	−37.5161			−1.53031			−0.765156	−	0.643845i	$-0.777338\pi$
$601$	−37.5161			−1.53031			−0.765156	+	0.643845i	$0.777338\pi$
$602$	−7.91560			−0.322616
$603$		0			0
$604$	7.76020	+	7.76020i	0.315758	+	0.315758i
$605$	−5.55527	−	5.55527i	−0.225854	−	0.225854i
$606$		0			0
$607$	−11.2703			−0.457448			−0.228724	−	0.973491i	$-0.573455\pi$
$607$	−11.2703			−0.457448			−0.228724	+	0.973491i	$0.573455\pi$
$608$	6.73064			0.272963
$609$		0			0
$610$			52.0484i			2.10738i
$611$	0.980320	+	7.86409i	0.0396595	+	0.318147i
$612$		0			0
$613$	−15.9175	−	15.9175i	−0.642902	−	0.642902i	0.308366	−	0.951268i	$-0.400218\pi$
$613$	−15.9175	−	15.9175i	−0.642902	−	0.642902i	−0.951268	+	0.308366i	$0.900218\pi$
$614$		−	19.8445i		−	0.800858i
$615$		0			0
$616$	2.53473	+	2.53473i	0.102127	+	0.102127i
$617$	24.0948	−	24.0948i	0.970021	−	0.970021i	−0.0295423	−	0.999564i	$-0.509405\pi$
$617$	24.0948	−	24.0948i	0.970021	−	0.970021i	0.999564	+	0.0295423i	$0.00940499\pi$
$618$		0			0
$619$	−31.9301	+	31.9301i	−1.28338	+	1.28338i	−0.344648	+	0.938732i	$0.612002\pi$
$619$	−31.9301	+	31.9301i	−1.28338	+	1.28338i	−0.938732	+	0.344648i	$0.887998\pi$
$620$			0.501831i			0.0201540i
$621$		0			0
$622$	−15.8729	+	15.8729i	−0.636445	+	0.636445i
$623$	−12.2987			−0.492738
$624$		0			0
$625$	−79.8520			−3.19408
$626$	−16.5171	+	16.5171i	−0.660156	+	0.660156i
$627$		0			0
$628$		−	11.7086i		−	0.467223i
$629$	13.3423	−	13.3423i	0.531991	−	0.531991i
$630$		0			0
$631$	32.0392	−	32.0392i	1.27546	−	1.27546i	0.332277	−	0.943182i	$-0.392183\pi$
$631$	32.0392	−	32.0392i	1.27546	−	1.27546i	0.943182	−	0.332277i	$-0.107817\pi$
$632$	−6.42210	−	6.42210i	−0.255457	−	0.255457i
$633$		0			0
$634$		−	2.63814i		−	0.104774i
$635$	59.4141	+	59.4141i	2.35778	+	2.35778i
$636$		0			0
$637$	2.21455	−	2.84530i	0.0877438	−	0.112735i
$638$		−	13.4826i		−	0.533780i
$639$		0			0
$640$	4.24741			0.167894
$641$	−45.8530			−1.81108			−0.905542	−	0.424256i	$-0.860536\pi$
$641$	−45.8530			−1.81108			−0.905542	+	0.424256i	$0.860536\pi$
$642$		0			0
$643$	−23.0966	−	23.0966i	−0.910841	−	0.910841i	0.0854970	−	0.996338i	$-0.472752\pi$
$643$	−23.0966	−	23.0966i	−0.910841	−	0.910841i	−0.996338	+	0.0854970i	$0.972752\pi$
$644$	−4.03725	−	4.03725i	−0.159090	−	0.159090i
$645$		0			0
$646$	48.2193			1.89716
$647$	−22.8866			−0.899765			−0.449882	−	0.893088i	$-0.648534\pi$
$647$	−22.8866			−0.899765			−0.449882	+	0.893088i	$0.648534\pi$
$648$		0			0
$649$		−	21.9556i		−	0.861832i
$650$	5.81617	+	46.6571i	0.228129	+	1.83004i
$651$		0			0
$652$	−2.18476	−	2.18476i	−0.0855618	−	0.0855618i
$653$		−	6.65079i		−	0.260266i	−0.991497	−	0.130133i	$-0.958460\pi$
$653$		−	6.65079i		−	0.260266i	0.991497	−	0.130133i	$-0.0415404\pi$
$654$		0			0
$655$	−38.8799	−	38.8799i	−1.51916	−	1.51916i
$656$	−3.47354	+	3.47354i	−0.135619	+	0.135619i
$657$		0			0
$658$	−1.55421	+	1.55421i	−0.0605895	+	0.0605895i
$659$		−	26.9997i		−	1.05176i	−0.850559	−	0.525880i	$-0.823736\pi$
$659$		−	26.9997i		−	1.05176i	0.850559	−	0.525880i	$-0.176264\pi$
$660$		0			0
$661$	23.4722	−	23.4722i	0.912962	−	0.912962i	−0.0835420	−	0.996504i	$-0.526623\pi$
$661$	23.4722	−	23.4722i	0.912962	−	0.912962i	0.996504	+	0.0835420i	$0.0266233\pi$
$662$	22.2009			0.862862
$663$		0			0
$664$	−11.6501			−0.452112
$665$	−20.2146	+	20.2146i	−0.783889	+	0.783889i
$666$		0			0
$667$			21.4747i			0.831505i
$668$	2.11815	−	2.11815i	0.0819538	−	0.0819538i
$669$		0			0
$670$	2.41535	−	2.41535i	0.0933132	−	0.0933132i
$671$	−31.0609	−	31.0609i	−1.19909	−	1.19909i
$672$		0			0
$673$			23.1428i			0.892090i	0.895011	+	0.446045i	$0.147168\pi$
$673$			23.1428i			0.892090i	−0.895011	+	0.446045i	$0.852832\pi$
$674$	−3.58880	−	3.58880i	−0.138236	−	0.138236i
$675$		0			0
$676$	12.6022	−	3.19151i	0.484698	−	0.122750i
$677$		−	30.5270i		−	1.17325i	−0.809859	−	0.586625i	$-0.800456\pi$
$677$		−	30.5270i		−	1.17325i	0.809859	−	0.586625i	$-0.199544\pi$
$678$		0			0
$679$	−13.3296			−0.511541
$680$	30.4291			1.16690
$681$		0			0
$682$	−0.299477	−	0.299477i	−0.0114676	−	0.0114676i
$683$	−28.1274	−	28.1274i	−1.07627	−	1.07627i	−0.996841	−	0.0794250i	$-0.974692\pi$
$683$	−28.1274	−	28.1274i	−1.07627	−	1.07627i	−0.0794250	−	0.996841i	$-0.525308\pi$
$684$		0			0
$685$	−38.1852			−1.45898
$686$	1.00000			0.0381802
$687$		0			0
$688$		−	7.91560i		−	0.301779i
$689$	−11.5621	+	14.8552i	−0.440480	+	0.565938i
$690$		0			0
$691$	−33.4816	−	33.4816i	−1.27370	−	1.27370i	−0.944132	−	0.329569i	$-0.893097\pi$
$691$	−33.4816	−	33.4816i	−1.27370	−	1.27370i	−0.329569	−	0.944132i	$-0.606903\pi$
$692$			5.06663i			0.192604i
$693$		0			0
$694$	−10.0933	−	10.0933i	−0.383138	−	0.383138i
$695$	4.35651	−	4.35651i	0.165252	−	0.165252i
$696$		0			0
$697$	−24.8850	+	24.8850i	−0.942586	+	0.942586i
$698$		−	8.73805i		−	0.330740i
$699$		0			0
$700$	−9.22102	+	9.22102i	−0.348522	+	0.348522i
$701$	10.1952			0.385067			0.192534	−	0.981290i	$-0.438330\pi$
$701$	10.1952			0.385067			0.192534	+	0.981290i	$0.438330\pi$
$702$		0			0
$703$	−17.7270			−0.668588
$704$	−2.53473	+	2.53473i	−0.0955311	+	0.0955311i
$705$		0			0
$706$		−	25.8979i		−	0.974679i
$707$	−6.32221	+	6.32221i	−0.237771	+	0.237771i
$708$		0			0
$709$	2.94732	−	2.94732i	0.110689	−	0.110689i	−0.649593	−	0.760282i	$-0.725061\pi$
$709$	2.94732	−	2.94732i	0.110689	−	0.110689i	0.760282	+	0.649593i	$0.225061\pi$
$710$	−36.5828	−	36.5828i	−1.37293	−	1.37293i
$711$		0			0
$712$		−	12.2987i		−	0.460914i
$713$	0.477001	+	0.477001i	0.0178638	+	0.0178638i
$714$		0			0
$715$	−43.3211	−	33.7176i	−1.62012	−	1.26097i
$716$			0.961445i			0.0359309i
$717$		0			0
$718$	1.17634			0.0439008
$719$	−9.34789			−0.348618			−0.174309	−	0.984691i	$-0.555769\pi$
$719$	−9.34789			−0.348618			−0.174309	+	0.984691i	$0.555769\pi$
$720$		0			0
$721$	12.3202	+	12.3202i	0.458830	+	0.458830i
$722$	−18.5980	−	18.5980i	−0.692144	−	0.692144i
$723$		0			0
$724$	14.9714			0.556408
$725$	49.0479			1.82159
$726$		0			0
$727$			8.92965i			0.331182i	0.986194	+	0.165591i	$0.0529532\pi$
$727$			8.92965i			0.331182i	−0.986194	+	0.165591i	$0.947047\pi$
$728$	2.84530	+	2.21455i	0.105454	+	0.0820768i
$729$		0			0
$730$	16.7638	+	16.7638i	0.620457	+	0.620457i
$731$		−	56.7086i		−	2.09744i
$732$		0			0
$733$	20.3409	+	20.3409i	0.751309	+	0.751309i	0.974724	−	0.223414i	$-0.0717202\pi$
$733$	20.3409	+	20.3409i	0.751309	+	0.751309i	−0.223414	+	0.974724i	$0.571720\pi$
$734$	−23.4614	+	23.4614i	−0.865975	+	0.865975i
$735$		0			0
$736$	4.03725	−	4.03725i	0.148815	−	0.148815i
$737$			2.88282i			0.106190i
$738$		0			0
$739$	12.0731	−	12.0731i	0.444118	−	0.444118i	−0.449276	−	0.893393i	$-0.648318\pi$
$739$	12.0731	−	12.0731i	0.444118	−	0.444118i	0.893393	+	0.449276i	$0.148318\pi$
$740$	−11.1868			−0.411233
$741$		0			0
$742$	−5.22095			−0.191667
$743$	−8.18690	+	8.18690i	−0.300348	+	0.300348i	−0.841150	−	0.540802i	$-0.818121\pi$
$743$	−8.18690	+	8.18690i	−0.300348	+	0.300348i	0.540802	+	0.841150i	$0.318121\pi$
$744$		0			0
$745$		−	31.2406i		−	1.14457i
$746$	21.5144	−	21.5144i	0.787697	−	0.787697i
$747$		0			0
$748$	−18.1592	+	18.1592i	−0.663965	+	0.663965i
$749$	−4.50079	−	4.50079i	−0.164455	−	0.164455i
$750$		0			0
$751$		−	10.8054i		−	0.394294i	−0.980374	−	0.197147i	$-0.936832\pi$
$751$		−	10.8054i		−	0.394294i	0.980374	−	0.197147i	$-0.0631676\pi$
$752$	−1.55421	−	1.55421i	−0.0566763	−	0.0566763i
$753$		0			0
$754$	−1.67753	−	13.4571i	−0.0610919	−	0.490077i
$755$		−	46.6135i		−	1.69644i
$756$		0			0
$757$	−11.4815			−0.417304			−0.208652	−	0.977990i	$-0.566908\pi$
$757$	−11.4815			−0.417304			−0.208652	+	0.977990i	$0.566908\pi$
$758$	−17.2049			−0.624911
$759$		0			0
$760$	−20.2146	−	20.2146i	−0.733261	−	0.733261i
$761$	10.1139	+	10.1139i	0.366629	+	0.366629i	0.866246	−	0.499617i	$-0.166526\pi$
$761$	10.1139	+	10.1139i	0.366629	+	0.366629i	−0.499617	+	0.866246i	$0.666526\pi$
$762$		0			0
$763$	3.49111			0.126387
$764$	−19.5793			−0.708356
$765$		0			0
$766$			37.4942i			1.35472i
$767$	−2.73175	−	21.9140i	−0.0986379	−	0.791269i
$768$		0			0
$769$	20.2565	+	20.2565i	0.730468	+	0.730468i	0.970712	−	0.240245i	$-0.0772276\pi$
$769$	20.2565	+	20.2565i	0.730468	+	0.730468i	−0.240245	+	0.970712i	$0.577228\pi$
$770$		−	15.2255i		−	0.548687i
$771$		0			0
$772$	4.06219	+	4.06219i	0.146201	+	0.146201i
$773$	−24.7776	+	24.7776i	−0.891189	+	0.891189i	−0.994635	−	0.103446i	$-0.967013\pi$
$773$	−24.7776	+	24.7776i	−0.891189	+	0.891189i	0.103446	+	0.994635i	$0.467013\pi$
$774$		0			0
$775$	1.08946	−	1.08946i	0.0391346	−	0.0391346i
$776$		−	13.3296i		−	0.478503i
$777$		0			0
$778$	−2.64286	+	2.64286i	−0.0947512	+	0.0947512i
$779$	33.0631			1.18461
$780$		0			0
$781$	43.6631			1.56239
$782$	28.9235	−	28.9235i	1.03430	−	1.03430i
$783$		0			0
$784$			1.00000i			0.0357143i
$785$	−35.1652	+	35.1652i	−1.25510	+	1.25510i
$786$		0			0
$787$	9.17773	−	9.17773i	0.327151	−	0.327151i	−0.524351	−	0.851502i	$-0.675692\pi$
$787$	9.17773	−	9.17773i	0.327151	−	0.327151i	0.851502	+	0.524351i	$0.175692\pi$
$788$	−4.34520	−	4.34520i	−0.154791	−	0.154791i
$789$		0			0
$790$			38.5759i			1.37247i
$791$	−2.90691	−	2.90691i	−0.103358	−	0.103358i
$792$		0			0
$793$	−34.8668	−	27.1375i	−1.23816	−	0.963680i
$794$		−	32.4700i		−	1.15232i
$795$		0			0
$796$	0.564211			0.0199979
$797$	46.2995			1.64001			0.820006	−	0.572355i	$-0.193970\pi$
$797$	46.2995			1.64001			0.820006	+	0.572355i	$0.193970\pi$
$798$		0			0
$799$	−11.1346	−	11.1346i	−0.393914	−	0.393914i
$800$	−9.22102	−	9.22102i	−0.326012	−	0.326012i
$801$		0			0
$802$	−22.0989			−0.780341
$803$	−20.0083			−0.706077
$804$		0			0
$805$			24.2508i			0.854727i
$806$	−0.336172	−	0.261649i	−0.0118412	−	0.00921619i
$807$		0			0
$808$	−6.32221	−	6.32221i	−0.222415	−	0.222415i
$809$			45.9996i			1.61726i	0.588317	+	0.808630i	$0.299791\pi$
$809$			45.9996i			1.61726i	−0.588317	+	0.808630i	$0.700209\pi$
$810$		0			0
$811$	−2.08788	−	2.08788i	−0.0733153	−	0.0733153i	0.669498	−	0.742814i	$-0.266509\pi$
$811$	−2.08788	−	2.08788i	−0.0733153	−	0.0733153i	−0.742814	+	0.669498i	$0.766509\pi$
$812$	2.65957	−	2.65957i	0.0933327	−	0.0933327i
$813$		0			0
$814$	6.67592	−	6.67592i	0.233991	−	0.233991i
$815$			13.1233i			0.459689i
$816$		0			0
$817$	−37.6725	+	37.6725i	−1.31800	+	1.31800i
$818$	−27.4435			−0.959539
$819$		0			0
$820$	20.8647			0.728627
$821$	−2.68463	+	2.68463i	−0.0936941	+	0.0936941i	−0.752400	−	0.658706i	$-0.771104\pi$
$821$	−2.68463	+	2.68463i	−0.0936941	+	0.0936941i	0.658706	+	0.752400i	$0.271104\pi$
$822$		0			0
$823$			3.92051i			0.136660i	0.997663	+	0.0683301i	$0.0217671\pi$
$823$			3.92051i			0.136660i	−0.997663	+	0.0683301i	$0.978233\pi$
$824$	−12.3202	+	12.3202i	−0.429196	+	0.429196i
$825$		0			0
$826$	4.33095	−	4.33095i	0.150693	−	0.150693i
$827$	14.9222	+	14.9222i	0.518896	+	0.518896i	0.917237	−	0.398341i	$-0.130414\pi$
$827$	14.9222	+	14.9222i	0.518896	+	0.518896i	−0.398341	+	0.917237i	$0.630414\pi$
$828$		0			0
$829$			14.0509i			0.488007i	0.969774	+	0.244003i	$0.0784608\pi$
$829$			14.0509i			0.488007i	−0.969774	+	0.244003i	$0.921539\pi$
$830$	34.9896	+	34.9896i	1.21451	+	1.21451i
$831$		0			0
$832$	−2.21455	+	2.84530i	−0.0767758	+	0.0986432i
$833$			7.16415i			0.248223i
$834$		0			0
$835$	−12.7232			−0.440304
$836$	24.1269			0.834448
$837$		0			0
$838$	6.25136	+	6.25136i	0.215950	+	0.215950i
$839$	23.9852	+	23.9852i	0.828061	+	0.828061i	0.987248	−	0.159187i	$-0.0508873\pi$
$839$	23.9852	+	23.9852i	0.828061	+	0.828061i	−0.159187	+	0.987248i	$0.550887\pi$
$840$		0			0
$841$	14.8534			0.512185
$842$	−10.7407			−0.370147
$843$		0			0
$844$		−	13.3542i		−	0.459671i
$845$	−47.4343	−	28.2637i	−1.63179	−	0.972300i
$846$		0			0
$847$	1.30792	+	1.30792i	0.0449407	+	0.0449407i
$848$		−	5.22095i		−	0.179288i
$849$		0			0
$850$	−66.0608	−	66.0608i	−2.26587	−	2.26587i
$851$	−10.6332	+	10.6332i	−0.364503	+	0.364503i
$852$		0			0
$853$	22.4555	−	22.4555i	0.768860	−	0.768860i	−0.209046	−	0.977906i	$-0.567036\pi$
$853$	22.4555	−	22.4555i	0.768860	−	0.768860i	0.977906	+	0.209046i	$0.0670357\pi$
$854$		−	12.2542i		−	0.419329i
$855$		0			0
$856$	4.50079	−	4.50079i	0.153834	−	0.153834i
$857$	22.3940			0.764966			0.382483	−	0.923963i	$-0.375069\pi$
$857$	22.3940			0.764966			0.382483	+	0.923963i	$0.375069\pi$
$858$		0			0
$859$	−49.2514			−1.68044			−0.840219	−	0.542247i	$-0.817574\pi$
$859$	−49.2514			−1.68044			−0.840219	+	0.542247i	$0.817574\pi$
$860$	−23.7735	+	23.7735i	−0.810669	+	0.810669i
$861$		0			0
$862$		−	4.24061i		−	0.144436i
$863$	−25.4860	+	25.4860i	−0.867552	+	0.867552i	−0.992201	−	0.124649i	$-0.960220\pi$
$863$	−25.4860	+	25.4860i	−0.867552	+	0.867552i	0.124649	+	0.992201i	$0.460220\pi$
$864$		0			0
$865$	15.2170	−	15.2170i	0.517393	−	0.517393i
$866$	−7.37727	−	7.37727i	−0.250690	−	0.250690i
$867$		0			0
$868$		−	0.118150i		−	0.00401026i
$869$	−23.0209	−	23.0209i	−0.780932	−	0.780932i
$870$		0			0
$871$	0.358686	+	2.87736i	0.0121536	+	0.0974956i
$872$			3.49111i			0.118224i
$873$		0			0
$874$	−38.4288			−1.29987
$875$	34.1513			1.15452
$876$		0			0
$877$	5.47083	+	5.47083i	0.184737	+	0.184737i	0.793416	−	0.608679i	$-0.208300\pi$
$877$	5.47083	+	5.47083i	0.184737	+	0.184737i	−0.608679	+	0.793416i	$0.708300\pi$
$878$	−8.18860	−	8.18860i	−0.276352	−	0.276352i
$879$		0			0
$880$	15.2255			0.513250
$881$	−5.90254			−0.198862			−0.0994308	−	0.995044i	$-0.531702\pi$
$881$	−5.90254			−0.198862			−0.0994308	+	0.995044i	$0.531702\pi$
$882$		0			0
$883$			9.29205i			0.312702i	0.987702	+	0.156351i	$0.0499732\pi$
$883$			9.29205i			0.312702i	−0.987702	+	0.156351i	$0.950027\pi$
$884$	−15.8654	+	20.3842i	−0.533611	+	0.685594i
$885$		0			0
$886$	−8.28934	−	8.28934i	−0.278486	−	0.278486i
$887$		−	32.2058i		−	1.08137i	−0.841226	−	0.540683i	$-0.818166\pi$
$887$		−	32.2058i		−	1.08137i	0.841226	−	0.540683i	$-0.181834\pi$
$888$		0			0
$889$	−13.9883	−	13.9883i	−0.469153	−	0.469153i
$890$	−36.9377	+	36.9377i	−1.23815	+	1.23815i
$891$		0			0
$892$	3.80927	−	3.80927i	0.127544	−	0.127544i
$893$			14.7939i			0.495058i
$894$		0			0
$895$	2.88758	−	2.88758i	0.0965211	−	0.0965211i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	−26.9101			−0.898000
$899$	−0.314228	+	0.314228i	−0.0104801	+	0.0104801i
$900$		0			0
$901$		−	37.4037i		−	1.24610i
$902$	−12.4514	+	12.4514i	−0.414587	+	0.414587i
$903$		0			0
$904$	2.90691	−	2.90691i	0.0966824	−	0.0966824i
$905$	−44.9647	−	44.9647i	−1.49468	−	1.49468i
$906$		0			0
$907$		−	9.25441i		−	0.307288i	−0.988126	−	0.153644i	$-0.950899\pi$
$907$		−	9.25441i		−	0.307288i	0.988126	−	0.153644i	$-0.0491009\pi$
$908$	9.92499	+	9.92499i	0.329372	+	0.329372i
$909$		0			0
$910$	−1.89438	−	15.1966i	−0.0627981	−	0.503764i
$911$			43.8510i			1.45285i	0.687246	+	0.726425i	$0.258819\pi$
$911$			43.8510i			1.45285i	−0.687246	+	0.726425i	$0.741181\pi$
$912$		0			0
$913$	−41.7615			−1.38210
$914$	35.5363			1.17544
$915$		0			0
$916$	−6.50911	−	6.50911i	−0.215067	−	0.215067i
$917$	9.15378	+	9.15378i	0.302284	+	0.302284i
$918$		0			0
$919$	20.2837			0.669099			0.334549	−	0.942378i	$-0.391416\pi$
$919$	20.2837			0.669099			0.334549	+	0.942378i	$0.391416\pi$
$920$	−24.2508			−0.799524
$921$		0			0
$922$			30.4302i			1.00216i
$923$	43.5804	−	5.43264i	1.43447	−	0.178817i
$924$		0			0
$925$	24.2862	+	24.2862i	0.798524	+	0.798524i
$926$			17.9314i			0.589263i
$927$		0			0
$928$	2.65957	+	2.65957i	0.0873047	+	0.0873047i
$929$	−25.3806	+	25.3806i	−0.832712	+	0.832712i	−0.987887	−	0.155175i	$-0.950406\pi$
$929$	−25.3806	+	25.3806i	−0.832712	+	0.832712i	0.155175	+	0.987887i	$0.450406\pi$
$930$		0			0
$931$	4.75928	−	4.75928i	0.155979	−	0.155979i
$932$			17.0947i			0.559955i
$933$		0			0
$934$	−8.95001	+	8.95001i	−0.292853	+	0.292853i
$935$	109.077			3.56722
$936$		0			0
$937$	−23.5723			−0.770072			−0.385036	−	0.922901i	$-0.625811\pi$
$937$	−23.5723			−0.770072			−0.385036	+	0.922901i	$0.625811\pi$
$938$	−0.568664	+	0.568664i	−0.0185676	+	0.0185676i
$939$		0			0
$940$			9.33576i			0.304499i
$941$	10.3997	−	10.3997i	0.339020	−	0.339020i	−0.516979	−	0.855998i	$-0.672943\pi$
$941$	10.3997	−	10.3997i	0.339020	−	0.339020i	0.855998	+	0.516979i	$0.172943\pi$
$942$		0			0
$943$	19.8323	−	19.8323i	0.645830	−	0.645830i
$944$	4.33095	+	4.33095i	0.140961	+	0.140961i
$945$		0			0
$946$		−	28.3746i		−	0.922538i
$947$	8.62956	+	8.62956i	0.280423	+	0.280423i	0.833278	−	0.552855i	$-0.186462\pi$
$947$	8.62956	+	8.62956i	0.280423	+	0.280423i	−0.552855	+	0.833278i	$0.686462\pi$
$948$		0			0
$949$	−19.9704	+	2.48947i	−0.648267	+	0.0808115i
$950$			87.7708i			2.84766i
$951$		0			0
$952$	−7.16415			−0.232191
$953$	25.6873			0.832094			0.416047	−	0.909343i	$-0.363415\pi$
$953$	25.6873			0.832094			0.416047	+	0.909343i	$0.363415\pi$
$954$		0			0
$955$	58.8041	+	58.8041i	1.90285	+	1.90285i
$956$	−5.78149	−	5.78149i	−0.186987	−	0.186987i
$957$		0			0
$958$	−9.63454			−0.311278
$959$	8.99023			0.290310
$960$		0			0
$961$		−	30.9860i		−	0.999550i
$962$	5.83265	−	7.49391i	0.188052	−	0.241613i
$963$		0			0
$964$	−14.4471	−	14.4471i	−0.465311	−	0.465311i
$965$		−	24.4005i		−	0.785481i
$966$		0			0
$967$	7.91958	+	7.91958i	0.254677	+	0.254677i	0.822885	−	0.568208i	$-0.192363\pi$
$967$	7.91958	+	7.91958i	0.254677	+	0.254677i	−0.568208	+	0.822885i	$0.692363\pi$
$968$	−1.30792	+	1.30792i	−0.0420382	+	0.0420382i
$969$		0			0
$970$	−40.0336	+	40.0336i	−1.28540	+	1.28540i
$971$			15.3284i			0.491912i	0.969281	+	0.245956i	$0.0791019\pi$
$971$			15.3284i			0.491912i	−0.969281	+	0.245956i	$0.920898\pi$
$972$		0			0
$973$	−1.02569	+	1.02569i	−0.0328820	+	0.0328820i
$974$	−28.9315			−0.927026
$975$		0			0
$976$	12.2542			0.392246
$977$	29.0548	−	29.0548i	0.929546	−	0.929546i	−0.0681300	−	0.997676i	$-0.521703\pi$
$977$	29.0548	−	29.0548i	0.929546	−	0.929546i	0.997676	+	0.0681300i	$0.0217033\pi$
$978$		0			0
$979$		−	44.0866i		−	1.40901i
$980$	3.00337	−	3.00337i	0.0959392	−	0.0959392i
$981$		0			0
$982$	28.9913	−	28.9913i	0.925148	−	0.925148i
$983$	36.0946	+	36.0946i	1.15124	+	1.15124i	0.986305	+	0.164934i	$0.0527410\pi$
$983$	36.0946	+	36.0946i	1.15124	+	1.15124i	0.164934	+	0.986305i	$0.447259\pi$
$984$		0			0
$985$			26.1005i			0.831632i
$986$	19.0536	+	19.0536i	0.606790	+	0.606790i
$987$		0			0
$988$	24.0813	−	3.00192i	0.766127	−	0.0955038i
$989$			45.1944i			1.43710i
$990$		0			0
$991$	−25.4375			−0.808047			−0.404024	−	0.914748i	$-0.632389\pi$
$991$	−25.4375			−0.808047			−0.404024	+	0.914748i	$0.632389\pi$
$992$	0.118150			0.00375126
$993$		0			0
$994$	8.61297	+	8.61297i	0.273187	+	0.273187i
$995$	−1.69454	−	1.69454i	−0.0537204	−	0.0537204i
$996$		0			0
$997$	−9.63659			−0.305194			−0.152597	−	0.988289i	$-0.548764\pi$
$997$	−9.63659			−0.305194			−0.152597	+	0.988289i	$0.548764\pi$
$998$	−4.41426			−0.139731
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.y.c.827.1	✓	16
3.2	odd	2		1638.2.y.d.827.8	yes	16
13.5	odd	4		1638.2.y.d.1331.8	yes	16
39.5	even	4	inner	1638.2.y.c.1331.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.y.c.827.1	✓	16	1.1	even	1	trivial
1638.2.y.c.1331.1	yes	16	39.5	even	4	inner
1638.2.y.d.827.8	yes	16	3.2	odd	2
1638.2.y.d.1331.8	yes	16	13.5	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{2} -1.00000i q^{4} +(-3.00337 + 3.00337i) q^{5} +(0.707107 - 0.707107i) q^{7} +(0.707107 + 0.707107i) q^{8} -4.24741i q^{10} +(2.53473 + 2.53473i) q^{11} +(2.84530 + 2.21455i) q^{13} +1.00000i q^{14} -1.00000 q^{16} -7.16415 q^{17} +(4.75928 + 4.75928i) q^{19} +(3.00337 + 3.00337i) q^{20} -3.58464 q^{22} +5.70954 q^{23} -13.0405i q^{25} +(-3.57786 + 0.446008i) q^{26} +(-0.707107 - 0.707107i) q^{28} +3.76120i q^{29} +(0.0835445 + 0.0835445i) q^{31} +(0.707107 - 0.707107i) q^{32} +(5.06582 - 5.06582i) q^{34} +4.24741i q^{35} +(-1.86236 + 1.86236i) q^{37} -6.73064 q^{38} -4.24741 q^{40} +(3.47354 - 3.47354i) q^{41} +7.91560i q^{43} +(2.53473 - 2.53473i) q^{44} +(-4.03725 + 4.03725i) q^{46} +(1.55421 + 1.55421i) q^{47} -1.00000i q^{49} +(9.22102 + 9.22102i) q^{50} +(2.21455 - 2.84530i) q^{52} +5.22095i q^{53} -15.2255 q^{55} +1.00000 q^{56} +(-2.65957 - 2.65957i) q^{58} +(-4.33095 - 4.33095i) q^{59} -12.2542 q^{61} -0.118150 q^{62} +1.00000i q^{64} +(-15.1966 + 1.89438i) q^{65} +(0.568664 + 0.568664i) q^{67} +7.16415i q^{68} +(-3.00337 - 3.00337i) q^{70} +(8.61297 - 8.61297i) q^{71} +(-3.94683 + 3.94683i) q^{73} -2.63378i q^{74} +(4.75928 - 4.75928i) q^{76} +3.58464 q^{77} -9.08222 q^{79} +(3.00337 - 3.00337i) q^{80} +4.91233i q^{82} +(-8.23786 + 8.23786i) q^{83} +(21.5166 - 21.5166i) q^{85} +(-5.59717 - 5.59717i) q^{86} +3.58464i q^{88} +(-8.69652 - 8.69652i) q^{89} +(3.57786 - 0.446008i) q^{91} -5.70954i q^{92} -2.19799 q^{94} -28.5878 q^{95} +(-9.42542 - 9.42542i) q^{97} +(0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{5} + 8 q^{11} + 4 q^{13} - 16 q^{16} + 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} - 24 q^{23} - 8 q^{26} - 8 q^{31} + 4 q^{34} - 4 q^{37} - 16 q^{38} + 16 q^{41} + 8 q^{44} - 4 q^{47} + 16 q^{50}+ \cdots - 28 q^{97}+O(q^{100}) \) 16 * q - 4 * q^5 + 8 * q^11 + 4 * q^13 - 16 * q^16 + 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 - 24 * q^23 - 8 * q^26 - 8 * q^31 + 4 * q^34 - 4 * q^37 - 16 * q^38 + 16 * q^41 + 8 * q^44 - 4 * q^47 + 16 * q^50 - 16 * q^55 + 16 * q^56 + 4 * q^58 + 8 * q^59 - 40 * q^61 - 24 * q^62 - 56 * q^65 - 4 * q^67 - 4 * q^70 + 24 * q^71 - 12 * q^73 + 16 * q^76 + 8 * q^77 + 16 * q^79 + 4 * q^80 + 8 * q^85 - 24 * q^86 - 16 * q^89 + 8 * q^91 - 8 * q^94 + 40 * q^95 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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