LMFDB - Embedded newform 201.3.o.a.35.1

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			35.1
Root			$-0.327068 - 0.945001i$ of defining polynomial
Character	$\chi$	$=$	201.35
Dual form			201.3.o.a.23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.426945 + 2.96946i) q^{3} +(-3.55534 - 1.83291i) q^{4} +(6.63942 + 6.33068i) q^{7} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})$	$q+(0.426945 + 2.96946i) q^{3} +(-3.55534 - 1.83291i) q^{4} +(6.63942 + 6.33068i) q^{7} +(-8.63544 + 2.53559i) q^{9} +(3.92482 - 11.3400i) q^{12} +(-25.0236 + 4.82291i) q^{13} +(9.28091 + 13.0332i) q^{16} +(-22.5305 + 21.4828i) q^{19} +(-15.9641 + 22.4184i) q^{21} +(-16.3715 - 18.8937i) q^{25} +(-11.2162 - 24.5601i) q^{27} +(-12.0019 - 34.6772i) q^{28} +(48.8442 + 9.41395i) q^{31} +(35.3494 + 6.81304i) q^{36} +(-17.6667 + 30.5997i) q^{37} +(-25.0052 - 72.2477i) q^{39} +(72.1471 + 46.3661i) q^{43} +(-34.7392 + 33.1238i) q^{48} +(1.67295 + 35.1196i) q^{49} +(97.8075 + 28.7189i) q^{52} +(-73.4117 - 57.7316i) q^{57} +(117.591 + 11.2286i) q^{61} +(-73.3864 - 37.8333i) q^{63} +(-9.10815 - 63.3486i) q^{64} +(6.23749 + 66.7090i) q^{67} +(-85.7981 - 8.19272i) q^{73} +(49.1146 - 56.6812i) q^{75} +(119.480 - 35.0824i) q^{76} +(49.8928 - 144.156i) q^{79} +(68.1415 - 43.7919i) q^{81} +(97.8485 - 50.4444i) q^{84} +(-196.675 - 126.395i) q^{91} +(-7.10062 + 149.060i) q^{93} +(-36.9777 + 64.0473i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{19}{33}\right)$

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		0.235759	−	0.971812i	$-0.424242\pi$
$2$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$3$	0.426945	+	2.96946i	0.142315	+	0.989821i
$3$	0.426945	+	2.96946i	0.142315	+	0.989821i
$4$	−3.55534	−	1.83291i	−0.888835	−	0.458227i
$5$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$5$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$6$		0			0
$7$	6.63942	+	6.33068i	0.948489	+	0.904383i	0.995524	−	0.0945106i	$-0.0301286\pi$
$7$	6.63942	+	6.33068i	0.948489	+	0.904383i	−0.0470346	+	0.998893i	$0.514977\pi$
$8$		0			0
$9$	−8.63544	+	2.53559i	−0.959493	+	0.281733i
$10$		0			0
$11$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$11$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$12$	3.92482	−	11.3400i	0.327068	−	0.945001i
$13$	−25.0236	+	4.82291i	−1.92489	+	0.370993i	−0.999958	−	0.00912873i	$-0.997094\pi$
$13$	−25.0236	+	4.82291i	−1.92489	+	0.370993i	−0.924936	+	0.380122i	$0.875882\pi$
$14$		0			0
$15$		0			0
$16$	9.28091	+	13.0332i	0.580057	+	0.814576i
$17$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$17$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$18$		0			0
$19$	−22.5305	+	21.4828i	−1.18582	+	1.13067i	−0.197177	+	0.980368i	$0.563177\pi$
$19$	−22.5305	+	21.4828i	−1.18582	+	1.13067i	−0.988640	+	0.150305i	$0.951974\pi$
$20$		0			0
$21$	−15.9641	+	22.4184i	−0.760193	+	1.06754i
$22$		0			0
$23$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$23$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$24$		0			0
$25$	−16.3715	−	18.8937i	−0.654861	−	0.755750i
$26$		0			0
$27$	−11.2162	−	24.5601i	−0.415415	−	0.909632i
$28$	−12.0019	−	34.6772i	−0.428639	−	1.23847i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	48.8442	+	9.41395i	1.57562	+	0.303676i	0.900775	−	0.434285i	$-0.142999\pi$
$31$	48.8442	+	9.41395i	1.57562	+	0.303676i	0.674844	+	0.737961i	$0.264211\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	35.3494	+	6.81304i	0.981929	+	0.189251i
$37$	−17.6667	+	30.5997i	−0.477480	+	0.827019i	−0.999667	−	0.0258120i	$-0.991783\pi$
$37$	−17.6667	+	30.5997i	−0.477480	+	0.827019i	0.522187	+	0.852831i	$0.325116\pi$
$38$		0			0
$39$	−25.0052	−	72.2477i	−0.641158	−	1.85250i
$40$		0			0
$41$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$41$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$42$		0			0
$43$	72.1471	+	46.3661i	1.67784	+	1.07828i	0.879165	+	0.476517i	$0.158101\pi$
$43$	72.1471	+	46.3661i	1.67784	+	1.07828i	0.798674	+	0.601764i	$0.205535\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$47$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$48$	−34.7392	+	33.1238i	−0.723734	+	0.690079i
$49$	1.67295	+	35.1196i	0.0341419	+	0.716727i
$50$		0			0
$51$		0			0
$52$	97.8075	+	28.7189i	1.88091	+	0.552286i
$53$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$53$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−73.4117	−	57.7316i	−1.28792	−	1.01283i
$58$		0			0
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$	117.591	+	11.2286i	1.92773	+	0.184076i	0.987626	−	0.156825i	$-0.0501258\pi$
$61$	117.591	+	11.2286i	1.92773	+	0.184076i	0.940099	+	0.340900i	$0.110732\pi$
$62$		0			0
$63$	−73.3864	−	37.8333i	−1.16486	−	0.600529i
$64$	−9.10815	−	63.3486i	−0.142315	−	0.989821i
$65$		0			0
$66$		0			0
$67$	6.23749	+	66.7090i	0.0930968	+	0.995657i
$67$	6.23749	+	66.7090i	0.0930968	+	0.995657i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$71$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$72$		0			0
$73$	−85.7981	−	8.19272i	−1.17532	−	0.112229i	−0.510932	−	0.859621i	$-0.670700\pi$
$73$	−85.7981	−	8.19272i	−1.17532	−	0.112229i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	49.1146	−	56.6812i	0.654861	−	0.755750i
$76$	119.480	−	35.0824i	1.57210	−	0.461610i
$77$		0			0
$78$		0			0
$79$	49.8928	−	144.156i	0.631554	−	1.82476i	0.0696203	−	0.997574i	$-0.477821\pi$
$79$	49.8928	−	144.156i	0.631554	−	1.82476i	0.561934	−	0.827182i	$-0.310058\pi$
$80$		0			0
$81$	68.1415	−	43.7919i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$83$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$84$	97.8485	−	50.4444i	1.16486	−	0.600529i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$89$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$90$		0			0
$91$	−196.675	−	126.395i	−2.16126	−	1.38896i
$92$		0			0
$93$	−7.10062	+	149.060i	−0.0763507	+	1.60280i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−36.9777	+	64.0473i	−0.381214	+	0.660281i	−0.991236	−	0.132103i	$-0.957827\pi$
$97$	−36.9777	+	64.0473i	−0.381214	+	0.660281i	0.610022	+	0.792384i	$0.291160\pi$
$98$		0			0
$99$		0			0
$100$	23.5759	+	97.1812i	0.235759	+	0.971812i
$101$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$101$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$102$		0			0
$103$	−163.612	−	31.5335i	−1.58846	−	0.306151i	−0.682948	−	0.730467i	$-0.739302\pi$
$103$	−163.612	−	31.5335i	−1.58846	−	0.306151i	−0.905514	+	0.424316i	$0.860514\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$	−5.13885	+	107.878i	−0.0475819	+	0.998867i
$109$	103.464	+	119.403i	0.949208	+	1.09544i	0.995332	+	0.0965092i	$0.0307677\pi$
$109$	103.464	+	119.403i	0.949208	+	1.09544i	−0.0461237	+	0.998936i	$0.514687\pi$
$110$		0			0
$111$	−98.4074	−	39.3964i	−0.886553	−	0.354922i
$112$	−20.8892	+	145.288i	−0.186511	+	1.29721i
$113$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$113$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	203.861	−	105.098i	1.74240	−	0.898271i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	118.813	−	22.8994i	0.981929	−	0.189251i
$122$		0			0
$123$		0			0
$124$	−156.403	−	122.997i	−1.26131	−	0.991908i
$125$		0			0
$126$		0			0
$127$	107.592	+	102.589i	0.847184	+	0.807788i	0.983126	−	0.182929i	$-0.0585577\pi$
$127$	107.592	+	102.589i	0.847184	+	0.807788i	−0.135942	+	0.990717i	$0.543406\pi$
$128$		0			0
$129$	−106.880	+	234.034i	−0.828525	+	1.81422i
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$	−285.590			−2.14730
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$137$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$138$		0			0
$139$	−51.2714	+	112.269i	−0.368859	+	0.807689i	0.630641	+	0.776075i	$0.282792\pi$
$139$	−51.2714	+	112.269i	−0.368859	+	0.807689i	−0.999500	+	0.0316144i	$0.989935\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−113.192	−	89.0149i	−0.786053	−	0.618159i
$145$		0			0
$146$		0			0
$147$	−103.572	+	19.9619i	−0.704573	+	0.135795i
$148$	118.898	−	76.4109i	0.803363	−	0.516290i
$149$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$149$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$150$		0			0
$151$	−261.251	+	134.684i	−1.73014	+	0.891948i	−0.759814	+	0.650140i	$0.774710\pi$
$151$	−261.251	+	134.684i	−1.73014	+	0.891948i	−0.970324	+	0.241808i	$0.922260\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−43.5213	+	302.697i	−0.278983	+	1.94037i
$157$	29.0025	+	11.6109i	0.184730	+	0.0739546i	0.462183	−	0.886784i	$-0.347066\pi$
$157$	29.0025	+	11.6109i	0.184730	+	0.0739546i	−0.277454	+	0.960739i	$0.589490\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−71.9918	−	124.694i	−0.441668	−	0.764991i	0.556146	−	0.831085i	$-0.312280\pi$
$163$	−71.9918	−	124.694i	−0.441668	−	0.764991i	−0.997813	+	0.0660939i	$0.978946\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$167$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$168$		0			0
$169$	446.028	−	178.562i	2.63922	−	1.05658i
$170$		0			0
$171$	140.089	−	242.642i	0.819235	−	1.41896i
$172$	−171.523	−	297.086i	−0.997225	−	1.72725i
$173$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$173$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$174$		0			0
$175$	10.9127	−	229.086i	0.0623585	−	1.30907i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$179$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$180$		0			0
$181$	−176.515	+	138.813i	−0.975222	+	0.766923i	−0.972459	−	0.233073i	$-0.925122\pi$
$181$	−176.515	+	138.813i	−0.975222	+	0.766923i	−0.00276243	+	0.999996i	$0.500879\pi$
$182$		0			0
$183$	16.8620	+	353.977i	0.0921421	+	1.93430i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	81.0127	−	234.071i	0.428639	−	1.23847i
$190$		0			0
$191$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$191$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$192$	184.223	−	54.0927i	0.959493	−	0.281733i
$193$	23.7038	−	27.3556i	0.122817	−	0.141739i	−0.691011	−	0.722845i	$-0.742834\pi$
$193$	23.7038	−	27.3556i	0.122817	−	0.141739i	0.813828	+	0.581106i	$0.197380\pi$
$194$		0			0
$195$		0			0
$196$	58.4231	−	127.929i	0.298077	−	0.652697i
$197$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$197$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$198$		0			0
$199$	−18.5114	+	76.3051i	−0.0930221	+	0.383442i	−0.999223	−	0.0394043i	$-0.987454\pi$
$199$	−18.5114	+	76.3051i	−0.0930221	+	0.383442i	0.906201	+	0.422847i	$0.138969\pi$
$200$		0			0
$201$	−195.427	+	47.0030i	−0.972274	+	0.233846i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−295.100	−	281.377i	−1.41875	−	1.35278i
$209$		0			0
$210$		0			0
$211$	324.112	+	254.884i	1.53607	+	1.20798i	0.899624	+	0.436665i	$0.143841\pi$
$211$	324.112	+	254.884i	1.53607	+	1.20798i	0.636451	+	0.771317i	$0.280402\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	264.701	+	371.720i	1.21982	+	1.71300i
$218$		0			0
$219$	−12.3030	−	258.272i	−0.0561782	−	1.17932i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−60.0478	+	417.642i	−0.269273	+	1.87283i	0.186099	+	0.982531i	$0.440416\pi$
$223$	−60.0478	+	417.642i	−0.269273	+	1.87283i	−0.455371	+	0.890302i	$0.650493\pi$
$224$		0			0
$225$	189.282	+	121.644i	0.841254	+	0.540641i
$226$		0			0
$227$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$227$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$228$	155.187	+	339.812i	0.680645	+	1.49040i
$229$	−36.6537	−	105.904i	−0.160060	−	0.462462i	0.836245	−	0.548357i	$-0.184746\pi$
$229$	−36.6537	−	105.904i	−0.160060	−	0.462462i	−0.996304	+	0.0858946i	$0.972625\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$233$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	449.367	+	86.6083i	1.89606	+	0.365436i
$238$		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
$241$	−147.164	−	322.244i	−0.610638	−	1.33711i	−0.922137	−	0.386864i	$-0.873558\pi$
$241$	−147.164	−	322.244i	−0.610638	−	1.33711i	0.311499	−	0.950247i	$-0.399169\pi$
$242$		0			0
$243$	159.131	+	183.647i	0.654861	+	0.755750i
$244$	−397.496	−	255.455i	−1.62908	−	1.04695i
$245$		0			0
$246$		0			0
$247$	460.186	−	646.240i	1.86310	−	2.61636i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$251$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$252$	191.569	+	269.021i	0.760193	+	1.06754i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−83.7294	+	241.920i	−0.327068	+	0.945001i
$257$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$257$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$258$		0			0
$259$	−311.014	+	91.3219i	−1.20083	+	0.352594i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	100.095	−	248.606i	0.373489	−	0.927635i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−75.1931	−	522.979i	−0.277465	−	1.92981i	−0.359418	−	0.933177i	$-0.617025\pi$
$271$	−75.1931	−	522.979i	−0.277465	−	1.92981i	0.0819526	−	0.996636i	$-0.473884\pi$
$272$		0			0
$273$	291.357	−	637.983i	1.06724	−	2.33693i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	341.640	−	100.315i	1.23336	−	0.362147i	0.400842	−	0.916147i	$-0.368718\pi$
$277$	341.640	−	100.315i	1.23336	−	0.362147i	0.832517	+	0.554000i	$0.186899\pi$
$278$		0			0
$279$	−445.661	+	42.5555i	−1.59735	+	0.152529i
$280$		0			0
$281$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$281$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$282$		0			0
$283$	107.695	+	31.6221i	0.380547	+	0.111739i	0.466411	−	0.884568i	$-0.345547\pi$
$283$	107.695	+	31.6221i	0.380547	+	0.111739i	−0.0858637	+	0.996307i	$0.527365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	167.636	−	235.412i	0.580057	−	0.814576i
$290$		0			0
$291$	−205.974	−	82.4594i	−0.707813	−	0.283365i
$292$	290.025	+	186.388i	0.993236	+	0.638314i
$293$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$293$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−278.510	+	111.499i	−0.928368	+	0.371662i
$301$	185.486	+	764.584i	0.616233	+	2.54015i
$302$		0			0
$303$		0			0
$304$	−489.094	−	94.2651i	−1.60886	−	0.310082i
$305$		0			0
$306$		0			0
$307$	−157.395	−	454.764i	−0.512688	−	1.48132i	−0.843245	−	0.537530i	$-0.819358\pi$
$307$	−157.395	−	454.764i	−0.512688	−	1.48132i	0.330556	−	0.943786i	$-0.392764\pi$
$308$		0			0
$309$	23.7847	−	499.302i	0.0769730	−	1.61586i
$310$		0			0
$311$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$311$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$312$		0			0
$313$	−43.8351	+	304.880i	−0.140048	+	0.974058i	0.791689	+	0.610925i	$0.209202\pi$
$313$	−43.8351	+	304.880i	−0.140048	+	0.974058i	−0.931737	+	0.363133i	$0.881707\pi$
$314$		0			0
$315$		0			0
$316$	−441.610	+	421.074i	−1.39750	+	1.33251i
$317$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$317$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−322.533	+	30.7982i	−0.995472	+	0.0950560i
$325$	500.798	+	393.832i	1.54092	+	1.21179i
$326$		0			0
$327$	−310.391	+	358.210i	−0.949208	+	1.09544i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−51.6122	−	26.6080i	−0.155928	−	0.0803866i	0.378496	−	0.925603i	$-0.376441\pi$
$331$	−51.6122	−	26.6080i	−0.155928	−	0.0803866i	−0.534424	+	0.845216i	$0.679472\pi$
$332$		0			0
$333$	74.9717	−	309.037i	0.225140	−	0.928040i
$334$		0			0
$335$		0			0
$336$	−440.345			−1.31055
$337$	−142.128	+	585.860i	−0.421745	+	1.73846i	0.223707	+	0.974656i	$0.428184\pi$
$337$	−142.128	+	585.860i	−0.421745	+	1.73846i	−0.645452	+	0.763801i	$0.723331\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	83.1484	−	95.9583i	0.242415	−	0.279762i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$347$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$348$		0			0
$349$	−587.182	+	377.359i	−1.68247	+	1.08126i	−0.837694	+	0.546140i	$0.816097\pi$
$349$	−587.182	+	377.359i	−1.68247	+	1.08126i	−0.844777	+	0.535118i	$0.820267\pi$
$350$		0			0
$351$	399.121	+	560.487i	1.13710	+	1.59683i
$352$		0			0
$353$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$353$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$359$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$360$		0			0
$361$	28.9362	−	607.446i	0.0801557	−	1.68267i
$362$		0			0
$363$	118.726	+	343.035i	0.327068	+	0.945001i
$364$	467.576	+	809.865i	1.28455	+	2.22490i
$365$		0			0
$366$		0			0
$367$	−148.187	+	59.3253i	−0.403780	+	0.161649i	−0.564658	−	0.825325i	$-0.690992\pi$
$367$	−148.187	+	59.3253i	−0.403780	+	0.161649i	0.160877	+	0.986974i	$0.448568\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	298.459	−	516.946i	0.802308	−	1.38964i
$373$	−210.205	−	364.086i	−0.563553	−	0.976102i	−0.997183	−	0.0750108i	$-0.976101\pi$
$373$	−210.205	−	364.086i	−0.563553	−	0.976102i	0.433630	−	0.901091i	$-0.357232\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	123.131	+	49.2941i	0.324883	+	0.130063i	0.528370	−	0.849014i	$-0.322803\pi$
$379$	123.131	+	49.2941i	0.324883	+	0.130063i	−0.203487	+	0.979078i	$0.565228\pi$
$380$		0			0
$381$	−258.699	+	363.291i	−0.678999	+	0.953521i
$382$		0			0
$383$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$383$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−740.587	−	217.456i	−1.91366	−	0.561902i
$388$	248.861	−	159.933i	0.641394	−	0.412199i
$389$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$389$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−286.076	+	626.420i	−0.720595	+	1.57788i	0.0924735	+	0.995715i	$0.470523\pi$
$397$	−286.076	+	626.420i	−0.720595	+	1.57788i	−0.813069	+	0.582168i	$0.802205\pi$
$398$		0			0
$399$	−121.931	−	848.050i	−0.305592	−	2.12544i
$400$	94.3036	−	388.725i	0.235759	−	0.971812i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−1267.66			−3.14556
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	534.847	+	509.976i	1.30769	+	1.24688i	0.949700	+	0.313161i	$0.101388\pi$
$409$	534.847	+	509.976i	1.30769	+	1.24688i	0.357994	+	0.933724i	$0.383461\pi$
$410$		0			0
$411$		0			0
$412$	523.897	+	411.997i	1.27159	+	0.999993i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−355.268	−	104.316i	−0.851962	−	0.250159i
$418$		0			0
$419$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$419$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$420$		0			0
$421$	605.966	−	577.788i	1.43935	−	1.37242i	0.646691	−	0.762752i	$-0.276152\pi$
$421$	605.966	−	577.788i	1.43935	−	1.37242i	0.792659	−	0.609666i	$-0.208696\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	709.654	+	818.984i	1.66195	+	1.91800i
$428$		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
$433$	492.592	+	94.9394i	1.13763	+	0.219260i	0.723077	−	0.690767i	$-0.242727\pi$
$433$	492.592	+	94.9394i	1.13763	+	0.219260i	0.414550	+	0.910027i	$0.363939\pi$
$434$		0			0
$435$		0			0
$436$	−148.993	−	614.160i	−0.341728	−	1.40862i
$437$		0			0
$438$		0			0
$439$	−241.781	+	418.776i	−0.550753	+	0.953932i	0.447468	+	0.894300i	$0.352326\pi$
$439$	−241.781	+	418.776i	−0.550753	+	0.953932i	−0.998220	+	0.0596318i	$0.981007\pi$
$440$		0			0
$441$	−103.496	−	299.031i	−0.234684	−	0.678076i
$442$		0			0
$443$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$443$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$444$	277.662	+	320.439i	0.625365	+	0.721710i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	340.567	−	478.259i	0.760193	−	1.06754i
$449$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$449$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−511.480	−	718.273i	−1.12909	−	1.58559i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	98.8593	−	285.635i	0.216322	−	0.625022i	−0.783670	−	0.621177i	$-0.786655\pi$
$457$	98.8593	−	285.635i	0.216322	−	0.625022i	0.999993	−	0.00384511i	$-0.00122394\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	−288.671	−	27.5648i	−0.623480	−	0.0595351i	−0.221465	−	0.975168i	$-0.571084\pi$
$463$	−288.671	−	27.5648i	−0.623480	−	0.0595351i	−0.402015	+	0.915633i	$0.631690\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$467$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$468$	−917.430			−1.96032
$469$	−380.900	+	482.397i	−0.812154	+	1.02857i
$470$		0			0
$471$	−22.0956	+	91.0792i	−0.0469121	+	0.193374i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	774.749	+	73.9796i	1.63105	+	0.155746i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$479$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$480$		0			0
$481$	294.507	−	850.921i	0.612280	−	1.76907i
$482$		0			0
$483$		0			0
$484$	−464.395	−	136.359i	−0.959493	−	0.281733i
$485$		0			0
$486$		0			0
$487$	−37.3621	−	784.327i	−0.0767189	−	1.61053i	−0.629363	−	0.777111i	$-0.716684\pi$
$487$	−37.3621	−	784.327i	−0.0767189	−	1.61053i	0.552645	−	0.833417i	$-0.313619\pi$
$488$		0			0
$489$	339.536	−	267.014i	0.694349	−	0.546042i
$490$		0			0
$491$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$491$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	330.625	+	723.967i	0.666582	+	1.45961i
$497$		0			0
$498$		0			0
$499$	−368.478	+	638.222i	−0.738433	+	1.27900i	0.214768	+	0.976665i	$0.431100\pi$
$499$	−368.478	+	638.222i	−0.738433	+	1.27900i	−0.953201	+	0.302338i	$0.902233\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$503$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	720.664	+	1248.23i	1.42143	+	2.46199i
$508$	−194.491	−	561.946i	−0.382857	−	1.10619i
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$	−517.784	−	597.555i	−1.01328	−	1.16938i
$512$		0			0
$513$	780.326	+	312.395i	1.52110	+	0.608958i
$514$		0			0
$515$		0			0
$516$	808.956	−	636.170i	1.56774	−	1.23289i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$521$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$522$		0			0
$523$	825.530	−	159.108i	1.57845	−	0.304222i	0.676628	−	0.736325i	$-0.263441\pi$
$523$	825.530	−	159.108i	1.57845	−	0.304222i	0.901824	+	0.432104i	$0.142228\pi$
$524$		0			0
$525$	684.923	−	65.4022i	1.30462	−	0.124576i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	382.855	+	365.052i	0.723734	+	0.690079i
$530$		0			0
$531$		0			0
$532$	1015.37	+	523.460i	1.90859	+	0.983948i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−442.455	+	968.841i	−0.817847	+	1.79083i	−0.248740	+	0.968570i	$0.580016\pi$
$541$	−442.455	+	968.841i	−0.817847	+	1.79083i	−0.569107	+	0.822264i	$0.692711\pi$
$542$		0			0
$543$	−487.562	−	464.890i	−0.897905	−	0.856151i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.27066	−	0.503287i	0.00963557	−	0.000920085i	−0.0902371	−	0.995920i	$-0.528763\pi$
$547$	5.27066	−	0.503287i	0.00963557	−	0.000920085i	0.0998727	+	0.995000i	$0.468156\pi$
$548$		0			0
$549$	−1043.92	+	201.200i	−1.90150	+	0.366484i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1243.86	−	641.256i	2.24930	−	1.15959i
$554$		0			0
$555$		0			0
$556$	388.066	−	305.178i	0.697960	−	0.548882i
$557$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$557$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$558$		0			0
$559$	−2029.00	−	812.290i	−3.62970	−	1.45311i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$563$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	729.653	+	140.629i	1.28687	+	0.248023i
$568$		0			0
$569$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$569$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$570$		0			0
$571$	233.057	−	93.3018i	0.408155	−	0.163401i	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	233.057	−	93.3018i	0.408155	−	0.163401i	0.566649	+	0.823959i	$0.308240\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	239.279	+	523.948i	0.415415	+	0.909632i
$577$	1.84658	−	38.7644i	0.00320030	−	0.0671827i	−0.996702	−	0.0811463i	$-0.974142\pi$
$577$	1.84658	−	38.7644i	0.00320030	−	0.0671827i	0.999903	+	0.0139637i	$0.00444492\pi$
$578$		0			0
$579$	91.3517	+	58.7082i	0.157775	+	0.101396i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$587$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$588$	404.823	+	118.867i	0.688474	+	0.202154i
$589$	−1302.72	+	837.209i	−2.21175	+	1.42141i
$590$		0			0
$591$		0			0
$592$	−562.776	+	53.7386i	−0.950635	+	0.0907746i
$593$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$593$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−234.488	−	22.3909i	−0.392778	−	0.0375058i
$598$		0			0
$599$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$599$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$600$		0			0
$601$	147.669	−	608.699i	0.245705	−	1.01281i	−0.706588	−	0.707625i	$-0.749767\pi$
$601$	147.669	−	608.699i	0.245705	−	1.01281i	0.952293	−	0.305185i	$-0.0987183\pi$
$602$		0			0
$603$	−223.010	−	560.246i	−0.369835	−	0.929098i
$604$	1175.70			1.94652
$605$		0			0
$606$		0			0
$607$	−607.404	−	313.138i	−1.00067	−	0.515879i	−0.121599	−	0.992579i	$-0.538802\pi$
$607$	−607.404	−	313.138i	−1.00067	−	0.515879i	−0.879066	+	0.476701i	$0.841833\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−338.669	−	266.332i	−0.552478	−	0.434474i	0.302491	−	0.953152i	$-0.402182\pi$
$613$	−338.669	−	266.332i	−0.552478	−	0.434474i	−0.854969	+	0.518679i	$0.826424\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$617$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$	56.3369	+	79.1141i	0.0910128	+	0.127810i	0.857572	−	0.514364i	$-0.171972\pi$
$619$	56.3369	+	79.1141i	0.0910128	+	0.127810i	−0.766559	+	0.642174i	$0.778033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	709.549	−	996.422i	1.13710	−	1.59683i
$625$	−88.9468	+	618.638i	−0.142315	+	0.989821i
$626$		0			0
$627$		0			0
$628$	−81.8323	−	94.4395i	−0.130306	−	0.150381i
$629$		0			0
$630$		0			0
$631$	−314.039	−	907.355i	−0.497684	−	1.43796i	−0.862105	−	0.506729i	$-0.830854\pi$
$631$	−314.039	−	907.355i	−0.497684	−	1.43796i	0.364421	−	0.931234i	$-0.381267\pi$
$632$		0			0
$633$	−618.492	+	1071.26i	−0.977081	+	1.69235i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−211.242	−	870.752i	−0.331620	−	1.36696i
$638$		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
$643$	389.815	+	853.575i	0.606244	+	1.32749i	0.925114	+	0.379690i	$0.123969\pi$
$643$	389.815	+	853.575i	0.606244	+	1.32749i	−0.318870	+	0.947798i	$0.603303\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$647$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−990.797	+	944.723i	−1.52196	+	1.45119i
$652$	27.4041	+	575.282i	0.0420308	+	0.882335i
$653$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$653$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	761.677	−	146.801i	1.15933	−	0.223442i
$658$		0			0
$659$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$659$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$660$		0			0
$661$	−474.712	+	139.388i	−0.718172	+	0.210874i	−0.620345	−	0.784329i	$-0.713007\pi$
$661$	−474.712	+	139.388i	−0.718172	+	0.210874i	−0.0978269	+	0.995203i	$0.531189\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1265.81			−1.89209
$670$		0			0
$671$		0			0
$672$		0			0
$673$	154.115	+	1071.90i	0.228997	+	1.59271i	0.702345	+	0.711837i	$0.252136\pi$
$673$	154.115	+	1071.90i	0.228997	+	1.59271i	−0.473348	+	0.880876i	$0.656955\pi$
$674$		0			0
$675$	−280.405	+	614.002i	−0.415415	+	0.909632i
$676$	−1913.07	−	182.676i	−2.82998	−	0.270231i
$677$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$677$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$678$		0			0
$679$	−650.973	+	191.143i	−0.958724	+	0.281507i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$683$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$684$	−942.804	+	605.903i	−1.37837	+	0.885824i
$685$		0			0
$686$		0			0
$687$	298.829	−	154.057i	0.434976	−	0.224246i
$688$	65.2911	+	1370.63i	0.0948998	+	1.99219i
$689$		0			0
$690$		0			0
$691$	682.881	−	958.972i	0.988250	−	1.38780i	0.0674705	−	0.997721i	$-0.478507\pi$
$691$	682.881	−	958.972i	0.988250	−	1.38780i	0.920780	−	0.390082i	$-0.127553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−458.692	+	794.478i	−0.655275	+	1.13497i
$701$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$701$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$702$		0			0
$703$	−259.326	−	1068.96i	−0.368885	−	1.52057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−388.886	−	1123.61i	−0.548499	−	1.58478i	−0.789169	−	0.614176i	$-0.789488\pi$
$709$	−388.886	−	1123.61i	−0.548499	−	1.58478i	0.240670	−	0.970607i	$-0.422633\pi$
$710$		0			0
$711$	−65.3257	+	1371.36i	−0.0918786	+	1.92877i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$719$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$720$		0			0
$721$	−886.658	−	1245.14i	−1.22976	−	1.72696i
$722$		0			0
$723$	894.060	−	574.578i	1.23660	−	0.794713i
$724$	882.003	−	169.992i	1.21824	−	0.234796i
$725$		0			0
$726$		0			0
$727$	1059.05	+	832.843i	1.45673	+	1.14559i	0.960805	+	0.277226i	$0.0894152\pi$
$727$	1059.05	+	832.843i	1.45673	+	1.14559i	0.495930	+	0.868362i	$0.334827\pi$
$728$		0			0
$729$	−477.393	+	550.941i	−0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$	588.857	−	1289.42i	0.804449	−	1.76150i
$733$	621.590	+	320.452i	0.848008	+	0.437179i	0.826733	−	0.562595i	$-0.190197\pi$
$733$	621.590	+	320.452i	0.848008	+	0.437179i	0.0212755	+	0.999774i	$0.493227\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	182.076	−	750.528i	0.246382	−	1.01560i	−0.705397	−	0.708813i	$-0.749231\pi$
$739$	182.076	−	750.528i	0.246382	−	1.01560i	0.951778	−	0.306786i	$-0.0992537\pi$
$740$		0			0
$741$	2115.46	+	1090.60i	2.85487	+	1.47179i
$742$		0			0
$743$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$743$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1260.30	−	809.947i	1.67817	−	1.07849i	0.800266	−	0.599645i	$-0.204691\pi$
$751$	1260.30	−	809.947i	1.67817	−	1.07849i	0.877899	−	0.478846i	$-0.158945\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−717.058	+	683.713i	−0.948489	+	0.904383i
$757$	−561.514	+	441.580i	−0.741762	+	0.583328i	−0.915906	−	0.401392i	$-0.868527\pi$
$757$	−561.514	+	441.580i	−0.741762	+	0.583328i	0.174144	+	0.984720i	$0.444284\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$	−68.9656	+	1447.77i	−0.0903874	+	1.89747i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−754.121	−	145.345i	−0.981929	−	0.189251i
$769$	368.532	−	147.538i	0.479236	−	0.191857i	−0.119451	−	0.992840i	$-0.538114\pi$
$769$	368.532	−	147.538i	0.479236	−	0.191857i	0.598687	+	0.800983i	$0.295689\pi$
$770$		0			0
$771$		0			0
$772$	−134.415	+	53.8117i	−0.174113	+	0.0697043i
$773$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$773$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$774$		0			0
$775$	−621.789	−	1076.97i	−0.802308	−	1.38964i
$776$		0			0
$777$	−403.963	−	884.555i	−0.519901	−	1.13842i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−442.195	+	347.746i	−0.564024	+	0.443554i
$785$		0			0
$786$		0			0
$787$	1039.49	−	535.896i	1.32083	−	0.680935i	0.353734	−	0.935346i	$-0.384912\pi$
$787$	1039.49	−	535.896i	1.32083	−	0.680935i	0.967096	+	0.254411i	$0.0818816\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2996.72	+	286.152i	−3.77896	+	0.360847i
$794$		0			0
$795$		0			0
$796$	205.674	−	237.361i	0.258385	−	0.298192i
$797$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$797$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	780.962	+	191.087i	0.971346	+	0.237671i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$809$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$810$		0			0
$811$	996.221	+	949.895i	1.22839	+	1.17126i	0.979220	+	0.202803i	$0.0650051\pi$
$811$	996.221	+	949.895i	1.22839	+	1.17126i	0.249167	+	0.968461i	$0.419843\pi$
$812$		0			0
$813$	1520.87	−	446.566i	1.87068	−	0.549282i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2621.58	+	505.269i	−3.20879	+	0.618444i
$818$		0			0
$819$	2018.86	+	592.791i	2.46503	+	0.723798i
$820$		0			0
$821$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$821$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$822$		0			0
$823$	1067.38	−	1017.74i	1.29694	−	1.23663i	0.342041	−	0.939685i	$-0.388882\pi$
$823$	1067.38	−	1017.74i	1.29694	−	1.23663i	0.954896	−	0.296942i	$-0.0959668\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$827$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$828$		0			0
$829$	−211.580	−	244.176i	−0.255223	−	0.294543i	0.613650	−	0.789578i	$-0.289701\pi$
$829$	−211.580	−	244.176i	−0.255223	−	0.294543i	−0.868873	+	0.495035i	$0.835155\pi$
$830$		0			0
$831$	443.742	+	971.660i	0.533986	+	1.16927i
$832$	533.443	+	1541.28i	0.641158	+	1.85250i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−316.639	−	1305.21i	−0.378303	−	1.55938i
$838$		0			0
$839$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$839$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−685.149	−	1500.27i	−0.811788	−	1.77757i
$845$		0			0
$846$		0			0
$847$	933.821	+	600.130i	1.10250	+	0.708537i
$848$		0			0
$849$	−47.9209	+	333.297i	−0.0564439	+	0.392576i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−7.73413	−	162.359i	−0.00906698	−	0.190339i	−0.998857	−	0.0478042i	$-0.984778\pi$
$853$	−7.73413	−	162.359i	−0.00906698	−	0.190339i	0.989790	−	0.142535i	$-0.0455254\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$857$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$	451.480	−	1304.46i	0.525587	−	1.51858i	−0.299791	−	0.954005i	$-0.596917\pi$
$859$	451.480	−	1304.46i	0.525587	−	1.51858i	0.825378	−	0.564580i	$-0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$863$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	770.620	+	397.282i	0.888835	+	0.458227i
$868$	−259.773	−	1806.76i	−0.299278	−	2.08152i
$869$		0			0
$870$		0			0
$871$	−477.816	−	1639.22i	−0.548583	−	1.88200i
$872$		0			0
$873$	156.921	−	646.837i	0.179749	−	0.740935i
$874$		0			0
$875$		0			0
$876$	−429.647	+	940.796i	−0.490465	+	1.07397i
$877$	−701.894	−	67.0228i	−0.800335	−	0.0764228i	−0.313130	−	0.949710i	$-0.601378\pi$
$877$	−701.894	−	67.0228i	−0.800335	−	0.0764228i	−0.487205	+	0.873287i	$0.661984\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$881$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$882$		0			0
$883$	73.2968	−	211.777i	0.0830088	−	0.239838i	−0.895857	−	0.444343i	$-0.853437\pi$
$883$	73.2968	−	211.777i	0.0830088	−	0.239838i	0.978866	+	0.204505i	$0.0655584\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$887$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$888$		0			0
$889$	64.8927	+	1362.26i	0.0729951	+	1.53236i
$890$		0			0
$891$		0			0
$892$	978.988	−	1374.80i	1.09752	−	1.54125i
$893$		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
$903$	−2191.21	+	877.229i	−2.42659	+	0.971461i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1521.65	+	293.273i	1.67767	+	0.323344i	0.937213	−	0.348759i	$-0.113397\pi$
$907$	1521.65	+	293.273i	1.67767	+	0.323344i	0.740458	+	0.672103i	$0.234609\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$911$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$912$	71.1009	−	1492.59i	0.0779615	−	1.63661i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−63.7954	+	443.707i	−0.0696457	+	0.484396i
$917$		0			0
$918$		0			0
$919$	1327.65	−	1265.91i	1.44467	−	1.37749i	0.679431	−	0.733739i	$-0.262227\pi$
$919$	1327.65	−	1265.91i	1.44467	−	1.37749i	0.765237	−	0.643749i	$-0.222622\pi$
$920$		0			0
$921$	1283.21	−	661.539i	1.39328	−	0.718283i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	867.374	−	167.173i	0.937702	−	0.180727i
$926$		0			0
$927$	1492.81	−	142.546i	1.61037	−	0.153772i
$928$		0			0
$929$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$929$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$930$		0			0
$931$	−792.160	−	755.323i	−0.850870	−	0.811303i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1382.43			1.47538			0.737690	−	0.675140i	$-0.235917\pi$
$937$	1382.43			1.47538			0.737690	+	0.675140i	$0.235917\pi$
$938$		0			0
$939$	−924.046			−0.984074
$940$		0			0
$941$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$941$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$947$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$948$	−1438.91	−	1131.57i	−1.51783	−	1.19364i
$949$	2186.49	−	208.785i	2.30400	−	0.220005i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$953$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1404.97	+	562.466i	1.46199	+	0.585292i
$962$		0			0
$963$		0			0
$964$	−67.4249	+	1415.42i	−0.0699429	+	1.46828i
$965$		0			0
$966$		0			0
$967$	930.929	+	1612.42i	0.962698	+	1.66744i	0.715676	+	0.698432i	$0.246119\pi$
$967$	930.929	+	1612.42i	0.962698	+	1.66744i	0.247022	+	0.969010i	$0.420548\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$971$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$972$	−229.158	−	944.601i	−0.235759	−	0.971812i
$973$	−1051.15	+	420.817i	−1.08032	+	0.432495i
$974$		0			0
$975$	−955.656	+	1655.25i	−0.980160	+	1.69769i
$976$	945.009	+	1636.80i	0.968247	+	1.67705i
$977$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$977$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1196.21	−	768.759i	−1.21938	−	0.783649i
$982$		0			0
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2820.61	+	1454.13i	−2.85487	+	1.47179i
$989$		0			0
$990$		0			0
$991$	−1532.36	+	984.785i	−1.54627	+	0.993728i	−0.560020	+	0.828479i	$0.689206\pi$
$991$	−1532.36	+	984.785i	−1.54627	+	0.993728i	−0.986252	+	0.165249i	$0.947157\pi$
$992$		0			0
$993$	56.9758	−	164.621i	0.0573775	−	0.165781i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	227.661	−	262.734i	0.228346	−	0.263525i	−0.630002	−	0.776594i	$-0.716946\pi$
$997$	227.661	−	262.734i	0.228346	−	0.263525i	0.858348	+	0.513069i	$0.171491\pi$
$998$		0			0
$999$	949.684	+	90.6839i	0.950635	+	0.0907746i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.35.1	yes	20
3.2	odd	2	CM	201.3.o.a.35.1	yes	20
67.23	even	33	inner	201.3.o.a.23.1	✓	20
201.23	odd	66	inner	201.3.o.a.23.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.23.1	✓	20	67.23	even	33	inner
201.3.o.a.23.1	✓	20	201.23	odd	66	inner
201.3.o.a.35.1	yes	20	1.1	even	1	trivial
201.3.o.a.35.1	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)

\(f(q)\)	\(=\)	\(q+(0.426945 + 2.96946i) q^{3} +(-3.55534 - 1.83291i) q^{4} +(6.63942 + 6.33068i) q^{7} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)	\(q+(0.426945 + 2.96946i) q^{3} +(-3.55534 - 1.83291i) q^{4} +(6.63942 + 6.33068i) q^{7} +(-8.63544 + 2.53559i) q^{9} +(3.92482 - 11.3400i) q^{12} +(-25.0236 + 4.82291i) q^{13} +(9.28091 + 13.0332i) q^{16} +(-22.5305 + 21.4828i) q^{19} +(-15.9641 + 22.4184i) q^{21} +(-16.3715 - 18.8937i) q^{25} +(-11.2162 - 24.5601i) q^{27} +(-12.0019 - 34.6772i) q^{28} +(48.8442 + 9.41395i) q^{31} +(35.3494 + 6.81304i) q^{36} +(-17.6667 + 30.5997i) q^{37} +(-25.0052 - 72.2477i) q^{39} +(72.1471 + 46.3661i) q^{43} +(-34.7392 + 33.1238i) q^{48} +(1.67295 + 35.1196i) q^{49} +(97.8075 + 28.7189i) q^{52} +(-73.4117 - 57.7316i) q^{57} +(117.591 + 11.2286i) q^{61} +(-73.3864 - 37.8333i) q^{63} +(-9.10815 - 63.3486i) q^{64} +(6.23749 + 66.7090i) q^{67} +(-85.7981 - 8.19272i) q^{73} +(49.1146 - 56.6812i) q^{75} +(119.480 - 35.0824i) q^{76} +(49.8928 - 144.156i) q^{79} +(68.1415 - 43.7919i) q^{81} +(97.8485 - 50.4444i) q^{84} +(-196.675 - 126.395i) q^{91} +(-7.10062 + 149.060i) q^{93} +(-36.9777 + 64.0473i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 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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