LMFDB - Embedded newform 1584.3.i.b.881.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,3,Mod(881,1584)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.65306824704.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 $ x^8 - 4x^7 - 2x^6 + 20x^5 + x^4 - 40x^3 + 36x^2 - 12x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.6
Root			$2.75726 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1584.881
Dual form			1584.3.i.b.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.10332i q^{5} -2.61086 q^{7} +O(q^{10})$	$q+6.10332i q^{5} -2.61086 q^{7} +3.31662i q^{11} -7.69890 q^{13} +27.5859i q^{17} -3.63254 q^{19} -22.4470i q^{23} -12.2506 q^{25} +16.9284i q^{29} -3.26034 q^{31} -15.9349i q^{35} +15.0843 q^{37} -40.2542i q^{41} -69.4624 q^{43} +21.7183i q^{47} -42.1834 q^{49} -12.1729i q^{53} -20.2424 q^{55} +34.0467i q^{59} +61.3863 q^{61} -46.9889i q^{65} -54.6101 q^{67} -11.5967i q^{71} +41.7131 q^{73} -8.65924i q^{77} -96.9294 q^{79} -89.8729i q^{83} -168.366 q^{85} -117.316i q^{89} +20.1007 q^{91} -22.1705i q^{95} -7.47681 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^7	$ 8 q - 16 q^{7} - 8 q^{13} - 40 q^{19} - 112 q^{25} + 56 q^{31} + 136 q^{37} + 104 q^{43} - 96 q^{49} - 8 q^{61} - 112 q^{67} + 448 q^{73} - 448 q^{79} + 48 q^{85} + 544 q^{91} - 152 q^{97}+O(q^{100}) $ 8 * q - 16 * q^7 - 8 * q^13 - 40 * q^19 - 112 * q^25 + 56 * q^31 + 136 * q^37 + 104 * q^43 - 96 * q^49 - 8 * q^61 - 112 * q^67 + 448 * q^73 - 448 * q^79 + 48 * q^85 + 544 * q^91 - 152 * q^97

Display 100 coefficients 10 coefficients

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
$5$	6.10332i	1.22066i	0.792145	+	0.610332i	$0.208964\pi$
$5$	6.10332i	1.22066i	−0.792145	+	0.610332i	$0.791036\pi$
$6$	0	0
$7$	−2.61086	−0.372980	−0.186490	−	0.982457i	$-0.559711\pi$
$7$	−2.61086	−0.372980	−0.186490	+	0.982457i	$0.559711\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	−7.69890	−0.592223	−0.296111	−	0.955153i	$-0.595690\pi$
$13$	−7.69890	−0.592223	−0.296111	+	0.955153i	$0.595690\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	27.5859i	1.62270i	0.584559	+	0.811351i	$0.301268\pi$
$17$	27.5859i	1.62270i	−0.584559	+	0.811351i	$0.698732\pi$
$18$	0	0
$19$	−3.63254	−0.191186	−0.0955930	−	0.995420i	$-0.530475\pi$
$19$	−3.63254	−0.191186	−0.0955930	+	0.995420i	$0.530475\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 22.4470i	− 0.975957i	−0.872856	−	0.487979i	$-0.837734\pi$
$23$	− 22.4470i	− 0.975957i	0.872856	−	0.487979i	$-0.162266\pi$
$24$	0	0
$25$	−12.2506	−0.490023
$26$	0	0
$27$	0	0
$28$	0	0
$29$	16.9284i	0.583738i	0.956458	+	0.291869i	$0.0942771\pi$
$29$	16.9284i	0.583738i	−0.956458	+	0.291869i	$0.905723\pi$
$30$	0	0
$31$	−3.26034	−0.105172	−0.0525862	−	0.998616i	$-0.516746\pi$
$31$	−3.26034	−0.105172	−0.0525862	+	0.998616i	$0.516746\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 15.9349i	− 0.455284i
$36$	0	0
$37$	15.0843	0.407683	0.203841	−	0.979004i	$-0.434657\pi$
$37$	15.0843	0.407683	0.203841	+	0.979004i	$0.434657\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 40.2542i	− 0.981809i	−0.871214	−	0.490904i	$-0.836666\pi$
$41$	− 40.2542i	− 0.981809i	0.871214	−	0.490904i	$-0.163334\pi$
$42$	0	0
$43$	−69.4624	−1.61540	−0.807702	−	0.589590i	$-0.799289\pi$
$43$	−69.4624	−1.61540	−0.807702	+	0.589590i	$0.799289\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	21.7183i	0.462091i	0.972943	+	0.231045i	$0.0742146\pi$
$47$	21.7183i	0.462091i	−0.972943	+	0.231045i	$0.925785\pi$
$48$	0	0
$49$	−42.1834	−0.860886
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.1729i	− 0.229677i	−0.993384	−	0.114839i	$-0.963365\pi$
$53$	− 12.1729i	− 0.229677i	0.993384	−	0.114839i	$-0.0366351\pi$
$54$	0	0
$55$	−20.2424	−0.368044
$56$	0	0
$57$	0	0
$58$	0	0
$59$	34.0467i	0.577063i	0.957470	+	0.288532i	$0.0931670\pi$
$59$	34.0467i	0.577063i	−0.957470	+	0.288532i	$0.906833\pi$
$60$	0	0
$61$	61.3863	1.00633	0.503166	−	0.864190i	$-0.332168\pi$
$61$	61.3863	1.00633	0.503166	+	0.864190i	$0.332168\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 46.9889i	− 0.722906i
$66$	0	0
$67$	−54.6101	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$67$	−54.6101	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 11.5967i	− 0.163334i	−0.996660	−	0.0816670i	$-0.973976\pi$
$71$	− 11.5967i	− 0.163334i	0.996660	−	0.0816670i	$-0.0260244\pi$
$72$	0	0
$73$	41.7131	0.571412	0.285706	−	0.958317i	$-0.407772\pi$
$73$	41.7131	0.571412	0.285706	+	0.958317i	$0.407772\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.65924i	− 0.112458i
$78$	0	0
$79$	−96.9294	−1.22695	−0.613477	−	0.789712i	$-0.710230\pi$
$79$	−96.9294	−1.22695	−0.613477	+	0.789712i	$0.710230\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 89.8729i	− 1.08281i	−0.840763	−	0.541403i	$-0.817893\pi$
$83$	− 89.8729i	− 1.08281i	0.840763	−	0.541403i	$-0.182107\pi$
$84$	0	0
$85$	−168.366	−1.98078
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 117.316i	− 1.31816i	−0.752074	−	0.659078i	$-0.770947\pi$
$89$	− 117.316i	− 1.31816i	0.752074	−	0.659078i	$-0.229053\pi$
$90$	0	0
$91$	20.1007	0.220887
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 22.1705i	− 0.233374i
$96$	0	0
$97$	−7.47681	−0.0770806	−0.0385403	−	0.999257i	$-0.512271\pi$
$97$	−7.47681	−0.0770806	−0.0385403	+	0.999257i	$0.512271\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	55.4534i	0.549044i	0.961581	+	0.274522i	$0.0885196\pi$
$101$	55.4534i	0.549044i	−0.961581	+	0.274522i	$0.911480\pi$
$102$	0	0
$103$	167.207	1.62337	0.811683	−	0.584098i	$-0.198552\pi$
$103$	167.207	1.62337	0.811683	+	0.584098i	$0.198552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 37.0923i	− 0.346657i	−0.984864	−	0.173329i	$-0.944548\pi$
$107$	− 37.0923i	− 0.346657i	0.984864	−	0.173329i	$-0.0554523\pi$
$108$	0	0
$109$	−97.9327	−0.898465	−0.449232	−	0.893415i	$-0.648303\pi$
$109$	−97.9327	−0.898465	−0.449232	+	0.893415i	$0.648303\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 207.091i	− 1.83267i	−0.400416	−	0.916334i	$-0.631134\pi$
$113$	− 207.091i	− 1.83267i	0.400416	−	0.916334i	$-0.368866\pi$
$114$	0	0
$115$	137.001	1.19132
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 72.0230i	− 0.605235i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	77.8139i	0.622511i
$126$	0	0
$127$	−182.507	−1.43707	−0.718533	−	0.695493i	$-0.755186\pi$
$127$	−182.507	−1.43707	−0.718533	+	0.695493i	$0.755186\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 163.461i	− 1.24780i	−0.781506	−	0.623898i	$-0.785548\pi$
$131$	− 163.461i	− 1.24780i	0.781506	−	0.623898i	$-0.214452\pi$
$132$	0	0
$133$	9.48404	0.0713086
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 199.563i	− 1.45666i	−0.685224	−	0.728332i	$-0.740296\pi$
$137$	− 199.563i	− 1.45666i	0.685224	−	0.728332i	$-0.259704\pi$
$138$	0	0
$139$	−2.75894	−0.0198485	−0.00992425	−	0.999951i	$-0.503159\pi$
$139$	−2.75894	−0.0198485	−0.00992425	+	0.999951i	$0.503159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 25.5344i	− 0.178562i
$144$	0	0
$145$	−103.320	−0.712549
$146$	0	0
$147$	0	0
$148$	0	0
$149$	79.7092i	0.534961i	0.963563	+	0.267481i	$0.0861911\pi$
$149$	79.7092i	0.534961i	−0.963563	+	0.267481i	$0.913809\pi$
$150$	0	0
$151$	149.776	0.991894	0.495947	−	0.868353i	$-0.334821\pi$
$151$	149.776	0.991894	0.495947	+	0.868353i	$0.334821\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 19.8989i	− 0.128380i
$156$	0	0
$157$	−273.723	−1.74346	−0.871730	−	0.489986i	$-0.837002\pi$
$157$	−273.723	−1.74346	−0.871730	+	0.489986i	$0.837002\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	58.6060i	0.364012i
$162$	0	0
$163$	−289.639	−1.77693	−0.888464	−	0.458946i	$-0.848227\pi$
$163$	−289.639	−1.77693	−0.888464	+	0.458946i	$0.848227\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	47.9894i	0.287362i	0.989624	+	0.143681i	$0.0458939\pi$
$167$	47.9894i	0.287362i	−0.989624	+	0.143681i	$0.954106\pi$
$168$	0	0
$169$	−109.727	−0.649272
$170$	0	0
$171$	0	0
$172$	0	0
$173$	35.0195i	0.202425i	0.994865	+	0.101212i	$0.0322722\pi$
$173$	35.0195i	0.202425i	−0.994865	+	0.101212i	$0.967728\pi$
$174$	0	0
$175$	31.9845	0.182769
$176$	0	0
$177$	0	0
$178$	0	0
$179$	190.174i	1.06242i	0.847239	+	0.531212i	$0.178263\pi$
$179$	190.174i	1.06242i	−0.847239	+	0.531212i	$0.821737\pi$
$180$	0	0
$181$	−78.2856	−0.432517	−0.216258	−	0.976336i	$-0.569385\pi$
$181$	−78.2856	−0.432517	−0.216258	+	0.976336i	$0.569385\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	92.0642i	0.497644i
$186$	0	0
$187$	−91.4922	−0.489263
$188$	0	0
$189$	0	0
$190$	0	0
$191$	99.6608i	0.521784i	0.965368	+	0.260892i	$0.0840167\pi$
$191$	99.6608i	0.521784i	−0.965368	+	0.260892i	$0.915983\pi$
$192$	0	0
$193$	29.9880	0.155378	0.0776891	−	0.996978i	$-0.475246\pi$
$193$	29.9880	0.155378	0.0776891	+	0.996978i	$0.475246\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	82.2920i	0.417726i	0.977945	+	0.208863i	$0.0669763\pi$
$197$	82.2920i	0.417726i	−0.977945	+	0.208863i	$0.933024\pi$
$198$	0	0
$199$	−217.671	−1.09383	−0.546913	−	0.837189i	$-0.684197\pi$
$199$	−217.671	−1.09383	−0.546913	+	0.837189i	$0.684197\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 44.1977i	− 0.217723i
$204$	0	0
$205$	245.684	1.19846
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 12.0478i	− 0.0576448i
$210$	0	0
$211$	105.008	0.497669	0.248834	−	0.968546i	$-0.419952\pi$
$211$	105.008	0.497669	0.248834	+	0.968546i	$0.419952\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 423.952i	− 1.97187i
$216$	0	0
$217$	8.51230	0.0392272
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 212.381i	− 0.961002i
$222$	0	0
$223$	−238.589	−1.06991	−0.534953	−	0.844882i	$-0.679671\pi$
$223$	−238.589	−1.06991	−0.534953	+	0.844882i	$0.679671\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	326.217i	1.43708i	0.695485	+	0.718540i	$0.255190\pi$
$227$	326.217i	1.43708i	−0.695485	+	0.718540i	$0.744810\pi$
$228$	0	0
$229$	233.208	1.01837	0.509187	−	0.860656i	$-0.329946\pi$
$229$	233.208	1.01837	0.509187	+	0.860656i	$0.329946\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	65.6551i	0.281781i	0.990025	+	0.140891i	$0.0449966\pi$
$233$	65.6551i	0.281781i	−0.990025	+	0.140891i	$0.955003\pi$
$234$	0	0
$235$	−132.554	−0.564058
$236$	0	0
$237$	0	0
$238$	0	0
$239$	214.400i	0.897072i	0.893765	+	0.448536i	$0.148054\pi$
$239$	214.400i	0.897072i	−0.893765	+	0.448536i	$0.851946\pi$
$240$	0	0
$241$	−404.958	−1.68032	−0.840161	−	0.542337i	$-0.817540\pi$
$241$	−404.958	−1.68032	−0.840161	+	0.542337i	$0.817540\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 257.459i	− 1.05085i
$246$	0	0
$247$	27.9665	0.113225
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 300.918i	− 1.19888i	−0.800420	−	0.599439i	$-0.795390\pi$
$251$	− 300.918i	− 1.19888i	0.800420	−	0.599439i	$-0.204610\pi$
$252$	0	0
$253$	74.4483	0.294262
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 424.503i	− 1.65176i	−0.563844	−	0.825881i	$-0.690678\pi$
$257$	− 424.503i	− 1.65176i	0.563844	−	0.825881i	$-0.309322\pi$
$258$	0	0
$259$	−39.3829	−0.152058
$260$	0	0
$261$	0	0
$262$	0	0
$263$	306.813i	1.16659i	0.812261	+	0.583294i	$0.198236\pi$
$263$	306.813i	1.16659i	−0.812261	+	0.583294i	$0.801764\pi$
$264$	0	0
$265$	74.2951	0.280359
$266$	0	0
$267$	0	0
$268$	0	0
$269$	248.773i	0.924809i	0.886669	+	0.462404i	$0.153013\pi$
$269$	248.773i	0.924809i	−0.886669	+	0.462404i	$0.846987\pi$
$270$	0	0
$271$	−230.792	−0.851631	−0.425816	−	0.904810i	$-0.640013\pi$
$271$	−230.792	−0.851631	−0.425816	+	0.904810i	$0.640013\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 40.6306i	− 0.147747i
$276$	0	0
$277$	442.379	1.59703	0.798517	−	0.601972i	$-0.205618\pi$
$277$	442.379	1.59703	0.798517	+	0.601972i	$0.205618\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	219.013i	0.779407i	0.920940	+	0.389704i	$0.127423\pi$
$281$	219.013i	0.779407i	−0.920940	+	0.389704i	$0.872577\pi$
$282$	0	0
$283$	144.620	0.511023	0.255512	−	0.966806i	$-0.417756\pi$
$283$	144.620	0.511023	0.255512	+	0.966806i	$0.417756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	105.098i	0.366195i
$288$	0	0
$289$	−471.984	−1.63316
$290$	0	0
$291$	0	0
$292$	0	0
$293$	103.257i	0.352412i	0.984353	+	0.176206i	$0.0563824\pi$
$293$	103.257i	0.352412i	−0.984353	+	0.176206i	$0.943618\pi$
$294$	0	0
$295$	−207.798	−0.704401
$296$	0	0
$297$	0	0
$298$	0	0
$299$	172.817i	0.577984i
$300$	0	0
$301$	181.357	0.602514
$302$	0	0
$303$	0	0
$304$	0	0
$305$	374.660i	1.22839i
$306$	0	0
$307$	−332.933	−1.08447	−0.542237	−	0.840226i	$-0.682422\pi$
$307$	−332.933	−1.08447	−0.542237	+	0.840226i	$0.682422\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 534.673i	− 1.71921i	−0.510962	−	0.859603i	$-0.670711\pi$
$311$	− 534.673i	− 1.71921i	0.510962	−	0.859603i	$-0.329289\pi$
$312$	0	0
$313$	−312.885	−0.999631	−0.499816	−	0.866132i	$-0.666599\pi$
$313$	−312.885	−0.999631	−0.499816	+	0.866132i	$0.666599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	260.003i	0.820200i	0.912041	+	0.410100i	$0.134506\pi$
$317$	260.003i	0.820200i	−0.912041	+	0.410100i	$0.865494\pi$
$318$	0	0
$319$	−56.1452	−0.176004
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 100.207i	− 0.310238i
$324$	0	0
$325$	94.3159	0.290203
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 56.7034i	− 0.172351i
$330$	0	0
$331$	545.194	1.64711	0.823555	−	0.567236i	$-0.191987\pi$
$331$	545.194	1.64711	0.823555	+	0.567236i	$0.191987\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 333.303i	− 0.994934i
$336$	0	0
$337$	321.319	0.953468	0.476734	−	0.879048i	$-0.341820\pi$
$337$	321.319	0.953468	0.476734	+	0.879048i	$0.341820\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 10.8133i	− 0.0317107i
$342$	0	0
$343$	238.067	0.694073
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 307.133i	− 0.885111i	−0.896741	−	0.442555i	$-0.854072\pi$
$347$	− 307.133i	− 0.885111i	0.896741	−	0.442555i	$-0.145928\pi$
$348$	0	0
$349$	483.011	1.38399	0.691993	−	0.721904i	$-0.256733\pi$
$349$	483.011	1.38399	0.691993	+	0.721904i	$0.256733\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 137.108i	− 0.388408i	−0.980961	−	0.194204i	$-0.937788\pi$
$353$	− 137.108i	− 0.388408i	0.980961	−	0.194204i	$-0.0622124\pi$
$354$	0	0
$355$	70.7785	0.199376
$356$	0	0
$357$	0	0
$358$	0	0
$359$	236.029i	0.657463i	0.944423	+	0.328731i	$0.106621\pi$
$359$	236.029i	0.657463i	−0.944423	+	0.328731i	$0.893379\pi$
$360$	0	0
$361$	−347.805	−0.963448
$362$	0	0
$363$	0	0
$364$	0	0
$365$	254.589i	0.697503i
$366$	0	0
$367$	332.474	0.905924	0.452962	−	0.891530i	$-0.350367\pi$
$367$	332.474	0.905924	0.452962	+	0.891530i	$0.350367\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.7817i	0.0856650i
$372$	0	0
$373$	−191.412	−0.513169	−0.256585	−	0.966522i	$-0.582597\pi$
$373$	−191.412	−0.513169	−0.256585	+	0.966522i	$0.582597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 130.330i	− 0.345703i
$378$	0	0
$379$	387.997	1.02374	0.511870	−	0.859063i	$-0.328953\pi$
$379$	387.997	1.02374	0.511870	+	0.859063i	$0.328953\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 119.163i	− 0.311131i	−0.987826	−	0.155566i	$-0.950280\pi$
$383$	− 119.163i	− 0.311131i	0.987826	−	0.155566i	$-0.0497200\pi$
$384$	0	0
$385$	52.8502	0.137273
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 204.409i	− 0.525473i	−0.964868	−	0.262736i	$-0.915375\pi$
$389$	− 204.409i	− 0.525473i	0.964868	−	0.262736i	$-0.0846249\pi$
$390$	0	0
$391$	619.222	1.58369
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 591.592i	− 1.49770i
$396$	0	0
$397$	543.945	1.37014	0.685069	−	0.728478i	$-0.259772\pi$
$397$	543.945	1.37014	0.685069	+	0.728478i	$0.259772\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	112.010i	0.279326i	0.990199	+	0.139663i	$0.0446020\pi$
$401$	112.010i	0.279326i	−0.990199	+	0.139663i	$0.955398\pi$
$402$	0	0
$403$	25.1010	0.0622855
$404$	0	0
$405$	0	0
$406$	0	0
$407$	50.0289i	0.122921i
$408$	0	0
$409$	−156.376	−0.382337	−0.191168	−	0.981557i	$-0.561228\pi$
$409$	−156.376	−0.382337	−0.191168	+	0.981557i	$0.561228\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 88.8912i	− 0.215233i
$414$	0	0
$415$	548.524	1.32174
$416$	0	0
$417$	0	0
$418$	0	0
$419$	618.122i	1.47523i	0.675221	+	0.737616i	$0.264048\pi$
$419$	618.122i	1.47523i	−0.675221	+	0.737616i	$0.735952\pi$
$420$	0	0
$421$	289.351	0.687295	0.343647	−	0.939099i	$-0.388338\pi$
$421$	289.351	0.687295	0.343647	+	0.939099i	$0.388338\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 337.944i	− 0.795161i
$426$	0	0
$427$	−160.271	−0.375342
$428$	0	0
$429$	0	0
$430$	0	0
$431$	608.054i	1.41080i	0.708811	+	0.705399i	$0.249232\pi$
$431$	608.054i	1.41080i	−0.708811	+	0.705399i	$0.750768\pi$
$432$	0	0
$433$	−39.7517	−0.0918054	−0.0459027	−	0.998946i	$-0.514616\pi$
$433$	−39.7517	−0.0918054	−0.0459027	+	0.998946i	$0.514616\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	81.5396i	0.186589i
$438$	0	0
$439$	474.376	1.08058	0.540292	−	0.841478i	$-0.318314\pi$
$439$	474.376	1.08058	0.540292	+	0.841478i	$0.318314\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	140.746i	0.317711i	0.987302	+	0.158855i	$0.0507804\pi$
$443$	140.746i	0.317711i	−0.987302	+	0.158855i	$0.949220\pi$
$444$	0	0
$445$	716.017	1.60903
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 193.508i	− 0.430976i	−0.976506	−	0.215488i	$-0.930866\pi$
$449$	− 193.508i	− 0.430976i	0.976506	−	0.215488i	$-0.0691342\pi$
$450$	0	0
$451$	133.508	0.296026
$452$	0	0
$453$	0	0
$454$	0	0
$455$	122.681i	0.269629i
$456$	0	0
$457$	−31.5272	−0.0689874	−0.0344937	−	0.999405i	$-0.510982\pi$
$457$	−31.5272	−0.0689874	−0.0344937	+	0.999405i	$0.510982\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	502.003i	1.08894i	0.838779	+	0.544471i	$0.183270\pi$
$461$	502.003i	1.08894i	−0.838779	+	0.544471i	$0.816730\pi$
$462$	0	0
$463$	−786.087	−1.69781	−0.848906	−	0.528544i	$-0.822738\pi$
$463$	−786.087	−1.69781	−0.848906	+	0.528544i	$0.822738\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	538.345i	1.15277i	0.817177	+	0.576387i	$0.195538\pi$
$467$	538.345i	1.15277i	−0.817177	+	0.576387i	$0.804462\pi$
$468$	0	0
$469$	142.579	0.304007
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 230.381i	− 0.487063i
$474$	0	0
$475$	44.5006	0.0936856
$476$	0	0
$477$	0	0
$478$	0	0
$479$	54.9462i	0.114710i	0.998354	+	0.0573551i	$0.0182667\pi$
$479$	54.9462i	0.114710i	−0.998354	+	0.0573551i	$0.981733\pi$
$480$	0	0
$481$	−116.132	−0.241439
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 45.6334i	− 0.0940895i
$486$	0	0
$487$	−281.751	−0.578543	−0.289272	−	0.957247i	$-0.593413\pi$
$487$	−281.751	−0.578543	−0.289272	+	0.957247i	$0.593413\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	575.037i	1.17116i	0.810616	+	0.585578i	$0.199132\pi$
$491$	575.037i	1.17116i	−0.810616	+	0.585578i	$0.800868\pi$
$492$	0	0
$493$	−466.986	−0.947233
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.2774i	0.0609203i
$498$	0	0
$499$	173.598	0.347892	0.173946	−	0.984755i	$-0.444348\pi$
$499$	173.598	0.347892	0.173946	+	0.984755i	$0.444348\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	195.789i	0.389242i	0.980879	+	0.194621i	$0.0623477\pi$
$503$	195.789i	0.389242i	−0.980879	+	0.194621i	$0.937652\pi$
$504$	0	0
$505$	−338.450	−0.670199
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 176.019i	− 0.345813i	−0.984938	−	0.172906i	$-0.944684\pi$
$509$	− 176.019i	− 0.345813i	0.984938	−	0.172906i	$-0.0553158\pi$
$510$	0	0
$511$	−108.907	−0.213125
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1020.52i	1.98159i
$516$	0	0
$517$	−72.0314	−0.139326
$518$	0	0
$519$	0	0
$520$	0	0
$521$	907.916i	1.74264i	0.490714	+	0.871321i	$0.336736\pi$
$521$	907.916i	1.74264i	−0.490714	+	0.871321i	$0.663264\pi$
$522$	0	0
$523$	439.027	0.839441	0.419720	−	0.907654i	$-0.362128\pi$
$523$	439.027	0.839441	0.419720	+	0.907654i	$0.362128\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 89.9396i	− 0.170663i
$528$	0	0
$529$	25.1315	0.0475076
$530$	0	0
$531$	0	0
$532$	0	0
$533$	309.913i	0.581450i
$534$	0	0
$535$	226.387	0.423152
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 139.907i	− 0.259567i
$540$	0	0
$541$	−40.4729	−0.0748113	−0.0374057	−	0.999300i	$-0.511909\pi$
$541$	−40.4729	−0.0748113	−0.0374057	+	0.999300i	$0.511909\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 597.715i	− 1.09672i
$546$	0	0
$547$	−178.824	−0.326918	−0.163459	−	0.986550i	$-0.552265\pi$
$547$	−178.824	−0.326918	−0.163459	+	0.986550i	$0.552265\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 61.4930i	− 0.111603i
$552$	0	0
$553$	253.069	0.457630
$554$	0	0
$555$	0	0
$556$	0	0
$557$	358.907i	0.644357i	0.946679	+	0.322179i	$0.104415\pi$
$557$	358.907i	0.644357i	−0.946679	+	0.322179i	$0.895585\pi$
$558$	0	0
$559$	534.784	0.956680
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 205.504i	− 0.365016i	−0.983204	−	0.182508i	$-0.941578\pi$
$563$	− 205.504i	− 0.365016i	0.983204	−	0.182508i	$-0.0584216\pi$
$564$	0	0
$565$	1263.95	2.23707
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 63.9111i	− 0.112322i	−0.998422	−	0.0561609i	$-0.982114\pi$
$569$	− 63.9111i	− 0.112322i	0.998422	−	0.0561609i	$-0.0178860\pi$
$570$	0	0
$571$	−645.953	−1.13127	−0.565633	−	0.824657i	$-0.691368\pi$
$571$	−645.953	−1.13127	−0.565633	+	0.824657i	$0.691368\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	274.989i	0.478241i
$576$	0	0
$577$	−30.5030	−0.0528648	−0.0264324	−	0.999651i	$-0.508415\pi$
$577$	−30.5030	−0.0528648	−0.0264324	+	0.999651i	$0.508415\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	234.646i	0.403865i
$582$	0	0
$583$	40.3729	0.0692503
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 433.560i	− 0.738604i	−0.929309	−	0.369302i	$-0.879597\pi$
$587$	− 433.560i	− 0.738604i	0.929309	−	0.369302i	$-0.120403\pi$
$588$	0	0
$589$	11.8433	0.0201075
$590$	0	0
$591$	0	0
$592$	0	0
$593$	583.923i	0.984693i	0.870399	+	0.492347i	$0.163861\pi$
$593$	583.923i	0.984693i	−0.870399	+	0.492347i	$0.836139\pi$
$594$	0	0
$595$	439.580	0.738790
$596$	0	0
$597$	0	0
$598$	0	0
$599$	62.6790i	0.104639i	0.998630	+	0.0523197i	$0.0166615\pi$
$599$	62.6790i	0.104639i	−0.998630	+	0.0523197i	$0.983339\pi$
$600$	0	0
$601$	−456.794	−0.760057	−0.380028	−	0.924975i	$-0.624086\pi$
$601$	−456.794	−0.760057	−0.380028	+	0.924975i	$0.624086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 67.1366i	− 0.110970i
$606$	0	0
$607$	−849.069	−1.39880	−0.699398	−	0.714732i	$-0.746549\pi$
$607$	−849.069	−1.39880	−0.699398	+	0.714732i	$0.746549\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 167.207i	− 0.273661i
$612$	0	0
$613$	217.267	0.354433	0.177216	−	0.984172i	$-0.443291\pi$
$613$	217.267	0.354433	0.177216	+	0.984172i	$0.443291\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 417.676i	− 0.676947i	−0.940976	−	0.338473i	$-0.890090\pi$
$617$	− 417.676i	− 0.676947i	0.940976	−	0.338473i	$-0.109910\pi$
$618$	0	0
$619$	−542.307	−0.876101	−0.438051	−	0.898950i	$-0.644331\pi$
$619$	−542.307	−0.876101	−0.438051	+	0.898950i	$0.644331\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	306.295i	0.491646i
$624$	0	0
$625$	−781.188	−1.24990
$626$	0	0
$627$	0	0
$628$	0	0
$629$	416.114i	0.661548i
$630$	0	0
$631$	660.757	1.04716	0.523579	−	0.851977i	$-0.324596\pi$
$631$	660.757	1.04716	0.523579	+	0.851977i	$0.324596\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1113.90i	− 1.75418i
$636$	0	0
$637$	324.766	0.509836
$638$	0	0
$639$	0	0
$640$	0	0
$641$	656.483i	1.02416i	0.858939	+	0.512078i	$0.171124\pi$
$641$	656.483i	1.02416i	−0.858939	+	0.512078i	$0.828876\pi$
$642$	0	0
$643$	−193.650	−0.301166	−0.150583	−	0.988597i	$-0.548115\pi$
$643$	−193.650	−0.301166	−0.150583	+	0.988597i	$0.548115\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	484.835i	0.749359i	0.927154	+	0.374679i	$0.122247\pi$
$647$	484.835i	0.749359i	−0.927154	+	0.374679i	$0.877753\pi$
$648$	0	0
$649$	−112.920	−0.173991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 530.535i	− 0.812457i	−0.913771	−	0.406229i	$-0.866844\pi$
$653$	− 530.535i	− 0.812457i	0.913771	−	0.406229i	$-0.133156\pi$
$654$	0	0
$655$	997.658	1.52314
$656$	0	0
$657$	0	0
$658$	0	0
$659$	513.861i	0.779759i	0.920866	+	0.389880i	$0.127483\pi$
$659$	513.861i	0.779759i	−0.920866	+	0.389880i	$0.872517\pi$
$660$	0	0
$661$	356.601	0.539487	0.269743	−	0.962932i	$-0.413061\pi$
$661$	356.601	0.539487	0.269743	+	0.962932i	$0.413061\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	57.8842i	0.0870439i
$666$	0	0
$667$	379.992	0.569703
$668$	0	0
$669$	0	0
$670$	0	0
$671$	203.595i	0.303421i
$672$	0	0
$673$	−551.643	−0.819677	−0.409839	−	0.912158i	$-0.634415\pi$
$673$	−551.643	−0.819677	−0.409839	+	0.912158i	$0.634415\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	673.629i	0.995021i	0.867458	+	0.497510i	$0.165752\pi$
$677$	673.629i	0.995021i	−0.867458	+	0.497510i	$0.834248\pi$
$678$	0	0
$679$	19.5209	0.0287495
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.2083i	0.0325158i	0.999868	+	0.0162579i	$0.00517528\pi$
$683$	22.2083i	0.0325158i	−0.999868	+	0.0162579i	$0.994825\pi$
$684$	0	0
$685$	1218.00	1.77810
$686$	0	0
$687$	0	0
$688$	0	0
$689$	93.7179i	0.136020i
$690$	0	0
$691$	932.929	1.35011	0.675057	−	0.737766i	$-0.264119\pi$
$691$	932.929	1.35011	0.675057	+	0.737766i	$0.264119\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 16.8387i	− 0.0242284i
$696$	0	0
$697$	1110.45	1.59318
$698$	0	0
$699$	0	0
$700$	0	0
$701$	690.617i	0.985188i	0.870259	+	0.492594i	$0.163951\pi$
$701$	690.617i	0.985188i	−0.870259	+	0.492594i	$0.836049\pi$
$702$	0	0
$703$	−54.7941	−0.0779433
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 144.781i	− 0.204782i
$708$	0	0
$709$	−432.626	−0.610192	−0.305096	−	0.952322i	$-0.598689\pi$
$709$	−432.626	−0.610192	−0.305096	+	0.952322i	$0.598689\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	73.1850i	0.102644i
$714$	0	0
$715$	155.844	0.217964
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1085.00i	− 1.50904i	−0.656274	−	0.754522i	$-0.727869\pi$
$719$	− 1085.00i	− 1.50904i	0.656274	−	0.754522i	$-0.272131\pi$
$720$	0	0
$721$	−436.553	−0.605483
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 207.383i	− 0.286045i
$726$	0	0
$727$	−688.568	−0.947137	−0.473568	−	0.880757i	$-0.657034\pi$
$727$	−688.568	−0.947137	−0.473568	+	0.880757i	$0.657034\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1916.19i	− 2.62132i
$732$	0	0
$733$	−259.491	−0.354012	−0.177006	−	0.984210i	$-0.556641\pi$
$733$	−259.491	−0.354012	−0.177006	+	0.984210i	$0.556641\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 181.121i	− 0.245755i
$738$	0	0
$739$	−195.434	−0.264457	−0.132228	−	0.991219i	$-0.542213\pi$
$739$	−195.434	−0.264457	−0.132228	+	0.991219i	$0.542213\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	322.257i	0.433725i	0.976202	+	0.216862i	$0.0695823\pi$
$743$	322.257i	0.433725i	−0.976202	+	0.216862i	$0.930418\pi$
$744$	0	0
$745$	−486.491	−0.653008
$746$	0	0
$747$	0	0
$748$	0	0
$749$	96.8429i	0.129296i
$750$	0	0
$751$	−407.900	−0.543142	−0.271571	−	0.962418i	$-0.587543\pi$
$751$	−407.900	−0.543142	−0.271571	+	0.962418i	$0.587543\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	914.131i	1.21077i
$756$	0	0
$757$	596.605	0.788118	0.394059	−	0.919085i	$-0.371071\pi$
$757$	596.605	0.788118	0.394059	+	0.919085i	$0.371071\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	671.334i	0.882174i	0.897464	+	0.441087i	$0.145407\pi$
$761$	671.334i	0.882174i	−0.897464	+	0.441087i	$0.854593\pi$
$762$	0	0
$763$	255.689	0.335109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 262.122i	− 0.341750i
$768$	0	0
$769$	947.641	1.23230	0.616151	−	0.787628i	$-0.288691\pi$
$769$	947.641	1.23230	0.616151	+	0.787628i	$0.288691\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1038.33i	1.34325i	0.740893	+	0.671623i	$0.234403\pi$
$773$	1038.33i	1.34325i	−0.740893	+	0.671623i	$0.765597\pi$
$774$	0	0
$775$	39.9411	0.0515369
$776$	0	0
$777$	0	0
$778$	0	0
$779$	146.225i	0.187708i
$780$	0	0
$781$	38.4620	0.0492471
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1670.62i	− 2.12818i
$786$	0	0
$787$	12.9336	0.0164340	0.00821702	−	0.999966i	$-0.497384\pi$
$787$	12.9336	0.0164340	0.00821702	+	0.999966i	$0.497384\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	540.687i	0.683548i
$792$	0	0
$793$	−472.607	−0.595973
$794$	0	0
$795$	0	0
$796$	0	0
$797$	459.455i	0.576480i	0.957558	+	0.288240i	$0.0930701\pi$
$797$	459.455i	0.576480i	−0.957558	+	0.288240i	$0.906930\pi$
$798$	0	0
$799$	−599.119	−0.749836
$800$	0	0
$801$	0	0
$802$	0	0
$803$	138.347i	0.172287i
$804$	0	0
$805$	−357.692	−0.444337
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1546.11i	− 1.91113i	−0.294773	−	0.955567i	$-0.595244\pi$
$809$	− 1546.11i	− 1.91113i	0.294773	−	0.955567i	$-0.404756\pi$
$810$	0	0
$811$	−1461.97	−1.80268	−0.901340	−	0.433112i	$-0.857415\pi$
$811$	−1461.97	−1.80268	−0.901340	+	0.433112i	$0.857415\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1767.76i	− 2.16903i
$816$	0	0
$817$	252.325	0.308843
$818$	0	0
$819$	0	0
$820$	0	0
$821$	795.413i	0.968834i	0.874837	+	0.484417i	$0.160968\pi$
$821$	795.413i	0.968834i	−0.874837	+	0.484417i	$0.839032\pi$
$822$	0	0
$823$	73.3941	0.0891788	0.0445894	−	0.999005i	$-0.485802\pi$
$823$	73.3941	0.0891788	0.0445894	+	0.999005i	$0.485802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	858.149i	1.03766i	0.854876	+	0.518832i	$0.173633\pi$
$827$	858.149i	1.03766i	−0.854876	+	0.518832i	$0.826367\pi$
$828$	0	0
$829$	116.737	0.140816	0.0704081	−	0.997518i	$-0.477570\pi$
$829$	116.737	0.140816	0.0704081	+	0.997518i	$0.477570\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1163.67i	− 1.39696i
$834$	0	0
$835$	−292.895	−0.350772
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 119.850i	− 0.142849i	−0.997446	−	0.0714243i	$-0.977246\pi$
$839$	− 119.850i	− 0.142849i	0.997446	−	0.0714243i	$-0.0227544\pi$
$840$	0	0
$841$	554.429	0.659250
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 669.699i	− 0.792544i
$846$	0	0
$847$	28.7195	0.0339073
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 338.597i	− 0.397881i
$852$	0	0
$853$	−480.900	−0.563775	−0.281887	−	0.959447i	$-0.590960\pi$
$853$	−480.900	−0.563775	−0.281887	+	0.959447i	$0.590960\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 898.929i	− 1.04893i	−0.851433	−	0.524463i	$-0.824266\pi$
$857$	− 898.929i	− 1.04893i	0.851433	−	0.524463i	$-0.175734\pi$
$858$	0	0
$859$	892.879	1.03944	0.519720	−	0.854337i	$-0.326036\pi$
$859$	892.879	1.03944	0.519720	+	0.854337i	$0.326036\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1385.04i	1.60491i	0.596712	+	0.802455i	$0.296473\pi$
$863$	1385.04i	1.60491i	−0.596712	+	0.802455i	$0.703527\pi$
$864$	0	0
$865$	−213.735	−0.247093
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 321.479i	− 0.369941i
$870$	0	0
$871$	420.437	0.482706
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 203.161i	− 0.232184i
$876$	0	0
$877$	−8.82028	−0.0100573	−0.00502867	−	0.999987i	$-0.501601\pi$
$877$	−8.82028	−0.0100573	−0.00502867	+	0.999987i	$0.501601\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 871.048i	− 0.988704i	−0.869262	−	0.494352i	$-0.835405\pi$
$881$	− 871.048i	− 0.988704i	0.869262	−	0.494352i	$-0.164595\pi$
$882$	0	0
$883$	426.150	0.482616	0.241308	−	0.970449i	$-0.422424\pi$
$883$	426.150	0.482616	0.241308	+	0.970449i	$0.422424\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1032.71i	− 1.16427i	−0.813091	−	0.582136i	$-0.802217\pi$
$887$	− 1032.71i	− 1.16427i	0.813091	−	0.582136i	$-0.197783\pi$
$888$	0	0
$889$	476.501	0.535997
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 78.8924i	− 0.0883454i
$894$	0	0
$895$	−1160.69	−1.29686
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 55.1924i	− 0.0613931i
$900$	0	0
$901$	335.801	0.372698
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 477.802i	− 0.527958i
$906$	0	0
$907$	−1283.21	−1.41479	−0.707393	−	0.706820i	$-0.750129\pi$
$907$	−1283.21	−1.41479	−0.707393	+	0.706820i	$0.750129\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1125.40i	1.23535i	0.786433	+	0.617675i	$0.211925\pi$
$911$	1125.40i	1.23535i	−0.786433	+	0.617675i	$0.788075\pi$
$912$	0	0
$913$	298.075	0.326478
$914$	0	0
$915$	0	0
$916$	0	0
$917$	426.775i	0.465403i
$918$	0	0
$919$	1149.37	1.25067	0.625336	−	0.780356i	$-0.284962\pi$
$919$	1149.37	1.25067	0.625336	+	0.780356i	$0.284962\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	89.2819i	0.0967301i
$924$	0	0
$925$	−184.791	−0.199774
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 250.860i	− 0.270032i	−0.990843	−	0.135016i	$-0.956891\pi$
$929$	− 250.860i	− 0.270032i	0.990843	−	0.135016i	$-0.0431087\pi$
$930$	0	0
$931$	153.233	0.164589
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 558.407i	− 0.597226i
$936$	0	0
$937$	−817.848	−0.872837	−0.436418	−	0.899744i	$-0.643753\pi$
$937$	−817.848	−0.872837	−0.436418	+	0.899744i	$0.643753\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1289.31i	− 1.37015i	−0.728472	−	0.685076i	$-0.759769\pi$
$941$	− 1289.31i	− 1.37015i	0.728472	−	0.685076i	$-0.240231\pi$
$942$	0	0
$943$	−903.586	−0.958203
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.5616i	0.0174885i	0.999962	+	0.00874423i	$0.00278341\pi$
$947$	16.5616i	0.0174885i	−0.999962	+	0.00874423i	$0.997217\pi$
$948$	0	0
$949$	−321.145	−0.338403
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 905.921i	− 0.950599i	−0.879824	−	0.475300i	$-0.842340\pi$
$953$	− 905.921i	− 0.950599i	0.879824	−	0.475300i	$-0.157660\pi$
$954$	0	0
$955$	−608.262	−0.636924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	521.031i	0.543307i
$960$	0	0
$961$	−950.370	−0.988939
$962$	0	0
$963$	0	0
$964$	0	0
$965$	183.027i	0.189665i
$966$	0	0
$967$	1034.62	1.06993	0.534965	−	0.844874i	$-0.320325\pi$
$967$	1034.62	1.06993	0.534965	+	0.844874i	$0.320325\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1733.52i	1.78529i	0.450757	+	0.892647i	$0.351154\pi$
$971$	1733.52i	1.78529i	−0.450757	+	0.892647i	$0.648846\pi$
$972$	0	0
$973$	7.20321	0.00740309
$974$	0	0
$975$	0	0
$976$	0	0
$977$	835.320i	0.854985i	0.904019	+	0.427492i	$0.140603\pi$
$977$	835.320i	0.854985i	−0.904019	+	0.427492i	$0.859397\pi$
$978$	0	0
$979$	389.093	0.397439
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1505.21i	− 1.53124i	−0.643294	−	0.765619i	$-0.722433\pi$
$983$	− 1505.21i	− 1.53124i	0.643294	−	0.765619i	$-0.277567\pi$
$984$	0	0
$985$	−502.255	−0.509904
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1559.22i	1.57657i
$990$	0	0
$991$	141.776	0.143064	0.0715318	−	0.997438i	$-0.477211\pi$
$991$	141.776	0.143064	0.0715318	+	0.997438i	$0.477211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1328.52i	− 1.33520i
$996$	0	0
$997$	−1566.45	−1.57116	−0.785579	−	0.618761i	$-0.787635\pi$
$997$	−1566.45	−1.57116	−0.785579	+	0.618761i	$0.787635\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.i.b.881.6		8
3.2	odd	2	inner	1584.3.i.b.881.3		8
4.3	odd	2		99.3.b.a.89.8	yes	8
12.11	even	2		99.3.b.a.89.1	✓	8
44.43	even	2		1089.3.b.g.485.1		8
132.131	odd	2		1089.3.b.g.485.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.1	✓	8	12.11	even	2
99.3.b.a.89.8	yes	8	4.3	odd	2
1089.3.b.g.485.1		8	44.43	even	2
1089.3.b.g.485.8		8	132.131	odd	2
1584.3.i.b.881.3		8	3.2	odd	2	inner
1584.3.i.b.881.6		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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+6.10332i q^{5} -2.61086 q^{7} +O(q^{10})\)	\(q+6.10332i q^{5} -2.61086 q^{7} +3.31662i q^{11} -7.69890 q^{13} +27.5859i q^{17} -3.63254 q^{19} -22.4470i q^{23} -12.2506 q^{25} +16.9284i q^{29} -3.26034 q^{31} -15.9349i q^{35} +15.0843 q^{37} -40.2542i q^{41} -69.4624 q^{43} +21.7183i q^{47} -42.1834 q^{49} -12.1729i q^{53} -20.2424 q^{55} +34.0467i q^{59} +61.3863 q^{61} -46.9889i q^{65} -54.6101 q^{67} -11.5967i q^{71} +41.7131 q^{73} -8.65924i q^{77} -96.9294 q^{79} -89.8729i q^{83} -168.366 q^{85} -117.316i q^{89} +20.1007 q^{91} -22.1705i q^{95} -7.47681 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^7	\( 8 q - 16 q^{7} - 8 q^{13} - 40 q^{19} - 112 q^{25} + 56 q^{31} + 136 q^{37} + 104 q^{43} - 96 q^{49} - 8 q^{61} - 112 q^{67} + 448 q^{73} - 448 q^{79} + 48 q^{85} + 544 q^{91} - 152 q^{97}+O(q^{100}) \) 8 * q - 16 * q^7 - 8 * q^13 - 40 * q^19 - 112 * q^25 + 56 * q^31 + 136 * q^37 + 104 * q^43 - 96 * q^49 - 8 * q^61 - 112 * q^67 + 448 * q^73 - 448 * q^79 + 48 * q^85 + 544 * q^91 - 152 * q^97

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