LMFDB - Embedded newform 177.8.a.c.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,8,Mod(1,177)]

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([0, 0]))

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

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

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:	yes
Analytic conductor:	$55.2921495107$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 $ x^17 - 2x^16 - 1669x^15 + 2385x^14 + 1108684x^13 - 848131x^12 - 377920980x^11 + 12724944x^10 + 71331230512x^9 + 50741131904x^8 - 7480805165760x^7 - 10751966150272x^6 + 413177144536320x^5 + 886760582981376x^4 - 10454479722123264x^3 - 29180140031461376x^2 + 79787300207378432x + 248723246810300416
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.56813$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} -750.136 q^{10} -1820.60 q^{11} +3112.25 q^{12} -12376.9 q^{13} +5510.17 q^{14} +5676.27 q^{15} +11657.2 q^{16} -36030.6 q^{17} +2601.17 q^{18} +18266.5 q^{19} +24233.1 q^{20} -41695.4 q^{21} -6496.13 q^{22} -99780.9 q^{23} +23436.4 q^{24} -33927.4 q^{25} -44162.4 q^{26} -19683.0 q^{27} -178006. q^{28} +14104.4 q^{29} +20253.7 q^{30} +55263.0 q^{31} +152700. q^{32} +49156.2 q^{33} -128562. q^{34} -324656. q^{35} -84030.7 q^{36} +532023. q^{37} +65177.1 q^{38} +334176. q^{39} +182484. q^{40} -27816.2 q^{41} -148774. q^{42} +784533. q^{43} +209858. q^{44} -153259. q^{45} -356031. q^{46} +115560. q^{47} -314744. q^{48} +1.56124e6 q^{49} -121058. q^{50} +972825. q^{51} +1.42667e6 q^{52} +566369. q^{53} -70231.5 q^{54} +382748. q^{55} -1.34045e6 q^{56} -493195. q^{57} +50326.3 q^{58} -205379. q^{59} -654295. q^{60} -126809. q^{61} +197186. q^{62} +1.12577e6 q^{63} -947265. q^{64} +2.60202e6 q^{65} +175396. q^{66} -4.67077e6 q^{67} +4.15319e6 q^{68} +2.69408e6 q^{69} -1.15841e6 q^{70} +1.27912e6 q^{71} -632782. q^{72} +55292.5 q^{73} +1.89833e6 q^{74} +916041. q^{75} -2.10555e6 q^{76} -2.81150e6 q^{77} +1.19238e6 q^{78} -1.32950e6 q^{79} -2.45071e6 q^{80} +531441. q^{81} -99251.9 q^{82} +2.54431e6 q^{83} +4.80616e6 q^{84} +7.57478e6 q^{85} +2.79932e6 q^{86} -380819. q^{87} +1.58030e6 q^{88} +8.04237e6 q^{89} -546849. q^{90} -1.91133e7 q^{91} +1.15016e7 q^{92} -1.49210e6 q^{93} +412332. q^{94} -3.84020e6 q^{95} -4.12290e6 q^{96} +1.17167e7 q^{97} +5.57069e6 q^{98} -1.32722e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9}+O(q^{10}) $ 17 * q + 2 * q^2 - 459 * q^3 + 1166 * q^4 - 318 * q^5 - 54 * q^6 + 3145 * q^7 + 2355 * q^8 + 12393 * q^9	\( 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9} + 6521 q^{10} - 1764 q^{11} - 31482 q^{12} + 18192 q^{13} - 7827 q^{14} + 8586 q^{15} + 139226 q^{16} - 15507 q^{17} + 1458 q^{18} + 52083 q^{19} + 721 q^{20} - 84915 q^{21} - 234434 q^{22} + 63823 q^{23} - 63585 q^{24} + 202153 q^{25} - 367956 q^{26} - 334611 q^{27} + 182306 q^{28} - 502955 q^{29} - 176067 q^{30} + 347531 q^{31} - 243908 q^{32} + 47628 q^{33} - 330872 q^{34} + 92641 q^{35} + 850014 q^{36} + 447615 q^{37} + 775669 q^{38} - 491184 q^{39} + 2203270 q^{40} + 940335 q^{41} + 211329 q^{42} + 478562 q^{43} - 596924 q^{44} - 231822 q^{45} - 3078663 q^{46} + 703121 q^{47} - 3759102 q^{48} + 1895082 q^{49} - 876967 q^{50} + 418689 q^{51} + 6278296 q^{52} - 1005974 q^{53} - 39366 q^{54} + 5212846 q^{55} + 3425294 q^{56} - 1406241 q^{57} + 6710166 q^{58} - 3491443 q^{59} - 19467 q^{60} + 11510749 q^{61} + 5996234 q^{62} + 2292705 q^{63} + 29496941 q^{64} + 11094180 q^{65} + 6329718 q^{66} + 14007144 q^{67} + 19688159 q^{68} - 1723221 q^{69} + 30909708 q^{70} + 5229074 q^{71} + 1716795 q^{72} + 5452211 q^{73} + 12819662 q^{74} - 5458131 q^{75} + 41929340 q^{76} + 9930777 q^{77} + 9934812 q^{78} + 15275654 q^{79} + 36576105 q^{80} + 9034497 q^{81} + 32025935 q^{82} + 7826609 q^{83} - 4922262 q^{84} + 11836945 q^{85} + 51649136 q^{86} + 13579785 q^{87} + 30223741 q^{88} - 6436185 q^{89} + 4753809 q^{90} + 11633535 q^{91} + 43357972 q^{92} - 9383337 q^{93} - 4494252 q^{94} + 23741055 q^{95} + 6585516 q^{96} + 26377540 q^{97} + 26517816 q^{98} - 1285956 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 459 * q^3 + 1166 * q^4 - 318 * q^5 - 54 * q^6 + 3145 * q^7 + 2355 * q^8 + 12393 * q^9 + 6521 * q^10 - 1764 * q^11 - 31482 * q^12 + 18192 * q^13 - 7827 * q^14 + 8586 * q^15 + 139226 * q^16 - 15507 * q^17 + 1458 * q^18 + 52083 * q^19 + 721 * q^20 - 84915 * q^21 - 234434 * q^22 + 63823 * q^23 - 63585 * q^24 + 202153 * q^25 - 367956 * q^26 - 334611 * q^27 + 182306 * q^28 - 502955 * q^29 - 176067 * q^30 + 347531 * q^31 - 243908 * q^32 + 47628 * q^33 - 330872 * q^34 + 92641 * q^35 + 850014 * q^36 + 447615 * q^37 + 775669 * q^38 - 491184 * q^39 + 2203270 * q^40 + 940335 * q^41 + 211329 * q^42 + 478562 * q^43 - 596924 * q^44 - 231822 * q^45 - 3078663 * q^46 + 703121 * q^47 - 3759102 * q^48 + 1895082 * q^49 - 876967 * q^50 + 418689 * q^51 + 6278296 * q^52 - 1005974 * q^53 - 39366 * q^54 + 5212846 * q^55 + 3425294 * q^56 - 1406241 * q^57 + 6710166 * q^58 - 3491443 * q^59 - 19467 * q^60 + 11510749 * q^61 + 5996234 * q^62 + 2292705 * q^63 + 29496941 * q^64 + 11094180 * q^65 + 6329718 * q^66 + 14007144 * q^67 + 19688159 * q^68 - 1723221 * q^69 + 30909708 * q^70 + 5229074 * q^71 + 1716795 * q^72 + 5452211 * q^73 + 12819662 * q^74 - 5458131 * q^75 + 41929340 * q^76 + 9930777 * q^77 + 9934812 * q^78 + 15275654 * q^79 + 36576105 * q^80 + 9034497 * q^81 + 32025935 * q^82 + 7826609 * q^83 - 4922262 * q^84 + 11836945 * q^85 + 51649136 * q^86 + 13579785 * q^87 + 30223741 * q^88 - 6436185 * q^89 + 4753809 * q^90 + 11633535 * q^91 + 43357972 * q^92 - 9383337 * q^93 - 4494252 * q^94 + 23741055 * q^95 + 6585516 * q^96 + 26377540 * q^97 + 26517816 * q^98 - 1285956 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	3.56813	0.315381	0.157691	−	0.987489i	$-0.449595\pi$
$2$	3.56813	0.315381	0.157691	+	0.987489i	$0.449595\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−115.268	−0.900535
$5$	−210.232	−0.752149	−0.376075	−	0.926589i	$-0.622726\pi$
$5$	−210.232	−0.752149	−0.376075	+	0.926589i	$0.622726\pi$
$6$	−96.3395	−0.182085
$7$	1544.27	1.70169	0.850846	−	0.525415i	$-0.176090\pi$
$7$	1544.27	1.70169	0.850846	+	0.525415i	$0.176090\pi$
$8$	−868.013	−0.599393
$9$	729.000	0.333333
$10$	−750.136	−0.237214
$11$	−1820.60	−0.412420	−0.206210	−	0.978508i	$-0.566113\pi$
$11$	−1820.60	−0.412420	−0.206210	+	0.978508i	$0.566113\pi$
$12$	3112.25	0.519924
$13$	−12376.9	−1.56246	−0.781232	−	0.624241i	$-0.785408\pi$
$13$	−12376.9	−1.56246	−0.781232	+	0.624241i	$0.785408\pi$
$14$	5510.17	0.536681
$15$	5676.27	0.434254
$16$	11657.2	0.711498
$17$	−36030.6	−1.77869	−0.889344	−	0.457238i	$-0.848839\pi$
$17$	−36030.6	−1.77869	−0.889344	+	0.457238i	$0.848839\pi$
$18$	2601.17	0.105127
$19$	18266.5	0.610966	0.305483	−	0.952198i	$-0.401182\pi$
$19$	18266.5	0.610966	0.305483	+	0.952198i	$0.401182\pi$
$20$	24233.1	0.677337
$21$	−41695.4	−0.982472
$22$	−6496.13	−0.130069
$23$	−99780.9	−1.71001	−0.855007	−	0.518616i	$-0.826447\pi$
$23$	−99780.9	−1.71001	−0.855007	+	0.518616i	$0.826447\pi$
$24$	23436.4	0.346060
$25$	−33927.4	−0.434271
$26$	−44162.4	−0.492771
$27$	−19683.0	−0.192450
$28$	−178006.	−1.53243
$29$	14104.4	0.107389	0.0536947	−	0.998557i	$-0.482900\pi$
$29$	14104.4	0.107389	0.0536947	+	0.998557i	$0.482900\pi$
$30$	20253.7	0.136955
$31$	55263.0	0.333172	0.166586	−	0.986027i	$-0.446726\pi$
$31$	55263.0	0.333172	0.166586	+	0.986027i	$0.446726\pi$
$32$	152700.	0.823786
$33$	49156.2	0.238111
$34$	−128562.	−0.560965
$35$	−324656.	−1.27993
$36$	−84030.7	−0.300178
$37$	532023.	1.72673	0.863364	−	0.504581i	$-0.168353\pi$
$37$	532023.	1.72673	0.863364	+	0.504581i	$0.168353\pi$
$38$	65177.1	0.192687
$39$	334176.	0.902089
$40$	182484.	0.450833
$41$	−27816.2	−0.0630311	−0.0315155	−	0.999503i	$-0.510033\pi$
$41$	−27816.2	−0.0630311	−0.0315155	+	0.999503i	$0.510033\pi$
$42$	−148774.	−0.309853
$43$	784533.	1.50478	0.752388	−	0.658720i	$-0.228902\pi$
$43$	784533.	1.50478	0.752388	+	0.658720i	$0.228902\pi$
$44$	209858.	0.371399
$45$	−153259.	−0.250716
$46$	−356031.	−0.539306
$47$	115560.	0.162355	0.0811773	−	0.996700i	$-0.474132\pi$
$47$	115560.	0.162355	0.0811773	+	0.996700i	$0.474132\pi$
$48$	−314744.	−0.410783
$49$	1.56124e6	1.89575
$50$	−121058.	−0.136961
$51$	972825.	1.02693
$52$	1.42667e6	1.40705
$53$	566369.	0.522558	0.261279	−	0.965263i	$-0.415856\pi$
$53$	566369.	0.522558	0.261279	+	0.965263i	$0.415856\pi$
$54$	−70231.5	−0.0606951
$55$	382748.	0.310202
$56$	−1.34045e6	−1.01998
$57$	−493195.	−0.352741
$58$	50326.3	0.0338686
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	−654295.	−0.391061
$61$	−126809.	−0.0715313	−0.0357656	−	0.999360i	$-0.511387\pi$
$61$	−126809.	−0.0715313	−0.0357656	+	0.999360i	$0.511387\pi$
$62$	197186.	0.105076
$63$	1.12577e6	0.567231
$64$	−947265.	−0.451691
$65$	2.60202e6	1.17521
$66$	175396.	0.0750957
$67$	−4.67077e6	−1.89726	−0.948629	−	0.316391i	$-0.897529\pi$
$67$	−4.67077e6	−1.89726	−0.948629	+	0.316391i	$0.897529\pi$
$68$	4.15319e6	1.60177
$69$	2.69408e6	0.987277
$70$	−1.15841e6	−0.403665
$71$	1.27912e6	0.424137	0.212068	−	0.977255i	$-0.431980\pi$
$71$	1.27912e6	0.424137	0.212068	+	0.977255i	$0.431980\pi$
$72$	−632782.	−0.199798
$73$	55292.5	0.0166355	0.00831775	−	0.999965i	$-0.497352\pi$
$73$	55292.5	0.0166355	0.00831775	+	0.999965i	$0.497352\pi$
$74$	1.89833e6	0.544578
$75$	916041.	0.250727
$76$	−2.10555e6	−0.550196
$77$	−2.81150e6	−0.701812
$78$	1.19238e6	0.284502
$79$	−1.32950e6	−0.303385	−0.151693	−	0.988428i	$-0.548472\pi$
$79$	−1.32950e6	−0.303385	−0.151693	+	0.988428i	$0.548472\pi$
$80$	−2.45071e6	−0.535153
$81$	531441.	0.111111
$82$	−99251.9	−0.0198788
$83$	2.54431e6	0.488423	0.244211	−	0.969722i	$-0.421471\pi$
$83$	2.54431e6	0.488423	0.244211	+	0.969722i	$0.421471\pi$
$84$	4.80616e6	0.884750
$85$	7.57478e6	1.33784
$86$	2.79932e6	0.474578
$87$	−380819.	−0.0620013
$88$	1.58030e6	0.247202
$89$	8.04237e6	1.20926	0.604629	−	0.796507i	$-0.293321\pi$
$89$	8.04237e6	1.20926	0.604629	+	0.796507i	$0.293321\pi$
$90$	−546849.	−0.0790712
$91$	−1.91133e7	−2.65883
$92$	1.15016e7	1.53993
$93$	−1.49210e6	−0.192357
$94$	412332.	0.0512035
$95$	−3.84020e6	−0.459538
$96$	−4.12290e6	−0.475613
$97$	1.17167e7	1.30348	0.651739	−	0.758443i	$-0.274040\pi$
$97$	1.17167e7	1.30348	0.651739	+	0.758443i	$0.274040\pi$
$98$	5.57069e6	0.597885
$99$	−1.32722e6	−0.137473
$100$	3.91076e6	0.391076
$101$	−826632.	−0.0798339	−0.0399169	−	0.999203i	$-0.512709\pi$
$101$	−826632.	−0.0798339	−0.0399169	+	0.999203i	$0.512709\pi$
$102$	3.47117e6	0.323873
$103$	9.34032e6	0.842232	0.421116	−	0.907007i	$-0.361639\pi$
$103$	9.34032e6	0.842232	0.421116	+	0.907007i	$0.361639\pi$
$104$	1.07433e7	0.936529
$105$	8.76571e6	0.738966
$106$	2.02088e6	0.164805
$107$	−1.21889e7	−0.961884	−0.480942	−	0.876752i	$-0.659705\pi$
$107$	−1.21889e7	−0.961884	−0.480942	+	0.876752i	$0.659705\pi$
$108$	2.26883e6	0.173308
$109$	−1.32306e7	−0.978556	−0.489278	−	0.872128i	$-0.662740\pi$
$109$	−1.32306e7	−0.978556	−0.489278	+	0.872128i	$0.662740\pi$
$110$	1.36570e6	0.0978317
$111$	−1.43646e7	−0.996927
$112$	1.80019e7	1.21075
$113$	1.51047e7	0.984777	0.492389	−	0.870375i	$-0.336124\pi$
$113$	1.51047e7	0.984777	0.492389	+	0.870375i	$0.336124\pi$
$114$	−1.75978e6	−0.111248
$115$	2.09771e7	1.28619
$116$	−1.62579e6	−0.0967079
$117$	−9.02275e6	−0.520821
$118$	−732819.	−0.0410591
$119$	−5.56410e7	−3.02678
$120$	−4.92708e6	−0.260288
$121$	−1.61726e7	−0.829910
$122$	−452471.	−0.0225596
$123$	751038.	0.0363910
$124$	−6.37008e6	−0.300033
$125$	2.35570e7	1.07879
$126$	4.01691e6	0.178894
$127$	567728.	0.0245939	0.0122969	−	0.999924i	$-0.496086\pi$
$127$	567728.	0.0245939	0.0122969	+	0.999924i	$0.496086\pi$
$128$	−2.29256e7	−0.966240
$129$	−2.11824e7	−0.868783
$130$	9.28435e6	0.370638
$131$	−4.38980e6	−0.170606	−0.0853032	−	0.996355i	$-0.527186\pi$
$131$	−4.38980e6	−0.170606	−0.0853032	+	0.996355i	$0.527186\pi$
$132$	−5.66615e6	−0.214427
$133$	2.82084e7	1.03968
$134$	−1.66659e7	−0.598359
$135$	4.13800e6	0.144751
$136$	3.12750e7	1.06613
$137$	5.41462e7	1.79906	0.899530	−	0.436858i	$-0.143909\pi$
$137$	5.41462e7	1.79906	0.899530	+	0.436858i	$0.143909\pi$
$138$	9.61284e6	0.311369
$139$	5.04738e7	1.59410	0.797048	−	0.603917i	$-0.206394\pi$
$139$	5.04738e7	1.59410	0.797048	+	0.603917i	$0.206394\pi$
$140$	3.74226e7	1.15262
$141$	−3.12012e6	−0.0937354
$142$	4.56405e6	0.133765
$143$	2.25334e7	0.644391
$144$	8.49808e6	0.237166
$145$	−2.96520e6	−0.0807729
$146$	197291.	0.00524652
$147$	−4.21533e7	−1.09451
$148$	−6.13254e7	−1.55498
$149$	5.85576e7	1.45021	0.725106	−	0.688638i	$-0.241791\pi$
$149$	5.85576e7	1.45021	0.725106	+	0.688638i	$0.241791\pi$
$150$	3.26855e6	0.0790744
$151$	−1.16831e6	−0.0276147	−0.0138074	−	0.999905i	$-0.504395\pi$
$151$	−1.16831e6	−0.0276147	−0.0138074	+	0.999905i	$0.504395\pi$
$152$	−1.58555e7	−0.366209
$153$	−2.62663e7	−0.592896
$154$	−1.00318e7	−0.221338
$155$	−1.16181e7	−0.250595
$156$	−3.85200e7	−0.812362
$157$	2.26283e7	0.466663	0.233332	−	0.972397i	$-0.425037\pi$
$157$	2.26283e7	0.466663	0.233332	+	0.972397i	$0.425037\pi$
$158$	−4.74384e6	−0.0956820
$159$	−1.52920e7	−0.301699
$160$	−3.21025e7	−0.619610
$161$	−1.54089e8	−2.90992
$162$	1.89625e6	0.0350423
$163$	8.98984e7	1.62591	0.812953	−	0.582329i	$-0.197858\pi$
$163$	8.98984e7	1.62591	0.812953	+	0.582329i	$0.197858\pi$
$164$	3.20633e6	0.0567617
$165$	−1.03342e7	−0.179095
$166$	9.07841e6	0.154039
$167$	8.00457e7	1.32994	0.664968	−	0.746872i	$-0.268445\pi$
$167$	8.00457e7	1.32994	0.664968	+	0.746872i	$0.268445\pi$
$168$	3.61921e7	0.588887
$169$	9.04390e7	1.44129
$170$	2.70278e7	0.421929
$171$	1.33163e7	0.203655
$172$	−9.04320e7	−1.35510
$173$	−2.61935e7	−0.384620	−0.192310	−	0.981334i	$-0.561598\pi$
$173$	−2.61935e7	−0.384620	−0.192310	+	0.981334i	$0.561598\pi$
$174$	−1.35881e6	−0.0195540
$175$	−5.23932e7	−0.738996
$176$	−2.12230e7	−0.293436
$177$	5.54523e6	0.0751646
$178$	2.86962e7	0.381377
$179$	−1.04938e8	−1.36756	−0.683781	−	0.729687i	$-0.739666\pi$
$179$	−1.04938e8	−1.36756	−0.683781	+	0.729687i	$0.739666\pi$
$180$	1.76660e7	0.225779
$181$	320272.	0.00401461	0.00200731	−	0.999998i	$-0.499361\pi$
$181$	320272.	0.00401461	0.00200731	+	0.999998i	$0.499361\pi$
$182$	−6.81987e7	−0.838545
$183$	3.42385e6	0.0412986
$184$	8.66111e7	1.02497
$185$	−1.11848e8	−1.29876
$186$	−5.32401e6	−0.0606658
$187$	6.55972e7	0.733567
$188$	−1.33204e7	−0.146206
$189$	−3.03959e7	−0.327491
$190$	−1.37023e7	−0.144929
$191$	−4.56002e7	−0.473533	−0.236767	−	0.971567i	$-0.576088\pi$
$191$	−4.56002e7	−0.473533	−0.236767	+	0.971567i	$0.576088\pi$
$192$	2.55762e7	0.260784
$193$	−4.97679e6	−0.0498309	−0.0249154	−	0.999690i	$-0.507932\pi$
$193$	−4.97679e6	−0.0498309	−0.0249154	+	0.999690i	$0.507932\pi$
$194$	4.18066e7	0.411092
$195$	−7.02546e7	−0.678506
$196$	−1.79961e8	−1.70719
$197$	−1.34591e8	−1.25425	−0.627126	−	0.778917i	$-0.715769\pi$
$197$	−1.34591e8	−1.25425	−0.627126	+	0.778917i	$0.715769\pi$
$198$	−4.73568e6	−0.0433565
$199$	−696482.	−0.00626504	−0.00313252	−	0.999995i	$-0.500997\pi$
$199$	−696482.	−0.00626504	−0.00313252	+	0.999995i	$0.500997\pi$
$200$	2.94495e7	0.260299
$201$	1.26111e8	1.09538
$202$	−2.94953e6	−0.0251781
$203$	2.17810e7	0.182744
$204$	−1.12136e8	−0.924783
$205$	5.84786e6	0.0474088
$206$	3.33275e7	0.265624
$207$	−7.27402e7	−0.570005
$208$	−1.44280e8	−1.11169
$209$	−3.32559e7	−0.251975
$210$	3.12772e7	0.233056
$211$	−7.56254e7	−0.554216	−0.277108	−	0.960839i	$-0.589376\pi$
$211$	−7.56254e7	−0.554216	−0.277108	+	0.960839i	$0.589376\pi$
$212$	−6.52845e7	−0.470581
$213$	−3.45361e7	−0.244875
$214$	−4.34917e7	−0.303360
$215$	−1.64934e8	−1.13182
$216$	1.70851e7	0.115353
$217$	8.53412e7	0.566956
$218$	−4.72084e7	−0.308618
$219$	−1.49290e6	−0.00960451
$220$	−4.41188e7	−0.279347
$221$	4.45946e8	2.77914
$222$	−5.12548e7	−0.314412
$223$	2.78187e8	1.67984	0.839922	−	0.542706i	$-0.182600\pi$
$223$	2.78187e8	1.67984	0.839922	+	0.542706i	$0.182600\pi$
$224$	2.35810e8	1.40183
$225$	−2.47331e7	−0.144757
$226$	5.38956e7	0.310580
$227$	−2.26384e8	−1.28457	−0.642283	−	0.766468i	$-0.722012\pi$
$227$	−2.26384e8	−1.28457	−0.642283	+	0.766468i	$0.722012\pi$
$228$	5.68498e7	0.317656
$229$	−1.60387e8	−0.882559	−0.441279	−	0.897370i	$-0.645475\pi$
$229$	−1.60387e8	−0.882559	−0.441279	+	0.897370i	$0.645475\pi$
$230$	7.48492e7	0.405639
$231$	7.59105e7	0.405191
$232$	−1.22428e7	−0.0643684
$233$	7.39393e7	0.382939	0.191469	−	0.981499i	$-0.438675\pi$
$233$	7.39393e7	0.382939	0.191469	+	0.981499i	$0.438675\pi$
$234$	−3.21944e7	−0.164257
$235$	−2.42944e7	−0.122115
$236$	2.36737e7	0.117240
$237$	3.58966e7	0.175160
$238$	−1.98534e8	−0.954589
$239$	9.87669e7	0.467971	0.233985	−	0.972240i	$-0.424823\pi$
$239$	9.87669e7	0.467971	0.233985	+	0.972240i	$0.424823\pi$
$240$	6.61693e7	0.308970
$241$	−1.16325e8	−0.535321	−0.267661	−	0.963513i	$-0.586251\pi$
$241$	−1.16325e8	−0.535321	−0.267661	+	0.963513i	$0.586251\pi$
$242$	−5.77059e7	−0.261738
$243$	−1.43489e7	−0.0641500
$244$	1.46171e7	0.0644164
$245$	−3.28222e8	−1.42589
$246$	2.67980e6	0.0114770
$247$	−2.26082e8	−0.954612
$248$	−4.79690e7	−0.199701
$249$	−6.86962e7	−0.281991
$250$	8.40545e7	0.340229
$251$	−3.82452e8	−1.52658	−0.763290	−	0.646056i	$-0.776417\pi$
$251$	−3.82452e8	−1.52658	−0.763290	+	0.646056i	$0.776417\pi$
$252$	−1.29766e8	−0.510811
$253$	1.81661e8	0.705244
$254$	2.02573e6	0.00775645
$255$	−2.04519e8	−0.772402
$256$	3.94485e7	0.146957
$257$	−1.87540e8	−0.689174	−0.344587	−	0.938754i	$-0.611981\pi$
$257$	−1.87540e8	−0.689174	−0.344587	+	0.938754i	$0.611981\pi$
$258$	−7.55816e7	−0.273998
$259$	8.21588e8	2.93836
$260$	−2.99931e8	−1.05831
$261$	1.02821e7	0.0357965
$262$	−1.56634e7	−0.0538061
$263$	−3.36862e8	−1.14185	−0.570923	−	0.821004i	$-0.693414\pi$
$263$	−3.36862e8	−1.14185	−0.570923	+	0.821004i	$0.693414\pi$
$264$	−4.26682e7	−0.142722
$265$	−1.19069e8	−0.393041
$266$	1.00651e8	0.327894
$267$	−2.17144e8	−0.698165
$268$	5.38392e8	1.70855
$269$	−6.01783e7	−0.188498	−0.0942490	−	0.995549i	$-0.530045\pi$
$269$	−6.01783e7	−0.188498	−0.0942490	+	0.995549i	$0.530045\pi$
$270$	1.47649e7	0.0456518
$271$	−1.24433e8	−0.379788	−0.189894	−	0.981805i	$-0.560814\pi$
$271$	−1.24433e8	−0.379788	−0.189894	+	0.981805i	$0.560814\pi$
$272$	−4.20015e8	−1.26553
$273$	5.16059e8	1.53508
$274$	1.93201e8	0.567390
$275$	6.17683e7	0.179102
$276$	−3.10543e8	−0.889078
$277$	−4.48164e8	−1.26695	−0.633473	−	0.773765i	$-0.718371\pi$
$277$	−4.48164e8	−1.26695	−0.633473	+	0.773765i	$0.718371\pi$
$278$	1.80097e8	0.502747
$279$	4.02867e7	0.111057
$280$	2.81806e8	0.767179
$281$	5.57459e8	1.49879	0.749395	−	0.662124i	$-0.230345\pi$
$281$	5.57459e8	1.49879	0.749395	+	0.662124i	$0.230345\pi$
$282$	−1.11330e7	−0.0295624
$283$	6.19049e8	1.62358	0.811788	−	0.583952i	$-0.198495\pi$
$283$	6.19049e8	1.62358	0.811788	+	0.583952i	$0.198495\pi$
$284$	−1.47442e8	−0.381950
$285$	1.03685e8	0.265314
$286$	8.04019e7	0.203229
$287$	−4.29558e7	−0.107259
$288$	1.11318e8	0.274595
$289$	8.87863e8	2.16373
$290$	−1.05802e7	−0.0254742
$291$	−3.16350e8	−0.752563
$292$	−6.37348e6	−0.0149809
$293$	5.08839e8	1.18180	0.590899	−	0.806745i	$-0.298773\pi$
$293$	5.08839e8	1.18180	0.590899	+	0.806745i	$0.298773\pi$
$294$	−1.50409e8	−0.345189
$295$	4.31773e7	0.0979215
$296$	−4.61803e8	−1.03499
$297$	3.58348e7	0.0793703
$298$	2.08941e8	0.457369
$299$	1.23498e9	2.67184
$300$	−1.05591e8	−0.225788
$301$	1.21153e9	2.56066
$302$	−4.16870e6	−0.00870915
$303$	2.23191e7	0.0460921
$304$	2.12935e8	0.434701
$305$	2.66593e7	0.0538022
$306$	−9.37215e7	−0.186988
$307$	−2.00181e8	−0.394855	−0.197428	−	0.980317i	$-0.563259\pi$
$307$	−2.00181e8	−0.394855	−0.197428	+	0.980317i	$0.563259\pi$
$308$	3.24077e8	0.632006
$309$	−2.52189e8	−0.486263
$310$	−4.14548e7	−0.0790330
$311$	−9.52557e7	−0.179568	−0.0897841	−	0.995961i	$-0.528618\pi$
$311$	−9.52557e7	−0.179568	−0.0897841	+	0.995961i	$0.528618\pi$
$312$	−2.90069e8	−0.540705
$313$	−3.67649e8	−0.677686	−0.338843	−	0.940843i	$-0.610035\pi$
$313$	−3.67649e8	−0.677686	−0.338843	+	0.940843i	$0.610035\pi$
$314$	8.07408e7	0.147177
$315$	−2.36674e8	−0.426642
$316$	1.53250e8	0.273209
$317$	3.80959e8	0.671693	0.335847	−	0.941917i	$-0.390978\pi$
$317$	3.80959e8	0.671693	0.335847	+	0.941917i	$0.390978\pi$
$318$	−5.45637e7	−0.0951501
$319$	−2.56784e7	−0.0442895
$320$	1.99146e8	0.339739
$321$	3.29101e8	0.555344
$322$	−5.49809e8	−0.917733
$323$	−6.58151e8	−1.08672
$324$	−6.12584e7	−0.100059
$325$	4.19916e8	0.678533
$326$	3.20769e8	0.512780
$327$	3.57225e8	0.564970
$328$	2.41449e7	0.0377804
$329$	1.78456e8	0.276277
$330$	−3.68738e7	−0.0564832
$331$	−8.19344e8	−1.24185	−0.620924	−	0.783871i	$-0.713242\pi$
$331$	−8.19344e8	−1.24185	−0.620924	+	0.783871i	$0.713242\pi$
$332$	−2.93278e8	−0.439842
$333$	3.87845e8	0.575576
$334$	2.85614e8	0.419436
$335$	9.81945e8	1.42702
$336$	−4.86050e8	−0.699027
$337$	1.23100e8	0.175208	0.0876039	−	0.996155i	$-0.472079\pi$
$337$	1.23100e8	0.175208	0.0876039	+	0.996155i	$0.472079\pi$
$338$	3.22698e8	0.454556
$339$	−4.07827e8	−0.568561
$340$	−8.73134e8	−1.20477
$341$	−1.00612e8	−0.137407
$342$	4.75141e7	0.0642290
$343$	1.13920e9	1.52430
$344$	−6.80986e8	−0.901952
$345$	−5.66383e8	−0.742580
$346$	−9.34617e7	−0.121302
$347$	−5.04884e8	−0.648692	−0.324346	−	0.945939i	$-0.605144\pi$
$347$	−5.04884e8	−0.648692	−0.324346	+	0.945939i	$0.605144\pi$
$348$	4.38964e7	0.0558343
$349$	8.90989e8	1.12197	0.560987	−	0.827824i	$-0.310422\pi$
$349$	8.90989e8	1.12197	0.560987	+	0.827824i	$0.310422\pi$
$350$	−1.86946e8	−0.233065
$351$	2.43614e8	0.300696
$352$	−2.78005e8	−0.339746
$353$	−7.69217e8	−0.930759	−0.465380	−	0.885111i	$-0.654082\pi$
$353$	−7.69217e8	−0.930759	−0.465380	+	0.885111i	$0.654082\pi$
$354$	1.97861e7	0.0237055
$355$	−2.68911e8	−0.319014
$356$	−9.27031e8	−1.08898
$357$	1.50231e9	1.74751
$358$	−3.74432e8	−0.431303
$359$	−8.50524e8	−0.970188	−0.485094	−	0.874462i	$-0.661215\pi$
$359$	−8.50524e8	−0.970188	−0.485094	+	0.874462i	$0.661215\pi$
$360$	1.33031e8	0.150278
$361$	−5.60208e8	−0.626721
$362$	1.14277e6	0.00126613
$363$	4.36660e8	0.479149
$364$	2.20316e9	2.39437
$365$	−1.16243e7	−0.0125124
$366$	1.22167e7	0.0130248
$367$	8.45450e8	0.892805	0.446402	−	0.894832i	$-0.352705\pi$
$367$	8.45450e8	0.892805	0.446402	+	0.894832i	$0.352705\pi$
$368$	−1.16316e9	−1.21667
$369$	−2.02780e7	−0.0210104
$370$	−3.99089e8	−0.409604
$371$	8.74629e8	0.889232
$372$	1.71992e8	0.173224
$373$	−7.54053e7	−0.0752352	−0.0376176	−	0.999292i	$-0.511977\pi$
$373$	−7.54053e7	−0.0752352	−0.0376176	+	0.999292i	$0.511977\pi$
$374$	2.34059e8	0.231353
$375$	−6.36040e8	−0.622838
$376$	−1.00307e8	−0.0973141
$377$	−1.74569e8	−0.167792
$378$	−1.08457e8	−0.103284
$379$	2.52216e8	0.237977	0.118988	−	0.992896i	$-0.462035\pi$
$379$	2.52216e8	0.237977	0.118988	+	0.992896i	$0.462035\pi$
$380$	4.42654e8	0.413830
$381$	−1.53287e7	−0.0141993
$382$	−1.62708e8	−0.149343
$383$	5.53340e8	0.503265	0.251632	−	0.967823i	$-0.419033\pi$
$383$	5.53340e8	0.503265	0.251632	+	0.967823i	$0.419033\pi$
$384$	6.18990e8	0.557859
$385$	5.91068e8	0.527867
$386$	−1.77578e7	−0.0157157
$387$	5.71925e8	0.501592
$388$	−1.35056e9	−1.17383
$389$	−1.67614e9	−1.44373	−0.721865	−	0.692034i	$-0.756715\pi$
$389$	−1.67614e9	−1.44373	−0.721865	+	0.692034i	$0.756715\pi$
$390$	−2.50677e8	−0.213988
$391$	3.59516e9	3.04158
$392$	−1.35517e9	−1.13630
$393$	1.18525e8	0.0984997
$394$	−4.80239e8	−0.395568
$395$	2.79504e8	0.228191
$396$	1.52986e8	0.123800
$397$	6.56873e7	0.0526883	0.0263442	−	0.999653i	$-0.491613\pi$
$397$	6.56873e7	0.0526883	0.0263442	+	0.999653i	$0.491613\pi$
$398$	−2.48514e6	−0.00197588
$399$	−7.61627e8	−0.600257
$400$	−3.95498e8	−0.308983
$401$	2.32525e9	1.80080	0.900398	−	0.435067i	$-0.143275\pi$
$401$	2.32525e9	1.80080	0.900398	+	0.435067i	$0.143275\pi$
$402$	4.49979e8	0.345463
$403$	−6.83985e8	−0.520569
$404$	9.52846e7	0.0718932
$405$	−1.11726e8	−0.0835722
$406$	7.77175e7	0.0576339
$407$	−9.68600e8	−0.712138
$408$	−8.44425e8	−0.615532
$409$	1.53360e9	1.10836	0.554178	−	0.832398i	$-0.313033\pi$
$409$	1.53360e9	1.10836	0.554178	+	0.832398i	$0.313033\pi$
$410$	2.08659e7	0.0149518
$411$	−1.46195e9	−1.03869
$412$	−1.07664e9	−0.758459
$413$	−3.17161e8	−0.221541
$414$	−2.59547e8	−0.179769
$415$	−5.34895e8	−0.367367
$416$	−1.88995e9	−1.28714
$417$	−1.36279e9	−0.920351
$418$	−1.18661e8	−0.0794680
$419$	7.21485e8	0.479158	0.239579	−	0.970877i	$-0.422991\pi$
$419$	7.21485e8	0.479158	0.239579	+	0.970877i	$0.422991\pi$
$420$	−1.01041e9	−0.665464
$421$	−4.02560e8	−0.262932	−0.131466	−	0.991321i	$-0.541968\pi$
$421$	−4.02560e8	−0.262932	−0.131466	+	0.991321i	$0.541968\pi$
$422$	−2.69841e8	−0.174789
$423$	8.42431e7	0.0541182
$424$	−4.91616e8	−0.313217
$425$	1.22243e9	0.772433
$426$	−1.23229e8	−0.0772291
$427$	−1.95828e8	−0.121724
$428$	1.40500e9	0.866210
$429$	−6.08401e8	−0.372040
$430$	−5.88506e8	−0.356953
$431$	−7.13192e8	−0.429078	−0.214539	−	0.976715i	$-0.568825\pi$
$431$	−7.13192e8	−0.429078	−0.214539	+	0.976715i	$0.568825\pi$
$432$	−2.29448e8	−0.136928
$433$	−1.48657e9	−0.879989	−0.439995	−	0.898000i	$-0.645020\pi$
$433$	−1.48657e9	−0.879989	−0.439995	+	0.898000i	$0.645020\pi$
$434$	3.04508e8	0.178807
$435$	8.00603e7	0.0466342
$436$	1.52507e9	0.881224
$437$	−1.82264e9	−1.04476
$438$	−5.32685e6	−0.00302908
$439$	2.80769e9	1.58388	0.791942	−	0.610597i	$-0.209070\pi$
$439$	2.80769e9	1.58388	0.791942	+	0.610597i	$0.209070\pi$
$440$	−3.32231e8	−0.185933
$441$	1.13814e9	0.631918
$442$	1.59119e9	0.876487
$443$	−2.23536e9	−1.22161	−0.610807	−	0.791780i	$-0.709155\pi$
$443$	−2.23536e9	−1.22161	−0.610807	+	0.791780i	$0.709155\pi$
$444$	1.65579e9	0.897768
$445$	−1.69076e9	−0.909542
$446$	9.92606e8	0.529791
$447$	−1.58105e9	−0.837280
$448$	−1.46284e9	−0.768639
$449$	−2.11217e9	−1.10120	−0.550600	−	0.834769i	$-0.685601\pi$
$449$	−2.11217e9	−1.10120	−0.550600	+	0.834769i	$0.685601\pi$
$450$	−8.82509e7	−0.0456536
$451$	5.06422e7	0.0259953
$452$	−1.74110e9	−0.886826
$453$	3.15445e7	0.0159434
$454$	−8.07769e8	−0.405128
$455$	4.01823e9	1.99984
$456$	4.28100e8	0.211431
$457$	−2.25521e9	−1.10530	−0.552650	−	0.833413i	$-0.686383\pi$
$457$	−2.25521e9	−1.10530	−0.552650	+	0.833413i	$0.686383\pi$
$458$	−5.72280e8	−0.278342
$459$	7.09190e8	0.342309
$460$	−2.41800e9	−1.15826
$461$	1.54873e9	0.736244	0.368122	−	0.929778i	$-0.380001\pi$
$461$	1.54873e9	0.736244	0.368122	+	0.929778i	$0.380001\pi$
$462$	2.70859e8	0.127790
$463$	2.79861e9	1.31041	0.655207	−	0.755449i	$-0.272581\pi$
$463$	2.79861e9	1.31041	0.655207	+	0.755449i	$0.272581\pi$
$464$	1.64417e8	0.0764073
$465$	3.13688e8	0.144681
$466$	2.63825e8	0.120772
$467$	3.32921e9	1.51263	0.756314	−	0.654209i	$-0.226998\pi$
$467$	3.32921e9	1.51263	0.756314	+	0.654209i	$0.226998\pi$
$468$	1.04004e9	0.469018
$469$	−7.21294e9	−3.22855
$470$	−8.66855e7	−0.0385127
$471$	−6.10965e8	−0.269428
$472$	1.78272e8	0.0780343
$473$	−1.42832e9	−0.620600
$474$	1.28084e8	0.0552420
$475$	−6.19735e8	−0.265325
$476$	6.41365e9	2.72572
$477$	4.12883e8	0.174186
$478$	3.52413e8	0.147589
$479$	−2.33364e9	−0.970196	−0.485098	−	0.874460i	$-0.661216\pi$
$479$	−2.33364e9	−0.970196	−0.485098	+	0.874460i	$0.661216\pi$
$480$	8.66766e8	0.357732
$481$	−6.58479e9	−2.69795
$482$	−4.15064e8	−0.168830
$483$	4.16040e9	1.68004
$484$	1.86419e9	0.747363
$485$	−2.46322e9	−0.980410
$486$	−5.11988e7	−0.0202317
$487$	9.35274e8	0.366934	0.183467	−	0.983026i	$-0.441268\pi$
$487$	9.35274e8	0.366934	0.183467	+	0.983026i	$0.441268\pi$
$488$	1.10072e8	0.0428753
$489$	−2.42726e9	−0.938718
$490$	−1.17114e9	−0.449699
$491$	2.77010e9	1.05611	0.528056	−	0.849210i	$-0.322921\pi$
$491$	2.77010e9	1.05611	0.528056	+	0.849210i	$0.322921\pi$
$492$	−8.65710e7	−0.0327714
$493$	−5.08189e8	−0.191012
$494$	−8.06690e8	−0.301067
$495$	2.79024e8	0.103401
$496$	6.44211e8	0.237051
$497$	1.97530e9	0.721750
$498$	−2.45117e8	−0.0889346
$499$	−1.78899e9	−0.644547	−0.322274	−	0.946647i	$-0.604447\pi$
$499$	−1.78899e9	−0.644547	−0.322274	+	0.946647i	$0.604447\pi$
$500$	−2.71538e9	−0.971485
$501$	−2.16123e9	−0.767839
$502$	−1.36464e9	−0.481454
$503$	2.69835e9	0.945388	0.472694	−	0.881227i	$-0.343282\pi$
$503$	2.69835e9	0.945388	0.472694	+	0.881227i	$0.343282\pi$
$504$	−9.77188e8	−0.339994
$505$	1.73785e8	0.0600470
$506$	6.48190e8	0.222421
$507$	−2.44185e9	−0.832131
$508$	−6.54411e7	−0.0221477
$509$	2.29043e8	0.0769848	0.0384924	−	0.999259i	$-0.487744\pi$
$509$	2.29043e8	0.0769848	0.0384924	+	0.999259i	$0.487744\pi$
$510$	−7.29751e8	−0.243601
$511$	8.53867e7	0.0283085
$512$	3.07523e9	1.01259
$513$	−3.59539e8	−0.117580
$514$	−6.69169e8	−0.217353
$515$	−1.96364e9	−0.633484
$516$	2.44166e9	0.782369
$517$	−2.10388e8	−0.0669583
$518$	2.93153e9	0.926703
$519$	7.07223e8	0.222060
$520$	−2.25859e9	−0.704410
$521$	−1.11723e9	−0.346107	−0.173054	−	0.984912i	$-0.555363\pi$
$521$	−1.11723e9	−0.346107	−0.173054	+	0.984912i	$0.555363\pi$
$522$	3.66879e7	0.0112895
$523$	−3.64533e8	−0.111425	−0.0557123	−	0.998447i	$-0.517743\pi$
$523$	−3.64533e8	−0.111425	−0.0557123	+	0.998447i	$0.517743\pi$
$524$	5.06006e8	0.153637
$525$	1.41462e9	0.426659
$526$	−1.20197e9	−0.360116
$527$	−1.99116e9	−0.592609
$528$	5.73022e8	0.169415
$529$	6.55140e9	1.92415
$530$	−4.24854e8	−0.123958
$531$	−1.49721e8	−0.0433963
$532$	−3.25154e9	−0.936264
$533$	3.44278e8	0.0984837
$534$	−7.74798e8	−0.220188
$535$	2.56251e9	0.723481
$536$	4.05429e9	1.13720
$537$	2.83333e9	0.789562
$538$	−2.14724e8	−0.0594487
$539$	−2.84238e9	−0.781847
$540$	−4.76981e8	−0.130354
$541$	−3.21998e9	−0.874305	−0.437153	−	0.899387i	$-0.644013\pi$
$541$	−3.21998e9	−0.874305	−0.437153	+	0.899387i	$0.644013\pi$
$542$	−4.43991e8	−0.119778
$543$	−8.64734e6	−0.00231784
$544$	−5.50187e9	−1.46526
$545$	2.78149e9	0.736020
$546$	1.84137e9	0.484134
$547$	3.19865e9	0.835625	0.417812	−	0.908533i	$-0.362797\pi$
$547$	3.19865e9	0.835625	0.417812	+	0.908533i	$0.362797\pi$
$548$	−6.24135e9	−1.62012
$549$	−9.24438e7	−0.0238438
$550$	2.20397e8	0.0564854
$551$	2.57637e8	0.0656113
$552$	−2.33850e9	−0.591767
$553$	−2.05312e9	−0.516268
$554$	−1.59911e9	−0.399571
$555$	3.01990e9	0.749838
$556$	−5.81804e9	−1.43554
$557$	−4.79806e9	−1.17645	−0.588223	−	0.808699i	$-0.700172\pi$
$557$	−4.79806e9	−1.17645	−0.588223	+	0.808699i	$0.700172\pi$
$558$	1.43748e8	0.0350254
$559$	−9.71009e9	−2.35116
$560$	−3.78457e9	−0.910665
$561$	−1.77112e9	−0.423525
$562$	1.98909e9	0.472690
$563$	2.61139e9	0.616726	0.308363	−	0.951269i	$-0.400219\pi$
$563$	2.61139e9	0.616726	0.308363	+	0.951269i	$0.400219\pi$
$564$	3.59651e8	0.0844120
$565$	−3.17550e9	−0.740700
$566$	2.20885e9	0.512045
$567$	8.20690e8	0.189077
$568$	−1.11029e9	−0.254224
$569$	8.32085e9	1.89354	0.946771	−	0.321909i	$-0.104324\pi$
$569$	8.32085e9	1.89354	0.946771	+	0.321909i	$0.104324\pi$
$570$	3.69963e8	0.0836751
$571$	−3.91195e8	−0.0879360	−0.0439680	−	0.999033i	$-0.514000\pi$
$571$	−3.91195e8	−0.0879360	−0.0439680	+	0.999033i	$0.514000\pi$
$572$	−2.59738e9	−0.580297
$573$	1.23121e9	0.273395
$574$	−1.53272e8	−0.0338276
$575$	3.38531e9	0.742610
$576$	−6.90556e8	−0.150564
$577$	6.50587e9	1.40991	0.704953	−	0.709254i	$-0.250968\pi$
$577$	6.50587e9	1.40991	0.704953	+	0.709254i	$0.250968\pi$
$578$	3.16801e9	0.682400
$579$	1.34373e8	0.0287699
$580$	3.41794e8	0.0727388
$581$	3.92910e9	0.831145
$582$	−1.12878e9	−0.237344
$583$	−1.03113e9	−0.215513
$584$	−4.79946e7	−0.00997120
$585$	1.89687e9	0.391735
$586$	1.81560e9	0.372717
$587$	−7.65590e9	−1.56230	−0.781148	−	0.624346i	$-0.785365\pi$
$587$	−7.65590e9	−1.56230	−0.781148	+	0.624346i	$0.785365\pi$
$588$	4.85895e9	0.985648
$589$	1.00946e9	0.203557
$590$	1.54062e8	0.0308826
$591$	3.63396e9	0.724143
$592$	6.20188e9	1.22856
$593$	2.64479e9	0.520835	0.260418	−	0.965496i	$-0.416140\pi$
$593$	2.64479e9	0.520835	0.260418	+	0.965496i	$0.416140\pi$
$594$	1.27863e8	0.0250319
$595$	1.16975e10	2.27659
$596$	−6.74984e9	−1.30597
$597$	1.88050e7	0.00361712
$598$	4.40656e9	0.842646
$599$	−3.92980e8	−0.0747096	−0.0373548	−	0.999302i	$-0.511893\pi$
$599$	−3.92980e8	−0.0747096	−0.0373548	+	0.999302i	$0.511893\pi$
$600$	−7.95136e8	−0.150284
$601$	3.38890e9	0.636792	0.318396	−	0.947958i	$-0.396856\pi$
$601$	3.38890e9	0.636792	0.318396	+	0.947958i	$0.396856\pi$
$602$	4.32291e9	0.807585
$603$	−3.40499e9	−0.632419
$604$	1.34670e8	0.0248680
$605$	3.40000e9	0.624216
$606$	7.96373e7	0.0145366
$607$	6.03144e9	1.09461	0.547307	−	0.836932i	$-0.315653\pi$
$607$	6.03144e9	1.09461	0.547307	+	0.836932i	$0.315653\pi$
$608$	2.78929e9	0.503305
$609$	−5.88088e8	−0.105507
$610$	9.51240e7	0.0169682
$611$	−1.43027e9	−0.253673
$612$	3.02767e9	0.533924
$613$	4.12690e9	0.723623	0.361812	−	0.932251i	$-0.382158\pi$
$613$	4.12690e9	0.723623	0.361812	+	0.932251i	$0.382158\pi$
$614$	−7.14271e8	−0.124530
$615$	−1.57892e8	−0.0273715
$616$	2.44042e9	0.420661
$617$	1.09713e10	1.88045	0.940224	−	0.340557i	$-0.110615\pi$
$617$	1.09713e10	1.88045	0.940224	+	0.340557i	$0.110615\pi$
$618$	−8.99842e8	−0.153358
$619$	6.38450e9	1.08196	0.540978	−	0.841037i	$-0.318054\pi$
$619$	6.38450e9	1.08196	0.540978	+	0.841037i	$0.318054\pi$
$620$	1.33920e9	0.225670
$621$	1.96399e9	0.329092
$622$	−3.39885e8	−0.0566324
$623$	1.24196e10	2.05778
$624$	3.89555e9	0.641834
$625$	−2.30186e9	−0.377137
$626$	−1.31182e9	−0.213729
$627$	8.97910e8	0.145478
$628$	−2.60833e9	−0.420246
$629$	−1.91691e10	−3.07131
$630$	−8.44484e8	−0.134555
$631$	4.25053e9	0.673504	0.336752	−	0.941593i	$-0.390672\pi$
$631$	4.25053e9	0.673504	0.336752	+	0.941593i	$0.390672\pi$
$632$	1.15403e9	0.181847
$633$	2.04189e9	0.319977
$634$	1.35931e9	0.211839
$635$	−1.19355e8	−0.0184983
$636$	1.76268e9	0.271690
$637$	−1.93232e10	−2.96205
$638$	−9.16240e7	−0.0139681
$639$	9.32476e8	0.141379
$640$	4.81969e9	0.726757
$641$	7.36203e9	1.10406	0.552032	−	0.833823i	$-0.313853\pi$
$641$	7.36203e9	1.10406	0.552032	+	0.833823i	$0.313853\pi$
$642$	1.17428e9	0.175145
$643$	4.96388e8	0.0736348	0.0368174	−	0.999322i	$-0.488278\pi$
$643$	4.96388e8	0.0736348	0.0368174	+	0.999322i	$0.488278\pi$
$644$	1.77616e10	2.62048
$645$	4.45322e9	0.653455
$646$	−2.34837e9	−0.342730
$647$	1.80994e8	0.0262724	0.0131362	−	0.999914i	$-0.495819\pi$
$647$	1.80994e8	0.0262724	0.0131362	+	0.999914i	$0.495819\pi$
$648$	−4.61298e8	−0.0665992
$649$	3.73913e8	0.0536925
$650$	1.49832e9	0.213996
$651$	−2.30421e9	−0.327332
$652$	−1.03625e10	−1.46419
$653$	−9.58608e9	−1.34724	−0.673620	−	0.739078i	$-0.735262\pi$
$653$	−9.58608e9	−1.34724	−0.673620	+	0.739078i	$0.735262\pi$
$654$	1.27463e9	0.178181
$655$	9.22878e8	0.128322
$656$	−3.24259e8	−0.0448465
$657$	4.03082e7	0.00554517
$658$	6.36754e8	0.0871326
$659$	−6.06150e9	−0.825051	−0.412526	−	0.910946i	$-0.635353\pi$
$659$	−6.06150e9	−0.825051	−0.412526	+	0.910946i	$0.635353\pi$
$660$	1.19121e9	0.161281
$661$	−9.42282e9	−1.26904	−0.634520	−	0.772906i	$-0.718802\pi$
$661$	−9.42282e9	−1.26904	−0.634520	+	0.772906i	$0.718802\pi$
$662$	−2.92352e9	−0.391655
$663$	−1.20406e10	−1.60453
$664$	−2.20849e9	−0.292757
$665$	−5.93031e9	−0.781991
$666$	1.38388e9	0.181526
$667$	−1.40735e9	−0.183637
$668$	−9.22675e9	−1.19765
$669$	−7.51104e9	−0.969859
$670$	3.50371e9	0.450056
$671$	2.30868e8	0.0295009
$672$	−6.36688e9	−0.809346
$673$	5.79599e9	0.732951	0.366475	−	0.930428i	$-0.380564\pi$
$673$	5.79599e9	0.732951	0.366475	+	0.930428i	$0.380564\pi$
$674$	4.39237e8	0.0552572
$675$	6.67794e8	0.0835755
$676$	−1.04248e10	−1.29793
$677$	−1.36618e10	−1.69219	−0.846093	−	0.533036i	$-0.821051\pi$
$677$	−1.36618e10	−1.69219	−0.846093	+	0.533036i	$0.821051\pi$
$678$	−1.45518e9	−0.179314
$679$	1.80938e10	2.21812
$680$	−6.57501e9	−0.801891
$681$	6.11238e9	0.741644
$682$	−3.58996e8	−0.0433355
$683$	1.32763e10	1.59443	0.797214	−	0.603697i	$-0.206306\pi$
$683$	1.32763e10	1.59443	0.797214	+	0.603697i	$0.206306\pi$
$684$	−1.53494e9	−0.183399
$685$	−1.13833e10	−1.35316
$686$	4.06481e9	0.480735
$687$	4.33044e9	0.509546
$688$	9.14545e9	1.07064
$689$	−7.00989e9	−0.816477
$690$	−2.02093e9	−0.234196
$691$	−1.61989e10	−1.86773	−0.933863	−	0.357630i	$-0.883585\pi$
$691$	−1.61989e10	−1.86773	−0.933863	+	0.357630i	$0.883585\pi$
$692$	3.01928e9	0.346363
$693$	−2.04958e9	−0.233937
$694$	−1.80149e9	−0.204585
$695$	−1.06112e10	−1.19900
$696$	3.30556e8	0.0371631
$697$	1.00223e9	0.112113
$698$	3.17916e9	0.353850
$699$	−1.99636e9	−0.221090
$700$	6.03929e9	0.665491
$701$	1.52630e10	1.67351	0.836754	−	0.547579i	$-0.184451\pi$
$701$	1.52630e10	1.67351	0.836754	+	0.547579i	$0.184451\pi$
$702$	8.69248e8	0.0948339
$703$	9.71818e9	1.05497
$704$	1.72459e9	0.186287
$705$	6.55949e8	0.0705030
$706$	−2.74467e9	−0.293544
$707$	−1.27654e9	−0.135853
$708$	−6.39190e8	−0.0676883
$709$	5.05134e9	0.532286	0.266143	−	0.963934i	$-0.414251\pi$
$709$	5.05134e9	0.532286	0.266143	+	0.963934i	$0.414251\pi$
$710$	−9.59511e8	−0.100611
$711$	−9.69208e8	−0.101128
$712$	−6.98088e9	−0.724820
$713$	−5.51419e9	−0.569729
$714$	5.36043e9	0.551132
$715$	−4.73724e9	−0.484679
$716$	1.20960e10	1.23154
$717$	−2.66671e9	−0.270183
$718$	−3.03478e9	−0.305979
$719$	1.20715e10	1.21118	0.605592	−	0.795775i	$-0.292936\pi$
$719$	1.20715e10	1.21118	0.605592	+	0.795775i	$0.292936\pi$
$720$	−1.78657e9	−0.178384
$721$	1.44240e10	1.43322
$722$	−1.99889e9	−0.197656
$723$	3.14078e9	0.309068
$724$	−3.69172e7	−0.00361530
$725$	−4.78526e8	−0.0466361
$726$	1.55806e9	0.151114
$727$	−1.02032e10	−0.984845	−0.492423	−	0.870356i	$-0.663889\pi$
$727$	−1.02032e10	−0.984845	−0.492423	+	0.870356i	$0.663889\pi$
$728$	1.65906e10	1.59368
$729$	3.87420e8	0.0370370
$730$	−4.14768e7	−0.00394617
$731$	−2.82672e10	−2.67653
$732$	−3.94661e8	−0.0371908
$733$	1.27539e10	1.19613	0.598067	−	0.801446i	$-0.295936\pi$
$733$	1.27539e10	1.19613	0.598067	+	0.801446i	$0.295936\pi$
$734$	3.01667e9	0.281574
$735$	8.86199e9	0.823238
$736$	−1.52365e10	−1.40869
$737$	8.50359e9	0.782467
$738$	−7.23546e7	−0.00662627
$739$	1.16465e10	1.06155	0.530774	−	0.847514i	$-0.321901\pi$
$739$	1.16465e10	1.06155	0.530774	+	0.847514i	$0.321901\pi$
$740$	1.28926e10	1.16958
$741$	6.10422e9	0.551146
$742$	3.12079e9	0.280447
$743$	−1.66695e10	−1.49095	−0.745475	−	0.666534i	$-0.767777\pi$
$743$	−1.66695e10	−1.49095	−0.745475	+	0.666534i	$0.767777\pi$
$744$	1.29516e9	0.115297
$745$	−1.23107e10	−1.09078
$746$	−2.69056e8	−0.0237277
$747$	1.85480e9	0.162808
$748$	−7.56129e9	−0.660602
$749$	−1.88230e10	−1.63683
$750$	−2.26947e9	−0.196431
$751$	−1.52168e10	−1.31094	−0.655471	−	0.755220i	$-0.727530\pi$
$751$	−1.52168e10	−1.31094	−0.655471	+	0.755220i	$0.727530\pi$
$752$	1.34710e9	0.115515
$753$	1.03262e10	0.881371
$754$	−6.22883e8	−0.0529184
$755$	2.45617e8	0.0207704
$756$	3.50369e9	0.294917
$757$	−8.51162e9	−0.713143	−0.356572	−	0.934268i	$-0.616054\pi$
$757$	−8.51162e9	−0.713143	−0.356572	+	0.934268i	$0.616054\pi$
$758$	8.99938e8	0.0750534
$759$	−4.90484e9	−0.407173
$760$	3.33334e9	0.275444
$761$	1.77193e10	1.45748	0.728738	−	0.684793i	$-0.240107\pi$
$761$	1.77193e10	1.45748	0.728738	+	0.684793i	$0.240107\pi$
$762$	−5.46946e7	−0.00447819
$763$	−2.04316e10	−1.66520
$764$	5.25627e9	0.426433
$765$	5.52202e9	0.445947
$766$	1.97439e9	0.158720
$767$	2.54195e9	0.203415
$768$	−1.06511e9	−0.0848458
$769$	−1.41872e10	−1.12500	−0.562502	−	0.826796i	$-0.690161\pi$
$769$	−1.41872e10	−1.12500	−0.562502	+	0.826796i	$0.690161\pi$
$770$	2.10901e9	0.166479
$771$	5.06359e9	0.397895
$772$	5.73667e8	0.0448744
$773$	−5.86760e9	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$773$	−5.86760e9	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$774$	2.04070e9	0.158193
$775$	−1.87493e9	−0.144687
$776$	−1.01702e10	−0.781295
$777$	−2.21829e10	−1.69646
$778$	−5.98067e9	−0.455325
$779$	−5.08104e8	−0.0385098
$780$	8.09813e9	0.611018
$781$	−2.32876e9	−0.174923
$782$	1.28280e10	0.959258
$783$	−2.77617e8	−0.0206671
$784$	1.81996e10	1.34882
$785$	−4.75720e9	−0.351001
$786$	4.22911e8	0.0310649
$787$	6.68562e9	0.488911	0.244456	−	0.969660i	$-0.421391\pi$
$787$	6.68562e9	0.488911	0.244456	+	0.969660i	$0.421391\pi$
$788$	1.55141e10	1.12950
$789$	9.09528e9	0.659245
$790$	9.97308e8	0.0719672
$791$	2.33258e10	1.67579
$792$	1.15204e9	0.0824005
$793$	1.56950e9	0.111765
$794$	2.34381e8	0.0166169
$795$	3.21486e9	0.226923
$796$	8.02824e7	0.00564189
$797$	1.99892e10	1.39860	0.699298	−	0.714831i	$-0.253496\pi$
$797$	1.99892e10	1.39860	0.699298	+	0.714831i	$0.253496\pi$
$798$	−2.71758e9	−0.189310
$799$	−4.16369e9	−0.288778
$800$	−5.18072e9	−0.357746
$801$	5.86289e9	0.403086
$802$	8.29680e9	0.567937
$803$	−1.00665e8	−0.00686082
$804$	−1.45366e10	−0.986430
$805$	3.23944e10	2.18869
$806$	−2.44055e9	−0.164178
$807$	1.62481e9	0.108829
$808$	7.17527e8	0.0478519
$809$	−7.22991e9	−0.480079	−0.240040	−	0.970763i	$-0.577160\pi$
$809$	−7.22991e9	−0.480079	−0.240040	+	0.970763i	$0.577160\pi$
$810$	−3.98653e8	−0.0263571
$811$	−1.55913e10	−1.02639	−0.513193	−	0.858273i	$-0.671537\pi$
$811$	−1.55913e10	−1.02639	−0.513193	+	0.858273i	$0.671537\pi$
$812$	−2.51067e9	−0.164567
$813$	3.35968e9	0.219271
$814$	−3.45609e9	−0.224595
$815$	−1.88995e10	−1.22292
$816$	1.13404e10	0.730656
$817$	1.43307e10	0.919367
$818$	5.47207e9	0.349555
$819$	−1.39336e10	−0.886277
$820$	−6.74074e8	−0.0426933
$821$	3.10602e8	0.0195886	0.00979431	−	0.999952i	$-0.496882\pi$
$821$	3.10602e8	0.0195886	0.00979431	+	0.999952i	$0.496882\pi$
$822$	−5.21642e9	−0.327583
$823$	−2.52102e10	−1.57644	−0.788218	−	0.615396i	$-0.788996\pi$
$823$	−2.52102e10	−1.57644	−0.788218	+	0.615396i	$0.788996\pi$
$824$	−8.10752e9	−0.504828
$825$	−1.66774e9	−0.103405
$826$	−1.13167e9	−0.0698700
$827$	1.09598e8	0.00673804	0.00336902	−	0.999994i	$-0.498928\pi$
$827$	1.09598e8	0.00673804	0.00336902	+	0.999994i	$0.498928\pi$
$828$	8.38466e9	0.513309
$829$	−2.09237e10	−1.27555	−0.637776	−	0.770222i	$-0.720145\pi$
$829$	−2.09237e10	−1.27555	−0.637776	+	0.770222i	$0.720145\pi$
$830$	−1.90857e9	−0.115861
$831$	1.21004e10	0.731472
$832$	1.17242e10	0.705751
$833$	−5.62522e10	−3.37196
$834$	−4.86262e9	−0.290261
$835$	−1.68282e10	−1.00031
$836$	3.83336e9	0.226912
$837$	−1.08774e9	−0.0641190
$838$	2.57435e9	0.151117
$839$	2.64073e9	0.154368	0.0771839	−	0.997017i	$-0.475407\pi$
$839$	2.64073e9	0.154368	0.0771839	+	0.997017i	$0.475407\pi$
$840$	−7.60875e9	−0.442931
$841$	−1.70509e10	−0.988468
$842$	−1.43639e9	−0.0829237
$843$	−1.50514e10	−0.865326
$844$	8.71722e9	0.499091
$845$	−1.90132e10	−1.08407
$846$	3.00590e8	0.0170678
$847$	−2.49749e10	−1.41225
$848$	6.60227e9	0.371798
$849$	−1.67143e10	−0.937372
$850$	4.36177e9	0.243611
$851$	−5.30857e10	−2.95273
$852$	3.98093e9	0.220519
$853$	1.11033e10	0.612534	0.306267	−	0.951946i	$-0.400920\pi$
$853$	1.11033e10	0.612534	0.306267	+	0.951946i	$0.400920\pi$
$854$	−6.98739e8	−0.0383895
$855$	−2.79951e9	−0.153179
$856$	1.05802e10	0.576546
$857$	−2.50546e9	−0.135974	−0.0679868	−	0.997686i	$-0.521658\pi$
$857$	−2.50546e9	−0.135974	−0.0679868	+	0.997686i	$0.521658\pi$
$858$	−2.17085e9	−0.117334
$859$	−1.03144e10	−0.555225	−0.277612	−	0.960693i	$-0.589543\pi$
$859$	−1.03144e10	−0.555225	−0.277612	+	0.960693i	$0.589543\pi$
$860$	1.90117e10	1.01924
$861$	1.15981e9	0.0619263
$862$	−2.54476e9	−0.135323
$863$	−1.04059e10	−0.551112	−0.275556	−	0.961285i	$-0.588862\pi$
$863$	−1.04059e10	−0.551112	−0.275556	+	0.961285i	$0.588862\pi$
$864$	−3.00559e9	−0.158538
$865$	5.50671e9	0.289291
$866$	−5.30427e9	−0.277532
$867$	−2.39723e10	−1.24923
$868$	−9.83715e9	−0.510564
$869$	2.42049e9	0.125122
$870$	2.85666e8	0.0147076
$871$	5.78096e10	2.96440
$872$	1.14843e10	0.586539
$873$	8.54146e9	0.434493
$874$	−6.50343e9	−0.329498
$875$	3.63785e10	1.83576
$876$	1.72084e8	0.00864920
$877$	3.14032e10	1.57208	0.786041	−	0.618175i	$-0.212128\pi$
$877$	3.14032e10	1.57208	0.786041	+	0.618175i	$0.212128\pi$
$878$	1.00182e10	0.499527
$879$	−1.37386e10	−0.682312
$880$	4.46177e9	0.220708
$881$	−9.24452e9	−0.455479	−0.227740	−	0.973722i	$-0.573133\pi$
$881$	−9.24452e9	−0.455479	−0.227740	+	0.973722i	$0.573133\pi$
$882$	4.06103e9	0.199295
$883$	2.47182e10	1.20824	0.604121	−	0.796893i	$-0.293524\pi$
$883$	2.47182e10	1.20824	0.604121	+	0.796893i	$0.293524\pi$
$884$	−5.14036e10	−2.50271
$885$	−1.16579e9	−0.0565350
$886$	−7.97605e9	−0.385274
$887$	1.21845e10	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$887$	1.21845e10	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$888$	1.24687e10	0.597551
$889$	8.76727e8	0.0418512
$890$	−6.03287e9	−0.286852
$891$	−9.67541e8	−0.0458245
$892$	−3.20661e10	−1.51276
$893$	2.11087e9	0.0991931
$894$	−5.64141e9	−0.264062
$895$	2.20613e10	1.02861
$896$	−3.54033e10	−1.64424
$897$	−3.33444e10	−1.54259
$898$	−7.53649e9	−0.347298
$899$	7.79451e8	0.0357792
$900$	2.85095e9	0.130359
$901$	−2.04066e10	−0.929467
$902$	1.80698e8	0.00819842
$903$	−3.27114e10	−1.47840
$904$	−1.31111e10	−0.590268
$905$	−6.73314e7	−0.00301959
$906$	1.12555e8	0.00502823
$907$	4.11727e10	1.83225	0.916124	−	0.400895i	$-0.131301\pi$
$907$	4.11727e10	1.83225	0.916124	+	0.400895i	$0.131301\pi$
$908$	2.60950e10	1.15680
$909$	−6.02615e8	−0.0266113
$910$	1.43376e10	0.630711
$911$	3.78466e10	1.65849	0.829244	−	0.558887i	$-0.188771\pi$
$911$	3.78466e10	1.65849	0.829244	+	0.558887i	$0.188771\pi$
$912$	−5.74926e9	−0.250975
$913$	−4.63216e9	−0.201435
$914$	−8.04688e9	−0.348591
$915$	−7.19802e8	−0.0310627
$916$	1.84875e10	0.794775
$917$	−6.77905e9	−0.290320
$918$	2.53048e9	0.107958
$919$	−5.50991e9	−0.234175	−0.117087	−	0.993122i	$-0.537356\pi$
$919$	−5.50991e9	−0.234175	−0.117087	+	0.993122i	$0.537356\pi$
$920$	−1.82084e10	−0.770931
$921$	5.40488e9	0.227970
$922$	5.52606e9	0.232197
$923$	−1.58315e10	−0.662698
$924$	−8.75009e9	−0.364889
$925$	−1.80502e10	−0.749869
$926$	9.98580e9	0.413280
$927$	6.80909e9	0.280744
$928$	2.15374e9	0.0884658
$929$	3.71808e10	1.52147	0.760736	−	0.649062i	$-0.224838\pi$
$929$	3.71808e10	1.52147	0.760736	+	0.649062i	$0.224838\pi$
$930$	1.11928e9	0.0456297
$931$	2.85183e10	1.15824
$932$	−8.52286e9	−0.344850
$933$	2.57190e9	0.103674
$934$	1.18791e10	0.477054
$935$	−1.37906e10	−0.551752
$936$	7.83187e9	0.312176
$937$	4.61321e10	1.83195	0.915977	−	0.401230i	$-0.131417\pi$
$937$	4.61321e10	1.83195	0.915977	+	0.401230i	$0.131417\pi$
$938$	−2.57367e10	−1.01822
$939$	9.92652e9	0.391262
$940$	2.80038e9	0.109969
$941$	−4.30417e10	−1.68394	−0.841968	−	0.539528i	$-0.818603\pi$
$941$	−4.30417e10	−1.68394	−0.841968	+	0.539528i	$0.818603\pi$
$942$	−2.18000e9	−0.0849725
$943$	2.77553e9	0.107784
$944$	−2.39414e9	−0.0926291
$945$	6.39020e9	0.246322
$946$	−5.09643e9	−0.195725
$947$	−2.96342e10	−1.13388	−0.566942	−	0.823758i	$-0.691874\pi$
$947$	−2.96342e10	−1.13388	−0.566942	+	0.823758i	$0.691874\pi$
$948$	−4.13775e9	−0.157737
$949$	−6.84349e8	−0.0259924
$950$	−2.21129e9	−0.0836785
$951$	−1.02859e10	−0.387802
$952$	4.82972e10	1.81423
$953$	−1.01449e10	−0.379683	−0.189841	−	0.981815i	$-0.560797\pi$
$953$	−1.01449e10	−0.379683	−0.189841	+	0.981815i	$0.560797\pi$
$954$	1.47322e9	0.0549349
$955$	9.58664e9	0.356168
$956$	−1.13847e10	−0.421424
$957$	6.93318e8	0.0255706
$958$	−8.32673e9	−0.305981
$959$	8.36165e10	3.06145
$960$	−5.37693e9	−0.196149
$961$	−2.44586e10	−0.888996
$962$	−2.34954e10	−0.850883
$963$	−8.88574e9	−0.320628
$964$	1.34086e10	0.482075
$965$	1.04628e9	0.0374803
$966$	1.48448e10	0.529853
$967$	−4.46512e9	−0.158796	−0.0793982	−	0.996843i	$-0.525300\pi$
$967$	−4.46512e9	−0.158796	−0.0793982	+	0.996843i	$0.525300\pi$
$968$	1.40380e10	0.497442
$969$	1.77701e10	0.627417
$970$	−8.78910e9	−0.309203
$971$	−7.60821e9	−0.266695	−0.133348	−	0.991069i	$-0.542573\pi$
$971$	−7.60821e9	−0.266695	−0.133348	+	0.991069i	$0.542573\pi$
$972$	1.65398e9	0.0577693
$973$	7.79453e10	2.71266
$974$	3.33718e9	0.115724
$975$	−1.13377e10	−0.391751
$976$	−1.47824e9	−0.0508943
$977$	−4.60674e9	−0.158038	−0.0790192	−	0.996873i	$-0.525179\pi$
$977$	−4.60674e9	−0.158038	−0.0790192	+	0.996873i	$0.525179\pi$
$978$	−8.66077e9	−0.296054
$979$	−1.46419e10	−0.498722
$980$	3.78336e10	1.28406
$981$	−9.64508e9	−0.326185
$982$	9.88407e9	0.333078
$983$	3.68529e10	1.23747	0.618735	−	0.785600i	$-0.287645\pi$
$983$	3.68529e10	1.23747	0.618735	+	0.785600i	$0.287645\pi$
$984$	−6.51911e8	−0.0218125
$985$	2.82954e10	0.943386
$986$	−1.81329e9	−0.0602417
$987$	−4.81831e9	−0.159509
$988$	2.60601e10	0.859661
$989$	−7.82814e10	−2.57319
$990$	9.95592e8	0.0326106
$991$	−8.00041e9	−0.261129	−0.130564	−	0.991440i	$-0.541679\pi$
$991$	−8.00041e9	−0.261129	−0.130564	+	0.991440i	$0.541679\pi$
$992$	8.43867e9	0.274462
$993$	2.21223e10	0.716981
$994$	7.04814e9	0.227626
$995$	1.46423e8	0.00471225
$996$	7.91851e9	0.253943
$997$	5.99071e9	0.191445	0.0957227	−	0.995408i	$-0.469484\pi$
$997$	5.99071e9	0.191445	0.0957227	+	0.995408i	$0.469484\pi$
$998$	−6.38333e9	−0.203278
$999$	−1.04718e10	−0.332309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.c.1.10	✓	17
3.2	odd	2		531.8.a.c.1.8		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.c.1.10	✓	17	1.1	even	1	trivial
531.8.a.c.1.8		17	3.2	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} -750.136 q^{10} -1820.60 q^{11} +3112.25 q^{12} -12376.9 q^{13} +5510.17 q^{14} +5676.27 q^{15} +11657.2 q^{16} -36030.6 q^{17} +2601.17 q^{18} +18266.5 q^{19} +24233.1 q^{20} -41695.4 q^{21} -6496.13 q^{22} -99780.9 q^{23} +23436.4 q^{24} -33927.4 q^{25} -44162.4 q^{26} -19683.0 q^{27} -178006. q^{28} +14104.4 q^{29} +20253.7 q^{30} +55263.0 q^{31} +152700. q^{32} +49156.2 q^{33} -128562. q^{34} -324656. q^{35} -84030.7 q^{36} +532023. q^{37} +65177.1 q^{38} +334176. q^{39} +182484. q^{40} -27816.2 q^{41} -148774. q^{42} +784533. q^{43} +209858. q^{44} -153259. q^{45} -356031. q^{46} +115560. q^{47} -314744. q^{48} +1.56124e6 q^{49} -121058. q^{50} +972825. q^{51} +1.42667e6 q^{52} +566369. q^{53} -70231.5 q^{54} +382748. q^{55} -1.34045e6 q^{56} -493195. q^{57} +50326.3 q^{58} -205379. q^{59} -654295. q^{60} -126809. q^{61} +197186. q^{62} +1.12577e6 q^{63} -947265. q^{64} +2.60202e6 q^{65} +175396. q^{66} -4.67077e6 q^{67} +4.15319e6 q^{68} +2.69408e6 q^{69} -1.15841e6 q^{70} +1.27912e6 q^{71} -632782. q^{72} +55292.5 q^{73} +1.89833e6 q^{74} +916041. q^{75} -2.10555e6 q^{76} -2.81150e6 q^{77} +1.19238e6 q^{78} -1.32950e6 q^{79} -2.45071e6 q^{80} +531441. q^{81} -99251.9 q^{82} +2.54431e6 q^{83} +4.80616e6 q^{84} +7.57478e6 q^{85} +2.79932e6 q^{86} -380819. q^{87} +1.58030e6 q^{88} +8.04237e6 q^{89} -546849. q^{90} -1.91133e7 q^{91} +1.15016e7 q^{92} -1.49210e6 q^{93} +412332. q^{94} -3.84020e6 q^{95} -4.12290e6 q^{96} +1.17167e7 q^{97} +5.57069e6 q^{98} -1.32722e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9}+O(q^{10}) \) 17 * q + 2 * q^2 - 459 * q^3 + 1166 * q^4 - 318 * q^5 - 54 * q^6 + 3145 * q^7 + 2355 * q^8 + 12393 * q^9	\( 17 q + 2 q^{2} - 459 q^{3} + 1166 q^{4} - 318 q^{5} - 54 q^{6} + 3145 q^{7} + 2355 q^{8} + 12393 q^{9} + 6521 q^{10} - 1764 q^{11} - 31482 q^{12} + 18192 q^{13} - 7827 q^{14} + 8586 q^{15} + 139226 q^{16} - 15507 q^{17} + 1458 q^{18} + 52083 q^{19} + 721 q^{20} - 84915 q^{21} - 234434 q^{22} + 63823 q^{23} - 63585 q^{24} + 202153 q^{25} - 367956 q^{26} - 334611 q^{27} + 182306 q^{28} - 502955 q^{29} - 176067 q^{30} + 347531 q^{31} - 243908 q^{32} + 47628 q^{33} - 330872 q^{34} + 92641 q^{35} + 850014 q^{36} + 447615 q^{37} + 775669 q^{38} - 491184 q^{39} + 2203270 q^{40} + 940335 q^{41} + 211329 q^{42} + 478562 q^{43} - 596924 q^{44} - 231822 q^{45} - 3078663 q^{46} + 703121 q^{47} - 3759102 q^{48} + 1895082 q^{49} - 876967 q^{50} + 418689 q^{51} + 6278296 q^{52} - 1005974 q^{53} - 39366 q^{54} + 5212846 q^{55} + 3425294 q^{56} - 1406241 q^{57} + 6710166 q^{58} - 3491443 q^{59} - 19467 q^{60} + 11510749 q^{61} + 5996234 q^{62} + 2292705 q^{63} + 29496941 q^{64} + 11094180 q^{65} + 6329718 q^{66} + 14007144 q^{67} + 19688159 q^{68} - 1723221 q^{69} + 30909708 q^{70} + 5229074 q^{71} + 1716795 q^{72} + 5452211 q^{73} + 12819662 q^{74} - 5458131 q^{75} + 41929340 q^{76} + 9930777 q^{77} + 9934812 q^{78} + 15275654 q^{79} + 36576105 q^{80} + 9034497 q^{81} + 32025935 q^{82} + 7826609 q^{83} - 4922262 q^{84} + 11836945 q^{85} + 51649136 q^{86} + 13579785 q^{87} + 30223741 q^{88} - 6436185 q^{89} + 4753809 q^{90} + 11633535 q^{91} + 43357972 q^{92} - 9383337 q^{93} - 4494252 q^{94} + 23741055 q^{95} + 6585516 q^{96} + 26377540 q^{97} + 26517816 q^{98} - 1285956 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 459 * q^3 + 1166 * q^4 - 318 * q^5 - 54 * q^6 + 3145 * q^7 + 2355 * q^8 + 12393 * q^9 + 6521 * q^10 - 1764 * q^11 - 31482 * q^12 + 18192 * q^13 - 7827 * q^14 + 8586 * q^15 + 139226 * q^16 - 15507 * q^17 + 1458 * q^18 + 52083 * q^19 + 721 * q^20 - 84915 * q^21 - 234434 * q^22 + 63823 * q^23 - 63585 * q^24 + 202153 * q^25 - 367956 * q^26 - 334611 * q^27 + 182306 * q^28 - 502955 * q^29 - 176067 * q^30 + 347531 * q^31 - 243908 * q^32 + 47628 * q^33 - 330872 * q^34 + 92641 * q^35 + 850014 * q^36 + 447615 * q^37 + 775669 * q^38 - 491184 * q^39 + 2203270 * q^40 + 940335 * q^41 + 211329 * q^42 + 478562 * q^43 - 596924 * q^44 - 231822 * q^45 - 3078663 * q^46 + 703121 * q^47 - 3759102 * q^48 + 1895082 * q^49 - 876967 * q^50 + 418689 * q^51 + 6278296 * q^52 - 1005974 * q^53 - 39366 * q^54 + 5212846 * q^55 + 3425294 * q^56 - 1406241 * q^57 + 6710166 * q^58 - 3491443 * q^59 - 19467 * q^60 + 11510749 * q^61 + 5996234 * q^62 + 2292705 * q^63 + 29496941 * q^64 + 11094180 * q^65 + 6329718 * q^66 + 14007144 * q^67 + 19688159 * q^68 - 1723221 * q^69 + 30909708 * q^70 + 5229074 * q^71 + 1716795 * q^72 + 5452211 * q^73 + 12819662 * q^74 - 5458131 * q^75 + 41929340 * q^76 + 9930777 * q^77 + 9934812 * q^78 + 15275654 * q^79 + 36576105 * q^80 + 9034497 * q^81 + 32025935 * q^82 + 7826609 * q^83 - 4922262 * q^84 + 11836945 * q^85 + 51649136 * q^86 + 13579785 * q^87 + 30223741 * q^88 - 6436185 * q^89 + 4753809 * q^90 + 11633535 * q^91 + 43357972 * q^92 - 9383337 * q^93 - 4494252 * q^94 + 23741055 * q^95 + 6585516 * q^96 + 26377540 * q^97 + 26517816 * q^98 - 1285956 * q^99

Introduction

L-functions

Modular forms

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