LMFDB - Embedded newform 201.3.o.a.173.1

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			173.1
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	201.173
Dual form			201.3.o.a.122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.96458 + 2.26725i) q^{3} +(2.89494 - 2.76032i) q^{4} +(4.57702 + 3.59941i) q^{7} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})$	$q+(1.96458 + 2.26725i) q^{3} +(2.89494 - 2.76032i) q^{4} +(4.57702 + 3.59941i) q^{7} +(-1.28083 + 8.90839i) q^{9} +(11.9457 + 1.14067i) q^{12} +(5.06257 - 7.10938i) q^{13} +(0.761311 - 15.9819i) q^{16} +(-3.07942 + 2.42168i) q^{19} +(0.831178 + 17.4486i) q^{21} +(10.3854 + 22.7408i) q^{25} +(-22.7138 + 14.5973i) q^{27} +(23.1857 - 2.21396i) q^{28} +(12.6675 + 17.7890i) q^{31} +(20.8820 + 29.3247i) q^{36} +(-36.9110 - 63.9317i) q^{37} +(26.0646 - 2.48886i) q^{39} +(-28.5960 + 8.39653i) q^{43} +(37.7305 - 29.6716i) q^{48} +(-3.55882 - 14.6696i) q^{49} +(-4.96833 - 34.5555i) q^{52} +(-11.5403 - 2.22421i) q^{57} +(-48.2694 - 24.8846i) q^{61} +(-37.9273 + 36.1636i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-58.0895 - 33.3857i) q^{67} +(-121.076 - 62.4189i) q^{73} +(-31.1561 + 68.2224i) q^{75} +(-2.23011 + 15.5108i) q^{76} +(25.8152 + 2.46505i) q^{79} +(-77.7189 - 22.8203i) q^{81} +(50.5698 + 48.2182i) q^{84} +(48.7610 - 14.3175i) q^{91} +(-15.4457 + 63.6682i) q^{93} +(91.9873 + 159.327i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{29}{33}\right)$

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		0.928368	−	0.371662i	$-0.121212\pi$
$2$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$3$	1.96458	+	2.26725i	0.654861	+	0.755750i
$3$	1.96458	+	2.26725i	0.654861	+	0.755750i
$4$	2.89494	−	2.76032i	0.723734	−	0.690079i
$5$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$5$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$6$		0			0
$7$	4.57702	+	3.59941i	0.653860	+	0.514201i	0.889086	−	0.457739i	$-0.151341\pi$
$7$	4.57702	+	3.59941i	0.653860	+	0.514201i	−0.235226	+	0.971941i	$0.575583\pi$
$8$		0			0
$9$	−1.28083	+	8.90839i	−0.142315	+	0.989821i
$10$		0			0
$11$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$11$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$12$	11.9457	+	1.14067i	0.995472	+	0.0950560i
$13$	5.06257	−	7.10938i	0.389428	−	0.546875i	−0.572597	−	0.819837i	$-0.694064\pi$
$13$	5.06257	−	7.10938i	0.389428	−	0.546875i	0.962025	+	0.272962i	$0.0880033\pi$
$14$		0			0
$15$		0			0
$16$	0.761311	−	15.9819i	0.0475819	−	0.998867i
$17$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$17$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$18$		0			0
$19$	−3.07942	+	2.42168i	−0.162075	+	0.127457i	−0.695902	−	0.718136i	$-0.744995\pi$
$19$	−3.07942	+	2.42168i	−0.162075	+	0.127457i	0.533828	+	0.845593i	$0.320753\pi$
$20$		0			0
$21$	0.831178	+	17.4486i	0.0395799	+	0.830885i
$22$		0			0
$23$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$23$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$24$		0			0
$25$	10.3854	+	22.7408i	0.415415	+	0.909632i
$26$		0			0
$27$	−22.7138	+	14.5973i	−0.841254	+	0.540641i
$28$	23.1857	−	2.21396i	0.828060	−	0.0790702i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	12.6675	+	17.7890i	0.408628	+	0.573838i	0.966686	−	0.255964i	$-0.0823929\pi$
$31$	12.6675	+	17.7890i	0.408628	+	0.573838i	−0.558058	+	0.829802i	$0.688454\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	20.8820	+	29.3247i	0.580057	+	0.814576i
$37$	−36.9110	−	63.9317i	−0.997594	−	1.72788i	−0.558838	−	0.829277i	$-0.688753\pi$
$37$	−36.9110	−	63.9317i	−0.997594	−	1.72788i	−0.438756	−	0.898606i	$-0.644581\pi$
$38$		0			0
$39$	26.0646	−	2.48886i	0.668322	−	0.0638170i
$40$		0			0
$41$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$41$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$42$		0			0
$43$	−28.5960	+	8.39653i	−0.665022	+	0.195268i	−0.596785	−	0.802401i	$-0.703556\pi$
$43$	−28.5960	+	8.39653i	−0.665022	+	0.195268i	−0.0682370	+	0.997669i	$0.521737\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$47$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$48$	37.7305	−	29.6716i	0.786053	−	0.618159i
$49$	−3.55882	−	14.6696i	−0.0726289	−	0.299381i
$50$		0			0
$51$		0			0
$52$	−4.96833	−	34.5555i	−0.0955448	−	0.664528i
$53$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$53$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−11.5403	−	2.22421i	−0.202462	−	0.0390213i
$58$		0			0
$59$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$59$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$	−48.2694	−	24.8846i	−0.791301	−	0.407944i	0.0146746	−	0.999892i	$-0.495329\pi$
$61$	−48.2694	−	24.8846i	−0.791301	−	0.407944i	−0.805976	+	0.591948i	$0.798359\pi$
$62$		0			0
$63$	−37.9273	+	36.1636i	−0.602021	+	0.574026i
$64$	−41.9111	−	48.3680i	−0.654861	−	0.755750i
$65$		0			0
$66$		0			0
$67$	−58.0895	−	33.3857i	−0.867008	−	0.498295i
$67$	−58.0895	−	33.3857i	−0.867008	−	0.498295i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$71$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$72$		0			0
$73$	−121.076	−	62.4189i	−1.65857	−	0.855053i	−0.994188	−	0.107661i	$-0.965664\pi$
$73$	−121.076	−	62.4189i	−1.65857	−	0.855053i	−0.664384	−	0.747392i	$-0.731306\pi$
$74$		0			0
$75$	−31.1561	+	68.2224i	−0.415415	+	0.909632i
$76$	−2.23011	+	15.5108i	−0.0293436	+	0.204089i
$77$		0			0
$78$		0			0
$79$	25.8152	+	2.46505i	0.326774	+	0.0312032i	0.257154	−	0.966370i	$-0.417215\pi$
$79$	25.8152	+	2.46505i	0.326774	+	0.0312032i	0.0696203	+	0.997574i	$0.477821\pi$
$80$		0			0
$81$	−77.7189	−	22.8203i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$83$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$84$	50.5698	+	48.2182i	0.602021	+	0.574026i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$89$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$	48.7610	−	14.3175i	0.535835	−	0.157335i
$92$		0			0
$93$	−15.4457	+	63.6682i	−0.166083	+	0.684604i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	91.9873	+	159.327i	0.948322	+	1.64254i	0.748958	+	0.662617i	$0.230554\pi$
$97$	91.9873	+	159.327i	0.948322	+	1.64254i	0.199364	+	0.979925i	$0.436112\pi$
$98$		0			0
$99$		0			0
$100$	92.8368	+	37.1662i	0.928368	+	0.371662i
$101$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$101$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$102$		0			0
$103$	−55.7933	−	78.3507i	−0.541683	−	0.760687i	0.449487	−	0.893287i	$-0.351607\pi$
$103$	−55.7933	−	78.3507i	−0.541683	−	0.760687i	−0.991170	+	0.132600i	$0.957667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$107$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$108$	−25.4620	+	104.956i	−0.235759	+	0.971812i
$109$	−21.4789	−	47.0322i	−0.197054	−	0.431488i	0.785150	−	0.619306i	$-0.212586\pi$
$109$	−21.4789	−	47.0322i	−0.197054	−	0.431488i	−0.982204	+	0.187818i	$0.939859\pi$
$110$		0			0
$111$	72.4344	−	209.285i	0.652562	−	1.88545i
$112$	61.0098	−	70.4091i	0.544731	−	0.628653i
$113$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$113$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	56.8488	+	54.2053i	0.485888	+	0.463293i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	70.1869	−	98.5637i	0.580057	−	0.814576i
$122$		0			0
$123$		0			0
$124$	85.7747	+	16.5317i	0.691731	+	0.133320i
$125$		0			0
$126$		0			0
$127$	98.7140	+	77.6296i	0.777276	+	0.611256i	0.925957	−	0.377629i	$-0.123260\pi$
$127$	98.7140	+	77.6296i	0.777276	+	0.611256i	−0.148681	+	0.988885i	$0.547503\pi$
$128$		0			0
$129$	−75.2161	−	48.3385i	−0.583071	−	0.374717i
$130$		0			0
$131$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$131$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$	−22.8112			−0.171512
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$137$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$138$		0			0
$139$	226.366	+	145.477i	1.62853	+	1.04659i	0.950107	+	0.311925i	$0.100974\pi$
$139$	226.366	+	145.477i	1.62853	+	1.04659i	0.678426	+	0.734669i	$0.262662\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	141.398	+	27.2522i	0.981929	+	0.189251i
$145$		0			0
$146$		0			0
$147$	26.2681	−	36.8884i	0.178695	−	0.250942i
$148$	−283.327	−	83.1922i	−1.91437	−	0.562109i
$149$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$149$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$150$		0			0
$151$	151.811	+	144.752i	1.00537	+	0.958622i	0.999144	−	0.0413588i	$-0.0131687\pi$
$151$	151.811	+	144.752i	1.00537	+	0.958622i	0.00622924	+	0.999981i	$0.498017\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	68.5852	−	79.1515i	0.439649	−	0.507382i
$157$	−99.6955	+	288.051i	−0.635003	+	1.83472i	−0.0906186	+	0.995886i	$0.528884\pi$
$157$	−99.6955	+	288.051i	−0.635003	+	1.83472i	−0.544385	+	0.838836i	$0.683237\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	27.8751	−	48.2810i	0.171013	−	0.296202i	−0.767762	−	0.640736i	$-0.778629\pi$
$163$	27.8751	−	48.2810i	0.171013	−	0.296202i	0.938774	+	0.344533i	$0.111963\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$167$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$168$		0			0
$169$	30.3608	+	87.7217i	0.179650	+	0.519064i
$170$		0			0
$171$	−17.6291	−	30.5344i	−0.103094	−	0.178564i
$172$	−59.6064	+	103.241i	−0.346549	+	0.600240i
$173$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$173$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$174$		0			0
$175$	−34.3194	+	141.466i	−0.196111	+	0.808379i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$179$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$180$		0			0
$181$	66.3065	−	12.7795i	0.366334	−	0.0706051i	−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	66.3065	−	12.7795i	0.366334	−	0.0706051i	0.369096	+	0.929391i	$0.379667\pi$
$182$		0			0
$183$	−38.4096	−	158.327i	−0.209889	−	0.865172i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−156.503	−	14.9443i	−0.828060	−	0.0790702i
$190$		0			0
$191$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$191$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$192$	27.3244	−	190.046i	0.142315	−	0.989821i
$193$	−155.879	+	341.327i	−0.807662	+	1.76853i	−0.190516	+	0.981684i	$0.561016\pi$
$193$	−155.879	+	341.327i	−0.807662	+	1.76853i	−0.617146	+	0.786849i	$0.711711\pi$
$194$		0			0
$195$		0			0
$196$	−50.7954	−	32.6442i	−0.259160	−	0.166552i
$197$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$197$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$198$		0			0
$199$	171.993	−	68.8555i	0.864286	−	0.346008i	0.103201	−	0.994661i	$-0.467091\pi$
$199$	171.993	−	68.8555i	0.864286	−	0.346008i	0.761085	+	0.648653i	$0.224667\pi$
$200$		0			0
$201$	−38.4278	−	197.292i	−0.191183	−	0.981554i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−109.767	−	86.3218i	−0.527726	−	0.415008i
$209$		0			0
$210$		0			0
$211$	−388.292	−	74.8372i	−1.84025	−	0.354679i	−0.854043	−	0.520203i	$-0.825856\pi$
$211$	−388.292	−	74.8372i	−1.84025	−	0.354679i	−0.986206	+	0.165524i	$0.947068\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.05051	+	127.016i	−0.0278825	+	0.585326i
$218$		0			0
$219$	−96.3441	−	397.136i	−0.439927	−	1.81341i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	252.468	−	291.364i	1.13214	−	1.30656i	0.186099	−	0.982531i	$-0.440416\pi$
$223$	252.468	−	291.364i	1.13214	−	1.30656i	0.946046	−	0.324033i	$-0.105039\pi$
$224$		0			0
$225$	−215.886	+	63.3898i	−0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$227$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$228$	−39.5480	+	25.4160i	−0.173456	+	0.111474i
$229$	355.774	−	33.9723i	1.55360	−	0.148351i	0.717355	−	0.696707i	$-0.245352\pi$
$229$	355.774	−	33.9723i	1.55360	−	0.148351i	0.836245	+	0.548357i	$0.184746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$233$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	45.1272	+	63.3722i	0.190410	+	0.267393i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	126.308	−	81.1731i	0.524099	−	0.336818i	−0.251693	−	0.967807i	$-0.580987\pi$
$241$	126.308	−	81.1731i	0.524099	−	0.336818i	0.775792	+	0.630989i	$0.217351\pi$
$242$		0			0
$243$	−100.946	−	221.041i	−0.415415	−	0.909632i
$244$	−208.426	+	61.1994i	−0.854206	+	0.250817i
$245$		0			0
$246$		0			0
$247$	1.62689	+	34.1526i	0.00658660	+	0.138270i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$251$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$252$	−9.97414	+	209.383i	−0.0395799	+	0.830885i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−254.841	−	24.3343i	−0.995472	−	0.0950560i
$257$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$257$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$258$		0			0
$259$	61.1740	−	425.474i	0.236193	−	1.64276i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−260.321	+	63.6958i	−0.971346	+	0.237671i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−29.0878	−	33.5691i	−0.107335	−	0.123871i	0.699540	−	0.714594i	$-0.253388\pi$
$271$	−29.0878	−	33.5691i	−0.107335	−	0.123871i	−0.806875	+	0.590722i	$0.798843\pi$
$272$		0			0
$273$	128.256	+	82.4254i	0.469804	+	0.301925i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−65.6376	+	456.520i	−0.236959	+	1.64809i	0.429882	+	0.902885i	$0.358555\pi$
$277$	−65.6376	+	456.520i	−0.236959	+	1.64809i	−0.666841	+	0.745200i	$0.732354\pi$
$278$		0			0
$279$	−174.696	+	90.0620i	−0.626151	+	0.322803i
$280$		0			0
$281$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$281$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$282$		0			0
$283$	−29.2460	−	203.410i	−0.103343	−	0.718764i	−0.973946	−	0.226779i	$-0.927180\pi$
$283$	−29.2460	−	203.410i	−0.103343	−	0.718764i	0.870604	−	0.491985i	$-0.163729\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.7512	+	288.673i	0.0475819	+	0.998867i
$290$		0			0
$291$	−180.517	+	521.568i	−0.620332	+	1.79233i
$292$	−522.802	+	153.509i	−1.79042	+	0.525714i
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	98.1204	+	283.500i	0.327068	+	0.945001i
$301$	−161.107	−	64.4974i	−0.535238	−	0.214277i
$302$		0			0
$303$		0			0
$304$	36.3586	+	51.0585i	0.119601	+	0.167956i
$305$		0			0
$306$		0			0
$307$	566.594	−	54.1032i	1.84558	−	0.176232i	0.887137	−	0.461507i	$-0.152691\pi$
$307$	566.594	−	54.1032i	1.84558	−	0.176232i	0.958446	+	0.285275i	$0.0920849\pi$
$308$		0			0
$309$	68.0301	−	280.424i	0.220162	−	0.907520i
$310$		0			0
$311$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$311$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$312$		0			0
$313$	381.959	−	440.804i	1.22032	−	1.40832i	0.335720	−	0.941962i	$-0.391020\pi$
$313$	381.959	−	440.804i	1.22032	−	1.40832i	0.884596	−	0.466358i	$-0.154434\pi$
$314$		0			0
$315$		0			0
$316$	81.5376	−	64.1219i	0.258030	−	0.202917i
$317$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$317$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−287.983	+	148.465i	−0.888835	+	0.458227i
$325$	214.250	+	41.2932i	0.659230	+	0.127056i
$326$		0			0
$327$	64.4367	−	141.097i	0.197054	−	0.431488i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	333.172	−	317.679i	1.00656	−	0.959756i	0.00737194	−	0.999973i	$-0.497653\pi$
$331$	333.172	−	317.679i	1.00656	−	0.959756i	0.999191	+	0.0402170i	$0.0128049\pi$
$332$		0			0
$333$	616.805	−	246.932i	1.85227	−	0.741536i
$334$		0			0
$335$		0			0
$336$	279.494			0.831827
$337$	−487.022	+	194.974i	−1.44517	+	0.578558i	−0.955931	−	0.293592i	$-0.905149\pi$
$337$	−487.022	+	194.974i	−1.44517	+	0.578558i	−0.489238	+	0.872150i	$0.662725\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	155.038	−	339.486i	0.452006	−	0.989755i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$347$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$348$		0			0
$349$	−90.9671	−	26.7104i	−0.260651	−	0.0765340i	0.148796	−	0.988868i	$-0.452460\pi$
$349$	−90.9671	−	26.7104i	−0.260651	−	0.0765340i	−0.409447	+	0.912334i	$0.634278\pi$
$350$		0			0
$351$	−11.2126	+	235.381i	−0.0319447	+	0.670602i
$352$		0			0
$353$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$353$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−81.4907	+	335.909i	−0.225736	+	0.930497i
$362$		0			0
$363$	361.356	−	34.5053i	0.995472	−	0.0950560i
$364$	101.639	−	176.044i	0.279228	−	0.483638i
$365$		0			0
$366$		0			0
$367$	170.604	+	492.927i	0.464860	+	1.34313i	0.897284	+	0.441454i	$0.145537\pi$
$367$	170.604	+	492.927i	0.464860	+	1.34313i	−0.432424	+	0.901671i	$0.642341\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	131.030	+	226.950i	0.352231	+	0.610082i
$373$	−275.076	+	476.446i	−0.737470	+	1.27734i	0.216161	+	0.976358i	$0.430646\pi$
$373$	−275.076	+	476.446i	−0.737470	+	1.27734i	−0.953631	+	0.300978i	$0.902687\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	247.782	−	715.919i	0.653778	−	1.88897i	0.295633	−	0.955302i	$-0.404469\pi$
$379$	247.782	−	715.919i	0.653778	−	1.88897i	0.358145	−	0.933666i	$-0.383409\pi$
$380$		0			0
$381$	17.9263	+	376.319i	0.0470506	+	0.987714i
$382$		0			0
$383$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$383$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−38.1729	−	265.499i	−0.0986381	−	0.686043i
$388$	706.089	+	207.326i	1.81982	+	0.534347i
$389$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$389$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	649.533	+	417.430i	1.63610	+	1.05146i	0.944149	+	0.329518i	$0.106886\pi$
$397$	649.533	+	417.430i	1.63610	+	1.05146i	0.691954	+	0.721942i	$0.256750\pi$
$398$		0			0
$399$	−44.8144	−	51.7186i	−0.112317	−	0.129620i
$400$	371.347	−	148.665i	0.928368	−	0.371662i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	190.598			0.472949
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−637.721	−	501.510i	−1.55922	−	1.22618i	−0.858592	−	0.512659i	$-0.828661\pi$
$409$	−637.721	−	501.510i	−1.55922	−	1.22618i	−0.700629	−	0.713526i	$-0.747097\pi$
$410$		0			0
$411$		0			0
$412$	−377.791	−	72.8132i	−0.916968	−	0.176731i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	114.883	+	799.029i	0.275499	+	1.91614i
$418$		0			0
$419$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$419$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$420$		0			0
$421$	−337.076	+	265.080i	−0.800656	+	0.629643i	−0.932313	−	0.361652i	$-0.882213\pi$
$421$	−337.076	+	265.080i	−0.800656	+	0.629643i	0.131657	+	0.991295i	$0.457970\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−131.360	−	287.639i	−0.307635	−	0.673627i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	216.000	+	374.123i	0.500000	+	0.866025i
$433$	−251.578	−	353.292i	−0.581012	−	0.815918i	0.414550	−	0.910027i	$-0.363939\pi$
$433$	−251.578	−	353.292i	−0.581012	−	0.815918i	−0.995562	+	0.0941092i	$0.970000\pi$
$434$		0			0
$435$		0			0
$436$	−192.004	−	76.8667i	−0.440376	−	0.176300i
$437$		0			0
$438$		0			0
$439$	335.219	+	580.616i	0.763597	+	1.32259i	0.940985	+	0.338447i	$0.109902\pi$
$439$	335.219	+	580.616i	0.763597	+	1.32259i	−0.177389	+	0.984141i	$0.556765\pi$
$440$		0			0
$441$	135.241	−	12.9140i	0.306669	−	0.0292834i
$442$		0			0
$443$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$443$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$444$	−368.001	−	805.810i	−0.828831	−	1.81489i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.7318	−	372.236i	−0.0395799	−	0.830885i
$449$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$449$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−29.9425	+	628.571i	−0.0660983	+	1.38757i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−892.750	−	85.2473i	−1.95350	−	0.186537i	−0.958403	−	0.285420i	$-0.907867\pi$
$457$	−892.750	−	85.2473i	−1.95350	−	0.186537i	−0.995099	+	0.0988830i	$0.968473\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	−737.025	−	379.963i	−1.59185	−	0.820654i	−0.999892	−	0.0146823i	$-0.995326\pi$
$463$	−737.025	−	379.963i	−1.59185	−	0.820654i	−0.591954	−	0.805972i	$-0.701643\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$467$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$468$	314.197			0.671362
$469$	−145.708	−	361.895i	−0.310678	−	0.771631i
$470$		0			0
$471$	−848.944	+	339.866i	−1.80243	+	0.721584i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−87.0518	−	44.8783i	−0.183267	−	0.0944807i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$479$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$480$		0			0
$481$	−641.379	−	61.2442i	−1.33343	−	0.127327i
$482$		0			0
$483$		0			0
$484$	−68.8804	−	479.074i	−0.142315	−	0.989821i
$485$		0			0
$486$		0			0
$487$	139.345	+	574.389i	0.286130	+	1.17944i	0.915493	+	0.402333i	$0.131801\pi$
$487$	139.345	+	574.389i	0.286130	+	1.17944i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	164.228	−	31.6523i	0.335844	−	0.0647287i
$490$		0			0
$491$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$491$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	293.945	−	188.907i	0.592631	−	0.380861i
$497$		0			0
$498$		0			0
$499$	−459.153	−	795.276i	−0.920146	−	1.59374i	−0.799188	−	0.601081i	$-0.794737\pi$
$499$	−459.153	−	795.276i	−0.920146	−	1.59374i	−0.120958	−	0.992658i	$-0.538597\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$503$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−139.241	+	241.172i	−0.274637	+	0.475684i
$508$	500.053	−	47.7493i	0.984356	−	0.0939946i
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$	−329.495	−	721.493i	−0.644804	−	1.41192i
$512$		0			0
$513$	34.5954	−	99.9568i	0.0674374	−	0.194848i
$514$		0			0
$515$		0			0
$516$	−351.175	+	67.6835i	−0.680572	+	0.131170i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$521$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$522$		0			0
$523$	287.913	−	404.317i	0.550503	−	0.773073i	−0.441773	−	0.897127i	$-0.645650\pi$
$523$	287.913	−	404.317i	0.550503	−	0.773073i	0.992275	+	0.124054i	$0.0395897\pi$
$524$		0			0
$525$	−388.162	+	200.112i	−0.739357	+	0.381165i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−415.822	−	327.006i	−0.786053	−	0.618159i
$530$		0			0
$531$		0			0
$532$	−66.0369	+	62.9660i	−0.124129	+	0.118357i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	814.558	+	523.485i	1.50565	+	0.967624i	0.994111	+	0.108366i	$0.0345619\pi$
$541$	814.558	+	523.485i	1.50565	+	0.967624i	0.511542	+	0.859258i	$0.329074\pi$
$542$		0			0
$543$	159.239	+	125.227i	0.293258	+	0.230620i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	269.434	−	138.903i	0.492567	−	0.253936i	−0.194001	−	0.981001i	$-0.562147\pi$
$547$	269.434	−	138.903i	0.492567	−	0.253936i	0.686568	+	0.727065i	$0.259116\pi$
$548$		0			0
$549$	283.507	−	398.130i	0.516406	−	0.725191i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	109.284	+	104.202i	0.197620	+	0.188430i
$554$		0			0
$555$		0			0
$556$	1056.88	−	203.696i	1.90086	−	0.366360i
$557$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$557$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$558$		0			0
$559$	−85.0748	+	245.807i	−0.152191	+	0.439727i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$563$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−273.581	−	384.191i	−0.482507	−	0.677586i
$568$		0			0
$569$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$569$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$570$		0			0
$571$	329.145	+	951.002i	0.576436	+	1.66550i	0.734930	+	0.678143i	$0.237215\pi$
$571$	329.145	+	951.002i	0.576436	+	1.66550i	−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	484.562	−	311.409i	0.841254	−	0.540641i
$577$	−139.310	+	574.242i	−0.241438	+	0.995221i	0.714027	+	0.700118i	$0.246869\pi$
$577$	−139.310	+	574.242i	−0.241438	+	0.995221i	−0.955465	+	0.295103i	$0.904646\pi$
$578$		0			0
$579$	−1080.11	+	317.149i	−1.86547	+	0.547753i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$587$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$588$	−25.7792	−	179.298i	−0.0438421	−	0.304929i
$589$	−82.0876	−	24.1031i	−0.139368	−	0.0409220i
$590$		0			0
$591$		0			0
$592$	−1049.85	+	541.235i	−1.77339	+	0.914248i
$593$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$593$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	494.007	+	254.678i	0.827482	+	0.426597i
$598$		0			0
$599$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$599$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$600$		0			0
$601$	−25.7377	+	10.3038i	−0.0428249	+	0.0171445i	−0.392976	−	0.919549i	$-0.628554\pi$
$601$	−25.7377	+	10.3038i	−0.0428249	+	0.0171445i	0.350151	+	0.936693i	$0.386130\pi$
$602$		0			0
$603$	371.816	−	474.723i	0.616611	−	0.787268i
$604$	839.045			1.38915
$605$		0			0
$606$		0			0
$607$	−789.185	+	752.487i	−1.30014	+	1.23968i	−0.346744	+	0.937960i	$0.612713\pi$
$607$	−789.185	+	752.487i	−1.30014	+	1.23968i	−0.953396	+	0.301722i	$0.902439\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−723.454	−	139.434i	−1.18019	−	0.227462i	−0.438827	−	0.898571i	$-0.644606\pi$
$613$	−723.454	−	139.434i	−1.18019	−	0.227462i	−0.741358	+	0.671109i	$0.765818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$617$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$	35.6367	−	748.107i	0.0575715	−	1.20857i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	35.6367	−	748.107i	0.0575715	−	1.20857i	0.824130	−	0.566400i	$-0.191664\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−19.9335	−	418.455i	−0.0319447	−	0.670602i
$625$	−409.288	+	472.343i	−0.654861	+	0.755750i
$626$		0			0
$627$		0			0
$628$	506.500	+	1109.08i	0.806529	+	1.76605i
$629$		0			0
$630$		0			0
$631$	1242.07	−	118.603i	1.96841	−	0.187961i	0.969946	−	0.243320i	$-0.0782365\pi$
$631$	1242.07	−	118.603i	1.96841	−	0.187961i	0.998466	+	0.0553595i	$0.0176305\pi$
$632$		0			0
$633$	−593.158	−	1027.38i	−0.937058	−	1.62303i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−122.309	−	48.9651i	−0.192008	−	0.0768682i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−549.021	+	352.835i	−0.853843	+	0.548732i	−0.892771	−	0.450510i	$-0.851242\pi$
$643$	−549.021	+	352.835i	−0.853843	+	0.548732i	0.0389279	+	0.999242i	$0.487606\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$647$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−299.863	+	235.815i	−0.460619	+	0.362235i
$652$	−52.5743	−	216.714i	−0.0806355	−	0.332384i
$653$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$653$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	711.130	−	998.642i	1.08239	−	1.52000i
$658$		0			0
$659$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$659$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$660$		0			0
$661$	116.712	−	811.749i	0.176569	−	1.22806i	−0.688062	−	0.725652i	$-0.741538\pi$
$661$	116.712	−	811.749i	0.176569	−	1.22806i	0.864630	−	0.502409i	$-0.167553\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1156.59			1.72883
$670$		0			0
$671$		0			0
$672$		0			0
$673$	879.599	+	1015.11i	1.30698	+	1.50834i	0.702345	+	0.711837i	$0.252136\pi$
$673$	879.599	+	1015.11i	1.30698	+	1.50834i	0.604637	+	0.796501i	$0.293318\pi$
$674$		0			0
$675$	−567.846	−	364.933i	−0.841254	−	0.540641i
$676$	330.032	+	170.143i	0.488213	+	0.251691i
$677$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$677$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$678$		0			0
$679$	−152.454	+	1060.34i	−0.224527	+	1.56162i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$683$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$684$	−135.320	−	39.7334i	−0.197836	−	0.0580898i
$685$		0			0
$686$		0			0
$687$	775.972	+	739.888i	1.12951	+	1.07698i
$688$	112.422	+	463.409i	0.163404	+	0.673560i
$689$		0			0
$690$		0			0
$691$	−8.05990	−	169.198i	−0.0116641	−	0.244860i	−0.997168	−	0.0752068i	$-0.976038\pi$
$691$	−8.05990	−	169.198i	−0.0116641	−	0.244860i	0.985504	−	0.169653i	$-0.0542647\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	291.139	+	504.268i	0.415913	+	0.720383i
$701$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$701$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$702$		0			0
$703$	268.486	+	107.486i	0.381915	+	0.152896i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	193.820	−	18.5076i	0.273371	−	0.0261038i	0.0425300	−	0.999095i	$-0.486458\pi$
$709$	193.820	−	18.5076i	0.273371	−	0.0261038i	0.230841	+	0.972991i	$0.425852\pi$
$710$		0			0
$711$	−55.0246	+	226.814i	−0.0773904	+	0.319008i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$719$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$720$		0			0
$721$	26.6492	−	559.436i	0.0369615	−	0.775916i
$722$		0			0
$723$	432.182	+	126.900i	0.597762	+	0.175519i
$724$	156.677	−	220.023i	0.216405	−	0.303899i
$725$		0			0
$726$		0			0
$727$	1420.61	+	273.800i	1.95407	+	0.376616i	0.995907	+	0.0903832i	$0.0288092\pi$
$727$	1420.61	+	273.800i	1.95407	+	0.376616i	0.958160	+	0.286232i	$0.0924029\pi$
$728$		0			0
$729$	302.838	−	663.122i	0.415415	−	0.909632i
$730$		0			0
$731$		0			0
$732$	−548.225	−	352.323i	−0.748941	−	0.481315i
$733$	−850.971	+	811.399i	−1.16094	+	1.10696i	−0.168317	+	0.985733i	$0.553833\pi$
$733$	−850.971	+	811.399i	−1.16094	+	1.10696i	−0.992625	+	0.121224i	$0.961318\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1017.53	+	407.359i	−1.37691	+	0.551230i	−0.937791	−	0.347200i	$-0.887133\pi$
$739$	−1017.53	+	407.359i	−1.37691	+	0.551230i	−0.439116	+	0.898430i	$0.644708\pi$
$740$		0			0
$741$	−74.2364	+	70.7843i	−0.100184	+	0.0955253i
$742$		0			0
$743$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$743$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1350.06	+	396.414i	1.79769	+	0.527848i	0.997419	−	0.0717966i	$-0.0228733\pi$
$751$	1350.06	+	396.414i	1.79769	+	0.527848i	0.800266	+	0.599645i	$0.204691\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−494.318	+	388.736i	−0.653860	+	0.514201i
$757$	1298.74	−	250.311i	1.71563	−	0.330661i	0.765721	−	0.643173i	$-0.222383\pi$
$757$	1298.74	−	250.311i	1.71563	−	0.330661i	0.949914	+	0.312512i	$0.101170\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$761$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$	70.9788	−	292.579i	0.0930260	−	0.383458i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−445.484	−	625.594i	−0.580057	−	0.814576i
$769$	−496.004	−	1433.11i	−0.644999	−	1.86360i	−0.479896	−	0.877325i	$-0.659326\pi$
$769$	−496.004	−	1433.11i	−0.644999	−	1.86360i	−0.165103	−	0.986276i	$-0.552796\pi$
$770$		0			0
$771$		0			0
$772$	490.911	+	1418.39i	0.635895	+	1.83730i
$773$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$773$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$774$		0			0
$775$	−272.979	+	472.813i	−0.352231	+	0.610082i
$776$		0			0
$777$	1084.84	−	697.182i	1.39619	−	0.897275i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−237.158	+	45.7084i	−0.302497	+	0.0583016i
$785$		0			0
$786$		0			0
$787$	341.906	+	326.006i	0.434442	+	0.414239i	0.875484	−	0.483247i	$-0.160543\pi$
$787$	341.906	+	326.006i	0.434442	+	0.414239i	−0.441042	+	0.897486i	$0.645391\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−421.281	+	217.185i	−0.531250	+	0.273878i
$794$		0			0
$795$		0			0
$796$	307.845	−	674.087i	0.386740	−	0.846843i
$797$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$797$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−655.836	−	465.076i	−0.815716	−	0.578453i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$809$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$	−1062.95	−	835.914i	−1.31067	−	1.03072i	−0.996691	−	0.0812899i	$-0.974096\pi$
$811$	−1062.95	−	835.914i	−1.31067	−	1.03072i	−0.313977	−	0.949431i	$-0.601662\pi$
$812$		0			0
$813$	18.9642	−	131.899i	0.0233262	−	0.162237i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	67.7251	−	95.1066i	0.0828949	−	0.116410i
$818$		0			0
$819$	65.0914	+	452.721i	0.0794767	+	0.552773i
$820$		0			0
$821$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$821$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$822$		0			0
$823$	−403.695	+	317.469i	−0.490516	+	0.385746i	−0.832557	−	0.553939i	$-0.813124\pi$
$823$	−403.695	+	317.469i	−0.490516	+	0.385746i	0.342041	+	0.939685i	$0.388882\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$827$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$828$		0			0
$829$	−319.106	−	698.744i	−0.384929	−	0.842876i	−0.998579	−	0.0532980i	$-0.983027\pi$
$829$	−319.106	−	698.744i	−0.384929	−	0.842876i	0.613650	−	0.789578i	$-0.289701\pi$
$830$		0			0
$831$	−1163.99	+	748.054i	−1.40071	+	0.900185i
$832$	−556.044	+	53.0958i	−0.668322	+	0.0638170i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−547.398	−	219.145i	−0.654000	−	0.261822i
$838$		0			0
$839$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$839$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	−1330.66	+	855.161i	−1.57661	+	1.01322i
$845$		0			0
$846$		0			0
$847$	676.018	−	198.497i	0.798132	−	0.234353i
$848$		0			0
$849$	403.726	−	465.924i	0.475531	−	0.548792i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−380.113	−	1566.85i	−0.445619	−	1.83687i	−0.540454	−	0.841374i	$-0.681747\pi$
$853$	−380.113	−	1566.85i	−0.445619	−	1.83687i	0.0948345	−	0.995493i	$-0.469768\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$857$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$858$		0			0
$859$	1496.97	+	142.943i	1.74269	+	0.166407i	0.917310	−	0.398173i	$-0.130356\pi$
$859$	1496.97	+	142.943i	1.74269	+	0.166407i	0.825378	+	0.564580i	$0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−627.477	+	598.299i	−0.723734	+	0.690079i
$868$	333.088	+	384.404i	0.383742	+	0.442862i
$869$		0			0
$870$		0			0
$871$	−531.434	+	243.963i	−0.610142	+	0.280095i
$872$		0			0
$873$	−1537.16	+	615.388i	−1.76078	+	0.704912i
$874$		0			0
$875$		0			0
$876$	−1375.13	−	883.742i	−1.56978	−	1.00884i
$877$	−1323.79	−	682.460i	−1.50945	−	0.778175i	−0.512686	−	0.858576i	$-0.671350\pi$
$877$	−1323.79	−	682.460i	−1.50945	−	0.778175i	−0.996763	+	0.0804008i	$0.974380\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$881$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$882$		0			0
$883$	−1463.96	−	139.791i	−1.65794	−	0.158314i	−0.776341	−	0.630313i	$-0.782926\pi$
$883$	−1463.96	−	139.791i	−1.65794	−	0.158314i	−0.881599	+	0.471999i	$0.843533\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$887$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$888$		0			0
$889$	172.396	+	710.624i	0.193921	+	0.799352i
$890$		0			0
$891$		0			0
$892$	−73.3769	−	1540.37i	−0.0822612	−	1.72687i
$893$		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
$903$	−170.276	−	491.980i	−0.188567	−	0.544828i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	460.476	+	646.648i	0.507691	+	0.712952i	0.986279	−	0.165087i	$-0.0527906\pi$
$907$	460.476	+	646.648i	0.507691	+	0.712952i	−0.478588	+	0.878040i	$0.658851\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$911$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$912$	−44.3329	+	182.743i	−0.0486106	+	0.200376i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	936.170	−	1080.40i	1.02202	−	1.17947i
$917$		0			0
$918$		0			0
$919$	−1261.62	+	992.153i	−1.37282	+	1.07960i	−0.385187	+	0.922839i	$0.625863\pi$
$919$	−1261.62	+	992.153i	−1.37282	+	1.07960i	−0.987636	+	0.156762i	$0.949894\pi$
$920$		0			0
$921$	1235.79	+	1178.32i	1.34179	+	1.27939i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1070.52	−	1503.34i	1.15732	−	1.62523i
$926$		0			0
$927$	769.441	−	396.674i	0.830034	−	0.427912i
$928$		0			0
$929$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$929$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$930$		0			0
$931$	46.4843	+	36.5556i	0.0499294	+	0.0392649i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1682.88			−1.79603			−0.898017	−	0.439961i	$-0.854992\pi$
$937$	−1682.88			−1.79603			−0.898017	+	0.439961i	$0.854992\pi$
$938$		0			0
$939$	1749.80			1.86347
$940$		0			0
$941$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$941$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$	305.568	+	58.8933i	0.322329	+	0.0621238i
$949$	−1056.71	+	544.773i	−1.11350	+	0.574050i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$953$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	158.330	−	457.464i	0.164755	−	0.476029i
$962$		0			0
$963$		0			0
$964$	141.590	−	583.640i	0.146877	−	0.605436i
$965$		0			0
$966$		0			0
$967$	−807.370	+	1398.41i	−0.834922	+	1.44613i	0.0591720	+	0.998248i	$0.481154\pi$
$967$	−807.370	+	1398.41i	−0.834922	+	1.44613i	−0.894094	+	0.447879i	$0.852179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$971$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$972$	−902.374	−	361.256i	−0.928368	−	0.371662i
$973$	512.452	+	1480.63i	0.526672	+	1.52172i
$974$		0			0
$975$	327.289	+	566.881i	0.335681	+	0.581417i
$976$	−434.451	+	752.491i	−0.445134	+	0.770994i
$977$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$977$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	446.492	−	131.102i	0.455140	−	0.133641i
$982$		0			0
$983$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$983$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	98.9819	+	94.3790i	0.100184	+	0.0955253i
$989$		0			0
$990$		0			0
$991$	1065.00	+	312.712i	1.07467	+	0.315552i	0.770745	−	0.637144i	$-0.219884\pi$
$991$	1065.00	+	312.712i	1.07467	+	0.315552i	0.303925	+	0.952696i	$0.401703\pi$
$992$		0			0
$993$	1374.80	+	131.278i	1.38449	+	0.132203i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	711.002	−	1556.88i	0.713141	−	1.56156i	−0.110133	−	0.993917i	$-0.535128\pi$
$997$	711.002	−	1556.88i	0.713141	−	1.56156i	0.823274	−	0.567644i	$-0.192145\pi$
$998$		0			0
$999$	1771.62	+	913.334i	1.77339	+	0.914248i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.173.1	yes	20
3.2	odd	2	CM	201.3.o.a.173.1	yes	20
67.55	even	33	inner	201.3.o.a.122.1	✓	20
201.122	odd	66	inner	201.3.o.a.122.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.122.1	✓	20	67.55	even	33	inner
201.3.o.a.122.1	✓	20	201.122	odd	66	inner
201.3.o.a.173.1	yes	20	1.1	even	1	trivial
201.3.o.a.173.1	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)

\(f(q)\)	\(=\)	\(q+(1.96458 + 2.26725i) q^{3} +(2.89494 - 2.76032i) q^{4} +(4.57702 + 3.59941i) q^{7} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})\)	\(q+(1.96458 + 2.26725i) q^{3} +(2.89494 - 2.76032i) q^{4} +(4.57702 + 3.59941i) q^{7} +(-1.28083 + 8.90839i) q^{9} +(11.9457 + 1.14067i) q^{12} +(5.06257 - 7.10938i) q^{13} +(0.761311 - 15.9819i) q^{16} +(-3.07942 + 2.42168i) q^{19} +(0.831178 + 17.4486i) q^{21} +(10.3854 + 22.7408i) q^{25} +(-22.7138 + 14.5973i) q^{27} +(23.1857 - 2.21396i) q^{28} +(12.6675 + 17.7890i) q^{31} +(20.8820 + 29.3247i) q^{36} +(-36.9110 - 63.9317i) q^{37} +(26.0646 - 2.48886i) q^{39} +(-28.5960 + 8.39653i) q^{43} +(37.7305 - 29.6716i) q^{48} +(-3.55882 - 14.6696i) q^{49} +(-4.96833 - 34.5555i) q^{52} +(-11.5403 - 2.22421i) q^{57} +(-48.2694 - 24.8846i) q^{61} +(-37.9273 + 36.1636i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-58.0895 - 33.3857i) q^{67} +(-121.076 - 62.4189i) q^{73} +(-31.1561 + 68.2224i) q^{75} +(-2.23011 + 15.5108i) q^{76} +(25.8152 + 2.46505i) q^{79} +(-77.7189 - 22.8203i) q^{81} +(50.5698 + 48.2182i) q^{84} +(48.7610 - 14.3175i) q^{91} +(-15.4457 + 63.6682i) q^{93} +(91.9873 + 159.327i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 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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