LMFDB - Embedded newform 1045.2.b.b.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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 $ x^16 + 19x^14 + 144x^12 + 552x^10 + 1119x^8 + 1146x^6 + 524x^4 + 83*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.15
Root			$2.10564i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.b.419.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10564i q^{2} +1.90142i q^{3} -2.43371 q^{4} +(1.75301 - 1.38815i) q^{5} -4.00370 q^{6} -0.116917i q^{7} -0.913237i q^{8} -0.615392 q^{9} +O(q^{10})$	\(q+2.10564i q^{2} +1.90142i q^{3} -2.43371 q^{4} +(1.75301 - 1.38815i) q^{5} -4.00370 q^{6} -0.116917i q^{7} -0.913237i q^{8} -0.615392 q^{9} +(2.92295 + 3.69120i) q^{10} -1.00000 q^{11} -4.62750i q^{12} +2.04770i q^{13} +0.246186 q^{14} +(2.63946 + 3.33320i) q^{15} -2.94447 q^{16} +6.68947i q^{17} -1.29579i q^{18} -1.00000 q^{19} +(-4.26631 + 3.37836i) q^{20} +0.222309 q^{21} -2.10564i q^{22} +0.393825i q^{23} +1.73645 q^{24} +(1.14606 - 4.86688i) q^{25} -4.31171 q^{26} +4.53414i q^{27} +0.284543i q^{28} -7.51619 q^{29} +(-7.01851 + 5.55775i) q^{30} +0.564887 q^{31} -8.02647i q^{32} -1.90142i q^{33} -14.0856 q^{34} +(-0.162299 - 0.204957i) q^{35} +1.49769 q^{36} +7.53547i q^{37} -2.10564i q^{38} -3.89353 q^{39} +(-1.26771 - 1.60091i) q^{40} -4.20665 q^{41} +0.468102i q^{42} +2.69721i q^{43} +2.43371 q^{44} +(-1.07879 + 0.854258i) q^{45} -0.829252 q^{46} -13.2536i q^{47} -5.59868i q^{48} +6.98633 q^{49} +(10.2479 + 2.41319i) q^{50} -12.7195 q^{51} -4.98351i q^{52} +10.7714i q^{53} -9.54725 q^{54} +(-1.75301 + 1.38815i) q^{55} -0.106773 q^{56} -1.90142i q^{57} -15.8264i q^{58} +10.7569 q^{59} +(-6.42368 - 8.11204i) q^{60} -7.51273 q^{61} +1.18945i q^{62} +0.0719500i q^{63} +11.0119 q^{64} +(2.84252 + 3.58963i) q^{65} +4.00370 q^{66} -4.18544i q^{67} -16.2802i q^{68} -0.748825 q^{69} +(0.431565 - 0.341744i) q^{70} +1.58482 q^{71} +0.561999i q^{72} +0.494002i q^{73} -15.8670 q^{74} +(9.25398 + 2.17915i) q^{75} +2.43371 q^{76} +0.116917i q^{77} -8.19837i q^{78} +13.3260 q^{79} +(-5.16168 + 4.08738i) q^{80} -10.4675 q^{81} -8.85768i q^{82} +2.11441i q^{83} -0.541036 q^{84} +(9.28600 + 11.7267i) q^{85} -5.67934 q^{86} -14.2914i q^{87} +0.913237i q^{88} +10.6303 q^{89} +(-1.79876 - 2.27153i) q^{90} +0.239412 q^{91} -0.958455i q^{92} +1.07409i q^{93} +27.9072 q^{94} +(-1.75301 + 1.38815i) q^{95} +15.2617 q^{96} +10.3322i q^{97} +14.7107i q^{98} +0.615392 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * 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 $

$2$			2.10564i			1.48891i	0.667672	+	0.744455i	$0.267291\pi$
$2$			2.10564i			1.48891i	−0.667672	+	0.744455i	$0.732709\pi$
$3$			1.90142i			1.09778i	0.835893	+	0.548892i	$0.184950\pi$
$3$			1.90142i			1.09778i	−0.835893	+	0.548892i	$0.815050\pi$
$4$	−2.43371			−1.21686
$5$	1.75301	−	1.38815i	0.783968	−	0.620801i
$5$	1.75301	−	1.38815i	0.783968	−	0.620801i
$6$	−4.00370			−1.63450
$7$		−	0.116917i		−	0.0441906i	−0.999756	−	0.0220953i	$-0.992966\pi$
$7$		−	0.116917i		−	0.0441906i	0.999756	−	0.0220953i	$-0.00703373\pi$
$8$		−	0.913237i		−	0.322878i
$9$	−0.615392			−0.205131
$10$	2.92295	+	3.69120i	0.924317	+	1.16726i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		−	4.62750i		−	1.33584i
$13$			2.04770i			0.567930i	0.958835	+	0.283965i	$0.0916499\pi$
$13$			2.04770i			0.567930i	−0.958835	+	0.283965i	$0.908350\pi$
$14$	0.246186			0.0657959
$15$	2.63946	+	3.33320i	0.681505	+	0.860628i
$16$	−2.94447			−0.736119
$17$			6.68947i			1.62243i	0.584745	+	0.811217i	$0.301194\pi$
$17$			6.68947i			1.62243i	−0.584745	+	0.811217i	$0.698806\pi$
$18$		−	1.29579i		−	0.305421i
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	−4.26631	+	3.37836i	−0.953976	+	0.755425i
$21$	0.222309			0.0485118
$22$		−	2.10564i		−	0.448923i
$23$			0.393825i			0.0821181i	0.999157	+	0.0410590i	$0.0130732\pi$
$23$			0.393825i			0.0821181i	−0.999157	+	0.0410590i	$0.986927\pi$
$24$	1.73645			0.354451
$25$	1.14606	−	4.86688i	0.229213	−	0.973376i
$26$	−4.31171			−0.845597
$27$			4.53414i			0.872595i
$28$			0.284543i			0.0537736i
$29$	−7.51619			−1.39572			−0.697861	−	0.716233i	$-0.745865\pi$
$29$	−7.51619			−1.39572			−0.697861	+	0.716233i	$0.745865\pi$
$30$	−7.01851	+	5.55775i	−1.28140	+	1.01470i
$31$	0.564887			0.101457			0.0507283	−	0.998712i	$-0.483846\pi$
$31$	0.564887			0.101457			0.0507283	+	0.998712i	$0.483846\pi$
$32$		−	8.02647i		−	1.41889i
$33$		−	1.90142i		−	0.330994i
$34$	−14.0856			−2.41566
$35$	−0.162299	−	0.204957i	−0.0274336	−	0.0346441i
$36$	1.49769			0.249614
$37$			7.53547i			1.23882i	0.785066	+	0.619412i	$0.212629\pi$
$37$			7.53547i			1.23882i	−0.785066	+	0.619412i	$0.787371\pi$
$38$		−	2.10564i		−	0.341580i
$39$	−3.89353			−0.623464
$40$	−1.26771	−	1.60091i	−0.200443	−	0.253126i
$41$	−4.20665			−0.656969			−0.328484	−	0.944509i	$-0.606538\pi$
$41$	−4.20665			−0.656969			−0.328484	+	0.944509i	$0.606538\pi$
$42$			0.468102i			0.0722297i
$43$			2.69721i			0.411320i	0.978624	+	0.205660i	$0.0659341\pi$
$43$			2.69721i			0.411320i	−0.978624	+	0.205660i	$0.934066\pi$
$44$	2.43371			0.366896
$45$	−1.07879	+	0.854258i	−0.160816	+	0.127345i
$46$	−0.829252			−0.122267
$47$		−	13.2536i		−	1.93323i	−0.256232	−	0.966615i	$-0.582481\pi$
$47$		−	13.2536i		−	1.93323i	0.256232	−	0.966615i	$-0.417519\pi$
$48$		−	5.59868i		−	0.808099i
$49$	6.98633			0.998047
$50$	10.2479	+	2.41319i	1.44927	+	0.341277i
$51$	−12.7195			−1.78108
$52$		−	4.98351i		−	0.691088i
$53$			10.7714i			1.47957i	0.672844	+	0.739785i	$0.265073\pi$
$53$			10.7714i			1.47957i	−0.672844	+	0.739785i	$0.734927\pi$
$54$	−9.54725			−1.29922
$55$	−1.75301	+	1.38815i	−0.236375	+	0.187178i
$56$	−0.106773			−0.0142682
$57$		−	1.90142i		−	0.251849i
$58$		−	15.8264i		−	2.07811i
$59$	10.7569			1.40043			0.700217	−	0.713930i	$-0.253086\pi$
$59$	10.7569			1.40043			0.700217	+	0.713930i	$0.253086\pi$
$60$	−6.42368	−	8.11204i	−0.829293	−	1.04726i
$61$	−7.51273			−0.961907			−0.480954	−	0.876746i	$-0.659709\pi$
$61$	−7.51273			−0.961907			−0.480954	+	0.876746i	$0.659709\pi$
$62$			1.18945i			0.151060i
$63$			0.0719500i			0.00906485i
$64$	11.0119			1.37649
$65$	2.84252	+	3.58963i	0.352571	+	0.445239i
$66$	4.00370			0.492821
$67$		−	4.18544i		−	0.511332i	−0.966765	−	0.255666i	$-0.917705\pi$
$67$		−	4.18544i		−	0.511332i	0.966765	−	0.255666i	$-0.0822948\pi$
$68$		−	16.2802i		−	1.97427i
$69$	−0.748825			−0.0901480
$70$	0.431565	−	0.341744i	0.0515819	−	0.0408462i
$71$	1.58482			0.188084			0.0940420	−	0.995568i	$-0.470021\pi$
$71$	1.58482			0.188084			0.0940420	+	0.995568i	$0.470021\pi$
$72$			0.561999i			0.0662322i
$73$			0.494002i			0.0578186i	0.999582	+	0.0289093i	$0.00920340\pi$
$73$			0.494002i			0.0578186i	−0.999582	+	0.0289093i	$0.990797\pi$
$74$	−15.8670			−1.84450
$75$	9.25398	+	2.17915i	1.06856	+	0.251626i
$76$	2.43371			0.279166
$77$			0.116917i			0.0133240i
$78$		−	8.19837i		−	0.928283i
$79$	13.3260			1.49930			0.749648	−	0.661837i	$-0.230223\pi$
$79$	13.3260			1.49930			0.749648	+	0.661837i	$0.230223\pi$
$80$	−5.16168	+	4.08738i	−0.577094	+	0.456983i
$81$	−10.4675			−1.16305
$82$		−	8.85768i		−	0.978167i
$83$			2.11441i			0.232087i	0.993244	+	0.116043i	$0.0370212\pi$
$83$			2.11441i			0.232087i	−0.993244	+	0.116043i	$0.962979\pi$
$84$	−0.541036			−0.0590318
$85$	9.28600	+	11.7267i	1.00721	+	1.27194i
$86$	−5.67934			−0.612419
$87$		−	14.2914i		−	1.53220i
$88$			0.913237i			0.0973514i
$89$	10.6303			1.12681			0.563406	−	0.826180i	$-0.309491\pi$
$89$	10.6303			1.12681			0.563406	+	0.826180i	$0.309491\pi$
$90$	−1.79876	−	2.27153i	−0.189606	−	0.239440i
$91$	0.239412			0.0250972
$92$		−	0.958455i		−	0.0999258i
$93$			1.07409i			0.111378i
$94$	27.9072			2.87841
$95$	−1.75301	+	1.38815i	−0.179855	+	0.142421i
$96$	15.2617			1.55764
$97$			10.3322i			1.04907i	0.851387	+	0.524537i	$0.175762\pi$
$97$			10.3322i			1.04907i	−0.851387	+	0.524537i	$0.824238\pi$
$98$			14.7107i			1.48600i
$99$	0.615392			0.0618492
$100$	−2.78919	+	11.8446i	−0.278919	+	1.18446i
$101$	7.52522			0.748788			0.374394	−	0.927270i	$-0.377851\pi$
$101$	7.52522			0.748788			0.374394	+	0.927270i	$0.377851\pi$
$102$		−	26.7826i		−	2.65187i
$103$			4.52064i			0.445432i	0.974883	+	0.222716i	$0.0714923\pi$
$103$			4.52064i			0.445432i	−0.974883	+	0.222716i	$0.928508\pi$
$104$	1.87004			0.183372
$105$	0.389709	−	0.308599i	0.0380317	−	0.0301162i
$106$	−22.6807			−2.20295
$107$		−	1.36972i		−	0.132416i	−0.997806	−	0.0662078i	$-0.978910\pi$
$107$		−	1.36972i		−	0.132416i	0.997806	−	0.0662078i	$-0.0210900\pi$
$108$		−	11.0348i		−	1.06182i
$109$	17.6124			1.68697			0.843483	−	0.537156i	$-0.180501\pi$
$109$	17.6124			1.68697			0.843483	+	0.537156i	$0.180501\pi$
$110$	−2.92295	−	3.69120i	−0.278692	−	0.351942i
$111$	−14.3281			−1.35996
$112$			0.344260i			0.0325296i
$113$		−	17.9806i		−	1.69147i	−0.533601	−	0.845736i	$-0.679162\pi$
$113$		−	17.9806i		−	1.69147i	0.533601	−	0.845736i	$-0.320838\pi$
$114$	4.00370			0.374981
$115$	0.546689	+	0.690377i	0.0509790	+	0.0643780i
$116$	18.2922			1.69839
$117$		−	1.26014i		−	0.116500i
$118$			22.6502i			2.08512i
$119$	0.782115			0.0716964
$120$	3.04400	−	2.41045i	0.277878	−	0.220043i
$121$	1.00000			0.0909091
$122$		−	15.8191i		−	1.43219i
$123$		−	7.99860i		−	0.721210i
$124$	−1.37477			−0.123458
$125$	−4.74692	−	10.1226i	−0.424577	−	0.905392i
$126$	−0.151501			−0.0134968
$127$		−	15.8670i		−	1.40797i	−0.710215	−	0.703985i	$-0.751402\pi$
$127$		−	15.8670i		−	1.40797i	0.710215	−	0.703985i	$-0.248598\pi$
$128$			7.13411i			0.630573i
$129$	−5.12852			−0.451541
$130$	−7.55846	+	5.98532i	−0.662921	+	0.524947i
$131$	−3.15763			−0.275883			−0.137942	−	0.990440i	$-0.544049\pi$
$131$	−3.15763			−0.275883			−0.137942	+	0.990440i	$0.544049\pi$
$132$			4.62750i			0.402772i
$133$			0.116917i			0.0101380i
$134$	8.81301			0.761328
$135$	6.29408	+	7.94837i	0.541708	+	0.684087i
$136$	6.10907			0.523848
$137$		−	8.75488i		−	0.747980i	−0.927433	−	0.373990i	$-0.877990\pi$
$137$		−	8.75488i		−	0.747980i	0.927433	−	0.373990i	$-0.122010\pi$
$138$		−	1.57675i		−	0.134222i
$139$	6.83435			0.579682			0.289841	−	0.957075i	$-0.406398\pi$
$139$	6.83435			0.579682			0.289841	+	0.957075i	$0.406398\pi$
$140$	0.394989	+	0.498806i	0.0333827	+	0.0421568i
$141$	25.2006			2.12227
$142$			3.33707i			0.280040i
$143$		−	2.04770i		−	0.171237i
$144$	1.81200			0.151000
$145$	−13.1759	+	10.4336i	−1.09420	+	0.866466i
$146$	−1.04019			−0.0860867
$147$			13.2839i			1.09564i
$148$		−	18.3392i		−	1.50747i
$149$	−13.8446			−1.13419			−0.567096	−	0.823652i	$-0.691933\pi$
$149$	−13.8446			−1.13419			−0.567096	+	0.823652i	$0.691933\pi$
$150$	−4.58849	+	19.4855i	−0.374649	+	1.59099i
$151$	11.6957			0.951779			0.475889	−	0.879505i	$-0.342126\pi$
$151$	11.6957			0.951779			0.475889	+	0.879505i	$0.342126\pi$
$152$			0.913237i			0.0740733i
$153$		−	4.11664i		−	0.332811i
$154$	−0.246186			−0.0198382
$155$	0.990251	−	0.784149i	0.0795388	−	0.0629844i
$156$	9.47574			0.758666
$157$			7.99822i			0.638327i	0.947700	+	0.319164i	$0.103402\pi$
$157$			7.99822i			0.638327i	−0.947700	+	0.319164i	$0.896598\pi$
$158$			28.0598i			2.23232i
$159$	−20.4810			−1.62425
$160$	−11.1420	−	14.0705i	−0.880850	−	1.11237i
$161$	0.0460450			0.00362885
$162$		−	22.0407i		−	1.73168i
$163$		−	22.0225i		−	1.72494i	−0.506110	−	0.862469i	$-0.668917\pi$
$163$		−	22.0225i		−	1.72494i	0.506110	−	0.862469i	$-0.331083\pi$
$164$	10.2378			0.799436
$165$	−2.63946	−	3.33320i	−0.205482	−	0.259489i
$166$	−4.45218			−0.345556
$167$			17.3899i			1.34567i	0.739793	+	0.672835i	$0.234923\pi$
$167$			17.3899i			1.34567i	−0.739793	+	0.672835i	$0.765077\pi$
$168$		−	0.203021i		−	0.0156634i
$169$	8.80693			0.677456
$170$	−24.6921	+	19.5530i	−1.89380	+	1.49964i
$171$	0.615392			0.0470602
$172$		−	6.56422i		−	0.500517i
$173$			13.5190i			1.02783i	0.857842	+	0.513914i	$0.171805\pi$
$173$			13.5190i			1.02783i	−0.857842	+	0.513914i	$0.828195\pi$
$174$	30.0926			2.28131
$175$	−0.569023	−	0.133995i	−0.0430141	−	0.0101291i
$176$	2.94447			0.221948
$177$			20.4535i			1.53738i
$178$			22.3836i			1.67772i
$179$	13.3448			0.997439			0.498720	−	0.866763i	$-0.333804\pi$
$179$	13.3448			0.997439			0.498720	+	0.866763i	$0.333804\pi$
$180$	2.62545	−	2.07902i	0.195690	−	0.154961i
$181$	−13.7327			−1.02075			−0.510373	−	0.859953i	$-0.670493\pi$
$181$	−13.7327			−1.02075			−0.510373	+	0.859953i	$0.670493\pi$
$182$			0.504115i			0.0373675i
$183$		−	14.2849i		−	1.05597i
$184$	0.359655			0.0265141
$185$	10.4604	+	13.2097i	0.769063	+	0.971199i
$186$	−2.26164			−0.165831
$187$		−	6.68947i		−	0.489182i
$188$			32.2553i			2.35246i
$189$	0.530120			0.0385605
$190$	−2.92295	−	3.69120i	−0.212053	−	0.267788i
$191$	−24.2373			−1.75375			−0.876874	−	0.480720i	$-0.840375\pi$
$191$	−24.2373			−1.75375			−0.876874	+	0.480720i	$0.840375\pi$
$192$			20.9382i			1.51109i
$193$		−	10.0718i		−	0.724981i	−0.931988	−	0.362490i	$-0.881927\pi$
$193$		−	10.0718i		−	0.724981i	0.931988	−	0.362490i	$-0.118073\pi$
$194$	−21.7558			−1.56198
$195$	−6.82539	+	5.40482i	−0.488776	+	0.387047i
$196$	−17.0027			−1.21448
$197$			12.5470i			0.893937i	0.894550	+	0.446969i	$0.147496\pi$
$197$			12.5470i			0.893937i	−0.894550	+	0.446969i	$0.852504\pi$
$198$			1.29579i			0.0920879i
$199$	−9.37419			−0.664519			−0.332259	−	0.943188i	$-0.607811\pi$
$199$	−9.37419			−0.664519			−0.332259	+	0.943188i	$0.607811\pi$
$200$	−4.44462	−	1.04663i	−0.314282	−	0.0740078i
$201$	7.95826			0.561333
$202$			15.8454i			1.11488i
$203$			0.878774i			0.0616779i
$204$	30.9555			2.16732
$205$	−7.37429	+	5.83947i	−0.515043	+	0.407847i
$206$	−9.51884			−0.663209
$207$		−	0.242356i		−	0.0168449i
$208$		−	6.02940i		−	0.418064i
$209$	1.00000			0.0691714
$210$	0.649797	+	0.820586i	0.0448403	+	0.0566258i
$211$	3.48973			0.240243			0.120122	−	0.992759i	$-0.461672\pi$
$211$	3.48973			0.240243			0.120122	+	0.992759i	$0.461672\pi$
$212$		−	26.2145i		−	1.80042i
$213$			3.01341i			0.206476i
$214$	2.88413			0.197155
$215$	3.74413	+	4.72822i	0.255348	+	0.322462i
$216$	4.14074			0.281742
$217$		−	0.0660451i		−	0.00448344i
$218$			37.0854i			2.51174i
$219$	−0.939305			−0.0634723
$220$	4.26631	−	3.37836i	0.287635	−	0.227769i
$221$	−13.6980			−0.921428
$222$		−	30.1698i		−	2.02486i
$223$		−	23.2293i		−	1.55555i	−0.628543	−	0.777775i	$-0.716348\pi$
$223$		−	23.2293i		−	1.55555i	0.628543	−	0.777775i	$-0.283652\pi$
$224$	−0.938434			−0.0627018
$225$	−0.705278	+	2.99504i	−0.0470185	+	0.199669i
$226$	37.8606			2.51845
$227$		−	3.17791i		−	0.210925i	−0.994423	−	0.105463i	$-0.966368\pi$
$227$		−	3.17791i		−	0.210925i	0.994423	−	0.105463i	$-0.0336323\pi$
$228$			4.62750i			0.306464i
$229$	19.0993			1.26212			0.631060	−	0.775734i	$-0.282620\pi$
$229$	19.0993			1.26212			0.631060	+	0.775734i	$0.282620\pi$
$230$	−1.45368	+	1.15113i	−0.0958531	+	0.0759031i
$231$	−0.222309			−0.0146269
$232$			6.86407i			0.450648i
$233$			17.9852i			1.17825i	0.808043	+	0.589123i	$0.200527\pi$
$233$			17.9852i			1.17825i	−0.808043	+	0.589123i	$0.799473\pi$
$234$	2.65339			0.173458
$235$	−18.3980	−	23.2336i	−1.20015	−	1.51559i
$236$	−26.1793			−1.70413
$237$			25.3384i			1.64590i
$238$			1.64685i			0.106750i
$239$	5.73127			0.370725			0.185363	−	0.982670i	$-0.440654\pi$
$239$	5.73127			0.370725			0.185363	+	0.982670i	$0.440654\pi$
$240$	−7.77182	−	9.81452i	−0.501669	−	0.633524i
$241$	−19.5364			−1.25845			−0.629226	−	0.777222i	$-0.716628\pi$
$241$	−19.5364			−1.25845			−0.629226	+	0.777222i	$0.716628\pi$
$242$			2.10564i			0.135356i
$243$		−	6.30062i		−	0.404185i
$244$	18.2838			1.17050
$245$	12.2471	−	9.69809i	0.782437	−	0.619588i
$246$	16.8422			1.07382
$247$		−	2.04770i		−	0.130292i
$248$		−	0.515876i		−	0.0327581i
$249$	−4.02038			−0.254781
$250$	21.3145	−	9.99529i	1.34805	−	0.632158i
$251$	−6.94500			−0.438365			−0.219182	−	0.975684i	$-0.570339\pi$
$251$	−6.94500			−0.438365			−0.219182	+	0.975684i	$0.570339\pi$
$252$		−	0.175106i		−	0.0110306i
$253$		−	0.393825i		−	0.0247595i
$254$	33.4102			2.09634
$255$	−22.2973	+	17.6566i	−1.39631	+	1.10570i
$256$	7.00192			0.437620
$257$			20.8527i			1.30075i	0.759611	+	0.650377i	$0.225389\pi$
$257$			20.8527i			1.30075i	−0.759611	+	0.650377i	$0.774611\pi$
$258$		−	10.7988i		−	0.672304i
$259$	0.881029			0.0547444
$260$	−6.91787	−	8.73612i	−0.429028	−	0.541791i
$261$	4.62540			0.286305
$262$		−	6.64882i		−	0.410765i
$263$			28.8912i			1.78151i	0.454485	+	0.890755i	$0.349823\pi$
$263$			28.8912i			1.78151i	−0.454485	+	0.890755i	$0.650177\pi$
$264$	−1.73645			−0.106871
$265$	14.9524	+	18.8824i	0.918518	+	1.15994i
$266$	−0.246186			−0.0150946
$267$			20.2127i			1.23700i
$268$			10.1861i			0.622217i
$269$	−6.26369			−0.381904			−0.190952	−	0.981599i	$-0.561157\pi$
$269$	−6.26369			−0.381904			−0.190952	+	0.981599i	$0.561157\pi$
$270$	−16.7364	+	13.2530i	−1.01854	+	0.806555i
$271$	24.6553			1.49770			0.748851	−	0.662738i	$-0.230606\pi$
$271$	24.6553			1.49770			0.748851	+	0.662738i	$0.230606\pi$
$272$		−	19.6970i		−	1.19430i
$273$			0.455222i			0.0275513i
$274$	18.4346			1.11367
$275$	−1.14606	+	4.86688i	−0.0691102	+	0.293484i
$276$	1.82242			0.109697
$277$		−	8.10059i		−	0.486718i	−0.969936	−	0.243359i	$-0.921751\pi$
$277$		−	8.10059i		−	0.486718i	0.969936	−	0.243359i	$-0.0782492\pi$
$278$			14.3907i			0.863095i
$279$	−0.347627			−0.0208119
$280$	−0.187174	+	0.148218i	−0.0111858	+	0.00885771i
$281$	3.29255			0.196417			0.0982086	−	0.995166i	$-0.468689\pi$
$281$	3.29255			0.196417			0.0982086	+	0.995166i	$0.468689\pi$
$282$			53.0633i			3.15987i
$283$		−	19.5633i		−	1.16291i	−0.813577	−	0.581457i	$-0.802483\pi$
$283$		−	19.5633i		−	1.16291i	0.813577	−	0.581457i	$-0.197517\pi$
$284$	−3.85700			−0.228871
$285$	−2.63946	−	3.33320i	−0.156348	−	0.197442i
$286$	4.31171			0.254957
$287$			0.491831i			0.0290319i
$288$			4.93942i			0.291058i
$289$	−27.7489			−1.63229
$290$	−21.9694	−	27.7438i	−1.29009	−	1.62917i
$291$	−19.6458			−1.15166
$292$		−	1.20226i		−	0.0703568i
$293$			13.0393i			0.761764i	0.924624	+	0.380882i	$0.124380\pi$
$293$			13.0393i			0.761764i	−0.924624	+	0.380882i	$0.875620\pi$
$294$	−27.9712			−1.63131
$295$	18.8570	−	14.9323i	1.09790	−	0.869391i
$296$	6.88168			0.399989
$297$		−	4.53414i		−	0.263097i
$298$		−	29.1517i		−	1.68871i
$299$	−0.806434			−0.0466373
$300$	−22.5215	−	5.30341i	−1.30028	−	0.306193i
$301$	0.315350			0.0181765
$302$			24.6268i			1.41711i
$303$			14.3086i			0.822007i
$304$	2.94447			0.168877
$305$	−13.1699	+	10.4288i	−0.754105	+	0.597153i
$306$	8.66816			0.495525
$307$		−	31.1098i		−	1.77553i	−0.460297	−	0.887765i	$-0.652257\pi$
$307$		−	31.1098i		−	1.77553i	0.460297	−	0.887765i	$-0.347743\pi$
$308$		−	0.284543i		−	0.0162134i
$309$	−8.59563			−0.488989
$310$	1.65113	+	2.08511i	0.0937781	+	0.118426i
$311$	23.0925			1.30945			0.654726	−	0.755866i	$-0.272784\pi$
$311$	23.0925			1.30945			0.654726	+	0.755866i	$0.272784\pi$
$312$			3.55572i			0.201303i
$313$		−	8.05494i		−	0.455292i	−0.973744	−	0.227646i	$-0.926897\pi$
$313$		−	8.05494i		−	0.455292i	0.973744	−	0.227646i	$-0.0731029\pi$
$314$	−16.8414			−0.950413
$315$	0.0998776	+	0.126129i	0.00562747	+	0.00710656i
$316$	−32.4317			−1.82443
$317$			9.19526i			0.516457i	0.966084	+	0.258229i	$0.0831388\pi$
$317$			9.19526i			0.516457i	−0.966084	+	0.258229i	$0.916861\pi$
$318$		−	43.1256i		−	2.41836i
$319$	7.51619			0.420826
$320$	19.3039	−	15.2862i	1.07912	−	0.854524i
$321$	2.60441			0.145364
$322$			0.0969540i			0.00540304i
$323$		−	6.68947i		−	0.372212i
$324$	25.4748			1.41527
$325$	9.96591	+	2.34679i	0.552809	+	0.130177i
$326$	46.3715			2.56828
$327$			33.4886i			1.85193i
$328$			3.84167i			0.212121i
$329$	−1.54957			−0.0854307
$330$	7.01851	−	5.55775i	0.386356	−	0.305944i
$331$	1.86485			0.102501			0.0512506	−	0.998686i	$-0.483679\pi$
$331$	1.86485			0.102501			0.0512506	+	0.998686i	$0.483679\pi$
$332$		−	5.14586i		−	0.282416i
$333$		−	4.63727i		−	0.254121i
$334$	−36.6168			−2.00358
$335$	−5.81002	−	7.33709i	−0.317435	−	0.400868i
$336$	−0.654583			−0.0357104
$337$		−	6.11130i		−	0.332904i	−0.986050	−	0.166452i	$-0.946769\pi$
$337$		−	6.11130i		−	0.332904i	0.986050	−	0.166452i	$-0.0532310\pi$
$338$			18.5442i			1.00867i
$339$	34.1886			1.85687
$340$	−22.5994	−	28.5393i	−1.22563	−	1.54776i
$341$	−0.564887			−0.0305903
$342$			1.29579i			0.0700684i
$343$		−	1.63525i		−	0.0882950i
$344$	2.46319			0.132806
$345$	−1.31270	+	1.03948i	−0.0706731	+	0.0559639i
$346$	−28.4660			−1.53034
$347$		−	17.4672i		−	0.937688i	−0.883281	−	0.468844i	$-0.844671\pi$
$347$		−	17.4672i		−	0.937688i	0.883281	−	0.468844i	$-0.155329\pi$
$348$			34.7812i			1.86447i
$349$	−13.3892			−0.716709			−0.358355	−	0.933586i	$-0.616662\pi$
$349$	−13.3892			−0.716709			−0.358355	+	0.933586i	$0.616662\pi$
$350$	0.282145	−	1.19816i	0.0150813	−	0.0640442i
$351$	−9.28455			−0.495573
$352$			8.02647i			0.427812i
$353$		−	6.13709i		−	0.326644i	−0.986573	−	0.163322i	$-0.947779\pi$
$353$		−	6.13709i		−	0.326644i	0.986573	−	0.163322i	$-0.0522210\pi$
$354$	−43.0676			−2.28901
$355$	2.77821	−	2.19998i	0.147452	−	0.116763i
$356$	−25.8711			−1.37117
$357$			1.48713i			0.0787072i
$358$			28.0994i			1.48510i
$359$	25.2627			1.33332			0.666658	−	0.745364i	$-0.267724\pi$
$359$	25.2627			1.33332			0.666658	+	0.745364i	$0.267724\pi$
$360$	0.780140	+	0.985187i	0.0411170	+	0.0519239i
$361$	1.00000			0.0526316
$362$		−	28.9162i		−	1.51980i
$363$			1.90142i			0.0997986i
$364$	−0.582659			−0.0305396
$365$	0.685751	+	0.865989i	0.0358938	+	0.0453279i
$366$	30.0787			1.57224
$367$			0.742688i			0.0387680i	0.999812	+	0.0193840i	$0.00617051\pi$
$367$			0.742688i			0.0387680i	−0.999812	+	0.0193840i	$0.993829\pi$
$368$		−	1.15961i		−	0.0604486i
$369$	2.58874			0.134764
$370$	−27.8149	+	22.0258i	−1.44603	+	1.14507i
$371$	1.25937			0.0653831
$372$		−	2.61402i		−	0.135530i
$373$			27.0435i			1.40026i	0.714017	+	0.700128i	$0.246874\pi$
$373$			27.0435i			1.40026i	−0.714017	+	0.700128i	$0.753126\pi$
$374$	14.0856			0.728349
$375$	19.2473	−	9.02588i	0.993925	−	0.466094i
$376$	−12.1036			−0.624198
$377$		−	15.3909i		−	0.792672i
$378$			1.11624i			0.0574132i
$379$	27.4829			1.41170			0.705851	−	0.708361i	$-0.250565\pi$
$379$	27.4829			1.41170			0.705851	+	0.708361i	$0.250565\pi$
$380$	4.26631	−	3.37836i	0.218857	−	0.173306i
$381$	30.1698			1.54565
$382$		−	51.0349i		−	2.61118i
$383$			24.1700i			1.23503i	0.786560	+	0.617514i	$0.211860\pi$
$383$			24.1700i			1.23503i	−0.786560	+	0.617514i	$0.788140\pi$
$384$	−13.5649			−0.692233
$385$	0.162299	+	0.204957i	0.00827154	+	0.0104456i
$386$	21.2075			1.07943
$387$		−	1.65984i		−	0.0843743i
$388$		−	25.1456i		−	1.27657i
$389$	14.0544			0.712588			0.356294	−	0.934374i	$-0.384040\pi$
$389$	14.0544			0.712588			0.356294	+	0.934374i	$0.384040\pi$
$390$	−11.3806	−	14.3718i	−0.576279	−	0.727744i
$391$	−2.63448			−0.133231
$392$		−	6.38018i		−	0.322248i
$393$		−	6.00397i		−	0.302860i
$394$	−26.4195			−1.33099
$395$	23.3606	−	18.4986i	1.17540	−	0.930764i
$396$	−1.49769			−0.0752615
$397$			4.21882i			0.211736i	0.994380	+	0.105868i	$0.0337621\pi$
$397$			4.21882i			0.211736i	−0.994380	+	0.105868i	$0.966238\pi$
$398$		−	19.7386i		−	0.989409i
$399$	−0.222309			−0.0111294
$400$	−3.37455	+	14.3304i	−0.168728	+	0.716520i
$401$	−5.80128			−0.289702			−0.144851	−	0.989453i	$-0.546270\pi$
$401$	−5.80128			−0.289702			−0.144851	+	0.989453i	$0.546270\pi$
$402$			16.7572i			0.835774i
$403$			1.15672i			0.0576203i
$404$	−18.3142			−0.911166
$405$	−18.3495	+	14.5304i	−0.911796	+	0.722024i
$406$	−1.85038			−0.0918328
$407$		−	7.53547i		−	0.373520i
$408$			11.6159i			0.575073i
$409$	31.7764			1.57124			0.785620	−	0.618709i	$-0.212344\pi$
$409$	31.7764			1.57124			0.785620	+	0.618709i	$0.212344\pi$
$410$	−12.2958	−	15.5276i	−0.607247	−	0.766852i
$411$	16.6467			0.821120
$412$		−	11.0019i		−	0.542027i
$413$		−	1.25767i		−	0.0618861i
$414$	0.510315			0.0250806
$415$	2.93513	+	3.70658i	0.144080	+	0.181949i
$416$	16.4358			0.805832
$417$			12.9950i			0.636366i
$418$			2.10564i			0.102990i
$419$	9.72052			0.474879			0.237439	−	0.971402i	$-0.423692\pi$
$419$	9.72052			0.474879			0.237439	+	0.971402i	$0.423692\pi$
$420$	−0.948439	+	0.751040i	−0.0462791	+	0.0366470i
$421$	33.0149			1.60905			0.804523	−	0.593922i	$-0.202421\pi$
$421$	33.0149			1.60905			0.804523	+	0.593922i	$0.202421\pi$
$422$			7.34811i			0.357700i
$423$			8.15613i			0.396565i
$424$	9.83687			0.477721
$425$	32.5568	+	7.66655i	1.57924	+	0.371882i
$426$	−6.34516			−0.307424
$427$			0.878370i			0.0425073i
$428$			3.33349i			0.161131i
$429$	3.89353			0.187982
$430$	−9.95592	+	7.88379i	−0.480117	+	0.380190i
$431$	−12.6973			−0.611607			−0.305803	−	0.952095i	$-0.598925\pi$
$431$	−12.6973			−0.611607			−0.305803	+	0.952095i	$0.598925\pi$
$432$		−	13.3507i		−	0.642334i
$433$		−	32.1290i		−	1.54402i	−0.635611	−	0.772009i	$-0.719252\pi$
$433$		−	32.1290i		−	1.54402i	0.635611	−	0.772009i	$-0.280748\pi$
$434$	0.139067			0.00667544
$435$	−19.8387	−	25.0530i	−0.951192	−	1.20120i
$436$	−42.8636			−2.05279
$437$		−	0.393825i		−	0.0188392i
$438$		−	1.97784i		−	0.0945046i
$439$	−0.750055			−0.0357982			−0.0178991	−	0.999840i	$-0.505698\pi$
$439$	−0.750055			−0.0357982			−0.0178991	+	0.999840i	$0.505698\pi$
$440$	1.26771	+	1.60091i	0.0604358	+	0.0763204i
$441$	−4.29933			−0.204730
$442$		−	28.8431i		−	1.37192i
$443$		−	32.5195i		−	1.54505i	−0.634986	−	0.772524i	$-0.718994\pi$
$443$		−	32.5195i		−	1.54505i	0.634986	−	0.772524i	$-0.281006\pi$
$444$	34.8704			1.65488
$445$	18.6350	−	14.7565i	0.883386	−	0.699526i
$446$	48.9125			2.31608
$447$		−	26.3243i		−	1.24510i
$448$		−	1.28748i		−	0.0608278i
$449$	−20.3546			−0.960591			−0.480295	−	0.877107i	$-0.659471\pi$
$449$	−20.3546			−0.960591			−0.480295	+	0.877107i	$0.659471\pi$
$450$	−6.30647	−	1.48506i	−0.297290	−	0.0700064i
$451$	4.20665			0.198083
$452$			43.7596i			2.05828i
$453$			22.2383i			1.04485i
$454$	6.69153			0.314049
$455$	0.419691	−	0.332340i	0.0196754	−	0.0155804i
$456$	−1.73645			−0.0813165
$457$			34.4250i			1.61034i	0.593047	+	0.805168i	$0.297925\pi$
$457$			34.4250i			1.61034i	−0.593047	+	0.805168i	$0.702075\pi$
$458$			40.2163i			1.87918i
$459$	−30.3310			−1.41573
$460$	−1.33048	−	1.68018i	−0.0620340	−	0.0783387i
$461$	−4.43693			−0.206648			−0.103324	−	0.994648i	$-0.532948\pi$
$461$	−4.43693			−0.206648			−0.103324	+	0.994648i	$0.532948\pi$
$462$		−	0.468102i		−	0.0217781i
$463$			6.72718i			0.312638i	0.987707	+	0.156319i	$0.0499629\pi$
$463$			6.72718i			0.312638i	−0.987707	+	0.156319i	$0.950037\pi$
$464$	22.1312			1.02742
$465$	1.49100	+	1.88288i	0.0691433	+	0.0873165i
$466$	−37.8702			−1.75430
$467$			9.98326i			0.461970i	0.972957	+	0.230985i	$0.0741949\pi$
$467$			9.98326i			0.461970i	−0.972957	+	0.230985i	$0.925805\pi$
$468$			3.06681i			0.141763i
$469$	−0.489350			−0.0225961
$470$	48.9215	−	38.7395i	2.25658	−	1.78692i
$471$	−15.2080			−0.700746
$472$		−	9.82364i		−	0.452170i
$473$		−	2.69721i		−	0.124018i
$474$	−53.3534			−2.45060
$475$	−1.14606	+	4.86688i	−0.0525850	+	0.223308i
$476$	−1.90344			−0.0872441
$477$		−	6.62865i		−	0.303505i
$478$			12.0680i			0.551977i
$479$	−30.2381			−1.38161			−0.690806	−	0.723040i	$-0.742744\pi$
$479$	−30.2381			−1.38161			−0.690806	+	0.723040i	$0.742744\pi$
$480$	26.7538	−	21.1855i	1.22114	−	0.966983i
$481$	−15.4304			−0.703565
$482$		−	41.1367i		−	1.87372i
$483$			0.0875507i			0.00398370i
$484$	−2.43371			−0.110623
$485$	14.3427	+	18.1124i	0.651266	+	0.822441i
$486$	13.2668			0.601796
$487$		−	10.4627i		−	0.474108i	−0.971496	−	0.237054i	$-0.923818\pi$
$487$		−	10.4627i		−	0.474108i	0.971496	−	0.237054i	$-0.0761818\pi$
$488$			6.86091i			0.310579i
$489$	41.8740			1.89361
$490$	20.4207	+	25.7879i	0.922512	+	1.16498i
$491$	−31.0301			−1.40037			−0.700186	−	0.713961i	$-0.746899\pi$
$491$	−31.0301			−1.40037			−0.700186	+	0.713961i	$0.746899\pi$
$492$			19.4663i			0.877608i
$493$		−	50.2793i		−	2.26447i
$494$	4.31171			0.193993
$495$	1.07879	−	0.854258i	0.0484878	−	0.0383960i
$496$	−1.66330			−0.0746842
$497$		−	0.185294i		−	0.00831155i
$498$		−	8.46546i		−	0.379347i
$499$	−19.5675			−0.875963			−0.437982	−	0.898984i	$-0.644307\pi$
$499$	−19.5675			−0.875963			−0.437982	+	0.898984i	$0.644307\pi$
$500$	11.5526	+	24.6354i	0.516649	+	1.10173i
$501$	−33.0654			−1.47725
$502$		−	14.6237i		−	0.652686i
$503$		−	4.55406i		−	0.203056i	−0.994833	−	0.101528i	$-0.967627\pi$
$503$		−	4.55406i		−	0.203056i	0.994833	−	0.101528i	$-0.0323731\pi$
$504$	0.0657075			0.00292684
$505$	13.1918	−	10.4462i	0.587026	−	0.464848i
$506$	0.829252			0.0368647
$507$			16.7456i			0.743700i
$508$			38.6157i			1.71330i
$509$	−1.79753			−0.0796743			−0.0398371	−	0.999206i	$-0.512684\pi$
$509$	−1.79753			−0.0796743			−0.0398371	+	0.999206i	$0.512684\pi$
$510$	−37.1783	−	46.9501i	−1.64628	−	2.07898i
$511$	0.0577575			0.00255504
$512$			29.0117i			1.28215i
$513$		−	4.53414i		−	0.200187i
$514$	−43.9082			−1.93671
$515$	6.27534	+	7.92472i	0.276525	+	0.349205i
$516$	12.4813			0.549460
$517$			13.2536i			0.582891i
$518$			1.85513i			0.0815096i
$519$	−25.7052			−1.12833
$520$	3.27819	−	2.59590i	0.143758	−	0.113838i
$521$	29.9998			1.31432			0.657158	−	0.753753i	$-0.271758\pi$
$521$	29.9998			1.31432			0.657158	+	0.753753i	$0.271758\pi$
$522$			9.73943i			0.426283i
$523$		−	9.53637i		−	0.416997i	−0.978023	−	0.208498i	$-0.933142\pi$
$523$		−	9.53637i		−	0.416997i	0.978023	−	0.208498i	$-0.0668576\pi$
$524$	7.68475			0.335710
$525$	0.254780	−	1.08195i	0.0111195	−	0.0472202i
$526$	−60.8345			−2.65251
$527$			3.77879i			0.164607i
$528$			5.59868i			0.243651i
$529$	22.8449			0.993257
$530$	−39.7595	+	31.4843i	−1.72704	+	1.36759i
$531$	−6.61973			−0.287272
$532$		−	0.284543i		−	0.0123365i
$533$		−	8.61396i		−	0.373112i
$534$	−42.5606			−1.84178
$535$	−1.90138	−	2.40112i	−0.0822037	−	0.103810i
$536$	−3.82230			−0.165098
$537$			25.3741i			1.09497i
$538$		−	13.1891i		−	0.568621i
$539$	−6.98633			−0.300923
$540$	−15.3180	−	19.3440i	−0.659180	−	0.832435i
$541$	5.80247			0.249467			0.124734	−	0.992190i	$-0.460192\pi$
$541$	5.80247			0.249467			0.124734	+	0.992190i	$0.460192\pi$
$542$			51.9151i			2.22994i
$543$		−	26.1117i		−	1.12056i
$544$	53.6928			2.30206
$545$	30.8747	−	24.4488i	1.32253	−	1.04727i
$546$	−0.958533			−0.0410214
$547$			21.9221i			0.937322i	0.883378	+	0.468661i	$0.155263\pi$
$547$			21.9221i			0.937322i	−0.883378	+	0.468661i	$0.844737\pi$
$548$			21.3068i			0.910183i
$549$	4.62327			0.197317
$550$	−10.2479	−	2.41319i	−0.436972	−	0.102899i
$551$	7.51619			0.320201
$552$			0.683855i			0.0291068i
$553$		−	1.55805i		−	0.0662548i
$554$	17.0569			0.724679
$555$	−25.1172	+	19.8896i	−1.06617	+	0.844266i
$556$	−16.6328			−0.705389
$557$			25.6558i			1.08707i	0.839386	+	0.543535i	$0.182915\pi$
$557$			25.6558i			1.08707i	−0.839386	+	0.543535i	$0.817085\pi$
$558$		−	0.731976i		−	0.0309870i
$559$	−5.52307			−0.233601
$560$	0.477886	+	0.603491i	0.0201944	+	0.0255021i
$561$	12.7195			0.537017
$562$			6.93292i			0.292448i
$563$			1.30126i			0.0548414i	0.999624	+	0.0274207i	$0.00872938\pi$
$563$			1.30126i			0.0548414i	−0.999624	+	0.0274207i	$0.991271\pi$
$564$	−61.3309			−2.58250
$565$	−24.9598	−	31.5201i	−1.05007	−	1.32606i
$566$	41.1931			1.73148
$567$			1.22383i			0.0513960i
$568$		−	1.44732i		−	0.0607282i
$569$	−13.6728			−0.573194			−0.286597	−	0.958051i	$-0.592524\pi$
$569$	−13.6728			−0.573194			−0.286597	+	0.958051i	$0.592524\pi$
$570$	7.01851	−	5.55775i	0.293973	−	0.232788i
$571$	2.19690			0.0919372			0.0459686	−	0.998943i	$-0.485363\pi$
$571$	2.19690			0.0919372			0.0459686	+	0.998943i	$0.485363\pi$
$572$			4.98351i			0.208371i
$573$		−	46.0852i		−	1.92524i
$574$	−1.03562			−0.0432259
$575$	1.91670	+	0.451348i	0.0799318	+	0.0188225i
$576$	−6.77663			−0.282359
$577$		−	43.7434i		−	1.82106i	−0.413443	−	0.910530i	$-0.635674\pi$
$577$		−	43.7434i		−	1.82106i	0.413443	−	0.910530i	$-0.364326\pi$
$578$		−	58.4292i		−	2.43034i
$579$	19.1506			0.795873
$580$	32.0664	−	25.3924i	1.33149	−	1.05436i
$581$	0.247212			0.0102561
$582$		−	41.3670i		−	1.71472i
$583$		−	10.7714i		−	0.446107i
$584$	0.451141			0.0186684
$585$	−1.74926	−	2.20903i	−0.0723231	−	0.0913321i
$586$	−27.4560			−1.13420
$587$		−	3.87706i		−	0.160023i	−0.996794	−	0.0800116i	$-0.974504\pi$
$587$		−	3.87706i		−	0.160023i	0.996794	−	0.0800116i	$-0.0254957\pi$
$588$		−	32.3293i		−	1.33324i
$589$	−0.564887			−0.0232758
$590$	31.4420	+	39.7060i	1.29445	+	1.63467i
$591$	−23.8571			−0.981350
$592$		−	22.1880i		−	0.911922i
$593$		−	16.9214i		−	0.694880i	−0.937702	−	0.347440i	$-0.887051\pi$
$593$		−	16.9214i		−	0.694880i	0.937702	−	0.347440i	$-0.112949\pi$
$594$	9.54725			0.391729
$595$	1.37105	−	1.08570i	0.0562077	−	0.0445092i
$596$	33.6937			1.38015
$597$		−	17.8243i		−	0.729498i
$598$		−	1.69806i		−	0.0694388i
$599$	−26.4428			−1.08042			−0.540211	−	0.841529i	$-0.681656\pi$
$599$	−26.4428			−1.08042			−0.540211	+	0.841529i	$0.681656\pi$
$600$	1.99008	−	8.45108i	0.0812446	−	0.345014i
$601$	−3.54017			−0.144406			−0.0722032	−	0.997390i	$-0.523003\pi$
$601$	−3.54017			−0.144406			−0.0722032	+	0.997390i	$0.523003\pi$
$602$			0.664014i			0.0270632i
$603$			2.57568i			0.104890i
$604$	−28.4638			−1.15818
$605$	1.75301	−	1.38815i	0.0712698	−	0.0564364i
$606$	−30.1287			−1.22390
$607$		−	13.1372i		−	0.533222i	−0.963804	−	0.266611i	$-0.914096\pi$
$607$		−	13.1372i		−	0.533222i	0.963804	−	0.266611i	$-0.0859039\pi$
$608$			8.02647i			0.325516i
$609$	−1.67092			−0.0677090
$610$	−21.9593	−	27.7310i	−0.889107	−	1.12279i
$611$	27.1393			1.09794
$612$			10.0187i			0.404983i
$613$			1.24598i			0.0503246i	0.999683	+	0.0251623i	$0.00801026\pi$
$613$			1.24598i			0.0503246i	−0.999683	+	0.0251623i	$0.991990\pi$
$614$	65.5060			2.64361
$615$	−11.1033	−	14.0216i	−0.447728	−	0.565406i
$616$	0.106773			0.00430202
$617$		−	21.2834i		−	0.856838i	−0.903580	−	0.428419i	$-0.859071\pi$
$617$		−	21.2834i		−	0.856838i	0.903580	−	0.428419i	$-0.140929\pi$
$618$		−	18.0993i		−	0.728060i
$619$	8.68908			0.349243			0.174622	−	0.984636i	$-0.444130\pi$
$619$	8.68908			0.349243			0.174622	+	0.984636i	$0.444130\pi$
$620$	−2.40998	+	1.90839i	−0.0967873	+	0.0766429i
$621$	−1.78565			−0.0716559
$622$			48.6243i			1.94966i
$623$		−	1.24287i		−	0.0497946i
$624$	11.4644			0.458944
$625$	−22.3731	−	11.1555i	−0.894923	−	0.446221i
$626$	16.9608			0.677890
$627$			1.90142i			0.0759353i
$628$		−	19.4654i		−	0.776752i
$629$	−50.4083			−2.00991
$630$	−0.265582	+	0.210306i	−0.0105810	+	0.00837880i
$631$	−45.0264			−1.79247			−0.896236	−	0.443578i	$-0.853709\pi$
$631$	−45.0264			−1.79247			−0.896236	+	0.443578i	$0.853709\pi$
$632$		−	12.1698i		−	0.484090i
$633$			6.63544i			0.263735i
$634$	−19.3619			−0.768959
$635$	−22.0258	−	27.8150i	−0.874069	−	1.10380i
$636$	49.8448			1.97648
$637$			14.3059i			0.566821i
$638$			15.8264i			0.626573i
$639$	−0.975288			−0.0385818
$640$	9.90324	+	12.5061i	0.391460	+	0.494349i
$641$	−33.8126			−1.33552			−0.667759	−	0.744378i	$-0.732746\pi$
$641$	−33.8126			−1.33552			−0.667759	+	0.744378i	$0.732746\pi$
$642$			5.48393i			0.216434i
$643$			21.7079i			0.856078i	0.903760	+	0.428039i	$0.140795\pi$
$643$			21.7079i			0.856078i	−0.903760	+	0.428039i	$0.859205\pi$
$644$	−0.112060			−0.00441579
$645$	−8.99032	+	7.11916i	−0.353994	+	0.280317i
$646$	14.0856			0.554190
$647$			6.48463i			0.254937i	0.991843	+	0.127469i	$0.0406852\pi$
$647$			6.48463i			0.254937i	−0.991843	+	0.127469i	$0.959315\pi$
$648$			9.55928i			0.375524i
$649$	−10.7569			−0.422247
$650$	−4.94150	+	20.9846i	−0.193822	+	0.823084i
$651$	0.125579			0.00492185
$652$			53.5965i			2.09900i
$653$		−	9.95275i		−	0.389481i	−0.980855	−	0.194741i	$-0.937613\pi$
$653$		−	9.95275i		−	0.389481i	0.980855	−	0.194741i	$-0.0623865\pi$
$654$	−70.5149			−2.75735
$655$	−5.53534	+	4.38327i	−0.216284	+	0.171268i
$656$	12.3864			0.483607
$657$		−	0.304005i		−	0.0118604i
$658$		−	3.26284i		−	0.127199i
$659$	8.97665			0.349681			0.174840	−	0.984597i	$-0.444059\pi$
$659$	8.97665			0.349681			0.174840	+	0.984597i	$0.444059\pi$
$660$	6.42368	+	8.11204i	0.250041	+	0.315761i
$661$	10.9663			0.426540			0.213270	−	0.976993i	$-0.431589\pi$
$661$	10.9663			0.426540			0.213270	+	0.976993i	$0.431589\pi$
$662$			3.92669i			0.152615i
$663$		−	26.0457i		−	1.01153i
$664$	1.93096			0.0749357
$665$	0.162299	+	0.204957i	0.00629370	+	0.00794789i
$666$	9.76441			0.378363
$667$		−	2.96006i		−	0.114614i
$668$		−	42.3219i		−	1.63748i
$669$	44.1687			1.70766
$670$	15.4493	−	12.2338i	0.596857	−	0.472633i
$671$	7.51273			0.290026
$672$		−	1.78436i		−	0.0688330i
$673$		−	16.1788i		−	0.623648i	−0.950140	−	0.311824i	$-0.899060\pi$
$673$		−	16.1788i		−	0.623648i	0.950140	−	0.311824i	$-0.100940\pi$
$674$	12.8682			0.495664
$675$	22.0671	+	5.19641i	0.849364	+	0.200010i
$676$	−21.4335			−0.824366
$677$			23.8569i			0.916893i	0.888722	+	0.458447i	$0.151594\pi$
$677$			23.8569i			0.916893i	−0.888722	+	0.458447i	$0.848406\pi$
$678$			71.9889i			2.76472i
$679$	1.20801			0.0463593
$680$	10.7092	−	8.48032i	0.410681	−	0.325205i
$681$	6.04253			0.231550
$682$		−	1.18945i		−	0.0455463i
$683$		−	6.10611i		−	0.233644i	−0.993153	−	0.116822i	$-0.962729\pi$
$683$		−	6.10611i		−	0.233644i	0.993153	−	0.116822i	$-0.0372707\pi$
$684$	−1.49769			−0.0572654
$685$	−12.1531	−	15.3474i	−0.464346	−	0.586392i
$686$	3.44324			0.131463
$687$			36.3158i			1.38554i
$688$		−	7.94185i		−	0.302780i
$689$	−22.0567			−0.840292
$690$	−2.18878	−	2.76406i	−0.0833253	−	0.105226i
$691$	−17.3732			−0.660909			−0.330454	−	0.943822i	$-0.607202\pi$
$691$	−17.3732			−0.660909			−0.330454	+	0.943822i	$0.607202\pi$
$692$		−	32.9013i		−	1.25072i
$693$		−	0.0719500i		−	0.00273316i
$694$	36.7796			1.39613
$695$	11.9807	−	9.48713i	0.454453	−	0.359867i
$696$	−13.0515			−0.494715
$697$		−	28.1402i		−	1.06589i
$698$		−	28.1929i		−	1.06712i
$699$	−34.1973			−1.29346
$700$	1.38484	+	0.326105i	0.0523420	+	0.0123256i
$701$	−14.3275			−0.541141			−0.270571	−	0.962700i	$-0.587212\pi$
$701$	−14.3275			−0.541141			−0.270571	+	0.962700i	$0.587212\pi$
$702$		−	19.5499i		−	0.737864i
$703$		−	7.53547i		−	0.284206i
$704$	−11.0119			−0.415026
$705$	44.1767	−	34.9822i	1.66379	−	1.31751i
$706$	12.9225			0.486344
$707$		−	0.879830i		−	0.0330894i
$708$		−	49.7778i		−	1.87076i
$709$	18.9182			0.710488			0.355244	−	0.934774i	$-0.384398\pi$
$709$	18.9182			0.710488			0.355244	+	0.934774i	$0.384398\pi$
$710$	4.63236	+	5.84990i	0.173849	+	0.219543i
$711$	−8.20073			−0.307551
$712$		−	9.70801i		−	0.363823i
$713$			0.222466i			0.00833143i
$714$	−3.13135			−0.117188
$715$	−2.84252	−	3.58963i	−0.106304	−	0.134245i
$716$	−32.4775			−1.21374
$717$			10.8975i			0.406976i
$718$			53.1941i			1.98519i
$719$	45.9520			1.71372			0.856860	−	0.515550i	$-0.172412\pi$
$719$	45.9520			1.71372			0.856860	+	0.515550i	$0.172412\pi$
$720$	3.17646	−	2.51534i	0.118380	−	0.0937412i
$721$	0.528542			0.0196839
$722$			2.10564i			0.0783637i
$723$		−	37.1469i		−	1.38151i
$724$	33.4215			1.24210
$725$	−8.61404	+	36.5804i	−0.319917	+	1.35856i
$726$	−4.00370			−0.148591
$727$			28.4946i			1.05681i	0.848993	+	0.528404i	$0.177209\pi$
$727$			28.4946i			1.05681i	−0.848993	+	0.528404i	$0.822791\pi$
$728$		−	0.218640i		−	0.00810333i
$729$	−19.4223			−0.719344
$730$	−1.82346	+	1.44394i	−0.0674893	+	0.0534427i
$731$	−18.0429			−0.667339
$732$			34.7652i			1.28496i
$733$		−	30.7740i		−	1.13666i	−0.822799	−	0.568332i	$-0.807589\pi$
$733$		−	30.7740i		−	1.13666i	0.822799	−	0.568332i	$-0.192411\pi$
$734$	−1.56383			−0.0577221
$735$	18.4401	+	23.2868i	0.680175	+	0.858948i
$736$	3.16102			0.116517
$737$			4.18544i			0.154172i
$738$			5.45095i			0.200652i
$739$	−9.29374			−0.341876			−0.170938	−	0.985282i	$-0.554680\pi$
$739$	−9.29374			−0.341876			−0.170938	+	0.985282i	$0.554680\pi$
$740$	−25.4576	−	32.1487i	−0.935839	−	1.18181i
$741$	3.89353			0.143033
$742$			2.65177i			0.0973496i
$743$		−	6.83474i		−	0.250742i	−0.992110	−	0.125371i	$-0.959988\pi$
$743$		−	6.83474i		−	0.250742i	0.992110	−	0.125371i	$-0.0400122\pi$
$744$	0.980896			0.0359614
$745$	−24.2696	+	19.2184i	−0.889170	+	0.704107i
$746$	−56.9437			−2.08486
$747$		−	1.30119i		−	0.0476081i
$748$			16.2802i			0.595264i
$749$	−0.160144			−0.00585153
$750$	19.0052	+	40.5278i	0.693973	+	1.47987i
$751$	−36.5000			−1.33190			−0.665952	−	0.745994i	$-0.731975\pi$
$751$	−36.5000			−1.33190			−0.665952	+	0.745994i	$0.731975\pi$
$752$			39.0248i			1.42309i
$753$		−	13.2054i		−	0.481230i
$754$	32.4077			1.18022
$755$	20.5026	−	16.2354i	0.746165	−	0.590865i
$756$	−1.29016			−0.0469226
$757$			9.95333i			0.361760i	0.983505	+	0.180880i	$0.0578945\pi$
$757$			9.95333i			0.361760i	−0.983505	+	0.180880i	$0.942105\pi$
$758$			57.8690i			2.10190i
$759$	0.748825			0.0271806
$760$	1.26771	+	1.60091i	0.0459848	+	0.0580711i
$761$	−25.2847			−0.916569			−0.458284	−	0.888806i	$-0.651536\pi$
$761$	−25.2847			−0.916569			−0.458284	+	0.888806i	$0.651536\pi$
$762$			63.5268i			2.30133i
$763$		−	2.05920i		−	0.0745481i
$764$	58.9865			2.13406
$765$	−5.71453	−	7.21650i	−0.206609	−	0.260913i
$766$	−50.8932			−1.83885
$767$			22.0270i			0.795349i
$768$			13.3136i			0.480413i
$769$	42.2585			1.52388			0.761940	−	0.647647i	$-0.224247\pi$
$769$	42.2585			1.52388			0.761940	+	0.647647i	$0.224247\pi$
$770$	−0.431565	+	0.341744i	−0.0155525	+	0.0123156i
$771$	−39.6497			−1.42795
$772$			24.5117i			0.882197i
$773$		−	10.2118i		−	0.367293i	−0.982992	−	0.183647i	$-0.941210\pi$
$773$		−	10.2118i		−	0.367293i	0.982992	−	0.183647i	$-0.0587902\pi$
$774$	3.49502			0.125626
$775$	0.647396	−	2.74924i	0.0232552	−	0.0987555i
$776$	9.43574			0.338723
$777$			1.67520i			0.0600976i
$778$			29.5936i			1.06098i
$779$	4.20665			0.150719
$780$	16.6110	−	13.1538i	0.594770	−	0.470980i
$781$	−1.58482			−0.0567095
$782$		−	5.54725i		−	0.198369i
$783$		−	34.0795i		−	1.21790i
$784$	−20.5711			−0.734681
$785$	11.1028	+	14.0209i	0.396274	+	0.500428i
$786$	12.6422			0.450932
$787$			40.0515i			1.42768i	0.700308	+	0.713841i	$0.253046\pi$
$787$			40.0515i			1.42768i	−0.700308	+	0.713841i	$0.746954\pi$
$788$		−	30.5358i		−	1.08779i
$789$	−54.9343			−1.95571
$790$	38.9513	+	49.1890i	1.38582	+	1.75007i
$791$	−2.10225			−0.0747473
$792$		−	0.561999i		−	0.0199698i
$793$		−	15.3838i		−	0.546296i
$794$	−8.88330			−0.315256
$795$	−35.9033	+	28.4307i	−1.27336	+	1.00833i
$796$	22.8141			0.808623
$797$		−	7.72747i		−	0.273721i	−0.990590	−	0.136861i	$-0.956299\pi$
$797$		−	7.72747i		−	0.273721i	0.990590	−	0.136861i	$-0.0437012\pi$
$798$		−	0.468102i		−	0.0165706i
$799$	88.6592			3.13654
$800$	−39.0639	−	9.19885i	−1.38112	−	0.325228i
$801$	−6.54182			−0.231144
$802$		−	12.2154i		−	0.431340i
$803$		−	0.494002i		−	0.0174330i
$804$	−19.3681			−0.683061
$805$	0.0807171	−	0.0639174i	0.00284490	−	0.00225279i
$806$	−2.43563			−0.0857915
$807$		−	11.9099i		−	0.419248i
$808$		−	6.87231i		−	0.241767i
$809$	8.60128			0.302405			0.151202	−	0.988503i	$-0.451685\pi$
$809$	8.60128			0.302405			0.151202	+	0.988503i	$0.451685\pi$
$810$	−30.5959	−	38.6375i	−1.07503	−	1.35758i
$811$	47.6882			1.67456			0.837280	−	0.546775i	$-0.184145\pi$
$811$	47.6882			1.67456			0.837280	+	0.546775i	$0.184145\pi$
$812$		−	2.13868i		−	0.0750530i
$813$			46.8800i			1.64415i
$814$	15.8670			0.556137
$815$	−30.5706	−	38.6056i	−1.07084	−	1.35230i
$816$	37.4522			1.31109
$817$		−	2.69721i		−	0.0943633i
$818$			66.9095i			2.33944i
$819$	−0.147332			−0.00514820
$820$	17.9469	−	14.2116i	0.626732	−	0.496290i
$821$	15.0329			0.524651			0.262326	−	0.964979i	$-0.415511\pi$
$821$	15.0329			0.524651			0.262326	+	0.964979i	$0.415511\pi$
$822$			35.0519i			1.22257i
$823$			24.0929i			0.839825i	0.907564	+	0.419913i	$0.137939\pi$
$823$			24.0929i			0.839825i	−0.907564	+	0.419913i	$0.862061\pi$
$824$	4.12842			0.143820
$825$	−9.25398	−	2.17915i	−0.322182	−	0.0758681i
$826$	2.64821			0.0921429
$827$		−	29.5434i		−	1.02733i	−0.857992	−	0.513663i	$-0.828288\pi$
$827$		−	29.5434i		−	1.02733i	0.857992	−	0.513663i	$-0.171712\pi$
$828$			0.589825i			0.0204978i
$829$	−39.6130			−1.37581			−0.687907	−	0.725798i	$-0.741471\pi$
$829$	−39.6130			−1.37581			−0.687907	+	0.725798i	$0.741471\pi$
$830$	−7.80471	+	6.18031i	−0.270905	+	0.214522i
$831$	15.4026			0.534311
$832$			22.5490i			0.781748i
$833$			46.7348i			1.61927i
$834$	−27.3627			−0.947492
$835$	24.1398	+	30.4846i	0.835393	+	1.05496i
$836$	−2.43371			−0.0841716
$837$			2.56128i			0.0885306i
$838$			20.4679i			0.707052i
$839$	−39.6207			−1.36786			−0.683929	−	0.729549i	$-0.739730\pi$
$839$	−39.6207			−1.36786			−0.683929	+	0.729549i	$0.739730\pi$
$840$	−0.281824	−	0.355897i	−0.00972385	−	0.0122796i
$841$	27.4932			0.948041
$842$			69.5173i			2.39572i
$843$			6.26052i			0.215624i
$844$	−8.49300			−0.292341
$845$	15.4386	−	12.2254i	0.531104	−	0.420565i
$846$	−17.1739			−0.590449
$847$		−	0.116917i		−	0.00401733i
$848$		−	31.7162i		−	1.08914i
$849$	37.1979			1.27663
$850$	−16.1430	+	68.5529i	−0.553700	+	2.35135i
$851$	−2.96765			−0.101730
$852$		−	7.33378i		−	0.251251i
$853$		−	10.0735i		−	0.344911i	−0.985017	−	0.172455i	$-0.944830\pi$
$853$		−	10.0735i		−	0.344911i	0.985017	−	0.172455i	$-0.0551700\pi$
$854$	−1.84953			−0.0632896
$855$	1.07879	−	0.854258i	0.0368937	−	0.0292150i
$856$	−1.25088			−0.0427541
$857$		−	23.8751i		−	0.815558i	−0.913081	−	0.407779i	$-0.866303\pi$
$857$		−	23.8751i		−	0.815558i	0.913081	−	0.407779i	$-0.133697\pi$
$858$			8.19837i			0.279888i
$859$	−7.40798			−0.252757			−0.126378	−	0.991982i	$-0.540335\pi$
$859$	−7.40798			−0.252757			−0.126378	+	0.991982i	$0.540335\pi$
$860$	−9.11214	−	11.5071i	−0.310721	−	0.392389i
$861$	−0.935176			−0.0318707
$862$		−	26.7359i		−	0.910628i
$863$			39.1661i			1.33323i	0.745402	+	0.666615i	$0.232257\pi$
$863$			39.1661i			1.33323i	−0.745402	+	0.666615i	$0.767743\pi$
$864$	36.3931			1.23812
$865$	18.7664	+	23.6988i	0.638076	+	0.805785i
$866$	67.6519			2.29891
$867$		−	52.7624i		−	1.79190i
$868$			0.160735i			0.00545569i
$869$	−13.3260			−0.452055
$870$	52.7525	−	41.7731i	1.78848	−	1.41624i
$871$	8.57051			0.290401
$872$		−	16.0843i		−	0.544684i
$873$		−	6.35834i		−	0.215197i
$874$	0.829252			0.0280499
$875$	−1.18351	+	0.554998i	−0.0400098	+	0.0187623i
$876$	2.28600			0.0772366
$877$			29.9446i			1.01116i	0.862781	+	0.505578i	$0.168721\pi$
$877$			29.9446i			1.01116i	−0.862781	+	0.505578i	$0.831279\pi$
$878$		−	1.57934i		−	0.0533003i
$879$	−24.7932			−0.836252
$880$	5.16168	−	4.08738i	0.174000	−	0.137786i
$881$	48.7322			1.64183			0.820914	−	0.571051i	$-0.193464\pi$
$881$	48.7322			1.64183			0.820914	+	0.571051i	$0.193464\pi$
$882$		−	9.05283i		−	0.304825i
$883$			19.5753i			0.658760i	0.944197	+	0.329380i	$0.106840\pi$
$883$			19.5753i			0.658760i	−0.944197	+	0.329380i	$0.893160\pi$
$884$	33.3370			1.12124
$885$	28.3925	+	35.8550i	0.954404	+	1.20525i
$886$	68.4743			2.30044
$887$		−	30.3740i		−	1.01986i	−0.860216	−	0.509929i	$-0.829672\pi$
$887$		−	30.3740i		−	1.01986i	0.860216	−	0.509929i	$-0.170328\pi$
$888$			13.0849i			0.439102i
$889$	−1.85513			−0.0622191
$890$	31.0719	+	39.2386i	1.04153	+	1.31528i
$891$	10.4675			0.350673
$892$			56.5334i			1.89288i
$893$			13.2536i			0.443513i
$894$	55.4295			1.85384
$895$	23.3936	−	18.5247i	0.781961	−	0.619211i
$896$	0.834102			0.0278654
$897$		−	1.53337i		−	0.0511977i
$898$		−	42.8593i		−	1.43023i
$899$	−4.24580			−0.141605
$900$	1.71644	−	7.28906i	0.0572148	−	0.242969i
$901$	−72.0551			−2.40050
$902$			8.85768i			0.294929i
$903$			0.599613i			0.0199539i
$904$	−16.4206			−0.546140
$905$	−24.0736	+	19.0631i	−0.800232	+	0.633680i
$906$	−46.8259			−1.55569
$907$			56.0209i			1.86014i	0.367380	+	0.930071i	$0.380255\pi$
$907$			56.0209i			1.86014i	−0.367380	+	0.930071i	$0.619745\pi$
$908$			7.73411i			0.256665i
$909$	−4.63096			−0.153599
$910$	0.699788	+	0.883716i	0.0231978	+	0.0292949i
$911$	22.0274			0.729802			0.364901	−	0.931046i	$-0.381103\pi$
$911$	22.0274			0.729802			0.364901	+	0.931046i	$0.381103\pi$
$912$			5.59868i			0.185391i
$913$		−	2.11441i		−	0.0699768i
$914$	−72.4866			−2.39765
$915$	−19.8296	−	25.0414i	−0.655545	−	0.827844i
$916$	−46.4822			−1.53582
$917$			0.369182i			0.0121914i
$918$		−	63.8660i		−	2.10789i
$919$	21.0610			0.694738			0.347369	−	0.937729i	$-0.387075\pi$
$919$	21.0610			0.694738			0.347369	+	0.937729i	$0.387075\pi$
$920$	0.630478	−	0.499256i	0.0207862	−	0.0164600i
$921$	59.1527			1.94915
$922$		−	9.34257i		−	0.307681i
$923$			3.24524i			0.106819i
$924$	0.541036			0.0177988
$925$	36.6743	+	8.63613i	1.20584	+	0.283954i
$926$	−14.1650			−0.465491
$927$		−	2.78197i		−	0.0913718i
$928$			60.3285i			1.98038i
$929$	15.0905			0.495103			0.247551	−	0.968875i	$-0.420374\pi$
$929$	15.0905			0.495103			0.247551	+	0.968875i	$0.420374\pi$
$930$	−3.96466	+	3.13950i	−0.130006	+	0.102948i
$931$	−6.98633			−0.228968
$932$		−	43.7707i		−	1.43376i
$933$			43.9084i			1.43750i
$934$	−21.0211			−0.687832
$935$	−9.28600	−	11.7267i	−0.303685	−	0.383503i
$936$	−1.15080			−0.0376152
$937$		−	54.0837i		−	1.76684i	−0.468585	−	0.883418i	$-0.655236\pi$
$937$		−	54.0837i		−	1.76684i	0.468585	−	0.883418i	$-0.344764\pi$
$938$		−	1.03039i		−	0.0336436i
$939$	15.3158			0.499813
$940$	44.7753	+	56.5438i	1.46041	+	1.84426i
$941$	14.2518			0.464596			0.232298	−	0.972645i	$-0.425376\pi$
$941$	14.2518			0.464596			0.232298	+	0.972645i	$0.425376\pi$
$942$		−	32.0225i		−	1.04335i
$943$		−	1.65668i		−	0.0539490i
$944$	−31.6735			−1.03089
$945$	0.929304	−	0.735887i	0.0302302	−	0.0239384i
$946$	5.67934			0.184651
$947$		−	14.1563i		−	0.460018i	−0.973188	−	0.230009i	$-0.926124\pi$
$947$		−	14.1563i		−	0.460018i	0.973188	−	0.230009i	$-0.0738756\pi$
$948$		−	61.6662i		−	2.00283i
$949$	−1.01157			−0.0328369
$950$	−10.2479	−	2.41319i	−0.332485	−	0.0782944i
$951$	−17.4840			−0.566959
$952$		−	0.714257i		−	0.0231492i
$953$		−	40.7319i		−	1.31943i	−0.751514	−	0.659717i	$-0.770676\pi$
$953$		−	40.7319i		−	1.31943i	0.751514	−	0.659717i	$-0.229324\pi$
$954$	13.9575			0.451892
$955$	−42.4881	+	33.6451i	−1.37488	+	1.08873i
$956$	−13.9483			−0.451119
$957$			14.2914i			0.461976i
$958$		−	63.6704i		−	2.05710i
$959$	−1.02360			−0.0330537
$960$	29.0654	+	36.7048i	0.938083	+	1.18464i
$961$	−30.6809			−0.989707
$962$		−	32.4908i		−	1.04755i
$963$			0.842913i			0.0271625i
$964$	47.5460			1.53135
$965$	−13.9811	−	17.6559i	−0.450069	−	0.568362i
$966$	−0.184350			−0.00593137
$967$		−	48.0781i		−	1.54609i	−0.634352	−	0.773044i	$-0.718733\pi$
$967$		−	48.0781i		−	1.54609i	0.634352	−	0.773044i	$-0.281267\pi$
$968$		−	0.913237i		−	0.0293526i
$969$	12.7195			0.408608
$970$	−38.1381	+	30.2004i	−1.22454	+	0.969677i
$971$	20.5967			0.660980			0.330490	−	0.943809i	$-0.392786\pi$
$971$	20.5967			0.660980			0.330490	+	0.943809i	$0.392786\pi$
$972$			15.3339i			0.491835i
$973$		−	0.799055i		−	0.0256165i
$974$	22.0306			0.705905
$975$	−4.46224	+	18.9494i	−0.142906	+	0.606866i
$976$	22.1211			0.708078
$977$		−	31.5003i		−	1.00778i	−0.863767	−	0.503892i	$-0.831901\pi$
$977$		−	31.5003i		−	1.00778i	0.863767	−	0.503892i	$-0.168099\pi$
$978$			88.1716i			2.81942i
$979$	−10.6303			−0.339747
$980$	−29.8059	+	23.6024i	−0.952113	+	0.753950i
$981$	−10.8386			−0.346048
$982$		−	65.3383i		−	2.08503i
$983$			44.7716i			1.42799i	0.700149	+	0.713996i	$0.253117\pi$
$983$			44.7716i			1.42799i	−0.700149	+	0.713996i	$0.746883\pi$
$984$	−7.30462			−0.232863
$985$	17.4172	+	21.9950i	0.554957	+	0.700819i
$986$	105.870			3.37159
$987$		−	2.94639i		−	0.0937845i
$988$			4.98351i			0.158547i
$989$	−1.06223			−0.0337768
$990$	1.79876	+	2.27153i	0.0571683	+	0.0721940i
$991$	−51.7016			−1.64235			−0.821177	−	0.570673i	$-0.806682\pi$
$991$	−51.7016			−1.64235			−0.821177	+	0.570673i	$0.806682\pi$
$992$		−	4.53405i		−	0.143956i
$993$			3.54585i			0.112524i
$994$	0.390161			0.0123752
$995$	−16.4330	+	13.0128i	−0.520962	+	0.412534i
$996$	9.78444			0.310032
$997$		−	23.2094i		−	0.735049i	−0.930014	−	0.367524i	$-0.880205\pi$
$997$		−	23.2094i		−	0.735049i	0.930014	−	0.367524i	$-0.119795\pi$
$998$		−	41.2021i		−	1.30423i
$999$	−34.1669			−1.08099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.b.419.15	yes	16
5.2	odd	4		5225.2.a.z.1.2		16
5.3	odd	4		5225.2.a.z.1.15		16
5.4	even	2	inner	1045.2.b.b.419.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.b.419.2	✓	16	5.4	even	2	inner
1045.2.b.b.419.15	yes	16	1.1	even	1	trivial
5225.2.a.z.1.2		16	5.2	odd	4
5225.2.a.z.1.15		16	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)

\(f(q)\)	\(=\)	\(q+2.10564i q^{2} +1.90142i q^{3} -2.43371 q^{4} +(1.75301 - 1.38815i) q^{5} -4.00370 q^{6} -0.116917i q^{7} -0.913237i q^{8} -0.615392 q^{9} +O(q^{10})\)	\(q+2.10564i q^{2} +1.90142i q^{3} -2.43371 q^{4} +(1.75301 - 1.38815i) q^{5} -4.00370 q^{6} -0.116917i q^{7} -0.913237i q^{8} -0.615392 q^{9} +(2.92295 + 3.69120i) q^{10} -1.00000 q^{11} -4.62750i q^{12} +2.04770i q^{13} +0.246186 q^{14} +(2.63946 + 3.33320i) q^{15} -2.94447 q^{16} +6.68947i q^{17} -1.29579i q^{18} -1.00000 q^{19} +(-4.26631 + 3.37836i) q^{20} +0.222309 q^{21} -2.10564i q^{22} +0.393825i q^{23} +1.73645 q^{24} +(1.14606 - 4.86688i) q^{25} -4.31171 q^{26} +4.53414i q^{27} +0.284543i q^{28} -7.51619 q^{29} +(-7.01851 + 5.55775i) q^{30} +0.564887 q^{31} -8.02647i q^{32} -1.90142i q^{33} -14.0856 q^{34} +(-0.162299 - 0.204957i) q^{35} +1.49769 q^{36} +7.53547i q^{37} -2.10564i q^{38} -3.89353 q^{39} +(-1.26771 - 1.60091i) q^{40} -4.20665 q^{41} +0.468102i q^{42} +2.69721i q^{43} +2.43371 q^{44} +(-1.07879 + 0.854258i) q^{45} -0.829252 q^{46} -13.2536i q^{47} -5.59868i q^{48} +6.98633 q^{49} +(10.2479 + 2.41319i) q^{50} -12.7195 q^{51} -4.98351i q^{52} +10.7714i q^{53} -9.54725 q^{54} +(-1.75301 + 1.38815i) q^{55} -0.106773 q^{56} -1.90142i q^{57} -15.8264i q^{58} +10.7569 q^{59} +(-6.42368 - 8.11204i) q^{60} -7.51273 q^{61} +1.18945i q^{62} +0.0719500i q^{63} +11.0119 q^{64} +(2.84252 + 3.58963i) q^{65} +4.00370 q^{66} -4.18544i q^{67} -16.2802i q^{68} -0.748825 q^{69} +(0.431565 - 0.341744i) q^{70} +1.58482 q^{71} +0.561999i q^{72} +0.494002i q^{73} -15.8670 q^{74} +(9.25398 + 2.17915i) q^{75} +2.43371 q^{76} +0.116917i q^{77} -8.19837i q^{78} +13.3260 q^{79} +(-5.16168 + 4.08738i) q^{80} -10.4675 q^{81} -8.85768i q^{82} +2.11441i q^{83} -0.541036 q^{84} +(9.28600 + 11.7267i) q^{85} -5.67934 q^{86} -14.2914i q^{87} +0.913237i q^{88} +10.6303 q^{89} +(-1.79876 - 2.27153i) q^{90} +0.239412 q^{91} -0.958455i q^{92} +1.07409i q^{93} +27.9072 q^{94} +(-1.75301 + 1.38815i) q^{95} +15.2617 q^{96} +10.3322i q^{97} +14.7107i q^{98} +0.615392 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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