LMFDB - Embedded newform 201.3.o.a.26.1

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			26.1
Root			$0.235759 - 0.971812i$ of defining polynomial
Character	$\chi$	$=$	201.26
Dual form			201.3.o.a.116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.52376 + 1.62192i) q^{3} +(-1.30827 + 3.78000i) q^{4} +(-9.12076 + 0.870927i) q^{7} +(3.73874 - 8.18669i) q^{9} +O(q^{10})$	$q+(-2.52376 + 1.62192i) q^{3} +(-1.30827 + 3.78000i) q^{4} +(-9.12076 + 0.870927i) q^{7} +(3.73874 - 8.18669i) q^{9} +(-2.82911 - 11.6617i) q^{12} +(14.8968 - 14.2041i) q^{13} +(-12.5768 - 9.89054i) q^{16} +(-6.85872 - 0.654928i) q^{19} +(21.6060 - 16.9912i) q^{21} +(-23.9873 - 7.04331i) q^{25} +(3.84250 + 26.7252i) q^{27} +(8.64033 - 35.6159i) q^{28} +(-42.0885 - 40.1313i) q^{31} +(26.0544 + 24.8428i) q^{36} +(2.71391 - 4.70062i) q^{37} +(-14.5581 + 60.0091i) q^{39} +(-49.5128 + 57.1409i) q^{43} +(47.7827 + 4.56269i) q^{48} +(34.3153 - 6.61374i) q^{49} +(34.2024 + 74.8927i) q^{52} +(18.3720 - 9.47142i) q^{57} +(-19.8004 - 7.92690i) q^{61} +(-26.9701 + 77.9250i) q^{63} +(53.8402 - 34.6010i) q^{64} +(41.3129 + 52.7470i) q^{67} +(-45.9507 - 18.3959i) q^{73} +(71.9620 - 21.1299i) q^{75} +(11.4487 - 25.0691i) q^{76} +(-35.5007 - 146.336i) q^{79} +(-53.0437 - 61.2157i) q^{81} +(35.9601 + 103.900i) q^{84} +(-123.499 + 142.526i) q^{91} +(171.311 + 33.0175i) q^{93} +(-96.9326 + 167.892i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{10}{33}\right)$

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		−0.580057	−	0.814576i	$-0.696970\pi$
$2$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$3$	−2.52376	+	1.62192i	−0.841254	+	0.540641i
$3$	−2.52376	+	1.62192i	−0.841254	+	0.540641i
$4$	−1.30827	+	3.78000i	−0.327068	+	0.945001i
$5$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$5$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$6$		0			0
$7$	−9.12076	+	0.870927i	−1.30297	+	0.124418i	−0.723356	−	0.690475i	$-0.757401\pi$
$7$	−9.12076	+	0.870927i	−1.30297	+	0.124418i	−0.579611	+	0.814894i	$0.696795\pi$
$8$		0			0
$9$	3.73874	−	8.18669i	0.415415	−	0.909632i
$10$		0			0
$11$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$11$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$12$	−2.82911	−	11.6617i	−0.235759	−	0.971812i
$13$	14.8968	−	14.2041i	1.14591	−	1.09262i	0.151345	−	0.988481i	$-0.451640\pi$
$13$	14.8968	−	14.2041i	1.14591	−	1.09262i	0.994563	−	0.104139i	$-0.0332088\pi$
$14$		0			0
$15$		0			0
$16$	−12.5768	−	9.89054i	−0.786053	−	0.618159i
$17$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$17$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$18$		0			0
$19$	−6.85872	−	0.654928i	−0.360985	−	0.0344699i	−0.0870120	−	0.996207i	$-0.527732\pi$
$19$	−6.85872	−	0.654928i	−0.360985	−	0.0344699i	−0.273973	+	0.961737i	$0.588338\pi$
$20$		0			0
$21$	21.6060	−	16.9912i	1.02886	−	0.809104i
$22$		0			0
$23$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$23$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$24$		0			0
$25$	−23.9873	−	7.04331i	−0.959493	−	0.281733i
$26$		0			0
$27$	3.84250	+	26.7252i	0.142315	+	0.989821i
$28$	8.64033	−	35.6159i	0.308583	−	1.27200i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−42.0885	−	40.1313i	−1.35769	−	1.29456i	−0.918093	−	0.396364i	$-0.870272\pi$
$31$	−42.0885	−	40.1313i	−1.35769	−	1.29456i	−0.439600	−	0.898194i	$-0.644880\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	26.0544	+	24.8428i	0.723734	+	0.690079i
$37$	2.71391	−	4.70062i	0.0733488	−	0.127044i	−0.827018	−	0.562175i	$-0.809965\pi$
$37$	2.71391	−	4.70062i	0.0733488	−	0.127044i	0.900367	+	0.435131i	$0.143298\pi$
$38$		0			0
$39$	−14.5581	+	60.0091i	−0.373283	+	1.53870i
$40$		0			0
$41$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$41$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$42$		0			0
$43$	−49.5128	+	57.1409i	−1.15146	+	1.32886i	−0.215603	+	0.976481i	$0.569172\pi$
$43$	−49.5128	+	57.1409i	−1.15146	+	1.32886i	−0.935858	+	0.352376i	$0.885374\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$47$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$48$	47.7827	+	4.56269i	0.995472	+	0.0950560i
$49$	34.3153	−	6.61374i	0.700313	−	0.134974i
$50$		0			0
$51$		0			0
$52$	34.2024	+	74.8927i	0.657738	+	1.44024i
$53$		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
$57$	18.3720	−	9.47142i	0.322316	−	0.166165i
$58$		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
$61$	−19.8004	−	7.92690i	−0.324597	−	0.129949i	0.203640	−	0.979046i	$-0.434723\pi$
$61$	−19.8004	−	7.92690i	−0.324597	−	0.129949i	−0.528238	+	0.849097i	$0.677147\pi$
$62$		0			0
$63$	−26.9701	+	77.9250i	−0.428097	+	1.23691i
$64$	53.8402	−	34.6010i	0.841254	−	0.540641i
$65$		0			0
$66$		0			0
$67$	41.3129	+	52.7470i	0.616611	+	0.787268i
$67$	41.3129	+	52.7470i	0.616611	+	0.787268i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$71$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$72$		0			0
$73$	−45.9507	−	18.3959i	−0.629461	−	0.251998i	0.0349224	−	0.999390i	$-0.488882\pi$
$73$	−45.9507	−	18.3959i	−0.629461	−	0.251998i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	71.9620	−	21.1299i	0.959493	−	0.281733i
$76$	11.4487	−	25.0691i	0.150641	−	0.329857i
$77$		0			0
$78$		0			0
$79$	−35.5007	−	146.336i	−0.449375	−	1.85235i	−0.518996	−	0.854777i	$-0.673694\pi$
$79$	−35.5007	−	146.336i	−0.449375	−	1.85235i	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		−0.786053	−	0.618159i	$-0.787879\pi$
$83$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$84$	35.9601	+	103.900i	0.428097	+	1.23691i
$85$		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
$91$	−123.499	+	142.526i	−1.35714	+	1.56622i
$92$		0			0
$93$	171.311	+	33.0175i	1.84206	+	0.355027i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−96.9326	+	167.892i	−0.999305	+	1.73085i	−0.467365	+	0.884064i	$0.654797\pi$
$97$	−96.9326	+	167.892i	−0.999305	+	1.73085i	−0.531939	+	0.846782i	$0.678537\pi$
$98$		0			0
$99$		0			0
$100$	58.0057	−	81.4576i	0.580057	−	0.814576i
$101$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$101$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$102$		0			0
$103$	12.7155	+	12.1242i	0.123452	+	0.117711i	0.749291	−	0.662241i	$-0.230395\pi$
$103$	12.7155	+	12.1242i	0.123452	+	0.117711i	−0.625839	+	0.779953i	$0.715243\pi$
$104$		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	$0.454545\pi$
$108$	−106.048	−	20.4391i	−0.981929	−	0.189251i
$109$	−186.057	−	54.6313i	−1.70694	−	0.501204i	−0.724741	−	0.689022i	$-0.758040\pi$
$109$	−186.057	−	54.6313i	−1.70694	−	0.501204i	−0.982204	+	0.187818i	$0.939859\pi$
$110$		0			0
$111$	0.774797	+	16.2650i	0.00698016	+	0.146531i
$112$	123.324	+	79.2558i	1.10111	+	0.707641i
$113$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$113$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−60.5891	−	175.061i	−0.517855	−	1.49625i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	87.5718	−	83.4996i	0.723734	−	0.690079i
$122$		0			0
$123$		0			0
$124$	206.760	−	106.592i	1.66742	−	0.859613i
$125$		0			0
$126$		0			0
$127$	−235.338	+	22.4720i	−1.85305	+	0.176945i	−0.961286	−	0.275552i	$-0.911139\pi$
$127$	−235.338	+	22.4720i	−1.85305	+	0.176945i	−0.891766	+	0.452497i	$0.850533\pi$
$128$		0			0
$129$	32.2805	−	224.516i	0.250236	−	1.74043i
$130$		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
$133$	63.1271			0.474640
$134$		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
$139$	39.5437	−	275.033i	0.284487	−	1.97865i	0.110951	−	0.993826i	$-0.464610\pi$
$139$	39.5437	−	275.033i	0.284487	−	1.97865i	0.173536	−	0.984827i	$-0.444481\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−127.992	+	65.9846i	−0.888835	+	0.458227i
$145$		0			0
$146$		0			0
$147$	−75.8767	+	72.3483i	−0.516168	+	0.492165i
$148$	14.2178	+	16.4083i	0.0960665	+	0.110867i
$149$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$149$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$150$		0			0
$151$	98.6315	+	284.977i	0.653188	+	1.88726i	0.377438	+	0.926035i	$0.376805\pi$
$151$	98.6315	+	284.977i	0.653188	+	1.88726i	0.275750	+	0.961229i	$0.411074\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−207.789	−	133.538i	−1.33198	−	0.856011i
$157$	10.2620	+	215.426i	0.0653630	+	1.37214i	0.758661	+	0.651486i	$0.225854\pi$
$157$	10.2620	+	215.426i	0.0653630	+	1.37214i	−0.693298	+	0.720651i	$0.743843\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−134.506	−	232.971i	−0.825191	−	1.42927i	−0.901773	−	0.432209i	$-0.857734\pi$
$163$	−134.506	−	232.971i	−0.825191	−	1.42927i	0.0765825	−	0.997063i	$-0.475599\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$167$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$168$		0			0
$169$	12.1177	−	254.382i	0.0717023	−	1.50522i
$170$		0			0
$171$	−31.0046	+	53.7016i	−0.181314	+	0.314044i
$172$	−151.216	−	261.914i	−0.879165	−	1.52276i
$173$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$173$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$174$		0			0
$175$	224.917	+	43.3492i	1.28524	+	0.247710i
$176$		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
$181$	146.648	+	75.6021i	0.810208	+	0.417691i	0.812970	−	0.582305i	$-0.197849\pi$
$181$	146.648	+	75.6021i	0.810208	+	0.417691i	−0.00276243	+	0.999996i	$0.500879\pi$
$182$		0			0
$183$	62.8284	−	12.1092i	0.343324	−	0.0661703i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−58.3222	−	240.408i	−0.308583	−	1.27200i
$190$		0			0
$191$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$191$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$192$	−79.7597	+	174.649i	−0.415415	+	0.909632i
$193$	255.926	−	75.1466i	1.32604	−	0.389360i	0.459371	−	0.888244i	$-0.348075\pi$
$193$	255.926	−	75.1466i	1.32604	−	0.389360i	0.866669	+	0.498884i	$0.166257\pi$
$194$		0			0
$195$		0			0
$196$	−19.8938	+	138.365i	−0.101499	+	0.705942i
$197$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$197$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$198$		0			0
$199$	230.504	+	323.697i	1.15831	+	1.62662i	0.624572	+	0.780967i	$0.285273\pi$
$199$	230.504	+	323.697i	1.15831	+	1.62662i	0.533737	+	0.845651i	$0.320787\pi$
$200$		0			0
$201$	−189.815	−	66.1143i	−0.944355	−	0.328927i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−327.841	+	31.3050i	−1.57616	+	0.150505i
$209$		0			0
$210$		0			0
$211$	−329.152	+	169.689i	−1.55996	+	0.804215i	−0.999709	−	0.0241079i	$-0.992325\pi$
$211$	−329.152	+	169.689i	−1.55996	+	0.804215i	−0.560251	+	0.828323i	$0.689295\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	418.831	+	329.372i	1.93010	+	1.51784i
$218$		0			0
$219$	145.805	−	28.1016i	0.665777	−	0.128318i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	258.044	+	165.835i	1.15715	+	0.743654i	0.971050	−	0.238877i	$-0.0767792\pi$
$223$	258.044	+	165.835i	1.15715	+	0.743654i	0.186099	+	0.982531i	$0.440416\pi$
$224$		0			0
$225$	−147.344	+	170.044i	−0.654861	+	0.755750i
$226$		0			0
$227$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$227$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$228$	11.7664	+	81.8374i	0.0516072	+	0.358936i
$229$	78.8292	−	324.938i	0.344232	−	1.41894i	−0.492012	−	0.870588i	$-0.663738\pi$
$229$	78.8292	−	324.938i	0.344232	−	1.41894i	0.836245	−	0.548357i	$-0.184746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$233$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	326.940	+	311.737i	1.37949	+	1.31535i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	67.5603	+	469.892i	0.280333	+	1.94976i	0.311499	+	0.950247i	$0.399169\pi$
$241$	67.5603	+	469.892i	0.280333	+	1.94976i	−0.0311655	+	0.999514i	$0.509922\pi$
$242$		0			0
$243$	233.157	+	68.4610i	0.959493	+	0.281733i
$244$	55.8680	−	64.4752i	0.228967	−	0.264242i
$245$		0			0
$246$		0			0
$247$	−111.476	+	87.6653i	−0.451318	+	0.354920i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$251$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$252$	−259.273	−	203.894i	−1.02886	−	0.809104i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	60.3543	+	248.784i	0.235759	+	0.971812i
$257$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$257$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$258$		0			0
$259$	−20.6590	+	45.2369i	−0.0797645	+	0.174660i
$260$		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
$268$	−253.432	+	87.1556i	−0.945643	+	0.325208i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−214.246	+	137.687i	−0.790575	+	0.508072i	−0.872528	−	0.488564i	$-0.837521\pi$
$271$	−214.246	+	137.687i	−0.790575	+	0.508072i	0.0819526	+	0.996636i	$0.473884\pi$
$272$		0			0
$273$	80.5170	−	560.008i	0.294934	−	2.05131i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	195.567	−	428.233i	0.706020	−	1.54597i	−0.126497	−	0.991967i	$-0.540373\pi$
$277$	195.567	−	428.233i	0.706020	−	1.54597i	0.832517	−	0.554000i	$-0.186899\pi$
$278$		0			0
$279$	−485.900	+	194.525i	−1.74158	+	0.697222i
$280$		0			0
$281$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$281$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$282$		0			0
$283$	109.665	+	240.133i	0.387508	+	0.848525i	0.998386	+	0.0567995i	$0.0180896\pi$
$283$	109.665	+	240.133i	0.387508	+	0.848525i	−0.610877	+	0.791725i	$0.709183\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−227.169	+	178.648i	−0.786053	+	0.618159i
$290$		0			0
$291$	−27.6734	−	580.937i	−0.0950977	−	1.99635i
$292$	129.652	−	149.627i	0.444015	−	0.512421i
$293$		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
$300$	−14.2746	+	299.660i	−0.0475819	+	0.998867i
$301$	401.829	−	564.290i	1.33498	−	1.87472i
$302$		0			0
$303$		0			0
$304$	79.7834	+	76.0734i	0.262446	+	0.250241i
$305$		0			0
$306$		0			0
$307$	−105.133	+	433.365i	−0.342453	+	1.41161i	0.496798	+	0.867866i	$0.334509\pi$
$307$	−105.133	+	433.365i	−0.342453	+	1.41161i	−0.839251	+	0.543745i	$0.817006\pi$
$308$		0			0
$309$	−51.7556	−	9.97507i	−0.167494	−	0.0322818i
$310$		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
$313$	−516.172	−	331.724i	−1.64911	−	1.05982i	−0.931737	−	0.363133i	$-0.881707\pi$
$313$	−516.172	−	331.724i	−1.64911	−	1.05982i	−0.717375	−	0.696687i	$-0.754657\pi$
$314$		0			0
$315$		0			0
$316$	599.594	+	57.2543i	1.89745	+	0.181184i
$317$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$317$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	300.791	−	120.419i	0.928368	−	0.371662i
$325$	−457.378	+	235.795i	−1.40732	+	0.725522i
$326$		0			0
$327$	558.171	−	163.894i	1.70694	−	0.501204i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	188.304	−	544.068i	0.568894	−	1.64371i	−0.182007	−	0.983297i	$-0.558259\pi$
$331$	188.304	−	544.068i	0.568894	−	1.64371i	0.750901	−	0.660414i	$-0.229619\pi$
$332$		0			0
$333$	−28.3360	−	39.7923i	−0.0850930	−	0.119496i
$334$		0			0
$335$		0			0
$336$	−439.788			−1.30889
$337$	−199.657	−	280.379i	−0.592453	−	0.831984i	0.404119	−	0.914706i	$-0.367578\pi$
$337$	−199.657	−	280.379i	−0.592453	−	0.831984i	−0.996573	+	0.0827222i	$0.973639\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	123.543	−	36.2754i	0.360183	−	0.105759i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$347$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$348$		0			0
$349$	−297.063	−	342.829i	−0.851182	−	0.982317i	0.148796	−	0.988868i	$-0.452460\pi$
$349$	−297.063	−	342.829i	−0.851182	−	0.982317i	−0.999979	+	0.00655113i	$0.997915\pi$
$350$		0			0
$351$	436.847	+	343.540i	1.24458	+	0.978748i
$352$		0			0
$353$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$353$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$354$		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
$361$	−307.863	−	59.3358i	−0.852807	−	0.164365i
$362$		0			0
$363$	−85.5805	+	352.768i	−0.235759	+	0.971812i
$364$	−377.178	−	653.291i	−1.03620	−	1.79476i
$365$		0			0
$366$		0			0
$367$	−2.31662	+	48.6319i	−0.00631232	+	0.132512i	0.993511	+	0.113733i	$0.0362809\pi$
$367$	−2.31662	+	48.6319i	−0.00631232	+	0.132512i	−0.999824	+	0.0187788i	$0.994022\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−348.928	+	604.361i	−0.937978	+	1.62463i
$373$	−95.2125	−	164.913i	−0.255261	−	0.442125i	0.709705	−	0.704499i	$-0.248828\pi$
$373$	−95.2125	−	164.913i	−0.255261	−	0.442125i	−0.964966	+	0.262373i	$0.915495\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−30.9699	−	650.138i	−0.0817148	−	1.71540i	−0.552797	−	0.833316i	$-0.686440\pi$
$379$	−30.9699	−	650.138i	−0.0817148	−	1.71540i	0.471083	−	0.882089i	$-0.343863\pi$
$380$		0			0
$381$	557.488	−	438.413i	1.46322	−	1.15069i
$382$		0			0
$383$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$383$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	282.679	+	618.981i	0.730437	+	1.59943i
$388$	−507.819	−	586.054i	−1.30881	−	1.51045i
$389$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$389$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−69.6201	+	484.218i	−0.175365	+	1.21969i	0.691954	+	0.721942i	$0.256750\pi$
$397$	−69.6201	+	484.218i	−0.175365	+	1.21969i	−0.867319	+	0.497752i	$0.834159\pi$
$398$		0			0
$399$	−159.318	+	102.387i	−0.399293	+	0.256610i
$400$	232.023	+	325.830i	0.580057	+	0.814576i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−1197.01			−2.97025
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	607.510	−	58.0101i	1.48535	−	0.141834i	0.679381	−	0.733786i	$-0.262249\pi$
$409$	607.510	−	58.0101i	1.48535	−	0.141834i	0.805973	+	0.591952i	$0.201642\pi$
$410$		0			0
$411$		0			0
$412$	−62.4651	+	32.2030i	−0.151614	+	0.0781626i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	346.283	+	758.254i	0.830415	+	1.81835i
$418$		0			0
$419$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$419$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$420$		0			0
$421$	−653.247	−	62.3775i	−1.55166	−	0.148165i	−0.716267	−	0.697826i	$-0.754151\pi$
$421$	−653.247	−	62.3775i	−1.55166	−	0.148165i	−0.835388	+	0.549661i	$0.814757\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	187.499	+	55.0546i	0.439107	+	0.128934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	216.000	−	374.123i	0.500000	−	0.866025i
$433$	581.636	+	554.589i	1.34327	+	1.28081i	0.928721	+	0.370779i	$0.120909\pi$
$433$	581.636	+	554.589i	1.34327	+	1.28081i	0.414550	+	0.910027i	$0.363939\pi$
$434$		0			0
$435$		0			0
$436$	449.919	−	631.823i	1.03193	−	1.44914i
$437$		0			0
$438$		0			0
$439$	397.099	−	687.795i	0.904553	−	1.56673i	0.0830367	−	0.996546i	$-0.473538\pi$
$439$	397.099	−	687.795i	0.904553	−	1.56673i	0.821516	−	0.570185i	$-0.193129\pi$
$440$		0			0
$441$	74.1513	−	305.656i	0.168144	−	0.693097i
$442$		0			0
$443$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$443$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$444$	−62.4954	−	18.3503i	−0.140755	−	0.0413295i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−460.929	+	362.479i	−1.02886	+	0.809104i
$449$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$449$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−711.133	−	559.241i	−1.56983	−	1.23453i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	200.355	+	825.873i	0.438413	+	1.80716i	0.576920	+	0.816800i	$0.304254\pi$
$457$	200.355	+	825.873i	0.438413	+	1.80716i	−0.138508	+	0.990361i	$0.544231\pi$
$458$		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
$463$	440.719	+	176.437i	0.951877	+	0.381074i	0.795044	−	0.606551i	$-0.207448\pi$
$463$	440.719	+	176.437i	0.951877	+	0.381074i	0.156832	+	0.987625i	$0.449872\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$467$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$468$	740.997			1.58333
$469$	−422.744	−	445.112i	−0.901374	−	0.949066i
$470$		0			0
$471$	−375.302	−	527.038i	−0.796820	−	1.11898i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	159.909	+	64.0181i	0.336651	+	0.134775i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$479$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$480$		0			0
$481$	−26.3395	−	108.573i	−0.0547598	−	0.225723i
$482$		0			0
$483$		0			0
$484$	201.061	+	440.262i	0.415415	+	0.909632i
$485$		0			0
$486$		0			0
$487$	−450.388	+	86.8052i	−0.924821	+	0.178245i	−0.629363	−	0.777111i	$-0.716684\pi$
$487$	−450.388	+	86.8052i	−0.924821	+	0.178245i	−0.295458	+	0.955356i	$0.595472\pi$
$488$		0			0
$489$	717.323	+	369.806i	1.46692	+	0.756249i
$490$		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
$496$	132.420	+	921.003i	0.266976	+	1.85686i
$497$		0			0
$498$		0			0
$499$	−128.063	+	221.812i	−0.256640	+	0.444514i	−0.965340	−	0.260997i	$-0.915949\pi$
$499$	−128.063	+	221.812i	−0.256640	+	0.444514i	0.708700	+	0.705510i	$0.249282\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$503$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	382.005	+	661.652i	0.753462	+	1.30503i
$508$	222.941	−	918.976i	0.438861	−	1.80901i
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$	435.127	+	127.765i	0.851520	+	0.250029i
$512$		0			0
$513$	−8.85155	−	185.817i	−0.0172545	−	0.362216i
$514$		0			0
$515$		0			0
$516$	806.439	+	415.748i	1.56287	+	0.805714i
$517$		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
$523$	−58.0634	+	55.3633i	−0.111020	+	0.105857i	−0.743556	−	0.668673i	$-0.766862\pi$
$523$	−58.0634	+	55.3633i	−0.111020	+	0.105857i	0.632536	+	0.774531i	$0.282014\pi$
$524$		0			0
$525$	−637.946	+	255.395i	−1.21513	+	0.486466i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−526.605	+	50.2846i	−0.995472	+	0.0950560i
$530$		0			0
$531$		0			0
$532$	−82.5874	+	238.621i	−0.155240	+	0.448535i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	5.26836	−	36.6422i	0.00973818	−	0.0677305i	−0.984373	−	0.176097i	$-0.943653\pi$
$541$	5.26836	−	36.6422i	0.00973818	−	0.0677305i	0.994111	+	0.108366i	$0.0345619\pi$
$542$		0			0
$543$	−492.724	+	47.0495i	−0.907411	+	0.0866473i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1004.59	+	402.175i	−1.83654	+	0.735238i	−0.863595	+	0.504185i	$0.831793\pi$
$547$	−1004.59	+	402.175i	−1.83654	+	0.735238i	−0.972941	+	0.231053i	$0.925783\pi$
$548$		0			0
$549$	−138.924	+	132.463i	−0.253048	+	0.241281i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	451.241	+	1303.77i	0.815987	+	2.35764i
$554$		0			0
$555$		0			0
$556$	987.891	+	509.293i	1.77678	+	0.915995i
$557$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$557$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$558$		0			0
$559$	74.0499	+	1554.50i	0.132469	+	2.78086i
$560$		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
$567$	537.114	+	512.137i	0.947291	+	0.903240i
$568$		0			0
$569$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$569$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$570$		0			0
$571$	53.1811	−	1116.41i	0.0931369	−	1.95518i	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	53.1811	−	1116.41i	0.0931369	−	1.95518i	0.251631	−	0.967823i	$-0.419033\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−81.9733	−	570.137i	−0.142315	−	0.989821i
$577$	−355.627	−	68.5415i	−0.616338	−	0.118789i	−0.128479	−	0.991712i	$-0.541010\pi$
$577$	−355.627	−	68.5415i	−0.616338	−	0.118789i	−0.487858	+	0.872923i	$0.662222\pi$
$578$		0			0
$579$	−524.013	+	604.744i	−0.905032	+	1.04446i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$587$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$588$	−174.209	−	381.465i	−0.296274	−	0.648751i
$589$	262.390	+	302.814i	0.445484	+	0.514116i
$590$		0			0
$591$		0			0
$592$	−80.6241	+	32.2770i	−0.136189	+	0.0545220i
$593$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$593$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1106.75	−	443.075i	−1.85385	−	0.742169i
$598$		0			0
$599$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$599$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$600$		0			0
$601$	−516.259	−	724.984i	−0.859000	−	1.20630i	−0.976991	−	0.213282i	$-0.931585\pi$
$601$	−516.259	−	724.984i	−0.859000	−	1.20630i	0.117991	−	0.993015i	$-0.462355\pi$
$602$		0			0
$603$	586.281	−	141.009i	0.972274	−	0.233846i
$604$	−1206.25			−1.99710
$605$		0			0
$606$		0			0
$607$	394.388	−	1139.51i	0.649732	−	1.87728i	0.215399	−	0.976526i	$-0.430895\pi$
$607$	394.388	−	1139.51i	0.649732	−	1.87728i	0.434333	−	0.900752i	$-0.356984\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1064.32	−	548.696i	1.73625	−	0.895099i	0.769785	−	0.638304i	$-0.220364\pi$
$613$	1064.32	−	548.696i	1.73625	−	0.895099i	0.966466	−	0.256796i	$-0.0826667\pi$
$614$		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
$619$	−200.067	−	157.334i	−0.323210	−	0.254175i	0.443349	−	0.896349i	$-0.353790\pi$
$619$	−200.067	−	157.334i	−0.323210	−	0.254175i	−0.766559	+	0.642174i	$0.778033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	776.617	−	610.739i	1.24458	−	0.978748i
$625$	525.783	+	337.901i	0.841254	+	0.540641i
$626$		0			0
$627$		0			0
$628$	−827.735	−	243.045i	−1.31805	−	0.387014i
$629$		0			0
$630$		0			0
$631$	269.670	−	1111.59i	0.427369	−	1.76164i	−0.196893	−	0.980425i	$-0.563085\pi$
$631$	269.670	−	1111.59i	0.427369	−	1.76164i	0.624262	−	0.781215i	$-0.285400\pi$
$632$		0			0
$633$	555.477	−	962.114i	0.877530	−	1.51993i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	417.246	−	585.941i	0.655018	−	0.919844i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−104.865	−	729.351i	−0.163087	−	1.13429i	−0.892771	−	0.450510i	$-0.851242\pi$
$643$	−104.865	−	729.351i	−0.163087	−	1.13429i	0.729684	−	0.683784i	$-0.239667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$647$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1591.24	−	151.945i	−2.44431	−	0.233403i
$652$	1056.60	−	203.644i	1.62056	−	0.312337i
$653$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$653$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−322.399	+	307.406i	−0.490713	+	0.467894i
$658$		0			0
$659$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$659$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$660$		0			0
$661$	−533.336	+	1167.84i	−0.806862	+	1.76678i	−0.186517	+	0.982452i	$0.559720\pi$
$661$	−533.336	+	1167.84i	−0.806862	+	1.76678i	−0.620345	+	0.784329i	$0.713007\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−920.213			−1.37551
$670$		0			0
$671$		0			0
$672$		0			0
$673$	684.647	−	439.996i	1.01731	−	0.653783i	0.0780319	−	0.996951i	$-0.475136\pi$
$673$	684.647	−	439.996i	1.01731	−	0.653783i	0.939274	+	0.343168i	$0.111500\pi$
$674$		0			0
$675$	96.0625	−	668.129i	0.142315	−	0.989821i
$676$	945.710	+	378.605i	1.39898	+	0.560067i
$677$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$677$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$678$		0			0
$679$	737.877	−	1615.73i	1.08671	−	2.37957i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$683$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$684$	−162.430	−	187.454i	−0.237470	−	0.274055i
$685$		0			0
$686$		0			0
$687$	328.079	+	947.922i	0.477553	+	1.37980i
$688$	1187.87	−	228.943i	1.72655	−	0.332766i
$689$		0			0
$690$		0			0
$691$	−1086.11	+	854.126i	−1.57179	+	1.23607i	−0.773583	+	0.633695i	$0.781537\pi$
$691$	−1086.11	+	854.126i	−1.57179	+	1.23607i	−0.798211	+	0.602378i	$0.794220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−458.113	+	793.474i	−0.654447	+	1.13353i
$701$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$701$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$702$		0			0
$703$	−21.6925	+	30.4628i	−0.0308570	+	0.0433326i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−49.2495	+	203.009i	−0.0694633	+	0.286332i	−0.995940	−	0.0900207i	$-0.971307\pi$
$709$	−49.2495	+	203.009i	−0.0694633	+	0.286332i	0.926477	+	0.376352i	$0.122822\pi$
$710$		0			0
$711$	−1330.73	−	256.478i	−1.87163	−	0.360728i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$719$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$720$		0			0
$721$	−126.535	−	99.5081i	−0.175499	−	0.138014i
$722$		0			0
$723$	−932.635	−	1076.32i	−1.28995	−	1.48868i
$724$	−477.631	+	455.420i	−0.659712	+	0.629034i
$725$		0			0
$726$		0			0
$727$	−651.429	+	335.835i	−0.896051	+	0.461946i	−0.843782	−	0.536687i	$-0.819676\pi$
$727$	−651.429	+	335.835i	−0.896051	+	0.461946i	−0.0522691	+	0.998633i	$0.516645\pi$
$728$		0			0
$729$	−699.470	+	205.383i	−0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$	−36.4239	+	253.333i	−0.0497594	+	0.346084i
$733$	168.695	−	487.411i	0.230143	−	0.664953i	−0.769511	−	0.638634i	$-0.779500\pi$
$733$	168.695	−	487.411i	0.230143	−	0.664953i	0.999654	−	0.0263198i	$-0.00837881\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−787.287	−	1105.59i	−1.06534	−	1.49606i	−0.855136	−	0.518404i	$-0.826526\pi$
$739$	−787.287	−	1105.59i	−1.06534	−	1.49606i	−0.210204	−	0.977657i	$-0.567413\pi$
$740$		0			0
$741$	139.151	−	402.051i	0.187788	−	0.542579i
$742$		0			0
$743$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$743$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	69.7193	+	80.4604i	0.0928353	+	0.107138i	0.800266	−	0.599645i	$-0.204691\pi$
$751$	69.7193	+	80.4604i	0.0928353	+	0.107138i	−0.707431	+	0.706782i	$0.750146\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	985.043	+	94.0602i	1.30297	+	0.124418i
$757$	274.944	+	141.744i	0.363202	+	0.187244i	0.630162	−	0.776464i	$-0.282989\pi$
$757$	274.944	+	141.744i	0.363202	+	0.187244i	−0.266959	+	0.963708i	$0.586019\pi$
$758$		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
$763$	1744.56	+	336.237i	2.28645	+	0.440677i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−555.828	−	529.981i	−0.723734	−	0.690079i
$769$	−22.3340	+	468.849i	−0.0290430	+	0.609687i	0.936692	+	0.350155i	$0.113871\pi$
$769$	−22.3340	+	468.849i	−0.0290430	+	0.609687i	−0.965735	+	0.259531i	$0.916432\pi$
$770$		0			0
$771$		0			0
$772$	−50.7661	+	1065.71i	−0.0657592	+	1.38046i
$773$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$773$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$774$		0			0
$775$	726.933	+	1259.08i	0.937978	+	1.62463i
$776$		0			0
$777$	−21.2324	−	147.674i	−0.0273261	−	0.190057i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−496.992	−	256.217i	−0.633919	−	0.326808i
$785$		0			0
$786$		0			0
$787$	508.061	+	1467.95i	0.645567	+	1.86524i	0.475241	+	0.879856i	$0.342361\pi$
$787$	508.061	+	1467.95i	0.645567	+	1.86524i	0.170326	+	0.985388i	$0.445518\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−407.557	+	163.161i	−0.513943	+	0.205752i
$794$		0			0
$795$		0			0
$796$	−1525.14	+	447.821i	−1.91600	+	0.562589i
$797$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$797$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	498.243	−	631.007i	0.619705	−	0.784835i
$805$		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
$811$	−690.986	+	65.9811i	−0.852017	+	0.0813577i	−0.511920	−	0.859033i	$-0.671066\pi$
$811$	−690.986	+	65.9811i	−0.852017	+	0.0813577i	−0.340096	+	0.940391i	$0.610460\pi$
$812$		0			0
$813$	317.387	−	694.980i	0.390390	−	0.854834i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	377.018	−	359.486i	0.461466	−	0.440007i
$818$		0			0
$819$	705.084	+	1543.92i	0.860908	+	1.88513i
$820$		0			0
$821$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$821$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$822$		0			0
$823$	411.553	+	39.2986i	0.500065	+	0.0477504i	0.342041	−	0.939685i	$-0.388882\pi$
$823$	411.553	+	39.2986i	0.500065	+	0.0477504i	0.158024	+	0.987435i	$0.449488\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$827$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$828$		0			0
$829$	1588.58	+	466.449i	1.91626	+	0.562664i	0.973145	+	0.230193i	$0.0739358\pi$
$829$	1588.58	+	466.449i	1.91626	+	0.562664i	0.943113	+	0.332471i	$0.107882\pi$
$830$		0			0
$831$	200.995	+	1397.95i	0.241872	+	1.68225i
$832$	310.572	−	1280.19i	0.373283	−	1.53870i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	910.791	−	1279.03i	1.08816	−	1.52811i
$838$		0			0
$839$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$839$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−210.807	−	1466.19i	−0.249771	−	1.73720i
$845$		0			0
$846$		0			0
$847$	−726.000	+	837.849i	−0.857143	+	0.989195i
$848$		0			0
$849$	−666.244	−	428.169i	−0.784740	−	0.504322i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	767.276	−	147.880i	0.899503	−	0.173365i	0.281519	−	0.959556i	$-0.409162\pi$
$853$	767.276	−	147.880i	0.899503	−	0.173365i	0.617984	+	0.786191i	$0.287950\pi$
$854$		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
$859$	−143.412	−	591.154i	−0.166953	−	0.688188i	−0.992331	−	0.123608i	$-0.960553\pi$
$859$	−143.412	−	591.154i	−0.166953	−	0.688188i	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
$867$	283.568	−	819.316i	0.327068	−	0.945001i
$868$	−1792.97	+	1152.27i	−2.06564	+	1.32750i
$869$		0			0
$870$		0			0
$871$	1364.65	+	198.949i	1.56676	+	0.228415i
$872$		0			0
$873$	1012.08	+	1421.26i	1.15931	+	1.62802i
$874$		0			0
$875$		0			0
$876$	−84.5285	+	587.909i	−0.0964937	+	0.671128i
$877$	698.162	+	279.502i	0.796080	+	0.318702i	0.733808	−	0.679357i	$-0.237741\pi$
$877$	698.162	+	279.502i	0.796080	+	0.318702i	0.0622717	+	0.998059i	$0.480165\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$881$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$882$		0			0
$883$	−397.612	−	1638.98i	−0.450297	−	1.85615i	−0.513413	−	0.858142i	$-0.671619\pi$
$883$	−397.612	−	1638.98i	−0.450297	−	1.85615i	0.0631161	−	0.998006i	$-0.479896\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$887$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$888$		0			0
$889$	2126.89	−	409.924i	2.39245	−	0.461107i
$890$		0			0
$891$		0			0
$892$	−964.448	+	758.451i	−1.08122	+	0.850281i
$893$		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
$903$	−98.8859	+	2075.87i	−0.109508	+	2.29886i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1268.37	−	1209.39i	−1.39843	−	1.33340i	−0.877318	−	0.479910i	$-0.840669\pi$
$907$	−1268.37	−	1209.39i	−1.39843	−	1.33340i	−0.521111	−	0.853489i	$-0.674482\pi$
$908$		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	$0.454545\pi$
$912$	−324.739	−	62.5884i	−0.356074	−	0.0686276i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1125.14	+	723.082i	1.22832	+	0.789391i
$917$		0			0
$918$		0			0
$919$	1720.12	+	164.251i	1.87173	+	0.178728i	0.968263	−	0.249933i	$-0.0804085\pi$
$919$	1720.12	+	164.251i	1.87173	+	0.178728i	0.903465	+	0.428661i	$0.141015\pi$
$920$		0			0
$921$	−437.553	−	1264.23i	−0.475085	−	1.37267i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−98.2073	+	93.6405i	−0.106170	+	0.101233i
$926$		0			0
$927$	146.797	−	58.7688i	0.158358	−	0.0633968i
$928$		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
$931$	−239.691	+	22.8877i	−0.257455	+	0.0245840i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1449.07			−1.54650			−0.773252	−	0.634099i	$-0.781371\pi$
$937$	−1449.07			−1.54650			−0.773252	+	0.634099i	$0.781371\pi$
$938$		0			0
$939$	1840.73			1.96030
$940$		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	$0.636364\pi$
$948$	−1606.09	+	827.999i	−1.69419	+	0.873416i
$949$	−945.814	+	378.647i	−0.996643	+	0.398996i
$950$		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
$961$	115.194	+	2418.22i	0.119869	+	2.51636i
$962$		0			0
$963$		0			0
$964$	−1864.58	−	359.369i	−1.93421	−	0.372789i
$965$		0			0
$966$		0			0
$967$	624.726	+	1082.06i	0.646045	+	1.11898i	0.984059	+	0.177842i	$0.0569115\pi$
$967$	624.726	+	1082.06i	0.646045	+	1.11898i	−0.338014	+	0.941141i	$0.609755\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$971$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$972$	−563.815	+	791.768i	−0.580057	+	0.814576i
$973$	−121.136	+	2542.95i	−0.124497	+	2.61351i
$974$		0			0
$975$	771.872	−	1336.92i	0.791663	−	1.37120i
$976$	170.626	+	295.532i	0.174821	+	0.302800i
$977$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$977$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1142.87	+	1318.94i	−1.16500	+	1.34448i
$982$		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
$988$	−185.535	−	536.068i	−0.187788	−	0.542579i
$989$		0			0
$990$		0			0
$991$	−1288.66	−	1487.20i	−1.30037	−	1.50070i	−0.740347	−	0.672224i	$-0.765339\pi$
$991$	−1288.66	−	1487.20i	−1.30037	−	1.50070i	−0.560020	−	0.828479i	$-0.689206\pi$
$992$		0			0
$993$	407.203	+	1678.51i	0.410073	+	1.69035i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1852.25	−	543.869i	1.85782	−	0.545506i	0.858348	−	0.513069i	$-0.171491\pi$
$997$	1852.25	−	543.869i	1.85782	−	0.545506i	0.999474	−	0.0324369i	$-0.0103268\pi$
$998$		0			0
$999$	136.053	+	54.4675i	0.136189	+	0.0545220i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.o.a.26.1	✓	20
3.2	odd	2	CM	201.3.o.a.26.1	✓	20
67.49	even	33	inner	201.3.o.a.116.1	yes	20
201.116	odd	66	inner	201.3.o.a.116.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.o.a.26.1	✓	20	1.1	even	1	trivial
201.3.o.a.26.1	✓	20	3.2	odd	2	CM
201.3.o.a.116.1	yes	20	67.49	even	33	inner
201.3.o.a.116.1	yes	20	201.116	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)

\(f(q)\)	\(=\)	\(q+(-2.52376 + 1.62192i) q^{3} +(-1.30827 + 3.78000i) q^{4} +(-9.12076 + 0.870927i) q^{7} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)	\(q+(-2.52376 + 1.62192i) q^{3} +(-1.30827 + 3.78000i) q^{4} +(-9.12076 + 0.870927i) q^{7} +(3.73874 - 8.18669i) q^{9} +(-2.82911 - 11.6617i) q^{12} +(14.8968 - 14.2041i) q^{13} +(-12.5768 - 9.89054i) q^{16} +(-6.85872 - 0.654928i) q^{19} +(21.6060 - 16.9912i) q^{21} +(-23.9873 - 7.04331i) q^{25} +(3.84250 + 26.7252i) q^{27} +(8.64033 - 35.6159i) q^{28} +(-42.0885 - 40.1313i) q^{31} +(26.0544 + 24.8428i) q^{36} +(2.71391 - 4.70062i) q^{37} +(-14.5581 + 60.0091i) q^{39} +(-49.5128 + 57.1409i) q^{43} +(47.7827 + 4.56269i) q^{48} +(34.3153 - 6.61374i) q^{49} +(34.2024 + 74.8927i) q^{52} +(18.3720 - 9.47142i) q^{57} +(-19.8004 - 7.92690i) q^{61} +(-26.9701 + 77.9250i) q^{63} +(53.8402 - 34.6010i) q^{64} +(41.3129 + 52.7470i) q^{67} +(-45.9507 - 18.3959i) q^{73} +(71.9620 - 21.1299i) q^{75} +(11.4487 - 25.0691i) q^{76} +(-35.5007 - 146.336i) q^{79} +(-53.0437 - 61.2157i) q^{81} +(35.9601 + 103.900i) q^{84} +(-123.499 + 142.526i) q^{91} +(171.311 + 33.0175i) q^{93} +(-96.9326 + 167.892i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9}+O(q^{10}) \) 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9	\( 20 q + 6 q^{3} + 4 q^{4} + 2 q^{7} - 18 q^{9} - 12 q^{12} + 23 q^{13} + 16 q^{16} + 26 q^{19} - 6 q^{21} - 50 q^{25} + 54 q^{27} + 8 q^{28} - 13 q^{31} + 36 q^{36} + 26 q^{37} - 69 q^{39} + 122 q^{43} - 48 q^{48} - 45 q^{49} + 828 q^{52} - 441 q^{57} + 47 q^{61} - 1269 q^{63} - 128 q^{64} + 109 q^{67} - 924 q^{73} + 150 q^{75} - 208 q^{76} + 252 q^{79} - 162 q^{81} + 1692 q^{84} - 92 q^{91} + 39 q^{93} + 167 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 4 * q^4 + 2 * q^7 - 18 * q^9 - 12 * q^12 + 23 * q^13 + 16 * q^16 + 26 * q^19 - 6 * q^21 - 50 * q^25 + 54 * q^27 + 8 * q^28 - 13 * q^31 + 36 * q^36 + 26 * q^37 - 69 * q^39 + 122 * q^43 - 48 * q^48 - 45 * q^49 + 828 * q^52 - 441 * q^57 + 47 * q^61 - 1269 * q^63 - 128 * q^64 + 109 * q^67 - 924 * q^73 + 150 * q^75 - 208 * q^76 + 252 * q^79 - 162 * q^81 + 1692 * q^84 - 92 * q^91 + 39 * q^93 + 167 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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