LMFDB - Embedded newform 576.5.m.a.559.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,5,Mod(271,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("576.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	576.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$59.5410987363$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 $ x^14 - 4x^13 + 15x^12 - 34x^11 + 62x^10 - 312x^9 + 1432x^8 - 4960x^7 + 11456x^6 - 19968x^5 + 31744x^4 - 139264x^3 + 491520x^2 - 1048576*x + 2097152
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			559.5
Root			$-2.40693 + 1.48549i$ of defining polynomial
Character	$\chi$	$=$	576.559
Dual form			576.5.m.a.271.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.04297 - 8.04297i) q^{5} +49.8797 q^{7} +O(q^{10})$	$q+(8.04297 - 8.04297i) q^{5} +49.8797 q^{7} +(-84.2573 - 84.2573i) q^{11} +(19.4838 + 19.4838i) q^{13} -437.855 q^{17} +(-349.021 + 349.021i) q^{19} -404.840 q^{23} +495.621i q^{25} +(1031.65 + 1031.65i) q^{29} -1506.15i q^{31} +(401.181 - 401.181i) q^{35} +(-434.262 + 434.262i) q^{37} +696.847i q^{41} +(-917.612 - 917.612i) q^{43} -111.917i q^{47} +86.9810 q^{49} +(-1041.19 + 1041.19i) q^{53} -1355.36 q^{55} +(-1711.60 - 1711.60i) q^{59} +(3711.24 + 3711.24i) q^{61} +313.415 q^{65} +(1854.18 - 1854.18i) q^{67} -1161.89 q^{71} +905.295i q^{73} +(-4202.72 - 4202.72i) q^{77} +5869.63i q^{79} +(-7560.06 + 7560.06i) q^{83} +(-3521.65 + 3521.65i) q^{85} +6439.80i q^{89} +(971.844 + 971.844i) q^{91} +5614.33i q^{95} -413.032 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 14 * q + 2 * q^5 + 4 * q^7	$ 14 q + 2 q^{5} + 4 q^{7} + 94 q^{11} - 2 q^{13} + 4 q^{17} + 706 q^{19} + 1148 q^{23} - 862 q^{29} + 1340 q^{35} - 1826 q^{37} - 1694 q^{43} + 682 q^{49} + 482 q^{53} + 11780 q^{55} - 2786 q^{59} - 3778 q^{61} + 2020 q^{65} - 7998 q^{67} + 19964 q^{71} + 9508 q^{77} - 17282 q^{83} + 9948 q^{85} + 28036 q^{91} - 4 q^{97}+O(q^{100}) $ 14 * q + 2 * q^5 + 4 * q^7 + 94 * q^11 - 2 * q^13 + 4 * q^17 + 706 * q^19 + 1148 * q^23 - 862 * q^29 + 1340 * q^35 - 1826 * q^37 - 1694 * q^43 + 682 * q^49 + 482 * q^53 + 11780 * q^55 - 2786 * q^59 - 3778 * q^61 + 2020 * q^65 - 7998 * q^67 + 19964 * q^71 + 9508 * q^77 - 17282 * q^83 + 9948 * q^85 + 28036 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$	8.04297	−	8.04297i	0.321719	−	0.321719i	−0.527707	−	0.849426i	$-0.676948\pi$
$5$	8.04297	−	8.04297i	0.321719	−	0.321719i	0.849426	+	0.527707i	$0.176948\pi$
$6$		0			0
$7$	49.8797			1.01795			0.508976	−	0.860781i	$-0.330024\pi$
$7$	49.8797			1.01795			0.508976	+	0.860781i	$0.330024\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−84.2573	−	84.2573i	−0.696341	−	0.696341i	0.267278	−	0.963619i	$-0.413876\pi$
$11$	−84.2573	−	84.2573i	−0.696341	−	0.696341i	−0.963619	+	0.267278i	$0.913876\pi$
$12$		0			0
$13$	19.4838	+	19.4838i	0.115289	+	0.115289i	0.762398	−	0.647109i	$-0.224022\pi$
$13$	19.4838	+	19.4838i	0.115289	+	0.115289i	−0.647109	+	0.762398i	$0.724022\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−437.855			−1.51507			−0.757535	−	0.652795i	$-0.773596\pi$
$17$	−437.855			−1.51507			−0.757535	+	0.652795i	$0.773596\pi$
$18$		0			0
$19$	−349.021	+	349.021i	−0.966817	+	0.966817i	−0.999467	−	0.0326494i	$-0.989606\pi$
$19$	−349.021	+	349.021i	−0.966817	+	0.966817i	0.0326494	+	0.999467i	$0.489606\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−404.840			−0.765293			−0.382647	−	0.923895i	$-0.624987\pi$
$23$	−404.840			−0.765293			−0.382647	+	0.923895i	$0.624987\pi$
$24$		0			0
$25$			495.621i			0.792994i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1031.65	+	1031.65i	1.22670	+	1.22670i	0.965206	+	0.261490i	$0.0842139\pi$
$29$	1031.65	+	1031.65i	1.22670	+	1.22670i	0.261490	+	0.965206i	$0.415786\pi$
$30$		0			0
$31$		−	1506.15i		−	1.56728i	−0.621217	−	0.783638i	$-0.713362\pi$
$31$		−	1506.15i		−	1.56728i	0.621217	−	0.783638i	$-0.286638\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	401.181	−	401.181i	0.327494	−	0.327494i
$36$		0			0
$37$	−434.262	+	434.262i	−0.317211	+	0.317211i	−0.847695	−	0.530484i	$-0.822010\pi$
$37$	−434.262	+	434.262i	−0.317211	+	0.317211i	0.530484	+	0.847695i	$0.322010\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			696.847i			0.414543i	0.978283	+	0.207272i	$0.0664584\pi$
$41$			696.847i			0.414543i	−0.978283	+	0.207272i	$0.933542\pi$
$42$		0			0
$43$	−917.612	−	917.612i	−0.496275	−	0.496275i	0.414002	−	0.910276i	$-0.364131\pi$
$43$	−917.612	−	917.612i	−0.496275	−	0.496275i	−0.910276	+	0.414002i	$0.864131\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	111.917i		−	0.0506641i	−0.999679	−	0.0253321i	$-0.991936\pi$
$47$		−	111.917i		−	0.0506641i	0.999679	−	0.0253321i	$-0.00806431\pi$
$48$		0			0
$49$	86.9810			0.0362270
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1041.19	+	1041.19i	−0.370663	+	0.370663i	−0.867719	−	0.497056i	$-0.834414\pi$
$53$	−1041.19	+	1041.19i	−0.370663	+	0.370663i	0.497056	+	0.867719i	$0.334414\pi$
$54$		0			0
$55$	−1355.36			−0.448052
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1711.60	−	1711.60i	−0.491697	−	0.491697i	0.417144	−	0.908841i	$-0.363031\pi$
$59$	−1711.60	−	1711.60i	−0.491697	−	0.491697i	−0.908841	+	0.417144i	$0.863031\pi$
$60$		0			0
$61$	3711.24	+	3711.24i	0.997376	+	0.997376i	0.999997	−	0.00262076i	$-0.000834216\pi$
$61$	3711.24	+	3711.24i	0.997376	+	0.997376i	−0.00262076	+	0.999997i	$0.500834\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	313.415			0.0741810
$66$		0			0
$67$	1854.18	−	1854.18i	0.413049	−	0.413049i	−0.469750	−	0.882799i	$-0.655656\pi$
$67$	1854.18	−	1854.18i	0.413049	−	0.413049i	0.882799	+	0.469750i	$0.155656\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1161.89			−0.230489			−0.115244	−	0.993337i	$-0.536765\pi$
$71$	−1161.89			−0.230489			−0.115244	+	0.993337i	$0.536765\pi$
$72$		0			0
$73$			905.295i			0.169881i	0.996386	+	0.0849404i	$0.0270700\pi$
$73$			905.295i			0.169881i	−0.996386	+	0.0849404i	$0.972930\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4202.72	−	4202.72i	−0.708842	−	0.708842i
$78$		0			0
$79$			5869.63i			0.940495i	0.882535	+	0.470247i	$0.155835\pi$
$79$			5869.63i			0.940495i	−0.882535	+	0.470247i	$0.844165\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7560.06	+	7560.06i	−1.09741	+	1.09741i	−0.102698	+	0.994713i	$0.532748\pi$
$83$	−7560.06	+	7560.06i	−1.09741	+	1.09741i	−0.994713	+	0.102698i	$0.967252\pi$
$84$		0			0
$85$	−3521.65	+	3521.65i	−0.487426	+	0.487426i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6439.80i			0.813004i	0.913650	+	0.406502i	$0.133252\pi$
$89$			6439.80i			0.813004i	−0.913650	+	0.406502i	$0.866748\pi$
$90$		0			0
$91$	971.844	+	971.844i	0.117358	+	0.117358i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5614.33i			0.622087i
$96$		0			0
$97$	−413.032			−0.0438976			−0.0219488	−	0.999759i	$-0.506987\pi$
$97$	−413.032			−0.0438976			−0.0219488	+	0.999759i	$0.506987\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8460.63	+	8460.63i	−0.829392	+	0.829392i	−0.987433	−	0.158041i	$-0.949482\pi$
$101$	−8460.63	+	8460.63i	−0.829392	+	0.829392i	0.158041	+	0.987433i	$0.449482\pi$
$102$		0			0
$103$	−17007.9			−1.60316			−0.801578	−	0.597891i	$-0.796006\pi$
$103$	−17007.9			−1.60316			−0.801578	+	0.597891i	$0.796006\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9368.65	+	9368.65i	0.818294	+	0.818294i	0.985861	−	0.167567i	$-0.0535909\pi$
$107$	9368.65	+	9368.65i	0.818294	+	0.818294i	−0.167567	+	0.985861i	$0.553591\pi$
$108$		0			0
$109$	−8308.73	−	8308.73i	−0.699329	−	0.699329i	0.264937	−	0.964266i	$-0.414649\pi$
$109$	−8308.73	−	8308.73i	−0.699329	−	0.699329i	−0.964266	+	0.264937i	$0.914649\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5814.63			−0.455371			−0.227685	−	0.973735i	$-0.573116\pi$
$113$	−5814.63			−0.455371			−0.227685	+	0.973735i	$0.573116\pi$
$114$		0			0
$115$	−3256.12	+	3256.12i	−0.246209	+	0.246209i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−21840.1			−1.54227
$120$		0			0
$121$		−	442.428i		−	0.0302185i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9013.12	+	9013.12i	0.576840	+	0.576840i
$126$		0			0
$127$		−	20367.1i		−	1.26276i	−0.775472	−	0.631382i	$-0.782488\pi$
$127$		−	20367.1i		−	1.26276i	0.775472	−	0.631382i	$-0.217512\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4414.52	−	4414.52i	0.257242	−	0.257242i	−0.566690	−	0.823931i	$-0.691776\pi$
$131$	4414.52	−	4414.52i	0.257242	−	0.257242i	0.823931	+	0.566690i	$0.191776\pi$
$132$		0			0
$133$	−17409.1	+	17409.1i	−0.984174	+	0.984174i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	11018.7i		−	0.587067i	−0.955949	−	0.293533i	$-0.905169\pi$
$137$		−	11018.7i		−	0.587067i	0.955949	−	0.293533i	$-0.0948312\pi$
$138$		0			0
$139$	14957.2	+	14957.2i	0.774140	+	0.774140i	0.978827	−	0.204688i	$-0.0656178\pi$
$139$	14957.2	+	14957.2i	0.774140	+	0.774140i	−0.204688	+	0.978827i	$0.565618\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3283.30i		−	0.160560i
$144$		0			0
$145$	16595.1			0.789302
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15393.9	+	15393.9i	−0.693388	+	0.693388i	−0.962976	−	0.269588i	$-0.913112\pi$
$149$	−15393.9	+	15393.9i	−0.693388	+	0.693388i	0.269588	+	0.962976i	$0.413112\pi$
$150$		0			0
$151$	16971.9			0.744347			0.372174	−	0.928163i	$-0.378613\pi$
$151$	16971.9			0.744347			0.372174	+	0.928163i	$0.378613\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12113.9	−	12113.9i	−0.504222	−	0.504222i
$156$		0			0
$157$	1167.73	+	1167.73i	0.0473744	+	0.0473744i	0.730397	−	0.683023i	$-0.239335\pi$
$157$	1167.73	+	1167.73i	0.0473744	+	0.0473744i	−0.683023	+	0.730397i	$0.739335\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20193.3			−0.779032
$162$		0			0
$163$	−28076.2	+	28076.2i	−1.05673	+	1.05673i	−0.0584383	+	0.998291i	$0.518612\pi$
$163$	−28076.2	+	28076.2i	−1.05673	+	1.05673i	−0.998291	+	0.0584383i	$0.981388\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2929.82			−0.105053			−0.0525264	−	0.998620i	$-0.516727\pi$
$167$	−2929.82			−0.105053			−0.0525264	+	0.998620i	$0.516727\pi$
$168$		0			0
$169$		−	27801.8i		−	0.973417i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8560.97	−	8560.97i	−0.286043	−	0.286043i	0.549470	−	0.835513i	$-0.314829\pi$
$173$	−8560.97	−	8560.97i	−0.286043	−	0.286043i	−0.835513	+	0.549470i	$0.814829\pi$
$174$		0			0
$175$			24721.4i			0.807230i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24420.1	+	24420.1i	−0.762153	+	0.762153i	−0.976711	−	0.214558i	$-0.931169\pi$
$179$	−24420.1	+	24420.1i	−0.762153	+	0.762153i	0.214558	+	0.976711i	$0.431169\pi$
$180$		0			0
$181$	−10946.5	+	10946.5i	−0.334133	+	0.334133i	−0.854154	−	0.520021i	$-0.825924\pi$
$181$	−10946.5	+	10946.5i	−0.334133	+	0.334133i	0.520021	+	0.854154i	$0.325924\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			6985.52i			0.204106i
$186$		0			0
$187$	36892.5	+	36892.5i	1.05501	+	1.05501i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			22384.9i			0.613604i	0.951773	+	0.306802i	$0.0992590\pi$
$191$			22384.9i			0.613604i	−0.951773	+	0.306802i	$0.900741\pi$
$192$		0			0
$193$	−30429.5			−0.816920			−0.408460	−	0.912776i	$-0.633934\pi$
$193$	−30429.5			−0.816920			−0.408460	+	0.912776i	$0.633934\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−19093.9	+	19093.9i	−0.491997	+	0.491997i	−0.908935	−	0.416938i	$-0.863103\pi$
$197$	−19093.9	+	19093.9i	−0.491997	+	0.491997i	0.416938	+	0.908935i	$0.363103\pi$
$198$		0			0
$199$	−67963.8			−1.71621			−0.858107	−	0.513470i	$-0.828360\pi$
$199$	−67963.8			−1.71621			−0.858107	+	0.513470i	$0.828360\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	51458.4	+	51458.4i	1.24872	+	1.24872i
$204$		0			0
$205$	5604.72	+	5604.72i	0.133366	+	0.133366i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	58815.1			1.34647
$210$		0			0
$211$	55219.8	−	55219.8i	1.24031	−	1.24031i	0.280438	−	0.959872i	$-0.409520\pi$
$211$	55219.8	−	55219.8i	1.24031	−	1.24031i	0.959872	−	0.280438i	$-0.0904797\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14760.6			−0.319322
$216$		0			0
$217$		−	75126.4i		−	1.59541i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8531.07	−	8531.07i	−0.174670	−	0.174670i
$222$		0			0
$223$		−	40417.5i		−	0.812754i	−0.913705	−	0.406377i	$-0.866792\pi$
$223$		−	40417.5i		−	0.812754i	0.913705	−	0.406377i	$-0.133208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1672.85	+	1672.85i	−0.0324643	+	0.0324643i	−0.723153	−	0.690688i	$-0.757308\pi$
$227$	−1672.85	+	1672.85i	−0.0324643	+	0.0324643i	0.690688	+	0.723153i	$0.257308\pi$
$228$		0			0
$229$	−26519.0	+	26519.0i	−0.505691	+	0.505691i	−0.913201	−	0.407510i	$-0.866397\pi$
$229$	−26519.0	+	26519.0i	−0.505691	+	0.505691i	0.407510	+	0.913201i	$0.366397\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	24163.3i		−	0.445087i	−0.974923	−	0.222543i	$-0.928564\pi$
$233$		−	24163.3i		−	0.445087i	0.974923	−	0.222543i	$-0.0714359\pi$
$234$		0			0
$235$	−900.145	−	900.145i	−0.0162996	−	0.0162996i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	76356.4i		−	1.33675i	−0.743825	−	0.668374i	$-0.766990\pi$
$239$		−	76356.4i		−	1.33675i	0.743825	−	0.668374i	$-0.233010\pi$
$240$		0			0
$241$	40548.1			0.698130			0.349065	−	0.937099i	$-0.386499\pi$
$241$	40548.1			0.698130			0.349065	+	0.937099i	$0.386499\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	699.586	−	699.586i	0.0116549	−	0.0116549i
$246$		0			0
$247$	−13600.5			−0.222926
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10536.0	−	10536.0i	−0.167235	−	0.167235i	0.618528	−	0.785763i	$-0.287729\pi$
$251$	−10536.0	−	10536.0i	−0.167235	−	0.167235i	−0.785763	+	0.618528i	$0.787729\pi$
$252$		0			0
$253$	34110.7	+	34110.7i	0.532905	+	0.532905i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−37983.9			−0.575086			−0.287543	−	0.957768i	$-0.592838\pi$
$257$	−37983.9			−0.575086			−0.287543	+	0.957768i	$0.592838\pi$
$258$		0			0
$259$	−21660.9	+	21660.9i	−0.322906	+	0.322906i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	37545.0			0.542801			0.271400	−	0.962467i	$-0.412513\pi$
$263$	37545.0			0.542801			0.271400	+	0.962467i	$0.412513\pi$
$264$		0			0
$265$			16748.6i			0.238498i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4676.63	−	4676.63i	−0.0646291	−	0.0646291i	0.674053	−	0.738683i	$-0.264552\pi$
$269$	−4676.63	−	4676.63i	−0.0646291	−	0.0646291i	−0.738683	+	0.674053i	$0.764552\pi$
$270$		0			0
$271$			2746.12i			0.0373922i	0.999825	+	0.0186961i	$0.00595149\pi$
$271$			2746.12i			0.0373922i	−0.999825	+	0.0186961i	$0.994049\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	41759.7	−	41759.7i	0.552194	−	0.552194i
$276$		0			0
$277$	−33056.5	+	33056.5i	−0.430822	+	0.430822i	−0.888908	−	0.458086i	$-0.848535\pi$
$277$	−33056.5	+	33056.5i	−0.430822	+	0.430822i	0.458086	+	0.888908i	$0.348535\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	80033.0i		−	1.01358i	−0.862071	−	0.506788i	$-0.830833\pi$
$281$		−	80033.0i		−	1.01358i	0.862071	−	0.506788i	$-0.169167\pi$
$282$		0			0
$283$	72284.3	+	72284.3i	0.902549	+	0.902549i	0.995656	−	0.0931068i	$-0.0296798\pi$
$283$	72284.3	+	72284.3i	0.902549	+	0.902549i	−0.0931068	+	0.995656i	$0.529680\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			34758.5i			0.421985i
$288$		0			0
$289$	108196.			1.29544
$290$		0			0
$291$		0			0
$292$		0			0
$293$	84911.3	−	84911.3i	0.989077	−	0.989077i	−0.0108641	−	0.999941i	$-0.503458\pi$
$293$	84911.3	−	84911.3i	0.989077	−	0.989077i	0.999941	+	0.0108641i	$0.00345823\pi$
$294$		0			0
$295$	−27532.6			−0.316376
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7887.81	−	7887.81i	−0.0882296	−	0.0882296i
$300$		0			0
$301$	−45770.2	−	45770.2i	−0.505184	−	0.505184i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	59698.7			0.641749
$306$		0			0
$307$	55472.5	−	55472.5i	0.588574	−	0.588574i	−0.348671	−	0.937245i	$-0.613367\pi$
$307$	55472.5	−	55472.5i	0.588574	−	0.588574i	0.937245	+	0.348671i	$0.113367\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−127048.			−1.31355			−0.656777	−	0.754084i	$-0.728081\pi$
$311$	−127048.			−1.31355			−0.656777	+	0.754084i	$0.728081\pi$
$312$		0			0
$313$			25469.3i			0.259974i	0.991516	+	0.129987i	$0.0414935\pi$
$313$			25469.3i			0.259974i	−0.991516	+	0.129987i	$0.958507\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	94218.4	+	94218.4i	0.937599	+	0.937599i	0.998164	−	0.0605656i	$-0.0192904\pi$
$317$	94218.4	+	94218.4i	0.937599	+	0.937599i	−0.0605656	+	0.998164i	$0.519290\pi$
$318$		0			0
$319$		−	173848.i		−	1.70840i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	152821.	−	152821.i	1.46480	−	1.46480i
$324$		0			0
$325$	−9656.57	+	9656.57i	−0.0914232	+	0.0914232i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	5582.38i		−	0.0515737i
$330$		0			0
$331$	−65141.3	−	65141.3i	−0.594567	−	0.594567i	0.344295	−	0.938862i	$-0.388118\pi$
$331$	−65141.3	−	65141.3i	−0.594567	−	0.594567i	−0.938862	+	0.344295i	$0.888118\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	29826.2i		−	0.265771i
$336$		0			0
$337$	135004.			1.18874			0.594369	−	0.804193i	$-0.297402\pi$
$337$	135004.			1.18874			0.594369	+	0.804193i	$0.297402\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−126904.	+	126904.i	−1.09136	+	1.09136i
$342$		0			0
$343$	−115422.			−0.981075
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10849.2	+	10849.2i	0.0901025	+	0.0901025i	0.750721	−	0.660619i	$-0.229706\pi$
$347$	10849.2	+	10849.2i	0.0901025	+	0.0901025i	−0.660619	+	0.750721i	$0.729706\pi$
$348$		0			0
$349$	−6073.15	−	6073.15i	−0.0498612	−	0.0498612i	0.681737	−	0.731598i	$-0.261225\pi$
$349$	−6073.15	−	6073.15i	−0.0498612	−	0.0498612i	−0.731598	+	0.681737i	$0.761225\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	143445.			1.15116			0.575579	−	0.817746i	$-0.304777\pi$
$353$	143445.			1.15116			0.575579	+	0.817746i	$0.304777\pi$
$354$		0			0
$355$	−9345.08	+	9345.08i	−0.0741526	+	0.0741526i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	124076.			0.962718			0.481359	−	0.876523i	$-0.340143\pi$
$359$	124076.			0.962718			0.481359	+	0.876523i	$0.340143\pi$
$360$		0			0
$361$		−	113310.i		−	0.869472i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7281.26	+	7281.26i	0.0546538	+	0.0546538i
$366$		0			0
$367$		−	126240.i		−	0.937268i	−0.883392	−	0.468634i	$-0.844746\pi$
$367$		−	126240.i		−	0.937268i	0.883392	−	0.468634i	$-0.155254\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−51934.3	+	51934.3i	−0.377317	+	0.377317i
$372$		0			0
$373$	−95513.7	+	95513.7i	−0.686512	+	0.686512i	−0.961459	−	0.274948i	$-0.911339\pi$
$373$	−95513.7	+	95513.7i	−0.686512	+	0.686512i	0.274948	+	0.961459i	$0.411339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			40200.9i			0.282848i
$378$		0			0
$379$	31220.0	+	31220.0i	0.217347	+	0.217347i	0.807380	−	0.590032i	$-0.200885\pi$
$379$	31220.0	+	31220.0i	0.217347	+	0.217347i	−0.590032	+	0.807380i	$0.700885\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	253102.i		−	1.72543i	−0.505689	−	0.862716i	$-0.668762\pi$
$383$		−	253102.i		−	1.72543i	0.505689	−	0.862716i	$-0.331238\pi$
$384$		0			0
$385$	−67604.7			−0.456095
$386$		0			0
$387$		0			0
$388$		0			0
$389$	43279.4	−	43279.4i	0.286011	−	0.286011i	−0.549490	−	0.835500i	$-0.685178\pi$
$389$	43279.4	−	43279.4i	0.286011	−	0.286011i	0.835500	+	0.549490i	$0.185178\pi$
$390$		0			0
$391$	177261.			1.15947
$392$		0			0
$393$		0			0
$394$		0			0
$395$	47209.2	+	47209.2i	0.302575	+	0.302575i
$396$		0			0
$397$	198533.	+	198533.i	1.25965	+	1.25965i	0.951258	+	0.308395i	$0.0997920\pi$
$397$	198533.	+	198533.i	1.25965	+	1.25965i	0.308395	+	0.951258i	$0.400208\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	89330.8			0.555536			0.277768	−	0.960648i	$-0.410405\pi$
$401$	89330.8			0.555536			0.277768	+	0.960648i	$0.410405\pi$
$402$		0			0
$403$	29345.5	−	29345.5i	0.180689	−	0.180689i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	73179.5			0.441775
$408$		0			0
$409$		−	26716.3i		−	0.159709i	−0.996807	−	0.0798545i	$-0.974554\pi$
$409$		−	26716.3i		−	0.159709i	0.996807	−	0.0798545i	$-0.0254456\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−85373.9	−	85373.9i	−0.500524	−	0.500524i
$414$		0			0
$415$			121611.i			0.706115i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−119011.	+	119011.i	−0.677889	+	0.677889i	−0.959522	−	0.281633i	$-0.909124\pi$
$419$	−119011.	+	119011.i	−0.677889	+	0.677889i	0.281633	+	0.959522i	$0.409124\pi$
$420$		0			0
$421$	126967.	−	126967.i	0.716355	−	0.716355i	−0.251502	−	0.967857i	$-0.580925\pi$
$421$	126967.	−	126967.i	0.716355	−	0.716355i	0.967857	+	0.251502i	$0.0809245\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	217010.i		−	1.20144i
$426$		0			0
$427$	185115.	+	185115.i	1.01528	+	1.01528i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	82986.1i		−	0.446736i	−0.974734	−	0.223368i	$-0.928295\pi$
$431$		−	82986.1i		−	0.446736i	0.974734	−	0.223368i	$-0.0717051\pi$
$432$		0			0
$433$	−153228.			−0.817265			−0.408633	−	0.912699i	$-0.633994\pi$
$433$	−153228.			−0.817265			−0.408633	+	0.912699i	$0.633994\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	141298.	−	141298.i	0.739899	−	0.739899i
$438$		0			0
$439$	−48984.2			−0.254172			−0.127086	−	0.991892i	$-0.540562\pi$
$439$	−48984.2			−0.254172			−0.127086	+	0.991892i	$0.540562\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5464.01	+	5464.01i	0.0278422	+	0.0278422i	0.720891	−	0.693049i	$-0.243733\pi$
$443$	5464.01	+	5464.01i	0.0278422	+	0.0278422i	−0.693049	+	0.720891i	$0.743733\pi$
$444$		0			0
$445$	51795.1	+	51795.1i	0.261559	+	0.261559i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−140068.			−0.694776			−0.347388	−	0.937721i	$-0.612931\pi$
$449$	−140068.			−0.694776			−0.347388	+	0.937721i	$0.612931\pi$
$450$		0			0
$451$	58714.4	−	58714.4i	0.288664	−	0.288664i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15633.0			0.0755127
$456$		0			0
$457$		−	219788.i		−	1.05238i	−0.850368	−	0.526188i	$-0.823621\pi$
$457$		−	219788.i		−	1.05238i	0.850368	−	0.526188i	$-0.176379\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−124163.	−	124163.i	−0.584239	−	0.584239i	0.351826	−	0.936065i	$-0.385561\pi$
$461$	−124163.	−	124163.i	−0.584239	−	0.584239i	−0.936065	+	0.351826i	$0.885561\pi$
$462$		0			0
$463$			236566.i			1.10354i	0.833995	+	0.551772i	$0.186048\pi$
$463$			236566.i			1.10354i	−0.833995	+	0.551772i	$0.813952\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	73415.5	−	73415.5i	0.336631	−	0.336631i	−0.518467	−	0.855098i	$-0.673497\pi$
$467$	73415.5	−	73415.5i	0.336631	−	0.336631i	0.855098	+	0.518467i	$0.173497\pi$
$468$		0			0
$469$	92485.8	−	92485.8i	0.420465	−	0.420465i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			154631.i			0.691153i
$474$		0			0
$475$	−172982.	−	172982.i	−0.766681	−	0.766681i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			212004.i			0.924003i	0.886879	+	0.462002i	$0.152869\pi$
$479$			212004.i			0.924003i	−0.886879	+	0.462002i	$0.847131\pi$
$480$		0			0
$481$	−16922.1			−0.0731417
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3322.01	+	3322.01i	−0.0141227	+	0.0141227i
$486$		0			0
$487$	−26716.3			−0.112647			−0.0563234	−	0.998413i	$-0.517938\pi$
$487$	−26716.3			−0.112647			−0.0563234	+	0.998413i	$0.517938\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	99544.8	+	99544.8i	0.412910	+	0.412910i	0.882751	−	0.469841i	$-0.155689\pi$
$491$	99544.8	+	99544.8i	0.412910	+	0.412910i	−0.469841	+	0.882751i	$0.655689\pi$
$492$		0			0
$493$	−451714.	−	451714.i	−1.85853	−	1.85853i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−57954.9			−0.234627
$498$		0			0
$499$	−189887.	+	189887.i	−0.762597	+	0.762597i	−0.976791	−	0.214194i	$-0.931287\pi$
$499$	−189887.	+	189887.i	−0.762597	+	0.762597i	0.214194	+	0.976791i	$0.431287\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	188872.			0.746502			0.373251	−	0.927730i	$-0.378243\pi$
$503$	188872.			0.746502			0.373251	+	0.927730i	$0.378243\pi$
$504$		0			0
$505$			136097.i			0.533662i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14343.5	+	14343.5i	0.0553629	+	0.0553629i	0.734246	−	0.678883i	$-0.237536\pi$
$509$	14343.5	+	14343.5i	0.0553629	+	0.0553629i	−0.678883	+	0.734246i	$0.737536\pi$
$510$		0			0
$511$			45155.8i			0.172931i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−136794.	+	136794.i	−0.515765	+	0.515765i
$516$		0			0
$517$	−9429.82	+	9429.82i	−0.0352795	+	0.0352795i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	65377.4i		−	0.240853i	−0.992722	−	0.120427i	$-0.961574\pi$
$521$		−	65377.4i		−	0.240853i	0.992722	−	0.120427i	$-0.0384262\pi$
$522$		0			0
$523$	143634.	+	143634.i	0.525113	+	0.525113i	0.919111	−	0.393998i	$-0.128908\pi$
$523$	143634.	+	143634.i	0.525113	+	0.525113i	−0.393998	+	0.919111i	$0.628908\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			659477.i			2.37453i
$528$		0			0
$529$	−115946.			−0.414327
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13577.2	+	13577.2i	−0.0477921	+	0.0477921i
$534$		0			0
$535$	150704.			0.526521
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7328.78	−	7328.78i	−0.0252263	−	0.0252263i
$540$		0			0
$541$	−275122.	−	275122.i	−0.940007	−	0.940007i	0.0582928	−	0.998300i	$-0.481434\pi$
$541$	−275122.	−	275122.i	−0.940007	−	0.940007i	−0.998300	+	0.0582928i	$0.981434\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−133654.			−0.449975
$546$		0			0
$547$	1032.53	−	1032.53i	0.00345087	−	0.00345087i	−0.705379	−	0.708830i	$-0.749223\pi$
$547$	1032.53	−	1032.53i	0.00345087	−	0.00345087i	0.708830	+	0.705379i	$0.249223\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−720136.			−2.37198
$552$		0			0
$553$			292775.i			0.957379i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	223266.	+	223266.i	0.719635	+	0.719635i	0.968530	−	0.248895i	$-0.0800674\pi$
$557$	223266.	+	223266.i	0.719635	+	0.719635i	−0.248895	+	0.968530i	$0.580067\pi$
$558$		0			0
$559$		−	35757.1i		−	0.114430i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	191551.	−	191551.i	0.604322	−	0.604322i	−0.337134	−	0.941457i	$-0.609458\pi$
$563$	191551.	−	191551.i	0.604322	−	0.604322i	0.941457	+	0.337134i	$0.109458\pi$
$564$		0			0
$565$	−46766.9	+	46766.9i	−0.146501	+	0.146501i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			119746.i			0.369858i	0.982752	+	0.184929i	$0.0592055\pi$
$569$			119746.i			0.369858i	−0.982752	+	0.184929i	$0.940795\pi$
$570$		0			0
$571$	−375516.	−	375516.i	−1.15175	−	1.15175i	−0.986202	−	0.165544i	$-0.947062\pi$
$571$	−375516.	−	375516.i	−1.15175	−	1.15175i	−0.165544	−	0.986202i	$-0.552938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	200647.i		−	0.606873i
$576$		0			0
$577$	185281.			0.556518			0.278259	−	0.960506i	$-0.410243\pi$
$577$	185281.			0.556518			0.278259	+	0.960506i	$0.410243\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−377093.	+	377093.i	−1.11711	+	1.11711i
$582$		0			0
$583$	175456.			0.516216
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−483071.	−	483071.i	−1.40196	−	1.40196i	−0.793875	−	0.608081i	$-0.791940\pi$
$587$	−483071.	−	483071.i	−1.40196	−	1.40196i	−0.608081	−	0.793875i	$-0.708060\pi$
$588$		0			0
$589$	525679.	+	525679.i	1.51527	+	1.51527i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−636299.			−1.80947			−0.904736	−	0.425973i	$-0.859932\pi$
$593$	−636299.			−1.80947			−0.904736	+	0.425973i	$0.859932\pi$
$594$		0			0
$595$	−175659.	+	175659.i	−0.496177	+	0.496177i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	410280.			1.14347			0.571737	−	0.820437i	$-0.306270\pi$
$599$	410280.			1.14347			0.571737	+	0.820437i	$0.306270\pi$
$600$		0			0
$601$			531872.i			1.47251i	0.676704	+	0.736255i	$0.263408\pi$
$601$			531872.i			1.47251i	−0.676704	+	0.736255i	$0.736592\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3558.44	−	3558.44i	−0.00972184	−	0.00972184i
$606$		0			0
$607$			505448.i			1.37182i	0.727684	+	0.685912i	$0.240597\pi$
$607$			505448.i			1.37182i	−0.727684	+	0.685912i	$0.759403\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2180.57	−	2180.57i	0.00584099	−	0.00584099i
$612$		0			0
$613$	−31334.4	+	31334.4i	−0.0833873	+	0.0833873i	−0.747570	−	0.664183i	$-0.768780\pi$
$613$	−31334.4	+	31334.4i	−0.0833873	+	0.0833873i	0.664183	+	0.747570i	$0.268780\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			481833.i			1.26569i	0.774280	+	0.632844i	$0.218112\pi$
$617$			481833.i			1.26569i	−0.774280	+	0.632844i	$0.781888\pi$
$618$		0			0
$619$	−55035.5	−	55035.5i	−0.143635	−	0.143635i	0.631632	−	0.775268i	$-0.282385\pi$
$619$	−55035.5	−	55035.5i	−0.143635	−	0.143635i	−0.775268	+	0.631632i	$0.782385\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			321215.i			0.827599i
$624$		0			0
$625$	−164779.			−0.421834
$626$		0			0
$627$		0			0
$628$		0			0
$629$	190144.	−	190144.i	0.480597	−	0.480597i
$630$		0			0
$631$	188802.			0.474184			0.237092	−	0.971487i	$-0.423806\pi$
$631$	188802.			0.474184			0.237092	+	0.971487i	$0.423806\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−163812.	−	163812.i	−0.406255	−	0.406255i
$636$		0			0
$637$	1694.72	+	1694.72i	0.00417656	+	0.00417656i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	442081.			1.07593			0.537967	−	0.842966i	$-0.319193\pi$
$641$	442081.			1.07593			0.537967	+	0.842966i	$0.319193\pi$
$642$		0			0
$643$	246339.	−	246339.i	0.595814	−	0.595814i	−0.343382	−	0.939196i	$-0.611573\pi$
$643$	246339.	−	246339.i	0.595814	−	0.595814i	0.939196	+	0.343382i	$0.111573\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−54549.4			−0.130311			−0.0651555	−	0.997875i	$-0.520754\pi$
$647$	−54549.4			−0.130311			−0.0651555	+	0.997875i	$0.520754\pi$
$648$		0			0
$649$			288429.i			0.684778i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−231745.	−	231745.i	−0.543481	−	0.543481i	0.381066	−	0.924548i	$-0.375557\pi$
$653$	−231745.	−	231745.i	−0.543481	−	0.543481i	−0.924548	+	0.381066i	$0.875557\pi$
$654$		0			0
$655$		−	71011.8i		−	0.165519i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−479757.	+	479757.i	−1.10472	+	1.10472i	−0.110883	+	0.993833i	$0.535368\pi$
$659$	−479757.	+	479757.i	−1.10472	+	1.10472i	−0.993833	+	0.110883i	$0.964632\pi$
$660$		0			0
$661$	515798.	−	515798.i	1.18053	−	1.18053i	0.200922	−	0.979607i	$-0.435606\pi$
$661$	515798.	−	515798.i	1.18053	−	1.18053i	0.979607	−	0.200922i	$-0.0643937\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			280041.i			0.633254i
$666$		0			0
$667$	−417654.	−	417654.i	−0.938782	−	0.938782i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	625397.i		−	1.38903i
$672$		0			0
$673$	−505551.			−1.11618			−0.558090	−	0.829780i	$-0.688466\pi$
$673$	−505551.			−1.11618			−0.558090	+	0.829780i	$0.688466\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	460825.	−	460825.i	1.00545	−	1.00545i	0.00546007	−	0.999985i	$-0.498262\pi$
$677$	460825.	−	460825.i	1.00545	−	1.00545i	0.999985	−	0.00546007i	$-0.00173800\pi$
$678$		0			0
$679$	−20601.9			−0.0446857
$680$		0			0
$681$		0			0
$682$		0			0
$683$	120242.	+	120242.i	0.257761	+	0.257761i	0.824143	−	0.566382i	$-0.191657\pi$
$683$	120242.	+	120242.i	0.257761	+	0.257761i	−0.566382	+	0.824143i	$0.691657\pi$
$684$		0			0
$685$	−88622.7	−	88622.7i	−0.188870	−	0.188870i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−40572.7			−0.0854664
$690$		0			0
$691$	−166965.	+	166965.i	−0.349678	+	0.349678i	−0.859990	−	0.510311i	$-0.829530\pi$
$691$	−166965.	+	166965.i	−0.349678	+	0.349678i	0.510311	+	0.859990i	$0.329530\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	240600.			0.498111
$696$		0			0
$697$		−	305118.i		−	0.628062i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39689.8	−	39689.8i	−0.0807687	−	0.0807687i	0.665568	−	0.746337i	$-0.268189\pi$
$701$	−39689.8	−	39689.8i	−0.0807687	−	0.0807687i	−0.746337	+	0.665568i	$0.768189\pi$
$702$		0			0
$703$		−	303133.i		−	0.613371i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−422013.	+	422013.i	−0.844281	+	0.844281i
$708$		0			0
$709$	250742.	−	250742.i	0.498809	−	0.498809i	−0.412258	−	0.911067i	$-0.635260\pi$
$709$	250742.	−	250742.i	0.498809	−	0.498809i	0.911067	+	0.412258i	$0.135260\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			609751.i			1.19943i
$714$		0			0
$715$	−26407.5	−	26407.5i	−0.0516553	−	0.0516553i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			133454.i			0.258152i	0.991635	+	0.129076i	$0.0412011\pi$
$719$			133454.i			0.258152i	−0.991635	+	0.129076i	$0.958799\pi$
$720$		0			0
$721$	−848347.			−1.63194
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−511309.	+	511309.i	−0.972763	+	0.972763i
$726$		0			0
$727$	−583370.			−1.10376			−0.551881	−	0.833923i	$-0.686090\pi$
$727$	−583370.			−1.10376			−0.551881	+	0.833923i	$0.686090\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	401781.	+	401781.i	0.751891	+	0.751891i
$732$		0			0
$733$	534516.	+	534516.i	0.994838	+	0.994838i	0.999987	−	0.00514857i	$-0.00163885\pi$
$733$	534516.	+	534516.i	0.994838	+	0.994838i	−0.00514857	+	0.999987i	$0.501639\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−312456.			−0.575247
$738$		0			0
$739$	91929.4	−	91929.4i	0.168332	−	0.168332i	−0.617914	−	0.786246i	$-0.712022\pi$
$739$	91929.4	−	91929.4i	0.168332	−	0.168332i	0.786246	+	0.617914i	$0.212022\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	83329.2			0.150945			0.0754727	−	0.997148i	$-0.475953\pi$
$743$	83329.2			0.150945			0.0754727	+	0.997148i	$0.475953\pi$
$744$		0			0
$745$			247625.i			0.446151i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	467305.	+	467305.i	0.832985	+	0.832985i
$750$		0			0
$751$			318689.i			0.565051i	0.959260	+	0.282525i	$0.0911722\pi$
$751$			318689.i			0.565051i	−0.959260	+	0.282525i	$0.908828\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	136504.	−	136504.i	0.239470	−	0.239470i
$756$		0			0
$757$	−428881.	+	428881.i	−0.748419	+	0.748419i	−0.974182	−	0.225763i	$-0.927512\pi$
$757$	−428881.	+	428881.i	−0.748419	+	0.748419i	0.225763	+	0.974182i	$0.427512\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	718933.i		−	1.24142i	−0.784040	−	0.620711i	$-0.786844\pi$
$761$		−	718933.i		−	1.24142i	0.784040	−	0.620711i	$-0.213156\pi$
$762$		0			0
$763$	−414437.	−	414437.i	−0.711884	−	0.711884i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	66696.8i		−	0.113374i
$768$		0			0
$769$	421326.			0.712468			0.356234	−	0.934397i	$-0.384061\pi$
$769$	421326.			0.712468			0.356234	+	0.934397i	$0.384061\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−455325.	+	455325.i	−0.762013	+	0.762013i	−0.976686	−	0.214673i	$-0.931132\pi$
$773$	−455325.	+	455325.i	−0.762013	+	0.762013i	0.214673	+	0.976686i	$0.431132\pi$
$774$		0			0
$775$	746482.			1.24284
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−243214.	−	243214.i	−0.400788	−	0.400788i
$780$		0			0
$781$	97898.1	+	97898.1i	0.160499	+	0.160499i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18784.0			0.0304825
$786$		0			0
$787$	−541211.	+	541211.i	−0.873811	+	0.873811i	−0.992885	−	0.119074i	$-0.962007\pi$
$787$	−541211.	+	541211.i	−0.873811	+	0.873811i	0.119074	+	0.992885i	$0.462007\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−290032.			−0.463546
$792$		0			0
$793$			144618.i			0.229972i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−481291.	−	481291.i	−0.757690	−	0.757690i	0.218212	−	0.975901i	$-0.429978\pi$
$797$	−481291.	−	481291.i	−0.757690	−	0.757690i	−0.975901	+	0.218212i	$0.929978\pi$
$798$		0			0
$799$			49003.4i			0.0767597i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	76277.7	−	76277.7i	0.118295	−	0.118295i
$804$		0			0
$805$	−162414.	+	162414.i	−0.250629	+	0.250629i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			34438.5i			0.0526195i	0.999654	+	0.0263097i	$0.00837562\pi$
$809$			34438.5i			0.0526195i	−0.999654	+	0.0263097i	$0.991624\pi$
$810$		0			0
$811$	−227480.	−	227480.i	−0.345862	−	0.345862i	0.512704	−	0.858566i	$-0.328644\pi$
$811$	−227480.	−	227480.i	−0.345862	−	0.345862i	−0.858566	+	0.512704i	$0.828644\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			451633.i			0.679939i
$816$		0			0
$817$	640532.			0.959614
$818$		0			0
$819$		0			0
$820$		0			0
$821$	807677.	−	807677.i	1.19826	−	1.19826i	0.223573	−	0.974687i	$-0.428228\pi$
$821$	807677.	−	807677.i	1.19826	−	1.19826i	0.974687	−	0.223573i	$-0.0717722\pi$
$822$		0			0
$823$	703593.			1.03878			0.519388	−	0.854539i	$-0.326160\pi$
$823$	703593.			1.03878			0.519388	+	0.854539i	$0.326160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−775918.	−	775918.i	−1.13450	−	1.13450i	−0.989420	−	0.145081i	$-0.953656\pi$
$827$	−775918.	−	775918.i	−1.13450	−	1.13450i	−0.145081	−	0.989420i	$-0.546344\pi$
$828$		0			0
$829$	−804056.	−	804056.i	−1.16998	−	1.16998i	−0.982214	−	0.187763i	$-0.939876\pi$
$829$	−804056.	−	804056.i	−1.16998	−	1.16998i	−0.187763	−	0.982214i	$-0.560124\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−38085.1			−0.0548864
$834$		0			0
$835$	−23564.4	+	23564.4i	−0.0337974	+	0.0337974i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−248652.			−0.353239			−0.176619	−	0.984279i	$-0.556516\pi$
$839$	−248652.			−0.353239			−0.176619	+	0.984279i	$0.556516\pi$
$840$		0			0
$841$			1.42133e6i			2.00957i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−223609.	−	223609.i	−0.313166	−	0.313166i
$846$		0			0
$847$		−	22068.2i		−	0.0307610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	175807.	−	175807.i	0.242760	−	0.242760i
$852$		0			0
$853$	779867.	−	779867.i	1.07182	−	1.07182i	0.0746081	−	0.997213i	$-0.476229\pi$
$853$	779867.	−	779867.i	1.07182	−	1.07182i	0.997213	−	0.0746081i	$-0.0237706\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	481792.i		−	0.655992i	−0.944679	−	0.327996i	$-0.893627\pi$
$857$		−	481792.i		−	0.655992i	0.944679	−	0.327996i	$-0.106373\pi$
$858$		0			0
$859$	947889.	+	947889.i	1.28461	+	1.28461i	0.938015	+	0.346594i	$0.112662\pi$
$859$	947889.	+	947889.i	1.28461	+	1.28461i	0.346594	+	0.938015i	$0.387338\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.00011e6i			1.34284i	0.741077	+	0.671421i	$0.234316\pi$
$863$			1.00011e6i			1.34284i	−0.741077	+	0.671421i	$0.765684\pi$
$864$		0			0
$865$	−137711.			−0.184051
$866$		0			0
$867$		0			0
$868$		0			0
$869$	494559.	−	494559.i	0.654905	−	0.654905i
$870$		0			0
$871$	72252.8			0.0952398
$872$		0			0
$873$		0			0
$874$		0			0
$875$	449571.	+	449571.i	0.587195	+	0.587195i
$876$		0			0
$877$	−747906.	−	747906.i	−0.972406	−	0.972406i	0.0272234	−	0.999629i	$-0.491333\pi$
$877$	−747906.	−	747906.i	−0.972406	−	0.972406i	−0.999629	+	0.0272234i	$0.991333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	53961.8			0.0695240			0.0347620	−	0.999396i	$-0.488933\pi$
$881$	53961.8			0.0695240			0.0347620	+	0.999396i	$0.488933\pi$
$882$		0			0
$883$	629494.	−	629494.i	0.807366	−	0.807366i	−0.176869	−	0.984234i	$-0.556597\pi$
$883$	629494.	−	629494.i	0.807366	−	0.807366i	0.984234	+	0.176869i	$0.0565968\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	880807.			1.11952			0.559762	−	0.828653i	$-0.310893\pi$
$887$	880807.			1.11952			0.559762	+	0.828653i	$0.310893\pi$
$888$		0			0
$889$		−	1.01590e6i		−	1.28543i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	39061.4	+	39061.4i	0.0489829	+	0.0489829i
$894$		0			0
$895$			392821.i			0.490398i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.55383e6	−	1.55383e6i	1.92257	−	1.92257i
$900$		0			0
$901$	455891.	−	455891.i	0.561580	−	0.561580i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			176085.i			0.214994i
$906$		0			0
$907$	−662568.	−	662568.i	−0.805408	−	0.805408i	0.178527	−	0.983935i	$-0.442867\pi$
$907$	−662568.	−	662568.i	−0.805408	−	0.805408i	−0.983935	+	0.178527i	$0.942867\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	817906.i		−	0.985522i	−0.870165	−	0.492761i	$-0.835988\pi$
$911$		−	817906.i		−	0.985522i	0.870165	−	0.492761i	$-0.164012\pi$
$912$		0			0
$913$	1.27398e6			1.52834
$914$		0			0
$915$		0			0
$916$		0			0
$917$	220195.	−	220195.i	0.261860	−	0.261860i
$918$		0			0
$919$	33891.9			0.0401296			0.0200648	−	0.999799i	$-0.493613\pi$
$919$	33891.9			0.0401296			0.0200648	+	0.999799i	$0.493613\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−22638.1	−	22638.1i	−0.0265728	−	0.0265728i
$924$		0			0
$925$	−215230.	−	215230.i	−0.251547	−	0.251547i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	207783.			0.240757			0.120378	−	0.992728i	$-0.461589\pi$
$929$	207783.			0.240757			0.120378	+	0.992728i	$0.461589\pi$
$930$		0			0
$931$	−30358.2	+	30358.2i	−0.0350249	+	0.0350249i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	593450.			0.678830
$936$		0			0
$937$		−	1.12361e6i		−	1.27979i	−0.768464	−	0.639893i	$-0.778978\pi$
$937$		−	1.12361e6i		−	1.27979i	0.768464	−	0.639893i	$-0.221022\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	856765.	+	856765.i	0.967570	+	0.967570i	0.999490	−	0.0319200i	$-0.0101622\pi$
$941$	856765.	+	856765.i	0.967570	+	0.967570i	−0.0319200	+	0.999490i	$0.510162\pi$
$942$		0			0
$943$		−	282112.i		−	0.317247i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−246666.	+	246666.i	−0.275048	+	0.275048i	−0.831129	−	0.556080i	$-0.812305\pi$
$947$	−246666.	+	246666.i	−0.275048	+	0.275048i	0.556080	+	0.831129i	$0.312305\pi$
$948$		0			0
$949$	−17638.6	+	17638.6i	−0.0195853	+	0.0195853i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	219446.i		−	0.241625i	−0.992675	−	0.120812i	$-0.961450\pi$
$953$		−	219446.i		−	0.241625i	0.992675	−	0.120812i	$-0.0385500\pi$
$954$		0			0
$955$	180041.	+	180041.i	0.197408	+	0.197408i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	549607.i		−	0.597606i
$960$		0			0
$961$	−1.34498e6			−1.45636
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−244743.	+	244743.i	−0.262819	+	0.262819i
$966$		0			0
$967$	−650799.			−0.695975			−0.347988	−	0.937499i	$-0.613135\pi$
$967$	−650799.			−0.695975			−0.347988	+	0.937499i	$0.613135\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	790825.	+	790825.i	0.838768	+	0.838768i	0.988697	−	0.149929i	$-0.0479044\pi$
$971$	790825.	+	790825.i	0.838768	+	0.838768i	−0.149929	+	0.988697i	$0.547904\pi$
$972$		0			0
$973$	746058.	+	746058.i	0.788037	+	0.788037i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.40812e6			1.47520			0.737600	−	0.675237i	$-0.235959\pi$
$977$	1.40812e6			1.47520			0.737600	+	0.675237i	$0.235959\pi$
$978$		0			0
$979$	542600.	−	542600.i	0.566128	−	0.566128i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−208227.			−0.215491			−0.107746	−	0.994179i	$-0.534363\pi$
$983$	−208227.			−0.215491			−0.107746	+	0.994179i	$0.534363\pi$
$984$		0			0
$985$			307144.i			0.316569i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	371486.	+	371486.i	0.379796	+	0.379796i
$990$		0			0
$991$			170063.i			0.173166i	0.996245	+	0.0865831i	$0.0275948\pi$
$991$			170063.i			0.173166i	−0.996245	+	0.0865831i	$0.972405\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−546631.	+	546631.i	−0.552138	+	0.552138i
$996$		0			0
$997$	−917799.	+	917799.i	−0.923331	+	0.923331i	−0.997263	−	0.0739326i	$-0.976445\pi$
$997$	−917799.	+	917799.i	−0.923331	+	0.923331i	0.0739326	+	0.997263i	$0.476445\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.5.m.a.559.5		14
3.2	odd	2		64.5.f.a.47.4		14
4.3	odd	2		144.5.m.a.19.1		14
12.11	even	2		16.5.f.a.3.7	✓	14
16.5	even	4		144.5.m.a.91.1		14
16.11	odd	4	inner	576.5.m.a.271.5		14
24.5	odd	2		128.5.f.a.95.4		14
24.11	even	2		128.5.f.b.95.4		14
48.5	odd	4		16.5.f.a.11.7	yes	14
48.11	even	4		64.5.f.a.15.4		14
48.29	odd	4		128.5.f.b.31.4		14
48.35	even	4		128.5.f.a.31.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.7	✓	14	12.11	even	2
16.5.f.a.11.7	yes	14	48.5	odd	4
64.5.f.a.15.4		14	48.11	even	4
64.5.f.a.47.4		14	3.2	odd	2
128.5.f.a.31.4		14	48.35	even	4
128.5.f.a.95.4		14	24.5	odd	2
128.5.f.b.31.4		14	48.29	odd	4
128.5.f.b.95.4		14	24.11	even	2
144.5.m.a.19.1		14	4.3	odd	2
144.5.m.a.91.1		14	16.5	even	4
576.5.m.a.271.5		14	16.11	odd	4	inner
576.5.m.a.559.5		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	576.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(8.04297 - 8.04297i) q^{5} +49.8797 q^{7} +O(q^{10})\)	\(q+(8.04297 - 8.04297i) q^{5} +49.8797 q^{7} +(-84.2573 - 84.2573i) q^{11} +(19.4838 + 19.4838i) q^{13} -437.855 q^{17} +(-349.021 + 349.021i) q^{19} -404.840 q^{23} +495.621i q^{25} +(1031.65 + 1031.65i) q^{29} -1506.15i q^{31} +(401.181 - 401.181i) q^{35} +(-434.262 + 434.262i) q^{37} +696.847i q^{41} +(-917.612 - 917.612i) q^{43} -111.917i q^{47} +86.9810 q^{49} +(-1041.19 + 1041.19i) q^{53} -1355.36 q^{55} +(-1711.60 - 1711.60i) q^{59} +(3711.24 + 3711.24i) q^{61} +313.415 q^{65} +(1854.18 - 1854.18i) q^{67} -1161.89 q^{71} +905.295i q^{73} +(-4202.72 - 4202.72i) q^{77} +5869.63i q^{79} +(-7560.06 + 7560.06i) q^{83} +(-3521.65 + 3521.65i) q^{85} +6439.80i q^{89} +(971.844 + 971.844i) q^{91} +5614.33i q^{95} -413.032 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 14 * q + 2 * q^5 + 4 * q^7	\( 14 q + 2 q^{5} + 4 q^{7} + 94 q^{11} - 2 q^{13} + 4 q^{17} + 706 q^{19} + 1148 q^{23} - 862 q^{29} + 1340 q^{35} - 1826 q^{37} - 1694 q^{43} + 682 q^{49} + 482 q^{53} + 11780 q^{55} - 2786 q^{59} - 3778 q^{61} + 2020 q^{65} - 7998 q^{67} + 19964 q^{71} + 9508 q^{77} - 17282 q^{83} + 9948 q^{85} + 28036 q^{91} - 4 q^{97}+O(q^{100}) \) 14 * q + 2 * q^5 + 4 * q^7 + 94 * q^11 - 2 * q^13 + 4 * q^17 + 706 * q^19 + 1148 * q^23 - 862 * q^29 + 1340 * q^35 - 1826 * q^37 - 1694 * q^43 + 682 * q^49 + 482 * q^53 + 11780 * q^55 - 2786 * q^59 - 3778 * q^61 + 2020 * q^65 - 7998 * q^67 + 19964 * q^71 + 9508 * q^77 - 17282 * q^83 + 9948 * q^85 + 28036 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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