LMFDB - Embedded newform 252.9.bg.a.29.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(29,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.29");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bg (of order $6$, degree $2$, minimal)

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:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.4
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-79.7801 - 14.0049i) q^{3} +(225.541 - 130.216i) q^{5} +(-453.746 + 785.912i) q^{7} +(6168.73 + 2234.62i) q^{9} +O(q^{10})$	\(q+(-79.7801 - 14.0049i) q^{3} +(225.541 - 130.216i) q^{5} +(-453.746 + 785.912i) q^{7} +(6168.73 + 2234.62i) q^{9} +(-6578.61 - 3798.16i) q^{11} +(7035.50 + 12185.8i) q^{13} +(-19817.4 + 7229.99i) q^{15} +5681.88i q^{17} -252476. q^{19} +(47206.5 - 56345.5i) q^{21} +(431834. - 249319. i) q^{23} +(-161400. + 279553. i) q^{25} +(-460846. - 264670. i) q^{27} +(564444. + 325882. i) q^{29} +(114193. + 197788. i) q^{31} +(471649. + 395150. i) q^{33} +236341. i q^{35} +1.99504e6 q^{37} +(-390632. - 1.07072e6i) q^{39} +(-3.01123e6 + 1.73853e6i) q^{41} +(2.38564e6 - 4.13206e6i) q^{43} +(1.68229e6 - 299270. i) q^{45} +(3.50541e6 + 2.02385e6i) q^{47} +(-411772. - 713209. i) q^{49} +(79574.0 - 453301. i) q^{51} +3.57890e6i q^{53} -1.97833e6 q^{55} +(2.01426e7 + 3.53589e6i) q^{57} +(7.16289e6 - 4.13550e6i) q^{59} +(-7.07891e6 + 1.22610e7i) q^{61} +(-4.55525e6 + 3.83412e6i) q^{63} +(3.17359e6 + 1.83227e6i) q^{65} +(-9.17106e6 - 1.58847e7i) q^{67} +(-3.79434e7 + 1.38429e7i) q^{69} -1.45330e7i q^{71} +1.94210e7 q^{73} +(1.67916e7 - 2.00424e7i) q^{75} +(5.97004e6 - 3.44680e6i) q^{77} +(-2.59991e7 + 4.50317e7i) q^{79} +(3.30597e7 + 2.75695e7i) q^{81} +(6.00862e7 + 3.46908e7i) q^{83} +(739874. + 1.28150e6i) q^{85} +(-4.04675e7 - 3.39039e7i) q^{87} -7.53685e7i q^{89} -1.27693e7 q^{91} +(-6.34032e6 - 1.73788e7i) q^{93} +(-5.69437e7 + 3.28765e7i) q^{95} +(-1.06245e7 + 1.84021e7i) q^{97} +(-3.20942e7 - 3.81305e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−79.7801	−	14.0049i	−0.984939	−	0.172900i
$3$	−79.7801	−	14.0049i	−0.984939	−	0.172900i
$4$		0			0
$5$	225.541	−	130.216i	0.360866	−	0.208346i	−0.308595	−	0.951194i	$-0.599859\pi$
$5$	225.541	−	130.216i	0.360866	−	0.208346i	0.669461	+	0.742848i	$0.266525\pi$
$6$		0			0
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$8$		0			0
$9$	6168.73	+	2234.62i	0.940211	+	0.340591i
$10$		0			0
$11$	−6578.61	−	3798.16i	−0.449328	−	0.259419i	0.258219	−	0.966087i	$-0.416865\pi$
$11$	−6578.61	−	3798.16i	−0.449328	−	0.259419i	−0.707546	+	0.706667i	$0.750198\pi$
$12$		0			0
$13$	7035.50	+	12185.8i	0.246332	+	0.426660i	0.962505	−	0.271262i	$-0.0874411\pi$
$13$	7035.50	+	12185.8i	0.246332	+	0.426660i	−0.716173	+	0.697923i	$0.754108\pi$
$14$		0			0
$15$	−19817.4	+	7229.99i	−0.391454	+	0.142815i
$16$		0			0
$17$			5681.88i			0.0680294i	0.999421	+	0.0340147i	$0.0108293\pi$
$17$			5681.88i			0.0680294i	−0.999421	+	0.0340147i	$0.989171\pi$
$18$		0			0
$19$	−252476.			−1.93734			−0.968669	−	0.248354i	$-0.920111\pi$
$19$	−252476.			−1.93734			−0.968669	+	0.248354i	$0.920111\pi$
$20$		0			0
$21$	47206.5	−	56345.5i	0.242731	−	0.289722i
$22$		0			0
$23$	431834.	−	249319.i	1.54314	−	0.890932i	0.544501	−	0.838760i	$-0.316719\pi$
$23$	431834.	−	249319.i	1.54314	−	0.890932i	0.998638	−	0.0521713i	$-0.0166142\pi$
$24$		0			0
$25$	−161400.	+	279553.i	−0.413184	+	0.715655i
$26$		0			0
$27$	−460846.	−	264670.i	−0.867163	−	0.498024i
$28$		0			0
$29$	564444.	+	325882.i	0.798048	+	0.460753i	0.842788	−	0.538245i	$-0.180913\pi$
$29$	564444.	+	325882.i	0.798048	+	0.460753i	−0.0447399	+	0.998999i	$0.514246\pi$
$30$		0			0
$31$	114193.	+	197788.i	0.123649	+	0.214167i	0.921204	−	0.389080i	$-0.127207\pi$
$31$	114193.	+	197788.i	0.123649	+	0.214167i	−0.797555	+	0.603246i	$0.793874\pi$
$32$		0			0
$33$	471649.	+	395150.i	0.397707	+	0.333201i
$34$		0			0
$35$			236341.i			0.157495i
$36$		0			0
$37$	1.99504e6			1.06450			0.532250	−	0.846587i	$-0.321347\pi$
$37$	1.99504e6			1.06450			0.532250	+	0.846587i	$0.321347\pi$
$38$		0			0
$39$	−390632.	−	1.07072e6i	−0.168853	−	0.462825i
$40$		0			0
$41$	−3.01123e6	+	1.73853e6i	−1.06564	+	0.615245i	−0.926986	−	0.375097i	$-0.877609\pi$
$41$	−3.01123e6	+	1.73853e6i	−1.06564	+	0.615245i	−0.138650	+	0.990342i	$0.544276\pi$
$42$		0			0
$43$	2.38564e6	−	4.13206e6i	0.697802	−	1.20863i	−0.271426	−	0.962459i	$-0.587495\pi$
$43$	2.38564e6	−	4.13206e6i	0.697802	−	1.20863i	0.969227	−	0.246168i	$-0.0791716\pi$
$44$		0			0
$45$	1.68229e6	−	299270.i	0.410251	−	0.0729815i
$46$		0			0
$47$	3.50541e6	+	2.02385e6i	0.718368	+	0.414750i	0.814152	−	0.580652i	$-0.197202\pi$
$47$	3.50541e6	+	2.02385e6i	0.718368	+	0.414750i	−0.0957835	+	0.995402i	$0.530536\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	79574.0	−	453301.i	0.0117623	−	0.0670048i
$52$		0			0
$53$			3.57890e6i			0.453572i	0.973945	+	0.226786i	$0.0728219\pi$
$53$			3.57890e6i			0.453572i	−0.973945	+	0.226786i	$0.927178\pi$
$54$		0			0
$55$	−1.97833e6			−0.216196
$56$		0			0
$57$	2.01426e7	+	3.53589e6i	1.90816	+	0.334965i
$58$		0			0
$59$	7.16289e6	−	4.13550e6i	0.591126	−	0.341287i	−0.174416	−	0.984672i	$-0.555804\pi$
$59$	7.16289e6	−	4.13550e6i	0.591126	−	0.341287i	0.765543	+	0.643385i	$0.222471\pi$
$60$		0			0
$61$	−7.07891e6	+	1.22610e7i	−0.511266	+	0.885538i	0.488649	+	0.872481i	$0.337490\pi$
$61$	−7.07891e6	+	1.22610e7i	−0.511266	+	0.885538i	−0.999915	+	0.0130579i	$0.995843\pi$
$62$		0			0
$63$	−4.55525e6	+	3.83412e6i	−0.289168	+	0.243391i
$64$		0			0
$65$	3.17359e6	+	1.83227e6i	0.177786	+	0.102645i
$66$		0			0
$67$	−9.17106e6	−	1.58847e7i	−0.455114	−	0.788281i	0.543581	−	0.839357i	$-0.317068\pi$
$67$	−9.17106e6	−	1.58847e7i	−0.455114	−	0.788281i	−0.998695	+	0.0510761i	$0.983735\pi$
$68$		0			0
$69$	−3.79434e7	+	1.38429e7i	−1.67394	+	0.610706i
$70$		0			0
$71$		−	1.45330e7i		−	0.571904i	−0.958244	−	0.285952i	$-0.907690\pi$
$71$		−	1.45330e7i		−	0.571904i	0.958244	−	0.285952i	$-0.0923097\pi$
$72$		0			0
$73$	1.94210e7			0.683881			0.341940	−	0.939722i	$-0.388916\pi$
$73$	1.94210e7			0.683881			0.341940	+	0.939722i	$0.388916\pi$
$74$		0			0
$75$	1.67916e7	−	2.00424e7i	0.530698	−	0.633438i
$76$		0			0
$77$	5.97004e6	−	3.44680e6i	0.169830	−	0.0980513i
$78$		0			0
$79$	−2.59991e7	+	4.50317e7i	−0.667497	+	1.15614i	0.311105	+	0.950376i	$0.399301\pi$
$79$	−2.59991e7	+	4.50317e7i	−0.667497	+	1.15614i	−0.978602	+	0.205763i	$0.934032\pi$
$80$		0			0
$81$	3.30597e7	+	2.75695e7i	0.767995	+	0.640456i
$82$		0			0
$83$	6.00862e7	+	3.46908e7i	1.26608	+	0.730973i	0.974245	−	0.225494i	$-0.0723996\pi$
$83$	6.00862e7	+	3.46908e7i	1.26608	+	0.730973i	0.291839	+	0.956468i	$0.405733\pi$
$84$		0			0
$85$	739874.	+	1.28150e6i	0.0141737	+	0.0245495i
$86$		0			0
$87$	−4.04675e7	−	3.39039e7i	−0.706365	−	0.591797i
$88$		0			0
$89$		−	7.53685e7i		−	1.20124i	−0.799534	−	0.600620i	$-0.794920\pi$
$89$		−	7.53685e7i		−	1.20124i	0.799534	−	0.600620i	$-0.205080\pi$
$90$		0			0
$91$	−1.27693e7			−0.186210
$92$		0			0
$93$	−6.34032e6	−	1.73788e7i	−0.0847577	−	0.232320i
$94$		0			0
$95$	−5.69437e7	+	3.28765e7i	−0.699120	+	0.403637i
$96$		0			0
$97$	−1.06245e7	+	1.84021e7i	−0.120011	+	0.207865i	−0.919772	−	0.392454i	$-0.871626\pi$
$97$	−1.06245e7	+	1.84021e7i	−0.120011	+	0.207865i	0.799761	+	0.600319i	$0.204960\pi$
$98$		0			0
$99$	−3.20942e7	−	3.81305e7i	−0.334107	−	0.396946i
$100$		0			0
$101$	−1.33402e8	−	7.70194e7i	−1.28196	−	0.740141i	−0.304755	−	0.952431i	$-0.598575\pi$
$101$	−1.33402e8	−	7.70194e7i	−1.28196	−	0.740141i	−0.977207	+	0.212289i	$0.931908\pi$
$102$		0			0
$103$	−6.42554e7	−	1.11294e8i	−0.570901	−	0.988829i	−0.996474	−	0.0839050i	$-0.973261\pi$
$103$	−6.42554e7	−	1.11294e8i	−0.570901	−	0.988829i	0.425573	−	0.904924i	$-0.360073\pi$
$104$		0			0
$105$	3.30992e6	−	1.88553e7i	0.0272308	−	0.155123i
$106$		0			0
$107$		−	1.77311e8i		−	1.35270i	−0.736580	−	0.676350i	$-0.763561\pi$
$107$		−	1.77311e8i		−	1.35270i	0.736580	−	0.676350i	$-0.236439\pi$
$108$		0			0
$109$	−2.19151e8			−1.55252			−0.776262	−	0.630411i	$-0.782886\pi$
$109$	−2.19151e8			−1.55252			−0.776262	+	0.630411i	$0.782886\pi$
$110$		0			0
$111$	−1.59165e8	−	2.79403e7i	−1.04847	−	0.184052i
$112$		0			0
$113$	−2.04453e8	+	1.18041e8i	−1.25395	+	0.723966i	−0.971891	−	0.235432i	$-0.924349\pi$
$113$	−2.04453e8	+	1.18041e8i	−1.25395	+	0.723966i	−0.282055	+	0.959398i	$0.591016\pi$
$114$		0			0
$115$	6.49309e7	−	1.12464e8i	0.371244	−	0.643014i
$116$		0			0
$117$	1.61694e7	+	9.08928e7i	0.0862878	+	0.485050i
$118$		0			0
$119$	−4.46546e6	−	2.57813e6i	−0.0222678	−	0.0128563i
$120$		0			0
$121$	−7.83274e7	−	1.35667e8i	−0.365403	−	0.632897i
$122$		0			0
$123$	2.64584e8	−	9.65286e7i	1.15596	−	0.421731i
$124$		0			0
$125$			1.85799e8i			0.761033i
$126$		0			0
$127$	−2.78190e8			−1.06937			−0.534683	−	0.845053i	$-0.679569\pi$
$127$	−2.78190e8			−1.06937			−0.534683	+	0.845053i	$0.679569\pi$
$128$		0			0
$129$	−2.48196e8	+	2.96245e8i	−0.896264	+	1.06978i
$130$		0			0
$131$	4.93321e7	−	2.84819e7i	0.167511	−	0.0967128i	−0.413900	−	0.910322i	$-0.635834\pi$
$131$	4.93321e7	−	2.84819e7i	0.167511	−	0.0967128i	0.581412	+	0.813609i	$0.302501\pi$
$132$		0			0
$133$	1.14560e8	−	1.98424e8i	0.366123	−	0.634143i
$134$		0			0
$135$	−1.38404e8	+	315569.i	−0.416691	+	0.000950079i
$136$		0			0
$137$	−4.39215e8	−	2.53581e8i	−1.24679	−	0.719837i	−0.276326	−	0.961064i	$-0.589117\pi$
$137$	−4.39215e8	−	2.53581e8i	−1.24679	−	0.719837i	−0.970469	+	0.241227i	$0.922450\pi$
$138$		0			0
$139$	−1.75384e8	−	3.03775e8i	−0.469820	−	0.813752i	0.529584	−	0.848257i	$-0.322348\pi$
$139$	−1.75384e8	−	3.03775e8i	−0.469820	−	0.813752i	−0.999405	+	0.0345049i	$0.989015\pi$
$140$		0			0
$141$	−2.51318e8	−	2.10556e8i	−0.635839	−	0.532709i
$142$		0			0
$143$		−	1.06888e8i		−	0.255614i
$144$		0			0
$145$	1.69741e8			0.383985
$146$		0			0
$147$	2.28628e7	+	6.26667e7i	0.0489620	+	0.134205i
$148$		0			0
$149$	6.70170e8	−	3.86923e8i	1.35969	−	0.785017i	0.370108	−	0.928989i	$-0.379321\pi$
$149$	6.70170e8	−	3.86923e8i	1.35969	−	0.785017i	0.989582	+	0.143971i	$0.0459873\pi$
$150$		0			0
$151$	−3.32609e8	+	5.76096e8i	−0.639774	+	1.10812i	0.345708	+	0.938342i	$0.387639\pi$
$151$	−3.32609e8	+	5.76096e8i	−0.639774	+	1.10812i	−0.985482	+	0.169779i	$0.945695\pi$
$152$		0			0
$153$	−1.26968e7	+	3.50500e7i	−0.0231702	+	0.0639620i
$154$		0			0
$155$	5.15104e7	+	2.97395e7i	0.0892417	+	0.0515237i
$156$		0			0
$157$	2.79789e8	+	4.84609e8i	0.460503	+	0.797615i	0.998986	−	0.0450218i	$-0.0143357\pi$
$157$	2.79789e8	+	4.84609e8i	0.460503	+	0.797615i	−0.538483	+	0.842636i	$0.681002\pi$
$158$		0			0
$159$	5.01221e7	−	2.85525e8i	0.0784225	−	0.446741i
$160$		0			0
$161$			4.52511e8i			0.673481i
$162$		0			0
$163$	8.56084e8			1.21274			0.606368	−	0.795184i	$-0.292626\pi$
$163$	8.56084e8			1.21274			0.606368	+	0.795184i	$0.292626\pi$
$164$		0			0
$165$	1.57831e8	+	2.77063e7i	0.212940	+	0.0373802i
$166$		0			0
$167$	−9.05711e8	+	5.22912e8i	−1.16446	+	0.672300i	−0.952368	−	0.304950i	$-0.901360\pi$
$167$	−9.05711e8	+	5.22912e8i	−1.16446	+	0.672300i	−0.212089	+	0.977250i	$0.568027\pi$
$168$		0			0
$169$	3.08869e8	−	5.34976e8i	0.378641	−	0.655825i
$170$		0			0
$171$	−1.55746e9	−	5.64188e8i	−1.82151	−	0.659841i
$172$		0			0
$173$	−9.24064e8	−	5.33508e8i	−1.03161	−	0.595603i	−0.114168	−	0.993461i	$-0.536420\pi$
$173$	−9.24064e8	−	5.33508e8i	−1.03161	−	0.595603i	−0.917447	+	0.397859i	$0.869753\pi$
$174$		0			0
$175$	−1.46469e8	−	2.53692e8i	−0.156169	−	0.270492i
$176$		0			0
$177$	−6.29373e8	+	2.29615e8i	−0.641232	+	0.233941i
$178$		0			0
$179$		−	3.98735e8i		−	0.388394i	−0.980963	−	0.194197i	$-0.937790\pi$
$179$		−	3.98735e8i		−	0.388394i	0.980963	−	0.194197i	$-0.0622101\pi$
$180$		0			0
$181$	−1.26613e9			−1.17968			−0.589838	−	0.807522i	$-0.700808\pi$
$181$	−1.26613e9			−1.17968			−0.589838	+	0.807522i	$0.700808\pi$
$182$		0			0
$183$	7.36470e8	−	8.79047e8i	0.656675	−	0.783804i
$184$		0			0
$185$	4.49965e8	−	2.59787e8i	0.384142	−	0.221784i
$186$		0			0
$187$	2.15807e7	−	3.73789e7i	0.0176481	−	0.0305675i
$188$		0			0
$189$	4.17115e8	−	2.42091e8i	0.326895	−	0.189728i
$190$		0			0
$191$	−1.51063e9	−	8.72161e8i	−1.13507	−	0.655334i	−0.189867	−	0.981810i	$-0.560806\pi$
$191$	−1.51063e9	−	8.72161e8i	−1.13507	−	0.655334i	−0.945206	+	0.326475i	$0.894139\pi$
$192$		0			0
$193$	−8.23237e8	−	1.42589e9i	−0.593329	−	1.02768i	−0.993780	−	0.111358i	$-0.964480\pi$
$193$	−8.23237e8	−	1.42589e9i	−0.593329	−	1.02768i	0.400451	−	0.916318i	$-0.368853\pi$
$194$		0			0
$195$	−2.27529e8	−	1.90625e8i	−0.157361	−	0.131838i
$196$		0			0
$197$			1.06214e9i			0.705211i	0.935772	+	0.352605i	$0.114704\pi$
$197$			1.06214e9i			0.705211i	−0.935772	+	0.352605i	$0.885296\pi$
$198$		0			0
$199$	5.69615e8			0.363220			0.181610	−	0.983371i	$-0.441869\pi$
$199$	5.69615e8			0.363220			0.181610	+	0.983371i	$0.441869\pi$
$200$		0			0
$201$	5.09204e8	+	1.39573e9i	0.311966	+	0.855098i
$202$		0			0
$203$	−5.12229e8	+	2.95736e8i	−0.301634	+	0.174148i
$204$		0			0
$205$	−4.52771e8	+	7.84223e8i	−0.256368	+	0.444042i
$206$		0			0
$207$	3.22100e9	−	5.72998e8i	1.75432	−	0.312084i
$208$		0			0
$209$	1.66094e9	+	9.58944e8i	0.870500	+	0.502583i
$210$		0			0
$211$	−1.27874e9	−	2.21484e9i	−0.645136	−	1.11741i	−0.984270	−	0.176670i	$-0.943468\pi$
$211$	−1.27874e9	−	2.21484e9i	−0.645136	−	1.11741i	0.339135	−	0.940738i	$-0.389866\pi$
$212$		0			0
$213$	−2.03533e8	+	1.15945e9i	−0.0988819	+	0.563290i
$214$		0			0
$215$		−	1.24260e9i		−	0.581537i
$216$		0			0
$217$	−2.07258e8			−0.0934701
$218$		0			0
$219$	−1.54941e9	−	2.71989e8i	−0.673581	−	0.118243i
$220$		0			0
$221$	−6.92385e7	+	3.99749e7i	−0.0290254	+	0.0167578i
$222$		0			0
$223$	8.64498e8	−	1.49735e9i	0.349578	−	0.605487i	−0.636596	−	0.771197i	$-0.719658\pi$
$223$	8.64498e8	−	1.49735e9i	0.349578	−	0.605487i	0.986175	+	0.165710i	$0.0529915\pi$
$224$		0			0
$225$	−1.62033e9	+	1.36382e9i	−0.632226	+	0.532140i
$226$		0			0
$227$	−3.26056e9	−	1.88249e9i	−1.22797	−	0.708971i	−0.261368	−	0.965239i	$-0.584174\pi$
$227$	−3.26056e9	−	1.88249e9i	−1.22797	−	0.708971i	−0.966606	+	0.256269i	$0.917507\pi$
$228$		0			0
$229$	1.04444e9	+	1.80903e9i	0.379789	+	0.657815i	0.991031	−	0.133629i	$-0.0426631\pi$
$229$	1.04444e9	+	1.80903e9i	0.379789	+	0.657815i	−0.611242	+	0.791444i	$0.709330\pi$
$230$		0			0
$231$	−5.24562e8	+	1.91377e8i	−0.184225	+	0.0672111i
$232$		0			0
$233$			1.52253e9i			0.516585i	0.966067	+	0.258292i	$0.0831598\pi$
$233$			1.52253e9i			0.516585i	−0.966067	+	0.258292i	$0.916840\pi$
$234$		0			0
$235$	1.05415e9			0.345646
$236$		0			0
$237$	2.70487e9	−	3.22852e9i	0.857340	−	1.02332i
$238$		0			0
$239$	−3.22085e9	+	1.85956e9i	−0.987140	+	0.569926i	−0.904418	−	0.426647i	$-0.859695\pi$
$239$	−3.22085e9	+	1.85956e9i	−0.987140	+	0.569926i	−0.0827221	+	0.996573i	$0.526361\pi$
$240$		0			0
$241$	−1.55246e9	+	2.68893e9i	−0.460205	+	0.797098i	−0.998971	−	0.0453574i	$-0.985557\pi$
$241$	−1.55246e9	+	2.68893e9i	−0.460205	+	0.797098i	0.538766	+	0.842455i	$0.318891\pi$
$242$		0			0
$243$	−2.25140e9	−	2.66250e9i	−0.645694	−	0.763596i
$244$		0			0
$245$	−1.85743e8	−	1.07239e8i	−0.0515523	−	0.0297637i
$246$		0			0
$247$	−1.77630e9	−	3.07663e9i	−0.477230	−	0.826586i
$248$		0			0
$249$	−4.30784e9	−	3.60913e9i	−1.12063	−	0.938870i
$250$		0			0
$251$			2.27552e8i			0.0573306i	0.999589	+	0.0286653i	$0.00912569\pi$
$251$			2.27552e8i			0.0573306i	−0.999589	+	0.0286653i	$0.990874\pi$
$252$		0			0
$253$	−3.78782e9			−0.924500
$254$		0			0
$255$	−4.10800e7	−	1.12600e8i	−0.00971559	−	0.0266304i
$256$		0			0
$257$	−1.40335e9	+	8.10222e8i	−0.321686	+	0.185726i	−0.652144	−	0.758095i	$-0.726130\pi$
$257$	−1.40335e9	+	8.10222e8i	−0.321686	+	0.185726i	0.330458	+	0.943821i	$0.392797\pi$
$258$		0			0
$259$	−9.05244e8	+	1.56793e9i	−0.201172	+	0.348439i
$260$		0			0
$261$	2.75368e9	+	3.27160e9i	0.593406	+	0.705014i
$262$		0			0
$263$	4.34161e9	+	2.50663e9i	0.907460	+	0.523922i	0.879613	−	0.475690i	$-0.157802\pi$
$263$	4.34161e9	+	2.50663e9i	0.907460	+	0.523922i	0.0278469	+	0.999612i	$0.491135\pi$
$264$		0			0
$265$	4.66032e8	+	8.07191e8i	0.0945001	+	0.163679i
$266$		0			0
$267$	−1.05553e9	+	6.01291e9i	−0.207694	+	1.18315i
$268$		0			0
$269$			1.54039e9i			0.294185i	0.989123	+	0.147092i	$0.0469915\pi$
$269$			1.54039e9i			0.294185i	−0.989123	+	0.147092i	$0.953009\pi$
$270$		0			0
$271$	3.39430e9			0.629322			0.314661	−	0.949204i	$-0.398109\pi$
$271$	3.39430e9			0.629322			0.314661	+	0.949204i	$0.398109\pi$
$272$		0			0
$273$	1.01874e9	+	1.78833e8i	0.183405	+	0.0321956i
$274$		0			0
$275$	2.12357e9	−	1.22605e9i	0.371310	−	0.214376i
$276$		0			0
$277$	4.44778e9	−	7.70377e9i	0.755481	−	1.30853i	−0.189653	−	0.981851i	$-0.560736\pi$
$277$	4.44778e9	−	7.70377e9i	0.755481	−	1.30853i	0.945135	−	0.326681i	$-0.105930\pi$
$278$		0			0
$279$	2.62444e8	+	1.47528e9i	0.0433131	+	0.243476i
$280$		0			0
$281$	−3.74036e9	−	2.15950e9i	−0.599912	−	0.346359i	0.169095	−	0.985600i	$-0.445916\pi$
$281$	−3.74036e9	−	2.15950e9i	−0.599912	−	0.346359i	−0.769007	+	0.639240i	$0.779249\pi$
$282$		0			0
$283$	−6.29939e8	−	1.09109e9i	−0.0982094	−	0.170104i	0.812734	−	0.582635i	$-0.197978\pi$
$283$	−6.29939e8	−	1.09109e9i	−0.0982094	−	0.170104i	−0.910944	+	0.412531i	$0.864645\pi$
$284$		0			0
$285$	5.00341e9	−	1.82540e9i	0.758379	−	0.276680i
$286$		0			0
$287$		−	3.15542e9i		−	0.465081i
$288$		0			0
$289$	6.94347e9			0.995372
$290$		0			0
$291$	1.10534e9	−	1.31933e9i	0.154143	−	0.183984i
$292$		0			0
$293$	−1.28361e8	+	7.41095e7i	−0.0174166	+	0.0100555i	−0.508683	−	0.860954i	$-0.669867\pi$
$293$	−1.28361e8	+	7.41095e7i	−0.0174166	+	0.0100555i	0.491266	+	0.871009i	$0.336534\pi$
$294$		0			0
$295$	1.07702e9	−	1.86545e9i	0.142212	−	0.246318i
$296$		0			0
$297$	2.02646e9	+	3.49153e9i	0.260443	+	0.448735i
$298$		0			0
$299$	6.07633e9	+	3.50817e9i	0.760251	+	0.438931i
$300$		0			0
$301$	2.16496e9	+	3.74981e9i	0.263744	+	0.456818i
$302$		0			0
$303$	9.56414e9	+	8.01289e9i	1.13469	+	0.950645i
$304$		0			0
$305$			3.68716e9i			0.426081i
$306$		0			0
$307$	−3.66723e9			−0.412843			−0.206421	−	0.978463i	$-0.566182\pi$
$307$	−3.66723e9			−0.412843			−0.206421	+	0.978463i	$0.566182\pi$
$308$		0			0
$309$	3.56765e9	+	9.77890e9i	0.391334	+	1.07265i
$310$		0			0
$311$	−4.09681e9	+	2.36529e9i	−0.437930	+	0.252839i	−0.702719	−	0.711467i	$-0.748031\pi$
$311$	−4.09681e9	+	2.36529e9i	−0.437930	+	0.252839i	0.264789	+	0.964306i	$0.414698\pi$
$312$		0			0
$313$	5.63124e9	−	9.75359e9i	0.586714	−	1.01622i	−0.407945	−	0.913006i	$-0.633755\pi$
$313$	5.63124e9	−	9.75359e9i	0.586714	−	1.01622i	0.994659	−	0.103212i	$-0.0329121\pi$
$314$		0			0
$315$	−5.28132e8	+	1.45792e9i	−0.0536414	+	0.148078i
$316$		0			0
$317$	−2.05799e9	−	1.18818e9i	−0.203801	−	0.117665i	0.394626	−	0.918842i	$-0.370874\pi$
$317$	−2.05799e9	−	1.18818e9i	−0.203801	−	0.117665i	−0.598427	+	0.801177i	$0.704208\pi$
$318$		0			0
$319$	−2.47551e9	−	4.28770e9i	−0.239057	−	0.414059i
$320$		0			0
$321$	−2.48322e9	+	1.41459e10i	−0.233881	+	1.33233i
$322$		0			0
$323$		−	1.43454e9i		−	0.131796i
$324$		0			0
$325$	−4.54212e9			−0.407122
$326$		0			0
$327$	1.74839e10	+	3.06919e9i	1.52914	+	0.268431i
$328$		0			0
$329$	−3.18113e9	+	1.83663e9i	−0.271518	+	0.156761i
$330$		0			0
$331$	6.27509e9	−	1.08688e10i	0.522767	−	0.905458i	−0.476882	−	0.878967i	$-0.658233\pi$
$331$	6.27509e9	−	1.08688e10i	0.522767	−	0.905458i	0.999649	−	0.0264913i	$-0.00843342\pi$
$332$		0			0
$333$	1.23069e10	+	4.45816e9i	1.00085	+	0.362559i
$334$		0			0
$335$	−4.13691e9	−	2.38844e9i	−0.328471	−	0.189643i
$336$		0			0
$337$	1.00334e10	+	1.73784e10i	0.777909	+	1.34738i	0.933145	+	0.359501i	$0.117053\pi$
$337$	1.00334e10	+	1.73784e10i	0.777909	+	1.34738i	−0.155235	+	0.987878i	$0.549614\pi$
$338$		0			0
$339$	1.79644e10	−	6.55397e9i	1.36023	−	0.496256i
$340$		0			0
$341$		−	1.73489e9i		−	0.128308i
$342$		0			0
$343$	7.47359e8			0.0539949
$344$		0			0
$345$	−6.75523e9	+	8.06300e9i	−0.476830	+	0.569142i
$346$		0			0
$347$	8.24004e9	−	4.75739e9i	0.568344	−	0.328134i	−0.188143	−	0.982142i	$-0.560247\pi$
$347$	8.24004e9	−	4.75739e9i	0.568344	−	0.328134i	0.756488	+	0.654008i	$0.226914\pi$
$348$		0			0
$349$	−4.30765e9	+	7.46106e9i	−0.290361	+	0.502920i	−0.973895	−	0.226999i	$-0.927109\pi$
$349$	−4.30765e9	+	7.46106e9i	−0.290361	+	0.502920i	0.683534	+	0.729919i	$0.260442\pi$
$350$		0			0
$351$	−1.70500e7	−	7.47789e9i	−0.00112330	−	0.492664i
$352$		0			0
$353$	2.01239e9	+	1.16185e9i	0.129602	+	0.0748260i	0.563400	−	0.826185i	$-0.309493\pi$
$353$	2.01239e9	+	1.16185e9i	0.129602	+	0.0748260i	−0.433797	+	0.901011i	$0.642827\pi$
$354$		0			0
$355$	−1.89244e9	−	3.27780e9i	−0.119154	−	0.206381i
$356$		0			0
$357$	3.20148e8	+	2.68222e8i	0.0197096	+	0.0165128i
$358$		0			0
$359$			2.83930e10i			1.70936i	0.519154	+	0.854681i	$0.326247\pi$
$359$			2.83930e10i			1.70936i	−0.519154	+	0.854681i	$0.673753\pi$
$360$		0			0
$361$	4.67605e10			2.75328
$362$		0			0
$363$	4.34897e9	+	1.19205e10i	0.250472	+	0.686543i
$364$		0			0
$365$	4.38024e9	−	2.52893e9i	0.246789	−	0.142484i
$366$		0			0
$367$	−7.69828e9	+	1.33338e10i	−0.424355	+	0.735005i	−0.996360	−	0.0852454i	$-0.972833\pi$
$367$	−7.69828e9	+	1.33338e10i	−0.424355	+	0.735005i	0.572005	+	0.820250i	$0.306166\pi$
$368$		0			0
$369$	−2.24604e10	+	3.99559e9i	−1.21147	+	0.215514i
$370$		0			0
$371$	−2.81270e9	−	1.62391e9i	−0.148466	−	0.0857171i
$372$		0			0
$373$	−1.87704e10	−	3.25112e10i	−0.969699	−	1.67957i	−0.696421	−	0.717633i	$-0.745225\pi$
$373$	−1.87704e10	−	3.25112e10i	−0.969699	−	1.67957i	−0.273278	−	0.961935i	$-0.588108\pi$
$374$		0			0
$375$	2.60209e9	−	1.48231e10i	0.131582	−	0.749572i
$376$		0			0
$377$			9.17098e9i			0.453994i
$378$		0			0
$379$	7.05281e9			0.341826			0.170913	−	0.985286i	$-0.445328\pi$
$379$	7.05281e9			0.341826			0.170913	+	0.985286i	$0.445328\pi$
$380$		0			0
$381$	2.21940e10	+	3.89601e9i	1.05326	+	0.184893i
$382$		0			0
$383$	−2.21753e10	+	1.28029e10i	−1.03056	+	0.594997i	−0.917146	−	0.398551i	$-0.869513\pi$
$383$	−2.21753e10	+	1.28029e10i	−1.03056	+	0.594997i	−0.113418	+	0.993547i	$0.536180\pi$
$384$		0			0
$385$	8.97660e8	−	1.55479e9i	0.0408572	−	0.0707668i
$386$		0			0
$387$	2.39500e10	−	2.01585e10i	1.06773	−	0.898700i
$388$		0			0
$389$	−2.03324e8	−	1.17389e8i	−0.00887955	−	0.00512661i	0.495554	−	0.868577i	$-0.334965\pi$
$389$	−2.03324e8	−	1.17389e8i	−0.00887955	−	0.00512661i	−0.504433	+	0.863451i	$0.668299\pi$
$390$		0			0
$391$	1.41660e9	+	2.45363e9i	0.0606095	+	0.104979i
$392$		0			0
$393$	−4.33461e9	+	1.58140e9i	−0.181710	+	0.0662936i
$394$		0			0
$395$			1.35420e10i			0.556282i
$396$		0			0
$397$	−1.62246e10			−0.653150			−0.326575	−	0.945171i	$-0.605895\pi$
$397$	−1.62246e10			−0.653150			−0.326575	+	0.945171i	$0.605895\pi$
$398$		0			0
$399$	−1.19185e10	+	1.42259e10i	−0.470252	+	0.561290i
$400$		0			0
$401$	3.08823e10	−	1.78299e10i	1.19435	−	0.689559i	0.235061	−	0.971981i	$-0.424471\pi$
$401$	3.08823e10	−	1.78299e10i	1.19435	−	0.689559i	0.959290	+	0.282422i	$0.0911378\pi$
$402$		0			0
$403$	−1.60681e9	+	2.78307e9i	−0.0609177	+	0.105513i
$404$		0			0
$405$	1.10463e10	+	1.91316e9i	0.410580	+	0.0711100i
$406$		0			0
$407$	−1.31246e10	−	7.57749e9i	−0.478309	−	0.276152i
$408$		0			0
$409$	−2.28712e10	−	3.96141e10i	−0.817328	−	1.41565i	−0.907644	−	0.419740i	$-0.862121\pi$
$409$	−2.28712e10	−	3.96141e10i	−0.817328	−	1.41565i	0.0903169	−	0.995913i	$-0.471212\pi$
$410$		0			0
$411$	3.14892e10	+	2.63819e10i	1.10356	+	0.924566i
$412$		0			0
$413$			7.50587e9i			0.257989i
$414$		0			0
$415$	1.80692e10			0.609182
$416$		0			0
$417$	9.73786e9	+	2.66914e10i	0.322047	+	0.882729i
$418$		0			0
$419$	−3.94622e10	+	2.27835e10i	−1.28034	+	0.739204i	−0.976910	−	0.213651i	$-0.931465\pi$
$419$	−3.94622e10	+	2.27835e10i	−1.28034	+	0.739204i	−0.303428	+	0.952854i	$0.598131\pi$
$420$		0			0
$421$	1.55855e10	−	2.69950e10i	0.496128	−	0.859319i	−0.503862	−	0.863784i	$-0.668088\pi$
$421$	1.55855e10	−	2.69950e10i	0.496128	−	0.859319i	0.999990	+	0.00446539i	$0.00142138\pi$
$422$		0			0
$423$	1.71014e10	+	2.03178e10i	0.534158	+	0.634623i
$424$		0			0
$425$	−1.58839e9	−	9.17055e8i	−0.0486856	−	0.0281086i
$426$		0			0
$427$	−6.42406e9	−	1.11268e10i	−0.193240	−	0.334702i
$428$		0			0
$429$	−1.49695e9	+	8.52752e9i	−0.0441955	+	0.251764i
$430$		0			0
$431$		−	6.19634e10i		−	1.79567i	−0.440333	−	0.897834i	$-0.645140\pi$
$431$		−	6.19634e10i		−	1.79567i	0.440333	−	0.897834i	$-0.354860\pi$
$432$		0			0
$433$	−2.91612e10			−0.829571			−0.414786	−	0.909919i	$-0.636143\pi$
$433$	−2.91612e10			−0.829571			−0.414786	+	0.909919i	$0.636143\pi$
$434$		0			0
$435$	−1.35419e10	−	2.37720e9i	−0.378202	−	0.0663908i
$436$		0			0
$437$	−1.09028e11	+	6.29471e10i	−2.98958	+	1.72604i
$438$		0			0
$439$	2.67670e10	−	4.63618e10i	0.720678	−	1.24825i	−0.240050	−	0.970760i	$-0.577164\pi$
$439$	2.67670e10	−	4.63618e10i	0.720678	−	1.24825i	0.960728	−	0.277490i	$-0.0895027\pi$
$440$		0			0
$441$	−9.46355e8	−	5.31975e9i	−0.0250207	−	0.140649i
$442$		0			0
$443$	−4.83089e10	−	2.78911e10i	−1.25433	−	0.724188i	−0.282364	−	0.959307i	$-0.591119\pi$
$443$	−4.83089e10	−	2.78911e10i	−1.25433	−	0.724188i	−0.971966	+	0.235120i	$0.924452\pi$
$444$		0			0
$445$	−9.81421e9	−	1.69987e10i	−0.250274	−	0.433487i
$446$		0			0
$447$	−5.88850e10	+	2.14831e10i	−1.47494	+	0.538105i
$448$		0			0
$449$			4.88666e10i			1.20234i	0.799121	+	0.601170i	$0.205298\pi$
$449$			4.88666e10i			1.20234i	−0.799121	+	0.601170i	$0.794702\pi$
$450$		0			0
$451$	2.64129e10			0.638426
$452$		0			0
$453$	3.46038e10	−	4.13029e10i	0.821732	−	0.980815i
$454$		0			0
$455$	−2.88001e9	+	1.66278e9i	−0.0671968	+	0.0387961i
$456$		0			0
$457$	−9.62490e9	+	1.66708e10i	−0.220664	+	0.382201i	−0.955010	−	0.296574i	$-0.904156\pi$
$457$	−9.62490e9	+	1.66708e10i	−0.220664	+	0.382201i	0.734346	+	0.678776i	$0.237489\pi$
$458$		0			0
$459$	1.50383e9	−	2.61847e9i	0.0338803	−	0.0589926i
$460$		0			0
$461$	−3.25638e9	−	1.88007e9i	−0.0720994	−	0.0416266i	0.463517	−	0.886088i	$-0.346587\pi$
$461$	−3.25638e9	−	1.88007e9i	−0.0720994	−	0.0416266i	−0.535616	+	0.844461i	$0.679921\pi$
$462$		0			0
$463$	−2.02800e10	−	3.51260e10i	−0.441310	−	0.764371i	0.556477	−	0.830863i	$-0.312153\pi$
$463$	−2.02800e10	−	3.51260e10i	−0.441310	−	0.764371i	−0.997787	+	0.0664917i	$0.978819\pi$
$464$		0			0
$465$	−3.69300e9	−	3.09402e9i	−0.0789892	−	0.0661776i
$466$		0			0
$467$		−	3.61731e10i		−	0.760534i	−0.924877	−	0.380267i	$-0.875832\pi$
$467$		−	3.61731e10i		−	0.760534i	0.924877	−	0.380267i	$-0.124168\pi$
$468$		0			0
$469$	1.66453e10			0.344034
$470$		0			0
$471$	−1.55347e10	−	4.25806e10i	−0.315660	−	0.865223i
$472$		0			0
$473$	−3.13884e10	+	1.81221e10i	−0.627083	+	0.362047i
$474$		0			0
$475$	4.07496e10	−	7.05804e10i	0.800477	−	1.38647i
$476$		0			0
$477$	−7.99749e9	+	2.20773e10i	−0.154483	+	0.426454i
$478$		0			0
$479$	2.39968e10	+	1.38546e10i	0.455839	+	0.263179i	0.710293	−	0.703906i	$-0.248562\pi$
$479$	2.39968e10	+	1.38546e10i	0.455839	+	0.263179i	−0.254454	+	0.967085i	$0.581896\pi$
$480$		0			0
$481$	1.40361e10	+	2.43113e10i	0.262221	+	0.454180i
$482$		0			0
$483$	6.33735e9	−	3.61013e10i	0.116445	−	0.663338i
$484$		0			0
$485$			5.53391e9i			0.100015i
$486$		0			0
$487$	−3.05051e10			−0.542322			−0.271161	−	0.962534i	$-0.587408\pi$
$487$	−3.05051e10			−0.542322			−0.271161	+	0.962534i	$0.587408\pi$
$488$		0			0
$489$	−6.82985e10	−	1.19894e10i	−1.19447	−	0.209682i
$490$		0			0
$491$	−3.07196e10	+	1.77360e10i	−0.528555	+	0.305161i	−0.740428	−	0.672136i	$-0.765377\pi$
$491$	−3.07196e10	+	1.77360e10i	−0.528555	+	0.305161i	0.211873	+	0.977297i	$0.432044\pi$
$492$		0			0
$493$	−1.85162e9	+	3.20711e9i	−0.0313448	+	0.0542907i
$494$		0			0
$495$	−1.22038e10	−	4.42082e9i	−0.203270	−	0.0736345i
$496$		0			0
$497$	1.14217e10	+	6.59431e9i	0.187199	+	0.108080i
$498$		0			0
$499$	1.16384e10	+	2.01582e10i	0.187711	+	0.325125i	0.944487	−	0.328550i	$-0.106560\pi$
$499$	1.16384e10	+	2.01582e10i	0.187711	+	0.325125i	−0.756776	+	0.653675i	$0.773226\pi$
$500$		0			0
$501$	7.95810e10	−	2.90336e10i	1.26316	−	0.460840i
$502$		0			0
$503$			5.03920e10i			0.787209i	0.919280	+	0.393605i	$0.128772\pi$
$503$			5.03920e10i			0.787209i	−0.919280	+	0.393605i	$0.871228\pi$
$504$		0			0
$505$	−4.01167e10			−0.616822
$506$		0			0
$507$	−3.21339e10	+	3.83548e10i	−0.486330	+	0.580481i
$508$		0			0
$509$	−3.05489e10	+	1.76374e10i	−0.455118	+	0.262763i	−0.709989	−	0.704212i	$-0.751300\pi$
$509$	−3.05489e10	+	1.76374e10i	−0.455118	+	0.262763i	0.254871	+	0.966975i	$0.417967\pi$
$510$		0			0
$511$	−8.81221e9	+	1.52632e10i	−0.129241	+	0.223853i
$512$		0			0
$513$	1.16353e11	+	6.68229e10i	1.67999	+	0.964842i
$514$		0			0
$515$	−2.89845e10	−	1.67342e10i	−0.412037	−	0.237890i
$516$		0			0
$517$	−1.53738e10	−	2.66282e10i	−0.215189	−	0.372717i
$518$		0			0
$519$	6.62502e10	+	5.55047e10i	0.913098	+	0.764999i
$520$		0			0
$521$		−	1.29960e11i		−	1.76384i	−0.471399	−	0.881920i	$-0.656251\pi$
$521$		−	1.29960e11i		−	1.76384i	0.471399	−	0.881920i	$-0.343749\pi$
$522$		0			0
$523$	−8.63319e10			−1.15389			−0.576945	−	0.816783i	$-0.695755\pi$
$523$	−8.63319e10			−1.15389			−0.576945	+	0.816783i	$0.695755\pi$
$524$		0			0
$525$	8.13240e9	+	2.22909e10i	0.107049	+	0.293420i
$526$		0			0
$527$	−1.12381e9	+	6.48830e8i	−0.0145696	+	0.00841179i
$528$		0			0
$529$	8.51646e10	−	1.47509e11i	1.08752	−	1.88364i
$530$		0			0
$531$	5.34272e10	−	9.50441e9i	0.672023	−	0.119549i
$532$		0			0
$533$	−4.23710e10	−	2.44629e10i	−0.525001	−	0.303110i
$534$		0			0
$535$	−2.30888e10	−	3.99911e10i	−0.281830	−	0.488144i
$536$		0			0
$537$	−5.58423e9	+	3.18111e10i	−0.0671532	+	0.382544i
$538$		0			0
$539$			6.25590e9i			0.0741198i
$540$		0			0
$541$	−5.40024e10			−0.630411			−0.315205	−	0.949023i	$-0.602073\pi$
$541$	−5.40024e10			−0.630411			−0.315205	+	0.949023i	$0.602073\pi$
$542$		0			0
$543$	1.01012e11	+	1.77319e10i	1.16191	+	0.203966i
$544$		0			0
$545$	−4.94277e10	+	2.85371e10i	−0.560253	+	0.323462i
$546$		0			0
$547$	4.54150e10	−	7.86611e10i	0.507282	−	0.878639i	−0.492682	−	0.870209i	$-0.663983\pi$
$547$	4.54150e10	−	7.86611e10i	0.507282	−	0.878639i	0.999964	−	0.00842960i	$-0.00268326\pi$
$548$		0			0
$549$	−7.10666e10	+	5.98163e10i	−0.782305	+	0.658461i
$550$		0			0
$551$	−1.42509e11	−	8.22774e10i	−1.54609	−	0.892636i
$552$		0			0
$553$	−2.35940e10	−	4.08659e10i	−0.252290	−	0.436979i
$554$		0			0
$555$	−3.95365e10	+	1.44242e10i	−0.416703	+	0.152026i
$556$		0			0
$557$			9.31032e10i			0.967262i	0.875272	+	0.483631i	$0.160682\pi$
$557$			9.31032e10i			0.967262i	−0.875272	+	0.483631i	$0.839318\pi$
$558$		0			0
$559$	6.71368e10			0.687565
$560$		0			0
$561$	−2.24520e9	+	2.67985e9i	−0.0226675	+	0.0270558i
$562$		0			0
$563$	9.85997e10	−	5.69265e10i	0.981391	−	0.566606i	0.0787009	−	0.996898i	$-0.474923\pi$
$563$	9.85997e10	−	5.69265e10i	0.981391	−	0.566606i	0.902690	+	0.430292i	$0.141589\pi$
$564$		0			0
$565$	−3.07417e10	+	5.32461e10i	−0.301671	+	0.522509i
$566$		0			0
$567$	−3.66679e10	+	1.34724e10i	−0.354776	+	0.130351i
$568$		0			0
$569$	−1.20212e11	−	6.94043e10i	−1.14683	−	0.662121i	−0.198715	−	0.980057i	$-0.563677\pi$
$569$	−1.20212e11	−	6.94043e10i	−1.14683	−	0.662121i	−0.948112	+	0.317936i	$0.897010\pi$
$570$		0			0
$571$	−4.38994e10	−	7.60360e10i	−0.412966	−	0.715278i	0.582247	−	0.813012i	$-0.302174\pi$
$571$	−4.38994e10	−	7.60360e10i	−0.412966	−	0.715278i	−0.995213	+	0.0977343i	$0.968840\pi$
$572$		0			0
$573$	1.08303e11	+	9.07372e10i	1.00467	+	0.841718i
$574$		0			0
$575$			1.60960e11i			1.47247i
$576$		0			0
$577$	1.01033e11			0.911509			0.455754	−	0.890106i	$-0.349370\pi$
$577$	1.01033e11			0.911509			0.455754	+	0.890106i	$0.349370\pi$
$578$		0			0
$579$	4.57086e10	+	1.25287e11i	0.406708	+	1.11479i
$580$		0			0
$581$	−5.45278e10	+	3.14816e10i	−0.478534	+	0.276282i
$582$		0			0
$583$	1.35933e10	−	2.35442e10i	0.117666	−	0.203803i
$584$		0			0
$585$	1.54826e10	+	1.83946e10i	0.132197	+	0.157060i
$586$		0			0
$587$	−1.85989e11	−	1.07381e11i	−1.56652	−	0.904430i	−0.996570	−	0.0827487i	$-0.973630\pi$
$587$	−1.85989e11	−	1.07381e11i	−1.56652	−	0.904430i	−0.569948	−	0.821681i	$-0.693037\pi$
$588$		0			0
$589$	−2.88309e10	−	4.99366e10i	−0.239551	−	0.414914i
$590$		0			0
$591$	1.48752e10	−	8.47380e10i	0.121931	−	0.694590i
$592$		0			0
$593$			1.26488e11i			1.02289i	0.859316	+	0.511445i	$0.170890\pi$
$593$			1.26488e11i			1.02289i	−0.859316	+	0.511445i	$0.829110\pi$
$594$		0			0
$595$	−1.34286e9			−0.0107143
$596$		0			0
$597$	−4.54440e10	−	7.97739e9i	−0.357749	−	0.0628006i
$598$		0			0
$599$	1.43714e10	−	8.29731e9i	0.111632	−	0.0644511i	−0.443144	−	0.896450i	$-0.646137\pi$
$599$	1.43714e10	−	8.29731e9i	0.111632	−	0.0644511i	0.554777	+	0.831999i	$0.312804\pi$
$600$		0			0
$601$	1.28307e10	−	2.22235e10i	0.0983453	−	0.170339i	−0.812655	−	0.582746i	$-0.801978\pi$
$601$	1.28307e10	−	2.22235e10i	0.0983453	−	0.170339i	0.911000	+	0.412407i	$0.135312\pi$
$602$		0			0
$603$	−2.10774e10	−	1.18482e11i	−0.159422	−	0.896159i
$604$		0			0
$605$	−3.53321e10	−	2.03990e10i	−0.263723	−	0.152261i
$606$		0			0
$607$	−4.68301e10	−	8.11120e10i	−0.344961	−	0.597490i	0.640386	−	0.768053i	$-0.278774\pi$
$607$	−4.68301e10	−	8.11120e10i	−0.344961	−	0.597490i	−0.985347	+	0.170564i	$0.945441\pi$
$608$		0			0
$609$	4.50074e10	−	1.64201e10i	0.327201	−	0.119373i
$610$		0			0
$611$			5.69552e10i			0.408666i
$612$		0			0
$613$	−1.90554e10			−0.134951			−0.0674756	−	0.997721i	$-0.521494\pi$
$613$	−1.90554e10			−0.134951			−0.0674756	+	0.997721i	$0.521494\pi$
$614$		0			0
$615$	4.71051e10	−	5.62244e10i	0.329281	−	0.393029i
$616$		0			0
$617$	−1.15005e10	+	6.63979e9i	−0.0793550	+	0.0458157i	−0.539152	−	0.842208i	$-0.681255\pi$
$617$	−1.15005e10	+	6.63979e9i	−0.0793550	+	0.0458157i	0.459797	+	0.888024i	$0.347922\pi$
$618$		0			0
$619$	3.49176e10	−	6.04791e10i	0.237839	−	0.411948i	−0.722255	−	0.691627i	$-0.756894\pi$
$619$	3.49176e10	−	6.04791e10i	0.237839	−	0.411948i	0.960094	+	0.279678i	$0.0902278\pi$
$620$		0			0
$621$	−2.64996e11	+	6.04206e8i	−1.78186	+	0.00406274i
$622$		0			0
$623$	5.92330e10	+	3.41982e10i	0.393198	+	0.227013i
$624$		0			0
$625$	−3.88528e10	−	6.72950e10i	−0.254625	−	0.441024i
$626$		0			0
$627$	−1.19080e11	−	9.97659e10i	−0.770493	−	0.645523i
$628$		0			0
$629$			1.13356e10i			0.0724172i
$630$		0			0
$631$	−1.33712e11			−0.843440			−0.421720	−	0.906726i	$-0.638573\pi$
$631$	−1.33712e11			−0.843440			−0.421720	+	0.906726i	$0.638573\pi$
$632$		0			0
$633$	7.09992e10	+	1.94608e11i	0.442220	+	1.21212i
$634$		0			0
$635$	−6.27432e10	+	3.62248e10i	−0.385898	+	0.222798i
$636$		0			0
$637$	5.79404e9	−	1.00356e10i	0.0351904	−	0.0609515i
$638$		0			0
$639$	3.24758e10	−	8.96503e10i	0.194785	−	0.537710i
$640$		0			0
$641$	1.10030e11	+	6.35256e10i	0.651744	+	0.376285i	0.789124	−	0.614234i	$-0.210535\pi$
$641$	1.10030e11	+	6.35256e10i	0.651744	+	0.376285i	−0.137380	+	0.990518i	$0.543868\pi$
$642$		0			0
$643$	−5.23993e10	−	9.07582e10i	−0.306536	−	0.530936i	0.671066	−	0.741397i	$-0.265837\pi$
$643$	−5.23993e10	−	9.07582e10i	−0.306536	−	0.530936i	−0.977602	+	0.210462i	$0.932503\pi$
$644$		0			0
$645$	−1.74024e10	+	9.91347e10i	−0.100548	+	0.572779i
$646$		0			0
$647$		−	3.20267e11i		−	1.82766i	−0.406097	−	0.913830i	$-0.633110\pi$
$647$		−	3.20267e11i		−	1.82766i	0.406097	−	0.913830i	$-0.366890\pi$
$648$		0			0
$649$	−6.28291e10			−0.354146
$650$		0			0
$651$	1.65351e10	+	2.90262e9i	0.0920624	+	0.0161609i
$652$		0			0
$653$	−2.13839e10	+	1.23460e10i	−0.117607	+	0.0679004i	−0.557650	−	0.830076i	$-0.688297\pi$
$653$	−2.13839e10	+	1.23460e10i	−0.117607	+	0.0679004i	0.440043	+	0.897977i	$0.354963\pi$
$654$		0			0
$655$	7.41762e9	−	1.28477e10i	0.0402995	−	0.0698007i
$656$		0			0
$657$	1.19803e11	+	4.33986e10i	0.642993	+	0.232924i
$658$		0			0
$659$	−5.38397e10	−	3.10843e10i	−0.285470	−	0.164816i	0.350427	−	0.936590i	$-0.386036\pi$
$659$	−5.38397e10	−	3.10843e10i	−0.285470	−	0.164816i	−0.635897	+	0.771774i	$0.719370\pi$
$660$		0			0
$661$	1.63665e11	+	2.83475e11i	0.857331	+	1.48494i	0.874465	+	0.485088i	$0.161213\pi$
$661$	1.63665e11	+	2.83475e11i	0.857331	+	1.48494i	−0.0171339	+	0.999853i	$0.505454\pi$
$662$		0			0
$663$	6.08370e9	−	2.21952e9i	0.0314857	−	0.0114870i
$664$		0			0
$665$		−	5.96703e10i		−	0.305121i
$666$		0			0
$667$	3.24995e11			1.64200
$668$		0			0
$669$	−8.99400e10	+	1.07352e11i	−0.449002	+	0.535926i
$670$		0			0
$671$	9.31387e10	−	5.37736e10i	0.459452	−	0.265265i
$672$		0			0
$673$	−8.35206e10	+	1.44662e11i	−0.407130	+	0.705170i	−0.994567	−	0.104100i	$-0.966804\pi$
$673$	−8.35206e10	+	1.44662e11i	−0.407130	+	0.705170i	0.587437	+	0.809270i	$0.300137\pi$
$674$		0			0
$675$	1.48370e11	−	8.61130e10i	0.714711	−	0.414814i
$676$		0			0
$677$	1.82090e11	+	1.05130e11i	0.866826	+	0.500462i	0.866292	−	0.499537i	$-0.166497\pi$
$677$	1.82090e11	+	1.05130e11i	0.866826	+	0.500462i	0.000534051		1.00000i	$0.499830\pi$
$678$		0			0
$679$	−9.64162e9	−	1.66998e10i	−0.0453598	−	0.0785655i
$680$		0			0
$681$	2.33764e11	+	1.95849e11i	1.08690	+	0.910609i
$682$		0			0
$683$			1.45013e11i			0.666381i	0.942859	+	0.333191i	$0.108125\pi$
$683$			1.45013e11i			0.666381i	−0.942859	+	0.333191i	$0.891875\pi$
$684$		0			0
$685$	−1.32081e11			−0.599901
$686$		0			0
$687$	−5.79906e10	−	1.58952e11i	−0.260334	−	0.713573i
$688$		0			0
$689$	−4.36120e10	+	2.51794e10i	−0.193521	+	0.111730i
$690$		0			0
$691$	9.09386e10	−	1.57510e11i	0.398874	−	0.690871i	−0.594713	−	0.803938i	$-0.702734\pi$
$691$	9.09386e10	−	1.57510e11i	0.398874	−	0.690871i	0.993587	+	0.113067i	$0.0360676\pi$
$692$		0			0
$693$	4.45298e10	−	7.92162e9i	0.193071	−	0.0343464i
$694$		0			0
$695$	−7.91128e10	−	4.56758e10i	−0.339084	−	0.195770i
$696$		0			0
$697$	−9.87815e9	−	1.71095e10i	−0.0418547	−	0.0724945i
$698$		0			0
$699$	2.13228e10	−	1.21467e11i	0.0893173	−	0.508805i
$700$		0			0
$701$			2.19516e10i			0.0909065i	0.998966	+	0.0454532i	$0.0144732\pi$
$701$			2.19516e10i			0.0909065i	−0.998966	+	0.0454532i	$0.985527\pi$
$702$		0			0
$703$	−5.03701e11			−2.06230
$704$		0			0
$705$	−8.41004e10	−	1.47633e10i	−0.340441	−	0.0597621i
$706$		0			0
$707$	1.21061e11	−	6.98945e10i	0.484536	−	0.279747i
$708$		0			0
$709$	3.13571e10	−	5.43120e10i	0.124094	−	0.214937i	−0.797285	−	0.603604i	$-0.793731\pi$
$709$	3.13571e10	−	5.43120e10i	0.124094	−	0.214937i	0.921378	+	0.388667i	$0.127064\pi$
$710$		0			0
$711$	−2.61010e11	+	2.19690e11i	−1.02136	+	0.859671i
$712$		0			0
$713$	9.86245e10	+	5.69409e10i	0.381616	+	0.220326i
$714$		0			0
$715$	−1.39185e10	−	2.41076e10i	−0.0532561	−	0.0922423i
$716$		0			0
$717$	2.83003e11	−	1.03248e11i	1.07081	−	0.390666i
$718$		0			0
$719$			1.45102e11i			0.542948i	0.962446	+	0.271474i	$0.0875111\pi$
$719$			1.45102e11i			0.542948i	−0.962446	+	0.271474i	$0.912489\pi$
$720$		0			0
$721$	1.16623e11			0.431560
$722$		0			0
$723$	1.61513e11	−	1.92781e11i	0.591092	−	0.705524i
$724$		0			0
$725$	−1.82203e11	+	1.05195e11i	−0.659481	+	0.380752i
$726$		0			0
$727$	3.07456e10	−	5.32530e10i	0.110064	−	0.190637i	−0.805732	−	0.592281i	$-0.798228\pi$
$727$	3.07456e10	−	5.32530e10i	0.110064	−	0.190637i	0.915796	+	0.401644i	$0.131561\pi$
$728$		0			0
$729$	1.42329e11	+	2.43945e11i	0.503944	+	0.863736i
$730$		0			0
$731$	2.34779e10	+	1.35550e10i	0.0822222	+	0.0474710i
$732$		0			0
$733$	1.54868e11	+	2.68239e11i	0.536470	+	0.929193i	0.999091	+	0.0426369i	$0.0135759\pi$
$733$	1.54868e11	+	2.68239e11i	0.536470	+	0.929193i	−0.462621	+	0.886556i	$0.653091\pi$
$734$		0			0
$735$	1.33167e10	+	1.11568e10i	0.0456297	+	0.0382288i
$736$		0			0
$737$			1.39333e11i			0.472262i
$738$		0			0
$739$	3.95821e11			1.32715			0.663577	−	0.748108i	$-0.269037\pi$
$739$	3.95821e11			1.32715			0.663577	+	0.748108i	$0.269037\pi$
$740$		0			0
$741$	9.86251e10	+	2.70331e11i	0.327126	+	0.896650i
$742$		0			0
$743$	−2.66940e11	+	1.54118e11i	−0.875908	+	0.505706i	−0.869307	−	0.494272i	$-0.835435\pi$
$743$	−2.66940e11	+	1.54118e11i	−0.875908	+	0.505706i	−0.00660130	+	0.999978i	$0.502101\pi$
$744$		0			0
$745$	1.00767e11	−	1.74534e11i	0.327111	−	0.566572i
$746$		0			0
$747$	2.93135e11	+	3.48268e11i	0.941423	+	1.11849i
$748$		0			0
$749$	1.39351e11	+	8.04544e10i	0.442775	+	0.255636i
$750$		0			0
$751$	7.38760e10	+	1.27957e11i	0.232244	+	0.402258i	0.958468	−	0.285200i	$-0.0920600\pi$
$751$	7.38760e10	+	1.27957e11i	0.232244	+	0.402258i	−0.726224	+	0.687458i	$0.758727\pi$
$752$		0			0
$753$	3.18684e9	−	1.81541e10i	0.00991243	−	0.0564671i
$754$		0			0
$755$			1.73245e11i			0.533178i
$756$		0			0
$757$	5.79202e11			1.76379			0.881894	−	0.471447i	$-0.156268\pi$
$757$	5.79202e11			1.76379			0.881894	+	0.471447i	$0.156268\pi$
$758$		0			0
$759$	3.02192e11	+	5.30479e10i	0.910576	+	0.159846i
$760$		0			0
$761$	−1.26846e11	+	7.32344e10i	−0.378214	+	0.218362i	−0.677041	−	0.735946i	$-0.736738\pi$
$761$	−1.26846e11	+	7.32344e10i	−0.378214	+	0.218362i	0.298827	+	0.954307i	$0.403405\pi$
$762$		0			0
$763$	9.94391e10	−	1.72234e11i	0.293399	−	0.508182i
$764$		0			0
$765$	1.70042e9	+	9.55855e9i	0.00496489	+	0.0279091i
$766$		0			0
$767$	1.00789e11	+	5.81906e10i	0.291227	+	0.168140i
$768$		0			0
$769$	2.79280e11	+	4.83728e11i	0.798611	+	1.38323i	0.920521	+	0.390693i	$0.127765\pi$
$769$	2.79280e11	+	4.83728e11i	0.798611	+	1.38323i	−0.121910	+	0.992541i	$0.538902\pi$
$770$		0			0
$771$	1.23306e11	−	4.49859e10i	0.348953	−	0.127309i
$772$		0			0
$773$			5.27012e11i			1.47605i	0.674771	+	0.738027i	$0.264242\pi$
$773$			5.27012e11i			1.47605i	−0.674771	+	0.738027i	$0.735758\pi$
$774$		0			0
$775$	−7.37228e10			−0.204360
$776$		0			0
$777$	9.41791e10	−	1.12412e11i	0.258387	−	0.308409i
$778$		0			0
$779$	7.60263e11	−	4.38938e11i	2.06450	−	1.19194i
$780$		0			0
$781$	−5.51988e10	+	9.56071e10i	−0.148363	+	0.256972i
$782$		0			0
$783$	−1.73871e11	−	2.99573e11i	−0.462572	−	0.796996i
$784$		0			0
$785$	1.26208e11	+	7.28663e10i	0.332360	+	0.191888i
$786$		0			0
$787$	−7.75783e10	−	1.34370e11i	−0.202228	−	0.350269i	0.747018	−	0.664804i	$-0.231485\pi$
$787$	−7.75783e10	−	1.34370e11i	−0.202228	−	0.350269i	−0.949246	+	0.314535i	$0.898152\pi$
$788$		0			0
$789$	−3.11269e11	−	2.60783e11i	−0.803207	−	0.672931i
$790$		0			0
$791$		−	2.14242e11i		−	0.547267i
$792$		0			0
$793$	−1.99215e11			−0.503766
$794$		0			0
$795$	−2.58755e10	−	7.09245e10i	−0.0647768	−	0.177553i
$796$		0			0
$797$	3.16687e11	−	1.82839e11i	0.784868	−	0.453144i	−0.0532849	−	0.998579i	$-0.516969\pi$
$797$	3.16687e11	−	1.82839e11i	0.784868	−	0.453144i	0.838153	+	0.545436i	$0.183636\pi$
$798$		0			0
$799$	−1.14993e10	+	1.99173e10i	−0.0282152	+	0.0488702i
$800$		0			0
$801$	1.68420e11	−	4.64928e11i	0.409132	−	1.12942i
$802$		0			0
$803$	−1.27763e11	−	7.37641e10i	−0.307287	−	0.177412i
$804$		0			0
$805$	5.89243e10	+	1.02060e11i	0.140317	+	0.243036i
$806$		0			0
$807$	2.15729e10	−	1.22892e11i	0.0508645	−	0.289754i
$808$		0			0
$809$			1.81421e11i			0.423539i	0.977320	+	0.211769i	$0.0679225\pi$
$809$			1.81421e11i			0.423539i	−0.977320	+	0.211769i	$0.932077\pi$
$810$		0			0
$811$	2.07236e11			0.479052			0.239526	−	0.970890i	$-0.423008\pi$
$811$	2.07236e11			0.479052			0.239526	+	0.970890i	$0.423008\pi$
$812$		0			0
$813$	−2.70798e11	−	4.75367e10i	−0.619844	−	0.108810i
$814$		0			0
$815$	1.93082e11	−	1.11476e11i	0.437635	−	0.252669i
$816$		0			0
$817$	−6.02318e11	+	1.04325e12i	−1.35188	+	2.34152i
$818$		0			0
$819$	−7.87705e10	−	2.85346e10i	−0.175077	−	0.0634215i
$820$		0			0
$821$	−1.52433e11	−	8.80073e10i	−0.335511	−	0.193707i	0.322774	−	0.946476i	$-0.395385\pi$
$821$	−1.52433e11	−	8.80073e10i	−0.335511	−	0.193707i	−0.658285	+	0.752769i	$0.728718\pi$
$822$		0			0
$823$	1.68245e11	+	2.91409e11i	0.366727	+	0.635189i	0.989052	−	0.147570i	$-0.0471451\pi$
$823$	1.68245e11	+	2.91409e11i	0.366727	+	0.635189i	−0.622325	+	0.782759i	$0.713812\pi$
$824$		0			0
$825$	−1.86589e11	+	6.80737e10i	−0.402783	+	0.146948i
$826$		0			0
$827$		−	7.73945e11i		−	1.65458i	−0.561774	−	0.827291i	$-0.689881\pi$
$827$		−	7.73945e11i		−	1.65458i	0.561774	−	0.827291i	$-0.310119\pi$
$828$		0			0
$829$	3.98366e11			0.843460			0.421730	−	0.906722i	$-0.361423\pi$
$829$	3.98366e11			0.843460			0.421730	+	0.906722i	$0.361423\pi$
$830$		0			0
$831$	−4.62734e11	+	5.52317e11i	−0.970348	+	1.15820i
$832$		0			0
$833$	4.05237e9	−	2.33964e9i	0.00841645	−	0.00485924i
$834$		0			0
$835$	−1.36183e11	+	2.35877e11i	−0.280142	+	0.485220i
$836$		0			0
$837$	−2.76737e8	−	1.21373e11i	−0.000563853	−	0.247298i
$838$		0			0
$839$	3.09960e11	+	1.78955e11i	0.625543	+	0.361158i	0.779024	−	0.626994i	$-0.215715\pi$
$839$	3.09960e11	+	1.78955e11i	0.625543	+	0.361158i	−0.153481	+	0.988152i	$0.549048\pi$
$840$		0			0
$841$	−3.77248e10	−	6.53413e10i	−0.0754125	−	0.130618i
$842$		0			0
$843$	2.68162e11	+	2.24668e11i	0.530992	+	0.444868i
$844$		0			0
$845$		−	1.60879e11i		−	0.315553i
$846$		0			0
$847$	1.42163e11			0.276219
$848$		0			0
$849$	3.49761e10	+	9.58692e10i	0.0673194	+	0.184522i
$850$		0			0
$851$	8.61527e11	−	4.97403e11i	1.64267	−	0.948396i
$852$		0			0
$853$	3.64383e10	−	6.31130e10i	0.0688275	−	0.119213i	−0.829558	−	0.558421i	$-0.811408\pi$
$853$	3.64383e10	−	6.31130e10i	0.0688275	−	0.119213i	0.898385	+	0.439208i	$0.144741\pi$
$854$		0			0
$855$	−4.24737e11	+	7.55584e10i	−0.794796	+	0.141390i
$856$		0			0
$857$	9.47165e10	+	5.46846e10i	0.175591	+	0.101378i	0.585220	−	0.810875i	$-0.301008\pi$
$857$	9.47165e10	+	5.46846e10i	0.175591	+	0.101378i	−0.409629	+	0.912252i	$0.634342\pi$
$858$		0			0
$859$	−5.02645e11	−	8.70607e11i	−0.923185	−	1.59900i	−0.794454	−	0.607324i	$-0.792243\pi$
$859$	−5.02645e11	−	8.70607e11i	−0.923185	−	1.59900i	−0.128731	−	0.991680i	$-0.541090\pi$
$860$		0			0
$861$	−4.41912e10	+	2.51739e11i	−0.0804124	+	0.458077i
$862$		0			0
$863$			3.24854e11i			0.585659i	0.956165	+	0.292830i	$0.0945969\pi$
$863$			3.24854e11i			0.585659i	−0.956165	+	0.292830i	$0.905403\pi$
$864$		0			0
$865$	−2.77886e11			−0.496366
$866$		0			0
$867$	−5.53951e11	−	9.72425e10i	−0.980381	−	0.172099i
$868$		0			0
$869$	3.42075e11	−	1.97497e11i	0.599850	−	0.346323i
$870$		0			0
$871$	1.29046e11	−	2.23514e11i	0.224219	−	0.388358i
$872$		0			0
$873$	−1.06661e11	+	8.97759e10i	−0.183632	+	0.154562i
$874$		0			0
$875$	−1.46022e11	−	8.43057e10i	−0.249107	−	0.143822i
$876$		0			0
$877$	1.41326e11	+	2.44784e11i	0.238904	+	0.413794i	0.960400	−	0.278624i	$-0.0898784\pi$
$877$	1.41326e11	+	2.44784e11i	0.238904	+	0.413794i	−0.721496	+	0.692419i	$0.756545\pi$
$878$		0			0
$879$	1.12786e10	−	4.11478e9i	0.0188929	−	0.00689273i
$880$		0			0
$881$			1.29442e11i			0.214869i	0.994212	+	0.107434i	$0.0342635\pi$
$881$			1.29442e11i			0.214869i	−0.994212	+	0.107434i	$0.965736\pi$
$882$		0			0
$883$	4.20917e10			0.0692395			0.0346197	−	0.999401i	$-0.488978\pi$
$883$	4.20917e10			0.0692395			0.0346197	+	0.999401i	$0.488978\pi$
$884$		0			0
$885$	−1.12050e11	+	1.33742e11i	−0.182658	+	0.218020i
$886$		0			0
$887$	−9.29321e11	+	5.36544e11i	−1.50131	+	0.866784i	−0.501315	+	0.865265i	$0.667150\pi$
$887$	−9.29321e11	+	5.36544e11i	−1.50131	+	0.866784i	−0.999999	+	0.00151901i	$0.999516\pi$
$888$		0			0
$889$	1.26228e11	−	2.18632e11i	0.202091	−	0.350032i
$890$		0			0
$891$	−1.12773e11	−	3.06935e11i	−0.178935	−	0.487007i
$892$		0			0
$893$	−8.85031e11	−	5.10973e11i	−1.39172	−	0.803512i
$894$		0			0
$895$	−5.19218e10	−	8.99312e10i	−0.0809204	−	0.140158i
$896$		0			0
$897$	−4.35639e11	−	3.64981e11i	−0.672910	−	0.563767i
$898$		0			0
$899$			1.48854e11i			0.227887i
$900$		0			0
$901$	−2.03349e10			−0.0308562
$902$		0			0
$903$	−1.20205e11	−	3.29480e11i	−0.180788	−	0.495540i
$904$		0			0
$905$	−2.85564e11	+	1.64870e11i	−0.425705	+	0.245781i
$906$		0			0
$907$	−2.36535e11	+	4.09691e11i	−0.349515	+	0.605378i	−0.986163	−	0.165776i	$-0.946987\pi$
$907$	−2.36535e11	+	4.09691e11i	−0.349515	+	0.605378i	0.636648	+	0.771154i	$0.280320\pi$
$908$		0			0
$909$	−6.50808e11	−	7.73213e11i	−0.953230	−	1.13251i
$910$		0			0
$911$	2.96237e11	+	1.71032e11i	0.430096	+	0.248316i	0.699387	−	0.714743i	$-0.253456\pi$
$911$	2.96237e11	+	1.71032e11i	0.430096	+	0.248316i	−0.269292	+	0.963059i	$0.586790\pi$
$912$		0			0
$913$	−2.63522e11	−	4.56434e11i	−0.379257	−	0.656893i
$914$		0			0
$915$	5.16382e10	−	2.94162e11i	0.0736693	−	0.419664i
$916$		0			0
$917$			5.16943e10i			0.0731080i
$918$		0			0
$919$	−3.14418e11			−0.440803			−0.220402	−	0.975409i	$-0.570737\pi$
$919$	−3.14418e11			−0.440803			−0.220402	+	0.975409i	$0.570737\pi$
$920$		0			0
$921$	2.92572e11	+	5.13591e10i	0.406625	+	0.0713804i
$922$		0			0
$923$	1.77097e11	−	1.02247e11i	0.244009	−	0.140878i
$924$		0			0
$925$	−3.22000e11	+	5.57720e11i	−0.439834	+	0.761815i
$926$		0			0
$927$	−1.47675e11	−	8.30126e11i	−0.199981	−	1.12415i
$928$		0			0
$929$	−1.10912e12	−	6.40353e11i	−1.48908	−	0.859719i	−0.489154	−	0.872197i	$-0.662695\pi$
$929$	−1.10912e12	−	6.40353e11i	−1.48908	−	0.859719i	−0.999922	+	0.0124784i	$0.996028\pi$
$930$		0			0
$931$	1.03962e11	+	1.80068e11i	0.138381	+	0.239684i
$932$		0			0
$933$	3.59970e11	−	1.31328e11i	0.475050	−	0.173313i
$934$		0			0
$935$		−	1.12406e10i		−	0.0147077i
$936$		0			0
$937$	−4.00575e11			−0.519667			−0.259834	−	0.965653i	$-0.583668\pi$
$937$	−4.00575e11			−0.519667			−0.259834	+	0.965653i	$0.583668\pi$
$938$		0			0
$939$	−5.85859e11	+	6.99278e11i	−0.753582	+	0.899471i
$940$		0			0
$941$	1.31746e12	−	7.60637e11i	1.68027	−	0.970106i	0.718795	−	0.695222i	$-0.244694\pi$
$941$	1.31746e12	−	7.60637e11i	1.68027	−	0.970106i	0.961477	−	0.274884i	$-0.0886396\pi$
$942$		0			0
$943$	−8.66900e11	+	1.50152e12i	−1.09628	+	1.89882i
$944$		0			0
$945$	6.25524e10	−	1.08917e11i	0.0784362	−	0.136574i
$946$		0			0
$947$	8.11856e11	+	4.68725e11i	1.00944	+	0.582798i	0.911027	−	0.412348i	$-0.135291\pi$
$947$	8.11856e11	+	4.68725e11i	1.00944	+	0.582798i	0.0984097	+	0.995146i	$0.468624\pi$
$948$		0			0
$949$	1.36637e11	+	2.36662e11i	0.168462	+	0.291785i
$950$		0			0
$951$	1.47547e11	+	1.23615e11i	0.180388	+	0.151130i
$952$		0			0
$953$			1.17927e12i			1.42969i	0.699284	+	0.714844i	$0.253502\pi$
$953$			1.17927e12i			1.42969i	−0.699284	+	0.714844i	$0.746498\pi$
$954$		0			0
$955$	−4.54278e11			−0.546146
$956$		0			0
$957$	1.37447e11	+	3.76742e11i	0.163866	+	0.449155i
$958$		0			0
$959$	3.98584e11	−	2.30123e11i	0.471244	−	0.272073i
$960$		0			0
$961$	4.00366e11	−	6.93453e11i	0.469422	−	0.813062i
$962$		0			0
$963$	3.96224e11	−	1.09379e12i	0.460718	−	1.27182i
$964$		0			0
$965$	−3.71348e11	−	2.14398e11i	−0.428225	−	0.247236i
$966$		0			0
$967$	1.63149e10	+	2.82582e10i	0.0186585	+	0.0323175i	0.875204	−	0.483754i	$-0.160727\pi$
$967$	1.63149e10	+	2.82582e10i	0.0186585	+	0.0323175i	−0.856545	+	0.516072i	$0.827394\pi$
$968$		0			0
$969$	−2.00905e10	+	1.14448e11i	−0.0227875	+	0.129811i
$970$		0			0
$971$			1.01296e12i			1.13950i	0.821817	+	0.569752i	$0.192961\pi$
$971$			1.01296e12i			1.13950i	−0.821817	+	0.569752i	$0.807039\pi$
$972$		0			0
$973$	3.18320e11			0.355151
$974$		0			0
$975$	3.62371e11	+	6.36118e10i	0.400991	+	0.0703913i
$976$		0			0
$977$	1.09168e12	−	6.30282e11i	1.19817	−	0.691762i	0.238020	−	0.971260i	$-0.423502\pi$
$977$	1.09168e12	−	6.30282e11i	1.19817	−	0.691762i	0.960146	+	0.279499i	$0.0901683\pi$
$978$		0			0
$979$	−2.86262e11	+	4.95820e11i	−0.311625	+	0.539751i
$980$		0			0
$981$	−1.35188e12	−	4.89720e11i	−1.45970	−	0.528776i
$982$		0			0
$983$	−2.86675e11	−	1.65512e11i	−0.307026	−	0.177262i	0.338569	−	0.940942i	$-0.390057\pi$
$983$	−2.86675e11	−	1.65512e11i	−0.307026	−	0.177262i	−0.645595	+	0.763680i	$0.723391\pi$
$984$		0			0
$985$	1.38309e11	+	2.39558e11i	0.146928	+	0.254487i
$986$		0			0
$987$	2.79513e11	−	1.01975e11i	0.294532	−	0.107455i
$988$		0			0
$989$		−	2.37915e12i		−	2.48677i
$990$		0			0
$991$	−1.22452e12			−1.26961			−0.634804	−	0.772673i	$-0.718919\pi$
$991$	−1.22452e12			−1.26961			−0.634804	+	0.772673i	$0.718919\pi$
$992$		0			0
$993$	−6.52843e11	+	7.79230e11i	−0.671447	+	0.801436i
$994$		0			0
$995$	1.28472e11	−	7.41732e10i	0.131074	−	0.0756754i
$996$		0			0
$997$	−4.03252e11	+	6.98453e11i	−0.408127	+	0.706897i	−0.994680	−	0.103014i	$-0.967151\pi$
$997$	−4.03252e11	+	6.98453e11i	−0.408127	+	0.706897i	0.586553	+	0.809911i	$0.300485\pi$
$998$		0			0
$999$	−9.19408e11	−	5.28029e11i	−0.923095	−	0.530146i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.4	✓	96
9.5	odd	6	inner	252.9.bg.a.113.4	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.4	✓	96	1.1	even	1	trivial
252.9.bg.a.113.4	yes	96	9.5	odd	6	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-79.7801 - 14.0049i) q^{3} +(225.541 - 130.216i) q^{5} +(-453.746 + 785.912i) q^{7} +(6168.73 + 2234.62i) q^{9} +O(q^{10})\)	\(q+(-79.7801 - 14.0049i) q^{3} +(225.541 - 130.216i) q^{5} +(-453.746 + 785.912i) q^{7} +(6168.73 + 2234.62i) q^{9} +(-6578.61 - 3798.16i) q^{11} +(7035.50 + 12185.8i) q^{13} +(-19817.4 + 7229.99i) q^{15} +5681.88i q^{17} -252476. q^{19} +(47206.5 - 56345.5i) q^{21} +(431834. - 249319. i) q^{23} +(-161400. + 279553. i) q^{25} +(-460846. - 264670. i) q^{27} +(564444. + 325882. i) q^{29} +(114193. + 197788. i) q^{31} +(471649. + 395150. i) q^{33} +236341. i q^{35} +1.99504e6 q^{37} +(-390632. - 1.07072e6i) q^{39} +(-3.01123e6 + 1.73853e6i) q^{41} +(2.38564e6 - 4.13206e6i) q^{43} +(1.68229e6 - 299270. i) q^{45} +(3.50541e6 + 2.02385e6i) q^{47} +(-411772. - 713209. i) q^{49} +(79574.0 - 453301. i) q^{51} +3.57890e6i q^{53} -1.97833e6 q^{55} +(2.01426e7 + 3.53589e6i) q^{57} +(7.16289e6 - 4.13550e6i) q^{59} +(-7.07891e6 + 1.22610e7i) q^{61} +(-4.55525e6 + 3.83412e6i) q^{63} +(3.17359e6 + 1.83227e6i) q^{65} +(-9.17106e6 - 1.58847e7i) q^{67} +(-3.79434e7 + 1.38429e7i) q^{69} -1.45330e7i q^{71} +1.94210e7 q^{73} +(1.67916e7 - 2.00424e7i) q^{75} +(5.97004e6 - 3.44680e6i) q^{77} +(-2.59991e7 + 4.50317e7i) q^{79} +(3.30597e7 + 2.75695e7i) q^{81} +(6.00862e7 + 3.46908e7i) q^{83} +(739874. + 1.28150e6i) q^{85} +(-4.04675e7 - 3.39039e7i) q^{87} -7.53685e7i q^{89} -1.27693e7 q^{91} +(-6.34032e6 - 1.73788e7i) q^{93} +(-5.69437e7 + 3.28765e7i) q^{95} +(-1.06245e7 + 1.84021e7i) q^{97} +(-3.20942e7 - 3.81305e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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