LMFDB - Embedded newform 7200.2.k.r.3601.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7200,2,Mod(3601,7200)] 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([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7200.3601"); 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.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0, -8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == 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:	8.0.214798336.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.1
Root			$-0.565036 + 1.29643i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.r.3601.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.72294 q^{7} -3.93012i q^{11} -3.46733i q^{13} -3.51575 q^{17} +5.44133i q^{19} -7.11585 q^{23} +3.66998i q^{29} -5.23414 q^{31} -0.414376i q^{37} -3.00454 q^{41} -5.34450i q^{43} -0.925579 q^{47} +15.3061 q^{49} +0.233196i q^{53} +14.3805i q^{59} -0.118290i q^{61} -13.4504i q^{67} +2.19027 q^{71} -0.563219 q^{73} +18.5617i q^{77} +10.2746 q^{79} +11.3490i q^{83} -8.88265 q^{89} +16.3760i q^{91} +7.27462 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} + 8 q^{23} - 8 q^{31} - 40 q^{71} - 16 q^{73} + 16 q^{79} - 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^23 - 8 * q^31 - 40 * q^71 - 16 * q^73 + 16 * q^79 - 8 * q^97

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
$7$	−4.72294	−1.78510	−0.892551	−	0.450947i	$-0.851086\pi$
$7$	−4.72294	−1.78510	−0.892551	+	0.450947i	$0.851086\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.93012i	− 1.18498i	−0.805579	−	0.592488i	$-0.798146\pi$
$11$	− 3.93012i	− 1.18498i	0.805579	−	0.592488i	$-0.201854\pi$
$12$	0	0
$13$	− 3.46733i	− 0.961665i	−0.876812	−	0.480833i	$-0.840334\pi$
$13$	− 3.46733i	− 0.961665i	0.876812	−	0.480833i	$-0.159666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.51575	−0.852694	−0.426347	−	0.904560i	$-0.640200\pi$
$17$	−3.51575	−0.852694	−0.426347	+	0.904560i	$0.640200\pi$
$18$	0	0
$19$	5.44133i	1.24833i	0.781294	+	0.624163i	$0.214560\pi$
$19$	5.44133i	1.24833i	−0.781294	+	0.624163i	$0.785440\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.11585	−1.48376	−0.741878	−	0.670534i	$-0.766065\pi$
$23$	−7.11585	−1.48376	−0.741878	+	0.670534i	$0.766065\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.66998i	0.681498i	0.940154	+	0.340749i	$0.110681\pi$
$29$	3.66998i	0.681498i	−0.940154	+	0.340749i	$0.889319\pi$
$30$	0	0
$31$	−5.23414	−0.940079	−0.470039	−	0.882645i	$-0.655760\pi$
$31$	−5.23414	−0.940079	−0.470039	+	0.882645i	$0.655760\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.414376i	− 0.0681231i	−0.999420	−	0.0340615i	$-0.989156\pi$
$37$	− 0.414376i	− 0.0681231i	0.999420	−	0.0340615i	$-0.0108442\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00454	−0.469231	−0.234616	−	0.972088i	$-0.575383\pi$
$41$	−3.00454	−0.469231	−0.234616	+	0.972088i	$0.575383\pi$
$42$	0	0
$43$	− 5.34450i	− 0.815029i	−0.913199	−	0.407514i	$-0.866396\pi$
$43$	− 5.34450i	− 0.815029i	0.913199	−	0.407514i	$-0.133604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.925579	−0.135010	−0.0675048	−	0.997719i	$-0.521504\pi$
$47$	−0.925579	−0.135010	−0.0675048	+	0.997719i	$0.521504\pi$
$48$	0	0
$49$	15.3061	2.18659
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.233196i	0.0320320i	0.999872	+	0.0160160i	$0.00509826\pi$
$53$	0.233196i	0.0320320i	−0.999872	+	0.0160160i	$0.994902\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.3805i	1.87219i	0.351752	+	0.936093i	$0.385586\pi$
$59$	14.3805i	1.87219i	−0.351752	+	0.936093i	$0.614414\pi$
$60$	0	0
$61$	− 0.118290i	− 0.0151454i	−0.999971	−	0.00757271i	$-0.997590\pi$
$61$	− 0.118290i	− 0.0151454i	0.999971	−	0.00757271i	$-0.00241049\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.4504i	− 1.64323i	−0.570043	−	0.821615i	$-0.693073\pi$
$67$	− 13.4504i	− 1.64323i	0.570043	−	0.821615i	$-0.306927\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.19027	0.259937	0.129969	−	0.991518i	$-0.458512\pi$
$71$	2.19027	0.259937	0.129969	+	0.991518i	$0.458512\pi$
$72$	0	0
$73$	−0.563219	−0.0659197	−0.0329599	−	0.999457i	$-0.510493\pi$
$73$	−0.563219	−0.0659197	−0.0329599	+	0.999457i	$0.510493\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.5617i	2.11530i
$78$	0	0
$79$	10.2746	1.15599	0.577993	−	0.816042i	$-0.303836\pi$
$79$	10.2746	1.15599	0.577993	+	0.816042i	$0.303836\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3490i	1.24572i	0.782334	+	0.622860i	$0.214029\pi$
$83$	11.3490i	1.24572i	−0.782334	+	0.622860i	$0.785971\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.88265	−0.941559	−0.470780	−	0.882251i	$-0.656027\pi$
$89$	−8.88265	−0.941559	−0.470780	+	0.882251i	$0.656027\pi$
$90$	0	0
$91$	16.3760i	1.71667i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.27462	0.738626	0.369313	−	0.929305i	$-0.379593\pi$
$97$	7.27462	0.738626	0.369313	+	0.929305i	$0.379593\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.23320i	− 0.421219i	−0.977570	−	0.210609i	$-0.932455\pi$
$101$	− 4.23320i	− 0.421219i	0.977570	−	0.210609i	$-0.0675448\pi$
$102$	0	0
$103$	−0.0429270	−0.00422972	−0.00211486	−	0.999998i	$-0.500673\pi$
$103$	−0.0429270	−0.00422972	−0.00211486	+	0.999998i	$0.500673\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.4728i	− 1.49581i	−0.663804	−	0.747907i	$-0.731059\pi$
$107$	− 15.4728i	− 1.49581i	0.663804	−	0.747907i	$-0.268941\pi$
$108$	0	0
$109$	− 12.9561i	− 1.24097i	−0.784217	−	0.620486i	$-0.786935\pi$
$109$	− 12.9561i	− 1.24097i	0.784217	−	0.620486i	$-0.213065\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.86025	−0.363141	−0.181571	−	0.983378i	$-0.558118\pi$
$113$	−3.86025	−0.363141	−0.181571	+	0.983378i	$0.558118\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.6046	1.52215
$120$	0	0
$121$	−4.44587	−0.404170
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.3805	1.63101	0.815505	−	0.578751i	$-0.196460\pi$
$127$	18.3805	1.63101	0.815505	+	0.578751i	$0.196460\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.41892i	− 0.298713i	−0.988783	−	0.149356i	$-0.952280\pi$
$131$	− 3.41892i	− 0.298713i	0.988783	−	0.149356i	$-0.0477201\pi$
$132$	0	0
$133$	− 25.6990i	− 2.22839i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.3714	1.39871	0.699354	−	0.714776i	$-0.253471\pi$
$137$	16.3714	1.39871	0.699354	+	0.714776i	$0.253471\pi$
$138$	0	0
$139$	1.95707i	0.165997i	0.996550	+	0.0829984i	$0.0264496\pi$
$139$	1.95707i	0.165997i	−0.996550	+	0.0829984i	$0.973550\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.6271	−1.13955
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.0968i	− 0.991011i	−0.868605	−	0.495505i	$-0.834983\pi$
$149$	− 12.0968i	− 0.991011i	0.868605	−	0.495505i	$-0.165017\pi$
$150$	0	0
$151$	4.87178	0.396460	0.198230	−	0.980156i	$-0.436481\pi$
$151$	4.87178	0.396460	0.198230	+	0.980156i	$0.436481\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.6561i	1.72835i	0.503195	+	0.864173i	$0.332158\pi$
$157$	21.6561i	1.72835i	−0.503195	+	0.864173i	$0.667842\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	33.6077	2.64866
$162$	0	0
$163$	− 16.2362i	− 1.27172i	−0.771804	−	0.635860i	$-0.780645\pi$
$163$	− 16.2362i	− 1.27172i	0.771804	−	0.635860i	$-0.219355\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.69238	−0.517872	−0.258936	−	0.965894i	$-0.583372\pi$
$167$	−6.69238	−0.517872	−0.258936	+	0.965894i	$0.583372\pi$
$168$	0	0
$169$	0.977595	0.0751996
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.4220i	1.70471i	0.522963	+	0.852355i	$0.324827\pi$
$173$	22.4220i	1.70471i	−0.522963	+	0.852355i	$0.675173\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 0.148842i	− 0.0111250i	−0.999985	−	0.00556249i	$-0.998229\pi$
$179$	− 0.148842i	− 0.0111250i	0.999985	−	0.00556249i	$-0.00177060\pi$
$180$	0	0
$181$	10.3929i	0.772499i	0.922394	+	0.386250i	$0.126230\pi$
$181$	10.3929i	0.772499i	−0.922394	+	0.386250i	$0.873770\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.8173i	1.01042i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.23320	0.451018	0.225509	−	0.974241i	$-0.427595\pi$
$191$	6.23320	0.451018	0.225509	+	0.974241i	$0.427595\pi$
$192$	0	0
$193$	−0.391971	−0.0282147	−0.0141074	−	0.999900i	$-0.504491\pi$
$193$	−0.391971	−0.0282147	−0.0141074	+	0.999900i	$0.504491\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.96616i	0.425071i	0.977153	+	0.212536i	$0.0681722\pi$
$197$	5.96616i	0.425071i	−0.977153	+	0.212536i	$0.931828\pi$
$198$	0	0
$199$	17.9322	1.27118	0.635591	−	0.772026i	$-0.280757\pi$
$199$	17.9322	1.27118	0.635591	+	0.772026i	$0.280757\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 17.3331i	− 1.21654i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.3851	1.47924
$210$	0	0
$211$	− 6.51575i	− 0.448563i	−0.974524	−	0.224281i	$-0.927997\pi$
$211$	− 6.51575i	− 0.448563i	0.974524	−	0.224281i	$-0.0720034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.7205	1.67814
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.1903i	0.820006i
$222$	0	0
$223$	−3.14640	−0.210699	−0.105349	−	0.994435i	$-0.533596\pi$
$223$	−3.14640	−0.210699	−0.105349	+	0.994435i	$0.533596\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.92103i	0.260248i	0.991498	+	0.130124i	$0.0415376\pi$
$227$	3.92103i	0.260248i	−0.991498	+	0.130124i	$0.958462\pi$
$228$	0	0
$229$	25.6899i	1.69764i	0.528683	+	0.848820i	$0.322686\pi$
$229$	25.6899i	1.69764i	−0.528683	+	0.848820i	$0.677314\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.7565	−1.68737	−0.843683	−	0.536841i	$-0.819617\pi$
$233$	−25.7565	−1.68737	−0.843683	+	0.536841i	$0.819617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.8727	−1.02672	−0.513360	−	0.858173i	$-0.671600\pi$
$239$	−15.8727	−1.02672	−0.513360	+	0.858173i	$0.671600\pi$
$240$	0	0
$241$	28.1664	1.81436	0.907178	−	0.420748i	$-0.138232\pi$
$241$	28.1664	1.81436	0.907178	+	0.420748i	$0.138232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.8669	1.20047
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.66004i	− 0.294139i	−0.989126	−	0.147070i	$-0.953016\pi$
$251$	− 4.66004i	− 0.294139i	0.989126	−	0.147070i	$-0.0469842\pi$
$252$	0	0
$253$	27.9662i	1.75822i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.33996	0.208341	0.104170	−	0.994559i	$-0.466781\pi$
$257$	3.33996	0.208341	0.104170	+	0.994559i	$0.466781\pi$
$258$	0	0
$259$	1.95707i	0.121607i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.5932	1.70147	0.850735	−	0.525595i	$-0.176157\pi$
$263$	27.5932	1.70147	0.850735	+	0.525595i	$0.176157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.8727i	1.33360i	0.745235	+	0.666802i	$0.232337\pi$
$269$	21.8727i	1.33360i	−0.745235	+	0.666802i	$0.767663\pi$
$270$	0	0
$271$	8.78583	0.533701	0.266850	−	0.963738i	$-0.414017\pi$
$271$	8.78583	0.533701	0.266850	+	0.963738i	$0.414017\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3838i	0.623904i	0.950098	+	0.311952i	$0.100983\pi$
$277$	10.3838i	0.623904i	−0.950098	+	0.311952i	$0.899017\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.0584	−1.01762	−0.508811	−	0.860878i	$-0.669915\pi$
$281$	−17.0584	−1.01762	−0.508811	+	0.860878i	$0.669915\pi$
$282$	0	0
$283$	− 3.54724i	− 0.210862i	−0.994427	−	0.105431i	$-0.966378\pi$
$283$	− 3.54724i	− 0.210862i	0.994427	−	0.105431i	$-0.0336222\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.1903	0.837625
$288$	0	0
$289$	−4.63952	−0.272913
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.72538i	0.334480i	0.985916	+	0.167240i	$0.0534855\pi$
$293$	5.72538i	0.334480i	−0.985916	+	0.167240i	$0.946515\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.6730i	1.42688i
$300$	0	0
$301$	25.2417i	1.45491i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3760i	0.706335i	0.935560	+	0.353168i	$0.114895\pi$
$307$	12.3760i	0.706335i	−0.935560	+	0.353168i	$0.885105\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.2746	−1.03626	−0.518129	−	0.855302i	$-0.673371\pi$
$311$	−18.2746	−1.03626	−0.518129	+	0.855302i	$0.673371\pi$
$312$	0	0
$313$	−12.7114	−0.718491	−0.359246	−	0.933243i	$-0.616966\pi$
$313$	−12.7114	−0.718491	−0.359246	+	0.933243i	$0.616966\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.8602i	0.890800i	0.895332	+	0.445400i	$0.146939\pi$
$317$	15.8602i	0.890800i	−0.895332	+	0.445400i	$0.853061\pi$
$318$	0	0
$319$	14.4235	0.807559
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 19.1303i	− 1.06444i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.37145	0.241006
$330$	0	0
$331$	− 23.0315i	− 1.26593i	−0.774182	−	0.632963i	$-0.781839\pi$
$331$	− 23.0315i	− 1.26593i	0.774182	−	0.632963i	$-0.218161\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.860247	−0.0468606	−0.0234303	−	0.999725i	$-0.507459\pi$
$337$	−0.860247	−0.0468606	−0.0234303	+	0.999725i	$0.507459\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.5708i	1.11397i
$342$	0	0
$343$	−39.2293	−2.11818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.2856i	1.73318i	0.499019	+	0.866591i	$0.333694\pi$
$347$	32.2856i	1.73318i	−0.499019	+	0.866591i	$0.666306\pi$
$348$	0	0
$349$	0.742899i	0.0397665i	0.999802	+	0.0198832i	$0.00632945\pi$
$349$	0.742899i	0.0397665i	−0.999802	+	0.0198832i	$0.993671\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.6392	1.73721	0.868604	−	0.495506i	$-0.165018\pi$
$353$	32.6392	1.73721	0.868604	+	0.495506i	$0.165018\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.71056	−0.143058	−0.0715290	−	0.997439i	$-0.522788\pi$
$359$	−2.71056	−0.143058	−0.0715290	+	0.997439i	$0.522788\pi$
$360$	0	0
$361$	−10.6080	−0.558317
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.680008	−0.0354961	−0.0177481	−	0.999842i	$-0.505650\pi$
$367$	−0.680008	−0.0354961	−0.0177481	+	0.999842i	$0.505650\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.10137i	− 0.0571803i
$372$	0	0
$373$	3.47642i	0.180002i	0.995942	+	0.0900012i	$0.0286871\pi$
$373$	3.47642i	0.180002i	−0.995942	+	0.0900012i	$0.971313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.7250	0.655373
$378$	0	0
$379$	23.0650i	1.18477i	0.805655	+	0.592385i	$0.201813\pi$
$379$	23.0650i	1.18477i	−0.805655	+	0.592385i	$0.798187\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.81544	−0.501545	−0.250773	−	0.968046i	$-0.580685\pi$
$383$	−9.81544	−0.501545	−0.250773	+	0.968046i	$0.580685\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.86175i	− 0.500010i	−0.968244	−	0.250005i	$-0.919568\pi$
$389$	− 9.86175i	− 0.500010i	0.968244	−	0.250005i	$-0.0804323\pi$
$390$	0	0
$391$	25.0175	1.26519
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.7783i	− 0.641326i	−0.947193	−	0.320663i	$-0.896094\pi$
$397$	− 12.7783i	− 0.641326i	0.947193	−	0.320663i	$-0.103906\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.17325	−0.158465	−0.0792323	−	0.996856i	$-0.525247\pi$
$401$	−3.17325	−0.158465	−0.0792323	+	0.996856i	$0.525247\pi$
$402$	0	0
$403$	18.1485i	0.904041i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.62855	−0.0807243
$408$	0	0
$409$	12.1125	0.598923	0.299461	−	0.954108i	$-0.403193\pi$
$409$	12.1125	0.598923	0.299461	+	0.954108i	$0.403193\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 67.9184i	− 3.34204i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.24767i	− 0.207512i	−0.994603	−	0.103756i	$-0.966914\pi$
$419$	− 4.24767i	− 0.207512i	0.994603	−	0.103756i	$-0.0330862\pi$
$420$	0	0
$421$	3.77928i	0.184191i	0.995750	+	0.0920953i	$0.0293564\pi$
$421$	3.77928i	0.184191i	−0.995750	+	0.0920953i	$0.970644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.558674i	0.0270361i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.6271	1.23441	0.617206	−	0.786802i	$-0.288265\pi$
$431$	25.6271	1.23441	0.617206	+	0.786802i	$0.288265\pi$
$432$	0	0
$433$	−2.03149	−0.0976274	−0.0488137	−	0.998808i	$-0.515544\pi$
$433$	−2.03149	−0.0976274	−0.0488137	+	0.998808i	$0.515544\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 38.7196i	− 1.85221i
$438$	0	0
$439$	22.4864	1.07322	0.536608	−	0.843832i	$-0.319706\pi$
$439$	22.4864	1.07322	0.536608	+	0.843832i	$0.319706\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.39385i	0.161247i	0.996745	+	0.0806234i	$0.0256911\pi$
$443$	3.39385i	0.161247i	−0.996745	+	0.0806234i	$0.974309\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.17780	0.102777	0.0513883	−	0.998679i	$-0.483635\pi$
$449$	2.17780	0.102777	0.0513883	+	0.998679i	$0.483635\pi$
$450$	0	0
$451$	11.8082i	0.556028i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.57653	−0.167303	−0.0836516	−	0.996495i	$-0.526658\pi$
$457$	−3.57653	−0.167303	−0.0836516	+	0.996495i	$0.526658\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.1158i	0.610866i	0.952214	+	0.305433i	$0.0988012\pi$
$461$	13.1158i	0.610866i	−0.952214	+	0.305433i	$0.901199\pi$
$462$	0	0
$463$	3.21417	0.149375	0.0746877	−	0.997207i	$-0.476204\pi$
$463$	3.21417	0.149375	0.0746877	+	0.997207i	$0.476204\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 30.3016i	− 1.40219i	−0.713068	−	0.701095i	$-0.752695\pi$
$467$	− 30.3016i	− 1.40219i	0.713068	−	0.701095i	$-0.247305\pi$
$468$	0	0
$469$	63.5254i	2.93333i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.0045	−0.965790
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.0896	1.60328	0.801642	−	0.597804i	$-0.203960\pi$
$479$	35.0896	1.60328	0.801642	+	0.597804i	$0.203960\pi$
$480$	0	0
$481$	−1.43678	−0.0655116
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.9117	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$487$	−36.9117	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.9867i	0.947116i	0.880763	+	0.473558i	$0.157031\pi$
$491$	20.9867i	0.947116i	−0.880763	+	0.473558i	$0.842969\pi$
$492$	0	0
$493$	− 12.9027i	− 0.581109i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.3445	−0.464014
$498$	0	0
$499$	15.9906i	0.715836i	0.933753	+	0.357918i	$0.116513\pi$
$499$	15.9906i	0.715836i	−0.933753	+	0.357918i	$0.883487\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.6023	−1.67660	−0.838302	−	0.545206i	$-0.816451\pi$
$503$	−37.6023	−1.67660	−0.838302	+	0.545206i	$0.816451\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.8579i	− 0.569917i	−0.958540	−	0.284958i	$-0.908020\pi$
$509$	− 12.8579i	− 0.569917i	0.958540	−	0.284958i	$-0.0919797\pi$
$510$	0	0
$511$	2.66004	0.117673
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.63764i	0.159983i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−37.0015	−1.62107	−0.810534	−	0.585692i	$-0.800823\pi$
$521$	−37.0015	−1.62107	−0.810534	+	0.585692i	$0.800823\pi$
$522$	0	0
$523$	− 17.4952i	− 0.765013i	−0.923953	−	0.382506i	$-0.875061\pi$
$523$	− 17.4952i	− 0.765013i	0.923953	−	0.382506i	$-0.124939\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.4019	0.801600
$528$	0	0
$529$	27.6353	1.20153
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.4178i	0.451243i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 60.1549i	− 2.59106i
$540$	0	0
$541$	− 38.1225i	− 1.63901i	−0.573069	−	0.819507i	$-0.694247\pi$
$541$	− 38.1225i	− 1.63901i	0.573069	−	0.819507i	$-0.305753\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.7406i	1.52816i	0.645124	+	0.764078i	$0.276806\pi$
$547$	35.7406i	1.52816i	−0.645124	+	0.764078i	$0.723194\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.9695	−0.850731
$552$	0	0
$553$	−48.5264	−2.06355
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.65516i	− 0.112503i	−0.998417	−	0.0562514i	$-0.982085\pi$
$557$	− 2.65516i	− 0.112503i	0.998417	−	0.0562514i	$-0.0179148\pi$
$558$	0	0
$559$	−18.5312	−0.783785
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 20.3107i	− 0.855992i	−0.903781	−	0.427996i	$-0.859220\pi$
$563$	− 20.3107i	− 0.855992i	0.903781	−	0.427996i	$-0.140780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.4274	1.19174	0.595868	−	0.803082i	$-0.296808\pi$
$569$	28.4274	1.19174	0.595868	+	0.803082i	$0.296808\pi$
$570$	0	0
$571$	− 16.1485i	− 0.675794i	−0.941183	−	0.337897i	$-0.890284\pi$
$571$	− 16.1485i	− 0.675794i	0.941183	−	0.337897i	$-0.109716\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.3600	−1.26390	−0.631952	−	0.775008i	$-0.717746\pi$
$577$	−30.3600	−1.26390	−0.631952	+	0.775008i	$0.717746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 53.6008i	− 2.22374i
$582$	0	0
$583$	0.916490	0.0379571
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.74070i	− 0.113121i	−0.998399	−	0.0565604i	$-0.981987\pi$
$587$	− 2.74070i	− 0.113121i	0.998399	−	0.0565604i	$-0.0180133\pi$
$588$	0	0
$589$	− 28.4807i	− 1.17352i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.2586	0.914053	0.457027	−	0.889453i	$-0.348914\pi$
$593$	22.2586	0.914053	0.457027	+	0.889453i	$0.348914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	48.4526	1.97972	0.989860	−	0.142046i	$-0.0453680\pi$
$599$	48.4526	1.97972	0.989860	+	0.142046i	$0.0453680\pi$
$600$	0	0
$601$	5.26553	0.214786	0.107393	−	0.994217i	$-0.465750\pi$
$601$	5.26553	0.214786	0.107393	+	0.994217i	$0.465750\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.38288	0.299662	0.149831	−	0.988712i	$-0.452127\pi$
$607$	7.38288	0.299662	0.149831	+	0.988712i	$0.452127\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.20929i	0.129834i
$612$	0	0
$613$	2.64607i	0.106874i	0.998571	+	0.0534369i	$0.0170176\pi$
$613$	2.64607i	0.106874i	−0.998571	+	0.0534369i	$0.982982\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.0136	0.845977	0.422989	−	0.906135i	$-0.360981\pi$
$617$	21.0136	0.845977	0.422989	+	0.906135i	$0.360981\pi$
$618$	0	0
$619$	24.0874i	0.968154i	0.875025	+	0.484077i	$0.160845\pi$
$619$	24.0874i	0.968154i	−0.875025	+	0.484077i	$0.839155\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	41.9522	1.68078
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.45684i	0.0580881i
$630$	0	0
$631$	−25.2094	−1.00357	−0.501785	−	0.864992i	$-0.667323\pi$
$631$	−25.2094	−1.00357	−0.501785	+	0.864992i	$0.667323\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 53.0714i	− 2.10277i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.4755	0.729738	0.364869	−	0.931059i	$-0.381114\pi$
$641$	18.4755	0.729738	0.364869	+	0.931059i	$0.381114\pi$
$642$	0	0
$643$	− 0.636984i	− 0.0251202i	−0.999921	−	0.0125601i	$-0.996002\pi$
$643$	− 0.636984i	− 0.0251202i	0.999921	−	0.0125601i	$-0.00399811\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0182	−1.25876	−0.629382	−	0.777096i	$-0.716692\pi$
$647$	−32.0182	−1.25876	−0.629382	+	0.777096i	$0.716692\pi$
$648$	0	0
$649$	56.5173	2.21850
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25.9769i	− 1.01656i	−0.861193	−	0.508278i	$-0.830282\pi$
$653$	− 25.9769i	− 1.01656i	0.861193	−	0.508278i	$-0.169718\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.3422i	0.831372i	0.909508	+	0.415686i	$0.136459\pi$
$659$	21.3422i	0.831372i	−0.909508	+	0.415686i	$0.863541\pi$
$660$	0	0
$661$	− 14.8397i	− 0.577198i	−0.957450	−	0.288599i	$-0.906810\pi$
$661$	− 14.8397i	− 0.577198i	0.957450	−	0.288599i	$-0.0931895\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 26.1150i	− 1.01118i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.464893	−0.0179470
$672$	0	0
$673$	−18.1167	−0.698347	−0.349174	−	0.937058i	$-0.613538\pi$
$673$	−18.1167	−0.698347	−0.349174	+	0.937058i	$0.613538\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.5617i	− 1.17458i	−0.809376	−	0.587291i	$-0.800194\pi$
$677$	− 30.5617i	− 1.17458i	0.809376	−	0.587291i	$-0.199806\pi$
$678$	0	0
$679$	−34.3576	−1.31852
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.2027i	0.849564i	0.905296	+	0.424782i	$0.139649\pi$
$683$	22.2027i	0.849564i	−0.905296	+	0.424782i	$0.860351\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.808569	0.0308040
$690$	0	0
$691$	− 12.6890i	− 0.482712i	−0.970437	−	0.241356i	$-0.922408\pi$
$691$	− 12.6890i	− 0.482712i	0.970437	−	0.241356i	$-0.0775922\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.5632	0.400110
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.9241i	1.31906i	0.751676	+	0.659532i	$0.229246\pi$
$701$	34.9241i	1.31906i	−0.751676	+	0.659532i	$0.770754\pi$
$702$	0	0
$703$	2.25476	0.0850398
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.9931i	0.751918i
$708$	0	0
$709$	− 5.65610i	− 0.212419i	−0.994344	−	0.106210i	$-0.966129\pi$
$709$	− 5.65610i	− 0.212419i	0.994344	−	0.106210i	$-0.0338715\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37.2453	1.39485
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.0844	−0.749020	−0.374510	−	0.927223i	$-0.622189\pi$
$719$	−20.0844	−0.749020	−0.374510	+	0.927223i	$0.622189\pi$
$720$	0	0
$721$	0.202741	0.00755048
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.0424	0.483715	0.241857	−	0.970312i	$-0.422243\pi$
$727$	13.0424	0.483715	0.241857	+	0.970312i	$0.422243\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.7899i	0.694970i
$732$	0	0
$733$	34.8917i	1.28876i	0.764707	+	0.644378i	$0.222884\pi$
$733$	34.8917i	1.28876i	−0.764707	+	0.644378i	$0.777116\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−52.8618	−1.94719
$738$	0	0
$739$	− 41.5040i	− 1.52675i	−0.645957	−	0.763374i	$-0.723541\pi$
$739$	− 41.5040i	− 1.52675i	0.645957	−	0.763374i	$-0.276459\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.63764	−0.280198	−0.140099	−	0.990138i	$-0.544742\pi$
$743$	−7.63764	−0.280198	−0.140099	+	0.990138i	$0.544742\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	73.0771i	2.67018i
$750$	0	0
$751$	19.4029	0.708023	0.354012	−	0.935241i	$-0.384817\pi$
$751$	19.4029	0.708023	0.354012	+	0.935241i	$0.384817\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.7959i	1.59179i	0.605436	+	0.795894i	$0.292999\pi$
$757$	43.7959i	1.59179i	−0.605436	+	0.795894i	$0.707001\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.95519	−0.360875	−0.180438	−	0.983586i	$-0.557751\pi$
$761$	−9.95519	−0.360875	−0.180438	+	0.983586i	$0.557751\pi$
$762$	0	0
$763$	61.1910i	2.21526i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	49.8621	1.80042
$768$	0	0
$769$	−17.9008	−0.645520	−0.322760	−	0.946481i	$-0.604611\pi$
$769$	−17.9008	−0.645520	−0.322760	+	0.946481i	$0.604611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.8182i	0.748777i	0.927272	+	0.374389i	$0.122147\pi$
$773$	20.8182i	0.748777i	−0.927272	+	0.374389i	$0.877853\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 16.3487i	− 0.585753i
$780$	0	0
$781$	− 8.60803i	− 0.308019i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.2047i	0.969744i	0.874585	+	0.484872i	$0.161134\pi$
$787$	27.2047i	0.969744i	−0.874585	+	0.484872i	$0.838866\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.2317	0.648244
$792$	0	0
$793$	−0.410149	−0.0145648
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.4344i	− 0.582138i	−0.956702	−	0.291069i	$-0.905989\pi$
$797$	− 16.4344i	− 0.582138i	0.956702	−	0.291069i	$-0.0940109\pi$
$798$	0	0
$799$	3.25410	0.115122
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.21352i	0.0781134i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28.7698	−1.01149	−0.505747	−	0.862682i	$-0.668783\pi$
$809$	−28.7698	−1.01149	−0.505747	+	0.862682i	$0.668783\pi$
$810$	0	0
$811$	14.3512i	0.503940i	0.967735	+	0.251970i	$0.0810785\pi$
$811$	14.3512i	0.503940i	−0.967735	+	0.251970i	$0.918922\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	29.0812	1.01742
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.97271i	0.173549i	0.996228	+	0.0867744i	$0.0276559\pi$
$821$	4.97271i	0.173549i	−0.996228	+	0.0867744i	$0.972344\pi$
$822$	0	0
$823$	−20.3625	−0.709791	−0.354895	−	0.934906i	$-0.615483\pi$
$823$	−20.3625	−0.709791	−0.354895	+	0.934906i	$0.615483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0039i	1.25198i	0.779832	+	0.625989i	$0.215304\pi$
$827$	36.0039i	1.25198i	−0.779832	+	0.625989i	$0.784696\pi$
$828$	0	0
$829$	7.68666i	0.266969i	0.991051	+	0.133484i	$0.0426166\pi$
$829$	7.68666i	0.266969i	−0.991051	+	0.133484i	$0.957383\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−53.8124	−1.86449
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.3725	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$839$	−25.3725	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$840$	0	0
$841$	15.5313	0.535561
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.9976	0.721485
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.94864i	0.101078i
$852$	0	0
$853$	− 20.1365i	− 0.689460i	−0.938702	−	0.344730i	$-0.887971\pi$
$853$	− 20.1365i	− 0.689460i	0.938702	−	0.344730i	$-0.112029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0380	−1.43599	−0.717996	−	0.696047i	$-0.754940\pi$
$857$	−42.0380	−1.43599	−0.717996	+	0.696047i	$0.754940\pi$
$858$	0	0
$859$	− 12.4095i	− 0.423406i	−0.977334	−	0.211703i	$-0.932099\pi$
$859$	− 12.4095i	− 0.423406i	0.977334	−	0.211703i	$-0.0679010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.8154	0.742606	0.371303	−	0.928512i	$-0.378911\pi$
$863$	21.8154	0.742606	0.371303	+	0.928512i	$0.378911\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 40.3805i	− 1.36982i
$870$	0	0
$871$	−46.6371	−1.58024
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.83690i	− 0.197098i	−0.995132	−	0.0985491i	$-0.968580\pi$
$877$	− 5.83690i	− 0.197098i	0.995132	−	0.0985491i	$-0.0314201\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.9613	0.402986	0.201493	−	0.979490i	$-0.435421\pi$
$881$	11.9613	0.402986	0.201493	+	0.979490i	$0.435421\pi$
$882$	0	0
$883$	− 3.74810i	− 0.126134i	−0.998009	−	0.0630668i	$-0.979912\pi$
$883$	− 3.74810i	− 0.126134i	0.998009	−	0.0630668i	$-0.0200881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.8101	1.43742	0.718711	−	0.695309i	$-0.244732\pi$
$887$	42.8101	1.43742	0.718711	+	0.695309i	$0.244732\pi$
$888$	0	0
$889$	−86.8101	−2.91152
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.03638i	− 0.168536i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 19.2092i	− 0.640662i
$900$	0	0
$901$	− 0.819859i	− 0.0273135i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.97759i	0.231687i	0.993267	+	0.115844i	$0.0369571\pi$
$907$	6.97759i	0.231687i	−0.993267	+	0.115844i	$0.963043\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.2837	0.804555	0.402278	−	0.915518i	$-0.368219\pi$
$911$	24.2837	0.804555	0.402278	+	0.915518i	$0.368219\pi$
$912$	0	0
$913$	44.6031	1.47615
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.1473i	0.533232i
$918$	0	0
$919$	1.25720	0.0414712	0.0207356	−	0.999785i	$-0.493399\pi$
$919$	1.25720	0.0414712	0.0207356	+	0.999785i	$0.493399\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 7.59440i	− 0.249973i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.4933	−0.344275	−0.172138	−	0.985073i	$-0.555067\pi$
$929$	−10.4933	−0.344275	−0.172138	+	0.985073i	$0.555067\pi$
$930$	0	0
$931$	83.2856i	2.72957i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.9680	0.423648	0.211824	−	0.977308i	$-0.432060\pi$
$937$	12.9680	0.423648	0.211824	+	0.977308i	$0.432060\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 44.9947i	− 1.46678i	−0.679806	−	0.733392i	$-0.737936\pi$
$941$	− 44.9947i	− 1.46678i	0.679806	−	0.733392i	$-0.262064\pi$
$942$	0	0
$943$	21.3799	0.696225
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.6282i	0.995283i	0.867383	+	0.497642i	$0.165801\pi$
$947$	30.6282i	0.995283i	−0.867383	+	0.497642i	$0.834199\pi$
$948$	0	0
$949$	1.95287i	0.0633927i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.5717	−0.569202	−0.284601	−	0.958646i	$-0.591861\pi$
$953$	−17.5717	−0.569202	−0.284601	+	0.958646i	$0.591861\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−77.3213	−2.49683
$960$	0	0
$961$	−3.60380	−0.116252
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.50932	−0.145010	−0.0725050	−	0.997368i	$-0.523099\pi$
$967$	−4.50932	−0.145010	−0.0725050	+	0.997368i	$0.523099\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.8512i	1.15052i	0.817971	+	0.575259i	$0.195099\pi$
$971$	35.8512i	1.15052i	−0.817971	+	0.575259i	$0.804901\pi$
$972$	0	0
$973$	− 9.24313i	− 0.296321i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.9704	1.40674	0.703368	−	0.710826i	$-0.251678\pi$
$977$	43.9704	1.40674	0.703368	+	0.710826i	$0.251678\pi$
$978$	0	0
$979$	34.9099i	1.11573i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.64187	−0.148053	−0.0740263	−	0.997256i	$-0.523585\pi$
$983$	−4.64187	−0.148053	−0.0740263	+	0.997256i	$0.523585\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.0307i	1.20930i
$990$	0	0
$991$	52.9117	1.68080	0.840398	−	0.541970i	$-0.182321\pi$
$991$	52.9117	1.68080	0.840398	+	0.541970i	$0.182321\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.4228i	1.24853i	0.781211	+	0.624266i	$0.214602\pi$
$997$	39.4228i	1.24853i	−0.781211	+	0.624266i	$0.785398\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.r.3601.1		8
3.2	odd	2		2400.2.k.d.1201.1		8
4.3	odd	2		1800.2.k.t.901.5		8
5.2	odd	4		7200.2.d.s.2449.1		8
5.3	odd	4		7200.2.d.t.2449.8		8
5.4	even	2		7200.2.k.s.3601.7		8
8.3	odd	2		1800.2.k.t.901.6		8
8.5	even	2	inner	7200.2.k.r.3601.2		8
12.11	even	2		600.2.k.d.301.4	yes	8
15.2	even	4		2400.2.d.g.49.1		8
15.8	even	4		2400.2.d.h.49.8		8
15.14	odd	2		2400.2.k.e.1201.8		8
20.3	even	4		1800.2.d.t.1549.1		8
20.7	even	4		1800.2.d.s.1549.8		8
20.19	odd	2		1800.2.k.q.901.4		8
24.5	odd	2		2400.2.k.d.1201.5		8
24.11	even	2		600.2.k.d.301.3	✓	8
40.3	even	4		1800.2.d.s.1549.7		8
40.13	odd	4		7200.2.d.s.2449.8		8
40.19	odd	2		1800.2.k.q.901.3		8
40.27	even	4		1800.2.d.t.1549.2		8
40.29	even	2		7200.2.k.s.3601.8		8
40.37	odd	4		7200.2.d.t.2449.1		8
60.23	odd	4		600.2.d.g.349.8		8
60.47	odd	4		600.2.d.h.349.1		8
60.59	even	2		600.2.k.e.301.5	yes	8
120.29	odd	2		2400.2.k.e.1201.4		8
120.53	even	4		2400.2.d.g.49.8		8
120.59	even	2		600.2.k.e.301.6	yes	8
120.77	even	4		2400.2.d.h.49.1		8
120.83	odd	4		600.2.d.h.349.2		8
120.107	odd	4		600.2.d.g.349.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.7		8	120.107	odd	4
600.2.d.g.349.8		8	60.23	odd	4
600.2.d.h.349.1		8	60.47	odd	4
600.2.d.h.349.2		8	120.83	odd	4
600.2.k.d.301.3	✓	8	24.11	even	2
600.2.k.d.301.4	yes	8	12.11	even	2
600.2.k.e.301.5	yes	8	60.59	even	2
600.2.k.e.301.6	yes	8	120.59	even	2
1800.2.d.s.1549.7		8	40.3	even	4
1800.2.d.s.1549.8		8	20.7	even	4
1800.2.d.t.1549.1		8	20.3	even	4
1800.2.d.t.1549.2		8	40.27	even	4
1800.2.k.q.901.3		8	40.19	odd	2
1800.2.k.q.901.4		8	20.19	odd	2
1800.2.k.t.901.5		8	4.3	odd	2
1800.2.k.t.901.6		8	8.3	odd	2
2400.2.d.g.49.1		8	15.2	even	4
2400.2.d.g.49.8		8	120.53	even	4
2400.2.d.h.49.1		8	120.77	even	4
2400.2.d.h.49.8		8	15.8	even	4
2400.2.k.d.1201.1		8	3.2	odd	2
2400.2.k.d.1201.5		8	24.5	odd	2
2400.2.k.e.1201.4		8	120.29	odd	2
2400.2.k.e.1201.8		8	15.14	odd	2
7200.2.d.s.2449.1		8	5.2	odd	4
7200.2.d.s.2449.8		8	40.13	odd	4
7200.2.d.t.2449.1		8	40.37	odd	4
7200.2.d.t.2449.8		8	5.3	odd	4
7200.2.k.r.3601.1		8	1.1	even	1	trivial
7200.2.k.r.3601.2		8	8.5	even	2	inner
7200.2.k.s.3601.7		8	5.4	even	2
7200.2.k.s.3601.8		8	40.29	even	2

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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.72294 q^{7} -3.93012i q^{11} -3.46733i q^{13} -3.51575 q^{17} +5.44133i q^{19} -7.11585 q^{23} +3.66998i q^{29} -5.23414 q^{31} -0.414376i q^{37} -3.00454 q^{41} -5.34450i q^{43} -0.925579 q^{47} +15.3061 q^{49} +0.233196i q^{53} +14.3805i q^{59} -0.118290i q^{61} -13.4504i q^{67} +2.19027 q^{71} -0.563219 q^{73} +18.5617i q^{77} +10.2746 q^{79} +11.3490i q^{83} -8.88265 q^{89} +16.3760i q^{91} +7.27462 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} + 8 q^{23} - 8 q^{31} - 40 q^{71} - 16 q^{73} + 16 q^{79} - 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^23 - 8 * q^31 - 40 * q^71 - 16 * q^73 + 16 * q^79 - 8 * q^97

Introduction

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