LMFDB - Embedded newform 1008.3.cg.q.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(145,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1008.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cg (of order $6$, 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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + \cdots + 1148023744 $ x^16 - 2x^15 + 81x^14 - 118x^13 + 1960x^12 - 366x^11 + 37625x^10 - 83714x^9 + 623931x^8 + 289492x^7 + 241286x^6 - 4777608x^5 + 98362220x^4 - 239083824x^3 + 523977616x^2 - 403737152*x + 1148023744
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Root			$-3.20903 - 3.57433i$ of defining polynomial
Character	$\chi$	$=$	1008.145
Dual form			1008.3.cg.q.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.51611 - 2.60738i) q^{5} +(-6.91807 - 1.06788i) q^{7} +O(q^{10})$	$q+(-4.51611 - 2.60738i) q^{5} +(-6.91807 - 1.06788i) q^{7} +(4.56040 + 7.89885i) q^{11} +3.43129i q^{13} +(-4.73709 + 2.73496i) q^{17} +(9.45229 + 5.45728i) q^{19} +(-9.95199 + 17.2374i) q^{23} +(1.09685 + 1.89981i) q^{25} -36.7480 q^{29} +(17.8110 - 10.2832i) q^{31} +(28.4584 + 22.8607i) q^{35} +(23.5754 - 40.8338i) q^{37} -41.5146i q^{41} +73.9986 q^{43} +(51.2578 + 29.5937i) q^{47} +(46.7193 + 14.7754i) q^{49} +(-31.7014 - 54.9084i) q^{53} -47.5628i q^{55} +(44.8630 - 25.9017i) q^{59} +(21.2053 + 12.2429i) q^{61} +(8.94668 - 15.4961i) q^{65} +(9.58916 + 16.6089i) q^{67} +2.54627 q^{71} +(102.444 - 59.1459i) q^{73} +(-23.1141 - 59.5147i) q^{77} +(5.67414 - 9.82791i) q^{79} -30.4340i q^{83} +28.5243 q^{85} +(139.698 + 80.6547i) q^{89} +(3.66422 - 23.7379i) q^{91} +(-28.4584 - 49.2914i) q^{95} +101.257i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.51611	−	2.60738i	−0.903223	−	0.521476i	−0.0249783	−	0.999688i	$-0.507952\pi$
$5$	−4.51611	−	2.60738i	−0.903223	−	0.521476i	−0.878244	+	0.478212i	$0.841285\pi$
$6$		0			0
$7$	−6.91807	−	1.06788i	−0.988295	−	0.152555i
$7$	−6.91807	−	1.06788i	−0.988295	−	0.152555i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.56040	+	7.89885i	0.414582	+	0.718077i	0.995384	−	0.0959675i	$-0.0305945\pi$
$11$	4.56040	+	7.89885i	0.414582	+	0.718077i	−0.580802	+	0.814045i	$0.697261\pi$
$12$		0			0
$13$			3.43129i			0.263946i	0.991253	+	0.131973i	$0.0421312\pi$
$13$			3.43129i			0.263946i	−0.991253	+	0.131973i	$0.957869\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.73709	+	2.73496i	−0.278652	+	0.160880i	−0.632813	−	0.774305i	$-0.718100\pi$
$17$	−4.73709	+	2.73496i	−0.278652	+	0.160880i	0.354161	+	0.935184i	$0.384767\pi$
$18$		0			0
$19$	9.45229	+	5.45728i	0.497489	+	0.287225i	0.727676	−	0.685921i	$-0.240601\pi$
$19$	9.45229	+	5.45728i	0.497489	+	0.287225i	−0.230187	+	0.973146i	$0.573934\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.95199	+	17.2374i	−0.432695	+	0.749450i	−0.997104	−	0.0760449i	$-0.975771\pi$
$23$	−9.95199	+	17.2374i	−0.432695	+	0.749450i	0.564409	+	0.825495i	$0.309104\pi$
$24$		0			0
$25$	1.09685	+	1.89981i	0.0438741	+	0.0759922i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−36.7480			−1.26717			−0.633586	−	0.773672i	$-0.718418\pi$
$29$	−36.7480			−1.26717			−0.633586	+	0.773672i	$0.718418\pi$
$30$		0			0
$31$	17.8110	−	10.2832i	0.574549	−	0.331716i	−0.184415	−	0.982848i	$-0.559039\pi$
$31$	17.8110	−	10.2832i	0.574549	−	0.331716i	0.758964	+	0.651132i	$0.225706\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	28.4584	+	22.8607i	0.813097	+	0.653163i
$36$		0			0
$37$	23.5754	−	40.8338i	0.637173	−	1.10362i	−0.348877	−	0.937168i	$-0.613437\pi$
$37$	23.5754	−	40.8338i	0.637173	−	1.10362i	0.986050	−	0.166448i	$-0.0532297\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	41.5146i		−	1.01255i	−0.862372	−	0.506276i	$-0.831022\pi$
$41$		−	41.5146i		−	1.01255i	0.862372	−	0.506276i	$-0.168978\pi$
$42$		0			0
$43$	73.9986			1.72090			0.860448	−	0.509538i	$-0.170184\pi$
$43$	73.9986			1.72090			0.860448	+	0.509538i	$0.170184\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	51.2578	+	29.5937i	1.09059	+	0.629654i	0.933734	−	0.357967i	$-0.116530\pi$
$47$	51.2578	+	29.5937i	1.09059	+	0.629654i	0.156858	+	0.987621i	$0.449863\pi$
$48$		0			0
$49$	46.7193	+	14.7754i	0.953454	+	0.301538i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−31.7014	−	54.9084i	−0.598139	−	1.03601i	−0.993096	−	0.117307i	$-0.962574\pi$
$53$	−31.7014	−	54.9084i	−0.598139	−	1.03601i	0.394957	−	0.918700i	$-0.370760\pi$
$54$		0			0
$55$		−	47.5628i		−	0.864778i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	44.8630	−	25.9017i	0.760390	−	0.439012i	−0.0690456	−	0.997614i	$-0.521995\pi$
$59$	44.8630	−	25.9017i	0.760390	−	0.439012i	0.829436	+	0.558602i	$0.188662\pi$
$60$		0			0
$61$	21.2053	+	12.2429i	0.347628	+	0.200703i	0.663640	−	0.748052i	$-0.269011\pi$
$61$	21.2053	+	12.2429i	0.347628	+	0.200703i	−0.316012	+	0.948755i	$0.602344\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.94668	−	15.4961i	0.137641	−	0.238402i
$66$		0			0
$67$	9.58916	+	16.6089i	0.143122	+	0.247894i	0.928671	−	0.370905i	$-0.120953\pi$
$67$	9.58916	+	16.6089i	0.143122	+	0.247894i	−0.785549	+	0.618800i	$0.787619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.54627			0.0358630			0.0179315	−	0.999839i	$-0.494292\pi$
$71$	2.54627			0.0358630			0.0179315	+	0.999839i	$0.494292\pi$
$72$		0			0
$73$	102.444	−	59.1459i	1.40334	−	0.810218i	0.408605	−	0.912711i	$-0.366015\pi$
$73$	102.444	−	59.1459i	1.40334	−	0.810218i	0.994734	+	0.102494i	$0.0326821\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−23.1141	−	59.5147i	−0.300183	−	0.772918i
$78$		0			0
$79$	5.67414	−	9.82791i	0.0718246	−	0.124404i	−0.827876	−	0.560911i	$-0.810451\pi$
$79$	5.67414	−	9.82791i	0.0718246	−	0.124404i	0.899701	+	0.436507i	$0.143784\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	30.4340i		−	0.366675i	−0.983050	−	0.183337i	$-0.941310\pi$
$83$		−	30.4340i		−	0.366675i	0.983050	−	0.183337i	$-0.0586901\pi$
$84$		0			0
$85$	28.5243			0.335580
$86$		0			0
$87$		0			0
$88$		0			0
$89$	139.698	+	80.6547i	1.56964	+	0.906233i	0.996210	+	0.0869797i	$0.0277215\pi$
$89$	139.698	+	80.6547i	1.56964	+	0.906233i	0.573432	+	0.819253i	$0.305612\pi$
$90$		0			0
$91$	3.66422	−	23.7379i	0.0402661	−	0.260856i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−28.4584	−	49.2914i	−0.299562	−	0.518857i
$96$		0			0
$97$			101.257i			1.04389i	0.852979	+	0.521945i	$0.174793\pi$
$97$			101.257i			1.04389i	−0.852979	+	0.521945i	$0.825207\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−58.1423	+	33.5685i	−0.575667	+	0.332361i	−0.759409	−	0.650613i	$-0.774512\pi$
$101$	−58.1423	+	33.5685i	−0.575667	+	0.332361i	0.183743	+	0.982974i	$0.441179\pi$
$102$		0			0
$103$	96.0523	+	55.4558i	0.932547	+	0.538406i	0.887616	−	0.460584i	$-0.152360\pi$
$103$	96.0523	+	55.4558i	0.932547	+	0.538406i	0.0449308	+	0.998990i	$0.485693\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−76.4341	+	132.388i	−0.714337	+	1.23727i	0.248877	+	0.968535i	$0.419938\pi$
$107$	−76.4341	+	132.388i	−0.714337	+	1.23727i	−0.963215	+	0.268734i	$0.913395\pi$
$108$		0			0
$109$	−57.1676	−	99.0172i	−0.524474	−	0.908415i	−0.999594	−	0.0284941i	$-0.990929\pi$
$109$	−57.1676	−	99.0172i	−0.524474	−	0.908415i	0.475120	−	0.879921i	$-0.342405\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−71.0257			−0.628546			−0.314273	−	0.949333i	$-0.601761\pi$
$113$	−71.0257			−0.628546			−0.314273	+	0.949333i	$0.601761\pi$
$114$		0			0
$115$	89.8887	−	51.8972i	0.781641	−	0.451280i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	35.6921	−	13.8620i	0.299933	−	0.116487i
$120$		0			0
$121$	18.9055	−	32.7452i	0.156244	−	0.270622i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			118.929i			0.951435i
$126$		0			0
$127$	73.4312			0.578198			0.289099	−	0.957299i	$-0.406644\pi$
$127$	73.4312			0.578198			0.289099	+	0.957299i	$0.406644\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−116.654	−	67.3503i	−0.890490	−	0.514125i	−0.0163873	−	0.999866i	$-0.505216\pi$
$131$	−116.654	−	67.3503i	−0.890490	−	0.514125i	−0.874103	+	0.485741i	$0.838550\pi$
$132$		0			0
$133$	−59.5638	−	47.8477i	−0.447848	−	0.359757i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	59.1992	+	102.536i	0.432111	+	0.748438i	0.997055	−	0.0766911i	$-0.0244355\pi$
$137$	59.1992	+	102.536i	0.432111	+	0.748438i	−0.564944	+	0.825129i	$0.691102\pi$
$138$		0			0
$139$			24.3343i			0.175067i	0.996162	+	0.0875336i	$0.0278985\pi$
$139$			24.3343i			0.175067i	−0.996162	+	0.0875336i	$0.972101\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−27.1033	+	15.6481i	−0.189533	+	0.109427i
$144$		0			0
$145$	165.958	+	95.8160i	1.14454	+	0.660800i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.19935	+	7.27348i	−0.0281835	+	0.0488153i	−0.879773	−	0.475394i	$-0.842306\pi$
$149$	−4.19935	+	7.27348i	−0.0281835	+	0.0488153i	0.851590	+	0.524209i	$0.175639\pi$
$150$		0			0
$151$	−31.4658	−	54.5003i	−0.208383	−	0.360929i	0.742823	−	0.669488i	$-0.233487\pi$
$151$	−31.4658	−	54.5003i	−0.208383	−	0.360929i	−0.951205	+	0.308559i	$0.900153\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−107.249			−0.691928
$156$		0			0
$157$	63.0893	−	36.4246i	0.401843	−	0.232004i	−0.285436	−	0.958398i	$-0.592138\pi$
$157$	63.0893	−	36.4246i	0.401843	−	0.232004i	0.687279	+	0.726394i	$0.258805\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	87.2560	−	108.622i	0.541963	−	0.674669i
$162$		0			0
$163$	15.2579	−	26.4274i	0.0936065	−	0.162131i	−0.815420	−	0.578870i	$-0.803494\pi$
$163$	15.2579	−	26.4274i	0.0936065	−	0.162131i	0.909026	+	0.416739i	$0.136827\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	259.382i		−	1.55319i	−0.630003	−	0.776593i	$-0.716946\pi$
$167$		−	259.382i		−	1.55319i	0.630003	−	0.776593i	$-0.283054\pi$
$168$		0			0
$169$	157.226			0.930333
$170$		0			0
$171$		0			0
$172$		0			0
$173$	295.907	+	170.842i	1.71044	+	0.987525i	0.933953	+	0.357397i	$0.116336\pi$
$173$	295.907	+	170.842i	1.71044	+	0.987525i	0.776491	+	0.630129i	$0.216998\pi$
$174$		0			0
$175$	−5.55933	−	14.3143i	−0.0317676	−	0.0817959i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−140.437	−	243.243i	−0.784562	−	1.35890i	−0.929261	−	0.369425i	$-0.879555\pi$
$179$	−140.437	−	243.243i	−0.784562	−	1.35890i	0.144699	−	0.989476i	$-0.453779\pi$
$180$		0			0
$181$		−	213.356i		−	1.17876i	−0.807855	−	0.589382i	$-0.799371\pi$
$181$		−	213.356i		−	1.17876i	0.807855	−	0.589382i	$-0.200629\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−212.938	+	122.940i	−1.15102	+	0.664541i
$186$		0			0
$187$	−43.2060	−	24.9450i	−0.231048	−	0.133396i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−166.145	+	287.771i	−0.869869	+	1.50666i	−0.00773872	+	0.999970i	$0.502463\pi$
$191$	−166.145	+	287.771i	−0.869869	+	1.50666i	−0.862130	+	0.506687i	$0.830870\pi$
$192$		0			0
$193$	82.6792	+	143.205i	0.428390	+	0.741993i	0.996730	−	0.0808007i	$-0.0257477\pi$
$193$	82.6792	+	143.205i	0.428390	+	0.741993i	−0.568341	+	0.822793i	$0.692414\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	200.474			1.01764			0.508818	−	0.860874i	$-0.330082\pi$
$197$	200.474			1.01764			0.508818	+	0.860874i	$0.330082\pi$
$198$		0			0
$199$	127.958	−	73.8764i	0.643003	−	0.371238i	−0.142767	−	0.989756i	$-0.545600\pi$
$199$	127.958	−	73.8764i	0.643003	−	0.371238i	0.785770	+	0.618518i	$0.212267\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	254.225	+	39.2425i	1.25234	+	0.193313i
$204$		0			0
$205$	−108.244	+	187.485i	−0.528021	+	0.914559i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			99.5495i			0.476314i
$210$		0			0
$211$	−208.573			−0.988495			−0.494248	−	0.869321i	$-0.664556\pi$
$211$	−208.573			−0.988495			−0.494248	+	0.869321i	$0.664556\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−334.186	−	192.942i	−1.55435	−	0.897406i
$216$		0			0
$217$	−134.199	+	52.1198i	−0.618429	+	0.240183i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.38444	−	16.2543i	−0.0424635	−	0.0735490i
$222$		0			0
$223$			413.973i			1.85638i	0.372104	+	0.928191i	$0.378636\pi$
$223$			413.973i			1.85638i	−0.372104	+	0.928191i	$0.621364\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	234.861	−	135.597i	1.03463	−	0.597344i	0.116322	−	0.993212i	$-0.462889\pi$
$227$	234.861	−	135.597i	1.03463	−	0.597344i	0.918308	+	0.395868i	$0.129556\pi$
$228$		0			0
$229$	−306.643	−	177.040i	−1.33905	−	0.773101i	−0.352384	−	0.935856i	$-0.614629\pi$
$229$	−306.643	−	177.040i	−1.33905	−	0.773101i	−0.986667	+	0.162754i	$0.947962\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	136.823	−	236.984i	0.587221	−	1.01710i	−0.407373	−	0.913262i	$-0.633555\pi$
$233$	136.823	−	236.984i	0.587221	−	1.01710i	0.994595	−	0.103835i	$-0.0331114\pi$
$234$		0			0
$235$	−154.324	−	267.297i	−0.656698	−	1.13744i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	362.362			1.51616			0.758080	−	0.652162i	$-0.226138\pi$
$239$	362.362			1.51616			0.758080	+	0.652162i	$0.226138\pi$
$240$		0			0
$241$	−42.4378	+	24.5015i	−0.176090	+	0.101666i	−0.585454	−	0.810705i	$-0.699084\pi$
$241$	−42.4378	+	24.5015i	−0.176090	+	0.101666i	0.409364	+	0.912371i	$0.365751\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−172.465	−	188.542i	−0.703937	−	0.769559i
$246$		0			0
$247$	−18.7255	+	32.4336i	−0.0758118	+	0.131310i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			189.207i			0.753813i	0.926251	+	0.376907i	$0.123012\pi$
$251$			189.207i			0.753813i	−0.926251	+	0.376907i	$0.876988\pi$
$252$		0			0
$253$	−181.540			−0.717551
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.7505	−	10.8256i	−0.0729591	−	0.0421229i	0.463077	−	0.886318i	$-0.346745\pi$
$257$	−18.7505	−	10.8256i	−0.0729591	−	0.0421229i	−0.536036	+	0.844195i	$0.680079\pi$
$258$		0			0
$259$	−206.702	+	257.315i	−0.798077	+	0.993495i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	27.9586	+	48.4258i	0.106307	+	0.184128i	0.914271	−	0.405103i	$-0.132764\pi$
$263$	27.9586	+	48.4258i	0.106307	+	0.184128i	−0.807965	+	0.589231i	$0.799431\pi$
$264$		0			0
$265$			330.630i			1.24766i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	54.8437	−	31.6640i	0.203880	−	0.117710i	−0.394584	−	0.918860i	$-0.629111\pi$
$269$	54.8437	−	31.6640i	0.203880	−	0.117710i	0.598464	+	0.801150i	$0.295778\pi$
$270$		0			0
$271$	−58.2356	−	33.6223i	−0.214891	−	0.124068i	0.388691	−	0.921368i	$-0.372927\pi$
$271$	−58.2356	−	33.6223i	−0.214891	−	0.124068i	−0.603583	+	0.797300i	$0.706261\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.0042	+	17.3277i	−0.0363788	+	0.0630100i
$276$		0			0
$277$	−220.081	−	381.191i	−0.794515	−	1.37614i	−0.923147	−	0.384447i	$-0.874392\pi$
$277$	−220.081	−	381.191i	−0.794515	−	1.37614i	0.128632	−	0.991692i	$-0.458941\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−265.924			−0.946350			−0.473175	−	0.880969i	$-0.656892\pi$
$281$	−265.924			−0.946350			−0.473175	+	0.880969i	$0.656892\pi$
$282$		0			0
$283$	108.996	−	62.9289i	0.385145	−	0.222364i	−0.294909	−	0.955525i	$-0.595289\pi$
$283$	108.996	−	62.9289i	0.385145	−	0.222364i	0.680054	+	0.733162i	$0.261956\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−44.3327	+	287.201i	−0.154469	+	1.00070i
$288$		0			0
$289$	−129.540	+	224.370i	−0.448235	+	0.776366i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			219.939i			0.750646i	0.926894	+	0.375323i	$0.122468\pi$
$293$			219.939i			0.750646i	−0.926894	+	0.375323i	$0.877532\pi$
$294$		0			0
$295$	−270.142			−0.915736
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−59.1464	−	34.1482i	−0.197814	−	0.114208i
$300$		0			0
$301$	−511.927	−	79.0217i	−1.70075	−	0.262531i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−63.8437	−	110.581i	−0.209324	−	0.362559i
$306$		0			0
$307$			233.059i			0.759150i	0.925161	+	0.379575i	$0.123930\pi$
$307$			233.059i			0.759150i	−0.925161	+	0.379575i	$0.876070\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	358.462	−	206.958i	1.15261	−	0.665461i	0.203089	−	0.979160i	$-0.434902\pi$
$311$	358.462	−	206.958i	1.15261	−	0.665461i	0.949522	+	0.313699i	$0.101568\pi$
$312$		0			0
$313$	360.041	+	207.870i	1.15029	+	0.664121i	0.948958	−	0.315402i	$-0.102139\pi$
$313$	360.041	+	207.870i	1.15029	+	0.664121i	0.201333	+	0.979523i	$0.435473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	251.731	−	436.011i	0.794105	−	1.37543i	−0.129301	−	0.991605i	$-0.541273\pi$
$317$	251.731	−	436.011i	0.794105	−	1.37543i	0.923406	−	0.383825i	$-0.125393\pi$
$318$		0			0
$319$	−167.586	−	290.267i	−0.525347	−	0.909928i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−59.7017			−0.184835
$324$		0			0
$325$	−6.51879	+	3.76362i	−0.0200578	+	0.0115804i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−323.002	−	259.469i	−0.981770	−	0.788659i
$330$		0			0
$331$	−120.274	+	208.321i	−0.363366	+	0.629368i	−0.988513	−	0.151139i	$-0.951706\pi$
$331$	−120.274	+	208.321i	−0.363366	+	0.629368i	0.625147	+	0.780507i	$0.285039\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	100.010i		−	0.298538i
$336$		0			0
$337$	202.444			0.600725			0.300362	−	0.953825i	$-0.402892\pi$
$337$	202.444			0.600725			0.300362	+	0.953825i	$0.402892\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	162.451	+	93.7910i	0.476395	+	0.275047i
$342$		0			0
$343$	−307.429	−	152.108i	−0.896293	−	0.443462i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	210.502	+	364.601i	0.606635	+	1.05072i	0.991791	+	0.127871i	$0.0408144\pi$
$347$	210.502	+	364.601i	0.606635	+	1.05072i	−0.385156	+	0.922852i	$0.625852\pi$
$348$		0			0
$349$			214.402i			0.614332i	0.951656	+	0.307166i	$0.0993807\pi$
$349$			214.402i			0.614332i	−0.951656	+	0.307166i	$0.900619\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−540.855	+	312.263i	−1.53217	+	0.884597i	−0.532906	+	0.846174i	$0.678900\pi$
$353$	−540.855	+	312.263i	−1.53217	+	0.884597i	−0.999262	+	0.0384229i	$0.987767\pi$
$354$		0			0
$355$	−11.4993	−	6.63910i	−0.0323923	−	0.0187017i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−290.906	+	503.864i	−0.810323	+	1.40352i	0.102316	+	0.994752i	$0.467375\pi$
$359$	−290.906	+	503.864i	−0.810323	+	1.40352i	−0.912638	+	0.408768i	$0.865959\pi$
$360$		0			0
$361$	−120.936	−	209.468i	−0.335003	−	0.580243i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−616.863			−1.69004
$366$		0			0
$367$	−311.562	+	179.880i	−0.848942	+	0.490137i	−0.860294	−	0.509799i	$-0.829720\pi$
$367$	−311.562	+	179.880i	−0.848942	+	0.490137i	0.0113519	+	0.999936i	$0.496387\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	160.676	+	413.713i	0.433090	+	1.11513i
$372$		0			0
$373$	67.2930	−	116.555i	0.180410	−	0.312480i	−0.761610	−	0.648036i	$-0.775591\pi$
$373$	67.2930	−	116.555i	0.180410	−	0.312480i	0.942020	+	0.335556i	$0.108924\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	126.093i		−	0.334465i
$378$		0			0
$379$	229.181			0.604699			0.302349	−	0.953197i	$-0.402229\pi$
$379$	229.181			0.604699			0.302349	+	0.953197i	$0.402229\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.8010	−	17.2056i	−0.0778093	−	0.0449232i	0.460591	−	0.887613i	$-0.347638\pi$
$383$	−29.8010	−	17.2056i	−0.0778093	−	0.0449232i	−0.538400	+	0.842689i	$0.680971\pi$
$384$		0			0
$385$	−50.7914	+	329.042i	−0.131926	+	0.854656i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	44.9912	+	77.9270i	0.115659	+	0.200327i	0.918043	−	0.396481i	$-0.129769\pi$
$389$	44.9912	+	77.9270i	0.115659	+	0.200327i	−0.802384	+	0.596808i	$0.796435\pi$
$390$		0			0
$391$		−	108.873i		−	0.278448i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−51.2502	+	29.5893i	−0.129747	+	0.0749096i
$396$		0			0
$397$	−197.880	−	114.246i	−0.498439	−	0.287774i	0.229629	−	0.973278i	$-0.426249\pi$
$397$	−197.880	−	114.246i	−0.498439	−	0.287774i	−0.728069	+	0.685504i	$0.759582\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	214.759	−	371.974i	0.535559	−	0.927616i	−0.463577	−	0.886057i	$-0.653434\pi$
$401$	214.759	−	371.974i	0.535559	−	0.927616i	0.999136	−	0.0415591i	$-0.0132325\pi$
$402$		0			0
$403$	35.2847	+	61.1148i	0.0875550	+	0.151650i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	430.053			1.05664
$408$		0			0
$409$	373.831	−	215.831i	0.914011	−	0.527705i	0.0322917	−	0.999478i	$-0.489719\pi$
$409$	373.831	−	215.831i	0.914011	−	0.527705i	0.881720	+	0.471774i	$0.156386\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−338.025	+	131.281i	−0.818463	+	0.317872i
$414$		0			0
$415$	−79.3530	+	137.443i	−0.191212	+	0.331189i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			144.735i			0.345430i	0.984972	+	0.172715i	$0.0552540\pi$
$419$			144.735i			0.345430i	−0.984972	+	0.172715i	$0.944746\pi$
$420$		0			0
$421$	−730.197			−1.73444			−0.867218	−	0.497929i	$-0.834094\pi$
$421$	−730.197			−1.73444			−0.867218	+	0.497929i	$0.834094\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.3918	−	5.99969i	−0.0244512	−	0.0141169i
$426$		0			0
$427$	−133.626	−	107.342i	−0.312941	−	0.251386i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	145.116	+	251.348i	0.336695	+	0.583173i	0.983809	−	0.179220i	$-0.0573574\pi$
$431$	145.116	+	251.348i	0.336695	+	0.583173i	−0.647114	+	0.762394i	$0.724024\pi$
$432$		0			0
$433$			439.027i			1.01392i	0.861970	+	0.506960i	$0.169231\pi$
$433$			439.027i			1.01392i	−0.861970	+	0.506960i	$0.830769\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−188.138	+	108.622i	−0.430522	+	0.248562i
$438$		0			0
$439$	252.777	+	145.941i	0.575802	+	0.332439i	0.759463	−	0.650550i	$-0.225462\pi$
$439$	252.777	+	145.941i	0.575802	+	0.332439i	−0.183662	+	0.982990i	$0.558795\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	268.639	−	465.297i	0.606410	−	1.05033i	−0.385417	−	0.922742i	$-0.625943\pi$
$443$	268.639	−	465.297i	0.606410	−	1.05033i	0.991827	−	0.127590i	$-0.0407241\pi$
$444$		0			0
$445$	−420.595	−	728.492i	−0.945157	−	1.63706i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	577.008			1.28509			0.642547	−	0.766246i	$-0.277878\pi$
$449$	577.008			1.28509			0.642547	+	0.766246i	$0.277878\pi$
$450$		0			0
$451$	327.917	−	189.323i	0.727090	−	0.419785i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−78.4417	+	97.6491i	−0.172399	+	0.214613i
$456$		0			0
$457$	−294.230	+	509.621i	−0.643830	+	1.11515i	0.340741	+	0.940157i	$0.389322\pi$
$457$	−294.230	+	509.621i	−0.643830	+	1.11515i	−0.984571	+	0.174988i	$0.944011\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	216.419i		−	0.469455i	−0.972061	−	0.234728i	$-0.924580\pi$
$461$		−	216.419i		−	0.469455i	0.972061	−	0.234728i	$-0.0754198\pi$
$462$		0			0
$463$	135.169			0.291941			0.145971	−	0.989289i	$-0.453370\pi$
$463$	135.169			0.291941			0.145971	+	0.989289i	$0.453370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−676.706	−	390.696i	−1.44905	−	0.836609i	−0.450624	−	0.892714i	$-0.648798\pi$
$467$	−676.706	−	390.696i	−1.44905	−	0.836609i	−0.998425	+	0.0561055i	$0.982132\pi$
$468$		0			0
$469$	−48.6021	−	125.142i	−0.103629	−	0.266827i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	337.463	+	584.503i	0.713453	+	1.23574i
$474$		0			0
$475$			23.9433i			0.0504070i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	200.349	−	115.672i	0.418266	−	0.241486i	−0.276069	−	0.961138i	$-0.589032\pi$
$479$	200.349	−	115.672i	0.418266	−	0.241486i	0.694335	+	0.719652i	$0.255699\pi$
$480$		0			0
$481$	140.113	+	80.8941i	0.291295	+	0.168179i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	264.016	−	457.289i	0.544363	−	0.942865i
$486$		0			0
$487$	−213.293	−	369.434i	−0.437973	−	0.758592i	0.559560	−	0.828790i	$-0.310970\pi$
$487$	−213.293	−	369.434i	−0.437973	−	0.758592i	−0.997533	+	0.0701980i	$0.977637\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−259.716			−0.528952			−0.264476	−	0.964392i	$-0.585199\pi$
$491$	−259.716			−0.528952			−0.264476	+	0.964392i	$0.585199\pi$
$492$		0			0
$493$	174.078	−	100.504i	0.353100	−	0.203863i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−17.6153	−	2.71912i	−0.0354432	−	0.00547107i
$498$		0			0
$499$	−452.741	+	784.170i	−0.907296	+	1.57148i	−0.0894906	+	0.995988i	$0.528524\pi$
$499$	−452.741	+	784.170i	−0.907296	+	1.57148i	−0.817805	+	0.575495i	$0.804809\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	981.208i		−	1.95071i	−0.220636	−	0.975356i	$-0.570813\pi$
$503$		−	981.208i		−	1.95071i	0.220636	−	0.975356i	$-0.429187\pi$
$504$		0			0
$505$	350.103			0.693274
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−48.8362	−	28.1956i	−0.0959453	−	0.0553940i	0.451260	−	0.892393i	$-0.350975\pi$
$509$	−48.8362	−	28.1956i	−0.0959453	−	0.0553940i	−0.547205	+	0.836999i	$0.684308\pi$
$510$		0			0
$511$	−771.873	+	299.777i	−1.51051	+	0.586649i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−289.189	−	500.890i	−0.561532	−	0.972601i
$516$		0			0
$517$			539.837i			1.04417i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	634.605	−	366.389i	1.21805	−	0.703242i	0.253550	−	0.967322i	$-0.418402\pi$
$521$	634.605	−	366.389i	1.21805	−	0.703242i	0.964501	+	0.264080i	$0.0850683\pi$
$522$		0			0
$523$	9.83785	+	5.67988i	0.0188104	+	0.0108602i	0.509376	−	0.860544i	$-0.329876\pi$
$523$	9.83785	+	5.67988i	0.0188104	+	0.0108602i	−0.490565	+	0.871404i	$0.663210\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−56.2482	+	97.4248i	−0.106733	+	0.184867i
$528$		0			0
$529$	66.4156	+	115.035i	0.125549	+	0.217458i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	142.449			0.267258
$534$		0			0
$535$	690.370	−	398.585i	1.29041	−	0.745019i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	96.3503	+	436.410i	0.178758	+	0.809666i
$540$		0			0
$541$	312.987	−	542.109i	0.578534	−	1.00205i	−0.417114	−	0.908854i	$-0.636958\pi$
$541$	312.987	−	542.109i	0.578534	−	1.00205i	0.995648	−	0.0931960i	$-0.0297083\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			596.231i			1.09400i
$546$		0			0
$547$	528.430			0.966051			0.483026	−	0.875606i	$-0.339538\pi$
$547$	528.430			0.966051			0.483026	+	0.875606i	$0.339538\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−347.353	−	200.544i	−0.630404	−	0.363964i
$552$		0			0
$553$	−49.7491	+	61.9308i	−0.0899623	+	0.111991i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	133.581	+	231.369i	0.239822	+	0.415384i	0.960663	−	0.277716i	$-0.0895776\pi$
$557$	133.581	+	231.369i	0.239822	+	0.415384i	−0.720841	+	0.693100i	$0.756244\pi$
$558$		0			0
$559$			253.911i			0.454223i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	718.302	−	414.712i	1.27585	−	0.736610i	0.299765	−	0.954013i	$-0.403092\pi$
$563$	718.302	−	414.712i	1.27585	−	0.736610i	0.976082	+	0.217403i	$0.0697586\pi$
$564$		0			0
$565$	320.760	+	185.191i	0.567717	+	0.327771i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.3916	−	21.4628i	0.0217778	−	0.0377203i	−0.854931	−	0.518742i	$-0.826401\pi$
$569$	12.3916	−	21.4628i	0.0217778	−	0.0377203i	0.876709	+	0.481021i	$0.159734\pi$
$570$		0			0
$571$	−548.479	−	949.994i	−0.960559	−	1.66374i	−0.721100	−	0.692831i	$-0.756363\pi$
$571$	−548.479	−	949.994i	−0.960559	−	1.66374i	−0.239459	−	0.970907i	$-0.576970\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−43.6635			−0.0759365
$576$		0			0
$577$	883.145	−	509.884i	1.53058	−	0.883681i	0.531245	−	0.847218i	$-0.321724\pi$
$577$	883.145	−	509.884i	1.53058	−	0.883681i	0.999335	−	0.0364622i	$-0.0116089\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.4999	+	210.544i	−0.0559379	+	0.362383i
$582$		0			0
$583$	289.142	−	500.809i	0.495955	−	0.859020i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			351.405i			0.598646i	0.954152	+	0.299323i	$0.0967609\pi$
$587$			351.405i			0.598646i	−0.954152	+	0.299323i	$0.903239\pi$
$588$		0			0
$589$	224.473			0.381109
$590$		0			0
$591$		0			0
$592$		0			0
$593$	624.862	+	360.764i	1.05373	+	0.608371i	0.923691	−	0.383138i	$-0.125156\pi$
$593$	624.862	+	360.764i	1.05373	+	0.608371i	0.130039	+	0.991509i	$0.458490\pi$
$594$		0			0
$595$	−197.333	−	30.4606i	−0.331652	−	0.0511942i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	70.5809	+	122.250i	0.117831	+	0.204090i	0.918908	−	0.394472i	$-0.129072\pi$
$599$	70.5809	+	122.250i	0.117831	+	0.204090i	−0.801077	+	0.598562i	$0.795739\pi$
$600$		0			0
$601$			478.176i			0.795634i	0.917465	+	0.397817i	$0.130232\pi$
$601$			478.176i			0.795634i	−0.917465	+	0.397817i	$0.869768\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−170.759	+	98.5875i	−0.282245	+	0.162954i
$606$		0			0
$607$	961.350	+	555.036i	1.58377	+	0.914391i	0.994302	+	0.106602i	$0.0339970\pi$
$607$	961.350	+	555.036i	1.58377	+	0.914391i	0.589471	+	0.807790i	$0.299336\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−101.545	+	175.881i	−0.166194	+	0.287857i
$612$		0			0
$613$	−298.361	−	516.777i	−0.486723	−	0.843029i	0.513160	−	0.858293i	$-0.328475\pi$
$613$	−298.361	−	516.777i	−0.486723	−	0.843029i	−0.999884	+	0.0152634i	$0.995141\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.9891			−0.0340181			−0.0170090	−	0.999855i	$-0.505414\pi$
$617$	−20.9891			−0.0340181			−0.0170090	+	0.999855i	$0.505414\pi$
$618$		0			0
$619$	867.158	−	500.654i	1.40090	−	0.808811i	0.406416	−	0.913688i	$-0.366778\pi$
$619$	867.158	−	500.654i	1.40090	−	0.808811i	0.994485	+	0.104877i	$0.0334450\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−880.311	−	707.156i	−1.41302	−	1.13508i
$624$		0			0
$625$	337.515	−	584.593i	0.540024	−	0.935349i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			257.911i			0.410033i
$630$		0			0
$631$	442.571			0.701381			0.350691	−	0.936491i	$-0.385947\pi$
$631$	442.571			0.701381			0.350691	+	0.936491i	$0.385947\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−331.624	−	191.463i	−0.522242	−	0.301516i
$636$		0			0
$637$	−50.6986	+	160.307i	−0.0795896	+	0.251660i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.9968	−	43.2958i	−0.0389966	−	0.0675441i	0.845868	−	0.533392i	$-0.179083\pi$
$641$	−24.9968	−	43.2958i	−0.0389966	−	0.0675441i	−0.884865	+	0.465848i	$0.845749\pi$
$642$		0			0
$643$			603.575i			0.938687i	0.883016	+	0.469343i	$0.155509\pi$
$643$			603.575i			0.938687i	−0.883016	+	0.469343i	$0.844491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−922.823	+	532.792i	−1.42631	+	0.823481i	−0.996827	−	0.0795944i	$-0.974637\pi$
$647$	−922.823	+	532.792i	−1.42631	+	0.823481i	−0.429483	+	0.903075i	$0.641304\pi$
$648$		0			0
$649$	409.187	+	236.244i	0.630488	+	0.364013i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−437.990	+	758.620i	−0.670734	+	1.16175i	0.306962	+	0.951722i	$0.400688\pi$
$653$	−437.990	+	758.620i	−0.670734	+	1.16175i	−0.977696	+	0.210024i	$0.932646\pi$
$654$		0			0
$655$	351.216	+	608.323i	0.536207	+	0.928738i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−229.670			−0.348513			−0.174256	−	0.984700i	$-0.555752\pi$
$659$	−229.670			−0.348513			−0.174256	+	0.984700i	$0.555752\pi$
$660$		0			0
$661$	649.011	−	374.707i	0.981863	−	0.566879i	0.0790306	−	0.996872i	$-0.474818\pi$
$661$	649.011	−	374.707i	0.981863	−	0.566879i	0.902832	+	0.429994i	$0.141484\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	144.240	+	371.391i	0.216902	+	0.558483i
$666$		0			0
$667$	365.716	−	633.439i	0.548300	−	0.949683i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			223.330i			0.332832i
$672$		0			0
$673$	229.438			0.340919			0.170459	−	0.985365i	$-0.445475\pi$
$673$	229.438			0.340919			0.170459	+	0.985365i	$0.445475\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−239.778	−	138.436i	−0.354177	−	0.204484i	0.312347	−	0.949968i	$-0.398885\pi$
$677$	−239.778	−	138.436i	−0.354177	−	0.204484i	−0.666523	+	0.745484i	$0.732218\pi$
$678$		0			0
$679$	108.131	−	700.504i	0.159250	−	1.03167i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−328.026	−	568.157i	−0.480272	−	0.831855i	0.519472	−	0.854487i	$-0.326129\pi$
$683$	−328.026	−	568.157i	−0.480272	−	0.831855i	−0.999744	+	0.0226323i	$0.992795\pi$
$684$		0			0
$685$		−	617.419i		−	0.901342i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	188.407	−	108.777i	0.273450	−	0.157876i
$690$		0			0
$691$	−185.761	−	107.249i	−0.268829	−	0.155208i	0.359526	−	0.933135i	$-0.382938\pi$
$691$	−185.761	−	107.249i	−0.268829	−	0.155208i	−0.628355	+	0.777926i	$0.716272\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	63.4489	−	109.897i	0.0912933	−	0.158125i
$696$		0			0
$697$	113.541	+	196.658i	0.162899	+	0.282150i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	492.902			0.703142			0.351571	−	0.936161i	$-0.385648\pi$
$701$	492.902			0.703142			0.351571	+	0.936161i	$0.385648\pi$
$702$		0			0
$703$	445.683	−	257.315i	0.633973	−	0.366024i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	438.080	−	170.140i	0.619632	−	0.240651i
$708$		0			0
$709$	−418.012	+	724.018i	−0.589580	+	1.02118i	0.404708	+	0.914446i	$0.367373\pi$
$709$	−418.012	+	724.018i	−0.589580	+	1.02118i	−0.994287	+	0.106736i	$0.965960\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			409.353i			0.574128i
$714$		0			0
$715$	163.202			0.228254
$716$		0			0
$717$		0			0
$718$		0			0
$719$	363.072	+	209.620i	0.504968	+	0.291544i	0.730763	−	0.682631i	$-0.239164\pi$
$719$	363.072	+	209.620i	0.504968	+	0.291544i	−0.225795	+	0.974175i	$0.572498\pi$
$720$		0			0
$721$	−605.276	−	486.220i	−0.839495	−	0.674369i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.3072	−	69.8141i	−0.0555961	−	0.0962952i
$726$		0			0
$727$			872.570i			1.20023i	0.799912	+	0.600117i	$0.204879\pi$
$727$			872.570i			1.20023i	−0.799912	+	0.600117i	$0.795121\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−350.538	+	202.383i	−0.479532	+	0.276858i
$732$		0			0
$733$	81.7339	+	47.1891i	0.111506	+	0.0643780i	0.554716	−	0.832040i	$-0.312827\pi$
$733$	81.7339	+	47.1891i	0.111506	+	0.0643780i	−0.443210	+	0.896418i	$0.646160\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−87.4609	+	151.487i	−0.118671	+	0.205545i
$738$		0			0
$739$	−91.8991	−	159.174i	−0.124356	−	0.215391i	0.797125	−	0.603814i	$-0.206353\pi$
$739$	−91.8991	−	159.174i	−0.124356	−	0.215391i	−0.921481	+	0.388423i	$0.873020\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	243.953			0.328335			0.164167	−	0.986432i	$-0.447506\pi$
$743$	243.953			0.328335			0.164167	+	0.986432i	$0.447506\pi$
$744$		0			0
$745$	37.9295	−	21.8986i	0.0509120	−	0.0293941i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	670.151	−	834.244i	0.894727	−	1.11381i
$750$		0			0
$751$	120.946	−	209.484i	0.161046	−	0.278940i	−0.774198	−	0.632943i	$-0.781847\pi$
$751$	120.946	−	209.484i	0.161046	−	0.278940i	0.935244	+	0.354003i	$0.115180\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			328.173i			0.434666i
$756$		0			0
$757$	1368.88			1.80829			0.904147	−	0.427221i	$-0.140508\pi$
$757$	1368.88			1.80829			0.904147	+	0.427221i	$0.140508\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−402.629	−	232.458i	−0.529079	−	0.305464i	0.211562	−	0.977365i	$-0.432145\pi$
$761$	−402.629	−	232.458i	−0.529079	−	0.305464i	−0.740641	+	0.671900i	$0.765478\pi$
$762$		0			0
$763$	289.751	+	746.056i	0.379752	+	0.977793i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	88.8762	+	153.938i	0.115875	+	0.200702i
$768$		0			0
$769$		−	2.47014i		−	0.00321215i	−0.999999	−	0.00160608i	$-0.999489\pi$
$769$		−	2.47014i		−	0.00321215i	0.999999	−	0.00160608i	$-0.000511230\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−667.371	+	385.307i	−0.863352	+	0.498456i	−0.865133	−	0.501542i	$-0.832766\pi$
$773$	−667.371	+	385.307i	−0.863352	+	0.498456i	0.00178154	+	0.999998i	$0.499433\pi$
$774$		0			0
$775$	39.0721	+	22.5583i	0.0504157	+	0.0291075i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	226.557	−	392.408i	0.290830	−	0.503733i
$780$		0			0
$781$	11.6120	+	20.1126i	0.0148682	+	0.0257524i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−379.891			−0.483938
$786$		0			0
$787$	404.529	−	233.555i	0.514014	−	0.296766i	−0.220468	−	0.975394i	$-0.570758\pi$
$787$	404.529	−	233.555i	0.514014	−	0.296766i	0.734482	+	0.678628i	$0.237425\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	491.360	+	75.8470i	0.621189	+	0.0958875i
$792$		0			0
$793$	−42.0090	+	72.7617i	−0.0529747	+	0.0917549i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	511.558i		−	0.641855i	−0.947104	−	0.320927i	$-0.896005\pi$
$797$		−	511.558i		−	0.641855i	0.947104	−	0.320927i	$-0.103995\pi$
$798$		0			0
$799$	−323.750			−0.405195
$800$		0			0
$801$		0			0
$802$		0			0
$803$	934.369	+	539.458i	1.16360	+	0.671803i
$804$		0			0
$805$	−677.276	+	263.038i	−0.841336	+	0.326755i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−695.722	−	1205.03i	−0.859978	−	1.48953i	−0.871949	−	0.489598i	$-0.837144\pi$
$809$	−695.722	−	1205.03i	−0.859978	−	1.48953i	0.0119703	−	0.999928i	$-0.496190\pi$
$810$		0			0
$811$		−	1203.34i		−	1.48377i	−0.670527	−	0.741885i	$-0.733932\pi$
$811$		−	1203.34i		−	1.48377i	0.670527	−	0.741885i	$-0.266068\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−137.812	+	79.5660i	−0.169095	+	0.0976271i
$816$		0			0
$817$	699.455	+	403.831i	0.856127	+	0.494285i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	134.906	−	233.663i	0.164319	−	0.284608i	−0.772094	−	0.635508i	$-0.780791\pi$
$821$	134.906	−	233.663i	0.164319	−	0.284608i	0.936413	+	0.350900i	$0.114124\pi$
$822$		0			0
$823$	−568.375	−	984.454i	−0.690613	−	1.19618i	−0.971637	−	0.236476i	$-0.924007\pi$
$823$	−568.375	−	984.454i	−0.690613	−	1.19618i	0.281024	−	0.959701i	$-0.409326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−39.3524			−0.0475845			−0.0237922	−	0.999717i	$-0.507574\pi$
$827$	−39.3524			−0.0475845			−0.0237922	+	0.999717i	$0.507574\pi$
$828$		0			0
$829$	896.538	−	517.616i	1.08147	−	0.624386i	0.150177	−	0.988659i	$-0.452016\pi$
$829$	896.538	−	517.616i	1.08147	−	0.624386i	0.931292	+	0.364273i	$0.118682\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−261.723	+	57.7831i	−0.314193	+	0.0693674i
$834$		0			0
$835$	−676.307	+	1171.40i	−0.809949	+	1.40287i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			836.458i			0.996970i	0.866898	+	0.498485i	$0.166110\pi$
$839$			836.458i			0.996970i	−0.866898	+	0.498485i	$0.833890\pi$
$840$		0			0
$841$	509.416			0.605726
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−710.051	−	409.948i	−0.840298	−	0.485146i
$846$		0			0
$847$	−165.757	+	206.345i	−0.195699	+	0.243619i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	469.245	+	812.756i	0.551404	+	0.955059i
$852$		0			0
$853$			782.794i			0.917695i	0.888515	+	0.458847i	$0.151738\pi$
$853$			782.794i			0.917695i	−0.888515	+	0.458847i	$0.848262\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	236.681	−	136.648i	0.276174	−	0.159449i	−0.355516	−	0.934670i	$-0.615695\pi$
$857$	236.681	−	136.648i	0.276174	−	0.159449i	0.631690	+	0.775221i	$0.282362\pi$
$858$		0			0
$859$	−8.26065	−	4.76929i	−0.00961659	−	0.00555214i	0.495184	−	0.868788i	$-0.335101\pi$
$859$	−8.26065	−	4.76929i	−0.00961659	−	0.00555214i	−0.504801	+	0.863236i	$0.668434\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	496.338	−	859.683i	0.575131	−	0.996156i	−0.420896	−	0.907109i	$-0.638284\pi$
$863$	496.338	−	859.683i	0.575131	−	0.996156i	0.996027	−	0.0890475i	$-0.0283823\pi$
$864$		0			0
$865$	−890.899	−	1543.08i	−1.02994	−	1.78391i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	103.506			0.119109
$870$		0			0
$871$	−56.9901	+	32.9032i	−0.0654306	+	0.0377764i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	127.002	−	822.761i	0.145146	−	0.940298i
$876$		0			0
$877$	−347.810	+	602.425i	−0.396591	+	0.686915i	−0.993303	−	0.115540i	$-0.963140\pi$
$877$	−347.810	+	602.425i	−0.396591	+	0.686915i	0.596712	+	0.802455i	$0.296473\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1431.80i		−	1.62520i	−0.582819	−	0.812602i	$-0.698051\pi$
$881$		−	1431.80i		−	1.62520i	0.582819	−	0.812602i	$-0.301949\pi$
$882$		0			0
$883$	−455.669			−0.516046			−0.258023	−	0.966139i	$-0.583071\pi$
$883$	−455.669			−0.516046			−0.258023	+	0.966139i	$0.583071\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1133.60	+	654.482i	1.27801	+	0.737860i	0.976482	−	0.215597i	$-0.0691697\pi$
$887$	1133.60	+	654.482i	1.27801	+	0.737860i	0.301529	+	0.953457i	$0.402503\pi$
$888$		0			0
$889$	−508.002	−	78.4158i	−0.571431	−	0.0882068i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	323.002	+	559.457i	0.361705	+	0.626491i
$894$		0			0
$895$			1464.69i			1.63652i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−654.520	+	377.887i	−0.728053	+	0.420342i
$900$		0			0
$901$	300.344	+	173.404i	0.333345	+	0.192457i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−556.301	+	963.541i	−0.614697	+	1.06469i
$906$		0			0
$907$	−233.845	−	405.031i	−0.257822	−	0.446561i	0.707836	−	0.706377i	$-0.249672\pi$
$907$	−233.845	−	405.031i	−0.257822	−	0.446561i	−0.965658	+	0.259816i	$0.916338\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	260.111			0.285522			0.142761	−	0.989757i	$-0.454402\pi$
$911$	260.111			0.285522			0.142761	+	0.989757i	$0.454402\pi$
$912$		0			0
$913$	240.394	−	138.791i	0.263301	−	0.152017i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	735.099	+	590.507i	0.801635	+	0.643955i
$918$		0			0
$919$	848.620	−	1469.85i	0.923417	−	1.59940i	0.129329	−	0.991602i	$-0.458718\pi$
$919$	848.620	−	1469.85i	0.923417	−	1.59940i	0.794088	−	0.607803i	$-0.207949\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			8.73701i			0.00946588i
$924$		0			0
$925$	103.435			0.111822
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−645.941	−	372.934i	−0.695308	−	0.401436i	0.110289	−	0.993900i	$-0.464822\pi$
$929$	−645.941	−	372.934i	−0.695308	−	0.401436i	−0.805598	+	0.592463i	$0.798156\pi$
$930$		0			0
$931$	360.970	+	394.621i	0.387723	+	0.423868i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	130.082	+	225.309i	0.139125	+	0.240972i
$936$		0			0
$937$		−	404.967i		−	0.432195i	−0.976372	−	0.216098i	$-0.930667\pi$
$937$		−	404.967i		−	0.432195i	0.976372	−	0.216098i	$-0.0693330\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1420.33	−	820.028i	1.50938	−	0.871443i	0.509444	−	0.860504i	$-0.329851\pi$
$941$	1420.33	−	820.028i	1.50938	−	0.871443i	0.999940	−	0.0109394i	$-0.00348220\pi$
$942$		0			0
$943$	715.602	+	413.153i	0.758857	+	0.438126i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−332.389	+	575.715i	−0.350992	+	0.607935i	−0.986423	−	0.164222i	$-0.947489\pi$
$947$	−332.389	+	575.715i	−0.350992	+	0.607935i	0.635432	+	0.772157i	$0.280822\pi$
$948$		0			0
$949$	202.947	+	351.514i	0.213853	+	0.370405i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−671.848			−0.704982			−0.352491	−	0.935815i	$-0.614665\pi$
$953$	−671.848			−0.704982			−0.352491	+	0.935815i	$0.614665\pi$
$954$		0			0
$955$	1500.66	−	866.406i	1.57137	−	0.907231i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−300.048	−	772.569i	−0.312875	−	0.805598i
$960$		0			0
$961$	−269.012	+	465.942i	−0.279929	+	0.484851i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	862.304i		−	0.893580i
$966$		0			0
$967$	333.134			0.344503			0.172252	−	0.985053i	$-0.444896\pi$
$967$	333.134			0.344503			0.172252	+	0.985053i	$0.444896\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1536.10	+	886.865i	1.58197	+	0.913353i	0.994571	+	0.104057i	$0.0331824\pi$
$971$	1536.10	+	886.865i	1.58197	+	0.913353i	0.587402	+	0.809296i	$0.300151\pi$
$972$		0			0
$973$	25.9862	−	168.347i	0.0267073	−	0.173018i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−727.729	−	1260.46i	−0.744860	−	1.29014i	−0.950260	−	0.311458i	$-0.899183\pi$
$977$	−727.729	−	1260.46i	−0.744860	−	1.29014i	0.205400	−	0.978678i	$-0.434151\pi$
$978$		0			0
$979$			1471.27i			1.50283i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	721.681	−	416.663i	0.734162	−	0.423868i	−0.0857811	−	0.996314i	$-0.527339\pi$
$983$	721.681	−	416.663i	0.734162	−	0.423868i	0.819943	+	0.572446i	$0.194005\pi$
$984$		0			0
$985$	−905.365	−	522.713i	−0.919153	−	0.530673i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−736.433	+	1275.54i	−0.744624	+	1.28973i
$990$		0			0
$991$	−381.647	−	661.032i	−0.385113	−	0.667036i	0.606672	−	0.794953i	$-0.292504\pi$
$991$	−381.647	−	661.032i	−0.385113	−	0.667036i	−0.991785	+	0.127917i	$0.959171\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−770.495			−0.774367
$996$		0			0
$997$	−232.890	+	134.459i	−0.233591	+	0.134864i	−0.612228	−	0.790682i	$-0.709726\pi$
$997$	−232.890	+	134.459i	−0.233591	+	0.134864i	0.378637	+	0.925545i	$0.376393\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cg.q.145.2		16
3.2	odd	2	inner	1008.3.cg.q.145.7		16
4.3	odd	2		504.3.by.d.145.2	yes	16
7.3	odd	6	inner	1008.3.cg.q.577.2		16
12.11	even	2		504.3.by.d.145.7	yes	16
21.17	even	6	inner	1008.3.cg.q.577.7		16
28.3	even	6		504.3.by.d.73.2	✓	16
28.19	even	6		3528.3.f.i.2449.4		16
28.23	odd	6		3528.3.f.i.2449.14		16
84.23	even	6		3528.3.f.i.2449.3		16
84.47	odd	6		3528.3.f.i.2449.13		16
84.59	odd	6		504.3.by.d.73.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.by.d.73.2	✓	16	28.3	even	6
504.3.by.d.73.7	yes	16	84.59	odd	6
504.3.by.d.145.2	yes	16	4.3	odd	2
504.3.by.d.145.7	yes	16	12.11	even	2
1008.3.cg.q.145.2		16	1.1	even	1	trivial
1008.3.cg.q.145.7		16	3.2	odd	2	inner
1008.3.cg.q.577.2		16	7.3	odd	6	inner
1008.3.cg.q.577.7		16	21.17	even	6	inner
3528.3.f.i.2449.3		16	84.23	even	6
3528.3.f.i.2449.4		16	28.19	even	6
3528.3.f.i.2449.13		16	84.47	odd	6
3528.3.f.i.2449.14		16	28.23	odd	6

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.51611 - 2.60738i) q^{5} +(-6.91807 - 1.06788i) q^{7} +O(q^{10})\)	\(q+(-4.51611 - 2.60738i) q^{5} +(-6.91807 - 1.06788i) q^{7} +(4.56040 + 7.89885i) q^{11} +3.43129i q^{13} +(-4.73709 + 2.73496i) q^{17} +(9.45229 + 5.45728i) q^{19} +(-9.95199 + 17.2374i) q^{23} +(1.09685 + 1.89981i) q^{25} -36.7480 q^{29} +(17.8110 - 10.2832i) q^{31} +(28.4584 + 22.8607i) q^{35} +(23.5754 - 40.8338i) q^{37} -41.5146i q^{41} +73.9986 q^{43} +(51.2578 + 29.5937i) q^{47} +(46.7193 + 14.7754i) q^{49} +(-31.7014 - 54.9084i) q^{53} -47.5628i q^{55} +(44.8630 - 25.9017i) q^{59} +(21.2053 + 12.2429i) q^{61} +(8.94668 - 15.4961i) q^{65} +(9.58916 + 16.6089i) q^{67} +2.54627 q^{71} +(102.444 - 59.1459i) q^{73} +(-23.1141 - 59.5147i) q^{77} +(5.67414 - 9.82791i) q^{79} -30.4340i q^{83} +28.5243 q^{85} +(139.698 + 80.6547i) q^{89} +(3.66422 - 23.7379i) q^{91} +(-28.4584 - 49.2914i) q^{95} +101.257i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 4 * q^7 + 24 * q^19 + 36 * q^25 + 84 * q^31 - 68 * q^37 - 80 * q^43 - 184 * q^49 + 216 * q^61 - 56 * q^67 + 156 * q^73 - 28 * q^79 + 448 * q^85 - 48 * q^91

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