LMFDB - Embedded newform 1008.3.o.b.1007.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(1007,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.4
Character	$\chi$	$=$	1008.1007
Dual form			1008.3.o.b.1007.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.41321 q^{5} +(-3.32249 + 6.16125i) q^{7} -17.6630 q^{11} +17.9225i q^{13} -13.3835 q^{17} -14.6302 q^{19} +30.1907 q^{23} +63.6085 q^{25} -25.9412i q^{29} -11.5500 q^{31} +(31.2753 - 57.9972i) q^{35} -17.9221 q^{37} -12.6503 q^{41} +19.2507i q^{43} +48.3171i q^{47} +(-26.9221 - 40.9414i) q^{49} -3.42395i q^{53} +166.265 q^{55} -53.9239i q^{59} -59.8685i q^{61} -168.709i q^{65} +104.977i q^{67} +76.5385 q^{71} -142.830i q^{73} +(58.6851 - 108.826i) q^{77} -47.9662i q^{79} +90.9483i q^{83} +125.981 q^{85} -122.663 q^{89} +(-110.425 - 59.5474i) q^{91} +137.717 q^{95} +94.7826i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) $ 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Character values

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

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	−9.41321			−1.88264			−0.941321	−	0.337512i	$-0.890414\pi$
$5$	−9.41321			−1.88264			−0.941321	+	0.337512i	$0.890414\pi$
$6$		0			0
$7$	−3.32249	+	6.16125i	−0.474642	+	0.880179i
$7$	−3.32249	+	6.16125i	−0.474642	+	0.880179i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.6630			−1.60573			−0.802863	−	0.596163i	$-0.796691\pi$
$11$	−17.6630			−1.60573			−0.802863	+	0.596163i	$0.796691\pi$
$12$		0			0
$13$			17.9225i			1.37866i	0.724449	+	0.689328i	$0.242094\pi$
$13$			17.9225i			1.37866i	−0.724449	+	0.689328i	$0.757906\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−13.3835			−0.787263			−0.393631	−	0.919268i	$-0.628781\pi$
$17$	−13.3835			−0.787263			−0.393631	+	0.919268i	$0.628781\pi$
$18$		0			0
$19$	−14.6302			−0.770008			−0.385004	−	0.922915i	$-0.625800\pi$
$19$	−14.6302			−0.770008			−0.385004	+	0.922915i	$0.625800\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	30.1907			1.31264			0.656319	−	0.754484i	$-0.272113\pi$
$23$	30.1907			1.31264			0.656319	+	0.754484i	$0.272113\pi$
$24$		0			0
$25$	63.6085			2.54434
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	25.9412i		−	0.894525i	−0.894403	−	0.447262i	$-0.852399\pi$
$29$		−	25.9412i		−	0.894525i	0.894403	−	0.447262i	$-0.147601\pi$
$30$		0			0
$31$	−11.5500			−0.372582			−0.186291	−	0.982495i	$-0.559647\pi$
$31$	−11.5500			−0.372582			−0.186291	+	0.982495i	$0.559647\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	31.2753	−	57.9972i	0.893580	−	1.65706i
$36$		0			0
$37$	−17.9221			−0.484381			−0.242191	−	0.970229i	$-0.577866\pi$
$37$	−17.9221			−0.484381			−0.242191	+	0.970229i	$0.577866\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.6503			−0.308544			−0.154272	−	0.988028i	$-0.549303\pi$
$41$	−12.6503			−0.308544			−0.154272	+	0.988028i	$0.549303\pi$
$42$		0			0
$43$			19.2507i			0.447691i	0.974625	+	0.223845i	$0.0718611\pi$
$43$			19.2507i			0.447691i	−0.974625	+	0.223845i	$0.928139\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			48.3171i			1.02802i	0.857783	+	0.514012i	$0.171841\pi$
$47$			48.3171i			1.02802i	−0.857783	+	0.514012i	$0.828159\pi$
$48$		0			0
$49$	−26.9221	−	40.9414i	−0.549431	−	0.835539i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.42395i		−	0.0646028i	−0.999478	−	0.0323014i	$-0.989716\pi$
$53$		−	3.42395i		−	0.0646028i	0.999478	−	0.0323014i	$-0.0102837\pi$
$54$		0			0
$55$	166.265			3.02301
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	53.9239i		−	0.913964i	−0.889476	−	0.456982i	$-0.848930\pi$
$59$		−	53.9239i		−	0.913964i	0.889476	−	0.456982i	$-0.151070\pi$
$60$		0			0
$61$		−	59.8685i		−	0.981451i	−0.871314	−	0.490725i	$-0.836732\pi$
$61$		−	59.8685i		−	0.981451i	0.871314	−	0.490725i	$-0.163268\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	168.709i		−	2.59552i
$66$		0			0
$67$			104.977i			1.56682i	0.621504	+	0.783411i	$0.286522\pi$
$67$			104.977i			1.56682i	−0.621504	+	0.783411i	$0.713478\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	76.5385			1.07801			0.539004	−	0.842303i	$-0.318801\pi$
$71$	76.5385			1.07801			0.539004	+	0.842303i	$0.318801\pi$
$72$		0			0
$73$		−	142.830i		−	1.95657i	−0.207268	−	0.978284i	$-0.566457\pi$
$73$		−	142.830i		−	1.95657i	0.207268	−	0.978284i	$-0.433543\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	58.6851	−	108.826i	0.762145	−	1.41333i
$78$		0			0
$79$		−	47.9662i		−	0.607167i	−0.952805	−	0.303584i	$-0.901817\pi$
$79$		−	47.9662i		−	0.607167i	0.952805	−	0.303584i	$-0.0981832\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			90.9483i			1.09576i	0.836556	+	0.547881i	$0.184565\pi$
$83$			90.9483i			1.09576i	−0.836556	+	0.547881i	$0.815435\pi$
$84$		0			0
$85$	125.981			1.48213
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−122.663			−1.37824			−0.689120	−	0.724647i	$-0.742003\pi$
$89$	−122.663			−1.37824			−0.689120	+	0.724647i	$0.742003\pi$
$90$		0			0
$91$	−110.425	−	59.5474i	−1.21346	−	0.654367i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	137.717			1.44965
$96$		0			0
$97$			94.7826i			0.977140i	0.872525	+	0.488570i	$0.162481\pi$
$97$			94.7826i			0.977140i	−0.872525	+	0.488570i	$0.837519\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	40.5919			0.401900			0.200950	−	0.979602i	$-0.435597\pi$
$101$	40.5919			0.401900			0.200950	+	0.979602i	$0.435597\pi$
$102$		0			0
$103$	120.550			1.17039			0.585195	−	0.810893i	$-0.301018\pi$
$103$	120.550			1.17039			0.585195	+	0.810893i	$0.301018\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−113.946			−1.06492			−0.532458	−	0.846456i	$-0.678732\pi$
$107$	−113.946			−1.06492			−0.532458	+	0.846456i	$0.678732\pi$
$108$		0			0
$109$	62.6864			0.575105			0.287552	−	0.957765i	$-0.407158\pi$
$109$	62.6864			0.575105			0.287552	+	0.957765i	$0.407158\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.15197i		−	0.0721413i	−0.999349	−	0.0360706i	$-0.988516\pi$
$113$		−	8.15197i		−	0.0721413i	0.999349	−	0.0360706i	$-0.0114841\pi$
$114$		0			0
$115$	−284.191			−2.47123
$116$		0			0
$117$		0			0
$118$		0			0
$119$	44.4664	−	82.4589i	0.373668	−	0.692932i
$120$		0			0
$121$	190.981			1.57836
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−363.430			−2.90744
$126$		0			0
$127$			209.693i			1.65112i	0.564312	+	0.825562i	$0.309142\pi$
$127$			209.693i			1.65112i	−0.564312	+	0.825562i	$0.690858\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	59.6098i		−	0.455037i	−0.973774	−	0.227518i	$-0.926939\pi$
$131$		−	59.6098i		−	0.455037i	0.973774	−	0.227518i	$-0.0730612\pi$
$132$		0			0
$133$	48.6085	−	90.1401i	0.365478	−	0.677745i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	155.981i		−	1.13854i	−0.822149	−	0.569272i	$-0.807225\pi$
$137$		−	155.981i		−	1.13854i	0.822149	−	0.569272i	$-0.192775\pi$
$138$		0			0
$139$	247.430			1.78008			0.890038	−	0.455887i	$-0.150678\pi$
$139$	247.430			1.78008			0.890038	+	0.455887i	$0.150678\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	316.566i		−	2.21375i
$144$		0			0
$145$			244.190i			1.68407i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			11.5788i			0.0777100i	0.999245	+	0.0388550i	$0.0123710\pi$
$149$			11.5788i			0.0777100i	−0.999245	+	0.0388550i	$0.987629\pi$
$150$		0			0
$151$		−	126.764i		−	0.839499i	−0.907640	−	0.419750i	$-0.862118\pi$
$151$		−	126.764i		−	0.839499i	0.907640	−	0.419750i	$-0.137882\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	108.723			0.701438
$156$		0			0
$157$			45.6576i			0.290813i	0.989372	+	0.145406i	$0.0464489\pi$
$157$			45.6576i			0.290813i	−0.989372	+	0.145406i	$0.953551\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−100.308	+	186.012i	−0.623032	+	1.15536i
$162$		0			0
$163$			34.6326i			0.212470i	0.994341	+	0.106235i	$0.0338796\pi$
$163$			34.6326i			0.212470i	−0.994341	+	0.106235i	$0.966120\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			252.799i			1.51377i	0.653550	+	0.756884i	$0.273279\pi$
$167$			252.799i			1.51377i	−0.653550	+	0.756884i	$0.726721\pi$
$168$		0			0
$169$	−152.217			−0.900693
$170$		0			0
$171$		0			0
$172$		0			0
$173$	112.219			0.648665			0.324332	−	0.945943i	$-0.394860\pi$
$173$	112.219			0.648665			0.324332	+	0.945943i	$0.394860\pi$
$174$		0			0
$175$	−211.339	+	391.908i	−1.20765	+	2.23948i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−266.447			−1.48853			−0.744267	−	0.667883i	$-0.767201\pi$
$179$	−266.447			−1.48853			−0.744267	+	0.667883i	$0.767201\pi$
$180$		0			0
$181$			80.3276i			0.443799i	0.975070	+	0.221899i	$0.0712256\pi$
$181$			80.3276i			0.443799i	−0.975070	+	0.221899i	$0.928774\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	168.705			0.911917
$186$		0			0
$187$	236.392			1.26413
$188$		0			0
$189$		0			0
$190$		0			0
$191$	60.0268			0.314277			0.157138	−	0.987577i	$-0.449773\pi$
$191$	60.0268			0.314277			0.157138	+	0.987577i	$0.449773\pi$
$192$		0			0
$193$	−160.297			−0.830554			−0.415277	−	0.909695i	$-0.636315\pi$
$193$	−160.297			−0.830554			−0.415277	+	0.909695i	$0.636315\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	43.5521i		−	0.221077i	−0.993872	−	0.110538i	$-0.964743\pi$
$197$		−	43.5521i		−	0.221077i	0.993872	−	0.110538i	$-0.0352575\pi$
$198$		0			0
$199$	−27.0647			−0.136003			−0.0680017	−	0.997685i	$-0.521662\pi$
$199$	−27.0647			−0.136003			−0.0680017	+	0.997685i	$0.521662\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	159.830	+	86.1895i	0.787342	+	0.424579i
$204$		0			0
$205$	119.080			0.580878
$206$		0			0
$207$		0			0
$208$		0			0
$209$	258.412			1.23642
$210$		0			0
$211$		−	328.308i		−	1.55596i	−0.628289	−	0.777980i	$-0.716244\pi$
$211$		−	328.308i		−	1.55596i	0.628289	−	0.777980i	$-0.283756\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	181.211i		−	0.842842i
$216$		0			0
$217$	38.3749	−	71.1627i	0.176843	−	0.327939i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	239.866i		−	1.08536i
$222$		0			0
$223$	1.28423			0.00575888			0.00287944	−	0.999996i	$-0.499083\pi$
$223$	1.28423			0.00575888			0.00287944	+	0.999996i	$0.499083\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			194.200i			0.855505i	0.903896	+	0.427753i	$0.140695\pi$
$227$			194.200i			0.855505i	−0.903896	+	0.427753i	$0.859305\pi$
$228$		0			0
$229$		−	289.223i		−	1.26298i	−0.775382	−	0.631492i	$-0.782443\pi$
$229$		−	289.223i		−	1.26298i	0.775382	−	0.631492i	$-0.217557\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			121.042i			0.519495i	0.965677	+	0.259748i	$0.0836394\pi$
$233$			121.042i			0.519495i	−0.965677	+	0.259748i	$0.916361\pi$
$234$		0			0
$235$		−	454.819i		−	1.93540i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	163.878			0.685681			0.342841	−	0.939394i	$-0.388611\pi$
$239$	163.878			0.685681			0.342841	+	0.939394i	$0.388611\pi$
$240$		0			0
$241$		−	421.554i		−	1.74919i	−0.484859	−	0.874593i	$-0.661129\pi$
$241$		−	421.554i		−	1.74919i	0.484859	−	0.874593i	$-0.338871\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	253.424	+	385.390i	1.03438	+	1.57302i
$246$		0			0
$247$		−	262.209i		−	1.06158i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			335.240i			1.33562i	0.744333	+	0.667809i	$0.232768\pi$
$251$			335.240i			1.33562i	−0.744333	+	0.667809i	$0.767232\pi$
$252$		0			0
$253$	−533.258			−2.10774
$254$		0			0
$255$		0			0
$256$		0			0
$257$	149.430			0.581441			0.290721	−	0.956808i	$-0.406105\pi$
$257$	149.430			0.581441			0.290721	+	0.956808i	$0.406105\pi$
$258$		0			0
$259$	59.5461	−	110.423i	0.229908	−	0.426342i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	75.2116			0.285976			0.142988	−	0.989724i	$-0.454329\pi$
$263$	75.2116			0.285976			0.142988	+	0.989724i	$0.454329\pi$
$264$		0			0
$265$			32.2304i			0.121624i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−168.115			−0.624962			−0.312481	−	0.949924i	$-0.601160\pi$
$269$	−168.115			−0.624962			−0.312481	+	0.949924i	$0.601160\pi$
$270$		0			0
$271$	−282.822			−1.04362			−0.521812	−	0.853061i	$-0.674744\pi$
$271$	−282.822			−1.04362			−0.521812	+	0.853061i	$0.674744\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1123.52			−4.08552
$276$		0			0
$277$	−306.571			−1.10676			−0.553378	−	0.832930i	$-0.686661\pi$
$277$	−306.571			−1.10676			−0.553378	+	0.832930i	$0.686661\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			325.990i			1.16011i	0.814578	+	0.580054i	$0.196969\pi$
$281$			325.990i			1.16011i	−0.814578	+	0.580054i	$0.803031\pi$
$282$		0			0
$283$	361.362			1.27690			0.638450	−	0.769664i	$-0.279576\pi$
$283$	361.362			1.27690			0.638450	+	0.769664i	$0.279576\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	42.0305	−	77.9417i	0.146448	−	0.271574i
$288$		0			0
$289$	−109.883			−0.380217
$290$		0			0
$291$		0			0
$292$		0			0
$293$	130.162			0.444240			0.222120	−	0.975019i	$-0.428702\pi$
$293$	130.162			0.444240			0.222120	+	0.975019i	$0.428702\pi$
$294$		0			0
$295$			507.597i			1.72067i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			541.093i			1.80968i
$300$		0			0
$301$	−118.609	−	63.9603i	−0.394048	−	0.212493i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			563.555i			1.84772i
$306$		0			0
$307$	−386.919			−1.26032			−0.630161	−	0.776464i	$-0.717011\pi$
$307$	−386.919			−1.26032			−0.630161	+	0.776464i	$0.717011\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			234.449i			0.753857i	0.926242	+	0.376928i	$0.123020\pi$
$311$			234.449i			0.753857i	−0.926242	+	0.376928i	$0.876980\pi$
$312$		0			0
$313$			427.938i			1.36721i	0.729851	+	0.683607i	$0.239590\pi$
$313$			427.938i			1.36721i	−0.729851	+	0.683607i	$0.760410\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	301.716i		−	0.951785i	−0.879504	−	0.475892i	$-0.842125\pi$
$317$		−	301.716i		−	0.951785i	0.879504	−	0.475892i	$-0.157875\pi$
$318$		0			0
$319$			458.200i			1.43636i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	195.802			0.606198
$324$		0			0
$325$			1140.03i			3.50777i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−297.694	−	160.533i	−0.904845	−	0.487943i
$330$		0			0
$331$		−	525.382i		−	1.58726i	−0.608403	−	0.793628i	$-0.708189\pi$
$331$		−	525.382i		−	1.58726i	0.608403	−	0.793628i	$-0.291811\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	988.171i		−	2.94976i
$336$		0			0
$337$	385.024			1.14250			0.571252	−	0.820775i	$-0.306458\pi$
$337$	385.024			1.14250			0.571252	+	0.820775i	$0.306458\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	204.008			0.598264
$342$		0			0
$343$	341.699	−	29.8465i	0.996207	−	0.0870161i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−126.470			−0.364468			−0.182234	−	0.983255i	$-0.558333\pi$
$347$	−126.470			−0.364468			−0.182234	+	0.983255i	$0.558333\pi$
$348$		0			0
$349$			74.0794i			0.212262i	0.994352	+	0.106131i	$0.0338463\pi$
$349$			74.0794i			0.212262i	−0.994352	+	0.106131i	$0.966154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	92.9638			0.263354			0.131677	−	0.991293i	$-0.457964\pi$
$353$	92.9638			0.263354			0.131677	+	0.991293i	$0.457964\pi$
$354$		0			0
$355$	−720.473			−2.02950
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−330.371			−0.920252			−0.460126	−	0.887854i	$-0.652196\pi$
$359$	−330.371			−0.920252			−0.460126	+	0.887854i	$0.652196\pi$
$360$		0			0
$361$	−146.959			−0.407088
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1344.48i			3.68352i
$366$		0			0
$367$	−2.30755			−0.00628759			−0.00314380	−	0.999995i	$-0.501001\pi$
$367$	−2.30755			−0.00628759			−0.00314380	+	0.999995i	$0.501001\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	21.0958	+	11.3760i	0.0568621	+	0.0306632i
$372$		0			0
$373$	332.901			0.892497			0.446249	−	0.894909i	$-0.352760\pi$
$373$	332.901			0.892497			0.446249	+	0.894909i	$0.352760\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	464.932			1.23324
$378$		0			0
$379$			144.379i			0.380946i	0.981692	+	0.190473i	$0.0610022\pi$
$379$			144.379i			0.380946i	−0.981692	+	0.190473i	$0.938998\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	255.101i		−	0.666061i	−0.942916	−	0.333031i	$-0.891929\pi$
$383$		−	255.101i		−	0.666061i	0.942916	−	0.333031i	$-0.108071\pi$
$384$		0			0
$385$	−552.416	+	1024.40i	−1.43485	+	2.66079i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			452.430i			1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$389$			452.430i			1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$390$		0			0
$391$	−404.056			−1.03339
$392$		0			0
$393$		0			0
$394$		0			0
$395$			451.516i			1.14308i
$396$		0			0
$397$		−	405.310i		−	1.02093i	−0.859898	−	0.510466i	$-0.829473\pi$
$397$		−	405.310i		−	1.02093i	0.859898	−	0.510466i	$-0.170527\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	18.3273i		−	0.0457040i	−0.999739	−	0.0228520i	$-0.992725\pi$
$401$		−	18.3273i		−	0.0457040i	0.999739	−	0.0228520i	$-0.00727464\pi$
$402$		0			0
$403$		−	207.006i		−	0.513662i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	316.558			0.777784
$408$		0			0
$409$		−	301.328i		−	0.736744i	−0.929678	−	0.368372i	$-0.879915\pi$
$409$		−	301.328i		−	0.736744i	0.929678	−	0.368372i	$-0.120085\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	332.239	+	179.162i	0.804453	+	0.433805i
$414$		0			0
$415$		−	856.115i		−	2.06293i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	515.599i		−	1.23055i	−0.788314	−	0.615273i	$-0.789046\pi$
$419$		−	515.599i		−	1.23055i	0.788314	−	0.615273i	$-0.210954\pi$
$420$		0			0
$421$	−461.666			−1.09659			−0.548297	−	0.836284i	$-0.684724\pi$
$421$	−461.666			−1.09659			−0.548297	+	0.836284i	$0.684724\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−851.303			−2.00307
$426$		0			0
$427$	368.865	+	198.913i	0.863853	+	0.465837i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−462.604			−1.07333			−0.536664	−	0.843796i	$-0.680316\pi$
$431$	−462.604			−1.07333			−0.536664	+	0.843796i	$0.680316\pi$
$432$		0			0
$433$		−	579.695i		−	1.33879i	−0.742907	−	0.669394i	$-0.766554\pi$
$433$		−	579.695i		−	1.33879i	0.742907	−	0.669394i	$-0.233446\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−441.694			−1.01074
$438$		0			0
$439$	331.988			0.756237			0.378119	−	0.925757i	$-0.376571\pi$
$439$	331.988			0.756237			0.378119	+	0.925757i	$0.376571\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−559.102			−1.26208			−0.631040	−	0.775750i	$-0.717372\pi$
$443$	−559.102			−1.26208			−0.631040	+	0.775750i	$0.717372\pi$
$444$		0			0
$445$	1154.66			2.59473
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	569.104i		−	1.26749i	−0.773541	−	0.633747i	$-0.781516\pi$
$449$		−	569.104i		−	1.26749i	0.773541	−	0.633747i	$-0.218484\pi$
$450$		0			0
$451$	223.442			0.495437
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1039.46	+	560.533i	2.28452	+	1.23194i
$456$		0			0
$457$	−413.774			−0.905413			−0.452707	−	0.891660i	$-0.649541\pi$
$457$	−413.774			−0.905413			−0.452707	+	0.891660i	$0.649541\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	475.662			1.03180			0.515902	−	0.856647i	$-0.327457\pi$
$461$	475.662			1.03180			0.515902	+	0.856647i	$0.327457\pi$
$462$		0			0
$463$			630.680i			1.36216i	0.732209	+	0.681080i	$0.238489\pi$
$463$			630.680i			1.36216i	−0.732209	+	0.681080i	$0.761511\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			124.466i			0.266522i	0.991081	+	0.133261i	$0.0425449\pi$
$467$			124.466i			0.266522i	−0.991081	+	0.133261i	$0.957455\pi$
$468$		0			0
$469$	−646.790	−	348.785i	−1.37908	−	0.743679i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	340.025i		−	0.718869i
$474$		0			0
$475$	−930.602			−1.95916
$476$		0			0
$477$		0			0
$478$		0			0
$479$			516.970i			1.07927i	0.841899	+	0.539635i	$0.181438\pi$
$479$			516.970i			1.07927i	−0.841899	+	0.539635i	$0.818562\pi$
$480$		0			0
$481$		−	321.210i		−	0.667795i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	892.209i		−	1.83961i
$486$		0			0
$487$		−	317.163i		−	0.651260i	−0.945497	−	0.325630i	$-0.894424\pi$
$487$		−	317.163i		−	0.651260i	0.945497	−	0.325630i	$-0.105576\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−225.238			−0.458733			−0.229367	−	0.973340i	$-0.573666\pi$
$491$	−225.238			−0.458733			−0.229367	+	0.973340i	$0.573666\pi$
$492$		0			0
$493$			347.183i			0.704226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−254.299	+	471.573i	−0.511667	+	0.948840i
$498$		0			0
$499$		−	376.347i		−	0.754202i	−0.926172	−	0.377101i	$-0.876921\pi$
$499$		−	376.347i		−	0.754202i	0.926172	−	0.377101i	$-0.123079\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			35.7324i			0.0710386i	0.999369	+	0.0355193i	$0.0113085\pi$
$503$			35.7324i			0.0710386i	−0.999369	+	0.0355193i	$0.988691\pi$
$504$		0			0
$505$	−382.100			−0.756634
$506$		0			0
$507$		0			0
$508$		0			0
$509$	924.267			1.81585			0.907924	−	0.419134i	$-0.137666\pi$
$509$	924.267			1.81585			0.907924	+	0.419134i	$0.137666\pi$
$510$		0			0
$511$	880.009	+	474.550i	1.72213	+	0.928669i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1134.76			−2.20342
$516$		0			0
$517$		−	853.425i		−	1.65072i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1024.09			1.96563			0.982816	−	0.184590i	$-0.0590956\pi$
$521$	1024.09			1.96563			0.982816	+	0.184590i	$0.0590956\pi$
$522$		0			0
$523$	−141.910			−0.271339			−0.135669	−	0.990754i	$-0.543319\pi$
$523$	−141.910			−0.271339			−0.135669	+	0.990754i	$0.543319\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	154.579			0.293320
$528$		0			0
$529$	382.477			0.723019
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	226.725i		−	0.425376i
$534$		0			0
$535$	1072.60			2.00486
$536$		0			0
$537$		0			0
$538$		0			0
$539$	475.525	+	723.148i	0.882236	+	1.34165i
$540$		0			0
$541$	−271.944			−0.502669			−0.251335	−	0.967900i	$-0.580869\pi$
$541$	−271.944			−0.502669			−0.251335	+	0.967900i	$0.580869\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−590.081			−1.08272
$546$		0			0
$547$		−	167.266i		−	0.305789i	−0.988243	−	0.152894i	$-0.951141\pi$
$547$		−	167.266i		−	0.305789i	0.988243	−	0.152894i	$-0.0488594\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			379.524i			0.688791i
$552$		0			0
$553$	295.532	+	159.367i	0.534416	+	0.288187i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	594.093i		−	1.06659i	−0.845928	−	0.533297i	$-0.820953\pi$
$557$		−	594.093i		−	1.06659i	0.845928	−	0.533297i	$-0.179047\pi$
$558$		0			0
$559$	−345.021			−0.617212
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	379.480i		−	0.674032i	−0.941499	−	0.337016i	$-0.890582\pi$
$563$		−	379.480i		−	0.674032i	0.941499	−	0.337016i	$-0.109418\pi$
$564$		0			0
$565$			76.7362i			0.135816i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	831.502i		−	1.46134i	−0.682731	−	0.730670i	$-0.739208\pi$
$569$		−	831.502i		−	1.46134i	0.682731	−	0.730670i	$-0.260792\pi$
$570$		0			0
$571$			18.8903i			0.0330828i	0.999863	+	0.0165414i	$0.00526554\pi$
$571$			18.8903i			0.0330828i	−0.999863	+	0.0165414i	$0.994734\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1920.38			3.33980
$576$		0			0
$577$		−	571.465i		−	0.990408i	−0.868777	−	0.495204i	$-0.835093\pi$
$577$		−	571.465i		−	0.990408i	0.868777	−	0.495204i	$-0.164907\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−560.355	−	302.175i	−0.964467	−	0.520094i
$582$		0			0
$583$			60.4772i			0.103735i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	883.232i		−	1.50465i	−0.658790	−	0.752327i	$-0.728931\pi$
$587$		−	883.232i		−	1.50465i	0.658790	−	0.752327i	$-0.271069\pi$
$588$		0			0
$589$	168.979			0.286891
$590$		0			0
$591$		0			0
$592$		0			0
$593$	498.733			0.841033			0.420517	−	0.907285i	$-0.361849\pi$
$593$	498.733			0.841033			0.420517	+	0.907285i	$0.361849\pi$
$594$		0			0
$595$	−418.572	+	776.203i	−0.703482	+	1.30454i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	438.918			0.732751			0.366375	−	0.930467i	$-0.380599\pi$
$599$	438.918			0.732751			0.366375	+	0.930467i	$0.380599\pi$
$600$		0			0
$601$		−	555.392i		−	0.924114i	−0.886850	−	0.462057i	$-0.847112\pi$
$601$		−	555.392i		−	0.924114i	0.886850	−	0.462057i	$-0.152888\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1797.75			−2.97148
$606$		0			0
$607$	615.415			1.01386			0.506931	−	0.861986i	$-0.330780\pi$
$607$	615.415			1.01386			0.506931	+	0.861986i	$0.330780\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−865.965			−1.41729
$612$		0			0
$613$	−284.085			−0.463435			−0.231717	−	0.972783i	$-0.574434\pi$
$613$	−284.085			−0.463435			−0.231717	+	0.972783i	$0.574434\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			453.948i			0.735734i	0.929878	+	0.367867i	$0.119912\pi$
$617$			453.948i			0.735734i	−0.929878	+	0.367867i	$0.880088\pi$
$618$		0			0
$619$	−359.393			−0.580602			−0.290301	−	0.956935i	$-0.593755\pi$
$619$	−359.393			−0.580602			−0.290301	+	0.956935i	$0.593755\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	407.548	−	755.761i	0.654170	−	1.21310i
$624$		0			0
$625$	1830.83			2.92933
$626$		0			0
$627$		0			0
$628$		0			0
$629$	239.860			0.381335
$630$		0			0
$631$		−	152.887i		−	0.242293i	−0.992635	−	0.121147i	$-0.961343\pi$
$631$		−	152.887i		−	0.242293i	0.992635	−	0.121147i	$-0.0386572\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1973.88i		−	3.10848i
$636$		0			0
$637$	733.774	−	482.512i	1.15192	−	0.757476i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			26.7862i			0.0417882i	0.999782	+	0.0208941i	$0.00665128\pi$
$641$			26.7862i			0.0417882i	−0.999782	+	0.0208941i	$0.993349\pi$
$642$		0			0
$643$	−184.314			−0.286646			−0.143323	−	0.989676i	$-0.545779\pi$
$643$	−184.314			−0.286646			−0.143323	+	0.989676i	$0.545779\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	393.760i		−	0.608594i	−0.952577	−	0.304297i	$-0.901578\pi$
$647$		−	393.760i		−	0.608594i	0.952577	−	0.304297i	$-0.0984216\pi$
$648$		0			0
$649$			952.458i			1.46758i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	128.089i		−	0.196155i	−0.995179	−	0.0980776i	$-0.968731\pi$
$653$		−	128.089i		−	0.196155i	0.995179	−	0.0980776i	$-0.0312693\pi$
$654$		0			0
$655$			561.120i			0.856672i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	983.280			1.49208			0.746040	−	0.665902i	$-0.231953\pi$
$659$	983.280			1.49208			0.746040	+	0.665902i	$0.231953\pi$
$660$		0			0
$661$			668.474i			1.01131i	0.862737	+	0.505654i	$0.168749\pi$
$661$			668.474i			1.01131i	−0.862737	+	0.505654i	$0.831251\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−457.562	+	848.508i	−0.688064	+	1.27595i
$666$		0			0
$667$		−	783.183i		−	1.17419i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1057.46i			1.57594i
$672$		0			0
$673$	16.9201			0.0251413			0.0125706	−	0.999921i	$-0.495999\pi$
$673$	16.9201			0.0251413			0.0125706	+	0.999921i	$0.495999\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1200.92			−1.77388			−0.886942	−	0.461881i	$-0.847175\pi$
$677$	−1200.92			−1.77388			−0.886942	+	0.461881i	$0.847175\pi$
$678$		0			0
$679$	−583.980	−	314.914i	−0.860059	−	0.463791i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1327.68			1.94390			0.971948	−	0.235197i	$-0.0755734\pi$
$683$	1327.68			1.94390			0.971948	+	0.235197i	$0.0755734\pi$
$684$		0			0
$685$			1468.28i			2.14347i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	61.3659			0.0890651
$690$		0			0
$691$	207.213			0.299874			0.149937	−	0.988696i	$-0.452093\pi$
$691$	207.213			0.299874			0.149937	+	0.988696i	$0.452093\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2329.12			−3.35124
$696$		0			0
$697$	169.305			0.242905
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	19.3505i		−	0.0276042i	−0.999905	−	0.0138021i	$-0.995607\pi$
$701$		−	19.3505i		−	0.0276042i	0.999905	−	0.0138021i	$-0.00439348\pi$
$702$		0			0
$703$	262.203			0.372978
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−134.866	+	250.097i	−0.190758	+	0.353744i
$708$		0			0
$709$	1329.52			1.87521			0.937603	−	0.347707i	$-0.113040\pi$
$709$	1329.52			1.87521			0.937603	+	0.347707i	$0.113040\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−348.703			−0.489065
$714$		0			0
$715$			2979.90i			4.16769i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	984.181i		−	1.36882i	−0.729098	−	0.684410i	$-0.760060\pi$
$719$		−	984.181i		−	1.36882i	0.729098	−	0.684410i	$-0.239940\pi$
$720$		0			0
$721$	−400.527	+	742.740i	−0.555515	+	1.03015i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1650.08i		−	2.27598i
$726$		0			0
$727$	166.808			0.229447			0.114724	−	0.993397i	$-0.463402\pi$
$727$	166.808			0.229447			0.114724	+	0.993397i	$0.463402\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	257.641i		−	0.352450i
$732$		0			0
$733$		−	989.176i		−	1.34949i	−0.738051	−	0.674745i	$-0.764254\pi$
$733$		−	989.176i		−	1.34949i	0.738051	−	0.674745i	$-0.235746\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1854.21i		−	2.51589i
$738$		0			0
$739$		−	254.767i		−	0.344745i	−0.985032	−	0.172373i	$-0.944857\pi$
$739$		−	254.767i		−	0.344745i	0.985032	−	0.172373i	$-0.0551433\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−159.188			−0.214250			−0.107125	−	0.994246i	$-0.534165\pi$
$743$	−159.188			−0.214250			−0.107125	+	0.994246i	$0.534165\pi$
$744$		0			0
$745$		−	108.994i		−	0.146300i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	378.584	−	702.050i	0.505453	−	0.937317i
$750$		0			0
$751$			854.873i			1.13831i	0.822229	+	0.569156i	$0.192730\pi$
$751$			854.873i			1.13831i	−0.822229	+	0.569156i	$0.807270\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1193.26i			1.58048i
$756$		0			0
$757$	233.833			0.308894			0.154447	−	0.988001i	$-0.450640\pi$
$757$	233.833			0.308894			0.154447	+	0.988001i	$0.450640\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−508.165			−0.667760			−0.333880	−	0.942616i	$-0.608358\pi$
$761$	−508.165			−0.667760			−0.333880	+	0.942616i	$0.608358\pi$
$762$		0			0
$763$	−208.275	+	386.227i	−0.272969	+	0.506195i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	966.453			1.26004
$768$		0			0
$769$		−	176.433i		−	0.229432i	−0.993398	−	0.114716i	$-0.963404\pi$
$769$		−	176.433i		−	0.229432i	0.993398	−	0.114716i	$-0.0365958\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−213.829			−0.276623			−0.138311	−	0.990389i	$-0.544167\pi$
$773$	−213.829			−0.276623			−0.138311	+	0.990389i	$0.544167\pi$
$774$		0			0
$775$	−734.681			−0.947975
$776$		0			0
$777$		0			0
$778$		0			0
$779$	185.076			0.237581
$780$		0			0
$781$	−1351.90			−1.73099
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	429.784i		−	0.547496i
$786$		0			0
$787$	118.300			0.150318			0.0751589	−	0.997172i	$-0.476054\pi$
$787$	118.300			0.150318			0.0751589	+	0.997172i	$0.476054\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	50.2263	+	27.0848i	0.0634973	+	0.0342413i
$792$		0			0
$793$	1072.99			1.35308
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1102.38			−1.38316			−0.691578	−	0.722302i	$-0.743084\pi$
$797$	−1102.38			−1.38316			−0.691578	+	0.722302i	$0.743084\pi$
$798$		0			0
$799$		−	646.650i		−	0.809325i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			2522.80i			3.14171i
$804$		0			0
$805$	944.223	−	1750.97i	1.17295	−	2.17512i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	319.116i		−	0.394458i	−0.980357	−	0.197229i	$-0.936806\pi$
$809$		−	319.116i		−	0.394458i	0.980357	−	0.197229i	$-0.0631942\pi$
$810$		0			0
$811$	1376.15			1.69686			0.848430	−	0.529307i	$-0.177548\pi$
$811$	1376.15			1.69686			0.848430	+	0.529307i	$0.177548\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	326.004i		−	0.400005i
$816$		0			0
$817$		−	281.641i		−	0.344726i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			962.109i			1.17187i	0.810357	+	0.585937i	$0.199273\pi$
$821$			962.109i			1.17187i	−0.810357	+	0.585937i	$0.800727\pi$
$822$		0			0
$823$		−	1224.55i		−	1.48791i	−0.668228	−	0.743957i	$-0.732947\pi$
$823$		−	1224.55i		−	1.48791i	0.668228	−	0.743957i	$-0.267053\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	443.864			0.536716			0.268358	−	0.963319i	$-0.413519\pi$
$827$	443.864			0.536716			0.268358	+	0.963319i	$0.413519\pi$
$828$		0			0
$829$		−	916.205i		−	1.10519i	−0.833449	−	0.552597i	$-0.813637\pi$
$829$		−	916.205i		−	1.10519i	0.833449	−	0.552597i	$-0.186363\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	360.311	+	547.938i	0.432546	+	0.657789i
$834$		0			0
$835$		−	2379.65i		−	2.84988i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1067.05i		−	1.27182i	−0.771765	−	0.635908i	$-0.780626\pi$
$839$		−	1067.05i		−	1.27182i	0.771765	−	0.635908i	$-0.219374\pi$
$840$		0			0
$841$	168.053			0.199825
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1432.85			1.69568
$846$		0			0
$847$	−634.534	+	1176.68i	−0.749154	+	1.38924i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−541.081			−0.635817
$852$		0			0
$853$			75.2815i			0.0882550i	0.999026	+	0.0441275i	$0.0140508\pi$
$853$			75.2815i			0.0882550i	−0.999026	+	0.0441275i	$0.985949\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1247.35			1.45549			0.727744	−	0.685849i	$-0.240569\pi$
$857$	1247.35			1.45549			0.727744	+	0.685849i	$0.240569\pi$
$858$		0			0
$859$	946.808			1.10222			0.551111	−	0.834432i	$-0.314204\pi$
$859$	946.808			1.10222			0.551111	+	0.834432i	$0.314204\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−664.907			−0.770460			−0.385230	−	0.922821i	$-0.625878\pi$
$863$	−664.907			−0.770460			−0.385230	+	0.922821i	$0.625878\pi$
$864$		0			0
$865$	−1056.34			−1.22120
$866$		0			0
$867$		0			0
$868$		0			0
$869$			847.227i			0.974945i
$870$		0			0
$871$	−1881.45			−2.16011
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1207.49	−	2239.19i	1.37999	−	2.55907i
$876$		0			0
$877$	10.7335			0.0122389			0.00611945	−	0.999981i	$-0.498052\pi$
$877$	10.7335			0.0122389			0.00611945	+	0.999981i	$0.498052\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−947.934			−1.07597			−0.537987	−	0.842953i	$-0.680815\pi$
$881$	−947.934			−1.07597			−0.537987	+	0.842953i	$0.680815\pi$
$882$		0			0
$883$		−	1073.67i		−	1.21594i	−0.793960	−	0.607970i	$-0.791984\pi$
$883$		−	1073.67i		−	1.21594i	0.793960	−	0.607970i	$-0.208016\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			89.2166i			0.100582i	0.998735	+	0.0502912i	$0.0160149\pi$
$887$			89.2166i			0.100582i	−0.998735	+	0.0502912i	$0.983985\pi$
$888$		0			0
$889$	−1291.97	−	696.702i	−1.45328	−	0.783692i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	706.886i		−	0.791586i
$894$		0			0
$895$	2508.13			2.80238
$896$		0			0
$897$		0			0
$898$		0			0
$899$			299.622i			0.333284i
$900$		0			0
$901$			45.8243i			0.0508594i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	756.140i		−	0.835514i
$906$		0			0
$907$		−	249.600i		−	0.275193i	−0.990488	−	0.137597i	$-0.956062\pi$
$907$		−	249.600i		−	0.275193i	0.990488	−	0.137597i	$-0.0439377\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1059.12			1.16259			0.581293	−	0.813694i	$-0.302547\pi$
$911$	1059.12			1.16259			0.581293	+	0.813694i	$0.302547\pi$
$912$		0			0
$913$		−	1606.42i		−	1.75949i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	367.271	+	198.053i	0.400514	+	0.215979i
$918$		0			0
$919$			565.363i			0.615194i	0.951517	+	0.307597i	$0.0995248\pi$
$919$			565.363i			0.615194i	−0.951517	+	0.307597i	$0.900475\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1371.76i			1.48620i
$924$		0			0
$925$	−1140.00			−1.23243
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1100.73			−1.18485			−0.592427	−	0.805624i	$-0.701830\pi$
$929$	−1100.73			−1.18485			−0.592427	+	0.805624i	$0.701830\pi$
$930$		0			0
$931$	393.875	+	598.979i	0.423066	+	0.643372i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2225.21			−2.37990
$936$		0			0
$937$			216.886i			0.231468i	0.993280	+	0.115734i	$0.0369220\pi$
$937$			216.886i			0.231468i	−0.993280	+	0.115734i	$0.963078\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1217.64			1.29399			0.646994	−	0.762495i	$-0.276026\pi$
$941$	1217.64			1.29399			0.646994	+	0.762495i	$0.276026\pi$
$942$		0			0
$943$	−381.921			−0.405006
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1191.48			1.25816			0.629081	−	0.777340i	$-0.283431\pi$
$947$	1191.48			1.25816			0.629081	+	0.777340i	$0.283431\pi$
$948$		0			0
$949$	2559.87			2.69744
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	581.578i		−	0.610260i	−0.952311	−	0.305130i	$-0.901300\pi$
$953$		−	581.578i		−	0.610260i	0.952311	−	0.305130i	$-0.0986999\pi$
$954$		0			0
$955$	−565.045			−0.591670
$956$		0			0
$957$		0			0
$958$		0			0
$959$	961.036	+	518.244i	1.00212	+	0.540401i
$960$		0			0
$961$	−827.597			−0.861183
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1508.91			1.56364
$966$		0			0
$967$			1017.88i			1.05262i	0.850294	+	0.526309i	$0.176424\pi$
$967$			1017.88i			1.05262i	−0.850294	+	0.526309i	$0.823576\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	191.299i		−	0.197013i	−0.995136	−	0.0985063i	$-0.968594\pi$
$971$		−	191.299i		−	0.197013i	0.995136	−	0.0985063i	$-0.0314065\pi$
$972$		0			0
$973$	−822.085	+	1524.48i	−0.844898	+	1.56679i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			212.822i			0.217832i	0.994051	+	0.108916i	$0.0347380\pi$
$977$			212.822i			0.217832i	−0.994051	+	0.108916i	$0.965262\pi$
$978$		0			0
$979$	2166.60			2.21308
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	625.538i		−	0.636357i	−0.948031	−	0.318178i	$-0.896929\pi$
$983$		−	625.538i		−	0.636357i	0.948031	−	0.318178i	$-0.103071\pi$
$984$		0			0
$985$			409.965i			0.416208i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			581.192i			0.587656i
$990$		0			0
$991$		−	528.316i		−	0.533114i	−0.963819	−	0.266557i	$-0.914114\pi$
$991$		−	528.316i		−	0.533114i	0.963819	−	0.266557i	$-0.0858861\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	254.765			0.256046
$996$		0			0
$997$			164.192i			0.164687i	0.996604	+	0.0823433i	$0.0262404\pi$
$997$			164.192i			0.164687i	−0.996604	+	0.0823433i	$0.973760\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.o.b.1007.4	yes	24
3.2	odd	2	inner	1008.3.o.b.1007.24	yes	24
4.3	odd	2	inner	1008.3.o.b.1007.2	yes	24
7.6	odd	2	inner	1008.3.o.b.1007.23	yes	24
12.11	even	2	inner	1008.3.o.b.1007.22	yes	24
21.20	even	2	inner	1008.3.o.b.1007.3	yes	24
28.27	even	2	inner	1008.3.o.b.1007.21	yes	24
84.83	odd	2	inner	1008.3.o.b.1007.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.o.b.1007.1	✓	24	84.83	odd	2	inner
1008.3.o.b.1007.2	yes	24	4.3	odd	2	inner
1008.3.o.b.1007.3	yes	24	21.20	even	2	inner
1008.3.o.b.1007.4	yes	24	1.1	even	1	trivial
1008.3.o.b.1007.21	yes	24	28.27	even	2	inner
1008.3.o.b.1007.22	yes	24	12.11	even	2	inner
1008.3.o.b.1007.23	yes	24	7.6	odd	2	inner
1008.3.o.b.1007.24	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.41321 q^{5} +(-3.32249 + 6.16125i) q^{7} -17.6630 q^{11} +17.9225i q^{13} -13.3835 q^{17} -14.6302 q^{19} +30.1907 q^{23} +63.6085 q^{25} -25.9412i q^{29} -11.5500 q^{31} +(31.2753 - 57.9972i) q^{35} -17.9221 q^{37} -12.6503 q^{41} +19.2507i q^{43} +48.3171i q^{47} +(-26.9221 - 40.9414i) q^{49} -3.42395i q^{53} +166.265 q^{55} -53.9239i q^{59} -59.8685i q^{61} -168.709i q^{65} +104.977i q^{67} +76.5385 q^{71} -142.830i q^{73} +(58.6851 - 108.826i) q^{77} -47.9662i q^{79} +90.9483i q^{83} +125.981 q^{85} -122.663 q^{89} +(-110.425 - 59.5474i) q^{91} +137.717 q^{95} +94.7826i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 264 q^{25} - 96 q^{37} - 312 q^{49} - 96 q^{85}+O(q^{100}) \) 24 * q + 264 * q^25 - 96 * q^37 - 312 * q^49 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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