LMFDB - Embedded newform 864.4.p.b.719.15

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)

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);

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")

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			719.15
Character	$\chi$	$=$	864.719
Dual form			864.4.p.b.143.15

$q$ -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 $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$ .

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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	$-0.875109\pi$
$5$	−1.46312	+	2.53420i	−0.130866	+	0.226666i	0.793145	+	0.609033i	$0.208442\pi$
$6$		0			0
$7$	−4.48829	+	2.59132i	−0.242345	+	0.139918i	−0.616254	−	0.787547i	$-0.711351\pi$
$7$	−4.48829	+	2.59132i	−0.242345	+	0.139918i	0.373909	+	0.927465i	$0.378017\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.6440	+	7.87738i	−0.373984	+	0.215920i	−0.675198	−	0.737637i	$-0.735942\pi$
$11$	−13.6440	+	7.87738i	−0.373984	+	0.215920i	0.301213	+	0.953557i	$0.402608\pi$
$12$		0			0
$13$	37.3474	+	21.5625i	0.796793	+	0.460028i	0.842348	−	0.538933i	$-0.181173\pi$
$13$	37.3474	+	21.5625i	0.796793	+	0.460028i	−0.0455558	+	0.998962i	$0.514506\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.31600i			0.0473087i	0.999720	+	0.0236544i	$0.00753012\pi$
$17$			3.31600i			0.0473087i	−0.999720	+	0.0236544i	$0.992470\pi$
$18$		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	−0.437972	−	0.898988i	$-0.644303\pi$
$23$	61.7219	−	106.905i	0.559561	−	0.969188i	0.997533	−	0.0701991i	$-0.0223635\pi$
$24$		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	$-0.142761\pi$
$29$	11.7175	+	20.2953i	0.0750306	+	0.129957i	−0.826069	+	0.563569i	$0.809428\pi$
$30$		0			0
$31$	−274.752	−	158.628i	−1.59184	−	0.919048i	−0.992992	−	0.118183i	$-0.962293\pi$
$31$	−274.752	−	158.628i	−1.59184	−	0.919048i	−0.598846	−	0.800865i	$-0.704374\pi$
$32$		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	−0.153068	−	0.988216i	$-0.548915\pi$
$41$	−284.954	−	164.518i	−1.08542	−	0.626669i	−0.932354	+	0.361547i	$0.882249\pi$
$42$		0			0
$43$	175.008	+	303.123i	0.620663	+	1.07502i	0.989363	+	0.145471i	$0.0464697\pi$
$43$	175.008	+	303.123i	0.620663	+	1.07502i	−0.368700	+	0.929548i	$0.620197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−165.936	−	287.410i	−0.514984	−	0.891979i	−0.999849	−	0.0173897i	$-0.994464\pi$
$47$	−165.936	−	287.410i	−0.514984	−	0.891979i	0.484864	−	0.874589i	$-0.338869\pi$
$48$		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	$-0.263964\pi$
$53$	521.214			1.35083			0.675417	+	0.737436i	$0.263964\pi$
$54$		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	0.348760	−	0.937212i	$-0.386603\pi$
$59$	−130.749	−	75.4878i	−0.288509	−	0.166571i	−0.637269	+	0.770641i	$0.719936\pi$
$60$		0			0
$61$	−650.051	+	375.307i	−1.36443	+	0.787756i	−0.990210	−	0.139582i	$-0.955424\pi$
$61$	−650.051	+	375.307i	−1.36443	+	0.787756i	−0.374223	+	0.927339i	$0.622091\pi$
$62$		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	−0.957930	−	0.287001i	$-0.907342\pi$
$67$	−126.364	+	218.868i	−0.230415	+	0.399090i	0.727515	+	0.686091i	$0.240675\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1083.28			−1.81072			−0.905362	−	0.424640i	$-0.860401\pi$
$71$	−1083.28			−1.81072			−0.905362	+	0.424640i	$0.860401\pi$
$72$		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	$-0.641095\pi$
$79$	97.5980	−	56.3482i	0.138995	−	0.0802490i	0.567885	+	0.823108i	$0.307762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−857.418	+	495.030i	−1.13390	+	0.654658i	−0.944913	−	0.327321i	$-0.893854\pi$
$83$	−857.418	+	495.030i	−1.13390	+	0.654658i	−0.188988	+	0.981979i	$0.560521\pi$
$84$		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	$0.109563\pi$
$89$			566.656i			0.674893i	−0.941345	+	0.337446i	$0.890437\pi$
$90$		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	0.902099	−	0.431529i	$-0.142026\pi$
$97$	73.8806	+	127.965i	0.0773344	+	0.133947i	−0.824765	+	0.565476i	$0.808692\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−733.536	−	1270.52i	−0.722669	−	1.25170i	−0.959926	−	0.280252i	$-0.909582\pi$
$101$	−733.536	−	1270.52i	−0.722669	−	1.25170i	0.237258	−	0.971447i	$-0.423751\pi$
$102$		0			0
$103$	−1005.07	−	580.278i	−0.961482	−	0.555112i	−0.0648531	−	0.997895i	$-0.520658\pi$
$103$	−1005.07	−	580.278i	−0.961482	−	0.555112i	−0.896629	+	0.442783i	$0.853991\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1244.09i			1.12402i	0.827130	+	0.562011i	$0.189972\pi$
$107$			1244.09i			1.12402i	−0.827130	+	0.562011i	$0.810028\pi$
$108$		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	0.802327	−	0.596885i	$-0.203595\pi$
$113$	824.716	+	476.150i	0.686573	+	0.396393i	−0.115754	+	0.993278i	$0.536928\pi$
$114$		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	0.334828	−	0.942279i	$-0.391322\pi$
$131$	−470.494	−	271.640i	−0.313796	−	0.181170i	−0.648624	+	0.761109i	$0.724655\pi$
$132$		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	−0.990002	−	0.141053i	$-0.954951\pi$
$137$	−2185.39	+	1261.73i	−1.36285	+	0.786840i	−0.372846	+	0.927893i	$0.621618\pi$
$138$		0			0
$139$	−937.946	+	1624.57i	−0.572342	+	0.991326i	0.423983	+	0.905670i	$0.360632\pi$
$139$	−937.946	+	1624.57i	−0.572342	+	0.991326i	−0.996325	+	0.0856555i	$0.972702\pi$
$140$		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	0.449263	+	0.893400i	$0.351687\pi$
$149$	−998.646	+	1729.71i	−0.549076	+	0.951027i	−0.998338	+	0.0576270i	$0.981647\pi$
$150$		0			0
$151$	−35.3646	+	20.4177i	−0.0190591	+	0.0110038i	−0.509499	−	0.860471i	$-0.670169\pi$
$151$	−35.3646	+	20.4177i	−0.0190591	+	0.0110038i	0.490440	+	0.871475i	$0.336836\pi$
$152$		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	$-0.444911\pi$
$157$	−1170.05	−	675.530i	−0.594779	−	0.343396i	−0.766985	+	0.641665i	$0.778244\pi$
$158$		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	−0.384074	−	0.923302i	$-0.625479\pi$
$167$	1311.20	−	2271.06i	0.607566	−	1.05234i	0.991640	−	0.129033i	$-0.0411873\pi$
$168$		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	−0.991404	−	0.130834i	$-0.958234\pi$
$173$	−1385.77	−	2400.23i	−0.609008	−	1.05483i	0.382396	−	0.923998i	$-0.375099\pi$
$174$		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	$0.145090\pi$
$179$			2108.40i			0.880389i	−0.897903	+	0.440194i	$0.854910\pi$
$180$		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	$-0.0908674\pi$
$191$	622.663	+	1078.48i	0.235886	+	0.408567i	−0.723644	+	0.690174i	$0.757534\pi$
$192$		0			0
$193$	−1316.15	+	2279.63i	−0.490872	+	0.850215i	−0.999945	−	0.0105083i	$-0.996655\pi$
$193$	−1316.15	+	2279.63i	−0.490872	+	0.850215i	0.509073	+	0.860723i	$0.329988\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1210.94			−0.437948			−0.218974	−	0.975731i	$-0.570271\pi$
$197$	−1210.94			−0.437948			−0.218974	+	0.975731i	$0.570271\pi$
$198$		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	−0.751112	−	0.660175i	$-0.770482\pi$
$211$	601.260	−	1041.41i	0.196173	−	0.339781i	0.947284	+	0.320394i	$0.103815\pi$
$212$		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	$-0.729759\pi$
$223$	−1135.80	+	655.752i	−0.341070	+	0.196917i	0.319675	+	0.947527i	$0.396426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2494.70	+	1440.32i	−0.729423	+	0.421133i	−0.818211	−	0.574918i	$-0.805034\pi$
$227$	−2494.70	+	1440.32i	−0.729423	+	0.421133i	0.0887881	+	0.996051i	$0.471701\pi$
$228$		0			0
$229$	−3858.07	−	2227.46i	−1.11331	−	0.642771i	−0.173627	−	0.984812i	$-0.555549\pi$
$229$	−3858.07	−	2227.46i	−1.11331	−	0.642771i	−0.939685	+	0.342040i	$0.888882\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1085.86i		−	0.305309i	−0.988280	−	0.152654i	$-0.951218\pi$
$233$		−	1085.86i		−	0.305309i	0.988280	−	0.152654i	$-0.0487821\pi$
$234$		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	$-0.689093\pi$
$239$	1617.59	−	2801.74i	0.437795	−	0.758283i	0.997519	+	0.0703960i	$0.0224263\pi$
$240$		0			0
$241$	715.242	+	1238.84i	0.191173	+	0.331122i	0.945639	−	0.325217i	$-0.105437\pi$
$241$	715.242	+	1238.84i	0.191173	+	0.331122i	−0.754466	+	0.656339i	$0.772104\pi$
$242$		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	$-0.733725\pi$
$251$		−	5903.80i		−	1.48464i	0.670046	−	0.742320i	$-0.266275\pi$
$252$		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	0.343825	−	0.939034i	$-0.388277\pi$
$257$	−1225.66	−	707.636i	−0.297489	−	0.171755i	−0.641314	+	0.767278i	$0.721611\pi$
$258$		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	0.981904	+	0.189379i	$0.0606476\pi$
$263$	2793.50	+	4838.48i	0.654959	+	1.13442i	−0.326945	+	0.945043i	$0.606019\pi$
$264$		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	$-0.306903\pi$
$269$	5030.52			1.14021			0.570105	+	0.821572i	$0.306903\pi$
$270$		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	$-0.559981\pi$
$277$	2626.49	−	1516.41i	0.569714	−	0.328924i	0.757035	+	0.653374i	$0.226647\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2118.05	−	1222.86i	0.449653	−	0.259607i	−0.258031	−	0.966137i	$-0.583074\pi$
$281$	2118.05	−	1222.86i	0.449653	−	0.259607i	0.707683	+	0.706530i	$0.249740\pi$
$282$		0			0
$283$	−4229.51	+	7325.72i	−0.888403	+	1.53876i	−0.0466401	+	0.998912i	$0.514851\pi$
$283$	−4229.51	+	7325.72i	−0.888403	+	1.53876i	−0.841763	+	0.539847i	$0.818482\pi$
$284$		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	$-0.847261\pi$
$293$	−219.370	+	379.960i	−0.0437397	+	0.0757593i	0.843327	+	0.537401i	$0.180594\pi$
$294$		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	$-0.864313\pi$
$311$	−532.946	+	923.090i	−0.0971724	+	0.168308i	0.813341	+	0.581788i	$0.197646\pi$
$312$		0			0
$313$	−4954.30	−	8581.10i	−0.894676	−	1.54962i	−0.834206	−	0.551454i	$-0.814073\pi$
$313$	−4954.30	−	8581.10i	−0.894676	−	1.54962i	−0.0604700	−	0.998170i	$-0.519260\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	35.1444	+	60.8718i	0.00622683	+	0.0107852i	0.869122	−	0.494598i	$-0.164685\pi$
$317$	35.1444	+	60.8718i	0.00622683	+	0.0107852i	−0.862895	+	0.505383i	$0.831351\pi$
$318$		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	−0.933168	−	0.359440i	$-0.882968\pi$
$331$	−4684.34	−	8113.51i	−0.777868	−	1.34731i	0.155300	−	0.987867i	$-0.450366\pi$
$332$		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	−0.501055	−	0.865416i	$-0.667055\pi$
$337$	3086.72	−	5346.35i	0.498944	−	0.864197i	0.999999	+	0.00121842i	$0.000387836\pi$
$338$		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	0.839724	−	0.543014i	$-0.182717\pi$
$347$	5102.09	+	2945.69i	0.789321	+	0.455715i	−0.0504023	+	0.998729i	$0.516050\pi$
$348$		0			0
$349$	8150.29	−	4705.57i	1.25007	−	0.721729i	0.278948	−	0.960306i	$-0.410014\pi$
$349$	8150.29	−	4705.57i	1.25007	−	0.721729i	0.971123	+	0.238577i	$0.0766811\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6050.47	−	3493.24i	0.912278	−	0.526704i	0.0311148	−	0.999516i	$-0.490094\pi$
$353$	6050.47	−	3493.24i	0.912278	−	0.526704i	0.881163	+	0.472812i	$0.156761\pi$
$354$		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	$-0.591199\pi$
$359$	−3844.64			−0.565215			−0.282607	+	0.959236i	$0.591199\pi$
$360$		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	0.685624	−	0.727956i	$-0.259530\pi$
$367$	11663.0	−	6733.63i	1.65886	−	0.957746i	0.973240	−	0.229789i	$-0.0738038\pi$
$368$		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	$-0.191956\pi$
$373$	5361.43	+	3095.42i	0.744248	+	0.429692i	−0.0793639	+	0.996846i	$0.525289\pi$
$374$		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	0.273912	+	0.961755i	$0.411682\pi$
$383$	−5216.45	+	9035.15i	−0.695948	+	1.20542i	−0.969860	+	0.243662i	$0.921651\pi$
$384$		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	0.949216	+	0.314626i	$0.101879\pi$
$389$	5731.82	+	9927.81i	0.747082	+	1.29398i	−0.202133	+	0.979358i	$0.564787\pi$
$390$		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	0.860312	−	0.509768i	$-0.170269\pi$
$401$	6817.45	+	3936.06i	0.848996	+	0.490168i	−0.0113159	+	0.999936i	$0.503602\pi$
$402$		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	−0.855506	−	0.517793i	$-0.826754\pi$
$409$	170.965	−	296.120i	0.0206691	−	0.0358000i	0.876175	+	0.481993i	$0.160087\pi$
$410$		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	0.565784	−	0.824554i	$-0.308574\pi$
$419$	1154.35	+	666.463i	0.134591	+	0.0777060i	−0.431193	+	0.902260i	$0.641907\pi$
$420$		0			0
$421$	5457.42	−	3150.84i	0.631777	−	0.364757i	−0.149663	−	0.988737i	$-0.547819\pi$
$421$	5457.42	−	3150.84i	0.631777	−	0.364757i	0.781440	+	0.623980i	$0.214485\pi$
$422$		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	$-0.531916\pi$
$431$	−1791.33			−0.200198			−0.100099	+	0.994977i	$0.531916\pi$
$432$		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	$-0.448951\pi$
$439$	10066.8	−	5812.07i	1.09445	−	0.631879i	0.934757	+	0.355288i	$0.115617\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12771.4	−	7373.55i	1.36972	−	0.790808i	0.378827	−	0.925467i	$-0.376328\pi$
$443$	12771.4	−	7373.55i	1.36972	−	0.790808i	0.990892	+	0.134660i	$0.0429942\pi$
$444$		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	$-0.854213\pi$
$449$		−	8413.52i		−	0.884317i	0.896937	−	0.442159i	$-0.145787\pi$
$450$		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	−0.787584	−	0.616208i	$-0.788668\pi$
$457$	−9060.70	−	15693.6i	−0.927443	−	1.60638i	−0.139860	−	0.990171i	$-0.544665\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8300.14	+	14376.3i	0.838560	+	1.45243i	0.891099	+	0.453809i	$0.149935\pi$
$461$	8300.14	+	14376.3i	0.838560	+	1.45243i	−0.0525393	+	0.998619i	$0.516731\pi$
$462$		0			0
$463$	−9006.61	−	5199.97i	−0.904045	−	0.521950i	−0.0255344	−	0.999674i	$-0.508129\pi$
$463$	−9006.61	−	5199.97i	−0.904045	−	0.521950i	−0.878510	+	0.477724i	$0.841462\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4926.33i			0.488144i	0.969757	+	0.244072i	$0.0784833\pi$
$467$			4926.33i			0.488144i	−0.969757	+	0.244072i	$0.921517\pi$
$468$		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	$-0.195072\pi$
$479$	−934.274	−	1618.21i	−0.0891191	−	0.154359i	−0.907139	+	0.420831i	$0.861739\pi$
$480$		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	0.444338	−	0.895859i	$-0.353439\pi$
$491$	−1189.49	−	686.750i	−0.109329	−	0.0631214i	−0.553668	+	0.832738i	$0.686772\pi$
$492$		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	−0.821545	−	0.570144i	$-0.806887\pi$
$499$	925.039	−	1602.21i	0.0829868	−	0.143737i	0.904532	+	0.426407i	$0.140221\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17948.3			1.59100			0.795502	−	0.605951i	$-0.207207\pi$
$503$	17948.3			1.59100			0.795502	+	0.605951i	$0.207207\pi$
$504$		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	$-0.829698\pi$
$509$	131.163	−	227.180i	0.0114218	−	0.0197831i	0.871680	+	0.490076i	$0.163031\pi$
$510$		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	$0.0125648\pi$
$521$			938.595i			0.0789263i	−0.999221	+	0.0394631i	$0.987435\pi$
$522$		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	0.899141	−	0.437658i	$-0.144192\pi$
$547$	902.529	+	1563.23i	0.0705473	+	0.122192i	−0.828594	+	0.559850i	$0.810859\pi$
$548$		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	$-0.925246\pi$
$557$	−25569.6			−1.94510			−0.972550	+	0.232694i	$0.925246\pi$
$558$		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	0.332251	−	0.943191i	$-0.392192\pi$
$563$	−4254.07	−	2456.09i	−0.318451	−	0.183858i	−0.650702	+	0.759333i	$0.725525\pi$
$564$		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	−0.898710	−	0.438544i	$-0.855494\pi$
$569$	−13142.2	+	7587.64i	−0.968275	+	0.559034i	−0.0695650	+	0.997577i	$0.522161\pi$
$570$		0			0
$571$	−6558.52	+	11359.7i	−0.480675	+	0.832554i	−0.999754	−	0.0221726i	$-0.992942\pi$
$571$	−6558.52	+	11359.7i	−0.480675	+	0.832554i	0.519079	+	0.854726i	$0.326275\pi$
$572$		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	−0.890765	−	0.454464i	$-0.849831\pi$
$587$	−13405.1	+	7739.45i	−0.942570	+	0.544193i	−0.0518051	+	0.998657i	$0.516497\pi$
$588$		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	$0.154813\pi$
$593$			13499.3i			0.934824i	−0.884040	+	0.467412i	$0.845187\pi$
$594$		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	$-0.994199\pi$
$599$	−7097.54	+	12293.3i	−0.484136	+	0.838548i	0.515698	+	0.856771i	$0.327533\pi$
$600$		0			0
$601$	505.327	+	875.253i	0.0342974	+	0.0594048i	0.882665	−	0.470003i	$-0.155747\pi$
$601$	505.327	+	875.253i	0.0342974	+	0.0594048i	−0.848367	+	0.529408i	$0.822414\pi$
$602$		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	$-0.236369\pi$
$607$	7769.04	+	4485.46i	0.519498	+	0.299933i	−0.217231	+	0.976120i	$0.569702\pi$
$608$		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	−0.227764	−	0.973716i	$-0.573142\pi$
$617$	−18159.9	−	10484.6i	−1.18491	−	0.684108i	−0.957145	+	0.289608i	$0.906475\pi$
$618$		0			0
$619$	3168.60	+	5488.18i	0.205746	+	0.356363i	0.950370	−	0.311121i	$-0.100705\pi$
$619$	3168.60	+	5488.18i	0.205746	+	0.356363i	−0.744624	+	0.667484i	$0.767371\pi$
$620$		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	0.357647	−	0.933857i	$-0.383579\pi$
$641$	21831.2	−	12604.3i	1.34521	−	0.776660i	0.987567	+	0.157197i	$0.0502458\pi$
$642$		0			0
$643$	−3931.26	+	6809.13i	−0.241110	+	0.417614i	−0.961031	−	0.276442i	$-0.910845\pi$
$643$	−3931.26	+	6809.13i	−0.241110	+	0.417614i	0.719921	+	0.694056i	$0.244178\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−188.733			−0.0114681			−0.00573404	−	0.999984i	$-0.501825\pi$
$647$	−188.733			−0.0114681			−0.00573404	+	0.999984i	$0.501825\pi$
$648$		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	0.106790	−	0.994282i	$-0.465943\pi$
$653$	15259.4	−	26430.1i	0.914468	−	1.58390i	0.807678	−	0.589623i	$-0.200724\pi$
$654$		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	−0.906292	−	0.422651i	$-0.861100\pi$
$659$	−16805.7	+	9702.80i	−0.993412	+	0.573547i	−0.0871197	+	0.996198i	$0.527766\pi$
$660$		0			0
$661$	−26953.4	−	15561.6i	−1.58603	−	0.915696i	−0.993952	−	0.109816i	$-0.964974\pi$
$661$	−26953.4	−	15561.6i	−1.58603	−	0.915696i	−0.592079	−	0.805880i	$-0.701693\pi$
$662$		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	0.747254	+	0.664539i	$0.231372\pi$
$673$	16571.1	+	28701.9i	0.949134	+	1.64395i	0.201880	+	0.979410i	$0.435295\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3831.16	−	6635.77i	−0.217494	−	0.376711i	0.736547	−	0.676386i	$-0.236455\pi$
$677$	−3831.16	−	6635.77i	−0.217494	−	0.376711i	−0.954041	+	0.299675i	$0.903122\pi$
$678$		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	$0.0740795\pi$
$683$			8233.45i			0.461265i	−0.973041	+	0.230633i	$0.925920\pi$
$684$		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	0.999971	+	0.00760808i	$0.00242175\pi$
$691$	9201.53	+	15937.5i	0.506574	+	0.877412i	−0.493397	+	0.869804i	$0.664245\pi$
$692$		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	$-0.477539\pi$
$701$	2617.11			0.141008			0.0705041	+	0.997511i	$0.477539\pi$
$702$		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	$-0.821790\pi$
$709$	−15311.9	+	8840.30i	−0.811070	+	0.468271i	0.0362573	+	0.999342i	$0.488456\pi$
$710$		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	$-0.564562\pi$
$719$	−7767.21			−0.402876			−0.201438	+	0.979501i	$0.564562\pi$
$720$		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	$-0.753886\pi$
$727$	−9187.03	+	5304.13i	−0.468677	+	0.270591i	0.247009	+	0.969013i	$0.420552\pi$
$728$		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	$-0.261050\pi$
$733$	7738.67	+	4467.92i	0.389951	+	0.225138i	−0.292188	+	0.956361i	$0.594383\pi$
$734$		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	$-0.844524\pi$
$743$	−711.865	+	1232.99i	−0.0351491	+	0.0608801i	0.847916	+	0.530131i	$0.177857\pi$
$744$		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	$-0.484612\pi$
$751$	−16310.8	−	9417.02i	−0.792527	−	0.457566i	−0.840851	+	0.541266i	$0.817945\pi$
$752$		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	0.0737237	−	0.997279i	$-0.476512\pi$
$761$	−15809.6	−	9127.66i	−0.753083	−	0.434793i	−0.826807	+	0.562486i	$0.809845\pi$
$762$		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	−0.990324	−	0.138774i	$-0.955684\pi$
$769$	−7996.45	+	13850.3i	−0.374980	+	0.649484i	0.615344	+	0.788259i	$0.289017\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−23733.1			−1.10429			−0.552147	−	0.833747i	$-0.686191\pi$
$773$	−23733.1			−1.10429			−0.552147	+	0.833747i	$0.686191\pi$
$774$		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	−0.938761	−	0.344569i	$-0.888025\pi$
$787$	−3774.80	+	6538.15i	−0.170975	+	0.296137i	0.767786	+	0.640706i	$0.221358\pi$
$788$		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	0.393831	+	0.919183i	$0.371149\pi$
$797$	−13480.4	+	23348.7i	−0.599120	+	1.03771i	−0.992951	+	0.118524i	$0.962184\pi$
$798$		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	$0.261941\pi$
$809$			33739.1i			1.46626i	−0.680090	+	0.733129i	$0.738059\pi$
$810$		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	$-0.0376716\pi$
$821$	9274.38	+	16063.7i	0.394248	+	0.682858i	−0.598756	+	0.800931i	$0.704338\pi$
$822$		0			0
$823$	−1262.62	−	728.972i	−0.0534775	−	0.0308753i	0.473023	−	0.881050i	$-0.343163\pi$
$823$	−1262.62	−	728.972i	−0.0534775	−	0.0308753i	−0.526500	+	0.850175i	$0.676496\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			42187.5i			1.77388i	0.461881	+	0.886942i	$0.347175\pi$
$827$			42187.5i			1.77388i	−0.461881	+	0.886942i	$0.652825\pi$
$828$		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	−0.622476	−	0.782639i	$-0.713873\pi$
$839$	−24035.3	−	41630.3i	−0.989023	−	1.71304i	−0.366548	−	0.930399i	$-0.619460\pi$
$840$		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	$-0.555950\pi$
$853$	14708.0	−	8491.65i	0.590377	−	0.340854i	0.765246	+	0.643737i	$0.222617\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22340.4	+	12898.3i	−0.890473	+	0.514115i	−0.874097	−	0.485751i	$-0.838546\pi$
$857$	−22340.4	+	12898.3i	−0.890473	+	0.514115i	−0.0163755	+	0.999866i	$0.505213\pi$
$858$		0			0
$859$	9914.38	−	17172.2i	0.393800	−	0.682082i	−0.599147	−	0.800639i	$-0.704494\pi$
$859$	9914.38	−	17172.2i	0.393800	−	0.682082i	0.992947	+	0.118557i	$0.0378269\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−42561.4			−1.67880			−0.839401	−	0.543513i	$-0.817094\pi$
$863$	−42561.4			−1.67880			−0.839401	+	0.543513i	$0.817094\pi$
$864$		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	$-0.237508\pi$
$877$	13338.6	+	7701.03i	0.513582	+	0.296517i	−0.220723	+	0.975337i	$0.570842\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			408.896i			0.0156368i	0.999969	+	0.00781842i	$0.00248871\pi$
$881$			408.896i			0.0156368i	−0.999969	+	0.00781842i	$0.997511\pi$
$882$		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	$-0.772538\pi$
$887$	5014.93	−	8686.11i	0.189836	−	0.328806i	0.945196	+	0.326505i	$0.105871\pi$
$888$		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	0.788788	−	0.614666i	$-0.210709\pi$
$907$	−3767.44	−	6525.40i	−0.137923	−	0.238889i	−0.926710	+	0.375777i	$0.877376\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11159.9	+	19329.6i	0.405867	+	0.702983i	0.994422	−	0.105475i	$-0.0336362\pi$
$911$	11159.9	+	19329.6i	0.405867	+	0.702983i	−0.588555	+	0.808457i	$0.700303\pi$
$912$		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	−0.0989336	−	0.995094i	$-0.531543\pi$
$929$	20199.6	−	11662.2i	0.713376	−	0.411868i	0.812310	+	0.583226i	$0.198210\pi$
$930$		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	$-0.848680\pi$
$941$	−1391.16	+	2409.56i	−0.0481939	+	0.0834743i	0.840922	+	0.541156i	$0.182013\pi$
$942$		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	−0.211167	−	0.977450i	$-0.567726\pi$
$947$	15438.1	−	8913.16i	0.529746	−	0.305849i	0.740913	+	0.671601i	$0.234393\pi$
$948$		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	$-0.880880\pi$
$953$		−	21508.9i		−	0.731104i	0.930791	−	0.365552i	$-0.119120\pi$
$954$		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	$-0.281707\pi$
$967$	8410.42	+	4855.76i	0.279691	+	0.161479i	−0.353593	+	0.935399i	$0.615040\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1908.56i			0.0630777i	0.999503	+	0.0315389i	$0.0100408\pi$
$971$			1908.56i			0.0630777i	−0.999503	+	0.0315389i	$0.989959\pi$
$972$		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	0.986704	−	0.162531i	$-0.0519656\pi$
$977$	40899.7	+	23613.4i	1.33930	+	0.773245i	0.352596	+	0.935776i	$0.385299\pi$
$978$		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	$-0.0418520\pi$
$983$	11777.7	+	20399.5i	0.382146	+	0.661895i	−0.609223	+	0.792999i	$0.708519\pi$
$984$		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	$-0.687699\pi$
$997$	−3600.15	+	2078.55i	−0.114361	+	0.0660263i	0.441729	+	0.897149i	$0.354365\pi$
$998$		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.719.15		64
3.2	odd	2		288.4.p.b.239.8		64
4.3	odd	2		216.4.l.b.179.13		64
8.3	odd	2	inner	864.4.p.b.719.18		64
8.5	even	2		216.4.l.b.179.3		64
9.2	odd	6	inner	864.4.p.b.143.18		64
9.7	even	3		288.4.p.b.47.7		64
12.11	even	2		72.4.l.b.59.20	yes	64
24.5	odd	2		72.4.l.b.59.30	yes	64
24.11	even	2		288.4.p.b.239.7		64
36.7	odd	6		72.4.l.b.11.30	yes	64
36.11	even	6		216.4.l.b.35.3		64
72.11	even	6	inner	864.4.p.b.143.15		64
72.29	odd	6		216.4.l.b.35.13		64
72.43	odd	6		288.4.p.b.47.8		64
72.61	even	6		72.4.l.b.11.20	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.b.11.20	✓	64	72.61	even	6
72.4.l.b.11.30	yes	64	36.7	odd	6
72.4.l.b.59.20	yes	64	12.11	even	2
72.4.l.b.59.30	yes	64	24.5	odd	2
216.4.l.b.35.3		64	36.11	even	6
216.4.l.b.35.13		64	72.29	odd	6
216.4.l.b.179.3		64	8.5	even	2
216.4.l.b.179.13		64	4.3	odd	2
288.4.p.b.47.7		64	9.7	even	3
288.4.p.b.47.8		64	72.43	odd	6
288.4.p.b.239.7		64	24.11	even	2
288.4.p.b.239.8		64	3.2	odd	2
864.4.p.b.143.15		64	72.11	even	6	inner
864.4.p.b.143.18		64	9.2	odd	6	inner
864.4.p.b.719.15		64	1.1	even	1	trivial
864.4.p.b.719.18		64	8.3	odd	2	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.719.15

Level	$864$
Weight	$4$
Character	864.719
Analytic conductor	$50.978$
Analytic rank	$0$
Dimension	$64$
Inner twists	$4$