LMFDB - Embedded newform 1620.4.i.q.541.2

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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} - 6x + 36 $ x^4 - x^3 - 5x^2 - 6x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.2
Root			$-1.82666 + 1.63197i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.q.1081.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(9.95994 - 17.2511i) q^{7} +O(q^{10})$	$q+(2.50000 + 4.33013i) q^{5} +(9.95994 - 17.2511i) q^{7} +(16.9599 - 29.3755i) q^{11} +(3.95994 + 6.85881i) q^{13} +12.9199 q^{17} +38.9199 q^{19} +(-42.3397 - 73.3346i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(51.7997 - 89.7197i) q^{29} +(-151.220 - 261.920i) q^{31} +99.5994 q^{35} +74.0000 q^{37} +(56.6394 + 98.1024i) q^{41} +(-166.800 + 288.906i) q^{43} +(121.599 - 210.616i) q^{47} +(-26.9006 - 46.5933i) q^{49} +447.638 q^{53} +169.599 q^{55} +(-175.720 - 304.355i) q^{59} +(-134.740 + 233.377i) q^{61} +(-19.7997 + 34.2940i) q^{65} +(-235.240 - 407.448i) q^{67} -903.837 q^{71} -80.3173 q^{73} +(-337.840 - 585.156i) q^{77} +(-326.380 + 565.306i) q^{79} +(-62.3782 + 108.042i) q^{83} +(32.2997 + 55.9447i) q^{85} -149.760 q^{89} +157.763 q^{91} +(97.2997 + 168.528i) q^{95} +(-395.359 + 684.782i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{5} - 10 q^{7}+O(q^{10}) $ 4 * q + 10 * q^5 - 10 * q^7	$ 4 q + 10 q^{5} - 10 q^{7} + 18 q^{11} - 34 q^{13} - 48 q^{17} + 56 q^{19} + 30 q^{23} - 50 q^{25} - 42 q^{29} - 256 q^{31} - 100 q^{35} + 296 q^{37} - 222 q^{41} - 418 q^{43} - 12 q^{47} - 606 q^{49} + 96 q^{53} + 180 q^{55} - 354 q^{59} - 838 q^{61} + 170 q^{65} - 1240 q^{67} - 924 q^{71} + 1772 q^{73} - 1152 q^{77} - 1156 q^{79} + 1146 q^{83} - 120 q^{85} - 300 q^{89} + 2824 q^{91} + 140 q^{95} - 784 q^{97}+O(q^{100}) $ 4 * q + 10 * q^5 - 10 * q^7 + 18 * q^11 - 34 * q^13 - 48 * q^17 + 56 * q^19 + 30 * q^23 - 50 * q^25 - 42 * q^29 - 256 * q^31 - 100 * q^35 + 296 * q^37 - 222 * q^41 - 418 * q^43 - 12 * q^47 - 606 * q^49 + 96 * q^53 + 180 * q^55 - 354 * q^59 - 838 * q^61 + 170 * q^65 - 1240 * q^67 - 924 * q^71 + 1772 * q^73 - 1152 * q^77 - 1156 * q^79 + 1146 * q^83 - 120 * q^85 - 300 * q^89 + 2824 * q^91 + 140 * q^95 - 784 * 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
$7$	9.95994	−	17.2511i	0.537786	−	0.931473i	−0.461237	−	0.887277i	$-0.652594\pi$
$7$	9.95994	−	17.2511i	0.537786	−	0.931473i	0.999023	−	0.0441956i	$-0.0140725\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	16.9599	−	29.3755i	0.464874	−	0.805185i	−0.534322	−	0.845281i	$-0.679433\pi$
$11$	16.9599	−	29.3755i	0.464874	−	0.805185i	0.999196	+	0.0400958i	$0.0127663\pi$
$12$		0			0
$13$	3.95994	+	6.85881i	0.0844837	+	0.146330i	0.905171	−	0.425047i	$-0.139743\pi$
$13$	3.95994	+	6.85881i	0.0844837	+	0.146330i	−0.820687	+	0.571377i	$0.806409\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.9199			0.184325			0.0921626	−	0.995744i	$-0.470622\pi$
$17$	12.9199			0.184325			0.0921626	+	0.995744i	$0.470622\pi$
$18$		0			0
$19$	38.9199			0.469938			0.234969	−	0.972003i	$-0.424501\pi$
$19$	38.9199			0.469938			0.234969	+	0.972003i	$0.424501\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−42.3397	−	73.3346i	−0.383846	−	0.664840i	0.607763	−	0.794119i	$-0.292067\pi$
$23$	−42.3397	−	73.3346i	−0.383846	−	0.664840i	−0.991608	+	0.129279i	$0.958734\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	51.7997	−	89.7197i	0.331688	−	0.574501i	−0.651155	−	0.758945i	$-0.725715\pi$
$29$	51.7997	−	89.7197i	0.331688	−	0.574501i	0.982843	+	0.184444i	$0.0590485\pi$
$30$		0			0
$31$	−151.220	−	261.920i	−0.876124	−	1.51749i	−0.855561	−	0.517702i	$-0.826788\pi$
$31$	−151.220	−	261.920i	−0.876124	−	1.51749i	−0.0205627	−	0.999789i	$-0.506546\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	99.5994			0.481010
$36$		0			0
$37$	74.0000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	74.0000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	56.6394	+	98.1024i	0.215746	+	0.373683i	0.953503	−	0.301383i	$-0.0974483\pi$
$41$	56.6394	+	98.1024i	0.215746	+	0.373683i	−0.737757	+	0.675066i	$0.764115\pi$
$42$		0			0
$43$	−166.800	+	288.906i	−0.591551	+	1.02460i	0.402472	+	0.915432i	$0.368151\pi$
$43$	−166.800	+	288.906i	−0.591551	+	1.02460i	−0.994024	+	0.109165i	$0.965182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	121.599	−	210.616i	0.377385	−	0.653650i	−0.613296	−	0.789853i	$-0.710157\pi$
$47$	121.599	−	210.616i	0.377385	−	0.653650i	0.990681	+	0.136203i	$0.0434901\pi$
$48$		0			0
$49$	−26.9006	−	46.5933i	−0.0784275	−	0.135840i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	447.638			1.16015			0.580073	−	0.814564i	$-0.303024\pi$
$53$	447.638			1.16015			0.580073	+	0.814564i	$0.303024\pi$
$54$		0			0
$55$	169.599			0.415796
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−175.720	−	304.355i	−0.387741	−	0.671588i	0.604404	−	0.796678i	$-0.293411\pi$
$59$	−175.720	−	304.355i	−0.387741	−	0.671588i	−0.992145	+	0.125090i	$0.960078\pi$
$60$		0			0
$61$	−134.740	+	233.377i	−0.282815	+	0.489851i	−0.972077	−	0.234662i	$-0.924602\pi$
$61$	−134.740	+	233.377i	−0.282815	+	0.489851i	0.689262	+	0.724512i	$0.257935\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−19.7997	+	34.2940i	−0.0377823	+	0.0654408i
$66$		0			0
$67$	−235.240	−	407.448i	−0.428943	−	0.742951i	0.567836	−	0.823141i	$-0.307781\pi$
$67$	−235.240	−	407.448i	−0.428943	−	0.742951i	−0.996780	+	0.0801901i	$0.974447\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−903.837			−1.51078			−0.755392	−	0.655273i	$-0.772554\pi$
$71$	−903.837			−1.51078			−0.755392	+	0.655273i	$0.772554\pi$
$72$		0			0
$73$	−80.3173			−0.128773			−0.0643865	−	0.997925i	$-0.520509\pi$
$73$	−80.3173			−0.128773			−0.0643865	+	0.997925i	$0.520509\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−337.840	−	585.156i	−0.500005	−	0.866035i
$78$		0			0
$79$	−326.380	+	565.306i	−0.464818	+	0.805088i	−0.999193	−	0.0401593i	$-0.987213\pi$
$79$	−326.380	+	565.306i	−0.464818	+	0.805088i	0.534376	+	0.845247i	$0.320547\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−62.3782	+	108.042i	−0.0824927	+	0.142882i	−0.904320	−	0.426855i	$-0.859621\pi$
$83$	−62.3782	+	108.042i	−0.0824927	+	0.142882i	0.821827	+	0.569737i	$0.192955\pi$
$84$		0			0
$85$	32.2997	+	55.9447i	0.0412164	+	0.0713889i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−149.760			−0.178365			−0.0891825	−	0.996015i	$-0.528425\pi$
$89$	−149.760			−0.178365			−0.0891825	+	0.996015i	$0.528425\pi$
$90$		0			0
$91$	157.763			0.181737
$92$		0			0
$93$		0			0
$94$		0			0
$95$	97.2997	+	168.528i	0.105081	+	0.182006i
$96$		0			0
$97$	−395.359	+	684.782i	−0.413841	+	0.716794i	−0.995306	−	0.0967773i	$-0.969147\pi$
$97$	−395.359	+	684.782i	−0.413841	+	0.716794i	0.581465	+	0.813572i	$0.302480\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	398.321	−	689.911i	0.392420	−	0.679691i	−0.600349	−	0.799738i	$-0.704971\pi$
$101$	398.321	−	689.911i	0.392420	−	0.679691i	0.992768	+	0.120048i	$0.0383048\pi$
$102$		0			0
$103$	−968.077	−	1676.76i	−0.926092	−	1.60404i	−0.789796	−	0.613370i	$-0.789814\pi$
$103$	−968.077	−	1676.76i	−0.926092	−	1.60404i	−0.136296	−	0.990668i	$-0.543520\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−676.474			−0.611189			−0.305595	−	0.952162i	$-0.598855\pi$
$107$	−676.474			−0.611189			−0.305595	+	0.952162i	$0.598855\pi$
$108$		0			0
$109$	1903.55			1.67273			0.836363	−	0.548176i	$-0.184678\pi$
$109$	1903.55			1.67273			0.836363	+	0.548176i	$0.184678\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	65.4776	+	113.410i	0.0545098	+	0.0944138i	0.891993	−	0.452050i	$-0.149307\pi$
$113$	65.4776	+	113.410i	0.0545098	+	0.0944138i	−0.837483	+	0.546464i	$0.815974\pi$
$114$		0			0
$115$	211.699	−	366.673i	0.171661	−	0.297325i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	128.681	−	222.882i	0.0991275	−	0.171694i
$120$		0			0
$121$	90.2212	+	156.268i	0.0677845	+	0.117406i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	997.519			0.696973			0.348486	−	0.937314i	$-0.386696\pi$
$127$	997.519			0.696973			0.348486	+	0.937314i	$0.386696\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1004.44	−	1739.73i	−0.669908	−	1.16031i	−0.977929	−	0.208936i	$-0.933000\pi$
$131$	−1004.44	−	1739.73i	−0.669908	−	1.16031i	0.308021	−	0.951379i	$-0.400333\pi$
$132$		0			0
$133$	387.639	−	671.411i	0.252726	−	0.437735i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−586.976	+	1016.67i	−0.366049	+	0.634016i	−0.988944	−	0.148290i	$-0.952623\pi$
$137$	−586.976	+	1016.67i	−0.366049	+	0.634016i	0.622895	+	0.782306i	$0.285956\pi$
$138$		0			0
$139$	826.154	+	1430.94i	0.504126	+	0.873171i	0.999989	+	0.00477036i	$0.00151846\pi$
$139$	826.154	+	1430.94i	0.504126	+	0.873171i	−0.495863	+	0.868401i	$0.665148\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	268.641			0.157097
$144$		0			0
$145$	517.997			0.296671
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−754.563	−	1306.94i	−0.414874	−	0.718582i	0.580542	−	0.814231i	$-0.302841\pi$
$149$	−754.563	−	1306.94i	−0.414874	−	0.718582i	−0.995415	+	0.0956484i	$0.969508\pi$
$150$		0			0
$151$	−322.000	+	557.720i	−0.173536	+	0.300574i	−0.939654	−	0.342127i	$-0.888853\pi$
$151$	−322.000	+	557.720i	−0.173536	+	0.300574i	0.766117	+	0.642701i	$0.222186\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	756.098	−	1309.60i	0.391814	−	0.678642i
$156$		0			0
$157$	487.441	+	844.272i	0.247783	+	0.429174i	0.962911	−	0.269821i	$-0.0869645\pi$
$157$	487.441	+	844.272i	0.247783	+	0.429174i	−0.715127	+	0.698994i	$0.753631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1686.80			−0.825707
$162$		0			0
$163$	−1377.67			−0.662007			−0.331004	−	0.943629i	$-0.607387\pi$
$163$	−1377.67			−0.662007			−0.331004	+	0.943629i	$0.607387\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1949.86	−	3377.25i	−0.903501	−	1.56491i	−0.822917	−	0.568161i	$-0.807655\pi$
$167$	−1949.86	−	3377.25i	−0.903501	−	1.56491i	−0.0805835	−	0.996748i	$-0.525678\pi$
$168$		0			0
$169$	1067.14	−	1848.34i	0.485725	−	0.841300i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2148.45	−	3721.23i	0.944183	−	1.63537i	0.186804	−	0.982397i	$-0.440187\pi$
$173$	2148.45	−	3721.23i	0.944183	−	1.63537i	0.757379	−	0.652976i	$-0.226480\pi$
$174$		0			0
$175$	248.998	+	431.278i	0.107557	+	0.186295i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1381.28			0.576768			0.288384	−	0.957515i	$-0.406882\pi$
$179$	1381.28			0.576768			0.288384	+	0.957515i	$0.406882\pi$
$180$		0			0
$181$	−2735.47			−1.12335			−0.561675	−	0.827358i	$-0.689843\pi$
$181$	−2735.47			−1.12335			−0.561675	+	0.827358i	$0.689843\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	185.000	+	320.429i	0.0735215	+	0.127343i
$186$		0			0
$187$	219.120	−	379.527i	0.0856880	−	0.148416i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2185.16	−	3784.81i	0.827815	−	1.43382i	−0.0719339	−	0.997409i	$-0.522917\pi$
$191$	2185.16	−	3784.81i	0.827815	−	1.43382i	0.899749	−	0.436408i	$-0.143750\pi$
$192$		0			0
$193$	796.524	+	1379.62i	0.297073	+	0.514545i	0.975465	−	0.220155i	$-0.0706563\pi$
$193$	796.524	+	1379.62i	0.297073	+	0.514545i	−0.678392	+	0.734700i	$0.737323\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4091.06			1.47957			0.739787	−	0.672841i	$-0.234926\pi$
$197$	4091.06			1.47957			0.739787	+	0.672841i	$0.234926\pi$
$198$		0			0
$199$	1817.26			0.647348			0.323674	−	0.946169i	$-0.395082\pi$
$199$	1817.26			0.647348			0.323674	+	0.946169i	$0.395082\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1031.84	−	1787.20i	−0.356755	−	0.617917i
$204$		0			0
$205$	−283.197	+	490.512i	−0.0964846	+	0.167116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	660.079	−	1143.29i	0.218462	−	0.378388i
$210$		0			0
$211$	−1184.93	−	2052.37i	−0.386608	−	0.669624i	0.605383	−	0.795934i	$-0.293020\pi$
$211$	−1184.93	−	2052.37i	−0.386608	−	0.669624i	−0.991991	+	0.126310i	$0.959687\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1668.00			−0.529100
$216$		0			0
$217$	−6024.55			−1.88467
$218$		0			0
$219$		0			0
$220$		0			0
$221$	51.1619	+	88.6149i	0.0155725	+	0.0269723i
$222$		0			0
$223$	−161.474	+	279.682i	−0.0484893	+	0.0839860i	−0.889251	−	0.457419i	$-0.848774\pi$
$223$	−161.474	+	279.682i	−0.0484893	+	0.0839860i	0.840762	+	0.541405i	$0.182107\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1493.93	−	2587.57i	0.436810	−	0.756577i	−0.560632	−	0.828065i	$-0.689442\pi$
$227$	1493.93	−	2587.57i	0.436810	−	0.756577i	0.997441	+	0.0714887i	$0.0227750\pi$
$228$		0			0
$229$	1970.98	+	3413.83i	0.568759	+	0.985120i	0.996689	+	0.0813079i	$0.0259097\pi$
$229$	1970.98	+	3413.83i	0.568759	+	0.985120i	−0.427930	+	0.903812i	$0.640757\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1996.08			−0.561233			−0.280617	−	0.959820i	$-0.590539\pi$
$233$	−1996.08			−0.561233			−0.280617	+	0.959820i	$0.590539\pi$
$234$		0			0
$235$	1215.99			0.337543
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2979.07	−	5159.91i	−0.806277	−	1.39651i	−0.915425	−	0.402488i	$-0.868146\pi$
$239$	−2979.07	−	5159.91i	−0.806277	−	1.39651i	0.109148	−	0.994025i	$-0.465188\pi$
$240$		0			0
$241$	649.304	−	1124.63i	0.173549	−	0.300596i	−0.766109	−	0.642711i	$-0.777810\pi$
$241$	649.304	−	1124.63i	0.173549	−	0.300596i	0.939658	+	0.342114i	$0.111143\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	134.503	−	232.966i	0.0350739	−	0.0607497i
$246$		0			0
$247$	154.120	+	266.944i	0.0397022	+	0.0687662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1809.93			−0.455146			−0.227573	−	0.973761i	$-0.573079\pi$
$251$	−1809.93			−0.455146			−0.227573	+	0.973761i	$0.573079\pi$
$252$		0			0
$253$	−2872.32			−0.713759
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2279.38	−	3948.01i	−0.553245	−	0.958249i	−0.998038	−	0.0626151i	$-0.980056\pi$
$257$	−2279.38	−	3948.01i	−0.553245	−	0.958249i	0.444793	−	0.895634i	$-0.353277\pi$
$258$		0			0
$259$	737.035	−	1276.58i	0.176823	−	0.306266i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−406.151	+	703.474i	−0.0952256	+	0.164935i	−0.909703	−	0.415260i	$-0.863691\pi$
$263$	−406.151	+	703.474i	−0.0952256	+	0.164935i	0.814477	+	0.580196i	$0.197024\pi$
$264$		0			0
$265$	1119.09	+	1938.33i	0.259417	+	0.449323i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2325.66			−0.527130			−0.263565	−	0.964642i	$-0.584898\pi$
$269$	−2325.66			−0.527130			−0.263565	+	0.964642i	$0.584898\pi$
$270$		0			0
$271$	−1442.20			−0.323275			−0.161638	−	0.986850i	$-0.551678\pi$
$271$	−1442.20			−0.323275			−0.161638	+	0.986850i	$0.551678\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	423.998	+	734.387i	0.0929748	+	0.161037i
$276$		0			0
$277$	1864.95	−	3230.19i	0.404528	−	0.700663i	−0.589739	−	0.807594i	$-0.700769\pi$
$277$	1864.95	−	3230.19i	0.404528	−	0.700663i	0.994266	+	0.106932i	$0.0341026\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3774.28	−	6537.24i	0.801262	−	1.38783i	−0.117525	−	0.993070i	$-0.537496\pi$
$281$	3774.28	−	6537.24i	0.801262	−	1.38783i	0.918786	−	0.394756i	$-0.129171\pi$
$282$		0			0
$283$	−519.200	−	899.281i	−0.109057	−	0.188893i	0.806331	−	0.591464i	$-0.201450\pi$
$283$	−519.200	−	899.281i	−0.109057	−	0.188893i	−0.915389	+	0.402571i	$0.868117\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2256.50			0.464101
$288$		0			0
$289$	−4746.08			−0.966024
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4678.73	−	8103.79i	−0.932881	−	1.61580i	−0.778370	−	0.627806i	$-0.783953\pi$
$293$	−4678.73	−	8103.79i	−0.932881	−	1.61580i	−0.154511	−	0.987991i	$-0.549380\pi$
$294$		0			0
$295$	878.598	−	1521.78i	0.173403	−	0.300343i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	335.325	−	580.801i	0.0648574	−	0.112336i
$300$		0			0
$301$	3322.63	+	5754.96i	0.636256	+	1.10203i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1347.40			−0.252958
$306$		0			0
$307$	1843.27			0.342674			0.171337	−	0.985212i	$-0.445191\pi$
$307$	1843.27			0.342674			0.171337	+	0.985212i	$0.445191\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5144.82	+	8911.09i	0.938058	+	1.62476i	0.769089	+	0.639142i	$0.220710\pi$
$311$	5144.82	+	8911.09i	0.938058	+	1.62476i	0.168969	+	0.985621i	$0.445956\pi$
$312$		0			0
$313$	4081.11	−	7068.69i	0.736991	−	1.27651i	−0.216854	−	0.976204i	$-0.569579\pi$
$313$	4081.11	−	7068.69i	0.736991	−	1.27651i	0.953844	−	0.300301i	$-0.0970872\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3276.14	+	5674.43i	−0.580461	+	1.00539i	0.414964	+	0.909838i	$0.363794\pi$
$317$	−3276.14	+	5674.43i	−0.580461	+	1.00539i	−0.995425	+	0.0955500i	$0.969539\pi$
$318$		0			0
$319$	−1757.04	−	3043.28i	−0.308386	−	0.534141i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	502.840			0.0866215
$324$		0			0
$325$	−197.997			−0.0337935
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2422.24	−	4195.45i	−0.405905	−	0.703047i
$330$		0			0
$331$	−1243.13	+	2153.16i	−0.206430	+	0.357547i	−0.950587	−	0.310457i	$-0.899518\pi$
$331$	−1243.13	+	2153.16i	−0.206430	+	0.357547i	0.744157	+	0.668004i	$0.232851\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1176.20	−	2037.24i	0.191829	−	0.332258i
$336$		0			0
$337$	−1082.58	−	1875.09i	−0.174992	−	0.303094i	0.765167	−	0.643832i	$-0.222656\pi$
$337$	−1082.58	−	1875.09i	−0.174992	−	0.303094i	−0.940158	+	0.340738i	$0.889323\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10258.7			−1.62915
$342$		0			0
$343$	5760.80			0.906863
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4338.48	−	7514.47i	−0.671187	−	1.16253i	−0.977568	−	0.210621i	$-0.932451\pi$
$347$	−4338.48	−	7514.47i	−0.671187	−	1.16253i	0.306381	−	0.951909i	$-0.400882\pi$
$348$		0			0
$349$	−1358.73	+	2353.39i	−0.208399	+	0.360957i	−0.951210	−	0.308543i	$-0.900159\pi$
$349$	−1358.73	+	2353.39i	−0.208399	+	0.360957i	0.742811	+	0.669501i	$0.233492\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2493.22	+	4318.39i	−0.375923	+	0.651117i	−0.990465	−	0.137767i	$-0.956008\pi$
$353$	−2493.22	+	4318.39i	−0.375923	+	0.651117i	0.614542	+	0.788884i	$0.289341\pi$
$354$		0			0
$355$	−2259.59	−	3913.73i	−0.337822	−	0.585124i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6477.49			0.952281			0.476141	−	0.879369i	$-0.342035\pi$
$359$	6477.49			0.952281			0.476141	+	0.879369i	$0.342035\pi$
$360$		0			0
$361$	−5344.24			−0.779158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−200.793	−	347.784i	−0.0287945	−	0.0498736i
$366$		0			0
$367$	−5272.59	+	9132.40i	−0.749937	+	1.29893i	0.197915	+	0.980219i	$0.436583\pi$
$367$	−5272.59	+	9132.40i	−0.749937	+	1.29893i	−0.947852	+	0.318710i	$0.896750\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4458.44	−	7722.25i	0.623911	−	1.08065i
$372$		0			0
$373$	−1888.64	−	3271.22i	−0.262171	−	0.454094i	0.704647	−	0.709558i	$-0.251105\pi$
$373$	−1888.64	−	3271.22i	−0.262171	−	0.454094i	−0.966819	+	0.255463i	$0.917772\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	820.494			0.112089
$378$		0			0
$379$	2741.57			0.371570			0.185785	−	0.982590i	$-0.440517\pi$
$379$	2741.57			0.371570			0.185785	+	0.982590i	$0.440517\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3749.37	−	6494.10i	−0.500219	−	0.866404i	−1.00000		0.000252619i	$-0.999920\pi$
$383$	−3749.37	−	6494.10i	−0.500219	−	0.866404i	0.499781	−	0.866152i	$-0.333414\pi$
$384$		0			0
$385$	1689.20	−	2925.78i	0.223609	−	0.387302i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1774.49	+	3073.50i	−0.231286	+	0.400599i	−0.958187	−	0.286144i	$-0.907627\pi$
$389$	−1774.49	+	3073.50i	−0.231286	+	0.400599i	0.726901	+	0.686742i	$0.240960\pi$
$390$		0			0
$391$	−547.024	−	947.473i	−0.0707524	−	0.122547i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3263.80			−0.415746
$396$		0			0
$397$	5807.90			0.734232			0.367116	−	0.930175i	$-0.380345\pi$
$397$	5807.90			0.734232			0.367116	+	0.930175i	$0.380345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4374.12	+	7576.19i	0.544720	+	0.943483i	0.998624	+	0.0524329i	$0.0166976\pi$
$401$	4374.12	+	7576.19i	0.544720	+	0.943483i	−0.453904	+	0.891051i	$0.649969\pi$
$402$		0			0
$403$	1197.64	−	2074.37i	0.148036	−	0.256407i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1255.04	−	2173.78i	0.152850	−	0.264743i
$408$		0			0
$409$	−1082.65	−	1875.20i	−0.130889	−	0.226706i	0.793131	−	0.609051i	$-0.208450\pi$
$409$	−1082.65	−	1875.20i	−0.130889	−	0.226706i	−0.924019	+	0.382346i	$0.875116\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7000.62			−0.834087
$414$		0			0
$415$	−623.782			−0.0737837
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3122.22	+	5407.85i	0.364035	+	0.630527i	0.988621	−	0.150429i	$-0.0480656\pi$
$419$	3122.22	+	5407.85i	0.364035	+	0.630527i	−0.624586	+	0.780956i	$0.714732\pi$
$420$		0			0
$421$	−7573.40	+	13117.5i	−0.876734	+	1.51855i	−0.0218305	+	0.999762i	$0.506949\pi$
$421$	−7573.40	+	13117.5i	−0.876734	+	1.51855i	−0.854904	+	0.518787i	$0.826384\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−161.498	+	279.723i	−0.0184325	+	0.0319261i
$426$		0			0
$427$	2684.01	+	4648.84i	0.304188	+	0.526870i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−736.587			−0.0823205			−0.0411602	−	0.999153i	$-0.513105\pi$
$431$	−736.587			−0.0823205			−0.0411602	+	0.999153i	$0.513105\pi$
$432$		0			0
$433$	14484.7			1.60760			0.803798	−	0.594902i	$-0.202809\pi$
$433$	14484.7			1.60760			0.803798	+	0.594902i	$0.202809\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1647.86	−	2854.17i	−0.180384	−	0.312434i
$438$		0			0
$439$	−5438.25	+	9419.33i	−0.591239	+	1.02406i	0.402827	+	0.915276i	$0.368028\pi$
$439$	−5438.25	+	9419.33i	−0.591239	+	1.02406i	−0.994066	+	0.108779i	$0.965306\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2568.48	−	4448.73i	0.275467	−	0.477124i	−0.694786	−	0.719217i	$-0.744501\pi$
$443$	2568.48	−	4448.73i	0.275467	−	0.477124i	0.970253	+	0.242093i	$0.0778341\pi$
$444$		0			0
$445$	−374.399	−	648.478i	−0.0398836	−	0.0690805i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5808.39			0.610501			0.305250	−	0.952272i	$-0.401260\pi$
$449$	5808.39			0.610501			0.305250	+	0.952272i	$0.401260\pi$
$450$		0			0
$451$	3842.40			0.401179
$452$		0			0
$453$		0			0
$454$		0			0
$455$	394.407	+	683.133i	0.0406376	+	0.0703863i
$456$		0			0
$457$	2470.14	−	4278.42i	0.252841	−	0.437934i	−0.711466	−	0.702721i	$-0.751968\pi$
$457$	2470.14	−	4278.42i	0.252841	−	0.437934i	0.964307	+	0.264787i	$0.0853016\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3525.47	−	6106.30i	0.356177	−	0.616917i	−0.631142	−	0.775668i	$-0.717413\pi$
$461$	3525.47	−	6106.30i	0.356177	−	0.616917i	0.987319	+	0.158751i	$0.0507467\pi$
$462$		0			0
$463$	723.298	+	1252.79i	0.0726015	+	0.125750i	0.900041	−	0.435806i	$-0.143537\pi$
$463$	723.298	+	1252.79i	0.0726015	+	0.125750i	−0.827439	+	0.561555i	$0.810203\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17875.7			1.77129			0.885644	−	0.464366i	$-0.153718\pi$
$467$	17875.7			1.77129			0.885644	+	0.464366i	$0.153718\pi$
$468$		0			0
$469$	−9371.92			−0.922718
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5657.82	+	9799.64i	0.549994	+	0.952617i
$474$		0			0
$475$	−486.498	+	842.640i	−0.0469938	+	0.0813957i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1782.55	−	3087.47i	0.170035	−	0.294509i	−0.768397	−	0.639974i	$-0.778945\pi$
$479$	1782.55	−	3087.47i	0.170035	−	0.294509i	0.938432	+	0.345464i	$0.112279\pi$
$480$		0			0
$481$	293.035	+	507.552i	0.0277781	+	0.0481130i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3953.59			−0.370151
$486$		0			0
$487$	−2707.17			−0.251896			−0.125948	−	0.992037i	$-0.540197\pi$
$487$	−2707.17			−0.251896			−0.125948	+	0.992037i	$0.540197\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−218.093	−	377.748i	−0.0200456	−	0.0347200i	0.855829	−	0.517260i	$-0.173048\pi$
$491$	−218.093	−	377.748i	−0.0200456	−	0.0347200i	−0.875874	+	0.482540i	$0.839714\pi$
$492$		0			0
$493$	669.245	−	1159.17i	0.0611385	−	0.105895i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9002.15	+	15592.2i	−0.812478	+	1.40725i
$498$		0			0
$499$	−6019.27	−	10425.7i	−0.539999	−	0.935306i	−0.998903	−	0.0468205i	$-0.985091\pi$
$499$	−6019.27	−	10425.7i	−0.539999	−	0.935306i	0.458904	−	0.888486i	$-0.348242\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14860.0			1.31725			0.658624	−	0.752472i	$-0.271139\pi$
$503$	14860.0			1.31725			0.658624	+	0.752472i	$0.271139\pi$
$504$		0			0
$505$	3983.21			0.350991
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8755.88	−	15165.6i	−0.762471	−	1.32064i	−0.941573	−	0.336808i	$-0.890653\pi$
$509$	−8755.88	−	15165.6i	−0.762471	−	1.32064i	0.179102	−	0.983830i	$-0.442681\pi$
$510$		0			0
$511$	−799.955	+	1385.56i	−0.0692523	+	0.119949i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4840.38	−	8383.79i	0.414161	−	0.717348i
$516$		0			0
$517$	−4124.63	−	7144.08i	−0.350873	−	0.607729i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10372.8			−0.872248			−0.436124	−	0.899887i	$-0.643649\pi$
$521$	−10372.8			−0.872248			−0.436124	+	0.899887i	$0.643649\pi$
$522$		0			0
$523$	20951.5			1.75172			0.875858	−	0.482569i	$-0.160296\pi$
$523$	20951.5			1.75172			0.875858	+	0.482569i	$0.160296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1953.74	−	3383.97i	−0.161492	−	0.279712i
$528$		0			0
$529$	2498.19	−	4327.00i	0.205325	−	0.355634i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−448.577	+	776.958i	−0.0364541	+	0.0631403i
$534$		0			0
$535$	−1691.19	−	2929.22i	−0.136666	−	0.236713i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1824.93			−0.145836
$540$		0			0
$541$	13085.4			1.03990			0.519951	−	0.854196i	$-0.325950\pi$
$541$	13085.4			1.03990			0.519951	+	0.854196i	$0.325950\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4758.88	+	8242.62i	0.374033	+	0.647844i
$546$		0			0
$547$	−8074.93	+	13986.2i	−0.631186	+	1.09325i	0.356123	+	0.934439i	$0.384098\pi$
$547$	−8074.93	+	13986.2i	−0.631186	+	1.09325i	−0.987309	+	0.158808i	$0.949235\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2016.04	−	3491.88i	0.155873	−	0.269980i
$552$		0			0
$553$	6501.44	+	11260.8i	0.499945	+	0.865930i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1305.37			0.0993000			0.0496500	−	0.998767i	$-0.484189\pi$
$557$	1305.37			0.0993000			0.0496500	+	0.998767i	$0.484189\pi$
$558$		0			0
$559$	−2642.06			−0.199906
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5912.76	+	10241.2i	0.442617	+	0.766635i	0.997883	−	0.0650381i	$-0.0207169\pi$
$563$	5912.76	+	10241.2i	0.442617	+	0.766635i	−0.555266	+	0.831673i	$0.687384\pi$
$564$		0			0
$565$	−327.388	+	567.052i	−0.0243775	+	0.0422231i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9446.11	+	16361.1i	−0.695960	+	1.20544i	0.273896	+	0.961759i	$0.411688\pi$
$569$	−9446.11	+	16361.1i	−0.695960	+	1.20544i	−0.969856	+	0.243679i	$0.921646\pi$
$570$		0			0
$571$	−7417.57	−	12847.6i	−0.543635	−	0.941603i	−0.998691	−	0.0511409i	$-0.983714\pi$
$571$	−7417.57	−	12847.6i	−0.543635	−	0.941603i	0.455056	−	0.890463i	$-0.349619\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2116.99			0.153538
$576$		0			0
$577$	−7354.74			−0.530644			−0.265322	−	0.964160i	$-0.585478\pi$
$577$	−7354.74			−0.530644			−0.265322	+	0.964160i	$0.585478\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1242.57	+	2152.19i	0.0887269	+	0.153679i
$582$		0			0
$583$	7591.91	−	13149.6i	0.539322	−	0.934133i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7660.48	+	13268.3i	−0.538640	+	0.932952i	0.460337	+	0.887744i	$0.347728\pi$
$587$	−7660.48	+	13268.3i	−0.538640	+	0.932952i	−0.998978	+	0.0452083i	$0.985605\pi$
$588$		0			0
$589$	−5885.45	−	10193.9i	−0.411724	−	0.713127i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7057.40			−0.488723			−0.244361	−	0.969684i	$-0.578578\pi$
$593$	−7057.40			−0.488723			−0.244361	+	0.969684i	$0.578578\pi$
$594$		0			0
$595$	1286.81			0.0886624
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12019.5	+	20818.4i	0.819875	+	1.42006i	0.905774	+	0.423761i	$0.139290\pi$
$599$	12019.5	+	20818.4i	0.819875	+	1.42006i	−0.0858998	+	0.996304i	$0.527376\pi$
$600$		0			0
$601$	−3007.38	+	5208.94i	−0.204116	+	0.353539i	−0.949851	−	0.312704i	$-0.898765\pi$
$601$	−3007.38	+	5208.94i	−0.204116	+	0.353539i	0.745735	+	0.666243i	$0.232099\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−451.106	+	781.338i	−0.0303141	+	0.0525056i
$606$		0			0
$607$	2293.51	+	3972.48i	0.153362	+	0.265631i	0.932461	−	0.361269i	$-0.117657\pi$
$607$	2293.51	+	3972.48i	0.153362	+	0.265631i	−0.779099	+	0.626901i	$0.784323\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1926.10			0.127532
$612$		0			0
$613$	17944.6			1.18234			0.591171	−	0.806546i	$-0.298666\pi$
$613$	17944.6			1.18234			0.591171	+	0.806546i	$0.298666\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8166.15	+	14144.2i	0.532831	+	0.922891i	0.999265	+	0.0383344i	$0.0122052\pi$
$617$	8166.15	+	14144.2i	0.532831	+	0.922891i	−0.466434	+	0.884556i	$0.654461\pi$
$618$		0			0
$619$	−480.144	+	831.634i	−0.0311771	+	0.0540003i	−0.881193	−	0.472757i	$-0.843259\pi$
$619$	−480.144	+	831.634i	−0.0311771	+	0.0540003i	0.850016	+	0.526757i	$0.176592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1491.60	+	2583.52i	−0.0959222	+	0.166142i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	956.070			0.0606058
$630$		0			0
$631$	−26358.4			−1.66293			−0.831467	−	0.555574i	$-0.812499\pi$
$631$	−26358.4			−1.66293			−0.831467	+	0.555574i	$0.812499\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2493.80	+	4319.38i	0.155848	+	0.269936i
$636$		0			0
$637$	213.050	−	369.013i	0.0132517	−	0.0229526i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10527.8	+	18234.7i	−0.648710	+	1.12360i	0.334721	+	0.942317i	$0.391358\pi$
$641$	−10527.8	+	18234.7i	−0.648710	+	1.12360i	−0.983431	+	0.181282i	$0.941975\pi$
$642$		0			0
$643$	1568.59	+	2716.88i	0.0962040	+	0.166630i	0.910110	−	0.414366i	$-0.135997\pi$
$643$	1568.59	+	2716.88i	0.0962040	+	0.166630i	−0.813906	+	0.580996i	$0.802663\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	773.506			0.0470010			0.0235005	−	0.999724i	$-0.492519\pi$
$647$	773.506			0.0470010			0.0235005	+	0.999724i	$0.492519\pi$
$648$		0			0
$649$	−11920.8			−0.721003
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10899.1	+	18877.9i	0.653164	+	1.13131i	0.982351	+	0.187049i	$0.0598922\pi$
$653$	10899.1	+	18877.9i	0.653164	+	1.13131i	−0.329187	+	0.944265i	$0.606775\pi$
$654$		0			0
$655$	5022.18	−	8698.67i	0.299592	−	0.518909i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6020.87	+	10428.5i	−0.355903	+	0.616442i	−0.987272	−	0.159041i	$-0.949160\pi$
$659$	−6020.87	+	10428.5i	−0.355903	+	0.616442i	0.631369	+	0.775482i	$0.282493\pi$
$660$		0			0
$661$	13271.4	+	22986.7i	0.780935	+	1.35262i	0.931398	+	0.364003i	$0.118590\pi$
$661$	13271.4	+	22986.7i	0.780935	+	1.35262i	−0.150463	+	0.988616i	$0.548077\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3876.39			0.226045
$666$		0			0
$667$	−8772.74			−0.509268
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4570.38	+	7916.12i	0.262947	+	0.455438i
$672$		0			0
$673$	13496.6	−	23376.8i	0.773040	−	1.33894i	−0.162850	−	0.986651i	$-0.552069\pi$
$673$	13496.6	−	23376.8i	0.773040	−	1.33894i	0.935890	−	0.352293i	$-0.114598\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−751.022	+	1300.81i	−0.0426354	+	0.0738466i	−0.886556	−	0.462622i	$-0.846909\pi$
$677$	−751.022	+	1300.81i	−0.0426354	+	0.0738466i	0.843920	+	0.536469i	$0.180242\pi$
$678$		0			0
$679$	7875.50	+	13640.8i	0.445116	+	0.770964i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8724.95			0.488801			0.244400	−	0.969674i	$-0.421409\pi$
$683$	8724.95			0.488801			0.244400	+	0.969674i	$0.421409\pi$
$684$		0			0
$685$	−5869.76			−0.327404
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1772.62	+	3070.26i	0.0980135	+	0.169764i
$690$		0			0
$691$	−5978.85	+	10355.7i	−0.329155	+	0.570113i	−0.982344	−	0.187081i	$-0.940097\pi$
$691$	−5978.85	+	10355.7i	−0.329155	+	0.570113i	0.653189	+	0.757195i	$0.273431\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4130.77	+	7154.70i	−0.225452	+	0.390494i
$696$		0			0
$697$	731.774	+	1267.47i	0.0397675	+	0.0688793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22726.6			−1.22450			−0.612248	−	0.790666i	$-0.709735\pi$
$701$	−22726.6			−1.22450			−0.612248	+	0.790666i	$0.709735\pi$
$702$		0			0
$703$	2880.07			0.154515
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7934.49	−	13742.9i	−0.422075	−	0.731056i
$708$		0			0
$709$	−2752.76	+	4767.93i	−0.145814	+	0.252557i	−0.929676	−	0.368377i	$-0.879913\pi$
$709$	−2752.76	+	4767.93i	−0.145814	+	0.252557i	0.783862	+	0.620935i	$0.213247\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12805.2	+	22179.2i	−0.672592	+	1.16496i
$714$		0			0
$715$	671.603	+	1163.25i	0.0351280	+	0.0608435i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3683.19			−0.191043			−0.0955215	−	0.995427i	$-0.530452\pi$
$719$	−3683.19			−0.191043			−0.0955215	+	0.995427i	$0.530452\pi$
$720$		0			0
$721$	−38567.9			−1.99216
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1294.99	+	2242.99i	0.0663376	+	0.114900i
$726$		0			0
$727$	−5077.57	+	8794.60i	−0.259032	+	0.448657i	−0.965983	−	0.258606i	$-0.916737\pi$
$727$	−5077.57	+	8794.60i	−0.259032	+	0.448657i	0.706951	+	0.707263i	$0.250070\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2155.03	+	3732.62i	−0.109038	+	0.188859i
$732$		0			0
$733$	−13812.1	−	23923.3i	−0.695992	−	1.20549i	−0.969845	−	0.243722i	$-0.921632\pi$
$733$	−13812.1	−	23923.3i	−0.695992	−	1.20549i	0.273853	−	0.961772i	$-0.411702\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15958.6			−0.797618
$738$		0			0
$739$	−16046.0			−0.798729			−0.399364	−	0.916792i	$-0.630769\pi$
$739$	−16046.0			−0.798729			−0.399364	+	0.916792i	$0.630769\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	216.519	+	375.022i	0.0106909	+	0.0185171i	0.871321	−	0.490713i	$-0.163264\pi$
$743$	216.519	+	375.022i	0.0106909	+	0.0185171i	−0.860630	+	0.509230i	$0.829930\pi$
$744$		0			0
$745$	3772.81	−	6534.70i	0.185537	−	0.321360i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6737.64	+	11669.9i	−0.328689	+	0.569306i
$750$		0			0
$751$	5795.56	+	10038.2i	0.281602	+	0.487749i	0.971779	−	0.235891i	$-0.0758009\pi$
$751$	5795.56	+	10038.2i	0.281602	+	0.487749i	−0.690178	+	0.723640i	$0.742468\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3220.00			−0.155216
$756$		0			0
$757$	−13898.7			−0.667317			−0.333658	−	0.942694i	$-0.608283\pi$
$757$	−13898.7			−0.667317			−0.333658	+	0.942694i	$0.608283\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19021.1	+	32945.5i	0.906062	+	1.56935i	0.819485	+	0.573101i	$0.194260\pi$
$761$	19021.1	+	32945.5i	0.906062	+	1.56935i	0.0865779	+	0.996245i	$0.472407\pi$
$762$		0			0
$763$	18959.2	−	32838.4i	0.899569	−	1.55810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1391.68	−	2410.45i	0.0655157	−	0.113476i
$768$		0			0
$769$	−2104.93	−	3645.84i	−0.0987070	−	0.170966i	0.812443	−	0.583041i	$-0.198137\pi$
$769$	−2104.93	−	3645.84i	−0.0987070	−	0.170966i	−0.911150	+	0.412075i	$0.864804\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16134.1			0.750714			0.375357	−	0.926880i	$-0.377520\pi$
$773$	16134.1			0.750714			0.375357	+	0.926880i	$0.377520\pi$
$774$		0			0
$775$	7560.98			0.350449
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2204.40	+	3818.13i	0.101387	+	0.175608i
$780$		0			0
$781$	−15329.0	+	26550.6i	−0.702324	+	1.21646i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2437.20	+	4221.36i	−0.110812	+	0.191932i
$786$		0			0
$787$	18085.7	+	31325.4i	0.819169	+	1.41884i	0.906295	+	0.422645i	$0.138898\pi$
$787$	18085.7	+	31325.4i	0.819169	+	1.41884i	−0.0871259	+	0.996197i	$0.527768\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2608.61			0.117258
$792$		0			0
$793$	−2134.25			−0.0955732
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13095.1	+	22681.4i	0.581998	+	1.00805i	0.995242	+	0.0974294i	$0.0310620\pi$
$797$	13095.1	+	22681.4i	0.581998	+	1.00805i	−0.413245	+	0.910620i	$0.635605\pi$
$798$		0			0
$799$	1571.05	−	2721.14i	0.0695616	−	0.120484i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1362.18	+	2359.36i	−0.0598632	+	0.103686i
$804$		0			0
$805$	−4217.01	−	7304.08i	−0.184634	−	0.319795i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16807.6			0.730436			0.365218	−	0.930922i	$-0.380994\pi$
$809$	16807.6			0.730436			0.365218	+	0.930922i	$0.380994\pi$
$810$		0			0
$811$	−16738.0			−0.724722			−0.362361	−	0.932038i	$-0.618029\pi$
$811$	−16738.0			−0.724722			−0.362361	+	0.932038i	$0.618029\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3444.17	−	5965.47i	−0.148029	−	0.256394i
$816$		0			0
$817$	−6491.82	+	11244.2i	−0.277993	+	0.481498i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12479.3	+	21614.7i	−0.530486	+	0.918829i	0.468881	+	0.883261i	$0.344657\pi$
$821$	−12479.3	+	21614.7i	−0.530486	+	0.918829i	−0.999367	+	0.0355676i	$0.988676\pi$
$822$		0			0
$823$	−416.377	−	721.185i	−0.0176354	−	0.0305455i	0.857073	−	0.515195i	$-0.172280\pi$
$823$	−416.377	−	721.185i	−0.0176354	−	0.0305455i	−0.874708	+	0.484649i	$0.838947\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16016.1			0.673441			0.336721	−	0.941605i	$-0.390682\pi$
$827$	16016.1			0.673441			0.336721	+	0.941605i	$0.390682\pi$
$828$		0			0
$829$	33169.9			1.38967			0.694835	−	0.719169i	$-0.255477\pi$
$829$	33169.9			1.38967			0.694835	+	0.719169i	$0.255477\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−347.553	−	601.979i	−0.0144562	−	0.0250388i
$834$		0			0
$835$	9749.29	−	16886.3i	0.404058	−	0.699849i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19443.3	−	33676.8i	0.800069	−	1.38576i	−0.119501	−	0.992834i	$-0.538130\pi$
$839$	19443.3	−	33676.8i	0.800069	−	1.38576i	0.919570	−	0.392926i	$-0.128537\pi$
$840$		0			0
$841$	6828.09	+	11826.6i	0.279966	+	0.484915i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10671.4			0.434446
$846$		0			0
$847$	3594.39			0.145814
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3133.14	−	5426.76i	−0.126208	−	0.218598i
$852$		0			0
$853$	15153.2	−	26246.0i	0.608247	−	1.05351i	−0.383283	−	0.923631i	$-0.625206\pi$
$853$	15153.2	−	26246.0i	0.608247	−	1.05351i	0.991529	−	0.129883i	$-0.0414602\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6939.27	+	12019.2i	−0.276594	+	0.479075i	−0.970536	−	0.240956i	$-0.922539\pi$
$857$	−6939.27	+	12019.2i	−0.276594	+	0.479075i	0.693942	+	0.720031i	$0.255872\pi$
$858$		0			0
$859$	20634.4	+	35739.8i	0.819601	+	1.41959i	0.905977	+	0.423327i	$0.139138\pi$
$859$	20634.4	+	35739.8i	0.819601	+	1.41959i	−0.0863762	+	0.996263i	$0.527529\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34667.6			1.36744			0.683719	−	0.729746i	$-0.260362\pi$
$863$	34667.6			1.36744			0.683719	+	0.729746i	$0.260362\pi$
$864$		0			0
$865$	21484.5			0.844503
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11070.8	+	19175.1i	0.432163	+	0.748529i
$870$		0			0
$871$	1863.07	−	3226.94i	0.0724774	−	0.125535i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1244.99	+	2156.39i	−0.0481010	+	0.0833134i
$876$		0			0
$877$	−17612.8	−	30506.3i	−0.678155	−	1.17460i	−0.975536	−	0.219840i	$-0.929446\pi$
$877$	−17612.8	−	30506.3i	−0.678155	−	1.17460i	0.297381	−	0.954759i	$-0.403887\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	38011.3			1.45361			0.726806	−	0.686843i	$-0.241004\pi$
$881$	38011.3			1.45361			0.726806	+	0.686843i	$0.241004\pi$
$882$		0			0
$883$	12891.8			0.491330			0.245665	−	0.969355i	$-0.420994\pi$
$883$	12891.8			0.491330			0.245665	+	0.969355i	$0.420994\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13511.9	+	23403.2i	0.511481	+	0.885912i	0.999911	+	0.0133086i	$0.00423639\pi$
$887$	13511.9	+	23403.2i	0.511481	+	0.885912i	−0.488430	+	0.872603i	$0.662430\pi$
$888$		0			0
$889$	9935.23	−	17208.3i	0.374822	−	0.649211i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4732.63	−	8197.16i	0.177348	−	0.307175i
$894$		0			0
$895$	3453.19	+	5981.10i	0.128969	+	0.223381i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31332.5			−1.16240
$900$		0			0
$901$	5783.42			0.213844
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6838.69	−	11845.0i	−0.251189	−	0.435071i
$906$		0			0
$907$	20544.9	−	35584.9i	0.752132	−	1.30273i	−0.194656	−	0.980872i	$-0.562359\pi$
$907$	20544.9	−	35584.9i	0.752132	−	1.30273i	0.946788	−	0.321859i	$-0.104308\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7295.55	−	12636.3i	0.265327	−	0.459559i	−0.702323	−	0.711859i	$-0.747854\pi$
$911$	7295.55	−	12636.3i	0.265327	−	0.459559i	0.967649	+	0.252300i	$0.0811869\pi$
$912$		0			0
$913$	2115.86	+	3664.78i	0.0766974	+	0.132844i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−40016.5			−1.44107
$918$		0			0
$919$	−28402.2			−1.01948			−0.509740	−	0.860329i	$-0.670258\pi$
$919$	−28402.2			−1.01948			−0.509740	+	0.860329i	$0.670258\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3579.13	−	6199.24i	−0.127637	−	0.221073i
$924$		0			0
$925$	−925.000	+	1602.15i	−0.0328798	+	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21281.4	+	36860.5i	−0.751583	+	1.30178i	0.195472	+	0.980709i	$0.437376\pi$
$929$	−21281.4	+	36860.5i	−0.751583	+	1.30178i	−0.947055	+	0.321070i	$0.895957\pi$
$930$		0			0
$931$	−1046.97	−	1813.40i	−0.0368561	−	0.0638367i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2191.20			0.0766417
$936$		0			0
$937$	−35567.9			−1.24008			−0.620039	−	0.784571i	$-0.712883\pi$
$937$	−35567.9			−1.24008			−0.620039	+	0.784571i	$0.712883\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12392.4	−	21464.3i	−0.429311	−	0.743588i	0.567502	−	0.823372i	$-0.307910\pi$
$941$	−12392.4	−	21464.3i	−0.429311	−	0.743588i	−0.996812	+	0.0797846i	$0.974577\pi$
$942$		0			0
$943$	4796.20	−	8307.26i	0.165626	−	0.286873i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22287.9	−	38603.8i	0.764794	−	1.32466i	−0.175561	−	0.984468i	$-0.556174\pi$
$947$	22287.9	−	38603.8i	0.764794	−	1.32466i	0.940355	−	0.340194i	$-0.110493\pi$
$948$		0			0
$949$	−318.051	−	550.881i	−0.0108792	−	0.0188434i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−56696.4			−1.92715			−0.963577	−	0.267431i	$-0.913825\pi$
$953$	−56696.4			−1.92715			−0.963577	+	0.267431i	$0.913825\pi$
$954$		0			0
$955$	21851.6			0.740420
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11692.5	+	20252.0i	0.393712	+	0.681930i
$960$		0			0
$961$	−30839.2	+	53415.1i	−1.03519	+	1.79299i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3982.62	+	6898.10i	−0.132855	+	0.230112i
$966$		0			0
$967$	1083.58	+	1876.81i	0.0360346	+	0.0624138i	0.883480	−	0.468468i	$-0.155194\pi$
$967$	1083.58	+	1876.81i	0.0360346	+	0.0624138i	−0.847446	+	0.530882i	$0.821861\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41957.3			1.38669			0.693343	−	0.720608i	$-0.256137\pi$
$971$	41957.3			1.38669			0.693343	+	0.720608i	$0.256137\pi$
$972$		0			0
$973$	32913.8			1.08445
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8289.47	+	14357.8i	0.271447	+	0.470160i	0.969233	−	0.246147i	$-0.0791645\pi$
$977$	8289.47	+	14357.8i	0.271447	+	0.470160i	−0.697786	+	0.716307i	$0.745831\pi$
$978$		0			0
$979$	−2539.91	+	4399.26i	−0.0829172	+	0.143617i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13719.1	+	23762.2i	−0.445140	+	0.771005i	−0.998062	−	0.0622280i	$-0.980179\pi$
$983$	−13719.1	+	23762.2i	−0.445140	+	0.771005i	0.552922	+	0.833233i	$0.313513\pi$
$984$		0			0
$985$	10227.7	+	17714.8i	0.330843	+	0.573036i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28249.0			0.908257
$990$		0			0
$991$	−33143.5			−1.06240			−0.531200	−	0.847246i	$-0.678259\pi$
$991$	−33143.5			−1.06240			−0.531200	+	0.847246i	$0.678259\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4543.16	+	7868.98i	0.144752	+	0.250717i
$996$		0			0
$997$	−18890.4	+	32719.0i	−0.600064	+	1.03934i	0.392747	+	0.919646i	$0.371525\pi$
$997$	−18890.4	+	32719.0i	−0.600064	+	1.03934i	−0.992811	+	0.119694i	$0.961809\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.q.541.2		4
3.2	odd	2		1620.4.i.n.541.2		4
9.2	odd	6		540.4.a.i.1.1	yes	2
9.4	even	3	inner	1620.4.i.q.1081.2		4
9.5	odd	6		1620.4.i.n.1081.2		4
9.7	even	3		540.4.a.f.1.1	✓	2
36.7	odd	6		2160.4.a.v.1.2		2
36.11	even	6		2160.4.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.f.1.1	✓	2	9.7	even	3
540.4.a.i.1.1	yes	2	9.2	odd	6
1620.4.i.n.541.2		4	3.2	odd	2
1620.4.i.n.1081.2		4	9.5	odd	6
1620.4.i.q.541.2		4	1.1	even	1	trivial
1620.4.i.q.1081.2		4	9.4	even	3	inner
2160.4.a.v.1.2		2	36.7	odd	6
2160.4.a.ba.1.2		2	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)

\(f(q)\)	\(=\)	\(q+(2.50000 + 4.33013i) q^{5} +(9.95994 - 17.2511i) q^{7} +O(q^{10})\)	\(q+(2.50000 + 4.33013i) q^{5} +(9.95994 - 17.2511i) q^{7} +(16.9599 - 29.3755i) q^{11} +(3.95994 + 6.85881i) q^{13} +12.9199 q^{17} +38.9199 q^{19} +(-42.3397 - 73.3346i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(51.7997 - 89.7197i) q^{29} +(-151.220 - 261.920i) q^{31} +99.5994 q^{35} +74.0000 q^{37} +(56.6394 + 98.1024i) q^{41} +(-166.800 + 288.906i) q^{43} +(121.599 - 210.616i) q^{47} +(-26.9006 - 46.5933i) q^{49} +447.638 q^{53} +169.599 q^{55} +(-175.720 - 304.355i) q^{59} +(-134.740 + 233.377i) q^{61} +(-19.7997 + 34.2940i) q^{65} +(-235.240 - 407.448i) q^{67} -903.837 q^{71} -80.3173 q^{73} +(-337.840 - 585.156i) q^{77} +(-326.380 + 565.306i) q^{79} +(-62.3782 + 108.042i) q^{83} +(32.2997 + 55.9447i) q^{85} -149.760 q^{89} +157.763 q^{91} +(97.2997 + 168.528i) q^{95} +(-395.359 + 684.782i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{5} - 10 q^{7}+O(q^{10}) \) 4 * q + 10 * q^5 - 10 * q^7	\( 4 q + 10 q^{5} - 10 q^{7} + 18 q^{11} - 34 q^{13} - 48 q^{17} + 56 q^{19} + 30 q^{23} - 50 q^{25} - 42 q^{29} - 256 q^{31} - 100 q^{35} + 296 q^{37} - 222 q^{41} - 418 q^{43} - 12 q^{47} - 606 q^{49} + 96 q^{53} + 180 q^{55} - 354 q^{59} - 838 q^{61} + 170 q^{65} - 1240 q^{67} - 924 q^{71} + 1772 q^{73} - 1152 q^{77} - 1156 q^{79} + 1146 q^{83} - 120 q^{85} - 300 q^{89} + 2824 q^{91} + 140 q^{95} - 784 q^{97}+O(q^{100}) \) 4 * q + 10 * q^5 - 10 * q^7 + 18 * q^11 - 34 * q^13 - 48 * q^17 + 56 * q^19 + 30 * q^23 - 50 * q^25 - 42 * q^29 - 256 * q^31 - 100 * q^35 + 296 * q^37 - 222 * q^41 - 418 * q^43 - 12 * q^47 - 606 * q^49 + 96 * q^53 + 180 * q^55 - 354 * q^59 - 838 * q^61 + 170 * q^65 - 1240 * q^67 - 924 * q^71 + 1772 * q^73 - 1152 * q^77 - 1156 * q^79 + 1146 * q^83 - 120 * q^85 - 300 * q^89 + 2824 * q^91 + 140 * q^95 - 784 * q^97

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