LMFDB - Embedded newform 177.8.a.a.1.11

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			1.11
Root			$-7.02227$ 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)

$f(q)$	$=$	$q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} -1873.35 q^{10} +2997.05 q^{11} +2124.57 q^{12} +11110.8 q^{13} +4675.13 q^{14} +7202.88 q^{15} -120.197 q^{16} +17329.7 q^{17} +5119.23 q^{18} -7569.90 q^{19} +20991.8 q^{20} -17975.5 q^{21} +21046.1 q^{22} -84837.6 q^{23} +39188.2 q^{24} -6956.93 q^{25} +78023.0 q^{26} -19683.0 q^{27} -52387.0 q^{28} +59721.5 q^{29} +50580.6 q^{30} -56202.0 q^{31} +184937. q^{32} -80920.4 q^{33} +121694. q^{34} -177607. q^{35} -57363.4 q^{36} -375672. q^{37} -53157.8 q^{38} -299992. q^{39} +387199. q^{40} +560321. q^{41} -126228. q^{42} -599895. q^{43} -235831. q^{44} -194478. q^{45} -595752. q^{46} -508735. q^{47} +3245.32 q^{48} -380309. q^{49} -48853.4 q^{50} -467903. q^{51} -874284. q^{52} +95112.1 q^{53} -138219. q^{54} -799533. q^{55} -966292. q^{56} +204387. q^{57} +419380. q^{58} +205379. q^{59} -566779. q^{60} +1.08522e6 q^{61} -394665. q^{62} +485337. q^{63} +1.31406e6 q^{64} -2.96407e6 q^{65} -568244. q^{66} -731050. q^{67} -1.36364e6 q^{68} +2.29062e6 q^{69} -1.24720e6 q^{70} +4.30967e6 q^{71} -1.05808e6 q^{72} -804495. q^{73} -2.63807e6 q^{74} +187837. q^{75} +595658. q^{76} +1.99531e6 q^{77} -2.10662e6 q^{78} -3.53260e6 q^{79} +32065.4 q^{80} +531441. q^{81} +3.93472e6 q^{82} -8.87854e6 q^{83} +1.41445e6 q^{84} -4.62312e6 q^{85} -4.21262e6 q^{86} -1.61248e6 q^{87} -4.34997e6 q^{88} -5.83811e6 q^{89} -1.36568e6 q^{90} +7.39710e6 q^{91} +6.67568e6 q^{92} +1.51745e6 q^{93} -3.57247e6 q^{94} +2.01945e6 q^{95} -4.99331e6 q^{96} +1.47129e7 q^{97} -2.67063e6 q^{98} +2.18485e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	7.02227	0.620686	0.310343	−	0.950625i	$-0.399556\pi$
$2$	7.02227	0.620686	0.310343	+	0.950625i	$0.399556\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−78.6878	−0.614748
$5$	−266.773	−0.954438	−0.477219	−	0.878784i	$-0.658355\pi$
$5$	−266.773	−0.954438	−0.477219	+	0.878784i	$0.658355\pi$
$6$	−189.601	−0.358354
$7$	665.758	0.733623	0.366812	−	0.930295i	$-0.380449\pi$
$7$	665.758	0.733623	0.366812	+	0.930295i	$0.380449\pi$
$8$	−1451.42	−1.00225
$9$	729.000	0.333333
$10$	−1873.35	−0.592407
$11$	2997.05	0.678922	0.339461	−	0.940620i	$-0.389755\pi$
$11$	2997.05	0.678922	0.339461	+	0.940620i	$0.389755\pi$
$12$	2124.57	0.354925
$13$	11110.8	1.40263	0.701316	−	0.712851i	$-0.252596\pi$
$13$	11110.8	1.40263	0.701316	+	0.712851i	$0.252596\pi$
$14$	4675.13	0.455350
$15$	7202.88	0.551045
$16$	−120.197	−0.00733625
$17$	17329.7	0.855502	0.427751	−	0.903897i	$-0.359306\pi$
$17$	17329.7	0.855502	0.427751	+	0.903897i	$0.359306\pi$
$18$	5119.23	0.206895
$19$	−7569.90	−0.253193	−0.126597	−	0.991954i	$-0.540405\pi$
$19$	−7569.90	−0.253193	−0.126597	+	0.991954i	$0.540405\pi$
$20$	20991.8	0.586739
$21$	−17975.5	−0.423558
$22$	21046.1	0.421397
$23$	−84837.6	−1.45392	−0.726961	−	0.686679i	$-0.759068\pi$
$23$	−84837.6	−1.45392	−0.726961	+	0.686679i	$0.759068\pi$
$24$	39188.2	0.578651
$25$	−6956.93	−0.0890487
$26$	78023.0	0.870594
$27$	−19683.0	−0.192450
$28$	−52387.0	−0.450994
$29$	59721.5	0.454713	0.227357	−	0.973812i	$-0.426992\pi$
$29$	59721.5	0.454713	0.227357	+	0.973812i	$0.426992\pi$
$30$	50580.6	0.342026
$31$	−56202.0	−0.338833	−0.169416	−	0.985545i	$-0.554188\pi$
$31$	−56202.0	−0.338833	−0.169416	+	0.985545i	$0.554188\pi$
$32$	184937.	0.997699
$33$	−80920.4	−0.391976
$34$	121694.	0.530998
$35$	−177607.	−0.700198
$36$	−57363.4	−0.204916
$37$	−375672.	−1.21928	−0.609639	−	0.792679i	$-0.708686\pi$
$37$	−375672.	−1.21928	−0.609639	+	0.792679i	$0.708686\pi$
$38$	−53157.8	−0.157154
$39$	−299992.	−0.809810
$40$	387199.	0.956587
$41$	560321.	1.26968	0.634839	−	0.772645i	$-0.281066\pi$
$41$	560321.	1.26968	0.634839	+	0.772645i	$0.281066\pi$
$42$	−126228.	−0.262897
$43$	−599895.	−1.15063	−0.575315	−	0.817932i	$-0.695120\pi$
$43$	−599895.	−1.15063	−0.575315	+	0.817932i	$0.695120\pi$
$44$	−235831.	−0.417366
$45$	−194478.	−0.318146
$46$	−595752.	−0.902429
$47$	−508735.	−0.714741	−0.357371	−	0.933963i	$-0.616327\pi$
$47$	−508735.	−0.714741	−0.357371	+	0.933963i	$0.616327\pi$
$48$	3245.32	0.00423558
$49$	−380309.	−0.461797
$50$	−48853.4	−0.0552713
$51$	−467903.	−0.493924
$52$	−874284.	−0.862265
$53$	95112.1	0.0877547	0.0438773	−	0.999037i	$-0.486029\pi$
$53$	95112.1	0.0877547	0.0438773	+	0.999037i	$0.486029\pi$
$54$	−138219.	−0.119451
$55$	−799533.	−0.647988
$56$	−966292.	−0.735276
$57$	204387.	0.146181
$58$	419380.	0.282234
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−566779.	−0.338754
$61$	1.08522e6	0.612160	0.306080	−	0.952006i	$-0.400982\pi$
$61$	1.08522e6	0.612160	0.306080	+	0.952006i	$0.400982\pi$
$62$	−394665.	−0.210309
$63$	485337.	0.244541
$64$	1.31406e6	0.626594
$65$	−2.96407e6	−1.33872
$66$	−568244.	−0.243294
$67$	−731050.	−0.296951	−0.148476	−	0.988916i	$-0.547437\pi$
$67$	−731050.	−0.296951	−0.148476	+	0.988916i	$0.547437\pi$
$68$	−1.36364e6	−0.525918
$69$	2.29062e6	0.839422
$70$	−1.24720e6	−0.434603
$71$	4.30967e6	1.42902	0.714512	−	0.699623i	$-0.246649\pi$
$71$	4.30967e6	1.42902	0.714512	+	0.699623i	$0.246649\pi$
$72$	−1.05808e6	−0.334084
$73$	−804495.	−0.242043	−0.121022	−	0.992650i	$-0.538617\pi$
$73$	−804495.	−0.242043	−0.121022	+	0.992650i	$0.538617\pi$
$74$	−2.63807e6	−0.756790
$75$	187837.	0.0514123
$76$	595658.	0.155650
$77$	1.99531e6	0.498073
$78$	−2.10662e6	−0.502638
$79$	−3.53260e6	−0.806121	−0.403060	−	0.915173i	$-0.632054\pi$
$79$	−3.53260e6	−0.806121	−0.403060	+	0.915173i	$0.632054\pi$
$80$	32065.4	0.00700199
$81$	531441.	0.111111
$82$	3.93472e6	0.788072
$83$	−8.87854e6	−1.70439	−0.852193	−	0.523227i	$-0.824728\pi$
$83$	−8.87854e6	−1.70439	−0.852193	+	0.523227i	$0.824728\pi$
$84$	1.41445e6	0.260381
$85$	−4.62312e6	−0.816523
$86$	−4.21262e6	−0.714180
$87$	−1.61248e6	−0.262529
$88$	−4.34997e6	−0.680451
$89$	−5.83811e6	−0.877823	−0.438912	−	0.898530i	$-0.644636\pi$
$89$	−5.83811e6	−0.877823	−0.438912	+	0.898530i	$0.644636\pi$
$90$	−1.36568e6	−0.197469
$91$	7.39710e6	1.02900
$92$	6.67568e6	0.893796
$93$	1.51745e6	0.195625
$94$	−3.57247e6	−0.443630
$95$	2.01945e6	0.241657
$96$	−4.99331e6	−0.576022
$97$	1.47129e7	1.63681	0.818405	−	0.574642i	$-0.194859\pi$
$97$	1.47129e7	1.63681	0.818405	+	0.574642i	$0.194859\pi$
$98$	−2.67063e6	−0.286631
$99$	2.18485e6	0.226307
$100$	547426.	0.0547426
$101$	−4.85985e6	−0.469352	−0.234676	−	0.972074i	$-0.575403\pi$
$101$	−4.85985e6	−0.469352	−0.234676	+	0.972074i	$0.575403\pi$
$102$	−3.28574e6	−0.306572
$103$	−4.63498e6	−0.417944	−0.208972	−	0.977922i	$-0.567012\pi$
$103$	−4.63498e6	−0.417944	−0.208972	+	0.977922i	$0.567012\pi$
$104$	−1.61264e7	−1.40579
$105$	4.79538e6	0.404259
$106$	667902.	0.0544681
$107$	−1.55941e7	−1.23060	−0.615301	−	0.788292i	$-0.710965\pi$
$107$	−1.55941e7	−1.23060	−0.615301	+	0.788292i	$0.710965\pi$
$108$	1.54881e6	0.118308
$109$	−554937.	−0.0410442	−0.0205221	−	0.999789i	$-0.506533\pi$
$109$	−554937.	−0.0410442	−0.0205221	+	0.999789i	$0.506533\pi$
$110$	−5.61454e6	−0.402198
$111$	1.01431e7	0.703951
$112$	−80022.2	−0.00538204
$113$	−812959.	−0.0530022	−0.0265011	−	0.999649i	$-0.508437\pi$
$113$	−812959.	−0.0530022	−0.0265011	+	0.999649i	$0.508437\pi$
$114$	1.43526e6	0.0907327
$115$	2.26324e7	1.38768
$116$	−4.69935e6	−0.279534
$117$	8.09977e6	0.467544
$118$	1.44223e6	0.0808065
$119$	1.15374e7	0.627616
$120$	−1.04544e7	−0.552286
$121$	−1.05049e7	−0.539066
$122$	7.62074e6	0.379960
$123$	−1.51287e7	−0.733049
$124$	4.42241e6	0.208297
$125$	2.26976e7	1.03943
$126$	3.40817e6	0.151783
$127$	−3.99070e7	−1.72877	−0.864383	−	0.502834i	$-0.832291\pi$
$127$	−3.99070e7	−1.72877	−0.864383	+	0.502834i	$0.832291\pi$
$128$	−1.44443e7	−0.608780
$129$	1.61972e7	0.664316
$130$	−2.08145e7	−0.830928
$131$	4.33560e7	1.68500	0.842500	−	0.538696i	$-0.181083\pi$
$131$	4.33560e7	1.68500	0.842500	+	0.538696i	$0.181083\pi$
$132$	6.36744e6	0.240966
$133$	−5.03972e6	−0.185749
$134$	−5.13363e6	−0.184314
$135$	5.25090e6	0.183682
$136$	−2.51527e7	−0.857429
$137$	−3.31553e7	−1.10162	−0.550809	−	0.834632i	$-0.685681\pi$
$137$	−3.31553e7	−1.10162	−0.550809	+	0.834632i	$0.685681\pi$
$138$	1.60853e7	0.521018
$139$	−4.77383e7	−1.50770	−0.753850	−	0.657046i	$-0.771806\pi$
$139$	−4.77383e7	−1.50770	−0.753850	+	0.657046i	$0.771806\pi$
$140$	1.39755e7	0.430445
$141$	1.37358e7	0.412656
$142$	3.02636e7	0.886976
$143$	3.32996e7	0.952277
$144$	−87623.7	−0.00244542
$145$	−1.59321e7	−0.433996
$146$	−5.64938e6	−0.150233
$147$	1.02684e7	0.266618
$148$	2.95608e7	0.749549
$149$	−7.91243e7	−1.95956	−0.979778	−	0.200086i	$-0.935878\pi$
$149$	−7.91243e7	−1.95956	−0.979778	+	0.200086i	$0.935878\pi$
$150$	1.31904e6	0.0319109
$151$	−5.25091e7	−1.24112	−0.620562	−	0.784157i	$-0.713096\pi$
$151$	−5.25091e7	−1.24112	−0.620562	+	0.784157i	$0.713096\pi$
$152$	1.09871e7	0.253764
$153$	1.26334e7	0.285167
$154$	1.40116e7	0.309147
$155$	1.49932e7	0.323395
$156$	2.36057e7	0.497829
$157$	−5.32097e7	−1.09734	−0.548671	−	0.836038i	$-0.684866\pi$
$157$	−5.32097e7	−1.09734	−0.548671	+	0.836038i	$0.684866\pi$
$158$	−2.48069e7	−0.500348
$159$	−2.56803e6	−0.0506652
$160$	−4.93364e7	−0.952241
$161$	−5.64813e7	−1.06663
$162$	3.73192e6	0.0689652
$163$	6.98231e7	1.26282	0.631412	−	0.775448i	$-0.282476\pi$
$163$	6.98231e7	1.26282	0.631412	+	0.775448i	$0.282476\pi$
$164$	−4.40904e7	−0.780532
$165$	2.15874e7	0.374116
$166$	−6.23475e7	−1.05789
$167$	8.36408e7	1.38967	0.694833	−	0.719171i	$-0.255478\pi$
$167$	8.36408e7	1.38967	0.694833	+	0.719171i	$0.255478\pi$
$168$	2.60899e7	0.424512
$169$	6.07013e7	0.967375
$170$	−3.24647e7	−0.506805
$171$	−5.51845e6	−0.0843978
$172$	4.72044e7	0.707347
$173$	−2.42624e7	−0.356265	−0.178132	−	0.984007i	$-0.557006\pi$
$173$	−2.42624e7	−0.356265	−0.178132	+	0.984007i	$0.557006\pi$
$174$	−1.13233e7	−0.162948
$175$	−4.63163e6	−0.0653282
$176$	−360237.	−0.00498074
$177$	−5.54523e6	−0.0751646
$178$	−4.09967e7	−0.544853
$179$	−1.12557e7	−0.146686	−0.0733430	−	0.997307i	$-0.523367\pi$
$179$	−1.12557e7	−0.146686	−0.0733430	+	0.997307i	$0.523367\pi$
$180$	1.53030e7	0.195580
$181$	1.11658e7	0.139963	0.0699816	−	0.997548i	$-0.477706\pi$
$181$	1.11658e7	0.139963	0.0699816	+	0.997548i	$0.477706\pi$
$182$	5.19444e7	0.638688
$183$	−2.93011e7	−0.353431
$184$	1.23135e8	1.45720
$185$	1.00219e8	1.16373
$186$	1.06560e7	0.121422
$187$	5.19381e7	0.580819
$188$	4.00312e7	0.439386
$189$	−1.31041e7	−0.141186
$190$	1.41811e7	0.149993
$191$	−3.59428e7	−0.373246	−0.186623	−	0.982432i	$-0.559754\pi$
$191$	−3.59428e7	−0.373246	−0.186623	+	0.982432i	$0.559754\pi$
$192$	−3.54797e7	−0.361764
$193$	2.78406e7	0.278758	0.139379	−	0.990239i	$-0.455489\pi$
$193$	2.78406e7	0.278758	0.139379	+	0.990239i	$0.455489\pi$
$194$	1.03318e8	1.01595
$195$	8.00298e7	0.772913
$196$	2.99257e7	0.283889
$197$	8.97822e7	0.836679	0.418339	−	0.908291i	$-0.362612\pi$
$197$	8.97822e7	0.836679	0.418339	+	0.908291i	$0.362612\pi$
$198$	1.53426e7	0.140466
$199$	1.97256e7	0.177437	0.0887183	−	0.996057i	$-0.471723\pi$
$199$	1.97256e7	0.177437	0.0887183	+	0.996057i	$0.471723\pi$
$200$	1.00974e7	0.0892493
$201$	1.97383e7	0.171445
$202$	−3.41272e7	−0.291320
$203$	3.97601e7	0.333588
$204$	3.68183e7	0.303639
$205$	−1.49479e8	−1.21183
$206$	−3.25481e7	−0.259412
$207$	−6.18466e7	−0.484641
$208$	−1.33549e6	−0.0102901
$209$	−2.26874e7	−0.171898
$210$	3.36744e7	0.250918
$211$	1.59619e8	1.16976	0.584881	−	0.811119i	$-0.301141\pi$
$211$	1.59619e8	1.16976	0.584881	+	0.811119i	$0.301141\pi$
$212$	−7.48416e6	−0.0539470
$213$	−1.16361e8	−0.825048
$214$	−1.09506e8	−0.763818
$215$	1.60036e8	1.09820
$216$	2.85682e7	0.192884
$217$	−3.74169e7	−0.248576
$218$	−3.89692e6	−0.0254756
$219$	2.17214e7	0.139744
$220$	6.29135e7	0.398350
$221$	1.92547e8	1.19995
$222$	7.12279e7	0.436933
$223$	−1.22127e8	−0.737471	−0.368736	−	0.929534i	$-0.620209\pi$
$223$	−1.22127e8	−0.737471	−0.368736	+	0.929534i	$0.620209\pi$
$224$	1.23123e8	0.731935
$225$	−5.07160e6	−0.0296829
$226$	−5.70881e6	−0.0328977
$227$	−1.05761e7	−0.0600118	−0.0300059	−	0.999550i	$-0.509553\pi$
$227$	−1.05761e7	−0.0600118	−0.0300059	+	0.999550i	$0.509553\pi$
$228$	−1.60828e7	−0.0898647
$229$	5.44892e7	0.299838	0.149919	−	0.988698i	$-0.452099\pi$
$229$	5.44892e7	0.299838	0.149919	+	0.988698i	$0.452099\pi$
$230$	1.58931e8	0.861313
$231$	−5.38734e7	−0.287562
$232$	−8.66808e7	−0.455738
$233$	1.03393e8	0.535484	0.267742	−	0.963491i	$-0.413722\pi$
$233$	1.03393e8	0.535484	0.267742	+	0.963491i	$0.413722\pi$
$234$	5.68788e7	0.290198
$235$	1.35717e8	0.682176
$236$	−1.61608e7	−0.0800334
$237$	9.53803e7	0.465414
$238$	8.10188e7	0.389553
$239$	−9.19771e7	−0.435800	−0.217900	−	0.975971i	$-0.569921\pi$
$239$	−9.19771e7	−0.435800	−0.217900	+	0.975971i	$0.569921\pi$
$240$	−865766.	−0.00404260
$241$	−3.22890e8	−1.48592	−0.742958	−	0.669338i	$-0.766578\pi$
$241$	−3.22890e8	−1.48592	−0.742958	+	0.669338i	$0.766578\pi$
$242$	−7.37679e7	−0.334591
$243$	−1.43489e7	−0.0641500
$244$	−8.53939e7	−0.376325
$245$	1.01456e8	0.440756
$246$	−1.06238e8	−0.454993
$247$	−8.41076e7	−0.355137
$248$	8.15725e7	0.339596
$249$	2.39721e8	0.984028
$250$	1.59389e8	0.645160
$251$	2.78313e8	1.11090	0.555451	−	0.831549i	$-0.312546\pi$
$251$	2.78313e8	1.11090	0.555451	+	0.831549i	$0.312546\pi$
$252$	−3.81901e7	−0.150331
$253$	−2.54263e8	−0.987099
$254$	−2.80238e8	−1.07302
$255$	1.24824e8	0.471420
$256$	−2.69632e8	−1.00446
$257$	−1.56960e8	−0.576795	−0.288398	−	0.957511i	$-0.593123\pi$
$257$	−1.56960e8	−0.576795	−0.288398	+	0.957511i	$0.593123\pi$
$258$	1.13741e8	0.412332
$259$	−2.50107e8	−0.894491
$260$	2.33236e8	0.822978
$261$	4.35370e7	0.151571
$262$	3.04458e8	1.04586
$263$	2.42199e8	0.820971	0.410486	−	0.911867i	$-0.365359\pi$
$263$	2.42199e8	0.820971	0.410486	+	0.911867i	$0.365359\pi$
$264$	1.17449e8	0.392858
$265$	−2.53734e7	−0.0837563
$266$	−3.53902e7	−0.115292
$267$	1.57629e8	0.506811
$268$	5.75247e7	0.182550
$269$	5.70264e7	0.178625	0.0893126	−	0.996004i	$-0.471533\pi$
$269$	5.70264e7	0.178625	0.0893126	+	0.996004i	$0.471533\pi$
$270$	3.68732e7	0.114009
$271$	2.57235e8	0.785124	0.392562	−	0.919726i	$-0.371589\pi$
$271$	2.57235e8	0.785124	0.392562	+	0.919726i	$0.371589\pi$
$272$	−2.08298e6	−0.00627617
$273$	−1.99722e8	−0.594095
$274$	−2.32825e8	−0.683759
$275$	−2.08503e7	−0.0604571
$276$	−1.80243e8	−0.516033
$277$	−6.22571e8	−1.75999	−0.879994	−	0.474984i	$-0.842454\pi$
$277$	−6.22571e8	−1.75999	−0.879994	+	0.474984i	$0.842454\pi$
$278$	−3.35231e8	−0.935809
$279$	−4.09712e7	−0.112944
$280$	2.57781e8	0.701775
$281$	3.00467e8	0.807838	0.403919	−	0.914795i	$-0.367648\pi$
$281$	3.00467e8	0.807838	0.403919	+	0.914795i	$0.367648\pi$
$282$	9.64567e7	0.256130
$283$	−4.37742e8	−1.14806	−0.574032	−	0.818833i	$-0.694621\pi$
$283$	−4.37742e8	−1.14806	−0.574032	+	0.818833i	$0.694621\pi$
$284$	−3.39118e8	−0.878490
$285$	−5.45251e7	−0.139521
$286$	2.33839e8	0.591065
$287$	3.73038e8	0.931465
$288$	1.34819e8	0.332566
$289$	−1.10019e8	−0.268117
$290$	−1.11880e8	−0.269375
$291$	−3.97249e8	−0.945013
$292$	6.33039e7	0.148796
$293$	−3.94385e8	−0.915976	−0.457988	−	0.888958i	$-0.651430\pi$
$293$	−3.94385e8	−0.915976	−0.457988	+	0.888958i	$0.651430\pi$
$294$	7.21071e7	0.165486
$295$	−5.47897e7	−0.124257
$296$	5.45257e8	1.22203
$297$	−5.89909e7	−0.130659
$298$	−5.55632e8	−1.21627
$299$	−9.42614e8	−2.03932
$300$	−1.47805e7	−0.0316056
$301$	−3.99385e8	−0.844129
$302$	−3.68733e8	−0.770349
$303$	1.31216e8	0.270980
$304$	909879.	0.00185749
$305$	−2.89509e8	−0.584269
$306$	8.87150e7	0.176999
$307$	−2.75368e8	−0.543162	−0.271581	−	0.962416i	$-0.587546\pi$
$307$	−2.75368e8	−0.543162	−0.271581	+	0.962416i	$0.587546\pi$
$308$	−1.57006e8	−0.306189
$309$	1.25145e8	0.241300
$310$	1.05286e8	0.200727
$311$	−2.81733e8	−0.531100	−0.265550	−	0.964097i	$-0.585554\pi$
$311$	−2.81733e8	−0.531100	−0.265550	+	0.964097i	$0.585554\pi$
$312$	4.35413e8	0.811634
$313$	−8.03742e8	−1.48153	−0.740767	−	0.671762i	$-0.765538\pi$
$313$	−8.03742e8	−1.48153	−0.740767	+	0.671762i	$0.765538\pi$
$314$	−3.73653e8	−0.681105
$315$	−1.29475e8	−0.233399
$316$	2.77973e8	0.495561
$317$	2.22203e8	0.391780	0.195890	−	0.980626i	$-0.437240\pi$
$317$	2.22203e8	0.391780	0.195890	+	0.980626i	$0.437240\pi$
$318$	−1.80334e7	−0.0314472
$319$	1.78988e8	0.308715
$320$	−3.50557e8	−0.598045
$321$	4.21041e8	0.710488
$322$	−3.96627e8	−0.662043
$323$	−1.31184e8	−0.216607
$324$	−4.18179e7	−0.0683054
$325$	−7.72971e7	−0.124903
$326$	4.90317e8	0.783818
$327$	1.49833e7	0.0236969
$328$	−8.13259e8	−1.27254
$329$	−3.38694e8	−0.524351
$330$	1.51592e8	0.232209
$331$	−2.85912e8	−0.433346	−0.216673	−	0.976244i	$-0.569521\pi$
$331$	−2.85912e8	−0.433346	−0.216673	+	0.976244i	$0.569521\pi$
$332$	6.98633e8	1.04777
$333$	−2.73865e8	−0.406426
$334$	5.87348e8	0.862547
$335$	1.95025e8	0.283421
$336$	2.16060e6	0.00310732
$337$	−4.83392e8	−0.688010	−0.344005	−	0.938968i	$-0.611784\pi$
$337$	−4.83392e8	−0.688010	−0.344005	+	0.938968i	$0.611784\pi$
$338$	4.26261e8	0.600436
$339$	2.19499e7	0.0306008
$340$	3.63783e8	0.501956
$341$	−1.68440e8	−0.230041
$342$	−3.87521e7	−0.0523846
$343$	−8.01474e8	−1.07241
$344$	8.70697e8	1.15322
$345$	−6.11075e8	−0.801176
$346$	−1.70377e8	−0.221129
$347$	6.40094e8	0.822414	0.411207	−	0.911542i	$-0.365107\pi$
$347$	6.40094e8	0.822414	0.411207	+	0.911542i	$0.365107\pi$
$348$	1.26883e8	0.161389
$349$	4.24832e7	0.0534968	0.0267484	−	0.999642i	$-0.491485\pi$
$349$	4.24832e7	0.0534968	0.0267484	+	0.999642i	$0.491485\pi$
$350$	−3.25246e7	−0.0405483
$351$	−2.18694e8	−0.269937
$352$	5.54266e8	0.677359
$353$	1.12511e9	1.36139	0.680694	−	0.732568i	$-0.261678\pi$
$353$	1.12511e9	1.36139	0.680694	+	0.732568i	$0.261678\pi$
$354$	−3.89401e7	−0.0466537
$355$	−1.14971e9	−1.36391
$356$	4.59388e8	0.539640
$357$	−3.11510e8	−0.362354
$358$	−7.90408e7	−0.0910460
$359$	1.35745e7	0.0154843	0.00774215	−	0.999970i	$-0.497536\pi$
$359$	1.35745e7	0.0154843	0.00774215	+	0.999970i	$0.497536\pi$
$360$	2.82268e8	0.318862
$361$	−8.36568e8	−0.935893
$362$	7.84090e7	0.0868733
$363$	2.83631e8	0.311230
$364$	−5.82061e8	−0.632578
$365$	2.14618e8	0.231015
$366$	−2.05760e8	−0.219370
$367$	8.41482e8	0.888615	0.444308	−	0.895874i	$-0.353450\pi$
$367$	8.41482e8	0.888615	0.444308	+	0.895874i	$0.353450\pi$
$368$	1.01972e7	0.0106663
$369$	4.08474e8	0.423226
$370$	7.03767e8	0.722309
$371$	6.33216e7	0.0643789
$372$	−1.19405e8	−0.120260
$373$	−1.40652e9	−1.40334	−0.701672	−	0.712500i	$-0.747563\pi$
$373$	−1.40652e9	−1.40334	−0.701672	+	0.712500i	$0.747563\pi$
$374$	3.64723e8	0.360506
$375$	−6.12835e8	−0.600115
$376$	7.38386e8	0.716351
$377$	6.63554e8	0.637795
$378$	−9.20206e7	−0.0876322
$379$	−7.42481e8	−0.700564	−0.350282	−	0.936644i	$-0.613914\pi$
$379$	−7.42481e8	−0.700564	−0.350282	+	0.936644i	$0.613914\pi$
$380$	−1.58906e8	−0.148558
$381$	1.07749e9	0.998104
$382$	−2.52400e8	−0.231669
$383$	−7.94805e8	−0.722878	−0.361439	−	0.932396i	$-0.617714\pi$
$383$	−7.94805e8	−0.722878	−0.361439	+	0.932396i	$0.617714\pi$
$384$	3.89995e8	0.351479
$385$	−5.32296e8	−0.475379
$386$	1.95504e8	0.173021
$387$	−4.37323e8	−0.383543
$388$	−1.15773e9	−1.00623
$389$	−8.55397e8	−0.736790	−0.368395	−	0.929669i	$-0.620093\pi$
$389$	−8.55397e8	−0.736790	−0.368395	+	0.929669i	$0.620093\pi$
$390$	5.61990e8	0.479736
$391$	−1.47021e9	−1.24383
$392$	5.51987e8	0.462837
$393$	−1.17061e9	−0.972835
$394$	6.30475e8	0.519315
$395$	9.42405e8	0.769392
$396$	−1.71921e8	−0.139122
$397$	3.23507e8	0.259487	0.129744	−	0.991548i	$-0.458585\pi$
$397$	3.23507e8	0.259487	0.129744	+	0.991548i	$0.458585\pi$
$398$	1.38518e8	0.110133
$399$	1.36072e8	0.107242
$400$	836203.	0.000653284 0
$401$	−2.15538e8	−0.166924	−0.0834621	−	0.996511i	$-0.526598\pi$
$401$	−2.15538e8	−0.166924	−0.0834621	+	0.996511i	$0.526598\pi$
$402$	1.38608e8	0.106414
$403$	−6.24449e8	−0.475258
$404$	3.82411e8	0.288533
$405$	−1.41774e8	−0.106049
$406$	2.79206e8	0.207054
$407$	−1.12591e9	−0.827795
$408$	6.79122e8	0.495037
$409$	2.07548e9	1.49998	0.749992	−	0.661447i	$-0.230057\pi$
$409$	2.07548e9	1.49998	0.749992	+	0.661447i	$0.230057\pi$
$410$	−1.04968e9	−0.752165
$411$	8.95193e8	0.636019
$412$	3.64716e8	0.256930
$413$	1.36733e8	0.0955096
$414$	−4.34303e8	−0.300810
$415$	2.36856e9	1.62673
$416$	2.05480e9	1.39940
$417$	1.28893e9	0.870471
$418$	−1.59317e8	−0.106695
$419$	−2.46186e9	−1.63499	−0.817495	−	0.575936i	$-0.804638\pi$
$419$	−2.46186e9	−1.63499	−0.817495	+	0.575936i	$0.804638\pi$
$420$	−3.77337e8	−0.248518
$421$	2.15291e9	1.40617	0.703086	−	0.711105i	$-0.251805\pi$
$421$	2.15291e9	1.40617	0.703086	+	0.711105i	$0.251805\pi$
$422$	1.12089e9	0.726055
$423$	−3.70868e8	−0.238247
$424$	−1.38047e8	−0.0879523
$425$	−1.20562e8	−0.0761814
$426$	−8.17118e8	−0.512096
$427$	7.22497e8	0.449095
$428$	1.22707e9	0.756510
$429$	−8.99090e8	−0.549797
$430$	1.12382e9	0.681640
$431$	−2.75746e9	−1.65897	−0.829486	−	0.558528i	$-0.811366\pi$
$431$	−2.75746e9	−1.65897	−0.829486	+	0.558528i	$0.811366\pi$
$432$	2.36584e6	0.00141186
$433$	1.22443e9	0.724814	0.362407	−	0.932020i	$-0.381955\pi$
$433$	1.22443e9	0.724814	0.362407	+	0.932020i	$0.381955\pi$
$434$	−2.62751e8	−0.154288
$435$	4.30167e8	0.250567
$436$	4.36668e7	0.0252318
$437$	6.42212e8	0.368123
$438$	1.52533e8	0.0867371
$439$	−1.80512e9	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$439$	−1.80512e9	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$440$	1.16046e9	0.649448
$441$	−2.77246e8	−0.153932
$442$	1.35212e9	0.744795
$443$	2.60973e9	1.42621	0.713103	−	0.701059i	$-0.247289\pi$
$443$	2.60973e9	1.42621	0.713103	+	0.701059i	$0.247289\pi$
$444$	−7.98142e8	−0.432753
$445$	1.55745e9	0.837828
$446$	−8.57609e8	−0.457738
$447$	2.13636e9	1.13135
$448$	8.74848e8	0.459684
$449$	2.04391e9	1.06561	0.532805	−	0.846238i	$-0.321138\pi$
$449$	2.04391e9	1.06561	0.532805	+	0.846238i	$0.321138\pi$
$450$	−3.56141e7	−0.0184238
$451$	1.67931e9	0.862012
$452$	6.39699e7	0.0325830
$453$	1.41775e9	0.716563
$454$	−7.42685e7	−0.0372485
$455$	−1.97335e9	−0.982119
$456$	−2.96651e8	−0.146511
$457$	−1.13009e9	−0.553866	−0.276933	−	0.960889i	$-0.589318\pi$
$457$	−1.13009e9	−0.553866	−0.276933	+	0.960889i	$0.589318\pi$
$458$	3.82638e8	0.186105
$459$	−3.41101e8	−0.164641
$460$	−1.78089e9	−0.853072
$461$	1.04379e9	0.496202	0.248101	−	0.968734i	$-0.420194\pi$
$461$	1.04379e9	0.496202	0.248101	+	0.968734i	$0.420194\pi$
$462$	−3.78313e8	−0.178486
$463$	3.87197e8	0.181300	0.0906501	−	0.995883i	$-0.471106\pi$
$463$	3.87197e8	0.181300	0.0906501	+	0.995883i	$0.471106\pi$
$464$	−7.17835e6	−0.00333589
$465$	−4.04816e8	−0.186712
$466$	7.26055e8	0.332368
$467$	3.01337e8	0.136913	0.0684563	−	0.997654i	$-0.478193\pi$
$467$	3.01337e8	0.136913	0.0684563	+	0.997654i	$0.478193\pi$
$468$	−6.37353e8	−0.287422
$469$	−4.86702e8	−0.217850
$470$	9.53040e8	0.423417
$471$	1.43666e9	0.633550
$472$	−2.98090e8	−0.130482
$473$	−1.79791e9	−0.781187
$474$	6.69786e8	0.288876
$475$	5.26633e7	0.0225465
$476$	−9.07853e8	−0.385826
$477$	6.93367e7	0.0292516
$478$	−6.45888e8	−0.270495
$479$	−4.36417e9	−1.81438	−0.907189	−	0.420723i	$-0.861776\pi$
$479$	−4.36417e9	−1.81438	−0.907189	+	0.420723i	$0.861776\pi$
$480$	1.33208e9	0.549777
$481$	−4.17402e9	−1.71020
$482$	−2.26742e9	−0.922288
$483$	1.52500e9	0.615820
$484$	8.26604e8	0.331390
$485$	−3.92502e9	−1.56223
$486$	−1.00762e8	−0.0398171
$487$	3.93974e9	1.54567	0.772835	−	0.634608i	$-0.218838\pi$
$487$	3.93974e9	1.54567	0.772835	+	0.634608i	$0.218838\pi$
$488$	−1.57511e9	−0.613539
$489$	−1.88522e9	−0.729092
$490$	7.12454e8	0.273571
$491$	−1.95275e9	−0.744495	−0.372248	−	0.928133i	$-0.621413\pi$
$491$	−1.95275e9	−0.744495	−0.372248	+	0.928133i	$0.621413\pi$
$492$	1.19044e9	0.450640
$493$	1.03496e9	0.389008
$494$	−5.90626e8	−0.220429
$495$	−5.82860e8	−0.215996
$496$	6.75531e6	0.00248576
$497$	2.86920e9	1.04837
$498$	1.68338e9	0.610773
$499$	1.78306e9	0.642414	0.321207	−	0.947009i	$-0.395912\pi$
$499$	1.78306e9	0.642414	0.321207	+	0.947009i	$0.395912\pi$
$500$	−1.78602e9	−0.638987
$501$	−2.25830e9	−0.802325
$502$	1.95439e9	0.689522
$503$	1.58061e8	0.0553781	0.0276890	−	0.999617i	$-0.491185\pi$
$503$	1.58061e8	0.0553781	0.0276890	+	0.999617i	$0.491185\pi$
$504$	−7.04427e8	−0.245092
$505$	1.29648e9	0.447967
$506$	−1.78550e9	−0.612679
$507$	−1.63894e9	−0.558514
$508$	3.14019e9	1.06276
$509$	3.07905e9	1.03492	0.517458	−	0.855709i	$-0.326878\pi$
$509$	3.07905e9	1.03492	0.517458	+	0.855709i	$0.326878\pi$
$510$	8.76548e8	0.292604
$511$	−5.35599e8	−0.177569
$512$	−4.45593e7	−0.0146721
$513$	1.48998e8	0.0487271
$514$	−1.10221e9	−0.358009
$515$	1.23649e9	0.398901
$516$	−1.27452e9	−0.408387
$517$	−1.52470e9	−0.485253
$518$	−1.75632e9	−0.555199
$519$	6.55086e8	0.205690
$520$	4.30209e9	1.34174
$521$	1.45110e9	0.449538	0.224769	−	0.974412i	$-0.427837\pi$
$521$	1.45110e9	0.449538	0.224769	+	0.974412i	$0.427837\pi$
$522$	3.05728e8	0.0940782
$523$	−6.11321e8	−0.186859	−0.0934293	−	0.995626i	$-0.529783\pi$
$523$	−6.11321e8	−0.186859	−0.0934293	+	0.995626i	$0.529783\pi$
$524$	−3.41159e9	−1.03585
$525$	1.25054e8	0.0377173
$526$	1.70079e9	0.509566
$527$	−9.73966e8	−0.289872
$528$	9.72639e6	0.00287563
$529$	3.79259e9	1.11389
$530$	−1.78179e8	−0.0519864
$531$	1.49721e8	0.0433963
$532$	3.96564e8	0.114189
$533$	6.22561e9	1.78089
$534$	1.10691e9	0.314571
$535$	4.16010e9	1.17453
$536$	1.06106e9	0.297620
$537$	3.03905e8	0.0846892
$538$	4.00454e8	0.110870
$539$	−1.13981e9	−0.313524
$540$	−4.13182e8	−0.112918
$541$	4.85916e9	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$541$	4.85916e9	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$542$	1.80637e9	0.487316
$543$	−3.01476e8	−0.0808078
$544$	3.20492e9	0.853533
$545$	1.48043e8	0.0391741
$546$	−1.40250e9	−0.368747
$547$	−4.87899e9	−1.27460	−0.637300	−	0.770616i	$-0.719949\pi$
$547$	−4.87899e9	−1.27460	−0.637300	+	0.770616i	$0.719949\pi$
$548$	2.60892e9	0.677217
$549$	7.91129e8	0.204053
$550$	−1.46416e8	−0.0375249
$551$	−4.52086e8	−0.115130
$552$	−3.32464e9	−0.841313
$553$	−2.35186e9	−0.591389
$554$	−4.37186e9	−1.09240
$555$	−2.70592e9	−0.671877
$556$	3.75642e9	0.926856
$557$	3.74696e9	0.918725	0.459362	−	0.888249i	$-0.348078\pi$
$557$	3.74696e9	0.918725	0.459362	+	0.888249i	$0.348078\pi$
$558$	−2.87711e8	−0.0701030
$559$	−6.66531e9	−1.61391
$560$	2.13478e7	0.00513682
$561$	−1.40233e9	−0.335336
$562$	2.10996e9	0.501414
$563$	1.61119e9	0.380512	0.190256	−	0.981735i	$-0.439068\pi$
$563$	1.61119e9	0.380512	0.190256	+	0.981735i	$0.439068\pi$
$564$	−1.08084e9	−0.253680
$565$	2.16876e8	0.0505873
$566$	−3.07394e9	−0.712588
$567$	3.53811e8	0.0815137
$568$	−6.25513e9	−1.43224
$569$	1.15056e9	0.261828	0.130914	−	0.991394i	$-0.458209\pi$
$569$	1.15056e9	0.261828	0.130914	+	0.991394i	$0.458209\pi$
$570$	−3.82890e8	−0.0865987
$571$	−4.25054e9	−0.955472	−0.477736	−	0.878503i	$-0.658543\pi$
$571$	−4.25054e9	−0.955472	−0.477736	+	0.878503i	$0.658543\pi$
$572$	−2.62027e9	−0.585410
$573$	9.70456e8	0.215494
$574$	2.61957e9	0.578148
$575$	5.90209e8	0.129470
$576$	9.57953e8	0.208865
$577$	2.31080e9	0.500780	0.250390	−	0.968145i	$-0.419441\pi$
$577$	2.31080e9	0.500780	0.250390	+	0.968145i	$0.419441\pi$
$578$	−7.72580e8	−0.166416
$579$	−7.51695e8	−0.160941
$580$	1.25366e9	0.266798
$581$	−5.91096e9	−1.25038
$582$	−2.78959e9	−0.586557
$583$	2.85056e8	0.0595785
$584$	1.16766e9	0.242589
$585$	−2.16080e9	−0.446241
$586$	−2.76948e9	−0.568534
$587$	−4.19562e9	−0.856175	−0.428087	−	0.903737i	$-0.640812\pi$
$587$	−4.19562e9	−0.856175	−0.428087	+	0.903737i	$0.640812\pi$
$588$	−8.07994e8	−0.163903
$589$	4.25443e8	0.0857902
$590$	−3.84748e8	−0.0771248
$591$	−2.42412e9	−0.483057
$592$	4.51547e7	0.00894493
$593$	−7.48887e9	−1.47477	−0.737386	−	0.675472i	$-0.763940\pi$
$593$	−7.48887e9	−1.47477	−0.737386	+	0.675472i	$0.763940\pi$
$594$	−4.14250e8	−0.0810980
$595$	−3.07788e9	−0.599020
$596$	6.22611e9	1.20463
$597$	−5.32590e8	−0.102443
$598$	−6.61928e9	−1.26578
$599$	2.12227e9	0.403467	0.201733	−	0.979440i	$-0.435343\pi$
$599$	2.12227e9	0.403467	0.201733	+	0.979440i	$0.435343\pi$
$600$	−2.72630e8	−0.0515281
$601$	−6.18903e8	−0.116295	−0.0581477	−	0.998308i	$-0.518519\pi$
$601$	−6.18903e8	−0.116295	−0.0581477	+	0.998308i	$0.518519\pi$
$602$	−2.80459e9	−0.523939
$603$	−5.32935e8	−0.0989838
$604$	4.13183e9	0.762979
$605$	2.80242e9	0.514504
$606$	9.21434e8	0.168194
$607$	−4.99735e8	−0.0906942	−0.0453471	−	0.998971i	$-0.514439\pi$
$607$	−4.99735e8	−0.0906942	−0.0453471	+	0.998971i	$0.514439\pi$
$608$	−1.39996e9	−0.252611
$609$	−1.07352e9	−0.192597
$610$	−2.03301e9	−0.362648
$611$	−5.65245e9	−1.00252
$612$	−9.94093e8	−0.175306
$613$	7.99751e9	1.40231	0.701154	−	0.713010i	$-0.252669\pi$
$613$	7.99751e9	1.40231	0.701154	+	0.713010i	$0.252669\pi$
$614$	−1.93371e9	−0.337133
$615$	4.03593e9	0.699649
$616$	−2.89603e9	−0.499195
$617$	1.58063e9	0.270914	0.135457	−	0.990783i	$-0.456750\pi$
$617$	1.58063e9	0.270914	0.135457	+	0.990783i	$0.456750\pi$
$618$	8.78798e8	0.149772
$619$	6.43846e9	1.09110	0.545550	−	0.838078i	$-0.316321\pi$
$619$	6.43846e9	1.09110	0.545550	+	0.838078i	$0.316321\pi$
$620$	−1.17978e9	−0.198806
$621$	1.66986e9	0.279807
$622$	−1.97840e9	−0.329646
$623$	−3.88677e9	−0.643992
$624$	3.60581e7	0.00594096
$625$	−5.51161e9	−0.903022
$626$	−5.64409e9	−0.919568
$627$	6.12559e8	0.0992456
$628$	4.18695e9	0.674589
$629$	−6.51030e9	−1.04310
$630$	−9.09209e8	−0.144868
$631$	−1.07401e10	−1.70179	−0.850896	−	0.525334i	$-0.823940\pi$
$631$	−1.07401e10	−1.70179	−0.850896	+	0.525334i	$0.823940\pi$
$632$	5.12728e9	0.807936
$633$	−4.30973e9	−0.675362
$634$	1.56037e9	0.243173
$635$	1.06461e10	1.65000
$636$	2.02072e8	0.0311463
$637$	−4.22554e9	−0.647731
$638$	1.25690e9	0.191615
$639$	3.14175e9	0.476342
$640$	3.85335e9	0.581043
$641$	4.76542e9	0.714658	0.357329	−	0.933978i	$-0.383687\pi$
$641$	4.76542e9	0.714658	0.357329	+	0.933978i	$0.383687\pi$
$642$	2.95666e9	0.440990
$643$	−5.32986e9	−0.790637	−0.395319	−	0.918544i	$-0.629366\pi$
$643$	−5.32986e9	−0.790637	−0.395319	+	0.918544i	$0.629366\pi$
$644$	4.44439e9	0.655709
$645$	−4.32097e9	−0.634048
$646$	−9.21211e8	−0.134445
$647$	3.64298e9	0.528801	0.264400	−	0.964413i	$-0.414826\pi$
$647$	3.64298e9	0.528801	0.264400	+	0.964413i	$0.414826\pi$
$648$	−7.71342e8	−0.111361
$649$	6.15531e8	0.0883881
$650$	−5.42801e8	−0.0775253
$651$	1.01026e9	0.143515
$652$	−5.49423e9	−0.776319
$653$	3.76641e9	0.529337	0.264668	−	0.964340i	$-0.414738\pi$
$653$	3.76641e9	0.529337	0.264668	+	0.964340i	$0.414738\pi$
$654$	1.05217e8	0.0147083
$655$	−1.15662e10	−1.60823
$656$	−6.73490e7	−0.00931467
$657$	−5.86477e8	−0.0806811
$658$	−2.37840e9	−0.325458
$659$	−4.55405e9	−0.619867	−0.309934	−	0.950758i	$-0.600307\pi$
$659$	−4.55405e9	−0.619867	−0.309934	+	0.950758i	$0.600307\pi$
$660$	−1.69866e9	−0.229987
$661$	−1.46628e9	−0.197475	−0.0987377	−	0.995113i	$-0.531480\pi$
$661$	−1.46628e9	−0.197475	−0.0987377	+	0.995113i	$0.531480\pi$
$662$	−2.00775e9	−0.268972
$663$	−5.19878e9	−0.692794
$664$	1.28865e10	1.70823
$665$	1.34446e9	0.177285
$666$	−1.92315e9	−0.252263
$667$	−5.06663e9	−0.661118
$668$	−6.58151e9	−0.854295
$669$	3.29743e9	0.425779
$670$	1.36952e9	0.175916
$671$	3.25247e9	0.415609
$672$	−3.32433e9	−0.422583
$673$	1.29610e10	1.63903	0.819514	−	0.573059i	$-0.194244\pi$
$673$	1.29610e10	1.63903	0.819514	+	0.573059i	$0.194244\pi$
$674$	−3.39451e9	−0.427039
$675$	1.36933e8	0.0171374
$676$	−4.77645e9	−0.594692
$677$	2.05035e9	0.253962	0.126981	−	0.991905i	$-0.459471\pi$
$677$	2.05035e9	0.253962	0.126981	+	0.991905i	$0.459471\pi$
$678$	1.54138e8	0.0189935
$679$	9.79525e9	1.20080
$680$	6.71007e9	0.818362
$681$	2.85556e8	0.0346478
$682$	−1.18283e9	−0.142783
$683$	1.07567e9	0.129184	0.0645918	−	0.997912i	$-0.479425\pi$
$683$	1.07567e9	0.129184	0.0645918	+	0.997912i	$0.479425\pi$
$684$	4.34235e8	0.0518834
$685$	8.84495e9	1.05143
$686$	−5.62817e9	−0.665629
$687$	−1.47121e9	−0.173111
$688$	7.21056e7	0.00844130
$689$	1.05677e9	0.123087
$690$	−4.29113e9	−0.497279
$691$	4.59832e9	0.530183	0.265092	−	0.964223i	$-0.414598\pi$
$691$	4.59832e9	0.530183	0.265092	+	0.964223i	$0.414598\pi$
$692$	1.90916e9	0.219013
$693$	1.45458e9	0.166024
$694$	4.49491e9	0.510461
$695$	1.27353e10	1.43901
$696$	2.34038e9	0.263120
$697$	9.71022e9	1.08621
$698$	2.98328e8	0.0332048
$699$	−2.79162e9	−0.309162
$700$	3.64453e8	0.0401604
$701$	9.38304e9	1.02880	0.514400	−	0.857551i	$-0.328015\pi$
$701$	9.38304e9	1.02880	0.514400	+	0.857551i	$0.328015\pi$
$702$	−1.53573e9	−0.167546
$703$	2.84380e9	0.308713
$704$	3.93832e9	0.425409
$705$	−3.66436e9	−0.393854
$706$	7.90079e9	0.844995
$707$	−3.23549e9	−0.344327
$708$	4.36342e8	0.0462073
$709$	−1.12461e10	−1.18506	−0.592528	−	0.805550i	$-0.701870\pi$
$709$	−1.12461e10	−1.18506	−0.592528	+	0.805550i	$0.701870\pi$
$710$	−8.07354e9	−0.846564
$711$	−2.57527e9	−0.268707
$712$	8.47353e9	0.879800
$713$	4.76804e9	0.492636
$714$	−2.18751e9	−0.224908
$715$	−8.88345e9	−0.908889
$716$	8.85689e8	0.0901750
$717$	2.48338e9	0.251609
$718$	9.53234e7	0.00961090
$719$	−6.98063e9	−0.700396	−0.350198	−	0.936676i	$-0.613886\pi$
$719$	−6.98063e9	−0.700396	−0.350198	+	0.936676i	$0.613886\pi$
$720$	2.33757e7	0.00233400
$721$	−3.08578e9	−0.306613
$722$	−5.87461e9	−0.580896
$723$	8.71802e9	0.857894
$724$	−8.78610e8	−0.0860421
$725$	−4.15478e8	−0.0404917
$726$	1.99173e9	0.193176
$727$	7.76937e9	0.749921	0.374960	−	0.927041i	$-0.377656\pi$
$727$	7.76937e9	0.749921	0.374960	+	0.927041i	$0.377656\pi$
$728$	−1.07363e10	−1.03132
$729$	3.87420e8	0.0370370
$730$	1.50710e9	0.143388
$731$	−1.03960e10	−0.984366
$732$	2.30564e9	0.217271
$733$	−1.54005e10	−1.44434	−0.722172	−	0.691713i	$-0.756856\pi$
$733$	−1.54005e10	−1.44434	−0.722172	+	0.691713i	$0.756856\pi$
$734$	5.90911e9	0.551551
$735$	−2.73932e9	−0.254471
$736$	−1.56896e10	−1.45058
$737$	−2.19099e9	−0.201607
$738$	2.86841e9	0.262691
$739$	8.56257e9	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$739$	8.56257e9	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$740$	−7.88604e9	−0.715398
$741$	2.27090e9	0.205038
$742$	4.44661e8	0.0399591
$743$	−3.55238e9	−0.317730	−0.158865	−	0.987300i	$-0.550783\pi$
$743$	−3.55238e9	−0.317730	−0.158865	+	0.987300i	$0.550783\pi$
$744$	−2.20246e9	−0.196066
$745$	2.11083e10	1.87027
$746$	−9.87694e9	−0.871037
$747$	−6.47246e9	−0.568129
$748$	−4.08689e9	−0.357057
$749$	−1.03819e10	−0.902798
$750$	−4.30349e9	−0.372483
$751$	1.09272e10	0.941391	0.470696	−	0.882296i	$-0.344003\pi$
$751$	1.09272e10	0.941391	0.470696	+	0.882296i	$0.344003\pi$
$752$	6.11484e7	0.00524352
$753$	−7.51445e9	−0.641379
$754$	4.65965e9	0.395871
$755$	1.40080e10	1.18458
$756$	1.03113e9	0.0867938
$757$	6.79624e9	0.569421	0.284710	−	0.958614i	$-0.408103\pi$
$757$	6.79624e9	0.569421	0.284710	+	0.958614i	$0.408103\pi$
$758$	−5.21390e9	−0.434831
$759$	6.86509e9	0.569902
$760$	−2.93106e9	−0.242202
$761$	1.52139e10	1.25139	0.625696	−	0.780067i	$-0.284815\pi$
$761$	1.52139e10	1.25139	0.625696	+	0.780067i	$0.284815\pi$
$762$	7.56642e9	0.619509
$763$	−3.69454e8	−0.0301109
$764$	2.82826e9	0.229452
$765$	−3.37025e9	−0.272174
$766$	−5.58133e9	−0.448680
$767$	2.28192e9	0.182607
$768$	7.28005e9	0.579923
$769$	−1.81575e10	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$769$	−1.81575e10	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$770$	−3.73792e9	−0.295062
$771$	4.23791e9	0.333013
$772$	−2.19071e9	−0.171366
$773$	1.26937e10	0.988459	0.494230	−	0.869331i	$-0.335450\pi$
$773$	1.26937e10	0.988459	0.494230	+	0.869331i	$0.335450\pi$
$774$	−3.07100e9	−0.238060
$775$	3.90993e8	0.0301726
$776$	−2.13546e10	−1.64050
$777$	6.75288e9	0.516435
$778$	−6.00682e9	−0.457316
$779$	−4.24157e9	−0.321474
$780$	−6.29737e9	−0.475147
$781$	1.29163e10	0.970196
$782$	−1.03242e10	−0.772030
$783$	−1.17550e9	−0.0875096
$784$	4.57121e7	0.00338786
$785$	1.41949e10	1.04734
$786$	−8.22035e9	−0.603826
$787$	−2.24532e10	−1.64198	−0.820988	−	0.570946i	$-0.806577\pi$
$787$	−2.24532e10	−1.64198	−0.820988	+	0.570946i	$0.806577\pi$
$788$	−7.06477e9	−0.514347
$789$	−6.53938e9	−0.473988
$790$	6.61782e9	0.477551
$791$	−5.41234e8	−0.0388837
$792$	−3.17113e9	−0.226817
$793$	1.20577e10	0.858635
$794$	2.27175e9	0.161060
$795$	6.85081e8	0.0483567
$796$	−1.55216e9	−0.109079
$797$	4.23290e9	0.296165	0.148083	−	0.988975i	$-0.452690\pi$
$797$	4.23290e9	0.296165	0.148083	+	0.988975i	$0.452690\pi$
$798$	9.55536e8	0.0665636
$799$	−8.81624e9	−0.611462
$800$	−1.28660e9	−0.0888438
$801$	−4.25598e9	−0.292608
$802$	−1.51357e9	−0.103608
$803$	−2.41111e9	−0.164328
$804$	−1.55317e9	−0.105395
$805$	1.50677e10	1.01803
$806$	−4.38504e9	−0.294986
$807$	−1.53971e9	−0.103129
$808$	7.05367e9	0.470409
$809$	−5.59705e9	−0.371654	−0.185827	−	0.982582i	$-0.559496\pi$
$809$	−5.59705e9	−0.371654	−0.185827	+	0.982582i	$0.559496\pi$
$810$	−9.95577e8	−0.0658230
$811$	2.20508e10	1.45162	0.725808	−	0.687897i	$-0.241466\pi$
$811$	2.20508e10	1.45162	0.725808	+	0.687897i	$0.241466\pi$
$812$	−3.12863e9	−0.205073
$813$	−6.94535e9	−0.453291
$814$	−7.90643e9	−0.513801
$815$	−1.86270e10	−1.20529
$816$	5.62406e7	0.00362355
$817$	4.54114e9	0.291332
$818$	1.45746e10	0.931020
$819$	5.39249e9	0.343001
$820$	1.17622e10	0.744969
$821$	1.92550e10	1.21435	0.607173	−	0.794569i	$-0.292303\pi$
$821$	1.92550e10	1.21435	0.607173	+	0.794569i	$0.292303\pi$
$822$	6.28628e9	0.394768
$823$	−2.63648e10	−1.64864	−0.824318	−	0.566127i	$-0.808441\pi$
$823$	−2.63648e10	−1.64864	−0.824318	+	0.566127i	$0.808441\pi$
$824$	6.72729e9	0.418885
$825$	5.62957e8	0.0349049
$826$	9.60173e8	0.0592815
$827$	2.32586e10	1.42993	0.714965	−	0.699160i	$-0.246443\pi$
$827$	2.32586e10	1.42993	0.714965	+	0.699160i	$0.246443\pi$
$828$	4.86657e9	0.297932
$829$	1.16870e10	0.712464	0.356232	−	0.934398i	$-0.384061\pi$
$829$	1.16870e10	0.712464	0.356232	+	0.934398i	$0.384061\pi$
$830$	1.66326e10	1.00969
$831$	1.68094e10	1.01613
$832$	1.46003e10	0.878881
$833$	−6.59067e9	−0.395068
$834$	9.05124e9	0.540290
$835$	−2.23132e10	−1.32635
$836$	1.78522e9	0.105674
$837$	1.10622e9	0.0652084
$838$	−1.72879e10	−1.01482
$839$	3.29655e10	1.92705	0.963524	−	0.267623i	$-0.0862383\pi$
$839$	3.29655e10	1.92705	0.963524	+	0.267623i	$0.0862383\pi$
$840$	−6.96009e9	−0.405170
$841$	−1.36832e10	−0.793236
$842$	1.51183e10	0.872791
$843$	−8.11260e9	−0.466405
$844$	−1.25601e10	−0.719109
$845$	−1.61935e10	−0.923299
$846$	−2.60433e9	−0.147877
$847$	−6.99369e9	−0.395471
$848$	−1.14322e7	−0.000643790 0
$849$	1.18190e10	0.662835
$850$	−8.46617e8	−0.0472847
$851$	3.18711e10	1.77274
$852$	9.15619e9	0.507197
$853$	−6.75557e9	−0.372684	−0.186342	−	0.982485i	$-0.559663\pi$
$853$	−6.75557e9	−0.372684	−0.186342	+	0.982485i	$0.559663\pi$
$854$	5.07356e9	0.278747
$855$	1.47218e9	0.0805524
$856$	2.26336e10	1.23337
$857$	3.29149e10	1.78632	0.893162	−	0.449735i	$-0.148482\pi$
$857$	3.29149e10	1.78632	0.893162	+	0.449735i	$0.148482\pi$
$858$	−6.31365e9	−0.341252
$859$	1.99567e10	1.07427	0.537135	−	0.843496i	$-0.319506\pi$
$859$	1.99567e10	1.07427	0.537135	+	0.843496i	$0.319506\pi$
$860$	−1.25929e10	−0.675119
$861$	−1.00720e10	−0.537782
$862$	−1.93636e10	−1.02970
$863$	1.08133e10	0.572689	0.286344	−	0.958127i	$-0.407560\pi$
$863$	1.08133e10	0.572689	0.286344	+	0.958127i	$0.407560\pi$
$864$	−3.64012e9	−0.192007
$865$	6.47257e9	0.340033
$866$	8.59828e9	0.449882
$867$	2.97050e9	0.154797
$868$	2.94425e9	0.152811
$869$	−1.05874e10	−0.547293
$870$	3.02075e9	0.155524
$871$	−8.12255e9	−0.416513
$872$	8.05445e8	0.0411366
$873$	1.07257e10	0.545603
$874$	4.50978e9	0.228489
$875$	1.51111e10	0.762549
$876$	−1.70921e9	−0.0859073
$877$	3.04464e10	1.52418	0.762091	−	0.647470i	$-0.224173\pi$
$877$	3.04464e10	1.52418	0.762091	+	0.647470i	$0.224173\pi$
$878$	−1.26760e10	−0.632051
$879$	1.06484e10	0.528839
$880$	9.61016e7	0.00475380
$881$	−1.83036e10	−0.901821	−0.450910	−	0.892569i	$-0.648901\pi$
$881$	−1.83036e10	−0.901821	−0.450910	+	0.892569i	$0.648901\pi$
$882$	−1.94689e9	−0.0955437
$883$	−2.58285e10	−1.26251	−0.631257	−	0.775574i	$-0.717461\pi$
$883$	−2.58285e10	−1.26251	−0.631257	+	0.775574i	$0.717461\pi$
$884$	−1.51511e10	−0.737669
$885$	1.47932e9	0.0717399
$886$	1.83262e10	0.885227
$887$	−1.56417e10	−0.752578	−0.376289	−	0.926502i	$-0.622800\pi$
$887$	−1.56417e10	−0.752578	−0.376289	+	0.926502i	$0.622800\pi$
$888$	−1.47219e10	−0.705536
$889$	−2.65684e10	−1.26826
$890$	1.09368e10	0.520028
$891$	1.59276e9	0.0754357
$892$	9.60991e9	0.453359
$893$	3.85107e9	0.180968
$894$	1.50021e10	0.702214
$895$	3.00273e9	0.140003
$896$	−9.61638e9	−0.446615
$897$	2.54506e10	1.17740
$898$	1.43528e10	0.661410
$899$	−3.35647e9	−0.154072
$900$	3.99073e8	0.0182475
$901$	1.64827e9	0.0750743
$902$	1.17926e10	0.535039
$903$	1.07834e10	0.487358
$904$	1.17994e9	0.0531216
$905$	−2.97873e9	−0.133586
$906$	9.95579e9	0.444761
$907$	3.32082e10	1.47781	0.738906	−	0.673808i	$-0.235343\pi$
$907$	3.32082e10	1.47781	0.738906	+	0.673808i	$0.235343\pi$
$908$	8.32213e8	0.0368922
$909$	−3.54283e9	−0.156451
$910$	−1.38574e10	−0.609588
$911$	−6.12880e9	−0.268572	−0.134286	−	0.990943i	$-0.542874\pi$
$911$	−6.12880e9	−0.268572	−0.134286	+	0.990943i	$0.542874\pi$
$912$	−2.45667e7	−0.00107242
$913$	−2.66094e10	−1.15715
$914$	−7.93576e9	−0.343777
$915$	7.81675e9	0.337328
$916$	−4.28764e9	−0.184325
$917$	2.88646e10	1.23616
$918$	−2.39530e9	−0.102191
$919$	1.60841e10	0.683585	0.341793	−	0.939775i	$-0.388966\pi$
$919$	1.60841e10	0.683585	0.341793	+	0.939775i	$0.388966\pi$
$920$	−3.28491e10	−1.39080
$921$	7.43494e9	0.313595
$922$	7.32974e9	0.307986
$923$	4.78839e10	2.00439
$924$	4.23918e9	0.176779
$925$	2.61353e9	0.108575
$926$	2.71900e9	0.112531
$927$	−3.37890e9	−0.139315
$928$	1.10447e10	0.453667
$929$	−1.02473e10	−0.419327	−0.209664	−	0.977774i	$-0.567237\pi$
$929$	−1.02473e10	−0.419327	−0.209664	+	0.977774i	$0.567237\pi$
$930$	−2.84273e9	−0.115890
$931$	2.87890e9	0.116924
$932$	−8.13579e9	−0.329188
$933$	7.60679e9	0.306631
$934$	2.11607e9	0.0849798
$935$	−1.38557e10	−0.554355
$936$	−1.17561e10	−0.468597
$937$	−1.95793e10	−0.777516	−0.388758	−	0.921340i	$-0.627096\pi$
$937$	−1.95793e10	−0.777516	−0.388758	+	0.921340i	$0.627096\pi$
$938$	−3.41775e9	−0.135217
$939$	2.17010e10	0.855364
$940$	−1.06793e10	−0.419367
$941$	7.05944e8	0.0276189	0.0138095	−	0.999905i	$-0.495604\pi$
$941$	7.05944e8	0.0276189	0.0138095	+	0.999905i	$0.495604\pi$
$942$	1.00886e10	0.393236
$943$	−4.75363e10	−1.84601
$944$	−2.46860e7	−0.000955098 0
$945$	3.49583e9	0.134753
$946$	−1.26254e10	−0.484872
$947$	1.93347e10	0.739798	0.369899	−	0.929072i	$-0.379392\pi$
$947$	1.93347e10	0.739798	0.369899	+	0.929072i	$0.379392\pi$
$948$	−7.50526e9	−0.286112
$949$	−8.93858e9	−0.339498
$950$	3.69815e8	0.0139943
$951$	−5.99948e9	−0.226194
$952$	−1.67456e10	−0.629030
$953$	−2.96357e10	−1.10915	−0.554574	−	0.832134i	$-0.687119\pi$
$953$	−2.96357e10	−1.10915	−0.554574	+	0.832134i	$0.687119\pi$
$954$	4.86901e8	0.0181560
$955$	9.58859e9	0.356240
$956$	7.23747e9	0.267907
$957$	−4.83269e9	−0.178237
$958$	−3.06464e10	−1.12616
$959$	−2.20734e10	−0.808172
$960$	9.46505e9	0.345282
$961$	−2.43540e10	−0.885192
$962$	−2.93111e10	−1.06150
$963$	−1.13681e10	−0.410201
$964$	2.54075e10	0.913465
$965$	−7.42712e9	−0.266057
$966$	1.07089e10	0.382231
$967$	−2.65671e10	−0.944824	−0.472412	−	0.881378i	$-0.656617\pi$
$967$	−2.65671e10	−0.944824	−0.472412	+	0.881378i	$0.656617\pi$
$968$	1.52469e10	0.540280
$969$	3.54198e9	0.125058
$970$	−2.75625e10	−0.969657
$971$	−1.88208e10	−0.659736	−0.329868	−	0.944027i	$-0.607004\pi$
$971$	−1.88208e10	−0.659736	−0.329868	+	0.944027i	$0.607004\pi$
$972$	1.12908e9	0.0394361
$973$	−3.17821e10	−1.10608
$974$	2.76659e10	0.959376
$975$	2.08702e9	0.0721125
$976$	−1.30441e8	−0.00449096
$977$	1.00691e10	0.345430	0.172715	−	0.984972i	$-0.444746\pi$
$977$	1.00691e10	0.345430	0.172715	+	0.984972i	$0.444746\pi$
$978$	−1.32386e10	−0.452537
$979$	−1.74971e10	−0.595973
$980$	−7.98338e9	−0.270954
$981$	−4.04549e8	−0.0136814
$982$	−1.37127e10	−0.462098
$983$	−4.43932e10	−1.49066	−0.745332	−	0.666694i	$-0.767709\pi$
$983$	−4.43932e10	−1.49066	−0.745332	+	0.666694i	$0.767709\pi$
$984$	2.19580e10	0.734700
$985$	−2.39515e10	−0.798558
$986$	7.26775e9	0.241452
$987$	9.14474e9	0.302734
$988$	6.61824e9	0.218320
$989$	5.08936e10	1.67292
$990$	−4.09300e9	−0.134066
$991$	1.29610e10	0.423040	0.211520	−	0.977374i	$-0.432159\pi$
$991$	1.29610e10	0.423040	0.211520	+	0.977374i	$0.432159\pi$
$992$	−1.03938e10	−0.338053
$993$	7.71963e9	0.250193
$994$	2.01483e10	0.650707
$995$	−5.26225e9	−0.169352
$996$	−1.88631e10	−0.604930
$997$	−5.75699e10	−1.83977	−0.919883	−	0.392194i	$-0.871716\pi$
$997$	−5.75699e10	−1.83977	−0.919883	+	0.392194i	$0.871716\pi$
$998$	1.25211e10	0.398737
$999$	7.39436e9	0.234650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.11	✓	16
3.2	odd	2		531.8.a.b.1.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.11	✓	16	1.1	even	1	trivial
531.8.a.b.1.6		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)

\(f(q)\)	\(=\)	\(q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} -1873.35 q^{10} +2997.05 q^{11} +2124.57 q^{12} +11110.8 q^{13} +4675.13 q^{14} +7202.88 q^{15} -120.197 q^{16} +17329.7 q^{17} +5119.23 q^{18} -7569.90 q^{19} +20991.8 q^{20} -17975.5 q^{21} +21046.1 q^{22} -84837.6 q^{23} +39188.2 q^{24} -6956.93 q^{25} +78023.0 q^{26} -19683.0 q^{27} -52387.0 q^{28} +59721.5 q^{29} +50580.6 q^{30} -56202.0 q^{31} +184937. q^{32} -80920.4 q^{33} +121694. q^{34} -177607. q^{35} -57363.4 q^{36} -375672. q^{37} -53157.8 q^{38} -299992. q^{39} +387199. q^{40} +560321. q^{41} -126228. q^{42} -599895. q^{43} -235831. q^{44} -194478. q^{45} -595752. q^{46} -508735. q^{47} +3245.32 q^{48} -380309. q^{49} -48853.4 q^{50} -467903. q^{51} -874284. q^{52} +95112.1 q^{53} -138219. q^{54} -799533. q^{55} -966292. q^{56} +204387. q^{57} +419380. q^{58} +205379. q^{59} -566779. q^{60} +1.08522e6 q^{61} -394665. q^{62} +485337. q^{63} +1.31406e6 q^{64} -2.96407e6 q^{65} -568244. q^{66} -731050. q^{67} -1.36364e6 q^{68} +2.29062e6 q^{69} -1.24720e6 q^{70} +4.30967e6 q^{71} -1.05808e6 q^{72} -804495. q^{73} -2.63807e6 q^{74} +187837. q^{75} +595658. q^{76} +1.99531e6 q^{77} -2.10662e6 q^{78} -3.53260e6 q^{79} +32065.4 q^{80} +531441. q^{81} +3.93472e6 q^{82} -8.87854e6 q^{83} +1.41445e6 q^{84} -4.62312e6 q^{85} -4.21262e6 q^{86} -1.61248e6 q^{87} -4.34997e6 q^{88} -5.83811e6 q^{89} -1.36568e6 q^{90} +7.39710e6 q^{91} +6.67568e6 q^{92} +1.51745e6 q^{93} -3.57247e6 q^{94} +2.01945e6 q^{95} -4.99331e6 q^{96} +1.47129e7 q^{97} -2.67063e6 q^{98} +2.18485e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) 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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