LMFDB - Embedded newform 1045.2.b.c.419.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.b (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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + \cdots + 4 $ x^20 + 26x^18 + 281x^16 + 1640x^14 + 5623x^12 + 11551x^10 + 13894x^8 + 9095x^6 + 2753x^4 + 276*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.15
Root			$1.23709i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.c.419.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23709i q^{2} +2.07277i q^{3} +0.469598 q^{4} +(2.13362 + 0.669071i) q^{5} -2.56421 q^{6} +0.314194i q^{7} +3.05513i q^{8} -1.29637 q^{9} +O(q^{10})$	\(q+1.23709i q^{2} +2.07277i q^{3} +0.469598 q^{4} +(2.13362 + 0.669071i) q^{5} -2.56421 q^{6} +0.314194i q^{7} +3.05513i q^{8} -1.29637 q^{9} +(-0.827703 + 2.63949i) q^{10} +1.00000 q^{11} +0.973368i q^{12} +0.897047i q^{13} -0.388688 q^{14} +(-1.38683 + 4.42251i) q^{15} -2.84028 q^{16} -0.623404i q^{17} -1.60374i q^{18} +1.00000 q^{19} +(1.00194 + 0.314194i) q^{20} -0.651252 q^{21} +1.23709i q^{22} -7.17386i q^{23} -6.33257 q^{24} +(4.10469 + 2.85509i) q^{25} -1.10973 q^{26} +3.53123i q^{27} +0.147545i q^{28} -3.13576 q^{29} +(-5.47106 - 1.71564i) q^{30} -1.16919 q^{31} +2.59655i q^{32} +2.07277i q^{33} +0.771209 q^{34} +(-0.210218 + 0.670372i) q^{35} -0.608774 q^{36} -1.42638i q^{37} +1.23709i q^{38} -1.85937 q^{39} +(-2.04409 + 6.51848i) q^{40} -1.02303 q^{41} -0.805660i q^{42} -5.18696i q^{43} +0.469598 q^{44} +(-2.76597 - 0.867365i) q^{45} +8.87474 q^{46} +4.33573i q^{47} -5.88725i q^{48} +6.90128 q^{49} +(-3.53201 + 5.07789i) q^{50} +1.29217 q^{51} +0.421252i q^{52} +3.29122i q^{53} -4.36846 q^{54} +(2.13362 + 0.669071i) q^{55} -0.959903 q^{56} +2.07277i q^{57} -3.87922i q^{58} +1.12009 q^{59} +(-0.651252 + 2.07680i) q^{60} -5.74190 q^{61} -1.44640i q^{62} -0.407313i q^{63} -8.89275 q^{64} +(-0.600188 + 1.91396i) q^{65} -2.56421 q^{66} -14.3904i q^{67} -0.292749i q^{68} +14.8698 q^{69} +(-0.829313 - 0.260060i) q^{70} -5.19451 q^{71} -3.96058i q^{72} -14.1336i q^{73} +1.76457 q^{74} +(-5.91794 + 8.50807i) q^{75} +0.469598 q^{76} +0.314194i q^{77} -2.30022i q^{78} -4.00598 q^{79} +(-6.06009 - 1.90035i) q^{80} -11.2085 q^{81} -1.26559i q^{82} -1.65116i q^{83} -0.305827 q^{84} +(0.417101 - 1.33011i) q^{85} +6.41676 q^{86} -6.49970i q^{87} +3.05513i q^{88} +13.9446 q^{89} +(1.07301 - 3.42177i) q^{90} -0.281847 q^{91} -3.36883i q^{92} -2.42347i q^{93} -5.36371 q^{94} +(2.13362 + 0.669071i) q^{95} -5.38206 q^{96} -0.809173i q^{97} +8.53754i q^{98} -1.29637 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			1.23709i			0.874758i	0.899277	+	0.437379i	$0.144093\pi$
$2$			1.23709i			0.874758i	−0.899277	+	0.437379i	$0.855907\pi$
$3$			2.07277i			1.19671i	0.801230	+	0.598357i	$0.204180\pi$
$3$			2.07277i			1.19671i	−0.801230	+	0.598357i	$0.795820\pi$
$4$	0.469598			0.234799
$5$	2.13362	+	0.669071i	0.954185	+	0.299217i
$5$	2.13362	+	0.669071i	0.954185	+	0.299217i
$6$	−2.56421			−1.04683
$7$			0.314194i			0.118754i	0.998236	+	0.0593771i	$0.0189115\pi$
$7$			0.314194i			0.118754i	−0.998236	+	0.0593771i	$0.981089\pi$
$8$			3.05513i			1.08015i
$9$	−1.29637			−0.432125
$10$	−0.827703	+	2.63949i	−0.261743	+	0.834681i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$			0.973368i			0.280987i
$13$			0.897047i			0.248796i	0.992232	+	0.124398i	$0.0397000\pi$
$13$			0.897047i			0.248796i	−0.992232	+	0.124398i	$0.960300\pi$
$14$	−0.388688			−0.103881
$15$	−1.38683	+	4.42251i	−0.358078	+	1.14189i
$16$	−2.84028			−0.710070
$17$		−	0.623404i		−	0.151198i	−0.997138	−	0.0755988i	$-0.975913\pi$
$17$		−	0.623404i		−	0.151198i	0.997138	−	0.0755988i	$-0.0240868\pi$
$18$		−	1.60374i		−	0.378004i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	1.00194	+	0.314194i	0.224042	+	0.0702560i
$21$	−0.651252			−0.142115
$22$			1.23709i			0.263749i
$23$		−	7.17386i		−	1.49585i	−0.663782	−	0.747926i	$-0.731050\pi$
$23$		−	7.17386i		−	1.49585i	0.663782	−	0.747926i	$-0.268950\pi$
$24$	−6.33257			−1.29263
$25$	4.10469	+	2.85509i	0.820938	+	0.571018i
$26$	−1.10973			−0.217636
$27$			3.53123i			0.679585i
$28$			0.147545i			0.0278834i
$29$	−3.13576			−0.582295			−0.291148	−	0.956678i	$-0.594037\pi$
$29$	−3.13576			−0.582295			−0.291148	+	0.956678i	$0.594037\pi$
$30$	−5.47106	−	1.71564i	−0.998874	−	0.313231i
$31$	−1.16919			−0.209993			−0.104997	−	0.994473i	$-0.533483\pi$
$31$	−1.16919			−0.209993			−0.104997	+	0.994473i	$0.533483\pi$
$32$			2.59655i			0.459010i
$33$			2.07277i			0.360823i
$34$	0.771209			0.132261
$35$	−0.210218	+	0.670372i	−0.0355334	+	0.113314i
$36$	−0.608774			−0.101462
$37$		−	1.42638i		−	0.234496i	−0.993103	−	0.117248i	$-0.962593\pi$
$37$		−	1.42638i		−	0.234496i	0.993103	−	0.117248i	$-0.0374072\pi$
$38$			1.23709i			0.200683i
$39$	−1.85937			−0.297738
$40$	−2.04409	+	6.51848i	−0.323200	+	1.03066i
$41$	−1.02303			−0.159771			−0.0798855	−	0.996804i	$-0.525455\pi$
$41$	−1.02303			−0.159771			−0.0798855	+	0.996804i	$0.525455\pi$
$42$		−	0.805660i		−	0.124316i
$43$		−	5.18696i		−	0.791004i	−0.918465	−	0.395502i	$-0.870571\pi$
$43$		−	5.18696i		−	0.791004i	0.918465	−	0.395502i	$-0.129429\pi$
$44$	0.469598			0.0707946
$45$	−2.76597	−	0.867365i	−0.412327	−	0.129299i
$46$	8.87474			1.30851
$47$			4.33573i			0.632431i	0.948687	+	0.316216i	$0.102412\pi$
$47$			4.33573i			0.632431i	−0.948687	+	0.316216i	$0.897588\pi$
$48$		−	5.88725i		−	0.849751i
$49$	6.90128			0.985897
$50$	−3.53201	+	5.07789i	−0.499502	+	0.718122i
$51$	1.29217			0.180940
$52$			0.421252i			0.0584171i
$53$			3.29122i			0.452084i	0.974118	+	0.226042i	$0.0725786\pi$
$53$			3.29122i			0.452084i	−0.974118	+	0.226042i	$0.927421\pi$
$54$	−4.36846			−0.594472
$55$	2.13362	+	0.669071i	0.287698	+	0.0902175i
$56$	−0.959903			−0.128272
$57$			2.07277i			0.274545i
$58$		−	3.87922i		−	0.509367i
$59$	1.12009			0.145823			0.0729116	−	0.997338i	$-0.476771\pi$
$59$	1.12009			0.145823			0.0729116	+	0.997338i	$0.476771\pi$
$60$	−0.651252	+	2.07680i	−0.0840763	+	0.268114i
$61$	−5.74190			−0.735175			−0.367587	−	0.929989i	$-0.619816\pi$
$61$	−5.74190			−0.735175			−0.367587	+	0.929989i	$0.619816\pi$
$62$		−	1.44640i		−	0.183693i
$63$		−	0.407313i		−	0.0513166i
$64$	−8.89275			−1.11159
$65$	−0.600188	+	1.91396i	−0.0744442	+	0.237398i
$66$	−2.56421			−0.315633
$67$		−	14.3904i		−	1.75807i	−0.476759	−	0.879034i	$-0.658188\pi$
$67$		−	14.3904i		−	1.75807i	0.476759	−	0.879034i	$-0.341812\pi$
$68$		−	0.292749i		−	0.0355010i
$69$	14.8698			1.79011
$70$	−0.829313	−	0.260060i	−0.0991219	−	0.0310831i
$71$	−5.19451			−0.616475			−0.308237	−	0.951310i	$-0.599739\pi$
$71$	−5.19451			−0.616475			−0.308237	+	0.951310i	$0.599739\pi$
$72$		−	3.96058i		−	0.466759i
$73$		−	14.1336i		−	1.65422i	−0.562044	−	0.827108i	$-0.689985\pi$
$73$		−	14.1336i		−	1.65422i	0.562044	−	0.827108i	$-0.310015\pi$
$74$	1.76457			0.205127
$75$	−5.91794	+	8.50807i	−0.683345	+	0.982428i
$76$	0.469598			0.0538666
$77$			0.314194i			0.0358058i
$78$		−	2.30022i		−	0.260449i
$79$	−4.00598			−0.450708			−0.225354	−	0.974277i	$-0.572354\pi$
$79$	−4.00598			−0.450708			−0.225354	+	0.974277i	$0.572354\pi$
$80$	−6.06009	−	1.90035i	−0.677538	−	0.212465i
$81$	−11.2085			−1.24539
$82$		−	1.26559i		−	0.139761i
$83$		−	1.65116i		−	0.181239i	−0.995886	−	0.0906194i	$-0.971115\pi$
$83$		−	1.65116i		−	0.181239i	0.995886	−	0.0906194i	$-0.0288847\pi$
$84$	−0.305827			−0.0333684
$85$	0.417101	−	1.33011i	0.0452410	−	0.144270i
$86$	6.41676			0.691937
$87$		−	6.49970i		−	0.696841i
$88$			3.05513i			0.325677i
$89$	13.9446			1.47812			0.739062	−	0.673638i	$-0.235269\pi$
$89$	13.9446			1.47812			0.739062	+	0.673638i	$0.235269\pi$
$90$	1.07301	−	3.42177i	0.113105	−	0.360686i
$91$	−0.281847			−0.0295456
$92$		−	3.36883i		−	0.351225i
$93$		−	2.42347i		−	0.251302i
$94$	−5.36371			−0.553224
$95$	2.13362	+	0.669071i	0.218905	+	0.0686452i
$96$	−5.38206			−0.549304
$97$		−	0.809173i		−	0.0821591i	−0.999156	−	0.0410795i	$-0.986920\pi$
$97$		−	0.809173i		−	0.0821591i	0.999156	−	0.0410795i	$-0.0130797\pi$
$98$			8.53754i			0.862421i
$99$	−1.29637			−0.130290
$100$	1.92755	+	1.34074i	0.192755	+	0.134074i
$101$	7.88444			0.784531			0.392266	−	0.919852i	$-0.371691\pi$
$101$	7.88444			0.784531			0.392266	+	0.919852i	$0.371691\pi$
$102$			1.59854i			0.158279i
$103$		−	9.45660i		−	0.931787i	−0.884841	−	0.465893i	$-0.845733\pi$
$103$		−	9.45660i		−	0.931787i	0.884841	−	0.465893i	$-0.154267\pi$
$104$	−2.74059			−0.268737
$105$	−1.38953	−	0.435734i	−0.135604	−	0.0425233i
$106$	−4.07155			−0.395464
$107$		−	4.43836i		−	0.429072i	−0.976716	−	0.214536i	$-0.931176\pi$
$107$		−	4.43836i		−	0.429072i	0.976716	−	0.214536i	$-0.0688240\pi$
$108$			1.65826i			0.159566i
$109$	−11.4134			−1.09321			−0.546605	−	0.837391i	$-0.684080\pi$
$109$	−11.4134			−1.09321			−0.546605	+	0.837391i	$0.684080\pi$
$110$	−0.827703	+	2.63949i	−0.0789184	+	0.251666i
$111$	2.95656			0.280624
$112$		−	0.892400i		−	0.0843239i
$113$			9.60198i			0.903278i	0.892201	+	0.451639i	$0.149160\pi$
$113$			9.60198i			0.903278i	−0.892201	+	0.451639i	$0.850840\pi$
$114$	−2.56421			−0.240160
$115$	4.79982	−	15.3063i	0.447585	−	1.42732i
$116$	−1.47254			−0.136722
$117$		−	1.16291i		−	0.107511i
$118$			1.38566i			0.127560i
$119$	0.195870			0.0179554
$120$	−13.5113	−	4.23694i	−1.23341	−	0.386778i
$121$	1.00000			0.0909091
$122$		−	7.10327i		−	0.643100i
$123$		−	2.12051i		−	0.191200i
$124$	−0.549051			−0.0493062
$125$	6.84760	+	8.83801i	0.612468	+	0.790495i
$126$	0.503885			0.0448896
$127$			1.63584i			0.145158i	0.997363	+	0.0725788i	$0.0231229\pi$
$127$			1.63584i			0.145158i	−0.997363	+	0.0725788i	$0.976877\pi$
$128$		−	5.80805i		−	0.513364i
$129$	10.7514			0.946606
$130$	−2.36775	−	0.742489i	−0.207665	−	0.0651206i
$131$	−19.2059			−1.67803			−0.839014	−	0.544110i	$-0.816867\pi$
$131$	−19.2059			−1.67803			−0.839014	+	0.544110i	$0.816867\pi$
$132$			0.973368i			0.0847209i
$133$			0.314194i			0.0272441i
$134$	17.8023			1.53788
$135$	−2.36264	+	7.53430i	−0.203344	+	0.648449i
$136$	1.90458			0.163316
$137$		−	16.2421i		−	1.38766i	−0.720140	−	0.693829i	$-0.755923\pi$
$137$		−	16.2421i		−	1.38766i	0.720140	−	0.693829i	$-0.244077\pi$
$138$			18.3953i			1.56591i
$139$	0.527173			0.0447142			0.0223571	−	0.999750i	$-0.492883\pi$
$139$	0.527173			0.0447142			0.0223571	+	0.999750i	$0.492883\pi$
$140$	−0.0987180	+	0.314805i	−0.00834320	+	0.0266059i
$141$	−8.98697			−0.756839
$142$		−	6.42609i		−	0.539266i
$143$			0.897047i			0.0750149i
$144$	3.68207			0.306839
$145$	−6.69052	−	2.09804i	−0.555617	−	0.174233i
$146$	17.4846			1.44704
$147$			14.3048i			1.17984i
$148$		−	0.669826i		−	0.0550594i
$149$	6.51164			0.533454			0.266727	−	0.963772i	$-0.414058\pi$
$149$	6.51164			0.533454			0.266727	+	0.963772i	$0.414058\pi$
$150$	−10.5253	−	7.32105i	−0.859386	−	0.597761i
$151$	−15.0375			−1.22373			−0.611867	−	0.790960i	$-0.709581\pi$
$151$	−15.0375			−1.22373			−0.611867	+	0.790960i	$0.709581\pi$
$152$			3.05513i			0.247803i
$153$			0.808164i			0.0653362i
$154$	−0.388688			−0.0313214
$155$	−2.49462	−	0.782273i	−0.200372	−	0.0628337i
$156$	−0.873158			−0.0699086
$157$			5.33516i			0.425792i	0.977075	+	0.212896i	$0.0682895\pi$
$157$			5.33516i			0.425792i	−0.977075	+	0.212896i	$0.931710\pi$
$158$		−	4.95578i		−	0.394260i
$159$	−6.82194			−0.541015
$160$	−1.73728	+	5.54007i	−0.137344	+	0.437981i
$161$	2.25399			0.177639
$162$		−	13.8660i		−	1.08942i
$163$			0.764022i			0.0598428i	0.999552	+	0.0299214i	$0.00952570\pi$
$163$			0.764022i			0.0598428i	−0.999552	+	0.0299214i	$0.990474\pi$
$164$	−0.480414			−0.0375141
$165$	−1.38683	+	4.42251i	−0.107965	+	0.344292i
$166$	2.04265			0.158540
$167$			1.61649i			0.125087i	0.998042	+	0.0625437i	$0.0199213\pi$
$167$			1.61649i			0.125087i	−0.998042	+	0.0625437i	$0.980079\pi$
$168$		−	1.98966i		−	0.153505i
$169$	12.1953			0.938100
$170$	1.64547	+	0.515993i	0.126202	+	0.0395749i
$171$	−1.29637			−0.0991362
$172$		−	2.43579i		−	0.185727i
$173$			11.0950i			0.843537i	0.906704	+	0.421769i	$0.138591\pi$
$173$			11.0950i			0.843537i	−0.906704	+	0.421769i	$0.861409\pi$
$174$	8.04074			0.609567
$175$	−0.897052	+	1.28967i	−0.0678108	+	0.0974899i
$176$	−2.84028			−0.214094
$177$			2.32169i			0.174509i
$178$			17.2508i			1.29300i
$179$	24.8558			1.85781			0.928905	−	0.370317i	$-0.120751\pi$
$179$	24.8558			1.85781			0.928905	+	0.370317i	$0.120751\pi$
$180$	−1.29889	−	0.407313i	−0.0968139	−	0.0303593i
$181$	20.1163			1.49524			0.747618	−	0.664129i	$-0.231197\pi$
$181$	20.1163			1.49524			0.747618	+	0.664129i	$0.231197\pi$
$182$		−	0.348671i		−	0.0258452i
$183$		−	11.9016i		−	0.879794i
$184$	21.9170			1.61575
$185$	0.954350	−	3.04336i	0.0701652	−	0.223752i
$186$	2.99806			0.219828
$187$		−	0.623404i		−	0.0455878i
$188$			2.03605i			0.148494i
$189$	−1.10949			−0.0807036
$190$	−0.827703	+	2.63949i	−0.0600479	+	0.191489i
$191$	10.2621			0.742542			0.371271	−	0.928524i	$-0.378922\pi$
$191$	10.2621			0.742542			0.371271	+	0.928524i	$0.378922\pi$
$192$		−	18.4326i		−	1.33026i
$193$			15.1459i			1.09023i	0.838363	+	0.545113i	$0.183513\pi$
$193$			15.1459i			1.09023i	−0.838363	+	0.545113i	$0.816487\pi$
$194$	1.00102			0.0718693
$195$	−3.96720	−	1.24405i	−0.284097	−	0.0890884i
$196$	3.24083			0.231488
$197$			23.2367i			1.65554i	0.561065	+	0.827772i	$0.310392\pi$
$197$			23.2367i			1.65554i	−0.561065	+	0.827772i	$0.689608\pi$
$198$		−	1.60374i		−	0.113973i
$199$	1.07244			0.0760232			0.0380116	−	0.999277i	$-0.487898\pi$
$199$	1.07244			0.0760232			0.0380116	+	0.999277i	$0.487898\pi$
$200$	−8.72265	+	12.5403i	−0.616785	+	0.886736i
$201$	29.8280			2.10391
$202$			9.75380i			0.686275i
$203$		−	0.985236i		−	0.0691500i
$204$	0.606801			0.0424846
$205$	−2.18277	−	0.684481i	−0.152451	−	0.0478063i
$206$	11.6987			0.815087
$207$			9.30000i			0.646395i
$208$		−	2.54787i		−	0.176663i
$209$	1.00000			0.0691714
$210$	0.539044	−	1.71898i	0.0371975	−	0.118621i
$211$	−11.4154			−0.785868			−0.392934	−	0.919567i	$-0.628540\pi$
$211$	−11.4154			−0.785868			−0.392934	+	0.919567i	$0.628540\pi$
$212$			1.54555i			0.106149i
$213$		−	10.7670i		−	0.737744i
$214$	5.49067			0.375334
$215$	3.47044	−	11.0670i	0.236682	−	0.754764i
$216$	−10.7883			−0.734053
$217$		−	0.367354i		−	0.0249376i
$218$		−	14.1195i		−	0.956294i
$219$	29.2957			1.97962
$220$	1.00194	+	0.314194i	0.0675511	+	0.0211830i
$221$	0.559223			0.0376174
$222$			3.65755i			0.245478i
$223$			14.5284i			0.972893i	0.873710	+	0.486447i	$0.161707\pi$
$223$			14.5284i			0.972893i	−0.873710	+	0.486447i	$0.838293\pi$
$224$	−0.815823			−0.0545094
$225$	−5.32121	−	3.70126i	−0.354747	−	0.246751i
$226$	−11.8785			−0.790149
$227$			12.3529i			0.819890i	0.912110	+	0.409945i	$0.134452\pi$
$227$			12.3529i			0.819890i	−0.912110	+	0.409945i	$0.865548\pi$
$228$			0.973368i			0.0644629i
$229$	−1.41187			−0.0932990			−0.0466495	−	0.998911i	$-0.514854\pi$
$229$	−1.41187			−0.0932990			−0.0466495	+	0.998911i	$0.514854\pi$
$230$	18.9353	+	5.93783i	1.24856	+	0.391529i
$231$	−0.651252			−0.0428493
$232$		−	9.58013i		−	0.628966i
$233$			2.09358i			0.137155i	0.997646	+	0.0685774i	$0.0218460\pi$
$233$			2.09358i			0.137155i	−0.997646	+	0.0685774i	$0.978154\pi$
$234$	1.43863			0.0940460
$235$	−2.90091	+	9.25081i	−0.189234	+	0.603456i
$236$	0.525992			0.0342392
$237$		−	8.30348i		−	0.539369i
$238$			0.242309i			0.0157066i
$239$	24.3275			1.57361			0.786807	−	0.617199i	$-0.211733\pi$
$239$	24.3275			1.57361			0.786807	+	0.617199i	$0.211733\pi$
$240$	3.93899	−	12.5612i	0.254260	−	0.810820i
$241$	8.11039			0.522436			0.261218	−	0.965280i	$-0.415876\pi$
$241$	8.11039			0.522436			0.261218	+	0.965280i	$0.415876\pi$
$242$			1.23709i			0.0795234i
$243$		−	12.6390i		−	0.810795i
$244$	−2.69638			−0.172618
$245$	14.7247	+	4.61745i	0.940728	+	0.294998i
$246$	2.62327			0.167254
$247$			0.897047i			0.0570778i
$248$		−	3.57203i		−	0.226824i
$249$	3.42248			0.216891
$250$	−10.9334	+	8.47113i	−0.691492	+	0.535761i
$251$	−15.8691			−1.00165			−0.500824	−	0.865549i	$-0.666970\pi$
$251$	−15.8691			−1.00165			−0.500824	+	0.865549i	$0.666970\pi$
$252$		−	0.191273i		−	0.0120491i
$253$		−	7.17386i		−	0.451017i
$254$	−2.02369			−0.126978
$255$	2.75701	+	0.864554i	0.172650	+	0.0541405i
$256$	−10.6004			−0.662524
$257$		−	24.7918i		−	1.54647i	−0.634120	−	0.773234i	$-0.718638\pi$
$257$		−	24.7918i		−	1.54647i	0.634120	−	0.773234i	$-0.281362\pi$
$258$			13.3005i			0.828051i
$259$	0.448161			0.0278474
$260$	−0.281847	+	0.898792i	−0.0174794	+	0.0557407i
$261$	4.06511			0.251624
$262$		−	23.7595i		−	1.46787i
$263$		−	0.927458i		−	0.0571895i	−0.999591	−	0.0285948i	$-0.990897\pi$
$263$		−	0.927458i		−	0.0571895i	0.999591	−	0.0285948i	$-0.00910323\pi$
$264$	−6.33257			−0.389743
$265$	−2.20206	+	7.02222i	−0.135271	+	0.431371i
$266$	−0.388688			−0.0238320
$267$			28.9039i			1.76889i
$268$		−	6.75771i		−	0.412793i
$269$	19.3925			1.18238			0.591190	−	0.806533i	$-0.298658\pi$
$269$	19.3925			1.18238			0.591190	+	0.806533i	$0.298658\pi$
$270$	−9.32064	−	2.92281i	−0.567236	−	0.177876i
$271$	−22.8304			−1.38685			−0.693424	−	0.720529i	$-0.743899\pi$
$271$	−22.8304			−1.38685			−0.693424	+	0.720529i	$0.743899\pi$
$272$			1.77064i			0.107361i
$273$		−	0.584204i		−	0.0353576i
$274$	20.0930			1.21386
$275$	4.10469	+	2.85509i	0.247522	+	0.172168i
$276$	6.98281			0.420316
$277$			10.5885i			0.636203i	0.948057	+	0.318102i	$0.103045\pi$
$277$			10.5885i			0.636203i	−0.948057	+	0.318102i	$0.896955\pi$
$278$			0.652162i			0.0391141i
$279$	1.51571			0.0907432
$280$	−2.04807	−	0.642243i	−0.122396	−	0.0383813i
$281$	−6.43575			−0.383925			−0.191962	−	0.981402i	$-0.561485\pi$
$281$	−6.43575			−0.383925			−0.191962	+	0.981402i	$0.561485\pi$
$282$		−	11.1177i		−	0.662051i
$283$		−	10.8999i		−	0.647930i	−0.946069	−	0.323965i	$-0.894984\pi$
$283$		−	10.8999i		−	0.647930i	0.946069	−	0.323965i	$-0.105016\pi$
$284$	−2.43933			−0.144748
$285$	−1.38683	+	4.42251i	−0.0821487	+	0.261967i
$286$	−1.10973			−0.0656198
$287$		−	0.321431i		−	0.0189735i
$288$		−	3.36610i		−	0.198350i
$289$	16.6114			0.977139
$290$	2.59548	−	8.27680i	0.152412	−	0.486030i
$291$	1.67723			0.0983209
$292$		−	6.63712i		−	0.388408i
$293$			6.58240i			0.384548i	0.981341	+	0.192274i	$0.0615862\pi$
$293$			6.58240i			0.384548i	−0.981341	+	0.192274i	$0.938414\pi$
$294$	−17.6963			−1.03207
$295$	2.38985	+	0.749419i	0.139142	+	0.0436329i
$296$	4.35778			0.253291
$297$			3.53123i			0.204902i
$298$			8.05551i			0.466643i
$299$	6.43529			0.372163
$300$	−2.77905	+	3.99537i	−0.160449	+	0.230673i
$301$	1.62971			0.0939351
$302$		−	18.6028i		−	1.07047i
$303$			16.3426i			0.938860i
$304$	−2.84028			−0.162901
$305$	−12.2510	−	3.84173i	−0.701493	−	0.219977i
$306$	−0.999775			−0.0571533
$307$			7.65811i			0.437071i	0.975829	+	0.218536i	$0.0701280\pi$
$307$			7.65811i			0.437071i	−0.975829	+	0.218536i	$0.929872\pi$
$308$			0.147545i			0.00840716i
$309$	19.6014			1.11508
$310$	0.967745	−	3.08608i	0.0549642	−	0.175277i
$311$	−2.44749			−0.138785			−0.0693923	−	0.997589i	$-0.522106\pi$
$311$	−2.44749			−0.138785			−0.0693923	+	0.997589i	$0.522106\pi$
$312$		−	5.68062i		−	0.321602i
$313$			9.40612i			0.531665i	0.964019	+	0.265833i	$0.0856469\pi$
$313$			9.40612i			0.531665i	−0.964019	+	0.265833i	$0.914353\pi$
$314$	−6.60009			−0.372465
$315$	0.272521	−	0.869052i	0.0153548	−	0.0489656i
$316$	−1.88120			−0.105826
$317$			15.3518i			0.862243i	0.902294	+	0.431121i	$0.141882\pi$
$317$			15.3518i			0.862243i	−0.902294	+	0.431121i	$0.858118\pi$
$318$		−	8.43938i		−	0.473257i
$319$	−3.13576			−0.175569
$320$	−18.9738	−	5.94988i	−1.06067	−	0.332608i
$321$	9.19969			0.513477
$322$			2.78839i			0.155391i
$323$		−	0.623404i		−	0.0346871i
$324$	−5.26351			−0.292417
$325$	−2.56115	+	3.68210i	−0.142067	+	0.204246i
$326$	−0.945167			−0.0523480
$327$		−	23.6574i		−	1.30826i
$328$		−	3.12549i		−	0.172577i
$329$	−1.36226			−0.0751039
$330$	−5.47106	−	1.71564i	−0.301172	−	0.0944428i
$331$	−30.7373			−1.68947			−0.844737	−	0.535182i	$-0.820243\pi$
$331$	−30.7373			−1.68947			−0.844737	+	0.535182i	$0.820243\pi$
$332$		−	0.775384i		−	0.0425547i
$333$			1.84912i			0.101331i
$334$	−1.99975			−0.109421
$335$	9.62820	−	30.7037i	0.526045	−	1.67752i
$336$	1.84974			0.100912
$337$		−	12.0108i		−	0.654272i	−0.944977	−	0.327136i	$-0.893917\pi$
$337$		−	12.0108i		−	0.654272i	0.944977	−	0.327136i	$-0.106083\pi$
$338$			15.0867i			0.820611i
$339$	−19.9027			−1.08097
$340$	0.195870	−	0.624616i	0.0106225	−	0.0338746i
$341$	−1.16919			−0.0633154
$342$		−	1.60374i		−	0.0867201i
$343$			4.36770i			0.235834i
$344$	15.8468			0.854403
$345$	31.7264	+	9.94892i	1.70809	+	0.535632i
$346$	−13.7256			−0.737891
$347$		−	6.01606i		−	0.322959i	−0.986876	−	0.161480i	$-0.948373\pi$
$347$		−	6.01606i		−	0.322959i	0.986876	−	0.161480i	$-0.0516266\pi$
$348$		−	3.05225i		−	0.163618i
$349$	31.5085			1.68661			0.843304	−	0.537436i	$-0.180607\pi$
$349$	31.5085			1.68661			0.843304	+	0.537436i	$0.180607\pi$
$350$	−1.59544	−	1.10974i	−0.0852800	−	0.0593180i
$351$	−3.16768			−0.169078
$352$			2.59655i			0.138397i
$353$			13.6680i			0.727472i	0.931502	+	0.363736i	$0.118499\pi$
$353$			13.6680i			0.727472i	−0.931502	+	0.363736i	$0.881501\pi$
$354$	−2.87215			−0.152653
$355$	−11.0831	−	3.47549i	−0.588231	−	0.184460i
$356$	6.54835			0.347062
$357$			0.405993i			0.0214874i
$358$			30.7490i			1.62513i
$359$	16.9144			0.892708			0.446354	−	0.894857i	$-0.352722\pi$
$359$	16.9144			0.892708			0.446354	+	0.894857i	$0.352722\pi$
$360$	2.64991	−	8.45039i	0.139663	−	0.445375i
$361$	1.00000			0.0526316
$362$			24.8858i			1.30797i
$363$			2.07277i			0.108792i
$364$	−0.132355			−0.00693728
$365$	9.45639	−	30.1558i	0.494970	−	1.57843i
$366$	14.7234			0.769606
$367$		−	21.8818i		−	1.14222i	−0.820874	−	0.571110i	$-0.806513\pi$
$367$		−	21.8818i		−	1.14222i	0.820874	−	0.571110i	$-0.193487\pi$
$368$			20.3758i			1.06216i
$369$	1.32623			0.0690409
$370$	3.76492	+	1.18062i	0.195729	+	0.0613776i
$371$	−1.03408			−0.0536869
$372$		−	1.13806i		−	0.0590054i
$373$		−	21.9840i		−	1.13829i	−0.822239	−	0.569143i	$-0.807275\pi$
$373$		−	21.9840i		−	1.13829i	0.822239	−	0.569143i	$-0.192725\pi$
$374$	0.771209			0.0398783
$375$	−18.3192	+	14.1935i	−0.945997	+	0.732949i
$376$	−13.2462			−0.683120
$377$		−	2.81292i		−	0.144873i
$378$		−	1.37254i		−	0.0705961i
$379$	−16.3858			−0.841682			−0.420841	−	0.907134i	$-0.638265\pi$
$379$	−16.3858			−0.841682			−0.420841	+	0.907134i	$0.638265\pi$
$380$	1.00194	+	0.314194i	0.0513987	+	0.0161178i
$381$	−3.39073			−0.173712
$382$			12.6952i			0.649545i
$383$			7.01836i			0.358621i	0.983792	+	0.179311i	$0.0573867\pi$
$383$			7.01836i			0.358621i	−0.983792	+	0.179311i	$0.942613\pi$
$384$	12.0388			0.614350
$385$	−0.210218	+	0.670372i	−0.0107137	+	0.0341653i
$386$	−18.7369			−0.953684
$387$			6.72424i			0.341812i
$388$		−	0.379986i		−	0.0192909i
$389$	−24.4290			−1.23860			−0.619300	−	0.785154i	$-0.712584\pi$
$389$	−24.4290			−1.23860			−0.619300	+	0.785154i	$0.712584\pi$
$390$	1.53901	−	4.90780i	0.0779307	−	0.248516i
$391$	−4.47221			−0.226169
$392$			21.0843i			1.06492i
$393$		−	39.8094i		−	2.00812i
$394$	−28.7459			−1.44820
$395$	−8.54725	−	2.68028i	−0.430059	−	0.134860i
$396$	−0.608774			−0.0305921
$397$			20.2563i			1.01663i	0.861170	+	0.508317i	$0.169732\pi$
$397$			20.2563i			1.01663i	−0.861170	+	0.508317i	$0.830268\pi$
$398$			1.32671i			0.0665019i
$399$	−0.651252			−0.0326034
$400$	−11.6585	−	8.10925i	−0.582924	−	0.405463i
$401$	3.36148			0.167864			0.0839321	−	0.996471i	$-0.473252\pi$
$401$	3.36148			0.167864			0.0839321	+	0.996471i	$0.473252\pi$
$402$			36.9001i			1.84041i
$403$		−	1.04882i		−	0.0522455i
$404$	3.70252			0.184207
$405$	−23.9148	−	7.49930i	−1.18834	−	0.372643i
$406$	1.21883			0.0604895
$407$		−	1.42638i		−	0.0707031i
$408$			3.94775i			0.195443i
$409$	−33.5373			−1.65832			−0.829158	−	0.559015i	$-0.811179\pi$
$409$	−33.5373			−1.65832			−0.829158	+	0.559015i	$0.811179\pi$
$410$	0.846768	−	2.70029i	0.0418189	−	0.133358i
$411$	33.6662			1.66063
$412$		−	4.44080i		−	0.218783i
$413$			0.351926i			0.0173171i
$414$	−11.5050			−0.565439
$415$	1.10475	−	3.52296i	0.0542298	−	0.172935i
$416$	−2.32923			−0.114200
$417$			1.09271i			0.0535101i
$418$			1.23709i			0.0605083i
$419$	−35.9922			−1.75834			−0.879168	−	0.476513i	$-0.841901\pi$
$419$	−35.9922			−1.75834			−0.879168	+	0.476513i	$0.841901\pi$
$420$	−0.652519	−	0.204620i	−0.0318397	−	0.00998442i
$421$	−15.5715			−0.758907			−0.379453	−	0.925211i	$-0.623888\pi$
$421$	−15.5715			−0.758907			−0.379453	+	0.925211i	$0.623888\pi$
$422$		−	14.1219i		−	0.687444i
$423$		−	5.62072i		−	0.273289i
$424$	−10.0551			−0.488318
$425$	1.77987	−	2.55888i	0.0863365	−	0.124124i
$426$	13.3198			0.645347
$427$		−	1.80407i		−	0.0873051i
$428$		−	2.08424i		−	0.100746i
$429$	−1.85937			−0.0897714
$430$	13.6909	+	4.29327i	0.660236	+	0.207040i
$431$	19.7186			0.949810			0.474905	−	0.880037i	$-0.342482\pi$
$431$	19.7186			0.949810			0.474905	+	0.880037i	$0.342482\pi$
$432$		−	10.0297i		−	0.482553i
$433$		−	21.7682i		−	1.04611i	−0.852298	−	0.523056i	$-0.824792\pi$
$433$		−	21.7682i		−	1.04611i	0.852298	−	0.523056i	$-0.175208\pi$
$434$	0.454451			0.0218144
$435$	4.34876	−	13.8679i	0.208507	−	0.664915i
$436$	−5.35973			−0.256685
$437$		−	7.17386i		−	0.343172i
$438$			36.2416i			1.73169i
$439$	11.0246			0.526174			0.263087	−	0.964772i	$-0.415259\pi$
$439$	11.0246			0.526174			0.263087	+	0.964772i	$0.415259\pi$
$440$	−2.04409	+	6.51848i	−0.0974484	+	0.310757i
$441$	−8.94664			−0.426030
$442$			0.691811i			0.0329061i
$443$			14.9049i			0.708154i	0.935216	+	0.354077i	$0.115205\pi$
$443$			14.9049i			0.708154i	−0.935216	+	0.354077i	$0.884795\pi$
$444$	1.38840			0.0658903
$445$	29.7525	+	9.32992i	1.41040	+	0.442280i
$446$	−17.9730			−0.851046
$447$			13.4971i			0.638392i
$448$		−	2.79405i		−	0.132006i
$449$	−6.75645			−0.318857			−0.159428	−	0.987209i	$-0.550965\pi$
$449$	−6.75645			−0.318857			−0.159428	+	0.987209i	$0.550965\pi$
$450$	4.57881	−	6.58284i	0.215847	−	0.310318i
$451$	−1.02303			−0.0481728
$452$			4.50907i			0.212089i
$453$		−	31.1693i		−	1.46446i
$454$	−15.2817			−0.717205
$455$	−0.601355	−	0.188576i	−0.0281920	−	0.00884056i
$456$	−6.33257			−0.296550
$457$		−	9.76392i		−	0.456737i	−0.973575	−	0.228369i	$-0.926661\pi$
$457$		−	9.76392i		−	0.456737i	0.973575	−	0.228369i	$-0.0733391\pi$
$458$		−	1.74662i		−	0.0816140i
$459$	2.20138			0.102752
$460$	2.25399	−	7.18781i	0.105093	−	0.335133i
$461$	24.5389			1.14289			0.571445	−	0.820641i	$-0.306383\pi$
$461$	24.5389			1.14289			0.571445	+	0.820641i	$0.306383\pi$
$462$		−	0.805660i		−	0.0374827i
$463$		−	31.8776i		−	1.48148i	−0.671793	−	0.740739i	$-0.734476\pi$
$463$		−	31.8776i		−	1.48148i	0.671793	−	0.740739i	$-0.265524\pi$
$464$	8.90643			0.413471
$465$	1.62147	−	5.17077i	0.0751939	−	0.239788i
$466$	−2.58995			−0.119977
$467$			2.47677i			0.114611i	0.998357	+	0.0573056i	$0.0182509\pi$
$467$			2.47677i			0.114611i	−0.998357	+	0.0573056i	$0.981749\pi$
$468$		−	0.546100i		−	0.0252435i
$469$	4.52138			0.208778
$470$	−11.4441	−	3.58870i	−0.527878	−	0.165534i
$471$	−11.0586			−0.509551
$472$			3.42201i			0.157511i
$473$		−	5.18696i		−	0.238497i
$474$	10.2722			0.471817
$475$	4.10469	+	2.85509i	0.188336	+	0.131000i
$476$	0.0919801			0.00421590
$477$		−	4.26665i		−	0.195356i
$478$			30.0954i			1.37653i
$479$	11.2179			0.512559			0.256280	−	0.966603i	$-0.417503\pi$
$479$	11.2179			0.512559			0.256280	+	0.966603i	$0.417503\pi$
$480$	−11.4833	−	3.60098i	−0.524138	−	0.164361i
$481$	1.27953			0.0583417
$482$			10.0333i			0.457005i
$483$			4.67199i			0.212583i
$484$	0.469598			0.0213454
$485$	0.541394	−	1.72647i	0.0245834	−	0.0783949i
$486$	15.6357			0.709249
$487$		−	10.7546i		−	0.487339i	−0.969858	−	0.243669i	$-0.921649\pi$
$487$		−	10.7546i		−	0.487339i	0.969858	−	0.243669i	$-0.0783512\pi$
$488$		−	17.5422i		−	0.794099i
$489$	−1.58364			−0.0716147
$490$	−5.71221	+	18.2159i	−0.258052	+	0.822909i
$491$	−19.2326			−0.867954			−0.433977	−	0.900924i	$-0.642890\pi$
$491$	−19.2326			−0.867954			−0.433977	+	0.900924i	$0.642890\pi$
$492$		−	0.995788i		−	0.0448936i
$493$			1.95484i			0.0880416i
$494$	−1.10973			−0.0499292
$495$	−2.76597	−	0.867365i	−0.124321	−	0.0389852i
$496$	3.32084			0.149110
$497$		−	1.63208i		−	0.0732090i
$498$			4.23394i			0.189727i
$499$	−11.2354			−0.502967			−0.251484	−	0.967862i	$-0.580918\pi$
$499$	−11.2354			−0.502967			−0.251484	+	0.967862i	$0.580918\pi$
$500$	3.21562	+	4.15031i	0.143807	+	0.185608i
$501$	−3.35060			−0.149694
$502$		−	19.6316i		−	0.876199i
$503$		−	24.5392i		−	1.09415i	−0.837084	−	0.547074i	$-0.815742\pi$
$503$		−	24.5392i		−	1.09415i	0.837084	−	0.547074i	$-0.184258\pi$
$504$	1.24439			0.0554297
$505$	16.8224	+	5.27525i	0.748588	+	0.234746i
$506$	8.87474			0.394530
$507$			25.2781i			1.12264i
$508$			0.768189i			0.0340829i
$509$	−2.49264			−0.110484			−0.0552421	−	0.998473i	$-0.517593\pi$
$509$	−2.49264			−0.110484			−0.0552421	+	0.998473i	$0.517593\pi$
$510$	−1.06954	+	3.41068i	−0.0473598	+	0.151027i
$511$	4.44070			0.196445
$512$		−	24.7298i		−	1.09291i
$513$			3.53123i			0.155907i
$514$	30.6698			1.35279
$515$	6.32713	−	20.1768i	0.278807	−	0.889097i
$516$	5.04883			0.222262
$517$			4.33573i			0.190685i
$518$			0.554417i			0.0243597i
$519$	−22.9974			−1.00947
$520$	−5.84739	−	1.83365i	−0.256425	−	0.0804109i
$521$	42.6121			1.86687			0.933435	−	0.358747i	$-0.116796\pi$
$521$	42.6121			1.86687			0.933435	+	0.358747i	$0.116796\pi$
$522$			5.02892i			0.220110i
$523$			26.0936i			1.14099i	0.821300	+	0.570497i	$0.193249\pi$
$523$			26.0936i			1.14099i	−0.821300	+	0.570497i	$0.806751\pi$
$524$	−9.01906			−0.393999
$525$	−2.67319	−	1.85938i	−0.116667	−	0.0811501i
$526$	1.14735			0.0500270
$527$			0.728879i			0.0317505i
$528$		−	5.88725i		−	0.256210i
$529$	−28.4642			−1.23758
$530$	−8.68715	−	2.72415i	−0.377345	−	0.118330i
$531$	−1.45205			−0.0630138
$532$			0.147545i			0.00639689i
$533$		−	0.917709i		−	0.0397504i
$534$	−35.7569			−1.54735
$535$	2.96957	−	9.46978i	0.128386	−	0.409414i
$536$	43.9645			1.89898
$537$			51.5204i			2.22327i
$538$			23.9903i			1.03430i
$539$	6.90128			0.297259
$540$	−1.10949	+	3.53809i	−0.0477449	+	0.152255i
$541$	−45.1164			−1.93971			−0.969853	−	0.243691i	$-0.921642\pi$
$541$	−45.1164			−1.93971			−0.969853	+	0.243691i	$0.921642\pi$
$542$		−	28.2434i		−	1.21316i
$543$			41.6966i			1.78937i
$544$	1.61870			0.0694013
$545$	−24.3520	−	7.63640i	−1.04312	−	0.327108i
$546$	0.722716			0.0309294
$547$		−	22.8147i		−	0.975485i	−0.872988	−	0.487742i	$-0.837821\pi$
$547$		−	22.8147i		−	0.975485i	0.872988	−	0.487742i	$-0.162179\pi$
$548$		−	7.62726i		−	0.325821i
$549$	7.44364			0.317687
$550$	−3.53201	+	5.07789i	−0.150606	+	0.216522i
$551$	−3.13576			−0.133588
$552$			45.4290i			1.93359i
$553$		−	1.25866i		−	0.0535235i
$554$	−13.0990			−0.556524
$555$	6.30819	+	1.97815i	0.267768	+	0.0839677i
$556$	0.247559			0.0104988
$557$			30.7731i			1.30390i	0.758262	+	0.651950i	$0.226049\pi$
$557$			30.7731i			1.30390i	−0.758262	+	0.651950i	$0.773951\pi$
$558$			1.87508i			0.0793783i
$559$	4.65295			0.196799
$560$	0.597079	−	1.90404i	0.0252312	−	0.0804606i
$561$	1.29217			0.0545555
$562$		−	7.96163i		−	0.335841i
$563$		−	12.1120i		−	0.510462i	−0.966880	−	0.255231i	$-0.917848\pi$
$563$		−	12.1120i		−	0.510462i	0.966880	−	0.255231i	$-0.0821515\pi$
$564$	−4.22026			−0.177705
$565$	−6.42440	+	20.4870i	−0.270277	+	0.861894i
$566$	13.4842			0.566782
$567$		−	3.52166i		−	0.147896i
$568$		−	15.8699i		−	0.665885i
$569$	25.5023			1.06911			0.534555	−	0.845134i	$-0.320479\pi$
$569$	25.5023			1.06911			0.534555	+	0.845134i	$0.320479\pi$
$570$	−5.47106	−	1.71564i	−0.229157	−	0.0718602i
$571$	−16.5347			−0.691955			−0.345978	−	0.938243i	$-0.612453\pi$
$571$	−16.5347			−0.691955			−0.345978	+	0.938243i	$0.612453\pi$
$572$			0.421252i			0.0176134i
$573$			21.2710i			0.888611i
$574$	0.397641			0.0165972
$575$	20.4820	−	29.4465i	0.854158	−	1.22800i
$576$	11.5283			0.480347
$577$			30.0804i			1.25226i	0.779717	+	0.626132i	$0.215363\pi$
$577$			30.0804i			1.25226i	−0.779717	+	0.626132i	$0.784637\pi$
$578$			20.5498i			0.854760i
$579$	−31.3940			−1.30469
$580$	−3.14185	−	0.985236i	−0.130458	−	0.0409097i
$581$	0.518787			0.0215229
$582$			2.07489i			0.0860070i
$583$			3.29122i			0.136308i
$584$	43.1800			1.78680
$585$	0.778068	−	2.48121i	0.0321692	−	0.102585i
$586$	−8.14304			−0.336386
$587$		−	33.4079i		−	1.37889i	−0.724337	−	0.689446i	$-0.757854\pi$
$587$		−	33.4079i		−	1.37889i	0.724337	−	0.689446i	$-0.242146\pi$
$588$			6.71749i			0.277025i
$589$	−1.16919			−0.0481758
$590$	−0.927102	+	2.95647i	−0.0381682	+	0.121716i
$591$	−48.1643			−1.98121
$592$			4.05133i			0.166509i
$593$		−	19.5162i		−	0.801435i	−0.916202	−	0.400717i	$-0.868761\pi$
$593$		−	19.5162i		−	0.801435i	0.916202	−	0.400717i	$-0.131239\pi$
$594$	−4.36846			−0.179240
$595$	0.417912	+	0.131051i	0.0171327	+	0.00537256i
$596$	3.05785			0.125254
$597$			2.22292i			0.0909780i
$598$			7.96106i			0.325552i
$599$	−23.8023			−0.972535			−0.486267	−	0.873810i	$-0.661642\pi$
$599$	−23.8023			−0.972535			−0.486267	+	0.873810i	$0.661642\pi$
$600$	−25.9932	−	18.0800i	−1.06117	−	0.738115i
$601$	−19.7433			−0.805346			−0.402673	−	0.915344i	$-0.631919\pi$
$601$	−19.7433			−0.805346			−0.402673	+	0.915344i	$0.631919\pi$
$602$			2.01611i			0.0821705i
$603$			18.6553i			0.759704i
$604$	−7.06158			−0.287332
$605$	2.13362	+	0.669071i	0.0867441	+	0.0272016i
$606$	−20.2174			−0.821275
$607$		−	21.1111i		−	0.856875i	−0.903572	−	0.428437i	$-0.859064\pi$
$607$		−	21.1111i		−	0.856875i	0.903572	−	0.428437i	$-0.140936\pi$
$608$			2.59655i			0.105304i
$609$	2.04217			0.0827528
$610$	4.75259	−	15.1557i	0.192427	−	0.613636i
$611$	−3.88935			−0.157346
$612$			0.379512i			0.0153409i
$613$		−	29.3006i		−	1.18344i	−0.806143	−	0.591720i	$-0.798449\pi$
$613$		−	29.3006i		−	1.18344i	0.806143	−	0.591720i	$-0.201551\pi$
$614$	−9.47380			−0.382332
$615$	1.41877	−	4.52437i	0.0572104	−	0.182440i
$616$	−0.959903			−0.0386756
$617$			13.3644i			0.538030i	0.963136	+	0.269015i	$0.0866982\pi$
$617$			13.3644i			0.538030i	−0.963136	+	0.269015i	$0.913302\pi$
$618$			24.2487i			0.975427i
$619$	−5.07280			−0.203893			−0.101947	−	0.994790i	$-0.532507\pi$
$619$	−5.07280			−0.203893			−0.101947	+	0.994790i	$0.532507\pi$
$620$	−1.17147	−	0.367354i	−0.0470472	−	0.0147533i
$621$	25.3325			1.01656
$622$		−	3.02778i		−	0.121403i
$623$			4.38131i			0.175533i
$624$	5.28114			0.211415
$625$	8.69694	+	23.4385i	0.347878	+	0.937540i
$626$	−11.6363			−0.465078
$627$			2.07277i			0.0827784i
$628$			2.50538i			0.0999755i
$629$	−0.889212			−0.0354552
$630$	1.07510	+	0.337134i	0.0428330	+	0.0134318i
$631$	−10.8359			−0.431371			−0.215686	−	0.976463i	$-0.569199\pi$
$631$	−10.8359			−0.431371			−0.215686	+	0.976463i	$0.569199\pi$
$632$		−	12.2388i		−	0.486832i
$633$		−	23.6615i		−	0.940459i
$634$	−18.9916			−0.754253
$635$	−1.09449	+	3.49027i	−0.0434337	+	0.138507i
$636$	−3.20357			−0.127030
$637$			6.19078i			0.245288i
$638$		−	3.87922i		−	0.153580i
$639$	6.73402			0.266394
$640$	3.88600	−	12.3922i	0.153608	−	0.489845i
$641$	−3.60462			−0.142374			−0.0711869	−	0.997463i	$-0.522679\pi$
$641$	−3.60462			−0.142374			−0.0711869	+	0.997463i	$0.522679\pi$
$642$			11.3809i			0.449168i
$643$		−	10.1882i		−	0.401785i	−0.979613	−	0.200893i	$-0.935616\pi$
$643$		−	10.1882i		−	0.401785i	0.979613	−	0.200893i	$-0.0643842\pi$
$644$	1.05847			0.0417094
$645$	22.9394	+	7.19343i	0.903237	+	0.283241i
$646$	0.771209			0.0303428
$647$			17.0298i			0.669509i	0.942305	+	0.334754i	$0.108653\pi$
$647$			17.0298i			0.669509i	−0.942305	+	0.334754i	$0.891347\pi$
$648$		−	34.2435i		−	1.34521i
$649$	1.12009			0.0439674
$650$	−4.55511	−	3.16838i	−0.178666	−	0.124274i
$651$	0.761440			0.0298432
$652$			0.358783i			0.0140510i
$653$			8.11964i			0.317746i	0.987299	+	0.158873i	$0.0507860\pi$
$653$			8.11964i			0.317746i	−0.987299	+	0.158873i	$0.949214\pi$
$654$	29.2665			1.14441
$655$	−40.9782	−	12.8501i	−1.60115	−	0.502095i
$656$	2.90570			0.113449
$657$			18.3225i			0.714827i
$658$		−	1.68525i		−	0.0656977i
$659$	−30.5197			−1.18888			−0.594439	−	0.804141i	$-0.702626\pi$
$659$	−30.5197			−1.18888			−0.594439	+	0.804141i	$0.702626\pi$
$660$	−0.651252	+	2.07680i	−0.0253500	+	0.0808394i
$661$	−5.45074			−0.212009			−0.106005	−	0.994366i	$-0.533806\pi$
$661$	−5.45074			−0.212009			−0.106005	+	0.994366i	$0.533806\pi$
$662$		−	38.0249i		−	1.47788i
$663$			1.15914i			0.0450173i
$664$	5.04452			0.195765
$665$	−0.210218	+	0.670372i	−0.00815191	+	0.0259959i
$666$	−2.28754			−0.0886404
$667$			22.4955i			0.871028i
$668$			0.759099i			0.0293704i
$669$	−30.1140			−1.16428
$670$	37.9834	+	11.9110i	1.46743	+	0.460162i
$671$	−5.74190			−0.221663
$672$		−	1.69101i		−	0.0652322i
$673$		−	12.2864i		−	0.473605i	−0.971558	−	0.236802i	$-0.923901\pi$
$673$		−	12.2864i		−	0.473605i	0.971558	−	0.236802i	$-0.0760994\pi$
$674$	14.8585			0.572329
$675$	−10.0820	+	14.4946i	−0.388055	+	0.557897i
$676$	5.72689			0.220265
$677$			7.06832i			0.271658i	0.990732	+	0.135829i	$0.0433697\pi$
$677$			7.06832i			0.271658i	−0.990732	+	0.135829i	$0.956630\pi$
$678$		−	24.6215i		−	0.945583i
$679$	0.254237			0.00975674
$680$	4.06365	+	1.27430i	0.155834	+	0.0488670i
$681$	−25.6047			−0.981174
$682$		−	1.44640i		−	0.0553856i
$683$			41.6548i			1.59388i	0.604062	+	0.796938i	$0.293548\pi$
$683$			41.6548i			1.59388i	−0.604062	+	0.796938i	$0.706452\pi$
$684$	−0.608774			−0.0232771
$685$	10.8671	−	34.6545i	0.415211	−	1.32408i
$686$	−5.40326			−0.206297
$687$		−	2.92648i		−	0.111652i
$688$			14.7324i			0.561669i
$689$	−2.95238			−0.112477
$690$	−12.3077	+	39.2486i	−0.468548	+	1.49417i
$691$	−19.7340			−0.750717			−0.375358	−	0.926880i	$-0.622480\pi$
$691$	−19.7340			−0.750717			−0.375358	+	0.926880i	$0.622480\pi$
$692$			5.21019i			0.198062i
$693$		−	0.407313i		−	0.0154725i
$694$	7.44244			0.282511
$695$	1.12479	+	0.352716i	0.0426656	+	0.0133793i
$696$	19.8574			0.752692
$697$			0.637763i			0.0241570i
$698$			38.9789i			1.47537i
$699$	−4.33950			−0.164135
$700$	−0.421254	+	0.605626i	−0.0159219	+	0.0228905i
$701$	−37.0168			−1.39810			−0.699052	−	0.715070i	$-0.746395\pi$
$701$	−37.0168			−1.39810			−0.699052	+	0.715070i	$0.746395\pi$
$702$		−	3.91871i		−	0.147902i
$703$		−	1.42638i		−	0.0537970i
$704$	−8.89275			−0.335158
$705$	−19.1748	−	6.01292i	−0.722165	−	0.226460i
$706$	−16.9086			−0.636362
$707$			2.47725i			0.0931665i
$708$			1.09026i			0.0409745i
$709$	−30.0213			−1.12748			−0.563738	−	0.825954i	$-0.690637\pi$
$709$	−30.0213			−1.12748			−0.563738	+	0.825954i	$0.690637\pi$
$710$	4.29951	−	13.7109i	0.161358	−	0.514559i
$711$	5.19325			0.194762
$712$			42.6025i			1.59660i
$713$			8.38763i			0.314119i
$714$	−0.502252			−0.0187963
$715$	−0.600188	+	1.91396i	−0.0224458	+	0.0715781i
$716$	11.6722			0.436212
$717$			50.4252i			1.88317i
$718$			20.9247i			0.780903i
$719$	−19.9674			−0.744659			−0.372330	−	0.928101i	$-0.621441\pi$
$719$	−19.9674			−0.744659			−0.372330	+	0.928101i	$0.621441\pi$
$720$	7.85614	+	2.46356i	0.292781	+	0.0918115i
$721$	2.97121			0.110654
$722$			1.23709i			0.0460399i
$723$			16.8110i			0.625207i
$724$	9.44660			0.351080
$725$	−12.8713	−	8.95286i	−0.478028	−	0.332501i
$726$	−2.56421			−0.0951668
$727$		−	27.8780i		−	1.03394i	−0.856005	−	0.516968i	$-0.827061\pi$
$727$		−	27.8780i		−	1.03394i	0.856005	−	0.516968i	$-0.172939\pi$
$728$		−	0.861078i		−	0.0319137i
$729$	−7.42780			−0.275104
$730$	37.3056	+	11.6984i	1.38074	+	0.432979i
$731$	−3.23357			−0.119598
$732$		−	5.58898i		−	0.206575i
$733$		−	6.72664i		−	0.248454i	−0.992254	−	0.124227i	$-0.960355\pi$
$733$		−	6.72664i		−	0.248454i	0.992254	−	0.124227i	$-0.0396451\pi$
$734$	27.0698			0.999166
$735$	−9.57090	+	30.5210i	−0.353028	+	1.12578i
$736$	18.6273			0.686612
$737$		−	14.3904i		−	0.530078i
$738$			1.64068i			0.0603941i
$739$	32.6139			1.19972			0.599862	−	0.800104i	$-0.295222\pi$
$739$	32.6139			1.19972			0.599862	+	0.800104i	$0.295222\pi$
$740$	0.448161	−	1.42916i	0.0164747	−	0.0525368i
$741$	−1.85937			−0.0683058
$742$		−	1.27926i		−	0.0469630i
$743$			1.17457i			0.0430907i	0.999768	+	0.0215454i	$0.00685863\pi$
$743$			1.17457i			0.0430907i	−0.999768	+	0.0215454i	$0.993141\pi$
$744$	7.40400			0.271444
$745$	13.8934	+	4.35674i	0.509014	+	0.159619i
$746$	27.1962			0.995724
$747$			2.14053i			0.0783178i
$748$		−	0.292749i		−	0.0107040i
$749$	1.39451			0.0509542
$750$	−17.5587	−	22.6625i	−0.641153	−	0.827518i
$751$	29.1972			1.06542			0.532711	−	0.846297i	$-0.321173\pi$
$751$	29.1972			1.06542			0.532711	+	0.846297i	$0.321173\pi$
$752$		−	12.3147i		−	0.449071i
$753$		−	32.8930i		−	1.19869i
$754$	3.47985			0.126729
$755$	−32.0844	−	10.0612i	−1.16767	−	0.366163i
$756$	−0.521015			−0.0189491
$757$			5.16599i			0.187761i	0.995583	+	0.0938806i	$0.0299272\pi$
$757$			5.16599i			0.187761i	−0.995583	+	0.0938806i	$0.970073\pi$
$758$		−	20.2708i		−	0.736268i
$759$	14.8698			0.539738
$760$	−2.04409	+	6.51848i	−0.0741471	+	0.236450i
$761$	−16.5957			−0.601595			−0.300798	−	0.953688i	$-0.597253\pi$
$761$	−16.5957			−0.601595			−0.300798	+	0.953688i	$0.597253\pi$
$762$		−	4.19465i		−	0.151956i
$763$		−	3.58604i		−	0.129823i
$764$	4.81908			0.174348
$765$	−0.540719	+	1.72432i	−0.0195497	+	0.0623428i
$766$	−8.68237			−0.313707
$767$			1.00477i			0.0362803i
$768$		−	21.9721i		−	0.792852i
$769$	11.8283			0.426538			0.213269	−	0.976993i	$-0.431589\pi$
$769$	11.8283			0.426538			0.213269	+	0.976993i	$0.431589\pi$
$770$	−0.829313	−	0.260060i	−0.0298864	−	0.00937190i
$771$	51.3877			1.85068
$772$			7.11249i			0.255984i
$773$		−	14.7844i		−	0.531757i	−0.964007	−	0.265878i	$-0.914338\pi$
$773$		−	14.7844i		−	0.531757i	0.964007	−	0.265878i	$-0.0856619\pi$
$774$	−8.31852			−0.299003
$775$	−4.79917	−	3.33815i	−0.172391	−	0.119910i
$776$	2.47212			0.0887441
$777$			0.928935i			0.0333253i
$778$		−	30.2210i		−	1.08347i
$779$	−1.02303			−0.0366540
$780$	−1.86299	−	0.584204i	−0.0667057	−	0.0209179i
$781$	−5.19451			−0.185874
$782$		−	5.53254i		−	0.197843i
$783$		−	11.0731i		−	0.395719i
$784$	−19.6016			−0.700057
$785$	−3.56960	+	11.3832i	−0.127404	+	0.406284i
$786$	49.2480			1.75662
$787$		−	53.9408i		−	1.92278i	−0.275183	−	0.961392i	$-0.588739\pi$
$787$		−	53.9408i		−	1.92278i	0.275183	−	0.961392i	$-0.411261\pi$
$788$			10.9119i			0.388720i
$789$	1.92241			0.0684395
$790$	3.31576	−	10.5738i	0.117970	−	0.376197i
$791$	−3.01689			−0.107268
$792$		−	3.96058i		−	0.140733i
$793$		−	5.15075i		−	0.182909i
$794$	−25.0589			−0.889309
$795$	−14.5554	−	4.56436i	−0.516228	−	0.161881i
$796$	0.503615			0.0178502
$797$		−	8.19451i		−	0.290264i	−0.989412	−	0.145132i	$-0.953639\pi$
$797$		−	8.19451i		−	0.290264i	0.989412	−	0.145132i	$-0.0463607\pi$
$798$		−	0.805660i		−	0.0285201i
$799$	2.70291			0.0956221
$800$	−7.41339	+	10.6581i	−0.262103	+	0.376819i
$801$	−18.0774			−0.638733
$802$			4.15846i			0.146840i
$803$		−	14.1336i		−	0.498765i
$804$	14.0072			0.493995
$805$	4.80915	+	1.50808i	0.169500	+	0.0531527i
$806$	1.29749			0.0457022
$807$			40.1961i			1.41497i
$808$			24.0880i			0.847412i
$809$	−12.5113			−0.439872			−0.219936	−	0.975514i	$-0.570585\pi$
$809$	−12.5113			−0.439872			−0.219936	+	0.975514i	$0.570585\pi$
$810$	9.27734	−	29.5848i	0.325973	−	1.03951i
$811$	38.8266			1.36339			0.681693	−	0.731638i	$-0.261244\pi$
$811$	38.8266			1.36339			0.681693	+	0.731638i	$0.261244\pi$
$812$		−	0.462665i		−	0.0162364i
$813$		−	47.3222i		−	1.65966i
$814$	1.76457			0.0618481
$815$	−0.511185	+	1.63013i	−0.0179060	+	0.0571011i
$816$	−3.67013			−0.128480
$817$		−	5.18696i		−	0.181469i
$818$		−	41.4889i		−	1.45062i
$819$	0.365379			0.0127674
$820$	−1.02502	−	0.321431i	−0.0357954	−	0.0112249i
$821$	48.3052			1.68586			0.842932	−	0.538021i	$-0.180828\pi$
$821$	48.3052			1.68586			0.842932	+	0.538021i	$0.180828\pi$
$822$			41.6482i			1.45265i
$823$		−	29.2789i		−	1.02060i	−0.859997	−	0.510299i	$-0.829535\pi$
$823$		−	29.2789i		−	1.02060i	0.859997	−	0.510299i	$-0.170465\pi$
$824$	28.8911			1.00647
$825$	−5.91794	+	8.50807i	−0.206036	+	0.296213i
$826$	−0.435365			−0.0151483
$827$			2.34864i			0.0816702i	0.999166	+	0.0408351i	$0.0130018\pi$
$827$			2.34864i			0.0816702i	−0.999166	+	0.0408351i	$0.986998\pi$
$828$			4.36726i			0.151773i
$829$	25.9906			0.902691			0.451346	−	0.892349i	$-0.350944\pi$
$829$	25.9906			0.902691			0.451346	+	0.892349i	$0.350944\pi$
$830$	4.35824	+	1.36667i	0.151277	+	0.0474380i
$831$	−21.9476			−0.761354
$832$		−	7.97722i		−	0.276560i
$833$		−	4.30228i		−	0.149065i
$834$	−1.35178			−0.0468084
$835$	−1.08154	+	3.44897i	−0.0374284	+	0.119357i
$836$	0.469598			0.0162414
$837$		−	4.12868i		−	0.142708i
$838$		−	44.5258i		−	1.53812i
$839$	5.06723			0.174940			0.0874701	−	0.996167i	$-0.472122\pi$
$839$	5.06723			0.174940			0.0874701	+	0.996167i	$0.472122\pi$
$840$	1.33122	−	4.24518i	0.0459315	−	0.146473i
$841$	−19.1670			−0.660932
$842$		−	19.2634i		−	0.663859i
$843$		−	13.3398i		−	0.459448i
$844$	−5.36065			−0.184521
$845$	26.0202	+	8.15952i	0.895121	+	0.280696i
$846$	6.95337			0.239062
$847$			0.314194i			0.0107958i
$848$		−	9.34799i		−	0.321011i
$849$	22.5929			0.775387
$850$	3.16557	+	2.20187i	0.108578	+	0.0755235i
$851$	−10.2327			−0.350771
$852$		−	5.05617i		−	0.173221i
$853$			1.16933i			0.0400371i	0.999800	+	0.0200186i	$0.00637253\pi$
$853$			1.16933i			0.0400371i	−0.999800	+	0.0200186i	$0.993627\pi$
$854$	2.23181			0.0763708
$855$	−2.76597	−	0.867365i	−0.0945942	−	0.0296633i
$856$	13.5597			0.463462
$857$		−	20.5020i		−	0.700336i	−0.936687	−	0.350168i	$-0.886124\pi$
$857$		−	20.5020i		−	0.700336i	0.936687	−	0.350168i	$-0.113876\pi$
$858$		−	2.30022i		−	0.0785282i
$859$	51.2221			1.74768			0.873838	−	0.486216i	$-0.161623\pi$
$859$	51.2221			1.74768			0.873838	+	0.486216i	$0.161623\pi$
$860$	1.62971	−	5.19705i	0.0555728	−	0.177218i
$861$	0.666253			0.0227058
$862$			24.3937i			0.830854i
$863$			40.4511i			1.37697i	0.725250	+	0.688486i	$0.241724\pi$
$863$			40.4511i			1.37697i	−0.725250	+	0.688486i	$0.758276\pi$
$864$	−9.16902			−0.311936
$865$	−7.42334	+	23.6725i	−0.252401	+	0.804890i
$866$	26.9293			0.915095
$867$			34.4315i			1.16936i
$868$		−	0.172509i		−	0.00585532i
$869$	−4.00598			−0.135894
$870$	17.1559	+	5.37982i	0.581639	+	0.182393i
$871$	12.9089			0.437401
$872$		−	34.8695i		−	1.18083i
$873$			1.04899i			0.0355029i
$874$	8.87474			0.300192
$875$	−2.77685	+	2.15148i	−0.0938747	+	0.0727332i
$876$	13.7572			0.464813
$877$		−	18.9754i		−	0.640754i	−0.947290	−	0.320377i	$-0.896190\pi$
$877$		−	18.9754i		−	0.640754i	0.947290	−	0.320377i	$-0.103810\pi$
$878$			13.6384i			0.460275i
$879$	−13.6438			−0.460194
$880$	−6.06009	−	1.90035i	−0.204286	−	0.0640608i
$881$	7.37496			0.248469			0.124234	−	0.992253i	$-0.460353\pi$
$881$	7.37496			0.248469			0.124234	+	0.992253i	$0.460353\pi$
$882$		−	11.0678i		−	0.372673i
$883$		−	31.6616i		−	1.06550i	−0.846274	−	0.532748i	$-0.821159\pi$
$883$		−	31.6616i		−	1.06550i	0.846274	−	0.532748i	$-0.178841\pi$
$884$	0.262610			0.00883253
$885$	−1.55337	+	4.95360i	−0.0522161	+	0.166514i
$886$	−18.4388			−0.619463
$887$			44.5979i			1.49745i	0.662880	+	0.748726i	$0.269334\pi$
$887$			44.5979i			1.49745i	−0.662880	+	0.748726i	$0.730666\pi$
$888$			9.03267i			0.303116i
$889$	−0.513973			−0.0172381
$890$	−11.5420	+	36.8066i	−0.386888	+	1.23376i
$891$	−11.2085			−0.375500
$892$			6.82251i			0.228434i
$893$			4.33573i			0.145090i
$894$	−16.6972			−0.558438
$895$	53.0329	+	16.6303i	1.77269	+	0.555889i
$896$	1.82486			0.0609642
$897$			13.3389i			0.445372i
$898$		−	8.35837i		−	0.278922i
$899$	3.66630			0.122278
$900$	−2.49883	−	1.73810i	−0.0832943	−	0.0579368i
$901$	2.05176			0.0683540
$902$		−	1.26559i		−	0.0421395i
$903$			3.37802i			0.112414i
$904$	−29.3352			−0.975676
$905$	42.9207	+	13.4593i	1.42673	+	0.447401i
$906$	38.5593			1.28105
$907$			19.8230i			0.658211i	0.944293	+	0.329105i	$0.106747\pi$
$907$			19.8230i			0.658211i	−0.944293	+	0.329105i	$0.893253\pi$
$908$			5.80089i			0.192509i
$909$	−10.2212			−0.339015
$910$	0.233286	−	0.743933i	0.00773335	−	0.0246611i
$911$	−46.0464			−1.52559			−0.762793	−	0.646642i	$-0.776173\pi$
$911$	−46.0464			−1.52559			−0.762793	+	0.646642i	$0.776173\pi$
$912$		−	5.88725i		−	0.194946i
$913$		−	1.65116i		−	0.0546456i
$914$	12.0789			0.399534
$915$	7.96303	−	25.3936i	0.263250	−	0.839486i
$916$	−0.663011			−0.0219065
$917$		−	6.03439i		−	0.199273i
$918$			2.72331i			0.0898827i
$919$	−1.40785			−0.0464406			−0.0232203	−	0.999730i	$-0.507392\pi$
$919$	−1.40785			−0.0464406			−0.0232203	+	0.999730i	$0.507392\pi$
$920$	46.7627	+	14.6640i	1.54172	+	0.483459i
$921$	−15.8735			−0.523050
$922$			30.3569i			0.999751i
$923$		−	4.65972i		−	0.153377i
$924$	−0.305827			−0.0100610
$925$	4.07245	−	5.85486i	0.133901	−	0.192506i
$926$	39.4356			1.29593
$927$			12.2593i			0.402648i
$928$		−	8.14216i		−	0.267280i
$929$	−2.06693			−0.0678137			−0.0339068	−	0.999425i	$-0.510795\pi$
$929$	−2.06693			−0.0678137			−0.0339068	+	0.999425i	$0.510795\pi$
$930$	6.39672	+	2.00591i	0.209757	+	0.0657765i
$931$	6.90128			0.226180
$932$			0.983139i			0.0322038i
$933$		−	5.07309i		−	0.166085i
$934$	−3.06400			−0.100257
$935$	0.417101	−	1.33011i	0.0136407	−	0.0434992i
$936$	3.55283			0.116128
$937$		−	56.0643i		−	1.83154i	−0.401704	−	0.915770i	$-0.631582\pi$
$937$		−	56.0643i		−	1.83154i	0.401704	−	0.915770i	$-0.368418\pi$
$938$			5.59338i			0.182630i
$939$	−19.4967			−0.636251
$940$	−1.36226	+	4.34416i	−0.0444321	+	0.141691i
$941$	13.0112			0.424152			0.212076	−	0.977253i	$-0.431978\pi$
$941$	13.0112			0.424152			0.212076	+	0.977253i	$0.431978\pi$
$942$		−	13.6805i		−	0.445734i
$943$			7.33910i			0.238994i
$944$	−3.18137			−0.103545
$945$	−2.36723	−	0.742328i	−0.0770061	−	0.0241479i
$946$	6.41676			0.208627
$947$		−	12.2916i		−	0.399422i	−0.979855	−	0.199711i	$-0.936000\pi$
$947$		−	12.2916i		−	0.399422i	0.979855	−	0.199711i	$-0.0640003\pi$
$948$		−	3.89930i		−	0.126643i
$949$	12.6785			0.411562
$950$	−3.53201	+	5.07789i	−0.114594	+	0.164748i
$951$	−31.8207			−1.03186
$952$			0.598407i			0.0193945i
$953$			3.46550i			0.112259i	0.998424	+	0.0561293i	$0.0178759\pi$
$953$			3.46550i			0.112259i	−0.998424	+	0.0561293i	$0.982124\pi$
$954$	5.27825			0.170890
$955$	21.8955	+	6.86610i	0.708523	+	0.222182i
$956$	11.4241			0.369483
$957$		−	6.49970i		−	0.210105i
$958$			13.8776i			0.448365i
$959$	5.10318			0.164790
$960$	12.3327	−	39.3282i	0.398037	−	1.26931i
$961$	−29.6330			−0.955903
$962$			1.58290i			0.0510348i
$963$			5.75377i			0.185413i
$964$	3.80862			0.122668
$965$	−10.1337	+	32.3156i	−0.326215	+	1.04028i
$966$	−5.77969			−0.185959
$967$			50.4829i			1.62342i	0.584061	+	0.811710i	$0.301463\pi$
$967$			50.4829i			1.62342i	−0.584061	+	0.811710i	$0.698537\pi$
$968$			3.05513i			0.0981954i
$969$	1.29217			0.0415105
$970$	2.13581	+	0.669755i	0.0685766	+	0.0215045i
$971$	47.0493			1.50989			0.754943	−	0.655791i	$-0.227665\pi$
$971$	47.0493			1.50989			0.754943	+	0.655791i	$0.227665\pi$
$972$		−	5.93527i		−	0.190374i
$973$			0.165635i			0.00531000i
$974$	13.3045			0.426303
$975$	−7.63215	−	5.30867i	−0.244424	−	0.170014i
$976$	16.3086			0.522026
$977$		−	39.3093i		−	1.25762i	−0.777560	−	0.628808i	$-0.783543\pi$
$977$		−	39.3093i		−	1.25762i	0.777560	−	0.628808i	$-0.216457\pi$
$978$		−	1.95911i		−	0.0626455i
$979$	13.9446			0.445671
$980$	6.91470	+	2.16834i	0.220882	+	0.0692652i
$981$	14.7961			0.472403
$982$		−	23.7925i		−	0.759250i
$983$			8.58356i			0.273773i	0.990587	+	0.136887i	$0.0437096\pi$
$983$			8.58356i			0.273773i	−0.990587	+	0.136887i	$0.956290\pi$
$984$	6.47843			0.206525
$985$	−15.5470	+	49.5783i	−0.495368	+	1.57970i
$986$	−2.41832			−0.0770151
$987$		−	2.82365i		−	0.0898779i
$988$			0.421252i			0.0134018i
$989$	−37.2105			−1.18323
$990$	1.07301	−	3.42177i	0.0341026	−	0.108751i
$991$	−50.1272			−1.59234			−0.796172	−	0.605070i	$-0.793145\pi$
$991$	−50.1272			−1.59234			−0.796172	+	0.605070i	$0.793145\pi$
$992$		−	3.03587i		−	0.0963891i
$993$		−	63.7113i		−	2.02182i
$994$	2.01904			0.0640401
$995$	2.28818	+	0.717538i	0.0725402	+	0.0227475i
$996$	1.60719			0.0509258
$997$			58.8564i			1.86400i	0.362457	+	0.932001i	$0.381938\pi$
$997$			58.8564i			1.86400i	−0.362457	+	0.932001i	$0.618062\pi$
$998$		−	13.8993i		−	0.439974i
$999$	5.03688			0.159360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.c.419.15	yes	20
5.2	odd	4		5225.2.a.ba.1.6		20
5.3	odd	4		5225.2.a.ba.1.15		20
5.4	even	2	inner	1045.2.b.c.419.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.c.419.6	✓	20	5.4	even	2	inner
1045.2.b.c.419.15	yes	20	1.1	even	1	trivial
5225.2.a.ba.1.6		20	5.2	odd	4
5225.2.a.ba.1.15		20	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.23709i q^{2} +2.07277i q^{3} +0.469598 q^{4} +(2.13362 + 0.669071i) q^{5} -2.56421 q^{6} +0.314194i q^{7} +3.05513i q^{8} -1.29637 q^{9} +O(q^{10})\)	\(q+1.23709i q^{2} +2.07277i q^{3} +0.469598 q^{4} +(2.13362 + 0.669071i) q^{5} -2.56421 q^{6} +0.314194i q^{7} +3.05513i q^{8} -1.29637 q^{9} +(-0.827703 + 2.63949i) q^{10} +1.00000 q^{11} +0.973368i q^{12} +0.897047i q^{13} -0.388688 q^{14} +(-1.38683 + 4.42251i) q^{15} -2.84028 q^{16} -0.623404i q^{17} -1.60374i q^{18} +1.00000 q^{19} +(1.00194 + 0.314194i) q^{20} -0.651252 q^{21} +1.23709i q^{22} -7.17386i q^{23} -6.33257 q^{24} +(4.10469 + 2.85509i) q^{25} -1.10973 q^{26} +3.53123i q^{27} +0.147545i q^{28} -3.13576 q^{29} +(-5.47106 - 1.71564i) q^{30} -1.16919 q^{31} +2.59655i q^{32} +2.07277i q^{33} +0.771209 q^{34} +(-0.210218 + 0.670372i) q^{35} -0.608774 q^{36} -1.42638i q^{37} +1.23709i q^{38} -1.85937 q^{39} +(-2.04409 + 6.51848i) q^{40} -1.02303 q^{41} -0.805660i q^{42} -5.18696i q^{43} +0.469598 q^{44} +(-2.76597 - 0.867365i) q^{45} +8.87474 q^{46} +4.33573i q^{47} -5.88725i q^{48} +6.90128 q^{49} +(-3.53201 + 5.07789i) q^{50} +1.29217 q^{51} +0.421252i q^{52} +3.29122i q^{53} -4.36846 q^{54} +(2.13362 + 0.669071i) q^{55} -0.959903 q^{56} +2.07277i q^{57} -3.87922i q^{58} +1.12009 q^{59} +(-0.651252 + 2.07680i) q^{60} -5.74190 q^{61} -1.44640i q^{62} -0.407313i q^{63} -8.89275 q^{64} +(-0.600188 + 1.91396i) q^{65} -2.56421 q^{66} -14.3904i q^{67} -0.292749i q^{68} +14.8698 q^{69} +(-0.829313 - 0.260060i) q^{70} -5.19451 q^{71} -3.96058i q^{72} -14.1336i q^{73} +1.76457 q^{74} +(-5.91794 + 8.50807i) q^{75} +0.469598 q^{76} +0.314194i q^{77} -2.30022i q^{78} -4.00598 q^{79} +(-6.06009 - 1.90035i) q^{80} -11.2085 q^{81} -1.26559i q^{82} -1.65116i q^{83} -0.305827 q^{84} +(0.417101 - 1.33011i) q^{85} +6.41676 q^{86} -6.49970i q^{87} +3.05513i q^{88} +13.9446 q^{89} +(1.07301 - 3.42177i) q^{90} -0.281847 q^{91} -3.36883i q^{92} -2.42347i q^{93} -5.36371 q^{94} +(2.13362 + 0.669071i) q^{95} -5.38206 q^{96} -0.809173i q^{97} +8.53754i q^{98} -1.29637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Introduction

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