LMFDB - Embedded newform 864.4.p.b.143.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,4,Mod(143,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("864.143"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$864 = 2^{5} \cdot 3^{3}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	864.p (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$32$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			143.18
Character	$\chi$	$=$	864.143
Dual form			864.4.p.b.719.18

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.46312 + 2.53420i) q^{5} +(4.48829 + 2.59132i) q^{7} +(-13.6440 - 7.87738i) q^{11} +(-37.3474 + 21.5625i) q^{13} -3.31600i q^{17} +15.0818 q^{19} +(-61.7219 - 106.905i) q^{23} +(58.2185 - 100.837i) q^{25} +(-11.7175 + 20.2953i) q^{29} +(274.752 - 158.628i) q^{31} +15.1657i q^{35} +286.660i q^{37} +(-284.954 + 164.518i) q^{41} +(175.008 - 303.123i) q^{43} +(165.936 - 287.410i) q^{47} +(-158.070 - 273.786i) q^{49} -521.214 q^{53} -46.1023i q^{55} +(-130.749 + 75.4878i) q^{59} +(650.051 + 375.307i) q^{61} +(-109.288 - 63.0972i) q^{65} +(-126.364 - 218.868i) q^{67} +1083.28 q^{71} +678.405 q^{73} +(-40.8256 - 70.7119i) q^{77} +(-97.5980 - 56.3482i) q^{79} +(-857.418 - 495.030i) q^{83} +(8.40342 - 4.85171i) q^{85} -566.656i q^{89} -223.501 q^{91} +(22.0666 + 38.2204i) q^{95} +(73.8806 - 127.965i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$64 q + 48 q^{11} + 220 q^{19} - 902 q^{25} - 1620 q^{41} + 292 q^{43} + 1762 q^{49} + 5592 q^{59} + 6 q^{65} - 68 q^{67} - 868 q^{73} + 3654 q^{83} + 1380 q^{91} - 1912 q^{97}+O(q^{100})$ 64 * q + 48 * q^11 + 220 * q^19 - 902 * q^25 - 1620 * q^41 + 292 * q^43 + 1762 * q^49 + 5592 * q^59 + 6 * q^65 - 68 * q^67 - 868 * q^73 + 3654 * q^83 + 1380 * q^91 - 1912 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$ .

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.46312	+	2.53420i	0.130866	+	0.226666i	0.924011	−	0.382367i	$-0.124891\pi$
$5$	1.46312	+	2.53420i	0.130866	+	0.226666i	−0.793145	+	0.609033i	$0.791558\pi$
$6$		0			0
$7$	4.48829	+	2.59132i	0.242345	+	0.139918i	0.616254	−	0.787547i	$-0.288649\pi$
$7$	4.48829	+	2.59132i	0.242345	+	0.139918i	−0.373909	+	0.927465i	$0.621983\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.6440	−	7.87738i	−0.373984	−	0.215920i	0.301213	−	0.953557i	$-0.402608\pi$
$11$	−13.6440	−	7.87738i	−0.373984	−	0.215920i	−0.675198	+	0.737637i	$0.735942\pi$
$12$		0			0
$13$	−37.3474	+	21.5625i	−0.796793	+	0.460028i	−0.842348	−	0.538933i	$-0.818827\pi$
$13$	−37.3474	+	21.5625i	−0.796793	+	0.460028i	0.0455558	+	0.998962i	$0.485494\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.31600i		−	0.0473087i	−0.999720	−	0.0236544i	$-0.992470\pi$
$17$		−	3.31600i		−	0.0473087i	0.999720	−	0.0236544i	$-0.00753012\pi$
$18$		0			0
$19$	15.0818			0.182106			0.0910528	−	0.995846i	$-0.470977\pi$
$19$	15.0818			0.182106			0.0910528	+	0.995846i	$0.470977\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−61.7219	−	106.905i	−0.559561	−	0.969188i	−0.997533	−	0.0701991i	$-0.977637\pi$
$23$	−61.7219	−	106.905i	−0.559561	−	0.969188i	0.437972	−	0.898988i	$-0.355697\pi$
$24$		0			0
$25$	58.2185	−	100.837i	0.465748	−	0.806700i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−11.7175	+	20.2953i	−0.0750306	+	0.129957i	−0.901100	−	0.433612i	$-0.857239\pi$
$29$	−11.7175	+	20.2953i	−0.0750306	+	0.129957i	0.826069	+	0.563569i	$0.190572\pi$
$30$		0			0
$31$	274.752	−	158.628i	1.59184	−	0.919048i	0.598846	−	0.800865i	$-0.295626\pi$
$31$	274.752	−	158.628i	1.59184	−	0.919048i	0.992992	−	0.118183i	$-0.0377070\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			15.1657i			0.0732418i
$36$		0			0
$37$			286.660i			1.27369i	0.770991	+	0.636846i	$0.219761\pi$
$37$			286.660i			1.27369i	−0.770991	+	0.636846i	$0.780239\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−284.954	+	164.518i	−1.08542	+	0.626669i	−0.932354	−	0.361547i	$-0.882249\pi$
$41$	−284.954	+	164.518i	−1.08542	+	0.626669i	−0.153068	+	0.988216i	$0.548915\pi$
$42$		0			0
$43$	175.008	−	303.123i	0.620663	−	1.07502i	−0.368700	−	0.929548i	$-0.620197\pi$
$43$	175.008	−	303.123i	0.620663	−	1.07502i	0.989363	−	0.145471i	$-0.0464697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	165.936	−	287.410i	0.514984	−	0.891979i	−0.484864	−	0.874589i	$-0.661131\pi$
$47$	165.936	−	287.410i	0.514984	−	0.891979i	0.999849	−	0.0173897i	$-0.00553559\pi$
$48$		0			0
$49$	−158.070	−	273.786i	−0.460846	−	0.798209i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−521.214			−1.35083			−0.675417	−	0.737436i	$-0.736036\pi$
$53$	−521.214			−1.35083			−0.675417	+	0.737436i	$0.736036\pi$
$54$		0			0
$55$		−	46.1023i		−	0.113026i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−130.749	+	75.4878i	−0.288509	+	0.166571i	−0.637269	−	0.770641i	$-0.719936\pi$
$59$	−130.749	+	75.4878i	−0.288509	+	0.166571i	0.348760	+	0.937212i	$0.386603\pi$
$60$		0			0
$61$	650.051	+	375.307i	1.36443	+	0.787756i	0.990210	−	0.139582i	$-0.0445760\pi$
$61$	650.051	+	375.307i	1.36443	+	0.787756i	0.374223	+	0.927339i	$0.377909\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−109.288	−	63.0972i	−0.208546	−	0.120404i
$66$		0			0
$67$	−126.364	−	218.868i	−0.230415	−	0.399090i	0.727515	−	0.686091i	$-0.240675\pi$
$67$	−126.364	−	218.868i	−0.230415	−	0.399090i	−0.957930	+	0.287001i	$0.907342\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1083.28			1.81072			0.905362	−	0.424640i	$-0.139599\pi$
$71$	1083.28			1.81072			0.905362	+	0.424640i	$0.139599\pi$
$72$		0			0
$73$	678.405			1.08769			0.543845	−	0.839186i	$-0.316968\pi$
$73$	678.405			1.08769			0.543845	+	0.839186i	$0.316968\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−40.8256	−	70.7119i	−0.0604221	−	0.104654i
$78$		0			0
$79$	−97.5980	−	56.3482i	−0.138995	−	0.0802490i	0.428890	−	0.903357i	$-0.358905\pi$
$79$	−97.5980	−	56.3482i	−0.138995	−	0.0802490i	−0.567885	+	0.823108i	$0.692238\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−857.418	−	495.030i	−1.13390	−	0.654658i	−0.188988	−	0.981979i	$-0.560521\pi$
$83$	−857.418	−	495.030i	−1.13390	−	0.654658i	−0.944913	+	0.327321i	$0.893854\pi$
$84$		0			0
$85$	8.40342	−	4.85171i	0.0107233	−	0.00619109i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	566.656i		−	0.674893i	−0.941345	−	0.337446i	$-0.890437\pi$
$89$		−	566.656i		−	0.674893i	0.941345	−	0.337446i	$-0.109563\pi$
$90$		0			0
$91$	−223.501			−0.257465
$92$		0			0
$93$		0			0
$94$		0			0
$95$	22.0666	+	38.2204i	0.0238314	+	0.0412771i
$96$		0			0
$97$	73.8806	−	127.965i	0.0773344	−	0.133947i	−0.824765	−	0.565476i	$-0.808692\pi$
$97$	73.8806	−	127.965i	0.0773344	−	0.133947i	0.902099	+	0.431529i	$0.142026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	733.536	−	1270.52i	0.722669	−	1.25170i	−0.237258	−	0.971447i	$-0.576249\pi$
$101$	733.536	−	1270.52i	0.722669	−	1.25170i	0.959926	−	0.280252i	$-0.0904181\pi$
$102$		0			0
$103$	1005.07	−	580.278i	0.961482	−	0.555112i	0.0648531	−	0.997895i	$-0.479342\pi$
$103$	1005.07	−	580.278i	0.961482	−	0.555112i	0.896629	+	0.442783i	$0.146009\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1244.09i		−	1.12402i	−0.827130	−	0.562011i	$-0.810028\pi$
$107$		−	1244.09i		−	1.12402i	0.827130	−	0.562011i	$-0.189972\pi$
$108$		0			0
$109$		−	569.937i		−	0.500826i	−0.968139	−	0.250413i	$-0.919434\pi$
$109$		−	569.937i		−	0.500826i	0.968139	−	0.250413i	$-0.0805664\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	824.716	−	476.150i	0.686573	−	0.396393i	−0.115754	−	0.993278i	$-0.536928\pi$
$113$	824.716	−	476.150i	0.686573	−	0.396393i	0.802327	+	0.596885i	$0.203595\pi$
$114$		0			0
$115$	180.613	−	312.831i	0.146455	−	0.253667i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.59280	−	14.8832i	0.00661934	−	0.0114650i
$120$		0			0
$121$	−541.394	−	937.722i	−0.406757	−	0.704524i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	706.504			0.505533
$126$		0			0
$127$			1274.79i			0.890701i	0.895356	+	0.445350i	$0.146921\pi$
$127$			1274.79i			0.890701i	−0.895356	+	0.445350i	$0.853079\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−470.494	+	271.640i	−0.313796	+	0.181170i	−0.648624	−	0.761109i	$-0.724655\pi$
$131$	−470.494	+	271.640i	−0.313796	+	0.181170i	0.334828	+	0.942279i	$0.391322\pi$
$132$		0			0
$133$	67.6916	+	39.0818i	0.0441324	+	0.0254798i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2185.39	−	1261.73i	−1.36285	−	0.786840i	−0.372846	−	0.927893i	$-0.621618\pi$
$137$	−2185.39	−	1261.73i	−1.36285	−	0.786840i	−0.990002	+	0.141053i	$0.954951\pi$
$138$		0			0
$139$	−937.946	−	1624.57i	−0.572342	−	0.991326i	−0.996325	−	0.0856555i	$-0.972702\pi$
$139$	−937.946	−	1624.57i	−0.572342	−	0.991326i	0.423983	−	0.905670i	$-0.360632\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	679.425			0.397317
$144$		0			0
$145$	−68.5766			−0.0392757
$146$		0			0
$147$		0			0
$148$		0			0
$149$	998.646	+	1729.71i	0.549076	+	0.951027i	0.998338	+	0.0576270i	$0.0183534\pi$
$149$	998.646	+	1729.71i	0.549076	+	0.951027i	−0.449263	+	0.893400i	$0.648313\pi$
$150$		0			0
$151$	35.3646	+	20.4177i	0.0190591	+	0.0110038i	0.509499	−	0.860471i	$-0.329831\pi$
$151$	35.3646	+	20.4177i	0.0190591	+	0.0110038i	−0.490440	+	0.871475i	$0.663164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	803.992	+	464.185i	0.416634	+	0.240544i
$156$		0			0
$157$	1170.05	−	675.530i	0.594779	−	0.343396i	−0.172206	−	0.985061i	$-0.555089\pi$
$157$	1170.05	−	675.530i	0.594779	−	0.343396i	0.766985	+	0.641665i	$0.221756\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	639.763i		−	0.313170i
$162$		0			0
$163$	1922.43			0.923779			0.461890	−	0.886937i	$-0.347172\pi$
$163$	1922.43			0.923779			0.461890	+	0.886937i	$0.347172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1311.20	−	2271.06i	−0.607566	−	1.05234i	−0.991640	−	0.129033i	$-0.958813\pi$
$167$	−1311.20	−	2271.06i	−0.607566	−	1.05234i	0.384074	−	0.923302i	$-0.374521\pi$
$168$		0			0
$169$	−168.615	+	292.049i	−0.0767477	+	0.132931i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1385.77	−	2400.23i	0.609008	−	1.05483i	−0.382396	−	0.923998i	$-0.624901\pi$
$173$	1385.77	−	2400.23i	0.609008	−	1.05483i	0.991404	−	0.130834i	$-0.0417656\pi$
$174$		0			0
$175$	522.603	−	301.725i	0.225743	−	0.130333i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2108.40i		−	0.880389i	−0.897903	−	0.440194i	$-0.854910\pi$
$179$		−	2108.40i		−	0.880389i	0.897903	−	0.440194i	$-0.145090\pi$
$180$		0			0
$181$			583.190i			0.239493i	0.992805	+	0.119746i	$0.0382081\pi$
$181$			583.190i			0.239493i	−0.992805	+	0.119746i	$0.961792\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−726.455	+	419.419i	−0.288703	+	0.166683i
$186$		0			0
$187$	−26.1214	+	45.2436i	−0.0102149	+	0.0176927i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−622.663	+	1078.48i	−0.235886	+	0.408567i	−0.959530	−	0.281607i	$-0.909133\pi$
$191$	−622.663	+	1078.48i	−0.235886	+	0.408567i	0.723644	+	0.690174i	$0.242466\pi$
$192$		0			0
$193$	−1316.15	−	2279.63i	−0.490872	−	0.850215i	0.509073	−	0.860723i	$-0.329988\pi$
$193$	−1316.15	−	2279.63i	−0.490872	−	0.850215i	−0.999945	+	0.0105083i	$0.996655\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1210.94			0.437948			0.218974	−	0.975731i	$-0.429729\pi$
$197$	1210.94			0.437948			0.218974	+	0.975731i	$0.429729\pi$
$198$		0			0
$199$			2015.72i			0.718043i	0.933329	+	0.359021i	$0.116890\pi$
$199$			2015.72i			0.718043i	−0.933329	+	0.359021i	$0.883110\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−105.183	+	60.7275i	−0.0363666	+	0.0209962i
$204$		0			0
$205$	−833.845	−	481.421i	−0.284089	−	0.164019i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−205.777	−	118.805i	−0.0681046	−	0.0393202i
$210$		0			0
$211$	601.260	+	1041.41i	0.196173	+	0.339781i	0.947284	−	0.320394i	$-0.103815\pi$
$211$	601.260	+	1041.41i	0.196173	+	0.339781i	−0.751112	+	0.660175i	$0.770482\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1024.23			0.324894
$216$		0			0
$217$	1644.22			0.514365
$218$		0			0
$219$		0			0
$220$		0			0
$221$	71.5013	+	123.844i	0.0217634	+	0.0376952i
$222$		0			0
$223$	1135.80	+	655.752i	0.341070	+	0.196917i	0.660745	−	0.750611i	$-0.270241\pi$
$223$	1135.80	+	655.752i	0.341070	+	0.196917i	−0.319675	+	0.947527i	$0.603574\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2494.70	−	1440.32i	−0.729423	−	0.421133i	0.0887881	−	0.996051i	$-0.471701\pi$
$227$	−2494.70	−	1440.32i	−0.729423	−	0.421133i	−0.818211	+	0.574918i	$0.805034\pi$
$228$		0			0
$229$	3858.07	−	2227.46i	1.11331	−	0.642771i	0.173627	−	0.984812i	$-0.444451\pi$
$229$	3858.07	−	2227.46i	1.11331	−	0.642771i	0.939685	+	0.342040i	$0.111118\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1085.86i			0.305309i	0.988280	+	0.152654i	$0.0487821\pi$
$233$			1085.86i			0.305309i	−0.988280	+	0.152654i	$0.951218\pi$
$234$		0			0
$235$	971.139			0.269575
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1617.59	−	2801.74i	−0.437795	−	0.758283i	0.559724	−	0.828679i	$-0.310907\pi$
$239$	−1617.59	−	2801.74i	−0.437795	−	0.758283i	−0.997519	+	0.0703960i	$0.977574\pi$
$240$		0			0
$241$	715.242	−	1238.84i	0.191173	−	0.331122i	−0.754466	−	0.656339i	$-0.772104\pi$
$241$	715.242	−	1238.84i	0.191173	−	0.331122i	0.945639	+	0.325217i	$0.105437\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	462.552	−	801.164i	0.120618	−	0.208916i
$246$		0			0
$247$	−563.267	+	325.202i	−0.145100	+	0.0837738i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5903.80i			1.48464i	0.670046	+	0.742320i	$0.266275\pi$
$251$			5903.80i			1.48464i	−0.670046	+	0.742320i	$0.733725\pi$
$252$		0			0
$253$			1944.83i			0.483281i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1225.66	+	707.636i	−0.297489	+	0.171755i	−0.641314	−	0.767278i	$-0.721611\pi$
$257$	−1225.66	+	707.636i	−0.297489	+	0.171755i	0.343825	+	0.939034i	$0.388277\pi$
$258$		0			0
$259$	−742.827	+	1286.61i	−0.178212	+	0.308673i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2793.50	+	4838.48i	−0.654959	+	1.13442i	0.326945	+	0.945043i	$0.393981\pi$
$263$	−2793.50	+	4838.48i	−0.654959	+	1.13442i	−0.981904	+	0.189379i	$0.939352\pi$
$264$		0			0
$265$	−762.599	−	1320.86i	−0.176778	−	0.306188i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5030.52			−1.14021			−0.570105	−	0.821572i	$-0.693097\pi$
$269$	−5030.52			−1.14021			−0.570105	+	0.821572i	$0.693097\pi$
$270$		0			0
$271$			1317.52i			0.295328i	0.989038	+	0.147664i	$0.0471754\pi$
$271$			1317.52i			0.295328i	−0.989038	+	0.147664i	$0.952825\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1588.67	+	917.219i	−0.348365	+	0.201129i
$276$		0			0
$277$	−2626.49	−	1516.41i	−0.569714	−	0.328924i	0.187321	−	0.982299i	$-0.440019\pi$
$277$	−2626.49	−	1516.41i	−0.569714	−	0.328924i	−0.757035	+	0.653374i	$0.773353\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2118.05	+	1222.86i	0.449653	+	0.259607i	0.707683	−	0.706530i	$-0.249740\pi$
$281$	2118.05	+	1222.86i	0.449653	+	0.259607i	−0.258031	+	0.966137i	$0.583074\pi$
$282$		0			0
$283$	−4229.51	−	7325.72i	−0.888403	−	1.53876i	−0.841763	−	0.539847i	$-0.818482\pi$
$283$	−4229.51	−	7325.72i	−0.888403	−	1.53876i	−0.0466401	−	0.998912i	$-0.514851\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1705.27			−0.350729
$288$		0			0
$289$	4902.00			0.997762
$290$		0			0
$291$		0			0
$292$		0			0
$293$	219.370	+	379.960i	0.0437397	+	0.0757593i	0.887066	−	0.461642i	$-0.152739\pi$
$293$	219.370	+	379.960i	0.0437397	+	0.0757593i	−0.843327	+	0.537401i	$0.819406\pi$
$294$		0			0
$295$	−382.603	−	220.896i	−0.0755118	−	0.0435968i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4610.30	+	2661.76i	0.891708	+	0.514828i
$300$		0			0
$301$	1570.97	−	907.003i	0.300829	−	0.173684i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2196.48i			0.412361i
$306$		0			0
$307$	7875.21			1.46405			0.732023	−	0.681280i	$-0.238577\pi$
$307$	7875.21			1.46405			0.732023	+	0.681280i	$0.238577\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	532.946	+	923.090i	0.0971724	+	0.168308i	0.910513	−	0.413480i	$-0.135687\pi$
$311$	532.946	+	923.090i	0.0971724	+	0.168308i	−0.813341	+	0.581788i	$0.802354\pi$
$312$		0			0
$313$	−4954.30	+	8581.10i	−0.894676	+	1.54962i	−0.0604700	+	0.998170i	$0.519260\pi$
$313$	−4954.30	+	8581.10i	−0.894676	+	1.54962i	−0.834206	+	0.551454i	$0.814073\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−35.1444	+	60.8718i	−0.00622683	+	0.0107852i	−0.869122	−	0.494598i	$-0.835315\pi$
$317$	−35.1444	+	60.8718i	−0.00622683	+	0.0107852i	0.862895	+	0.505383i	$0.168649\pi$
$318$		0			0
$319$	319.748	−	184.607i	0.0561205	−	0.0324012i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	50.0113i		−	0.00861518i
$324$		0			0
$325$			5021.36i			0.857030i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1489.54	−	859.986i	0.249608	−	0.144111i
$330$		0			0
$331$	−4684.34	+	8113.51i	−0.777868	+	1.34731i	0.155300	+	0.987867i	$0.450366\pi$
$331$	−4684.34	+	8113.51i	−0.777868	+	1.34731i	−0.933168	+	0.359440i	$0.882968\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	369.771	−	640.463i	0.0603067	−	0.104454i
$336$		0			0
$337$	3086.72	+	5346.35i	0.498944	+	0.864197i	0.999999	−	0.00121842i	$-0.000387836\pi$
$337$	3086.72	+	5346.35i	0.498944	+	0.864197i	−0.501055	+	0.865416i	$0.667055\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4998.30			−0.793763
$342$		0			0
$343$		−	3416.08i		−	0.537758i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5102.09	−	2945.69i	0.789321	−	0.455715i	−0.0504023	−	0.998729i	$-0.516050\pi$
$347$	5102.09	−	2945.69i	0.789321	−	0.455715i	0.839724	+	0.543014i	$0.182717\pi$
$348$		0			0
$349$	−8150.29	−	4705.57i	−1.25007	−	0.721729i	−0.278948	−	0.960306i	$-0.589986\pi$
$349$	−8150.29	−	4705.57i	−1.25007	−	0.721729i	−0.971123	+	0.238577i	$0.923319\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6050.47	+	3493.24i	0.912278	+	0.526704i	0.881163	−	0.472812i	$-0.156761\pi$
$353$	6050.47	+	3493.24i	0.912278	+	0.526704i	0.0311148	+	0.999516i	$0.490094\pi$
$354$		0			0
$355$	1584.97	+	2745.25i	0.236962	+	0.410430i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3844.64			0.565215			0.282607	−	0.959236i	$-0.408801\pi$
$359$	3844.64			0.565215			0.282607	+	0.959236i	$0.408801\pi$
$360$		0			0
$361$	−6631.54			−0.966838
$362$		0			0
$363$		0			0
$364$		0			0
$365$	992.590	+	1719.22i	0.142341	+	0.246542i
$366$		0			0
$367$	−11663.0	−	6733.63i	−1.65886	−	0.957746i	−0.973240	−	0.229789i	$-0.926196\pi$
$367$	−11663.0	−	6733.63i	−1.65886	−	0.957746i	−0.685624	−	0.727956i	$-0.740470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2339.36	−	1350.63i	−0.327368	−	0.189006i
$372$		0			0
$373$	−5361.43	+	3095.42i	−0.744248	+	0.429692i	−0.823612	−	0.567154i	$-0.808044\pi$
$373$	−5361.43	+	3095.42i	−0.744248	+	0.429692i	0.0793639	+	0.996846i	$0.474711\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1010.64i		−	0.138065i
$378$		0			0
$379$	−7854.31			−1.06451			−0.532255	−	0.846584i	$-0.678655\pi$
$379$	−7854.31			−1.06451			−0.532255	+	0.846584i	$0.678655\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5216.45	+	9035.15i	0.695948	+	1.20542i	0.969860	+	0.243662i	$0.0783489\pi$
$383$	5216.45	+	9035.15i	0.695948	+	1.20542i	−0.273912	+	0.961755i	$0.588318\pi$
$384$		0			0
$385$	119.466	−	206.920i	0.0158144	−	0.0273913i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5731.82	+	9927.81i	−0.747082	+	1.29398i	0.202133	+	0.979358i	$0.435213\pi$
$389$	−5731.82	+	9927.81i	−0.747082	+	1.29398i	−0.949216	+	0.314626i	$0.898121\pi$
$390$		0			0
$391$	−354.498	+	204.670i	−0.0458510	+	0.0264721i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	329.777i		−	0.0420074i
$396$		0			0
$397$			8809.54i			1.11370i	0.830614	+	0.556849i	$0.187990\pi$
$397$			8809.54i			1.11370i	−0.830614	+	0.556849i	$0.812010\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6817.45	−	3936.06i	0.848996	−	0.490168i	−0.0113159	−	0.999936i	$-0.503602\pi$
$401$	6817.45	−	3936.06i	0.848996	−	0.490168i	0.860312	+	0.509768i	$0.170269\pi$
$402$		0			0
$403$	−6840.85	+	11848.7i	−0.845576	+	1.46458i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2258.13	−	3911.20i	0.275016	−	0.476341i
$408$		0			0
$409$	170.965	+	296.120i	0.0206691	+	0.0358000i	0.876175	−	0.481993i	$-0.160087\pi$
$409$	170.965	+	296.120i	0.0206691	+	0.0358000i	−0.855506	+	0.517793i	$0.826754\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−782.451			−0.0932249
$414$		0			0
$415$		−	2897.16i		−	0.342689i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1154.35	−	666.463i	0.134591	−	0.0777060i	−0.431193	−	0.902260i	$-0.641907\pi$
$419$	1154.35	−	666.463i	0.134591	−	0.0777060i	0.565784	+	0.824554i	$0.308574\pi$
$420$		0			0
$421$	−5457.42	−	3150.84i	−0.631777	−	0.364757i	0.149663	−	0.988737i	$-0.452181\pi$
$421$	−5457.42	−	3150.84i	−0.631777	−	0.364757i	−0.781440	+	0.623980i	$0.785515\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−334.377	−	193.053i	−0.0381639	−	0.0220340i
$426$		0			0
$427$	1945.08	+	3368.97i	0.220442	+	0.381817i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1791.33			0.200198			0.100099	−	0.994977i	$-0.468084\pi$
$431$	1791.33			0.200198			0.100099	+	0.994977i	$0.468084\pi$
$432$		0			0
$433$	7947.77			0.882092			0.441046	−	0.897485i	$-0.354608\pi$
$433$	7947.77			0.882092			0.441046	+	0.897485i	$0.354608\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−930.878	−	1612.33i	−0.101899	−	0.176495i
$438$		0			0
$439$	−10066.8	−	5812.07i	−1.09445	−	0.631879i	−0.159690	−	0.987167i	$-0.551049\pi$
$439$	−10066.8	−	5812.07i	−1.09445	−	0.631879i	−0.934757	+	0.355288i	$0.884383\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12771.4	+	7373.55i	1.36972	+	0.790808i	0.990892	−	0.134660i	$-0.0429942\pi$
$443$	12771.4	+	7373.55i	1.36972	+	0.790808i	0.378827	+	0.925467i	$0.376328\pi$
$444$		0			0
$445$	1436.02	−	829.088i	0.152975	−	0.0883203i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			8413.52i			0.884317i	0.896937	+	0.442159i	$0.145787\pi$
$449$			8413.52i			0.884317i	−0.896937	+	0.442159i	$0.854213\pi$
$450$		0			0
$451$	5183.89			0.541241
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−327.010	−	566.398i	−0.0336933	−	0.0583585i
$456$		0			0
$457$	−9060.70	+	15693.6i	−0.927443	+	1.60638i	−0.139860	+	0.990171i	$0.544665\pi$
$457$	−9060.70	+	15693.6i	−0.927443	+	1.60638i	−0.787584	+	0.616208i	$0.788668\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8300.14	+	14376.3i	−0.838560	+	1.45243i	0.0525393	+	0.998619i	$0.483269\pi$
$461$	−8300.14	+	14376.3i	−0.838560	+	1.45243i	−0.891099	+	0.453809i	$0.850065\pi$
$462$		0			0
$463$	9006.61	−	5199.97i	0.904045	−	0.521950i	0.0255344	−	0.999674i	$-0.491871\pi$
$463$	9006.61	−	5199.97i	0.904045	−	0.521950i	0.878510	+	0.477724i	$0.158538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4926.33i		−	0.488144i	−0.969757	−	0.244072i	$-0.921517\pi$
$467$		−	4926.33i		−	0.488144i	0.969757	−	0.244072i	$-0.0784833\pi$
$468$		0			0
$469$		−	1309.79i		−	0.128957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4775.63	+	2757.21i	−0.464236	+	0.268027i
$474$		0			0
$475$	878.042	−	1520.81i	0.0848154	−	0.146905i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	934.274	−	1618.21i	0.0891191	−	0.154359i	−0.818020	−	0.575190i	$-0.804928\pi$
$479$	934.274	−	1618.21i	0.0891191	−	0.154359i	0.907139	+	0.420831i	$0.138261\pi$
$480$		0			0
$481$	−6181.12	−	10706.0i	−0.585935	−	1.01487i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	432.385			0.0404817
$486$		0			0
$487$			3289.20i			0.306053i	0.988222	+	0.153026i	$0.0489020\pi$
$487$			3289.20i			0.306053i	−0.988222	+	0.153026i	$0.951098\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1189.49	+	686.750i	−0.109329	+	0.0631214i	−0.553668	−	0.832738i	$-0.686772\pi$
$491$	−1189.49	+	686.750i	−0.109329	+	0.0631214i	0.444338	+	0.895859i	$0.353439\pi$
$492$		0			0
$493$	67.2993	+	38.8553i	0.00614809	+	0.00354960i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4862.07	+	2807.11i	0.438820	+	0.253353i
$498$		0			0
$499$	925.039	+	1602.21i	0.0829868	+	0.143737i	0.904532	−	0.426407i	$-0.140221\pi$
$499$	925.039	+	1602.21i	0.0829868	+	0.143737i	−0.821545	+	0.570144i	$0.806887\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17948.3			−1.59100			−0.795502	−	0.605951i	$-0.792793\pi$
$503$	−17948.3			−1.59100			−0.795502	+	0.605951i	$0.792793\pi$
$504$		0			0
$505$	4293.01			0.378290
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−131.163	−	227.180i	−0.0114218	−	0.0197831i	0.860258	−	0.509859i	$-0.170302\pi$
$509$	−131.163	−	227.180i	−0.0114218	−	0.0197831i	−0.871680	+	0.490076i	$0.836969\pi$
$510$		0			0
$511$	3044.88	+	1757.96i	0.263596	+	0.152187i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2941.09	+	1698.04i	0.251650	+	0.145290i
$516$		0			0
$517$	−4528.07	+	2614.28i	−0.385192	+	0.222391i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	938.595i		−	0.0789263i	−0.999221	−	0.0394631i	$-0.987435\pi$
$521$		−	938.595i		−	0.0789263i	0.999221	−	0.0394631i	$-0.0125648\pi$
$522$		0			0
$523$	7118.07			0.595127			0.297563	−	0.954702i	$-0.403826\pi$
$523$	7118.07			0.595127			0.297563	+	0.954702i	$0.403826\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−526.011	−	911.078i	−0.0434790	−	0.0753078i
$528$		0			0
$529$	−1535.67	+	2659.87i	−0.126216	+	0.218613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7094.86	−	12288.7i	0.576571	−	0.998650i
$534$		0			0
$535$	3152.77	−	1820.25i	0.254777	−	0.147096i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			4980.71i			0.398023i
$540$		0			0
$541$		−	6529.45i		−	0.518896i	−0.965757	−	0.259448i	$-0.916459\pi$
$541$		−	6529.45i		−	0.518896i	0.965757	−	0.259448i	$-0.0835407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1444.33	−	833.887i	0.113520	−	0.0655409i
$546$		0			0
$547$	902.529	−	1563.23i	0.0705473	−	0.122192i	−0.828594	−	0.559850i	$-0.810859\pi$
$547$	902.529	−	1563.23i	0.0705473	−	0.122192i	0.899141	+	0.437658i	$0.144192\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−176.721	+	306.090i	−0.0136635	+	0.0236659i
$552$		0			0
$553$	−292.032	−	505.814i	−0.0224565	−	0.0388959i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25569.6			1.94510			0.972550	−	0.232694i	$-0.0747540\pi$
$557$	25569.6			1.94510			0.972550	+	0.232694i	$0.0747540\pi$
$558$		0			0
$559$			15094.5i			1.14209i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4254.07	+	2456.09i	−0.318451	+	0.183858i	−0.650702	−	0.759333i	$-0.725525\pi$
$563$	−4254.07	+	2456.09i	−0.318451	+	0.183858i	0.332251	+	0.943191i	$0.392192\pi$
$564$		0			0
$565$	2413.32	+	1393.33i	0.179698	+	0.103748i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13142.2	−	7587.64i	−0.968275	−	0.559034i	−0.0695650	−	0.997577i	$-0.522161\pi$
$569$	−13142.2	−	7587.64i	−0.968275	−	0.559034i	−0.898710	+	0.438544i	$0.855494\pi$
$570$		0			0
$571$	−6558.52	−	11359.7i	−0.480675	−	0.832554i	0.519079	−	0.854726i	$-0.326275\pi$
$571$	−6558.52	−	11359.7i	−0.480675	−	0.832554i	−0.999754	+	0.0221726i	$0.992942\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14373.4			−1.04246
$576$		0			0
$577$	9122.28			0.658173			0.329086	−	0.944300i	$-0.393259\pi$
$577$	9122.28			0.658173			0.329086	+	0.944300i	$0.393259\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2565.56	−	4443.68i	−0.183197	−	0.317306i
$582$		0			0
$583$	7111.45	+	4105.80i	0.505191	+	0.291672i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13405.1	−	7739.45i	−0.942570	−	0.544193i	−0.0518051	−	0.998657i	$-0.516497\pi$
$587$	−13405.1	−	7739.45i	−0.942570	−	0.544193i	−0.890765	+	0.454464i	$0.849831\pi$
$588$		0			0
$589$	4143.76	−	2392.40i	0.289883	−	0.167364i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	13499.3i		−	0.934824i	−0.884040	−	0.467412i	$-0.845187\pi$
$593$		−	13499.3i		−	0.934824i	0.884040	−	0.467412i	$-0.154813\pi$
$594$		0			0
$595$	50.2893			0.00346498
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7097.54	+	12293.3i	0.484136	+	0.838548i	0.999834	−	0.0182222i	$-0.00580063\pi$
$599$	7097.54	+	12293.3i	0.484136	+	0.838548i	−0.515698	+	0.856771i	$0.672467\pi$
$600$		0			0
$601$	505.327	−	875.253i	0.0342974	−	0.0594048i	−0.848367	−	0.529408i	$-0.822414\pi$
$601$	505.327	−	875.253i	0.0342974	−	0.0594048i	0.882665	+	0.470003i	$0.155747\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1584.25	−	2744.00i	0.106461	−	0.184396i
$606$		0			0
$607$	−7769.04	+	4485.46i	−0.519498	+	0.299933i	−0.736729	−	0.676188i	$-0.763631\pi$
$607$	−7769.04	+	4485.46i	−0.519498	+	0.299933i	0.217231	+	0.976120i	$0.430298\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14312.0i			0.947630i
$612$		0			0
$613$		−	14462.3i		−	0.952897i	−0.879202	−	0.476449i	$-0.841924\pi$
$613$		−	14462.3i		−	0.952897i	0.879202	−	0.476449i	$-0.158076\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18159.9	+	10484.6i	−1.18491	+	0.684108i	−0.957145	−	0.289608i	$-0.906475\pi$
$617$	−18159.9	+	10484.6i	−1.18491	+	0.684108i	−0.227764	+	0.973716i	$0.573142\pi$
$618$		0			0
$619$	3168.60	−	5488.18i	0.205746	−	0.356363i	−0.744624	−	0.667484i	$-0.767371\pi$
$619$	3168.60	−	5488.18i	0.205746	−	0.356363i	0.950370	+	0.311121i	$0.100705\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1468.39	−	2543.32i	0.0944296	−	0.163557i
$624$		0			0
$625$	−6243.62	−	10814.3i	−0.399591	−	0.692113i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	950.565			0.0602568
$630$		0			0
$631$			26190.2i			1.65232i	0.563436	+	0.826160i	$0.309479\pi$
$631$			26190.2i			1.65232i	−0.563436	+	0.826160i	$0.690521\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3230.57	+	1865.17i	−0.201892	+	0.116562i
$636$		0			0
$637$	11807.0	+	6816.79i	0.734397	+	0.424004i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21831.2	+	12604.3i	1.34521	+	0.776660i	0.987567	−	0.157197i	$-0.0502458\pi$
$641$	21831.2	+	12604.3i	1.34521	+	0.776660i	0.357647	+	0.933857i	$0.383579\pi$
$642$		0			0
$643$	−3931.26	−	6809.13i	−0.241110	−	0.417614i	0.719921	−	0.694056i	$-0.244178\pi$
$643$	−3931.26	−	6809.13i	−0.241110	−	0.417614i	−0.961031	+	0.276442i	$0.910845\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	188.733			0.0114681			0.00573404	−	0.999984i	$-0.498175\pi$
$647$	188.733			0.0114681			0.00573404	+	0.999984i	$0.498175\pi$
$648$		0			0
$649$	2378.58			0.143864
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15259.4	−	26430.1i	−0.914468	−	1.58390i	−0.807678	−	0.589623i	$-0.799276\pi$
$653$	−15259.4	−	26430.1i	−0.914468	−	1.58390i	−0.106790	−	0.994282i	$-0.534057\pi$
$654$		0			0
$655$	−1376.78	−	794.884i	−0.0821301	−	0.0474178i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16805.7	−	9702.80i	−0.993412	−	0.573547i	−0.0871197	−	0.996198i	$-0.527766\pi$
$659$	−16805.7	−	9702.80i	−0.993412	−	0.573547i	−0.906292	+	0.422651i	$0.861100\pi$
$660$		0			0
$661$	26953.4	−	15561.6i	1.58603	−	0.915696i	0.592079	−	0.805880i	$-0.298307\pi$
$661$	26953.4	−	15561.6i	1.58603	−	0.915696i	0.993952	−	0.109816i	$-0.0350262\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			228.726i			0.0133377i
$666$		0			0
$667$	2892.91			0.167937
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5912.87	−	10241.4i	−0.340185	−	0.589217i
$672$		0			0
$673$	16571.1	−	28701.9i	0.949134	−	1.64395i	0.201880	−	0.979410i	$-0.435295\pi$
$673$	16571.1	−	28701.9i	0.949134	−	1.64395i	0.747254	−	0.664539i	$-0.231372\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3831.16	−	6635.77i	0.217494	−	0.376711i	−0.736547	−	0.676386i	$-0.763545\pi$
$677$	3831.16	−	6635.77i	0.217494	−	0.376711i	0.954041	+	0.299675i	$0.0968783\pi$
$678$		0			0
$679$	663.195	−	382.896i	0.0374832	−	0.0216409i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8233.45i		−	0.461265i	−0.973041	−	0.230633i	$-0.925920\pi$
$683$		−	8233.45i		−	0.461265i	0.973041	−	0.230633i	$-0.0740795\pi$
$684$		0			0
$685$		−	7384.28i		−	0.411882i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	19466.0	−	11238.7i	1.07633	−	0.621422i
$690$		0			0
$691$	9201.53	−	15937.5i	0.506574	−	0.877412i	−0.493397	−	0.869804i	$-0.664245\pi$
$691$	9201.53	−	15937.5i	0.506574	−	0.877412i	0.999971	−	0.00760808i	$-0.00242175\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2744.66	−	4753.89i	0.149800	−	0.259461i
$696$		0			0
$697$	545.542	+	944.907i	0.0296469	+	0.0513499i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2617.11			−0.141008			−0.0705041	−	0.997511i	$-0.522461\pi$
$701$	−2617.11			−0.141008			−0.0705041	+	0.997511i	$0.522461\pi$
$702$		0			0
$703$			4323.36i			0.231947i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6584.64	−	3801.65i	0.350270	−	0.202229i
$708$		0			0
$709$	15311.9	+	8840.30i	0.811070	+	0.468271i	0.847327	−	0.531071i	$-0.178210\pi$
$709$	15311.9	+	8840.30i	0.811070	+	0.468271i	−0.0362573	+	0.999342i	$0.511544\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33916.4	−	19581.7i	−1.78146	−	1.02853i
$714$		0			0
$715$	994.082	+	1721.80i	0.0519952	+	0.0900583i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7767.21			0.402876			0.201438	−	0.979501i	$-0.435438\pi$
$719$	7767.21			0.402876			0.201438	+	0.979501i	$0.435438\pi$
$720$		0			0
$721$	6014.74			0.310680
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1364.35	+	2363.13i	0.0698907	+	0.121054i
$726$		0			0
$727$	9187.03	+	5304.13i	0.468677	+	0.270591i	0.715686	−	0.698423i	$-0.246114\pi$
$727$	9187.03	+	5304.13i	0.468677	+	0.270591i	−0.247009	+	0.969013i	$0.579448\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1005.16	−	580.327i	−0.0508578	−	0.0293628i
$732$		0			0
$733$	−7738.67	+	4467.92i	−0.389951	+	0.225138i	−0.682139	−	0.731223i	$-0.738950\pi$
$733$	−7738.67	+	4467.92i	−0.389951	+	0.225138i	0.292188	+	0.956361i	$0.405617\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3981.66i			0.199005i
$738$		0			0
$739$	−5767.08			−0.287071			−0.143535	−	0.989645i	$-0.545847\pi$
$739$	−5767.08			−0.287071			−0.143535	+	0.989645i	$0.545847\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	711.865	+	1232.99i	0.0351491	+	0.0608801i	0.883065	−	0.469251i	$-0.155476\pi$
$743$	711.865	+	1232.99i	0.0351491	+	0.0608801i	−0.847916	+	0.530131i	$0.822143\pi$
$744$		0			0
$745$	−2922.28	+	5061.54i	−0.143710	+	0.248913i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3223.82	−	5583.82i	0.157271	−	0.272401i
$750$		0			0
$751$	16310.8	−	9417.02i	0.792527	−	0.457566i	−0.0483243	−	0.998832i	$-0.515388\pi$
$751$	16310.8	−	9417.02i	0.792527	−	0.457566i	0.840851	+	0.541266i	$0.182055\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			119.495i			0.00576007i
$756$		0			0
$757$			21381.1i			1.02657i	0.858219	+	0.513283i	$0.171571\pi$
$757$			21381.1i			1.02657i	−0.858219	+	0.513283i	$0.828429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15809.6	+	9127.66i	−0.753083	+	0.434793i	−0.826807	−	0.562486i	$-0.809845\pi$
$761$	−15809.6	+	9127.66i	−0.753083	+	0.434793i	0.0737237	+	0.997279i	$0.476512\pi$
$762$		0			0
$763$	1476.89	−	2558.04i	0.0700745	−	0.121373i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3255.42	−	5638.55i	0.153255	−	0.265445i
$768$		0			0
$769$	−7996.45	−	13850.3i	−0.374980	−	0.649484i	0.615344	−	0.788259i	$-0.289017\pi$
$769$	−7996.45	−	13850.3i	−0.374980	−	0.649484i	−0.990324	+	0.138774i	$0.955684\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23733.1			1.10429			0.552147	−	0.833747i	$-0.313809\pi$
$773$	23733.1			1.10429			0.552147	+	0.833747i	$0.313809\pi$
$774$		0			0
$775$		−	36940.4i		−	1.71218i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4297.62	+	2481.23i	−0.197662	+	0.114120i
$780$		0			0
$781$	−14780.3	−	8533.39i	−0.677182	−	0.390971i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3423.86	+	1976.77i	0.155672	+	0.0898775i
$786$		0			0
$787$	−3774.80	−	6538.15i	−0.170975	−	0.296137i	0.767786	−	0.640706i	$-0.221358\pi$
$787$	−3774.80	−	6538.15i	−0.170975	−	0.296137i	−0.938761	+	0.344569i	$0.888025\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4935.42			0.221850
$792$		0			0
$793$	−32370.3			−1.44956
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13480.4	+	23348.7i	0.599120	+	1.03771i	0.992951	+	0.118524i	$0.0378161\pi$
$797$	13480.4	+	23348.7i	0.599120	+	1.03771i	−0.393831	+	0.919183i	$0.628851\pi$
$798$		0			0
$799$	−953.051	−	550.244i	−0.0421984	−	0.0243632i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9256.18	−	5344.06i	−0.406779	−	0.234854i
$804$		0			0
$805$	1621.29	−	936.052i	0.0709850	−	0.0409832i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	33739.1i		−	1.46626i	−0.680090	−	0.733129i	$-0.738059\pi$
$809$		−	33739.1i		−	1.46626i	0.680090	−	0.733129i	$-0.261941\pi$
$810$		0			0
$811$	9809.12			0.424716			0.212358	−	0.977192i	$-0.431886\pi$
$811$	9809.12			0.424716			0.212358	+	0.977192i	$0.431886\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2812.74	+	4871.82i	0.120891	+	0.209389i
$816$		0			0
$817$	2639.44	−	4571.65i	0.113026	−	0.195767i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9274.38	+	16063.7i	−0.394248	+	0.682858i	−0.993005	−	0.118073i	$-0.962328\pi$
$821$	−9274.38	+	16063.7i	−0.394248	+	0.682858i	0.598756	+	0.800931i	$0.295662\pi$
$822$		0			0
$823$	1262.62	−	728.972i	0.0534775	−	0.0308753i	−0.473023	−	0.881050i	$-0.656837\pi$
$823$	1262.62	−	728.972i	0.0534775	−	0.0308753i	0.526500	+	0.850175i	$0.323504\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	42187.5i		−	1.77388i	−0.461881	−	0.886942i	$-0.652825\pi$
$827$		−	42187.5i		−	1.77388i	0.461881	−	0.886942i	$-0.347175\pi$
$828$		0			0
$829$			22370.3i			0.937218i	0.883406	+	0.468609i	$0.155245\pi$
$829$			22370.3i			0.937218i	−0.883406	+	0.468609i	$0.844755\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−907.873	+	524.161i	−0.0377622	+	0.0218020i
$834$		0			0
$835$	3836.89	−	6645.68i	0.159019	−	0.275429i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24035.3	−	41630.3i	0.989023	−	1.71304i	0.366548	−	0.930399i	$-0.380540\pi$
$839$	24035.3	−	41630.3i	0.989023	−	1.71304i	0.622476	−	0.782639i	$-0.286127\pi$
$840$		0			0
$841$	11919.9	+	20645.9i	0.488741	+	0.846524i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−986.816			−0.0401745
$846$		0			0
$847$		−	5611.69i		−	0.227650i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30645.5	−	17693.2i	1.23445	−	0.712708i
$852$		0			0
$853$	−14708.0	−	8491.65i	−0.590377	−	0.340854i	0.174870	−	0.984592i	$-0.444050\pi$
$853$	−14708.0	−	8491.65i	−0.590377	−	0.340854i	−0.765246	+	0.643737i	$0.777383\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22340.4	−	12898.3i	−0.890473	−	0.514115i	−0.0163755	−	0.999866i	$-0.505213\pi$
$857$	−22340.4	−	12898.3i	−0.890473	−	0.514115i	−0.874097	+	0.485751i	$0.838546\pi$
$858$		0			0
$859$	9914.38	+	17172.2i	0.393800	+	0.682082i	0.992947	−	0.118557i	$-0.0378269\pi$
$859$	9914.38	+	17172.2i	0.393800	+	0.682082i	−0.599147	+	0.800639i	$0.704494\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	42561.4			1.67880			0.839401	−	0.543513i	$-0.182906\pi$
$863$	42561.4			1.67880			0.839401	+	0.543513i	$0.182906\pi$
$864$		0			0
$865$	8110.22			0.318793
$866$		0			0
$867$		0			0
$868$		0			0
$869$	887.753	+	1537.63i	0.0346547	+	0.0600237i
$870$		0			0
$871$	9438.71	+	5449.44i	0.367185	+	0.211995i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3171.00	+	1830.78i	0.122513	+	0.0707331i
$876$		0			0
$877$	−13338.6	+	7701.03i	−0.513582	+	0.296517i	−0.734305	−	0.678820i	$-0.762492\pi$
$877$	−13338.6	+	7701.03i	−0.513582	+	0.296517i	0.220723	+	0.975337i	$0.429158\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	408.896i		−	0.0156368i	−0.999969	−	0.00781842i	$-0.997511\pi$
$881$		−	408.896i		−	0.0156368i	0.999969	−	0.00781842i	$-0.00248871\pi$
$882$		0			0
$883$	−24945.2			−0.950704			−0.475352	−	0.879796i	$-0.657679\pi$
$883$	−24945.2			−0.950704			−0.475352	+	0.879796i	$0.657679\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5014.93	−	8686.11i	−0.189836	−	0.328806i	0.755359	−	0.655311i	$-0.227462\pi$
$887$	−5014.93	−	8686.11i	−0.189836	−	0.328806i	−0.945196	+	0.326505i	$0.894129\pi$
$888$		0			0
$889$	−3303.37	+	5721.61i	−0.124625	+	0.215857i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2502.62	−	4334.66i	0.0937815	−	0.162434i
$894$		0			0
$895$	5343.12	−	3084.85i	0.199554	−	0.115213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			7434.91i			0.275827i
$900$		0			0
$901$			1728.34i			0.0639062i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1477.92	+	853.278i	−0.0542848	+	0.0313413i
$906$		0			0
$907$	−3767.44	+	6525.40i	−0.137923	+	0.238889i	−0.926710	−	0.375777i	$-0.877376\pi$
$907$	−3767.44	+	6525.40i	−0.137923	+	0.238889i	0.788788	+	0.614666i	$0.210709\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11159.9	+	19329.6i	−0.405867	+	0.702983i	−0.994422	−	0.105475i	$-0.966364\pi$
$911$	−11159.9	+	19329.6i	−0.405867	+	0.702983i	0.588555	+	0.808457i	$0.299697\pi$
$912$		0			0
$913$	7799.08	+	13508.4i	0.282708	+	0.489664i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2815.62			−0.101396
$918$		0			0
$919$		−	40992.7i		−	1.47141i	−0.677303	−	0.735704i	$-0.736851\pi$
$919$		−	40992.7i		−	1.47141i	0.677303	−	0.735704i	$-0.263149\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40457.6	+	23358.2i	−1.44277	+	0.832985i
$924$		0			0
$925$	28906.1	+	16688.9i	1.02749	+	0.593220i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20199.6	+	11662.2i	0.713376	+	0.411868i	0.812310	−	0.583226i	$-0.198210\pi$
$929$	20199.6	+	11662.2i	0.713376	+	0.411868i	−0.0989336	+	0.995094i	$0.531543\pi$
$930$		0			0
$931$	−2383.99	−	4129.18i	−0.0839226	−	0.145358i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−152.875			−0.00534712
$936$		0			0
$937$	−29355.3			−1.02348			−0.511738	−	0.859142i	$-0.670998\pi$
$937$	−29355.3			−1.02348			−0.511738	+	0.859142i	$0.670998\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1391.16	+	2409.56i	0.0481939	+	0.0834743i	0.889116	−	0.457682i	$-0.151320\pi$
$941$	1391.16	+	2409.56i	0.0481939	+	0.0834743i	−0.840922	+	0.541156i	$0.817987\pi$
$942$		0			0
$943$	35175.8	+	20308.7i	1.21472	+	0.701319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15438.1	+	8913.16i	0.529746	+	0.305849i	0.740913	−	0.671601i	$-0.234393\pi$
$947$	15438.1	+	8913.16i	0.529746	+	0.305849i	−0.211167	+	0.977450i	$0.567726\pi$
$948$		0			0
$949$	−25336.7	+	14628.1i	−0.866663	+	0.500368i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21508.9i			0.731104i	0.930791	+	0.365552i	$0.119120\pi$
$953$			21508.9i			0.731104i	−0.930791	+	0.365552i	$0.880880\pi$
$954$		0			0
$955$	−3644.13			−0.123478
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6539.10	−	11326.1i	−0.220186	−	0.381374i
$960$		0			0
$961$	35430.4	−	61367.2i	1.18930	−	2.05992i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3851.37	−	6670.76i	0.128477	−	0.222528i
$966$		0			0
$967$	−8410.42	+	4855.76i	−0.279691	+	0.161479i	−0.633283	−	0.773920i	$-0.718293\pi$
$967$	−8410.42	+	4855.76i	−0.279691	+	0.161479i	0.353593	+	0.935399i	$0.384960\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1908.56i		−	0.0630777i	−0.999503	−	0.0315389i	$-0.989959\pi$
$971$		−	1908.56i		−	0.0630777i	0.999503	−	0.0315389i	$-0.0100408\pi$
$972$		0			0
$973$		−	9722.06i		−	0.320324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	40899.7	−	23613.4i	1.33930	−	0.773245i	0.352596	−	0.935776i	$-0.385299\pi$
$977$	40899.7	−	23613.4i	1.33930	−	0.773245i	0.986704	+	0.162531i	$0.0519656\pi$
$978$		0			0
$979$	−4463.77	+	7731.47i	−0.145723	+	0.252399i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11777.7	+	20399.5i	−0.382146	+	0.661895i	−0.991369	−	0.131103i	$-0.958148\pi$
$983$	−11777.7	+	20399.5i	−0.382146	+	0.661895i	0.609223	+	0.792999i	$0.291481\pi$
$984$		0			0
$985$	1771.75	+	3068.76i	0.0573123	+	0.0992678i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−43207.3			−1.38919
$990$		0			0
$991$		−	31768.2i		−	1.01831i	−0.860673	−	0.509157i	$-0.829957\pi$
$991$		−	31768.2i		−	1.01831i	0.860673	−	0.509157i	$-0.170043\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5108.24	+	2949.24i	−0.162756	+	0.0939672i
$996$		0			0
$997$	3600.15	+	2078.55i	0.114361	+	0.0660263i	0.556089	−	0.831122i	$-0.312301\pi$
$997$	3600.15	+	2078.55i	0.114361	+	0.0660263i	−0.441729	+	0.897149i	$0.645635\pi$
$998$		0			0
$999$		0			0

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.p.b.143.18		64
3.2	odd	2		288.4.p.b.47.7		64
4.3	odd	2		216.4.l.b.35.3		64
8.3	odd	2	inner	864.4.p.b.143.15		64
8.5	even	2		216.4.l.b.35.13		64
9.4	even	3		288.4.p.b.239.8		64
9.5	odd	6	inner	864.4.p.b.719.15		64
12.11	even	2		72.4.l.b.11.30	yes	64
24.5	odd	2		72.4.l.b.11.20	✓	64
24.11	even	2		288.4.p.b.47.8		64
36.23	even	6		216.4.l.b.179.13		64
36.31	odd	6		72.4.l.b.59.20	yes	64
72.5	odd	6		216.4.l.b.179.3		64
72.13	even	6		72.4.l.b.59.30	yes	64
72.59	even	6	inner	864.4.p.b.719.18		64
72.67	odd	6		288.4.p.b.239.7		64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.b.11.20	✓	64	24.5	odd	2
72.4.l.b.11.30	yes	64	12.11	even	2
72.4.l.b.59.20	yes	64	36.31	odd	6
72.4.l.b.59.30	yes	64	72.13	even	6
216.4.l.b.35.3		64	4.3	odd	2
216.4.l.b.35.13		64	8.5	even	2
216.4.l.b.179.3		64	72.5	odd	6
216.4.l.b.179.13		64	36.23	even	6
288.4.p.b.47.7		64	3.2	odd	2
288.4.p.b.47.8		64	24.11	even	2
288.4.p.b.239.7		64	72.67	odd	6
288.4.p.b.239.8		64	9.4	even	3
864.4.p.b.143.15		64	8.3	odd	2	inner
864.4.p.b.143.18		64	1.1	even	1	trivial
864.4.p.b.719.15		64	9.5	odd	6	inner
864.4.p.b.719.18		64	72.59	even	6	inner

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

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

Label	864.4.p.b.143.18

Level	$864$
Weight	$4$
Character	864.143
Analytic conductor	$50.978$
Analytic rank	$0$
Dimension	$64$
Inner twists	$4$