LMFDB - Embedded newform 252.9.bg.a.29.20

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			29.20
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-21.6670 + 78.0483i) q^{3} +(514.619 - 297.115i) q^{5} +(453.746 - 785.912i) q^{7} +(-5622.08 - 3382.15i) q^{9} +O(q^{10})$	\(q+(-21.6670 + 78.0483i) q^{3} +(514.619 - 297.115i) q^{5} +(453.746 - 785.912i) q^{7} +(-5622.08 - 3382.15i) q^{9} +(8562.71 + 4943.68i) q^{11} +(-3001.24 - 5198.30i) q^{13} +(12039.1 + 46602.7i) q^{15} -133107. i q^{17} -81901.3 q^{19} +(51507.7 + 52442.5i) q^{21} +(14225.3 - 8213.00i) q^{23} +(-18757.6 + 32489.2i) q^{25} +(385785. - 365513. i) q^{27} +(-741234. - 427951. i) q^{29} +(62105.4 + 107570. i) q^{31} +(-571374. + 561190. i) q^{33} -539260. i q^{35} +8039.17 q^{37} +(470747. - 121610. i) q^{39} +(-3.40614e6 + 1.96654e6i) q^{41} +(-2.13673e6 + 3.70093e6i) q^{43} +(-3.89812e6 - 70113.2i) q^{45} +(1.24040e6 + 716147. i) q^{47} +(-411772. - 713209. i) q^{49} +(1.03888e7 + 2.88403e6i) q^{51} +1.05532e7i q^{53} +5.87537e6 q^{55} +(1.77456e6 - 6.39226e6i) q^{57} +(-3.43449e6 + 1.98290e6i) q^{59} +(899212. - 1.55748e6i) q^{61} +(-5.20907e6 + 2.88382e6i) q^{63} +(-3.08899e6 - 1.78343e6i) q^{65} +(1.42781e7 + 2.47304e7i) q^{67} +(332790. + 1.28821e6i) q^{69} +2.40246e7i q^{71} -8.45579e6 q^{73} +(-2.12930e6 - 2.16795e6i) q^{75} +(7.77059e6 - 4.48635e6i) q^{77} +(-2.02624e7 + 3.50954e7i) q^{79} +(2.01688e7 + 3.80294e7i) q^{81} +(-6.82136e7 - 3.93831e7i) q^{83} +(-3.95481e7 - 6.84993e7i) q^{85} +(4.94612e7 - 4.85796e7i) q^{87} -8.93985e7i q^{89} -5.44721e6 q^{91} +(-9.74128e6 + 2.51651e6i) q^{93} +(-4.21479e7 + 2.43341e7i) q^{95} +(-6.60071e7 + 1.14328e8i) q^{97} +(-3.14199e7 - 5.67541e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−21.6670	+	78.0483i	−0.267494	+	0.963559i
$3$	−21.6670	+	78.0483i	−0.267494	+	0.963559i
$4$		0			0
$5$	514.619	−	297.115i	0.823390	−	0.475384i	−0.0281942	−	0.999602i	$-0.508976\pi$
$5$	514.619	−	297.115i	0.823390	−	0.475384i	0.851584	+	0.524218i	$0.175642\pi$
$6$		0			0
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$8$		0			0
$9$	−5622.08	−	3382.15i	−0.856894	−	0.515493i
$10$		0			0
$11$	8562.71	+	4943.68i	0.584844	+	0.337660i	0.763056	−	0.646332i	$-0.223698\pi$
$11$	8562.71	+	4943.68i	0.584844	+	0.337660i	−0.178212	+	0.983992i	$0.557031\pi$
$12$		0			0
$13$	−3001.24	−	5198.30i	−0.105082	−	0.182007i	0.808690	−	0.588235i	$-0.200177\pi$
$13$	−3001.24	−	5198.30i	−0.105082	−	0.182007i	−0.913772	+	0.406228i	$0.866844\pi$
$14$		0			0
$15$	12039.1	+	46602.7i	0.237809	+	0.920548i
$16$		0			0
$17$		−	133107.i		−	1.59369i	−0.604181	−	0.796847i	$-0.706500\pi$
$17$		−	133107.i		−	1.59369i	0.604181	−	0.796847i	$-0.293500\pi$
$18$		0			0
$19$	−81901.3			−0.628458			−0.314229	−	0.949347i	$-0.601746\pi$
$19$	−81901.3			−0.628458			−0.314229	+	0.949347i	$0.601746\pi$
$20$		0			0
$21$	51507.7	+	52442.5i	0.264847	+	0.269654i
$22$		0			0
$23$	14225.3	−	8213.00i	0.0508336	−	0.0293488i	−0.474368	−	0.880327i	$-0.657323\pi$
$23$	14225.3	−	8213.00i	0.0508336	−	0.0293488i	0.525201	+	0.850978i	$0.323990\pi$
$24$		0			0
$25$	−18757.6	+	32489.2i	−0.0480195	+	0.0831723i
$26$		0			0
$27$	385785.	−	365513.i	0.725923	−	0.687776i
$28$		0			0
$29$	−741234.	−	427951.i	−1.04800	−	0.605066i	−0.125914	−	0.992041i	$-0.540186\pi$
$29$	−741234.	−	427951.i	−1.04800	−	0.605066i	−0.922090	+	0.386975i	$0.873520\pi$
$30$		0			0
$31$	62105.4	+	107570.i	0.0672485	+	0.116478i	0.897689	−	0.440629i	$-0.145245\pi$
$31$	62105.4	+	107570.i	0.0672485	+	0.116478i	−0.830441	+	0.557107i	$0.811911\pi$
$32$		0			0
$33$	−571374.	+	561190.i	−0.481798	+	0.473210i
$34$		0			0
$35$		−	539260.i		−	0.359357i
$36$		0			0
$37$	8039.17			0.00428948			0.00214474	−	0.999998i	$-0.499317\pi$
$37$	8039.17			0.00428948			0.00214474	+	0.999998i	$0.499317\pi$
$38$		0			0
$39$	470747.	−	121610.i	0.203483	−	0.0525667i
$40$		0			0
$41$	−3.40614e6	+	1.96654e6i	−1.20539	+	0.695932i	−0.961749	−	0.273934i	$-0.911675\pi$
$41$	−3.40614e6	+	1.96654e6i	−1.20539	+	0.695932i	−0.243641	+	0.969866i	$0.578342\pi$
$42$		0			0
$43$	−2.13673e6	+	3.70093e6i	−0.624995	+	1.08252i	0.363547	+	0.931576i	$0.381566\pi$
$43$	−2.13673e6	+	3.70093e6i	−0.624995	+	1.08252i	−0.988542	+	0.150947i	$0.951768\pi$
$44$		0			0
$45$	−3.89812e6	−	70113.2i	−0.950615	−	0.0170982i
$46$		0			0
$47$	1.24040e6	+	716147.i	0.254197	+	0.146761i	0.621685	−	0.783268i	$-0.286449\pi$
$47$	1.24040e6	+	716147.i	0.254197	+	0.146761i	−0.367487	+	0.930029i	$0.619782\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	1.03888e7	+	2.88403e6i	1.53562	+	0.426304i
$52$		0			0
$53$			1.05532e7i			1.33747i	0.743503	+	0.668733i	$0.233163\pi$
$53$			1.05532e7i			1.33747i	−0.743503	+	0.668733i	$0.766837\pi$
$54$		0			0
$55$	5.87537e6			0.642073
$56$		0			0
$57$	1.77456e6	−	6.39226e6i	0.168109	−	0.605557i
$58$		0			0
$59$	−3.43449e6	+	1.98290e6i	−0.283435	+	0.163641i	−0.634978	−	0.772531i	$-0.718991\pi$
$59$	−3.43449e6	+	1.98290e6i	−0.283435	+	0.163641i	0.351542	+	0.936172i	$0.385657\pi$
$60$		0			0
$61$	899212.	−	1.55748e6i	0.0649445	−	0.112487i	−0.831725	−	0.555188i	$-0.812646\pi$
$61$	899212.	−	1.55748e6i	0.0649445	−	0.112487i	0.896669	+	0.442701i	$0.145980\pi$
$62$		0			0
$63$	−5.20907e6	+	2.88382e6i	−0.330672	+	0.183065i
$64$		0			0
$65$	−3.08899e6	−	1.78343e6i	−0.173047	−	0.0999084i
$66$		0			0
$67$	1.42781e7	+	2.47304e7i	0.708553	+	1.22725i	0.965394	+	0.260795i	$0.0839848\pi$
$67$	1.42781e7	+	2.47304e7i	0.708553	+	1.22725i	−0.256842	+	0.966454i	$0.582682\pi$
$68$		0			0
$69$	332790.	+	1.28821e6i	0.0146816	+	0.0568319i
$70$		0			0
$71$			2.40246e7i			0.945415i	0.881219	+	0.472707i	$0.156723\pi$
$71$			2.40246e7i			0.945415i	−0.881219	+	0.472707i	$0.843277\pi$
$72$		0			0
$73$	−8.45579e6			−0.297758			−0.148879	−	0.988855i	$-0.547566\pi$
$73$	−8.45579e6			−0.297758			−0.148879	+	0.988855i	$0.547566\pi$
$74$		0			0
$75$	−2.12930e6	−	2.16795e6i	−0.0672965	−	0.0685178i
$76$		0			0
$77$	7.77059e6	−	4.48635e6i	0.221050	−	0.127624i
$78$		0			0
$79$	−2.02624e7	+	3.50954e7i	−0.520213	+	0.901036i	0.479510	+	0.877536i	$0.340814\pi$
$79$	−2.02624e7	+	3.50954e7i	−0.520213	+	0.901036i	−0.999724	+	0.0234999i	$0.992519\pi$
$80$		0			0
$81$	2.01688e7	+	3.80294e7i	0.468533	+	0.883446i
$82$		0			0
$83$	−6.82136e7	−	3.93831e7i	−1.43734	−	0.829847i	−0.439673	−	0.898158i	$-0.644906\pi$
$83$	−6.82136e7	−	3.93831e7i	−1.43734	−	0.829847i	−0.997664	+	0.0683109i	$0.978239\pi$
$84$		0			0
$85$	−3.95481e7	−	6.84993e7i	−0.757617	−	1.31223i
$86$		0			0
$87$	4.94612e7	−	4.85796e7i	0.863352	−	0.847963i
$88$		0			0
$89$		−	8.93985e7i		−	1.42485i	−0.701747	−	0.712427i	$-0.747596\pi$
$89$		−	8.93985e7i		−	1.42485i	0.701747	−	0.712427i	$-0.252404\pi$
$90$		0			0
$91$	−5.44721e6			−0.0794344
$92$		0			0
$93$	−9.74128e6	+	2.51651e6i	−0.130222	+	0.0336408i
$94$		0			0
$95$	−4.21479e7	+	2.43341e7i	−0.517466	+	0.298759i
$96$		0			0
$97$	−6.60071e7	+	1.14328e8i	−0.745597	+	1.29141i	0.204319	+	0.978904i	$0.434502\pi$
$97$	−6.60071e7	+	1.14328e8i	−0.745597	+	1.29141i	−0.949916	+	0.312507i	$0.898831\pi$
$98$		0			0
$99$	−3.14199e7	−	5.67541e7i	−0.327088	−	0.590822i
$100$		0			0
$101$	−1.63756e7	−	9.45444e6i	−0.157366	−	0.0908553i	0.419249	−	0.907871i	$-0.362293\pi$
$101$	−1.63756e7	−	9.45444e6i	−0.157366	−	0.0908553i	−0.576615	+	0.817016i	$0.695627\pi$
$102$		0			0
$103$	−5.75824e7	−	9.97357e7i	−0.511613	−	0.886139i	−0.999909	−	0.0134614i	$-0.995715\pi$
$103$	−5.75824e7	−	9.97357e7i	−0.511613	−	0.886139i	0.488297	−	0.872678i	$-0.337618\pi$
$104$		0			0
$105$	4.20883e7	+	1.16842e7i	0.346262	+	0.0961259i
$106$		0			0
$107$		−	2.15403e8i		−	1.64330i	−0.569990	−	0.821651i	$-0.693053\pi$
$107$		−	2.15403e8i		−	1.64330i	0.569990	−	0.821651i	$-0.306947\pi$
$108$		0			0
$109$	−6.56564e7			−0.465127			−0.232563	−	0.972581i	$-0.574711\pi$
$109$	−6.56564e7			−0.465127			−0.232563	+	0.972581i	$0.574711\pi$
$110$		0			0
$111$	−174185.	+	627444.i	−0.00114741	+	0.00413317i
$112$		0			0
$113$	9.83422e6	−	5.67779e6i	0.0603151	−	0.0348230i	−0.469539	−	0.882912i	$-0.655580\pi$
$113$	9.83422e6	−	5.67779e6i	0.0603151	−	0.0348230i	0.529854	+	0.848089i	$0.322247\pi$
$114$		0			0
$115$	4.88041e6	−	8.45312e6i	0.0279039	−	0.0483310i
$116$		0			0
$117$	−708232.	+	3.93759e7i	−0.00377948	+	0.210130i
$118$		0			0
$119$	−1.04610e8	−	6.03968e7i	−0.521659	−	0.301180i
$120$		0			0
$121$	−5.82995e7	−	1.00978e8i	−0.271971	−	0.471068i
$122$		0			0
$123$	−7.96839e7	−	3.08453e8i	−0.348137	−	1.34762i
$124$		0			0
$125$			2.54414e8i			1.04208i
$126$		0			0
$127$	3.36357e8			1.29296			0.646481	−	0.762930i	$-0.276240\pi$
$127$	3.36357e8			1.29296			0.646481	+	0.762930i	$0.276240\pi$
$128$		0			0
$129$	−2.42555e8	−	2.46957e8i	−0.875892	−	0.891788i
$130$		0			0
$131$	−1.68634e8	+	9.73608e7i	−0.572611	+	0.330597i	−0.758192	−	0.652032i	$-0.773917\pi$
$131$	−1.68634e8	+	9.73608e7i	−0.572611	+	0.330597i	0.185580	+	0.982629i	$0.440583\pi$
$132$		0			0
$133$	−3.71624e7	+	6.43672e7i	−0.118767	+	0.205711i
$134$		0			0
$135$	8.99328e7	−	3.02722e8i	0.270759	−	0.911400i
$136$		0			0
$137$	−4.79846e8	−	2.77039e8i	−1.36213	−	0.786429i	−0.372227	−	0.928142i	$-0.621406\pi$
$137$	−4.79846e8	−	2.77039e8i	−1.36213	−	0.786429i	−0.989908	+	0.141713i	$0.954739\pi$
$138$		0			0
$139$	−1.43169e7	−	2.47975e7i	−0.0383521	−	0.0664277i	0.846212	−	0.532846i	$-0.178878\pi$
$139$	−1.43169e7	−	2.47975e7i	−0.0383521	−	0.0664277i	−0.884564	+	0.466418i	$0.845544\pi$
$140$		0			0
$141$	−8.27699e7	+	8.12945e7i	−0.209409	+	0.205677i
$142$		0			0
$143$		−	5.93487e7i		−	0.141928i
$144$		0			0
$145$	−5.08603e8			−1.15055
$146$		0			0
$147$	6.45866e7	−	1.66849e7i	0.138316	−	0.0357318i
$148$		0			0
$149$	−2.88085e8	+	1.66326e8i	−0.584487	+	0.337454i	−0.762915	−	0.646499i	$-0.776232\pi$
$149$	−2.88085e8	+	1.66326e8i	−0.584487	+	0.337454i	0.178428	+	0.983953i	$0.442899\pi$
$150$		0			0
$151$	−1.40823e8	+	2.43912e8i	−0.270873	+	0.469165i	−0.969086	−	0.246725i	$-0.920646\pi$
$151$	−1.40823e8	+	2.43912e8i	−0.270873	+	0.469165i	0.698213	+	0.715890i	$0.253979\pi$
$152$		0			0
$153$	−4.50188e8	+	7.48338e8i	−0.821539	+	1.36563i
$154$		0			0
$155$	6.39212e7	+	3.69049e7i	0.110744	+	0.0639378i
$156$		0			0
$157$	3.38942e8	+	5.87064e8i	0.557861	+	0.966244i	0.997675	+	0.0681551i	$0.0217113\pi$
$157$	3.38942e8	+	5.87064e8i	0.557861	+	0.966244i	−0.439813	+	0.898089i	$0.644955\pi$
$158$		0			0
$159$	−8.23663e8	−	2.28658e8i	−1.28873	−	0.357765i
$160$		0			0
$161$		−	1.49065e7i		−	0.0221856i
$162$		0			0
$163$	−7.48701e8			−1.06061			−0.530307	−	0.847805i	$-0.677924\pi$
$163$	−7.48701e8			−1.06061			−0.530307	+	0.847805i	$0.677924\pi$
$164$		0			0
$165$	−1.27302e8	+	4.58563e8i	−0.171751	+	0.618676i
$166$		0			0
$167$	2.31847e8	−	1.33857e8i	0.298082	−	0.172098i	−0.343499	−	0.939153i	$-0.611612\pi$
$167$	2.31847e8	−	1.33857e8i	0.298082	−	0.172098i	0.641581	+	0.767055i	$0.278279\pi$
$168$		0			0
$169$	3.89850e8	−	6.75241e8i	0.477916	−	0.827774i
$170$		0			0
$171$	4.60455e8	+	2.77003e8i	0.538522	+	0.323966i
$172$		0			0
$173$	8.70006e8	+	5.02298e8i	0.971265	+	0.560760i	0.899622	−	0.436670i	$-0.143842\pi$
$173$	8.70006e8	+	5.02298e8i	0.971265	+	0.560760i	0.0716434	+	0.997430i	$0.477176\pi$
$174$		0			0
$175$	1.70224e7	+	2.94837e7i	0.0181497	+	0.0314362i
$176$		0			0
$177$	−8.03470e7	−	3.11020e8i	−0.0818609	−	0.316880i
$178$		0			0
$179$		−	2.57600e8i		−	0.250919i	−0.992099	−	0.125460i	$-0.959959\pi$
$179$		−	2.57600e8i		−	0.250919i	0.992099	−	0.125460i	$-0.0400406\pi$
$180$		0			0
$181$	−1.81792e9			−1.69380			−0.846899	−	0.531754i	$-0.821533\pi$
$181$	−1.81792e9			−1.69380			−0.846899	+	0.531754i	$0.821533\pi$
$182$		0			0
$183$	1.02075e8	+	1.03928e8i	0.0910159	+	0.0926676i
$184$		0			0
$185$	4.13711e6	−	2.38856e6i	0.00353191	−	0.00203915i
$186$		0			0
$187$	6.58038e8	−	1.13976e9i	0.538127	−	0.932063i
$188$		0			0
$189$	−1.12212e8	−	4.69043e8i	−0.0879412	−	0.367591i
$190$		0			0
$191$	−1.08191e9	−	6.24644e8i	−0.812942	−	0.469352i	0.0350344	−	0.999386i	$-0.488846\pi$
$191$	−1.08191e9	−	6.24644e8i	−0.812942	−	0.469352i	−0.847977	+	0.530034i	$0.822179\pi$
$192$		0			0
$193$	−4.48788e8	−	7.77324e8i	−0.323454	−	0.560238i	0.657745	−	0.753241i	$-0.271511\pi$
$193$	−4.48788e8	−	7.77324e8i	−0.323454	−	0.560238i	−0.981198	+	0.193003i	$0.938177\pi$
$194$		0			0
$195$	2.06123e8	−	2.02449e8i	0.142557	−	0.140016i
$196$		0			0
$197$			6.59192e8i			0.437670i	0.975762	+	0.218835i	$0.0702257\pi$
$197$			6.59192e8i			0.437670i	−0.975762	+	0.218835i	$0.929774\pi$
$198$		0			0
$199$	−1.65053e9			−1.05248			−0.526238	−	0.850338i	$-0.676398\pi$
$199$	−1.65053e9			−1.05248			−0.526238	+	0.850338i	$0.676398\pi$
$200$		0			0
$201$	−2.23953e9	+	5.78548e8i	−1.37206	+	0.354450i
$202$		0			0
$203$	−6.72664e8	+	3.88363e8i	−0.396108	+	0.228693i
$204$		0			0
$205$	−1.16858e9	+	2.02403e9i	−0.661670	+	1.14605i
$206$		0			0
$207$	−1.07754e8	−	1.93810e6i	−0.0586881	−	0.00105559i
$208$		0			0
$209$	−7.01297e8	−	4.04894e8i	−0.367550	−	0.212205i
$210$		0			0
$211$	−7.08803e8	−	1.22768e9i	−0.357599	−	0.619379i	0.629960	−	0.776627i	$-0.283071\pi$
$211$	−7.08803e8	−	1.22768e9i	−0.357599	−	0.619379i	−0.987559	+	0.157248i	$0.949738\pi$
$212$		0			0
$213$	−1.87508e9	−	5.20542e8i	−0.910963	−	0.252893i
$214$		0			0
$215$			2.53942e9i			1.18845i
$216$		0			0
$217$	1.12720e8			0.0508351
$218$		0			0
$219$	1.83212e8	−	6.59960e8i	0.0796485	−	0.286907i
$220$		0			0
$221$	−6.91930e8	+	3.99486e8i	−0.290063	+	0.167468i
$222$		0			0
$223$	1.90981e9	−	3.30788e9i	0.772271	−	1.33761i	−0.164044	−	0.986453i	$-0.552454\pi$
$223$	1.90981e9	−	3.30788e9i	0.772271	−	1.33761i	0.936315	−	0.351160i	$-0.114213\pi$
$224$		0			0
$225$	2.15340e8	−	1.19216e8i	0.0840224	−	0.0465160i
$226$		0			0
$227$	2.38992e9	+	1.37982e9i	0.900079	+	0.519661i	0.877226	−	0.480078i	$-0.159392\pi$
$227$	2.38992e9	+	1.37982e9i	0.900079	+	0.519661i	0.0228534	+	0.999739i	$0.492725\pi$
$228$		0			0
$229$	−1.98580e9	−	3.43950e9i	−0.722093	−	1.25070i	−0.960159	−	0.279453i	$-0.909847\pi$
$229$	−1.98580e9	−	3.43950e9i	−0.722093	−	1.25070i	0.238066	−	0.971249i	$-0.423486\pi$
$230$		0			0
$231$	1.81787e8	+	7.03688e8i	0.0638431	+	0.247134i
$232$		0			0
$233$		−	2.11890e8i		−	0.0718929i	−0.999354	−	0.0359465i	$-0.988555\pi$
$233$		−	2.11890e8i		−	0.0718929i	0.999354	−	0.0359465i	$-0.0114446\pi$
$234$		0			0
$235$	8.51112e8			0.279071
$236$		0			0
$237$	−2.30011e9	−	2.34186e9i	−0.729048	−	0.742279i
$238$		0			0
$239$	3.97416e9	−	2.29448e9i	1.21802	−	0.703224i	0.253525	−	0.967329i	$-0.418410\pi$
$239$	3.97416e9	−	2.29448e9i	1.21802	−	0.703224i	0.964494	+	0.264105i	$0.0850765\pi$
$240$		0			0
$241$	1.02686e9	−	1.77857e9i	0.304398	−	0.527233i	−0.672729	−	0.739889i	$-0.734878\pi$
$241$	1.02686e9	−	1.77857e9i	0.304398	−	0.527233i	0.977127	+	0.212656i	$0.0682113\pi$
$242$		0			0
$243$	−3.40513e9	+	7.50157e8i	−0.976583	+	0.215143i
$244$		0			0
$245$	−4.23811e8	−	2.44687e8i	−0.117627	−	0.0679120i
$246$		0			0
$247$	2.45805e8	+	4.25747e8i	0.0660395	+	0.114384i
$248$		0			0
$249$	4.55177e9	−	4.47064e9i	1.18409	−	1.16298i
$250$		0			0
$251$			1.03512e9i			0.260794i	0.991462	+	0.130397i	$0.0416252\pi$
$251$			1.03512e9i			0.260794i	−0.991462	+	0.130397i	$0.958375\pi$
$252$		0			0
$253$	1.62410e8			0.0396397
$254$		0			0
$255$	6.20314e9	−	1.60248e9i	1.46707	−	0.378995i
$256$		0			0
$257$	−1.09871e9	+	6.34343e8i	−0.251856	+	0.145409i	−0.620614	−	0.784116i	$-0.713117\pi$
$257$	−1.09871e9	+	6.34343e8i	−0.251856	+	0.145409i	0.368758	+	0.929526i	$0.379783\pi$
$258$		0			0
$259$	3.64774e6	−	6.31808e6i	0.000810635	−	0.00140406i
$260$		0			0
$261$	2.71988e9	+	4.91294e9i	0.586121	+	1.05872i
$262$		0			0
$263$	−3.04740e9	−	1.75942e9i	−0.636951	−	0.367744i	0.146488	−	0.989212i	$-0.453203\pi$
$263$	−3.04740e9	−	1.75942e9i	−0.636951	−	0.367744i	−0.783439	+	0.621468i	$0.786536\pi$
$264$		0			0
$265$	3.13553e9	+	5.43090e9i	0.635810	+	1.10126i
$266$		0			0
$267$	6.97740e9	+	1.93700e9i	1.37293	+	0.381140i
$268$		0			0
$269$			3.44608e8i			0.0658137i	0.999458	+	0.0329069i	$0.0104765\pi$
$269$			3.44608e8i			0.0658137i	−0.999458	+	0.0329069i	$0.989524\pi$
$270$		0			0
$271$	−2.01501e9			−0.373595			−0.186797	−	0.982398i	$-0.559811\pi$
$271$	−2.01501e9			−0.373595			−0.186797	+	0.982398i	$0.559811\pi$
$272$		0			0
$273$	1.18025e8	−	4.25145e8i	0.0212482	−	0.0765397i
$274$		0			0
$275$	−3.21232e8	+	1.85463e8i	−0.0561679	+	0.0324286i
$276$		0			0
$277$	−1.96540e9	+	3.40418e9i	−0.333835	+	0.578220i	−0.983260	−	0.182206i	$-0.941676\pi$
$277$	−1.96540e9	+	3.40418e9i	−0.333835	+	0.578220i	0.649425	+	0.760426i	$0.275010\pi$
$278$		0			0
$279$	1.46556e7	−	8.14816e8i	0.00241873	−	0.134475i
$280$		0			0
$281$	4.18693e9	+	2.41733e9i	0.671538	+	0.387713i	0.796659	−	0.604429i	$-0.206599\pi$
$281$	4.18693e9	+	2.41733e9i	0.671538	+	0.387713i	−0.125121	+	0.992141i	$0.539932\pi$
$282$		0			0
$283$	4.02858e9	+	6.97771e9i	0.628068	+	1.08785i	0.987939	+	0.154844i	$0.0494874\pi$
$283$	4.02858e9	+	6.97771e9i	0.628068	+	1.08785i	−0.359871	+	0.933002i	$0.617179\pi$
$284$		0			0
$285$	−9.86016e8	−	3.81682e9i	−0.149453	−	0.578525i
$286$		0			0
$287$			3.56924e9i			0.526075i
$288$		0			0
$289$	−1.07417e10			−1.53986
$290$		0			0
$291$	−7.49291e9	−	7.62889e9i	−1.04491	−	1.06387i
$292$		0			0
$293$	2.02174e9	−	1.16725e9i	0.274319	−	0.158378i	−0.356530	−	0.934284i	$-0.616040\pi$
$293$	2.02174e9	−	1.16725e9i	0.274319	−	0.158378i	0.630849	+	0.775906i	$0.282707\pi$
$294$		0			0
$295$	−1.17830e9	+	2.04088e9i	−0.155585	+	0.269481i
$296$		0			0
$297$	5.11034e9	−	1.22258e9i	0.656786	−	0.157127i
$298$		0			0
$299$	−8.53873e7	−	4.92984e7i	−0.0106834	−	0.00616805i
$300$		0			0
$301$	1.93907e9	+	3.35857e9i	0.236226	+	0.409155i
$302$		0			0
$303$	1.09271e9	−	1.07324e9i	0.129639	−	0.127328i
$304$		0			0
$305$		−	1.06868e9i		−	0.123494i
$306$		0			0
$307$	−8.03489e8			−0.0904537			−0.0452269	−	0.998977i	$-0.514401\pi$
$307$	−8.03489e8			−0.0904537			−0.0452269	+	0.998977i	$0.514401\pi$
$308$		0			0
$309$	9.03185e9	−	2.33324e9i	0.990701	−	0.255932i
$310$		0			0
$311$	1.68181e9	−	9.70992e8i	0.179777	−	0.103794i	−0.407411	−	0.913245i	$-0.633568\pi$
$311$	1.68181e9	−	9.70992e8i	0.179777	−	0.103794i	0.587188	+	0.809451i	$0.300235\pi$
$312$		0			0
$313$	4.73088e8	−	8.19412e8i	0.0492906	−	0.0853739i	−0.840327	−	0.542079i	$-0.817637\pi$
$313$	4.73088e8	−	8.19412e8i	0.0492906	−	0.0853739i	0.889618	+	0.456705i	$0.150971\pi$
$314$		0			0
$315$	−1.82386e9	+	3.03176e9i	−0.185246	+	0.307930i
$316$		0			0
$317$	9.96466e8	+	5.75310e8i	0.0986792	+	0.0569725i	0.548527	−	0.836133i	$-0.315189\pi$
$317$	9.96466e8	+	5.75310e8i	0.0986792	+	0.0569725i	−0.449848	+	0.893105i	$0.648522\pi$
$318$		0			0
$319$	−4.23131e9	−	7.32884e9i	−0.408613	−	0.707738i
$320$		0			0
$321$	1.68119e10	+	4.66716e9i	1.58342	+	0.439574i
$322$		0			0
$323$			1.09016e10i			1.00157i
$324$		0			0
$325$	2.25185e8			0.0201839
$326$		0			0
$327$	1.42258e9	−	5.12437e9i	0.124419	−	0.448177i
$328$		0			0
$329$	1.12566e9	−	6.49898e8i	0.0960776	−	0.0554704i
$330$		0			0
$331$	2.16938e9	−	3.75748e9i	0.180728	−	0.313029i	−0.761401	−	0.648281i	$-0.775488\pi$
$331$	2.16938e9	−	3.75748e9i	0.180728	−	0.313029i	0.942129	+	0.335252i	$0.108821\pi$
$332$		0			0
$333$	−4.51968e7	−	2.71897e7i	−0.00367563	−	0.00221120i
$334$		0			0
$335$	1.46956e10	+	8.48450e9i	1.16683	+	0.673670i
$336$		0			0
$337$	6.96899e9	+	1.20707e10i	0.540319	+	0.935861i	0.998885	+	0.0472003i	$0.0150299\pi$
$337$	6.96899e9	+	1.20707e10i	0.540319	+	0.935861i	−0.458566	+	0.888660i	$0.651637\pi$
$338$		0			0
$339$	2.30064e8	+	8.90566e8i	0.0174200	+	0.0674322i
$340$		0			0
$341$			1.22812e9i			0.0908286i
$342$		0			0
$343$	−7.47359e8			−0.0539949
$344$		0			0
$345$	5.54008e8	+	5.64062e8i	0.0391057	+	0.0398154i
$346$		0			0
$347$	−7.42767e9	+	4.28837e9i	−0.512312	+	0.295783i	−0.733784	−	0.679383i	$-0.762247\pi$
$347$	−7.42767e9	+	4.28837e9i	−0.512312	+	0.295783i	0.221471	+	0.975167i	$0.428914\pi$
$348$		0			0
$349$	−1.14890e10	+	1.98995e10i	−0.774427	+	1.34135i	0.160690	+	0.987005i	$0.448628\pi$
$349$	−1.14890e10	+	1.98995e10i	−0.774427	+	1.34135i	−0.935116	+	0.354341i	$0.884705\pi$
$350$		0			0
$351$	−3.05788e9	−	9.08436e8i	−0.201461	−	0.0598502i
$352$		0			0
$353$	2.15762e10	+	1.24570e10i	1.38956	+	0.802261i	0.993265	−	0.115863i	$-0.0369634\pi$
$353$	2.15762e10	+	1.24570e10i	1.38956	+	0.802261i	0.396292	+	0.918124i	$0.370297\pi$
$354$		0			0
$355$	7.13807e9	+	1.23635e10i	0.449435	+	0.778445i
$356$		0			0
$357$	6.98046e9	−	6.85604e9i	0.429745	−	0.422085i
$358$		0			0
$359$		−	3.14990e9i		−	0.189635i	−0.995495	−	0.0948177i	$-0.969773\pi$
$359$		−	3.14990e9i		−	0.189635i	0.995495	−	0.0948177i	$-0.0302268\pi$
$360$		0			0
$361$	−1.02757e10			−0.605041
$362$		0			0
$363$	9.14431e9	−	2.36229e9i	0.526653	−	0.136053i
$364$		0			0
$365$	−4.35151e9	+	2.51234e9i	−0.245171	+	0.141549i
$366$		0			0
$367$	1.16512e10	−	2.01805e10i	0.642254	−	1.11242i	−0.342675	−	0.939454i	$-0.611333\pi$
$367$	1.16512e10	−	2.01805e10i	0.642254	−	1.11242i	0.984929	−	0.172962i	$-0.0553337\pi$
$368$		0			0
$369$	2.58007e10	+	4.64063e8i	1.39164	+	0.0250306i
$370$		0			0
$371$	8.29392e9	+	4.78850e9i	0.437788	+	0.252757i
$372$		0			0
$373$	−1.34073e9	−	2.32221e9i	−0.0692637	−	0.119968i	0.829314	−	0.558783i	$-0.188732\pi$
$373$	−1.34073e9	−	2.32221e9i	−0.0692637	−	0.119968i	−0.898577	+	0.438815i	$0.855398\pi$
$374$		0			0
$375$	−1.98566e10	−	5.51240e9i	−1.00411	−	0.278750i
$376$		0			0
$377$			5.13754e9i			0.254325i
$378$		0			0
$379$	1.37893e10			0.668319			0.334160	−	0.942517i	$-0.391548\pi$
$379$	1.37893e10			0.668319			0.334160	+	0.942517i	$0.391548\pi$
$380$		0			0
$381$	−7.28786e9	+	2.62521e10i	−0.345860	+	1.24585i
$382$		0			0
$383$	−6.54150e9	+	3.77674e9i	−0.304006	+	0.175518i	−0.644241	−	0.764822i	$-0.722827\pi$
$383$	−6.54150e9	+	3.77674e9i	−0.304006	+	0.175518i	0.340235	+	0.940340i	$0.389493\pi$
$384$		0			0
$385$	2.66593e9	−	4.61752e9i	0.121340	−	0.210168i
$386$		0			0
$387$	2.45300e10	−	1.35802e10i	1.09359	−	0.605426i
$388$		0			0
$389$	1.64952e10	+	9.52349e9i	0.720374	+	0.415908i	0.814890	−	0.579615i	$-0.196797\pi$
$389$	1.64952e10	+	9.52349e9i	0.720374	+	0.415908i	−0.0945160	+	0.995523i	$0.530130\pi$
$390$		0			0
$391$	−1.09321e9	−	1.89349e9i	−0.0467730	−	0.0810132i
$392$		0			0
$393$	−3.94505e9	−	1.52711e10i	−0.165380	−	0.640178i
$394$		0			0
$395$			2.40810e10i			0.989205i
$396$		0			0
$397$	−3.87314e10			−1.55920			−0.779600	−	0.626278i	$-0.784577\pi$
$397$	−3.87314e10			−1.55920			−0.779600	+	0.626278i	$0.784577\pi$
$398$		0			0
$399$	−4.21855e9	−	4.29511e9i	−0.166445	−	0.169466i
$400$		0			0
$401$	2.26577e10	−	1.30814e10i	0.876272	−	0.505916i	0.00684468	−	0.999977i	$-0.497821\pi$
$401$	2.26577e10	−	1.30814e10i	0.876272	−	0.505916i	0.869427	+	0.494061i	$0.164488\pi$
$402$		0			0
$403$	3.72787e8	−	6.45686e8i	0.0141332	−	0.0244794i
$404$		0			0
$405$	2.16784e10	+	1.35782e10i	0.805762	+	0.504687i
$406$		0			0
$407$	6.88370e7	+	3.97431e7i	0.00250868	+	0.00144838i
$408$		0			0
$409$	−1.45021e10	−	2.51183e10i	−0.518247	−	0.897630i	−0.999775	−	0.0211992i	$-0.993252\pi$
$409$	−1.45021e10	−	2.51183e10i	−0.518247	−	0.897630i	0.481529	−	0.876430i	$-0.340082\pi$
$410$		0			0
$411$	3.20193e10	−	3.14486e10i	1.12213	−	1.10213i
$412$		0			0
$413$			3.59894e9i			0.123701i
$414$		0			0
$415$	−4.68053e10			−1.57798
$416$		0			0
$417$	2.24561e9	−	5.80118e8i	0.0742660	−	0.0191855i
$418$		0			0
$419$	3.10856e9	−	1.79473e9i	0.100856	−	0.0582293i	−0.448724	−	0.893671i	$-0.648121\pi$
$419$	3.10856e9	−	1.79473e9i	0.100856	−	0.0582293i	0.549580	+	0.835441i	$0.314788\pi$
$420$		0			0
$421$	−1.83233e10	+	3.17369e10i	−0.583278	+	1.01027i	0.411809	+	0.911270i	$0.364897\pi$
$421$	−1.83233e10	+	3.17369e10i	−0.583278	+	1.01027i	−0.995088	+	0.0989978i	$0.968436\pi$
$422$		0			0
$423$	−4.55152e9	−	8.22146e9i	−0.142166	−	0.256796i
$424$		0			0
$425$	4.32453e9	+	2.49677e9i	0.132551	+	0.0765284i
$426$		0			0
$427$	−8.16028e8	−	1.41340e9i	−0.0245467	−	0.0425162i
$428$		0			0
$429$	4.63207e9	+	1.28591e9i	0.136756	+	0.0379648i
$430$		0			0
$431$			4.18065e10i			1.21153i	0.795643	+	0.605766i	$0.207133\pi$
$431$			4.18065e10i			1.21153i	−0.795643	+	0.605766i	$0.792867\pi$
$432$		0			0
$433$	6.79636e9			0.193341			0.0966707	−	0.995316i	$-0.469181\pi$
$433$	6.79636e9			0.193341			0.0966707	+	0.995316i	$0.469181\pi$
$434$		0			0
$435$	1.10199e10	−	3.96956e10i	0.307767	−	1.10863i
$436$		0			0
$437$	−1.16507e9	+	6.72655e8i	−0.0319468	+	0.0184445i
$438$		0			0
$439$	1.40172e10	−	2.42786e10i	0.377402	−	0.653679i	−0.613281	−	0.789864i	$-0.710151\pi$
$439$	1.40172e10	−	2.42786e10i	0.377402	−	0.653679i	0.990683	+	0.136185i	$0.0434842\pi$
$440$		0			0
$441$	−9.71698e7	+	5.40239e9i	−0.00256907	+	0.142834i
$442$		0			0
$443$	1.65239e10	+	9.54008e9i	0.429040	+	0.247706i	0.698937	−	0.715183i	$-0.253657\pi$
$443$	1.65239e10	+	9.54008e9i	0.429040	+	0.247706i	−0.269898	+	0.962889i	$0.586990\pi$
$444$		0			0
$445$	−2.65616e10	−	4.60061e10i	−0.677353	−	1.17321i
$446$		0			0
$447$	−6.73950e9	−	2.60883e10i	−0.168810	−	0.653455i
$448$		0			0
$449$			1.30348e10i			0.320714i	0.987059	+	0.160357i	$0.0512645\pi$
$449$			1.30348e10i			0.320714i	−0.987059	+	0.160357i	$0.948735\pi$
$450$		0			0
$451$	−3.88877e10			−0.939954
$452$		0			0
$453$	−1.59857e10	−	1.62758e10i	−0.379612	−	0.386501i
$454$		0			0
$455$	−2.80323e9	+	1.61845e9i	−0.0654054	+	0.0377618i
$456$		0			0
$457$	2.05947e10	−	3.56711e10i	0.472162	−	0.817809i	−0.527330	−	0.849660i	$-0.676807\pi$
$457$	2.05947e10	−	3.56711e10i	0.472162	−	0.817809i	0.999493	+	0.0318513i	$0.0101403\pi$
$458$		0			0
$459$	−4.86522e10	−	5.13507e10i	−1.09611	−	1.15690i
$460$		0			0
$461$	2.06707e10	+	1.19342e10i	0.457668	+	0.264235i	0.711063	−	0.703128i	$-0.248214\pi$
$461$	2.06707e10	+	1.19342e10i	0.457668	+	0.264235i	−0.253395	+	0.967363i	$0.581547\pi$
$462$		0			0
$463$	−3.95079e10	−	6.84297e10i	−0.859726	−	1.48909i	−0.872191	−	0.489166i	$-0.837301\pi$
$463$	−3.95079e10	−	6.84297e10i	−0.859726	−	1.48909i	0.0124652	−	0.999922i	$-0.496032\pi$
$464$		0			0
$465$	−4.26535e9	+	4.18932e9i	−0.0912311	+	0.0896050i
$466$		0			0
$467$		−	5.37777e10i		−	1.13067i	−0.824863	−	0.565333i	$-0.808748\pi$
$467$		−	5.37777e10i		−	1.13067i	0.824863	−	0.565333i	$-0.191252\pi$
$468$		0			0
$469$	2.59146e10			0.535615
$470$		0			0
$471$	−5.31632e10	+	1.37339e10i	−1.08026	+	0.279068i
$472$		0			0
$473$	−3.65924e10	+	2.11266e10i	−0.731049	+	0.422071i
$474$		0			0
$475$	1.53627e9	−	2.66090e9i	0.0301783	−	0.0522703i
$476$		0			0
$477$	3.56927e10	−	5.93312e10i	0.689455	−	1.14607i
$478$		0			0
$479$	4.51023e10	+	2.60398e10i	0.856754	+	0.494647i	0.862924	−	0.505334i	$-0.168631\pi$
$479$	4.51023e10	+	2.60398e10i	0.856754	+	0.494647i	−0.00616981	+	0.999981i	$0.501964\pi$
$480$		0			0
$481$	−2.41275e7	−	4.17900e7i	−0.000450746	−	0.000780715i
$482$		0			0
$483$	1.16343e9	+	3.22979e8i	0.0213772	+	0.00593452i
$484$		0			0
$485$			7.84469e10i			1.41778i
$486$		0			0
$487$	−4.82299e10			−0.857433			−0.428716	−	0.903439i	$-0.641034\pi$
$487$	−4.82299e10			−0.857433			−0.428716	+	0.903439i	$0.641034\pi$
$488$		0			0
$489$	1.62221e10	−	5.84348e10i	0.283708	−	1.02197i
$490$		0			0
$491$	−5.12165e10	+	2.95698e10i	−0.881219	+	0.508772i	−0.871060	−	0.491177i	$-0.836567\pi$
$491$	−5.12165e10	+	2.95698e10i	−0.881219	+	0.508772i	−0.0101586	+	0.999948i	$0.503234\pi$
$492$		0			0
$493$	−5.69633e10	+	9.86633e10i	−0.964289	+	1.67020i
$494$		0			0
$495$	−3.30318e10	−	1.98714e10i	−0.550188	−	0.330984i
$496$		0			0
$497$	1.88812e10	+	1.09011e10i	0.309460	+	0.178667i
$498$		0			0
$499$	−2.22282e10	−	3.85003e10i	−0.358510	−	0.620957i	0.629202	−	0.777242i	$-0.283382\pi$
$499$	−2.22282e10	−	3.85003e10i	−0.358510	−	0.620957i	−0.987712	+	0.156284i	$0.950048\pi$
$500$		0			0
$501$	5.42387e9	+	2.09955e10i	0.0860910	+	0.333254i
$502$		0			0
$503$			8.14177e10i			1.27188i	0.771737	+	0.635941i	$0.219388\pi$
$503$			8.14177e10i			1.27188i	−0.771737	+	0.635941i	$0.780612\pi$
$504$		0			0
$505$	−1.12362e10			−0.172765
$506$		0			0
$507$	4.42545e10	+	4.50576e10i	0.669770	+	0.681925i
$508$		0			0
$509$	9.98458e9	−	5.76460e9i	0.148751	−	0.0858812i	−0.423777	−	0.905766i	$-0.639296\pi$
$509$	9.98458e9	−	5.76460e9i	0.148751	−	0.0858812i	0.572528	+	0.819885i	$0.305963\pi$
$510$		0			0
$511$	−3.83678e9	+	6.64551e9i	−0.0562709	+	0.0974640i
$512$		0			0
$513$	−3.15963e10	+	2.99359e10i	−0.456212	+	0.432239i
$514$		0			0
$515$	−5.92660e10	−	3.42172e10i	−0.842513	−	0.486425i
$516$		0			0
$517$	7.08080e9	+	1.22643e10i	0.0991106	+	0.171665i
$518$		0			0
$519$	−5.80540e10	+	5.70192e10i	−0.800134	+	0.785871i
$520$		0			0
$521$			2.72129e10i			0.369338i	0.982801	+	0.184669i	$0.0591213\pi$
$521$			2.72129e10i			0.369338i	−0.982801	+	0.184669i	$0.940879\pi$
$522$		0			0
$523$	−5.74199e10			−0.767460			−0.383730	−	0.923445i	$-0.625361\pi$
$523$	−5.74199e10			−0.767460			−0.383730	+	0.923445i	$0.625361\pi$
$524$		0			0
$525$	−2.66998e9	+	6.89746e8i	−0.0351455	+	0.00907930i
$526$		0			0
$527$	1.43183e10	−	8.26666e9i	0.185630	−	0.107174i
$528$		0			0
$529$	−3.90206e10	+	6.75856e10i	−0.498277	+	0.863042i
$530$		0			0
$531$	2.60154e10	+	4.67925e8i	0.327230	+	0.00588570i
$532$		0			0
$533$	2.04453e10	+	1.18041e10i	0.253329	+	0.146260i
$534$		0			0
$535$	−6.39996e10	−	1.10851e11i	−0.781200	−	1.35308i
$536$		0			0
$537$	2.01053e10	+	5.58143e9i	0.241776	+	0.0671195i
$538$		0			0
$539$		−	8.14267e9i		−	0.0964743i
$540$		0			0
$541$	−1.03093e11			−1.20348			−0.601742	−	0.798691i	$-0.705526\pi$
$541$	−1.03093e11			−1.20348			−0.601742	+	0.798691i	$0.705526\pi$
$542$		0			0
$543$	3.93890e10	−	1.41886e11i	0.453081	−	1.63207i
$544$		0			0
$545$	−3.37880e10	+	1.95075e10i	−0.382981	+	0.221114i
$546$		0			0
$547$	−8.21564e10	+	1.42299e11i	−0.917682	+	1.58947i	−0.114755	+	0.993394i	$0.536608\pi$
$547$	−8.21564e10	+	1.42299e11i	−0.917682	+	1.58947i	−0.802927	+	0.596077i	$0.796725\pi$
$548$		0			0
$549$	−1.03231e10	+	5.71501e9i	−0.113637	+	0.0629111i
$550$		0			0
$551$	6.07080e10	+	3.50498e10i	0.658627	+	0.380258i
$552$		0			0
$553$	1.83879e10	+	3.18488e10i	0.196622	+	0.340560i
$554$		0			0
$555$	9.67842e7	+	3.74647e8i	0.00102008	+	0.00394867i
$556$		0			0
$557$		−	8.00993e10i		−	0.832162i	−0.909328	−	0.416081i	$-0.863403\pi$
$557$		−	8.00993e10i		−	0.832162i	0.909328	−	0.416081i	$-0.136597\pi$
$558$		0			0
$559$	2.56514e10			0.262702
$560$		0			0
$561$	7.46982e10	+	7.60539e10i	0.754152	+	0.767839i
$562$		0			0
$563$	1.16237e11	−	6.71097e10i	1.15694	−	0.667962i	0.206375	−	0.978473i	$-0.433833\pi$
$563$	1.16237e11	−	6.71097e10i	1.15694	−	0.667962i	0.950570	+	0.310511i	$0.100500\pi$
$564$		0			0
$565$	3.37392e9	−	5.84379e9i	0.0331086	−	0.0573457i
$566$		0			0
$567$	3.90393e10	+	1.40481e9i	0.377720	+	0.0135921i
$568$		0			0
$569$	2.31448e10	+	1.33626e10i	0.220802	+	0.127480i	0.606322	−	0.795219i	$-0.292644\pi$
$569$	2.31448e10	+	1.33626e10i	0.220802	+	0.127480i	−0.385519	+	0.922700i	$0.625978\pi$
$570$		0			0
$571$	−1.28139e10	−	2.21943e10i	−0.120541	−	0.208784i	0.799440	−	0.600746i	$-0.205130\pi$
$571$	−1.28139e10	−	2.21943e10i	−0.120541	−	0.208784i	−0.919981	+	0.391962i	$0.871796\pi$
$572$		0			0
$573$	7.21943e10	−	7.09075e10i	0.669706	−	0.657769i
$574$		0			0
$575$			6.16226e8i			0.00563726i
$576$		0			0
$577$	−1.20916e11			−1.09089			−0.545443	−	0.838148i	$-0.683639\pi$
$577$	−1.20916e11			−1.09089			−0.545443	+	0.838148i	$0.683639\pi$
$578$		0			0
$579$	7.03927e10	−	1.81848e10i	0.626345	−	0.161806i
$580$		0			0
$581$	−6.19033e10	+	3.57399e10i	−0.543262	+	0.313653i
$582$		0			0
$583$	−5.21719e10	+	9.03644e10i	−0.451609	+	0.782209i
$584$		0			0
$585$	1.13347e10	+	2.04740e10i	0.0967803	+	0.174815i
$586$		0			0
$587$	−2.85495e10	−	1.64830e10i	−0.240461	−	0.138830i	0.374927	−	0.927054i	$-0.377668\pi$
$587$	−2.85495e10	−	1.64830e10i	−0.240461	−	0.138830i	−0.615389	+	0.788224i	$0.711001\pi$
$588$		0			0
$589$	−5.08651e9	−	8.81010e9i	−0.0422629	−	0.0732015i
$590$		0			0
$591$	−5.14489e10	−	1.42827e10i	−0.421722	−	0.117074i
$592$		0			0
$593$			1.39596e11i			1.12890i	0.825467	+	0.564450i	$0.190912\pi$
$593$			1.39596e11i			1.12890i	−0.825467	+	0.564450i	$0.809088\pi$
$594$		0			0
$595$	−7.17792e10			−0.572705
$596$		0			0
$597$	3.57622e10	−	1.28821e11i	0.281531	−	1.01412i
$598$		0			0
$599$	−2.09193e11	+	1.20778e11i	−1.62495	+	0.938164i	−0.639378	+	0.768892i	$0.720808\pi$
$599$	−2.09193e11	+	1.20778e11i	−1.62495	+	0.938164i	−0.985569	+	0.169272i	$0.945858\pi$
$600$		0			0
$601$	6.02379e10	−	1.04335e11i	0.461713	−	0.799710i	−0.537334	−	0.843370i	$-0.680568\pi$
$601$	6.02379e10	−	1.04335e11i	0.461713	−	0.799710i	0.999046	+	0.0436600i	$0.0139018\pi$
$602$		0			0
$603$	3.36935e9	−	1.87327e11i	0.0254845	−	1.41688i
$604$		0			0
$605$	−6.00040e10	−	3.46433e10i	−0.447877	−	0.258582i
$606$		0			0
$607$	−6.17480e10	−	1.06951e11i	−0.454850	−	0.787823i	0.543830	−	0.839196i	$-0.316974\pi$
$607$	−6.17480e10	−	1.06951e11i	−0.454850	−	0.787823i	−0.998680	+	0.0513726i	$0.983640\pi$
$608$		0			0
$609$	−1.57364e10	−	6.09150e10i	−0.114403	−	0.442848i
$610$		0			0
$611$		−	8.59731e9i		−	0.0616876i
$612$		0			0
$613$	7.00170e9			0.0495863			0.0247931	−	0.999693i	$-0.492107\pi$
$613$	7.00170e9			0.0495863			0.0247931	+	0.999693i	$0.492107\pi$
$614$		0			0
$615$	−1.32653e11	−	1.35060e11i	−0.927291	−	0.944119i
$616$		0			0
$617$	−6.43040e10	+	3.71259e10i	−0.443708	+	0.256175i	−0.705169	−	0.709039i	$-0.749129\pi$
$617$	−6.43040e10	+	3.71259e10i	−0.443708	+	0.256175i	0.261461	+	0.965214i	$0.415796\pi$
$618$		0			0
$619$	−6.89891e10	+	1.19493e11i	−0.469914	+	0.813914i	−0.999408	−	0.0343990i	$-0.989048\pi$
$619$	−6.89891e10	+	1.19493e11i	−0.469914	+	0.813914i	0.529495	+	0.848313i	$0.322382\pi$
$620$		0			0
$621$	2.48597e9	−	8.36799e9i	0.0167159	−	0.0562671i
$622$		0			0
$623$	−7.02593e10	−	4.05642e10i	−0.466393	−	0.269272i
$624$		0			0
$625$	6.82630e10	+	1.18235e11i	0.447369	+	0.774865i
$626$		0			0
$627$	4.67963e10	−	4.59622e10i	0.302790	−	0.297393i
$628$		0			0
$629$		−	1.07007e9i		−	0.00683611i
$630$		0			0
$631$	2.55167e11			1.60956			0.804781	−	0.593571i	$-0.202283\pi$
$631$	2.55167e11			1.60956			0.804781	+	0.593571i	$0.202283\pi$
$632$		0			0
$633$	1.11176e11	−	2.87206e10i	0.692464	−	0.178887i
$634$		0			0
$635$	1.73096e11	−	9.99368e10i	1.06461	−	0.614654i
$636$		0			0
$637$	−2.47165e9	+	4.28102e9i	−0.0150117	+	0.0260010i
$638$		0			0
$639$	8.12548e10	−	1.35068e11i	0.487355	−	0.810120i
$640$		0			0
$641$	1.50962e11	+	8.71577e10i	0.894199	+	0.516266i	0.875314	−	0.483556i	$-0.160655\pi$
$641$	1.50962e11	+	8.71577e10i	0.894199	+	0.516266i	0.0188852	+	0.999822i	$0.493988\pi$
$642$		0			0
$643$	−3.64222e10	−	6.30852e10i	−0.213070	−	0.369048i	0.739604	−	0.673043i	$-0.235013\pi$
$643$	−3.64222e10	−	6.30852e10i	−0.213070	−	0.369048i	−0.952674	+	0.303994i	$0.901680\pi$
$644$		0			0
$645$	−1.98198e11	−	5.50218e10i	−1.14514	−	0.317904i
$646$		0			0
$647$		−	1.94102e11i		−	1.10767i	−0.832625	−	0.553837i	$-0.813163\pi$
$647$		−	1.94102e11i		−	1.10767i	0.832625	−	0.553837i	$-0.186837\pi$
$648$		0			0
$649$	−3.92113e10			−0.221021
$650$		0			0
$651$	−2.44232e9	+	8.79764e9i	−0.0135981	+	0.0489827i
$652$		0			0
$653$	−1.05355e11	+	6.08269e10i	−0.579433	+	0.334536i	−0.760908	−	0.648860i	$-0.775246\pi$
$653$	−1.05355e11	+	6.08269e10i	−0.579433	+	0.334536i	0.181475	+	0.983396i	$0.441913\pi$
$654$		0			0
$655$	−5.78548e10	+	1.00207e11i	−0.314321	+	0.544421i
$656$		0			0
$657$	4.75391e10	+	2.85988e10i	0.255147	+	0.153492i
$658$		0			0
$659$	−6.68307e10	−	3.85847e10i	−0.354351	−	0.204585i	0.312249	−	0.950000i	$-0.398918\pi$
$659$	−6.68307e10	−	3.85847e10i	−0.354351	−	0.204585i	−0.666600	+	0.745416i	$0.732251\pi$
$660$		0			0
$661$	−1.58533e11	−	2.74588e11i	−0.830452	−	1.43839i	−0.897680	−	0.440648i	$-0.854749\pi$
$661$	−1.58533e11	−	2.74588e11i	−0.830452	−	1.43839i	0.0672279	−	0.997738i	$-0.478585\pi$
$662$		0			0
$663$	−1.61871e10	−	6.26596e10i	−0.0837752	−	0.324290i
$664$		0			0
$665$			4.41661e10i			0.225841i
$666$		0			0
$667$	−1.40591e10			−0.0710318
$668$		0			0
$669$	2.16795e11	+	2.20729e11i	1.08229	+	1.10193i
$670$		0			0
$671$	1.53994e10	−	8.89083e9i	0.0759649	−	0.0438584i
$672$		0			0
$673$	1.55624e11	−	2.69550e11i	0.758608	−	1.31395i	−0.184952	−	0.982748i	$-0.559213\pi$
$673$	1.55624e11	−	2.69550e11i	0.758608	−	1.31395i	0.943560	−	0.331201i	$-0.107454\pi$
$674$		0			0
$675$	4.63879e9	+	1.93900e10i	0.0223455	+	0.0934033i
$676$		0			0
$677$	3.12593e11	+	1.80475e11i	1.48807	+	0.859139i	0.999907	−	0.0136123i	$-0.00433308\pi$
$677$	3.12593e11	+	1.80475e11i	1.48807	+	0.859139i	0.488165	+	0.872751i	$0.337666\pi$
$678$		0			0
$679$	5.99010e10	+	1.03752e11i	0.281809	+	0.488108i
$680$		0			0
$681$	−1.59476e11	+	1.56633e11i	−0.741490	+	0.728274i
$682$		0			0
$683$			1.05524e11i			0.484917i	0.970162	+	0.242458i	$0.0779538\pi$
$683$			1.05524e11i			0.484917i	−0.970162	+	0.242458i	$0.922046\pi$
$684$		0			0
$685$	−3.29251e11			−1.49542
$686$		0			0
$687$	3.11474e11	−	8.04643e10i	1.39828	−	0.361224i
$688$		0			0
$689$	5.48590e10	−	3.16728e10i	0.243428	−	0.140543i
$690$		0			0
$691$	1.38096e11	−	2.39189e11i	0.605716	−	1.04913i	−0.386222	−	0.922406i	$-0.626220\pi$
$691$	1.38096e11	−	2.39189e11i	0.605716	−	1.04913i	0.991938	−	0.126725i	$-0.0404466\pi$
$692$		0			0
$693$	−5.88604e10	−	1.05869e9i	−0.255206	−	0.00459024i
$694$		0			0
$695$	−1.47354e10	−	8.50752e9i	−0.0631574	−	0.0364639i
$696$		0			0
$697$	2.61760e11	+	4.53381e11i	1.10910	+	1.92102i
$698$		0			0
$699$	1.65376e10	+	4.59102e9i	0.0692731	+	0.0192309i
$700$		0			0
$701$			3.48466e11i			1.44307i	0.692377	+	0.721536i	$0.256564\pi$
$701$			3.48466e11i			1.44307i	−0.692377	+	0.721536i	$0.743436\pi$
$702$		0			0
$703$	−6.58418e8			−0.00269576
$704$		0			0
$705$	−1.84411e10	+	6.64279e10i	−0.0746500	+	0.268902i
$706$		0			0
$707$	−1.48607e10	+	8.57984e9i	−0.0594788	+	0.0343401i
$708$		0			0
$709$	−1.46635e10	+	2.53980e10i	−0.0580301	+	0.100511i	−0.893581	−	0.448902i	$-0.851815\pi$
$709$	−1.46635e10	+	2.53980e10i	−0.0580301	+	0.100511i	0.835551	+	0.549413i	$0.185149\pi$
$710$		0			0
$711$	2.32615e11	−	1.28779e11i	0.910246	−	0.503925i
$712$		0			0
$713$	1.76694e9	+	1.02014e9i	0.00683697	+	0.00394733i
$714$		0			0
$715$	−1.76334e10	−	3.05419e10i	−0.0674702	−	0.116862i
$716$		0			0
$717$	9.29723e10	+	3.59891e11i	0.351785	+	1.36174i
$718$		0			0
$719$		−	3.64694e11i		−	1.36463i	−0.731060	−	0.682313i	$-0.760974\pi$
$719$		−	3.64694e11i		−	1.36463i	0.731060	−	0.682313i	$-0.239026\pi$
$720$		0			0
$721$	−1.04511e11			−0.386743
$722$		0			0
$723$	1.16565e11	+	1.18681e11i	0.426596	+	0.434338i
$724$		0			0
$725$	2.78076e10	−	1.60547e10i	0.100649	−	0.0581099i
$726$		0			0
$727$	−3.58925e9	+	6.21677e9i	−0.0128489	+	0.0222550i	−0.872378	−	0.488831i	$-0.837423\pi$
$727$	−3.58925e9	+	6.21677e9i	−0.0128489	+	0.0222550i	0.859529	+	0.511086i	$0.170757\pi$
$728$		0			0
$729$	1.52307e10	−	2.82019e11i	0.0539273	−	0.998545i
$730$		0			0
$731$	4.92619e11	+	2.84414e11i	1.72521	+	0.996050i
$732$		0			0
$733$	2.92996e10	+	5.07484e10i	0.101495	+	0.175795i	0.912301	−	0.409521i	$-0.134304\pi$
$733$	2.92996e10	+	5.07484e10i	0.101495	+	0.175795i	−0.810806	+	0.585315i	$0.800971\pi$
$734$		0			0
$735$	2.82801e10	−	2.77761e10i	0.0969019	−	0.0951746i
$736$		0			0
$737$			2.82346e11i			0.957000i
$738$		0			0
$739$	3.16452e11			1.06104			0.530518	−	0.847673i	$-0.321997\pi$
$739$	3.16452e11			1.06104			0.530518	+	0.847673i	$0.321997\pi$
$740$		0			0
$741$	−3.85547e10	+	9.96001e9i	−0.127881	+	0.0330360i
$742$		0			0
$743$	3.15082e11	−	1.81913e11i	1.03388	−	0.596909i	0.115784	−	0.993274i	$-0.463062\pi$
$743$	3.15082e11	−	1.81913e11i	1.03388	−	0.596909i	0.918093	+	0.396365i	$0.129729\pi$
$744$		0			0
$745$	−9.88358e10	+	1.71189e11i	−0.320840	+	0.555712i
$746$		0			0
$747$	2.50303e11	+	4.52124e11i	0.803864	+	1.45203i
$748$		0			0
$749$	−1.69288e11	−	9.77385e10i	−0.537897	−	0.310555i
$750$		0			0
$751$	−1.91338e11	−	3.31407e11i	−0.601508	−	1.04184i	−0.992593	−	0.121488i	$-0.961233\pi$
$751$	−1.91338e11	−	3.31407e11i	−0.601508	−	1.04184i	0.391084	−	0.920355i	$-0.372100\pi$
$752$		0			0
$753$	−8.07897e10	−	2.24281e10i	−0.251291	−	0.0697609i
$754$		0			0
$755$			1.67362e11i			0.515074i
$756$		0			0
$757$	3.70198e11			1.12733			0.563665	−	0.826004i	$-0.309391\pi$
$757$	3.70198e11			1.12733			0.563665	+	0.826004i	$0.309391\pi$
$758$		0			0
$759$	−3.51894e9	+	1.26758e10i	−0.0106034	+	0.0381952i
$760$		0			0
$761$	8.93743e10	−	5.16003e10i	0.266486	−	0.153856i	−0.360804	−	0.932642i	$-0.617498\pi$
$761$	8.93743e10	−	5.16003e10i	0.266486	−	0.153856i	0.627290	+	0.778786i	$0.284164\pi$
$762$		0			0
$763$	−2.97914e10	+	5.16002e10i	−0.0879007	+	0.152248i
$764$		0			0
$765$	−9.33255e9	+	5.18866e11i	−0.0272493	+	1.51499i
$766$		0			0
$767$	2.06154e10	+	1.19023e10i	0.0595677	+	0.0343915i
$768$		0			0
$769$	1.38983e11	+	2.40726e11i	0.397427	+	0.688364i	0.993408	−	0.114635i	$-0.0365697\pi$
$769$	1.38983e11	+	2.40726e11i	0.397427	+	0.688364i	−0.595980	+	0.802999i	$0.703236\pi$
$770$		0			0
$771$	−2.57035e10	−	9.94972e10i	−0.0727403	−	0.281574i
$772$		0			0
$773$		−	2.32574e11i		−	0.651393i	−0.945474	−	0.325696i	$-0.894401\pi$
$773$		−	2.32574e11i		−	0.651393i	0.945474	−	0.325696i	$-0.105599\pi$
$774$		0			0
$775$	−4.65980e9			−0.0129170
$776$		0			0
$777$	4.14080e8	+	4.21594e8i	0.00113606	+	0.00115667i
$778$		0			0
$779$	2.78967e11	−	1.61062e11i	0.757537	−	0.437364i
$780$		0			0
$781$	−1.18770e11	+	2.05715e11i	−0.319229	+	0.552921i
$782$		0			0
$783$	−4.42378e11	+	1.05833e11i	−1.17692	+	0.281562i
$784$		0			0
$785$	3.48851e11	+	2.01409e11i	0.918675	+	0.530397i
$786$		0			0
$787$	−8.91296e10	−	1.54377e11i	−0.232339	−	0.402424i	0.726157	−	0.687529i	$-0.241305\pi$
$787$	−8.91296e10	−	1.54377e11i	−0.232339	−	0.402424i	−0.958496	+	0.285106i	$0.907971\pi$
$788$		0			0
$789$	2.03348e11	−	1.99723e11i	0.524724	−	0.515371i
$790$		0			0
$791$		−	1.03051e10i		−	0.0263237i
$792$		0			0
$793$	−1.07950e10			−0.0272980
$794$		0			0
$795$	−4.91810e11	+	1.27051e11i	−1.23120	+	0.318061i
$796$		0			0
$797$	−2.40983e11	+	1.39131e11i	−0.597245	+	0.344819i	−0.767957	−	0.640502i	$-0.778727\pi$
$797$	−2.40983e11	+	1.39131e11i	−0.597245	+	0.344819i	0.170712	+	0.985321i	$0.445393\pi$
$798$		0			0
$799$	9.53241e10	−	1.65106e11i	0.233892	−	0.405113i
$800$		0			0
$801$	−3.02359e11	+	5.02605e11i	−0.734502	+	1.22095i
$802$		0			0
$803$	−7.24045e10	−	4.18027e10i	−0.174142	−	0.100541i
$804$		0			0
$805$	−4.42894e9	−	7.67115e9i	−0.0105467	−	0.0182674i
$806$		0			0
$807$	−2.68961e10	−	7.46664e9i	−0.0634154	−	0.0176048i
$808$		0			0
$809$			6.67843e11i			1.55912i	0.626326	+	0.779562i	$0.284558\pi$
$809$			6.67843e11i			1.55912i	−0.626326	+	0.779562i	$0.715442\pi$
$810$		0			0
$811$	8.47558e11			1.95923			0.979616	−	0.200878i	$-0.0643795\pi$
$811$	8.47558e11			1.95923			0.979616	+	0.200878i	$0.0643795\pi$
$812$		0			0
$813$	4.36594e10	−	1.57268e11i	0.0999344	−	0.359981i
$814$		0			0
$815$	−3.85295e11	+	2.22450e11i	−0.873300	+	0.504200i
$816$		0			0
$817$	1.75001e11	−	3.03111e11i	0.392783	−	0.680320i
$818$		0			0
$819$	3.06246e10	+	1.84233e10i	0.0680668	+	0.0409479i
$820$		0			0
$821$	3.26727e11	+	1.88636e11i	0.719138	+	0.415195i	0.814435	−	0.580254i	$-0.197047\pi$
$821$	3.26727e11	+	1.88636e11i	0.719138	+	0.415195i	−0.0952973	+	0.995449i	$0.530380\pi$
$822$		0			0
$823$	−1.04019e11	−	1.80166e11i	−0.226732	−	0.392711i	0.730106	−	0.683334i	$-0.239471\pi$
$823$	−1.04019e11	−	1.80166e11i	−0.226732	−	0.392711i	−0.956838	+	0.290623i	$0.906137\pi$
$824$		0			0
$825$	−7.51496e9	−	2.90901e10i	−0.0162222	−	0.0627956i
$826$		0			0
$827$		−	6.48957e11i		−	1.38738i	−0.720276	−	0.693688i	$-0.755985\pi$
$827$		−	6.48957e11i		−	1.38738i	0.720276	−	0.693688i	$-0.244015\pi$
$828$		0			0
$829$	1.01707e11			0.215345			0.107672	−	0.994186i	$-0.465660\pi$
$829$	1.01707e11			0.215345			0.107672	+	0.994186i	$0.465660\pi$
$830$		0			0
$831$	−2.23106e11	−	2.27155e11i	−0.467850	−	0.476341i
$832$		0			0
$833$	−9.49331e10	+	5.48096e10i	−0.197168	+	0.113835i
$834$		0			0
$835$	7.95418e10	−	1.37770e11i	0.163625	−	0.283407i
$836$		0			0
$837$	6.32775e10	+	1.87985e10i	0.128928	+	0.0383020i
$838$		0			0
$839$	−2.15932e11	−	1.24668e11i	−0.435782	−	0.251599i	0.266025	−	0.963966i	$-0.414290\pi$
$839$	−2.15932e11	−	1.24668e11i	−0.435782	−	0.251599i	−0.701807	+	0.712367i	$0.747623\pi$
$840$		0			0
$841$	1.16162e11	+	2.01198e11i	0.232209	+	0.402197i
$842$		0			0
$843$	−2.79387e11	+	2.74407e11i	−0.553217	+	0.543356i
$844$		0			0
$845$		−	4.63322e11i		−	0.908774i
$846$		0			0
$847$	−1.05813e11			−0.205591
$848$		0			0
$849$	−6.31886e11	+	1.63238e11i	−1.21621	+	0.314188i
$850$		0			0
$851$	1.14360e8	−	6.60257e7i	0.000218050	−	0.000125891i
$852$		0			0
$853$	−1.81832e11	+	3.14943e11i	−0.343459	+	0.594889i	−0.985073	−	0.172139i	$-0.944932\pi$
$853$	−1.81832e11	+	3.14943e11i	−0.343459	+	0.594889i	0.641613	+	0.767028i	$0.278265\pi$
$854$		0			0
$855$	3.19261e11	+	5.74236e9i	0.597421	+	0.0107455i
$856$		0			0
$857$	−3.63191e11	−	2.09689e11i	−0.673305	−	0.388733i	0.124023	−	0.992279i	$-0.460420\pi$
$857$	−3.63191e11	−	2.09689e11i	−0.673305	−	0.388733i	−0.797328	+	0.603546i	$0.793754\pi$
$858$		0			0
$859$	1.35782e11	+	2.35181e11i	0.249384	+	0.431946i	0.963355	−	0.268229i	$-0.0864384\pi$
$859$	1.35782e11	+	2.35181e11i	0.249384	+	0.431946i	−0.713971	+	0.700175i	$0.753105\pi$
$860$		0			0
$861$	−2.78573e11	−	7.73348e10i	−0.506905	−	0.140722i
$862$		0			0
$863$		−	7.28162e11i		−	1.31276i	−0.754431	−	0.656380i	$-0.772087\pi$
$863$		−	7.28162e11i		−	1.31276i	0.754431	−	0.656380i	$-0.227913\pi$
$864$		0			0
$865$	5.96962e11			1.06631
$866$		0			0
$867$	2.32741e11	−	8.38371e11i	0.411904	−	1.48375i
$868$		0			0
$869$	−3.47001e11	+	2.00341e11i	−0.608488	+	0.351311i
$870$		0			0
$871$	8.57042e10	−	1.48444e11i	0.148912	−	0.257923i
$872$		0			0
$873$	7.57771e11	−	4.19513e11i	1.30461	−	0.722252i
$874$		0			0
$875$	1.99947e11	+	1.15439e11i	0.341101	+	0.196935i
$876$		0			0
$877$	4.30314e11	+	7.45326e11i	0.727423	+	1.25993i	0.957969	+	0.286872i	$0.0926155\pi$
$877$	4.30314e11	+	7.45326e11i	0.727423	+	1.25993i	−0.230546	+	0.973061i	$0.574051\pi$
$878$		0			0
$879$	4.72970e10	+	1.83085e11i	0.0792280	+	0.306688i
$880$		0			0
$881$		−	9.04398e11i		−	1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$881$		−	9.04398e11i		−	1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$882$		0			0
$883$	−1.87461e11			−0.308368			−0.154184	−	0.988042i	$-0.549275\pi$
$883$	−1.87461e11			−0.308368			−0.154184	+	0.988042i	$0.549275\pi$
$884$		0			0
$885$	−1.33757e11	−	1.36184e11i	−0.218043	−	0.222000i
$886$		0			0
$887$	4.18621e11	−	2.41691e11i	0.676280	−	0.390450i	−0.122172	−	0.992509i	$-0.538986\pi$
$887$	4.18621e11	−	2.41691e11i	0.676280	−	0.390450i	0.798452	+	0.602058i	$0.205653\pi$
$888$		0			0
$889$	1.52621e11	−	2.64347e11i	0.244347	−	0.423221i
$890$		0			0
$891$	−1.53058e10	+	4.25343e11i	−0.0242853	+	0.674883i
$892$		0			0
$893$	−1.01591e11	−	5.86533e10i	−0.159752	−	0.0922331i
$894$		0			0
$895$	−7.65369e10	−	1.32566e11i	−0.119283	−	0.206604i
$896$		0			0
$897$	5.69774e9	−	5.59618e9i	0.00880102	−	0.00864415i
$898$		0			0
$899$		−	1.06312e11i		−	0.162759i
$900$		0			0
$901$	1.40471e12			2.13151
$902$		0			0
$903$	−3.04144e11	+	7.85709e10i	−0.457434	+	0.118171i
$904$		0			0
$905$	−9.35538e11	+	5.40133e11i	−1.39466	+	0.805205i
$906$		0			0
$907$	−2.30399e11	+	3.99063e11i	−0.340449	+	0.589675i	−0.984516	−	0.175294i	$-0.943912\pi$
$907$	−2.30399e11	+	3.99063e11i	−0.340449	+	0.589675i	0.644067	+	0.764969i	$0.277246\pi$
$908$		0			0
$909$	6.00884e10	+	1.08538e11i	0.0880106	+	0.158975i
$910$		0			0
$911$	−1.03916e12	−	5.99957e11i	−1.50872	−	0.871057i	−0.999948	−	0.0101528i	$-0.996768\pi$
$911$	−1.03916e12	−	5.99957e11i	−1.50872	−	0.871057i	−0.508767	−	0.860904i	$-0.669898\pi$
$912$		0			0
$913$	−3.89395e11	−	6.74453e11i	−0.560412	−	0.970663i
$914$		0			0
$915$	8.34085e10	+	2.31551e10i	0.118994	+	0.0330341i
$916$		0			0
$917$			1.76709e11i			0.249908i
$918$		0			0
$919$	−1.31696e12			−1.84633			−0.923165	−	0.384404i	$-0.874407\pi$
$919$	−1.31696e12			−1.84633			−0.923165	+	0.384404i	$0.874407\pi$
$920$		0			0
$921$	1.74092e10	−	6.27110e10i	0.0241959	−	0.0871576i
$922$		0			0
$923$	1.24887e11	−	7.21035e10i	0.172072	−	0.0993459i
$924$		0			0
$925$	−1.50796e8	+	2.61186e8i	−0.000205979	+	0.000356765i
$926$		0			0
$927$	−1.35883e10	+	7.55475e11i	−0.0184012	+	1.02306i
$928$		0			0
$929$	2.09405e11	+	1.20900e11i	0.281142	+	0.162317i	0.633940	−	0.773382i	$-0.281437\pi$
$929$	2.09405e11	+	1.20900e11i	0.281142	+	0.162317i	−0.352799	+	0.935699i	$0.614770\pi$
$930$		0			0
$931$	3.37246e10	+	5.84127e10i	0.0448899	+	0.0777515i
$932$		0			0
$933$	3.93445e10	+	1.52301e11i	0.0519227	+	0.200991i
$934$		0			0
$935$		−	7.82052e11i		−	1.02327i
$936$		0			0
$937$	−5.20230e11			−0.674896			−0.337448	−	0.941344i	$-0.609564\pi$
$937$	−5.20230e11			−0.674896			−0.337448	+	0.941344i	$0.609564\pi$
$938$		0			0
$939$	5.37033e10	+	5.46779e10i	0.0690778	+	0.0703315i
$940$		0			0
$941$	6.34114e11	−	3.66106e11i	0.808740	−	0.466926i	−0.0377779	−	0.999286i	$-0.512028\pi$
$941$	6.34114e11	−	3.66106e11i	0.808740	−	0.466926i	0.846518	+	0.532360i	$0.178695\pi$
$942$		0			0
$943$	−3.23023e10	+	5.59493e10i	−0.0408495	+	0.0707535i
$944$		0			0
$945$	−1.97106e11	−	2.08038e11i	−0.247157	−	0.260865i
$946$		0			0
$947$	−1.97039e11	−	1.13761e11i	−0.244992	−	0.141446i	0.372477	−	0.928042i	$-0.378509\pi$
$947$	−1.97039e11	−	1.13761e11i	−0.244992	−	0.141446i	−0.617469	+	0.786595i	$0.711842\pi$
$948$		0			0
$949$	2.53779e10	+	4.39558e10i	0.0312889	+	0.0541940i
$950$		0			0
$951$	−6.64925e10	+	6.53073e10i	−0.0812925	+	0.0798435i
$952$		0			0
$953$		−	1.54557e12i		−	1.87377i	−0.349635	−	0.936886i	$-0.613694\pi$
$953$		−	1.54557e12i		−	1.87377i	0.349635	−	0.936886i	$-0.386306\pi$
$954$		0			0
$955$	−7.42365e11			−0.892491
$956$		0			0
$957$	6.63684e11	−	1.71452e11i	0.791250	−	0.204407i
$958$		0			0
$959$	−4.35457e11	+	2.51411e11i	−0.514838	+	0.297242i
$960$		0			0
$961$	4.18731e11	−	7.25264e11i	0.490955	−	0.850359i
$962$		0			0
$963$	−7.28527e11	+	1.21102e12i	−0.847112	+	1.40814i
$964$		0			0
$965$	−4.61909e11	−	2.66684e11i	−0.532657	−	0.307530i
$966$		0			0
$967$	7.67424e11	+	1.32922e12i	0.877666	+	1.52016i	0.853895	+	0.520445i	$0.174234\pi$
$967$	7.67424e11	+	1.32922e12i	0.877666	+	1.52016i	0.0237712	+	0.999717i	$0.492433\pi$
$968$		0			0
$969$	−8.50853e11	−	2.36206e11i	−0.965072	−	0.267914i
$970$		0			0
$971$		−	6.80715e11i		−	0.765753i	−0.923800	−	0.382876i	$-0.874934\pi$
$971$		−	6.80715e11i		−	0.765753i	0.923800	−	0.382876i	$-0.125066\pi$
$972$		0			0
$973$	−2.59849e10			−0.0289914
$974$		0			0
$975$	−4.87908e9	+	1.75753e10i	−0.00539908	+	0.0194484i
$976$		0			0
$977$	−1.10085e12	+	6.35574e11i	−1.20823	+	0.697570i	−0.962372	−	0.271737i	$-0.912402\pi$
$977$	−1.10085e12	+	6.35574e11i	−1.20823	+	0.697570i	−0.245855	+	0.969307i	$0.579069\pi$
$978$		0			0
$979$	4.41958e11	−	7.65493e11i	0.481116	−	0.833317i
$980$		0			0
$981$	3.69126e11	+	2.22060e11i	0.398564	+	0.239770i
$982$		0			0
$983$	−9.45427e11	−	5.45843e11i	−1.01254	−	0.584593i	−0.100609	−	0.994926i	$-0.532079\pi$
$983$	−9.45427e11	−	5.45843e11i	−1.01254	−	0.584593i	−0.911936	+	0.410333i	$0.865412\pi$
$984$		0			0
$985$	1.95856e11	+	3.39233e11i	0.208062	+	0.360373i
$986$		0			0
$987$	2.63338e10	+	1.01937e11i	0.0277488	+	0.107414i
$988$		0			0
$989$			7.01959e10i			0.0733714i
$990$		0			0
$991$	1.17966e11			0.122310			0.0611549	−	0.998128i	$-0.480522\pi$
$991$	1.17966e11			0.122310			0.0611549	+	0.998128i	$0.480522\pi$
$992$		0			0
$993$	2.46261e11	+	2.50730e11i	0.253279	+	0.257875i
$994$		0			0
$995$	−8.49395e11	+	4.90398e11i	−0.866597	+	0.500330i
$996$		0			0
$997$	3.82091e11	−	6.61800e11i	0.386710	−	0.669802i	−0.605295	−	0.796002i	$-0.706945\pi$
$997$	3.82091e11	−	6.61800e11i	0.386710	−	0.669802i	0.992005	+	0.126200i	$0.0402780\pi$
$998$		0			0
$999$	3.10139e9	−	2.93842e9i	0.00311383	−	0.00295020i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.20	✓	96
9.5	odd	6	inner	252.9.bg.a.113.20	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.20	✓	96	1.1	even	1	trivial
252.9.bg.a.113.20	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)

\(f(q)\)	\(=\)	\(q+(-21.6670 + 78.0483i) q^{3} +(514.619 - 297.115i) q^{5} +(453.746 - 785.912i) q^{7} +(-5622.08 - 3382.15i) q^{9} +O(q^{10})\)	\(q+(-21.6670 + 78.0483i) q^{3} +(514.619 - 297.115i) q^{5} +(453.746 - 785.912i) q^{7} +(-5622.08 - 3382.15i) q^{9} +(8562.71 + 4943.68i) q^{11} +(-3001.24 - 5198.30i) q^{13} +(12039.1 + 46602.7i) q^{15} -133107. i q^{17} -81901.3 q^{19} +(51507.7 + 52442.5i) q^{21} +(14225.3 - 8213.00i) q^{23} +(-18757.6 + 32489.2i) q^{25} +(385785. - 365513. i) q^{27} +(-741234. - 427951. i) q^{29} +(62105.4 + 107570. i) q^{31} +(-571374. + 561190. i) q^{33} -539260. i q^{35} +8039.17 q^{37} +(470747. - 121610. i) q^{39} +(-3.40614e6 + 1.96654e6i) q^{41} +(-2.13673e6 + 3.70093e6i) q^{43} +(-3.89812e6 - 70113.2i) q^{45} +(1.24040e6 + 716147. i) q^{47} +(-411772. - 713209. i) q^{49} +(1.03888e7 + 2.88403e6i) q^{51} +1.05532e7i q^{53} +5.87537e6 q^{55} +(1.77456e6 - 6.39226e6i) q^{57} +(-3.43449e6 + 1.98290e6i) q^{59} +(899212. - 1.55748e6i) q^{61} +(-5.20907e6 + 2.88382e6i) q^{63} +(-3.08899e6 - 1.78343e6i) q^{65} +(1.42781e7 + 2.47304e7i) q^{67} +(332790. + 1.28821e6i) q^{69} +2.40246e7i q^{71} -8.45579e6 q^{73} +(-2.12930e6 - 2.16795e6i) q^{75} +(7.77059e6 - 4.48635e6i) q^{77} +(-2.02624e7 + 3.50954e7i) q^{79} +(2.01688e7 + 3.80294e7i) q^{81} +(-6.82136e7 - 3.93831e7i) q^{83} +(-3.95481e7 - 6.84993e7i) q^{85} +(4.94612e7 - 4.85796e7i) q^{87} -8.93985e7i q^{89} -5.44721e6 q^{91} +(-9.74128e6 + 2.51651e6i) q^{93} +(-4.21479e7 + 2.43341e7i) q^{95} +(-6.60071e7 + 1.14328e8i) q^{97} +(-3.14199e7 - 5.67541e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 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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