LMFDB - Embedded newform 117.4.g.f.100.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (of order $3$, 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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
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} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + \cdots + 24920064 $ x^16 + 52x^14 + 1899x^12 + 33440x^10 + 424113x^8 + 2869882x^6 + 13705540x^4 + 21016320*x^2 + 24920064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			100.2
Root			$-1.84606 + 3.19747i$ of defining polynomial
Character	$\chi$	$=$	117.100
Dual form			117.4.g.f.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.84606 + 3.19747i) q^{2} +(-2.81586 - 4.87721i) q^{4} -18.7574 q^{5} +(12.0691 + 20.9044i) q^{7} -8.74396 q^{8} +O(q^{10})$	\(q+(-1.84606 + 3.19747i) q^{2} +(-2.81586 - 4.87721i) q^{4} -18.7574 q^{5} +(12.0691 + 20.9044i) q^{7} -8.74396 q^{8} +(34.6273 - 59.9763i) q^{10} +(24.6371 - 42.6726i) q^{11} +(-34.3114 - 31.9331i) q^{13} -89.1214 q^{14} +(38.6687 - 66.9762i) q^{16} +(32.6560 + 56.5619i) q^{17} +(-54.9701 - 95.2109i) q^{19} +(52.8183 + 91.4840i) q^{20} +(90.9629 + 157.552i) q^{22} +(41.6487 - 72.1377i) q^{23} +226.841 q^{25} +(165.446 - 50.7593i) q^{26} +(67.9701 - 117.728i) q^{28} +(-2.49797 + 4.32660i) q^{29} -255.810 q^{31} +(107.794 + 186.704i) q^{32} -241.140 q^{34} +(-226.386 - 392.112i) q^{35} +(-46.8178 + 81.0908i) q^{37} +405.912 q^{38} +164.014 q^{40} +(-33.9744 + 58.8453i) q^{41} +(-71.2716 - 123.446i) q^{43} -277.498 q^{44} +(153.772 + 266.341i) q^{46} -379.275 q^{47} +(-119.828 + 207.549i) q^{49} +(-418.762 + 725.317i) q^{50} +(-59.1284 + 257.263i) q^{52} -389.560 q^{53} +(-462.128 + 800.429i) q^{55} +(-105.532 - 182.787i) q^{56} +(-9.22278 - 15.9743i) q^{58} +(-66.9404 - 115.944i) q^{59} +(-310.396 - 537.621i) q^{61} +(472.241 - 817.945i) q^{62} -177.273 q^{64} +(643.595 + 598.983i) q^{65} +(-59.5010 + 103.059i) q^{67} +(183.910 - 318.541i) q^{68} +1671.69 q^{70} +(-180.693 - 312.970i) q^{71} -748.241 q^{73} +(-172.857 - 299.397i) q^{74} +(-309.576 + 536.201i) q^{76} +1189.39 q^{77} +514.165 q^{79} +(-725.326 + 1256.30i) q^{80} +(-125.437 - 217.264i) q^{82} +260.260 q^{83} +(-612.543 - 1060.96i) q^{85} +526.286 q^{86} +(-215.425 + 373.128i) q^{88} +(416.815 - 721.945i) q^{89} +(253.432 - 1102.66i) q^{91} -469.108 q^{92} +(700.163 - 1212.72i) q^{94} +(1031.10 + 1785.91i) q^{95} +(740.993 + 1283.44i) q^{97} +(-442.421 - 766.295i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) $ 16 * q - 40 * q^4 + 22 * q^7	$ 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) $ 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−1.84606	+	3.19747i	−0.652680	+	1.13048i	0.329790	+	0.944054i	$0.393022\pi$
$2$	−1.84606	+	3.19747i	−0.652680	+	1.13048i	−0.982470	+	0.186421i	$0.940311\pi$
$3$		0			0
$3$		0			0
$4$	−2.81586	−	4.87721i	−0.351983	−	0.609652i
$5$	−18.7574			−1.67772			−0.838858	−	0.544351i	$-0.816776\pi$
$5$	−18.7574			−1.67772			−0.838858	+	0.544351i	$0.816776\pi$
$6$		0			0
$7$	12.0691	+	20.9044i	0.651672	+	1.12873i	0.982717	+	0.185115i	$0.0592657\pi$
$7$	12.0691	+	20.9044i	0.651672	+	1.12873i	−0.331044	+	0.943615i	$0.607401\pi$
$8$	−8.74396			−0.386432
$9$		0			0
$10$	34.6273	−	59.9763i	1.09501	−	1.89662i
$11$	24.6371	−	42.6726i	0.675305	−	1.16966i	−0.301075	−	0.953600i	$-0.597345\pi$
$11$	24.6371	−	42.6726i	0.675305	−	1.16966i	0.976380	−	0.216062i	$-0.0693212\pi$
$12$		0			0
$13$	−34.3114	−	31.9331i	−0.732022	−	0.681281i
$13$	−34.3114	−	31.9331i	−0.732022	−	0.681281i
$14$	−89.1214			−1.70133
$15$		0			0
$16$	38.6687	−	66.9762i	0.604199	−	1.04650i
$17$	32.6560	+	56.5619i	0.465897	+	0.806958i	0.999242	−	0.0389405i	$-0.0123983\pi$
$17$	32.6560	+	56.5619i	0.465897	+	0.806958i	−0.533344	+	0.845898i	$0.679065\pi$
$18$		0			0
$19$	−54.9701	−	95.2109i	−0.663737	−	1.14963i	−0.979626	−	0.200830i	$-0.935636\pi$
$19$	−54.9701	−	95.2109i	−0.663737	−	1.14963i	0.315890	−	0.948796i	$-0.397697\pi$
$20$	52.8183	+	91.4840i	0.590527	+	1.02282i
$21$		0			0
$22$	90.9629	+	157.552i	0.881516	+	1.52683i
$23$	41.6487	−	72.1377i	0.377581	−	0.653989i	−0.613129	−	0.789983i	$-0.710089\pi$
$23$	41.6487	−	72.1377i	0.377581	−	0.653989i	0.990710	+	0.135994i	$0.0434227\pi$
$24$		0			0
$25$	226.841			1.81473
$26$	165.446	−	50.7593i	1.24795	−	0.382874i
$27$		0			0
$28$	67.9701	−	117.728i	0.458755	−	0.794586i
$29$	−2.49797	+	4.32660i	−0.0159952	+	0.0277045i	−0.873912	−	0.486084i	$-0.838425\pi$
$29$	−2.49797	+	4.32660i	−0.0159952	+	0.0277045i	0.857917	+	0.513788i	$0.171758\pi$
$30$		0			0
$31$	−255.810			−1.48209			−0.741047	−	0.671453i	$-0.765670\pi$
$31$	−255.810			−1.48209			−0.741047	+	0.671453i	$0.765670\pi$
$32$	107.794	+	186.704i	0.595481	+	1.03140i
$33$		0			0
$34$	−241.140			−1.21633
$35$	−226.386	−	392.112i	−1.09332	−	1.89369i
$36$		0			0
$37$	−46.8178	+	81.0908i	−0.208022	+	0.360304i	−0.951091	−	0.308910i	$-0.900036\pi$
$37$	−46.8178	+	81.0908i	−0.208022	+	0.360304i	0.743070	+	0.669214i	$0.233369\pi$
$38$	405.912			1.73283
$39$		0			0
$40$	164.014			0.648323
$41$	−33.9744	+	58.8453i	−0.129412	+	0.224149i	−0.923449	−	0.383721i	$-0.874642\pi$
$41$	−33.9744	+	58.8453i	−0.129412	+	0.224149i	0.794037	+	0.607870i	$0.207976\pi$
$42$		0			0
$43$	−71.2716	−	123.446i	−0.252763	−	0.437799i	0.711522	−	0.702664i	$-0.248006\pi$
$43$	−71.2716	−	123.446i	−0.252763	−	0.437799i	−0.964286	+	0.264865i	$0.914673\pi$
$44$	−277.498			−0.950782
$45$		0			0
$46$	153.772	+	266.341i	0.492879	+	0.853691i
$47$	−379.275			−1.17708			−0.588541	−	0.808467i	$-0.700298\pi$
$47$	−379.275			−1.17708			−0.588541	+	0.808467i	$0.700298\pi$
$48$		0			0
$49$	−119.828	+	207.549i	−0.349354	+	0.605099i
$50$	−418.762	+	725.317i	−1.18444	+	2.05151i
$51$		0			0
$52$	−59.1284	+	257.263i	−0.157685	+	0.686077i
$53$	−389.560			−1.00963			−0.504813	−	0.863229i	$-0.668438\pi$
$53$	−389.560			−1.00963			−0.504813	+	0.863229i	$0.668438\pi$
$54$		0			0
$55$	−462.128	+	800.429i	−1.13297	+	1.96236i
$56$	−105.532	−	182.787i	−0.251827	−	0.436177i
$57$		0			0
$58$	−9.22278	−	15.9743i	−0.0208795	−	0.0361643i
$59$	−66.9404	−	115.944i	−0.147710	−	0.255842i	0.782671	−	0.622436i	$-0.213857\pi$
$59$	−66.9404	−	115.944i	−0.147710	−	0.255842i	−0.930381	+	0.366595i	$0.880524\pi$
$60$		0			0
$61$	−310.396	−	537.621i	−0.651510	−	1.12845i	−0.982757	−	0.184904i	$-0.940803\pi$
$61$	−310.396	−	537.621i	−0.651510	−	1.12845i	0.331247	−	0.943544i	$-0.392531\pi$
$62$	472.241	−	817.945i	0.967333	−	1.67547i
$63$		0			0
$64$	−177.273			−0.346237
$65$	643.595	+	598.983i	1.22812	+	1.14300i
$66$		0			0
$67$	−59.5010	+	103.059i	−0.108496	+	0.187920i	−0.915161	−	0.403089i	$-0.867937\pi$
$67$	−59.5010	+	103.059i	−0.108496	+	0.187920i	0.806665	+	0.591008i	$0.201270\pi$
$68$	183.910	−	318.541i	0.327975	−	0.568070i
$69$		0			0
$70$	1671.69			2.85436
$71$	−180.693	−	312.970i	−0.302033	−	0.523136i	0.674564	−	0.738217i	$-0.264332\pi$
$71$	−180.693	−	312.970i	−0.302033	−	0.523136i	−0.976596	+	0.215081i	$0.930999\pi$
$72$		0			0
$73$	−748.241			−1.19966			−0.599829	−	0.800128i	$-0.704765\pi$
$73$	−748.241			−1.19966			−0.599829	+	0.800128i	$0.704765\pi$
$74$	−172.857	−	299.397i	−0.271543	−	0.470327i
$75$		0			0
$76$	−309.576	+	536.201i	−0.467247	+	0.809296i
$77$	1189.39			1.76031
$78$		0			0
$79$	514.165			0.732254			0.366127	−	0.930565i	$-0.380684\pi$
$79$	514.165			0.732254			0.366127	+	0.930565i	$0.380684\pi$
$80$	−725.326	+	1256.30i	−1.01367	+	1.75574i
$81$		0			0
$82$	−125.437	−	217.264i	−0.168930	−	0.292595i
$83$	260.260			0.344183			0.172092	−	0.985081i	$-0.444947\pi$
$83$	260.260			0.344183			0.172092	+	0.985081i	$0.444947\pi$
$84$		0			0
$85$	−612.543	−	1060.96i	−0.781643	−	1.35385i
$86$	526.286			0.659894
$87$		0			0
$88$	−215.425	+	373.128i	−0.260959	+	0.451995i
$89$	416.815	−	721.945i	0.496431	−	0.859843i	−0.503561	−	0.863960i	$-0.667977\pi$
$89$	416.815	−	721.945i	0.496431	−	0.859843i	0.999992	+	0.00411656i	$0.00131035\pi$
$90$		0			0
$91$	253.432	−	1102.66i	0.291944	−	1.27023i
$92$	−469.108			−0.531607
$93$		0			0
$94$	700.163	−	1212.72i	0.768258	−	1.33066i
$95$	1031.10	+	1785.91i	1.11356	+	1.92874i
$96$		0			0
$97$	740.993	+	1283.44i	0.775634	+	1.34344i	0.934438	+	0.356127i	$0.115903\pi$
$97$	740.993	+	1283.44i	0.775634	+	1.34344i	−0.158804	+	0.987310i	$0.550764\pi$
$98$	−442.421	−	766.295i	−0.456033	−	0.789872i
$99$		0			0
$100$	−638.753	−	1106.35i	−0.638753	−	1.10635i
$101$	−870.107	+	1507.07i	−0.857217	+	1.48474i	0.0173567	+	0.999849i	$0.494475\pi$
$101$	−870.107	+	1507.07i	−0.857217	+	1.48474i	−0.874573	+	0.484893i	$0.838858\pi$
$102$		0			0
$103$	1020.16			0.975917			0.487959	−	0.872867i	$-0.337742\pi$
$103$	1020.16			0.975917			0.487959	+	0.872867i	$0.337742\pi$
$104$	300.018	+	279.222i	0.282877	+	0.263269i
$105$		0			0
$106$	719.150	−	1245.60i	0.658962	−	1.14136i
$107$	753.980	−	1305.93i	0.681215	−	1.17990i	−0.293396	−	0.955991i	$-0.594785\pi$
$107$	753.980	−	1305.93i	0.681215	−	1.17990i	0.974610	−	0.223907i	$-0.0718813\pi$
$108$		0			0
$109$	−1074.08			−0.943840			−0.471920	−	0.881641i	$-0.656439\pi$
$109$	−1074.08			−0.943840			−0.471920	+	0.881641i	$0.656439\pi$
$110$	−1706.23	−	2955.28i	−1.47893	−	2.56159i
$111$		0			0
$112$	1866.79			1.57496
$113$	−445.669	−	771.921i	−0.371018	−	0.642621i	0.618705	−	0.785624i	$-0.287658\pi$
$113$	−445.669	−	771.921i	−0.371018	−	0.642621i	−0.989722	+	0.143002i	$0.954324\pi$
$114$		0			0
$115$	−781.223	+	1353.12i	−0.633473	+	1.09721i
$116$	28.1357			0.0225201
$117$		0			0
$118$	494.304			0.385630
$119$	−788.261	+	1365.31i	−0.607225	+	1.05174i
$120$		0			0
$121$	−548.469	−	949.976i	−0.412073	−	0.713731i
$122$	2292.03			1.70091
$123$		0			0
$124$	720.326	+	1247.64i	0.521671	+	0.903561i
$125$	−1910.28			−1.36688
$126$		0			0
$127$	3.27997	−	5.68107i	0.00229173	−	0.00396940i	−0.864877	−	0.501983i	$-0.832604\pi$
$127$	3.27997	−	5.68107i	0.00229173	−	0.00396940i	0.867169	+	0.498014i	$0.165937\pi$
$128$	−535.092	+	926.807i	−0.369499	+	0.639992i
$129$		0			0
$130$	−3103.34	+	952.114i	−2.09370	+	0.642353i
$131$	−1267.44			−0.845320			−0.422660	−	0.906288i	$-0.638903\pi$
$131$	−1267.44			−0.845320			−0.422660	+	0.906288i	$0.638903\pi$
$132$		0			0
$133$	1326.88	−	2298.23i	0.865078	−	1.49836i
$134$	−219.685	−	380.505i	−0.141626	−	0.245303i
$135$		0			0
$136$	−285.543	−	494.575i	−0.180038	−	0.311834i
$137$	675.425	+	1169.87i	0.421208	+	0.729553i	0.996058	−	0.0887053i	$-0.0282729\pi$
$137$	675.425	+	1169.87i	0.421208	+	0.729553i	−0.574850	+	0.818259i	$0.694940\pi$
$138$		0			0
$139$	495.303	+	857.890i	0.302238	+	0.523491i	0.976643	−	0.214871i	$-0.0689330\pi$
$139$	495.303	+	857.890i	0.302238	+	0.523491i	−0.674405	+	0.738362i	$0.735600\pi$
$140$	−1274.94	+	2208.27i	−0.769660	+	1.33309i
$141$		0			0
$142$	1334.28			0.788523
$143$	−2208.00	+	677.421i	−1.29121	+	0.396146i
$144$		0			0
$145$	46.8554	−	81.1560i	0.0268354	−	0.0464802i
$146$	1381.30	−	2392.48i	0.782993	−	1.35618i
$147$		0			0
$148$	527.330			0.292880
$149$	−1066.19	−	1846.70i	−0.586214	−	1.01535i	−0.994723	−	0.102598i	$-0.967285\pi$
$149$	−1066.19	−	1846.70i	−0.586214	−	1.01535i	0.408509	−	0.912754i	$-0.366049\pi$
$150$		0			0
$151$	−251.734			−0.135668			−0.0678338	−	0.997697i	$-0.521609\pi$
$151$	−251.734			−0.135668			−0.0678338	+	0.997697i	$0.521609\pi$
$152$	480.656	+	832.521i	0.256489	+	0.444252i
$153$		0			0
$154$	−2195.69	+	3803.04i	−1.14892	+	1.98999i
$155$	4798.35			2.48653
$156$		0			0
$157$	−3686.27			−1.87386			−0.936931	−	0.349515i	$-0.886346\pi$
$157$	−3686.27			−1.87386			−0.936931	+	0.349515i	$0.886346\pi$
$158$	−949.178	+	1644.03i	−0.477928	+	0.827795i
$159$		0			0
$160$	−2021.93	−	3502.09i	−0.999048	−	1.73040i
$161$	2010.66			0.984236
$162$		0			0
$163$	633.894	+	1097.94i	0.304604	+	0.527589i	0.977173	−	0.212445i	$-0.0681427\pi$
$163$	633.894	+	1097.94i	0.304604	+	0.527589i	−0.672569	+	0.740034i	$0.734809\pi$
$164$	382.668			0.182204
$165$		0			0
$166$	−480.455	+	832.172i	−0.224642	+	0.389091i
$167$	469.498	−	813.195i	0.217550	−	0.376808i	−0.736508	−	0.676429i	$-0.763527\pi$
$167$	469.498	−	813.195i	0.217550	−	0.376808i	0.954058	+	0.299621i	$0.0968601\pi$
$168$		0			0
$169$	157.551	+	2191.34i	0.0717117	+	0.997425i
$170$	4523.16			2.04065
$171$		0			0
$172$	−401.382	+	695.214i	−0.177937	+	0.308195i
$173$	−910.720	−	1577.41i	−0.400236	−	0.693229i	0.593518	−	0.804820i	$-0.297738\pi$
$173$	−910.720	−	1577.41i	−0.400236	−	0.693229i	−0.993754	+	0.111592i	$0.964405\pi$
$174$		0			0
$175$	2737.78	+	4741.97i	1.18261	+	2.04834i
$176$	−1905.37	−	3300.19i	−0.816037	−	1.41342i
$177$		0			0
$178$	1538.93	+	2665.51i	0.648021	+	1.12241i
$179$	322.870	−	559.227i	0.134818	−	0.233512i	−0.790710	−	0.612191i	$-0.790288\pi$
$179$	322.870	−	559.227i	0.134818	−	0.233512i	0.925528	+	0.378679i	$0.123622\pi$
$180$		0			0
$181$	2387.31			0.980370			0.490185	−	0.871618i	$-0.336929\pi$
$181$	2387.31			0.980370			0.490185	+	0.871618i	$0.336929\pi$
$182$	3057.88	+	2845.92i	1.24541	+	1.15909i
$183$		0			0
$184$	−364.175	+	630.769i	−0.145909	+	0.252722i
$185$	878.182	−	1521.06i	0.349001	−	0.604488i
$186$		0			0
$187$	3218.19			1.25849
$188$	1067.98	+	1849.80i	0.414313	+	0.717610i
$189$		0			0
$190$	−7613.86			−2.90720
$191$	563.178	+	975.452i	0.213351	+	0.369535i	0.952761	−	0.303720i	$-0.0982288\pi$
$191$	563.178	+	975.452i	0.213351	+	0.369535i	−0.739410	+	0.673255i	$0.764895\pi$
$192$		0			0
$193$	1629.32	−	2822.06i	0.607672	−	1.05252i	−0.383951	−	0.923353i	$-0.625437\pi$
$193$	1629.32	−	2822.06i	0.607672	−	1.05252i	0.991623	−	0.129165i	$-0.0412297\pi$
$194$	−5471.67			−2.02496
$195$		0			0
$196$	1349.68			0.491866
$197$	89.7471	−	155.447i	0.0324580	−	0.0562188i	−0.849340	−	0.527846i	$-0.823000\pi$
$197$	89.7471	−	155.447i	0.0324580	−	0.0562188i	0.881798	+	0.471627i	$0.156333\pi$
$198$		0			0
$199$	−1476.71	−	2557.73i	−0.526036	−	0.911121i	−0.999540	−	0.0303291i	$-0.990344\pi$
$199$	−1476.71	−	2557.73i	−0.526036	−	0.911121i	0.473504	−	0.880792i	$-0.342989\pi$
$200$	−1983.49			−0.701270
$201$		0			0
$202$	−3212.54	−	5564.28i	−1.11898	−	1.93812i
$203$	−120.593			−0.0416945
$204$		0			0
$205$	637.272	−	1103.79i	0.217117	−	0.376058i
$206$	−1883.28	+	3261.93i	−0.636962	+	1.10325i
$207$		0			0
$208$	−3465.54	+	1063.24i	−1.15525	+	0.354434i
$209$	−5417.20			−1.79290
$210$		0			0
$211$	−1564.20	+	2709.28i	−0.510352	+	0.883956i	0.489576	+	0.871961i	$0.337152\pi$
$211$	−1564.20	+	2709.28i	−0.510352	+	0.883956i	−0.999928	+	0.0119952i	$0.996182\pi$
$212$	1096.95	+	1899.97i	0.355370	+	0.615520i
$213$		0			0
$214$	2783.78	+	4821.65i	0.889231	+	1.54019i
$215$	1336.87	+	2315.53i	0.424065	+	0.734502i
$216$		0			0
$217$	−3087.41	−	5347.56i	−0.965840	−	1.67288i
$218$	1982.82	−	3434.35i	0.616026	−	1.06699i
$219$		0			0
$220$	5205.15			1.59514
$221$	685.723	−	2983.53i	0.208718	−	0.908118i
$222$		0			0
$223$	1877.06	−	3251.16i	0.563665	−	0.976296i	−0.433508	−	0.901150i	$-0.642724\pi$
$223$	1877.06	−	3251.16i	0.563665	−	0.976296i	0.997173	−	0.0751459i	$-0.0239423\pi$
$224$	−2601.95	+	4506.72i	−0.776118	+	1.34428i
$225$		0			0
$226$	3290.92			0.968623
$227$	−774.758	−	1341.92i	−0.226531	−	0.392363i	0.730247	−	0.683183i	$-0.239405\pi$
$227$	−774.758	−	1341.92i	−0.226531	−	0.392363i	−0.956778	+	0.290821i	$0.906072\pi$
$228$		0			0
$229$	1364.84			0.393847			0.196924	−	0.980419i	$-0.436905\pi$
$229$	1364.84			0.393847			0.196924	+	0.980419i	$0.436905\pi$
$230$	−2884.37	−	4995.87i	−0.826910	−	1.43225i
$231$		0			0
$232$	21.8421	−	37.8317i	0.00618106	−	0.0107059i
$233$	−1665.69			−0.468339			−0.234170	−	0.972196i	$-0.575237\pi$
$233$	−1665.69			−0.468339			−0.234170	+	0.972196i	$0.575237\pi$
$234$		0			0
$235$	7114.22			1.97481
$236$	−376.990	+	652.966i	−0.103983	+	0.180104i
$237$		0			0
$238$	−2910.35	−	5040.88i	−0.792647	−	1.37291i
$239$	−5950.06			−1.61037			−0.805183	−	0.593026i	$-0.797933\pi$
$239$	−5950.06			−1.61037			−0.805183	+	0.593026i	$0.797933\pi$
$240$		0			0
$241$	853.503	+	1478.31i	0.228128	+	0.395130i	0.957253	−	0.289250i	$-0.0934060\pi$
$241$	853.503	+	1478.31i	0.228128	+	0.395130i	−0.729125	+	0.684381i	$0.760073\pi$
$242$	4050.02			1.07581
$243$		0			0
$244$	−1748.06	+	3027.73i	−0.458640	+	0.794388i
$245$	2247.67	−	3893.09i	0.586117	−	1.01518i
$246$		0			0
$247$	−1154.28	+	5022.19i	−0.297349	+	1.29374i
$248$	2236.80			0.572729
$249$		0			0
$250$	3526.49	−	6108.05i	0.892138	−	1.54523i
$251$	153.195	+	265.341i	0.0385242	+	0.0667259i	0.884645	−	0.466266i	$-0.154401\pi$
$251$	153.195	+	265.341i	0.0385242	+	0.0667259i	−0.846120	+	0.532992i	$0.821068\pi$
$252$		0			0
$253$	−2052.20	−	3554.52i	−0.509964	−	0.883284i
$254$	12.1100	+	20.9752i	0.00299154	+	0.00518149i
$255$		0			0
$256$	−2684.72	−	4650.06i	−0.655448	−	1.13527i
$257$	1778.08	−	3079.72i	0.431570	−	0.747502i	−0.565439	−	0.824790i	$-0.691293\pi$
$257$	1778.08	−	3079.72i	0.431570	−	0.747502i	0.997009	+	0.0772889i	$0.0246264\pi$
$258$		0			0
$259$	−2260.20			−0.542248
$260$	1109.10	−	4825.60i	0.264551	−	1.15104i
$261$		0			0
$262$	2339.77	−	4052.60i	0.551723	−	0.955613i
$263$	−3560.68	+	6167.27i	−0.834831	+	1.44597i	0.0593362	+	0.998238i	$0.481102\pi$
$263$	−3560.68	+	6167.27i	−0.834831	+	1.44597i	−0.894168	+	0.447732i	$0.852232\pi$
$264$		0			0
$265$	7307.14			1.69386
$266$	4899.01	+	8485.33i	1.12924	+	1.95590i
$267$		0			0
$268$	670.186			0.152754
$269$	3209.53	+	5559.07i	0.727467	+	1.26001i	0.957951	+	0.286933i	$0.0926359\pi$
$269$	3209.53	+	5559.07i	0.727467	+	1.26001i	−0.230484	+	0.973076i	$0.574031\pi$
$270$		0			0
$271$	−4315.66	+	7474.95i	−0.967372	+	1.67554i	−0.264270	+	0.964449i	$0.585131\pi$
$271$	−4315.66	+	7474.95i	−0.967372	+	1.67554i	−0.703102	+	0.711089i	$0.748202\pi$
$272$	5051.07			1.12598
$273$		0			0
$274$	−4987.50			−1.09966
$275$	5588.70	−	9679.91i	1.22550	−	2.12262i
$276$		0			0
$277$	−363.737	−	630.011i	−0.0788983	−	0.136656i	0.823877	−	0.566769i	$-0.191807\pi$
$277$	−363.737	−	630.011i	−0.0788983	−	0.136656i	−0.902775	+	0.430113i	$0.858474\pi$
$278$	−3657.43			−0.789059
$279$		0			0
$280$	1979.51	+	3428.61i	0.422494	+	0.731782i
$281$	−5588.39			−1.18639			−0.593194	−	0.805059i	$-0.702133\pi$
$281$	−5588.39			−1.18639			−0.593194	+	0.805059i	$0.702133\pi$
$282$		0			0
$283$	716.378	−	1240.80i	0.150474	−	0.260629i	−0.780928	−	0.624622i	$-0.785253\pi$
$283$	716.378	−	1240.80i	0.150474	−	0.260629i	0.931402	+	0.363992i	$0.118587\pi$
$284$	−1017.61	+	1762.56i	−0.212621	+	0.368270i
$285$		0			0
$286$	1910.07	−	8310.58i	0.394912	−	1.71823i
$287$	−1640.17			−0.337338
$288$		0			0
$289$	323.666	−	560.606i	0.0658795	−	0.114107i
$290$	172.996	+	299.637i	0.0350298	+	0.0606735i
$291$		0			0
$292$	2106.94	+	3649.33i	0.422259	+	0.731373i
$293$	4137.25	+	7165.93i	0.824918	+	1.42880i	0.901982	+	0.431773i	$0.142112\pi$
$293$	4137.25	+	7165.93i	0.824918	+	1.42880i	−0.0770645	+	0.997026i	$0.524555\pi$
$294$		0			0
$295$	1255.63	+	2174.82i	0.247816	+	0.429229i
$296$	409.373	−	709.055i	0.0803862	−	0.139233i
$297$		0			0
$298$	7873.01			1.53044
$299$	−3732.61	+	1145.17i	−0.721948	+	0.221495i
$300$		0			0
$301$	1720.38	−	2979.78i	0.329438	−	0.570603i
$302$	464.715	−	804.910i	0.0885475	−	0.153369i
$303$		0			0
$304$	−8502.49			−1.60412
$305$	5822.23	+	10084.4i	1.09305	+	1.89322i
$306$		0			0
$307$	628.477			0.116837			0.0584187	−	0.998292i	$-0.481394\pi$
$307$	628.477			0.116837			0.0584187	+	0.998292i	$0.481394\pi$
$308$	−3349.16	−	5800.92i	−0.619598	−	1.07318i
$309$		0			0
$310$	−8858.03	+	15342.6i	−1.62291	+	2.81096i
$311$	−85.8693			−0.0156566			−0.00782830	−	0.999969i	$-0.502492\pi$
$311$	−85.8693			−0.0156566			−0.00782830	+	0.999969i	$0.502492\pi$
$312$		0			0
$313$	−2279.49			−0.411643			−0.205821	−	0.978590i	$-0.565987\pi$
$313$	−2279.49			−0.411643			−0.205821	+	0.978590i	$0.565987\pi$
$314$	6805.07	−	11786.7i	1.22303	−	2.11835i
$315$		0			0
$316$	−1447.82	−	2507.69i	−0.257741	−	0.446420i
$317$	6576.19			1.16516			0.582580	−	0.812774i	$-0.302043\pi$
$317$	6576.19			1.16516			0.582580	+	0.812774i	$0.302043\pi$
$318$		0			0
$319$	123.085	+	213.190i	0.0216033	+	0.0374179i
$320$	3325.19			0.580887
$321$		0			0
$322$	−3711.79	+	6429.01i	−0.642391	+	1.11265i
$323$	3590.21	−	6218.42i	0.618466	−	1.07121i
$324$		0			0
$325$	−7783.25	−	7243.75i	−1.32842	−	1.23634i
$326$	−4680.82			−0.795235
$327$		0			0
$328$	297.071	−	514.541i	0.0500091	−	0.0866183i
$329$	−4577.52	−	7928.50i	−0.767072	−	1.32861i
$330$		0			0
$331$	−1246.68	−	2159.31i	−0.207020	−	0.358568i	0.743755	−	0.668453i	$-0.233043\pi$
$331$	−1246.68	−	2159.31i	−0.207020	−	0.358568i	−0.950774	+	0.309884i	$0.899710\pi$
$332$	−732.855	−	1269.34i	−0.121147	−	0.209832i
$333$		0			0
$334$	1733.44	+	3002.41i	0.283981	+	0.491870i
$335$	1116.09	−	1933.12i	0.182025	−	0.315276i
$336$		0			0
$337$	−2089.60			−0.337767			−0.168884	−	0.985636i	$-0.554016\pi$
$337$	−2089.60			−0.337767			−0.168884	+	0.985636i	$0.554016\pi$
$338$	−7297.59	−	3541.58i	−1.17437	−	0.569931i
$339$		0			0
$340$	−3449.67	+	5975.01i	−0.550250	+	0.953060i
$341$	−6302.42	+	10916.1i	−1.00086	+	1.73355i
$342$		0			0
$343$	2494.53			0.392687
$344$	623.196	+	1079.41i	0.0976759	+	0.169180i
$345$		0			0
$346$	6724.97			1.04490
$347$	3234.22	+	5601.83i	0.500351	+	0.866634i	1.00000		0.000405575i	$0.000129099\pi$
$347$	3234.22	+	5601.83i	0.500351	+	0.866634i	−0.499649	+	0.866228i	$0.666538\pi$
$348$		0			0
$349$	−2623.47	+	4543.98i	−0.402382	+	0.696945i	−0.994013	−	0.109263i	$-0.965151\pi$
$349$	−2623.47	+	4543.98i	−0.402382	+	0.696945i	0.591631	+	0.806209i	$0.298484\pi$
$350$	−20216.4			−3.08746
$351$		0			0
$352$	10622.9			1.60853
$353$	−5134.93	+	8893.96i	−0.774235	+	1.34101i	0.160989	+	0.986956i	$0.448532\pi$
$353$	−5134.93	+	8893.96i	−0.774235	+	1.34101i	−0.935224	+	0.354058i	$0.884802\pi$
$354$		0			0
$355$	3389.34	+	5870.51i	0.506725	+	0.877674i
$356$	−4694.78			−0.698940
$357$		0			0
$358$	1192.07	+	2064.73i	0.175986	+	0.304817i
$359$	3200.22			0.470477			0.235238	−	0.971938i	$-0.424413\pi$
$359$	3200.22			0.470477			0.235238	+	0.971938i	$0.424413\pi$
$360$		0			0
$361$	−2613.91	+	4527.43i	−0.381092	+	0.660071i
$362$	−4407.10	+	7633.33i	−0.639868	+	1.10828i
$363$		0			0
$364$	−6091.56	+	1868.91i	−0.877155	+	0.269114i
$365$	14035.1			2.01268
$366$		0			0
$367$	−11.4790	+	19.8823i	−0.00163270	+	0.00282792i	−0.866841	−	0.498585i	$-0.833853\pi$
$367$	−11.4790	+	19.8823i	−0.00163270	+	0.00282792i	0.865208	+	0.501413i	$0.167186\pi$
$368$	−3221.01	−	5578.95i	−0.456268	−	0.790279i
$369$		0			0
$370$	3242.35	+	5615.91i	0.455572	+	0.789074i
$371$	−4701.65	−	8143.50i	−0.657945	−	1.13959i
$372$		0			0
$373$	−3097.45	−	5364.95i	−0.429973	−	0.744736i	0.566897	−	0.823789i	$-0.308144\pi$
$373$	−3097.45	−	5364.95i	−0.429973	−	0.744736i	−0.996870	+	0.0790529i	$0.974810\pi$
$374$	−5940.97	+	10290.1i	−0.821392	+	1.42269i
$375$		0			0
$376$	3316.36			0.454863
$377$	223.871	−	68.6842i	0.0305834	−	0.00938306i
$378$		0			0
$379$	−3477.62	+	6023.42i	−0.471328	+	0.816364i	−0.999462	−	0.0327968i	$-0.989559\pi$
$379$	−3477.62	+	6023.42i	−0.471328	+	0.816364i	0.528134	+	0.849161i	$0.322892\pi$
$380$	5806.85	−	10057.8i	0.783908	−	1.35777i
$381$		0			0
$382$	−4158.64			−0.557001
$383$	−610.530	−	1057.47i	−0.0814534	−	0.141081i	0.822421	−	0.568879i	$-0.192623\pi$
$383$	−610.530	−	1057.47i	−0.0814534	−	0.141081i	−0.903874	+	0.427798i	$0.859290\pi$
$384$		0			0
$385$	−22309.9			−2.95330
$386$	6015.62	+	10419.4i	0.793231	+	1.37392i
$387$		0			0
$388$	4173.07	−	7227.96i	0.546019	−	0.945733i
$389$	−6318.58			−0.823560			−0.411780	−	0.911283i	$-0.635093\pi$
$389$	−6318.58			−0.823560			−0.411780	+	0.911283i	$0.635093\pi$
$390$		0			0
$391$	5440.33			0.703655
$392$	1047.78	−	1814.80i	0.135002	−	0.233830i
$393$		0			0
$394$	331.357	+	573.927i	0.0423693	+	0.0733858i
$395$	−9644.41			−1.22851
$396$		0			0
$397$	−1542.87	−	2672.33i	−0.195049	−	0.337835i	0.751867	−	0.659314i	$-0.229153\pi$
$397$	−1542.87	−	2672.33i	−0.195049	−	0.337835i	−0.946917	+	0.321479i	$0.895820\pi$
$398$	10904.4			1.37333
$399$		0			0
$400$	8771.66	−	15193.0i	1.09646	−	1.89912i
$401$	5390.00	−	9335.75i	0.671231	−	1.16261i	−0.306324	−	0.951927i	$-0.599099\pi$
$401$	5390.00	−	9335.75i	0.671231	−	1.16261i	0.977555	−	0.210679i	$-0.0675674\pi$
$402$		0			0
$403$	8777.23	+	8168.83i	1.08492	+	1.00972i
$404$	9800.40			1.20690
$405$		0			0
$406$	222.622	−	385.593i	0.0272132	−	0.0471346i
$407$	2306.91	+	3995.68i	0.280956	+	0.486630i
$408$		0			0
$409$	−3816.63	−	6610.59i	−0.461418	−	0.799200i	0.537614	−	0.843191i	$-0.319326\pi$
$409$	−3816.63	−	6610.59i	−0.461418	−	0.799200i	−0.999032	+	0.0439913i	$0.985993\pi$
$410$	2352.88	+	4075.31i	0.283416	+	0.490891i
$411$		0			0
$412$	−2872.63	−	4975.54i	−0.343506	−	0.594970i
$413$	1615.83	−	2798.69i	0.192517	−	0.333450i
$414$		0			0
$415$	−4881.81			−0.577442
$416$	2263.49	−	9848.27i	0.266771	−	1.16070i
$417$		0			0
$418$	10000.5	−	17321.3i	1.17019	−	2.02683i
$419$	−1754.35	+	3038.61i	−0.204547	+	0.354287i	−0.949988	−	0.312285i	$-0.898906\pi$
$419$	−1754.35	+	3038.61i	−0.204547	+	0.354287i	0.745441	+	0.666572i	$0.232239\pi$
$420$		0			0
$421$	12477.7			1.44448			0.722241	−	0.691642i	$-0.243112\pi$
$421$	12477.7			1.44448			0.722241	+	0.691642i	$0.243112\pi$
$422$	−5775.23	−	10003.0i	−0.666193	−	1.15388i
$423$		0			0
$424$	3406.29			0.390152
$425$	7407.74	+	12830.6i	0.845478	+	1.46441i
$426$		0			0
$427$	7492.42	−	12977.3i	0.849142	−	1.47076i
$428$	−8492.41			−0.959103
$429$		0			0
$430$	−9871.78			−1.10711
$431$	8872.74	−	15368.0i	0.991612	−	1.71752i	0.383873	−	0.923386i	$-0.374590\pi$
$431$	8872.74	−	15368.0i	0.991612	−	1.71752i	0.607739	−	0.794137i	$-0.292077\pi$
$432$		0			0
$433$	−4848.08	−	8397.13i	−0.538069	−	0.931964i	−0.999008	−	0.0445316i	$-0.985820\pi$
$433$	−4848.08	−	8397.13i	−0.538069	−	0.931964i	0.460939	−	0.887432i	$-0.347513\pi$
$434$	22798.2			2.52154
$435$		0			0
$436$	3024.47	+	5238.54i	0.332215	+	0.575414i
$437$	−9157.73			−1.00246
$438$		0			0
$439$	−6699.03	+	11603.1i	−0.728308	+	1.26147i	0.229290	+	0.973358i	$0.426360\pi$
$439$	−6699.03	+	11603.1i	−0.728308	+	1.26147i	−0.957598	+	0.288108i	$0.906974\pi$
$440$	4040.83	−	6998.92i	0.437816	−	0.758319i
$441$		0			0
$442$	8273.85	+	7700.35i	0.890378	+	0.828661i
$443$	13630.2			1.46183			0.730913	−	0.682471i	$-0.239095\pi$
$443$	13630.2			1.46183			0.730913	+	0.682471i	$0.239095\pi$
$444$		0			0
$445$	−7818.38	+	13541.8i	−0.832870	+	1.44257i
$446$	6930.32	+	12003.7i	0.735785	+	1.27442i
$447$		0			0
$448$	−2139.54	−	3705.79i	−0.225633	−	0.390808i
$449$	−690.894	−	1196.66i	−0.0726176	−	0.125777i	0.827430	−	0.561569i	$-0.189802\pi$
$449$	−690.894	−	1196.66i	−0.0726176	−	0.125777i	−0.900048	+	0.435791i	$0.856469\pi$
$450$		0			0
$451$	1674.06	+	2899.55i	0.174786	+	0.302737i
$452$	−2509.88	+	4347.24i	−0.261183	+	0.452383i
$453$		0			0
$454$	5720.99			0.591409
$455$	−4753.74	+	20683.2i	−0.489799	+	2.13108i
$456$		0			0
$457$	4130.80	−	7154.75i	0.422824	−	0.732353i	−0.573390	−	0.819282i	$-0.694372\pi$
$457$	4130.80	−	7154.75i	0.422824	−	0.732353i	0.996214	+	0.0869295i	$0.0277055\pi$
$458$	−2519.57	+	4364.03i	−0.257056	+	0.445235i
$459$		0			0
$460$	8799.26			0.891886
$461$	−8437.12	−	14613.5i	−0.852399	−	1.47640i	−0.879037	−	0.476753i	$-0.841814\pi$
$461$	−8437.12	−	14613.5i	−0.852399	−	1.47640i	0.0266383	−	0.999645i	$-0.491520\pi$
$462$		0			0
$463$	4466.93			0.448371			0.224186	−	0.974546i	$-0.428028\pi$
$463$	4466.93			0.448371			0.224186	+	0.974546i	$0.428028\pi$
$464$	193.186	+	334.609i	0.0193286	+	0.0334781i
$465$		0			0
$466$	3074.96	−	5325.99i	0.305676	−	0.529446i
$467$	−2731.37			−0.270649			−0.135324	−	0.990801i	$-0.543208\pi$
$467$	−2731.37			−0.270649			−0.135324	+	0.990801i	$0.543208\pi$
$468$		0			0
$469$	−2872.50			−0.282814
$470$	−13133.3	+	22747.5i	−1.28892	+	2.23247i
$471$		0			0
$472$	585.324	+	1013.81i	0.0570800	+	0.0988654i
$473$	−7023.69			−0.682769
$474$		0			0
$475$	−12469.5	−	21597.8i	−1.20450	−	2.08626i
$476$	8878.53			0.854930
$477$		0			0
$478$	10984.2	−	19025.1i	1.05105	−	1.82048i
$479$	−2815.45	+	4876.51i	−0.268562	+	0.465163i	−0.968491	−	0.249049i	$-0.919882\pi$
$479$	−2815.45	+	4876.51i	−0.268562	+	0.465163i	0.699929	+	0.714213i	$0.253215\pi$
$480$		0			0
$481$	4195.87	−	1287.30i	0.397745	−	0.122029i
$482$	−6302.46			−0.595580
$483$		0			0
$484$	−3088.82	+	5350.00i	−0.290085	+	0.502442i
$485$	−13899.1	−	24074.0i	−1.30129	−	2.25391i
$486$		0			0
$487$	6455.24	+	11180.8i	0.600647	+	1.04035i	0.992723	+	0.120418i	$0.0384236\pi$
$487$	6455.24	+	11180.8i	0.600647	+	1.04035i	−0.392076	+	0.919933i	$0.628243\pi$
$488$	2714.09	+	4700.94i	0.251764	+	0.436069i
$489$		0			0
$490$	8298.67	+	14373.7i	0.765094	+	1.32518i
$491$	−1557.33	+	2697.38i	−0.143139	+	0.247925i	−0.928677	−	0.370889i	$-0.879053\pi$
$491$	−1557.33	+	2697.38i	−0.143139	+	0.247925i	0.785538	+	0.618814i	$0.212386\pi$
$492$		0			0
$493$	−326.295			−0.0298085
$494$	−13927.4	−	12962.0i	−1.26847	−	1.18054i
$495$		0			0
$496$	−9891.87	+	17133.2i	−0.895480	+	1.55102i
$497$	4361.62	−	7554.55i	0.393653	−	0.681827i
$498$		0			0
$499$	−11637.7			−1.04404			−0.522018	−	0.852934i	$-0.674821\pi$
$499$	−11637.7			−1.04404			−0.522018	+	0.852934i	$0.674821\pi$
$500$	5379.08	+	9316.84i	0.481120	+	0.833324i
$501$		0			0
$502$	−1131.23			−0.100576
$503$	−1416.47	−	2453.39i	−0.125561	−	0.217478i	0.796391	−	0.604782i	$-0.206740\pi$
$503$	−1416.47	−	2453.39i	−0.125561	−	0.217478i	−0.921952	+	0.387304i	$0.873406\pi$
$504$		0			0
$505$	16321.0	−	28268.7i	1.43817	−	2.49098i
$506$	15153.9			1.33137
$507$		0			0
$508$	−36.9437			−0.00322660
$509$	3849.97	−	6668.35i	0.335260	−	0.580687i	−0.648275	−	0.761406i	$-0.724509\pi$
$509$	3849.97	−	6668.35i	0.335260	−	0.580687i	0.983535	+	0.180720i	$0.0578427\pi$
$510$		0			0
$511$	−9030.63	−	15641.5i	−0.781784	−	1.35409i
$512$	11263.1			0.972193
$513$		0			0
$514$	6564.88	+	11370.7i	0.563355	+	0.975759i
$515$	−19135.6			−1.63731
$516$		0			0
$517$	−9344.21	+	16184.6i	−0.794889	+	1.37679i
$518$	4172.47	−	7226.92i	0.353914	−	0.612998i
$519$		0			0
$520$	−5627.57	−	5237.49i	−0.474587	−	0.441690i
$521$	−14688.2			−1.23513			−0.617565	−	0.786520i	$-0.711881\pi$
$521$	−14688.2			−1.23513			−0.617565	+	0.786520i	$0.711881\pi$
$522$		0			0
$523$	4447.74	−	7703.70i	0.371866	−	0.644091i	−0.617987	−	0.786189i	$-0.712051\pi$
$523$	4447.74	−	7703.70i	0.371866	−	0.644091i	0.989853	+	0.142098i	$0.0453848\pi$
$524$	3568.94	+	6181.58i	0.297538	+	0.515351i
$525$		0			0
$526$	−13146.4	−	22770.3i	−1.08976	−	1.88751i
$527$	−8353.76	−	14469.1i	−0.690503	−	1.19599i
$528$		0			0
$529$	2614.27	+	4528.05i	0.214866	+	0.372158i
$530$	−13489.4	+	23364.3i	−1.10555	+	1.91487i
$531$		0			0
$532$	−14945.3			−1.21797
$533$	3044.83	−	934.161i	0.247441	−	0.0759156i
$534$		0			0
$535$	−14142.7	+	24495.9i	−1.14288	+	1.97953i
$536$	520.274	−	901.142i	0.0419262	−	0.0726183i
$537$		0			0
$538$	−23699.9			−1.89921
$539$	5904.44	+	10226.8i	0.471841	+	0.817252i
$540$		0			0
$541$	20667.7			1.64246			0.821232	−	0.570594i	$-0.193287\pi$
$541$	20667.7			1.64246			0.821232	+	0.570594i	$0.193287\pi$
$542$	−15933.9	−	27598.4i	−1.26277	−	2.18718i
$543$		0			0
$544$	−7040.23	+	12194.0i	−0.554866	+	0.961057i
$545$	20147.1			1.58350
$546$		0			0
$547$	−903.226			−0.0706018			−0.0353009	−	0.999377i	$-0.511239\pi$
$547$	−903.226			−0.0706018			−0.0353009	+	0.999377i	$0.511239\pi$
$548$	3803.81	−	6588.39i	0.296516	−	0.513580i
$549$		0			0
$550$	20634.1	+	35739.4i	1.59971	+	2.77078i
$551$	549.253			0.0424664
$552$		0			0
$553$	6205.53	+	10748.3i	0.477190	+	0.826517i
$554$	2685.92			0.205981
$555$		0			0
$556$	2789.41	−	4831.40i	0.212765	−	0.368520i
$557$	−9634.49	+	16687.4i	−0.732902	+	1.26942i	0.222735	+	0.974879i	$0.428501\pi$
$557$	−9634.49	+	16687.4i	−0.732902	+	1.26942i	−0.955638	+	0.294545i	$0.904832\pi$
$558$		0			0
$559$	−1496.59	+	6511.54i	−0.113236	+	0.492681i
$560$	−35016.3			−2.64233
$561$		0			0
$562$	10316.5	−	17868.7i	0.774332	−	1.34118i
$563$	11314.2	+	19596.7i	0.846953	+	1.46697i	0.883914	+	0.467649i	$0.154899\pi$
$563$	11314.2	+	19596.7i	0.846953	+	1.46697i	−0.0369607	+	0.999317i	$0.511768\pi$
$564$		0			0
$565$	8359.60	+	14479.3i	0.622462	+	1.07814i
$566$	2644.95	+	4581.19i	0.196423	+	0.340215i
$567$		0			0
$568$	1579.97	+	2736.59i	0.116715	+	0.202157i
$569$	12841.3	−	22241.8i	0.946109	−	1.63871i	0.192592	−	0.981279i	$-0.438311\pi$
$569$	12841.3	−	22241.8i	0.946109	−	1.63871i	0.753516	−	0.657429i	$-0.228356\pi$
$570$		0			0
$571$	301.979			0.0221321			0.0110660	−	0.999939i	$-0.496477\pi$
$571$	301.979			0.0221321			0.0110660	+	0.999939i	$0.496477\pi$
$572$	9521.36	+	8861.38i	0.695993	+	0.647750i
$573$		0			0
$574$	3027.84	−	5244.38i	0.220174	−	0.381352i
$575$	9447.64	−	16363.8i	0.685207	−	1.18681i
$576$		0			0
$577$	−17642.4			−1.27290			−0.636449	−	0.771319i	$-0.719598\pi$
$577$	−17642.4			−1.27290			−0.636449	+	0.771319i	$0.719598\pi$
$578$	1195.01	+	2069.82i	0.0859965	+	0.148950i
$579$		0			0
$580$	−527.753			−0.0377824
$581$	3141.11	+	5440.57i	0.224295	+	0.388490i
$582$		0			0
$583$	−9597.60	+	16623.5i	−0.681805	+	1.18092i
$584$	6542.59			0.463586
$585$		0			0
$586$	−30550.4			−2.15363
$587$	−11410.1	+	19762.8i	−0.802288	+	1.38960i	0.115818	+	0.993270i	$0.463051\pi$
$587$	−11410.1	+	19762.8i	−0.802288	+	1.38960i	−0.918107	+	0.396334i	$0.870282\pi$
$588$		0			0
$589$	14061.9	+	24355.9i	0.983720	+	1.70385i
$590$	−9271.87			−0.646977
$591$		0			0
$592$	3620.77	+	6271.36i	0.251373	+	0.435391i
$593$	13688.7			0.947940			0.473970	−	0.880541i	$-0.342821\pi$
$593$	13688.7			0.947940			0.473970	+	0.880541i	$0.342821\pi$
$594$		0			0
$595$	14785.7	−	25609.7i	1.01875	−	1.76453i
$596$	−6004.50	+	10400.1i	−0.412674	+	0.714772i
$597$		0			0
$598$	3228.96	−	14049.0i	0.220806	−	0.960710i
$599$	20276.7			1.38311			0.691556	−	0.722323i	$-0.256925\pi$
$599$	20276.7			1.38311			0.691556	+	0.722323i	$0.256925\pi$
$600$		0			0
$601$	−1798.39	+	3114.90i	−0.122060	+	0.211414i	−0.920580	−	0.390555i	$-0.872283\pi$
$601$	−1798.39	+	3114.90i	−0.122060	+	0.211414i	0.798520	+	0.601968i	$0.205617\pi$
$602$	6351.82	+	11001.7i	0.430035	+	0.744842i
$603$		0			0
$604$	708.847	+	1227.76i	0.0477526	+	0.0827099i
$605$	10287.9	+	17819.1i	0.691341	+	1.19744i
$606$		0			0
$607$	−2778.19	−	4811.97i	−0.185772	−	0.321766i	0.758065	−	0.652179i	$-0.226145\pi$
$607$	−2778.19	−	4811.97i	−0.185772	−	0.321766i	−0.943836	+	0.330413i	$0.892812\pi$
$608$	11850.8	−	20526.3i	0.790486	−	1.36916i
$609$		0			0
$610$	−42992.7			−2.85364
$611$	13013.5	+	12111.4i	0.861650	+	0.801924i
$612$		0			0
$613$	−2804.40	+	4857.36i	−0.184777	+	0.320044i	−0.943501	−	0.331368i	$-0.892490\pi$
$613$	−2804.40	+	4857.36i	−0.184777	+	0.320044i	0.758724	+	0.651412i	$0.225823\pi$
$614$	−1160.21	+	2009.54i	−0.0762575	+	0.132082i
$615$		0			0
$616$	−10400.0			−0.680240
$617$	−11562.7	−	20027.2i	−0.754454	−	1.30675i	−0.945645	−	0.325200i	$-0.894568\pi$
$617$	−11562.7	−	20027.2i	−0.754454	−	1.30675i	0.191191	−	0.981553i	$-0.438765\pi$
$618$		0			0
$619$	11536.0			0.749065			0.374533	−	0.927214i	$-0.377803\pi$
$619$	11536.0			0.749065			0.374533	+	0.927214i	$0.377803\pi$
$620$	−13511.5	−	23402.6i	−0.875216	−	1.51592i
$621$		0			0
$622$	158.520	−	274.564i	0.0102187	−	0.0176994i
$623$	20122.4			1.29404
$624$		0			0
$625$	7476.78			0.478514
$626$	4208.06	−	7288.58i	0.268671	−	0.465352i
$627$		0			0
$628$	10380.0	+	17978.7i	0.659567	+	1.14240i
$629$	−6115.54			−0.387667
$630$		0			0
$631$	−11840.6	−	20508.5i	−0.747016	−	1.29387i	−0.949247	−	0.314532i	$-0.898152\pi$
$631$	−11840.6	−	20508.5i	−0.747016	−	1.29387i	0.202231	−	0.979338i	$-0.435181\pi$
$632$	−4495.84			−0.282967
$633$		0			0
$634$	−12140.0	+	21027.1i	−0.760476	+	1.31718i
$635$	−61.5238	+	106.562i	−0.00384488	+	0.00665952i
$636$		0			0
$637$	10739.2	−	3294.81i	0.667977	−	0.204937i
$638$	−908.888			−0.0564001
$639$		0			0
$640$	10037.0	−	17384.5i	0.619915	−	1.07372i
$641$	−7110.75	−	12316.2i	−0.438156	−	0.758908i	0.559392	−	0.828904i	$-0.311035\pi$
$641$	−7110.75	−	12316.2i	−0.438156	−	0.758908i	−0.997547	+	0.0699955i	$0.977702\pi$
$642$		0			0
$643$	−8011.21	−	13875.8i	−0.491340	−	0.851025i	0.508611	−	0.860997i	$-0.330159\pi$
$643$	−8011.21	−	13875.8i	−0.491340	−	0.851025i	−0.999950	+	0.00997134i	$0.996826\pi$
$644$	−5661.73	−	9806.40i	−0.346434	−	0.600041i
$645$		0			0
$646$	13255.5	+	22959.1i	0.807321	+	1.39832i
$647$	12769.9	−	22118.1i	0.775945	−	1.34398i	−0.158318	−	0.987388i	$-0.550607\pi$
$647$	12769.9	−	22118.1i	0.775945	−	1.34398i	0.934262	−	0.356587i	$-0.116060\pi$
$648$		0			0
$649$	−6596.86			−0.398998
$650$	37530.0	−	11514.3i	2.26469	−	0.694812i
$651$		0			0
$652$	3569.91	−	6183.27i	0.214430	−	0.371404i
$653$	−5603.36	+	9705.30i	−0.335798	+	0.581620i	−0.983638	−	0.180157i	$-0.942339\pi$
$653$	−5603.36	+	9705.30i	−0.335798	+	0.581620i	0.647839	+	0.761777i	$0.275673\pi$
$654$		0			0
$655$	23773.9			1.41821
$656$	2627.49	+	4550.95i	0.156382	+	0.270861i
$657$		0			0
$658$	33801.5			2.00261
$659$	1380.45	+	2391.00i	0.0816002	+	0.141336i	0.903937	−	0.427665i	$-0.140664\pi$
$659$	1380.45	+	2391.00i	0.0816002	+	0.141336i	−0.822337	+	0.569000i	$0.807330\pi$
$660$		0			0
$661$	14934.2	−	25866.8i	0.878781	−	1.52209i	0.0261006	−	0.999659i	$-0.491691\pi$
$661$	14934.2	−	25866.8i	0.878781	−	1.52209i	0.852680	−	0.522433i	$-0.174976\pi$
$662$	9205.74			0.540470
$663$		0			0
$664$	−2275.70			−0.133004
$665$	−24888.9	+	43108.9i	−1.45135	+	2.51382i
$666$		0			0
$667$	208.074	+	360.395i	0.0120790	+	0.0209214i
$668$	−5288.17			−0.306296
$669$		0			0
$670$	4120.72	+	7137.29i	0.237608	+	0.411549i
$671$	−30589.0			−1.75987
$672$		0			0
$673$	1119.32	−	1938.72i	0.0641110	−	0.111044i	−0.832188	−	0.554493i	$-0.812912\pi$
$673$	1119.32	−	1938.72i	0.0641110	−	0.111044i	0.896299	+	0.443450i	$0.146245\pi$
$674$	3857.52	−	6681.41i	0.220454	−	0.381837i
$675$		0			0
$676$	10244.0	−	6938.93i	0.582841	−	0.394796i
$677$	−27574.0			−1.56537			−0.782685	−	0.622418i	$-0.786151\pi$
$677$	−27574.0			−1.56537			−0.782685	+	0.622418i	$0.786151\pi$
$678$		0			0
$679$	−17886.3	+	30980.0i	−1.01092	+	1.75096i
$680$	5356.06	+	9276.96i	0.302052	+	0.523169i
$681$		0			0
$682$	−23269.2	−	40303.5i	−1.30649	−	2.26291i
$683$	−931.299	−	1613.06i	−0.0521745	−	0.0903689i	0.838759	−	0.544503i	$-0.183282\pi$
$683$	−931.299	−	1613.06i	−0.0521745	−	0.0903689i	−0.890933	+	0.454135i	$0.849949\pi$
$684$		0			0
$685$	−12669.2	−	21943.8i	−0.706667	−	1.22398i
$686$	−4605.04	+	7976.16i	−0.256299	+	0.443923i
$687$		0			0
$688$	−11023.9			−0.610877
$689$	13366.4	+	12439.9i	0.739067	+	0.687839i
$690$		0			0
$691$	2999.81	−	5195.82i	0.165149	−	0.286047i	−0.771559	−	0.636158i	$-0.780523\pi$
$691$	2999.81	−	5195.82i	0.165149	−	0.286047i	0.936708	+	0.350111i	$0.113856\pi$
$692$	−5128.92	+	8883.56i	−0.281752	+	0.488009i
$693$		0			0
$694$	−23882.2			−1.30628
$695$	−9290.62	−	16091.8i	−0.507069	−	0.878270i
$696$		0			0
$697$	−4437.87			−0.241171
$698$	−9686.16	−	16776.9i	−0.525253	−	0.909765i
$699$		0			0
$700$	15418.4	−	26705.5i	0.832516	−	1.44196i
$701$	13737.7			0.740177			0.370088	−	0.928997i	$-0.379327\pi$
$701$	13737.7			0.740177			0.370088	+	0.928997i	$0.379327\pi$
$702$		0			0
$703$	10294.3			0.552286
$704$	−4367.49	+	7564.72i	−0.233816	+	0.404980i
$705$		0			0
$706$	−18958.8	−	32837.5i	−1.01066	−	1.75051i
$707$	−42005.8			−2.23450
$708$		0			0
$709$	778.277	+	1348.01i	0.0412254	+	0.0714044i	0.885902	−	0.463873i	$-0.153541\pi$
$709$	778.277	+	1348.01i	0.0412254	+	0.0714044i	−0.844676	+	0.535277i	$0.820207\pi$
$710$	−25027.7			−1.32292
$711$		0			0
$712$	−3644.62	+	6312.66i	−0.191837	+	0.332271i
$713$	−10654.2	+	18453.6i	−0.559610	+	0.969273i
$714$		0			0
$715$	41416.5	−	12706.7i	2.16628	−	0.664620i
$716$	−3636.63			−0.189814
$717$		0			0
$718$	−5907.79	+	10232.6i	−0.307071	+	0.531862i
$719$	7870.12	+	13631.4i	0.408214	+	0.707047i	0.994690	−	0.102919i	$-0.0328184\pi$
$719$	7870.12	+	13631.4i	0.408214	+	0.707047i	−0.586476	+	0.809967i	$0.699485\pi$
$720$		0			0
$721$	12312.5	+	21325.8i	0.635978	+	1.10155i
$722$	−9650.87	−	16715.8i	−0.497463	−	0.861631i
$723$		0			0
$724$	−6722.32	−	11643.4i	−0.345073	−	0.597684i
$725$	−566.642	+	981.452i	−0.0290270	+	0.0502762i
$726$		0			0
$727$	17533.2			0.894459			0.447230	−	0.894419i	$-0.352411\pi$
$727$	17533.2			0.894459			0.447230	+	0.894419i	$0.352411\pi$
$728$	−2216.00	+	9641.66i	−0.112817	+	0.490857i
$729$		0			0
$730$	−25909.6	+	44876.7i	−1.31364	+	2.27529i
$731$	4654.90	−	8062.52i	0.235523	−	0.407939i
$732$		0			0
$733$	32766.4			1.65110			0.825549	−	0.564330i	$-0.190865\pi$
$733$	32766.4			1.65110			0.825549	+	0.564330i	$0.190865\pi$
$734$	−42.3819	−	73.4077i	−0.00213126	−	0.00369145i
$735$		0			0
$736$	17957.9			0.899369
$737$	2931.86	+	5078.13i	0.146535	+	0.253806i
$738$		0			0
$739$	15841.6	−	27438.5i	0.788557	−	1.36582i	−0.138294	−	0.990391i	$-0.544162\pi$
$739$	15841.6	−	27438.5i	0.788557	−	1.36582i	0.926851	−	0.375429i	$-0.122505\pi$
$740$	−9891.35			−0.491369
$741$		0			0
$742$	34718.1			1.71771
$743$	9878.90	−	17110.8i	0.487781	−	0.844862i	−0.512120	−	0.858914i	$-0.671140\pi$
$743$	9878.90	−	17110.8i	0.487781	−	0.844862i	0.999901	+	0.0140519i	$0.00447299\pi$
$744$		0			0
$745$	19999.0	+	34639.3i	0.983500	+	1.70347i
$746$	22872.3			1.12254
$747$		0			0
$748$	−9061.99	−	15695.8i	−0.442967	−	0.767241i
$749$	36399.6			1.77572
$750$		0			0
$751$	−85.4386	+	147.984i	−0.00415140	+	0.00719043i	−0.868094	−	0.496400i	$-0.834655\pi$
$751$	−85.4386	+	147.984i	−0.00415140	+	0.00719043i	0.863942	+	0.503591i	$0.167988\pi$
$752$	−14666.1	+	25402.4i	−0.711192	+	1.23182i
$753$		0			0
$754$	−193.663	+	842.614i	−0.00935384	+	0.0406979i
$755$	4721.88			0.227612
$756$		0			0
$757$	−16221.0	+	28095.6i	−0.778815	+	1.34895i	0.153810	+	0.988100i	$0.450846\pi$
$757$	−16221.0	+	28095.6i	−0.778815	+	1.34895i	−0.932625	+	0.360847i	$0.882488\pi$
$758$	−12839.8	−	22239.2i	−0.615253	−	1.06565i
$759$		0			0
$760$	−9015.87	−	15615.9i	−0.430316	−	0.745329i
$761$	−2297.48	−	3979.35i	−0.109439	−	0.189555i	0.806104	−	0.591774i	$-0.201572\pi$
$761$	−2297.48	−	3979.35i	−0.109439	−	0.189555i	−0.915543	+	0.402219i	$0.868239\pi$
$762$		0			0
$763$	−12963.3	−	22453.0i	−0.615075	−	1.06534i
$764$	3171.66	−	5493.48i	0.150192	−	0.260140i
$765$		0			0
$766$	4508.30			0.212652
$767$	−1405.64	+	6115.83i	−0.0661730	+	0.287914i
$768$		0			0
$769$	10606.9	−	18371.6i	0.497391	−	0.861506i	−0.502604	−	0.864516i	$-0.667625\pi$
$769$	10606.9	−	18371.6i	0.497391	−	0.861506i	0.999995	+	0.00301003i	$0.000958124\pi$
$770$	41185.5	−	71335.3i	1.92756	−	3.33863i
$771$		0			0
$772$	−18351.7			−0.855560
$773$	1408.30	+	2439.24i	0.0655278	+	0.113497i	0.896928	−	0.442177i	$-0.145794\pi$
$773$	1408.30	+	2439.24i	0.0655278	+	0.113497i	−0.831400	+	0.555674i	$0.812460\pi$
$774$		0			0
$775$	−58028.3			−2.68960
$776$	−6479.22	−	11222.3i	−0.299730	−	0.519147i
$777$		0			0
$778$	11664.5	−	20203.4i	0.537521	−	0.931014i
$779$	7470.29			0.343583
$780$		0			0
$781$	−17807.0			−0.815857
$782$	−10043.2	+	17395.3i	−0.459262	+	0.795465i
$783$		0			0
$784$	9267.23	+	16051.3i	0.422159	+	0.731201i
$785$	69144.9			3.14381
$786$		0			0
$787$	2156.07	+	3734.41i	0.0976562	+	0.169146i	0.910714	−	0.413037i	$-0.135532\pi$
$787$	2156.07	+	3734.41i	0.0976562	+	0.169146i	−0.813058	+	0.582183i	$0.802199\pi$
$788$	−1010.86			−0.0456985
$789$		0			0
$790$	17804.1	−	30837.7i	0.801827	−	1.38880i
$791$	10757.7	−	18632.8i	0.483564	−	0.837557i
$792$		0			0
$793$	−6517.80	+	28358.5i	−0.291871	+	1.26991i
$794$	11392.9			0.509219
$795$		0			0
$796$	−8316.41	+	14404.4i	−0.370311	+	0.641397i
$797$	504.809	+	874.355i	0.0224357	+	0.0388598i	0.877025	−	0.480444i	$-0.159525\pi$
$797$	504.809	+	874.355i	0.0224357	+	0.0388598i	−0.854590	+	0.519304i	$0.826191\pi$
$798$		0			0
$799$	−12385.6	−	21452.5i	−0.548400	−	0.949856i
$800$	24452.0	+	42352.2i	1.08064	+	1.87172i
$801$		0			0
$802$	19900.5	+	34468.7i	0.876198	+	1.51762i
$803$	−18434.5	+	31929.4i	−0.810134	+	1.40319i
$804$		0			0
$805$	−37714.8			−1.65127
$806$	−42322.8	+	12984.8i	−1.84958	+	0.567455i
$807$		0			0
$808$	7608.18	−	13177.8i	0.331256	−	0.573752i
$809$	−3684.82	+	6382.30i	−0.160138	+	0.277367i	−0.934918	−	0.354864i	$-0.884527\pi$
$809$	−3684.82	+	6382.30i	−0.160138	+	0.277367i	0.774780	+	0.632231i	$0.217860\pi$
$810$		0			0
$811$	6713.40			0.290678			0.145339	−	0.989382i	$-0.453573\pi$
$811$	6713.40			0.290678			0.145339	+	0.989382i	$0.453573\pi$
$812$	339.574	+	588.159i	0.0146757	+	0.0254191i
$813$		0			0
$814$	−17034.7			−0.733497
$815$	−11890.2	−	20594.5i	−0.511038	−	0.885144i
$816$		0			0
$817$	−7835.61	+	13571.7i	−0.335536	+	0.581166i
$818$	28182.9			1.20463
$819$		0			0
$820$	−7177.88			−0.305686
$821$	−1204.55	+	2086.35i	−0.0512049	+	0.0886894i	−0.890492	−	0.454999i	$-0.849639\pi$
$821$	−1204.55	+	2086.35i	−0.0512049	+	0.0886894i	0.839287	+	0.543689i	$0.182973\pi$
$822$		0			0
$823$	1185.39	+	2053.16i	0.0502069	+	0.0869608i	0.890037	−	0.455889i	$-0.150679\pi$
$823$	1185.39	+	2053.16i	0.0502069	+	0.0869608i	−0.839830	+	0.542850i	$0.817345\pi$
$824$	−8920.25			−0.377126
$825$		0			0
$826$	5965.82	+	10333.1i	0.251304	+	0.435272i
$827$	−10168.6			−0.427567			−0.213783	−	0.976881i	$-0.568579\pi$
$827$	−10168.6			−0.427567			−0.213783	+	0.976881i	$0.568579\pi$
$828$		0			0
$829$	9918.65	−	17179.6i	0.415547	−	0.719749i	−0.579938	−	0.814660i	$-0.696923\pi$
$829$	9918.65	−	17179.6i	0.415547	−	0.719749i	0.995486	+	0.0949111i	$0.0302567\pi$
$830$	9012.10	−	15609.4i	0.376885	−	0.652784i
$831$		0			0
$832$	6082.51	+	5660.89i	0.253453	+	0.235885i
$833$	−15652.5			−0.651052
$834$		0			0
$835$	−8806.59	+	15253.5i	−0.364987	+	0.632177i
$836$	15254.1	+	26420.8i	0.631069	+	1.09304i
$837$		0			0
$838$	−6477.25	−	11218.9i	−0.267008	−	0.462471i
$839$	−735.038	−	1273.12i	−0.0302459	−	0.0523875i	0.850506	−	0.525965i	$-0.176296\pi$
$839$	−735.038	−	1273.12i	−0.0302459	−	0.0523875i	−0.880752	+	0.473577i	$0.842962\pi$
$840$		0			0
$841$	12182.0	+	21099.9i	0.499488	+	0.865139i
$842$	−23034.6	+	39897.1i	−0.942784	+	1.63295i
$843$		0			0
$844$	17618.3			0.718540
$845$	−2955.24	−	41104.0i	−0.120312	−	1.67340i
$846$		0			0
$847$	13239.1	−	22930.8i	0.537073	−	0.930238i
$848$	−15063.8	+	26091.2i	−0.610015	+	1.05658i
$849$		0			0
$850$	−54700.4			−2.20731
$851$	3899.80	+	6754.66i	0.157090	+	0.272088i
$852$		0			0
$853$	−10200.0			−0.409426			−0.204713	−	0.978822i	$-0.565626\pi$
$853$	−10200.0			−0.409426			−0.204713	+	0.978822i	$0.565626\pi$
$854$	27662.9	+	47913.5i	1.10844	+	1.91987i
$855$		0			0
$856$	−6592.77	+	11419.0i	−0.263243	+	0.455951i
$857$	2379.48			0.0948443			0.0474222	−	0.998875i	$-0.484899\pi$
$857$	2379.48			0.0948443			0.0474222	+	0.998875i	$0.484899\pi$
$858$		0			0
$859$	9651.19			0.383346			0.191673	−	0.981459i	$-0.438609\pi$
$859$	9651.19			0.383346			0.191673	+	0.981459i	$0.438609\pi$
$860$	7528.89	−	13040.4i	0.298527	−	0.517064i
$861$		0			0
$862$	32759.2	+	56740.6i	1.29441	+	2.24199i
$863$	19568.7			0.771872			0.385936	−	0.922526i	$-0.373879\pi$
$863$	19568.7			0.771872			0.385936	+	0.922526i	$0.373879\pi$
$864$		0			0
$865$	17082.8	+	29588.2i	0.671482	+	1.16304i
$866$	35799.4			1.40475
$867$		0			0
$868$	−17387.4	+	30115.9i	−0.679918	+	1.17765i
$869$	12667.5	−	21940.8i	0.494495	−	0.856490i
$870$		0			0
$871$	5332.55	−	1636.04i	0.207447	−	0.0636454i
$872$	9391.75			0.364730
$873$		0			0
$874$	16905.7	−	29281.5i	0.654283	−	1.13325i
$875$	−23055.4	−	39933.2i	−0.890761	−	1.54284i
$876$		0			0
$877$	−989.333	−	1713.57i	−0.0380928	−	0.0659787i	0.846350	−	0.532626i	$-0.178795\pi$
$877$	−989.333	−	1713.57i	−0.0380928	−	0.0659787i	−0.884443	+	0.466648i	$0.845462\pi$
$878$	−24733.6	−	42839.8i	−0.950704	−	1.64667i
$879$		0			0
$880$	35739.8	+	61903.2i	1.36908	+	2.37131i
$881$	−2401.38	+	4159.32i	−0.0918327	+	0.159059i	−0.908282	−	0.418358i	$-0.862606\pi$
$881$	−2401.38	+	4159.32i	−0.0918327	+	0.159059i	0.816450	+	0.577417i	$0.195939\pi$
$882$		0			0
$883$	−49531.7			−1.88774			−0.943871	−	0.330315i	$-0.892845\pi$
$883$	−49531.7			−1.88774			−0.943871	+	0.330315i	$0.892845\pi$
$884$	−16482.2	+	5056.79i	−0.627101	+	0.192396i
$885$		0			0
$886$	−25162.1	+	43582.0i	−0.954104	+	1.65256i
$887$	3859.37	−	6684.62i	0.146093	−	0.253041i	−0.783687	−	0.621156i	$-0.786663\pi$
$887$	3859.37	−	6684.62i	0.146093	−	0.253041i	0.929780	+	0.368115i	$0.119997\pi$
$888$		0			0
$889$	158.346			0.00597384
$890$	−28866.4	−	49998.0i	−1.08719	−	1.88308i
$891$		0			0
$892$	−21142.2			−0.793600
$893$	20848.7	+	36111.1i	0.781273	+	1.35320i
$894$		0			0
$895$	−6056.21	+	10489.7i	−0.226186	+	0.391766i
$896$	−25832.4			−0.963170
$897$		0			0
$898$	5101.72			0.189584
$899$	639.006	−	1106.79i	0.0237064	−	0.0410606i
$900$		0			0
$901$	−12721.5	−	22034.2i	−0.470382	−	0.814725i
$902$	−12361.6			−0.456316
$903$		0			0
$904$	3896.91	+	6749.64i	0.143373	+	0.248329i
$905$	−44779.7			−1.64478
$906$		0			0
$907$	−894.290	+	1548.96i	−0.0327392	+	0.0567059i	−0.881931	−	0.471379i	$-0.843756\pi$
$907$	−894.290	+	1548.96i	−0.0327392	+	0.0567059i	0.849192	+	0.528085i	$0.177090\pi$
$908$	−4363.22	+	7557.32i	−0.159470	+	0.276210i
$909$		0			0
$910$	−57358.0	−	53382.2i	−2.08945	−	1.94462i
$911$	−8239.52			−0.299657			−0.149828	−	0.988712i	$-0.547872\pi$
$911$	−8239.52			−0.299657			−0.149828	+	0.988712i	$0.547872\pi$
$912$		0			0
$913$	6412.04	−	11106.0i	0.232429	−	0.402578i
$914$	15251.4	+	26416.2i	0.551938	+	0.955984i
$915$		0			0
$916$	−3843.19	−	6656.61i	−0.138627	−	0.240110i
$917$	−15296.9	−	26495.1i	−0.550872	−	0.954138i
$918$		0			0
$919$	−10526.9	−	18233.1i	−0.377857	−	0.654467i	0.612893	−	0.790166i	$-0.290005\pi$
$919$	−10526.9	−	18233.1i	−0.377857	−	0.654467i	−0.990750	+	0.135699i	$0.956672\pi$
$920$	6830.98	−	11831.6i	0.244794	−	0.423996i
$921$		0			0
$922$	62301.7			2.22538
$923$	−3794.26	+	16508.5i	−0.135308	+	0.588716i
$924$		0			0
$925$	−10620.2	+	18394.7i	−0.377503	+	0.653854i
$926$	−8246.22	+	14282.9i	−0.292643	+	0.506873i
$927$		0			0
$928$	−1077.06			−0.0380994
$929$	−17269.5	−	29911.6i	−0.609896	−	1.05637i	−0.991257	−	0.131945i	$-0.957878\pi$
$929$	−17269.5	−	29911.6i	−0.609896	−	1.05637i	0.381361	−	0.924426i	$-0.375456\pi$
$930$		0			0
$931$	26347.9			0.927516
$932$	4690.35	+	8123.93i	0.164847	+	0.285524i
$933$		0			0
$934$	5042.28	−	8733.48i	0.176647	−	0.305962i
$935$	−60365.1			−2.11139
$936$		0			0
$937$	−6368.78			−0.222048			−0.111024	−	0.993818i	$-0.535413\pi$
$937$	−6368.78			−0.222048			−0.111024	+	0.993818i	$0.535413\pi$
$938$	5302.81	−	9184.74i	0.184587	−	0.319715i
$939$		0			0
$940$	−20032.6	−	34697.6i	−0.695099	−	1.20395i
$941$	4504.64			0.156054			0.0780272	−	0.996951i	$-0.475138\pi$
$941$	4504.64			0.156054			0.0780272	+	0.996951i	$0.475138\pi$
$942$		0			0
$943$	2829.98	+	4901.67i	0.0977272	+	0.169268i
$944$	−10354.0			−0.356985
$945$		0			0
$946$	12966.1	−	22458.0i	0.445630	−	0.771853i
$947$	19382.6	−	33571.7i	0.665101	−	1.15199i	−0.314156	−	0.949371i	$-0.601722\pi$
$947$	19382.6	−	33571.7i	0.665101	−	1.15199i	0.979258	−	0.202618i	$-0.0649451\pi$
$948$		0			0
$949$	25673.2	+	23893.7i	0.878176	+	0.817304i
$950$	92077.5			3.14462
$951$		0			0
$952$	6892.52	−	11938.2i	0.234651	−	0.406428i
$953$	10972.6	+	19005.2i	0.372968	+	0.645999i	0.990021	−	0.140923i	$-0.0450068\pi$
$953$	10972.6	+	19005.2i	0.372968	+	0.645999i	−0.617053	+	0.786922i	$0.711674\pi$
$954$		0			0
$955$	−10563.8	−	18297.0i	−0.357943	−	0.619975i
$956$	16754.5	+	29019.7i	0.566821	+	0.981762i
$957$		0			0
$958$	−10395.0	−	18004.6i	−0.350570	−	0.607206i
$959$	−16303.6	+	28238.7i	−0.548979	+	0.950860i
$960$		0			0
$961$	35648.0			1.19660
$962$	−3629.71	+	15792.6i	−0.121649	+	0.529287i
$963$		0			0
$964$	4806.69	−	8325.43i	0.160595	−	0.278158i
$965$	−30561.8	+	52934.5i	−1.01950	+	1.76583i
$966$		0			0
$967$	−58962.4			−1.96081			−0.980405	−	0.196991i	$-0.936883\pi$
$967$	−58962.4			−1.96081			−0.980405	+	0.196991i	$0.936883\pi$
$968$	4795.79	+	8306.55i	0.159238	+	0.275809i
$969$		0			0
$970$	102634.			3.39731
$971$	−25832.2	−	44742.6i	−0.853753	−	1.47874i	−0.877797	−	0.479032i	$-0.840988\pi$
$971$	−25832.2	−	44742.6i	−0.853753	−	1.47874i	0.0240449	−	0.999711i	$-0.492346\pi$
$972$		0			0
$973$	−11955.8	+	20708.0i	−0.393920	+	0.682290i
$974$	−47667.0			−1.56812
$975$		0			0
$976$	−48010.5			−1.57457
$977$	8944.42	−	15492.2i	0.292894	−	0.507307i	−0.681599	−	0.731726i	$-0.738715\pi$
$977$	8944.42	−	15492.2i	0.292894	−	0.507307i	0.974493	+	0.224419i	$0.0720484\pi$
$978$		0			0
$979$	−20538.2	−	35573.2i	−0.670484	−	1.16131i
$980$	−25316.5			−0.825212
$981$		0			0
$982$	−5749.86	−	9959.04i	−0.186849	−	0.323631i
$983$	40916.4			1.32760			0.663800	−	0.747910i	$-0.268942\pi$
$983$	40916.4			1.32760			0.663800	+	0.747910i	$0.268942\pi$
$984$		0			0
$985$	−1683.43	+	2915.78i	−0.0544552	+	0.0943192i
$986$	602.359	−	1043.32i	0.0194554	−	0.0336977i
$987$		0			0
$988$	27744.6	−	8512.11i	0.893394	−	0.274096i
$989$	−11873.5			−0.381754
$990$		0			0
$991$	−13902.1	+	24079.1i	−0.445624	+	0.771844i	−0.998095	−	0.0616881i	$-0.980352\pi$
$991$	−13902.1	+	24079.1i	−0.445624	+	0.771844i	0.552471	+	0.833532i	$0.313685\pi$
$992$	−27574.7	−	47760.8i	−0.882559	−	1.52864i
$993$		0			0
$994$	16103.6	+	27892.3i	0.513859	+	0.890030i
$995$	27699.3	+	47976.5i	0.882538	+	1.52860i
$996$		0			0
$997$	−20802.7	−	36031.4i	−0.660812	−	1.14456i	−0.980403	−	0.197004i	$-0.936879\pi$
$997$	−20802.7	−	36031.4i	−0.660812	−	1.14456i	0.319591	−	0.947556i	$-0.396454\pi$
$998$	21483.8	−	37211.1i	0.681422	−	1.18026i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.g.f.100.2	yes	16
3.2	odd	2	inner	117.4.g.f.100.7	yes	16
13.3	even	3	inner	117.4.g.f.55.2	✓	16
13.4	even	6		1521.4.a.bd.1.2		8
13.9	even	3		1521.4.a.bc.1.7		8
39.17	odd	6		1521.4.a.bd.1.7		8
39.29	odd	6	inner	117.4.g.f.55.7	yes	16
39.35	odd	6		1521.4.a.bc.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.2	✓	16	13.3	even	3	inner
117.4.g.f.55.7	yes	16	39.29	odd	6	inner
117.4.g.f.100.2	yes	16	1.1	even	1	trivial
117.4.g.f.100.7	yes	16	3.2	odd	2	inner
1521.4.a.bc.1.2		8	39.35	odd	6
1521.4.a.bc.1.7		8	13.9	even	3
1521.4.a.bd.1.2		8	13.4	even	6
1521.4.a.bd.1.7		8	39.17	odd	6

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.84606 + 3.19747i) q^{2} +(-2.81586 - 4.87721i) q^{4} -18.7574 q^{5} +(12.0691 + 20.9044i) q^{7} -8.74396 q^{8} +O(q^{10})\)	\(q+(-1.84606 + 3.19747i) q^{2} +(-2.81586 - 4.87721i) q^{4} -18.7574 q^{5} +(12.0691 + 20.9044i) q^{7} -8.74396 q^{8} +(34.6273 - 59.9763i) q^{10} +(24.6371 - 42.6726i) q^{11} +(-34.3114 - 31.9331i) q^{13} -89.1214 q^{14} +(38.6687 - 66.9762i) q^{16} +(32.6560 + 56.5619i) q^{17} +(-54.9701 - 95.2109i) q^{19} +(52.8183 + 91.4840i) q^{20} +(90.9629 + 157.552i) q^{22} +(41.6487 - 72.1377i) q^{23} +226.841 q^{25} +(165.446 - 50.7593i) q^{26} +(67.9701 - 117.728i) q^{28} +(-2.49797 + 4.32660i) q^{29} -255.810 q^{31} +(107.794 + 186.704i) q^{32} -241.140 q^{34} +(-226.386 - 392.112i) q^{35} +(-46.8178 + 81.0908i) q^{37} +405.912 q^{38} +164.014 q^{40} +(-33.9744 + 58.8453i) q^{41} +(-71.2716 - 123.446i) q^{43} -277.498 q^{44} +(153.772 + 266.341i) q^{46} -379.275 q^{47} +(-119.828 + 207.549i) q^{49} +(-418.762 + 725.317i) q^{50} +(-59.1284 + 257.263i) q^{52} -389.560 q^{53} +(-462.128 + 800.429i) q^{55} +(-105.532 - 182.787i) q^{56} +(-9.22278 - 15.9743i) q^{58} +(-66.9404 - 115.944i) q^{59} +(-310.396 - 537.621i) q^{61} +(472.241 - 817.945i) q^{62} -177.273 q^{64} +(643.595 + 598.983i) q^{65} +(-59.5010 + 103.059i) q^{67} +(183.910 - 318.541i) q^{68} +1671.69 q^{70} +(-180.693 - 312.970i) q^{71} -748.241 q^{73} +(-172.857 - 299.397i) q^{74} +(-309.576 + 536.201i) q^{76} +1189.39 q^{77} +514.165 q^{79} +(-725.326 + 1256.30i) q^{80} +(-125.437 - 217.264i) q^{82} +260.260 q^{83} +(-612.543 - 1060.96i) q^{85} +526.286 q^{86} +(-215.425 + 373.128i) q^{88} +(416.815 - 721.945i) q^{89} +(253.432 - 1102.66i) q^{91} -469.108 q^{92} +(700.163 - 1212.72i) q^{94} +(1031.10 + 1785.91i) q^{95} +(740.993 + 1283.44i) q^{97} +(-442.421 - 766.295i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) \) 16 * q - 40 * q^4 + 22 * q^7	\( 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Introduction

L-functions

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