LMFDB - Embedded newform 354.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} + \cdots - 22\!\cdots\!40 $ x^8 - x^7 - 103558x^6 + 5805883x^5 + 2559087821x^4 - 196601024266x^3 - 17055507548256x^2 + 1662873307981032x - 22179668074797840
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{7}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$77.9355$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -497.674 q^{5} -216.000 q^{6} -1339.98 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -497.674 q^{5} -216.000 q^{6} -1339.98 q^{7} +512.000 q^{8} +729.000 q^{9} -3981.39 q^{10} +2888.81 q^{11} -1728.00 q^{12} +9492.97 q^{13} -10719.9 q^{14} +13437.2 q^{15} +4096.00 q^{16} +32871.9 q^{17} +5832.00 q^{18} -36447.6 q^{19} -31851.1 q^{20} +36179.5 q^{21} +23110.5 q^{22} +23666.3 q^{23} -13824.0 q^{24} +169554. q^{25} +75943.7 q^{26} -19683.0 q^{27} -85758.9 q^{28} -60927.3 q^{29} +107498. q^{30} +289298. q^{31} +32768.0 q^{32} -77997.9 q^{33} +262975. q^{34} +666874. q^{35} +46656.0 q^{36} -320206. q^{37} -291581. q^{38} -256310. q^{39} -254809. q^{40} +142533. q^{41} +289436. q^{42} -757230. q^{43} +184884. q^{44} -362804. q^{45} +189330. q^{46} +187839. q^{47} -110592. q^{48} +972009. q^{49} +1.35644e6 q^{50} -887541. q^{51} +607550. q^{52} -1.18577e6 q^{53} -157464. q^{54} -1.43769e6 q^{55} -686071. q^{56} +984086. q^{57} -487418. q^{58} -205379. q^{59} +859981. q^{60} -2.98179e6 q^{61} +2.31438e6 q^{62} -976847. q^{63} +262144. q^{64} -4.72440e6 q^{65} -623983. q^{66} -1.28373e6 q^{67} +2.10380e6 q^{68} -638990. q^{69} +5.33499e6 q^{70} +3.16875e6 q^{71} +373248. q^{72} -187529. q^{73} -2.56165e6 q^{74} -4.57797e6 q^{75} -2.33265e6 q^{76} -3.87095e6 q^{77} -2.05048e6 q^{78} +4.42566e6 q^{79} -2.03847e6 q^{80} +531441. q^{81} +1.14027e6 q^{82} -1.07294e6 q^{83} +2.31549e6 q^{84} -1.63595e7 q^{85} -6.05784e6 q^{86} +1.64504e6 q^{87} +1.47907e6 q^{88} -4.04768e6 q^{89} -2.90243e6 q^{90} -1.27204e7 q^{91} +1.51464e6 q^{92} -7.81104e6 q^{93} +1.50271e6 q^{94} +1.81390e7 q^{95} -884736. q^{96} +5.04076e6 q^{97} +7.77607e6 q^{98} +2.10594e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * 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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	−497.674	−1.78053	−0.890266	−	0.455440i	$-0.849482\pi$
$5$	−497.674	−1.78053	−0.890266	+	0.455440i	$0.849482\pi$
$6$	−216.000	−0.408248
$7$	−1339.98	−1.47658	−0.738288	−	0.674485i	$-0.764366\pi$
$7$	−1339.98	−1.47658	−0.738288	+	0.674485i	$0.764366\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−3981.39	−1.25903
$11$	2888.81	0.654402	0.327201	−	0.944955i	$-0.393895\pi$
$11$	2888.81	0.654402	0.327201	+	0.944955i	$0.393895\pi$
$12$	−1728.00	−0.288675
$13$	9492.97	1.19840	0.599198	−	0.800601i	$-0.295486\pi$
$13$	9492.97	1.19840	0.599198	+	0.800601i	$0.295486\pi$
$14$	−10719.9	−1.04410
$15$	13437.2	1.02799
$16$	4096.00	0.250000
$17$	32871.9	1.62276	0.811379	−	0.584521i	$-0.198717\pi$
$17$	32871.9	1.62276	0.811379	+	0.584521i	$0.198717\pi$
$18$	5832.00	0.235702
$19$	−36447.6	−1.21908	−0.609540	−	0.792756i	$-0.708646\pi$
$19$	−36447.6	−1.21908	−0.609540	+	0.792756i	$0.708646\pi$
$20$	−31851.1	−0.890266
$21$	36179.5	0.852502
$22$	23110.5	0.462732
$23$	23666.3	0.405586	0.202793	−	0.979222i	$-0.434998\pi$
$23$	23666.3	0.405586	0.202793	+	0.979222i	$0.434998\pi$
$24$	−13824.0	−0.204124
$25$	169554.	2.17030
$26$	75943.7	0.847394
$27$	−19683.0	−0.192450
$28$	−85758.9	−0.738288
$29$	−60927.3	−0.463894	−0.231947	−	0.972728i	$-0.574510\pi$
$29$	−60927.3	−0.463894	−0.231947	+	0.972728i	$0.574510\pi$
$30$	107498.	0.726899
$31$	289298.	1.74413	0.872065	−	0.489390i	$-0.162780\pi$
$31$	289298.	1.74413	0.872065	+	0.489390i	$0.162780\pi$
$32$	32768.0	0.176777
$33$	−77997.9	−0.377819
$34$	262975.	1.14746
$35$	666874.	2.62909
$36$	46656.0	0.166667
$37$	−320206.	−1.03926	−0.519629	−	0.854392i	$-0.673930\pi$
$37$	−320206.	−1.03926	−0.519629	+	0.854392i	$0.673930\pi$
$38$	−291581.	−0.862019
$39$	−256310.	−0.691894
$40$	−254809.	−0.629513
$41$	142533.	0.322978	0.161489	−	0.986874i	$-0.448370\pi$
$41$	142533.	0.322978	0.161489	+	0.986874i	$0.448370\pi$
$42$	289436.	0.602810
$43$	−757230.	−1.45241	−0.726203	−	0.687480i	$-0.758717\pi$
$43$	−757230.	−1.45241	−0.726203	+	0.687480i	$0.758717\pi$
$44$	184884.	0.327201
$45$	−362804.	−0.593511
$46$	189330.	0.286792
$47$	187839.	0.263902	0.131951	−	0.991256i	$-0.457876\pi$
$47$	187839.	0.263902	0.131951	+	0.991256i	$0.457876\pi$
$48$	−110592.	−0.144338
$49$	972009.	1.18028
$50$	1.35644e6	1.53463
$51$	−887541.	−0.936899
$52$	607550.	0.599198
$53$	−1.18577e6	−1.09404	−0.547020	−	0.837120i	$-0.684238\pi$
$53$	−1.18577e6	−1.09404	−0.547020	+	0.837120i	$0.684238\pi$
$54$	−157464.	−0.136083
$55$	−1.43769e6	−1.16518
$56$	−686071.	−0.522049
$57$	984086.	0.703836
$58$	−487418.	−0.328023
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	859981.	0.513995
$61$	−2.98179e6	−1.68199	−0.840994	−	0.541044i	$-0.818029\pi$
$61$	−2.98179e6	−1.68199	−0.840994	+	0.541044i	$0.818029\pi$
$62$	2.31438e6	1.23329
$63$	−976847.	−0.492192
$64$	262144.	0.125000
$65$	−4.72440e6	−2.13378
$66$	−623983.	−0.267158
$67$	−1.28373e6	−0.521448	−0.260724	−	0.965413i	$-0.583961\pi$
$67$	−1.28373e6	−0.521448	−0.260724	+	0.965413i	$0.583961\pi$
$68$	2.10380e6	0.811379
$69$	−638990.	−0.234165
$70$	5.33499e6	1.85905
$71$	3.16875e6	1.05071	0.525356	−	0.850883i	$-0.323932\pi$
$71$	3.16875e6	1.05071	0.525356	+	0.850883i	$0.323932\pi$
$72$	373248.	0.117851
$73$	−187529.	−0.0564207	−0.0282104	−	0.999602i	$-0.508981\pi$
$73$	−187529.	−0.0564207	−0.0282104	+	0.999602i	$0.508981\pi$
$74$	−2.56165e6	−0.734867
$75$	−4.57797e6	−1.25302
$76$	−2.33265e6	−0.609540
$77$	−3.87095e6	−0.966274
$78$	−2.05048e6	−0.489243
$79$	4.42566e6	1.00991	0.504955	−	0.863145i	$-0.331509\pi$
$79$	4.42566e6	1.00991	0.504955	+	0.863145i	$0.331509\pi$
$80$	−2.03847e6	−0.445133
$81$	531441.	0.111111
$82$	1.14027e6	0.228380
$83$	−1.07294e6	−0.205970	−0.102985	−	0.994683i	$-0.532839\pi$
$83$	−1.07294e6	−0.205970	−0.102985	+	0.994683i	$0.532839\pi$
$84$	2.31549e6	0.426251
$85$	−1.63595e7	−2.88937
$86$	−6.05784e6	−1.02701
$87$	1.64504e6	0.267829
$88$	1.47907e6	0.231366
$89$	−4.04768e6	−0.608613	−0.304307	−	0.952574i	$-0.598425\pi$
$89$	−4.04768e6	−0.608613	−0.304307	+	0.952574i	$0.598425\pi$
$90$	−2.90243e6	−0.419676
$91$	−1.27204e7	−1.76952
$92$	1.51464e6	0.202793
$93$	−7.81104e6	−1.00697
$94$	1.50271e6	0.186607
$95$	1.81390e7	2.17061
$96$	−884736.	−0.102062
$97$	5.04076e6	0.560783	0.280392	−	0.959886i	$-0.409536\pi$
$97$	5.04076e6	0.560783	0.280392	+	0.959886i	$0.409536\pi$
$98$	7.77607e6	0.834582
$99$	2.10594e6	0.218134
$100$	1.08515e7	1.08515
$101$	−3.61963e6	−0.349574	−0.174787	−	0.984606i	$-0.555924\pi$
$101$	−3.61963e6	−0.349574	−0.174787	+	0.984606i	$0.555924\pi$
$102$	−7.10033e6	−0.662488
$103$	1.61190e7	1.45348	0.726739	−	0.686914i	$-0.241035\pi$
$103$	1.61190e7	1.45348	0.726739	+	0.686914i	$0.241035\pi$
$104$	4.86040e6	0.423697
$105$	−1.80056e7	−1.51791
$106$	−9.48612e6	−0.773603
$107$	2.08801e7	1.64774	0.823872	−	0.566776i	$-0.191810\pi$
$107$	2.08801e7	1.64774	0.823872	+	0.566776i	$0.191810\pi$
$108$	−1.25971e6	−0.0962250
$109$	2.22588e7	1.64630	0.823151	−	0.567822i	$-0.192214\pi$
$109$	2.22588e7	1.64630	0.823151	+	0.567822i	$0.192214\pi$
$110$	−1.15015e7	−0.823910
$111$	8.64556e6	0.600016
$112$	−5.48857e6	−0.369144
$113$	−3.94290e6	−0.257064	−0.128532	−	0.991705i	$-0.541026\pi$
$113$	−3.94290e6	−0.257064	−0.128532	+	0.991705i	$0.541026\pi$
$114$	7.87269e6	0.497687
$115$	−1.17781e7	−0.722158
$116$	−3.89935e6	−0.231947
$117$	6.92037e6	0.399465
$118$	−1.64303e6	−0.0920575
$119$	−4.40478e7	−2.39613
$120$	6.87984e6	0.363450
$121$	−1.11419e7	−0.571758
$122$	−2.38543e7	−1.18935
$123$	−3.84840e6	−0.186472
$124$	1.85150e7	0.872065
$125$	−4.55020e7	−2.08375
$126$	−7.81478e6	−0.348032
$127$	1.52784e7	0.661859	0.330929	−	0.943656i	$-0.392638\pi$
$127$	1.52784e7	0.661859	0.330929	+	0.943656i	$0.392638\pi$
$128$	2.09715e6	0.0883883
$129$	2.04452e7	0.838547
$130$	−3.77952e7	−1.50881
$131$	−3.99997e7	−1.55456	−0.777279	−	0.629156i	$-0.783401\pi$
$131$	−3.99997e7	−1.55456	−0.777279	+	0.629156i	$0.783401\pi$
$132$	−4.99186e6	−0.188910
$133$	4.88392e7	1.80006
$134$	−1.02698e7	−0.368719
$135$	9.79572e6	0.342664
$136$	1.68304e7	0.573731
$137$	−4.43999e7	−1.47523	−0.737616	−	0.675221i	$-0.764048\pi$
$137$	−4.43999e7	−1.47523	−0.737616	+	0.675221i	$0.764048\pi$
$138$	−5.11192e6	−0.165580
$139$	−2.68900e7	−0.849255	−0.424628	−	0.905368i	$-0.639595\pi$
$139$	−2.68900e7	−0.849255	−0.424628	+	0.905368i	$0.639595\pi$
$140$	4.26800e7	1.31455
$141$	−5.07165e6	−0.152364
$142$	2.53500e7	0.742965
$143$	2.74234e7	0.784232
$144$	2.98598e6	0.0833333
$145$	3.03219e7	0.825978
$146$	−1.50023e6	−0.0398955
$147$	−2.62442e7	−0.681433
$148$	−2.04932e7	−0.519629
$149$	5.16405e6	0.127891	0.0639453	−	0.997953i	$-0.479632\pi$
$149$	5.16405e6	0.127891	0.0639453	+	0.997953i	$0.479632\pi$
$150$	−3.66237e7	−0.886020
$151$	−1.98488e7	−0.469153	−0.234577	−	0.972098i	$-0.575370\pi$
$151$	−1.98488e7	−0.469153	−0.234577	+	0.972098i	$0.575370\pi$
$152$	−1.86612e7	−0.431010
$153$	2.39636e7	0.540919
$154$	−3.09676e7	−0.683259
$155$	−1.43976e8	−3.10548
$156$	−1.64038e7	−0.345947
$157$	5.80193e7	1.19653	0.598265	−	0.801298i	$-0.295857\pi$
$157$	5.80193e7	1.19653	0.598265	+	0.801298i	$0.295857\pi$
$158$	3.54053e7	0.714115
$159$	3.20157e7	0.631644
$160$	−1.63078e7	−0.314757
$161$	−3.17124e7	−0.598878
$162$	4.25153e6	0.0785674
$163$	5.55168e7	1.00408	0.502039	−	0.864845i	$-0.332583\pi$
$163$	5.55168e7	1.00408	0.502039	+	0.864845i	$0.332583\pi$
$164$	9.12214e6	0.161489
$165$	3.88175e7	0.672719
$166$	−8.58355e6	−0.145643
$167$	−7.43688e7	−1.23562	−0.617808	−	0.786329i	$-0.711979\pi$
$167$	−7.43688e7	−1.23562	−0.617808	+	0.786329i	$0.711979\pi$
$168$	1.85239e7	0.301405
$169$	2.73679e7	0.436152
$170$	−1.30876e8	−2.04309
$171$	−2.65703e7	−0.406360
$172$	−4.84627e7	−0.726203
$173$	−3.00030e7	−0.440558	−0.220279	−	0.975437i	$-0.570697\pi$
$173$	−3.00030e7	−0.440558	−0.220279	+	0.975437i	$0.570697\pi$
$174$	1.31603e7	0.189384
$175$	−2.27200e8	−3.20461
$176$	1.18326e7	0.163600
$177$	5.54523e6	0.0751646
$178$	−3.23815e7	−0.430354
$179$	−824966.	−0.0107510	−0.00537552	−	0.999986i	$-0.501711\pi$
$179$	−824966.	−0.0107510	−0.00537552	+	0.999986i	$0.501711\pi$
$180$	−2.32195e7	−0.296755
$181$	−1.25776e8	−1.57660	−0.788302	−	0.615289i	$-0.789039\pi$
$181$	−1.25776e8	−1.57660	−0.788302	+	0.615289i	$0.789039\pi$
$182$	−1.01763e8	−1.25124
$183$	8.05084e7	0.971096
$184$	1.21171e7	0.143396
$185$	1.59358e8	1.85043
$186$	−6.24883e7	−0.712038
$187$	9.49607e7	1.06194
$188$	1.20217e7	0.131951
$189$	2.63749e7	0.284167
$190$	1.45112e8	1.53485
$191$	1.31758e8	1.36824	0.684118	−	0.729372i	$-0.260188\pi$
$191$	1.31758e8	1.36824	0.684118	+	0.729372i	$0.260188\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.02935e8	−1.03066	−0.515328	−	0.856993i	$-0.672330\pi$
$193$	−1.02935e8	−1.03066	−0.515328	+	0.856993i	$0.672330\pi$
$194$	4.03261e7	0.396534
$195$	1.27559e8	1.23194
$196$	6.22086e7	0.590139
$197$	−1.38077e8	−1.28674	−0.643368	−	0.765557i	$-0.722464\pi$
$197$	−1.38077e8	−1.28674	−0.643368	+	0.765557i	$0.722464\pi$
$198$	1.68475e7	0.154244
$199$	−1.11025e8	−0.998703	−0.499351	−	0.866400i	$-0.666428\pi$
$199$	−1.11025e8	−0.998703	−0.499351	+	0.866400i	$0.666428\pi$
$200$	8.68118e7	0.767316
$201$	3.46606e7	0.301058
$202$	−2.89571e7	−0.247186
$203$	8.16414e7	0.684975
$204$	−5.68026e7	−0.468450
$205$	−7.09352e7	−0.575073
$206$	1.28952e8	1.02776
$207$	1.72527e7	0.135195
$208$	3.88832e7	0.299599
$209$	−1.05290e8	−0.797768
$210$	−1.44045e8	−1.07332
$211$	−2.26682e8	−1.66122	−0.830611	−	0.556853i	$-0.812009\pi$
$211$	−2.26682e8	−1.66122	−0.830611	+	0.556853i	$0.812009\pi$
$212$	−7.58890e7	−0.547020
$213$	−8.55562e7	−0.606629
$214$	1.67041e8	1.16513
$215$	3.76854e8	2.58606
$216$	−1.00777e7	−0.0680414
$217$	−3.87654e8	−2.57534
$218$	1.78071e8	1.16411
$219$	5.06329e6	0.0325745
$220$	−9.20119e7	−0.582592
$221$	3.12052e8	1.94471
$222$	6.91645e7	0.424275
$223$	2.28248e8	1.37829	0.689144	−	0.724625i	$-0.257987\pi$
$223$	2.28248e8	1.37829	0.689144	+	0.724625i	$0.257987\pi$
$224$	−4.39085e7	−0.261024
$225$	1.23605e8	0.723432
$226$	−3.15432e7	−0.181772
$227$	−1.29674e8	−0.735805	−0.367902	−	0.929864i	$-0.619924\pi$
$227$	−1.29674e8	−0.735805	−0.367902	+	0.929864i	$0.619924\pi$
$228$	6.29815e7	0.351918
$229$	1.08288e8	0.595876	0.297938	−	0.954585i	$-0.403701\pi$
$229$	1.08288e8	0.595876	0.297938	+	0.954585i	$0.403701\pi$
$230$	−9.42247e7	−0.510643
$231$	1.04516e8	0.557879
$232$	−3.11948e7	−0.164011
$233$	1.43924e8	0.745398	0.372699	−	0.927952i	$-0.378432\pi$
$233$	1.43924e8	0.745398	0.372699	+	0.927952i	$0.378432\pi$
$234$	5.53630e7	0.282465
$235$	−9.34826e7	−0.469887
$236$	−1.31443e7	−0.0650945
$237$	−1.19493e8	−0.583072
$238$	−3.52382e8	−1.69432
$239$	2.86252e8	1.35630	0.678151	−	0.734923i	$-0.262782\pi$
$239$	2.86252e8	1.35630	0.678151	+	0.734923i	$0.262782\pi$
$240$	5.50388e7	0.256998
$241$	5.43549e7	0.250138	0.125069	−	0.992148i	$-0.460085\pi$
$241$	5.43549e7	0.250138	0.125069	+	0.992148i	$0.460085\pi$
$242$	−8.91356e7	−0.404294
$243$	−1.43489e7	−0.0641500
$244$	−1.90835e8	−0.840994
$245$	−4.83744e8	−2.10152
$246$	−3.07872e7	−0.131855
$247$	−3.45996e8	−1.46094
$248$	1.48120e8	0.616643
$249$	2.89695e7	0.118917
$250$	−3.64016e8	−1.47343
$251$	−3.39700e8	−1.35593	−0.677965	−	0.735094i	$-0.737138\pi$
$251$	−3.39700e8	−1.35593	−0.677965	+	0.735094i	$0.737138\pi$
$252$	−6.25182e7	−0.246096
$253$	6.83674e7	0.265416
$254$	1.22227e8	0.468005
$255$	4.41706e8	1.66818
$256$	1.67772e7	0.0625000
$257$	−5.32775e7	−0.195784	−0.0978921	−	0.995197i	$-0.531210\pi$
$257$	−5.32775e7	−0.195784	−0.0978921	+	0.995197i	$0.531210\pi$
$258$	1.63562e8	0.592943
$259$	4.29070e8	1.53454
$260$	−3.02362e8	−1.06689
$261$	−4.44160e7	−0.154631
$262$	−3.19997e8	−1.09924
$263$	−4.77522e8	−1.61863	−0.809316	−	0.587373i	$-0.800162\pi$
$263$	−4.77522e8	−1.61863	−0.809316	+	0.587373i	$0.800162\pi$
$264$	−3.99349e7	−0.133579
$265$	5.90125e8	1.94797
$266$	3.90714e8	1.27284
$267$	1.09287e8	0.351383
$268$	−8.21585e7	−0.260724
$269$	−1.69788e8	−0.531833	−0.265916	−	0.963996i	$-0.585675\pi$
$269$	−1.69788e8	−0.531833	−0.265916	+	0.963996i	$0.585675\pi$
$270$	7.83657e7	0.242300
$271$	−2.74198e8	−0.836896	−0.418448	−	0.908241i	$-0.637426\pi$
$271$	−2.74198e8	−0.836896	−0.418448	+	0.908241i	$0.637426\pi$
$272$	1.34643e8	0.405689
$273$	3.43451e8	1.02163
$274$	−3.55199e8	−1.04315
$275$	4.89810e8	1.42025
$276$	−4.08953e7	−0.117082
$277$	5.78735e7	0.163606	0.0818032	−	0.996648i	$-0.473932\pi$
$277$	5.78735e7	0.163606	0.0818032	+	0.996648i	$0.473932\pi$
$278$	−2.15120e8	−0.600514
$279$	2.10898e8	0.581377
$280$	3.41440e8	0.929524
$281$	2.53874e8	0.682569	0.341284	−	0.939960i	$-0.389138\pi$
$281$	2.53874e8	0.682569	0.341284	+	0.939960i	$0.389138\pi$
$282$	−4.05732e7	−0.107738
$283$	−5.54246e8	−1.45362	−0.726809	−	0.686840i	$-0.758997\pi$
$283$	−5.54246e8	−1.45362	−0.726809	+	0.686840i	$0.758997\pi$
$284$	2.02800e8	0.525356
$285$	−4.89754e8	−1.25320
$286$	2.19387e8	0.554536
$287$	−1.90992e8	−0.476902
$288$	2.38879e7	0.0589256
$289$	6.70223e8	1.63334
$290$	2.42575e8	0.584055
$291$	−1.36101e8	−0.323768
$292$	−1.20019e7	−0.0282104
$293$	−7.65695e8	−1.77836	−0.889179	−	0.457560i	$-0.848724\pi$
$293$	−7.65695e8	−1.77836	−0.889179	+	0.457560i	$0.848724\pi$
$294$	−2.09954e8	−0.481846
$295$	1.02212e8	0.231806
$296$	−1.63945e8	−0.367433
$297$	−5.68605e7	−0.125940
$298$	4.13124e7	0.0904323
$299$	2.24663e8	0.486052
$300$	−2.92990e8	−0.626511
$301$	1.01467e9	2.14459
$302$	−1.58790e8	−0.331742
$303$	9.77301e7	0.201827
$304$	−1.49290e8	−0.304770
$305$	1.48396e9	2.99483
$306$	1.91709e8	0.382488
$307$	3.10619e8	0.612694	0.306347	−	0.951920i	$-0.400893\pi$
$307$	3.10619e8	0.612694	0.306347	+	0.951920i	$0.400893\pi$
$308$	−2.47741e8	−0.483137
$309$	−4.35213e8	−0.839165
$310$	−1.15181e9	−2.19591
$311$	−9.21261e8	−1.73669	−0.868343	−	0.495963i	$-0.834815\pi$
$311$	−9.21261e8	−1.73669	−0.868343	+	0.495963i	$0.834815\pi$
$312$	−1.31231e8	−0.244621
$313$	−7.98682e8	−1.47221	−0.736103	−	0.676869i	$-0.763336\pi$
$313$	−7.98682e8	−1.47221	−0.736103	+	0.676869i	$0.763336\pi$
$314$	4.64154e8	0.846075
$315$	4.86151e8	0.876364
$316$	2.83242e8	0.504955
$317$	−5.61807e8	−0.990558	−0.495279	−	0.868734i	$-0.664934\pi$
$317$	−5.61807e8	−0.990558	−0.495279	+	0.868734i	$0.664934\pi$
$318$	2.56125e8	0.446640
$319$	−1.76007e8	−0.303573
$320$	−1.30462e8	−0.222567
$321$	−5.63763e8	−0.951325
$322$	−2.53699e8	−0.423471
$323$	−1.19810e9	−1.97827
$324$	3.40122e7	0.0555556
$325$	1.60957e9	2.60087
$326$	4.44134e8	0.709991
$327$	−6.00989e8	−0.950493
$328$	7.29771e7	0.114190
$329$	−2.51701e8	−0.389672
$330$	3.10540e8	0.475684
$331$	−9.41103e8	−1.42639	−0.713197	−	0.700964i	$-0.752753\pi$
$331$	−9.41103e8	−1.42639	−0.713197	+	0.700964i	$0.752753\pi$
$332$	−6.86684e7	−0.102985
$333$	−2.33430e8	−0.346419
$334$	−5.94951e8	−0.873712
$335$	6.38878e8	0.928455
$336$	1.48191e8	0.213125
$337$	−1.00470e9	−1.42999	−0.714995	−	0.699129i	$-0.753571\pi$
$337$	−1.00470e9	−1.42999	−0.714995	+	0.699129i	$0.753571\pi$
$338$	2.18943e8	0.308406
$339$	1.06458e8	0.148416
$340$	−1.04701e9	−1.44469
$341$	8.35726e8	1.14136
$342$	−2.12563e8	−0.287340
$343$	−1.98942e8	−0.266193
$344$	−3.87702e8	−0.513503
$345$	3.18008e8	0.416938
$346$	−2.40024e8	−0.311522
$347$	8.13475e8	1.04518	0.522590	−	0.852584i	$-0.324966\pi$
$347$	8.13475e8	1.04518	0.522590	+	0.852584i	$0.324966\pi$
$348$	1.05282e8	0.133915
$349$	−7.32703e8	−0.922655	−0.461327	−	0.887230i	$-0.652627\pi$
$349$	−7.32703e8	−0.922655	−0.461327	+	0.887230i	$0.652627\pi$
$350$	−1.81760e9	−2.26600
$351$	−1.86850e8	−0.230631
$352$	9.46605e7	0.115683
$353$	−5.51873e8	−0.667771	−0.333885	−	0.942614i	$-0.608360\pi$
$353$	−5.51873e8	−0.667771	−0.333885	+	0.942614i	$0.608360\pi$
$354$	4.43619e7	0.0531494
$355$	−1.57700e9	−1.87083
$356$	−2.59052e8	−0.304307
$357$	1.18929e9	1.38340
$358$	−6.59973e6	−0.00760213
$359$	3.09482e8	0.353024	0.176512	−	0.984298i	$-0.443518\pi$
$359$	3.09482e8	0.353024	0.176512	+	0.984298i	$0.443518\pi$
$360$	−1.85756e8	−0.209838
$361$	4.34559e8	0.486154
$362$	−1.00621e9	−1.11483
$363$	3.00833e8	0.330105
$364$	−8.14106e8	−0.884761
$365$	9.33284e7	0.100459
$366$	6.44067e8	0.686669
$367$	8.80262e8	0.929567	0.464784	−	0.885424i	$-0.346132\pi$
$367$	8.80262e8	0.929567	0.464784	+	0.885424i	$0.346132\pi$
$368$	9.69371e7	0.101396
$369$	1.03907e8	0.107659
$370$	1.27487e9	1.30845
$371$	1.58890e9	1.61543
$372$	−4.99906e8	−0.503487
$373$	1.59253e9	1.58894	0.794469	−	0.607304i	$-0.207749\pi$
$373$	1.59253e9	1.58894	0.794469	+	0.607304i	$0.207749\pi$
$374$	7.59685e8	0.750902
$375$	1.22855e9	1.20305
$376$	9.61736e7	0.0933036
$377$	−5.78380e8	−0.555928
$378$	2.10999e8	0.200937
$379$	1.35674e9	1.28014	0.640072	−	0.768315i	$-0.278905\pi$
$379$	1.35674e9	1.28014	0.640072	+	0.768315i	$0.278905\pi$
$380$	1.16090e9	1.08530
$381$	−4.12517e8	−0.382124
$382$	1.05407e9	0.967488
$383$	7.71400e8	0.701591	0.350795	−	0.936452i	$-0.385911\pi$
$383$	7.71400e8	0.701591	0.350795	+	0.936452i	$0.385911\pi$
$384$	−5.66231e7	−0.0510310
$385$	1.92647e9	1.72048
$386$	−8.23482e8	−0.728784
$387$	−5.52021e8	−0.484136
$388$	3.22609e8	0.280392
$389$	1.65560e9	1.42604	0.713018	−	0.701146i	$-0.247328\pi$
$389$	1.65560e9	1.42604	0.713018	+	0.701146i	$0.247328\pi$
$390$	1.02047e9	0.871113
$391$	7.77956e8	0.658167
$392$	4.97669e8	0.417291
$393$	1.07999e9	0.897525
$394$	−1.10462e9	−0.909860
$395$	−2.20253e9	−1.79818
$396$	1.34780e8	0.109067
$397$	−4.80970e8	−0.385790	−0.192895	−	0.981219i	$-0.561788\pi$
$397$	−4.80970e8	−0.385790	−0.192895	+	0.981219i	$0.561788\pi$
$398$	−8.88203e8	−0.706190
$399$	−1.31866e9	−1.03927
$400$	6.94495e8	0.542574
$401$	−2.25105e9	−1.74333	−0.871667	−	0.490099i	$-0.836961\pi$
$401$	−2.25105e9	−1.74333	−0.871667	+	0.490099i	$0.836961\pi$
$402$	2.77285e8	0.212880
$403$	2.74629e9	2.09016
$404$	−2.31656e8	−0.174787
$405$	−2.64484e8	−0.197837
$406$	6.53132e8	0.484350
$407$	−9.25014e8	−0.680093
$408$	−4.54421e8	−0.331244
$409$	1.10454e9	0.798273	0.399137	−	0.916891i	$-0.369310\pi$
$409$	1.10454e9	0.798273	0.399137	+	0.916891i	$0.369310\pi$
$410$	−5.67482e8	−0.406638
$411$	1.19880e9	0.851725
$412$	1.03162e9	0.726739
$413$	2.75204e8	0.192234
$414$	1.38022e8	0.0955975
$415$	5.33976e8	0.366736
$416$	3.11065e8	0.211848
$417$	7.26029e8	0.490318
$418$	−8.42323e8	−0.564107
$419$	−1.06754e9	−0.708983	−0.354492	−	0.935059i	$-0.615346\pi$
$419$	−1.06754e9	−0.708983	−0.354492	+	0.935059i	$0.615346\pi$
$420$	−1.15236e9	−0.758953
$421$	4.38024e8	0.286095	0.143048	−	0.989716i	$-0.454310\pi$
$421$	4.38024e8	0.286095	0.143048	+	0.989716i	$0.454310\pi$
$422$	−1.81345e9	−1.17466
$423$	1.36935e8	0.0879675
$424$	−6.07112e8	−0.386802
$425$	5.57358e9	3.52186
$426$	−6.84450e8	−0.428951
$427$	3.99555e9	2.48358
$428$	1.33633e9	0.823872
$429$	−7.40431e8	−0.452777
$430$	3.01483e9	1.82862
$431$	1.75922e9	1.05840	0.529201	−	0.848497i	$-0.322492\pi$
$431$	1.75922e9	1.05840	0.529201	+	0.848497i	$0.322492\pi$
$432$	−8.06216e7	−0.0481125
$433$	2.17078e9	1.28501	0.642507	−	0.766279i	$-0.277894\pi$
$433$	2.17078e9	1.28501	0.642507	+	0.766279i	$0.277894\pi$
$434$	−3.10123e9	−1.82104
$435$	−8.18692e8	−0.476879
$436$	1.42457e9	0.823151
$437$	−8.62580e8	−0.494441
$438$	4.05063e7	0.0230337
$439$	−2.64981e8	−0.149482	−0.0747411	−	0.997203i	$-0.523813\pi$
$439$	−2.64981e8	−0.149482	−0.0747411	+	0.997203i	$0.523813\pi$
$440$	−7.36095e8	−0.411955
$441$	7.08595e8	0.393426
$442$	2.49641e9	1.37511
$443$	1.46985e9	0.803267	0.401633	−	0.915801i	$-0.368443\pi$
$443$	1.46985e9	0.803267	0.401633	+	0.915801i	$0.368443\pi$
$444$	5.53316e8	0.300008
$445$	2.01443e9	1.08366
$446$	1.82598e9	0.974597
$447$	−1.39429e8	−0.0738377
$448$	−3.51268e8	−0.184572
$449$	−1.72485e9	−0.899269	−0.449634	−	0.893213i	$-0.648446\pi$
$449$	−1.72485e9	−0.899269	−0.449634	+	0.893213i	$0.648446\pi$
$450$	9.88841e8	0.511544
$451$	4.11752e8	0.211358
$452$	−2.52346e8	−0.128532
$453$	5.35918e8	0.270866
$454$	−1.03739e9	−0.520293
$455$	6.33061e9	3.15069
$456$	5.03852e8	0.248843
$457$	2.26442e9	1.10982	0.554908	−	0.831912i	$-0.312754\pi$
$457$	2.26442e9	1.10982	0.554908	+	0.831912i	$0.312754\pi$
$458$	8.66303e8	0.421348
$459$	−6.47018e8	−0.312300
$460$	−7.53798e8	−0.361079
$461$	−2.43731e9	−1.15866	−0.579332	−	0.815092i	$-0.696686\pi$
$461$	−2.43731e9	−1.15866	−0.579332	+	0.815092i	$0.696686\pi$
$462$	8.36126e8	0.394480
$463$	1.81050e9	0.847744	0.423872	−	0.905722i	$-0.360671\pi$
$463$	1.81050e9	0.847744	0.423872	+	0.905722i	$0.360671\pi$
$464$	−2.49558e8	−0.115973
$465$	3.88735e9	1.79295
$466$	1.15139e9	0.527076
$467$	−2.71915e9	−1.23545	−0.617724	−	0.786395i	$-0.711945\pi$
$467$	−2.71915e9	−1.23545	−0.617724	+	0.786395i	$0.711945\pi$
$468$	4.42904e8	0.199733
$469$	1.72017e9	0.769958
$470$	−7.47861e8	−0.332260
$471$	−1.56652e9	−0.690817
$472$	−1.05154e8	−0.0460287
$473$	−2.18749e9	−0.950458
$474$	−9.55942e8	−0.412294
$475$	−6.17986e9	−2.64576
$476$	−2.81906e9	−1.19806
$477$	−8.64423e8	−0.364680
$478$	2.29002e9	0.959050
$479$	−1.19423e9	−0.496495	−0.248248	−	0.968697i	$-0.579855\pi$
$479$	−1.19423e9	−0.496495	−0.248248	+	0.968697i	$0.579855\pi$
$480$	4.40310e8	0.181725
$481$	−3.03970e9	−1.24544
$482$	4.34839e8	0.176874
$483$	8.56235e8	0.345762
$484$	−7.13085e8	−0.285879
$485$	−2.50866e9	−0.998493
$486$	−1.14791e8	−0.0453609
$487$	−2.85516e9	−1.12016	−0.560078	−	0.828440i	$-0.689229\pi$
$487$	−2.85516e9	−1.12016	−0.560078	+	0.828440i	$0.689229\pi$
$488$	−1.52668e9	−0.594673
$489$	−1.49895e9	−0.579705
$490$	−3.86995e9	−1.48600
$491$	2.36395e9	0.901268	0.450634	−	0.892709i	$-0.351198\pi$
$491$	2.36395e9	0.901268	0.450634	+	0.892709i	$0.351198\pi$
$492$	−2.46298e8	−0.0932358
$493$	−2.00280e9	−0.752787
$494$	−2.76797e9	−1.03304
$495$	−1.04807e9	−0.388395
$496$	1.18496e9	0.436033
$497$	−4.24607e9	−1.55146
$498$	2.31756e8	0.0840869
$499$	6.11806e8	0.220425	0.110213	−	0.993908i	$-0.464847\pi$
$499$	6.11806e8	0.220425	0.110213	+	0.993908i	$0.464847\pi$
$500$	−2.91213e9	−1.04188
$501$	2.00796e9	0.713383
$502$	−2.71760e9	−0.958788
$503$	−3.99783e9	−1.40067	−0.700335	−	0.713814i	$-0.746966\pi$
$503$	−3.99783e9	−1.40067	−0.700335	+	0.713814i	$0.746966\pi$
$504$	−5.00146e8	−0.174016
$505$	1.80140e9	0.622429
$506$	5.46939e8	0.187677
$507$	−7.38933e8	−0.251812
$508$	9.77818e8	0.330929
$509$	6.08052e8	0.204375	0.102188	−	0.994765i	$-0.467416\pi$
$509$	6.08052e8	0.204375	0.102188	+	0.994765i	$0.467416\pi$
$510$	3.53365e9	1.17958
$511$	2.51286e8	0.0833095
$512$	1.34218e8	0.0441942
$513$	7.17399e8	0.234612
$514$	−4.26220e8	−0.138440
$515$	−8.02201e9	−2.58796
$516$	1.30849e9	0.419274
$517$	5.42631e8	0.172698
$518$	3.43256e9	1.08509
$519$	8.10081e8	0.254356
$520$	−2.41889e9	−0.754406
$521$	3.21544e8	0.0996112	0.0498056	−	0.998759i	$-0.484140\pi$
$521$	3.21544e8	0.0996112	0.0498056	+	0.998759i	$0.484140\pi$
$522$	−3.55328e8	−0.109341
$523$	1.19368e9	0.364864	0.182432	−	0.983218i	$-0.441603\pi$
$523$	1.19368e9	0.364864	0.182432	+	0.983218i	$0.441603\pi$
$524$	−2.55998e9	−0.777279
$525$	6.13440e9	1.85018
$526$	−3.82018e9	−1.14455
$527$	9.50976e9	2.83030
$528$	−3.19479e8	−0.0944548
$529$	−2.84473e9	−0.835500
$530$	4.72100e9	1.37743
$531$	−1.49721e8	−0.0433963
$532$	3.12571e9	0.900032
$533$	1.35307e9	0.387056
$534$	8.74299e8	0.248465
$535$	−1.03915e10	−2.93386
$536$	−6.57268e8	−0.184360
$537$	2.22741e7	0.00620712
$538$	−1.35831e9	−0.376063
$539$	2.80795e9	0.772376
$540$	6.26926e8	0.171332
$541$	5.26791e9	1.43037	0.715184	−	0.698937i	$-0.246343\pi$
$541$	5.26791e9	1.43037	0.715184	+	0.698937i	$0.246343\pi$
$542$	−2.19358e9	−0.591775
$543$	3.39595e9	0.910252
$544$	1.07715e9	0.286866
$545$	−1.10776e10	−2.93130
$546$	2.74761e9	0.722404
$547$	2.35374e9	0.614899	0.307449	−	0.951564i	$-0.400525\pi$
$547$	2.35374e9	0.614899	0.307449	+	0.951564i	$0.400525\pi$
$548$	−2.84160e9	−0.737616
$549$	−2.17373e9	−0.560663
$550$	3.91848e9	1.00427
$551$	2.22066e9	0.565523
$552$	−3.27163e8	−0.0827898
$553$	−5.93030e9	−1.49121
$554$	4.62988e8	0.115687
$555$	−4.30267e9	−1.06835
$556$	−1.72096e9	−0.424628
$557$	−3.70811e8	−0.0909201	−0.0454600	−	0.998966i	$-0.514475\pi$
$557$	−3.70811e8	−0.0909201	−0.0454600	+	0.998966i	$0.514475\pi$
$558$	1.68718e9	0.411095
$559$	−7.18836e9	−1.74056
$560$	2.73152e9	0.657273
$561$	−2.56394e9	−0.613109
$562$	2.03099e9	0.482649
$563$	−1.70136e9	−0.401806	−0.200903	−	0.979611i	$-0.564388\pi$
$563$	−1.70136e9	−0.401806	−0.200903	+	0.979611i	$0.564388\pi$
$564$	−3.24586e8	−0.0761821
$565$	1.96228e9	0.457711
$566$	−4.43397e9	−1.02786
$567$	−7.12121e8	−0.164064
$568$	1.62240e9	0.371483
$569$	−4.74738e9	−1.08034	−0.540171	−	0.841556i	$-0.681640\pi$
$569$	−4.74738e9	−1.08034	−0.540171	+	0.841556i	$0.681640\pi$
$570$	−3.91803e9	−0.886148
$571$	7.29826e9	1.64056	0.820282	−	0.571960i	$-0.193817\pi$
$571$	7.29826e9	1.64056	0.820282	+	0.571960i	$0.193817\pi$
$572$	1.75510e9	0.392116
$573$	−3.55747e9	−0.789951
$574$	−1.52794e9	−0.337221
$575$	4.01272e9	0.880241
$576$	1.91103e8	0.0416667
$577$	−1.42252e9	−0.308278	−0.154139	−	0.988049i	$-0.549260\pi$
$577$	−1.42252e9	−0.308278	−0.154139	+	0.988049i	$0.549260\pi$
$578$	5.36179e9	1.15495
$579$	2.77925e9	0.595050
$580$	1.94060e9	0.412989
$581$	1.43773e9	0.304130
$582$	−1.08880e9	−0.228939
$583$	−3.42545e9	−0.715942
$584$	−9.60149e7	−0.0199477
$585$	−3.44409e9	−0.711261
$586$	−6.12556e9	−1.25749
$587$	7.72276e9	1.57594	0.787969	−	0.615714i	$-0.211132\pi$
$587$	7.72276e9	1.57594	0.787969	+	0.615714i	$0.211132\pi$
$588$	−1.67963e9	−0.340717
$589$	−1.05442e10	−2.12623
$590$	8.17694e8	0.163911
$591$	3.72808e9	0.742898
$592$	−1.31156e9	−0.259815
$593$	−4.13781e9	−0.814854	−0.407427	−	0.913238i	$-0.633574\pi$
$593$	−4.13781e9	−0.814854	−0.407427	+	0.913238i	$0.633574\pi$
$594$	−4.54884e8	−0.0890528
$595$	2.19214e10	4.26638
$596$	3.30499e8	0.0639453
$597$	2.99768e9	0.576601
$598$	1.79731e9	0.343691
$599$	2.00397e9	0.380976	0.190488	−	0.981690i	$-0.438993\pi$
$599$	2.00397e9	0.380976	0.190488	+	0.981690i	$0.438993\pi$
$600$	−2.34392e9	−0.443010
$601$	−4.90149e9	−0.921017	−0.460509	−	0.887655i	$-0.652333\pi$
$601$	−4.90149e9	−0.921017	−0.460509	+	0.887655i	$0.652333\pi$
$602$	8.11740e9	1.51645
$603$	−9.35837e8	−0.173816
$604$	−1.27032e9	−0.234577
$605$	5.54506e9	1.01803
$606$	7.81840e8	0.142713
$607$	−6.60944e9	−1.19951	−0.599755	−	0.800184i	$-0.704735\pi$
$607$	−6.60944e9	−1.19951	−0.599755	+	0.800184i	$0.704735\pi$
$608$	−1.19432e9	−0.215505
$609$	−2.20432e9	−0.395470
$610$	1.18717e10	2.11767
$611$	1.78315e9	0.316259
$612$	1.53367e9	0.270460
$613$	−4.08971e9	−0.717101	−0.358551	−	0.933510i	$-0.616729\pi$
$613$	−4.08971e9	−0.717101	−0.358551	+	0.933510i	$0.616729\pi$
$614$	2.48495e9	0.433240
$615$	1.91525e9	0.332019
$616$	−1.98193e9	−0.341630
$617$	−7.01266e9	−1.20195	−0.600973	−	0.799269i	$-0.705220\pi$
$617$	−7.01266e9	−1.20195	−0.600973	+	0.799269i	$0.705220\pi$
$618$	−3.48171e9	−0.593380
$619$	−2.74641e8	−0.0465423	−0.0232712	−	0.999729i	$-0.507408\pi$
$619$	−2.74641e8	−0.0465423	−0.0232712	+	0.999729i	$0.507408\pi$
$620$	−9.21446e9	−1.55274
$621$	−4.65823e8	−0.0780550
$622$	−7.37009e9	−1.22802
$623$	5.42382e9	0.898664
$624$	−1.04985e9	−0.172973
$625$	9.39874e9	1.53989
$626$	−6.38946e9	−1.04101
$627$	2.84284e9	0.460591
$628$	3.71324e9	0.598265
$629$	−1.05258e10	−1.68646
$630$	3.88921e9	0.619683
$631$	−7.75033e9	−1.22805	−0.614027	−	0.789285i	$-0.710451\pi$
$631$	−7.75033e9	−1.22805	−0.614027	+	0.789285i	$0.710451\pi$
$632$	2.26594e9	0.357057
$633$	6.12040e9	0.959107
$634$	−4.49446e9	−0.700430
$635$	−7.60367e9	−1.17846
$636$	2.04900e9	0.315822
$637$	9.22725e9	1.41444
$638$	−1.40806e9	−0.214659
$639$	2.31002e9	0.350237
$640$	−1.04370e9	−0.157378
$641$	−1.24194e10	−1.86250	−0.931250	−	0.364382i	$-0.881280\pi$
$641$	−1.24194e10	−1.86250	−0.931250	+	0.364382i	$0.881280\pi$
$642$	−4.51010e9	−0.672688
$643$	−3.09246e7	−0.00458739	−0.00229370	−	0.999997i	$-0.500730\pi$
$643$	−3.09246e7	−0.00458739	−0.00229370	+	0.999997i	$0.500730\pi$
$644$	−2.02959e9	−0.299439
$645$	−1.01751e10	−1.49306
$646$	−9.58483e9	−1.39885
$647$	−3.71885e9	−0.539814	−0.269907	−	0.962886i	$-0.586993\pi$
$647$	−3.71885e9	−0.539814	−0.269907	+	0.962886i	$0.586993\pi$
$648$	2.72098e8	0.0392837
$649$	−5.93301e8	−0.0851959
$650$	1.28766e10	1.83910
$651$	1.04666e10	1.48687
$652$	3.55307e9	0.502039
$653$	2.28677e9	0.321385	0.160693	−	0.987004i	$-0.448627\pi$
$653$	2.28677e9	0.321385	0.160693	+	0.987004i	$0.448627\pi$
$654$	−4.80791e9	−0.672100
$655$	1.99068e10	2.76794
$656$	5.83817e8	0.0807446
$657$	−1.36709e8	−0.0188069
$658$	−2.01361e9	−0.275540
$659$	−4.25753e9	−0.579508	−0.289754	−	0.957101i	$-0.593573\pi$
$659$	−4.25753e9	−0.579508	−0.289754	+	0.957101i	$0.593573\pi$
$660$	2.48432e9	0.336360
$661$	1.40311e10	1.88967	0.944837	−	0.327540i	$-0.106220\pi$
$661$	1.40311e10	1.88967	0.944837	+	0.327540i	$0.106220\pi$
$662$	−7.52883e9	−1.00861
$663$	−8.42540e9	−1.12278
$664$	−5.49347e8	−0.0728214
$665$	−2.43060e10	−3.20507
$666$	−1.86744e9	−0.244956
$667$	−1.44192e9	−0.188149
$668$	−4.75961e9	−0.617808
$669$	−6.16269e9	−0.795755
$670$	5.11102e9	0.656517
$671$	−8.61383e9	−1.10070
$672$	1.18553e9	0.150702
$673$	3.09621e7	0.00391542	0.00195771	−	0.999998i	$-0.499377\pi$
$673$	3.09621e7	0.00391542	0.00195771	+	0.999998i	$0.499377\pi$
$674$	−8.03763e9	−1.01116
$675$	−3.33734e9	−0.417674
$676$	1.75154e9	0.218076
$677$	−3.97564e9	−0.492433	−0.246216	−	0.969215i	$-0.579187\pi$
$677$	−3.97564e9	−0.492433	−0.246216	+	0.969215i	$0.579187\pi$
$678$	8.51666e8	0.104946
$679$	−6.75453e9	−0.828039
$680$	−8.37606e9	−1.02155
$681$	3.50120e9	0.424817
$682$	6.68581e9	0.807065
$683$	1.06792e9	0.128252	0.0641261	−	0.997942i	$-0.479574\pi$
$683$	1.06792e9	0.128252	0.0641261	+	0.997942i	$0.479574\pi$
$684$	−1.70050e9	−0.203180
$685$	2.20967e10	2.62670
$686$	−1.59154e9	−0.188227
$687$	−2.92377e9	−0.344029
$688$	−3.10161e9	−0.363102
$689$	−1.12564e10	−1.31109
$690$	2.54407e9	0.294820
$691$	−9.48026e9	−1.09307	−0.546534	−	0.837437i	$-0.684053\pi$
$691$	−9.48026e9	−1.09307	−0.546534	+	0.837437i	$0.684053\pi$
$692$	−1.92019e9	−0.220279
$693$	−2.82193e9	−0.322091
$694$	6.50780e9	0.739054
$695$	1.33824e10	1.51213
$696$	8.42259e8	0.0946920
$697$	4.68535e9	0.524115
$698$	−5.86163e9	−0.652415
$699$	−3.88595e9	−0.430356
$700$	−1.45408e10	−1.60230
$701$	6.94657e8	0.0761654	0.0380827	−	0.999275i	$-0.487875\pi$
$701$	6.94657e8	0.0761654	0.0380827	+	0.999275i	$0.487875\pi$
$702$	−1.49480e9	−0.163081
$703$	1.16708e10	1.26694
$704$	7.57284e8	0.0818002
$705$	2.52403e9	0.271289
$706$	−4.41498e9	−0.472185
$707$	4.85024e9	0.516173
$708$	3.54895e8	0.0375823
$709$	3.44289e9	0.362795	0.181397	−	0.983410i	$-0.441938\pi$
$709$	3.44289e9	0.362795	0.181397	+	0.983410i	$0.441938\pi$
$710$	−1.26160e10	−1.32287
$711$	3.22630e9	0.336637
$712$	−2.07241e9	−0.215177
$713$	6.84660e9	0.707394
$714$	9.51432e9	0.978214
$715$	−1.36479e10	−1.39635
$716$	−5.27978e7	−0.00537552
$717$	−7.72881e9	−0.783061
$718$	2.47586e9	0.249626
$719$	8.82183e9	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$719$	8.82183e9	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$720$	−1.48605e9	−0.148378
$721$	−2.15992e10	−2.14617
$722$	3.47647e9	0.343763
$723$	−1.46758e9	−0.144417
$724$	−8.04966e9	−0.788302
$725$	−1.03305e10	−1.00679
$726$	2.40666e9	0.233419
$727$	−1.08220e10	−1.04457	−0.522283	−	0.852772i	$-0.674920\pi$
$727$	−1.08220e10	−1.04457	−0.522283	+	0.852772i	$0.674920\pi$
$728$	−6.51285e9	−0.625621
$729$	3.87420e8	0.0370370
$730$	7.46627e8	0.0710352
$731$	−2.48916e10	−2.35690
$732$	5.15254e9	0.485548
$733$	7.50244e9	0.703621	0.351810	−	0.936071i	$-0.385566\pi$
$733$	7.50244e9	0.703621	0.351810	+	0.936071i	$0.385566\pi$
$734$	7.04210e9	0.657303
$735$	1.30611e10	1.21331
$736$	7.75497e8	0.0716981
$737$	−3.70844e9	−0.341237
$738$	8.31255e8	0.0761267
$739$	−2.10887e9	−0.192218	−0.0961089	−	0.995371i	$-0.530640\pi$
$739$	−2.10887e9	−0.192218	−0.0961089	+	0.995371i	$0.530640\pi$
$740$	1.01989e10	0.925217
$741$	9.34190e9	0.843473
$742$	1.27112e10	1.14228
$743$	1.17637e10	1.05216	0.526082	−	0.850434i	$-0.323660\pi$
$743$	1.17637e10	1.05216	0.526082	+	0.850434i	$0.323660\pi$
$744$	−3.99925e9	−0.356019
$745$	−2.57001e9	−0.227713
$746$	1.27403e10	1.12355
$747$	−7.82176e8	−0.0686566
$748$	6.07748e9	0.530968
$749$	−2.79790e10	−2.43302
$750$	9.82844e9	0.850687
$751$	5.59405e9	0.481933	0.240966	−	0.970533i	$-0.422536\pi$
$751$	5.59405e9	0.481933	0.240966	+	0.970533i	$0.422536\pi$
$752$	7.69389e8	0.0659756
$753$	9.17190e9	0.782847
$754$	−4.62704e9	−0.393101
$755$	9.87823e9	0.835343
$756$	1.68799e9	0.142084
$757$	−1.20065e9	−0.100596	−0.0502979	−	0.998734i	$-0.516017\pi$
$757$	−1.20065e9	−0.100596	−0.0502979	+	0.998734i	$0.516017\pi$
$758$	1.08539e10	0.905198
$759$	−1.84592e9	−0.153238
$760$	9.28719e9	0.767427
$761$	−1.35518e9	−0.111468	−0.0557340	−	0.998446i	$-0.517750\pi$
$761$	−1.35518e9	−0.111468	−0.0557340	+	0.998446i	$0.517750\pi$
$762$	−3.30014e9	−0.270203
$763$	−2.98264e10	−2.43089
$764$	8.43252e9	0.684118
$765$	−1.19261e10	−0.963124
$766$	6.17120e9	0.496100
$767$	−1.94966e9	−0.156018
$768$	−4.52985e8	−0.0360844
$769$	4.64233e9	0.368124	0.184062	−	0.982915i	$-0.441075\pi$
$769$	4.64233e9	0.368124	0.184062	+	0.982915i	$0.441075\pi$
$770$	1.54118e10	1.21657
$771$	1.43849e9	0.113036
$772$	−6.58786e9	−0.515328
$773$	−2.07357e10	−1.61470	−0.807348	−	0.590075i	$-0.799098\pi$
$773$	−2.07357e10	−1.61470	−0.807348	+	0.590075i	$0.799098\pi$
$774$	−4.41617e9	−0.342336
$775$	4.90517e10	3.78528
$776$	2.58087e9	0.198267
$777$	−1.15849e10	−0.885969
$778$	1.32448e10	1.00836
$779$	−5.19501e9	−0.393736
$780$	8.16377e9	0.615970
$781$	9.15391e9	0.687588
$782$	6.22365e9	0.465394
$783$	1.19923e9	0.0892764
$784$	3.98135e9	0.295069
$785$	−2.88747e10	−2.13046
$786$	8.63993e9	0.634646
$787$	1.50377e10	1.09969	0.549846	−	0.835266i	$-0.314687\pi$
$787$	1.50377e10	1.09969	0.549846	+	0.835266i	$0.314687\pi$
$788$	−8.83693e9	−0.643368
$789$	1.28931e10	0.934518
$790$	−1.76203e10	−1.27150
$791$	5.28342e9	0.379575
$792$	1.07824e9	0.0771220
$793$	−2.83060e10	−2.01569
$794$	−3.84776e9	−0.272795
$795$	−1.59334e10	−1.12466
$796$	−7.10562e9	−0.499351
$797$	6.00916e9	0.420445	0.210223	−	0.977654i	$-0.432581\pi$
$797$	6.00916e9	0.420445	0.210223	+	0.977654i	$0.432581\pi$
$798$	−1.05493e10	−0.734873
$799$	6.17463e9	0.428250
$800$	5.55596e9	0.383658
$801$	−2.95076e9	−0.202871
$802$	−1.80084e10	−1.23272
$803$	−5.41736e8	−0.0369218
$804$	2.21828e9	0.150529
$805$	1.57824e10	1.06632
$806$	2.19703e10	1.47796
$807$	4.58429e9	0.307054
$808$	−1.85325e9	−0.123593
$809$	6.21481e9	0.412675	0.206337	−	0.978481i	$-0.433846\pi$
$809$	6.21481e9	0.412675	0.206337	+	0.978481i	$0.433846\pi$
$810$	−2.11587e9	−0.139892
$811$	8.53696e9	0.561992	0.280996	−	0.959709i	$-0.409335\pi$
$811$	8.53696e9	0.561992	0.280996	+	0.959709i	$0.409335\pi$
$812$	5.22505e9	0.342487
$813$	7.40334e9	0.483182
$814$	−7.40012e9	−0.480898
$815$	−2.76293e10	−1.78779
$816$	−3.63537e9	−0.234225
$817$	2.75993e10	1.77060
$818$	8.83636e9	0.564465
$819$	−9.27317e9	−0.589841
$820$	−4.53985e9	−0.287537
$821$	9.32112e9	0.587850	0.293925	−	0.955828i	$-0.405038\pi$
$821$	9.32112e9	0.587850	0.293925	+	0.955828i	$0.405038\pi$
$822$	9.59038e9	0.602261
$823$	−1.14106e10	−0.713524	−0.356762	−	0.934195i	$-0.616119\pi$
$823$	−1.14106e10	−0.713524	−0.356762	+	0.934195i	$0.616119\pi$
$824$	8.25294e9	0.513882
$825$	−1.32249e10	−0.819979
$826$	2.20163e9	0.135930
$827$	−2.16763e10	−1.33265	−0.666324	−	0.745662i	$-0.732133\pi$
$827$	−2.16763e10	−1.33265	−0.666324	+	0.745662i	$0.732133\pi$
$828$	1.10417e9	0.0675976
$829$	−2.60486e9	−0.158798	−0.0793988	−	0.996843i	$-0.525300\pi$
$829$	−2.60486e9	−0.158798	−0.0793988	+	0.996843i	$0.525300\pi$
$830$	4.27181e9	0.259322
$831$	−1.56258e9	−0.0944582
$832$	2.48852e9	0.149799
$833$	3.19518e10	1.91530
$834$	5.80823e9	0.346707
$835$	3.70114e10	2.20005
$836$	−6.73858e9	−0.398884
$837$	−5.69425e9	−0.335658
$838$	−8.54034e9	−0.501327
$839$	−1.55847e10	−0.911030	−0.455515	−	0.890228i	$-0.650545\pi$
$839$	−1.55847e10	−0.911030	−0.455515	+	0.890228i	$0.650545\pi$
$840$	−9.21887e9	−0.536661
$841$	−1.35377e10	−0.784802
$842$	3.50419e9	0.202300
$843$	−6.85460e9	−0.394081
$844$	−1.45076e10	−0.830611
$845$	−1.36203e10	−0.776582
$846$	1.09548e9	0.0622024
$847$	1.49300e10	0.844244
$848$	−4.85689e9	−0.273510
$849$	1.49646e10	0.839247
$850$	4.45886e10	2.49033
$851$	−7.57809e9	−0.421508
$852$	−5.47560e9	−0.303314
$853$	−6.88235e9	−0.379678	−0.189839	−	0.981815i	$-0.560797\pi$
$853$	−6.88235e9	−0.379678	−0.189839	+	0.981815i	$0.560797\pi$
$854$	3.19644e10	1.75616
$855$	1.32234e10	0.723537
$856$	1.06906e10	0.582565
$857$	−2.31566e10	−1.25673	−0.628365	−	0.777918i	$-0.716276\pi$
$857$	−2.31566e10	−1.25673	−0.628365	+	0.777918i	$0.716276\pi$
$858$	−5.92345e9	−0.320162
$859$	2.78434e10	1.49881	0.749404	−	0.662113i	$-0.230340\pi$
$859$	2.78434e10	1.49881	0.749404	+	0.662113i	$0.230340\pi$
$860$	2.41186e10	1.29303
$861$	5.15679e9	0.275340
$862$	1.40738e10	0.748403
$863$	3.11349e10	1.64896	0.824479	−	0.565892i	$-0.191468\pi$
$863$	3.11349e10	1.64896	0.824479	+	0.565892i	$0.191468\pi$
$864$	−6.44973e8	−0.0340207
$865$	1.49317e10	0.784428
$866$	1.73662e10	0.908643
$867$	−1.80960e10	−0.943010
$868$	−2.48098e10	−1.28767
$869$	1.27849e10	0.660888
$870$	−6.54953e9	−0.337204
$871$	−1.21864e10	−0.624901
$872$	1.13965e10	0.582056
$873$	3.67472e9	0.186928
$874$	−6.90064e9	−0.349623
$875$	6.09719e10	3.07682
$876$	3.24050e8	0.0162873
$877$	−1.12801e10	−0.564697	−0.282348	−	0.959312i	$-0.591113\pi$
$877$	−1.12801e10	−0.564697	−0.282348	+	0.959312i	$0.591113\pi$
$878$	−2.11985e9	−0.105700
$879$	2.06738e10	1.02674
$880$	−5.88876e9	−0.291296
$881$	−2.91926e10	−1.43832	−0.719162	−	0.694842i	$-0.755474\pi$
$881$	−2.91926e10	−1.43832	−0.719162	+	0.694842i	$0.755474\pi$
$882$	5.66876e9	0.278194
$883$	−3.39796e10	−1.66095	−0.830474	−	0.557058i	$-0.811930\pi$
$883$	−3.39796e10	−1.66095	−0.830474	+	0.557058i	$0.811930\pi$
$884$	1.99713e10	0.972353
$885$	−2.75972e9	−0.133833
$886$	1.17588e10	0.567995
$887$	2.49412e10	1.20001	0.600005	−	0.799996i	$-0.295165\pi$
$887$	2.49412e10	1.20001	0.600005	+	0.799996i	$0.295165\pi$
$888$	4.42653e9	0.212138
$889$	−2.04728e10	−0.977285
$890$	1.61154e10	0.766260
$891$	1.53523e9	0.0727113
$892$	1.46079e10	0.689144
$893$	−6.84629e9	−0.321718
$894$	−1.11543e9	−0.0522111
$895$	4.10564e8	0.0191426
$896$	−2.81015e9	−0.130512
$897$	−6.06591e9	−0.280622
$898$	−1.37988e10	−0.635879
$899$	−1.76261e10	−0.809092
$900$	7.91073e9	0.361716
$901$	−3.89784e10	−1.77536
$902$	3.29402e9	0.149452
$903$	−2.73962e10	−1.23818
$904$	−2.01876e9	−0.0908858
$905$	6.25954e10	2.80719
$906$	4.28734e9	0.191531
$907$	8.53428e9	0.379788	0.189894	−	0.981805i	$-0.439186\pi$
$907$	8.53428e9	0.379788	0.189894	+	0.981805i	$0.439186\pi$
$908$	−8.29914e9	−0.367902
$909$	−2.63871e9	−0.116525
$910$	5.06449e10	2.22788
$911$	−2.08882e10	−0.915347	−0.457674	−	0.889120i	$-0.651317\pi$
$911$	−2.08882e10	−0.915347	−0.457674	+	0.889120i	$0.651317\pi$
$912$	4.03082e9	0.175959
$913$	−3.09953e9	−0.134787
$914$	1.81154e10	0.784758
$915$	−4.00669e10	−1.72907
$916$	6.93042e9	0.297938
$917$	5.35989e10	2.29542
$918$	−5.17614e9	−0.220829
$919$	8.47165e9	0.360051	0.180025	−	0.983662i	$-0.442382\pi$
$919$	8.47165e9	0.360051	0.180025	+	0.983662i	$0.442382\pi$
$920$	−6.03038e9	−0.255322
$921$	−8.38672e9	−0.353739
$922$	−1.94985e10	−0.819299
$923$	3.00808e10	1.25917
$924$	6.68901e9	0.278939
$925$	−5.42923e10	−2.25550
$926$	1.44840e10	0.599446
$927$	1.17508e10	0.484492
$928$	−1.99646e9	−0.0820056
$929$	−2.85037e10	−1.16640	−0.583198	−	0.812330i	$-0.698199\pi$
$929$	−2.85037e10	−1.16640	−0.583198	+	0.812330i	$0.698199\pi$
$930$	3.10988e10	1.26781
$931$	−3.54274e10	−1.43885
$932$	9.21115e9	0.372699
$933$	2.48741e10	1.00268
$934$	−2.17532e10	−0.873593
$935$	−4.72595e10	−1.89081
$936$	3.54323e9	0.141232
$937$	2.42977e10	0.964887	0.482443	−	0.875927i	$-0.339749\pi$
$937$	2.42977e10	0.964887	0.482443	+	0.875927i	$0.339749\pi$
$938$	1.37614e10	0.544442
$939$	2.15644e10	0.849979
$940$	−5.98289e9	−0.234943
$941$	−4.21978e8	−0.0165092	−0.00825461	−	0.999966i	$-0.502628\pi$
$941$	−4.21978e8	−0.0165092	−0.00825461	+	0.999966i	$0.502628\pi$
$942$	−1.25322e10	−0.488482
$943$	3.37324e9	0.130995
$944$	−8.41232e8	−0.0325472
$945$	−1.31261e10	−0.505969
$946$	−1.75000e10	−0.672075
$947$	−3.91604e10	−1.49838	−0.749189	−	0.662356i	$-0.769557\pi$
$947$	−3.91604e10	−1.49838	−0.749189	+	0.662356i	$0.769557\pi$
$948$	−7.64754e9	−0.291536
$949$	−1.78021e9	−0.0676144
$950$	−4.94389e10	−1.87084
$951$	1.51688e10	0.571899
$952$	−2.25525e10	−0.847158
$953$	−3.73482e10	−1.39780	−0.698899	−	0.715221i	$-0.746326\pi$
$953$	−3.73482e10	−1.39780	−0.698899	+	0.715221i	$0.746326\pi$
$954$	−6.91538e9	−0.257868
$955$	−6.55726e10	−2.43619
$956$	1.83201e10	0.678151
$957$	4.75220e9	0.175268
$958$	−9.55387e9	−0.351075
$959$	5.94951e10	2.17829
$960$	3.52248e9	0.128499
$961$	5.61805e10	2.04199
$962$	−2.43176e10	−0.880661
$963$	1.52216e10	0.549248
$964$	3.47871e9	0.125069
$965$	5.12282e10	1.83512
$966$	6.84988e9	0.244491
$967$	1.77790e10	0.632289	0.316144	−	0.948711i	$-0.397612\pi$
$967$	1.77790e10	0.632289	0.316144	+	0.948711i	$0.397612\pi$
$968$	−5.70468e9	−0.202147
$969$	3.23488e10	1.14215
$970$	−2.00692e10	−0.706041
$971$	−2.36795e10	−0.830050	−0.415025	−	0.909810i	$-0.636227\pi$
$971$	−2.36795e10	−0.830050	−0.415025	+	0.909810i	$0.636227\pi$
$972$	−9.18330e8	−0.0320750
$973$	3.60321e10	1.25399
$974$	−2.28412e10	−0.792070
$975$	−4.34585e10	−1.50161
$976$	−1.22134e10	−0.420497
$977$	1.52997e10	0.524869	0.262435	−	0.964950i	$-0.415475\pi$
$977$	1.52997e10	0.524869	0.262435	+	0.964950i	$0.415475\pi$
$978$	−1.19916e10	−0.409913
$979$	−1.16930e10	−0.398278
$980$	−3.09596e10	−1.05076
$981$	1.62267e10	0.548768
$982$	1.89116e10	0.637293
$983$	−5.55638e9	−0.186575	−0.0932877	−	0.995639i	$-0.529738\pi$
$983$	−5.55638e9	−0.186575	−0.0932877	+	0.995639i	$0.529738\pi$
$984$	−1.97038e9	−0.0659277
$985$	6.87173e10	2.29108
$986$	−1.60224e10	−0.532301
$987$	6.79593e9	0.224977
$988$	−2.21438e10	−0.730469
$989$	−1.79208e10	−0.589075
$990$	−8.38458e9	−0.274637
$991$	−8.14619e9	−0.265887	−0.132943	−	0.991124i	$-0.542443\pi$
$991$	−8.14619e9	−0.265887	−0.132943	+	0.991124i	$0.542443\pi$
$992$	9.47970e9	0.308322
$993$	2.54098e10	0.823529
$994$	−3.39685e10	−1.09704
$995$	5.52544e10	1.77822
$996$	1.85405e9	0.0594584
$997$	−3.86942e10	−1.23655	−0.618277	−	0.785961i	$-0.712169\pi$
$997$	−3.86942e10	−1.23655	−0.618277	+	0.785961i	$0.712169\pi$
$998$	4.89444e9	0.155864
$999$	6.30262e9	0.200005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.d.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.d.1.1	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -497.674 q^{5} -216.000 q^{6} -1339.98 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -497.674 q^{5} -216.000 q^{6} -1339.98 q^{7} +512.000 q^{8} +729.000 q^{9} -3981.39 q^{10} +2888.81 q^{11} -1728.00 q^{12} +9492.97 q^{13} -10719.9 q^{14} +13437.2 q^{15} +4096.00 q^{16} +32871.9 q^{17} +5832.00 q^{18} -36447.6 q^{19} -31851.1 q^{20} +36179.5 q^{21} +23110.5 q^{22} +23666.3 q^{23} -13824.0 q^{24} +169554. q^{25} +75943.7 q^{26} -19683.0 q^{27} -85758.9 q^{28} -60927.3 q^{29} +107498. q^{30} +289298. q^{31} +32768.0 q^{32} -77997.9 q^{33} +262975. q^{34} +666874. q^{35} +46656.0 q^{36} -320206. q^{37} -291581. q^{38} -256310. q^{39} -254809. q^{40} +142533. q^{41} +289436. q^{42} -757230. q^{43} +184884. q^{44} -362804. q^{45} +189330. q^{46} +187839. q^{47} -110592. q^{48} +972009. q^{49} +1.35644e6 q^{50} -887541. q^{51} +607550. q^{52} -1.18577e6 q^{53} -157464. q^{54} -1.43769e6 q^{55} -686071. q^{56} +984086. q^{57} -487418. q^{58} -205379. q^{59} +859981. q^{60} -2.98179e6 q^{61} +2.31438e6 q^{62} -976847. q^{63} +262144. q^{64} -4.72440e6 q^{65} -623983. q^{66} -1.28373e6 q^{67} +2.10380e6 q^{68} -638990. q^{69} +5.33499e6 q^{70} +3.16875e6 q^{71} +373248. q^{72} -187529. q^{73} -2.56165e6 q^{74} -4.57797e6 q^{75} -2.33265e6 q^{76} -3.87095e6 q^{77} -2.05048e6 q^{78} +4.42566e6 q^{79} -2.03847e6 q^{80} +531441. q^{81} +1.14027e6 q^{82} -1.07294e6 q^{83} +2.31549e6 q^{84} -1.63595e7 q^{85} -6.05784e6 q^{86} +1.64504e6 q^{87} +1.47907e6 q^{88} -4.04768e6 q^{89} -2.90243e6 q^{90} -1.27204e7 q^{91} +1.51464e6 q^{92} -7.81104e6 q^{93} +1.50271e6 q^{94} +1.81390e7 q^{95} -884736. q^{96} +5.04076e6 q^{97} +7.77607e6 q^{98} +2.10594e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * q^99

Introduction

L-functions

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