LMFDB - Embedded newform 201.3.o.a.194.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,3,Mod(17,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 64]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("201.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	201.o (of order $66$, degree $20$, 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:	$5.47685331364$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			194.1
Root			$-0.786053 - 0.618159i$ of defining polynomial
Character	$\chi$	$=$	201.194
Dual form			201.3.o.a.86.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.87848 + 0.845198i) q^{3} +(2.32023 - 3.25830i) q^{4} +(-0.560201 + 11.7601i) q^{7} +(7.57128 + 4.86577i) q^{9} +O(q^{10})$	$q+(2.87848 + 0.845198i) q^{3} +(2.32023 - 3.25830i) q^{4} +(-0.560201 + 11.7601i) q^{7} +(7.57128 + 4.86577i) q^{9} +(9.43264 - 7.41791i) q^{12} +(11.8775 + 4.75502i) q^{13} +(-5.23309 - 15.1200i) q^{16} +(-1.55852 - 32.7173i) q^{19} +(-11.5521 + 33.3776i) q^{21} +(-3.55787 - 24.7455i) q^{25} +(17.6812 + 20.4052i) q^{27} +(37.0181 + 29.1113i) q^{28} +(-34.6297 + 13.8636i) q^{31} +(33.4212 - 13.3798i) q^{36} +(-32.4383 + 56.1848i) q^{37} +(30.1701 + 23.7260i) q^{39} +(28.5332 - 62.4789i) q^{43} +(-2.28393 - 47.9456i) q^{48} +(-89.2072 - 8.51825i) q^{49} +(43.0517 - 27.6676i) q^{52} +(23.1664 - 95.4934i) q^{57} +(-75.4264 + 14.5372i) q^{61} +(-61.4632 + 86.3130i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(-30.8183 + 59.4914i) q^{67} +(-113.804 + 21.9338i) q^{73} +(10.6736 - 74.2366i) q^{75} +(-110.219 - 70.8335i) q^{76} +(83.3835 - 65.5735i) q^{79} +(33.6486 + 73.6802i) q^{81} +(81.9509 + 115.084i) q^{84} +(-62.5731 + 137.016i) q^{91} +(-111.398 + 10.6373i) q^{93} +(66.2104 - 114.680i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) $ 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9	$ 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9} - 12 q^{12} + 23 q^{13} + 16 q^{16} + 26 q^{19} - 6 q^{21} - 50 q^{25} + 54 q^{27} + 8 q^{28} - 13 q^{31} + 36 q^{36} + 26 q^{37} - 69 q^{39} + 122 q^{43} - 48 q^{48} - 45 q^{49} + 828 q^{52} - 441 q^{57} + 47 q^{61} - 1269 q^{63} - 128 q^{64} + 109 q^{67} - 924 q^{73} + 150 q^{75} - 208 q^{76} + 252 q^{79} - 162 q^{81} + 1692 q^{84} - 92 q^{91} + 39 q^{93} + 167 q^{97}+O(q^{100}) $ 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9 - 12 * q^12 + 23 * q^13 + 16 * q^16 + 26 * q^19 - 6 * q^21 - 50 * q^25 + 54 * q^27 + 8 * q^28 - 13 * q^31 + 36 * q^36 + 26 * q^37 - 69 * q^39 + 122 * q^43 - 48 * q^48 - 45 * q^49 + 828 * q^52 - 441 * q^57 + 47 * q^61 - 1269 * q^63 - 128 * q^64 + 109 * q^67 - 924 * q^73 + 150 * q^75 - 208 * q^76 + 252 * q^79 - 162 * q^81 + 1692 * q^84 - 92 * q^91 + 39 * q^93 + 167 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{28}{33}\right)$

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 $

$2$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$2$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$3$	2.87848	+	0.845198i	0.959493	+	0.281733i
$3$	2.87848	+	0.845198i	0.959493	+	0.281733i
$4$	2.32023	−	3.25830i	0.580057	−	0.814576i
$5$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$5$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$6$		0			0
$7$	−0.560201	+	11.7601i	−0.0800287	+	1.68001i	0.500474	+	0.865751i	$0.333159\pi$
$7$	−0.560201	+	11.7601i	−0.0800287	+	1.68001i	−0.580503	+	0.814258i	$0.697144\pi$
$8$		0			0
$9$	7.57128	+	4.86577i	0.841254	+	0.540641i
$10$		0			0
$11$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$11$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$12$	9.43264	−	7.41791i	0.786053	−	0.618159i
$13$	11.8775	+	4.75502i	0.913650	+	0.365770i	0.780377	−	0.625309i	$-0.215027\pi$
$13$	11.8775	+	4.75502i	0.913650	+	0.365770i	0.133273	+	0.991079i	$0.457451\pi$
$14$		0			0
$15$		0			0
$16$	−5.23309	−	15.1200i	−0.327068	−	0.945001i
$17$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$17$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$18$		0			0
$19$	−1.55852	−	32.7173i	−0.0820273	−	1.72196i	−0.547418	−	0.836859i	$-0.684389\pi$
$19$	−1.55852	−	32.7173i	−0.0820273	−	1.72196i	0.465391	−	0.885105i	$-0.345914\pi$
$20$		0			0
$21$	−11.5521	+	33.3776i	−0.550100	+	1.58941i
$22$		0			0
$23$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$23$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$24$		0			0
$25$	−3.55787	−	24.7455i	−0.142315	−	0.989821i
$26$		0			0
$27$	17.6812	+	20.4052i	0.654861	+	0.755750i
$28$	37.0181	+	29.1113i	1.32207	+	1.03969i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−34.6297	+	13.8636i	−1.11709	+	0.447214i	−0.855415	−	0.517943i	$-0.826698\pi$
$31$	−34.6297	+	13.8636i	−1.11709	+	0.447214i	−0.261672	+	0.965157i	$0.584274\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	33.4212	−	13.3798i	0.928368	−	0.371662i
$37$	−32.4383	+	56.1848i	−0.876712	+	1.51851i	−0.0217833	+	0.999763i	$0.506934\pi$
$37$	−32.4383	+	56.1848i	−0.876712	+	1.51851i	−0.854928	+	0.518746i	$0.826399\pi$
$38$		0			0
$39$	30.1701	+	23.7260i	0.773592	+	0.608359i
$40$		0			0
$41$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$41$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$42$		0			0
$43$	28.5332	−	62.4789i	0.663562	−	1.45300i	−0.215603	−	0.976481i	$-0.569172\pi$
$43$	28.5332	−	62.4789i	0.663562	−	1.45300i	0.879165	−	0.476517i	$-0.158101\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$47$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$48$	−2.28393	−	47.9456i	−0.0475819	−	0.998867i
$49$	−89.2072	−	8.51825i	−1.82055	−	0.173842i
$50$		0			0
$51$		0			0
$52$	43.0517	−	27.6676i	0.827917	−	0.532070i
$53$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$53$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	23.1664	−	95.4934i	0.406429	−	1.67532i
$58$		0			0
$59$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$59$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$60$		0			0
$61$	−75.4264	+	14.5372i	−1.23650	+	0.238316i	−0.765278	−	0.643700i	$-0.777399\pi$
$61$	−75.4264	+	14.5372i	−1.23650	+	0.238316i	−0.471221	+	0.882015i	$0.656186\pi$
$62$		0			0
$63$	−61.4632	+	86.3130i	−0.975606	+	1.37005i
$64$	−61.4076	−	18.0309i	−0.959493	−	0.281733i
$65$		0			0
$66$		0			0
$67$	−30.8183	+	59.4914i	−0.459975	+	0.887932i
$67$	−30.8183	+	59.4914i	−0.459975	+	0.887932i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$71$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$72$		0			0
$73$	−113.804	+	21.9338i	−1.55895	+	0.300464i	−0.894570	−	0.446928i	$-0.852518\pi$
$73$	−113.804	+	21.9338i	−1.55895	+	0.300464i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	10.6736	−	74.2366i	0.142315	−	0.989821i
$76$	−110.219	−	70.8335i	−1.45025	−	0.932020i
$77$		0			0
$78$		0			0
$79$	83.3835	−	65.5735i	1.05549	−	0.830044i	0.0696203	−	0.997574i	$-0.477821\pi$
$79$	83.3835	−	65.5735i	1.05549	−	0.830044i	0.985867	+	0.167529i	$0.0535788\pi$
$80$		0			0
$81$	33.6486	+	73.6802i	0.415415	+	0.909632i
$82$		0			0
$83$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$83$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$84$	81.9509	+	115.084i	0.975606	+	1.37005i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$89$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$90$		0			0
$91$	−62.5731	+	137.016i	−0.687616	+	1.50567i
$92$		0			0
$93$	−111.398	+	10.6373i	−1.19783	+	0.114379i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	66.2104	−	114.680i	0.682581	−	1.18227i	−0.291609	−	0.956538i	$-0.594191\pi$
$97$	66.2104	−	114.680i	0.682581	−	1.18227i	0.974190	−	0.225728i	$-0.0724760\pi$
$98$		0			0
$99$		0			0
$100$	−88.8835	−	45.8227i	−0.888835	−	0.458227i
$101$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$101$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$102$		0			0
$103$	186.286	−	74.5777i	1.80860	−	0.724056i	0.820226	−	0.572040i	$-0.193848\pi$
$103$	186.286	−	74.5777i	1.80860	−	0.724056i	0.988378	−	0.152016i	$-0.0485765\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$	107.511	−	10.2661i	0.995472	−	0.0950560i
$109$	−15.5515	−	108.163i	−0.142675	−	0.992323i	−0.927824	−	0.373017i	$-0.878323\pi$
$109$	−15.5515	−	108.163i	−0.142675	−	0.992323i	0.785150	−	0.619306i	$-0.212586\pi$
$110$		0			0
$111$	−140.860	+	134.310i	−1.26901	+	1.21000i
$112$	180.744	−	53.0712i	1.61379	−	0.473850i
$113$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$113$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	66.7908	+	93.7945i	0.570861	+	0.801662i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	112.333	+	44.9712i	0.928368	+	0.371662i
$122$		0			0
$123$		0			0
$124$	−35.1768	+	145.001i	−0.283684	+	1.16936i
$125$		0			0
$126$		0			0
$127$	0.652737	−	13.7026i	0.00513966	−	0.107895i	−0.994840	−	0.101460i	$-0.967649\pi$
$127$	0.652737	−	13.7026i	0.00513966	−	0.107895i	0.999979	−	0.00643494i	$-0.00204832\pi$
$128$		0			0
$129$	134.939	−	155.728i	1.04604	−	1.20719i
$130$		0			0
$131$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$131$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$132$		0			0
$133$	385.631			2.89948
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$137$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$138$		0			0
$139$	−114.809	+	132.497i	−0.825964	+	0.953213i	−0.999500	−	0.0316144i	$-0.989935\pi$
$139$	−114.809	+	132.497i	−0.825964	+	0.953213i	0.173536	+	0.984827i	$0.444481\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	33.9493	−	139.941i	0.235759	−	0.971812i
$145$		0			0
$146$		0			0
$147$	−249.581	−	99.9173i	−1.69783	−	0.679710i
$148$	107.803	+	236.056i	0.728398	+	1.59497i
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$	−152.251	−	213.806i	−1.00828	−	1.41593i	−0.907027	−	0.421073i	$-0.861654\pi$
$151$	−152.251	−	213.806i	−1.00828	−	1.41593i	−0.101255	−	0.994860i	$-0.532286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	147.308	−	43.2535i	0.944282	−	0.277266i
$157$	−185.700	+	177.065i	−1.18280	+	1.12780i	−0.193625	+	0.981076i	$0.562025\pi$
$157$	−185.700	+	177.065i	−1.18280	+	1.12780i	−0.989177	+	0.146725i	$0.953127\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	154.997	+	268.463i	0.950903	+	1.64701i	0.743478	+	0.668761i	$0.233175\pi$
$163$	154.997	+	268.463i	0.950903	+	1.64701i	0.207425	+	0.978251i	$0.433492\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$167$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$168$		0			0
$169$	−3.84728	−	3.66837i	−0.0227650	−	0.0217064i
$170$		0			0
$171$	147.395	−	255.295i	0.861958	−	1.49296i
$172$	−137.372	−	237.935i	−0.798674	−	1.38334i
$173$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$173$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$174$		0			0
$175$	293.002	−	27.9783i	1.67430	−	0.159876i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$179$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$180$		0			0
$181$	82.8831	+	341.649i	0.457918	+	1.88756i	0.460680	+	0.887566i	$0.347606\pi$
$181$	82.8831	+	341.649i	0.457918	+	1.88756i	−0.00276243	+	0.999996i	$0.500879\pi$
$182$		0			0
$183$	−229.400	−	21.9051i	−1.25355	−	0.119700i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−249.872	+	196.502i	−1.32207	+	1.03969i
$190$		0			0
$191$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$191$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$192$	−161.521	−	103.803i	−0.841254	−	0.540641i
$193$	−44.7064	+	310.940i	−0.231640	+	1.61109i	0.459371	+	0.888244i	$0.348075\pi$
$193$	−44.7064	+	310.940i	−0.231640	+	1.61109i	−0.691011	+	0.722845i	$0.742834\pi$
$194$		0			0
$195$		0			0
$196$	−234.736	+	270.900i	−1.19763	+	1.38214i
$197$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$197$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$198$		0			0
$199$	333.342	−	171.850i	1.67509	−	0.863566i	0.684132	−	0.729358i	$-0.260181\pi$
$199$	333.342	−	171.850i	1.67509	−	0.863566i	0.990953	−	0.134208i	$-0.0428490\pi$
$200$		0			0
$201$	−138.992	+	145.197i	−0.691502	+	0.722375i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	9.74014	−	204.471i	0.0468276	−	0.983032i
$209$		0			0
$210$		0			0
$211$	99.2390	−	409.069i	0.470327	−	1.93871i	0.166136	−	0.986103i	$-0.446871\pi$
$211$	99.2390	−	409.069i	0.470327	−	1.93871i	0.304191	−	0.952611i	$-0.401614\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−143.638	−	415.014i	−0.661925	−	1.91251i
$218$		0			0
$219$	−346.120	−	33.0504i	−1.58046	−	0.150915i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−42.0448	+	12.3455i	−0.188542	+	0.0553608i	−0.374640	−	0.927170i	$-0.622234\pi$
$223$	−42.0448	+	12.3455i	−0.188542	+	0.0553608i	0.186099	+	0.982531i	$0.440416\pi$
$224$		0			0
$225$	93.4684	−	204.667i	0.415415	−	0.909632i
$226$		0			0
$227$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$227$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$228$	−257.395	−	297.050i	−1.12893	−	1.30285i
$229$	114.614	+	90.1333i	0.500497	+	0.393595i	0.836245	−	0.548357i	$-0.184746\pi$
$229$	114.614	+	90.1333i	0.500497	+	0.393595i	−0.335747	+	0.941952i	$0.608989\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$233$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	295.440	−	118.276i	1.24658	−	0.499057i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	79.4450	+	91.6844i	0.329647	+	0.380433i	0.896244	−	0.443562i	$-0.146286\pi$
$241$	79.4450	+	91.6844i	0.329647	+	0.380433i	−0.566596	+	0.823996i	$0.691740\pi$
$242$		0			0
$243$	34.5825	+	240.527i	0.142315	+	0.989821i
$244$	−127.640	+	279.492i	−0.523113	+	1.14546i
$245$		0			0
$246$		0			0
$247$	137.060	−	396.009i	0.554899	−	1.60328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$251$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$252$	138.625	+	400.531i	0.550100	+	1.58941i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−201.230	+	158.249i	−0.786053	+	0.618159i
$257$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$257$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$258$		0			0
$259$	−642.565	−	412.952i	−2.48095	−	1.59441i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	122.336	+	238.449i	0.456476	+	0.889736i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	419.611	+	123.209i	1.54838	+	0.454646i	0.940617	−	0.339471i	$-0.110248\pi$
$271$	419.611	+	123.209i	1.54838	+	0.454646i	0.607765	+	0.794117i	$0.292066\pi$
$272$		0			0
$273$	−295.921	+	341.511i	−1.08396	+	1.25096i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−310.784	−	199.729i	−1.12196	−	0.721043i	−0.158096	−	0.987424i	$-0.550536\pi$
$277$	−310.784	−	199.729i	−1.12196	−	0.721043i	−0.963868	+	0.266381i	$0.914172\pi$
$278$		0			0
$279$	−329.648	−	63.5345i	−1.18154	−	0.227722i
$280$		0			0
$281$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$281$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$282$		0			0
$283$	−40.8840	+	26.2745i	−0.144466	+	0.0928428i	−0.610877	−	0.791725i	$-0.709183\pi$
$283$	−40.8840	+	26.2745i	−0.144466	+	0.0928428i	0.466411	+	0.884568i	$0.345547\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−94.5226	+	273.105i	−0.327068	+	0.945001i
$290$		0			0
$291$	287.512	−	274.142i	0.988014	−	0.942070i
$292$	−192.583	+	421.698i	−0.659531	+	1.44417i
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−217.120	−	207.024i	−0.723734	−	0.690079i
$301$	718.772	+	370.553i	2.38795	+	1.23107i
$302$		0			0
$303$		0			0
$304$	−486.530	+	194.777i	−1.60043	+	0.640715i
$305$		0			0
$306$		0			0
$307$	−242.861	−	190.988i	−0.791077	−	0.622110i	0.138658	−	0.990340i	$-0.455721\pi$
$307$	−242.861	−	190.988i	−0.791077	−	0.622110i	−0.929734	+	0.368231i	$0.879964\pi$
$308$		0			0
$309$	599.254	−	57.2218i	1.93933	−	0.185184i
$310$		0			0
$311$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$311$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$312$		0			0
$313$	−201.648	+	59.2092i	−0.644243	+	0.189167i	−0.587503	−	0.809222i	$-0.699889\pi$
$313$	−201.648	+	59.2092i	−0.644243	+	0.189167i	−0.0567400	+	0.998389i	$0.518071\pi$
$314$		0			0
$315$		0			0
$316$	−20.1897	−	423.834i	−0.0638915	−	1.34125i
$317$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$317$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	318.145	+	61.3174i	0.981929	+	0.189251i
$325$	75.4070	−	310.832i	0.232021	−	0.956405i
$326$		0			0
$327$	46.6546	−	324.490i	0.142675	−	0.992323i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	361.942	−	508.277i	1.09348	−	1.53558i	0.274652	−	0.961544i	$-0.411437\pi$
$331$	361.942	−	508.277i	1.09348	−	1.53558i	0.818830	−	0.574036i	$-0.194623\pi$
$332$		0			0
$333$	−518.982	+	267.554i	−1.55850	+	0.803465i
$334$		0			0
$335$		0			0
$336$	565.123			1.68191
$337$	595.612	−	307.059i	1.76739	−	0.911155i	0.834494	−	0.551016i	$-0.185760\pi$
$337$	595.612	−	307.059i	1.76739	−	0.911155i	0.932899	−	0.360138i	$-0.117270\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	68.0482	−	473.286i	0.198391	−	1.37984i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$347$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$348$		0			0
$349$	191.315	+	418.921i	0.548180	+	1.20035i	0.957627	+	0.288012i	$0.0929944\pi$
$349$	191.315	+	418.921i	0.548180	+	1.20035i	−0.409447	+	0.912334i	$0.634278\pi$
$350$		0			0
$351$	112.981	+	326.437i	0.321883	+	0.930020i
$352$		0			0
$353$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$353$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	−708.629	+	67.6658i	−1.96296	+	0.187440i
$362$		0			0
$363$	285.337	+	224.392i	0.786053	+	0.618159i
$364$	301.256	+	521.790i	0.827625	+	1.43349i
$365$		0			0
$366$		0			0
$367$	256.924	+	244.977i	0.700066	+	0.667512i	0.954033	−	0.299701i	$-0.0968869\pi$
$367$	256.924	+	244.977i	0.700066	+	0.667512i	−0.253967	+	0.967213i	$0.581735\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−223.810	+	387.651i	−0.601640	+	1.04207i
$373$	370.523	+	641.764i	0.993358	+	1.72055i	0.596326	+	0.802742i	$0.296626\pi$
$373$	370.523	+	641.764i	0.993358	+	1.72055i	0.397032	+	0.917805i	$0.370040\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	345.341	−	329.282i	0.911190	−	0.868818i	−0.0805934	−	0.996747i	$-0.525682\pi$
$379$	345.341	−	329.282i	0.911190	−	0.868818i	0.991783	+	0.127929i	$0.0408330\pi$
$380$		0			0
$381$	13.4603	−	38.8911i	0.0353290	−	0.102076i
$382$		0			0
$383$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$383$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	520.041	−	334.210i	1.34377	−	0.863591i
$388$	−220.038	−	481.817i	−0.567109	−	1.24180i
$389$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$389$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	191.997	−	221.576i	0.483619	−	0.558126i	−0.460531	−	0.887644i	$-0.652341\pi$
$397$	191.997	−	221.576i	0.483619	−	0.558126i	0.944149	+	0.329518i	$0.106886\pi$
$398$		0			0
$399$	1110.03	+	325.934i	2.78203	+	0.816878i
$400$	−355.534	+	183.291i	−0.888835	+	0.458227i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−477.234			−1.18420
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−38.4404	+	806.963i	−0.0939862	+	1.97301i	0.109659	+	0.993969i	$0.465024\pi$
$409$	−38.4404	+	806.963i	−0.0939862	+	1.97301i	−0.203645	+	0.979045i	$0.565279\pi$
$410$		0			0
$411$		0			0
$412$	189.229	−	780.014i	0.459295	−	1.89324i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−442.461	+	284.352i	−1.06106	+	0.681900i
$418$		0			0
$419$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$419$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$420$		0			0
$421$	35.8057	+	751.655i	0.0850493	+	1.78540i	0.490653	+	0.871355i	$0.336758\pi$
$421$	35.8057	+	751.655i	0.0850493	+	1.78540i	−0.405604	+	0.914049i	$0.632939\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−128.705	−	895.163i	−0.301417	−	2.09640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	216.000	−	374.123i	0.500000	−	0.866025i
$433$	37.4905	−	15.0089i	0.0865830	−	0.0346626i	−0.327967	−	0.944689i	$-0.606363\pi$
$433$	37.4905	−	15.0089i	0.0865830	−	0.0346626i	0.414550	+	0.910027i	$0.363939\pi$
$434$		0			0
$435$		0			0
$436$	−388.512	−	200.292i	−0.891082	−	0.459385i
$437$		0			0
$438$		0			0
$439$	−328.281	+	568.599i	−0.747792	+	1.29521i	0.201086	+	0.979574i	$0.435553\pi$
$439$	−328.281	+	568.599i	−0.747792	+	1.29521i	−0.948879	+	0.315641i	$0.897781\pi$
$440$		0			0
$441$	−633.965	−	498.556i	−1.43756	−	1.13051i
$442$		0			0
$443$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$443$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$444$	110.795	+	770.596i	0.249538	+	1.73558i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	246.445	−	712.056i	0.550100	−	1.58941i
$449$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$449$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−257.542	−	744.118i	−0.568525	−	1.64264i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	356.831	−	280.615i	0.780812	−	0.614038i	−0.146120	−	0.989267i	$-0.546678\pi$
$457$	356.831	−	280.615i	0.780812	−	0.614038i	0.926932	+	0.375229i	$0.122436\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$461$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$	−839.081	+	161.720i	−1.81227	+	0.349286i	−0.979044	−	0.203648i	$-0.934720\pi$
$463$	−839.081	+	161.720i	−1.81227	+	0.349286i	−0.833225	+	0.552934i	$0.813508\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$467$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$468$	460.581			0.984147
$469$	−682.359	−	395.753i	−1.45492	−	0.843822i
$470$		0			0
$471$	−684.188	+	352.724i	−1.45263	+	0.748882i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−804.063	+	154.970i	−1.69276	+	0.326253i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$479$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$480$		0			0
$481$	−652.445	+	513.088i	−1.35643	+	1.06671i
$482$		0			0
$483$		0			0
$484$	407.167	−	261.670i	0.841254	−	0.540641i
$485$		0			0
$486$		0			0
$487$	−679.084	−	64.8446i	−1.39442	−	0.133151i	−0.629363	−	0.777111i	$-0.716684\pi$
$487$	−679.084	−	64.8446i	−1.39442	−	0.133151i	−0.765059	+	0.643960i	$0.777290\pi$
$488$		0			0
$489$	219.252	+	903.768i	0.448368	+	1.84820i
$490$		0			0
$491$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$491$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	390.839	+	451.052i	0.787981	+	0.909379i
$497$		0			0
$498$		0			0
$499$	−491.903	+	852.001i	−0.985778	+	1.70742i	−0.347350	+	0.937735i	$0.612919\pi$
$499$	−491.903	+	852.001i	−0.985778	+	1.70742i	−0.638428	+	0.769682i	$0.720415\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$503$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.97381	−	13.8110i	−0.0157274	−	0.0272407i
$508$	−43.1329	−	33.9201i	−0.0849072	−	0.0667718i
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$	−194.191	−	1350.63i	−0.380021	−	2.64310i
$512$		0			0
$513$	640.048	−	610.285i	1.24766	−	1.18964i
$514$		0			0
$515$		0			0
$516$	−194.320	−	800.998i	−0.376589	−	1.55232i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$521$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$522$		0			0
$523$	−947.758	−	379.425i	−1.81216	−	0.725478i	−0.987031	−	0.160527i	$-0.948681\pi$
$523$	−947.758	−	379.425i	−1.81216	−	0.725478i	−0.825125	−	0.564951i	$-0.808895\pi$
$524$		0			0
$525$	867.048	+	167.110i	1.65152	+	0.318304i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	25.1708	−	528.401i	0.0475819	−	0.998867i
$530$		0			0
$531$		0			0
$532$	894.751	−	1256.50i	1.68186	−	2.36185i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−222.768	+	257.088i	−0.411770	+	0.475208i	−0.923312	−	0.384050i	$-0.874529\pi$
$541$	−222.768	+	257.088i	−0.411770	+	0.475208i	0.511542	+	0.859258i	$0.329074\pi$
$542$		0			0
$543$	−50.1835	+	1053.48i	−0.0924190	+	1.94011i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1064.02	+	205.074i	1.94520	+	0.374906i	0.998625	+	0.0524156i	$0.0166920\pi$
$547$	1064.02	+	205.074i	1.94520	+	0.374906i	0.946573	+	0.322491i	$0.104520\pi$
$548$		0			0
$549$	−641.809	−	256.942i	−1.16905	−	0.468018i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	724.437	+	1017.33i	1.31001	+	1.83966i
$554$		0			0
$555$		0			0
$556$	165.331	+	681.505i	0.297358	+	1.22573i
$557$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$557$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$558$		0			0
$559$	635.990	−	606.415i	1.13773	−	1.08482i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$563$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−885.334	+	354.434i	−1.56144	+	0.625104i
$568$		0			0
$569$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$569$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$570$		0			0
$571$	−657.950	−	627.354i	−1.15228	−	1.09869i	−0.993783	−	0.111334i	$-0.964488\pi$
$571$	−657.950	−	627.354i	−1.15228	−	1.09869i	−0.158494	−	0.987360i	$-0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−377.200	−	435.312i	−0.654861	−	0.755750i
$577$	−286.396	+	27.3475i	−0.496354	+	0.0473961i	−0.340232	−	0.940342i	$-0.610506\pi$
$577$	−286.396	+	27.3475i	−0.496354	+	0.0473961i	−0.156123	+	0.987738i	$0.549899\pi$
$578$		0			0
$579$	−391.492	+	857.249i	−0.676153	+	1.48057i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$587$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$588$	−904.647	+	581.381i	−1.53851	+	0.988743i
$589$	507.552	+	1111.38i	0.861718	+	1.88690i
$590$		0			0
$591$		0			0
$592$	1019.27	+	196.448i	1.72174	+	0.331838i
$593$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$593$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1104.76	−	212.926i	1.85053	−	0.356660i
$598$		0			0
$599$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$599$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$600$		0			0
$601$	277.274	−	142.945i	0.461354	−	0.237845i	−0.211849	−	0.977302i	$-0.567948\pi$
$601$	277.274	−	142.945i	0.461354	−	0.237845i	0.673203	+	0.739458i	$0.264918\pi$
$602$		0			0
$603$	−522.806	+	300.472i	−0.867008	+	0.498295i
$604$	−1049.90			−1.73825
$605$		0			0
$606$		0			0
$607$	180.283	−	253.172i	0.297007	−	0.417088i	−0.638925	−	0.769269i	$-0.720620\pi$
$607$	180.283	−	253.172i	0.297007	−	0.417088i	0.935932	+	0.352181i	$0.114560\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	253.394	−	1044.50i	0.413367	−	1.70392i	−0.260841	−	0.965382i	$-0.584000\pi$
$613$	253.394	−	1044.50i	0.413367	−	1.70392i	0.674208	−	0.738541i	$-0.264485\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$617$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$618$		0			0
$619$	144.204	+	416.650i	0.232963	+	0.673101i	0.999522	+	0.0309275i	$0.00984610\pi$
$619$	144.204	+	416.650i	0.232963	+	0.673101i	−0.766559	+	0.642174i	$0.778033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	200.855	−	580.332i	0.321883	−	0.930020i
$625$	−599.683	+	176.083i	−0.959493	+	0.281733i
$626$		0			0
$627$		0			0
$628$	146.064	+	1015.90i	0.232586	+	1.61767i
$629$		0			0
$630$		0			0
$631$	272.057	+	213.948i	0.431153	+	0.339062i	0.810034	−	0.586383i	$-0.199449\pi$
$631$	272.057	+	213.948i	0.431153	+	0.339062i	−0.378881	+	0.925445i	$0.623691\pi$
$632$		0			0
$633$	631.401	−	1093.62i	0.997474	−	1.72768i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1019.05	−	525.357i	−1.59976	−	0.824736i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	657.441	+	758.728i	1.02246	+	1.17998i	0.983531	+	0.180739i	$0.0578488\pi$
$643$	657.441	+	758.728i	1.02246	+	1.17998i	0.0389279	+	0.999242i	$0.487606\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$647$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−62.6893	−	1316.01i	−0.0962970	−	2.02152i
$652$	1234.36	+	117.867i	1.89319	+	0.180778i
$653$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$653$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−968.364	−	387.675i	−1.47392	−	0.590068i
$658$		0			0
$659$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$659$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$660$		0			0
$661$	961.587	+	617.975i	1.45475	+	0.934909i	0.998996	+	0.0448010i	$0.0142654\pi$
$661$	961.587	+	617.975i	1.45475	+	0.934909i	0.455750	+	0.890108i	$0.349371\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−131.459			−0.196501
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1213.05	−	356.184i	−1.80245	−	0.529248i	−0.804545	−	0.593891i	$-0.797591\pi$
$673$	−1213.05	−	356.184i	−1.80245	−	0.529248i	−0.997908	+	0.0646433i	$0.979409\pi$
$674$		0			0
$675$	442.031	−	510.131i	0.654861	−	0.755750i
$676$	−20.8792	+	4.02414i	−0.0308864	+	0.00595287i
$677$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$677$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$678$		0			0
$679$	1311.55	+	842.882i	1.93159	+	1.24136i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$683$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$684$	−489.840	−	1072.60i	−0.716141	−	1.56813i
$685$		0			0
$686$		0			0
$687$	253.733	+	356.318i	0.369335	+	0.518658i
$688$	−1094.00	−	104.464i	−1.59011	−	0.151838i
$689$		0			0
$690$		0			0
$691$	−195.924	+	566.086i	−0.283537	+	0.819227i	0.709844	+	0.704359i	$0.248766\pi$
$691$	−195.924	+	566.086i	−0.283537	+	0.819227i	−0.993381	+	0.114868i	$0.963356\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	588.670	−	1019.61i	0.840957	−	1.45658i
$701$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$701$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$702$		0			0
$703$	1888.77	+	973.730i	2.68673	+	1.38511i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−940.716	−	739.787i	−1.32682	−	1.04342i	−0.994978	−	0.100089i	$-0.968087\pi$
$709$	−940.716	−	739.787i	−1.32682	−	1.04342i	−0.331843	−	0.943335i	$-0.607670\pi$
$710$		0			0
$711$	950.385	−	90.7508i	1.33669	−	0.127638i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$719$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$720$		0			0
$721$	772.681	+	2232.52i	1.07168	+	3.09642i
$722$		0			0
$723$	151.189	+	331.058i	0.209114	+	0.457895i
$724$	1305.50	+	522.645i	1.80318	+	0.721885i
$725$		0			0
$726$		0			0
$727$	−82.3791	+	339.571i	−0.113314	+	0.467086i	0.886675	+	0.462393i	$0.153009\pi$
$727$	−82.3791	+	339.571i	−0.113314	+	0.467086i	−0.999989	+	0.00469307i	$0.998506\pi$
$728$		0			0
$729$	−103.748	+	721.580i	−0.142315	+	0.989821i
$730$		0			0
$731$		0			0
$732$	−603.634	+	696.631i	−0.824637	+	0.951681i
$733$	830.471	−	1166.23i	1.13298	−	1.59104i	0.391330	−	0.920251i	$-0.372015\pi$
$733$	830.471	−	1166.23i	1.13298	−	1.59104i	0.741646	−	0.670791i	$-0.234045\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1011.00	+	521.205i	−1.36806	+	0.705284i	−0.976631	−	0.214925i	$-0.931049\pi$
$739$	−1011.00	+	521.205i	−1.36806	+	0.705284i	−0.391429	+	0.920208i	$0.628019\pi$
$740$		0			0
$741$	729.231	−	1024.06i	0.984117	−	1.38200i
$742$		0			0
$743$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$743$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−132.910	−	291.033i	−0.176978	−	0.387528i	0.800266	−	0.599645i	$-0.204691\pi$
$751$	−132.910	−	291.033i	−0.176978	−	0.387528i	−0.977244	+	0.212117i	$0.931964\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	60.5017	+	1270.09i	0.0800287	+	1.68001i
$757$	−250.256	−	1031.57i	−0.330589	−	1.36270i	−0.858242	−	0.513245i	$-0.828443\pi$
$757$	−250.256	−	1031.57i	−0.330589	−	1.36270i	0.527653	−	0.849460i	$-0.323072\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$761$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$762$		0			0
$763$	1280.72	−	122.294i	1.67853	−	0.160280i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−712.987	+	285.437i	−0.928368	+	0.371662i
$769$	823.090	+	784.814i	1.07034	+	1.02056i	0.999734	+	0.0230602i	$0.00734096\pi$
$769$	823.090	+	784.814i	1.07034	+	1.02056i	0.0706034	+	0.997504i	$0.477508\pi$
$770$		0			0
$771$		0			0
$772$	909.408	+	867.119i	1.17799	+	1.12321i
$773$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$773$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$774$		0			0
$775$	466.271	+	807.605i	0.601640	+	1.04207i
$776$		0			0
$777$	−1500.58	−	1731.77i	−1.93125	−	2.22879i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	338.033	+	1393.39i	0.431164	+	1.77728i
$785$		0			0
$786$		0			0
$787$	−240.324	−	337.488i	−0.305367	−	0.428828i	0.633167	−	0.774015i	$-0.281755\pi$
$787$	−240.324	−	337.488i	−0.305367	−	0.428828i	−0.938534	+	0.345187i	$0.887815\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−964.999	−	185.988i	−1.21690	−	0.234538i
$794$		0			0
$795$		0			0
$796$	213.491	−	1484.86i	0.268204	−	1.86540i
$797$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$797$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	150.604	+	789.769i	0.187318	+	0.982299i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$809$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$810$		0			0
$811$	77.1705	−	1620.01i	0.0951547	−	1.99754i	0.0337733	−	0.999430i	$-0.489248\pi$
$811$	77.1705	−	1620.01i	0.0951547	−	1.99754i	0.0613814	−	0.998114i	$-0.480449\pi$
$812$		0			0
$813$	1103.71	+	709.309i	1.35757	+	0.872459i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2088.61	−	836.154i	−2.55644	−	1.02344i
$818$		0			0
$819$	−1140.45	+	732.920i	−1.39249	+	0.894896i
$820$		0			0
$821$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$821$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$822$		0			0
$823$	−72.2380	−	1516.46i	−0.0877739	−	1.84260i	−0.429815	−	0.902917i	$-0.641421\pi$
$823$	−72.2380	−	1516.46i	−0.0877739	−	1.84260i	0.342041	−	0.939685i	$-0.388882\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$827$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$828$		0			0
$829$	−222.535	−	1547.77i	−0.268438	−	1.86703i	−0.463306	−	0.886198i	$-0.653337\pi$
$829$	−222.535	−	1547.77i	−0.268438	−	1.86703i	0.194868	−	0.980829i	$-0.437572\pi$
$830$		0			0
$831$	−725.775	−	837.589i	−0.873376	−	1.00793i
$832$	−643.628	−	506.155i	−0.773592	−	0.608359i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−895.187	−	461.501i	−1.06952	−	0.551375i
$838$		0			0
$839$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$839$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−1102.61	−	1272.48i	−1.30641	−	1.50768i
$845$		0			0
$846$		0			0
$847$	−591.793	+	1295.84i	−0.698692	+	1.52992i
$848$		0			0
$849$	−139.891	+	41.0756i	−0.164771	+	0.0483812i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1050.10	+	100.272i	1.23107	+	0.117553i	0.690240	−	0.723580i	$-0.257505\pi$
$853$	1050.10	+	100.272i	1.23107	+	0.117553i	0.540828	+	0.841133i	$0.318111\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$857$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$858$		0			0
$859$	1347.46	−	1059.65i	1.56863	−	1.23359i	0.743256	−	0.669007i	$-0.233281\pi$
$859$	1347.46	−	1059.65i	1.56863	−	1.23359i	0.825378	−	0.564580i	$-0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−502.909	+	706.237i	−0.580057	+	0.814576i
$868$	−1685.51	−	494.911i	−1.94184	−	0.570174i
$869$		0			0
$870$		0			0
$871$	−648.926	+	560.065i	−0.745035	+	0.643014i
$872$		0			0
$873$	1059.30	−	546.108i	1.21340	−	0.625554i
$874$		0			0
$875$		0			0
$876$	−910.765	+	1051.08i	−1.03969	+	1.19986i
$877$	−1645.22	+	317.090i	−1.87596	+	0.361562i	−0.993966	−	0.109691i	$-0.965014\pi$
$877$	−1645.22	+	317.090i	−1.87596	+	0.361562i	−0.881998	+	0.471253i	$0.843802\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$881$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$882$		0			0
$883$	−1179.34	+	927.441i	−1.33560	+	1.05033i	−0.341730	+	0.939798i	$0.611013\pi$
$883$	−1179.34	+	927.441i	−1.33560	+	1.05033i	−0.993873	+	0.110532i	$0.964745\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$887$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$888$		0			0
$889$	160.778	+	15.3525i	0.180853	+	0.0172694i
$890$		0			0
$891$		0			0
$892$	−57.3282	+	165.639i	−0.0642693	+	0.185694i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−450.000	−	779.423i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	1755.78	+	1674.13i	1.94439	+	1.85397i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1133.74	+	453.880i	−1.24999	+	0.500419i	−0.899817	−	0.436267i	$-0.856300\pi$
$907$	−1133.74	+	453.880i	−1.24999	+	0.500419i	−0.350170	+	0.936686i	$0.613876\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$	−1565.09	+	149.448i	−1.71611	+	0.163869i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	559.612	−	164.317i	0.610930	−	0.179385i
$917$		0			0
$918$		0			0
$919$	−85.2825	−	1790.30i	−0.0927992	−	1.94810i	−0.267683	−	0.963507i	$-0.586258\pi$
$919$	−85.2825	−	1790.30i	−0.0927992	−	1.94810i	0.174884	−	0.984589i	$-0.444045\pi$
$920$		0			0
$921$	−537.647	−	755.019i	−0.583764	−	0.819782i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1505.74	+	602.805i	1.62782	+	0.651682i
$926$		0			0
$927$	1773.30	+	341.776i	1.91295	+	0.368691i
$928$		0			0
$929$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$929$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$930$		0			0
$931$	−139.663	+	2931.90i	−0.150014	+	3.14919i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1055.59			1.12657			0.563284	−	0.826264i	$-0.309538\pi$
$937$	1055.59			1.12657			0.563284	+	0.826264i	$0.309538\pi$
$938$		0			0
$939$	−630.483			−0.671441
$940$		0			0
$941$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$941$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$947$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$948$	300.108	−	1237.06i	0.316570	−	1.30492i
$949$	−1455.99	−	280.620i	−1.53424	−	0.295700i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$953$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	311.507	−	297.021i	0.324149	−	0.309075i
$962$		0			0
$963$		0			0
$964$	483.066	−	46.1272i	0.501106	−	0.0478498i
$965$		0			0
$966$		0			0
$967$	148.718	+	257.588i	0.153794	+	0.266378i	0.932619	−	0.360862i	$-0.117518\pi$
$967$	148.718	+	257.588i	0.153794	+	0.266378i	−0.778826	+	0.627241i	$0.784184\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$971$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$972$	863.948	+	445.396i	0.888835	+	0.458227i
$973$	−1493.85	−	1424.39i	−1.53531	−	1.46391i
$974$		0			0
$975$	479.772	−	830.989i	0.492073	−	0.852296i
$976$	614.516	+	1064.37i	0.629627	+	1.09055i
$977$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$977$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	408.552	−	894.604i	0.416465	−	0.911931i
$982$		0			0
$983$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$983$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−972.308	−	1365.42i	−0.984117	−	1.38200i
$989$		0			0
$990$		0			0
$991$	634.595	+	1389.57i	0.640358	+	1.40219i	0.899746	+	0.436415i	$0.143752\pi$
$991$	634.595	+	1389.57i	0.640358	+	1.40219i	−0.259388	+	0.965773i	$0.583521\pi$
$992$		0			0
$993$	1471.44	−	1157.15i	1.48181	−	1.16531i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−233.625	+	1624.90i	−0.234328	+	1.62979i	0.444702	+	0.895679i	$0.353310\pi$
$997$	−233.625	+	1624.90i	−0.234328	+	1.62979i	−0.679030	+	0.734110i	$0.737600\pi$
$998$		0			0
$999$	−1720.01	+	331.506i	−1.72174	+	0.331838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.194.1	yes	20
3.2	odd	2	CM	201.3.o.a.194.1	yes	20
67.19	even	33	inner	201.3.o.a.86.1	✓	20
201.86	odd	66	inner	201.3.o.a.86.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.86.1	✓	20	67.19	even	33	inner
201.3.o.a.86.1	✓	20	201.86	odd	66	inner
201.3.o.a.194.1	yes	20	1.1	even	1	trivial
201.3.o.a.194.1	yes	20	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.o (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(2.87848 + 0.845198i) q^{3} +(2.32023 - 3.25830i) q^{4} +(-0.560201 + 11.7601i) q^{7} +(7.57128 + 4.86577i) q^{9} +O(q^{10})\)	\(q+(2.87848 + 0.845198i) q^{3} +(2.32023 - 3.25830i) q^{4} +(-0.560201 + 11.7601i) q^{7} +(7.57128 + 4.86577i) q^{9} +(9.43264 - 7.41791i) q^{12} +(11.8775 + 4.75502i) q^{13} +(-5.23309 - 15.1200i) q^{16} +(-1.55852 - 32.7173i) q^{19} +(-11.5521 + 33.3776i) q^{21} +(-3.55787 - 24.7455i) q^{25} +(17.6812 + 20.4052i) q^{27} +(37.0181 + 29.1113i) q^{28} +(-34.6297 + 13.8636i) q^{31} +(33.4212 - 13.3798i) q^{36} +(-32.4383 + 56.1848i) q^{37} +(30.1701 + 23.7260i) q^{39} +(28.5332 - 62.4789i) q^{43} +(-2.28393 - 47.9456i) q^{48} +(-89.2072 - 8.51825i) q^{49} +(43.0517 - 27.6676i) q^{52} +(23.1664 - 95.4934i) q^{57} +(-75.4264 + 14.5372i) q^{61} +(-61.4632 + 86.3130i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(-30.8183 + 59.4914i) q^{67} +(-113.804 + 21.9338i) q^{73} +(10.6736 - 74.2366i) q^{75} +(-110.219 - 70.8335i) q^{76} +(83.3835 - 65.5735i) q^{79} +(33.6486 + 73.6802i) q^{81} +(81.9509 + 115.084i) q^{84} +(-62.5731 + 137.016i) q^{91} +(-111.398 + 10.6373i) q^{93} +(66.2104 - 114.680i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9} - 12 q^{12} + 23 q^{13} + 16 q^{16} + 26 q^{19} - 6 q^{21} - 50 q^{25} + 54 q^{27} + 8 q^{28} - 13 q^{31} + 36 q^{36} + 26 q^{37} - 69 q^{39} + 122 q^{43} - 48 q^{48} - 45 q^{49} + 828 q^{52} - 441 q^{57} + 47 q^{61} - 1269 q^{63} - 128 q^{64} + 109 q^{67} - 924 q^{73} + 150 q^{75} - 208 q^{76} + 252 q^{79} - 162 q^{81} + 1692 q^{84} - 92 q^{91} + 39 q^{93} + 167 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9 - 12 * q^12 + 23 * q^13 + 16 * q^16 + 26 * q^19 - 6 * q^21 - 50 * q^25 + 54 * q^27 + 8 * q^28 - 13 * q^31 + 36 * q^36 + 26 * q^37 - 69 * q^39 + 122 * q^43 - 48 * q^48 - 45 * q^49 + 828 * q^52 - 441 * q^57 + 47 * q^61 - 1269 * q^63 - 128 * q^64 + 109 * q^67 - 924 * q^73 + 150 * q^75 - 208 * q^76 + 252 * q^79 - 162 * q^81 + 1692 * q^84 - 92 * q^91 + 39 * q^93 + 167 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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