LMFDB - Embedded newform 378.4.g.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.11184604443.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 $ x^6 + 43x^4 - 210x^3 + 1849x^2 - 4515x + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-3.77219 - 6.53362i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.b.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.90212 - 6.75867i) q^{5} +(16.3412 + 8.71578i) q^{7} -8.00000 q^{8} +(7.80425 - 13.5173i) q^{10} +(2.79192 - 4.83575i) q^{11} +12.8042 q^{13} +(1.24504 + 37.0196i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(23.9251 - 41.4394i) q^{17} +(23.4637 + 40.6404i) q^{19} +31.2170 q^{20} +11.1677 q^{22} +(19.0130 + 32.9315i) q^{23} +(32.0469 - 55.5068i) q^{25} +(12.8042 + 22.1776i) q^{26} +(-62.8748 + 39.1761i) q^{28} +290.356 q^{29} +(-126.303 + 218.763i) q^{31} +(16.0000 - 27.7128i) q^{32} +95.7002 q^{34} +(-4.85832 - 144.455i) q^{35} +(205.336 + 355.652i) q^{37} +(-46.9275 + 81.2808i) q^{38} +(31.2170 + 54.0694i) q^{40} +11.7824 q^{41} +242.586 q^{43} +(11.1677 + 19.3430i) q^{44} +(-38.0261 + 65.8631i) q^{46} +(28.4874 + 49.3416i) q^{47} +(191.070 + 284.853i) q^{49} +128.188 q^{50} +(-25.6085 + 44.3552i) q^{52} +(-115.369 + 199.825i) q^{53} -43.5777 q^{55} +(-130.730 - 69.7262i) q^{56} +(290.356 + 502.911i) q^{58} +(259.028 - 448.650i) q^{59} +(129.473 + 224.253i) q^{61} -505.212 q^{62} +64.0000 q^{64} +(-49.9637 - 86.5397i) q^{65} +(-123.139 + 213.283i) q^{67} +(95.7002 + 165.758i) q^{68} +(245.345 - 152.870i) q^{70} -545.724 q^{71} +(143.956 - 249.339i) q^{73} +(-410.672 + 711.304i) q^{74} -187.710 q^{76} +(87.7707 - 54.6883i) q^{77} +(45.1287 + 78.1652i) q^{79} +(-62.4340 + 108.139i) q^{80} +(11.7824 + 20.4077i) q^{82} -372.935 q^{83} -373.434 q^{85} +(242.586 + 420.171i) q^{86} +(-22.3354 + 38.6860i) q^{88} +(-559.424 - 968.951i) q^{89} +(209.237 + 111.599i) q^{91} -152.104 q^{92} +(-56.9747 + 98.6832i) q^{94} +(183.117 - 317.168i) q^{95} +393.762 q^{97} +(-302.309 + 615.796i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−3.90212	−	6.75867i	−0.349016	−	0.604514i	0.637059	−	0.770815i	$-0.280151\pi$
$5$	−3.90212	−	6.75867i	−0.349016	−	0.604514i	−0.986075	+	0.166301i	$0.946818\pi$
$6$		0			0
$7$	16.3412	+	8.71578i	0.882343	+	0.470608i
$7$	16.3412	+	8.71578i	0.882343	+	0.470608i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	7.80425	−	13.5173i	0.246792	−	0.427456i
$11$	2.79192	−	4.83575i	0.0765269	−	0.132549i	−0.825222	−	0.564808i	$-0.808950\pi$
$11$	2.79192	−	4.83575i	0.0765269	−	0.132549i	0.901749	+	0.432259i	$0.142284\pi$
$12$		0			0
$13$	12.8042			0.273174			0.136587	−	0.990628i	$-0.456387\pi$
$13$	12.8042			0.273174			0.136587	+	0.990628i	$0.456387\pi$
$14$	1.24504	+	37.0196i	0.0237680	+	0.706707i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	23.9251	−	41.4394i	0.341334	−	0.591208i	−0.643347	−	0.765575i	$-0.722455\pi$
$17$	23.9251	−	41.4394i	0.341334	−	0.591208i	0.984681	+	0.174367i	$0.0557879\pi$
$18$		0			0
$19$	23.4637	+	40.6404i	0.283313	+	0.490713i	0.972199	−	0.234157i	$-0.0752331\pi$
$19$	23.4637	+	40.6404i	0.283313	+	0.490713i	−0.688886	+	0.724870i	$0.741900\pi$
$20$	31.2170			0.349016
$21$		0			0
$22$	11.1677			0.108225
$23$	19.0130	+	32.9315i	0.172369	+	0.298552i	0.939248	−	0.343240i	$-0.111524\pi$
$23$	19.0130	+	32.9315i	0.172369	+	0.298552i	−0.766879	+	0.641792i	$0.778191\pi$
$24$		0			0
$25$	32.0469	−	55.5068i	0.256375	−	0.444055i
$26$	12.8042	+	22.1776i	0.0965815	+	0.167284i
$27$		0			0
$28$	−62.8748	+	39.1761i	−0.424365	+	0.264414i
$29$	290.356			1.85923			0.929616	−	0.368528i	$-0.120138\pi$
$29$	290.356			1.85923			0.929616	+	0.368528i	$0.120138\pi$
$30$		0			0
$31$	−126.303	+	218.763i	−0.731764	+	1.26745i	0.224365	+	0.974505i	$0.427969\pi$
$31$	−126.303	+	218.763i	−0.731764	+	1.26745i	−0.956129	+	0.292947i	$0.905364\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	95.7002			0.482719
$35$	−4.85832	−	144.455i	−0.0234630	−	0.697638i
$36$		0			0
$37$	205.336	+	355.652i	0.912351	+	1.58024i	0.810734	+	0.585415i	$0.199068\pi$
$37$	205.336	+	355.652i	0.912351	+	1.58024i	0.101617	+	0.994824i	$0.467598\pi$
$38$	−46.9275	+	81.2808i	−0.200333	+	0.346986i
$39$		0			0
$40$	31.2170	+	54.0694i	0.123396	+	0.213728i
$41$	11.7824			0.0448806			0.0224403	−	0.999748i	$-0.492856\pi$
$41$	11.7824			0.0448806			0.0224403	+	0.999748i	$0.492856\pi$
$42$		0			0
$43$	242.586			0.860326			0.430163	−	0.902751i	$-0.358456\pi$
$43$	242.586			0.860326			0.430163	+	0.902751i	$0.358456\pi$
$44$	11.1677	+	19.3430i	0.0382635	+	0.0662743i
$45$		0			0
$46$	−38.0261	+	65.8631i	−0.121883	+	0.211108i
$47$	28.4874	+	49.3416i	0.0884109	+	0.153132i	0.906840	−	0.421476i	$-0.138488\pi$
$47$	28.4874	+	49.3416i	0.0884109	+	0.153132i	−0.818429	+	0.574608i	$0.805154\pi$
$48$		0			0
$49$	191.070	+	284.853i	0.557057	+	0.830474i
$50$	128.188			0.362569
$51$		0			0
$52$	−25.6085	+	44.3552i	−0.0682934	+	0.118288i
$53$	−115.369	+	199.825i	−0.299003	+	0.517889i	−0.975908	−	0.218181i	$-0.929988\pi$
$53$	−115.369	+	199.825i	−0.299003	+	0.517889i	0.676905	+	0.736071i	$0.263321\pi$
$54$		0			0
$55$	−43.5777			−0.106837
$56$	−130.730	−	69.7262i	−0.311955	−	0.166385i
$57$		0			0
$58$	290.356	+	502.911i	0.657338	+	1.13854i
$59$	259.028	−	448.650i	0.571570	−	0.989988i	−0.424835	−	0.905271i	$-0.639668\pi$
$59$	259.028	−	448.650i	0.571570	−	0.989988i	0.996405	−	0.0847176i	$-0.0269988\pi$
$60$		0			0
$61$	129.473	+	224.253i	0.271759	+	0.470700i	0.969312	−	0.245833i	$-0.0790614\pi$
$61$	129.473	+	224.253i	0.271759	+	0.470700i	−0.697554	+	0.716533i	$0.745728\pi$
$62$	−505.212			−1.03487
$63$		0			0
$64$	64.0000			0.125000
$65$	−49.9637	−	86.5397i	−0.0953421	−	0.165137i
$66$		0			0
$67$	−123.139	+	213.283i	−0.224535	+	0.388906i	−0.956180	−	0.292780i	$-0.905420\pi$
$67$	−123.139	+	213.283i	−0.224535	+	0.388906i	0.731645	+	0.681686i	$0.238753\pi$
$68$	95.7002	+	165.758i	0.170667	+	0.295604i
$69$		0			0
$70$	245.345	−	152.870i	0.418919	−	0.261021i
$71$	−545.724			−0.912191			−0.456096	−	0.889931i	$-0.650752\pi$
$71$	−545.724			−0.912191			−0.456096	+	0.889931i	$0.650752\pi$
$72$		0			0
$73$	143.956	−	249.339i	0.230805	−	0.399767i	−0.727240	−	0.686383i	$-0.759197\pi$
$73$	143.956	−	249.339i	0.230805	−	0.399767i	0.958045	+	0.286617i	$0.0925306\pi$
$74$	−410.672	+	711.304i	−0.645130	+	1.11740i
$75$		0			0
$76$	−187.710			−0.283313
$77$	87.7707	−	54.6883i	0.129901	−	0.0809391i
$78$		0			0
$79$	45.1287	+	78.1652i	0.0642706	+	0.111320i	0.896370	−	0.443306i	$-0.146195\pi$
$79$	45.1287	+	78.1652i	0.0642706	+	0.111320i	−0.832100	+	0.554626i	$0.812861\pi$
$80$	−62.4340	+	108.139i	−0.0872541	+	0.151129i
$81$		0			0
$82$	11.7824	+	20.4077i	0.0158677	+	0.0274836i
$83$	−372.935			−0.493192			−0.246596	−	0.969118i	$-0.579312\pi$
$83$	−372.935			−0.493192			−0.246596	+	0.969118i	$0.579312\pi$
$84$		0			0
$85$	−373.434			−0.476525
$86$	242.586	+	420.171i	0.304171	+	0.526840i
$87$		0			0
$88$	−22.3354	+	38.6860i	−0.0270564	+	0.0468630i
$89$	−559.424	−	968.951i	−0.666279	−	1.15403i	−0.978937	−	0.204164i	$-0.934553\pi$
$89$	−559.424	−	968.951i	−0.666279	−	1.15403i	0.312658	−	0.949866i	$-0.398781\pi$
$90$		0			0
$91$	209.237	+	111.599i	0.241033	+	0.128558i
$92$	−152.104			−0.172369
$93$		0			0
$94$	−56.9747	+	98.6832i	−0.0625159	+	0.108281i
$95$	183.117	−	317.168i	0.197762	−	0.342534i
$96$		0			0
$97$	393.762			0.412170			0.206085	−	0.978534i	$-0.433928\pi$
$97$	393.762			0.412170			0.206085	+	0.978534i	$0.433928\pi$
$98$	−302.309	+	615.796i	−0.311610	+	0.634743i
$99$		0			0
$100$	128.188	+	222.027i	0.128188	+	0.222027i
$101$	547.800	−	948.818i	0.539685	−	0.934762i	−0.459236	−	0.888314i	$-0.651877\pi$
$101$	547.800	−	948.818i	0.539685	−	0.934762i	0.998921	−	0.0464472i	$-0.0147899\pi$
$102$		0			0
$103$	121.111	+	209.771i	0.115859	+	0.200673i	0.918123	−	0.396296i	$-0.129705\pi$
$103$	121.111	+	209.771i	0.115859	+	0.200673i	−0.802264	+	0.596969i	$0.796371\pi$
$104$	−102.434			−0.0965815
$105$		0			0
$106$	−461.477			−0.422855
$107$	−49.5638	−	85.8470i	−0.0447805	−	0.0775621i	0.842766	−	0.538279i	$-0.180926\pi$
$107$	−49.5638	−	85.8470i	−0.0447805	−	0.0775621i	−0.887547	+	0.460717i	$0.847592\pi$
$108$		0			0
$109$	459.570	−	795.998i	0.403842	−	0.699475i	−0.590344	−	0.807152i	$-0.701008\pi$
$109$	459.570	−	795.998i	0.403842	−	0.699475i	0.994186	+	0.107677i	$0.0343412\pi$
$110$	−43.5777	−	75.4788i	−0.0377725	−	0.0654238i
$111$		0			0
$112$	−9.96036	−	296.157i	−0.00840326	−	0.249859i
$113$	2121.07			1.76578			0.882889	−	0.469581i	$-0.155595\pi$
$113$	2121.07			1.76578			0.882889	+	0.469581i	$0.155595\pi$
$114$		0			0
$115$	148.382	−	257.006i	0.120319	−	0.208399i
$116$	−580.712	+	1005.82i	−0.464808	+	0.805071i
$117$		0			0
$118$	1036.11			0.808322
$119$	752.141	−	468.645i	0.579401	−	0.361013i
$120$		0			0
$121$	649.910	+	1125.68i	0.488287	+	0.845738i
$122$	−258.945	+	448.507i	−0.192162	+	0.332835i
$123$		0			0
$124$	−505.212	−	875.052i	−0.365882	−	0.633726i
$125$	−1475.73			−1.05595
$126$		0			0
$127$	−1250.33			−0.873611			−0.436805	−	0.899556i	$-0.643890\pi$
$127$	−1250.33			−0.873611			−0.436805	+	0.899556i	$0.643890\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	99.9275	−	173.079i	0.0674171	−	0.116770i
$131$	56.7324	+	98.2634i	0.0378376	+	0.0655367i	0.884324	−	0.466874i	$-0.154620\pi$
$131$	56.7324	+	98.2634i	0.0378376	+	0.0655367i	−0.846486	+	0.532410i	$0.821286\pi$
$132$		0			0
$133$	29.2134	+	868.618i	0.0190460	+	0.566306i
$134$	−492.556			−0.317540
$135$		0			0
$136$	−191.400	+	331.515i	−0.120680	+	0.209024i
$137$	−7.65748	+	13.2631i	−0.00477535	+	0.00827115i	−0.868403	−	0.495859i	$-0.834853\pi$
$137$	−7.65748	+	13.2631i	−0.00477535	+	0.00827115i	0.863628	+	0.504130i	$0.168187\pi$
$138$		0			0
$139$	−2539.81			−1.54981			−0.774905	−	0.632077i	$-0.782203\pi$
$139$	−2539.81			−1.54981			−0.774905	+	0.632077i	$0.782203\pi$
$140$	510.123	+	272.080i	0.307952	+	0.164250i
$141$		0			0
$142$	−545.724	−	945.222i	−0.322508	−	0.558601i
$143$	35.7485	−	61.9182i	0.0209052	−	0.0362088i
$144$		0			0
$145$	−1133.00	−	1962.42i	−0.648903	−	1.12393i
$146$	575.825			0.326408
$147$		0			0
$148$	−1642.69			−0.912351
$149$	−43.5846	−	75.4907i	−0.0239637	−	0.0415063i	0.853795	−	0.520610i	$-0.174295\pi$
$149$	−43.5846	−	75.4907i	−0.0239637	−	0.0415063i	−0.877759	+	0.479103i	$0.840962\pi$
$150$		0			0
$151$	498.030	−	862.613i	0.268405	−	0.464890i	−0.700045	−	0.714098i	$-0.746837\pi$
$151$	498.030	−	862.613i	0.268405	−	0.464890i	0.968450	+	0.249208i	$0.0801703\pi$
$152$	−187.710	−	325.123i	−0.100166	−	0.173493i
$153$		0			0
$154$	182.494	+	97.3351i	0.0954919	+	0.0509317i
$155$	1971.40			1.02159
$156$		0			0
$157$	1081.66	−	1873.48i	0.549844	−	0.952358i	−0.448441	−	0.893813i	$-0.648021\pi$
$157$	1081.66	−	1873.48i	0.549844	−	0.952358i	0.998285	−	0.0585451i	$-0.0186461\pi$
$158$	−90.2574	+	156.330i	−0.0454462	+	0.0787151i
$159$		0			0
$160$	−249.736			−0.123396
$161$	23.6721	+	703.854i	0.0115877	+	0.344543i
$162$		0			0
$163$	−661.783	−	1146.24i	−0.318005	−	0.550801i	0.662066	−	0.749445i	$-0.269680\pi$
$163$	−661.783	−	1146.24i	−0.318005	−	0.550801i	−0.980072	+	0.198644i	$0.936346\pi$
$164$	−23.5648	+	40.8155i	−0.0112201	+	0.0194339i
$165$		0			0
$166$	−372.935	−	645.942i	−0.174370	−	0.302017i
$167$	2071.29			0.959766			0.479883	−	0.877332i	$-0.340679\pi$
$167$	2071.29			0.959766			0.479883	+	0.877332i	$0.340679\pi$
$168$		0			0
$169$	−2033.05			−0.925376
$170$	−373.434	−	646.807i	−0.168477	−	0.291811i
$171$		0			0
$172$	−485.172	+	840.342i	−0.215081	+	0.372532i
$173$	−1542.64	−	2671.93i	−0.677946	−	1.17424i	−0.975598	−	0.219562i	$-0.929537\pi$
$173$	−1542.64	−	2671.93i	−0.677946	−	1.17424i	0.297653	−	0.954674i	$-0.403796\pi$
$174$		0			0
$175$	1007.47	−	627.735i	0.435186	−	0.271156i
$176$	−89.3415			−0.0382635
$177$		0			0
$178$	1118.85	−	1937.90i	0.471131	−	0.816022i
$179$	−723.786	+	1253.63i	−0.302225	+	0.523470i	−0.976640	−	0.214883i	$-0.931063\pi$
$179$	−723.786	+	1253.63i	−0.302225	+	0.523470i	0.674414	+	0.738353i	$0.264396\pi$
$180$		0			0
$181$	−1683.16			−0.691208			−0.345604	−	0.938380i	$-0.612326\pi$
$181$	−1683.16			−0.691208			−0.345604	+	0.938380i	$0.612326\pi$
$182$	15.9419	+	474.008i	0.00649280	+	0.193054i
$183$		0			0
$184$	−152.104	−	263.452i	−0.0609417	−	0.105554i
$185$	1602.49	−	2775.60i	0.636851	−	1.10306i
$186$		0			0
$187$	−133.594	−	231.391i	−0.0522425	−	0.0904867i
$188$	−227.899			−0.0884109
$189$		0			0
$190$	732.467			0.279678
$191$	1047.11	+	1813.66i	0.396684	+	0.687076i	0.993314	−	0.115440i	$-0.0368277\pi$
$191$	1047.11	+	1813.66i	0.396684	+	0.687076i	−0.596631	+	0.802516i	$0.703494\pi$
$192$		0			0
$193$	1709.71	−	2961.31i	0.637657	−	1.10445i	−0.348289	−	0.937387i	$-0.613237\pi$
$193$	1709.71	−	2961.31i	0.637657	−	1.10445i	0.985946	−	0.167067i	$-0.0534296\pi$
$194$	393.762	+	682.016i	0.145724	+	0.252401i
$195$		0			0
$196$	−1368.90	+	92.1821i	−0.498870	+	0.0335941i
$197$	−3517.08			−1.27199			−0.635993	−	0.771695i	$-0.719409\pi$
$197$	−3517.08			−1.27199			−0.635993	+	0.771695i	$0.719409\pi$
$198$		0			0
$199$	1566.03	−	2712.45i	0.557855	−	0.966233i	−0.439820	−	0.898086i	$-0.644958\pi$
$199$	1566.03	−	2712.45i	0.557855	−	0.966233i	0.997675	−	0.0681473i	$-0.0217088\pi$
$200$	−256.375	+	444.055i	−0.0906423	+	0.156997i
$201$		0			0
$202$	2191.20			0.763230
$203$	4744.77	+	2530.68i	1.64048	+	0.874969i
$204$		0			0
$205$	−45.9764	−	79.6335i	−0.0156641	−	0.0271309i
$206$	−242.222	+	419.542i	−0.0819245	+	0.141897i
$207$		0			0
$208$	−102.434	−	177.421i	−0.0341467	−	0.0591439i
$209$	262.036			0.0867244
$210$		0			0
$211$	−3088.38			−1.00764			−0.503822	−	0.863807i	$-0.668073\pi$
$211$	−3088.38			−1.00764			−0.503822	+	0.863807i	$0.668073\pi$
$212$	−461.477	−	799.302i	−0.149502	−	0.258945i
$213$		0			0
$214$	99.1276	−	171.694i	0.0316646	−	0.0548447i
$215$	−946.600	−	1639.56i	−0.300268	−	0.520079i
$216$		0			0
$217$	−3970.63	+	2474.03i	−1.24214	+	0.773953i
$218$	1838.28			0.571119
$219$		0			0
$220$	87.1554	−	150.958i	0.0267092	−	0.0462616i
$221$	306.342	−	530.600i	0.0932435	−	0.161502i
$222$		0			0
$223$	4399.79			1.32122			0.660609	−	0.750730i	$-0.270298\pi$
$223$	4399.79			1.32122			0.660609	+	0.750730i	$0.270298\pi$
$224$	502.998	−	313.409i	0.150036	−	0.0934843i
$225$		0			0
$226$	2121.07	+	3673.79i	0.624297	+	1.08131i
$227$	−1643.69	+	2846.96i	−0.480597	+	0.832419i	−0.999752	−	0.0222611i	$-0.992913\pi$
$227$	−1643.69	+	2846.96i	−0.480597	+	0.832419i	0.519155	+	0.854680i	$0.326247\pi$
$228$		0			0
$229$	−1506.76	−	2609.78i	−0.434800	−	0.753096i	0.562479	−	0.826811i	$-0.309848\pi$
$229$	−1506.76	−	2609.78i	−0.434800	−	0.753096i	−0.997279	+	0.0737158i	$0.976514\pi$
$230$	593.529			0.170157
$231$		0			0
$232$	−2322.85			−0.657338
$233$	−2540.80	−	4400.80i	−0.714393	−	1.23737i	−0.963193	−	0.268811i	$-0.913369\pi$
$233$	−2540.80	−	4400.80i	−0.714393	−	1.23737i	0.248800	−	0.968555i	$-0.419964\pi$
$234$		0			0
$235$	222.322	−	385.074i	0.0617137	−	0.106891i
$236$	1036.11	+	1794.60i	0.285785	+	0.494994i
$237$		0			0
$238$	1563.86	+	834.102i	0.425924	+	0.227171i
$239$	3087.38			0.835590			0.417795	−	0.908541i	$-0.362803\pi$
$239$	3087.38			0.835590			0.417795	+	0.908541i	$0.362803\pi$
$240$		0			0
$241$	608.325	−	1053.65i	0.162596	−	0.281625i	−0.773203	−	0.634159i	$-0.781347\pi$
$241$	608.325	−	1053.65i	0.162596	−	0.281625i	0.935799	+	0.352534i	$0.114680\pi$
$242$	−1299.82	+	2251.36i	−0.345271	+	0.598027i
$243$		0			0
$244$	−1035.78			−0.271759
$245$	1179.65	−	2402.91i	0.307612	−	0.626598i
$246$		0			0
$247$	300.435	+	520.369i	0.0773937	+	0.134050i
$248$	1010.42	−	1750.10i	0.258718	−	0.448112i
$249$		0			0
$250$	−1475.73	−	2556.05i	−0.373334	−	0.646634i
$251$	−2649.43			−0.666257			−0.333128	−	0.942881i	$-0.608104\pi$
$251$	−2649.43			−0.666257			−0.333128	+	0.942881i	$0.608104\pi$
$252$		0			0
$253$	212.332			0.0527635
$254$	−1250.33	−	2165.63i	−0.308868	−	0.534975i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−357.989	−	620.055i	−0.0868901	−	0.150498i	0.819305	−	0.573358i	$-0.194360\pi$
$257$	−357.989	−	620.055i	−0.0868901	−	0.150498i	−0.906195	+	0.422860i	$0.861026\pi$
$258$		0			0
$259$	255.652	+	7601.45i	0.0613338	+	1.82367i
$260$	399.710			0.0953421
$261$		0			0
$262$	−113.465	+	196.527i	−0.0267553	+	0.0463415i
$263$	−3173.49	+	5496.65i	−0.744053	+	1.28874i	0.206582	+	0.978429i	$0.433766\pi$
$263$	−3173.49	+	5496.65i	−0.744053	+	1.28874i	−0.950636	+	0.310309i	$0.899567\pi$
$264$		0			0
$265$	1800.74			0.417429
$266$	−1475.28	+	919.217i	−0.340056	+	0.211883i
$267$		0			0
$268$	−492.556	−	853.132i	−0.112267	−	0.194453i
$269$	−3919.92	+	6789.50i	−0.888482	+	1.53890i	−0.0468125	+	0.998904i	$0.514906\pi$
$269$	−3919.92	+	6789.50i	−0.888482	+	1.53890i	−0.841670	+	0.539993i	$0.818427\pi$
$270$		0			0
$271$	−1640.28	−	2841.04i	−0.367674	−	0.636831i	0.621527	−	0.783393i	$-0.286513\pi$
$271$	−1640.28	−	2841.04i	−0.367674	−	0.636831i	−0.989202	+	0.146562i	$0.953179\pi$
$272$	−765.602			−0.170667
$273$		0			0
$274$	−30.6299			−0.00675336
$275$	−178.945	−	309.942i	−0.0392392	−	0.0679643i
$276$		0			0
$277$	−3766.56	+	6523.88i	−0.817006	+	1.41510i	0.0908720	+	0.995863i	$0.471035\pi$
$277$	−3766.56	+	6523.88i	−0.817006	+	1.41510i	−0.907878	+	0.419234i	$0.862299\pi$
$278$	−2539.81	−	4399.08i	−0.547941	−	0.949062i
$279$		0			0
$280$	38.8665	+	1155.64i	0.00829543	+	0.246652i
$281$	−3454.53			−0.733380			−0.366690	−	0.930343i	$-0.619509\pi$
$281$	−3454.53			−0.733380			−0.366690	+	0.930343i	$0.619509\pi$
$282$		0			0
$283$	1322.28	−	2290.26i	0.277744	−	0.481066i	−0.693080	−	0.720861i	$-0.743747\pi$
$283$	1322.28	−	2290.26i	0.277744	−	0.481066i	0.970824	+	0.239794i	$0.0770800\pi$
$284$	1091.45	−	1890.44i	0.228048	−	0.394990i
$285$		0			0
$286$	142.994			0.0295643
$287$	192.539	+	102.693i	0.0396000	+	0.0211211i
$288$		0			0
$289$	1311.68	+	2271.90i	0.266982	+	0.462427i
$290$	2266.01	−	3924.84i	0.458844	−	0.794741i
$291$		0			0
$292$	575.825	+	997.358i	0.115403	+	0.199883i
$293$	−8580.20			−1.71079			−0.855394	−	0.517978i	$-0.826685\pi$
$293$	−8580.20			−1.71079			−0.855394	+	0.517978i	$0.826685\pi$
$294$		0			0
$295$	−4043.04			−0.797949
$296$	−1642.69	−	2845.22i	−0.322565	−	0.558699i
$297$		0			0
$298$	87.1692	−	150.981i	0.0169449	−	0.0293494i
$299$	243.447	+	421.663i	0.0470867	+	0.0815566i
$300$		0			0
$301$	3964.15	+	2114.33i	0.759102	+	0.404876i
$302$	1992.12			0.379581
$303$		0			0
$304$	375.420	−	650.246i	0.0708283	−	0.122678i
$305$	1010.44	−	1750.13i	0.189697	−	0.328564i
$306$		0			0
$307$	9077.13			1.68749			0.843745	−	0.536745i	$-0.180346\pi$
$307$	9077.13			1.68749			0.843745	+	0.536745i	$0.180346\pi$
$308$	13.9043	+	413.423i	0.00257230	+	0.0764837i
$309$		0			0
$310$	1971.40	+	3414.56i	0.361187	+	0.625594i
$311$	−3620.21	+	6270.39i	−0.660075	+	1.14328i	0.320521	+	0.947241i	$0.396142\pi$
$311$	−3620.21	+	6270.39i	−0.660075	+	1.14328i	−0.980596	+	0.196041i	$0.937191\pi$
$312$		0			0
$313$	4977.05	+	8620.50i	0.898784	+	1.55674i	0.829051	+	0.559173i	$0.188881\pi$
$313$	4977.05	+	8620.50i	0.898784	+	1.55674i	0.0697331	+	0.997566i	$0.477785\pi$
$314$	4326.62			0.777597
$315$		0			0
$316$	−361.030			−0.0642706
$317$	2275.80	+	3941.80i	0.403223	+	0.698402i	0.994113	−	0.108350i	$-0.0345567\pi$
$317$	2275.80	+	3941.80i	0.403223	+	0.698402i	−0.590890	+	0.806752i	$0.701223\pi$
$318$		0			0
$319$	810.652	−	1404.09i	0.142281	−	0.246439i
$320$	−249.736	−	432.555i	−0.0436271	−	0.0755643i
$321$		0			0
$322$	−1195.44	+	744.856i	−0.206892	+	0.128910i
$323$	2245.48			0.386818
$324$		0			0
$325$	410.336	−	710.723i	0.0700349	−	0.121304i
$326$	1323.57	−	2292.48i	0.224864	−	0.389475i
$327$		0			0
$328$	−94.2593			−0.0158677
$329$	35.4681	+	1054.59i	0.00594352	+	0.176722i
$330$		0			0
$331$	4953.58	+	8579.85i	0.822578	+	1.42475i	0.903756	+	0.428047i	$0.140798\pi$
$331$	4953.58	+	8579.85i	0.822578	+	1.42475i	−0.0811782	+	0.996700i	$0.525868\pi$
$332$	745.870	−	1291.88i	0.123298	−	0.213558i
$333$		0			0
$334$	2071.29	+	3587.57i	0.339329	+	0.587734i
$335$	1922.01			0.313465
$336$		0			0
$337$	−8206.45			−1.32651			−0.663255	−	0.748393i	$-0.730826\pi$
$337$	−8206.45			−1.32651			−0.663255	+	0.748393i	$0.730826\pi$
$338$	−2033.05	−	3521.35i	−0.327170	−	0.566675i
$339$		0			0
$340$	746.868	−	1293.61i	0.119131	−	0.206341i
$341$	705.256	+	1221.54i	0.111999	+	0.193988i
$342$		0			0
$343$	639.611	+	6320.17i	0.100687	+	0.994918i
$344$	−1940.69			−0.304171
$345$		0			0
$346$	3085.28	−	5343.86i	0.479380	−	0.830311i
$347$	−980.017	+	1697.44i	−0.151614	+	0.262603i	−0.931821	−	0.362918i	$-0.881780\pi$
$347$	−980.017	+	1697.44i	−0.151614	+	0.262603i	0.780207	+	0.625522i	$0.215114\pi$
$348$		0			0
$349$	3808.47			0.584134			0.292067	−	0.956398i	$-0.405657\pi$
$349$	3808.47			0.584134			0.292067	+	0.956398i	$0.405657\pi$
$350$	2094.74	+	1117.25i	0.319910	+	0.170628i
$351$		0			0
$352$	−89.3415	−	154.744i	−0.0135282	−	0.0234315i
$353$	4695.51	−	8132.86i	0.707979	−	1.22626i	−0.257626	−	0.966245i	$-0.582940\pi$
$353$	4695.51	−	8132.86i	0.707979	−	1.22626i	0.965606	−	0.260011i	$-0.0837263\pi$
$354$		0			0
$355$	2129.48	+	3688.37i	0.318370	+	0.551433i
$356$	4475.39			0.666279
$357$		0			0
$358$	−2895.15			−0.427411
$359$	6027.50	+	10439.9i	0.886126	+	1.53481i	0.844418	+	0.535685i	$0.179946\pi$
$359$	6027.50	+	10439.9i	0.886126	+	1.53481i	0.0417077	+	0.999130i	$0.486720\pi$
$360$		0			0
$361$	2328.41	−	4032.92i	0.339467	−	0.587975i
$362$	−1683.16	−	2915.33i	−0.244379	−	0.423277i
$363$		0			0
$364$	−805.064	+	501.620i	−0.115925	+	0.0722309i
$365$	−2246.94			−0.322220
$366$		0			0
$367$	2923.66	−	5063.94i	0.415842	−	0.720260i	−0.579674	−	0.814848i	$-0.696820\pi$
$367$	2923.66	−	5063.94i	0.415842	−	0.720260i	0.995516	+	0.0945886i	$0.0301536\pi$
$368$	304.208	−	526.904i	0.0430923	−	0.0746380i
$369$		0			0
$370$	6409.96			0.900644
$371$	−3626.91	+	2259.86i	−0.507546	+	0.316242i
$372$		0			0
$373$	4066.66	+	7043.66i	0.564513	+	0.977766i	0.997095	+	0.0761709i	$0.0242695\pi$
$373$	4066.66	+	7043.66i	0.564513	+	0.977766i	−0.432581	+	0.901595i	$0.642397\pi$
$374$	267.188	−	462.783i	0.0369410	−	0.0639837i
$375$		0			0
$376$	−227.899	−	394.733i	−0.0312580	−	0.0541404i
$377$	3717.79			0.507894
$378$		0			0
$379$	−7913.69			−1.07256			−0.536278	−	0.844041i	$-0.680170\pi$
$379$	−7913.69			−1.07256			−0.536278	+	0.844041i	$0.680170\pi$
$380$	732.467	+	1268.67i	0.0988810	+	0.171267i
$381$		0			0
$382$	−2094.23	+	3627.31i	−0.280498	+	0.485836i
$383$	949.004	+	1643.72i	0.126611	+	0.219296i	0.922361	−	0.386328i	$-0.126257\pi$
$383$	949.004	+	1643.72i	0.126611	+	0.219296i	−0.795751	+	0.605624i	$0.792923\pi$
$384$		0			0
$385$	−712.112	−	379.813i	−0.0942665	−	0.0502782i
$386$	6838.85			0.901783
$387$		0			0
$388$	−787.524	+	1364.03i	−0.103042	+	0.178475i
$389$	751.212	−	1301.14i	0.0979124	−	0.169589i	−0.812908	−	0.582392i	$-0.802117\pi$
$389$	751.212	−	1301.14i	0.0979124	−	0.169589i	0.910820	+	0.412803i	$0.135450\pi$
$390$		0			0
$391$	1819.55			0.235342
$392$	−1528.56	−	2278.82i	−0.196949	−	0.293617i
$393$		0			0
$394$	−3517.08	−	6091.76i	−0.449715	−	0.778930i
$395$	352.195	−	610.020i	0.0448630	−	0.0777050i
$396$		0			0
$397$	−7327.84	−	12692.2i	−0.926382	−	1.60454i	−0.789323	−	0.613978i	$-0.789568\pi$
$397$	−7327.84	−	12692.2i	−0.926382	−	1.60454i	−0.137059	−	0.990563i	$-0.543765\pi$
$398$	6264.13			0.788926
$399$		0			0
$400$	−1025.50			−0.128188
$401$	85.1443	+	147.474i	0.0106033	+	0.0183654i	0.871278	−	0.490789i	$-0.163291\pi$
$401$	85.1443	+	147.474i	0.0106033	+	0.0183654i	−0.860675	+	0.509155i	$0.829958\pi$
$402$		0			0
$403$	−1617.21	+	2801.10i	−0.199899	+	0.346235i
$404$	2191.20	+	3795.27i	0.269842	+	0.467381i
$405$		0			0
$406$	361.506	+	10748.9i	0.0441903	+	1.31393i
$407$	2293.13			0.279278
$408$		0			0
$409$	5733.15	−	9930.11i	0.693120	−	1.20052i	−0.277690	−	0.960671i	$-0.589569\pi$
$409$	5733.15	−	9930.11i	0.693120	−	1.20052i	0.970810	−	0.239849i	$-0.0770978\pi$
$410$	91.9528	−	159.267i	0.0110762	−	0.0191845i
$411$		0			0
$412$	−968.890			−0.115859
$413$	8143.18	−	5073.86i	0.970217	−	0.604524i
$414$		0			0
$415$	1455.24	+	2520.55i	0.172132	+	0.298142i
$416$	204.868	−	354.842i	0.0241454	−	0.0418210i
$417$		0			0
$418$	262.036	+	453.859i	0.0306617	+	0.0531076i
$419$	−9329.18			−1.08773			−0.543866	−	0.839172i	$-0.683040\pi$
$419$	−9329.18			−1.08773			−0.543866	+	0.839172i	$0.683040\pi$
$420$		0			0
$421$	4184.24			0.484388			0.242194	−	0.970228i	$-0.422133\pi$
$421$	4184.24			0.484388			0.242194	+	0.970228i	$0.422133\pi$
$422$	−3088.38	−	5349.23i	−0.356256	−	0.617053i
$423$		0			0
$424$	922.954	−	1598.60i	0.105714	−	0.183101i
$425$	−1533.45	−	2656.01i	−0.175019	−	0.303142i
$426$		0			0
$427$	161.199	+	4793.03i	0.0182693	+	0.543210i
$428$	396.510			0.0447805
$429$		0			0
$430$	1893.20	−	3279.12i	0.212321	−	0.367752i
$431$	3643.40	−	6310.56i	0.407185	−	0.705265i	−0.587388	−	0.809305i	$-0.699844\pi$
$431$	3643.40	−	6310.56i	0.407185	−	0.705265i	0.994573	+	0.104041i	$0.0331772\pi$
$432$		0			0
$433$	−5314.89			−0.589879			−0.294939	−	0.955516i	$-0.595299\pi$
$433$	−5314.89			−0.589879			−0.294939	+	0.955516i	$0.595299\pi$
$434$	−8255.77	−	4403.31i	−0.913110	−	0.487018i
$435$		0			0
$436$	1838.28	+	3183.99i	0.201921	+	0.349738i
$437$	−892.233	+	1545.39i	−0.0976689	+	0.169167i
$438$		0			0
$439$	−4273.05	−	7401.14i	−0.464560	−	0.804641i	0.534622	−	0.845091i	$-0.320454\pi$
$439$	−4273.05	−	7401.14i	−0.464560	−	0.804641i	−0.999182	+	0.0404505i	$0.987121\pi$
$440$	348.622			0.0377725
$441$		0			0
$442$	1225.37			0.131866
$443$	5398.26	+	9350.06i	0.578960	+	1.00279i	0.995599	+	0.0937156i	$0.0298745\pi$
$443$	5398.26	+	9350.06i	0.578960	+	1.00279i	−0.416639	+	0.909072i	$0.636792\pi$
$444$		0			0
$445$	−4365.88	+	7561.93i	−0.465085	+	0.805551i
$446$	4399.79	+	7620.66i	0.467121	+	0.809078i
$447$		0			0
$448$	1045.84	+	557.810i	0.110293	+	0.0588260i
$449$	−3617.92			−0.380267			−0.190134	−	0.981758i	$-0.560892\pi$
$449$	−3617.92			−0.380267			−0.190134	+	0.981758i	$0.560892\pi$
$450$		0			0
$451$	32.8956	−	56.9768i	0.00343457	−	0.00594886i
$452$	−4242.13	+	7347.59i	−0.441445	+	0.764605i
$453$		0			0
$454$	−6574.76			−0.679667
$455$	−62.2071	−	1849.64i	−0.00640948	−	0.190577i
$456$		0			0
$457$	3549.67	+	6148.21i	0.363340	+	0.629324i	0.988508	−	0.151166i	$-0.0483029\pi$
$457$	3549.67	+	6148.21i	0.363340	+	0.629324i	−0.625168	+	0.780490i	$0.714970\pi$
$458$	3013.51	−	5219.55i	0.307450	−	0.532519i
$459$		0			0
$460$	593.529	+	1028.02i	0.0601597	+	0.104200i
$461$	−16652.8			−1.68243			−0.841215	−	0.540700i	$-0.818159\pi$
$461$	−16652.8			−1.68243			−0.841215	+	0.540700i	$0.818159\pi$
$462$		0			0
$463$	−15394.4			−1.54523			−0.772614	−	0.634876i	$-0.781051\pi$
$463$	−15394.4			−1.54523			−0.772614	+	0.634876i	$0.781051\pi$
$464$	−2322.85	−	4023.29i	−0.232404	−	0.402536i
$465$		0			0
$466$	5081.61	−	8801.61i	0.505152	−	0.874950i
$467$	8307.83	+	14389.6i	0.823214	+	1.42585i	0.903277	+	0.429058i	$0.141154\pi$
$467$	8307.83	+	14389.6i	0.823214	+	1.42585i	−0.0800634	+	0.996790i	$0.525512\pi$
$468$		0			0
$469$	−3871.17	+	2412.05i	−0.381139	+	0.237480i
$470$	889.290			0.0872763
$471$		0			0
$472$	−2072.23	+	3589.20i	−0.202081	+	0.350014i
$473$	677.281	−	1173.09i	0.0658381	−	0.114035i
$474$		0			0
$475$	3007.76			0.290538
$476$	119.151	+	3542.78i	0.0114733	+	0.341141i
$477$		0			0
$478$	3087.38	+	5347.50i	0.295426	+	0.511693i
$479$	839.186	−	1453.51i	0.0800489	−	0.138649i	−0.823222	−	0.567720i	$-0.807826\pi$
$479$	839.186	−	1453.51i	0.0800489	−	0.138649i	0.903271	+	0.429071i	$0.141159\pi$
$480$		0			0
$481$	2629.17	+	4553.86i	0.249230	+	0.431680i
$482$	2433.30			0.229946
$483$		0			0
$484$	−5199.28			−0.488287
$485$	−1536.51	−	2661.31i	−0.143854	−	0.249163i
$486$		0			0
$487$	4729.44	−	8191.64i	0.440065	−	0.762214i	−0.557629	−	0.830090i	$-0.688289\pi$
$487$	4729.44	−	8191.64i	0.440065	−	0.762214i	0.997694	+	0.0678758i	$0.0216222\pi$
$488$	−1035.78	−	1794.03i	−0.0960812	−	0.166418i
$489$		0			0
$490$	5341.61	−	359.706i	0.492468	−	0.0331630i
$491$	−12084.6			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$491$	−12084.6			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$492$		0			0
$493$	6946.78	−	12032.2i	0.634620	−	1.09919i
$494$	−600.871	+	1040.74i	−0.0547256	+	0.0947876i
$495$		0			0
$496$	4041.69			0.365882
$497$	−8917.80	−	4756.41i	−0.804865	−	0.429284i
$498$		0			0
$499$	−5576.33	−	9658.48i	−0.500262	−	0.866479i	−1.00000		0.000302529i	$-0.999904\pi$
$499$	−5576.33	−	9658.48i	−0.500262	−	0.866479i	0.499738	−	0.866177i	$-0.333430\pi$
$500$	2951.47	−	5112.09i	0.263987	−	0.457239i
$501$		0			0
$502$	−2649.43	−	4588.95i	−0.235557	−	0.407997i
$503$	18353.8			1.62695			0.813476	−	0.581598i	$-0.197572\pi$
$503$	18353.8			1.62695			0.813476	+	0.581598i	$0.197572\pi$
$504$		0			0
$505$	−8550.34			−0.753436
$506$	212.332	+	367.769i	0.0186547	+	0.0323109i
$507$		0			0
$508$	2500.65	−	4331.26i	0.218403	−	0.378284i
$509$	−8839.43	−	15310.3i	−0.769746	−	1.33324i	−0.937700	−	0.347445i	$-0.887049\pi$
$509$	−8839.43	−	15310.3i	−0.769746	−	1.33324i	0.167954	−	0.985795i	$-0.446284\pi$
$510$		0			0
$511$	4525.61	−	2819.82i	0.391783	−	0.244112i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	715.978	−	1240.11i	0.0614405	−	0.106418i
$515$	945.182	−	1637.10i	0.0808732	−	0.140076i
$516$		0			0
$517$	318.138			0.0270633
$518$	−12910.4	+	8044.25i	−1.09508	+	0.682324i
$519$		0			0
$520$	399.710	+	692.318i	0.0337085	+	0.0583849i
$521$	−3844.84	+	6659.46i	−0.323312	+	0.559993i	−0.981169	−	0.193150i	$-0.938130\pi$
$521$	−3844.84	+	6659.46i	−0.323312	+	0.559993i	0.657857	+	0.753143i	$0.271463\pi$
$522$		0			0
$523$	−10654.7	−	18454.5i	−0.890819	−	1.54294i	−0.838896	−	0.544292i	$-0.816798\pi$
$523$	−10654.7	−	18454.5i	−0.890819	−	1.54294i	−0.0519230	−	0.998651i	$-0.516535\pi$
$524$	−453.859			−0.0378376
$525$		0			0
$526$	−12694.0			−1.05225
$527$	6043.61	+	10467.8i	0.499552	+	0.865249i
$528$		0			0
$529$	5360.51	−	9284.68i	0.440578	−	0.763103i
$530$	1800.74	+	3118.97i	0.147583	+	0.255622i
$531$		0			0
$532$	−3067.41	−	1636.04i	−0.249979	−	0.133329i
$533$	150.865			0.0122602
$534$		0			0
$535$	−386.808	+	669.971i	−0.0312583	+	0.0541409i
$536$	985.112	−	1706.26i	0.0793850	−	0.137499i
$537$		0			0
$538$	−15679.7			−1.25650
$539$	1910.93	−	128.683i	0.152708	−	0.0102834i
$540$		0			0
$541$	2262.65	+	3919.03i	0.179813	+	0.311446i	0.941816	−	0.336128i	$-0.109117\pi$
$541$	2262.65	+	3919.03i	0.179813	+	0.311446i	−0.762003	+	0.647573i	$0.775784\pi$
$542$	3280.56	−	5682.09i	0.259985	−	0.450307i
$543$		0			0
$544$	−765.602	−	1326.06i	−0.0603399	−	0.104512i
$545$	−7173.19			−0.563790
$546$		0			0
$547$	−15313.6			−1.19701			−0.598504	−	0.801120i	$-0.704238\pi$
$547$	−15313.6			−1.19701			−0.598504	+	0.801120i	$0.704238\pi$
$548$	−30.6299	−	53.0526i	−0.00238767	−	0.00413557i
$549$		0			0
$550$	357.890	−	619.883i	0.0277463	−	0.0480580i
$551$	6812.84	+	11800.2i	0.526745	+	0.912349i
$552$		0			0
$553$	56.1873	+	1670.65i	0.00432066	+	0.128469i
$554$	−15066.2			−1.15542
$555$		0			0
$556$	5079.62	−	8798.15i	0.387453	−	0.671088i
$557$	12455.6	−	21573.7i	0.947503	−	1.64112i	0.196844	−	0.980435i	$-0.436931\pi$
$557$	12455.6	−	21573.7i	0.947503	−	1.64112i	0.750660	−	0.660689i	$-0.229736\pi$
$558$		0			0
$559$	3106.13			0.235018
$560$	−1962.76	+	1222.96i	−0.148110	+	0.0922847i
$561$		0			0
$562$	−3454.53	−	5983.41i	−0.259289	−	0.449102i
$563$	−3212.07	+	5563.48i	−0.240449	+	0.416470i	−0.960842	−	0.277096i	$-0.910628\pi$
$563$	−3212.07	+	5563.48i	−0.240449	+	0.416470i	0.720393	+	0.693566i	$0.243961\pi$
$564$		0			0
$565$	−8276.66	−	14335.6i	−0.616286	−	1.06744i
$566$	5289.13			0.392789
$567$		0			0
$568$	4365.80			0.322508
$569$	4793.27	+	8302.19i	0.353153	+	0.611680i	0.986800	−	0.161943i	$-0.0517760\pi$
$569$	4793.27	+	8302.19i	0.353153	+	0.611680i	−0.633647	+	0.773623i	$0.718443\pi$
$570$		0			0
$571$	1521.81	−	2635.86i	0.111534	−	0.193183i	−0.804855	−	0.593472i	$-0.797757\pi$
$571$	1521.81	−	2635.86i	0.111534	−	0.193183i	0.916389	+	0.400289i	$0.131090\pi$
$572$	142.994	+	247.673i	0.0104526	+	0.0181044i
$573$		0			0
$574$	14.6696	+	436.180i	0.00106672	+	0.0317174i
$575$	2437.23			0.176765
$576$		0			0
$577$	2121.12	−	3673.89i	0.153039	−	0.265071i	−0.779304	−	0.626646i	$-0.784427\pi$
$577$	2121.12	−	3673.89i	0.153039	−	0.265071i	0.932343	+	0.361574i	$0.117761\pi$
$578$	−2623.37	+	4543.80i	−0.188785	+	0.326985i
$579$		0			0
$580$	9064.04			0.648903
$581$	−6094.21	−	3250.42i	−0.435164	−	0.232100i
$582$		0			0
$583$	644.204	+	1115.79i	0.0457636	+	0.0792650i
$584$	−1151.65	+	1994.72i	−0.0816021	+	0.141339i
$585$		0			0
$586$	−8580.20	−	14861.3i	−0.604855	−	1.04764i
$587$	−13992.3			−0.983854			−0.491927	−	0.870637i	$-0.663707\pi$
$587$	−13992.3			−0.983854			−0.491927	+	0.870637i	$0.663707\pi$
$588$		0			0
$589$	−11854.2			−0.829273
$590$	−4043.04	−	7002.76i	−0.282118	−	0.488642i
$591$		0			0
$592$	3285.37	−	5690.43i	0.228088	−	0.395060i
$593$	−9155.07	−	15857.0i	−0.633986	−	1.09810i	−0.986729	−	0.162375i	$-0.948084\pi$
$593$	−9155.07	−	15857.0i	−0.633986	−	1.09810i	0.352743	−	0.935720i	$-0.385249\pi$
$594$		0			0
$595$	−6102.37	−	3254.77i	−0.420458	−	0.224256i
$596$	348.677			0.0239637
$597$		0			0
$598$	−486.895	+	843.327i	−0.0332953	+	0.0576692i
$599$	4911.78	−	8507.45i	0.335042	−	0.580309i	−0.648451	−	0.761256i	$-0.724583\pi$
$599$	4911.78	−	8507.45i	0.335042	−	0.580309i	0.983493	+	0.180947i	$0.0579163\pi$
$600$		0			0
$601$	9284.95			0.630184			0.315092	−	0.949061i	$-0.397965\pi$
$601$	9284.95			0.630184			0.315092	+	0.949061i	$0.397965\pi$
$602$	302.030	+	8980.43i	0.0204482	+	0.607999i
$603$		0			0
$604$	1992.12	+	3450.45i	0.134202	+	0.232445i
$605$	5072.06	−	8785.07i	0.340841	−	0.590353i
$606$		0			0
$607$	−1297.19	−	2246.80i	−0.0867403	−	0.150239i	0.819391	−	0.573235i	$-0.194312\pi$
$607$	−1297.19	−	2246.80i	−0.0867403	−	0.150239i	−0.906131	+	0.422996i	$0.860978\pi$
$608$	1501.68			0.100166
$609$		0			0
$610$	4041.75			0.268271
$611$	364.759	+	631.782i	0.0241515	+	0.0418317i
$612$		0			0
$613$	−242.268	+	419.621i	−0.0159627	+	0.0276482i	−0.873896	−	0.486112i	$-0.838415\pi$
$613$	−242.268	+	419.621i	−0.0159627	+	0.0276482i	0.857934	+	0.513760i	$0.171748\pi$
$614$	9077.13	+	15722.1i	0.596618	+	1.03337i
$615$		0			0
$616$	−702.166	+	437.506i	−0.0459271	+	0.0286163i
$617$	−14454.7			−0.943154			−0.471577	−	0.881825i	$-0.656315\pi$
$617$	−14454.7			−0.943154			−0.471577	+	0.881825i	$0.656315\pi$
$618$		0			0
$619$	−8395.18	+	14540.9i	−0.545122	+	0.944179i	0.453477	+	0.891268i	$0.350183\pi$
$619$	−8395.18	+	14540.9i	−0.545122	+	0.944179i	−0.998599	+	0.0529114i	$0.983150\pi$
$620$	−3942.80	+	6829.12i	−0.255398	+	0.442362i
$621$		0			0
$622$	−14480.8			−0.933487
$623$	−696.508	−	20709.7i	−0.0447914	−	1.33181i
$624$		0			0
$625$	1752.64	+	3035.65i	0.112169	+	0.194282i
$626$	−9954.09	+	17241.0i	−0.635536	+	1.10078i
$627$		0			0
$628$	4326.62	+	7493.93i	0.274922	+	0.476179i
$629$	19650.7			1.24567
$630$		0			0
$631$	28338.7			1.78787			0.893935	−	0.448197i	$-0.147934\pi$
$631$	28338.7			1.78787			0.893935	+	0.448197i	$0.147934\pi$
$632$	−361.030	−	625.322i	−0.0227231	−	0.0393575i
$633$		0			0
$634$	−4551.60	+	7883.60i	−0.285121	+	0.493845i
$635$	4878.93	+	8450.55i	0.304904	+	0.528110i
$636$		0			0
$637$	2446.51	+	3647.32i	0.152173	+	0.226864i
$638$	3242.61			0.201216
$639$		0			0
$640$	499.472	−	865.110i	0.0308490	−	0.0534320i
$641$	5183.34	−	8977.81i	0.319391	−	0.553201i	−0.660970	−	0.750412i	$-0.729855\pi$
$641$	5183.34	−	8977.81i	0.319391	−	0.553201i	0.980361	+	0.197211i	$0.0631883\pi$
$642$		0			0
$643$	−6466.39			−0.396593			−0.198297	−	0.980142i	$-0.563541\pi$
$643$	−6466.39			−0.396593			−0.198297	+	0.980142i	$0.563541\pi$
$644$	−2485.57	−	1325.71i	−0.152089	−	0.0811182i
$645$		0			0
$646$	2245.48	+	3889.29i	0.136761	+	0.236877i
$647$	−8156.73	+	14127.9i	−0.495632	+	0.858460i	−0.999987	−	0.00503632i	$-0.998397\pi$
$647$	−8156.73	+	14127.9i	−0.495632	+	0.858460i	0.504355	+	0.863496i	$0.331730\pi$
$648$		0			0
$649$	−1446.37	−	2505.19i	−0.0874810	−	0.151522i
$650$	1641.34			0.0990443
$651$		0			0
$652$	5294.27			0.318005
$653$	−5052.60	−	8751.35i	−0.302792	−	0.524452i	0.673975	−	0.738754i	$-0.264586\pi$
$653$	−5052.60	−	8751.35i	−0.302792	−	0.524452i	−0.976767	+	0.214302i	$0.931252\pi$
$654$		0			0
$655$	442.753	−	766.872i	0.0264119	−	0.0457468i
$656$	−94.2593	−	163.262i	−0.00561007	−	0.00971693i
$657$		0			0
$658$	−1791.14	+	1116.02i	−0.106118	+	0.0661202i
$659$	13873.0			0.820055			0.410028	−	0.912073i	$-0.365519\pi$
$659$	13873.0			0.820055			0.410028	+	0.912073i	$0.365519\pi$
$660$		0			0
$661$	11937.6	−	20676.6i	0.702452	−	1.21668i	−0.265152	−	0.964207i	$-0.585422\pi$
$661$	11937.6	−	20676.6i	0.702452	−	1.21668i	0.967603	−	0.252475i	$-0.0812446\pi$
$662$	−9907.16	+	17159.7i	−0.581651	+	1.00745i
$663$		0			0
$664$	2983.48			0.174370
$665$	5756.71	−	3586.90i	0.335693	−	0.209164i
$666$		0			0
$667$	5520.55	+	9561.87i	0.320474	+	0.555078i
$668$	−4142.57	+	7175.15i	−0.239942	+	0.415591i
$669$		0			0
$670$	1922.01	+	3329.03i	0.110827	+	0.191958i
$671$	1445.91			0.0831875
$672$		0			0
$673$	−19177.0			−1.09839			−0.549196	−	0.835694i	$-0.685066\pi$
$673$	−19177.0			−1.09839			−0.549196	+	0.835694i	$0.685066\pi$
$674$	−8206.45	−	14214.0i	−0.468992	−	0.812318i
$675$		0			0
$676$	4066.10	−	7042.70i	0.231344	−	0.400700i
$677$	−3088.77	−	5349.90i	−0.175349	−	0.303713i	0.764933	−	0.644109i	$-0.222772\pi$
$677$	−3088.77	−	5349.90i	−0.175349	−	0.303713i	−0.940282	+	0.340397i	$0.889439\pi$
$678$		0			0
$679$	6434.55	+	3431.94i	0.363675	+	0.193970i
$680$	2987.47			0.168477
$681$		0			0
$682$	−1410.51	+	2443.08i	−0.0791955	+	0.137171i
$683$	11887.8	−	20590.2i	0.665991	−	1.15353i	−0.313024	−	0.949745i	$-0.601342\pi$
$683$	11887.8	−	20590.2i	0.665991	−	1.15353i	0.979015	−	0.203786i	$-0.0653246\pi$
$684$		0			0
$685$	119.522			0.00666670
$686$	−10307.2	+	7428.00i	−0.573662	+	0.413415i
$687$		0			0
$688$	−1940.69	−	3361.37i	−0.107541	−	0.186266i
$689$	−1477.22	+	2558.61i	−0.0816799	+	0.141474i
$690$		0			0
$691$	4217.58	+	7305.06i	0.232192	+	0.402168i	0.958453	−	0.285251i	$-0.0920771\pi$
$691$	4217.58	+	7305.06i	0.232192	+	0.402168i	−0.726261	+	0.687419i	$0.758744\pi$
$692$	12341.1			0.677946
$693$		0			0
$694$	−3920.07			−0.214415
$695$	9910.64	+	17165.7i	0.540910	+	0.936883i
$696$		0			0
$697$	281.895	−	488.256i	0.0153193	−	0.0265338i
$698$	3808.47	+	6596.46i	0.206522	+	0.357707i
$699$		0			0
$700$	159.599	+	4745.45i	0.00861755	+	0.256230i
$701$	−9383.79			−0.505594			−0.252797	−	0.967519i	$-0.581350\pi$
$701$	−9383.79			−0.505594			−0.252797	+	0.967519i	$0.581350\pi$
$702$		0			0
$703$	−9635.89	+	16689.8i	−0.516962	+	0.895405i
$704$	178.683	−	309.488i	0.00956587	−	0.0165686i
$705$		0			0
$706$	18782.0			1.00123
$707$	17221.4	−	10730.3i	0.916093	−	0.570800i
$708$		0			0
$709$	11425.9	+	19790.3i	0.605233	+	1.04829i	0.992015	+	0.126122i	$0.0402532\pi$
$709$	11425.9	+	19790.3i	0.605233	+	1.04829i	−0.386782	+	0.922171i	$0.626413\pi$
$710$	−4258.97	+	7376.75i	−0.225121	+	0.389922i
$711$		0			0
$712$	4475.39	+	7751.61i	0.235565	+	0.408011i
$713$	−9605.60			−0.504534
$714$		0			0
$715$	−557.980			−0.0291850
$716$	−2895.15	−	5014.54i	−0.151113	−	0.261735i
$717$		0			0
$718$	−12055.0	+	20879.9i	−0.626585	+	1.08528i
$719$	264.286	+	457.757i	0.0137082	+	0.0237433i	0.872798	−	0.488081i	$-0.162303\pi$
$719$	264.286	+	457.757i	0.0137082	+	0.0237433i	−0.859090	+	0.511825i	$0.828970\pi$
$720$		0			0
$721$	150.789	+	4483.49i	0.00778873	+	0.231586i
$722$	9313.63			0.480079
$723$		0			0
$724$	3366.33	−	5830.65i	0.172802	−	0.299302i
$725$	9305.00	−	16116.7i	0.476661	−	0.825601i
$726$		0			0
$727$	−19525.3			−0.996083			−0.498041	−	0.867153i	$-0.665947\pi$
$727$	−19525.3			−0.996083			−0.498041	+	0.867153i	$0.665947\pi$
$728$	−1673.90	−	892.792i	−0.0852180	−	0.0454520i
$729$		0			0
$730$	−2246.94	−	3891.81i	−0.113922	−	0.197318i
$731$	5803.88	−	10052.6i	0.293659	−	0.508631i
$732$		0			0
$733$	13680.3	+	23695.0i	0.689351	+	1.19399i	0.972048	+	0.234782i	$0.0754377\pi$
$733$	13680.3	+	23695.0i	0.689351	+	1.19399i	−0.282697	+	0.959209i	$0.591229\pi$
$734$	11694.7			0.588090
$735$		0			0
$736$	1216.83			0.0609417
$737$	687.589	+	1190.94i	0.0343659	+	0.0595235i
$738$		0			0
$739$	2265.66	−	3924.24i	0.112779	−	0.195339i	−0.804111	−	0.594480i	$-0.797358\pi$
$739$	2265.66	−	3924.24i	0.112779	−	0.195339i	0.916890	+	0.399140i	$0.130691\pi$
$740$	6409.96	+	11102.4i	0.318426	+	0.551529i
$741$		0			0
$742$	−7541.09	−	4022.13i	−0.373103	−	0.198999i
$743$	21343.8			1.05387			0.526937	−	0.849904i	$-0.323340\pi$
$743$	21343.8			1.05387			0.526937	+	0.849904i	$0.323340\pi$
$744$		0			0
$745$	−340.145	+	589.148i	−0.0167274	+	0.0289728i
$746$	−8133.31	+	14087.3i	−0.399171	+	0.691385i
$747$		0			0
$748$	1068.75			0.0522425
$749$	−61.7092	−	1834.83i	−0.00301042	−	0.0895104i
$750$		0			0
$751$	−5657.26	−	9798.67i	−0.274882	−	0.476110i	0.695223	−	0.718794i	$-0.255305\pi$
$751$	−5657.26	−	9798.67i	−0.274882	−	0.476110i	−0.970105	+	0.242684i	$0.921972\pi$
$752$	455.798	−	789.465i	0.0221027	−	0.0382830i
$753$		0			0
$754$	3717.79	+	6439.40i	0.179568	+	0.311020i
$755$	−7773.49			−0.374711
$756$		0			0
$757$	−10211.6			−0.490285			−0.245142	−	0.969487i	$-0.578835\pi$
$757$	−10211.6			−0.490285			−0.245142	+	0.969487i	$0.578835\pi$
$758$	−7913.69	−	13706.9i	−0.379206	−	0.656804i
$759$		0			0
$760$	−1464.93	+	2537.34i	−0.0699194	+	0.121104i
$761$	2493.48	+	4318.84i	0.118776	+	0.205726i	0.919283	−	0.393597i	$-0.128770\pi$
$761$	2493.48	+	4318.84i	0.118776	+	0.205726i	−0.800507	+	0.599324i	$0.795436\pi$
$762$		0			0
$763$	14447.7	−	9002.07i	0.685505	−	0.427125i
$764$	−8376.92			−0.396684
$765$		0			0
$766$	−1898.01	+	3287.45i	−0.0895272	+	0.155066i
$767$	3316.66	−	5744.63i	0.156138	−	0.270439i
$768$		0			0
$769$	−10481.8			−0.491526			−0.245763	−	0.969330i	$-0.579039\pi$
$769$	−10481.8			−0.491526			−0.245763	+	0.969330i	$0.579039\pi$
$770$	−54.2562	−	1613.23i	−0.00253930	−	0.0755022i
$771$		0			0
$772$	6838.85	+	11845.2i	0.318828	+	0.552227i
$773$	−4818.10	+	8345.19i	−0.224185	+	0.388300i	−0.956075	−	0.293123i	$-0.905305\pi$
$773$	−4818.10	+	8345.19i	−0.224185	+	0.388300i	0.731890	+	0.681423i	$0.238639\pi$
$774$		0			0
$775$	8095.23	+	14021.3i	0.375212	+	0.649886i
$776$	−3150.10			−0.145724
$777$		0			0
$778$	3004.85			0.138469
$779$	276.459	+	478.842i	0.0127153	+	0.0220235i
$780$		0			0
$781$	−1523.62	+	2638.99i	−0.0698072	+	0.120910i
$782$	1819.55	+	3151.55i	0.0832059	+	0.144117i
$783$		0			0
$784$	2418.47	−	4926.37i	0.110171	−	0.224416i
$785$	−16883.0			−0.767618
$786$		0			0
$787$	2946.89	−	5104.17i	0.133476	−	0.231187i	−0.791538	−	0.611119i	$-0.790720\pi$
$787$	2946.89	−	5104.17i	0.133476	−	0.231187i	0.925014	+	0.379933i	$0.124053\pi$
$788$	7034.15	−	12183.5i	0.317997	−	0.550786i
$789$		0			0
$790$	1408.78			0.0634458
$791$	34660.8	+	18486.7i	1.55802	+	0.830989i
$792$		0			0
$793$	1657.80	+	2871.39i	0.0742374	+	0.128583i
$794$	14655.7	−	25384.4i	0.655051	−	1.13458i
$795$		0			0
$796$	6264.13	+	10849.8i	0.278927	+	0.483117i
$797$	32641.4			1.45071			0.725357	−	0.688373i	$-0.241675\pi$
$797$	32641.4			1.45071			0.725357	+	0.688373i	$0.241675\pi$
$798$		0			0
$799$	2726.25			0.120711
$800$	−1025.50	−	1776.22i	−0.0453211	−	0.0784985i
$801$		0			0
$802$	−170.289	+	294.949i	−0.00749763	+	0.0129863i
$803$	−803.829	−	1392.27i	−0.0353257	−	0.0611859i
$804$		0			0
$805$	4664.75	−	2906.52i	0.204237	−	0.127256i
$806$	−6468.86			−0.282699
$807$		0			0
$808$	−4382.40	+	7590.54i	−0.190807	+	0.330488i
$809$	9017.11	−	15618.1i	0.391872	−	0.678743i	−0.600824	−	0.799381i	$-0.705161\pi$
$809$	9017.11	−	15618.1i	0.391872	−	0.678743i	0.992697	+	0.120638i	$0.0384942\pi$
$810$		0			0
$811$	39379.6			1.70506			0.852531	−	0.522677i	$-0.175067\pi$
$811$	39379.6			1.70506			0.852531	+	0.522677i	$0.175067\pi$
$812$	−18256.1	+	11375.0i	−0.788993	+	0.491606i
$813$		0			0
$814$	2293.13	+	3971.81i	0.0987396	+	0.171022i
$815$	−5164.72	+	8945.56i	−0.221978	+	0.384477i
$816$		0			0
$817$	5691.97	+	9858.79i	0.243742	+	0.422173i
$818$	22932.6			0.980220
$819$		0			0
$820$	367.811			0.0156641
$821$	13818.4	+	23934.2i	0.587413	+	1.01743i	0.994570	+	0.104071i	$0.0331870\pi$
$821$	13818.4	+	23934.2i	0.587413	+	1.01743i	−0.407157	+	0.913358i	$0.633480\pi$
$822$		0			0
$823$	−4696.30	+	8134.24i	−0.198910	+	0.344522i	−0.948175	−	0.317748i	$-0.897073\pi$
$823$	−4696.30	+	8134.24i	−0.198910	+	0.344522i	0.749265	+	0.662270i	$0.230407\pi$
$824$	−968.890	−	1678.17i	−0.0409622	−	0.0709487i
$825$		0			0
$826$	16931.4	+	9030.54i	0.713217	+	0.380403i
$827$	−40070.2			−1.68486			−0.842428	−	0.538809i	$-0.818874\pi$
$827$	−40070.2			−1.68486			−0.842428	+	0.538809i	$0.818874\pi$
$828$		0			0
$829$	−14119.9	+	24456.5i	−0.591563	+	1.02462i	0.402459	+	0.915438i	$0.368156\pi$
$829$	−14119.9	+	24456.5i	−0.591563	+	1.02462i	−0.994022	+	0.109180i	$0.965178\pi$
$830$	−2910.48	+	5041.09i	−0.121716	+	0.210818i
$831$		0			0
$832$	819.472			0.0341467
$833$	16375.5	−	1102.73i	0.681126	−	0.0458672i
$834$		0			0
$835$	−8082.41	−	13999.1i	−0.334974	−	0.580192i
$836$	−524.071	+	907.718i	−0.0216811	+	0.0375527i
$837$		0			0
$838$	−9329.18	−	16158.6i	−0.384572	−	0.666098i
$839$	−24586.0			−1.01168			−0.505842	−	0.862626i	$-0.668818\pi$
$839$	−24586.0			−1.01168			−0.505842	+	0.862626i	$0.668818\pi$
$840$		0			0
$841$	59917.6			2.45675
$842$	4184.24	+	7247.32i	0.171257	+	0.296626i
$843$		0			0
$844$	6176.76	−	10698.5i	0.251911	−	0.436323i
$845$	7933.22	+	13740.7i	0.322971	+	0.559403i
$846$		0			0
$847$	809.168	+	24059.4i	0.0328256	+	0.976023i
$848$	3691.82			0.149502
$849$		0			0
$850$	3066.89	−	5312.02i	0.123757	−	0.214354i
$851$	−7808.11	+	13524.0i	−0.314522	+	0.544769i
$852$		0			0
$853$	−150.549			−0.00604303			−0.00302151	−	0.999995i	$-0.500962\pi$
$853$	−150.549			−0.00604303			−0.00302151	+	0.999995i	$0.500962\pi$
$854$	−8140.57	+	5072.23i	−0.326188	+	0.203241i
$855$		0			0
$856$	396.510	+	686.776i	0.0158323	+	0.0274223i
$857$	4332.63	−	7504.33i	0.172695	−	0.299117i	−0.766666	−	0.642046i	$-0.778086\pi$
$857$	4332.63	−	7504.33i	0.172695	−	0.299117i	0.939361	+	0.342929i	$0.111419\pi$
$858$		0			0
$859$	8006.96	+	13868.5i	0.318037	+	0.550856i	0.980078	−	0.198611i	$-0.0636430\pi$
$859$	8006.96	+	13868.5i	0.318037	+	0.550856i	−0.662041	+	0.749467i	$0.730310\pi$
$860$	7572.80			0.300268
$861$		0			0
$862$	14573.6			0.575846
$863$	−19339.0	−	33496.2i	−0.762814	−	1.32123i	−0.941394	−	0.337308i	$-0.890484\pi$
$863$	−19339.0	−	33496.2i	−0.762814	−	1.32123i	0.178580	−	0.983925i	$-0.442850\pi$
$864$		0			0
$865$	−12039.1	+	20852.4i	−0.473229	+	0.819656i
$866$	−5314.89	−	9205.66i	−0.208554	−	0.361225i
$867$		0			0
$868$	−629.011	−	18702.7i	−0.0245968	−	0.731350i
$869$	503.983			0.0196737
$870$		0			0
$871$	−1576.70	+	2730.93i	−0.0613370	+	0.106239i
$872$	−3676.56	+	6367.98i	−0.142780	+	0.247302i
$873$		0			0
$874$	−3568.93			−0.138125
$875$	−24115.3	−	12862.2i	−0.931709	−	0.496938i
$876$		0			0
$877$	−14169.4	−	24542.2i	−0.545573	−	0.944961i	−0.998571	−	0.0534489i	$-0.982979\pi$
$877$	−14169.4	−	24542.2i	−0.545573	−	0.944961i	0.452997	−	0.891512i	$-0.350355\pi$
$878$	8546.10	−	14802.3i	0.328493	−	0.568967i
$879$		0			0
$880$	348.622	+	603.830i	0.0133546	+	0.0231308i
$881$	−37656.8			−1.44006			−0.720028	−	0.693945i	$-0.755871\pi$
$881$	−37656.8			−1.44006			−0.720028	+	0.693945i	$0.755871\pi$
$882$		0			0
$883$	41858.7			1.59531			0.797655	−	0.603114i	$-0.206074\pi$
$883$	41858.7			1.59531			0.797655	+	0.603114i	$0.206074\pi$
$884$	1225.37	+	2122.40i	0.0466218	+	0.0807512i
$885$		0			0
$886$	−10796.5	+	18700.1i	−0.409386	+	0.709078i
$887$	−20573.8	−	35634.8i	−0.778805	−	1.34893i	−0.932631	−	0.360832i	$-0.882493\pi$
$887$	−20573.8	−	35634.8i	−0.778805	−	1.34893i	0.153826	−	0.988098i	$-0.450841\pi$
$888$		0			0
$889$	−20431.9	−	10897.6i	−0.770824	−	0.411128i
$890$	−17463.5			−0.657729
$891$		0			0
$892$	−8799.58	+	15241.3i	−0.330305	+	0.572104i
$893$	−1336.84	+	2315.48i	−0.0500959	+	0.0867687i
$894$		0			0
$895$	11297.2			0.421926
$896$	79.6829	+	2369.25i	0.00297100	+	0.0883384i
$897$		0			0
$898$	−3617.92	−	6266.41i	−0.134445	−	0.232865i
$899$	−36672.8	+	63519.2i	−1.36052	+	2.35649i
$900$		0			0
$901$	5520.43	+	9561.67i	0.204120	+	0.353546i
$902$	131.582			0.00485722
$903$		0			0
$904$	−16968.5			−0.624297
$905$	6567.91	+	11376.0i	0.241243	+	0.417845i
$906$		0			0
$907$	−5874.59	+	10175.1i	−0.215064	+	0.372501i	−0.953292	−	0.302050i	$-0.902329\pi$
$907$	−5874.59	+	10175.1i	−0.215064	+	0.372501i	0.738229	+	0.674551i	$0.235663\pi$
$908$	−6574.76	−	11387.8i	−0.240299	−	0.416210i
$909$		0			0
$910$	3141.46	−	1957.38i	0.114438	−	0.0713040i
$911$	48408.6			1.76054			0.880268	−	0.474477i	$-0.157363\pi$
$911$	48408.6			1.76054			0.880268	+	0.474477i	$0.157363\pi$
$912$		0			0
$913$	−1041.21	+	1803.42i	−0.0377425	+	0.0653719i
$914$	−7099.34	+	12296.4i	−0.256920	+	0.444999i
$915$		0			0
$916$	12054.0			0.434800
$917$	70.6344	+	2100.21i	0.00254368	+	0.0756325i
$918$		0			0
$919$	5070.66	+	8782.64i	0.182008	+	0.315248i	0.942564	−	0.334025i	$-0.108407\pi$
$919$	5070.66	+	8782.64i	0.182008	+	0.315248i	−0.760556	+	0.649272i	$0.775074\pi$
$920$	−1187.06	+	2056.05i	−0.0425393	+	0.0736802i
$921$		0			0
$922$	−16652.8	−	28843.6i	−0.594829	−	1.03027i
$923$	−6987.59			−0.249187
$924$		0			0
$925$	26321.5			0.935616
$926$	−15394.4	−	26664.0i	−0.546321	−	0.946255i
$927$		0			0
$928$	4645.70	−	8046.58i	0.164335	−	0.284636i
$929$	6798.05	+	11774.6i	0.240083	+	0.415836i	0.960738	−	0.277458i	$-0.0894921\pi$
$929$	6798.05	+	11774.6i	0.240083	+	0.415836i	−0.720655	+	0.693294i	$0.756159\pi$
$930$		0			0
$931$	−7093.30	+	14448.9i	−0.249703	+	0.508639i
$932$	20326.4			0.714393
$933$		0			0
$934$	−16615.7	+	28779.2i	−0.582100	+	1.00823i
$935$	−1042.60	+	1805.83i	−0.0364670	+	0.0631627i
$936$		0			0
$937$	−46027.4			−1.60475			−0.802374	−	0.596822i	$-0.796430\pi$
$937$	−46027.4			−1.60475			−0.802374	+	0.596822i	$0.796430\pi$
$938$	−8048.97	−	4293.01i	−0.280179	−	0.149437i
$939$		0			0
$940$	889.290	+	1540.30i	0.0308568	+	0.0534456i
$941$	−253.733	+	439.478i	−0.00879006	+	0.0152248i	−0.870387	−	0.492368i	$-0.836131\pi$
$941$	−253.733	+	439.478i	−0.00879006	+	0.0152248i	0.861597	+	0.507593i	$0.169465\pi$
$942$		0			0
$943$	224.019	+	388.013i	0.00773603	+	0.0133992i
$944$	−8288.91			−0.285785
$945$		0			0
$946$	2709.13			0.0931091
$947$	−602.414	−	1043.41i	−0.0206714	−	0.0358039i	0.855505	−	0.517795i	$-0.173247\pi$
$947$	−602.414	−	1043.41i	−0.0206714	−	0.0358039i	−0.876176	+	0.481991i	$0.839914\pi$
$948$		0			0
$949$	1843.25	−	3192.60i	0.0630500	−	0.109206i
$950$	3007.76	+	5209.59i	0.102721	+	0.177917i
$951$		0			0
$952$	−6017.13	+	3749.16i	−0.204849	+	0.127638i
$953$	−18606.2			−0.632437			−0.316219	−	0.948686i	$-0.602413\pi$
$953$	−18606.2			−0.632437			−0.316219	+	0.948686i	$0.602413\pi$
$954$		0			0
$955$	8171.94	−	14154.2i	0.276898	−	0.479602i
$956$	−6174.76	+	10695.0i	−0.208898	+	0.361821i
$957$		0			0
$958$	3356.75			0.113206
$959$	−240.731	+	149.995i	−0.00810596	+	0.00505067i
$960$		0			0
$961$	−17009.4	−	29461.1i	−0.570956	−	0.988926i
$962$	−5258.34	+	9107.71i	−0.176233	+	0.305244i
$963$		0			0
$964$	2433.30	+	4214.60i	0.0812981	+	0.140812i
$965$	−26686.0			−0.890211
$966$		0			0
$967$	−39645.1			−1.31841			−0.659204	−	0.751964i	$-0.729107\pi$
$967$	−39645.1			−1.31841			−0.659204	+	0.751964i	$0.729107\pi$
$968$	−5199.28	−	9005.42i	−0.172636	−	0.299014i
$969$		0			0
$970$	3073.02	−	5322.62i	0.101720	−	0.176185i
$971$	−26704.0	−	46252.7i	−0.882567	−	1.52865i	−0.848477	−	0.529232i	$-0.822480\pi$
$971$	−26704.0	−	46252.7i	−0.882567	−	1.52865i	−0.0340898	−	0.999419i	$-0.510853\pi$
$972$		0			0
$973$	−41503.5	−	22136.4i	−1.36746	−	0.729353i
$974$	18917.8			0.622345
$975$		0			0
$976$	2071.56	−	3588.05i	0.0679397	−	0.117675i
$977$	−11856.7	+	20536.4i	−0.388259	+	0.672484i	−0.992215	−	0.124533i	$-0.960257\pi$
$977$	−11856.7	+	20536.4i	−0.388259	+	0.672484i	0.603957	+	0.797017i	$0.293590\pi$
$978$		0			0
$979$	−6247.48			−0.203953
$980$	5964.64	+	8892.24i	0.194422	+	0.289849i
$981$		0			0
$982$	−12084.6	−	20931.1i	−0.392703	−	0.680181i
$983$	26074.2	−	45161.8i	0.846020	−	1.46535i	−0.0387129	−	0.999250i	$-0.512326\pi$
$983$	26074.2	−	45161.8i	0.846020	−	1.46535i	0.884733	−	0.466099i	$-0.154341\pi$
$984$		0			0
$985$	13724.1	+	23770.8i	0.443944	+	0.768934i
$986$	27787.1			0.897488
$987$		0			0
$988$	−2403.48			−0.0773937
$989$	4612.29	+	7988.73i	0.148294	+	0.256852i
$990$		0			0
$991$	6416.26	−	11113.3i	0.205670	−	0.356231i	−0.744676	−	0.667426i	$-0.767396\pi$
$991$	6416.26	−	11113.3i	0.205670	−	0.356231i	0.950346	+	0.311195i	$0.100729\pi$
$992$	4041.69	+	7000.42i	0.129359	+	0.224056i
$993$		0			0
$994$	−679.451	−	20202.5i	−0.0216810	−	0.644652i
$995$	−24443.4			−0.778802
$996$		0			0
$997$	−11795.0	+	20429.5i	−0.374675	+	0.648955i	−0.990278	−	0.139100i	$-0.955579\pi$
$997$	−11795.0	+	20429.5i	−0.374675	+	0.648955i	0.615604	+	0.788056i	$0.288912\pi$
$998$	11152.7	−	19317.0i	0.353739	−	0.612693i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.b.109.1	yes	6
3.2	odd	2		378.4.g.a.109.3	✓	6
7.2	even	3	inner	378.4.g.b.163.1	yes	6
21.2	odd	6		378.4.g.a.163.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.a.109.3	✓	6	3.2	odd	2
378.4.g.a.163.3	yes	6	21.2	odd	6
378.4.g.b.109.1	yes	6	1.1	even	1	trivial
378.4.g.b.163.1	yes	6	7.2	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.90212 - 6.75867i) q^{5} +(16.3412 + 8.71578i) q^{7} -8.00000 q^{8} +(7.80425 - 13.5173i) q^{10} +(2.79192 - 4.83575i) q^{11} +12.8042 q^{13} +(1.24504 + 37.0196i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(23.9251 - 41.4394i) q^{17} +(23.4637 + 40.6404i) q^{19} +31.2170 q^{20} +11.1677 q^{22} +(19.0130 + 32.9315i) q^{23} +(32.0469 - 55.5068i) q^{25} +(12.8042 + 22.1776i) q^{26} +(-62.8748 + 39.1761i) q^{28} +290.356 q^{29} +(-126.303 + 218.763i) q^{31} +(16.0000 - 27.7128i) q^{32} +95.7002 q^{34} +(-4.85832 - 144.455i) q^{35} +(205.336 + 355.652i) q^{37} +(-46.9275 + 81.2808i) q^{38} +(31.2170 + 54.0694i) q^{40} +11.7824 q^{41} +242.586 q^{43} +(11.1677 + 19.3430i) q^{44} +(-38.0261 + 65.8631i) q^{46} +(28.4874 + 49.3416i) q^{47} +(191.070 + 284.853i) q^{49} +128.188 q^{50} +(-25.6085 + 44.3552i) q^{52} +(-115.369 + 199.825i) q^{53} -43.5777 q^{55} +(-130.730 - 69.7262i) q^{56} +(290.356 + 502.911i) q^{58} +(259.028 - 448.650i) q^{59} +(129.473 + 224.253i) q^{61} -505.212 q^{62} +64.0000 q^{64} +(-49.9637 - 86.5397i) q^{65} +(-123.139 + 213.283i) q^{67} +(95.7002 + 165.758i) q^{68} +(245.345 - 152.870i) q^{70} -545.724 q^{71} +(143.956 - 249.339i) q^{73} +(-410.672 + 711.304i) q^{74} -187.710 q^{76} +(87.7707 - 54.6883i) q^{77} +(45.1287 + 78.1652i) q^{79} +(-62.4340 + 108.139i) q^{80} +(11.7824 + 20.4077i) q^{82} -372.935 q^{83} -373.434 q^{85} +(242.586 + 420.171i) q^{86} +(-22.3354 + 38.6860i) q^{88} +(-559.424 - 968.951i) q^{89} +(209.237 + 111.599i) q^{91} -152.104 q^{92} +(-56.9747 + 98.6832i) q^{94} +(183.117 - 317.168i) q^{95} +393.762 q^{97} +(-302.309 + 615.796i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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