LMFDB - Embedded newform 177.5.b.a.119.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(119,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("177.119");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.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:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			119.18
Character	$\chi$	$=$	177.119
Dual form			177.5.b.a.119.61

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.04906i q^{2} +(-3.85410 - 8.13301i) q^{3} -9.49305 q^{4} +33.9624i q^{5} +(-41.0641 + 19.4596i) q^{6} +88.4932 q^{7} -32.8540i q^{8} +(-51.2919 + 62.6909i) q^{9} +O(q^{10})$	\(q-5.04906i q^{2} +(-3.85410 - 8.13301i) q^{3} -9.49305 q^{4} +33.9624i q^{5} +(-41.0641 + 19.4596i) q^{6} +88.4932 q^{7} -32.8540i q^{8} +(-51.2919 + 62.6909i) q^{9} +171.478 q^{10} -111.430i q^{11} +(36.5871 + 77.2071i) q^{12} +6.16984 q^{13} -446.808i q^{14} +(276.217 - 130.894i) q^{15} -317.771 q^{16} -288.710i q^{17} +(316.530 + 258.976i) q^{18} +261.438 q^{19} -322.407i q^{20} +(-341.062 - 719.717i) q^{21} -562.615 q^{22} -401.655i q^{23} +(-267.202 + 126.623i) q^{24} -528.444 q^{25} -31.1519i q^{26} +(707.550 + 175.541i) q^{27} -840.071 q^{28} -696.014i q^{29} +(-660.894 - 1394.64i) q^{30} +1230.65 q^{31} +1078.78i q^{32} +(-906.259 + 429.461i) q^{33} -1457.71 q^{34} +3005.44i q^{35} +(486.916 - 595.128i) q^{36} -1133.89 q^{37} -1320.02i q^{38} +(-23.7791 - 50.1794i) q^{39} +1115.80 q^{40} +1509.67i q^{41} +(-3633.90 + 1722.04i) q^{42} -1234.10 q^{43} +1057.81i q^{44} +(-2129.13 - 1741.99i) q^{45} -2027.98 q^{46} +148.995i q^{47} +(1224.72 + 2584.43i) q^{48} +5430.05 q^{49} +2668.15i q^{50} +(-2348.08 + 1112.72i) q^{51} -58.5706 q^{52} -4971.34i q^{53} +(886.317 - 3572.46i) q^{54} +3784.42 q^{55} -2907.36i q^{56} +(-1007.61 - 2126.28i) q^{57} -3514.22 q^{58} -453.188i q^{59} +(-2622.14 + 1242.59i) q^{60} +2566.52 q^{61} -6213.62i q^{62} +(-4538.98 + 5547.72i) q^{63} +362.503 q^{64} +209.542i q^{65} +(2168.37 + 4575.76i) q^{66} -7694.85 q^{67} +2740.74i q^{68} +(-3266.67 + 1548.02i) q^{69} +15174.7 q^{70} -7593.28i q^{71} +(2059.65 + 1685.14i) q^{72} -4759.98 q^{73} +5725.10i q^{74} +(2036.68 + 4297.84i) q^{75} -2481.85 q^{76} -9860.77i q^{77} +(-253.359 + 120.062i) q^{78} +8166.54 q^{79} -10792.3i q^{80} +(-1299.29 - 6431.06i) q^{81} +7622.44 q^{82} +7869.62i q^{83} +(3237.71 + 6832.31i) q^{84} +9805.27 q^{85} +6231.06i q^{86} +(-5660.69 + 2682.50i) q^{87} -3660.91 q^{88} +5351.59i q^{89} +(-8795.44 + 10750.1i) q^{90} +545.989 q^{91} +3812.94i q^{92} +(-4743.04 - 10008.9i) q^{93} +752.283 q^{94} +8879.07i q^{95} +(8773.74 - 4157.73i) q^{96} -7987.92 q^{97} -27416.7i q^{98} +(6985.62 + 5715.43i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9}+O(q^{10}) $ 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9	$ 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9} - 156 q^{10} - 4 q^{13} + 13 q^{15} + 4948 q^{16} + 22 q^{18} + 812 q^{19} - 173 q^{21} - 1644 q^{22} - 678 q^{24} - 8238 q^{25} + 777 q^{27} - 3764 q^{28} + 1374 q^{30} + 4664 q^{31} - 1042 q^{33} + 3244 q^{34} + 3648 q^{36} - 3960 q^{37} - 7078 q^{39} - 1576 q^{40} + 4934 q^{42} - 1492 q^{43} - 2063 q^{45} - 2036 q^{46} - 2620 q^{48} + 24274 q^{49} + 7300 q^{51} + 8408 q^{52} - 14766 q^{54} + 9780 q^{55} + 6939 q^{57} - 3856 q^{58} + 4712 q^{60} - 212 q^{61} - 7438 q^{63} - 45760 q^{64} + 3048 q^{66} - 12972 q^{67} + 21672 q^{69} + 5828 q^{70} - 866 q^{72} - 5240 q^{73} - 20922 q^{75} + 12368 q^{76} - 16508 q^{78} - 14976 q^{79} + 25524 q^{81} - 14484 q^{82} + 9540 q^{84} + 11572 q^{85} + 5695 q^{87} + 62160 q^{88} - 31672 q^{90} + 8284 q^{91} - 9590 q^{93} - 10992 q^{94} + 34102 q^{96} - 55000 q^{97} - 14254 q^{99}+O(q^{100}) $ 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9 - 156 * q^10 - 4 * q^13 + 13 * q^15 + 4948 * q^16 + 22 * q^18 + 812 * q^19 - 173 * q^21 - 1644 * q^22 - 678 * q^24 - 8238 * q^25 + 777 * q^27 - 3764 * q^28 + 1374 * q^30 + 4664 * q^31 - 1042 * q^33 + 3244 * q^34 + 3648 * q^36 - 3960 * q^37 - 7078 * q^39 - 1576 * q^40 + 4934 * q^42 - 1492 * q^43 - 2063 * q^45 - 2036 * q^46 - 2620 * q^48 + 24274 * q^49 + 7300 * q^51 + 8408 * q^52 - 14766 * q^54 + 9780 * q^55 + 6939 * q^57 - 3856 * q^58 + 4712 * q^60 - 212 * q^61 - 7438 * q^63 - 45760 * q^64 + 3048 * q^66 - 12972 * q^67 + 21672 * q^69 + 5828 * q^70 - 866 * q^72 - 5240 * q^73 - 20922 * q^75 + 12368 * q^76 - 16508 * q^78 - 14976 * q^79 + 25524 * q^81 - 14484 * q^82 + 9540 * q^84 + 11572 * q^85 + 5695 * q^87 + 62160 * q^88 - 31672 * q^90 + 8284 * q^91 - 9590 * q^93 - 10992 * q^94 + 34102 * q^96 - 55000 * q^97 - 14254 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	5.04906i		−	1.26227i	−0.775675	−	0.631133i	$-0.782590\pi$
$2$		−	5.04906i		−	1.26227i	0.775675	−	0.631133i	$-0.217410\pi$
$3$	−3.85410	−	8.13301i	−0.428233	−	0.903668i
$3$	−3.85410	−	8.13301i	−0.428233	−	0.903668i
$4$	−9.49305			−0.593316
$5$			33.9624i			1.35850i	0.733909	+	0.679248i	$0.237694\pi$
$5$			33.9624i			1.35850i	−0.733909	+	0.679248i	$0.762306\pi$
$6$	−41.0641	+	19.4596i	−1.14067	+	0.540544i
$7$	88.4932			1.80598			0.902992	−	0.429657i	$-0.141366\pi$
$7$	88.4932			1.80598			0.902992	+	0.429657i	$0.141366\pi$
$8$		−	32.8540i		−	0.513344i
$9$	−51.2919	+	62.6909i	−0.633233	+	0.773961i
$10$	171.478			1.71478
$11$		−	111.430i		−	0.920906i	−0.887684	−	0.460453i	$-0.847687\pi$
$11$		−	111.430i		−	0.920906i	0.887684	−	0.460453i	$-0.152313\pi$
$12$	36.5871	+	77.2071i	0.254077	+	0.536161i
$13$	6.16984			0.0365079			0.0182539	−	0.999833i	$-0.494189\pi$
$13$	6.16984			0.0365079			0.0182539	+	0.999833i	$0.494189\pi$
$14$		−	446.808i		−	2.27963i
$15$	276.217	−	130.894i	1.22763	−	0.581753i
$16$	−317.771			−1.24129
$17$		−	288.710i		−	0.998996i	−0.866315	−	0.499498i	$-0.833518\pi$
$17$		−	288.710i		−	0.998996i	0.866315	−	0.499498i	$-0.166482\pi$
$18$	316.530	+	258.976i	0.976945	+	0.799308i
$19$	261.438			0.724205			0.362103	−	0.932138i	$-0.382059\pi$
$19$	261.438			0.724205			0.362103	+	0.932138i	$0.382059\pi$
$20$		−	322.407i		−	0.806017i
$21$	−341.062	−	719.717i	−0.773382	−	1.63201i
$22$	−562.615			−1.16243
$23$		−	401.655i		−	0.759273i	−0.925136	−	0.379636i	$-0.876049\pi$
$23$		−	401.655i		−	0.759273i	0.925136	−	0.379636i	$-0.123951\pi$
$24$	−267.202	+	126.623i	−0.463893	+	0.219831i
$25$	−528.444			−0.845511
$26$		−	31.1519i		−	0.0460827i
$27$	707.550	+	175.541i	0.970576	+	0.240797i
$28$	−840.071			−1.07152
$29$		−	696.014i		−	0.827603i	−0.910367	−	0.413801i	$-0.864201\pi$
$29$		−	696.014i		−	0.827603i	0.910367	−	0.413801i	$-0.135799\pi$
$30$	−660.894	−	1394.64i	−0.734327	−	1.54960i
$31$	1230.65			1.28059			0.640295	−	0.768129i	$-0.278812\pi$
$31$	1230.65			1.28059			0.640295	+	0.768129i	$0.278812\pi$
$32$			1078.78i			1.05350i
$33$	−906.259	+	429.461i	−0.832194	+	0.394362i
$34$	−1457.71			−1.26100
$35$			3005.44i			2.45342i
$36$	486.916	−	595.128i	0.375707	−	0.459203i
$37$	−1133.89			−0.828264			−0.414132	−	0.910217i	$-0.635915\pi$
$37$	−1133.89			−0.828264			−0.414132	+	0.910217i	$0.635915\pi$
$38$		−	1320.02i		−	0.914140i
$39$	−23.7791	−	50.1794i	−0.0156339	−	0.0329910i
$40$	1115.80			0.697375
$41$			1509.67i			0.898081i	0.893511	+	0.449040i	$0.148234\pi$
$41$			1509.67i			0.898081i	−0.893511	+	0.449040i	$0.851766\pi$
$42$	−3633.90	+	1722.04i	−2.06003	+	0.976214i
$43$	−1234.10			−0.667443			−0.333722	−	0.942672i	$-0.608305\pi$
$43$	−1234.10			−0.667443			−0.333722	+	0.942672i	$0.608305\pi$
$44$			1057.81i			0.546388i
$45$	−2129.13	−	1741.99i	−1.05142	−	0.860244i
$46$	−2027.98			−0.958405
$47$			148.995i			0.0674489i	0.999431	+	0.0337244i	$0.0107369\pi$
$47$			148.995i			0.0674489i	−0.999431	+	0.0337244i	$0.989263\pi$
$48$	1224.72	+	2584.43i	0.531562	+	1.12172i
$49$	5430.05			2.26158
$50$			2668.15i			1.06726i
$51$	−2348.08	+	1112.72i	−0.902761	+	0.427803i
$52$	−58.5706			−0.0216607
$53$		−	4971.34i		−	1.76979i	−0.465792	−	0.884894i	$-0.654231\pi$
$53$		−	4971.34i		−	1.76979i	0.465792	−	0.884894i	$-0.345769\pi$
$54$	886.317	−	3572.46i	0.303950	−	1.22512i
$55$	3784.42			1.25105
$56$		−	2907.36i		−	0.927091i
$57$	−1007.61	−	2126.28i	−0.310129	−	0.654441i
$58$	−3514.22			−1.04465
$59$		−	453.188i		−	0.130189i
$59$		−	453.188i		−	0.130189i
$60$	−2622.14	+	1242.59i	−0.728372	+	0.345163i
$61$	2566.52			0.689740			0.344870	−	0.938651i	$-0.387923\pi$
$61$	2566.52			0.689740			0.344870	+	0.938651i	$0.387923\pi$
$62$		−	6213.62i		−	1.61645i
$63$	−4538.98	+	5547.72i	−1.14361	+	1.39776i
$64$	362.503			0.0885017
$65$			209.542i			0.0495958i
$66$	2168.37	+	4575.76i	0.497790	+	1.05045i
$67$	−7694.85			−1.71416			−0.857079	−	0.515185i	$-0.827723\pi$
$67$	−7694.85			−1.71416			−0.857079	+	0.515185i	$0.827723\pi$
$68$			2740.74i			0.592720i
$69$	−3266.67	+	1548.02i	−0.686131	+	0.325146i
$70$	15174.7			3.09687
$71$		−	7593.28i		−	1.50630i	−0.657846	−	0.753152i	$-0.728532\pi$
$71$		−	7593.28i		−	1.50630i	0.657846	−	0.753152i	$-0.271468\pi$
$72$	2059.65	+	1685.14i	0.397308	+	0.325066i
$73$	−4759.98			−0.893221			−0.446611	−	0.894728i	$-0.647369\pi$
$73$	−4759.98			−0.893221			−0.446611	+	0.894728i	$0.647369\pi$
$74$			5725.10i			1.04549i
$75$	2036.68	+	4297.84i	0.362076	+	0.764061i
$76$	−2481.85			−0.429682
$77$		−	9860.77i		−	1.66314i
$78$	−253.359	+	120.062i	−0.0416435	+	0.0197341i
$79$	8166.54			1.30853			0.654266	−	0.756265i	$-0.272978\pi$
$79$	8166.54			1.30853			0.654266	+	0.756265i	$0.272978\pi$
$80$		−	10792.3i		−	1.68629i
$81$	−1299.29	−	6431.06i	−0.198032	−	0.980196i
$82$	7622.44			1.13362
$83$			7869.62i			1.14235i	0.820830	+	0.571173i	$0.193511\pi$
$83$			7869.62i			1.14235i	−0.820830	+	0.571173i	$0.806489\pi$
$84$	3237.71	+	6832.31i	0.458860	+	0.968298i
$85$	9805.27			1.35713
$86$			6231.06i			0.842491i
$87$	−5660.69	+	2682.50i	−0.747878	+	0.354407i
$88$	−3660.91			−0.472741
$89$			5351.59i			0.675620i	0.941214	+	0.337810i	$0.109686\pi$
$89$			5351.59i			0.675620i	−0.941214	+	0.337810i	$0.890314\pi$
$90$	−8795.44	+	10750.1i	−1.08586	+	1.32718i
$91$	545.989			0.0659327
$92$			3812.94i			0.450489i
$93$	−4743.04	−	10008.9i	−0.548391	−	1.15723i
$94$	752.283			0.0851384
$95$			8879.07i			0.983830i
$96$	8773.74	−	4157.73i	0.952012	−	0.451142i
$97$	−7987.92			−0.848966			−0.424483	−	0.905436i	$-0.639544\pi$
$97$	−7987.92			−0.848966			−0.424483	+	0.905436i	$0.639544\pi$
$98$		−	27416.7i		−	2.85472i
$99$	6985.62	+	5715.43i	0.712746	+	0.583148i
$100$	5016.55			0.501655
$101$		−	3277.52i		−	0.321294i	−0.987012	−	0.160647i	$-0.948642\pi$
$101$		−	3277.52i		−	0.321294i	0.987012	−	0.160647i	$-0.0513580\pi$
$102$	5618.17	+	11855.6i	0.540001	+	1.13952i
$103$	14838.0			1.39862			0.699310	−	0.714819i	$-0.253491\pi$
$103$	14838.0			1.39862			0.699310	+	0.714819i	$0.253491\pi$
$104$		−	202.704i		−	0.0187411i
$105$	24443.3	−	11583.3i	2.21708	−	1.05064i
$106$	−25100.6			−2.23394
$107$			3771.28i			0.329398i	0.986344	+	0.164699i	$0.0526653\pi$
$107$			3771.28i			0.329398i	−0.986344	+	0.164699i	$0.947335\pi$
$108$	−6716.80	−	1666.42i	−0.575858	−	0.142868i
$109$	17882.7			1.50515			0.752577	−	0.658504i	$-0.228810\pi$
$109$	17882.7			1.50515			0.752577	+	0.658504i	$0.228810\pi$
$110$		−	19107.8i		−	1.57915i
$111$	4370.14	+	9221.97i	0.354690	+	0.748476i
$112$	−28120.6			−2.24175
$113$		−	2411.73i		−	0.188874i	−0.995531	−	0.0944370i	$-0.969895\pi$
$113$		−	2411.73i		−	0.188874i	0.995531	−	0.0944370i	$-0.0301051\pi$
$114$	−10735.7	+	5087.48i	−0.826079	+	0.391465i
$115$	13641.2			1.03147
$116$			6607.29i			0.491030i
$117$	−316.462	+	386.792i	−0.0231180	+	0.0282557i
$118$	−2288.17			−0.164333
$119$		−	25548.9i		−	1.80417i
$120$	−4300.40	−	9074.82i	−0.298639	−	0.630196i
$121$	2224.44			0.151932
$122$		−	12958.5i		−	0.870635i
$123$	12278.2	−	5818.43i	0.811567	−	0.384588i
$124$	−11682.6			−0.759794
$125$			3279.27i			0.209873i
$126$	28010.8	+	22917.6i	1.76435	+	1.44354i
$127$	3307.85			0.205087			0.102543	−	0.994729i	$-0.467302\pi$
$127$	3307.85			0.205087			0.102543	+	0.994729i	$0.467302\pi$
$128$			15430.2i			0.941785i
$129$	4756.35	+	10037.0i	0.285821	+	0.603147i
$130$	1057.99			0.0626031
$131$			24056.8i			1.40183i	0.713245	+	0.700915i	$0.247225\pi$
$131$			24056.8i			1.40183i	−0.713245	+	0.700915i	$0.752775\pi$
$132$	8603.16	−	4076.89i	0.493754	−	0.233981i
$133$	23135.5			1.30790
$134$			38851.8i			2.16372i
$135$	−5961.79	+	24030.1i	−0.327121	+	1.31852i
$136$	−9485.27			−0.512828
$137$			15669.2i			0.834845i	0.908713	+	0.417422i	$0.137066\pi$
$137$			15669.2i			0.834845i	−0.908713	+	0.417422i	$0.862934\pi$
$138$	7816.05	+	16493.6i	0.410421	+	0.866080i
$139$	22577.2			1.16853			0.584267	−	0.811562i	$-0.301382\pi$
$139$	22577.2			1.16853			0.584267	+	0.811562i	$0.301382\pi$
$140$		−	28530.8i		−	1.45565i
$141$	1211.77	−	574.239i	0.0609514	−	0.0288838i
$142$	−38339.0			−1.90136
$143$		−	687.502i		−	0.0336203i
$144$	16299.1	−	19921.3i	0.786027	−	0.960712i
$145$	23638.3			1.12429
$146$			24033.4i			1.12748i
$147$	−20928.0	−	44162.7i	−0.968483	−	2.04372i
$148$	10764.1			0.491422
$149$		−	1485.69i		−	0.0669201i	−0.999440	−	0.0334600i	$-0.989347\pi$
$149$		−	1485.69i		−	0.0669201i	0.999440	−	0.0334600i	$-0.0106526\pi$
$150$	21700.1	−	10283.3i	0.964449	−	0.457036i
$151$	−20410.8			−0.895170			−0.447585	−	0.894241i	$-0.647716\pi$
$151$	−20410.8			−0.895170			−0.447585	+	0.894241i	$0.647716\pi$
$152$		−	8589.29i		−	0.371766i
$153$	18099.5	+	14808.5i	0.773184	+	0.632597i
$154$	−49787.7			−2.09933
$155$			41795.7i			1.73968i
$156$	225.737	+	476.355i	0.00927583	+	0.0195741i
$157$	−45940.4			−1.86378			−0.931891	−	0.362738i	$-0.881842\pi$
$157$	−45940.4			−1.86378			−0.931891	+	0.362738i	$0.881842\pi$
$158$		−	41233.4i		−	1.65171i
$159$	−40431.9	+	19160.0i	−1.59930	+	0.757882i
$160$	−36638.0			−1.43117
$161$		−	35543.8i		−	1.37124i
$162$	−32470.9	+	6560.19i	−1.23727	+	0.249969i
$163$	25058.4			0.943145			0.471573	−	0.881827i	$-0.343687\pi$
$163$	25058.4			0.943145			0.471573	+	0.881827i	$0.343687\pi$
$164$		−	14331.4i		−	0.532846i
$165$	−14585.5	−	30778.7i	−0.535740	−	1.13053i
$166$	39734.2			1.44194
$167$		−	32413.6i		−	1.16224i	−0.813819	−	0.581118i	$-0.802615\pi$
$167$		−	32413.6i		−	1.16224i	0.813819	−	0.581118i	$-0.197385\pi$
$168$	−23645.6	+	11205.2i	−0.837783	+	0.397011i
$169$	−28522.9			−0.998667
$170$		−	49507.5i		−	1.71306i
$171$	−13409.7	+	16389.8i	−0.458591	+	0.560507i
$172$	11715.4			0.396005
$173$			55290.6i			1.84739i	0.383126	+	0.923696i	$0.374847\pi$
$173$			55290.6i			1.84739i	−0.383126	+	0.923696i	$0.625153\pi$
$174$	13544.1	+	28581.2i	0.447356	+	0.944021i
$175$	−46763.7			−1.52698
$176$			35409.1i			1.14311i
$177$	−3685.78	+	1746.63i	−0.117648	+	0.0557512i
$178$	27020.5			0.852812
$179$		−	17012.3i		−	0.530955i	−0.964117	−	0.265477i	$-0.914470\pi$
$179$		−	17012.3i		−	0.530955i	0.964117	−	0.265477i	$-0.0855295\pi$
$180$	20212.0	+	16536.8i	0.623826	+	0.510396i
$181$	17714.0			0.540703			0.270352	−	0.962762i	$-0.412860\pi$
$181$	17714.0			0.540703			0.270352	+	0.962762i	$0.412860\pi$
$182$		−	2756.73i		−	0.0832246i
$183$	−9891.62	−	20873.6i	−0.295369	−	0.623296i
$184$	−13196.0			−0.389768
$185$		−	38509.7i		−	1.12519i
$186$	−50535.5	+	23947.9i	−1.46073	+	0.692216i
$187$	−32170.8			−0.919981
$188$		−	1414.41i		−	0.0400185i
$189$	62613.4	+	15534.2i	1.75284	+	0.434875i
$190$	44831.0			1.24186
$191$			48580.3i			1.33166i	0.746103	+	0.665830i	$0.231923\pi$
$191$			48580.3i			1.33166i	−0.746103	+	0.665830i	$0.768077\pi$
$192$	−1397.12	−	2948.24i	−0.0378993	−	0.0799761i
$193$	32320.8			0.867697			0.433848	−	0.900986i	$-0.357155\pi$
$193$	32320.8			0.867697			0.433848	+	0.900986i	$0.357155\pi$
$194$			40331.5i			1.07162i
$195$	1704.21	−	807.597i	0.0448182	−	0.0212386i
$196$	−51547.8			−1.34183
$197$		−	24102.9i		−	0.621066i	−0.950563	−	0.310533i	$-0.899493\pi$
$197$		−	24102.9i		−	0.621066i	0.950563	−	0.310533i	$-0.100507\pi$
$198$	28857.6	−	35270.8i	0.736088	−	0.899675i
$199$	42180.4			1.06513			0.532567	−	0.846388i	$-0.321228\pi$
$199$	42180.4			1.06513			0.532567	+	0.846388i	$0.321228\pi$
$200$			17361.5i			0.434038i
$201$	29656.7	+	62582.4i	0.734059	+	1.54903i
$202$	−16548.4			−0.405558
$203$		−	61592.5i		−	1.49464i
$204$	22290.5	−	10563.1i	0.535622	−	0.253822i
$205$	−51272.1			−1.22004
$206$		−	74917.8i		−	1.76543i
$207$	25180.1	+	20601.7i	0.587648	+	0.480797i
$208$	−1960.59			−0.0453170
$209$		−	29132.0i		−	0.666925i
$210$	−58484.7	−	123416.i	−1.32618	−	2.79854i
$211$	−40533.0			−0.910425			−0.455212	−	0.890383i	$-0.650437\pi$
$211$	−40533.0			−0.910425			−0.455212	+	0.890383i	$0.650437\pi$
$212$			47193.1i			1.05004i
$213$	−61756.3	+	29265.2i	−1.36120	+	0.645049i
$214$	19041.4			0.415788
$215$		−	41913.1i		−	0.906719i
$216$	5767.22	−	23245.8i	0.123611	−	0.498239i
$217$	108904.			2.31273
$218$		−	90291.1i		−	1.89991i
$219$	18345.4	+	38713.0i	0.382507	+	0.807176i
$220$	−35925.7			−0.742266
$221$		−	1781.29i		−	0.0364712i
$222$	46562.3	−	22065.1i	0.944776	−	0.447713i
$223$	16360.3			0.328989			0.164494	−	0.986378i	$-0.447401\pi$
$223$	16360.3			0.328989			0.164494	+	0.986378i	$0.447401\pi$
$224$			95464.8i			1.90260i
$225$	27104.9	−	33128.6i	0.535405	−	0.654393i
$226$	−12177.0			−0.238409
$227$		−	87068.2i		−	1.68969i	−0.535009	−	0.844846i	$-0.679692\pi$
$227$		−	87068.2i		−	1.68969i	0.535009	−	0.844846i	$-0.320308\pi$
$228$	9565.28	+	20184.9i	0.184004	+	0.388290i
$229$	−16660.9			−0.317707			−0.158854	−	0.987302i	$-0.550780\pi$
$229$	−16660.9			−0.317707			−0.158854	+	0.987302i	$0.550780\pi$
$230$		−	68875.2i		−	1.30199i
$231$	−80197.8	+	38004.4i	−1.50293	+	0.712212i
$232$	−22866.8			−0.424845
$233$			26333.3i			0.485059i	0.970144	+	0.242529i	$0.0779771\pi$
$233$			26333.3i			0.485059i	−0.970144	+	0.242529i	$0.922023\pi$
$234$	1952.94	+	1597.84i	0.0356662	+	0.0291811i
$235$	−5060.21			−0.0916290
$236$			4302.13i			0.0772431i
$237$	−31474.7	−	66418.6i	−0.560356	−	1.18248i
$238$	−128998.			−2.27734
$239$			66473.4i			1.16373i	0.813285	+	0.581865i	$0.197677\pi$
$239$			66473.4i			1.16373i	−0.813285	+	0.581865i	$0.802323\pi$
$240$	−87773.6	+	41594.4i	−1.52385	+	0.722125i
$241$	15975.1			0.275048			0.137524	−	0.990498i	$-0.456086\pi$
$241$	15975.1			0.275048			0.137524	+	0.990498i	$0.456086\pi$
$242$		−	11231.3i		−	0.191779i
$243$	−47296.3	+	35353.1i	−0.800968	+	0.598708i
$244$	−24364.1			−0.409233
$245$			184418.i			3.07235i
$246$	−29377.6	−	61993.4i	−0.485452	−	1.02441i
$247$	1613.03			0.0264392
$248$		−	40431.7i		−	0.657383i
$249$	64003.7	−	30330.3i	1.03230	−	0.489190i
$250$	16557.2			0.264916
$251$			60374.2i			0.958306i	0.877732	+	0.479153i	$0.159056\pi$
$251$			60374.2i			0.958306i	−0.877732	+	0.479153i	$0.840944\pi$
$252$	43088.8	−	52664.8i	0.678521	−	0.829314i
$253$	−44756.3			−0.699219
$254$		−	16701.5i		−	0.258874i
$255$	−37790.5	−	79746.4i	−0.581169	−	1.22640i
$256$	83708.1			1.27728
$257$			42564.2i			0.644434i	0.946666	+	0.322217i	$0.104428\pi$
$257$			42564.2i			0.644434i	−0.946666	+	0.322217i	$0.895572\pi$
$258$	50677.3	−	24015.1i	0.761333	−	0.360783i
$259$	−100342.			−1.49583
$260$		−	1989.20i		−	0.0294260i
$261$	43633.7	+	35699.8i	0.640532	+	0.524065i
$262$	121464.			1.76948
$263$			75835.7i			1.09638i	0.836353	+	0.548191i	$0.184683\pi$
$263$			75835.7i			1.09638i	−0.836353	+	0.548191i	$0.815317\pi$
$264$	14109.5	+	29774.2i	0.202444	+	0.427201i
$265$	168838.			2.40425
$266$		−	116813.i		−	1.65092i
$267$	43524.5	−	20625.5i	0.610536	−	0.289323i
$268$	73047.6			1.01704
$269$		−	107497.i		−	1.48557i	−0.669533	−	0.742783i	$-0.733506\pi$
$269$		−	107497.i		−	1.48557i	0.669533	−	0.742783i	$-0.266494\pi$
$270$	121329.	+	30101.4i	1.66433	+	0.412914i
$271$	13069.1			0.177953			0.0889767	−	0.996034i	$-0.471640\pi$
$271$	13069.1			0.177953			0.0889767	+	0.996034i	$0.471640\pi$
$272$			91743.5i			1.24005i
$273$	−2104.29	−	4440.53i	−0.0282346	−	0.0595813i
$274$	79114.8			1.05380
$275$			58884.3i			0.778636i
$276$	31010.7	−	14695.4i	0.407092	−	0.192914i
$277$	36791.8			0.479503			0.239751	−	0.970834i	$-0.422934\pi$
$277$	36791.8			0.479503			0.239751	+	0.970834i	$0.422934\pi$
$278$		−	113994.i		−	1.47500i
$279$	−63122.2	+	77150.4i	−0.810912	+	0.991127i
$280$	98740.8			1.25945
$281$			116425.i			1.47447i	0.675638	+	0.737233i	$0.263868\pi$
$281$			116425.i			1.47447i	−0.675638	+	0.737233i	$0.736132\pi$
$282$	−2899.37	−	6118.33i	−0.0364591	−	0.0769369i
$283$	−75649.7			−0.944570			−0.472285	−	0.881446i	$-0.656571\pi$
$283$	−75649.7			−0.944570			−0.472285	+	0.881446i	$0.656571\pi$
$284$			72083.4i			0.893714i
$285$	72213.6	−	34220.8i	0.889056	−	0.421309i
$286$	−3471.24			−0.0424378
$287$			133596.i			1.62192i
$288$	−67629.7	−	55332.7i	−0.815366	−	0.667109i
$289$	167.676			0.00200759
$290$		−	119351.i		−	1.41916i
$291$	30786.2	+	64965.9i	0.363555	+	0.767184i
$292$	45186.7			0.529962
$293$			136522.i			1.59026i	0.606439	+	0.795130i	$0.292598\pi$
$293$			136522.i			1.59026i	−0.606439	+	0.795130i	$0.707402\pi$
$294$	−222980.	+	105667.i	−2.57972	+	1.22248i
$295$	15391.3			0.176861
$296$			37252.9i			0.425184i
$297$	19560.4	−	78842.0i	0.221751	−	0.893809i
$298$	−7501.36			−0.0844709
$299$		−	2478.15i		−	0.0277195i
$300$	−19334.3	−	40799.7i	−0.214825	−	0.453330i
$301$	−109210.			−1.20539
$302$			103055.i			1.12994i
$303$	−26656.1	+	12631.9i	−0.290343	+	0.137589i
$304$	−83077.4			−0.898951
$305$			87165.2i			0.937008i
$306$	74768.9	−	91385.4i	0.798506	−	0.975964i
$307$	40342.3			0.428039			0.214020	−	0.976829i	$-0.431344\pi$
$307$	40342.3			0.428039			0.214020	+	0.976829i	$0.431344\pi$
$308$			93608.8i			0.986768i
$309$	−57186.9	−	120677.i	−0.598935	−	1.26389i
$310$	211029.			2.19594
$311$		−	2156.57i		−	0.0222968i	−0.999938	−	0.0111484i	$-0.996451\pi$
$311$		−	2156.57i		−	0.0222968i	0.999938	−	0.0111484i	$-0.00354872\pi$
$312$	−1648.59	+	781.240i	−0.0169357	+	0.00802556i
$313$	140844.			1.43763			0.718817	−	0.695199i	$-0.244684\pi$
$313$	140844.			1.43763			0.718817	+	0.695199i	$0.244684\pi$
$314$			231956.i			2.35259i
$315$	−188414.	−	154155.i	−1.89885	−	1.55359i
$316$	−77525.4			−0.776372
$317$			101071.i			1.00579i	0.864346	+	0.502897i	$0.167732\pi$
$317$			101071.i			1.00579i	−0.864346	+	0.502897i	$0.832268\pi$
$318$	96740.1	+	204143.i	0.956649	+	2.01874i
$319$	−77556.6			−0.762144
$320$			12311.5i			0.120229i
$321$	30671.9	−	14534.9i	0.297667	−	0.141059i
$322$	−179463.			−1.73086
$323$		−	75479.7i		−	0.723478i
$324$	12334.2	+	61050.4i	0.117496	+	0.581565i
$325$	−3260.41			−0.0308678
$326$		−	126522.i		−	1.19050i
$327$	−68921.8	−	145441.i	−0.644557	−	1.36016i
$328$	49598.8			0.461024
$329$			13185.0i			0.121812i
$330$	−155404.	+	73643.2i	−1.42703	+	0.676246i
$331$	40959.7			0.373853			0.186926	−	0.982374i	$-0.440147\pi$
$331$	40959.7			0.373853			0.186926	+	0.982374i	$0.440147\pi$
$332$		−	74706.7i		−	0.677771i
$333$	58159.5	−	71084.8i	0.524484	−	0.641044i
$334$	−163658.			−1.46705
$335$		−	261336.i		−	2.32868i
$336$	108379.	+	228705.i	0.959993	+	2.02580i
$337$	−105275.			−0.926965			−0.463483	−	0.886106i	$-0.653400\pi$
$337$	−105275.			−0.926965			−0.463483	+	0.886106i	$0.653400\pi$
$338$			144014.i			1.26058i
$339$	−19614.7	+	9295.06i	−0.170680	+	0.0808821i
$340$	−93082.0			−0.805207
$341$		−	137131.i		−	1.17930i
$342$	82753.1	+	67706.2i	0.707509	+	0.578863i
$343$	268051.			2.27839
$344$			40545.2i			0.342628i
$345$	−52574.4	−	110944.i	−0.441709	−	0.932106i
$346$	279166.			2.33190
$347$			36624.9i			0.304171i	0.988367	+	0.152085i	$0.0485989\pi$
$347$			36624.9i			0.304171i	−0.988367	+	0.152085i	$0.951401\pi$
$348$	53737.2	−	25465.2i	0.443728	−	0.210275i
$349$	11886.3			0.0975879			0.0487939	−	0.998809i	$-0.484462\pi$
$349$	11886.3			0.0975879			0.0487939	+	0.998809i	$0.484462\pi$
$350$			236113.i			1.92745i
$351$	4365.46	+	1083.06i	0.0354337	+	0.00879098i
$352$	120208.			0.970172
$353$			128488.i			1.03113i	0.856850	+	0.515565i	$0.172418\pi$
$353$			128488.i			1.03113i	−0.856850	+	0.515565i	$0.827582\pi$
$354$	8818.84	+	18609.7i	0.0703728	+	0.148503i
$355$	257886.			2.04631
$356$		−	50802.9i		−	0.400856i
$357$	−207789.	+	98467.8i	−1.63037	+	0.772605i
$358$	−85896.3			−0.670206
$359$		−	207156.i		−	1.60735i	−0.595071	−	0.803673i	$-0.702876\pi$
$359$		−	207156.i		−	1.60735i	0.595071	−	0.803673i	$-0.297124\pi$
$360$	−57231.5	+	69950.5i	−0.441601	+	0.539742i
$361$	−61971.1			−0.475527
$362$		−	89439.0i		−	0.682511i
$363$	−8573.19	−	18091.4i	−0.0650623	−	0.137296i
$364$	−5183.10			−0.0391189
$365$		−	161660.i		−	1.21344i
$366$	−105392.	+	49943.4i	−0.786765	+	0.372835i
$367$	−123091.			−0.913892			−0.456946	−	0.889494i	$-0.651057\pi$
$367$	−123091.			−0.913892			−0.456946	+	0.889494i	$0.651057\pi$
$368$			127634.i			0.942480i
$369$	−94642.8	−	77434.0i	−0.695080	−	0.568694i
$370$	−194438.			−1.42029
$371$		−	439930.i		−	3.19621i
$372$	45025.9	+	95014.8i	0.325369	+	0.686602i
$373$	−215834.			−1.55132			−0.775662	−	0.631149i	$-0.782584\pi$
$373$	−215834.			−1.55132			−0.775662	+	0.631149i	$0.782584\pi$
$374$			162433.i			1.16126i
$375$	26670.3	−	12638.6i	0.189656	−	0.0898745i
$376$	4895.07			0.0346245
$377$		−	4294.29i		−	0.0302140i
$378$	78433.0	−	316139.i	0.548928	−	2.21256i
$379$	170916.			1.18988			0.594942	−	0.803769i	$-0.297175\pi$
$379$	170916.			1.18988			0.594942	+	0.803769i	$0.297175\pi$
$380$		−	84289.4i		−	0.583722i
$381$	−12748.8	−	26902.8i	−0.0878250	−	0.185331i
$382$	245285.			1.68091
$383$			41171.9i			0.280675i	0.990104	+	0.140337i	$0.0448187\pi$
$383$			41171.9i			0.280675i	−0.990104	+	0.140337i	$0.955181\pi$
$384$	125494.	−	59469.5i	0.851061	−	0.403303i
$385$	334895.			2.25937
$386$		−	163190.i		−	1.09526i
$387$	63299.4	−	77367.0i	0.422647	−	0.516575i
$388$	75829.8			0.503705
$389$			11993.0i			0.0792552i	0.999215	+	0.0396276i	$0.0126172\pi$
$389$			11993.0i			0.0792552i	−0.999215	+	0.0396276i	$0.987383\pi$
$390$	−4077.61	−	8604.67i	−0.0268087	−	0.0565725i
$391$	−115962.			−0.758510
$392$		−	178399.i		−	1.16097i
$393$	195654.	−	92717.3i	1.26679	−	0.600310i
$394$	−121697.			−0.783950
$395$			277355.i			1.77763i
$396$	−66314.9	−	54256.9i	−0.422883	−	0.345991i
$397$	−65587.3			−0.416139			−0.208070	−	0.978114i	$-0.566718\pi$
$397$	−65587.3			−0.416139			−0.208070	+	0.978114i	$0.566718\pi$
$398$		−	212971.i		−	1.34448i
$399$	−89166.5	−	188161.i	−0.560088	−	1.18191i
$400$	167924.			1.04953
$401$			155422.i			0.966551i	0.875468	+	0.483275i	$0.160553\pi$
$401$			155422.i			0.966551i	−0.875468	+	0.483275i	$0.839447\pi$
$402$	315982.	−	149739.i	1.95529	−	0.926578i
$403$	7592.89			0.0467517
$404$			31113.6i			0.190629i
$405$	218414.	−	44127.0i	1.33159	−	0.269026i
$406$	−310985.			−1.88663
$407$			126349.i			0.762753i
$408$	36557.2	+	77143.9i	0.219610	+	0.463427i
$409$	−4099.29			−0.0245054			−0.0122527	−	0.999925i	$-0.503900\pi$
$409$	−4099.29			−0.0245054			−0.0122527	+	0.999925i	$0.503900\pi$
$410$			258876.i			1.54001i
$411$	127438.	−	60390.6i	0.754423	−	0.357508i
$412$	−140857.			−0.829823
$413$		−	40104.0i		−	0.235119i
$414$	104019.	−	127136.i	0.606893	−	0.741768i
$415$	−267271.			−1.55187
$416$			6655.90i			0.0384610i
$417$	−87014.9	−	183621.i	−0.500405	−	1.05597i
$418$	−147089.			−0.841837
$419$			13136.9i			0.0748281i	0.999300	+	0.0374141i	$0.0119120\pi$
$419$			13136.9i			0.0748281i	−0.999300	+	0.0374141i	$0.988088\pi$
$420$	−232042.	+	109961.i	−1.31543	+	0.623359i
$421$	87174.3			0.491840			0.245920	−	0.969290i	$-0.420910\pi$
$421$	87174.3			0.491840			0.245920	+	0.969290i	$0.420910\pi$
$422$			204654.i			1.14920i
$423$	−9340.60	−	7642.21i	−0.0522028	−	0.0427108i
$424$	−163328.			−0.908510
$425$			152567.i			0.844662i
$426$	147762.	+	311811.i	0.814224	+	1.71820i
$427$	227120.			1.24566
$428$		−	35800.9i		−	0.195437i
$429$	−5591.47	+	2649.70i	−0.0303816	+	0.0143973i
$430$	−211622.			−1.14452
$431$			32749.0i			0.176297i	0.996107	+	0.0881483i	$0.0280949\pi$
$431$			32749.0i			0.176297i	−0.996107	+	0.0881483i	$0.971905\pi$
$432$	−224839.	−	55781.7i	−1.20477	−	0.298899i
$433$	−48.9144			−0.000260892			−0.000130446	−	1.00000i	$-0.500042\pi$
$433$	−48.9144			−0.000260892			−0.000130446		1.00000i	$0.500042\pi$
$434$		−	549863.i		−	2.91928i
$435$	−91104.3	−	192251.i	−0.481460	−	1.01599i
$436$	−169762.			−0.893032
$437$		−	105008.i		−	0.549870i
$438$	195464.	−	92627.2i	1.01887	−	0.482825i
$439$	−359966.			−1.86781			−0.933905	−	0.357522i	$-0.883622\pi$
$439$	−359966.			−1.86781			−0.933905	+	0.357522i	$0.883622\pi$
$440$		−	124333.i		−	0.642217i
$441$	−278518.	+	340415.i	−1.43211	+	1.75038i
$442$	−8993.86			−0.0460364
$443$			271249.i			1.38217i	0.722775	+	0.691083i	$0.242866\pi$
$443$			271249.i			1.38217i	−0.722775	+	0.691083i	$0.757134\pi$
$444$	−41485.9	−	87544.7i	−0.210443	−	0.444083i
$445$	−181753.			−0.917827
$446$		−	82604.1i		−	0.415271i
$447$	−12083.2	+	5726.00i	−0.0604736	+	0.0286574i
$448$	32079.0			0.159833
$449$		−	12238.5i		−	0.0607064i	−0.999539	−	0.0303532i	$-0.990337\pi$
$449$		−	12238.5i		−	0.0607064i	0.999539	−	0.0303532i	$-0.00966321\pi$
$450$	−167269.	−	136854.i	−0.826018	−	0.675824i
$451$	168222.			0.827048
$452$			22894.7i			0.112062i
$453$	78665.1	+	166001.i	0.383341	+	0.808937i
$454$	−439613.			−2.13284
$455$			18543.1i			0.0895693i
$456$	−69856.8	+	33104.0i	−0.335953	+	0.159203i
$457$	−364846.			−1.74694			−0.873469	−	0.486879i	$-0.838135\pi$
$457$	−364846.			−1.74694			−0.873469	+	0.486879i	$0.838135\pi$
$458$			84121.9i			0.401031i
$459$	50680.3	−	204276.i	0.240555	−	0.969601i
$460$	−129496.			−0.611987
$461$			219775.i			1.03413i	0.855945	+	0.517066i	$0.172976\pi$
$461$			219775.i			1.03413i	−0.855945	+	0.517066i	$0.827024\pi$
$462$	191886.	+	404924.i	0.899002	+	1.89710i
$463$	190085.			0.886717			0.443359	−	0.896344i	$-0.353787\pi$
$463$	190085.			0.886717			0.443359	+	0.896344i	$0.353787\pi$
$464$			221173.i			1.02730i
$465$	339925.	−	161085.i	1.57209	−	0.744987i
$466$	132959.			0.612273
$467$		−	427960.i		−	1.96232i	−0.193207	−	0.981158i	$-0.561889\pi$
$467$		−	427960.i		−	1.96232i	0.193207	−	0.981158i	$-0.438111\pi$
$468$	3004.19	−	3671.84i	0.0137163	−	0.0167646i
$469$	−680943.			−3.09574
$470$			25549.3i			0.115660i
$471$	177059.	+	373634.i	0.798133	+	1.68424i
$472$	−14889.0			−0.0668317
$473$			137516.i			0.614653i
$474$	−335352.	+	158918.i	−1.49260	+	0.707319i
$475$	−138155.			−0.612323
$476$			242537.i			1.07044i
$477$	311657.	+	254989.i	1.36975	+	1.12069i
$478$	335629.			1.46894
$479$			62417.6i			0.272042i	0.990706	+	0.136021i	$0.0434315\pi$
$479$			62417.6i			0.272042i	−0.990706	+	0.136021i	$0.956569\pi$
$480$	141206.	+	297977.i	0.612875	+	1.29330i
$481$	−6995.94			−0.0302382
$482$		−	80659.2i		−	0.347184i
$483$	−289078.	+	136989.i	−1.23914	+	0.587208i
$484$	−21116.7			−0.0901436
$485$		−	271289.i		−	1.15332i
$486$	178500.	+	238802.i	0.755728	+	1.01103i
$487$	469157.			1.97816			0.989078	−	0.147395i	$-0.0470888\pi$
$487$	469157.			1.97816			0.989078	+	0.147395i	$0.0470888\pi$
$488$		−	84320.5i		−	0.354074i
$489$	−96577.6	−	203801.i	−0.403886	−	0.852290i
$490$	931136.			3.87812
$491$		−	106092.i		−	0.440068i	−0.975492	−	0.220034i	$-0.929383\pi$
$491$		−	106092.i		−	0.440068i	0.975492	−	0.220034i	$-0.0706168\pi$
$492$	−116558.	+	55234.7i	−0.481516	+	0.228182i
$493$	−200946.			−0.826771
$494$		−	8144.29i		−	0.0333733i
$495$	−194110.	+	237248.i	−0.792204	+	0.968262i
$496$	−391064.			−1.58959
$497$		−	671954.i		−	2.72036i
$498$	−153139.	−	323159.i	−0.617488	−	1.30304i
$499$	−384086.			−1.54251			−0.771254	−	0.636527i	$-0.780370\pi$
$499$	−384086.			−1.54251			−0.771254	+	0.636527i	$0.780370\pi$
$500$		−	31130.2i		−	0.124521i
$501$	−263620.	+	124925.i	−1.05028	+	0.497708i
$502$	304833.			1.20964
$503$			136712.i			0.540343i	0.962812	+	0.270172i	$0.0870804\pi$
$503$			136712.i			0.540343i	−0.962812	+	0.270172i	$0.912920\pi$
$504$	182265.	+	149124.i	0.717532	+	0.587064i
$505$	111312.			0.436476
$506$			225978.i			0.882601i
$507$	109930.	+	231977.i	0.427662	+	0.902464i
$508$	−31401.6			−0.121681
$509$			282297.i			1.08961i	0.838563	+	0.544805i	$0.183396\pi$
$509$			282297.i			1.08961i	−0.838563	+	0.544805i	$0.816604\pi$
$510$	−402645.	+	190807.i	−1.54804	+	0.733589i
$511$	−421226.			−1.61314
$512$		−	175764.i		−	0.670488i
$513$	184980.	+	45893.1i	0.702896	+	0.174386i
$514$	214910.			0.813447
$515$			503933.i			1.90002i
$516$	−45152.3	−	95281.5i	−0.169582	−	0.357857i
$517$	16602.4			0.0621141
$518$			506633.i			1.88814i
$519$	449679.	−	213095.i	1.66943	−	0.791114i
$520$	6884.31			0.0254597
$521$		−	337551.i		−	1.24355i	−0.783195	−	0.621776i	$-0.786411\pi$
$521$		−	337551.i		−	1.24355i	0.783195	−	0.621776i	$-0.213589\pi$
$522$	180251.	−	220309.i	0.661510	−	0.808522i
$523$	−316302.			−1.15637			−0.578187	−	0.815904i	$-0.696240\pi$
$523$	−316302.			−1.15637			−0.578187	+	0.815904i	$0.696240\pi$
$524$		−	228372.i		−	0.831728i
$525$	180232.	+	380330.i	0.653903	+	1.37988i
$526$	382899.			1.38393
$527$		−	355300.i		−	1.27930i
$528$	287983.	−	136470.i	1.03300	−	0.489519i
$529$	118514.			0.423505
$530$		−	852476.i		−	3.03480i
$531$	28410.7	+	23244.8i	0.100761	+	0.0824399i
$532$	−219627.			−0.776000
$533$			9314.44i			0.0327870i
$534$	−104140.	−	219758.i	−0.365202	−	0.770659i
$535$	−128082.			−0.447486
$536$			252807.i			0.879952i
$537$	−138361.	+	65567.1i	−0.479807	+	0.227372i
$538$	−542759.			−1.87518
$539$		−	605069.i		−	2.08270i
$540$	56595.5	−	228119.i	0.194086	−	0.782300i
$541$	66158.9			0.226044			0.113022	−	0.993592i	$-0.463947\pi$
$541$	66158.9			0.226044			0.113022	+	0.993592i	$0.463947\pi$
$542$		−	65986.6i		−	0.224624i
$543$	−68271.4	−	144068.i	−0.231547	−	0.488616i
$544$	311455.			1.05244
$545$			607341.i			2.04475i
$546$	−22420.5	+	10624.7i	−0.0752074	+	0.0356395i
$547$	−167080.			−0.558407			−0.279204	−	0.960232i	$-0.590070\pi$
$547$	−167080.			−0.558407			−0.279204	+	0.960232i	$0.590070\pi$
$548$		−	148748.i		−	0.495326i
$549$	−131642.	+	160897.i	−0.436766	+	0.533832i
$550$	297311.			0.982846
$551$		−	181965.i		−	0.599354i
$552$	50858.6	+	107323.i	0.166912	+	0.352221i
$553$	722684.			2.36319
$554$		−	185764.i		−	0.605260i
$555$	−313200.	+	148420.i	−1.01680	+	0.481845i
$556$	−214327.			−0.693309
$557$		−	302225.i		−	0.974135i	−0.873364	−	0.487068i	$-0.838067\pi$
$557$		−	302225.i		−	0.974135i	0.873364	−	0.487068i	$-0.161933\pi$
$558$	389537.	+	318708.i	1.25107	+	1.02359i
$559$	−7614.21			−0.0243670
$560$		−	955042.i		−	3.04541i
$561$	123989.	+	261646.i	0.393966	+	0.831358i
$562$	587839.			1.86117
$563$		−	33389.3i		−	0.105339i	−0.998612	−	0.0526697i	$-0.983227\pi$
$563$		−	33389.3i		−	0.105339i	0.998612	−	0.0526697i	$-0.0167730\pi$
$564$	−11503.4	+	5451.28i	−0.0361634	+	0.0171372i
$565$	81908.2			0.256585
$566$			381960.i			1.19230i
$567$	−114978.	−	569106.i	−0.357643	−	1.77022i
$568$	−249470.			−0.773252
$569$			462649.i			1.42898i	0.699644	+	0.714491i	$0.253342\pi$
$569$			462649.i			1.42898i	−0.699644	+	0.714491i	$0.746658\pi$
$570$	−172783.	−	364611.i	−0.531803	−	1.12223i
$571$	−149876.			−0.459685			−0.229842	−	0.973228i	$-0.573821\pi$
$571$	−149876.			−0.459685			−0.229842	+	0.973228i	$0.573821\pi$
$572$			6526.50i			0.0199475i
$573$	395104.	−	187233.i	1.20338	−	0.570261i
$574$	674535.			2.04729
$575$			212252.i			0.641973i
$576$	−18593.4	+	22725.6i	−0.0560422	+	0.0684969i
$577$	−101628.			−0.305253			−0.152627	−	0.988284i	$-0.548773\pi$
$577$	−101628.			−0.305253			−0.152627	+	0.988284i	$0.548773\pi$
$578$		−	846.608i		−	0.00253412i
$579$	−124568.	−	262866.i	−0.371577	−	0.784110i
$580$	−224400.			−0.667062
$581$			696408.i			2.06306i
$582$	328017.	−	155442.i	0.968390	−	0.458904i
$583$	−553954.			−1.62981
$584$			156384.i			0.458530i
$585$	−13136.4	−	10747.8i	−0.0383853	−	0.0314057i
$586$	689310.			2.00733
$587$			163854.i			0.475533i	0.971322	+	0.237766i	$0.0764153\pi$
$587$			163854.i			0.475533i	−0.971322	+	0.237766i	$0.923585\pi$
$588$	198670.	+	419239.i	0.574616	+	1.21257i
$589$	321738.			0.927411
$590$		−	77711.8i		−	0.223246i
$591$	−196030.	+	92895.1i	−0.561237	+	0.265961i
$592$	360318.			1.02812
$593$		−	366216.i		−	1.04142i	−0.853733	−	0.520712i	$-0.825667\pi$
$593$		−	366216.i		−	1.04142i	0.853733	−	0.520712i	$-0.174333\pi$
$594$	−398078.	−	98762.0i	−1.12822	−	0.279909i
$595$	867700.			2.45096
$596$			14103.8i			0.0397047i
$597$	−162567.	−	343053.i	−0.456125	−	0.962528i
$598$	−12512.3			−0.0349893
$599$			630279.i			1.75663i	0.478087	+	0.878313i	$0.341330\pi$
$599$			630279.i			1.75663i	−0.478087	+	0.878313i	$0.658670\pi$
$600$	141201.	−	66912.9i	0.392226	−	0.185869i
$601$	494937.			1.37025			0.685127	−	0.728424i	$-0.259747\pi$
$601$	494937.			1.37025			0.685127	+	0.728424i	$0.259747\pi$
$602$			551407.i			1.52153i
$603$	394683.	−	482397.i	1.08546	−	1.32669i
$604$	193761.			0.531119
$605$			75547.2i			0.206399i
$606$	63779.1	+	134588.i	0.173673	+	0.366490i
$607$	465849.			1.26435			0.632175	−	0.774825i	$-0.282162\pi$
$607$	465849.			1.26435			0.632175	+	0.774825i	$0.282162\pi$
$608$			282035.i			0.762948i
$609$	−500933.	+	237384.i	−1.35066	+	0.640053i
$610$	440103.			1.18275
$611$			919.272i			0.00246242i
$612$	−171819.	−	140577.i	−0.458742	−	0.375330i
$613$	453160.			1.20595			0.602976	−	0.797759i	$-0.293981\pi$
$613$	453160.			1.20595			0.602976	+	0.797759i	$0.293981\pi$
$614$		−	203691.i		−	0.540299i
$615$	197608.	+	416997.i	0.522461	+	1.10251i
$616$	−323966.			−0.853764
$617$		−	382747.i		−	1.00541i	−0.864459	−	0.502703i	$-0.832339\pi$
$617$		−	382747.i		−	1.00541i	0.864459	−	0.502703i	$-0.167661\pi$
$618$	−609308.	+	288741.i	−1.59536	+	0.756016i
$619$	−514375.			−1.34245			−0.671225	−	0.741253i	$-0.734232\pi$
$619$	−514375.			−1.34245			−0.671225	+	0.741253i	$0.734232\pi$
$620$		−	396769.i		−	1.03218i
$621$	70506.9	−	284191.i	0.182830	−	0.736932i
$622$	−10888.7			−0.0281445
$623$			473579.i			1.22016i
$624$	7556.32	+	15945.5i	0.0194062	+	0.0409515i
$625$	−441649.			−1.13062
$626$		−	711128.i		−	1.81468i
$627$	−236931.	+	112277.i	−0.602679	+	0.285599i
$628$	436114.			1.10581
$629$			327366.i			0.827432i
$630$	−778337.	+	951313.i	−1.96104	+	2.39686i
$631$	683875.			1.71758			0.858792	−	0.512324i	$-0.171215\pi$
$631$	683875.			1.71758			0.858792	+	0.512324i	$0.171215\pi$
$632$		−	268304.i		−	0.671726i
$633$	156218.	+	329656.i	0.389874	+	0.822722i
$634$	510315.			1.26958
$635$			112342.i			0.278610i
$636$	383823.	−	181887.i	0.948891	−	0.449663i
$637$	33502.5			0.0825655
$638$			391588.i			0.962029i
$639$	476029.	+	389473.i	1.16582	+	0.953841i
$640$	−524046.			−1.27941
$641$		−	218772.i		−	0.532447i	−0.963911	−	0.266223i	$-0.914224\pi$
$641$		−	218772.i		−	0.532447i	0.963911	−	0.266223i	$-0.0857759\pi$
$642$	−73387.5	−	154864.i	−0.178054	−	0.375735i
$643$	−187495.			−0.453490			−0.226745	−	0.973954i	$-0.572808\pi$
$643$	−187495.			−0.453490			−0.226745	+	0.973954i	$0.572808\pi$
$644$			337419.i			0.813575i
$645$	−340880.	+	161537.i	−0.819373	+	0.388287i
$646$	−381102.			−0.913222
$647$		−	390577.i		−	0.933035i	−0.884512	−	0.466518i	$-0.845508\pi$
$647$		−	390577.i		−	0.933035i	0.884512	−	0.466518i	$-0.154492\pi$
$648$	−211286.	+	42686.8i	−0.503177	+	0.101659i
$649$	−50498.5			−0.119892
$650$			16462.0i			0.0389634i
$651$	−419727.	−	885718.i	−0.990386	−	2.08994i
$652$	−237881.			−0.559583
$653$		−	801944.i		−	1.88069i	−0.340219	−	0.940346i	$-0.610501\pi$
$653$		−	801944.i		−	1.88069i	0.340219	−	0.940346i	$-0.389499\pi$
$654$	−734339.	+	347991.i	−1.71688	+	0.813603i
$655$	−817027.			−1.90438
$656$		−	479730.i		−	1.11478i
$657$	244148.	−	298407.i	0.565617	−	0.691319i
$658$	66571.9			0.153759
$659$		−	892.461i		−	0.00205503i	−0.999999	−	0.00102752i	$-0.999673\pi$
$659$		−	892.461i		−	0.00205503i	0.999999	−	0.00102752i	$-0.000327069\pi$
$660$	138461.	+	292184.i	0.317863	+	0.670762i
$661$	610212.			1.39662			0.698309	−	0.715796i	$-0.253936\pi$
$661$	610212.			1.39662			0.698309	+	0.715796i	$0.253936\pi$
$662$		−	206808.i		−	0.471902i
$663$	−14487.3	+	6865.27i	−0.0329579	+	0.0156182i
$664$	258548.			0.586416
$665$			785737.i			1.77678i
$666$	−358912.	−	293651.i	−0.809169	−	0.662038i
$667$	−279558.			−0.628376
$668$			307704.i			0.689573i
$669$	−63054.1	−	133058.i	−0.140884	−	0.297297i
$670$	−1.31950e6			−2.93941
$671$		−	285987.i		−	0.635185i
$672$	776417.	−	367931.i	1.71932	−	0.814756i
$673$	385805.			0.851801			0.425900	−	0.904770i	$-0.359957\pi$
$673$	385805.			0.851801			0.425900	+	0.904770i	$0.359957\pi$
$674$			531538.i			1.17008i
$675$	−373901.	−	92763.5i	−0.820632	−	0.203596i
$676$	270770.			0.592525
$677$			106731.i			0.232871i	0.993198	+	0.116435i	$0.0371468\pi$
$677$			106731.i			0.232871i	−0.993198	+	0.116435i	$0.962853\pi$
$678$	46931.3	+	99035.7i	0.102095	+	0.215443i
$679$	−706877.			−1.53322
$680$		−	322143.i		−	0.696675i
$681$	−708127.	+	335569.i	−1.52692	+	0.723582i
$682$	−692381.			−1.48859
$683$		−	115229.i		−	0.247013i	−0.992344	−	0.123507i	$-0.960586\pi$
$683$		−	115229.i		−	0.247013i	0.992344	−	0.123507i	$-0.0394140\pi$
$684$	127298.	−	155589.i	0.272089	−	0.332558i
$685$	−532163.			−1.13413
$686$		−	1.35340e6i		−	2.87594i
$687$	64212.7	+	135503.i	0.136053	+	0.287102i
$688$	392162.			0.828492
$689$		−	30672.3i		−	0.0646112i
$690$	−560163.	+	265452.i	−1.17657	+	0.557555i
$691$	−735740.			−1.54088			−0.770439	−	0.637513i	$-0.779963\pi$
$691$	−735740.			−1.54088			−0.770439	+	0.637513i	$0.779963\pi$
$692$		−	524876.i		−	1.09609i
$693$	618180.	+	505777.i	1.28721	+	1.05316i
$694$	184922.			0.383945
$695$			766777.i			1.58745i
$696$	88131.0	+	185976.i	0.181933	+	0.383919i
$697$	435858.			0.897179
$698$		−	60014.7i		−	0.123182i
$699$	214169.	−	101491.i	0.438332	−	0.207718i
$700$	443931.			0.905981
$701$		−	12876.8i		−	0.0262043i	−0.999914	−	0.0131022i	$-0.995829\pi$
$701$		−	12876.8i		−	0.0262043i	0.999914	−	0.0131022i	$-0.00417067\pi$
$702$	5468.43	−	22041.5i	0.0110966	−	0.0447267i
$703$	−296443.			−0.599833
$704$		−	40393.6i		−	0.0815017i
$705$	19502.5	+	41154.8i	0.0392386	+	0.0828022i
$706$	648745.			1.30156
$707$		−	290038.i		−	0.580251i
$708$	34989.3	−	16580.8i	0.0698022	−	0.0330781i
$709$	495300.			0.985316			0.492658	−	0.870223i	$-0.336025\pi$
$709$	495300.			0.985316			0.492658	+	0.870223i	$0.336025\pi$
$710$		−	1.30208e6i		−	2.58299i
$711$	−418877.	+	511968.i	−0.828605	+	1.01275i
$712$	175821.			0.346825
$713$		−	494296.i		−	0.972318i
$714$	497170.	+	1.04914e6i	0.975234	+	2.05796i
$715$	23349.2			0.0456731
$716$			161499.i			0.315024i
$717$	540629.	−	256195.i	1.05163	−	0.498348i
$718$	−1.04595e6			−2.02890
$719$			670622.i			1.29724i	0.761113	+	0.648619i	$0.224653\pi$
$719$			670622.i			1.29724i	−0.761113	+	0.648619i	$0.775347\pi$
$720$	676576.	+	553555.i	1.30512	+	1.06781i
$721$	1.31306e6			2.52589
$722$			312896.i			0.600241i
$723$	−61569.5	−	129926.i	−0.117785	−	0.248552i
$724$	−168160.			−0.320808
$725$			367804.i			0.699747i
$726$	−91344.5	+	43286.6i	−0.173304	+	0.0821259i
$727$	−191652.			−0.362614			−0.181307	−	0.983427i	$-0.558033\pi$
$727$	−191652.			−0.362614			−0.181307	+	0.983427i	$0.558033\pi$
$728$		−	17937.9i		−	0.0338461i
$729$	469812.	+	248408.i	0.884034	+	0.467423i
$730$	−816233.			−1.53168
$731$			356298.i			0.666773i
$732$	93901.7	+	198154.i	0.175247	+	0.369811i
$733$	−2889.84			−0.00537856			−0.00268928	−	0.999996i	$-0.500856\pi$
$733$	−2889.84			−0.00537856			−0.00268928	+	0.999996i	$0.500856\pi$
$734$			621496.i			1.15358i
$735$	1.49987e6	−	710763.i	2.77638	−	1.31568i
$736$	433298.			0.799892
$737$			857435.i			1.57858i
$738$	−390969.	+	477857.i	−0.717844	+	0.877376i
$739$	−779522.			−1.42738			−0.713690	−	0.700462i	$-0.752977\pi$
$739$	−779522.			−1.42738			−0.713690	+	0.700462i	$0.752977\pi$
$740$			365575.i			0.667595i
$741$	−6216.78	−	13118.8i	−0.0113221	−	0.0238923i
$742$	−2.22123e6			−4.03447
$743$			341811.i			0.619168i	0.950872	+	0.309584i	$0.100190\pi$
$743$			341811.i			0.619168i	−0.950872	+	0.309584i	$0.899810\pi$
$744$	−328832.	+	155828.i	−0.594056	+	0.281513i
$745$	50457.7			0.0909106
$746$			1.08976e6i			1.95818i
$747$	−493353.	−	403647.i	−0.884131	−	0.723371i
$748$	305399.			0.545839
$749$			333733.i			0.594888i
$750$	−63813.1	−	134660.i	−0.113446	−	0.239396i
$751$	110915.			0.196658			0.0983290	−	0.995154i	$-0.468650\pi$
$751$	110915.			0.196658			0.0983290	+	0.995154i	$0.468650\pi$
$752$		−	47346.1i		−	0.0837237i
$753$	491024.	−	232688.i	0.865990	−	0.410378i
$754$	−21682.1			−0.0381381
$755$		−	693199.i		−	1.21609i
$756$	−594392.	−	147467.i	−1.03999	−	0.258018i
$757$	−1.02325e6			−1.78563			−0.892813	−	0.450427i	$-0.851272\pi$
$757$	−1.02325e6			−1.78563			−0.892813	+	0.450427i	$0.851272\pi$
$758$		−	862966.i		−	1.50195i
$759$	172495.	+	364004.i	0.299429	+	0.631862i
$760$	291713.			0.505043
$761$		−	191598.i		−	0.330842i	−0.986223	−	0.165421i	$-0.947102\pi$
$761$		−	191598.i		−	0.330842i	0.986223	−	0.165421i	$-0.0528983\pi$
$762$	−135834.	+	64369.3i	−0.233936	+	0.110858i
$763$	1.58250e6			2.71829
$764$		−	461175.i		−	0.790095i
$765$	−502931.	+	614701.i	−0.859380	+	1.05037i
$766$	207880.			0.354286
$767$		−	2796.09i		−	0.00475292i
$768$	−322619.	−	680799.i	−0.546975	−	1.15424i
$769$	−423911.			−0.716839			−0.358419	−	0.933561i	$-0.616684\pi$
$769$	−423911.			−0.716839			−0.358419	+	0.933561i	$0.616684\pi$
$770$		−	1.69091e6i		−	2.85193i
$771$	346176.	−	164047.i	0.582355	−	0.275968i
$772$	−306823.			−0.514818
$773$			413018.i			0.691210i	0.938380	+	0.345605i	$0.112326\pi$
$773$			413018.i			0.691210i	−0.938380	+	0.345605i	$0.887674\pi$
$774$	−390631.	−	319603.i	−0.652056	−	0.533493i
$775$	−650329.			−1.08275
$776$			262435.i			0.435812i
$777$	386727.	+	816082.i	0.640565	+	1.35174i
$778$	60553.3			0.100041
$779$			394686.i			0.650395i
$780$	−16178.2	+	7666.56i	−0.0265913	+	0.0126012i
$781$	−846116.			−1.38716
$782$			585499.i			0.957442i
$783$	122179.	−	492464.i	0.199284	−	0.803251i
$784$	−1.72551e6			−2.80728
$785$		−	1.56024e6i		−	2.53194i
$786$	−468135.	−	987871.i	−0.757751	−	1.59903i
$787$	682596.			1.10208			0.551041	−	0.834478i	$-0.314231\pi$
$787$	682596.			1.10208			0.551041	+	0.834478i	$0.314231\pi$
$788$			228810.i			0.368488i
$789$	616773.	−	292278.i	0.990767	−	0.469507i
$790$	1.40038e6			2.24385
$791$		−	213422.i		−	0.341104i
$792$	187775.	−	229506.i	0.299355	−	0.365884i
$793$	15835.0			0.0251809
$794$			331155.i			0.525279i
$795$	−650720.	−	1.37317e6i	−1.02958	−	2.17264i
$796$	−400420.			−0.631960
$797$			753632.i			1.18643i	0.805043	+	0.593216i	$0.202142\pi$
$797$			753632.i			1.18643i	−0.805043	+	0.593216i	$0.797858\pi$
$798$	−950039.	+	450207.i	−1.49189	+	0.706980i
$799$	43016.2			0.0673811
$800$		−	570076.i		−	0.890743i
$801$	−335496.	−	274493.i	−0.522904	−	0.427825i
$802$	784737.			1.22004
$803$			530402.i			0.822573i
$804$	−281533.	−	594098.i	−0.435529	−	0.919064i
$805$	1.20715e6			1.86282
$806$		−	38337.0i		−	0.0590130i
$807$	−874275.	+	414304.i	−1.34246	+	0.636168i
$808$	−107680.			−0.164934
$809$		−	682052.i		−	1.04213i	−0.853518	−	0.521063i	$-0.825536\pi$
$809$		−	682052.i		−	1.04213i	0.853518	−	0.521063i	$-0.174464\pi$
$810$	−222800.	−	1.10279e6i	−0.339582	−	1.68082i
$811$	784718.			1.19309			0.596543	−	0.802581i	$-0.296541\pi$
$811$	784718.			1.19309			0.596543	+	0.802581i	$0.296541\pi$
$812$			584701.i			0.886792i
$813$	−50369.5	−	106291.i	−0.0762055	−	0.160811i
$814$	637946.			0.962798
$815$			851044.i			1.28126i
$816$	746152.	−	353589.i	1.12059	−	0.531028i
$817$	−322642.			−0.483366
$818$			20697.6i			0.0309324i
$819$	−28004.8	+	34228.5i	−0.0417508	+	0.0510294i
$820$	486729.			0.723868
$821$		−	579986.i		−	0.860461i	−0.902719	−	0.430231i	$-0.858432\pi$
$821$		−	579986.i		−	0.860461i	0.902719	−	0.430231i	$-0.141568\pi$
$822$	−304916.	−	643442.i	−0.451270	−	0.952282i
$823$	−1.12641e6			−1.66302			−0.831512	−	0.555507i	$-0.812524\pi$
$823$	−1.12641e6			−1.66302			−0.831512	+	0.555507i	$0.812524\pi$
$824$		−	487486.i		−	0.717973i
$825$	478907.	−	226946.i	0.703629	−	0.333438i
$826$	−202488.			−0.296783
$827$			380772.i			0.556743i	0.960474	+	0.278371i	$0.0897946\pi$
$827$			380772.i			0.556743i	−0.960474	+	0.278371i	$0.910205\pi$
$828$	−239036.	−	195573.i	−0.348661	−	0.285264i
$829$	292205.			0.425186			0.212593	−	0.977141i	$-0.431809\pi$
$829$	292205.			0.425186			0.212593	+	0.977141i	$0.431809\pi$
$830$			1.34947e6i			1.95887i
$831$	−141799.	−	299228.i	−0.205339	−	0.433311i
$832$	2236.58			0.00323101
$833$		−	1.56771e6i		−	2.25931i
$834$	−927114.	+	439344.i	−1.33291	+	0.631644i
$835$	1.10084e6			1.57889
$836$			276551.i			0.395697i
$837$	870744.	+	216029.i	1.24291	+	0.308362i
$838$	66329.0			0.0944530
$839$		−	546728.i		−	0.776689i	−0.921514	−	0.388344i	$-0.873047\pi$
$839$		−	546728.i		−	0.776689i	0.921514	−	0.388344i	$-0.126953\pi$
$840$	−380557.	−	803060.i	−0.539338	−	1.13812i
$841$	222846.			0.315074
$842$		−	440149.i		−	0.620833i
$843$	946889.	−	448715.i	1.33243	−	0.631415i
$844$	384782.			0.540169
$845$		−	968707.i		−	1.35669i
$846$	−38586.0	+	47161.3i	−0.0539124	+	0.0658938i
$847$	196848.			0.274387
$848$			1.57975e6i			2.19682i
$849$	291561.	+	615260.i	0.404496	+	0.853578i
$850$	770321.			1.06619
$851$			455434.i			0.628879i
$852$	586255.	−	277816.i	0.807621	−	0.382718i
$853$	−474069.			−0.651543			−0.325772	−	0.945449i	$-0.605624\pi$
$853$	−474069.			−0.651543			−0.325772	+	0.945449i	$0.605624\pi$
$854$		−	1.14674e6i		−	1.57235i
$855$	−556636.	−	455424.i	−0.761446	−	0.622994i
$856$	123902.			0.169095
$857$			686889.i			0.935245i	0.883928	+	0.467622i	$0.154889\pi$
$857$			686889.i			0.935245i	−0.883928	+	0.467622i	$0.845111\pi$
$858$	13378.5	+	28231.7i	0.0181733	+	0.0383497i
$859$	−251544.			−0.340900			−0.170450	−	0.985366i	$-0.554522\pi$
$859$	−251544.			−0.340900			−0.170450	+	0.985366i	$0.554522\pi$
$860$			397883.i			0.537971i
$861$	1.08654e6	−	514892.i	1.46568	−	0.694560i
$862$	165352.			0.222533
$863$			779321.i			1.04639i	0.852212	+	0.523196i	$0.175260\pi$
$863$			779321.i			1.04639i	−0.852212	+	0.523196i	$0.824740\pi$
$864$	−189370.	+	763291.i	−0.253679	+	1.02250i
$865$	−1.87780e6			−2.50967
$866$			246.972i			0.000329315i
$867$	−646.240	−	1363.71i	−0.000859717	−	0.00181420i
$868$	−1.03383e6			−1.37218
$869$		−	909995.i		−	1.20503i
$870$	−970686.	+	459991.i	−1.28245	+	0.607731i
$871$	−47476.0			−0.0625803
$872$		−	587520.i		−	0.772662i
$873$	409715.	−	500770.i	0.537593	−	0.657067i
$874$	−530192.			−0.694082
$875$			290193.i			0.379027i
$876$	−174154.	−	367504.i	−0.226947	−	0.478910i
$877$	−1.13477e6			−1.47540			−0.737698	−	0.675131i	$-0.764087\pi$
$877$	−1.13477e6			−1.47540			−0.737698	+	0.675131i	$0.764087\pi$
$878$			1.81749e6i			2.35767i
$879$	1.11034e6	−	526170.i	1.43707	−	0.681002i
$880$	−1.20258e6			−1.55291
$881$			651746.i			0.839705i	0.907592	+	0.419853i	$0.137918\pi$
$881$			651746.i			0.839705i	−0.907592	+	0.419853i	$0.862082\pi$
$882$	1.71878e6	+	1.40625e6i	2.20944	+	1.80770i
$883$	−740498.			−0.949736			−0.474868	−	0.880057i	$-0.657504\pi$
$883$	−740498.			−0.949736			−0.474868	+	0.880057i	$0.657504\pi$
$884$			16909.9i			0.0216390i
$885$	−59319.7	−	125178.i	−0.0757378	−	0.159824i
$886$	1.36955e6			1.74466
$887$		−	846445.i		−	1.07585i	−0.842993	−	0.537925i	$-0.819208\pi$
$887$		−	846445.i		−	1.07585i	0.842993	−	0.537925i	$-0.180792\pi$
$888$	302979.	−	143576.i	0.384226	−	0.182078i
$889$	292722.			0.370384
$890$			917681.i			1.15854i
$891$	−716611.	+	144779.i	−0.902668	+	0.182369i
$892$	−155309.			−0.195194
$893$			38952.9i			0.0488468i
$894$	28911.0	+	61008.7i	0.0361732	+	0.0763337i
$895$	577779.			0.721300
$896$			1.36547e6i			1.70085i
$897$	−20154.8	+	9551.02i	−0.0250492	+	0.0118704i
$898$	−61792.8			−0.0766276
$899$		−	856548.i		−	1.05982i
$900$	−257308.	+	314492.i	−0.317664	+	0.388261i
$901$	−1.43527e6			−1.76801
$902$		−	849366.i		−	1.04395i
$903$	420905.	+	888204.i	0.516189	+	1.08927i
$904$	−79235.1			−0.0969573
$905$			601609.i			0.734543i
$906$	838151.	−	397185.i	1.02109	−	0.483879i
$907$	−1.01416e6			−1.23280			−0.616402	−	0.787432i	$-0.711410\pi$
$907$	−1.01416e6			−1.23280			−0.616402	+	0.787432i	$0.711410\pi$
$908$			826543.i			1.00252i
$909$	205470.	+	168110.i	0.248669	+	0.203454i
$910$	93625.2			0.113060
$911$		−	353146.i		−	0.425518i	−0.977105	−	0.212759i	$-0.931755\pi$
$911$		−	353146.i		−	0.425518i	0.977105	−	0.212759i	$-0.0682449\pi$
$912$	320188.	+	675670.i	0.384960	+	0.812353i
$913$	876908.			1.05199
$914$			1.84213e6i			2.20510i
$915$	708916.	−	335943.i	0.846745	−	0.401258i
$916$	158163.			0.188501
$917$			2.12886e6i			2.53168i
$918$	−1.03141e6	−	255888.i	−1.22389	−	0.303644i
$919$	571593.			0.676793			0.338396	−	0.941004i	$-0.390116\pi$
$919$	571593.			0.676793			0.338396	+	0.941004i	$0.390116\pi$
$920$		−	448167.i		−	0.529498i
$921$	−155483.	−	328104.i	−0.183301	−	0.386806i
$922$	1.10966e6			1.30535
$923$		−	46849.3i		−	0.0549920i
$924$	761322.	−	360777.i	0.891711	−	0.422567i
$925$	599200.			0.700306
$926$		−	959750.i		−	1.11927i
$927$	−761067.	+	930204.i	−0.885652	+	1.08248i
$928$	750847.			0.871877
$929$		−	613192.i		−	0.710502i	−0.934771	−	0.355251i	$-0.884395\pi$
$929$		−	613192.i		−	0.710502i	0.934771	−	0.355251i	$-0.115605\pi$
$930$	−813328.	−	1.71630e6i	−0.940372	−	1.98440i
$931$	1.41962e6			1.63785
$932$		−	249984.i		−	0.287793i
$933$	−17539.4	+	8311.63i	−0.0201489	+	0.00954823i
$934$	−2.16080e6			−2.47697
$935$		−	1.09260e6i		−	1.24979i
$936$	12707.7	+	10397.1i	0.0145049	+	0.0118675i
$937$	1.14653e6			1.30589			0.652944	−	0.757406i	$-0.273534\pi$
$937$	1.14653e6			1.30589			0.652944	+	0.757406i	$0.273534\pi$
$938$			3.43812e6i			3.90765i
$939$	−542825.	−	1.14548e6i	−0.615642	−	1.29914i
$940$	48036.8			0.0543649
$941$		−	1.17604e6i		−	1.32814i	−0.747670	−	0.664070i	$-0.768828\pi$
$941$		−	1.17604e6i		−	1.32814i	0.747670	−	0.664070i	$-0.231172\pi$
$942$	1.88650e6	−	893981.i	2.12596	−	1.00746i
$943$	606369.			0.681889
$944$			144010.i			0.161602i
$945$	−527578.	+	2.12650e6i	−0.590776	+	2.38123i
$946$	694325.			0.775855
$947$		−	777766.i		−	0.867259i	−0.901091	−	0.433629i	$-0.857233\pi$
$947$		−	777766.i		−	0.867259i	0.901091	−	0.433629i	$-0.142767\pi$
$948$	298790.	+	630515.i	0.332468	+	0.701583i
$949$	−29368.3			−0.0326096
$950$			697556.i			0.772915i
$951$	822014.	−	389538.i	0.908904	−	0.430714i
$952$	−839382.			−0.926160
$953$			1840.55i			0.00202657i	0.999999	+	0.00101329i	$0.000322539\pi$
$953$			1840.55i			0.00202657i	−0.999999	+	0.00101329i	$0.999677\pi$
$954$	1.28746e6	−	1.57358e6i	1.41461	−	1.72899i
$955$	−1.64990e6			−1.80906
$956$		−	631036.i		−	0.690459i
$957$	298911.	+	630769.i	0.326375	+	0.688726i
$958$	315151.			0.343390
$959$			1.38662e6i			1.50772i
$960$	100129.	−	47449.6i	0.108647	−	0.0514861i
$961$	590972.			0.639912
$962$			35322.9i			0.0381686i
$963$	−236425.	−	193436.i	−0.254941	−	0.208586i
$964$	−151652.			−0.163190
$965$			1.09769e6i			1.17876i
$966$	691667.	+	1.45957e6i	0.741213	+	1.56413i
$967$	408089.			0.436417			0.218209	−	0.975902i	$-0.429979\pi$
$967$	408089.			0.436417			0.218209	+	0.975902i	$0.429979\pi$
$968$		−	73081.6i		−	0.0779933i
$969$	−613878.	+	290906.i	−0.653784	+	0.309817i
$970$	−1.36976e6			−1.45579
$971$			109032.i			0.115642i	0.998327	+	0.0578209i	$0.0184152\pi$
$971$			109032.i			0.115642i	−0.998327	+	0.0578209i	$0.981585\pi$
$972$	448987.	−	335609.i	0.475227	−	0.355223i
$973$	1.99793e6			2.11035
$974$		−	2.36880e6i		−	2.49696i
$975$	12566.0	+	26517.0i	0.0132186	+	0.0278943i
$976$	−815566.			−0.856168
$977$		−	874095.i		−	0.915735i	−0.889020	−	0.457867i	$-0.848613\pi$
$977$		−	874095.i		−	0.915735i	0.889020	−	0.457867i	$-0.151387\pi$
$978$	−1.02900e6	+	487627.i	−1.07582	+	0.509811i
$979$	596325.			0.622183
$980$		−	1.75069e6i		−	1.82287i
$981$	−917239.	+	1.12108e6i	−0.953114	+	1.16493i
$982$	−535665.			−0.555482
$983$		−	641128.i		−	0.663495i	−0.943368	−	0.331747i	$-0.892362\pi$
$983$		−	641128.i		−	0.663495i	0.943368	−	0.331747i	$-0.107638\pi$
$984$	−191159.	−	403388.i	−0.197426	−	0.416613i
$985$	818593.			0.843715
$986$			1.01459e6i			1.04361i
$987$	107234.	−	50816.3i	0.110077	−	0.0521637i
$988$	−15312.6			−0.0156868
$989$			495684.i			0.506772i
$990$	1.19788e6	+	980073.i	1.22220	+	0.999972i
$991$	−290313.			−0.295610			−0.147805	−	0.989017i	$-0.547221\pi$
$991$	−290313.			−0.295610			−0.147805	+	0.989017i	$0.547221\pi$
$992$			1.32760e6i			1.34910i
$993$	−157863.	−	333126.i	−0.160096	−	0.337839i
$994$	−3.39274e6			−3.43382
$995$			1.43255e6i			1.44698i
$996$	−607590.	+	287927.i	−0.612480	+	0.290244i
$997$	996017.			1.00202			0.501010	−	0.865442i	$-0.332962\pi$
$997$	996017.			1.00202			0.501010	+	0.865442i	$0.332962\pi$
$998$			1.93928e6i			1.94706i
$999$	−802286.	−	199045.i	−0.803893	−	0.199443i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.b.a.119.18	✓	78
3.2	odd	2	inner	177.5.b.a.119.61	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.b.a.119.18	✓	78	1.1	even	1	trivial
177.5.b.a.119.61	yes	78	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.04906i q^{2} +(-3.85410 - 8.13301i) q^{3} -9.49305 q^{4} +33.9624i q^{5} +(-41.0641 + 19.4596i) q^{6} +88.4932 q^{7} -32.8540i q^{8} +(-51.2919 + 62.6909i) q^{9} +O(q^{10})\)	\(q-5.04906i q^{2} +(-3.85410 - 8.13301i) q^{3} -9.49305 q^{4} +33.9624i q^{5} +(-41.0641 + 19.4596i) q^{6} +88.4932 q^{7} -32.8540i q^{8} +(-51.2919 + 62.6909i) q^{9} +171.478 q^{10} -111.430i q^{11} +(36.5871 + 77.2071i) q^{12} +6.16984 q^{13} -446.808i q^{14} +(276.217 - 130.894i) q^{15} -317.771 q^{16} -288.710i q^{17} +(316.530 + 258.976i) q^{18} +261.438 q^{19} -322.407i q^{20} +(-341.062 - 719.717i) q^{21} -562.615 q^{22} -401.655i q^{23} +(-267.202 + 126.623i) q^{24} -528.444 q^{25} -31.1519i q^{26} +(707.550 + 175.541i) q^{27} -840.071 q^{28} -696.014i q^{29} +(-660.894 - 1394.64i) q^{30} +1230.65 q^{31} +1078.78i q^{32} +(-906.259 + 429.461i) q^{33} -1457.71 q^{34} +3005.44i q^{35} +(486.916 - 595.128i) q^{36} -1133.89 q^{37} -1320.02i q^{38} +(-23.7791 - 50.1794i) q^{39} +1115.80 q^{40} +1509.67i q^{41} +(-3633.90 + 1722.04i) q^{42} -1234.10 q^{43} +1057.81i q^{44} +(-2129.13 - 1741.99i) q^{45} -2027.98 q^{46} +148.995i q^{47} +(1224.72 + 2584.43i) q^{48} +5430.05 q^{49} +2668.15i q^{50} +(-2348.08 + 1112.72i) q^{51} -58.5706 q^{52} -4971.34i q^{53} +(886.317 - 3572.46i) q^{54} +3784.42 q^{55} -2907.36i q^{56} +(-1007.61 - 2126.28i) q^{57} -3514.22 q^{58} -453.188i q^{59} +(-2622.14 + 1242.59i) q^{60} +2566.52 q^{61} -6213.62i q^{62} +(-4538.98 + 5547.72i) q^{63} +362.503 q^{64} +209.542i q^{65} +(2168.37 + 4575.76i) q^{66} -7694.85 q^{67} +2740.74i q^{68} +(-3266.67 + 1548.02i) q^{69} +15174.7 q^{70} -7593.28i q^{71} +(2059.65 + 1685.14i) q^{72} -4759.98 q^{73} +5725.10i q^{74} +(2036.68 + 4297.84i) q^{75} -2481.85 q^{76} -9860.77i q^{77} +(-253.359 + 120.062i) q^{78} +8166.54 q^{79} -10792.3i q^{80} +(-1299.29 - 6431.06i) q^{81} +7622.44 q^{82} +7869.62i q^{83} +(3237.71 + 6832.31i) q^{84} +9805.27 q^{85} +6231.06i q^{86} +(-5660.69 + 2682.50i) q^{87} -3660.91 q^{88} +5351.59i q^{89} +(-8795.44 + 10750.1i) q^{90} +545.989 q^{91} +3812.94i q^{92} +(-4743.04 - 10008.9i) q^{93} +752.283 q^{94} +8879.07i q^{95} +(8773.74 - 4157.73i) q^{96} -7987.92 q^{97} -27416.7i q^{98} +(6985.62 + 5715.43i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9}+O(q^{10}) \) 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9	\( 78 q - 612 q^{4} + 64 q^{6} + 76 q^{7} - 100 q^{9} - 156 q^{10} - 4 q^{13} + 13 q^{15} + 4948 q^{16} + 22 q^{18} + 812 q^{19} - 173 q^{21} - 1644 q^{22} - 678 q^{24} - 8238 q^{25} + 777 q^{27} - 3764 q^{28} + 1374 q^{30} + 4664 q^{31} - 1042 q^{33} + 3244 q^{34} + 3648 q^{36} - 3960 q^{37} - 7078 q^{39} - 1576 q^{40} + 4934 q^{42} - 1492 q^{43} - 2063 q^{45} - 2036 q^{46} - 2620 q^{48} + 24274 q^{49} + 7300 q^{51} + 8408 q^{52} - 14766 q^{54} + 9780 q^{55} + 6939 q^{57} - 3856 q^{58} + 4712 q^{60} - 212 q^{61} - 7438 q^{63} - 45760 q^{64} + 3048 q^{66} - 12972 q^{67} + 21672 q^{69} + 5828 q^{70} - 866 q^{72} - 5240 q^{73} - 20922 q^{75} + 12368 q^{76} - 16508 q^{78} - 14976 q^{79} + 25524 q^{81} - 14484 q^{82} + 9540 q^{84} + 11572 q^{85} + 5695 q^{87} + 62160 q^{88} - 31672 q^{90} + 8284 q^{91} - 9590 q^{93} - 10992 q^{94} + 34102 q^{96} - 55000 q^{97} - 14254 q^{99}+O(q^{100}) \) 78 * q - 612 * q^4 + 64 * q^6 + 76 * q^7 - 100 * q^9 - 156 * q^10 - 4 * q^13 + 13 * q^15 + 4948 * q^16 + 22 * q^18 + 812 * q^19 - 173 * q^21 - 1644 * q^22 - 678 * q^24 - 8238 * q^25 + 777 * q^27 - 3764 * q^28 + 1374 * q^30 + 4664 * q^31 - 1042 * q^33 + 3244 * q^34 + 3648 * q^36 - 3960 * q^37 - 7078 * q^39 - 1576 * q^40 + 4934 * q^42 - 1492 * q^43 - 2063 * q^45 - 2036 * q^46 - 2620 * q^48 + 24274 * q^49 + 7300 * q^51 + 8408 * q^52 - 14766 * q^54 + 9780 * q^55 + 6939 * q^57 - 3856 * q^58 + 4712 * q^60 - 212 * q^61 - 7438 * q^63 - 45760 * q^64 + 3048 * q^66 - 12972 * q^67 + 21672 * q^69 + 5828 * q^70 - 866 * q^72 - 5240 * q^73 - 20922 * q^75 + 12368 * q^76 - 16508 * q^78 - 14976 * q^79 + 25524 * q^81 - 14484 * q^82 + 9540 * q^84 + 11572 * q^85 + 5695 * q^87 + 62160 * q^88 - 31672 * q^90 + 8284 * q^91 - 9590 * q^93 - 10992 * q^94 + 34102 * q^96 - 55000 * q^97 - 14254 * q^99

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