LMFDB - Embedded newform 2268.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,4,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, 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("2268.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2268.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:	$133.816331893$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 180x^{7} + 566x^{6} + 8543x^{5} - 33702x^{4} - 75401x^{3} + 385978x^{2} - 453885x + 167250 $ x^9 - 2x^8 - 180x^7 + 566x^6 + 8543x^5 - 33702x^4 - 75401x^3 + 385978x^2 - 453885x + 167250
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41731$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.3368 q^{5} +7.00000 q^{7} +O(q^{10})$	$q-15.3368 q^{5} +7.00000 q^{7} +67.0375 q^{11} +81.5617 q^{13} +101.984 q^{17} +86.0055 q^{19} +129.102 q^{23} +110.216 q^{25} +90.4455 q^{29} -95.7316 q^{31} -107.357 q^{35} -207.875 q^{37} -130.143 q^{41} +73.2845 q^{43} +331.041 q^{47} +49.0000 q^{49} +120.105 q^{53} -1028.14 q^{55} -195.307 q^{59} -59.7514 q^{61} -1250.89 q^{65} -805.962 q^{67} +391.162 q^{71} -729.424 q^{73} +469.262 q^{77} +655.667 q^{79} -12.3077 q^{83} -1564.11 q^{85} +716.679 q^{89} +570.932 q^{91} -1319.05 q^{95} +751.388 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{5} + 63 q^{7}+O(q^{10}) $ 9 * q + 6 * q^5 + 63 * q^7	$ 9 q + 6 q^{5} + 63 q^{7} + 93 q^{11} + 18 q^{13} - 27 q^{17} - 45 q^{19} + 246 q^{23} + 315 q^{25} + 318 q^{29} + 18 q^{31} + 42 q^{35} - 36 q^{37} + 57 q^{41} + 171 q^{43} + 1056 q^{47} + 441 q^{49} + 756 q^{53} - 900 q^{55} + 411 q^{59} + 198 q^{61} + 1326 q^{65} + 441 q^{67} + 1758 q^{71} + 27 q^{73} + 651 q^{77} - 72 q^{79} + 558 q^{83} + 1008 q^{85} + 1392 q^{89} + 126 q^{91} + 156 q^{95} - 909 q^{97}+O(q^{100}) $ 9 * q + 6 * q^5 + 63 * q^7 + 93 * q^11 + 18 * q^13 - 27 * q^17 - 45 * q^19 + 246 * q^23 + 315 * q^25 + 318 * q^29 + 18 * q^31 + 42 * q^35 - 36 * q^37 + 57 * q^41 + 171 * q^43 + 1056 * q^47 + 441 * q^49 + 756 * q^53 - 900 * q^55 + 411 * q^59 + 198 * q^61 + 1326 * q^65 + 441 * q^67 + 1758 * q^71 + 27 * q^73 + 651 * q^77 - 72 * q^79 + 558 * q^83 + 1008 * q^85 + 1392 * q^89 + 126 * q^91 + 156 * q^95 - 909 * 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
$5$	−15.3368	−1.37176	−0.685881	−	0.727714i	$-0.740583\pi$
$5$	−15.3368	−1.37176	−0.685881	+	0.727714i	$0.740583\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	67.0375	1.83751	0.918753	−	0.394834i	$-0.129198\pi$
$11$	67.0375	1.83751	0.918753	+	0.394834i	$0.129198\pi$
$12$	0	0
$13$	81.5617	1.74009	0.870044	−	0.492975i	$-0.164091\pi$
$13$	81.5617	1.74009	0.870044	+	0.492975i	$0.164091\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	101.984	1.45499	0.727496	−	0.686112i	$-0.240684\pi$
$17$	101.984	1.45499	0.727496	+	0.686112i	$0.240684\pi$
$18$	0	0
$19$	86.0055	1.03848	0.519238	−	0.854630i	$-0.326216\pi$
$19$	86.0055	1.03848	0.519238	+	0.854630i	$0.326216\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	129.102	1.17042	0.585208	−	0.810883i	$-0.301013\pi$
$23$	129.102	1.17042	0.585208	+	0.810883i	$0.301013\pi$
$24$	0	0
$25$	110.216	0.881732
$26$	0	0
$27$	0	0
$28$	0	0
$29$	90.4455	0.579149	0.289574	−	0.957156i	$-0.406486\pi$
$29$	90.4455	0.579149	0.289574	+	0.957156i	$0.406486\pi$
$30$	0	0
$31$	−95.7316	−0.554642	−0.277321	−	0.960777i	$-0.589447\pi$
$31$	−95.7316	−0.554642	−0.277321	+	0.960777i	$0.589447\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−107.357	−0.518477
$36$	0	0
$37$	−207.875	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$37$	−207.875	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−130.143	−0.495729	−0.247864	−	0.968795i	$-0.579729\pi$
$41$	−130.143	−0.495729	−0.247864	+	0.968795i	$0.579729\pi$
$42$	0	0
$43$	73.2845	0.259902	0.129951	−	0.991520i	$-0.458518\pi$
$43$	73.2845	0.259902	0.129951	+	0.991520i	$0.458518\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	331.041	1.02739	0.513695	−	0.857973i	$-0.328276\pi$
$47$	331.041	1.02739	0.513695	+	0.857973i	$0.328276\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	120.105	0.311276	0.155638	−	0.987814i	$-0.450257\pi$
$53$	120.105	0.311276	0.155638	+	0.987814i	$0.450257\pi$
$54$	0	0
$55$	−1028.14	−2.52062
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−195.307	−0.430962	−0.215481	−	0.976508i	$-0.569132\pi$
$59$	−195.307	−0.430962	−0.215481	+	0.976508i	$0.569132\pi$
$60$	0	0
$61$	−59.7514	−0.125416	−0.0627080	−	0.998032i	$-0.519974\pi$
$61$	−59.7514	−0.125416	−0.0627080	+	0.998032i	$0.519974\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1250.89	−2.38699
$66$	0	0
$67$	−805.962	−1.46961	−0.734806	−	0.678277i	$-0.762727\pi$
$67$	−805.962	−1.46961	−0.734806	+	0.678277i	$0.762727\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	391.162	0.653837	0.326918	−	0.945053i	$-0.393990\pi$
$71$	391.162	0.653837	0.326918	+	0.945053i	$0.393990\pi$
$72$	0	0
$73$	−729.424	−1.16949	−0.584744	−	0.811218i	$-0.698805\pi$
$73$	−729.424	−1.16949	−0.584744	+	0.811218i	$0.698805\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	469.262	0.694512
$78$	0	0
$79$	655.667	0.933775	0.466888	−	0.884317i	$-0.345375\pi$
$79$	655.667	0.933775	0.466888	+	0.884317i	$0.345375\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.3077	−0.0162764	−0.00813822	−	0.999967i	$-0.502591\pi$
$83$	−12.3077	−0.0162764	−0.00813822	+	0.999967i	$0.502591\pi$
$84$	0	0
$85$	−1564.11	−1.99590
$86$	0	0
$87$	0	0
$88$	0	0
$89$	716.679	0.853571	0.426785	−	0.904353i	$-0.359646\pi$
$89$	716.679	0.853571	0.426785	+	0.904353i	$0.359646\pi$
$90$	0	0
$91$	570.932	0.657691
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1319.05	−1.42454
$96$	0	0
$97$	751.388	0.786515	0.393257	−	0.919428i	$-0.371348\pi$
$97$	751.388	0.786515	0.393257	+	0.919428i	$0.371348\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	266.002	0.262062	0.131031	−	0.991378i	$-0.458171\pi$
$101$	266.002	0.262062	0.131031	+	0.991378i	$0.458171\pi$
$102$	0	0
$103$	−1161.91	−1.11151	−0.555757	−	0.831345i	$-0.687572\pi$
$103$	−1161.91	−1.11151	−0.555757	+	0.831345i	$0.687572\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1672.22	1.51084	0.755418	−	0.655243i	$-0.227434\pi$
$107$	1672.22	1.51084	0.755418	+	0.655243i	$0.227434\pi$
$108$	0	0
$109$	686.800	0.603519	0.301759	−	0.953384i	$-0.402426\pi$
$109$	686.800	0.603519	0.301759	+	0.953384i	$0.402426\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1642.64	1.36749	0.683744	−	0.729722i	$-0.260350\pi$
$113$	1642.64	1.36749	0.683744	+	0.729722i	$0.260350\pi$
$114$	0	0
$115$	−1980.00	−1.60553
$116$	0	0
$117$	0	0
$118$	0	0
$119$	713.891	0.549935
$120$	0	0
$121$	3163.02	2.37643
$122$	0	0
$123$	0	0
$124$	0	0
$125$	226.732	0.162236
$126$	0	0
$127$	−1149.15	−0.802917	−0.401459	−	0.915877i	$-0.631497\pi$
$127$	−1149.15	−0.802917	−0.401459	+	0.915877i	$0.631497\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2120.84	−1.41449	−0.707247	−	0.706967i	$-0.750063\pi$
$131$	−2120.84	−1.41449	−0.707247	+	0.706967i	$0.750063\pi$
$132$	0	0
$133$	602.039	0.392507
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2545.36	−1.58734	−0.793669	−	0.608350i	$-0.791832\pi$
$137$	−2545.36	−1.58734	−0.793669	+	0.608350i	$0.791832\pi$
$138$	0	0
$139$	−2237.55	−1.36537	−0.682685	−	0.730713i	$-0.739188\pi$
$139$	−2237.55	−1.36537	−0.682685	+	0.730713i	$0.739188\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5467.69	3.19742
$144$	0	0
$145$	−1387.14	−0.794454
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2713.45	−1.49191	−0.745955	−	0.665996i	$-0.768007\pi$
$149$	−2713.45	−1.49191	−0.745955	+	0.665996i	$0.768007\pi$
$150$	0	0
$151$	−1003.00	−0.540552	−0.270276	−	0.962783i	$-0.587115\pi$
$151$	−1003.00	−0.540552	−0.270276	+	0.962783i	$0.587115\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1468.21	0.760837
$156$	0	0
$157$	2511.02	1.27644	0.638222	−	0.769853i	$-0.279670\pi$
$157$	2511.02	1.27644	0.638222	+	0.769853i	$0.279670\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	903.712	0.442376
$162$	0	0
$163$	3600.11	1.72995	0.864977	−	0.501812i	$-0.167333\pi$
$163$	3600.11	1.72995	0.864977	+	0.501812i	$0.167333\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2244.99	1.04025	0.520127	−	0.854089i	$-0.325885\pi$
$167$	2244.99	1.04025	0.520127	+	0.854089i	$0.325885\pi$
$168$	0	0
$169$	4455.30	2.02790
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1468.65	0.645430	0.322715	−	0.946496i	$-0.395404\pi$
$173$	1468.65	0.645430	0.322715	+	0.946496i	$0.395404\pi$
$174$	0	0
$175$	771.515	0.333263
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2121.48	−0.885847	−0.442923	−	0.896559i	$-0.646059\pi$
$179$	−2121.48	−0.885847	−0.442923	+	0.896559i	$0.646059\pi$
$180$	0	0
$181$	3103.09	1.27431	0.637157	−	0.770734i	$-0.280110\pi$
$181$	3103.09	1.27431	0.637157	+	0.770734i	$0.280110\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3188.13	1.26701
$186$	0	0
$187$	6836.78	2.67356
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1935.87	0.733373	0.366687	−	0.930345i	$-0.380492\pi$
$191$	1935.87	0.733373	0.366687	+	0.930345i	$0.380492\pi$
$192$	0	0
$193$	−2721.72	−1.01510	−0.507549	−	0.861623i	$-0.669448\pi$
$193$	−2721.72	−1.01510	−0.507549	+	0.861623i	$0.669448\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1198.22	−0.433350	−0.216675	−	0.976244i	$-0.569521\pi$
$197$	−1198.22	−0.433350	−0.216675	+	0.976244i	$0.569521\pi$
$198$	0	0
$199$	−4258.24	−1.51688	−0.758439	−	0.651744i	$-0.774038\pi$
$199$	−4258.24	−1.51688	−0.758439	+	0.651744i	$0.774038\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	633.119	0.218898
$204$	0	0
$205$	1995.97	0.680022
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5765.59	1.90820
$210$	0	0
$211$	−585.730	−0.191106	−0.0955528	−	0.995424i	$-0.530462\pi$
$211$	−585.730	−0.191106	−0.0955528	+	0.995424i	$0.530462\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1123.95	−0.356523
$216$	0	0
$217$	−670.121	−0.209635
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8318.02	2.53181
$222$	0	0
$223$	−2186.96	−0.656725	−0.328363	−	0.944552i	$-0.606497\pi$
$223$	−2186.96	−0.656725	−0.328363	+	0.944552i	$0.606497\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4843.20	−1.41610	−0.708050	−	0.706163i	$-0.750425\pi$
$227$	−4843.20	−1.41610	−0.708050	+	0.706163i	$0.750425\pi$
$228$	0	0
$229$	5256.57	1.51687	0.758436	−	0.651747i	$-0.225964\pi$
$229$	5256.57	1.51687	0.758436	+	0.651747i	$0.225964\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2434.53	−0.684511	−0.342256	−	0.939607i	$-0.611191\pi$
$233$	−2434.53	−0.684511	−0.342256	+	0.939607i	$0.611191\pi$
$234$	0	0
$235$	−5077.11	−1.40934
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2218.40	−0.600404	−0.300202	−	0.953876i	$-0.597054\pi$
$239$	−2218.40	−0.600404	−0.300202	+	0.953876i	$0.597054\pi$
$240$	0	0
$241$	−1658.56	−0.443309	−0.221654	−	0.975125i	$-0.571146\pi$
$241$	−1658.56	−0.443309	−0.221654	+	0.975125i	$0.571146\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−751.502	−0.195966
$246$	0	0
$247$	7014.75	1.80704
$248$	0	0
$249$	0	0
$250$	0	0
$251$	297.180	0.0747323	0.0373662	−	0.999302i	$-0.488103\pi$
$251$	297.180	0.0747323	0.0373662	+	0.999302i	$0.488103\pi$
$252$	0	0
$253$	8654.65	2.15065
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2320.52	−0.563229	−0.281614	−	0.959528i	$-0.590870\pi$
$257$	−2320.52	−0.563229	−0.281614	+	0.959528i	$0.590870\pi$
$258$	0	0
$259$	−1455.13	−0.349101
$260$	0	0
$261$	0	0
$262$	0	0
$263$	94.9064	0.0222516	0.0111258	−	0.999938i	$-0.496458\pi$
$263$	94.9064	0.0222516	0.0111258	+	0.999938i	$0.496458\pi$
$264$	0	0
$265$	−1842.02	−0.426997
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2243.74	0.508563	0.254281	−	0.967130i	$-0.418161\pi$
$269$	2243.74	0.508563	0.254281	+	0.967130i	$0.418161\pi$
$270$	0	0
$271$	−1740.35	−0.390105	−0.195053	−	0.980793i	$-0.562488\pi$
$271$	−1740.35	−0.390105	−0.195053	+	0.980793i	$0.562488\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7388.63	1.62019
$276$	0	0
$277$	−6497.92	−1.40947	−0.704733	−	0.709472i	$-0.748933\pi$
$277$	−6497.92	−1.40947	−0.704733	+	0.709472i	$0.748933\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3939.23	0.836280	0.418140	−	0.908382i	$-0.362682\pi$
$281$	3939.23	0.836280	0.418140	+	0.908382i	$0.362682\pi$
$282$	0	0
$283$	−6574.28	−1.38092	−0.690460	−	0.723370i	$-0.742592\pi$
$283$	−6574.28	−1.38092	−0.690460	+	0.723370i	$0.742592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−910.999	−0.187368
$288$	0	0
$289$	5487.83	1.11700
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.2643	−0.00483800	−0.00241900	−	0.999997i	$-0.500770\pi$
$293$	−24.2643	−0.00483800	−0.00241900	+	0.999997i	$0.500770\pi$
$294$	0	0
$295$	2995.38	0.591178
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10529.7	2.03663
$300$	0	0
$301$	512.991	0.0982336
$302$	0	0
$303$	0	0
$304$	0	0
$305$	916.393	0.172041
$306$	0	0
$307$	−8877.90	−1.65045	−0.825226	−	0.564803i	$-0.808952\pi$
$307$	−8877.90	−1.65045	−0.825226	+	0.564803i	$0.808952\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−763.447	−0.139200	−0.0695999	−	0.997575i	$-0.522172\pi$
$311$	−763.447	−0.139200	−0.0695999	+	0.997575i	$0.522172\pi$
$312$	0	0
$313$	−10025.9	−1.81053	−0.905264	−	0.424849i	$-0.860327\pi$
$313$	−10025.9	−1.81053	−0.905264	+	0.424849i	$0.860327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1270.33	−0.225075	−0.112537	−	0.993647i	$-0.535898\pi$
$317$	−1270.33	−0.225075	−0.112537	+	0.993647i	$0.535898\pi$
$318$	0	0
$319$	6063.24	1.06419
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8771.23	1.51097
$324$	0	0
$325$	8989.44	1.53429
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2317.29	0.388317
$330$	0	0
$331$	−6064.31	−1.00702	−0.503511	−	0.863989i	$-0.667959\pi$
$331$	−6064.31	−1.00702	−0.503511	+	0.863989i	$0.667959\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12360.9	2.01596
$336$	0	0
$337$	−8864.59	−1.43289	−0.716447	−	0.697642i	$-0.754233\pi$
$337$	−8864.59	−1.43289	−0.716447	+	0.697642i	$0.754233\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6417.60	−1.01916
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3354.82	−0.519010	−0.259505	−	0.965742i	$-0.583559\pi$
$347$	−3354.82	−0.519010	−0.259505	+	0.965742i	$0.583559\pi$
$348$	0	0
$349$	−5383.99	−0.825784	−0.412892	−	0.910780i	$-0.635481\pi$
$349$	−5383.99	−0.825784	−0.412892	+	0.910780i	$0.635481\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4561.70	0.687804	0.343902	−	0.939006i	$-0.388251\pi$
$353$	4561.70	0.687804	0.343902	+	0.939006i	$0.388251\pi$
$354$	0	0
$355$	−5999.17	−0.896909
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4104.70	0.603448	0.301724	−	0.953395i	$-0.402438\pi$
$359$	4104.70	0.603448	0.301724	+	0.953395i	$0.402438\pi$
$360$	0	0
$361$	537.954	0.0784303
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11187.0	1.60426
$366$	0	0
$367$	−8192.00	−1.16517	−0.582587	−	0.812768i	$-0.697959\pi$
$367$	−8192.00	−1.16517	−0.582587	+	0.812768i	$0.697959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	840.732	0.117651
$372$	0	0
$373$	5600.76	0.777470	0.388735	−	0.921350i	$-0.372912\pi$
$373$	5600.76	0.777470	0.388735	+	0.921350i	$0.372912\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7376.89	1.00777
$378$	0	0
$379$	6164.97	0.835549	0.417774	−	0.908551i	$-0.362810\pi$
$379$	6164.97	0.835549	0.417774	+	0.908551i	$0.362810\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6240.72	0.832600	0.416300	−	0.909227i	$-0.363327\pi$
$383$	6240.72	0.832600	0.416300	+	0.909227i	$0.363327\pi$
$384$	0	0
$385$	−7196.97	−0.952705
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4557.10	−0.593970	−0.296985	−	0.954882i	$-0.595981\pi$
$389$	−4557.10	−0.593970	−0.296985	+	0.954882i	$0.595981\pi$
$390$	0	0
$391$	13166.4	1.70295
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10055.8	−1.28092
$396$	0	0
$397$	−1518.01	−0.191906	−0.0959531	−	0.995386i	$-0.530590\pi$
$397$	−1518.01	−0.191906	−0.0959531	+	0.995386i	$0.530590\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9556.91	1.19015	0.595074	−	0.803671i	$-0.297123\pi$
$401$	9556.91	1.19015	0.595074	+	0.803671i	$0.297123\pi$
$402$	0	0
$403$	−7808.03	−0.965125
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13935.4	−1.69718
$408$	0	0
$409$	−9054.74	−1.09469	−0.547345	−	0.836907i	$-0.684361\pi$
$409$	−9054.74	−1.09469	−0.547345	+	0.836907i	$0.684361\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1367.15	−0.162889
$414$	0	0
$415$	188.760	0.0223274
$416$	0	0
$417$	0	0
$418$	0	0
$419$	907.299	0.105786	0.0528931	−	0.998600i	$-0.483156\pi$
$419$	907.299	0.105786	0.0528931	+	0.998600i	$0.483156\pi$
$420$	0	0
$421$	8073.31	0.934605	0.467303	−	0.884097i	$-0.345226\pi$
$421$	8073.31	0.934605	0.467303	+	0.884097i	$0.345226\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11240.4	1.28291
$426$	0	0
$427$	−418.260	−0.0474028
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13087.2	1.46261	0.731307	−	0.682049i	$-0.238911\pi$
$431$	13087.2	1.46261	0.731307	+	0.682049i	$0.238911\pi$
$432$	0	0
$433$	184.294	0.0204540	0.0102270	−	0.999948i	$-0.496745\pi$
$433$	184.294	0.0204540	0.0102270	+	0.999948i	$0.496745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11103.5	1.21545
$438$	0	0
$439$	2476.82	0.269276	0.134638	−	0.990895i	$-0.457013\pi$
$439$	2476.82	0.269276	0.134638	+	0.990895i	$0.457013\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6841.58	0.733755	0.366878	−	0.930269i	$-0.380427\pi$
$443$	6841.58	0.733755	0.366878	+	0.930269i	$0.380427\pi$
$444$	0	0
$445$	−10991.5	−1.17090
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8977.61	0.943607	0.471803	−	0.881704i	$-0.343603\pi$
$449$	8977.61	0.943607	0.471803	+	0.881704i	$0.343603\pi$
$450$	0	0
$451$	−8724.44	−0.910904
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8756.25	−0.902196
$456$	0	0
$457$	10994.2	1.12535	0.562676	−	0.826678i	$-0.309772\pi$
$457$	10994.2	1.12535	0.562676	+	0.826678i	$0.309772\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11708.9	−1.18295	−0.591474	−	0.806324i	$-0.701454\pi$
$461$	−11708.9	−1.18295	−0.591474	+	0.806324i	$0.701454\pi$
$462$	0	0
$463$	14852.8	1.49086	0.745431	−	0.666582i	$-0.232244\pi$
$463$	14852.8	1.49086	0.745431	+	0.666582i	$0.232244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9614.01	−0.952641	−0.476321	−	0.879272i	$-0.658030\pi$
$467$	−9614.01	−0.952641	−0.476321	+	0.879272i	$0.658030\pi$
$468$	0	0
$469$	−5641.74	−0.555461
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4912.81	0.477571
$474$	0	0
$475$	9479.23	0.915656
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4262.95	0.406637	0.203318	−	0.979113i	$-0.434827\pi$
$479$	4262.95	0.406637	0.203318	+	0.979113i	$0.434827\pi$
$480$	0	0
$481$	−16954.6	−1.60720
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11523.9	−1.07891
$486$	0	0
$487$	12531.7	1.16605	0.583025	−	0.812454i	$-0.301869\pi$
$487$	12531.7	1.16605	0.583025	+	0.812454i	$0.301869\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12259.1	1.12677	0.563386	−	0.826194i	$-0.309499\pi$
$491$	12259.1	1.12677	0.563386	+	0.826194i	$0.309499\pi$
$492$	0	0
$493$	9224.04	0.842657
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2738.14	0.247127
$498$	0	0
$499$	9400.08	0.843298	0.421649	−	0.906759i	$-0.361452\pi$
$499$	9400.08	0.843298	0.421649	+	0.906759i	$0.361452\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10688.3	−0.947455	−0.473728	−	0.880671i	$-0.657092\pi$
$503$	−10688.3	−0.947455	−0.473728	+	0.880671i	$0.657092\pi$
$504$	0	0
$505$	−4079.62	−0.359486
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16855.7	−1.46781	−0.733906	−	0.679251i	$-0.762305\pi$
$509$	−16855.7	−1.46781	−0.733906	+	0.679251i	$0.762305\pi$
$510$	0	0
$511$	−5105.97	−0.442025
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17819.9	1.52473
$516$	0	0
$517$	22192.2	1.88784
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19657.4	−1.65299	−0.826494	−	0.562946i	$-0.809668\pi$
$521$	−19657.4	−1.65299	−0.826494	+	0.562946i	$0.809668\pi$
$522$	0	0
$523$	−17874.8	−1.49448	−0.747238	−	0.664557i	$-0.768620\pi$
$523$	−17874.8	−1.49448	−0.747238	+	0.664557i	$0.768620\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9763.14	−0.807000
$528$	0	0
$529$	4500.25	0.369874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10614.7	−0.862611
$534$	0	0
$535$	−25646.4	−2.07251
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3284.84	0.262501
$540$	0	0
$541$	−2697.58	−0.214377	−0.107189	−	0.994239i	$-0.534185\pi$
$541$	−2697.58	−0.214377	−0.107189	+	0.994239i	$0.534185\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10533.3	−0.827884
$546$	0	0
$547$	13190.0	1.03101	0.515505	−	0.856887i	$-0.327604\pi$
$547$	13190.0	1.03101	0.515505	+	0.856887i	$0.327604\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7778.82	0.601432
$552$	0	0
$553$	4589.67	0.352934
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20815.7	1.58346	0.791731	−	0.610869i	$-0.209180\pi$
$557$	20815.7	1.58346	0.791731	+	0.610869i	$0.209180\pi$
$558$	0	0
$559$	5977.20	0.452252
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7430.92	−0.556263	−0.278132	−	0.960543i	$-0.589715\pi$
$563$	−7430.92	−0.556263	−0.278132	+	0.960543i	$0.589715\pi$
$564$	0	0
$565$	−25192.7	−1.87587
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11396.7	0.839675	0.419838	−	0.907599i	$-0.362087\pi$
$569$	11396.7	0.839675	0.419838	+	0.907599i	$0.362087\pi$
$570$	0	0
$571$	6212.79	0.455337	0.227668	−	0.973739i	$-0.426890\pi$
$571$	6212.79	0.455337	0.227668	+	0.973739i	$0.426890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14229.1	1.03199
$576$	0	0
$577$	15646.5	1.12890	0.564448	−	0.825469i	$-0.309089\pi$
$577$	15646.5	1.12890	0.564448	+	0.825469i	$0.309089\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−86.1538	−0.00615192
$582$	0	0
$583$	8051.51	0.571972
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11853.9	0.833498	0.416749	−	0.909022i	$-0.363169\pi$
$587$	11853.9	0.833498	0.416749	+	0.909022i	$0.363169\pi$
$588$	0	0
$589$	−8233.45	−0.575982
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24339.6	1.68551	0.842755	−	0.538297i	$-0.180932\pi$
$593$	24339.6	1.68551	0.842755	+	0.538297i	$0.180932\pi$
$594$	0	0
$595$	−10948.8	−0.754381
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12187.1	0.831305	0.415652	−	0.909524i	$-0.363553\pi$
$599$	12187.1	0.831305	0.415652	+	0.909524i	$0.363553\pi$
$600$	0	0
$601$	−1322.56	−0.0897646	−0.0448823	−	0.998992i	$-0.514291\pi$
$601$	−1322.56	−0.0897646	−0.0448823	+	0.998992i	$0.514291\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−48510.5	−3.25989
$606$	0	0
$607$	18384.2	1.22931	0.614657	−	0.788795i	$-0.289295\pi$
$607$	18384.2	1.22931	0.614657	+	0.788795i	$0.289295\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27000.3	1.78775
$612$	0	0
$613$	−14174.8	−0.933959	−0.466979	−	0.884268i	$-0.654658\pi$
$613$	−14174.8	−0.933959	−0.466979	+	0.884268i	$0.654658\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17273.0	−1.12704	−0.563519	−	0.826103i	$-0.690553\pi$
$617$	−17273.0	−1.12704	−0.563519	+	0.826103i	$0.690553\pi$
$618$	0	0
$619$	17563.7	1.14046	0.570229	−	0.821485i	$-0.306854\pi$
$619$	17563.7	1.14046	0.570229	+	0.821485i	$0.306854\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5016.75	0.322619
$624$	0	0
$625$	−17254.4	−1.10428
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21200.0	−1.34388
$630$	0	0
$631$	−7749.44	−0.488907	−0.244454	−	0.969661i	$-0.578609\pi$
$631$	−7749.44	−0.488907	−0.244454	+	0.969661i	$0.578609\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17624.2	1.10141
$636$	0	0
$637$	3996.52	0.248584
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1961.32	−0.120854	−0.0604272	−	0.998173i	$-0.519246\pi$
$641$	−1961.32	−0.120854	−0.0604272	+	0.998173i	$0.519246\pi$
$642$	0	0
$643$	9630.70	0.590665	0.295333	−	0.955394i	$-0.404570\pi$
$643$	9630.70	0.590665	0.295333	+	0.955394i	$0.404570\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10126.3	0.615313	0.307656	−	0.951498i	$-0.400455\pi$
$647$	10126.3	0.615313	0.307656	+	0.951498i	$0.400455\pi$
$648$	0	0
$649$	−13092.9	−0.791896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19568.1	−1.17268	−0.586338	−	0.810066i	$-0.699431\pi$
$653$	−19568.1	−1.17268	−0.586338	+	0.810066i	$0.699431\pi$
$654$	0	0
$655$	32526.8	1.94035
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26070.8	−1.54108	−0.770542	−	0.637390i	$-0.780014\pi$
$659$	−26070.8	−1.54108	−0.770542	+	0.637390i	$0.780014\pi$
$660$	0	0
$661$	622.201	0.0366124	0.0183062	−	0.999832i	$-0.494173\pi$
$661$	622.201	0.0366124	0.0183062	+	0.999832i	$0.494173\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9233.33	−0.538426
$666$	0	0
$667$	11676.7	0.677845
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4005.58	−0.230453
$672$	0	0
$673$	−23944.6	−1.37146	−0.685732	−	0.727854i	$-0.740518\pi$
$673$	−23944.6	−1.37146	−0.685732	+	0.727854i	$0.740518\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4299.22	0.244066	0.122033	−	0.992526i	$-0.461059\pi$
$677$	4299.22	0.244066	0.122033	+	0.992526i	$0.461059\pi$
$678$	0	0
$679$	5259.72	0.297275
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19177.7	1.07440	0.537200	−	0.843455i	$-0.319482\pi$
$683$	19177.7	1.07440	0.537200	+	0.843455i	$0.319482\pi$
$684$	0	0
$685$	39037.7	2.17745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9795.93	0.541648
$690$	0	0
$691$	−27500.3	−1.51398	−0.756990	−	0.653427i	$-0.773331\pi$
$691$	−27500.3	−1.51398	−0.756990	+	0.653427i	$0.773331\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34316.8	1.87296
$696$	0	0
$697$	−13272.5	−0.721281
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17920.6	0.965550	0.482775	−	0.875744i	$-0.339629\pi$
$701$	17920.6	0.965550	0.482775	+	0.875744i	$0.339629\pi$
$702$	0	0
$703$	−17878.4	−0.959171
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1862.02	0.0990500
$708$	0	0
$709$	20518.6	1.08687	0.543436	−	0.839451i	$-0.317123\pi$
$709$	20518.6	1.08687	0.543436	+	0.839451i	$0.317123\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12359.1	−0.649162
$714$	0	0
$715$	−83856.7	−4.38610
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14773.2	0.766268	0.383134	−	0.923693i	$-0.374845\pi$
$719$	14773.2	0.766268	0.383134	+	0.923693i	$0.374845\pi$
$720$	0	0
$721$	−8133.34	−0.420113
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9968.59	0.510654
$726$	0	0
$727$	18640.9	0.950965	0.475482	−	0.879725i	$-0.342274\pi$
$727$	18640.9	0.950965	0.475482	+	0.879725i	$0.342274\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7473.88	0.378155
$732$	0	0
$733$	36290.7	1.82869	0.914344	−	0.404937i	$-0.132707\pi$
$733$	36290.7	1.82869	0.914344	+	0.404937i	$0.132707\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−54029.7	−2.70042
$738$	0	0
$739$	15137.4	0.753503	0.376751	−	0.926314i	$-0.377041\pi$
$739$	15137.4	0.753503	0.376751	+	0.926314i	$0.377041\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11162.3	0.551152	0.275576	−	0.961279i	$-0.411132\pi$
$743$	11162.3	0.551152	0.275576	+	0.961279i	$0.411132\pi$
$744$	0	0
$745$	41615.6	2.04655
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11705.5	0.571043
$750$	0	0
$751$	25269.1	1.22781	0.613903	−	0.789382i	$-0.289599\pi$
$751$	25269.1	1.22781	0.613903	+	0.789382i	$0.289599\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15382.8	0.741509
$756$	0	0
$757$	27731.0	1.33144	0.665721	−	0.746200i	$-0.268124\pi$
$757$	27731.0	1.33144	0.665721	+	0.746200i	$0.268124\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2438.08	0.116137	0.0580686	−	0.998313i	$-0.481506\pi$
$761$	2438.08	0.116137	0.0580686	+	0.998313i	$0.481506\pi$
$762$	0	0
$763$	4807.60	0.228109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15929.6	−0.749912
$768$	0	0
$769$	28373.8	1.33054	0.665269	−	0.746603i	$-0.268317\pi$
$769$	28373.8	1.33054	0.665269	+	0.746603i	$0.268317\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40257.5	1.87317	0.936585	−	0.350439i	$-0.113968\pi$
$773$	40257.5	1.87317	0.936585	+	0.350439i	$0.113968\pi$
$774$	0	0
$775$	−10551.2	−0.489045
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11193.0	−0.514802
$780$	0	0
$781$	26222.5	1.20143
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−38511.0	−1.75098
$786$	0	0
$787$	−5077.63	−0.229985	−0.114992	−	0.993366i	$-0.536684\pi$
$787$	−5077.63	−0.229985	−0.114992	+	0.993366i	$0.536684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11498.5	0.516862
$792$	0	0
$793$	−4873.42	−0.218235
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25840.5	−1.14845	−0.574226	−	0.818697i	$-0.694697\pi$
$797$	−25840.5	−1.14845	−0.574226	+	0.818697i	$0.694697\pi$
$798$	0	0
$799$	33761.1	1.49485
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−48898.8	−2.14894
$804$	0	0
$805$	−13860.0	−0.606834
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8475.12	−0.368318	−0.184159	−	0.982896i	$-0.558956\pi$
$809$	−8475.12	−0.368318	−0.184159	+	0.982896i	$0.558956\pi$
$810$	0	0
$811$	−25028.1	−1.08367	−0.541833	−	0.840486i	$-0.682270\pi$
$811$	−25028.1	−1.08367	−0.541833	+	0.840486i	$0.682270\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55214.1	−2.37308
$816$	0	0
$817$	6302.87	0.269901
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38666.2	−1.64368	−0.821840	−	0.569719i	$-0.807052\pi$
$821$	−38666.2	−1.64368	−0.821840	+	0.569719i	$0.807052\pi$
$822$	0	0
$823$	−2551.70	−0.108076	−0.0540380	−	0.998539i	$-0.517209\pi$
$823$	−2551.70	−0.108076	−0.0540380	+	0.998539i	$0.517209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20418.3	0.858540	0.429270	−	0.903176i	$-0.358771\pi$
$827$	20418.3	0.858540	0.429270	+	0.903176i	$0.358771\pi$
$828$	0	0
$829$	6122.36	0.256500	0.128250	−	0.991742i	$-0.459064\pi$
$829$	6122.36	0.256500	0.128250	+	0.991742i	$0.459064\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4997.24	0.207856
$834$	0	0
$835$	−34430.9	−1.42698
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39101.8	1.60899	0.804496	−	0.593958i	$-0.202435\pi$
$839$	39101.8	1.60899	0.804496	+	0.593958i	$0.202435\pi$
$840$	0	0
$841$	−16208.6	−0.664587
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−68330.0	−2.78180
$846$	0	0
$847$	22141.2	0.898205
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−26837.0	−1.08104
$852$	0	0
$853$	−20542.6	−0.824579	−0.412289	−	0.911053i	$-0.635271\pi$
$853$	−20542.6	−0.824579	−0.412289	+	0.911053i	$0.635271\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15778.9	−0.628933	−0.314466	−	0.949269i	$-0.601826\pi$
$857$	−15778.9	−0.628933	−0.314466	+	0.949269i	$0.601826\pi$
$858$	0	0
$859$	37416.7	1.48620	0.743098	−	0.669182i	$-0.233355\pi$
$859$	37416.7	1.48620	0.743098	+	0.669182i	$0.233355\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26182.8	−1.03276	−0.516380	−	0.856360i	$-0.672721\pi$
$863$	−26182.8	−1.03276	−0.516380	+	0.856360i	$0.672721\pi$
$864$	0	0
$865$	−22524.4	−0.885377
$866$	0	0
$867$	0	0
$868$	0	0
$869$	43954.2	1.71582
$870$	0	0
$871$	−65735.6	−2.55725
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1587.12	0.0613195
$876$	0	0
$877$	20166.9	0.776498	0.388249	−	0.921555i	$-0.373080\pi$
$877$	20166.9	0.776498	0.388249	+	0.921555i	$0.373080\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3159.11	0.120809	0.0604047	−	0.998174i	$-0.480761\pi$
$881$	3159.11	0.120809	0.0604047	+	0.998174i	$0.480761\pi$
$882$	0	0
$883$	32379.2	1.23403	0.617015	−	0.786952i	$-0.288342\pi$
$883$	32379.2	1.23403	0.617015	+	0.786952i	$0.288342\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15063.5	−0.570217	−0.285109	−	0.958495i	$-0.592030\pi$
$887$	−15063.5	−0.570217	−0.285109	+	0.958495i	$0.592030\pi$
$888$	0	0
$889$	−8044.04	−0.303474
$890$	0	0
$891$	0	0
$892$	0	0
$893$	28471.4	1.06692
$894$	0	0
$895$	32536.6	1.21517
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8658.50	−0.321220
$900$	0	0
$901$	12248.8	0.452904
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−47591.3	−1.74805
$906$	0	0
$907$	−14359.2	−0.525679	−0.262839	−	0.964840i	$-0.584659\pi$
$907$	−14359.2	−0.525679	−0.262839	+	0.964840i	$0.584659\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16573.3	−0.602743	−0.301371	−	0.953507i	$-0.597444\pi$
$911$	−16573.3	−0.602743	−0.301371	+	0.953507i	$0.597444\pi$
$912$	0	0
$913$	−825.077	−0.0299080
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14845.9	−0.534628
$918$	0	0
$919$	−9972.35	−0.357952	−0.178976	−	0.983853i	$-0.557278\pi$
$919$	−9972.35	−0.357952	−0.178976	+	0.983853i	$0.557278\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31903.8	1.13773
$924$	0	0
$925$	−22911.3	−0.814398
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17659.5	0.623671	0.311835	−	0.950136i	$-0.399056\pi$
$929$	17659.5	0.623671	0.311835	+	0.950136i	$0.399056\pi$
$930$	0	0
$931$	4214.27	0.148354
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−104854.	−3.66748
$936$	0	0
$937$	−16521.2	−0.576012	−0.288006	−	0.957629i	$-0.592992\pi$
$937$	−16521.2	−0.576012	−0.288006	+	0.957629i	$0.592992\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40296.1	−1.39598	−0.697988	−	0.716109i	$-0.745921\pi$
$941$	−40296.1	−1.39598	−0.697988	+	0.716109i	$0.745921\pi$
$942$	0	0
$943$	−16801.6	−0.580209
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22031.5	−0.755994	−0.377997	−	0.925807i	$-0.623387\pi$
$947$	−22031.5	−0.755994	−0.377997	+	0.925807i	$0.623387\pi$
$948$	0	0
$949$	−59493.0	−2.03501
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31578.2	−1.07337	−0.536683	−	0.843784i	$-0.680323\pi$
$953$	−31578.2	−1.07337	−0.536683	+	0.843784i	$0.680323\pi$
$954$	0	0
$955$	−29689.9	−1.00601
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17817.6	−0.599957
$960$	0	0
$961$	−20626.5	−0.692372
$962$	0	0
$963$	0	0
$964$	0	0
$965$	41742.4	1.39247
$966$	0	0
$967$	−47651.5	−1.58466	−0.792331	−	0.610092i	$-0.791132\pi$
$967$	−47651.5	−1.58466	−0.792331	+	0.610092i	$0.791132\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11343.0	−0.374885	−0.187443	−	0.982276i	$-0.560020\pi$
$971$	−11343.0	−0.374885	−0.187443	+	0.982276i	$0.560020\pi$
$972$	0	0
$973$	−15662.8	−0.516061
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16077.5	−0.526473	−0.263237	−	0.964731i	$-0.584790\pi$
$977$	−16077.5	−0.526473	−0.263237	+	0.964731i	$0.584790\pi$
$978$	0	0
$979$	48044.3	1.56844
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47324.9	−1.53553	−0.767767	−	0.640729i	$-0.778632\pi$
$983$	−47324.9	−1.53553	−0.767767	+	0.640729i	$0.778632\pi$
$984$	0	0
$985$	18376.9	0.594453
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9461.15	0.304193
$990$	0	0
$991$	−12416.7	−0.398011	−0.199005	−	0.979998i	$-0.563771\pi$
$991$	−12416.7	−0.398011	−0.199005	+	0.979998i	$0.563771\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	65307.7	2.08080
$996$	0	0
$997$	−25637.0	−0.814376	−0.407188	−	0.913344i	$-0.633491\pi$
$997$	−25637.0	−0.814376	−0.407188	+	0.913344i	$0.633491\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.4.a.i.1.2		9
3.2	odd	2		2268.4.a.h.1.8		9
9.2	odd	6		756.4.j.b.253.2		18
9.4	even	3		252.4.j.b.169.6	yes	18
9.5	odd	6		756.4.j.b.505.2		18
9.7	even	3		252.4.j.b.85.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.j.b.85.6	✓	18	9.7	even	3
252.4.j.b.169.6	yes	18	9.4	even	3
756.4.j.b.253.2		18	9.2	odd	6
756.4.j.b.505.2		18	9.5	odd	6
2268.4.a.h.1.8		9	3.2	odd	2
2268.4.a.i.1.2		9	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q-15.3368 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q-15.3368 q^{5} +7.00000 q^{7} +67.0375 q^{11} +81.5617 q^{13} +101.984 q^{17} +86.0055 q^{19} +129.102 q^{23} +110.216 q^{25} +90.4455 q^{29} -95.7316 q^{31} -107.357 q^{35} -207.875 q^{37} -130.143 q^{41} +73.2845 q^{43} +331.041 q^{47} +49.0000 q^{49} +120.105 q^{53} -1028.14 q^{55} -195.307 q^{59} -59.7514 q^{61} -1250.89 q^{65} -805.962 q^{67} +391.162 q^{71} -729.424 q^{73} +469.262 q^{77} +655.667 q^{79} -12.3077 q^{83} -1564.11 q^{85} +716.679 q^{89} +570.932 q^{91} -1319.05 q^{95} +751.388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{5} + 63 q^{7}+O(q^{10}) \) 9 * q + 6 * q^5 + 63 * q^7	\( 9 q + 6 q^{5} + 63 q^{7} + 93 q^{11} + 18 q^{13} - 27 q^{17} - 45 q^{19} + 246 q^{23} + 315 q^{25} + 318 q^{29} + 18 q^{31} + 42 q^{35} - 36 q^{37} + 57 q^{41} + 171 q^{43} + 1056 q^{47} + 441 q^{49} + 756 q^{53} - 900 q^{55} + 411 q^{59} + 198 q^{61} + 1326 q^{65} + 441 q^{67} + 1758 q^{71} + 27 q^{73} + 651 q^{77} - 72 q^{79} + 558 q^{83} + 1008 q^{85} + 1392 q^{89} + 126 q^{91} + 156 q^{95} - 909 q^{97}+O(q^{100}) \) 9 * q + 6 * q^5 + 63 * q^7 + 93 * q^11 + 18 * q^13 - 27 * q^17 - 45 * q^19 + 246 * q^23 + 315 * q^25 + 318 * q^29 + 18 * q^31 + 42 * q^35 - 36 * q^37 + 57 * q^41 + 171 * q^43 + 1056 * q^47 + 441 * q^49 + 756 * q^53 - 900 * q^55 + 411 * q^59 + 198 * q^61 + 1326 * q^65 + 441 * q^67 + 1758 * q^71 + 27 * q^73 + 651 * q^77 - 72 * q^79 + 558 * q^83 + 1008 * q^85 + 1392 * q^89 + 126 * q^91 + 156 * q^95 - 909 * q^97

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