LMFDB - Embedded newform 1584.3.j.h.1297.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(1297,1584)] 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("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.4
Root			$0.500000 - 3.07253i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.h.1297.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.69042 q^{5} +4.24264i q^{7} +(4.69042 + 9.94987i) q^{11} -12.7279i q^{13} -19.8997i q^{17} -33.9411i q^{19} -32.8329 q^{23} -3.00000 q^{25} -39.7995i q^{29} +10.0000 q^{31} +19.8997i q^{35} +2.00000 q^{37} -39.7995i q^{41} +16.9706i q^{43} -32.8329 q^{47} +31.0000 q^{49} +89.1179 q^{53} +(22.0000 + 46.6690i) q^{55} +65.6658 q^{59} -29.6985i q^{61} -59.6992i q^{65} -38.0000 q^{67} +79.7371 q^{71} +110.309i q^{73} +(-42.2137 + 19.8997i) q^{77} -55.1543i q^{79} -39.7995i q^{83} -93.3381i q^{85} +103.189 q^{89} +54.0000 q^{91} -159.198i q^{95} -112.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{25} + 40 q^{31} + 8 q^{37} + 124 q^{49} + 88 q^{55} - 152 q^{67} + 216 q^{91} - 448 q^{97}+O(q^{100}) $ 4 * q - 12 * q^25 + 40 * q^31 + 8 * q^37 + 124 * q^49 + 88 * q^55 - 152 * q^67 + 216 * q^91 - 448 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	4.69042			0.938083			0.469042	−	0.883176i	$-0.344599\pi$
$5$	4.69042			0.938083			0.469042	+	0.883176i	$0.344599\pi$
$6$		0			0
$7$			4.24264i			0.606092i	0.952976	+	0.303046i	$0.0980035\pi$
$7$			4.24264i			0.606092i	−0.952976	+	0.303046i	$0.901996\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.69042	+	9.94987i	0.426401	+	0.904534i
$11$	4.69042	+	9.94987i	0.426401	+	0.904534i
$12$		0			0
$13$		−	12.7279i		−	0.979071i	−0.871983	−	0.489535i	$-0.837166\pi$
$13$		−	12.7279i		−	0.979071i	0.871983	−	0.489535i	$-0.162834\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	19.8997i		−	1.17057i	−0.810826	−	0.585287i	$-0.800982\pi$
$17$		−	19.8997i		−	1.17057i	0.810826	−	0.585287i	$-0.199018\pi$
$18$		0			0
$19$		−	33.9411i		−	1.78638i	−0.449684	−	0.893188i	$-0.648464\pi$
$19$		−	33.9411i		−	1.78638i	0.449684	−	0.893188i	$-0.351536\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32.8329			−1.42752			−0.713759	−	0.700391i	$-0.753009\pi$
$23$	−32.8329			−1.42752			−0.713759	+	0.700391i	$0.753009\pi$
$24$		0			0
$25$	−3.00000			−0.120000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	39.7995i		−	1.37240i	−0.727415	−	0.686198i	$-0.759278\pi$
$29$		−	39.7995i		−	1.37240i	0.727415	−	0.686198i	$-0.240722\pi$
$30$		0			0
$31$	10.0000			0.322581			0.161290	−	0.986907i	$-0.448434\pi$
$31$	10.0000			0.322581			0.161290	+	0.986907i	$0.448434\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			19.8997i			0.568564i
$36$		0			0
$37$	2.00000			0.0540541			0.0270270	−	0.999635i	$-0.491396\pi$
$37$	2.00000			0.0540541			0.0270270	+	0.999635i	$0.491396\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	39.7995i		−	0.970719i	−0.874315	−	0.485360i	$-0.838689\pi$
$41$		−	39.7995i		−	0.970719i	0.874315	−	0.485360i	$-0.161311\pi$
$42$		0			0
$43$			16.9706i			0.394664i	0.980337	+	0.197332i	$0.0632277\pi$
$43$			16.9706i			0.394664i	−0.980337	+	0.197332i	$0.936772\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.8329			−0.698573			−0.349286	−	0.937016i	$-0.613576\pi$
$47$	−32.8329			−0.698573			−0.349286	+	0.937016i	$0.613576\pi$
$48$		0			0
$49$	31.0000			0.632653
$50$		0			0
$51$		0			0
$52$		0			0
$53$	89.1179			1.68147			0.840735	−	0.541447i	$-0.182123\pi$
$53$	89.1179			1.68147			0.840735	+	0.541447i	$0.182123\pi$
$54$		0			0
$55$	22.0000	+	46.6690i	0.400000	+	0.848528i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	65.6658			1.11298			0.556490	−	0.830854i	$-0.312148\pi$
$59$	65.6658			1.11298			0.556490	+	0.830854i	$0.312148\pi$
$60$		0			0
$61$		−	29.6985i		−	0.486860i	−0.969918	−	0.243430i	$-0.921727\pi$
$61$		−	29.6985i		−	0.486860i	0.969918	−	0.243430i	$-0.0782727\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	59.6992i		−	0.918450i
$66$		0			0
$67$	−38.0000			−0.567164			−0.283582	−	0.958948i	$-0.591523\pi$
$67$	−38.0000			−0.567164			−0.283582	+	0.958948i	$0.591523\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	79.7371			1.12306			0.561529	−	0.827457i	$-0.310213\pi$
$71$	79.7371			1.12306			0.561529	+	0.827457i	$0.310213\pi$
$72$		0			0
$73$			110.309i			1.51108i	0.655104	+	0.755539i	$0.272625\pi$
$73$			110.309i			1.51108i	−0.655104	+	0.755539i	$0.727375\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−42.2137	+	19.8997i	−0.548230	+	0.258438i
$78$		0			0
$79$		−	55.1543i		−	0.698156i	−0.937094	−	0.349078i	$-0.886495\pi$
$79$		−	55.1543i		−	0.698156i	0.937094	−	0.349078i	$-0.113505\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	39.7995i		−	0.479512i	−0.970833	−	0.239756i	$-0.922933\pi$
$83$		−	39.7995i		−	0.479512i	0.970833	−	0.239756i	$-0.0770674\pi$
$84$		0			0
$85$		−	93.3381i		−	1.09810i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	103.189			1.15943			0.579714	−	0.814820i	$-0.303164\pi$
$89$	103.189			1.15943			0.579714	+	0.814820i	$0.303164\pi$
$90$		0			0
$91$	54.0000			0.593407
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	159.198i		−	1.67577i
$96$		0			0
$97$	−112.000			−1.15464			−0.577320	−	0.816518i	$-0.695901\pi$
$97$	−112.000			−1.15464			−0.577320	+	0.816518i	$0.695901\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−26.0000			−0.252427			−0.126214	−	0.992003i	$-0.540282\pi$
$103$	−26.0000			−0.252427			−0.126214	+	0.992003i	$0.540282\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	139.298i		−	1.30185i	−0.759141	−	0.650926i	$-0.774381\pi$
$107$		−	139.298i		−	1.30185i	0.759141	−	0.650926i	$-0.225619\pi$
$108$		0			0
$109$			80.6102i			0.739543i	0.929123	+	0.369771i	$0.120564\pi$
$109$			80.6102i			0.739543i	−0.929123	+	0.369771i	$0.879436\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−65.6658			−0.581113			−0.290557	−	0.956858i	$-0.593840\pi$
$113$	−65.6658			−0.581113			−0.290557	+	0.956858i	$0.593840\pi$
$114$		0			0
$115$	−154.000			−1.33913
$116$		0			0
$117$		0			0
$118$		0			0
$119$	84.4275			0.709475
$120$		0			0
$121$	−77.0000	+	93.3381i	−0.636364	+	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−131.332			−1.05065
$126$		0			0
$127$			72.1249i			0.567913i	0.958837	+	0.283956i	$0.0916471\pi$
$127$			72.1249i			0.567913i	−0.958837	+	0.283956i	$0.908353\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	198.997i		−	1.51906i	−0.650469	−	0.759532i	$-0.725428\pi$
$131$		−	198.997i		−	1.51906i	0.650469	−	0.759532i	$-0.274572\pi$
$132$		0			0
$133$	144.000			1.08271
$134$		0			0
$135$		0			0
$136$		0			0
$137$	75.0467			0.547786			0.273893	−	0.961760i	$-0.411689\pi$
$137$	75.0467			0.547786			0.273893	+	0.961760i	$0.411689\pi$
$138$		0			0
$139$			161.220i			1.15986i	0.814667	+	0.579929i	$0.196920\pi$
$139$			161.220i			1.15986i	−0.814667	+	0.579929i	$0.803080\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	126.641	−	59.6992i	0.885603	−	0.417477i
$144$		0			0
$145$		−	186.676i		−	1.28742i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			79.5990i			0.534221i	0.963666	+	0.267111i	$0.0860689\pi$
$149$			79.5990i			0.534221i	−0.963666	+	0.267111i	$0.913931\pi$
$150$		0			0
$151$		−	89.0955i		−	0.590036i	−0.955492	−	0.295018i	$-0.904674\pi$
$151$		−	89.0955i		−	0.590036i	0.955492	−	0.295018i	$-0.0953257\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	46.9042			0.302607
$156$		0			0
$157$	−70.0000			−0.445860			−0.222930	−	0.974834i	$-0.571562\pi$
$157$	−70.0000			−0.445860			−0.222930	+	0.974834i	$0.571562\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	139.298i		−	0.865206i
$162$		0			0
$163$	196.000			1.20245			0.601227	−	0.799078i	$-0.294679\pi$
$163$	196.000			1.20245			0.601227	+	0.799078i	$0.294679\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	278.596i		−	1.66824i	−0.551581	−	0.834121i	$-0.685975\pi$
$167$		−	278.596i		−	1.66824i	0.551581	−	0.834121i	$-0.314025\pi$
$168$		0			0
$169$	7.00000			0.0414201
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	79.5990i		−	0.460110i	−0.973178	−	0.230055i	$-0.926109\pi$
$173$		−	79.5990i		−	0.460110i	0.973178	−	0.230055i	$-0.0738906\pi$
$174$		0			0
$175$		−	12.7279i		−	0.0727310i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.38083			0.0524069			0.0262034	−	0.999657i	$-0.491658\pi$
$179$	9.38083			0.0524069			0.0262034	+	0.999657i	$0.491658\pi$
$180$		0			0
$181$	302.000			1.66851			0.834254	−	0.551380i	$-0.185899\pi$
$181$	302.000			1.66851			0.834254	+	0.551380i	$0.185899\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.38083			0.0507072
$186$		0			0
$187$	198.000	−	93.3381i	1.05882	−	0.499134i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−229.830			−1.20330			−0.601650	−	0.798760i	$-0.705490\pi$
$191$	−229.830			−1.20330			−0.601650	+	0.798760i	$0.705490\pi$
$192$		0			0
$193$			305.470i			1.58275i	0.611333	+	0.791373i	$0.290634\pi$
$193$			305.470i			1.58275i	−0.611333	+	0.791373i	$0.709366\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−182.000			−0.914573			−0.457286	−	0.889319i	$-0.651179\pi$
$199$	−182.000			−0.914573			−0.457286	+	0.889319i	$0.651179\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	168.855			0.831798
$204$		0			0
$205$		−	186.676i		−	0.910616i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	337.710	−	159.198i	1.61584	−	0.761713i
$210$		0			0
$211$		−	212.132i		−	1.00537i	−0.864471	−	0.502683i	$-0.832346\pi$
$211$		−	212.132i		−	1.00537i	0.864471	−	0.502683i	$-0.167654\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			79.5990i			0.370228i
$216$		0			0
$217$			42.4264i			0.195513i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−253.282			−1.14607
$222$		0			0
$223$	250.000			1.12108			0.560538	−	0.828129i	$-0.310594\pi$
$223$	250.000			1.12108			0.560538	+	0.828129i	$0.310594\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			139.298i			0.613649i	0.951766	+	0.306824i	$0.0992664\pi$
$227$			139.298i			0.613649i	−0.951766	+	0.306824i	$0.900734\pi$
$228$		0			0
$229$	−418.000			−1.82533			−0.912664	−	0.408711i	$-0.865978\pi$
$229$	−418.000			−1.82533			−0.912664	+	0.408711i	$0.865978\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	358.195i		−	1.53732i	−0.639658	−	0.768660i	$-0.720924\pi$
$233$		−	358.195i		−	1.53732i	0.639658	−	0.768660i	$-0.279076\pi$
$234$		0			0
$235$	−154.000			−0.655319
$236$		0			0
$237$		0			0
$238$		0			0
$239$			39.7995i			0.166525i	0.996528	+	0.0832625i	$0.0265340\pi$
$239$			39.7995i			0.166525i	−0.996528	+	0.0832625i	$0.973466\pi$
$240$		0			0
$241$		−	356.382i		−	1.47876i	−0.673287	−	0.739381i	$-0.735118\pi$
$241$		−	356.382i		−	1.47876i	0.673287	−	0.739381i	$-0.264882\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	145.403			0.593481
$246$		0			0
$247$	−432.000			−1.74899
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.7617			−0.0747477			−0.0373738	−	0.999301i	$-0.511899\pi$
$251$	−18.7617			−0.0747477			−0.0373738	+	0.999301i	$0.511899\pi$
$252$		0			0
$253$	−154.000	−	326.683i	−0.608696	−	1.29124i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	103.189			0.401514			0.200757	−	0.979641i	$-0.435660\pi$
$257$	103.189			0.401514			0.200757	+	0.979641i	$0.435660\pi$
$258$		0			0
$259$			8.48528i			0.0327617i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			397.995i			1.51329i	0.653827	+	0.756644i	$0.273163\pi$
$263$			397.995i			1.51329i	−0.653827	+	0.756644i	$0.726837\pi$
$264$		0			0
$265$	418.000			1.57736
$266$		0			0
$267$		0			0
$268$		0			0
$269$	426.828			1.58672			0.793360	−	0.608752i	$-0.208330\pi$
$269$	426.828			1.58672			0.793360	+	0.608752i	$0.208330\pi$
$270$		0			0
$271$		−	207.889i		−	0.767120i	−0.923516	−	0.383560i	$-0.874698\pi$
$271$		−	207.889i		−	0.767120i	0.923516	−	0.383560i	$-0.125302\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.0712	−	29.8496i	−0.0511682	−	0.108544i
$276$		0			0
$277$			318.198i			1.14873i	0.818600	+	0.574365i	$0.194751\pi$
$277$			318.198i			1.14873i	−0.818600	+	0.574365i	$0.805249\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	417.895i		−	1.48717i	−0.668642	−	0.743585i	$-0.733124\pi$
$281$		−	417.895i		−	1.48717i	0.668642	−	0.743585i	$-0.266876\pi$
$282$		0			0
$283$			364.867i			1.28928i	0.764485	+	0.644642i	$0.222993\pi$
$283$			364.867i			1.28928i	−0.764485	+	0.644642i	$0.777007\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	168.855			0.588345
$288$		0			0
$289$	−107.000			−0.370242
$290$		0			0
$291$		0			0
$292$		0			0
$293$			278.596i			0.950841i	0.879759	+	0.475421i	$0.157704\pi$
$293$			278.596i			0.950841i	−0.879759	+	0.475421i	$0.842296\pi$
$294$		0			0
$295$	308.000			1.04407
$296$		0			0
$297$		0			0
$298$		0			0
$299$			417.895i			1.39764i
$300$		0			0
$301$	−72.0000			−0.239203
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	139.298i		−	0.456716i
$306$		0			0
$307$		−	42.4264i		−	0.138197i	−0.997610	−	0.0690984i	$-0.977988\pi$
$307$		−	42.4264i		−	0.138197i	0.997610	−	0.0690984i	$-0.0220122\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	23.4521			0.0754086			0.0377043	−	0.999289i	$-0.487996\pi$
$311$	23.4521			0.0754086			0.0377043	+	0.999289i	$0.487996\pi$
$312$		0			0
$313$	506.000			1.61661			0.808307	−	0.588762i	$-0.200384\pi$
$313$	506.000			1.61661			0.808307	+	0.588762i	$0.200384\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−333.020			−1.05053			−0.525267	−	0.850937i	$-0.676035\pi$
$317$	−333.020			−1.05053			−0.525267	+	0.850937i	$0.676035\pi$
$318$		0			0
$319$	396.000	−	186.676i	1.24138	−	0.585192i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−675.420			−2.09108
$324$		0			0
$325$			38.1838i			0.117489i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	139.298i		−	0.423399i
$330$		0			0
$331$	−296.000			−0.894260			−0.447130	−	0.894469i	$-0.647554\pi$
$331$	−296.000			−0.894260			−0.447130	+	0.894469i	$0.647554\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−178.236			−0.532047
$336$		0			0
$337$			76.3675i			0.226610i	0.993560	+	0.113305i	$0.0361437\pi$
$337$			76.3675i			0.226610i	−0.993560	+	0.113305i	$0.963856\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	46.9042	+	99.4987i	0.137549	+	0.291785i
$342$		0			0
$343$			339.411i			0.989537i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	437.794i		−	1.26166i	−0.775923	−	0.630828i	$-0.782715\pi$
$347$		−	437.794i		−	1.26166i	0.775923	−	0.630828i	$-0.217285\pi$
$348$		0			0
$349$			292.742i			0.838803i	0.907801	+	0.419401i	$0.137760\pi$
$349$			292.742i			0.838803i	−0.907801	+	0.419401i	$0.862240\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	131.332			0.372044			0.186022	−	0.982546i	$-0.440440\pi$
$353$	131.332			0.372044			0.186022	+	0.982546i	$0.440440\pi$
$354$		0			0
$355$	374.000			1.05352
$356$		0			0
$357$		0			0
$358$		0			0
$359$			278.596i			0.776035i	0.921652	+	0.388017i	$0.126840\pi$
$359$			278.596i			0.776035i	−0.921652	+	0.388017i	$0.873160\pi$
$360$		0			0
$361$	−791.000			−2.19114
$362$		0			0
$363$		0			0
$364$		0			0
$365$			517.393i			1.41752i
$366$		0			0
$367$	−206.000			−0.561308			−0.280654	−	0.959809i	$-0.590551\pi$
$367$	−206.000			−0.561308			−0.280654	+	0.959809i	$0.590551\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			378.095i			1.01912i
$372$		0			0
$373$			657.609i			1.76303i	0.472158	+	0.881514i	$0.343475\pi$
$373$			657.609i			1.76303i	−0.472158	+	0.881514i	$0.656525\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−506.565			−1.34367
$378$		0			0
$379$	−692.000			−1.82586			−0.912929	−	0.408119i	$-0.866185\pi$
$379$	−692.000			−1.82586			−0.912929	+	0.408119i	$0.866185\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	333.020			0.869503			0.434751	−	0.900551i	$-0.356836\pi$
$383$	333.020			0.869503			0.434751	+	0.900551i	$0.356836\pi$
$384$		0			0
$385$	−198.000	+	93.3381i	−0.514286	+	0.242437i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−558.159			−1.43486			−0.717429	−	0.696632i	$-0.754681\pi$
$389$	−558.159			−1.43486			−0.717429	+	0.696632i	$0.754681\pi$
$390$		0			0
$391$			653.367i			1.67101i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	258.697i		−	0.654928i
$396$		0			0
$397$	362.000			0.911839			0.455919	−	0.890021i	$-0.349311\pi$
$397$	362.000			0.911839			0.455919	+	0.890021i	$0.349311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−65.6658			−0.163755			−0.0818776	−	0.996642i	$-0.526092\pi$
$401$	−65.6658			−0.163755			−0.0818776	+	0.996642i	$0.526092\pi$
$402$		0			0
$403$		−	127.279i		−	0.315829i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.38083	+	19.8997i	0.0230487	+	0.0488937i
$408$		0			0
$409$		−	585.484i		−	1.43150i	−0.698355	−	0.715751i	$-0.746085\pi$
$409$		−	585.484i		−	1.43150i	0.698355	−	0.715751i	$-0.253915\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			278.596i			0.674568i
$414$		0			0
$415$		−	186.676i		−	0.449822i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	150.093			0.358218			0.179109	−	0.983829i	$-0.442679\pi$
$419$	150.093			0.358218			0.179109	+	0.983829i	$0.442679\pi$
$420$		0			0
$421$	−22.0000			−0.0522565			−0.0261283	−	0.999659i	$-0.508318\pi$
$421$	−22.0000			−0.0522565			−0.0261283	+	0.999659i	$0.508318\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			59.6992i			0.140469i
$426$		0			0
$427$	126.000			0.295082
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	278.596i		−	0.646396i	−0.946331	−	0.323198i	$-0.895242\pi$
$431$		−	278.596i		−	0.646396i	0.946331	−	0.323198i	$-0.104758\pi$
$432$		0			0
$433$	62.0000			0.143187			0.0715935	−	0.997434i	$-0.477192\pi$
$433$	62.0000			0.143187			0.0715935	+	0.997434i	$0.477192\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1114.39i			2.55008i
$438$		0			0
$439$			89.0955i			0.202951i	0.994838	+	0.101475i	$0.0323563\pi$
$439$			89.0955i			0.202951i	−0.994838	+	0.101475i	$0.967644\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−750.467			−1.69406			−0.847028	−	0.531549i	$-0.821610\pi$
$443$	−750.467			−1.69406			−0.847028	+	0.531549i	$0.821610\pi$
$444$		0			0
$445$	484.000			1.08764
$446$		0			0
$447$		0			0
$448$		0			0
$449$	637.897			1.42070			0.710352	−	0.703846i	$-0.248536\pi$
$449$	637.897			1.42070			0.710352	+	0.703846i	$0.248536\pi$
$450$		0			0
$451$	396.000	−	186.676i	0.878049	−	0.413916i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	253.282			0.556665
$456$		0			0
$457$		−	381.838i		−	0.835531i	−0.908555	−	0.417765i	$-0.862813\pi$
$457$		−	381.838i		−	0.835531i	0.908555	−	0.417765i	$-0.137187\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			835.789i			1.81299i	0.422214	+	0.906496i	$0.361253\pi$
$461$			835.789i			1.81299i	−0.422214	+	0.906496i	$0.638747\pi$
$462$		0			0
$463$	−482.000			−1.04104			−0.520518	−	0.853850i	$-0.674261\pi$
$463$	−482.000			−1.04104			−0.520518	+	0.853850i	$0.674261\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	684.801			1.46638			0.733191	−	0.680022i	$-0.238030\pi$
$467$	684.801			1.46638			0.733191	+	0.680022i	$0.238030\pi$
$468$		0			0
$469$		−	161.220i		−	0.343753i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−168.855	+	79.5990i	−0.356987	+	0.168285i
$474$		0			0
$475$			101.823i			0.214365i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			397.995i			0.830887i	0.909619	+	0.415444i	$0.136374\pi$
$479$			397.995i			0.830887i	−0.909619	+	0.415444i	$0.863626\pi$
$480$		0			0
$481$		−	25.4558i		−	0.0529228i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−525.327			−1.08315
$486$		0			0
$487$	682.000			1.40041			0.700205	−	0.713942i	$-0.253092\pi$
$487$	682.000			1.40041			0.700205	+	0.713942i	$0.253092\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	179.098i		−	0.364761i	−0.983228	−	0.182381i	$-0.941620\pi$
$491$		−	179.098i		−	0.364761i	0.983228	−	0.182381i	$-0.0583803\pi$
$492$		0			0
$493$	−792.000			−1.60649
$494$		0			0
$495$		0			0
$496$		0			0
$497$			338.296i			0.680676i
$498$		0			0
$499$	−152.000			−0.304609			−0.152305	−	0.988334i	$-0.548669\pi$
$499$	−152.000			−0.304609			−0.152305	+	0.988334i	$0.548669\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	39.7995i		−	0.0791242i	−0.999217	−	0.0395621i	$-0.987404\pi$
$503$		−	39.7995i		−	0.0791242i	0.999217	−	0.0395621i	$-0.0125963\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−558.159			−1.09658			−0.548290	−	0.836288i	$-0.684721\pi$
$509$	−558.159			−1.09658			−0.548290	+	0.836288i	$0.684721\pi$
$510$		0			0
$511$	−468.000			−0.915851
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−121.951			−0.236798
$516$		0			0
$517$	−154.000	−	326.683i	−0.297872	−	0.631883i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−121.951			−0.234071			−0.117035	−	0.993128i	$-0.537339\pi$
$521$	−121.951			−0.234071			−0.117035	+	0.993128i	$0.537339\pi$
$522$		0			0
$523$		−	916.410i		−	1.75222i	−0.482112	−	0.876109i	$-0.660130\pi$
$523$		−	916.410i		−	1.75222i	0.482112	−	0.876109i	$-0.339870\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	198.997i		−	0.377604i
$528$		0			0
$529$	549.000			1.03781
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−506.565			−0.950403
$534$		0			0
$535$		−	653.367i		−	1.22125i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	145.403	+	308.446i	0.269764	+	0.572256i
$540$		0			0
$541$			946.109i			1.74881i	0.485192	+	0.874407i	$0.338750\pi$
$541$			946.109i			1.74881i	−0.485192	+	0.874407i	$0.661250\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			378.095i			0.693753i
$546$		0			0
$547$		−	237.588i		−	0.434347i	−0.976133	−	0.217174i	$-0.930316\pi$
$547$		−	237.588i		−	0.434347i	0.976133	−	0.217174i	$-0.0696838\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1350.84			−2.45161
$552$		0			0
$553$	234.000			0.423146
$554$		0			0
$555$		0			0
$556$		0			0
$557$			676.591i			1.21471i	0.794432	+	0.607353i	$0.207769\pi$
$557$			676.591i			1.21471i	−0.794432	+	0.607353i	$0.792231\pi$
$558$		0			0
$559$	216.000			0.386404
$560$		0			0
$561$		0			0
$562$		0			0
$563$			417.895i			0.742264i	0.928580	+	0.371132i	$0.121030\pi$
$563$			417.895i			0.742264i	−0.928580	+	0.371132i	$0.878970\pi$
$564$		0			0
$565$	−308.000			−0.545133
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	417.895i		−	0.734437i	−0.930135	−	0.367219i	$-0.880310\pi$
$569$		−	417.895i		−	0.734437i	0.930135	−	0.367219i	$-0.119690\pi$
$570$		0			0
$571$			704.278i			1.23341i	0.787193	+	0.616706i	$0.211533\pi$
$571$			704.278i			1.23341i	−0.787193	+	0.616706i	$0.788467\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	98.4987			0.171302
$576$		0			0
$577$	536.000			0.928943			0.464471	−	0.885588i	$-0.346244\pi$
$577$	536.000			0.928943			0.464471	+	0.885588i	$0.346244\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	168.855			0.290628
$582$		0			0
$583$	418.000	+	886.712i	0.716981	+	1.52095i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	853.656			1.45427			0.727134	−	0.686495i	$-0.240852\pi$
$587$	853.656			1.45427			0.727134	+	0.686495i	$0.240852\pi$
$588$		0			0
$589$		−	339.411i		−	0.576250i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	696.491i		−	1.17452i	−0.809398	−	0.587261i	$-0.800206\pi$
$593$		−	696.491i		−	1.17452i	0.809398	−	0.587261i	$-0.199794\pi$
$594$		0			0
$595$	396.000			0.665546
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−567.540			−0.947480			−0.473740	−	0.880665i	$-0.657096\pi$
$599$	−567.540			−0.947480			−0.473740	+	0.880665i	$0.657096\pi$
$600$		0			0
$601$		−	42.4264i		−	0.0705930i	−0.999377	−	0.0352965i	$-0.988762\pi$
$601$		−	42.4264i		−	0.0705930i	0.999377	−	0.0352965i	$-0.0112376\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−361.162	+	437.794i	−0.596962	+	0.723627i
$606$		0			0
$607$		−	292.742i		−	0.482277i	−0.970491	−	0.241139i	$-0.922479\pi$
$607$		−	292.742i		−	0.482277i	0.970491	−	0.241139i	$-0.0775208\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			417.895i			0.683952i
$612$		0			0
$613$		−	801.859i		−	1.30809i	−0.756456	−	0.654045i	$-0.773071\pi$
$613$		−	801.859i		−	1.30809i	0.756456	−	0.654045i	$-0.226929\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	609.754			0.988256			0.494128	−	0.869389i	$-0.335487\pi$
$617$	609.754			0.988256			0.494128	+	0.869389i	$0.335487\pi$
$618$		0			0
$619$	346.000			0.558966			0.279483	−	0.960151i	$-0.409837\pi$
$619$	346.000			0.558966			0.279483	+	0.960151i	$0.409837\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			437.794i			0.702720i
$624$		0			0
$625$	−541.000			−0.865600
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	39.7995i		−	0.0632742i
$630$		0			0
$631$	−770.000			−1.22029			−0.610143	−	0.792292i	$-0.708888\pi$
$631$	−770.000			−1.22029			−0.610143	+	0.792292i	$0.708888\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			338.296i			0.532749i
$636$		0			0
$637$		−	394.566i		−	0.619412i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	722.324			1.12687			0.563435	−	0.826160i	$-0.309479\pi$
$641$	722.324			1.12687			0.563435	+	0.826160i	$0.309479\pi$
$642$		0			0
$643$	−302.000			−0.469673			−0.234837	−	0.972035i	$-0.575456\pi$
$643$	−302.000			−0.469673			−0.234837	+	0.972035i	$0.575456\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−229.830			−0.355225			−0.177612	−	0.984101i	$-0.556837\pi$
$647$	−229.830			−0.355225			−0.177612	+	0.984101i	$0.556837\pi$
$648$		0			0
$649$	308.000	+	653.367i	0.474576	+	1.00673i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	60.9754			0.0933773			0.0466887	−	0.998909i	$-0.485133\pi$
$653$	60.9754			0.0933773			0.0466887	+	0.998909i	$0.485133\pi$
$654$		0			0
$655$		−	933.381i		−	1.42501i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1293.48i			1.96280i	0.191980	+	0.981399i	$0.438509\pi$
$659$			1293.48i			1.96280i	−0.191980	+	0.981399i	$0.561491\pi$
$660$		0			0
$661$	710.000			1.07413			0.537065	−	0.843541i	$-0.319533\pi$
$661$	710.000			1.07413			0.537065	+	0.843541i	$0.319533\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	675.420			1.01567
$666$		0			0
$667$			1306.73i			1.95912i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	295.496	−	139.298i	0.440382	−	0.207598i
$672$		0			0
$673$		−	161.220i		−	0.239555i	−0.992801	−	0.119777i	$-0.961782\pi$
$673$		−	161.220i		−	0.239555i	0.992801	−	0.119777i	$-0.0382181\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			835.789i			1.23455i	0.786748	+	0.617274i	$0.211763\pi$
$677$			835.789i			1.23455i	−0.786748	+	0.617274i	$0.788237\pi$
$678$		0			0
$679$		−	475.176i		−	0.699817i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−131.332			−0.192286			−0.0961432	−	0.995368i	$-0.530651\pi$
$683$	−131.332			−0.192286			−0.0961432	+	0.995368i	$0.530651\pi$
$684$		0			0
$685$	352.000			0.513869
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1134.29i		−	1.64628i
$690$		0			0
$691$	−530.000			−0.767004			−0.383502	−	0.923540i	$-0.625282\pi$
$691$	−530.000			−0.767004			−0.383502	+	0.923540i	$0.625282\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			756.190i			1.08804i
$696$		0			0
$697$	−792.000			−1.13630
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	795.990i		−	1.13551i	−0.823199	−	0.567753i	$-0.807813\pi$
$701$		−	795.990i		−	1.13551i	0.823199	−	0.567753i	$-0.192187\pi$
$702$		0			0
$703$		−	67.8823i		−	0.0965608i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−154.000			−0.217207			−0.108604	−	0.994085i	$-0.534638\pi$
$709$	−154.000			−0.217207			−0.108604	+	0.994085i	$0.534638\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−328.329			−0.460490
$714$		0			0
$715$	594.000	−	280.014i	0.830769	−	0.391628i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−623.825			−0.867629			−0.433815	−	0.901002i	$-0.642833\pi$
$719$	−623.825			−0.867629			−0.433815	+	0.901002i	$0.642833\pi$
$720$		0			0
$721$		−	110.309i		−	0.152994i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			119.398i			0.164688i
$726$		0			0
$727$	22.0000			0.0302613			0.0151307	−	0.999886i	$-0.495184\pi$
$727$	22.0000			0.0302613			0.0151307	+	0.999886i	$0.495184\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	337.710			0.461983
$732$		0			0
$733$			352.139i			0.480408i	0.970722	+	0.240204i	$0.0772144\pi$
$733$			352.139i			0.480408i	−0.970722	+	0.240204i	$0.922786\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−178.236	−	378.095i	−0.241840	−	0.513019i
$738$		0			0
$739$			347.897i			0.470767i	0.971903	+	0.235383i	$0.0756345\pi$
$739$			347.897i			0.470767i	−0.971903	+	0.235383i	$0.924365\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1154.19i			1.55341i	0.629863	+	0.776706i	$0.283111\pi$
$743$			1154.19i			1.55341i	−0.629863	+	0.776706i	$0.716889\pi$
$744$		0			0
$745$			373.352i			0.501144i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	590.992			0.789042
$750$		0			0
$751$	730.000			0.972037			0.486019	−	0.873948i	$-0.338449\pi$
$751$	730.000			0.972037			0.486019	+	0.873948i	$0.338449\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	417.895i		−	0.553503i
$756$		0			0
$757$	−826.000			−1.09115			−0.545575	−	0.838062i	$-0.683689\pi$
$757$	−826.000			−1.09115			−0.545575	+	0.838062i	$0.683689\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1054.69i		−	1.38592i	−0.720975	−	0.692961i	$-0.756306\pi$
$761$		−	1054.69i		−	1.38592i	0.720975	−	0.692961i	$-0.243694\pi$
$762$		0			0
$763$	−342.000			−0.448231
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	835.789i		−	1.08969i
$768$		0			0
$769$			967.322i			1.25790i	0.777447	+	0.628948i	$0.216514\pi$
$769$			967.322i			1.25790i	−0.777447	+	0.628948i	$0.783486\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−164.165			−0.212373			−0.106187	−	0.994346i	$-0.533864\pi$
$773$	−164.165			−0.212373			−0.106187	+	0.994346i	$0.533864\pi$
$774$		0			0
$775$	−30.0000			−0.0387097
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1350.84			−1.73407
$780$		0			0
$781$	374.000	+	793.374i	0.478873	+	1.01584i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−328.329			−0.418254
$786$		0			0
$787$			475.176i			0.603781i	0.953343	+	0.301891i	$0.0976177\pi$
$787$			475.176i			0.603781i	−0.953343	+	0.301891i	$0.902382\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	278.596i		−	0.352208i
$792$		0			0
$793$	−378.000			−0.476671
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−952.154			−1.19467			−0.597337	−	0.801991i	$-0.703774\pi$
$797$	−952.154			−1.19467			−0.597337	+	0.801991i	$0.703774\pi$
$798$		0			0
$799$			653.367i			0.817730i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1097.56	+	517.393i	−1.36682	+	0.644326i
$804$		0			0
$805$		−	653.367i		−	0.811636i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	835.789i		−	1.03311i	−0.856253	−	0.516557i	$-0.827213\pi$
$809$		−	835.789i		−	1.03311i	0.856253	−	0.516557i	$-0.172787\pi$
$810$		0			0
$811$			475.176i			0.585913i	0.956126	+	0.292957i	$0.0946392\pi$
$811$			475.176i			0.585913i	−0.956126	+	0.292957i	$0.905361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	919.321			1.12800
$816$		0			0
$817$	576.000			0.705018
$818$		0			0
$819$		0			0
$820$		0			0
$821$			79.5990i			0.0969537i	0.998824	+	0.0484769i	$0.0154367\pi$
$821$			79.5990i			0.0969537i	−0.998824	+	0.0484769i	$0.984563\pi$
$822$		0			0
$823$	−1274.00			−1.54800			−0.773998	−	0.633189i	$-0.781746\pi$
$823$	−1274.00			−1.54800			−0.773998	+	0.633189i	$0.781746\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1392.98i		−	1.68438i	−0.539181	−	0.842190i	$-0.681266\pi$
$827$		−	1392.98i		−	1.68438i	0.539181	−	0.842190i	$-0.318734\pi$
$828$		0			0
$829$	158.000			0.190591			0.0952955	−	0.995449i	$-0.469620\pi$
$829$	158.000			0.190591			0.0952955	+	0.995449i	$0.469620\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	616.892i		−	0.740567i
$834$		0			0
$835$		−	1306.73i		−	1.56495i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	248.592			0.296296			0.148148	−	0.988965i	$-0.452669\pi$
$839$	248.592			0.296296			0.148148	+	0.988965i	$0.452669\pi$
$840$		0			0
$841$	−743.000			−0.883472
$842$		0			0
$843$		0			0
$844$		0			0
$845$	32.8329			0.0388555
$846$		0			0
$847$	−396.000	−	326.683i	−0.467532	−	0.385695i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−65.6658			−0.0771631
$852$		0			0
$853$			309.713i			0.363086i	0.983383	+	0.181543i	$0.0581092\pi$
$853$			309.713i			0.363086i	−0.983383	+	0.181543i	$0.941891\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			835.789i			0.975250i	0.873053	+	0.487625i	$0.162137\pi$
$857$			835.789i			0.975250i	−0.873053	+	0.487625i	$0.837863\pi$
$858$		0			0
$859$	340.000			0.395809			0.197905	−	0.980221i	$-0.436586\pi$
$859$	340.000			0.395809			0.197905	+	0.980221i	$0.436586\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−595.683			−0.690247			−0.345123	−	0.938557i	$-0.612163\pi$
$863$	−595.683			−0.690247			−0.345123	+	0.938557i	$0.612163\pi$
$864$		0			0
$865$		−	373.352i		−	0.431621i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	548.779	−	258.697i	0.631506	−	0.297695i
$870$		0			0
$871$			483.661i			0.555294i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	557.193i		−	0.636792i
$876$		0			0
$877$		−	504.874i		−	0.575683i	−0.957678	−	0.287842i	$-0.907062\pi$
$877$		−	504.874i		−	0.575683i	0.957678	−	0.287842i	$-0.0929377\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−769.228			−0.873131			−0.436565	−	0.899673i	$-0.643805\pi$
$881$	−769.228			−0.873131			−0.436565	+	0.899673i	$0.643805\pi$
$882$		0			0
$883$	106.000			0.120045			0.0600227	−	0.998197i	$-0.480883\pi$
$883$	106.000			0.120045			0.0600227	+	0.998197i	$0.480883\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	278.596i		−	0.314088i	−0.987592	−	0.157044i	$-0.949803\pi$
$887$		−	278.596i		−	0.314088i	0.987592	−	0.157044i	$-0.0501965\pi$
$888$		0			0
$889$	−306.000			−0.344207
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1114.39i			1.24791i
$894$		0			0
$895$	44.0000			0.0491620
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	397.995i		−	0.442709i
$900$		0			0
$901$		−	1773.42i		−	1.96828i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1416.51			1.56520
$906$		0			0
$907$	94.0000			0.103638			0.0518192	−	0.998656i	$-0.483498\pi$
$907$	94.0000			0.103638			0.0518192	+	0.998656i	$0.483498\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1740.14			1.91015			0.955074	−	0.296368i	$-0.0957756\pi$
$911$	1740.14			1.91015			0.955074	+	0.296368i	$0.0957756\pi$
$912$		0			0
$913$	396.000	−	186.676i	0.433735	−	0.204465i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	844.275			0.920692
$918$		0			0
$919$			776.403i			0.844835i	0.906401	+	0.422417i	$0.138818\pi$
$919$			776.403i			0.844835i	−0.906401	+	0.422417i	$0.861182\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1014.89i		−	1.09955i
$924$		0			0
$925$	−6.00000			−0.00648649
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−93.8083			−0.100978			−0.0504889	−	0.998725i	$-0.516078\pi$
$929$	−93.8083			−0.100978			−0.0504889	+	0.998725i	$0.516078\pi$
$930$		0			0
$931$		−	1052.17i		−	1.13016i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	928.702	−	437.794i	0.993265	−	0.468229i
$936$		0			0
$937$			1204.91i			1.28592i	0.765898	+	0.642962i	$0.222295\pi$
$937$			1204.91i			1.28592i	−0.765898	+	0.642962i	$0.777705\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	278.596i		−	0.296064i	−0.988983	−	0.148032i	$-0.952706\pi$
$941$		−	278.596i		−	0.296064i	0.988983	−	0.148032i	$-0.0472939\pi$
$942$		0			0
$943$			1306.73i			1.38572i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1838.64			1.94154			0.970772	−	0.240002i	$-0.0771481\pi$
$947$	1838.64			1.94154			0.970772	+	0.240002i	$0.0771481\pi$
$948$		0			0
$949$	1404.00			1.47945
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	517.393i		−	0.542910i	−0.962451	−	0.271455i	$-0.912495\pi$
$953$		−	517.393i		−	0.542910i	0.962451	−	0.271455i	$-0.0875049\pi$
$954$		0			0
$955$	−1078.00			−1.12880
$956$		0			0
$957$		0			0
$958$		0			0
$959$			318.396i			0.332008i
$960$		0			0
$961$	−861.000			−0.895942
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1432.78i			1.48475i
$966$		0			0
$967$		−	666.095i		−	0.688826i	−0.938818	−	0.344413i	$-0.888078\pi$
$967$		−	666.095i		−	0.688826i	0.938818	−	0.344413i	$-0.111922\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1838.64			1.89356			0.946778	−	0.321887i	$-0.104317\pi$
$971$	1838.64			1.89356			0.946778	+	0.321887i	$0.104317\pi$
$972$		0			0
$973$	−684.000			−0.702980
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1191.37			−1.21941			−0.609706	−	0.792628i	$-0.708713\pi$
$977$	−1191.37			−1.21941			−0.609706	+	0.792628i	$0.708713\pi$
$978$		0			0
$979$	484.000	+	1026.72i	0.494382	+	1.04874i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	304.877			0.310150			0.155075	−	0.987903i	$-0.450438\pi$
$983$	304.877			0.310150			0.155075	+	0.987903i	$0.450438\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	557.193i		−	0.563390i
$990$		0			0
$991$	1474.00			1.48739			0.743693	−	0.668521i	$-0.233072\pi$
$991$	1474.00			1.48739			0.743693	+	0.668521i	$0.233072\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−853.656			−0.857945
$996$		0			0
$997$			1463.71i			1.46812i	0.679087	+	0.734058i	$0.262376\pi$
$997$			1463.71i			1.46812i	−0.679087	+	0.734058i	$0.737624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.h.1297.4		4
3.2	odd	2	inner	1584.3.j.h.1297.2		4
4.3	odd	2		396.3.f.c.109.3	yes	4
11.10	odd	2	inner	1584.3.j.h.1297.3		4
12.11	even	2		396.3.f.c.109.1	✓	4
33.32	even	2	inner	1584.3.j.h.1297.1		4
44.43	even	2		396.3.f.c.109.4	yes	4
132.131	odd	2		396.3.f.c.109.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.3.f.c.109.1	✓	4	12.11	even	2
396.3.f.c.109.2	yes	4	132.131	odd	2
396.3.f.c.109.3	yes	4	4.3	odd	2
396.3.f.c.109.4	yes	4	44.43	even	2
1584.3.j.h.1297.1		4	33.32	even	2	inner
1584.3.j.h.1297.2		4	3.2	odd	2	inner
1584.3.j.h.1297.3		4	11.10	odd	2	inner
1584.3.j.h.1297.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.69042 q^{5} +4.24264i q^{7} +(4.69042 + 9.94987i) q^{11} -12.7279i q^{13} -19.8997i q^{17} -33.9411i q^{19} -32.8329 q^{23} -3.00000 q^{25} -39.7995i q^{29} +10.0000 q^{31} +19.8997i q^{35} +2.00000 q^{37} -39.7995i q^{41} +16.9706i q^{43} -32.8329 q^{47} +31.0000 q^{49} +89.1179 q^{53} +(22.0000 + 46.6690i) q^{55} +65.6658 q^{59} -29.6985i q^{61} -59.6992i q^{65} -38.0000 q^{67} +79.7371 q^{71} +110.309i q^{73} +(-42.2137 + 19.8997i) q^{77} -55.1543i q^{79} -39.7995i q^{83} -93.3381i q^{85} +103.189 q^{89} +54.0000 q^{91} -159.198i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{25} + 40 q^{31} + 8 q^{37} + 124 q^{49} + 88 q^{55} - 152 q^{67} + 216 q^{91} - 448 q^{97}+O(q^{100}) \) 4 * q - 12 * q^25 + 40 * q^31 + 8 * q^37 + 124 * q^49 + 88 * q^55 - 152 * q^67 + 216 * q^91 - 448 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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