LMFDB - Embedded newform 405.4.e.f.136.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,4,Mod(136,405)] 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("405.136"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,7,-5,0,6] 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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.4.e.f.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(3.00000 - 5.19615i) q^{7} -15.0000 q^{8} +5.00000 q^{10} +(23.5000 - 40.7032i) q^{11} +(2.50000 + 4.33013i) q^{13} +(3.00000 + 5.19615i) q^{14} +(-20.5000 + 35.5070i) q^{16} -131.000 q^{17} -56.0000 q^{19} +(17.5000 - 30.3109i) q^{20} +(23.5000 + 40.7032i) q^{22} +(-1.50000 - 2.59808i) q^{23} +(-12.5000 + 21.6506i) q^{25} -5.00000 q^{26} +42.0000 q^{28} +(78.5000 - 135.966i) q^{29} +(-112.500 - 194.856i) q^{31} +(-80.5000 - 139.430i) q^{32} +(65.5000 - 113.449i) q^{34} -30.0000 q^{35} -70.0000 q^{37} +(28.0000 - 48.4974i) q^{38} +(37.5000 + 64.9519i) q^{40} +(-70.0000 - 121.244i) q^{41} +(-198.500 + 343.812i) q^{43} +329.000 q^{44} +3.00000 q^{46} +(173.500 - 300.511i) q^{47} +(153.500 + 265.870i) q^{49} +(-12.5000 - 21.6506i) q^{50} +(-17.5000 + 30.3109i) q^{52} +4.00000 q^{53} -235.000 q^{55} +(-45.0000 + 77.9423i) q^{56} +(78.5000 + 135.966i) q^{58} +(-374.000 - 647.787i) q^{59} +(169.000 - 292.717i) q^{61} +225.000 q^{62} -167.000 q^{64} +(12.5000 - 21.6506i) q^{65} +(-246.000 - 426.084i) q^{67} +(-458.500 - 794.145i) q^{68} +(15.0000 - 25.9808i) q^{70} +32.0000 q^{71} +970.000 q^{73} +(35.0000 - 60.6218i) q^{74} +(-196.000 - 339.482i) q^{76} +(-141.000 - 244.219i) q^{77} +(628.500 - 1088.59i) q^{79} +205.000 q^{80} +140.000 q^{82} +(51.0000 - 88.3346i) q^{83} +(327.500 + 567.247i) q^{85} +(-198.500 - 343.812i) q^{86} +(-352.500 + 610.548i) q^{88} -1488.00 q^{89} +30.0000 q^{91} +(10.5000 - 18.1865i) q^{92} +(173.500 + 300.511i) q^{94} +(140.000 + 242.487i) q^{95} +(-487.000 + 843.509i) q^{97} -307.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 6 q^{7} - 30 q^{8} + 10 q^{10} + 47 q^{11} + 5 q^{13} + 6 q^{14} - 41 q^{16} - 262 q^{17} - 112 q^{19} + 35 q^{20} + 47 q^{22} - 3 q^{23} - 25 q^{25} - 10 q^{26} + 84 q^{28}+ \cdots - 614 q^{98}+O(q^{100}) $ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 6 * q^7 - 30 * q^8 + 10 * q^10 + 47 * q^11 + 5 * q^13 + 6 * q^14 - 41 * q^16 - 262 * q^17 - 112 * q^19 + 35 * q^20 + 47 * q^22 - 3 * q^23 - 25 * q^25 - 10 * q^26 + 84 * q^28 + 157 * q^29 - 225 * q^31 - 161 * q^32 + 131 * q^34 - 60 * q^35 - 140 * q^37 + 56 * q^38 + 75 * q^40 - 140 * q^41 - 397 * q^43 + 658 * q^44 + 6 * q^46 + 347 * q^47 + 307 * q^49 - 25 * q^50 - 35 * q^52 + 8 * q^53 - 470 * q^55 - 90 * q^56 + 157 * q^58 - 748 * q^59 + 338 * q^61 + 450 * q^62 - 334 * q^64 + 25 * q^65 - 492 * q^67 - 917 * q^68 + 30 * q^70 + 64 * q^71 + 1940 * q^73 + 70 * q^74 - 392 * q^76 - 282 * q^77 + 1257 * q^79 + 410 * q^80 + 280 * q^82 + 102 * q^83 + 655 * q^85 - 397 * q^86 - 705 * q^88 - 2976 * q^89 + 60 * q^91 + 21 * q^92 + 347 * q^94 + 280 * q^95 - 974 * q^97 - 614 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.176777	+	0.306186i	−0.940775	−	0.339032i	$-0.889900\pi$
$2$	−0.500000	+	0.866025i	−0.176777	+	0.306186i	0.763998	+	0.645219i	$0.223234\pi$
$3$		0			0
$3$		0			0
$4$	3.50000	+	6.06218i	0.437500	+	0.757772i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$6$		0			0
$7$	3.00000	−	5.19615i	0.161985	−	0.280566i	−0.773596	−	0.633680i	$-0.781544\pi$
$7$	3.00000	−	5.19615i	0.161985	−	0.280566i	0.935580	+	0.353114i	$0.114877\pi$
$8$	−15.0000			−0.662913
$9$		0			0
$10$	5.00000			0.158114
$11$	23.5000	−	40.7032i	0.644138	−	1.11568i	−0.340362	−	0.940294i	$-0.610550\pi$
$11$	23.5000	−	40.7032i	0.644138	−	1.11568i	0.984500	−	0.175385i	$-0.0561170\pi$
$12$		0			0
$13$	2.50000	+	4.33013i	0.0533366	+	0.0923816i	0.891461	−	0.453097i	$-0.149681\pi$
$13$	2.50000	+	4.33013i	0.0533366	+	0.0923816i	−0.838124	+	0.545479i	$0.816348\pi$
$14$	3.00000	+	5.19615i	0.0572703	+	0.0991950i
$15$		0			0
$16$	−20.5000	+	35.5070i	−0.320312	+	0.554798i
$17$	−131.000			−1.86895			−0.934475	−	0.356027i	$-0.884131\pi$
$17$	−131.000			−1.86895			−0.934475	+	0.356027i	$0.884131\pi$
$18$		0			0
$19$	−56.0000			−0.676173			−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000			−0.676173			−0.338086	+	0.941115i	$0.609780\pi$
$20$	17.5000	−	30.3109i	0.195656	−	0.338886i
$21$		0			0
$22$	23.5000	+	40.7032i	0.227737	+	0.394452i
$23$	−1.50000	−	2.59808i	−0.0135988	−	0.0235538i	0.859146	−	0.511731i	$-0.170995\pi$
$23$	−1.50000	−	2.59808i	−0.0135988	−	0.0235538i	−0.872745	+	0.488177i	$0.837662\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	−5.00000			−0.0377146
$27$		0			0
$28$	42.0000			0.283473
$29$	78.5000	−	135.966i	0.502658	−	0.870629i	−0.497337	−	0.867557i	$-0.665689\pi$
$29$	78.5000	−	135.966i	0.502658	−	0.870629i	0.999995	−	0.00307200i	$-0.000977849\pi$
$30$		0			0
$31$	−112.500	−	194.856i	−0.651793	−	1.12894i	−0.982687	−	0.185271i	$-0.940684\pi$
$31$	−112.500	−	194.856i	−0.651793	−	1.12894i	0.330894	−	0.943668i	$-0.392650\pi$
$32$	−80.5000	−	139.430i	−0.444704	−	0.770250i
$33$		0			0
$34$	65.5000	−	113.449i	0.330387	−	0.572247i
$35$	−30.0000			−0.144884
$36$		0			0
$37$	−70.0000			−0.311025			−0.155513	−	0.987834i	$-0.549703\pi$
$37$	−70.0000			−0.311025			−0.155513	+	0.987834i	$0.549703\pi$
$38$	28.0000	−	48.4974i	0.119532	−	0.207035i
$39$		0			0
$40$	37.5000	+	64.9519i	0.148232	+	0.256745i
$41$	−70.0000	−	121.244i	−0.266638	−	0.461831i	0.701353	−	0.712814i	$-0.252580\pi$
$41$	−70.0000	−	121.244i	−0.266638	−	0.461831i	−0.967992	+	0.250983i	$0.919246\pi$
$42$		0			0
$43$	−198.500	+	343.812i	−0.703976	+	1.21932i	0.263084	+	0.964773i	$0.415260\pi$
$43$	−198.500	+	343.812i	−0.703976	+	1.21932i	−0.967060	+	0.254549i	$0.918073\pi$
$44$	329.000			1.12724
$45$		0			0
$46$	3.00000			0.00961578
$47$	173.500	−	300.511i	0.538459	−	0.932638i	−0.460528	−	0.887645i	$-0.652340\pi$
$47$	173.500	−	300.511i	0.538459	−	0.932638i	0.998987	−	0.0449934i	$-0.0143267\pi$
$48$		0			0
$49$	153.500	+	265.870i	0.447522	+	0.775131i
$50$	−12.5000	−	21.6506i	−0.0353553	−	0.0612372i
$51$		0			0
$52$	−17.5000	+	30.3109i	−0.0466695	+	0.0808339i
$53$	4.00000			0.0103668			0.00518342	−	0.999987i	$-0.498350\pi$
$53$	4.00000			0.0103668			0.00518342	+	0.999987i	$0.498350\pi$
$54$		0			0
$55$	−235.000			−0.576134
$56$	−45.0000	+	77.9423i	−0.107382	+	0.185991i
$57$		0			0
$58$	78.5000	+	135.966i	0.177716	+	0.307814i
$59$	−374.000	−	647.787i	−0.825265	−	1.42940i	−0.901716	−	0.432328i	$-0.857692\pi$
$59$	−374.000	−	647.787i	−0.825265	−	1.42940i	0.0764511	−	0.997073i	$-0.475641\pi$
$60$		0			0
$61$	169.000	−	292.717i	0.354725	−	0.614402i	−0.632346	−	0.774686i	$-0.717908\pi$
$61$	169.000	−	292.717i	0.354725	−	0.614402i	0.987071	+	0.160284i	$0.0512411\pi$
$62$	225.000			0.460888
$63$		0			0
$64$	−167.000			−0.326172
$65$	12.5000	−	21.6506i	0.0238528	−	0.0413143i
$66$		0			0
$67$	−246.000	−	426.084i	−0.448562	−	0.776933i	0.549730	−	0.835342i	$-0.314730\pi$
$67$	−246.000	−	426.084i	−0.448562	−	0.776933i	−0.998293	+	0.0584093i	$0.981397\pi$
$68$	−458.500	−	794.145i	−0.817666	−	1.41624i
$69$		0			0
$70$	15.0000	−	25.9808i	0.0256120	−	0.0443614i
$71$	32.0000			0.0534888			0.0267444	−	0.999642i	$-0.491486\pi$
$71$	32.0000			0.0534888			0.0267444	+	0.999642i	$0.491486\pi$
$72$		0			0
$73$	970.000			1.55520			0.777602	−	0.628757i	$-0.216436\pi$
$73$	970.000			1.55520			0.777602	+	0.628757i	$0.216436\pi$
$74$	35.0000	−	60.6218i	0.0549820	−	0.0952316i
$75$		0			0
$76$	−196.000	−	339.482i	−0.295826	−	0.512385i
$77$	−141.000	−	244.219i	−0.208681	−	0.361446i
$78$		0			0
$79$	628.500	−	1088.59i	0.895086	−	1.55033i	0.0613873	−	0.998114i	$-0.480448\pi$
$79$	628.500	−	1088.59i	0.895086	−	1.55033i	0.833698	−	0.552220i	$-0.186219\pi$
$80$	205.000			0.286496
$81$		0			0
$82$	140.000			0.188542
$83$	51.0000	−	88.3346i	0.0674455	−	0.116819i	−0.830331	−	0.557271i	$-0.811848\pi$
$83$	51.0000	−	88.3346i	0.0674455	−	0.116819i	0.897776	+	0.440452i	$0.145182\pi$
$84$		0			0
$85$	327.500	+	567.247i	0.417910	+	0.723842i
$86$	−198.500	−	343.812i	−0.248893	−	0.431095i
$87$		0			0
$88$	−352.500	+	610.548i	−0.427007	+	0.739598i
$89$	−1488.00			−1.77222			−0.886111	−	0.463474i	$-0.846603\pi$
$89$	−1488.00			−1.77222			−0.886111	+	0.463474i	$0.846603\pi$
$90$		0			0
$91$	30.0000			0.0345588
$92$	10.5000	−	18.1865i	0.0118989	−	0.0206095i
$93$		0			0
$94$	173.500	+	300.511i	0.190374	+	0.329737i
$95$	140.000	+	242.487i	0.151197	+	0.261881i
$96$		0			0
$97$	−487.000	+	843.509i	−0.509767	+	0.882942i	0.490169	+	0.871627i	$0.336935\pi$
$97$	−487.000	+	843.509i	−0.509767	+	0.882942i	−0.999936	+	0.0113144i	$0.996398\pi$
$98$	−307.000			−0.316446
$99$		0			0
$100$	−175.000			−0.175000
$101$	667.500	−	1156.14i	0.657611	−	1.13902i	−0.323621	−	0.946187i	$-0.604900\pi$
$101$	667.500	−	1156.14i	0.657611	−	1.13902i	0.981232	−	0.192829i	$-0.0617664\pi$
$102$		0			0
$103$	−343.000	−	594.093i	−0.328124	−	0.568328i	0.654015	−	0.756481i	$-0.273083\pi$
$103$	−343.000	−	594.093i	−0.328124	−	0.568328i	−0.982140	+	0.188153i	$0.939750\pi$
$104$	−37.5000	−	64.9519i	−0.0353575	−	0.0612409i
$105$		0			0
$106$	−2.00000	+	3.46410i	−0.00183261	+	0.00317418i
$107$	−1098.00			−0.992034			−0.496017	−	0.868313i	$-0.665205\pi$
$107$	−1098.00			−0.992034			−0.496017	+	0.868313i	$0.665205\pi$
$108$		0			0
$109$	−700.000			−0.615118			−0.307559	−	0.951529i	$-0.599512\pi$
$109$	−700.000			−0.615118			−0.307559	+	0.951529i	$0.599512\pi$
$110$	117.500	−	203.516i	0.101847	−	0.176404i
$111$		0			0
$112$	123.000	+	213.042i	0.103771	+	0.179738i
$113$	527.500	+	913.657i	0.439142	+	0.760616i	0.997624	−	0.0689009i	$-0.0219492\pi$
$113$	527.500	+	913.657i	0.439142	+	0.760616i	−0.558482	+	0.829517i	$0.688616\pi$
$114$		0			0
$115$	−7.50000	+	12.9904i	−0.00608155	+	0.0105336i
$116$	1099.00			0.879652
$117$		0			0
$118$	748.000			0.583551
$119$	−393.000	+	680.696i	−0.302742	+	0.524364i
$120$		0			0
$121$	−439.000	−	760.370i	−0.329827	−	0.571277i
$122$	169.000	+	292.717i	0.125414	+	0.217224i
$123$		0			0
$124$	787.500	−	1363.99i	0.570319	−	0.987822i
$125$	125.000			0.0894427
$126$		0			0
$127$	−1646.00			−1.15007			−0.575035	−	0.818129i	$-0.695012\pi$
$127$	−1646.00			−1.15007			−0.575035	+	0.818129i	$0.695012\pi$
$128$	727.500	−	1260.07i	0.502363	−	0.870119i
$129$		0			0
$130$	12.5000	+	21.6506i	0.00843325	+	0.0146068i
$131$	916.500	+	1587.42i	0.611259	+	1.05873i	0.991028	+	0.133651i	$0.0426701\pi$
$131$	916.500	+	1587.42i	0.611259	+	1.05873i	−0.379769	+	0.925081i	$0.623997\pi$
$132$		0			0
$133$	−168.000	+	290.985i	−0.109530	+	0.189711i
$134$	492.000			0.317182
$135$		0			0
$136$	1965.00			1.23895
$137$	−549.000	+	950.896i	−0.342367	+	0.592996i	−0.984872	−	0.173285i	$-0.944562\pi$
$137$	−549.000	+	950.896i	−0.342367	+	0.592996i	0.642505	+	0.766281i	$0.277895\pi$
$138$		0			0
$139$	521.000	+	902.398i	0.317918	+	0.550651i	0.980054	−	0.198734i	$-0.0636829\pi$
$139$	521.000	+	902.398i	0.317918	+	0.550651i	−0.662135	+	0.749384i	$0.730350\pi$
$140$	−105.000	−	181.865i	−0.0633866	−	0.109789i
$141$		0			0
$142$	−16.0000	+	27.7128i	−0.00945556	+	0.0163775i
$143$	235.000			0.137424
$144$		0			0
$145$	−785.000			−0.449591
$146$	−485.000	+	840.045i	−0.274924	+	0.476182i
$147$		0			0
$148$	−245.000	−	424.352i	−0.136073	−	0.235686i
$149$	1470.50	+	2546.98i	0.808510	+	1.40038i	0.913895	+	0.405950i	$0.133059\pi$
$149$	1470.50	+	2546.98i	0.808510	+	1.40038i	−0.105385	+	0.994431i	$0.533607\pi$
$150$		0			0
$151$	−255.500	+	442.539i	−0.137697	+	0.238499i	−0.926625	−	0.375988i	$-0.877303\pi$
$151$	−255.500	+	442.539i	−0.137697	+	0.238499i	0.788927	+	0.614487i	$0.210637\pi$
$152$	840.000			0.448243
$153$		0			0
$154$	282.000			0.147560
$155$	−562.500	+	974.279i	−0.291491	+	0.504877i
$156$		0			0
$157$	285.500	+	494.501i	0.145130	+	0.251372i	0.929421	−	0.369020i	$-0.120307\pi$
$157$	285.500	+	494.501i	0.145130	+	0.251372i	−0.784292	+	0.620392i	$0.786973\pi$
$158$	628.500	+	1088.59i	0.316461	+	0.548126i
$159$		0			0
$160$	−402.500	+	697.150i	−0.198878	+	0.344466i
$161$	−18.0000			−0.00881117
$162$		0			0
$163$	713.000			0.342616			0.171308	−	0.985217i	$-0.445201\pi$
$163$	713.000			0.342616			0.171308	+	0.985217i	$0.445201\pi$
$164$	490.000	−	848.705i	0.233308	−	0.404102i
$165$		0			0
$166$	51.0000	+	88.3346i	0.0238456	+	0.0413018i
$167$	−798.000	−	1382.18i	−0.369767	−	0.640455i	0.619762	−	0.784790i	$-0.287229\pi$
$167$	−798.000	−	1382.18i	−0.369767	−	0.640455i	−0.989529	+	0.144335i	$0.953896\pi$
$168$		0			0
$169$	1086.00	−	1881.01i	0.494310	−	0.856171i
$170$	−655.000			−0.295507
$171$		0			0
$172$	−2779.00			−1.23196
$173$	−2067.00	+	3580.15i	−0.908388	+	1.57337i	−0.0920840	+	0.995751i	$0.529353\pi$
$173$	−2067.00	+	3580.15i	−0.908388	+	1.57337i	−0.816304	+	0.577623i	$0.803981\pi$
$174$		0			0
$175$	75.0000	+	129.904i	0.0323970	+	0.0561132i
$176$	963.500	+	1668.83i	0.412651	+	0.714732i
$177$		0			0
$178$	744.000	−	1288.65i	0.313287	−	0.542630i
$179$	1828.00			0.763302			0.381651	−	0.924306i	$-0.375356\pi$
$179$	1828.00			0.763302			0.381651	+	0.924306i	$0.375356\pi$
$180$		0			0
$181$	−520.000			−0.213543			−0.106772	−	0.994284i	$-0.534051\pi$
$181$	−520.000			−0.213543			−0.106772	+	0.994284i	$0.534051\pi$
$182$	−15.0000	+	25.9808i	−0.00610920	+	0.0105814i
$183$		0			0
$184$	22.5000	+	38.9711i	0.00901479	+	0.0156141i
$185$	175.000	+	303.109i	0.0695473	+	0.120460i
$186$		0			0
$187$	−3078.50	+	5332.12i	−1.20386	+	2.08515i
$188$	2429.00			0.942303
$189$		0			0
$190$	−280.000			−0.106912
$191$	−2413.00	+	4179.44i	−0.914129	+	1.58332i	−0.105956	+	0.994371i	$0.533790\pi$
$191$	−2413.00	+	4179.44i	−0.914129	+	1.58332i	−0.808172	+	0.588946i	$0.799543\pi$
$192$		0			0
$193$	−835.000	−	1446.26i	−0.311423	−	0.539400i	0.667248	−	0.744836i	$-0.267472\pi$
$193$	−835.000	−	1446.26i	−0.311423	−	0.539400i	−0.978671	+	0.205436i	$0.934139\pi$
$194$	−487.000	−	843.509i	−0.180230	−	0.312167i
$195$		0			0
$196$	−1074.50	+	1861.09i	−0.391582	+	0.678239i
$197$	−1380.00			−0.499091			−0.249546	−	0.968363i	$-0.580281\pi$
$197$	−1380.00			−0.499091			−0.249546	+	0.968363i	$0.580281\pi$
$198$		0			0
$199$	4357.00			1.55206			0.776029	−	0.630697i	$-0.217231\pi$
$199$	4357.00			1.55206			0.776029	+	0.630697i	$0.217231\pi$
$200$	187.500	−	324.760i	0.0662913	−	0.114820i
$201$		0			0
$202$	667.500	+	1156.14i	0.232501	+	0.402703i
$203$	−471.000	−	815.796i	−0.162846	−	0.282057i
$204$		0			0
$205$	−350.000	+	606.218i	−0.119244	+	0.206537i
$206$	686.000			0.232019
$207$		0			0
$208$	−205.000			−0.0683375
$209$	−1316.00	+	2279.38i	−0.435548	+	0.754392i
$210$		0			0
$211$	2081.00	+	3604.40i	0.678967	+	1.17600i	0.975292	+	0.220918i	$0.0709052\pi$
$211$	2081.00	+	3604.40i	0.678967	+	1.17600i	−0.296326	+	0.955087i	$0.595761\pi$
$212$	14.0000	+	24.2487i	0.00453549	+	0.00785570i
$213$		0			0
$214$	549.000	−	950.896i	0.175369	−	0.303747i
$215$	1985.00			0.629655
$216$		0			0
$217$	−1350.00			−0.422322
$218$	350.000	−	606.218i	0.108738	−	0.188341i
$219$		0			0
$220$	−822.500	−	1424.61i	−0.252059	−	0.436579i
$221$	−327.500	−	567.247i	−0.0996834	−	0.172657i
$222$		0			0
$223$	2978.00	−	5158.05i	0.894267	−	1.54892i	0.0595587	−	0.998225i	$-0.481031\pi$
$223$	2978.00	−	5158.05i	0.894267	−	1.54892i	0.834709	−	0.550692i	$-0.185636\pi$
$224$	−966.000			−0.288141
$225$		0			0
$226$	−1055.00			−0.310520
$227$	−2470.00	+	4278.17i	−0.722201	+	1.25089i	0.237915	+	0.971286i	$0.423536\pi$
$227$	−2470.00	+	4278.17i	−0.722201	+	1.25089i	−0.960116	+	0.279603i	$0.909797\pi$
$228$		0			0
$229$	−2172.00	−	3762.01i	−0.626768	−	1.08559i	−0.988196	−	0.153194i	$-0.951044\pi$
$229$	−2172.00	−	3762.01i	−0.626768	−	1.08559i	0.361429	−	0.932400i	$-0.382289\pi$
$230$	−7.50000	−	12.9904i	−0.00215015	−	0.00372418i
$231$		0			0
$232$	−1177.50	+	2039.49i	−0.333218	+	0.577151i
$233$	−5202.00			−1.46264			−0.731318	−	0.682036i	$-0.761095\pi$
$233$	−5202.00			−1.46264			−0.731318	+	0.682036i	$0.761095\pi$
$234$		0			0
$235$	−1735.00			−0.481612
$236$	2618.00	−	4534.51i	0.722107	−	1.25073i
$237$		0			0
$238$	−393.000	−	680.696i	−0.107035	−	0.185391i
$239$	−773.000	−	1338.88i	−0.209210	−	0.362363i	0.742256	−	0.670117i	$-0.233756\pi$
$239$	−773.000	−	1338.88i	−0.209210	−	0.362363i	−0.951466	+	0.307754i	$0.900423\pi$
$240$		0			0
$241$	1829.50	−	3168.79i	0.488998	−	0.846969i	−0.510922	−	0.859627i	$-0.670696\pi$
$241$	1829.50	−	3168.79i	0.488998	−	0.846969i	0.999920	+	0.0126581i	$0.00402929\pi$
$242$	878.000			0.233223
$243$		0			0
$244$	2366.00			0.620769
$245$	767.500	−	1329.35i	0.200138	−	0.346649i
$246$		0			0
$247$	−140.000	−	242.487i	−0.0360647	−	0.0624659i
$248$	1687.50	+	2922.84i	0.432082	+	0.748388i
$249$		0			0
$250$	−62.5000	+	108.253i	−0.0158114	+	0.0273861i
$251$	1221.00			0.307047			0.153524	−	0.988145i	$-0.450938\pi$
$251$	1221.00			0.307047			0.153524	+	0.988145i	$0.450938\pi$
$252$		0			0
$253$	−141.000			−0.0350379
$254$	823.000	−	1425.48i	0.203306	−	0.352136i
$255$		0			0
$256$	59.5000	+	103.057i	0.0145264	+	0.0251604i
$257$	3127.50	+	5416.99i	0.759098	+	1.31480i	0.943311	+	0.331909i	$0.107693\pi$
$257$	3127.50	+	5416.99i	0.759098	+	1.31480i	−0.184214	+	0.982886i	$0.558974\pi$
$258$		0			0
$259$	−210.000	+	363.731i	−0.0503813	+	0.0872630i
$260$	175.000			0.0417425
$261$		0			0
$262$	−1833.00			−0.432226
$263$	−418.000	+	723.997i	−0.0980037	+	0.169747i	−0.910858	−	0.412719i	$-0.864579\pi$
$263$	−418.000	+	723.997i	−0.0980037	+	0.169747i	0.812855	+	0.582467i	$0.197912\pi$
$264$		0			0
$265$	−10.0000	−	17.3205i	−0.00231809	−	0.00401506i
$266$	−168.000	−	290.985i	−0.0387246	−	0.0670730i
$267$		0			0
$268$	1722.00	−	2982.59i	0.392492	−	0.679816i
$269$	−2231.00			−0.505675			−0.252837	−	0.967509i	$-0.581364\pi$
$269$	−2231.00			−0.505675			−0.252837	+	0.967509i	$0.581364\pi$
$270$		0			0
$271$	−4832.00			−1.08311			−0.541556	−	0.840665i	$-0.682164\pi$
$271$	−4832.00			−1.08311			−0.541556	+	0.840665i	$0.682164\pi$
$272$	2685.50	−	4651.42i	0.598648	−	1.03689i
$273$		0			0
$274$	−549.000	−	950.896i	−0.121045	−	0.209656i
$275$	587.500	+	1017.58i	0.128828	+	0.223136i
$276$		0			0
$277$	−3225.00	+	5585.86i	−0.699536	+	1.21163i	0.269091	+	0.963115i	$0.413277\pi$
$277$	−3225.00	+	5585.86i	−0.699536	+	1.21163i	−0.968627	+	0.248517i	$0.920057\pi$
$278$	−1042.00			−0.224802
$279$		0			0
$280$	450.000			0.0960452
$281$	525.000	−	909.327i	0.111455	−	0.193046i	−0.804902	−	0.593408i	$-0.797782\pi$
$281$	525.000	−	909.327i	0.111455	−	0.193046i	0.916357	+	0.400362i	$0.131116\pi$
$282$		0			0
$283$	792.000	+	1371.78i	0.166359	+	0.288142i	0.937137	−	0.348962i	$-0.113466\pi$
$283$	792.000	+	1371.78i	0.166359	+	0.288142i	−0.770778	+	0.637104i	$0.780132\pi$
$284$	112.000	+	193.990i	0.0234013	+	0.0405323i
$285$		0			0
$286$	−117.500	+	203.516i	−0.0242934	+	0.0420775i
$287$	−840.000			−0.172765
$288$		0			0
$289$	12248.0			2.49298
$290$	392.500	−	679.830i	0.0794772	−	0.137659i
$291$		0			0
$292$	3395.00	+	5880.31i	0.680402	+	1.17849i
$293$	−3297.00	−	5710.57i	−0.657382	−	1.13862i	−0.981291	−	0.192530i	$-0.938331\pi$
$293$	−3297.00	−	5710.57i	−0.657382	−	1.13862i	0.323909	−	0.946088i	$-0.395003\pi$
$294$		0			0
$295$	−1870.00	+	3238.94i	−0.369070	+	0.639248i
$296$	1050.00			0.206182
$297$		0			0
$298$	−2941.00			−0.571703
$299$	7.50000	−	12.9904i	0.00145062	−	0.00251255i
$300$		0			0
$301$	1191.00	+	2062.87i	0.228067	+	0.395023i
$302$	−255.500	−	442.539i	−0.0486834	−	0.0843220i
$303$		0			0
$304$	1148.00	−	1988.39i	0.216587	−	0.375139i
$305$	−1690.00			−0.317276
$306$		0			0
$307$	−4343.00			−0.807388			−0.403694	−	0.914894i	$-0.632274\pi$
$307$	−4343.00			−0.807388			−0.403694	+	0.914894i	$0.632274\pi$
$308$	987.000	−	1709.53i	0.182596	−	0.316265i
$309$		0			0
$310$	−562.500	−	974.279i	−0.103058	−	0.178501i
$311$	1062.00	+	1839.44i	0.193635	+	0.335386i	0.946452	−	0.322844i	$-0.104639\pi$
$311$	1062.00	+	1839.44i	0.193635	+	0.335386i	−0.752817	+	0.658230i	$0.771306\pi$
$312$		0			0
$313$	3758.00	−	6509.05i	0.678641	−	1.17544i	−0.296749	−	0.954956i	$-0.595902\pi$
$313$	3758.00	−	6509.05i	0.678641	−	1.17544i	0.975390	−	0.220486i	$-0.0707642\pi$
$314$	−571.000			−0.102622
$315$		0			0
$316$	8799.00			1.56640
$317$	3440.00	−	5958.25i	0.609494	−	1.05567i	−0.381830	−	0.924233i	$-0.624706\pi$
$317$	3440.00	−	5958.25i	0.609494	−	1.05567i	0.991324	−	0.131442i	$-0.0419608\pi$
$318$		0			0
$319$	−3689.50	−	6390.40i	−0.647562	−	1.12161i
$320$	417.500	+	723.131i	0.0729342	+	0.126326i
$321$		0			0
$322$	9.00000	−	15.5885i	0.00155761	−	0.00269786i
$323$	7336.00			1.26373
$324$		0			0
$325$	−125.000			−0.0213346
$326$	−356.500	+	617.476i	−0.0605666	+	0.104904i
$327$		0			0
$328$	1050.00	+	1818.65i	0.176758	+	0.306153i
$329$	−1041.00	−	1803.06i	−0.174444	−	0.302146i
$330$		0			0
$331$	2493.00	−	4318.00i	0.413981	−	0.717036i	−0.581340	−	0.813661i	$-0.697471\pi$
$331$	2493.00	−	4318.00i	0.413981	−	0.717036i	0.995321	+	0.0966248i	$0.0308047\pi$
$332$	714.000			0.118030
$333$		0			0
$334$	1596.00			0.261465
$335$	−1230.00	+	2130.42i	−0.200603	+	0.347455i
$336$		0			0
$337$	−452.000	−	782.887i	−0.0730623	−	0.126548i	0.827180	−	0.561938i	$-0.189944\pi$
$337$	−452.000	−	782.887i	−0.0730623	−	0.126548i	−0.900242	+	0.435390i	$0.856611\pi$
$338$	1086.00	+	1881.01i	0.174765	+	0.302702i
$339$		0			0
$340$	−2292.50	+	3970.73i	−0.365671	+	0.633361i
$341$	−10575.0			−1.67938
$342$		0			0
$343$	3900.00			0.613936
$344$	2977.50	−	5157.18i	0.466675	−	0.808304i
$345$		0			0
$346$	−2067.00	−	3580.15i	−0.321164	−	0.556272i
$347$	−4430.00	−	7672.99i	−0.685345	−	1.18705i	−0.973328	−	0.229417i	$-0.926318\pi$
$347$	−4430.00	−	7672.99i	−0.685345	−	1.18705i	0.287983	−	0.957636i	$-0.407015\pi$
$348$		0			0
$349$	2227.00	−	3857.28i	0.341572	−	0.591620i	−0.643153	−	0.765738i	$-0.722374\pi$
$349$	2227.00	−	3857.28i	0.341572	−	0.591620i	0.984725	+	0.174118i	$0.0557074\pi$
$350$	−150.000			−0.0229081
$351$		0			0
$352$	−7567.00			−1.14580
$353$	4390.50	−	7604.57i	0.661991	−	1.14660i	−0.318101	−	0.948057i	$-0.603045\pi$
$353$	4390.50	−	7604.57i	0.661991	−	1.14660i	0.980092	−	0.198545i	$-0.0636215\pi$
$354$		0			0
$355$	−80.0000	−	138.564i	−0.0119604	−	0.0207161i
$356$	−5208.00	−	9020.52i	−0.775347	−	1.34294i
$357$		0			0
$358$	−914.000	+	1583.09i	−0.134934	+	0.233713i
$359$	2928.00			0.430457			0.215228	−	0.976564i	$-0.430950\pi$
$359$	2928.00			0.430457			0.215228	+	0.976564i	$0.430950\pi$
$360$		0			0
$361$	−3723.00			−0.542790
$362$	260.000	−	450.333i	0.0377494	−	0.0653839i
$363$		0			0
$364$	105.000	+	181.865i	0.0151195	+	0.0261877i
$365$	−2425.00	−	4200.22i	−0.347754	−	0.602328i
$366$		0			0
$367$	−4551.00	+	7882.56i	−0.647303	+	1.12116i	0.336461	+	0.941697i	$0.390770\pi$
$367$	−4551.00	+	7882.56i	−0.647303	+	1.12116i	−0.983764	+	0.179465i	$0.942563\pi$
$368$	123.000			0.0174234
$369$		0			0
$370$	−350.000			−0.0491774
$371$	12.0000	−	20.7846i	0.00167927	−	0.00290858i
$372$		0			0
$373$	4091.50	+	7086.69i	0.567962	+	0.983739i	0.996767	+	0.0803422i	$0.0256013\pi$
$373$	4091.50	+	7086.69i	0.567962	+	0.983739i	−0.428805	+	0.903397i	$0.641065\pi$
$374$	−3078.50	−	5332.12i	−0.425630	−	0.737212i
$375$		0			0
$376$	−2602.50	+	4507.66i	−0.356951	+	0.618258i
$377$	785.000			0.107240
$378$		0			0
$379$	6136.00			0.831623			0.415812	−	0.909451i	$-0.363498\pi$
$379$	6136.00			0.831623			0.415812	+	0.909451i	$0.363498\pi$
$380$	−980.000	+	1697.41i	−0.132297	+	0.229145i
$381$		0			0
$382$	−2413.00	−	4179.44i	−0.323193	−	0.559787i
$383$	−2821.50	−	4886.98i	−0.376428	−	0.651992i	0.614112	−	0.789219i	$-0.289514\pi$
$383$	−2821.50	−	4886.98i	−0.376428	−	0.651992i	−0.990540	+	0.137227i	$0.956181\pi$
$384$		0			0
$385$	−705.000	+	1221.10i	−0.0933250	+	0.161644i
$386$	1670.00			0.220209
$387$		0			0
$388$	−6818.00			−0.892092
$389$	−4495.50	+	7786.43i	−0.585941	+	1.01488i	0.408817	+	0.912616i	$0.365941\pi$
$389$	−4495.50	+	7786.43i	−0.585941	+	1.01488i	−0.994757	+	0.102262i	$0.967392\pi$
$390$		0			0
$391$	196.500	+	340.348i	0.0254154	+	0.0440208i
$392$	−2302.50	−	3988.05i	−0.296668	−	0.513844i
$393$		0			0
$394$	690.000	−	1195.12i	0.0882277	−	0.152815i
$395$	−6285.00			−0.800589
$396$		0			0
$397$	−12449.0			−1.57380			−0.786898	−	0.617082i	$-0.788314\pi$
$397$	−12449.0			−1.57380			−0.786898	+	0.617082i	$0.788314\pi$
$398$	−2178.50	+	3773.27i	−0.274368	+	0.475219i
$399$		0			0
$400$	−512.500	−	887.676i	−0.0640625	−	0.110960i
$401$	−4038.00	−	6994.02i	−0.502863	−	0.870984i	−0.999995	−	0.00330917i	$-0.998947\pi$
$401$	−4038.00	−	6994.02i	−0.502863	−	0.870984i	0.497131	−	0.867675i	$-0.334387\pi$
$402$		0			0
$403$	562.500	−	974.279i	0.0695288	−	0.120427i
$404$	9345.00			1.15082
$405$		0			0
$406$	942.000			0.115149
$407$	−1645.00	+	2849.22i	−0.200343	+	0.347004i
$408$		0			0
$409$	1416.50	+	2453.45i	0.171250	+	0.296614i	0.938857	−	0.344307i	$-0.111886\pi$
$409$	1416.50	+	2453.45i	0.171250	+	0.296614i	−0.767607	+	0.640921i	$0.778553\pi$
$410$	−350.000	−	606.218i	−0.0421592	−	0.0730219i
$411$		0			0
$412$	2401.00	−	4158.65i	0.287109	−	0.497287i
$413$	−4488.00			−0.534722
$414$		0			0
$415$	−510.000			−0.0603251
$416$	402.500	−	697.150i	0.0474379	−	0.0821649i
$417$		0			0
$418$	−1316.00	−	2279.38i	−0.153990	−	0.266718i
$419$	2388.50	+	4137.00i	0.278487	+	0.482353i	0.971009	−	0.239044i	$-0.0768339\pi$
$419$	2388.50	+	4137.00i	0.278487	+	0.482353i	−0.692522	+	0.721396i	$0.743501\pi$
$420$		0			0
$421$	3232.00	−	5597.99i	0.374152	−	0.648050i	−0.616048	−	0.787709i	$-0.711267\pi$
$421$	3232.00	−	5597.99i	0.374152	−	0.648050i	0.990200	+	0.139658i	$0.0446005\pi$
$422$	−4162.00			−0.480102
$423$		0			0
$424$	−60.0000			−0.00687231
$425$	1637.50	−	2836.23i	0.186895	−	0.323712i
$426$		0			0
$427$	−1014.00	−	1756.30i	−0.114920	−	0.199048i
$428$	−3843.00	−	6656.27i	−0.434015	−	0.751736i
$429$		0			0
$430$	−992.500	+	1719.06i	−0.111308	+	0.192792i
$431$	10680.0			1.19359			0.596795	−	0.802394i	$-0.296440\pi$
$431$	10680.0			1.19359			0.596795	+	0.802394i	$0.296440\pi$
$432$		0			0
$433$	11566.0			1.28366			0.641832	−	0.766845i	$-0.278175\pi$
$433$	11566.0			1.28366			0.641832	+	0.766845i	$0.278175\pi$
$434$	675.000	−	1169.13i	0.0746568	−	0.129309i
$435$		0			0
$436$	−2450.00	−	4243.52i	−0.269114	−	0.466119i
$437$	84.0000	+	145.492i	0.00919511	+	0.0159264i
$438$		0			0
$439$	724.000	−	1254.00i	0.0787122	−	0.136333i	−0.823982	−	0.566615i	$-0.808253\pi$
$439$	724.000	−	1254.00i	0.0787122	−	0.136333i	0.902695	+	0.430282i	$0.141586\pi$
$440$	3525.00			0.381927
$441$		0			0
$442$	655.000			0.0704868
$443$	1188.00	−	2057.68i	0.127412	−	0.220684i	−0.795261	−	0.606267i	$-0.792666\pi$
$443$	1188.00	−	2057.68i	0.127412	−	0.220684i	0.922673	+	0.385583i	$0.126000\pi$
$444$		0			0
$445$	3720.00	+	6443.23i	0.396281	+	0.686378i
$446$	2978.00	+	5158.05i	0.316171	+	0.547625i
$447$		0			0
$448$	−501.000	+	867.757i	−0.0528349	+	0.0915127i
$449$	−14894.0			−1.56546			−0.782730	−	0.622362i	$-0.786173\pi$
$449$	−14894.0			−1.56546			−0.782730	+	0.622362i	$0.786173\pi$
$450$		0			0
$451$	−6580.00			−0.687007
$452$	−3692.50	+	6395.60i	−0.384249	+	0.665539i
$453$		0			0
$454$	−2470.00	−	4278.17i	−0.255337	−	0.442256i
$455$	−75.0000	−	129.904i	−0.00772759	−	0.0133846i
$456$		0			0
$457$	−8102.00	+	14033.1i	−0.829312	+	1.43641i	0.0692668	+	0.997598i	$0.477934\pi$
$457$	−8102.00	+	14033.1i	−0.829312	+	1.43641i	−0.898579	+	0.438812i	$0.855399\pi$
$458$	4344.00			0.443192
$459$		0			0
$460$	−105.000			−0.0106427
$461$	2541.00	−	4401.14i	0.256716	−	0.444646i	−0.708644	−	0.705566i	$-0.750693\pi$
$461$	2541.00	−	4401.14i	0.256716	−	0.444646i	0.965360	+	0.260921i	$0.0840261\pi$
$462$		0			0
$463$	5163.00	+	8942.58i	0.518240	+	0.897617i	0.999775	+	0.0211910i	$0.00674580\pi$
$463$	5163.00	+	8942.58i	0.518240	+	0.897617i	−0.481536	+	0.876426i	$0.659921\pi$
$464$	3218.50	+	5574.61i	0.322015	+	0.557747i
$465$		0			0
$466$	2601.00	−	4505.06i	0.258560	−	0.447839i
$467$	4184.00			0.414588			0.207294	−	0.978279i	$-0.433534\pi$
$467$	4184.00			0.414588			0.207294	+	0.978279i	$0.433534\pi$
$468$		0			0
$469$	−2952.00			−0.290641
$470$	867.500	−	1502.55i	0.0851379	−	0.147463i
$471$		0			0
$472$	5610.00	+	9716.81i	0.547079	+	0.947568i
$473$	9329.50	+	16159.2i	0.906915	+	1.57082i
$474$		0			0
$475$	700.000	−	1212.44i	0.0676173	−	0.117117i
$476$	−5502.00			−0.529798
$477$		0			0
$478$	1546.00			0.147934
$479$	7788.00	−	13489.2i	0.742887	−	1.28672i	−0.208289	−	0.978067i	$-0.566789\pi$
$479$	7788.00	−	13489.2i	0.742887	−	1.28672i	0.951176	−	0.308650i	$-0.0998773\pi$
$480$		0			0
$481$	−175.000	−	303.109i	−0.0165890	−	0.0287330i
$482$	1829.50	+	3168.79i	0.172887	+	0.299449i
$483$		0			0
$484$	3073.00	−	5322.59i	0.288599	−	0.499868i
$485$	4870.00			0.455949
$486$		0			0
$487$	10220.0			0.950949			0.475475	−	0.879729i	$-0.342276\pi$
$487$	10220.0			0.950949			0.475475	+	0.879729i	$0.342276\pi$
$488$	−2535.00	+	4390.75i	−0.235152	+	0.407295i
$489$		0			0
$490$	767.500	+	1329.35i	0.0707594	+	0.122559i
$491$	1346.00	+	2331.34i	0.123715	+	0.214281i	0.921230	−	0.389018i	$-0.127186\pi$
$491$	1346.00	+	2331.34i	0.123715	+	0.214281i	−0.797515	+	0.603299i	$0.793852\pi$
$492$		0			0
$493$	−10283.5	+	17811.5i	−0.939443	+	1.62716i
$494$	280.000			0.0255016
$495$		0			0
$496$	9225.00			0.835110
$497$	96.0000	−	166.277i	0.00866436	−	0.0150071i
$498$		0			0
$499$	−2882.00	−	4991.77i	−0.258549	−	0.447820i	0.707304	−	0.706909i	$-0.249911\pi$
$499$	−2882.00	−	4991.77i	−0.258549	−	0.447820i	−0.965853	+	0.259089i	$0.916578\pi$
$500$	437.500	+	757.772i	0.0391312	+	0.0677772i
$501$		0			0
$502$	−610.500	+	1057.42i	−0.0542788	+	0.0940136i
$503$	−2437.00			−0.216025			−0.108012	−	0.994150i	$-0.534449\pi$
$503$	−2437.00			−0.216025			−0.108012	+	0.994150i	$0.534449\pi$
$504$		0			0
$505$	−6675.00			−0.588185
$506$	70.5000	−	122.110i	0.00619389	−	0.0107281i
$507$		0			0
$508$	−5761.00	−	9978.34i	−0.503156	−	0.871491i
$509$	2924.50	+	5065.38i	0.254668	+	0.441098i	0.964805	−	0.262965i	$-0.0847004\pi$
$509$	2924.50	+	5065.38i	0.254668	+	0.441098i	−0.710137	+	0.704063i	$0.751367\pi$
$510$		0			0
$511$	2910.00	−	5040.27i	0.251919	−	0.436337i
$512$	11521.0			0.994455
$513$		0			0
$514$	−6255.00			−0.536763
$515$	−1715.00	+	2970.47i	−0.146742	+	0.254164i
$516$		0			0
$517$	−8154.50	−	14124.0i	−0.693684	−	1.20150i
$518$	−210.000	−	363.731i	−0.0178125	−	0.0308521i
$519$		0			0
$520$	−187.500	+	324.760i	−0.0158123	+	0.0273878i
$521$	−17032.0			−1.43222			−0.716109	−	0.697989i	$-0.754079\pi$
$521$	−17032.0			−1.43222			−0.716109	+	0.697989i	$0.754079\pi$
$522$		0			0
$523$	4147.00			0.346722			0.173361	−	0.984858i	$-0.444537\pi$
$523$	4147.00			0.346722			0.173361	+	0.984858i	$0.444537\pi$
$524$	−6415.50	+	11112.0i	−0.534852	+	0.926391i
$525$		0			0
$526$	−418.000	−	723.997i	−0.0346496	−	0.0600148i
$527$	14737.5	+	25526.1i	1.21817	+	2.10993i
$528$		0			0
$529$	6079.00	−	10529.1i	0.499630	−	0.865385i
$530$	20.0000			0.00163914
$531$		0			0
$532$	−2352.00			−0.191677
$533$	350.000	−	606.218i	0.0284431	−	0.0492649i
$534$		0			0
$535$	2745.00	+	4754.48i	0.221826	+	0.384213i
$536$	3690.00	+	6391.27i	0.297358	+	0.515039i
$537$		0			0
$538$	1115.50	−	1932.10i	0.0893915	−	0.154831i
$539$	14429.0			1.15306
$540$		0			0
$541$	−3942.00			−0.313271			−0.156636	−	0.987656i	$-0.550065\pi$
$541$	−3942.00			−0.313271			−0.156636	+	0.987656i	$0.550065\pi$
$542$	2416.00	−	4184.63i	0.191469	−	0.331634i
$543$		0			0
$544$	10545.5	+	18265.3i	0.831130	+	1.43956i
$545$	1750.00	+	3031.09i	0.137545	+	0.238234i
$546$		0			0
$547$	6875.50	−	11908.7i	0.537432	−	0.930859i	−0.461610	−	0.887083i	$-0.652728\pi$
$547$	6875.50	−	11908.7i	0.537432	−	0.930859i	0.999041	−	0.0437759i	$-0.0139388\pi$
$548$	−7686.00			−0.599142
$549$		0			0
$550$	−1175.00			−0.0910949
$551$	−4396.00	+	7614.10i	−0.339884	+	0.588696i
$552$		0			0
$553$	−3771.00	−	6531.56i	−0.289981	−	0.502261i
$554$	−3225.00	−	5585.86i	−0.247323	−	0.428377i
$555$		0			0
$556$	−3647.00	+	6316.79i	−0.278179	+	0.481819i
$557$	7944.00			0.604305			0.302153	−	0.953260i	$-0.402295\pi$
$557$	7944.00			0.604305			0.302153	+	0.953260i	$0.402295\pi$
$558$		0			0
$559$	−1985.00			−0.150191
$560$	615.000	−	1065.21i	0.0464080	−	0.0803811i
$561$		0			0
$562$	525.000	+	909.327i	0.0394053	+	0.0682520i
$563$	3351.00	+	5804.10i	0.250849	+	0.434483i	0.963760	−	0.266772i	$-0.0859570\pi$
$563$	3351.00	+	5804.10i	0.250849	+	0.434483i	−0.712911	+	0.701254i	$0.752624\pi$
$564$		0			0
$565$	2637.50	−	4568.28i	0.196390	−	0.340158i
$566$	−1584.00			−0.117633
$567$		0			0
$568$	−480.000			−0.0354584
$569$	1380.00	−	2390.23i	0.101674	−	0.176105i	−0.810700	−	0.585461i	$-0.800913\pi$
$569$	1380.00	−	2390.23i	0.101674	−	0.176105i	0.912375	+	0.409356i	$0.134247\pi$
$570$		0			0
$571$	−4465.00	−	7733.61i	−0.327241	−	0.566798i	0.654723	−	0.755869i	$-0.272785\pi$
$571$	−4465.00	−	7733.61i	−0.327241	−	0.566798i	−0.981963	+	0.189072i	$0.939452\pi$
$572$	822.500	+	1424.61i	0.0601232	+	0.104136i
$573$		0			0
$574$	420.000	−	727.461i	0.0305409	−	0.0528983i
$575$	75.0000			0.00543951
$576$		0			0
$577$	6944.00			0.501010			0.250505	−	0.968115i	$-0.419403\pi$
$577$	6944.00			0.501010			0.250505	+	0.968115i	$0.419403\pi$
$578$	−6124.00	+	10607.1i	−0.440700	+	0.763315i
$579$		0			0
$580$	−2747.50	−	4758.81i	−0.196696	−	0.340688i
$581$	−306.000	−	530.008i	−0.0218503	−	0.0378458i
$582$		0			0
$583$	94.0000	−	162.813i	0.00667767	−	0.0115661i
$584$	−14550.0			−1.03096
$585$		0			0
$586$	6594.00			0.464839
$587$	2103.00	−	3642.50i	0.147871	−	0.256120i	−0.782570	−	0.622563i	$-0.786091\pi$
$587$	2103.00	−	3642.50i	0.147871	−	0.256120i	0.930440	+	0.366444i	$0.119425\pi$
$588$		0			0
$589$	6300.00	+	10911.9i	0.440725	+	0.763358i
$590$	−1870.00	−	3238.94i	−0.130486	−	0.226008i
$591$		0			0
$592$	1435.00	−	2485.49i	0.0996252	−	0.172556i
$593$	−6571.00			−0.455040			−0.227520	−	0.973773i	$-0.573062\pi$
$593$	−6571.00			−0.455040			−0.227520	+	0.973773i	$0.573062\pi$
$594$		0			0
$595$	3930.00			0.270780
$596$	−10293.5	+	17828.9i	−0.707447	+	1.22533i
$597$		0			0
$598$	7.50000	+	12.9904i	0.000512873	+	0.000888321i
$599$	4745.00	+	8218.58i	0.323665	+	0.560605i	0.981241	−	0.192784i	$-0.0617515\pi$
$599$	4745.00	+	8218.58i	0.323665	+	0.560605i	−0.657576	+	0.753388i	$0.728418\pi$
$600$		0			0
$601$	−5930.50	+	10271.9i	−0.402513	+	0.697172i	−0.994028	−	0.109121i	$-0.965196\pi$
$601$	−5930.50	+	10271.9i	−0.402513	+	0.697172i	0.591516	+	0.806293i	$0.298530\pi$
$602$	−2382.00			−0.161268
$603$		0			0
$604$	−3577.00			−0.240970
$605$	−2195.00	+	3801.85i	−0.147503	+	0.255483i
$606$		0			0
$607$	259.000	+	448.601i	0.0173188	+	0.0299970i	0.874555	−	0.484927i	$-0.161154\pi$
$607$	259.000	+	448.601i	0.0173188	+	0.0299970i	−0.857236	+	0.514923i	$0.827820\pi$
$608$	4508.00	+	7808.09i	0.300697	+	0.520822i
$609$		0			0
$610$	845.000	−	1463.58i	0.0560870	−	0.0971455i
$611$	1735.00			0.114878
$612$		0			0
$613$	15163.0			0.999067			0.499533	−	0.866295i	$-0.333505\pi$
$613$	15163.0			0.999067			0.499533	+	0.866295i	$0.333505\pi$
$614$	2171.50	−	3761.15i	0.142727	−	0.247211i
$615$		0			0
$616$	2115.00	+	3663.29i	0.138337	+	0.239607i
$617$	−9505.50	−	16464.0i	−0.620222	−	1.07426i	−0.989444	−	0.144915i	$-0.953709\pi$
$617$	−9505.50	−	16464.0i	−0.620222	−	1.07426i	0.369222	−	0.929341i	$-0.379624\pi$
$618$		0			0
$619$	3953.00	−	6846.80i	0.256679	−	0.444582i	−0.708671	−	0.705539i	$-0.750705\pi$
$619$	3953.00	−	6846.80i	0.256679	−	0.444582i	0.965350	+	0.260958i	$0.0840383\pi$
$620$	−7875.00			−0.510109
$621$		0			0
$622$	−2124.00			−0.136921
$623$	−4464.00	+	7731.87i	−0.287073	+	0.497225i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	3758.00	+	6509.05i	0.239936	+	0.415581i
$627$		0			0
$628$	−1998.50	+	3461.50i	−0.126989	+	0.219951i
$629$	9170.00			0.581291
$630$		0			0
$631$	−3416.00			−0.215513			−0.107757	−	0.994177i	$-0.534367\pi$
$631$	−3416.00			−0.215513			−0.107757	+	0.994177i	$0.534367\pi$
$632$	−9427.50	+	16328.9i	−0.593364	+	1.02774i
$633$		0			0
$634$	3440.00	+	5958.25i	0.215489	+	0.373237i
$635$	4115.00	+	7127.39i	0.257163	+	0.445420i
$636$		0			0
$637$	−767.500	+	1329.35i	−0.0477386	+	0.0826856i
$638$	7379.00			0.457896
$639$		0			0
$640$	−7275.00			−0.449328
$641$	−2415.00	+	4182.90i	−0.148809	+	0.257745i	−0.930788	−	0.365560i	$-0.880877\pi$
$641$	−2415.00	+	4182.90i	−0.148809	+	0.257745i	0.781978	+	0.623306i	$0.214211\pi$
$642$		0			0
$643$	−6274.50	−	10867.8i	−0.384824	−	0.666536i	0.606920	−	0.794763i	$-0.292405\pi$
$643$	−6274.50	−	10867.8i	−0.384824	−	0.666536i	−0.991745	+	0.128227i	$0.959071\pi$
$644$	−63.0000	−	109.119i	−0.00385489	−	0.00667686i
$645$		0			0
$646$	−3668.00	+	6353.16i	−0.223399	+	0.386938i
$647$	8164.00			0.496074			0.248037	−	0.968751i	$-0.420215\pi$
$647$	8164.00			0.496074			0.248037	+	0.968751i	$0.420215\pi$
$648$		0			0
$649$	−35156.0			−2.12634
$650$	62.5000	−	108.253i	0.00377146	−	0.00653237i
$651$		0			0
$652$	2495.50	+	4322.33i	0.149895	+	0.259625i
$653$	−11884.0	−	20583.7i	−0.712185	−	1.23354i	−0.964035	−	0.265774i	$-0.914372\pi$
$653$	−11884.0	−	20583.7i	−0.712185	−	1.23354i	0.251850	−	0.967766i	$-0.418961\pi$
$654$		0			0
$655$	4582.50	−	7937.12i	0.273363	−	0.473479i
$656$	5740.00			0.341630
$657$		0			0
$658$	2082.00			0.123351
$659$	−10736.0	+	18595.3i	−0.634621	+	1.09920i	0.351974	+	0.936010i	$0.385510\pi$
$659$	−10736.0	+	18595.3i	−0.634621	+	1.09920i	−0.986595	+	0.163186i	$0.947823\pi$
$660$		0			0
$661$	−6491.00	−	11242.7i	−0.381953	−	0.661561i	0.609389	−	0.792872i	$-0.291415\pi$
$661$	−6491.00	−	11242.7i	−0.381953	−	0.661561i	−0.991341	+	0.131310i	$0.958082\pi$
$662$	2493.00	+	4318.00i	0.146364	+	0.253510i
$663$		0			0
$664$	−765.000	+	1325.02i	−0.0447105	+	0.0774408i
$665$	1680.00			0.0979663
$666$		0			0
$667$	−471.000			−0.0273421
$668$	5586.00	−	9675.24i	0.323546	−	0.560398i
$669$		0			0
$670$	−1230.00	−	2130.42i	−0.0709239	−	0.122844i
$671$	−7943.00	−	13757.7i	−0.456984	−	0.791519i
$672$		0			0
$673$	3003.00	−	5201.35i	0.172002	−	0.297916i	−0.767118	−	0.641506i	$-0.778310\pi$
$673$	3003.00	−	5201.35i	0.172002	−	0.297916i	0.939120	+	0.343590i	$0.111643\pi$
$674$	904.000			0.0516629
$675$		0			0
$676$	15204.0			0.865043
$677$	−582.000	+	1008.05i	−0.0330400	+	0.0572270i	−0.882073	−	0.471114i	$-0.843852\pi$
$677$	−582.000	+	1008.05i	−0.0330400	+	0.0572270i	0.849033	+	0.528340i	$0.177186\pi$
$678$		0			0
$679$	2922.00	+	5061.05i	0.165149	+	0.286046i
$680$	−4912.50	−	8508.70i	−0.277038	−	0.479844i
$681$		0			0
$682$	5287.50	−	9158.22i	0.296875	−	0.514203i
$683$	26496.0			1.48439			0.742197	−	0.670182i	$-0.233784\pi$
$683$	26496.0			1.48439			0.742197	+	0.670182i	$0.233784\pi$
$684$		0			0
$685$	5490.00			0.306222
$686$	−1950.00	+	3377.50i	−0.108530	+	0.187979i
$687$		0			0
$688$	−8138.50	−	14096.3i	−0.450985	−	0.781128i
$689$	10.0000	+	17.3205i	0.000552931	+	0.000957705i
$690$		0			0
$691$	−8555.00	+	14817.7i	−0.470981	+	0.815762i	−0.999449	−	0.0331907i	$-0.989433\pi$
$691$	−8555.00	+	14817.7i	−0.470981	+	0.815762i	0.528469	+	0.848953i	$0.322766\pi$
$692$	−28938.0			−1.58968
$693$		0			0
$694$	8860.00			0.484612
$695$	2605.00	−	4511.99i	0.142177	−	0.246258i
$696$		0			0
$697$	9170.00	+	15882.9i	0.498334	+	0.863139i
$698$	2227.00	+	3857.28i	0.120764	+	0.209169i
$699$		0			0
$700$	−525.000	+	909.327i	−0.0283473	+	0.0490990i
$701$	30251.0			1.62991			0.814953	−	0.579527i	$-0.196763\pi$
$701$	30251.0			1.62991			0.814953	+	0.579527i	$0.196763\pi$
$702$		0			0
$703$	3920.00			0.210307
$704$	−3924.50	+	6797.43i	−0.210100	+	0.363903i
$705$		0			0
$706$	4390.50	+	7604.57i	0.234049	+	0.405385i
$707$	−4005.00	−	6936.86i	−0.213046	−	0.369007i
$708$		0			0
$709$	−9410.00	+	16298.6i	−0.498448	+	0.863338i	−0.999998	−	0.00179062i	$-0.999430\pi$
$709$	−9410.00	+	16298.6i	−0.498448	+	0.863338i	0.501550	+	0.865129i	$0.332763\pi$
$710$	160.000			0.00845731
$711$		0			0
$712$	22320.0			1.17483
$713$	−337.500	+	584.567i	−0.0177272	+	0.0307044i
$714$		0			0
$715$	−587.500	−	1017.58i	−0.0307290	−	0.0532242i
$716$	6398.00	+	11081.7i	0.333945	+	0.578409i
$717$		0			0
$718$	−1464.00	+	2535.72i	−0.0760947	+	0.131800i
$719$	31890.0			1.65410			0.827049	−	0.562130i	$-0.190018\pi$
$719$	31890.0			1.65410			0.827049	+	0.562130i	$0.190018\pi$
$720$		0			0
$721$	−4116.00			−0.212605
$722$	1861.50	−	3224.21i	0.0959527	−	0.166195i
$723$		0			0
$724$	−1820.00	−	3152.33i	−0.0934251	−	0.161817i
$725$	1962.50	+	3399.15i	0.100532	+	0.174126i
$726$		0			0
$727$	5726.00	−	9917.72i	0.292112	−	0.505953i	−0.682197	−	0.731169i	$-0.738975\pi$
$727$	5726.00	−	9917.72i	0.292112	−	0.505953i	0.974309	+	0.225215i	$0.0723086\pi$
$728$	−450.000			−0.0229095
$729$		0			0
$730$	4850.00			0.245899
$731$	26003.5	−	45039.4i	1.31570	−	2.27885i
$732$		0			0
$733$	−3547.00	−	6143.58i	−0.178733	−	0.309575i	0.762714	−	0.646736i	$-0.223867\pi$
$733$	−3547.00	−	6143.58i	−0.178733	−	0.309575i	−0.941447	+	0.337161i	$0.890533\pi$
$734$	−4551.00	−	7882.56i	−0.228856	−	0.396391i
$735$		0			0
$736$	−241.500	+	418.290i	−0.0120948	+	0.0209489i
$737$	−23124.0			−1.15574
$738$		0			0
$739$	−3200.00			−0.159288			−0.0796440	−	0.996823i	$-0.525378\pi$
$739$	−3200.00			−0.159288			−0.0796440	+	0.996823i	$0.525378\pi$
$740$	−1225.00	+	2121.76i	−0.0608539	+	0.105402i
$741$		0			0
$742$	12.0000	+	20.7846i	0.000593711	+	0.00102834i
$743$	10415.5	+	18040.2i	0.514277	+	0.890753i	0.999863	+	0.0165646i	$0.00527292\pi$
$743$	10415.5	+	18040.2i	0.514277	+	0.890753i	−0.485586	+	0.874189i	$0.661394\pi$
$744$		0			0
$745$	7352.50	−	12734.9i	0.361577	−	0.626269i
$746$	−8183.00			−0.401610
$747$		0			0
$748$	−43099.0			−2.10676
$749$	−3294.00	+	5705.38i	−0.160694	+	0.278331i
$750$		0			0
$751$	7802.50	+	13514.3i	0.379118	+	0.656651i	0.990934	−	0.134349i	$-0.0428943\pi$
$751$	7802.50	+	13514.3i	0.379118	+	0.656651i	−0.611817	+	0.791000i	$0.709561\pi$
$752$	7113.50	+	12320.9i	0.344950	+	0.597472i
$753$		0			0
$754$	−392.500	+	679.830i	−0.0189576	+	0.0328355i
$755$	2555.00			0.123160
$756$		0			0
$757$	21349.0			1.02502			0.512512	−	0.858680i	$-0.328715\pi$
$757$	21349.0			1.02502			0.512512	+	0.858680i	$0.328715\pi$
$758$	−3068.00	+	5313.93i	−0.147012	+	0.254632i
$759$		0			0
$760$	−2100.00	−	3637.31i	−0.100230	−	0.173604i
$761$	−1851.00	−	3206.03i	−0.0881717	−	0.152718i	0.818567	−	0.574412i	$-0.194769\pi$
$761$	−1851.00	−	3206.03i	−0.0881717	−	0.152718i	−0.906738	+	0.421694i	$0.861436\pi$
$762$		0			0
$763$	−2100.00	+	3637.31i	−0.0996397	+	0.172581i
$764$	−33782.0			−1.59972
$765$		0			0
$766$	5643.00			0.266175
$767$	1870.00	−	3238.94i	0.0880336	−	0.152479i
$768$		0			0
$769$	696.500	+	1206.37i	0.0326612	+	0.0565708i	0.881894	−	0.471448i	$-0.156268\pi$
$769$	696.500	+	1206.37i	0.0326612	+	0.0565708i	−0.849233	+	0.528019i	$0.822935\pi$
$770$	−705.000	−	1221.10i	−0.0329954	−	0.0571497i
$771$		0			0
$772$	5845.00	−	10123.8i	0.272495	−	0.471975i
$773$	−6906.00			−0.321334			−0.160667	−	0.987009i	$-0.551365\pi$
$773$	−6906.00			−0.321334			−0.160667	+	0.987009i	$0.551365\pi$
$774$		0			0
$775$	5625.00			0.260717
$776$	7305.00	−	12652.6i	0.337931	−	0.585313i
$777$		0			0
$778$	−4495.50	−	7786.43i	−0.207161	−	0.358814i
$779$	3920.00	+	6789.64i	0.180293	+	0.312277i
$780$		0			0
$781$	752.000	−	1302.50i	0.0344541	−	0.0596763i
$782$	−393.000			−0.0179714
$783$		0			0
$784$	−12587.0			−0.573387
$785$	1427.50	−	2472.50i	0.0649040	−	0.112417i
$786$		0			0
$787$	−15246.5	−	26407.7i	−0.690571	−	1.19610i	−0.971651	−	0.236419i	$-0.924026\pi$
$787$	−15246.5	−	26407.7i	−0.690571	−	1.19610i	0.281081	−	0.959684i	$-0.409307\pi$
$788$	−4830.00	−	8365.81i	−0.218352	−	0.378197i
$789$		0			0
$790$	3142.50	−	5442.97i	0.141525	−	0.245129i
$791$	6330.00			0.284537
$792$		0			0
$793$	1690.00			0.0756793
$794$	6224.50	−	10781.2i	0.278211	−	0.481875i
$795$		0			0
$796$	15249.5	+	26412.9i	0.679025	+	1.17611i
$797$	−16744.0	−	29001.5i	−0.744169	−	1.28894i	−0.950582	−	0.310474i	$-0.899512\pi$
$797$	−16744.0	−	29001.5i	−0.744169	−	1.28894i	0.206413	−	0.978465i	$-0.433821\pi$
$798$		0			0
$799$	−22728.5	+	39366.9i	−1.00635	+	1.74306i
$800$	4025.00			0.177882
$801$		0			0
$802$	8076.00			0.355578
$803$	22795.0	−	39482.1i	1.00177	−	1.73511i
$804$		0			0
$805$	45.0000	+	77.9423i	0.00197024	+	0.00341255i
$806$	562.500	+	974.279i	0.0245822	+	0.0425775i
$807$		0			0
$808$	−10012.5	+	17342.2i	−0.435939	+	0.755068i
$809$	−15304.0			−0.665093			−0.332546	−	0.943087i	$-0.607908\pi$
$809$	−15304.0			−0.665093			−0.332546	+	0.943087i	$0.607908\pi$
$810$		0			0
$811$	−40122.0			−1.73721			−0.868603	−	0.495509i	$-0.834982\pi$
$811$	−40122.0			−1.73721			−0.868603	+	0.495509i	$0.834982\pi$
$812$	3297.00	−	5710.57i	0.142490	−	0.246800i
$813$		0			0
$814$	−1645.00	−	2849.22i	−0.0708320	−	0.122685i
$815$	−1782.50	−	3087.38i	−0.0766114	−	0.132695i
$816$		0			0
$817$	11116.0	−	19253.5i	0.476009	−	0.824472i
$818$	−2833.00			−0.121092
$819$		0			0
$820$	−4900.00			−0.208677
$821$	−12549.0	+	21735.5i	−0.533451	+	0.923964i	0.465786	+	0.884898i	$0.345772\pi$
$821$	−12549.0	+	21735.5i	−0.533451	+	0.923964i	−0.999237	+	0.0390664i	$0.987562\pi$
$822$		0			0
$823$	21746.0	+	37665.2i	0.921042	+	1.59529i	0.797805	+	0.602915i	$0.205994\pi$
$823$	21746.0	+	37665.2i	0.921042	+	1.59529i	0.123237	+	0.992377i	$0.460672\pi$
$824$	5145.00	+	8911.40i	0.217518	+	0.376752i
$825$		0			0
$826$	2244.00	−	3886.72i	0.0945263	−	0.163724i
$827$	11206.0			0.471186			0.235593	−	0.971852i	$-0.424297\pi$
$827$	11206.0			0.471186			0.235593	+	0.971852i	$0.424297\pi$
$828$		0			0
$829$	−23964.0			−1.00399			−0.501993	−	0.864872i	$-0.667400\pi$
$829$	−23964.0			−1.00399			−0.501993	+	0.864872i	$0.667400\pi$
$830$	255.000	−	441.673i	0.0106641	−	0.0184707i
$831$		0			0
$832$	−417.500	−	723.131i	−0.0173969	−	0.0301323i
$833$	−20108.5	−	34828.9i	−0.836396	−	1.44868i
$834$		0			0
$835$	−3990.00	+	6910.88i	−0.165365	+	0.286420i
$836$	−18424.0			−0.762210
$837$		0			0
$838$	−4777.00			−0.196920
$839$	17303.0	−	29969.7i	0.711997	−	1.23322i	−0.252109	−	0.967699i	$-0.581124\pi$
$839$	17303.0	−	29969.7i	0.711997	−	1.23322i	0.964106	−	0.265517i	$-0.0855426\pi$
$840$		0			0
$841$	−130.000	−	225.167i	−0.00533027	−	0.00923230i
$842$	3232.00	+	5597.99i	0.132283	+	0.229120i
$843$		0			0
$844$	−14567.0	+	25230.8i	−0.594096	+	1.02900i
$845$	−10860.0			−0.442125
$846$		0			0
$847$	−5268.00			−0.213708
$848$	−82.0000	+	142.028i	−0.00332063	+	0.00575149i
$849$		0			0
$850$	1637.50	+	2836.23i	0.0660774	+	0.114449i
$851$	105.000	+	181.865i	0.00422956	+	0.00732581i
$852$		0			0
$853$	9238.50	−	16001.6i	0.370833	−	0.642301i	−0.618861	−	0.785500i	$-0.712406\pi$
$853$	9238.50	−	16001.6i	0.370833	−	0.642301i	0.989694	+	0.143199i	$0.0457390\pi$
$854$	2028.00			0.0812608
$855$		0			0
$856$	16470.0			0.657632
$857$	20671.0	−	35803.2i	0.823930	−	1.42709i	−0.0788045	−	0.996890i	$-0.525110\pi$
$857$	20671.0	−	35803.2i	0.823930	−	1.42709i	0.902734	−	0.430198i	$-0.141556\pi$
$858$		0			0
$859$	−10949.0	−	18964.2i	−0.434895	−	0.753261i	0.562392	−	0.826871i	$-0.309881\pi$
$859$	−10949.0	−	18964.2i	−0.434895	−	0.753261i	−0.997287	+	0.0736101i	$0.976548\pi$
$860$	6947.50	+	12033.4i	0.275474	+	0.477135i
$861$		0			0
$862$	−5340.00	+	9249.15i	−0.210999	+	0.365461i
$863$	−18487.0			−0.729206			−0.364603	−	0.931163i	$-0.618795\pi$
$863$	−18487.0			−0.729206			−0.364603	+	0.931163i	$0.618795\pi$
$864$		0			0
$865$	20670.0			0.812487
$866$	−5783.00	+	10016.4i	−0.226922	+	0.393040i
$867$		0			0
$868$	−4725.00	−	8183.94i	−0.184766	−	0.320024i
$869$	−29539.5	−	51163.9i	−1.15312	−	1.99726i
$870$		0			0
$871$	1230.00	−	2130.42i	0.0478496	−	0.0828779i
$872$	10500.0			0.407769
$873$		0			0
$874$	−168.000			−0.00650193
$875$	375.000	−	649.519i	0.0144884	−	0.0250946i
$876$		0			0
$877$	−3796.50	−	6575.73i	−0.146179	−	0.253189i	0.783633	−	0.621224i	$-0.213364\pi$
$877$	−3796.50	−	6575.73i	−0.146179	−	0.253189i	−0.929812	+	0.368035i	$0.880031\pi$
$878$	724.000	+	1254.00i	0.0278289	+	0.0482012i
$879$		0			0
$880$	4817.50	−	8344.15i	0.184543	−	0.319638i
$881$	−3038.00			−0.116178			−0.0580890	−	0.998311i	$-0.518501\pi$
$881$	−3038.00			−0.116178			−0.0580890	+	0.998311i	$0.518501\pi$
$882$		0			0
$883$	−16732.0			−0.637686			−0.318843	−	0.947808i	$-0.603294\pi$
$883$	−16732.0			−0.637686			−0.318843	+	0.947808i	$0.603294\pi$
$884$	2292.50	−	3970.73i	0.0872230	−	0.151075i
$885$		0			0
$886$	1188.00	+	2057.68i	0.0450470	+	0.0780237i
$887$	4015.50	+	6955.05i	0.152004	+	0.263278i	0.931964	−	0.362551i	$-0.118094\pi$
$887$	4015.50	+	6955.05i	0.152004	+	0.263278i	−0.779960	+	0.625829i	$0.784761\pi$
$888$		0			0
$889$	−4938.00	+	8552.87i	−0.186294	+	0.322670i
$890$	−7440.00			−0.280213
$891$		0			0
$892$	41692.0			1.56497
$893$	−9716.00	+	16828.6i	−0.364091	+	0.630625i
$894$		0			0
$895$	−4570.00	−	7915.47i	−0.170680	−	0.295626i
$896$	−4365.00	−	7560.40i	−0.162750	−	0.281892i
$897$		0			0
$898$	7447.00	−	12898.6i	0.276737	−	0.479322i
$899$	−35325.0			−1.31052
$900$		0			0
$901$	−524.000			−0.0193751
$902$	3290.00	−	5698.45i	0.121447	−	0.210352i
$903$		0			0
$904$	−7912.50	−	13704.9i	−0.291113	−	0.504222i
$905$	1300.00	+	2251.67i	0.0477497	+	0.0827049i
$906$		0			0
$907$	19243.5	−	33330.7i	0.704487	−	1.22021i	−0.262389	−	0.964962i	$-0.584510\pi$
$907$	19243.5	−	33330.7i	0.704487	−	1.22021i	0.966876	−	0.255246i	$-0.0821563\pi$
$908$	−34580.0			−1.26385
$909$		0			0
$910$	150.000			0.00546423
$911$	2560.00	−	4434.05i	0.0931027	−	0.161259i	−0.815712	−	0.578458i	$-0.803655\pi$
$911$	2560.00	−	4434.05i	0.0931027	−	0.161259i	0.908815	+	0.417199i	$0.136988\pi$
$912$		0			0
$913$	−2397.00	−	4151.73i	−0.0868884	−	0.150495i
$914$	−8102.00	−	14033.1i	−0.293206	−	0.507848i
$915$		0			0
$916$	15204.0	−	26334.1i	0.548422	−	0.949894i
$917$	10998.0			0.396059
$918$		0			0
$919$	28075.0			1.00774			0.503868	−	0.863781i	$-0.331910\pi$
$919$	28075.0			1.00774			0.503868	+	0.863781i	$0.331910\pi$
$920$	112.500	−	194.856i	0.00403154	−	0.00698283i
$921$		0			0
$922$	2541.00	+	4401.14i	0.0907629	+	0.157206i
$923$	80.0000	+	138.564i	0.00285291	+	0.00494138i
$924$		0			0
$925$	875.000	−	1515.54i	0.0311025	−	0.0538711i
$926$	−10326.0			−0.366451
$927$		0			0
$928$	−25277.0			−0.894136
$929$	6428.00	−	11133.6i	0.227014	−	0.393199i	−0.729908	−	0.683546i	$-0.760437\pi$
$929$	6428.00	−	11133.6i	0.227014	−	0.393199i	0.956922	+	0.290346i	$0.0937704\pi$
$930$		0			0
$931$	−8596.00	−	14888.7i	−0.302602	−	0.524122i
$932$	−18207.0	−	31535.4i	−0.639904	−	1.10835i
$933$		0			0
$934$	−2092.00	+	3623.45i	−0.0732894	+	0.126941i
$935$	30785.0			1.07677
$936$		0			0
$937$	−1374.00			−0.0479046			−0.0239523	−	0.999713i	$-0.507625\pi$
$937$	−1374.00			−0.0479046			−0.0239523	+	0.999713i	$0.507625\pi$
$938$	1476.00	−	2556.51i	0.0513786	−	0.0889903i
$939$		0			0
$940$	−6072.50	−	10517.9i	−0.210705	−	0.364953i
$941$	−4271.50	−	7398.46i	−0.147978	−	0.256305i	0.782502	−	0.622648i	$-0.213943\pi$
$941$	−4271.50	−	7398.46i	−0.147978	−	0.256305i	−0.930480	+	0.366343i	$0.880610\pi$
$942$		0			0
$943$	−210.000	+	363.731i	−0.00725190	+	0.0125607i
$944$	30668.0			1.05737
$945$		0			0
$946$	−18659.0			−0.641286
$947$	−6753.00	+	11696.5i	−0.231724	+	0.401358i	−0.958316	−	0.285712i	$-0.907770\pi$
$947$	−6753.00	+	11696.5i	−0.231724	+	0.401358i	0.726591	+	0.687070i	$0.241103\pi$
$948$		0			0
$949$	2425.00	+	4200.22i	0.0829492	+	0.143672i
$950$	700.000	+	1212.44i	0.0239063	+	0.0414070i
$951$		0			0
$952$	5895.00	−	10210.4i	0.200691	−	0.347607i
$953$	21775.0			0.740148			0.370074	−	0.929002i	$-0.379332\pi$
$953$	21775.0			0.740148			0.370074	+	0.929002i	$0.379332\pi$
$954$		0			0
$955$	24130.0			0.817621
$956$	5411.00	−	9372.13i	0.183059	−	0.317067i
$957$		0			0
$958$	7788.00	+	13489.2i	0.262650	+	0.454923i
$959$	3294.00	+	5705.38i	0.110916	+	0.192113i
$960$		0			0
$961$	−10417.0	+	18042.8i	−0.349669	+	0.605645i
$962$	350.000			0.0117302
$963$		0			0
$964$	25613.0			0.855746
$965$	−4175.00	+	7231.31i	−0.139273	+	0.241227i
$966$		0			0
$967$	−1927.00	−	3337.66i	−0.0640829	−	0.110995i	0.832204	−	0.554470i	$-0.187079\pi$
$967$	−1927.00	−	3337.66i	−0.0640829	−	0.110995i	−0.896287	+	0.443475i	$0.853746\pi$
$968$	6585.00	+	11405.6i	0.218647	+	0.378707i
$969$		0			0
$970$	−2435.00	+	4217.54i	−0.0806012	+	0.139605i
$971$	12933.0			0.427435			0.213718	−	0.976895i	$-0.431443\pi$
$971$	12933.0			0.427435			0.213718	+	0.976895i	$0.431443\pi$
$972$		0			0
$973$	6252.00			0.205992
$974$	−5110.00	+	8850.78i	−0.168106	+	0.291168i
$975$		0			0
$976$	6929.00	+	12001.4i	0.227246	+	0.393601i
$977$	−8760.50	−	15173.6i	−0.286871	−	0.496876i	0.686190	−	0.727422i	$-0.259282\pi$
$977$	−8760.50	−	15173.6i	−0.286871	−	0.496876i	−0.973061	+	0.230547i	$0.925949\pi$
$978$		0			0
$979$	−34968.0	+	60566.4i	−1.14155	+	1.97723i
$980$	10745.0			0.350241
$981$		0			0
$982$	−2692.00			−0.0874798
$983$	6286.50	−	10888.5i	0.203976	−	0.353296i	−0.745830	−	0.666136i	$-0.767947\pi$
$983$	6286.50	−	10888.5i	0.203976	−	0.353296i	0.949806	+	0.312840i	$0.101280\pi$
$984$		0			0
$985$	3450.00	+	5975.58i	0.111600	+	0.193297i
$986$	−10283.5	−	17811.5i	−0.332143	−	0.575289i
$987$		0			0
$988$	980.000	−	1697.41i	0.0315566	−	0.0546577i
$989$	1191.00			0.0382928
$990$		0			0
$991$	8945.00			0.286728			0.143364	−	0.989670i	$-0.454208\pi$
$991$	8945.00			0.286728			0.143364	+	0.989670i	$0.454208\pi$
$992$	−18112.5	+	31371.8i	−0.579710	+	1.00409i
$993$		0			0
$994$	96.0000	+	166.277i	0.00306331	+	0.00530582i
$995$	−10892.5	−	18866.4i	−0.347051	−	0.601109i
$996$		0			0
$997$	29089.5	−	50384.5i	0.924046	−	1.60049i	0.130957	−	0.991388i	$-0.458195\pi$
$997$	29089.5	−	50384.5i	0.924046	−	1.60049i	0.793089	−	0.609106i	$-0.208472\pi$
$998$	5764.00			0.182822
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.f.136.1		2
3.2	odd	2		405.4.e.h.136.1		2
9.2	odd	6		135.4.a.b.1.1	✓	1
9.4	even	3	inner	405.4.e.f.271.1		2
9.5	odd	6		405.4.e.h.271.1		2
9.7	even	3		135.4.a.c.1.1	yes	1
36.7	odd	6		2160.4.a.p.1.1		1
36.11	even	6		2160.4.a.f.1.1		1
45.2	even	12		675.4.b.f.649.1		2
45.7	odd	12		675.4.b.e.649.2		2
45.29	odd	6		675.4.a.h.1.1		1
45.34	even	6		675.4.a.c.1.1		1
45.38	even	12		675.4.b.f.649.2		2
45.43	odd	12		675.4.b.e.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.b.1.1	✓	1	9.2	odd	6
135.4.a.c.1.1	yes	1	9.7	even	3
405.4.e.f.136.1		2	1.1	even	1	trivial
405.4.e.f.271.1		2	9.4	even	3	inner
405.4.e.h.136.1		2	3.2	odd	2
405.4.e.h.271.1		2	9.5	odd	6
675.4.a.c.1.1		1	45.34	even	6
675.4.a.h.1.1		1	45.29	odd	6
675.4.b.e.649.1		2	45.43	odd	12
675.4.b.e.649.2		2	45.7	odd	12
675.4.b.f.649.1		2	45.2	even	12
675.4.b.f.649.2		2	45.38	even	12
2160.4.a.f.1.1		1	36.11	even	6
2160.4.a.p.1.1		1	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(3.50000 + 6.06218i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(3.00000 - 5.19615i) q^{7} -15.0000 q^{8} +5.00000 q^{10} +(23.5000 - 40.7032i) q^{11} +(2.50000 + 4.33013i) q^{13} +(3.00000 + 5.19615i) q^{14} +(-20.5000 + 35.5070i) q^{16} -131.000 q^{17} -56.0000 q^{19} +(17.5000 - 30.3109i) q^{20} +(23.5000 + 40.7032i) q^{22} +(-1.50000 - 2.59808i) q^{23} +(-12.5000 + 21.6506i) q^{25} -5.00000 q^{26} +42.0000 q^{28} +(78.5000 - 135.966i) q^{29} +(-112.500 - 194.856i) q^{31} +(-80.5000 - 139.430i) q^{32} +(65.5000 - 113.449i) q^{34} -30.0000 q^{35} -70.0000 q^{37} +(28.0000 - 48.4974i) q^{38} +(37.5000 + 64.9519i) q^{40} +(-70.0000 - 121.244i) q^{41} +(-198.500 + 343.812i) q^{43} +329.000 q^{44} +3.00000 q^{46} +(173.500 - 300.511i) q^{47} +(153.500 + 265.870i) q^{49} +(-12.5000 - 21.6506i) q^{50} +(-17.5000 + 30.3109i) q^{52} +4.00000 q^{53} -235.000 q^{55} +(-45.0000 + 77.9423i) q^{56} +(78.5000 + 135.966i) q^{58} +(-374.000 - 647.787i) q^{59} +(169.000 - 292.717i) q^{61} +225.000 q^{62} -167.000 q^{64} +(12.5000 - 21.6506i) q^{65} +(-246.000 - 426.084i) q^{67} +(-458.500 - 794.145i) q^{68} +(15.0000 - 25.9808i) q^{70} +32.0000 q^{71} +970.000 q^{73} +(35.0000 - 60.6218i) q^{74} +(-196.000 - 339.482i) q^{76} +(-141.000 - 244.219i) q^{77} +(628.500 - 1088.59i) q^{79} +205.000 q^{80} +140.000 q^{82} +(51.0000 - 88.3346i) q^{83} +(327.500 + 567.247i) q^{85} +(-198.500 - 343.812i) q^{86} +(-352.500 + 610.548i) q^{88} -1488.00 q^{89} +30.0000 q^{91} +(10.5000 - 18.1865i) q^{92} +(173.500 + 300.511i) q^{94} +(140.000 + 242.487i) q^{95} +(-487.000 + 843.509i) q^{97} -307.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 6 q^{7} - 30 q^{8} + 10 q^{10} + 47 q^{11} + 5 q^{13} + 6 q^{14} - 41 q^{16} - 262 q^{17} - 112 q^{19} + 35 q^{20} + 47 q^{22} - 3 q^{23} - 25 q^{25} - 10 q^{26} + 84 q^{28}+ \cdots - 614 q^{98}+O(q^{100}) \) 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 6 * q^7 - 30 * q^8 + 10 * q^10 + 47 * q^11 + 5 * q^13 + 6 * q^14 - 41 * q^16 - 262 * q^17 - 112 * q^19 + 35 * q^20 + 47 * q^22 - 3 * q^23 - 25 * q^25 - 10 * q^26 + 84 * q^28 + 157 * q^29 - 225 * q^31 - 161 * q^32 + 131 * q^34 - 60 * q^35 - 140 * q^37 + 56 * q^38 + 75 * q^40 - 140 * q^41 - 397 * q^43 + 658 * q^44 + 6 * q^46 + 347 * q^47 + 307 * q^49 - 25 * q^50 - 35 * q^52 + 8 * q^53 - 470 * q^55 - 90 * q^56 + 157 * q^58 - 748 * q^59 + 338 * q^61 + 450 * q^62 - 334 * q^64 + 25 * q^65 - 492 * q^67 - 917 * q^68 + 30 * q^70 + 64 * q^71 + 1940 * q^73 + 70 * q^74 - 392 * q^76 - 282 * q^77 + 1257 * q^79 + 410 * q^80 + 280 * q^82 + 102 * q^83 + 655 * q^85 - 397 * q^86 - 705 * q^88 - 2976 * q^89 + 60 * q^91 + 21 * q^92 + 347 * q^94 + 280 * q^95 - 974 * q^97 - 614 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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