LMFDB - Embedded newform 252.9.bg.a.29.13

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			29.13
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-55.9608 - 58.5610i) q^{3} +(-883.740 + 510.228i) q^{5} +(-453.746 + 785.912i) q^{7} +(-297.778 + 6554.24i) q^{9} +O(q^{10})$	\(q+(-55.9608 - 58.5610i) q^{3} +(-883.740 + 510.228i) q^{5} +(-453.746 + 785.912i) q^{7} +(-297.778 + 6554.24i) q^{9} +(-617.947 - 356.772i) q^{11} +(8420.47 + 14584.7i) q^{13} +(79334.3 + 23200.0i) q^{15} -719.189i q^{17} +45497.0 q^{19} +(71415.8 - 17408.4i) q^{21} +(15945.2 - 9205.95i) q^{23} +(325352. - 563527. i) q^{25} +(400487. - 349342. i) q^{27} +(706132. + 407686. i) q^{29} +(214381. + 371318. i) q^{31} +(13687.9 + 56152.8i) q^{33} -926056. i q^{35} -1.14779e6 q^{37} +(382877. - 1.30928e6i) q^{39} +(728799. - 420772. i) q^{41} +(1.33406e6 - 2.31067e6i) q^{43} +(-3.08100e6 - 5.94418e6i) q^{45} +(5.82898e6 + 3.36536e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-42116.4 + 40246.4i) q^{51} -6.79223e6i q^{53} +728139. q^{55} +(-2.54605e6 - 2.66435e6i) q^{57} +(-1.55084e7 + 8.95379e6i) q^{59} +(590368. - 1.02255e6i) q^{61} +(-5.01594e6 - 3.20799e6i) q^{63} +(-1.48830e7 - 8.59271e6i) q^{65} +(-3.41114e6 - 5.90827e6i) q^{67} +(-1.43141e6 - 418593. i) q^{69} +3.00271e7i q^{71} +2.36409e7 q^{73} +(-5.12076e7 + 1.24824e7i) q^{75} +(560782. - 323768. i) q^{77} +(9.73416e6 - 1.68601e7i) q^{79} +(-4.28694e7 - 3.90342e6i) q^{81} +(4.39123e7 + 2.53528e7i) q^{83} +(366950. + 635576. i) q^{85} +(-1.56413e7 - 6.41662e7i) q^{87} +3.45879e7i q^{89} -1.52830e7 q^{91} +(9.74785e6 - 3.33336e7i) q^{93} +(-4.02075e7 + 2.32138e7i) q^{95} +(9.17589e6 - 1.58931e7i) q^{97} +(2.52238e6 - 3.94393e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−55.9608	−	58.5610i	−0.690874	−	0.722975i
$3$	−55.9608	−	58.5610i	−0.690874	−	0.722975i
$4$		0			0
$5$	−883.740	+	510.228i	−1.41398	+	0.816364i	−0.995761	−	0.0919797i	$-0.970681\pi$
$5$	−883.740	+	510.228i	−1.41398	+	0.816364i	−0.418224	+	0.908344i	$0.637347\pi$
$6$		0			0
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$8$		0			0
$9$	−297.778	+	6554.24i	−0.0453861	+	0.998970i
$10$		0			0
$11$	−617.947	−	356.772i	−0.0422066	−	0.0243680i	0.478748	−	0.877952i	$-0.341091\pi$
$11$	−617.947	−	356.772i	−0.0422066	−	0.0243680i	−0.520955	+	0.853584i	$0.674424\pi$
$12$		0			0
$13$	8420.47	+	14584.7i	0.294824	+	0.510650i	0.974944	−	0.222451i	$-0.0714057\pi$
$13$	8420.47	+	14584.7i	0.294824	+	0.510650i	−0.680120	+	0.733101i	$0.738072\pi$
$14$		0			0
$15$	79334.3	+	23200.0i	1.56710	+	0.458271i
$16$		0			0
$17$		−	719.189i		−	0.00861088i	−0.999991	−	0.00430544i	$-0.998630\pi$
$17$		−	719.189i		−	0.00861088i	0.999991	−	0.00430544i	$-0.00137047\pi$
$18$		0			0
$19$	45497.0			0.349115			0.174557	−	0.984647i	$-0.444151\pi$
$19$	45497.0			0.349115			0.174557	+	0.984647i	$0.444151\pi$
$20$		0			0
$21$	71415.8	−	17408.4i	0.367212	−	0.0895122i
$22$		0			0
$23$	15945.2	−	9205.95i	0.0569794	−	0.0328971i	−0.471240	−	0.882005i	$-0.656193\pi$
$23$	15945.2	−	9205.95i	0.0569794	−	0.0328971i	0.528219	+	0.849108i	$0.322860\pi$
$24$		0			0
$25$	325352.	−	563527.i	0.832902	−	1.44263i
$26$		0			0
$27$	400487.	−	349342.i	0.753586	−	0.657349i
$28$		0			0
$29$	706132.	+	407686.i	0.998376	+	0.576413i	0.907767	−	0.419474i	$-0.137785\pi$
$29$	706132.	+	407686.i	0.998376	+	0.576413i	0.0906086	+	0.995887i	$0.471119\pi$
$30$		0			0
$31$	214381.	+	371318.i	0.232134	+	0.402068i	0.958436	−	0.285308i	$-0.0920958\pi$
$31$	214381.	+	371318.i	0.232134	+	0.402068i	−0.726302	+	0.687376i	$0.758762\pi$
$32$		0			0
$33$	13687.9	+	56152.8i	0.0115420	+	0.0473495i
$34$		0			0
$35$		−	926056.i		−	0.617113i
$36$		0			0
$37$	−1.14779e6			−0.612430			−0.306215	−	0.951962i	$-0.599063\pi$
$37$	−1.14779e6			−0.612430			−0.306215	+	0.951962i	$0.599063\pi$
$38$		0			0
$39$	382877.	−	1.30928e6i	0.165501	−	0.565945i
$40$		0			0
$41$	728799.	−	420772.i	0.257912	−	0.148906i	−0.365470	−	0.930823i	$-0.619092\pi$
$41$	728799.	−	420772.i	0.257912	−	0.148906i	0.623382	+	0.781918i	$0.285758\pi$
$42$		0			0
$43$	1.33406e6	−	2.31067e6i	0.390214	−	0.675870i	−0.602264	−	0.798297i	$-0.705734\pi$
$43$	1.33406e6	−	2.31067e6i	0.390214	−	0.675870i	0.992478	+	0.122427i	$0.0390678\pi$
$44$		0			0
$45$	−3.08100e6	−	5.94418e6i	−0.751348	−	1.44958i
$46$		0			0
$47$	5.82898e6	+	3.36536e6i	1.19454	+	0.689668i	0.959333	−	0.282277i	$-0.0910898\pi$
$47$	5.82898e6	+	3.36536e6i	1.19454	+	0.689668i	0.235207	+	0.971945i	$0.424423\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	−42116.4	+	40246.4i	−0.00622545	+	0.00594903i
$52$		0			0
$53$		−	6.79223e6i		−	0.860814i	−0.902635	−	0.430407i	$-0.858370\pi$
$53$		−	6.79223e6i		−	0.860814i	0.902635	−	0.430407i	$-0.141630\pi$
$54$		0			0
$55$	728139.			0.0795726
$56$		0			0
$57$	−2.54605e6	−	2.66435e6i	−0.241194	−	0.252401i
$58$		0			0
$59$	−1.55084e7	+	8.95379e6i	−1.27985	+	0.738923i	−0.976821	−	0.214058i	$-0.931332\pi$
$59$	−1.55084e7	+	8.95379e6i	−1.27985	+	0.738923i	−0.303031	+	0.952981i	$0.597998\pi$
$60$		0			0
$61$	590368.	−	1.02255e6i	0.0426387	−	0.0738523i	−0.843918	−	0.536471i	$-0.819757\pi$
$61$	590368.	−	1.02255e6i	0.0426387	−	0.0738523i	0.886557	+	0.462619i	$0.153090\pi$
$62$		0			0
$63$	−5.01594e6	−	3.20799e6i	−0.318412	−	0.203644i
$64$		0			0
$65$	−1.48830e7	−	8.59271e6i	−0.833753	−	0.481367i
$66$		0			0
$67$	−3.41114e6	−	5.90827e6i	−0.169278	−	0.293198i	0.768888	−	0.639383i	$-0.220810\pi$
$67$	−3.41114e6	−	5.90827e6i	−0.169278	−	0.293198i	−0.938166	+	0.346185i	$0.887477\pi$
$68$		0			0
$69$	−1.43141e6	−	418593.i	−0.0631494	−	0.0184670i
$70$		0			0
$71$			3.00271e7i			1.18163i	0.806809	+	0.590813i	$0.201193\pi$
$71$			3.00271e7i			1.18163i	−0.806809	+	0.590813i	$0.798807\pi$
$72$		0			0
$73$	2.36409e7			0.832478			0.416239	−	0.909255i	$-0.363348\pi$
$73$	2.36409e7			0.832478			0.416239	+	0.909255i	$0.363348\pi$
$74$		0			0
$75$	−5.12076e7	+	1.24824e7i	−1.61841	+	0.394507i
$76$		0			0
$77$	560782.	−	323768.i	0.0159526	−	0.00921023i
$78$		0			0
$79$	9.73416e6	−	1.68601e7i	0.249914	−	0.432863i	−0.713588	−	0.700566i	$-0.752931\pi$
$79$	9.73416e6	−	1.68601e7i	0.249914	−	0.432863i	0.963502	+	0.267702i	$0.0862644\pi$
$80$		0			0
$81$	−4.28694e7	−	3.90342e6i	−0.995880	−	0.0906787i
$82$		0			0
$83$	4.39123e7	+	2.53528e7i	0.925282	+	0.534212i	0.885316	−	0.464989i	$-0.153942\pi$
$83$	4.39123e7	+	2.53528e7i	0.925282	+	0.534212i	0.0399654	+	0.999201i	$0.487275\pi$
$84$		0			0
$85$	366950.	+	635576.i	0.00702961	+	0.0121756i
$86$		0			0
$87$	−1.56413e7	−	6.41662e7i	−0.273020	−	1.12003i
$88$		0			0
$89$			3.45879e7i			0.551270i	0.961262	+	0.275635i	$0.0888882\pi$
$89$			3.45879e7i			0.551270i	−0.961262	+	0.275635i	$0.911112\pi$
$90$		0			0
$91$	−1.52830e7			−0.222866
$92$		0			0
$93$	9.74785e6	−	3.33336e7i	0.130310	−	0.445606i
$94$		0			0
$95$	−4.02075e7	+	2.32138e7i	−0.493643	+	0.285005i
$96$		0			0
$97$	9.17589e6	−	1.58931e7i	0.103648	−	0.179524i	−0.809537	−	0.587069i	$-0.800282\pi$
$97$	9.17589e6	−	1.58931e7i	0.103648	−	0.179524i	0.913185	+	0.407545i	$0.133615\pi$
$98$		0			0
$99$	2.52238e6	−	3.94393e6i	0.0262585	−	0.0410571i
$100$		0			0
$101$	3.46837e7	+	2.00246e7i	0.333303	+	0.192433i	0.657307	−	0.753623i	$-0.271696\pi$
$101$	3.46837e7	+	2.00246e7i	0.333303	+	0.192433i	−0.324003	+	0.946056i	$0.605029\pi$
$102$		0			0
$103$	8.82659e7	+	1.52881e8i	0.784231	+	1.35833i	0.929457	+	0.368930i	$0.120276\pi$
$103$	8.82659e7	+	1.52881e8i	0.784231	+	1.35833i	−0.145226	+	0.989398i	$0.546391\pi$
$104$		0			0
$105$	−5.42307e7	+	5.18228e7i	−0.446158	+	0.426348i
$106$		0			0
$107$			2.52810e8i			1.92868i	0.264670	+	0.964339i	$0.414737\pi$
$107$			2.52810e8i			1.92868i	−0.264670	+	0.964339i	$0.585263\pi$
$108$		0			0
$109$	1.74105e8			1.23340			0.616701	−	0.787198i	$-0.288469\pi$
$109$	1.74105e8			1.23340			0.616701	+	0.787198i	$0.288469\pi$
$110$		0			0
$111$	6.42313e7	+	6.72158e7i	0.423112	+	0.442771i
$112$		0			0
$113$	−1.64741e8	+	9.51134e7i	−1.01039	+	0.583348i	−0.911306	−	0.411731i	$-0.864924\pi$
$113$	−1.64741e8	+	9.51134e7i	−1.01039	+	0.583348i	−0.0990834	+	0.995079i	$0.531591\pi$
$114$		0			0
$115$	−9.39426e6	+	1.62713e7i	−0.0537120	+	0.0930319i
$116$		0			0
$117$	−9.80989e7	+	5.08467e7i	−0.523505	+	0.271344i
$118$		0			0
$119$	565219.	+	326329.i	0.00281857	+	0.00162730i
$120$		0			0
$121$	−1.06925e8	−	1.85199e8i	−0.498812	−	0.863968i
$122$		0			0
$123$	−6.54250e7	−	1.91324e7i	−0.285840	−	0.0835891i
$124$		0			0
$125$			2.65399e8i			1.08708i
$126$		0			0
$127$	9.46367e7			0.363785			0.181892	−	0.983318i	$-0.441778\pi$
$127$	9.46367e7			0.363785			0.181892	+	0.983318i	$0.441778\pi$
$128$		0			0
$129$	−2.09970e8	+	5.11826e7i	−0.758226	+	0.184826i
$130$		0			0
$131$	−5.57169e6	+	3.21682e6i	−0.0189192	+	0.0109230i	−0.509430	−	0.860512i	$-0.670144\pi$
$131$	−5.57169e6	+	3.21682e6i	−0.0189192	+	0.0109230i	0.490511	+	0.871435i	$0.336810\pi$
$132$		0			0
$133$	−2.06441e7	+	3.57566e7i	−0.0659765	+	0.114275i
$134$		0			0
$135$	−1.75682e8	+	5.13067e8i	−0.528923	+	1.54468i
$136$		0			0
$137$	−4.80116e6	−	2.77195e6i	−0.0136290	−	0.00786870i	0.493170	−	0.869933i	$-0.335838\pi$
$137$	−4.80116e6	−	2.77195e6i	−0.0136290	−	0.00786870i	−0.506799	+	0.862064i	$0.669171\pi$
$138$		0			0
$139$	3.52872e7	+	6.11193e7i	0.0945276	+	0.163727i	0.909411	−	0.415898i	$-0.136533\pi$
$139$	3.52872e7	+	6.11193e7i	0.0945276	+	0.163727i	−0.814884	+	0.579624i	$0.803199\pi$
$140$		0			0
$141$	−1.29115e8	−	5.29679e8i	−0.326664	−	1.34010i
$142$		0			0
$143$		−	1.20167e7i		−	0.0287371i
$144$		0			0
$145$	−8.32050e8			−1.88225
$146$		0			0
$147$	−1.87232e7	+	6.40255e7i	−0.0400968	+	0.137115i
$148$		0			0
$149$	−7.49693e8	+	4.32835e8i	−1.52103	+	0.878168i	−0.521340	+	0.853349i	$0.674568\pi$
$149$	−7.49693e8	+	4.32835e8i	−1.52103	+	0.878168i	−0.999692	+	0.0248195i	$0.992099\pi$
$150$		0			0
$151$	−1.74010e8	+	3.01393e8i	−0.334707	+	0.579730i	−0.983429	−	0.181296i	$-0.941971\pi$
$151$	−1.74010e8	+	3.01393e8i	−0.334707	+	0.579730i	0.648721	+	0.761026i	$0.275304\pi$
$152$		0			0
$153$	4.71374e6	+	214159.i	0.00860200	+	0.000390814i
$154$		0			0
$155$	−3.78914e8	−	2.18766e8i	−0.656468	−	0.379012i
$156$		0			0
$157$	1.59881e8	+	2.76923e8i	0.263148	+	0.455785i	0.967077	−	0.254485i	$-0.0819060\pi$
$157$	1.59881e8	+	2.76923e8i	0.263148	+	0.455785i	−0.703929	+	0.710270i	$0.748573\pi$
$158$		0			0
$159$	−3.97760e8	+	3.80099e8i	−0.622347	+	0.594714i
$160$		0			0
$161$			1.67087e7i			0.0248679i
$162$		0			0
$163$	−9.07875e8			−1.28610			−0.643052	−	0.765823i	$-0.722332\pi$
$163$	−9.07875e8			−1.28610			−0.643052	+	0.765823i	$0.722332\pi$
$164$		0			0
$165$	−4.07472e7	−	4.26405e7i	−0.0549746	−	0.0575290i
$166$		0			0
$167$	−1.75278e8	+	1.01197e8i	−0.225352	+	0.130107i	−0.608426	−	0.793611i	$-0.708199\pi$
$167$	−1.75278e8	+	1.01197e8i	−0.225352	+	0.130107i	0.383074	+	0.923718i	$0.374865\pi$
$168$		0			0
$169$	2.66057e8	−	4.60824e8i	0.326158	−	0.564922i
$170$		0			0
$171$	−1.35480e7	+	2.98198e8i	−0.0158449	+	0.348755i
$172$		0			0
$173$	−5.36138e8	−	3.09539e8i	−0.598539	−	0.345566i	0.169928	−	0.985457i	$-0.445647\pi$
$173$	−5.36138e8	−	3.09539e8i	−0.598539	−	0.345566i	−0.768466	+	0.639890i	$0.778980\pi$
$174$		0			0
$175$	2.95255e8	+	5.11396e8i	0.314807	+	0.545262i
$176$		0			0
$177$	1.39221e9	+	4.07127e8i	1.41844	+	0.414798i
$178$		0			0
$179$		−	5.52370e8i		−	0.538044i	−0.963134	−	0.269022i	$-0.913300\pi$
$179$		−	5.52370e8i		−	0.538044i	0.963134	−	0.269022i	$-0.0867005\pi$
$180$		0			0
$181$	−6.46104e8			−0.601989			−0.300994	−	0.953626i	$-0.597319\pi$
$181$	−6.46104e8			−0.601989			−0.300994	+	0.953626i	$0.597319\pi$
$182$		0			0
$183$	−9.29189e7	+	2.26500e7i	−0.0828513	+	0.0201960i
$184$		0			0
$185$	1.01435e9	−	5.85635e8i	0.865966	−	0.499966i
$186$		0			0
$187$	−256586.	+	444420.i	−0.000209830	+	0.000363436i
$188$		0			0
$189$	9.28328e7	+	4.73260e8i	0.0727536	+	0.370896i
$190$		0			0
$191$	−1.48409e8	−	8.56837e7i	−0.111513	−	0.0643821i	0.443206	−	0.896420i	$-0.353841\pi$
$191$	−1.48409e8	−	8.56837e7i	−0.111513	−	0.0643821i	−0.554719	+	0.832038i	$0.687174\pi$
$192$		0			0
$193$	3.24633e7	+	5.62281e7i	0.0233972	+	0.0405251i	0.877487	−	0.479601i	$-0.159218\pi$
$193$	3.24633e7	+	5.62281e7i	0.0233972	+	0.0405251i	−0.854090	+	0.520126i	$0.825885\pi$
$194$		0			0
$195$	3.29668e8	+	1.35242e9i	0.228002	+	0.935347i
$196$		0			0
$197$		−	1.99565e9i		−	1.32501i	−0.749057	−	0.662506i	$-0.769493\pi$
$197$		−	1.99565e9i		−	1.32501i	0.749057	−	0.662506i	$-0.230507\pi$
$198$		0			0
$199$	−9.98111e8			−0.636453			−0.318227	−	0.948015i	$-0.603087\pi$
$199$	−9.98111e8			−0.636453			−0.318227	+	0.948015i	$0.603087\pi$
$200$		0			0
$201$	−1.55104e8	+	5.30391e8i	−0.0950251	+	0.324947i
$202$		0			0
$203$	−6.40810e8	+	3.69972e8i	−0.377351	+	0.217863i
$204$		0			0
$205$	−4.29379e8	+	7.43707e8i	−0.243123	+	0.421101i
$206$		0			0
$207$	5.55899e7	+	1.07250e8i	0.0302771	+	0.0584138i
$208$		0			0
$209$	−2.81147e7	−	1.62320e7i	−0.0147349	−	0.00850722i
$210$		0			0
$211$	−1.01355e9	−	1.75553e9i	−0.511349	−	0.885682i	−0.999913	−	0.0131541i	$-0.995813\pi$
$211$	−1.01355e9	−	1.75553e9i	−0.511349	−	0.885682i	0.488565	−	0.872528i	$-0.337521\pi$
$212$		0			0
$213$	1.75842e9	−	1.68034e9i	0.854286	−	0.816354i
$214$		0			0
$215$			2.72270e9i			1.27423i
$216$		0			0
$217$	−3.89098e8			−0.175477
$218$		0			0
$219$	−1.32296e9	−	1.38443e9i	−0.575137	−	0.601861i
$220$		0			0
$221$	1.04891e7	−	6.05591e6i	0.00439714	−	0.00253869i
$222$		0			0
$223$	−1.96295e9	+	3.39993e9i	−0.793760	+	1.37483i	0.129863	+	0.991532i	$0.458546\pi$
$223$	−1.96295e9	+	3.39993e9i	−0.793760	+	1.37483i	−0.923624	+	0.383301i	$0.874787\pi$
$224$		0			0
$225$	3.59660e9	+	2.30024e9i	1.40334	+	0.897519i
$226$		0			0
$227$	−5.10409e8	−	2.94685e8i	−0.192227	−	0.110983i	0.400797	−	0.916167i	$-0.368733\pi$
$227$	−5.10409e8	−	2.94685e8i	−0.192227	−	0.110983i	−0.593025	+	0.805184i	$0.702066\pi$
$228$		0			0
$229$	1.49792e9	+	2.59448e9i	0.544688	+	0.943428i	0.998626	+	0.0523947i	$0.0166854\pi$
$229$	1.49792e9	+	2.59448e9i	0.544688	+	0.943428i	−0.453938	+	0.891033i	$0.649981\pi$
$230$		0			0
$231$	−5.03420e7	−	1.47217e7i	−0.0176800	−	0.00517022i
$232$		0			0
$233$			1.66503e7i			0.00564934i	0.999996	+	0.00282467i	$0.000899121\pi$
$233$			1.66503e7i			0.00564934i	−0.999996	+	0.00282467i	$0.999101\pi$
$234$		0			0
$235$	−6.86840e9			−2.25208
$236$		0			0
$237$	−1.53207e9	+	3.73460e8i	−0.485608	+	0.118373i
$238$		0			0
$239$	3.28213e9	−	1.89494e9i	1.00592	−	0.580769i	0.0959268	−	0.995388i	$-0.469419\pi$
$239$	3.28213e9	−	1.89494e9i	1.00592	−	0.580769i	0.909995	+	0.414619i	$0.136085\pi$
$240$		0			0
$241$	1.19734e9	−	2.07385e9i	0.354935	−	0.614765i	−0.632172	−	0.774828i	$-0.717836\pi$
$241$	1.19734e9	−	2.07385e9i	0.354935	−	0.614765i	0.987107	+	0.160063i	$0.0511698\pi$
$242$		0			0
$243$	2.17042e9	+	2.72891e9i	0.622469	+	0.782644i
$244$		0			0
$245$	7.27798e8	+	4.20194e8i	0.201998	+	0.116623i
$246$		0			0
$247$	3.83106e8	+	6.63558e8i	0.102927	+	0.178275i
$248$		0			0
$249$	−9.72684e8	−	3.99031e9i	−0.253031	−	1.03803i
$250$		0			0
$251$			4.86303e9i			1.22521i	0.790388	+	0.612607i	$0.209879\pi$
$251$			4.86303e9i			1.22521i	−0.790388	+	0.612607i	$0.790121\pi$
$252$		0			0
$253$	−1.31377e7			−0.00320654
$254$		0			0
$255$	1.66852e7	−	5.70563e7i	0.00394611	−	0.0134941i
$256$		0			0
$257$	−5.50883e9	+	3.18053e9i	−1.26278	+	0.729065i	−0.973611	−	0.228213i	$-0.926712\pi$
$257$	−5.50883e9	+	3.18053e9i	−1.26278	+	0.729065i	−0.289167	+	0.957279i	$0.593378\pi$
$258$		0			0
$259$	5.20806e8	−	9.02063e8i	0.115738	−	0.200465i
$260$		0			0
$261$	−2.88234e9	+	4.50676e9i	−0.621131	+	0.971186i
$262$		0			0
$263$	−4.84134e9	−	2.79515e9i	−1.01191	−	0.584227i	−0.100160	−	0.994971i	$-0.531936\pi$
$263$	−4.84134e9	−	2.79515e9i	−1.01191	−	0.584227i	−0.911751	+	0.410744i	$0.865269\pi$
$264$		0			0
$265$	3.46559e9	+	6.00257e9i	0.702738	+	1.21718i
$266$		0			0
$267$	2.02550e9	−	1.93557e9i	0.398555	−	0.380858i
$268$		0			0
$269$			5.91981e9i			1.13057i	0.824895	+	0.565286i	$0.191234\pi$
$269$			5.91981e9i			1.13057i	−0.824895	+	0.565286i	$0.808766\pi$
$270$		0			0
$271$	−4.09931e9			−0.760035			−0.380018	−	0.924979i	$-0.624082\pi$
$271$	−4.09931e9			−0.760035			−0.380018	+	0.924979i	$0.624082\pi$
$272$		0			0
$273$	8.55250e8	+	8.94989e8i	0.153972	+	0.161127i
$274$		0			0
$275$	−4.02101e8	+	2.32153e8i	−0.0703079	+	0.0405923i
$276$		0			0
$277$	5.32996e9	−	9.23176e9i	0.905326	−	1.56807i	0.0848458	−	0.996394i	$-0.472960\pi$
$277$	5.32996e9	−	9.23176e9i	0.905326	−	1.56807i	0.820480	−	0.571676i	$-0.193706\pi$
$278$		0			0
$279$	−2.49755e9	+	1.29453e9i	−0.412189	+	0.213647i
$280$		0			0
$281$	3.23647e8	+	1.86858e8i	0.0519095	+	0.0299700i	0.525730	−	0.850651i	$-0.323792\pi$
$281$	3.23647e8	+	1.86858e8i	0.0519095	+	0.0299700i	−0.473821	+	0.880621i	$0.657126\pi$
$282$		0			0
$283$	−4.83833e9	−	8.38023e9i	−0.754309	−	1.30650i	−0.945717	−	0.324992i	$-0.894639\pi$
$283$	−4.83833e9	−	8.38023e9i	−0.754309	−	1.30650i	0.191407	−	0.981511i	$-0.438695\pi$
$284$		0			0
$285$	3.60947e9	+	1.05553e9i	0.547096	+	0.159989i
$286$		0			0
$287$			7.63695e8i			0.112562i
$288$		0			0
$289$	6.97524e9			0.999926
$290$		0			0
$291$	−1.44421e9	+	3.52042e8i	−0.201399	+	0.0490933i
$292$		0			0
$293$	1.99357e9	−	1.15099e9i	0.270496	−	0.156171i	−0.358617	−	0.933485i	$-0.616752\pi$
$293$	1.99357e9	−	1.15099e9i	0.270496	−	0.156171i	0.629113	+	0.777314i	$0.283418\pi$
$294$		0			0
$295$	9.13695e9	−	1.58257e10i	1.20646	−	2.08965i
$296$		0			0
$297$	−3.72115e8	+	7.29926e7i	−0.0478246	+	0.00938108i
$298$		0			0
$299$	2.68532e8	+	1.55037e8i	0.0335978	+	0.0193977i
$300$		0			0
$301$	1.21065e9	+	2.09691e9i	0.147487	+	0.255455i
$302$		0			0
$303$	−7.68264e8	−	3.15170e9i	−0.0911465	−	0.373917i
$304$		0			0
$305$			1.20489e9i			0.139235i
$306$		0			0
$307$	−1.85786e9			−0.209151			−0.104575	−	0.994517i	$-0.533348\pi$
$307$	−1.85786e9			−0.209151			−0.104575	+	0.994517i	$0.533348\pi$
$308$		0			0
$309$	4.01343e9	−	1.37243e10i	0.440233	−	1.50541i
$310$		0			0
$311$	4.15323e9	−	2.39787e9i	0.443961	−	0.256321i	−0.261315	−	0.965253i	$-0.584156\pi$
$311$	4.15323e9	−	2.39787e9i	0.443961	−	0.256321i	0.705276	+	0.708933i	$0.250823\pi$
$312$		0			0
$313$	5.76113e9	−	9.97857e9i	0.600247	−	1.03966i	−0.392536	−	0.919737i	$-0.628402\pi$
$313$	5.76113e9	−	9.97857e9i	0.600247	−	1.03966i	0.992783	−	0.119922i	$-0.0382646\pi$
$314$		0			0
$315$	6.06959e9	+	2.75759e8i	0.616478	+	0.0280084i
$316$		0			0
$317$	5.99109e9	+	3.45896e9i	0.593292	+	0.342538i	0.766398	−	0.642366i	$-0.222047\pi$
$317$	5.99109e9	+	3.45896e9i	0.593292	+	0.342538i	−0.173106	+	0.984903i	$0.555380\pi$
$318$		0			0
$319$	−2.90901e8	−	5.03856e8i	−0.0280920	−	0.0486568i
$320$		0			0
$321$	1.48048e10	−	1.41475e10i	1.39439	−	1.33247i
$322$		0			0
$323$		−	3.27209e7i		−	0.00300618i
$324$		0			0
$325$	1.09585e10			0.982237
$326$		0			0
$327$	−9.74304e9	−	1.01957e10i	−0.852125	−	0.891719i
$328$		0			0
$329$	−5.28975e9	+	3.05404e9i	−0.451494	+	0.260670i
$330$		0			0
$331$	6.21702e9	−	1.07682e10i	0.517929	−	0.897079i	−0.481854	−	0.876251i	$-0.660037\pi$
$331$	6.21702e9	−	1.07682e10i	0.517929	−	0.897079i	0.999783	−	0.0208277i	$-0.00663015\pi$
$332$		0			0
$333$	3.41787e8	−	7.52290e9i	0.0277958	−	0.611799i
$334$		0			0
$335$	6.02912e9	+	3.48092e9i	0.478713	+	0.276385i
$336$		0			0
$337$	9.58976e9	+	1.66100e10i	0.743513	+	1.28780i	0.950886	+	0.309540i	$0.100175\pi$
$337$	9.58976e9	+	1.66100e10i	0.743513	+	1.28780i	−0.207374	+	0.978262i	$0.566492\pi$
$338$		0			0
$339$	1.47890e10	+	4.32479e9i	1.11980	+	0.327466i
$340$		0			0
$341$		−	3.05940e8i		−	0.0226266i
$342$		0			0
$343$	7.47359e8			0.0539949
$344$		0			0
$345$	1.47858e9	−	3.60420e8i	0.104368	−	0.0254409i
$346$		0			0
$347$	−8.45573e9	+	4.88192e9i	−0.583221	+	0.336723i	−0.762412	−	0.647092i	$-0.775985\pi$
$347$	−8.45573e9	+	4.88192e9i	−0.583221	+	0.336723i	0.179192	+	0.983814i	$0.442652\pi$
$348$		0			0
$349$	8.59130e9	−	1.48806e10i	0.579105	−	1.00304i	−0.416478	−	0.909146i	$-0.636736\pi$
$349$	8.59130e9	−	1.48806e10i	0.579105	−	1.00304i	0.995582	−	0.0938928i	$-0.0299311\pi$
$350$		0			0
$351$	8.46733e9	+	2.89934e9i	0.557851	+	0.191017i
$352$		0			0
$353$	1.44621e10	+	8.34968e9i	0.931391	+	0.537739i	0.887251	−	0.461287i	$-0.152612\pi$
$353$	1.44621e10	+	8.34968e9i	0.931391	+	0.537739i	0.0441394	+	0.999025i	$0.485945\pi$
$354$		0			0
$355$	−1.53206e10	−	2.65361e10i	−0.964637	−	1.67080i
$356$		0			0
$357$	−1.25199e7	−	5.13614e7i	−0.000770778	−	0.00316202i
$358$		0			0
$359$		−	1.08607e10i		−	0.653850i	−0.945050	−	0.326925i	$-0.893987\pi$
$359$		−	1.08607e10i		−	0.653850i	0.945050	−	0.326925i	$-0.106013\pi$
$360$		0			0
$361$	−1.49136e10			−0.878119
$362$		0			0
$363$	−4.86185e9	+	1.66255e10i	−0.280011	+	0.957522i
$364$		0			0
$365$	−2.08924e10	+	1.20622e10i	−1.17711	+	0.679605i
$366$		0			0
$367$	1.10453e10	−	1.91309e10i	0.608852	−	1.05456i	−0.382578	−	0.923923i	$-0.624964\pi$
$367$	1.10453e10	−	1.91309e10i	0.608852	−	1.05456i	0.991430	−	0.130639i	$-0.0417029\pi$
$368$		0			0
$369$	2.54082e9	+	4.90202e9i	0.137047	+	0.264405i
$370$		0			0
$371$	5.33810e9	+	3.08195e9i	0.281767	+	0.162679i
$372$		0			0
$373$	−1.43244e10	−	2.48106e10i	−0.740016	−	1.28174i	−0.952488	−	0.304578i	$-0.901485\pi$
$373$	−1.43244e10	−	2.48106e10i	−0.740016	−	1.28174i	0.212472	−	0.977167i	$-0.431849\pi$
$374$		0			0
$375$	1.55421e10	−	1.48520e10i	0.785929	−	0.751033i
$376$		0			0
$377$			1.37316e10i			0.679761i
$378$		0			0
$379$	−1.55676e10			−0.754511			−0.377255	−	0.926109i	$-0.623132\pi$
$379$	−1.55676e10			−0.754511			−0.377255	+	0.926109i	$0.623132\pi$
$380$		0			0
$381$	−5.29594e9	−	5.54202e9i	−0.251329	−	0.263007i
$382$		0			0
$383$	2.45143e9	−	1.41533e9i	0.113926	−	0.0657754i	−0.441954	−	0.897038i	$-0.645715\pi$
$383$	2.45143e9	−	1.41533e9i	0.113926	−	0.0657754i	0.555880	+	0.831262i	$0.312381\pi$
$384$		0			0
$385$	−3.30390e8	+	5.72253e8i	−0.0150378	+	0.0260462i
$386$		0			0
$387$	1.47474e10	+	9.43183e9i	0.657463	+	0.420487i
$388$		0			0
$389$	−2.52038e10	−	1.45514e10i	−1.10069	−	0.635486i	−0.164291	−	0.986412i	$-0.552534\pi$
$389$	−2.52038e10	−	1.45514e10i	−1.10069	−	0.635486i	−0.936403	+	0.350926i	$0.885867\pi$
$390$		0			0
$391$	−6.62082e6	−	1.14676e7i	−0.000283273	−	0.000490643i
$392$		0			0
$393$	5.00176e8	+	1.46268e8i	0.0209678	+	0.00613167i
$394$		0			0
$395$			1.98665e10i			0.816083i
$396$		0			0
$397$	4.74504e9			0.191020			0.0955099	−	0.995428i	$-0.469552\pi$
$397$	4.74504e9			0.191020			0.0955099	+	0.995428i	$0.469552\pi$
$398$		0			0
$399$	3.24920e9	−	7.92030e8i	0.128199	−	0.0312500i
$400$		0			0
$401$	−3.66984e10	+	2.11878e10i	−1.41929	+	0.819425i	−0.996236	−	0.0866816i	$-0.972374\pi$
$401$	−3.66984e10	+	2.11878e10i	−1.41929	+	0.819425i	−0.423050	+	0.906107i	$0.639040\pi$
$402$		0			0
$403$	−3.61037e9	+	6.25335e9i	−0.136877	+	0.237079i
$404$		0			0
$405$	3.98770e10	−	1.84235e10i	1.48219	−	0.684783i
$406$		0			0
$407$	7.09274e8	+	4.09500e8i	0.0258486	+	0.0149237i
$408$		0			0
$409$	4.83893e9	+	8.38128e9i	0.172925	+	0.299514i	0.939441	−	0.342711i	$-0.111345\pi$
$409$	4.83893e9	+	8.38128e9i	0.172925	+	0.299514i	−0.766517	+	0.642225i	$0.778012\pi$
$410$		0			0
$411$	1.06349e8	+	4.36281e8i	0.00372704	+	0.0152897i
$412$		0			0
$413$		−	1.62510e10i		−	0.558573i
$414$		0			0
$415$	−5.17428e10			−1.74445
$416$		0			0
$417$	1.60450e9	−	5.48674e9i	0.0530636	−	0.181456i
$418$		0			0
$419$	2.73252e10	−	1.57762e10i	0.886559	−	0.511855i	0.0137433	−	0.999906i	$-0.495625\pi$
$419$	2.73252e10	−	1.57762e10i	0.886559	−	0.511855i	0.872815	+	0.488051i	$0.162292\pi$
$420$		0			0
$421$	−2.45940e10	+	4.25980e10i	−0.782890	+	1.35601i	0.147362	+	0.989083i	$0.452922\pi$
$421$	−2.45940e10	+	4.25980e10i	−0.782890	+	1.35601i	−0.930252	+	0.366922i	$0.880411\pi$
$422$		0			0
$423$	−2.37931e10	+	3.72024e10i	−0.743173	+	1.16201i
$424$		0			0
$425$	−4.05282e8	−	2.33990e8i	−0.0124223	−	0.00717201i
$426$		0			0
$427$	5.35755e8	+	9.27954e8i	0.0161159	+	0.0279136i
$428$		0			0
$429$	−7.03712e8	+	6.72466e8i	−0.0207762	+	0.0198537i
$430$		0			0
$431$			4.34872e9i			0.126024i	0.998013	+	0.0630119i	$0.0200706\pi$
$431$			4.34872e9i			0.126024i	−0.998013	+	0.0630119i	$0.979929\pi$
$432$		0			0
$433$	−1.25557e10			−0.357182			−0.178591	−	0.983923i	$-0.557154\pi$
$433$	−1.25557e10			−0.357182			−0.178591	+	0.983923i	$0.557154\pi$
$434$		0			0
$435$	4.65622e10	+	4.87257e10i	1.30040	+	1.36082i
$436$		0			0
$437$	7.25457e8	−	4.18843e8i	0.0198923	−	0.0114848i
$438$		0			0
$439$	−1.42697e10	+	2.47158e10i	−0.384198	+	0.665451i	−0.991658	−	0.128900i	$-0.958855\pi$
$439$	−1.42697e10	+	2.47158e10i	−0.384198	+	0.665451i	0.607459	+	0.794351i	$0.292189\pi$
$440$		0			0
$441$	4.79716e9	−	2.48647e9i	0.126832	−	0.0657399i
$442$		0			0
$443$	1.62558e10	+	9.38529e9i	0.422079	+	0.243687i	0.695966	−	0.718074i	$-0.254976\pi$
$443$	1.62558e10	+	9.38529e9i	0.422079	+	0.243687i	−0.273888	+	0.961762i	$0.588310\pi$
$444$		0			0
$445$	−1.76477e10	−	3.05668e10i	−0.450038	−	0.779488i
$446$		0			0
$447$	6.73007e10	+	1.96809e10i	1.68574	+	0.492965i
$448$		0			0
$449$			4.33533e10i			1.06669i	0.845899	+	0.533344i	$0.179065\pi$
$449$			4.33533e10i			1.06669i	−0.845899	+	0.533344i	$0.820935\pi$
$450$		0			0
$451$	−6.00478e8			−0.0145141
$452$		0			0
$453$	2.73876e10	−	6.67604e9i	0.650371	−	0.158535i
$454$		0			0
$455$	1.35062e10	−	7.79782e9i	0.315129	−	0.181940i
$456$		0			0
$457$	2.40982e10	−	4.17392e10i	0.552483	−	0.956929i	−0.445611	−	0.895226i	$-0.647014\pi$
$457$	2.40982e10	−	4.17392e10i	0.552483	−	0.956929i	0.998095	−	0.0617024i	$-0.0196530\pi$
$458$		0			0
$459$	−2.51243e8	−	2.88026e8i	−0.00566035	−	0.00648904i
$460$		0			0
$461$	−4.81078e10	−	2.77751e10i	−1.06515	−	0.614967i	−0.138300	−	0.990390i	$-0.544164\pi$
$461$	−4.81078e10	−	2.77751e10i	−1.06515	−	0.614967i	−0.926853	+	0.375424i	$0.877497\pi$
$462$		0			0
$463$	7.55890e9	+	1.30924e10i	0.164488	+	0.284902i	0.936473	−	0.350739i	$-0.114069\pi$
$463$	7.55890e9	+	1.30924e10i	0.164488	+	0.284902i	−0.771985	+	0.635640i	$0.780736\pi$
$464$		0			0
$465$	8.39317e9	+	3.44319e10i	0.179521	+	0.736460i
$466$		0			0
$467$		−	5.74609e9i		−	0.120811i	−0.998174	−	0.0604053i	$-0.980761\pi$
$467$		−	5.74609e9i		−	0.120811i	0.998174	−	0.0604053i	$-0.0192393\pi$
$468$		0			0
$469$	6.19117e9			0.127962
$470$		0			0
$471$	7.26978e9	−	2.48596e10i	0.147719	−	0.505139i
$472$		0			0
$473$	−1.64876e9	+	9.51912e8i	−0.0329392	+	0.0190174i
$474$		0			0
$475$	1.48025e10	−	2.56387e10i	0.290778	−	0.503642i
$476$		0			0
$477$	4.45179e10	+	2.02258e9i	0.859927	+	0.0390690i
$478$		0			0
$479$	4.77601e10	+	2.75743e10i	0.907243	+	0.523797i	0.879543	−	0.475819i	$-0.157848\pi$
$479$	4.77601e10	+	2.75743e10i	0.907243	+	0.523797i	0.0276997	+	0.999616i	$0.491182\pi$
$480$		0			0
$481$	−9.66494e9	−	1.67402e10i	−0.180559	−	0.312737i
$482$		0			0
$483$	9.78476e8	−	9.35030e8i	0.0179788	−	0.0171806i
$484$		0			0
$485$			1.87272e10i			0.338458i
$486$		0			0
$487$	−6.37547e10			−1.13343			−0.566717	−	0.823913i	$-0.691787\pi$
$487$	−6.37547e10			−1.13343			−0.566717	+	0.823913i	$0.691787\pi$
$488$		0			0
$489$	5.08054e10	+	5.31661e10i	0.888535	+	0.929821i
$490$		0			0
$491$	−9.43363e10	+	5.44651e10i	−1.62313	+	0.937113i	−0.637051	+	0.770821i	$0.719846\pi$
$491$	−9.43363e10	+	5.44651e10i	−1.62313	+	0.937113i	−0.986077	+	0.166292i	$0.946821\pi$
$492$		0			0
$493$	2.93203e8	−	5.07843e8i	0.00496342	−	0.00859689i
$494$		0			0
$495$	−2.16824e8	+	4.77240e9i	−0.00361149	+	0.0794906i
$496$		0			0
$497$	−2.35986e10	−	1.36247e10i	−0.386778	−	0.223306i
$498$		0			0
$499$	−5.27256e10	−	9.13235e10i	−0.850393	−	1.47292i	−0.880854	−	0.473387i	$-0.843031\pi$
$499$	−5.27256e10	−	9.13235e10i	−0.850393	−	1.47292i	0.0304616	−	0.999536i	$-0.490302\pi$
$500$		0			0
$501$	1.57348e10	+	4.60139e9i	0.249754	+	0.0730362i
$502$		0			0
$503$			3.96999e10i			0.620179i	0.950707	+	0.310090i	$0.100359\pi$
$503$			3.96999e10i			0.620179i	−0.950707	+	0.310090i	$0.899641\pi$
$504$		0			0
$505$	−4.08685e10			−0.628381
$506$		0			0
$507$	−4.18751e10	+	1.02075e10i	−0.633758	+	0.154486i
$508$		0			0
$509$	−3.52328e10	+	2.03417e10i	−0.524899	+	0.303051i	−0.738937	−	0.673775i	$-0.764672\pi$
$509$	−3.52328e10	+	2.03417e10i	−0.524899	+	0.303051i	0.214038	+	0.976825i	$0.431338\pi$
$510$		0			0
$511$	−1.07270e10	+	1.85797e10i	−0.157323	+	0.272492i
$512$		0			0
$513$	1.82209e10	−	1.58940e10i	0.263088	−	0.229490i
$514$		0			0
$515$	−1.56008e11	−	9.00714e10i	−2.21778	−	1.28044i
$516$		0			0
$517$	−2.40133e9	−	4.15923e9i	−0.0336116	−	0.0582171i
$518$		0			0
$519$	1.18758e10	+	4.87188e10i	0.163679	+	0.671471i
$520$		0			0
$521$			1.22293e11i			1.65978i	0.557924	+	0.829892i	$0.311598\pi$
$521$			1.22293e11i			1.65978i	−0.557924	+	0.829892i	$0.688402\pi$
$522$		0			0
$523$	−3.92729e10			−0.524911			−0.262456	−	0.964944i	$-0.584532\pi$
$523$	−3.92729e10			−0.524911			−0.262456	+	0.964944i	$0.584532\pi$
$524$		0			0
$525$	1.34252e10	−	4.59085e10i	0.176719	−	0.604305i
$526$		0			0
$527$	2.67048e8	−	1.54180e8i	0.00346216	−	0.00199888i
$528$		0			0
$529$	−3.89860e10	+	6.75257e10i	−0.497836	+	0.862276i
$530$		0			0
$531$	−5.40672e10	−	1.04312e11i	−0.680074	−	1.31207i
$532$		0			0
$533$	1.22736e10	+	7.08619e9i	0.152077	+	0.0878020i
$534$		0			0
$535$	−1.28991e11	−	2.23419e11i	−1.57450	−	2.72712i
$536$		0			0
$537$	−3.23473e10	+	3.09111e10i	−0.388993	+	0.371721i
$538$		0			0
$539$			5.87634e8i			0.00696228i
$540$		0			0
$541$	7.12326e10			0.831552			0.415776	−	0.909467i	$-0.363510\pi$
$541$	7.12326e10			0.831552			0.415776	+	0.909467i	$0.363510\pi$
$542$		0			0
$543$	3.61565e10	+	3.78365e10i	0.415898	+	0.435223i
$544$		0			0
$545$	−1.53863e11	+	8.88331e10i	−1.74401	+	1.00691i
$546$		0			0
$547$	−5.62243e10	+	9.73834e10i	−0.628022	+	1.08777i	0.359926	+	0.932981i	$0.382802\pi$
$547$	−5.62243e10	+	9.73834e10i	−0.628022	+	1.08777i	−0.987948	+	0.154785i	$0.950531\pi$
$548$		0			0
$549$	6.52622e9	+	4.17391e9i	0.0718410	+	0.0459466i
$550$		0			0
$551$	3.21269e10	+	1.85485e10i	0.348548	+	0.201234i
$552$		0			0
$553$	8.83368e9	+	1.53004e10i	0.0944585	+	0.163607i
$554$		0			0
$555$	−9.10592e10	−	2.66287e10i	−0.959736	−	0.280659i
$556$		0			0
$557$			8.32007e10i			0.864383i	0.901782	+	0.432192i	$0.142260\pi$
$557$			8.32007e10i			0.864383i	−0.901782	+	0.432192i	$0.857740\pi$
$558$		0			0
$559$	4.49337e10			0.460177
$560$		0			0
$561$	4.03845e7	−	9.84418e6i	0.000407721	−	9.93866e-5i
$562$		0			0
$563$	−1.51580e10	+	8.75150e9i	−0.150872	+	0.0871062i	−0.573536	−	0.819181i	$-0.694429\pi$
$563$	−1.51580e10	+	8.75150e9i	−0.150872	+	0.0871062i	0.422663	+	0.906287i	$0.361095\pi$
$564$		0			0
$565$	9.70590e10	−	1.68111e11i	0.952450	−	1.64969i
$566$		0			0
$567$	2.25196e10	−	3.19204e10i	0.217885	−	0.308842i
$568$		0			0
$569$	8.93411e10	+	5.15811e10i	0.852319	+	0.492087i	0.861433	−	0.507872i	$-0.169568\pi$
$569$	8.93411e10	+	5.15811e10i	0.852319	+	0.492087i	−0.00911368	+	0.999958i	$0.502901\pi$
$570$		0			0
$571$	−1.73741e10	−	3.00928e10i	−0.163440	−	0.283086i	0.772660	−	0.634820i	$-0.218926\pi$
$571$	−1.73741e10	−	3.00928e10i	−0.163440	−	0.283086i	−0.936100	+	0.351734i	$0.885592\pi$
$572$		0			0
$573$	3.28734e9	+	1.34859e10i	0.0304948	+	0.125101i
$574$		0			0
$575$		−	1.19807e10i		−	0.109600i
$576$		0			0
$577$	−1.07803e10			−0.0972584			−0.0486292	−	0.998817i	$-0.515485\pi$
$577$	−1.07803e10			−0.0972584			−0.0486292	+	0.998817i	$0.515485\pi$
$578$		0			0
$579$	1.47610e9	−	5.04765e9i	0.0131341	−	0.0449133i
$580$		0			0
$581$	−3.98501e10	+	2.30075e10i	−0.349724	+	0.201913i
$582$		0			0
$583$	−2.42328e9	+	4.19724e9i	−0.0209763	+	0.0363320i
$584$		0			0
$585$	6.07505e10	−	9.49881e10i	0.518712	−	0.811046i
$586$		0			0
$587$	−8.29697e10	−	4.79026e10i	−0.698823	−	0.403466i	0.108086	−	0.994142i	$-0.465528\pi$
$587$	−8.29697e10	−	4.79026e10i	−0.698823	−	0.403466i	−0.806909	+	0.590676i	$0.798861\pi$
$588$		0			0
$589$	9.75367e9	+	1.68939e10i	0.0810414	+	0.140368i
$590$		0			0
$591$	−1.16867e11	+	1.11678e11i	−0.957950	+	0.915416i
$592$		0			0
$593$			2.07271e11i			1.67617i	0.545537	+	0.838087i	$0.316326\pi$
$593$			2.07271e11i			1.67617i	−0.545537	+	0.838087i	$0.683674\pi$
$594$		0			0
$595$	−6.66009e8			−0.00531389
$596$		0			0
$597$	5.58551e10	+	5.84503e10i	0.439709	+	0.460140i
$598$		0			0
$599$	−1.46693e11	+	8.46935e10i	−1.13947	+	0.657874i	−0.946300	−	0.323291i	$-0.895211\pi$
$599$	−1.46693e11	+	8.46935e10i	−1.13947	+	0.657874i	−0.193172	+	0.981165i	$0.561877\pi$
$600$		0			0
$601$	−6.17684e10	+	1.06986e11i	−0.473444	+	0.820028i	−0.999538	−	0.0303979i	$-0.990323\pi$
$601$	−6.17684e10	+	1.06986e11i	−0.473444	+	0.820028i	0.526094	+	0.850426i	$0.323656\pi$
$602$		0			0
$603$	3.97400e10	−	2.05981e10i	0.300579	−	0.155796i
$604$		0			0
$605$	1.88988e11	+	1.09112e11i	1.41063	+	0.814425i
$606$		0			0
$607$	−2.96840e10	−	5.14141e10i	−0.218659	−	0.378728i	0.735739	−	0.677265i	$-0.236835\pi$
$607$	−2.96840e10	−	5.14141e10i	−0.218659	−	0.378728i	−0.954398	+	0.298537i	$0.903502\pi$
$608$		0			0
$609$	5.75261e10	+	1.68225e10i	0.418212	+	0.122299i
$610$		0			0
$611$			1.13352e11i			0.813323i
$612$		0			0
$613$	2.10962e11			1.49404			0.747019	−	0.664802i	$-0.231484\pi$
$613$	2.10962e11			1.49404			0.747019	+	0.664802i	$0.231484\pi$
$614$		0			0
$615$	6.75806e10	−	1.64735e10i	0.472413	−	0.115156i
$616$		0			0
$617$	1.61309e10	−	9.31318e9i	0.111306	−	0.0642624i	−0.443314	−	0.896367i	$-0.646197\pi$
$617$	1.61309e10	−	9.31318e9i	0.111306	−	0.0642624i	0.554619	+	0.832104i	$0.312864\pi$
$618$		0			0
$619$	1.39714e11	−	2.41992e11i	0.951652	−	1.64831i	0.209801	−	0.977744i	$-0.432718\pi$
$619$	1.39714e11	−	2.41992e11i	0.951652	−	1.64831i	0.741851	−	0.670565i	$-0.233948\pi$
$620$		0			0
$621$	3.16980e9	−	9.25718e9i	0.0213140	−	0.0622461i
$622$		0			0
$623$	−2.71831e10	−	1.56942e10i	−0.180446	−	0.104180i
$624$		0			0
$625$	−8.32345e9	−	1.44166e10i	−0.0545485	−	0.0944808i
$626$		0			0
$627$	6.22757e8	+	2.55478e9i	0.00402948	+	0.0165304i
$628$		0			0
$629$			8.25479e8i			0.00527356i
$630$		0			0
$631$	1.82543e11			1.15146			0.575728	−	0.817642i	$-0.304719\pi$
$631$	1.82543e11			1.15146			0.575728	+	0.817642i	$0.304719\pi$
$632$		0			0
$633$	−4.60861e10	+	1.57595e11i	−0.287048	+	0.981587i
$634$		0			0
$635$	−8.36342e10	+	4.82862e10i	−0.514386	+	0.296981i
$636$		0			0
$637$	6.93462e9	−	1.20111e10i	0.0421177	−	0.0729500i
$638$		0			0
$639$	−1.96805e11	−	8.94141e9i	−1.18041	−	0.0536294i
$640$		0			0
$641$	−3.15642e10	−	1.82236e10i	−0.186966	−	0.107945i	0.403595	−	0.914938i	$-0.367760\pi$
$641$	−3.15642e10	−	1.82236e10i	−0.186966	−	0.107945i	−0.590561	+	0.806993i	$0.701094\pi$
$642$		0			0
$643$	−3.54335e10	−	6.13726e10i	−0.207286	−	0.359030i	0.743573	−	0.668655i	$-0.233130\pi$
$643$	−3.54335e10	−	6.13726e10i	−0.207286	−	0.359030i	−0.950859	+	0.309625i	$0.899796\pi$
$644$		0			0
$645$	1.59444e11	−	1.52365e11i	0.921234	−	0.880330i
$646$		0			0
$647$			2.10799e11i			1.20296i	0.798888	+	0.601480i	$0.205422\pi$
$647$			2.10799e11i			1.20296i	−0.798888	+	0.601480i	$0.794578\pi$
$648$		0			0
$649$	1.27778e10			0.0720242
$650$		0			0
$651$	2.17742e10	+	2.27860e10i	0.121232	+	0.126865i
$652$		0			0
$653$	1.77144e11	−	1.02274e11i	0.974259	−	0.562489i	0.0737272	−	0.997278i	$-0.476511\pi$
$653$	1.77144e11	−	1.02274e11i	0.974259	−	0.562489i	0.900532	+	0.434790i	$0.143177\pi$
$654$		0			0
$655$	3.28262e9	−	5.68566e9i	0.0178343	−	0.0308898i
$656$		0			0
$657$	−7.03975e9	+	1.54948e11i	−0.0377829	+	0.831620i
$658$		0			0
$659$	−2.16350e11	−	1.24910e11i	−1.14714	−	0.662300i	−0.198950	−	0.980010i	$-0.563753\pi$
$659$	−2.16350e11	−	1.24910e11i	−1.14714	−	0.662300i	−0.948188	+	0.317709i	$0.897086\pi$
$660$		0			0
$661$	−1.37099e11	−	2.37463e11i	−0.718174	−	1.24391i	−0.961722	−	0.274026i	$-0.911645\pi$
$661$	−1.37099e11	−	2.37463e11i	−0.718174	−	1.24391i	0.243548	−	0.969889i	$-0.421689\pi$
$662$		0			0
$663$	−9.41620e8	−	2.75361e8i	−0.00487328	−	0.00142511i
$664$		0			0
$665$		−	4.21327e10i		−	0.215443i
$666$		0			0
$667$	1.50125e10			0.0758492
$668$		0			0
$669$	3.08951e11	−	7.53104e10i	1.54236	−	0.375968i
$670$		0			0
$671$	−7.29632e8	+	4.21253e8i	−0.00359926	+	0.00207804i
$672$		0			0
$673$	−7.29802e10	+	1.26405e11i	−0.355750	+	0.616177i	−0.987246	−	0.159202i	$-0.949108\pi$
$673$	−7.29802e10	+	1.26405e11i	−0.355750	+	0.616177i	0.631496	+	0.775379i	$0.282441\pi$
$674$		0			0
$675$	−6.65644e10	−	3.39344e11i	−0.320647	−	1.63465i
$676$		0			0
$677$	2.08276e11	+	1.20248e11i	0.991481	+	0.572432i	0.905717	−	0.423884i	$-0.139333\pi$
$677$	2.08276e11	+	1.20248e11i	0.991481	+	0.572432i	0.0857643	+	0.996315i	$0.472667\pi$
$678$		0			0
$679$	8.32706e9	+	1.44229e10i	0.0391753	+	0.0678536i
$680$		0			0
$681$	1.13059e10	+	4.63809e10i	0.0525673	+	0.215651i
$682$		0			0
$683$		−	8.65685e10i		−	0.397811i	−0.980019	−	0.198906i	$-0.936261\pi$
$683$		−	8.65685e10i		−	0.397811i	0.980019	−	0.198906i	$-0.0637387\pi$
$684$		0			0
$685$	5.65730e9			0.0256949
$686$		0			0
$687$	6.81104e10	−	2.32909e11i	0.305764	−	1.04559i
$688$		0			0
$689$	9.90625e10	−	5.71938e10i	0.439575	−	0.253788i
$690$		0			0
$691$	−2.16063e11	+	3.74231e11i	−0.947692	+	1.64145i	−0.197424	+	0.980318i	$0.563257\pi$
$691$	−2.16063e11	+	3.74231e11i	−0.947692	+	1.64145i	−0.750269	+	0.661133i	$0.770076\pi$
$692$		0			0
$693$	1.95506e9	+	3.77191e9i	0.00847671	+	0.0163542i
$694$		0			0
$695$	−6.23695e10	−	3.60091e10i	−0.267321	−	0.154338i
$696$		0			0
$697$	−3.02615e8	−	5.24144e8i	−0.00128221	−	0.00222085i
$698$		0			0
$699$	9.75056e8	−	9.31762e8i	0.00408433	−	0.00390298i
$700$		0			0
$701$		−	2.61857e11i		−	1.08441i	−0.840248	−	0.542203i	$-0.817590\pi$
$701$		−	2.61857e11i		−	1.08441i	0.840248	−	0.542203i	$-0.182410\pi$
$702$		0			0
$703$	−5.22210e10			−0.213808
$704$		0			0
$705$	3.84361e11	+	4.02220e11i	1.55590	+	1.62820i
$706$		0			0
$707$	−3.14752e10	+	1.81722e10i	−0.125977	+	0.0727327i
$708$		0			0
$709$	9.91803e10	−	1.71785e11i	0.392501	−	0.679831i	−0.600278	−	0.799791i	$-0.704943\pi$
$709$	9.91803e10	−	1.71785e11i	0.392501	−	0.679831i	0.992779	+	0.119960i	$0.0382768\pi$
$710$		0			0
$711$	1.07606e11	+	6.88206e10i	0.421075	+	0.269302i
$712$		0			0
$713$	6.83668e9	+	3.94716e9i	0.0264537	+	0.0152731i
$714$		0			0
$715$	6.13127e9	+	1.06197e10i	0.0234599	+	0.0406338i
$716$		0			0
$717$	−2.94640e11	−	8.61625e10i	−1.11485	−	0.326018i
$718$		0			0
$719$			4.16778e11i			1.55951i	0.626082	+	0.779757i	$0.284658\pi$
$719$			4.16778e11i			1.55951i	−0.626082	+	0.779757i	$0.715342\pi$
$720$		0			0
$721$	−1.60201e11			−0.592823
$722$		0			0
$723$	−1.88451e11	+	4.59370e10i	−0.689675	+	0.168116i
$724$		0			0
$725$	4.59483e11	−	2.65283e11i	1.66310	−	0.960190i
$726$		0			0
$727$	−2.09398e11	+	3.62688e11i	−0.749609	+	1.29836i	0.198401	+	0.980121i	$0.436425\pi$
$727$	−2.09398e11	+	3.62688e11i	−0.749609	+	1.29836i	−0.948010	+	0.318240i	$0.896908\pi$
$728$		0			0
$729$	3.83495e10	−	2.79814e11i	0.135784	−	0.990738i
$730$		0			0
$731$	−1.66181e9	−	9.59444e8i	−0.00581983	−	0.00336008i
$732$		0			0
$733$	1.36861e11	+	2.37051e11i	0.474094	+	0.821156i	0.999560	−	0.0296592i	$-0.00944219\pi$
$733$	1.36861e11	+	2.37051e11i	0.474094	+	0.821156i	−0.525466	+	0.850815i	$0.676109\pi$
$734$		0			0
$735$	−1.61212e10	−	6.61350e10i	−0.0552392	−	0.226612i
$736$		0			0
$737$			4.86799e9i			0.0164998i
$738$		0			0
$739$	−2.36847e11			−0.794128			−0.397064	−	0.917791i	$-0.629971\pi$
$739$	−2.36847e11			−0.794128			−0.397064	+	0.917791i	$0.629971\pi$
$740$		0			0
$741$	1.74197e10	−	5.95683e10i	0.0577788	−	0.197580i
$742$		0			0
$743$	−8.79269e10	+	5.07646e10i	−0.288514	+	0.166573i	−0.637271	−	0.770640i	$-0.719937\pi$
$743$	−8.79269e10	+	5.07646e10i	−0.288514	+	0.166573i	0.348758	+	0.937213i	$0.386604\pi$
$744$		0			0
$745$	4.41689e11	−	7.65028e11i	1.43381	−	2.48343i
$746$		0			0
$747$	−1.79244e11	+	2.80262e11i	−0.575656	+	0.900082i
$748$		0			0
$749$	−1.98687e11	−	1.14712e11i	−0.631308	−	0.364486i
$750$		0			0
$751$	−2.75936e11	−	4.77935e11i	−0.867457	−	1.50248i	−0.864587	−	0.502484i	$-0.832420\pi$
$751$	−2.75936e11	−	4.77935e11i	−0.867457	−	1.50248i	−0.00287060	−	0.999996i	$-0.500914\pi$
$752$		0			0
$753$	2.84784e11	−	2.72139e11i	0.885799	−	0.846468i
$754$		0			0
$755$		−	3.55138e11i		−	1.09297i
$756$		0			0
$757$	−8.78820e10			−0.267619			−0.133809	−	0.991007i	$-0.542721\pi$
$757$	−8.78820e10			−0.267619			−0.133809	+	0.991007i	$0.542721\pi$
$758$		0			0
$759$	7.35195e8	+	7.69356e8i	0.00221532	+	0.00231825i
$760$		0			0
$761$	5.26464e11	−	3.03954e11i	1.56975	−	0.906294i	0.573550	−	0.819171i	$-0.305566\pi$
$761$	5.26464e11	−	3.03954e11i	1.56975	−	0.906294i	0.996198	−	0.0871233i	$-0.0277674\pi$
$762$		0			0
$763$	−7.89994e10	+	1.36831e11i	−0.233091	+	0.403725i
$764$		0			0
$765$	−4.27499e9	+	2.21582e9i	−0.0124821	+	0.00646976i
$766$		0			0
$767$	−2.61176e11	−	1.50790e11i	−0.754662	−	0.435704i
$768$		0			0
$769$	5.43447e10	+	9.41278e10i	0.155400	+	0.269161i	0.933205	−	0.359345i	$-0.117000\pi$
$769$	5.43447e10	+	9.41278e10i	0.155400	+	0.269161i	−0.777804	+	0.628506i	$0.783667\pi$
$770$		0			0
$771$	4.94533e11	+	1.44618e11i	1.39952	+	0.409265i
$772$		0			0
$773$		−	5.50494e11i		−	1.54182i	−0.636942	−	0.770912i	$-0.719801\pi$
$773$		−	5.50494e11i		−	1.54182i	0.636942	−	0.770912i	$-0.280199\pi$
$774$		0			0
$775$	2.78997e11			0.773380
$776$		0			0
$777$	−8.19704e10	+	1.99812e10i	−0.224892	+	0.0548199i
$778$		0			0
$779$	3.31581e10	−	1.91439e10i	0.0900410	−	0.0519852i
$780$		0			0
$781$	1.07128e10	−	1.85551e10i	0.0287938	−	0.0498724i
$782$		0			0
$783$	4.25218e11	−	8.34092e10i	1.13127	−	0.221905i
$784$		0			0
$785$	−2.82587e11	−	1.63152e11i	−0.744173	−	0.429649i
$786$		0			0
$787$	−4.40043e10	−	7.62177e10i	−0.114709	−	0.198681i	0.802955	−	0.596040i	$-0.203260\pi$
$787$	−4.40043e10	−	7.62177e10i	−0.114709	−	0.198681i	−0.917663	+	0.397359i	$0.869927\pi$
$788$		0			0
$789$	1.07238e11	+	4.39932e11i	0.276721	+	1.13521i
$790$		0			0
$791$		−	1.72629e11i		−	0.440970i
$792$		0			0
$793$	1.98847e10			0.0502836
$794$		0			0
$795$	1.57580e11	−	5.38857e11i	0.394486	−	1.34898i
$796$		0			0
$797$	−1.55939e11	+	9.00316e10i	−0.386476	+	0.223132i	−0.680632	−	0.732625i	$-0.738295\pi$
$797$	−1.55939e11	+	9.00316e10i	−0.386476	+	0.223132i	0.294156	+	0.955757i	$0.404961\pi$
$798$		0			0
$799$	2.42033e9	−	4.19214e9i	0.00593865	−	0.0102860i
$800$		0			0
$801$	−2.26698e11	−	1.02995e10i	−0.550702	−	0.0250200i
$802$		0			0
$803$	−1.46088e10	−	8.43440e9i	−0.0351360	−	0.0202858i
$804$		0			0
$805$	−8.52522e9	−	1.47661e10i	−0.0203012	−	0.0351628i
$806$		0			0
$807$	3.46670e11	−	3.31277e11i	0.817376	−	0.781083i
$808$		0			0
$809$		−	5.54208e11i		−	1.29383i	−0.762560	−	0.646917i	$-0.776058\pi$
$809$		−	5.54208e11i		−	1.29383i	0.762560	−	0.646917i	$-0.223942\pi$
$810$		0			0
$811$	1.12131e11			0.259205			0.129603	−	0.991566i	$-0.458630\pi$
$811$	1.12131e11			0.259205			0.129603	+	0.991566i	$0.458630\pi$
$812$		0			0
$813$	2.29401e11	+	2.40060e11i	0.525089	+	0.549487i
$814$		0			0
$815$	8.02326e11	−	4.63223e11i	1.81853	−	1.04993i
$816$		0			0
$817$	6.06958e10	−	1.05128e11i	0.136229	−	0.235956i
$818$		0			0
$819$	4.55095e9	−	1.00169e11i	0.0101150	−	0.222636i
$820$		0			0
$821$	4.95112e10	+	2.85853e10i	0.108976	+	0.0629173i	0.553497	−	0.832851i	$-0.313293\pi$
$821$	4.95112e10	+	2.85853e10i	0.108976	+	0.0629173i	−0.444521	+	0.895768i	$0.646626\pi$
$822$		0			0
$823$	3.87278e11	+	6.70785e11i	0.844157	+	1.46212i	0.886351	+	0.463014i	$0.153232\pi$
$823$	3.87278e11	+	6.70785e11i	0.844157	+	1.46212i	−0.0421935	+	0.999109i	$0.513435\pi$
$824$		0			0
$825$	3.60970e10	+	1.05559e10i	0.0779211	+	0.0227867i
$826$		0			0
$827$		−	2.54310e11i		−	0.543678i	−0.962343	−	0.271839i	$-0.912368\pi$
$827$		−	2.54310e11i		−	0.543678i	0.962343	−	0.271839i	$-0.0876318\pi$
$828$		0			0
$829$	−5.21968e11			−1.10516			−0.552581	−	0.833459i	$-0.686357\pi$
$829$	−5.21968e11			−1.10516			−0.552581	+	0.833459i	$0.686357\pi$
$830$		0			0
$831$	−8.38890e11	+	2.04489e11i	−1.75914	+	0.428811i
$832$		0			0
$833$	−5.12932e8	+	2.96142e8i	−0.00106532	+	0.000615063i
$834$		0			0
$835$	1.03267e11	−	1.78863e11i	0.212429	−	0.367938i
$836$		0			0
$837$	2.15574e11	+	7.38158e10i	0.439232	+	0.150400i
$838$		0			0
$839$	1.91533e11	+	1.10582e11i	0.386541	+	0.223170i	0.680660	−	0.732599i	$-0.261693\pi$
$839$	1.91533e11	+	1.10582e11i	0.386541	+	0.223170i	−0.294119	+	0.955769i	$0.595026\pi$
$840$		0			0
$841$	8.22920e10	+	1.42534e11i	0.164503	+	0.284928i
$842$		0			0
$843$	−7.16898e9	−	2.94098e10i	−0.0141954	−	0.0582347i
$844$		0			0
$845$			5.42998e11i			1.06505i
$846$		0			0
$847$	1.94067e11			0.377067
$848$		0			0
$849$	−2.19998e11	+	7.52302e11i	−0.423436	+	1.44798i
$850$		0			0
$851$	−1.83017e10	+	1.05665e10i	−0.0348959	+	0.0201471i
$852$		0			0
$853$	−7.41088e10	+	1.28360e11i	−0.139982	+	0.242457i	−0.927490	−	0.373849i	$-0.878038\pi$
$853$	−7.41088e10	+	1.28360e11i	−0.139982	+	0.242457i	0.787507	+	0.616305i	$0.211371\pi$
$854$		0			0
$855$	−1.40176e11	−	2.70442e11i	−0.262306	−	0.506069i
$856$		0			0
$857$	3.86953e11	+	2.23407e11i	0.717356	+	0.414166i	0.813779	−	0.581175i	$-0.197407\pi$
$857$	3.86953e11	+	2.23407e11i	0.717356	+	0.414166i	−0.0964228	+	0.995340i	$0.530740\pi$
$858$		0			0
$859$	3.42493e11	+	5.93216e11i	0.629042	+	1.08953i	0.987744	+	0.156080i	$0.0498858\pi$
$859$	3.42493e11	+	5.93216e11i	0.629042	+	1.08953i	−0.358703	+	0.933452i	$0.616781\pi$
$860$		0			0
$861$	4.47228e10	−	4.27370e10i	0.0813797	−	0.0777663i
$862$		0			0
$863$			3.70546e11i			0.668035i	0.942567	+	0.334017i	$0.108404\pi$
$863$			3.70546e11i			0.668035i	−0.942567	+	0.334017i	$0.891596\pi$
$864$		0			0
$865$	6.31742e11			1.12843
$866$		0			0
$867$	−3.90340e11	−	4.08477e11i	−0.690823	−	0.722922i
$868$		0			0
$869$	−1.20304e10	+	6.94574e9i	−0.0210960	+	0.0121798i
$870$		0			0
$871$	5.74468e10	−	9.95007e10i	0.0998144	−	0.172884i
$872$		0			0
$873$	1.01435e11	+	6.48736e10i	0.174635	+	0.111689i
$874$		0			0
$875$	−2.08581e11	−	1.20424e11i	−0.355829	−	0.205438i
$876$		0			0
$877$	9.87976e9	+	1.71122e10i	0.0167012	+	0.0289273i	0.874255	−	0.485467i	$-0.161350\pi$
$877$	9.87976e9	+	1.71122e10i	0.0167012	+	0.0289273i	−0.857554	+	0.514394i	$0.828017\pi$
$878$		0			0
$879$	−1.78965e11	−	5.23353e10i	−0.299787	−	0.0876676i
$880$		0			0
$881$		−	7.55594e10i		−	0.125425i	−0.998032	−	0.0627126i	$-0.980025\pi$
$881$		−	7.55594e10i		−	0.125425i	0.998032	−	0.0627126i	$-0.0199751\pi$
$882$		0			0
$883$	−3.31377e10			−0.0545104			−0.0272552	−	0.999629i	$-0.508677\pi$
$883$	−3.31377e10			−0.0545104			−0.0272552	+	0.999629i	$0.508677\pi$
$884$		0			0
$885$	−1.43808e12	+	3.50548e11i	−2.34428	+	0.571444i
$886$		0			0
$887$	−2.79375e10	+	1.61297e10i	−0.0451329	+	0.0260575i	−0.522397	−	0.852703i	$-0.674962\pi$
$887$	−2.79375e10	+	1.61297e10i	−0.0451329	+	0.0260575i	0.477264	+	0.878760i	$0.341629\pi$
$888$		0			0
$889$	−4.29410e10	+	7.43761e10i	−0.0687489	+	0.119077i
$890$		0			0
$891$	2.50984e10	+	1.77067e10i	0.0398230	+	0.0280948i
$892$		0			0
$893$	2.65201e11	+	1.53114e11i	0.417031	+	0.240773i
$894$		0			0
$895$	2.81834e11	+	4.88152e11i	0.439240	+	0.760786i
$896$		0			0
$897$	−5.94813e9	−	2.44015e10i	−0.00918779	−	0.0376917i
$898$		0			0
$899$			3.49600e11i			0.535220i
$900$		0			0
$901$	−4.88490e9			−0.00741236
$902$		0			0
$903$	5.50481e10	−	1.88242e11i	0.0827926	−	0.283117i
$904$		0			0
$905$	5.70988e11	−	3.29660e11i	0.851203	−	0.491442i
$906$		0			0
$907$	−1.97817e11	+	3.42630e11i	−0.292304	+	0.506286i	−0.974354	−	0.225020i	$-0.927755\pi$
$907$	−1.97817e11	+	3.42630e11i	−0.292304	+	0.506286i	0.682050	+	0.731306i	$0.261089\pi$
$908$		0			0
$909$	−1.41574e11	+	2.21362e11i	−0.207362	+	0.324226i
$910$		0			0
$911$	−1.73594e11	−	1.00224e11i	−0.252035	−	0.145512i	0.368661	−	0.929564i	$-0.379816\pi$
$911$	−1.73594e11	−	1.00224e11i	−0.252035	−	0.145512i	−0.620696	+	0.784052i	$0.713150\pi$
$912$		0			0
$913$	−1.80903e10	−	3.13333e10i	−0.0260353	−	0.0450945i
$914$		0			0
$915$	7.05595e10	−	6.74265e10i	0.100663	−	0.0961937i
$916$		0			0
$917$		−	5.83847e9i		−	0.00825699i
$918$		0			0
$919$	3.89225e11			0.545680			0.272840	−	0.962059i	$-0.412037\pi$
$919$	3.89225e11			0.545680			0.272840	+	0.962059i	$0.412037\pi$
$920$		0			0
$921$	1.03967e11	+	1.08798e11i	0.144497	+	0.151211i
$922$		0			0
$923$	−4.37935e11	+	2.52842e11i	−0.603397	+	0.348371i
$924$		0			0
$925$	−3.73437e11	+	6.46811e11i	−0.510094	+	0.883508i
$926$		0			0
$927$	−1.02830e12	+	5.32991e11i	−1.39252	+	0.721774i
$928$		0			0
$929$	−3.48400e11	−	2.01149e11i	−0.467752	−	0.270057i	0.247546	−	0.968876i	$-0.420376\pi$
$929$	−3.48400e11	−	2.01149e11i	−0.467752	−	0.270057i	−0.715298	+	0.698819i	$0.753709\pi$
$930$		0			0
$931$	−1.87344e10	−	3.24488e10i	−0.0249368	−	0.0431917i
$932$		0			0
$933$	−3.72840e11	−	1.09031e11i	−0.492035	−	0.143887i
$934$		0			0
$935$		−	5.23670e8i		−	0.000685190i
$936$		0			0
$937$	−9.81855e11			−1.27376			−0.636882	−	0.770961i	$-0.719776\pi$
$937$	−9.81855e11			−1.27376			−0.636882	+	0.770961i	$0.719776\pi$
$938$		0			0
$939$	−9.06752e11	+	2.21031e11i	−1.16634	+	0.284309i
$940$		0			0
$941$	−6.15338e11	+	3.55265e11i	−0.784793	+	0.453100i	−0.838126	−	0.545476i	$-0.816349\pi$
$941$	−6.15338e11	+	3.55265e11i	−0.784793	+	0.453100i	0.0533333	+	0.998577i	$0.483015\pi$
$942$		0			0
$943$	7.74722e9	−	1.34186e10i	0.00979713	−	0.0169691i
$944$		0			0
$945$	−3.23510e11	−	3.70873e11i	−0.405659	−	0.465048i
$946$		0			0
$947$	−7.93177e11	−	4.57941e11i	−0.986211	−	0.569389i	−0.0820717	−	0.996626i	$-0.526154\pi$
$947$	−7.93177e11	−	4.57941e11i	−0.986211	−	0.569389i	−0.904140	+	0.427237i	$0.859487\pi$
$948$		0			0
$949$	1.99067e11	+	3.44795e11i	0.245434	+	0.425105i
$950$		0			0
$951$	−1.32706e11	−	5.44410e11i	−0.162244	−	0.665586i
$952$		0			0
$953$			1.60089e12i			1.94084i	0.241430	+	0.970418i	$0.422384\pi$
$953$			1.60089e12i			1.94084i	−0.241430	+	0.970418i	$0.577616\pi$
$954$		0			0
$955$	1.74873e11			0.210237
$956$		0			0
$957$	−1.32272e10	+	4.52317e10i	−0.0157696	+	0.0539256i
$958$		0			0
$959$	4.35702e9	−	2.51552e9i	0.00515128	−	0.00297409i
$960$		0			0
$961$	3.34527e11	−	5.79418e11i	0.392227	−	0.679358i
$962$		0			0
$963$	−1.65698e12	−	7.52814e10i	−1.92669	−	0.0875352i
$964$		0			0
$965$	−5.73783e10	−	3.31274e10i	−0.0661665	−	0.0382013i
$966$		0			0
$967$	7.30773e11	+	1.26574e12i	0.835750	+	1.44756i	0.893418	+	0.449226i	$0.148300\pi$
$967$	7.30773e11	+	1.26574e12i	0.835750	+	1.44756i	−0.0576679	+	0.998336i	$0.518366\pi$
$968$		0			0
$969$	−1.91617e9	+	1.83109e9i	−0.00217340	+	0.00207689i
$970$		0			0
$971$			9.28819e11i			1.04485i	0.852685	+	0.522425i	$0.174973\pi$
$971$			9.28819e11i			1.04485i	−0.852685	+	0.522425i	$0.825027\pi$
$972$		0			0
$973$	−6.40458e10			−0.0714561
$974$		0			0
$975$	−6.13245e11	−	6.41739e11i	−0.678602	−	0.710133i
$976$		0			0
$977$	2.92208e11	−	1.68707e11i	0.320711	−	0.185163i	−0.330998	−	0.943631i	$-0.607385\pi$
$977$	2.92208e11	−	1.68707e11i	0.320711	−	0.185163i	0.651710	+	0.758469i	$0.274052\pi$
$978$		0			0
$979$	1.23400e10	−	2.13735e10i	0.0134333	−	0.0232672i
$980$		0			0
$981$	−5.18446e10	+	1.14112e12i	−0.0559793	+	1.23213i
$982$		0			0
$983$	6.63290e11	+	3.82951e11i	0.710378	+	0.410137i	0.811201	−	0.584767i	$-0.198814\pi$
$983$	6.63290e11	+	3.82951e11i	0.710378	+	0.410137i	−0.100823	+	0.994904i	$0.532148\pi$
$984$		0			0
$985$	1.01824e12	+	1.76364e12i	1.08169	+	1.87355i
$986$		0			0
$987$	4.74866e11	+	1.38867e11i	0.500383	+	0.146329i
$988$		0			0
$989$		−	4.91253e10i		−	0.0513476i
$990$		0			0
$991$	−1.35084e12			−1.40059			−0.700294	−	0.713855i	$-0.746948\pi$
$991$	−1.35084e12			−1.40059			−0.700294	+	0.713855i	$0.746948\pi$
$992$		0			0
$993$	−9.78505e11	+	2.38522e11i	−1.00639	+	0.245319i
$994$		0			0
$995$	8.82071e11	−	5.09264e11i	0.899935	−	0.519578i
$996$		0			0
$997$	9.14799e10	−	1.58448e11i	0.0925860	−	0.160364i	−0.816013	−	0.578034i	$-0.803820\pi$
$997$	9.14799e10	−	1.58448e11i	0.0925860	−	0.160364i	0.908599	+	0.417671i	$0.137153\pi$
$998$		0			0
$999$	−4.59675e11	+	4.00972e11i	−0.461519	+	0.402580i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.13	✓	96
9.5	odd	6	inner	252.9.bg.a.113.13	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.13	✓	96	1.1	even	1	trivial
252.9.bg.a.113.13	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)

\(f(q)\)	\(=\)	\(q+(-55.9608 - 58.5610i) q^{3} +(-883.740 + 510.228i) q^{5} +(-453.746 + 785.912i) q^{7} +(-297.778 + 6554.24i) q^{9} +O(q^{10})\)	\(q+(-55.9608 - 58.5610i) q^{3} +(-883.740 + 510.228i) q^{5} +(-453.746 + 785.912i) q^{7} +(-297.778 + 6554.24i) q^{9} +(-617.947 - 356.772i) q^{11} +(8420.47 + 14584.7i) q^{13} +(79334.3 + 23200.0i) q^{15} -719.189i q^{17} +45497.0 q^{19} +(71415.8 - 17408.4i) q^{21} +(15945.2 - 9205.95i) q^{23} +(325352. - 563527. i) q^{25} +(400487. - 349342. i) q^{27} +(706132. + 407686. i) q^{29} +(214381. + 371318. i) q^{31} +(13687.9 + 56152.8i) q^{33} -926056. i q^{35} -1.14779e6 q^{37} +(382877. - 1.30928e6i) q^{39} +(728799. - 420772. i) q^{41} +(1.33406e6 - 2.31067e6i) q^{43} +(-3.08100e6 - 5.94418e6i) q^{45} +(5.82898e6 + 3.36536e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-42116.4 + 40246.4i) q^{51} -6.79223e6i q^{53} +728139. q^{55} +(-2.54605e6 - 2.66435e6i) q^{57} +(-1.55084e7 + 8.95379e6i) q^{59} +(590368. - 1.02255e6i) q^{61} +(-5.01594e6 - 3.20799e6i) q^{63} +(-1.48830e7 - 8.59271e6i) q^{65} +(-3.41114e6 - 5.90827e6i) q^{67} +(-1.43141e6 - 418593. i) q^{69} +3.00271e7i q^{71} +2.36409e7 q^{73} +(-5.12076e7 + 1.24824e7i) q^{75} +(560782. - 323768. i) q^{77} +(9.73416e6 - 1.68601e7i) q^{79} +(-4.28694e7 - 3.90342e6i) q^{81} +(4.39123e7 + 2.53528e7i) q^{83} +(366950. + 635576. i) q^{85} +(-1.56413e7 - 6.41662e7i) q^{87} +3.45879e7i q^{89} -1.52830e7 q^{91} +(9.74785e6 - 3.33336e7i) q^{93} +(-4.02075e7 + 2.32138e7i) q^{95} +(9.17589e6 - 1.58931e7i) q^{97} +(2.52238e6 - 3.94393e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 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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