LMFDB - Embedded newform 201.3.o.a.17.1

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			17.1
Root			$0.723734 + 0.690079i$ of defining polynomial
Character	$\chi$	$=$	201.17
Dual form			201.3.o.a.71.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.52376 + 1.62192i) q^{3} +(3.92771 - 0.757005i) q^{4} +(-6.83405 - 9.59709i) q^{7} +(3.73874 - 8.18669i) q^{9} +O(q^{10})$	$q+(-2.52376 + 1.62192i) q^{3} +(3.92771 - 0.757005i) q^{4} +(-6.83405 - 9.59709i) q^{7} +(3.73874 - 8.18669i) q^{9} +(-8.68481 + 8.28095i) q^{12} +(-0.347546 - 1.43260i) q^{13} +(14.8539 - 5.94660i) q^{16} +(21.3782 - 30.0214i) q^{19} +(32.8132 + 13.1364i) q^{21} +(-23.9873 - 7.04331i) q^{25} +(3.84250 + 26.7252i) q^{27} +(-34.1073 - 32.5212i) q^{28} +(14.5828 - 60.1112i) q^{31} +(8.48732 - 34.9852i) q^{36} +(22.2329 + 38.5085i) q^{37} +(3.20069 + 3.05185i) q^{39} +(-27.1439 + 31.3258i) q^{43} +(-27.8427 + 39.0996i) q^{48} +(-29.3734 + 84.8689i) q^{49} +(-2.44955 - 5.36376i) q^{52} +(-5.26093 + 110.441i) q^{57} +(-78.9245 + 62.0669i) q^{61} +(-104.119 + 20.0673i) q^{63} +(53.8402 - 34.6010i) q^{64} +(63.2718 - 22.0381i) q^{67} +(-6.91266 + 5.43617i) q^{73} +(71.9620 - 21.1299i) q^{75} +(61.2409 - 134.099i) q^{76} +(-72.9573 + 69.5647i) q^{79} +(-53.0437 - 61.2157i) q^{81} +(138.825 + 26.7564i) q^{84} +(-11.3737 + 13.1259i) q^{91} +(60.6922 + 175.359i) q^{93} +(-43.5568 - 75.4425i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{32}{33}\right)$

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		0.995472	−	0.0950560i	$-0.0303030\pi$
$2$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$3$	−2.52376	+	1.62192i	−0.841254	+	0.540641i
$3$	−2.52376	+	1.62192i	−0.841254	+	0.540641i
$4$	3.92771	−	0.757005i	0.981929	−	0.189251i
$5$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$5$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$6$		0			0
$7$	−6.83405	−	9.59709i	−0.976293	−	1.37101i	−0.928164	−	0.372171i	$-0.878613\pi$
$7$	−6.83405	−	9.59709i	−0.976293	−	1.37101i	−0.0481292	−	0.998841i	$-0.515326\pi$
$8$		0			0
$9$	3.73874	−	8.18669i	0.415415	−	0.909632i
$10$		0			0
$11$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$11$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$12$	−8.68481	+	8.28095i	−0.723734	+	0.690079i
$13$	−0.347546	−	1.43260i	−0.0267343	−	0.110200i	0.956881	−	0.290480i	$-0.0938149\pi$
$13$	−0.347546	−	1.43260i	−0.0267343	−	0.110200i	−0.983615	+	0.180280i	$0.942300\pi$
$14$		0			0
$15$		0			0
$16$	14.8539	−	5.94660i	0.928368	−	0.371662i
$17$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$17$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$18$		0			0
$19$	21.3782	−	30.0214i	1.12517	−	1.58007i	0.364151	−	0.931340i	$-0.381359\pi$
$19$	21.3782	−	30.0214i	1.12517	−	1.58007i	0.761015	−	0.648735i	$-0.224702\pi$
$20$		0			0
$21$	32.8132	+	13.1364i	1.56254	+	0.625545i
$22$		0			0
$23$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$23$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$24$		0			0
$25$	−23.9873	−	7.04331i	−0.959493	−	0.281733i
$26$		0			0
$27$	3.84250	+	26.7252i	0.142315	+	0.989821i
$28$	−34.1073	−	32.5212i	−1.21812	−	1.16147i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	14.5828	−	60.1112i	0.470414	−	1.93907i	0.168743	−	0.985660i	$-0.446029\pi$
$31$	14.5828	−	60.1112i	0.470414	−	1.93907i	0.301671	−	0.953412i	$-0.402456\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	8.48732	−	34.9852i	0.235759	−	0.971812i
$37$	22.2329	+	38.5085i	0.600889	+	1.04077i	0.992687	+	0.120719i	$0.0385202\pi$
$37$	22.2329	+	38.5085i	0.600889	+	1.04077i	−0.391797	+	0.920052i	$0.628147\pi$
$38$		0			0
$39$	3.20069	+	3.05185i	0.0820690	+	0.0782527i
$40$		0			0
$41$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$41$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$42$		0			0
$43$	−27.1439	+	31.3258i	−0.631254	+	0.728506i	−0.977803	−	0.209526i	$-0.932808\pi$
$43$	−27.1439	+	31.3258i	−0.631254	+	0.728506i	0.346549	+	0.938032i	$0.387354\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$47$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$48$	−27.8427	+	39.0996i	−0.580057	+	0.814576i
$49$	−29.3734	+	84.8689i	−0.599458	+	1.73202i
$50$		0			0
$51$		0			0
$52$	−2.44955	−	5.36376i	−0.0471067	−	0.103149i
$53$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$53$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.26093	+	110.441i	−0.0922971	+	1.93755i
$58$		0			0
$59$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$59$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$60$		0			0
$61$	−78.9245	+	62.0669i	−1.29384	+	1.01749i	−0.295782	+	0.955255i	$0.595580\pi$
$61$	−78.9245	+	62.0669i	−1.29384	+	1.01749i	−0.998062	+	0.0622348i	$0.980177\pi$
$62$		0			0
$63$	−104.119	+	20.0673i	−1.65268	+	0.318529i
$64$	53.8402	−	34.6010i	0.841254	−	0.540641i
$65$		0			0
$66$		0			0
$67$	63.2718	−	22.0381i	0.944355	−	0.328927i
$67$	63.2718	−	22.0381i	0.944355	−	0.328927i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$71$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$72$		0			0
$73$	−6.91266	+	5.43617i	−0.0946939	+	0.0744681i	−0.664384	−	0.747392i	$-0.731306\pi$
$73$	−6.91266	+	5.43617i	−0.0946939	+	0.0744681i	0.569690	+	0.821860i	$0.307063\pi$
$74$		0			0
$75$	71.9620	−	21.1299i	0.959493	−	0.281733i
$76$	61.2409	−	134.099i	0.805802	−	1.76446i
$77$		0			0
$78$		0			0
$79$	−72.9573	+	69.5647i	−0.923511	+	0.880566i	−0.993131	−	0.117008i	$-0.962670\pi$
$79$	−72.9573	+	69.5647i	−0.923511	+	0.880566i	0.0696203	+	0.997574i	$0.477821\pi$
$80$		0			0
$81$	−53.0437	−	61.2157i	−0.654861	−	0.755750i
$82$		0			0
$83$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$83$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$84$	138.825	+	26.7564i	1.65268	+	0.318529i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$89$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$	−11.3737	+	13.1259i	−0.124985	+	0.144241i
$92$		0			0
$93$	60.6922	+	175.359i	0.652604	+	1.88558i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−43.5568	−	75.4425i	−0.449039	−	0.777758i	0.549285	−	0.835635i	$-0.314900\pi$
$97$	−43.5568	−	75.4425i	−0.449039	−	0.777758i	−0.998324	+	0.0578770i	$0.981567\pi$
$98$		0			0
$99$		0			0
$100$	−99.5472	−	9.50560i	−0.995472	−	0.0950560i
$101$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$101$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$102$		0			0
$103$	33.8577	−	139.563i	0.328716	−	1.35499i	−0.532364	−	0.846516i	$-0.678696\pi$
$103$	33.8577	−	139.563i	0.328716	−	1.35499i	0.861080	−	0.508470i	$-0.169789\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$107$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$108$	35.3233	+	102.060i	0.327068	+	0.945001i
$109$	205.447	+	60.3247i	1.88484	+	0.553438i	0.995332	+	0.0965092i	$0.0307677\pi$
$109$	205.447	+	60.3247i	1.88484	+	0.553438i	0.889503	+	0.456928i	$0.151050\pi$
$110$		0			0
$111$	−118.568	−	61.1263i	−1.06818	−	0.550687i
$112$	−158.582	−	101.915i	−1.41591	−	0.909952i
$113$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$113$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.0276	−	2.51087i	−0.111347	−	0.0214605i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	28.5268	+	117.589i	0.235759	+	0.971812i
$122$		0			0
$123$		0			0
$124$	11.7727	−	247.139i	0.0949410	−	1.99306i
$125$		0			0
$126$		0			0
$127$	144.490	+	202.907i	1.13771	+	1.59769i	0.728160	+	0.685408i	$0.240376\pi$
$127$	144.490	+	202.907i	1.13771	+	1.59769i	0.409553	+	0.912286i	$0.365685\pi$
$128$		0			0
$129$	17.6968	−	123.084i	0.137185	−	0.954141i
$130$		0			0
$131$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$131$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$132$		0			0
$133$	−434.218			−3.26479
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$137$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$138$		0			0
$139$	15.2893	−	106.340i	0.109995	−	0.765033i	−0.857925	−	0.513775i	$-0.828247\pi$
$139$	15.2893	−	106.340i	0.109995	−	0.765033i	0.967920	−	0.251258i	$-0.0808442\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.85180	−	143.837i	0.0475819	−	0.998867i
$145$		0			0
$146$		0			0
$147$	−63.5193	−	261.830i	−0.432104	−	1.78116i
$148$	116.476	+	134.420i	0.786998	+	0.908244i
$149$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$149$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$150$		0			0
$151$	279.623	+	53.8930i	1.85181	+	0.356907i	0.988916	−	0.148478i	$-0.0474374\pi$
$151$	279.623	+	53.8930i	1.85181	+	0.356907i	0.862894	+	0.505385i	$0.168650\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	14.8817	+	9.56387i	0.0953953	+	0.0613069i
$157$	278.835	+	143.749i	1.77602	+	0.915600i	0.907772	+	0.419465i	$0.137782\pi$
$157$	278.835	+	143.749i	1.77602	+	0.915600i	0.868245	+	0.496135i	$0.165248\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	110.276	−	191.004i	0.676541	−	1.17180i	−0.299474	−	0.954104i	$-0.596811\pi$
$163$	110.276	−	191.004i	0.676541	−	1.17180i	0.976016	−	0.217700i	$-0.0698554\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$167$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$168$		0			0
$169$	148.282	−	76.4445i	0.877406	−	0.452334i
$170$		0			0
$171$	−165.849	−	287.258i	−0.969876	−	1.67987i
$172$	−82.8999	+	143.587i	−0.481976	+	0.834807i
$173$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$173$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$174$		0			0
$175$	96.3354	+	278.343i	0.550488	+	1.59053i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$179$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$180$		0			0
$181$	−17.2073	−	361.227i	−0.0950682	−	1.99573i	−0.0923058	−	0.995731i	$-0.529424\pi$
$181$	−17.2073	−	361.227i	−0.0950682	−	1.99573i	−0.00276243	−	0.999996i	$-0.500879\pi$
$182$		0			0
$183$	98.5188	−	284.651i	0.538354	−	1.55547i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	230.224	−	219.518i	1.21812	−	1.16147i
$190$		0			0
$191$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$191$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$192$	−79.7597	+	174.649i	−0.415415	+	0.909632i
$193$	−369.922	+	108.619i	−1.91669	+	0.562792i	−0.944579	+	0.328285i	$0.893529\pi$
$193$	−369.922	+	108.619i	−1.91669	+	0.562792i	−0.972114	+	0.234507i	$0.924652\pi$
$194$		0			0
$195$		0			0
$196$	−51.1242	+	355.577i	−0.260838	+	1.81417i
$197$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$197$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$198$		0			0
$199$	34.4314	−	3.28780i	0.173022	−	0.0165216i	−0.00818534	−	0.999966i	$-0.502606\pi$
$199$	34.4314	−	3.28780i	0.173022	−	0.0165216i	0.181207	+	0.983445i	$0.441999\pi$
$200$		0			0
$201$	−123.939	+	158.241i	−0.616611	+	0.787268i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−13.6815	−	19.2130i	−0.0657765	−	0.0923702i
$209$		0			0
$210$		0			0
$211$	−14.1940	+	297.969i	−0.0672701	+	1.41217i	0.672890	+	0.739743i	$0.265053\pi$
$211$	−14.1940	+	297.969i	−0.0672701	+	1.41217i	−0.740160	+	0.672431i	$0.765250\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−676.553	+	270.851i	−3.11775	+	1.24816i
$218$		0			0
$219$	8.62884	−	24.9314i	0.0394011	−	0.113842i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−140.565	−	90.3354i	−0.630335	−	0.405092i	0.186099	−	0.982531i	$-0.440416\pi$
$223$	−140.565	−	90.3354i	−0.630335	−	0.405092i	−0.816434	+	0.577439i	$0.804052\pi$
$224$		0			0
$225$	−147.344	+	170.044i	−0.654861	+	0.755750i
$226$		0			0
$227$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$227$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$228$	62.9406	+	437.762i	0.276055	+	1.92001i
$229$	326.043	+	310.882i	1.42377	+	1.35756i	0.836245	+	0.548357i	$0.184746\pi$
$229$	326.043	+	310.882i	1.42377	+	1.35756i	0.587526	+	0.809205i	$0.300102\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$233$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	71.2983	−	293.896i	0.300837	−	1.24007i
$238$		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
$241$	63.2547	+	439.946i	0.262467	+	1.82550i	0.514160	+	0.857694i	$0.328104\pi$
$241$	63.2547	+	439.946i	0.262467	+	1.82550i	−0.251693	+	0.967807i	$0.580987\pi$
$242$		0			0
$243$	233.157	+	68.4610i	0.959493	+	0.281733i
$244$	−263.008	+	303.527i	−1.07790	+	1.24396i
$245$		0			0
$246$		0			0
$247$	−50.4386	−	20.1926i	−0.204205	−	0.0817513i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$251$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$252$	−393.759	+	157.637i	−1.56254	+	0.625545i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	185.276	−	176.660i	0.723734	−	0.690079i
$257$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$257$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$258$		0			0
$259$	217.629	−	476.540i	0.840265	−	1.83992i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$263$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	231.831	−	134.456i	0.865040	−	0.501703i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−163.880	+	105.319i	−0.604723	+	0.388632i	−0.806875	−	0.590722i	$-0.798843\pi$
$271$	−163.880	+	105.319i	−0.604723	+	0.388632i	0.202152	+	0.979354i	$0.435207\pi$
$272$		0			0
$273$	7.41520	−	51.5738i	0.0271619	−	0.188915i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	92.2498	−	201.999i	0.333032	−	0.729238i	−0.666841	−	0.745200i	$-0.732354\pi$
$277$	92.2498	−	201.999i	0.333032	−	0.729238i	0.999873	+	0.0159622i	$0.00508116\pi$
$278$		0			0
$279$	−437.591	−	344.125i	−1.56843	−	1.23342i
$280$		0			0
$281$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$281$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$282$		0			0
$283$	−163.819	−	358.712i	−0.578864	−	1.26754i	−0.941942	−	0.335776i	$-0.891001\pi$
$283$	−163.819	−	358.712i	−0.578864	−	1.26754i	0.363078	−	0.931759i	$-0.381726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	268.298	+	107.410i	0.928368	+	0.371662i
$290$		0			0
$291$	232.289	+	119.753i	0.798243	+	0.411523i
$292$	−23.0357	+	26.5847i	−0.0788895	+	0.0910433i
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	266.651	−	137.468i	0.888835	−	0.458227i
$301$	486.139	+	46.4207i	1.61508	+	0.154222i
$302$		0			0
$303$		0			0
$304$	139.023	−	573.062i	0.457314	−	1.88507i
$305$		0			0
$306$		0			0
$307$	−256.423	−	244.498i	−0.835253	−	0.796412i	0.145970	−	0.989289i	$-0.453370\pi$
$307$	−256.423	−	244.498i	−0.835253	−	0.796412i	−0.981223	+	0.192877i	$0.938218\pi$
$308$		0			0
$309$	140.912	+	407.139i	0.456027	+	1.31760i
$310$		0			0
$311$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$311$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$	416.923	+	267.940i	1.33202	+	0.856039i	0.996302	−	0.0859231i	$-0.0273839\pi$
$313$	416.923	+	267.940i	1.33202	+	0.856039i	0.335720	+	0.941962i	$0.391020\pi$
$314$		0			0
$315$		0			0
$316$	−233.895	+	328.459i	−0.740174	+	1.03943i
$317$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$317$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−254.681	−	200.284i	−0.786053	−	0.618159i
$325$	−1.75358	+	36.8122i	−0.00539563	+	0.113268i
$326$		0			0
$327$	−616.341	+	180.974i	−1.88484	+	0.553438i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−526.710	+	101.515i	−1.59127	+	0.306692i	−0.906545	−	0.422109i	$-0.861290\pi$
$331$	−526.710	+	101.515i	−1.59127	+	0.306692i	−0.684725	+	0.728802i	$0.740078\pi$
$332$		0			0
$333$	398.380	−	38.0407i	1.19634	−	0.114236i
$334$		0			0
$335$		0			0
$336$	565.521			1.68310
$337$	−23.6238	+	2.25580i	−0.0701004	+	0.00669378i	−0.130047	−	0.991508i	$-0.541513\pi$
$337$	−23.6238	+	2.25580i	−0.0701004	+	0.00669378i	0.0599469	+	0.998202i	$0.480907\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	461.316	−	135.455i	1.34494	−	0.394911i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$347$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$348$		0			0
$349$	−68.0137	−	78.4920i	−0.194882	−	0.224905i	0.649896	−	0.760023i	$-0.274813\pi$
$349$	−68.0137	−	78.4920i	−0.194882	−	0.224905i	−0.844777	+	0.535118i	$0.820267\pi$
$350$		0			0
$351$	36.9511	−	14.7930i	0.105274	−	0.0421453i
$352$		0			0
$353$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$353$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$359$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	−326.188	−	942.460i	−0.903569	−	2.61069i
$362$		0			0
$363$	−262.715	−	250.499i	−0.723734	−	0.690079i
$364$	−34.7361	+	60.1647i	−0.0954289	+	0.165288i
$365$		0			0
$366$		0			0
$367$	−224.921	+	115.955i	−0.612863	+	0.315953i	−0.736565	−	0.676366i	$-0.763554\pi$
$367$	−224.921	+	115.955i	−0.612863	+	0.315953i	0.123702	+	0.992319i	$0.460523\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	371.129	+	642.814i	0.997659	+	1.72800i
$373$	334.907	−	580.076i	0.897874	−	1.55516i	0.0676666	−	0.997708i	$-0.478445\pi$
$373$	334.907	−	580.076i	0.897874	−	1.55516i	0.830207	−	0.557455i	$-0.188222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−639.814	−	329.847i	−1.68816	−	0.870309i	−0.987651	−	0.156670i	$-0.949924\pi$
$379$	−639.814	−	329.847i	−1.68816	−	0.870309i	−0.700514	−	0.713639i	$-0.747046\pi$
$380$		0			0
$381$	−693.757	−	277.738i	−1.82088	−	0.728972i
$382$		0			0
$383$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$383$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	154.970	+	339.338i	0.400440	+	0.876842i
$388$	−228.189	−	263.344i	−0.588116	−	0.678722i
$389$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$389$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−78.1894	+	543.819i	−0.196951	+	1.36982i	0.616118	+	0.787654i	$0.288704\pi$
$397$	−78.1894	+	543.819i	−0.196951	+	1.36982i	−0.813069	+	0.582168i	$0.802205\pi$
$398$		0			0
$399$	1095.86	−	704.267i	2.74652	−	1.76508i
$400$	−398.189	+	38.0224i	−0.995472	+	0.0950560i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−91.1837			−0.226262
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	250.640	+	351.974i	0.612811	+	0.860572i	0.998061	−	0.0622397i	$-0.0198243\pi$
$409$	250.640	+	351.974i	0.612811	+	0.860572i	−0.385250	+	0.922812i	$0.625885\pi$
$410$		0			0
$411$		0			0
$412$	27.3333	−	573.796i	0.0663429	−	1.39271i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	133.888	+	293.174i	0.321074	+	0.703054i
$418$		0			0
$419$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$419$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$420$		0			0
$421$	154.767	−	217.340i	0.367618	−	0.516248i	−0.588788	−	0.808288i	$-0.700395\pi$
$421$	154.767	−	217.340i	0.367618	−	0.516248i	0.956406	+	0.292040i	$0.0943340\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1135.04	+	333.276i	2.65816	+	0.780507i
$428$		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
$433$	200.514	−	826.531i	0.463082	−	1.90885i	0.0485319	−	0.998822i	$-0.484546\pi$
$433$	200.514	−	826.531i	0.463082	−	1.90885i	0.414550	−	0.910027i	$-0.363939\pi$
$434$		0			0
$435$		0			0
$436$	852.604	+	81.4138i	1.95551	+	0.186729i
$437$		0			0
$438$		0			0
$439$	128.754	+	223.009i	0.293290	+	0.507994i	0.974586	−	0.224015i	$-0.0719164\pi$
$439$	128.754	+	223.009i	0.293290	+	0.507994i	−0.681295	+	0.732009i	$0.738583\pi$
$440$		0			0
$441$	584.976	+	557.774i	1.32648	+	1.26479i
$442$		0			0
$443$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$443$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$444$	−511.976	−	150.330i	−1.15310	−	0.338580i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−700.016	−	280.244i	−1.56254	−	0.625545i
$449$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$449$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−793.112	+	317.514i	−1.75080	+	0.700915i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−586.788	+	559.502i	−1.28400	+	1.22429i	−0.323432	+	0.946251i	$0.604837\pi$
$457$	−586.788	+	559.502i	−1.28400	+	1.22429i	−0.960569	+	0.278041i	$0.910315\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$461$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$	716.677	−	563.601i	1.54790	−	1.21728i	0.665886	−	0.746053i	$-0.268054\pi$
$463$	716.677	−	563.601i	1.54790	−	1.21728i	0.882012	−	0.471227i	$-0.156189\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$467$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$468$	−53.0696			−0.113397
$469$	−643.905	−	456.615i	−1.37293	−	0.973593i
$470$		0			0
$471$	−936.862	+	89.4595i	−1.98909	+	0.189935i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−724.255	+	569.560i	−1.52475	+	1.19907i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$479$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$480$		0			0
$481$	47.4405	−	45.2344i	0.0986288	−	0.0940424i
$482$		0			0
$483$		0			0
$484$	201.061	+	440.262i	0.415415	+	0.909632i
$485$		0			0
$486$		0			0
$487$	−299.519	+	865.403i	−0.615029	+	1.77701i	0.0143349	+	0.999897i	$0.495437\pi$
$487$	−299.519	+	865.403i	−0.615029	+	1.77701i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	31.4829	+	660.908i	0.0643823	+	1.35155i
$490$		0			0
$491$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$491$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−140.846	−	979.604i	−0.283963	−	1.97501i
$497$		0			0
$498$		0			0
$499$	153.010	+	265.022i	0.306634	+	0.531105i	0.977624	−	0.210361i	$-0.0674640\pi$
$499$	153.010	+	265.022i	0.306634	+	0.531105i	−0.670990	+	0.741466i	$0.734131\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$503$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−250.240	+	433.429i	−0.493571	+	0.854889i
$508$	721.115	+	687.582i	1.41952	+	1.35351i
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$	99.4129	+	29.1903i	0.194546	+	0.0571238i
$512$		0			0
$513$	884.473	+	455.978i	1.72412	+	0.888845i
$514$		0			0
$515$		0			0
$516$	−23.6672	−	496.836i	−0.0458667	−	0.962860i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$521$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$522$		0			0
$523$	−173.435	−	714.911i	−0.331617	−	1.36694i	−0.856666	−	0.515871i	$-0.827468\pi$
$523$	−173.435	−	714.911i	−0.331617	−	1.36694i	0.525050	−	0.851071i	$-0.324047\pi$
$524$		0			0
$525$	−694.578	−	546.222i	−1.32301	−	1.04042i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	306.850	+	430.911i	0.580057	+	0.814576i
$530$		0			0
$531$		0			0
$532$	−1705.48	+	328.705i	−3.20579	+	0.617866i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−153.078	+	1064.68i	−0.282954	+	1.96798i	−0.0342135	+	0.999415i	$0.510893\pi$
$541$	−153.078	+	1064.68i	−0.282954	+	1.96798i	−0.248740	+	0.968570i	$0.580016\pi$
$542$		0			0
$543$	629.309	+	883.740i	1.15895	+	1.62751i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−246.264	−	193.664i	−0.450209	−	0.354048i	0.367165	−	0.930156i	$-0.380328\pi$
$547$	−246.264	−	193.664i	−0.450209	−	0.354048i	−0.817374	+	0.576108i	$0.804571\pi$
$548$		0			0
$549$	213.045	+	878.182i	0.388059	+	1.59960i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1166.21	+	224.769i	2.10888	+	0.406454i
$554$		0			0
$555$		0			0
$556$	−20.4475	−	429.246i	−0.0367761	−	0.772024i
$557$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$557$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$558$		0			0
$559$	54.3111	+	27.9993i	0.0971577	+	0.0500883i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$563$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−224.989	+	927.417i	−0.396806	+	1.63566i
$568$		0			0
$569$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$569$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$570$		0			0
$571$	−602.239	+	310.476i	−1.05471	+	0.543740i	−0.896215	−	0.443620i	$-0.853694\pi$
$571$	−602.239	+	310.476i	−1.05471	+	0.543740i	−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−81.9733	−	570.137i	−0.142315	−	0.989821i
$577$	−214.620	−	620.104i	−0.371959	−	1.07470i	−0.963333	−	0.268307i	$-0.913536\pi$
$577$	−214.620	−	620.104i	−0.371959	−	1.07470i	0.591375	−	0.806397i	$-0.298585\pi$
$578$		0			0
$579$	757.423	−	874.112i	1.30816	−	1.50969i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$587$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$588$	−447.693	−	980.310i	−0.761382	−	1.66719i
$589$	−1492.87	−	1722.86i	−2.53458	−	2.92507i
$590$		0			0
$591$		0			0
$592$	559.240	+	439.791i	0.944662	+	0.742890i
$593$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$593$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−81.5641	+	64.1427i	−0.136623	+	0.107442i
$598$		0			0
$599$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$599$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$600$		0			0
$601$	−1076.67	+	102.810i	−1.79147	+	0.171064i	−0.937206	−	0.348777i	$-0.886597\pi$
$601$	−1076.67	+	102.810i	−1.79147	+	0.171064i	−0.854263	+	0.519842i	$0.825991\pi$
$602$		0			0
$603$	56.1374	−	600.381i	0.0930968	−	0.995657i
$604$	1139.08			1.88589
$605$		0			0
$606$		0			0
$607$	879.737	−	169.555i	1.44932	−	0.279334i	0.596953	−	0.802277i	$-0.296378\pi$
$607$	879.737	−	169.555i	1.44932	−	0.279334i	0.852368	+	0.522943i	$0.175166\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.28344	+	89.9205i	−0.00698767	+	0.146689i	0.992677	+	0.120798i	$0.0385454\pi$
$613$	−4.28344	+	89.9205i	−0.00698767	+	0.146689i	−0.999665	+	0.0258912i	$0.991758\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$617$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$618$		0			0
$619$	−543.601	+	217.625i	−0.878193	+	0.351575i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	−543.601	+	217.625i	−0.878193	+	0.351575i	−0.111634	+	0.993749i	$0.535608\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	65.6909	+	26.2987i	0.105274	+	0.0421453i
$625$	525.783	+	337.901i	0.841254	+	0.540641i
$626$		0			0
$627$		0			0
$628$	1204.00	+	353.527i	1.91720	+	0.562941i
$629$		0			0
$630$		0			0
$631$	−7.11301	−	6.78225i	−0.0112726	−	0.0107484i	0.684422	−	0.729086i	$-0.260055\pi$
$631$	−7.11301	−	6.78225i	−0.0112726	−	0.0107484i	−0.695694	+	0.718338i	$0.744903\pi$
$632$		0			0
$633$	−447.460	−	775.023i	−0.706887	−	1.22436i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	131.792	+	12.5846i	0.206895	+	0.0197561i
$638$		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
$643$	163.392	+	1136.42i	0.254109	+	1.76737i	0.572979	+	0.819570i	$0.305788\pi$
$643$	163.392	+	1136.42i	0.254109	+	1.76737i	−0.318870	+	0.947798i	$0.603303\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$647$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1268.16	−	1780.88i	1.94801	−	2.73560i
$652$	288.543	−	833.689i	0.442550	−	1.27866i
$653$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$653$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	18.6597	+	76.9162i	0.0284013	+	0.117072i
$658$		0			0
$659$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$659$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$660$		0			0
$661$	−53.7245	+	117.640i	−0.0812776	+	0.177973i	−0.945908	−	0.324436i	$-0.894826\pi$
$661$	−53.7245	+	117.640i	−0.0812776	+	0.177973i	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
$669$	501.269			0.749280
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−402.819	+	258.876i	−0.598543	+	0.384660i	−0.804545	−	0.593891i	$-0.797591\pi$
$673$	−402.819	+	258.876i	−0.598543	+	0.384660i	0.206002	+	0.978551i	$0.433955\pi$
$674$		0			0
$675$	96.0625	−	668.129i	0.142315	−	0.989821i
$676$	524.539	−	412.502i	0.775945	−	0.610210i
$677$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$677$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$678$		0			0
$679$	−426.359	+	933.596i	−0.627922	+	1.37496i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$683$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$684$	−868.862	−	1002.72i	−1.27027	−	1.46597i
$685$		0			0
$686$		0			0
$687$	−1327.08	−	255.774i	−1.93171	−	0.372306i
$688$	−216.911	+	626.724i	−0.315278	+	0.910936i
$689$		0			0
$690$		0			0
$691$	1065.30	+	426.482i	1.54168	+	0.617195i	0.977964	−	0.208774i	$-0.0669474\pi$
$691$	1065.30	+	426.482i	1.54168	+	0.617195i	0.563715	+	0.825969i	$0.309372\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	589.085	+	1020.32i	0.841550	+	1.45761i
$701$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$701$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$702$		0			0
$703$	1631.38	+	155.778i	2.32060	+	0.221590i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−986.134	−	940.277i	−1.39088	−	1.32620i	−0.886507	−	0.462716i	$-0.846875\pi$
$709$	−986.134	−	940.277i	−1.39088	−	1.32620i	−0.504373	−	0.863486i	$-0.668276\pi$
$710$		0			0
$711$	296.736	+	857.363i	0.417351	+	1.20586i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$719$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$720$		0			0
$721$	−1570.79	+	628.849i	−2.17862	+	0.872190i
$722$		0			0
$723$	−873.198	−	1007.72i	−1.20774	−	1.39381i
$724$	−341.036	−	1405.77i	−0.471044	−	1.94167i
$725$		0			0
$726$		0			0
$727$	54.1811	−	1137.40i	0.0745269	−	1.56451i	−0.583873	−	0.811845i	$-0.698464\pi$
$727$	54.1811	−	1137.40i	0.0745269	−	1.56451i	0.658400	−	0.752668i	$-0.271233\pi$
$728$		0			0
$729$	−699.470	+	205.383i	−0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$	171.471	−	1192.61i	0.234250	−	1.62925i
$733$	1350.01	−	260.193i	1.84176	−	0.354970i	0.855192	−	0.518312i	$-0.173439\pi$
$733$	1350.01	−	260.193i	1.84176	−	0.354970i	0.986570	+	0.163342i	$0.0522273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	384.234	−	36.6899i	0.519938	−	0.0496481i	0.168212	−	0.985751i	$-0.446201\pi$
$739$	384.234	−	36.6899i	0.519938	−	0.0496481i	0.351726	+	0.936103i	$0.385595\pi$
$740$		0			0
$741$	160.046	−	30.8463i	0.215986	−	0.0416279i
$742$		0			0
$743$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$743$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	961.218	+	1109.30i	1.27992	+	1.47710i	0.800266	+	0.599645i	$0.204691\pi$
$751$	961.218	+	1109.30i	1.27992	+	1.47710i	0.479651	+	0.877459i	$0.340763\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	738.078	−	1036.49i	0.976293	−	1.37101i
$757$	−5.74526	−	120.608i	−0.00758951	−	0.159323i	−0.999480	−	0.0322313i	$-0.989739\pi$
$757$	−5.74526	−	120.608i	−0.00758951	−	0.159323i	0.991891	−	0.127092i	$-0.0405644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$761$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$	−825.095	−	2383.96i	−1.08138	−	3.12445i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−181.063	+	746.351i	−0.235759	+	0.971812i
$769$	1054.78	−	543.779i	1.37163	−	0.707125i	0.394328	−	0.918970i	$-0.370977\pi$
$769$	1054.78	−	543.779i	1.37163	−	0.707125i	0.977303	+	0.211844i	$0.0679470\pi$
$770$		0			0
$771$		0			0
$772$	−1370.72	+	706.656i	−1.77555	+	0.915358i
$773$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$773$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$774$		0			0
$775$	−773.185	+	1339.20i	−0.997659	+	1.72800i
$776$		0			0
$777$	223.669	+	1555.65i	0.287862	+	2.00212i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	68.3720	+	1435.31i	0.0872092	+	1.83075i
$785$		0			0
$786$		0			0
$787$	−1417.98	−	273.293i	−1.80175	−	0.347259i	−0.825584	−	0.564279i	$-0.809154\pi$
$787$	−1417.98	−	273.293i	−1.80175	−	0.347259i	−0.976167	+	0.217020i	$0.930366\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	116.347	+	91.4963i	0.146718	+	0.115380i
$794$		0			0
$795$		0			0
$796$	132.748	−	38.9783i	0.166769	−	0.0489677i
$797$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$797$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−367.007	+	715.347i	−0.456476	+	0.889736i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$809$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$810$		0			0
$811$	−121.015	−	169.941i	−0.149216	−	0.209545i	0.733202	−	0.680011i	$-0.238025\pi$
$811$	−121.015	−	169.941i	−0.149216	−	0.209545i	−0.882418	+	0.470466i	$0.844086\pi$
$812$		0			0
$813$	242.774	−	531.601i	0.298615	−	0.653876i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	360.157	+	1484.59i	0.440828	+	1.81712i
$818$		0			0
$819$	64.9346	+	142.187i	0.0792852	+	0.173610i
$820$		0			0
$821$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$821$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$822$		0			0
$823$	−541.396	+	760.284i	−0.657832	+	0.923796i	−0.999874	−	0.0158885i	$-0.994942\pi$
$823$	−541.396	+	760.284i	−0.657832	+	0.923796i	0.342041	+	0.939685i	$0.388882\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$827$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$828$		0			0
$829$	−142.153	−	41.7398i	−0.171475	−	0.0503496i	0.194868	−	0.980829i	$-0.437572\pi$
$829$	−142.153	−	41.7398i	−0.171475	−	0.0503496i	−0.366343	+	0.930480i	$0.619390\pi$
$830$		0			0
$831$	94.8101	+	659.419i	0.114092	+	0.793525i
$832$	−68.2814	−	65.1062i	−0.0820690	−	0.0782527i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1662.52	+	158.751i	1.98628	+	0.189667i
$838$		0			0
$839$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$839$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	169.814	+	1181.08i	0.201201	+	1.39938i
$845$		0			0
$846$		0			0
$847$	933.560	−	1077.39i	1.10220	−	1.27200i
$848$		0			0
$849$	995.243	+	639.604i	1.17225	+	0.753361i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−483.161	+	1396.00i	−0.566426	+	1.63658i	0.189470	+	0.981887i	$0.439323\pi$
$853$	−483.161	+	1396.00i	−0.566426	+	1.63658i	−0.755895	+	0.654693i	$0.772798\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$857$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$858$		0			0
$859$	258.311	−	246.299i	0.300711	−	0.286727i	−0.524667	−	0.851307i	$-0.675810\pi$
$859$	258.311	−	246.299i	0.300711	−	0.286727i	0.825378	+	0.564580i	$0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$863$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−851.332	+	164.081i	−0.981929	+	0.189251i
$868$	−2452.27	+	1575.98i	−2.82520	+	1.81564i
$869$		0			0
$870$		0			0
$871$	−53.5617	−	82.9841i	−0.0614945	−	0.0952745i
$872$		0			0
$873$	−780.472	+	74.5260i	−0.894011	+	0.0853677i
$874$		0			0
$875$		0			0
$876$	15.0184	−	104.455i	0.0171443	−	0.119241i
$877$	1328.46	−	1044.71i	1.51478	−	1.19123i	0.591978	−	0.805954i	$-0.298347\pi$
$877$	1328.46	−	1044.71i	1.51478	−	1.19123i	0.922800	−	0.385280i	$-0.125895\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$881$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$882$		0			0
$883$	−851.913	+	812.298i	−0.964794	+	0.919929i	−0.996830	−	0.0795575i	$-0.974649\pi$
$883$	−851.913	+	812.298i	−0.964794	+	0.919929i	0.0320363	+	0.999487i	$0.489801\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$887$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$888$		0			0
$889$	959.868	−	2773.36i	1.07972	−	3.11964i
$890$		0			0
$891$		0			0
$892$	−620.482	−	248.404i	−0.695608	−	0.278479i
$893$		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
$903$	−1302.19	+	671.326i	−1.44207	+	0.743439i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	427.538	−	1762.33i	0.471376	−	1.94304i	0.211829	−	0.977307i	$-0.432058\pi$
$907$	427.538	−	1762.33i	0.471376	−	1.94304i	0.259546	−	0.965731i	$-0.416427\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$911$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$912$	578.601	+	1671.76i	0.634430	+	1.83307i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1515.94	+	974.238i	1.65496	+	1.06358i
$917$		0			0
$918$		0			0
$919$	859.327	−	1206.76i	0.935068	−	1.31312i	−0.0146109	−	0.999893i	$-0.504651\pi$
$919$	859.327	−	1206.76i	0.935068	−	1.31312i	0.949679	−	0.313226i	$-0.101410\pi$
$920$		0			0
$921$	1043.71	+	201.158i	1.13323	+	0.218413i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−262.080	−	1080.31i	−0.283330	−	1.16790i
$926$		0			0
$927$	−1015.98	−	798.974i	−1.09598	−	0.861892i
$928$		0			0
$929$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$929$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$930$		0			0
$931$	1919.94	+	2696.17i	2.06223	+	2.89600i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−969.981			−1.03520			−0.517599	−	0.855623i	$-0.673174\pi$
$937$	−969.981			−1.03520			−0.517599	+	0.855623i	$0.673174\pi$
$938$		0			0
$939$	−1486.79			−1.58338
$940$		0			0
$941$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$941$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$947$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$948$	57.5590	−	1208.31i	0.0607162	−	1.27459i
$949$	10.1903	+	8.01377i	0.0107380	+	0.00844444i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$953$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2546.53	−	1312.83i	−2.64988	−	1.36611i
$962$		0			0
$963$		0			0
$964$	581.487	+	1680.10i	0.603203	+	1.74284i
$965$		0			0
$966$		0			0
$967$	−411.888	+	713.412i	−0.425945	+	0.737758i	−0.996508	−	0.0834958i	$-0.973391\pi$
$967$	−411.888	+	713.412i	−0.425945	+	0.737758i	0.570564	+	0.821253i	$0.306725\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$971$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$972$	967.599	+	92.3945i	0.995472	+	0.0950560i
$973$	−1125.04	+	579.997i	−1.15626	+	0.596092i
$974$		0			0
$975$	−55.2809	−	95.7493i	−0.0566983	−	0.0982044i
$976$	−803.248	+	1391.27i	−0.823000	+	1.42548i
$977$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$977$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1261.97	−	1456.39i	1.28641	−	1.48460i
$982$		0			0
$983$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$983$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−213.394	−	41.1284i	−0.215986	−	0.0416279i
$989$		0			0
$990$		0			0
$991$	1280.09	+	1477.30i	1.29172	+	1.49072i	0.770745	+	0.637144i	$0.219884\pi$
$991$	1280.09	+	1477.30i	1.29172	+	1.49072i	0.520971	+	0.853575i	$0.325570\pi$
$992$		0			0
$993$	1164.64	−	1110.48i	1.17285	−	1.11831i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1205.34	−	353.919i	1.20896	−	0.354984i	0.385691	−	0.922628i	$-0.373963\pi$
$997$	1205.34	−	353.919i	1.20896	−	0.354984i	0.823274	+	0.567644i	$0.192145\pi$
$998$		0			0
$999$	−943.717	+	742.148i	−0.944662	+	0.742890i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.17.1	✓	20
3.2	odd	2	CM	201.3.o.a.17.1	✓	20
67.4	even	33	inner	201.3.o.a.71.1	yes	20
201.71	odd	66	inner	201.3.o.a.71.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.17.1	✓	20	1.1	even	1	trivial
201.3.o.a.17.1	✓	20	3.2	odd	2	CM
201.3.o.a.71.1	yes	20	67.4	even	33	inner
201.3.o.a.71.1	yes	20	201.71	odd	66	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-2.52376 + 1.62192i) q^{3} +(3.92771 - 0.757005i) q^{4} +(-6.83405 - 9.59709i) q^{7} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)	\(q+(-2.52376 + 1.62192i) q^{3} +(3.92771 - 0.757005i) q^{4} +(-6.83405 - 9.59709i) q^{7} +(3.73874 - 8.18669i) q^{9} +(-8.68481 + 8.28095i) q^{12} +(-0.347546 - 1.43260i) q^{13} +(14.8539 - 5.94660i) q^{16} +(21.3782 - 30.0214i) q^{19} +(32.8132 + 13.1364i) q^{21} +(-23.9873 - 7.04331i) q^{25} +(3.84250 + 26.7252i) q^{27} +(-34.1073 - 32.5212i) q^{28} +(14.5828 - 60.1112i) q^{31} +(8.48732 - 34.9852i) q^{36} +(22.2329 + 38.5085i) q^{37} +(3.20069 + 3.05185i) q^{39} +(-27.1439 + 31.3258i) q^{43} +(-27.8427 + 39.0996i) q^{48} +(-29.3734 + 84.8689i) q^{49} +(-2.44955 - 5.36376i) q^{52} +(-5.26093 + 110.441i) q^{57} +(-78.9245 + 62.0669i) q^{61} +(-104.119 + 20.0673i) q^{63} +(53.8402 - 34.6010i) q^{64} +(63.2718 - 22.0381i) q^{67} +(-6.91266 + 5.43617i) q^{73} +(71.9620 - 21.1299i) q^{75} +(61.2409 - 134.099i) q^{76} +(-72.9573 + 69.5647i) q^{79} +(-53.0437 - 61.2157i) q^{81} +(138.825 + 26.7564i) q^{84} +(-11.3737 + 13.1259i) q^{91} +(60.6922 + 175.359i) q^{93} +(-43.5568 - 75.4425i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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