LMFDB - Embedded newform 1584.3.i.d.881.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(881,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.881"); S:= CuspForms(chi, 3); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,16,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.765751005184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 7x^{6} - 38x^{5} - 3x^{4} + 212x^{3} - 140x^{2} - 352x + 396 $ x^8 - 2x^7 + 7x^6 - 38x^5 - 3x^4 + 212x^3 - 140x^2 - 352*x + 396
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.7
Root			$1.31766 - 0.650677i$ of defining polynomial
Character	$\chi$	$=$	1584.881
Dual form			1584.3.i.d.881.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.14112i q^{5} -6.53081 q^{7} +3.31662i q^{11} +17.6314 q^{13} +11.2438i q^{17} +6.26101 q^{19} +11.7312i q^{23} -1.43108 q^{25} -11.5270i q^{29} -15.5120 q^{31} -33.5757i q^{35} +30.0340 q^{37} +49.6155i q^{41} +17.3813 q^{43} -31.5758i q^{47} -6.34853 q^{49} +81.4442i q^{53} -17.0512 q^{55} -3.76090i q^{59} -90.3789 q^{61} +90.6450i q^{65} -94.3890 q^{67} +54.0794i q^{71} +16.1593 q^{73} -21.6602i q^{77} -149.604 q^{79} -86.6168i q^{83} -57.8058 q^{85} +115.959i q^{89} -115.147 q^{91} +32.1886i q^{95} +61.8154 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} + 8 q^{13} - 8 q^{19} + 16 q^{25} - 8 q^{31} + 8 q^{37} + 40 q^{43} - 160 q^{49} + 72 q^{61} + 16 q^{67} - 64 q^{79} + 240 q^{85} - 416 q^{91} - 24 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 + 8 * q^13 - 8 * q^19 + 16 * q^25 - 8 * q^31 + 8 * q^37 + 40 * q^43 - 160 * q^49 + 72 * q^61 + 16 * q^67 - 64 * q^79 + 240 * q^85 - 416 * q^91 - 24 * q^97

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.14112i	1.02822i	0.857723	+	0.514112i	$0.171878\pi$
$5$	5.14112i	1.02822i	−0.857723	+	0.514112i	$0.828122\pi$
$6$	0	0
$7$	−6.53081	−0.932973	−0.466486	−	0.884528i	$-0.654480\pi$
$7$	−6.53081	−0.932973	−0.466486	+	0.884528i	$0.654480\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	17.6314	1.35626	0.678130	−	0.734942i	$-0.262791\pi$
$13$	17.6314	1.35626	0.678130	+	0.734942i	$0.262791\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.2438i	0.661402i	0.943736	+	0.330701i	$0.107285\pi$
$17$	11.2438i	0.661402i	−0.943736	+	0.330701i	$0.892715\pi$
$18$	0	0
$19$	6.26101	0.329527	0.164763	−	0.986333i	$-0.447314\pi$
$19$	6.26101	0.329527	0.164763	+	0.986333i	$0.447314\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.7312i	0.510050i	0.966934	+	0.255025i	$0.0820837\pi$
$23$	11.7312i	0.510050i	−0.966934	+	0.255025i	$0.917916\pi$
$24$	0	0
$25$	−1.43108	−0.0572431
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 11.5270i	− 0.397482i	−0.980052	−	0.198741i	$-0.936315\pi$
$29$	− 11.5270i	− 0.397482i	0.980052	−	0.198741i	$-0.0636853\pi$
$30$	0	0
$31$	−15.5120	−0.500386	−0.250193	−	0.968196i	$-0.580494\pi$
$31$	−15.5120	−0.500386	−0.250193	+	0.968196i	$0.580494\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 33.5757i	− 0.959304i
$36$	0	0
$37$	30.0340	0.811729	0.405864	−	0.913933i	$-0.366971\pi$
$37$	30.0340	0.811729	0.405864	+	0.913933i	$0.366971\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	49.6155i	1.21013i	0.796175	+	0.605067i	$0.206854\pi$
$41$	49.6155i	1.21013i	−0.796175	+	0.605067i	$0.793146\pi$
$42$	0	0
$43$	17.3813	0.404215	0.202108	−	0.979363i	$-0.435221\pi$
$43$	17.3813	0.404215	0.202108	+	0.979363i	$0.435221\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 31.5758i	− 0.671826i	−0.941893	−	0.335913i	$-0.890955\pi$
$47$	− 31.5758i	− 0.671826i	0.941893	−	0.335913i	$-0.109045\pi$
$48$	0	0
$49$	−6.34853	−0.129562
$50$	0	0
$51$	0	0
$52$	0	0
$53$	81.4442i	1.53668i	0.640040	+	0.768341i	$0.278918\pi$
$53$	81.4442i	1.53668i	−0.640040	+	0.768341i	$0.721082\pi$
$54$	0	0
$55$	−17.0512	−0.310021
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3.76090i	− 0.0637440i	−0.999492	−	0.0318720i	$-0.989853\pi$
$59$	− 3.76090i	− 0.0637440i	0.999492	−	0.0318720i	$-0.0101469\pi$
$60$	0	0
$61$	−90.3789	−1.48162	−0.740811	−	0.671714i	$-0.765559\pi$
$61$	−90.3789	−1.48162	−0.740811	+	0.671714i	$0.765559\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	90.6450i	1.39454i
$66$	0	0
$67$	−94.3890	−1.40879	−0.704395	−	0.709808i	$-0.748782\pi$
$67$	−94.3890	−1.40879	−0.704395	+	0.709808i	$0.748782\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	54.0794i	0.761681i	0.924641	+	0.380841i	$0.124365\pi$
$71$	54.0794i	0.761681i	−0.924641	+	0.380841i	$0.875635\pi$
$72$	0	0
$73$	16.1593	0.221360	0.110680	−	0.993856i	$-0.464697\pi$
$73$	16.1593	0.221360	0.110680	+	0.993856i	$0.464697\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 21.6602i	− 0.281302i
$78$	0	0
$79$	−149.604	−1.89372	−0.946858	−	0.321652i	$-0.895762\pi$
$79$	−149.604	−1.89372	−0.946858	+	0.321652i	$0.895762\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 86.6168i	− 1.04358i	−0.853075	−	0.521788i	$-0.825265\pi$
$83$	− 86.6168i	− 1.04358i	0.853075	−	0.521788i	$-0.174735\pi$
$84$	0	0
$85$	−57.8058	−0.680069
$86$	0	0
$87$	0	0
$88$	0	0
$89$	115.959i	1.30291i	0.758689	+	0.651453i	$0.225840\pi$
$89$	115.959i	1.30291i	−0.758689	+	0.651453i	$0.774160\pi$
$90$	0	0
$91$	−115.147	−1.26535
$92$	0	0
$93$	0	0
$94$	0	0
$95$	32.1886i	0.338827i
$96$	0	0
$97$	61.8154	0.637272	0.318636	−	0.947877i	$-0.396775\pi$
$97$	61.8154	0.637272	0.318636	+	0.947877i	$0.396775\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 54.2101i	− 0.536734i	−0.963317	−	0.268367i	$-0.913516\pi$
$101$	− 54.2101i	− 0.536734i	0.963317	−	0.268367i	$-0.0864840\pi$
$102$	0	0
$103$	−71.5138	−0.694308	−0.347154	−	0.937808i	$-0.612852\pi$
$103$	−71.5138	−0.694308	−0.347154	+	0.937808i	$0.612852\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 68.4553i	− 0.639769i	−0.947457	−	0.319884i	$-0.896356\pi$
$107$	− 68.4553i	− 0.639769i	0.947457	−	0.319884i	$-0.103644\pi$
$108$	0	0
$109$	−56.9124	−0.522132	−0.261066	−	0.965321i	$-0.584074\pi$
$109$	−56.9124	−0.522132	−0.261066	+	0.965321i	$0.584074\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.00225i	− 0.0442677i	−0.999755	−	0.0221339i	$-0.992954\pi$
$113$	− 5.00225i	− 0.0442677i	0.999755	−	0.0221339i	$-0.00704600\pi$
$114$	0	0
$115$	−60.3112	−0.524445
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 73.4313i	− 0.617070i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	121.171i	0.969365i
$126$	0	0
$127$	−103.416	−0.814297	−0.407148	−	0.913362i	$-0.633477\pi$
$127$	−103.416	−0.814297	−0.407148	+	0.913362i	$0.633477\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	224.687i	1.71517i	0.514346	+	0.857583i	$0.328035\pi$
$131$	224.687i	1.71517i	−0.514346	+	0.857583i	$0.671965\pi$
$132$	0	0
$133$	−40.8894	−0.307439
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 107.878i	− 0.787434i	−0.919232	−	0.393717i	$-0.871189\pi$
$137$	− 107.878i	− 0.787434i	0.919232	−	0.393717i	$-0.128811\pi$
$138$	0	0
$139$	244.702	1.76045	0.880223	−	0.474560i	$-0.157393\pi$
$139$	244.702	1.76045	0.880223	+	0.474560i	$0.157393\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	58.4767i	0.408928i
$144$	0	0
$145$	59.2616	0.408701
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 102.226i	− 0.686083i	−0.939320	−	0.343041i	$-0.888543\pi$
$149$	− 102.226i	− 0.686083i	0.939320	−	0.343041i	$-0.111457\pi$
$150$	0	0
$151$	15.8765	0.105142	0.0525711	−	0.998617i	$-0.483258\pi$
$151$	15.8765	0.105142	0.0525711	+	0.998617i	$0.483258\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 79.7488i	− 0.514508i
$156$	0	0
$157$	−289.531	−1.84415	−0.922075	−	0.387012i	$-0.873507\pi$
$157$	−289.531	−1.84415	−0.922075	+	0.387012i	$0.873507\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 76.6139i	− 0.475863i
$162$	0	0
$163$	−68.0738	−0.417630	−0.208815	−	0.977955i	$-0.566961\pi$
$163$	−68.0738	−0.417630	−0.208815	+	0.977955i	$0.566961\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	38.5311i	0.230725i	0.993323	+	0.115363i	$0.0368030\pi$
$167$	38.5311i	0.230725i	−0.993323	+	0.115363i	$0.963197\pi$
$168$	0	0
$169$	141.865	0.839440
$170$	0	0
$171$	0	0
$172$	0	0
$173$	70.6790i	0.408549i	0.978914	+	0.204274i	$0.0654835\pi$
$173$	70.6790i	0.408549i	−0.978914	+	0.204274i	$0.934516\pi$
$174$	0	0
$175$	9.34610	0.0534063
$176$	0	0
$177$	0	0
$178$	0	0
$179$	126.675i	0.707684i	0.935305	+	0.353842i	$0.115125\pi$
$179$	126.675i	0.707684i	−0.935305	+	0.353842i	$0.884875\pi$
$180$	0	0
$181$	−131.282	−0.725312	−0.362656	−	0.931923i	$-0.618130\pi$
$181$	−131.282	−0.725312	−0.362656	+	0.931923i	$0.618130\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	154.408i	0.834639i
$186$	0	0
$187$	−37.2916	−0.199420
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 69.1159i	− 0.361864i	−0.983496	−	0.180932i	$-0.942089\pi$
$191$	− 69.1159i	− 0.361864i	0.983496	−	0.180932i	$-0.0579113\pi$
$192$	0	0
$193$	29.6968	0.153869	0.0769346	−	0.997036i	$-0.475487\pi$
$193$	29.6968	0.153869	0.0769346	+	0.997036i	$0.475487\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	190.155i	0.965253i	0.875826	+	0.482626i	$0.160317\pi$
$197$	190.155i	0.965253i	−0.875826	+	0.482626i	$0.839683\pi$
$198$	0	0
$199$	324.697	1.63164	0.815821	−	0.578305i	$-0.196285\pi$
$199$	324.697	1.63164	0.815821	+	0.578305i	$0.196285\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	75.2806i	0.370840i
$204$	0	0
$205$	−255.079	−1.24429
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.7654i	0.0993560i
$210$	0	0
$211$	−137.160	−0.650048	−0.325024	−	0.945706i	$-0.605372\pi$
$211$	−137.160	−0.650048	−0.325024	+	0.945706i	$0.605372\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	89.3591i	0.415624i
$216$	0	0
$217$	101.306	0.466846
$218$	0	0
$219$	0	0
$220$	0	0
$221$	198.244i	0.897032i
$222$	0	0
$223$	−307.199	−1.37758	−0.688788	−	0.724963i	$-0.741857\pi$
$223$	−307.199	−1.37758	−0.688788	+	0.724963i	$0.741857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	209.890i	0.924625i	0.886717	+	0.462313i	$0.152980\pi$
$227$	209.890i	0.924625i	−0.886717	+	0.462313i	$0.847020\pi$
$228$	0	0
$229$	−223.991	−0.978127	−0.489064	−	0.872248i	$-0.662661\pi$
$229$	−223.991	−0.978127	−0.489064	+	0.872248i	$0.662661\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 103.049i	− 0.442269i	−0.975243	−	0.221134i	$-0.929024\pi$
$233$	− 103.049i	− 0.442269i	0.975243	−	0.221134i	$-0.0709759\pi$
$234$	0	0
$235$	162.335	0.690787
$236$	0	0
$237$	0	0
$238$	0	0
$239$	300.704i	1.25818i	0.777333	+	0.629089i	$0.216572\pi$
$239$	300.704i	1.25818i	−0.777333	+	0.629089i	$0.783428\pi$
$240$	0	0
$241$	−87.3178	−0.362314	−0.181157	−	0.983454i	$-0.557984\pi$
$241$	−87.3178	−0.362314	−0.181157	+	0.983454i	$0.557984\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 32.6385i	− 0.133219i
$246$	0	0
$247$	110.390	0.446924
$248$	0	0
$249$	0	0
$250$	0	0
$251$	257.973i	1.02778i	0.857856	+	0.513890i	$0.171796\pi$
$251$	257.973i	1.02778i	−0.857856	+	0.513890i	$0.828204\pi$
$252$	0	0
$253$	−38.9078	−0.153786
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 176.473i	− 0.686667i	−0.939214	−	0.343334i	$-0.888444\pi$
$257$	− 176.473i	− 0.686667i	0.939214	−	0.343334i	$-0.111556\pi$
$258$	0	0
$259$	−196.146	−0.757321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	430.468i	1.63676i	0.574677	+	0.818380i	$0.305128\pi$
$263$	430.468i	1.63676i	−0.574677	+	0.818380i	$0.694872\pi$
$264$	0	0
$265$	−418.714	−1.58005
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 317.393i	− 1.17990i	−0.807440	−	0.589950i	$-0.799148\pi$
$269$	− 317.393i	− 1.17990i	0.807440	−	0.589950i	$-0.200852\pi$
$270$	0	0
$271$	221.303	0.816617	0.408309	−	0.912844i	$-0.366119\pi$
$271$	221.303	0.816617	0.408309	+	0.912844i	$0.366119\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.74635i	− 0.0172595i
$276$	0	0
$277$	358.691	1.29491	0.647456	−	0.762103i	$-0.275833\pi$
$277$	358.691	1.29491	0.647456	+	0.762103i	$0.275833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 349.449i	− 1.24359i	−0.783179	−	0.621796i	$-0.786403\pi$
$281$	− 349.449i	− 1.24359i	0.783179	−	0.621796i	$-0.213597\pi$
$282$	0	0
$283$	32.9640	0.116480	0.0582402	−	0.998303i	$-0.481451\pi$
$283$	32.9640	0.116480	0.0582402	+	0.998303i	$0.481451\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 324.029i	− 1.12902i
$288$	0	0
$289$	162.576	0.562548
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 34.2107i	− 0.116760i	−0.998294	−	0.0583800i	$-0.981407\pi$
$293$	− 34.2107i	− 0.116760i	0.998294	−	0.0583800i	$-0.0185935\pi$
$294$	0	0
$295$	19.3352	0.0655431
$296$	0	0
$297$	0	0
$298$	0	0
$299$	206.836i	0.691760i
$300$	0	0
$301$	−113.514	−0.377122
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 464.649i	− 1.52344i
$306$	0	0
$307$	259.296	0.844612	0.422306	−	0.906453i	$-0.361221\pi$
$307$	259.296	0.844612	0.422306	+	0.906453i	$0.361221\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	203.443i	0.654158i	0.944997	+	0.327079i	$0.106064\pi$
$311$	203.443i	0.654158i	−0.944997	+	0.327079i	$0.893936\pi$
$312$	0	0
$313$	−260.512	−0.832308	−0.416154	−	0.909294i	$-0.636622\pi$
$313$	−260.512	−0.832308	−0.416154	+	0.909294i	$0.636622\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	430.399i	1.35772i	0.734265	+	0.678862i	$0.237527\pi$
$317$	430.399i	1.35772i	−0.734265	+	0.678862i	$0.762473\pi$
$318$	0	0
$319$	38.2307	0.119845
$320$	0	0
$321$	0	0
$322$	0	0
$323$	70.3977i	0.217950i
$324$	0	0
$325$	−25.2319	−0.0776365
$326$	0	0
$327$	0	0
$328$	0	0
$329$	206.216i	0.626795i
$330$	0	0
$331$	−281.803	−0.851368	−0.425684	−	0.904872i	$-0.639967\pi$
$331$	−281.803	−0.851368	−0.425684	+	0.904872i	$0.639967\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 485.265i	− 1.44855i
$336$	0	0
$337$	540.253	1.60313	0.801563	−	0.597911i	$-0.204002\pi$
$337$	540.253	1.60313	0.801563	+	0.597911i	$0.204002\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 51.4473i	− 0.150872i
$342$	0	0
$343$	361.471	1.05385
$344$	0	0
$345$	0	0
$346$	0	0
$347$	203.053i	0.585168i	0.956240	+	0.292584i	$0.0945151\pi$
$347$	203.053i	0.585168i	−0.956240	+	0.292584i	$0.905485\pi$
$348$	0	0
$349$	484.661	1.38871	0.694357	−	0.719631i	$-0.255689\pi$
$349$	484.661	1.38871	0.694357	+	0.719631i	$0.255689\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 667.489i	− 1.89090i	−0.325761	−	0.945452i	$-0.605621\pi$
$353$	− 667.489i	− 1.89090i	0.325761	−	0.945452i	$-0.394379\pi$
$354$	0	0
$355$	−278.028	−0.783178
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 357.533i	− 0.995915i	−0.867202	−	0.497957i	$-0.834084\pi$
$359$	− 357.533i	− 0.995915i	0.867202	−	0.497957i	$-0.165916\pi$
$360$	0	0
$361$	−321.800	−0.891412
$362$	0	0
$363$	0	0
$364$	0	0
$365$	83.0767i	0.227607i
$366$	0	0
$367$	55.6254	0.151568	0.0757839	−	0.997124i	$-0.475854\pi$
$367$	55.6254	0.151568	0.0757839	+	0.997124i	$0.475854\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 531.896i	− 1.43368i
$372$	0	0
$373$	−512.765	−1.37470	−0.687352	−	0.726324i	$-0.741227\pi$
$373$	−512.765	−1.37470	−0.687352	+	0.726324i	$0.741227\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 203.237i	− 0.539089i
$378$	0	0
$379$	414.573	1.09386	0.546930	−	0.837179i	$-0.315797\pi$
$379$	414.573	1.09386	0.546930	+	0.837179i	$0.315797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 218.336i	− 0.570067i	−0.958518	−	0.285033i	$-0.907995\pi$
$383$	− 218.336i	− 0.570067i	0.958518	−	0.285033i	$-0.0920047\pi$
$384$	0	0
$385$	111.358	0.289241
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 176.025i	− 0.452508i	−0.974068	−	0.226254i	$-0.927352\pi$
$389$	− 176.025i	− 0.452508i	0.974068	−	0.226254i	$-0.0726479\pi$
$390$	0	0
$391$	−131.903	−0.337348
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 769.129i	− 1.94716i
$396$	0	0
$397$	−453.253	−1.14169	−0.570847	−	0.821056i	$-0.693385\pi$
$397$	−453.253	−1.14169	−0.570847	+	0.821056i	$0.693385\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	485.460i	1.21062i	0.795988	+	0.605312i	$0.206952\pi$
$401$	485.460i	1.21062i	−0.795988	+	0.605312i	$0.793048\pi$
$402$	0	0
$403$	−273.497	−0.678653
$404$	0	0
$405$	0	0
$406$	0	0
$407$	99.6114i	0.244745i
$408$	0	0
$409$	723.039	1.76782	0.883911	−	0.467655i	$-0.154901\pi$
$409$	723.039	1.76782	0.883911	+	0.467655i	$0.154901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	24.5617i	0.0594714i
$414$	0	0
$415$	445.307	1.07303
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 409.844i	− 0.978149i	−0.872242	−	0.489074i	$-0.837335\pi$
$419$	− 409.844i	− 0.978149i	0.872242	−	0.489074i	$-0.162665\pi$
$420$	0	0
$421$	387.315	0.919988	0.459994	−	0.887922i	$-0.347852\pi$
$421$	387.315	0.919988	0.459994	+	0.887922i	$0.347852\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 16.0908i	− 0.0378607i
$426$	0	0
$427$	590.248	1.38231
$428$	0	0
$429$	0	0
$430$	0	0
$431$	415.264i	0.963490i	0.876311	+	0.481745i	$0.159997\pi$
$431$	415.264i	0.963490i	−0.876311	+	0.481745i	$0.840003\pi$
$432$	0	0
$433$	−109.402	−0.252659	−0.126330	−	0.991988i	$-0.540320\pi$
$433$	−109.402	−0.252659	−0.126330	+	0.991988i	$0.540320\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	73.4488i	0.168075i
$438$	0	0
$439$	−251.139	−0.572070	−0.286035	−	0.958219i	$-0.592337\pi$
$439$	−251.139	−0.572070	−0.286035	+	0.958219i	$0.592337\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	877.049i	1.97979i	0.141785	+	0.989897i	$0.454716\pi$
$443$	877.049i	1.97979i	−0.141785	+	0.989897i	$0.545284\pi$
$444$	0	0
$445$	−596.157	−1.33968
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 552.992i	− 1.23161i	−0.787900	−	0.615803i	$-0.788832\pi$
$449$	− 552.992i	− 1.23161i	0.787900	−	0.615803i	$-0.211168\pi$
$450$	0	0
$451$	−164.556	−0.364869
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 591.985i	− 1.30107i
$456$	0	0
$457$	470.246	1.02898	0.514492	−	0.857495i	$-0.327981\pi$
$457$	470.246	1.02898	0.514492	+	0.857495i	$0.327981\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	481.242i	1.04391i	0.852973	+	0.521955i	$0.174797\pi$
$461$	481.242i	1.04391i	−0.852973	+	0.521955i	$0.825203\pi$
$462$	0	0
$463$	390.401	0.843198	0.421599	−	0.906782i	$-0.361469\pi$
$463$	390.401	0.843198	0.421599	+	0.906782i	$0.361469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 841.146i	− 1.80117i	−0.434681	−	0.900584i	$-0.643139\pi$
$467$	− 841.146i	− 1.80117i	0.434681	−	0.900584i	$-0.356861\pi$
$468$	0	0
$469$	616.436	1.31436
$470$	0	0
$471$	0	0
$472$	0	0
$473$	57.6471i	0.121876i
$474$	0	0
$475$	−8.95999	−0.0188631
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 494.806i	− 1.03300i	−0.856288	−	0.516499i	$-0.827235\pi$
$479$	− 494.806i	− 1.03300i	0.856288	−	0.516499i	$-0.172765\pi$
$480$	0	0
$481$	529.540	1.10092
$482$	0	0
$483$	0	0
$484$	0	0
$485$	317.800i	0.655258i
$486$	0	0
$487$	−587.629	−1.20663	−0.603315	−	0.797503i	$-0.706154\pi$
$487$	−587.629	−1.20663	−0.603315	+	0.797503i	$0.706154\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.6397i	0.0399994i	0.999800	+	0.0199997i	$0.00636652\pi$
$491$	19.6397i	0.0399994i	−0.999800	+	0.0199997i	$0.993633\pi$
$492$	0	0
$493$	129.608	0.262896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 353.182i	− 0.710628i
$498$	0	0
$499$	206.961	0.414752	0.207376	−	0.978261i	$-0.433508\pi$
$499$	206.961	0.414752	0.207376	+	0.978261i	$0.433508\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	579.562i	1.15221i	0.817375	+	0.576106i	$0.195428\pi$
$503$	579.562i	1.15221i	−0.817375	+	0.576106i	$0.804572\pi$
$504$	0	0
$505$	278.701	0.551882
$506$	0	0
$507$	0	0
$508$	0	0
$509$	245.866i	0.483037i	0.970396	+	0.241518i	$0.0776454\pi$
$509$	245.866i	0.483037i	−0.970396	+	0.241518i	$0.922355\pi$
$510$	0	0
$511$	−105.533	−0.206523
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 367.661i	− 0.713904i
$516$	0	0
$517$	104.725	0.202563
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 666.364i	− 1.27901i	−0.768787	−	0.639505i	$-0.779139\pi$
$521$	− 666.364i	− 1.27901i	0.768787	−	0.639505i	$-0.220861\pi$
$522$	0	0
$523$	37.5819	0.0718583	0.0359292	−	0.999354i	$-0.488561\pi$
$523$	37.5819	0.0718583	0.0359292	+	0.999354i	$0.488561\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 174.414i	− 0.330956i
$528$	0	0
$529$	391.380	0.739849
$530$	0	0
$531$	0	0
$532$	0	0
$533$	874.789i	1.64126i
$534$	0	0
$535$	351.936	0.657825
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 21.0557i	− 0.0390644i
$540$	0	0
$541$	492.691	0.910705	0.455352	−	0.890311i	$-0.349513\pi$
$541$	492.691	0.910705	0.455352	+	0.890311i	$0.349513\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 292.593i	− 0.536868i
$546$	0	0
$547$	−514.210	−0.940055	−0.470027	−	0.882652i	$-0.655756\pi$
$547$	−514.210	−0.940055	−0.470027	+	0.882652i	$0.655756\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 72.1706i	− 0.130981i
$552$	0	0
$553$	977.032	1.76679
$554$	0	0
$555$	0	0
$556$	0	0
$557$	295.653i	0.530795i	0.964139	+	0.265398i	$0.0855032\pi$
$557$	295.653i	0.530795i	−0.964139	+	0.265398i	$0.914497\pi$
$558$	0	0
$559$	306.456	0.548221
$560$	0	0
$561$	0	0
$562$	0	0
$563$	746.059i	1.32515i	0.748996	+	0.662575i	$0.230536\pi$
$563$	746.059i	1.32515i	−0.748996	+	0.662575i	$0.769464\pi$
$564$	0	0
$565$	25.7172	0.0455171
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.8074i	0.0523856i	0.999657	+	0.0261928i	$0.00833838\pi$
$569$	29.8074i	0.0523856i	−0.999657	+	0.0261928i	$0.991662\pi$
$570$	0	0
$571$	853.099	1.49404	0.747022	−	0.664799i	$-0.231483\pi$
$571$	853.099	1.49404	0.747022	+	0.664799i	$0.231483\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 16.7882i	− 0.0291969i
$576$	0	0
$577$	694.635	1.20387	0.601937	−	0.798544i	$-0.294396\pi$
$577$	694.635	1.20387	0.601937	+	0.798544i	$0.294396\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	565.678i	0.973628i
$582$	0	0
$583$	−270.120	−0.463327
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 57.7063i	− 0.0983072i	−0.998791	−	0.0491536i	$-0.984348\pi$
$587$	− 57.7063i	− 0.0983072i	0.998791	−	0.0491536i	$-0.0156524\pi$
$588$	0	0
$589$	−97.1205	−0.164890
$590$	0	0
$591$	0	0
$592$	0	0
$593$	85.9372i	0.144919i	0.997371	+	0.0724597i	$0.0230849\pi$
$593$	85.9372i	0.144919i	−0.997371	+	0.0724597i	$0.976915\pi$
$594$	0	0
$595$	377.519	0.634486
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 309.516i	− 0.516721i	−0.966049	−	0.258360i	$-0.916818\pi$
$599$	− 309.516i	− 0.516721i	0.966049	−	0.258360i	$-0.0831822\pi$
$600$	0	0
$601$	685.624	1.14081	0.570403	−	0.821365i	$-0.306787\pi$
$601$	685.624	1.14081	0.570403	+	0.821365i	$0.306787\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 56.5523i	− 0.0934748i
$606$	0	0
$607$	72.9666	0.120209	0.0601043	−	0.998192i	$-0.480857\pi$
$607$	72.9666	0.120209	0.0601043	+	0.998192i	$0.480857\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 556.725i	− 0.911170i
$612$	0	0
$613$	−1048.28	−1.71008	−0.855039	−	0.518563i	$-0.826467\pi$
$613$	−1048.28	−1.71008	−0.855039	+	0.518563i	$0.826467\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	262.873i	0.426051i	0.977047	+	0.213025i	$0.0683317\pi$
$617$	262.873i	0.426051i	−0.977047	+	0.213025i	$0.931668\pi$
$618$	0	0
$619$	−518.070	−0.836946	−0.418473	−	0.908229i	$-0.637435\pi$
$619$	−518.070	−0.836946	−0.418473	+	0.908229i	$0.637435\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 757.303i	− 1.21558i
$624$	0	0
$625$	−658.729	−1.05397
$626$	0	0
$627$	0	0
$628$	0	0
$629$	337.697i	0.536879i
$630$	0	0
$631$	−1045.97	−1.65763	−0.828816	−	0.559521i	$-0.810985\pi$
$631$	−1045.97	−1.65763	−0.828816	+	0.559521i	$0.810985\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 531.672i	− 0.837279i
$636$	0	0
$637$	−111.933	−0.175720
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 150.474i	− 0.234749i	−0.993088	−	0.117374i	$-0.962552\pi$
$641$	− 150.474i	− 0.234749i	0.993088	−	0.117374i	$-0.0374477\pi$
$642$	0	0
$643$	532.270	0.827792	0.413896	−	0.910324i	$-0.364168\pi$
$643$	532.270	0.827792	0.413896	+	0.910324i	$0.364168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	554.583i	0.857161i	0.903504	+	0.428581i	$0.140986\pi$
$647$	554.583i	0.857161i	−0.903504	+	0.428581i	$0.859014\pi$
$648$	0	0
$649$	12.4735	0.0192195
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.2762i	− 0.0371765i	−0.999827	−	0.0185882i	$-0.994083\pi$
$653$	− 24.2762i	− 0.0371765i	0.999827	−	0.0185882i	$-0.00591716\pi$
$654$	0	0
$655$	−1155.14	−1.76357
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 502.822i	− 0.763007i	−0.924367	−	0.381504i	$-0.875406\pi$
$659$	− 502.822i	− 0.763007i	0.924367	−	0.381504i	$-0.124594\pi$
$660$	0	0
$661$	−1086.03	−1.64301	−0.821505	−	0.570202i	$-0.806865\pi$
$661$	−1086.03	−1.64301	−0.821505	+	0.570202i	$0.806865\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 210.217i	− 0.316116i
$666$	0	0
$667$	135.225	0.202736
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 299.753i	− 0.446726i
$672$	0	0
$673$	1261.06	1.87378	0.936891	−	0.349622i	$-0.113690\pi$
$673$	1261.06	1.87378	0.936891	+	0.349622i	$0.113690\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 315.493i	− 0.466017i	−0.972475	−	0.233008i	$-0.925143\pi$
$677$	− 315.493i	− 0.466017i	0.972475	−	0.233008i	$-0.0748569\pi$
$678$	0	0
$679$	−403.705	−0.594558
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 736.009i	− 1.07761i	−0.842430	−	0.538806i	$-0.818875\pi$
$683$	− 736.009i	− 1.07761i	0.842430	−	0.538806i	$-0.181125\pi$
$684$	0	0
$685$	554.616	0.809658
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1435.97i	2.08414i
$690$	0	0
$691$	1180.45	1.70832	0.854158	−	0.520014i	$-0.174073\pi$
$691$	1180.45	1.70832	0.854158	+	0.520014i	$0.174073\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1258.04i	1.81013i
$696$	0	0
$697$	−557.868	−0.800385
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 263.104i	− 0.375327i	−0.982233	−	0.187663i	$-0.939909\pi$
$701$	− 263.104i	− 0.375327i	0.982233	−	0.187663i	$-0.0600914\pi$
$702$	0	0
$703$	188.043	0.267486
$704$	0	0
$705$	0	0
$706$	0	0
$707$	354.036i	0.500758i
$708$	0	0
$709$	93.4100	0.131749	0.0658745	−	0.997828i	$-0.479016\pi$
$709$	93.4100	0.131749	0.0658745	+	0.997828i	$0.479016\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 181.973i	− 0.255222i
$714$	0	0
$715$	−300.635	−0.420469
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1244.91i	− 1.73145i	−0.500523	−	0.865723i	$-0.666859\pi$
$719$	− 1244.91i	− 1.73145i	0.500523	−	0.865723i	$-0.333141\pi$
$720$	0	0
$721$	467.043	0.647771
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.4960i	0.0227531i
$726$	0	0
$727$	1428.70	1.96520	0.982601	−	0.185727i	$-0.0594640\pi$
$727$	1428.70	1.96520	0.982601	+	0.185727i	$0.0594640\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	195.432i	0.267349i
$732$	0	0
$733$	−1022.10	−1.39440	−0.697202	−	0.716875i	$-0.745572\pi$
$733$	−1022.10	−1.39440	−0.697202	+	0.716875i	$0.745572\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 313.053i	− 0.424766i
$738$	0	0
$739$	−1068.37	−1.44569	−0.722846	−	0.691009i	$-0.757166\pi$
$739$	−1068.37	−1.44569	−0.722846	+	0.691009i	$0.757166\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	970.958i	1.30681i	0.757009	+	0.653404i	$0.226660\pi$
$743$	970.958i	1.30681i	−0.757009	+	0.653404i	$0.773340\pi$
$744$	0	0
$745$	525.558	0.705446
$746$	0	0
$747$	0	0
$748$	0	0
$749$	447.068i	0.596887i
$750$	0	0
$751$	1375.55	1.83163	0.915814	−	0.401603i	$-0.131547\pi$
$751$	1375.55	1.83163	0.915814	+	0.401603i	$0.131547\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	81.6228i	0.108110i
$756$	0	0
$757$	53.6621	0.0708878	0.0354439	−	0.999372i	$-0.488715\pi$
$757$	53.6621	0.0708878	0.0354439	+	0.999372i	$0.488715\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 946.461i	− 1.24371i	−0.783134	−	0.621853i	$-0.786380\pi$
$761$	− 946.461i	− 1.24371i	0.783134	−	0.621853i	$-0.213620\pi$
$762$	0	0
$763$	371.684	0.487135
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 66.3098i	− 0.0864534i
$768$	0	0
$769$	−145.151	−0.188754	−0.0943768	−	0.995537i	$-0.530086\pi$
$769$	−145.151	−0.188754	−0.0943768	+	0.995537i	$0.530086\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 615.110i	− 0.795743i	−0.917441	−	0.397872i	$-0.869749\pi$
$773$	− 615.110i	− 0.795743i	0.917441	−	0.397872i	$-0.130251\pi$
$774$	0	0
$775$	22.1988	0.0286436
$776$	0	0
$777$	0	0
$778$	0	0
$779$	310.643i	0.398772i
$780$	0	0
$781$	−179.361	−0.229655
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1488.51i	− 1.89620i
$786$	0	0
$787$	−55.7686	−0.0708622	−0.0354311	−	0.999372i	$-0.511280\pi$
$787$	−55.7686	−0.0708622	−0.0354311	+	0.999372i	$0.511280\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.6688i	0.0413006i
$792$	0	0
$793$	−1593.50	−2.00946
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 510.236i	− 0.640196i	−0.947385	−	0.320098i	$-0.896284\pi$
$797$	− 510.236i	− 0.640196i	0.947385	−	0.320098i	$-0.103716\pi$
$798$	0	0
$799$	355.033	0.444347
$800$	0	0
$801$	0	0
$802$	0	0
$803$	53.5943i	0.0667425i
$804$	0	0
$805$	393.881	0.489293
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 583.538i	− 0.721307i	−0.932700	−	0.360654i	$-0.882554\pi$
$809$	− 583.538i	− 0.721307i	0.932700	−	0.360654i	$-0.117446\pi$
$810$	0	0
$811$	398.888	0.491847	0.245923	−	0.969289i	$-0.420909\pi$
$811$	398.888	0.491847	0.245923	+	0.969289i	$0.420909\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 349.975i	− 0.429417i
$816$	0	0
$817$	108.824	0.133200
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 792.594i	− 0.965400i	−0.875786	−	0.482700i	$-0.839656\pi$
$821$	− 792.594i	− 0.965400i	0.875786	−	0.482700i	$-0.160344\pi$
$822$	0	0
$823$	51.7597	0.0628915	0.0314457	−	0.999505i	$-0.489989\pi$
$823$	51.7597	0.0628915	0.0314457	+	0.999505i	$0.489989\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	629.335i	0.760986i	0.924784	+	0.380493i	$0.124246\pi$
$827$	629.335i	0.760986i	−0.924784	+	0.380493i	$0.875754\pi$
$828$	0	0
$829$	−475.928	−0.574099	−0.287050	−	0.957916i	$-0.592674\pi$
$829$	−475.928	−0.574099	−0.287050	+	0.957916i	$0.592674\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 71.3818i	− 0.0856925i
$834$	0	0
$835$	−198.093	−0.237237
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 453.978i	− 0.541094i	−0.962707	−	0.270547i	$-0.912795\pi$
$839$	− 453.978i	− 0.541094i	0.962707	−	0.270547i	$-0.0872045\pi$
$840$	0	0
$841$	708.129	0.842008
$842$	0	0
$843$	0	0
$844$	0	0
$845$	729.346i	0.863132i
$846$	0	0
$847$	71.8389	0.0848157
$848$	0	0
$849$	0	0
$850$	0	0
$851$	352.333i	0.414022i
$852$	0	0
$853$	−118.608	−0.139048	−0.0695238	−	0.997580i	$-0.522148\pi$
$853$	−118.608	−0.139048	−0.0695238	+	0.997580i	$0.522148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	578.704i	0.675268i	0.941278	+	0.337634i	$0.109627\pi$
$857$	578.704i	0.675268i	−0.941278	+	0.337634i	$0.890373\pi$
$858$	0	0
$859$	−178.338	−0.207611	−0.103806	−	0.994598i	$-0.533102\pi$
$859$	−178.338	−0.207611	−0.103806	+	0.994598i	$0.533102\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	852.256i	0.987550i	0.869590	+	0.493775i	$0.164383\pi$
$863$	852.256i	0.987550i	−0.869590	+	0.493775i	$0.835617\pi$
$864$	0	0
$865$	−363.369	−0.420080
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 496.179i	− 0.570977i
$870$	0	0
$871$	−1664.21	−1.91069
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 791.342i	− 0.904391i
$876$	0	0
$877$	−348.288	−0.397136	−0.198568	−	0.980087i	$-0.563629\pi$
$877$	−348.288	−0.397136	−0.198568	+	0.980087i	$0.563629\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	557.938i	0.633301i	0.948542	+	0.316650i	$0.102558\pi$
$881$	557.938i	0.633301i	−0.948542	+	0.316650i	$0.897442\pi$
$882$	0	0
$883$	−721.194	−0.816755	−0.408377	−	0.912813i	$-0.633905\pi$
$883$	−721.194	−0.816755	−0.408377	+	0.912813i	$0.633905\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	979.843i	1.10467i	0.833622	+	0.552335i	$0.186263\pi$
$887$	979.843i	1.10467i	−0.833622	+	0.552335i	$0.813737\pi$
$888$	0	0
$889$	675.388	0.759717
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 197.696i	− 0.221385i
$894$	0	0
$895$	−651.253	−0.727657
$896$	0	0
$897$	0	0
$898$	0	0
$899$	178.806i	0.198894i
$900$	0	0
$901$	−915.744	−1.01636
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 674.934i	− 0.745783i
$906$	0	0
$907$	−181.575	−0.200193	−0.100096	−	0.994978i	$-0.531915\pi$
$907$	−181.575	−0.200193	−0.100096	+	0.994978i	$0.531915\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 331.415i	− 0.363793i	−0.983318	−	0.181896i	$-0.941776\pi$
$911$	− 331.415i	− 0.363793i	0.983318	−	0.181896i	$-0.0582236\pi$
$912$	0	0
$913$	287.275	0.314650
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1467.39i	− 1.60020i
$918$	0	0
$919$	−565.851	−0.615725	−0.307862	−	0.951431i	$-0.599614\pi$
$919$	−565.851	−0.615725	−0.307862	+	0.951431i	$0.599614\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	953.493i	1.03304i
$924$	0	0
$925$	−42.9810	−0.0464659
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1673.87i	1.80180i	0.434032	+	0.900898i	$0.357091\pi$
$929$	1673.87i	1.80180i	−0.434032	+	0.900898i	$0.642909\pi$
$930$	0	0
$931$	−39.7482	−0.0426941
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 191.720i	− 0.205048i
$936$	0	0
$937$	−161.303	−0.172149	−0.0860744	−	0.996289i	$-0.527432\pi$
$937$	−161.303	−0.172149	−0.0860744	+	0.996289i	$0.527432\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1263.86i	− 1.34311i	−0.740956	−	0.671553i	$-0.765627\pi$
$941$	− 1263.86i	− 1.34311i	0.740956	−	0.671553i	$-0.234373\pi$
$942$	0	0
$943$	−582.047	−0.617229
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1051.95i	1.11082i	0.831576	+	0.555411i	$0.187439\pi$
$947$	1051.95i	1.11082i	−0.831576	+	0.555411i	$0.812561\pi$
$948$	0	0
$949$	284.910	0.300222
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1367.82i	1.43528i	0.696414	+	0.717640i	$0.254778\pi$
$953$	1367.82i	1.43528i	−0.696414	+	0.717640i	$0.745222\pi$
$954$	0	0
$955$	355.333	0.372076
$956$	0	0
$957$	0	0
$958$	0	0
$959$	704.534i	0.734655i
$960$	0	0
$961$	−720.379	−0.749614
$962$	0	0
$963$	0	0
$964$	0	0
$965$	152.675i	0.158212i
$966$	0	0
$967$	879.133	0.909135	0.454567	−	0.890712i	$-0.349794\pi$
$967$	879.133	0.909135	0.454567	+	0.890712i	$0.349794\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 347.738i	− 0.358123i	−0.983838	−	0.179062i	$-0.942694\pi$
$971$	− 347.738i	− 0.358123i	0.983838	−	0.179062i	$-0.0573061\pi$
$972$	0	0
$973$	−1598.10	−1.64245
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1338.30i	1.36980i	0.728637	+	0.684900i	$0.240154\pi$
$977$	1338.30i	1.36980i	−0.728637	+	0.684900i	$0.759846\pi$
$978$	0	0
$979$	−384.591	−0.392841
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 266.954i	− 0.271571i	−0.990738	−	0.135785i	$-0.956644\pi$
$983$	− 266.954i	− 0.271571i	0.990738	−	0.135785i	$-0.0433558\pi$
$984$	0	0
$985$	−977.608	−0.992495
$986$	0	0
$987$	0	0
$988$	0	0
$989$	203.902i	0.206170i
$990$	0	0
$991$	1705.91	1.72141	0.860703	−	0.509107i	$-0.170024\pi$
$991$	1705.91	1.72141	0.860703	+	0.509107i	$0.170024\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1669.30i	1.67769i
$996$	0	0
$997$	−1760.53	−1.76583	−0.882913	−	0.469537i	$-0.844421\pi$
$997$	−1760.53	−1.76583	−0.882913	+	0.469537i	$0.844421\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.i.d.881.7		8
3.2	odd	2	inner	1584.3.i.d.881.2		8
4.3	odd	2		792.3.i.a.89.7	yes	8
12.11	even	2		792.3.i.a.89.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.3.i.a.89.2	✓	8	12.11	even	2
792.3.i.a.89.7	yes	8	4.3	odd	2
1584.3.i.d.881.2		8	3.2	odd	2	inner
1584.3.i.d.881.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+5.14112i q^{5} -6.53081 q^{7} +3.31662i q^{11} +17.6314 q^{13} +11.2438i q^{17} +6.26101 q^{19} +11.7312i q^{23} -1.43108 q^{25} -11.5270i q^{29} -15.5120 q^{31} -33.5757i q^{35} +30.0340 q^{37} +49.6155i q^{41} +17.3813 q^{43} -31.5758i q^{47} -6.34853 q^{49} +81.4442i q^{53} -17.0512 q^{55} -3.76090i q^{59} -90.3789 q^{61} +90.6450i q^{65} -94.3890 q^{67} +54.0794i q^{71} +16.1593 q^{73} -21.6602i q^{77} -149.604 q^{79} -86.6168i q^{83} -57.8058 q^{85} +115.959i q^{89} -115.147 q^{91} +32.1886i q^{95} +61.8154 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} + 8 q^{13} - 8 q^{19} + 16 q^{25} - 8 q^{31} + 8 q^{37} + 40 q^{43} - 160 q^{49} + 72 q^{61} + 16 q^{67} - 64 q^{79} + 240 q^{85} - 416 q^{91} - 24 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 + 8 * q^13 - 8 * q^19 + 16 * q^25 - 8 * q^31 + 8 * q^37 + 40 * q^43 - 160 * q^49 + 72 * q^61 + 16 * q^67 - 64 * q^79 + 240 * q^85 - 416 * q^91 - 24 * q^97

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