LMFDB - Embedded newform 45.9.g.b.37.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,9,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, 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:	$18.3320374528$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3006 x^{14} + 3660359 x^{12} + 2360769624 x^{10} + 888292333775 x^{8} + 201214811046486 x^{6} + \cdots + 60\!\cdots\!84 $ x^16 + 3006x^14 + 3660359x^12 + 2360769624x^10 + 888292333775x^8 + 201214811046486x^6 + 26986219156502761x^4 + 1971352965206494164*x^2 + 60370178541549816384
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{12}\cdot 5^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.7
Root			$12.3888i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.9.g.b.28.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.1900 + 15.1900i) q^{2} +205.471i q^{4} +(115.601 - 614.216i) q^{5} +(605.930 + 605.930i) q^{7} +(767.539 - 767.539i) q^{8} +O(q^{10})$	\(q+(15.1900 + 15.1900i) q^{2} +205.471i q^{4} +(115.601 - 614.216i) q^{5} +(605.930 + 605.930i) q^{7} +(767.539 - 767.539i) q^{8} +(11085.9 - 7573.95i) q^{10} +4284.92 q^{11} +(28977.7 - 28977.7i) q^{13} +18408.1i q^{14} +75918.3 q^{16} +(49977.7 + 49977.7i) q^{17} +12625.1i q^{19} +(126203. + 23752.6i) q^{20} +(65087.8 + 65087.8i) q^{22} +(104680. - 104680. i) q^{23} +(-363898. - 142008. i) q^{25} +880342. q^{26} +(-124501. + 124501. i) q^{28} +640943. i q^{29} -1.30632e6 q^{31} +(956707. + 956707. i) q^{32} +1.51832e6i q^{34} +(442218. - 302126. i) q^{35} +(311959. + 311959. i) q^{37} +(-191775. + 191775. i) q^{38} +(-382707. - 560163. i) q^{40} +3.20281e6 q^{41} +(4.12468e6 - 4.12468e6i) q^{43} +880425. i q^{44} +3.18019e6 q^{46} +(-3.49502e6 - 3.49502e6i) q^{47} -5.03050e6i q^{49} +(-3.37050e6 - 7.68470e6i) q^{50} +(5.95407e6 + 5.95407e6i) q^{52} +(-9.08340e6 + 9.08340e6i) q^{53} +(495340. - 2.63186e6i) q^{55} +930149. q^{56} +(-9.73590e6 + 9.73590e6i) q^{58} +1.10425e7i q^{59} -1.13283e7 q^{61} +(-1.98429e7 - 1.98429e7i) q^{62} +9.62963e6i q^{64} +(-1.44487e7 - 2.11484e7i) q^{65} +(-1.93890e7 - 1.93890e7i) q^{67} +(-1.02689e7 + 1.02689e7i) q^{68} +(1.13066e7 + 2.12800e6i) q^{70} +2.80078e7 q^{71} +(-2.13453e7 + 2.13453e7i) q^{73} +9.47731e6i q^{74} -2.59409e6 q^{76} +(2.59636e6 + 2.59636e6i) q^{77} -3.60404e7i q^{79} +(8.77622e6 - 4.66302e7i) q^{80} +(4.86506e7 + 4.86506e7i) q^{82} +(-4.88658e7 + 4.88658e7i) q^{83} +(3.64746e7 - 2.49196e7i) q^{85} +1.25308e8 q^{86} +(3.28884e6 - 3.28884e6i) q^{88} +9.25997e7i q^{89} +3.51169e7 q^{91} +(2.15088e7 + 2.15088e7i) q^{92} -1.06179e8i q^{94} +(7.75455e6 + 1.45947e6i) q^{95} +(5.68733e7 + 5.68733e7i) q^{97} +(7.64132e7 - 7.64132e7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4220 q^{7}+O(q^{10}) $ 16 * q - 4220 * q^7	$ 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) $ 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	15.1900	+	15.1900i	0.949373	+	0.949373i	0.998779	−	0.0494054i	$-0.0157326\pi$
$2$	15.1900	+	15.1900i	0.949373	+	0.949373i	−0.0494054	+	0.998779i	$0.515733\pi$
$3$		0			0
$3$		0			0
$4$			205.471i			0.802620i
$5$	115.601	−	614.216i	0.184961	−	0.982746i
$5$	115.601	−	614.216i	0.184961	−	0.982746i
$6$		0			0
$7$	605.930	+	605.930i	0.252366	+	0.252366i	0.821940	−	0.569574i	$-0.192892\pi$
$7$	605.930	+	605.930i	0.252366	+	0.252366i	−0.569574	+	0.821940i	$0.692892\pi$
$8$	767.539	−	767.539i	0.187387	−	0.187387i
$9$		0			0
$10$	11085.9	−	7573.95i	1.10859	−	0.757395i
$11$	4284.92			0.292666			0.146333	−	0.989235i	$-0.453253\pi$
$11$	4284.92			0.292666			0.146333	+	0.989235i	$0.453253\pi$
$12$		0			0
$13$	28977.7	−	28977.7i	1.01459	−	1.01459i	0.0146990	−	0.999892i	$-0.495321\pi$
$13$	28977.7	−	28977.7i	1.01459	−	1.01459i	0.999892	−	0.0146990i	$-0.00467901\pi$
$14$			18408.1i			0.479178i
$15$		0			0
$16$	75918.3			1.15842
$17$	49977.7	+	49977.7i	0.598385	+	0.598385i	0.939883	−	0.341498i	$-0.110934\pi$
$17$	49977.7	+	49977.7i	0.598385	+	0.598385i	−0.341498	+	0.939883i	$0.610934\pi$
$18$		0			0
$19$			12625.1i			0.0968771i	0.998826	+	0.0484385i	$0.0154245\pi$
$19$			12625.1i			0.0968771i	−0.998826	+	0.0484385i	$0.984576\pi$
$20$	126203.	+	23752.6i	0.788771	+	0.148454i
$21$		0			0
$22$	65087.8	+	65087.8i	0.277849	+	0.277849i
$23$	104680.	−	104680.i	0.374071	−	0.374071i	−0.494887	−	0.868958i	$-0.664790\pi$
$23$	104680.	−	104680.i	0.374071	−	0.374071i	0.868958	+	0.494887i	$0.164790\pi$
$24$		0			0
$25$	−363898.	−	142008.i	−0.931579	−	0.363540i
$26$	880342.			1.92645
$27$		0			0
$28$	−124501.	+	124501.i	−0.202554	+	0.202554i
$29$			640943.i			0.906207i	0.891458	+	0.453103i	$0.149683\pi$
$29$			640943.i			0.906207i	−0.891458	+	0.453103i	$0.850317\pi$
$30$		0			0
$31$	−1.30632e6			−1.41450			−0.707248	−	0.706966i	$-0.750063\pi$
$31$	−1.30632e6			−1.41450			−0.707248	+	0.706966i	$0.750063\pi$
$32$	956707.	+	956707.i	0.912387	+	0.912387i
$33$		0			0
$34$			1.51832e6i			1.13618i
$35$	442218.	−	302126.i	0.294689	−	0.201333i
$36$		0			0
$37$	311959.	+	311959.i	0.166453	+	0.166453i	0.785418	−	0.618965i	$-0.212448\pi$
$37$	311959.	+	311959.i	0.166453	+	0.166453i	−0.618965	+	0.785418i	$0.712448\pi$
$38$	−191775.	+	191775.i	−0.0919725	+	0.0919725i
$39$		0			0
$40$	−382707.	−	560163.i	−0.149495	−	0.218814i
$41$	3.20281e6			1.13343			0.566716	−	0.823913i	$-0.308214\pi$
$41$	3.20281e6			1.13343			0.566716	+	0.823913i	$0.308214\pi$
$42$		0			0
$43$	4.12468e6	−	4.12468e6i	1.20647	−	1.20647i	0.234308	−	0.972162i	$-0.424717\pi$
$43$	4.12468e6	−	4.12468e6i	1.20647	−	1.20647i	0.972162	−	0.234308i	$-0.0752825\pi$
$44$			880425.i			0.234899i
$45$		0			0
$46$	3.18019e6			0.710266
$47$	−3.49502e6	−	3.49502e6i	−0.716240	−	0.716240i	0.251593	−	0.967833i	$-0.419045\pi$
$47$	−3.49502e6	−	3.49502e6i	−0.716240	−	0.716240i	−0.967833	+	0.251593i	$0.919045\pi$
$48$		0			0
$49$		−	5.03050e6i		−	0.872623i
$50$	−3.37050e6	−	7.68470e6i	−0.539281	−	1.22955i
$51$		0			0
$52$	5.95407e6	+	5.95407e6i	0.814331	+	0.814331i
$53$	−9.08340e6	+	9.08340e6i	−1.15118	+	1.15118i	−0.164869	+	0.986316i	$0.552720\pi$
$53$	−9.08340e6	+	9.08340e6i	−1.15118	+	1.15118i	−0.986316	+	0.164869i	$0.947280\pi$
$54$		0			0
$55$	495340.	−	2.63186e6i	0.0541318	−	0.287616i
$56$	930149.			0.0945803
$57$		0			0
$58$	−9.73590e6	+	9.73590e6i	−0.860329	+	0.860329i
$59$			1.10425e7i			0.911295i	0.890160	+	0.455647i	$0.150592\pi$
$59$			1.10425e7i			0.911295i	−0.890160	+	0.455647i	$0.849408\pi$
$60$		0			0
$61$	−1.13283e7			−0.818173			−0.409086	−	0.912496i	$-0.634152\pi$
$61$	−1.13283e7			−0.818173			−0.409086	+	0.912496i	$0.634152\pi$
$62$	−1.98429e7	−	1.98429e7i	−1.34288	−	1.34288i
$63$		0			0
$64$			9.62963e6i			0.573971i
$65$	−1.44487e7	−	2.11484e7i	−0.809425	−	1.18475i
$66$		0			0
$67$	−1.93890e7	−	1.93890e7i	−0.962178	−	0.962178i	0.0371324	−	0.999310i	$-0.488178\pi$
$67$	−1.93890e7	−	1.93890e7i	−0.962178	−	0.962178i	−0.999310	+	0.0371324i	$0.988178\pi$
$68$	−1.02689e7	+	1.02689e7i	−0.480275	+	0.480275i
$69$		0			0
$70$	1.13066e7	+	2.12800e6i	0.470911	+	0.0886295i
$71$	2.80078e7			1.10216			0.551081	−	0.834452i	$-0.314216\pi$
$71$	2.80078e7			1.10216			0.551081	+	0.834452i	$0.314216\pi$
$72$		0			0
$73$	−2.13453e7	+	2.13453e7i	−0.751640	+	0.751640i	−0.974785	−	0.223145i	$-0.928368\pi$
$73$	−2.13453e7	+	2.13453e7i	−0.751640	+	0.751640i	0.223145	+	0.974785i	$0.428368\pi$
$74$			9.47731e6i			0.316052i
$75$		0			0
$76$	−2.59409e6			−0.0777555
$77$	2.59636e6	+	2.59636e6i	0.0738587	+	0.0738587i
$78$		0			0
$79$		−	3.60404e7i		−	0.925297i	−0.886542	−	0.462649i	$-0.846899\pi$
$79$		−	3.60404e7i		−	0.925297i	0.886542	−	0.462649i	$-0.153101\pi$
$80$	8.77622e6	−	4.66302e7i	0.214263	−	1.13843i
$81$		0			0
$82$	4.86506e7	+	4.86506e7i	1.07605	+	1.07605i
$83$	−4.88658e7	+	4.88658e7i	−1.02966	+	1.02966i	−0.0301106	+	0.999547i	$0.509586\pi$
$83$	−4.88658e7	+	4.88658e7i	−1.02966	+	1.02966i	−0.999547	+	0.0301106i	$0.990414\pi$
$84$		0			0
$85$	3.64746e7	−	2.49196e7i	0.698738	−	0.477382i
$86$	1.25308e8			2.29078
$87$		0			0
$88$	3.28884e6	−	3.28884e6i	0.0548418	−	0.0548418i
$89$			9.25997e7i			1.47588i	0.674869	+	0.737938i	$0.264200\pi$
$89$			9.25997e7i			1.47588i	−0.674869	+	0.737938i	$0.735800\pi$
$90$		0			0
$91$	3.51169e7			0.512096
$92$	2.15088e7	+	2.15088e7i	0.300237	+	0.300237i
$93$		0			0
$94$		−	1.06179e8i		−	1.35996i
$95$	7.75455e6	+	1.45947e6i	0.0952055	+	0.0179185i
$96$		0			0
$97$	5.68733e7	+	5.68733e7i	0.642424	+	0.642424i	0.951151	−	0.308727i	$-0.0999029\pi$
$97$	5.68733e7	+	5.68733e7i	0.642424	+	0.642424i	−0.308727	+	0.951151i	$0.599903\pi$
$98$	7.64132e7	−	7.64132e7i	0.828445	−	0.828445i
$99$		0			0
$100$	2.91785e7	−	7.47703e7i	0.291785	−	0.747703i
$101$	5.83352e7			0.560590			0.280295	−	0.959914i	$-0.409568\pi$
$101$	5.83352e7			0.560590			0.280295	+	0.959914i	$0.409568\pi$
$102$		0			0
$103$	−6.89632e6	+	6.89632e6i	−0.0612730	+	0.0612730i	−0.737079	−	0.675806i	$-0.763796\pi$
$103$	−6.89632e6	+	6.89632e6i	−0.0612730	+	0.0612730i	0.675806	+	0.737079i	$0.263796\pi$
$104$		−	4.44831e7i		−	0.380243i
$105$		0			0
$106$	−2.75953e8			−2.18581
$107$	4.16325e7	+	4.16325e7i	0.317612	+	0.317612i	0.847849	−	0.530237i	$-0.177897\pi$
$107$	4.16325e7	+	4.16325e7i	0.317612	+	0.317612i	−0.530237	+	0.847849i	$0.677897\pi$
$108$		0			0
$109$			2.43647e8i			1.72606i	0.505153	+	0.863030i	$0.331436\pi$
$109$			2.43647e8i			1.72606i	−0.505153	+	0.863030i	$0.668564\pi$
$110$	4.75022e7	−	3.24538e7i	0.324446	−	0.221663i
$111$		0			0
$112$	4.60012e7	+	4.60012e7i	0.292346	+	0.292346i
$113$	−8.19194e7	+	8.19194e7i	−0.502427	+	0.502427i	−0.912191	−	0.409764i	$-0.865611\pi$
$113$	−8.19194e7	+	8.19194e7i	−0.502427	+	0.502427i	0.409764	+	0.912191i	$0.365611\pi$
$114$		0			0
$115$	−5.21952e7	−	7.63975e7i	−0.298428	−	0.436805i
$116$	−1.31695e8			−0.727340
$117$		0			0
$118$	−1.67735e8	+	1.67735e8i	−0.865159	+	0.865159i
$119$			6.05659e7i			0.302023i
$120$		0			0
$121$	−1.95998e8			−0.914347
$122$	−1.72076e8	−	1.72076e8i	−0.776751	−	0.776751i
$123$		0			0
$124$		−	2.68410e8i		−	1.13530i
$125$	−1.29290e8	+	2.07096e8i	−0.529574	+	0.848264i
$126$		0			0
$127$	3.58603e8	+	3.58603e8i	1.37848	+	1.37848i	0.847197	+	0.531278i	$0.178288\pi$
$127$	3.58603e8	+	3.58603e8i	1.37848	+	1.37848i	0.531278	+	0.847197i	$0.321712\pi$
$128$	9.86432e7	−	9.86432e7i	0.367474	−	0.367474i
$129$		0			0
$130$	1.01768e8	−	5.40720e8i	0.356319	−	1.89321i
$131$	−5.79706e8			−1.96844			−0.984221	−	0.176945i	$-0.943379\pi$
$131$	−5.79706e8			−1.96844			−0.984221	+	0.176945i	$0.943379\pi$
$132$		0			0
$133$	−7.64993e6	+	7.64993e6i	−0.0244484	+	0.0244484i
$134$		−	5.89036e8i		−	1.82693i
$135$		0			0
$136$	7.67196e7			0.224259
$137$	1.35867e8	+	1.35867e8i	0.385683	+	0.385683i	0.873144	−	0.487462i	$-0.162077\pi$
$137$	1.35867e8	+	1.35867e8i	0.385683	+	0.385683i	−0.487462	+	0.873144i	$0.662077\pi$
$138$		0			0
$139$		−	3.73739e8i		−	1.00117i	−0.865686	−	0.500587i	$-0.833118\pi$
$139$		−	3.73739e8i		−	1.00117i	0.865686	−	0.500587i	$-0.166882\pi$
$140$	6.20780e7	+	9.08628e7i	0.161594	+	0.236523i
$141$		0			0
$142$	4.25438e8	+	4.25438e8i	1.04636	+	1.04636i
$143$	1.24167e8	−	1.24167e8i	0.296936	−	0.296936i
$144$		0			0
$145$	3.93677e8	+	7.40936e7i	0.890571	+	0.167613i
$146$	−6.48468e8			−1.42718
$147$		0			0
$148$	−6.40985e7	+	6.40985e7i	−0.133598	+	0.133598i
$149$		−	3.18508e8i		−	0.646212i	−0.946363	−	0.323106i	$-0.895273\pi$
$149$		−	3.18508e8i		−	0.646212i	0.946363	−	0.323106i	$-0.104727\pi$
$150$		0			0
$151$	−1.55101e8			−0.298337			−0.149169	−	0.988812i	$-0.547660\pi$
$151$	−1.55101e8			−0.298337			−0.149169	+	0.988812i	$0.547660\pi$
$152$	9.69027e6	+	9.69027e6i	0.0181535	+	0.0181535i
$153$		0			0
$154$			7.88772e7i			0.140239i
$155$	−1.51011e8	+	8.02360e8i	−0.261627	+	1.39009i
$156$		0			0
$157$	−2.35107e8	−	2.35107e8i	−0.386961	−	0.386961i	0.486641	−	0.873602i	$-0.338222\pi$
$157$	−2.35107e8	−	2.35107e8i	−0.386961	−	0.386961i	−0.873602	+	0.486641i	$0.838222\pi$
$158$	5.47453e8	−	5.47453e8i	0.878452	−	0.878452i
$159$		0			0
$160$	6.98221e8	−	4.77029e8i	1.06540	−	0.727888i
$161$	1.26858e8			0.188805
$162$		0			0
$163$	−6.62468e8	+	6.62468e8i	−0.938457	+	0.938457i	−0.998213	−	0.0597563i	$-0.980968\pi$
$163$	−6.62468e8	+	6.62468e8i	−0.938457	+	0.938457i	0.0597563	+	0.998213i	$0.480968\pi$
$164$			6.58083e8i			0.909715i
$165$		0			0
$166$	−1.48454e9			−1.95506
$167$	5.95302e8	+	5.95302e8i	0.765369	+	0.765369i	0.977287	−	0.211918i	$-0.0679710\pi$
$167$	5.95302e8	+	5.95302e8i	0.765369	+	0.765369i	−0.211918	+	0.977287i	$0.567971\pi$
$168$		0			0
$169$		−	8.63687e8i		−	1.05879i
$170$	9.32576e8	+	1.75519e8i	1.11658	+	0.210150i
$171$		0			0
$172$	8.47502e8	+	8.47502e8i	0.968337	+	0.968337i
$173$	−1.36775e8	+	1.36775e8i	−0.152694	+	0.152694i	−0.779320	−	0.626626i	$-0.784435\pi$
$173$	−1.36775e8	+	1.36775e8i	−0.152694	+	0.152694i	0.626626	+	0.779320i	$0.284435\pi$
$174$		0			0
$175$	−1.34450e8	−	3.06543e8i	−0.143353	−	0.326843i
$176$	3.25304e8			0.339030
$177$		0			0
$178$	−1.40659e9	+	1.40659e9i	−1.40116	+	1.40116i
$179$		−	1.22129e9i		−	1.18961i	−0.803869	−	0.594807i	$-0.797229\pi$
$179$		−	1.22129e9i		−	1.18961i	0.803869	−	0.594807i	$-0.202771\pi$
$180$		0			0
$181$	−2.84242e8			−0.264834			−0.132417	−	0.991194i	$-0.542274\pi$
$181$	−2.84242e8			−0.264834			−0.132417	+	0.991194i	$0.542274\pi$
$182$	5.33426e8	+	5.33426e8i	0.486170	+	0.486170i
$183$		0			0
$184$		−	1.60693e8i		−	0.140192i
$185$	2.27673e8	−	1.55548e8i	0.194368	−	0.132793i
$186$		0			0
$187$	2.14150e8	+	2.14150e8i	0.175127	+	0.175127i
$188$	7.18125e8	−	7.18125e8i	0.574868	−	0.574868i
$189$		0			0
$190$	9.56220e7	+	1.39961e8i	0.0733742	+	0.107397i
$191$	7.13822e8			0.536360			0.268180	−	0.963369i	$-0.413578\pi$
$191$	7.13822e8			0.536360			0.268180	+	0.963369i	$0.413578\pi$
$192$		0			0
$193$	5.56478e8	−	5.56478e8i	0.401069	−	0.401069i	−0.477541	−	0.878610i	$-0.658472\pi$
$193$	5.56478e8	−	5.56478e8i	0.401069	−	0.401069i	0.878610	+	0.477541i	$0.158472\pi$
$194$			1.72781e9i			1.21980i
$195$		0			0
$196$	1.03362e9			0.700385
$197$	2.96874e7	+	2.96874e7i	0.0197109	+	0.0197109i	0.716894	−	0.697183i	$-0.245563\pi$
$197$	2.96874e7	+	2.96874e7i	0.0197109	+	0.0197109i	−0.697183	+	0.716894i	$0.745563\pi$
$198$		0			0
$199$		−	6.20227e8i		−	0.395493i	−0.980253	−	0.197746i	$-0.936638\pi$
$199$		−	6.20227e8i		−	0.395493i	0.980253	−	0.197746i	$-0.0633623\pi$
$200$	−3.88302e8	+	1.70309e8i	−0.242689	+	0.106443i
$201$		0			0
$202$	8.86111e8	+	8.86111e8i	0.532209	+	0.532209i
$203$	−3.88366e8	+	3.88366e8i	−0.228695	+	0.228695i
$204$		0			0
$205$	3.70247e8	−	1.96722e9i	0.209641	−	1.11388i
$206$	−2.09510e8			−0.116342
$207$		0			0
$208$	2.19994e9	−	2.19994e9i	1.17532	−	1.17532i
$209$			5.40976e7i			0.0283526i
$210$		0			0
$211$	8.91245e8			0.449642			0.224821	−	0.974400i	$-0.427820\pi$
$211$	8.91245e8			0.449642			0.224821	+	0.974400i	$0.427820\pi$
$212$	−1.86637e9	−	1.86637e9i	−0.923963	−	0.923963i
$213$		0			0
$214$			1.26479e9i			0.603065i
$215$	−2.05663e9	−	3.01026e9i	−0.962503	−	1.40880i
$216$		0			0
$217$	−7.91536e8	−	7.91536e8i	−0.356970	−	0.356970i
$218$	−3.70100e9	+	3.70100e9i	−1.63867	+	1.63867i
$219$		0			0
$220$	5.40771e8	+	1.01778e8i	0.230846	+	0.0434473i
$221$	2.89648e9			1.21423
$222$		0			0
$223$	−3.96849e8	+	3.96849e8i	−0.160475	+	0.160475i	−0.782777	−	0.622302i	$-0.786197\pi$
$223$	−3.96849e8	+	3.96849e8i	−0.160475	+	0.160475i	0.622302	+	0.782777i	$0.286197\pi$
$224$			1.15939e9i			0.460510i
$225$		0			0
$226$	−2.48871e9			−0.953982
$227$	−1.46131e9	−	1.46131e9i	−0.550349	−	0.550349i	0.376193	−	0.926541i	$-0.377233\pi$
$227$	−1.46131e9	−	1.46131e9i	−0.550349	−	0.550349i	−0.926541	+	0.376193i	$0.877233\pi$
$228$		0			0
$229$			2.82283e9i			1.02646i	0.858251	+	0.513231i	$0.171551\pi$
$229$			2.82283e9i			1.02646i	−0.858251	+	0.513231i	$0.828449\pi$
$230$	3.67632e8	−	1.95332e9i	0.131372	−	0.698011i
$231$		0			0
$232$	4.91948e8	+	4.91948e8i	0.169812	+	0.169812i
$233$	6.82714e8	−	6.82714e8i	0.231641	−	0.231641i	−0.581736	−	0.813377i	$-0.697626\pi$
$233$	6.82714e8	−	6.82714e8i	0.231641	−	0.231641i	0.813377	+	0.581736i	$0.197626\pi$
$234$		0			0
$235$	−2.55073e9	+	1.74267e9i	−0.836358	+	0.571405i
$236$	−2.26891e9			−0.731423
$237$		0			0
$238$	−9.19995e8	+	9.19995e8i	−0.286733	+	0.286733i
$239$		−	3.33204e9i		−	1.02122i	−0.859813	−	0.510608i	$-0.829420\pi$
$239$		−	3.33204e9i		−	1.02122i	0.859813	−	0.510608i	$-0.170580\pi$
$240$		0			0
$241$	−2.42023e9			−0.717445			−0.358722	−	0.933444i	$-0.616788\pi$
$241$	−2.42023e9			−0.717445			−0.358722	+	0.933444i	$0.616788\pi$
$242$	−2.97721e9	−	2.97721e9i	−0.868057	−	0.868057i
$243$		0			0
$244$		−	2.32763e9i		−	0.656682i
$245$	−3.08981e9	−	5.81530e8i	−0.857567	−	0.161402i
$246$		0			0
$247$	3.65847e8	+	3.65847e8i	0.0982906	+	0.0982906i
$248$	−1.00265e9	+	1.00265e9i	−0.265059	+	0.265059i
$249$		0			0
$250$	−5.10970e9	+	1.18186e9i	−1.30808	+	0.302556i
$251$	5.01881e9			1.26446			0.632231	−	0.774780i	$-0.282139\pi$
$251$	5.01881e9			1.26446			0.632231	+	0.774780i	$0.282139\pi$
$252$		0			0
$253$	4.48547e8	−	4.48547e8i	0.109478	−	0.109478i
$254$			1.08943e10i			2.61738i
$255$		0			0
$256$	5.46196e9			1.27171
$257$	1.84074e9	+	1.84074e9i	0.421948	+	0.421948i	0.885874	−	0.463926i	$-0.153560\pi$
$257$	1.84074e9	+	1.84074e9i	0.421948	+	0.421948i	−0.463926	+	0.885874i	$0.653560\pi$
$258$		0			0
$259$			3.78051e8i			0.0840139i
$260$	4.34539e9	−	2.96879e9i	0.950900	−	0.649660i
$261$		0			0
$262$	−8.80572e9	−	8.80572e9i	−1.86879	−	1.86879i
$263$	−3.72394e9	+	3.72394e9i	−0.778358	+	0.778358i	−0.979552	−	0.201194i	$-0.935518\pi$
$263$	−3.72394e9	+	3.72394e9i	−0.778358	+	0.778358i	0.201194	+	0.979552i	$0.435518\pi$
$264$		0			0
$265$	4.52912e9	+	6.62922e9i	0.918397	+	1.34425i
$266$	−2.32405e8			−0.0464214
$267$		0			0
$268$	3.98386e9	−	3.98386e9i	0.772263	−	0.772263i
$269$		−	6.73435e9i		−	1.28614i	−0.765809	−	0.643068i	$-0.777661\pi$
$269$		−	6.73435e9i		−	1.28614i	0.765809	−	0.643068i	$-0.222339\pi$
$270$		0			0
$271$	7.06460e9			1.30982			0.654908	−	0.755708i	$-0.272707\pi$
$271$	7.06460e9			1.30982			0.654908	+	0.755708i	$0.272707\pi$
$272$	3.79422e9	+	3.79422e9i	0.693181	+	0.693181i
$273$		0			0
$274$			4.12762e9i			0.732314i
$275$	−1.55927e9	−	6.08492e8i	−0.272641	−	0.106396i
$276$		0			0
$277$	1.76284e9	+	1.76284e9i	0.299429	+	0.299429i	0.840790	−	0.541361i	$-0.182091\pi$
$277$	1.76284e9	+	1.76284e9i	0.299429	+	0.299429i	−0.541361	+	0.840790i	$0.682091\pi$
$278$	5.67709e9	−	5.67709e9i	0.950487	−	0.950487i
$279$		0			0
$280$	1.07526e8	−	5.71313e8i	0.0174937	−	0.0929484i
$281$	8.08318e9			1.29645			0.648227	−	0.761447i	$-0.275511\pi$
$281$	8.08318e9			1.29645			0.648227	+	0.761447i	$0.275511\pi$
$282$		0			0
$283$	1.25335e8	−	1.25335e8i	0.0195401	−	0.0195401i	−0.697269	−	0.716809i	$-0.745602\pi$
$283$	1.25335e8	−	1.25335e8i	0.0195401	−	0.0195401i	0.716809	+	0.697269i	$0.245602\pi$
$284$			5.75478e9i			0.884617i
$285$		0			0
$286$	3.77219e9			0.563806
$287$	1.94068e9	+	1.94068e9i	0.286039	+	0.286039i
$288$		0			0
$289$		−	1.98022e9i		−	0.283872i
$290$	4.85447e9	+	7.10543e9i	0.686357	+	1.00461i
$291$		0			0
$292$	−4.38583e9	−	4.38583e9i	−0.603282	−	0.603282i
$293$	4.13188e9	−	4.13188e9i	0.560631	−	0.560631i	−0.368855	−	0.929487i	$-0.620250\pi$
$293$	4.13188e9	−	4.13188e9i	0.560631	−	0.560631i	0.929487	+	0.368855i	$0.120250\pi$
$294$		0			0
$295$	6.78247e9	+	1.27652e9i	0.895571	+	0.168554i
$296$	4.78882e8			0.0623823
$297$		0			0
$298$	4.83813e9	−	4.83813e9i	0.613497	−	0.613497i
$299$		−	6.06680e9i		−	0.759058i
$300$		0			0
$301$	4.99854e9			0.608943
$302$	−2.35598e9	−	2.35598e9i	−0.283234	−	0.283234i
$303$		0			0
$304$			9.58477e8i			0.112224i
$305$	−1.30956e9	+	6.95802e9i	−0.151330	+	0.804056i
$306$		0			0
$307$	−1.15844e10	−	1.15844e10i	−1.30413	−	1.30413i	−0.925580	−	0.378553i	$-0.876422\pi$
$307$	−1.15844e10	−	1.15844e10i	−1.30413	−	1.30413i	−0.378553	−	0.925580i	$-0.623578\pi$
$308$	−5.33476e8	+	5.33476e8i	−0.0592805	+	0.0592805i
$309$		0			0
$310$	−1.44817e10	+	9.89398e9i	−1.56810	+	1.07133i
$311$	−1.41409e10			−1.51160			−0.755799	−	0.654804i	$-0.772751\pi$
$311$	−1.41409e10			−1.51160			−0.755799	+	0.654804i	$0.772751\pi$
$312$		0			0
$313$	−6.49310e9	+	6.49310e9i	−0.676511	+	0.676511i	−0.959209	−	0.282698i	$-0.908771\pi$
$313$	−6.49310e9	+	6.49310e9i	−0.676511	+	0.676511i	0.282698	+	0.959209i	$0.408771\pi$
$314$		−	7.14254e9i		−	0.734741i
$315$		0			0
$316$	7.40524e9			0.742662
$317$	1.31825e10	+	1.31825e10i	1.30545	+	1.30545i	0.924659	+	0.380796i	$0.124350\pi$
$317$	1.31825e10	+	1.31825e10i	1.30545	+	1.30545i	0.380796	+	0.924659i	$0.375650\pi$
$318$		0			0
$319$			2.74639e9i			0.265215i
$320$	5.91467e9	+	1.11319e9i	0.564067	+	0.106162i
$321$		0			0
$322$	1.92697e9	+	1.92697e9i	0.179247	+	0.179247i
$323$	−6.30974e8	+	6.30974e8i	−0.0579697	+	0.0579697i
$324$		0			0
$325$	−1.46600e10	+	6.42987e9i	−1.31402	+	0.576327i
$326$	−2.01257e10			−1.78189
$327$		0			0
$328$	2.45828e9	−	2.45828e9i	0.212391	−	0.212391i
$329$		−	4.23548e9i		−	0.361509i
$330$		0			0
$331$	−1.04163e10			−0.867761			−0.433880	−	0.900971i	$-0.642856\pi$
$331$	−1.04163e10			−0.867761			−0.433880	+	0.900971i	$0.642856\pi$
$332$	−1.00405e10	−	1.00405e10i	−0.826423	−	0.826423i
$333$		0			0
$334$			1.80852e10i			1.45324i
$335$	−1.41504e10	+	9.66763e9i	−1.12354	+	0.767610i
$336$		0			0
$337$	−5.51923e9	−	5.51923e9i	−0.427917	−	0.427917i	0.460002	−	0.887918i	$-0.347849\pi$
$337$	−5.51923e9	−	5.51923e9i	−0.427917	−	0.427917i	−0.887918	+	0.460002i	$0.847849\pi$
$338$	1.31194e10	−	1.31194e10i	1.00519	−	1.00519i
$339$		0			0
$340$	5.12025e9	+	7.49445e9i	0.383156	+	0.560821i
$341$	−5.59746e9			−0.413974
$342$		0			0
$343$	6.54119e9	−	6.54119e9i	0.472586	−	0.472586i
$344$		−	6.33171e9i		−	0.452155i
$345$		0			0
$346$	−4.15523e9			−0.289928
$347$	−7.83468e9	−	7.83468e9i	−0.540385	−	0.540385i	0.383257	−	0.923642i	$-0.374802\pi$
$347$	−7.83468e9	−	7.83468e9i	−0.540385	−	0.540385i	−0.923642	+	0.383257i	$0.874802\pi$
$348$		0			0
$349$		−	1.11380e10i		−	0.750765i	−0.926870	−	0.375383i	$-0.877511\pi$
$349$		−	1.11380e10i		−	0.750765i	0.926870	−	0.375383i	$-0.122489\pi$
$350$	2.61410e9	−	6.69867e9i	0.174201	−	0.446392i
$351$		0			0
$352$	4.09941e9	+	4.09941e9i	0.267024	+	0.267024i
$353$	−8.38048e9	+	8.38048e9i	−0.539722	+	0.539722i	−0.923447	−	0.383725i	$-0.874641\pi$
$353$	−8.38048e9	+	8.38048e9i	−0.539722	+	0.539722i	0.383725	+	0.923447i	$0.374641\pi$
$354$		0			0
$355$	3.23773e9	−	1.72028e10i	0.203857	−	1.08314i
$356$	−1.90265e10			−1.18457
$357$		0			0
$358$	1.85513e10	−	1.85513e10i	1.12939	−	1.12939i
$359$		−	1.42412e10i		−	0.857372i	−0.903454	−	0.428686i	$-0.858977\pi$
$359$		−	1.42412e10i		−	0.857372i	0.903454	−	0.428686i	$-0.141023\pi$
$360$		0			0
$361$	1.68242e10			0.990615
$362$	−4.31762e9	−	4.31762e9i	−0.251426	−	0.251426i
$363$		0			0
$364$			7.21550e9i			0.411018i
$365$	1.06431e10	+	1.55781e10i	0.599647	+	0.877696i
$366$		0			0
$367$	2.13528e8	+	2.13528e8i	0.0117704	+	0.0117704i	0.712967	−	0.701197i	$-0.247351\pi$
$367$	2.13528e8	+	2.13528e8i	0.0117704	+	0.0117704i	−0.701197	+	0.712967i	$0.747351\pi$
$368$	7.94716e9	−	7.94716e9i	0.433332	−	0.433332i
$369$		0			0
$370$	5.82112e9	+	1.09559e9i	0.310599	+	0.0584574i
$371$	−1.10078e10			−0.581039
$372$		0			0
$373$	9.79026e9	−	9.79026e9i	0.505777	−	0.505777i	−0.407450	−	0.913227i	$-0.633582\pi$
$373$	9.79026e9	−	9.79026e9i	0.505777	−	0.505777i	0.913227	+	0.407450i	$0.133582\pi$
$374$			6.50587e9i			0.332521i
$375$		0			0
$376$	−5.36513e9			−0.268429
$377$	1.85731e10	+	1.85731e10i	0.919429	+	0.919429i
$378$		0			0
$379$			1.29022e9i			0.0625326i	0.999511	+	0.0312663i	$0.00995399\pi$
$379$			1.29022e9i			0.0625326i	−0.999511	+	0.0312663i	$0.990046\pi$
$380$	−2.99879e8	+	1.59333e9i	−0.0143818	+	0.0764138i
$381$		0			0
$382$	1.08429e10	+	1.08429e10i	0.509206	+	0.509206i
$383$	−9.77735e9	+	9.77735e9i	−0.454387	+	0.454387i	−0.896808	−	0.442421i	$-0.854120\pi$
$383$	−9.77735e9	+	9.77735e9i	−0.454387	+	0.454387i	0.442421	+	0.896808i	$0.354120\pi$
$384$		0			0
$385$	1.89487e9	−	1.29458e9i	0.0862454	−	0.0589233i
$386$	1.69058e10			0.761528
$387$		0			0
$388$	−1.16858e10	+	1.16858e10i	−0.515622	+	0.515622i
$389$			8.38723e9i			0.366286i	0.983086	+	0.183143i	$0.0586271\pi$
$389$			8.38723e9i			0.366286i	−0.983086	+	0.183143i	$0.941373\pi$
$390$		0			0
$391$	1.04634e10			0.447677
$392$	−3.86110e9	−	3.86110e9i	−0.163519	−	0.163519i
$393$		0			0
$394$			9.01902e8i			0.0374261i
$395$	−2.21366e10	−	4.16630e9i	−0.909332	−	0.171144i
$396$		0			0
$397$	7.53135e8	+	7.53135e8i	0.0303187	+	0.0303187i	0.722104	−	0.691785i	$-0.243175\pi$
$397$	7.53135e8	+	7.53135e8i	0.0303187	+	0.0303187i	−0.691785	+	0.722104i	$0.743175\pi$
$398$	9.42124e9	−	9.42124e9i	0.375470	−	0.375470i
$399$		0			0
$400$	−2.76265e10	−	1.07810e10i	−1.07916	−	0.421133i
$401$	4.56245e10			1.76449			0.882247	−	0.470786i	$-0.156030\pi$
$401$	4.56245e10			1.76449			0.882247	+	0.470786i	$0.156030\pi$
$402$		0			0
$403$	−3.78541e10	+	3.78541e10i	−1.43513	+	1.43513i
$404$			1.19862e10i			0.449941i
$405$		0			0
$406$	−1.17985e10			−0.434235
$407$	1.33672e9	+	1.33672e9i	0.0487150	+	0.0487150i
$408$		0			0
$409$		−	3.94186e10i		−	1.40866i	−0.709871	−	0.704332i	$-0.751247\pi$
$409$		−	3.94186e10i		−	1.40866i	0.709871	−	0.704332i	$-0.248753\pi$
$410$	3.55060e10	−	2.42579e10i	1.25651	−	0.858456i
$411$		0			0
$412$	−1.41699e9	−	1.41699e9i	−0.0491789	−	0.0491789i
$413$	−6.69097e9	+	6.69097e9i	−0.229979	+	0.229979i
$414$		0			0
$415$	2.43652e10	+	3.56631e10i	0.821444	+	1.20234i
$416$	5.54464e10			1.85140
$417$		0			0
$418$	−8.21741e8	+	8.21741e8i	−0.0269172	+	0.0269172i
$419$			3.51993e10i			1.14203i	0.820940	+	0.571015i	$0.193450\pi$
$419$			3.51993e10i			1.14203i	−0.820940	+	0.571015i	$0.806550\pi$
$420$		0			0
$421$	−2.81133e10			−0.894920			−0.447460	−	0.894304i	$-0.647671\pi$
$421$	−2.81133e10			−0.894920			−0.447460	+	0.894304i	$0.647671\pi$
$422$	1.35380e10	+	1.35380e10i	0.426878	+	0.426878i
$423$		0			0
$424$			1.39437e10i			0.431435i
$425$	−1.10895e10	−	2.52840e10i	−0.339905	−	0.774979i
$426$		0			0
$427$	−6.86415e9	−	6.86415e9i	−0.206479	−	0.206479i
$428$	−8.55425e9	+	8.55425e9i	−0.254922	+	0.254922i
$429$		0			0
$430$	1.44857e10	−	7.69660e10i	0.423706	−	2.25126i
$431$	−2.83437e9			−0.0821386			−0.0410693	−	0.999156i	$-0.513076\pi$
$431$	−2.83437e9			−0.0821386			−0.0410693	+	0.999156i	$0.513076\pi$
$432$		0			0
$433$	−3.45461e10	+	3.45461e10i	−0.982759	+	0.982759i	−0.999854	−	0.0170951i	$-0.994558\pi$
$433$	−3.45461e10	+	3.45461e10i	−0.982759	+	0.982759i	0.0170951	+	0.999854i	$0.494558\pi$
$434$		−	2.40468e10i		−	0.677796i
$435$		0			0
$436$	−5.00624e10			−1.38537
$437$	1.32160e9	+	1.32160e9i	0.0362389	+	0.0362389i
$438$		0			0
$439$			3.87810e9i			0.104415i	0.998636	+	0.0522073i	$0.0166257\pi$
$439$			3.87810e9i			0.104415i	−0.998636	+	0.0522073i	$0.983374\pi$
$440$	−1.63987e9	−	2.40025e9i	−0.0437520	−	0.0640392i
$441$		0			0
$442$	4.39975e10	+	4.39975e10i	1.15276	+	1.15276i
$443$	−1.45277e10	+	1.45277e10i	−0.377208	+	0.377208i	−0.870094	−	0.492886i	$-0.835942\pi$
$443$	−1.45277e10	+	1.45277e10i	−0.377208	+	0.377208i	0.492886	+	0.870094i	$0.335942\pi$
$444$		0			0
$445$	5.68762e10	+	1.07046e10i	1.45041	+	0.272980i
$446$	−1.20563e10			−0.304701
$447$		0			0
$448$	−5.83488e9	+	5.83488e9i	−0.144850	+	0.144850i
$449$			4.42731e10i			1.08932i	0.838658	+	0.544659i	$0.183341\pi$
$449$			4.42731e10i			1.08932i	−0.838658	+	0.544659i	$0.816659\pi$
$450$		0			0
$451$	1.37238e10			0.331716
$452$	−1.68320e10	−	1.68320e10i	−0.403258	−	0.403258i
$453$		0			0
$454$		−	4.43944e10i		−	1.04497i
$455$	4.05955e9	−	2.15694e10i	0.0947180	−	0.503260i
$456$		0			0
$457$	1.96557e10	+	1.96557e10i	0.450633	+	0.450633i	0.895564	−	0.444932i	$-0.146772\pi$
$457$	1.96557e10	+	1.96557e10i	0.450633	+	0.450633i	−0.444932	+	0.895564i	$0.646772\pi$
$458$	−4.28787e10	+	4.28787e10i	−0.974495	+	0.974495i
$459$		0			0
$460$	1.56975e10	−	1.07246e10i	0.350589	−	0.239524i
$461$	6.79618e10			1.50474			0.752370	−	0.658741i	$-0.228911\pi$
$461$	6.79618e10			1.50474			0.752370	+	0.658741i	$0.228911\pi$
$462$		0			0
$463$	7.98171e9	−	7.98171e9i	0.173689	−	0.173689i	−0.614909	−	0.788598i	$-0.710807\pi$
$463$	7.98171e9	−	7.98171e9i	0.173689	−	0.173689i	0.788598	+	0.614909i	$0.210807\pi$
$464$			4.86593e10i			1.04977i
$465$		0			0
$466$	2.07408e10			0.439827
$467$	−5.78480e10	−	5.78480e10i	−1.21624	−	1.21624i	−0.968937	−	0.247307i	$-0.920454\pi$
$467$	−5.78480e10	−	5.78480e10i	−1.21624	−	1.21624i	−0.247307	−	0.968937i	$-0.579546\pi$
$468$		0			0
$469$		−	2.34967e10i		−	0.485641i
$470$	−6.52166e10	−	1.22743e10i	−1.33649	−	0.251540i
$471$		0			0
$472$	8.47554e9	+	8.47554e9i	0.170765	+	0.170765i
$473$	1.76739e10	−	1.76739e10i	0.353092	−	0.353092i
$474$		0			0
$475$	1.79287e9	−	4.59425e9i	0.0352187	−	0.0902486i
$476$	−1.24445e10			−0.242410
$477$		0			0
$478$	5.06135e10	−	5.06135e10i	0.969516	−	0.969516i
$479$		−	9.03281e10i		−	1.71586i	−0.513770	−	0.857928i	$-0.671752\pi$
$479$		−	9.03281e10i		−	1.71586i	0.513770	−	0.857928i	$-0.328248\pi$
$480$		0			0
$481$	1.80798e10			0.337763
$482$	−3.67632e10	−	3.67632e10i	−0.681123	−	0.681123i
$483$		0			0
$484$		−	4.02719e10i		−	0.733873i
$485$	4.15071e10	−	2.83579e10i	0.750163	−	0.512516i
$486$		0			0
$487$	2.93536e10	+	2.93536e10i	0.521850	+	0.521850i	0.918130	−	0.396280i	$-0.129699\pi$
$487$	2.93536e10	+	2.93536e10i	0.521850	+	0.521850i	−0.396280	+	0.918130i	$0.629699\pi$
$488$	−8.69490e9	+	8.69490e9i	−0.153315	+	0.153315i
$489$		0			0
$490$	−3.81008e10	−	5.57676e10i	−0.660921	−	0.967382i
$491$	−1.01085e11			−1.73924			−0.869619	−	0.493724i	$-0.835635\pi$
$491$	−1.01085e11			−1.73924			−0.869619	+	0.493724i	$0.835635\pi$
$492$		0			0
$493$	−3.20328e10	+	3.20328e10i	−0.542260	+	0.542260i
$494$			1.11144e10i			0.186629i
$495$		0			0
$496$	−9.91733e10			−1.63858
$497$	1.69708e10	+	1.69708e10i	0.278148	+	0.278148i
$498$		0			0
$499$			6.16998e9i			0.0995134i	0.998761	+	0.0497567i	$0.0158446\pi$
$499$			6.16998e9i			0.0995134i	−0.998761	+	0.0497567i	$0.984155\pi$
$500$	−4.25521e10	−	2.65654e10i	−0.680834	−	0.425046i
$501$		0			0
$502$	7.62356e10	+	7.62356e10i	1.20045	+	1.20045i
$503$	−1.79791e10	+	1.79791e10i	−0.280864	+	0.280864i	−0.833453	−	0.552590i	$-0.813640\pi$
$503$	−1.79791e10	+	1.79791e10i	−0.280864	+	0.280864i	0.552590	+	0.833453i	$0.313640\pi$
$504$		0			0
$505$	6.74360e9	−	3.58304e10i	0.103688	−	0.550917i
$506$	1.36268e10			0.207870
$507$		0			0
$508$	−7.36824e10	+	7.36824e10i	−1.10639	+	1.10639i
$509$			6.48919e10i			0.966762i	0.875410	+	0.483381i	$0.160591\pi$
$509$			6.48919e10i			0.966762i	−0.875410	+	0.483381i	$0.839409\pi$
$510$		0			0
$511$	−2.58675e10			−0.379376
$512$	5.77144e10	+	5.77144e10i	0.839855	+	0.839855i
$513$		0			0
$514$			5.59215e10i			0.801173i
$515$	3.43861e9	+	5.03306e9i	0.0488826	+	0.0715489i
$516$		0			0
$517$	−1.49759e10	−	1.49759e10i	−0.209619	−	0.209619i
$518$	−5.74258e9	+	5.74258e9i	−0.0797606	+	0.0797606i
$519$		0			0
$520$	−2.73222e10	−	5.14228e9i	−0.373682	−	0.0703303i
$521$	−1.34163e10			−0.182089			−0.0910443	−	0.995847i	$-0.529020\pi$
$521$	−1.34163e10			−0.182089			−0.0910443	+	0.995847i	$0.529020\pi$
$522$		0			0
$523$	−5.20405e10	+	5.20405e10i	−0.695560	+	0.695560i	−0.963450	−	0.267890i	$-0.913674\pi$
$523$	−5.20405e10	+	5.20405e10i	−0.695560	+	0.695560i	0.267890	+	0.963450i	$0.413674\pi$
$524$		−	1.19113e11i		−	1.57991i
$525$		0			0
$526$	−1.13133e11			−1.47790
$527$	−6.52866e10	−	6.52866e10i	−0.846412	−	0.846412i
$528$		0			0
$529$			5.63950e10i			0.720142i
$530$	−3.19004e10	+	1.69495e11i	−0.404290	+	2.14809i
$531$		0			0
$532$	−1.57184e9	−	1.57184e9i	−0.0196228	−	0.0196228i
$533$	9.28101e10	−	9.28101e10i	1.14997	−	1.14997i
$534$		0			0
$535$	3.03841e10	−	2.07586e10i	0.370878	−	0.253386i
$536$	−2.97636e10			−0.360600
$537$		0			0
$538$	1.02295e11	−	1.02295e11i	1.22102	−	1.22102i
$539$		−	2.15553e10i		−	0.255387i
$540$		0			0
$541$	8.89152e9			0.103797			0.0518987	−	0.998652i	$-0.483473\pi$
$541$	8.89152e9			0.103797			0.0518987	+	0.998652i	$0.483473\pi$
$542$	1.07311e11	+	1.07311e11i	1.24351	+	1.24351i
$543$		0			0
$544$			9.56280e10i			1.09192i
$545$	1.49652e11	+	2.81659e10i	1.69628	+	0.319254i
$546$		0			0
$547$	−8.56469e10	−	8.56469e10i	−0.956671	−	0.956671i	0.0424288	−	0.999099i	$-0.486490\pi$
$547$	−8.56469e10	−	8.56469e10i	−0.956671	−	0.956671i	−0.999099	+	0.0424288i	$0.986490\pi$
$548$	−2.79166e10	+	2.79166e10i	−0.309557	+	0.309557i
$549$		0			0
$550$	−1.44423e10	−	3.29283e10i	−0.157829	−	0.359847i
$551$	−8.09198e9			−0.0877906
$552$		0			0
$553$	2.18379e10	−	2.18379e10i	0.233513	−	0.233513i
$554$			5.35551e10i			0.568541i
$555$		0			0
$556$	7.67924e10			0.803562
$557$	7.17947e10	+	7.17947e10i	0.745884	+	0.745884i	0.973703	−	0.227819i	$-0.0731596\pi$
$557$	7.17947e10	+	7.17947e10i	0.745884	+	0.745884i	−0.227819	+	0.973703i	$0.573160\pi$
$558$		0			0
$559$		−	2.39048e11i		−	2.44815i
$560$	3.35724e10	−	2.29369e10i	0.341374	−	0.233229i
$561$		0			0
$562$	1.22783e11	+	1.22783e11i	1.23082	+	1.23082i
$563$	1.14328e11	−	1.14328e11i	1.13794	−	1.13794i	0.149117	−	0.988819i	$-0.452357\pi$
$563$	1.14328e11	−	1.14328e11i	1.13794	−	1.13794i	0.988819	−	0.149117i	$-0.0476432\pi$
$564$		0			0
$565$	4.08463e10	+	5.97862e10i	0.400828	+	0.586688i
$566$	3.80768e9			0.0371018
$567$		0			0
$568$	2.14971e10	−	2.14971e10i	0.206531	−	0.206531i
$569$		−	1.17871e11i		−	1.12449i	−0.826969	−	0.562247i	$-0.809937\pi$
$569$		−	1.17871e11i		−	1.12449i	0.826969	−	0.562247i	$-0.190063\pi$
$570$		0			0
$571$	1.31971e11			1.24147			0.620733	−	0.784022i	$-0.286835\pi$
$571$	1.31971e11			1.24147			0.620733	+	0.784022i	$0.286835\pi$
$572$	2.55127e10	+	2.55127e10i	0.238327	+	0.238327i
$573$		0			0
$574$			5.89576e10i			0.543116i
$575$	−5.29584e10	+	2.32275e10i	−0.484466	+	0.212487i
$576$		0			0
$577$	−1.04538e11	−	1.04538e11i	−0.943130	−	0.943130i	0.0553376	−	0.998468i	$-0.482376\pi$
$577$	−1.04538e11	−	1.04538e11i	−0.943130	−	0.943130i	−0.998468	+	0.0553376i	$0.982376\pi$
$578$	3.00795e10	−	3.00795e10i	0.269500	−	0.269500i
$579$		0			0
$580$	−1.52241e10	+	8.08892e10i	−0.134530	+	0.714790i
$581$	−5.92185e10			−0.519700
$582$		0			0
$583$	−3.89216e10	+	3.89216e10i	−0.336912	+	0.336912i
$584$			3.27666e10i			0.281696i
$585$		0			0
$586$	1.25526e11			1.06450
$587$	−2.48950e10	−	2.48950e10i	−0.209681	−	0.209681i	0.594451	−	0.804132i	$-0.297370\pi$
$587$	−2.48950e10	−	2.48950e10i	−0.209681	−	0.209681i	−0.804132	+	0.594451i	$0.797370\pi$
$588$		0			0
$589$		−	1.64924e10i		−	0.137032i
$590$	8.36353e10	+	1.22416e11i	0.690210	+	1.01025i
$591$		0			0
$592$	2.36834e10	+	2.36834e10i	0.192822	+	0.192822i
$593$	5.00144e10	−	5.00144e10i	0.404461	−	0.404461i	−0.475341	−	0.879802i	$-0.657675\pi$
$593$	5.00144e10	−	5.00144e10i	0.404461	−	0.404461i	0.879802	+	0.475341i	$0.157675\pi$
$594$		0			0
$595$	3.72006e10	+	7.00148e9i	0.296812	+	0.0558627i
$596$	6.54440e10			0.518663
$597$		0			0
$598$	9.21546e10	−	9.21546e10i	0.720630	−	0.720630i
$599$			1.89666e11i			1.47327i	0.676292	+	0.736634i	$0.263586\pi$
$599$			1.89666e11i			1.47327i	−0.676292	+	0.736634i	$0.736414\pi$
$600$		0			0
$601$	−5.55111e10			−0.425483			−0.212742	−	0.977109i	$-0.568239\pi$
$601$	−5.55111e10			−0.425483			−0.212742	+	0.977109i	$0.568239\pi$
$602$	7.59277e10	+	7.59277e10i	0.578115	+	0.578115i
$603$		0			0
$604$		−	3.18688e10i		−	0.239451i
$605$	−2.26576e10	+	1.20385e11i	−0.169119	+	0.898571i
$606$		0			0
$607$	8.12289e9	+	8.12289e9i	0.0598351	+	0.0598351i	0.736391	−	0.676556i	$-0.236528\pi$
$607$	8.12289e9	+	8.12289e9i	0.0598351	+	0.0598351i	−0.676556	+	0.736391i	$0.736528\pi$
$608$	−1.20785e10	+	1.20785e10i	−0.0883894	+	0.0883894i
$609$		0			0
$610$	−1.25584e11	+	8.57999e10i	−0.907018	+	0.619680i
$611$	−2.02556e11			−1.45338
$612$		0			0
$613$	−1.38370e11	+	1.38370e11i	−0.979939	+	0.979939i	−0.999803	−	0.0198637i	$-0.993677\pi$
$613$	−1.38370e11	+	1.38370e11i	−0.979939	+	0.979939i	0.0198637	+	0.999803i	$0.493677\pi$
$614$		−	3.51935e11i		−	2.47622i
$615$		0			0
$616$	3.98561e9			0.0276804
$617$	−6.81338e10	−	6.81338e10i	−0.470135	−	0.470135i	0.431824	−	0.901958i	$-0.357870\pi$
$617$	−6.81338e10	−	6.81338e10i	−0.470135	−	0.470135i	−0.901958	+	0.431824i	$0.857870\pi$
$618$		0			0
$619$		−	1.25414e11i		−	0.854249i	−0.904193	−	0.427125i	$-0.859527\pi$
$619$		−	1.25414e11i		−	0.854249i	0.904193	−	0.427125i	$-0.140473\pi$
$620$	−1.64862e11	−	3.10284e10i	−1.11571	−	0.209987i
$621$		0			0
$622$	−2.14800e11	−	2.14800e11i	−1.43507	−	1.43507i
$623$	−5.61089e10	+	5.61089e10i	−0.372460	+	0.372460i
$624$		0			0
$625$	1.12255e11	+	1.03353e11i	0.735677	+	0.677332i
$626$	−1.97260e11			−1.28452
$627$		0			0
$628$	4.83076e10	−	4.83076e10i	0.310583	−	0.310583i
$629$			3.11820e10i			0.199206i
$630$		0			0
$631$	−7.70795e10			−0.486207			−0.243104	−	0.970000i	$-0.578165\pi$
$631$	−7.70795e10			−0.486207			−0.243104	+	0.970000i	$0.578165\pi$
$632$	−2.76624e10	−	2.76624e10i	−0.173389	−	0.173389i
$633$		0			0
$634$			4.00485e11i			2.47873i
$635$	2.61715e11	−	1.78805e11i	1.60966	−	1.09973i
$636$		0			0
$637$	−1.45772e11	−	1.45772e11i	−0.885356	−	0.885356i
$638$	−4.17175e10	+	4.17175e10i	−0.251789	+	0.251789i
$639$		0			0
$640$	−4.91850e10	−	7.19915e10i	−0.293165	−	0.429103i
$641$	2.92061e10			0.172998			0.0864991	−	0.996252i	$-0.472432\pi$
$641$	2.92061e10			0.172998			0.0864991	+	0.996252i	$0.472432\pi$
$642$		0			0
$643$	−1.29504e10	+	1.29504e10i	−0.0757597	+	0.0757597i	−0.743971	−	0.668212i	$-0.767060\pi$
$643$	−1.29504e10	+	1.29504e10i	−0.0757597	+	0.0757597i	0.668212	+	0.743971i	$0.267060\pi$
$644$			2.60656e10i			0.151539i
$645$		0			0
$646$	−1.91690e10			−0.110070
$647$	−9.84373e10	−	9.84373e10i	−0.561750	−	0.561750i	0.368055	−	0.929804i	$-0.380024\pi$
$647$	−9.84373e10	−	9.84373e10i	−0.561750	−	0.561750i	−0.929804	+	0.368055i	$0.880024\pi$
$648$		0			0
$649$			4.73161e10i			0.266705i
$650$	−3.20355e11	−	1.25016e11i	−1.79464	−	0.700343i
$651$		0			0
$652$	−1.36118e11	−	1.36118e11i	−0.753224	−	0.753224i
$653$	−3.36757e10	+	3.36757e10i	−0.185210	+	0.185210i	−0.793622	−	0.608412i	$-0.791807\pi$
$653$	−3.36757e10	+	3.36757e10i	−0.185210	+	0.185210i	0.608412	+	0.793622i	$0.291807\pi$
$654$		0			0
$655$	−6.70145e10	+	3.56065e11i	−0.364086	+	1.93448i
$656$	2.43152e11			1.31299
$657$		0			0
$658$	6.43368e10	−	6.43368e10i	0.343207	−	0.343207i
$659$			5.43312e10i			0.288077i	0.989572	+	0.144038i	$0.0460089\pi$
$659$			5.43312e10i			0.288077i	−0.989572	+	0.144038i	$0.953991\pi$
$660$		0			0
$661$	2.69785e11			1.41323			0.706614	−	0.707599i	$-0.250222\pi$
$661$	2.69785e11			1.41323			0.706614	+	0.707599i	$0.250222\pi$
$662$	−1.58223e11	−	1.58223e11i	−0.823829	−	0.823829i
$663$		0			0
$664$			7.50128e10i			0.385890i
$665$	3.81437e9	+	5.58305e9i	0.0195046	+	0.0285486i
$666$		0			0
$667$	6.70941e10	+	6.70941e10i	0.338986	+	0.338986i
$668$	−1.22317e11	+	1.22317e11i	−0.614301	+	0.614301i
$669$		0			0
$670$	−3.61795e11	−	6.80931e10i	−1.79541	−	0.337912i
$671$	−4.85408e10			−0.239451
$672$		0			0
$673$	7.07886e10	−	7.07886e10i	0.345067	−	0.345067i	−0.513202	−	0.858268i	$-0.671541\pi$
$673$	7.07886e10	−	7.07886e10i	0.345067	−	0.345067i	0.858268	+	0.513202i	$0.171541\pi$
$674$		−	1.67674e11i		−	0.812505i
$675$		0			0
$676$	1.77462e11			0.849806
$677$	2.99576e9	+	2.99576e9i	0.0142611	+	0.0142611i	0.714201	−	0.699940i	$-0.246790\pi$
$677$	2.99576e9	+	2.99576e9i	0.0142611	+	0.0142611i	−0.699940	+	0.714201i	$0.746790\pi$
$678$		0			0
$679$			6.89225e10i			0.324251i
$680$	8.86886e9	−	4.71224e10i	0.0414794	−	0.220390i
$681$		0			0
$682$	−8.50252e10	−	8.50252e10i	−0.393016	−	0.393016i
$683$	−1.50976e11	+	1.50976e11i	−0.693786	+	0.693786i	−0.963063	−	0.269277i	$-0.913215\pi$
$683$	−1.50976e11	+	1.50976e11i	−0.693786	+	0.693786i	0.269277	+	0.963063i	$0.413215\pi$
$684$		0			0
$685$	9.91578e10	−	6.77451e10i	0.450365	−	0.307692i
$686$	1.98721e11			0.897321
$687$		0			0
$688$	3.13139e11	−	3.13139e11i	1.39760	−	1.39760i
$689$			5.26433e11i			2.33596i
$690$		0			0
$691$	3.51850e11			1.54328			0.771641	−	0.636058i	$-0.219436\pi$
$691$	3.51850e11			1.54328			0.771641	+	0.636058i	$0.219436\pi$
$692$	−2.81033e10	−	2.81033e10i	−0.122556	−	0.122556i
$693$		0			0
$694$		−	2.38017e11i		−	1.02605i
$695$	−2.29557e11	−	4.32046e10i	−0.983899	−	0.185178i
$696$		0			0
$697$	1.60069e11	+	1.60069e11i	0.678228	+	0.678228i
$698$	1.69185e11	−	1.69185e11i	0.712756	−	0.712756i
$699$		0			0
$700$	6.29857e10	−	2.76255e10i	0.262331	−	0.115058i
$701$	9.64074e10			0.399244			0.199622	−	0.979873i	$-0.436029\pi$
$701$	9.64074e10			0.399244			0.199622	+	0.979873i	$0.436029\pi$
$702$		0			0
$703$	−3.93852e9	+	3.93852e9i	−0.0161255	+	0.0161255i
$704$			4.12622e10i			0.167981i
$705$		0			0
$706$	−2.54599e11			−1.02480
$707$	3.53470e10	+	3.53470e10i	0.141474	+	0.141474i
$708$		0			0
$709$			8.40677e7i			0.000332693i	1.00000		0.000166347i	$5.29498e-5\pi$
$709$			8.40677e7i			0.000332693i	−1.00000		0.000166347i	$0.999947\pi$
$710$	3.10492e11	−	2.12130e11i	1.22185	−	0.834772i
$711$		0			0
$712$	7.10739e10	+	7.10739e10i	0.276560	+	0.276560i
$713$	−1.36746e11	+	1.36746e11i	−0.529122	+	0.529122i
$714$		0			0
$715$	−6.19116e10	−	9.06193e10i	−0.236891	−	0.346734i
$716$	2.50939e11			0.954807
$717$		0			0
$718$	2.16324e11	−	2.16324e11i	0.813966	−	0.813966i
$719$			1.00149e11i			0.374741i	0.982289	+	0.187371i	$0.0599965\pi$
$719$			1.00149e11i			0.374741i	−0.982289	+	0.187371i	$0.940003\pi$
$720$		0			0
$721$	−8.35738e9			−0.0309264
$722$	2.55559e11	+	2.55559e11i	0.940463	+	0.940463i
$723$		0			0
$724$		−	5.84033e10i		−	0.212561i
$725$	9.10189e10	−	2.33238e11i	0.329443	−	0.844203i
$726$		0			0
$727$	−1.90847e11	−	1.90847e11i	−0.683199	−	0.683199i	0.277520	−	0.960720i	$-0.410487\pi$
$727$	−1.90847e11	−	1.90847e11i	−0.683199	−	0.683199i	−0.960720	+	0.277520i	$0.910487\pi$
$728$	2.69536e10	−	2.69536e10i	0.0959603	−	0.0959603i
$729$		0			0
$730$	−7.49635e10	+	3.98300e11i	−0.263972	+	1.40255i
$731$	4.12284e11			1.44387
$732$		0			0
$733$	−8.74944e10	+	8.74944e10i	−0.303085	+	0.303085i	−0.842220	−	0.539135i	$-0.818751\pi$
$733$	−8.74944e10	+	8.74944e10i	−0.303085	+	0.303085i	0.539135	+	0.842220i	$0.318751\pi$
$734$			6.48698e9i			0.0223490i
$735$		0			0
$736$	2.00297e11			0.682595
$737$	−8.30801e10	−	8.30801e10i	−0.281596	−	0.281596i
$738$		0			0
$739$			5.50849e11i			1.84695i	0.383659	+	0.923475i	$0.374664\pi$
$739$			5.50849e11i			1.84695i	−0.383659	+	0.923475i	$0.625336\pi$
$740$	3.19605e10	+	4.67802e10i	0.106583	+	0.156004i
$741$		0			0
$742$	−1.67208e11	−	1.67208e11i	−0.551623	−	0.551623i
$743$	−1.58679e9	+	1.58679e9i	−0.00520673	+	0.00520673i	−0.709705	−	0.704499i	$-0.751172\pi$
$743$	−1.58679e9	+	1.58679e9i	−0.00520673	+	0.00520673i	0.704499	+	0.709705i	$0.251172\pi$
$744$		0			0
$745$	−1.95633e11	−	3.68198e10i	−0.635062	−	0.119524i
$746$	2.97428e11			0.960343
$747$		0			0
$748$	−4.40016e10	+	4.40016e10i	−0.140560	+	0.140560i
$749$			5.04527e10i			0.160309i
$750$		0			0
$751$	−4.67832e11			−1.47072			−0.735360	−	0.677677i	$-0.762987\pi$
$751$	−4.67832e11			−1.47072			−0.735360	+	0.677677i	$0.762987\pi$
$752$	−2.65336e11	−	2.65336e11i	−0.829707	−	0.829707i
$753$		0			0
$754$			5.64249e11i			1.74576i
$755$	−1.79299e10	+	9.52657e10i	−0.0551809	+	0.293190i
$756$		0			0
$757$	−1.00150e11	−	1.00150e11i	−0.304976	−	0.304976i	0.537981	−	0.842957i	$-0.319187\pi$
$757$	−1.00150e11	−	1.00150e11i	−0.304976	−	0.304976i	−0.842957	+	0.537981i	$0.819187\pi$
$758$	−1.95984e10	+	1.95984e10i	−0.0593668	+	0.0593668i
$759$		0			0
$760$	7.07212e9	−	4.83171e9i	0.0211980	−	0.0144826i
$761$	−1.53260e10			−0.0456974			−0.0228487	−	0.999739i	$-0.507274\pi$
$761$	−1.53260e10			−0.0456974			−0.0228487	+	0.999739i	$0.507274\pi$
$762$		0			0
$763$	−1.47633e11	+	1.47633e11i	−0.435598	+	0.435598i
$764$			1.46669e11i			0.430493i
$765$		0			0
$766$	−2.97035e11			−0.862766
$767$	3.19986e11	+	3.19986e11i	0.924591	+	0.924591i
$768$		0			0
$769$		−	6.35796e11i		−	1.81808i	−0.416710	−	0.909039i	$-0.636817\pi$
$769$		−	6.35796e11i		−	1.81808i	0.416710	−	0.909039i	$-0.363183\pi$
$770$	4.84477e10	+	9.11828e9i	0.137819	+	0.0259388i
$771$		0			0
$772$	1.14340e11	+	1.14340e11i	0.321906	+	0.321906i
$773$	−1.14853e11	+	1.14853e11i	−0.321679	+	0.321679i	−0.849411	−	0.527732i	$-0.823043\pi$
$773$	−1.14853e11	+	1.14853e11i	−0.321679	+	0.321679i	0.527732	+	0.849411i	$0.323043\pi$
$774$		0			0
$775$	4.75366e11	+	1.85507e11i	1.31771	+	0.514226i
$776$	8.73050e10			0.240764
$777$		0			0
$778$	−1.27402e11	+	1.27402e11i	−0.347742	+	0.347742i
$779$			4.04358e10i			0.109804i
$780$		0			0
$781$	1.20011e11			0.322565
$782$	1.58938e11	+	1.58938e11i	0.425012	+	0.425012i
$783$		0			0
$784$		−	3.81907e11i		−	1.01087i
$785$	−1.71585e11	+	1.17228e11i	−0.451857	+	0.308711i
$786$		0			0
$787$	−2.70517e11	−	2.70517e11i	−0.705173	−	0.705173i	0.260343	−	0.965516i	$-0.416164\pi$
$787$	−2.70517e11	−	2.70517e11i	−0.705173	−	0.705173i	−0.965516	+	0.260343i	$0.916164\pi$
$788$	−6.09989e9	+	6.09989e9i	−0.0158204	+	0.0158204i
$789$		0			0
$790$	−2.72968e11	−	3.99540e11i	−0.700816	−	1.02578i
$791$	−9.92748e10			−0.253591
$792$		0			0
$793$	−3.28268e11	+	3.28268e11i	−0.830111	+	0.830111i
$794$			2.28802e10i			0.0575676i
$795$		0			0
$796$	1.27439e11			0.317430
$797$	−1.54755e11	−	1.54755e11i	−0.383540	−	0.383540i	0.488836	−	0.872376i	$-0.337422\pi$
$797$	−1.54755e11	−	1.54755e11i	−0.383540	−	0.383540i	−0.872376	+	0.488836i	$0.837422\pi$
$798$		0			0
$799$		−	3.49346e11i		−	0.857174i
$800$	−2.12284e11	−	4.84004e11i	−0.518271	−	1.18165i
$801$		0			0
$802$	6.93035e11	+	6.93035e11i	1.67516	+	1.67516i
$803$	−9.14627e10	+	9.14627e10i	−0.219979	+	0.219979i
$804$		0			0
$805$	1.46649e10	−	7.79182e10i	0.0349217	−	0.185548i
$806$	−1.15001e12			−2.72496
$807$		0			0
$808$	4.47746e10	−	4.47746e10i	0.105048	−	0.105048i
$809$		−	7.92854e11i		−	1.85097i	−0.378786	−	0.925484i	$-0.623658\pi$
$809$		−	7.92854e11i		−	1.85097i	0.378786	−	0.925484i	$-0.376342\pi$
$810$		0			0
$811$	4.89034e11			1.13046			0.565230	−	0.824933i	$-0.308787\pi$
$811$	4.89034e11			1.13046			0.565230	+	0.824933i	$0.308787\pi$
$812$	−7.97979e10	−	7.97979e10i	−0.183555	−	0.183555i
$813$		0			0
$814$			4.06095e10i			0.0924975i
$815$	3.30316e11	+	4.83480e11i	0.748686	+	1.09584i
$816$		0			0
$817$	5.20746e10	+	5.20746e10i	0.116879	+	0.116879i
$818$	5.98767e11	−	5.98767e11i	1.33735	−	1.33735i
$819$		0			0
$820$	4.04205e11	+	7.60750e10i	0.894018	+	0.168262i
$821$	−4.29567e11			−0.945493			−0.472746	−	0.881199i	$-0.656737\pi$
$821$	−4.29567e11			−0.945493			−0.472746	+	0.881199i	$0.656737\pi$
$822$		0			0
$823$	1.85110e11	−	1.85110e11i	0.403489	−	0.403489i	−0.475972	−	0.879461i	$-0.657904\pi$
$823$	1.85110e11	−	1.85110e11i	0.403489	−	0.403489i	0.879461	+	0.475972i	$0.157904\pi$
$824$			1.05864e10i			0.0229636i
$825$		0			0
$826$	−2.03271e11			−0.436673
$827$	−1.55069e11	−	1.55069e11i	−0.331515	−	0.331515i	0.521646	−	0.853162i	$-0.325318\pi$
$827$	−1.55069e11	−	1.55069e11i	−0.331515	−	0.331515i	−0.853162	+	0.521646i	$0.825318\pi$
$828$		0			0
$829$		−	3.62675e11i		−	0.767891i	−0.923356	−	0.383945i	$-0.874565\pi$
$829$		−	3.62675e11i		−	0.767891i	0.923356	−	0.383945i	$-0.125435\pi$
$830$	−1.71614e11	+	9.11829e11i	−0.361610	+	1.92133i
$831$		0			0
$832$	2.79045e11	+	2.79045e11i	0.582345	+	0.582345i
$833$	2.51413e11	−	2.51413e11i	0.522164	−	0.522164i
$834$		0			0
$835$	4.34461e11	−	2.96826e11i	0.893727	−	0.610600i
$836$	−1.11155e10			−0.0227563
$837$		0			0
$838$	−5.34676e11	+	5.34676e11i	−1.08421	+	1.08421i
$839$		−	3.64884e11i		−	0.736388i	−0.929749	−	0.368194i	$-0.879976\pi$
$839$		−	3.64884e11i		−	0.736388i	0.929749	−	0.368194i	$-0.120024\pi$
$840$		0			0
$841$	8.94388e10			0.178789
$842$	−4.27041e11	−	4.27041e11i	−0.849613	−	0.849613i
$843$		0			0
$844$			1.83125e11i			0.360892i
$845$	−5.30491e11	−	9.98430e10i	−1.04052	−	0.195835i
$846$		0			0
$847$	−1.18761e11	−	1.18761e11i	−0.230750	−	0.230750i
$848$	−6.89596e11	+	6.89596e11i	−1.33356	+	1.33356i
$849$		0			0
$850$	2.15613e11	−	5.52513e11i	0.413047	−	1.05844i
$851$	6.53121e10			0.124530
$852$		0			0
$853$	1.28836e11	−	1.28836e11i	0.243355	−	0.243355i	−0.574881	−	0.818237i	$-0.694952\pi$
$853$	1.28836e11	−	1.28836e11i	0.243355	−	0.243355i	0.818237	+	0.574881i	$0.194952\pi$
$854$		−	2.08532e11i		−	0.392051i
$855$		0			0
$856$	6.39091e10			0.119033
$857$	2.22087e11	+	2.22087e11i	0.411718	+	0.411718i	0.882337	−	0.470619i	$-0.155969\pi$
$857$	2.22087e11	+	2.22087e11i	0.411718	+	0.411718i	−0.470619	+	0.882337i	$0.655969\pi$
$858$		0			0
$859$		−	2.11229e11i		−	0.387954i	−0.981006	−	0.193977i	$-0.937861\pi$
$859$		−	2.11229e11i		−	0.387954i	0.981006	−	0.193977i	$-0.0621387\pi$
$860$	6.18521e11	−	4.22577e11i	1.13073	−	0.772524i
$861$		0			0
$862$	−4.30540e10	−	4.30540e10i	−0.0779802	−	0.0779802i
$863$	4.98279e11	−	4.98279e11i	0.898316	−	0.898316i	−0.0969711	−	0.995287i	$-0.530915\pi$
$863$	4.98279e11	−	4.98279e11i	0.898316	−	0.898316i	0.995287	+	0.0969711i	$0.0309154\pi$
$864$		0			0
$865$	6.81982e10	+	9.98209e10i	0.121817	+	0.178302i
$866$	−1.04951e12			−1.86601
$867$		0			0
$868$	1.62637e11	−	1.62637e11i	0.286511	−	0.286511i
$869$		−	1.54430e11i		−	0.270803i
$870$		0			0
$871$	−1.12370e12			−1.95243
$872$	1.87009e11	+	1.87009e11i	0.323442	+	0.323442i
$873$		0			0
$874$			4.01502e10i			0.0688085i
$875$	−2.03826e11	+	4.71445e10i	−0.347719	+	0.0804264i
$876$		0			0
$877$	3.72052e11	+	3.72052e11i	0.628934	+	0.628934i	0.947800	−	0.318865i	$-0.103302\pi$
$877$	3.72052e11	+	3.72052e11i	0.628934	+	0.628934i	−0.318865	+	0.947800i	$0.603302\pi$
$878$	−5.89083e10	+	5.89083e10i	−0.0991284	+	0.0991284i
$879$		0			0
$880$	3.76054e10	−	1.99807e11i	0.0627075	−	0.333180i
$881$	−5.29284e11			−0.878588			−0.439294	−	0.898343i	$-0.644771\pi$
$881$	−5.29284e11			−0.878588			−0.439294	+	0.898343i	$0.644771\pi$
$882$		0			0
$883$	4.05100e10	−	4.05100e10i	0.0666376	−	0.0666376i	−0.673003	−	0.739640i	$-0.734996\pi$
$883$	4.05100e10	−	4.05100e10i	0.0666376	−	0.0666376i	0.739640	+	0.673003i	$0.234996\pi$
$884$			5.95142e11i			0.974566i
$885$		0			0
$886$	−4.41350e11			−0.716223
$887$	6.41779e11	+	6.41779e11i	1.03679	+	1.03679i	0.999297	+	0.0374934i	$0.0119373\pi$
$887$	6.41779e11	+	6.41779e11i	1.03679	+	1.03679i	0.0374934	+	0.999297i	$0.488063\pi$
$888$		0			0
$889$			4.34577e11i			0.695760i
$890$	7.01346e11	+	1.02655e12i	1.11782	+	1.63614i
$891$		0			0
$892$	−8.15409e10	−	8.15409e10i	−0.128800	−	0.128800i
$893$	4.41251e10	−	4.41251e10i	0.0693872	−	0.0693872i
$894$		0			0
$895$	−7.50135e11	−	1.41182e11i	−1.16909	−	0.220033i
$896$	1.19542e11			0.185476
$897$		0			0
$898$	−6.72507e11	+	6.72507e11i	−1.03417	+	1.03417i
$899$		−	8.37274e11i		−	1.28183i
$900$		0			0
$901$	−9.07934e11			−1.37770
$902$	2.08464e11	+	2.08464e11i	0.314923	+	0.314923i
$903$		0			0
$904$			1.25753e11i			0.188297i
$905$	−3.28586e10	+	1.74586e11i	−0.0489841	+	0.260264i
$906$		0			0
$907$	6.63662e11	+	6.63662e11i	0.980658	+	0.980658i	0.999816	−	0.0191584i	$-0.00609868\pi$
$907$	6.63662e11	+	6.63662e11i	0.980658	+	0.980658i	−0.0191584	+	0.999816i	$0.506099\pi$
$908$	3.00256e11	−	3.00256e11i	0.441721	−	0.441721i
$909$		0			0
$910$	3.89303e11	−	2.65974e11i	0.567704	−	0.387859i
$911$	6.46227e11			0.938235			0.469118	−	0.883136i	$-0.344572\pi$
$911$	6.46227e11			0.938235			0.469118	+	0.883136i	$0.344572\pi$
$912$		0			0
$913$	−2.09386e11	+	2.09386e11i	−0.301345	+	0.301345i
$914$			5.97138e11i			0.855637i
$915$		0			0
$916$	−5.80008e11			−0.823858
$917$	−3.51261e11	−	3.51261e11i	−0.496767	−	0.496767i
$918$		0			0
$919$		−	1.63185e11i		−	0.228780i	−0.993436	−	0.114390i	$-0.963509\pi$
$919$		−	1.63185e11i		−	0.228780i	0.993436	−	0.114390i	$-0.0364914\pi$
$920$	−9.87000e10	−	1.85762e10i	−0.137773	−	0.0259302i
$921$		0			0
$922$	1.03234e12	+	1.03234e12i	1.42856	+	1.42856i
$923$	8.11602e11	−	8.11602e11i	1.11824	−	1.11824i
$924$		0			0
$925$	−6.92207e10	−	1.57822e11i	−0.0945516	−	0.215576i
$926$	2.42484e11			0.329791
$927$		0			0
$928$	−6.13194e11	+	6.13194e11i	−0.826811	+	0.826811i
$929$		−	7.83337e11i		−	1.05169i	−0.850582	−	0.525843i	$-0.823750\pi$
$929$		−	7.83337e11i		−	1.05169i	0.850582	−	0.525843i	$-0.176250\pi$
$930$		0			0
$931$	6.35106e10			0.0845372
$932$	1.40278e11	+	1.40278e11i	0.185920	+	0.185920i
$933$		0			0
$934$		−	1.75742e12i		−	2.30934i
$935$	1.56290e11	−	1.06779e11i	0.204497	−	0.139713i
$936$		0			0
$937$	4.59654e8	+	4.59654e8i	0.000596311	+	0.000596311i	0.707405	−	0.706809i	$-0.249866\pi$
$937$	4.59654e8	+	4.59654e8i	0.000596311	+	0.000596311i	−0.706809	+	0.707405i	$0.749866\pi$
$938$	3.56914e11	−	3.56914e11i	0.461055	−	0.461055i
$939$		0			0
$940$	−3.58068e11	−	5.24100e11i	−0.458621	−	0.671278i
$941$	5.93683e11			0.757175			0.378587	−	0.925566i	$-0.376410\pi$
$941$	5.93683e11			0.757175			0.378587	+	0.925566i	$0.376410\pi$
$942$		0			0
$943$	3.35271e11	−	3.35271e11i	0.423984	−	0.423984i
$944$			8.38327e11i			1.05566i
$945$		0			0
$946$	5.36933e11			0.670433
$947$	7.81898e11	+	7.81898e11i	0.972187	+	0.972187i	0.999624	−	0.0274361i	$-0.00873427\pi$
$947$	7.81898e11	+	7.81898e11i	0.972187	+	0.972187i	−0.0274361	+	0.999624i	$0.508734\pi$
$948$		0			0
$949$			1.23707e12i			1.52522i
$950$	9.70202e10	−	4.25530e10i	0.119115	−	0.0522439i
$951$		0			0
$952$	4.64867e10	+	4.64867e10i	0.0565954	+	0.0565954i
$953$	−9.89132e11	+	9.89132e11i	−1.19917	+	1.19917i	−0.224761	+	0.974414i	$0.572160\pi$
$953$	−9.89132e11	+	9.89132e11i	−1.19917	+	1.19917i	−0.974414	+	0.224761i	$0.927840\pi$
$954$		0			0
$955$	8.25184e10	−	4.38441e11i	0.0992059	−	0.527105i
$956$	6.84636e11			0.819649
$957$		0			0
$958$	1.37208e12	−	1.37208e12i	1.62899	−	1.62899i
$959$			1.64651e11i			0.194666i
$960$		0			0
$961$	8.53571e11			1.00080
$962$	2.74631e11	+	2.74631e11i	0.320663	+	0.320663i
$963$		0			0
$964$		−	4.97286e11i		−	0.575835i
$965$	−2.77469e11	−	4.06127e11i	−0.319966	−	0.468331i
$966$		0			0
$967$	2.58888e11	+	2.58888e11i	0.296078	+	0.296078i	0.839475	−	0.543398i	$-0.182862\pi$
$967$	2.58888e11	+	2.58888e11i	0.296078	+	0.296078i	−0.543398	+	0.839475i	$0.682862\pi$
$968$	−1.50436e11	+	1.50436e11i	−0.171337	+	0.171337i
$969$		0			0
$970$	1.06125e12	+	1.99736e11i	1.19875	+	0.225616i
$971$	1.02230e12			1.15001			0.575005	−	0.818150i	$-0.305000\pi$
$971$	1.02230e12			1.15001			0.575005	+	0.818150i	$0.305000\pi$
$972$		0			0
$973$	2.26460e11	−	2.26460e11i	0.252662	−	0.252662i
$974$			8.91761e11i			0.990861i
$975$		0			0
$976$	−8.60024e11			−0.947788
$977$	−4.00291e11	−	4.00291e11i	−0.439337	−	0.439337i	0.452452	−	0.891789i	$-0.350549\pi$
$977$	−4.00291e11	−	4.00291e11i	−0.439337	−	0.439337i	−0.891789	+	0.452452i	$0.850549\pi$
$978$		0			0
$979$			3.96782e11i			0.431938i
$980$	1.19487e11	−	6.34866e11i	0.129544	−	0.688300i
$981$		0			0
$982$	−1.53547e12	−	1.53547e12i	−1.65119	−	1.65119i
$983$	−8.42757e11	+	8.42757e11i	−0.902586	+	0.902586i	−0.995659	−	0.0930734i	$-0.970331\pi$
$983$	−8.42757e11	+	8.42757e11i	−0.902586	+	0.902586i	0.0930734	+	0.995659i	$0.470331\pi$
$984$		0			0
$985$	2.16664e10	−	1.48026e10i	0.0230166	−	0.0157251i
$986$	−9.73156e11			−1.02961
$987$		0			0
$988$	−7.51709e10	+	7.51709e10i	−0.0788900	+	0.0788900i
$989$		−	8.63547e11i		−	0.902611i
$990$		0			0
$991$	1.85378e12			1.92205			0.961023	−	0.276468i	$-0.0891641\pi$
$991$	1.85378e12			1.92205			0.961023	+	0.276468i	$0.0891641\pi$
$992$	−1.24976e12	−	1.24976e12i	−1.29057	−	1.29057i
$993$		0			0
$994$			5.15571e11i			0.528132i
$995$	−3.80954e11	−	7.16988e10i	−0.388669	−	0.0731509i
$996$		0			0
$997$	9.39430e11	+	9.39430e11i	0.950788	+	0.950788i	0.998845	−	0.0480567i	$-0.0153028\pi$
$997$	9.39430e11	+	9.39430e11i	0.950788	+	0.950788i	−0.0480567	+	0.998845i	$0.515303\pi$
$998$	−9.37219e10	+	9.37219e10i	−0.0944754	+	0.0944754i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.9.g.b.37.7	yes	16
3.2	odd	2	inner	45.9.g.b.37.2	yes	16
5.3	odd	4	inner	45.9.g.b.28.7	yes	16
15.8	even	4	inner	45.9.g.b.28.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.g.b.28.2	✓	16	15.8	even	4	inner
45.9.g.b.28.7	yes	16	5.3	odd	4	inner
45.9.g.b.37.2	yes	16	3.2	odd	2	inner
45.9.g.b.37.7	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(15.1900 + 15.1900i) q^{2} +205.471i q^{4} +(115.601 - 614.216i) q^{5} +(605.930 + 605.930i) q^{7} +(767.539 - 767.539i) q^{8} +O(q^{10})\)	\(q+(15.1900 + 15.1900i) q^{2} +205.471i q^{4} +(115.601 - 614.216i) q^{5} +(605.930 + 605.930i) q^{7} +(767.539 - 767.539i) q^{8} +(11085.9 - 7573.95i) q^{10} +4284.92 q^{11} +(28977.7 - 28977.7i) q^{13} +18408.1i q^{14} +75918.3 q^{16} +(49977.7 + 49977.7i) q^{17} +12625.1i q^{19} +(126203. + 23752.6i) q^{20} +(65087.8 + 65087.8i) q^{22} +(104680. - 104680. i) q^{23} +(-363898. - 142008. i) q^{25} +880342. q^{26} +(-124501. + 124501. i) q^{28} +640943. i q^{29} -1.30632e6 q^{31} +(956707. + 956707. i) q^{32} +1.51832e6i q^{34} +(442218. - 302126. i) q^{35} +(311959. + 311959. i) q^{37} +(-191775. + 191775. i) q^{38} +(-382707. - 560163. i) q^{40} +3.20281e6 q^{41} +(4.12468e6 - 4.12468e6i) q^{43} +880425. i q^{44} +3.18019e6 q^{46} +(-3.49502e6 - 3.49502e6i) q^{47} -5.03050e6i q^{49} +(-3.37050e6 - 7.68470e6i) q^{50} +(5.95407e6 + 5.95407e6i) q^{52} +(-9.08340e6 + 9.08340e6i) q^{53} +(495340. - 2.63186e6i) q^{55} +930149. q^{56} +(-9.73590e6 + 9.73590e6i) q^{58} +1.10425e7i q^{59} -1.13283e7 q^{61} +(-1.98429e7 - 1.98429e7i) q^{62} +9.62963e6i q^{64} +(-1.44487e7 - 2.11484e7i) q^{65} +(-1.93890e7 - 1.93890e7i) q^{67} +(-1.02689e7 + 1.02689e7i) q^{68} +(1.13066e7 + 2.12800e6i) q^{70} +2.80078e7 q^{71} +(-2.13453e7 + 2.13453e7i) q^{73} +9.47731e6i q^{74} -2.59409e6 q^{76} +(2.59636e6 + 2.59636e6i) q^{77} -3.60404e7i q^{79} +(8.77622e6 - 4.66302e7i) q^{80} +(4.86506e7 + 4.86506e7i) q^{82} +(-4.88658e7 + 4.88658e7i) q^{83} +(3.64746e7 - 2.49196e7i) q^{85} +1.25308e8 q^{86} +(3.28884e6 - 3.28884e6i) q^{88} +9.25997e7i q^{89} +3.51169e7 q^{91} +(2.15088e7 + 2.15088e7i) q^{92} -1.06179e8i q^{94} +(7.75455e6 + 1.45947e6i) q^{95} +(5.68733e7 + 5.68733e7i) q^{97} +(7.64132e7 - 7.64132e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4220 q^{7}+O(q^{10}) \) 16 * q - 4220 * q^7	\( 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) \) 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

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