LMFDB - Embedded newform 201.3.o.a.77.1

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			77.1
Root			$0.981929 + 0.189251i$ of defining polynomial
Character	$\chi$	$=$	201.77
Dual form			201.3.o.a.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.426945 + 2.96946i) q^{3} +(0.190328 + 3.99547i) q^{4} +(-1.36948 + 5.64509i) q^{7} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})$	$q+(0.426945 + 2.96946i) q^{3} +(0.190328 + 3.99547i) q^{4} +(-1.36948 + 5.64509i) q^{7} +(-8.63544 + 2.53559i) q^{9} +(-11.7831 + 2.27101i) q^{12} +(6.10064 - 17.6266i) q^{13} +(-15.9276 + 1.52090i) q^{16} +(5.59469 + 23.0616i) q^{19} +(-17.3476 - 1.65649i) q^{21} +(-16.3715 - 18.8937i) q^{25} +(-11.2162 - 24.5601i) q^{27} +(-22.8154 - 4.39731i) q^{28} +(19.8020 + 57.2140i) q^{31} +(-11.7744 - 34.0200i) q^{36} +(34.6931 + 60.0903i) q^{37} +(54.9463 + 10.5900i) q^{39} +(-4.93679 - 3.17268i) q^{43} +(-11.3164 - 46.6470i) q^{48} +(13.5614 + 6.99138i) q^{49} +(71.5878 + 21.0201i) q^{52} +(-66.0920 + 26.4593i) q^{57} +(60.7601 - 85.3257i) q^{61} +(-2.48756 - 52.2203i) q^{63} +(-9.10815 - 63.3486i) q^{64} +(65.1423 + 15.6677i) q^{67} +(18.9217 - 26.5718i) q^{73} +(49.1146 - 56.6812i) q^{75} +(-91.0772 + 26.7427i) q^{76} +(-18.6841 + 3.60106i) q^{79} +(68.1415 - 43.7919i) q^{81} +(3.31674 - 69.6270i) q^{84} +(91.1492 + 58.5781i) q^{91} +(-161.441 + 83.2285i) q^{93} +(60.7443 + 105.212i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{8}{33}\right)$

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		−0.723734	−	0.690079i	$-0.757576\pi$
$2$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$3$	0.426945	+	2.96946i	0.142315	+	0.989821i
$3$	0.426945	+	2.96946i	0.142315	+	0.989821i
$4$	0.190328	+	3.99547i	0.0475819	+	0.998867i
$5$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$5$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$6$		0			0
$7$	−1.36948	+	5.64509i	−0.195641	+	0.806441i	0.786392	+	0.617728i	$0.211947\pi$
$7$	−1.36948	+	5.64509i	−0.195641	+	0.806441i	−0.982032	+	0.188713i	$0.939568\pi$
$8$		0			0
$9$	−8.63544	+	2.53559i	−0.959493	+	0.281733i
$10$		0			0
$11$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$11$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$12$	−11.7831	+	2.27101i	−0.981929	+	0.189251i
$13$	6.10064	−	17.6266i	0.469280	−	1.35590i	−0.423701	−	0.905802i	$-0.639269\pi$
$13$	6.10064	−	17.6266i	0.469280	−	1.35590i	0.892981	−	0.450093i	$-0.148609\pi$
$14$		0			0
$15$		0			0
$16$	−15.9276	+	1.52090i	−0.995472	+	0.0950560i
$17$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$17$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$18$		0			0
$19$	5.59469	+	23.0616i	0.294457	+	1.21377i	0.906247	+	0.422749i	$0.138935\pi$
$19$	5.59469	+	23.0616i	0.294457	+	1.21377i	−0.611789	+	0.791021i	$0.709550\pi$
$20$		0			0
$21$	−17.3476	−	1.65649i	−0.826076	−	0.0788807i
$22$		0			0
$23$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$23$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$24$		0			0
$25$	−16.3715	−	18.8937i	−0.654861	−	0.755750i
$26$		0			0
$27$	−11.2162	−	24.5601i	−0.415415	−	0.909632i
$28$	−22.8154	−	4.39731i	−0.814837	−	0.157047i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	19.8020	+	57.2140i	0.638773	+	1.84561i	0.522988	+	0.852340i	$0.324817\pi$
$31$	19.8020	+	57.2140i	0.638773	+	1.84561i	0.115785	+	0.993274i	$0.463062\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−11.7744	−	34.0200i	−0.327068	−	0.945001i
$37$	34.6931	+	60.0903i	0.937652	+	1.62406i	0.769835	+	0.638243i	$0.220338\pi$
$37$	34.6931	+	60.0903i	0.937652	+	1.62406i	0.167817	+	0.985818i	$0.446328\pi$
$38$		0			0
$39$	54.9463	+	10.5900i	1.40888	+	0.271539i
$40$		0			0
$41$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$41$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$42$		0			0
$43$	−4.93679	−	3.17268i	−0.114809	−	0.0737834i	0.481976	−	0.876184i	$-0.339919\pi$
$43$	−4.93679	−	3.17268i	−0.114809	−	0.0737834i	−0.596785	+	0.802401i	$0.703556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$47$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$48$	−11.3164	−	46.6470i	−0.235759	−	0.971812i
$49$	13.5614	+	6.99138i	0.276763	+	0.142681i
$50$		0			0
$51$		0			0
$52$	71.5878	+	21.0201i	1.37669	+	0.404232i
$53$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$53$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−66.0920	+	26.4593i	−1.15951	+	0.464198i
$58$		0			0
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$	60.7601	−	85.3257i	0.996068	−	1.39878i	0.0804376	−	0.996760i	$-0.474368\pi$
$61$	60.7601	−	85.3257i	0.996068	−	1.39878i	0.915630	−	0.402022i	$-0.131692\pi$
$62$		0			0
$63$	−2.48756	−	52.2203i	−0.0394850	−	0.828893i
$64$	−9.10815	−	63.3486i	−0.142315	−	0.989821i
$65$		0			0
$66$		0			0
$67$	65.1423	+	15.6677i	0.972274	+	0.233846i
$67$	65.1423	+	15.6677i	0.972274	+	0.233846i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$71$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$72$		0			0
$73$	18.9217	−	26.5718i	0.259201	−	0.363997i	−0.664384	−	0.747392i	$-0.731306\pi$
$73$	18.9217	−	26.5718i	0.259201	−	0.363997i	0.923584	+	0.383395i	$0.125245\pi$
$74$		0			0
$75$	49.1146	−	56.6812i	0.654861	−	0.755750i
$76$	−91.0772	+	26.7427i	−1.19838	+	0.351877i
$77$		0			0
$78$		0			0
$79$	−18.6841	+	3.60106i	−0.236507	+	0.0455830i	−0.306127	−	0.951991i	$-0.599033\pi$
$79$	−18.6841	+	3.60106i	−0.236507	+	0.0455830i	0.0696203	+	0.997574i	$0.477821\pi$
$80$		0			0
$81$	68.1415	−	43.7919i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$83$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$84$	3.31674	−	69.6270i	0.0394850	−	0.828893i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$89$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$90$		0			0
$91$	91.1492	+	58.5781i	1.00164	+	0.643715i
$92$		0			0
$93$	−161.441	+	83.2285i	−1.73592	+	0.894930i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	60.7443	+	105.212i	0.626230	+	1.08466i	0.988302	+	0.152512i	$0.0487361\pi$
$97$	60.7443	+	105.212i	0.626230	+	1.08466i	−0.362072	+	0.932150i	$0.617931\pi$
$98$		0			0
$99$		0			0
$100$	72.3734	−	69.0079i	0.723734	−	0.690079i
$101$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$101$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$102$		0			0
$103$	67.2650	+	194.349i	0.653058	+	1.88689i	0.380750	+	0.924678i	$0.375666\pi$
$103$	67.2650	+	194.349i	0.653058	+	1.88689i	0.272309	+	0.962210i	$0.412213\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$	95.9942	−	49.4885i	0.888835	−	0.458227i
$109$	−142.093	−	163.984i	−1.30361	−	1.50444i	−0.724741	−	0.689022i	$-0.758040\pi$
$109$	−142.093	−	163.984i	−1.30361	−	1.50444i	−0.578867	−	0.815422i	$-0.696505\pi$
$110$		0			0
$111$	−163.624	+	128.675i	−1.47409	+	1.15924i
$112$	13.2269	−	91.9953i	0.118098	−	0.821387i
$113$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$113$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.98770	+	167.682i	−0.0682709	+	1.43318i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−39.5752	+	114.345i	−0.327068	+	0.945001i
$122$		0			0
$123$		0			0
$124$	−224.828	+	90.0076i	−1.81313	+	0.725868i
$125$		0			0
$126$		0			0
$127$	43.0724	−	177.547i	0.339153	−	1.39801i	−0.505562	−	0.862790i	$-0.668715\pi$
$127$	43.0724	−	177.547i	0.339153	−	1.39801i	0.844715	−	0.535216i	$-0.179770\pi$
$128$		0			0
$129$	7.31344	−	16.0142i	0.0566933	−	0.124141i
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$	−137.847			−1.03644
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$137$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$138$		0			0
$139$	78.3483	−	171.559i	0.563657	−	1.23424i	−0.386450	−	0.922310i	$-0.626299\pi$
$139$	78.3483	−	171.559i	0.563657	−	1.23424i	0.950107	−	0.311925i	$-0.100974\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	133.685	−	53.5194i	0.928368	−	0.371662i
$145$		0			0
$146$		0			0
$147$	−14.9707	+	43.2550i	−0.101841	+	0.294252i
$148$	−233.486	+	150.052i	−1.57761	+	1.01387i
$149$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$149$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$150$		0			0
$151$	−11.6531	+	244.629i	−0.0771728	+	1.62006i	0.545871	+	0.837869i	$0.316199\pi$
$151$	−11.6531	+	244.629i	−0.0771728	+	1.62006i	−0.623044	+	0.782187i	$0.714104\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−31.8544	+	221.552i	−0.204195	+	1.42020i
$157$	−153.438	+	120.665i	−0.977309	+	0.768565i	−0.972853	−	0.231426i	$-0.925661\pi$
$157$	−153.438	+	120.665i	−0.977309	+	0.768565i	−0.00445665	+	0.999990i	$0.501419\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	103.118	−	178.606i	0.632627	−	1.09574i	−0.354386	−	0.935099i	$-0.615310\pi$
$163$	103.118	−	178.606i	0.632627	−	1.09574i	0.987013	−	0.160643i	$-0.0513567\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$167$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$168$		0			0
$169$	−140.638	−	110.599i	−0.832175	−	0.654430i
$170$		0			0
$171$	−106.787	−	184.961i	−0.624488	−	1.08165i
$172$	11.7368	−	20.3287i	0.0682370	−	0.118190i
$173$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$173$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$174$		0			0
$175$	129.077	−	66.5440i	0.737585	−	0.380252i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$179$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$180$		0			0
$181$	−125.766	−	50.3490i	−0.694838	−	0.278171i	−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	−125.766	−	50.3490i	−0.694838	−	0.278171i	−0.692076	+	0.721825i	$0.743304\pi$
$182$		0			0
$183$	279.313	+	143.996i	1.52630	+	0.786862i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	154.004	−	29.6819i	0.814837	−	0.157047i
$190$		0			0
$191$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$191$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$192$	184.223	−	54.0927i	0.959493	−	0.281733i
$193$	156.000	−	180.033i	0.808289	−	0.932816i	−0.190516	−	0.981684i	$-0.561016\pi$
$193$	156.000	−	180.033i	0.808289	−	0.932816i	0.998805	+	0.0488685i	$0.0155615\pi$
$194$		0			0
$195$		0			0
$196$	−25.3527	+	55.5147i	−0.129351	+	0.283238i
$197$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$197$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$198$		0			0
$199$	−280.473	−	267.431i	−1.40941	−	1.34387i	−0.861904	−	0.507072i	$-0.830728\pi$
$199$	−280.473	−	267.431i	−1.40941	−	1.34387i	−0.547509	−	0.836800i	$-0.684424\pi$
$200$		0			0
$201$	−18.7125	+	200.127i	−0.0930968	+	0.995657i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−70.3599	+	290.028i	−0.338269	+	1.39436i
$209$		0			0
$210$		0			0
$211$	187.649	−	75.1234i	0.889333	−	0.356035i	0.118411	−	0.992965i	$-0.462220\pi$
$211$	187.649	−	75.1234i	0.889333	−	0.356035i	0.770922	+	0.636929i	$0.219796\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−350.097	+	33.4302i	−1.61335	+	0.154056i
$218$		0			0
$219$	86.9824	+	44.8425i	0.397180	+	0.204760i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	63.4099	−	441.025i	0.284349	−	1.97769i	0.0982506	−	0.995162i	$-0.468675\pi$
$223$	63.4099	−	441.025i	0.284349	−	1.97769i	0.186099	−	0.982531i	$-0.440416\pi$
$224$		0			0
$225$	189.282	+	121.644i	0.841254	+	0.540641i
$226$		0			0
$227$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$227$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$228$	−118.296	−	259.033i	−0.518844	−	1.13611i
$229$	415.953	+	80.1685i	1.81639	+	0.350081i	0.980146	−	0.198276i	$-0.0635343\pi$
$229$	415.953	+	80.1685i	1.81639	+	0.350081i	0.836245	+	0.548357i	$0.184746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$233$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−18.6703	−	53.9442i	−0.0787775	−	0.227613i
$238$		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
$241$	−113.450	−	248.420i	−0.470745	−	1.03079i	−0.984905	−	0.173094i	$-0.944624\pi$
$241$	−113.450	−	248.420i	−0.470745	−	1.03079i	0.514160	−	0.857694i	$-0.328104\pi$
$242$		0			0
$243$	159.131	+	183.647i	0.654861	+	0.755750i
$244$	352.480	+	226.525i	1.44459	+	0.928383i
$245$		0			0
$246$		0			0
$247$	440.630	+	42.0751i	1.78393	+	0.170344i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$251$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$252$	208.171	−	19.8779i	0.826076	−	0.0788807i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	251.374	−	48.4483i	0.981929	−	0.189251i
$257$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$257$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$258$		0			0
$259$	−386.727	+	113.553i	−1.49315	+	0.438429i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−50.2013	+	263.256i	−0.187318	+	0.982299i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−72.5541	−	504.625i	−0.267727	−	1.86208i	−0.469879	−	0.882731i	$-0.655703\pi$
$271$	−72.5541	−	504.625i	−0.267727	−	1.86208i	0.202152	−	0.979354i	$-0.435207\pi$
$272$		0			0
$273$	−135.030	+	295.674i	−0.494615	+	1.08305i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	512.353	−	150.440i	1.84965	−	0.543106i	0.849776	−	0.527144i	$-0.176737\pi$
$277$	512.353	−	150.440i	1.84965	−	0.543106i	0.999873	−	0.0159622i	$-0.00508116\pi$
$278$		0			0
$279$	−316.070	−	443.858i	−1.13287	−	1.59089i
$280$		0			0
$281$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$281$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$282$		0			0
$283$	−472.801	−	138.827i	−1.67068	−	0.490555i	−0.696730	−	0.717334i	$-0.745362\pi$
$283$	−472.801	−	138.827i	−1.67068	−	0.490555i	−0.973946	+	0.226779i	$0.927180\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−287.691	−	27.4712i	−0.995472	−	0.0950560i
$290$		0			0
$291$	−286.489	+	225.298i	−0.984500	+	0.774219i
$292$	109.768	+	70.5436i	0.375918	+	0.241588i
$293$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$293$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	235.816	+	185.448i	0.786053	+	0.618159i
$301$	24.6710	−	23.5237i	0.0819633	−	0.0781518i
$302$		0			0
$303$		0			0
$304$	−124.184	−	358.806i	−0.408500	−	1.18028i
$305$		0			0
$306$		0			0
$307$	536.899	+	103.479i	1.74886	+	0.337065i	0.960527	−	0.278186i	$-0.0897333\pi$
$307$	536.899	+	103.479i	1.74886	+	0.337065i	0.788331	+	0.615251i	$0.210945\pi$
$308$		0			0
$309$	−548.395	+	282.717i	−1.77474	+	0.914943i
$310$		0			0
$311$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$311$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$312$		0			0
$313$	5.05491	−	35.1577i	0.0161499	−	0.112325i	−0.980152	−	0.198248i	$-0.936475\pi$
$313$	5.05491	−	35.1577i	0.0161499	−	0.112325i	0.996302	+	0.0859231i	$0.0273839\pi$
$314$		0			0
$315$		0			0
$316$	−17.9440	−	73.9662i	−0.0567849	−	0.234070i
$317$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$317$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	187.938	+	263.923i	0.580057	+	0.814576i
$325$	−432.910	+	173.311i	−1.33203	+	0.533265i
$326$		0			0
$327$	426.280	−	491.953i	1.30361	−	1.50444i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.18912	+	129.926i	0.0186982	+	0.392525i	0.988745	+	0.149608i	$0.0478011\pi$
$331$	6.18912	+	129.926i	0.0186982	+	0.392525i	−0.970047	+	0.242917i	$0.921896\pi$
$332$		0			0
$333$	−451.955	−	430.938i	−1.35722	−	1.29411i
$334$		0			0
$335$		0			0
$336$	278.824			0.829833
$337$	278.008	+	265.080i	0.824949	+	0.786588i	0.979510	−	0.201396i	$-0.0645477\pi$
$337$	278.008	+	265.080i	0.824949	+	0.786588i	−0.154561	+	0.987983i	$0.549396\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−244.434	+	282.092i	−0.712635	+	0.822425i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$347$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$348$		0			0
$349$	−496.049	+	318.791i	−1.42134	+	0.913442i	−0.421365	+	0.906891i	$0.638449\pi$
$349$	−496.049	+	318.791i	−1.42134	+	0.913442i	−0.999979	+	0.00655113i	$0.997915\pi$
$350$		0			0
$351$	−501.337	+	47.8719i	−1.42831	+	0.136387i
$352$		0			0
$353$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$353$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$359$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$360$		0			0
$361$	−179.668	+	92.6255i	−0.497696	+	0.256580i
$362$		0			0
$363$	−356.440	−	68.6982i	−0.981929	−	0.189251i
$364$	−216.699	+	375.333i	−0.595326	+	1.03113i
$365$		0			0
$366$		0			0
$367$	343.438	+	270.083i	0.935798	+	0.735920i	0.964614	−	0.263665i	$-0.0849313\pi$
$367$	343.438	+	270.083i	0.935798	+	0.735920i	−0.0288160	+	0.999585i	$0.509174\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−363.263	−	629.191i	−0.976515	−	1.69137i
$373$	−343.422	+	594.825i	−0.920703	+	1.59470i	−0.122374	+	0.992484i	$0.539051\pi$
$373$	−343.422	+	594.825i	−0.920703	+	1.59470i	−0.798330	+	0.602221i	$0.794283\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	229.454	−	180.445i	0.605421	−	0.476108i	−0.267773	−	0.963482i	$-0.586288\pi$
$379$	229.454	−	180.445i	0.605421	−	0.476108i	0.873193	+	0.487374i	$0.162045\pi$
$380$		0			0
$381$	545.608	+	52.0993i	1.43204	+	0.136744i
$382$		0			0
$383$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$383$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	50.6760	+	14.8798i	0.130946	+	0.0384491i
$388$	−408.811	+	262.727i	−1.05364	+	0.677131i
$389$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$389$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−268.182	+	587.237i	−0.675522	+	1.47919i	0.191798	+	0.981434i	$0.438568\pi$
$397$	−268.182	+	587.237i	−0.675522	+	1.47919i	−0.867319	+	0.497752i	$0.834159\pi$
$398$		0			0
$399$	−58.8529	−	409.331i	−0.147501	−	1.02589i
$400$	289.494	+	276.032i	0.723734	+	0.690079i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	1129.30			2.80222
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−51.6102	+	212.740i	−0.126186	+	0.520147i	0.873272	+	0.487233i	$0.161994\pi$
$409$	−51.6102	+	212.740i	−0.126186	+	0.520147i	−0.999458	+	0.0329139i	$0.989521\pi$
$410$		0			0
$411$		0			0
$412$	−763.715	+	305.745i	−1.85368	+	0.742100i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	542.888	+	159.406i	1.30189	+	0.382269i
$418$		0			0
$419$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$419$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$420$		0			0
$421$	−48.8732	−	201.458i	−0.116088	−	0.478523i	−0.999942	−	0.0107595i	$-0.996575\pi$
$421$	−48.8732	−	201.458i	−0.116088	−	0.478523i	0.883854	−	0.467763i	$-0.154940\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	398.461	+	459.849i	0.933164	+	1.07693i
$428$		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
$433$	−205.177	−	592.820i	−0.473850	−	1.36910i	−0.888399	−	0.459072i	$-0.848182\pi$
$433$	−205.177	−	592.820i	−0.473850	−	1.36910i	0.414550	−	0.910027i	$-0.363939\pi$
$434$		0			0
$435$		0			0
$436$	628.150	−	598.940i	1.44071	−	1.37372i
$437$		0			0
$438$		0			0
$439$	−433.746	−	751.270i	−0.988032	−	1.71132i	−0.627599	−	0.778537i	$-0.715962\pi$
$439$	−433.746	−	751.270i	−0.988032	−	1.71132i	−0.360433	−	0.932785i	$-0.617371\pi$
$440$		0			0
$441$	−134.836	−	25.9875i	−0.305750	−	0.0589285i
$442$		0			0
$443$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$443$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$444$	−545.260	−	629.264i	−1.22806	−	1.41726i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	370.082	+	35.3385i	0.826076	+	0.0788807i
$449$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$449$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−731.391	+	69.8394i	−1.61455	+	0.154171i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	208.234	−	40.1339i	0.455655	−	0.0878203i	0.0437408	−	0.999043i	$-0.486072\pi$
$457$	208.234	−	40.1339i	0.455655	−	0.0878203i	0.411914	+	0.911223i	$0.364860\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	417.294	−	586.008i	0.901284	−	1.26568i	−0.0622424	−	0.998061i	$-0.519825\pi$
$463$	417.294	−	586.008i	0.901284	−	1.26568i	0.963526	−	0.267615i	$-0.0862354\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$467$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$468$	−671.490			−1.43481
$469$	−177.657	+	346.278i	−0.378799	+	0.738332i
$470$		0			0
$471$	−423.819	−	404.110i	−0.899827	−	0.857984i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	344.127	−	483.258i	0.724477	−	1.01739i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$479$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$480$		0			0
$481$	1270.84	−	244.934i	2.64208	−	0.509219i
$482$		0			0
$483$		0			0
$484$	−464.395	−	136.359i	−0.959493	−	0.281733i
$485$		0			0
$486$		0			0
$487$	−176.009	−	90.7388i	−0.361414	−	0.186322i	0.267949	−	0.963433i	$-0.413654\pi$
$487$	−176.009	−	90.7388i	−0.361414	−	0.186322i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	574.390	+	229.951i	1.17462	+	0.470247i
$490$		0			0
$491$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$491$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−402.414	−	881.163i	−0.811318	−	1.77654i
$497$		0			0
$498$		0			0
$499$	385.504	+	667.713i	0.772554	+	1.33810i	0.936159	+	0.351576i	$0.114354\pi$
$499$	385.504	+	667.713i	0.772554	+	1.33810i	−0.163606	+	0.986526i	$0.552312\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$503$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	268.374	−	464.838i	0.529338	−	0.916840i
$508$	717.581	+	138.302i	1.41256	+	0.272249i
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$	124.087	+	143.204i	0.242832	+	0.280243i
$512$		0			0
$513$	503.644	−	396.070i	0.981762	−	0.772066i
$514$		0			0
$515$		0			0
$516$	65.3762	+	26.1727i	0.126698	+	0.0507222i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$521$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$522$		0			0
$523$	−341.368	+	986.317i	−0.652711	+	1.88588i	−0.264007	+	0.964521i	$0.585044\pi$
$523$	−341.368	+	986.317i	−0.652711	+	1.88588i	−0.388704	+	0.921363i	$0.627077\pi$
$524$		0			0
$525$	252.709	+	354.880i	0.481350	+	0.675962i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	124.716	−	514.088i	0.235759	−	0.971812i
$530$		0			0
$531$		0			0
$532$	−26.2361	−	550.763i	−0.0493159	−	1.03527i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−111.803	+	244.815i	−0.206661	+	0.452524i	−0.984373	−	0.176097i	$-0.943653\pi$
$541$	−111.803	+	244.815i	−0.206661	+	0.452524i	0.777712	+	0.628620i	$0.216380\pi$
$542$		0			0
$543$	95.8146	−	394.953i	0.176454	−	0.727354i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	575.954	+	808.814i	1.05293	+	1.47864i	0.868435	+	0.495803i	$0.165126\pi$
$547$	575.954	+	808.814i	1.05293	+	1.47864i	0.184497	+	0.982833i	$0.440934\pi$
$548$		0			0
$549$	−308.339	+	890.887i	−0.561638	+	1.62275i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.25923	−	110.405i	0.00951036	−	0.199647i
$554$		0			0
$555$		0			0
$556$	700.370	+	280.386i	1.25966	+	0.504291i
$557$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$557$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$558$		0			0
$559$	−86.0414	+	67.6637i	−0.153920	+	0.121044i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$563$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	153.891	+	444.637i	0.271412	+	0.784193i
$568$		0			0
$569$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$569$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$570$		0			0
$571$	436.054	+	342.917i	0.763668	+	0.600555i	0.922161	−	0.386805i	$-0.126421\pi$
$571$	436.054	+	342.917i	0.763668	+	0.600555i	−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	239.279	+	523.948i	0.415415	+	0.909632i
$577$	797.335	−	411.055i	1.38186	−	0.712400i	0.402673	−	0.915344i	$-0.368081\pi$
$577$	797.335	−	411.055i	1.38186	−	0.712400i	0.979190	+	0.202944i	$0.0650510\pi$
$578$		0			0
$579$	601.206	+	386.372i	1.03835	+	0.667309i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$587$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$588$	−175.673	−	51.5823i	−0.298764	−	0.0877250i
$589$	−1208.66	+	776.760i	−2.05206	+	1.31878i
$590$		0			0
$591$		0			0
$592$	−643.968	−	904.326i	−1.08778	−	1.52758i
$593$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$593$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	674.379	−	947.033i	1.12961	−	1.58632i
$598$		0			0
$599$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$599$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$600$		0			0
$601$	−857.989	−	818.091i	−1.42760	−	1.36122i	−0.827738	−	0.561115i	$-0.810373\pi$
$601$	−857.989	−	818.091i	−1.42760	−	1.36122i	−0.599865	−	0.800102i	$-0.704779\pi$
$602$		0			0
$603$	−602.259	+	29.8772i	−0.998772	+	0.0495476i
$604$	−979.624			−1.62189
$605$		0			0
$606$		0			0
$607$	1.54220	+	32.3748i	0.00254069	+	0.0533357i	0.999782	−	0.0208971i	$-0.00665224\pi$
$607$	1.54220	+	32.3748i	0.00254069	+	0.0533357i	−0.997241	+	0.0742328i	$0.976349\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1135.45	−	454.564i	1.85228	−	0.741540i	0.900402	−	0.435060i	$-0.143273\pi$
$613$	1135.45	−	454.564i	1.85228	−	0.741540i	0.951877	−	0.306481i	$-0.0991514\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$617$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$	−865.197	+	82.6163i	−1.39773	+	0.133467i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	−865.197	+	82.6163i	−1.39773	+	0.133467i	−0.631174	+	0.775641i	$0.717427\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−891.267	−	85.1056i	−1.42831	−	0.136387i
$625$	−88.9468	+	618.638i	−0.142315	+	0.989821i
$626$		0			0
$627$		0			0
$628$	−511.315	−	590.089i	−0.814196	−	0.939633i
$629$		0			0
$630$		0			0
$631$	−1131.19	−	218.018i	−1.79269	−	0.345512i	−0.819069	−	0.573696i	$-0.805509\pi$
$631$	−1131.19	−	218.018i	−1.79269	−	0.345512i	−0.973618	+	0.228184i	$0.926721\pi$
$632$		0			0
$633$	303.192	+	525.144i	0.478977	+	0.829612i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	205.968	−	196.390i	0.323340	−	0.308304i
$638$		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
$643$	−170.348	−	373.010i	−0.264927	−	0.580109i	0.729684	−	0.683784i	$-0.239667\pi$
$643$	−170.348	−	373.010i	−0.264927	−	0.580109i	−0.994611	+	0.103675i	$0.966940\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$647$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−248.742	−	1025.33i	−0.382092	−	1.57500i
$652$	733.241	+	378.012i	1.12460	+	0.579773i
$653$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$653$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−96.0217	+	277.436i	−0.146152	+	0.422278i
$658$		0			0
$659$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$659$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$660$		0			0
$661$	−1267.18	+	372.076i	−1.91706	+	0.562899i	−0.945908	+	0.324436i	$0.894826\pi$
$661$	−1267.18	+	372.076i	−1.91706	+	0.562899i	−0.971151	+	0.238463i	$0.923356\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1336.68			1.99803
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.9475	−	103.962i	−0.0222102	−	0.154475i	0.975698	−	0.219119i	$-0.0703181\pi$
$673$	−14.9475	−	103.962i	−0.0222102	−	0.154475i	−0.997908	+	0.0646433i	$0.979409\pi$
$674$		0			0
$675$	−280.405	+	614.002i	−0.415415	+	0.909632i
$676$	415.126	−	582.963i	0.614092	−	0.862372i
$677$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$677$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$678$		0			0
$679$	−677.121	+	198.821i	−0.997232	+	0.292814i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$683$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$684$	718.683	−	461.869i	1.05071	−	0.675248i
$685$		0			0
$686$		0			0
$687$	−60.4683	+	1269.39i	−0.0880180	+	1.84772i
$688$	83.4564	+	43.0247i	0.121303	+	0.0625360i
$689$		0			0
$690$		0			0
$691$	−546.462	−	52.1808i	−0.790827	−	0.0755148i	−0.308178	−	0.951329i	$-0.599719\pi$
$691$	−546.462	−	52.1808i	−0.790827	−	0.0755148i	−0.482649	+	0.875814i	$0.660325\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	290.442	+	503.060i	0.414917	+	0.718657i
$701$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$701$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$702$		0			0
$703$	−1191.68	+	1136.27i	−1.69514	+	1.61631i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−906.479	−	174.709i	−1.27853	−	0.246417i	−0.495614	−	0.868543i	$-0.665057\pi$
$709$	−906.479	−	174.709i	−1.27853	−	0.246417i	−0.782917	+	0.622126i	$0.786269\pi$
$710$		0			0
$711$	152.214	−	78.4719i	0.214085	−	0.110368i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$719$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$720$		0			0
$721$	−1189.24	+	113.559i	−1.64943	+	0.157501i
$722$		0			0
$723$	689.237	−	442.946i	0.953302	−	0.612650i
$724$	177.231	−	512.076i	0.244794	−	0.707287i
$725$		0			0
$726$		0			0
$727$	−981.291	+	392.850i	−1.34978	+	0.540371i	−0.930102	−	0.367301i	$-0.880282\pi$
$727$	−981.291	+	392.850i	−1.34978	+	0.540371i	−0.419679	+	0.907672i	$0.637857\pi$
$728$		0			0
$729$	−477.393	+	550.941i	−0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$	−522.170	+	1143.39i	−0.713346	+	1.56201i
$733$	−61.1380	−	1283.45i	−0.0834080	−	1.75095i	−0.522620	−	0.852566i	$-0.675045\pi$
$733$	−61.1380	−	1283.45i	−0.0834080	−	1.75095i	0.439212	−	0.898383i	$-0.355258\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1061.80	+	1012.43i	1.43681	+	1.37000i	0.801986	+	0.597342i	$0.203777\pi$
$739$	1061.80	+	1012.43i	1.43681	+	1.37000i	0.634826	+	0.772655i	$0.281072\pi$
$740$		0			0
$741$	63.1842	+	1326.40i	0.0852688	+	1.79001i
$742$		0			0
$743$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$743$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	441.027	−	283.431i	0.587253	−	0.377405i	−0.213013	−	0.977049i	$-0.568328\pi$
$751$	441.027	−	283.431i	0.587253	−	0.377405i	0.800266	+	0.599645i	$0.204691\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	147.904	+	609.670i	0.195641	+	0.806441i
$757$	502.282	+	201.084i	0.663517	+	0.265632i	0.678879	−	0.734250i	$-0.262466\pi$
$757$	502.282	+	201.084i	0.663517	+	0.265632i	−0.0153624	+	0.999882i	$0.504890\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$	1120.30	−	577.555i	1.46828	−	0.756953i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	251.188	+	725.761i	0.327068	+	0.945001i
$769$	1200.51	+	944.088i	1.56113	+	1.22768i	0.853497	+	0.521098i	$0.174477\pi$
$769$	1200.51	+	944.088i	1.56113	+	1.22768i	0.707628	+	0.706585i	$0.249765\pi$
$770$		0			0
$771$		0			0
$772$	749.009	+	589.027i	0.970219	+	0.762989i
$773$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$773$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$774$		0			0
$775$	756.799	−	1310.81i	0.976515	−	1.69137i
$776$		0			0
$777$	−502.303	−	1099.89i	−0.646464	−	1.41556i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−226.633	−	90.7301i	−0.289072	−	0.115727i
$785$		0			0
$786$		0			0
$787$	−41.6953	+	875.292i	−0.0529801	+	1.11219i	0.803266	+	0.595620i	$0.203094\pi$
$787$	−41.6953	+	875.292i	−0.0529801	+	1.11219i	−0.856246	+	0.516568i	$0.827209\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1133.33	−	1591.54i	−1.42917	−	2.00698i
$794$		0			0
$795$		0			0
$796$	1015.13	−	1171.52i	1.27529	−	1.47176i
$797$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$797$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−803.163	−	36.6753i	−0.998959	−	0.0456161i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$809$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$810$		0			0
$811$	−246.402	+	1015.68i	−0.303824	+	1.25238i	0.591259	+	0.806482i	$0.298631\pi$
$811$	−246.402	+	1015.68i	−0.303824	+	1.25238i	−0.895083	+	0.445899i	$0.852884\pi$
$812$		0			0
$813$	1467.49	−	430.894i	1.80503	−	0.530005i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	45.5474	−	131.601i	0.0557496	−	0.161078i
$818$		0			0
$819$	−935.644	−	274.730i	−1.14242	−	0.335445i
$820$		0			0
$821$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$821$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$822$		0			0
$823$	385.667	+	1589.74i	0.468611	+	1.93164i	0.342041	+	0.939685i	$0.388882\pi$
$823$	385.667	+	1589.74i	0.468611	+	1.93164i	0.126570	+	0.991958i	$0.459603\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$827$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$828$		0			0
$829$	−1056.60	−	1219.38i	−1.27455	−	1.47091i	−0.811243	−	0.584710i	$-0.801208\pi$
$829$	−1056.60	−	1219.38i	−1.27455	−	1.47091i	−0.463306	−	0.886198i	$-0.653337\pi$
$830$		0			0
$831$	665.473	+	1457.18i	0.800810	+	1.75353i
$832$	−1172.19	−	225.921i	−1.40888	−	0.271539i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1183.08	−	1128.06i	1.41347	−	1.34774i
$838$		0			0
$839$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$839$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	335.868	+	735.449i	0.397948	+	0.871385i
$845$		0			0
$846$		0			0
$847$	−591.291	−	379.999i	−0.698100	−	0.448642i
$848$		0			0
$849$	210.382	−	1463.24i	0.247800	−	1.72348i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1514.65	−	780.857i	−1.77568	−	0.915425i	−0.909540	−	0.415617i	$-0.863566\pi$
$853$	−1514.65	−	780.857i	−1.77568	−	0.915425i	−0.866137	−	0.499807i	$-0.833404\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$857$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$	1186.97	−	228.769i	1.38180	−	0.266320i	0.556422	−	0.830900i	$-0.312174\pi$
$859$	1186.97	−	228.769i	1.38180	−	0.266320i	0.825378	+	0.564580i	$0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$863$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−41.2535	−	866.018i	−0.0475819	−	0.998867i
$868$	−200.202	−	1392.44i	−0.230648	−	1.60419i
$869$		0			0
$870$		0			0
$871$	673.578	−	1052.66i	0.773339	−	1.20856i
$872$		0			0
$873$	−791.329	−	754.530i	−0.906448	−	0.864296i
$874$		0			0
$875$		0			0
$876$	−162.612	+	356.070i	−0.185630	+	0.406473i
$877$	847.723	−	1190.46i	0.966617	−	1.35742i	0.0328819	−	0.999459i	$-0.489531\pi$
$877$	847.723	−	1190.46i	0.966617	−	1.35742i	0.933735	−	0.357964i	$-0.116529\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$881$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$882$		0			0
$883$	695.738	−	134.093i	0.787925	−	0.151860i	0.220611	−	0.975362i	$-0.429195\pi$
$883$	695.738	−	134.093i	0.787925	−	0.151860i	0.567314	+	0.823502i	$0.307983\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$887$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$888$		0			0
$889$	943.281	+	486.295i	1.06106	+	0.547013i
$890$		0			0
$891$		0			0
$892$	1774.17	+	169.413i	1.98898	+	0.189925i
$893$		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
$903$	80.3859	+	63.2162i	0.0890210	+	0.0700069i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	564.993	+	1632.44i	0.622925	+	1.79982i	0.599880	+	0.800090i	$0.295215\pi$
$907$	564.993	+	1632.44i	0.622925	+	1.79982i	0.0230449	+	0.999734i	$0.492664\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$911$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$912$	1012.44	−	521.950i	1.11013	−	0.572314i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−241.143	+	1677.19i	−0.263257	+	1.83099i
$917$		0			0
$918$		0			0
$919$	−262.858	−	1083.52i	−0.286027	−	1.17902i	−0.915604	−	0.402080i	$-0.868287\pi$
$919$	−262.858	−	1083.52i	−0.286027	−	1.17902i	0.629578	−	0.776937i	$-0.283228\pi$
$920$		0			0
$921$	−78.0506	+	1638.48i	−0.0847455	+	1.77903i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	567.350	−	1639.25i	0.613352	−	1.77216i
$926$		0			0
$927$	−1073.65	−	1507.74i	−1.15820	−	1.62647i
$928$		0			0
$929$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$929$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$930$		0			0
$931$	−85.3608	+	351.862i	−0.0916872	+	0.377940i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1725.16			1.84115			0.920576	−	0.390563i	$-0.127720\pi$
$937$	1725.16			1.84115			0.920576	+	0.390563i	$0.127720\pi$
$938$		0			0
$939$	106.558			0.113480
$940$		0			0
$941$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$941$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$947$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$948$	211.979	−	84.8636i	0.223607	−	0.0895186i
$949$	−352.937	−	495.630i	−0.371904	−	0.522266i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$953$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2125.93	+	1671.85i	−2.21221	+	1.73970i
$962$		0			0
$963$		0			0
$964$	970.962	−	500.566i	1.00722	−	0.519259i
$965$		0			0
$966$		0			0
$967$	−966.935	+	1674.78i	−0.999933	+	1.73193i	−0.489915	+	0.871770i	$0.662972\pi$
$967$	−966.935	+	1674.78i	−0.999933	+	1.73193i	−0.510018	+	0.860164i	$0.670361\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$971$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$972$	−703.469	+	670.757i	−0.723734	+	0.690079i
$973$	861.168	+	677.230i	0.885065	+	0.696023i
$974$		0			0
$975$	−699.469	−	1211.52i	−0.717404	−	1.24258i
$976$	−837.989	+	1451.44i	−0.858595	+	1.48713i
$977$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$977$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1642.83	+	1055.79i	1.67465	+	1.07623i
$982$		0			0
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−84.2456	+	1768.53i	−0.0852688	+	1.79001i
$989$		0			0
$990$		0			0
$991$	1500.20	−	964.123i	1.51383	−	0.972878i	0.520971	−	0.853575i	$-0.325570\pi$
$991$	1500.20	−	964.123i	1.51383	−	0.972878i	0.992858	−	0.119304i	$-0.0380663\pi$
$992$		0			0
$993$	−383.167	+	73.8494i	−0.385868	+	0.0743700i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−580.689	+	670.150i	−0.582436	+	0.672167i	−0.968127	−	0.250461i	$-0.919418\pi$
$997$	−580.689	+	670.150i	−0.582436	+	0.672167i	0.385691	+	0.922628i	$0.373963\pi$
$998$		0			0
$999$	1086.70	−	1526.05i	1.08778	−	1.52758i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.77.1	yes	20
3.2	odd	2	CM	201.3.o.a.77.1	yes	20
67.47	even	33	inner	201.3.o.a.47.1	✓	20
201.47	odd	66	inner	201.3.o.a.47.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.47.1	✓	20	67.47	even	33	inner
201.3.o.a.47.1	✓	20	201.47	odd	66	inner
201.3.o.a.77.1	yes	20	1.1	even	1	trivial
201.3.o.a.77.1	yes	20	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.426945 + 2.96946i) q^{3} +(0.190328 + 3.99547i) q^{4} +(-1.36948 + 5.64509i) q^{7} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)	\(q+(0.426945 + 2.96946i) q^{3} +(0.190328 + 3.99547i) q^{4} +(-1.36948 + 5.64509i) q^{7} +(-8.63544 + 2.53559i) q^{9} +(-11.7831 + 2.27101i) q^{12} +(6.10064 - 17.6266i) q^{13} +(-15.9276 + 1.52090i) q^{16} +(5.59469 + 23.0616i) q^{19} +(-17.3476 - 1.65649i) q^{21} +(-16.3715 - 18.8937i) q^{25} +(-11.2162 - 24.5601i) q^{27} +(-22.8154 - 4.39731i) q^{28} +(19.8020 + 57.2140i) q^{31} +(-11.7744 - 34.0200i) q^{36} +(34.6931 + 60.0903i) q^{37} +(54.9463 + 10.5900i) q^{39} +(-4.93679 - 3.17268i) q^{43} +(-11.3164 - 46.6470i) q^{48} +(13.5614 + 6.99138i) q^{49} +(71.5878 + 21.0201i) q^{52} +(-66.0920 + 26.4593i) q^{57} +(60.7601 - 85.3257i) q^{61} +(-2.48756 - 52.2203i) q^{63} +(-9.10815 - 63.3486i) q^{64} +(65.1423 + 15.6677i) q^{67} +(18.9217 - 26.5718i) q^{73} +(49.1146 - 56.6812i) q^{75} +(-91.0772 + 26.7427i) q^{76} +(-18.6841 + 3.60106i) q^{79} +(68.1415 - 43.7919i) q^{81} +(3.31674 - 69.6270i) q^{84} +(91.1492 + 58.5781i) q^{91} +(-161.441 + 83.2285i) q^{93} +(60.7443 + 105.212i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more