LMFDB - Embedded newform 1080.4.q.e.721.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,-50,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1542 x^{18} + 976429 x^{16} + 327887620 x^{14} + 62946909772 x^{12} + 6953278937404 x^{10} + \cdots + 45\!\cdots\!76 $ x^20 + 1542x^18 + 976429x^16 + 327887620x^14 + 62946909772x^12 + 6953278937404x^10 + 426056944129744x^8 + 13516703885670016x^6 + 186027361852684288x^4 + 550048054374998016*x^2 + 452028941289112576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{25} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.9
Root			$-15.7380i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.4.q.e.361.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 - 4.33013i) q^{5} +(14.1295 - 24.4730i) q^{7} +(-35.4379 + 61.3802i) q^{11} +(-31.3500 - 54.2997i) q^{13} +116.633 q^{17} +81.1071 q^{19} +(50.4969 + 87.4632i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(19.6732 - 34.0749i) q^{29} +(-66.4223 - 115.047i) q^{31} -141.295 q^{35} +382.860 q^{37} +(-88.0191 - 152.454i) q^{41} +(-32.6094 + 56.4812i) q^{43} +(244.747 - 423.914i) q^{47} +(-227.786 - 394.538i) q^{49} -223.526 q^{53} +354.379 q^{55} +(-349.850 - 605.959i) q^{59} +(-102.596 + 177.701i) q^{61} +(-156.750 + 271.499i) q^{65} +(94.4948 + 163.670i) q^{67} -383.977 q^{71} -266.289 q^{73} +(1001.44 + 1734.54i) q^{77} +(167.183 - 289.569i) q^{79} +(-125.478 + 217.334i) q^{83} +(-291.583 - 505.036i) q^{85} -1004.55 q^{89} -1771.84 q^{91} +(-202.768 - 351.204i) q^{95} +(175.973 - 304.794i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 50 q^{5} + 22 q^{7} - 50 q^{11} - 16 q^{13} + 68 q^{17} + 212 q^{19} - 50 q^{23} - 250 q^{25} + 64 q^{29} - 22 q^{31} - 220 q^{35} + 600 q^{37} + 198 q^{41} - 382 q^{43} - 128 q^{47} - 1230 q^{49}+ \cdots - 1354 q^{97}+O(q^{100}) $ 20 * q - 50 * q^5 + 22 * q^7 - 50 * q^11 - 16 * q^13 + 68 * q^17 + 212 * q^19 - 50 * q^23 - 250 * q^25 + 64 * q^29 - 22 * q^31 - 220 * q^35 + 600 * q^37 + 198 * q^41 - 382 * q^43 - 128 * q^47 - 1230 * q^49 + 1128 * q^53 + 500 * q^55 + 200 * q^59 - 232 * q^61 - 80 * q^65 - 1316 * q^67 + 228 * q^71 + 828 * q^73 - 616 * q^77 - 300 * q^79 + 1082 * q^83 - 170 * q^85 + 768 * q^89 + 1972 * q^91 - 530 * q^95 - 1354 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$7$	14.1295	−	24.4730i	0.762922	−	1.32142i	−0.178416	−	0.983955i	$-0.557097\pi$
$7$	14.1295	−	24.4730i	0.762922	−	1.32142i	0.941338	−	0.337465i	$-0.109569\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−35.4379	+	61.3802i	−0.971356	+	1.68244i	−0.279885	+	0.960033i	$0.590296\pi$
$11$	−35.4379	+	61.3802i	−0.971356	+	1.68244i	−0.691471	+	0.722404i	$0.743037\pi$
$12$		0			0
$13$	−31.3500	−	54.2997i	−0.668839	−	1.15846i	−0.978229	−	0.207529i	$-0.933458\pi$
$13$	−31.3500	−	54.2997i	−0.668839	−	1.15846i	0.309390	−	0.950935i	$-0.399875\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	116.633			1.66398			0.831991	−	0.554790i	$-0.187201\pi$
$17$	116.633			1.66398			0.831991	+	0.554790i	$0.187201\pi$
$18$		0			0
$19$	81.1071			0.979328			0.489664	−	0.871911i	$-0.337119\pi$
$19$	81.1071			0.979328			0.489664	+	0.871911i	$0.337119\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	50.4969	+	87.4632i	0.457797	+	0.792928i	0.998844	−	0.0480645i	$-0.0153053\pi$
$23$	50.4969	+	87.4632i	0.457797	+	0.792928i	−0.541047	+	0.840992i	$0.681972\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	19.6732	−	34.0749i	0.125973	−	0.218192i	−0.796140	−	0.605113i	$-0.793128\pi$
$29$	19.6732	−	34.0749i	0.125973	−	0.218192i	0.922113	+	0.386921i	$0.126461\pi$
$30$		0			0
$31$	−66.4223	−	115.047i	−0.384832	−	0.666549i	0.606914	−	0.794768i	$-0.292407\pi$
$31$	−66.4223	−	115.047i	−0.384832	−	0.666549i	−0.991746	+	0.128219i	$0.959074\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−141.295			−0.682378
$36$		0			0
$37$	382.860			1.70113			0.850564	−	0.525871i	$-0.176261\pi$
$37$	382.860			1.70113			0.850564	+	0.525871i	$0.176261\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−88.0191	−	152.454i	−0.335275	−	0.580713i	0.648263	−	0.761417i	$-0.275496\pi$
$41$	−88.0191	−	152.454i	−0.335275	−	0.580713i	−0.983538	+	0.180703i	$0.942163\pi$
$42$		0			0
$43$	−32.6094	+	56.4812i	−0.115649	+	0.200309i	−0.918039	−	0.396490i	$-0.870228\pi$
$43$	−32.6094	+	56.4812i	−0.115649	+	0.200309i	0.802390	+	0.596800i	$0.203561\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	244.747	−	423.914i	0.759575	−	1.31562i	−0.183492	−	0.983021i	$-0.558740\pi$
$47$	244.747	−	423.914i	0.759575	−	1.31562i	0.943067	−	0.332602i	$-0.107926\pi$
$48$		0			0
$49$	−227.786	−	394.538i	−0.664100	−	1.15026i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−223.526			−0.579314			−0.289657	−	0.957131i	$-0.593541\pi$
$53$	−223.526			−0.579314			−0.289657	+	0.957131i	$0.593541\pi$
$54$		0			0
$55$	354.379			0.868807
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−349.850	−	605.959i	−0.771977	−	1.33710i	−0.936478	−	0.350727i	$-0.885935\pi$
$59$	−349.850	−	605.959i	−0.771977	−	1.33710i	0.164501	−	0.986377i	$-0.447399\pi$
$60$		0			0
$61$	−102.596	+	177.701i	−0.215345	+	0.372989i	−0.953379	−	0.301774i	$-0.902421\pi$
$61$	−102.596	+	177.701i	−0.215345	+	0.372989i	0.738034	+	0.674763i	$0.235754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−156.750	+	271.499i	−0.299114	+	0.518081i
$66$		0			0
$67$	94.4948	+	163.670i	0.172304	+	0.298440i	0.939225	−	0.343302i	$-0.111545\pi$
$67$	94.4948	+	163.670i	0.172304	+	0.298440i	−0.766921	+	0.641742i	$0.778212\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−383.977			−0.641827			−0.320913	−	0.947109i	$-0.603990\pi$
$71$	−383.977			−0.641827			−0.320913	+	0.947109i	$0.603990\pi$
$72$		0			0
$73$	−266.289			−0.426942			−0.213471	−	0.976949i	$-0.568477\pi$
$73$	−266.289			−0.426942			−0.213471	+	0.976949i	$0.568477\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1001.44	+	1734.54i	1.48214	+	2.56714i
$78$		0			0
$79$	167.183	−	289.569i	0.238096	−	0.412394i	−0.722072	−	0.691818i	$-0.756810\pi$
$79$	167.183	−	289.569i	0.238096	−	0.412394i	0.960168	+	0.279424i	$0.0901435\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−125.478	+	217.334i	−0.165940	+	0.287416i	−0.936989	−	0.349360i	$-0.886399\pi$
$83$	−125.478	+	217.334i	−0.165940	+	0.287416i	0.771049	+	0.636776i	$0.219732\pi$
$84$		0			0
$85$	−291.583	−	505.036i	−0.372078	−	0.644457i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1004.55			−1.19643			−0.598214	−	0.801336i	$-0.704123\pi$
$89$	−1004.55			−1.19643			−0.598214	+	0.801336i	$0.704123\pi$
$90$		0			0
$91$	−1771.84			−2.04109
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−202.768	−	351.204i	−0.218984	−	0.379292i
$96$		0			0
$97$	175.973	−	304.794i	0.184199	−	0.319043i	−0.759107	−	0.650966i	$-0.774364\pi$
$97$	175.973	−	304.794i	0.184199	−	0.319043i	0.943306	+	0.331923i	$0.107697\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	110.092	−	190.686i	0.108461	−	0.187861i	−0.806686	−	0.590981i	$-0.798741\pi$
$101$	110.092	−	190.686i	0.108461	−	0.187861i	0.915147	+	0.403120i	$0.132074\pi$
$102$		0			0
$103$	225.741	+	390.995i	0.215951	+	0.374038i	0.953566	−	0.301183i	$-0.0973816\pi$
$103$	225.741	+	390.995i	0.215951	+	0.374038i	−0.737616	+	0.675221i	$0.764048\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	258.255			0.233332			0.116666	−	0.993171i	$-0.462779\pi$
$107$	258.255			0.233332			0.116666	+	0.993171i	$0.462779\pi$
$108$		0			0
$109$	−491.615			−0.432002			−0.216001	−	0.976393i	$-0.569301\pi$
$109$	−491.615			−0.432002			−0.216001	+	0.976393i	$0.569301\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1003.95	−	1738.89i	−0.835785	−	1.44762i	−0.893390	−	0.449282i	$-0.851680\pi$
$113$	−1003.95	−	1738.89i	−0.835785	−	1.44762i	0.0576052	−	0.998339i	$-0.481654\pi$
$114$		0			0
$115$	252.485	−	437.316i	0.204733	−	0.354608i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1647.97	−	2854.37i	1.26949	−	2.19882i
$120$		0			0
$121$	−1846.18	−	3197.68i	−1.38706	−	2.40247i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	206.625			0.144370			0.0721852	−	0.997391i	$-0.477003\pi$
$127$	206.625			0.144370			0.0721852	+	0.997391i	$0.477003\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−89.5461	−	155.098i	−0.0597228	−	0.103443i	0.834618	−	0.550829i	$-0.185688\pi$
$131$	−89.5461	−	155.098i	−0.0597228	−	0.103443i	−0.894341	+	0.447386i	$0.852355\pi$
$132$		0			0
$133$	1146.00	−	1984.94i	0.747151	−	1.29410i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1335.46	−	2313.08i	0.832817	−	1.44248i	−0.0629778	−	0.998015i	$-0.520060\pi$
$137$	1335.46	−	2313.08i	0.832817	−	1.44248i	0.895795	−	0.444467i	$-0.146607\pi$
$138$		0			0
$139$	1256.27	+	2175.93i	0.766588	+	1.32777i	0.939403	+	0.342815i	$0.111381\pi$
$139$	1256.27	+	2175.93i	0.766588	+	1.32777i	−0.172815	+	0.984954i	$0.555286\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4443.90			2.59872
$144$		0			0
$145$	−196.732			−0.112674
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−535.464	−	927.451i	−0.294409	−	0.509931i	0.680438	−	0.732805i	$-0.261789\pi$
$149$	−535.464	−	927.451i	−0.294409	−	0.509931i	−0.974847	+	0.222874i	$0.928456\pi$
$150$		0			0
$151$	896.496	−	1552.78i	0.483151	−	0.836842i	−0.516662	−	0.856189i	$-0.672826\pi$
$151$	896.496	−	1552.78i	0.483151	−	0.836842i	0.999813	+	0.0193477i	$0.00615894\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−332.112	+	575.234i	−0.172102	+	0.298090i
$156$		0			0
$157$	−1311.31	−	2271.25i	−0.666583	−	1.15456i	−0.978853	−	0.204563i	$-0.934423\pi$
$157$	−1311.31	−	2271.25i	−0.666583	−	1.15456i	0.312270	−	0.949993i	$-0.398911\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2853.99			1.39705
$162$		0			0
$163$	−3806.29			−1.82903			−0.914514	−	0.404554i	$-0.867427\pi$
$163$	−3806.29			−1.82903			−0.914514	+	0.404554i	$0.867427\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−846.849	−	1466.78i	−0.392402	−	0.679660i	0.600364	−	0.799727i	$-0.295022\pi$
$167$	−846.849	−	1466.78i	−0.392402	−	0.679660i	−0.992766	+	0.120067i	$0.961689\pi$
$168$		0			0
$169$	−867.139	+	1501.93i	−0.394692	+	0.683627i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	419.904	−	727.295i	0.184536	−	0.319625i	−0.758884	−	0.651226i	$-0.774255\pi$
$173$	419.904	−	727.295i	0.184536	−	0.319625i	0.943420	+	0.331600i	$0.107588\pi$
$174$		0			0
$175$	353.238	+	611.826i	0.152584	+	0.264284i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3324.30			1.38810			0.694051	−	0.719926i	$-0.255824\pi$
$179$	3324.30			1.38810			0.694051	+	0.719926i	$0.255824\pi$
$180$		0			0
$181$	447.118			0.183613			0.0918066	−	0.995777i	$-0.470736\pi$
$181$	447.118			0.183613			0.0918066	+	0.995777i	$0.470736\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−957.149	−	1657.83i	−0.380384	−	0.658844i
$186$		0			0
$187$	−4133.23	+	7158.96i	−1.61632	+	2.79955i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−64.6834	+	112.035i	−0.0245043	+	0.0424427i	−0.878018	−	0.478628i	$-0.841134\pi$
$191$	−64.6834	+	112.035i	−0.0245043	+	0.0424427i	0.853513	+	0.521071i	$0.174467\pi$
$192$		0			0
$193$	−687.509	−	1190.80i	−0.256415	−	0.444123i	0.708864	−	0.705345i	$-0.249208\pi$
$193$	−687.509	−	1190.80i	−0.256415	−	0.444123i	−0.965279	+	0.261222i	$0.915875\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−657.111			−0.237651			−0.118825	−	0.992915i	$-0.537913\pi$
$197$	−657.111			−0.237651			−0.118825	+	0.992915i	$0.537913\pi$
$198$		0			0
$199$	3567.27			1.27074			0.635369	−	0.772209i	$-0.280848\pi$
$199$	3567.27			1.27074			0.635369	+	0.772209i	$0.280848\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−555.945	−	962.925i	−0.192215	−	0.332926i
$204$		0			0
$205$	−440.096	+	762.268i	−0.149940	+	0.259703i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2874.26	+	4978.37i	−0.951277	+	1.64766i
$210$		0			0
$211$	−1054.76	−	1826.89i	−0.344135	−	0.596060i	0.641061	−	0.767490i	$-0.278495\pi$
$211$	−1054.76	−	1826.89i	−0.344135	−	0.596060i	−0.985196	+	0.171430i	$0.945161\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	326.094			0.103439
$216$		0			0
$217$	−3754.06			−1.17439
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3656.44	−	6333.14i	−1.11294	−	1.92766i
$222$		0			0
$223$	2011.76	−	3484.47i	0.604114	−	1.04636i	−0.388077	−	0.921627i	$-0.626861\pi$
$223$	2011.76	−	3484.47i	0.604114	−	1.04636i	0.992191	−	0.124729i	$-0.0398061\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	124.012	−	214.796i	0.0362598	−	0.0628039i	−0.847326	−	0.531073i	$-0.821789\pi$
$227$	124.012	−	214.796i	0.0362598	−	0.0628039i	0.883586	+	0.468269i	$0.155122\pi$
$228$		0			0
$229$	401.644	+	695.668i	0.115901	+	0.200747i	0.918140	−	0.396257i	$-0.129691\pi$
$229$	401.644	+	695.668i	0.115901	+	0.200747i	−0.802238	+	0.597004i	$0.796358\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1970.32			0.553992			0.276996	−	0.960871i	$-0.410661\pi$
$233$	1970.32			0.553992			0.276996	+	0.960871i	$0.410661\pi$
$234$		0			0
$235$	−2447.47			−0.679385
$236$		0			0
$237$		0			0
$238$		0			0
$239$	100.058	+	173.306i	0.0270805	+	0.0469048i	0.879248	−	0.476364i	$-0.158046\pi$
$239$	100.058	+	173.306i	0.0270805	+	0.0469048i	−0.852168	+	0.523269i	$0.824712\pi$
$240$		0			0
$241$	−2813.18	+	4872.57i	−0.751921	+	1.30237i	0.194970	+	0.980809i	$0.437539\pi$
$241$	−2813.18	+	4872.57i	−0.751921	+	1.30237i	−0.946891	+	0.321556i	$0.895794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1138.93	+	1972.69i	−0.296995	+	0.514410i
$246$		0			0
$247$	−2542.70	−	4404.09i	−0.655013	−	1.13452i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2995.52			0.753289			0.376645	−	0.926358i	$-0.377078\pi$
$251$	2995.52			0.753289			0.376645	+	0.926358i	$0.377078\pi$
$252$		0			0
$253$	−7158.01			−1.77874
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2065.14	−	3576.92i	−0.501244	−	0.868180i	−0.999999	−	0.00143676i	$-0.999543\pi$
$257$	−2065.14	−	3576.92i	−0.501244	−	0.868180i	0.498755	−	0.866743i	$-0.333791\pi$
$258$		0			0
$259$	5409.62	−	9369.74i	1.29783	−	2.24790i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	469.302	−	812.856i	0.110032	−	0.190581i	−0.805751	−	0.592255i	$-0.798238\pi$
$263$	469.302	−	812.856i	0.110032	−	0.190581i	0.915783	+	0.401673i	$0.131571\pi$
$264$		0			0
$265$	558.815	+	967.895i	0.129539	+	0.224367i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3184.71			−0.721842			−0.360921	−	0.932596i	$-0.617538\pi$
$269$	−3184.71			−0.721842			−0.360921	+	0.932596i	$0.617538\pi$
$270$		0			0
$271$	7138.96			1.60022			0.800112	−	0.599850i	$-0.204773\pi$
$271$	7138.96			1.60022			0.800112	+	0.599850i	$0.204773\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−885.946	−	1534.50i	−0.194271	−	0.336488i
$276$		0			0
$277$	3552.99	−	6153.97i	0.770681	−	1.33486i	−0.166509	−	0.986040i	$-0.553249\pi$
$277$	3552.99	−	6153.97i	0.770681	−	1.33486i	0.937190	−	0.348819i	$-0.113417\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3180.76	+	5509.24i	−0.675261	+	1.16959i	0.301132	+	0.953582i	$0.402635\pi$
$281$	−3180.76	+	5509.24i	−0.675261	+	1.16959i	−0.976393	+	0.216003i	$0.930698\pi$
$282$		0			0
$283$	1146.48	+	1985.75i	0.240816	+	0.417106i	0.960947	−	0.276733i	$-0.0892516\pi$
$283$	1146.48	+	1985.75i	0.240816	+	0.417106i	−0.720131	+	0.693838i	$0.755918\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4974.67			−1.02315
$288$		0			0
$289$	8690.28			1.76883
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4142.31	+	7174.69i	0.825926	+	1.43055i	0.901209	+	0.433384i	$0.142681\pi$
$293$	4142.31	+	7174.69i	0.825926	+	1.43055i	−0.0752832	+	0.997162i	$0.523986\pi$
$294$		0			0
$295$	−1749.25	+	3029.79i	−0.345239	+	0.597971i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3166.15	−	5483.93i	0.612385	−	1.06068i
$300$		0			0
$301$	921.511	+	1596.10i	0.176462	+	0.305641i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1025.96			0.192611
$306$		0			0
$307$	4205.85			0.781891			0.390946	−	0.920414i	$-0.372148\pi$
$307$	4205.85			0.781891			0.390946	+	0.920414i	$0.372148\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1482.22	+	2567.28i	0.270254	+	0.468093i	0.968927	−	0.247348i	$-0.0795591\pi$
$311$	1482.22	+	2567.28i	0.270254	+	0.468093i	−0.698673	+	0.715441i	$0.746226\pi$
$312$		0			0
$313$	1464.28	−	2536.21i	0.264428	−	0.458004i	−0.702985	−	0.711204i	$-0.748150\pi$
$313$	1464.28	−	2536.21i	0.264428	−	0.458004i	0.967414	+	0.253201i	$0.0814834\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3831.67	+	6636.65i	−0.678890	+	1.17587i	0.296425	+	0.955056i	$0.404205\pi$
$317$	−3831.67	+	6636.65i	−0.678890	+	1.17587i	−0.975315	+	0.220816i	$0.929128\pi$
$318$		0			0
$319$	1394.35	+	2415.09i	0.244729	+	0.423883i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9459.77			1.62958
$324$		0			0
$325$	1567.50			0.267536
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6916.32	−	11979.4i	−1.15899	−	2.00744i
$330$		0			0
$331$	668.853	−	1158.49i	0.111068	−	0.192375i	−0.805133	−	0.593094i	$-0.797906\pi$
$331$	668.853	−	1158.49i	0.111068	−	0.192375i	0.916201	+	0.400719i	$0.131240\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	472.474	−	818.349i	0.0770568	−	0.133466i
$336$		0			0
$337$	−4054.65	−	7022.85i	−0.655402	−	1.13519i	−0.981793	−	0.189955i	$-0.939166\pi$
$337$	−4054.65	−	7022.85i	−0.655402	−	1.13519i	0.326390	−	0.945235i	$-0.394168\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9415.46			1.49524
$342$		0			0
$343$	−3181.20			−0.500783
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5494.87	+	9517.39i	0.850087	+	1.47239i	0.881129	+	0.472875i	$0.156784\pi$
$347$	5494.87	+	9517.39i	0.850087	+	1.47239i	−0.0310429	+	0.999518i	$0.509883\pi$
$348$		0			0
$349$	−445.146	+	771.016i	−0.0682755	+	0.118257i	−0.898142	−	0.439705i	$-0.855083\pi$
$349$	−445.146	+	771.016i	−0.0682755	+	0.118257i	0.829867	+	0.557962i	$0.188416\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−943.375	+	1633.97i	−0.142240	+	0.246367i	−0.928340	−	0.371733i	$-0.878764\pi$
$353$	−943.375	+	1633.97i	−0.142240	+	0.246367i	0.786100	+	0.618100i	$0.212097\pi$
$354$		0			0
$355$	959.943	+	1662.67i	0.143517	+	0.248579i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11935.2			−1.75464			−0.877321	−	0.479904i	$-0.840671\pi$
$359$	−11935.2			−1.75464			−0.877321	+	0.479904i	$0.840671\pi$
$360$		0			0
$361$	−280.641			−0.0409158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	665.722	+	1153.06i	0.0954671	+	0.165354i
$366$		0			0
$367$	−1338.71	+	2318.72i	−0.190409	+	0.329798i	−0.945386	−	0.325953i	$-0.894315\pi$
$367$	−1338.71	+	2318.72i	−0.190409	+	0.329798i	0.754977	+	0.655752i	$0.227648\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3158.31	+	5470.36i	−0.441971	+	0.765517i
$372$		0			0
$373$	−558.751	−	967.785i	−0.0775631	−	0.134343i	0.824635	−	0.565665i	$-0.191381\pi$
$373$	−558.751	−	967.785i	−0.0775631	−	0.134343i	−0.902198	+	0.431322i	$0.858047\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2467.01			−0.337023
$378$		0			0
$379$	−3561.46			−0.482691			−0.241345	−	0.970439i	$-0.577589\pi$
$379$	−3561.46			−0.482691			−0.241345	+	0.970439i	$0.577589\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	329.509	+	570.727i	0.0439612	+	0.0761431i	0.887169	−	0.461445i	$-0.152669\pi$
$383$	329.509	+	570.727i	0.0439612	+	0.0761431i	−0.843208	+	0.537588i	$0.819336\pi$
$384$		0			0
$385$	5007.20	−	8672.72i	0.662832	−	1.14806i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3781.21	−	6549.25i	0.492840	−	0.853625i	−0.507126	−	0.861872i	$-0.669292\pi$
$389$	3781.21	−	6549.25i	0.492840	−	0.853625i	0.999966	+	0.00824760i	$0.00262532\pi$
$390$		0			0
$391$	5889.61	+	10201.1i	0.761766	+	1.31942i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1671.83			−0.212959
$396$		0			0
$397$	−13458.6			−1.70143			−0.850714	−	0.525628i	$-0.823830\pi$
$397$	−13458.6			−1.70143			−0.850714	+	0.525628i	$0.823830\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1114.51	−	1930.39i	−0.138793	−	0.240397i	0.788247	−	0.615359i	$-0.210989\pi$
$401$	−1114.51	−	1930.39i	−0.138793	−	0.240397i	−0.927040	+	0.374962i	$0.877656\pi$
$402$		0			0
$403$	−4164.67	+	7213.42i	−0.514782	+	0.891628i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−13567.7	+	23500.0i	−1.65240	+	2.86204i
$408$		0			0
$409$	2783.71	+	4821.52i	0.336542	+	0.582907i	0.983780	−	0.179381i	$-0.0574094\pi$
$409$	2783.71	+	4821.52i	0.336542	+	0.582907i	−0.647238	+	0.762288i	$0.724076\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19772.9			−2.35583
$414$		0			0
$415$	1254.78			0.148421
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3253.15	−	5634.61i	−0.379300	−	0.656966i	0.611661	−	0.791120i	$-0.290502\pi$
$419$	−3253.15	−	5634.61i	−0.379300	−	0.656966i	−0.990961	+	0.134154i	$0.957168\pi$
$420$		0			0
$421$	−5586.70	+	9676.44i	−0.646743	+	1.12019i	0.337152	+	0.941450i	$0.390536\pi$
$421$	−5586.70	+	9676.44i	−0.646743	+	1.12019i	−0.983896	+	0.178742i	$0.942797\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1457.91	+	2525.18i	−0.166398	+	0.288210i
$426$		0			0
$427$	2899.26	+	5021.67i	0.328583	+	0.569123i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13021.8			1.45531			0.727653	−	0.685946i	$-0.240611\pi$
$431$	13021.8			1.45531			0.727653	+	0.685946i	$0.240611\pi$
$432$		0			0
$433$	12537.4			1.39148			0.695738	−	0.718295i	$-0.255077\pi$
$433$	12537.4			1.39148			0.695738	+	0.718295i	$0.255077\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4095.66	+	7093.88i	0.448334	+	0.776537i
$438$		0			0
$439$	−159.278	+	275.878i	−0.0173165	+	0.0299931i	−0.874554	−	0.484928i	$-0.838846\pi$
$439$	−159.278	+	275.878i	−0.0173165	+	0.0299931i	0.857237	+	0.514922i	$0.172179\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5308.70	−	9194.94i	0.569355	−	0.986151i	−0.427275	−	0.904122i	$-0.640527\pi$
$443$	5308.70	−	9194.94i	0.569355	−	0.986151i	0.996630	−	0.0820294i	$-0.0261401\pi$
$444$		0			0
$445$	2511.37	+	4349.83i	0.267529	+	0.463374i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4773.58			−0.501735			−0.250868	−	0.968021i	$-0.580716\pi$
$449$	−4773.58			−0.501735			−0.250868	+	0.968021i	$0.580716\pi$
$450$		0			0
$451$	12476.8			1.30269
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4429.60	+	7672.29i	0.456402	+	0.790511i
$456$		0			0
$457$	−4622.34	+	8006.12i	−0.473137	+	0.819498i	−0.999527	−	0.0307452i	$-0.990212\pi$
$457$	−4622.34	+	8006.12i	−0.473137	+	0.819498i	0.526390	+	0.850243i	$0.323545\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3082.48	−	5339.02i	0.311422	−	0.539399i	−0.667249	−	0.744835i	$-0.732528\pi$
$461$	3082.48	−	5339.02i	0.311422	−	0.539399i	0.978670	+	0.205437i	$0.0658614\pi$
$462$		0			0
$463$	8461.59	+	14655.9i	0.849338	+	1.47110i	0.881800	+	0.471623i	$0.156332\pi$
$463$	8461.59	+	14655.9i	0.849338	+	1.47110i	−0.0324625	+	0.999473i	$0.510335\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11457.5			1.13531			0.567655	−	0.823266i	$-0.307851\pi$
$467$	11457.5			1.13531			0.567655	+	0.823266i	$0.307851\pi$
$468$		0			0
$469$	5340.66			0.525819
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2311.22	−	4003.15i	−0.224672	−	0.389143i
$474$		0			0
$475$	−1013.84	+	1756.02i	−0.0979328	+	0.169625i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7196.00	+	12463.8i	−0.686416	+	1.18891i	0.286573	+	0.958058i	$0.407484\pi$
$479$	−7196.00	+	12463.8i	−0.686416	+	1.18891i	−0.972989	+	0.230850i	$0.925849\pi$
$480$		0			0
$481$	−12002.6	−	20789.2i	−1.13778	−	1.97070i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1759.73			−0.164753
$486$		0			0
$487$	−3589.81			−0.334024			−0.167012	−	0.985955i	$-0.553412\pi$
$487$	−3589.81			−0.334024			−0.167012	+	0.985955i	$0.553412\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1085.51	+	1880.15i	0.0997724	+	0.172811i	0.911590	−	0.411100i	$-0.134855\pi$
$491$	1085.51	+	1880.15i	0.0997724	+	0.172811i	−0.811818	+	0.583911i	$0.801522\pi$
$492$		0			0
$493$	2294.54	−	3974.27i	0.209617	−	0.363067i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5425.41	+	9397.09i	−0.489664	+	0.848123i
$498$		0			0
$499$	6665.42	+	11544.9i	0.597967	+	1.03571i	0.993121	+	0.117094i	$0.0373578\pi$
$499$	6665.42	+	11544.9i	0.597967	+	1.03571i	−0.395154	+	0.918615i	$0.629309\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13677.7			1.21244			0.606222	−	0.795295i	$-0.292684\pi$
$503$	13677.7			1.21244			0.606222	+	0.795295i	$0.292684\pi$
$504$		0			0
$505$	−1100.92			−0.0970109
$506$		0			0
$507$		0			0
$508$		0			0
$509$	793.895	+	1375.07i	0.0691332	+	0.119742i	0.898520	−	0.438933i	$-0.144643\pi$
$509$	793.895	+	1375.07i	0.0691332	+	0.119742i	−0.829387	+	0.558675i	$0.811310\pi$
$510$		0			0
$511$	−3762.53	+	6516.89i	−0.325723	+	0.564169i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1128.71	−	1954.98i	0.0965761	−	0.167275i
$516$		0			0
$517$	17346.6	+	30045.2i	1.47564	+	2.55588i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1472.45			−0.123818			−0.0619090	−	0.998082i	$-0.519719\pi$
$521$	−1472.45			−0.123818			−0.0619090	+	0.998082i	$0.519719\pi$
$522$		0			0
$523$	−4319.49			−0.361143			−0.180572	−	0.983562i	$-0.557795\pi$
$523$	−4319.49			−0.361143			−0.180572	+	0.983562i	$0.557795\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7747.04	−	13418.3i	−0.640354	−	1.10912i
$528$		0			0
$529$	983.626	−	1703.69i	0.0808437	−	0.140025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5518.79	+	9558.82i	−0.448490	+	0.776808i
$534$		0			0
$535$	−645.638	−	1118.28i	−0.0521745	−	0.0903690i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	32289.0			2.58031
$540$		0			0
$541$	8142.01			0.647047			0.323523	−	0.946220i	$-0.395133\pi$
$541$	8142.01			0.647047			0.323523	+	0.946220i	$0.395133\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1229.04	+	2128.76i	0.0965986	+	0.167314i
$546$		0			0
$547$	3033.11	−	5253.50i	0.237087	−	0.410646i	−0.722790	−	0.691067i	$-0.757141\pi$
$547$	3033.11	−	5253.50i	0.237087	−	0.410646i	0.959877	+	0.280421i	$0.0904741\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1595.63	−	2763.72i	0.123369	−	0.213681i
$552$		0			0
$553$	−4724.43	−	8182.95i	−0.363297	−	0.629248i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1792.03			0.136321			0.0681604	−	0.997674i	$-0.478287\pi$
$557$	1792.03			0.136321			0.0681604	+	0.997674i	$0.478287\pi$
$558$		0			0
$559$	4089.22			0.309402
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10238.8	−	17734.2i	−0.766458	−	1.32754i	−0.939472	−	0.342625i	$-0.888684\pi$
$563$	−10238.8	−	17734.2i	−0.766458	−	1.32754i	0.173014	−	0.984919i	$-0.444649\pi$
$564$		0			0
$565$	−5019.75	+	8694.46i	−0.373774	+	0.647396i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1287.83	+	2230.59i	−0.0948833	+	0.164343i	−0.909560	−	0.415573i	$-0.863581\pi$
$569$	−1287.83	+	2230.59i	−0.0948833	+	0.164343i	0.814677	+	0.579916i	$0.196914\pi$
$570$		0			0
$571$	−1335.22	−	2312.67i	−0.0978586	−	0.169496i	0.812939	−	0.582348i	$-0.197866\pi$
$571$	−1335.22	−	2312.67i	−0.0978586	−	0.169496i	−0.910798	+	0.412852i	$0.864533\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2524.85			−0.183119
$576$		0			0
$577$	−23938.6			−1.72717			−0.863584	−	0.504205i	$-0.831786\pi$
$577$	−23938.6			−1.72717			−0.863584	+	0.504205i	$0.831786\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3545.88	+	6141.65i	0.253198	+	0.438552i
$582$		0			0
$583$	7921.28	−	13720.1i	0.562720	−	0.974660i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1856.66	+	3215.83i	−0.130549	+	0.226118i	−0.923888	−	0.382662i	$-0.875007\pi$
$587$	−1856.66	+	3215.83i	−0.130549	+	0.226118i	0.793339	+	0.608780i	$0.208341\pi$
$588$		0			0
$589$	−5387.32	−	9331.11i	−0.376877	−	0.652770i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6634.76			0.459455			0.229728	−	0.973255i	$-0.426216\pi$
$593$	6634.76			0.459455			0.229728	+	0.973255i	$0.426216\pi$
$594$		0			0
$595$	−16479.7			−1.13546
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8064.71	−	13968.5i	−0.550109	−	0.952816i	−0.998266	−	0.0588615i	$-0.981253\pi$
$599$	−8064.71	−	13968.5i	−0.550109	−	0.952816i	0.448158	−	0.893955i	$-0.352080\pi$
$600$		0			0
$601$	−5289.56	+	9161.79i	−0.359011	+	0.621826i	−0.987796	−	0.155754i	$-0.950219\pi$
$601$	−5289.56	+	9161.79i	−0.359011	+	0.621826i	0.628785	+	0.777580i	$0.283553\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9230.92	+	15988.4i	−0.620314	+	1.07442i
$606$		0			0
$607$	−11811.4	−	20457.9i	−0.789800	−	1.36797i	−0.926089	−	0.377304i	$-0.876851\pi$
$607$	−11811.4	−	20457.9i	−0.789800	−	1.36797i	0.136289	−	0.990669i	$-0.456482\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−30691.2			−2.03214
$612$		0			0
$613$	17537.7			1.15553			0.577767	−	0.816201i	$-0.303924\pi$
$613$	17537.7			1.15553			0.577767	+	0.816201i	$0.303924\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4956.91	+	8585.61i	0.323432	+	0.560201i	0.981194	−	0.193025i	$-0.0618300\pi$
$617$	4956.91	+	8585.61i	0.323432	+	0.560201i	−0.657762	+	0.753226i	$0.728497\pi$
$618$		0			0
$619$	−8217.29	+	14232.8i	−0.533572	+	0.924173i	0.465659	+	0.884964i	$0.345817\pi$
$619$	−8217.29	+	14232.8i	−0.533572	+	0.924173i	−0.999231	+	0.0392091i	$0.987516\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14193.8	+	24584.4i	−0.912781	+	1.58098i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	44654.1			2.83065
$630$		0			0
$631$	−9675.24			−0.610404			−0.305202	−	0.952288i	$-0.598724\pi$
$631$	−9675.24			−0.610404			−0.305202	+	0.952288i	$0.598724\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−516.563	−	894.714i	−0.0322822	−	0.0559144i
$636$		0			0
$637$	−14282.2	+	24737.5i	−0.888353	+	1.53867i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3280.93	+	5682.73i	−0.202167	+	0.350163i	−0.949226	−	0.314594i	$-0.898132\pi$
$641$	−3280.93	+	5682.73i	−0.202167	+	0.350163i	0.747060	+	0.664757i	$0.231465\pi$
$642$		0			0
$643$	13340.9	+	23107.1i	0.818218	+	1.41720i	0.906994	+	0.421144i	$0.138371\pi$
$643$	13340.9	+	23107.1i	0.818218	+	1.41720i	−0.0887753	+	0.996052i	$0.528295\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18569.8			1.12837			0.564183	−	0.825650i	$-0.309191\pi$
$647$	18569.8			1.12837			0.564183	+	0.825650i	$0.309191\pi$
$648$		0			0
$649$	49591.8			2.99946
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4692.26	−	8127.23i	−0.281198	−	0.487049i	0.690482	−	0.723350i	$-0.257398\pi$
$653$	−4692.26	−	8127.23i	−0.281198	−	0.487049i	−0.971680	+	0.236300i	$0.924065\pi$
$654$		0			0
$655$	−447.731	+	775.492i	−0.0267088	+	0.0462611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3204.61	−	5550.55i	0.189429	−	0.328101i	−0.755631	−	0.654998i	$-0.772670\pi$
$659$	3204.61	−	5550.55i	0.189429	−	0.328101i	0.945060	+	0.326896i	$0.106003\pi$
$660$		0			0
$661$	−3994.79	−	6919.17i	−0.235067	−	0.407148i	0.724225	−	0.689563i	$-0.242198\pi$
$661$	−3994.79	−	6919.17i	−0.235067	−	0.407148i	−0.959292	+	0.282416i	$0.908864\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11460.0			−0.668272
$666$		0			0
$667$	3973.74			0.230680
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7271.56	−	12594.7i	−0.418354	−	0.724610i
$672$		0			0
$673$	8794.62	−	15232.7i	0.503726	−	0.872480i	−0.496264	−	0.868171i	$-0.665295\pi$
$673$	8794.62	−	15232.7i	0.503726	−	0.872480i	0.999991	−	0.00430813i	$-0.00137132\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8380.19	+	14514.9i	−0.475741	+	0.824008i	−0.999614	−	0.0277884i	$-0.991154\pi$
$677$	−8380.19	+	14514.9i	−0.475741	+	0.824008i	0.523872	+	0.851797i	$0.324487\pi$
$678$		0			0
$679$	−4972.82	−	8613.18i	−0.281060	−	0.486809i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19144.5			−1.07254			−0.536269	−	0.844047i	$-0.680167\pi$
$683$	−19144.5			−1.07254			−0.536269	+	0.844047i	$0.680167\pi$
$684$		0			0
$685$	−13354.6			−0.744894
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7007.52	+	12137.4i	0.387468	+	0.671114i
$690$		0			0
$691$	5618.53	−	9731.57i	0.309318	−	0.535755i	−0.668895	−	0.743357i	$-0.733233\pi$
$691$	5618.53	−	9731.57i	0.309318	−	0.535755i	0.978213	+	0.207602i	$0.0665659\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6281.37	−	10879.7i	0.342829	−	0.593797i
$696$		0			0
$697$	−10265.9	−	17781.1i	−0.557891	−	0.966296i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7341.40			−0.395550			−0.197775	−	0.980247i	$-0.563372\pi$
$701$	−7341.40			−0.395550			−0.197775	+	0.980247i	$0.563372\pi$
$702$		0			0
$703$	31052.6			1.66596
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3111.11	−	5388.59i	−0.165495	−	0.286646i
$708$		0			0
$709$	−1185.69	+	2053.67i	−0.0628060	+	0.108783i	−0.895719	−	0.444621i	$-0.853338\pi$
$709$	−1185.69	+	2053.67i	−0.0628060	+	0.108783i	0.832913	+	0.553404i	$0.186672\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6708.24	−	11619.0i	0.352350	−	0.610288i
$714$		0			0
$715$	−11109.8	−	19242.7i	−0.581093	−	1.00648i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18337.4			0.951141			0.475571	−	0.879678i	$-0.342242\pi$
$719$	18337.4			0.951141			0.475571	+	0.879678i	$0.342242\pi$
$720$		0			0
$721$	12758.4			0.659014
$722$		0			0
$723$		0			0
$724$		0			0
$725$	491.829	+	851.873i	0.0251946	+	0.0436383i
$726$		0			0
$727$	4542.93	−	7868.58i	0.231758	−	0.401416i	−0.726568	−	0.687095i	$-0.758886\pi$
$727$	4542.93	−	7868.58i	0.231758	−	0.401416i	0.958325	+	0.285679i	$0.0922190\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3803.34	+	6587.58i	−0.192437	+	0.333311i
$732$		0			0
$733$	−4452.27	−	7711.56i	−0.224350	−	0.388585i	0.731774	−	0.681547i	$-0.238692\pi$
$733$	−4452.27	−	7711.56i	−0.224350	−	0.388585i	−0.956124	+	0.292962i	$0.905359\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13394.8			−0.669475
$738$		0			0
$739$	7295.24			0.363139			0.181569	−	0.983378i	$-0.441882\pi$
$739$	7295.24			0.363139			0.181569	+	0.983378i	$0.441882\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1041.48	+	1803.89i	0.0514241	+	0.0890691i	0.890592	−	0.454804i	$-0.150291\pi$
$743$	1041.48	+	1803.89i	0.0514241	+	0.0890691i	−0.839168	+	0.543873i	$0.816957\pi$
$744$		0			0
$745$	−2677.32	+	4637.26i	−0.131664	+	0.228048i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3649.02	−	6320.29i	0.178014	−	0.308329i
$750$		0			0
$751$	3976.35	+	6887.23i	0.193208	+	0.334645i	0.946311	−	0.323256i	$-0.104778\pi$
$751$	3976.35	+	6887.23i	0.193208	+	0.334645i	−0.753104	+	0.657902i	$0.771444\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8964.96			−0.432143
$756$		0			0
$757$	−17476.8			−0.839108			−0.419554	−	0.907730i	$-0.637814\pi$
$757$	−17476.8			−0.839108			−0.419554	+	0.907730i	$0.637814\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11127.2	−	19272.8i	−0.530038	−	0.918053i	−0.999386	−	0.0350397i	$-0.988844\pi$
$761$	−11127.2	−	19272.8i	−0.530038	−	0.918053i	0.469348	−	0.883013i	$-0.344489\pi$
$762$		0			0
$763$	−6946.29	+	12031.3i	−0.329584	+	0.570856i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21935.6	+	37993.6i	−1.03266	+	1.78862i
$768$		0			0
$769$	4243.66	+	7350.23i	0.198999	+	0.344676i	0.948204	−	0.317662i	$-0.102898\pi$
$769$	4243.66	+	7350.23i	0.198999	+	0.344676i	−0.749205	+	0.662338i	$0.769564\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	34964.8			1.62690			0.813452	−	0.581632i	$-0.197586\pi$
$773$	34964.8			1.62690			0.813452	+	0.581632i	$0.197586\pi$
$774$		0			0
$775$	3321.12			0.153933
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7138.97	−	12365.1i	−0.328344	−	0.568709i
$780$		0			0
$781$	13607.3	−	23568.6i	0.623442	−	1.07983i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6556.53	+	11356.2i	−0.298105	+	0.516333i
$786$		0			0
$787$	18093.6	+	31339.0i	0.819524	+	1.41946i	0.906033	+	0.423207i	$0.139096\pi$
$787$	18093.6	+	31339.0i	0.819524	+	1.41946i	−0.0865087	+	0.996251i	$0.527571\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−56741.3			−2.55055
$792$		0			0
$793$	12865.5			0.576126
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2706.11	−	4687.12i	−0.120270	−	0.208314i	0.799604	−	0.600528i	$-0.205043\pi$
$797$	−2706.11	−	4687.12i	−0.120270	−	0.208314i	−0.919874	+	0.392214i	$0.871709\pi$
$798$		0			0
$799$	28545.6	−	49442.5i	1.26392	−	2.18917i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9436.70	−	16344.8i	0.414712	−	0.718303i
$804$		0			0
$805$	−7134.97	−	12358.1i	−0.312391	−	0.541077i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5626.82			−0.244534			−0.122267	−	0.992497i	$-0.539016\pi$
$809$	−5626.82			−0.244534			−0.122267	+	0.992497i	$0.539016\pi$
$810$		0			0
$811$	7559.74			0.327322			0.163661	−	0.986517i	$-0.447670\pi$
$811$	7559.74			0.327322			0.163661	+	0.986517i	$0.447670\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9515.72	+	16481.7i	0.408983	+	0.708380i
$816$		0			0
$817$	−2644.86	+	4581.03i	−0.113258	+	0.196169i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7242.20	+	12543.9i	−0.307862	+	0.533233i	−0.977894	−	0.209099i	$-0.932947\pi$
$821$	−7242.20	+	12543.9i	−0.307862	+	0.533233i	0.670032	+	0.742332i	$0.266280\pi$
$822$		0			0
$823$	15111.3	+	26173.5i	0.640032	+	1.10857i	0.985425	+	0.170110i	$0.0544123\pi$
$823$	15111.3	+	26173.5i	0.640032	+	1.10857i	−0.345393	+	0.938458i	$0.612254\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11032.1			−0.463875			−0.231937	−	0.972731i	$-0.574506\pi$
$827$	−11032.1			−0.463875			−0.231937	+	0.972731i	$0.574506\pi$
$828$		0			0
$829$	−12061.6			−0.505326			−0.252663	−	0.967554i	$-0.581306\pi$
$829$	−12061.6			−0.505326			−0.252663	+	0.967554i	$0.581306\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26567.4	−	46016.1i	−1.10505	−	1.91400i
$834$		0			0
$835$	−4234.24	+	7333.92i	−0.175487	+	0.303953i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22373.8	−	38752.5i	0.920655	−	1.59462i	0.122250	−	0.992499i	$-0.460989\pi$
$839$	22373.8	−	38752.5i	0.920655	−	1.59462i	0.798405	−	0.602121i	$-0.205678\pi$
$840$		0			0
$841$	11420.4	+	19780.8i	0.468262	+	0.811053i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8671.39			0.353024
$846$		0			0
$847$	−104343.			−4.23289
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19333.2	+	33486.1i	0.778771	+	1.34887i
$852$		0			0
$853$	−9823.29	+	17014.4i	−0.394306	+	0.682958i	−0.993012	−	0.118011i	$-0.962348\pi$
$853$	−9823.29	+	17014.4i	−0.394306	+	0.682958i	0.598706	+	0.800969i	$0.295682\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10923.3	−	18919.8i	0.435396	−	0.754127i	−0.561932	−	0.827183i	$-0.689942\pi$
$857$	10923.3	−	18919.8i	0.435396	−	0.754127i	0.997328	+	0.0730559i	$0.0232751\pi$
$858$		0			0
$859$	−10438.2	−	18079.5i	−0.414607	−	0.718120i	0.580780	−	0.814060i	$-0.302747\pi$
$859$	−10438.2	−	18079.5i	−0.414607	−	0.718120i	−0.995387	+	0.0959404i	$0.969414\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4812.37			0.189820			0.0949102	−	0.995486i	$-0.469744\pi$
$863$	4812.37			0.189820			0.0949102	+	0.995486i	$0.469744\pi$
$864$		0			0
$865$	−4199.04			−0.165054
$866$		0			0
$867$		0			0
$868$		0			0
$869$	11849.2	+	20523.4i	0.462551	+	0.801162i
$870$		0			0
$871$	5924.82	−	10262.1i	0.230488	−	0.399216i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1766.19	−	3059.13i	0.0682378	−	0.118191i
$876$		0			0
$877$	−7719.06	−	13369.8i	−0.297211	−	0.514785i	0.678286	−	0.734798i	$-0.262723\pi$
$877$	−7719.06	−	13369.8i	−0.297211	−	0.514785i	−0.975497	+	0.220014i	$0.929390\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43054.1			1.64646			0.823228	−	0.567711i	$-0.192171\pi$
$881$	43054.1			1.64646			0.823228	+	0.567711i	$0.192171\pi$
$882$		0			0
$883$	−47526.9			−1.81133			−0.905667	−	0.423990i	$-0.860629\pi$
$883$	−47526.9			−1.81133			−0.905667	+	0.423990i	$0.860629\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10033.6	+	17378.7i	0.379815	+	0.657858i	0.991035	−	0.133602i	$-0.0426545\pi$
$887$	10033.6	+	17378.7i	0.379815	+	0.657858i	−0.611220	+	0.791460i	$0.709321\pi$
$888$		0			0
$889$	2919.52	−	5056.75i	0.110143	−	0.190774i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19850.7	−	34382.5i	0.743874	−	1.28843i
$894$		0			0
$895$	−8310.76	−	14394.7i	−0.310389	−	0.537609i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5226.95			−0.193914
$900$		0			0
$901$	−26070.5			−0.963967
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1117.79	−	1936.08i	−0.0410572	−	0.0711131i
$906$		0			0
$907$	10457.5	−	18112.8i	0.382838	−	0.663095i	−0.608629	−	0.793455i	$-0.708280\pi$
$907$	10457.5	−	18112.8i	0.382838	−	0.663095i	0.991467	+	0.130360i	$0.0416134\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22762.3	−	39425.5i	0.827826	−	1.43384i	−0.0719144	−	0.997411i	$-0.522911\pi$
$911$	22762.3	−	39425.5i	0.827826	−	1.43384i	0.899740	−	0.436426i	$-0.143756\pi$
$912$		0			0
$913$	−8893.34	−	15403.7i	−0.322373	−	0.558366i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5060.97			−0.182255
$918$		0			0
$919$	−45257.7			−1.62450			−0.812249	−	0.583311i	$-0.801757\pi$
$919$	−45257.7			−1.62450			−0.812249	+	0.583311i	$0.801757\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12037.7	+	20849.9i	0.429279	+	0.743533i
$924$		0			0
$925$	−4785.75	+	8289.16i	−0.170113	+	0.294644i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12499.7	−	21650.1i	0.441445	−	0.764604i	−0.556352	−	0.830946i	$-0.687800\pi$
$929$	12499.7	−	21650.1i	0.441445	−	0.764604i	0.997797	+	0.0663420i	$0.0211328\pi$
$930$		0			0
$931$	−18475.1	−	31999.8i	−0.650372	−	1.12648i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	41332.3			1.44568
$936$		0			0
$937$	48386.6			1.68700			0.843502	−	0.537126i	$-0.180490\pi$
$937$	48386.6			1.68700			0.843502	+	0.537126i	$0.180490\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1881.68	−	3259.16i	−0.0651870	−	0.112907i	0.831590	−	0.555390i	$-0.187431\pi$
$941$	−1881.68	−	3259.16i	−0.0651870	−	0.112907i	−0.896777	+	0.442483i	$0.854098\pi$
$942$		0			0
$943$	8889.38	−	15396.9i	0.306976	−	0.531698i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13296.2	+	23029.7i	−0.456249	+	0.790247i	−0.998759	−	0.0498027i	$-0.984141\pi$
$947$	−13296.2	+	23029.7i	−0.456249	+	0.790247i	0.542510	+	0.840049i	$0.317474\pi$
$948$		0			0
$949$	8348.14	+	14459.4i	0.285555	+	0.494596i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	22556.7			0.766719			0.383360	−	0.923599i	$-0.374767\pi$
$953$	22556.7			0.766719			0.383360	+	0.923599i	$0.374767\pi$
$954$		0			0
$955$	646.834			0.0219173
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−37738.8	−	65365.5i	−1.27075	−	2.20100i
$960$		0			0
$961$	6071.66	−	10516.4i	0.203808	−	0.353006i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3437.55	+	5954.01i	−0.114672	+	0.198618i
$966$		0			0
$967$	15264.7	+	26439.2i	0.507631	+	0.879242i	0.999961	+	0.00883360i	$0.00281186\pi$
$967$	15264.7	+	26439.2i	0.507631	+	0.879242i	−0.492330	+	0.870408i	$0.663855\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3354.98			−0.110882			−0.0554409	−	0.998462i	$-0.517656\pi$
$971$	−3354.98			−0.110882			−0.0554409	+	0.998462i	$0.517656\pi$
$972$		0			0
$973$	71002.2			2.33939
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3111.02	+	5388.44i	0.101873	+	0.176450i	0.912456	−	0.409174i	$-0.134183\pi$
$977$	3111.02	+	5388.44i	0.101873	+	0.176450i	−0.810583	+	0.585624i	$0.800850\pi$
$978$		0			0
$979$	35599.1	−	61659.4i	1.16216	−	2.01292i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−10262.8	+	17775.6i	−0.332992	+	0.576760i	−0.983097	−	0.183085i	$-0.941392\pi$
$983$	−10262.8	+	17775.6i	−0.332992	+	0.576760i	0.650105	+	0.759845i	$0.274725\pi$
$984$		0			0
$985$	1642.78	+	2845.37i	0.0531404	+	0.0920418i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6586.70			−0.211774
$990$		0			0
$991$	−14110.0			−0.452289			−0.226145	−	0.974094i	$-0.572612\pi$
$991$	−14110.0			−0.452289			−0.226145	+	0.974094i	$0.572612\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8918.16	−	15446.7i	−0.284146	−	0.492154i
$996$		0			0
$997$	−8359.84	+	14479.7i	−0.265555	+	0.459955i	−0.967709	−	0.252070i	$-0.918889\pi$
$997$	−8359.84	+	14479.7i	−0.265555	+	0.459955i	0.702154	+	0.712026i	$0.252222\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.q.e.721.9		20
3.2	odd	2		360.4.q.e.241.8	yes	20
9.4	even	3	inner	1080.4.q.e.361.9		20
9.5	odd	6		360.4.q.e.121.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.q.e.121.8	✓	20	9.5	odd	6
360.4.q.e.241.8	yes	20	3.2	odd	2
1080.4.q.e.361.9		20	9.4	even	3	inner
1080.4.q.e.721.9		20	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 - 4.33013i) q^{5} +(14.1295 - 24.4730i) q^{7} +(-35.4379 + 61.3802i) q^{11} +(-31.3500 - 54.2997i) q^{13} +116.633 q^{17} +81.1071 q^{19} +(50.4969 + 87.4632i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(19.6732 - 34.0749i) q^{29} +(-66.4223 - 115.047i) q^{31} -141.295 q^{35} +382.860 q^{37} +(-88.0191 - 152.454i) q^{41} +(-32.6094 + 56.4812i) q^{43} +(244.747 - 423.914i) q^{47} +(-227.786 - 394.538i) q^{49} -223.526 q^{53} +354.379 q^{55} +(-349.850 - 605.959i) q^{59} +(-102.596 + 177.701i) q^{61} +(-156.750 + 271.499i) q^{65} +(94.4948 + 163.670i) q^{67} -383.977 q^{71} -266.289 q^{73} +(1001.44 + 1734.54i) q^{77} +(167.183 - 289.569i) q^{79} +(-125.478 + 217.334i) q^{83} +(-291.583 - 505.036i) q^{85} -1004.55 q^{89} -1771.84 q^{91} +(-202.768 - 351.204i) q^{95} +(175.973 - 304.794i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 50 q^{5} + 22 q^{7} - 50 q^{11} - 16 q^{13} + 68 q^{17} + 212 q^{19} - 50 q^{23} - 250 q^{25} + 64 q^{29} - 22 q^{31} - 220 q^{35} + 600 q^{37} + 198 q^{41} - 382 q^{43} - 128 q^{47} - 1230 q^{49}+ \cdots - 1354 q^{97}+O(q^{100}) \) 20 * q - 50 * q^5 + 22 * q^7 - 50 * q^11 - 16 * q^13 + 68 * q^17 + 212 * q^19 - 50 * q^23 - 250 * q^25 + 64 * q^29 - 22 * q^31 - 220 * q^35 + 600 * q^37 + 198 * q^41 - 382 * q^43 - 128 * q^47 - 1230 * q^49 + 1128 * q^53 + 500 * q^55 + 200 * q^59 - 232 * q^61 - 80 * q^65 - 1316 * q^67 + 228 * q^71 + 828 * q^73 - 616 * q^77 - 300 * q^79 + 1082 * q^83 - 170 * q^85 + 768 * q^89 + 1972 * q^91 - 530 * q^95 - 1354 * q^97

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