LMFDB - Embedded newform 76.8.i.a.73.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(5,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 16]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.i (of order $9$, degree $6$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.4
Character	$\chi$	$=$	76.73
Dual form			76.8.i.a.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-38.1246 + 13.8762i) q^{3} +(91.2548 - 517.532i) q^{5} +(78.5860 + 136.115i) q^{7} +(-414.403 + 347.726i) q^{9} +O(q^{10})$	\(q+(-38.1246 + 13.8762i) q^{3} +(91.2548 - 517.532i) q^{5} +(78.5860 + 136.115i) q^{7} +(-414.403 + 347.726i) q^{9} +(3457.19 - 5988.03i) q^{11} +(-1157.41 - 421.262i) q^{13} +(3702.33 + 20997.0i) q^{15} +(15020.1 + 12603.3i) q^{17} +(-16413.4 + 24989.4i) q^{19} +(-4884.82 - 4098.85i) q^{21} +(-9214.34 - 52257.1i) q^{23} +(-186098. - 67734.2i) q^{25} +(55338.6 - 95849.3i) q^{27} +(-156077. + 130964. i) q^{29} +(-121919. - 211169. i) q^{31} +(-48712.8 + 276264. i) q^{33} +(77615.2 - 28249.6i) q^{35} -398660. q^{37} +49971.3 q^{39} +(-490175. + 178409. i) q^{41} +(-120563. + 683749. i) q^{43} +(142143. + 246199. i) q^{45} +(407643. - 342053. i) q^{47} +(399420. - 691816. i) q^{49} +(-747521. - 272075. i) q^{51} +(213778. + 1.21240e6i) q^{53} +(-2.78351e6 - 2.33564e6i) q^{55} +(278998. - 1.18047e6i) q^{57} +(872339. + 731979. i) q^{59} +(116759. + 662173. i) q^{61} +(-79897.0 - 29080.1i) q^{63} +(-323636. + 560553. i) q^{65} +(-1.06156e6 + 890758. i) q^{67} +(1.07642e6 + 1.86442e6i) q^{69} +(-150438. + 853175. i) q^{71} +(901700. - 328192. i) q^{73} +8.03481e6 q^{75} +1.08675e6 q^{77} +(-2.21863e6 + 807516. i) q^{79} +(-574295. + 3.25699e6i) q^{81} +(-3.65434e6 - 6.32950e6i) q^{83} +(7.89328e6 - 6.62325e6i) q^{85} +(4.13309e6 - 7.15872e6i) q^{87} +(3.83076e6 + 1.39428e6i) q^{89} +(-33616.0 - 190646. i) q^{91} +(7.57833e6 + 6.35897e6i) q^{93} +(1.14350e7 + 1.07749e7i) q^{95} +(-9.06506e6 - 7.60649e6i) q^{97} +(649521. + 3.68362e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	$ 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) $ 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−38.1246	+	13.8762i	−0.815231	+	0.296720i	−0.715783	−	0.698323i	$-0.753930\pi$
$3$	−38.1246	+	13.8762i	−0.815231	+	0.296720i	−0.0994484	+	0.995043i	$0.531708\pi$
$4$		0			0
$5$	91.2548	−	517.532i	0.326483	−	1.85158i	−0.172558	−	0.984999i	$-0.555203\pi$
$5$	91.2548	−	517.532i	0.326483	−	1.85158i	0.499041	−	0.866579i	$-0.333686\pi$
$6$		0			0
$7$	78.5860	+	136.115i	0.0865969	+	0.149990i	0.906071	−	0.423127i	$-0.139067\pi$
$7$	78.5860	+	136.115i	0.0865969	+	0.149990i	−0.819474	+	0.573117i	$0.805734\pi$
$8$		0			0
$9$	−414.403	+	347.726i	−0.189485	+	0.158997i
$10$		0			0
$11$	3457.19	−	5988.03i	0.783158	−	1.35647i	−0.146936	−	0.989146i	$-0.546941\pi$
$11$	3457.19	−	5988.03i	0.783158	−	1.35647i	0.930094	−	0.367323i	$-0.119726\pi$
$12$		0			0
$13$	−1157.41	−	421.262i	−0.146112	−	0.0531803i	0.267929	−	0.963439i	$-0.413661\pi$
$13$	−1157.41	−	421.262i	−0.146112	−	0.0531803i	−0.414041	+	0.910258i	$0.635883\pi$
$14$		0			0
$15$	3702.33	+	20997.0i	0.283241	+	1.60634i
$16$		0			0
$17$	15020.1	+	12603.3i	0.741483	+	0.622178i	0.933235	−	0.359265i	$-0.116973\pi$
$17$	15020.1	+	12603.3i	0.741483	+	0.622178i	−0.191753	+	0.981443i	$0.561417\pi$
$18$		0			0
$19$	−16413.4	+	24989.4i	−0.548987	+	0.835831i
$19$	−16413.4	+	24989.4i	−0.548987	+	0.835831i
$20$		0			0
$21$	−4884.82	−	4098.85i	−0.115102	−	0.0965817i
$22$		0			0
$23$	−9214.34	−	52257.1i	−0.157913	−	0.895567i	−0.956074	−	0.293124i	$-0.905305\pi$
$23$	−9214.34	−	52257.1i	−0.157913	−	0.895567i	0.798162	−	0.602443i	$-0.205806\pi$
$24$		0			0
$25$	−186098.	−	67734.2i	−2.38206	−	0.866998i
$26$		0			0
$27$	55338.6	−	95849.3i	0.541072	−	0.937164i
$28$		0			0
$29$	−156077.	+	130964.i	−1.18836	+	0.997149i	−0.188469	+	0.982079i	$0.560353\pi$
$29$	−156077.	+	130964.i	−1.18836	+	0.997149i	−0.999886	+	0.0150699i	$0.995203\pi$
$30$		0			0
$31$	−121919.	−	211169.i	−0.735028	−	1.27311i	−0.954711	−	0.297535i	$-0.903835\pi$
$31$	−121919.	−	211169.i	−0.735028	−	1.27311i	0.219682	−	0.975571i	$-0.429498\pi$
$32$		0			0
$33$	−48712.8	+	276264.i	−0.235963	+	1.33821i
$34$		0			0
$35$	77615.2	−	28249.6i	0.305991	−	0.111372i
$36$		0			0
$37$	−398660.			−1.29389			−0.646945	−	0.762537i	$-0.723954\pi$
$37$	−398660.			−1.29389			−0.646945	+	0.762537i	$0.723954\pi$
$38$		0			0
$39$	49971.3			0.134894
$40$		0			0
$41$	−490175.	+	178409.i	−1.11073	+	0.404271i	−0.831260	−	0.555884i	$-0.812380\pi$
$41$	−490175.	+	178409.i	−1.11073	+	0.404271i	−0.279467	+	0.960155i	$0.590158\pi$
$42$		0			0
$43$	−120563.	+	683749.i	−0.231247	+	1.31147i	0.619129	+	0.785289i	$0.287486\pi$
$43$	−120563.	+	683749.i	−0.231247	+	1.31147i	−0.850376	+	0.526176i	$0.823625\pi$
$44$		0			0
$45$	142143.	+	246199.i	0.232531	+	0.402756i
$46$		0			0
$47$	407643.	−	342053.i	0.572714	−	0.480564i	−0.309831	−	0.950792i	$-0.600273\pi$
$47$	407643.	−	342053.i	0.572714	−	0.480564i	0.882545	+	0.470228i	$0.155828\pi$
$48$		0			0
$49$	399420.	−	691816.i	0.485002	−	0.840048i
$50$		0			0
$51$	−747521.	−	272075.i	−0.789093	−	0.287206i
$52$		0			0
$53$	213778.	+	1.21240e6i	0.197241	+	1.11861i	0.909191	+	0.416380i	$0.136701\pi$
$53$	213778.	+	1.21240e6i	0.197241	+	1.11861i	−0.711949	+	0.702231i	$0.752187\pi$
$54$		0			0
$55$	−2.78351e6	−	2.33564e6i	−2.25592	−	1.89294i
$56$		0			0
$57$	278998.	−	1.18047e6i	0.199544	−	0.844291i
$58$		0			0
$59$	872339.	+	731979.i	0.552972	+	0.463999i	0.875946	−	0.482409i	$-0.160238\pi$
$59$	872339.	+	731979.i	0.552972	+	0.463999i	−0.322974	+	0.946408i	$0.604683\pi$
$60$		0			0
$61$	116759.	+	662173.i	0.0658622	+	0.373523i	0.999868	+	0.0162645i	$0.00517736\pi$
$61$	116759.	+	662173.i	0.0658622	+	0.373523i	−0.934006	+	0.357258i	$0.883712\pi$
$62$		0			0
$63$	−79897.0	−	29080.1i	−0.0402567	−	0.0146523i
$64$		0			0
$65$	−323636.	+	560553.i	−0.146170	+	0.253175i
$66$		0			0
$67$	−1.06156e6	+	890758.i	−0.431206	+	0.361825i	−0.832406	−	0.554166i	$-0.813037\pi$
$67$	−1.06156e6	+	890758.i	−0.431206	+	0.361825i	0.401201	+	0.915990i	$0.368593\pi$
$68$		0			0
$69$	1.07642e6	+	1.86442e6i	0.394468	+	0.683239i
$70$		0			0
$71$	−150438.	+	853175.i	−0.0498830	+	0.282901i	−0.999538	−	0.0303973i	$-0.990323\pi$
$71$	−150438.	+	853175.i	−0.0498830	+	0.282901i	0.949655	+	0.313298i	$0.101434\pi$
$72$		0			0
$73$	901700.	−	328192.i	0.271289	−	0.0987411i	−0.202794	−	0.979222i	$-0.565002\pi$
$73$	901700.	−	328192.i	0.271289	−	0.0987411i	0.474083	+	0.880480i	$0.342780\pi$
$74$		0			0
$75$	8.03481e6			2.19918
$76$		0			0
$77$	1.08675e6			0.271276
$78$		0			0
$79$	−2.21863e6	+	807516.i	−0.506280	+	0.184271i	−0.582516	−	0.812819i	$-0.697932\pi$
$79$	−2.21863e6	+	807516.i	−0.506280	+	0.184271i	0.0762365	+	0.997090i	$0.475710\pi$
$80$		0			0
$81$	−574295.	+	3.25699e6i	−0.120071	+	0.680956i
$82$		0			0
$83$	−3.65434e6	−	6.32950e6i	−0.701513	−	1.21506i	−0.967935	−	0.251199i	$-0.919175\pi$
$83$	−3.65434e6	−	6.32950e6i	−0.701513	−	1.21506i	0.266423	−	0.963856i	$-0.414158\pi$
$84$		0			0
$85$	7.89328e6	−	6.62325e6i	1.39409	−	1.16978i
$86$		0			0
$87$	4.13309e6	−	7.15872e6i	0.672911	−	1.16552i
$88$		0			0
$89$	3.83076e6	+	1.39428e6i	0.575997	+	0.209646i	0.613560	−	0.789648i	$-0.289737\pi$
$89$	3.83076e6	+	1.39428e6i	0.575997	+	0.209646i	−0.0375623	+	0.999294i	$0.511959\pi$
$90$		0			0
$91$	−33616.0	−	190646.i	−0.00467629	−	0.0265206i
$92$		0			0
$93$	7.57833e6	+	6.35897e6i	0.976975	+	0.819779i
$94$		0			0
$95$	1.14350e7	+	1.07749e7i	1.36837	+	1.28938i
$96$		0			0
$97$	−9.06506e6	−	7.60649e6i	−1.00849	−	0.846220i	−0.0203485	−	0.999793i	$-0.506478\pi$
$97$	−9.06506e6	−	7.60649e6i	−1.00849	−	0.846220i	−0.988137	+	0.153573i	$0.950922\pi$
$98$		0			0
$99$	649521.	+	3.68362e6i	0.0672775	+	0.381550i
$100$		0			0
$101$	5.89617e6	+	2.14603e6i	0.569437	+	0.207258i	0.610661	−	0.791892i	$-0.290904\pi$
$101$	5.89617e6	+	2.14603e6i	0.569437	+	0.207258i	−0.0412246	+	0.999150i	$0.513126\pi$
$102$		0			0
$103$	−1.31279e6	+	2.27382e6i	−0.118376	+	0.205034i	−0.919124	−	0.393967i	$-0.871102\pi$
$103$	−1.31279e6	+	2.27382e6i	−0.118376	+	0.205034i	0.800748	+	0.599001i	$0.204436\pi$
$104$		0			0
$105$	−2.56705e6	+	2.15401e6i	−0.216407	+	0.181587i
$106$		0			0
$107$	−6.68047e6	−	1.15709e7i	−0.527186	−	0.913113i	−0.999498	−	0.0316814i	$-0.989914\pi$
$107$	−6.68047e6	−	1.15709e7i	−0.527186	−	0.913113i	0.472312	−	0.881431i	$-0.343420\pi$
$108$		0			0
$109$	−3.18038e6	+	1.80368e7i	−0.235226	+	1.33404i	0.606910	+	0.794771i	$0.292409\pi$
$109$	−3.18038e6	+	1.80368e7i	−0.235226	+	1.33404i	−0.842136	+	0.539265i	$0.818702\pi$
$110$		0			0
$111$	1.51988e7	−	5.53190e6i	1.05482	−	0.383923i
$112$		0			0
$113$	−8.03904e6			−0.524119			−0.262060	−	0.965052i	$-0.584402\pi$
$113$	−8.03904e6			−0.524119			−0.262060	+	0.965052i	$0.584402\pi$
$114$		0			0
$115$	−2.78856e7			−1.70977
$116$		0			0
$117$	626117.	−	227888.i	0.0361414	−	0.0131544i
$118$		0			0
$119$	−535136.	+	3.03491e6i	−0.0291105	+	0.165094i
$120$		0			0
$121$	−1.41608e7	−	2.45272e7i	−0.726672	−	1.25863i
$122$		0			0
$123$	1.62121e7	−	1.36035e7i	0.785544	−	0.659150i
$124$		0			0
$125$	−3.15090e7	+	5.45752e7i	−1.44294	+	2.49925i
$126$		0			0
$127$	1.90707e7	+	6.94115e6i	0.826138	+	0.300690i	0.720273	−	0.693691i	$-0.244017\pi$
$127$	1.90707e7	+	6.94115e6i	0.826138	+	0.300690i	0.105865	+	0.994380i	$0.466239\pi$
$128$		0			0
$129$	−4.89142e6	−	2.77406e7i	−0.200618	−	1.13776i
$130$		0			0
$131$	−2.66256e7	−	2.23415e7i	−1.03478	−	0.868286i	−0.0433708	−	0.999059i	$-0.513810\pi$
$131$	−2.66256e7	−	2.23415e7i	−1.03478	−	0.868286i	−0.991412	+	0.130773i	$0.958254\pi$
$132$		0			0
$133$	−4.69130e6	−	270298.i	−0.172907	−	0.00996236i
$134$		0			0
$135$	−4.45551e7	−	3.73862e7i	−1.55858	−	1.30781i
$136$		0			0
$137$	−6.68840e6	−	3.79318e7i	−0.222229	−	1.26032i	−0.867913	−	0.496717i	$-0.834539\pi$
$137$	−6.68840e6	−	3.79318e7i	−0.222229	−	1.26032i	0.645684	−	0.763605i	$-0.276572\pi$
$138$		0			0
$139$	1.03078e7	+	3.75172e6i	0.325546	+	0.118489i	0.499623	−	0.866243i	$-0.333472\pi$
$139$	1.03078e7	+	3.75172e6i	0.325546	+	0.118489i	−0.174077	+	0.984732i	$0.555694\pi$
$140$		0			0
$141$	−1.07948e7	+	1.86972e7i	−0.324301	+	0.561706i
$142$		0			0
$143$	−6.52391e6	+	5.47421e6i	−0.186566	+	0.156547i
$144$		0			0
$145$	5.35354e7	+	9.27260e7i	1.45832	+	2.52589i
$146$		0			0
$147$	−5.62794e6	+	3.19176e7i	−0.146130	+	0.828743i
$148$		0			0
$149$	4.88046e7	−	1.77634e7i	1.20867	−	0.439921i	0.342429	−	0.939544i	$-0.388750\pi$
$149$	4.88046e7	−	1.77634e7i	1.20867	−	0.439921i	0.866243	+	0.499623i	$0.166528\pi$
$150$		0			0
$151$	2.86848e7			0.678004			0.339002	−	0.940786i	$-0.389911\pi$
$151$	2.86848e7			0.678004			0.339002	+	0.940786i	$0.389911\pi$
$152$		0			0
$153$	−1.06069e7			−0.239424
$154$		0			0
$155$	−1.20412e8	+	4.38266e7i	−2.59723	+	0.945315i
$156$		0			0
$157$	−8.19176e6	+	4.64578e7i	−0.168938	+	0.958098i	0.775972	+	0.630768i	$0.217260\pi$
$157$	−8.19176e6	+	4.64578e7i	−0.168938	+	0.958098i	−0.944910	+	0.327330i	$0.893851\pi$
$158$		0			0
$159$	−2.49737e7	−	4.32557e7i	−0.492711	−	0.853401i
$160$		0			0
$161$	6.38886e6	−	5.36089e6i	0.120652	−	0.101239i
$162$		0			0
$163$	1.71782e7	−	2.97536e7i	0.310686	−	0.538125i	−0.667825	−	0.744319i	$-0.732774\pi$
$163$	1.71782e7	−	2.97536e7i	0.310686	−	0.538125i	0.978511	+	0.206194i	$0.0661078\pi$
$164$		0			0
$165$	1.38530e8	+	5.04209e7i	2.40077	+	0.873809i
$166$		0			0
$167$	−6.72215e6	−	3.81232e7i	−0.111686	−	0.633405i	−0.988338	−	0.152279i	$-0.951339\pi$
$167$	−6.72215e6	−	3.81232e7i	−0.111686	−	0.633405i	0.876651	−	0.481127i	$-0.159772\pi$
$168$		0			0
$169$	−4.69060e7	−	3.93588e7i	−0.747524	−	0.627247i
$170$		0			0
$171$	−1.88767e6	−	1.60631e7i	−0.0288696	−	0.245664i
$172$		0			0
$173$	−7.23392e7	−	6.06998e7i	−1.06221	−	0.891304i	−0.0678900	−	0.997693i	$-0.521627\pi$
$173$	−7.23392e7	−	6.06998e7i	−1.06221	−	0.891304i	−0.994325	+	0.106389i	$0.966071\pi$
$174$		0			0
$175$	−5.40508e6	−	3.06537e7i	−0.0762375	−	0.432365i
$176$		0			0
$177$	−4.34147e7	−	1.58016e7i	−0.588478	−	0.214188i
$178$		0			0
$179$	3.96481e7	−	6.86725e7i	0.516698	−	0.894947i	−0.483114	−	0.875557i	$-0.660494\pi$
$179$	3.96481e7	−	6.86725e7i	0.516698	−	0.894947i	0.999812	−	0.0193893i	$-0.00617219\pi$
$180$		0			0
$181$	2.41879e6	−	2.02961e6i	0.0303196	−	0.0254412i	−0.627502	−	0.778615i	$-0.715923\pi$
$181$	2.41879e6	−	2.02961e6i	0.0303196	−	0.0254412i	0.657822	+	0.753174i	$0.271478\pi$
$182$		0			0
$183$	−1.36399e7	−	2.36249e7i	−0.164525	−	0.284965i
$184$		0			0
$185$	−3.63797e7	+	2.06319e8i	−0.422433	+	2.39574i
$186$		0			0
$187$	1.27397e8	−	4.63685e7i	1.42466	−	0.518535i
$188$		0			0
$189$	1.73954e7			0.187421
$190$		0			0
$191$	2.02112e7			0.209882			0.104941	−	0.994478i	$-0.466535\pi$
$191$	2.02112e7			0.209882			0.104941	+	0.994478i	$0.466535\pi$
$192$		0			0
$193$	1.08164e8	−	3.93684e7i	1.08301	−	0.394182i	0.261981	−	0.965073i	$-0.415624\pi$
$193$	1.08164e8	−	3.93684e7i	1.08301	−	0.394182i	0.821026	+	0.570891i	$0.193402\pi$
$194$		0			0
$195$	4.56012e6	−	2.58617e7i	0.0440408	−	0.249768i
$196$		0			0
$197$	1.08367e7	+	1.87697e7i	0.100987	+	0.174915i	0.912092	−	0.409986i	$-0.134467\pi$
$197$	1.08367e7	+	1.87697e7i	0.100987	+	0.174915i	−0.811104	+	0.584901i	$0.801133\pi$
$198$		0			0
$199$	8.50963e7	−	7.14043e7i	0.765464	−	0.642301i	−0.174079	−	0.984732i	$-0.555695\pi$
$199$	8.50963e7	−	7.14043e7i	0.765464	−	0.642301i	0.939543	+	0.342431i	$0.111250\pi$
$200$		0			0
$201$	2.81114e7	−	4.86903e7i	0.244172	−	0.422918i
$202$		0			0
$203$	−3.00917e7	−	1.09525e7i	−0.252470	−	0.0918917i
$204$		0			0
$205$	4.76015e7	+	2.69962e8i	0.385907	+	2.18858i
$206$		0			0
$207$	2.19896e7	+	1.84515e7i	0.172314	+	0.144589i
$208$		0			0
$209$	9.28930e7	+	1.84677e8i	0.703835	+	1.39927i
$210$		0			0
$211$	−8.83867e7	−	7.41653e7i	−0.647737	−	0.543516i	0.258647	−	0.965972i	$-0.416724\pi$
$211$	−8.83867e7	−	7.41653e7i	−0.647737	−	0.543516i	−0.906383	+	0.422456i	$0.861168\pi$
$212$		0			0
$213$	−6.10346e6	−	3.46145e7i	−0.0432761	−	0.245431i
$214$		0			0
$215$	3.42860e8	+	1.24791e8i	2.35278	+	0.856343i
$216$		0			0
$217$	1.91622e7	−	3.31899e7i	0.127302	−	0.220494i
$218$		0			0
$219$	−2.98229e7	+	2.50244e7i	−0.191865	+	0.160994i
$220$		0			0
$221$	−1.20750e7	−	2.09146e7i	−0.0752516	−	0.130340i
$222$		0			0
$223$	1.32985e7	−	7.54198e7i	0.0803040	−	0.455426i	−0.917968	−	0.396656i	$-0.870171\pi$
$223$	1.32985e7	−	7.54198e7i	0.0803040	−	0.455426i	0.998271	−	0.0587709i	$-0.0187182\pi$
$224$		0			0
$225$	1.00673e8	−	3.66418e7i	0.589213	−	0.214456i
$226$		0			0
$227$	5.79738e7			0.328958			0.164479	−	0.986381i	$-0.447406\pi$
$227$	5.79738e7			0.328958			0.164479	+	0.986381i	$0.447406\pi$
$228$		0			0
$229$	−2.06343e8			−1.13544			−0.567722	−	0.823220i	$-0.692175\pi$
$229$	−2.06343e8			−1.13544			−0.567722	+	0.823220i	$0.692175\pi$
$230$		0			0
$231$	−4.14319e7	+	1.50800e7i	−0.221153	+	0.0804930i
$232$		0			0
$233$	3.02344e7	−	1.71468e8i	0.156587	−	0.888049i	−0.800733	−	0.599021i	$-0.795557\pi$
$233$	3.02344e7	−	1.71468e8i	0.156587	−	0.888049i	0.957320	−	0.289029i	$-0.0933323\pi$
$234$		0			0
$235$	−1.39824e8	−	2.42182e8i	−0.702820	−	1.21732i
$236$		0			0
$237$	7.33792e7	−	6.15725e7i	0.358058	−	0.300447i
$238$		0			0
$239$	8.60382e6	−	1.49022e7i	0.0407660	−	0.0706088i	−0.844922	−	0.534889i	$-0.820354\pi$
$239$	8.60382e6	−	1.49022e7i	0.0407660	−	0.0706088i	0.885689	+	0.464280i	$0.153687\pi$
$240$		0			0
$241$	−2.70799e8	−	9.85629e7i	−1.24620	−	0.453580i	−0.367084	−	0.930188i	$-0.619644\pi$
$241$	−2.70799e8	−	9.85629e7i	−1.24620	−	0.453580i	−0.879116	+	0.476608i	$0.841866\pi$
$242$		0			0
$243$	1.87318e7	+	1.06233e8i	0.0837447	+	0.474940i
$244$		0			0
$245$	−3.21588e8	−	2.69844e8i	−1.39707	−	1.17228i
$246$		0			0
$247$	2.95241e7	−	2.20086e7i	0.124663	−	0.0929293i
$248$		0			0
$249$	2.27150e8	+	1.90601e8i	0.932427	+	0.782399i
$250$		0			0
$251$	2.62459e7	+	1.48848e8i	0.104762	+	0.594133i	0.991315	+	0.131507i	$0.0419817\pi$
$251$	2.62459e7	+	1.48848e8i	0.104762	+	0.594133i	−0.886553	+	0.462626i	$0.846907\pi$
$252$		0			0
$253$	−3.44773e8	−	1.25487e8i	−1.33848	−	0.487167i
$254$		0			0
$255$	−2.09023e8	+	3.62038e8i	−0.789410	+	1.36730i
$256$		0			0
$257$	−3.78085e8	+	3.17251e8i	−1.38939	+	1.16583i	−0.423798	+	0.905757i	$0.639303\pi$
$257$	−3.78085e8	+	3.17251e8i	−1.38939	+	1.16583i	−0.965588	+	0.260077i	$0.916252\pi$
$258$		0			0
$259$	−3.13291e7	−	5.42637e7i	−0.112047	−	0.194071i
$260$		0			0
$261$	1.91393e7	−	1.08544e8i	0.0666321	−	0.377889i
$262$		0			0
$263$	−3.62021e8	+	1.31765e8i	−1.22713	+	0.446637i	−0.872612	−	0.488414i	$-0.837576\pi$
$263$	−3.62021e8	+	1.31765e8i	−1.22713	+	0.446637i	−0.354514	+	0.935051i	$0.615354\pi$
$264$		0			0
$265$	6.46962e8			2.13559
$266$		0			0
$267$	−1.65394e8			−0.531777
$268$		0			0
$269$	4.37401e8	−	1.59201e8i	1.37008	−	0.498670i	0.450927	−	0.892561i	$-0.351094\pi$
$269$	4.37401e8	−	1.59201e8i	1.37008	−	0.498670i	0.919157	+	0.393891i	$0.128871\pi$
$270$		0			0
$271$	6.74187e7	−	3.82351e8i	0.205773	−	1.16700i	−0.690446	−	0.723384i	$-0.742586\pi$
$271$	6.74187e7	−	3.82351e8i	0.205773	−	1.16700i	0.896219	−	0.443612i	$-0.146303\pi$
$272$		0			0
$273$	3.92704e6	+	6.80184e6i	0.0116814	+	0.0202328i
$274$		0			0
$275$	−1.04897e9	+	8.80192e8i	−3.04158	+	2.55219i
$276$		0			0
$277$	−6.02060e7	+	1.04280e8i	−0.170200	+	0.294796i	−0.938490	−	0.345307i	$-0.887775\pi$
$277$	−6.02060e7	+	1.04280e8i	−0.170200	+	0.294796i	0.768290	+	0.640103i	$0.221108\pi$
$278$		0			0
$279$	1.23952e8	+	4.51150e7i	0.341696	+	0.124367i
$280$		0			0
$281$	2.52205e7	+	1.43032e8i	0.0678080	+	0.384558i	0.999759	+	0.0219730i	$0.00699479\pi$
$281$	2.52205e7	+	1.43032e8i	0.0678080	+	0.384558i	−0.931951	+	0.362585i	$0.881894\pi$
$282$		0			0
$283$	−1.91953e7	−	1.61068e7i	−0.0503434	−	0.0422431i	0.617269	−	0.786752i	$-0.288239\pi$
$283$	−1.91953e7	−	1.61068e7i	−0.0503434	−	0.0422431i	−0.667612	+	0.744509i	$0.732684\pi$
$284$		0			0
$285$	−5.85470e8	−	2.52114e8i	−1.49812	−	0.645118i
$286$		0			0
$287$	−6.28050e7	−	5.26997e7i	−0.156822	−	0.131589i
$288$		0			0
$289$	−4.49605e6	−	2.54984e7i	−0.0109569	−	0.0621398i
$290$		0			0
$291$	4.51151e8	+	1.64206e8i	1.07324	+	0.390627i
$292$		0			0
$293$	3.66198e8	−	6.34274e8i	0.850510	−	1.47313i	−0.0302379	−	0.999543i	$-0.509627\pi$
$293$	3.66198e8	−	6.34274e8i	0.850510	−	1.47313i	0.880748	−	0.473585i	$-0.157040\pi$
$294$		0			0
$295$	4.58428e8	−	3.84666e8i	1.03967	−	0.872383i
$296$		0			0
$297$	−3.82633e8	−	6.62739e8i	−0.847490	−	1.46789i
$298$		0			0
$299$	−1.13492e7	+	6.43645e7i	−0.0245536	+	0.139251i
$300$		0			0
$301$	−1.02543e8	+	3.73226e7i	−0.216732	+	0.0788841i
$302$		0			0
$303$	−2.54568e8			−0.525720
$304$		0			0
$305$	3.53351e8			0.713110
$306$		0			0
$307$	3.07592e8	−	1.11954e8i	0.606724	−	0.220829i	−0.0203454	−	0.999793i	$-0.506477\pi$
$307$	3.07592e8	−	1.11954e8i	0.606724	−	0.220829i	0.627069	+	0.778964i	$0.284254\pi$
$308$		0			0
$309$	1.84976e7	−	1.04905e8i	0.0356665	−	0.202275i
$310$		0			0
$311$	2.64596e8	+	4.58295e8i	0.498796	+	0.863940i	0.999999	−	0.00139002i	$-0.000442456\pi$
$311$	2.64596e8	+	4.58295e8i	0.498796	+	0.863940i	−0.501203	+	0.865330i	$0.667109\pi$
$312$		0			0
$313$	−8.68446e7	+	7.28713e7i	−0.160080	+	0.134323i	−0.719309	−	0.694690i	$-0.755542\pi$
$313$	−8.68446e7	+	7.28713e7i	−0.160080	+	0.134323i	0.559229	+	0.829013i	$0.311097\pi$
$314$		0			0
$315$	−2.23409e7	+	3.86955e7i	−0.0402729	+	0.0697548i
$316$		0			0
$317$	−1.92041e8	−	6.98972e7i	−0.338600	−	0.123240i	0.167124	−	0.985936i	$-0.446552\pi$
$317$	−1.92041e8	−	6.98972e7i	−0.338600	−	0.123240i	−0.505723	+	0.862696i	$0.668774\pi$
$318$		0			0
$319$	2.44630e8	+	1.38736e9i	0.421931	+	2.39289i
$320$		0			0
$321$	4.15251e8	+	3.48437e8i	0.700717	+	0.587972i
$322$		0			0
$323$	−5.61481e8	+	1.68479e8i	−0.927100	+	0.278186i
$324$		0			0
$325$	1.86858e8	+	1.56792e8i	0.301939	+	0.253357i
$326$		0			0
$327$	−1.29032e8	−	7.31778e8i	−0.204071	−	1.15734i
$328$		0			0
$329$	7.85936e7	+	2.86057e7i	0.121675	+	0.0442861i
$330$		0			0
$331$	5.70809e8	−	9.88671e8i	0.865154	−	1.49849i	−0.00174115	−	0.999998i	$-0.500554\pi$
$331$	5.70809e8	−	9.88671e8i	0.865154	−	1.49849i	0.866895	−	0.498491i	$-0.166112\pi$
$332$		0			0
$333$	1.65206e8	−	1.38624e8i	0.245172	−	0.205724i
$334$		0			0
$335$	3.64123e8	+	6.30679e8i	0.529165	+	0.916541i
$336$		0			0
$337$	1.56560e6	−	8.87896e6i	0.00222831	−	0.0126374i	−0.983673	−	0.179963i	$-0.942402\pi$
$337$	1.56560e6	−	8.87896e6i	0.00222831	−	0.0126374i	0.985902	+	0.167326i	$0.0535132\pi$
$338$		0			0
$339$	3.06485e8	−	1.11552e8i	0.427278	−	0.155517i
$340$		0			0
$341$	−1.68598e9			−2.30257
$342$		0			0
$343$	2.54993e8			0.341192
$344$		0			0
$345$	1.06313e9	−	3.86947e8i	1.39386	−	0.507323i
$346$		0			0
$347$	6.23341e7	−	3.53514e8i	0.0800890	−	0.454207i	−0.918220	−	0.396071i	$-0.870373\pi$
$347$	6.23341e7	−	3.53514e8i	0.0800890	−	0.454207i	0.998309	−	0.0581357i	$-0.0185156\pi$
$348$		0			0
$349$	5.44499e8	+	9.43100e8i	0.685659	+	1.18760i	0.973229	+	0.229837i	$0.0738192\pi$
$349$	5.44499e8	+	9.43100e8i	0.685659	+	1.18760i	−0.287570	+	0.957760i	$0.592847\pi$
$350$		0			0
$351$	−1.04427e8	+	8.76247e7i	−0.128896	+	0.108156i
$352$		0			0
$353$	−2.04976e8	+	3.55029e8i	−0.248023	+	0.429588i	−0.962977	−	0.269583i	$-0.913114\pi$
$353$	−2.04976e8	+	3.55029e8i	−0.248023	+	0.429588i	0.714954	+	0.699171i	$0.246448\pi$
$354$		0			0
$355$	4.27817e8	+	1.55713e8i	0.507527	+	0.184725i
$356$		0			0
$357$	−2.17112e7	−	1.23130e8i	−0.0252548	−	0.143227i
$358$		0			0
$359$	2.61268e8	+	2.19230e8i	0.298027	+	0.250075i	0.779523	−	0.626374i	$-0.215462\pi$
$359$	2.61268e8	+	2.19230e8i	0.298027	+	0.250075i	−0.481495	+	0.876449i	$0.659906\pi$
$360$		0			0
$361$	−3.55069e8	−	8.20325e8i	−0.397226	−	0.917721i
$362$		0			0
$363$	8.80219e8	+	7.38591e8i	0.965867	+	0.810459i
$364$		0			0
$365$	−8.75653e7	−	4.96608e8i	−0.0942556	−	0.534550i
$366$		0			0
$367$	3.91872e8	+	1.42630e8i	0.413821	+	0.150619i	0.540537	−	0.841321i	$-0.318221\pi$
$367$	3.91872e8	+	1.42630e8i	0.413821	+	0.150619i	−0.126715	+	0.991939i	$0.540443\pi$
$368$		0			0
$369$	1.41093e8	−	2.44380e8i	0.146188	−	0.253205i
$370$		0			0
$371$	−1.48225e8	+	1.24376e8i	−0.150700	+	0.126452i
$372$		0			0
$373$	−3.37367e8	−	5.84337e8i	−0.336606	−	0.583018i	0.647186	−	0.762332i	$-0.275946\pi$
$373$	−3.37367e8	−	5.84337e8i	−0.336606	−	0.583018i	−0.983792	+	0.179314i	$0.942612\pi$
$374$		0			0
$375$	4.43971e8	−	2.51788e9i	0.434755	−	2.46562i
$376$		0			0
$377$	2.35815e8	−	8.58298e7i	0.226661	−	0.0824980i
$378$		0			0
$379$	1.19643e9			1.12889			0.564444	−	0.825471i	$-0.309091\pi$
$379$	1.19643e9			1.12889			0.564444	+	0.825471i	$0.309091\pi$
$380$		0			0
$381$	−8.23378e8			−0.762714
$382$		0			0
$383$	−7.66011e8	+	2.78805e8i	−0.696690	+	0.253574i	−0.665997	−	0.745955i	$-0.731994\pi$
$383$	−7.66011e8	+	2.78805e8i	−0.696690	+	0.253574i	−0.0306928	+	0.999529i	$0.509771\pi$
$384$		0			0
$385$	9.91710e7	−	5.62427e8i	0.0885671	−	0.502289i
$386$		0			0
$387$	−1.87795e8	−	3.25271e8i	−0.164701	−	0.285270i
$388$		0			0
$389$	−1.61512e9	+	1.35525e9i	−1.39118	+	1.16734i	−0.426317	+	0.904574i	$0.640189\pi$
$389$	−1.61512e9	+	1.35525e9i	−1.39118	+	1.16734i	−0.964861	+	0.262762i	$0.915367\pi$
$390$		0			0
$391$	5.20215e8	−	9.01038e8i	0.440113	−	0.762297i
$392$		0			0
$393$	1.32510e9	+	4.82299e8i	1.10123	+	0.400813i
$394$		0			0
$395$	2.15454e8	+	1.22190e9i	0.175900	+	0.997578i
$396$		0			0
$397$	9.33743e8	+	7.83504e8i	0.748964	+	0.628455i	0.935229	−	0.354044i	$-0.115194\pi$
$397$	9.33743e8	+	7.83504e8i	0.748964	+	0.628455i	−0.186265	+	0.982500i	$0.559638\pi$
$398$		0			0
$399$	1.82605e8	−	5.47925e7i	0.143915	−	0.0431833i
$400$		0			0
$401$	−4.49037e8	−	3.76787e8i	−0.347758	−	0.291803i	0.452131	−	0.891951i	$-0.350664\pi$
$401$	−4.49037e8	−	3.76787e8i	−0.347758	−	0.291803i	−0.799889	+	0.600148i	$0.795108\pi$
$402$		0			0
$403$	5.21520e7	+	2.95769e8i	0.0396920	+	0.225105i
$404$		0			0
$405$	1.63319e9	+	5.94432e8i	1.22164	+	0.444641i
$406$		0			0
$407$	−1.37825e9	+	2.38719e9i	−1.01332	+	1.75512i
$408$		0			0
$409$	1.07807e8	−	9.04606e7i	0.0779138	−	0.0653774i	−0.602999	−	0.797742i	$-0.706028\pi$
$409$	1.07807e8	−	9.04606e7i	0.0779138	−	0.0653774i	0.680913	+	0.732365i	$0.261583\pi$
$410$		0			0
$411$	7.81342e8	+	1.35332e9i	0.555130	+	0.961514i
$412$		0			0
$413$	−3.10797e7	+	1.76262e8i	−0.0217096	+	0.123121i
$414$		0			0
$415$	−3.60919e9	+	1.31364e9i	−2.47880	+	0.902210i
$416$		0			0
$417$	−4.45039e8			−0.300554
$418$		0			0
$419$	1.27860e9			0.849155			0.424577	−	0.905392i	$-0.360423\pi$
$419$	1.27860e9			0.849155			0.424577	+	0.905392i	$0.360423\pi$
$420$		0			0
$421$	1.99703e8	−	7.26859e7i	0.130436	−	0.0474747i	−0.275978	−	0.961164i	$-0.589002\pi$
$421$	1.99703e8	−	7.26859e7i	0.130436	−	0.0474747i	0.406413	+	0.913689i	$0.366779\pi$
$422$		0			0
$423$	−4.99880e7	+	2.83496e8i	−0.0321125	+	0.182119i
$424$		0			0
$425$	−1.94153e9	−	3.36283e9i	−1.22683	−	2.12493i
$426$		0			0
$427$	−8.09561e7	+	6.79302e7i	−0.0503213	+	0.0422246i
$428$		0			0
$429$	1.72760e8	−	2.99230e8i	0.105644	−	0.182980i
$430$		0			0
$431$	−1.00979e9	−	3.67532e8i	−0.607518	−	0.221118i	0.0198990	−	0.999802i	$-0.493666\pi$
$431$	−1.00979e9	−	3.67532e8i	−0.607518	−	0.221118i	−0.627417	+	0.778684i	$0.715888\pi$
$432$		0			0
$433$	−1.81922e8	−	1.03173e9i	−0.107690	−	0.610743i	−0.990112	−	0.140283i	$-0.955199\pi$
$433$	−1.81922e8	−	1.03173e9i	−0.107690	−	0.610743i	0.882421	−	0.470461i	$-0.155912\pi$
$434$		0			0
$435$	−3.32770e9	−	2.79227e9i	−1.93835	−	1.62647i
$436$		0			0
$437$	1.45711e9	+	6.27459e8i	0.835235	+	0.359667i
$438$		0			0
$439$	−7.07788e7	−	5.93904e7i	−0.0399280	−	0.0335035i	0.622605	−	0.782536i	$-0.286074\pi$
$439$	−7.07788e7	−	5.93904e7i	−0.0399280	−	0.0335035i	−0.662533	+	0.749032i	$0.730519\pi$
$440$		0			0
$441$	7.50411e7	+	4.25579e8i	0.0416643	+	0.236290i
$442$		0			0
$443$	−3.31393e9	−	1.20617e9i	−1.81105	−	0.659167i	−0.996915	−	0.0784868i	$-0.974991\pi$
$443$	−3.31393e9	−	1.20617e9i	−1.81105	−	0.659167i	−0.814132	−	0.580680i	$-0.802787\pi$
$444$		0			0
$445$	1.07116e9	−	1.85531e9i	0.576229	−	0.998058i
$446$		0			0
$447$	−1.61417e9	+	1.35445e9i	−0.854815	+	0.717275i
$448$		0			0
$449$	−9.94296e8	−	1.72217e9i	−0.518386	−	0.897871i	−0.999772	−	0.0213622i	$-0.993200\pi$
$449$	−9.94296e8	−	1.72217e9i	−0.518386	−	0.897871i	0.481386	−	0.876509i	$-0.340134\pi$
$450$		0			0
$451$	−6.26309e8	+	3.55198e9i	−0.321492	+	1.82327i
$452$		0			0
$453$	−1.09360e9	+	3.98036e8i	−0.552730	+	0.201177i
$454$		0			0
$455$	−1.01733e8			−0.0506316
$456$		0			0
$457$	−3.69655e9			−1.81172			−0.905858	−	0.423581i	$-0.860773\pi$
$457$	−3.69655e9			−1.81172			−0.905858	+	0.423581i	$0.860773\pi$
$458$		0			0
$459$	2.03921e9	−	7.42212e8i	0.984278	−	0.358248i
$460$		0			0
$461$	−6.67039e8	+	3.78297e9i	−0.317101	+	1.79837i	0.243086	+	0.970005i	$0.421840\pi$
$461$	−6.67039e8	+	3.78297e9i	−0.317101	+	1.79837i	−0.560188	+	0.828366i	$0.689271\pi$
$462$		0			0
$463$	−1.75238e9	−	3.03522e9i	−0.820533	−	1.42120i	−0.905286	−	0.424802i	$-0.860344\pi$
$463$	−1.75238e9	−	3.03522e9i	−0.820533	−	1.42120i	0.0847533	−	0.996402i	$-0.472990\pi$
$464$		0			0
$465$	3.98253e9	−	3.34174e9i	1.83685	−	1.54130i
$466$		0			0
$467$	−9.65454e8	+	1.67222e9i	−0.438654	+	0.759772i	−0.997586	−	0.0694421i	$-0.977878\pi$
$467$	−9.65454e8	+	1.67222e9i	−0.438654	+	0.759772i	0.558932	+	0.829214i	$0.311211\pi$
$468$		0			0
$469$	−2.04670e8	−	7.44937e7i	−0.0916112	−	0.0333438i
$470$		0			0
$471$	−3.32351e8	−	1.88486e9i	−0.146563	−	0.831199i
$472$		0			0
$473$	3.67750e9	+	3.08579e9i	1.59786	+	1.34076i
$474$		0			0
$475$	4.74715e9	−	3.53873e9i	2.03238	−	1.51503i
$476$		0			0
$477$	−5.10172e8	−	4.28085e8i	−0.215230	−	0.180599i
$478$		0			0
$479$	2.47627e8	+	1.40436e9i	0.102950	+	0.583856i	0.992020	+	0.126085i	$0.0402411\pi$
$479$	2.47627e8	+	1.40436e9i	0.102950	+	0.583856i	−0.889070	+	0.457771i	$0.848648\pi$
$480$		0			0
$481$	4.61413e8	+	1.67941e8i	0.189052	+	0.0688094i
$482$		0			0
$483$	−1.69184e8	+	2.93035e8i	−0.0683194	+	0.118333i
$484$		0			0
$485$	−4.76383e9	+	3.99733e9i	−1.89610	+	1.59101i
$486$		0			0
$487$	1.55548e9	+	2.69418e9i	0.610259	+	1.05700i	0.991196	+	0.132399i	$0.0422681\pi$
$487$	1.55548e9	+	2.69418e9i	0.610259	+	1.05700i	−0.380937	+	0.924601i	$0.624399\pi$
$488$		0			0
$489$	−2.42046e8	+	1.37271e9i	−0.0936090	+	0.530883i
$490$		0			0
$491$	1.25735e9	−	4.57639e8i	0.479371	−	0.174477i	−0.0910216	−	0.995849i	$-0.529013\pi$
$491$	1.25735e9	−	4.57639e8i	0.479371	−	0.174477i	0.570393	+	0.821372i	$0.306791\pi$
$492$		0			0
$493$	−3.99488e9			−1.50155
$494$		0			0
$495$	1.96566e9			0.728434
$496$		0			0
$497$	−1.27952e8	+	4.65708e7i	−0.0467520	+	0.0170163i
$498$		0			0
$499$	−2.08804e8	+	1.18418e9i	−0.0752291	+	0.426646i	0.923811	+	0.382848i	$0.125057\pi$
$499$	−2.08804e8	+	1.18418e9i	−0.0752291	+	0.426646i	−0.999040	+	0.0437975i	$0.986054\pi$
$500$		0			0
$501$	7.85285e8	+	1.36015e9i	0.278994	+	0.483232i
$502$		0			0
$503$	3.84094e9	−	3.22293e9i	1.34570	−	1.12918i	0.365584	−	0.930778i	$-0.380869\pi$
$503$	3.84094e9	−	3.22293e9i	1.34570	−	1.12918i	0.980121	−	0.198402i	$-0.0635752\pi$
$504$		0			0
$505$	1.64869e9	−	2.85562e9i	0.569666	−	0.986690i
$506$		0			0
$507$	2.33443e9	+	8.49661e8i	0.795522	+	0.289546i
$508$		0			0
$509$	5.02164e8	+	2.84791e9i	0.168785	+	0.957226i	0.945076	+	0.326851i	$0.105988\pi$
$509$	5.02164e8	+	2.84791e9i	0.168785	+	0.957226i	−0.776291	+	0.630375i	$0.782901\pi$
$510$		0			0
$511$	1.15533e8	+	9.69436e7i	0.0383030	+	0.0321400i
$512$		0			0
$513$	1.48692e9	+	2.95610e9i	0.486269	+	0.966736i
$514$		0			0
$515$	1.05698e9	+	8.86908e8i	0.340989	+	0.286123i
$516$		0			0
$517$	−6.38925e8	−	3.62352e9i	−0.203345	−	1.15323i
$518$		0			0
$519$	3.60019e9	+	1.31036e9i	1.13042	+	0.411439i
$520$		0			0
$521$	1.91403e9	−	3.31519e9i	0.592947	−	1.02701i	−0.400886	−	0.916128i	$-0.631298\pi$
$521$	1.91403e9	−	3.31519e9i	0.592947	−	1.02701i	0.993833	−	0.110886i	$-0.0353689\pi$
$522$		0			0
$523$	1.11730e8	−	9.37529e7i	0.0341519	−	0.0286569i	−0.625552	−	0.780183i	$-0.715126\pi$
$523$	1.11730e8	−	9.37529e7i	0.0341519	−	0.0286569i	0.659704	+	0.751526i	$0.270682\pi$
$524$		0			0
$525$	6.31424e8	+	1.09366e9i	0.190442	+	0.329856i
$526$		0			0
$527$	8.30211e8	−	4.70836e9i	0.247088	−	1.40130i
$528$		0			0
$529$	5.53584e8	−	2.01488e8i	0.162588	−	0.0591772i
$530$		0			0
$531$	−6.16028e8			−0.178554
$532$		0			0
$533$	6.42489e8			0.183789
$534$		0			0
$535$	−6.59794e9	+	2.40145e9i	−1.86282	+	0.678010i
$536$		0			0
$537$	−5.58653e8	+	3.16828e9i	−0.155680	+	0.882903i
$538$		0			0
$539$	−2.76174e9	−	4.78348e9i	−0.759666	−	1.31578i
$540$		0			0
$541$	1.08722e9	−	9.12287e8i	0.295208	−	0.247709i	−0.483139	−	0.875544i	$-0.660503\pi$
$541$	1.08722e9	−	9.12287e8i	0.295208	−	0.247709i	0.778346	+	0.627835i	$0.216059\pi$
$542$		0			0
$543$	−6.40523e7	+	1.10942e8i	−0.0171686	+	0.0297369i
$544$		0			0
$545$	9.04440e9	+	3.29189e9i	2.39327	+	0.871080i
$546$		0			0
$547$	−8.32038e8	−	4.71872e9i	−0.217364	−	1.23273i	−0.876757	−	0.480934i	$-0.840298\pi$
$547$	−8.32038e8	−	4.71872e9i	−0.217364	−	1.23273i	0.659393	−	0.751799i	$-0.270813\pi$
$548$		0			0
$549$	−2.78640e8	−	2.33807e8i	−0.0718688	−	0.0603051i
$550$		0			0
$551$	−7.10956e8	−	6.04985e9i	−0.181056	−	1.54069i
$552$		0			0
$553$	−2.84269e8	−	2.38530e8i	−0.0714811	−	0.0599798i
$554$		0			0
$555$	−1.47597e9	−	8.37066e9i	−0.366482	−	2.07842i
$556$		0			0
$557$	1.29477e9	+	4.71257e8i	0.317467	+	0.115549i	0.495839	−	0.868415i	$-0.334861\pi$
$557$	1.29477e9	+	4.71257e8i	0.317467	+	0.115549i	−0.178372	+	0.983963i	$0.557083\pi$
$558$		0			0
$559$	4.27578e8	−	7.40587e8i	0.103532	−	0.179323i
$560$		0			0
$561$	−4.21352e9	+	3.53556e9i	−1.00757	+	0.845452i
$562$		0			0
$563$	2.28602e9	+	3.95950e9i	0.539884	+	0.935107i	0.998910	+	0.0466839i	$0.0148653\pi$
$563$	2.28602e9	+	3.95950e9i	0.539884	+	0.935107i	−0.459025	+	0.888423i	$0.651801\pi$
$564$		0			0
$565$	−7.33602e8	+	4.16046e9i	−0.171116	+	0.970447i
$566$		0			0
$567$	−4.88457e8	+	1.77784e8i	−0.112534	+	0.0409592i
$568$		0			0
$569$	−1.01986e9			−0.232085			−0.116042	−	0.993244i	$-0.537021\pi$
$569$	−1.01986e9			−0.232085			−0.116042	+	0.993244i	$0.537021\pi$
$570$		0			0
$571$	5.79330e9			1.30227			0.651133	−	0.758963i	$-0.274294\pi$
$571$	5.79330e9			1.30227			0.651133	+	0.758963i	$0.274294\pi$
$572$		0			0
$573$	−7.70543e8	+	2.80455e8i	−0.171102	+	0.0622761i
$574$		0			0
$575$	−1.82482e9	+	1.03491e10i	−0.400298	+	2.27020i
$576$		0			0
$577$	−2.40579e9	−	4.16695e9i	−0.521365	−	0.903031i	−0.999691	−	0.0248489i	$-0.992090\pi$
$577$	−2.40579e9	−	4.16695e9i	−0.521365	−	0.903031i	0.478326	−	0.878182i	$-0.341244\pi$
$578$		0			0
$579$	−3.57742e9	+	3.00181e9i	−0.765940	+	0.642700i
$580$		0			0
$581$	5.74360e8	−	9.94821e8i	0.121498	−	0.210440i
$582$		0			0
$583$	7.99894e9	+	2.91138e9i	1.67183	+	0.608497i
$584$		0			0
$585$	−6.08031e7	−	3.44832e8i	−0.0125568	−	0.0712134i
$586$		0			0
$587$	−3.91757e9	−	3.28723e9i	−0.799436	−	0.670806i	0.148626	−	0.988894i	$-0.452515\pi$
$587$	−3.91757e9	−	3.28723e9i	−0.799436	−	0.670806i	−0.948061	+	0.318087i	$0.896959\pi$
$588$		0			0
$589$	7.27810e9	+	4.19341e8i	1.46762	+	0.0845598i
$590$		0			0
$591$	−6.73599e8	−	5.65216e8i	−0.134229	−	0.112631i
$592$		0			0
$593$	5.31884e8	+	3.01647e9i	0.104743	+	0.594028i	0.991323	+	0.131452i	$0.0419638\pi$
$593$	5.31884e8	+	3.01647e9i	0.104743	+	0.594028i	−0.886579	+	0.462576i	$0.846925\pi$
$594$		0			0
$595$	1.52183e9	+	5.53899e8i	0.296180	+	0.107801i
$596$		0			0
$597$	−2.25344e9	+	3.90307e9i	−0.433447	+	0.750752i
$598$		0			0
$599$	6.26041e9	−	5.25311e9i	1.19017	−	0.998671i	0.190313	−	0.981723i	$-0.439050\pi$
$599$	6.26041e9	−	5.25311e9i	1.19017	−	0.998671i	0.999856	−	0.0169477i	$-0.00539487\pi$
$600$		0			0
$601$	4.27694e9	+	7.40788e9i	0.803661	+	1.39198i	0.917191	+	0.398447i	$0.130451\pi$
$601$	4.27694e9	+	7.40788e9i	0.803661	+	1.39198i	−0.113530	+	0.993535i	$0.536216\pi$
$602$		0			0
$603$	1.30176e8	−	7.38266e8i	0.0241781	−	0.137121i
$604$		0			0
$605$	−1.39858e10	+	5.09043e9i	−2.56770	+	0.934567i
$606$		0			0
$607$	−6.39006e9			−1.15970			−0.579849	−	0.814724i	$-0.696888\pi$
$607$	−6.39006e9			−1.15970			−0.579849	+	0.814724i	$0.696888\pi$
$608$		0			0
$609$	1.29921e9			0.233088
$610$		0			0
$611$	−6.15904e8	+	2.24171e8i	−0.109237	+	0.0397589i
$612$		0			0
$613$	1.02222e9	−	5.79730e9i	0.179239	−	1.01652i	−0.753897	−	0.656993i	$-0.771828\pi$
$613$	1.02222e9	−	5.79730e9i	0.179239	−	1.01652i	0.933136	−	0.359523i	$-0.117061\pi$
$614$		0			0
$615$	−5.56083e9	−	9.63165e9i	−0.964000	−	1.66970i
$616$		0			0
$617$	5.93604e9	−	4.98093e9i	1.01742	−	0.853714i	0.0281159	−	0.999605i	$-0.491049\pi$
$617$	5.93604e9	−	4.98093e9i	1.01742	−	0.853714i	0.989301	+	0.145891i	$0.0466048\pi$
$618$		0			0
$619$	−4.11708e9	+	7.13099e9i	−0.697705	+	1.20846i	0.271555	+	0.962423i	$0.412462\pi$
$619$	−4.11708e9	+	7.13099e9i	−0.697705	+	1.20846i	−0.969260	+	0.246038i	$0.920871\pi$
$620$		0			0
$621$	−5.51872e9	−	2.00865e9i	−0.924736	−	0.336576i
$622$		0			0
$623$	1.11262e8	+	6.30996e8i	0.0184347	+	0.104549i
$624$		0			0
$625$	1.35168e10	+	1.13420e10i	2.21460	+	1.85827i
$626$		0			0
$627$	−6.10413e9	−	5.75175e9i	−0.988980	−	0.931888i
$628$		0			0
$629$	−5.98791e9	−	5.02445e9i	−0.959396	−	0.805029i
$630$		0			0
$631$	−9.77763e8	−	5.54517e9i	−0.154928	−	0.878642i	−0.958851	−	0.283909i	$-0.908369\pi$
$631$	−9.77763e8	−	5.54517e9i	−0.154928	−	0.878642i	0.803923	−	0.594733i	$-0.202742\pi$
$632$		0			0
$633$	4.39884e9	+	1.60105e9i	0.689327	+	0.250895i
$634$		0			0
$635$	5.33256e9	−	9.23626e9i	0.826470	−	1.43149i
$636$		0			0
$637$	−7.53728e8	+	6.32453e8i	−0.115538	+	0.0969482i
$638$		0			0
$639$	−2.34329e8	−	4.05870e8i	−0.0355282	−	0.0615366i
$640$		0			0
$641$	1.52709e9	−	8.66057e9i	0.229014	−	1.29880i	−0.625847	−	0.779946i	$-0.715247\pi$
$641$	1.52709e9	−	8.66057e9i	0.229014	−	1.29880i	0.854861	−	0.518857i	$-0.173642\pi$
$642$		0			0
$643$	−5.50973e9	+	2.00538e9i	−0.817319	+	0.297480i	−0.716644	−	0.697439i	$-0.754323\pi$
$643$	−5.50973e9	+	2.00538e9i	−0.817319	+	0.297480i	−0.100676	+	0.994919i	$0.532100\pi$
$644$		0			0
$645$	−1.48030e10			−2.17216
$646$		0			0
$647$	−8.58535e9			−1.24621			−0.623107	−	0.782136i	$-0.714130\pi$
$647$	−8.58535e9			−1.24621			−0.623107	+	0.782136i	$0.714130\pi$
$648$		0			0
$649$	7.39896e9	−	2.69300e9i	1.06246	−	0.386705i
$650$		0			0
$651$	−2.70001e8	+	1.53125e9i	−0.0383559	+	0.217527i
$652$		0			0
$653$	−7.09509e8	−	1.22891e9i	−0.0997153	−	0.172712i	0.811851	−	0.583864i	$-0.198460\pi$
$653$	−7.09509e8	−	1.22891e9i	−0.0997153	−	0.172712i	−0.911567	+	0.411152i	$0.865127\pi$
$654$		0			0
$655$	−1.39921e10	+	1.17408e10i	−1.94554	+	1.63250i
$656$		0			0
$657$	−2.59547e8	+	4.49548e8i	−0.0357056	+	0.0618440i
$658$		0			0
$659$	5.52426e9	+	2.01067e9i	0.751926	+	0.273679i	0.689416	−	0.724366i	$-0.257867\pi$
$659$	5.52426e9	+	2.01067e9i	0.751926	+	0.273679i	0.0625100	+	0.998044i	$0.480089\pi$
$660$		0			0
$661$	1.25366e9	+	7.10984e9i	0.168839	+	0.957536i	0.945017	+	0.327021i	$0.106045\pi$
$661$	1.25366e9	+	7.10984e9i	0.168839	+	0.957536i	−0.776178	+	0.630514i	$0.782844\pi$
$662$		0			0
$663$	7.50572e8	+	6.29805e8i	0.100022	+	0.0839283i
$664$		0			0
$665$	−5.67992e8	+	2.40323e9i	−0.0748973	+	0.316898i
$666$		0			0
$667$	8.28197e9	+	6.94940e9i	1.08067	+	0.906790i
$668$		0			0
$669$	5.39540e8	+	3.05988e9i	0.0696678	+	0.395106i
$670$		0			0
$671$	4.36877e9	+	1.59010e9i	0.558253	+	0.203187i
$672$		0			0
$673$	4.16758e9	−	7.21846e9i	0.527025	−	0.912834i	−0.472479	−	0.881342i	$-0.656641\pi$
$673$	4.16758e9	−	7.21846e9i	0.527025	−	0.912834i	0.999504	−	0.0314923i	$-0.0100260\pi$
$674$		0			0
$675$	−1.67907e10	+	1.40891e10i	−2.10138	+	1.76327i
$676$		0			0
$677$	−7.74120e9	−	1.34081e10i	−0.958844	−	1.66077i	−0.725316	−	0.688416i	$-0.758307\pi$
$677$	−7.74120e9	−	1.34081e10i	−0.958844	−	1.66077i	−0.233527	−	0.972350i	$-0.575027\pi$
$678$		0			0
$679$	3.22970e8	−	1.83166e9i	0.0395930	−	0.224543i
$680$		0			0
$681$	−2.21023e9	+	8.04457e8i	−0.268177	+	0.0976085i
$682$		0			0
$683$	3.71070e9			0.445639			0.222820	−	0.974860i	$-0.428474\pi$
$683$	3.71070e9			0.445639			0.222820	+	0.974860i	$0.428474\pi$
$684$		0			0
$685$	−2.02412e10			−2.40614
$686$		0			0
$687$	7.86675e9	−	2.86326e9i	0.925650	−	0.336909i
$688$		0			0
$689$	2.63308e8	−	1.49329e9i	0.0306688	−	0.173931i
$690$		0			0
$691$	1.91357e9	+	3.31440e9i	0.220633	+	0.382148i	0.955000	−	0.296605i	$-0.0958543\pi$
$691$	1.91357e9	+	3.31440e9i	0.220633	+	0.382148i	−0.734367	+	0.678752i	$0.762521\pi$
$692$		0			0
$693$	−4.50352e8	+	3.77890e8i	−0.0514027	+	0.0431320i
$694$		0			0
$695$	2.88227e9	−	4.99224e9i	0.325677	−	0.564090i
$696$		0			0
$697$	−9.61101e9	−	3.49812e9i	−1.07511	−	0.391309i
$698$		0			0
$699$	1.22665e9	+	6.95669e9i	0.135847	+	0.770428i
$700$		0			0
$701$	2.32840e9	+	1.95376e9i	0.255296	+	0.214219i	0.761449	−	0.648225i	$-0.224488\pi$
$701$	2.32840e9	+	1.95376e9i	0.255296	+	0.214219i	−0.506153	+	0.862444i	$0.668933\pi$
$702$		0			0
$703$	6.54339e9	−	9.96229e9i	0.710329	−	1.08147i
$704$		0			0
$705$	8.69131e9	+	7.29287e9i	0.934164	+	0.783857i
$706$		0			0
$707$	1.71250e8	+	9.71206e8i	0.0182248	+	0.103358i
$708$		0			0
$709$	−6.59900e9	−	2.40184e9i	−0.695370	−	0.253094i	−0.0299374	−	0.999552i	$-0.509531\pi$
$709$	−6.59900e9	−	2.40184e9i	−0.695370	−	0.253094i	−0.665433	+	0.746458i	$0.731753\pi$
$710$		0			0
$711$	6.38615e8	−	1.10611e9i	0.0666339	−	0.115413i
$712$		0			0
$713$	−9.91170e9	+	8.31691e9i	−1.02408	+	0.859307i
$714$		0			0
$715$	2.23774e9	+	3.87588e9i	0.228949	+	0.396551i
$716$		0			0
$717$	−1.21230e8	+	6.87531e8i	−0.0122827	+	0.0696587i
$718$		0			0
$719$	−2.50580e9	+	9.12035e8i	−0.251417	+	0.0915083i	−0.464654	−	0.885492i	$-0.653821\pi$
$719$	−2.50580e9	+	9.12035e8i	−0.251417	+	0.0915083i	0.213237	+	0.977000i	$0.431599\pi$
$720$		0			0
$721$	−4.12668e8			−0.0410041
$722$		0			0
$723$	1.16918e10			1.15053
$724$		0			0
$725$	3.79165e10	−	1.38005e10i	3.69526	−	1.34496i
$726$		0			0
$727$	1.96124e9	−	1.11228e10i	0.189305	−	1.07360i	−0.730994	−	0.682384i	$-0.760943\pi$
$727$	1.96124e9	−	1.11228e10i	0.189305	−	1.07360i	0.920299	−	0.391216i	$-0.127946\pi$
$728$		0			0
$729$	−5.80472e9	−	1.00541e10i	−0.554926	−	0.961160i
$730$		0			0
$731$	−1.04284e10	+	8.75046e9i	−0.987430	+	0.828552i
$732$		0			0
$733$	−4.63387e8	+	8.02610e8i	−0.0434590	+	0.0752732i	−0.886937	−	0.461891i	$-0.847171\pi$
$733$	−4.63387e8	+	8.02610e8i	−0.0434590	+	0.0752732i	0.843478	+	0.537164i	$0.180504\pi$
$734$		0			0
$735$	1.60048e10	+	5.82528e9i	1.48677	+	0.541141i
$736$		0			0
$737$	1.66386e9	+	9.43621e9i	0.153102	+	0.868283i
$738$		0			0
$739$	−1.02868e10	−	8.63166e9i	−0.937616	−	0.786753i	0.0395528	−	0.999217i	$-0.487407\pi$
$739$	−1.02868e10	−	8.63166e9i	−0.937616	−	0.786753i	−0.977169	+	0.212464i	$0.931851\pi$
$740$		0			0
$741$	−8.20200e8	+	1.24875e9i	−0.0740553	+	0.112749i
$742$		0			0
$743$	1.38130e10	+	1.15905e10i	1.23545	+	1.03667i	0.997865	+	0.0653032i	$0.0208015\pi$
$743$	1.38130e10	+	1.15905e10i	1.23545	+	1.03667i	0.237589	+	0.971366i	$0.423643\pi$
$744$		0			0
$745$	−4.73948e9	−	2.68789e10i	−0.419936	−	2.38158i
$746$		0			0
$747$	3.71530e9	+	1.35226e9i	0.326116	+	0.118696i
$748$		0			0
$749$	1.04998e9	−	1.81862e9i	0.0913053	−	0.158145i
$750$		0			0
$751$	7.71514e8	−	6.47377e8i	0.0664667	−	0.0557722i	−0.608949	−	0.793209i	$-0.708409\pi$
$751$	7.71514e8	−	6.47377e8i	0.0664667	−	0.0557722i	0.675416	+	0.737437i	$0.263964\pi$
$752$		0			0
$753$	−3.06606e9	−	5.31056e9i	−0.261696	−	0.453271i
$754$		0			0
$755$	2.61762e9	−	1.48453e10i	0.221357	−	1.25538i
$756$		0			0
$757$	−1.55551e10	+	5.66160e9i	−1.30328	+	0.474355i	−0.898064	−	0.439866i	$-0.855026\pi$
$757$	−1.55551e10	+	5.66160e9i	−1.30328	+	0.474355i	−0.405217	+	0.914221i	$0.632804\pi$
$758$		0			0
$759$	1.48856e10			1.23572
$760$		0			0
$761$	2.02210e10			1.66325			0.831624	−	0.555340i	$-0.187412\pi$
$761$	2.02210e10			1.66325			0.831624	+	0.555340i	$0.187412\pi$
$762$		0			0
$763$	−2.70502e9	+	9.84545e8i	−0.220462	+	0.0802417i
$764$		0			0
$765$	−9.67929e8	+	5.48940e9i	−0.0781679	+	0.443312i
$766$		0			0
$767$	−7.01297e8	−	1.21468e9i	−0.0561201	−	0.0972028i
$768$		0			0
$769$	−1.95127e9	+	1.63731e9i	−0.154731	+	0.129834i	−0.716866	−	0.697211i	$-0.754424\pi$
$769$	−1.95127e9	+	1.63731e9i	−0.154731	+	0.129834i	0.562136	+	0.827045i	$0.309980\pi$
$770$		0			0
$771$	1.00121e10	−	1.73414e10i	0.786745	−	1.36268i
$772$		0			0
$773$	5.51072e9	+	2.00574e9i	0.429122	+	0.156188i	0.547546	−	0.836776i	$-0.315562\pi$
$773$	5.51072e9	+	2.00574e9i	0.429122	+	0.156188i	−0.118424	+	0.992963i	$0.537784\pi$
$774$		0			0
$775$	8.38546e9	+	4.75563e10i	0.647099	+	3.66988i
$776$		0			0
$777$	1.94739e9	+	1.63405e9i	0.148929	+	0.124966i
$778$		0			0
$779$	3.58712e9	−	1.51775e10i	0.271872	−	1.15032i
$780$		0			0
$781$	4.58875e9	+	3.85042e9i	0.344680	+	0.289220i
$782$		0			0
$783$	3.91574e9	+	2.22073e10i	0.291506	+	1.65321i
$784$		0			0
$785$	2.32958e10	+	8.47900e9i	1.71884	+	0.625606i
$786$		0			0
$787$	−8.76603e9	+	1.51832e10i	−0.641049	+	1.11033i	0.344149	+	0.938915i	$0.388167\pi$
$787$	−8.76603e9	+	1.51832e10i	−0.641049	+	1.11033i	−0.985199	+	0.171415i	$0.945166\pi$
$788$		0			0
$789$	1.19735e10	−	1.00470e10i	0.867865	−	0.728226i
$790$		0			0
$791$	−6.31757e8	−	1.09423e9i	−0.0453871	−	0.0786127i
$792$		0			0
$793$	1.43811e8	−	8.15591e8i	0.0102408	−	0.0580786i
$794$		0			0
$795$	−2.46652e10	+	8.97738e9i	−1.74100	+	0.633672i
$796$		0			0
$797$	−9.00802e9			−0.630268			−0.315134	−	0.949047i	$-0.602049\pi$
$797$	−9.00802e9			−0.630268			−0.315134	+	0.949047i	$0.602049\pi$
$798$		0			0
$799$	1.04338e10			0.723654
$800$		0			0
$801$	−2.07231e9	+	7.54259e8i	−0.142476	+	0.0518569i
$802$		0			0
$803$	1.15213e9	−	6.53403e9i	0.0785228	−	0.445325i
$804$		0			0
$805$	−2.19142e9	−	3.79565e9i	−0.148061	−	0.256449i
$806$		0			0
$807$	−1.44666e10	+	1.21390e10i	−0.968970	+	0.813063i
$808$		0			0
$809$	1.82583e9	−	3.16243e9i	0.121239	−	0.209991i	−0.799018	−	0.601307i	$-0.794647\pi$
$809$	1.82583e9	−	3.16243e9i	0.121239	−	0.209991i	0.920256	+	0.391316i	$0.127980\pi$
$810$		0			0
$811$	−2.40207e9	−	8.74283e8i	−0.158130	−	0.0575545i	0.261743	−	0.965138i	$-0.415703\pi$
$811$	−2.40207e9	−	8.74283e8i	−0.158130	−	0.0575545i	−0.419872	+	0.907583i	$0.637925\pi$
$812$		0			0
$813$	2.73527e9	+	1.55125e10i	0.178518	+	1.01243i
$814$		0			0
$815$	−1.38308e10	−	1.16054e10i	−0.894946	−	0.750949i
$816$		0			0
$817$	−1.51076e10	−	1.42355e10i	−0.969212	−	0.913261i
$818$		0			0
$819$	8.02231e7	+	6.73152e7i	0.00510277	+	0.00428173i
$820$		0			0
$821$	−2.85794e9	−	1.62082e10i	−0.180241	−	1.02220i	−0.931919	−	0.362666i	$-0.881867\pi$
$821$	−2.85794e9	−	1.62082e10i	−0.180241	−	1.02220i	0.751679	−	0.659530i	$-0.229245\pi$
$822$		0			0
$823$	3.39821e8	+	1.23685e8i	0.0212496	+	0.00773421i	0.352623	−	0.935765i	$-0.385290\pi$
$823$	3.39821e8	+	1.23685e8i	0.0212496	+	0.00773421i	−0.331373	+	0.943500i	$0.607512\pi$
$824$		0			0
$825$	2.77779e10	−	4.81127e10i	1.72231	−	2.98312i
$826$		0			0
$827$	−5.87848e9	+	4.93263e9i	−0.361407	+	0.303256i	−0.805351	−	0.592798i	$-0.798023\pi$
$827$	−5.87848e9	+	4.93263e9i	−0.361407	+	0.303256i	0.443944	+	0.896054i	$0.353579\pi$
$828$		0			0
$829$	4.81875e9	+	8.34632e9i	0.293761	+	0.508808i	0.974696	−	0.223535i	$-0.0717598\pi$
$829$	4.81875e9	+	8.34632e9i	0.293761	+	0.508808i	−0.680935	+	0.732344i	$0.738427\pi$
$830$		0			0
$831$	8.48319e8	−	4.81106e9i	0.0512809	−	0.290828i
$832$		0			0
$833$	1.47185e10	−	5.35710e9i	0.882280	−	0.321124i
$834$		0			0
$835$	−2.03434e10			−1.20926
$836$		0			0
$837$	−2.69872e10			−1.59081
$838$		0			0
$839$	−2.78004e9	+	1.01185e9i	−0.162511	+	0.0591493i	−0.421995	−	0.906598i	$-0.638670\pi$
$839$	−2.78004e9	+	1.01185e9i	−0.162511	+	0.0591493i	0.259483	+	0.965748i	$0.416448\pi$
$840$		0			0
$841$	4.21303e9	−	2.38933e10i	0.244235	−	1.38513i
$842$		0			0
$843$	−2.94627e9	−	5.10309e9i	−0.169385	−	0.293384i
$844$		0			0
$845$	−2.46498e10	+	2.06837e10i	−1.40545	+	1.17931i
$846$		0			0
$847$	2.22568e9	−	3.85499e9i	0.125855	−	0.217987i
$848$		0			0
$849$	9.55314e8	+	3.47706e8i	0.0535759	+	0.0195000i
$850$		0			0
$851$	3.67339e9	+	2.08329e10i	0.204322	+	1.15877i
$852$		0			0
$853$	1.07112e9	+	8.98774e8i	0.0590902	+	0.0495825i	0.671854	−	0.740683i	$-0.265498\pi$
$853$	1.07112e9	+	8.98774e8i	0.0590902	+	0.0495825i	−0.612764	+	0.790266i	$0.709942\pi$
$854$		0			0
$855$	−8.48541e9	−	4.88903e8i	−0.464292	−	0.0267511i
$856$		0			0
$857$	−4.34876e9	−	3.64904e9i	−0.236011	−	0.198037i	0.517110	−	0.855919i	$-0.327008\pi$
$857$	−4.34876e9	−	3.64904e9i	−0.236011	−	0.198037i	−0.753121	+	0.657882i	$0.771452\pi$
$858$		0			0
$859$	−1.72259e9	−	9.76928e9i	−0.0927268	−	0.525880i	−0.995420	−	0.0955956i	$-0.969524\pi$
$859$	−1.72259e9	−	9.76928e9i	−0.0927268	−	0.525880i	0.902693	−	0.430284i	$-0.141587\pi$
$860$		0			0
$861$	3.12569e9	+	1.13766e9i	0.166892	+	0.0607436i
$862$		0			0
$863$	1.41869e10	−	2.45725e10i	0.751365	−	1.30140i	−0.195796	−	0.980645i	$-0.562729\pi$
$863$	1.41869e10	−	2.45725e10i	0.751365	−	1.30140i	0.947161	−	0.320758i	$-0.103938\pi$
$864$		0			0
$865$	−3.80154e10	+	3.18987e10i	−1.99711	+	1.67578i
$866$		0			0
$867$	5.25231e8	+	9.09727e8i	0.0273706	+	0.0474072i
$868$		0			0
$869$	−2.83481e9	+	1.60770e10i	−0.146539	+	0.831066i
$870$		0			0
$871$	1.60391e9	−	5.83774e8i	0.0822461	−	0.0299351i
$872$		0			0
$873$	6.40157e9			0.325639
$874$		0			0
$875$	−9.90467e9			−0.499818
$876$		0			0
$877$	2.36448e10	−	8.60599e9i	1.18369	−	0.430827i	0.326184	−	0.945306i	$-0.394237\pi$
$877$	2.36448e10	−	8.60599e9i	1.18369	−	0.430827i	0.857502	+	0.514480i	$0.172015\pi$
$878$		0			0
$879$	−5.15984e9	+	2.92629e10i	−0.256257	+	1.45330i
$880$		0			0
$881$	−1.16914e10	−	2.02500e10i	−0.576036	−	0.997724i	−0.995928	−	0.0901496i	$-0.971265\pi$
$881$	−1.16914e10	−	2.02500e10i	−0.576036	−	0.997724i	0.419892	−	0.907574i	$-0.362068\pi$
$882$		0			0
$883$	1.85623e10	−	1.55756e10i	0.907336	−	0.761345i	−0.0642743	−	0.997932i	$-0.520473\pi$
$883$	1.85623e10	−	1.55756e10i	0.907336	−	0.761345i	0.971610	+	0.236587i	$0.0760288\pi$
$884$		0			0
$885$	−1.21397e10	+	2.10265e10i	−0.588715	+	1.01968i
$886$		0			0
$887$	−4.67886e9	−	1.70297e9i	−0.225117	−	0.0819357i	0.226999	−	0.973895i	$-0.427108\pi$
$887$	−4.67886e9	−	1.70297e9i	−0.225117	−	0.0819357i	−0.452116	+	0.891959i	$0.649331\pi$
$888$		0			0
$889$	5.53892e8	+	3.14128e9i	0.0264405	+	0.149951i
$890$		0			0
$891$	1.75175e10	+	1.46989e10i	0.829661	+	0.696168i
$892$		0			0
$893$	1.85688e9	+	1.58010e10i	0.0872576	+	0.742515i
$894$		0			0
$895$	−3.19221e10	−	2.67858e10i	−1.48837	−	1.24889i
$896$		0			0
$897$	−4.60452e8	−	2.61135e9i	−0.0213015	−	0.120807i
$898$		0			0
$899$	4.66844e10	+	1.69917e10i	2.14295	+	0.779971i
$900$		0			0
$901$	−1.20693e10	+	2.09046e10i	−0.549724	+	0.952149i
$902$		0			0
$903$	3.39152e9	−	2.84582e9i	0.153280	−	0.128618i
$904$		0			0
$905$	−8.29661e8	−	1.43701e9i	−0.0372075	−	0.0644453i
$906$		0			0
$907$	−5.53706e8	+	3.14022e9i	−0.0246407	+	0.139745i	−0.994646	−	0.103338i	$-0.967048\pi$
$907$	−5.53706e8	+	3.14022e9i	−0.0246407	+	0.139745i	0.970006	+	0.243083i	$0.0781587\pi$
$908$		0			0
$909$	−3.18962e9	+	1.16093e9i	−0.140853	+	0.0512663i
$910$		0			0
$911$	−1.99310e10			−0.873403			−0.436702	−	0.899606i	$-0.643853\pi$
$911$	−1.99310e10			−0.873403			−0.436702	+	0.899606i	$0.643853\pi$
$912$		0			0
$913$	−5.05350e10			−2.19758
$914$		0			0
$915$	−1.34714e10	+	4.90317e9i	−0.581349	+	0.211594i
$916$		0			0
$917$	9.48616e8	−	5.37987e9i	0.0406254	−	0.230398i
$918$		0			0
$919$	−1.35878e9	−	2.35347e9i	−0.0577490	−	0.100024i	0.835706	−	0.549178i	$-0.185059\pi$
$919$	−1.35878e9	−	2.35347e9i	−0.0577490	−	0.100024i	−0.893455	+	0.449154i	$0.851726\pi$
$920$		0			0
$921$	−1.01733e10	+	8.53643e9i	−0.429096	+	0.360054i
$922$		0			0
$923$	5.33528e8	−	9.24098e8i	0.0223332	−	0.0386823i
$924$		0			0
$925$	7.41900e10	+	2.70029e10i	3.08212	+	1.12180i
$926$		0			0
$927$	−2.46641e8	−	1.39877e9i	−0.0101692	−	0.0576723i
$928$		0			0
$929$	9.26245e9	+	7.77212e9i	0.379028	+	0.318042i	0.812321	−	0.583211i	$-0.198204\pi$
$929$	9.26245e9	+	7.77212e9i	0.379028	+	0.318042i	−0.433293	+	0.901253i	$0.642648\pi$
$930$		0			0
$931$	1.07322e10	+	2.13363e10i	0.435878	+	0.866555i
$932$		0			0
$933$	−1.64470e10	−	1.38007e10i	−0.662982	−	0.556308i
$934$		0			0
$935$	−1.23716e10	−	7.01631e10i	−0.494979	−	2.80717i
$936$		0			0
$937$	−6.51126e9	−	2.36991e9i	−0.258569	−	0.0941115i	0.209483	−	0.977812i	$-0.432822\pi$
$937$	−6.51126e9	−	2.36991e9i	−0.258569	−	0.0941115i	−0.468052	+	0.883701i	$0.655044\pi$
$938$		0			0
$939$	2.29974e9	−	3.98326e9i	0.0906460	−	0.157004i
$940$		0			0
$941$	−2.11156e10	+	1.77181e10i	−0.826113	+	0.693192i	−0.954395	−	0.298546i	$-0.903498\pi$
$941$	−2.11156e10	+	1.77181e10i	−0.826113	+	0.693192i	0.128282	+	0.991738i	$0.459054\pi$
$942$		0			0
$943$	1.38398e10	+	2.39712e10i	0.537450	+	0.930891i
$944$		0			0
$945$	1.58741e9	−	9.00266e9i	0.0611897	−	0.347024i
$946$		0			0
$947$	−1.45164e10	+	5.28354e9i	−0.555436	+	0.202162i	−0.604460	−	0.796635i	$-0.706611\pi$
$947$	−1.45164e10	+	5.28354e9i	−0.555436	+	0.202162i	0.0490239	+	0.998798i	$0.484389\pi$
$948$		0			0
$949$	−1.18189e9			−0.0448896
$950$		0			0
$951$	8.29140e9			0.312605
$952$		0			0
$953$	−5.89110e9	+	2.14418e9i	−0.220481	+	0.0802485i	−0.449899	−	0.893080i	$-0.648540\pi$
$953$	−5.89110e9	+	2.14418e9i	−0.220481	+	0.0802485i	0.229418	+	0.973328i	$0.426318\pi$
$954$		0			0
$955$	1.84437e9	−	1.04599e10i	0.0685228	−	0.388612i
$956$		0			0
$957$	−2.85778e10	−	4.94982e10i	−1.05399	−	1.82557i
$958$		0			0
$959$	4.63747e9	−	3.89130e9i	0.169792	−	0.142472i
$960$		0			0
$961$	−1.59720e10	+	2.76643e10i	−0.580534	+	1.00551i
$962$		0			0
$963$	6.79191e9	+	2.47205e9i	0.245076	+	0.0892002i
$964$		0			0
$965$	−1.05039e10	−	5.95708e10i	−0.376276	−	2.13397i
$966$		0			0
$967$	−1.82917e10	−	1.53485e10i	−0.650521	−	0.545852i	0.256708	−	0.966489i	$-0.417362\pi$
$967$	−1.82917e10	−	1.53485e10i	−0.650521	−	0.545852i	−0.907229	+	0.420637i	$0.861807\pi$
$968$		0			0
$969$	1.90684e10	−	1.42144e10i	0.673257	−	0.501875i
$970$		0			0
$971$	8.94014e9	+	7.50166e9i	0.313384	+	0.262960i	0.785889	−	0.618368i	$-0.212206\pi$
$971$	8.94014e9	+	7.50166e9i	0.313384	+	0.262960i	−0.472505	+	0.881328i	$0.656650\pi$
$972$		0			0
$973$	2.99381e8	+	1.69788e9i	0.0104191	+	0.0590896i
$974$		0			0
$975$	−9.29956e9	−	3.38476e9i	−0.321326	−	0.116953i
$976$		0			0
$977$	7.12169e8	−	1.23351e9i	0.0244316	−	0.0423168i	−0.853551	−	0.521009i	$-0.825556\pi$
$977$	7.12169e8	−	1.23351e9i	0.0244316	−	0.0423168i	0.877983	+	0.478692i	$0.158889\pi$
$978$		0			0
$979$	2.15927e10	−	1.81184e10i	0.735475	−	0.617137i
$980$		0			0
$981$	−4.95391e9	−	8.58042e9i	−0.167535	−	0.290180i
$982$		0			0
$983$	−7.03510e9	+	3.98980e10i	−0.236229	+	1.33972i	0.603781	+	0.797150i	$0.293660\pi$
$983$	−7.03510e9	+	3.98980e10i	−0.236229	+	1.33972i	−0.840010	+	0.542571i	$0.817451\pi$
$984$		0			0
$985$	1.07028e10	−	3.89552e9i	0.356839	−	0.129879i
$986$		0			0
$987$	−3.39329e9			−0.112334
$988$		0			0
$989$	3.68417e10			1.21102
$990$		0			0
$991$	4.80327e10	−	1.74825e10i	1.56776	−	0.570617i	0.595261	−	0.803533i	$-0.297049\pi$
$991$	4.80327e10	−	1.74825e10i	1.56776	−	0.570617i	0.972497	+	0.232916i	$0.0748266\pi$
$992$		0			0
$993$	−8.04287e9	+	4.56134e10i	−0.260668	+	1.47832i
$994$		0			0
$995$	−2.91885e10	−	5.05560e10i	−0.939359	−	1.62702i
$996$		0			0
$997$	−2.12721e9	+	1.78494e9i	−0.0679792	+	0.0570414i	−0.676144	−	0.736769i	$-0.736350\pi$
$997$	−2.12721e9	+	1.78494e9i	−0.0679792	+	0.0570414i	0.608165	+	0.793811i	$0.291906\pi$
$998$		0			0
$999$	−2.20613e10	+	3.82113e10i	−0.700087	+	1.21259i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.i.a.73.4	yes	72
19.6	even	9	inner	76.8.i.a.25.4	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.i.a.25.4	✓	72	19.6	even	9	inner
76.8.i.a.73.4	yes	72	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-38.1246 + 13.8762i) q^{3} +(91.2548 - 517.532i) q^{5} +(78.5860 + 136.115i) q^{7} +(-414.403 + 347.726i) q^{9} +O(q^{10})\)	\(q+(-38.1246 + 13.8762i) q^{3} +(91.2548 - 517.532i) q^{5} +(78.5860 + 136.115i) q^{7} +(-414.403 + 347.726i) q^{9} +(3457.19 - 5988.03i) q^{11} +(-1157.41 - 421.262i) q^{13} +(3702.33 + 20997.0i) q^{15} +(15020.1 + 12603.3i) q^{17} +(-16413.4 + 24989.4i) q^{19} +(-4884.82 - 4098.85i) q^{21} +(-9214.34 - 52257.1i) q^{23} +(-186098. - 67734.2i) q^{25} +(55338.6 - 95849.3i) q^{27} +(-156077. + 130964. i) q^{29} +(-121919. - 211169. i) q^{31} +(-48712.8 + 276264. i) q^{33} +(77615.2 - 28249.6i) q^{35} -398660. q^{37} +49971.3 q^{39} +(-490175. + 178409. i) q^{41} +(-120563. + 683749. i) q^{43} +(142143. + 246199. i) q^{45} +(407643. - 342053. i) q^{47} +(399420. - 691816. i) q^{49} +(-747521. - 272075. i) q^{51} +(213778. + 1.21240e6i) q^{53} +(-2.78351e6 - 2.33564e6i) q^{55} +(278998. - 1.18047e6i) q^{57} +(872339. + 731979. i) q^{59} +(116759. + 662173. i) q^{61} +(-79897.0 - 29080.1i) q^{63} +(-323636. + 560553. i) q^{65} +(-1.06156e6 + 890758. i) q^{67} +(1.07642e6 + 1.86442e6i) q^{69} +(-150438. + 853175. i) q^{71} +(901700. - 328192. i) q^{73} +8.03481e6 q^{75} +1.08675e6 q^{77} +(-2.21863e6 + 807516. i) q^{79} +(-574295. + 3.25699e6i) q^{81} +(-3.65434e6 - 6.32950e6i) q^{83} +(7.89328e6 - 6.62325e6i) q^{85} +(4.13309e6 - 7.15872e6i) q^{87} +(3.83076e6 + 1.39428e6i) q^{89} +(-33616.0 - 190646. i) q^{91} +(7.57833e6 + 6.35897e6i) q^{93} +(1.14350e7 + 1.07749e7i) q^{95} +(-9.06506e6 - 7.60649e6i) q^{97} +(649521. + 3.68362e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9	\( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) \) 72 * q - 39 * q^3 + 591 * q^7 - 1677 * q^9 - 5793 * q^11 + 22083 * q^13 - 70707 * q^15 + 29409 * q^17 + 12756 * q^19 - 40419 * q^21 - 116796 * q^23 + 26934 * q^25 + 439347 * q^27 - 148005 * q^29 + 141366 * q^31 + 1206654 * q^33 - 531651 * q^35 - 1508004 * q^37 - 2140644 * q^39 + 494157 * q^41 + 923511 * q^43 + 1790028 * q^45 - 662955 * q^47 - 4191369 * q^49 + 3753363 * q^51 - 246513 * q^53 + 2650185 * q^55 + 5831286 * q^57 - 2244432 * q^59 - 10302306 * q^61 - 12102117 * q^63 + 3985635 * q^65 + 11688288 * q^67 + 12632535 * q^69 + 10541736 * q^71 + 560730 * q^73 - 17718750 * q^75 - 22266192 * q^77 + 9424767 * q^79 + 16585437 * q^81 + 2359839 * q^83 - 20859762 * q^85 + 2348574 * q^87 - 4397130 * q^89 - 12653112 * q^91 + 52259766 * q^93 + 46248669 * q^95 - 34112766 * q^97 - 93757926 * q^99

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