LMFDB - Embedded newform 177.8.a.a.1.15

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			1.15
Root			$-18.7189$ 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)

$f(q)$	$=$	$q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +8308.06 q^{10} -2763.59 q^{11} -6004.75 q^{12} -10956.4 q^{13} -31743.3 q^{14} -11983.5 q^{15} +4610.01 q^{16} -17168.5 q^{17} +13646.1 q^{18} -41280.8 q^{19} +98707.5 q^{20} +45786.3 q^{21} -51731.4 q^{22} +82535.2 q^{23} -47709.9 q^{24} +118862. q^{25} -205092. q^{26} -19683.0 q^{27} -377140. q^{28} +103089. q^{29} -224318. q^{30} -162172. q^{31} -139886. q^{32} +74616.9 q^{33} -321376. q^{34} -752645. q^{35} +162128. q^{36} +446923. q^{37} -772732. q^{38} +295822. q^{39} +784266. q^{40} +371488. q^{41} +857070. q^{42} -790401. q^{43} -614617. q^{44} +323554. q^{45} +1.54497e6 q^{46} -1.11011e6 q^{47} -124470. q^{48} +2.05215e6 q^{49} +2.22497e6 q^{50} +463549. q^{51} -2.43668e6 q^{52} -771292. q^{53} -368445. q^{54} -1.22657e6 q^{55} -2.99652e6 q^{56} +1.11458e6 q^{57} +1.92972e6 q^{58} +205379. q^{59} -2.66510e6 q^{60} -211813. q^{61} -3.03568e6 q^{62} -1.23623e6 q^{63} -3.20860e6 q^{64} -4.86279e6 q^{65} +1.39675e6 q^{66} +1.48116e6 q^{67} -3.81824e6 q^{68} -2.22845e6 q^{69} -1.40887e7 q^{70} +3.40336e6 q^{71} +1.28817e6 q^{72} -3.93442e6 q^{73} +8.36592e6 q^{74} -3.20927e6 q^{75} -9.18078e6 q^{76} +4.68646e6 q^{77} +5.53747e6 q^{78} +4.55385e6 q^{79} +2.04607e6 q^{80} +531441. q^{81} +6.95386e6 q^{82} -4.51953e6 q^{83} +1.01828e7 q^{84} -7.61992e6 q^{85} -1.47955e7 q^{86} -2.78342e6 q^{87} -4.88336e6 q^{88} -4.21568e6 q^{89} +6.05658e6 q^{90} +1.85797e7 q^{91} +1.83557e7 q^{92} +4.37864e6 q^{93} -2.07800e7 q^{94} -1.83217e7 q^{95} +3.77692e6 q^{96} -1.59925e6 q^{97} +3.84141e7 q^{98} -2.01466e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	18.7189	1.65454	0.827268	−	0.561808i	$-0.189894\pi$
$2$	18.7189	1.65454	0.827268	+	0.561808i	$0.189894\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	222.398	1.73749
$5$	443.832	1.58790	0.793951	−	0.607982i	$-0.208021\pi$
$5$	443.832	1.58790	0.793951	+	0.607982i	$0.208021\pi$
$6$	−505.411	−0.955246
$7$	−1695.79	−1.86865	−0.934326	−	0.356419i	$-0.883997\pi$
$7$	−1695.79	−1.86865	−0.934326	+	0.356419i	$0.883997\pi$
$8$	1767.03	1.22020
$9$	729.000	0.333333
$10$	8308.06	2.62724
$11$	−2763.59	−0.626036	−0.313018	−	0.949747i	$-0.601340\pi$
$11$	−2763.59	−0.626036	−0.313018	+	0.949747i	$0.601340\pi$
$12$	−6004.75	−1.00314
$13$	−10956.4	−1.38314	−0.691568	−	0.722311i	$-0.743080\pi$
$13$	−10956.4	−1.38314	−0.691568	+	0.722311i	$0.743080\pi$
$14$	−31743.3	−3.09175
$15$	−11983.5	−0.916775
$16$	4610.01	0.281373
$17$	−17168.5	−0.847541	−0.423771	−	0.905770i	$-0.639294\pi$
$17$	−17168.5	−0.847541	−0.423771	+	0.905770i	$0.639294\pi$
$18$	13646.1	0.551512
$19$	−41280.8	−1.38074	−0.690368	−	0.723459i	$-0.742551\pi$
$19$	−41280.8	−1.38074	−0.690368	+	0.723459i	$0.742551\pi$
$20$	98707.5	2.75896
$21$	45786.3	1.07887
$22$	−51731.4	−1.03580
$23$	82535.2	1.41446	0.707232	−	0.706982i	$-0.249944\pi$
$23$	82535.2	1.41446	0.707232	+	0.706982i	$0.249944\pi$
$24$	−47709.9	−0.704481
$25$	118862.	1.52143
$26$	−205092.	−2.28845
$27$	−19683.0	−0.192450
$28$	−377140.	−3.24676
$29$	103089.	0.784912	0.392456	−	0.919771i	$-0.371625\pi$
$29$	103089.	0.784912	0.392456	+	0.919771i	$0.371625\pi$
$30$	−224318.	−1.51684
$31$	−162172.	−0.977709	−0.488854	−	0.872365i	$-0.662585\pi$
$31$	−162172.	−0.977709	−0.488854	+	0.872365i	$0.662585\pi$
$32$	−139886.	−0.754656
$33$	74616.9	0.361442
$34$	−321376.	−1.40229
$35$	−752645.	−2.96724
$36$	162128.	0.579162
$37$	446923.	1.45053	0.725265	−	0.688470i	$-0.241717\pi$
$37$	446923.	1.45053	0.725265	+	0.688470i	$0.241717\pi$
$38$	−772732.	−2.28448
$39$	295822.	0.798554
$40$	784266.	1.93755
$41$	371488.	0.841785	0.420893	−	0.907111i	$-0.361717\pi$
$41$	371488.	0.841785	0.420893	+	0.907111i	$0.361717\pi$
$42$	857070.	1.78502
$43$	−790401.	−1.51603	−0.758015	−	0.652237i	$-0.773831\pi$
$43$	−790401.	−1.51603	−0.758015	+	0.652237i	$0.773831\pi$
$44$	−614617.	−1.08773
$45$	323554.	0.529301
$46$	1.54497e6	2.34028
$47$	−1.11011e6	−1.55964	−0.779818	−	0.626007i	$-0.784688\pi$
$47$	−1.11011e6	−1.55964	−0.779818	+	0.626007i	$0.784688\pi$
$48$	−124470.	−0.162451
$49$	2.05215e6	2.49186
$50$	2.22497e6	2.51726
$51$	463549.	0.489328
$52$	−2.43668e6	−2.40318
$53$	−771292.	−0.711628	−0.355814	−	0.934557i	$-0.615796\pi$
$53$	−771292.	−0.711628	−0.355814	+	0.934557i	$0.615796\pi$
$54$	−368445.	−0.318415
$55$	−1.22657e6	−0.994083
$56$	−2.99652e6	−2.28012
$57$	1.11458e6	0.797168
$58$	1.92972e6	1.29867
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−2.66510e6	−1.59288
$61$	−211813.	−0.119481	−0.0597403	−	0.998214i	$-0.519027\pi$
$61$	−211813.	−0.119481	−0.0597403	+	0.998214i	$0.519027\pi$
$62$	−3.03568e6	−1.61765
$63$	−1.23623e6	−0.622884
$64$	−3.20860e6	−1.52998
$65$	−4.86279e6	−2.19628
$66$	1.39675e6	0.598018
$67$	1.48116e6	0.601644	0.300822	−	0.953680i	$-0.402739\pi$
$67$	1.48116e6	0.601644	0.300822	+	0.953680i	$0.402739\pi$
$68$	−3.81824e6	−1.47259
$69$	−2.22845e6	−0.816641
$70$	−1.40887e7	−4.90940
$71$	3.40336e6	1.12851	0.564253	−	0.825602i	$-0.309164\pi$
$71$	3.40336e6	1.12851	0.564253	+	0.825602i	$0.309164\pi$
$72$	1.28817e6	0.406732
$73$	−3.93442e6	−1.18372	−0.591862	−	0.806040i	$-0.701607\pi$
$73$	−3.93442e6	−1.18372	−0.591862	+	0.806040i	$0.701607\pi$
$74$	8.36592e6	2.39995
$75$	−3.20927e6	−0.878399
$76$	−9.18078e6	−2.39901
$77$	4.68646e6	1.16984
$78$	5.53747e6	1.32124
$79$	4.55385e6	1.03916	0.519582	−	0.854421i	$-0.326088\pi$
$79$	4.55385e6	1.03916	0.519582	+	0.854421i	$0.326088\pi$
$80$	2.04607e6	0.446793
$81$	531441.	0.111111
$82$	6.95386e6	1.39276
$83$	−4.51953e6	−0.867602	−0.433801	−	0.901009i	$-0.642828\pi$
$83$	−4.51953e6	−0.867602	−0.433801	+	0.901009i	$0.642828\pi$
$84$	1.01828e7	1.87452
$85$	−7.61992e6	−1.34581
$86$	−1.47955e7	−2.50833
$87$	−2.78342e6	−0.453169
$88$	−4.88336e6	−0.763887
$89$	−4.21568e6	−0.633874	−0.316937	−	0.948447i	$-0.602654\pi$
$89$	−4.21568e6	−0.633874	−0.316937	+	0.948447i	$0.602654\pi$
$90$	6.05658e6	0.875746
$91$	1.85797e7	2.58460
$92$	1.83557e7	2.45761
$93$	4.37864e6	0.564480
$94$	−2.07800e7	−2.58047
$95$	−1.83217e7	−2.19247
$96$	3.77692e6	0.435701
$97$	−1.59925e6	−0.177916	−0.0889579	−	0.996035i	$-0.528354\pi$
$97$	−1.59925e6	−0.177916	−0.0889579	+	0.996035i	$0.528354\pi$
$98$	3.84141e7	4.12287
$99$	−2.01466e6	−0.208679
$100$	2.64347e7	2.64347
$101$	−1.38691e6	−0.133944	−0.0669719	−	0.997755i	$-0.521334\pi$
$101$	−1.38691e6	−0.133944	−0.0669719	+	0.997755i	$0.521334\pi$
$102$	8.67714e6	0.809611
$103$	−1.11202e6	−0.100273	−0.0501363	−	0.998742i	$-0.515966\pi$
$103$	−1.11202e6	−0.100273	−0.0501363	+	0.998742i	$0.515966\pi$
$104$	−1.93603e7	−1.68770
$105$	2.03214e7	1.71313
$106$	−1.44378e7	−1.17741
$107$	−2.40434e7	−1.89737	−0.948685	−	0.316223i	$-0.897585\pi$
$107$	−2.40434e7	−1.89737	−0.948685	+	0.316223i	$0.897585\pi$
$108$	−4.37747e6	−0.334379
$109$	1.06281e7	0.786070	0.393035	−	0.919524i	$-0.371425\pi$
$109$	1.06281e7	0.786070	0.393035	+	0.919524i	$0.371425\pi$
$110$	−2.29601e7	−1.64475
$111$	−1.20669e7	−0.837463
$112$	−7.81761e6	−0.525788
$113$	1.82364e7	1.18895	0.594477	−	0.804112i	$-0.297359\pi$
$113$	1.82364e7	1.18895	0.594477	+	0.804112i	$0.297359\pi$
$114$	2.08638e7	1.31894
$115$	3.66318e7	2.24603
$116$	2.29269e7	1.36377
$117$	−7.98719e6	−0.461045
$118$	3.84447e6	0.215402
$119$	2.91141e7	1.58376
$120$	−2.11752e7	−1.11865
$121$	−1.18497e7	−0.608079
$122$	−3.96491e6	−0.197685
$123$	−1.00302e7	−0.486005
$124$	−3.60667e7	−1.69876
$125$	1.80803e7	0.827982
$126$	−2.31409e7	−1.03058
$127$	1.28606e7	0.557119	0.278559	−	0.960419i	$-0.410143\pi$
$127$	1.28606e7	0.557119	0.278559	+	0.960419i	$0.410143\pi$
$128$	−4.21561e7	−1.77675
$129$	2.13408e7	0.875281
$130$	−9.10262e7	−3.63383
$131$	−1.03027e7	−0.400405	−0.200203	−	0.979754i	$-0.564160\pi$
$131$	−1.03027e7	−0.400405	−0.200203	+	0.979754i	$0.564160\pi$
$132$	1.65947e7	0.628000
$133$	7.00035e7	2.58011
$134$	2.77257e7	0.995442
$135$	−8.73595e6	−0.305592
$136$	−3.03373e7	−1.03417
$137$	−4.73008e7	−1.57161	−0.785807	−	0.618471i	$-0.787752\pi$
$137$	−4.73008e7	−1.57161	−0.785807	+	0.618471i	$0.787752\pi$
$138$	−4.17142e7	−1.35116
$139$	5.76637e7	1.82117	0.910585	−	0.413322i	$-0.135631\pi$
$139$	5.76637e7	1.82117	0.910585	+	0.413322i	$0.135631\pi$
$140$	−1.67387e8	−5.15553
$141$	2.99729e7	0.900456
$142$	6.37073e7	1.86715
$143$	3.02789e7	0.865893
$144$	3.36070e6	0.0937910
$145$	4.57544e7	1.24636
$146$	−7.36480e7	−1.95851
$147$	−5.54082e7	−1.43868
$148$	9.93949e7	2.52028
$149$	9.45541e6	0.234168	0.117084	−	0.993122i	$-0.462645\pi$
$149$	9.45541e6	0.234168	0.117084	+	0.993122i	$0.462645\pi$
$150$	−6.00741e7	−1.45334
$151$	−1.54572e7	−0.365353	−0.182677	−	0.983173i	$-0.558476\pi$
$151$	−1.54572e7	−0.365353	−0.182677	+	0.983173i	$0.558476\pi$
$152$	−7.29446e7	−1.68477
$153$	−1.25158e7	−0.282514
$154$	8.77255e7	1.93555
$155$	−7.19770e7	−1.55251
$156$	6.57903e7	1.38748
$157$	3.62627e7	0.747845	0.373922	−	0.927460i	$-0.378013\pi$
$157$	3.62627e7	0.747845	0.373922	+	0.927460i	$0.378013\pi$
$158$	8.52432e7	1.71933
$159$	2.08249e7	0.410859
$160$	−6.20858e7	−1.19832
$161$	−1.39962e8	−2.64314
$162$	9.94801e6	0.183837
$163$	5.27645e7	0.954301	0.477151	−	0.878822i	$-0.341670\pi$
$163$	5.27645e7	0.954301	0.477151	+	0.878822i	$0.341670\pi$
$164$	8.26183e7	1.46259
$165$	3.31174e7	0.573934
$166$	−8.46008e7	−1.43548
$167$	−7.36648e6	−0.122392	−0.0611959	−	0.998126i	$-0.519491\pi$
$167$	−7.36648e6	−0.122392	−0.0611959	+	0.998126i	$0.519491\pi$
$168$	8.09059e7	1.31643
$169$	5.72936e7	0.913066
$170$	−1.42637e8	−2.22669
$171$	−3.00937e7	−0.460245
$172$	−1.75784e8	−2.63408
$173$	7.87542e7	1.15641	0.578206	−	0.815891i	$-0.303753\pi$
$173$	7.87542e7	1.15641	0.578206	+	0.815891i	$0.303753\pi$
$174$	−5.21025e7	−0.749785
$175$	−2.01564e8	−2.84303
$176$	−1.27402e7	−0.176149
$177$	−5.54523e6	−0.0751646
$178$	−7.89130e7	−1.04877
$179$	2.13413e7	0.278121	0.139061	−	0.990284i	$-0.455592\pi$
$179$	2.13413e7	0.278121	0.139061	+	0.990284i	$0.455592\pi$
$180$	7.19577e7	0.919653
$181$	1.72243e7	0.215907	0.107953	−	0.994156i	$-0.465570\pi$
$181$	1.72243e7	0.215907	0.107953	+	0.994156i	$0.465570\pi$
$182$	3.47792e8	4.27631
$183$	5.71894e6	0.0689822
$184$	1.45843e8	1.72592
$185$	1.98359e8	2.30330
$186$	8.19634e7	0.933953
$187$	4.74466e7	0.530591
$188$	−2.46886e8	−2.70984
$189$	3.33782e7	0.359622
$190$	−3.42963e8	−3.62752
$191$	2.66378e7	0.276619	0.138309	−	0.990389i	$-0.455833\pi$
$191$	2.66378e7	0.276619	0.138309	+	0.990389i	$0.455833\pi$
$192$	8.66321e7	0.883333
$193$	−1.69658e8	−1.69873	−0.849365	−	0.527807i	$-0.823015\pi$
$193$	−1.69658e8	−1.69873	−0.849365	+	0.527807i	$0.823015\pi$
$194$	−2.99362e7	−0.294368
$195$	1.31295e8	1.26803
$196$	4.56396e8	4.32957
$197$	6.41433e7	0.597750	0.298875	−	0.954292i	$-0.403389\pi$
$197$	6.41433e7	0.597750	0.298875	+	0.954292i	$0.403389\pi$
$198$	−3.77122e7	−0.345266
$199$	−1.14787e8	−1.03254	−0.516269	−	0.856427i	$-0.672679\pi$
$199$	−1.14787e8	−1.03254	−0.516269	+	0.856427i	$0.672679\pi$
$200$	2.10033e8	1.85645
$201$	−3.99913e7	−0.347359
$202$	−2.59614e7	−0.221615
$203$	−1.74818e8	−1.46673
$204$	1.03093e8	0.850201
$205$	1.64878e8	1.33667
$206$	−2.08158e7	−0.165905
$207$	6.01682e7	0.471488
$208$	−5.05090e7	−0.389177
$209$	1.14083e8	0.864389
$210$	3.80395e8	2.83444
$211$	6.79484e6	0.0497956	0.0248978	−	0.999690i	$-0.492074\pi$
$211$	6.79484e6	0.0497956	0.0248978	+	0.999690i	$0.492074\pi$
$212$	−1.71534e8	−1.23644
$213$	−9.18908e7	−0.651543
$214$	−4.50066e8	−3.13926
$215$	−3.50805e8	−2.40731
$216$	−3.47805e7	−0.234827
$217$	2.75009e8	1.82700
$218$	1.98946e8	1.30058
$219$	1.06229e8	0.683423
$220$	−2.72787e8	−1.72721
$221$	1.88104e8	1.17227
$222$	−2.25880e8	−1.38561
$223$	2.78892e8	1.68410	0.842052	−	0.539396i	$-0.181347\pi$
$223$	2.78892e8	1.68410	0.842052	+	0.539396i	$0.181347\pi$
$224$	2.37217e8	1.41019
$225$	8.66503e7	0.507144
$226$	3.41367e8	1.96717
$227$	4.19942e7	0.238286	0.119143	−	0.992877i	$-0.461985\pi$
$227$	4.19942e7	0.238286	0.119143	+	0.992877i	$0.461985\pi$
$228$	2.47881e8	1.38507
$229$	−1.50761e7	−0.0829591	−0.0414795	−	0.999139i	$-0.513207\pi$
$229$	−1.50761e7	−0.0829591	−0.0414795	+	0.999139i	$0.513207\pi$
$230$	6.85707e8	3.71613
$231$	−1.26534e8	−0.675409
$232$	1.82163e8	0.957748
$233$	5.24867e7	0.271834	0.135917	−	0.990720i	$-0.456602\pi$
$233$	5.24867e7	0.271834	0.135917	+	0.990720i	$0.456602\pi$
$234$	−1.49512e8	−0.762816
$235$	−4.92702e8	−2.47655
$236$	4.56759e7	0.226201
$237$	−1.22954e8	−0.599962
$238$	5.44985e8	2.62039
$239$	1.82265e8	0.863597	0.431799	−	0.901970i	$-0.357879\pi$
$239$	1.82265e8	0.863597	0.431799	+	0.901970i	$0.357879\pi$
$240$	−5.52439e7	−0.257956
$241$	−5.51242e7	−0.253678	−0.126839	−	0.991923i	$-0.540483\pi$
$241$	−5.51242e7	−0.253678	−0.126839	+	0.991923i	$0.540483\pi$
$242$	−2.21815e8	−1.00609
$243$	−1.43489e7	−0.0641500
$244$	−4.71068e7	−0.207596
$245$	9.10812e8	3.95683
$246$	−1.87754e8	−0.804112
$247$	4.52288e8	1.90975
$248$	−2.86563e8	−1.19300
$249$	1.22027e8	0.500910
$250$	3.38444e8	1.36993
$251$	−3.50809e8	−1.40027	−0.700137	−	0.714009i	$-0.746878\pi$
$251$	−3.50809e8	−1.40027	−0.700137	+	0.714009i	$0.746878\pi$
$252$	−2.74935e8	−1.08225
$253$	−2.28093e8	−0.885505
$254$	2.40736e8	0.921772
$255$	2.05738e8	0.777005
$256$	−3.78416e8	−1.40971
$257$	−3.34359e8	−1.22871	−0.614353	−	0.789032i	$-0.710583\pi$
$257$	−3.34359e8	−1.22871	−0.614353	+	0.789032i	$0.710583\pi$
$258$	3.99477e8	1.44818
$259$	−7.57886e8	−2.71053
$260$	−1.08148e9	−3.81601
$261$	7.51522e7	0.261637
$262$	−1.92855e8	−0.662485
$263$	−3.99920e8	−1.35559	−0.677795	−	0.735251i	$-0.737064\pi$
$263$	−3.99920e8	−1.35559	−0.677795	+	0.735251i	$0.737064\pi$
$264$	1.31851e8	0.441030
$265$	−3.42324e8	−1.13000
$266$	1.31039e9	4.26889
$267$	1.13823e8	0.365967
$268$	3.29407e8	1.04535
$269$	1.62396e7	0.0508677	0.0254338	−	0.999677i	$-0.491903\pi$
$269$	1.62396e7	0.0508677	0.0254338	+	0.999677i	$0.491903\pi$
$270$	−1.63528e8	−0.505612
$271$	−4.27140e8	−1.30370	−0.651851	−	0.758347i	$-0.726007\pi$
$271$	−4.27140e8	−1.30370	−0.651851	+	0.758347i	$0.726007\pi$
$272$	−7.91470e7	−0.238475
$273$	−5.01652e8	−1.49222
$274$	−8.85419e8	−2.60029
$275$	−3.28485e8	−0.952470
$276$	−4.95604e8	−1.41890
$277$	−4.70787e8	−1.33090	−0.665449	−	0.746443i	$-0.731760\pi$
$277$	−4.70787e8	−1.33090	−0.665449	+	0.746443i	$0.731760\pi$
$278$	1.07940e9	3.01319
$279$	−1.18223e8	−0.325903
$280$	−1.32995e9	−3.62061
$281$	−3.44673e8	−0.926692	−0.463346	−	0.886177i	$-0.653351\pi$
$281$	−3.44673e8	−0.926692	−0.463346	+	0.886177i	$0.653351\pi$
$282$	5.61061e8	1.48984
$283$	3.05838e8	0.802120	0.401060	−	0.916052i	$-0.368642\pi$
$283$	3.05838e8	0.802120	0.401060	+	0.916052i	$0.368642\pi$
$284$	7.56902e8	1.96076
$285$	4.94687e8	1.26582
$286$	5.66789e8	1.43265
$287$	−6.29965e8	−1.57300
$288$	−1.01977e8	−0.251552
$289$	−1.15582e8	−0.281674
$290$	8.56473e8	2.06215
$291$	4.31797e7	0.102720
$292$	−8.75007e8	−2.05670
$293$	−8.35017e7	−0.193936	−0.0969680	−	0.995287i	$-0.530914\pi$
$293$	−8.35017e7	−0.193936	−0.0969680	+	0.995287i	$0.530914\pi$
$294$	−1.03718e9	−2.38034
$295$	9.11538e7	0.206727
$296$	7.89728e8	1.76993
$297$	5.43957e7	0.120481
$298$	1.76995e8	0.387440
$299$	−9.04286e8	−1.95640
$300$	−7.13736e8	−1.52621
$301$	1.34035e9	2.83293
$302$	−2.89343e8	−0.604489
$303$	3.74465e7	0.0773325
$304$	−1.90305e8	−0.388502
$305$	−9.40092e7	−0.189723
$306$	−2.34283e8	−0.467429
$307$	−7.34193e8	−1.44819	−0.724096	−	0.689699i	$-0.757743\pi$
$307$	−7.34193e8	−1.44819	−0.724096	+	0.689699i	$0.757743\pi$
$308$	1.04226e9	2.03259
$309$	3.00245e7	0.0578924
$310$	−1.34733e9	−2.56867
$311$	2.34224e7	0.0441540	0.0220770	−	0.999756i	$-0.492972\pi$
$311$	2.34224e7	0.0441540	0.0220770	+	0.999756i	$0.492972\pi$
$312$	5.22728e8	0.974394
$313$	4.22581e8	0.778941	0.389471	−	0.921039i	$-0.372658\pi$
$313$	4.22581e8	0.778941	0.389471	+	0.921039i	$0.372658\pi$
$314$	6.78799e8	1.23734
$315$	−5.48678e8	−0.989079
$316$	1.01277e9	1.80553
$317$	−1.79803e7	−0.0317023	−0.0158511	−	0.999874i	$-0.505046\pi$
$317$	−1.79803e7	−0.0317023	−0.0158511	+	0.999874i	$0.505046\pi$
$318$	3.89819e8	0.679780
$319$	−2.84897e8	−0.491383
$320$	−1.42408e9	−2.42945
$321$	6.49171e8	1.09545
$322$	−2.61994e9	−4.37317
$323$	7.08729e8	1.17023
$324$	1.18192e8	0.193054
$325$	−1.30229e9	−2.10435
$326$	9.87695e8	1.57892
$327$	−2.86958e8	−0.453838
$328$	6.56432e8	1.02714
$329$	1.88251e9	2.91442
$330$	6.19922e8	0.949594
$331$	−2.19639e8	−0.332899	−0.166449	−	0.986050i	$-0.553230\pi$
$331$	−2.19639e8	−0.332899	−0.166449	+	0.986050i	$0.553230\pi$
$332$	−1.00514e9	−1.50745
$333$	3.25807e8	0.483510
$334$	−1.37893e8	−0.202502
$335$	6.57386e8	0.955352
$336$	2.11075e8	0.303564
$337$	4.32909e7	0.0616158	0.0308079	−	0.999525i	$-0.490192\pi$
$337$	4.32909e7	0.0616158	0.0308079	+	0.999525i	$0.490192\pi$
$338$	1.07247e9	1.51070
$339$	−4.92384e8	−0.686443
$340$	−1.69466e9	−2.33833
$341$	4.48176e8	0.612080
$342$	−5.63322e8	−0.761492
$343$	−2.08346e9	−2.78777
$344$	−1.39667e9	−1.84986
$345$	−9.89058e8	−1.29675
$346$	1.47419e9	1.91332
$347$	2.07119e8	0.266114	0.133057	−	0.991108i	$-0.457521\pi$
$347$	2.07119e8	0.266114	0.133057	+	0.991108i	$0.457521\pi$
$348$	−6.19027e8	−0.787376
$349$	−1.35268e9	−1.70336	−0.851680	−	0.524062i	$-0.824416\pi$
$349$	−1.35268e9	−1.70336	−0.851680	+	0.524062i	$0.824416\pi$
$350$	−3.77307e9	−4.70389
$351$	2.15654e8	0.266185
$352$	3.86587e8	0.472442
$353$	−9.24750e8	−1.11896	−0.559478	−	0.828845i	$-0.688998\pi$
$353$	−9.24750e8	−1.11896	−0.559478	+	0.828845i	$0.688998\pi$
$354$	−1.03801e8	−0.124362
$355$	1.51052e9	1.79196
$356$	−9.37560e8	−1.10135
$357$	−7.86081e8	−0.914384
$358$	3.99486e8	0.460162
$359$	5.95223e8	0.678968	0.339484	−	0.940612i	$-0.389748\pi$
$359$	5.95223e8	0.678968	0.339484	+	0.940612i	$0.389748\pi$
$360$	5.71730e8	0.645851
$361$	8.10232e8	0.906430
$362$	3.22421e8	0.357226
$363$	3.19943e8	0.351075
$364$	4.13209e9	4.49071
$365$	−1.74622e9	−1.87964
$366$	1.07052e8	0.114133
$367$	5.60873e8	0.592289	0.296144	−	0.955143i	$-0.404299\pi$
$367$	5.60873e8	0.592289	0.296144	+	0.955143i	$0.404299\pi$
$368$	3.80488e8	0.397992
$369$	2.70815e8	0.280595
$370$	3.71306e9	3.81089
$371$	1.30795e9	1.32979
$372$	9.73802e8	0.980777
$373$	−6.85459e8	−0.683912	−0.341956	−	0.939716i	$-0.611089\pi$
$373$	−6.85459e8	−0.683912	−0.341956	+	0.939716i	$0.611089\pi$
$374$	8.88150e8	0.877881
$375$	−4.88168e8	−0.478036
$376$	−1.96160e9	−1.90306
$377$	−1.12949e9	−1.08564
$378$	6.24804e8	0.595008
$379$	−1.03369e9	−0.975337	−0.487668	−	0.873029i	$-0.662152\pi$
$379$	−1.03369e9	−0.975337	−0.487668	+	0.873029i	$0.662152\pi$
$380$	−4.07472e9	−3.80939
$381$	−3.47236e8	−0.321653
$382$	4.98632e8	0.457676
$383$	2.09792e9	1.90806	0.954032	−	0.299705i	$-0.0968882\pi$
$383$	2.09792e9	1.90806	0.954032	+	0.299705i	$0.0968882\pi$
$384$	1.13821e9	1.02580
$385$	2.08000e9	1.85760
$386$	−3.17582e9	−2.81061
$387$	−5.76202e8	−0.505343
$388$	−3.55670e8	−0.309126
$389$	−6.60600e7	−0.0569003	−0.0284502	−	0.999595i	$-0.509057\pi$
$389$	−6.60600e7	−0.0569003	−0.0284502	+	0.999595i	$0.509057\pi$
$390$	2.45771e9	2.09799
$391$	−1.41700e9	−1.19882
$392$	3.62623e9	3.04056
$393$	2.78172e8	0.231174
$394$	1.20069e9	0.988998
$395$	2.02115e9	1.65009
$396$	−4.48056e8	−0.362576
$397$	1.83981e9	1.47573	0.737863	−	0.674951i	$-0.235835\pi$
$397$	1.83981e9	1.47573	0.737863	+	0.674951i	$0.235835\pi$
$398$	−2.14868e9	−1.70837
$399$	−1.89009e9	−1.48963
$400$	5.47955e8	0.428090
$401$	1.61549e9	1.25112	0.625560	−	0.780176i	$-0.284870\pi$
$401$	1.61549e9	1.25112	0.625560	+	0.780176i	$0.284870\pi$
$402$	−7.48594e8	−0.574718
$403$	1.77681e9	1.35230
$404$	−3.08446e8	−0.232726
$405$	2.35871e8	0.176434
$406$	−3.27240e9	−2.42675
$407$	−1.23511e9	−0.908083
$408$	8.19107e8	0.597077
$409$	−1.07871e9	−0.779605	−0.389802	−	0.920899i	$-0.627457\pi$
$409$	−1.07871e9	−0.779605	−0.389802	+	0.920899i	$0.627457\pi$
$410$	3.08634e9	2.21157
$411$	1.27712e9	0.907372
$412$	−2.47311e8	−0.174222
$413$	−3.48279e8	−0.243278
$414$	1.12628e9	0.780093
$415$	−2.00591e9	−1.37767
$416$	1.53264e9	1.04379
$417$	−1.55692e9	−1.05145
$418$	2.13551e9	1.43016
$419$	−8.10783e8	−0.538463	−0.269231	−	0.963076i	$-0.586770\pi$
$419$	−8.10783e8	−0.538463	−0.269231	+	0.963076i	$0.586770\pi$
$420$	4.51945e9	2.97655
$421$	−1.13878e9	−0.743792	−0.371896	−	0.928274i	$-0.621292\pi$
$421$	−1.13878e9	−0.743792	−0.371896	+	0.928274i	$0.621292\pi$
$422$	1.27192e8	0.0823886
$423$	−8.09269e8	−0.519878
$424$	−1.36290e9	−0.868327
$425$	−2.04068e9	−1.28948
$426$	−1.72010e9	−1.07800
$427$	3.59189e8	0.223268
$428$	−5.34720e9	−3.29665
$429$	−8.17530e8	−0.499923
$430$	−6.56670e9	−3.98297
$431$	−2.09869e9	−1.26263	−0.631317	−	0.775525i	$-0.717485\pi$
$431$	−2.09869e9	−1.26263	−0.631317	+	0.775525i	$0.717485\pi$
$432$	−9.07389e7	−0.0541503
$433$	1.85715e9	1.09936	0.549679	−	0.835376i	$-0.314750\pi$
$433$	1.85715e9	1.09936	0.549679	+	0.835376i	$0.314750\pi$
$434$	5.14787e9	3.02283
$435$	−1.23537e9	−0.719588
$436$	2.36366e9	1.36579
$437$	−3.40712e9	−1.95300
$438$	1.98850e9	1.13075
$439$	6.58145e8	0.371275	0.185638	−	0.982618i	$-0.440565\pi$
$439$	6.58145e8	0.371275	0.185638	+	0.982618i	$0.440565\pi$
$440$	−2.16739e9	−1.21298
$441$	1.49602e9	0.830620
$442$	3.52111e9	1.93955
$443$	−2.56332e8	−0.140084	−0.0700422	−	0.997544i	$-0.522313\pi$
$443$	−2.56332e8	−0.140084	−0.0700422	+	0.997544i	$0.522313\pi$
$444$	−2.68366e9	−1.45508
$445$	−1.87105e9	−1.00653
$446$	5.22056e9	2.78641
$447$	−2.55296e8	−0.135197
$448$	5.44110e9	2.85900
$449$	5.70165e8	0.297261	0.148631	−	0.988893i	$-0.452513\pi$
$449$	5.70165e8	0.297261	0.148631	+	0.988893i	$0.452513\pi$
$450$	1.62200e9	0.839087
$451$	−1.02664e9	−0.526987
$452$	4.05575e9	2.06579
$453$	4.17346e8	0.210937
$454$	7.86086e8	0.394253
$455$	8.24626e9	4.10409
$456$	1.96950e9	0.972702
$457$	−1.42553e8	−0.0698669	−0.0349334	−	0.999390i	$-0.511122\pi$
$457$	−1.42553e8	−0.0698669	−0.0349334	+	0.999390i	$0.511122\pi$
$458$	−2.82208e8	−0.137259
$459$	3.37927e8	0.163109
$460$	8.14684e9	3.90245
$461$	2.44186e9	1.16083	0.580414	−	0.814321i	$-0.302891\pi$
$461$	2.44186e9	1.16083	0.580414	+	0.814321i	$0.302891\pi$
$462$	−2.36859e9	−1.11749
$463$	−2.57981e9	−1.20796	−0.603982	−	0.796998i	$-0.706420\pi$
$463$	−2.57981e9	−1.20796	−0.603982	+	0.796998i	$0.706420\pi$
$464$	4.75244e8	0.220853
$465$	1.94338e9	0.896339
$466$	9.82495e8	0.449759
$467$	−1.03284e9	−0.469272	−0.234636	−	0.972083i	$-0.575390\pi$
$467$	−1.03284e9	−0.469272	−0.234636	+	0.972083i	$0.575390\pi$
$468$	−1.77634e9	−0.801060
$469$	−2.51173e9	−1.12426
$470$	−9.22285e9	−4.09753
$471$	−9.79093e8	−0.431768
$472$	3.62912e8	0.158856
$473$	2.18434e9	0.949089
$474$	−2.30157e9	−0.992657
$475$	−4.90671e9	−2.10069
$476$	6.47493e9	2.75176
$477$	−5.62272e8	−0.237209
$478$	3.41181e9	1.42885
$479$	−2.54815e9	−1.05938	−0.529688	−	0.848193i	$-0.677691\pi$
$479$	−2.54815e9	−1.05938	−0.529688	+	0.848193i	$0.677691\pi$
$480$	1.67632e9	0.691850
$481$	−4.89665e9	−2.00628
$482$	−1.03187e9	−0.419719
$483$	3.77898e9	1.52602
$484$	−2.63536e9	−1.05653
$485$	−7.09797e8	−0.282513
$486$	−2.68596e8	−0.106138
$487$	−1.38050e9	−0.541608	−0.270804	−	0.962635i	$-0.587289\pi$
$487$	−1.38050e9	−0.541608	−0.270804	+	0.962635i	$0.587289\pi$
$488$	−3.74280e8	−0.145790
$489$	−1.42464e9	−0.550966
$490$	1.70494e10	6.54671
$491$	3.40284e9	1.29735	0.648674	−	0.761067i	$-0.275324\pi$
$491$	3.40284e9	1.29735	0.648674	+	0.761067i	$0.275324\pi$
$492$	−2.23069e9	−0.844427
$493$	−1.76989e9	−0.665246
$494$	8.46634e9	3.15974
$495$	−8.94169e8	−0.331361
$496$	−7.47614e8	−0.275101
$497$	−5.77138e9	−2.10879
$498$	2.28422e9	0.828773
$499$	−2.71570e9	−0.978430	−0.489215	−	0.872163i	$-0.662717\pi$
$499$	−2.71570e9	−0.978430	−0.489215	+	0.872163i	$0.662717\pi$
$500$	4.02103e9	1.43861
$501$	1.98895e8	0.0706630
$502$	−6.56677e9	−2.31680
$503$	−1.12999e9	−0.395899	−0.197950	−	0.980212i	$-0.563428\pi$
$503$	−1.12999e9	−0.395899	−0.197950	+	0.980212i	$0.563428\pi$
$504$	−2.18446e9	−0.760041
$505$	−6.15554e8	−0.212690
$506$	−4.26966e9	−1.46510
$507$	−1.54693e9	−0.527159
$508$	2.86017e9	0.967986
$509$	2.58498e9	0.868849	0.434425	−	0.900708i	$-0.356952\pi$
$509$	2.58498e9	0.868849	0.434425	+	0.900708i	$0.356952\pi$
$510$	3.85119e9	1.28558
$511$	6.67193e9	2.21197
$512$	−1.68757e9	−0.555670
$513$	8.12530e8	0.265723
$514$	−6.25885e9	−2.03294
$515$	−4.93550e8	−0.159223
$516$	4.74616e9	1.52079
$517$	3.06788e9	0.976387
$518$	−1.41868e10	−4.48467
$519$	−2.12636e9	−0.667655
$520$	−8.59272e9	−2.67990
$521$	−1.68355e9	−0.521546	−0.260773	−	0.965400i	$-0.583977\pi$
$521$	−1.68355e9	−0.521546	−0.260773	+	0.965400i	$0.583977\pi$
$522$	1.40677e9	0.432888
$523$	−2.47225e9	−0.755676	−0.377838	−	0.925872i	$-0.623332\pi$
$523$	−2.47225e9	−0.755676	−0.377838	+	0.925872i	$0.623332\pi$
$524$	−2.29129e9	−0.695699
$525$	5.44224e9	1.64142
$526$	−7.48608e9	−2.24287
$527$	2.78424e9	0.828648
$528$	3.43985e8	0.101700
$529$	3.40723e9	1.00071
$530$	−6.40794e9	−1.86962
$531$	1.49721e8	0.0433963
$532$	1.55687e10	4.48291
$533$	−4.07016e9	−1.16430
$534$	2.13065e9	0.605505
$535$	−1.06712e10	−3.01284
$536$	2.61726e9	0.734125
$537$	−5.76214e8	−0.160573
$538$	3.03988e8	0.0841624
$539$	−5.67131e9	−1.55999
$540$	−1.94286e9	−0.530962
$541$	−3.60201e9	−0.978034	−0.489017	−	0.872274i	$-0.662644\pi$
$541$	−3.60201e9	−0.978034	−0.489017	+	0.872274i	$0.662644\pi$
$542$	−7.99561e9	−2.15702
$543$	−4.65056e8	−0.124654
$544$	2.40163e9	0.639602
$545$	4.71707e9	1.24820
$546$	−9.39038e9	−2.46893
$547$	1.24077e9	0.324141	0.162071	−	0.986779i	$-0.448183\pi$
$547$	1.24077e9	0.324141	0.162071	+	0.986779i	$0.448183\pi$
$548$	−1.05196e10	−2.73066
$549$	−1.54411e8	−0.0398269
$550$	−6.14889e9	−1.57590
$551$	−4.25561e9	−1.08376
$552$	−3.93775e9	−0.996463
$553$	−7.72237e9	−1.94184
$554$	−8.81262e9	−2.20202
$555$	−5.35568e9	−1.32981
$556$	1.28243e10	3.16426
$557$	1.77982e9	0.436399	0.218200	−	0.975904i	$-0.429982\pi$
$557$	1.77982e9	0.436399	0.218200	+	0.975904i	$0.429982\pi$
$558$	−2.21301e9	−0.539218
$559$	8.65993e9	2.09688
$560$	−3.46970e9	−0.834900
$561$	−1.28106e9	−0.306337
$562$	−6.45192e9	−1.53325
$563$	−6.60782e9	−1.56056	−0.780278	−	0.625433i	$-0.784922\pi$
$563$	−6.60782e9	−1.56056	−0.780278	+	0.625433i	$0.784922\pi$
$564$	6.66593e9	1.56453
$565$	8.09391e9	1.88794
$566$	5.72496e9	1.32714
$567$	−9.01211e8	−0.207628
$568$	6.01386e9	1.37700
$569$	4.59260e9	1.04512	0.522560	−	0.852603i	$-0.324977\pi$
$569$	4.59260e9	1.04512	0.522560	+	0.852603i	$0.324977\pi$
$570$	9.26001e9	2.09435
$571$	3.76881e9	0.847183	0.423592	−	0.905853i	$-0.360769\pi$
$571$	3.76881e9	0.847183	0.423592	+	0.905853i	$0.360769\pi$
$572$	6.73398e9	1.50448
$573$	−7.19221e8	−0.159706
$574$	−1.17923e10	−2.60259
$575$	9.81029e9	2.15201
$576$	−2.33907e9	−0.509993
$577$	3.58209e9	0.776284	0.388142	−	0.921600i	$-0.373117\pi$
$577$	3.58209e9	0.776284	0.388142	+	0.921600i	$0.373117\pi$
$578$	−2.16357e9	−0.466040
$579$	4.58077e9	0.980762
$580$	1.01757e10	2.16554
$581$	7.66417e9	1.62125
$582$	8.08277e8	0.169953
$583$	2.13153e9	0.445504
$584$	−6.95225e9	−1.44438
$585$	−3.54497e9	−0.732095
$586$	−1.56306e9	−0.320874
$587$	3.85835e8	0.0787350	0.0393675	−	0.999225i	$-0.487466\pi$
$587$	3.85835e8	0.0787350	0.0393675	+	0.999225i	$0.487466\pi$
$588$	−1.23227e10	−2.49968
$589$	6.69458e9	1.34996
$590$	1.70630e9	0.342037
$591$	−1.73187e9	−0.345111
$592$	2.06032e9	0.408140
$593$	−1.84467e9	−0.363267	−0.181634	−	0.983366i	$-0.558139\pi$
$593$	−1.84467e9	−0.363267	−0.181634	+	0.983366i	$0.558139\pi$
$594$	1.01823e9	0.199339
$595$	1.29218e10	2.51485
$596$	2.10287e9	0.406864
$597$	3.09924e9	0.596136
$598$	−1.69273e10	−3.23693
$599$	4.66546e9	0.886953	0.443476	−	0.896286i	$-0.353745\pi$
$599$	4.66546e9	0.886953	0.443476	+	0.896286i	$0.353745\pi$
$600$	−5.67089e9	−1.07182
$601$	2.24604e9	0.422043	0.211021	−	0.977481i	$-0.432321\pi$
$601$	2.24604e9	0.422043	0.211021	+	0.977481i	$0.432321\pi$
$602$	2.50900e10	4.68719
$603$	1.07976e9	0.200548
$604$	−3.43766e9	−0.634796
$605$	−5.25930e9	−0.965570
$606$	7.00958e8	0.127949
$607$	9.43363e9	1.71206	0.856029	−	0.516928i	$-0.172925\pi$
$607$	9.43363e9	1.71206	0.856029	+	0.516928i	$0.172925\pi$
$608$	5.77460e9	1.04198
$609$	4.72008e9	0.846816
$610$	−1.75975e9	−0.313904
$611$	1.21628e10	2.15719
$612$	−2.78350e9	−0.490864
$613$	2.56081e9	0.449020	0.224510	−	0.974472i	$-0.427922\pi$
$613$	2.56081e9	0.449020	0.224510	+	0.974472i	$0.427922\pi$
$614$	−1.37433e10	−2.39608
$615$	−4.45171e9	−0.771728
$616$	8.28114e9	1.42744
$617$	2.42876e9	0.416281	0.208141	−	0.978099i	$-0.433259\pi$
$617$	2.42876e9	0.416281	0.208141	+	0.978099i	$0.433259\pi$
$618$	5.62027e8	0.0957850
$619$	6.53273e9	1.10708	0.553538	−	0.832824i	$-0.313277\pi$
$619$	6.53273e9	1.10708	0.553538	+	0.832824i	$0.313277\pi$
$620$	−1.60076e10	−2.69746
$621$	−1.62454e9	−0.272214
$622$	4.38443e8	0.0730544
$623$	7.14890e9	1.18449
$624$	1.36374e9	0.224692
$625$	−1.26146e9	−0.206678
$626$	7.91026e9	1.28879
$627$	−3.08024e9	−0.499055
$628$	8.06476e9	1.29937
$629$	−7.67299e9	−1.22938
$630$	−1.02707e10	−1.63647
$631$	−2.21286e9	−0.350631	−0.175315	−	0.984512i	$-0.556095\pi$
$631$	−2.21286e9	−0.350631	−0.175315	+	0.984512i	$0.556095\pi$
$632$	8.04681e9	1.26799
$633$	−1.83461e8	−0.0287495
$634$	−3.36573e8	−0.0524525
$635$	5.70794e9	0.884649
$636$	4.63142e9	0.713861
$637$	−2.24842e10	−3.44658
$638$	−5.33296e9	−0.813011
$639$	2.48105e9	0.376169
$640$	−1.87102e10	−2.82130
$641$	5.20588e9	0.780713	0.390356	−	0.920664i	$-0.372352\pi$
$641$	5.20588e9	0.780713	0.390356	+	0.920664i	$0.372352\pi$
$642$	1.21518e10	1.81246
$643$	−1.18888e10	−1.76359	−0.881796	−	0.471631i	$-0.843665\pi$
$643$	−1.18888e10	−1.76359	−0.881796	+	0.471631i	$0.843665\pi$
$644$	−3.11274e10	−4.59242
$645$	9.47174e9	1.38986
$646$	1.32666e10	1.93619
$647$	−6.02914e9	−0.875166	−0.437583	−	0.899178i	$-0.644165\pi$
$647$	−6.02914e9	−0.875166	−0.437583	+	0.899178i	$0.644165\pi$
$648$	9.39075e8	0.135577
$649$	−5.67583e8	−0.0815029
$650$	−2.43776e10	−3.48172
$651$	−7.42524e9	−1.05482
$652$	1.17347e10	1.65809
$653$	2.77068e8	0.0389395	0.0194697	−	0.999810i	$-0.493802\pi$
$653$	2.77068e8	0.0389395	0.0194697	+	0.999810i	$0.493802\pi$
$654$	−5.37154e9	−0.750890
$655$	−4.57265e9	−0.635804
$656$	1.71256e9	0.236856
$657$	−2.86819e9	−0.394574
$658$	3.52386e10	4.82200
$659$	−1.19887e10	−1.63182	−0.815909	−	0.578181i	$-0.803763\pi$
$659$	−1.19887e10	−1.63182	−0.815909	+	0.578181i	$0.803763\pi$
$660$	7.36525e9	0.997203
$661$	3.39523e9	0.457261	0.228631	−	0.973513i	$-0.426575\pi$
$661$	3.39523e9	0.457261	0.228631	+	0.973513i	$0.426575\pi$
$662$	−4.11142e9	−0.550793
$663$	−5.07882e9	−0.676808
$664$	−7.98617e9	−1.05865
$665$	3.10698e10	4.09697
$666$	6.09875e9	0.799984
$667$	8.50851e9	1.11023
$668$	−1.63829e9	−0.212654
$669$	−7.53008e9	−0.972318
$670$	1.23056e10	1.58066
$671$	5.85363e8	0.0747991
$672$	−6.40485e9	−0.814173
$673$	9.44306e9	1.19415	0.597077	−	0.802184i	$-0.296329\pi$
$673$	9.44306e9	1.19415	0.597077	+	0.802184i	$0.296329\pi$
$674$	8.10359e8	0.101945
$675$	−2.33956e9	−0.292800
$676$	1.27420e10	1.58644
$677$	2.31127e9	0.286279	0.143140	−	0.989703i	$-0.454280\pi$
$677$	2.31127e9	0.286279	0.143140	+	0.989703i	$0.454280\pi$
$678$	−9.21690e9	−1.13574
$679$	2.71199e9	0.332463
$680$	−1.34647e10	−1.64216
$681$	−1.13384e9	−0.137575
$682$	8.38938e9	1.01271
$683$	−1.31138e10	−1.57491	−0.787454	−	0.616374i	$-0.788601\pi$
$683$	−1.31138e10	−1.57491	−0.787454	+	0.616374i	$0.788601\pi$
$684$	−6.69279e9	−0.799670
$685$	−2.09936e10	−2.49557
$686$	−3.90002e10	−4.61246
$687$	4.07054e8	0.0478964
$688$	−3.64376e9	−0.426570
$689$	8.45056e9	0.984279
$690$	−1.85141e10	−2.14551
$691$	8.33408e9	0.960913	0.480457	−	0.877018i	$-0.340471\pi$
$691$	8.33408e9	0.960913	0.480457	+	0.877018i	$0.340471\pi$
$692$	1.75148e10	2.00925
$693$	3.41643e9	0.389948
$694$	3.87705e9	0.440295
$695$	2.55930e10	2.89184
$696$	−4.91839e9	−0.552956
$697$	−6.37789e9	−0.713448
$698$	−2.53207e10	−2.81827
$699$	−1.41714e9	−0.156943
$700$	−4.48276e10	−4.93972
$701$	−2.09612e9	−0.229829	−0.114914	−	0.993375i	$-0.536659\pi$
$701$	−2.09612e9	−0.229829	−0.114914	+	0.993375i	$0.536659\pi$
$702$	4.03682e9	0.440412
$703$	−1.84493e10	−2.00280
$704$	8.86724e9	0.957821
$705$	1.33029e10	1.42984
$706$	−1.73103e10	−1.85135
$707$	2.35190e9	0.250294
$708$	−1.23325e9	−0.130597
$709$	4.17621e8	0.0440069	0.0220035	−	0.999758i	$-0.492996\pi$
$709$	4.17621e8	0.0440069	0.0220035	+	0.999758i	$0.492996\pi$
$710$	2.82753e10	2.96486
$711$	3.31976e9	0.346388
$712$	−7.44925e9	−0.773451
$713$	−1.33849e10	−1.38293
$714$	−1.47146e10	−1.51288
$715$	1.34387e10	1.37495
$716$	4.74626e9	0.483232
$717$	−4.92116e9	−0.498598
$718$	1.11419e10	1.12338
$719$	−6.71535e9	−0.673779	−0.336890	−	0.941544i	$-0.609375\pi$
$719$	−6.71535e9	−0.673779	−0.336890	+	0.941544i	$0.609375\pi$
$720$	1.49159e9	0.148931
$721$	1.88575e9	0.187375
$722$	1.51667e10	1.49972
$723$	1.48835e9	0.146461
$724$	3.83066e9	0.375136
$725$	1.22534e10	1.19419
$726$	5.98899e9	0.580866
$727$	−8.79100e9	−0.848531	−0.424266	−	0.905538i	$-0.639468\pi$
$727$	−8.79100e9	−0.848531	−0.424266	+	0.905538i	$0.639468\pi$
$728$	3.28309e10	3.15372
$729$	3.87420e8	0.0370370
$730$	−3.26874e10	−3.10992
$731$	1.35700e10	1.28490
$732$	1.27188e9	0.119856
$733$	−1.87568e10	−1.75912	−0.879560	−	0.475787i	$-0.842163\pi$
$733$	−1.87568e10	−1.75912	−0.879560	+	0.475787i	$0.842163\pi$
$734$	1.04989e10	0.979963
$735$	−2.45919e10	−2.28448
$736$	−1.15455e10	−1.06743
$737$	−4.09331e9	−0.376651
$738$	5.06936e9	0.464254
$739$	−3.83197e9	−0.349274	−0.174637	−	0.984633i	$-0.555875\pi$
$739$	−3.83197e9	−0.349274	−0.174637	+	0.984633i	$0.555875\pi$
$740$	4.41146e10	4.00195
$741$	−1.22118e10	−1.10259
$742$	2.44834e10	2.20018
$743$	−1.74075e9	−0.155695	−0.0778475	−	0.996965i	$-0.524805\pi$
$743$	−1.74075e9	−0.155695	−0.0778475	+	0.996965i	$0.524805\pi$
$744$	7.73721e9	0.688777
$745$	4.19661e9	0.371836
$746$	−1.28311e10	−1.13156
$747$	−3.29474e9	−0.289201
$748$	1.05521e10	0.921895
$749$	4.07724e10	3.54552
$750$	−9.13799e9	−0.790927
$751$	1.87554e10	1.61580	0.807900	−	0.589320i	$-0.200604\pi$
$751$	1.87554e10	1.61580	0.807900	+	0.589320i	$0.200604\pi$
$752$	−5.11762e9	−0.438839
$753$	9.47185e9	0.808448
$754$	−2.11428e10	−1.79623
$755$	−6.86042e9	−0.580145
$756$	7.42325e9	0.624839
$757$	−1.48636e10	−1.24534	−0.622671	−	0.782483i	$-0.713953\pi$
$757$	−1.48636e10	−1.24534	−0.622671	+	0.782483i	$0.713953\pi$
$758$	−1.93496e10	−1.61373
$759$	6.15852e9	0.511246
$760$	−3.23751e10	−2.67525
$761$	2.55852e9	0.210446	0.105223	−	0.994449i	$-0.466444\pi$
$761$	2.55852e9	0.210446	0.105223	+	0.994449i	$0.466444\pi$
$762$	−6.49988e9	−0.532185
$763$	−1.80229e10	−1.46889
$764$	5.92421e9	0.480622
$765$	−5.55492e9	−0.448604
$766$	3.92708e10	3.15696
$767$	−2.25021e9	−0.180069
$768$	1.02172e10	0.813897
$769$	1.98268e10	1.57221	0.786104	−	0.618094i	$-0.212095\pi$
$769$	1.98268e10	1.57221	0.786104	+	0.618094i	$0.212095\pi$
$770$	3.89354e10	3.07346
$771$	9.02771e9	0.709393
$772$	−3.77317e10	−2.95152
$773$	−1.29417e10	−1.00777	−0.503886	−	0.863770i	$-0.668097\pi$
$773$	−1.29417e10	−1.00777	−0.503886	+	0.863770i	$0.668097\pi$
$774$	−1.07859e10	−0.836109
$775$	−1.92760e10	−1.48752
$776$	−2.82593e9	−0.217092
$777$	2.04629e10	1.56493
$778$	−1.23657e9	−0.0941436
$779$	−1.53353e10	−1.16228
$780$	2.91998e10	2.20318
$781$	−9.40549e9	−0.706485
$782$	−2.65248e10	−1.98348
$783$	−2.02911e9	−0.151056
$784$	9.46046e9	0.701142
$785$	1.60945e10	1.18750
$786$	5.20708e9	0.382486
$787$	9.60328e9	0.702276	0.351138	−	0.936324i	$-0.385795\pi$
$787$	9.60328e9	0.702276	0.351138	+	0.936324i	$0.385795\pi$
$788$	1.42654e10	1.03858
$789$	1.07978e10	0.782650
$790$	3.78337e10	2.73013
$791$	−3.09251e10	−2.22174
$792$	−3.55997e9	−0.254629
$793$	2.32070e9	0.165258
$794$	3.44392e10	2.44164
$795$	9.24274e9	0.652403
$796$	−2.55284e10	−1.79402
$797$	−9.10764e9	−0.637238	−0.318619	−	0.947883i	$-0.603219\pi$
$797$	−9.10764e9	−0.637238	−0.318619	+	0.947883i	$0.603219\pi$
$798$	−3.53805e10	−2.46464
$799$	1.90589e10	1.32185
$800$	−1.66271e10	−1.14816
$801$	−3.07323e9	−0.211291
$802$	3.02403e10	2.07002
$803$	1.08731e10	0.741053
$804$	−8.89399e9	−0.603532
$805$	−6.21197e10	−4.19705
$806$	3.32601e10	2.23744
$807$	−4.38469e8	−0.0293685
$808$	−2.45071e9	−0.163438
$809$	−1.78494e10	−1.18524	−0.592618	−	0.805484i	$-0.701906\pi$
$809$	−1.78494e10	−1.18524	−0.592618	+	0.805484i	$0.701906\pi$
$810$	4.41524e9	0.291915
$811$	6.82320e9	0.449174	0.224587	−	0.974454i	$-0.427897\pi$
$811$	6.82320e9	0.449174	0.224587	+	0.974454i	$0.427897\pi$
$812$	−3.88792e10	−2.54842
$813$	1.15328e10	0.752692
$814$	−2.31200e10	−1.50246
$815$	2.34186e10	1.51534
$816$	2.13697e9	0.137684
$817$	3.26284e10	2.09324
$818$	−2.01924e10	−1.28988
$819$	1.35446e10	0.861534
$820$	3.66686e10	2.32245
$821$	7.07763e9	0.446362	0.223181	−	0.974777i	$-0.428356\pi$
$821$	7.07763e9	0.446362	0.223181	+	0.974777i	$0.428356\pi$
$822$	2.39063e10	1.50128
$823$	−1.02079e10	−0.638317	−0.319159	−	0.947701i	$-0.603400\pi$
$823$	−1.02079e10	−0.638317	−0.319159	+	0.947701i	$0.603400\pi$
$824$	−1.96498e9	−0.122352
$825$	8.86910e9	0.549909
$826$	−6.51941e9	−0.402512
$827$	2.59808e9	0.159729	0.0798644	−	0.996806i	$-0.474551\pi$
$827$	2.59808e9	0.159729	0.0798644	+	0.996806i	$0.474551\pi$
$828$	1.33813e10	0.819204
$829$	2.93686e9	0.179037	0.0895184	−	0.995985i	$-0.471467\pi$
$829$	2.93686e9	0.179037	0.0895184	+	0.995985i	$0.471467\pi$
$830$	−3.75486e10	−2.27940
$831$	1.27112e10	0.768395
$832$	3.51546e10	2.11617
$833$	−3.52324e10	−2.11195
$834$	−2.91439e10	−1.73967
$835$	−3.26948e9	−0.194346
$836$	2.53719e10	1.50187
$837$	3.19203e9	0.188160
$838$	−1.51770e10	−0.890905
$839$	1.16450e10	0.680728	0.340364	−	0.940294i	$-0.389450\pi$
$839$	1.16450e10	0.680728	0.340364	+	0.940294i	$0.389450\pi$
$840$	3.59086e10	2.09036
$841$	−6.62244e9	−0.383912
$842$	−2.13167e10	−1.23063
$843$	9.30618e9	0.535026
$844$	1.51116e9	0.0865192
$845$	2.54287e10	1.44986
$846$	−1.51487e10	−0.860157
$847$	2.00947e10	1.13629
$848$	−3.55567e9	−0.200233
$849$	−8.25763e9	−0.463104
$850$	−3.81993e10	−2.13348
$851$	3.68869e10	2.05172
$852$	−2.04363e10	−1.13205
$853$	1.63671e10	0.902922	0.451461	−	0.892291i	$-0.350903\pi$
$853$	1.63671e10	0.902922	0.451461	+	0.892291i	$0.350903\pi$
$854$	6.72364e9	0.369404
$855$	−1.33565e10	−0.730824
$856$	−4.24854e10	−2.31517
$857$	−1.66058e10	−0.901210	−0.450605	−	0.892724i	$-0.648792\pi$
$857$	−1.66058e10	−0.901210	−0.450605	+	0.892724i	$0.648792\pi$
$858$	−1.53033e10	−0.827141
$859$	−1.88831e10	−1.01647	−0.508237	−	0.861217i	$-0.669703\pi$
$859$	−1.88831e10	−1.01647	−0.508237	+	0.861217i	$0.669703\pi$
$860$	−7.80185e10	−4.18266
$861$	1.70091e10	0.908174
$862$	−3.92852e10	−2.08907
$863$	2.09759e10	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$863$	2.09759e10	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$864$	2.75337e9	0.145234
$865$	3.49536e10	1.83627
$866$	3.47639e10	1.81893
$867$	3.12071e9	0.162625
$868$	6.11615e10	3.17438
$869$	−1.25850e10	−0.650554
$870$	−2.31248e10	−1.19058
$871$	−1.62281e10	−0.832156
$872$	1.87801e10	0.959161
$873$	−1.16585e9	−0.0593053
$874$	−6.37776e10	−3.23131
$875$	−3.06604e10	−1.54721
$876$	2.36252e10	1.18744
$877$	3.68896e10	1.84674	0.923370	−	0.383911i	$-0.125423\pi$
$877$	3.68896e10	1.84674	0.923370	+	0.383911i	$0.125423\pi$
$878$	1.23198e10	0.614288
$879$	2.25455e9	0.111969
$880$	−5.65450e9	−0.279708
$881$	−3.57144e10	−1.75966	−0.879829	−	0.475291i	$-0.842343\pi$
$881$	−3.57144e10	−1.75966	−0.879829	+	0.475291i	$0.842343\pi$
$882$	2.80039e10	1.37429
$883$	−3.90952e9	−0.191100	−0.0955501	−	0.995425i	$-0.530461\pi$
$883$	−3.90952e9	−0.191100	−0.0955501	+	0.995425i	$0.530461\pi$
$884$	4.18341e10	2.03679
$885$	−2.46115e9	−0.119354
$886$	−4.79826e9	−0.231775
$887$	−1.25223e10	−0.602494	−0.301247	−	0.953546i	$-0.597403\pi$
$887$	−1.25223e10	−0.602494	−0.301247	+	0.953546i	$0.597403\pi$
$888$	−2.13227e10	−1.02187
$889$	−2.18088e10	−1.04106
$890$	−3.50241e10	−1.66534
$891$	−1.46868e9	−0.0695595
$892$	6.20251e10	2.92611
$893$	4.58262e10	2.15344
$894$	−4.77887e9	−0.223688
$895$	9.47193e9	0.441630
$896$	7.14878e10	3.32012
$897$	2.44157e10	1.12953
$898$	1.06729e10	0.491829
$899$	−1.67182e10	−0.767416
$900$	1.92709e10	0.881156
$901$	1.32419e10	0.603134
$902$	−1.92176e10	−0.871919
$903$	−3.61895e10	−1.63559
$904$	3.22244e10	1.45076
$905$	7.64470e9	0.342839
$906$	7.81226e9	0.349002
$907$	2.27291e10	1.01148	0.505739	−	0.862687i	$-0.331220\pi$
$907$	2.27291e10	1.01148	0.505739	+	0.862687i	$0.331220\pi$
$908$	9.33944e9	0.414019
$909$	−1.01106e9	−0.0446479
$910$	1.54361e11	6.79036
$911$	−2.77093e10	−1.21426	−0.607129	−	0.794603i	$-0.707679\pi$
$911$	−2.77093e10	−1.21426	−0.607129	+	0.794603i	$0.707679\pi$
$912$	5.13824e9	0.224301
$913$	1.24901e10	0.543150
$914$	−2.66845e9	−0.115597
$915$	2.53825e9	0.109537
$916$	−3.35289e9	−0.144140
$917$	1.74711e10	0.748218
$918$	6.32564e9	0.269870
$919$	−1.24377e10	−0.528612	−0.264306	−	0.964439i	$-0.585143\pi$
$919$	−1.24377e10	−0.528612	−0.264306	+	0.964439i	$0.585143\pi$
$920$	6.47296e10	2.74060
$921$	1.98232e10	0.836114
$922$	4.57091e10	1.92063
$923$	−3.72885e10	−1.56088
$924$	−2.81410e10	−1.17351
$925$	5.31221e10	2.20688
$926$	−4.82913e10	−1.99862
$927$	−8.10663e8	−0.0334242
$928$	−1.44208e10	−0.592339
$929$	−3.14739e10	−1.28794	−0.643970	−	0.765051i	$-0.722714\pi$
$929$	−3.14739e10	−1.28794	−0.643970	+	0.765051i	$0.722714\pi$
$930$	3.63780e10	1.48302
$931$	−8.47146e10	−3.44060
$932$	1.16730e10	0.472308
$933$	−6.32405e8	−0.0254923
$934$	−1.93337e10	−0.776428
$935$	2.10583e10	0.842526
$936$	−1.41136e10	−0.562567
$937$	−3.85637e10	−1.53140	−0.765702	−	0.643195i	$-0.777608\pi$
$937$	−3.85637e10	−1.53140	−0.765702	+	0.643195i	$0.777608\pi$
$938$	−4.70169e10	−1.86013
$939$	−1.14097e10	−0.449722
$940$	−1.09576e11	−4.30297
$941$	2.15748e10	0.844078	0.422039	−	0.906578i	$-0.361314\pi$
$941$	2.15748e10	0.844078	0.422039	+	0.906578i	$0.361314\pi$
$942$	−1.83276e10	−0.714376
$943$	3.06608e10	1.19067
$944$	9.46800e8	0.0366316
$945$	1.48143e10	0.571045
$946$	4.08886e10	1.57030
$947$	−2.58130e10	−0.987675	−0.493838	−	0.869554i	$-0.664406\pi$
$947$	−2.58130e10	−0.987675	−0.493838	+	0.869554i	$0.664406\pi$
$948$	−2.73448e10	−1.04243
$949$	4.31069e10	1.63725
$950$	−9.18484e10	−3.47567
$951$	4.85469e8	0.0183033
$952$	5.14457e10	1.93250
$953$	3.92406e10	1.46862	0.734311	−	0.678813i	$-0.237505\pi$
$953$	3.92406e10	1.46862	0.734311	+	0.678813i	$0.237505\pi$
$954$	−1.05251e10	−0.392471
$955$	1.18227e10	0.439244
$956$	4.05355e10	1.50049
$957$	7.69222e9	0.283700
$958$	−4.76986e10	−1.75278
$959$	8.02121e10	2.93680
$960$	3.84501e10	1.40265
$961$	−1.21292e9	−0.0440859
$962$	−9.16601e10	−3.31946
$963$	−1.75276e10	−0.632457
$964$	−1.22595e10	−0.440762
$965$	−7.52997e10	−2.69741
$966$	7.07385e10	2.52485
$967$	−1.78901e9	−0.0636240	−0.0318120	−	0.999494i	$-0.510128\pi$
$967$	−1.78901e9	−0.0636240	−0.0318120	+	0.999494i	$0.510128\pi$
$968$	−2.09389e10	−0.741977
$969$	−1.91357e10	−0.675633
$970$	−1.32866e10	−0.467427
$971$	−2.44481e10	−0.856993	−0.428497	−	0.903543i	$-0.640957\pi$
$971$	−2.44481e10	−0.856993	−0.428497	+	0.903543i	$0.640957\pi$
$972$	−3.19117e9	−0.111460
$973$	−9.77854e10	−3.40313
$974$	−2.58415e10	−0.896109
$975$	3.51619e10	1.21495
$976$	−9.76459e8	−0.0336186
$977$	3.42307e10	1.17432	0.587158	−	0.809472i	$-0.300247\pi$
$977$	3.42307e10	1.17432	0.587158	+	0.809472i	$0.300247\pi$
$978$	−2.66678e10	−0.911593
$979$	1.16504e10	0.396827
$980$	2.02563e11	6.87494
$981$	7.74785e9	0.262023
$982$	6.36975e10	2.14651
$983$	−4.53628e10	−1.52322	−0.761610	−	0.648036i	$-0.775591\pi$
$983$	−4.53628e10	−1.52322	−0.761610	+	0.648036i	$0.775591\pi$
$984$	−1.77237e10	−0.593022
$985$	2.84688e10	0.949167
$986$	−3.31304e10	−1.10067
$987$	−5.08277e10	−1.68264
$988$	1.00588e11	3.31816
$989$	−6.52359e10	−2.14437
$990$	−1.67379e10	−0.548248
$991$	2.84050e10	0.927123	0.463561	−	0.886065i	$-0.346571\pi$
$991$	2.84050e10	0.927123	0.463561	+	0.886065i	$0.346571\pi$
$992$	2.26855e10	0.737834
$993$	5.93027e9	0.192199
$994$	−1.08034e11	−3.48906
$995$	−5.09460e10	−1.63957
$996$	2.71387e10	0.870324
$997$	−3.63738e10	−1.16240	−0.581201	−	0.813760i	$-0.697417\pi$
$997$	−3.63738e10	−1.16240	−0.581201	+	0.813760i	$0.697417\pi$
$998$	−5.08350e10	−1.61885
$999$	−8.79678e9	−0.279154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.15	✓	16
3.2	odd	2		531.8.a.b.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.15	✓	16	1.1	even	1	trivial
531.8.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+18.7189 q^{2} -27.0000 q^{3} +222.398 q^{4} +443.832 q^{5} -505.411 q^{6} -1695.79 q^{7} +1767.03 q^{8} +729.000 q^{9} +8308.06 q^{10} -2763.59 q^{11} -6004.75 q^{12} -10956.4 q^{13} -31743.3 q^{14} -11983.5 q^{15} +4610.01 q^{16} -17168.5 q^{17} +13646.1 q^{18} -41280.8 q^{19} +98707.5 q^{20} +45786.3 q^{21} -51731.4 q^{22} +82535.2 q^{23} -47709.9 q^{24} +118862. q^{25} -205092. q^{26} -19683.0 q^{27} -377140. q^{28} +103089. q^{29} -224318. q^{30} -162172. q^{31} -139886. q^{32} +74616.9 q^{33} -321376. q^{34} -752645. q^{35} +162128. q^{36} +446923. q^{37} -772732. q^{38} +295822. q^{39} +784266. q^{40} +371488. q^{41} +857070. q^{42} -790401. q^{43} -614617. q^{44} +323554. q^{45} +1.54497e6 q^{46} -1.11011e6 q^{47} -124470. q^{48} +2.05215e6 q^{49} +2.22497e6 q^{50} +463549. q^{51} -2.43668e6 q^{52} -771292. q^{53} -368445. q^{54} -1.22657e6 q^{55} -2.99652e6 q^{56} +1.11458e6 q^{57} +1.92972e6 q^{58} +205379. q^{59} -2.66510e6 q^{60} -211813. q^{61} -3.03568e6 q^{62} -1.23623e6 q^{63} -3.20860e6 q^{64} -4.86279e6 q^{65} +1.39675e6 q^{66} +1.48116e6 q^{67} -3.81824e6 q^{68} -2.22845e6 q^{69} -1.40887e7 q^{70} +3.40336e6 q^{71} +1.28817e6 q^{72} -3.93442e6 q^{73} +8.36592e6 q^{74} -3.20927e6 q^{75} -9.18078e6 q^{76} +4.68646e6 q^{77} +5.53747e6 q^{78} +4.55385e6 q^{79} +2.04607e6 q^{80} +531441. q^{81} +6.95386e6 q^{82} -4.51953e6 q^{83} +1.01828e7 q^{84} -7.61992e6 q^{85} -1.47955e7 q^{86} -2.78342e6 q^{87} -4.88336e6 q^{88} -4.21568e6 q^{89} +6.05658e6 q^{90} +1.85797e7 q^{91} +1.83557e7 q^{92} +4.37864e6 q^{93} -2.07800e7 q^{94} -1.83217e7 q^{95} +3.77692e6 q^{96} -1.59925e6 q^{97} +3.84141e7 q^{98} -2.01466e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) 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

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