LMFDB - Embedded newform 216.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,4,Mod(1,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	216.a (trivial)

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:	yes
Analytic conductor:	$12.7444125612$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.4164 q^{5} +23.8328 q^{7} +O(q^{10})$	$q-15.4164 q^{5} +23.8328 q^{7} +14.2492 q^{11} -13.8328 q^{13} -80.5836 q^{17} -144.331 q^{19} -141.082 q^{23} +112.666 q^{25} -251.331 q^{29} -16.6687 q^{31} -367.416 q^{35} +305.164 q^{37} +429.325 q^{41} -181.666 q^{43} -79.4164 q^{47} +225.003 q^{49} -663.830 q^{53} -219.672 q^{55} +220.255 q^{59} -473.158 q^{61} +213.252 q^{65} -647.663 q^{67} +14.4922 q^{71} +776.003 q^{73} +339.599 q^{77} -257.827 q^{79} +1285.15 q^{83} +1242.31 q^{85} -156.255 q^{89} -329.675 q^{91} +2225.07 q^{95} +1161.64 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 6 * q^7	$ 2 q - 4 q^{5} - 6 q^{7} - 52 q^{11} + 26 q^{13} - 188 q^{17} - 74 q^{19} - 148 q^{23} + 118 q^{25} - 288 q^{29} - 248 q^{31} - 708 q^{35} + 342 q^{37} - 256 q^{43} - 132 q^{47} + 772 q^{49} - 952 q^{53} - 976 q^{55} + 1004 q^{59} - 34 q^{61} + 668 q^{65} - 866 q^{67} - 776 q^{71} + 1874 q^{73} + 2316 q^{77} + 182 q^{79} + 1336 q^{83} + 16 q^{85} - 876 q^{89} - 1518 q^{91} + 3028 q^{95} - 38 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 6 * q^7 - 52 * q^11 + 26 * q^13 - 188 * q^17 - 74 * q^19 - 148 * q^23 + 118 * q^25 - 288 * q^29 - 248 * q^31 - 708 * q^35 + 342 * q^37 - 256 * q^43 - 132 * q^47 + 772 * q^49 - 952 * q^53 - 976 * q^55 + 1004 * q^59 - 34 * q^61 + 668 * q^65 - 866 * q^67 - 776 * q^71 + 1874 * q^73 + 2316 * q^77 + 182 * q^79 + 1336 * q^83 + 16 * q^85 - 876 * q^89 - 1518 * q^91 + 3028 * q^95 - 38 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−15.4164	−1.37889	−0.689443	−	0.724340i	$-0.742145\pi$
$5$	−15.4164	−1.37889	−0.689443	+	0.724340i	$0.742145\pi$
$6$	0	0
$7$	23.8328	1.28685	0.643426	−	0.765509i	$-0.277513\pi$
$7$	23.8328	1.28685	0.643426	+	0.765509i	$0.277513\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.2492	0.390573	0.195286	−	0.980746i	$-0.437436\pi$
$11$	14.2492	0.390573	0.195286	+	0.980746i	$0.437436\pi$
$12$	0	0
$13$	−13.8328	−0.295118	−0.147559	−	0.989053i	$-0.547142\pi$
$13$	−13.8328	−0.295118	−0.147559	+	0.989053i	$0.547142\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−80.5836	−1.14967	−0.574835	−	0.818269i	$-0.694934\pi$
$17$	−80.5836	−1.14967	−0.574835	+	0.818269i	$0.694934\pi$
$18$	0	0
$19$	−144.331	−1.74273	−0.871365	−	0.490636i	$-0.836765\pi$
$19$	−144.331	−1.74273	−0.871365	+	0.490636i	$0.836765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−141.082	−1.27903	−0.639514	−	0.768780i	$-0.720864\pi$
$23$	−141.082	−1.27903	−0.639514	+	0.768780i	$0.720864\pi$
$24$	0	0
$25$	112.666	0.901325
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−251.331	−1.60935	−0.804673	−	0.593718i	$-0.797659\pi$
$29$	−251.331	−1.60935	−0.804673	+	0.593718i	$0.797659\pi$
$30$	0	0
$31$	−16.6687	−0.0965740	−0.0482870	−	0.998834i	$-0.515376\pi$
$31$	−16.6687	−0.0965740	−0.0482870	+	0.998834i	$0.515376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−367.416	−1.77442
$36$	0	0
$37$	305.164	1.35591	0.677955	−	0.735103i	$-0.262866\pi$
$37$	305.164	1.35591	0.677955	+	0.735103i	$0.262866\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	429.325	1.63535	0.817674	−	0.575681i	$-0.195263\pi$
$41$	429.325	1.63535	0.817674	+	0.575681i	$0.195263\pi$
$42$	0	0
$43$	−181.666	−0.644273	−0.322137	−	0.946693i	$-0.604401\pi$
$43$	−181.666	−0.644273	−0.322137	+	0.946693i	$0.604401\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−79.4164	−0.246470	−0.123235	−	0.992378i	$-0.539327\pi$
$47$	−79.4164	−0.246470	−0.123235	+	0.992378i	$0.539327\pi$
$48$	0	0
$49$	225.003	0.655986
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−663.830	−1.72045	−0.860227	−	0.509912i	$-0.829678\pi$
$53$	−663.830	−1.72045	−0.860227	+	0.509912i	$0.829678\pi$
$54$	0	0
$55$	−219.672	−0.538555
$56$	0	0
$57$	0	0
$58$	0	0
$59$	220.255	0.486014	0.243007	−	0.970025i	$-0.421866\pi$
$59$	220.255	0.486014	0.243007	+	0.970025i	$0.421866\pi$
$60$	0	0
$61$	−473.158	−0.993142	−0.496571	−	0.867996i	$-0.665408\pi$
$61$	−473.158	−0.993142	−0.496571	+	0.867996i	$0.665408\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	213.252	0.406934
$66$	0	0
$67$	−647.663	−1.18096	−0.590482	−	0.807051i	$-0.701062\pi$
$67$	−647.663	−1.18096	−0.590482	+	0.807051i	$0.701062\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.4922	0.0242241	0.0121121	−	0.999927i	$-0.496145\pi$
$71$	14.4922	0.0242241	0.0121121	+	0.999927i	$0.496145\pi$
$72$	0	0
$73$	776.003	1.24417	0.622084	−	0.782950i	$-0.286286\pi$
$73$	776.003	1.24417	0.622084	+	0.782950i	$0.286286\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	339.599	0.502609
$78$	0	0
$79$	−257.827	−0.367187	−0.183593	−	0.983002i	$-0.558773\pi$
$79$	−257.827	−0.367187	−0.183593	+	0.983002i	$0.558773\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1285.15	1.69957	0.849784	−	0.527132i	$-0.176733\pi$
$83$	1285.15	1.69957	0.849784	+	0.527132i	$0.176733\pi$
$84$	0	0
$85$	1242.31	1.58526
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−156.255	−0.186102	−0.0930508	−	0.995661i	$-0.529662\pi$
$89$	−156.255	−0.186102	−0.0930508	+	0.995661i	$0.529662\pi$
$90$	0	0
$91$	−329.675	−0.379773
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2225.07	2.40302
$96$	0	0
$97$	1161.64	1.21595	0.607975	−	0.793956i	$-0.291982\pi$
$97$	1161.64	1.21595	0.607975	+	0.793956i	$0.291982\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1685.98	1.66100	0.830502	−	0.557016i	$-0.188054\pi$
$101$	1685.98	1.66100	0.830502	+	0.557016i	$0.188054\pi$
$102$	0	0
$103$	−765.820	−0.732607	−0.366304	−	0.930495i	$-0.619377\pi$
$103$	−765.820	−0.732607	−0.366304	+	0.930495i	$0.619377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1747.89	−1.57920	−0.789602	−	0.613619i	$-0.789713\pi$
$107$	−1747.89	−1.57920	−0.789602	+	0.613619i	$0.789713\pi$
$108$	0	0
$109$	−1211.64	−1.06472	−0.532358	−	0.846519i	$-0.678694\pi$
$109$	−1211.64	−1.06472	−0.532358	+	0.846519i	$0.678694\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	964.426	0.802881	0.401440	−	0.915885i	$-0.368510\pi$
$113$	964.426	0.802881	0.401440	+	0.915885i	$0.368510\pi$
$114$	0	0
$115$	2174.98	1.76363
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1920.53	−1.47945
$120$	0	0
$121$	−1127.96	−0.847453
$122$	0	0
$123$	0	0
$124$	0	0
$125$	190.152	0.136061
$126$	0	0
$127$	−2238.31	−1.56392	−0.781960	−	0.623329i	$-0.785780\pi$
$127$	−2238.31	−1.56392	−0.781960	+	0.623329i	$0.785780\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1674.31	1.11668	0.558340	−	0.829612i	$-0.311438\pi$
$131$	1674.31	1.11668	0.558340	+	0.829612i	$0.311438\pi$
$132$	0	0
$133$	−3439.82	−2.24263
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−95.0758	−0.0592911	−0.0296455	−	0.999560i	$-0.509438\pi$
$137$	−95.0758	−0.0592911	−0.0296455	+	0.999560i	$0.509438\pi$
$138$	0	0
$139$	1170.64	0.714334	0.357167	−	0.934041i	$-0.383743\pi$
$139$	1170.64	0.714334	0.357167	+	0.934041i	$0.383743\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−197.107	−0.115265
$144$	0	0
$145$	3874.63	2.21910
$146$	0	0
$147$	0	0
$148$	0	0
$149$	859.136	0.472370	0.236185	−	0.971708i	$-0.424103\pi$
$149$	859.136	0.472370	0.236185	+	0.971708i	$0.424103\pi$
$150$	0	0
$151$	2898.83	1.56227	0.781137	−	0.624359i	$-0.214640\pi$
$151$	2898.83	1.56227	0.781137	+	0.624359i	$0.214640\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	256.972	0.133164
$156$	0	0
$157$	−1309.02	−0.665419	−0.332710	−	0.943029i	$-0.607963\pi$
$157$	−1309.02	−0.665419	−0.332710	+	0.943029i	$0.607963\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3362.38	−1.64592
$162$	0	0
$163$	−190.988	−0.0917749	−0.0458874	−	0.998947i	$-0.514612\pi$
$163$	−190.988	−0.0917749	−0.0458874	+	0.998947i	$0.514612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1440.73	−0.667587	−0.333793	−	0.942646i	$-0.608329\pi$
$167$	−1440.73	−0.667587	−0.333793	+	0.942646i	$0.608329\pi$
$168$	0	0
$169$	−2005.65	−0.912905
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1581.33	−0.694947	−0.347474	−	0.937690i	$-0.612960\pi$
$173$	−1581.33	−0.694947	−0.347474	+	0.937690i	$0.612960\pi$
$174$	0	0
$175$	2685.14	1.15987
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−741.836	−0.309762	−0.154881	−	0.987933i	$-0.549499\pi$
$179$	−741.836	−0.309762	−0.154881	+	0.987933i	$0.549499\pi$
$180$	0	0
$181$	626.786	0.257396	0.128698	−	0.991684i	$-0.458920\pi$
$181$	626.786	0.257396	0.128698	+	0.991684i	$0.458920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4704.53	−1.86964
$186$	0	0
$187$	−1148.25	−0.449030
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−182.615	−0.0691808	−0.0345904	−	0.999402i	$-0.511013\pi$
$191$	−182.615	−0.0691808	−0.0345904	+	0.999402i	$0.511013\pi$
$192$	0	0
$193$	1718.66	0.640994	0.320497	−	0.947250i	$-0.396150\pi$
$193$	1718.66	0.640994	0.320497	+	0.947250i	$0.396150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	119.587	0.0432497	0.0216249	−	0.999766i	$-0.493116\pi$
$197$	119.587	0.0432497	0.0216249	+	0.999766i	$0.493116\pi$
$198$	0	0
$199$	707.467	0.252015	0.126008	−	0.992029i	$-0.459784\pi$
$199$	707.467	0.252015	0.126008	+	0.992029i	$0.459784\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5989.93	−2.07099
$204$	0	0
$205$	−6618.65	−2.25496
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2056.61	−0.680663
$210$	0	0
$211$	−2388.62	−0.779332	−0.389666	−	0.920956i	$-0.627410\pi$
$211$	−2388.62	−0.779332	−0.389666	+	0.920956i	$0.627410\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2800.63	0.888379
$216$	0	0
$217$	−397.263	−0.124276
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1114.70	0.339288
$222$	0	0
$223$	5622.60	1.68842	0.844209	−	0.536014i	$-0.180071\pi$
$223$	5622.60	1.68842	0.844209	+	0.536014i	$0.180071\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3464.46	1.01297	0.506485	−	0.862249i	$-0.330944\pi$
$227$	3464.46	1.01297	0.506485	+	0.862249i	$0.330944\pi$
$228$	0	0
$229$	−6082.62	−1.75524	−0.877622	−	0.479354i	$-0.840871\pi$
$229$	−6082.62	−1.75524	−0.877622	+	0.479354i	$0.840871\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5024.29	1.41267	0.706335	−	0.707877i	$-0.250347\pi$
$233$	5024.29	1.41267	0.706335	+	0.707877i	$0.250347\pi$
$234$	0	0
$235$	1224.32	0.339853
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2670.20	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$239$	−2670.20	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$240$	0	0
$241$	754.653	0.201707	0.100854	−	0.994901i	$-0.467843\pi$
$241$	754.653	0.201707	0.100854	+	0.994901i	$0.467843\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3468.74	−0.904529
$246$	0	0
$247$	1996.51	0.514311
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1588.91	−0.399566	−0.199783	−	0.979840i	$-0.564024\pi$
$251$	−1588.91	−0.399566	−0.199783	+	0.979840i	$0.564024\pi$
$252$	0	0
$253$	−2010.31	−0.499554
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4911.76	1.19217	0.596083	−	0.802923i	$-0.296723\pi$
$257$	4911.76	1.19217	0.596083	+	0.802923i	$0.296723\pi$
$258$	0	0
$259$	7272.92	1.74485
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5400.41	−1.26617	−0.633087	−	0.774081i	$-0.718212\pi$
$263$	−5400.41	−1.26617	−0.633087	+	0.774081i	$0.718212\pi$
$264$	0	0
$265$	10233.9	2.37231
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3234.09	0.733034	0.366517	−	0.930411i	$-0.380550\pi$
$269$	3234.09	0.733034	0.366517	+	0.930411i	$0.380550\pi$
$270$	0	0
$271$	−6205.83	−1.39106	−0.695530	−	0.718497i	$-0.744830\pi$
$271$	−6205.83	−1.39106	−0.695530	+	0.718497i	$0.744830\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1605.40	0.352033
$276$	0	0
$277$	1030.91	0.223615	0.111808	−	0.993730i	$-0.464336\pi$
$277$	1030.91	0.223615	0.111808	+	0.993730i	$0.464336\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3821.33	0.811250	0.405625	−	0.914040i	$-0.367054\pi$
$281$	3821.33	0.811250	0.405625	+	0.914040i	$0.367054\pi$
$282$	0	0
$283$	−4402.33	−0.924705	−0.462353	−	0.886696i	$-0.652995\pi$
$283$	−4402.33	−0.924705	−0.462353	+	0.886696i	$0.652995\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10232.0	2.10445
$288$	0	0
$289$	1580.72	0.321741
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5973.54	−1.19105	−0.595526	−	0.803336i	$-0.703056\pi$
$293$	−5973.54	−1.19105	−0.595526	+	0.803336i	$0.703056\pi$
$294$	0	0
$295$	−3395.55	−0.670157
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1951.56	0.377464
$300$	0	0
$301$	−4329.60	−0.829084
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7294.39	1.36943
$306$	0	0
$307$	−219.622	−0.0408290	−0.0204145	−	0.999792i	$-0.506499\pi$
$307$	−219.622	−0.0408290	−0.0204145	+	0.999792i	$0.506499\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−957.227	−0.174532	−0.0872659	−	0.996185i	$-0.527813\pi$
$311$	−957.227	−0.174532	−0.0872659	+	0.996185i	$0.527813\pi$
$312$	0	0
$313$	1874.05	0.338427	0.169214	−	0.985579i	$-0.445877\pi$
$313$	1874.05	0.338427	0.169214	+	0.985579i	$0.445877\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2919.78	0.517322	0.258661	−	0.965968i	$-0.416719\pi$
$317$	2919.78	0.517322	0.258661	+	0.965968i	$0.416719\pi$
$318$	0	0
$319$	−3581.28	−0.628567
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11630.7	2.00356
$324$	0	0
$325$	−1558.48	−0.265997
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1892.72	−0.317170
$330$	0	0
$331$	−6648.67	−1.10406	−0.552030	−	0.833824i	$-0.686147\pi$
$331$	−6648.67	−1.10406	−0.552030	+	0.833824i	$0.686147\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9984.63	1.62841
$336$	0	0
$337$	3886.34	0.628197	0.314098	−	0.949390i	$-0.398298\pi$
$337$	3886.34	0.628197	0.314098	+	0.949390i	$0.398298\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−237.517	−0.0377192
$342$	0	0
$343$	−2812.20	−0.442695
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4207.76	0.650963	0.325481	−	0.945548i	$-0.394474\pi$
$347$	4207.76	0.650963	0.325481	+	0.945548i	$0.394474\pi$
$348$	0	0
$349$	3011.42	0.461885	0.230942	−	0.972967i	$-0.425819\pi$
$349$	3011.42	0.461885	0.230942	+	0.972967i	$0.425819\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2499.79	0.376914	0.188457	−	0.982081i	$-0.439651\pi$
$353$	2499.79	0.376914	0.188457	+	0.982081i	$0.439651\pi$
$354$	0	0
$355$	−223.418	−0.0334023
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8319.17	−1.22303	−0.611517	−	0.791231i	$-0.709440\pi$
$359$	−8319.17	−1.22303	−0.611517	+	0.791231i	$0.709440\pi$
$360$	0	0
$361$	13972.5	2.03711
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11963.2	−1.71557
$366$	0	0
$367$	−4459.85	−0.634339	−0.317169	−	0.948369i	$-0.602732\pi$
$367$	−4459.85	−0.634339	−0.317169	+	0.948369i	$0.602732\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15820.9	−2.21397
$372$	0	0
$373$	−1179.41	−0.163719	−0.0818596	−	0.996644i	$-0.526086\pi$
$373$	−1179.41	−0.163719	−0.0818596	+	0.996644i	$0.526086\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3476.62	0.474947
$378$	0	0
$379$	8413.88	1.14035	0.570174	−	0.821524i	$-0.306876\pi$
$379$	8413.88	1.14035	0.570174	+	0.821524i	$0.306876\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1601.94	−0.213722	−0.106861	−	0.994274i	$-0.534080\pi$
$383$	−1601.94	−0.213722	−0.106861	+	0.994274i	$0.534080\pi$
$384$	0	0
$385$	−5235.40	−0.693041
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5202.43	0.678081	0.339041	−	0.940772i	$-0.389898\pi$
$389$	5202.43	0.678081	0.339041	+	0.940772i	$0.389898\pi$
$390$	0	0
$391$	11368.9	1.47046
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3974.76	0.506309
$396$	0	0
$397$	4310.87	0.544979	0.272489	−	0.962159i	$-0.412153\pi$
$397$	4310.87	0.544979	0.272489	+	0.962159i	$0.412153\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6240.59	−0.777157	−0.388579	−	0.921416i	$-0.627034\pi$
$401$	−6240.59	−0.777157	−0.388579	+	0.921416i	$0.627034\pi$
$402$	0	0
$403$	230.576	0.0285007
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4348.35	0.529582
$408$	0	0
$409$	2047.66	0.247556	0.123778	−	0.992310i	$-0.460499\pi$
$409$	2047.66	0.247556	0.123778	+	0.992310i	$0.460499\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5249.31	0.625427
$414$	0	0
$415$	−19812.5	−2.34351
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5960.12	−0.694919	−0.347459	−	0.937695i	$-0.612956\pi$
$419$	−5960.12	−0.694919	−0.347459	+	0.937695i	$0.612956\pi$
$420$	0	0
$421$	311.567	0.0360685	0.0180342	−	0.999837i	$-0.494259\pi$
$421$	311.567	0.0360685	0.0180342	+	0.999837i	$0.494259\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9079.00	−1.03623
$426$	0	0
$427$	−11276.7	−1.27803
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17105.0	−1.91164	−0.955821	−	0.293950i	$-0.905030\pi$
$431$	−17105.0	−1.91164	−0.955821	+	0.293950i	$0.905030\pi$
$432$	0	0
$433$	2582.96	0.286672	0.143336	−	0.989674i	$-0.454217\pi$
$433$	2582.96	0.286672	0.143336	+	0.989674i	$0.454217\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20362.5	2.22900
$438$	0	0
$439$	−6797.87	−0.739054	−0.369527	−	0.929220i	$-0.620480\pi$
$439$	−6797.87	−0.739054	−0.369527	+	0.929220i	$0.620480\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10454.1	−1.12119	−0.560596	−	0.828089i	$-0.689428\pi$
$443$	−10454.1	−1.12119	−0.560596	+	0.828089i	$0.689428\pi$
$444$	0	0
$445$	2408.90	0.256613
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17208.6	−1.80873	−0.904367	−	0.426755i	$-0.859657\pi$
$449$	−17208.6	−1.80873	−0.904367	+	0.426755i	$0.859657\pi$
$450$	0	0
$451$	6117.55	0.638723
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5082.40	0.523663
$456$	0	0
$457$	15788.1	1.61605	0.808025	−	0.589148i	$-0.200537\pi$
$457$	15788.1	1.61605	0.808025	+	0.589148i	$0.200537\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5183.17	−0.523654	−0.261827	−	0.965115i	$-0.584325\pi$
$461$	−5183.17	−0.523654	−0.261827	+	0.965115i	$0.584325\pi$
$462$	0	0
$463$	−5833.38	−0.585530	−0.292765	−	0.956184i	$-0.594575\pi$
$463$	−5833.38	−0.585530	−0.292765	+	0.956184i	$0.594575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6315.33	−0.625779	−0.312889	−	0.949790i	$-0.601297\pi$
$467$	−6315.33	−0.625779	−0.312889	+	0.949790i	$0.601297\pi$
$468$	0	0
$469$	−15435.6	−1.51972
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2588.59	−0.251636
$474$	0	0
$475$	−16261.2	−1.57077
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9735.88	−0.928692	−0.464346	−	0.885654i	$-0.653711\pi$
$479$	−9735.88	−0.928692	−0.464346	+	0.885654i	$0.653711\pi$
$480$	0	0
$481$	−4221.28	−0.400153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17908.4	−1.67665
$486$	0	0
$487$	13447.9	1.25130	0.625649	−	0.780104i	$-0.284834\pi$
$487$	13447.9	1.25130	0.625649	+	0.780104i	$0.284834\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7028.09	0.645974	0.322987	−	0.946403i	$-0.395313\pi$
$491$	7028.09	0.645974	0.322987	+	0.946403i	$0.395313\pi$
$492$	0	0
$493$	20253.2	1.85022
$494$	0	0
$495$	0	0
$496$	0	0
$497$	345.391	0.0311728
$498$	0	0
$499$	1655.44	0.148513	0.0742563	−	0.997239i	$-0.476342\pi$
$499$	1655.44	0.148513	0.0742563	+	0.997239i	$0.476342\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1909.16	−0.169235	−0.0846174	−	0.996414i	$-0.526967\pi$
$503$	−1909.16	−0.169235	−0.0846174	+	0.996414i	$0.526967\pi$
$504$	0	0
$505$	−25991.8	−2.29033
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8865.70	−0.772034	−0.386017	−	0.922492i	$-0.626149\pi$
$509$	−8865.70	−0.772034	−0.386017	+	0.922492i	$0.626149\pi$
$510$	0	0
$511$	18494.3	1.60106
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11806.2	1.01018
$516$	0	0
$517$	−1131.62	−0.0962644
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9256.80	−0.778403	−0.389201	−	0.921153i	$-0.627249\pi$
$521$	−9256.80	−0.778403	−0.389201	+	0.921153i	$0.627249\pi$
$522$	0	0
$523$	13607.5	1.13770	0.568849	−	0.822442i	$-0.307389\pi$
$523$	13607.5	1.13770	0.568849	+	0.822442i	$0.307389\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1343.23	0.111028
$528$	0	0
$529$	7737.14	0.635912
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5938.77	−0.482621
$534$	0	0
$535$	26946.2	2.17754
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3206.12	0.256210
$540$	0	0
$541$	−483.548	−0.0384276	−0.0192138	−	0.999815i	$-0.506116\pi$
$541$	−483.548	−0.0384276	−0.0192138	+	0.999815i	$0.506116\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18679.1	1.46812
$546$	0	0
$547$	−4711.28	−0.368263	−0.184131	−	0.982902i	$-0.558947\pi$
$547$	−4711.28	−0.368263	−0.184131	+	0.982902i	$0.558947\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	36275.0	2.80466
$552$	0	0
$553$	−6144.73	−0.472515
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21650.7	1.64698	0.823492	−	0.567327i	$-0.192023\pi$
$557$	21650.7	1.64698	0.823492	+	0.567327i	$0.192023\pi$
$558$	0	0
$559$	2512.95	0.190137
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14023.0	−1.04973	−0.524866	−	0.851185i	$-0.675884\pi$
$563$	−14023.0	−1.04973	−0.524866	+	0.851185i	$0.675884\pi$
$564$	0	0
$565$	−14868.0	−1.10708
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4875.33	−0.359200	−0.179600	−	0.983740i	$-0.557480\pi$
$569$	−4875.33	−0.359200	−0.179600	+	0.983740i	$0.557480\pi$
$570$	0	0
$571$	−25969.1	−1.90328	−0.951640	−	0.307215i	$-0.900603\pi$
$571$	−25969.1	−1.90328	−0.951640	+	0.307215i	$0.900603\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15895.1	−1.15282
$576$	0	0
$577$	−23113.1	−1.66761	−0.833806	−	0.552058i	$-0.813843\pi$
$577$	−23113.1	−1.66761	−0.833806	+	0.552058i	$0.813843\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30628.9	2.18709
$582$	0	0
$583$	−9459.06	−0.671963
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5471.66	0.384735	0.192368	−	0.981323i	$-0.438383\pi$
$587$	5471.66	0.384735	0.192368	+	0.981323i	$0.438383\pi$
$588$	0	0
$589$	2405.82	0.168302
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22609.2	−1.56568	−0.782841	−	0.622222i	$-0.786230\pi$
$593$	−22609.2	−1.56568	−0.782841	+	0.622222i	$0.786230\pi$
$594$	0	0
$595$	29607.7	2.04000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3708.89	0.252990	0.126495	−	0.991967i	$-0.459627\pi$
$599$	3708.89	0.252990	0.126495	+	0.991967i	$0.459627\pi$
$600$	0	0
$601$	−10478.2	−0.711170	−0.355585	−	0.934644i	$-0.615718\pi$
$601$	−10478.2	−0.711170	−0.355585	+	0.934644i	$0.615718\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17389.1	1.16854
$606$	0	0
$607$	10713.6	0.716392	0.358196	−	0.933646i	$-0.383392\pi$
$607$	10713.6	0.716392	0.358196	+	0.933646i	$0.383392\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1098.55	0.0727376
$612$	0	0
$613$	7860.62	0.517924	0.258962	−	0.965887i	$-0.416619\pi$
$613$	7860.62	0.517924	0.258962	+	0.965887i	$0.416619\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8528.00	0.556442	0.278221	−	0.960517i	$-0.410255\pi$
$617$	8528.00	0.556442	0.278221	+	0.960517i	$0.410255\pi$
$618$	0	0
$619$	−8022.89	−0.520949	−0.260474	−	0.965481i	$-0.583879\pi$
$619$	−8022.89	−0.520949	−0.260474	+	0.965481i	$0.583879\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3724.01	−0.239485
$624$	0	0
$625$	−17014.7	−1.08894
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24591.2	−1.55885
$630$	0	0
$631$	−3984.40	−0.251373	−0.125687	−	0.992070i	$-0.540113\pi$
$631$	−3984.40	−0.251373	−0.125687	+	0.992070i	$0.540113\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34506.7	2.15647
$636$	0	0
$637$	−3112.43	−0.193593
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14750.2	0.908892	0.454446	−	0.890774i	$-0.349837\pi$
$641$	14750.2	0.908892	0.454446	+	0.890774i	$0.349837\pi$
$642$	0	0
$643$	9810.74	0.601707	0.300854	−	0.953670i	$-0.402728\pi$
$643$	9810.74	0.601707	0.300854	+	0.953670i	$0.402728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5683.55	0.345353	0.172677	−	0.984979i	$-0.444758\pi$
$647$	5683.55	0.345353	0.172677	+	0.984979i	$0.444758\pi$
$648$	0	0
$649$	3138.47	0.189824
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3405.52	0.204086	0.102043	−	0.994780i	$-0.467462\pi$
$653$	3405.52	0.204086	0.102043	+	0.994780i	$0.467462\pi$
$654$	0	0
$655$	−25811.8	−1.53977
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19049.1	−1.12602	−0.563011	−	0.826449i	$-0.690357\pi$
$659$	−19049.1	−1.12602	−0.563011	+	0.826449i	$0.690357\pi$
$660$	0	0
$661$	20422.5	1.20173	0.600864	−	0.799351i	$-0.294823\pi$
$661$	20422.5	1.20173	0.600864	+	0.799351i	$0.294823\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	53029.7	3.09233
$666$	0	0
$667$	35458.3	2.05840
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6742.13	−0.387894
$672$	0	0
$673$	8613.45	0.493349	0.246675	−	0.969098i	$-0.420662\pi$
$673$	8613.45	0.493349	0.246675	+	0.969098i	$0.420662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10469.1	0.594329	0.297164	−	0.954826i	$-0.403959\pi$
$677$	10469.1	0.594329	0.297164	+	0.954826i	$0.403959\pi$
$678$	0	0
$679$	27685.2	1.56475
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9490.28	−0.531677	−0.265839	−	0.964018i	$-0.585649\pi$
$683$	−9490.28	−0.531677	−0.265839	+	0.964018i	$0.585649\pi$
$684$	0	0
$685$	1465.73	0.0817556
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9182.63	0.507737
$690$	0	0
$691$	15814.7	0.870652	0.435326	−	0.900273i	$-0.356633\pi$
$691$	15814.7	0.870652	0.435326	+	0.900273i	$0.356633\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18047.1	−0.984985
$696$	0	0
$697$	−34596.6	−1.88011
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15790.6	−0.850791	−0.425395	−	0.905008i	$-0.639865\pi$
$701$	−15790.6	−0.850791	−0.425395	+	0.905008i	$0.639865\pi$
$702$	0	0
$703$	−44044.7	−2.36298
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40181.7	2.13746
$708$	0	0
$709$	27542.9	1.45895	0.729474	−	0.684009i	$-0.239765\pi$
$709$	27542.9	1.45895	0.729474	+	0.684009i	$0.239765\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2351.66	0.123521
$714$	0	0
$715$	3038.68	0.158937
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19578.6	1.01552	0.507761	−	0.861498i	$-0.330473\pi$
$719$	19578.6	1.01552	0.507761	+	0.861498i	$0.330473\pi$
$720$	0	0
$721$	−18251.7	−0.942756
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−28316.4	−1.45054
$726$	0	0
$727$	20948.0	1.06866	0.534330	−	0.845276i	$-0.320564\pi$
$727$	20948.0	1.06866	0.534330	+	0.845276i	$0.320564\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14639.3	0.740702
$732$	0	0
$733$	−825.349	−0.0415893	−0.0207946	−	0.999784i	$-0.506620\pi$
$733$	−825.349	−0.0415893	−0.0207946	+	0.999784i	$0.506620\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9228.69	−0.461253
$738$	0	0
$739$	22821.1	1.13598	0.567988	−	0.823037i	$-0.307722\pi$
$739$	22821.1	1.13598	0.567988	+	0.823037i	$0.307722\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12056.5	0.595301	0.297650	−	0.954675i	$-0.403797\pi$
$743$	12056.5	0.595301	0.297650	+	0.954675i	$0.403797\pi$
$744$	0	0
$745$	−13244.8	−0.651345
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−41657.1	−2.03220
$750$	0	0
$751$	−7722.32	−0.375221	−0.187611	−	0.982243i	$-0.560074\pi$
$751$	−7722.32	−0.375221	−0.187611	+	0.982243i	$0.560074\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−44689.5	−2.15420
$756$	0	0
$757$	−1440.60	−0.0691671	−0.0345835	−	0.999402i	$-0.511010\pi$
$757$	−1440.60	−0.0691671	−0.0345835	+	0.999402i	$0.511010\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30722.6	−1.46346	−0.731729	−	0.681595i	$-0.761286\pi$
$761$	−30722.6	−1.46346	−0.731729	+	0.681595i	$0.761286\pi$
$762$	0	0
$763$	−28876.8	−1.37013
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3046.75	−0.143431
$768$	0	0
$769$	−40280.9	−1.88890	−0.944451	−	0.328652i	$-0.893406\pi$
$769$	−40280.9	−1.88890	−0.944451	+	0.328652i	$0.893406\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4282.45	−0.199261	−0.0996306	−	0.995024i	$-0.531766\pi$
$773$	−4282.45	−0.199261	−0.0996306	+	0.995024i	$0.531766\pi$
$774$	0	0
$775$	−1877.99	−0.0870446
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−61965.0	−2.84997
$780$	0	0
$781$	206.503	0.00946128
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20180.3	0.917537
$786$	0	0
$787$	−12571.6	−0.569416	−0.284708	−	0.958614i	$-0.591897\pi$
$787$	−12571.6	−0.569416	−0.284708	+	0.958614i	$0.591897\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22985.0	1.03319
$792$	0	0
$793$	6545.11	0.293094
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28074.0	−1.24772	−0.623860	−	0.781536i	$-0.714436\pi$
$797$	−28074.0	−1.24772	−0.623860	+	0.781536i	$0.714436\pi$
$798$	0	0
$799$	6399.66	0.283359
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11057.4	0.485939
$804$	0	0
$805$	51835.9	2.26953
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−293.254	−0.0127445	−0.00637223	−	0.999980i	$-0.502028\pi$
$809$	−293.254	−0.0127445	−0.00637223	+	0.999980i	$0.502028\pi$
$810$	0	0
$811$	−45634.2	−1.97588	−0.987938	−	0.154851i	$-0.950510\pi$
$811$	−45634.2	−1.97588	−0.987938	+	0.154851i	$0.950510\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2944.34	0.126547
$816$	0	0
$817$	26220.0	1.12279
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5406.68	0.229835	0.114917	−	0.993375i	$-0.463340\pi$
$821$	5406.68	0.229835	0.114917	+	0.993375i	$0.463340\pi$
$822$	0	0
$823$	1228.88	0.0520487	0.0260243	−	0.999661i	$-0.491715\pi$
$823$	1228.88	0.0520487	0.0260243	+	0.999661i	$0.491715\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45769.8	1.92451	0.962256	−	0.272147i	$-0.0877337\pi$
$827$	45769.8	1.92451	0.962256	+	0.272147i	$0.0877337\pi$
$828$	0	0
$829$	−13630.9	−0.571076	−0.285538	−	0.958367i	$-0.592172\pi$
$829$	−13630.9	−0.571076	−0.285538	+	0.958367i	$0.592172\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18131.6	−0.754167
$834$	0	0
$835$	22210.9	0.920525
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44987.6	−1.85119	−0.925593	−	0.378520i	$-0.876433\pi$
$839$	−44987.6	−1.85119	−0.925593	+	0.378520i	$0.876433\pi$
$840$	0	0
$841$	38778.4	1.59000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	30920.0	1.25879
$846$	0	0
$847$	−26882.5	−1.09055
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43053.2	−1.73425
$852$	0	0
$853$	26043.2	1.04537	0.522685	−	0.852526i	$-0.324930\pi$
$853$	26043.2	1.04537	0.522685	+	0.852526i	$0.324930\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13706.4	−0.546327	−0.273163	−	0.961968i	$-0.588070\pi$
$857$	−13706.4	−0.546327	−0.273163	+	0.961968i	$0.588070\pi$
$858$	0	0
$859$	−45319.1	−1.80008	−0.900039	−	0.435810i	$-0.856462\pi$
$859$	−45319.1	−1.80008	−0.900039	+	0.435810i	$0.856462\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28057.3	−1.10670	−0.553350	−	0.832949i	$-0.686651\pi$
$863$	−28057.3	−1.10670	−0.553350	+	0.832949i	$0.686651\pi$
$864$	0	0
$865$	24378.4	0.958253
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3673.83	−0.143413
$870$	0	0
$871$	8959.00	0.348524
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4531.85	0.175091
$876$	0	0
$877$	3923.18	0.151056	0.0755282	−	0.997144i	$-0.475936\pi$
$877$	3923.18	0.151056	0.0755282	+	0.997144i	$0.475936\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2441.97	0.0933849	0.0466924	−	0.998909i	$-0.485132\pi$
$881$	2441.97	0.0933849	0.0466924	+	0.998909i	$0.485132\pi$
$882$	0	0
$883$	44576.9	1.69890	0.849452	−	0.527667i	$-0.176933\pi$
$883$	44576.9	1.69890	0.849452	+	0.527667i	$0.176933\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17095.5	−0.647137	−0.323568	−	0.946205i	$-0.604883\pi$
$887$	−17095.5	−0.647137	−0.323568	+	0.946205i	$0.604883\pi$
$888$	0	0
$889$	−53345.2	−2.01253
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11462.3	0.429530
$894$	0	0
$895$	11436.4	0.427126
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4189.37	0.155421
$900$	0	0
$901$	53493.8	1.97795
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9662.79	−0.354919
$906$	0	0
$907$	−33142.7	−1.21332	−0.606662	−	0.794960i	$-0.707492\pi$
$907$	−33142.7	−1.21332	−0.606662	+	0.794960i	$0.707492\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32750.5	−1.19108	−0.595539	−	0.803326i	$-0.703061\pi$
$911$	−32750.5	−1.19108	−0.595539	+	0.803326i	$0.703061\pi$
$912$	0	0
$913$	18312.5	0.663805
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39903.5	1.43700
$918$	0	0
$919$	6095.76	0.218804	0.109402	−	0.993998i	$-0.465106\pi$
$919$	6095.76	0.218804	0.109402	+	0.993998i	$0.465106\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−200.468	−0.00714897
$924$	0	0
$925$	34381.5	1.22212
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24708.0	−0.872597	−0.436298	−	0.899802i	$-0.643711\pi$
$929$	−24708.0	−0.872597	−0.436298	+	0.899802i	$0.643711\pi$
$930$	0	0
$931$	−32475.0	−1.14321
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17701.9	0.619161
$936$	0	0
$937$	−44713.2	−1.55893	−0.779465	−	0.626446i	$-0.784509\pi$
$937$	−44713.2	−1.55893	−0.779465	+	0.626446i	$0.784509\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10172.7	0.352414	0.176207	−	0.984353i	$-0.443617\pi$
$941$	10172.7	0.352414	0.176207	+	0.984353i	$0.443617\pi$
$942$	0	0
$943$	−60570.1	−2.09166
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13346.6	0.457978	0.228989	−	0.973429i	$-0.426458\pi$
$947$	13346.6	0.457978	0.228989	+	0.973429i	$0.426458\pi$
$948$	0	0
$949$	−10734.3	−0.367176
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36232.4	−1.23157	−0.615783	−	0.787916i	$-0.711160\pi$
$953$	−36232.4	−1.23157	−0.615783	+	0.787916i	$0.711160\pi$
$954$	0	0
$955$	2815.26	0.0953924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2265.92	−0.0762988
$960$	0	0
$961$	−29513.2	−0.990673
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26495.6	−0.883857
$966$	0	0
$967$	5429.03	0.180544	0.0902718	−	0.995917i	$-0.471226\pi$
$967$	5429.03	0.180544	0.0902718	+	0.995917i	$0.471226\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5645.37	0.186579	0.0932897	−	0.995639i	$-0.470262\pi$
$971$	5645.37	0.186579	0.0932897	+	0.995639i	$0.470262\pi$
$972$	0	0
$973$	27899.7	0.919242
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51103.6	1.67344	0.836719	−	0.547632i	$-0.184471\pi$
$977$	51103.6	1.67344	0.836719	+	0.547632i	$0.184471\pi$
$978$	0	0
$979$	−2226.52	−0.0726863
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32082.5	1.04097	0.520485	−	0.853871i	$-0.325751\pi$
$983$	32082.5	1.04097	0.520485	+	0.853871i	$0.325751\pi$
$984$	0	0
$985$	−1843.60	−0.0596364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25629.8	0.824043
$990$	0	0
$991$	−27696.1	−0.887785	−0.443893	−	0.896080i	$-0.646403\pi$
$991$	−27696.1	−0.887785	−0.443893	+	0.896080i	$0.646403\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10906.6	−0.347500
$996$	0	0
$997$	−41284.9	−1.31144	−0.655720	−	0.755004i	$-0.727635\pi$
$997$	−41284.9	−1.31144	−0.655720	+	0.755004i	$0.727635\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.4.a.f.1.1	✓	2
3.2	odd	2		216.4.a.g.1.2	yes	2
4.3	odd	2		432.4.a.p.1.1		2
8.3	odd	2		1728.4.a.br.1.2		2
8.5	even	2		1728.4.a.bq.1.2		2
9.2	odd	6		648.4.i.o.433.1		4
9.4	even	3		648.4.i.r.217.2		4
9.5	odd	6		648.4.i.o.217.1		4
9.7	even	3		648.4.i.r.433.2		4
12.11	even	2		432.4.a.r.1.2		2
24.5	odd	2		1728.4.a.bi.1.1		2
24.11	even	2		1728.4.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.a.f.1.1	✓	2	1.1	even	1	trivial
216.4.a.g.1.2	yes	2	3.2	odd	2
432.4.a.p.1.1		2	4.3	odd	2
432.4.a.r.1.2		2	12.11	even	2
648.4.i.o.217.1		4	9.5	odd	6
648.4.i.o.433.1		4	9.2	odd	6
648.4.i.r.217.2		4	9.4	even	3
648.4.i.r.433.2		4	9.7	even	3
1728.4.a.bi.1.1		2	24.5	odd	2
1728.4.a.bj.1.1		2	24.11	even	2
1728.4.a.bq.1.2		2	8.5	even	2
1728.4.a.br.1.2		2	8.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	216.a (trivial)

\(f(q)\)	\(=\)	\(q-15.4164 q^{5} +23.8328 q^{7} +O(q^{10})\)	\(q-15.4164 q^{5} +23.8328 q^{7} +14.2492 q^{11} -13.8328 q^{13} -80.5836 q^{17} -144.331 q^{19} -141.082 q^{23} +112.666 q^{25} -251.331 q^{29} -16.6687 q^{31} -367.416 q^{35} +305.164 q^{37} +429.325 q^{41} -181.666 q^{43} -79.4164 q^{47} +225.003 q^{49} -663.830 q^{53} -219.672 q^{55} +220.255 q^{59} -473.158 q^{61} +213.252 q^{65} -647.663 q^{67} +14.4922 q^{71} +776.003 q^{73} +339.599 q^{77} -257.827 q^{79} +1285.15 q^{83} +1242.31 q^{85} -156.255 q^{89} -329.675 q^{91} +2225.07 q^{95} +1161.64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 6 * q^7	\( 2 q - 4 q^{5} - 6 q^{7} - 52 q^{11} + 26 q^{13} - 188 q^{17} - 74 q^{19} - 148 q^{23} + 118 q^{25} - 288 q^{29} - 248 q^{31} - 708 q^{35} + 342 q^{37} - 256 q^{43} - 132 q^{47} + 772 q^{49} - 952 q^{53} - 976 q^{55} + 1004 q^{59} - 34 q^{61} + 668 q^{65} - 866 q^{67} - 776 q^{71} + 1874 q^{73} + 2316 q^{77} + 182 q^{79} + 1336 q^{83} + 16 q^{85} - 876 q^{89} - 1518 q^{91} + 3028 q^{95} - 38 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 6 * q^7 - 52 * q^11 + 26 * q^13 - 188 * q^17 - 74 * q^19 - 148 * q^23 + 118 * q^25 - 288 * q^29 - 248 * q^31 - 708 * q^35 + 342 * q^37 - 256 * q^43 - 132 * q^47 + 772 * q^49 - 952 * q^53 - 976 * q^55 + 1004 * q^59 - 34 * q^61 + 668 * q^65 - 866 * q^67 - 776 * q^71 + 1874 * q^73 + 2316 * q^77 + 182 * q^79 + 1336 * q^83 + 16 * q^85 - 876 * q^89 - 1518 * q^91 + 3028 * q^95 - 38 * q^97

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