LMFDB - Embedded newform 576.5.m.a.271.5

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			271.5
Root			$-2.40693 - 1.48549i$ of defining polynomial
Character	$\chi$	$=$	576.271
Dual form			576.5.m.a.559.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.04297 + 8.04297i) q^{5} +49.8797 q^{7} +(-84.2573 + 84.2573i) q^{11} +(19.4838 - 19.4838i) q^{13} -437.855 q^{17} +(-349.021 - 349.021i) q^{19} -404.840 q^{23} -495.621i q^{25} +(1031.65 - 1031.65i) q^{29} +1506.15i q^{31} +(401.181 + 401.181i) q^{35} +(-434.262 - 434.262i) q^{37} -696.847i q^{41} +(-917.612 + 917.612i) q^{43} +111.917i q^{47} +86.9810 q^{49} +(-1041.19 - 1041.19i) q^{53} -1355.36 q^{55} +(-1711.60 + 1711.60i) q^{59} +(3711.24 - 3711.24i) q^{61} +313.415 q^{65} +(1854.18 + 1854.18i) q^{67} -1161.89 q^{71} -905.295i q^{73} +(-4202.72 + 4202.72i) q^{77} -5869.63i q^{79} +(-7560.06 - 7560.06i) q^{83} +(-3521.65 - 3521.65i) q^{85} -6439.80i q^{89} +(971.844 - 971.844i) q^{91} -5614.33i q^{95} -413.032 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} + 4 q^{7} + 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}+ \cdots - 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

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
$5$	8.04297	+	8.04297i	0.321719	+	0.321719i	0.849426	−	0.527707i	$-0.176948\pi$
$5$	8.04297	+	8.04297i	0.321719	+	0.321719i	−0.527707	+	0.849426i	$0.676948\pi$
$6$		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	−0.963619	−	0.267278i	$-0.913876\pi$
$11$	−84.2573	+	84.2573i	−0.696341	+	0.696341i	0.267278	+	0.963619i	$0.413876\pi$
$12$		0			0
$13$	19.4838	−	19.4838i	0.115289	−	0.115289i	−0.647109	−	0.762398i	$-0.724022\pi$
$13$	19.4838	−	19.4838i	0.115289	−	0.115289i	0.762398	+	0.647109i	$0.224022\pi$
$14$		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	0.0326494	−	0.999467i	$-0.489606\pi$
$19$	−349.021	−	349.021i	−0.966817	−	0.966817i	−0.999467	+	0.0326494i	$0.989606\pi$
$20$		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	0.261490	−	0.965206i	$-0.415786\pi$
$29$	1031.65	−	1031.65i	1.22670	−	1.22670i	0.965206	−	0.261490i	$-0.0842139\pi$
$30$		0			0
$31$			1506.15i			1.56728i	0.621217	+	0.783638i	$0.286638\pi$
$31$			1506.15i			1.56728i	−0.621217	+	0.783638i	$0.713362\pi$
$32$		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	0.530484	−	0.847695i	$-0.322010\pi$
$37$	−434.262	−	434.262i	−0.317211	−	0.317211i	−0.847695	+	0.530484i	$0.822010\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	696.847i		−	0.414543i	−0.978283	−	0.207272i	$-0.933542\pi$
$41$		−	696.847i		−	0.414543i	0.978283	−	0.207272i	$-0.0664584\pi$
$42$		0			0
$43$	−917.612	+	917.612i	−0.496275	+	0.496275i	−0.910276	−	0.414002i	$-0.864131\pi$
$43$	−917.612	+	917.612i	−0.496275	+	0.496275i	0.414002	+	0.910276i	$0.364131\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			111.917i			0.0506641i	0.999679	+	0.0253321i	$0.00806431\pi$
$47$			111.917i			0.0506641i	−0.999679	+	0.0253321i	$0.991936\pi$
$48$		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	0.497056	−	0.867719i	$-0.334414\pi$
$53$	−1041.19	−	1041.19i	−0.370663	−	0.370663i	−0.867719	+	0.497056i	$0.834414\pi$
$54$		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	−0.908841	−	0.417144i	$-0.863031\pi$
$59$	−1711.60	+	1711.60i	−0.491697	+	0.491697i	0.417144	+	0.908841i	$0.363031\pi$
$60$		0			0
$61$	3711.24	−	3711.24i	0.997376	−	0.997376i	−0.00262076	−	0.999997i	$-0.500834\pi$
$61$	3711.24	−	3711.24i	0.997376	−	0.997376i	0.999997	+	0.00262076i	$0.000834216\pi$
$62$		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	0.882799	−	0.469750i	$-0.155656\pi$
$67$	1854.18	+	1854.18i	0.413049	+	0.413049i	−0.469750	+	0.882799i	$0.655656\pi$
$68$		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	$-0.972930\pi$
$73$		−	905.295i		−	0.169881i	0.996386	−	0.0849404i	$-0.0270700\pi$
$74$		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	$-0.844165\pi$
$79$		−	5869.63i		−	0.940495i	0.882535	−	0.470247i	$-0.155835\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7560.06	−	7560.06i	−1.09741	−	1.09741i	−0.994713	−	0.102698i	$-0.967252\pi$
$83$	−7560.06	−	7560.06i	−1.09741	−	1.09741i	−0.102698	−	0.994713i	$-0.532748\pi$
$84$		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	$-0.866748\pi$
$89$		−	6439.80i		−	0.813004i	0.913650	−	0.406502i	$-0.133252\pi$
$90$		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	0.158041	−	0.987433i	$-0.449482\pi$
$101$	−8460.63	−	8460.63i	−0.829392	−	0.829392i	−0.987433	+	0.158041i	$0.949482\pi$
$102$		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	−0.167567	−	0.985861i	$-0.553591\pi$
$107$	9368.65	−	9368.65i	0.818294	−	0.818294i	0.985861	+	0.167567i	$0.0535909\pi$
$108$		0			0
$109$	−8308.73	+	8308.73i	−0.699329	+	0.699329i	−0.964266	−	0.264937i	$-0.914649\pi$
$109$	−8308.73	+	8308.73i	−0.699329	+	0.699329i	0.264937	+	0.964266i	$0.414649\pi$
$110$		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	$0.217512\pi$
$127$			20367.1i			1.26276i	−0.775472	+	0.631382i	$0.782488\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4414.52	+	4414.52i	0.257242	+	0.257242i	0.823931	−	0.566690i	$-0.191776\pi$
$131$	4414.52	+	4414.52i	0.257242	+	0.257242i	−0.566690	+	0.823931i	$0.691776\pi$
$132$		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	$0.0948312\pi$
$137$			11018.7i			0.587067i	−0.955949	+	0.293533i	$0.905169\pi$
$138$		0			0
$139$	14957.2	−	14957.2i	0.774140	−	0.774140i	−0.204688	−	0.978827i	$-0.565618\pi$
$139$	14957.2	−	14957.2i	0.774140	−	0.774140i	0.978827	+	0.204688i	$0.0656178\pi$
$140$		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	0.269588	−	0.962976i	$-0.413112\pi$
$149$	−15393.9	−	15393.9i	−0.693388	−	0.693388i	−0.962976	+	0.269588i	$0.913112\pi$
$150$		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	−0.683023	−	0.730397i	$-0.739335\pi$
$157$	1167.73	−	1167.73i	0.0473744	−	0.0473744i	0.730397	+	0.683023i	$0.239335\pi$
$158$		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	−0.998291	−	0.0584383i	$-0.981388\pi$
$163$	−28076.2	−	28076.2i	−1.05673	−	1.05673i	−0.0584383	−	0.998291i	$-0.518612\pi$
$164$		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	−0.835513	−	0.549470i	$-0.814829\pi$
$173$	−8560.97	+	8560.97i	−0.286043	+	0.286043i	0.549470	+	0.835513i	$0.314829\pi$
$174$		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	0.214558	−	0.976711i	$-0.431169\pi$
$179$	−24420.1	−	24420.1i	−0.762153	−	0.762153i	−0.976711	+	0.214558i	$0.931169\pi$
$180$		0			0
$181$	−10946.5	−	10946.5i	−0.334133	−	0.334133i	0.520021	−	0.854154i	$-0.325924\pi$
$181$	−10946.5	−	10946.5i	−0.334133	−	0.334133i	−0.854154	+	0.520021i	$0.825924\pi$
$182$		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	$-0.900741\pi$
$191$		−	22384.9i		−	0.613604i	0.951773	−	0.306802i	$-0.0992590\pi$
$192$		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	0.416938	−	0.908935i	$-0.363103\pi$
$197$	−19093.9	−	19093.9i	−0.491997	−	0.491997i	−0.908935	+	0.416938i	$0.863103\pi$
$198$		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	0.959872	+	0.280438i	$0.0904797\pi$
$211$	55219.8	+	55219.8i	1.24031	+	1.24031i	0.280438	+	0.959872i	$0.409520\pi$
$212$		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	$0.133208\pi$
$223$			40417.5i			0.812754i	−0.913705	+	0.406377i	$0.866792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1672.85	−	1672.85i	−0.0324643	−	0.0324643i	0.690688	−	0.723153i	$-0.257308\pi$
$227$	−1672.85	−	1672.85i	−0.0324643	−	0.0324643i	−0.723153	+	0.690688i	$0.757308\pi$
$228$		0			0
$229$	−26519.0	−	26519.0i	−0.505691	−	0.505691i	0.407510	−	0.913201i	$-0.366397\pi$
$229$	−26519.0	−	26519.0i	−0.505691	−	0.505691i	−0.913201	+	0.407510i	$0.866397\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24163.3i			0.445087i	0.974923	+	0.222543i	$0.0714359\pi$
$233$			24163.3i			0.445087i	−0.974923	+	0.222543i	$0.928564\pi$
$234$		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	$0.233010\pi$
$239$			76356.4i			1.33675i	−0.743825	+	0.668374i	$0.766990\pi$
$240$		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	−0.785763	−	0.618528i	$-0.787729\pi$
$251$	−10536.0	+	10536.0i	−0.167235	+	0.167235i	0.618528	+	0.785763i	$0.287729\pi$
$252$		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	−0.738683	−	0.674053i	$-0.764552\pi$
$269$	−4676.63	+	4676.63i	−0.0646291	+	0.0646291i	0.674053	+	0.738683i	$0.264552\pi$
$270$		0			0
$271$		−	2746.12i		−	0.0373922i	−0.999825	−	0.0186961i	$-0.994049\pi$
$271$		−	2746.12i		−	0.0373922i	0.999825	−	0.0186961i	$-0.00595149\pi$
$272$		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	0.458086	−	0.888908i	$-0.348535\pi$
$277$	−33056.5	−	33056.5i	−0.430822	−	0.430822i	−0.888908	+	0.458086i	$0.848535\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			80033.0i			1.01358i	0.862071	+	0.506788i	$0.169167\pi$
$281$			80033.0i			1.01358i	−0.862071	+	0.506788i	$0.830833\pi$
$282$		0			0
$283$	72284.3	−	72284.3i	0.902549	−	0.902549i	−0.0931068	−	0.995656i	$-0.529680\pi$
$283$	72284.3	−	72284.3i	0.902549	−	0.902549i	0.995656	+	0.0931068i	$0.0296798\pi$
$284$		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	0.999941	−	0.0108641i	$-0.00345823\pi$
$293$	84911.3	+	84911.3i	0.989077	+	0.989077i	−0.0108641	+	0.999941i	$0.503458\pi$
$294$		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	0.937245	−	0.348671i	$-0.113367\pi$
$307$	55472.5	+	55472.5i	0.588574	+	0.588574i	−0.348671	+	0.937245i	$0.613367\pi$
$308$		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	$-0.958507\pi$
$313$		−	25469.3i		−	0.259974i	0.991516	−	0.129987i	$-0.0414935\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	94218.4	−	94218.4i	0.937599	−	0.937599i	−0.0605656	−	0.998164i	$-0.519290\pi$
$317$	94218.4	−	94218.4i	0.937599	−	0.937599i	0.998164	+	0.0605656i	$0.0192904\pi$
$318$		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	−0.938862	−	0.344295i	$-0.888118\pi$
$331$	−65141.3	+	65141.3i	−0.594567	+	0.594567i	0.344295	+	0.938862i	$0.388118\pi$
$332$		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	−0.660619	−	0.750721i	$-0.729706\pi$
$347$	10849.2	−	10849.2i	0.0901025	−	0.0901025i	0.750721	+	0.660619i	$0.229706\pi$
$348$		0			0
$349$	−6073.15	+	6073.15i	−0.0498612	+	0.0498612i	−0.731598	−	0.681737i	$-0.761225\pi$
$349$	−6073.15	+	6073.15i	−0.0498612	+	0.0498612i	0.681737	+	0.731598i	$0.261225\pi$
$350$		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	$0.155254\pi$
$367$			126240.i			0.937268i	−0.883392	+	0.468634i	$0.844746\pi$
$368$		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	0.274948	−	0.961459i	$-0.411339\pi$
$373$	−95513.7	−	95513.7i	−0.686512	−	0.686512i	−0.961459	+	0.274948i	$0.911339\pi$
$374$		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	−0.590032	−	0.807380i	$-0.700885\pi$
$379$	31220.0	−	31220.0i	0.217347	−	0.217347i	0.807380	+	0.590032i	$0.200885\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			253102.i			1.72543i	0.505689	+	0.862716i	$0.331238\pi$
$383$			253102.i			1.72543i	−0.505689	+	0.862716i	$0.668762\pi$
$384$		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	0.835500	−	0.549490i	$-0.185178\pi$
$389$	43279.4	+	43279.4i	0.286011	+	0.286011i	−0.549490	+	0.835500i	$0.685178\pi$
$390$		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	0.308395	−	0.951258i	$-0.400208\pi$
$397$	198533.	−	198533.i	1.25965	−	1.25965i	0.951258	−	0.308395i	$-0.0997920\pi$
$398$		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	$0.0254456\pi$
$409$			26716.3i			0.159709i	−0.996807	+	0.0798545i	$0.974554\pi$
$410$		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	0.281633	−	0.959522i	$-0.409124\pi$
$419$	−119011.	−	119011.i	−0.677889	−	0.677889i	−0.959522	+	0.281633i	$0.909124\pi$
$420$		0			0
$421$	126967.	+	126967.i	0.716355	+	0.716355i	0.967857	−	0.251502i	$-0.0809245\pi$
$421$	126967.	+	126967.i	0.716355	+	0.716355i	−0.251502	+	0.967857i	$0.580925\pi$
$422$		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	$0.0717051\pi$
$431$			82986.1i			0.446736i	−0.974734	+	0.223368i	$0.928295\pi$
$432$		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	−0.693049	−	0.720891i	$-0.743733\pi$
$443$	5464.01	−	5464.01i	0.0278422	−	0.0278422i	0.720891	+	0.693049i	$0.243733\pi$
$444$		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	$0.176379\pi$
$457$			219788.i			1.05238i	−0.850368	+	0.526188i	$0.823621\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−124163.	+	124163.i	−0.584239	+	0.584239i	−0.936065	−	0.351826i	$-0.885561\pi$
$461$	−124163.	+	124163.i	−0.584239	+	0.584239i	0.351826	+	0.936065i	$0.385561\pi$
$462$		0			0
$463$		−	236566.i		−	1.10354i	−0.833995	−	0.551772i	$-0.813952\pi$
$463$		−	236566.i		−	1.10354i	0.833995	−	0.551772i	$-0.186048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	73415.5	+	73415.5i	0.336631	+	0.336631i	0.855098	−	0.518467i	$-0.173497\pi$
$467$	73415.5	+	73415.5i	0.336631	+	0.336631i	−0.518467	+	0.855098i	$0.673497\pi$
$468$		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	$-0.847131\pi$
$479$		−	212004.i		−	0.924003i	0.886879	−	0.462002i	$-0.152869\pi$
$480$		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	−0.469841	−	0.882751i	$-0.655689\pi$
$491$	99544.8	−	99544.8i	0.412910	−	0.412910i	0.882751	+	0.469841i	$0.155689\pi$
$492$		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	0.214194	−	0.976791i	$-0.431287\pi$
$499$	−189887.	−	189887.i	−0.762597	−	0.762597i	−0.976791	+	0.214194i	$0.931287\pi$
$500$		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	−0.678883	−	0.734246i	$-0.737536\pi$
$509$	14343.5	−	14343.5i	0.0553629	−	0.0553629i	0.734246	+	0.678883i	$0.237536\pi$
$510$		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	$0.0384262\pi$
$521$			65377.4i			0.240853i	−0.992722	+	0.120427i	$0.961574\pi$
$522$		0			0
$523$	143634.	−	143634.i	0.525113	−	0.525113i	−0.393998	−	0.919111i	$-0.628908\pi$
$523$	143634.	−	143634.i	0.525113	−	0.525113i	0.919111	+	0.393998i	$0.128908\pi$
$524$		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	−0.998300	−	0.0582928i	$-0.981434\pi$
$541$	−275122.	+	275122.i	−0.940007	+	0.940007i	0.0582928	+	0.998300i	$0.481434\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−133654.			−0.449975
$546$		0			0
$547$	1032.53	+	1032.53i	0.00345087	+	0.00345087i	0.708830	−	0.705379i	$-0.249223\pi$
$547$	1032.53	+	1032.53i	0.00345087	+	0.00345087i	−0.705379	+	0.708830i	$0.749223\pi$
$548$		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	−0.248895	−	0.968530i	$-0.580067\pi$
$557$	223266.	−	223266.i	0.719635	−	0.719635i	0.968530	+	0.248895i	$0.0800674\pi$
$558$		0			0
$559$			35757.1i			0.114430i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	191551.	+	191551.i	0.604322	+	0.604322i	0.941457	−	0.337134i	$-0.109458\pi$
$563$	191551.	+	191551.i	0.604322	+	0.604322i	−0.337134	+	0.941457i	$0.609458\pi$
$564$		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	$-0.940795\pi$
$569$		−	119746.i		−	0.369858i	0.982752	−	0.184929i	$-0.0592055\pi$
$570$		0			0
$571$	−375516.	+	375516.i	−1.15175	+	1.15175i	−0.165544	+	0.986202i	$0.552938\pi$
$571$	−375516.	+	375516.i	−1.15175	+	1.15175i	−0.986202	+	0.165544i	$0.947062\pi$
$572$		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	−0.608081	+	0.793875i	$0.708060\pi$
$587$	−483071.	+	483071.i	−1.40196	+	1.40196i	−0.793875	+	0.608081i	$0.791940\pi$
$588$		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	$-0.736592\pi$
$601$		−	531872.i		−	1.47251i	0.676704	−	0.736255i	$-0.263408\pi$
$602$		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	$-0.759403\pi$
$607$		−	505448.i		−	1.37182i	0.727684	−	0.685912i	$-0.240597\pi$
$608$		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	0.664183	−	0.747570i	$-0.268780\pi$
$613$	−31334.4	−	31334.4i	−0.0833873	−	0.0833873i	−0.747570	+	0.664183i	$0.768780\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	481833.i		−	1.26569i	−0.774280	−	0.632844i	$-0.781888\pi$
$617$		−	481833.i		−	1.26569i	0.774280	−	0.632844i	$-0.218112\pi$
$618$		0			0
$619$	−55035.5	+	55035.5i	−0.143635	+	0.143635i	−0.775268	−	0.631632i	$-0.782385\pi$
$619$	−55035.5	+	55035.5i	−0.143635	+	0.143635i	0.631632	+	0.775268i	$0.282385\pi$
$620$		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	0.939196	−	0.343382i	$-0.111573\pi$
$643$	246339.	+	246339.i	0.595814	+	0.595814i	−0.343382	+	0.939196i	$0.611573\pi$
$644$		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	−0.924548	−	0.381066i	$-0.875557\pi$
$653$	−231745.	+	231745.i	−0.543481	+	0.543481i	0.381066	+	0.924548i	$0.375557\pi$
$654$		0			0
$655$			71011.8i			0.165519i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−479757.	−	479757.i	−1.10472	−	1.10472i	−0.993833	−	0.110883i	$-0.964632\pi$
$659$	−479757.	−	479757.i	−1.10472	−	1.10472i	−0.110883	−	0.993833i	$-0.535368\pi$
$660$		0			0
$661$	515798.	+	515798.i	1.18053	+	1.18053i	0.979607	+	0.200922i	$0.0643937\pi$
$661$	515798.	+	515798.i	1.18053	+	1.18053i	0.200922	+	0.979607i	$0.435606\pi$
$662$		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	0.999985	+	0.00546007i	$0.00173800\pi$
$677$	460825.	+	460825.i	1.00545	+	1.00545i	0.00546007	+	0.999985i	$0.498262\pi$
$678$		0			0
$679$	−20601.9			−0.0446857
$680$		0			0
$681$		0			0
$682$		0			0
$683$	120242.	−	120242.i	0.257761	−	0.257761i	−0.566382	−	0.824143i	$-0.691657\pi$
$683$	120242.	−	120242.i	0.257761	−	0.257761i	0.824143	+	0.566382i	$0.191657\pi$
$684$		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	0.510311	−	0.859990i	$-0.329530\pi$
$691$	−166965.	−	166965.i	−0.349678	−	0.349678i	−0.859990	+	0.510311i	$0.829530\pi$
$692$		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	−0.746337	−	0.665568i	$-0.768189\pi$
$701$	−39689.8	+	39689.8i	−0.0807687	+	0.0807687i	0.665568	+	0.746337i	$0.268189\pi$
$702$		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	0.911067	−	0.412258i	$-0.135260\pi$
$709$	250742.	+	250742.i	0.498809	+	0.498809i	−0.412258	+	0.911067i	$0.635260\pi$
$710$		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	$-0.958799\pi$
$719$		−	133454.i		−	0.258152i	0.991635	−	0.129076i	$-0.0412011\pi$
$720$		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	−0.00514857	−	0.999987i	$-0.501639\pi$
$733$	534516.	−	534516.i	0.994838	−	0.994838i	0.999987	+	0.00514857i	$0.00163885\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−312456.			−0.575247
$738$		0			0
$739$	91929.4	+	91929.4i	0.168332	+	0.168332i	0.786246	−	0.617914i	$-0.212022\pi$
$739$	91929.4	+	91929.4i	0.168332	+	0.168332i	−0.617914	+	0.786246i	$0.712022\pi$
$740$		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	$-0.908828\pi$
$751$		−	318689.i		−	0.565051i	0.959260	−	0.282525i	$-0.0911722\pi$
$752$		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	0.225763	−	0.974182i	$-0.427512\pi$
$757$	−428881.	−	428881.i	−0.748419	−	0.748419i	−0.974182	+	0.225763i	$0.927512\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			718933.i			1.24142i	0.784040	+	0.620711i	$0.213156\pi$
$761$			718933.i			1.24142i	−0.784040	+	0.620711i	$0.786844\pi$
$762$		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	0.214673	−	0.976686i	$-0.431132\pi$
$773$	−455325.	−	455325.i	−0.762013	−	0.762013i	−0.976686	+	0.214673i	$0.931132\pi$
$774$		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	0.119074	−	0.992885i	$-0.462007\pi$
$787$	−541211.	−	541211.i	−0.873811	−	0.873811i	−0.992885	+	0.119074i	$0.962007\pi$
$788$		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	−0.975901	−	0.218212i	$-0.929978\pi$
$797$	−481291.	+	481291.i	−0.757690	+	0.757690i	0.218212	+	0.975901i	$0.429978\pi$
$798$		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	$-0.991624\pi$
$809$		−	34438.5i		−	0.0526195i	0.999654	−	0.0263097i	$-0.00837562\pi$
$810$		0			0
$811$	−227480.	+	227480.i	−0.345862	+	0.345862i	−0.858566	−	0.512704i	$-0.828644\pi$
$811$	−227480.	+	227480.i	−0.345862	+	0.345862i	0.512704	+	0.858566i	$0.328644\pi$
$812$		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	0.974687	+	0.223573i	$0.0717722\pi$
$821$	807677.	+	807677.i	1.19826	+	1.19826i	0.223573	+	0.974687i	$0.428228\pi$
$822$		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	−0.145081	+	0.989420i	$0.546344\pi$
$827$	−775918.	+	775918.i	−1.13450	+	1.13450i	−0.989420	+	0.145081i	$0.953656\pi$
$828$		0			0
$829$	−804056.	+	804056.i	−1.16998	+	1.16998i	−0.187763	+	0.982214i	$0.560124\pi$
$829$	−804056.	+	804056.i	−1.16998	+	1.16998i	−0.982214	+	0.187763i	$0.939876\pi$
$830$		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	0.997213	+	0.0746081i	$0.0237706\pi$
$853$	779867.	+	779867.i	1.07182	+	1.07182i	0.0746081	+	0.997213i	$0.476229\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			481792.i			0.655992i	0.944679	+	0.327996i	$0.106373\pi$
$857$			481792.i			0.655992i	−0.944679	+	0.327996i	$0.893627\pi$
$858$		0			0
$859$	947889.	−	947889.i	1.28461	−	1.28461i	0.346594	−	0.938015i	$-0.387338\pi$
$859$	947889.	−	947889.i	1.28461	−	1.28461i	0.938015	−	0.346594i	$-0.112662\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.00011e6i		−	1.34284i	−0.741077	−	0.671421i	$-0.765684\pi$
$863$		−	1.00011e6i		−	1.34284i	0.741077	−	0.671421i	$-0.234316\pi$
$864$		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	−0.999629	−	0.0272234i	$-0.991333\pi$
$877$	−747906.	+	747906.i	−0.972406	+	0.972406i	0.0272234	+	0.999629i	$0.491333\pi$
$878$		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	0.984234	−	0.176869i	$-0.0565968\pi$
$883$	629494.	+	629494.i	0.807366	+	0.807366i	−0.176869	+	0.984234i	$0.556597\pi$
$884$		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	−0.983935	−	0.178527i	$-0.942867\pi$
$907$	−662568.	+	662568.i	−0.805408	+	0.805408i	0.178527	+	0.983935i	$0.442867\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			817906.i			0.985522i	0.870165	+	0.492761i	$0.164012\pi$
$911$			817906.i			0.985522i	−0.870165	+	0.492761i	$0.835988\pi$
$912$		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	$0.221022\pi$
$937$			1.12361e6i			1.27979i	−0.768464	+	0.639893i	$0.778978\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	856765.	−	856765.i	0.967570	−	0.967570i	−0.0319200	−	0.999490i	$-0.510162\pi$
$941$	856765.	−	856765.i	0.967570	−	0.967570i	0.999490	+	0.0319200i	$0.0101622\pi$
$942$		0			0
$943$			282112.i			0.317247i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−246666.	−	246666.i	−0.275048	−	0.275048i	0.556080	−	0.831129i	$-0.312305\pi$
$947$	−246666.	−	246666.i	−0.275048	−	0.275048i	−0.831129	+	0.556080i	$0.812305\pi$
$948$		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	$0.0385500\pi$
$953$			219446.i			0.241625i	−0.992675	+	0.120812i	$0.961450\pi$
$954$		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	−0.149929	−	0.988697i	$-0.547904\pi$
$971$	790825.	−	790825.i	0.838768	−	0.838768i	0.988697	+	0.149929i	$0.0479044\pi$
$972$		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	$-0.972405\pi$
$991$		−	170063.i		−	0.173166i	0.996245	−	0.0865831i	$-0.0275948\pi$
$992$		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	0.0739326	−	0.997263i	$-0.476445\pi$
$997$	−917799.	−	917799.i	−0.923331	−	0.923331i	−0.997263	+	0.0739326i	$0.976445\pi$
$998$		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.271.5		14
3.2	odd	2		64.5.f.a.15.4		14
4.3	odd	2		144.5.m.a.91.1		14
12.11	even	2		16.5.f.a.11.7	yes	14
16.3	odd	4	inner	576.5.m.a.559.5		14
16.13	even	4		144.5.m.a.19.1		14
24.5	odd	2		128.5.f.a.31.4		14
24.11	even	2		128.5.f.b.31.4		14
48.5	odd	4		128.5.f.b.95.4		14
48.11	even	4		128.5.f.a.95.4		14
48.29	odd	4		16.5.f.a.3.7	✓	14
48.35	even	4		64.5.f.a.47.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.7	✓	14	48.29	odd	4
16.5.f.a.11.7	yes	14	12.11	even	2
64.5.f.a.15.4		14	3.2	odd	2
64.5.f.a.47.4		14	48.35	even	4
128.5.f.a.31.4		14	24.5	odd	2
128.5.f.a.95.4		14	48.11	even	4
128.5.f.b.31.4		14	24.11	even	2
128.5.f.b.95.4		14	48.5	odd	4
144.5.m.a.19.1		14	16.13	even	4
144.5.m.a.91.1		14	4.3	odd	2
576.5.m.a.271.5		14	1.1	even	1	trivial
576.5.m.a.559.5		14	16.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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