LMFDB - Embedded newform 1620.4.i.u.541.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,4,Mod(541,1620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [6,0,0,0,15,0,-15] 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864$ x^6 - x^5 + 91x^4 - 294x^3 + 8292x^2 - 17280x + 36864
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$1.09880 - 1.90317i$ of defining polynomial
Character	$\chi$	$=$	1620.541
Dual form			1620.4.i.u.1081.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 + 4.33013i) q^{5} +(-16.8420 + 29.1713i) q^{7} +(-4.74925 + 8.22594i) q^{11} +(-19.8420 - 34.3674i) q^{13} -106.365 q^{17} -96.5507 q^{19} +(20.0276 + 34.6889i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(126.209 - 218.600i) q^{29} +(-24.4363 - 42.3250i) q^{31} -168.420 q^{35} -136.476 q^{37} +(69.5522 + 120.468i) q^{41} +(101.180 - 175.248i) q^{43} +(150.025 - 259.850i) q^{47} +(-395.809 - 685.561i) q^{49} +344.142 q^{53} -47.4925 q^{55} +(-30.4669 - 52.7703i) q^{59} +(61.0612 - 105.761i) q^{61} +(99.2102 - 171.837i) q^{65} +(372.333 + 644.900i) q^{67} -436.971 q^{71} +586.353 q^{73} +(-159.974 - 277.083i) q^{77} +(-236.890 + 410.306i) q^{79} +(-499.432 + 865.041i) q^{83} +(-265.913 - 460.575i) q^{85} -204.463 q^{89} +1336.72 q^{91} +(-241.377 - 418.077i) q^{95} +(816.392 - 1414.03i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53}+ \cdots + 2556 q^{97}+O(q^{100})$ 6 * q + 15 * q^5 - 15 * q^7 + 24 * q^11 - 33 * q^13 - 84 * q^17 + 42 * q^19 - 33 * q^23 - 75 * q^25 + 222 * q^29 - 132 * q^31 - 150 * q^35 + 348 * q^37 - 99 * q^41 + 120 * q^43 + 537 * q^47 - 492 * q^49 - 534 * q^53 + 240 * q^55 - 225 * q^59 + 480 * q^61 + 165 * q^65 - 12 * q^67 - 1140 * q^71 + 2124 * q^73 + 312 * q^77 + 1026 * q^79 + 702 * q^83 - 210 * q^85 - 2280 * q^89 + 3222 * q^91 + 105 * q^95 + 2556 * q^97

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
$7$	−16.8420	+	29.1713i	−0.909385	+	1.57510i	−0.0944639	+	0.995528i	$0.530114\pi$
$7$	−16.8420	+	29.1713i	−0.909385	+	1.57510i	−0.814921	+	0.579572i	$0.803220\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.74925	+	8.22594i	−0.130178	+	0.225474i	−0.923745	−	0.383008i	$-0.874888\pi$
$11$	−4.74925	+	8.22594i	−0.130178	+	0.225474i	0.793567	+	0.608482i	$0.208221\pi$
$12$		0			0
$13$	−19.8420	−	34.3674i	−0.423322	−	0.733216i	0.572940	−	0.819598i	$-0.305803\pi$
$13$	−19.8420	−	34.3674i	−0.423322	−	0.733216i	−0.996262	+	0.0863815i	$0.972470\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−106.365			−1.51749			−0.758745	−	0.651387i	$-0.774187\pi$
$17$	−106.365			−1.51749			−0.758745	+	0.651387i	$0.774187\pi$
$18$		0			0
$19$	−96.5507			−1.16580			−0.582902	−	0.812543i	$-0.698083\pi$
$19$	−96.5507			−1.16580			−0.582902	+	0.812543i	$0.698083\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.0276	+	34.6889i	0.181567	+	0.314484i	0.942414	−	0.334447i	$-0.108550\pi$
$23$	20.0276	+	34.6889i	0.181567	+	0.314484i	−0.760847	+	0.648931i	$0.775216\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	126.209	−	218.600i	0.808151	−	1.39976i	−0.105993	−	0.994367i	$-0.533802\pi$
$29$	126.209	−	218.600i	0.808151	−	1.39976i	0.914143	−	0.405391i	$-0.132865\pi$
$30$		0			0
$31$	−24.4363	−	42.3250i	−0.141577	−	0.245219i	0.786513	−	0.617573i	$-0.211884\pi$
$31$	−24.4363	−	42.3250i	−0.141577	−	0.245219i	−0.928091	+	0.372354i	$0.878551\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−168.420			−0.813378
$36$		0			0
$37$	−136.476			−0.606391			−0.303195	−	0.952928i	$-0.598053\pi$
$37$	−136.476			−0.606391			−0.303195	+	0.952928i	$0.598053\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	69.5522	+	120.468i	0.264933	+	0.458877i	0.967546	−	0.252695i	$-0.0813170\pi$
$41$	69.5522	+	120.468i	0.264933	+	0.458877i	−0.702613	+	0.711572i	$0.747984\pi$
$42$		0			0
$43$	101.180	−	175.248i	0.358831	−	0.621514i	−0.628935	−	0.777458i	$-0.716509\pi$
$43$	101.180	−	175.248i	0.358831	−	0.621514i	0.987766	+	0.155944i	$0.0498420\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	150.025	−	259.850i	0.465603	−	0.806448i	−0.533626	−	0.845721i	$-0.679171\pi$
$47$	150.025	−	259.850i	0.465603	−	0.806448i	0.999229	+	0.0392728i	$0.0125041\pi$
$48$		0			0
$49$	−395.809	−	685.561i	−1.15396	−	1.99872i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	344.142			0.891915			0.445958	−	0.895054i	$-0.352863\pi$
$53$	344.142			0.891915			0.445958	+	0.895054i	$0.352863\pi$
$54$		0			0
$55$	−47.4925			−0.116434
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−30.4669	−	52.7703i	−0.0672281	−	0.116443i	0.830452	−	0.557090i	$-0.188082\pi$
$59$	−30.4669	−	52.7703i	−0.0672281	−	0.116443i	−0.897680	+	0.440648i	$0.854749\pi$
$60$		0			0
$61$	61.0612	−	105.761i	0.128165	−	0.221989i	−0.794800	−	0.606871i	$-0.792424\pi$
$61$	61.0612	−	105.761i	0.128165	−	0.221989i	0.922966	+	0.384882i	$0.125758\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	99.2102	−	171.837i	0.189316	−	0.327904i
$66$		0			0
$67$	372.333	+	644.900i	0.678922	+	1.17593i	0.975306	+	0.220858i	$0.0708858\pi$
$67$	372.333	+	644.900i	0.678922	+	1.17593i	−0.296384	+	0.955069i	$0.595781\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−436.971			−0.730408			−0.365204	−	0.930928i	$-0.619001\pi$
$71$	−436.971			−0.730408			−0.365204	+	0.930928i	$0.619001\pi$
$72$		0			0
$73$	586.353			0.940102			0.470051	−	0.882639i	$-0.344236\pi$
$73$	586.353			0.940102			0.470051	+	0.882639i	$0.344236\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−159.974	−	277.083i	−0.236763	−	0.410085i
$78$		0			0
$79$	−236.890	+	410.306i	−0.337370	+	0.584342i	−0.983937	−	0.178515i	$-0.942871\pi$
$79$	−236.890	+	410.306i	−0.337370	+	0.584342i	0.646567	+	0.762857i	$0.276204\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−499.432	+	865.041i	−0.660479	+	1.14398i	0.320011	+	0.947414i	$0.396313\pi$
$83$	−499.432	+	865.041i	−0.660479	+	1.14398i	−0.980490	+	0.196569i	$0.937020\pi$
$84$		0			0
$85$	−265.913	−	460.575i	−0.339321	−	0.587722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−204.463			−0.243517			−0.121759	−	0.992560i	$-0.538853\pi$
$89$	−204.463			−0.243517			−0.121759	+	0.992560i	$0.538853\pi$
$90$		0			0
$91$	1336.72			1.53985
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−241.377	−	418.077i	−0.260682	−	0.451514i
$96$		0			0
$97$	816.392	−	1414.03i	0.854557	−	1.48014i	−0.0224989	−	0.999747i	$-0.507162\pi$
$97$	816.392	−	1414.03i	0.854557	−	1.48014i	0.877056	−	0.480389i	$-0.159504\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−599.902	+	1039.06i	−0.591015	+	1.02367i	0.403081	+	0.915164i	$0.367939\pi$
$101$	−599.902	+	1039.06i	−0.591015	+	1.02367i	−0.994096	+	0.108504i	$0.965394\pi$
$102$		0			0
$103$	804.829	+	1394.00i	0.769924	+	1.33355i	0.937604	+	0.347705i	$0.113039\pi$
$103$	804.829	+	1394.00i	0.769924	+	1.33355i	−0.167680	+	0.985841i	$0.553628\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	206.586			0.186649			0.0933245	−	0.995636i	$-0.470251\pi$
$107$	206.586			0.186649			0.0933245	+	0.995636i	$0.470251\pi$
$108$		0			0
$109$	1604.44			1.40989			0.704944	−	0.709263i	$-0.250972\pi$
$109$	1604.44			1.40989			0.704944	+	0.709263i	$0.250972\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	947.514	+	1641.14i	0.788802	+	1.36625i	0.926701	+	0.375799i	$0.122632\pi$
$113$	947.514	+	1641.14i	0.788802	+	1.36625i	−0.137899	+	0.990446i	$0.544035\pi$
$114$		0			0
$115$	−100.138	+	173.444i	−0.0811994	+	0.140641i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1791.41	−	3102.81i	1.37998	−	2.39020i
$120$		0			0
$121$	620.389	+	1074.55i	0.466108	+	0.807322i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	946.537			0.661351			0.330675	−	0.943745i	$-0.392723\pi$
$127$	946.537			0.661351			0.330675	+	0.943745i	$0.392723\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	842.052	+	1458.48i	0.561606	+	0.972731i	0.997357	+	0.0726630i	$0.0231498\pi$
$131$	842.052	+	1458.48i	0.561606	+	0.972731i	−0.435750	+	0.900068i	$0.643517\pi$
$132$		0			0
$133$	1626.11	−	2816.51i	1.06016	−	1.83626i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−207.157	+	358.806i	−0.129187	+	0.223758i	−0.923362	−	0.383931i	$-0.874570\pi$
$137$	−207.157	+	358.806i	−0.129187	+	0.223758i	0.794175	+	0.607689i	$0.207903\pi$
$138$		0			0
$139$	158.970	+	275.345i	0.0970049	+	0.168017i	0.910444	−	0.413633i	$-0.135740\pi$
$139$	158.970	+	275.345i	0.0970049	+	0.168017i	−0.813439	+	0.581651i	$0.802407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	376.939			0.220428
$144$		0			0
$145$	1262.09			0.722832
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1518.77	−	2630.59i	−0.835050	−	1.44635i	−0.893990	−	0.448087i	$-0.852105\pi$
$149$	−1518.77	−	2630.59i	−0.835050	−	1.44635i	0.0589400	−	0.998262i	$-0.481228\pi$
$150$		0			0
$151$	1060.40	−	1836.67i	0.571485	−	0.989841i	−0.424929	−	0.905227i	$-0.639701\pi$
$151$	1060.40	−	1836.67i	0.571485	−	0.989841i	0.996414	−	0.0846140i	$-0.0269657\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	122.182	−	211.625i	0.0633153	−	0.109665i
$156$		0			0
$157$	−1254.17	−	2172.29i	−0.637539	−	1.10425i	−0.985971	−	0.166916i	$-0.946619\pi$
$157$	−1254.17	−	2172.29i	−0.637539	−	1.10425i	0.348432	−	0.937334i	$-0.386714\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1349.22			−0.660458
$162$		0			0
$163$	−1834.04			−0.881308			−0.440654	−	0.897677i	$-0.645253\pi$
$163$	−1834.04			−0.881308			−0.440654	+	0.897677i	$0.645253\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1594.31	+	2761.43i	0.738753	+	1.27956i	0.953057	+	0.302791i	$0.0979183\pi$
$167$	1594.31	+	2761.43i	0.738753	+	1.27956i	−0.214304	+	0.976767i	$0.568748\pi$
$168$		0			0
$169$	311.087	−	538.818i	0.141596	−	0.245252i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−112.474	+	194.811i	−0.0494293	+	0.0856141i	−0.889681	−	0.456582i	$-0.849074\pi$
$173$	−112.474	+	194.811i	−0.0494293	+	0.0856141i	0.840252	+	0.542196i	$0.182407\pi$
$174$		0			0
$175$	−421.051	−	729.282i	−0.181877	−	0.315020i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3815.09			−1.59303			−0.796517	−	0.604616i	$-0.793326\pi$
$179$	−3815.09			−1.59303			−0.796517	+	0.604616i	$0.793326\pi$
$180$		0			0
$181$	−3356.82			−1.37851			−0.689256	−	0.724518i	$-0.742062\pi$
$181$	−3356.82			−1.37851			−0.689256	+	0.724518i	$0.742062\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−341.189	−	590.957i	−0.135593	−	0.234854i
$186$		0			0
$187$	505.155	−	874.954i	0.197543	−	0.342155i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	415.584	−	719.812i	0.157438	−	0.272690i	−0.776506	−	0.630109i	$-0.783010\pi$
$191$	415.584	−	719.812i	0.157438	−	0.272690i	0.933944	+	0.357420i	$0.116343\pi$
$192$		0			0
$193$	−1519.21	−	2631.36i	−0.566609	−	0.981395i	−0.996898	−	0.0787038i	$-0.974922\pi$
$193$	−1519.21	−	2631.36i	−0.566609	−	0.981395i	0.430290	−	0.902691i	$-0.358411\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1288.06			−0.465839			−0.232919	−	0.972496i	$-0.574828\pi$
$197$	−1288.06			−0.465839			−0.232919	+	0.972496i	$0.574828\pi$
$198$		0			0
$199$	5069.77			1.80596			0.902981	−	0.429680i	$-0.141374\pi$
$199$	5069.77			1.80596			0.902981	+	0.429680i	$0.141374\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4251.22	+	7363.34i	1.46984	+	2.54584i
$204$		0			0
$205$	−347.761	+	602.340i	−0.118481	+	0.205216i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	458.544	−	794.221i	0.151761	−	0.262858i
$210$		0			0
$211$	−1261.82	−	2185.53i	−0.411692	−	0.713071i	0.583383	−	0.812197i	$-0.301729\pi$
$211$	−1261.82	−	2185.53i	−0.411692	−	0.713071i	−0.995075	+	0.0991259i	$0.968395\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1011.80			0.320948
$216$		0			0
$217$	1646.23			0.514993
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2110.50	+	3655.50i	0.642388	+	1.11265i
$222$		0			0
$223$	1202.85	−	2083.40i	0.361206	−	0.625627i	−0.626953	−	0.779057i	$-0.715698\pi$
$223$	1202.85	−	2083.40i	0.361206	−	0.625627i	0.988160	+	0.153429i	$0.0490317\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2694.15	−	4666.40i	0.787739	−	1.36440i	−0.139610	−	0.990207i	$-0.544585\pi$
$227$	2694.15	−	4666.40i	0.787739	−	1.36440i	0.927349	−	0.374197i	$-0.122082\pi$
$228$		0			0
$229$	−2357.06	−	4082.55i	−0.680171	−	1.17809i	−0.974929	−	0.222518i	$-0.928572\pi$
$229$	−2357.06	−	4082.55i	−0.680171	−	1.17809i	0.294758	−	0.955572i	$-0.404761\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5065.99			−1.42439			−0.712197	−	0.701979i	$-0.752300\pi$
$233$	−5065.99			−1.42439			−0.712197	+	0.701979i	$0.752300\pi$
$234$		0			0
$235$	1500.25			0.416448
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3318.96	−	5748.61i	−0.898267	−	1.55584i	−0.829709	−	0.558196i	$-0.811494\pi$
$239$	−3318.96	−	5748.61i	−0.898267	−	1.55584i	−0.0685578	−	0.997647i	$-0.521840\pi$
$240$		0			0
$241$	2529.41	−	4381.07i	0.676073	−	1.17099i	−0.300081	−	0.953914i	$-0.597014\pi$
$241$	2529.41	−	4381.07i	0.676073	−	1.17099i	0.976154	−	0.217079i	$-0.0696530\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1979.04	−	3427.80i	0.516067	−	0.893855i
$246$		0			0
$247$	1915.76	+	3318.20i	0.493511	+	0.854786i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2428.65			−0.610737			−0.305368	−	0.952234i	$-0.598780\pi$
$251$	−2428.65			−0.610737			−0.305368	+	0.952234i	$0.598780\pi$
$252$		0			0
$253$	−380.465			−0.0945439
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1860.48	+	3222.45i	0.451571	+	0.782144i	0.998484	−	0.0550455i	$-0.0175304\pi$
$257$	1860.48	+	3222.45i	0.451571	+	0.782144i	−0.546913	+	0.837190i	$0.684197\pi$
$258$		0			0
$259$	2298.53	−	3981.17i	0.551443	−	0.955127i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−108.507	+	187.939i	−0.0254403	+	0.0440639i	−0.878465	−	0.477806i	$-0.841432\pi$
$263$	−108.507	+	187.939i	−0.0254403	+	0.0440639i	0.853025	+	0.521870i	$0.174765\pi$
$264$		0			0
$265$	860.354	+	1490.18i	0.199438	+	0.345437i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	766.464			0.173725			0.0868627	−	0.996220i	$-0.472316\pi$
$269$	766.464			0.173725			0.0868627	+	0.996220i	$0.472316\pi$
$270$		0			0
$271$	4030.90			0.903541			0.451770	−	0.892134i	$-0.350793\pi$
$271$	4030.90			0.903541			0.451770	+	0.892134i	$0.350793\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−118.731	−	205.649i	−0.0260355	−	0.0450948i
$276$		0			0
$277$	591.446	−	1024.41i	0.128291	−	0.222206i	−0.794724	−	0.606971i	$-0.792384\pi$
$277$	591.446	−	1024.41i	0.128291	−	0.222206i	0.923014	+	0.384765i	$0.125718\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4030.41	−	6980.87i	0.855637	−	1.48201i	−0.0204158	−	0.999792i	$-0.506499\pi$
$281$	4030.41	−	6980.87i	0.855637	−	1.48201i	0.876053	−	0.482215i	$-0.160168\pi$
$282$		0			0
$283$	410.066	+	710.254i	0.0861338	+	0.149188i	0.905874	−	0.423548i	$-0.139215\pi$
$283$	410.066	+	710.254i	0.0861338	+	0.149188i	−0.819740	+	0.572736i	$0.805882\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4685.61			−0.963703
$288$		0			0
$289$	6400.55			1.30278
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−204.870	−	354.845i	−0.0408486	−	0.0707519i	0.844878	−	0.534959i	$-0.179673\pi$
$293$	−204.870	−	354.845i	−0.0408486	−	0.0707519i	−0.885727	+	0.464207i	$0.846339\pi$
$294$		0			0
$295$	152.335	−	263.852i	0.0300653	−	0.0520747i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	794.778	−	1376.60i	0.153723	−	0.266256i
$300$		0			0
$301$	3408.14	+	5903.07i	0.652631	+	1.13039i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	610.612			0.114635
$306$		0			0
$307$	4200.96			0.780982			0.390491	−	0.920607i	$-0.372305\pi$
$307$	4200.96			0.780982			0.390491	+	0.920607i	$0.372305\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4221.56	−	7311.96i	−0.769719	−	1.33319i	−0.937715	−	0.347405i	$-0.887063\pi$
$311$	−4221.56	−	7311.96i	−0.769719	−	1.33319i	0.167996	−	0.985788i	$-0.446271\pi$
$312$		0			0
$313$	−67.0176	+	116.078i	−0.0121024	+	0.0209620i	−0.872013	−	0.489482i	$-0.837186\pi$
$313$	−67.0176	+	116.078i	−0.0121024	+	0.0209620i	0.859911	+	0.510444i	$0.170519\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4790.31	+	8297.05i	−0.848740	+	1.47006i	0.0335941	+	0.999436i	$0.489305\pi$
$317$	−4790.31	+	8297.05i	−0.848740	+	1.47006i	−0.882334	+	0.470624i	$0.844029\pi$
$318$		0			0
$319$	1198.79	+	2076.37i	0.210406	+	0.364434i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10269.6			1.76910
$324$		0			0
$325$	992.102			0.169329
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5053.44	+	8752.82i	0.846825	+	1.46674i
$330$		0			0
$331$	4379.82	−	7586.07i	0.727301	−	1.25972i	−0.230718	−	0.973021i	$-0.574108\pi$
$331$	4379.82	−	7586.07i	0.727301	−	1.25972i	0.958020	−	0.286702i	$-0.0925590\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1861.67	+	3224.50i	−0.303623	+	0.525891i
$336$		0			0
$337$	−2779.33	−	4813.94i	−0.449257	−	0.778136i	0.549081	−	0.835769i	$-0.314978\pi$
$337$	−2779.33	−	4813.94i	−0.449257	−	0.778136i	−0.998338	+	0.0576333i	$0.981645\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	464.217			0.0737207
$342$		0			0
$343$	15111.3			2.37881
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5343.54	−	9255.28i	−0.826675	−	1.43184i	−0.900632	−	0.434582i	$-0.856896\pi$
$347$	−5343.54	−	9255.28i	−0.826675	−	1.43184i	0.0739571	−	0.997261i	$-0.476437\pi$
$348$		0			0
$349$	−1969.73	+	3411.67i	−0.302113	+	0.523274i	−0.976614	−	0.214999i	$-0.931025\pi$
$349$	−1969.73	+	3411.67i	−0.302113	+	0.523274i	0.674502	+	0.738273i	$0.264358\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2648.95	−	4588.11i	0.399403	−	0.691786i	−0.594249	−	0.804281i	$-0.702551\pi$
$353$	2648.95	−	4588.11i	0.399403	−	0.691786i	0.993652	+	0.112495i	$0.0358841\pi$
$354$		0			0
$355$	−1092.43	−	1892.14i	−0.163324	−	0.282886i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1103.06			0.162165			0.0810827	−	0.996707i	$-0.474162\pi$
$359$	1103.06			0.162165			0.0810827	+	0.996707i	$0.474162\pi$
$360$		0			0
$361$	2463.05			0.359097
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1465.88	+	2538.98i	0.210213	+	0.364100i
$366$		0			0
$367$	3657.86	−	6335.60i	0.520269	−	0.901133i	−0.479453	−	0.877567i	$-0.659165\pi$
$367$	3657.86	−	6335.60i	0.520269	−	0.901133i	0.999722	−	0.0235652i	$-0.00750172\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5796.05	+	10039.1i	−0.811094	+	1.40486i
$372$		0			0
$373$	942.902	+	1633.15i	0.130889	+	0.226706i	0.924020	−	0.382345i	$-0.124884\pi$
$373$	942.902	+	1633.15i	0.130889	+	0.226706i	−0.793131	+	0.609052i	$0.791550\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10017.0			−1.36843
$378$		0			0
$379$	−5371.24			−0.727973			−0.363987	−	0.931404i	$-0.618585\pi$
$379$	−5371.24			−0.727973			−0.363987	+	0.931404i	$0.618585\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	36.1306	+	62.5800i	0.00482033	+	0.00834906i	0.868426	−	0.495820i	$-0.165132\pi$
$383$	36.1306	+	62.5800i	0.00482033	+	0.00834906i	−0.863605	+	0.504169i	$0.831799\pi$
$384$		0			0
$385$	799.871	−	1385.42i	0.105884	−	0.183396i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	896.639	−	1553.02i	0.116867	−	0.202420i	−0.801657	−	0.597784i	$-0.796048\pi$
$389$	896.639	−	1553.02i	0.116867	−	0.202420i	0.918525	+	0.395364i	$0.129381\pi$
$390$		0			0
$391$	−2130.24	−	3689.69i	−0.275527	−	0.477226i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2368.90			−0.301753
$396$		0			0
$397$	596.085			0.0753568			0.0376784	−	0.999290i	$-0.488004\pi$
$397$	596.085			0.0753568			0.0376784	+	0.999290i	$0.488004\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	477.398	+	826.878i	0.0594517	+	0.102973i	0.894219	−	0.447629i	$-0.147731\pi$
$401$	477.398	+	826.878i	0.0594517	+	0.102973i	−0.834768	+	0.550602i	$0.814398\pi$
$402$		0			0
$403$	−969.733	+	1679.63i	−0.119866	+	0.207613i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	648.157	−	1122.64i	0.0789385	−	0.136725i
$408$		0			0
$409$	583.519	+	1010.68i	0.0705456	+	0.122189i	0.899141	−	0.437660i	$-0.144193\pi$
$409$	583.519	+	1010.68i	0.0705456	+	0.122189i	−0.828595	+	0.559849i	$0.810859\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2052.50			0.244545
$414$		0			0
$415$	−4994.32			−0.590750
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7428.50	+	12866.5i	0.866124	+	1.50017i	0.865926	+	0.500172i	$0.166730\pi$
$419$	7428.50	+	12866.5i	0.866124	+	1.50017i	0.000198097		1.00000i	$0.499937\pi$
$420$		0			0
$421$	−1080.53	+	1871.54i	−0.125088	+	0.216659i	−0.921767	−	0.387744i	$-0.873255\pi$
$421$	−1080.53	+	1871.54i	−0.125088	+	0.216659i	0.796679	+	0.604402i	$0.206588\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1329.56	−	2302.87i	0.151749	−	0.262837i
$426$		0			0
$427$	2056.79	+	3562.47i	0.233103	+	0.403747i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6031.12			−0.674035			−0.337017	−	0.941498i	$-0.609418\pi$
$431$	−6031.12			−0.674035			−0.337017	+	0.941498i	$0.609418\pi$
$432$		0			0
$433$	7477.24			0.829869			0.414934	−	0.909851i	$-0.363804\pi$
$433$	7477.24			0.829869			0.414934	+	0.909851i	$0.363804\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1933.68	−	3349.24i	−0.211672	−	0.366626i
$438$		0			0
$439$	3258.47	−	5643.83i	0.354256	−	0.613589i	−0.632735	−	0.774369i	$-0.718068\pi$
$439$	3258.47	−	5643.83i	0.354256	−	0.613589i	0.986990	+	0.160780i	$0.0514009\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3956.36	−	6852.61i	0.424317	−	0.734938i	−0.572040	−	0.820226i	$-0.693848\pi$
$443$	3956.36	−	6852.61i	0.424317	−	0.734938i	0.996356	+	0.0852879i	$0.0271810\pi$
$444$		0			0
$445$	−511.158	−	885.351i	−0.0544521	−	0.0943139i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8686.04			−0.912961			−0.456480	−	0.889733i	$-0.650890\pi$
$449$	−8686.04			−0.912961			−0.456480	+	0.889733i	$0.650890\pi$
$450$		0			0
$451$	−1321.28			−0.137953
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3341.80	+	5788.18i	0.344321	+	0.596382i
$456$		0			0
$457$	3393.69	−	5878.05i	0.347375	−	0.601671i	−0.638407	−	0.769699i	$-0.720406\pi$
$457$	3393.69	−	5878.05i	0.347375	−	0.601671i	0.985782	+	0.168028i	$0.0537398\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7565.43	−	13103.7i	0.764332	−	1.32386i	−0.176267	−	0.984342i	$-0.556402\pi$
$461$	7565.43	−	13103.7i	0.764332	−	1.32386i	0.940599	−	0.339520i	$-0.110265\pi$
$462$		0			0
$463$	−6158.34	−	10666.6i	−0.618147	−	1.07066i	−0.989824	−	0.142300i	$-0.954550\pi$
$463$	−6158.34	−	10666.6i	−0.618147	−	1.07066i	0.371676	−	0.928362i	$-0.378783\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16423.4			1.62737			0.813687	−	0.581303i	$-0.197457\pi$
$467$	16423.4			1.62737			0.813687	+	0.581303i	$0.197457\pi$
$468$		0			0
$469$	−25083.4			−2.46960
$470$		0			0
$471$		0			0
$472$		0			0
$473$	961.054	+	1664.60i	0.0934235	+	0.161814i
$474$		0			0
$475$	1206.88	−	2090.39i	0.116580	−	0.201923i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5880.92	+	10186.1i	−0.560973	+	0.971635i	0.436438	+	0.899734i	$0.356240\pi$
$479$	−5880.92	+	10186.1i	−0.560973	+	0.971635i	−0.997412	+	0.0719004i	$0.977094\pi$
$480$		0			0
$481$	2707.96	+	4690.32i	0.256699	+	0.444615i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8163.92			0.764339
$486$		0			0
$487$	−18620.0			−1.73255			−0.866277	−	0.499564i	$-0.833494\pi$
$487$	−18620.0			−1.73255			−0.866277	+	0.499564i	$0.833494\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5011.10	−	8679.47i	−0.460586	−	0.797758i	0.538405	−	0.842687i	$-0.319027\pi$
$491$	−5011.10	−	8679.47i	−0.460586	−	0.797758i	−0.998990	+	0.0449288i	$0.985694\pi$
$492$		0			0
$493$	−13424.2	+	23251.4i	−1.22636	+	2.12412i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7359.49	−	12747.0i	0.664221	−	1.15047i
$498$		0			0
$499$	1921.82	+	3328.69i	0.172410	+	0.298623i	0.939262	−	0.343201i	$-0.111511\pi$
$499$	1921.82	+	3328.69i	0.172410	+	0.298623i	−0.766852	+	0.641824i	$0.778178\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6782.43			0.601220			0.300610	−	0.953747i	$-0.402810\pi$
$503$	6782.43			0.601220			0.300610	+	0.953747i	$0.402810\pi$
$504$		0			0
$505$	−5999.02			−0.528620
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8509.53	+	14738.9i	0.741018	+	1.28348i	0.952032	+	0.305999i	$0.0989903\pi$
$509$	8509.53	+	14738.9i	0.741018	+	1.28348i	−0.211014	+	0.977483i	$0.567676\pi$
$510$		0			0
$511$	−9875.38	+	17104.7i	−0.854914	+	1.48076i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4024.14	+	6970.02i	−0.344320	+	0.596380i
$516$		0			0
$517$	1425.01	+	2468.19i	0.121222	+	0.209963i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19278.7			−1.62114			−0.810572	−	0.585639i	$-0.800844\pi$
$521$	−19278.7			−1.62114			−0.810572	+	0.585639i	$0.800844\pi$
$522$		0			0
$523$	8074.65			0.675105			0.337552	−	0.941307i	$-0.390401\pi$
$523$	8074.65			0.675105			0.337552	+	0.941307i	$0.390401\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2599.17	+	4501.90i	0.214842	+	0.372118i
$528$		0			0
$529$	5281.29	−	9147.46i	0.434067	−	0.751825i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2760.12	−	4780.66i	0.224304	−	0.388506i
$534$		0			0
$535$	516.465	+	894.544i	0.0417360	+	0.0722888i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7519.18			0.600879
$540$		0			0
$541$	−18260.8			−1.45119			−0.725594	−	0.688123i	$-0.758435\pi$
$541$	−18260.8			−1.45119			−0.725594	+	0.688123i	$0.758435\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4011.11	+	6947.44i	0.315260	+	0.546047i
$546$		0			0
$547$	−5353.71	+	9272.90i	−0.418479	+	0.724827i	−0.995787	−	0.0916997i	$-0.970770\pi$
$547$	−5353.71	+	9272.90i	−0.418479	+	0.724827i	0.577308	+	0.816527i	$0.304103\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12185.5	+	21106.0i	−0.942145	+	1.63184i
$552$		0			0
$553$	−7979.42	−	13820.8i	−0.613598	−	1.06278i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17348.5			−1.31971			−0.659857	−	0.751392i	$-0.729383\pi$
$557$	−17348.5			−1.31971			−0.659857	+	0.751392i	$0.729383\pi$
$558$		0			0
$559$	−8030.44			−0.607605
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7473.57	−	12944.6i	−0.559455	−	0.969005i	−0.997542	−	0.0700724i	$-0.977677\pi$
$563$	−7473.57	−	12944.6i	−0.559455	−	0.969005i	0.438086	−	0.898933i	$-0.355656\pi$
$564$		0			0
$565$	−4737.57	+	8205.71i	−0.352763	+	0.611003i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13361.5	+	23142.8i	−0.984435	+	1.70509i	−0.340013	+	0.940421i	$0.610432\pi$
$569$	−13361.5	+	23142.8i	−0.984435	+	1.70509i	−0.644422	+	0.764670i	$0.722902\pi$
$570$		0			0
$571$	−4209.97	−	7291.88i	−0.308549	−	0.534423i	0.669496	−	0.742816i	$-0.266510\pi$
$571$	−4209.97	−	7291.88i	−0.308549	−	0.534423i	−0.978045	+	0.208393i	$0.933177\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1001.38			−0.0726269
$576$		0			0
$577$	2921.42			0.210780			0.105390	−	0.994431i	$-0.466391\pi$
$577$	2921.42			0.210780			0.105390	+	0.994431i	$0.466391\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16822.9	−	29138.1i	−1.20126	−	2.08064i
$582$		0			0
$583$	−1634.42	+	2830.89i	−0.116107	+	0.201104i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−32.3681	+	56.0632i	−0.00227594	+	0.00394204i	−0.867161	−	0.498028i	$-0.834058\pi$
$587$	−32.3681	+	56.0632i	−0.00227594	+	0.00394204i	0.864885	+	0.501970i	$0.167391\pi$
$588$		0			0
$589$	2359.35	+	4086.51i	0.165051	+	0.285877i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20682.2			1.43223			0.716117	−	0.697980i	$-0.245918\pi$
$593$	20682.2			1.43223			0.716117	+	0.697980i	$0.245918\pi$
$594$		0			0
$595$	17914.1			1.23429
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10319.9	−	17874.6i	−0.703939	−	1.21926i	−0.967073	−	0.254499i	$-0.918089\pi$
$599$	−10319.9	−	17874.6i	−0.703939	−	1.21926i	0.263134	−	0.964759i	$-0.415244\pi$
$600$		0			0
$601$	9874.60	−	17103.3i	0.670205	−	1.16083i	−0.307640	−	0.951503i	$-0.599539\pi$
$601$	9874.60	−	17103.3i	0.670205	−	1.16083i	0.977846	−	0.209327i	$-0.0671273\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3101.95	+	5372.73i	−0.208450	+	0.361045i
$606$		0			0
$607$	7590.19	+	13146.6i	0.507539	+	0.879084i	0.999962	+	0.00872760i	$0.00277812\pi$
$607$	7590.19	+	13146.6i	0.507539	+	0.879084i	−0.492423	+	0.870356i	$0.663889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11907.2			−0.788401
$612$		0			0
$613$	−15998.0			−1.05408			−0.527042	−	0.849839i	$-0.676699\pi$
$613$	−15998.0			−1.05408			−0.527042	+	0.849839i	$0.676699\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	913.698	+	1582.57i	0.0596176	+	0.103261i	0.894294	−	0.447480i	$-0.147679\pi$
$617$	913.698	+	1582.57i	0.0596176	+	0.103261i	−0.834676	+	0.550741i	$0.814345\pi$
$618$		0			0
$619$	6048.28	−	10475.9i	0.392732	−	0.680232i	−0.600077	−	0.799942i	$-0.704863\pi$
$619$	6048.28	−	10475.9i	0.392732	−	0.680232i	0.992809	+	0.119711i	$0.0381967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3443.58	−	5964.45i	0.221451	−	0.383564i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14516.3			0.920192
$630$		0			0
$631$	20914.5			1.31948			0.659741	−	0.751493i	$-0.270666\pi$
$631$	20914.5			1.31948			0.659741	+	0.751493i	$0.270666\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2366.34	+	4098.62i	0.147883	+	0.256140i
$636$		0			0
$637$	−15707.3	+	27205.9i	−0.976995	+	1.69221i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7245.19	−	12549.0i	0.446440	−	0.773257i	−0.551711	−	0.834035i	$-0.686025\pi$
$641$	7245.19	−	12549.0i	0.446440	−	0.773257i	0.998151	+	0.0607786i	$0.0193583\pi$
$642$		0			0
$643$	−5758.42	−	9973.88i	−0.353173	−	0.611713i	0.633631	−	0.773636i	$-0.281564\pi$
$643$	−5758.42	−	9973.88i	−0.353173	−	0.611713i	−0.986803	+	0.161923i	$0.948231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18928.4			1.15016			0.575078	−	0.818099i	$-0.304972\pi$
$647$	18928.4			1.15016			0.575078	+	0.818099i	$0.304972\pi$
$648$		0			0
$649$	578.781			0.0350064
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2572.08	−	4454.97i	−0.154140	−	0.266978i	0.778606	−	0.627513i	$-0.215927\pi$
$653$	−2572.08	−	4454.97i	−0.154140	−	0.266978i	−0.932745	+	0.360536i	$0.882594\pi$
$654$		0			0
$655$	−4210.26	+	7292.38i	−0.251158	+	0.435018i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1494.22	+	2588.07i	−0.0883257	+	0.152985i	−0.906804	−	0.421554i	$-0.861485\pi$
$659$	−1494.22	+	2588.07i	−0.0883257	+	0.152985i	0.818478	+	0.574538i	$0.194818\pi$
$660$		0			0
$661$	6725.48	+	11648.9i	0.395750	+	0.685459i	0.993197	−	0.116449i	$-0.0371513\pi$
$661$	6725.48	+	11648.9i	0.395750	+	0.685459i	−0.597446	+	0.801909i	$0.703818\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16261.1			0.948239
$666$		0			0
$667$	10110.6			0.586935
$668$		0			0
$669$		0			0
$670$		0			0
$671$	579.990	+	1004.57i	0.0333685	+	0.0577960i
$672$		0			0
$673$	−5042.84	+	8734.46i	−0.288837	+	0.500280i	−0.973532	−	0.228549i	$-0.926602\pi$
$673$	−5042.84	+	8734.46i	−0.288837	+	0.500280i	0.684695	+	0.728829i	$0.259935\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10258.9	+	17769.0i	−0.582396	+	1.00874i	0.412798	+	0.910822i	$0.364551\pi$
$677$	−10258.9	+	17769.0i	−0.582396	+	1.00874i	−0.995195	+	0.0979175i	$0.968782\pi$
$678$		0			0
$679$	27499.4	+	47630.4i	1.55424	+	2.69203i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15006.6			0.840719			0.420359	−	0.907358i	$-0.361904\pi$
$683$	15006.6			0.840719			0.420359	+	0.907358i	$0.361904\pi$
$684$		0			0
$685$	−2071.57			−0.115548
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6828.48	−	11827.3i	−0.377568	−	0.653967i
$690$		0			0
$691$	−12859.3	+	22272.9i	−0.707944	+	1.22619i	0.257675	+	0.966232i	$0.417044\pi$
$691$	−12859.3	+	22272.9i	−0.707944	+	1.22619i	−0.965619	+	0.259963i	$0.916290\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−794.852	+	1376.72i	−0.0433819	+	0.0751397i
$696$		0			0
$697$	−7397.94	−	12813.6i	−0.402033	−	0.696341i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14908.1			0.803239			0.401619	−	0.915807i	$-0.368448\pi$
$701$	14908.1			0.803239			0.401619	+	0.915807i	$0.368448\pi$
$702$		0			0
$703$	13176.8			0.706932
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20207.2	−	34999.8i	−1.07492	−	1.86182i
$708$		0			0
$709$	−11696.0	+	20258.0i	−0.619537	+	1.07307i	0.370033	+	0.929019i	$0.379346\pi$
$709$	−11696.0	+	20258.0i	−0.619537	+	1.07307i	−0.989570	+	0.144052i	$0.953987\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	978.803	−	1695.34i	0.0514116	−	0.0890475i
$714$		0			0
$715$	942.348	+	1632.20i	0.0492893	+	0.0853715i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22561.1			1.17022			0.585110	−	0.810954i	$-0.301051\pi$
$719$	22561.1			1.17022			0.585110	+	0.810954i	$0.301051\pi$
$720$		0			0
$721$	−54219.8			−2.80063
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3155.22	+	5465.00i	0.161630	+	0.279952i
$726$		0			0
$727$	10432.3	−	18069.3i	0.532206	−	0.921809i	−0.467087	−	0.884212i	$-0.654696\pi$
$727$	10432.3	−	18069.3i	0.532206	−	0.921809i	0.999293	−	0.0375970i	$-0.0119703\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10762.0	+	18640.3i	−0.544523	+	0.943142i
$732$		0			0
$733$	−12277.9	−	21266.0i	−0.618683	−	1.07159i	−0.989726	−	0.142975i	$-0.954333\pi$
$733$	−12277.9	−	21266.0i	−0.618683	−	1.07159i	0.371043	−	0.928616i	$-0.379000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7073.22			−0.353521
$738$		0			0
$739$	14842.0			0.738801			0.369400	−	0.929270i	$-0.379563\pi$
$739$	14842.0			0.738801			0.369400	+	0.929270i	$0.379563\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7759.05	+	13439.1i	0.383111	+	0.663568i	0.991505	−	0.130067i	$-0.0415192\pi$
$743$	7759.05	+	13439.1i	0.383111	+	0.663568i	−0.608394	+	0.793635i	$0.708186\pi$
$744$		0			0
$745$	7593.85	−	13152.9i	0.373446	−	0.646827i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3479.33	+	6026.38i	−0.169736	+	0.293991i
$750$		0			0
$751$	19646.8	+	34029.3i	0.954624	+	1.65346i	0.735227	+	0.677821i	$0.237076\pi$
$751$	19646.8	+	34029.3i	0.954624	+	1.65346i	0.219397	+	0.975636i	$0.429591\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10604.0			0.511152
$756$		0			0
$757$	−30307.7			−1.45516			−0.727578	−	0.686025i	$-0.759354\pi$
$757$	−30307.7			−1.45516			−0.727578	+	0.686025i	$0.759354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13679.1	−	23693.0i	−0.651601	−	1.12861i	−0.982734	−	0.185022i	$-0.940764\pi$
$761$	−13679.1	−	23693.0i	−0.651601	−	1.12861i	0.331133	−	0.943584i	$-0.392569\pi$
$762$		0			0
$763$	−27022.1	+	46803.6i	−1.28213	+	2.22071i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1209.05	+	2094.14i	−0.0569183	+	0.0985855i
$768$		0			0
$769$	−8641.10	−	14966.8i	−0.405209	−	0.701843i	0.589136	−	0.808034i	$-0.299468\pi$
$769$	−8641.10	−	14966.8i	−0.405209	−	0.701843i	−0.994346	+	0.106190i	$0.966135\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30621.8			1.42482			0.712411	−	0.701762i	$-0.247603\pi$
$773$	30621.8			1.42482			0.712411	+	0.701762i	$0.247603\pi$
$774$		0			0
$775$	1221.82			0.0566309
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6715.32	−	11631.3i	−0.308859	−	0.534960i
$780$		0			0
$781$	2075.29	−	3594.50i	0.0950827	−	0.164688i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6270.85	−	10861.4i	0.285116	−	0.493836i
$786$		0			0
$787$	16393.3	+	28394.0i	0.742512	+	1.28607i	0.951348	+	0.308118i	$0.0996991\pi$
$787$	16393.3	+	28394.0i	0.742512	+	1.28607i	−0.208836	+	0.977951i	$0.566968\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−63832.3			−2.86930
$792$		0			0
$793$	−4846.32			−0.217021
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6846.03	−	11857.7i	−0.304265	−	0.527002i	0.672833	−	0.739795i	$-0.265077\pi$
$797$	−6846.03	−	11857.7i	−0.304265	−	0.527002i	−0.977097	+	0.212793i	$0.931744\pi$
$798$		0			0
$799$	−15957.4	+	27639.0i	−0.706548	+	1.22378i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2784.74	+	4823.31i	−0.122380	+	0.211969i
$804$		0			0
$805$	−3373.06	−	5842.31i	−0.147683	−	0.255794i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39429.7			−1.71357			−0.856783	−	0.515677i	$-0.827541\pi$
$809$	−39429.7			−1.71357			−0.856783	+	0.515677i	$0.827541\pi$
$810$		0			0
$811$	−10397.9			−0.450209			−0.225105	−	0.974335i	$-0.572272\pi$
$811$	−10397.9			−0.450209			−0.225105	+	0.974335i	$0.572272\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4585.10	−	7941.63i	−0.197066	−	0.341329i
$816$		0			0
$817$	−9768.97	+	16920.3i	−0.418327	+	0.724563i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1456.54	−	2522.80i	0.0619166	−	0.107243i	−0.833405	−	0.552662i	$-0.813612\pi$
$821$	1456.54	−	2522.80i	0.0619166	−	0.107243i	0.895322	+	0.445419i	$0.146945\pi$
$822$		0			0
$823$	8467.29	+	14665.8i	0.358628	+	0.621163i	0.987732	−	0.156159i	$-0.0499112\pi$
$823$	8467.29	+	14665.8i	0.358628	+	0.621163i	−0.629104	+	0.777321i	$0.716578\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1191.52			0.0501006			0.0250503	−	0.999686i	$-0.492025\pi$
$827$	1191.52			0.0501006			0.0250503	+	0.999686i	$0.492025\pi$
$828$		0			0
$829$	−5792.37			−0.242675			−0.121337	−	0.992611i	$-0.538718\pi$
$829$	−5792.37			−0.242675			−0.121337	+	0.992611i	$0.538718\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	42100.3	+	72919.8i	1.75113	+	3.03304i
$834$		0			0
$835$	−7971.57	+	13807.2i	−0.330380	+	0.572235i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1310.10	+	2269.16i	−0.0539091	+	0.0933734i	−0.891721	−	0.452586i	$-0.850501\pi$
$839$	−1310.10	+	2269.16i	−0.0539091	+	0.0933734i	0.837811	+	0.545960i	$0.183835\pi$
$840$		0			0
$841$	−19662.8	−	34056.9i	−0.806215	−	1.39641i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3110.87			0.126647
$846$		0			0
$847$	−41794.5			−1.69548
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2733.28	−	4734.18i	−0.110101	−	0.190700i
$852$		0			0
$853$	−742.553	+	1286.14i	−0.0298060	+	0.0516256i	−0.880544	−	0.473965i	$-0.842822\pi$
$853$	−742.553	+	1286.14i	−0.0298060	+	0.0516256i	0.850738	+	0.525591i	$0.176156\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6999.92	+	12124.2i	−0.279011	+	0.483262i	−0.971139	−	0.238513i	$-0.923340\pi$
$857$	−6999.92	+	12124.2i	−0.279011	+	0.483262i	0.692128	+	0.721775i	$0.256673\pi$
$858$		0			0
$859$	−23003.1	−	39842.6i	−0.913686	−	1.58255i	−0.808814	−	0.588064i	$-0.799890\pi$
$859$	−23003.1	−	39842.6i	−0.913686	−	1.58255i	−0.104872	−	0.994486i	$-0.533443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37731.8			1.48830			0.744152	−	0.668010i	$-0.232854\pi$
$863$	37731.8			1.48830			0.744152	+	0.668010i	$0.232854\pi$
$864$		0			0
$865$	−1124.74			−0.0442109
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2250.10	−	3897.29i	−0.0878360	−	0.152136i
$870$		0			0
$871$	14775.7	−	25592.3i	0.574806	−	0.995593i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2105.26	−	3646.41i	0.0813378	−	0.140881i
$876$		0			0
$877$	−20221.9	−	35025.3i	−0.778614	−	1.34860i	−0.932741	−	0.360547i	$-0.882590\pi$
$877$	−20221.9	−	35025.3i	−0.778614	−	1.34860i	0.154127	−	0.988051i	$-0.450743\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	35957.5			1.37507			0.687535	−	0.726151i	$-0.258693\pi$
$881$	35957.5			1.37507			0.687535	+	0.726151i	$0.258693\pi$
$882$		0			0
$883$	4208.35			0.160387			0.0801937	−	0.996779i	$-0.474446\pi$
$883$	4208.35			0.160387			0.0801937	+	0.996779i	$0.474446\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3307.76	+	5729.21i	0.125213	+	0.216875i	0.921816	−	0.387627i	$-0.126705\pi$
$887$	3307.76	+	5729.21i	0.125213	+	0.216875i	−0.796603	+	0.604502i	$0.793372\pi$
$888$		0			0
$889$	−15941.6	+	27611.7i	−0.601422	+	1.04169i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14485.0	+	25088.7i	−0.542801	+	0.940160i
$894$		0			0
$895$	−9537.72	−	16519.8i	−0.356213	−	0.616979i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12336.3			−0.457663
$900$		0			0
$901$	−36604.7			−1.35347
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8392.05	−	14535.5i	−0.308244	−	0.533895i
$906$		0			0
$907$	18176.4	−	31482.4i	0.665421	−	1.15254i	−0.313750	−	0.949506i	$-0.601585\pi$
$907$	18176.4	−	31482.4i	0.665421	−	1.15254i	0.979171	−	0.203037i	$-0.0650813\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1122.98	−	1945.06i	0.0408409	−	0.0707384i	−0.844882	−	0.534952i	$-0.820330\pi$
$911$	1122.98	−	1945.06i	0.0408409	−	0.0707384i	0.885723	+	0.464214i	$0.153663\pi$
$912$		0			0
$913$	−4743.85	−	8216.60i	−0.171959	−	0.297842i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−56727.5			−2.04286
$918$		0			0
$919$	−13665.8			−0.490527			−0.245263	−	0.969456i	$-0.578874\pi$
$919$	−13665.8			−0.490527			−0.245263	+	0.969456i	$0.578874\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8670.40	+	15017.6i	0.309198	+	0.535547i
$924$		0			0
$925$	1705.95	−	2954.78i	0.0606391	−	0.105030i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21754.8	−	37680.3i	0.768300	−	1.33073i	−0.170185	−	0.985412i	$-0.554436\pi$
$929$	21754.8	−	37680.3i	0.768300	−	1.33073i	0.938484	−	0.345322i	$-0.112230\pi$
$930$		0			0
$931$	38215.6	+	66191.4i	1.34529	+	2.33011i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5051.55			0.176688
$936$		0			0
$937$	−6545.89			−0.228223			−0.114111	−	0.993468i	$-0.536402\pi$
$937$	−6545.89			−0.228223			−0.114111	+	0.993468i	$0.536402\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−118.739	−	205.662i	−0.00411349	−	0.00712477i	0.863961	−	0.503558i	$-0.167976\pi$
$941$	−118.739	−	205.662i	−0.00411349	−	0.00712477i	−0.868075	+	0.496433i	$0.834643\pi$
$942$		0			0
$943$	−2785.93	+	4825.38i	−0.0962062	+	0.166634i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17431.3	+	30191.9i	−0.598143	+	1.03601i	0.394953	+	0.918701i	$0.370761\pi$
$947$	−17431.3	+	30191.9i	−0.598143	+	1.03601i	−0.993095	+	0.117312i	$0.962572\pi$
$948$		0			0
$949$	−11634.4	−	20151.4i	−0.397966	−	0.689298i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30830.7			1.04796			0.523979	−	0.851731i	$-0.324447\pi$
$953$	30830.7			1.04796			0.523979	+	0.851731i	$0.324447\pi$
$954$		0			0
$955$	4155.84			0.140816
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6977.88	−	12086.1i	−0.234961	−	0.406964i
$960$		0			0
$961$	13701.2	−	23731.2i	0.459912	−	0.796591i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7596.07	−	13156.8i	0.253395	−	0.438893i
$966$		0			0
$967$	8446.37	+	14629.5i	0.280886	+	0.486509i	0.971603	−	0.236616i	$-0.0760384\pi$
$967$	8446.37	+	14629.5i	0.280886	+	0.486509i	−0.690717	+	0.723125i	$0.742705\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3347.26			−0.110627			−0.0553135	−	0.998469i	$-0.517616\pi$
$971$	−3347.26			−0.110627			−0.0553135	+	0.998469i	$0.517616\pi$
$972$		0			0
$973$	−10709.5			−0.352859
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10665.8	−	18473.7i	−0.349261	−	0.604938i	0.636857	−	0.770982i	$-0.280234\pi$
$977$	−10665.8	−	18473.7i	−0.349261	−	0.604938i	−0.986118	+	0.166044i	$0.946901\pi$
$978$		0			0
$979$	971.047	−	1681.90i	0.0317005	−	0.0549069i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3765.57	+	6522.15i	−0.122180	+	0.211622i	−0.920627	−	0.390443i	$-0.872322\pi$
$983$	−3765.57	+	6522.15i	−0.122180	+	0.211622i	0.798447	+	0.602065i	$0.205655\pi$
$984$		0			0
$985$	−3220.14	−	5577.45i	−0.104165	−	0.180419i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8105.55			0.260608
$990$		0			0
$991$	−26751.6			−0.857510			−0.428755	−	0.903421i	$-0.641048\pi$
$991$	−26751.6			−0.857510			−0.428755	+	0.903421i	$0.641048\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12674.4	+	21952.8i	0.403825	+	0.699446i
$996$		0			0
$997$	−1419.27	+	2458.25i	−0.0450840	+	0.0780878i	−0.887687	−	0.460448i	$-0.847689\pi$
$997$	−1419.27	+	2458.25i	−0.0450840	+	0.0780878i	0.842603	+	0.538535i	$0.181022\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.u.541.1		6
3.2	odd	2		1620.4.i.s.541.1		6
9.2	odd	6		1620.4.a.f.1.3	yes	3
9.4	even	3	inner	1620.4.i.u.1081.1		6
9.5	odd	6		1620.4.i.s.1081.1		6
9.7	even	3		1620.4.a.d.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.3	✓	3	9.7	even	3
1620.4.a.f.1.3	yes	3	9.2	odd	6
1620.4.i.s.541.1		6	3.2	odd	2
1620.4.i.s.1081.1		6	9.5	odd	6
1620.4.i.u.541.1		6	1.1	even	1	trivial
1620.4.i.u.1081.1		6	9.4	even	3	inner

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1620.4.i.u.541.1

Level	$1620$
Weight	$4$
Character	1620.541
Analytic conductor	$95.583$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$