LMFDB - Embedded newform 252.9.bg.a.29.12

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			29.12
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-62.9498 - 50.9738i) q^{3} +(-401.865 + 232.017i) q^{5} +(453.746 - 785.912i) q^{7} +(1364.35 + 6417.57i) q^{9} +O(q^{10})$	\(q+(-62.9498 - 50.9738i) q^{3} +(-401.865 + 232.017i) q^{5} +(453.746 - 785.912i) q^{7} +(1364.35 + 6417.57i) q^{9} +(-9479.62 - 5473.06i) q^{11} +(3676.57 + 6368.00i) q^{13} +(37124.1 + 5879.15i) q^{15} -115536. i q^{17} -64661.6 q^{19} +(-68624.1 + 26343.8i) q^{21} +(-365155. + 210823. i) q^{23} +(-87648.8 + 151812. i) q^{25} +(241242. - 473531. i) q^{27} +(-169336. - 97765.9i) q^{29} +(435443. + 754209. i) q^{31} +(317758. + 827740. i) q^{33} +421107. i q^{35} -284175. q^{37} +(93161.8 - 588273. i) q^{39} +(-1.81338e6 + 1.04696e6i) q^{41} +(-2.19653e6 + 3.80450e6i) q^{43} +(-2.03727e6 - 2.26245e6i) q^{45} +(-2.41792e6 - 1.39599e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-5.88929e6 + 7.27295e6i) q^{51} +4.39346e6i q^{53} +5.07937e6 q^{55} +(4.07043e6 + 3.29604e6i) q^{57} +(2.01066e7 - 1.16086e7i) q^{59} +(-1.32993e7 + 2.30350e7i) q^{61} +(5.66272e6 + 1.83969e6i) q^{63} +(-2.95497e6 - 1.70605e6i) q^{65} +(-1.29323e7 - 2.23994e7i) q^{67} +(3.37329e7 + 5.34210e6i) q^{69} -1.20891e7i q^{71} -9.83010e6 q^{73} +(1.32559e7 - 5.08875e6i) q^{75} +(-8.60269e6 + 4.96677e6i) q^{77} +(8.41906e6 - 1.45822e7i) q^{79} +(-3.93238e7 + 1.75117e7i) q^{81} +(6.50930e7 + 3.75815e7i) q^{83} +(2.68062e7 + 4.64298e7i) q^{85} +(5.67614e6 + 1.47860e7i) q^{87} +9.52903e6i q^{89} +6.67292e6 q^{91} +(1.10338e7 - 6.96734e7i) q^{93} +(2.59852e7 - 1.50026e7i) q^{95} +(-1.77249e7 + 3.07005e7i) q^{97} +(2.21902e7 - 6.83034e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−62.9498	−	50.9738i	−0.777158	−	0.629306i
$3$	−62.9498	−	50.9738i	−0.777158	−	0.629306i
$4$		0			0
$5$	−401.865	+	232.017i	−0.642984	+	0.371227i	−0.785763	−	0.618527i	$-0.787730\pi$
$5$	−401.865	+	232.017i	−0.642984	+	0.371227i	0.142779	+	0.989755i	$0.454396\pi$
$6$		0			0
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$8$		0			0
$9$	1364.35	+	6417.57i	0.207949	+	0.978140i
$10$		0			0
$11$	−9479.62	−	5473.06i	−0.647471	−	0.373818i	0.140016	−	0.990149i	$-0.455285\pi$
$11$	−9479.62	−	5473.06i	−0.647471	−	0.373818i	−0.787487	+	0.616332i	$0.788618\pi$
$12$		0			0
$13$	3676.57	+	6368.00i	0.128727	+	0.222961i	0.923184	−	0.384359i	$-0.125578\pi$
$13$	3676.57	+	6368.00i	0.128727	+	0.222961i	−0.794457	+	0.607321i	$0.792244\pi$
$14$		0			0
$15$	37124.1	+	5879.15i	0.733316	+	0.116131i
$16$		0			0
$17$		−	115536.i		−	1.38331i	−0.722226	−	0.691657i	$-0.756881\pi$
$17$		−	115536.i		−	1.38331i	0.722226	−	0.691657i	$-0.243119\pi$
$18$		0			0
$19$	−64661.6			−0.496172			−0.248086	−	0.968738i	$-0.579801\pi$
$19$	−64661.6			−0.496172			−0.248086	+	0.968738i	$0.579801\pi$
$20$		0			0
$21$	−68624.1	+	26343.8i	−0.352858	+	0.135457i
$22$		0			0
$23$	−365155.	+	210823.i	−1.30487	+	0.753365i	−0.981235	−	0.192817i	$-0.938238\pi$
$23$	−365155.	+	210823.i	−1.30487	+	0.753365i	−0.323633	+	0.946183i	$0.604904\pi$
$24$		0			0
$25$	−87648.8	+	151812.i	−0.224381	+	0.388639i
$26$		0			0
$27$	241242.	−	473531.i	0.453939	−	0.891033i
$28$		0			0
$29$	−169336.	−	97765.9i	−0.239418	−	0.138228i	0.375491	−	0.926826i	$-0.377474\pi$
$29$	−169336.	−	97765.9i	−0.239418	−	0.138228i	−0.614909	+	0.788598i	$0.710807\pi$
$30$		0			0
$31$	435443.	+	754209.i	0.471503	+	0.816667i	0.999469	−	0.0325989i	$-0.0103784\pi$
$31$	435443.	+	754209.i	0.471503	+	0.816667i	−0.527966	+	0.849266i	$0.677045\pi$
$32$		0			0
$33$	317758.	+	827740.i	0.267942	+	0.697972i
$34$		0			0
$35$			421107.i			0.280621i
$36$		0			0
$37$	−284175.			−0.151628			−0.0758139	−	0.997122i	$-0.524155\pi$
$37$	−284175.			−0.151628			−0.0758139	+	0.997122i	$0.524155\pi$
$38$		0			0
$39$	93161.8	−	588273.i	0.0402698	−	0.254285i
$40$		0			0
$41$	−1.81338e6	+	1.04696e6i	−0.641733	+	0.370505i	−0.785282	−	0.619139i	$-0.787482\pi$
$41$	−1.81338e6	+	1.04696e6i	−0.641733	+	0.370505i	0.143549	+	0.989643i	$0.454149\pi$
$42$		0			0
$43$	−2.19653e6	+	3.80450e6i	−0.642485	+	1.11282i	0.342392	+	0.939557i	$0.388763\pi$
$43$	−2.19653e6	+	3.80450e6i	−0.642485	+	1.11282i	−0.984876	+	0.173259i	$0.944570\pi$
$44$		0			0
$45$	−2.03727e6	−	2.26245e6i	−0.496820	−	0.551732i
$46$		0			0
$47$	−2.41792e6	−	1.39599e6i	−0.495507	−	0.286081i	0.231349	−	0.972871i	$-0.425686\pi$
$47$	−2.41792e6	−	1.39599e6i	−0.495507	−	0.286081i	−0.726856	+	0.686789i	$0.759019\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	−5.88929e6	+	7.27295e6i	−0.870527	+	1.07505i
$52$		0			0
$53$			4.39346e6i			0.556805i	0.960465	+	0.278402i	$0.0898049\pi$
$53$			4.39346e6i			0.556805i	−0.960465	+	0.278402i	$0.910195\pi$
$54$		0			0
$55$	5.07937e6			0.555085
$56$		0			0
$57$	4.07043e6	+	3.29604e6i	0.385604	+	0.312244i
$58$		0			0
$59$	2.01066e7	−	1.16086e7i	1.65933	−	0.958012i	0.686298	−	0.727320i	$-0.259234\pi$
$59$	2.01066e7	−	1.16086e7i	1.65933	−	0.958012i	0.973027	−	0.230692i	$-0.0740989\pi$
$60$		0			0
$61$	−1.32993e7	+	2.30350e7i	−0.960524	+	1.66368i	−0.239335	+	0.970937i	$0.576930\pi$
$61$	−1.32993e7	+	2.30350e7i	−0.960524	+	1.66368i	−0.721188	+	0.692739i	$0.756404\pi$
$62$		0			0
$63$	5.66272e6	+	1.83969e6i	0.359470	+	0.116784i
$64$		0			0
$65$	−2.95497e6	−	1.70605e6i	−0.165539	−	0.0955738i
$66$		0			0
$67$	−1.29323e7	−	2.23994e7i	−0.641766	−	1.11157i	−0.985038	−	0.172336i	$-0.944869\pi$
$67$	−1.29323e7	−	2.23994e7i	−0.641766	−	1.11157i	0.343272	−	0.939236i	$-0.388465\pi$
$68$		0			0
$69$	3.37329e7	+	5.34210e6i	1.48819	+	0.235676i
$70$		0			0
$71$		−	1.20891e7i		−	0.475731i	−0.971298	−	0.237865i	$-0.923552\pi$
$71$		−	1.20891e7i		−	0.475731i	0.971298	−	0.237865i	$-0.0764477\pi$
$72$		0			0
$73$	−9.83010e6			−0.346152			−0.173076	−	0.984909i	$-0.555371\pi$
$73$	−9.83010e6			−0.346152			−0.173076	+	0.984909i	$0.555371\pi$
$74$		0			0
$75$	1.32559e7	−	5.08875e6i	0.418952	−	0.160830i
$76$		0			0
$77$	−8.60269e6	+	4.96677e6i	−0.244721	+	0.141290i
$78$		0			0
$79$	8.41906e6	−	1.45822e7i	0.216150	−	0.374383i	−0.737478	−	0.675371i	$-0.763983\pi$
$79$	8.41906e6	−	1.45822e7i	0.216150	−	0.374383i	0.953628	+	0.300989i	$0.0973166\pi$
$80$		0			0
$81$	−3.93238e7	+	1.75117e7i	−0.913514	+	0.406806i
$82$		0			0
$83$	6.50930e7	+	3.75815e7i	1.37158	+	0.791884i	0.991127	−	0.132915i	$-0.0424337\pi$
$83$	6.50930e7	+	3.75815e7i	1.37158	+	0.791884i	0.380456	+	0.924799i	$0.375767\pi$
$84$		0			0
$85$	2.68062e7	+	4.64298e7i	0.513523	+	0.889449i
$86$		0			0
$87$	5.67614e6	+	1.47860e7i	0.0990778	+	0.258092i
$88$		0			0
$89$			9.52903e6i			0.151876i	0.997113	+	0.0759379i	$0.0241951\pi$
$89$			9.52903e6i			0.151876i	−0.997113	+	0.0759379i	$0.975805\pi$
$90$		0			0
$91$	6.67292e6			0.0973084
$92$		0			0
$93$	1.10338e7	−	6.96734e7i	0.147501	−	0.931398i
$94$		0			0
$95$	2.59852e7	−	1.50026e7i	0.319031	−	0.184192i
$96$		0			0
$97$	−1.77249e7	+	3.07005e7i	−0.200216	+	0.346784i	−0.948598	−	0.316484i	$-0.897498\pi$
$97$	−1.77249e7	+	3.07005e7i	−0.200216	+	0.346784i	0.748382	+	0.663268i	$0.230831\pi$
$98$		0			0
$99$	2.21902e7	−	6.83034e7i	0.231005	−	0.711052i
$100$		0			0
$101$	−1.13260e8	−	6.53906e7i	−1.08841	−	0.628391i	−0.155254	−	0.987875i	$-0.549620\pi$
$101$	−1.13260e8	−	6.53906e7i	−1.08841	−	0.628391i	−0.933151	+	0.359483i	$0.882953\pi$
$102$		0			0
$103$	−7.26725e7	−	1.25872e8i	−0.645685	−	1.11836i	−0.984143	−	0.177378i	$-0.943238\pi$
$103$	−7.26725e7	−	1.25872e8i	−0.645685	−	1.11836i	0.338457	−	0.940982i	$-0.390095\pi$
$104$		0			0
$105$	2.14654e7	−	2.65086e7i	0.176597	−	0.218087i
$106$		0			0
$107$		−	1.06268e8i		−	0.810714i	−0.914159	−	0.405357i	$-0.867147\pi$
$107$		−	1.06268e8i		−	0.810714i	0.914159	−	0.405357i	$-0.132853\pi$
$108$		0			0
$109$	1.82831e8			1.29522			0.647612	−	0.761971i	$-0.275768\pi$
$109$	1.82831e8			1.29522			0.647612	+	0.761971i	$0.275768\pi$
$110$		0			0
$111$	1.78887e7	+	1.44855e7i	0.117839	+	0.0954202i
$112$		0			0
$113$	2.11580e8	−	1.22156e8i	1.29766	−	0.749205i	0.317661	−	0.948204i	$-0.397103\pi$
$113$	2.11580e8	−	1.22156e8i	1.29766	−	0.749205i	0.980000	+	0.199000i	$0.0637692\pi$
$114$		0			0
$115$	9.78288e7	−	1.69444e8i	0.559339	−	0.968804i
$116$		0			0
$117$	−3.58510e7	+	3.22829e7i	−0.191319	+	0.172277i
$118$		0			0
$119$	−9.08009e7	−	5.24239e7i	−0.452796	−	0.261422i
$120$		0			0
$121$	−4.72706e7	−	8.18751e7i	−0.220521	−	0.381953i
$122$		0			0
$123$	1.67519e8	+	2.65292e7i	0.731888	+	0.115905i
$124$		0			0
$125$		−	2.62607e8i		−	1.07564i
$126$		0			0
$127$	−3.29977e8			−1.26844			−0.634218	−	0.773155i	$-0.718678\pi$
$127$	−3.29977e8			−1.26844			−0.634218	+	0.773155i	$0.718678\pi$
$128$		0			0
$129$	3.32200e8	−	1.27527e8i	1.19961	−	0.460515i
$130$		0			0
$131$	3.26552e8	−	1.88535e8i	1.10884	−	0.640187i	0.170309	−	0.985391i	$-0.445523\pi$
$131$	3.26552e8	−	1.88535e8i	1.10884	−	0.640187i	0.938528	+	0.345203i	$0.112190\pi$
$132$		0			0
$133$	−2.93400e7	+	5.08183e7i	−0.0937676	+	0.162410i
$134$		0			0
$135$	1.29205e7	+	2.46268e8i	0.0388995	+	0.741435i
$136$		0			0
$137$	4.40125e8	+	2.54106e8i	1.24938	+	0.721329i	0.970985	−	0.239139i	$-0.0768651\pi$
$137$	4.40125e8	+	2.54106e8i	1.24938	+	0.721329i	0.278392	+	0.960467i	$0.410198\pi$
$138$		0			0
$139$	2.52590e8	+	4.37499e8i	0.676640	+	1.17197i	0.975987	+	0.217830i	$0.0698978\pi$
$139$	2.52590e8	+	4.37499e8i	0.676640	+	1.17197i	−0.299347	+	0.954144i	$0.596769\pi$
$140$		0			0
$141$	8.10488e7	+	2.11127e8i	0.205055	+	0.534156i
$142$		0			0
$143$		−	8.04884e7i		−	0.192481i
$144$		0			0
$145$	9.07334e7			0.205256
$146$		0			0
$147$	−1.04340e7	+	6.58859e7i	−0.0223451	+	0.141099i
$148$		0			0
$149$	−1.11577e8	+	6.44188e7i	−0.226375	+	0.130698i	−0.608899	−	0.793248i	$-0.708388\pi$
$149$	−1.11577e8	+	6.44188e7i	−0.226375	+	0.130698i	0.382524	+	0.923946i	$0.375055\pi$
$150$		0			0
$151$	4.44800e8	−	7.70416e8i	0.855573	−	1.48190i	−0.0205401	−	0.999789i	$-0.506539\pi$
$151$	4.44800e8	−	7.70416e8i	0.855573	−	1.48190i	0.876113	−	0.482106i	$-0.160128\pi$
$152$		0			0
$153$	7.41459e8	−	1.57632e8i	1.35307	−	0.287659i
$154$		0			0
$155$	−3.49978e8	−	2.02060e8i	−0.606338	−	0.350069i
$156$		0			0
$157$	3.44748e8	+	5.97120e8i	0.567417	+	0.982796i	0.996820	+	0.0796827i	$0.0253907\pi$
$157$	3.44748e8	+	5.97120e8i	0.567417	+	0.982796i	−0.429403	+	0.903113i	$0.641276\pi$
$158$		0			0
$159$	2.23951e8	−	2.76567e8i	0.350400	−	0.432725i
$160$		0			0
$161$			3.82640e8i			0.569491i
$162$		0			0
$163$	7.50269e8			1.06284			0.531418	−	0.847110i	$-0.321659\pi$
$163$	7.50269e8			1.06284			0.531418	+	0.847110i	$0.321659\pi$
$164$		0			0
$165$	−3.19746e8	−	2.58915e8i	−0.431389	−	0.349318i
$166$		0			0
$167$	2.94372e8	−	1.69956e8i	0.378469	−	0.218509i	−0.298683	−	0.954352i	$-0.596547\pi$
$167$	2.94372e8	−	1.69956e8i	0.378469	−	0.218509i	0.677152	+	0.735843i	$0.263214\pi$
$168$		0			0
$169$	3.80831e8	−	6.59619e8i	0.466859	−	0.808623i
$170$		0			0
$171$	−8.82213e7	−	4.14970e8i	−0.103178	−	0.485325i
$172$		0			0
$173$	−3.40547e8	−	1.96615e8i	−0.380183	−	0.219499i	0.297715	−	0.954655i	$-0.403775\pi$
$173$	−3.40547e8	−	1.96615e8i	−0.380183	−	0.219499i	−0.677898	+	0.735156i	$0.737109\pi$
$174$		0			0
$175$	7.95406e7	+	1.37768e8i	0.0848080	+	0.146892i
$176$		0			0
$177$	−1.85744e9	−	2.94154e8i	−1.89244	−	0.299696i
$178$		0			0
$179$			9.16399e8i			0.892632i	0.894875	+	0.446316i	$0.147264\pi$
$179$			9.16399e8i			0.892632i	−0.894875	+	0.446316i	$0.852736\pi$
$180$		0			0
$181$	−3.49821e8			−0.325935			−0.162968	−	0.986631i	$-0.552107\pi$
$181$	−3.49821e8			−0.325935			−0.162968	+	0.986631i	$0.552107\pi$
$182$		0			0
$183$	2.01137e9	−	7.72135e8i	1.79344	−	0.688476i
$184$		0			0
$185$	1.14200e8	−	6.59334e7i	0.0974943	−	0.0562883i
$186$		0			0
$187$	−6.32334e8	+	1.09524e9i	−0.517107	+	0.895655i
$188$		0			0
$189$	−2.62691e8	−	4.04458e8i	−0.205872	−	0.316976i
$190$		0			0
$191$	1.44808e9	+	8.36047e8i	1.08807	+	0.628199i	0.933063	−	0.359714i	$-0.117126\pi$
$191$	1.44808e9	+	8.36047e8i	1.08807	+	0.628199i	0.155010	+	0.987913i	$0.450459\pi$
$192$		0			0
$193$	−4.29853e8	−	7.44527e8i	−0.309806	−	0.536600i	0.668514	−	0.743700i	$-0.266931\pi$
$193$	−4.29853e8	−	7.44527e8i	−0.309806	−	0.536600i	−0.978320	+	0.207100i	$0.933598\pi$
$194$		0			0
$195$	9.90508e7	+	2.58022e8i	0.0685046	+	0.178450i
$196$		0			0
$197$		−	5.18396e8i		−	0.344189i	−0.985080	−	0.172094i	$-0.944947\pi$
$197$		−	5.18396e8i		−	0.344189i	0.985080	−	0.172094i	$-0.0550534\pi$
$198$		0			0
$199$	2.34050e9			1.49244			0.746220	−	0.665699i	$-0.231867\pi$
$199$	2.34050e9			1.49244			0.746220	+	0.665699i	$0.231867\pi$
$200$		0			0
$201$	−3.27696e8	+	2.06925e9i	−0.200765	+	1.26773i
$202$		0			0
$203$	−1.53671e8	+	8.87219e7i	−0.0904914	+	0.0522452i
$204$		0			0
$205$	4.85824e8	−	8.41472e8i	0.275083	−	0.476457i
$206$		0			0
$207$	−1.85117e9	−	2.05578e9i	−1.00824	−	1.11968i
$208$		0			0
$209$	6.12967e8	+	3.53897e8i	0.321257	+	0.185478i
$210$		0			0
$211$	−8.52805e7	−	1.47710e8i	−0.0430249	−	0.0745213i	0.843711	−	0.536798i	$-0.180366\pi$
$211$	−8.52805e7	−	1.47710e8i	−0.0430249	−	0.0745213i	−0.886736	+	0.462276i	$0.847033\pi$
$212$		0			0
$213$	−6.16228e8	+	7.61008e8i	−0.299380	+	0.369718i
$214$		0			0
$215$		−	2.03853e9i		−	0.954031i
$216$		0			0
$217$	7.90322e8			0.356423
$218$		0			0
$219$	6.18803e8	+	5.01077e8i	0.269015	+	0.217835i
$220$		0			0
$221$	7.35732e8	−	4.24775e8i	0.308426	−	0.178070i
$222$		0			0
$223$	−9.90698e8	+	1.71594e9i	−0.400610	+	0.693877i	−0.993800	−	0.111186i	$-0.964535\pi$
$223$	−9.90698e8	+	1.71594e9i	−0.400610	+	0.693877i	0.593189	+	0.805063i	$0.297868\pi$
$224$		0			0
$225$	−1.09385e9	−	3.55367e8i	−0.426803	−	0.138659i
$226$		0			0
$227$	1.83638e9	+	1.06023e9i	0.691606	+	0.399299i	0.804213	−	0.594341i	$-0.202587\pi$
$227$	1.83638e9	+	1.06023e9i	0.691606	+	0.399299i	−0.112608	+	0.993640i	$0.535920\pi$
$228$		0			0
$229$	−6.44922e8	−	1.11704e9i	−0.234512	−	0.406187i	0.724619	−	0.689150i	$-0.242016\pi$
$229$	−6.44922e8	−	1.11704e9i	−0.234512	−	0.406187i	−0.959131	+	0.282963i	$0.908683\pi$
$230$		0			0
$231$	7.94712e8	+	1.25855e8i	0.279101	+	0.0441999i
$232$		0			0
$233$		−	4.14000e8i		−	0.140468i	−0.997531	−	0.0702339i	$-0.977625\pi$
$233$		−	4.14000e8i		−	0.140468i	0.997531	−	0.0702339i	$-0.0223746\pi$
$234$		0			0
$235$	1.29557e9			0.424805
$236$		0			0
$237$	−1.27329e9	+	4.88798e8i	−0.403584	+	0.154930i
$238$		0			0
$239$	5.20640e9	−	3.00591e9i	1.59568	−	0.921266i	0.603373	−	0.797459i	$-0.293823\pi$
$239$	5.20640e9	−	3.00591e9i	1.59568	−	0.921266i	0.992306	−	0.123807i	$-0.0395105\pi$
$240$		0			0
$241$	1.05502e8	−	1.82735e8i	0.0312748	−	0.0541695i	−0.849964	−	0.526840i	$-0.823377\pi$
$241$	1.05502e8	−	1.82735e8i	0.0312748	−	0.0541695i	0.881239	+	0.472671i	$0.156710\pi$
$242$		0			0
$243$	3.36806e9	+	9.02125e8i	0.965951	+	0.258727i
$244$		0			0
$245$	3.30953e8	+	1.91076e8i	0.0918549	+	0.0530324i
$246$		0			0
$247$	−2.37733e8	−	4.11765e8i	−0.0638706	−	0.110627i
$248$		0			0
$249$	−2.18192e9	−	5.68378e9i	−0.567600	−	1.47856i
$250$		0			0
$251$			1.65866e9i			0.417890i	0.977927	+	0.208945i	$0.0670030\pi$
$251$			1.65866e9i			0.417890i	−0.977927	+	0.208945i	$0.932997\pi$
$252$		0			0
$253$	4.61538e9			1.12649
$254$		0			0
$255$	6.79252e8	−	4.28916e9i	0.160646	−	1.01441i
$256$		0			0
$257$	−1.98437e9	+	1.14568e9i	−0.454874	+	0.262622i	−0.709886	−	0.704316i	$-0.751254\pi$
$257$	−1.98437e9	+	1.14568e9i	−0.454874	+	0.262622i	0.255013	+	0.966938i	$0.417920\pi$
$258$		0			0
$259$	−1.28943e8	+	2.23336e8i	−0.0286550	+	0.0496318i
$260$		0			0
$261$	3.96387e8	−	1.22011e9i	0.0854195	−	0.262928i
$262$		0			0
$263$	−6.66403e9	−	3.84748e9i	−1.39288	−	0.804180i	−0.399248	−	0.916843i	$-0.630729\pi$
$263$	−6.66403e9	−	3.84748e9i	−1.39288	−	0.804180i	−0.993633	+	0.112663i	$0.964062\pi$
$264$		0			0
$265$	−1.01936e9	−	1.76558e9i	−0.206701	−	0.358017i
$266$		0			0
$267$	4.85731e8	−	5.99851e8i	0.0955763	−	0.118032i
$268$		0			0
$269$		−	4.73438e8i		−	0.0904178i	−0.998978	−	0.0452089i	$-0.985605\pi$
$269$		−	4.73438e8i		−	0.0904178i	0.998978	−	0.0452089i	$-0.0143954\pi$
$270$		0			0
$271$	2.15214e9			0.399019			0.199510	−	0.979896i	$-0.436065\pi$
$271$	2.15214e9			0.399019			0.199510	+	0.979896i	$0.436065\pi$
$272$		0			0
$273$	−4.20059e8	−	3.40144e8i	−0.0756240	−	0.0612367i
$274$		0			0
$275$	1.66175e9	−	9.59414e8i	0.290560	−	0.167755i
$276$		0			0
$277$	−3.31923e9	+	5.74908e9i	−0.563792	+	0.976516i	0.433369	+	0.901216i	$0.357325\pi$
$277$	−3.31923e9	+	5.74908e9i	−0.563792	+	0.976516i	−0.997161	+	0.0752994i	$0.976009\pi$
$278$		0			0
$279$	−4.24609e9	+	3.82349e9i	−0.700766	+	0.631021i
$280$		0			0
$281$	−2.17768e9	−	1.25728e9i	−0.349276	−	0.201654i	0.315091	−	0.949062i	$-0.397965\pi$
$281$	−2.17768e9	−	1.25728e9i	−0.349276	−	0.201654i	−0.664366	+	0.747407i	$0.731298\pi$
$282$		0			0
$283$	1.91553e9	+	3.31780e9i	0.298637	+	0.517255i	0.975824	−	0.218556i	$-0.0701345\pi$
$283$	1.91553e9	+	3.31780e9i	0.298637	+	0.517255i	−0.677187	+	0.735811i	$0.736801\pi$
$284$		0			0
$285$	−2.40050e9	−	3.80155e8i	−0.363850	−	0.0576211i
$286$		0			0
$287$			1.90021e9i			0.280075i
$288$		0			0
$289$	−6.37274e9			−0.913556
$290$		0			0
$291$	2.68070e9	−	1.02908e9i	0.373832	−	0.143509i
$292$		0			0
$293$	2.36828e9	−	1.36733e9i	0.321339	−	0.185525i	−0.330650	−	0.943753i	$-0.607268\pi$
$293$	2.36828e9	−	1.36733e9i	0.321339	−	0.185525i	0.651989	+	0.758228i	$0.273935\pi$
$294$		0			0
$295$	−5.38677e9	+	9.33016e9i	−0.711280	+	1.23197i
$296$		0			0
$297$	−4.87855e9	+	3.16856e9i	−0.626996	+	0.407227i
$298$		0			0
$299$	−2.68504e9	−	1.55021e9i	−0.335943	−	0.193957i
$300$		0			0
$301$	1.99333e9	+	3.45255e9i	0.242836	+	0.420605i
$302$		0			0
$303$	3.79648e9	+	9.88961e9i	0.450413	+	1.17330i
$304$		0			0
$305$		−	1.23426e10i		−	1.42629i
$306$		0			0
$307$	−2.86351e9			−0.322363			−0.161181	−	0.986925i	$-0.551530\pi$
$307$	−2.86351e9			−0.322363			−0.161181	+	0.986925i	$0.551530\pi$
$308$		0			0
$309$	−1.84147e9	+	1.16280e10i	−0.201991	+	1.27548i
$310$		0			0
$311$	−2.68966e9	+	1.55288e9i	−0.287512	+	0.165995i	−0.636819	−	0.771013i	$-0.719750\pi$
$311$	−2.68966e9	+	1.55288e9i	−0.287512	+	0.165995i	0.349307	+	0.937008i	$0.386417\pi$
$312$		0			0
$313$	−8.10579e9	+	1.40396e10i	−0.844536	+	1.46278i	0.0414874	+	0.999139i	$0.486790\pi$
$313$	−8.10579e9	+	1.40396e10i	−0.844536	+	1.46278i	−0.886023	+	0.463640i	$0.846543\pi$
$314$		0			0
$315$	−2.70249e9	+	5.74539e8i	−0.274487	+	0.0583549i
$316$		0			0
$317$	−3.19141e9	−	1.84256e9i	−0.316043	−	0.182467i	0.333585	−	0.942720i	$-0.391742\pi$
$317$	−3.19141e9	−	1.84256e9i	−0.316043	−	0.182467i	−0.649627	+	0.760253i	$0.725075\pi$
$318$		0			0
$319$	1.07016e9	+	1.85357e9i	0.103344	+	0.178997i
$320$		0			0
$321$	−5.41688e9	+	6.68955e9i	−0.510187	+	0.630053i
$322$		0			0
$323$			7.47072e9i			0.686361i
$324$		0			0
$325$	−1.28899e9			−0.115535
$326$		0			0
$327$	−1.15092e10	−	9.31960e9i	−1.00659	−	0.815091i
$328$		0			0
$329$	−2.19424e9	+	1.26685e9i	−0.187284	+	0.108129i
$330$		0			0
$331$	−3.25014e8	+	5.62941e8i	−0.0270764	+	0.0468977i	−0.879246	−	0.476368i	$-0.841953\pi$
$331$	−3.25014e8	+	5.62941e8i	−0.0270764	+	0.0468977i	0.852170	+	0.523266i	$0.175286\pi$
$332$		0			0
$333$	−3.87715e8	−	1.82371e9i	−0.0315308	−	0.148313i
$334$		0			0
$335$	1.03941e10	+	6.00103e9i	0.825291	+	0.476482i
$336$		0			0
$337$	−5.46476e9	−	9.46524e9i	−0.423693	−	0.733858i	0.572604	−	0.819832i	$-0.305933\pi$
$337$	−5.46476e9	−	9.46524e9i	−0.423693	−	0.733858i	−0.996297	+	0.0859737i	$0.972600\pi$
$338$		0			0
$339$	−1.95457e10	−	3.09535e9i	−1.47997	−	0.234375i
$340$		0			0
$341$		−	9.53282e9i		−	0.705024i
$342$		0			0
$343$	−7.47359e8			−0.0539949
$344$		0			0
$345$	−1.47955e10	+	5.67979e9i	−1.04437	+	0.400919i
$346$		0			0
$347$	−3.99311e9	+	2.30542e9i	−0.275418	+	0.159013i	−0.631347	−	0.775500i	$-0.717498\pi$
$347$	−3.99311e9	+	2.30542e9i	−0.275418	+	0.159013i	0.355929	+	0.934513i	$0.384164\pi$
$348$		0			0
$349$	−5.19922e9	+	9.00531e9i	−0.350458	+	0.607011i	−0.986330	−	0.164784i	$-0.947307\pi$
$349$	−5.19922e9	+	9.00531e9i	−0.350458	+	0.607011i	0.635872	+	0.771795i	$0.280641\pi$
$350$		0			0
$351$	3.90239e9	−	2.04740e8i	0.257100	−	0.0134888i
$352$		0			0
$353$	1.06158e10	+	6.12903e9i	0.683681	+	0.394723i	0.801240	−	0.598343i	$-0.204174\pi$
$353$	1.06158e10	+	6.12903e9i	0.683681	+	0.394723i	−0.117560	+	0.993066i	$0.537507\pi$
$354$		0			0
$355$	2.80488e9	+	4.85820e9i	0.176604	+	0.305887i
$356$		0			0
$357$	3.04365e9	+	7.92854e9i	0.187380	+	0.488113i
$358$		0			0
$359$		−	1.00100e10i		−	0.602635i	−0.953524	−	0.301317i	$-0.902574\pi$
$359$		−	1.00100e10i		−	0.602635i	0.953524	−	0.301317i	$-0.0974264\pi$
$360$		0			0
$361$	−1.28024e10			−0.753814
$362$		0			0
$363$	−1.19781e9	+	7.56358e9i	−0.0689858	+	0.435613i
$364$		0			0
$365$	3.95037e9	−	2.28075e9i	0.222570	−	0.128501i
$366$		0			0
$367$	−3.49978e9	+	6.06180e9i	−0.192920	+	0.334147i	−0.946217	−	0.323534i	$-0.895129\pi$
$367$	−3.49978e9	+	6.06180e9i	−0.192920	+	0.334147i	0.753297	+	0.657681i	$0.228462\pi$
$368$		0			0
$369$	−9.19302e9	−	1.02091e10i	−0.495853	−	0.550658i
$370$		0			0
$371$	3.45287e9	+	1.99352e9i	0.182257	+	0.105226i
$372$		0			0
$373$	−1.74497e9	−	3.02238e9i	−0.0901473	−	0.156140i	0.817426	−	0.576034i	$-0.195400\pi$
$373$	−1.74497e9	−	3.02238e9i	−0.0901473	−	0.156140i	−0.907573	+	0.419894i	$0.862067\pi$
$374$		0			0
$375$	−1.33861e10	+	1.65311e10i	−0.676906	+	0.835942i
$376$		0			0
$377$		−	1.43777e9i		−	0.0711746i
$378$		0			0
$379$	2.59135e10			1.25594			0.627971	−	0.778237i	$-0.283886\pi$
$379$	2.59135e10			1.25594			0.627971	+	0.778237i	$0.283886\pi$
$380$		0			0
$381$	2.07720e10	+	1.68201e10i	0.985775	+	0.798233i
$382$		0			0
$383$	−1.20523e10	+	6.95839e9i	−0.560111	+	0.323380i	−0.753190	−	0.657803i	$-0.771486\pi$
$383$	−1.20523e10	+	6.95839e9i	−0.560111	+	0.323380i	0.193079	+	0.981183i	$0.438153\pi$
$384$		0			0
$385$	2.30475e9	−	3.99194e9i	0.104901	−	0.181694i
$386$		0			0
$387$	−2.74125e10	−	8.90570e9i	−1.22209	−	0.397031i
$388$		0			0
$389$	−2.18814e10	−	1.26332e10i	−0.955600	−	0.551716i	−0.0607843	−	0.998151i	$-0.519360\pi$
$389$	−2.18814e10	−	1.26332e10i	−0.955600	−	0.551716i	−0.894816	+	0.446435i	$0.852693\pi$
$390$		0			0
$391$	2.43575e10	+	4.21885e10i	1.04214	+	1.80504i
$392$		0			0
$393$	−3.01667e10	−	4.77735e9i	−1.26461	−	0.200271i
$394$		0			0
$395$			7.81346e9i			0.320963i
$396$		0			0
$397$	1.41767e10			0.570705			0.285352	−	0.958423i	$-0.407889\pi$
$397$	1.41767e10			0.570705			0.285352	+	0.958423i	$0.407889\pi$
$398$		0			0
$399$	4.43734e9	−	1.70343e9i	0.175078	−	0.0672100i
$400$		0			0
$401$	3.55948e10	−	2.05507e10i	1.37660	−	0.794782i	0.384854	−	0.922977i	$-0.374252\pi$
$401$	3.55948e10	−	2.05507e10i	1.37660	−	0.794782i	0.991749	+	0.128195i	$0.0409184\pi$
$402$		0			0
$403$	−3.20187e9	+	5.54580e9i	−0.121390	+	0.210254i
$404$		0			0
$405$	1.17399e10	−	1.61611e10i	0.436358	−	0.600691i
$406$		0			0
$407$	2.69387e9	+	1.55531e9i	0.0981746	+	0.0566811i
$408$		0			0
$409$	9.02481e9	+	1.56314e10i	0.322511	+	0.558606i	0.981005	−	0.193980i	$-0.0621397\pi$
$409$	9.02481e9	+	1.56314e10i	0.322511	+	0.558606i	−0.658494	+	0.752586i	$0.728806\pi$
$410$		0			0
$411$	−1.47530e10	−	3.84308e10i	−0.517028	−	1.34683i
$412$		0			0
$413$		−	2.10694e10i		−	0.724189i
$414$		0			0
$415$	−3.48782e10			−1.17588
$416$		0			0
$417$	6.40047e9	−	4.04160e10i	0.211674	−	1.33662i
$418$		0			0
$419$	1.75602e10	−	1.01384e10i	0.569735	−	0.328937i	−0.187308	−	0.982301i	$-0.559976\pi$
$419$	1.75602e10	−	1.01384e10i	0.569735	−	0.328937i	0.757044	+	0.653364i	$0.226643\pi$
$420$		0			0
$421$	−2.86380e10	+	4.96025e10i	−0.911622	+	1.57898i	−0.0998489	+	0.995003i	$0.531836\pi$
$421$	−2.86380e10	+	4.96025e10i	−0.911622	+	1.57898i	−0.811773	+	0.583973i	$0.801497\pi$
$422$		0			0
$423$	5.65995e9	−	1.74218e10i	0.176787	−	0.544166i
$424$		0			0
$425$	1.75397e10	+	1.01266e10i	0.537609	+	0.310389i
$426$		0			0
$427$	1.20690e10	+	2.09041e10i	0.363044	+	0.628810i
$428$		0			0
$429$	−4.10279e9	+	5.06673e9i	−0.121130	+	0.149589i
$430$		0			0
$431$		−	5.44935e10i		−	1.57919i	−0.613626	−	0.789597i	$-0.710290\pi$
$431$		−	5.44935e10i		−	1.57919i	0.613626	−	0.789597i	$-0.289710\pi$
$432$		0			0
$433$	3.79008e10			1.07819			0.539097	−	0.842244i	$-0.318766\pi$
$433$	3.79008e10			1.07819			0.539097	+	0.842244i	$0.318766\pi$
$434$		0			0
$435$	−5.71165e9	−	4.62502e9i	−0.159516	−	0.129169i
$436$		0			0
$437$	2.36115e10	−	1.36321e10i	0.647438	−	0.373799i
$438$		0			0
$439$	2.45963e10	−	4.26020e10i	0.662234	−	1.14702i	−0.317794	−	0.948160i	$-0.602942\pi$
$439$	2.45963e10	−	4.26020e10i	0.662234	−	1.14702i	0.980027	−	0.198863i	$-0.0637248\pi$
$440$		0			0
$441$	4.01527e9	−	3.61564e9i	0.106160	−	0.0955941i
$442$		0			0
$443$	−8.70205e9	−	5.02413e9i	−0.225947	−	0.130451i	0.382754	−	0.923850i	$-0.374976\pi$
$443$	−8.70205e9	−	5.02413e9i	−0.225947	−	0.130451i	−0.608701	+	0.793400i	$0.708309\pi$
$444$		0			0
$445$	−2.21090e9	−	3.82939e9i	−0.0563805	−	0.0976538i
$446$		0			0
$447$	1.03074e10	+	1.63233e9i	0.258178	+	0.0408863i
$448$		0			0
$449$		−	6.27596e9i		−	0.154417i	−0.997015	−	0.0772084i	$-0.975399\pi$
$449$		−	6.27596e9i		−	0.154417i	0.997015	−	0.0772084i	$-0.0246007\pi$
$450$		0			0
$451$	2.29203e10			0.554005
$452$		0			0
$453$	−6.72711e10	+	2.58244e10i	−1.59748	+	0.613250i
$454$		0			0
$455$	−2.68161e9	+	1.54823e9i	−0.0625677	+	0.0361235i
$456$		0			0
$457$	1.24958e10	−	2.16433e10i	0.286483	−	0.496203i	−0.686485	−	0.727144i	$-0.740847\pi$
$457$	1.24958e10	−	2.16433e10i	0.286483	−	0.496203i	0.972968	+	0.230941i	$0.0741805\pi$
$458$		0			0
$459$	−5.47098e10	−	2.78721e10i	−1.23258	−	0.627941i
$460$		0			0
$461$	2.70783e10	+	1.56336e10i	0.599538	+	0.346144i	0.768860	−	0.639417i	$-0.220824\pi$
$461$	2.70783e10	+	1.56336e10i	0.599538	+	0.346144i	−0.169322	+	0.985561i	$0.554158\pi$
$462$		0			0
$463$	3.00717e10	+	5.20857e10i	0.654386	+	1.13343i	0.982047	+	0.188634i	$0.0604061\pi$
$463$	3.00717e10	+	5.20857e10i	0.654386	+	1.13343i	−0.327661	+	0.944795i	$0.606261\pi$
$464$		0			0
$465$	1.17313e10	+	3.05594e10i	0.250920	+	0.653631i
$466$		0			0
$467$			4.11906e10i			0.866025i	0.901388	+	0.433012i	$0.142549\pi$
$467$			4.11906e10i			0.866025i	−0.901388	+	0.433012i	$0.857451\pi$
$468$		0			0
$469$	−2.34720e10			−0.485130
$470$		0			0
$471$	8.73567e9	−	5.51617e10i	0.177506	−	1.12087i
$472$		0			0
$473$	4.16445e10	−	2.40435e10i	0.831981	−	0.480344i
$474$		0			0
$475$	5.66751e9	−	9.81641e9i	0.111331	−	0.192832i
$476$		0			0
$477$	−2.81953e10	+	5.99423e9i	−0.544633	+	0.115787i
$478$		0			0
$479$	3.38922e10	+	1.95677e10i	0.643809	+	0.371704i	0.786080	−	0.618124i	$-0.212107\pi$
$479$	3.38922e10	+	1.95677e10i	0.643809	+	0.371704i	−0.142271	+	0.989828i	$0.545440\pi$
$480$		0			0
$481$	−1.04479e9	−	1.80963e9i	−0.0195186	−	0.0338071i
$482$		0			0
$483$	1.95046e10	−	2.40871e10i	0.358384	−	0.442584i
$484$		0			0
$485$		−	1.64500e10i		−	0.297302i
$486$		0			0
$487$	−2.33538e10			−0.415186			−0.207593	−	0.978215i	$-0.566563\pi$
$487$	−2.33538e10			−0.415186			−0.207593	+	0.978215i	$0.566563\pi$
$488$		0			0
$489$	−4.72292e10	−	3.82440e10i	−0.825992	−	0.668849i
$490$		0			0
$491$	−3.25464e9	+	1.87907e9i	−0.0559986	+	0.0323308i	−0.527738	−	0.849407i	$-0.676960\pi$
$491$	−3.25464e9	+	1.87907e9i	−0.0559986	+	0.0323308i	0.471739	+	0.881738i	$0.343626\pi$
$492$		0			0
$493$	−1.12955e10	+	1.95643e10i	−0.191212	+	0.331190i
$494$		0			0
$495$	6.93006e9	+	3.25973e10i	0.115429	+	0.542951i
$496$		0			0
$497$	−9.50098e9	−	5.48539e9i	−0.155719	−	0.0899047i
$498$		0			0
$499$	2.97463e10	+	5.15222e10i	0.479768	+	0.830983i	0.999731	−	0.0232062i	$-0.00738741\pi$
$499$	2.97463e10	+	5.15222e10i	0.479768	+	0.830983i	−0.519962	+	0.854189i	$0.674054\pi$
$500$		0			0
$501$	−2.71939e10	−	4.30656e9i	−0.431639	−	0.0683565i
$502$		0			0
$503$		−	1.08459e11i		−	1.69431i	−0.531347	−	0.847155i	$-0.678314\pi$
$503$		−	1.08459e11i		−	1.69431i	0.531347	−	0.847155i	$-0.321686\pi$
$504$		0			0
$505$	6.06869e10			0.933103
$506$		0			0
$507$	−5.75965e10	+	2.21105e10i	−0.871694	+	0.334631i
$508$		0			0
$509$	6.52783e10	−	3.76884e10i	0.972518	−	0.561483i	0.0725147	−	0.997367i	$-0.476898\pi$
$509$	6.52783e10	−	3.76884e10i	0.972518	−	0.561483i	0.900003	+	0.435884i	$0.143564\pi$
$510$		0			0
$511$	−4.46037e9	+	7.72559e9i	−0.0654165	+	0.113305i
$512$		0			0
$513$	−1.55991e10	+	3.06193e10i	−0.225232	+	0.442105i
$514$		0			0
$515$	5.84091e10	+	3.37225e10i	0.830331	+	0.479392i
$516$		0			0
$517$	1.52806e10	+	2.64668e10i	0.213885	+	0.370459i
$518$		0			0
$519$	1.14152e10	+	2.97358e10i	0.157330	+	0.409836i
$520$		0			0
$521$		−	1.71310e10i		−	0.232505i	−0.993220	−	0.116253i	$-0.962912\pi$
$521$		−	1.71310e10i		−	0.232505i	0.993220	−	0.116253i	$-0.0370882\pi$
$522$		0			0
$523$	6.98615e10			0.933751			0.466876	−	0.884323i	$-0.345380\pi$
$523$	6.98615e10			0.933751			0.466876	+	0.884323i	$0.345380\pi$
$524$		0			0
$525$	2.01551e9	−	1.27270e10i	0.0265306	−	0.167528i
$526$		0			0
$527$	8.71381e10	−	5.03092e10i	1.12971	−	0.652236i
$528$		0			0
$529$	4.97368e10	−	8.61467e10i	0.635119	−	1.10006i
$530$		0			0
$531$	1.01931e11	+	1.13198e11i	1.28212	+	1.42383i
$532$		0			0
$533$	−1.33341e10	−	7.69842e9i	−0.165217	−	0.0953878i
$534$		0			0
$535$	2.46560e10	+	4.27054e10i	0.300959	+	0.521276i
$536$		0			0
$537$	4.67123e10	−	5.76871e10i	0.561738	−	0.693716i
$538$		0			0
$539$			9.01461e9i			0.106805i
$540$		0			0
$541$	8.96671e10			1.04675			0.523377	−	0.852101i	$-0.324672\pi$
$541$	8.96671e10			1.04675			0.523377	+	0.852101i	$0.324672\pi$
$542$		0			0
$543$	2.20212e10	+	1.78317e10i	0.253303	+	0.205113i
$544$		0			0
$545$	−7.34735e10	+	4.24200e10i	−0.832808	+	0.480822i
$546$		0			0
$547$	−7.92699e10	+	1.37300e11i	−0.885440	+	1.53363i	−0.0402311	+	0.999190i	$0.512809\pi$
$547$	−7.92699e10	+	1.37300e11i	−0.885440	+	1.53363i	−0.845209	+	0.534436i	$0.820524\pi$
$548$		0			0
$549$	−1.65974e11	−	5.39211e10i	−1.82705	−	0.593567i
$550$		0			0
$551$	1.09495e10	+	6.32170e9i	0.118792	+	0.0685847i
$552$		0			0
$553$	−7.64024e9	−	1.32333e10i	−0.0816970	−	0.141503i
$554$		0			0
$555$	−1.05497e10	−	1.67071e9i	−0.111191	−	0.0176087i
$556$		0			0
$557$		−	6.12382e10i		−	0.636212i	−0.948055	−	0.318106i	$-0.896953\pi$
$557$		−	6.12382e10i		−	0.636212i	0.948055	−	0.318106i	$-0.103047\pi$
$558$		0			0
$559$	−3.23027e10			−0.330820
$560$		0			0
$561$	9.56336e10	−	3.67124e10i	0.965515	−	0.370648i
$562$		0			0
$563$	1.00020e11	−	5.77463e10i	0.995523	−	0.574765i	0.0886023	−	0.996067i	$-0.471760\pi$
$563$	1.00020e11	−	5.77463e10i	0.995523	−	0.574765i	0.906921	+	0.421302i	$0.138427\pi$
$564$		0			0
$565$	−5.66845e10	+	9.81804e10i	−0.556250	+	0.963454i
$566$		0			0
$567$	−4.08040e9	+	3.88509e10i	−0.0394793	+	0.375897i
$568$		0			0
$569$	7.17163e10	+	4.14054e10i	0.684178	+	0.395010i	0.801427	−	0.598092i	$-0.204074\pi$
$569$	7.17163e10	+	4.14054e10i	0.684178	+	0.395010i	−0.117250	+	0.993102i	$0.537408\pi$
$570$		0			0
$571$	6.27960e10	+	1.08766e11i	0.590728	+	1.02317i	0.994135	+	0.108150i	$0.0344927\pi$
$571$	6.27960e10	+	1.08766e11i	0.590728	+	1.02317i	−0.403407	+	0.915021i	$0.632174\pi$
$572$		0			0
$573$	−4.85396e10	−	1.26443e11i	−0.450275	−	1.17294i
$574$		0			0
$575$		−	7.39133e10i		−	0.676163i
$576$		0			0
$577$	2.39184e10			0.215789			0.107894	−	0.994162i	$-0.465589\pi$
$577$	2.39184e10			0.215789			0.107894	+	0.994162i	$0.465589\pi$
$578$		0			0
$579$	−1.08922e10	+	6.87790e10i	−0.0969171	+	0.611986i
$580$		0			0
$581$	5.90715e10	−	3.41049e10i	0.518410	−	0.299304i
$582$		0			0
$583$	2.40457e10	−	4.16483e10i	0.208143	−	0.360515i
$584$		0			0
$585$	6.91709e9	−	2.12914e10i	0.0590609	−	0.181794i
$586$		0			0
$587$	−3.03722e10	−	1.75354e10i	−0.255813	−	0.147694i	0.366610	−	0.930375i	$-0.380518\pi$
$587$	−3.03722e10	−	1.75354e10i	−0.255813	−	0.147694i	−0.622423	+	0.782681i	$0.713852\pi$
$588$		0			0
$589$	−2.81564e10	−	4.87683e10i	−0.233946	−	0.405207i
$590$		0			0
$591$	−2.64246e10	+	3.26329e10i	−0.216600	+	0.267489i
$592$		0			0
$593$		−	2.08754e11i		−	1.68817i	−0.536211	−	0.844084i	$-0.680145\pi$
$593$		−	2.08754e11i		−	1.68817i	0.536211	−	0.844084i	$-0.319855\pi$
$594$		0			0
$595$	4.86529e10			0.388187
$596$		0			0
$597$	−1.47334e11	−	1.19304e11i	−1.15986	−	0.939201i
$598$		0			0
$599$	−1.23862e10	+	7.15117e9i	−0.0962123	+	0.0555482i	−0.547334	−	0.836914i	$-0.684357\pi$
$599$	−1.23862e10	+	7.15117e9i	−0.0962123	+	0.0555482i	0.451122	+	0.892462i	$0.351024\pi$
$600$		0			0
$601$	−6.10360e10	+	1.05717e11i	−0.467830	+	0.810305i	−0.999324	−	0.0367565i	$-0.988297\pi$
$601$	−6.10360e10	+	1.05717e11i	−0.467830	+	0.810305i	0.531494	+	0.847062i	$0.321631\pi$
$602$		0			0
$603$	1.26106e11	−	1.13555e11i	0.953818	−	0.858887i
$604$		0			0
$605$	3.79928e10	+	2.19352e10i	0.283583	+	0.163727i
$606$		0			0
$607$	4.01086e10	+	6.94702e10i	0.295449	+	0.511734i	0.975089	−	0.221812i	$-0.0711973\pi$
$607$	4.01086e10	+	6.94702e10i	0.295449	+	0.511734i	−0.679640	+	0.733546i	$0.737864\pi$
$608$		0			0
$609$	1.41960e10	+	2.24815e9i	0.103204	+	0.0163439i
$610$		0			0
$611$		−	2.05297e10i		−	0.147305i
$612$		0			0
$613$	−1.01416e11			−0.718235			−0.359117	−	0.933292i	$-0.616922\pi$
$613$	−1.01416e11			−0.718235			−0.359117	+	0.933292i	$0.616922\pi$
$614$		0			0
$615$	−7.34755e10	+	2.82062e10i	−0.513620	+	0.197171i
$616$		0			0
$617$	4.78866e10	−	2.76474e10i	0.330426	−	0.190771i	−0.325604	−	0.945506i	$-0.605568\pi$
$617$	4.78866e10	−	2.76474e10i	0.330426	−	0.190771i	0.656030	+	0.754735i	$0.272234\pi$
$618$		0			0
$619$	1.31727e11	−	2.28158e11i	0.897249	−	1.55408i	0.0662534	−	0.997803i	$-0.478895\pi$
$619$	1.31727e11	−	2.28158e11i	0.897249	−	1.55408i	0.830996	−	0.556279i	$-0.187771\pi$
$620$		0			0
$621$	1.17402e10	+	2.23772e11i	0.0789424	+	1.50466i
$622$		0			0
$623$	7.48898e9	+	4.32376e9i	0.0497131	+	0.0287018i
$624$		0			0
$625$	2.66915e10	+	4.62311e10i	0.174926	+	0.302980i
$626$		0			0
$627$	−2.05467e10	−	5.35230e10i	−0.132945	−	0.346314i
$628$		0			0
$629$			3.28323e10i			0.209749i
$630$		0			0
$631$	5.77715e10			0.364415			0.182207	−	0.983260i	$-0.441676\pi$
$631$	5.77715e10			0.364415			0.182207	+	0.983260i	$0.441676\pi$
$632$		0			0
$633$	−2.16095e9	+	1.36454e10i	−0.0134595	+	0.0849906i
$634$		0			0
$635$	1.32606e11	−	7.65602e10i	0.815584	−	0.470878i
$636$		0			0
$637$	3.02781e9	−	5.24432e9i	0.0183896	−	0.0318516i
$638$		0			0
$639$	7.75828e10	−	1.64938e10i	0.465331	−	0.0989278i
$640$		0			0
$641$	−1.86556e11	−	1.07708e11i	−1.10504	−	0.637992i	−0.167496	−	0.985873i	$-0.553568\pi$
$641$	−1.86556e11	−	1.07708e11i	−1.10504	−	0.637992i	−0.937539	+	0.347880i	$0.886902\pi$
$642$		0			0
$643$	8.53023e10	+	1.47748e11i	0.499019	+	0.864326i	0.999999	−	0.00113287i	$-0.000360605\pi$
$643$	8.53023e10	+	1.47748e11i	0.499019	+	0.864326i	−0.500981	+	0.865458i	$0.667027\pi$
$644$		0			0
$645$	−1.03911e11	+	1.28325e11i	−0.600377	+	0.741433i
$646$		0			0
$647$		−	8.29711e10i		−	0.473489i	−0.971572	−	0.236744i	$-0.923920\pi$
$647$		−	8.29711e10i		−	0.473489i	0.971572	−	0.236744i	$-0.0760804\pi$
$648$		0			0
$649$	−2.54138e11			−1.43249
$650$		0			0
$651$	−4.97506e10	−	4.02857e10i	−0.276997	−	0.224299i
$652$		0			0
$653$	−1.72048e11	+	9.93321e10i	−0.946231	+	0.546307i	−0.891908	−	0.452216i	$-0.850633\pi$
$653$	−1.72048e11	+	9.93321e10i	−0.946231	+	0.546307i	−0.0543231	+	0.998523i	$0.517300\pi$
$654$		0			0
$655$	−8.74867e10	+	1.51531e11i	−0.475310	+	0.823261i
$656$		0			0
$657$	−1.34117e10	−	6.30854e10i	−0.0719819	−	0.338585i
$658$		0			0
$659$	1.12002e11	+	6.46642e10i	0.593859	+	0.342864i	0.766622	−	0.642099i	$-0.221936\pi$
$659$	1.12002e11	+	6.46642e10i	0.593859	+	0.342864i	−0.172763	+	0.984963i	$0.555270\pi$
$660$		0			0
$661$	−1.80264e11	−	3.12227e11i	−0.944288	−	1.63555i	−0.757171	−	0.653217i	$-0.773419\pi$
$661$	−1.80264e11	−	3.12227e11i	−0.944288	−	1.63555i	−0.187117	−	0.982338i	$-0.559914\pi$
$662$		0			0
$663$	−6.79665e10	−	1.07635e10i	−0.351756	−	0.0557057i
$664$		0			0
$665$		−	2.72295e10i		−	0.139236i
$666$		0			0
$667$	8.24451e10			0.416544
$668$		0			0
$669$	1.49832e11	−	5.75185e10i	0.747998	−	0.287146i
$670$		0			0
$671$	2.52144e11	−	1.45575e11i	1.24382	−	0.718121i
$672$		0			0
$673$	−7.86932e10	+	1.36301e11i	−0.383599	+	0.664413i	−0.991574	−	0.129544i	$-0.958649\pi$
$673$	−7.86932e10	+	1.36301e11i	−0.383599	+	0.664413i	0.607975	+	0.793956i	$0.291982\pi$
$674$		0			0
$675$	5.07432e10	+	7.81279e10i	0.244435	+	0.376349i
$676$		0			0
$677$	1.65221e11	+	9.53904e10i	0.786522	+	0.454099i	0.838737	−	0.544537i	$-0.183295\pi$
$677$	1.65221e11	+	9.53904e10i	0.786522	+	0.454099i	−0.0522148	+	0.998636i	$0.516628\pi$
$678$		0			0
$679$	1.60853e10	+	2.78605e10i	0.0756744	+	0.131072i
$680$		0			0
$681$	−6.15555e10	−	1.60349e11i	−0.286206	−	0.745550i
$682$		0			0
$683$		−	6.98016e10i		−	0.320762i	−0.987055	−	0.160381i	$-0.948728\pi$
$683$		−	6.98016e10i		−	0.320762i	0.987055	−	0.160381i	$-0.0512723\pi$
$684$		0			0
$685$	−2.35828e11			−1.07111
$686$		0			0
$687$	−1.63419e10	+	1.03191e11i	−0.0733627	+	0.463251i
$688$		0			0
$689$	−2.79776e10	+	1.61528e10i	−0.124146	+	0.0716757i
$690$		0			0
$691$	−1.56532e11	+	2.71121e11i	−0.686579	+	1.18919i	0.286359	+	0.958122i	$0.407555\pi$
$691$	−1.56532e11	+	2.71121e11i	−0.686579	+	1.18919i	−0.972938	+	0.231067i	$0.925778\pi$
$692$		0			0
$693$	−4.36117e10	−	4.84320e10i	−0.189091	−	0.209990i
$694$		0			0
$695$	−2.03014e11	−	1.17210e11i	−0.870137	−	0.502374i
$696$		0			0
$697$	1.20961e11	+	2.09511e11i	0.512524	+	0.887718i
$698$		0			0
$699$	−2.11031e10	+	2.60612e10i	−0.0883971	+	0.109166i
$700$		0			0
$701$		−	1.71495e11i		−	0.710197i	−0.934829	−	0.355098i	$-0.884447\pi$
$701$		−	1.71495e11i		−	0.710197i	0.934829	−	0.355098i	$-0.115553\pi$
$702$		0			0
$703$	1.83752e10			0.0752334
$704$		0			0
$705$	−8.15558e10	−	6.60400e10i	−0.330140	−	0.267332i
$706$		0			0
$707$	−1.02783e11	+	5.93415e10i	−0.411379	+	0.237510i
$708$		0			0
$709$	6.43752e10	−	1.11501e11i	0.254761	−	0.441260i	−0.710069	−	0.704132i	$-0.751336\pi$
$709$	6.43752e10	−	1.11501e11i	0.254761	−	0.441260i	0.964831	+	0.262872i	$0.0846698\pi$
$710$		0			0
$711$	1.05069e11	+	3.41346e10i	0.411147	+	0.133572i
$712$		0			0
$713$	−3.18008e11	−	1.83602e11i	−1.23050	−	0.710428i
$714$		0			0
$715$	1.86747e10	+	3.23455e10i	0.0714543	+	0.123763i
$716$		0			0
$717$	−4.80964e11	−	7.61679e10i	−1.81985	−	0.288201i
$718$		0			0
$719$			5.22844e11i			1.95639i	0.207678	+	0.978197i	$0.433409\pi$
$719$			5.22844e11i			1.95639i	−0.207678	+	0.978197i	$0.566591\pi$
$720$		0			0
$721$	−1.31899e11			−0.488092
$722$		0			0
$723$	−1.59561e10	+	6.12531e9i	−0.0583946	+	0.0224169i
$724$		0			0
$725$	2.96841e10	−	1.71381e10i	0.107441	−	0.0620314i
$726$		0			0
$727$	2.26432e11	−	3.92192e11i	0.810589	−	1.40398i	−0.101864	−	0.994798i	$-0.532480\pi$
$727$	2.26432e11	−	3.92192e11i	0.810589	−	1.40398i	0.912452	−	0.409183i	$-0.134186\pi$
$728$		0			0
$729$	−1.66034e11	−	2.28471e11i	−0.587878	−	0.808950i
$730$		0			0
$731$	4.39555e11	+	2.53777e11i	1.53937	+	0.888758i
$732$		0			0
$733$	1.82318e11	+	3.15785e11i	0.631560	+	1.09389i	0.987233	+	0.159283i	$0.0509183\pi$
$733$	1.82318e11	+	3.15785e11i	0.631560	+	1.09389i	−0.355673	+	0.934610i	$0.615748\pi$
$734$		0			0
$735$	−1.10936e10	−	2.88981e10i	−0.0380121	−	0.0990194i
$736$		0			0
$737$			2.83117e11i			0.959614i
$738$		0			0
$739$	4.12802e11			1.38409			0.692045	−	0.721854i	$-0.256710\pi$
$739$	4.12802e11			1.38409			0.692045	+	0.721854i	$0.256710\pi$
$740$		0			0
$741$	−6.02399e9	+	3.80387e10i	−0.0199807	+	0.126169i
$742$		0			0
$743$	−3.10155e11	+	1.79068e11i	−1.01771	+	0.587574i	−0.913439	−	0.406976i	$-0.866583\pi$
$743$	−3.10155e11	+	1.79068e11i	−1.01771	+	0.587574i	−0.104268	+	0.994549i	$0.533250\pi$
$744$		0			0
$745$	2.98925e10	−	5.17754e10i	0.0970370	−	0.168073i
$746$		0			0
$747$	−1.52372e11	+	4.69014e11i	−0.489354	+	1.50627i
$748$		0			0
$749$	−8.35173e10	−	4.82187e10i	−0.265368	−	0.153211i
$750$		0			0
$751$	−3.99349e10	−	6.91692e10i	−0.125543	−	0.217447i	0.796402	−	0.604767i	$-0.206734\pi$
$751$	−3.99349e10	−	6.91692e10i	−0.125543	−	0.217447i	−0.921945	+	0.387321i	$0.873401\pi$
$752$		0			0
$753$	8.45480e10	−	1.04412e11i	0.262981	−	0.324767i
$754$		0			0
$755$			4.12804e11i			1.27045i
$756$		0			0
$757$	−9.59672e10			−0.292240			−0.146120	−	0.989267i	$-0.546679\pi$
$757$	−9.59672e10			−0.292240			−0.146120	+	0.989267i	$0.546679\pi$
$758$		0			0
$759$	−2.90537e11	−	2.35263e11i	−0.875457	−	0.708903i
$760$		0			0
$761$	−4.37522e11	+	2.52603e11i	−1.30455	+	0.753183i	−0.981181	−	0.193090i	$-0.938149\pi$
$761$	−4.37522e11	+	2.52603e11i	−1.30455	+	0.753183i	−0.323370	+	0.946273i	$0.604816\pi$
$762$		0			0
$763$	8.29590e10	−	1.43689e11i	0.244774	−	0.423961i
$764$		0			0
$765$	−2.61393e11	+	2.35378e11i	−0.763218	+	0.687258i
$766$		0			0
$767$	1.47847e11	+	8.53594e10i	0.427199	+	0.246644i
$768$		0			0
$769$	2.66295e11	+	4.61236e11i	0.761478	+	1.31892i	0.942089	+	0.335364i	$0.108859\pi$
$769$	2.66295e11	+	4.61236e11i	0.761478	+	1.31892i	−0.180610	+	0.983555i	$0.557807\pi$
$770$		0			0
$771$	1.83315e11	+	2.90307e10i	0.518778	+	0.0821562i
$772$		0			0
$773$		−	1.39459e11i		−	0.390598i	−0.980744	−	0.195299i	$-0.937432\pi$
$773$		−	1.39459e11i		−	0.390598i	0.980744	−	0.195299i	$-0.0625677\pi$
$774$		0			0
$775$	−1.52664e11			−0.423185
$776$		0			0
$777$	1.95012e10	−	7.48625e9i	0.0535030	−	0.0205391i
$778$		0			0
$779$	1.17256e11	−	6.76979e10i	0.318410	−	0.183834i
$780$		0			0
$781$	−6.61645e10	+	1.14600e11i	−0.177837	+	0.308022i
$782$		0			0
$783$	−8.71461e10	+	5.66004e10i	−0.231847	+	0.150582i
$784$		0			0
$785$	−2.77084e11	−	1.59975e11i	−0.729681	−	0.421281i
$786$		0			0
$787$	−1.80254e11	−	3.12210e11i	−0.469880	−	0.813856i	0.529527	−	0.848293i	$-0.322370\pi$
$787$	−1.80254e11	−	3.12210e11i	−0.469880	−	0.813856i	−0.999407	+	0.0344370i	$0.989036\pi$
$788$		0			0
$789$	2.23379e11	+	5.81889e11i	0.576414	+	1.50152i
$790$		0			0
$791$		−	2.21711e11i		−	0.566346i
$792$		0			0
$793$	−1.95583e11			−0.494581
$794$		0			0
$795$	−2.58298e10	+	1.63103e11i	−0.0646626	+	0.408314i
$796$		0			0
$797$	−5.45331e11	+	3.14847e11i	−1.35153	+	0.780308i	−0.988464	−	0.151455i	$-0.951604\pi$
$797$	−5.45331e11	+	3.14847e11i	−1.35153	+	0.780308i	−0.363069	+	0.931762i	$0.618271\pi$
$798$		0			0
$799$	−1.61286e11	+	2.79356e11i	−0.395740	+	0.685442i
$800$		0			0
$801$	−6.11533e10	+	1.30010e10i	−0.148556	+	0.0315824i
$802$		0			0
$803$	9.31856e10	+	5.38007e10i	0.224123	+	0.129398i
$804$		0			0
$805$	−8.87789e10	−	1.53770e11i	−0.211410	−	0.366174i
$806$		0			0
$807$	−2.41329e10	+	2.98028e10i	−0.0569005	+	0.0702689i
$808$		0			0
$809$			1.59866e11i			0.373218i	0.982434	+	0.186609i	$0.0597497\pi$
$809$			1.59866e11i			0.373218i	−0.982434	+	0.186609i	$0.940250\pi$
$810$		0			0
$811$	−6.54959e11			−1.51402			−0.757008	−	0.653406i	$-0.773340\pi$
$811$	−6.54959e11			−1.51402			−0.757008	+	0.653406i	$0.773340\pi$
$812$		0			0
$813$	−1.35477e11	−	1.09703e11i	−0.310101	−	0.251105i
$814$		0			0
$815$	−3.01507e11	+	1.74075e11i	−0.683387	+	0.394554i
$816$		0			0
$817$	1.42031e11	−	2.46005e11i	0.318783	−	0.552148i
$818$		0			0
$819$	9.10422e9	+	4.28239e10i	0.0202352	+	0.0951812i
$820$		0			0
$821$	1.45713e11	+	8.41272e10i	0.320719	+	0.185167i	0.651713	−	0.758466i	$-0.274051\pi$
$821$	1.45713e11	+	8.41272e10i	0.320719	+	0.185167i	−0.330994	+	0.943633i	$0.607384\pi$
$822$		0			0
$823$	−3.61049e11	−	6.25356e11i	−0.786986	−	1.36310i	−0.927805	−	0.373064i	$-0.878307\pi$
$823$	−3.61049e11	−	6.25356e11i	−0.786986	−	1.36310i	0.140819	−	0.990035i	$-0.455026\pi$
$824$		0			0
$825$	−1.53512e11	−	2.43109e10i	−0.331380	−	0.0524790i
$826$		0			0
$827$		−	1.67219e11i		−	0.357490i	−0.983895	−	0.178745i	$-0.942796\pi$
$827$		−	1.67219e11i		−	0.357490i	0.983895	−	0.178745i	$-0.0572037\pi$
$828$		0			0
$829$	6.63877e10			0.140563			0.0702813	−	0.997527i	$-0.477610\pi$
$829$	6.63877e10			0.140563			0.0702813	+	0.997527i	$0.477610\pi$
$830$		0			0
$831$	5.01997e11	−	1.92710e11i	1.05268	−	0.404110i
$832$		0			0
$833$	−8.24011e10	+	4.75743e10i	−0.171141	+	0.0988081i
$834$		0			0
$835$	−7.88652e10	+	1.36598e11i	−0.162233	+	0.280996i
$836$		0			0
$837$	4.62189e11	−	2.42488e10i	0.941710	−	0.0494070i
$838$		0			0
$839$	1.87815e11	+	1.08435e11i	0.379037	+	0.218837i	0.677399	−	0.735615i	$-0.263107\pi$
$839$	1.87815e11	+	1.08435e11i	0.379037	+	0.218837i	−0.298362	+	0.954453i	$0.596440\pi$
$840$		0			0
$841$	−2.31007e11	−	4.00116e11i	−0.461786	−	0.799837i
$842$		0			0
$843$	7.29959e10	+	1.90150e11i	0.144540	+	0.376518i
$844$		0			0
$845$			3.53437e11i			0.693243i
$846$		0			0
$847$	−8.57954e10			−0.166698
$848$		0			0
$849$	4.85384e10	−	3.06497e11i	0.0934231	−	0.589923i
$850$		0			0
$851$	1.03768e11	−	5.99105e10i	0.197854	−	0.114231i
$852$		0			0
$853$	−9.68338e10	+	1.67721e11i	−0.182907	+	0.316805i	−0.942869	−	0.333163i	$-0.891884\pi$
$853$	−9.68338e10	+	1.67721e11i	−0.182907	+	0.316805i	0.759962	+	0.649967i	$0.225217\pi$
$854$		0			0
$855$	1.31733e11	+	1.46293e11i	0.246508	+	0.273754i
$856$		0			0
$857$	−5.39413e10	−	3.11430e10i	−0.0999995	−	0.0577348i	0.449166	−	0.893448i	$-0.351721\pi$
$857$	−5.39413e10	−	3.11430e10i	−0.0999995	−	0.0577348i	−0.549166	+	0.835713i	$0.685054\pi$
$858$		0			0
$859$	2.17056e11	+	3.75952e11i	0.398657	+	0.690493i	0.993560	−	0.113303i	$-0.0361432\pi$
$859$	2.17056e11	+	3.75952e11i	0.398657	+	0.690493i	−0.594904	+	0.803797i	$0.702810\pi$
$860$		0			0
$861$	9.68610e10	−	1.19618e11i	0.176253	−	0.217663i
$862$		0			0
$863$		−	3.57344e11i		−	0.644233i	−0.946700	−	0.322116i	$-0.895606\pi$
$863$		−	3.57344e11i		−	0.644233i	0.946700	−	0.322116i	$-0.104394\pi$
$864$		0			0
$865$	1.82472e11			0.325935
$866$		0			0
$867$	4.01163e11	+	3.24843e11i	0.709977	+	0.574906i
$868$		0			0
$869$	−1.59619e11	+	9.21561e10i	−0.279902	+	0.161601i
$870$		0			0
$871$	9.50930e10	−	1.64706e11i	0.165225	−	0.286178i
$872$		0			0
$873$	−2.21206e11	−	7.18648e10i	−0.380837	−	0.123726i
$874$		0			0
$875$	−2.06386e11	−	1.19157e11i	−0.352086	−	0.203277i
$876$		0			0
$877$	−4.62900e11	−	8.01767e11i	−0.782508	−	1.35534i	−0.930476	−	0.366352i	$-0.880607\pi$
$877$	−4.62900e11	−	8.01767e11i	−0.782508	−	1.35534i	0.147968	−	0.988992i	$-0.452727\pi$
$878$		0			0
$879$	−2.18781e11	−	3.46472e10i	−0.366483	−	0.0580381i
$880$		0			0
$881$		−	8.74220e11i		−	1.45117i	−0.688135	−	0.725583i	$-0.741570\pi$
$881$		−	8.74220e11i		−	1.45117i	0.688135	−	0.725583i	$-0.258430\pi$
$882$		0			0
$883$	7.15593e11			1.17713			0.588563	−	0.808451i	$-0.299694\pi$
$883$	7.15593e11			1.17713			0.588563	+	0.808451i	$0.299694\pi$
$884$		0			0
$885$	8.14690e11	−	3.12748e11i	1.32806	−	0.509825i
$886$		0			0
$887$	−9.24005e11	+	5.33475e11i	−1.49273	+	0.861825i	−0.999965	−	0.00833996i	$-0.997345\pi$
$887$	−9.24005e11	+	5.33475e11i	−1.49273	+	0.861825i	−0.492760	+	0.870165i	$0.664012\pi$
$888$		0			0
$889$	−1.49726e11	+	2.59333e11i	−0.239712	+	0.415193i
$890$		0			0
$891$	4.68617e11	+	4.92175e10i	0.743546	+	0.0780924i
$892$		0			0
$893$	1.56346e11	+	9.02666e10i	0.245857	+	0.141945i
$894$		0			0
$895$	−2.12620e11	−	3.68269e11i	−0.331369	−	0.573948i
$896$		0			0
$897$	9.00027e10	+	2.34452e11i	0.139023	+	0.362146i
$898$		0			0
$899$		−	1.70286e11i		−	0.260699i
$900$		0			0
$901$	5.07601e11			0.770236
$902$		0			0
$903$	5.05097e10	−	3.18945e11i	0.0759668	−	0.479695i
$904$		0			0
$905$	1.40581e11	−	8.11644e10i	0.209571	−	0.120996i
$906$		0			0
$907$	7.91909e10	−	1.37163e11i	0.117016	−	0.202678i	−0.801568	−	0.597904i	$-0.796000\pi$
$907$	7.91909e10	−	1.37163e11i	0.117016	−	0.202678i	0.918584	+	0.395226i	$0.129334\pi$
$908$		0			0
$909$	2.65123e11	−	8.16070e11i	0.388321	−	1.19529i
$910$		0			0
$911$	−3.91658e11	−	2.26124e11i	−0.568635	−	0.328302i	0.187969	−	0.982175i	$-0.439810\pi$
$911$	−3.91658e11	−	2.26124e11i	−0.568635	−	0.328302i	−0.756604	+	0.653873i	$0.773143\pi$
$912$		0			0
$913$	−4.11372e11	−	7.12517e11i	−0.592040	−	1.02544i
$914$		0			0
$915$	−6.29149e11	+	7.76965e11i	−0.897572	+	1.10845i
$916$		0			0
$917$		−	3.42188e11i		−	0.483936i
$918$		0			0
$919$	8.94840e11			1.25454			0.627268	−	0.778803i	$-0.284173\pi$
$919$	8.94840e11			1.25454			0.627268	+	0.778803i	$0.284173\pi$
$920$		0			0
$921$	1.80257e11	+	1.45964e11i	0.250527	+	0.202865i
$922$		0			0
$923$	7.69835e10	−	4.44465e10i	0.106070	−	0.0612393i
$924$		0			0
$925$	2.49076e10	−	4.31412e10i	0.0340224	−	0.0589285i
$926$		0			0
$927$	7.08644e11	−	6.38115e11i	0.959642	−	0.864132i
$928$		0			0
$929$	8.46580e11	+	4.88773e11i	1.13659	+	0.656212i	0.945585	−	0.325376i	$-0.105491\pi$
$929$	8.46580e11	+	4.88773e11i	1.13659	+	0.656212i	0.191009	+	0.981588i	$0.438824\pi$
$930$		0			0
$931$	2.66258e10	+	4.61172e10i	0.0354408	+	0.0613853i
$932$		0			0
$933$	2.48469e11	+	3.93488e10i	0.327904	+	0.0519284i
$934$		0			0
$935$		−	5.86849e11i		−	0.767856i
$936$		0			0
$937$	6.02092e11			0.781096			0.390548	−	0.920583i	$-0.372286\pi$
$937$	6.02092e11			0.781096			0.390548	+	0.920583i	$0.372286\pi$
$938$		0			0
$939$	1.22591e12	−	4.70610e11i	1.57687	−	0.605339i
$940$		0			0
$941$	−2.34188e11	+	1.35209e11i	−0.298680	+	0.172443i	−0.641850	−	0.766830i	$-0.721833\pi$
$941$	−2.34188e11	+	1.35209e11i	−0.298680	+	0.172443i	0.343170	+	0.939273i	$0.388499\pi$
$942$		0			0
$943$	4.41444e11	−	7.64604e11i	0.558251	−	0.966919i
$944$		0			0
$945$	1.99407e11	+	1.01589e11i	0.250043	+	0.127385i
$946$		0			0
$947$	8.43376e11	+	4.86923e11i	1.04863	+	0.605425i	0.922265	−	0.386559i	$-0.126336\pi$
$947$	8.43376e11	+	4.86923e11i	1.04863	+	0.605425i	0.126363	+	0.991984i	$0.459670\pi$
$948$		0			0
$949$	−3.61410e10	−	6.25981e10i	−0.0445590	−	0.0771785i
$950$		0			0
$951$	1.06976e11	+	2.78667e11i	0.130787	+	0.340694i
$952$		0			0
$953$			1.31834e12i			1.59830i	0.601135	+	0.799148i	$0.294715\pi$
$953$			1.31834e12i			1.59830i	−0.601135	+	0.799148i	$0.705285\pi$
$954$		0			0
$955$	−7.75908e11			−0.932818
$956$		0			0
$957$	2.71171e10	−	1.71232e11i	0.0323292	−	0.204144i
$958$		0			0
$959$	3.99410e11	−	2.30600e11i	0.472220	−	0.272637i
$960$		0			0
$961$	4.72249e10	−	8.17958e10i	0.0553703	−	0.0959042i
$962$		0			0
$963$	6.81983e11	−	1.44987e11i	0.792991	−	0.168587i
$964$		0			0
$965$	3.45486e11	+	1.99466e11i	0.398401	+	0.230017i
$966$		0			0
$967$	2.83346e11	+	4.90770e11i	0.324050	+	0.561271i	0.981320	−	0.192385i	$-0.0616222\pi$
$967$	2.83346e11	+	4.90770e11i	0.324050	+	0.561271i	−0.657270	+	0.753655i	$0.728289\pi$
$968$		0			0
$969$	3.80811e11	−	4.70280e11i	0.431931	−	0.533411i
$970$		0			0
$971$			1.60703e12i			1.80778i	0.427763	+	0.903891i	$0.359302\pi$
$971$			1.60703e12i			1.80778i	−0.427763	+	0.903891i	$0.640698\pi$
$972$		0			0
$973$	4.58448e11			0.511491
$974$		0			0
$975$	8.11414e10	+	6.57045e10i	0.0897892	+	0.0727070i
$976$		0			0
$977$	2.62367e11	−	1.51478e11i	0.287959	−	0.166253i	−0.349062	−	0.937100i	$-0.613500\pi$
$977$	2.62367e11	−	1.51478e11i	0.287959	−	0.166253i	0.637021	+	0.770846i	$0.280166\pi$
$978$		0			0
$979$	5.21530e10	−	9.03317e10i	0.0567739	−	0.0983352i
$980$		0			0
$981$	2.49447e11	+	1.17333e12i	0.269340	+	1.26691i
$982$		0			0
$983$	−4.02707e11	−	2.32503e11i	−0.431296	−	0.249009i	0.268603	−	0.963251i	$-0.413438\pi$
$983$	−4.02707e11	−	2.32503e11i	−0.431296	−	0.249009i	−0.699898	+	0.714242i	$0.746771\pi$
$984$		0			0
$985$	1.20277e11	+	2.08325e11i	0.127772	+	0.221308i
$986$		0			0
$987$	2.02703e11	+	3.21011e10i	0.213595	+	0.0338260i
$988$		0			0
$989$		−	1.85231e12i		−	1.93610i
$990$		0			0
$991$	−1.45288e12			−1.50639			−0.753193	−	0.657800i	$-0.771487\pi$
$991$	−1.45288e12			−1.50639			−0.753193	+	0.657800i	$0.771487\pi$
$992$		0			0
$993$	4.91548e10	−	1.88698e10i	0.0505556	−	0.0194076i
$994$		0			0
$995$	−9.40567e11	+	5.43037e11i	−0.959616	+	0.554034i
$996$		0			0
$997$	2.91912e11	−	5.05606e11i	0.295441	−	0.511719i	−0.679646	−	0.733540i	$-0.737867\pi$
$997$	2.91912e11	−	5.05606e11i	0.295441	−	0.511719i	0.975087	+	0.221821i	$0.0712001\pi$
$998$		0			0
$999$	−6.85549e10	+	1.34566e11i	−0.0688298	+	0.135105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.12	✓	96
9.5	odd	6	inner	252.9.bg.a.113.12	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.12	✓	96	1.1	even	1	trivial
252.9.bg.a.113.12	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)

\(f(q)\)	\(=\)	\(q+(-62.9498 - 50.9738i) q^{3} +(-401.865 + 232.017i) q^{5} +(453.746 - 785.912i) q^{7} +(1364.35 + 6417.57i) q^{9} +O(q^{10})\)	\(q+(-62.9498 - 50.9738i) q^{3} +(-401.865 + 232.017i) q^{5} +(453.746 - 785.912i) q^{7} +(1364.35 + 6417.57i) q^{9} +(-9479.62 - 5473.06i) q^{11} +(3676.57 + 6368.00i) q^{13} +(37124.1 + 5879.15i) q^{15} -115536. i q^{17} -64661.6 q^{19} +(-68624.1 + 26343.8i) q^{21} +(-365155. + 210823. i) q^{23} +(-87648.8 + 151812. i) q^{25} +(241242. - 473531. i) q^{27} +(-169336. - 97765.9i) q^{29} +(435443. + 754209. i) q^{31} +(317758. + 827740. i) q^{33} +421107. i q^{35} -284175. q^{37} +(93161.8 - 588273. i) q^{39} +(-1.81338e6 + 1.04696e6i) q^{41} +(-2.19653e6 + 3.80450e6i) q^{43} +(-2.03727e6 - 2.26245e6i) q^{45} +(-2.41792e6 - 1.39599e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-5.88929e6 + 7.27295e6i) q^{51} +4.39346e6i q^{53} +5.07937e6 q^{55} +(4.07043e6 + 3.29604e6i) q^{57} +(2.01066e7 - 1.16086e7i) q^{59} +(-1.32993e7 + 2.30350e7i) q^{61} +(5.66272e6 + 1.83969e6i) q^{63} +(-2.95497e6 - 1.70605e6i) q^{65} +(-1.29323e7 - 2.23994e7i) q^{67} +(3.37329e7 + 5.34210e6i) q^{69} -1.20891e7i q^{71} -9.83010e6 q^{73} +(1.32559e7 - 5.08875e6i) q^{75} +(-8.60269e6 + 4.96677e6i) q^{77} +(8.41906e6 - 1.45822e7i) q^{79} +(-3.93238e7 + 1.75117e7i) q^{81} +(6.50930e7 + 3.75815e7i) q^{83} +(2.68062e7 + 4.64298e7i) q^{85} +(5.67614e6 + 1.47860e7i) q^{87} +9.52903e6i q^{89} +6.67292e6 q^{91} +(1.10338e7 - 6.96734e7i) q^{93} +(2.59852e7 - 1.50026e7i) q^{95} +(-1.77249e7 + 3.07005e7i) q^{97} +(2.21902e7 - 6.83034e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more