LMFDB - Embedded newform 576.3.m.b.271.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,3,Mod(271,576)] 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("576.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 $ x^16 - 6x^14 + 10x^12 + 88x^10 - 752x^8 + 1408x^6 + 2560x^4 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			271.8
Root			$1.99750 - 0.0999235i$ of defining polynomial
Character	$\chi$	$=$	576.271
Dual form			576.3.m.b.559.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.01265 + 6.01265i) q^{5} -8.23187 q^{7} +(6.51529 - 6.51529i) q^{11} +(8.82865 - 8.82865i) q^{13} +14.1753 q^{17} +(23.7488 + 23.7488i) q^{19} -9.42125 q^{23} +47.3040i q^{25} +(-23.7973 + 23.7973i) q^{29} +24.4148i q^{31} +(-49.4954 - 49.4954i) q^{35} +(24.2052 + 24.2052i) q^{37} +6.67771i q^{41} +(-0.897918 + 0.897918i) q^{43} +25.2401i q^{47} +18.7636 q^{49} +(-32.6251 - 32.6251i) q^{53} +78.3484 q^{55} +(-8.31871 + 8.31871i) q^{59} +(-68.4028 + 68.4028i) q^{61} +106.167 q^{65} +(7.71922 + 7.71922i) q^{67} +137.259 q^{71} -52.8655i q^{73} +(-53.6330 + 53.6330i) q^{77} +87.2269i q^{79} +(9.53893 + 9.53893i) q^{83} +(85.2314 + 85.2314i) q^{85} -146.488i q^{89} +(-72.6762 + 72.6762i) q^{91} +285.586i q^{95} +101.170 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{19} + 96 q^{37} + 32 q^{43} + 112 q^{49} + 256 q^{55} - 32 q^{61} + 256 q^{67} + 160 q^{85} - 288 q^{91}+O(q^{100}) $ 16 * q + 32 * q^19 + 96 * q^37 + 32 * q^43 + 112 * q^49 + 256 * q^55 - 32 * q^61 + 256 * q^67 + 160 * q^85 - 288 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.01265	+	6.01265i	1.20253	+	1.20253i	0.973394	+	0.229136i	$0.0735901\pi$
$5$	6.01265	+	6.01265i	1.20253	+	1.20253i	0.229136	+	0.973394i	$0.426410\pi$
$6$		0			0
$7$	−8.23187			−1.17598			−0.587990	−	0.808868i	$-0.700081\pi$
$7$	−8.23187			−1.17598			−0.587990	+	0.808868i	$0.700081\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.51529	−	6.51529i	0.592299	−	0.592299i	−0.345953	−	0.938252i	$-0.612444\pi$
$11$	6.51529	−	6.51529i	0.592299	−	0.592299i	0.938252	+	0.345953i	$0.112444\pi$
$12$		0			0
$13$	8.82865	−	8.82865i	0.679127	−	0.679127i	−0.280676	−	0.959803i	$-0.590559\pi$
$13$	8.82865	−	8.82865i	0.679127	−	0.679127i	0.959803	+	0.280676i	$0.0905586\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.1753			0.833844			0.416922	−	0.908942i	$-0.363109\pi$
$17$	14.1753			0.833844			0.416922	+	0.908942i	$0.363109\pi$
$18$		0			0
$19$	23.7488	+	23.7488i	1.24994	+	1.24994i	0.955748	+	0.294187i	$0.0950490\pi$
$19$	23.7488	+	23.7488i	1.24994	+	1.24994i	0.294187	+	0.955748i	$0.404951\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.42125			−0.409619			−0.204810	−	0.978802i	$-0.565658\pi$
$23$	−9.42125			−0.409619			−0.204810	+	0.978802i	$0.565658\pi$
$24$		0			0
$25$			47.3040i			1.89216i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−23.7973	+	23.7973i	−0.820598	+	0.820598i	−0.986194	−	0.165596i	$-0.947045\pi$
$29$	−23.7973	+	23.7973i	−0.820598	+	0.820598i	0.165596	+	0.986194i	$0.447045\pi$
$30$		0			0
$31$			24.4148i			0.787575i	0.919202	+	0.393787i	$0.128835\pi$
$31$			24.4148i			0.787575i	−0.919202	+	0.393787i	$0.871165\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−49.4954	−	49.4954i	−1.41415	−	1.41415i
$36$		0			0
$37$	24.2052	+	24.2052i	0.654193	+	0.654193i	0.954000	−	0.299807i	$-0.0969221\pi$
$37$	24.2052	+	24.2052i	0.654193	+	0.654193i	−0.299807	+	0.954000i	$0.596922\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.67771i			0.162871i	0.996679	+	0.0814355i	$0.0259505\pi$
$41$			6.67771i			0.162871i	−0.996679	+	0.0814355i	$0.974050\pi$
$42$		0			0
$43$	−0.897918	+	0.897918i	−0.0208818	+	0.0208818i	−0.717471	−	0.696589i	$-0.754700\pi$
$43$	−0.897918	+	0.897918i	−0.0208818	+	0.0208818i	0.696589	+	0.717471i	$0.254700\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			25.2401i			0.537023i	0.963276	+	0.268512i	$0.0865318\pi$
$47$			25.2401i			0.537023i	−0.963276	+	0.268512i	$0.913468\pi$
$48$		0			0
$49$	18.7636			0.382931
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−32.6251	−	32.6251i	−0.615568	−	0.615568i	0.328824	−	0.944391i	$-0.393348\pi$
$53$	−32.6251	−	32.6251i	−0.615568	−	0.615568i	−0.944391	+	0.328824i	$0.893348\pi$
$54$		0			0
$55$	78.3484			1.42452
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.31871	+	8.31871i	−0.140995	+	0.140995i	−0.774081	−	0.633086i	$-0.781788\pi$
$59$	−8.31871	+	8.31871i	−0.140995	+	0.140995i	0.633086	+	0.774081i	$0.281788\pi$
$60$		0			0
$61$	−68.4028	+	68.4028i	−1.12136	+	1.12136i	−0.129820	+	0.991538i	$0.541440\pi$
$61$	−68.4028	+	68.4028i	−1.12136	+	1.12136i	−0.991538	+	0.129820i	$0.958560\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	106.167			1.63334
$66$		0			0
$67$	7.71922	+	7.71922i	0.115212	+	0.115212i	0.762362	−	0.647150i	$-0.224039\pi$
$67$	7.71922	+	7.71922i	0.115212	+	0.115212i	−0.647150	+	0.762362i	$0.724039\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	137.259			1.93322			0.966612	−	0.256245i	$-0.0824853\pi$
$71$	137.259			1.93322			0.966612	+	0.256245i	$0.0824853\pi$
$72$		0			0
$73$		−	52.8655i		−	0.724185i	−0.932142	−	0.362093i	$-0.882062\pi$
$73$		−	52.8655i		−	0.724185i	0.932142	−	0.362093i	$-0.117938\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−53.6330	+	53.6330i	−0.696533	+	0.696533i
$78$		0			0
$79$			87.2269i			1.10414i	0.833798	+	0.552069i	$0.186162\pi$
$79$			87.2269i			1.10414i	−0.833798	+	0.552069i	$0.813838\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.53893	+	9.53893i	0.114927	+	0.114927i	0.762231	−	0.647305i	$-0.224104\pi$
$83$	9.53893	+	9.53893i	0.114927	+	0.114927i	−0.647305	+	0.762231i	$0.724104\pi$
$84$		0			0
$85$	85.2314	+	85.2314i	1.00272	+	1.00272i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	146.488i		−	1.64593i	−0.568089	−	0.822967i	$-0.692317\pi$
$89$		−	146.488i		−	1.64593i	0.568089	−	0.822967i	$-0.307683\pi$
$90$		0			0
$91$	−72.6762	+	72.6762i	−0.798640	+	0.798640i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			285.586i			3.00617i
$96$		0			0
$97$	101.170			1.04298			0.521492	−	0.853256i	$-0.325376\pi$
$97$	101.170			1.04298			0.521492	+	0.853256i	$0.325376\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−47.4912	−	47.4912i	−0.470210	−	0.470210i	0.431773	−	0.901982i	$-0.357888\pi$
$101$	−47.4912	−	47.4912i	−0.470210	−	0.470210i	−0.901982	+	0.431773i	$0.857888\pi$
$102$		0			0
$103$	−7.58518			−0.0736425			−0.0368213	−	0.999322i	$-0.511723\pi$
$103$	−7.58518			−0.0736425			−0.0368213	+	0.999322i	$0.511723\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	119.757	−	119.757i	1.11922	−	1.11922i	0.127367	−	0.991856i	$-0.459347\pi$
$107$	119.757	−	119.757i	1.11922	−	1.11922i	0.991856	−	0.127367i	$-0.0406525\pi$
$108$		0			0
$109$	61.0263	−	61.0263i	0.559874	−	0.559874i	−0.369397	−	0.929272i	$-0.620436\pi$
$109$	61.0263	−	61.0263i	0.559874	−	0.559874i	0.929272	+	0.369397i	$0.120436\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	31.6377			0.279980			0.139990	−	0.990153i	$-0.455293\pi$
$113$	31.6377			0.279980			0.139990	+	0.990153i	$0.455293\pi$
$114$		0			0
$115$	−56.6467	−	56.6467i	−0.492580	−	0.492580i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−116.690			−0.980585
$120$		0			0
$121$			36.1019i			0.298363i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−134.106	+	134.106i	−1.07285	+	1.07285i
$126$		0			0
$127$		−	87.8736i		−	0.691918i	−0.938250	−	0.345959i	$-0.887554\pi$
$127$		−	87.8736i		−	0.691918i	0.938250	−	0.345959i	$-0.112446\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−176.351	−	176.351i	−1.34619	−	1.34619i	−0.889761	−	0.456428i	$-0.849129\pi$
$131$	−176.351	−	176.351i	−1.34619	−	1.34619i	−0.456428	−	0.889761i	$-0.650871\pi$
$132$		0			0
$133$	−195.497	−	195.497i	−1.46990	−	1.46990i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	97.0224i		−	0.708192i	−0.935209	−	0.354096i	$-0.884789\pi$
$137$		−	97.0224i		−	0.708192i	0.935209	−	0.354096i	$-0.115211\pi$
$138$		0			0
$139$	130.221	−	130.221i	0.936841	−	0.936841i	−0.0612800	−	0.998121i	$-0.519518\pi$
$139$	130.221	−	130.221i	0.936841	−	0.936841i	0.998121	+	0.0612800i	$0.0195183\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	115.042i		−	0.804493i
$144$		0			0
$145$	−286.170			−1.97359
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−26.3592	−	26.3592i	−0.176907	−	0.176907i	0.613099	−	0.790006i	$-0.289923\pi$
$149$	−26.3592	−	26.3592i	−0.176907	−	0.176907i	−0.790006	+	0.613099i	$0.789923\pi$
$150$		0			0
$151$	−196.107			−1.29872			−0.649360	−	0.760481i	$-0.724963\pi$
$151$	−196.107			−1.29872			−0.649360	+	0.760481i	$0.724963\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−146.798	+	146.798i	−0.947083	+	0.947083i
$156$		0			0
$157$	98.5323	−	98.5323i	0.627594	−	0.627594i	−0.319868	−	0.947462i	$-0.603638\pi$
$157$	98.5323	−	98.5323i	0.627594	−	0.627594i	0.947462	+	0.319868i	$0.103638\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	77.5545			0.481705
$162$		0			0
$163$	90.6677	+	90.6677i	0.556244	+	0.556244i	0.928236	−	0.371992i	$-0.121325\pi$
$163$	90.6677	+	90.6677i	0.556244	+	0.556244i	−0.371992	+	0.928236i	$0.621325\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	181.602			1.08744			0.543719	−	0.839267i	$-0.317016\pi$
$167$	181.602			1.08744			0.543719	+	0.839267i	$0.317016\pi$
$168$		0			0
$169$			13.1100i			0.0775741i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.66532	+	2.66532i	−0.0154065	+	0.0154065i	−0.714768	−	0.699362i	$-0.753468\pi$
$173$	−2.66532	+	2.66532i	−0.0154065	+	0.0154065i	0.699362	+	0.714768i	$0.253468\pi$
$174$		0			0
$175$		−	389.400i		−	2.22514i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.7491	+	12.7491i	0.0712241	+	0.0712241i	0.741822	−	0.670597i	$-0.233962\pi$
$179$	12.7491	+	12.7491i	0.0712241	+	0.0712241i	−0.670597	+	0.741822i	$0.733962\pi$
$180$		0			0
$181$	−61.3009	−	61.3009i	−0.338679	−	0.338679i	0.517191	−	0.855870i	$-0.326978\pi$
$181$	−61.3009	−	61.3009i	−0.338679	−	0.338679i	−0.855870	+	0.517191i	$0.826978\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			291.074i			1.57338i
$186$		0			0
$187$	92.3566	−	92.3566i	0.493885	−	0.493885i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			80.1204i			0.419479i	0.977757	+	0.209739i	$0.0672615\pi$
$191$			80.1204i			0.419479i	−0.977757	+	0.209739i	$0.932738\pi$
$192$		0			0
$193$	28.7555			0.148992			0.0744961	−	0.997221i	$-0.476265\pi$
$193$	28.7555			0.148992			0.0744961	+	0.997221i	$0.476265\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	92.8030	+	92.8030i	0.471081	+	0.471081i	0.902264	−	0.431183i	$-0.141904\pi$
$197$	92.8030	+	92.8030i	0.471081	+	0.471081i	−0.431183	+	0.902264i	$0.641904\pi$
$198$		0			0
$199$	−165.555			−0.831934			−0.415967	−	0.909380i	$-0.636557\pi$
$199$	−165.555			−0.831934			−0.415967	+	0.909380i	$0.636557\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	195.897	−	195.897i	0.965008	−	0.965008i
$204$		0			0
$205$	−40.1507	+	40.1507i	−0.195857	+	0.195857i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	309.460			1.48067
$210$		0			0
$211$	−187.769	−	187.769i	−0.889902	−	0.889902i	0.104611	−	0.994513i	$-0.466640\pi$
$211$	−187.769	−	187.769i	−0.889902	−	0.889902i	−0.994513	+	0.104611i	$0.966640\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.7977			−0.0502220
$216$		0			0
$217$		−	200.980i		−	0.926173i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	125.149	−	125.149i	0.566286	−	0.566286i
$222$		0			0
$223$			307.192i			1.37754i	0.724979	+	0.688771i	$0.241849\pi$
$223$			307.192i			1.37754i	−0.724979	+	0.688771i	$0.758151\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−112.960	−	112.960i	−0.497621	−	0.497621i	0.413075	−	0.910697i	$-0.364455\pi$
$227$	−112.960	−	112.960i	−0.497621	−	0.497621i	−0.910697	+	0.413075i	$0.864455\pi$
$228$		0			0
$229$	−7.32210	−	7.32210i	−0.0319742	−	0.0319742i	0.690939	−	0.722913i	$-0.257197\pi$
$229$	−7.32210	−	7.32210i	−0.0319742	−	0.0319742i	−0.722913	+	0.690939i	$0.757197\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			115.408i			0.495315i	0.968848	+	0.247657i	$0.0796607\pi$
$233$			115.408i			0.495315i	−0.968848	+	0.247657i	$0.920339\pi$
$234$		0			0
$235$	−151.760	+	151.760i	−0.645787	+	0.645787i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	323.271i		−	1.35260i	−0.736627	−	0.676300i	$-0.763583\pi$
$239$		−	323.271i		−	1.35260i	0.736627	−	0.676300i	$-0.236417\pi$
$240$		0			0
$241$	118.056			0.489860			0.244930	−	0.969541i	$-0.421235\pi$
$241$	118.056			0.489860			0.244930	+	0.969541i	$0.421235\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	112.819	+	112.819i	0.460486	+	0.460486i
$246$		0			0
$247$	419.339			1.69773
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−166.229	+	166.229i	−0.662265	+	0.662265i	−0.955913	−	0.293648i	$-0.905130\pi$
$251$	−166.229	+	166.229i	−0.662265	+	0.662265i	0.293648	+	0.955913i	$0.405130\pi$
$252$		0			0
$253$	−61.3822	+	61.3822i	−0.242617	+	0.242617i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−342.745			−1.33364			−0.666820	−	0.745219i	$-0.732345\pi$
$257$	−342.745			−1.33364			−0.666820	+	0.745219i	$0.732345\pi$
$258$		0			0
$259$	−199.254	−	199.254i	−0.769319	−	0.769319i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−219.637			−0.835123			−0.417562	−	0.908649i	$-0.637115\pi$
$263$	−219.637			−0.835123			−0.417562	+	0.908649i	$0.637115\pi$
$264$		0			0
$265$		−	392.327i		−	1.48048i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−77.9061	+	77.9061i	−0.289614	+	0.289614i	−0.836928	−	0.547314i	$-0.815650\pi$
$269$	−77.9061	+	77.9061i	−0.289614	+	0.289614i	0.547314	+	0.836928i	$0.315650\pi$
$270$		0			0
$271$		−	155.092i		−	0.572294i	−0.958186	−	0.286147i	$-0.907625\pi$
$271$		−	155.092i		−	0.572294i	0.958186	−	0.286147i	$-0.0923746\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	308.199	+	308.199i	1.12072	+	1.12072i
$276$		0			0
$277$	−30.0601	−	30.0601i	−0.108520	−	0.108520i	0.650762	−	0.759282i	$-0.274450\pi$
$277$	−30.0601	−	30.0601i	−0.108520	−	0.108520i	−0.759282	+	0.650762i	$0.774450\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	265.590i		−	0.945162i	−0.881287	−	0.472581i	$-0.843323\pi$
$281$		−	265.590i		−	0.945162i	0.881287	−	0.472581i	$-0.156677\pi$
$282$		0			0
$283$	−160.221	+	160.221i	−0.566153	+	0.566153i	−0.931049	−	0.364895i	$-0.881105\pi$
$283$	−160.221	+	160.221i	−0.566153	+	0.566153i	0.364895	+	0.931049i	$0.381105\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	54.9700i		−	0.191533i
$288$		0			0
$289$	−88.0595			−0.304704
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−173.762	−	173.762i	−0.593044	−	0.593044i	0.345408	−	0.938453i	$-0.387740\pi$
$293$	−173.762	−	173.762i	−0.593044	−	0.593044i	−0.938453	+	0.345408i	$0.887740\pi$
$294$		0			0
$295$	−100.035			−0.339102
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−83.1769	+	83.1769i	−0.278183	+	0.278183i
$300$		0			0
$301$	7.39154	−	7.39154i	0.0245566	−	0.0245566i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−822.565			−2.69693
$306$		0			0
$307$	−136.842	−	136.842i	−0.445741	−	0.445741i	0.448195	−	0.893936i	$-0.352067\pi$
$307$	−136.842	−	136.842i	−0.445741	−	0.445741i	−0.893936	+	0.448195i	$0.852067\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−458.117			−1.47305			−0.736523	−	0.676412i	$-0.763534\pi$
$311$	−458.117			−1.47305			−0.736523	+	0.676412i	$0.763534\pi$
$312$		0			0
$313$			170.865i			0.545893i	0.962029	+	0.272947i	$0.0879982\pi$
$313$			170.865i			0.545893i	−0.962029	+	0.272947i	$0.912002\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	372.560	−	372.560i	1.17527	−	1.17527i	0.194334	−	0.980935i	$-0.437746\pi$
$317$	372.560	−	372.560i	1.17527	−	1.17527i	0.980935	−	0.194334i	$-0.0622544\pi$
$318$		0			0
$319$			310.093i			0.972079i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	336.647	+	336.647i	1.04225	+	1.04225i
$324$		0			0
$325$	417.630	+	417.630i	1.28502	+	1.28502i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	207.773i		−	0.631529i
$330$		0			0
$331$	177.685	−	177.685i	0.536812	−	0.536812i	−0.385779	−	0.922591i	$-0.626067\pi$
$331$	177.685	−	177.685i	0.536812	−	0.536812i	0.922591	+	0.385779i	$0.126067\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			92.8260i			0.277092i
$336$		0			0
$337$	491.680			1.45899			0.729496	−	0.683985i	$-0.239755\pi$
$337$	491.680			1.45899			0.729496	+	0.683985i	$0.239755\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	159.070	+	159.070i	0.466480	+	0.466480i
$342$		0			0
$343$	248.902			0.725661
$344$		0			0
$345$		0			0
$346$		0			0
$347$	176.788	−	176.788i	0.509477	−	0.509477i	−0.404889	−	0.914366i	$-0.632690\pi$
$347$	176.788	−	176.788i	0.509477	−	0.509477i	0.914366	+	0.404889i	$0.132690\pi$
$348$		0			0
$349$	−272.298	+	272.298i	−0.780223	+	0.780223i	−0.979868	−	0.199645i	$-0.936021\pi$
$349$	−272.298	+	272.298i	−0.780223	+	0.780223i	0.199645	+	0.979868i	$0.436021\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−168.099			−0.476200			−0.238100	−	0.971241i	$-0.576525\pi$
$353$	−168.099			−0.476200			−0.238100	+	0.971241i	$0.576525\pi$
$354$		0			0
$355$	825.290	+	825.290i	2.32476	+	2.32476i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	476.009			1.32593			0.662965	−	0.748651i	$-0.269298\pi$
$359$	476.009			1.32593			0.662965	+	0.748651i	$0.269298\pi$
$360$		0			0
$361$			767.008i			2.12468i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	317.862	−	317.862i	0.870855	−	0.870855i
$366$		0			0
$367$			104.894i			0.285814i	0.989736	+	0.142907i	$0.0456450\pi$
$367$			104.894i			0.285814i	−0.989736	+	0.142907i	$0.954355\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	268.565	+	268.565i	0.723896	+	0.723896i
$372$		0			0
$373$	−122.301	−	122.301i	−0.327884	−	0.327884i	0.523898	−	0.851781i	$-0.324477\pi$
$373$	−122.301	−	122.301i	−0.327884	−	0.327884i	−0.851781	+	0.523898i	$0.824477\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			420.197i			1.11458i
$378$		0			0
$379$	−75.5625	+	75.5625i	−0.199373	+	0.199373i	−0.799731	−	0.600358i	$-0.795025\pi$
$379$	−75.5625	+	75.5625i	−0.199373	+	0.199373i	0.600358	+	0.799731i	$0.295025\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	501.509i		−	1.30942i	−0.755878	−	0.654712i	$-0.772790\pi$
$383$		−	501.509i		−	1.30942i	0.755878	−	0.654712i	$-0.227210\pi$
$384$		0			0
$385$	−644.953			−1.67520
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.6053	−	22.6053i	−0.0581114	−	0.0581114i	0.677454	−	0.735565i	$-0.263083\pi$
$389$	−22.6053	−	22.6053i	−0.0581114	−	0.0581114i	−0.735565	+	0.677454i	$0.763083\pi$
$390$		0			0
$391$	−133.549			−0.341559
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−524.465	+	524.465i	−1.32776	+	1.32776i
$396$		0			0
$397$	−6.09563	+	6.09563i	−0.0153542	+	0.0153542i	−0.714742	−	0.699388i	$-0.753456\pi$
$397$	−6.09563	+	6.09563i	−0.0153542	+	0.0153542i	0.699388	+	0.714742i	$0.253456\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	598.130			1.49160			0.745798	−	0.666172i	$-0.232068\pi$
$401$	598.130			1.49160			0.745798	+	0.666172i	$0.232068\pi$
$402$		0			0
$403$	215.550	+	215.550i	0.534863	+	0.534863i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	315.407			0.774957
$408$		0			0
$409$			271.241i			0.663181i	0.943423	+	0.331591i	$0.107585\pi$
$409$			271.241i			0.663181i	−0.943423	+	0.331591i	$0.892415\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	68.4785	−	68.4785i	0.165808	−	0.165808i
$414$		0			0
$415$			114.709i			0.276406i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	412.163	+	412.163i	0.983681	+	0.983681i	0.999869	−	0.0161876i	$-0.00515289\pi$
$419$	412.163	+	412.163i	0.983681	+	0.983681i	−0.0161876	+	0.999869i	$0.505153\pi$
$420$		0			0
$421$	135.609	+	135.609i	0.322111	+	0.322111i	0.849576	−	0.527466i	$-0.176858\pi$
$421$	135.609	+	135.609i	0.322111	+	0.322111i	−0.527466	+	0.849576i	$0.676858\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			670.550i			1.57777i
$426$		0			0
$427$	563.083	−	563.083i	1.31869	−	1.31869i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	204.169i		−	0.473710i	−0.971545	−	0.236855i	$-0.923883\pi$
$431$		−	204.169i		−	0.473710i	0.971545	−	0.236855i	$-0.0761166\pi$
$432$		0			0
$433$	317.530			0.733325			0.366663	−	0.930354i	$-0.380500\pi$
$433$	317.530			0.733325			0.366663	+	0.930354i	$0.380500\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−223.743	−	223.743i	−0.511998	−	0.511998i
$438$		0			0
$439$	−226.050			−0.514921			−0.257461	−	0.966289i	$-0.582886\pi$
$439$	−226.050			−0.514921			−0.257461	+	0.966289i	$0.582886\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	157.388	−	157.388i	0.355278	−	0.355278i	−0.506791	−	0.862069i	$-0.669169\pi$
$443$	157.388	−	157.388i	0.355278	−	0.355278i	0.862069	+	0.506791i	$0.169169\pi$
$444$		0			0
$445$	880.782	−	880.782i	1.97929	−	1.97929i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−458.567			−1.02131			−0.510653	−	0.859787i	$-0.670596\pi$
$449$	−458.567			−1.02131			−0.510653	+	0.859787i	$0.670596\pi$
$450$		0			0
$451$	43.5072	+	43.5072i	0.0964684	+	0.0964684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−873.954			−1.92078
$456$		0			0
$457$			548.682i			1.20062i	0.799768	+	0.600309i	$0.204956\pi$
$457$			548.682i			1.20062i	−0.799768	+	0.600309i	$0.795044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	359.058	−	359.058i	0.778868	−	0.778868i	−0.200770	−	0.979638i	$-0.564344\pi$
$461$	359.058	−	359.058i	0.778868	−	0.778868i	0.979638	+	0.200770i	$0.0643444\pi$
$462$		0			0
$463$		−	126.697i		−	0.273643i	−0.990596	−	0.136822i	$-0.956311\pi$
$463$		−	126.697i		−	0.273643i	0.990596	−	0.136822i	$-0.0436887\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−526.834	−	526.834i	−1.12812	−	1.12812i	−0.990482	−	0.137643i	$-0.956047\pi$
$467$	−526.834	−	526.834i	−1.12812	−	1.12812i	−0.137643	−	0.990482i	$-0.543953\pi$
$468$		0			0
$469$	−63.5436	−	63.5436i	−0.135487	−	0.135487i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			11.7004i			0.0247366i
$474$		0			0
$475$	−1123.41	+	1123.41i	−2.36508	+	2.36508i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	712.831i		−	1.48817i	−0.668087	−	0.744083i	$-0.732887\pi$
$479$		−	712.831i		−	1.48817i	0.668087	−	0.744083i	$-0.267113\pi$
$480$		0			0
$481$	427.398			0.888560
$482$		0			0
$483$		0			0
$484$		0			0
$485$	608.297	+	608.297i	1.25422	+	1.25422i
$486$		0			0
$487$	−579.851			−1.19066			−0.595330	−	0.803481i	$-0.702979\pi$
$487$	−579.851			−1.19066			−0.595330	+	0.803481i	$0.702979\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−131.309	+	131.309i	−0.267432	+	0.267432i	−0.828065	−	0.560632i	$-0.810558\pi$
$491$	−131.309	+	131.309i	−0.267432	+	0.267432i	0.560632	+	0.828065i	$0.310558\pi$
$492$		0			0
$493$	−337.336	+	337.336i	−0.684251	+	0.684251i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1129.90			−2.27343
$498$		0			0
$499$	−329.208	−	329.208i	−0.659736	−	0.659736i	0.295582	−	0.955318i	$-0.404487\pi$
$499$	−329.208	−	329.208i	−0.659736	−	0.659736i	−0.955318	+	0.295582i	$0.904487\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	254.299			0.505564			0.252782	−	0.967523i	$-0.418654\pi$
$503$	254.299			0.505564			0.252782	+	0.967523i	$0.418654\pi$
$504$		0			0
$505$		−	571.096i		−	1.13088i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−326.512	+	326.512i	−0.641477	+	0.641477i	−0.950918	−	0.309442i	$-0.899858\pi$
$509$	−326.512	+	326.512i	−0.641477	+	0.641477i	0.309442	+	0.950918i	$0.399858\pi$
$510$		0			0
$511$			435.182i			0.851628i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−45.6071	−	45.6071i	−0.0885574	−	0.0885574i
$516$		0			0
$517$	164.447	+	164.447i	0.318079	+	0.318079i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	775.042i		−	1.48760i	−0.668400	−	0.743802i	$-0.733020\pi$
$521$		−	775.042i		−	1.48760i	0.668400	−	0.743802i	$-0.266980\pi$
$522$		0			0
$523$	461.875	−	461.875i	0.883126	−	0.883126i	−0.110725	−	0.993851i	$-0.535317\pi$
$523$	461.875	−	461.875i	0.883126	−	0.883126i	0.993851	+	0.110725i	$0.0353172\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			346.089i			0.656715i
$528$		0			0
$529$	−440.240			−0.832212
$530$		0			0
$531$		0			0
$532$		0			0
$533$	58.9551	+	58.9551i	0.110610	+	0.110610i
$534$		0			0
$535$	1440.11			2.69180
$536$		0			0
$537$		0			0
$538$		0			0
$539$	122.251	−	122.251i	0.226810	−	0.226810i
$540$		0			0
$541$	−426.071	+	426.071i	−0.787563	+	0.787563i	−0.981094	−	0.193531i	$-0.938006\pi$
$541$	−426.071	+	426.071i	−0.787563	+	0.787563i	0.193531	+	0.981094i	$0.438006\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	733.860			1.34653
$546$		0			0
$547$	111.138	+	111.138i	0.203178	+	0.203178i	0.801360	−	0.598182i	$-0.204110\pi$
$547$	111.138	+	111.138i	0.203178	+	0.203178i	−0.598182	+	0.801360i	$0.704110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1130.32			−2.05139
$552$		0			0
$553$		−	718.041i		−	1.29845i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−233.731	+	233.731i	−0.419625	+	0.419625i	−0.885075	−	0.465449i	$-0.845893\pi$
$557$	−233.731	+	233.731i	−0.419625	+	0.419625i	0.465449	+	0.885075i	$0.345893\pi$
$558$		0			0
$559$			15.8548i			0.0283628i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−261.362	−	261.362i	−0.464231	−	0.464231i	0.435809	−	0.900039i	$-0.356462\pi$
$563$	−261.362	−	261.362i	−0.464231	−	0.464231i	−0.900039	+	0.435809i	$0.856462\pi$
$564$		0			0
$565$	190.227	+	190.227i	0.336684	+	0.336684i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			453.881i			0.797682i	0.917020	+	0.398841i	$0.130587\pi$
$569$			453.881i			0.797682i	−0.917020	+	0.398841i	$0.869413\pi$
$570$		0			0
$571$	289.009	−	289.009i	0.506146	−	0.506146i	−0.407195	−	0.913341i	$-0.633493\pi$
$571$	289.009	−	289.009i	0.506146	−	0.506146i	0.913341	+	0.407195i	$0.133493\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	445.662i		−	0.775065i
$576$		0			0
$577$	588.041			1.01913			0.509567	−	0.860431i	$-0.329806\pi$
$577$	588.041			1.01913			0.509567	+	0.860431i	$0.329806\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−78.5232	−	78.5232i	−0.135152	−	0.135152i
$582$		0			0
$583$	−425.124			−0.729201
$584$		0			0
$585$		0			0
$586$		0			0
$587$	392.596	−	392.596i	0.668818	−	0.668818i	−0.288624	−	0.957442i	$-0.593198\pi$
$587$	392.596	−	392.596i	0.668818	−	0.668818i	0.957442	+	0.288624i	$0.0931978\pi$
$588$		0			0
$589$	−579.822	+	579.822i	−0.984417	+	0.984417i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1008.70			1.70101			0.850505	−	0.525967i	$-0.176297\pi$
$593$	1008.70			1.70101			0.850505	+	0.525967i	$0.176297\pi$
$594$		0			0
$595$	−701.614	−	701.614i	−1.17918	−	1.17918i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	526.819			0.879498			0.439749	−	0.898121i	$-0.355067\pi$
$599$	526.819			0.879498			0.439749	+	0.898121i	$0.355067\pi$
$600$		0			0
$601$		−	867.891i		−	1.44408i	−0.691853	−	0.722039i	$-0.743205\pi$
$601$		−	867.891i		−	1.44408i	0.691853	−	0.722039i	$-0.256795\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−217.068	+	217.068i	−0.358790	+	0.358790i
$606$		0			0
$607$		−	98.8497i		−	0.162850i	−0.996679	−	0.0814248i	$-0.974053\pi$
$607$		−	98.8497i		−	0.162850i	0.996679	−	0.0814248i	$-0.0259470\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	222.836	+	222.836i	0.364707	+	0.364707i
$612$		0			0
$613$	−616.976	−	616.976i	−1.00649	−	1.00649i	−0.999979	−	0.00650651i	$-0.997929\pi$
$613$	−616.976	−	616.976i	−1.00649	−	1.00649i	−0.00650651	−	0.999979i	$-0.502071\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1073.62i			1.74007i	0.492993	+	0.870033i	$0.335903\pi$
$617$			1073.62i			1.74007i	−0.492993	+	0.870033i	$0.664097\pi$
$618$		0			0
$619$	−397.119	+	397.119i	−0.641549	+	0.641549i	−0.950936	−	0.309387i	$-0.899876\pi$
$619$	−397.119	+	397.119i	−0.641549	+	0.641549i	0.309387	+	0.950936i	$0.399876\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1205.87i			1.93559i
$624$		0			0
$625$	−430.067			−0.688107
$626$		0			0
$627$		0			0
$628$		0			0
$629$	343.117	+	343.117i	0.545495	+	0.545495i
$630$		0			0
$631$	381.081			0.603931			0.301966	−	0.953319i	$-0.402357\pi$
$631$	381.081			0.603931			0.301966	+	0.953319i	$0.402357\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	528.354	−	528.354i	0.832053	−	0.832053i
$636$		0			0
$637$	165.657	−	165.657i	0.260059	−	0.260059i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	485.443			0.757321			0.378661	−	0.925536i	$-0.376385\pi$
$641$	485.443			0.757321			0.378661	+	0.925536i	$0.376385\pi$
$642$		0			0
$643$	−457.641	−	457.641i	−0.711728	−	0.711728i	0.255168	−	0.966897i	$-0.417869\pi$
$643$	−457.641	−	457.641i	−0.711728	−	0.711728i	−0.966897	+	0.255168i	$0.917869\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	845.780			1.30723			0.653617	−	0.756826i	$-0.273251\pi$
$647$	845.780			1.30723			0.653617	+	0.756826i	$0.273251\pi$
$648$		0			0
$649$			108.398i			0.167023i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.1039	−	11.1039i	0.0170044	−	0.0170044i	−0.698553	−	0.715558i	$-0.746173\pi$
$653$	11.1039	−	11.1039i	0.0170044	−	0.0170044i	0.715558	+	0.698553i	$0.246173\pi$
$654$		0			0
$655$		−	2120.67i		−	3.23766i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−296.711	−	296.711i	−0.450245	−	0.450245i	0.445191	−	0.895436i	$-0.353136\pi$
$659$	−296.711	−	296.711i	−0.450245	−	0.450245i	−0.895436	+	0.445191i	$0.853136\pi$
$660$		0			0
$661$	−0.564899	−	0.564899i	−0.000854613	−	0.000854613i	0.706679	−	0.707534i	$-0.250192\pi$
$661$	−0.564899	−	0.564899i	−0.000854613	−	0.000854613i	−0.707534	+	0.706679i	$0.750192\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2350.91i		−	3.53520i
$666$		0			0
$667$	224.201	−	224.201i	0.336133	−	0.336133i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			891.329i			1.32836i
$672$		0			0
$673$	−1091.24			−1.62146			−0.810731	−	0.585419i	$-0.800930\pi$
$673$	−1091.24			−1.62146			−0.810731	+	0.585419i	$0.800930\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−494.409	−	494.409i	−0.730294	−	0.730294i	0.240384	−	0.970678i	$-0.422727\pi$
$677$	−494.409	−	494.409i	−0.730294	−	0.730294i	−0.970678	+	0.240384i	$0.922727\pi$
$678$		0			0
$679$	−832.814			−1.22653
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−845.268	+	845.268i	−1.23758	+	1.23758i	−0.276594	+	0.960987i	$0.589206\pi$
$683$	−845.268	+	845.268i	−1.23758	+	1.23758i	−0.960987	+	0.276594i	$0.910794\pi$
$684$		0			0
$685$	583.362	−	583.362i	0.851623	−	0.851623i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−576.071			−0.836097
$690$		0			0
$691$	−457.715	−	457.715i	−0.662395	−	0.662395i	0.293549	−	0.955944i	$-0.405164\pi$
$691$	−457.715	−	457.715i	−0.662395	−	0.662395i	−0.955944	+	0.293549i	$0.905164\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1565.95			2.25316
$696$		0			0
$697$			94.6589i			0.135809i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	199.252	−	199.252i	0.284239	−	0.284239i	−0.550558	−	0.834797i	$-0.685585\pi$
$701$	199.252	−	199.252i	0.284239	−	0.284239i	0.834797	+	0.550558i	$0.185585\pi$
$702$		0			0
$703$			1149.69i			1.63540i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	390.941	+	390.941i	0.552958	+	0.552958i
$708$		0			0
$709$	−547.374	−	547.374i	−0.772036	−	0.772036i	0.206426	−	0.978462i	$-0.433817\pi$
$709$	−547.374	−	547.374i	−0.772036	−	0.772036i	−0.978462	+	0.206426i	$0.933817\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	230.018i		−	0.322606i
$714$		0			0
$715$	691.710	−	691.710i	0.967427	−	0.967427i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			90.5082i			0.125881i	0.998017	+	0.0629403i	$0.0200478\pi$
$719$			90.5082i			0.125881i	−0.998017	+	0.0629403i	$0.979952\pi$
$720$		0			0
$721$	62.4402			0.0866022
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1125.71	−	1125.71i	−1.55270	−	1.55270i
$726$		0			0
$727$	222.068			0.305459			0.152729	−	0.988268i	$-0.451194\pi$
$727$	222.068			0.305459			0.152729	+	0.988268i	$0.451194\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.7283	+	12.7283i	−0.0174122	+	0.0174122i
$732$		0			0
$733$	201.142	−	201.142i	0.274410	−	0.274410i	−0.556463	−	0.830873i	$-0.687842\pi$
$733$	201.142	−	201.142i	0.274410	−	0.274410i	0.830873	+	0.556463i	$0.187842\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	100.586			0.136480
$738$		0			0
$739$	771.595	+	771.595i	1.04411	+	1.04411i	0.998981	+	0.0451261i	$0.0143690\pi$
$739$	771.595	+	771.595i	1.04411	+	1.04411i	0.0451261	+	0.998981i	$0.485631\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−754.171			−1.01503			−0.507517	−	0.861641i	$-0.669437\pi$
$743$	−754.171			−1.01503			−0.507517	+	0.861641i	$0.669437\pi$
$744$		0			0
$745$		−	316.977i		−	0.425473i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−985.822	+	985.822i	−1.31618	+	1.31618i
$750$		0			0
$751$		−	606.477i		−	0.807559i	−0.914856	−	0.403780i	$-0.867696\pi$
$751$		−	606.477i		−	0.807559i	0.914856	−	0.403780i	$-0.132304\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1179.12	−	1179.12i	−1.56175	−	1.56175i
$756$		0			0
$757$	−852.254	−	852.254i	−1.12583	−	1.12583i	−0.990848	−	0.134983i	$-0.956902\pi$
$757$	−852.254	−	852.254i	−1.12583	−	1.12583i	−0.134983	−	0.990848i	$-0.543098\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1373.25i			1.80453i	0.431180	+	0.902266i	$0.358097\pi$
$761$			1373.25i			1.80453i	−0.431180	+	0.902266i	$0.641903\pi$
$762$		0			0
$763$	−502.360	+	502.360i	−0.658401	+	0.658401i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			146.886i			0.191507i
$768$		0			0
$769$	515.727			0.670647			0.335323	−	0.942103i	$-0.391154\pi$
$769$	515.727			0.670647			0.335323	+	0.942103i	$0.391154\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−448.985	−	448.985i	−0.580834	−	0.580834i	0.354299	−	0.935132i	$-0.384720\pi$
$773$	−448.985	−	448.985i	−0.580834	−	0.580834i	−0.935132	+	0.354299i	$0.884720\pi$
$774$		0			0
$775$	−1154.92			−1.49022
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−158.587	+	158.587i	−0.203578	+	0.203578i
$780$		0			0
$781$	894.282	−	894.282i	1.14505	−	1.14505i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1184.88			1.50940
$786$		0			0
$787$	−70.0485	−	70.0485i	−0.0890070	−	0.0890070i	0.661201	−	0.750208i	$-0.270047\pi$
$787$	−70.0485	−	70.0485i	−0.0890070	−	0.0890070i	−0.750208	+	0.661201i	$0.770047\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−260.437			−0.329251
$792$		0			0
$793$			1207.81i			1.52309i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−584.026	+	584.026i	−0.732780	+	0.732780i	−0.971170	−	0.238390i	$-0.923380\pi$
$797$	−584.026	+	584.026i	−0.732780	+	0.732780i	0.238390	+	0.971170i	$0.423380\pi$
$798$		0			0
$799$			357.787i			0.447794i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−344.434	−	344.434i	−0.428935	−	0.428935i
$804$		0			0
$805$	466.308	+	466.308i	0.579265	+	0.579265i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1252.15i		−	1.54778i	−0.633321	−	0.773889i	$-0.718309\pi$
$809$		−	1252.15i		−	1.54778i	0.633321	−	0.773889i	$-0.281691\pi$
$810$		0			0
$811$	255.775	−	255.775i	0.315382	−	0.315382i	−0.531608	−	0.846990i	$-0.678412\pi$
$811$	255.775	−	255.775i	0.315382	−	0.315382i	0.846990	+	0.531608i	$0.178412\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1090.31i			1.33780i
$816$		0			0
$817$	−42.6489			−0.0522018
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−228.380	−	228.380i	−0.278173	−	0.278173i	0.554206	−	0.832379i	$-0.313022\pi$
$821$	−228.380	−	228.380i	−0.278173	−	0.278173i	−0.832379	+	0.554206i	$0.813022\pi$
$822$		0			0
$823$	−1050.55			−1.27649			−0.638246	−	0.769832i	$-0.720340\pi$
$823$	−1050.55			−1.27649			−0.638246	+	0.769832i	$0.720340\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	78.0300	−	78.0300i	0.0943531	−	0.0943531i	−0.658355	−	0.752708i	$-0.728747\pi$
$827$	78.0300	−	78.0300i	0.0943531	−	0.0943531i	0.752708	+	0.658355i	$0.228747\pi$
$828$		0			0
$829$	−517.848	+	517.848i	−0.624665	+	0.624665i	−0.946721	−	0.322056i	$-0.895626\pi$
$829$	−517.848	+	517.848i	−0.624665	+	0.624665i	0.322056	+	0.946721i	$0.395626\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	265.981			0.319305
$834$		0			0
$835$	1091.91	+	1091.91i	1.30768	+	1.30768i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	649.581			0.774232			0.387116	−	0.922031i	$-0.373471\pi$
$839$	649.581			0.774232			0.387116	+	0.922031i	$0.373471\pi$
$840$		0			0
$841$		−	291.627i		−	0.346762i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−78.8260	+	78.8260i	−0.0932852	+	0.0932852i
$846$		0			0
$847$		−	297.186i		−	0.350869i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−228.043	−	228.043i	−0.267970	−	0.267970i
$852$		0			0
$853$	1124.04	+	1124.04i	1.31775	+	1.31775i	0.915551	+	0.402202i	$0.131755\pi$
$853$	1124.04	+	1124.04i	1.31775	+	1.31775i	0.402202	+	0.915551i	$0.368245\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	596.973i		−	0.696585i	−0.937386	−	0.348292i	$-0.886762\pi$
$857$		−	596.973i		−	0.696585i	0.937386	−	0.348292i	$-0.113238\pi$
$858$		0			0
$859$	−701.595	+	701.595i	−0.816758	+	0.816758i	−0.985637	−	0.168879i	$-0.945985\pi$
$859$	−701.595	+	701.595i	−0.816758	+	0.816758i	0.168879	+	0.985637i	$0.445985\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			437.703i			0.507187i	0.967311	+	0.253594i	$0.0816126\pi$
$863$			437.703i			0.507187i	−0.967311	+	0.253594i	$0.918387\pi$
$864$		0			0
$865$	−32.0513			−0.0370535
$866$		0			0
$867$		0			0
$868$		0			0
$869$	568.309	+	568.309i	0.653981	+	0.653981i
$870$		0			0
$871$	136.301			0.156487
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1103.94	−	1103.94i	1.26165	−	1.26165i
$876$		0			0
$877$	769.110	−	769.110i	0.876978	−	0.876978i	−0.116243	−	0.993221i	$-0.537085\pi$
$877$	769.110	−	769.110i	0.876978	−	0.876978i	0.993221	+	0.116243i	$0.0370850\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	351.923			0.399459			0.199729	−	0.979851i	$-0.435994\pi$
$881$	351.923			0.399459			0.199729	+	0.979851i	$0.435994\pi$
$882$		0			0
$883$	−661.014	−	661.014i	−0.748601	−	0.748601i	0.225616	−	0.974216i	$-0.427561\pi$
$883$	−661.014	−	661.014i	−0.748601	−	0.748601i	−0.974216	+	0.225616i	$0.927561\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−794.853			−0.896114			−0.448057	−	0.894005i	$-0.647884\pi$
$887$	−794.853			−0.896114			−0.448057	+	0.894005i	$0.647884\pi$
$888$		0			0
$889$			723.364i			0.813683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−599.421	+	599.421i	−0.671244	+	0.671244i
$894$		0			0
$895$			153.312i			0.171298i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−581.008	−	581.008i	−0.646282	−	0.646282i
$900$		0			0
$901$	−462.472	−	462.472i	−0.513288	−	0.513288i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	737.162i		−	0.814544i
$906$		0			0
$907$	943.805	−	943.805i	1.04058	−	1.04058i	0.0414379	−	0.999141i	$-0.486806\pi$
$907$	943.805	−	943.805i	1.04058	−	1.04058i	0.999141	−	0.0414379i	$-0.0131939\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			190.842i			0.209487i	0.994499	+	0.104743i	$0.0334021\pi$
$911$			190.842i			0.209487i	−0.994499	+	0.104743i	$0.966598\pi$
$912$		0			0
$913$	124.298			0.136142
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1451.70	+	1451.70i	1.58309	+	1.58309i
$918$		0			0
$919$	925.230			1.00678			0.503389	−	0.864060i	$-0.332086\pi$
$919$	925.230			1.00678			0.503389	+	0.864060i	$0.332086\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1211.81	−	1211.81i	1.31290	−	1.31290i
$924$		0			0
$925$	−1145.00	+	1145.00i	−1.23784	+	1.23784i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	91.8388			0.0988577			0.0494289	−	0.998778i	$-0.484260\pi$
$929$	91.8388			0.0988577			0.0494289	+	0.998778i	$0.484260\pi$
$930$		0			0
$931$	445.613	+	445.613i	0.478639	+	0.478639i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1110.62			1.18782
$936$		0			0
$937$		−	1615.44i		−	1.72406i	−0.506857	−	0.862030i	$-0.669193\pi$
$937$		−	1615.44i		−	1.72406i	0.506857	−	0.862030i	$-0.330807\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	484.931	−	484.931i	0.515336	−	0.515336i	−0.400820	−	0.916157i	$-0.631275\pi$
$941$	484.931	−	484.931i	0.515336	−	0.515336i	0.916157	+	0.400820i	$0.131275\pi$
$942$		0			0
$943$		−	62.9124i		−	0.0667151i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1053.02	+	1053.02i	1.11195	+	1.11195i	0.992887	+	0.119064i	$0.0379895\pi$
$947$	1053.02	+	1053.02i	1.11195	+	1.11195i	0.119064	+	0.992887i	$0.462010\pi$
$948$		0			0
$949$	−466.731	−	466.731i	−0.491814	−	0.491814i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1256.72i			1.31870i	0.751838	+	0.659348i	$0.229168\pi$
$953$			1256.72i			1.31870i	−0.751838	+	0.659348i	$0.770832\pi$
$954$		0			0
$955$	−481.736	+	481.736i	−0.504436	+	0.504436i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			798.675i			0.832821i
$960$		0			0
$961$	364.917			0.379726
$962$		0			0
$963$		0			0
$964$		0			0
$965$	172.897	+	172.897i	0.179168	+	0.179168i
$966$		0			0
$967$	1842.43			1.90530			0.952651	−	0.304066i	$-0.0983444\pi$
$967$	1842.43			1.90530			0.952651	+	0.304066i	$0.0983444\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−802.530	+	802.530i	−0.826498	+	0.826498i	−0.987031	−	0.160532i	$-0.948679\pi$
$971$	−802.530	+	802.530i	−0.826498	+	0.826498i	0.160532	+	0.987031i	$0.448679\pi$
$972$		0			0
$973$	−1071.96	+	1071.96i	−1.10171	+	1.10171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1505.64			1.54109			0.770544	−	0.637387i	$-0.219985\pi$
$977$	1505.64			1.54109			0.770544	+	0.637387i	$0.219985\pi$
$978$		0			0
$979$	−954.413	−	954.413i	−0.974886	−	0.974886i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−764.708			−0.777933			−0.388966	−	0.921252i	$-0.627168\pi$
$983$	−764.708			−0.777933			−0.388966	+	0.921252i	$0.627168\pi$
$984$		0			0
$985$			1115.98i			1.13298i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.45951	−	8.45951i	0.00855360	−	0.00855360i
$990$		0			0
$991$			1683.22i			1.69851i	0.527984	+	0.849254i	$0.322948\pi$
$991$			1683.22i			1.69851i	−0.527984	+	0.849254i	$0.677052\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−995.424	−	995.424i	−1.00043	−	1.00043i
$996$		0			0
$997$	942.082	+	942.082i	0.944917	+	0.944917i	0.998560	−	0.0536431i	$-0.0170833\pi$
$997$	942.082	+	942.082i	0.944917	+	0.944917i	−0.0536431	+	0.998560i	$0.517083\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.m.b.271.8		16
3.2	odd	2	inner	576.3.m.b.271.1		16
4.3	odd	2		144.3.m.b.91.5	yes	16
8.3	odd	2		1152.3.m.e.415.1		16
8.5	even	2		1152.3.m.d.415.1		16
12.11	even	2		144.3.m.b.91.4	yes	16
16.3	odd	4	inner	576.3.m.b.559.8		16
16.5	even	4		1152.3.m.e.991.1		16
16.11	odd	4		1152.3.m.d.991.1		16
16.13	even	4		144.3.m.b.19.5	yes	16
24.5	odd	2		1152.3.m.d.415.8		16
24.11	even	2		1152.3.m.e.415.8		16
48.5	odd	4		1152.3.m.e.991.8		16
48.11	even	4		1152.3.m.d.991.8		16
48.29	odd	4		144.3.m.b.19.4	✓	16
48.35	even	4	inner	576.3.m.b.559.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.m.b.19.4	✓	16	48.29	odd	4
144.3.m.b.19.5	yes	16	16.13	even	4
144.3.m.b.91.4	yes	16	12.11	even	2
144.3.m.b.91.5	yes	16	4.3	odd	2
576.3.m.b.271.1		16	3.2	odd	2	inner
576.3.m.b.271.8		16	1.1	even	1	trivial
576.3.m.b.559.1		16	48.35	even	4	inner
576.3.m.b.559.8		16	16.3	odd	4	inner
1152.3.m.d.415.1		16	8.5	even	2
1152.3.m.d.415.8		16	24.5	odd	2
1152.3.m.d.991.1		16	16.11	odd	4
1152.3.m.d.991.8		16	48.11	even	4
1152.3.m.e.415.1		16	8.3	odd	2
1152.3.m.e.415.8		16	24.11	even	2
1152.3.m.e.991.1		16	16.5	even	4
1152.3.m.e.991.8		16	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.01265 + 6.01265i) q^{5} -8.23187 q^{7} +(6.51529 - 6.51529i) q^{11} +(8.82865 - 8.82865i) q^{13} +14.1753 q^{17} +(23.7488 + 23.7488i) q^{19} -9.42125 q^{23} +47.3040i q^{25} +(-23.7973 + 23.7973i) q^{29} +24.4148i q^{31} +(-49.4954 - 49.4954i) q^{35} +(24.2052 + 24.2052i) q^{37} +6.67771i q^{41} +(-0.897918 + 0.897918i) q^{43} +25.2401i q^{47} +18.7636 q^{49} +(-32.6251 - 32.6251i) q^{53} +78.3484 q^{55} +(-8.31871 + 8.31871i) q^{59} +(-68.4028 + 68.4028i) q^{61} +106.167 q^{65} +(7.71922 + 7.71922i) q^{67} +137.259 q^{71} -52.8655i q^{73} +(-53.6330 + 53.6330i) q^{77} +87.2269i q^{79} +(9.53893 + 9.53893i) q^{83} +(85.2314 + 85.2314i) q^{85} -146.488i q^{89} +(-72.6762 + 72.6762i) q^{91} +285.586i q^{95} +101.170 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{19} + 96 q^{37} + 32 q^{43} + 112 q^{49} + 256 q^{55} - 32 q^{61} + 256 q^{67} + 160 q^{85} - 288 q^{91}+O(q^{100}) \) 16 * q + 32 * q^19 + 96 * q^37 + 32 * q^43 + 112 * q^49 + 256 * q^55 - 32 * q^61 + 256 * q^67 + 160 * q^85 - 288 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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