LMFDB - Embedded newform 7200.2.m.c.3599.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7200,2,Mod(3599,7200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7200.3599"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-32,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{53}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3599.1
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	7200.3599
Dual form			7200.2.m.c.3599.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	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
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.89898i	− 1.02151i	−0.859727	−	0.510754i	$-0.829366\pi$
$23$	− 4.89898i	− 1.02151i	0.859727	−	0.510754i	$-0.170634\pi$
$24$	0	0
$25$	0	0
$26$	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	$0.100693\pi$
$31$	3.46410i	0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	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
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.89898i	− 0.714590i	−0.933992	−	0.357295i	$-0.883699\pi$
$47$	− 4.89898i	− 0.714590i	0.933992	−	0.357295i	$-0.116301\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.34847i	1.00939i	0.863298	+	0.504695i	$0.168395\pi$
$53$	7.34847i	1.00939i	−0.863298	+	0.504695i	$0.831605\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	13.8564i	1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$	13.8564i	1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.6969	1.74421	0.872103	−	0.489323i	$-0.162756\pi$
$71$	14.6969	1.74421	0.872103	+	0.489323i	$0.162756\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.79796i	1.11658i
$78$	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
$83$	−14.1421	−1.55230	−0.776151	−	0.630548i	$-0.782830\pi$
$83$	−14.1421	−1.55230	−0.776151	+	0.630548i	$0.782830\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.07107i	− 0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$	− 7.07107i	− 0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.44949	0.243733	0.121867	−	0.992546i	$-0.461112\pi$
$101$	2.44949	0.243733	0.121867	+	0.992546i	$0.461112\pi$
$102$	0	0
$103$	−3.46410	−0.341328	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410	−0.341328	−0.170664	+	0.985329i	$0.554591\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	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
$113$	18.3848	1.72949	0.864747	−	0.502208i	$-0.167479\pi$
$113$	18.3848	1.72949	0.864747	+	0.502208i	$0.167479\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	−10.3923	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	−10.3923	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$128$	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
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.89949	−0.845771	−0.422885	−	0.906183i	$-0.638983\pi$
$137$	−9.89949	−0.845771	−0.422885	+	0.906183i	$0.638983\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 9.79796i	− 0.819346i
$144$	0	0
$145$	0	0
$146$	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	$-0.954984\pi$
$151$	− 3.46410i	− 0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.9706i	1.33747i
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.5959i	1.51638i	0.652035	+	0.758189i	$0.273915\pi$
$167$	19.5959i	1.51638i	−0.652035	+	0.758189i	$0.726085\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.44949i	0.186231i	0.995655	+	0.0931156i	$0.0296826\pi$
$173$	2.44949i	0.186231i	−0.995655	+	0.0931156i	$0.970317\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.65685i	− 0.422813i	−0.977398	−	0.211407i	$-0.932196\pi$
$179$	− 5.65685i	− 0.422813i	0.977398	−	0.211407i	$-0.0678044\pi$
$180$	0	0
$181$	10.3923i	0.772454i	0.922404	+	0.386227i	$0.126222\pi$
$181$	10.3923i	0.772454i	−0.922404	+	0.386227i	$0.873778\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.5959	1.41791	0.708955	−	0.705253i	$-0.249167\pi$
$191$	19.5959	1.41791	0.708955	+	0.705253i	$0.249167\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.34847i	0.523557i	0.965128	+	0.261778i	$0.0843089\pi$
$197$	7.34847i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$198$	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
$203$	8.48528	0.595550
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	0	0
$217$	− 12.0000i	− 0.814613i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.89898	−0.329541
$222$	0	0
$223$	−17.3205	−1.15987	−0.579934	−	0.814664i	$-0.696921\pi$
$223$	−17.3205	−1.15987	−0.579934	+	0.814664i	$0.696921\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	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
$233$	1.41421	0.0926482	0.0463241	−	0.998926i	$-0.485249\pi$
$233$	1.41421	0.0926482	0.0463241	+	0.998926i	$0.485249\pi$
$234$	0	0
$235$	0	0
$236$	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
$245$	0	0
$246$	0	0
$247$	−13.8564	−0.881662
$248$	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
$253$	−13.8564	−0.871145
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.0416	1.49968	0.749838	−	0.661622i	$-0.230131\pi$
$257$	24.0416	1.49968	0.749838	+	0.661622i	$0.230131\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 9.79796i	− 0.604168i	−0.953281	−	0.302084i	$-0.902318\pi$
$263$	− 9.79796i	− 0.604168i	0.953281	−	0.302084i	$-0.0976823\pi$
$264$	0	0
$265$	0	0
$266$	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	$0.395835\pi$
$271$	31.1769i	1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.3205	−1.04069	−0.520344	−	0.853957i	$-0.674196\pi$
$277$	−17.3205	−1.04069	−0.520344	+	0.853957i	$0.674196\pi$
$278$	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
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.89898i	0.289178i
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.9444i	1.57411i	0.616884	+	0.787054i	$0.288395\pi$
$293$	26.9444i	1.57411i	−0.616884	+	0.787054i	$0.711605\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.79796	0.555591	0.277796	−	0.960640i	$-0.410396\pi$
$311$	9.79796	0.555591	0.277796	+	0.960640i	$0.410396\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.2474i	0.687885i	0.938991	+	0.343943i	$0.111763\pi$
$317$	12.2474i	0.687885i	−0.938991	+	0.343943i	$0.888237\pi$
$318$	0	0
$319$	6.92820i	0.387905i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.65685	0.314756
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	4.00000i	0.217894i	0.994048	+	0.108947i	$0.0347479\pi$
$337$	4.00000i	0.217894i	−0.994048	+	0.108947i	$0.965252\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.79796	0.530589
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1421	0.759190	0.379595	−	0.925153i	$-0.376063\pi$
$347$	14.1421	0.759190	0.379595	+	0.925153i	$0.376063\pi$
$348$	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
$353$	−15.5563	−0.827981	−0.413990	−	0.910281i	$-0.635865\pi$
$353$	−15.5563	−0.827981	−0.413990	+	0.910281i	$0.635865\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	−24.2487	−1.26577	−0.632886	−	0.774245i	$-0.718130\pi$
$367$	−24.2487	−1.26577	−0.632886	+	0.774245i	$0.718130\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 25.4558i	− 1.32160i
$372$	0	0
$373$	13.8564	0.717458	0.358729	−	0.933442i	$-0.383210\pi$
$373$	13.8564	0.717458	0.358729	+	0.933442i	$0.383210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.48528	−0.437014
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.79796i	0.500652i	0.968162	+	0.250326i	$0.0805379\pi$
$383$	9.79796i	0.500652i	−0.968162	+	0.250326i	$0.919462\pi$
$384$	0	0
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	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
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 39.1918i	− 1.92850i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.7990i	0.967244i	0.875277	+	0.483622i	$0.160679\pi$
$419$	19.7990i	0.967244i	−0.875277	+	0.483622i	$0.839321\pi$
$420$	0	0
$421$	24.2487i	1.18181i	0.806741	+	0.590905i	$0.201229\pi$
$421$	24.2487i	1.18181i	−0.806741	+	0.590905i	$0.798771\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 48.0000i	− 2.32288i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.6969	−0.707927	−0.353963	−	0.935259i	$-0.615166\pi$
$431$	−14.6969	−0.707927	−0.353963	+	0.935259i	$0.615166\pi$
$432$	0	0
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.5959i	0.937400i
$438$	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
$443$	2.82843	0.134383	0.0671913	−	0.997740i	$-0.478596\pi$
$443$	2.82843	0.134383	0.0671913	+	0.997740i	$0.478596\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.0416i	− 1.13459i	−0.823513	−	0.567297i	$-0.807989\pi$
$449$	− 24.0416i	− 1.13459i	0.823513	−	0.567297i	$-0.192011\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.1464	0.798589	0.399294	−	0.916823i	$-0.369255\pi$
$461$	17.1464	0.798589	0.399294	+	0.916823i	$0.369255\pi$
$462$	0	0
$463$	17.3205	0.804952	0.402476	−	0.915430i	$-0.368150\pi$
$463$	17.3205	0.804952	0.402476	+	0.915430i	$0.368150\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.7696	−1.70149	−0.850746	−	0.525577i	$-0.823849\pi$
$467$	−36.7696	−1.70149	−0.850746	+	0.525577i	$0.823849\pi$
$468$	0	0
$469$	13.8564i	0.639829i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.6274	−1.04041
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	−10.3923	−0.470920	−0.235460	−	0.971884i	$-0.575660\pi$
$487$	−10.3923	−0.470920	−0.235460	+	0.971884i	$0.575660\pi$
$488$	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
$493$	3.46410	0.156015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−50.9117	−2.28370
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 14.6969i	− 0.655304i	−0.944798	−	0.327652i	$-0.893743\pi$
$503$	− 14.6969i	− 0.655304i	0.944798	−	0.327652i	$-0.106257\pi$
$504$	0	0
$505$	0	0
$506$	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
$515$	0	0
$516$	0	0
$517$	−13.8564	−0.609404
$518$	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
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.89898i	− 0.213403i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.89898i	− 0.212198i
$534$	0	0
$535$	0	0
$536$	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	$-0.928284\pi$
$541$	− 10.3923i	− 0.446800i	0.974727	−	0.223400i	$-0.0717156\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.79796	0.417407
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.34847i	− 0.311365i	−0.987807	−	0.155682i	$-0.950242\pi$
$557$	− 7.34847i	− 0.311365i	0.987807	−	0.155682i	$-0.0497576\pi$
$558$	0	0
$559$	− 27.7128i	− 1.17213i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.1421	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$563$	−14.1421	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.89949i	0.415008i	0.978234	+	0.207504i	$0.0665341\pi$
$569$	9.89949i	0.415008i	−0.978234	+	0.207504i	$0.933466\pi$
$570$	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
$575$	0	0
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	48.9898	2.03244
$582$	0	0
$583$	20.7846	0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.6274	0.933933	0.466967	−	0.884275i	$-0.345347\pi$
$587$	22.6274	0.933933	0.466967	+	0.884275i	$0.345347\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	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
$605$	0	0
$606$	0	0
$607$	45.0333	1.82785	0.913923	−	0.405887i	$-0.133038\pi$
$607$	45.0333	1.82785	0.913923	+	0.405887i	$0.133038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16.9706i	− 0.686555i
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.0416	0.967880	0.483940	−	0.875101i	$-0.339205\pi$
$617$	24.0416	0.967880	0.483940	+	0.875101i	$0.339205\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.4949i	0.981367i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31.1769i	1.24113i	0.784154	+	0.620567i	$0.213097\pi$
$631$	31.1769i	1.24113i	−0.784154	+	0.620567i	$0.786903\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17.3205	0.686264
$638$	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
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.6969i	0.577796i	0.957360	+	0.288898i	$0.0932889\pi$
$647$	14.6969i	0.577796i	−0.957360	+	0.288898i	$0.906711\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 17.1464i	− 0.670992i	−0.942042	−	0.335496i	$-0.891096\pi$
$653$	− 17.1464i	− 0.670992i	0.942042	−	0.335496i	$-0.108904\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.3137i	0.440720i	0.975419	+	0.220360i	$0.0707231\pi$
$659$	11.3137i	0.440720i	−0.975419	+	0.220360i	$0.929277\pi$
$660$	0	0
$661$	− 13.8564i	− 0.538952i	−0.963007	−	0.269476i	$-0.913150\pi$
$661$	− 13.8564i	− 0.538952i	0.963007	−	0.269476i	$-0.0868504\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.1918	1.51298
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.1464i	− 0.658991i	−0.944157	−	0.329495i	$-0.893121\pi$
$677$	− 17.1464i	− 0.658991i	0.944157	−	0.329495i	$-0.106879\pi$
$678$	0	0
$679$	27.7128i	1.06352i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.7696	1.40695	0.703474	−	0.710721i	$-0.251631\pi$
$683$	36.7696	1.40695	0.703474	+	0.710721i	$0.251631\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	0	0
$697$	2.00000i	0.0757554i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.34847	−0.277548	−0.138774	−	0.990324i	$-0.544316\pi$
$701$	−7.34847	−0.277548	−0.138774	+	0.990324i	$0.544316\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.48528	−0.319122
$708$	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
$713$	16.9706	0.635553
$714$	0	0
$715$	0	0
$716$	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
$725$	0	0
$726$	0	0
$727$	−24.2487	−0.899335	−0.449667	−	0.893196i	$-0.648458\pi$
$727$	−24.2487	−0.899335	−0.449667	+	0.893196i	$0.648458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.3137i	0.418453i
$732$	0	0
$733$	−38.1051	−1.40744	−0.703722	−	0.710475i	$-0.748480\pi$
$733$	−38.1051	−1.40744	−0.703722	+	0.710475i	$0.748480\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.3137	−0.416746
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.1918i	1.43781i	0.695109	+	0.718905i	$0.255356\pi$
$743$	39.1918i	1.43781i	−0.695109	+	0.718905i	$0.744644\pi$
$744$	0	0
$745$	0	0
$746$	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	$0.0201317\pi$
$751$	3.46410i	0.126407i	−0.998001	+	0.0632034i	$0.979868\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.3923	−0.377715	−0.188857	−	0.982005i	$-0.560478\pi$
$757$	−10.3923	−0.377715	−0.188857	+	0.982005i	$0.560478\pi$
$758$	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
$763$	− 36.0000i	− 1.30329i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.1918i	1.41514i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 22.0454i	− 0.792918i	−0.918052	−	0.396459i	$-0.870239\pi$
$773$	− 22.0454i	− 0.792918i	0.918052	−	0.396459i	$-0.129761\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−63.6867	−2.26444
$792$	0	0
$793$	48.0000i	1.70453i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.2474i	− 0.433827i	−0.976191	−	0.216913i	$-0.930401\pi$
$797$	− 12.2474i	− 0.433827i	0.976191	−	0.216913i	$-0.0695989\pi$
$798$	0	0
$799$	6.92820i	0.245102i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.3137	−0.399252
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.41421i	0.0497211i	0.999691	+	0.0248606i	$0.00791417\pi$
$809$	1.41421i	0.0497211i	−0.999691	+	0.0248606i	$0.992086\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	32.0000i	1.11954i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.44949	0.0854878	0.0427439	−	0.999086i	$-0.486390\pi$
$821$	2.44949	0.0854878	0.0427439	+	0.999086i	$0.486390\pi$
$822$	0	0
$823$	17.3205	0.603755	0.301877	−	0.953347i	$-0.402387\pi$
$823$	17.3205	0.603755	0.301877	+	0.953347i	$0.402387\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.65685	0.196708	0.0983540	−	0.995151i	$-0.468642\pi$
$827$	5.65685	0.196708	0.0983540	+	0.995151i	$0.468642\pi$
$828$	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
$833$	−7.07107	−0.244998
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	−10.3923	−0.357084
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−13.8564	−0.474434	−0.237217	−	0.971457i	$-0.576235\pi$
$853$	−13.8564	−0.474434	−0.237217	+	0.971457i	$0.576235\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.41421	−0.0483086	−0.0241543	−	0.999708i	$-0.507689\pi$
$857$	−1.41421	−0.0483086	−0.0241543	+	0.999708i	$0.507689\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	−55.4256	−1.87159	−0.935795	−	0.352544i	$-0.885317\pi$
$877$	−55.4256	−1.87159	−0.935795	+	0.352544i	$0.885317\pi$
$878$	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
$883$	40.0000i	1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$	40.0000i	1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.79796i	− 0.328983i	−0.986378	−	0.164492i	$-0.947402\pi$
$887$	− 9.79796i	− 0.328983i	0.986378	−	0.164492i	$-0.0525984\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.5959i	0.655752i
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.1918	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$911$	−39.1918	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$912$	0	0
$913$	40.0000i	1.32381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 19.5959i	− 0.647114i
$918$	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
$923$	50.9117	1.67578
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.41421i	0.0463988i	0.999731	+	0.0231994i	$0.00738527\pi$
$929$	1.41421i	0.0463988i	−0.999731	+	0.0231994i	$0.992615\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.44949	−0.0798511	−0.0399255	−	0.999203i	$-0.512712\pi$
$941$	−2.44949	−0.0798511	−0.0399255	+	0.999203i	$0.512712\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.6274	0.735292	0.367646	−	0.929966i	$-0.380164\pi$
$947$	22.6274	0.735292	0.367646	+	0.929966i	$0.380164\pi$
$948$	0	0
$949$	− 13.8564i	− 0.449798i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.8701	0.870407	0.435203	−	0.900332i	$-0.356677\pi$
$953$	26.8701	0.870407	0.435203	+	0.900332i	$0.356677\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	45.0333	1.44817	0.724087	−	0.689709i	$-0.242261\pi$
$967$	45.0333	1.44817	0.724087	+	0.689709i	$0.242261\pi$
$968$	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
$973$	13.8564	0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0416	0.769160	0.384580	−	0.923092i	$-0.374346\pi$
$977$	24.0416	0.769160	0.384580	+	0.923092i	$0.374346\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.9898i	1.56253i	0.624198	+	0.781266i	$0.285426\pi$
$983$	48.9898i	1.56253i	−0.624198	+	0.781266i	$0.714574\pi$
$984$	0	0
$985$	0	0
$986$	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	$-0.691003\pi$
$991$	− 51.9615i	− 1.65061i	0.564686	−	0.825306i	$-0.308997\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.8564	0.438837	0.219418	−	0.975631i	$-0.429584\pi$
$997$	13.8564	0.438837	0.219418	+	0.975631i	$0.429584\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.m.c.3599.1		8
3.2	odd	2	inner	7200.2.m.c.3599.4		8
4.3	odd	2		1800.2.m.c.899.2		8
5.2	odd	4		288.2.f.a.143.1		4
5.3	odd	4		7200.2.b.c.4751.3		4
5.4	even	2	inner	7200.2.m.c.3599.6		8
8.3	odd	2	inner	7200.2.m.c.3599.5		8
8.5	even	2		1800.2.m.c.899.3		8
12.11	even	2		1800.2.m.c.899.8		8
15.2	even	4		288.2.f.a.143.3		4
15.8	even	4		7200.2.b.c.4751.4		4
15.14	odd	2	inner	7200.2.m.c.3599.7		8
20.3	even	4		1800.2.b.c.251.2		4
20.7	even	4		72.2.f.a.35.3	yes	4
20.19	odd	2		1800.2.m.c.899.7		8
24.5	odd	2		1800.2.m.c.899.5		8
24.11	even	2	inner	7200.2.m.c.3599.8		8
40.3	even	4		7200.2.b.c.4751.1		4
40.13	odd	4		1800.2.b.c.251.4		4
40.19	odd	2	inner	7200.2.m.c.3599.2		8
40.27	even	4		288.2.f.a.143.4		4
40.29	even	2		1800.2.m.c.899.6		8
40.37	odd	4		72.2.f.a.35.1	✓	4
45.2	even	12		2592.2.p.a.2159.1		4
45.7	odd	12		2592.2.p.a.2159.2		4
45.22	odd	12		2592.2.p.c.431.2		4
45.32	even	12		2592.2.p.c.431.1		4
60.23	odd	4		1800.2.b.c.251.3		4
60.47	odd	4		72.2.f.a.35.2	yes	4
60.59	even	2		1800.2.m.c.899.1		8
80.27	even	4		2304.2.c.i.2303.1		8
80.37	odd	4		2304.2.c.i.2303.3		8
80.67	even	4		2304.2.c.i.2303.6		8
80.77	odd	4		2304.2.c.i.2303.8		8
120.29	odd	2		1800.2.m.c.899.4		8
120.53	even	4		1800.2.b.c.251.1		4
120.59	even	2	inner	7200.2.m.c.3599.3		8
120.77	even	4		72.2.f.a.35.4	yes	4
120.83	odd	4		7200.2.b.c.4751.2		4
120.107	odd	4		288.2.f.a.143.2		4
180.7	even	12		648.2.l.c.539.1		4
180.47	odd	12		648.2.l.c.539.2		4
180.67	even	12		648.2.l.a.107.2		4
180.167	odd	12		648.2.l.a.107.1		4
240.77	even	4		2304.2.c.i.2303.4		8
240.107	odd	4		2304.2.c.i.2303.5		8
240.197	even	4		2304.2.c.i.2303.7		8
240.227	odd	4		2304.2.c.i.2303.2		8
360.67	even	12		2592.2.p.a.431.1		4
360.77	even	12		648.2.l.c.107.1		4
360.157	odd	12		648.2.l.c.107.2		4
360.187	even	12		2592.2.p.c.2159.1		4
360.227	odd	12		2592.2.p.c.2159.2		4
360.277	odd	12		648.2.l.a.539.2		4
360.317	even	12		648.2.l.a.539.1		4
360.347	odd	12		2592.2.p.a.431.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.f.a.35.1	✓	4	40.37	odd	4
72.2.f.a.35.2	yes	4	60.47	odd	4
72.2.f.a.35.3	yes	4	20.7	even	4
72.2.f.a.35.4	yes	4	120.77	even	4
288.2.f.a.143.1		4	5.2	odd	4
288.2.f.a.143.2		4	120.107	odd	4
288.2.f.a.143.3		4	15.2	even	4
288.2.f.a.143.4		4	40.27	even	4
648.2.l.a.107.1		4	180.167	odd	12
648.2.l.a.107.2		4	180.67	even	12
648.2.l.a.539.1		4	360.317	even	12
648.2.l.a.539.2		4	360.277	odd	12
648.2.l.c.107.1		4	360.77	even	12
648.2.l.c.107.2		4	360.157	odd	12
648.2.l.c.539.1		4	180.7	even	12
648.2.l.c.539.2		4	180.47	odd	12
1800.2.b.c.251.1		4	120.53	even	4
1800.2.b.c.251.2		4	20.3	even	4
1800.2.b.c.251.3		4	60.23	odd	4
1800.2.b.c.251.4		4	40.13	odd	4
1800.2.m.c.899.1		8	60.59	even	2
1800.2.m.c.899.2		8	4.3	odd	2
1800.2.m.c.899.3		8	8.5	even	2
1800.2.m.c.899.4		8	120.29	odd	2
1800.2.m.c.899.5		8	24.5	odd	2
1800.2.m.c.899.6		8	40.29	even	2
1800.2.m.c.899.7		8	20.19	odd	2
1800.2.m.c.899.8		8	12.11	even	2
2304.2.c.i.2303.1		8	80.27	even	4
2304.2.c.i.2303.2		8	240.227	odd	4
2304.2.c.i.2303.3		8	80.37	odd	4
2304.2.c.i.2303.4		8	240.77	even	4
2304.2.c.i.2303.5		8	240.107	odd	4
2304.2.c.i.2303.6		8	80.67	even	4
2304.2.c.i.2303.7		8	240.197	even	4
2304.2.c.i.2303.8		8	80.77	odd	4
2592.2.p.a.431.1		4	360.67	even	12
2592.2.p.a.431.2		4	360.347	odd	12
2592.2.p.a.2159.1		4	45.2	even	12
2592.2.p.a.2159.2		4	45.7	odd	12
2592.2.p.c.431.1		4	45.32	even	12
2592.2.p.c.431.2		4	45.22	odd	12
2592.2.p.c.2159.1		4	360.187	even	12
2592.2.p.c.2159.2		4	360.227	odd	12
7200.2.b.c.4751.1		4	40.3	even	4
7200.2.b.c.4751.2		4	120.83	odd	4
7200.2.b.c.4751.3		4	5.3	odd	4
7200.2.b.c.4751.4		4	15.8	even	4
7200.2.m.c.3599.1		8	1.1	even	1	trivial
7200.2.m.c.3599.2		8	40.19	odd	2	inner
7200.2.m.c.3599.3		8	120.59	even	2	inner
7200.2.m.c.3599.4		8	3.2	odd	2	inner
7200.2.m.c.3599.5		8	8.3	odd	2	inner
7200.2.m.c.3599.6		8	5.4	even	2	inner
7200.2.m.c.3599.7		8	15.14	odd	2	inner
7200.2.m.c.3599.8		8	24.11	even	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.46410 q^{7} -2.82843i q^{11} +3.46410 q^{13} -1.41421 q^{17} -4.00000 q^{19} -4.89898i q^{23} -2.44949 q^{29} +3.46410i q^{31} -1.41421i q^{41} -8.00000i q^{43} -4.89898i q^{47} +5.00000 q^{49} +7.34847i q^{53} +11.3137i q^{59} +13.8564i q^{61} -4.00000i q^{67} +14.6969 q^{71} -4.00000i q^{73} +9.79796i q^{77} +3.46410i q^{79} -14.1421 q^{83} -7.07107i q^{89} -12.0000 q^{91} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{19} + 40 q^{49} - 96 q^{91}+O(q^{100}) \) 8 * q - 32 * q^19 + 40 * q^49 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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