LMFDB - Embedded newform 1620.4.i.k.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.k.1081.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(11.0000 - 19.0526i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(11.0000 - 19.0526i) q^{7} +(4.50000 - 7.79423i) q^{11} +(-8.50000 - 14.7224i) q^{13} -75.0000 q^{17} -4.00000 q^{19} +(-91.5000 - 158.483i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-64.5000 + 111.717i) q^{29} +(93.5000 + 161.947i) q^{31} +110.000 q^{35} -34.0000 q^{37} +(-132.000 - 228.631i) q^{41} +(-221.500 + 383.649i) q^{43} +(-304.500 + 527.409i) q^{47} +(-70.5000 - 122.110i) q^{49} -228.000 q^{53} +45.0000 q^{55} +(-30.0000 - 51.9615i) q^{59} +(227.000 - 393.176i) q^{61} +(42.5000 - 73.6122i) q^{65} +(122.000 + 211.310i) q^{67} +444.000 q^{71} +398.000 q^{73} +(-99.0000 - 171.473i) q^{77} +(174.500 - 302.243i) q^{79} +(-519.000 + 898.934i) q^{83} +(-187.500 - 324.760i) q^{85} +852.000 q^{89} -374.000 q^{91} +(-10.0000 - 17.3205i) q^{95} +(-457.000 + 791.547i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{5} + 22 q^{7}+O(q^{10}) $ 2 * q + 5 * q^5 + 22 * q^7	$ 2 q + 5 q^{5} + 22 q^{7} + 9 q^{11} - 17 q^{13} - 150 q^{17} - 8 q^{19} - 183 q^{23} - 25 q^{25} - 129 q^{29} + 187 q^{31} + 220 q^{35} - 68 q^{37} - 264 q^{41} - 443 q^{43} - 609 q^{47} - 141 q^{49} - 456 q^{53} + 90 q^{55} - 60 q^{59} + 454 q^{61} + 85 q^{65} + 244 q^{67} + 888 q^{71} + 796 q^{73} - 198 q^{77} + 349 q^{79} - 1038 q^{83} - 375 q^{85} + 1704 q^{89} - 748 q^{91} - 20 q^{95} - 914 q^{97}+O(q^{100}) $ 2 * q + 5 * q^5 + 22 * q^7 + 9 * q^11 - 17 * q^13 - 150 * q^17 - 8 * q^19 - 183 * q^23 - 25 * q^25 - 129 * q^29 + 187 * q^31 + 220 * q^35 - 68 * q^37 - 264 * q^41 - 443 * q^43 - 609 * q^47 - 141 * q^49 - 456 * q^53 + 90 * q^55 - 60 * q^59 + 454 * q^61 + 85 * q^65 + 244 * q^67 + 888 * q^71 + 796 * q^73 - 198 * q^77 + 349 * q^79 - 1038 * q^83 - 375 * q^85 + 1704 * q^89 - 748 * q^91 - 20 * q^95 - 914 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	11.0000	−	19.0526i	0.593944	−	1.02874i	−0.399751	−	0.916624i	$-0.630903\pi$
$7$	11.0000	−	19.0526i	0.593944	−	1.02874i	0.993695	−	0.112118i	$-0.0357633\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.50000	−	7.79423i	0.123346	−	0.213641i	−0.797739	−	0.603002i	$-0.793971\pi$
$11$	4.50000	−	7.79423i	0.123346	−	0.213641i	0.921085	+	0.389362i	$0.127304\pi$
$12$		0			0
$13$	−8.50000	−	14.7224i	−0.181344	−	0.314098i	0.760994	−	0.648759i	$-0.224712\pi$
$13$	−8.50000	−	14.7224i	−0.181344	−	0.314098i	−0.942339	+	0.334661i	$0.891378\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−75.0000			−1.07001			−0.535005	−	0.844849i	$-0.679690\pi$
$17$	−75.0000			−1.07001			−0.535005	+	0.844849i	$0.679690\pi$
$18$		0			0
$19$	−4.00000			−0.0482980			−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000			−0.0482980			−0.0241490	+	0.999708i	$0.507688\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−91.5000	−	158.483i	−0.829525	−	1.43678i	−0.898412	−	0.439155i	$-0.855278\pi$
$23$	−91.5000	−	158.483i	−0.829525	−	1.43678i	0.0688868	−	0.997624i	$-0.478055\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−64.5000	+	111.717i	−0.413012	+	0.715358i	−0.995217	−	0.0976839i	$-0.968857\pi$
$29$	−64.5000	+	111.717i	−0.413012	+	0.715358i	0.582205	+	0.813042i	$0.302190\pi$
$30$		0			0
$31$	93.5000	+	161.947i	0.541713	+	0.938274i	0.998806	+	0.0488552i	$0.0155573\pi$
$31$	93.5000	+	161.947i	0.541713	+	0.938274i	−0.457093	+	0.889419i	$0.651109\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	110.000			0.531240
$36$		0			0
$37$	−34.0000			−0.151069			−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000			−0.151069			−0.0755347	+	0.997143i	$0.524066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−132.000	−	228.631i	−0.502803	−	0.870881i	−0.999995	−	0.00324004i	$-0.998969\pi$
$41$	−132.000	−	228.631i	−0.502803	−	0.870881i	0.497191	−	0.867641i	$-0.334365\pi$
$42$		0			0
$43$	−221.500	+	383.649i	−0.785545	+	1.36060i	0.143128	+	0.989704i	$0.454284\pi$
$43$	−221.500	+	383.649i	−0.785545	+	1.36060i	−0.928673	+	0.370900i	$0.879049\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−304.500	+	527.409i	−0.945019	+	1.63682i	−0.189306	+	0.981918i	$0.560624\pi$
$47$	−304.500	+	527.409i	−0.945019	+	1.63682i	−0.755713	+	0.654903i	$0.772710\pi$
$48$		0			0
$49$	−70.5000	−	122.110i	−0.205539	−	0.356005i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−228.000			−0.590910			−0.295455	−	0.955357i	$-0.595471\pi$
$53$	−228.000			−0.590910			−0.295455	+	0.955357i	$0.595471\pi$
$54$		0			0
$55$	45.0000			0.110324
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−30.0000	−	51.9615i	−0.0661978	−	0.114658i	0.831027	−	0.556232i	$-0.187753\pi$
$59$	−30.0000	−	51.9615i	−0.0661978	−	0.114658i	−0.897225	+	0.441574i	$0.854420\pi$
$60$		0			0
$61$	227.000	−	393.176i	0.476465	−	0.825262i	−0.523171	−	0.852228i	$-0.675251\pi$
$61$	227.000	−	393.176i	0.476465	−	0.825262i	0.999636	+	0.0269658i	$0.00858452\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	42.5000	−	73.6122i	0.0810996	−	0.140469i
$66$		0			0
$67$	122.000	+	211.310i	0.222458	+	0.385308i	0.955554	−	0.294817i	$-0.0952587\pi$
$67$	122.000	+	211.310i	0.222458	+	0.385308i	−0.733096	+	0.680125i	$0.761925\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	444.000			0.742156			0.371078	−	0.928602i	$-0.378988\pi$
$71$	444.000			0.742156			0.371078	+	0.928602i	$0.378988\pi$
$72$		0			0
$73$	398.000			0.638115			0.319057	−	0.947735i	$-0.396634\pi$
$73$	398.000			0.638115			0.319057	+	0.947735i	$0.396634\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−99.0000	−	171.473i	−0.146521	−	0.253781i
$78$		0			0
$79$	174.500	−	302.243i	0.248516	−	0.430443i	−0.714598	−	0.699535i	$-0.753390\pi$
$79$	174.500	−	302.243i	0.248516	−	0.430443i	0.963114	+	0.269092i	$0.0867237\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−519.000	+	898.934i	−0.686357	+	1.18881i	0.286651	+	0.958035i	$0.407458\pi$
$83$	−519.000	+	898.934i	−0.686357	+	1.18881i	−0.973008	+	0.230771i	$0.925875\pi$
$84$		0			0
$85$	−187.500	−	324.760i	−0.239262	−	0.414413i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	852.000			1.01474			0.507370	−	0.861728i	$-0.330618\pi$
$89$	852.000			1.01474			0.507370	+	0.861728i	$0.330618\pi$
$90$		0			0
$91$	−374.000			−0.430834
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.0000	−	17.3205i	−0.0107998	−	0.0187058i
$96$		0			0
$97$	−457.000	+	791.547i	−0.478364	+	0.828551i	−0.999692	−	0.0248053i	$-0.992103\pi$
$97$	−457.000	+	791.547i	−0.478364	+	0.828551i	0.521328	+	0.853356i	$0.325437\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−577.500	+	1000.26i	−0.568945	+	0.985441i	0.427726	+	0.903908i	$0.359315\pi$
$101$	−577.500	+	1000.26i	−0.568945	+	0.985441i	−0.996671	+	0.0815325i	$0.974019\pi$
$102$		0			0
$103$	287.000	+	497.099i	0.274553	+	0.475540i	0.970022	−	0.243016i	$-0.0781368\pi$
$103$	287.000	+	497.099i	0.274553	+	0.475540i	−0.695469	+	0.718556i	$0.744803\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	294.000			0.265627			0.132813	−	0.991141i	$-0.457599\pi$
$107$	294.000			0.265627			0.132813	+	0.991141i	$0.457599\pi$
$108$		0			0
$109$	−340.000			−0.298772			−0.149386	−	0.988779i	$-0.547730\pi$
$109$	−340.000			−0.298772			−0.149386	+	0.988779i	$0.547730\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−340.500	−	589.763i	−0.283465	−	0.490976i	0.688771	−	0.724979i	$-0.258151\pi$
$113$	−340.500	−	589.763i	−0.283465	−	0.490976i	−0.972236	+	0.234003i	$0.924817\pi$
$114$		0			0
$115$	457.500	−	792.413i	0.370975	−	0.642547i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−825.000	+	1428.94i	−0.635526	+	1.10076i
$120$		0			0
$121$	625.000	+	1082.53i	0.469572	+	0.813322i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	578.000			0.403852			0.201926	−	0.979401i	$-0.435280\pi$
$127$	578.000			0.403852			0.201926	+	0.979401i	$0.435280\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	391.500	+	678.098i	0.261111	+	0.452257i	0.966537	−	0.256526i	$-0.0825779\pi$
$131$	391.500	+	678.098i	0.261111	+	0.452257i	−0.705427	+	0.708783i	$0.749245\pi$
$132$		0			0
$133$	−44.0000	+	76.2102i	−0.0286863	+	0.0496862i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−117.000	+	202.650i	−0.0729634	+	0.126376i	−0.900199	−	0.435479i	$-0.856579\pi$
$137$	−117.000	+	202.650i	−0.0729634	+	0.126376i	0.827235	+	0.561855i	$0.189912\pi$
$138$		0			0
$139$	−1585.00	−	2745.30i	−0.967179	−	1.67520i	−0.703642	−	0.710555i	$-0.748444\pi$
$139$	−1585.00	−	2745.30i	−0.967179	−	1.67520i	−0.263538	−	0.964649i	$-0.584889\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−153.000			−0.0894720
$144$		0			0
$145$	−645.000			−0.369409
$146$		0			0
$147$		0			0
$148$		0			0
$149$	631.500	+	1093.79i	0.347211	+	0.601388i	0.985753	−	0.168200i	$-0.0537953\pi$
$149$	631.500	+	1093.79i	0.347211	+	0.601388i	−0.638542	+	0.769587i	$0.720462\pi$
$150$		0			0
$151$	−425.500	+	736.988i	−0.229316	+	0.397187i	−0.957606	−	0.288083i	$-0.906982\pi$
$151$	−425.500	+	736.988i	−0.229316	+	0.397187i	0.728290	+	0.685269i	$0.240316\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−467.500	+	809.734i	−0.242261	+	0.419609i
$156$		0			0
$157$	−1139.50	−	1973.67i	−0.579248	−	1.00329i	−0.995566	−	0.0940683i	$-0.970013\pi$
$157$	−1139.50	−	1973.67i	−0.579248	−	1.00329i	0.416317	−	0.909219i	$-0.363321\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4026.00			−1.97077
$162$		0			0
$163$	−1297.00			−0.623245			−0.311622	−	0.950206i	$-0.600872\pi$
$163$	−1297.00			−0.623245			−0.311622	+	0.950206i	$0.600872\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−162.000	−	280.592i	−0.0750655	−	0.130017i	0.826049	−	0.563598i	$-0.190583\pi$
$167$	−162.000	−	280.592i	−0.0750655	−	0.130017i	−0.901115	+	0.433581i	$0.857250\pi$
$168$		0			0
$169$	954.000	−	1652.38i	0.434228	−	0.752106i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	87.0000	−	150.688i	0.0382340	−	0.0662233i	−0.846275	−	0.532746i	$-0.821160\pi$
$173$	87.0000	−	150.688i	0.0382340	−	0.0662233i	0.884509	+	0.466523i	$0.154493\pi$
$174$		0			0
$175$	275.000	+	476.314i	0.118789	+	0.205748i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4596.00			−1.91911			−0.959556	−	0.281517i	$-0.909163\pi$
$179$	−4596.00			−1.91911			−0.959556	+	0.281517i	$0.909163\pi$
$180$		0			0
$181$	548.000			0.225042			0.112521	−	0.993649i	$-0.464108\pi$
$181$	548.000			0.225042			0.112521	+	0.993649i	$0.464108\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−85.0000	−	147.224i	−0.0337801	−	0.0585089i
$186$		0			0
$187$	−337.500	+	584.567i	−0.131981	+	0.228598i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2319.00	+	4016.63i	−0.878518	+	1.52164i	−0.0255508	+	0.999674i	$0.508134\pi$
$191$	−2319.00	+	4016.63i	−0.878518	+	1.52164i	−0.852967	+	0.521964i	$0.825199\pi$
$192$		0			0
$193$	1721.00	+	2980.86i	0.641867	+	1.11175i	0.985016	+	0.172465i	$0.0551731\pi$
$193$	1721.00	+	2980.86i	0.641867	+	1.11175i	−0.343149	+	0.939281i	$0.611494\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3312.00			−1.19782			−0.598909	−	0.800817i	$-0.704399\pi$
$197$	−3312.00			−1.19782			−0.598909	+	0.800817i	$0.704399\pi$
$198$		0			0
$199$	1745.00			0.621607			0.310803	−	0.950474i	$-0.399402\pi$
$199$	1745.00			0.621607			0.310803	+	0.950474i	$0.399402\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1419.00	+	2457.78i	0.490612	+	0.849765i
$204$		0			0
$205$	660.000	−	1143.15i	0.224860	−	0.389470i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−18.0000	+	31.1769i	−0.00595735	+	0.0103184i
$210$		0			0
$211$	1679.00	+	2908.11i	0.547806	+	0.948828i	0.998425	+	0.0561116i	$0.0178702\pi$
$211$	1679.00	+	2908.11i	0.547806	+	0.948828i	−0.450618	+	0.892717i	$0.648796\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2215.00			−0.702613
$216$		0			0
$217$	4114.00			1.28699
$218$		0			0
$219$		0			0
$220$		0			0
$221$	637.500	+	1104.18i	0.194040	+	0.336088i
$222$		0			0
$223$	−1756.00	+	3041.48i	−0.527311	+	0.913330i	0.472182	+	0.881501i	$0.343467\pi$
$223$	−1756.00	+	3041.48i	−0.527311	+	0.913330i	−0.999493	+	0.0318291i	$0.989867\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1056.00	+	1829.05i	−0.308763	+	0.534793i	−0.978092	−	0.208173i	$-0.933248\pi$
$227$	−1056.00	+	1829.05i	−0.308763	+	0.534793i	0.669329	+	0.742966i	$0.266582\pi$
$228$		0			0
$229$	−1450.00	−	2511.47i	−0.418422	−	0.724729i	0.577359	−	0.816491i	$-0.304083\pi$
$229$	−1450.00	−	2511.47i	−0.418422	−	0.724729i	−0.995781	+	0.0917620i	$0.970750\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5178.00			−1.45589			−0.727944	−	0.685636i	$-0.759524\pi$
$233$	−5178.00			−1.45589			−0.727944	+	0.685636i	$0.759524\pi$
$234$		0			0
$235$	−3045.00			−0.845251
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2505.00	−	4338.79i	−0.677971	−	1.17428i	−0.975591	−	0.219595i	$-0.929526\pi$
$239$	−2505.00	−	4338.79i	−0.677971	−	1.17428i	0.297620	−	0.954684i	$-0.403807\pi$
$240$		0			0
$241$	2601.50	−	4505.93i	0.695342	−	1.20437i	−0.274724	−	0.961523i	$-0.588586\pi$
$241$	2601.50	−	4505.93i	0.695342	−	1.20437i	0.970065	−	0.242844i	$-0.0780803\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	352.500	−	610.548i	0.0919200	−	0.159210i
$246$		0			0
$247$	34.0000	+	58.8897i	0.00875858	+	0.0151703i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−549.000			−0.138058			−0.0690290	−	0.997615i	$-0.521990\pi$
$251$	−549.000			−0.138058			−0.0690290	+	0.997615i	$0.521990\pi$
$252$		0			0
$253$	−1647.00			−0.409273
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3547.50	+	6144.45i	0.861039	+	1.49136i	0.870927	+	0.491412i	$0.163519\pi$
$257$	3547.50	+	6144.45i	0.861039	+	1.49136i	−0.00988853	+	0.999951i	$0.503148\pi$
$258$		0			0
$259$	−374.000	+	647.787i	−0.0897268	+	0.155411i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2142.00	+	3710.05i	−0.502211	+	0.869854i	0.497786	+	0.867300i	$0.334146\pi$
$263$	−2142.00	+	3710.05i	−0.502211	+	0.869854i	−0.999997	+	0.00255442i	$0.999187\pi$
$264$		0			0
$265$	−570.000	−	987.269i	−0.132131	−	0.228858i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6669.00			−1.51158			−0.755792	−	0.654812i	$-0.772748\pi$
$269$	−6669.00			−1.51158			−0.755792	+	0.654812i	$0.772748\pi$
$270$		0			0
$271$	−5560.00			−1.24630			−0.623148	−	0.782104i	$-0.714146\pi$
$271$	−5560.00			−1.24630			−0.623148	+	0.782104i	$0.714146\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	112.500	+	194.856i	0.0246691	+	0.0427282i
$276$		0			0
$277$	−3367.00	+	5831.82i	−0.730337	+	1.26498i	0.226402	+	0.974034i	$0.427304\pi$
$277$	−3367.00	+	5831.82i	−0.730337	+	1.26498i	−0.956739	+	0.290947i	$0.906030\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3855.00	−	6677.06i	0.818399	−	1.41751i	−0.0884630	−	0.996079i	$-0.528195\pi$
$281$	3855.00	−	6677.06i	0.818399	−	1.41751i	0.906862	−	0.421429i	$-0.138471\pi$
$282$		0			0
$283$	−2368.00	−	4101.50i	−0.497396	−	0.861515i	0.502600	−	0.864519i	$-0.332377\pi$
$283$	−2368.00	−	4101.50i	−0.497396	−	0.861515i	−0.999995	+	0.00300456i	$0.999044\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5808.00			−1.19455
$288$		0			0
$289$	712.000			0.144922
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2775.00	+	4806.44i	0.553301	+	0.958346i	0.998034	+	0.0626822i	$0.0199654\pi$
$293$	2775.00	+	4806.44i	0.553301	+	0.958346i	−0.444732	+	0.895663i	$0.646701\pi$
$294$		0			0
$295$	150.000	−	259.808i	0.0296045	−	0.0512766i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1555.50	+	2694.21i	−0.300859	+	0.521103i
$300$		0			0
$301$	4873.00	+	8440.28i	0.933140	+	1.61625i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2270.00			0.426163
$306$		0			0
$307$	5663.00			1.05278			0.526392	−	0.850242i	$-0.323545\pi$
$307$	5663.00			1.05278			0.526392	+	0.850242i	$0.323545\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	960.000	+	1662.77i	0.175037	+	0.303174i	0.940174	−	0.340694i	$-0.110662\pi$
$311$	960.000	+	1662.77i	0.175037	+	0.303174i	−0.765137	+	0.643868i	$0.777329\pi$
$312$		0			0
$313$	−1408.00	+	2438.73i	−0.254265	+	0.440399i	−0.964696	−	0.263368i	$-0.915167\pi$
$313$	−1408.00	+	2438.73i	−0.254265	+	0.440399i	0.710431	+	0.703767i	$0.248500\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2892.00	+	5009.09i	−0.512400	+	0.887503i	0.487496	+	0.873125i	$0.337910\pi$
$317$	−2892.00	+	5009.09i	−0.512400	+	0.887503i	−0.999897	+	0.0143783i	$0.995423\pi$
$318$		0			0
$319$	580.500	+	1005.46i	0.101886	+	0.176472i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	300.000			0.0516794
$324$		0			0
$325$	425.000			0.0725377
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6699.00	+	11603.0i	1.12258	+	1.94436i
$330$		0			0
$331$	2177.00	−	3770.67i	0.361507	−	0.626148i	−0.626702	−	0.779259i	$-0.715596\pi$
$331$	2177.00	−	3770.67i	0.361507	−	0.626148i	0.988209	+	0.153111i	$0.0489291\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−610.000	+	1056.55i	−0.0994861	+	0.172315i
$336$		0			0
$337$	−1948.00	−	3374.03i	−0.314879	−	0.545387i	0.664533	−	0.747259i	$-0.268631\pi$
$337$	−1948.00	−	3374.03i	−0.314879	−	0.545387i	−0.979412	+	0.201872i	$0.935297\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1683.00			0.267271
$342$		0			0
$343$	4444.00			0.699573
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3804.00	−	6588.72i	−0.588500	−	1.01931i	−0.994429	−	0.105407i	$-0.966385\pi$
$347$	−3804.00	−	6588.72i	−0.588500	−	1.01931i	0.405929	−	0.913904i	$-0.366948\pi$
$348$		0			0
$349$	−2143.00	+	3711.78i	−0.328688	+	0.569305i	−0.982252	−	0.187567i	$-0.939940\pi$
$349$	−2143.00	+	3711.78i	−0.328688	+	0.569305i	0.653564	+	0.756872i	$0.273273\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1453.50	+	2517.54i	−0.219156	+	0.379589i	−0.954550	−	0.298050i	$-0.903664\pi$
$353$	−1453.50	+	2517.54i	−0.219156	+	0.379589i	0.735394	+	0.677639i	$0.236997\pi$
$354$		0			0
$355$	1110.00	+	1922.58i	0.165951	+	0.287436i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5988.00			0.880319			0.440160	−	0.897920i	$-0.354922\pi$
$359$	5988.00			0.880319			0.440160	+	0.897920i	$0.354922\pi$
$360$		0			0
$361$	−6843.00			−0.997667
$362$		0			0
$363$		0			0
$364$		0			0
$365$	995.000	+	1723.39i	0.142687	+	0.247141i
$366$		0			0
$367$	1607.00	−	2783.41i	0.228569	−	0.395893i	−0.728815	−	0.684710i	$-0.759929\pi$
$367$	1607.00	−	2783.41i	0.228569	−	0.395893i	0.957384	+	0.288818i	$0.0932621\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2508.00	+	4343.98i	−0.350967	+	0.607893i
$372$		0			0
$373$	−4709.50	−	8157.09i	−0.653750	−	1.13233i	−0.982206	−	0.187809i	$-0.939862\pi$
$373$	−4709.50	−	8157.09i	−0.653750	−	1.13233i	0.328456	−	0.944519i	$-0.393472\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2193.00			0.299590
$378$		0			0
$379$	4544.00			0.615856			0.307928	−	0.951410i	$-0.400364\pi$
$379$	4544.00			0.615856			0.307928	+	0.951410i	$0.400364\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5743.50	−	9948.03i	−0.766264	−	1.32721i	−0.939576	−	0.342341i	$-0.888780\pi$
$383$	−5743.50	−	9948.03i	−0.766264	−	1.32721i	0.173312	−	0.984867i	$-0.444553\pi$
$384$		0			0
$385$	495.000	−	857.365i	0.0655261	−	0.113494i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3922.50	+	6793.97i	−0.511256	+	0.885522i	0.488659	+	0.872475i	$0.337486\pi$
$389$	−3922.50	+	6793.97i	−0.511256	+	0.885522i	−0.999915	+	0.0130466i	$0.995847\pi$
$390$		0			0
$391$	6862.50	+	11886.2i	0.887600	+	1.53737i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1745.00			0.222280
$396$		0			0
$397$	4133.00			0.522492			0.261246	−	0.965272i	$-0.415867\pi$
$397$	4133.00			0.522492			0.261246	+	0.965272i	$0.415867\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1662.00	−	2878.67i	−0.206973	−	0.358488i	0.743786	−	0.668417i	$-0.233028\pi$
$401$	−1662.00	−	2878.67i	−0.206973	−	0.358488i	−0.950760	+	0.309929i	$0.899695\pi$
$402$		0			0
$403$	1589.50	−	2753.09i	0.196473	−	0.340301i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−153.000	+	265.004i	−0.0186337	+	0.0322746i
$408$		0			0
$409$	−935.500	−	1620.33i	−0.113099	−	0.195893i	0.803919	−	0.594738i	$-0.202744\pi$
$409$	−935.500	−	1620.33i	−0.113099	−	0.195893i	−0.917018	+	0.398845i	$0.869411\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1320.00			−0.157271
$414$		0			0
$415$	−5190.00			−0.613897
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2704.50	−	4684.33i	−0.315330	−	0.546168i	0.664177	−	0.747575i	$-0.268782\pi$
$419$	−2704.50	−	4684.33i	−0.315330	−	0.546168i	−0.979508	+	0.201407i	$0.935449\pi$
$420$		0			0
$421$	6542.00	−	11331.1i	0.757334	−	1.31174i	−0.186872	−	0.982384i	$-0.559835\pi$
$421$	6542.00	−	11331.1i	0.757334	−	1.31174i	0.944206	−	0.329356i	$-0.106832\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	937.500	−	1623.80i	0.107001	−	0.185331i
$426$		0			0
$427$	−4994.00	−	8649.86i	−0.565987	−	0.980319i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7848.00			−0.877088			−0.438544	−	0.898710i	$-0.644506\pi$
$431$	−7848.00			−0.877088			−0.438544	+	0.898710i	$0.644506\pi$
$432$		0			0
$433$	−10222.0			−1.13450			−0.567249	−	0.823546i	$-0.691992\pi$
$433$	−10222.0			−1.13450			−0.567249	+	0.823546i	$0.691992\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	366.000	+	633.931i	0.0400644	+	0.0693936i
$438$		0			0
$439$	−2836.00	+	4912.10i	−0.308326	+	0.534035i	−0.977996	−	0.208623i	$-0.933102\pi$
$439$	−2836.00	+	4912.10i	−0.308326	+	0.534035i	0.669671	+	0.742658i	$0.266435\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−222.000	+	384.515i	−0.0238093	+	0.0412390i	−0.877685	−	0.479239i	$-0.840913\pi$
$443$	−222.000	+	384.515i	−0.0238093	+	0.0412390i	0.853875	+	0.520478i	$0.174246\pi$
$444$		0			0
$445$	2130.00	+	3689.27i	0.226903	+	0.393007i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10434.0			1.09668			0.548342	−	0.836254i	$-0.315259\pi$
$449$	10434.0			1.09668			0.548342	+	0.836254i	$0.315259\pi$
$450$		0			0
$451$	−2376.00			−0.248074
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−935.000	−	1619.47i	−0.0963373	−	0.166861i
$456$		0			0
$457$	−5590.00	+	9682.16i	−0.572186	+	0.991056i	0.424155	+	0.905590i	$0.360571\pi$
$457$	−5590.00	+	9682.16i	−0.572186	+	0.991056i	−0.996341	+	0.0854661i	$0.972762\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7509.00	+	13006.0i	−0.758631	+	1.31399i	0.184917	+	0.982754i	$0.440798\pi$
$461$	−7509.00	+	13006.0i	−0.758631	+	1.31399i	−0.943549	+	0.331234i	$0.892535\pi$
$462$		0			0
$463$	−8083.00	−	14000.2i	−0.811337	−	1.40528i	−0.911929	−	0.410349i	$-0.865407\pi$
$463$	−8083.00	−	14000.2i	−0.811337	−	1.40528i	0.100592	−	0.994928i	$-0.467926\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2964.00			0.293699			0.146850	−	0.989159i	$-0.453087\pi$
$467$	2964.00			0.293699			0.146850	+	0.989159i	$0.453087\pi$
$468$		0			0
$469$	5368.00			0.528510
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1993.50	+	3452.84i	0.193787	+	0.335649i
$474$		0			0
$475$	50.0000	−	86.6025i	0.00482980	−	0.00836547i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2706.00	−	4686.93i	0.258122	−	0.447080i	−0.707617	−	0.706596i	$-0.750230\pi$
$479$	2706.00	−	4686.93i	0.258122	−	0.447080i	0.965739	+	0.259516i	$0.0835631\pi$
$480$		0			0
$481$	289.000	+	500.563i	0.0273956	+	0.0474505i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4570.00			−0.427862
$486$		0			0
$487$	12356.0			1.14970			0.574850	−	0.818259i	$-0.305060\pi$
$487$	12356.0			1.14970			0.574850	+	0.818259i	$0.305060\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8370.00	−	14497.3i	−0.769313	−	1.33249i	−0.937936	−	0.346809i	$-0.887265\pi$
$491$	−8370.00	−	14497.3i	−0.769313	−	1.33249i	0.168623	−	0.985681i	$-0.446068\pi$
$492$		0			0
$493$	4837.50	−	8378.80i	0.441927	−	0.765440i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4884.00	−	8459.34i	0.440799	−	0.763487i
$498$		0			0
$499$	7046.00	+	12204.0i	0.632109	+	1.09484i	0.987120	+	0.159982i	$0.0511438\pi$
$499$	7046.00	+	12204.0i	0.632109	+	1.09484i	−0.355011	+	0.934862i	$0.615523\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2313.00			−0.205033			−0.102516	−	0.994731i	$-0.532689\pi$
$503$	−2313.00			−0.205033			−0.102516	+	0.994731i	$0.532689\pi$
$504$		0			0
$505$	−5775.00			−0.508879
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7462.50	−	12925.4i	−0.649842	−	1.12556i	−0.983160	−	0.182745i	$-0.941502\pi$
$509$	−7462.50	−	12925.4i	−0.649842	−	1.12556i	0.333319	−	0.942814i	$-0.391832\pi$
$510$		0			0
$511$	4378.00	−	7582.92i	0.379005	−	0.656455i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1435.00	+	2485.49i	−0.122784	+	0.212668i
$516$		0			0
$517$	2740.50	+	4746.69i	0.233128	+	0.403789i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5820.00			0.489403			0.244701	−	0.969598i	$-0.421310\pi$
$521$	5820.00			0.489403			0.244701	+	0.969598i	$0.421310\pi$
$522$		0			0
$523$	−20875.0			−1.74532			−0.872658	−	0.488332i	$-0.837605\pi$
$523$	−20875.0			−1.74532			−0.872658	+	0.488332i	$0.837605\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7012.50	−	12146.0i	−0.579638	−	1.00396i
$528$		0			0
$529$	−10661.0	+	18465.4i	−0.876223	+	1.51766i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2244.00	+	3886.72i	−0.182361	+	0.315859i
$534$		0			0
$535$	735.000	+	1273.06i	0.0593959	+	0.102877i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1269.00			−0.101409
$540$		0			0
$541$	22442.0			1.78347			0.891735	−	0.452559i	$-0.149489\pi$
$541$	22442.0			1.78347			0.891735	+	0.452559i	$0.149489\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−850.000	−	1472.24i	−0.0668073	−	0.115714i
$546$		0			0
$547$	−4307.50	+	7460.81i	−0.336701	+	0.583183i	−0.983810	−	0.179214i	$-0.942644\pi$
$547$	−4307.50	+	7460.81i	−0.336701	+	0.583183i	0.647109	+	0.762397i	$0.275978\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	258.000	−	446.869i	0.0199477	−	0.0345504i
$552$		0			0
$553$	−3839.00	−	6649.34i	−0.295210	−	0.511318i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19524.0			1.48520			0.742602	−	0.669733i	$-0.233592\pi$
$557$	19524.0			1.48520			0.742602	+	0.669733i	$0.233592\pi$
$558$		0			0
$559$	7531.00			0.569816
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6507.00	−	11270.5i	−0.487100	−	0.843682i	0.512790	−	0.858514i	$-0.328612\pi$
$563$	−6507.00	−	11270.5i	−0.487100	−	0.843682i	−0.999890	+	0.0148320i	$0.995279\pi$
$564$		0			0
$565$	1702.50	−	2948.82i	0.126769	−	0.219571i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9168.00	−	15879.4i	0.675470	−	1.16995i	−0.300861	−	0.953668i	$-0.597274\pi$
$569$	9168.00	−	15879.4i	0.675470	−	1.16995i	0.976331	−	0.216281i	$-0.0693926\pi$
$570$		0			0
$571$	−1591.00	−	2755.69i	−0.116605	−	0.201965i	0.801815	−	0.597572i	$-0.203868\pi$
$571$	−1591.00	−	2755.69i	−0.116605	−	0.201965i	−0.918420	+	0.395607i	$0.870534\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4575.00			0.331810
$576$		0			0
$577$	7088.00			0.511399			0.255700	−	0.966756i	$-0.417694\pi$
$577$	7088.00			0.511399			0.255700	+	0.966756i	$0.417694\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11418.0	+	19776.6i	0.815316	+	1.41217i
$582$		0			0
$583$	−1026.00	+	1777.08i	−0.0728861	+	0.126242i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3171.00	+	5492.33i	−0.222966	+	0.386189i	−0.955707	−	0.294319i	$-0.904907\pi$
$587$	−3171.00	+	5492.33i	−0.222966	+	0.386189i	0.732741	+	0.680508i	$0.238241\pi$
$588$		0			0
$589$	−374.000	−	647.787i	−0.0261637	−	0.0453168i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26253.0			1.81801			0.909006	−	0.416782i	$-0.136842\pi$
$593$	26253.0			1.81801			0.909006	+	0.416782i	$0.136842\pi$
$594$		0			0
$595$	−8250.00			−0.568432
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2691.00	+	4660.95i	0.183558	+	0.317932i	0.943090	−	0.332538i	$-0.107905\pi$
$599$	2691.00	+	4660.95i	0.183558	+	0.317932i	−0.759532	+	0.650470i	$0.774572\pi$
$600$		0			0
$601$	14205.5	−	24604.6i	0.964150	−	1.66996i	0.252270	−	0.967657i	$-0.418823\pi$
$601$	14205.5	−	24604.6i	0.964150	−	1.66996i	0.711880	−	0.702301i	$-0.247844\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3125.00	+	5412.66i	−0.209999	+	0.363729i
$606$		0			0
$607$	−7687.00	−	13314.3i	−0.514013	−	0.890296i	−0.999868	−	0.0162568i	$-0.994825\pi$
$607$	−7687.00	−	13314.3i	−0.514013	−	0.890296i	0.485855	−	0.874039i	$-0.338508\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10353.0			0.685495
$612$		0			0
$613$	−23983.0			−1.58020			−0.790101	−	0.612976i	$-0.789972\pi$
$613$	−23983.0			−1.58020			−0.790101	+	0.612976i	$0.789972\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6886.50	+	11927.8i	0.449336	+	0.778272i	0.998343	−	0.0575455i	$-0.0183274\pi$
$617$	6886.50	+	11927.8i	0.449336	+	0.778272i	−0.549007	+	0.835818i	$0.684994\pi$
$618$		0			0
$619$	5573.00	−	9652.72i	0.361870	−	0.626778i	−0.626398	−	0.779503i	$-0.715472\pi$
$619$	5573.00	−	9652.72i	0.361870	−	0.626778i	0.988269	+	0.152725i	$0.0488050\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9372.00	−	16232.8i	0.602699	−	1.04390i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2550.00			0.161646
$630$		0			0
$631$	−2080.00			−0.131226			−0.0656129	−	0.997845i	$-0.520900\pi$
$631$	−2080.00			−0.131226			−0.0656129	+	0.997845i	$0.520900\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1445.00	+	2502.81i	0.0903041	+	0.156411i
$636$		0			0
$637$	−1198.50	+	2075.86i	−0.0745468	+	0.129119i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15525.0	−	26890.1i	0.956631	−	1.65693i	0.226041	−	0.974118i	$-0.427422\pi$
$641$	15525.0	−	26890.1i	0.956631	−	1.65693i	0.730590	−	0.682816i	$-0.239245\pi$
$642$		0			0
$643$	−4541.50	−	7866.11i	−0.278537	−	0.482440i	0.692484	−	0.721433i	$-0.256516\pi$
$643$	−4541.50	−	7866.11i	−0.278537	−	0.482440i	−0.971021	+	0.238993i	$0.923183\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4740.00			−0.288020			−0.144010	−	0.989576i	$-0.546000\pi$
$647$	−4740.00			−0.288020			−0.144010	+	0.989576i	$0.546000\pi$
$648$		0			0
$649$	−540.000			−0.0326608
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4098.00	+	7097.94i	0.245585	+	0.425366i	0.962296	−	0.272004i	$-0.0876866\pi$
$653$	4098.00	+	7097.94i	0.245585	+	0.425366i	−0.716711	+	0.697371i	$0.754353\pi$
$654$		0			0
$655$	−1957.50	+	3390.49i	−0.116772	+	0.202256i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3456.00	+	5985.97i	−0.204289	+	0.353839i	−0.949906	−	0.312536i	$-0.898822\pi$
$659$	−3456.00	+	5985.97i	−0.204289	+	0.353839i	0.745617	+	0.666375i	$0.232155\pi$
$660$		0			0
$661$	14219.0	+	24628.0i	0.836694	+	1.44920i	0.892643	+	0.450764i	$0.148848\pi$
$661$	14219.0	+	24628.0i	0.836694	+	1.44920i	−0.0559490	+	0.998434i	$0.517818\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−440.000			−0.0256578
$666$		0			0
$667$	23607.0			1.37041
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2043.00	−	3538.58i	−0.117540	−	0.203585i
$672$		0			0
$673$	−973.000	+	1685.29i	−0.0557302	+	0.0965275i	−0.892545	−	0.450959i	$-0.851082\pi$
$673$	−973.000	+	1685.29i	−0.0557302	+	0.0965275i	0.836814	+	0.547487i	$0.184415\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12978.0	−	22478.6i	0.736758	−	1.27610i	−0.217190	−	0.976129i	$-0.569689\pi$
$677$	12978.0	−	22478.6i	0.736758	−	1.27610i	0.953948	−	0.299973i	$-0.0969776\pi$
$678$		0			0
$679$	10054.0	+	17414.0i	0.568243	+	0.984226i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8184.00			−0.458495			−0.229247	−	0.973368i	$-0.573627\pi$
$683$	−8184.00			−0.458495			−0.229247	+	0.973368i	$0.573627\pi$
$684$		0			0
$685$	−1170.00			−0.0652604
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1938.00	+	3356.71i	0.107158	+	0.185603i
$690$		0			0
$691$	−15211.0	+	26346.2i	−0.837415	+	1.45045i	0.0546341	+	0.998506i	$0.482601\pi$
$691$	−15211.0	+	26346.2i	−0.837415	+	1.45045i	−0.892049	+	0.451939i	$0.850733\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7925.00	−	13726.5i	0.432536	−	0.749174i
$696$		0			0
$697$	9900.00	+	17147.3i	0.538005	+	0.931851i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27273.0			1.46945			0.734727	−	0.678363i	$-0.237310\pi$
$701$	27273.0			1.46945			0.734727	+	0.678363i	$0.237310\pi$
$702$		0			0
$703$	136.000			0.00729635
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12705.0	+	22005.7i	0.675843	+	1.17059i
$708$		0			0
$709$	−14470.0	+	25062.8i	−0.766477	+	1.32758i	0.172985	+	0.984924i	$0.444659\pi$
$709$	−14470.0	+	25062.8i	−0.766477	+	1.32758i	−0.939462	+	0.342653i	$0.888675\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17110.5	−	29636.3i	0.898728	−	1.55664i
$714$		0			0
$715$	−382.500	−	662.509i	−0.0200066	−	0.0346524i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−606.000			−0.0314325			−0.0157163	−	0.999876i	$-0.505003\pi$
$719$	−606.000			−0.0314325			−0.0157163	+	0.999876i	$0.505003\pi$
$720$		0			0
$721$	12628.0			0.652276
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1612.50	−	2792.93i	−0.0826024	−	0.143072i
$726$		0			0
$727$	6542.00	−	11331.1i	0.333741	−	0.578056i	−0.649502	−	0.760360i	$-0.725022\pi$
$727$	6542.00	−	11331.1i	0.333741	−	0.578056i	0.983242	+	0.182305i	$0.0583557\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16612.5	−	28773.7i	0.840541	−	1.45586i
$732$		0			0
$733$	−13537.0	−	23446.8i	−0.682129	−	1.18148i	−0.974330	−	0.225125i	$-0.927721\pi$
$733$	−13537.0	−	23446.8i	−0.682129	−	1.18148i	0.292201	−	0.956357i	$-0.405612\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2196.00			0.109757
$738$		0			0
$739$	−3580.00			−0.178204			−0.0891018	−	0.996023i	$-0.528400\pi$
$739$	−3580.00			−0.178204			−0.0891018	+	0.996023i	$0.528400\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6973.50	+	12078.5i	0.344324	+	0.596387i	0.985231	−	0.171232i	$-0.0547747\pi$
$743$	6973.50	+	12078.5i	0.344324	+	0.596387i	−0.640907	+	0.767619i	$0.721441\pi$
$744$		0			0
$745$	−3157.50	+	5468.95i	−0.155278	+	0.268949i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3234.00	−	5601.45i	0.157767	−	0.273261i
$750$		0			0
$751$	15636.5	+	27083.2i	0.759766	+	1.31595i	0.942970	+	0.332878i	$0.108020\pi$
$751$	15636.5	+	27083.2i	0.759766	+	1.31595i	−0.183204	+	0.983075i	$0.558647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4255.00			−0.205106
$756$		0			0
$757$	−31729.0			−1.52340			−0.761698	−	0.647933i	$-0.775634\pi$
$757$	−31729.0			−1.52340			−0.761698	+	0.647933i	$0.775634\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1569.00	+	2717.59i	0.0747388	+	0.129451i	0.900973	−	0.433876i	$-0.142854\pi$
$761$	1569.00	+	2717.59i	0.0747388	+	0.129451i	−0.826234	+	0.563327i	$0.809521\pi$
$762$		0			0
$763$	−3740.00	+	6477.87i	−0.177454	+	0.307359i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−510.000	+	883.346i	−0.0240092	+	0.0415851i
$768$		0			0
$769$	−10403.5	−	18019.4i	−0.487854	−	0.844988i	0.512048	−	0.858957i	$-0.328887\pi$
$769$	−10403.5	−	18019.4i	−0.487854	−	0.844988i	−0.999902	+	0.0139685i	$0.995554\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33078.0			1.53911			0.769556	−	0.638580i	$-0.220478\pi$
$773$	33078.0			1.53911			0.769556	+	0.638580i	$0.220478\pi$
$774$		0			0
$775$	−4675.00			−0.216685
$776$		0			0
$777$		0			0
$778$		0			0
$779$	528.000	+	914.523i	0.0242844	+	0.0420618i
$780$		0			0
$781$	1998.00	−	3460.64i	0.0915417	−	0.158555i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5697.50	−	9868.36i	0.259048	−	0.448684i
$786$		0			0
$787$	6270.50	+	10860.8i	0.284014	+	0.491927i	0.972370	−	0.233447i	$-0.0750004\pi$
$787$	6270.50	+	10860.8i	0.284014	+	0.491927i	−0.688355	+	0.725374i	$0.741667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14982.0			−0.673450
$792$		0			0
$793$	−7718.00			−0.345617
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15990.0	−	27695.5i	−0.710659	−	1.23090i	−0.964610	−	0.263680i	$-0.915064\pi$
$797$	−15990.0	−	27695.5i	−0.710659	−	1.23090i	0.253952	−	0.967217i	$-0.418270\pi$
$798$		0			0
$799$	22837.5	−	39555.7i	1.01118	−	1.75141i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1791.00	−	3102.10i	0.0787086	−	0.136327i
$804$		0			0
$805$	−10065.0	−	17433.1i	−0.440677	−	0.763274i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3792.00			−0.164796			−0.0823978	−	0.996600i	$-0.526258\pi$
$809$	−3792.00			−0.164796			−0.0823978	+	0.996600i	$0.526258\pi$
$810$		0			0
$811$	24086.0			1.04288			0.521439	−	0.853289i	$-0.325395\pi$
$811$	24086.0			1.04288			0.521439	+	0.853289i	$0.325395\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3242.50	−	5616.17i	−0.139362	−	0.241382i
$816$		0			0
$817$	886.000	−	1534.60i	0.0379403	−	0.0657145i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4887.00	+	8464.53i	−0.207744	+	0.359822i	−0.951003	−	0.309180i	$-0.899945\pi$
$821$	−4887.00	+	8464.53i	−0.207744	+	0.359822i	0.743260	+	0.669003i	$0.233279\pi$
$822$		0			0
$823$	−19318.0	−	33459.8i	−0.818206	−	1.41717i	−0.907003	−	0.421124i	$-0.861636\pi$
$823$	−19318.0	−	33459.8i	−0.818206	−	1.41717i	0.0887976	−	0.996050i	$-0.471698\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16458.0			−0.692020			−0.346010	−	0.938231i	$-0.612464\pi$
$827$	−16458.0			−0.692020			−0.346010	+	0.938231i	$0.612464\pi$
$828$		0			0
$829$	20252.0			0.848469			0.424235	−	0.905552i	$-0.360543\pi$
$829$	20252.0			0.848469			0.424235	+	0.905552i	$0.360543\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5287.50	+	9158.22i	0.219929	+	0.380929i
$834$		0			0
$835$	810.000	−	1402.96i	0.0335703	−	0.0581455i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20361.0	−	35266.3i	0.837830	−	1.45116i	−0.0538744	−	0.998548i	$-0.517157\pi$
$839$	20361.0	−	35266.3i	0.837830	−	1.45116i	0.891705	−	0.452617i	$-0.149510\pi$
$840$		0			0
$841$	3874.00	+	6709.96i	0.158842	+	0.275123i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9540.00			0.388386
$846$		0			0
$847$	27500.0			1.11560
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3111.00	+	5388.41i	0.125316	+	0.217053i
$852$		0			0
$853$	−6056.50	+	10490.2i	−0.243107	+	0.421074i	−0.961598	−	0.274463i	$-0.911500\pi$
$853$	−6056.50	+	10490.2i	−0.243107	+	0.421074i	0.718490	+	0.695537i	$0.244833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	963.000	−	1667.96i	0.0383844	−	0.0664838i	−0.846195	−	0.532873i	$-0.821112\pi$
$857$	963.000	−	1667.96i	0.0383844	−	0.0664838i	0.884579	+	0.466390i	$0.154446\pi$
$858$		0			0
$859$	−2197.00	−	3805.32i	−0.0872650	−	0.151147i	0.819089	−	0.573666i	$-0.194479\pi$
$859$	−2197.00	−	3805.32i	−0.0872650	−	0.151147i	−0.906354	+	0.422519i	$0.861146\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37491.0			−1.47880			−0.739402	−	0.673264i	$-0.764892\pi$
$863$	−37491.0			−1.47880			−0.739402	+	0.673264i	$0.764892\pi$
$864$		0			0
$865$	870.000			0.0341976
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1570.50	−	2720.19i	−0.0613067	−	0.106186i
$870$		0			0
$871$	2074.00	−	3592.27i	0.0806829	−	0.139747i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1375.00	+	2381.57i	−0.0531240	+	0.0920134i
$876$		0			0
$877$	10662.5	+	18468.0i	0.410544	+	0.711083i	0.994949	−	0.100379i	$-0.0320055\pi$
$877$	10662.5	+	18468.0i	0.410544	+	0.711083i	−0.584405	+	0.811462i	$0.698672\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17982.0			0.687661			0.343830	−	0.939032i	$-0.388276\pi$
$881$	17982.0			0.687661			0.343830	+	0.939032i	$0.388276\pi$
$882$		0			0
$883$	−25636.0			−0.977033			−0.488516	−	0.872555i	$-0.662462\pi$
$883$	−25636.0			−0.977033			−0.488516	+	0.872555i	$0.662462\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2110.50	−	3655.49i	−0.0798914	−	0.138376i	0.823312	−	0.567590i	$-0.192124\pi$
$887$	−2110.50	−	3655.49i	−0.0798914	−	0.138376i	−0.903203	+	0.429214i	$0.858791\pi$
$888$		0			0
$889$	6358.00	−	11012.4i	0.239866	−	0.415459i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1218.00	−	2109.64i	0.0456426	−	0.0790553i
$894$		0			0
$895$	−11490.0	−	19901.3i	−0.429127	−	0.743269i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24123.0			−0.894936
$900$		0			0
$901$	17100.0			0.632279
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1370.00	+	2372.91i	0.0503208	+	0.0871582i
$906$		0			0
$907$	14196.5	−	24589.1i	0.519721	−	0.900183i	−0.480016	−	0.877260i	$-0.659369\pi$
$907$	14196.5	−	24589.1i	0.519721	−	0.900183i	0.999737	−	0.0229237i	$-0.00729748\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5028.00	−	8708.75i	0.182860	−	0.316722i	−0.759994	−	0.649931i	$-0.774798\pi$
$911$	5028.00	−	8708.75i	0.182860	−	0.316722i	0.942853	+	0.333208i	$0.108131\pi$
$912$		0			0
$913$	4671.00	+	8090.41i	0.169318	+	0.293268i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	17226.0			0.620341
$918$		0			0
$919$	−37249.0			−1.33703			−0.668515	−	0.743698i	$-0.733070\pi$
$919$	−37249.0			−1.33703			−0.668515	+	0.743698i	$0.733070\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3774.00	−	6536.76i	−0.134586	−	0.233109i
$924$		0			0
$925$	425.000	−	736.122i	0.0151069	−	0.0261660i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	834.000	−	1444.53i	0.0294539	−	0.0510156i	−0.850923	−	0.525291i	$-0.823957\pi$
$929$	834.000	−	1444.53i	0.0294539	−	0.0510156i	0.880377	+	0.474275i	$0.157290\pi$
$930$		0			0
$931$	282.000	+	488.438i	0.00992715	+	0.0171943i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3375.00			−0.118047
$936$		0			0
$937$	2786.00			0.0971341			0.0485671	−	0.998820i	$-0.484535\pi$
$937$	2786.00			0.0971341			0.0485671	+	0.998820i	$0.484535\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4306.50	−	7459.08i	−0.149190	−	0.258405i	0.781738	−	0.623607i	$-0.214333\pi$
$941$	−4306.50	−	7459.08i	−0.149190	−	0.258405i	−0.930928	+	0.365202i	$0.881000\pi$
$942$		0			0
$943$	−24156.0	+	41839.4i	−0.834176	+	1.44483i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27801.0	−	48152.7i	0.953972	−	1.65233i	0.217267	−	0.976112i	$-0.430286\pi$
$947$	27801.0	−	48152.7i	0.953972	−	1.65233i	0.736704	−	0.676215i	$-0.236381\pi$
$948$		0			0
$949$	−3383.00	−	5859.53i	−0.115718	−	0.200430i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10785.0			−0.366590			−0.183295	−	0.983058i	$-0.558676\pi$
$953$	−10785.0			−0.366590			−0.183295	+	0.983058i	$0.558676\pi$
$954$		0			0
$955$	−23190.0			−0.785770
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2574.00	+	4458.30i	0.0866724	+	0.150121i
$960$		0			0
$961$	−2589.00	+	4484.28i	−0.0869054	+	0.150525i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8605.00	+	14904.3i	−0.287052	+	0.497188i
$966$		0			0
$967$	17531.0	+	30364.6i	0.582998	+	1.00978i	0.995122	+	0.0986541i	$0.0314538\pi$
$967$	17531.0	+	30364.6i	0.582998	+	1.00978i	−0.412124	+	0.911128i	$0.635213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−49701.0			−1.64262			−0.821308	−	0.570484i	$-0.806755\pi$
$971$	−49701.0			−1.64262			−0.821308	+	0.570484i	$0.806755\pi$
$972$		0			0
$973$	−69740.0			−2.29780
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3955.50	+	6851.13i	0.129527	+	0.224347i	0.923493	−	0.383614i	$-0.125321\pi$
$977$	3955.50	+	6851.13i	0.129527	+	0.224347i	−0.793967	+	0.607961i	$0.791988\pi$
$978$		0			0
$979$	3834.00	−	6640.68i	0.125164	−	0.216790i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18859.5	+	32665.6i	−0.611927	+	1.05989i	0.378988	+	0.925402i	$0.376272\pi$
$983$	−18859.5	+	32665.6i	−0.611927	+	1.05989i	−0.990915	+	0.134488i	$0.957061\pi$
$984$		0			0
$985$	−8280.00	−	14341.4i	−0.267840	−	0.463913i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	81069.0			2.60652
$990$		0			0
$991$	54725.0			1.75418			0.877092	−	0.480322i	$-0.159480\pi$
$991$	54725.0			1.75418			0.877092	+	0.480322i	$0.159480\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4362.50	+	7556.07i	0.138996	+	0.240747i
$996$		0			0
$997$	3252.50	−	5633.50i	0.103318	−	0.178951i	−0.809732	−	0.586800i	$-0.800388\pi$
$997$	3252.50	−	5633.50i	0.103318	−	0.178951i	0.913050	+	0.407848i	$0.133721\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.k.541.1		2
3.2	odd	2		1620.4.i.e.541.1		2
9.2	odd	6		540.4.a.c.1.1	yes	1
9.4	even	3	inner	1620.4.i.k.1081.1		2
9.5	odd	6		1620.4.i.e.1081.1		2
9.7	even	3		540.4.a.a.1.1	✓	1
36.7	odd	6		2160.4.a.h.1.1		1
36.11	even	6		2160.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.a.1.1	✓	1	9.7	even	3
540.4.a.c.1.1	yes	1	9.2	odd	6
1620.4.i.e.541.1		2	3.2	odd	2
1620.4.i.e.1081.1		2	9.5	odd	6
1620.4.i.k.541.1		2	1.1	even	1	trivial
1620.4.i.k.1081.1		2	9.4	even	3	inner
2160.4.a.h.1.1		1	36.7	odd	6
2160.4.a.s.1.1		1	36.11	even	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(11.0000 - 19.0526i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(11.0000 - 19.0526i) q^{7} +(4.50000 - 7.79423i) q^{11} +(-8.50000 - 14.7224i) q^{13} -75.0000 q^{17} -4.00000 q^{19} +(-91.5000 - 158.483i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-64.5000 + 111.717i) q^{29} +(93.5000 + 161.947i) q^{31} +110.000 q^{35} -34.0000 q^{37} +(-132.000 - 228.631i) q^{41} +(-221.500 + 383.649i) q^{43} +(-304.500 + 527.409i) q^{47} +(-70.5000 - 122.110i) q^{49} -228.000 q^{53} +45.0000 q^{55} +(-30.0000 - 51.9615i) q^{59} +(227.000 - 393.176i) q^{61} +(42.5000 - 73.6122i) q^{65} +(122.000 + 211.310i) q^{67} +444.000 q^{71} +398.000 q^{73} +(-99.0000 - 171.473i) q^{77} +(174.500 - 302.243i) q^{79} +(-519.000 + 898.934i) q^{83} +(-187.500 - 324.760i) q^{85} +852.000 q^{89} -374.000 q^{91} +(-10.0000 - 17.3205i) q^{95} +(-457.000 + 791.547i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{5} + 22 q^{7}+O(q^{10}) \) 2 * q + 5 * q^5 + 22 * q^7	\( 2 q + 5 q^{5} + 22 q^{7} + 9 q^{11} - 17 q^{13} - 150 q^{17} - 8 q^{19} - 183 q^{23} - 25 q^{25} - 129 q^{29} + 187 q^{31} + 220 q^{35} - 68 q^{37} - 264 q^{41} - 443 q^{43} - 609 q^{47} - 141 q^{49} - 456 q^{53} + 90 q^{55} - 60 q^{59} + 454 q^{61} + 85 q^{65} + 244 q^{67} + 888 q^{71} + 796 q^{73} - 198 q^{77} + 349 q^{79} - 1038 q^{83} - 375 q^{85} + 1704 q^{89} - 748 q^{91} - 20 q^{95} - 914 q^{97}+O(q^{100}) \) 2 * q + 5 * q^5 + 22 * q^7 + 9 * q^11 - 17 * q^13 - 150 * q^17 - 8 * q^19 - 183 * q^23 - 25 * q^25 - 129 * q^29 + 187 * q^31 + 220 * q^35 - 68 * q^37 - 264 * q^41 - 443 * q^43 - 609 * q^47 - 141 * q^49 - 456 * q^53 + 90 * q^55 - 60 * q^59 + 454 * q^61 + 85 * q^65 + 244 * q^67 + 888 * q^71 + 796 * q^73 - 198 * q^77 + 349 * q^79 - 1038 * q^83 - 375 * q^85 + 1704 * q^89 - 748 * q^91 - 20 * q^95 - 914 * q^97

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