LMFDB - Embedded newform 600.3.u.h.193.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(193,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.193"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.u (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-4,0,0,0,32,0,4,0,0,0,-52,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 $ x^8 - 4x^7 + 8x^6 + 269x^4 - 1116x^3 + 2312x^2 + 680x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$2.36263 - 2.36263i$ of defining polynomial
Character	$\chi$	$=$	600.193
Dual form			600.3.u.h.457.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(8.78825 - 8.78825i) q^{7} +3.00000i q^{9} +13.7482 q^{11} +(4.88927 + 4.88927i) q^{13} +(-5.99029 + 5.99029i) q^{17} +25.5765i q^{19} -21.5267 q^{21} +(-18.4257 - 18.4257i) q^{23} +(3.67423 - 3.67423i) q^{27} -37.2222i q^{29} +31.6955 q^{31} +(-16.8380 - 16.8380i) q^{33} +(32.5046 - 32.5046i) q^{37} -11.9762i q^{39} +36.7295 q^{41} +(24.5780 + 24.5780i) q^{43} +(-20.4063 + 20.4063i) q^{47} -105.467i q^{49} +14.6732 q^{51} +(-33.5559 - 33.5559i) q^{53} +(31.3247 - 31.3247i) q^{57} +7.54615i q^{59} +43.6183 q^{61} +(26.3648 + 26.3648i) q^{63} +(60.1739 - 60.1739i) q^{67} +45.1336i q^{69} -18.1632 q^{71} +(-68.8218 - 68.8218i) q^{73} +(120.823 - 120.823i) q^{77} -22.2628i q^{79} -9.00000 q^{81} +(0.00221809 + 0.00221809i) q^{83} +(-45.5877 + 45.5877i) q^{87} +77.1943i q^{89} +85.9363 q^{91} +(-38.8189 - 38.8189i) q^{93} +(-29.4309 + 29.4309i) q^{97} +41.2446i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} + 32 q^{11} + 4 q^{13} - 52 q^{17} - 24 q^{21} + 40 q^{23} + 96 q^{31} - 60 q^{33} + 60 q^{37} - 152 q^{41} + 88 q^{43} + 16 q^{47} - 168 q^{51} - 108 q^{53} + 24 q^{57} + 264 q^{61} - 12 q^{63}+ \cdots + 208 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 + 32 * q^11 + 4 * q^13 - 52 * q^17 - 24 * q^21 + 40 * q^23 + 96 * q^31 - 60 * q^33 + 60 * q^37 - 152 * q^41 + 88 * q^43 + 16 * q^47 - 168 * q^51 - 108 * q^53 + 24 * q^57 + 264 * q^61 - 12 * q^63 + 216 * q^67 - 240 * q^71 + 208 * q^73 - 168 * q^77 - 72 * q^81 - 336 * q^83 - 252 * q^87 + 592 * q^91 - 264 * q^93 + 208 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	8.78825	−	8.78825i	1.25546	−	1.25546i	0.302229	−	0.953235i	$-0.402269\pi$
$7$	8.78825	−	8.78825i	1.25546	−	1.25546i	0.953235	−	0.302229i	$-0.0977309\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	13.7482			1.24984			0.624918	−	0.780691i	$-0.285133\pi$
$11$	13.7482			1.24984			0.624918	+	0.780691i	$0.285133\pi$
$12$		0			0
$13$	4.88927	+	4.88927i	0.376098	+	0.376098i	0.869692	−	0.493594i	$-0.164317\pi$
$13$	4.88927	+	4.88927i	0.376098	+	0.376098i	−0.493594	+	0.869692i	$0.664317\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.99029	+	5.99029i	−0.352370	+	0.352370i	−0.860991	−	0.508621i	$-0.830156\pi$
$17$	−5.99029	+	5.99029i	−0.352370	+	0.352370i	0.508621	+	0.860991i	$0.330156\pi$
$18$		0			0
$19$			25.5765i			1.34613i	0.739583	+	0.673066i	$0.235023\pi$
$19$			25.5765i			1.34613i	−0.739583	+	0.673066i	$0.764977\pi$
$20$		0			0
$21$	−21.5267			−1.02508
$22$		0			0
$23$	−18.4257	−	18.4257i	−0.801118	−	0.801118i	0.182152	−	0.983270i	$-0.441694\pi$
$23$	−18.4257	−	18.4257i	−0.801118	−	0.801118i	−0.983270	+	0.182152i	$0.941694\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		−	37.2222i		−	1.28352i	−0.766904	−	0.641762i	$-0.778204\pi$
$29$		−	37.2222i		−	1.28352i	0.766904	−	0.641762i	$-0.221796\pi$
$30$		0			0
$31$	31.6955			1.02243			0.511217	−	0.859452i	$-0.329195\pi$
$31$	31.6955			1.02243			0.511217	+	0.859452i	$0.329195\pi$
$32$		0			0
$33$	−16.8380	−	16.8380i	−0.510243	−	0.510243i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	32.5046	−	32.5046i	0.878503	−	0.878503i	−0.114877	−	0.993380i	$-0.536647\pi$
$37$	32.5046	−	32.5046i	0.878503	−	0.878503i	0.993380	+	0.114877i	$0.0366474\pi$
$38$		0			0
$39$		−	11.9762i		−	0.307083i
$40$		0			0
$41$	36.7295			0.895842			0.447921	−	0.894073i	$-0.352165\pi$
$41$	36.7295			0.895842			0.447921	+	0.894073i	$0.352165\pi$
$42$		0			0
$43$	24.5780	+	24.5780i	0.571581	+	0.571581i	0.932570	−	0.360989i	$-0.117561\pi$
$43$	24.5780	+	24.5780i	0.571581	+	0.571581i	−0.360989	+	0.932570i	$0.617561\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−20.4063	+	20.4063i	−0.434177	+	0.434177i	−0.890046	−	0.455870i	$-0.849328\pi$
$47$	−20.4063	+	20.4063i	−0.434177	+	0.434177i	0.455870	+	0.890046i	$0.349328\pi$
$48$		0			0
$49$		−	105.467i		−	2.15238i
$50$		0			0
$51$	14.6732			0.287709
$52$		0			0
$53$	−33.5559	−	33.5559i	−0.633129	−	0.633129i	0.315722	−	0.948852i	$-0.397753\pi$
$53$	−33.5559	−	33.5559i	−0.633129	−	0.633129i	−0.948852	+	0.315722i	$0.897753\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	31.3247	−	31.3247i	0.549556	−	0.549556i
$58$		0			0
$59$			7.54615i			0.127901i	0.997953	+	0.0639504i	$0.0203699\pi$
$59$			7.54615i			0.127901i	−0.997953	+	0.0639504i	$0.979630\pi$
$60$		0			0
$61$	43.6183			0.715054			0.357527	−	0.933903i	$-0.383620\pi$
$61$	43.6183			0.715054			0.357527	+	0.933903i	$0.383620\pi$
$62$		0			0
$63$	26.3648	+	26.3648i	0.418488	+	0.418488i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	60.1739	−	60.1739i	0.898118	−	0.898118i	−0.0971517	−	0.995270i	$-0.530973\pi$
$67$	60.1739	−	60.1739i	0.898118	−	0.898118i	0.995270	+	0.0971517i	$0.0309732\pi$
$68$		0			0
$69$			45.1336i			0.654110i
$70$		0			0
$71$	−18.1632			−0.255820			−0.127910	−	0.991786i	$-0.540827\pi$
$71$	−18.1632			−0.255820			−0.127910	+	0.991786i	$0.540827\pi$
$72$		0			0
$73$	−68.8218	−	68.8218i	−0.942764	−	0.942764i	0.0556839	−	0.998448i	$-0.482266\pi$
$73$	−68.8218	−	68.8218i	−0.942764	−	0.942764i	−0.998448	+	0.0556839i	$0.982266\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	120.823	−	120.823i	1.56912	−	1.56912i
$78$		0			0
$79$		−	22.2628i		−	0.281807i	−0.990023	−	0.140904i	$-0.954999\pi$
$79$		−	22.2628i		−	0.281807i	0.990023	−	0.140904i	$-0.0450007\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	0.00221809	+	0.00221809i	2.67240e−5	+	2.67240e-5i	0.707120	−	0.707093i	$-0.249994\pi$
$83$	0.00221809	+	0.00221809i	2.67240e−5	+	2.67240e-5i	−0.707093	+	0.707120i	$0.749994\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−45.5877	+	45.5877i	−0.523996	+	0.523996i
$88$		0			0
$89$			77.1943i			0.867352i	0.901069	+	0.433676i	$0.142784\pi$
$89$			77.1943i			0.867352i	−0.901069	+	0.433676i	$0.857216\pi$
$90$		0			0
$91$	85.9363			0.944355
$92$		0			0
$93$	−38.8189	−	38.8189i	−0.417407	−	0.417407i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−29.4309	+	29.4309i	−0.303411	+	0.303411i	−0.842347	−	0.538936i	$-0.818827\pi$
$97$	−29.4309	+	29.4309i	−0.303411	+	0.303411i	0.538936	+	0.842347i	$0.318827\pi$
$98$		0			0
$99$			41.2446i			0.416612i
$100$		0			0
$101$	−178.867			−1.77096			−0.885479	−	0.464679i	$-0.846170\pi$
$101$	−178.867			−1.77096			−0.885479	+	0.464679i	$0.846170\pi$
$102$		0			0
$103$	−41.0972	−	41.0972i	−0.399002	−	0.399002i	0.478879	−	0.877881i	$-0.341043\pi$
$103$	−41.0972	−	41.0972i	−0.399002	−	0.399002i	−0.877881	+	0.478879i	$0.841043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	36.8104	−	36.8104i	0.344022	−	0.344022i	−0.513855	−	0.857877i	$-0.671783\pi$
$107$	36.8104	−	36.8104i	0.344022	−	0.344022i	0.857877	+	0.513855i	$0.171783\pi$
$108$		0			0
$109$		−	85.4905i		−	0.784316i	−0.919898	−	0.392158i	$-0.871729\pi$
$109$		−	85.4905i		−	0.784316i	0.919898	−	0.392158i	$-0.128271\pi$
$110$		0			0
$111$	−79.6197			−0.717295
$112$		0			0
$113$	15.6917	+	15.6917i	0.138864	+	0.138864i	0.773122	−	0.634258i	$-0.218694\pi$
$113$	15.6917	+	15.6917i	0.138864	+	0.138864i	−0.634258	+	0.773122i	$0.718694\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.6678	+	14.6678i	−0.125366	+	0.125366i
$118$		0			0
$119$			105.288i			0.884777i
$120$		0			0
$121$	68.0127			0.562088
$122$		0			0
$123$	−44.9843	−	44.9843i	−0.365726	−	0.365726i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−127.554	+	127.554i	−1.00436	+	1.00436i	−0.00436887	+	0.999990i	$0.501391\pi$
$127$	−127.554	+	127.554i	−1.00436	+	1.00436i	−0.999990	+	0.00436887i	$0.998609\pi$
$128$		0			0
$129$		−	60.2035i		−	0.466694i
$130$		0			0
$131$	123.810			0.945117			0.472559	−	0.881299i	$-0.343330\pi$
$131$	123.810			0.945117			0.472559	+	0.881299i	$0.343330\pi$
$132$		0			0
$133$	224.773	+	224.773i	1.69002	+	1.69002i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−173.145	+	173.145i	−1.26383	+	1.26383i	−0.314609	+	0.949221i	$0.601873\pi$
$137$	−173.145	+	173.145i	−1.26383	+	1.26383i	−0.949221	+	0.314609i	$0.898127\pi$
$138$		0			0
$139$			1.06696i			0.00767601i	0.999993	+	0.00383800i	$0.00122168\pi$
$139$			1.06696i			0.00767601i	−0.999993	+	0.00383800i	$0.998778\pi$
$140$		0			0
$141$	49.9850			0.354504
$142$		0			0
$143$	67.2186	+	67.2186i	0.470060	+	0.470060i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−129.170	+	129.170i	−0.878707	+	0.878707i
$148$		0			0
$149$		−	65.2505i		−	0.437923i	−0.975734	−	0.218961i	$-0.929733\pi$
$149$		−	65.2505i		−	0.437923i	0.975734	−	0.218961i	$-0.0702669\pi$
$150$		0			0
$151$	157.183			1.04095			0.520473	−	0.853878i	$-0.325756\pi$
$151$	157.183			1.04095			0.520473	+	0.853878i	$0.325756\pi$
$152$		0			0
$153$	−17.9709	−	17.9709i	−0.117457	−	0.117457i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	121.763	−	121.763i	0.775561	−	0.775561i	−0.203512	−	0.979073i	$-0.565235\pi$
$157$	121.763	−	121.763i	0.775561	−	0.775561i	0.979073	+	0.203512i	$0.0652355\pi$
$158$		0			0
$159$			82.1947i			0.516948i
$160$		0			0
$161$	−323.860			−2.01155
$162$		0			0
$163$	196.095	+	196.095i	1.20304	+	1.20304i	0.973238	+	0.229801i	$0.0738074\pi$
$163$	196.095	+	196.095i	1.20304	+	1.20304i	0.229801	+	0.973238i	$0.426193\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	26.4088	−	26.4088i	0.158136	−	0.158136i	−0.623604	−	0.781740i	$-0.714332\pi$
$167$	26.4088	−	26.4088i	0.158136	−	0.158136i	0.781740	+	0.623604i	$0.214332\pi$
$168$		0			0
$169$		−	121.190i		−	0.717101i
$170$		0			0
$171$	−76.7295			−0.448711
$172$		0			0
$173$	8.69052	+	8.69052i	0.0502342	+	0.0502342i	0.731778	−	0.681543i	$-0.238691\pi$
$173$	8.69052	+	8.69052i	0.0502342	+	0.0502342i	−0.681543	+	0.731778i	$0.738691\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	9.24211	−	9.24211i	0.0522153	−	0.0522153i
$178$		0			0
$179$			9.67002i			0.0540225i	0.999635	+	0.0270112i	$0.00859899\pi$
$179$			9.67002i			0.0540225i	−0.999635	+	0.0270112i	$0.991401\pi$
$180$		0			0
$181$	249.048			1.37595			0.687977	−	0.725733i	$-0.258499\pi$
$181$	249.048			1.37595			0.687977	+	0.725733i	$0.258499\pi$
$182$		0			0
$183$	−53.4213	−	53.4213i	−0.291920	−	0.291920i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−82.3557	+	82.3557i	−0.440405	+	0.440405i
$188$		0			0
$189$		−	64.5802i		−	0.341694i
$190$		0			0
$191$	−162.349			−0.849996			−0.424998	−	0.905194i	$-0.639725\pi$
$191$	−162.349			−0.849996			−0.424998	+	0.905194i	$0.639725\pi$
$192$		0			0
$193$	8.11418	+	8.11418i	0.0420424	+	0.0420424i	0.727815	−	0.685773i	$-0.240536\pi$
$193$	8.11418	+	8.11418i	0.0420424	+	0.0420424i	−0.685773	+	0.727815i	$0.740536\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−229.057	+	229.057i	−1.16272	+	1.16272i	−0.178847	+	0.983877i	$0.557237\pi$
$197$	−229.057	+	229.057i	−1.16272	+	1.16272i	−0.983877	+	0.178847i	$0.942763\pi$
$198$		0			0
$199$			296.892i			1.49192i	0.665992	+	0.745959i	$0.268008\pi$
$199$			296.892i			1.49192i	−0.665992	+	0.745959i	$0.731992\pi$
$200$		0			0
$201$	−147.395			−0.733310
$202$		0			0
$203$	−327.118	−	327.118i	−1.61142	−	1.61142i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	55.2771	−	55.2771i	0.267039	−	0.267039i
$208$		0			0
$209$			351.631i			1.68244i
$210$		0			0
$211$	−273.424			−1.29585			−0.647923	−	0.761706i	$-0.724362\pi$
$211$	−273.424			−1.29585			−0.647923	+	0.761706i	$0.724362\pi$
$212$		0			0
$213$	22.2453	+	22.2453i	0.104438	+	0.104438i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	278.548	−	278.548i	1.28363	−	1.28363i
$218$		0			0
$219$			168.578i			0.769764i
$220$		0			0
$221$	−58.5764			−0.265051
$222$		0			0
$223$	232.809	+	232.809i	1.04399	+	1.04399i	0.998987	+	0.0450002i	$0.0143289\pi$
$223$	232.809	+	232.809i	1.04399	+	1.04399i	0.0450002	+	0.998987i	$0.485671\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−105.206	+	105.206i	−0.463465	+	0.463465i	−0.899789	−	0.436325i	$-0.856280\pi$
$227$	−105.206	+	105.206i	−0.463465	+	0.463465i	0.436325	+	0.899789i	$0.356280\pi$
$228$		0			0
$229$			154.353i			0.674032i	0.941499	+	0.337016i	$0.109418\pi$
$229$			154.353i			0.674032i	−0.941499	+	0.337016i	$0.890582\pi$
$230$		0			0
$231$	−295.954			−1.28118
$232$		0			0
$233$	−16.6237	−	16.6237i	−0.0713464	−	0.0713464i	0.670533	−	0.741880i	$-0.266065\pi$
$233$	−16.6237	−	16.6237i	−0.0713464	−	0.0713464i	−0.741880	+	0.670533i	$0.766065\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−27.2662	+	27.2662i	−0.115047	+	0.115047i
$238$		0			0
$239$		−	97.2929i		−	0.407083i	−0.979066	−	0.203542i	$-0.934755\pi$
$239$		−	97.2929i		−	0.407083i	0.979066	−	0.203542i	$-0.0652452\pi$
$240$		0			0
$241$	−308.012			−1.27806			−0.639029	−	0.769183i	$-0.720663\pi$
$241$	−308.012			−1.27806			−0.639029	+	0.769183i	$0.720663\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−125.051	+	125.051i	−0.506277	+	0.506277i
$248$		0			0
$249$		−	0.00543319i		−	2.18200e-5i
$250$		0			0
$251$	155.213			0.618379			0.309190	−	0.951000i	$-0.399942\pi$
$251$	155.213			0.618379			0.309190	+	0.951000i	$0.399942\pi$
$252$		0			0
$253$	−253.320	−	253.320i	−1.00127	−	1.00127i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−206.538	+	206.538i	−0.803648	+	0.803648i	−0.983664	−	0.180015i	$-0.942385\pi$
$257$	−206.538	+	206.538i	−0.803648	+	0.803648i	0.180015	+	0.983664i	$0.442385\pi$
$258$		0			0
$259$		−	571.317i		−	2.20586i
$260$		0			0
$261$	111.667			0.427841
$262$		0			0
$263$	236.194	+	236.194i	0.898075	+	0.898075i	0.995266	−	0.0971908i	$-0.0309857\pi$
$263$	236.194	+	236.194i	0.898075	+	0.898075i	−0.0971908	+	0.995266i	$0.530986\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	94.5433	−	94.5433i	0.354095	−	0.354095i
$268$		0			0
$269$		−	376.885i		−	1.40106i	−0.713623	−	0.700530i	$-0.752947\pi$
$269$		−	376.885i		−	1.40106i	0.713623	−	0.700530i	$-0.247053\pi$
$270$		0			0
$271$	196.495			0.725074			0.362537	−	0.931969i	$-0.381911\pi$
$271$	196.495			0.725074			0.362537	+	0.931969i	$0.381911\pi$
$272$		0			0
$273$	−105.250	−	105.250i	−0.385531	−	0.385531i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−90.3781	+	90.3781i	−0.326275	+	0.326275i	−0.851168	−	0.524893i	$-0.824105\pi$
$277$	−90.3781	+	90.3781i	−0.326275	+	0.326275i	0.524893	+	0.851168i	$0.324105\pi$
$278$		0			0
$279$			95.0864i			0.340811i
$280$		0			0
$281$	124.506			0.443083			0.221542	−	0.975151i	$-0.428891\pi$
$281$	124.506			0.443083			0.221542	+	0.975151i	$0.428891\pi$
$282$		0			0
$283$	91.3284	+	91.3284i	0.322715	+	0.322715i	0.849808	−	0.527093i	$-0.176718\pi$
$283$	91.3284	+	91.3284i	0.322715	+	0.322715i	−0.527093	+	0.849808i	$0.676718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	322.788	−	322.788i	1.12470	−	1.12470i
$288$		0			0
$289$			217.233i			0.751670i
$290$		0			0
$291$	72.0907			0.247734
$292$		0			0
$293$	311.623	+	311.623i	1.06356	+	1.06356i	0.997838	+	0.0657234i	$0.0209355\pi$
$293$	311.623	+	311.623i	1.06356	+	1.06356i	0.0657234	+	0.997838i	$0.479065\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	50.5141	−	50.5141i	0.170081	−	0.170081i
$298$		0			0
$299$		−	180.177i		−	0.602598i
$300$		0			0
$301$	431.995			1.43520
$302$		0			0
$303$	219.066	+	219.066i	0.722991	+	0.722991i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	98.1938	−	98.1938i	0.319850	−	0.319850i	−0.528860	−	0.848709i	$-0.677380\pi$
$307$	98.1938	−	98.1938i	0.319850	−	0.319850i	0.848709	+	0.528860i	$0.177380\pi$
$308$		0			0
$309$			100.667i			0.325784i
$310$		0			0
$311$	429.889			1.38228			0.691141	−	0.722720i	$-0.257109\pi$
$311$	429.889			1.38228			0.691141	+	0.722720i	$0.257109\pi$
$312$		0			0
$313$	−65.4997	−	65.4997i	−0.209264	−	0.209264i	0.594691	−	0.803955i	$-0.297275\pi$
$313$	−65.4997	−	65.4997i	−0.209264	−	0.209264i	−0.803955	+	0.594691i	$0.797275\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−174.272	+	174.272i	−0.549755	+	0.549755i	−0.926370	−	0.376615i	$-0.877088\pi$
$317$	−174.272	+	174.272i	−0.549755	+	0.549755i	0.376615	+	0.926370i	$0.377088\pi$
$318$		0			0
$319$		−	511.738i		−	1.60419i
$320$		0			0
$321$	−90.1667			−0.280893
$322$		0			0
$323$	−153.211	−	153.211i	−0.474337	−	0.474337i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−104.704	+	104.704i	−0.320196	+	0.320196i
$328$		0			0
$329$			358.671i			1.09019i
$330$		0			0
$331$	−380.789			−1.15042			−0.575210	−	0.818006i	$-0.695080\pi$
$331$	−380.789			−1.15042			−0.575210	+	0.818006i	$0.695080\pi$
$332$		0			0
$333$	97.5138	+	97.5138i	0.292834	+	0.292834i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−245.960	+	245.960i	−0.729851	+	0.729851i	−0.970590	−	0.240739i	$-0.922610\pi$
$337$	−245.960	+	245.960i	−0.729851	+	0.729851i	0.240739	+	0.970590i	$0.422610\pi$
$338$		0			0
$339$		−	38.4366i		−	0.113382i
$340$		0			0
$341$	435.755			1.27787
$342$		0			0
$343$	−496.244	−	496.244i	−1.44678	−	1.44678i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	172.320	−	172.320i	0.496600	−	0.496600i	−0.413778	−	0.910378i	$-0.635791\pi$
$347$	172.320	−	172.320i	0.496600	−	0.496600i	0.910378	+	0.413778i	$0.135791\pi$
$348$		0			0
$349$		−	221.895i		−	0.635803i	−0.948124	−	0.317902i	$-0.897022\pi$
$349$		−	221.895i		−	0.635803i	0.948124	−	0.317902i	$-0.102978\pi$
$350$		0			0
$351$	35.9287			0.102361
$352$		0			0
$353$	−53.4623	−	53.4623i	−0.151451	−	0.151451i	0.627315	−	0.778766i	$-0.284154\pi$
$353$	−53.4623	−	53.4623i	−0.151451	−	0.151451i	−0.778766	+	0.627315i	$0.784154\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	128.951	−	128.951i	0.361209	−	0.361209i
$358$		0			0
$359$			49.7577i			0.138601i	0.997596	+	0.0693004i	$0.0220767\pi$
$359$			49.7577i			0.138601i	−0.997596	+	0.0693004i	$0.977923\pi$
$360$		0			0
$361$	−293.158			−0.812071
$362$		0			0
$363$	−83.2982	−	83.2982i	−0.229472	−	0.229472i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	56.5205	−	56.5205i	0.154007	−	0.154007i	−0.625898	−	0.779905i	$-0.715267\pi$
$367$	56.5205	−	56.5205i	0.154007	−	0.154007i	0.779905	+	0.625898i	$0.215267\pi$
$368$		0			0
$369$			110.189i			0.298614i
$370$		0			0
$371$	−589.795			−1.58974
$372$		0			0
$373$	143.430	+	143.430i	0.384531	+	0.384531i	0.872732	−	0.488200i	$-0.162346\pi$
$373$	143.430	+	143.430i	0.384531	+	0.384531i	−0.488200	+	0.872732i	$0.662346\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	181.989	−	181.989i	0.482731	−	0.482731i
$378$		0			0
$379$			659.728i			1.74071i	0.492427	+	0.870354i	$0.336110\pi$
$379$			659.728i			1.74071i	−0.492427	+	0.870354i	$0.663890\pi$
$380$		0			0
$381$	312.441			0.820056
$382$		0			0
$383$	−402.216	−	402.216i	−1.05017	−	1.05017i	−0.998673	−	0.0514981i	$-0.983600\pi$
$383$	−402.216	−	402.216i	−1.05017	−	1.05017i	−0.0514981	−	0.998673i	$-0.516400\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−73.7340	+	73.7340i	−0.190527	+	0.190527i
$388$		0			0
$389$			206.222i			0.530133i	0.964230	+	0.265067i	$0.0853940\pi$
$389$			206.222i			0.530133i	−0.964230	+	0.265067i	$0.914606\pi$
$390$		0			0
$391$	220.751			0.564580
$392$		0			0
$393$	−151.636	−	151.636i	−0.385843	−	0.385843i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	72.3916	−	72.3916i	0.182347	−	0.182347i	−0.610031	−	0.792378i	$-0.708843\pi$
$397$	72.3916	−	72.3916i	0.182347	−	0.182347i	0.792378	+	0.610031i	$0.208843\pi$
$398$		0			0
$399$		−	550.579i		−	1.37990i
$400$		0			0
$401$	−489.191			−1.21993			−0.609964	−	0.792429i	$-0.708816\pi$
$401$	−489.191			−1.21993			−0.609964	+	0.792429i	$0.708816\pi$
$402$		0			0
$403$	154.968	+	154.968i	0.384535	+	0.384535i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	446.879	−	446.879i	1.09798	−	1.09798i
$408$		0			0
$409$			724.079i			1.77037i	0.465244	+	0.885183i	$0.345967\pi$
$409$			724.079i			1.77037i	−0.465244	+	0.885183i	$0.654033\pi$
$410$		0			0
$411$	424.116			1.03191
$412$		0			0
$413$	66.3174	+	66.3174i	0.160575	+	0.160575i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.30676	−	1.30676i	0.00313372	−	0.00313372i
$418$		0			0
$419$			20.3800i			0.0486395i	0.999704	+	0.0243198i	$0.00774199\pi$
$419$			20.3800i			0.0486395i	−0.999704	+	0.0243198i	$0.992258\pi$
$420$		0			0
$421$	471.915			1.12094			0.560469	−	0.828176i	$-0.310621\pi$
$421$	471.915			1.12094			0.560469	+	0.828176i	$0.310621\pi$
$422$		0			0
$423$	−61.2189	−	61.2189i	−0.144726	−	0.144726i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	383.329	−	383.329i	0.897725	−	0.897725i
$428$		0			0
$429$		−	164.651i		−	0.383803i
$430$		0			0
$431$	−433.846			−1.00660			−0.503302	−	0.864110i	$-0.667882\pi$
$431$	−433.846			−1.00660			−0.503302	+	0.864110i	$0.667882\pi$
$432$		0			0
$433$	−143.893	−	143.893i	−0.332317	−	0.332317i	0.521149	−	0.853466i	$-0.325504\pi$
$433$	−143.893	−	143.893i	−0.332317	−	0.332317i	−0.853466	+	0.521149i	$0.825504\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	471.265	−	471.265i	1.07841	−	1.07841i
$438$		0			0
$439$		−	21.7905i		−	0.0496368i	−0.999692	−	0.0248184i	$-0.992099\pi$
$439$		−	21.7905i		−	0.0496368i	0.999692	−	0.0248184i	$-0.00790075\pi$
$440$		0			0
$441$	316.400			0.717461
$442$		0			0
$443$	−186.194	−	186.194i	−0.420302	−	0.420302i	0.465005	−	0.885308i	$-0.346052\pi$
$443$	−186.194	−	186.194i	−0.420302	−	0.420302i	−0.885308	+	0.465005i	$0.846052\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−79.9152	+	79.9152i	−0.178781	+	0.178781i
$448$		0			0
$449$		−	173.546i		−	0.386517i	−0.981148	−	0.193258i	$-0.938094\pi$
$449$		−	173.546i		−	0.386517i	0.981148	−	0.193258i	$-0.0619056\pi$
$450$		0			0
$451$	504.964			1.11965
$452$		0			0
$453$	−192.509	−	192.509i	−0.424964	−	0.424964i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	195.883	−	195.883i	0.428628	−	0.428628i	−0.459533	−	0.888161i	$-0.651983\pi$
$457$	195.883	−	195.883i	0.428628	−	0.428628i	0.888161	+	0.459533i	$0.151983\pi$
$458$		0			0
$459$			44.0195i			0.0959030i
$460$		0			0
$461$	710.784			1.54183			0.770916	−	0.636937i	$-0.219799\pi$
$461$	710.784			1.54183			0.770916	+	0.636937i	$0.219799\pi$
$462$		0			0
$463$	−407.194	−	407.194i	−0.879468	−	0.879468i	0.114011	−	0.993479i	$-0.463630\pi$
$463$	−407.194	−	407.194i	−0.879468	−	0.879468i	−0.993479	+	0.114011i	$0.963630\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	46.4766	−	46.4766i	0.0995215	−	0.0995215i	−0.655593	−	0.755114i	$-0.727581\pi$
$467$	46.4766	−	46.4766i	0.0995215	−	0.0995215i	0.755114	+	0.655593i	$0.227581\pi$
$468$		0			0
$469$		−	1057.65i		−	2.25511i
$470$		0			0
$471$	−298.257			−0.633243
$472$		0			0
$473$	337.903	+	337.903i	0.714382	+	0.714382i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	100.668	−	100.668i	0.211043	−	0.211043i
$478$		0			0
$479$		−	632.568i		−	1.32060i	−0.751001	−	0.660301i	$-0.770429\pi$
$479$		−	632.568i		−	1.32060i	0.751001	−	0.660301i	$-0.229571\pi$
$480$		0			0
$481$	317.848			0.660806
$482$		0			0
$483$	396.645	+	396.645i	0.821212	+	0.821212i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	24.5000	−	24.5000i	0.0503080	−	0.0503080i	−0.681505	−	0.731813i	$-0.738674\pi$
$487$	24.5000	−	24.5000i	0.0503080	−	0.0503080i	0.731813	+	0.681505i	$0.238674\pi$
$488$		0			0
$489$		−	480.333i		−	0.982277i
$490$		0			0
$491$	888.550			1.80967			0.904837	−	0.425759i	$-0.139993\pi$
$491$	888.550			1.80967			0.904837	+	0.425759i	$0.139993\pi$
$492$		0			0
$493$	222.972	+	222.972i	0.452276	+	0.452276i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−159.623	+	159.623i	−0.321173	+	0.321173i
$498$		0			0
$499$			313.223i			0.627702i	0.949472	+	0.313851i	$0.101619\pi$
$499$			313.223i			0.627702i	−0.949472	+	0.313851i	$0.898381\pi$
$500$		0			0
$501$	−64.6880			−0.129118
$502$		0			0
$503$	22.1824	+	22.1824i	0.0441001	+	0.0441001i	0.728813	−	0.684713i	$-0.240072\pi$
$503$	22.1824	+	22.1824i	0.0441001	+	0.0441001i	−0.684713	+	0.728813i	$0.740072\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−148.427	+	148.427i	−0.292755	+	0.292755i
$508$		0			0
$509$			510.456i			1.00286i	0.865198	+	0.501431i	$0.167193\pi$
$509$			510.456i			1.00286i	−0.865198	+	0.501431i	$0.832807\pi$
$510$		0			0
$511$	−1209.65			−2.36721
$512$		0			0
$513$	93.9741	+	93.9741i	0.183185	+	0.183185i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−280.550	+	280.550i	−0.542649	+	0.542649i
$518$		0			0
$519$		−	21.2874i		−	0.0410161i
$520$		0			0
$521$	−765.379			−1.46906			−0.734529	−	0.678578i	$-0.762597\pi$
$521$	−765.379			−1.46906			−0.734529	+	0.678578i	$0.762597\pi$
$522$		0			0
$523$	57.8350	+	57.8350i	0.110583	+	0.110583i	0.760233	−	0.649650i	$-0.225085\pi$
$523$	57.8350	+	57.8350i	0.110583	+	0.110583i	−0.649650	+	0.760233i	$0.725085\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−189.865	+	189.865i	−0.360275	+	0.360275i
$528$		0			0
$529$			150.014i			0.283580i
$530$		0			0
$531$	−22.6384			−0.0426336
$532$		0			0
$533$	179.581	+	179.581i	0.336924	+	0.336924i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	11.8433	−	11.8433i	0.0220546	−	0.0220546i
$538$		0			0
$539$		−	1449.98i		−	2.69012i
$540$		0			0
$541$	−650.947			−1.20323			−0.601615	−	0.798786i	$-0.705476\pi$
$541$	−650.947			−1.20323			−0.601615	+	0.798786i	$0.705476\pi$
$542$		0			0
$543$	−305.020	−	305.020i	−0.561731	−	0.561731i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−200.024	+	200.024i	−0.365674	+	0.365674i	−0.865897	−	0.500223i	$-0.833251\pi$
$547$	−200.024	+	200.024i	−0.365674	+	0.365674i	0.500223	+	0.865897i	$0.333251\pi$
$548$		0			0
$549$			130.855i			0.238351i
$550$		0			0
$551$	952.014			1.72779
$552$		0			0
$553$	−195.651	−	195.651i	−0.353799	−	0.353799i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−169.939	+	169.939i	−0.305097	+	0.305097i	−0.843004	−	0.537907i	$-0.819215\pi$
$557$	−169.939	+	169.939i	−0.305097	+	0.305097i	0.537907	+	0.843004i	$0.319215\pi$
$558$		0			0
$559$			240.337i			0.429941i
$560$		0			0
$561$	201.729			0.359589
$562$		0			0
$563$	288.641	+	288.641i	0.512684	+	0.512684i	0.915348	−	0.402664i	$-0.131916\pi$
$563$	288.641	+	288.641i	0.512684	+	0.512684i	−0.402664	+	0.915348i	$0.631916\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−79.0943	+	79.0943i	−0.139496	+	0.139496i
$568$		0			0
$569$		−	119.228i		−	0.209540i	−0.994496	−	0.104770i	$-0.966589\pi$
$569$		−	119.228i		−	0.209540i	0.994496	−	0.104770i	$-0.0334107\pi$
$570$		0			0
$571$	−1070.64			−1.87503			−0.937513	−	0.347951i	$-0.886878\pi$
$571$	−1070.64			−1.87503			−0.937513	+	0.347951i	$0.886878\pi$
$572$		0			0
$573$	198.836	+	198.836i	0.347010	+	0.347010i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−572.086	+	572.086i	−0.991484	+	0.991484i	−0.999964	−	0.00848018i	$-0.997301\pi$
$577$	−572.086	+	572.086i	−0.991484	+	0.991484i	0.00848018	+	0.999964i	$0.497301\pi$
$578$		0			0
$579$		−	19.8756i		−	0.0343275i
$580$		0			0
$581$	0.0389863			6.71020e−5
$582$		0			0
$583$	−461.332	−	461.332i	−0.791307	−	0.791307i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	314.886	−	314.886i	0.536433	−	0.536433i	−0.386047	−	0.922479i	$-0.626160\pi$
$587$	314.886	−	314.886i	0.536433	−	0.536433i	0.922479	+	0.386047i	$0.126160\pi$
$588$		0			0
$589$			810.659i			1.37633i
$590$		0			0
$591$	561.072			0.949360
$592$		0			0
$593$	−781.658	−	781.658i	−1.31814	−	1.31814i	−0.915246	−	0.402896i	$-0.868004\pi$
$593$	−781.658	−	781.658i	−1.31814	−	1.31814i	−0.402896	−	0.915246i	$-0.631996\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	363.616	−	363.616i	0.609073	−	0.609073i
$598$		0			0
$599$			423.280i			0.706645i	0.935502	+	0.353322i	$0.114948\pi$
$599$			423.280i			0.706645i	−0.935502	+	0.353322i	$0.885052\pi$
$600$		0			0
$601$	−68.1617			−0.113414			−0.0567069	−	0.998391i	$-0.518060\pi$
$601$	−68.1617			−0.113414			−0.0567069	+	0.998391i	$0.518060\pi$
$602$		0			0
$603$	180.522	+	180.522i	0.299373	+	0.299373i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−186.390	+	186.390i	−0.307068	+	0.307068i	−0.843771	−	0.536703i	$-0.819669\pi$
$607$	−186.390	+	186.390i	−0.307068	+	0.307068i	0.536703	+	0.843771i	$0.319669\pi$
$608$		0			0
$609$			801.272i			1.31572i
$610$		0			0
$611$	−199.544			−0.326586
$612$		0			0
$613$	−337.566	−	337.566i	−0.550679	−	0.550679i	0.375958	−	0.926637i	$-0.377314\pi$
$613$	−337.566	−	337.566i	−0.550679	−	0.550679i	−0.926637	+	0.375958i	$0.877314\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−858.797	+	858.797i	−1.39189	+	1.39189i	−0.570808	+	0.821084i	$0.693370\pi$
$617$	−858.797	+	858.797i	−1.39189	+	1.39189i	−0.821084	+	0.570808i	$0.806630\pi$
$618$		0			0
$619$			202.653i			0.327388i	0.986511	+	0.163694i	$0.0523410\pi$
$619$			202.653i			0.327388i	−0.986511	+	0.163694i	$0.947659\pi$
$620$		0			0
$621$	−135.401			−0.218037
$622$		0			0
$623$	678.403	+	678.403i	1.08893	+	1.08893i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	430.658	−	430.658i	0.686855	−	0.686855i
$628$		0			0
$629$			389.424i			0.619116i
$630$		0			0
$631$	−536.853			−0.850797			−0.425398	−	0.905006i	$-0.639866\pi$
$631$	−536.853			−0.850797			−0.425398	+	0.905006i	$0.639866\pi$
$632$		0			0
$633$	334.874	+	334.874i	0.529027	+	0.529027i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	515.656	−	515.656i	0.809507	−	0.809507i
$638$		0			0
$639$		−	54.4896i		−	0.0852733i
$640$		0			0
$641$	665.993			1.03899			0.519495	−	0.854473i	$-0.326120\pi$
$641$	665.993			1.03899			0.519495	+	0.854473i	$0.326120\pi$
$642$		0			0
$643$	7.83317	+	7.83317i	0.0121822	+	0.0121822i	0.713172	−	0.700989i	$-0.247258\pi$
$643$	7.83317	+	7.83317i	0.0121822	+	0.0121822i	−0.700989	+	0.713172i	$0.747258\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	287.426	−	287.426i	0.444244	−	0.444244i	−0.449191	−	0.893436i	$-0.648288\pi$
$647$	287.426	−	287.426i	0.444244	−	0.444244i	0.893436	+	0.449191i	$0.148288\pi$
$648$		0			0
$649$			103.746i			0.159855i
$650$		0			0
$651$	−682.300			−1.04808
$652$		0			0
$653$	587.685	+	587.685i	0.899976	+	0.899976i	0.995434	−	0.0954571i	$-0.0304313\pi$
$653$	587.685	+	587.685i	0.899976	+	0.899976i	−0.0954571	+	0.995434i	$0.530431\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	206.465	−	206.465i	0.314255	−	0.314255i
$658$		0			0
$659$			754.028i			1.14420i	0.820184	+	0.572101i	$0.193871\pi$
$659$			754.028i			1.14420i	−0.820184	+	0.572101i	$0.806129\pi$
$660$		0			0
$661$	−818.290			−1.23796			−0.618978	−	0.785408i	$-0.712453\pi$
$661$	−818.290			−1.23796			−0.618978	+	0.785408i	$0.712453\pi$
$662$		0			0
$663$	71.7411	+	71.7411i	0.108207	+	0.108207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−685.845	+	685.845i	−1.02825	+	1.02825i
$668$		0			0
$669$		−	570.264i		−	0.852412i
$670$		0			0
$671$	599.672			0.893700
$672$		0			0
$673$	−202.080	−	202.080i	−0.300268	−	0.300268i	0.540851	−	0.841119i	$-0.318102\pi$
$673$	−202.080	−	202.080i	−0.300268	−	0.300268i	−0.841119	+	0.540851i	$0.818102\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	47.2645	−	47.2645i	0.0698147	−	0.0698147i	−0.671337	−	0.741152i	$-0.734280\pi$
$677$	47.2645	−	47.2645i	0.0698147	−	0.0698147i	0.741152	+	0.671337i	$0.234280\pi$
$678$		0			0
$679$			517.292i			0.761844i
$680$		0			0
$681$	257.702			0.378417
$682$		0			0
$683$	126.090	+	126.090i	0.184612	+	0.184612i	0.793362	−	0.608750i	$-0.208329\pi$
$683$	126.090	+	126.090i	0.184612	+	0.184612i	−0.608750	+	0.793362i	$0.708329\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	189.043	−	189.043i	0.275172	−	0.275172i
$688$		0			0
$689$		−	328.127i		−	0.476237i
$690$		0			0
$691$	485.927			0.703223			0.351612	−	0.936146i	$-0.385634\pi$
$691$	485.927			0.703223			0.351612	+	0.936146i	$0.385634\pi$
$692$		0			0
$693$	362.468	+	362.468i	0.523041	+	0.523041i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−220.021	+	220.021i	−0.315668	+	0.315668i
$698$		0			0
$699$			40.7196i			0.0582541i
$700$		0			0
$701$	−94.5850			−0.134929			−0.0674643	−	0.997722i	$-0.521491\pi$
$701$	−94.5850			−0.134929			−0.0674643	+	0.997722i	$0.521491\pi$
$702$		0			0
$703$	831.354	+	831.354i	1.18258	+	1.18258i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1571.93	+	1571.93i	−2.22338	+	2.22338i
$708$		0			0
$709$			24.3499i			0.0343440i	0.999853	+	0.0171720i	$0.00546629\pi$
$709$			24.3499i			0.0343440i	−0.999853	+	0.0171720i	$0.994534\pi$
$710$		0			0
$711$	66.7883			0.0939357
$712$		0			0
$713$	−584.011	−	584.011i	−0.819090	−	0.819090i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−119.159	+	119.159i	−0.166191	+	0.166191i
$718$		0			0
$719$			534.841i			0.743868i	0.928259	+	0.371934i	$0.121305\pi$
$719$			534.841i			0.743868i	−0.928259	+	0.371934i	$0.878695\pi$
$720$		0			0
$721$	−722.346			−1.00187
$722$		0			0
$723$	377.236	+	377.236i	0.521765	+	0.521765i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−435.834	+	435.834i	−0.599497	+	0.599497i	−0.940179	−	0.340682i	$-0.889342\pi$
$727$	−435.834	+	435.834i	−0.599497	+	0.599497i	0.340682	+	0.940179i	$0.389342\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−294.459			−0.402816
$732$		0			0
$733$	−224.087	−	224.087i	−0.305712	−	0.305712i	0.537531	−	0.843244i	$-0.319357\pi$
$733$	−224.087	−	224.087i	−0.305712	−	0.305712i	−0.843244	+	0.537531i	$0.819357\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	827.282	−	827.282i	1.12250	−	1.12250i
$738$		0			0
$739$			730.510i			0.988512i	0.869317	+	0.494256i	$0.164559\pi$
$739$			730.510i			0.988512i	−0.869317	+	0.494256i	$0.835441\pi$
$740$		0			0
$741$	306.310			0.413374
$742$		0			0
$743$	941.961	+	941.961i	1.26778	+	1.26778i	0.947232	+	0.320549i	$0.103867\pi$
$743$	941.961	+	941.961i	1.26778	+	1.26778i	0.320549	+	0.947232i	$0.396133\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.00665427	+	0.00665427i	−8.90800e−6	+	8.90800e-6i
$748$		0			0
$749$		−	646.998i		−	0.863816i
$750$		0			0
$751$	231.169			0.307814			0.153907	−	0.988085i	$-0.450814\pi$
$751$	231.169			0.307814			0.153907	+	0.988085i	$0.450814\pi$
$752$		0			0
$753$	−190.096	−	190.096i	−0.252452	−	0.252452i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	382.911	−	382.911i	0.505826	−	0.505826i	−0.407416	−	0.913243i	$-0.633570\pi$
$757$	382.911	−	382.911i	0.505826	−	0.505826i	0.913243	+	0.407416i	$0.133570\pi$
$758$		0			0
$759$			620.505i			0.817530i
$760$		0			0
$761$	−941.999			−1.23784			−0.618922	−	0.785453i	$-0.712430\pi$
$761$	−941.999			−1.23784			−0.618922	+	0.785453i	$0.712430\pi$
$762$		0			0
$763$	−751.312	−	751.312i	−0.984681	−	0.984681i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.8952	+	36.8952i	−0.0481032	+	0.0481032i
$768$		0			0
$769$		−	1388.76i		−	1.80593i	−0.429718	−	0.902963i	$-0.641387\pi$
$769$		−	1388.76i		−	1.80593i	0.429718	−	0.902963i	$-0.358613\pi$
$770$		0			0
$771$	505.912			0.656176
$772$		0			0
$773$	185.158	+	185.158i	0.239532	+	0.239532i	0.816656	−	0.577124i	$-0.195825\pi$
$773$	185.158	+	185.158i	0.239532	+	0.239532i	−0.577124	+	0.816656i	$0.695825\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−699.718	+	699.718i	−0.900538	+	0.900538i
$778$		0			0
$779$			939.413i			1.20592i
$780$		0			0
$781$	−249.711			−0.319733
$782$		0			0
$783$	−136.763	−	136.763i	−0.174665	−	0.174665i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−194.815	+	194.815i	−0.247542	+	0.247542i	−0.819961	−	0.572419i	$-0.806005\pi$
$787$	−194.815	+	194.815i	−0.247542	+	0.247542i	0.572419	+	0.819961i	$0.306005\pi$
$788$		0			0
$789$		−	578.554i		−	0.733275i
$790$		0			0
$791$	275.805			0.348679
$792$		0			0
$793$	213.262	+	213.262i	0.268930	+	0.268930i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−694.112	+	694.112i	−0.870906	+	0.870906i	−0.992571	−	0.121665i	$-0.961177\pi$
$797$	−694.112	+	694.112i	−0.870906	+	0.870906i	0.121665	+	0.992571i	$0.461177\pi$
$798$		0			0
$799$		−	244.479i		−	0.305982i
$800$		0			0
$801$	−231.583			−0.289117
$802$		0			0
$803$	−946.175	−	946.175i	−1.17830	−	1.17830i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−461.588	+	461.588i	−0.571980	+	0.571980i
$808$		0			0
$809$		−	1492.22i		−	1.84453i	−0.386563	−	0.922263i	$-0.626338\pi$
$809$		−	1492.22i		−	1.84453i	0.386563	−	0.922263i	$-0.373662\pi$
$810$		0			0
$811$	−80.5721			−0.0993490			−0.0496745	−	0.998765i	$-0.515818\pi$
$811$	−80.5721			−0.0993490			−0.0496745	+	0.998765i	$0.515818\pi$
$812$		0			0
$813$	−240.656	−	240.656i	−0.296010	−	0.296010i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−628.619	+	628.619i	−0.769423	+	0.769423i
$818$		0			0
$819$			257.809i			0.314785i
$820$		0			0
$821$	−221.422			−0.269698			−0.134849	−	0.990866i	$-0.543055\pi$
$821$	−221.422			−0.269698			−0.134849	+	0.990866i	$0.543055\pi$
$822$		0			0
$823$	815.236	+	815.236i	0.990566	+	0.990566i	0.999956	−	0.00938979i	$-0.00298891\pi$
$823$	815.236	+	815.236i	0.990566	+	0.990566i	−0.00938979	+	0.999956i	$0.502989\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.5640	−	29.5640i	0.0357485	−	0.0357485i	−0.689007	−	0.724755i	$-0.741953\pi$
$827$	29.5640	−	29.5640i	0.0357485	−	0.0357485i	0.724755	+	0.689007i	$0.241953\pi$
$828$		0			0
$829$			558.092i			0.673211i	0.941646	+	0.336606i	$0.109279\pi$
$829$			558.092i			0.673211i	−0.941646	+	0.336606i	$0.890721\pi$
$830$		0			0
$831$	221.380			0.266402
$832$		0			0
$833$	631.777	+	631.777i	0.758436	+	0.758436i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	116.457	−	116.457i	0.139136	−	0.139136i
$838$		0			0
$839$			623.755i			0.743451i	0.928343	+	0.371725i	$0.121234\pi$
$839$			623.755i			0.743451i	−0.928343	+	0.371725i	$0.878766\pi$
$840$		0			0
$841$	−544.492			−0.647434
$842$		0			0
$843$	−152.489	−	152.489i	−0.180888	−	0.180888i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	597.713	−	597.713i	0.705682	−	0.705682i
$848$		0			0
$849$		−	223.708i		−	0.263496i
$850$		0			0
$851$	−1197.84			−1.40757
$852$		0			0
$853$	454.178	+	454.178i	0.532448	+	0.532448i	0.921300	−	0.388852i	$-0.127128\pi$
$853$	454.178	+	454.178i	0.532448	+	0.532448i	−0.388852	+	0.921300i	$0.627128\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	769.104	−	769.104i	0.897438	−	0.897438i	−0.0977709	−	0.995209i	$-0.531171\pi$
$857$	769.104	−	769.104i	0.897438	−	0.897438i	0.995209	+	0.0977709i	$0.0311712\pi$
$858$		0			0
$859$		−	536.071i		−	0.624064i	−0.950072	−	0.312032i	$-0.898990\pi$
$859$		−	536.071i		−	0.624064i	0.950072	−	0.312032i	$-0.101010\pi$
$860$		0			0
$861$	−790.666			−0.918312
$862$		0			0
$863$	−500.177	−	500.177i	−0.579579	−	0.579579i	0.355208	−	0.934787i	$-0.384410\pi$
$863$	−500.177	−	500.177i	−0.579579	−	0.579579i	−0.934787	+	0.355208i	$0.884410\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	266.055	−	266.055i	0.306868	−	0.306868i
$868$		0			0
$869$		−	306.073i		−	0.352212i
$870$		0			0
$871$	588.413			0.675561
$872$		0			0
$873$	−88.2927	−	88.2927i	−0.101137	−	0.101137i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	102.540	−	102.540i	0.116922	−	0.116922i	−0.646225	−	0.763147i	$-0.723653\pi$
$877$	102.540	−	102.540i	0.116922	−	0.116922i	0.763147	+	0.646225i	$0.223653\pi$
$878$		0			0
$879$		−	763.318i		−	0.868394i
$880$		0			0
$881$	649.515			0.737247			0.368624	−	0.929579i	$-0.379829\pi$
$881$	649.515			0.737247			0.368624	+	0.929579i	$0.379829\pi$
$882$		0			0
$883$	341.476	+	341.476i	0.386723	+	0.386723i	0.873517	−	0.486794i	$-0.161834\pi$
$883$	341.476	+	341.476i	0.386723	+	0.386723i	−0.486794	+	0.873517i	$0.661834\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−516.110	+	516.110i	−0.581860	+	0.581860i	−0.935414	−	0.353554i	$-0.884973\pi$
$887$	−516.110	+	516.110i	−0.581860	+	0.581860i	0.353554	+	0.935414i	$0.384973\pi$
$888$		0			0
$889$			2241.95i			2.52188i
$890$		0			0
$891$	−123.734			−0.138871
$892$		0			0
$893$	−521.922	−	521.922i	−0.584459	−	0.584459i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−220.670	+	220.670i	−0.246009	+	0.246009i
$898$		0			0
$899$		−	1179.77i		−	1.31232i
$900$		0			0
$901$	402.019			0.446192
$902$		0			0
$903$	−529.084	−	529.084i	−0.585918	−	0.585918i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	62.4116	−	62.4116i	0.0688111	−	0.0688111i	−0.671864	−	0.740675i	$-0.734506\pi$
$907$	62.4116	−	62.4116i	0.0688111	−	0.0688111i	0.740675	+	0.671864i	$0.234506\pi$
$908$		0			0
$909$		−	536.600i		−	0.590319i
$910$		0			0
$911$	−1140.91			−1.25237			−0.626183	−	0.779676i	$-0.715384\pi$
$911$	−1140.91			−1.25237			−0.626183	+	0.779676i	$0.715384\pi$
$912$		0			0
$913$	0.0304947	+	0.0304947i	3.34006e−5	+	3.34006e-5i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1088.08	−	1088.08i	1.18656	−	1.18656i
$918$		0			0
$919$		−	1200.77i		−	1.30661i	−0.757097	−	0.653303i	$-0.773383\pi$
$919$		−	1200.77i		−	1.30661i	0.757097	−	0.653303i	$-0.226617\pi$
$920$		0			0
$921$	−240.525			−0.261156
$922$		0			0
$923$	−88.8049	−	88.8049i	−0.0962133	−	0.0962133i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	123.292	−	123.292i	0.133001	−	0.133001i
$928$		0			0
$929$		−	87.8294i		−	0.0945419i	−0.998882	−	0.0472709i	$-0.984948\pi$
$929$		−	87.8294i		−	0.0945419i	0.998882	−	0.0472709i	$-0.0150524\pi$
$930$		0			0
$931$	2697.47			2.89739
$932$		0			0
$933$	−526.505	−	526.505i	−0.564314	−	0.564314i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	215.447	−	215.447i	0.229933	−	0.229933i	−0.582732	−	0.812665i	$-0.698016\pi$
$937$	215.447	−	215.447i	0.229933	−	0.229933i	0.812665	+	0.582732i	$0.198016\pi$
$938$		0			0
$939$			160.441i			0.170863i
$940$		0			0
$941$	805.908			0.856438			0.428219	−	0.903675i	$-0.359141\pi$
$941$	805.908			0.856438			0.428219	+	0.903675i	$0.359141\pi$
$942$		0			0
$943$	−676.767	−	676.767i	−0.717675	−	0.717675i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	739.218	−	739.218i	0.780589	−	0.780589i	−0.199341	−	0.979930i	$-0.563880\pi$
$947$	739.218	−	739.218i	0.780589	−	0.780589i	0.979930	+	0.199341i	$0.0638801\pi$
$948$		0			0
$949$		−	672.977i		−	0.709144i
$950$		0			0
$951$	426.879			0.448873
$952$		0			0
$953$	790.131	+	790.131i	0.829098	+	0.829098i	0.987392	−	0.158294i	$-0.0505993\pi$
$953$	790.131	+	790.131i	0.829098	+	0.829098i	−0.158294	+	0.987392i	$0.550599\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−626.748	+	626.748i	−0.654909	+	0.654909i
$958$		0			0
$959$			3043.28i			3.17339i
$960$		0			0
$961$	43.6021			0.0453716
$962$		0			0
$963$	110.431	+	110.431i	0.114674	+	0.114674i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	213.927	−	213.927i	0.221227	−	0.221227i	−0.587788	−	0.809015i	$-0.700001\pi$
$967$	213.927	−	213.927i	0.221227	−	0.221227i	0.809015	+	0.587788i	$0.200001\pi$
$968$		0			0
$969$			375.288i			0.387294i
$970$		0			0
$971$	−1356.34			−1.39685			−0.698423	−	0.715686i	$-0.746114\pi$
$971$	−1356.34			−1.39685			−0.698423	+	0.715686i	$0.746114\pi$
$972$		0			0
$973$	9.37676	+	9.37676i	0.00963695	+	0.00963695i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1047.93	−	1047.93i	1.07260	−	1.07260i	0.0754457	−	0.997150i	$-0.475962\pi$
$977$	1047.93	−	1047.93i	1.07260	−	1.07260i	0.997150	−	0.0754457i	$-0.0240380\pi$
$978$		0			0
$979$			1061.28i			1.08405i
$980$		0			0
$981$	256.471			0.261439
$982$		0			0
$983$	741.601	+	741.601i	0.754426	+	0.754426i	0.975302	−	0.220876i	$-0.0708915\pi$
$983$	741.601	+	741.601i	0.754426	+	0.754426i	−0.220876	+	0.975302i	$0.570891\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	439.281	−	439.281i	0.445067	−	0.445067i
$988$		0			0
$989$		−	905.734i		−	0.915808i
$990$		0			0
$991$	304.849			0.307618			0.153809	−	0.988101i	$-0.450846\pi$
$991$	304.849			0.307618			0.153809	+	0.988101i	$0.450846\pi$
$992$		0			0
$993$	466.369	+	466.369i	0.469657	+	0.469657i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	542.650	−	542.650i	0.544283	−	0.544283i	−0.380499	−	0.924782i	$-0.624248\pi$
$997$	542.650	−	542.650i	0.544283	−	0.544283i	0.924782	+	0.380499i	$0.124248\pi$
$998$		0			0
$999$		−	238.859i		−	0.239098i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.u.h.193.2		8
3.2	odd	2		1800.3.v.t.793.4		8
4.3	odd	2		1200.3.bg.q.193.3		8
5.2	odd	4	inner	600.3.u.h.457.2		8
5.3	odd	4		120.3.u.b.97.3	yes	8
5.4	even	2		120.3.u.b.73.3	✓	8
15.2	even	4		1800.3.v.t.1657.4		8
15.8	even	4		360.3.v.f.217.3		8
15.14	odd	2		360.3.v.f.73.3		8
20.3	even	4		240.3.bg.e.97.1		8
20.7	even	4		1200.3.bg.q.1057.3		8
20.19	odd	2		240.3.bg.e.193.1		8
40.3	even	4		960.3.bg.k.577.4		8
40.13	odd	4		960.3.bg.l.577.2		8
40.19	odd	2		960.3.bg.k.193.4		8
40.29	even	2		960.3.bg.l.193.2		8
60.23	odd	4		720.3.bh.o.577.3		8
60.59	even	2		720.3.bh.o.433.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.b.73.3	✓	8	5.4	even	2
120.3.u.b.97.3	yes	8	5.3	odd	4
240.3.bg.e.97.1		8	20.3	even	4
240.3.bg.e.193.1		8	20.19	odd	2
360.3.v.f.73.3		8	15.14	odd	2
360.3.v.f.217.3		8	15.8	even	4
600.3.u.h.193.2		8	1.1	even	1	trivial
600.3.u.h.457.2		8	5.2	odd	4	inner
720.3.bh.o.433.3		8	60.59	even	2
720.3.bh.o.577.3		8	60.23	odd	4
960.3.bg.k.193.4		8	40.19	odd	2
960.3.bg.k.577.4		8	40.3	even	4
960.3.bg.l.193.2		8	40.29	even	2
960.3.bg.l.577.2		8	40.13	odd	4
1200.3.bg.q.193.3		8	4.3	odd	2
1200.3.bg.q.1057.3		8	20.7	even	4
1800.3.v.t.793.4		8	3.2	odd	2
1800.3.v.t.1657.4		8	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(8.78825 - 8.78825i) q^{7} +3.00000i q^{9} +13.7482 q^{11} +(4.88927 + 4.88927i) q^{13} +(-5.99029 + 5.99029i) q^{17} +25.5765i q^{19} -21.5267 q^{21} +(-18.4257 - 18.4257i) q^{23} +(3.67423 - 3.67423i) q^{27} -37.2222i q^{29} +31.6955 q^{31} +(-16.8380 - 16.8380i) q^{33} +(32.5046 - 32.5046i) q^{37} -11.9762i q^{39} +36.7295 q^{41} +(24.5780 + 24.5780i) q^{43} +(-20.4063 + 20.4063i) q^{47} -105.467i q^{49} +14.6732 q^{51} +(-33.5559 - 33.5559i) q^{53} +(31.3247 - 31.3247i) q^{57} +7.54615i q^{59} +43.6183 q^{61} +(26.3648 + 26.3648i) q^{63} +(60.1739 - 60.1739i) q^{67} +45.1336i q^{69} -18.1632 q^{71} +(-68.8218 - 68.8218i) q^{73} +(120.823 - 120.823i) q^{77} -22.2628i q^{79} -9.00000 q^{81} +(0.00221809 + 0.00221809i) q^{83} +(-45.5877 + 45.5877i) q^{87} +77.1943i q^{89} +85.9363 q^{91} +(-38.8189 - 38.8189i) q^{93} +(-29.4309 + 29.4309i) q^{97} +41.2446i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} + 32 q^{11} + 4 q^{13} - 52 q^{17} - 24 q^{21} + 40 q^{23} + 96 q^{31} - 60 q^{33} + 60 q^{37} - 152 q^{41} + 88 q^{43} + 16 q^{47} - 168 q^{51} - 108 q^{53} + 24 q^{57} + 264 q^{61} - 12 q^{63}+ \cdots + 208 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 + 32 * q^11 + 4 * q^13 - 52 * q^17 - 24 * q^21 + 40 * q^23 + 96 * q^31 - 60 * q^33 + 60 * q^37 - 152 * q^41 + 88 * q^43 + 16 * q^47 - 168 * q^51 - 108 * q^53 + 24 * q^57 + 264 * q^61 - 12 * q^63 + 216 * q^67 - 240 * q^71 + 208 * q^73 - 168 * q^77 - 72 * q^81 - 336 * q^83 - 252 * q^87 + 592 * q^91 - 264 * q^93 + 208 * q^97

Introduction

L-functions

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