LMFDB - Embedded newform 117.8.g.d.55.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,8,Mod(55,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.55"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 8, names="a")

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	117.g (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [16,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.5490479816$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{15} + 796 x^{14} - 475 x^{13} + 449889 x^{12} - 92038 x^{11} + 116806037 x^{10} + \cdots + 11\!\cdots\!00$ x^16 - x^15 + 796x^14 - 475x^13 + 449889x^12 - 92038x^11 + 116806037x^10 + 198490961x^9 + 21546318357x^8 + 32373621530x^7 + 2116860322049x^6 + 2147931692741x^5 + 148294262896076x^4 + 71847528967407x^3 + 4924011367741489x^2 - 2411116235274240x + 113340278793830400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{12}\cdot 3\cdot 13^{2}$
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$-5.55707 - 9.62513i$ of defining polynomial
Character	$\chi$	$=$	117.55
Dual form			117.8.g.d.100.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.05707 - 8.75910i) q^{2} +(12.8521 - 22.2605i) q^{4} +163.930 q^{5} +(-123.469 + 213.855i) q^{7} -1554.59 q^{8} +(-829.005 - 1435.88i) q^{10} +(1015.56 + 1759.01i) q^{11} +(7174.32 - 3358.22i) q^{13} +2497.57 q^{14} +(6216.58 + 10767.4i) q^{16} +(-17484.9 + 30284.8i) q^{17} +(-10739.1 + 18600.7i) q^{19} +(2106.84 - 3649.16i) q^{20} +(10271.6 - 17790.9i) q^{22} +(-35146.7 - 60875.8i) q^{23} -51252.0 q^{25} +(-65696.0 - 45857.9i) q^{26} +(3173.67 + 5496.95i) q^{28} +(103098. + 178571. i) q^{29} -304382. q^{31} +(-36618.1 + 63424.4i) q^{32} +353690. q^{34} +(-20240.3 + 35057.2i) q^{35} +(61327.8 + 106223. i) q^{37} +217233. q^{38} -254843. q^{40} +(117759. + 203964. i) q^{41} +(-45675.1 + 79111.6i) q^{43} +52208.5 q^{44} +(-355478. + 615706. i) q^{46} +967683. q^{47} +(381282. + 660400. i) q^{49} +(259185. + 448921. i) q^{50} +(17449.5 - 202864. i) q^{52} -28768.9 q^{53} +(166482. + 288354. i) q^{55} +(191943. - 332455. i) q^{56} +(1.04275e6 - 1.80609e6i) q^{58} +(38248.5 - 66248.4i) q^{59} +(296556. - 513650. i) q^{61} +(1.53928e6 + 2.66612e6i) q^{62} +2.33217e6 q^{64} +(1.17609e6 - 550512. i) q^{65} +(915012. + 1.58485e6i) q^{67} +(449435. + 778444. i) q^{68} +409426. q^{70} +(-1.41939e6 + 2.45845e6i) q^{71} +1.74309e6 q^{73} +(620278. - 1.07435e6i) q^{74} +(276039. + 478114. i) q^{76} -501563. q^{77} +3.23539e6 q^{79} +(1.01908e6 + 1.76511e6i) q^{80} +(1.19103e6 - 2.06292e6i) q^{82} +4.95198e6 q^{83} +(-2.86630e6 + 4.96458e6i) q^{85} +923929. q^{86} +(-1.57878e6 - 2.73453e6i) q^{88} +(-2.54255e6 - 4.40382e6i) q^{89} +(-167636. + 1.94890e6i) q^{91} -1.80683e6 q^{92} +(-4.89364e6 - 8.47603e6i) q^{94} +(-1.76046e6 + 3.04920e6i) q^{95} +(-7.53784e6 + 1.30559e7i) q^{97} +(3.85634e6 - 6.67938e6i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 9 q^{2} - 577 q^{4} - 384 q^{5} + 196 q^{7} - 5922 q^{8} - 6813 q^{10} - 5052 q^{11} + 17064 q^{13} - 14484 q^{14} - 61377 q^{16} + 22824 q^{17} + 63692 q^{19} + 121971 q^{20} - 73726 q^{22} + 72468 q^{23}+ \cdots + 51546651 q^{98}+O(q^{100})$ 16 * q + 9 * q^2 - 577 * q^4 - 384 * q^5 + 196 * q^7 - 5922 * q^8 - 6813 * q^10 - 5052 * q^11 + 17064 * q^13 - 14484 * q^14 - 61377 * q^16 + 22824 * q^17 + 63692 * q^19 + 121971 * q^20 - 73726 * q^22 + 72468 * q^23 + 58488 * q^25 - 154443 * q^26 - 172816 * q^28 + 221772 * q^29 + 58432 * q^31 - 77823 * q^32 + 643334 * q^34 - 31560 * q^35 - 368884 * q^37 - 2364492 * q^38 + 766030 * q^40 + 385824 * q^41 + 391492 * q^43 + 6149616 * q^44 - 2599078 * q^46 - 1089984 * q^47 - 1108780 * q^49 - 1626894 * q^50 - 1312478 * q^52 - 3329328 * q^53 - 418216 * q^55 + 3609156 * q^56 + 3685081 * q^58 + 6318924 * q^59 + 1763548 * q^61 + 12998388 * q^62 + 32227074 * q^64 - 9500508 * q^65 - 8607964 * q^67 + 10118091 * q^68 - 25381840 * q^70 - 9473220 * q^71 + 5020888 * q^73 - 2206221 * q^74 + 18702876 * q^76 - 35263992 * q^77 + 12644416 * q^79 + 24671247 * q^80 - 9347751 * q^82 + 22158432 * q^83 - 36911824 * q^85 + 30103956 * q^86 - 36676404 * q^88 - 18469860 * q^89 - 9614540 * q^91 - 64127424 * q^92 - 6026128 * q^94 + 16267080 * q^95 + 19660916 * q^97 + 51546651 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$ .

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−5.05707	−	8.75910i	−0.446986	−	0.774203i	0.551202	−	0.834372i	$-0.314169\pi$
$2$	−5.05707	−	8.75910i	−0.446986	−	0.774203i	−0.998188	+	0.0601691i	$0.980836\pi$
$3$		0			0
$3$		0			0
$4$	12.8521	−	22.2605i	0.100407	−	0.173910i
$5$	163.930			0.586494			0.293247	−	0.956037i	$-0.405264\pi$
$5$	163.930			0.586494			0.293247	+	0.956037i	$0.405264\pi$
$6$		0			0
$7$	−123.469	+	213.855i	−0.136055	+	0.235654i	−0.926000	−	0.377524i	$-0.876776\pi$
$7$	−123.469	+	213.855i	−0.136055	+	0.235654i	0.789945	+	0.613178i	$0.210109\pi$
$8$	−1554.59			−1.07349
$9$		0			0
$10$	−829.005	−	1435.88i	−0.262155	−	0.454065i
$11$	1015.56	+	1759.01i	0.230056	+	0.398468i	0.957824	−	0.287355i	$-0.0927758\pi$
$11$	1015.56	+	1759.01i	0.230056	+	0.398468i	−0.727768	+	0.685823i	$0.759442\pi$
$12$		0			0
$13$	7174.32	−	3358.22i	0.905689	−	0.423943i
$13$	7174.32	−	3358.22i	0.905689	−	0.423943i
$14$	2497.57			0.243259
$15$		0			0
$16$	6216.58	+	10767.4i	0.379430	+	0.657192i
$17$	−17484.9	+	30284.8i	−0.863162	+	1.49504i	0.00569886	+	0.999984i	$0.498186\pi$
$17$	−17484.9	+	30284.8i	−0.863162	+	1.49504i	−0.868861	+	0.495057i	$0.835147\pi$
$18$		0			0
$19$	−10739.1	+	18600.7i	−0.359195	+	0.622144i	−0.987827	−	0.155560i	$-0.950282\pi$
$19$	−10739.1	+	18600.7i	−0.359195	+	0.622144i	0.628632	+	0.777703i	$0.283615\pi$
$20$	2106.84	−	3649.16i	0.0588880	−	0.101997i
$21$		0			0
$22$	10271.6	−	17790.9i	0.205663	−	0.356220i
$23$	−35146.7	−	60875.8i	−0.602333	−	1.04327i	−0.992467	−	0.122513i	$-0.960905\pi$
$23$	−35146.7	−	60875.8i	−0.602333	−	1.04327i	0.390134	−	0.920758i	$-0.372429\pi$
$24$		0			0
$25$	−51252.0			−0.656025
$26$	−65696.0	−	45857.9i	−0.733048	−	0.511690i
$27$		0			0
$28$	3173.67	+	5496.95i	0.0273217	+	0.0473226i
$29$	103098.	+	178571.i	0.784979	+	1.35962i	0.929012	+	0.370050i	$0.120659\pi$
$29$	103098.	+	178571.i	0.784979	+	1.35962i	−0.144033	+	0.989573i	$0.546007\pi$
$30$		0			0
$31$	−304382.			−1.83507			−0.917537	−	0.397651i	$-0.869825\pi$
$31$	−304382.			−1.83507			−0.917537	+	0.397651i	$0.869825\pi$
$32$	−36618.1	+	63424.4i	−0.197547	+	0.342162i
$33$		0			0
$34$	353690.			1.54329
$35$	−20240.3	+	35057.2i	−0.0797954	+	0.138210i
$36$		0			0
$37$	61327.8	+	106223.i	0.199045	+	0.344756i	0.948219	−	0.317617i	$-0.102883\pi$
$37$	61327.8	+	106223.i	0.199045	+	0.344756i	−0.749174	+	0.662373i	$0.769549\pi$
$38$	217233.			0.642220
$39$		0			0
$40$	−254843.			−0.629598
$41$	117759.	+	203964.i	0.266839	+	0.462179i	0.968044	−	0.250781i	$-0.0806874\pi$
$41$	117759.	+	203964.i	0.266839	+	0.462179i	−0.701205	+	0.712960i	$0.747354\pi$
$42$		0			0
$43$	−45675.1	+	79111.6i	−0.0876072	+	0.151740i	−0.906499	−	0.422207i	$-0.861255\pi$
$43$	−45675.1	+	79111.6i	−0.0876072	+	0.151740i	0.818892	+	0.573948i	$0.194589\pi$
$44$	52208.5			0.0923967
$45$		0			0
$46$	−355478.	+	615706.i	−0.538469	+	0.932655i
$47$	967683.			1.35954			0.679768	−	0.733427i	$-0.262081\pi$
$47$	967683.			1.35954			0.679768	+	0.733427i	$0.262081\pi$
$48$		0			0
$49$	381282.	+	660400.i	0.462978	+	0.801901i
$50$	259185.	+	448921.i	0.293234	+	0.507896i
$51$		0			0
$52$	17449.5	−	202864.i	0.0172096	−	0.200075i
$53$	−28768.9			−0.0265435			−0.0132717	−	0.999912i	$-0.504225\pi$
$53$	−28768.9			−0.0265435			−0.0132717	+	0.999912i	$0.504225\pi$
$54$		0			0
$55$	166482.	+	288354.i	0.134926	+	0.233699i
$56$	191943.	−	332455.i	0.146054	−	0.252973i
$57$		0			0
$58$	1.04275e6	−	1.80609e6i	0.701749	−	1.21547i
$59$	38248.5	−	66248.4i	0.0242456	−	0.0419946i	−0.853648	−	0.520850i	$-0.825615\pi$
$59$	38248.5	−	66248.4i	0.0242456	−	0.0419946i	0.877894	+	0.478856i	$0.158948\pi$
$60$		0			0
$61$	296556.	−	513650.i	0.167283	−	0.289743i	−0.770180	−	0.637826i	$-0.779834\pi$
$61$	296556.	−	513650.i	0.167283	−	0.289743i	0.937464	+	0.348083i	$0.113167\pi$
$62$	1.53928e6	+	2.66612e6i	0.820252	+	1.42072i
$63$		0			0
$64$	2.33217e6			1.11206
$65$	1.17609e6	−	550512.i	0.531181	−	0.248640i
$66$		0			0
$67$	915012.	+	1.58485e6i	0.371676	+	0.643762i	0.989824	−	0.142300i	$-0.0454498\pi$
$67$	915012.	+	1.58485e6i	0.371676	+	0.643762i	−0.618147	+	0.786062i	$0.712116\pi$
$68$	449435.	+	778444.i	0.173335	+	0.300225i
$69$		0			0
$70$	409426.			0.142670
$71$	−1.41939e6	+	2.45845e6i	−0.470649	+	0.815188i	−0.999436	−	0.0335663i	$-0.989314\pi$
$71$	−1.41939e6	+	2.45845e6i	−0.470649	+	0.815188i	0.528788	+	0.848754i	$0.322647\pi$
$72$		0			0
$73$	1.74309e6			0.524434			0.262217	−	0.965009i	$-0.415546\pi$
$73$	1.74309e6			0.524434			0.262217	+	0.965009i	$0.415546\pi$
$74$	620278.	−	1.07435e6i	0.177941	−	0.308202i
$75$		0			0
$76$	276039.	+	478114.i	0.0721312	+	0.124935i
$77$	−501563.			−0.125201
$78$		0			0
$79$	3.23539e6			0.738298			0.369149	−	0.929370i	$-0.379649\pi$
$79$	3.23539e6			0.738298			0.369149	+	0.929370i	$0.379649\pi$
$80$	1.01908e6	+	1.76511e6i	0.222533	+	0.385439i
$81$		0			0
$82$	1.19103e6	−	2.06292e6i	0.238547	−	0.413175i
$83$	4.95198e6			0.950617			0.475309	−	0.879819i	$-0.342336\pi$
$83$	4.95198e6			0.950617			0.475309	+	0.879819i	$0.342336\pi$
$84$		0			0
$85$	−2.86630e6	+	4.96458e6i	−0.506239	+	0.876832i
$86$	923929.			0.156637
$87$		0			0
$88$	−1.57878e6	−	2.73453e6i	−0.246964	−	0.427753i
$89$	−2.54255e6	−	4.40382e6i	−0.382300	−	0.662163i	0.609091	−	0.793101i	$-0.291535\pi$
$89$	−2.54255e6	−	4.40382e6i	−0.382300	−	0.662163i	−0.991391	+	0.130938i	$0.958201\pi$
$90$		0			0
$91$	−167636.	+	1.94890e6i	−0.0233197	+	0.271109i
$92$	−1.80683e6			−0.241913
$93$		0			0
$94$	−4.89364e6	−	8.47603e6i	−0.607694	−	1.05256i
$95$	−1.76046e6	+	3.04920e6i	−0.210665	+	0.364883i
$96$		0			0
$97$	−7.53784e6	+	1.30559e7i	−0.838583	+	1.45247i	0.0524965	+	0.998621i	$0.483282\pi$
$97$	−7.53784e6	+	1.30559e7i	−0.838583	+	1.45247i	−0.891080	+	0.453847i	$0.850051\pi$
$98$	3.85634e6	−	6.67938e6i	0.413889	−	0.716878i
$99$		0			0
$100$	−658694.	+	1.14089e6i	−0.0658694	+	0.114089i
$101$	5.83275e6	+	1.01026e7i	0.563312	+	0.975685i	0.997205	+	0.0747201i	$0.0238063\pi$
$101$	5.83275e6	+	1.01026e7i	0.563312	+	0.975685i	−0.433893	+	0.900965i	$0.642860\pi$
$102$		0			0
$103$	6.15835e6			0.555308			0.277654	−	0.960681i	$-0.410443\pi$
$103$	6.15835e6			0.555308			0.277654	+	0.960681i	$0.410443\pi$
$104$	−1.11531e7	+	5.22064e6i	−0.972252	+	0.455100i
$105$		0			0
$106$	145486.	+	251990.i	0.0118646	+	0.0205500i
$107$	−8.05811e6	−	1.39571e7i	−0.635902	−	1.10141i	−0.986323	−	0.164822i	$-0.947295\pi$
$107$	−8.05811e6	−	1.39571e7i	−0.635902	−	1.10141i	0.350422	−	0.936592i	$-0.386038\pi$
$108$		0			0
$109$	−2.48704e7			−1.83946			−0.919729	−	0.392553i	$-0.871592\pi$
$109$	−2.48704e7			−1.83946			−0.919729	+	0.392553i	$0.871592\pi$
$110$	1.68382e6	−	2.91646e6i	0.120620	−	0.208921i
$111$		0			0
$112$	−3.07022e6			−0.206494
$113$	−727777.	+	1.26055e6i	−0.0474486	+	0.0821834i	−0.888774	−	0.458345i	$-0.848442\pi$
$113$	−727777.	+	1.26055e6i	−0.0474486	+	0.0821834i	0.841326	+	0.540529i	$0.181776\pi$
$114$		0			0
$115$	−5.76159e6	−	9.97937e6i	−0.353264	−	0.611872i
$116$	5.30010e6			0.315269
$117$		0			0
$118$	−773702.			−0.0433497
$119$	−4.31769e6	−	7.47846e6i	−0.234875	−	0.406816i
$120$		0			0
$121$	7.68084e6	−	1.33036e7i	0.394149	−	0.682685i
$122$	−5.99882e6			−0.299093
$123$		0			0
$124$	−3.91195e6	+	6.77569e6i	−0.184254	+	0.319137i
$125$	−2.12088e7			−0.971248
$126$		0			0
$127$	6.95145e6	+	1.20403e7i	0.301136	+	0.521582i	0.976393	−	0.216000i	$-0.0693010\pi$
$127$	6.95145e6	+	1.20403e7i	0.301136	+	0.521582i	−0.675258	+	0.737582i	$0.735968\pi$
$128$	−7.10681e6	−	1.23094e7i	−0.299530	−	0.518801i
$129$		0			0
$130$	−1.07695e7	−	7.51748e6i	−0.429928	−	0.300103i
$131$	−2.68014e7			−1.04162			−0.520809	−	0.853673i	$-0.674369\pi$
$131$	−2.68014e7			−1.04162			−0.520809	+	0.853673i	$0.674369\pi$
$132$		0			0
$133$	−2.65189e6	−	4.59321e6i	−0.0977405	−	0.169292i
$134$	9.25456e6	−	1.60294e7i	0.332268	−	0.575506i
$135$		0			0
$136$	2.71818e7	−	4.70802e7i	0.926599	−	1.60492i
$137$	−3.59209e6	+	6.22169e6i	−0.119351	+	0.206722i	−0.919511	−	0.393065i	$-0.871415\pi$
$137$	−3.59209e6	+	6.22169e6i	−0.119351	+	0.206722i	0.800160	+	0.599787i	$0.204748\pi$
$138$		0			0
$139$	−9.16798e6	+	1.58794e7i	−0.289549	+	0.501514i	−0.973702	−	0.227825i	$-0.926839\pi$
$139$	−9.16798e6	+	1.58794e7i	−0.289549	+	0.501514i	0.684153	+	0.729338i	$0.260172\pi$
$140$	520259.	+	901115.i	0.0160240	+	0.0277544i
$141$		0			0
$142$	2.87118e7			0.841494
$143$	1.31931e7	+	9.20921e6i	0.377287	+	0.263358i
$144$		0			0
$145$	1.69009e7	+	2.92732e7i	0.460385	+	0.797410i
$146$	−8.81495e6	−	1.52679e7i	−0.234415	−	0.406018i
$147$		0			0
$148$	3.15276e6			0.0799419
$149$	5.10856e6	−	8.84829e6i	0.126516	−	0.219133i	−0.795808	−	0.605549i	$-0.792954\pi$
$149$	5.10856e6	−	8.84829e6i	0.126516	−	0.219133i	0.922325	+	0.386416i	$0.126287\pi$
$150$		0			0
$151$	1.08524e7			0.256512			0.128256	−	0.991741i	$-0.459062\pi$
$151$	1.08524e7			0.256512			0.128256	+	0.991741i	$0.459062\pi$
$152$	1.66948e7	−	2.89163e7i	0.385593	−	0.667867i
$153$		0			0
$154$	2.53644e6	+	4.39324e6i	0.0559631	+	0.0969310i
$155$	−4.98974e7			−1.07626
$156$		0			0
$157$	−3.58959e7			−0.740280			−0.370140	−	0.928976i	$-0.620690\pi$
$157$	−3.58959e7			−0.740280			−0.370140	+	0.928976i	$0.620690\pi$
$158$	−1.63616e7	−	2.83391e7i	−0.330009	−	0.571592i
$159$		0			0
$160$	−6.00280e6	+	1.03972e7i	−0.115860	+	0.200676i
$161$	1.73581e7			0.327802
$162$		0			0
$163$	2.59182e7	−	4.48917e7i	0.468758	−	0.811912i	−0.530605	−	0.847619i	$-0.678035\pi$
$163$	2.59182e7	−	4.48917e7i	0.468758	−	0.811912i	0.999362	+	0.0357073i	$0.0113684\pi$
$164$	6.05378e6			0.107170
$165$		0			0
$166$	−2.50425e7	−	4.33749e7i	−0.424913	−	0.735970i
$167$	3.48738e7	+	6.04031e7i	0.579417	+	1.00358i	0.995546	+	0.0942740i	$0.0300530\pi$
$167$	3.48738e7	+	6.04031e7i	0.579417	+	1.00358i	−0.416129	+	0.909305i	$0.636614\pi$
$168$		0			0
$169$	4.01933e7	−	4.81859e7i	0.640545	−	0.767920i
$170$	5.79804e7			0.905127
$171$		0			0
$172$	1.17404e6	+	2.03350e6i	0.0175927	+	0.0304715i
$173$	−2.17440e7	+	3.76617e7i	−0.319285	+	0.553018i	−0.980339	−	0.197320i	$-0.936776\pi$
$173$	−2.17440e7	+	3.76617e7i	−0.319285	+	0.553018i	0.661054	+	0.750338i	$0.270109\pi$
$174$		0			0
$175$	6.32803e6	−	1.09605e7i	0.0892555	−	0.154595i
$176$	−1.26267e7	+	2.18701e7i	−0.174580	+	0.302382i
$177$		0			0
$178$	−2.57157e7	+	4.45409e7i	−0.341766	+	0.591955i
$179$	−1.54367e7	−	2.67372e7i	−0.201173	−	0.348441i	0.747734	−	0.663999i	$-0.231142\pi$
$179$	−1.54367e7	−	2.67372e7i	−0.201173	−	0.348441i	−0.948907	+	0.315557i	$0.897809\pi$
$180$		0			0
$181$	−1.39507e8			−1.74872			−0.874360	−	0.485278i	$-0.838718\pi$
$181$	−1.39507e8			−1.74872			−0.874360	+	0.485278i	$0.838718\pi$
$182$	1.79183e7	−	8.38737e6i	0.220317	−	0.103128i
$183$		0			0
$184$	5.46385e7	+	9.46366e7i	0.646601	+	1.11995i
$185$	1.00535e7	+	1.74131e7i	0.116739	+	0.202197i
$186$		0			0
$187$	−7.10282e7			−0.794302
$188$	1.24367e7	−	2.15411e7i	0.136507	−	0.236437i
$189$		0			0
$190$	3.56111e7			0.376658
$191$	−2.06193e7	+	3.57136e7i	−0.214120	+	0.370866i	−0.953000	−	0.302971i	$-0.902022\pi$
$191$	−2.06193e7	+	3.57136e7i	−0.214120	+	0.370866i	0.738880	+	0.673837i	$0.235355\pi$
$192$		0			0
$193$	6.94001e7	+	1.20204e8i	0.694879	+	1.20357i	0.970221	+	0.242220i	$0.0778754\pi$
$193$	6.94001e7	+	1.20204e8i	0.694879	+	1.20357i	−0.275342	+	0.961346i	$0.588791\pi$
$194$	1.52478e8			1.49934
$195$		0			0
$196$	1.96011e7			0.185945
$197$	2.56717e6	+	4.44647e6i	0.0239234	+	0.0414366i	0.877739	−	0.479139i	$-0.159051\pi$
$197$	2.56717e6	+	4.44647e6i	0.0239234	+	0.0414366i	−0.853816	+	0.520575i	$0.825718\pi$
$198$		0			0
$199$	−4.81377e7	+	8.33770e7i	−0.433012	+	0.749999i	−0.997131	−	0.0756947i	$-0.975883\pi$
$199$	−4.81377e7	+	8.33770e7i	−0.433012	+	0.749999i	0.564119	+	0.825693i	$0.309216\pi$
$200$	7.96756e7			0.704239
$201$		0			0
$202$	5.89933e7	−	1.02179e8i	0.503585	−	0.872235i
$203$	−5.09177e7			−0.427201
$204$		0			0
$205$	1.93042e7	+	3.34358e7i	0.156499	+	0.271065i
$206$	−3.11432e7	−	5.39416e7i	−0.248215	−	0.429921i
$207$		0			0
$208$	8.07591e7	+	5.63724e7i	0.622257	+	0.434355i
$209$	−4.36250e7			−0.330539
$210$		0			0
$211$	−5.38980e7	−	9.33541e7i	−0.394988	−	0.684140i	0.598112	−	0.801413i	$-0.295918\pi$
$211$	−5.38980e7	−	9.33541e7i	−0.394988	−	0.684140i	−0.993100	+	0.117273i	$0.962585\pi$
$212$	−369740.	+	640409.i	−0.00266515	+	0.00461617i
$213$		0			0
$214$	−8.15009e7	+	1.41164e8i	−0.568478	+	0.984634i
$215$	−7.48752e6	+	1.29688e7i	−0.0513811	+	0.0889946i
$216$		0			0
$217$	3.75818e7	−	6.50935e7i	0.249671	−	0.432443i
$218$	1.25771e8	+	2.17842e8i	0.822212	+	1.42411i
$219$		0			0
$220$	8.55853e6			0.0541901
$221$	−2.37396e7	+	2.75991e8i	−0.147945	+	1.71997i
$222$		0			0
$223$	−3.77450e6	−	6.53762e6i	−0.0227925	−	0.0394778i	0.854404	−	0.519609i	$-0.173922\pi$
$223$	−3.77450e6	−	6.53762e6i	−0.0227925	−	0.0394778i	−0.877197	+	0.480131i	$0.840589\pi$
$224$	−9.04239e6	−	1.56619e7i	−0.0537546	−	0.0931056i
$225$		0			0
$226$	1.47217e7			0.0848355
$227$	8.05895e7	−	1.39585e8i	0.457286	−	0.792043i	−0.541530	−	0.840681i	$-0.682155\pi$
$227$	8.05895e7	−	1.39585e8i	0.457286	−	0.792043i	0.998816	+	0.0486385i	$0.0154882\pi$
$228$		0			0
$229$	4.77059e7			0.262511			0.131256	−	0.991349i	$-0.458099\pi$
$229$	4.77059e7			0.262511			0.131256	+	0.991349i	$0.458099\pi$
$230$	−5.82735e7	+	1.00933e8i	−0.315809	+	0.546997i
$231$		0			0
$232$	−1.60275e8	−	2.77604e8i	−0.842670	−	1.45955i
$233$	−1.71967e8			−0.890632			−0.445316	−	0.895373i	$-0.646909\pi$
$233$	−1.71967e8			−0.890632			−0.445316	+	0.895373i	$0.646909\pi$
$234$		0			0
$235$	1.58632e8			0.797359
$236$	−983146.	−	1.70286e6i	−0.00486885	−	0.00843309i
$237$		0			0
$238$	−4.36697e7	+	7.56382e7i	−0.209972	+	0.363682i
$239$	−2.50000e8			−1.18453			−0.592266	−	0.805743i	$-0.701766\pi$
$239$	−2.50000e8			−1.18453			−0.592266	+	0.805743i	$0.701766\pi$
$240$		0			0
$241$	3.56126e7	−	6.16829e7i	0.163887	−	0.283861i	−0.772373	−	0.635170i	$-0.780930\pi$
$241$	3.56126e7	−	6.16829e7i	0.163887	−	0.283861i	0.936259	+	0.351309i	$0.114263\pi$
$242$	−1.55370e8			−0.704716
$243$		0			0
$244$	−7.62273e6	−	1.32030e7i	−0.0335928	−	0.0581844i
$245$	6.25036e7	+	1.08259e8i	0.271534	+	0.470310i
$246$		0			0
$247$	−1.45807e7	+	1.69511e8i	−0.0615656	+	0.715747i
$248$	4.73188e8			1.96994
$249$		0			0
$250$	1.07254e8	+	1.85770e8i	0.434135	+	0.751943i
$251$	1.90407e8	−	3.29795e8i	0.760022	−	1.31640i	−0.182817	−	0.983147i	$-0.558522\pi$
$251$	1.90407e8	−	3.29795e8i	0.760022	−	1.31640i	0.942839	−	0.333249i	$-0.108145\pi$
$252$		0			0
$253$	7.13874e7	−	1.23647e8i	0.277140	−	0.480021i
$254$	7.03079e7	−	1.21777e8i	0.269207	−	0.466280i
$255$		0			0
$256$	7.73793e7	−	1.34025e8i	0.288260	−	0.499282i
$257$	2.42914e7	+	4.20740e7i	0.0892662	+	0.154614i	0.907201	−	0.420697i	$-0.138214\pi$
$257$	2.42914e7	+	4.20740e7i	0.0892662	+	0.154614i	−0.817935	+	0.575311i	$0.804881\pi$
$258$		0			0
$259$	−3.02883e7			−0.108324
$260$	2.86050e6	−	3.32554e7i	0.0100933	−	0.117343i
$261$		0			0
$262$	1.35537e8	+	2.34756e8i	0.465588	+	0.806423i
$263$	1.24702e8	+	2.15990e8i	0.422697	+	0.732132i	0.996202	−	0.0870697i	$-0.0277503\pi$
$263$	1.24702e8	+	2.15990e8i	0.422697	+	0.732132i	−0.573506	+	0.819202i	$0.694417\pi$
$264$		0			0
$265$	−4.71609e6			−0.0155676
$266$	−2.68216e7	+	4.64563e7i	−0.0873773	+	0.151342i
$267$		0			0
$268$	4.70392e7			0.149275
$269$	2.36778e8	−	4.10112e8i	0.741667	−	1.28460i	−0.210069	−	0.977687i	$-0.567369\pi$
$269$	2.36778e8	−	4.10112e8i	0.741667	−	1.28460i	0.951736	−	0.306918i	$-0.0992978\pi$
$270$		0			0
$271$	7.22227e7	+	1.25093e8i	0.220435	+	0.381805i	0.954940	−	0.296798i	$-0.0959188\pi$
$271$	7.22227e7	+	1.25093e8i	0.220435	+	0.381805i	−0.734505	+	0.678603i	$0.762585\pi$
$272$	−4.34786e8			−1.31004
$273$		0			0
$274$	7.26619e7			0.213393
$275$	−5.20497e7	−	9.01527e7i	−0.150922	−	0.261405i
$276$		0			0
$277$	−2.26085e8	+	3.91590e8i	−0.639134	+	1.10701i	0.346489	+	0.938054i	$0.387374\pi$
$277$	−2.26085e8	+	3.91590e8i	−0.639134	+	1.10701i	−0.985623	+	0.168959i	$0.945959\pi$
$278$	1.85453e8			0.517697
$279$		0			0
$280$	3.14652e7	−	5.44994e7i	0.0856599	−	0.148367i
$281$	−1.80819e8			−0.486151			−0.243075	−	0.970007i	$-0.578156\pi$
$281$	−1.80819e8			−0.486151			−0.243075	+	0.970007i	$0.578156\pi$
$282$		0			0
$283$	−2.79735e8	−	4.84514e8i	−0.733658	−	1.27073i	−0.955310	−	0.295607i	$-0.904478\pi$
$283$	−2.79735e8	−	4.84514e8i	−0.733658	−	1.27073i	0.221652	−	0.975126i	$-0.428855\pi$
$284$	3.64842e7	+	6.31925e7i	0.0945128	+	0.163701i
$285$		0			0
$286$	1.39459e7	−	1.62132e8i	0.0352505	−	0.409814i
$287$	−5.81582e7			−0.145219
$288$		0			0
$289$	−4.06275e8	−	7.03689e8i	−0.990097	−	1.71490i
$290$	1.70938e8	−	2.96073e8i	0.411571	−	0.712863i
$291$		0			0
$292$	2.24024e7	−	3.88021e7i	0.0526568	−	0.0912042i
$293$	−1.89521e8	+	3.28260e8i	−0.440170	+	0.762397i	−0.997702	−	0.0677587i	$-0.978415\pi$
$293$	−1.89521e8	+	3.28260e8i	−0.440170	+	0.762397i	0.557532	+	0.830156i	$0.311749\pi$
$294$		0			0
$295$	6.27008e6	−	1.08601e7i	0.0142199	−	0.0246296i
$296$	−9.53393e7	−	1.65132e8i	−0.213674	−	0.370094i
$297$		0			0
$298$	−1.03337e8			−0.226204
$299$	−4.56588e8	−	3.18712e8i	−0.987813	−	0.689525i
$300$		0			0
$301$	−1.12789e7	−	1.95357e7i	−0.0238388	−	0.0412900i
$302$	−5.48815e7	−	9.50575e7i	−0.114657	−	0.198592i
$303$		0			0
$304$	−2.67042e8			−0.545157
$305$	4.86145e7	−	8.42027e7i	0.0981106	−	0.169933i
$306$		0			0
$307$	1.00446e9			1.98130			0.990649	−	0.136433i	$-0.0435639\pi$
$307$	1.00446e9			1.98130			0.990649	+	0.136433i	$0.0435639\pi$
$308$	−6.44613e6	+	1.11650e7i	−0.0125710	+	0.0217737i
$309$		0			0
$310$	2.52335e8	+	4.37056e8i	0.481073	+	0.833243i
$311$	−4.57084e8			−0.861658			−0.430829	−	0.902434i	$-0.641779\pi$
$311$	−4.57084e8			−0.861658			−0.430829	+	0.902434i	$0.641779\pi$
$312$		0			0
$313$	−6.79628e8			−1.25275			−0.626377	−	0.779520i	$-0.715463\pi$
$313$	−6.79628e8			−1.25275			−0.626377	+	0.779520i	$0.715463\pi$
$314$	1.81528e8	+	3.14416e8i	0.330895	+	0.573127i
$315$		0			0
$316$	4.15815e7	−	7.20212e7i	0.0741302	−	0.128397i
$317$	−1.42499e8			−0.251249			−0.125625	−	0.992078i	$-0.540093\pi$
$317$	−1.42499e8			−0.251249			−0.125625	+	0.992078i	$0.540093\pi$
$318$		0			0
$319$	−2.09406e8	+	3.62701e8i	−0.361178	+	0.625578i
$320$	3.82312e8			0.652218
$321$		0			0
$322$	−8.77811e7	−	1.52041e8i	−0.146523	−	0.253785i
$323$	−3.75544e8	−	6.50462e8i	−0.620086	−	1.07402i
$324$		0			0
$325$	−3.67698e8	+	1.72115e8i	−0.594155	+	0.278117i
$326$	−5.24281e8			−0.838113
$327$		0			0
$328$	−1.83066e8	−	3.17080e8i	−0.286450	−	0.496146i
$329$	−1.19479e8	+	2.06943e8i	−0.184972	+	0.320380i
$330$		0			0
$331$	−5.95525e7	+	1.03148e8i	−0.0902614	+	0.156337i	−0.907621	−	0.419790i	$-0.862104\pi$
$331$	−5.95525e7	+	1.03148e8i	−0.0902614	+	0.156337i	0.817360	+	0.576128i	$0.195437\pi$
$332$	6.36433e7	−	1.10233e8i	0.0954485	−	0.165322i
$333$		0			0
$334$	3.52718e8	−	6.10926e8i	0.517982	−	0.897172i
$335$	1.49998e8	+	2.59804e8i	0.217986	+	0.377563i
$336$		0			0
$337$	−3.15254e8			−0.448700			−0.224350	−	0.974509i	$-0.572026\pi$
$337$	−3.15254e8			−0.448700			−0.224350	+	0.974509i	$0.572026\pi$
$338$	−6.25325e8	−	1.08378e8i	−0.880841	−	0.152662i
$339$		0			0
$340$	7.36759e7	+	1.27610e8i	0.101660	+	0.176080i
$341$	−3.09120e8	−	5.35411e8i	−0.422169	−	0.731219i
$342$		0			0
$343$	−3.91670e8			−0.524072
$344$	7.10058e7	−	1.22986e8i	0.0940458	−	0.162892i
$345$		0			0
$346$	4.39844e8			0.570864
$347$	−2.70320e8	+	4.68208e8i	−0.347316	+	0.601569i	−0.985772	−	0.168089i	$-0.946240\pi$
$347$	−2.70320e8	+	4.68208e8i	−0.347316	+	0.601569i	0.638456	+	0.769659i	$0.279574\pi$
$348$		0			0
$349$	1.38744e8	+	2.40312e8i	0.174713	+	0.302612i	0.940062	−	0.341004i	$-0.110767\pi$
$349$	1.38744e8	+	2.40312e8i	0.174713	+	0.302612i	−0.765349	+	0.643616i	$0.777434\pi$
$350$	−1.28005e8			−0.159584
$351$		0			0
$352$	−1.48752e8			−0.181787
$353$	3.29885e6	+	5.71377e6i	0.00399163	+	0.00691371i	0.868014	−	0.496539i	$-0.165396\pi$
$353$	3.29885e6	+	5.71377e6i	0.00399163	+	0.00691371i	−0.864023	+	0.503453i	$0.832063\pi$
$354$		0			0
$355$	−2.32680e8	+	4.03014e8i	−0.276033	+	0.478103i
$356$	−1.30708e8			−0.153542
$357$		0			0
$358$	−1.56129e8	+	2.70423e8i	−0.179843	+	0.311497i
$359$	−6.67727e8			−0.761672			−0.380836	−	0.924643i	$-0.624364\pi$
$359$	−6.67727e8			−0.761672			−0.380836	+	0.924643i	$0.624364\pi$
$360$		0			0
$361$	2.16280e8	+	3.74607e8i	0.241958	+	0.419084i
$362$	7.05495e8	+	1.22195e9i	0.781653	+	1.35386i
$363$		0			0
$364$	4.12289e7	+	2.87790e7i	0.0448071	+	0.0312767i
$365$	2.85745e8			0.307577
$366$		0			0
$367$	−4.05024e8	−	7.01522e8i	−0.427710	−	0.740816i	0.568959	−	0.822366i	$-0.307346\pi$
$367$	−4.05024e8	−	7.01522e8i	−0.427710	−	0.740816i	−0.996669	+	0.0815499i	$0.974013\pi$
$368$	4.36984e8	−	7.56879e8i	0.457086	−	0.791697i
$369$		0			0
$370$	1.01682e8	−	1.76119e8i	0.104361	−	0.180759i
$371$	3.55207e6	−	6.15236e6i	0.00361137	−	0.00625508i
$372$		0			0
$373$	3.26424e8	−	5.65382e8i	0.325687	−	0.564107i	−0.655964	−	0.754792i	$-0.727738\pi$
$373$	3.26424e8	−	5.65382e8i	0.325687	−	0.564107i	0.981651	+	0.190685i	$0.0610711\pi$
$374$	3.59195e8	+	6.22144e8i	0.355042	+	0.614950i
$375$		0			0
$376$	−1.50435e9			−1.45945
$377$	1.33934e9	+	9.34902e8i	1.28735	+	0.898610i
$378$		0			0
$379$	7.44867e8	+	1.29015e9i	0.702816	+	1.21731i	0.967474	+	0.252971i	$0.0814076\pi$
$379$	7.44867e8	+	1.29015e9i	0.702816	+	1.21731i	−0.264658	+	0.964342i	$0.585259\pi$
$380$	4.52511e7	+	7.83772e7i	0.0423045	+	0.0732736i
$381$		0			0
$382$	4.17093e8			0.382834
$383$	−2.20288e7	+	3.81550e7i	−0.0200353	+	0.0347021i	−0.875869	−	0.482549i	$-0.839711\pi$
$383$	−2.20288e7	+	3.81550e7i	−0.0200353	+	0.0347021i	0.855834	+	0.517251i	$0.173045\pi$
$384$		0			0
$385$	−8.22212e7			−0.0734296
$386$	7.01922e8	−	1.21576e9i	0.621203	−	1.07595i
$387$		0			0
$388$	1.93754e8	+	3.35592e8i	0.168399	+	0.291676i
$389$	742221.			0.000639307			0.000319654	−	1.00000i	$-0.499898\pi$
$389$	742221.			0.000639307			0.000319654		1.00000i	$0.499898\pi$
$390$		0			0
$391$	2.45814e9			2.07964
$392$	−5.92736e8	−	1.02665e9i	−0.497004	−	0.860837i
$393$		0			0
$394$	2.59647e7	−	4.49723e7i	0.0213869	−	0.0370431i
$395$	5.30377e8			0.433007
$396$		0			0
$397$	6.46952e8	−	1.12055e9i	0.518926	−	0.898806i	−0.480832	−	0.876813i	$-0.659665\pi$
$397$	6.46952e8	−	1.12055e9i	0.518926	−	0.898806i	0.999758	−	0.0219932i	$-0.00700121\pi$
$398$	9.73744e8			0.774201
$399$		0			0
$400$	−3.18612e8	−	5.51852e8i	−0.248916	−	0.431135i
$401$	1.13847e9	+	1.97188e9i	0.881688	+	1.52713i	0.849464	+	0.527647i	$0.176926\pi$
$401$	1.13847e9	+	1.97188e9i	0.881688	+	1.52713i	0.0322237	+	0.999481i	$0.489741\pi$
$402$		0			0
$403$	−2.18374e9	+	1.02218e9i	−1.66201	+	0.777966i
$404$	2.99852e8			0.226241
$405$		0			0
$406$	2.57494e8	+	4.45993e8i	0.190953	+	0.330740i
$407$	−1.24565e8	+	2.15752e8i	−0.0915829	+	0.158626i
$408$		0			0
$409$	−1.21746e8	+	2.10871e8i	−0.0879882	+	0.152400i	−0.906661	−	0.421861i	$-0.861377\pi$
$409$	−1.21746e8	+	2.10871e8i	−0.0879882	+	0.152400i	0.818672	+	0.574261i	$0.194710\pi$
$410$	1.95245e8	−	3.38175e8i	0.139906	−	0.242325i
$411$		0			0
$412$	7.91476e7	−	1.37088e8i	0.0557567	−	0.0965735i
$413$	9.44501e6	+	1.63592e7i	0.00659747	+	0.0114271i
$414$		0			0
$415$	8.11778e8			0.557531
$416$	−4.97171e7	+	5.77998e8i	−0.0338594	+	0.393641i
$417$		0			0
$418$	2.20615e8	+	3.82116e8i	0.147746	+	0.255904i
$419$	−4.30530e8	−	7.45700e8i	−0.285926	−	0.495239i	0.686907	−	0.726745i	$-0.258968\pi$
$419$	−4.30530e8	−	7.45700e8i	−0.285926	−	0.495239i	−0.972833	+	0.231506i	$0.925635\pi$
$420$		0			0
$421$	3.23521e8			0.211308			0.105654	−	0.994403i	$-0.466306\pi$
$421$	3.23521e8			0.211308			0.105654	+	0.994403i	$0.466306\pi$
$422$	−5.45132e8	+	9.44196e8i	−0.353108	+	0.611602i
$423$		0			0
$424$	4.47237e7			0.0284943
$425$	8.96136e8	−	1.55215e9i	0.566256	−	0.980784i
$426$		0			0
$427$	7.32310e7	+	1.26840e8i	0.0455195	+	0.0788421i
$428$	−4.14254e8			−0.255396
$429$		0			0
$430$	1.51460e8			0.0918665
$431$	1.11143e9	+	1.92505e9i	0.668667	+	1.15817i	0.978277	+	0.207302i	$0.0664682\pi$
$431$	1.11143e9	+	1.92505e9i	0.668667	+	1.15817i	−0.309610	+	0.950864i	$0.600198\pi$
$432$		0			0
$433$	1.02721e9	−	1.77918e9i	0.608066	−	1.05320i	−0.383493	−	0.923544i	$-0.625279\pi$
$433$	1.02721e9	−	1.77918e9i	0.608066	−	1.05320i	0.991559	−	0.129657i	$-0.0413876\pi$
$434$	−7.60215e8			−0.446398
$435$		0			0
$436$	−3.19636e8	+	5.53626e8i	−0.184694	+	0.319900i
$437$	1.50977e9			0.865419
$438$		0			0
$439$	1.17568e8	+	2.03634e8i	0.0663229	+	0.114875i	0.897280	−	0.441462i	$-0.145540\pi$
$439$	1.17568e8	+	2.03634e8i	0.0663229	+	0.114875i	−0.830957	+	0.556336i	$0.812207\pi$
$440$	−2.58810e8	−	4.48272e8i	−0.144843	−	0.250875i
$441$		0			0
$442$	2.53748e9	−	1.18777e9i	1.39774	−	0.654264i
$443$	1.35426e9			0.740100			0.370050	−	0.929012i	$-0.379341\pi$
$443$	1.35426e9			0.740100			0.370050	+	0.929012i	$0.379341\pi$
$444$		0			0
$445$	−4.16800e8	−	7.21919e8i	−0.224217	−	0.388354i
$446$	−3.81758e7	+	6.61224e7i	−0.0203759	+	0.0352920i
$447$		0			0
$448$	−2.87950e8	+	4.98744e8i	−0.151302	+	0.262062i
$449$	−9.56886e8	+	1.65738e9i	−0.498882	+	0.864089i	−0.999999	−	0.00129025i	$-0.999589\pi$
$449$	−9.56886e8	+	1.65738e9i	−0.498882	+	0.864089i	0.501117	+	0.865380i	$0.332923\pi$
$450$		0			0
$451$	−2.39183e8	+	4.14277e8i	−0.122776	+	0.212654i
$452$	1.87069e7	+	3.24013e7i	0.00952834	+	0.0165036i
$453$		0			0
$454$	−1.63019e9			−0.817602
$455$	−2.74806e7	+	3.19483e8i	−0.0136769	+	0.159004i
$456$		0			0
$457$	1.64555e9	+	2.85018e9i	0.806500	+	1.39690i	0.915274	+	0.402832i	$0.131974\pi$
$457$	1.64555e9	+	2.85018e9i	0.806500	+	1.39690i	−0.108774	+	0.994067i	$0.534692\pi$
$458$	−2.41252e8	−	4.17861e8i	−0.117339	−	0.203237i
$459$		0			0
$460$	−2.96194e8			−0.141881
$461$	8.70376e8	−	1.50754e9i	0.413765	−	0.716662i	−0.581533	−	0.813523i	$-0.697547\pi$
$461$	8.70376e8	−	1.50754e9i	0.413765	−	0.716662i	0.995298	+	0.0968608i	$0.0308802\pi$
$462$		0			0
$463$	5.88177e7			0.0275407			0.0137703	−	0.999905i	$-0.495617\pi$
$463$	5.88177e7			0.0275407			0.0137703	+	0.999905i	$0.495617\pi$
$464$	−1.28184e9	+	2.22021e9i	−0.595689	+	1.03176i
$465$		0			0
$466$	8.69647e8	+	1.50627e9i	0.398100	+	0.689530i
$467$	−1.47573e9			−0.670499			−0.335249	−	0.942129i	$-0.608821\pi$
$467$	−1.47573e9			−0.670499			−0.335249	+	0.942129i	$0.608821\pi$
$468$		0			0
$469$	−4.51902e8			−0.202274
$470$	−8.02214e8	−	1.38948e9i	−0.356408	−	0.617318i
$471$		0			0
$472$	−5.94606e7	+	1.02989e8i	−0.0260275	+	0.0450809i
$473$	−1.85544e8			−0.0806182
$474$		0			0
$475$	5.50399e8	−	9.53320e8i	0.235641	−	0.408142i
$476$	−2.21965e8			−0.0943323
$477$		0			0
$478$	1.26427e9	+	2.18977e9i	0.529469	+	0.917067i
$479$	5.75573e8	+	9.96921e8i	0.239291	+	0.414464i	0.960511	−	0.278242i	$-0.0897518\pi$
$479$	5.75573e8	+	9.96921e8i	0.239291	+	0.414464i	−0.721220	+	0.692706i	$0.756418\pi$
$480$		0			0
$481$	7.96705e8	+	5.56125e8i	0.326430	+	0.227858i
$482$	−7.20382e8			−0.293021
$483$		0			0
$484$	−1.97430e8	−	3.41958e8i	−0.0791505	−	0.137093i
$485$	−1.23568e9	+	2.14026e9i	−0.491824	+	0.851864i
$486$		0			0
$487$	2.44105e9	−	4.22803e9i	0.957692	−	1.65877i	0.229607	−	0.973283i	$-0.426256\pi$
$487$	2.44105e9	−	4.22803e9i	0.957692	−	1.65877i	0.728084	−	0.685487i	$-0.240411\pi$
$488$	−4.61022e8	+	7.98514e8i	−0.179578	+	0.311038i
$489$		0			0
$490$	6.32170e8	−	1.09495e9i	0.242744	−	0.420444i
$491$	−6.01834e8	−	1.04241e9i	−0.229452	−	0.397422i	0.728194	−	0.685371i	$-0.240360\pi$
$491$	−6.01834e8	−	1.04241e9i	−0.229452	−	0.397422i	−0.957646	+	0.287949i	$0.907027\pi$
$492$		0			0
$493$	−7.21065e9			−2.71025
$494$	1.55850e9	−	7.29517e8i	0.581652	−	0.272264i
$495$		0			0
$496$	−1.89222e9	−	3.27742e9i	−0.696282	−	1.20600i
$497$	−3.50501e8	−	6.07085e8i	−0.128068	−	0.221821i
$498$		0			0
$499$	−3.64043e9			−1.31160			−0.655799	−	0.754935i	$-0.727668\pi$
$499$	−3.64043e9			−1.31160			−0.655799	+	0.754935i	$0.727668\pi$
$500$	−2.72577e8	+	4.72117e8i	−0.0975200	+	0.168910i
$501$		0			0
$502$	−3.85162e9			−1.35888
$503$	1.65318e9	−	2.86340e9i	0.579206	−	1.00321i	−0.416365	−	0.909198i	$-0.636696\pi$
$503$	1.65318e9	−	2.86340e9i	0.579206	−	1.00321i	0.995571	−	0.0940160i	$-0.0299705\pi$
$504$		0			0
$505$	9.56163e8	+	1.65612e9i	0.330379	+	0.572233i
$506$	−1.44404e9			−0.495512
$507$		0			0
$508$	3.57362e8			0.120944
$509$	−5.50673e8	−	9.53794e8i	−0.185089	−	0.320584i	0.758517	−	0.651653i	$-0.225924\pi$
$509$	−5.50673e8	−	9.53794e8i	−0.185089	−	0.320584i	−0.943607	+	0.331069i	$0.892591\pi$
$510$		0			0
$511$	−2.15218e8	+	3.72769e8i	−0.0713519	+	0.123585i
$512$	−3.38459e9			−1.11445
$513$		0			0
$514$	2.45687e8	−	4.25542e8i	0.0798015	−	0.138220i
$515$	1.00954e9			0.325685
$516$		0			0
$517$	9.82745e8	+	1.70216e9i	0.312769	+	0.541732i
$518$	1.53170e8	+	2.65298e8i	0.0484195	+	0.0838650i
$519$		0			0
$520$	−1.82833e9	+	8.55819e8i	−0.570220	+	0.266913i
$521$	−4.27358e9			−1.32391			−0.661957	−	0.749541i	$-0.730274\pi$
$521$	−4.27358e9			−1.32391			−0.661957	+	0.749541i	$0.730274\pi$
$522$		0			0
$523$	1.50948e9	+	2.61450e9i	0.461395	+	0.799159i	0.999031	−	0.0440177i	$-0.0140158\pi$
$523$	1.50948e9	+	2.61450e9i	0.461395	+	0.799159i	−0.537636	+	0.843177i	$0.680682\pi$
$524$	−3.44454e8	+	5.96612e8i	−0.104586	+	0.181147i
$525$		0			0
$526$	1.26125e9	−	2.18456e9i	0.377879	−	0.654506i
$527$	5.32210e9	−	9.21814e9i	1.58397	−	2.74351i
$528$		0			0
$529$	−7.68162e8	+	1.33050e9i	−0.225610	+	0.390768i
$530$	2.38496e7	+	4.13087e7i	0.00695849	+	0.0120525i
$531$		0			0
$532$	−1.36329e8			−0.0392553
$533$	1.52979e9	+	1.06784e9i	0.437611	+	0.305466i
$534$		0			0
$535$	−1.32097e9	−	2.28798e9i	−0.372952	−	0.645973i
$536$	−1.42246e9	−	2.46378e9i	−0.398992	−	0.691075i
$537$		0			0
$538$	−4.78962e9			−1.32606
$539$	−7.74434e8	+	1.34136e9i	−0.213022	+	0.368964i
$540$		0			0
$541$	2.08205e9			0.565329			0.282664	−	0.959219i	$-0.408782\pi$
$541$	2.08205e9			0.565329			0.282664	+	0.959219i	$0.408782\pi$
$542$	7.30471e8	−	1.26521e9i	0.197063	−	0.341323i
$543$		0			0
$544$	−1.28053e9	−	2.21794e9i	−0.341030	−	0.590682i
$545$	−4.07700e9			−1.07883
$546$		0			0
$547$	−1.26947e9			−0.331641			−0.165820	−	0.986156i	$-0.553027\pi$
$547$	−1.26947e9			−0.331641			−0.165820	+	0.986156i	$0.553027\pi$
$548$	9.23317e7	+	1.59923e8i	0.0239673	+	0.0415126i
$549$		0			0
$550$	−5.26438e8	+	9.11817e8i	−0.134920	+	0.233689i
$551$	−4.42872e9			−1.12784
$552$		0			0
$553$	−3.99470e8	+	6.91903e8i	−0.100449	+	0.173983i
$554$	4.57331e9			1.14274
$555$		0			0
$556$	2.35655e8	+	4.08167e8i	0.0581454	+	0.100711i
$557$	1.46286e9	+	2.53375e9i	0.358682	+	0.621256i	0.987741	−	0.156102i	$-0.0498928\pi$
$557$	1.46286e9	+	2.53375e9i	0.358682	+	0.621256i	−0.629059	+	0.777358i	$0.716559\pi$
$558$		0			0
$559$	−6.20140e7	+	7.20959e8i	−0.0150158	+	0.174570i
$560$	−5.03301e8			−0.121107
$561$		0			0
$562$	9.14412e8	+	1.58381e9i	0.217303	+	0.376379i
$563$	4.08319e9	−	7.07229e9i	0.964318	−	1.67025i	0.252882	−	0.967497i	$-0.418621\pi$
$563$	4.08319e9	−	7.07229e9i	0.964318	−	1.67025i	0.711436	−	0.702751i	$-0.248045\pi$
$564$		0			0
$565$	−1.19304e8	+	2.06641e8i	−0.0278283	+	0.0482001i
$566$	−2.82927e9	+	4.90045e9i	−0.655870	+	1.13600i
$567$		0			0
$568$	2.20656e9	−	3.82188e9i	0.505239	−	0.875099i
$569$	−1.01027e9	−	1.74984e9i	−0.229903	−	0.398203i	0.727876	−	0.685708i	$-0.240507\pi$
$569$	−1.01027e9	−	1.74984e9i	−0.229903	−	0.398203i	−0.957779	+	0.287505i	$0.907174\pi$
$570$		0			0
$571$	4.65062e9			1.04540			0.522702	−	0.852515i	$-0.324924\pi$
$571$	4.65062e9			1.04540			0.522702	+	0.852515i	$0.324924\pi$
$572$	3.74560e8	−	1.75327e8i	0.0836827	−	0.0391709i
$573$		0			0
$574$	2.94110e8	+	5.09414e8i	0.0649110	+	0.112429i
$575$	1.80133e9	+	3.12000e9i	0.395145	+	0.684412i
$576$		0			0
$577$	5.94249e9			1.28781			0.643907	−	0.765104i	$-0.277312\pi$
$577$	5.94249e9			1.28781			0.643907	+	0.765104i	$0.277312\pi$
$578$	−4.10912e9	+	7.11721e9i	−0.885119	+	1.53307i
$579$		0			0
$580$	8.68846e8			0.184903
$581$	−6.11416e8	+	1.05900e9i	−0.129336	+	0.224017i
$582$		0			0
$583$	−2.92167e7	−	5.06048e7i	−0.00610648	−	0.0105767i
$584$	−2.70979e9			−0.562977
$585$		0			0
$586$	3.83368e9			0.787000
$587$	−3.98257e8	−	6.89802e8i	−0.0812700	−	0.140764i	0.822526	−	0.568728i	$-0.192564\pi$
$587$	−3.98257e8	−	6.89802e8i	−0.0812700	−	0.140764i	−0.903796	+	0.427964i	$0.859231\pi$
$588$		0			0
$589$	3.26879e9	−	5.66171e9i	0.659149	−	1.14168i
$590$	−1.26833e8			−0.0254244
$591$		0			0
$592$	−7.62498e8	+	1.32069e9i	−0.151047	+	0.261622i
$593$	6.11768e9			1.20475			0.602373	−	0.798215i	$-0.294222\pi$
$593$	6.11768e9			1.20475			0.602373	+	0.798215i	$0.294222\pi$
$594$		0			0
$595$	−7.07799e8	−	1.22594e9i	−0.137753	−	0.238595i
$596$	−1.31311e8	−	2.27438e8i	−0.0254062	−	0.0440049i
$597$		0			0
$598$	−4.82640e8	+	5.61105e9i	−0.0922930	+	1.07298i
$599$	−4.08018e9			−0.775685			−0.387843	−	0.921726i	$-0.626780\pi$
$599$	−4.08018e9			−0.775685			−0.387843	+	0.921726i	$0.626780\pi$
$600$		0			0
$601$	−5.06656e9	−	8.77554e9i	−0.952035	−	1.64897i	−0.741013	−	0.671491i	$-0.765654\pi$
$601$	−5.06656e9	−	8.77554e9i	−0.952035	−	1.64897i	−0.211022	−	0.977481i	$-0.567679\pi$
$602$	−1.14077e8	+	1.97586e8i	−0.0213112	+	0.0369121i
$603$		0			0
$604$	1.39476e8	−	2.41580e8i	0.0257556	−	0.0446099i
$605$	1.25912e9	−	2.18086e9i	0.231166	−	0.400391i
$606$		0			0
$607$	2.26264e9	−	3.91901e9i	0.410635	−	0.711240i	−0.584325	−	0.811520i	$-0.698640\pi$
$607$	2.26264e9	−	3.91901e9i	0.410635	−	0.711240i	0.994959	+	0.100280i	$0.0319738\pi$
$608$	−7.86490e8	−	1.36224e9i	−0.141916	−	0.245805i
$609$		0			0
$610$	−9.83387e8			−0.175416
$611$	6.94247e9	−	3.24969e9i	1.23132	−	0.576365i
$612$		0			0
$613$	−4.12211e7	−	7.13970e7i	−0.00722783	−	0.0125190i	0.862389	−	0.506246i	$-0.168967\pi$
$613$	−4.12211e7	−	7.13970e7i	−0.00722783	−	0.0125190i	−0.869617	+	0.493727i	$0.835634\pi$
$614$	−5.07964e9	−	8.79820e9i	−0.885613	−	1.53393i
$615$		0			0
$616$	7.79722e8			0.134403
$617$	−5.32900e8	+	9.23010e8i	−0.0913372	+	0.158201i	−0.908074	−	0.418810i	$-0.862447\pi$
$617$	−5.32900e8	+	9.23010e8i	−0.0913372	+	0.158201i	0.816737	+	0.577010i	$0.195781\pi$
$618$		0			0
$619$	−7.35459e8			−0.124635			−0.0623177	−	0.998056i	$-0.519849\pi$
$619$	−7.35459e8			−0.124635			−0.0623177	+	0.998056i	$0.519849\pi$
$620$	−6.41285e8	+	1.11074e9i	−0.108064	+	0.187172i
$621$		0			0
$622$	2.31151e9	+	4.00365e9i	0.385149	+	0.667098i
$623$	1.25570e9			0.208055
$624$		0			0
$625$	5.27307e8			0.0863940
$626$	3.43693e9	+	5.95293e9i	0.559964	+	0.969886i
$627$		0			0
$628$	−4.61337e8	+	7.99059e8i	−0.0743292	+	0.128742i
$629$	−4.28924e9			−0.687232
$630$		0			0
$631$	−2.07991e8	+	3.60252e8i	−0.0329566	+	0.0570825i	−0.882033	−	0.471187i	$-0.843826\pi$
$631$	−2.07991e8	+	3.60252e8i	−0.0329566	+	0.0570825i	0.849077	+	0.528270i	$0.177159\pi$
$632$	−5.02969e9			−0.792559
$633$		0			0
$634$	7.20628e8	+	1.24816e9i	0.112305	+	0.194518i
$635$	1.13955e9	+	1.97376e9i	0.176614	+	0.305905i
$636$		0			0
$637$	4.95321e9	+	3.45750e9i	0.759274	+	0.529997i
$638$	4.23592e9			0.645766
$639$		0			0
$640$	−1.16502e9	−	2.01787e9i	−0.175672	−	0.304273i
$641$	2.81157e9	−	4.86978e9i	0.421644	−	0.730309i	−0.574457	−	0.818535i	$-0.694787\pi$
$641$	2.81157e9	−	4.86978e9i	0.421644	−	0.730309i	0.996100	+	0.0882264i	$0.0281199\pi$
$642$		0			0
$643$	−4.47709e9	+	7.75455e9i	−0.664137	+	1.15032i	0.315382	+	0.948965i	$0.397867\pi$
$643$	−4.47709e9	+	7.75455e9i	−0.664137	+	1.15032i	−0.979519	+	0.201354i	$0.935466\pi$
$644$	2.23087e8	−	3.86399e8i	0.0329135	−	0.0570079i
$645$		0			0
$646$	−3.79831e9	+	6.57886e9i	−0.554340	+	0.960145i
$647$	4.36397e9	+	7.55861e9i	0.633456	+	1.09718i	0.986840	+	0.161699i	$0.0516975\pi$
$647$	4.36397e9	+	7.55861e9i	0.633456	+	1.09718i	−0.353384	+	0.935478i	$0.614969\pi$
$648$		0			0
$649$	1.55375e8			0.0223113
$650$	3.36705e9	+	2.35031e9i	0.480898	+	0.335682i
$651$		0			0
$652$	−6.66206e8	−	1.15390e9i	−0.0941330	−	0.163043i
$653$	2.01742e9	+	3.49427e9i	0.283531	+	0.491090i	0.972252	−	0.233937i	$-0.0751609\pi$
$653$	2.01742e9	+	3.49427e9i	0.283531	+	0.491090i	−0.688721	+	0.725026i	$0.741828\pi$
$654$		0			0
$655$	−4.39355e9			−0.610902
$656$	−1.46411e9	+	2.53592e9i	−0.202494	+	0.350729i
$657$		0			0
$658$	2.41685e9			0.330719
$659$	−2.23633e9	+	3.87343e9i	−0.304394	+	0.527226i	−0.977126	−	0.212660i	$-0.931787\pi$
$659$	−2.23633e9	+	3.87343e9i	−0.304394	+	0.527226i	0.672732	+	0.739886i	$0.265121\pi$
$660$		0			0
$661$	4.53934e9	+	7.86236e9i	0.611346	+	1.05888i	0.991014	+	0.133760i	$0.0427050\pi$
$661$	4.53934e9	+	7.86236e9i	0.611346	+	1.05888i	−0.379668	+	0.925123i	$0.623962\pi$
$662$	1.20465e9			0.161382
$663$		0			0
$664$	−7.69828e9			−1.02048
$665$	−4.34724e8	−	7.52964e8i	−0.0573242	−	0.0992884i
$666$		0			0
$667$	7.24711e9	−	1.25524e10i	0.945637	−	1.63789i
$668$	1.79280e9			0.232710
$669$		0			0
$670$	1.51710e9	−	2.62769e9i	0.194873	−	0.337530i
$671$	1.20469e9			0.153938
$672$		0			0
$673$	4.13568e9	+	7.16321e9i	0.522991	+	0.905848i	0.999642	+	0.0267549i	$0.00851736\pi$
$673$	4.13568e9	+	7.16321e9i	0.522991	+	0.905848i	−0.476651	+	0.879093i	$0.658149\pi$
$674$	1.59426e9	+	2.76134e9i	0.200563	+	0.347385i
$675$		0			0
$676$	−5.56072e8	−	1.51401e9i	−0.0692337	−	0.188502i
$677$	7.18333e9			0.889744			0.444872	−	0.895594i	$-0.353249\pi$
$677$	7.18333e9			0.889744			0.444872	+	0.895594i	$0.353249\pi$
$678$		0			0
$679$	−1.86138e9	−	3.22400e9i	−0.228187	−	0.395231i
$680$	4.45591e9	−	7.71786e9i	0.543445	−	0.941274i
$681$		0			0
$682$	−3.12648e9	+	5.41523e9i	−0.377408	+	0.653689i
$683$	−2.09930e9	+	3.63609e9i	−0.252117	+	0.436679i	−0.964108	−	0.265509i	$-0.914460\pi$
$683$	−2.09930e9	+	3.63609e9i	−0.252117	+	0.436679i	0.711992	+	0.702188i	$0.247793\pi$
$684$		0			0
$685$	−5.88852e8	+	1.01992e9i	−0.0699986	+	0.121241i
$686$	1.98070e9	+	3.43068e9i	0.234253	+	0.405738i
$687$		0			0
$688$	−1.13577e9			−0.132963
$689$	−2.06397e8	+	9.66122e7i	−0.0240401	+	0.0112529i
$690$		0			0
$691$	−6.95215e9	−	1.20415e10i	−0.801579	−	1.38837i	−0.918577	−	0.395243i	$-0.870661\pi$
$691$	−6.95215e9	−	1.20415e10i	−0.801579	−	1.38837i	0.116998	−	0.993132i	$-0.462673\pi$
$692$	5.58912e8	+	9.68064e8i	0.0641168	+	0.111054i
$693$		0			0
$694$	5.46811e9			0.620982
$695$	−1.50291e9	+	2.60311e9i	−0.169819	+	0.294135i
$696$		0			0
$697$	−8.23600e9			−0.921302
$698$	1.40328e9	−	2.43055e9i	0.156189	−	0.270527i
$699$		0			0
$700$	−1.62657e8	−	2.81729e8i	−0.0179237	−	0.0310448i
$701$	−9.20624e9			−1.00941			−0.504707	−	0.863291i	$-0.668399\pi$
$701$	−9.20624e9			−1.00941			−0.504707	+	0.863291i	$0.668399\pi$
$702$		0			0
$703$	−2.63442e9			−0.285984
$704$	2.36847e9	+	4.10230e9i	0.255837	+	0.443122i
$705$		0			0
$706$	3.33650e7	−	5.77899e7i	0.00356841	−	0.00618067i
$707$	−2.88066e9			−0.306566
$708$		0			0
$709$	−7.85494e9	+	1.36052e10i	−0.827716	+	1.43365i	0.0721104	+	0.997397i	$0.477027\pi$
$709$	−7.85494e9	+	1.36052e10i	−0.827716	+	1.43365i	−0.899826	+	0.436249i	$0.856307\pi$
$710$	4.70672e9			0.493531
$711$		0			0
$712$	3.95261e9	+	6.84612e9i	0.410397	+	0.710828i
$713$	1.06980e10	+	1.85295e10i	1.10532	+	1.91448i
$714$		0			0
$715$	2.16275e9	+	1.50967e9i	0.221276	+	0.154458i
$716$	−7.93575e8			−0.0807965
$717$		0			0
$718$	3.37674e9	+	5.84869e9i	0.340457	+	0.589689i
$719$	2.92888e9	−	5.07296e9i	0.293866	−	0.508992i	−0.680854	−	0.732419i	$-0.738391\pi$
$719$	2.92888e9	−	5.07296e9i	0.293866	−	0.508992i	0.974721	+	0.223427i	$0.0717246\pi$
$720$		0			0
$721$	−7.60365e8	+	1.31699e9i	−0.0755525	+	0.130861i
$722$	2.18748e9	−	3.78883e9i	0.216304	−	0.374650i
$723$		0			0
$724$	−1.79295e9	+	3.10548e9i	−0.175583	+	0.304119i
$725$	−5.28398e9	−	9.15213e9i	−0.514966	−	0.891947i
$726$		0			0
$727$	7.95539e8			0.0767876			0.0383938	−	0.999263i	$-0.487776\pi$
$727$	7.95539e8			0.0767876			0.0383938	+	0.999263i	$0.487776\pi$
$728$	2.60605e8	−	3.02973e9i	0.0250336	−	0.291034i
$729$		0			0
$730$	−1.44503e9	−	2.50287e9i	−0.137483	−	0.238127i
$731$	−1.59725e9	−	2.76652e9i	−0.151238	−	0.261953i
$732$		0			0
$733$	5.89067e9			0.552460			0.276230	−	0.961092i	$-0.410915\pi$
$733$	5.89067e9			0.552460			0.276230	+	0.961092i	$0.410915\pi$
$734$	−4.09647e9	+	7.09530e9i	−0.382361	+	0.662269i
$735$		0			0
$736$	5.14801e9			0.475957
$737$	−1.85851e9	+	3.21903e9i	−0.171013	+	0.296202i
$738$		0			0
$739$	−4.03340e9	−	6.98606e9i	−0.367634	−	0.636761i	0.621561	−	0.783366i	$-0.286499\pi$
$739$	−4.03340e9	−	6.98606e9i	−0.367634	−	0.636761i	−0.989195	+	0.146605i	$0.953165\pi$
$740$	5.16832e8			0.0468854
$741$		0			0
$742$	−7.18522e7			−0.00645693
$743$	−9.23486e9	−	1.59953e10i	−0.825980	−	1.43064i	−0.901168	−	0.433470i	$-0.857289\pi$
$743$	−9.23486e9	−	1.59953e10i	−0.825980	−	1.43064i	0.0751879	−	0.997169i	$-0.476044\pi$
$744$		0			0
$745$	8.37447e8	−	1.45050e9i	0.0742011	−	0.128520i
$746$	−6.60299e9			−0.582310
$747$		0			0
$748$	−9.12861e8	+	1.58112e9i	−0.0797533	+	0.138137i
$749$	3.97971e9			0.346071
$750$		0			0
$751$	−2.05763e9	−	3.56392e9i	−0.177267	−	0.307035i	0.763677	−	0.645599i	$-0.223392\pi$
$751$	−2.05763e9	−	3.56392e9i	−0.177267	−	0.307035i	−0.940943	+	0.338564i	$0.890059\pi$
$752$	6.01568e9	+	1.04195e10i	0.515849	+	0.893476i
$753$		0			0
$754$	1.41576e9	−	1.64593e10i	0.120279	−	1.39833i
$755$	1.77904e9			0.150443
$756$		0			0
$757$	5.00576e9	+	8.67023e9i	0.419406	+	0.726432i	0.995880	−	0.0906832i	$-0.0289051\pi$
$757$	5.00576e9	+	8.67023e9i	0.419406	+	0.726432i	−0.576474	+	0.817116i	$0.695572\pi$
$758$	7.53369e9	−	1.30487e10i	0.628298	−	1.08824i
$759$		0			0
$760$	2.73678e9	−	4.74025e9i	0.226148	−	0.391700i
$761$	−9.42141e9	+	1.63184e10i	−0.774942	+	1.34224i	0.159884	+	0.987136i	$0.448888\pi$
$761$	−9.42141e9	+	1.63184e10i	−0.774942	+	1.34224i	−0.934827	+	0.355104i	$0.884445\pi$
$762$		0			0
$763$	3.07072e9	−	5.31865e9i	0.250268	−	0.433476i
$764$	5.30001e8	+	9.17989e8i	0.0429982	+	0.0744751i
$765$		0			0
$766$	4.45605e8			0.0358219
$767$	5.19308e7	−	6.03734e8i	0.00415567	−	0.0483128i
$768$		0			0
$769$	3.92588e9	+	6.79983e9i	0.311312	+	0.539207i	0.978647	−	0.205550i	$-0.0658984\pi$
$769$	3.92588e9	+	6.79983e9i	0.311312	+	0.539207i	−0.667335	+	0.744758i	$0.732565\pi$
$770$	4.15798e8	+	7.20184e8i	0.0328220	+	0.0568494i
$771$		0			0
$772$	3.56774e9			0.279083
$773$	−9.70260e9	+	1.68054e10i	−0.755544	+	1.30864i	0.189560	+	0.981869i	$0.439294\pi$
$773$	−9.70260e9	+	1.68054e10i	−0.755544	+	1.30864i	−0.945103	+	0.326771i	$0.894039\pi$
$774$		0			0
$775$	1.56002e10			1.20385
$776$	1.17182e10	−	2.02966e10i	0.900214	−	1.55922i
$777$		0			0
$778$	−3.75347e6	−	6.50119e6i	−0.000285762	−	0.000494954i
$779$	−5.05849e9			−0.383389
$780$		0			0
$781$	−5.76592e9			−0.433102
$782$	−1.24310e10	−	2.15311e10i	−0.929572	−	1.61007i
$783$		0			0
$784$	−4.74055e9	+	8.21087e9i	−0.351336	+	0.608531i
$785$	−5.88442e9			−0.434170
$786$		0			0
$787$	1.26249e10	−	2.18670e10i	0.923246	−	1.59911i	0.128887	−	0.991659i	$-0.458859\pi$
$787$	1.26249e10	−	2.18670e10i	0.923246	−	1.59911i	0.794359	−	0.607449i	$-0.207807\pi$
$788$	1.31974e8			0.00960830
$789$		0			0
$790$	−2.68216e9	−	4.64563e9i	−0.193548	−	0.335235i
$791$	−1.79716e8	−	3.11277e8i	−0.0129112	−	0.0223629i
$792$		0			0
$793$	4.02640e8	−	4.68099e9i	0.0286722	−	0.333336i
$794$	−1.30867e10			−0.927810
$795$		0			0
$796$	1.23734e9	+	2.14314e9i	0.0869548	+	0.150610i
$797$	1.17485e9	−	2.03490e9i	0.0822011	−	0.142376i	−0.821994	−	0.569496i	$-0.807138\pi$
$797$	1.17485e9	−	2.03490e9i	0.0822011	−	0.142376i	0.904195	+	0.427120i	$0.140472\pi$
$798$		0			0
$799$	−1.69199e10	+	2.93060e10i	−1.17350	+	2.03256i
$800$	1.87675e9	−	3.25062e9i	0.129596	−	0.224467i
$801$		0			0
$802$	1.15146e10	−	1.99439e10i	0.788204	−	1.36521i
$803$	1.77023e9	+	3.06612e9i	0.120649	+	0.208970i
$804$		0			0
$805$	2.84551e9			0.192254
$806$	1.99967e10	+	1.39583e10i	1.34520	+	0.938989i
$807$		0			0
$808$	−9.06751e9	−	1.57054e10i	−0.604712	−	1.04739i
$809$	−5.29367e9	−	9.16891e9i	−0.351510	−	0.608832i	0.635005	−	0.772508i	$-0.280998\pi$
$809$	−5.29367e9	−	9.16891e9i	−0.351510	−	0.608832i	−0.986514	+	0.163676i	$0.947665\pi$
$810$		0			0
$811$	−8.44691e9			−0.556064			−0.278032	−	0.960572i	$-0.589682\pi$
$811$	−8.44691e9			−0.556064			−0.278032	+	0.960572i	$0.589682\pi$
$812$	−6.54398e8	+	1.13345e9i	−0.0428939	+	0.0742945i
$813$		0			0
$814$	2.51973e9			0.163745
$815$	4.24877e9	−	7.35909e9i	0.274923	−	0.476181i
$816$		0			0
$817$	−9.81018e8	−	1.69917e9i	−0.0629361	−	0.109009i
$818$	2.46272e9			0.157318
$819$		0			0
$820$	9.92396e8			0.0628545
$821$	−4.61864e8	−	7.99972e8i	−0.0291281	−	0.0504514i	0.851094	−	0.525014i	$-0.175940\pi$
$821$	−4.61864e8	−	7.99972e8i	−0.0291281	−	0.0504514i	−0.880222	+	0.474562i	$0.842606\pi$
$822$		0			0
$823$	2.97774e9	−	5.15760e9i	0.186203	−	0.322514i	−0.757778	−	0.652512i	$-0.773715\pi$
$823$	2.97774e9	−	5.15760e9i	0.186203	−	0.322514i	0.943981	+	0.329999i	$0.107048\pi$
$824$	−9.57368e9			−0.596120
$825$		0			0
$826$	9.55282e7	−	1.65460e8i	0.00589795	−	0.0102156i
$827$	7.33081e9			0.450695			0.225348	−	0.974278i	$-0.427648\pi$
$827$	7.33081e9			0.450695			0.225348	+	0.974278i	$0.427648\pi$
$828$		0			0
$829$	−1.01384e10	−	1.75603e10i	−0.618059	−	1.07051i	−0.989839	−	0.142189i	$-0.954586\pi$
$829$	−1.01384e10	−	1.75603e10i	−0.618059	−	1.07051i	0.371780	−	0.928321i	$-0.378748\pi$
$830$	−4.10522e9	−	7.11045e9i	−0.249209	−	0.431642i
$831$		0			0
$832$	1.67317e10	−	7.83192e9i	1.00718	−	0.471451i
$833$	−2.66668e10			−1.59850
$834$		0			0
$835$	5.71685e9	+	9.90188e9i	0.339824	+	0.588593i
$836$	−5.60672e8	+	9.71112e8i	−0.0331884	+	0.0574840i
$837$		0			0
$838$	−4.35444e9	+	7.54211e9i	−0.255610	+	0.442730i
$839$	5.98389e9	−	1.03644e10i	0.349798	−	0.605867i	−0.636416	−	0.771346i	$-0.719584\pi$
$839$	5.98389e9	−	1.03644e10i	0.349798	−	0.605867i	0.986213	+	0.165479i	$0.0529170\pi$
$840$		0			0
$841$	−1.26335e10	+	2.18819e10i	−0.732383	+	1.26852i
$842$	−1.63607e9	−	2.83376e9i	−0.0944516	−	0.163595i
$843$		0			0
$844$	−2.77081e9			−0.158638
$845$	6.58888e9	−	7.89911e9i	0.375676	−	0.450380i
$846$		0			0
$847$	1.89669e9	+	3.28517e9i	0.107252	+	0.185766i
$848$	−1.78844e8	−	3.09767e8i	−0.0100714	−	0.0174442i
$849$		0			0
$850$	−1.81273e10			−1.01243
$851$	4.31093e9	−	7.46675e9i	0.239783	−	0.415316i
$852$		0			0
$853$	−1.44625e10			−0.797853			−0.398927	−	0.916983i	$-0.630617\pi$
$853$	−1.44625e10			−0.797853			−0.398927	+	0.916983i	$0.630617\pi$
$854$	7.40668e8	−	1.28288e9i	0.0406931	−	0.0704826i
$855$		0			0
$856$	1.25270e10	+	2.16974e10i	0.682637	+	1.18236i
$857$	−1.74997e10			−0.949723			−0.474861	−	0.880061i	$-0.657502\pi$
$857$	−1.74997e10			−0.949723			−0.474861	+	0.880061i	$0.657502\pi$
$858$		0			0
$859$	−1.21484e10			−0.653945			−0.326973	−	0.945034i	$-0.606028\pi$
$859$	−1.21484e10			−0.653945			−0.326973	+	0.945034i	$0.606028\pi$
$860$	1.92460e8	+	3.33351e8i	0.0103180	+	0.0178713i
$861$		0			0
$862$	1.12411e10	−	1.94702e10i	0.597770	−	1.03537i
$863$	−5.06062e9			−0.268019			−0.134010	−	0.990980i	$-0.542785\pi$
$863$	−5.06062e9			−0.268019			−0.134010	+	0.990980i	$0.542785\pi$
$864$		0			0
$865$	−3.56450e9	+	6.17389e9i	−0.187259	+	0.324341i
$866$	−2.07786e10			−1.08719
$867$		0			0
$868$	−9.66008e8	−	1.67317e9i	−0.0501374	−	0.0868405i
$869$	3.28575e9	+	5.69108e9i	0.169850	+	0.294188i
$870$		0			0
$871$	1.18868e10	+	8.29739e9i	0.609541	+	0.425479i
$872$	3.86632e10			1.97465
$873$		0			0
$874$	−7.63503e9	−	1.32243e10i	−0.386830	−	0.670010i
$875$	2.61862e9	−	4.53559e9i	0.132143	−	0.228879i
$876$		0			0
$877$	−1.14200e10	+	1.97801e10i	−0.571700	+	0.990213i	0.424692	+	0.905338i	$0.360383\pi$
$877$	−1.14200e10	+	1.97801e10i	−0.571700	+	0.990213i	−0.996392	+	0.0848754i	$0.972951\pi$
$878$	1.18910e9	−	2.05958e9i	0.0592909	−	0.102695i
$879$		0			0
$880$	−2.06989e9	+	3.58516e9i	−0.102390	+	0.177345i
$881$	1.17368e10	+	2.03287e10i	0.578273	+	1.00160i	0.995677	+	0.0928780i	$0.0296067\pi$
$881$	1.17368e10	+	2.03287e10i	0.578273	+	1.00160i	−0.417404	+	0.908721i	$0.637060\pi$
$882$		0			0
$883$	−1.70477e10			−0.833302			−0.416651	−	0.909067i	$-0.636796\pi$
$883$	−1.70477e10			−0.833302			−0.416651	+	0.909067i	$0.636796\pi$
$884$	5.83858e9	+	4.07551e9i	0.284265	+	0.198426i
$885$		0			0
$886$	−6.84861e9	−	1.18621e10i	−0.330814	−	0.572987i
$887$	7.28432e9	+	1.26168e10i	0.350474	+	0.607039i	0.986333	−	0.164766i	$-0.0526870\pi$
$887$	7.28432e9	+	1.26168e10i	0.350474	+	0.607039i	−0.635858	+	0.771806i	$0.719354\pi$
$888$		0			0
$889$	−3.43315e9			−0.163884
$890$	−4.21557e9	+	7.30159e9i	−0.200443	+	0.347178i
$891$		0			0
$892$	−1.94040e8			−0.00915409
$893$	−1.03920e10	+	1.79995e10i	−0.488338	+	0.845826i
$894$		0			0
$895$	−2.53054e9	−	4.38302e9i	−0.117987	−	0.204359i
$896$	3.50988e9			0.163010
$897$		0			0
$898$	1.93562e10			0.891974
$899$	−3.13812e10	−	5.43539e10i	−1.44049	−	2.49501i
$900$		0			0
$901$	5.03022e8	−	8.71259e8i	0.0229113	−	0.0396836i
$902$	4.83826e9			0.219516
$903$		0			0
$904$	1.13139e9	−	1.95963e9i	0.0509358	−	0.0882234i
$905$	−2.28693e10			−1.02561
$906$		0			0
$907$	−9.66075e9	−	1.67329e10i	−0.429918	−	0.744640i	0.566948	−	0.823754i	$-0.308124\pi$
$907$	−9.66075e9	−	1.67329e10i	−0.429918	−	0.744640i	−0.996866	+	0.0791142i	$0.974791\pi$
$908$	−2.07149e9	−	3.58792e9i	−0.0918293	−	0.159053i
$909$		0			0
$910$	2.93735e9	−	1.37494e9i	0.129214	−	0.0604838i
$911$	−1.95446e10			−0.856471			−0.428235	−	0.903667i	$-0.640865\pi$
$911$	−1.95446e10			−0.856471			−0.428235	+	0.903667i	$0.640865\pi$
$912$		0			0
$913$	5.02906e9	+	8.71058e9i	0.218695	+	0.378791i
$914$	1.66433e10	−	2.88271e10i	0.720989	−	1.24879i
$915$		0			0
$916$	6.13120e8	−	1.06196e9i	0.0263579	−	0.0456533i
$917$	3.30914e9	−	5.73160e9i	0.141717	−	0.245462i
$918$		0			0
$919$	−9.81678e9	+	1.70032e10i	−0.417220	+	0.722646i	−0.995659	−	0.0930800i	$-0.970329\pi$
$919$	−9.81678e9	+	1.70032e10i	−0.417220	+	0.722646i	0.578439	+	0.815726i	$0.303662\pi$
$920$	8.95688e9	+	1.55138e10i	0.379227	+	0.656841i
$921$		0			0
$922$	−1.76062e10			−0.739789
$923$	−1.92713e9	+	2.24044e10i	−0.0806688	+	0.937835i
$924$		0			0
$925$	−3.14317e9	−	5.44413e9i	−0.130578	−	0.226169i
$926$	−2.97445e8	−	5.15190e8i	−0.0123103	−	0.0213221i
$927$		0			0
$928$	−1.51010e10			−0.620281
$929$	−1.33151e10	+	2.30625e10i	−0.544867	+	0.943737i	0.453748	+	0.891130i	$0.350087\pi$
$929$	−1.33151e10	+	2.30625e10i	−0.544867	+	0.943737i	−0.998615	+	0.0526074i	$0.983247\pi$
$930$		0			0
$931$	−1.63785e10			−0.665197
$932$	−2.21013e9	+	3.82805e9i	−0.0894256	+	0.154890i
$933$		0			0
$934$	7.46287e9	+	1.29261e10i	0.299704	+	0.519102i
$935$	−1.16437e10			−0.465853
$936$		0			0
$937$	1.47566e9			0.0585999			0.0292999	−	0.999571i	$-0.490672\pi$
$937$	1.47566e9			0.0585999			0.0292999	+	0.999571i	$0.490672\pi$
$938$	2.28530e9	+	3.95826e9i	0.0904136	+	0.156601i
$939$		0			0
$940$	2.03875e9	−	3.53123e9i	0.0800603	−	0.138669i
$941$	2.25755e10			0.883231			0.441616	−	0.897204i	$-0.354405\pi$
$941$	2.25755e10			0.883231			0.441616	+	0.897204i	$0.354405\pi$
$942$		0			0
$943$	8.27765e9	−	1.43373e10i	0.321452	−	0.556771i
$944$	9.51100e8			0.0367980
$945$		0			0
$946$	9.38309e8	+	1.62520e9i	0.0360352	+	0.0624148i
$947$	9.93974e9	+	1.72161e10i	0.380321	+	0.658735i	0.991108	−	0.133060i	$-0.0424802\pi$
$947$	9.93974e9	+	1.72161e10i	0.380321	+	0.658735i	−0.610787	+	0.791795i	$0.709147\pi$
$948$		0			0
$949$	1.25055e10	−	5.85369e9i	0.474974	−	0.222330i
$950$	−1.11336e10			−0.421313
$951$		0			0
$952$	6.71222e9	+	1.16259e10i	0.252137	+	0.436714i
$953$	1.23018e10	−	2.13073e10i	0.460408	−	0.797451i	−0.538573	−	0.842579i	$-0.681036\pi$
$953$	1.23018e10	−	2.13073e10i	0.460408	−	0.797451i	0.998981	+	0.0451281i	$0.0143696\pi$
$954$		0			0
$955$	−3.38012e9	+	5.85454e9i	−0.125580	+	0.217511i
$956$	−3.21301e9	+	5.56510e9i	−0.118935	+	0.206002i
$957$		0			0
$958$	5.82142e9	−	1.00830e10i	0.213919	−	0.370519i
$959$	−8.87024e8	−	1.53637e9i	−0.0324766	−	0.0562511i
$960$		0			0
$961$	6.51360e10			2.36749
$962$	8.42163e8	−	9.79078e9i	0.0304989	−	0.354572i
$963$		0			0
$964$	−9.15393e8	−	1.58551e9i	−0.0329108	−	0.0570031i
$965$	1.13767e10	+	1.97051e10i	0.407542	+	0.705884i
$966$		0			0
$967$	1.21598e10			0.432449			0.216224	−	0.976344i	$-0.430626\pi$
$967$	1.21598e10			0.432449			0.216224	+	0.976344i	$0.430626\pi$
$968$	−1.19405e10	+	2.06816e10i	−0.423116	+	0.732859i
$969$		0			0
$970$	2.49957e10			0.879353
$971$	1.72456e10	−	2.98703e10i	0.604522	−	1.04706i	−0.387605	−	0.921826i	$-0.626697\pi$
$971$	1.72456e10	−	2.98703e10i	0.604522	−	1.04706i	0.992127	−	0.125237i	$-0.0399693\pi$
$972$		0			0
$973$	−2.26392e9	−	3.92123e9i	−0.0787892	−	0.136467i
$974$	−4.93783e10			−1.71230
$975$		0			0
$976$	7.37426e9			0.253889
$977$	1.64824e10	+	2.85484e10i	0.565445	+	0.979380i	0.997008	+	0.0772970i	$0.0246290\pi$
$977$	1.64824e10	+	2.85484e10i	0.565445	+	0.979380i	−0.431563	+	0.902083i	$0.642038\pi$
$978$		0			0
$979$	5.16425e9	−	8.94474e9i	0.175901	−	0.304669i
$980$	3.21321e9			0.109055
$981$		0			0
$982$	−6.08703e9	+	1.05430e10i	−0.205123	+	0.355284i
$983$	4.71393e10			1.58287			0.791435	−	0.611253i	$-0.209334\pi$
$983$	4.71393e10			1.58287			0.791435	+	0.611253i	$0.209334\pi$
$984$		0			0
$985$	4.20837e8	+	7.28910e8i	0.0140309	+	0.0243023i
$986$	3.64648e10	+	6.31588e10i	1.21145	+	2.09829i
$987$		0			0
$988$	3.58601e9	+	2.50314e9i	0.118294	+	0.0825727i
$989$	6.42131e9			0.211075
$990$		0			0
$991$	1.18660e10	+	2.05525e10i	0.387300	+	0.670823i	0.992085	−	0.125565i	$-0.0400745\pi$
$991$	1.18660e10	+	2.05525e10i	0.387300	+	0.670823i	−0.604786	+	0.796388i	$0.706741\pi$
$992$	1.11459e10	−	1.93053e10i	0.362513	−	0.627892i
$993$		0			0
$994$	−3.54502e9	+	6.14015e9i	−0.114490	+	0.198302i
$995$	−7.89122e9	+	1.36680e10i	−0.253959	+	0.439870i
$996$		0			0
$997$	−6.00395e9	+	1.03991e10i	−0.191869	+	0.332326i	−0.945870	−	0.324547i	$-0.894788\pi$
$997$	−6.00395e9	+	1.03991e10i	−0.191869	+	0.332326i	0.754001	+	0.656873i	$0.228121\pi$
$998$	1.84099e10	+	3.18869e10i	0.586266	+	1.01544i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.8.g.d.55.2		16
3.2	odd	2		13.8.c.a.3.7	✓	16
12.11	even	2		208.8.i.d.81.1		16
13.9	even	3	inner	117.8.g.d.100.2		16
39.2	even	12		169.8.b.e.168.12		16
39.11	even	12		169.8.b.e.168.5		16
39.23	odd	6		169.8.a.e.1.7		8
39.29	odd	6		169.8.a.f.1.2		8
39.35	odd	6		13.8.c.a.9.7	yes	16
156.35	even	6		208.8.i.d.113.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.c.a.3.7	✓	16	3.2	odd	2
13.8.c.a.9.7	yes	16	39.35	odd	6
117.8.g.d.55.2		16	1.1	even	1	trivial
117.8.g.d.100.2		16	13.9	even	3	inner
169.8.a.e.1.7		8	39.23	odd	6
169.8.a.f.1.2		8	39.29	odd	6
169.8.b.e.168.5		16	39.11	even	12
169.8.b.e.168.12		16	39.2	even	12
208.8.i.d.81.1		16	12.11	even	2
208.8.i.d.113.1		16	156.35	even	6

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Database

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	117.8.g.d.55.2

Level	$117$
Weight	$8$
Character	117.55
Analytic conductor	$36.549$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$