LMFDB - Embedded newform 177.8.a.a.1.5

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:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 $ x^16 - 6x^15 - 1493x^14 + 8791x^13 + 890490x^12 - 5107725x^11 - 269092298x^10 + 1488374176x^9 + 42885295136x^8 - 226132003872x^7 - 3353576629440x^6 + 16796366777600x^5 + 99470801612800x^4 - 494039551757568x^3 - 493048066650624x^2 + 3193975642099712*x - 2385018853548032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$14.7989$ 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)

$f(q)$	$=$	$q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +4388.42 q^{10} -7538.45 q^{11} -2457.23 q^{12} +7066.47 q^{13} +20877.7 q^{14} +8006.47 q^{15} -19750.5 q^{16} +14832.8 q^{17} -10788.4 q^{18} -24023.8 q^{19} -26987.3 q^{20} +38090.5 q^{21} +111561. q^{22} +41590.5 q^{23} -14780.8 q^{24} +9808.63 q^{25} -104576. q^{26} -19683.0 q^{27} -128391. q^{28} +15791.6 q^{29} -118487. q^{30} +108246. q^{31} +222215. q^{32} +203538. q^{33} -219509. q^{34} +418341. q^{35} +66345.1 q^{36} +331221. q^{37} +355527. q^{38} -190795. q^{39} -162335. q^{40} +224396. q^{41} -563699. q^{42} -134226. q^{43} -686063. q^{44} -216175. q^{45} -615494. q^{46} +816259. q^{47} +533265. q^{48} +1.16670e6 q^{49} -145157. q^{50} -400485. q^{51} +643108. q^{52} +1.21786e6 q^{53} +291287. q^{54} +2.23542e6 q^{55} -772301. q^{56} +648644. q^{57} -233699. q^{58} +205379. q^{59} +728656. q^{60} +1.88156e6 q^{61} -1.60193e6 q^{62} -1.02844e6 q^{63} -760476. q^{64} -2.09546e6 q^{65} -3.01215e6 q^{66} -2.09166e6 q^{67} +1.34991e6 q^{68} -1.12294e6 q^{69} -6.19100e6 q^{70} -1.80830e6 q^{71} +399081. q^{72} -4.93066e6 q^{73} -4.90172e6 q^{74} -264833. q^{75} -2.18637e6 q^{76} +1.06349e7 q^{77} +2.82356e6 q^{78} -8.26089e6 q^{79} +5.85675e6 q^{80} +531441. q^{81} -3.32083e6 q^{82} -2.95933e6 q^{83} +3.46655e6 q^{84} -4.39846e6 q^{85} +1.98640e6 q^{86} -426373. q^{87} -4.12683e6 q^{88} +3.82628e6 q^{89} +3.19916e6 q^{90} -9.96908e6 q^{91} +3.78508e6 q^{92} -2.92265e6 q^{93} -1.20798e7 q^{94} +7.12393e6 q^{95} -5.99981e6 q^{96} -6.24098e6 q^{97} -1.72659e7 q^{98} -5.49553e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) $ 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * 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 $

$2$	−14.7989	−1.30805	−0.654027	−	0.756472i	$-0.726922\pi$
$2$	−14.7989	−1.30805	−0.654027	+	0.756472i	$0.726922\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	91.0084	0.711003
$5$	−296.536	−1.06092	−0.530460	−	0.847710i	$-0.677981\pi$
$5$	−296.536	−1.06092	−0.530460	+	0.847710i	$0.677981\pi$
$6$	399.571	0.755205
$7$	−1410.76	−1.55457	−0.777284	−	0.629150i	$-0.783403\pi$
$7$	−1410.76	−1.55457	−0.777284	+	0.629150i	$0.783403\pi$
$8$	547.437	0.378024
$9$	729.000	0.333333
$10$	4388.42	1.38774
$11$	−7538.45	−1.70769	−0.853843	−	0.520531i	$-0.825734\pi$
$11$	−7538.45	−1.70769	−0.853843	+	0.520531i	$0.825734\pi$
$12$	−2457.23	−0.410498
$13$	7066.47	0.892073	0.446037	−	0.895015i	$-0.352835\pi$
$13$	7066.47	0.892073	0.446037	+	0.895015i	$0.352835\pi$
$14$	20877.7	2.03346
$15$	8006.47	0.612522
$16$	−19750.5	−1.20548
$17$	14832.8	0.732237	0.366118	−	0.930568i	$-0.380687\pi$
$17$	14832.8	0.732237	0.366118	+	0.930568i	$0.380687\pi$
$18$	−10788.4	−0.436018
$19$	−24023.8	−0.803535	−0.401767	−	0.915742i	$-0.631604\pi$
$19$	−24023.8	−0.803535	−0.401767	+	0.915742i	$0.631604\pi$
$20$	−26987.3	−0.754317
$21$	38090.5	0.897530
$22$	111561.	2.23374
$23$	41590.5	0.712765	0.356382	−	0.934340i	$-0.384010\pi$
$23$	41590.5	0.712765	0.356382	+	0.934340i	$0.384010\pi$
$24$	−14780.8	−0.218252
$25$	9808.63	0.125550
$26$	−104576.	−1.16688
$27$	−19683.0	−0.192450
$28$	−128391.	−1.10530
$29$	15791.6	0.120235	0.0601177	−	0.998191i	$-0.480852\pi$
$29$	15791.6	0.120235	0.0601177	+	0.998191i	$0.480852\pi$
$30$	−118487.	−0.801212
$31$	108246.	0.652601	0.326300	−	0.945266i	$-0.394198\pi$
$31$	108246.	0.652601	0.326300	+	0.945266i	$0.394198\pi$
$32$	222215.	1.19881
$33$	203538.	0.985933
$34$	−219509.	−0.957805
$35$	418341.	1.64927
$36$	66345.1	0.237001
$37$	331221.	1.07501	0.537504	−	0.843261i	$-0.319367\pi$
$37$	331221.	1.07501	0.537504	+	0.843261i	$0.319367\pi$
$38$	355527.	1.05107
$39$	−190795.	−0.515039
$40$	−162335.	−0.401053
$41$	224396.	0.508478	0.254239	−	0.967141i	$-0.418175\pi$
$41$	224396.	0.508478	0.254239	+	0.967141i	$0.418175\pi$
$42$	−563699.	−1.17402
$43$	−134226.	−0.257453	−0.128726	−	0.991680i	$-0.541089\pi$
$43$	−134226.	−0.257453	−0.128726	+	0.991680i	$0.541089\pi$
$44$	−686063.	−1.21417
$45$	−216175.	−0.353640
$46$	−615494.	−0.932334
$47$	816259.	1.14679	0.573397	−	0.819278i	$-0.305625\pi$
$47$	816259.	1.14679	0.573397	+	0.819278i	$0.305625\pi$
$48$	533265.	0.695983
$49$	1.16670e6	1.41668
$50$	−145157.	−0.164227
$51$	−400485.	−0.422757
$52$	643108.	0.634267
$53$	1.21786e6	1.12365	0.561825	−	0.827256i	$-0.310100\pi$
$53$	1.21786e6	1.12365	0.561825	+	0.827256i	$0.310100\pi$
$54$	291287.	0.251735
$55$	2.23542e6	1.81172
$56$	−772301.	−0.587663
$57$	648644.	0.463921
$58$	−233699.	−0.157274
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	728656.	0.435505
$61$	1.88156e6	1.06136	0.530681	−	0.847571i	$-0.321936\pi$
$61$	1.88156e6	1.06136	0.530681	+	0.847571i	$0.321936\pi$
$62$	−1.60193e6	−0.853637
$63$	−1.02844e6	−0.518189
$64$	−760476.	−0.362623
$65$	−2.09546e6	−0.946418
$66$	−3.01215e6	−1.28965
$67$	−2.09166e6	−0.849627	−0.424814	−	0.905281i	$-0.639660\pi$
$67$	−2.09166e6	−0.849627	−0.424814	+	0.905281i	$0.639660\pi$
$68$	1.34991e6	0.520622
$69$	−1.12294e6	−0.411515
$70$	−6.19100e6	−2.15734
$71$	−1.80830e6	−0.599605	−0.299802	−	0.954001i	$-0.596921\pi$
$71$	−1.80830e6	−0.599605	−0.299802	+	0.954001i	$0.596921\pi$
$72$	399081.	0.126008
$73$	−4.93066e6	−1.48346	−0.741729	−	0.670699i	$-0.765994\pi$
$73$	−4.93066e6	−1.48346	−0.741729	+	0.670699i	$0.765994\pi$
$74$	−4.90172e6	−1.40617
$75$	−264833.	−0.0724866
$76$	−2.18637e6	−0.571316
$77$	1.06349e7	2.65471
$78$	2.82356e6	0.673698
$79$	−8.26089e6	−1.88509	−0.942545	−	0.334078i	$-0.891575\pi$
$79$	−8.26089e6	−1.88509	−0.942545	+	0.334078i	$0.891575\pi$
$80$	5.85675e6	1.27892
$81$	531441.	0.111111
$82$	−3.32083e6	−0.665116
$83$	−2.95933e6	−0.568093	−0.284047	−	0.958810i	$-0.591677\pi$
$83$	−2.95933e6	−0.568093	−0.284047	+	0.958810i	$0.591677\pi$
$84$	3.46655e6	0.638147
$85$	−4.39846e6	−0.776844
$86$	1.98640e6	0.336762
$87$	−426373.	−0.0694180
$88$	−4.12683e6	−0.645545
$89$	3.82628e6	0.575323	0.287661	−	0.957732i	$-0.407122\pi$
$89$	3.82628e6	0.575323	0.287661	+	0.957732i	$0.407122\pi$
$90$	3.19916e6	0.462580
$91$	−9.96908e6	−1.38679
$92$	3.78508e6	0.506778
$93$	−2.92265e6	−0.376779
$94$	−1.20798e7	−1.50007
$95$	7.12393e6	0.852486
$96$	−5.99981e6	−0.692131
$97$	−6.24098e6	−0.694308	−0.347154	−	0.937808i	$-0.612852\pi$
$97$	−6.24098e6	−0.694308	−0.347154	+	0.937808i	$0.612852\pi$
$98$	−1.72659e7	−1.85310
$99$	−5.49553e6	−0.569229
$100$	892667.	0.0892667
$101$	1.50527e7	1.45375	0.726873	−	0.686772i	$-0.240973\pi$
$101$	1.50527e7	1.45375	0.726873	+	0.686772i	$0.240973\pi$
$102$	5.92675e6	0.552989
$103$	1.06416e7	0.959572	0.479786	−	0.877386i	$-0.340714\pi$
$103$	1.06416e7	0.959572	0.479786	+	0.877386i	$0.340714\pi$
$104$	3.86844e6	0.337225
$105$	−1.12952e7	−0.952207
$106$	−1.80230e7	−1.46979
$107$	−1.71933e7	−1.35680	−0.678402	−	0.734691i	$-0.737327\pi$
$107$	−1.71933e7	−1.35680	−0.678402	+	0.734691i	$0.737327\pi$
$108$	−1.79132e6	−0.136833
$109$	−2.61258e7	−1.93231	−0.966154	−	0.257968i	$-0.916947\pi$
$109$	−2.61258e7	−1.93231	−0.966154	+	0.257968i	$0.916947\pi$
$110$	−3.30819e7	−2.36982
$111$	−8.94296e6	−0.620656
$112$	2.78633e7	1.87400
$113$	1.49158e7	0.972463	0.486231	−	0.873830i	$-0.338371\pi$
$113$	1.49158e7	0.972463	0.486231	+	0.873830i	$0.338371\pi$
$114$	−9.59923e6	−0.606833
$115$	−1.23331e7	−0.756186
$116$	1.43717e6	0.0854878
$117$	5.15145e6	0.297358
$118$	−3.03939e6	−0.170294
$119$	−2.09255e7	−1.13831
$120$	4.38304e6	0.231548
$121$	3.73411e7	1.91619
$122$	−2.78451e7	−1.38832
$123$	−6.05870e6	−0.293570
$124$	9.85133e6	0.464001
$125$	2.02583e7	0.927721
$126$	1.52199e7	0.677819
$127$	−1.99083e7	−0.862426	−0.431213	−	0.902250i	$-0.641914\pi$
$127$	−1.99083e7	−0.862426	−0.431213	+	0.902250i	$0.641914\pi$
$128$	−1.71893e7	−0.724475
$129$	3.62410e6	0.148640
$130$	3.10106e7	1.23797
$131$	2.18462e7	0.849038	0.424519	−	0.905419i	$-0.360443\pi$
$131$	2.18462e7	0.849038	0.424519	+	0.905419i	$0.360443\pi$
$132$	1.85237e7	0.701001
$133$	3.38918e7	1.24915
$134$	3.09543e7	1.11136
$135$	5.83672e6	0.204174
$136$	8.12001e6	0.276803
$137$	−1.96904e7	−0.654233	−0.327117	−	0.944984i	$-0.606077\pi$
$137$	−1.96904e7	−0.654233	−0.327117	+	0.944984i	$0.606077\pi$
$138$	1.66183e7	0.538284
$139$	2.42247e6	0.0765078	0.0382539	−	0.999268i	$-0.487820\pi$
$139$	2.42247e6	0.0765078	0.0382539	+	0.999268i	$0.487820\pi$
$140$	3.80725e7	1.17264
$141$	−2.20390e7	−0.662102
$142$	2.67608e7	0.784315
$143$	−5.32702e7	−1.52338
$144$	−1.43981e7	−0.401826
$145$	−4.68277e6	−0.127560
$146$	7.29686e7	1.94044
$147$	−3.15009e7	−0.817922
$148$	3.01439e7	0.764334
$149$	2.09513e7	0.518871	0.259436	−	0.965760i	$-0.416463\pi$
$149$	2.09513e7	0.518871	0.259436	+	0.965760i	$0.416463\pi$
$150$	3.91924e6	0.0948163
$151$	4.77834e6	0.112943	0.0564713	−	0.998404i	$-0.482015\pi$
$151$	4.77834e6	0.112943	0.0564713	+	0.998404i	$0.482015\pi$
$152$	−1.31515e7	−0.303755
$153$	1.08131e7	0.244079
$154$	−1.57386e8	−3.47251
$155$	−3.20990e7	−0.692357
$156$	−1.73639e7	−0.366194
$157$	−6.17818e7	−1.27412	−0.637062	−	0.770813i	$-0.719850\pi$
$157$	−6.17818e7	−1.27412	−0.637062	+	0.770813i	$0.719850\pi$
$158$	1.22252e8	2.46580
$159$	−3.28821e7	−0.648739
$160$	−6.58948e7	−1.27184
$161$	−5.86741e7	−1.10804
$162$	−7.86476e6	−0.145339
$163$	8.27535e7	1.49668	0.748342	−	0.663313i	$-0.230850\pi$
$163$	8.27535e7	1.49668	0.748342	+	0.663313i	$0.230850\pi$
$164$	2.04219e7	0.361529
$165$	−6.03564e7	−1.04600
$166$	4.37949e7	0.743096
$167$	−1.39381e7	−0.231578	−0.115789	−	0.993274i	$-0.536940\pi$
$167$	−1.39381e7	−0.231578	−0.115789	+	0.993274i	$0.536940\pi$
$168$	2.08521e7	0.339288
$169$	−1.28136e7	−0.204205
$170$	6.50925e7	1.01615
$171$	−1.75134e7	−0.267845
$172$	−1.22157e7	−0.183049
$173$	9.03037e7	1.32600	0.663001	−	0.748618i	$-0.269282\pi$
$173$	9.03037e7	1.32600	0.663001	+	0.748618i	$0.269282\pi$
$174$	6.30986e6	0.0908024
$175$	−1.38376e7	−0.195177
$176$	1.48889e8	2.05858
$177$	−5.54523e6	−0.0751646
$178$	−5.66248e7	−0.752553
$179$	9.61633e7	1.25321	0.626605	−	0.779337i	$-0.284444\pi$
$179$	9.61633e7	1.25321	0.626605	+	0.779337i	$0.284444\pi$
$180$	−1.96737e7	−0.251439
$181$	1.27453e8	1.59763	0.798816	−	0.601575i	$-0.205460\pi$
$181$	1.27453e8	1.59763	0.798816	+	0.601575i	$0.205460\pi$
$182$	1.47532e8	1.81399
$183$	−5.08022e7	−0.612778
$184$	2.27681e7	0.269442
$185$	−9.82189e7	−1.14050
$186$	4.32522e7	0.492847
$187$	−1.11816e8	−1.25043
$188$	7.42864e7	0.815374
$189$	2.77680e7	0.299177
$190$	−1.05427e8	−1.11510
$191$	8.72604e6	0.0906151	0.0453075	−	0.998973i	$-0.485573\pi$
$191$	8.72604e6	0.0906151	0.0453075	+	0.998973i	$0.485573\pi$
$192$	2.05329e7	0.209361
$193$	6.92616e7	0.693492	0.346746	−	0.937959i	$-0.387287\pi$
$193$	6.92616e7	0.693492	0.346746	+	0.937959i	$0.387287\pi$
$194$	9.23599e7	0.908191
$195$	5.65775e7	0.546415
$196$	1.06179e8	1.00726
$197$	1.99367e7	0.185790	0.0928949	−	0.995676i	$-0.470388\pi$
$197$	1.99367e7	0.185790	0.0928949	+	0.995676i	$0.470388\pi$
$198$	8.13280e7	0.744581
$199$	−1.15995e8	−1.04341	−0.521705	−	0.853126i	$-0.674704\pi$
$199$	−1.15995e8	−1.04341	−0.521705	+	0.853126i	$0.674704\pi$
$200$	5.36960e6	0.0474610
$201$	5.64747e7	0.490533
$202$	−2.22763e8	−1.90158
$203$	−2.22781e7	−0.186914
$204$	−3.64475e7	−0.300582
$205$	−6.65416e7	−0.539454
$206$	−1.57485e8	−1.25517
$207$	3.03194e7	0.237588
$208$	−1.39567e8	−1.07537
$209$	1.81103e8	1.37218
$210$	1.67157e8	1.24554
$211$	6.67529e7	0.489195	0.244597	−	0.969625i	$-0.421344\pi$
$211$	6.67529e7	0.489195	0.244597	+	0.969625i	$0.421344\pi$
$212$	1.10835e8	0.798918
$213$	4.88240e7	0.346182
$214$	2.54443e8	1.77477
$215$	3.98029e7	0.273136
$216$	−1.07752e7	−0.0727507
$217$	−1.52710e8	−1.01451
$218$	3.86633e8	2.52756
$219$	1.33128e8	0.856475
$220$	2.03442e8	1.28814
$221$	1.04815e8	0.653209
$222$	1.32346e8	0.811851
$223$	1.76404e8	1.06522	0.532611	−	0.846360i	$-0.321211\pi$
$223$	1.76404e8	1.06522	0.532611	+	0.846360i	$0.321211\pi$
$224$	−3.13492e8	−1.86362
$225$	7.15049e6	0.0418501
$226$	−2.20738e8	−1.27203
$227$	−1.70858e8	−0.969491	−0.484745	−	0.874655i	$-0.661088\pi$
$227$	−1.70858e8	−0.969491	−0.484745	+	0.874655i	$0.661088\pi$
$228$	5.90320e7	0.329849
$229$	−1.06631e8	−0.586757	−0.293379	−	0.955996i	$-0.594780\pi$
$229$	−1.06631e8	−0.586757	−0.293379	+	0.955996i	$0.594780\pi$
$230$	1.82516e8	0.989132
$231$	−2.87143e8	−1.53270
$232$	8.64489e6	0.0454518
$233$	−2.36647e8	−1.22562	−0.612809	−	0.790231i	$-0.709960\pi$
$233$	−2.36647e8	−1.22562	−0.612809	+	0.790231i	$0.709960\pi$
$234$	−7.62360e7	−0.388960
$235$	−2.42050e8	−1.21666
$236$	1.86912e7	0.0925647
$237$	2.23044e8	1.08836
$238$	3.09675e8	1.48897
$239$	4.70420e6	0.0222891	0.0111446	−	0.999938i	$-0.496453\pi$
$239$	4.70420e6	0.0222891	0.0111446	+	0.999938i	$0.496453\pi$
$240$	−1.58132e8	−0.738382
$241$	−1.57232e8	−0.723573	−0.361786	−	0.932261i	$-0.617833\pi$
$241$	−1.57232e8	−0.723573	−0.361786	+	0.932261i	$0.617833\pi$
$242$	−5.52609e8	−2.50648
$243$	−1.43489e7	−0.0641500
$244$	1.71238e8	0.754632
$245$	−3.45968e8	−1.50299
$246$	8.96623e7	0.384005
$247$	−1.69764e8	−0.716812
$248$	5.92581e7	0.246699
$249$	7.99018e7	0.327989
$250$	−2.99801e8	−1.21351
$251$	1.41210e7	0.0563647	0.0281823	−	0.999603i	$-0.491028\pi$
$251$	1.41210e7	0.0563647	0.0281823	+	0.999603i	$0.491028\pi$
$252$	−9.35970e7	−0.368434
$253$	−3.13528e8	−1.21718
$254$	2.94622e8	1.12810
$255$	1.18758e8	0.448511
$256$	3.51724e8	1.31027
$257$	−3.85019e8	−1.41487	−0.707434	−	0.706780i	$-0.750147\pi$
$257$	−3.85019e8	−1.41487	−0.707434	+	0.706780i	$0.750147\pi$
$258$	−5.36328e7	−0.194429
$259$	−4.67273e8	−1.67117
$260$	−1.90705e8	−0.672906
$261$	1.15121e7	0.0400785
$262$	−3.23301e8	−1.11059
$263$	1.74319e8	0.590882	0.295441	−	0.955361i	$-0.404533\pi$
$263$	1.74319e8	0.590882	0.295441	+	0.955361i	$0.404533\pi$
$264$	1.11424e8	0.372706
$265$	−3.61138e8	−1.19210
$266$	−5.01563e8	−1.63395
$267$	−1.03310e8	−0.332163
$268$	−1.90358e8	−0.604087
$269$	2.39784e7	0.0751083	0.0375541	−	0.999295i	$-0.488043\pi$
$269$	2.39784e7	0.0751083	0.0375541	+	0.999295i	$0.488043\pi$
$270$	−8.63772e7	−0.267071
$271$	5.61954e8	1.71518	0.857588	−	0.514338i	$-0.171962\pi$
$271$	5.61954e8	1.71518	0.857588	+	0.514338i	$0.171962\pi$
$272$	−2.92956e8	−0.882695
$273$	2.69165e8	0.800663
$274$	2.91397e8	0.855772
$275$	−7.39419e7	−0.214401
$276$	−1.02197e8	−0.292588
$277$	6.23564e7	0.176280	0.0881398	−	0.996108i	$-0.471908\pi$
$277$	6.23564e7	0.176280	0.0881398	+	0.996108i	$0.471908\pi$
$278$	−3.58499e7	−0.100076
$279$	7.89117e7	0.217534
$280$	2.29015e8	0.623463
$281$	5.33351e8	1.43397	0.716986	−	0.697087i	$-0.245521\pi$
$281$	5.33351e8	1.43397	0.716986	+	0.697087i	$0.245521\pi$
$282$	3.26153e8	0.866064
$283$	−7.38516e8	−1.93690	−0.968450	−	0.249207i	$-0.919830\pi$
$283$	−7.38516e8	−1.93690	−0.968450	+	0.249207i	$0.919830\pi$
$284$	−1.64570e8	−0.426321
$285$	−1.92346e8	−0.492183
$286$	7.88343e8	1.99266
$287$	−3.16569e8	−0.790464
$288$	1.61995e8	0.399602
$289$	−1.90327e8	−0.463829
$290$	6.93001e7	0.166856
$291$	1.68507e8	0.400859
$292$	−4.48732e8	−1.05474
$293$	4.24434e8	0.985764	0.492882	−	0.870096i	$-0.335943\pi$
$293$	4.24434e8	0.985764	0.492882	+	0.870096i	$0.335943\pi$
$294$	4.66179e8	1.06988
$295$	−6.09023e7	−0.138120
$296$	1.81322e8	0.406378
$297$	1.48379e8	0.328644
$298$	−3.10057e8	−0.678711
$299$	2.93898e8	0.635839
$300$	−2.41020e7	−0.0515382
$301$	1.89361e8	0.400227
$302$	−7.07143e7	−0.147735
$303$	−4.06422e8	−0.839320
$304$	4.74484e8	0.968643
$305$	−5.57951e8	−1.12602
$306$	−1.60022e8	−0.319268
$307$	5.81873e8	1.14774	0.573871	−	0.818946i	$-0.305441\pi$
$307$	5.81873e8	1.14774	0.573871	+	0.818946i	$0.305441\pi$
$308$	9.67869e8	1.88751
$309$	−2.87324e8	−0.554009
$310$	4.75031e8	0.905640
$311$	4.40429e8	0.830260	0.415130	−	0.909762i	$-0.363736\pi$
$311$	4.40429e8	0.830260	0.415130	+	0.909762i	$0.363736\pi$
$312$	−1.04448e8	−0.194697
$313$	1.33282e8	0.245678	0.122839	−	0.992427i	$-0.460800\pi$
$313$	1.33282e8	0.245678	0.122839	+	0.992427i	$0.460800\pi$
$314$	9.14304e8	1.66662
$315$	3.04971e8	0.549757
$316$	−7.51811e8	−1.34031
$317$	−5.31810e8	−0.937668	−0.468834	−	0.883286i	$-0.655326\pi$
$317$	−5.31810e8	−0.937668	−0.468834	+	0.883286i	$0.655326\pi$
$318$	4.86620e8	0.848585
$319$	−1.19044e8	−0.205324
$320$	2.25509e8	0.384714
$321$	4.64220e8	0.783351
$322$	8.68314e8	1.44938
$323$	−3.56340e8	−0.588378
$324$	4.83656e7	0.0790003
$325$	6.93123e7	0.112000
$326$	−1.22466e9	−1.95774
$327$	7.05395e8	1.11562
$328$	1.22843e8	0.192217
$329$	−1.15154e9	−1.78277
$330$	8.93211e8	1.36822
$331$	−1.41095e8	−0.213853	−0.106926	−	0.994267i	$-0.534101\pi$
$331$	−1.41095e8	−0.213853	−0.106926	+	0.994267i	$0.534101\pi$
$332$	−2.69323e8	−0.403916
$333$	2.41460e8	0.358336
$334$	2.06269e8	0.302916
$335$	6.20251e8	0.901386
$336$	−7.52308e8	−1.08195
$337$	−1.15891e9	−1.64948	−0.824739	−	0.565513i	$-0.808678\pi$
$337$	−1.15891e9	−1.64948	−0.824739	+	0.565513i	$0.808678\pi$
$338$	1.89627e8	0.267111
$339$	−4.02728e8	−0.561452
$340$	−4.00296e8	−0.552339
$341$	−8.16011e8	−1.11444
$342$	2.59179e8	0.350355
$343$	−4.84110e8	−0.647760
$344$	−7.34802e7	−0.0973231
$345$	3.32993e8	0.436584
$346$	−1.33640e9	−1.73448
$347$	4.17462e8	0.536369	0.268184	−	0.963368i	$-0.413576\pi$
$347$	4.17462e8	0.536369	0.268184	+	0.963368i	$0.413576\pi$
$348$	−3.88035e7	−0.0493564
$349$	−9.78142e7	−0.123172	−0.0615861	−	0.998102i	$-0.519616\pi$
$349$	−9.78142e7	−0.123172	−0.0615861	+	0.998102i	$0.519616\pi$
$350$	2.04782e8	0.255301
$351$	−1.39089e8	−0.171680
$352$	−1.67516e9	−2.04718
$353$	−1.27204e9	−1.53918	−0.769590	−	0.638539i	$-0.779539\pi$
$353$	−1.27204e9	−1.53918	−0.769590	+	0.638539i	$0.779539\pi$
$354$	8.20635e7	0.0983193
$355$	5.36225e8	0.636133
$356$	3.48223e8	0.409056
$357$	5.64988e8	0.657205
$358$	−1.42311e9	−1.63927
$359$	−2.70429e8	−0.308477	−0.154238	−	0.988034i	$-0.549292\pi$
$359$	−2.70429e8	−0.308477	−0.154238	+	0.988034i	$0.549292\pi$
$360$	−1.18342e8	−0.133684
$361$	−3.16727e8	−0.354332
$362$	−1.88618e9	−2.08979
$363$	−1.00821e9	−1.10631
$364$	−9.07270e8	−0.986011
$365$	1.46212e9	1.57383
$366$	7.51818e8	0.801546
$367$	1.26226e9	1.33296	0.666481	−	0.745522i	$-0.267800\pi$
$367$	1.26226e9	1.33296	0.666481	+	0.745522i	$0.267800\pi$
$368$	−8.21434e8	−0.859222
$369$	1.63585e8	0.169493
$370$	1.45354e9	1.49183
$371$	−1.71810e9	−1.74679
$372$	−2.65986e8	−0.267891
$373$	−4.77047e7	−0.0475971	−0.0237985	−	0.999717i	$-0.507576\pi$
$373$	−4.77047e7	−0.0475971	−0.0237985	+	0.999717i	$0.507576\pi$
$374$	1.65476e9	1.63563
$375$	−5.46973e8	−0.535620
$376$	4.46850e8	0.433515
$377$	1.11591e8	0.107259
$378$	−4.10936e8	−0.391339
$379$	1.65053e9	1.55735	0.778674	−	0.627429i	$-0.215893\pi$
$379$	1.65053e9	1.55735	0.778674	+	0.627429i	$0.215893\pi$
$380$	6.48338e8	0.606120
$381$	5.37525e8	0.497922
$382$	−1.29136e8	−0.118529
$383$	4.84522e7	0.0440674	0.0220337	−	0.999757i	$-0.492986\pi$
$383$	4.84522e7	0.0440674	0.0220337	+	0.999757i	$0.492986\pi$
$384$	4.64111e8	0.418276
$385$	−3.15364e9	−2.81644
$386$	−1.02500e9	−0.907125
$387$	−9.78508e7	−0.0858175
$388$	−5.67982e8	−0.493655
$389$	2.13932e9	1.84269	0.921345	−	0.388745i	$-0.127091\pi$
$389$	2.13932e9	1.84269	0.921345	+	0.388745i	$0.127091\pi$
$390$	−8.37286e8	−0.714740
$391$	6.16902e8	0.521913
$392$	6.38693e8	0.535539
$393$	−5.89848e8	−0.490192
$394$	−2.95042e8	−0.243023
$395$	2.44965e9	1.99993
$396$	−5.00140e8	−0.404723
$397$	1.67380e9	1.34257	0.671284	−	0.741200i	$-0.265743\pi$
$397$	1.67380e9	1.34257	0.671284	+	0.741200i	$0.265743\pi$
$398$	1.71661e9	1.36484
$399$	−9.15080e8	−0.721197
$400$	−1.93726e8	−0.151348
$401$	−5.90271e8	−0.457137	−0.228568	−	0.973528i	$-0.573404\pi$
$401$	−5.90271e8	−0.457137	−0.228568	+	0.973528i	$0.573404\pi$
$402$	−8.35765e8	−0.641643
$403$	7.64920e8	0.582168
$404$	1.36992e9	1.03362
$405$	−1.57591e8	−0.117880
$406$	3.29692e8	0.244494
$407$	−2.49689e9	−1.83578
$408$	−2.19240e8	−0.159812
$409$	−2.01375e8	−0.145537	−0.0727685	−	0.997349i	$-0.523183\pi$
$409$	−2.01375e8	−0.145537	−0.0727685	+	0.997349i	$0.523183\pi$
$410$	9.84745e8	0.705635
$411$	5.31641e8	0.377722
$412$	9.68476e8	0.682258
$413$	−2.89740e8	−0.202388
$414$	−4.48695e8	−0.310778
$415$	8.77547e8	0.602701
$416$	1.57028e9	1.06942
$417$	−6.54066e7	−0.0441718
$418$	−2.68012e9	−1.79489
$419$	1.53310e9	1.01817	0.509086	−	0.860716i	$-0.329983\pi$
$419$	1.53310e9	1.01817	0.509086	+	0.860716i	$0.329983\pi$
$420$	−1.02796e9	−0.677022
$421$	1.10242e9	0.720044	0.360022	−	0.932944i	$-0.382769\pi$
$421$	1.10242e9	0.720044	0.360022	+	0.932944i	$0.382769\pi$
$422$	−9.87872e8	−0.639892
$423$	5.95053e8	0.382265
$424$	6.66699e8	0.424766
$425$	1.45489e8	0.0919326
$426$	−7.22543e8	−0.452825
$427$	−2.65443e9	−1.64996
$428$	−1.56474e9	−0.964691
$429$	1.43830e9	0.879524
$430$	−5.89040e8	−0.357277
$431$	−2.15146e9	−1.29438	−0.647192	−	0.762327i	$-0.724057\pi$
$431$	−2.15146e9	−1.29438	−0.647192	+	0.762327i	$0.724057\pi$
$432$	3.88750e8	0.231994
$433$	4.65835e8	0.275756	0.137878	−	0.990449i	$-0.455972\pi$
$433$	4.65835e8	0.275756	0.137878	+	0.990449i	$0.455972\pi$
$434$	2.25994e9	1.32704
$435$	1.26435e8	0.0736469
$436$	−2.37766e9	−1.37388
$437$	−9.99162e8	−0.572732
$438$	−1.97015e9	−1.12032
$439$	2.25997e9	1.27490	0.637451	−	0.770491i	$-0.279989\pi$
$439$	2.25997e9	1.27490	0.637451	+	0.770491i	$0.279989\pi$
$440$	1.22375e9	0.684872
$441$	8.50523e8	0.472227
$442$	−1.55116e9	−0.854432
$443$	−1.51398e9	−0.827383	−0.413691	−	0.910417i	$-0.635761\pi$
$443$	−1.51398e9	−0.827383	−0.413691	+	0.910417i	$0.635761\pi$
$444$	−8.13885e8	−0.441288
$445$	−1.13463e9	−0.610371
$446$	−2.61058e9	−1.39337
$447$	−5.65686e8	−0.299570
$448$	1.07285e9	0.563723
$449$	−2.99785e9	−1.56296	−0.781480	−	0.623930i	$-0.785535\pi$
$449$	−2.99785e9	−1.56296	−0.781480	+	0.623930i	$0.785535\pi$
$450$	−1.05820e8	−0.0547422
$451$	−1.69160e9	−0.868321
$452$	1.35747e9	0.691424
$453$	−1.29015e8	−0.0652074
$454$	2.52851e9	1.26815
$455$	2.95619e9	1.47127
$456$	3.55091e8	0.175373
$457$	2.58428e9	1.26658	0.633289	−	0.773915i	$-0.281704\pi$
$457$	2.58428e9	1.26658	0.633289	+	0.773915i	$0.281704\pi$
$458$	1.57802e9	0.767510
$459$	−2.91954e8	−0.140919
$460$	−1.12241e9	−0.537651
$461$	−1.75068e9	−0.832247	−0.416124	−	0.909308i	$-0.636612\pi$
$461$	−1.75068e9	−0.832247	−0.416124	+	0.909308i	$0.636612\pi$
$462$	4.24942e9	2.00485
$463$	−3.28077e9	−1.53618	−0.768091	−	0.640341i	$-0.778793\pi$
$463$	−3.28077e9	−1.53618	−0.768091	+	0.640341i	$0.778793\pi$
$464$	−3.11892e8	−0.144941
$465$	8.66672e8	0.399733
$466$	3.50212e9	1.60317
$467$	2.15325e9	0.978329	0.489164	−	0.872192i	$-0.337302\pi$
$467$	2.15325e9	0.978329	0.489164	+	0.872192i	$0.337302\pi$
$468$	4.68825e8	0.211422
$469$	2.95082e9	1.32080
$470$	3.58208e9	1.59145
$471$	1.66811e9	0.735616
$472$	1.12432e8	0.0492145
$473$	1.01186e9	0.439648
$474$	−3.30082e9	−1.42363
$475$	−2.35641e8	−0.100884
$476$	−1.90439e9	−0.809343
$477$	8.87818e8	0.374550
$478$	−6.96171e7	−0.0291554
$479$	−4.22659e9	−1.75718	−0.878589	−	0.477578i	$-0.841515\pi$
$479$	−4.22659e9	−1.75718	−0.878589	+	0.477578i	$0.841515\pi$
$480$	1.77916e9	0.734295
$481$	2.34056e9	0.958986
$482$	2.32687e9	0.946471
$483$	1.58420e9	0.639728
$484$	3.39836e9	1.36242
$485$	1.85068e9	0.736605
$486$	2.12348e8	0.0839116
$487$	−3.71716e9	−1.45835	−0.729173	−	0.684330i	$-0.760095\pi$
$487$	−3.71716e9	−1.45835	−0.729173	+	0.684330i	$0.760095\pi$
$488$	1.03004e9	0.401220
$489$	−2.23435e9	−0.864111
$490$	5.11996e9	1.96599
$491$	3.64965e9	1.39144	0.695722	−	0.718311i	$-0.255085\pi$
$491$	3.64965e9	1.39144	0.695722	+	0.718311i	$0.255085\pi$
$492$	−5.51392e8	−0.208729
$493$	2.34233e8	0.0880409
$494$	2.51232e9	0.937628
$495$	1.62962e9	0.603906
$496$	−2.13793e9	−0.786696
$497$	2.55107e9	0.932127
$498$	−1.18246e9	−0.429027
$499$	−8.31205e8	−0.299472	−0.149736	−	0.988726i	$-0.547842\pi$
$499$	−8.31205e8	−0.299472	−0.149736	+	0.988726i	$0.547842\pi$
$500$	1.84367e9	0.659612
$501$	3.76330e8	0.133702
$502$	−2.08976e8	−0.0737280
$503$	−3.38642e9	−1.18646	−0.593230	−	0.805033i	$-0.702147\pi$
$503$	−3.38642e9	−1.18646	−0.593230	+	0.805033i	$0.702147\pi$
$504$	−5.63007e8	−0.195888
$505$	−4.46366e9	−1.54231
$506$	4.63988e9	1.59213
$507$	3.45966e8	0.117898
$508$	−1.81183e9	−0.613188
$509$	−2.52186e9	−0.847633	−0.423817	−	0.905748i	$-0.639310\pi$
$509$	−2.52186e9	−0.847633	−0.423817	+	0.905748i	$0.639310\pi$
$510$	−1.75750e9	−0.586677
$511$	6.95598e9	2.30614
$512$	−3.00491e9	−0.989434
$513$	4.72861e8	0.154640
$514$	5.69786e9	1.85072
$515$	−3.15562e9	−1.01803
$516$	3.29824e8	0.105684
$517$	−6.15333e9	−1.95836
$518$	6.91514e9	2.18598
$519$	−2.43820e9	−0.765568
$520$	−1.14713e9	−0.357768
$521$	4.55029e9	1.40964	0.704818	−	0.709388i	$-0.251029\pi$
$521$	4.55029e9	1.40964	0.704818	+	0.709388i	$0.251029\pi$
$522$	−1.70366e8	−0.0524248
$523$	−1.39010e9	−0.424903	−0.212451	−	0.977172i	$-0.568145\pi$
$523$	−1.39010e9	−0.424903	−0.212451	+	0.977172i	$0.568145\pi$
$524$	1.98819e9	0.603668
$525$	3.73616e8	0.112685
$526$	−2.57974e9	−0.772905
$527$	1.60560e9	0.477858
$528$	−4.01999e9	−1.18852
$529$	−1.67506e9	−0.491966
$530$	5.34446e9	1.55933
$531$	1.49721e8	0.0433963
$532$	3.08444e9	0.888149
$533$	1.58569e9	0.453600
$534$	1.52887e9	0.434487
$535$	5.09844e9	1.43946
$536$	−1.14505e9	−0.321179
$537$	−2.59641e9	−0.723541
$538$	−3.54855e8	−0.0982456
$539$	−8.79510e9	−2.41925
$540$	5.31190e8	0.145168
$541$	3.45589e9	0.938360	0.469180	−	0.883103i	$-0.344550\pi$
$541$	3.45589e9	0.938360	0.469180	+	0.883103i	$0.344550\pi$
$542$	−8.31632e9	−2.24354
$543$	−3.44124e9	−0.922393
$544$	3.29607e9	0.877809
$545$	7.74723e9	2.05002
$546$	−3.98336e9	−1.04731
$547$	5.69696e9	1.48829	0.744145	−	0.668018i	$-0.232857\pi$
$547$	5.69696e9	1.48829	0.744145	+	0.668018i	$0.232857\pi$
$548$	−1.79199e9	−0.465162
$549$	1.37166e9	0.353788
$550$	1.09426e9	0.280447
$551$	−3.79374e8	−0.0966134
$552$	−6.14740e8	−0.155562
$553$	1.16541e10	2.93050
$554$	−9.22809e8	−0.230583
$555$	2.65191e9	0.658466
$556$	2.20465e8	0.0543973
$557$	−7.56400e9	−1.85464	−0.927318	−	0.374275i	$-0.877892\pi$
$557$	−7.56400e9	−1.85464	−0.927318	+	0.374275i	$0.877892\pi$
$558$	−1.16781e9	−0.284546
$559$	−9.48504e8	−0.229667
$560$	−8.26246e9	−1.98816
$561$	3.01904e9	0.721936
$562$	−7.89302e9	−1.87571
$563$	−5.67154e9	−1.33943	−0.669717	−	0.742616i	$-0.733585\pi$
$563$	−5.67154e9	−1.33943	−0.669717	+	0.742616i	$0.733585\pi$
$564$	−2.00573e9	−0.470756
$565$	−4.42308e9	−1.03170
$566$	1.09292e10	2.53357
$567$	−7.49735e8	−0.172730
$568$	−9.89927e8	−0.226665
$569$	1.36290e9	0.310149	0.155074	−	0.987903i	$-0.450438\pi$
$569$	1.36290e9	0.310149	0.155074	+	0.987903i	$0.450438\pi$
$570$	2.84652e9	0.643801
$571$	−2.64928e9	−0.595528	−0.297764	−	0.954640i	$-0.596241\pi$
$571$	−2.64928e9	−0.595528	−0.297764	+	0.954640i	$0.596241\pi$
$572$	−4.84804e9	−1.08313
$573$	−2.35603e8	−0.0523166
$574$	4.68489e9	1.03397
$575$	4.07945e8	0.0894880
$576$	−5.54387e8	−0.120874
$577$	7.60073e9	1.64718	0.823588	−	0.567188i	$-0.191969\pi$
$577$	7.60073e9	1.64718	0.823588	+	0.567188i	$0.191969\pi$
$578$	2.81664e9	0.606713
$579$	−1.87006e9	−0.400388
$580$	−4.26172e8	−0.0906957
$581$	4.17490e9	0.883139
$582$	−2.49372e9	−0.524344
$583$	−9.18076e9	−1.91884
$584$	−2.69923e9	−0.560782
$585$	−1.52759e9	−0.315473
$586$	−6.28116e9	−1.28943
$587$	−3.30015e9	−0.673443	−0.336721	−	0.941604i	$-0.609318\pi$
$587$	−3.30015e9	−0.673443	−0.336721	+	0.941604i	$0.609318\pi$
$588$	−2.86684e9	−0.581545
$589$	−2.60049e9	−0.524388
$590$	9.01289e8	0.180668
$591$	−5.38292e8	−0.107266
$592$	−6.54179e9	−1.29590
$593$	5.80009e9	1.14220	0.571102	−	0.820879i	$-0.306516\pi$
$593$	5.80009e9	1.14220	0.571102	+	0.820879i	$0.306516\pi$
$594$	−2.19586e9	−0.429884
$595$	6.20516e9	1.20766
$596$	1.90675e9	0.368919
$597$	3.13188e9	0.602413
$598$	−4.34937e9	−0.831711
$599$	3.53769e9	0.672551	0.336276	−	0.941764i	$-0.390833\pi$
$599$	3.53769e9	0.672551	0.336276	+	0.941764i	$0.390833\pi$
$600$	−1.44979e8	−0.0274016
$601$	−7.08661e9	−1.33161	−0.665806	−	0.746125i	$-0.731912\pi$
$601$	−7.08661e9	−1.33161	−0.665806	+	0.746125i	$0.731912\pi$
$602$	−2.80233e9	−0.523519
$603$	−1.52482e9	−0.283209
$604$	4.34869e8	0.0803025
$605$	−1.10730e10	−2.03292
$606$	6.01461e9	1.09788
$607$	−9.13138e8	−0.165720	−0.0828602	−	0.996561i	$-0.526406\pi$
$607$	−9.13138e8	−0.165720	−0.0828602	+	0.996561i	$0.526406\pi$
$608$	−5.33846e9	−0.963282
$609$	6.01509e8	0.107915
$610$	8.25707e9	1.47289
$611$	5.76807e9	1.02302
$612$	9.84083e8	0.173541
$613$	6.24314e8	0.109469	0.0547346	−	0.998501i	$-0.482569\pi$
$613$	6.24314e8	0.109469	0.0547346	+	0.998501i	$0.482569\pi$
$614$	−8.61110e9	−1.50131
$615$	1.79662e9	0.311454
$616$	5.82196e9	1.00354
$617$	−1.00319e10	−1.71943	−0.859714	−	0.510775i	$-0.829358\pi$
$617$	−1.00319e10	−1.71943	−0.859714	+	0.510775i	$0.829358\pi$
$618$	4.25208e9	0.724673
$619$	−1.02470e10	−1.73652	−0.868258	−	0.496114i	$-0.834760\pi$
$619$	−1.02470e10	−1.73652	−0.868258	+	0.496114i	$0.834760\pi$
$620$	−2.92128e9	−0.492268
$621$	−8.18625e8	−0.137172
$622$	−6.51787e9	−1.08602
$623$	−5.39796e9	−0.894378
$624$	3.76830e9	0.620868
$625$	−6.77361e9	−1.10979
$626$	−1.97243e9	−0.321360
$627$	−4.88977e9	−0.792231
$628$	−5.62266e9	−0.905906
$629$	4.91293e9	0.787160
$630$	−4.51324e9	−0.719112
$631$	3.58757e9	0.568457	0.284228	−	0.958757i	$-0.408263\pi$
$631$	3.58757e9	0.568457	0.284228	+	0.958757i	$0.408263\pi$
$632$	−4.52232e9	−0.712609
$633$	−1.80233e9	−0.282437
$634$	7.87022e9	1.22652
$635$	5.90354e9	0.914965
$636$	−2.99255e9	−0.461255
$637$	8.24444e9	1.26378
$638$	1.76173e9	0.268575
$639$	−1.31825e9	−0.199868
$640$	5.09725e9	0.768609
$641$	−2.92622e9	−0.438837	−0.219419	−	0.975631i	$-0.570416\pi$
$641$	−2.92622e9	−0.438837	−0.219419	+	0.975631i	$0.570416\pi$
$642$	−6.86996e9	−1.02466
$643$	5.85273e9	0.868200	0.434100	−	0.900865i	$-0.357066\pi$
$643$	5.85273e9	0.868200	0.434100	+	0.900865i	$0.357066\pi$
$644$	−5.33984e9	−0.787821
$645$	−1.07468e9	−0.157695
$646$	5.27346e9	0.769629
$647$	−7.40954e9	−1.07554	−0.537770	−	0.843092i	$-0.680733\pi$
$647$	−7.40954e9	−1.07554	−0.537770	+	0.843092i	$0.680733\pi$
$648$	2.90930e8	0.0420026
$649$	−1.54824e9	−0.222322
$650$	−1.02575e9	−0.146502
$651$	4.12316e9	0.585729
$652$	7.53127e9	1.06415
$653$	−4.89548e9	−0.688018	−0.344009	−	0.938966i	$-0.611785\pi$
$653$	−4.89548e9	−0.688018	−0.344009	+	0.938966i	$0.611785\pi$
$654$	−1.04391e10	−1.45929
$655$	−6.47820e9	−0.900761
$656$	−4.43195e9	−0.612959
$657$	−3.59445e9	−0.494486
$658$	1.70416e10	2.33196
$659$	−1.26032e10	−1.71547	−0.857736	−	0.514091i	$-0.828129\pi$
$659$	−1.26032e10	−1.71547	−0.857736	+	0.514091i	$0.828129\pi$
$660$	−5.49294e9	−0.743706
$661$	3.60198e8	0.0485106	0.0242553	−	0.999706i	$-0.492279\pi$
$661$	3.60198e8	0.0485106	0.0242553	+	0.999706i	$0.492279\pi$
$662$	2.08806e9	0.279731
$663$	−2.83002e9	−0.377130
$664$	−1.62004e9	−0.214753
$665$	−1.00502e10	−1.32525
$666$	−3.57335e9	−0.468723
$667$	6.56779e8	0.0856996
$668$	−1.26849e9	−0.164652
$669$	−4.76290e9	−0.615007
$670$	−9.17906e9	−1.17906
$671$	−1.41841e10	−1.81247
$672$	8.46428e9	1.07596
$673$	1.40565e9	0.177756	0.0888782	−	0.996043i	$-0.471672\pi$
$673$	1.40565e9	0.177756	0.0888782	+	0.996043i	$0.471672\pi$
$674$	1.71507e10	2.15761
$675$	−1.93063e8	−0.0241622
$676$	−1.16614e9	−0.145190
$677$	5.12306e9	0.634555	0.317278	−	0.948333i	$-0.397231\pi$
$677$	5.12306e9	0.634555	0.317278	+	0.948333i	$0.397231\pi$
$678$	5.95994e9	0.734409
$679$	8.80452e9	1.07935
$680$	−2.40788e9	−0.293665
$681$	4.61315e9	0.559736
$682$	1.20761e10	1.45774
$683$	−1.70981e9	−0.205341	−0.102671	−	0.994715i	$-0.532739\pi$
$683$	−1.70981e9	−0.205341	−0.102671	+	0.994715i	$0.532739\pi$
$684$	−1.59386e9	−0.190439
$685$	5.83891e9	0.694089
$686$	7.16431e9	0.847305
$687$	2.87903e9	0.338764
$688$	2.65104e9	0.310353
$689$	8.60594e9	1.00238
$690$	−4.92794e9	−0.571076
$691$	−1.38466e10	−1.59651	−0.798253	−	0.602323i	$-0.794242\pi$
$691$	−1.38466e10	−1.59651	−0.798253	+	0.602323i	$0.794242\pi$
$692$	8.21839e9	0.942791
$693$	7.75287e9	0.884905
$694$	−6.17799e9	−0.701599
$695$	−7.18348e8	−0.0811686
$696$	−2.33412e8	−0.0262416
$697$	3.32842e9	0.372326
$698$	1.44755e9	0.161116
$699$	6.38947e9	0.707611
$700$	−1.25934e9	−0.138771
$701$	−1.16579e10	−1.27823	−0.639115	−	0.769111i	$-0.720699\pi$
$701$	−1.16579e10	−1.27823	−0.639115	+	0.769111i	$0.720699\pi$
$702$	2.05837e9	0.224566
$703$	−7.95720e9	−0.863806
$704$	5.73282e9	0.619247
$705$	6.53535e9	0.702437
$706$	1.88248e10	2.01333
$707$	−2.12357e10	−2.25995
$708$	−5.04663e8	−0.0534423
$709$	−1.12659e10	−1.18714	−0.593572	−	0.804781i	$-0.702283\pi$
$709$	−1.12659e10	−1.18714	−0.593572	+	0.804781i	$0.702283\pi$
$710$	−7.93555e9	−0.832095
$711$	−6.02219e9	−0.628364
$712$	2.09464e9	0.217486
$713$	4.50202e9	0.465151
$714$	−8.36122e9	−0.859659
$715$	1.57965e10	1.61618
$716$	8.75167e9	0.891036
$717$	−1.27013e8	−0.0128686
$718$	4.00206e9	0.403504
$719$	−9.57915e9	−0.961116	−0.480558	−	0.876963i	$-0.659566\pi$
$719$	−9.57915e9	−0.961116	−0.480558	+	0.876963i	$0.659566\pi$
$720$	4.26957e9	0.426305
$721$	−1.50128e10	−1.49172
$722$	4.68722e9	0.463485
$723$	4.24527e9	0.417755
$724$	1.15993e10	1.13592
$725$	1.54894e8	0.0150956
$726$	1.49204e10	1.44712
$727$	1.06396e10	1.02697	0.513484	−	0.858099i	$-0.328355\pi$
$727$	1.06396e10	1.02697	0.513484	+	0.858099i	$0.328355\pi$
$728$	−5.45744e9	−0.524239
$729$	3.87420e8	0.0370370
$730$	−2.16378e10	−2.05865
$731$	−1.99095e9	−0.188516
$732$	−4.62342e9	−0.435687
$733$	−1.85210e10	−1.73700	−0.868500	−	0.495689i	$-0.834916\pi$
$733$	−1.85210e10	−1.73700	−0.868500	+	0.495689i	$0.834916\pi$
$734$	−1.86801e10	−1.74359
$735$	9.34114e9	0.867749
$736$	9.24203e9	0.854467
$737$	1.57679e10	1.45090
$738$	−2.42088e9	−0.221705
$739$	2.17412e9	0.198166	0.0990828	−	0.995079i	$-0.468409\pi$
$739$	2.17412e9	0.198166	0.0990828	+	0.995079i	$0.468409\pi$
$740$	−8.93875e9	−0.810897
$741$	4.58362e9	0.413852
$742$	2.54261e10	2.28489
$743$	1.86262e10	1.66595	0.832977	−	0.553307i	$-0.186634\pi$
$743$	1.86262e10	1.66595	0.832977	+	0.553307i	$0.186634\pi$
$744$	−1.59997e9	−0.142431
$745$	−6.21282e9	−0.550481
$746$	7.05979e8	0.0622595
$747$	−2.15735e9	−0.189364
$748$	−1.01762e10	−0.889060
$749$	2.42557e10	2.10924
$750$	8.09462e9	0.700619
$751$	−2.24532e9	−0.193437	−0.0967185	−	0.995312i	$-0.530835\pi$
$751$	−2.24532e9	−0.193437	−0.0967185	+	0.995312i	$0.530835\pi$
$752$	−1.61216e10	−1.38243
$753$	−3.81267e8	−0.0325422
$754$	−1.65142e9	−0.140300
$755$	−1.41695e9	−0.119823
$756$	2.52712e9	0.212716
$757$	1.91646e10	1.60570	0.802852	−	0.596179i	$-0.203315\pi$
$757$	1.91646e10	1.60570	0.802852	+	0.596179i	$0.203315\pi$
$758$	−2.44260e10	−2.03709
$759$	8.46525e9	0.702738
$760$	3.89990e9	0.322260
$761$	1.33554e10	1.09853	0.549265	−	0.835648i	$-0.314908\pi$
$761$	1.33554e10	1.09853	0.549265	+	0.835648i	$0.314908\pi$
$762$	−7.95480e9	−0.651309
$763$	3.68572e10	3.00390
$764$	7.94143e8	0.0644276
$765$	−3.20647e9	−0.258948
$766$	−7.17041e8	−0.0576426
$767$	1.45130e9	0.116138
$768$	−9.49655e9	−0.756488
$769$	−5.20020e9	−0.412361	−0.206181	−	0.978514i	$-0.566103\pi$
$769$	−5.20020e9	−0.412361	−0.206181	+	0.978514i	$0.566103\pi$
$770$	4.66706e10	3.68405
$771$	1.03955e10	0.816874
$772$	6.30338e9	0.493075
$773$	2.37920e10	1.85269	0.926347	−	0.376672i	$-0.122932\pi$
$773$	2.37920e10	1.85269	0.926347	+	0.376672i	$0.122932\pi$
$774$	1.44809e9	0.112254
$775$	1.06175e9	0.0819343
$776$	−3.41654e9	−0.262465
$777$	1.26164e10	0.964852
$778$	−3.16597e10	−2.41034
$779$	−5.39086e9	−0.408580
$780$	5.14902e9	0.388502
$781$	1.36318e10	1.02394
$782$	−9.12950e9	−0.682690
$783$	−3.10826e8	−0.0231393
$784$	−2.30429e10	−1.70778
$785$	1.83205e10	1.35174
$786$	8.72913e9	0.641198
$787$	5.45988e9	0.399275	0.199637	−	0.979870i	$-0.436024\pi$
$787$	5.45988e9	0.399275	0.199637	+	0.979870i	$0.436024\pi$
$788$	1.81441e9	0.132097
$789$	−4.70662e9	−0.341146
$790$	−3.62522e10	−2.61601
$791$	−2.10427e10	−1.51176
$792$	−3.00846e9	−0.215182
$793$	1.32960e10	0.946814
$794$	−2.47704e10	−1.75615
$795$	9.75074e9	0.688260
$796$	−1.05566e10	−0.741868
$797$	−4.12638e9	−0.288712	−0.144356	−	0.989526i	$-0.546111\pi$
$797$	−4.12638e9	−0.288712	−0.144356	+	0.989526i	$0.546111\pi$
$798$	1.35422e10	0.943364
$799$	1.21074e10	0.839725
$800$	2.17963e9	0.150511
$801$	2.78936e9	0.191774
$802$	8.73538e9	0.597959
$803$	3.71696e10	2.53328
$804$	5.13967e9	0.348770
$805$	1.73990e10	1.17554
$806$	−1.13200e10	−0.761507
$807$	−6.47418e8	−0.0433638
$808$	8.24037e9	0.549550
$809$	4.36775e9	0.290027	0.145013	−	0.989430i	$-0.453678\pi$
$809$	4.36775e9	0.290027	0.145013	+	0.989430i	$0.453678\pi$
$810$	2.33218e9	0.154193
$811$	−1.81321e10	−1.19364	−0.596822	−	0.802374i	$-0.703570\pi$
$811$	−1.81321e10	−1.19364	−0.596822	+	0.802374i	$0.703570\pi$
$812$	−2.02750e9	−0.132897
$813$	−1.51728e10	−0.990257
$814$	3.69514e10	2.40129
$815$	−2.45394e10	−1.58786
$816$	7.90980e9	0.509624
$817$	3.22462e9	0.206872
$818$	2.98013e9	0.190370
$819$	−7.26746e9	−0.462263
$820$	−6.05584e9	−0.383554
$821$	1.18439e10	0.746951	0.373475	−	0.927640i	$-0.378166\pi$
$821$	1.18439e10	0.746951	0.373475	+	0.927640i	$0.378166\pi$
$822$	−7.86772e9	−0.494080
$823$	7.82921e8	0.0489574	0.0244787	−	0.999700i	$-0.492207\pi$
$823$	7.82921e8	0.0489574	0.0244787	+	0.999700i	$0.492207\pi$
$824$	5.82561e9	0.362741
$825$	1.99643e9	0.123784
$826$	4.28785e9	0.264734
$827$	−1.09138e10	−0.670974	−0.335487	−	0.942045i	$-0.608901\pi$
$827$	−1.09138e10	−0.670974	−0.335487	+	0.942045i	$0.608901\pi$
$828$	2.75932e9	0.168926
$829$	−2.91147e10	−1.77489	−0.887446	−	0.460911i	$-0.847523\pi$
$829$	−2.91147e10	−1.77489	−0.887446	+	0.460911i	$0.847523\pi$
$830$	−1.29868e10	−0.788365
$831$	−1.68362e9	−0.101775
$832$	−5.37388e9	−0.323487
$833$	1.73054e10	1.03735
$834$	9.67947e8	0.0577791
$835$	4.13316e9	0.245685
$836$	1.64819e10	0.975628
$837$	−2.13061e9	−0.125593
$838$	−2.26882e10	−1.33182
$839$	−3.47596e9	−0.203193	−0.101596	−	0.994826i	$-0.532395\pi$
$839$	−3.47596e9	−0.203193	−0.101596	+	0.994826i	$0.532395\pi$
$840$	−6.18341e9	−0.359957
$841$	−1.70005e10	−0.985543
$842$	−1.63146e10	−0.941855
$843$	−1.44005e10	−0.827905
$844$	6.07507e9	0.347819
$845$	3.79969e9	0.216645
$846$	−8.80614e9	−0.500022
$847$	−5.26793e10	−2.97885
$848$	−2.40533e10	−1.35453
$849$	1.99399e10	1.11827
$850$	−2.15309e9	−0.120253
$851$	1.37756e10	0.766228
$852$	4.44339e9	0.246136
$853$	−2.90613e10	−1.60322	−0.801610	−	0.597847i	$-0.796023\pi$
$853$	−2.90613e10	−1.60322	−0.801610	+	0.597847i	$0.796023\pi$
$854$	3.92827e10	2.15824
$855$	5.19335e9	0.284162
$856$	−9.41226e9	−0.512904
$857$	3.15674e10	1.71319	0.856597	−	0.515987i	$-0.172575\pi$
$857$	3.15674e10	1.71319	0.856597	+	0.515987i	$0.172575\pi$
$858$	−2.12852e10	−1.15046
$859$	4.24010e9	0.228244	0.114122	−	0.993467i	$-0.463594\pi$
$859$	4.24010e9	0.228244	0.114122	+	0.993467i	$0.463594\pi$
$860$	3.62239e9	0.194201
$861$	8.54737e9	0.456374
$862$	3.18393e10	1.69312
$863$	−2.94013e9	−0.155715	−0.0778573	−	0.996965i	$-0.524808\pi$
$863$	−2.94013e9	−0.155715	−0.0778573	+	0.996965i	$0.524808\pi$
$864$	−4.37386e9	−0.230710
$865$	−2.67783e10	−1.40678
$866$	−6.89387e9	−0.360703
$867$	5.13883e9	0.267792
$868$	−1.38979e10	−0.721321
$869$	6.22744e10	3.21914
$870$	−1.87110e9	−0.0963341
$871$	−1.47806e10	−0.757930
$872$	−1.43022e10	−0.730458
$873$	−4.54968e9	−0.231436
$874$	1.47865e10	0.749163
$875$	−2.85795e10	−1.44220
$876$	1.21158e10	0.608956
$877$	2.78040e10	1.39190	0.695952	−	0.718088i	$-0.254983\pi$
$877$	2.78040e10	1.39190	0.695952	+	0.718088i	$0.254983\pi$
$878$	−3.34451e10	−1.66764
$879$	−1.14597e10	−0.569131
$880$	−4.41508e10	−2.18398
$881$	2.13975e10	1.05426	0.527131	−	0.849784i	$-0.323268\pi$
$881$	2.13975e10	1.05426	0.527131	+	0.849784i	$0.323268\pi$
$882$	−1.25868e10	−0.617698
$883$	2.73260e10	1.33572	0.667858	−	0.744289i	$-0.267211\pi$
$883$	2.73260e10	1.33572	0.667858	+	0.744289i	$0.267211\pi$
$884$	9.53908e9	0.464433
$885$	1.64436e9	0.0797436
$886$	2.24053e10	1.08226
$887$	2.35865e10	1.13483	0.567416	−	0.823431i	$-0.307943\pi$
$887$	2.35865e10	1.13483	0.567416	+	0.823431i	$0.307943\pi$
$888$	−4.89571e9	−0.234623
$889$	2.80859e10	1.34070
$890$	1.67913e10	0.798398
$891$	−4.00624e9	−0.189743
$892$	1.60542e10	0.757376
$893$	−1.96097e10	−0.921489
$894$	8.37154e9	0.391854
$895$	−2.85159e10	−1.32955
$896$	2.42500e10	1.12625
$897$	−7.93524e9	−0.367102
$898$	4.43650e10	2.04444
$899$	1.70938e9	0.0784658
$900$	6.50754e8	0.0297556
$901$	1.80642e10	0.822777
$902$	2.50339e10	1.13581
$903$	−5.11274e9	−0.231071
$904$	8.16547e9	0.367614
$905$	−3.77946e10	−1.69496
$906$	1.90929e9	0.0852948
$907$	−1.48021e9	−0.0658716	−0.0329358	−	0.999457i	$-0.510486\pi$
$907$	−1.48021e9	−0.0658716	−0.0329358	+	0.999457i	$0.510486\pi$
$908$	−1.55495e10	−0.689311
$909$	1.09734e10	0.484582
$910$	−4.37485e10	−1.92450
$911$	1.06953e10	0.468681	0.234341	−	0.972155i	$-0.424707\pi$
$911$	1.06953e10	0.468681	0.234341	+	0.972155i	$0.424707\pi$
$912$	−1.28111e10	−0.559247
$913$	2.23087e10	0.970124
$914$	−3.82445e10	−1.65675
$915$	1.50647e10	0.650108
$916$	−9.70430e9	−0.417186
$917$	−3.08198e10	−1.31989
$918$	4.32060e9	0.184330
$919$	−3.15248e9	−0.133983	−0.0669913	−	0.997754i	$-0.521340\pi$
$919$	−3.15248e9	−0.133983	−0.0669913	+	0.997754i	$0.521340\pi$
$920$	−6.75157e9	−0.285856
$921$	−1.57106e10	−0.662649
$922$	2.59081e10	1.08862
$923$	−1.27783e10	−0.534892
$924$	−2.61325e10	−1.08975
$925$	3.24882e9	0.134968
$926$	4.85519e10	2.00941
$927$	7.75774e9	0.319857
$928$	3.50913e9	0.144139
$929$	4.53386e10	1.85529	0.927647	−	0.373458i	$-0.121828\pi$
$929$	4.53386e10	1.85529	0.927647	+	0.373458i	$0.121828\pi$
$930$	−1.28258e10	−0.522871
$931$	−2.80286e10	−1.13835
$932$	−2.15368e10	−0.871418
$933$	−1.18916e10	−0.479351
$934$	−3.18658e10	−1.27971
$935$	3.31576e10	1.32661
$936$	2.82009e9	0.112408
$937$	4.84194e9	0.192279	0.0961393	−	0.995368i	$-0.469351\pi$
$937$	4.84194e9	0.192279	0.0961393	+	0.995368i	$0.469351\pi$
$938$	−4.36690e10	−1.72768
$939$	−3.59861e9	−0.141842
$940$	−2.20286e10	−0.865046
$941$	−8.67653e9	−0.339455	−0.169728	−	0.985491i	$-0.554289\pi$
$941$	−8.67653e9	−0.339455	−0.169728	+	0.985491i	$0.554289\pi$
$942$	−2.46862e10	−0.962224
$943$	9.33275e9	0.362425
$944$	−4.05635e9	−0.156940
$945$	−8.23420e9	−0.317402
$946$	−1.49744e10	−0.575083
$947$	1.85330e10	0.709121	0.354561	−	0.935033i	$-0.384630\pi$
$947$	1.85330e10	0.709121	0.354561	+	0.935033i	$0.384630\pi$
$948$	2.02989e10	0.773825
$949$	−3.48424e10	−1.32335
$950$	3.48723e9	0.131962
$951$	1.43589e10	0.541363
$952$	−1.14554e10	−0.430309
$953$	1.33156e10	0.498352	0.249176	−	0.968458i	$-0.419840\pi$
$953$	1.33156e10	0.498352	0.249176	+	0.968458i	$0.419840\pi$
$954$	−1.31388e10	−0.489931
$955$	−2.58759e9	−0.0961353
$956$	4.28122e8	0.0158476
$957$	3.21419e9	0.118544
$958$	6.25490e10	2.29848
$959$	2.77784e10	1.01705
$960$	−6.08873e9	−0.222115
$961$	−1.57953e10	−0.574112
$962$	−3.46378e10	−1.25440
$963$	−1.25339e10	−0.452268
$964$	−1.43095e10	−0.514462
$965$	−2.05385e10	−0.735740
$966$	−2.34445e10	−0.836798
$967$	1.13617e10	0.404064	0.202032	−	0.979379i	$-0.435246\pi$
$967$	1.13617e10	0.404064	0.202032	+	0.979379i	$0.435246\pi$
$968$	2.04419e10	0.724365
$969$	9.62119e9	0.339700
$970$	−2.73880e10	−0.963518
$971$	1.97070e10	0.690802	0.345401	−	0.938455i	$-0.387743\pi$
$971$	1.97070e10	0.690802	0.345401	+	0.938455i	$0.387743\pi$
$972$	−1.30587e9	−0.0456109
$973$	−3.41751e9	−0.118937
$974$	5.50100e10	1.90759
$975$	−1.87143e9	−0.0646633
$976$	−3.71619e10	−1.27945
$977$	−4.83923e10	−1.66014	−0.830072	−	0.557657i	$-0.811701\pi$
$977$	−4.83923e10	−1.66014	−0.830072	+	0.557657i	$0.811701\pi$
$978$	3.30659e10	1.13030
$979$	−2.88442e10	−0.982470
$980$	−3.14860e10	−1.06863
$981$	−1.90457e10	−0.644102
$982$	−5.40109e10	−1.82008
$983$	2.12104e10	0.712214	0.356107	−	0.934445i	$-0.384104\pi$
$983$	2.12104e10	0.712214	0.356107	+	0.934445i	$0.384104\pi$
$984$	−3.31675e9	−0.110976
$985$	−5.91196e9	−0.197108
$986$	−3.46640e9	−0.115162
$987$	3.10917e10	1.02928
$988$	−1.54499e10	−0.509655
$989$	−5.58252e9	−0.183503
$990$	−2.41167e10	−0.789941
$991$	3.64631e10	1.19013	0.595067	−	0.803676i	$-0.297125\pi$
$991$	3.64631e10	1.19013	0.595067	+	0.803676i	$0.297125\pi$
$992$	2.40540e10	0.782342
$993$	3.80958e9	0.123468
$994$	−3.77531e10	−1.21927
$995$	3.43968e10	1.10697
$996$	7.27173e9	0.233201
$997$	−3.41980e10	−1.09287	−0.546434	−	0.837502i	$-0.684015\pi$
$997$	−3.41980e10	−1.09287	−0.546434	+	0.837502i	$0.684015\pi$
$998$	1.23009e10	0.391725
$999$	−6.51942e9	−0.206885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.5	✓	16
3.2	odd	2		531.8.a.b.1.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.5	✓	16	1.1	even	1	trivial
531.8.a.b.1.12		16	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)

\(f(q)\)	\(=\)	\(q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +4388.42 q^{10} -7538.45 q^{11} -2457.23 q^{12} +7066.47 q^{13} +20877.7 q^{14} +8006.47 q^{15} -19750.5 q^{16} +14832.8 q^{17} -10788.4 q^{18} -24023.8 q^{19} -26987.3 q^{20} +38090.5 q^{21} +111561. q^{22} +41590.5 q^{23} -14780.8 q^{24} +9808.63 q^{25} -104576. q^{26} -19683.0 q^{27} -128391. q^{28} +15791.6 q^{29} -118487. q^{30} +108246. q^{31} +222215. q^{32} +203538. q^{33} -219509. q^{34} +418341. q^{35} +66345.1 q^{36} +331221. q^{37} +355527. q^{38} -190795. q^{39} -162335. q^{40} +224396. q^{41} -563699. q^{42} -134226. q^{43} -686063. q^{44} -216175. q^{45} -615494. q^{46} +816259. q^{47} +533265. q^{48} +1.16670e6 q^{49} -145157. q^{50} -400485. q^{51} +643108. q^{52} +1.21786e6 q^{53} +291287. q^{54} +2.23542e6 q^{55} -772301. q^{56} +648644. q^{57} -233699. q^{58} +205379. q^{59} +728656. q^{60} +1.88156e6 q^{61} -1.60193e6 q^{62} -1.02844e6 q^{63} -760476. q^{64} -2.09546e6 q^{65} -3.01215e6 q^{66} -2.09166e6 q^{67} +1.34991e6 q^{68} -1.12294e6 q^{69} -6.19100e6 q^{70} -1.80830e6 q^{71} +399081. q^{72} -4.93066e6 q^{73} -4.90172e6 q^{74} -264833. q^{75} -2.18637e6 q^{76} +1.06349e7 q^{77} +2.82356e6 q^{78} -8.26089e6 q^{79} +5.85675e6 q^{80} +531441. q^{81} -3.32083e6 q^{82} -2.95933e6 q^{83} +3.46655e6 q^{84} -4.39846e6 q^{85} +1.98640e6 q^{86} -426373. q^{87} -4.12683e6 q^{88} +3.82628e6 q^{89} +3.19916e6 q^{90} -9.96908e6 q^{91} +3.78508e6 q^{92} -2.92265e6 q^{93} -1.20798e7 q^{94} +7.12393e6 q^{95} -5.99981e6 q^{96} -6.24098e6 q^{97} -1.72659e7 q^{98} -5.49553e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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