LMFDB - Embedded newform 45.4.f.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(8,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.8");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 $ x^12 - 16x^10 - 14x^8 - 512x^6 + 3889x^4 + 126224*x^2 + 506944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.2
Root			$-0.347140 - 2.27426i$ of defining polynomial
Character	$\chi$	$=$	45.17
Dual form			45.4.f.a.8.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.62140 + 2.62140i) q^{2} -5.74350i q^{4} +(-8.38994 + 7.38978i) q^{5} +(-10.8652 - 10.8652i) q^{7} +(-5.91519 - 5.91519i) q^{8} +O(q^{10})$	\(q+(-2.62140 + 2.62140i) q^{2} -5.74350i q^{4} +(-8.38994 + 7.38978i) q^{5} +(-10.8652 - 10.8652i) q^{7} +(-5.91519 - 5.91519i) q^{8} +(2.62183 - 41.3650i) q^{10} -37.8905i q^{11} +(-48.1085 + 48.1085i) q^{13} +56.9640 q^{14} +76.9602 q^{16} +(-60.3458 + 60.3458i) q^{17} +109.678i q^{19} +(42.4432 + 48.1877i) q^{20} +(99.3262 + 99.3262i) q^{22} +(39.7660 + 39.7660i) q^{23} +(15.7823 - 124.000i) q^{25} -252.224i q^{26} +(-62.4042 + 62.4042i) q^{28} +90.3553 q^{29} -233.678 q^{31} +(-154.422 + 154.422i) q^{32} -316.381i q^{34} +(171.449 + 10.8670i) q^{35} +(-19.0042 - 19.0042i) q^{37} +(-287.509 - 287.509i) q^{38} +(93.3400 + 5.91616i) q^{40} +260.783i q^{41} +(176.216 - 176.216i) q^{43} -217.624 q^{44} -208.485 q^{46} +(-145.788 + 145.788i) q^{47} -106.896i q^{49} +(283.681 + 366.425i) q^{50} +(276.311 + 276.311i) q^{52} +(-183.285 - 183.285i) q^{53} +(280.002 + 317.899i) q^{55} +128.539i q^{56} +(-236.858 + 236.858i) q^{58} +279.564 q^{59} -390.267 q^{61} +(612.563 - 612.563i) q^{62} -193.924i q^{64} +(48.1164 - 759.139i) q^{65} +(150.444 + 150.444i) q^{67} +(346.596 + 346.596i) q^{68} +(-477.925 + 420.951i) q^{70} -470.042i q^{71} +(480.765 - 480.765i) q^{73} +99.6355 q^{74} +629.934 q^{76} +(-411.687 + 411.687i) q^{77} +1322.44i q^{79} +(-645.692 + 568.719i) q^{80} +(-683.618 - 683.618i) q^{82} +(456.192 + 456.192i) q^{83} +(60.3558 - 952.240i) q^{85} +923.868i q^{86} +(-224.129 + 224.129i) q^{88} -1364.85 q^{89} +1045.41 q^{91} +(228.396 - 228.396i) q^{92} -764.340i q^{94} +(-810.494 - 920.190i) q^{95} +(-785.308 - 785.308i) q^{97} +(280.218 + 280.218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−2.62140	+	2.62140i	−0.926806	+	0.926806i	−0.997498	−	0.0706924i	$-0.977479\pi$
$2$	−2.62140	+	2.62140i	−0.926806	+	0.926806i	0.0706924	+	0.997498i	$0.477479\pi$
$3$		0			0
$3$		0			0
$4$		−	5.74350i		−	0.717938i
$5$	−8.38994	+	7.38978i	−0.750419	+	0.660962i
$5$	−8.38994	+	7.38978i	−0.750419	+	0.660962i
$6$		0			0
$7$	−10.8652	−	10.8652i	−0.586664	−	0.586664i	0.350062	−	0.936726i	$-0.386160\pi$
$7$	−10.8652	−	10.8652i	−0.586664	−	0.586664i	−0.936726	+	0.350062i	$0.886160\pi$
$8$	−5.91519	−	5.91519i	−0.261417	−	0.261417i
$9$		0			0
$10$	2.62183	−	41.3650i	0.0829097	−	1.30808i
$11$		−	37.8905i		−	1.03858i	−0.854597	−	0.519291i	$-0.826196\pi$
$11$		−	37.8905i		−	1.03858i	0.854597	−	0.519291i	$-0.173804\pi$
$12$		0			0
$13$	−48.1085	+	48.1085i	−1.02638	+	1.02638i	−0.0267344	+	0.999643i	$0.508511\pi$
$13$	−48.1085	+	48.1085i	−1.02638	+	1.02638i	−0.999643	+	0.0267344i	$0.991489\pi$
$14$	56.9640			1.08745
$15$		0			0
$16$	76.9602			1.20250
$17$	−60.3458	+	60.3458i	−0.860942	+	0.860942i	−0.991448	−	0.130506i	$-0.958340\pi$
$17$	−60.3458	+	60.3458i	−0.860942	+	0.860942i	0.130506	+	0.991448i	$0.458340\pi$
$18$		0			0
$19$			109.678i			1.32430i	0.749369	+	0.662152i	$0.230357\pi$
$19$			109.678i			1.32430i	−0.749369	+	0.662152i	$0.769643\pi$
$20$	42.4432	+	48.1877i	0.474530	+	0.538755i
$21$		0			0
$22$	99.3262	+	99.3262i	0.962564	+	0.962564i
$23$	39.7660	+	39.7660i	0.360512	+	0.360512i	0.864002	−	0.503489i	$-0.167951\pi$
$23$	39.7660	+	39.7660i	0.360512	+	0.360512i	−0.503489	+	0.864002i	$0.667951\pi$
$24$		0			0
$25$	15.7823	−	124.000i	0.126259	−	0.991997i
$26$		−	252.224i		−	1.90250i
$27$		0			0
$28$	−62.4042	+	62.4042i	−0.421188	+	0.421188i
$29$	90.3553			0.578571			0.289285	−	0.957243i	$-0.406582\pi$
$29$	90.3553			0.578571			0.289285	+	0.957243i	$0.406582\pi$
$30$		0			0
$31$	−233.678			−1.35386			−0.676932	−	0.736046i	$-0.736691\pi$
$31$	−233.678			−1.35386			−0.676932	+	0.736046i	$0.736691\pi$
$32$	−154.422	+	154.422i	−0.853070	+	0.853070i
$33$		0			0
$34$		−	316.381i		−	1.59585i
$35$	171.449	+	10.8670i	0.828007	+	0.0524815i
$36$		0			0
$37$	−19.0042	−	19.0042i	−0.0844399	−	0.0844399i	0.663625	−	0.748065i	$-0.269017\pi$
$37$	−19.0042	−	19.0042i	−0.0844399	−	0.0844399i	−0.748065	+	0.663625i	$0.769017\pi$
$38$	−287.509	−	287.509i	−1.22737	−	1.22737i
$39$		0			0
$40$	93.3400	+	5.91616i	0.368959	+	0.0233857i
$41$			260.783i			0.993354i	0.867935	+	0.496677i	$0.165447\pi$
$41$			260.783i			0.993354i	−0.867935	+	0.496677i	$0.834553\pi$
$42$		0			0
$43$	176.216	−	176.216i	0.624948	−	0.624948i	−0.321845	−	0.946792i	$-0.604303\pi$
$43$	176.216	−	176.216i	0.624948	−	0.624948i	0.946792	+	0.321845i	$0.104303\pi$
$44$	−217.624			−0.745638
$45$		0			0
$46$	−208.485			−0.668250
$47$	−145.788	+	145.788i	−0.452456	+	0.452456i	−0.896169	−	0.443713i	$-0.853661\pi$
$47$	−145.788	+	145.788i	−0.452456	+	0.452456i	0.443713	+	0.896169i	$0.353661\pi$
$48$		0			0
$49$		−	106.896i		−	0.311650i
$50$	283.681	+	366.425i	0.802372	+	1.03641i
$51$		0			0
$52$	276.311	+	276.311i	0.736875	+	0.736875i
$53$	−183.285	−	183.285i	−0.475022	−	0.475022i	0.428513	−	0.903535i	$-0.359038\pi$
$53$	−183.285	−	183.285i	−0.475022	−	0.475022i	−0.903535	+	0.428513i	$0.859038\pi$
$54$		0			0
$55$	280.002	+	317.899i	0.686464	+	0.779373i
$56$			128.539i			0.306728i
$57$		0			0
$58$	−236.858	+	236.858i	−0.536223	+	0.536223i
$59$	279.564			0.616883			0.308441	−	0.951243i	$-0.400193\pi$
$59$	279.564			0.616883			0.308441	+	0.951243i	$0.400193\pi$
$60$		0			0
$61$	−390.267			−0.819157			−0.409578	−	0.912275i	$-0.634324\pi$
$61$	−390.267			−0.819157			−0.409578	+	0.912275i	$0.634324\pi$
$62$	612.563	−	612.563i	1.25477	−	1.25477i
$63$		0			0
$64$		−	193.924i		−	0.378757i
$65$	48.1164	−	759.139i	0.0918170	−	1.44861i
$66$		0			0
$67$	150.444	+	150.444i	0.274324	+	0.274324i	0.830838	−	0.556514i	$-0.187861\pi$
$67$	150.444	+	150.444i	0.274324	+	0.274324i	−0.556514	+	0.830838i	$0.687861\pi$
$68$	346.596	+	346.596i	0.618103	+	0.618103i
$69$		0			0
$70$	−477.925	+	420.951i	−0.816042	+	0.718761i
$71$		−	470.042i		−	0.785686i	−0.919605	−	0.392843i	$-0.871492\pi$
$71$		−	470.042i		−	0.785686i	0.919605	−	0.392843i	$-0.128508\pi$
$72$		0			0
$73$	480.765	−	480.765i	0.770812	−	0.770812i	−0.207436	−	0.978249i	$-0.566512\pi$
$73$	480.765	−	480.765i	0.770812	−	0.770812i	0.978249	+	0.207436i	$0.0665119\pi$
$74$	99.6355			0.156519
$75$		0			0
$76$	629.934			0.950769
$77$	−411.687	+	411.687i	−0.609299	+	0.609299i
$78$		0			0
$79$			1322.44i			1.88337i	0.336497	+	0.941685i	$0.390758\pi$
$79$			1322.44i			1.88337i	−0.336497	+	0.941685i	$0.609242\pi$
$80$	−645.692	+	568.719i	−0.902382	+	0.794809i
$81$		0			0
$82$	−683.618	−	683.618i	−0.920646	−	0.920646i
$83$	456.192	+	456.192i	0.603296	+	0.603296i	0.941186	−	0.337890i	$-0.109713\pi$
$83$	456.192	+	456.192i	0.603296	+	0.603296i	−0.337890	+	0.941186i	$0.609713\pi$
$84$		0			0
$85$	60.3558	−	952.240i	0.0770177	−	1.21512i
$86$			923.868i			1.15841i
$87$		0			0
$88$	−224.129	+	224.129i	−0.271503	+	0.271503i
$89$	−1364.85			−1.62555			−0.812774	−	0.582580i	$-0.802043\pi$
$89$	−1364.85			−1.62555			−0.812774	+	0.582580i	$0.802043\pi$
$90$		0			0
$91$	1045.41			1.20428
$92$	228.396	−	228.396i	0.258825	−	0.258825i
$93$		0			0
$94$		−	764.340i		−	0.838677i
$95$	−810.494	−	920.190i	−0.875315	−	0.993784i
$96$		0			0
$97$	−785.308	−	785.308i	−0.822020	−	0.822020i	0.164377	−	0.986398i	$-0.447439\pi$
$97$	−785.308	−	785.308i	−0.822020	−	0.822020i	−0.986398	+	0.164377i	$0.947439\pi$
$98$	280.218	+	280.218i	0.288839	+	0.288839i
$99$		0			0
$100$	−712.193	−	90.6458i	−0.712193	−	0.0906458i
$101$		−	111.555i		−	0.109902i	−0.998489	−	0.0549510i	$-0.982500\pi$
$101$		−	111.555i		−	0.109902i	0.998489	−	0.0549510i	$-0.0175003\pi$
$102$		0			0
$103$	13.8820	−	13.8820i	0.0132800	−	0.0132800i	−0.700436	−	0.713716i	$-0.747011\pi$
$103$	13.8820	−	13.8820i	0.0132800	−	0.0132800i	0.713716	+	0.700436i	$0.247011\pi$
$104$	569.142			0.536624
$105$		0			0
$106$	960.929			0.880507
$107$	776.165	−	776.165i	0.701259	−	0.701259i	−0.263422	−	0.964681i	$-0.584851\pi$
$107$	776.165	−	776.165i	0.701259	−	0.701259i	0.964681	+	0.263422i	$0.0848510\pi$
$108$		0			0
$109$			410.579i			0.360792i	0.983594	+	0.180396i	$0.0577379\pi$
$109$			410.579i			0.360792i	−0.983594	+	0.180396i	$0.942262\pi$
$110$	−1567.34	−	99.3425i	−1.35855	−	0.0861085i
$111$		0			0
$112$	−836.186	−	836.186i	−0.705465	−	0.705465i
$113$	521.328	+	521.328i	0.434004	+	0.434004i	0.889988	−	0.455984i	$-0.150713\pi$
$113$	521.328	+	521.328i	0.434004	+	0.434004i	−0.455984	+	0.889988i	$0.650713\pi$
$114$		0			0
$115$	−627.496	−	39.7725i	−0.508820	−	0.0322505i
$116$		−	518.956i		−	0.415378i
$117$		0			0
$118$	−732.849	+	732.849i	−0.571730	+	0.571730i
$119$	1311.34			1.01017
$120$		0			0
$121$	−104.688			−0.0786538
$122$	1023.05	−	1023.05i	0.759199	−	0.759199i
$123$		0			0
$124$			1342.13i			0.971990i
$125$	783.917	+	1156.98i	0.560926	+	0.827866i
$126$		0			0
$127$	786.555	+	786.555i	0.549571	+	0.549571i	0.926317	−	0.376746i	$-0.122957\pi$
$127$	786.555	+	786.555i	0.549571	+	0.549571i	−0.376746	+	0.926317i	$0.622957\pi$
$128$	−727.025	−	727.025i	−0.502036	−	0.502036i
$129$		0			0
$130$	1863.88	+	2116.14i	1.25748	+	1.42768i
$131$			448.364i			0.299036i	0.988759	+	0.149518i	$0.0477722\pi$
$131$			448.364i			0.299036i	−0.988759	+	0.149518i	$0.952228\pi$
$132$		0			0
$133$	1191.67	−	1191.67i	0.776922	−	0.776922i
$134$	−788.751			−0.508490
$135$		0			0
$136$	713.914			0.450129
$137$	−950.120	+	950.120i	−0.592512	+	0.592512i	−0.938309	−	0.345797i	$-0.887609\pi$
$137$	−950.120	+	950.120i	−0.592512	+	0.592512i	0.345797	+	0.938309i	$0.387609\pi$
$138$		0			0
$139$			112.735i			0.0687918i	0.999408	+	0.0343959i	$0.0109507\pi$
$139$			112.735i			0.0687918i	−0.999408	+	0.0343959i	$0.989049\pi$
$140$	62.4144	−	984.720i	0.0376784	−	0.594458i
$141$		0			0
$142$	1232.17	+	1232.17i	0.728179	+	0.728179i
$143$	1822.85	+	1822.85i	1.06598	+	1.06598i
$144$		0			0
$145$	−758.076	+	667.706i	−0.434171	+	0.382413i
$146$			2520.56i			1.42879i
$147$		0			0
$148$	−109.151	+	109.151i	−0.0606226	+	0.0606226i
$149$	−1492.95			−0.820852			−0.410426	−	0.911894i	$-0.634620\pi$
$149$	−1492.95			−0.820852			−0.410426	+	0.911894i	$0.634620\pi$
$150$		0			0
$151$	−2939.54			−1.58422			−0.792108	−	0.610381i	$-0.791016\pi$
$151$	−2939.54			−1.58422			−0.792108	+	0.610381i	$0.791016\pi$
$152$	648.764	−	648.764i	0.346196	−	0.346196i
$153$		0			0
$154$		−	2158.39i		−	1.12940i
$155$	1960.54	−	1726.83i	1.01597	−	0.894852i
$156$		0			0
$157$	156.386	+	156.386i	0.0794967	+	0.0794967i	0.745737	−	0.666240i	$-0.232098\pi$
$157$	156.386	+	156.386i	0.0794967	+	0.0794967i	−0.666240	+	0.745737i	$0.732098\pi$
$158$	−3466.65	−	3466.65i	−1.74552	−	1.74552i
$159$		0			0
$160$	154.448	−	2436.74i	0.0763135	−	1.20401i
$161$		−	864.129i		−	0.422999i
$162$		0			0
$163$	−239.292	+	239.292i	−0.114986	+	0.114986i	−0.762259	−	0.647272i	$-0.775910\pi$
$163$	−239.292	+	239.292i	−0.114986	+	0.114986i	0.647272	+	0.762259i	$0.275910\pi$
$164$	1497.81			0.713167
$165$		0			0
$166$	−2391.72			−1.11828
$167$	−677.159	+	677.159i	−0.313773	+	0.313773i	−0.846369	−	0.532596i	$-0.821216\pi$
$167$	−677.159	+	677.159i	−0.313773	+	0.313773i	0.532596	+	0.846369i	$0.321216\pi$
$168$		0			0
$169$		−	2431.86i		−	1.10690i
$170$	2337.99	+	2654.42i	1.05480	+	1.19756i
$171$		0			0
$172$	−1012.10	−	1012.10i	−0.448674	−	0.448674i
$173$	−441.523	−	441.523i	−0.194037	−	0.194037i	0.603401	−	0.797438i	$-0.293812\pi$
$173$	−441.523	−	441.523i	−0.194037	−	0.194037i	−0.797438	+	0.603401i	$0.793812\pi$
$174$		0			0
$175$	−1518.76	+	1175.80i	−0.656041	+	0.507898i
$176$		−	2916.06i		−	1.24890i
$177$		0			0
$178$	3577.82	−	3577.82i	1.50657	−	1.50657i
$179$	−4724.27			−1.97267			−0.986337	−	0.164737i	$-0.947322\pi$
$179$	−4724.27			−1.97267			−0.986337	+	0.164737i	$0.947322\pi$
$180$		0			0
$181$	−1548.35			−0.635847			−0.317923	−	0.948116i	$-0.602985\pi$
$181$	−1548.35			−0.635847			−0.317923	+	0.948116i	$0.602985\pi$
$182$	−2740.45	+	2740.45i	−1.11613	+	1.11613i
$183$		0			0
$184$		−	470.447i		−	0.188488i
$185$	299.882	+	19.0074i	0.119177	+	0.00755378i
$186$		0			0
$187$	2286.53	+	2286.53i	0.894159	+	0.894159i
$188$	837.336	+	837.336i	0.324835	+	0.324835i
$189$		0			0
$190$	4536.82	+	287.557i	1.73229	+	0.109798i
$191$			1090.71i			0.413200i	0.978426	+	0.206600i	$0.0662398\pi$
$191$			1090.71i			0.413200i	−0.978426	+	0.206600i	$0.933760\pi$
$192$		0			0
$193$	−2825.69	+	2825.69i	−1.05387	+	1.05387i	−0.0554112	+	0.998464i	$0.517647\pi$
$193$	−2825.69	+	2825.69i	−1.05387	+	1.05387i	−0.998464	+	0.0554112i	$0.982353\pi$
$194$	4117.22			1.52371
$195$		0			0
$196$	−613.958			−0.223746
$197$	−1163.99	+	1163.99i	−0.420969	+	0.420969i	−0.885537	−	0.464569i	$-0.846209\pi$
$197$	−1163.99	+	1163.99i	−0.420969	+	0.420969i	0.464569	+	0.885537i	$0.346209\pi$
$198$		0			0
$199$		−	520.257i		−	0.185327i	−0.995697	−	0.0926633i	$-0.970462\pi$
$199$		−	520.257i		−	0.185327i	0.995697	−	0.0926633i	$-0.0295380\pi$
$200$	−826.837	+	640.126i	−0.292331	+	0.226319i
$201$		0			0
$202$	292.430	+	292.430i	0.101858	+	0.101858i
$203$	−981.726	−	981.726i	−0.339427	−	0.339427i
$204$		0			0
$205$	−1927.13	−	2187.96i	−0.656569	−	0.745432i
$206$			72.7808i			0.0246159i
$207$		0			0
$208$	−3702.44	+	3702.44i	−1.23422	+	1.23422i
$209$	4155.74			1.37540
$210$		0			0
$211$	5524.38			1.80244			0.901219	−	0.433365i	$-0.142674\pi$
$211$	5524.38			1.80244			0.901219	+	0.433365i	$0.142674\pi$
$212$	−1052.70	+	1052.70i	−0.341036	+	0.341036i
$213$		0			0
$214$			4069.28i			1.29986i
$215$	−176.245	+	2780.65i	−0.0559062	+	0.882039i
$216$		0			0
$217$	2538.95	+	2538.95i	0.794263	+	0.794263i
$218$	−1076.29	−	1076.29i	−0.334384	−	0.334384i
$219$		0			0
$220$	1825.85	−	1608.19i	0.559541	−	0.492838i
$221$		−	5806.30i		−	1.76730i
$222$		0			0
$223$	323.198	−	323.198i	0.0970537	−	0.0970537i	−0.656913	−	0.753967i	$-0.728138\pi$
$223$	323.198	−	323.198i	0.0970537	−	0.0970537i	0.753967	+	0.656913i	$0.228138\pi$
$224$	3355.65			1.00093
$225$		0			0
$226$	−2733.22			−0.804474
$227$	2739.76	−	2739.76i	0.801077	−	0.801077i	−0.182187	−	0.983264i	$-0.558318\pi$
$227$	2739.76	−	2739.76i	0.801077	−	0.801077i	0.983264	+	0.182187i	$0.0583177\pi$
$228$		0			0
$229$			230.334i			0.0664667i	0.999448	+	0.0332333i	$0.0105804\pi$
$229$			230.334i			0.0664667i	−0.999448	+	0.0332333i	$0.989420\pi$
$230$	1749.18	−	1540.66i	0.501468	−	0.441688i
$231$		0			0
$232$	−534.468	−	534.468i	−0.151248	−	0.151248i
$233$	350.156	+	350.156i	0.0984526	+	0.0984526i	0.754618	−	0.656165i	$-0.227822\pi$
$233$	350.156	+	350.156i	0.0984526	+	0.0984526i	−0.656165	+	0.754618i	$0.727822\pi$
$234$		0			0
$235$	145.812	−	2300.50i	0.0404755	−	0.638587i
$236$		−	1605.67i		−	0.442883i
$237$		0			0
$238$	−3437.54	+	3437.54i	−0.936229	+	0.936229i
$239$	−1013.20			−0.274218			−0.137109	−	0.990556i	$-0.543781\pi$
$239$	−1013.20			−0.274218			−0.137109	+	0.990556i	$0.543781\pi$
$240$		0			0
$241$	2584.30			0.690744			0.345372	−	0.938466i	$-0.387753\pi$
$241$	2584.30			0.690744			0.345372	+	0.938466i	$0.387753\pi$
$242$	274.430	−	274.430i	0.0728968	−	0.0728968i
$243$		0			0
$244$			2241.50i			0.588104i
$245$	789.938	+	896.852i	0.205989	+	0.233868i
$246$		0			0
$247$	−5276.43	−	5276.43i	−1.35924	−	1.35924i
$248$	1382.25	+	1382.25i	0.353923	+	0.353923i
$249$		0			0
$250$	−5087.87	−	977.942i	−1.28714	−	0.247402i
$251$		−	1708.34i		−	0.429599i	−0.976658	−	0.214800i	$-0.931090\pi$
$251$		−	1708.34i		−	0.429599i	0.976658	−	0.214800i	$-0.0689099\pi$
$252$		0			0
$253$	1506.75	−	1506.75i	0.374422	−	0.374422i
$254$	−4123.76			−1.01869
$255$		0			0
$256$	5363.04			1.30934
$257$	−2659.48	+	2659.48i	−0.645500	+	0.645500i	−0.951902	−	0.306402i	$-0.900875\pi$
$257$	−2659.48	+	2659.48i	−0.645500	+	0.645500i	0.306402	+	0.951902i	$0.400875\pi$
$258$		0			0
$259$			412.969i			0.0990757i
$260$	−4360.12	−	276.357i	−1.04001	−	0.0659189i
$261$		0			0
$262$	−1175.34	−	1175.34i	−0.277148	−	0.277148i
$263$	2307.03	+	2307.03i	0.540903	+	0.540903i	0.923794	−	0.382891i	$-0.125071\pi$
$263$	2307.03	+	2307.03i	0.540903	+	0.540903i	−0.382891	+	0.923794i	$0.625071\pi$
$264$		0			0
$265$	2892.19	+	183.315i	0.670437	+	0.0424943i
$266$			6247.68i			1.44011i
$267$		0			0
$268$	864.078	−	864.078i	0.196948	−	0.196948i
$269$	447.584			0.101449			0.0507243	−	0.998713i	$-0.483847\pi$
$269$	447.584			0.101449			0.0507243	+	0.998713i	$0.483847\pi$
$270$		0			0
$271$	1573.15			0.352627			0.176314	−	0.984334i	$-0.443583\pi$
$271$	1573.15			0.352627			0.176314	+	0.984334i	$0.443583\pi$
$272$	−4644.23	+	4644.23i	−1.03529	+	1.03529i
$273$		0			0
$274$		−	4981.29i		−	1.09829i
$275$	−4698.41	−	598.000i	−1.03027	−	0.131130i
$276$		0			0
$277$	5816.91	+	5816.91i	1.26175	+	1.26175i	0.950245	+	0.311504i	$0.100833\pi$
$277$	5816.91	+	5816.91i	1.26175	+	1.26175i	0.311504	+	0.950245i	$0.399167\pi$
$278$	−295.524	−	295.524i	−0.0637566	−	0.0637566i
$279$		0			0
$280$	−949.875	−	1078.44i	−0.202735	−	0.230174i
$281$			140.032i			0.0297281i	0.999890	+	0.0148640i	$0.00473155\pi$
$281$			140.032i			0.0297281i	−0.999890	+	0.0148640i	$0.995268\pi$
$282$		0			0
$283$	4431.85	−	4431.85i	0.930905	−	0.930905i	−0.0668572	−	0.997763i	$-0.521297\pi$
$283$	4431.85	−	4431.85i	0.930905	−	0.930905i	0.997763	+	0.0668572i	$0.0212972\pi$
$284$	−2699.69			−0.564074
$285$		0			0
$286$	−9556.87			−1.97591
$287$	2833.46	−	2833.46i	0.582765	−	0.582765i
$288$		0			0
$289$		−	2370.24i		−	0.482442i
$290$	236.897	−	3737.55i	0.0479691	−	0.756815i
$291$		0			0
$292$	−2761.28	−	2761.28i	−0.553395	−	0.553395i
$293$	1244.45	+	1244.45i	0.248127	+	0.248127i	0.820202	−	0.572074i	$-0.193861\pi$
$293$	1244.45	+	1244.45i	0.248127	+	0.248127i	−0.572074	+	0.820202i	$0.693861\pi$
$294$		0			0
$295$	−2345.52	+	2065.91i	−0.462921	+	0.407736i
$296$			224.827i			0.0441480i
$297$		0			0
$298$	3913.61	−	3913.61i	0.760771	−	0.760771i
$299$	−3826.17			−0.740043
$300$		0			0
$301$	−3829.24			−0.733269
$302$	7705.73	−	7705.73i	1.46826	−	1.46826i
$303$		0			0
$304$			8440.82i			1.59248i
$305$	3274.32	−	2883.99i	0.614711	−	0.541432i
$306$		0			0
$307$	−7327.60	−	7327.60i	−1.36224	−	1.36224i	−0.871053	−	0.491189i	$-0.836563\pi$
$307$	−7327.60	−	7327.60i	−1.36224	−	1.36224i	−0.491189	−	0.871053i	$-0.663437\pi$
$308$	2364.52	+	2364.52i	0.437439	+	0.437439i
$309$		0			0
$310$	−612.664	+	9666.08i	−0.112248	+	1.77096i
$311$			7489.38i			1.36554i	0.730632	+	0.682771i	$0.239225\pi$
$311$			7489.38i			1.36554i	−0.730632	+	0.682771i	$0.760775\pi$
$312$		0			0
$313$	−2456.04	+	2456.04i	−0.443525	+	0.443525i	−0.893195	−	0.449670i	$-0.851542\pi$
$313$	−2456.04	+	2456.04i	−0.443525	+	0.443525i	0.449670	+	0.893195i	$0.351542\pi$
$314$	−819.903			−0.147356
$315$		0			0
$316$	7595.44			1.35214
$317$	6724.30	−	6724.30i	1.19140	−	1.19140i	0.214728	−	0.976674i	$-0.431113\pi$
$317$	6724.30	−	6724.30i	1.19140	−	1.19140i	0.976674	−	0.214728i	$-0.0688866\pi$
$318$		0			0
$319$		−	3423.60i		−	0.600894i
$320$	1433.05	+	1627.01i	0.250344	+	0.284227i
$321$		0			0
$322$	2265.23	+	2265.23i	0.392038	+	0.392038i
$323$	−6618.59	−	6618.59i	−1.14015	−	1.14015i
$324$		0			0
$325$	5206.18	+	6724.70i	0.888574	+	1.14775i
$326$		−	1254.56i		−	0.213140i
$327$		0			0
$328$	1542.58	−	1542.58i	0.259679	−	0.259679i
$329$	3168.03			0.530879
$330$		0			0
$331$	−199.080			−0.0330586			−0.0165293	−	0.999863i	$-0.505262\pi$
$331$	−199.080			−0.0330586			−0.0165293	+	0.999863i	$0.505262\pi$
$332$	2620.14	−	2620.14i	0.433129	−	0.433129i
$333$		0			0
$334$		−	3550.21i		−	0.581614i
$335$	−2373.97	−	150.469i	−0.387176	−	0.0245403i
$336$		0			0
$337$	3308.77	+	3308.77i	0.534838	+	0.534838i	0.922008	−	0.387170i	$-0.126547\pi$
$337$	3308.77	+	3308.77i	0.534838	+	0.534838i	−0.387170	+	0.922008i	$0.626547\pi$
$338$	6374.88	+	6374.88i	1.02588	+	1.02588i
$339$		0			0
$340$	−5469.20	−	346.653i	−0.872379	−	0.0552939i
$341$			8854.16i			1.40610i
$342$		0			0
$343$	−4888.20	+	4888.20i	−0.769498	+	0.769498i
$344$	−2084.71			−0.326744
$345$		0			0
$346$	2314.82			0.359669
$347$	−2371.68	+	2371.68i	−0.366912	+	0.366912i	−0.866350	−	0.499438i	$-0.833540\pi$
$347$	−2371.68	+	2371.68i	−0.366912	+	0.366912i	0.499438	+	0.866350i	$0.333540\pi$
$348$		0			0
$349$			8208.92i			1.25906i	0.776975	+	0.629532i	$0.216753\pi$
$349$			8208.92i			1.25906i	−0.776975	+	0.629532i	$0.783247\pi$
$350$	899.024	−	7063.52i	0.137300	−	1.07875i
$351$		0			0
$352$	5851.13	+	5851.13i	0.885984	+	0.885984i
$353$	−6800.98	−	6800.98i	−1.02544	−	1.02544i	−0.999668	−	0.0257707i	$-0.991796\pi$
$353$	−6800.98	−	6800.98i	−1.02544	−	1.02544i	−0.0257707	−	0.999668i	$-0.508204\pi$
$354$		0			0
$355$	3473.51	+	3943.63i	0.519309	+	0.589594i
$356$			7839.01i			1.16704i
$357$		0			0
$358$	12384.2	−	12384.2i	1.82829	−	1.82829i
$359$	1778.14			0.261411			0.130705	−	0.991421i	$-0.458276\pi$
$359$	1778.14			0.261411			0.130705	+	0.991421i	$0.458276\pi$
$360$		0			0
$361$	−5170.20			−0.753783
$362$	4058.86	−	4058.86i	0.589306	−	0.589306i
$363$		0			0
$364$		−	6004.34i		−	0.864596i
$365$	−480.844	+	7586.34i	−0.0689549	+	1.08791i
$366$		0			0
$367$	−4673.56	−	4673.56i	−0.664735	−	0.664735i	0.291757	−	0.956492i	$-0.405760\pi$
$367$	−4673.56	−	4673.56i	−0.664735	−	0.664735i	−0.956492	+	0.291757i	$0.905760\pi$
$368$	3060.40	+	3060.40i	0.433517	+	0.433517i
$369$		0			0
$370$	−835.936	+	736.284i	−0.117455	+	0.103453i
$371$			3982.85i			0.557357i
$372$		0			0
$373$	−5729.37	+	5729.37i	−0.795323	+	0.795323i	−0.982354	−	0.187031i	$-0.940114\pi$
$373$	−5729.37	+	5729.37i	−0.795323	+	0.795323i	0.187031	+	0.982354i	$0.440114\pi$
$374$	−11987.8			−1.65742
$375$		0			0
$376$	1724.73			0.236559
$377$	−4346.86	+	4346.86i	−0.593832	+	0.593832i
$378$		0			0
$379$			153.883i			0.0208561i	0.999946	+	0.0104280i	$0.00331941\pi$
$379$			153.883i			0.0208561i	−0.999946	+	0.0104280i	$0.996681\pi$
$380$	−5285.11	+	4655.08i	−0.713475	+	0.628422i
$381$		0			0
$382$	−2859.19	−	2859.19i	−0.382956	−	0.382956i
$383$	4297.76	+	4297.76i	0.573382	+	0.573382i	0.933072	−	0.359690i	$-0.117118\pi$
$383$	4297.76	+	4297.76i	0.573382	+	0.573382i	−0.359690	+	0.933072i	$0.617118\pi$
$384$		0			0
$385$	411.754	−	6496.30i	0.0545063	−	0.859954i
$386$		−	14814.6i		−	1.95347i
$387$		0			0
$388$	−4510.42	+	4510.42i	−0.590160	+	0.590160i
$389$	7833.00			1.02095			0.510474	−	0.859893i	$-0.329470\pi$
$389$	7833.00			1.02095			0.510474	+	0.859893i	$0.329470\pi$
$390$		0			0
$391$	−4799.42			−0.620760
$392$	−632.310	+	632.310i	−0.0814706	+	0.0814706i
$393$		0			0
$394$		−	6102.57i		−	0.780312i
$395$	−9772.54	−	11095.2i	−1.24484	−	1.41332i
$396$		0			0
$397$	−10195.4	−	10195.4i	−1.28889	−	1.28889i	−0.935459	−	0.353435i	$-0.885013\pi$
$397$	−10195.4	−	10195.4i	−1.28889	−	1.28889i	−0.353435	−	0.935459i	$-0.614987\pi$
$398$	1363.80	+	1363.80i	0.171762	+	0.171762i
$399$		0			0
$400$	1214.61	−	9543.04i	0.151826	−	1.19288i
$401$		−	10778.3i		−	1.34226i	−0.741341	−	0.671128i	$-0.765810\pi$
$401$		−	10778.3i		−	1.34226i	0.741341	−	0.671128i	$-0.234190\pi$
$402$		0			0
$403$	11241.9	−	11241.9i	1.38957	−	1.38957i
$404$	−640.714			−0.0789028
$405$		0			0
$406$	5147.00			0.629165
$407$	−720.079	+	720.079i	−0.0876978	+	0.0876978i
$408$		0			0
$409$			4629.92i			0.559743i	0.960037	+	0.279872i	$0.0902919\pi$
$409$			4629.92i			0.559743i	−0.960037	+	0.279872i	$0.909708\pi$
$410$	10787.3	+	683.731i	1.29938	+	0.0823587i
$411$		0			0
$412$	−79.7315	−	79.7315i	−0.00953420	−	0.00953420i
$413$	−3037.51	−	3037.51i	−0.361903	−	0.361903i
$414$		0			0
$415$	−7198.58	−	456.267i	−0.851480	−	0.0539693i
$416$		−	14858.0i		−	1.75114i
$417$		0			0
$418$	−10893.9	+	10893.9i	−1.27473	+	1.27473i
$419$	11276.5			1.31478			0.657389	−	0.753551i	$-0.271661\pi$
$419$	11276.5			1.31478			0.657389	+	0.753551i	$0.271661\pi$
$420$		0			0
$421$	3387.58			0.392163			0.196082	−	0.980588i	$-0.437178\pi$
$421$	3387.58			0.392163			0.196082	+	0.980588i	$0.437178\pi$
$422$	−14481.6	+	14481.6i	−1.67051	+	1.67051i
$423$		0			0
$424$			2168.33i			0.248358i
$425$	6530.47	+	8435.26i	0.745351	+	0.962753i
$426$		0			0
$427$	4240.32	+	4240.32i	0.480570	+	0.480570i
$428$	−4457.91	−	4457.91i	−0.503461	−	0.503461i
$429$		0			0
$430$	−6827.18	−	7751.20i	−0.765665	−	0.869293i
$431$			16161.7i			1.80623i	0.429402	+	0.903113i	$0.358724\pi$
$431$			16161.7i			1.80623i	−0.429402	+	0.903113i	$0.641276\pi$
$432$		0			0
$433$	−3808.79	+	3808.79i	−0.422722	+	0.422722i	−0.886140	−	0.463418i	$-0.846623\pi$
$433$	−3808.79	+	3808.79i	−0.422722	+	0.422722i	0.463418	+	0.886140i	$0.346623\pi$
$434$	−13311.2			−1.47226
$435$		0			0
$436$	2358.16			0.259026
$437$	−4361.44	+	4361.44i	−0.477428	+	0.477428i
$438$		0			0
$439$			1607.30i			0.174744i	0.996176	+	0.0873718i	$0.0278468\pi$
$439$			1607.30i			0.174744i	−0.996176	+	0.0873718i	$0.972153\pi$
$440$	224.166	−	3536.70i	0.0242880	−	0.383194i
$441$		0			0
$442$	15220.6	+	15220.6i	1.63795	+	1.63795i
$443$	8296.60	+	8296.60i	0.889805	+	0.889805i	0.994504	−	0.104699i	$-0.0333880\pi$
$443$	8296.60	+	8296.60i	0.889805	+	0.889805i	−0.104699	+	0.994504i	$0.533388\pi$
$444$		0			0
$445$	11451.0	−	10085.9i	1.21984	−	1.07442i
$446$			1694.47i			0.179900i
$447$		0			0
$448$	−2107.02	+	2107.02i	−0.222203	+	0.222203i
$449$	15061.8			1.58309			0.791547	−	0.611108i	$-0.209276\pi$
$449$	15061.8			1.58309			0.791547	+	0.611108i	$0.209276\pi$
$450$		0			0
$451$	9881.20			1.03168
$452$	2994.25	−	2994.25i	0.311588	−	0.311588i
$453$		0			0
$454$			14364.0i			1.48489i
$455$	−8770.97	+	7725.38i	−0.903713	+	0.795981i
$456$		0			0
$457$	2671.27	+	2671.27i	0.273428	+	0.273428i	0.830479	−	0.557050i	$-0.188067\pi$
$457$	2671.27	+	2671.27i	0.273428	+	0.273428i	−0.557050	+	0.830479i	$0.688067\pi$
$458$	−603.797	−	603.797i	−0.0616017	−	0.0616017i
$459$		0			0
$460$	−228.434	+	3604.03i	−0.0231539	+	0.365301i
$461$		−	12809.1i		−	1.29410i	−0.762449	−	0.647049i	$-0.776003\pi$
$461$		−	12809.1i		−	1.29410i	0.762449	−	0.647049i	$-0.223997\pi$
$462$		0			0
$463$	−11154.1	+	11154.1i	−1.11960	+	1.11960i	−0.127804	+	0.991799i	$0.540793\pi$
$463$	−11154.1	+	11154.1i	−1.11960	+	1.11960i	−0.991799	+	0.127804i	$0.959207\pi$
$464$	6953.76			0.695733
$465$		0			0
$466$	−1835.80			−0.182493
$467$	7905.44	−	7905.44i	0.783341	−	0.783341i	−0.197052	−	0.980393i	$-0.563137\pi$
$467$	7905.44	−	7905.44i	0.783341	−	0.783341i	0.980393	+	0.197052i	$0.0631368\pi$
$468$		0			0
$469$		−	3269.21i		−	0.321872i
$470$	5648.30	+	6412.77i	0.554334	+	0.629360i
$471$		0			0
$472$	−1653.67	−	1653.67i	−0.161263	−	0.161263i
$473$	−6676.92	−	6676.92i	−0.649060	−	0.649060i
$474$		0			0
$475$	13600.0	+	1730.97i	1.31371	+	0.167205i
$476$		−	7531.66i		−	0.725238i
$477$		0			0
$478$	2655.99	−	2655.99i	0.254147	−	0.254147i
$479$	−10715.8			−1.02217			−0.511085	−	0.859530i	$-0.670756\pi$
$479$	−10715.8			−1.02217			−0.511085	+	0.859530i	$0.670756\pi$
$480$		0			0
$481$	1828.53			0.173334
$482$	−6774.48	+	6774.48i	−0.640185	+	0.640185i
$483$		0			0
$484$			601.277i			0.0564685i
$485$	12391.9	+	785.437i	1.16018	+	0.0735358i
$486$		0			0
$487$	−4489.84	−	4489.84i	−0.417770	−	0.417770i	0.466664	−	0.884435i	$-0.345456\pi$
$487$	−4489.84	−	4489.84i	−0.417770	−	0.417770i	−0.884435	+	0.466664i	$0.845456\pi$
$488$	2308.50	+	2308.50i	0.214141	+	0.214141i
$489$		0			0
$490$	−4421.76	−	280.264i	−0.407662	−	0.0258388i
$491$		−	15174.1i		−	1.39470i	−0.716729	−	0.697352i	$-0.754362\pi$
$491$		−	15174.1i		−	1.39470i	0.716729	−	0.697352i	$-0.245638\pi$
$492$		0			0
$493$	−5452.56	+	5452.56i	−0.498116	+	0.498116i
$494$	27663.3			2.51950
$495$		0			0
$496$	−17983.9			−1.62802
$497$	−5107.09	+	5107.09i	−0.460934	+	0.460934i
$498$		0			0
$499$		−	8887.20i		−	0.797286i	−0.917106	−	0.398643i	$-0.869481\pi$
$499$		−	8887.20i		−	0.797286i	0.917106	−	0.398643i	$-0.130519\pi$
$500$	6645.11	−	4502.43i	0.594357	−	0.402710i
$501$		0			0
$502$	4478.25	+	4478.25i	0.398155	+	0.398155i
$503$	−12740.5	−	12740.5i	−1.12936	−	1.12936i	−0.990281	−	0.139084i	$-0.955584\pi$
$503$	−12740.5	−	12740.5i	−1.12936	−	1.12936i	−0.139084	−	0.990281i	$-0.544416\pi$
$504$		0			0
$505$	824.364	+	935.937i	0.0726410	+	0.0824726i
$506$			7899.61i			0.694033i
$507$		0			0
$508$	4517.58	−	4517.58i	0.394558	−	0.394558i
$509$	−9257.00			−0.806108			−0.403054	−	0.915176i	$-0.632051\pi$
$509$	−9257.00			−0.806108			−0.403054	+	0.915176i	$0.632051\pi$
$510$		0			0
$511$	−10447.2			−0.904416
$512$	−8242.49	+	8242.49i	−0.711465	+	0.711465i
$513$		0			0
$514$		−	13943.1i		−	1.19651i
$515$	−13.8843	+	219.055i	−0.00118799	+	0.0187431i
$516$		0			0
$517$	5523.99	+	5523.99i	0.469913	+	0.469913i
$518$	−1082.56	−	1082.56i	−0.0918240	−	0.0918240i
$519$		0			0
$520$	−4775.07	+	4205.83i	−0.402693	+	0.354688i
$521$			13563.3i			1.14054i	0.821458	+	0.570268i	$0.193161\pi$
$521$			13563.3i			1.14054i	−0.821458	+	0.570268i	$0.806839\pi$
$522$		0			0
$523$	11193.8	−	11193.8i	0.935894	−	0.935894i	−0.0621714	−	0.998065i	$-0.519803\pi$
$523$	11193.8	−	11193.8i	0.935894	−	0.935894i	0.998065	+	0.0621714i	$0.0198026\pi$
$524$	2575.18			0.214689
$525$		0			0
$526$	−12095.3			−1.00262
$527$	14101.5	−	14101.5i	1.16560	−	1.16560i
$528$		0			0
$529$		−	9004.33i		−	0.740062i
$530$	−8062.14	+	7101.05i	−0.660749	+	0.581981i
$531$		0			0
$532$	−6844.34	−	6844.34i	−0.557782	−	0.557782i
$533$	−12545.9	−	12545.9i	−1.01956	−	1.01956i
$534$		0			0
$535$	−776.293	+	12247.7i	−0.0627329	+	0.989744i
$536$		−	1779.81i		−	0.143426i
$537$		0			0
$538$	−1173.30	+	1173.30i	−0.0940231	+	0.0940231i
$539$	−4050.34			−0.323675
$540$		0			0
$541$	1383.33			0.109934			0.0549668	−	0.998488i	$-0.482495\pi$
$541$	1383.33			0.109934			0.0549668	+	0.998488i	$0.482495\pi$
$542$	−4123.86	+	4123.86i	−0.326817	+	0.326817i
$543$		0			0
$544$		−	18637.5i		−	1.46889i
$545$	−3034.09	−	3444.73i	−0.238470	−	0.270745i
$546$		0			0
$547$	2181.76	+	2181.76i	0.170540	+	0.170540i	0.787217	−	0.616677i	$-0.211521\pi$
$547$	2181.76	+	2181.76i	0.170540	+	0.170540i	−0.616677	+	0.787217i	$0.711521\pi$
$548$	5457.01	+	5457.01i	0.425387	+	0.425387i
$549$		0			0
$550$	13884.0	−	10748.8i	1.07639	−	0.833329i
$551$			9909.96i			0.766204i
$552$		0			0
$553$	14368.5	−	14368.5i	1.10491	−	1.10491i
$554$	−30496.9			−2.33879
$555$		0			0
$556$	647.494			0.0493882
$557$	−15674.9	+	15674.9i	−1.19240	+	1.19240i	−0.216007	+	0.976392i	$0.569303\pi$
$557$	−15674.9	+	15674.9i	−1.19240	+	1.19240i	−0.976392	+	0.216007i	$0.930697\pi$
$558$		0			0
$559$			16955.0i			1.28286i
$560$	13194.8	+	836.323i	0.995681	+	0.0631091i
$561$		0			0
$562$	−367.080	−	367.080i	−0.0275522	−	0.0275522i
$563$	8213.60	+	8213.60i	0.614852	+	0.614852i	0.944206	−	0.329354i	$-0.106831\pi$
$563$	8213.60	+	8213.60i	0.614852	+	0.614852i	−0.329354	+	0.944206i	$0.606831\pi$
$564$		0			0
$565$	−8226.41	−	521.414i	−0.612545	−	0.0388248i
$566$			23235.3i			1.72554i
$567$		0			0
$568$	−2780.39	+	2780.39i	−0.205392	+	0.205392i
$569$	−16902.1			−1.24530			−0.622648	−	0.782502i	$-0.713943\pi$
$569$	−16902.1			−1.24530			−0.622648	+	0.782502i	$0.713943\pi$
$570$		0			0
$571$	6507.12			0.476908			0.238454	−	0.971154i	$-0.423359\pi$
$571$	6507.12			0.476908			0.238454	+	0.971154i	$0.423359\pi$
$572$	10469.6	−	10469.6i	0.765305	−	0.765305i
$573$		0			0
$574$			14855.3i			1.08022i
$575$	5558.57	−	4303.37i	0.403145	−	0.312109i
$576$		0			0
$577$	7884.54	+	7884.54i	0.568870	+	0.568870i	0.931812	−	0.362942i	$-0.118228\pi$
$577$	7884.54	+	7884.54i	0.568870	+	0.568870i	−0.362942	+	0.931812i	$0.618228\pi$
$578$	6213.35	+	6213.35i	0.447130	+	0.447130i
$579$		0			0
$580$	3834.97	+	4354.01i	0.274549	+	0.311708i
$581$		−	9913.20i		−	0.707864i
$582$		0			0
$583$	−6944.77	+	6944.77i	−0.493350	+	0.493350i
$584$	−5687.63			−0.403007
$585$		0			0
$586$	−6524.39			−0.459932
$587$	9548.44	−	9548.44i	0.671390	−	0.671390i	−0.286646	−	0.958037i	$-0.592540\pi$
$587$	9548.44	−	9548.44i	0.671390	−	0.671390i	0.958037	+	0.286646i	$0.0925404\pi$
$588$		0			0
$589$		−	25629.2i		−	1.79293i
$590$	732.969	−	11564.1i	0.0511455	−	0.806930i
$591$		0			0
$592$	−1462.57	−	1462.57i	−0.101539	−	0.101539i
$593$	3491.97	+	3491.97i	0.241818	+	0.241818i	0.817602	−	0.575784i	$-0.195303\pi$
$593$	3491.97	+	3491.97i	0.241818	+	0.241818i	−0.575784	+	0.817602i	$0.695303\pi$
$594$		0			0
$595$	−11002.0	+	9690.48i	−0.758049	+	0.667682i
$596$			8574.75i			0.589321i
$597$		0			0
$598$	10029.9	−	10029.9i	0.685876	−	0.685876i
$599$	11203.0			0.764175			0.382087	−	0.924126i	$-0.375205\pi$
$599$	11203.0			0.764175			0.382087	+	0.924126i	$0.375205\pi$
$600$		0			0
$601$	−13953.5			−0.947048			−0.473524	−	0.880781i	$-0.657018\pi$
$601$	−13953.5			−0.947048			−0.473524	+	0.880781i	$0.657018\pi$
$602$	10038.0	−	10038.0i	0.679598	−	0.679598i
$603$		0			0
$604$			16883.3i			1.13737i
$605$	878.328	−	773.622i	0.0590233	−	0.0519871i
$606$		0			0
$607$	−1292.54	−	1292.54i	−0.0864293	−	0.0864293i	0.662570	−	0.749000i	$-0.269466\pi$
$607$	−1292.54	−	1292.54i	−0.0864293	−	0.0864293i	−0.749000	+	0.662570i	$0.769466\pi$
$608$	−16936.7	−	16936.7i	−1.12972	−	1.12972i
$609$		0			0
$610$	−1023.22	+	16143.4i	−0.0679160	+	1.07152i
$611$		−	14027.3i		−	0.928780i
$612$		0			0
$613$	225.964	−	225.964i	0.0148884	−	0.0148884i	−0.699623	−	0.714512i	$-0.746649\pi$
$613$	225.964	−	225.964i	0.0148884	−	0.0148884i	0.714512	+	0.699623i	$0.246649\pi$
$614$	38417.2			2.52507
$615$		0			0
$616$	4870.41			0.318562
$617$	2717.41	−	2717.41i	0.177308	−	0.177308i	−0.612873	−	0.790181i	$-0.709986\pi$
$617$	2717.41	−	2717.41i	0.177308	−	0.177308i	0.790181	+	0.612873i	$0.209986\pi$
$618$		0			0
$619$			8870.55i			0.575989i	0.957632	+	0.287995i	$0.0929886\pi$
$619$			8870.55i			0.575989i	−0.957632	+	0.287995i	$0.907011\pi$
$620$	−9918.04	−	11260.4i	−0.642448	−	0.729400i
$621$		0			0
$622$	−19632.7	−	19632.7i	−1.26559	−	1.26559i
$623$	14829.3	+	14829.3i	0.953650	+	0.953650i
$624$		0			0
$625$	−15126.8	−	3914.00i	−0.968118	−	0.250496i
$626$		−	12876.5i		−	0.822123i
$627$		0			0
$628$	898.205	−	898.205i	0.0570737	−	0.0570737i
$629$	2293.65			0.145396
$630$		0			0
$631$	29762.7			1.87771			0.938853	−	0.344318i	$-0.111890\pi$
$631$	29762.7			1.87771			0.938853	+	0.344318i	$0.111890\pi$
$632$	7822.48	−	7822.48i	0.492344	−	0.492344i
$633$		0			0
$634$			35254.2i			2.20840i
$635$	−12411.6	−	786.685i	−0.775654	−	0.0491632i
$636$		0			0
$637$	5142.61	+	5142.61i	0.319871	+	0.319871i
$638$	8974.65	+	8974.65i	0.556912	+	0.556912i
$639$		0			0
$640$	11472.3	+	727.145i	0.708564	+	0.0449108i
$641$			5168.75i			0.318492i	0.987239	+	0.159246i	$0.0509063\pi$
$641$			5168.75i			0.318492i	−0.987239	+	0.159246i	$0.949094\pi$
$642$		0			0
$643$	−2078.46	+	2078.46i	−0.127475	+	0.127475i	−0.767966	−	0.640491i	$-0.778731\pi$
$643$	−2078.46	+	2078.46i	−0.127475	+	0.127475i	0.640491	+	0.767966i	$0.278731\pi$
$644$	−4963.13			−0.303687
$645$		0			0
$646$	34700.0			2.11339
$647$	10765.7	−	10765.7i	0.654164	−	0.654164i	−0.299829	−	0.953993i	$-0.596930\pi$
$647$	10765.7	−	10765.7i	0.654164	−	0.654164i	0.953993	+	0.299829i	$0.0969297\pi$
$648$		0			0
$649$		−	10592.8i		−	0.640684i
$650$	−31275.6	−	3980.67i	−1.88728	−	0.240207i
$651$		0			0
$652$	1374.37	+	1374.37i	0.0825530	+	0.0825530i
$653$	12815.1	+	12815.1i	0.767984	+	0.767984i	0.977751	−	0.209767i	$-0.0672707\pi$
$653$	12815.1	+	12815.1i	0.767984	+	0.767984i	−0.209767	+	0.977751i	$0.567271\pi$
$654$		0			0
$655$	−3313.31	−	3761.75i	−0.197651	−	0.224402i
$656$			20069.9i			1.19451i
$657$		0			0
$658$	−8304.68	+	8304.68i	−0.492022	+	0.492022i
$659$	−16634.8			−0.983307			−0.491654	−	0.870791i	$-0.663607\pi$
$659$	−16634.8			−0.983307			−0.491654	+	0.870791i	$0.663607\pi$
$660$		0			0
$661$	−10835.6			−0.637601			−0.318800	−	0.947822i	$-0.603280\pi$
$661$	−10835.6			−0.637601			−0.318800	+	0.947822i	$0.603280\pi$
$662$	521.868	−	521.868i	0.0306389	−	0.0306389i
$663$		0			0
$664$		−	5396.92i		−	0.315423i
$665$	−1191.86	+	18804.2i	−0.0695015	+	1.09653i
$666$		0			0
$667$	3593.07	+	3593.07i	0.208582	+	0.208582i
$668$	3889.27	+	3889.27i	0.225270	+	0.225270i
$669$		0			0
$670$	6617.58	−	5828.70i	0.381581	−	0.336093i
$671$			14787.4i			0.850762i
$672$		0			0
$673$	−16220.3	+	16220.3i	−0.929045	+	0.929045i	−0.997644	−	0.0685992i	$-0.978147\pi$
$673$	−16220.3	+	16220.3i	−0.929045	+	0.929045i	0.0685992	+	0.997644i	$0.478147\pi$
$674$	−17347.3			−0.991382
$675$		0			0
$676$	−13967.4			−0.794685
$677$	−11196.6	+	11196.6i	−0.635626	+	0.635626i	−0.949473	−	0.313847i	$-0.898382\pi$
$677$	−11196.6	+	11196.6i	−0.635626	+	0.635626i	0.313847	+	0.949473i	$0.398382\pi$
$678$		0			0
$679$			17065.0i			0.964500i
$680$	−5989.70	+	5275.66i	−0.337786	+	0.297518i
$681$		0			0
$682$	−23210.3	−	23210.3i	−1.30318	−	1.30318i
$683$	15428.1	+	15428.1i	0.864331	+	0.864331i	0.991838	−	0.127507i	$-0.0406975\pi$
$683$	15428.1	+	15428.1i	0.864331	+	0.864331i	−0.127507	+	0.991838i	$0.540697\pi$
$684$		0			0
$685$	950.276	−	14992.6i	0.0530046	−	0.836261i
$686$		−	25627.9i		−	1.42635i
$687$		0			0
$688$	13561.6	−	13561.6i	0.751501	−	0.751501i
$689$	17635.2			0.975104
$690$		0			0
$691$	21858.6			1.20339			0.601693	−	0.798728i	$-0.294493\pi$
$691$	21858.6			1.20339			0.601693	+	0.798728i	$0.294493\pi$
$692$	−2535.89	+	2535.89i	−0.139307	+	0.139307i
$693$		0			0
$694$		−	12434.2i		−	0.680112i
$695$	−833.087	−	945.841i	−0.0454688	−	0.0516227i
$696$		0			0
$697$	−15737.2	−	15737.2i	−0.855220	−	0.855220i
$698$	−21518.9	−	21518.9i	−1.16691	−	1.16691i
$699$		0			0
$700$	6753.21	+	8722.98i	0.364639	+	0.470996i
$701$		−	8846.82i		−	0.476662i	−0.971184	−	0.238331i	$-0.923400\pi$
$701$		−	8846.82i		−	0.476662i	0.971184	−	0.238331i	$-0.0766002\pi$
$702$		0			0
$703$	2084.34	−	2084.34i	0.111824	−	0.111824i
$704$	−7347.86			−0.393371
$705$		0			0
$706$	35656.2			1.90076
$707$	−1212.06	+	1212.06i	−0.0644755	+	0.0644755i
$708$		0			0
$709$		−	23189.8i		−	1.22837i	−0.789163	−	0.614184i	$-0.789485\pi$
$709$		−	23189.8i		−	1.22837i	0.789163	−	0.614184i	$-0.210515\pi$
$710$	−19443.3	−	1232.37i	−1.02774	−	0.0651410i
$711$		0			0
$712$	8073.34	+	8073.34i	0.424945	+	0.424945i
$713$	−9292.43	−	9292.43i	−0.488084	−	0.488084i
$714$		0			0
$715$	−28764.1	−	1823.15i	−1.50450	−	0.0953596i
$716$			27133.9i			1.41626i
$717$		0			0
$718$	−4661.21	+	4661.21i	−0.242277	+	0.242277i
$719$	−7005.29			−0.363356			−0.181678	−	0.983358i	$-0.558153\pi$
$719$	−7005.29			−0.363356			−0.181678	+	0.983358i	$0.558153\pi$
$720$		0			0
$721$	−301.661			−0.0155818
$722$	13553.2	−	13553.2i	0.698611	−	0.698611i
$723$		0			0
$724$			8892.98i			0.456498i
$725$	1426.02	−	11204.0i	0.0730495	−	0.573941i
$726$		0			0
$727$	−16025.7	−	16025.7i	−0.817553	−	0.817553i	0.168200	−	0.985753i	$-0.446205\pi$
$727$	−16025.7	−	16025.7i	−0.817553	−	0.817553i	−0.985753	+	0.168200i	$0.946205\pi$
$728$	−6183.82	−	6183.82i	−0.314818	−	0.314818i
$729$		0			0
$730$	−18626.4	−	21147.3i	−0.944374	−	1.07219i
$731$			21267.8i			1.07609i
$732$		0			0
$733$	−3319.43	+	3319.43i	−0.167266	+	0.167266i	−0.785777	−	0.618511i	$-0.787736\pi$
$733$	−3319.43	+	3319.43i	−0.167266	+	0.167266i	0.618511	+	0.785777i	$0.287736\pi$
$734$	24502.5			1.23216
$735$		0			0
$736$	−12281.5			−0.615084
$737$	5700.41	−	5700.41i	0.284908	−	0.284908i
$738$		0			0
$739$			19979.4i			0.994526i	0.867600	+	0.497263i	$0.165662\pi$
$739$			19979.4i			0.994526i	−0.867600	+	0.497263i	$0.834338\pi$
$740$	109.169	−	1722.37i	0.00542314	−	0.0855616i
$741$		0			0
$742$	−10440.7	−	10440.7i	−0.516562	−	0.516562i
$743$	14574.6	+	14574.6i	0.719637	+	0.719637i	0.968531	−	0.248894i	$-0.0800670\pi$
$743$	14574.6	+	14574.6i	0.719637	+	0.719637i	−0.248894	+	0.968531i	$0.580067\pi$
$744$		0			0
$745$	12525.7	−	11032.5i	0.615983	−	0.542552i
$746$		−	30038.0i		−	1.47422i
$747$		0			0
$748$	13132.7	−	13132.7i	0.641951	−	0.641951i
$749$	−16866.3			−0.822807
$750$		0			0
$751$	−11620.1			−0.564612			−0.282306	−	0.959324i	$-0.591099\pi$
$751$	−11620.1			−0.564612			−0.282306	+	0.959324i	$0.591099\pi$
$752$	−11219.9	+	11219.9i	−0.544079	+	0.544079i
$753$		0			0
$754$		−	22789.7i		−	1.10073i
$755$	24662.6	−	21722.6i	1.18883	−	1.04711i
$756$		0			0
$757$	10502.0	+	10502.0i	0.504229	+	0.504229i	0.912749	−	0.408520i	$-0.133955\pi$
$757$	10502.0	+	10502.0i	0.504229	+	0.504229i	−0.408520	+	0.912749i	$0.633955\pi$
$758$	−403.390	−	403.390i	−0.0193296	−	0.0193296i
$759$		0			0
$760$	−648.871	+	10237.3i	−0.0309698	+	0.488614i
$761$		−	36489.6i		−	1.73817i	−0.494664	−	0.869085i	$-0.664709\pi$
$761$		−	36489.6i		−	1.73817i	0.494664	−	0.869085i	$-0.335291\pi$
$762$		0			0
$763$	4461.01	−	4461.01i	0.211664	−	0.211664i
$764$	6264.50			0.296652
$765$		0			0
$766$	−22532.3			−1.06283
$767$	−13449.4	+	13449.4i	−0.633154	+	0.633154i
$768$		0			0
$769$			30323.9i			1.42199i	0.703198	+	0.710994i	$0.251755\pi$
$769$			30323.9i			1.42199i	−0.703198	+	0.710994i	$0.748245\pi$
$770$	15950.0	+	18108.8i	0.746493	+	0.847527i
$771$		0			0
$772$	16229.4	+	16229.4i	0.756617	+	0.756617i
$773$	10327.2	+	10327.2i	0.480522	+	0.480522i	0.905298	−	0.424777i	$-0.139647\pi$
$773$	10327.2	+	10327.2i	0.480522	+	0.480522i	−0.424777	+	0.905298i	$0.639647\pi$
$774$		0			0
$775$	−3687.98	+	28976.0i	−0.170937	+	1.34303i
$776$			9290.49i			0.429780i
$777$		0			0
$778$	−20533.4	+	20533.4i	−0.946220	+	0.946220i
$779$	−28602.1			−1.31550
$780$		0			0
$781$	−17810.1			−0.816000
$782$	12581.2	−	12581.2i	0.575324	−	0.575324i
$783$		0			0
$784$		−	8226.74i		−	0.374760i
$785$	−2467.73	−	156.412i	−0.112200	−	0.00711157i
$786$		0			0
$787$	21521.8	+	21521.8i	0.974802	+	0.974802i	0.999690	−	0.0248882i	$-0.00792299\pi$
$787$	21521.8	+	21521.8i	0.974802	+	0.974802i	−0.0248882	+	0.999690i	$0.507923\pi$
$788$	6685.38	+	6685.38i	0.302229	+	0.302229i
$789$		0			0
$790$	54702.8	+	3467.22i	2.46359	+	0.156150i
$791$		−	11328.6i		−	0.509229i
$792$		0			0
$793$	18775.2	−	18775.2i	0.840764	−	0.840764i
$794$	53452.3			2.38911
$795$		0			0
$796$	−2988.10			−0.133053
$797$	−17277.8	+	17277.8i	−0.767895	+	0.767895i	−0.977736	−	0.209841i	$-0.932706\pi$
$797$	−17277.8	+	17277.8i	−0.767895	+	0.767895i	0.209841	+	0.977736i	$0.432706\pi$
$798$		0			0
$799$		−	17595.4i		−	0.779076i
$800$	16711.2	+	21585.4i	0.738536	+	0.953951i
$801$		0			0
$802$	28254.4	+	28254.4i	1.24401	+	1.24401i
$803$	−18216.4	−	18216.4i	−0.800552	−	0.800552i
$804$		0			0
$805$	6385.72	+	7249.99i	0.279586	+	0.317427i
$806$			58939.0i			2.57573i
$807$		0			0
$808$	−659.866	+	659.866i	−0.0287302	+	0.0287302i
$809$	−41260.2			−1.79312			−0.896558	−	0.442926i	$-0.853941\pi$
$809$	−41260.2			−1.79312			−0.896558	+	0.442926i	$0.853941\pi$
$810$		0			0
$811$	−23357.1			−1.01132			−0.505659	−	0.862733i	$-0.668751\pi$
$811$	−23357.1			−1.01132			−0.505659	+	0.862733i	$0.668751\pi$
$812$	−5638.54	+	5638.54i	−0.243687	+	0.243687i
$813$		0			0
$814$		−	3775.24i		−	0.162558i
$815$	239.331	−	3775.96i	0.0102864	−	0.162290i
$816$		0			0
$817$	19327.0	+	19327.0i	0.827621	+	0.827621i
$818$	−12136.9	−	12136.9i	−0.518773	−	0.518773i
$819$		0			0
$820$	−12566.5	+	11068.5i	−0.535174	+	0.471376i
$821$		−	8296.46i		−	0.352678i	−0.984330	−	0.176339i	$-0.943575\pi$
$821$		−	8296.46i		−	0.352678i	0.984330	−	0.176339i	$-0.0564254\pi$
$822$		0			0
$823$	17980.5	−	17980.5i	0.761555	−	0.761555i	−0.215048	−	0.976603i	$-0.568991\pi$
$823$	17980.5	−	17980.5i	0.761555	−	0.761555i	0.976603	+	0.215048i	$0.0689910\pi$
$824$	−164.230			−0.00694322
$825$		0			0
$826$	15925.1			0.670827
$827$	9128.21	−	9128.21i	0.383820	−	0.383820i	−0.488656	−	0.872476i	$-0.662513\pi$
$827$	9128.21	−	9128.21i	0.383820	−	0.383820i	0.872476	+	0.488656i	$0.162513\pi$
$828$		0			0
$829$			24401.3i			1.02231i	0.859489	+	0.511154i	$0.170782\pi$
$829$			24401.3i			1.02231i	−0.859489	+	0.511154i	$0.829218\pi$
$830$	20066.4	−	17674.3i	0.839176	−	0.739138i
$831$		0			0
$832$	9329.38	+	9329.38i	0.388748	+	0.388748i
$833$	6450.73	+	6450.73i	0.268313	+	0.268313i
$834$		0			0
$835$	677.271	−	10685.4i	0.0280694	−	0.442854i
$836$		−	23868.5i		−	0.987452i
$837$		0			0
$838$	−29560.2	+	29560.2i	−1.21854	+	1.21854i
$839$	38174.8			1.57084			0.785422	−	0.618960i	$-0.212446\pi$
$839$	38174.8			1.57084			0.785422	+	0.618960i	$0.212446\pi$
$840$		0			0
$841$	−16224.9			−0.665256
$842$	−8880.22	+	8880.22i	−0.363459	+	0.363459i
$843$		0			0
$844$		−	31729.3i		−	1.29404i
$845$	17970.9	+	20403.1i	0.731618	+	0.830639i
$846$		0			0
$847$	1137.45	+	1137.45i	0.0461433	+	0.0461433i
$848$	−14105.7	−	14105.7i	−0.571216	−	0.571216i
$849$		0			0
$850$	−39231.2	−	4993.23i	−1.58308	−	0.201490i
$851$		−	1511.44i		−	0.0608833i
$852$		0			0
$853$	1757.73	−	1757.73i	0.0705550	−	0.0705550i	−0.670949	−	0.741504i	$-0.734113\pi$
$853$	1757.73	−	1757.73i	0.0705550	−	0.0705550i	0.741504	+	0.670949i	$0.234113\pi$
$854$	−22231.2			−0.890790
$855$		0			0
$856$	−9182.33			−0.366642
$857$	−7091.77	+	7091.77i	−0.282672	+	0.282672i	−0.834174	−	0.551501i	$-0.814055\pi$
$857$	−7091.77	+	7091.77i	−0.282672	+	0.282672i	0.551501	+	0.834174i	$0.314055\pi$
$858$		0			0
$859$		−	16842.4i		−	0.668982i	−0.942399	−	0.334491i	$-0.891436\pi$
$859$		−	16842.4i		−	0.668982i	0.942399	−	0.334491i	$-0.108564\pi$
$860$	15970.6	+	1012.27i	0.633249	+	0.0401372i
$861$		0			0
$862$	−42366.4	−	42366.4i	−1.67402	−	1.67402i
$863$	−33056.1	−	33056.1i	−1.30387	−	1.30387i	−0.925759	−	0.378114i	$-0.876573\pi$
$863$	−33056.1	−	33056.1i	−1.30387	−	1.30387i	−0.378114	−	0.925759i	$-0.623427\pi$
$864$		0			0
$865$	6967.12	+	441.596i	0.273860	+	0.0173581i
$866$		−	19968.7i		−	0.783562i
$867$		0			0
$868$	14582.5	−	14582.5i	0.570231	−	0.570231i
$869$	50107.9			1.95603
$870$		0			0
$871$	−14475.3			−0.563120
$872$	2428.65	−	2428.65i	0.0943171	−	0.0943171i
$873$		0			0
$874$		−	22866.2i		−	0.884966i
$875$	4053.37	−	21088.2i	0.156604	−	0.814754i
$876$		0			0
$877$	1780.97	+	1780.97i	0.0685738	+	0.0685738i	0.740562	−	0.671988i	$-0.234559\pi$
$877$	1780.97	+	1780.97i	0.0685738	+	0.0685738i	−0.671988	+	0.740562i	$0.734559\pi$
$878$	−4213.39	−	4213.39i	−0.161953	−	0.161953i
$879$		0			0
$880$	21549.0	+	24465.6i	0.825475	+	0.937198i
$881$			2698.82i			0.103207i	0.998668	+	0.0516036i	$0.0164332\pi$
$881$			2698.82i			0.103207i	−0.998668	+	0.0516036i	$0.983567\pi$
$882$		0			0
$883$	6744.66	−	6744.66i	0.257051	−	0.257051i	−0.566803	−	0.823854i	$-0.691820\pi$
$883$	6744.66	−	6744.66i	0.257051	−	0.257051i	0.823854	+	0.566803i	$0.191820\pi$
$884$	−33348.5			−1.26881
$885$		0			0
$886$	−43497.5			−1.64935
$887$	1302.99	−	1302.99i	0.0493238	−	0.0493238i	−0.682015	−	0.731338i	$-0.738896\pi$
$887$	1302.99	−	1302.99i	0.0493238	−	0.0493238i	0.731338	+	0.682015i	$0.238896\pi$
$888$		0			0
$889$		−	17092.1i		−	0.644827i
$890$	−3578.41	+	56457.0i	−0.134774	+	2.12634i
$891$		0			0
$892$	−1856.29	−	1856.29i	−0.0696785	−	0.0696785i
$893$	−15989.7	−	15989.7i	−0.599189	−	0.599189i
$894$		0			0
$895$	39636.4	−	34911.3i	1.48033	−	1.30386i
$896$			15798.5i			0.589052i
$897$		0			0
$898$	−39483.0	+	39483.0i	−1.46722	+	1.46722i
$899$	−21114.0			−0.783306
$900$		0			0
$901$	22121.0			0.817933
$902$	−25902.6	+	25902.6i	−0.956167	+	0.956167i
$903$		0			0
$904$		−	6167.50i		−	0.226912i
$905$	12990.6	−	11442.0i	0.477152	−	0.420271i
$906$		0			0
$907$	−3070.33	−	3070.33i	−0.112402	−	0.112402i	0.648669	−	0.761071i	$-0.275326\pi$
$907$	−3070.33	−	3070.33i	−0.112402	−	0.112402i	−0.761071	+	0.648669i	$0.775326\pi$
$908$	−15735.8	−	15735.8i	−0.575123	−	0.575123i
$909$		0			0
$910$	2740.90	−	43243.6i	0.0998462	−	1.57529i
$911$			19102.6i			0.694727i	0.937731	+	0.347364i	$0.112923\pi$
$911$			19102.6i			0.694727i	−0.937731	+	0.347364i	$0.887077\pi$
$912$		0			0
$913$	17285.3	−	17285.3i	0.626572	−	0.626572i
$914$	−14004.9			−0.506830
$915$		0			0
$916$	1322.92			0.0477189
$917$	4871.55	−	4871.55i	0.175434	−	0.175434i
$918$		0			0
$919$		−	29786.7i		−	1.06917i	−0.845113	−	0.534587i	$-0.820467\pi$
$919$		−	29786.7i		−	1.06917i	0.845113	−	0.534587i	$-0.179533\pi$
$920$	3476.50	+	3947.02i	0.124583	+	0.141445i
$921$		0			0
$922$	33577.8	+	33577.8i	1.19938	+	1.19938i
$923$	22613.0	+	22613.0i	0.806410	+	0.806410i
$924$		0			0
$925$	−2656.45	+	2056.59i	−0.0944254	+	0.0731029i
$926$		−	58478.9i		−	2.07531i
$927$		0			0
$928$	−13952.9	+	13952.9i	−0.493561	+	0.493561i
$929$	7724.87			0.272815			0.136407	−	0.990653i	$-0.456444\pi$
$929$	7724.87			0.272815			0.136407	+	0.990653i	$0.456444\pi$
$930$		0			0
$931$	11724.1			0.412720
$932$	2011.12	−	2011.12i	0.0706829	−	0.0706829i
$933$		0			0
$934$			41446.7i			1.45201i
$935$	−36080.8	−	2286.91i	−1.26200	−	0.0799892i
$936$		0			0
$937$	−20722.5	−	20722.5i	−0.722492	−	0.722492i	0.246620	−	0.969112i	$-0.420680\pi$
$937$	−20722.5	−	20722.5i	−0.722492	−	0.722492i	−0.969112	+	0.246620i	$0.920680\pi$
$938$	8569.92	+	8569.92i	0.298313	+	0.298313i
$939$		0			0
$940$	−13212.9	−	837.474i	−0.458466	−	0.0290589i
$941$			21780.7i			0.754548i	0.926102	+	0.377274i	$0.123138\pi$
$941$			21780.7i			0.754548i	−0.926102	+	0.377274i	$0.876862\pi$
$942$		0			0
$943$	−10370.3	+	10370.3i	−0.358116	+	0.358116i
$944$	21515.3			0.741803
$945$		0			0
$946$	35005.8			1.20310
$947$	−29938.4	+	29938.4i	−1.02731	+	1.02731i	−0.0276975	+	0.999616i	$0.508818\pi$
$947$	−29938.4	+	29938.4i	−1.02731	+	1.02731i	−0.999616	+	0.0276975i	$0.991182\pi$
$948$		0			0
$949$			46257.8i			1.58229i
$950$	−40188.6	+	31113.5i	−1.37252	+	1.06258i
$951$		0			0
$952$	−7756.79	−	7756.79i	−0.264075	−	0.264075i
$953$	29572.1	+	29572.1i	1.00518	+	1.00518i	0.999987	+	0.00519163i	$0.00165255\pi$
$953$	29572.1	+	29572.1i	1.00518	+	1.00518i	0.00519163	+	0.999987i	$0.498347\pi$
$954$		0			0
$955$	−8060.12	−	9151.01i	−0.273109	−	0.310073i
$956$			5819.29i			0.196872i
$957$		0			0
$958$	28090.5	−	28090.5i	0.947353	−	0.947353i
$959$	20646.4			0.695212
$960$		0			0
$961$	24814.3			0.832945
$962$	−4793.32	+	4793.32i	−0.160647	+	0.160647i
$963$		0			0
$964$		−	14842.9i		−	0.495911i
$965$	2826.16	−	44588.7i	0.0942769	−	1.48742i
$966$		0			0
$967$	−9478.37	−	9478.37i	−0.315206	−	0.315206i	0.531717	−	0.846922i	$-0.321547\pi$
$967$	−9478.37	−	9478.37i	−0.315206	−	0.315206i	−0.846922	+	0.531717i	$0.821547\pi$
$968$	619.250	+	619.250i	0.0205614	+	0.0205614i
$969$		0			0
$970$	−34543.2	+	30425.3i	−1.14342	+	1.00711i
$971$		−	8382.79i		−	0.277051i	−0.990359	−	0.138525i	$-0.955764\pi$
$971$		−	8382.79i		−	0.277051i	0.990359	−	0.138525i	$-0.0442363\pi$
$972$		0			0
$973$	1224.89	−	1224.89i	0.0403577	−	0.0403577i
$974$	23539.4			0.774384
$975$		0			0
$976$	−30035.0			−0.985039
$977$	−24277.3	+	24277.3i	−0.794986	+	0.794986i	−0.982300	−	0.187314i	$-0.940022\pi$
$977$	−24277.3	+	24277.3i	−0.794986	+	0.794986i	0.187314	+	0.982300i	$0.440022\pi$
$978$		0			0
$979$			51714.8i			1.68826i
$980$	5151.07	−	4537.01i	0.167903	−	0.147887i
$981$		0			0
$982$	39777.5	+	39777.5i	1.29262	+	1.29262i
$983$	14666.4	+	14666.4i	0.475876	+	0.475876i	0.903810	−	0.427934i	$-0.140758\pi$
$983$	14666.4	+	14666.4i	0.475876	+	0.475876i	−0.427934	+	0.903810i	$0.640758\pi$
$984$		0			0
$985$	1164.18	−	18367.4i	0.0376588	−	0.594147i
$986$		−	28586.7i		−	0.923313i
$987$		0			0
$988$	−30305.2	+	30305.2i	−0.975847	+	0.975847i
$989$	14014.8			0.450603
$990$		0			0
$991$	17192.0			0.551080			0.275540	−	0.961290i	$-0.411143\pi$
$991$	17192.0			0.551080			0.275540	+	0.961290i	$0.411143\pi$
$992$	36085.0	−	36085.0i	1.15494	−	1.15494i
$993$		0			0
$994$		−	26775.5i		−	0.854393i
$995$	3844.58	+	4364.92i	0.122494	+	0.139073i
$996$		0			0
$997$	13617.4	+	13617.4i	0.432566	+	0.432566i	0.889500	−	0.456935i	$-0.151053\pi$
$997$	13617.4	+	13617.4i	0.432566	+	0.432566i	−0.456935	+	0.889500i	$0.651053\pi$
$998$	23296.9	+	23296.9i	0.738929	+	0.738929i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.f.a.17.2	yes	12
3.2	odd	2	inner	45.4.f.a.17.5	yes	12
4.3	odd	2		720.4.w.d.17.2		12
5.2	odd	4		225.4.f.c.143.2		12
5.3	odd	4	inner	45.4.f.a.8.5	yes	12
5.4	even	2		225.4.f.c.107.5		12
12.11	even	2		720.4.w.d.17.5		12
15.2	even	4		225.4.f.c.143.5		12
15.8	even	4	inner	45.4.f.a.8.2	✓	12
15.14	odd	2		225.4.f.c.107.2		12
20.3	even	4		720.4.w.d.593.5		12
60.23	odd	4		720.4.w.d.593.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.f.a.8.2	✓	12	15.8	even	4	inner
45.4.f.a.8.5	yes	12	5.3	odd	4	inner
45.4.f.a.17.2	yes	12	1.1	even	1	trivial
45.4.f.a.17.5	yes	12	3.2	odd	2	inner
225.4.f.c.107.2		12	15.14	odd	2
225.4.f.c.107.5		12	5.4	even	2
225.4.f.c.143.2		12	5.2	odd	4
225.4.f.c.143.5		12	15.2	even	4
720.4.w.d.17.2		12	4.3	odd	2
720.4.w.d.17.5		12	12.11	even	2
720.4.w.d.593.2		12	60.23	odd	4
720.4.w.d.593.5		12	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.62140 + 2.62140i) q^{2} -5.74350i q^{4} +(-8.38994 + 7.38978i) q^{5} +(-10.8652 - 10.8652i) q^{7} +(-5.91519 - 5.91519i) q^{8} +O(q^{10})\)	\(q+(-2.62140 + 2.62140i) q^{2} -5.74350i q^{4} +(-8.38994 + 7.38978i) q^{5} +(-10.8652 - 10.8652i) q^{7} +(-5.91519 - 5.91519i) q^{8} +(2.62183 - 41.3650i) q^{10} -37.8905i q^{11} +(-48.1085 + 48.1085i) q^{13} +56.9640 q^{14} +76.9602 q^{16} +(-60.3458 + 60.3458i) q^{17} +109.678i q^{19} +(42.4432 + 48.1877i) q^{20} +(99.3262 + 99.3262i) q^{22} +(39.7660 + 39.7660i) q^{23} +(15.7823 - 124.000i) q^{25} -252.224i q^{26} +(-62.4042 + 62.4042i) q^{28} +90.3553 q^{29} -233.678 q^{31} +(-154.422 + 154.422i) q^{32} -316.381i q^{34} +(171.449 + 10.8670i) q^{35} +(-19.0042 - 19.0042i) q^{37} +(-287.509 - 287.509i) q^{38} +(93.3400 + 5.91616i) q^{40} +260.783i q^{41} +(176.216 - 176.216i) q^{43} -217.624 q^{44} -208.485 q^{46} +(-145.788 + 145.788i) q^{47} -106.896i q^{49} +(283.681 + 366.425i) q^{50} +(276.311 + 276.311i) q^{52} +(-183.285 - 183.285i) q^{53} +(280.002 + 317.899i) q^{55} +128.539i q^{56} +(-236.858 + 236.858i) q^{58} +279.564 q^{59} -390.267 q^{61} +(612.563 - 612.563i) q^{62} -193.924i q^{64} +(48.1164 - 759.139i) q^{65} +(150.444 + 150.444i) q^{67} +(346.596 + 346.596i) q^{68} +(-477.925 + 420.951i) q^{70} -470.042i q^{71} +(480.765 - 480.765i) q^{73} +99.6355 q^{74} +629.934 q^{76} +(-411.687 + 411.687i) q^{77} +1322.44i q^{79} +(-645.692 + 568.719i) q^{80} +(-683.618 - 683.618i) q^{82} +(456.192 + 456.192i) q^{83} +(60.3558 - 952.240i) q^{85} +923.868i q^{86} +(-224.129 + 224.129i) q^{88} -1364.85 q^{89} +1045.41 q^{91} +(228.396 - 228.396i) q^{92} -764.340i q^{94} +(-810.494 - 920.190i) q^{95} +(-785.308 - 785.308i) q^{97} +(280.218 + 280.218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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