LMFDB - Embedded newform 45.9.g.b.37.6

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			37.6
Root			$11.5567i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.9.g.b.28.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.7931 + 10.7931i) q^{2} -23.0166i q^{4} +(-489.683 + 388.375i) q^{5} +(2354.79 + 2354.79i) q^{7} +(3011.46 - 3011.46i) q^{8} +O(q^{10})$	\(q+(10.7931 + 10.7931i) q^{2} -23.0166i q^{4} +(-489.683 + 388.375i) q^{5} +(2354.79 + 2354.79i) q^{7} +(3011.46 - 3011.46i) q^{8} +(-9477.00 - 1093.43i) q^{10} -11916.9 q^{11} +(-37189.2 + 37189.2i) q^{13} +50831.1i q^{14} +59114.0 q^{16} +(-77040.9 - 77040.9i) q^{17} +172555. i q^{19} +(8939.09 + 11270.9i) q^{20} +(-128621. - 128621. i) q^{22} +(-303832. + 303832. i) q^{23} +(88954.1 - 380362. i) q^{25} -802776. q^{26} +(54199.3 - 54199.3i) q^{28} +361997. i q^{29} +432913. q^{31} +(-132910. - 132910. i) q^{32} -1.66303e6i q^{34} +(-2.06764e6 - 238558. i) q^{35} +(698415. + 698415. i) q^{37} +(-1.86241e6 + 1.86241e6i) q^{38} +(-305084. + 2.64424e6i) q^{40} +2.69766e6 q^{41} +(337059. - 337059. i) q^{43} +274287. i q^{44} -6.55859e6 q^{46} +(-1.97588e6 - 1.97588e6i) q^{47} +5.32525e6i q^{49} +(5.06539e6 - 3.14520e6i) q^{50} +(855970. + 855970. i) q^{52} +(3.56064e6 - 3.56064e6i) q^{53} +(5.83552e6 - 4.62824e6i) q^{55} +1.41827e7 q^{56} +(-3.90708e6 + 3.90708e6i) q^{58} +9.30238e6i q^{59} +1.35077e7 q^{61} +(4.67249e6 + 4.67249e6i) q^{62} -1.80022e7i q^{64} +(3.76755e6 - 3.26543e7i) q^{65} +(269805. + 269805. i) q^{67} +(-1.77322e6 + 1.77322e6i) q^{68} +(-1.97415e7 - 2.48911e7i) q^{70} +4.06937e7 q^{71} +(-718506. + 718506. i) q^{73} +1.50762e7i q^{74} +3.97165e6 q^{76} +(-2.80618e7 - 2.80618e7i) q^{77} +2.64669e7i q^{79} +(-2.89471e7 + 2.29584e7i) q^{80} +(2.91161e7 + 2.91161e7i) q^{82} +(-5.69550e7 + 5.69550e7i) q^{83} +(6.76464e7 + 7.80484e6i) q^{85} +7.27584e6 q^{86} +(-3.58874e7 + 3.58874e7i) q^{88} -1.47189e7i q^{89} -1.75145e8 q^{91} +(6.99318e6 + 6.99318e6i) q^{92} -4.26519e7i q^{94} +(-6.70163e7 - 8.44975e7i) q^{95} +(5.88484e7 + 5.88484e7i) q^{97} +(-5.74761e7 + 5.74761e7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	10.7931	+	10.7931i	0.674571	+	0.674571i	0.958766	−	0.284196i	$-0.0917265\pi$
$2$	10.7931	+	10.7931i	0.674571	+	0.674571i	−0.284196	+	0.958766i	$0.591727\pi$
$3$		0			0
$3$		0			0
$4$		−	23.0166i		−	0.0899087i
$5$	−489.683	+	388.375i	−0.783493	+	0.621401i
$5$	−489.683	+	388.375i	−0.783493	+	0.621401i
$6$		0			0
$7$	2354.79	+	2354.79i	0.980753	+	0.980753i	0.999818	−	0.0190653i	$-0.00606904\pi$
$7$	2354.79	+	2354.79i	0.980753	+	0.980753i	−0.0190653	+	0.999818i	$0.506069\pi$
$8$	3011.46	−	3011.46i	0.735220	−	0.735220i
$9$		0			0
$10$	−9477.00	−	1093.43i	−0.947700	−	0.109343i
$11$	−11916.9			−0.813942			−0.406971	−	0.913441i	$-0.633415\pi$
$11$	−11916.9			−0.813942			−0.406971	+	0.913441i	$0.633415\pi$
$12$		0			0
$13$	−37189.2	+	37189.2i	−1.30210	+	1.30210i	−0.375122	+	0.926975i	$0.622399\pi$
$13$	−37189.2	+	37189.2i	−1.30210	+	1.30210i	−0.926975	+	0.375122i	$0.877601\pi$
$14$			50831.1i			1.32317i
$15$		0			0
$16$	59114.0			0.902008
$17$	−77040.9	−	77040.9i	−0.922414	−	0.922414i	0.0747857	−	0.997200i	$-0.476173\pi$
$17$	−77040.9	−	77040.9i	−0.922414	−	0.922414i	−0.997200	+	0.0747857i	$0.976173\pi$
$18$		0			0
$19$			172555.i			1.32408i	0.749468	+	0.662040i	$0.230309\pi$
$19$			172555.i			1.32408i	−0.749468	+	0.662040i	$0.769691\pi$
$20$	8939.09	+	11270.9i	0.0558693	+	0.0704428i
$21$		0			0
$22$	−128621.	−	128621.i	−0.549061	−	0.549061i
$23$	−303832.	+	303832.i	−1.08573	+	1.08573i	−0.0897669	+	0.995963i	$0.528612\pi$
$23$	−303832.	+	303832.i	−1.08573	+	1.08573i	−0.995963	+	0.0897669i	$0.971388\pi$
$24$		0			0
$25$	88954.1	−	380362.i	0.227722	−	0.973726i
$26$	−802776.			−1.75671
$27$		0			0
$28$	54199.3	−	54199.3i	0.0881782	−	0.0881782i
$29$			361997.i			0.511815i	0.966701	+	0.255907i	$0.0823742\pi$
$29$			361997.i			0.511815i	−0.966701	+	0.255907i	$0.917626\pi$
$30$		0			0
$31$	432913.			0.468763			0.234382	−	0.972145i	$-0.424693\pi$
$31$	432913.			0.468763			0.234382	+	0.972145i	$0.424693\pi$
$32$	−132910.	−	132910.i	−0.126752	−	0.126752i
$33$		0			0
$34$		−	1.66303e6i		−	1.24447i
$35$	−2.06764e6	−	238558.i	−1.37785	−	0.158973i
$36$		0			0
$37$	698415.	+	698415.i	0.372655	+	0.372655i	0.868443	−	0.495788i	$-0.165121\pi$
$37$	698415.	+	698415.i	0.372655	+	0.372655i	−0.495788	+	0.868443i	$0.665121\pi$
$38$	−1.86241e6	+	1.86241e6i	−0.893186	+	0.893186i
$39$		0			0
$40$	−305084.	+	2.64424e6i	−0.119174	+	1.03291i
$41$	2.69766e6			0.954665			0.477333	−	0.878723i	$-0.341604\pi$
$41$	2.69766e6			0.954665			0.477333	+	0.878723i	$0.341604\pi$
$42$		0			0
$43$	337059.	−	337059.i	0.0985898	−	0.0985898i	−0.656092	−	0.754681i	$-0.727791\pi$
$43$	337059.	−	337059.i	0.0985898	−	0.0985898i	0.754681	+	0.656092i	$0.227791\pi$
$44$			274287.i			0.0731805i
$45$		0			0
$46$	−6.55859e6			−1.46480
$47$	−1.97588e6	−	1.97588e6i	−0.404920	−	0.404920i	0.475042	−	0.879963i	$-0.342433\pi$
$47$	−1.97588e6	−	1.97588e6i	−0.404920	−	0.404920i	−0.879963	+	0.475042i	$0.842433\pi$
$48$		0			0
$49$			5.32525e6i			0.923753i
$50$	5.06539e6	−	3.14520e6i	0.810462	−	0.503232i
$51$		0			0
$52$	855970.	+	855970.i	0.117070	+	0.117070i
$53$	3.56064e6	−	3.56064e6i	0.451257	−	0.451257i	−0.444514	−	0.895772i	$-0.646624\pi$
$53$	3.56064e6	−	3.56064e6i	0.451257	−	0.451257i	0.895772	+	0.444514i	$0.146624\pi$
$54$		0			0
$55$	5.83552e6	−	4.62824e6i	0.637718	−	0.505784i
$56$	1.41827e7			1.44214
$57$		0			0
$58$	−3.90708e6	+	3.90708e6i	−0.345255	+	0.345255i
$59$			9.30238e6i			0.767690i	0.923398	+	0.383845i	$0.125400\pi$
$59$			9.30238e6i			0.767690i	−0.923398	+	0.383845i	$0.874600\pi$
$60$		0			0
$61$	1.35077e7			0.975581			0.487791	−	0.872961i	$-0.337803\pi$
$61$	1.35077e7			0.975581			0.487791	+	0.872961i	$0.337803\pi$
$62$	4.67249e6	+	4.67249e6i	0.316214	+	0.316214i
$63$		0			0
$64$		−	1.80022e7i		−	1.07301i
$65$	3.76755e6	−	3.26543e7i	0.211060	−	1.82931i
$66$		0			0
$67$	269805.	+	269805.i	0.0133891	+	0.0133891i	0.713770	−	0.700381i	$-0.246986\pi$
$67$	269805.	+	269805.i	0.0133891	+	0.0133891i	−0.700381	+	0.713770i	$0.746986\pi$
$68$	−1.77322e6	+	1.77322e6i	−0.0829330	+	0.0829330i
$69$		0			0
$70$	−1.97415e7	−	2.48911e7i	−0.822221	−	1.03670i
$71$	4.06937e7			1.60138			0.800688	−	0.599081i	$-0.204467\pi$
$71$	4.06937e7			1.60138			0.800688	+	0.599081i	$0.204467\pi$
$72$		0			0
$73$	−718506.	+	718506.i	−0.0253011	+	0.0253011i	−0.719644	−	0.694343i	$-0.755695\pi$
$73$	−718506.	+	718506.i	−0.0253011	+	0.0253011i	0.694343	+	0.719644i	$0.255695\pi$
$74$			1.50762e7i			0.502764i
$75$		0			0
$76$	3.97165e6			0.119046
$77$	−2.80618e7	−	2.80618e7i	−0.798276	−	0.798276i
$78$		0			0
$79$			2.64669e7i			0.679507i	0.940515	+	0.339754i	$0.110344\pi$
$79$			2.64669e7i			0.679507i	−0.940515	+	0.339754i	$0.889656\pi$
$80$	−2.89471e7	+	2.29584e7i	−0.706717	+	0.560508i
$81$		0			0
$82$	2.91161e7	+	2.91161e7i	0.643989	+	0.643989i
$83$	−5.69550e7	+	5.69550e7i	−1.20011	+	1.20011i	−0.225973	+	0.974134i	$0.572556\pi$
$83$	−5.69550e7	+	5.69550e7i	−1.20011	+	1.20011i	−0.974134	+	0.225973i	$0.927444\pi$
$84$		0			0
$85$	6.76464e7	+	7.80484e6i	1.29589	+	0.149516i
$86$	7.27584e6			0.133012
$87$		0			0
$88$	−3.58874e7	+	3.58874e7i	−0.598427	+	0.598427i
$89$		−	1.47189e7i		−	0.234594i	−0.993097	−	0.117297i	$-0.962577\pi$
$89$		−	1.47189e7i		−	0.234594i	0.993097	−	0.117297i	$-0.0374229\pi$
$90$		0			0
$91$	−1.75145e8			−2.55407
$92$	6.99318e6	+	6.99318e6i	0.0976165	+	0.0976165i
$93$		0			0
$94$		−	4.26519e7i		−	0.546295i
$95$	−6.70163e7	−	8.44975e7i	−0.822784	−	1.03741i
$96$		0			0
$97$	5.88484e7	+	5.88484e7i	0.664733	+	0.664733i	0.956492	−	0.291759i	$-0.0942404\pi$
$97$	5.88484e7	+	5.88484e7i	0.664733	+	0.664733i	−0.291759	+	0.956492i	$0.594240\pi$
$98$	−5.74761e7	+	5.74761e7i	−0.623136	+	0.623136i
$99$		0			0
$100$	−8.75464e6	−	2.04742e6i	−0.0875464	−	0.0204742i
$101$	−9.72062e7			−0.934133			−0.467066	−	0.884222i	$-0.654689\pi$
$101$	−9.72062e7			−0.934133			−0.467066	+	0.884222i	$0.654689\pi$
$102$		0			0
$103$	1.40220e7	−	1.40220e7i	0.124584	−	0.124584i	−0.642066	−	0.766650i	$-0.721922\pi$
$103$	1.40220e7	−	1.40220e7i	0.124584	−	0.124584i	0.766650	+	0.642066i	$0.221922\pi$
$104$			2.23988e8i			1.91466i
$105$		0			0
$106$	7.68609e7			0.608810
$107$	−7.68807e7	−	7.68807e7i	−0.586519	−	0.586519i	0.350168	−	0.936687i	$-0.386124\pi$
$107$	−7.68807e7	−	7.68807e7i	−0.586519	−	0.586519i	−0.936687	+	0.350168i	$0.886124\pi$
$108$		0			0
$109$		−	1.90017e8i		−	1.34613i	−0.739585	−	0.673063i	$-0.764978\pi$
$109$		−	1.90017e8i		−	1.34613i	0.739585	−	0.673063i	$-0.235022\pi$
$110$	1.12937e8	+	1.30303e7i	0.771373	+	0.0889987i
$111$		0			0
$112$	1.39201e8	+	1.39201e8i	0.884647	+	0.884647i
$113$	8.91224e7	−	8.91224e7i	0.546605	−	0.546605i	−0.378852	−	0.925457i	$-0.623681\pi$
$113$	8.91224e7	−	8.91224e7i	0.546605	−	0.546605i	0.925457	+	0.378852i	$0.123681\pi$
$114$		0			0
$115$	3.07805e7	−	2.66782e8i	0.175988	−	1.52533i
$116$	8.33194e6			0.0460166
$117$		0			0
$118$	−1.00402e8	+	1.00402e8i	−0.517861	+	0.517861i
$119$		−	3.62830e8i		−	1.80932i
$120$		0			0
$121$	−7.23458e7			−0.337498
$122$	1.45791e8	+	1.45791e8i	0.658098	+	0.658098i
$123$		0			0
$124$		−	9.96419e6i		−	0.0421459i
$125$	1.04164e8	+	2.20804e8i	0.426655	+	0.904414i
$126$		0			0
$127$	1.24116e7	+	1.24116e7i	0.0477105	+	0.0477105i	0.730560	−	0.682849i	$-0.239259\pi$
$127$	1.24116e7	+	1.24116e7i	0.0477105	+	0.0477105i	−0.682849	+	0.730560i	$0.739259\pi$
$128$	1.60275e8	−	1.60275e8i	0.597072	−	0.597072i
$129$		0			0
$130$	3.93106e8	−	3.11778e8i	1.37637	−	1.09162i
$131$	1.88066e8			0.638594			0.319297	−	0.947655i	$-0.396553\pi$
$131$	1.88066e8			0.638594			0.319297	+	0.947655i	$0.396553\pi$
$132$		0			0
$133$	−4.06332e8	+	4.06332e8i	−1.29860	+	1.29860i
$134$			5.82407e6i			0.0180637i
$135$		0			0
$136$	−4.64012e8			−1.35636
$137$	2.41115e7	+	2.41115e7i	0.0684450	+	0.0684450i	0.740501	−	0.672056i	$-0.234588\pi$
$137$	2.41115e7	+	2.41115e7i	0.0684450	+	0.0684450i	−0.672056	+	0.740501i	$0.734588\pi$
$138$		0			0
$139$			2.96522e8i			0.794323i	0.917749	+	0.397162i	$0.130005\pi$
$139$			2.96522e8i			0.794323i	−0.917749	+	0.397162i	$0.869995\pi$
$140$	−5.49080e6	+	4.75901e7i	−0.0142930	+	0.123881i
$141$		0			0
$142$	4.39212e8	+	4.39212e8i	1.08024	+	1.08024i
$143$	4.43181e8	−	4.43181e8i	1.05983	−	1.05983i
$144$		0			0
$145$	−1.40591e8	−	1.77264e8i	−0.318042	−	0.401003i
$146$	−1.55099e7			−0.0341347
$147$		0			0
$148$	1.60752e7	−	1.60752e7i	0.0335049	−	0.0335049i
$149$		−	5.45181e8i		−	1.10610i	−0.833147	−	0.553051i	$-0.813463\pi$
$149$		−	5.45181e8i		−	1.10610i	0.833147	−	0.553051i	$-0.186537\pi$
$150$		0			0
$151$	−1.40613e8			−0.270469			−0.135234	−	0.990814i	$-0.543179\pi$
$151$	−1.40613e8			−0.270469			−0.135234	+	0.990814i	$0.543179\pi$
$152$	5.19644e8	+	5.19644e8i	0.973491	+	0.973491i
$153$		0			0
$154$		−	6.05750e8i		−	1.07699i
$155$	−2.11990e8	+	1.68133e8i	−0.367273	+	0.291290i
$156$		0			0
$157$	7.57340e8	+	7.57340e8i	1.24650	+	1.24650i	0.957255	+	0.289244i	$0.0934038\pi$
$157$	7.57340e8	+	7.57340e8i	1.24650	+	1.24650i	0.289244	+	0.957255i	$0.406596\pi$
$158$	−2.85660e8	+	2.85660e8i	−0.458376	+	0.458376i
$159$		0			0
$160$	1.16702e8	+	1.34648e7i	0.178074	+	0.0205456i
$161$	−1.43092e9			−2.12967
$162$		0			0
$163$	−1.69602e8	+	1.69602e8i	−0.240259	+	0.240259i	−0.816957	−	0.576698i	$-0.804341\pi$
$163$	−1.69602e8	+	1.69602e8i	−0.240259	+	0.240259i	0.576698	+	0.816957i	$0.304341\pi$
$164$		−	6.20909e7i		−	0.0858327i
$165$		0			0
$166$	−1.22945e9			−1.61911
$167$	−9.80537e7	−	9.80537e7i	−0.126066	−	0.126066i	0.641259	−	0.767325i	$-0.278413\pi$
$167$	−9.80537e7	−	9.80537e7i	−0.126066	−	0.126066i	−0.767325	+	0.641259i	$0.778413\pi$
$168$		0			0
$169$		−	1.95034e9i		−	2.39092i
$170$	6.45878e8	+	8.14356e8i	0.773312	+	0.975031i
$171$		0			0
$172$	−7.75796e6	−	7.75796e6i	−0.00886408	−	0.00886408i
$173$	−2.38374e8	+	2.38374e8i	−0.266118	+	0.266118i	−0.827534	−	0.561416i	$-0.810257\pi$
$173$	−2.38374e8	+	2.38374e8i	−0.266118	+	0.266118i	0.561416	+	0.827534i	$0.310257\pi$
$174$		0			0
$175$	1.10514e9	−	6.86203e8i	1.17832	−	0.731645i
$176$	−7.04457e8			−0.734182
$177$		0			0
$178$	1.58864e8	−	1.58864e8i	0.158250	−	0.158250i
$179$			7.48826e8i			0.729405i	0.931124	+	0.364702i	$0.118829\pi$
$179$			7.48826e8i			0.729405i	−0.931124	+	0.364702i	$0.881171\pi$
$180$		0			0
$181$	−4.91277e7			−0.0457733			−0.0228866	−	0.999738i	$-0.507286\pi$
$181$	−4.91277e7			−0.0457733			−0.0228866	+	0.999738i	$0.507286\pi$
$182$	−1.89037e9	−	1.89037e9i	−1.72290	−	1.72290i
$183$		0			0
$184$			1.82996e9i			1.59650i
$185$	−6.13250e8	−	7.07549e7i	−0.523541	−	0.0604045i
$186$		0			0
$187$	9.18091e8	+	9.18091e8i	0.750791	+	0.750791i
$188$	−4.54782e7	+	4.54782e7i	−0.0364059	+	0.0364059i
$189$		0			0
$190$	1.88677e8	−	1.63531e9i	0.144779	−	1.25483i
$191$	5.04885e8			0.379367			0.189683	−	0.981845i	$-0.439254\pi$
$191$	5.04885e8			0.379367			0.189683	+	0.981845i	$0.439254\pi$
$192$		0			0
$193$	−3.87670e8	+	3.87670e8i	−0.279404	+	0.279404i	−0.832871	−	0.553467i	$-0.813305\pi$
$193$	−3.87670e8	+	3.87670e8i	−0.279404	+	0.279404i	0.553467	+	0.832871i	$0.313305\pi$
$194$			1.27032e9i			0.896819i
$195$		0			0
$196$	1.22569e8			0.0830534
$197$	1.89522e9	+	1.89522e9i	1.25833	+	1.25833i	0.951888	+	0.306445i	$0.0991397\pi$
$197$	1.89522e9	+	1.89522e9i	1.25833	+	1.25833i	0.306445	+	0.951888i	$0.400860\pi$
$198$		0			0
$199$			1.04114e9i			0.663891i	0.943299	+	0.331946i	$0.107705\pi$
$199$			1.04114e9i			0.663891i	−0.943299	+	0.331946i	$0.892295\pi$
$200$	−8.77563e8	−	1.41333e9i	−0.548477	−	0.883330i
$201$		0			0
$202$	−1.04916e9	−	1.04916e9i	−0.630139	−	0.630139i
$203$	−8.52425e8	+	8.52425e8i	−0.501964	+	0.501964i
$204$		0			0
$205$	−1.32100e9	+	1.04770e9i	−0.747973	+	0.593229i
$206$	3.02683e8			0.168081
$207$		0			0
$208$	−2.19840e9	+	2.19840e9i	−1.17450	+	1.17450i
$209$		−	2.05633e9i		−	1.07772i
$210$		0			0
$211$	3.91930e8			0.197733			0.0988665	−	0.995101i	$-0.468478\pi$
$211$	3.91930e8			0.197733			0.0988665	+	0.995101i	$0.468478\pi$
$212$	−8.19539e7	−	8.19539e7i	−0.0405720	−	0.0405720i
$213$		0			0
$214$		−	1.65957e9i		−	0.791297i
$215$	−3.41467e7	+	2.95958e8i	−0.0159807	+	0.138508i
$216$		0			0
$217$	1.01942e9	+	1.01942e9i	0.459741	+	0.459741i
$218$	2.05087e9	−	2.05087e9i	0.908057	−	0.908057i
$219$		0			0
$220$	−1.06526e8	−	1.34314e8i	−0.0454744	−	0.0573364i
$221$	5.73018e9			2.40215
$222$		0			0
$223$	1.39072e8	−	1.39072e8i	0.0562366	−	0.0562366i	−0.678429	−	0.734666i	$-0.737339\pi$
$223$	1.39072e8	−	1.39072e8i	0.0562366	−	0.0562366i	0.734666	+	0.678429i	$0.237339\pi$
$224$		−	6.25948e8i		−	0.248626i
$225$		0			0
$226$	1.92382e9			0.737447
$227$	−1.95817e7	−	1.95817e7i	−0.00737473	−	0.00737473i	0.703410	−	0.710785i	$-0.251660\pi$
$227$	−1.95817e7	−	1.95817e7i	−0.00737473	−	0.00737473i	−0.710785	+	0.703410i	$0.751660\pi$
$228$		0			0
$229$		−	5.91417e8i		−	0.215056i	−0.994202	−	0.107528i	$-0.965706\pi$
$229$		−	5.91417e8i		−	0.215056i	0.994202	−	0.107528i	$-0.0342936\pi$
$230$	3.21163e9	−	2.54720e9i	1.14766	−	0.910229i
$231$		0			0
$232$	1.09014e9	+	1.09014e9i	0.376297	+	0.376297i
$233$	−3.09790e9	+	3.09790e9i	−1.05110	+	1.05110i	−0.0524788	+	0.998622i	$0.516712\pi$
$233$	−3.09790e9	+	3.09790e9i	−1.05110	+	1.05110i	−0.998622	+	0.0524788i	$0.983288\pi$
$234$		0			0
$235$	1.73494e9	+	2.00172e8i	0.568870	+	0.0656345i
$236$	2.14109e8			0.0690220
$237$		0			0
$238$	3.91607e9	−	3.91607e9i	1.22051	−	1.22051i
$239$			3.11329e9i			0.954175i	0.878856	+	0.477087i	$0.158307\pi$
$239$			3.11329e9i			0.954175i	−0.878856	+	0.477087i	$0.841693\pi$
$240$		0			0
$241$	−2.81919e9			−0.835710			−0.417855	−	0.908514i	$-0.637218\pi$
$241$	−2.81919e9			−0.835710			−0.417855	+	0.908514i	$0.637218\pi$
$242$	−7.80838e8	−	7.80838e8i	−0.227667	−	0.227667i
$243$		0			0
$244$		−	3.10903e8i		−	0.0877132i
$245$	−2.06820e9	−	2.60769e9i	−0.574020	−	0.723754i
$246$		0			0
$247$	−6.41720e9	−	6.41720e9i	−1.72408	−	1.72408i
$248$	1.30370e9	−	1.30370e9i	0.344644	−	0.344644i
$249$		0			0
$250$	−1.25892e9	+	3.50742e9i	−0.322283	+	0.897900i
$251$	6.95898e9			1.75328			0.876638	−	0.481150i	$-0.159781\pi$
$251$	6.95898e9			1.75328			0.876638	+	0.481150i	$0.159781\pi$
$252$		0			0
$253$	3.62074e9	−	3.62074e9i	0.883721	−	0.883721i
$254$			2.67921e8i			0.0643682i
$255$		0			0
$256$	−1.14882e9			−0.267480
$257$	−1.15022e9	−	1.15022e9i	−0.263662	−	0.263662i	0.562878	−	0.826540i	$-0.309694\pi$
$257$	−1.15022e9	−	1.15022e9i	−0.263662	−	0.263662i	−0.826540	+	0.562878i	$0.809694\pi$
$258$		0			0
$259$			3.28924e9i			0.730965i
$260$	−7.51592e8	−	8.67164e7i	−0.164471	−	0.0189761i
$261$		0			0
$262$	2.02982e9	+	2.02982e9i	0.430777	+	0.430777i
$263$	1.11400e9	−	1.11400e9i	0.232842	−	0.232842i	−0.581036	−	0.813878i	$-0.697352\pi$
$263$	1.11400e9	−	1.11400e9i	0.232842	−	0.232842i	0.813878	+	0.581036i	$0.197352\pi$
$264$		0			0
$265$	−3.60720e8	+	3.12645e9i	−0.0731454	+	0.633969i
$266$	−8.77118e9			−1.75199
$267$		0			0
$268$	6.20999e6	−	6.20999e6i	0.00120379	−	0.00120379i
$269$			7.89077e9i			1.50699i	0.657454	+	0.753494i	$0.271633\pi$
$269$			7.89077e9i			1.50699i	−0.657454	+	0.753494i	$0.728367\pi$
$270$		0			0
$271$	7.24107e9			1.34254			0.671268	−	0.741215i	$-0.265750\pi$
$271$	7.24107e9			1.34254			0.671268	+	0.741215i	$0.265750\pi$
$272$	−4.55420e9	−	4.55420e9i	−0.832024	−	0.832024i
$273$		0			0
$274$			5.20477e8i			0.0923420i
$275$	−1.06006e9	+	4.53274e9i	−0.185353	+	0.792557i
$276$		0			0
$277$	−5.00538e9	−	5.00538e9i	−0.850194	−	0.850194i	0.139963	−	0.990157i	$-0.455302\pi$
$277$	−5.00538e9	−	5.00538e9i	−0.850194	−	0.850194i	−0.990157	+	0.139963i	$0.955302\pi$
$278$	−3.20040e9	+	3.20040e9i	−0.535827	+	0.535827i
$279$		0			0
$280$	−6.94504e9	+	5.50822e9i	−1.12991	+	0.896146i
$281$	−9.19706e9			−1.47511			−0.737554	−	0.675288i	$-0.764019\pi$
$281$	−9.19706e9			−1.47511			−0.737554	+	0.675288i	$0.764019\pi$
$282$		0			0
$283$	−7.65446e9	+	7.65446e9i	−1.19335	+	1.19335i	−0.217232	+	0.976120i	$0.569703\pi$
$283$	−7.65446e9	+	7.65446e9i	−1.19335	+	1.19335i	−0.976120	+	0.217232i	$0.930297\pi$
$284$		−	9.36631e8i		−	0.143978i
$285$		0			0
$286$	9.56662e9			1.42986
$287$	6.35241e9	+	6.35241e9i	0.936291	+	0.936291i
$288$		0			0
$289$			4.89485e9i			0.701695i
$290$	3.95817e8	−	3.43064e9i	0.0559632	−	0.485047i
$291$		0			0
$292$	1.65376e7	+	1.65376e7i	0.00227479	+	0.00227479i
$293$	−7.03347e9	+	7.03347e9i	−0.954331	+	0.954331i	−0.999002	−	0.0446703i	$-0.985776\pi$
$293$	−7.03347e9	+	7.03347e9i	−0.954331	+	0.954331i	0.0446703	+	0.999002i	$0.485776\pi$
$294$		0			0
$295$	−3.61282e9	−	4.55522e9i	−0.477043	−	0.601480i
$296$	4.20650e9			0.547967
$297$		0			0
$298$	5.88420e9	−	5.88420e9i	0.746144	−	0.746144i
$299$		−	2.25985e10i		−	2.82745i
$300$		0			0
$301$	1.58740e9			0.193385
$302$	−1.51765e9	−	1.51765e9i	−0.182450	−	0.182450i
$303$		0			0
$304$			1.02004e10i			1.19433i
$305$	−6.61451e9	+	5.24607e9i	−0.764361	+	0.606227i
$306$		0			0
$307$	−7.27595e9	−	7.27595e9i	−0.819098	−	0.819098i	0.166879	−	0.985977i	$-0.446631\pi$
$307$	−7.27595e9	−	7.27595e9i	−0.819098	−	0.819098i	−0.985977	+	0.166879i	$0.946631\pi$
$308$	−6.45889e8	+	6.45889e8i	−0.0717720	+	0.0717720i
$309$		0			0
$310$	−4.10272e9	−	4.73359e8i	−0.444247	−	0.0512559i
$311$	−2.14163e9			−0.228930			−0.114465	−	0.993427i	$-0.536515\pi$
$311$	−2.14163e9			−0.228930			−0.114465	+	0.993427i	$0.536515\pi$
$312$		0			0
$313$	8.45857e9	−	8.45857e9i	0.881292	−	0.881292i	−0.112374	−	0.993666i	$-0.535845\pi$
$313$	8.45857e9	−	8.45857e9i	0.881292	−	0.881292i	0.993666	+	0.112374i	$0.0358455\pi$
$314$			1.63481e10i			1.68170i
$315$		0			0
$316$	6.09178e8			0.0610936
$317$	3.33452e9	+	3.33452e9i	0.330214	+	0.330214i	0.852668	−	0.522454i	$-0.174983\pi$
$317$	3.33452e9	+	3.33452e9i	0.330214	+	0.330214i	−0.522454	+	0.852668i	$0.674983\pi$
$318$		0			0
$319$		−	4.31389e9i		−	0.416587i
$320$	6.99161e9	+	8.81537e9i	0.666772	+	0.840700i
$321$		0			0
$322$	−1.54441e10	−	1.54441e10i	−1.43661	−	1.43661i
$323$	1.32938e10	−	1.32938e10i	1.22135	−	1.22135i
$324$		0			0
$325$	1.08372e10	+	1.74535e10i	0.971369	+	1.56440i
$326$	−3.66106e9			−0.324143
$327$		0			0
$328$	8.12389e9	−	8.12389e9i	0.701889	−	0.701889i
$329$		−	9.30557e9i		−	0.794254i
$330$		0			0
$331$	1.16750e10			0.972626			0.486313	−	0.873785i	$-0.338342\pi$
$331$	1.16750e10			0.972626			0.486313	+	0.873785i	$0.338342\pi$
$332$	1.31091e9	+	1.31091e9i	0.107900	+	0.107900i
$333$		0			0
$334$		−	2.11661e9i		−	0.170081i
$335$	−2.36904e8	−	2.73333e7i	−0.0188102	−	0.00217026i
$336$		0			0
$337$	6.19588e9	+	6.19588e9i	0.480379	+	0.480379i	0.905253	−	0.424874i	$-0.139682\pi$
$337$	6.19588e9	+	6.19588e9i	0.480379	+	0.480379i	−0.424874	+	0.905253i	$0.639682\pi$
$338$	2.10503e10	−	2.10503e10i	1.61284	−	1.61284i
$339$		0			0
$340$	1.79641e8	−	1.55699e9i	0.0134428	−	0.116512i
$341$	−5.15899e9			−0.381546
$342$		0			0
$343$	1.03505e9	−	1.03505e9i	0.0747798	−	0.0747798i
$344$		−	2.03008e9i		−	0.144971i
$345$		0			0
$346$	−5.14560e9			−0.359031
$347$	7.73980e9	+	7.73980e9i	0.533841	+	0.533841i	0.921713	−	0.387872i	$-0.126790\pi$
$347$	7.73980e9	+	7.73980e9i	0.533841	+	0.533841i	−0.387872	+	0.921713i	$0.626790\pi$
$348$		0			0
$349$		−	7.72219e9i		−	0.520522i	−0.965538	−	0.260261i	$-0.916191\pi$
$349$		−	7.72219e9i		−	0.520522i	0.965538	−	0.260261i	$-0.0838086\pi$
$350$	1.93342e10	+	4.52163e9i	1.28841	+	0.301317i
$351$		0			0
$352$	1.58387e9	+	1.58387e9i	0.103169	+	0.103169i
$353$	4.22696e9	−	4.22696e9i	0.272226	−	0.272226i	−0.557770	−	0.829996i	$-0.688343\pi$
$353$	4.22696e9	−	4.22696e9i	0.272226	−	0.272226i	0.829996	+	0.557770i	$0.188343\pi$
$354$		0			0
$355$	−1.99270e10	+	1.58044e10i	−1.25467	+	0.995096i
$356$	−3.38781e8			−0.0210920
$357$		0			0
$358$	−8.08218e9	+	8.08218e9i	−0.492035	+	0.492035i
$359$		−	1.49369e10i		−	0.899252i	−0.893217	−	0.449626i	$-0.851557\pi$
$359$		−	1.49369e10i		−	0.899252i	0.893217	−	0.449626i	$-0.148443\pi$
$360$		0			0
$361$	−1.27918e10			−0.753189
$362$	−5.30242e8	−	5.30242e8i	−0.0308773	−	0.0308773i
$363$		0			0
$364$			4.03126e9i			0.229633i
$365$	7.27902e7	−	6.30890e8i	0.00410111	−	0.0355453i
$366$		0			0
$367$	8.83565e9	+	8.83565e9i	0.487051	+	0.487051i	0.907374	−	0.420323i	$-0.138083\pi$
$367$	8.83565e9	+	8.83565e9i	0.487051	+	0.487051i	−0.420323	+	0.907374i	$0.638083\pi$
$368$	−1.79607e10	+	1.79607e10i	−0.979337	+	0.979337i
$369$		0			0
$370$	−5.85522e9	−	7.38255e9i	−0.312418	−	0.393912i
$371$	1.67691e10			0.885144
$372$		0			0
$373$	−3.68942e9	+	3.68942e9i	−0.190600	+	0.190600i	−0.795955	−	0.605355i	$-0.793031\pi$
$373$	−3.68942e9	+	3.68942e9i	−0.190600	+	0.190600i	0.605355	+	0.795955i	$0.293031\pi$
$374$			1.98182e10i			1.01292i
$375$		0			0
$376$	−1.19006e10			−0.595412
$377$	−1.34624e10	−	1.34624e10i	−0.666433	−	0.666433i
$378$		0			0
$379$		−	1.18084e9i		−	0.0572314i	−0.999590	−	0.0286157i	$-0.990890\pi$
$379$		−	1.18084e9i		−	0.0572314i	0.999590	−	0.0286157i	$-0.00910991\pi$
$380$	−1.94485e9	+	1.54249e9i	−0.0932720	+	0.0739755i
$381$		0			0
$382$	5.44929e9	+	5.44929e9i	0.255910	+	0.255910i
$383$	2.58376e9	−	2.58376e9i	0.120076	−	0.120076i	−0.644515	−	0.764591i	$-0.722941\pi$
$383$	2.58376e9	−	2.58376e9i	0.120076	−	0.120076i	0.764591	+	0.644515i	$0.222941\pi$
$384$		0			0
$385$	2.46399e10	+	2.84288e9i	1.12149	+	0.129394i
$386$	−8.36835e9			−0.376956
$387$		0			0
$388$	1.35449e9	−	1.35449e9i	0.0597653	−	0.0597653i
$389$		−	1.34028e10i		−	0.585324i	−0.956216	−	0.292662i	$-0.905459\pi$
$389$		−	1.34028e10i		−	0.585324i	0.956216	−	0.292662i	$-0.0945410\pi$
$390$		0			0
$391$	4.68150e10			2.00298
$392$	1.60368e10	+	1.60368e10i	0.679162	+	0.679162i
$393$		0			0
$394$			4.09108e10i			1.69767i
$395$	−1.02791e10	−	1.29604e10i	−0.422246	−	0.532389i
$396$		0			0
$397$	−2.56117e10	−	2.56117e10i	−1.03104	−	1.03104i	−0.999503	−	0.0315396i	$-0.989959\pi$
$397$	−2.56117e10	−	2.56117e10i	−1.03104	−	1.03104i	−0.0315396	−	0.999503i	$-0.510041\pi$
$398$	−1.12372e10	+	1.12372e10i	−0.447842	+	0.447842i
$399$		0			0
$400$	5.25843e9	−	2.24847e10i	0.205407	−	0.878308i
$401$	−3.48774e10			−1.34886			−0.674430	−	0.738339i	$-0.735610\pi$
$401$	−3.48774e10			−1.34886			−0.674430	+	0.738339i	$0.735610\pi$
$402$		0			0
$403$	−1.60997e10	+	1.60997e10i	−0.610376	+	0.610376i
$404$			2.23736e9i			0.0839866i
$405$		0			0
$406$	−1.84007e10			−0.677220
$407$	−8.32296e9	−	8.32296e9i	−0.303319	−	0.303319i
$408$		0			0
$409$		−	3.13433e10i		−	1.12009i	−0.828463	−	0.560043i	$-0.810784\pi$
$409$		−	3.13433e10i		−	1.12009i	0.828463	−	0.560043i	$-0.189216\pi$
$410$	−2.55657e10	−	2.94969e9i	−0.904736	−	0.104386i
$411$		0			0
$412$	−3.22739e8	−	3.22739e8i	−0.0112012	−	0.0112012i
$413$	−2.19051e10	+	2.19051e10i	−0.752914	+	0.752914i
$414$		0			0
$415$	5.76999e9	−	5.00099e10i	0.194528	−	1.68602i
$416$	9.88561e9			0.330088
$417$		0			0
$418$	2.21942e10	−	2.21942e10i	0.727001	−	0.727001i
$419$			2.32098e10i			0.753035i	0.926410	+	0.376517i	$0.122879\pi$
$419$			2.32098e10i			0.753035i	−0.926410	+	0.376517i	$0.877121\pi$
$420$		0			0
$421$	−2.28451e10			−0.727218			−0.363609	−	0.931552i	$-0.618456\pi$
$421$	−2.28451e10			−0.727218			−0.363609	+	0.931552i	$0.618456\pi$
$422$	4.23015e9	+	4.23015e9i	0.133385	+	0.133385i
$423$		0			0
$424$		−	2.14455e10i		−	0.663547i
$425$	−3.61565e10	+	2.24503e10i	−1.10823	+	0.688124i
$426$		0			0
$427$	3.18079e10	+	3.18079e10i	0.956804	+	0.956804i
$428$	−1.76953e9	+	1.76953e9i	−0.0527331	+	0.0527331i
$429$		0			0
$430$	−3.56286e9	+	2.82576e9i	−0.104214	+	0.0826535i
$431$	5.00451e10			1.45028			0.725141	−	0.688600i	$-0.241774\pi$
$431$	5.00451e10			1.45028			0.725141	+	0.688600i	$0.241774\pi$
$432$		0			0
$433$	−4.69119e10	+	4.69119e10i	−1.33454	+	1.33454i	−0.433281	+	0.901259i	$0.642644\pi$
$433$	−4.69119e10	+	4.69119e10i	−1.33454	+	1.33454i	−0.901259	+	0.433281i	$0.857356\pi$
$434$			2.20054e10i			0.620256i
$435$		0			0
$436$	−4.37354e9			−0.121028
$437$	−5.24278e10	−	5.24278e10i	−1.43759	−	1.43759i
$438$		0			0
$439$		−	3.71350e10i		−	0.999829i	−0.866075	−	0.499914i	$-0.833365\pi$
$439$		−	3.71350e10i		−	0.999829i	0.866075	−	0.499914i	$-0.166635\pi$
$440$	3.63567e9	−	3.15112e10i	0.0970004	−	0.840726i
$441$		0			0
$442$	6.18466e10	+	6.18466e10i	1.62042	+	1.62042i
$443$	−2.25929e10	+	2.25929e10i	−0.586621	+	0.586621i	−0.936715	−	0.350094i	$-0.886150\pi$
$443$	−2.25929e10	+	2.25929e10i	−0.586621	+	0.586621i	0.350094	+	0.936715i	$0.386150\pi$
$444$		0			0
$445$	5.71648e9	+	7.20762e9i	0.145777	+	0.183803i
$446$	3.00204e9			0.0758711
$447$		0			0
$448$	4.23914e10	−	4.23914e10i	1.05236	−	1.05236i
$449$			1.01376e10i			0.249431i	0.992193	+	0.124716i	$0.0398019\pi$
$449$			1.01376e10i			0.249431i	−0.992193	+	0.124716i	$0.960198\pi$
$450$		0			0
$451$	−3.21478e10			−0.777042
$452$	−2.05130e9	−	2.05130e9i	−0.0491445	−	0.0491445i
$453$		0			0
$454$		−	4.22695e8i		−	0.00994955i
$455$	8.57657e10	−	6.80222e10i	2.00110	−	1.58710i
$456$		0			0
$457$	9.96852e9	+	9.96852e9i	0.228542	+	0.228542i	0.812083	−	0.583541i	$-0.198333\pi$
$457$	9.96852e9	+	9.96852e9i	0.228542	+	0.228542i	−0.583541	+	0.812083i	$0.698333\pi$
$458$	6.38324e9	−	6.38324e9i	0.145071	−	0.145071i
$459$		0			0
$460$	−6.14042e9	−	7.08463e8i	−0.137141	−	0.0158229i
$461$	3.39307e10			0.751259			0.375629	−	0.926770i	$-0.377427\pi$
$461$	3.39307e10			0.751259			0.375629	+	0.926770i	$0.377427\pi$
$462$		0			0
$463$	−5.43575e10	+	5.43575e10i	−1.18287	+	1.18287i	−0.203868	+	0.978998i	$0.565351\pi$
$463$	−5.43575e10	+	5.43575e10i	−1.18287	+	1.18287i	−0.978998	+	0.203868i	$0.934649\pi$
$464$			2.13991e10i			0.461661i
$465$		0			0
$466$	−6.68722e10			−1.41808
$467$	3.83599e10	+	3.83599e10i	0.806510	+	0.806510i	0.984104	−	0.177594i	$-0.0568313\pi$
$467$	3.83599e10	+	3.83599e10i	0.806510	+	0.806510i	−0.177594	+	0.984104i	$0.556831\pi$
$468$		0			0
$469$			1.27067e9i			0.0262627i
$470$	1.65650e10	+	2.08859e10i	0.339468	+	0.428018i
$471$		0			0
$472$	2.80138e10	+	2.80138e10i	0.564422	+	0.564422i
$473$	−4.01671e9	+	4.01671e9i	−0.0802464	+	0.0802464i
$474$		0			0
$475$	6.56335e10	+	1.53495e10i	1.28929	+	0.301523i
$476$	−8.35112e9			−0.162674
$477$		0			0
$478$	−3.36021e10	+	3.36021e10i	−0.643658	+	0.643658i
$479$			1.52155e10i			0.289032i	0.989502	+	0.144516i	$0.0461625\pi$
$479$			1.52155e10i			0.289032i	−0.989502	+	0.144516i	$0.953838\pi$
$480$		0			0
$481$	−5.19470e10			−0.970466
$482$	−3.04279e10	−	3.04279e10i	−0.563746	−	0.563746i
$483$		0			0
$484$			1.66516e9i			0.0303440i
$485$	−5.16723e10	−	5.96179e9i	−0.933880	−	0.107748i
$486$		0			0
$487$	−5.90807e10	−	5.90807e10i	−1.05034	−	1.05034i	−0.998664	−	0.0516753i	$-0.983544\pi$
$487$	−5.90807e10	−	5.90807e10i	−1.05034	−	1.05034i	−0.0516753	−	0.998664i	$-0.516456\pi$
$488$	4.06781e10	−	4.06781e10i	0.717267	−	0.717267i
$489$		0			0
$490$	5.82277e9	−	5.04674e10i	0.101006	−	0.875440i
$491$	4.79401e10			0.824846			0.412423	−	0.910992i	$-0.364682\pi$
$491$	4.79401e10			0.824846			0.412423	+	0.910992i	$0.364682\pi$
$492$		0			0
$493$	2.78886e10	−	2.78886e10i	0.472105	−	0.472105i
$494$		−	1.38523e11i		−	2.32603i
$495$		0			0
$496$	2.55912e10			0.422828
$497$	9.58250e10	+	9.58250e10i	1.57055	+	1.57055i
$498$		0			0
$499$			1.52158e10i			0.245410i	0.992443	+	0.122705i	$0.0391569\pi$
$499$			1.52158e10i			0.245410i	−0.992443	+	0.122705i	$0.960843\pi$
$500$	5.08217e9	−	2.39750e9i	0.0813147	−	0.0383600i
$501$		0			0
$502$	7.51092e10	+	7.51092e10i	1.18271	+	1.18271i
$503$	8.15505e10	−	8.15505e10i	1.27396	−	1.27396i	0.329962	−	0.943994i	$-0.392964\pi$
$503$	8.15505e10	−	8.15505e10i	1.27396	−	1.27396i	0.943994	−	0.329962i	$-0.107036\pi$
$504$		0			0
$505$	4.76002e10	−	3.77525e10i	0.731886	−	0.580471i
$506$	7.81582e10			1.19226
$507$		0			0
$508$	2.85674e8	−	2.85674e8i	0.00428959	−	0.00428959i
$509$			1.00335e11i			1.49480i	0.664376	+	0.747399i	$0.268697\pi$
$509$			1.00335e11i			1.49480i	−0.664376	+	0.747399i	$0.731303\pi$
$510$		0			0
$511$	−3.38386e9			−0.0496282
$512$	−5.34298e10	−	5.34298e10i	−0.777506	−	0.777506i
$513$		0			0
$514$		−	2.48289e10i		−	0.355718i
$515$	−1.42054e9	+	1.23121e10i	−0.0201941	+	0.175027i
$516$		0			0
$517$	2.35464e10	+	2.35464e10i	0.329582	+	0.329582i
$518$	−3.55012e10	+	3.55012e10i	−0.493087	+	0.493087i
$519$		0			0
$520$	−8.69914e10	−	1.09683e11i	−1.18977	−	1.50012i
$521$	−1.09771e11			−1.48983			−0.744914	−	0.667161i	$-0.767509\pi$
$521$	−1.09771e11			−1.48983			−0.744914	+	0.667161i	$0.767509\pi$
$522$		0			0
$523$	1.04269e10	−	1.04269e10i	0.139364	−	0.139364i	−0.633983	−	0.773347i	$-0.718581\pi$
$523$	1.04269e10	−	1.04269e10i	0.139364	−	0.139364i	0.773347	+	0.633983i	$0.218581\pi$
$524$		−	4.32864e9i		−	0.0574152i
$525$		0			0
$526$	2.40470e10			0.314136
$527$	−3.33520e10	−	3.33520e10i	−0.432394	−	0.432394i
$528$		0			0
$529$		−	1.06316e11i		−	1.35762i
$530$	−3.76375e10	+	2.98509e10i	−0.476998	+	0.378315i
$531$		0			0
$532$	9.35238e9	+	9.35238e9i	0.116755	+	0.116755i
$533$	−1.00324e11	+	1.00324e11i	−1.24307	+	1.24307i
$534$		0			0
$535$	6.75057e10	+	7.78860e9i	0.823997	+	0.0950702i
$536$	1.62501e9			0.0196878
$537$		0			0
$538$	−8.51661e10	+	8.51661e10i	−1.01657	+	1.01657i
$539$		−	6.34606e10i		−	0.751881i
$540$		0			0
$541$	1.22410e11			1.42898			0.714490	−	0.699646i	$-0.246659\pi$
$541$	1.22410e11			1.42898			0.714490	+	0.699646i	$0.246659\pi$
$542$	7.81539e10	+	7.81539e10i	0.905635	+	0.905635i
$543$		0			0
$544$			2.04790e10i			0.233837i
$545$	7.37978e10	+	9.30479e10i	0.836483	+	1.05468i
$546$		0			0
$547$	−2.50031e10	−	2.50031e10i	−0.279283	−	0.279283i	0.553540	−	0.832823i	$-0.313277\pi$
$547$	−2.50031e10	−	2.50031e10i	−0.279283	−	0.279283i	−0.832823	+	0.553540i	$0.813277\pi$
$548$	5.54965e8	−	5.54965e8i	0.00615380	−	0.00615380i
$549$		0			0
$550$	−6.03638e10	+	3.74811e10i	−0.659669	+	0.409602i
$551$	−6.24645e10			−0.677684
$552$		0			0
$553$	−6.23238e10	+	6.23238e10i	−0.666429	+	0.666429i
$554$		−	1.08047e11i		−	1.14703i
$555$		0			0
$556$	6.82493e9			0.0714166
$557$	−7.92533e10	−	7.92533e10i	−0.823373	−	0.823373i	0.163217	−	0.986590i	$-0.447813\pi$
$557$	−7.92533e10	−	7.92533e10i	−0.823373	−	0.823373i	−0.986590	+	0.163217i	$0.947813\pi$
$558$		0			0
$559$			2.50699e10i			0.256747i
$560$	−1.22227e11	−	1.41021e10i	−1.24283	−	0.143394i
$561$		0			0
$562$	−9.92651e10	−	9.92651e10i	−0.995065	−	0.995065i
$563$	8.82437e10	−	8.82437e10i	0.878314	−	0.878314i	−0.115046	−	0.993360i	$-0.536702\pi$
$563$	8.82437e10	−	8.82437e10i	0.878314	−	0.878314i	0.993360	+	0.115046i	$0.0367015\pi$
$564$		0			0
$565$	−9.02879e9	+	7.82547e10i	−0.0886004	+	0.767921i
$566$	−1.65231e11			−1.61000
$567$		0			0
$568$	1.22547e11	−	1.22547e11i	1.17736	−	1.17736i
$569$			1.02792e11i			0.980644i	0.871541	+	0.490322i	$0.163121\pi$
$569$			1.02792e11i			0.980644i	−0.871541	+	0.490322i	$0.836879\pi$
$570$		0			0
$571$	−1.56890e11			−1.47588			−0.737940	−	0.674867i	$-0.764201\pi$
$571$	−1.56890e11			−1.47588			−0.737940	+	0.674867i	$0.764201\pi$
$572$	−1.02005e10	−	1.02005e10i	−0.0952881	−	0.0952881i
$573$		0			0
$574$			1.37125e11i			1.26319i
$575$	8.85389e10	+	1.42593e11i	0.809958	+	1.30445i
$576$		0			0
$577$	8.23424e10	+	8.23424e10i	0.742883	+	0.742883i	0.973132	−	0.230249i	$-0.0739539\pi$
$577$	8.23424e10	+	8.23424e10i	0.742883	+	0.742883i	−0.230249	+	0.973132i	$0.573954\pi$
$578$	−5.28308e10	+	5.28308e10i	−0.473343	+	0.473343i
$579$		0			0
$580$	−4.08001e9	+	3.23592e9i	−0.0360537	+	0.0285947i
$581$	−2.68234e11			−2.35402
$582$		0			0
$583$	−4.24319e10	+	4.24319e10i	−0.367297	+	0.367297i
$584$			4.32751e9i			0.0372037i
$585$		0			0
$586$	−1.51826e11			−1.28753
$587$	1.04017e11	+	1.04017e11i	0.876095	+	0.876095i	0.993128	−	0.117033i	$-0.0373384\pi$
$587$	1.04017e11	+	1.04017e11i	0.876095	+	0.876095i	−0.117033	+	0.993128i	$0.537338\pi$
$588$		0			0
$589$			7.47015e10i			0.620680i
$590$	1.01715e10	−	8.81587e10i	0.0839414	−	0.727540i
$591$		0			0
$592$	4.12861e10	+	4.12861e10i	0.336138	+	0.336138i
$593$	5.75919e10	−	5.75919e10i	0.465739	−	0.465739i	−0.434792	−	0.900531i	$-0.643178\pi$
$593$	5.75919e10	−	5.75919e10i	0.465739	−	0.465739i	0.900531	+	0.434792i	$0.143178\pi$
$594$		0			0
$595$	1.40914e11	+	1.77672e11i	1.12431	+	1.41759i
$596$	−1.25482e10			−0.0994482
$597$		0			0
$598$	2.43909e11	−	2.43909e11i	1.90732	−	1.90732i
$599$			1.44757e11i			1.12443i	0.826991	+	0.562215i	$0.190051\pi$
$599$			1.44757e11i			1.12443i	−0.826991	+	0.562215i	$0.809949\pi$
$600$		0			0
$601$	2.15714e11			1.65341			0.826705	−	0.562636i	$-0.190213\pi$
$601$	2.15714e11			1.65341			0.826705	+	0.562636i	$0.190213\pi$
$602$	1.71331e10	+	1.71331e10i	0.130452	+	0.130452i
$603$		0			0
$604$			3.23643e9i			0.0243175i
$605$	3.54265e10	−	2.80973e10i	0.264428	−	0.209722i
$606$		0			0
$607$	−1.13943e11	−	1.13943e11i	−0.839327	−	0.839327i	0.149443	−	0.988770i	$-0.452252\pi$
$607$	−1.13943e11	−	1.13943e11i	−0.839327	−	0.839327i	−0.988770	+	0.149443i	$0.952252\pi$
$608$	2.29343e10	−	2.29343e10i	0.167830	−	0.167830i
$609$		0			0
$610$	−1.28013e11	−	1.47697e10i	−0.924558	−	0.106673i
$611$	1.46963e11			1.05449
$612$		0			0
$613$	−1.23541e11	+	1.23541e11i	−0.874925	+	0.874925i	−0.993004	−	0.118079i	$-0.962326\pi$
$613$	−1.23541e11	+	1.23541e11i	−0.874925	+	0.874925i	0.118079	+	0.993004i	$0.462326\pi$
$614$		−	1.57061e11i		−	1.10508i
$615$		0			0
$616$	−1.69014e11			−1.17382
$617$	−1.22575e11	−	1.22575e11i	−0.845784	−	0.845784i	0.143820	−	0.989604i	$-0.454061\pi$
$617$	−1.22575e11	−	1.22575e11i	−0.845784	−	0.845784i	−0.989604	+	0.143820i	$0.954061\pi$
$618$		0			0
$619$		−	2.49963e11i		−	1.70260i	−0.524679	−	0.851300i	$-0.675814\pi$
$619$		−	2.49963e11i		−	1.70260i	0.524679	−	0.851300i	$-0.324186\pi$
$620$	3.86985e9	+	4.87930e9i	0.0261895	+	0.0330210i
$621$		0			0
$622$	−2.31149e10	−	2.31149e10i	−0.154429	−	0.154429i
$623$	3.46600e10	−	3.46600e10i	0.230079	−	0.230079i
$624$		0			0
$625$	−1.36762e11	−	6.76695e10i	−0.896285	−	0.443479i
$626$	1.82589e11			1.18899
$627$		0			0
$628$	1.74314e10	−	1.74314e10i	0.112071	−	0.112071i
$629$		−	1.07613e11i		−	0.687484i
$630$		0			0
$631$	2.93773e11			1.85308			0.926541	−	0.376193i	$-0.122767\pi$
$631$	2.93773e11			1.85308			0.926541	+	0.376193i	$0.122767\pi$
$632$	7.97040e10	+	7.97040e10i	0.499588	+	0.499588i
$633$		0			0
$634$			7.19798e10i			0.445506i
$635$	−1.08981e10	−	1.25739e9i	−0.0670282	−	0.00773351i
$636$		0			0
$637$	−1.98042e11	−	1.98042e11i	−1.20282	−	1.20282i
$638$	4.65604e10	−	4.65604e10i	0.281018	−	0.281018i
$639$		0			0
$640$	−1.62371e10	+	1.40731e11i	−0.0967808	+	0.838822i
$641$	2.44222e11			1.44661			0.723307	−	0.690527i	$-0.242621\pi$
$641$	2.44222e11			1.44661			0.723307	+	0.690527i	$0.242621\pi$
$642$		0			0
$643$	9.99311e10	−	9.99311e10i	0.584597	−	0.584597i	−0.351566	−	0.936163i	$-0.614351\pi$
$643$	9.99311e10	−	9.99311e10i	0.584597	−	0.584597i	0.936163	+	0.351566i	$0.114351\pi$
$644$			3.29349e10i			0.191475i
$645$		0			0
$646$	2.86964e11			1.64777
$647$	−1.59162e10	−	1.59162e10i	−0.0908286	−	0.0908286i	0.660233	−	0.751061i	$-0.270458\pi$
$647$	−1.59162e10	−	1.59162e10i	−0.0908286	−	0.0908286i	−0.751061	+	0.660233i	$0.770458\pi$
$648$		0			0
$649$		−	1.10856e11i		−	0.624855i
$650$	−7.14102e10	+	3.05345e11i	−0.400043	+	1.71056i
$651$		0			0
$652$	3.90366e9	+	3.90366e9i	0.0216014	+	0.0216014i
$653$	−8.73181e9	+	8.73181e9i	−0.0480233	+	0.0480233i	−0.730711	−	0.682687i	$-0.760811\pi$
$653$	−8.73181e9	+	8.73181e9i	−0.0480233	+	0.0480233i	0.682687	+	0.730711i	$0.260811\pi$
$654$		0			0
$655$	−9.20927e10	+	7.30402e10i	−0.500334	+	0.396823i
$656$	1.59469e11			0.861115
$657$		0			0
$658$	1.00436e11	−	1.00436e11i	0.535780	−	0.535780i
$659$		−	2.65013e11i		−	1.40516i	−0.711605	−	0.702580i	$-0.752031\pi$
$659$		−	2.65013e11i		−	1.40516i	0.711605	−	0.702580i	$-0.247969\pi$
$660$		0			0
$661$	2.25047e11			1.17887			0.589437	−	0.807814i	$-0.299349\pi$
$661$	2.25047e11			1.17887			0.589437	+	0.807814i	$0.299349\pi$
$662$	1.26010e11	+	1.26010e11i	0.656105	+	0.656105i
$663$		0			0
$664$			3.43036e11i			1.76469i
$665$	4.11645e10	−	3.56783e11i	0.210492	−	1.82439i
$666$		0			0
$667$	−1.09986e11	−	1.09986e11i	−0.555692	−	0.555692i
$668$	−2.25687e9	+	2.25687e9i	−0.0113344	+	0.0113344i
$669$		0			0
$670$	−2.26193e9	−	2.85195e9i	−0.0112248	−	0.0141528i
$671$	−1.60971e11			−0.794066
$672$		0			0
$673$	−7.27218e10	+	7.27218e10i	−0.354490	+	0.354490i	−0.861777	−	0.507287i	$-0.830648\pi$
$673$	−7.27218e10	+	7.27218e10i	−0.354490	+	0.354490i	0.507287	+	0.861777i	$0.330648\pi$
$674$			1.33746e11i			0.648099i
$675$		0			0
$676$	−4.48903e10			−0.214964
$677$	−3.96382e9	−	3.96382e9i	−0.0188694	−	0.0188694i	0.697609	−	0.716479i	$-0.254247\pi$
$677$	−3.96382e9	−	3.96382e9i	−0.0188694	−	0.0188694i	−0.716479	+	0.697609i	$0.754247\pi$
$678$		0			0
$679$			2.77151e11i			1.30388i
$680$	2.27219e11	−	1.80211e11i	1.06269	−	0.842840i
$681$		0			0
$682$	−5.56817e10	−	5.56817e10i	−0.257380	−	0.257380i
$683$	−8.82124e10	+	8.82124e10i	−0.405366	+	0.405366i	−0.880119	−	0.474753i	$-0.842537\pi$
$683$	−8.82124e10	+	8.82124e10i	−0.405366	+	0.405366i	0.474753	+	0.880119i	$0.342537\pi$
$684$		0			0
$685$	−2.11713e10	−	2.44268e9i	−0.0961579	−	0.0110944i
$686$	2.23428e10			0.100889
$687$		0			0
$688$	1.99249e10	−	1.99249e10i	0.0889288	−	0.0889288i
$689$			2.64835e11i			1.17516i
$690$		0			0
$691$	−2.61168e11			−1.14553			−0.572766	−	0.819719i	$-0.694130\pi$
$691$	−2.61168e11			−1.14553			−0.572766	+	0.819719i	$0.694130\pi$
$692$	5.48656e9	+	5.48656e9i	0.0239263	+	0.0239263i
$693$		0			0
$694$			1.67073e11i			0.720227i
$695$	−1.15162e11	−	1.45202e11i	−0.493593	−	0.622347i
$696$		0			0
$697$	−2.07830e11	−	2.07830e11i	−0.880596	−	0.880596i
$698$	8.33467e10	−	8.33467e10i	0.351129	−	0.351129i
$699$		0			0
$700$	−1.57941e10	−	2.54366e10i	−0.0657813	−	0.105942i
$701$	1.23222e11			0.510287			0.255144	−	0.966903i	$-0.417877\pi$
$701$	1.23222e11			0.510287			0.255144	+	0.966903i	$0.417877\pi$
$702$		0			0
$703$	−1.20515e11	+	1.20515e11i	−0.493425	+	0.493425i
$704$			2.14531e11i			0.873372i
$705$		0			0
$706$	9.12443e10			0.367271
$707$	−2.28900e11	−	2.28900e11i	−0.916153	−	0.916153i
$708$		0			0
$709$		−	1.95997e10i		−	0.0775648i	−0.999248	−	0.0387824i	$-0.987652\pi$
$709$		−	1.95997e10i		−	0.0775648i	0.999248	−	0.0387824i	$-0.0123479\pi$
$710$	−3.85654e11	−	4.44956e10i	−1.51762	−	0.175099i
$711$		0			0
$712$	−4.43256e10	−	4.43256e10i	−0.172478	−	0.172478i
$713$	−1.31533e11	+	1.31533e11i	−0.508950	+	0.508950i
$714$		0			0
$715$	−4.48977e10	+	3.89139e11i	−0.171791	+	1.48895i
$716$	1.72354e10			0.0655799
$717$		0			0
$718$	1.61215e11	−	1.61215e11i	0.606609	−	0.606609i
$719$			1.94089e11i			0.726250i	0.931740	+	0.363125i	$0.118290\pi$
$719$			1.94089e11i			0.726250i	−0.931740	+	0.363125i	$0.881710\pi$
$720$		0			0
$721$	6.60377e10			0.244372
$722$	−1.38064e11	−	1.38064e11i	−0.508079	−	0.508079i
$723$		0			0
$724$			1.13075e9i			0.00411542i
$725$	1.37690e11	+	3.22011e10i	0.498367	+	0.116552i
$726$		0			0
$727$	1.39853e11	+	1.39853e11i	0.500649	+	0.500649i	0.911639	−	0.410991i	$-0.134817\pi$
$727$	1.39853e11	+	1.39853e11i	0.500649	+	0.500649i	−0.410991	+	0.911639i	$0.634817\pi$
$728$	−5.27444e11	+	5.27444e11i	−1.87781	+	1.87781i
$729$		0			0
$730$	7.59491e9	−	6.02365e9i	0.0267443	−	0.0212113i
$731$	−5.19347e10			−0.181881
$732$		0			0
$733$	5.55699e9	−	5.55699e9i	0.0192497	−	0.0192497i	−0.697416	−	0.716666i	$-0.745667\pi$
$733$	5.55699e9	−	5.55699e9i	0.0192497	−	0.0192497i	0.716666	+	0.697416i	$0.245667\pi$
$734$			1.90729e11i			0.657101i
$735$		0			0
$736$	8.07643e10			0.275238
$737$	−3.21524e9	−	3.21524e9i	−0.0108979	−	0.0108979i
$738$		0			0
$739$		−	5.93312e11i		−	1.98932i	−0.103197	−	0.994661i	$-0.532907\pi$
$739$		−	5.93312e11i		−	1.98932i	0.103197	−	0.994661i	$-0.467093\pi$
$740$	−1.62854e9	+	1.41149e10i	−0.00543089	+	0.0470708i
$741$		0			0
$742$	1.80991e11	+	1.80991e11i	0.597092	+	0.597092i
$743$	9.21623e10	−	9.21623e10i	0.302411	−	0.302411i	−0.539545	−	0.841957i	$-0.681404\pi$
$743$	9.21623e10	−	9.21623e10i	0.302411	−	0.302411i	0.841957	+	0.539545i	$0.181404\pi$
$744$		0			0
$745$	2.11735e11	+	2.66966e11i	0.687333	+	0.866623i
$746$	−7.96407e10			−0.257146
$747$		0			0
$748$	2.11314e10	−	2.11314e10i	0.0675027	−	0.0675027i
$749$		−	3.62075e11i		−	1.15046i
$750$		0			0
$751$	3.75368e10			0.118004			0.0590020	−	0.998258i	$-0.481208\pi$
$751$	3.75368e10			0.118004			0.0590020	+	0.998258i	$0.481208\pi$
$752$	−1.16802e11	−	1.16802e11i	−0.365241	−	0.365241i
$753$		0			0
$754$		−	2.90602e11i		−	0.899112i
$755$	6.88557e10	−	5.46106e10i	0.211910	−	0.168069i
$756$		0			0
$757$	−2.36814e11	−	2.36814e11i	−0.721146	−	0.721146i	0.247693	−	0.968839i	$-0.420328\pi$
$757$	−2.36814e11	−	2.36814e11i	−0.721146	−	0.721146i	−0.968839	+	0.247693i	$0.920328\pi$
$758$	1.27450e10	−	1.27450e10i	0.0386067	−	0.0386067i
$759$		0			0
$760$	−4.56278e11	−	5.26440e10i	−1.36765	−	0.157795i
$761$	−4.40653e11			−1.31389			−0.656943	−	0.753940i	$-0.728151\pi$
$761$	−4.40653e11			−1.31389			−0.656943	+	0.753940i	$0.728151\pi$
$762$		0			0
$763$	4.47449e11	−	4.47449e11i	1.32022	−	1.32022i
$764$		−	1.16208e10i		−	0.0341084i
$765$		0			0
$766$	5.57737e10			0.162000
$767$	−3.45948e11	−	3.45948e11i	−0.999608	−	0.999608i
$768$		0			0
$769$			8.47485e10i			0.242341i	0.992632	+	0.121170i	$0.0386648\pi$
$769$			8.47485e10i			0.242341i	−0.992632	+	0.121170i	$0.961335\pi$
$770$	2.35258e11	+	2.96626e11i	0.669241	+	0.843812i
$771$		0			0
$772$	8.92286e9	+	8.92286e9i	0.0251209	+	0.0251209i
$773$	4.85507e10	−	4.85507e10i	0.135981	−	0.135981i	−0.635840	−	0.771821i	$-0.719346\pi$
$773$	4.85507e10	−	4.85507e10i	0.135981	−	0.135981i	0.771821	+	0.635840i	$0.219346\pi$
$774$		0			0
$775$	3.85094e10	−	1.64663e11i	0.106748	−	0.456447i
$776$	3.54439e11			0.977451
$777$		0			0
$778$	1.44658e11	−	1.44658e11i	0.394842	−	0.394842i
$779$			4.65495e11i			1.26405i
$780$		0			0
$781$	−4.84943e11			−1.30343
$782$	5.05280e11	+	5.05280e11i	1.35115	+	1.35115i
$783$		0			0
$784$			3.14797e11i			0.833232i
$785$	−6.64988e11	−	7.67243e10i	−1.75120	−	0.202048i
$786$		0			0
$787$	3.69680e11	+	3.69680e11i	0.963667	+	0.963667i	0.999363	−	0.0356957i	$-0.0113647\pi$
$787$	3.69680e11	+	3.69680e11i	0.963667	+	0.963667i	−0.0356957	+	0.999363i	$0.511365\pi$
$788$	4.36217e10	−	4.36217e10i	0.113135	−	0.113135i
$789$		0			0
$790$	2.89396e10	−	2.50826e11i	0.0742992	−	0.643969i
$791$	4.19729e11			1.07217
$792$		0			0
$793$	−5.02342e11	+	5.02342e11i	−1.27030	+	1.27030i
$794$		−	5.52861e11i		−	1.39102i
$795$		0			0
$796$	2.39635e10			0.0596896
$797$	3.03220e11	+	3.03220e11i	0.751493	+	0.751493i	0.974758	−	0.223265i	$-0.0716715\pi$
$797$	3.03220e11	+	3.03220e11i	0.751493	+	0.751493i	−0.223265	+	0.974758i	$0.571672\pi$
$798$		0			0
$799$			3.04448e11i			0.747009i
$800$	−6.23766e10	+	3.87309e10i	−0.152287	+	0.0945578i
$801$		0			0
$802$	−3.76436e11	−	3.76436e11i	−0.909901	−	0.909901i
$803$	8.56238e9	−	8.56238e9i	0.0205936	−	0.0205936i
$804$		0			0
$805$	7.00696e11	−	5.55733e11i	1.66858	−	1.32338i
$806$	−3.47532e11			−0.823483
$807$		0			0
$808$	−2.92733e11	+	2.92733e11i	−0.686793	+	0.686793i
$809$		−	2.84151e11i		−	0.663369i	−0.943390	−	0.331684i	$-0.892383\pi$
$809$		−	2.84151e11i		−	0.663369i	0.943390	−	0.331684i	$-0.107617\pi$
$810$		0			0
$811$	2.66222e11			0.615404			0.307702	−	0.951483i	$-0.400440\pi$
$811$	2.66222e11			0.615404			0.307702	+	0.951483i	$0.400440\pi$
$812$	1.96200e10	+	1.96200e10i	0.0451309	+	0.0451309i
$813$		0			0
$814$		−	1.79662e11i		−	0.409221i
$815$	1.71819e10	−	1.48920e11i	0.0389441	−	0.337538i
$816$		0			0
$817$	5.81614e10	+	5.81614e10i	0.130541	+	0.130541i
$818$	3.38292e11	−	3.38292e11i	0.755578	−	0.755578i
$819$		0			0
$820$	2.41146e10	+	3.04049e10i	0.0533365	+	0.0672493i
$821$	4.28852e11			0.943920			0.471960	−	0.881620i	$-0.343547\pi$
$821$	4.28852e11			0.943920			0.471960	+	0.881620i	$0.343547\pi$
$822$		0			0
$823$	−3.88192e11	+	3.88192e11i	−0.846150	+	0.846150i	−0.989650	−	0.143500i	$-0.954164\pi$
$823$	−3.88192e11	+	3.88192e11i	−0.846150	+	0.846150i	0.143500	+	0.989650i	$0.454164\pi$
$824$		−	8.44535e10i		−	0.183193i
$825$		0			0
$826$	−4.72850e11			−1.01579
$827$	3.15974e11	+	3.15974e11i	0.675507	+	0.675507i	0.958980	−	0.283473i	$-0.0914866\pi$
$827$	3.15974e11	+	3.15974e11i	0.675507	+	0.675507i	−0.283473	+	0.958980i	$0.591487\pi$
$828$		0			0
$829$			4.60293e11i			0.974578i	0.873241	+	0.487289i	$0.162014\pi$
$829$			4.60293e11i			0.974578i	−0.873241	+	0.487289i	$0.837986\pi$
$830$	6.02039e11	−	4.77487e11i	1.26856	−	1.00612i
$831$		0			0
$832$	6.69488e11	+	6.69488e11i	1.39717	+	1.39717i
$833$	4.10262e11	−	4.10262e11i	0.852082	−	0.852082i
$834$		0			0
$835$	8.60969e10	+	9.93360e9i	0.177109	+	0.0204343i
$836$	−4.73298e10			−0.0968968
$837$		0			0
$838$	−2.50506e11	+	2.50506e11i	−0.507975	+	0.507975i
$839$		−	6.30877e11i		−	1.27320i	−0.771193	−	0.636601i	$-0.780340\pi$
$839$		−	6.30877e11i		−	1.27320i	0.771193	−	0.636601i	$-0.219660\pi$
$840$		0			0
$841$	3.69205e11			0.738046
$842$	−2.46570e11	−	2.46570e11i	−0.490560	−	0.490560i
$843$		0			0
$844$		−	9.02091e9i		−	0.0177779i
$845$	7.57466e11	+	9.55050e11i	1.48572	+	1.87327i
$846$		0			0
$847$	−1.70359e11	−	1.70359e11i	−0.331003	−	0.331003i
$848$	2.10483e11	−	2.10483e11i	0.407038	−	0.407038i
$849$		0			0
$850$	−6.32551e11	−	1.47933e11i	−1.21177	−	0.283393i
$851$	−4.24401e11			−0.809205
$852$		0			0
$853$	2.33231e11	−	2.33231e11i	0.440544	−	0.440544i	−0.451650	−	0.892195i	$-0.649165\pi$
$853$	2.33231e11	−	2.33231e11i	0.440544	−	0.440544i	0.892195	+	0.451650i	$0.149165\pi$
$854$			6.86613e11i			1.29086i
$855$		0			0
$856$	−4.63047e11			−0.862441
$857$	3.65561e11	+	3.65561e11i	0.677699	+	0.677699i	0.959479	−	0.281780i	$-0.0909248\pi$
$857$	3.65561e11	+	3.65561e11i	0.677699	+	0.677699i	−0.281780	+	0.959479i	$0.590925\pi$
$858$		0			0
$859$		−	4.19242e11i		−	0.770002i	−0.922916	−	0.385001i	$-0.874201\pi$
$859$		−	4.19242e11i		−	0.770002i	0.922916	−	0.385001i	$-0.125799\pi$
$860$	6.81194e9	+	7.85941e8i	0.0124531	+	0.00143680i
$861$		0			0
$862$	5.40144e11	+	5.40144e11i	0.978318	+	0.978318i
$863$	−2.23940e11	+	2.23940e11i	−0.403728	+	0.403728i	−0.879544	−	0.475817i	$-0.842153\pi$
$863$	−2.23940e11	+	2.23940e11i	−0.403728	+	0.403728i	0.475817	+	0.879544i	$0.342153\pi$
$864$		0			0
$865$	2.41491e10	−	2.09306e11i	0.0431357	−	0.373868i
$866$	−1.01265e12			−1.80048
$867$		0			0
$868$	2.34636e10	−	2.34636e10i	0.0413347	−	0.0413347i
$869$		−	3.15404e11i		−	0.553079i
$870$		0			0
$871$	−2.00676e10			−0.0348677
$872$	−5.72228e11	−	5.72228e11i	−0.989699	−	0.989699i
$873$		0			0
$874$		−	1.13172e12i		−	1.93952i
$875$	−2.74664e11	+	7.65231e11i	−0.468564	+	1.30545i
$876$		0			0
$877$	2.87476e11	+	2.87476e11i	0.485963	+	0.485963i	0.907030	−	0.421067i	$-0.138344\pi$
$877$	2.87476e11	+	2.87476e11i	0.485963	+	0.485963i	−0.421067	+	0.907030i	$0.638344\pi$
$878$	4.00803e11	−	4.00803e11i	0.674455	−	0.674455i
$879$		0			0
$880$	3.44961e11	−	2.73594e11i	0.575226	−	0.456221i
$881$	3.93551e11			0.653278			0.326639	−	0.945149i	$-0.394084\pi$
$881$	3.93551e11			0.653278			0.326639	+	0.945149i	$0.394084\pi$
$882$		0			0
$883$	−2.31269e11	+	2.31269e11i	−0.380430	+	0.380430i	−0.871257	−	0.490827i	$-0.836695\pi$
$883$	−2.31269e11	+	2.31269e11i	−0.380430	+	0.380430i	0.490827	+	0.871257i	$0.336695\pi$
$884$		−	1.31889e11i		−	0.215974i
$885$		0			0
$886$	−4.87697e11			−0.791435
$887$	9.92586e10	+	9.92586e10i	0.160352	+	0.160352i	0.782723	−	0.622371i	$-0.213830\pi$
$887$	9.92586e10	+	9.92586e10i	0.160352	+	0.160352i	−0.622371	+	0.782723i	$0.713830\pi$
$888$		0			0
$889$			5.84535e10i			0.0935845i
$890$	−1.60941e10	+	1.39491e11i	−0.0256511	+	0.222325i
$891$		0			0
$892$	−3.20096e9	−	3.20096e9i	−0.00505616	−	0.00505616i
$893$	3.40949e11	−	3.40949e11i	0.536147	−	0.536147i
$894$		0			0
$895$	−2.90826e11	−	3.66687e11i	−0.453253	−	0.571484i
$896$	7.54828e11			1.17116
$897$		0			0
$898$	−1.09417e11	+	1.09417e11i	−0.168259	+	0.168259i
$899$			1.56713e11i			0.239920i
$900$		0			0
$901$	−5.48630e11			−0.832492
$902$	−3.46975e11	−	3.46975e11i	−0.524170	−	0.524170i
$903$		0			0
$904$		−	5.36778e11i		−	0.803750i
$905$	2.40570e10	−	1.90800e10i	0.0358631	−	0.0284436i
$906$		0			0
$907$	1.17851e11	+	1.17851e11i	0.174142	+	0.174142i	0.788797	−	0.614654i	$-0.210705\pi$
$907$	1.17851e11	+	1.17851e11i	0.174142	+	0.174142i	−0.614654	+	0.788797i	$0.710705\pi$
$908$	−4.50704e8	+	4.50704e8i	−0.000663052	+	0.000663052i
$909$		0			0
$910$	1.65985e12	+	1.91509e11i	2.42049	+	0.279269i
$911$	−1.23532e12			−1.79352			−0.896762	−	0.442514i	$-0.854087\pi$
$911$	−1.23532e12			−1.79352			−0.896762	+	0.442514i	$0.854087\pi$
$912$		0			0
$913$	6.78729e11	−	6.78729e11i	0.976817	−	0.976817i
$914$			2.15183e11i			0.308335i
$915$		0			0
$916$	−1.36124e10			−0.0193354
$917$	4.42855e11	+	4.42855e11i	0.626303	+	0.626303i
$918$		0			0
$919$		−	4.72598e11i		−	0.662567i	−0.943531	−	0.331283i	$-0.892518\pi$
$919$		−	4.72598e11i		−	0.662567i	0.943531	−	0.331283i	$-0.107482\pi$
$920$	−7.10710e11	−	8.96098e11i	−0.992067	−	1.25085i
$921$		0			0
$922$	3.66219e11	+	3.66219e11i	0.506777	+	0.506777i
$923$	−1.51337e12	+	1.51337e12i	−2.08515	+	2.08515i
$924$		0			0
$925$	3.27777e11	−	2.03524e11i	0.447726	−	0.278002i
$926$	−1.17338e12			−1.59585
$927$		0			0
$928$	4.81128e10	−	4.81128e10i	0.0648738	−	0.0648738i
$929$		−	1.97121e11i		−	0.264649i	−0.991206	−	0.132325i	$-0.957756\pi$
$929$		−	1.97121e11i		−	0.264649i	0.991206	−	0.132325i	$-0.0422441\pi$
$930$		0			0
$931$	−9.18901e11			−1.22312
$932$	7.13033e10	+	7.13033e10i	0.0945031	+	0.0945031i
$933$		0			0
$934$			8.28047e11i			1.08810i
$935$	−8.06138e11	−	9.30097e10i	−1.05478	−	0.121698i
$936$		0			0
$937$	9.21042e11	+	9.21042e11i	1.19487	+	1.19487i	0.975682	+	0.219189i	$0.0703412\pi$
$937$	9.21042e11	+	9.21042e11i	1.19487	+	1.19487i	0.219189	+	0.975682i	$0.429659\pi$
$938$	−1.37145e10	+	1.37145e10i	−0.0177161	+	0.0177161i
$939$		0			0
$940$	4.60729e9	−	3.99325e10i	0.00590111	−	0.0511464i
$941$	3.96460e11			0.505640			0.252820	−	0.967513i	$-0.418642\pi$
$941$	3.96460e11			0.505640			0.252820	+	0.967513i	$0.418642\pi$
$942$		0			0
$943$	−8.19633e11	+	8.19633e11i	−1.03651	+	1.03651i
$944$			5.49901e11i			0.692462i
$945$		0			0
$946$	−8.67057e10			−0.108264
$947$	4.69199e11	+	4.69199e11i	0.583388	+	0.583388i	0.935833	−	0.352445i	$-0.114650\pi$
$947$	4.69199e11	+	4.69199e11i	0.583388	+	0.583388i	−0.352445	+	0.935833i	$0.614650\pi$
$948$		0			0
$949$		−	5.34413e10i		−	0.0658889i
$950$	5.42722e11	+	8.74060e11i	0.666320	+	1.07312i
$951$		0			0
$952$	−1.09265e12	−	1.09265e12i	−1.33025	−	1.33025i
$953$	3.06016e11	−	3.06016e11i	0.370999	−	0.370999i	−0.496842	−	0.867841i	$-0.665507\pi$
$953$	3.06016e11	−	3.06016e11i	0.370999	−	0.370999i	0.867841	+	0.496842i	$0.165507\pi$
$954$		0			0
$955$	−2.47234e11	+	1.96085e11i	−0.297231	+	0.235739i
$956$	7.16574e10			0.0857886
$957$		0			0
$958$	−1.64223e11	+	1.64223e11i	−0.194972	+	0.194972i
$959$			1.13555e11i			0.134255i
$960$		0			0
$961$	−6.65478e11			−0.780261
$962$	−5.60671e11	−	5.60671e11i	−0.654648	−	0.654648i
$963$		0			0
$964$			6.48882e10i			0.0751376i
$965$	3.92740e10	−	3.40397e11i	0.0452893	−	0.392533i
$966$		0			0
$967$	−2.20508e11	−	2.20508e11i	−0.252184	−	0.252184i	0.569681	−	0.821866i	$-0.307067\pi$
$967$	−2.20508e11	−	2.20508e11i	−0.252184	−	0.252184i	−0.821866	+	0.569681i	$0.807067\pi$
$968$	−2.17867e11	+	2.17867e11i	−0.248136	+	0.248136i
$969$		0			0
$970$	−4.93360e11	−	6.22052e11i	−0.557284	−	0.702652i
$971$	1.44240e10			0.0162259			0.00811296	−	0.999967i	$-0.497418\pi$
$971$	1.44240e10			0.0162259			0.00811296	+	0.999967i	$0.497418\pi$
$972$		0			0
$973$	−6.98246e11	+	6.98246e11i	−0.779035	+	0.779035i
$974$		−	1.27533e12i		−	1.41706i
$975$		0			0
$976$	7.98496e11			0.879982
$977$	−5.07897e11	−	5.07897e11i	−0.557439	−	0.557439i	0.371138	−	0.928578i	$-0.378968\pi$
$977$	−5.07897e11	−	5.07897e11i	−0.557439	−	0.557439i	−0.928578	+	0.371138i	$0.878968\pi$
$978$		0			0
$979$			1.75405e11i			0.190946i
$980$	−6.00201e10	+	4.76029e10i	−0.0650718	+	0.0516094i
$981$		0			0
$982$	5.17424e11	+	5.17424e11i	0.556417	+	0.556417i
$983$	−4.38651e11	+	4.38651e11i	−0.469791	+	0.469791i	−0.901847	−	0.432056i	$-0.857788\pi$
$983$	−4.38651e11	+	4.38651e11i	−0.469791	+	0.469791i	0.432056	+	0.901847i	$0.357788\pi$
$984$		0			0
$985$	−1.66412e12	−	1.92001e11i	−1.76782	−	0.203966i
$986$	6.02010e11			0.636936
$987$		0			0
$988$	−1.47702e11	+	1.47702e11i	−0.155010	+	0.155010i
$989$			2.04818e11i			0.214084i
$990$		0			0
$991$	1.25893e12			1.30529			0.652645	−	0.757664i	$-0.273659\pi$
$991$	1.25893e12			1.30529			0.652645	+	0.757664i	$0.273659\pi$
$992$	−5.75383e10	−	5.75383e10i	−0.0594169	−	0.0594169i
$993$		0			0
$994$			2.06850e12i			2.11890i
$995$	−4.04353e11	−	5.09829e11i	−0.412542	−	0.520154i
$996$		0			0
$997$	8.69429e11	+	8.69429e11i	0.879940	+	0.879940i	0.993528	−	0.113588i	$-0.0362342\pi$
$997$	8.69429e11	+	8.69429e11i	0.879940	+	0.879940i	−0.113588	+	0.993528i	$0.536234\pi$
$998$	−1.64226e11	+	1.64226e11i	−0.165546	+	0.165546i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.9.g.b.37.6	yes	16
3.2	odd	2	inner	45.9.g.b.37.3	yes	16
5.3	odd	4	inner	45.9.g.b.28.6	yes	16
15.8	even	4	inner	45.9.g.b.28.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.g.b.28.3	✓	16	15.8	even	4	inner
45.9.g.b.28.6	yes	16	5.3	odd	4	inner
45.9.g.b.37.3	yes	16	3.2	odd	2	inner
45.9.g.b.37.6	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)

\(f(q)\)	\(=\)	\(q+(10.7931 + 10.7931i) q^{2} -23.0166i q^{4} +(-489.683 + 388.375i) q^{5} +(2354.79 + 2354.79i) q^{7} +(3011.46 - 3011.46i) q^{8} +O(q^{10})\)	\(q+(10.7931 + 10.7931i) q^{2} -23.0166i q^{4} +(-489.683 + 388.375i) q^{5} +(2354.79 + 2354.79i) q^{7} +(3011.46 - 3011.46i) q^{8} +(-9477.00 - 1093.43i) q^{10} -11916.9 q^{11} +(-37189.2 + 37189.2i) q^{13} +50831.1i q^{14} +59114.0 q^{16} +(-77040.9 - 77040.9i) q^{17} +172555. i q^{19} +(8939.09 + 11270.9i) q^{20} +(-128621. - 128621. i) q^{22} +(-303832. + 303832. i) q^{23} +(88954.1 - 380362. i) q^{25} -802776. q^{26} +(54199.3 - 54199.3i) q^{28} +361997. i q^{29} +432913. q^{31} +(-132910. - 132910. i) q^{32} -1.66303e6i q^{34} +(-2.06764e6 - 238558. i) q^{35} +(698415. + 698415. i) q^{37} +(-1.86241e6 + 1.86241e6i) q^{38} +(-305084. + 2.64424e6i) q^{40} +2.69766e6 q^{41} +(337059. - 337059. i) q^{43} +274287. i q^{44} -6.55859e6 q^{46} +(-1.97588e6 - 1.97588e6i) q^{47} +5.32525e6i q^{49} +(5.06539e6 - 3.14520e6i) q^{50} +(855970. + 855970. i) q^{52} +(3.56064e6 - 3.56064e6i) q^{53} +(5.83552e6 - 4.62824e6i) q^{55} +1.41827e7 q^{56} +(-3.90708e6 + 3.90708e6i) q^{58} +9.30238e6i q^{59} +1.35077e7 q^{61} +(4.67249e6 + 4.67249e6i) q^{62} -1.80022e7i q^{64} +(3.76755e6 - 3.26543e7i) q^{65} +(269805. + 269805. i) q^{67} +(-1.77322e6 + 1.77322e6i) q^{68} +(-1.97415e7 - 2.48911e7i) q^{70} +4.06937e7 q^{71} +(-718506. + 718506. i) q^{73} +1.50762e7i q^{74} +3.97165e6 q^{76} +(-2.80618e7 - 2.80618e7i) q^{77} +2.64669e7i q^{79} +(-2.89471e7 + 2.29584e7i) q^{80} +(2.91161e7 + 2.91161e7i) q^{82} +(-5.69550e7 + 5.69550e7i) q^{83} +(6.76464e7 + 7.80484e6i) q^{85} +7.27584e6 q^{86} +(-3.58874e7 + 3.58874e7i) q^{88} -1.47189e7i q^{89} -1.75145e8 q^{91} +(6.99318e6 + 6.99318e6i) q^{92} -4.26519e7i q^{94} +(-6.70163e7 - 8.44975e7i) q^{95} +(5.88484e7 + 5.88484e7i) q^{97} +(-5.74761e7 + 5.74761e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4220 q^{7}+O(q^{10}) \) 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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