LMFDB - Embedded newform 180.8.e.a.71.38

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,8,Mod(71,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("180.71");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	180.e (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:	$56.2293045871$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			71.38
Character	$\chi$	$=$	180.71
Dual form			180.8.e.a.71.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.90287 + 9.65174i) q^{2} +(-58.3123 + 113.946i) q^{4} +125.000i q^{5} -122.396i q^{7} +(-1443.99 + 109.792i) q^{8} +O(q^{10})$	\(q+(5.90287 + 9.65174i) q^{2} +(-58.3123 + 113.946i) q^{4} +125.000i q^{5} -122.396i q^{7} +(-1443.99 + 109.792i) q^{8} +(-1206.47 + 737.858i) q^{10} +7266.48 q^{11} +13662.9 q^{13} +(1181.34 - 722.489i) q^{14} +(-9583.34 - 13288.9i) q^{16} -15039.4i q^{17} -26490.9i q^{19} +(-14243.2 - 7289.04i) q^{20} +(42893.0 + 70134.2i) q^{22} +75846.1 q^{23} -15625.0 q^{25} +(80650.5 + 131871. i) q^{26} +(13946.6 + 7137.21i) q^{28} +204032. i q^{29} +63009.3i q^{31} +(71692.0 - 170939. i) q^{32} +(145157. - 88775.7i) q^{34} +15299.5 q^{35} -310172. q^{37} +(255683. - 156372. i) q^{38} +(-13723.9 - 180498. i) q^{40} -728345. i q^{41} +788528. i q^{43} +(-423725. + 827985. i) q^{44} +(447709. + 732047. i) q^{46} -125284. q^{47} +808562. q^{49} +(-92232.3 - 150809. i) q^{50} +(-796718. + 1.55684e6i) q^{52} +375202. i q^{53} +908310. i q^{55} +(13438.1 + 176739. i) q^{56} +(-1.96926e6 + 1.20437e6i) q^{58} +2.62863e6 q^{59} -895367. q^{61} +(-608149. + 371935. i) q^{62} +(2.07304e6 - 317075. i) q^{64} +1.70787e6i q^{65} +6449.82i q^{67} +(1.71368e6 + 876984. i) q^{68} +(90311.1 + 147667. i) q^{70} -3.46044e6 q^{71} +5.24470e6 q^{73} +(-1.83091e6 - 2.99370e6i) q^{74} +(3.01853e6 + 1.54475e6i) q^{76} -889390. i q^{77} +7.25957e6i q^{79} +(1.66111e6 - 1.19792e6i) q^{80} +(7.02980e6 - 4.29933e6i) q^{82} -8.71203e6 q^{83} +1.87993e6 q^{85} +(-7.61067e6 + 4.65457e6i) q^{86} +(-1.04927e7 + 797798. i) q^{88} +2.87191e6i q^{89} -1.67229e6i q^{91} +(-4.42276e6 + 8.64235e6i) q^{92} +(-739536. - 1.20921e6i) q^{94} +3.31136e6 q^{95} +8.16010e6 q^{97} +(4.77283e6 + 7.80404e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 52 q^{4}+O(q^{10}) $ 56 * q + 52 * q^4	$ 56 q + 52 q^{4} - 6500 q^{10} + 14128 q^{13} - 8060 q^{16} + 259088 q^{22} - 875000 q^{25} - 490976 q^{28} + 40912 q^{34} + 1268048 q^{37} - 266500 q^{40} + 3108200 q^{46} - 3522056 q^{49} - 8882216 q^{52} + 8807592 q^{58} - 3944912 q^{61} - 16633580 q^{64} + 4533000 q^{70} + 7602384 q^{73} - 38876976 q^{76} + 33213064 q^{82} - 15068000 q^{85} - 56145472 q^{88} + 29409456 q^{94} - 45595824 q^{97}+O(q^{100}) $ 56 * q + 52 * q^4 - 6500 * q^10 + 14128 * q^13 - 8060 * q^16 + 259088 * q^22 - 875000 * q^25 - 490976 * q^28 + 40912 * q^34 + 1268048 * q^37 - 266500 * q^40 + 3108200 * q^46 - 3522056 * q^49 - 8882216 * q^52 + 8807592 * q^58 - 3944912 * q^61 - 16633580 * q^64 + 4533000 * q^70 + 7602384 * q^73 - 38876976 * q^76 + 33213064 * q^82 - 15068000 * q^85 - 56145472 * q^88 + 29409456 * q^94 - 45595824 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	5.90287	+	9.65174i	0.521745	+	0.853102i
$2$	5.90287	+	9.65174i	0.521745	+	0.853102i
$3$		0			0
$3$		0			0
$4$	−58.3123	+	113.946i	−0.455565	+	0.890202i
$5$			125.000i			0.447214i
$5$			125.000i			0.447214i
$6$		0			0
$7$		−	122.396i		−	0.134873i	−0.997724	−	0.0674365i	$-0.978518\pi$
$7$		−	122.396i		−	0.134873i	0.997724	−	0.0674365i	$-0.0214820\pi$
$8$	−1443.99	+	109.792i	−0.997122	+	0.0758148i
$9$		0			0
$10$	−1206.47	+	737.858i	−0.381519	+	0.233331i
$11$	7266.48			1.64607			0.823037	−	0.567987i	$-0.192278\pi$
$11$	7266.48			1.64607			0.823037	+	0.567987i	$0.192278\pi$
$12$		0			0
$13$	13662.9			1.72482			0.862408	−	0.506215i	$-0.168956\pi$
$13$	13662.9			1.72482			0.862408	+	0.506215i	$0.168956\pi$
$14$	1181.34	−	722.489i	0.115060	−	0.0703693i
$15$		0			0
$16$	−9583.34	−	13288.9i	−0.584921	−	0.811091i
$17$		−	15039.4i		−	0.742438i	−0.928545	−	0.371219i	$-0.878940\pi$
$17$		−	15039.4i		−	0.742438i	0.928545	−	0.371219i	$-0.121060\pi$
$18$		0			0
$19$		−	26490.9i		−	0.886052i	−0.896509	−	0.443026i	$-0.853905\pi$
$19$		−	26490.9i		−	0.886052i	0.896509	−	0.443026i	$-0.146095\pi$
$20$	−14243.2	−	7289.04i	−0.398111	−	0.203735i
$21$		0			0
$22$	42893.0	+	70134.2i	0.858830	+	1.40427i
$23$	75846.1			1.29983			0.649914	−	0.760008i	$-0.274805\pi$
$23$	75846.1			1.29983			0.649914	+	0.760008i	$0.274805\pi$
$24$		0			0
$25$	−15625.0			−0.200000
$26$	80650.5	+	131871.i	0.899913	+	1.47144i
$27$		0			0
$28$	13946.6	+	7137.21i	0.120064	+	0.0614435i
$29$			204032.i			1.55348i	0.629824	+	0.776738i	$0.283127\pi$
$29$			204032.i			1.55348i	−0.629824	+	0.776738i	$0.716873\pi$
$30$		0			0
$31$			63009.3i			0.379873i	0.981796	+	0.189937i	$0.0608282\pi$
$31$			63009.3i			0.379873i	−0.981796	+	0.189937i	$0.939172\pi$
$32$	71692.0	−	170939.i	0.386764	−	0.922179i
$33$		0			0
$34$	145157.	−	88775.7i	0.633375	−	0.387363i
$35$	15299.5			0.0603170
$36$		0			0
$37$	−310172.			−1.00669			−0.503346	−	0.864085i	$-0.667898\pi$
$37$	−310172.			−1.00669			−0.503346	+	0.864085i	$0.667898\pi$
$38$	255683.	−	156372.i	0.755893	−	0.462293i
$39$		0			0
$40$	−13723.9	−	180498.i	−0.0339054	−	0.445926i
$41$		−	728345.i		−	1.65042i	−0.564828	−	0.825209i	$-0.691057\pi$
$41$		−	728345.i		−	1.65042i	0.564828	−	0.825209i	$-0.308943\pi$
$42$		0			0
$43$			788528.i			1.51244i	0.654319	+	0.756219i	$0.272955\pi$
$43$			788528.i			1.51244i	−0.654319	+	0.756219i	$0.727045\pi$
$44$	−423725.	+	827985.i	−0.749894	+	1.46534i
$45$		0			0
$46$	447709.	+	732047.i	0.678178	+	1.10889i
$47$	−125284.			−0.176017			−0.0880083	−	0.996120i	$-0.528050\pi$
$47$	−125284.			−0.176017			−0.0880083	+	0.996120i	$0.528050\pi$
$48$		0			0
$49$	808562.			0.981809
$50$	−92232.3	−	150809.i	−0.104349	−	0.170620i
$51$		0			0
$52$	−796718.	+	1.55684e6i	−0.785766	+	1.53543i
$53$			375202.i			0.346178i	0.984906	+	0.173089i	$0.0553749\pi$
$53$			375202.i			0.346178i	−0.984906	+	0.173089i	$0.944625\pi$
$54$		0			0
$55$			908310.i			0.736147i
$56$	13438.1	+	176739.i	0.0102254	+	0.134485i
$57$		0			0
$58$	−1.96926e6	+	1.20437e6i	−1.32527	+	0.810518i
$59$	2.62863e6			1.66628			0.833139	−	0.553064i	$-0.186542\pi$
$59$	2.62863e6			1.66628			0.833139	+	0.553064i	$0.186542\pi$
$60$		0			0
$61$	−895367.			−0.505064			−0.252532	−	0.967589i	$-0.581263\pi$
$61$	−895367.			−0.505064			−0.252532	+	0.967589i	$0.581263\pi$
$62$	−608149.	+	371935.i	−0.324070	+	0.198197i
$63$		0			0
$64$	2.07304e6	−	317075.i	0.988504	−	0.151193i
$65$			1.70787e6i			0.771361i
$66$		0			0
$67$			6449.82i			0.00261991i	0.999999	+	0.00130995i	$0.000416971\pi$
$67$			6449.82i			0.00261991i	−0.999999	+	0.00130995i	$0.999583\pi$
$68$	1.71368e6	+	876984.i	0.660920	+	0.338229i
$69$		0			0
$70$	90311.1	+	147667.i	0.0314701	+	0.0514566i
$71$	−3.46044e6			−1.14743			−0.573716	−	0.819054i	$-0.694499\pi$
$71$	−3.46044e6			−1.14743			−0.573716	+	0.819054i	$0.694499\pi$
$72$		0			0
$73$	5.24470e6			1.57794			0.788970	−	0.614432i	$-0.210615\pi$
$73$	5.24470e6			1.57794			0.788970	+	0.614432i	$0.210615\pi$
$74$	−1.83091e6	−	2.99370e6i	−0.525236	−	0.858811i
$75$		0			0
$76$	3.01853e6	+	1.54475e6i	0.788766	+	0.403654i
$77$		−	889390.i		−	0.222011i
$78$		0			0
$79$			7.25957e6i			1.65659i	0.560290	+	0.828297i	$0.310690\pi$
$79$			7.25957e6i			1.65659i	−0.560290	+	0.828297i	$0.689310\pi$
$80$	1.66111e6	−	1.19792e6i	0.362731	−	0.261584i
$81$		0			0
$82$	7.02980e6	−	4.29933e6i	1.40797	−	0.861097i
$83$	−8.71203e6			−1.67242			−0.836211	−	0.548407i	$-0.815234\pi$
$83$	−8.71203e6			−1.67242			−0.836211	+	0.548407i	$0.815234\pi$
$84$		0			0
$85$	1.87993e6			0.332028
$86$	−7.61067e6	+	4.65457e6i	−1.29026	+	0.789106i
$87$		0			0
$88$	−1.04927e7	+	797798.i	−1.64134	+	0.124797i
$89$			2.87191e6i			0.431822i	0.976413	+	0.215911i	$0.0692722\pi$
$89$			2.87191e6i			0.431822i	−0.976413	+	0.215911i	$0.930728\pi$
$90$		0			0
$91$		−	1.67229e6i		−	0.232631i
$92$	−4.42276e6	+	8.64235e6i	−0.592156	+	1.15711i
$93$		0			0
$94$	−739536.	−	1.20921e6i	−0.0918357	−	0.150160i
$95$	3.31136e6			0.396255
$96$		0			0
$97$	8.16010e6			0.907809			0.453904	−	0.891050i	$-0.350031\pi$
$97$	8.16010e6			0.907809			0.453904	+	0.891050i	$0.350031\pi$
$98$	4.77283e6	+	7.80404e6i	0.512254	+	0.837583i
$99$		0			0
$100$	911130.	−	1.78040e6i	0.0911130	−	0.178040i
$101$		−	1.15096e7i		−	1.11156i	−0.831329	−	0.555781i	$-0.812419\pi$
$101$		−	1.15096e7i		−	1.11156i	0.831329	−	0.555781i	$-0.187581\pi$
$102$		0			0
$103$			8.61359e6i			0.776701i	0.921512	+	0.388350i	$0.126955\pi$
$103$			8.61359e6i			0.776701i	−0.921512	+	0.388350i	$0.873045\pi$
$104$	−1.97291e7	+	1.50008e6i	−1.71985	+	0.130766i
$105$		0			0
$106$	−3.62136e6	+	2.21477e6i	−0.295325	+	0.180617i
$107$	−8.11361e6			−0.640281			−0.320141	−	0.947370i	$-0.603730\pi$
$107$	−8.11361e6			−0.640281			−0.320141	+	0.947370i	$0.603730\pi$
$108$		0			0
$109$	−5.67600e6			−0.419807			−0.209904	−	0.977722i	$-0.567315\pi$
$109$	−5.67600e6			−0.419807			−0.209904	+	0.977722i	$0.567315\pi$
$110$	−8.76677e6	+	5.36163e6i	−0.628008	+	0.384081i
$111$		0			0
$112$	−1.62651e6	+	1.17297e6i	−0.109394	+	0.0788900i
$113$			1.85736e6i			0.121093i	0.998165	+	0.0605467i	$0.0192844\pi$
$113$			1.85736e6i			0.121093i	−0.998165	+	0.0605467i	$0.980716\pi$
$114$		0			0
$115$			9.48076e6i			0.581300i
$116$	−2.32486e7	−	1.18976e7i	−1.38291	−	0.707710i
$117$		0			0
$118$	1.55164e7	+	2.53709e7i	0.869371	+	1.42150i
$119$	−1.84077e6			−0.100135
$120$		0			0
$121$	3.33145e7			1.70956
$122$	−5.28523e6	−	8.64185e6i	−0.263514	−	0.430871i
$123$		0			0
$124$	−7.17965e6	−	3.67422e6i	−0.338164	−	0.173057i
$125$		−	1.95312e6i		−	0.0894427i
$126$		0			0
$127$			4.16468e6i			0.180414i	0.995923	+	0.0902068i	$0.0287528\pi$
$127$			4.16468e6i			0.180414i	−0.995923	+	0.0902068i	$0.971247\pi$
$128$	1.52972e7	+	1.81368e7i	0.644730	+	0.764410i
$129$		0			0
$130$	−1.64839e7	+	1.00813e7i	−0.658049	+	0.402453i
$131$	−2.64488e7			−1.02791			−0.513956	−	0.857817i	$-0.671820\pi$
$131$	−2.64488e7			−1.02791			−0.513956	+	0.857817i	$0.671820\pi$
$132$		0			0
$133$	−3.24239e6			−0.119505
$134$	−62252.0	+	38072.4i	−0.00223505	+	0.00136692i
$135$		0			0
$136$	1.65120e6	+	2.17167e7i	0.0562877	+	0.740301i
$137$		−	3.52261e7i		−	1.17042i	−0.810881	−	0.585211i	$-0.801012\pi$
$137$		−	3.52261e7i		−	1.17042i	0.810881	−	0.585211i	$-0.198988\pi$
$138$		0			0
$139$		−	8.44276e6i		−	0.266645i	−0.991073	−	0.133322i	$-0.957435\pi$
$139$		−	8.44276e6i		−	0.266645i	0.991073	−	0.133322i	$-0.0425645\pi$
$140$	−892152.	+	1.74332e6i	−0.0274783	+	0.0536944i
$141$		0			0
$142$	−2.04265e7	−	3.33993e7i	−0.598667	−	0.978877i
$143$	9.92815e7			2.83917
$144$		0			0
$145$	−2.55040e7			−0.694736
$146$	3.09587e7	+	5.06205e7i	0.823281	+	1.34614i
$147$		0			0
$148$	1.80869e7	−	3.53429e7i	0.458614	−	0.896160i
$149$		−	4.47453e7i		−	1.10814i	−0.832469	−	0.554071i	$-0.813074\pi$
$149$		−	4.47453e7i		−	1.10814i	0.832469	−	0.554071i	$-0.186926\pi$
$150$		0			0
$151$			6.68446e7i			1.57996i	0.613130	+	0.789982i	$0.289910\pi$
$151$			6.68446e7i			1.57996i	−0.613130	+	0.789982i	$0.710090\pi$
$152$	2.90848e6	+	3.82525e7i	0.0671758	+	0.883502i
$153$		0			0
$154$	8.58416e6	−	5.24995e6i	0.189398	−	0.115833i
$155$	−7.87616e6			−0.169884
$156$		0			0
$157$	2.43859e7			0.502909			0.251455	−	0.967869i	$-0.419091\pi$
$157$	2.43859e7			0.502909			0.251455	+	0.967869i	$0.419091\pi$
$158$	−7.00675e7	+	4.28522e7i	−1.41324	+	0.864318i
$159$		0			0
$160$	2.13673e7	+	8.96150e6i	0.412411	+	0.172966i
$161$		−	9.28328e6i		−	0.175312i
$162$		0			0
$163$			1.04790e7i			0.189523i	0.995500	+	0.0947615i	$0.0302088\pi$
$163$			1.04790e7i			0.189523i	−0.995500	+	0.0947615i	$0.969791\pi$
$164$	8.29920e7	+	4.24715e7i	1.46921	+	0.751873i
$165$		0			0
$166$	−5.14260e7	−	8.40863e7i	−0.872578	−	1.42675i
$167$	7.93340e7			1.31811			0.659055	−	0.752095i	$-0.270956\pi$
$167$	7.93340e7			1.31811			0.659055	+	0.752095i	$0.270956\pi$
$168$		0			0
$169$	1.23928e8			1.97499
$170$	1.10970e7	+	1.81446e7i	0.173234	+	0.283254i
$171$		0			0
$172$	−8.98495e7	−	4.59809e7i	−1.34638	−	0.689014i
$173$		−	6.58167e7i		−	0.966440i	−0.875499	−	0.483220i	$-0.839467\pi$
$173$		−	6.58167e7i		−	0.966440i	0.875499	−	0.483220i	$-0.160533\pi$
$174$		0			0
$175$			1.91244e6i			0.0269746i
$176$	−6.96371e7	−	9.65635e7i	−0.962823	−	1.33512i
$177$		0			0
$178$	−2.77189e7	+	1.69525e7i	−0.368388	+	0.225301i
$179$	1.03616e8			1.35034			0.675168	−	0.737664i	$-0.264071\pi$
$179$	1.03616e8			1.35034			0.675168	+	0.737664i	$0.264071\pi$
$180$		0			0
$181$	−5.20727e6			−0.0652733			−0.0326366	−	0.999467i	$-0.510390\pi$
$181$	−5.20727e6			−0.0652733			−0.0326366	+	0.999467i	$0.510390\pi$
$182$	1.61405e7	−	9.87132e6i	0.198458	−	0.121374i
$183$		0			0
$184$	−1.09521e8	+	8.32726e6i	−1.29609	+	0.0985461i
$185$		−	3.87715e7i		−	0.450207i
$186$		0			0
$187$		−	1.09284e8i		−	1.22211i
$188$	7.30562e6	−	1.42756e7i	0.0801871	−	0.156690i
$189$		0			0
$190$	1.95465e7	+	3.19604e7i	0.206744	+	0.338045i
$191$	6.54048e7			0.679193			0.339596	−	0.940571i	$-0.389709\pi$
$191$	6.54048e7			0.679193			0.339596	+	0.940571i	$0.389709\pi$
$192$		0			0
$193$	−6.48093e7			−0.648914			−0.324457	−	0.945900i	$-0.605182\pi$
$193$	−6.48093e7			−0.648914			−0.324457	+	0.945900i	$0.605182\pi$
$194$	4.81680e7	+	7.87592e7i	0.473644	+	0.774453i
$195$		0			0
$196$	−4.71492e7	+	9.21323e7i	−0.447278	+	0.874009i
$197$			3.03985e7i			0.283283i	0.989918	+	0.141641i	$0.0452380\pi$
$197$			3.03985e7i			0.283283i	−0.989918	+	0.141641i	$0.954762\pi$
$198$		0			0
$199$		−	8.28741e7i		−	0.745475i	−0.927937	−	0.372737i	$-0.878419\pi$
$199$		−	8.28741e7i		−	0.745475i	0.927937	−	0.372737i	$-0.121581\pi$
$200$	2.25623e7	−	1.71549e6i	0.199424	−	0.0151630i
$201$		0			0
$202$	1.11087e8	−	6.79394e7i	0.948276	−	0.579952i
$203$	2.49727e7			0.209522
$204$		0			0
$205$	9.10432e7			0.738089
$206$	−8.31361e7	+	5.08448e7i	−0.662605	+	0.405239i
$207$		0			0
$208$	−1.30937e8	−	1.81566e8i	−1.00888	−	1.39898i
$209$		−	1.92496e8i		−	1.45851i
$210$		0			0
$211$			1.98873e8i			1.45743i	0.684819	+	0.728713i	$0.259881\pi$
$211$			1.98873e8i			1.45743i	−0.684819	+	0.728713i	$0.740119\pi$
$212$	−4.27528e7	−	2.18789e7i	−0.308169	−	0.157707i
$213$		0			0
$214$	−4.78936e7	−	7.83105e7i	−0.334063	−	0.546225i
$215$	−9.85660e7			−0.676383
$216$		0			0
$217$	7.71210e6			0.0512346
$218$	−3.35047e7	−	5.47833e7i	−0.219032	−	0.358138i
$219$		0			0
$220$	−1.03498e8	−	5.29657e7i	−0.655320	−	0.335363i
$221$		−	2.05483e8i		−	1.28057i
$222$		0			0
$223$		−	1.61749e7i		−	0.0976729i	−0.998807	−	0.0488365i	$-0.984449\pi$
$223$		−	1.61749e7i		−	0.0976729i	0.998807	−	0.0488365i	$-0.0155513\pi$
$224$	−2.09222e7	−	8.77483e6i	−0.124377	−	0.0521640i
$225$		0			0
$226$	−1.79267e7	+	1.09637e7i	−0.103305	+	0.0631798i
$227$	1.24221e8			0.704863			0.352432	−	0.935838i	$-0.385355\pi$
$227$	1.24221e8			0.704863			0.352432	+	0.935838i	$0.385355\pi$
$228$		0			0
$229$	−7.40834e7			−0.407659			−0.203829	−	0.979006i	$-0.565339\pi$
$229$	−7.40834e7			−0.407659			−0.203829	+	0.979006i	$0.565339\pi$
$230$	−9.15059e7	+	5.59637e7i	−0.495908	+	0.303290i
$231$		0			0
$232$	−2.24010e7	−	2.94619e8i	−0.117776	−	1.54901i
$233$		−	2.19916e8i		−	1.13897i	−0.822003	−	0.569484i	$-0.807143\pi$
$233$		−	2.19916e8i		−	1.13897i	0.822003	−	0.569484i	$-0.192857\pi$
$234$		0			0
$235$		−	1.56605e7i		−	0.0787171i
$236$	−1.53282e8	+	2.99522e8i	−0.759098	+	1.48332i
$237$		0			0
$238$	−1.08658e7	−	1.77666e7i	−0.0522448	−	0.0854252i
$239$	2.05230e7			0.0972409			0.0486204	−	0.998817i	$-0.484518\pi$
$239$	2.05230e7			0.0972409			0.0486204	+	0.998817i	$0.484518\pi$
$240$		0			0
$241$	−1.18857e8			−0.546972			−0.273486	−	0.961876i	$-0.588177\pi$
$241$	−1.18857e8			−0.546972			−0.273486	+	0.961876i	$0.588177\pi$
$242$	1.96651e8	+	3.21543e8i	0.891954	+	1.45843i
$243$		0			0
$244$	5.22109e7	−	1.02023e8i	0.230090	−	0.449609i
$245$			1.01070e8i			0.439078i
$246$		0			0
$247$		−	3.61944e8i		−	1.52828i
$248$	−6.91788e6	−	9.09845e7i	−0.0288000	−	0.378780i
$249$		0			0
$250$	1.88511e7	−	1.15290e7i	0.0763037	−	0.0466663i
$251$	5.63044e7			0.224742			0.112371	−	0.993666i	$-0.464155\pi$
$251$	5.63044e7			0.224742			0.112371	+	0.993666i	$0.464155\pi$
$252$		0			0
$253$	5.51134e8			2.13961
$254$	−4.01965e7	+	2.45836e7i	−0.153911	+	0.0941298i
$255$		0			0
$256$	−8.47546e7	+	2.54704e8i	−0.315736	+	0.948847i
$257$		−	1.60324e8i		−	0.589159i	−0.955627	−	0.294579i	$-0.904820\pi$
$257$		−	1.60324e8i		−	0.589159i	0.955627	−	0.294579i	$-0.0951796\pi$
$258$		0			0
$259$			3.79639e7i			0.135776i
$260$	−1.94605e8	−	9.95898e7i	−0.686667	−	0.351405i
$261$		0			0
$262$	−1.56123e8	−	2.55277e8i	−0.536307	−	0.876913i
$263$	−1.18513e8			−0.401717			−0.200859	−	0.979620i	$-0.564373\pi$
$263$	−1.18513e8			−0.401717			−0.200859	+	0.979620i	$0.564373\pi$
$264$		0			0
$265$	−4.69003e7			−0.154816
$266$	−1.91394e7	−	3.12947e7i	−0.0623508	−	0.101950i
$267$		0			0
$268$	−734930.	−	376104.i	−0.00233225	−	0.00119354i
$269$			2.22164e8i			0.695889i	0.937515	+	0.347944i	$0.113120\pi$
$269$			2.22164e8i			0.695889i	−0.937515	+	0.347944i	$0.886880\pi$
$270$		0			0
$271$		−	3.63210e7i		−	0.110857i	−0.998463	−	0.0554287i	$-0.982347\pi$
$271$		−	3.63210e7i		−	0.110857i	0.998463	−	0.0554287i	$-0.0176526\pi$
$272$	−1.99858e8	+	1.44128e8i	−0.602184	+	0.434267i
$273$		0			0
$274$	3.39993e8	−	2.07935e8i	0.998489	−	0.610662i
$275$	−1.13539e8			−0.329215
$276$		0			0
$277$	−2.65288e8			−0.749960			−0.374980	−	0.927033i	$-0.622351\pi$
$277$	−2.65288e8			−0.749960			−0.374980	+	0.927033i	$0.622351\pi$
$278$	8.14874e7	−	4.98365e7i	0.227475	−	0.139120i
$279$		0			0
$280$	−2.20923e7	+	1.67976e6i	−0.0601434	+	0.00457292i
$281$			9.85782e7i			0.265038i	0.991180	+	0.132519i	$0.0423066\pi$
$281$			9.85782e7i			0.265038i	−0.991180	+	0.132519i	$0.957693\pi$
$282$		0			0
$283$		−	3.80088e7i		−	0.0996855i	−0.998757	−	0.0498428i	$-0.984128\pi$
$283$		−	3.80088e7i		−	0.0996855i	0.998757	−	0.0498428i	$-0.0158720\pi$
$284$	2.01786e8	−	3.94303e8i	0.522730	−	1.02145i
$285$		0			0
$286$	5.86045e8	+	9.58239e8i	1.48132	+	2.42210i
$287$	−8.91468e7			−0.222597
$288$		0			0
$289$	1.84154e8			0.448786
$290$	−1.50546e8	−	2.46158e8i	−0.362475	−	0.592680i
$291$		0			0
$292$	−3.05831e8	+	5.97612e8i	−0.718854	+	1.40469i
$293$			2.02747e8i			0.470888i	0.971888	+	0.235444i	$0.0756545\pi$
$293$			2.02747e8i			0.470888i	−0.971888	+	0.235444i	$0.924346\pi$
$294$		0			0
$295$			3.28579e8i			0.745182i
$296$	4.47885e8	−	3.40543e7i	1.00380	−	0.0763222i
$297$		0			0
$298$	4.31870e8	−	2.64126e8i	0.945358	−	0.578167i
$299$	1.03628e9			2.24196
$300$		0			0
$301$	9.65129e7			0.203987
$302$	−6.45167e8	+	3.94575e8i	−1.34787	+	0.824338i
$303$		0			0
$304$	−3.52035e8	+	2.53871e8i	−0.718668	+	0.518270i
$305$		−	1.11921e8i		−	0.225872i
$306$		0			0
$307$			1.18928e8i			0.234585i	0.993097	+	0.117293i	$0.0374216\pi$
$307$			1.18928e8i			0.234585i	−0.993097	+	0.117293i	$0.962578\pi$
$308$	1.01342e8	+	5.18624e7i	0.197635	+	0.101141i
$309$		0			0
$310$	−4.64919e7	−	7.60187e7i	−0.0886363	−	0.144929i
$311$	−7.43833e8			−1.40221			−0.701107	−	0.713056i	$-0.747310\pi$
$311$	−7.43833e8			−1.40221			−0.701107	+	0.713056i	$0.747310\pi$
$312$		0			0
$313$	−2.14208e8			−0.394849			−0.197424	−	0.980318i	$-0.563258\pi$
$313$	−2.14208e8			−0.394849			−0.197424	+	0.980318i	$0.563258\pi$
$314$	1.43947e8	+	2.35366e8i	0.262390	+	0.429033i
$315$		0			0
$316$	−8.27198e8	−	4.23322e8i	−1.47470	−	0.754686i
$317$			3.91909e8i			0.691000i	0.938419	+	0.345500i	$0.112291\pi$
$317$			3.91909e8i			0.691000i	−0.938419	+	0.345500i	$0.887709\pi$
$318$		0			0
$319$			1.48259e9i			2.55714i
$320$	3.96344e7	+	2.59130e8i	0.0676156	+	0.442073i
$321$		0			0
$322$	8.95998e7	−	5.47979e7i	0.149559	−	0.0914679i
$323$	−3.98408e8			−0.657838
$324$		0			0
$325$	−2.13483e8			−0.344963
$326$	−1.01140e8	+	6.18559e7i	−0.161682	+	0.0988825i
$327$		0			0
$328$	7.99662e7	+	1.05172e9i	0.125126	+	1.64567i
$329$			1.53343e7i			0.0237399i
$330$		0			0
$331$			1.25506e9i			1.90225i	0.308811	+	0.951123i	$0.400069\pi$
$331$			1.25506e9i			1.90225i	−0.308811	+	0.951123i	$0.599931\pi$
$332$	5.08019e8	−	9.92700e8i	0.761898	−	1.48879i
$333$		0			0
$334$	4.68298e8	+	7.65712e8i	0.687717	+	1.12448i
$335$	−806227.			−0.00117166
$336$		0			0
$337$	5.51380e8			0.784777			0.392389	−	0.919800i	$-0.371649\pi$
$337$	5.51380e8			0.784777			0.392389	+	0.919800i	$0.371649\pi$
$338$	7.31527e8	+	1.19612e9i	1.03044	+	1.68486i
$339$		0			0
$340$	−1.09623e8	+	2.14210e8i	−0.151261	+	0.295572i
$341$			4.57855e8i			0.625299i
$342$		0			0
$343$		−	1.99764e8i		−	0.267293i
$344$	−8.65737e7	−	1.13862e9i	−0.114665	−	1.50808i
$345$		0			0
$346$	6.35246e8	−	3.88507e8i	0.824471	−	0.504235i
$347$	−1.30786e9			−1.68038			−0.840189	−	0.542294i	$-0.817556\pi$
$347$	−1.30786e9			−1.68038			−0.840189	+	0.542294i	$0.817556\pi$
$348$		0			0
$349$	1.36783e7			0.0172243			0.00861215	−	0.999963i	$-0.497259\pi$
$349$	1.36783e7			0.0172243			0.00861215	+	0.999963i	$0.497259\pi$
$350$	−1.84584e7	+	1.12889e7i	−0.0230121	+	0.0140739i
$351$		0			0
$352$	5.20948e8	−	1.24212e9i	0.636642	−	1.51798i
$353$		−	4.25605e8i		−	0.514985i	−0.966280	−	0.257493i	$-0.917104\pi$
$353$		−	4.25605e8i		−	0.514985i	0.966280	−	0.257493i	$-0.0828963\pi$
$354$		0			0
$355$		−	4.32555e8i		−	0.513147i
$356$	−3.27242e8	−	1.67468e8i	−0.384409	−	0.196723i
$357$		0			0
$358$	6.11632e8	+	1.00008e9i	0.704530	+	1.15197i
$359$	−5.85463e8			−0.667835			−0.333918	−	0.942602i	$-0.608371\pi$
$359$	−5.85463e8			−0.667835			−0.333918	+	0.942602i	$0.608371\pi$
$360$		0			0
$361$	1.92104e8			0.214912
$362$	−3.07378e7	−	5.02593e7i	−0.0340560	−	0.0556847i
$363$		0			0
$364$	1.90551e8	+	9.75154e7i	0.207089	+	0.105979i
$365$			6.55587e8i			0.705676i
$366$		0			0
$367$		−	1.72932e9i		−	1.82618i	−0.407754	−	0.913092i	$-0.633688\pi$
$367$		−	1.72932e9i		−	1.82618i	0.407754	−	0.913092i	$-0.366312\pi$
$368$	−7.26859e8	−	1.00791e9i	−0.760296	−	1.05428i
$369$		0			0
$370$	3.74213e8	−	2.28863e8i	0.384072	−	0.234893i
$371$	4.59234e7			0.0466901
$372$		0			0
$373$	−1.41965e9			−1.41644			−0.708222	−	0.705990i	$-0.750502\pi$
$373$	−1.41965e9			−1.41644			−0.708222	+	0.705990i	$0.750502\pi$
$374$	1.05478e9	−	6.45087e8i	1.04258	−	0.637628i
$375$		0			0
$376$	1.80909e8	−	1.37551e7i	0.175510	−	0.0133447i
$377$			2.78767e9i			2.67946i
$378$		0			0
$379$		−	1.23438e9i		−	1.16470i	−0.812939	−	0.582348i	$-0.802134\pi$
$379$		−	1.23438e9i		−	1.16470i	0.812939	−	0.582348i	$-0.197866\pi$
$380$	−1.93093e8	+	3.77316e8i	−0.180520	+	0.352747i
$381$		0			0
$382$	3.86076e8	+	6.31271e8i	0.354365	+	0.579421i
$383$	1.81044e9			1.64660			0.823300	−	0.567606i	$-0.192130\pi$
$383$	1.81044e9			1.64660			0.823300	+	0.567606i	$0.192130\pi$
$384$		0			0
$385$	1.11174e8			0.0992863
$386$	−3.82561e8	−	6.25523e8i	−0.338567	−	0.553589i
$387$		0			0
$388$	−4.75835e8	+	9.29810e8i	−0.413566	+	0.808133i
$389$			7.13353e8i			0.614442i	0.951638	+	0.307221i	$0.0993991\pi$
$389$			7.13353e8i			0.614442i	−0.951638	+	0.307221i	$0.900601\pi$
$390$		0			0
$391$		−	1.14068e9i		−	0.965041i
$392$	−1.16755e9	+	8.87733e7i	−0.978984	+	0.0744357i
$393$		0			0
$394$	−2.93398e8	+	1.79438e8i	−0.241669	+	0.147801i
$395$	−9.07446e8			−0.740851
$396$		0			0
$397$	−1.79257e9			−1.43783			−0.718916	−	0.695097i	$-0.755362\pi$
$397$	−1.79257e9			−1.43783			−0.718916	+	0.695097i	$0.755362\pi$
$398$	7.99880e8	−	4.89195e8i	0.635966	−	0.388948i
$399$		0			0
$400$	1.49740e8	+	2.07639e8i	0.116984	+	0.162218i
$401$			4.40430e7i			0.0341092i	0.999855	+	0.0170546i	$0.00542892\pi$
$401$			4.40430e7i			0.0341092i	−0.999855	+	0.0170546i	$0.994571\pi$
$402$		0			0
$403$			8.60892e8i			0.655211i
$404$	1.31147e9	+	6.71150e8i	0.989516	+	0.506389i
$405$		0			0
$406$	1.47411e8	+	2.41030e8i	0.109317	+	0.178744i
$407$	−2.25386e9			−1.65709
$408$		0			0
$409$	1.77072e9			1.27973			0.639867	−	0.768486i	$-0.278990\pi$
$409$	1.77072e9			1.27973			0.639867	+	0.768486i	$0.278990\pi$
$410$	5.37416e8	+	8.78726e8i	0.385094	+	0.629665i
$411$		0			0
$412$	−9.81483e8	−	5.02278e8i	−0.691421	−	0.353838i
$413$		−	3.21734e8i		−	0.224736i
$414$		0			0
$415$		−	1.08900e9i		−	0.747930i
$416$	9.79523e8	−	2.33552e9i	0.667096	−	1.59059i
$417$		0			0
$418$	1.85792e9	−	1.13628e9i	1.24426	−	0.760968i
$419$	−1.12316e9			−0.745920			−0.372960	−	0.927847i	$-0.621657\pi$
$419$	−1.12316e9			−0.745920			−0.372960	+	0.927847i	$0.621657\pi$
$420$		0			0
$421$	−3.82924e8			−0.250107			−0.125053	−	0.992150i	$-0.539910\pi$
$421$	−3.82924e8			−0.250107			−0.125053	+	0.992150i	$0.539910\pi$
$422$	−1.91947e9	+	1.17392e9i	−1.24333	+	0.760404i
$423$		0			0
$424$	−4.11940e7	−	5.41787e8i	−0.0262454	−	0.345182i
$425$			2.34991e8i			0.148488i
$426$		0			0
$427$			1.09590e8i			0.0681195i
$428$	4.73124e8	−	9.24513e8i	0.291690	−	0.569980i
$429$		0			0
$430$	−5.81822e8	−	9.51334e8i	−0.352899	−	0.577023i
$431$	6.28232e7			0.0377964			0.0188982	−	0.999821i	$-0.493984\pi$
$431$	6.28232e7			0.0377964			0.0188982	+	0.999821i	$0.493984\pi$
$432$		0			0
$433$	−2.47147e9			−1.46301			−0.731507	−	0.681834i	$-0.761183\pi$
$433$	−2.47147e9			−1.46301			−0.731507	+	0.681834i	$0.761183\pi$
$434$	4.55235e7	+	7.44352e7i	0.0267314	+	0.0437083i
$435$		0			0
$436$	3.30981e8	−	6.46757e8i	0.191250	−	0.373713i
$437$		−	2.00923e9i		−	1.15171i
$438$		0			0
$439$			1.40661e9i			0.793501i	0.917926	+	0.396750i	$0.129862\pi$
$439$			1.40661e9i			0.793501i	−0.917926	+	0.396750i	$0.870138\pi$
$440$	−9.97247e7	−	1.31159e9i	−0.0558108	−	0.734028i
$441$		0			0
$442$	1.98327e9	−	1.21294e9i	1.09245	−	0.668129i
$443$	1.75653e9			0.959935			0.479967	−	0.877286i	$-0.340649\pi$
$443$	1.75653e9			0.959935			0.479967	+	0.877286i	$0.340649\pi$
$444$		0			0
$445$	−3.58988e8			−0.193117
$446$	1.56116e8	−	9.54782e7i	0.0833249	−	0.0509603i
$447$		0			0
$448$	−3.88088e7	−	2.53733e8i	−0.0203919	−	0.133323i
$449$		−	2.29530e9i		−	1.19668i	−0.801243	−	0.598339i	$-0.795828\pi$
$449$		−	2.29530e9i		−	1.19668i	0.801243	−	0.598339i	$-0.204172\pi$
$450$		0			0
$451$		−	5.29251e9i		−	2.71671i
$452$	−2.11638e8	−	1.08307e8i	−0.107798	−	0.0551659i
$453$		0			0
$454$	7.33261e8	+	1.19895e9i	0.367759	+	0.601320i
$455$	2.09037e8			0.104036
$456$		0			0
$457$	−2.26736e9			−1.11125			−0.555627	−	0.831431i	$-0.687522\pi$
$457$	−2.26736e9			−1.11125			−0.555627	+	0.831431i	$0.687522\pi$
$458$	−4.37305e8	−	7.15034e8i	−0.212694	−	0.347775i
$459$		0			0
$460$	−1.08029e9	−	5.52845e8i	−0.517475	−	0.264820i
$461$			1.93243e8i			0.0918650i	0.998945	+	0.0459325i	$0.0146259\pi$
$461$			1.93243e8i			0.0918650i	−0.998945	+	0.0459325i	$0.985374\pi$
$462$		0			0
$463$			1.20718e9i			0.565245i	0.959231	+	0.282623i	$0.0912044\pi$
$463$			1.20718e9i			0.565245i	−0.959231	+	0.282623i	$0.908796\pi$
$464$	2.71136e9	−	1.95531e9i	1.26001	−	0.908660i
$465$		0			0
$466$	2.12257e9	−	1.29813e9i	0.971655	−	0.594250i
$467$	−3.88723e9			−1.76617			−0.883083	−	0.469218i	$-0.844536\pi$
$467$	−3.88723e9			−1.76617			−0.883083	+	0.469218i	$0.844536\pi$
$468$		0			0
$469$	789434.			0.000353355
$470$	1.51151e8	−	9.24420e7i	0.0671537	−	0.0410702i
$471$		0			0
$472$	−3.79571e9	+	2.88601e8i	−1.66148	+	0.126328i
$473$			5.72982e9i			2.48959i
$474$		0			0
$475$			4.13920e8i			0.177210i
$476$	1.07340e8	−	2.09748e8i	0.0456179	−	0.0891402i
$477$		0			0
$478$	1.21145e8	+	1.98083e8i	0.0507349	+	0.0829563i
$479$	1.94948e9			0.810486			0.405243	−	0.914209i	$-0.367187\pi$
$479$	1.94948e9			0.810486			0.405243	+	0.914209i	$0.367187\pi$
$480$		0			0
$481$	−4.23787e9			−1.73636
$482$	−7.01596e8	−	1.14718e9i	−0.285379	−	0.466622i
$483$		0			0
$484$	−1.94265e9	+	3.79605e9i	−0.778817	+	1.52186i
$485$			1.02001e9i			0.405984i
$486$		0			0
$487$		−	1.41576e9i		−	0.555441i	−0.960662	−	0.277721i	$-0.910421\pi$
$487$		−	1.41576e9i		−	0.555441i	0.960662	−	0.277721i	$-0.0895789\pi$
$488$	1.29290e9	−	9.83037e7i	0.503610	−	0.0382913i
$489$		0			0
$490$	−9.75504e8	+	5.96604e8i	−0.374579	+	0.229087i
$491$	3.86165e9			1.47227			0.736136	−	0.676834i	$-0.236648\pi$
$491$	3.86165e9			1.47227			0.736136	+	0.676834i	$0.236648\pi$
$492$		0			0
$493$	3.06852e9			1.15336
$494$	3.49339e9	−	2.13651e9i	1.30377	−	0.797370i
$495$		0			0
$496$	8.37324e8	−	6.03839e8i	0.308111	−	0.222196i
$497$			4.23545e8i			0.154758i
$498$		0			0
$499$			9.37019e8i			0.337595i	0.985651	+	0.168798i	$0.0539885\pi$
$499$			9.37019e8i			0.337595i	−0.985651	+	0.168798i	$0.946012\pi$
$500$	2.22551e8	+	1.13891e8i	0.0796221	+	0.0407470i
$501$		0			0
$502$	3.32357e8	+	5.43436e8i	0.117258	+	0.191728i
$503$	8.25008e8			0.289048			0.144524	−	0.989501i	$-0.453835\pi$
$503$	8.25008e8			0.289048			0.144524	+	0.989501i	$0.453835\pi$
$504$		0			0
$505$	1.43870e9			0.497106
$506$	3.25327e9	+	5.31940e9i	1.11633	+	1.82531i
$507$		0			0
$508$	−4.74549e8	−	2.42853e8i	−0.160605	−	0.0821901i
$509$			4.42818e9i			1.48838i	0.667970	+	0.744188i	$0.267163\pi$
$509$			4.42818e9i			1.48838i	−0.667970	+	0.744188i	$0.732837\pi$
$510$		0			0
$511$		−	6.41931e8i		−	0.212821i
$512$	−2.95864e9	+	6.85455e8i	−0.974197	+	0.225701i
$513$		0			0
$514$	1.54741e9	−	9.46371e8i	0.502612	−	0.307390i
$515$	−1.07670e9			−0.347351
$516$		0			0
$517$	−9.10375e8			−0.289737
$518$	−3.66418e8	+	2.24096e8i	−0.115830	+	0.0708402i
$519$		0			0
$520$	−1.87509e8	−	2.46614e9i	−0.0584805	−	0.769141i
$521$		−	3.25464e9i		−	1.00826i	−0.863629	−	0.504129i	$-0.831814\pi$
$521$		−	3.25464e9i		−	1.00826i	0.863629	−	0.504129i	$-0.168186\pi$
$522$		0			0
$523$		−	4.40423e9i		−	1.34621i	−0.739546	−	0.673106i	$-0.764960\pi$
$523$		−	4.40423e9i		−	1.34621i	0.739546	−	0.673106i	$-0.235040\pi$
$524$	1.54229e9	−	3.01373e9i	0.468281	−	0.915049i
$525$		0			0
$526$	−6.99566e8	−	1.14386e9i	−0.209594	−	0.342706i
$527$	9.47623e8			0.282032
$528$		0			0
$529$	2.34780e9			0.689551
$530$	−2.76846e8	−	4.52669e8i	−0.0807742	−	0.132073i
$531$		0			0
$532$	1.89071e8	−	3.69457e8i	0.0544421	−	0.106383i
$533$		−	9.95134e9i		−	2.84667i
$534$		0			0
$535$		−	1.01420e9i		−	0.286343i
$536$	−708135.	−	9.31345e6i	−0.000198628	−	0.00261237i
$537$		0			0
$538$	−2.14427e9	+	1.31140e9i	−0.593664	+	0.363076i
$539$	5.87540e9			1.61613
$540$		0			0
$541$	4.72422e9			1.28274			0.641372	−	0.767230i	$-0.278366\pi$
$541$	4.72422e9			1.28274			0.641372	+	0.767230i	$0.278366\pi$
$542$	3.50561e8	−	2.14398e8i	0.0945727	−	0.0578393i
$543$		0			0
$544$	−2.57082e9	−	1.07821e9i	−0.684660	−	0.287148i
$545$		−	7.09500e8i		−	0.187744i
$546$		0			0
$547$			9.77663e8i			0.255407i	0.991812	+	0.127704i	$0.0407606\pi$
$547$			9.77663e8i			0.255407i	−0.991812	+	0.127704i	$0.959239\pi$
$548$	4.01387e9	+	2.05412e9i	1.04191	+	0.533204i
$549$		0			0
$550$	−6.70204e8	−	1.09585e9i	−0.171766	−	0.280854i
$551$	5.40499e9			1.37646
$552$		0			0
$553$	8.88544e8			0.223430
$554$	−1.56596e9	−	2.56049e9i	−0.391288	−	0.639793i
$555$		0			0
$556$	9.62018e8	+	4.92317e8i	0.237368	+	0.121474i
$557$			6.68206e9i			1.63839i	0.573516	+	0.819195i	$0.305579\pi$
$557$			6.68206e9i			1.63839i	−0.573516	+	0.819195i	$0.694421\pi$
$558$		0			0
$559$			1.07736e10i			2.60868i
$560$	−1.46621e8	−	2.03314e8i	−0.0352807	−	0.0489226i
$561$		0			0
$562$	−9.51452e8	+	5.81894e8i	−0.226105	+	0.138282i
$563$	−3.30473e9			−0.780471			−0.390236	−	0.920715i	$-0.627606\pi$
$563$	−3.30473e9			−0.780471			−0.390236	+	0.920715i	$0.627606\pi$
$564$		0			0
$565$	−2.32169e8			−0.0541546
$566$	3.66852e8	−	2.24361e8i	0.0850419	−	0.0520104i
$567$		0			0
$568$	4.99683e9	−	3.79927e8i	1.14413	−	0.0869924i
$569$		−	4.82172e9i		−	1.09726i	−0.836065	−	0.548630i	$-0.815150\pi$
$569$		−	4.82172e9i		−	1.09726i	0.836065	−	0.548630i	$-0.184850\pi$
$570$		0			0
$571$			3.39568e9i			0.763308i	0.924305	+	0.381654i	$0.124645\pi$
$571$			3.39568e9i			0.763308i	−0.924305	+	0.381654i	$0.875355\pi$
$572$	−5.78933e9	+	1.13127e10i	−1.29343	+	2.52744i
$573$		0			0
$574$	−5.26221e8	−	8.60422e8i	−0.116139	−	0.189898i
$575$	−1.18509e9			−0.259965
$576$		0			0
$577$	−1.22604e9			−0.265700			−0.132850	−	0.991136i	$-0.542413\pi$
$577$	−1.22604e9			−0.265700			−0.132850	+	0.991136i	$0.542413\pi$
$578$	1.08704e9	+	1.77741e9i	0.234152	+	0.382860i
$579$		0			0
$580$	1.48720e9	−	2.90607e9i	0.316497	−	0.618455i
$581$			1.06632e9i			0.225565i
$582$		0			0
$583$			2.72640e9i			0.569835i
$584$	−7.57327e9	+	5.75823e8i	−1.57340	+	0.119631i
$585$		0			0
$586$	−1.95686e9	+	1.19679e9i	−0.401716	+	0.245683i
$587$	−5.14677e9			−1.05027			−0.525135	−	0.851019i	$-0.675985\pi$
$587$	−5.14677e9			−1.05027			−0.525135	+	0.851019i	$0.675985\pi$
$588$		0			0
$589$	1.66917e9			0.336587
$590$	−3.17136e9	+	1.93956e9i	−0.635716	+	0.388795i
$591$		0			0
$592$	2.97249e9	+	4.12185e9i	0.588835	+	0.816519i
$593$		−	7.22217e9i		−	1.42225i	−0.703065	−	0.711125i	$-0.748186\pi$
$593$		−	7.22217e9i		−	1.42225i	0.703065	−	0.711125i	$-0.251814\pi$
$594$		0			0
$595$		−	2.30096e8i		−	0.0447816i
$596$	5.09854e9	+	2.60920e9i	0.986471	+	0.504831i
$597$		0			0
$598$	6.11703e9	+	1.00019e10i	1.16973	+	1.91262i
$599$	−4.36431e9			−0.829702			−0.414851	−	0.909889i	$-0.636166\pi$
$599$	−4.36431e9			−0.829702			−0.414851	+	0.909889i	$0.636166\pi$
$600$		0			0
$601$	−5.60457e9			−1.05313			−0.526565	−	0.850135i	$-0.676520\pi$
$601$	−5.60457e9			−1.05313			−0.526565	+	0.850135i	$0.676520\pi$
$602$	5.69703e8	+	9.31518e8i	0.106429	+	0.174022i
$603$		0			0
$604$	−7.61667e9	−	3.89787e9i	−1.40649	−	0.719777i
$605$			4.16431e9i			0.764539i
$606$		0			0
$607$		−	4.10127e9i		−	0.744317i	−0.928169	−	0.372158i	$-0.878618\pi$
$607$		−	4.10127e9i		−	0.744317i	0.928169	−	0.372158i	$-0.121382\pi$
$608$	−4.52832e9	−	1.89919e9i	−0.817099	−	0.342693i
$609$		0			0
$610$	1.08023e9	−	6.60654e8i	0.192691	−	0.117847i
$611$	−1.71175e9			−0.303596
$612$		0			0
$613$	−3.39639e9			−0.595532			−0.297766	−	0.954639i	$-0.596242\pi$
$613$	−3.39639e9			−0.595532			−0.297766	+	0.954639i	$0.596242\pi$
$614$	−1.14787e9	+	7.02018e8i	−0.200125	+	0.122394i
$615$		0			0
$616$	9.76475e7	+	1.28427e9i	0.0168317	+	0.221372i
$617$		−	5.66110e9i		−	0.970292i	−0.874433	−	0.485146i	$-0.838766\pi$
$617$		−	5.66110e9i		−	0.970292i	0.874433	−	0.485146i	$-0.161234\pi$
$618$		0			0
$619$			8.65699e9i			1.46707i	0.679654	+	0.733533i	$0.262130\pi$
$619$			8.65699e9i			1.46707i	−0.679654	+	0.733533i	$0.737870\pi$
$620$	4.59277e8	−	8.97456e8i	0.0773934	−	0.151231i
$621$		0			0
$622$	−4.39075e9	−	7.17929e9i	−0.731597	−	1.19623i
$623$	3.51511e8			0.0582412
$624$		0			0
$625$	2.44141e8			0.0400000
$626$	−1.26444e9	−	2.06748e9i	−0.206010	−	0.336846i
$627$		0			0
$628$	−1.42200e9	+	2.77867e9i	−0.229108	+	0.447691i
$629$			4.66481e9i			0.747407i
$630$		0			0
$631$		−	1.65268e9i		−	0.261870i	−0.991391	−	0.130935i	$-0.958202\pi$
$631$		−	1.65268e9i		−	0.261870i	0.991391	−	0.130935i	$-0.0417980\pi$
$632$	−7.97039e8	−	1.04827e10i	−0.125594	−	1.65183i
$633$		0			0
$634$	−3.78261e9	+	2.31339e9i	−0.589493	+	0.360525i
$635$	−5.20586e8			−0.0806834
$636$		0			0
$637$	1.10473e10			1.69344
$638$	−1.43096e10	+	8.75154e9i	−2.18150	+	1.33417i
$639$		0			0
$640$	−2.26710e9	+	1.91215e9i	−0.341855	+	0.288332i
$641$			5.16385e8i			0.0774410i	0.999250	+	0.0387205i	$0.0123282\pi$
$641$			5.16385e8i			0.0774410i	−0.999250	+	0.0387205i	$0.987672\pi$
$642$		0			0
$643$		−	8.70158e9i		−	1.29080i	−0.763844	−	0.645401i	$-0.776690\pi$
$643$		−	8.70158e9i		−	1.29080i	0.763844	−	0.645401i	$-0.223310\pi$
$644$	1.05779e9	+	5.41330e8i	0.156063	+	0.0798659i
$645$		0			0
$646$	−2.35175e9	−	3.84533e9i	−0.343224	−	0.561203i
$647$	−3.72995e9			−0.541424			−0.270712	−	0.962660i	$-0.587259\pi$
$647$	−3.72995e9			−0.541424			−0.270712	+	0.962660i	$0.587259\pi$
$648$		0			0
$649$	1.91009e10			2.74282
$650$	−1.26016e9	−	2.06049e9i	−0.179983	−	0.294289i
$651$		0			0
$652$	−1.19403e9	−	6.11053e8i	−0.168714	−	0.0863400i
$653$		−	1.21371e10i		−	1.70577i	−0.522101	−	0.852883i	$-0.674852\pi$
$653$		−	1.21371e10i		−	1.70577i	0.522101	−	0.852883i	$-0.325148\pi$
$654$		0			0
$655$		−	3.30609e9i		−	0.459696i
$656$	−9.67892e9	+	6.97998e9i	−1.33864	+	0.965364i
$657$		0			0
$658$	−1.48003e8	+	9.05164e7i	−0.0202526	+	0.0123862i
$659$	6.12282e9			0.833398			0.416699	−	0.909045i	$-0.363187\pi$
$659$	6.12282e9			0.833398			0.416699	+	0.909045i	$0.363187\pi$
$660$		0			0
$661$	1.02760e10			1.38395			0.691975	−	0.721922i	$-0.256741\pi$
$661$	1.02760e10			1.38395			0.691975	+	0.721922i	$0.256741\pi$
$662$	−1.21135e10	+	7.40846e9i	−1.62281	+	0.992487i
$663$		0			0
$664$	1.25801e10	−	9.56507e8i	1.66761	−	0.126794i
$665$		−	4.05299e8i		−	0.0534440i
$666$		0			0
$667$			1.54750e10i			2.01925i
$668$	−4.62615e9	+	9.03979e9i	−0.600485	+	1.17338i
$669$		0			0
$670$	−4.75905e6	−	7.78150e6i	−0.000611306	−	0.000999543i
$671$	−6.50616e9			−0.831373
$672$		0			0
$673$	−1.12302e10			−1.42015			−0.710077	−	0.704124i	$-0.751340\pi$
$673$	−1.12302e10			−1.42015			−0.710077	+	0.704124i	$0.751340\pi$
$674$	3.25472e9	+	5.32178e9i	0.409453	+	0.669495i
$675$		0			0
$676$	−7.22650e9	+	1.41210e10i	−0.899735	+	1.75814i
$677$		−	3.70388e9i		−	0.458772i	−0.973336	−	0.229386i	$-0.926328\pi$
$677$		−	3.70388e9i		−	0.458772i	0.973336	−	0.229386i	$-0.0736718\pi$
$678$		0			0
$679$		−	9.98766e8i		−	0.122439i
$680$	−2.71459e9	+	2.06400e8i	−0.331073	+	0.0251726i
$681$		0			0
$682$	−4.41910e9	+	2.70266e9i	−0.533444	+	0.326247i
$683$	1.52937e10			1.83671			0.918356	−	0.395755i	$-0.129517\pi$
$683$	1.52937e10			1.83671			0.918356	+	0.395755i	$0.129517\pi$
$684$		0			0
$685$	4.40326e9			0.523429
$686$	1.92807e9	−	1.17918e9i	0.228028	−	0.139458i
$687$		0			0
$688$	1.04787e10	−	7.55673e9i	1.22672	−	0.884656i
$689$			5.12637e9i			0.597094i
$690$		0			0
$691$			4.05950e9i			0.468057i	0.972230	+	0.234029i	$0.0751910\pi$
$691$			4.05950e9i			0.468057i	−0.972230	+	0.234029i	$0.924809\pi$
$692$	7.49954e9	+	3.83793e9i	0.860327	+	0.440276i
$693$		0			0
$694$	−7.72010e9	−	1.26231e10i	−0.876728	−	1.43353i
$695$	1.05535e9			0.119247
$696$		0			0
$697$	−1.09539e10			−1.22533
$698$	8.07409e7	+	1.32019e8i	0.00898669	+	0.0146941i
$699$		0			0
$700$	−2.17915e8	−	1.11519e8i	−0.0240129	−	0.0122887i
$701$		−	1.35077e10i		−	1.48104i	−0.672033	−	0.740521i	$-0.734579\pi$
$701$		−	1.35077e10i		−	1.48104i	0.672033	−	0.740521i	$-0.265421\pi$
$702$		0			0
$703$			8.21675e9i			0.891982i
$704$	1.50637e10	−	2.30402e9i	1.62715	−	0.248875i
$705$		0			0
$706$	4.10783e9	−	2.51229e9i	0.439335	−	0.268691i
$707$	−1.40873e9			−0.149920
$708$		0			0
$709$	1.23522e10			1.30162			0.650808	−	0.759243i	$-0.274430\pi$
$709$	1.23522e10			1.30162			0.650808	+	0.759243i	$0.274430\pi$
$710$	4.17491e9	−	2.55331e9i	0.437767	−	0.267732i
$711$		0			0
$712$	−3.15311e8	−	4.14699e9i	−0.0327385	−	0.430579i
$713$			4.77901e9i			0.493769i
$714$		0			0
$715$			1.24102e10i			1.26972i
$716$	−6.04210e9	+	1.18066e10i	−0.615166	+	1.20207i
$717$		0			0
$718$	−3.45591e9	−	5.65074e9i	−0.348439	−	0.569731i
$719$	−7.93879e9			−0.796532			−0.398266	−	0.917270i	$-0.630388\pi$
$719$	−7.93879e9			−0.796532			−0.398266	+	0.917270i	$0.630388\pi$
$720$		0			0
$721$	1.05427e9			0.104756
$722$	1.13396e9	+	1.85413e9i	0.112129	+	0.183342i
$723$		0			0
$724$	3.03648e8	−	5.93347e8i	0.0297362	−	0.0581064i
$725$		−	3.18800e9i		−	0.310695i
$726$		0			0
$727$		−	2.06794e10i		−	1.99603i	−0.0629759	−	0.998015i	$-0.520059\pi$
$727$		−	2.06794e10i		−	1.99603i	0.0629759	−	0.998015i	$-0.479941\pi$
$728$	1.83604e8	+	2.41477e9i	0.0176369	+	0.231961i
$729$		0			0
$730$	−6.32756e9	+	3.86984e9i	−0.602013	+	0.368183i
$731$	1.18590e10			1.12289
$732$		0			0
$733$	3.23498e8			0.0303394			0.0151697	−	0.999885i	$-0.495171\pi$
$733$	3.23498e8			0.0303394			0.0151697	+	0.999885i	$0.495171\pi$
$734$	1.66910e10	−	1.02080e10i	1.55792	−	0.952801i
$735$		0			0
$736$	5.43755e9	−	1.29650e10i	0.502726	−	1.19867i
$737$			4.68675e7i			0.00431256i
$738$		0			0
$739$			1.48518e9i			0.135371i	0.997707	+	0.0676853i	$0.0215614\pi$
$739$			1.48518e9i			0.135371i	−0.997707	+	0.0676853i	$0.978439\pi$
$740$	4.41786e9	+	2.26086e9i	0.400775	+	0.205099i
$741$		0			0
$742$	2.71079e8	+	4.43240e8i	0.0243603	+	0.0398314i
$743$	−5.11300e9			−0.457314			−0.228657	−	0.973507i	$-0.573433\pi$
$743$	−5.11300e9			−0.457314			−0.228657	+	0.973507i	$0.573433\pi$
$744$		0			0
$745$	5.59316e9			0.495576
$746$	−8.37998e9	−	1.37021e10i	−0.739022	−	1.20837i
$747$		0			0
$748$	1.24524e10	+	6.37258e9i	1.08792	+	0.556750i
$749$			9.93076e8i			0.0863567i
$750$		0			0
$751$			1.20760e10i			1.04036i	0.854057	+	0.520179i	$0.174135\pi$
$751$			1.20760e10i			1.04036i	−0.854057	+	0.520179i	$0.825865\pi$
$752$	1.20064e9	+	1.66489e9i	0.102956	+	0.142765i
$753$		0			0
$754$	−2.69059e10	+	1.64553e10i	−2.28585	+	1.39799i
$755$	−8.35558e9			−0.706581
$756$		0			0
$757$	−3.89527e9			−0.326364			−0.163182	−	0.986596i	$-0.552176\pi$
$757$	−3.89527e9			−0.326364			−0.163182	+	0.986596i	$0.552176\pi$
$758$	1.19140e10	−	7.28640e9i	0.993605	−	0.607674i
$759$		0			0
$760$	−4.78156e9	+	3.63560e8i	−0.395114	+	0.0300419i
$761$			4.40927e9i			0.362678i	0.983421	+	0.181339i	$0.0580431\pi$
$761$			4.40927e9i			0.362678i	−0.983421	+	0.181339i	$0.941957\pi$
$762$		0			0
$763$			6.94722e8i			0.0566207i
$764$	−3.81391e9	+	7.45261e9i	−0.309417	+	0.604619i
$765$		0			0
$766$	1.06868e10	+	1.74739e10i	0.859105	+	1.40472i
$767$	3.59148e10			2.87402
$768$		0			0
$769$	3.27594e9			0.259773			0.129887	−	0.991529i	$-0.458539\pi$
$769$	3.27594e9			0.259773			0.129887	+	0.991529i	$0.458539\pi$
$770$	6.56244e8	+	1.07302e9i	0.0518021	+	0.0847014i
$771$		0			0
$772$	3.77918e9	−	7.38476e9i	0.295623	−	0.577665i
$773$			1.02453e10i			0.797805i	0.916993	+	0.398902i	$0.130609\pi$
$773$			1.02453e10i			0.797805i	−0.916993	+	0.398902i	$0.869391\pi$
$774$		0			0
$775$		−	9.84520e8i		−	0.0759746i
$776$	−1.17831e10	+	8.95910e8i	−0.905196	+	0.0688253i
$777$		0			0
$778$	−6.88510e9	+	4.21083e9i	−0.524182	+	0.320582i
$779$	−1.92945e10			−1.46236
$780$		0			0
$781$	−2.51452e10			−1.88876
$782$	1.10096e10	−	6.73329e9i	0.823278	−	0.503505i
$783$		0			0
$784$	−7.74873e9	−	1.07449e10i	−0.574281	−	0.796336i
$785$			3.04823e9i			0.224908i
$786$		0			0
$787$		−	6.04359e9i		−	0.441961i	−0.975278	−	0.220980i	$-0.929074\pi$
$787$		−	6.04359e9i		−	0.441961i	0.975278	−	0.220980i	$-0.0709256\pi$
$788$	−3.46378e9	−	1.77261e9i	−0.252179	−	0.129054i
$789$		0			0
$790$	−5.35653e9	−	8.75843e9i	−0.386535	−	0.632021i
$791$	2.27333e8			0.0163322
$792$		0			0
$793$	−1.22333e10			−0.871142
$794$	−1.05813e10	−	1.73014e10i	−0.750181	−	1.22662i
$795$		0			0
$796$	9.44317e9	+	4.83258e9i	0.663624	+	0.339612i
$797$		−	7.64453e9i		−	0.534868i	−0.963576	−	0.267434i	$-0.913824\pi$
$797$		−	7.64453e9i		−	0.534868i	0.963576	−	0.267434i	$-0.0861758\pi$
$798$		0			0
$799$			1.88420e9i			0.130681i
$800$	−1.12019e9	+	2.67092e9i	−0.0773527	+	0.184436i
$801$		0			0
$802$	−4.25092e8	+	2.59980e8i	−0.0290986	+	0.0177963i
$803$	3.81105e10			2.59741
$804$		0			0
$805$	1.16041e9			0.0784017
$806$	−8.30911e9	+	5.08173e9i	−0.558961	+	0.341853i
$807$		0			0
$808$	1.26365e9	+	1.66197e10i	0.0842729	+	1.10836i
$809$		−	6.58451e9i		−	0.437224i	−0.975812	−	0.218612i	$-0.929847\pi$
$809$		−	6.58451e9i		−	0.437224i	0.975812	−	0.218612i	$-0.0701529\pi$
$810$		0			0
$811$		−	1.77639e10i		−	1.16940i	−0.811248	−	0.584702i	$-0.801211\pi$
$811$		−	1.77639e10i		−	1.16940i	0.811248	−	0.584702i	$-0.198789\pi$
$812$	−1.45622e9	+	2.84554e9i	−0.0954510	+	0.186517i
$813$		0			0
$814$	−1.33042e10	−	2.17537e10i	−0.864578	−	1.41367i
$815$	−1.30987e9			−0.0847572
$816$		0			0
$817$	2.08888e10			1.34010
$818$	1.04524e10	+	1.70906e10i	0.667694	+	1.09174i
$819$		0			0
$820$	−5.30894e9	+	1.03740e10i	−0.336248	+	0.657049i
$821$		−	2.15306e10i		−	1.35786i	−0.734202	−	0.678931i	$-0.762444\pi$
$821$		−	2.15306e10i		−	1.35786i	0.734202	−	0.678931i	$-0.237556\pi$
$822$		0			0
$823$		−	1.92135e10i		−	1.20146i	−0.799453	−	0.600728i	$-0.794877\pi$
$823$		−	1.92135e10i		−	1.20146i	0.799453	−	0.600728i	$-0.205123\pi$
$824$	−9.45699e8	−	1.24379e10i	−0.0588854	−	0.774465i
$825$		0			0
$826$	3.10530e9	−	1.89916e9i	0.191723	−	0.117255i
$827$	3.43998e9			0.211488			0.105744	−	0.994393i	$-0.466278\pi$
$827$	3.43998e9			0.211488			0.105744	+	0.994393i	$0.466278\pi$
$828$		0			0
$829$	−3.30449e9			−0.201448			−0.100724	−	0.994914i	$-0.532116\pi$
$829$	−3.30449e9			−0.201448			−0.100724	+	0.994914i	$0.532116\pi$
$830$	1.05108e10	−	6.42824e9i	0.638061	−	0.390229i
$831$		0			0
$832$	2.83239e10	−	4.33218e9i	1.70499	−	0.260780i
$833$		−	1.21603e10i		−	0.728932i
$834$		0			0
$835$			9.91675e9i			0.589477i
$836$	2.19341e10	+	1.12249e10i	1.29837	+	0.664445i
$837$		0			0
$838$	−6.62986e9	−	1.08404e10i	−0.389180	−	0.636345i
$839$	−5.81927e9			−0.340174			−0.170087	−	0.985429i	$-0.554405\pi$
$839$	−5.81927e9			−0.340174			−0.170087	+	0.985429i	$0.554405\pi$
$840$		0			0
$841$	−2.43791e10			−1.41329
$842$	−2.26035e9	−	3.69589e9i	−0.130492	−	0.213367i
$843$		0			0
$844$	−2.26607e10	−	1.15967e10i	−1.29740	−	0.663953i
$845$			1.54909e10i			0.883241i
$846$		0			0
$847$		−	4.07757e9i		−	0.230574i
$848$	4.98603e9	−	3.59569e9i	0.280782	−	0.202487i
$849$		0			0
$850$	−2.26807e9	+	1.38712e9i	−0.126675	+	0.0774726i
$851$	−2.35254e10			−1.30853
$852$		0			0
$853$	−8.90685e9			−0.491363			−0.245682	−	0.969351i	$-0.579012\pi$
$853$	−8.90685e9			−0.491363			−0.245682	+	0.969351i	$0.579012\pi$
$854$	−1.05773e9	+	6.46892e8i	−0.0581129	+	0.0355410i
$855$		0			0
$856$	1.17159e10	−	8.90806e8i	0.638439	−	0.0485428i
$857$		−	4.57533e9i		−	0.248307i	−0.992263	−	0.124154i	$-0.960378\pi$
$857$		−	4.57533e9i		−	0.248307i	0.992263	−	0.124154i	$-0.0396216\pi$
$858$		0			0
$859$		−	3.49444e10i		−	1.88105i	−0.339719	−	0.940527i	$-0.610332\pi$
$859$		−	3.49444e10i		−	1.88105i	0.339719	−	0.940527i	$-0.389668\pi$
$860$	5.74761e9	−	1.12312e10i	0.308136	−	0.602118i
$861$		0			0
$862$	3.70837e8	+	6.06354e8i	0.0197200	+	0.0322441i
$863$	−8.89734e9			−0.471219			−0.235609	−	0.971848i	$-0.575709\pi$
$863$	−8.89734e9			−0.471219			−0.235609	+	0.971848i	$0.575709\pi$
$864$		0			0
$865$	8.22709e9			0.432205
$866$	−1.45888e10	−	2.38540e10i	−0.763319	−	1.24810i
$867$		0			0
$868$	−4.49711e8	+	8.78762e8i	−0.0233407	+	0.0456092i
$869$			5.27515e10i			2.72688i
$870$		0			0
$871$			8.81235e7i			0.00451885i
$872$	8.19607e9	−	6.23177e8i	0.418599	−	0.0318276i
$873$		0			0
$874$	1.93926e10	−	1.18602e10i	0.982530	−	0.600901i
$875$	−2.39055e8			−0.0120634
$876$		0			0
$877$	−1.21015e10			−0.605816			−0.302908	−	0.953020i	$-0.597958\pi$
$877$	−1.21015e10			−0.605816			−0.302908	+	0.953020i	$0.597958\pi$
$878$	−1.35762e10	+	8.30302e9i	−0.676937	+	0.414005i
$879$		0			0
$880$	1.20704e10	−	8.70464e9i	0.597082	−	0.430588i
$881$			3.45097e8i			0.0170030i	0.999964	+	0.00850149i	$0.00270614\pi$
$881$			3.45097e8i			0.0170030i	−0.999964	+	0.00850149i	$0.997294\pi$
$882$		0			0
$883$		−	1.23229e10i		−	0.602350i	−0.953569	−	0.301175i	$-0.902621\pi$
$883$		−	1.23229e10i		−	0.602350i	0.953569	−	0.301175i	$-0.0973789\pi$
$884$	2.34139e10	+	1.19822e10i	1.13996	+	0.583382i
$885$		0			0
$886$	1.03685e10	+	1.69535e10i	0.500841	+	0.818922i
$887$	1.02180e10			0.491626			0.245813	−	0.969317i	$-0.420945\pi$
$887$	1.02180e10			0.491626			0.245813	+	0.969317i	$0.420945\pi$
$888$		0			0
$889$	5.09742e8			0.0243329
$890$	−2.11906e9	−	3.46486e9i	−0.100758	−	0.164748i
$891$		0			0
$892$	1.84306e9	+	9.43196e8i	0.0869487	+	0.0444964i
$893$			3.31889e9i			0.155960i
$894$		0			0
$895$			1.29520e10i			0.603889i
$896$	2.21988e9	−	1.87232e9i	0.103098	−	0.0869567i
$897$		0			0
$898$	2.21536e10	−	1.35488e10i	1.02089	−	0.624360i
$899$	−1.28559e10			−0.590124
$900$		0			0
$901$	5.64283e9			0.257016
$902$	5.10819e10	−	3.12409e10i	2.31763	−	1.41743i
$903$		0			0
$904$	−2.03922e8	−	2.68200e9i	−0.00918067	−	0.120745i
$905$		−	6.50909e8i		−	0.0291911i
$906$		0			0
$907$		−	3.09780e10i		−	1.37857i	−0.724491	−	0.689284i	$-0.757925\pi$
$907$		−	3.09780e10i		−	1.37857i	0.724491	−	0.689284i	$-0.242075\pi$
$908$	−7.24362e9	+	1.41545e10i	−0.321111	+	0.627471i
$909$		0			0
$910$	1.23392e9	+	2.01757e9i	0.0542801	+	0.0887531i
$911$	−1.25152e10			−0.548432			−0.274216	−	0.961668i	$-0.588418\pi$
$911$	−1.25152e10			−0.548432			−0.274216	+	0.961668i	$0.588418\pi$
$912$		0			0
$913$	−6.33058e10			−2.75293
$914$	−1.33839e10	−	2.18840e10i	−0.579791	−	0.948013i
$915$		0			0
$916$	4.31998e9	−	8.44150e9i	0.185715	−	0.362899i
$917$			3.23723e9i			0.138638i
$918$		0			0
$919$			3.59558e10i			1.52815i	0.645130	+	0.764073i	$0.276803\pi$
$919$			3.59558e10i			1.52815i	−0.645130	+	0.764073i	$0.723197\pi$
$920$	−1.04091e9	−	1.36901e10i	−0.0440712	−	0.579627i
$921$		0			0
$922$	−1.86513e9	+	1.14069e9i	−0.0783702	+	0.0479301i
$923$	−4.72798e10			−1.97911
$924$		0			0
$925$	4.84644e9			0.201339
$926$	−1.16514e10	+	7.12580e9i	−0.482212	+	0.294914i
$927$		0			0
$928$	3.48769e10	+	1.46274e10i	1.43258	+	0.600828i
$929$			2.85222e10i			1.16715i	0.812058	+	0.583577i	$0.198347\pi$
$929$			2.85222e10i			1.16715i	−0.812058	+	0.583577i	$0.801653\pi$
$930$		0			0
$931$		−	2.14195e10i		−	0.869934i
$932$	2.50585e10	+	1.28238e10i	1.01391	+	0.518874i
$933$		0			0
$934$	−2.29458e10	−	3.75186e10i	−0.921487	−	1.50672i
$935$	1.36605e10			0.546543
$936$		0			0
$937$	−3.91756e10			−1.55570			−0.777852	−	0.628448i	$-0.783691\pi$
$937$	−3.91756e10			−1.55570			−0.777852	+	0.628448i	$0.783691\pi$
$938$	4.65992e6	+	7.61941e6i	0.000184361	+	0.000301447i
$939$		0			0
$940$	1.78445e9	+	9.13202e8i	0.0700741	+	0.0358608i
$941$			3.32167e10i			1.29955i	0.760128	+	0.649774i	$0.225136\pi$
$941$			3.32167e10i			1.29955i	−0.760128	+	0.649774i	$0.774864\pi$
$942$		0			0
$943$		−	5.52421e10i		−	2.14526i
$944$	−2.51910e10	−	3.49316e10i	−0.974640	−	1.35150i
$945$		0			0
$946$	−5.53028e10	+	3.38224e10i	−2.12387	+	1.29893i
$947$	−1.71236e10			−0.655196			−0.327598	−	0.944817i	$-0.606239\pi$
$947$	−1.71236e10			−0.655196			−0.327598	+	0.944817i	$0.606239\pi$
$948$		0			0
$949$	7.16580e10			2.72165
$950$	−3.99505e9	+	2.44332e9i	−0.151179	+	0.0924586i
$951$		0			0
$952$	2.65805e9	−	2.02101e8i	0.0998466	−	0.00759170i
$953$		−	3.38111e10i		−	1.26542i	−0.774389	−	0.632710i	$-0.781943\pi$
$953$		−	3.38111e10i		−	1.26542i	0.774389	−	0.632710i	$-0.218057\pi$
$954$		0			0
$955$			8.17561e9i			0.303744i
$956$	−1.19675e9	+	2.33852e9i	−0.0442996	+	0.0865640i
$957$		0			0
$958$	1.15075e10	+	1.88159e10i	0.422866	+	0.691427i
$959$	−4.31154e9			−0.157858
$960$		0			0
$961$	2.35424e10			0.855696
$962$	−2.50156e10	−	4.09028e10i	−0.905936	−	1.48129i
$963$		0			0
$964$	6.93083e9	−	1.35433e10i	0.249181	−	0.486915i
$965$		−	8.10117e9i		−	0.290203i
$966$		0			0
$967$			4.31003e10i			1.53281i	0.642360	+	0.766403i	$0.277956\pi$
$967$			4.31003e10i			1.53281i	−0.642360	+	0.766403i	$0.722044\pi$
$968$	−4.81057e10	+	3.65765e9i	−1.70464	+	0.129610i
$969$		0			0
$970$	−9.84490e9	+	6.02100e9i	−0.346346	+	0.211820i
$971$	−1.66939e10			−0.585180			−0.292590	−	0.956238i	$-0.594517\pi$
$971$	−1.66939e10			−0.585180			−0.292590	+	0.956238i	$0.594517\pi$
$972$		0			0
$973$	−1.03336e9			−0.0359632
$974$	1.36645e10	−	8.35704e9i	0.473848	−	0.289798i
$975$		0			0
$976$	8.58060e9	+	1.18984e10i	0.295422	+	0.409653i
$977$			1.34962e9i			0.0462998i	0.999732	+	0.0231499i	$0.00736951\pi$
$977$			1.34962e9i			0.0462998i	−0.999732	+	0.0231499i	$0.992630\pi$
$978$		0			0
$979$			2.08686e10i			0.710812i
$980$	−1.15165e10	−	5.89364e9i	−0.390869	−	0.200029i
$981$		0			0
$982$	2.27948e10	+	3.72717e10i	0.768150	+	1.25600i
$983$	−4.83211e9			−0.162256			−0.0811278	−	0.996704i	$-0.525852\pi$
$983$	−4.83211e9			−0.162256			−0.0811278	+	0.996704i	$0.525852\pi$
$984$		0			0
$985$	−3.79981e9			−0.126688
$986$	1.81131e10	+	2.96166e10i	0.601759	+	0.983933i
$987$		0			0
$988$	4.12420e10	+	2.11058e10i	1.36047	+	0.696229i
$989$			5.98067e10i			1.96591i
$990$		0			0
$991$			3.39066e9i			0.110669i	0.998468	+	0.0553345i	$0.0176225\pi$
$991$			3.39066e9i			0.110669i	−0.998468	+	0.0553345i	$0.982377\pi$
$992$	1.07707e10	+	4.51726e9i	0.350311	+	0.146921i
$993$		0			0
$994$	−4.08795e9	+	2.50013e9i	−0.132024	+	0.0807440i
$995$	1.03593e10			0.333387
$996$		0			0
$997$	2.94937e10			0.942531			0.471265	−	0.881991i	$-0.343797\pi$
$997$	2.94937e10			0.942531			0.471265	+	0.881991i	$0.343797\pi$
$998$	−9.04387e9	+	5.53110e9i	−0.288003	+	0.176139i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.8.e.a.71.38	yes	56
3.2	odd	2	inner	180.8.e.a.71.19	✓	56
4.3	odd	2	inner	180.8.e.a.71.20	yes	56
12.11	even	2	inner	180.8.e.a.71.37	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.8.e.a.71.19	✓	56	3.2	odd	2	inner
180.8.e.a.71.20	yes	56	4.3	odd	2	inner
180.8.e.a.71.37	yes	56	12.11	even	2	inner
180.8.e.a.71.38	yes	56	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	180.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(5.90287 + 9.65174i) q^{2} +(-58.3123 + 113.946i) q^{4} +125.000i q^{5} -122.396i q^{7} +(-1443.99 + 109.792i) q^{8} +O(q^{10})\)	\(q+(5.90287 + 9.65174i) q^{2} +(-58.3123 + 113.946i) q^{4} +125.000i q^{5} -122.396i q^{7} +(-1443.99 + 109.792i) q^{8} +(-1206.47 + 737.858i) q^{10} +7266.48 q^{11} +13662.9 q^{13} +(1181.34 - 722.489i) q^{14} +(-9583.34 - 13288.9i) q^{16} -15039.4i q^{17} -26490.9i q^{19} +(-14243.2 - 7289.04i) q^{20} +(42893.0 + 70134.2i) q^{22} +75846.1 q^{23} -15625.0 q^{25} +(80650.5 + 131871. i) q^{26} +(13946.6 + 7137.21i) q^{28} +204032. i q^{29} +63009.3i q^{31} +(71692.0 - 170939. i) q^{32} +(145157. - 88775.7i) q^{34} +15299.5 q^{35} -310172. q^{37} +(255683. - 156372. i) q^{38} +(-13723.9 - 180498. i) q^{40} -728345. i q^{41} +788528. i q^{43} +(-423725. + 827985. i) q^{44} +(447709. + 732047. i) q^{46} -125284. q^{47} +808562. q^{49} +(-92232.3 - 150809. i) q^{50} +(-796718. + 1.55684e6i) q^{52} +375202. i q^{53} +908310. i q^{55} +(13438.1 + 176739. i) q^{56} +(-1.96926e6 + 1.20437e6i) q^{58} +2.62863e6 q^{59} -895367. q^{61} +(-608149. + 371935. i) q^{62} +(2.07304e6 - 317075. i) q^{64} +1.70787e6i q^{65} +6449.82i q^{67} +(1.71368e6 + 876984. i) q^{68} +(90311.1 + 147667. i) q^{70} -3.46044e6 q^{71} +5.24470e6 q^{73} +(-1.83091e6 - 2.99370e6i) q^{74} +(3.01853e6 + 1.54475e6i) q^{76} -889390. i q^{77} +7.25957e6i q^{79} +(1.66111e6 - 1.19792e6i) q^{80} +(7.02980e6 - 4.29933e6i) q^{82} -8.71203e6 q^{83} +1.87993e6 q^{85} +(-7.61067e6 + 4.65457e6i) q^{86} +(-1.04927e7 + 797798. i) q^{88} +2.87191e6i q^{89} -1.67229e6i q^{91} +(-4.42276e6 + 8.64235e6i) q^{92} +(-739536. - 1.20921e6i) q^{94} +3.31136e6 q^{95} +8.16010e6 q^{97} +(4.77283e6 + 7.80404e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 52 q^{4}+O(q^{10}) \) 56 * q + 52 * q^4	\( 56 q + 52 q^{4} - 6500 q^{10} + 14128 q^{13} - 8060 q^{16} + 259088 q^{22} - 875000 q^{25} - 490976 q^{28} + 40912 q^{34} + 1268048 q^{37} - 266500 q^{40} + 3108200 q^{46} - 3522056 q^{49} - 8882216 q^{52} + 8807592 q^{58} - 3944912 q^{61} - 16633580 q^{64} + 4533000 q^{70} + 7602384 q^{73} - 38876976 q^{76} + 33213064 q^{82} - 15068000 q^{85} - 56145472 q^{88} + 29409456 q^{94} - 45595824 q^{97}+O(q^{100}) \) 56 * q + 52 * q^4 - 6500 * q^10 + 14128 * q^13 - 8060 * q^16 + 259088 * q^22 - 875000 * q^25 - 490976 * q^28 + 40912 * q^34 + 1268048 * q^37 - 266500 * q^40 + 3108200 * q^46 - 3522056 * q^49 - 8882216 * q^52 + 8807592 * q^58 - 3944912 * q^61 - 16633580 * q^64 + 4533000 * q^70 + 7602384 * q^73 - 38876976 * q^76 + 33213064 * q^82 - 15068000 * q^85 - 56145472 * q^88 + 29409456 * q^94 - 45595824 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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