LMFDB - Embedded newform 288.2.f.a.143.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,Mod(143,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("288.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	288.f (of order $2$, degree $1$, 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:	$2.29969157821$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.4
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	288.143
Dual form			288.2.f.a.143.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{5} +3.46410i q^{7} +O(q^{10})$	$q+2.44949 q^{5} +3.46410i q^{7} -2.82843i q^{11} +3.46410i q^{13} -1.41421i q^{17} +4.00000 q^{19} +4.89898 q^{23} +1.00000 q^{25} -2.44949 q^{29} -3.46410i q^{31} +8.48528i q^{35} -1.41421i q^{41} -8.00000 q^{43} -4.89898 q^{47} -5.00000 q^{49} -7.34847 q^{53} -6.92820i q^{55} -11.3137i q^{59} -13.8564i q^{61} +8.48528i q^{65} +4.00000 q^{67} -14.6969 q^{71} -4.00000 q^{73} +9.79796 q^{77} +3.46410i q^{79} +14.1421i q^{83} -3.46410i q^{85} +7.07107i q^{89} -12.0000 q^{91} +9.79796 q^{95} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{19} + 4 q^{25} - 32 q^{43} - 20 q^{49} + 16 q^{67} - 16 q^{73} - 48 q^{91} + 32 q^{97}+O(q^{100}) $ 4 * q + 16 * q^19 + 4 * q^25 - 32 * q^43 - 20 * q^49 + 16 * q^67 - 16 * q^73 - 48 * q^91 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	3.46410i	1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$	3.46410i	1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.41421i	− 0.342997i	−0.985184	−	0.171499i	$-0.945139\pi$
$17$	− 1.41421i	− 0.342997i	0.985184	−	0.171499i	$-0.0548609\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.89898	1.02151	0.510754	−	0.859727i	$-0.329366\pi$
$23$	4.89898	1.02151	0.510754	+	0.859727i	$0.329366\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.48528i	1.43427i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.89898	−0.714590	−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898	−0.714590	−0.357295	+	0.933992i	$0.616301\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.34847	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$53$	−7.34847	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$54$	0	0
$55$	− 6.92820i	− 0.934199i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.3137i	− 1.47292i	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	− 11.3137i	− 1.47292i	0.676481	−	0.736460i	$-0.263504\pi$
$60$	0	0
$61$	− 13.8564i	− 1.77413i	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	− 13.8564i	− 1.77413i	0.461644	−	0.887066i	$-0.347260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.48528i	1.05247i
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.6969	−1.74421	−0.872103	−	0.489323i	$-0.837244\pi$
$71$	−14.6969	−1.74421	−0.872103	+	0.489323i	$0.837244\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.79796	1.11658
$78$	0	0
$79$	3.46410i	0.389742i	0.980829	+	0.194871i	$0.0624288\pi$
$79$	3.46410i	0.389742i	−0.980829	+	0.194871i	$0.937571\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.1421i	1.55230i	0.630548	+	0.776151i	$0.282830\pi$
$83$	14.1421i	1.55230i	−0.630548	+	0.776151i	$0.717170\pi$
$84$	0	0
$85$	− 3.46410i	− 0.375735i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.07107i	0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$	7.07107i	0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.79796	1.00525
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.44949	−0.243733	−0.121867	−	0.992546i	$-0.538888\pi$
$101$	−2.44949	−0.243733	−0.121867	+	0.992546i	$0.538888\pi$
$102$	0	0
$103$	− 3.46410i	− 0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	− 3.46410i	− 0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.3137i	− 1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	− 11.3137i	− 1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$	0	0
$109$	10.3923i	0.995402i	0.867349	+	0.497701i	$0.165822\pi$
$109$	10.3923i	0.995402i	−0.867349	+	0.497701i	$0.834178\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 18.3848i	− 1.72949i	−0.502208	−	0.864747i	$-0.667479\pi$
$113$	− 18.3848i	− 1.72949i	0.502208	−	0.864747i	$-0.332521\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.89898	0.449089
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.79796	−0.876356
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685i	0.494242i	0.968985	+	0.247121i	$0.0794845\pi$
$131$	5.65685i	0.494242i	−0.968985	+	0.247121i	$0.920516\pi$
$132$	0	0
$133$	13.8564i	1.20150i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.89949i	− 0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$	− 9.89949i	− 0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.79796	0.819346
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.1464	1.40469	0.702345	−	0.711837i	$-0.252136\pi$
$149$	17.1464	1.40469	0.702345	+	0.711837i	$0.252136\pi$
$150$	0	0
$151$	3.46410i	0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$	3.46410i	0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.48528i	− 0.681554i
$156$	0	0
$157$	13.8564i	1.10586i	0.833227	+	0.552931i	$0.186491\pi$
$157$	13.8564i	1.10586i	−0.833227	+	0.552931i	$0.813509\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.9706i	1.33747i
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.5959	1.51638	0.758189	−	0.652035i	$-0.226085\pi$
$167$	19.5959	1.51638	0.758189	+	0.652035i	$0.226085\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.44949	−0.186231	−0.0931156	−	0.995655i	$-0.529683\pi$
$173$	−2.44949	−0.186231	−0.0931156	+	0.995655i	$0.529683\pi$
$174$	0	0
$175$	3.46410i	0.261861i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.65685i	0.422813i	0.977398	+	0.211407i	$0.0678044\pi$
$179$	5.65685i	0.422813i	−0.977398	+	0.211407i	$0.932196\pi$
$180$	0	0
$181$	− 10.3923i	− 0.772454i	−0.922404	−	0.386227i	$-0.873778\pi$
$181$	− 10.3923i	− 0.772454i	0.922404	−	0.386227i	$-0.126222\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.5959	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.34847	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$197$	7.34847	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$198$	0	0
$199$	10.3923i	0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$	10.3923i	0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.48528i	− 0.595550i
$204$	0	0
$205$	− 3.46410i	− 0.241943i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 11.3137i	− 0.782586i
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.5959	−1.33643
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.89898	0.329541
$222$	0	0
$223$	− 17.3205i	− 1.15987i	−0.814664	−	0.579934i	$-0.803079\pi$
$223$	− 17.3205i	− 1.15987i	0.814664	−	0.579934i	$-0.196921\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.82843i	− 0.187729i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$	− 2.82843i	− 0.187729i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$	0	0
$229$	− 17.3205i	− 1.14457i	−0.820054	−	0.572286i	$-0.806057\pi$
$229$	− 17.3205i	− 1.14457i	0.820054	−	0.572286i	$-0.193943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.41421i	− 0.0926482i	−0.998926	−	0.0463241i	$-0.985249\pi$
$233$	− 1.41421i	− 0.0926482i	0.998926	−	0.0463241i	$-0.0147507\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.79796	−0.633777	−0.316889	−	0.948463i	$-0.602638\pi$
$239$	−9.79796	−0.633777	−0.316889	+	0.948463i	$0.602638\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.2474	−0.782461
$246$	0	0
$247$	13.8564i	0.881662i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.1421i	0.892644i	0.894873	+	0.446322i	$0.147266\pi$
$251$	14.1421i	0.892644i	−0.894873	+	0.446322i	$0.852734\pi$
$252$	0	0
$253$	− 13.8564i	− 0.871145i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.0416i	1.49968i	0.661622	+	0.749838i	$0.269869\pi$
$257$	24.0416i	1.49968i	−0.661622	+	0.749838i	$0.730131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.34847	−0.448044	−0.224022	−	0.974584i	$-0.571919\pi$
$269$	−7.34847	−0.448044	−0.224022	+	0.974584i	$0.571919\pi$
$270$	0	0
$271$	− 31.1769i	− 1.89386i	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	− 31.1769i	− 1.89386i	0.321436	−	0.946931i	$-0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.82843i	− 0.170561i
$276$	0	0
$277$	17.3205i	1.04069i	0.853957	+	0.520344i	$0.174196\pi$
$277$	17.3205i	1.04069i	−0.853957	+	0.520344i	$0.825804\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.0416i	1.43420i	0.696969	+	0.717102i	$0.254532\pi$
$281$	24.0416i	1.43420i	−0.696969	+	0.717102i	$0.745468\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.89898	0.289178
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.9444	−1.57411	−0.787054	−	0.616884i	$-0.788395\pi$
$293$	−26.9444	−1.57411	−0.787054	+	0.616884i	$0.788395\pi$
$294$	0	0
$295$	− 27.7128i	− 1.61350i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.9706i	0.981433i
$300$	0	0
$301$	− 27.7128i	− 1.59734i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 33.9411i	− 1.94346i
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.79796	−0.555591	−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796	−0.555591	−0.277796	+	0.960640i	$0.589604\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.2474	0.687885	0.343943	−	0.938991i	$-0.388237\pi$
$317$	12.2474	0.687885	0.343943	+	0.938991i	$0.388237\pi$
$318$	0	0
$319$	6.92820i	0.387905i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.65685i	− 0.314756i
$324$	0	0
$325$	3.46410i	0.192154i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 16.9706i	− 0.935617i
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.79796	0.535320
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.79796	−0.530589
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1421i	0.759190i	0.925153	+	0.379595i	$0.123937\pi$
$347$	14.1421i	0.759190i	−0.925153	+	0.379595i	$0.876063\pi$
$348$	0	0
$349$	27.7128i	1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$	27.7128i	1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5563i	0.827981i	0.910281	+	0.413990i	$0.135865\pi$
$353$	15.5563i	0.827981i	−0.910281	+	0.413990i	$0.864135\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.6969	0.775675	0.387837	−	0.921728i	$-0.373222\pi$
$359$	14.6969	0.775675	0.387837	+	0.921728i	$0.373222\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.79796	−0.512849
$366$	0	0
$367$	24.2487i	1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$	24.2487i	1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 25.4558i	− 1.32160i
$372$	0	0
$373$	13.8564i	0.717458i	0.933442	+	0.358729i	$0.116790\pi$
$373$	13.8564i	0.717458i	−0.933442	+	0.358729i	$0.883210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.48528i	− 0.437014i
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.79796	−0.500652	−0.250326	−	0.968162i	$-0.580538\pi$
$383$	−9.79796	−0.500652	−0.250326	+	0.968162i	$0.580538\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.9444	1.36613	0.683067	−	0.730355i	$-0.260646\pi$
$389$	26.9444	1.36613	0.683067	+	0.730355i	$0.260646\pi$
$390$	0	0
$391$	− 6.92820i	− 0.350374i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.48528i	0.426941i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.41421i	− 0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$	− 1.41421i	− 0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.1918	1.92850
$414$	0	0
$415$	34.6410i	1.70046i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 19.7990i	− 0.967244i	−0.875277	−	0.483622i	$-0.839321\pi$
$419$	− 19.7990i	− 0.967244i	0.875277	−	0.483622i	$-0.160679\pi$
$420$	0	0
$421$	− 24.2487i	− 1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$	− 24.2487i	− 1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.41421i	− 0.0685994i
$426$	0	0
$427$	48.0000	2.32288
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.6969	0.707927	0.353963	−	0.935259i	$-0.384834\pi$
$431$	14.6969	0.707927	0.353963	+	0.935259i	$0.384834\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.5959	0.937400
$438$	0	0
$439$	3.46410i	0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$	3.46410i	0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.82843i	− 0.134383i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	− 2.82843i	− 0.134383i	0.997740	−	0.0671913i	$-0.0214038\pi$
$444$	0	0
$445$	17.3205i	0.821071i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.0416i	1.13459i	0.823513	+	0.567297i	$0.192011\pi$
$449$	24.0416i	1.13459i	−0.823513	+	0.567297i	$0.807989\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−29.3939	−1.37801
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.1464	−0.798589	−0.399294	−	0.916823i	$-0.630745\pi$
$461$	−17.1464	−0.798589	−0.399294	+	0.916823i	$0.630745\pi$
$462$	0	0
$463$	17.3205i	0.804952i	0.915430	+	0.402476i	$0.131850\pi$
$463$	17.3205i	0.804952i	−0.915430	+	0.402476i	$0.868150\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.7696i	− 1.70149i	−0.525577	−	0.850746i	$-0.676151\pi$
$467$	− 36.7696i	− 1.70149i	0.525577	−	0.850746i	$-0.323849\pi$
$468$	0	0
$469$	13.8564i	0.639829i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.6274i	1.04041i
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.4949	1.11920	0.559600	−	0.828763i	$-0.310955\pi$
$479$	24.4949	1.11920	0.559600	+	0.828763i	$0.310955\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.5959	0.889805
$486$	0	0
$487$	10.3923i	0.470920i	0.971884	+	0.235460i	$0.0756597\pi$
$487$	10.3923i	0.470920i	−0.971884	+	0.235460i	$0.924340\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.5980i	1.78703i	0.449032	+	0.893516i	$0.351769\pi$
$491$	39.5980i	1.78703i	−0.449032	+	0.893516i	$0.648231\pi$
$492$	0	0
$493$	3.46410i	0.156015i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 50.9117i	− 2.28370i
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.6969	0.655304	0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969	0.655304	0.327652	+	0.944798i	$0.393743\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−41.6413	−1.84572	−0.922860	−	0.385136i	$-0.874154\pi$
$509$	−41.6413	−1.84572	−0.922860	+	0.385136i	$0.874154\pi$
$510$	0	0
$511$	− 13.8564i	− 0.612971i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8.48528i	− 0.373906i
$516$	0	0
$517$	13.8564i	0.609404i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 26.8701i	− 1.17720i	−0.808425	−	0.588599i	$-0.799680\pi$
$521$	− 26.8701i	− 1.17720i	0.808425	−	0.588599i	$-0.200320\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.89898	−0.213403
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.89898	0.212198
$534$	0	0
$535$	− 27.7128i	− 1.19813i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.1421i	0.609145i
$540$	0	0
$541$	10.3923i	0.446800i	0.974727	+	0.223400i	$0.0717156\pi$
$541$	10.3923i	0.446800i	−0.974727	+	0.223400i	$0.928284\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.4558i	1.09041i
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.79796	−0.417407
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.34847	−0.311365	−0.155682	−	0.987807i	$-0.549758\pi$
$557$	−7.34847	−0.311365	−0.155682	+	0.987807i	$0.549758\pi$
$558$	0	0
$559$	− 27.7128i	− 1.17213i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.1421i	0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$	14.1421i	0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$	0	0
$565$	− 45.0333i	− 1.89457i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.89949i	− 0.415008i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	− 9.89949i	− 0.415008i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.89898	0.204302
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−48.9898	−2.03244
$582$	0	0
$583$	20.7846i	0.860811i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.6274i	0.933933i	0.884275	+	0.466967i	$0.154653\pi$
$587$	22.6274i	0.933933i	−0.884275	+	0.466967i	$0.845347\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.07107i	0.290374i	0.989404	+	0.145187i	$0.0463784\pi$
$593$	7.07107i	0.290374i	−0.989404	+	0.145187i	$0.953622\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.89898	0.200167	0.100083	−	0.994979i	$-0.468089\pi$
$599$	4.89898	0.200167	0.100083	+	0.994979i	$0.468089\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.34847	0.298758
$606$	0	0
$607$	− 45.0333i	− 1.82785i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	− 45.0333i	− 1.82785i	0.405887	−	0.913923i	$-0.366962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16.9706i	− 0.686555i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0416i	0.967880i	0.875101	+	0.483940i	$0.160795\pi$
$617$	24.0416i	0.967880i	−0.875101	+	0.483940i	$0.839205\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.4949	−0.981367
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 31.1769i	− 1.24113i	−0.784154	−	0.620567i	$-0.786903\pi$
$631$	− 31.1769i	− 1.24113i	0.784154	−	0.620567i	$-0.213097\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	25.4558i	1.01018i
$636$	0	0
$637$	− 17.3205i	− 0.686264i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.5563i	0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$	15.5563i	0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.6969	0.577796	0.288898	−	0.957360i	$-0.406711\pi$
$647$	14.6969	0.577796	0.288898	+	0.957360i	$0.406711\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.1464	0.670992	0.335496	−	0.942042i	$-0.391096\pi$
$653$	17.1464	0.670992	0.335496	+	0.942042i	$0.391096\pi$
$654$	0	0
$655$	13.8564i	0.541415i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.3137i	− 0.440720i	−0.975419	−	0.220360i	$-0.929277\pi$
$659$	− 11.3137i	− 0.440720i	0.975419	−	0.220360i	$-0.0707231\pi$
$660$	0	0
$661$	13.8564i	0.538952i	0.963007	+	0.269476i	$0.0868504\pi$
$661$	13.8564i	0.538952i	−0.963007	+	0.269476i	$0.913150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.9411i	1.31618i
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39.1918	−1.51298
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.1464	−0.658991	−0.329495	−	0.944157i	$-0.606879\pi$
$677$	−17.1464	−0.658991	−0.329495	+	0.944157i	$0.606879\pi$
$678$	0	0
$679$	27.7128i	1.06352i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.7696i	− 1.40695i	−0.710721	−	0.703474i	$-0.751631\pi$
$683$	− 36.7696i	− 1.40695i	0.710721	−	0.703474i	$-0.248369\pi$
$684$	0	0
$685$	− 24.2487i	− 0.926496i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 25.4558i	− 0.969790i
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.79796	0.371658
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.34847	0.277548	0.138774	−	0.990324i	$-0.455684\pi$
$701$	7.34847	0.277548	0.138774	+	0.990324i	$0.455684\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.48528i	− 0.319122i
$708$	0	0
$709$	− 3.46410i	− 0.130097i	−0.997882	−	0.0650485i	$-0.979280\pi$
$709$	− 3.46410i	− 0.130097i	0.997882	−	0.0650485i	$-0.0207202\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.9706i	− 0.635553i
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.0908	−1.64431	−0.822155	−	0.569264i	$-0.807228\pi$
$719$	−44.0908	−1.64431	−0.822155	+	0.569264i	$0.807228\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.44949	−0.0909718
$726$	0	0
$727$	24.2487i	0.899335i	0.893196	+	0.449667i	$0.148458\pi$
$727$	24.2487i	0.899335i	−0.893196	+	0.449667i	$0.851542\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.3137i	0.418453i
$732$	0	0
$733$	− 38.1051i	− 1.40744i	−0.710475	−	0.703722i	$-0.751520\pi$
$733$	− 38.1051i	− 1.40744i	0.710475	−	0.703722i	$-0.248480\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.3137i	− 0.416746i
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.1918	−1.43781	−0.718905	−	0.695109i	$-0.755356\pi$
$743$	−39.1918	−1.43781	−0.718905	+	0.695109i	$0.755356\pi$
$744$	0	0
$745$	42.0000	1.53876
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.1918	1.43204
$750$	0	0
$751$	− 3.46410i	− 0.126407i	−0.998001	−	0.0632034i	$-0.979868\pi$
$751$	− 3.46410i	− 0.126407i	0.998001	−	0.0632034i	$-0.0201317\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.48528i	0.308811i
$756$	0	0
$757$	10.3923i	0.377715i	0.982005	+	0.188857i	$0.0604784\pi$
$757$	10.3923i	0.377715i	−0.982005	+	0.188857i	$0.939522\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.3848i	− 0.666448i	−0.942848	−	0.333224i	$-0.891864\pi$
$761$	− 18.3848i	− 0.666448i	0.942848	−	0.333224i	$-0.108136\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.1918	1.41514
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0454	0.792918	0.396459	−	0.918052i	$-0.370239\pi$
$773$	22.0454	0.792918	0.396459	+	0.918052i	$0.370239\pi$
$774$	0	0
$775$	− 3.46410i	− 0.124434i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.65685i	− 0.202678i
$780$	0	0
$781$	41.5692i	1.48746i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	33.9411i	1.21141i
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	63.6867	2.26444
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.2474	−0.433827	−0.216913	−	0.976191i	$-0.569599\pi$
$797$	−12.2474	−0.433827	−0.216913	+	0.976191i	$0.569599\pi$
$798$	0	0
$799$	6.92820i	0.245102i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3137i	0.399252i
$804$	0	0
$805$	41.5692i	1.46512i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.41421i	− 0.0497211i	−0.999691	−	0.0248606i	$-0.992086\pi$
$809$	− 1.41421i	− 0.0497211i	0.999691	−	0.0248606i	$-0.00791417\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	39.1918	1.37283
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.44949	−0.0854878	−0.0427439	−	0.999086i	$-0.513610\pi$
$821$	−2.44949	−0.0854878	−0.0427439	+	0.999086i	$0.513610\pi$
$822$	0	0
$823$	17.3205i	0.603755i	0.953347	+	0.301877i	$0.0976134\pi$
$823$	17.3205i	0.603755i	−0.953347	+	0.301877i	$0.902387\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.65685i	0.196708i	0.995151	+	0.0983540i	$0.0313578\pi$
$827$	5.65685i	0.196708i	−0.995151	+	0.0983540i	$0.968642\pi$
$828$	0	0
$829$	10.3923i	0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$	10.3923i	0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.07107i	0.244998i
$834$	0	0
$835$	48.0000	1.66111
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.89898	−0.169132	−0.0845658	−	0.996418i	$-0.526950\pi$
$839$	−4.89898	−0.169132	−0.0845658	+	0.996418i	$0.526950\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.44949	0.0842650
$846$	0	0
$847$	10.3923i	0.357084i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 13.8564i	− 0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$	− 13.8564i	− 0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.41421i	− 0.0483086i	−0.999708	−	0.0241543i	$-0.992311\pi$
$857$	− 1.41421i	− 0.0483086i	0.999708	−	0.0241543i	$-0.00768930\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.79796	0.332373
$870$	0	0
$871$	13.8564i	0.469506i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 33.9411i	− 1.14742i
$876$	0	0
$877$	55.4256i	1.87159i	0.352544	+	0.935795i	$0.385317\pi$
$877$	55.4256i	1.87159i	−0.352544	+	0.935795i	$0.614683\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43.8406i	− 1.47703i	−0.674238	−	0.738514i	$-0.735528\pi$
$881$	− 43.8406i	− 1.47703i	0.674238	−	0.738514i	$-0.264472\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.79796	−0.328983	−0.164492	−	0.986378i	$-0.552598\pi$
$887$	−9.79796	−0.328983	−0.164492	+	0.986378i	$0.552598\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−19.5959	−0.655752
$894$	0	0
$895$	13.8564i	0.463169i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.48528i	0.283000i
$900$	0	0
$901$	10.3923i	0.346218i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 25.4558i	− 0.846181i
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.1918	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$911$	39.1918	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.5959	−0.647114
$918$	0	0
$919$	− 10.3923i	− 0.342811i	−0.985201	−	0.171405i	$-0.945169\pi$
$919$	− 10.3923i	− 0.342811i	0.985201	−	0.171405i	$-0.0548307\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 50.9117i	− 1.67578i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.41421i	− 0.0463988i	−0.999731	−	0.0231994i	$-0.992615\pi$
$929$	− 1.41421i	− 0.0463988i	0.999731	−	0.0231994i	$-0.00738527\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.79796	−0.320428
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.44949	0.0798511	0.0399255	−	0.999203i	$-0.487288\pi$
$941$	2.44949	0.0798511	0.0399255	+	0.999203i	$0.487288\pi$
$942$	0	0
$943$	− 6.92820i	− 0.225613i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.6274i	0.735292i	0.929966	+	0.367646i	$0.119836\pi$
$947$	22.6274i	0.735292i	−0.929966	+	0.367646i	$0.880164\pi$
$948$	0	0
$949$	− 13.8564i	− 0.449798i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 26.8701i	− 0.870407i	−0.900332	−	0.435203i	$-0.856677\pi$
$953$	− 26.8701i	− 0.870407i	0.900332	−	0.435203i	$-0.143323\pi$
$954$	0	0
$955$	−48.0000	−1.55324
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.2929	1.10737
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	34.2929	1.10393
$966$	0	0
$967$	− 45.0333i	− 1.44817i	−0.689709	−	0.724087i	$-0.742261\pi$
$967$	− 45.0333i	− 1.44817i	0.689709	−	0.724087i	$-0.257739\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1127i	0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$	31.1127i	0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	13.8564i	0.444216i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0416i	0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$977$	24.0416i	0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.9898	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$983$	−48.9898	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−39.1918	−1.24623
$990$	0	0
$991$	51.9615i	1.65061i	0.564686	+	0.825306i	$0.308997\pi$
$991$	51.9615i	1.65061i	−0.564686	+	0.825306i	$0.691003\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.4558i	0.807005i
$996$	0	0
$997$	− 13.8564i	− 0.438837i	−0.975631	−	0.219418i	$-0.929584\pi$
$997$	− 13.8564i	− 0.438837i	0.975631	−	0.219418i	$-0.0704160\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.2.f.a.143.4		4
3.2	odd	2	inner	288.2.f.a.143.2		4
4.3	odd	2		72.2.f.a.35.1	✓	4
5.2	odd	4		7200.2.m.c.3599.2		8
5.3	odd	4		7200.2.m.c.3599.5		8
5.4	even	2		7200.2.b.c.4751.1		4
8.3	odd	2	inner	288.2.f.a.143.1		4
8.5	even	2		72.2.f.a.35.3	yes	4
9.2	odd	6		2592.2.p.c.2159.2		4
9.4	even	3		2592.2.p.a.431.1		4
9.5	odd	6		2592.2.p.a.431.2		4
9.7	even	3		2592.2.p.c.2159.1		4
12.11	even	2		72.2.f.a.35.4	yes	4
15.2	even	4		7200.2.m.c.3599.3		8
15.8	even	4		7200.2.m.c.3599.8		8
15.14	odd	2		7200.2.b.c.4751.2		4
16.3	odd	4		2304.2.c.i.2303.3		8
16.5	even	4		2304.2.c.i.2303.6		8
16.11	odd	4		2304.2.c.i.2303.8		8
16.13	even	4		2304.2.c.i.2303.1		8
20.3	even	4		1800.2.m.c.899.3		8
20.7	even	4		1800.2.m.c.899.6		8
20.19	odd	2		1800.2.b.c.251.4		4
24.5	odd	2		72.2.f.a.35.2	yes	4
24.11	even	2	inner	288.2.f.a.143.3		4
36.7	odd	6		648.2.l.a.539.2		4
36.11	even	6		648.2.l.a.539.1		4
36.23	even	6		648.2.l.c.107.1		4
36.31	odd	6		648.2.l.c.107.2		4
40.3	even	4		7200.2.m.c.3599.1		8
40.13	odd	4		1800.2.m.c.899.2		8
40.19	odd	2		7200.2.b.c.4751.3		4
40.27	even	4		7200.2.m.c.3599.6		8
40.29	even	2		1800.2.b.c.251.2		4
40.37	odd	4		1800.2.m.c.899.7		8
48.5	odd	4		2304.2.c.i.2303.2		8
48.11	even	4		2304.2.c.i.2303.4		8
48.29	odd	4		2304.2.c.i.2303.5		8
48.35	even	4		2304.2.c.i.2303.7		8
60.23	odd	4		1800.2.m.c.899.5		8
60.47	odd	4		1800.2.m.c.899.4		8
60.59	even	2		1800.2.b.c.251.1		4
72.5	odd	6		648.2.l.a.107.1		4
72.11	even	6		2592.2.p.a.2159.1		4
72.13	even	6		648.2.l.a.107.2		4
72.29	odd	6		648.2.l.c.539.2		4
72.43	odd	6		2592.2.p.a.2159.2		4
72.59	even	6		2592.2.p.c.431.1		4
72.61	even	6		648.2.l.c.539.1		4
72.67	odd	6		2592.2.p.c.431.2		4
120.29	odd	2		1800.2.b.c.251.3		4
120.53	even	4		1800.2.m.c.899.8		8
120.59	even	2		7200.2.b.c.4751.4		4
120.77	even	4		1800.2.m.c.899.1		8
120.83	odd	4		7200.2.m.c.3599.4		8
120.107	odd	4		7200.2.m.c.3599.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.f.a.35.1	✓	4	4.3	odd	2
72.2.f.a.35.2	yes	4	24.5	odd	2
72.2.f.a.35.3	yes	4	8.5	even	2
72.2.f.a.35.4	yes	4	12.11	even	2
288.2.f.a.143.1		4	8.3	odd	2	inner
288.2.f.a.143.2		4	3.2	odd	2	inner
288.2.f.a.143.3		4	24.11	even	2	inner
288.2.f.a.143.4		4	1.1	even	1	trivial
648.2.l.a.107.1		4	72.5	odd	6
648.2.l.a.107.2		4	72.13	even	6
648.2.l.a.539.1		4	36.11	even	6
648.2.l.a.539.2		4	36.7	odd	6
648.2.l.c.107.1		4	36.23	even	6
648.2.l.c.107.2		4	36.31	odd	6
648.2.l.c.539.1		4	72.61	even	6
648.2.l.c.539.2		4	72.29	odd	6
1800.2.b.c.251.1		4	60.59	even	2
1800.2.b.c.251.2		4	40.29	even	2
1800.2.b.c.251.3		4	120.29	odd	2
1800.2.b.c.251.4		4	20.19	odd	2
1800.2.m.c.899.1		8	120.77	even	4
1800.2.m.c.899.2		8	40.13	odd	4
1800.2.m.c.899.3		8	20.3	even	4
1800.2.m.c.899.4		8	60.47	odd	4
1800.2.m.c.899.5		8	60.23	odd	4
1800.2.m.c.899.6		8	20.7	even	4
1800.2.m.c.899.7		8	40.37	odd	4
1800.2.m.c.899.8		8	120.53	even	4
2304.2.c.i.2303.1		8	16.13	even	4
2304.2.c.i.2303.2		8	48.5	odd	4
2304.2.c.i.2303.3		8	16.3	odd	4
2304.2.c.i.2303.4		8	48.11	even	4
2304.2.c.i.2303.5		8	48.29	odd	4
2304.2.c.i.2303.6		8	16.5	even	4
2304.2.c.i.2303.7		8	48.35	even	4
2304.2.c.i.2303.8		8	16.11	odd	4
2592.2.p.a.431.1		4	9.4	even	3
2592.2.p.a.431.2		4	9.5	odd	6
2592.2.p.a.2159.1		4	72.11	even	6
2592.2.p.a.2159.2		4	72.43	odd	6
2592.2.p.c.431.1		4	72.59	even	6
2592.2.p.c.431.2		4	72.67	odd	6
2592.2.p.c.2159.1		4	9.7	even	3
2592.2.p.c.2159.2		4	9.2	odd	6
7200.2.b.c.4751.1		4	5.4	even	2
7200.2.b.c.4751.2		4	15.14	odd	2
7200.2.b.c.4751.3		4	40.19	odd	2
7200.2.b.c.4751.4		4	120.59	even	2
7200.2.m.c.3599.1		8	40.3	even	4
7200.2.m.c.3599.2		8	5.2	odd	4
7200.2.m.c.3599.3		8	15.2	even	4
7200.2.m.c.3599.4		8	120.83	odd	4
7200.2.m.c.3599.5		8	5.3	odd	4
7200.2.m.c.3599.6		8	40.27	even	4
7200.2.m.c.3599.7		8	120.107	odd	4
7200.2.m.c.3599.8		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{5} +3.46410i q^{7} +O(q^{10})\)	\(q+2.44949 q^{5} +3.46410i q^{7} -2.82843i q^{11} +3.46410i q^{13} -1.41421i q^{17} +4.00000 q^{19} +4.89898 q^{23} +1.00000 q^{25} -2.44949 q^{29} -3.46410i q^{31} +8.48528i q^{35} -1.41421i q^{41} -8.00000 q^{43} -4.89898 q^{47} -5.00000 q^{49} -7.34847 q^{53} -6.92820i q^{55} -11.3137i q^{59} -13.8564i q^{61} +8.48528i q^{65} +4.00000 q^{67} -14.6969 q^{71} -4.00000 q^{73} +9.79796 q^{77} +3.46410i q^{79} +14.1421i q^{83} -3.46410i q^{85} +7.07107i q^{89} -12.0000 q^{91} +9.79796 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{19} + 4 q^{25} - 32 q^{43} - 20 q^{49} + 16 q^{67} - 16 q^{73} - 48 q^{91} + 32 q^{97}+O(q^{100}) \) 4 * q + 16 * q^19 + 4 * q^25 - 32 * q^43 - 20 * q^49 + 16 * q^67 - 16 * q^73 - 48 * q^91 + 32 * q^97

Introduction

L-functions

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