LMFDB - Embedded newform 1728.4.a.cd.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,6,0,9,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 864)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.83638 q^{5} -17.1016 q^{7} +48.7123 q^{11} -64.6502 q^{13} +118.791 q^{17} -99.4428 q^{19} +57.9337 q^{23} -116.955 q^{25} +27.3455 q^{29} +58.0366 q^{31} -48.5066 q^{35} +133.378 q^{37} -195.255 q^{41} +351.459 q^{43} -568.504 q^{47} -50.5363 q^{49} +208.201 q^{53} +138.167 q^{55} +810.077 q^{59} -583.852 q^{61} -183.373 q^{65} -22.5520 q^{67} -218.914 q^{71} +273.373 q^{73} -833.057 q^{77} +830.315 q^{79} +650.502 q^{83} +336.937 q^{85} -633.902 q^{89} +1105.62 q^{91} -282.058 q^{95} -290.255 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} + 9 q^{7} + 18 q^{11} - 3 q^{13} - 18 q^{17} - 27 q^{19} - 90 q^{23} + 21 q^{25} + 72 q^{29} + 144 q^{31} + 450 q^{35} + 159 q^{37} + 168 q^{41} + 180 q^{43} - 522 q^{47} + 150 q^{49} - 84 q^{53}+ \cdots - 117 q^{97}+O(q^{100}) $ 3 * q + 6 * q^5 + 9 * q^7 + 18 * q^11 - 3 * q^13 - 18 * q^17 - 27 * q^19 - 90 * q^23 + 21 * q^25 + 72 * q^29 + 144 * q^31 + 450 * q^35 + 159 * q^37 + 168 * q^41 + 180 * q^43 - 522 * q^47 + 150 * q^49 - 84 * q^53 + 324 * q^55 + 1458 * q^59 - 33 * q^61 - 246 * q^65 + 837 * q^67 - 1908 * q^71 - 165 * q^73 - 954 * q^77 + 531 * q^79 + 2268 * q^83 - 708 * q^85 - 618 * q^89 + 1719 * q^91 - 2934 * q^95 - 117 * q^97

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
$5$	2.83638	0.253694	0.126847	−	0.991922i	$-0.459514\pi$
$5$	2.83638	0.253694	0.126847	+	0.991922i	$0.459514\pi$
$6$	0	0
$7$	−17.1016	−0.923398	−0.461699	−	0.887037i	$-0.652760\pi$
$7$	−17.1016	−0.923398	−0.461699	+	0.887037i	$0.652760\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	48.7123	1.33521	0.667605	−	0.744516i	$-0.267320\pi$
$11$	48.7123	1.33521	0.667605	+	0.744516i	$0.267320\pi$
$12$	0	0
$13$	−64.6502	−1.37929	−0.689644	−	0.724148i	$-0.742233\pi$
$13$	−64.6502	−1.37929	−0.689644	+	0.724148i	$0.742233\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	118.791	1.69477	0.847386	−	0.530977i	$-0.178175\pi$
$17$	118.791	1.69477	0.847386	+	0.530977i	$0.178175\pi$
$18$	0	0
$19$	−99.4428	−1.20072	−0.600362	−	0.799728i	$-0.704977\pi$
$19$	−99.4428	−1.20072	−0.600362	+	0.799728i	$0.704977\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	57.9337	0.525218	0.262609	−	0.964902i	$-0.415417\pi$
$23$	57.9337	0.525218	0.262609	+	0.964902i	$0.415417\pi$
$24$	0	0
$25$	−116.955	−0.935640
$26$	0	0
$27$	0	0
$28$	0	0
$29$	27.3455	0.175101	0.0875506	−	0.996160i	$-0.472096\pi$
$29$	27.3455	0.175101	0.0875506	+	0.996160i	$0.472096\pi$
$30$	0	0
$31$	58.0366	0.336248	0.168124	−	0.985766i	$-0.446229\pi$
$31$	58.0366	0.336248	0.168124	+	0.985766i	$0.446229\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−48.5066	−0.234260
$36$	0	0
$37$	133.378	0.592627	0.296313	−	0.955091i	$-0.404243\pi$
$37$	133.378	0.592627	0.296313	+	0.955091i	$0.404243\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−195.255	−0.743751	−0.371875	−	0.928283i	$-0.621285\pi$
$41$	−195.255	−0.743751	−0.371875	+	0.928283i	$0.621285\pi$
$42$	0	0
$43$	351.459	1.24644	0.623220	−	0.782047i	$-0.285824\pi$
$43$	351.459	1.24644	0.623220	+	0.782047i	$0.285824\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−568.504	−1.76436	−0.882179	−	0.470914i	$-0.843924\pi$
$47$	−568.504	−1.76436	−0.882179	+	0.470914i	$0.843924\pi$
$48$	0	0
$49$	−50.5363	−0.147336
$50$	0	0
$51$	0	0
$52$	0	0
$53$	208.201	0.539595	0.269798	−	0.962917i	$-0.413043\pi$
$53$	208.201	0.539595	0.269798	+	0.962917i	$0.413043\pi$
$54$	0	0
$55$	138.167	0.338734
$56$	0	0
$57$	0	0
$58$	0	0
$59$	810.077	1.78751	0.893755	−	0.448555i	$-0.148061\pi$
$59$	810.077	1.78751	0.893755	+	0.448555i	$0.148061\pi$
$60$	0	0
$61$	−583.852	−1.22549	−0.612743	−	0.790283i	$-0.709934\pi$
$61$	−583.852	−1.22549	−0.612743	+	0.790283i	$0.709934\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−183.373	−0.349917
$66$	0	0
$67$	−22.5520	−0.0411219	−0.0205609	−	0.999789i	$-0.506545\pi$
$67$	−22.5520	−0.0411219	−0.0205609	+	0.999789i	$0.506545\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−218.914	−0.365919	−0.182960	−	0.983120i	$-0.558568\pi$
$71$	−218.914	−0.365919	−0.182960	+	0.983120i	$0.558568\pi$
$72$	0	0
$73$	273.373	0.438300	0.219150	−	0.975691i	$-0.429672\pi$
$73$	273.373	0.438300	0.219150	+	0.975691i	$0.429672\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−833.057	−1.23293
$78$	0	0
$79$	830.315	1.18250	0.591252	−	0.806487i	$-0.298634\pi$
$79$	830.315	1.18250	0.591252	+	0.806487i	$0.298634\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	650.502	0.860264	0.430132	−	0.902766i	$-0.358467\pi$
$83$	650.502	0.860264	0.430132	+	0.902766i	$0.358467\pi$
$84$	0	0
$85$	336.937	0.429953
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−633.902	−0.754983	−0.377492	−	0.926013i	$-0.623213\pi$
$89$	−633.902	−0.754983	−0.377492	+	0.926013i	$0.623213\pi$
$90$	0	0
$91$	1105.62	1.27363
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−282.058	−0.304616
$96$	0	0
$97$	−290.255	−0.303824	−0.151912	−	0.988394i	$-0.548543\pi$
$97$	−290.255	−0.303824	−0.151912	+	0.988394i	$0.548543\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	507.001	0.499490	0.249745	−	0.968312i	$-0.419653\pi$
$101$	507.001	0.499490	0.249745	+	0.968312i	$0.419653\pi$
$102$	0	0
$103$	−752.431	−0.719798	−0.359899	−	0.932991i	$-0.617189\pi$
$103$	−752.431	−0.719798	−0.359899	+	0.932991i	$0.617189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1460.58	1.31962	0.659811	−	0.751432i	$-0.270636\pi$
$107$	1460.58	1.31962	0.659811	+	0.751432i	$0.270636\pi$
$108$	0	0
$109$	1017.99	0.894552	0.447276	−	0.894396i	$-0.352394\pi$
$109$	1017.99	0.894552	0.447276	+	0.894396i	$0.352394\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2071.65	1.72464	0.862320	−	0.506363i	$-0.169010\pi$
$113$	2071.65	1.72464	0.862320	+	0.506363i	$0.169010\pi$
$114$	0	0
$115$	164.322	0.133244
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2031.52	−1.56495
$120$	0	0
$121$	1041.89	0.782785
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−686.276	−0.491059
$126$	0	0
$127$	899.943	0.628796	0.314398	−	0.949291i	$-0.398197\pi$
$127$	899.943	0.628796	0.314398	+	0.949291i	$0.398197\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1147.63	0.765414	0.382707	−	0.923870i	$-0.374992\pi$
$131$	1147.63	0.765414	0.382707	+	0.923870i	$0.374992\pi$
$132$	0	0
$133$	1700.63	1.10875
$134$	0	0
$135$	0	0
$136$	0	0
$137$	154.596	0.0964090	0.0482045	−	0.998837i	$-0.484650\pi$
$137$	154.596	0.0964090	0.0482045	+	0.998837i	$0.484650\pi$
$138$	0	0
$139$	−715.023	−0.436313	−0.218156	−	0.975914i	$-0.570004\pi$
$139$	−715.023	−0.436313	−0.218156	+	0.975914i	$0.570004\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3149.26	−1.84164
$144$	0	0
$145$	77.5623	0.0444221
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1166.14	0.641165	0.320583	−	0.947221i	$-0.396121\pi$
$149$	1166.14	0.641165	0.320583	+	0.947221i	$0.396121\pi$
$150$	0	0
$151$	2559.09	1.37918	0.689588	−	0.724202i	$-0.257792\pi$
$151$	2559.09	1.37918	0.689588	+	0.724202i	$0.257792\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	164.614	0.0853039
$156$	0	0
$157$	−2233.25	−1.13524	−0.567620	−	0.823290i	$-0.692136\pi$
$157$	−2233.25	−1.13524	−0.567620	+	0.823290i	$0.692136\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−990.757	−0.484985
$162$	0	0
$163$	1886.95	0.906734	0.453367	−	0.891324i	$-0.350223\pi$
$163$	1886.95	0.906734	0.453367	+	0.891324i	$0.350223\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3645.85	−1.68937	−0.844683	−	0.535267i	$-0.820211\pi$
$167$	−3645.85	−1.68937	−0.844683	+	0.535267i	$0.820211\pi$
$168$	0	0
$169$	1982.65	0.902436
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3741.77	1.64440	0.822201	−	0.569197i	$-0.192746\pi$
$173$	3741.77	1.64440	0.822201	+	0.569197i	$0.192746\pi$
$174$	0	0
$175$	2000.11	0.863968
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1620.08	0.676485	0.338242	−	0.941059i	$-0.390168\pi$
$179$	1620.08	0.676485	0.338242	+	0.941059i	$0.390168\pi$
$180$	0	0
$181$	−2497.77	−1.02573	−0.512867	−	0.858468i	$-0.671417\pi$
$181$	−2497.77	−1.02573	−0.512867	+	0.858468i	$0.671417\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	378.310	0.150346
$186$	0	0
$187$	5786.60	2.26288
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4667.12	1.76807	0.884035	−	0.467421i	$-0.154817\pi$
$191$	4667.12	1.76807	0.884035	+	0.467421i	$0.154817\pi$
$192$	0	0
$193$	−5225.17	−1.94879	−0.974394	−	0.224848i	$-0.927811\pi$
$193$	−5225.17	−1.94879	−0.974394	+	0.224848i	$0.927811\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1773.26	−0.641319	−0.320659	−	0.947195i	$-0.603905\pi$
$197$	−1773.26	−0.641319	−0.320659	+	0.947195i	$0.603905\pi$
$198$	0	0
$199$	4645.00	1.65465	0.827325	−	0.561724i	$-0.189862\pi$
$199$	4645.00	1.65465	0.827325	+	0.561724i	$0.189862\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−467.651	−0.161688
$204$	0	0
$205$	−553.819	−0.188685
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4844.09	−1.60322
$210$	0	0
$211$	4957.52	1.61749	0.808744	−	0.588160i	$-0.200148\pi$
$211$	4957.52	1.61749	0.808744	+	0.588160i	$0.200148\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	996.870	0.316214
$216$	0	0
$217$	−992.516	−0.310490
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7679.89	−2.33758
$222$	0	0
$223$	−1009.14	−0.303037	−0.151519	−	0.988454i	$-0.548416\pi$
$223$	−1009.14	−0.303037	−0.151519	+	0.988454i	$0.548416\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3780.29	1.10532	0.552658	−	0.833408i	$-0.313614\pi$
$227$	3780.29	1.10532	0.552658	+	0.833408i	$0.313614\pi$
$228$	0	0
$229$	1947.59	0.562012	0.281006	−	0.959706i	$-0.409332\pi$
$229$	1947.59	0.562012	0.281006	+	0.959706i	$0.409332\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1144.46	0.321786	0.160893	−	0.986972i	$-0.448563\pi$
$233$	1144.46	0.321786	0.160893	+	0.986972i	$0.448563\pi$
$234$	0	0
$235$	−1612.49	−0.447606
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1682.61	−0.455394	−0.227697	−	0.973732i	$-0.573120\pi$
$239$	−1682.61	−0.455394	−0.227697	+	0.973732i	$0.573120\pi$
$240$	0	0
$241$	−834.555	−0.223064	−0.111532	−	0.993761i	$-0.535576\pi$
$241$	−834.555	−0.223064	−0.111532	+	0.993761i	$0.535576\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−143.340	−0.0373782
$246$	0	0
$247$	6429.00	1.65614
$248$	0	0
$249$	0	0
$250$	0	0
$251$	736.754	0.185273	0.0926364	−	0.995700i	$-0.470471\pi$
$251$	736.754	0.185273	0.0926364	+	0.995700i	$0.470471\pi$
$252$	0	0
$253$	2822.08	0.701276
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5048.80	1.22543	0.612715	−	0.790304i	$-0.290077\pi$
$257$	5048.80	1.22543	0.612715	+	0.790304i	$0.290077\pi$
$258$	0	0
$259$	−2280.97	−0.547230
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2988.83	−0.700757	−0.350379	−	0.936608i	$-0.613947\pi$
$263$	−2988.83	−0.700757	−0.350379	+	0.936608i	$0.613947\pi$
$264$	0	0
$265$	590.536	0.136892
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8325.70	1.88709	0.943545	−	0.331246i	$-0.107469\pi$
$269$	8325.70	1.88709	0.943545	+	0.331246i	$0.107469\pi$
$270$	0	0
$271$	−3810.55	−0.854149	−0.427075	−	0.904216i	$-0.640456\pi$
$271$	−3810.55	−0.854149	−0.427075	+	0.904216i	$0.640456\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5697.14	−1.24927
$276$	0	0
$277$	−2789.54	−0.605081	−0.302540	−	0.953137i	$-0.597835\pi$
$277$	−2789.54	−0.605081	−0.302540	+	0.953137i	$0.597835\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1509.74	−0.320510	−0.160255	−	0.987076i	$-0.551232\pi$
$281$	−1509.74	−0.320510	−0.160255	+	0.987076i	$0.551232\pi$
$282$	0	0
$283$	5958.51	1.25158	0.625789	−	0.779992i	$-0.284777\pi$
$283$	5958.51	1.25158	0.625789	+	0.779992i	$0.284777\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3339.17	0.686778
$288$	0	0
$289$	9198.38	1.87225
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9255.97	−1.84553	−0.922764	−	0.385365i	$-0.874075\pi$
$293$	−9255.97	−1.84553	−0.922764	+	0.385365i	$0.874075\pi$
$294$	0	0
$295$	2297.69	0.453480
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3745.43	−0.724427
$300$	0	0
$301$	−6010.49	−1.15096
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1656.03	−0.310898
$306$	0	0
$307$	6520.16	1.21213	0.606067	−	0.795414i	$-0.292746\pi$
$307$	6520.16	1.21213	0.606067	+	0.795414i	$0.292746\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10271.9	1.87288	0.936439	−	0.350831i	$-0.114101\pi$
$311$	10271.9	1.87288	0.936439	+	0.350831i	$0.114101\pi$
$312$	0	0
$313$	−6053.49	−1.09317	−0.546587	−	0.837403i	$-0.684073\pi$
$313$	−6053.49	−1.09317	−0.546587	+	0.837403i	$0.684073\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7954.44	1.40936	0.704678	−	0.709527i	$-0.251091\pi$
$317$	7954.44	1.40936	0.704678	+	0.709527i	$0.251091\pi$
$318$	0	0
$319$	1332.06	0.233797
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−11812.9	−2.03495
$324$	0	0
$325$	7561.16	1.29052
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9722.31	1.62920
$330$	0	0
$331$	604.526	0.100386	0.0501930	−	0.998740i	$-0.484016\pi$
$331$	604.526	0.100386	0.0501930	+	0.998740i	$0.484016\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−63.9660	−0.0104324
$336$	0	0
$337$	2682.87	0.433666	0.216833	−	0.976209i	$-0.430427\pi$
$337$	2682.87	0.433666	0.216833	+	0.976209i	$0.430427\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2827.09	0.448961
$342$	0	0
$343$	6730.09	1.05945
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2586.72	0.400180	0.200090	−	0.979777i	$-0.435876\pi$
$347$	2586.72	0.400180	0.200090	+	0.979777i	$0.435876\pi$
$348$	0	0
$349$	1518.59	0.232918	0.116459	−	0.993196i	$-0.462846\pi$
$349$	1518.59	0.232918	0.116459	+	0.993196i	$0.462846\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7663.44	−1.15548	−0.577739	−	0.816221i	$-0.696065\pi$
$353$	−7663.44	−1.15548	−0.577739	+	0.816221i	$0.696065\pi$
$354$	0	0
$355$	−620.923	−0.0928314
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9532.18	1.40136	0.700681	−	0.713474i	$-0.252879\pi$
$359$	9532.18	1.40136	0.700681	+	0.713474i	$0.252879\pi$
$360$	0	0
$361$	3029.88	0.441738
$362$	0	0
$363$	0	0
$364$	0	0
$365$	775.390	0.111194
$366$	0	0
$367$	10773.2	1.53231	0.766153	−	0.642658i	$-0.222168\pi$
$367$	10773.2	1.53231	0.766153	+	0.642658i	$0.222168\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3560.56	−0.498261
$372$	0	0
$373$	−735.143	−0.102049	−0.0510245	−	0.998697i	$-0.516249\pi$
$373$	−735.143	−0.102049	−0.0510245	+	0.998697i	$0.516249\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1767.89	−0.241515
$378$	0	0
$379$	8788.41	1.19111	0.595554	−	0.803315i	$-0.296933\pi$
$379$	8788.41	1.19111	0.595554	+	0.803315i	$0.296933\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5259.76	0.701726	0.350863	−	0.936427i	$-0.385888\pi$
$383$	5259.76	0.701726	0.350863	+	0.936427i	$0.385888\pi$
$384$	0	0
$385$	−2362.87	−0.312786
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10853.8	1.41467	0.707337	−	0.706876i	$-0.249896\pi$
$389$	10853.8	1.41467	0.707337	+	0.706876i	$0.249896\pi$
$390$	0	0
$391$	6882.02	0.890125
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2355.09	0.299994
$396$	0	0
$397$	12182.8	1.54014	0.770071	−	0.637959i	$-0.220221\pi$
$397$	12182.8	1.54014	0.770071	+	0.637959i	$0.220221\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3537.82	0.440574	0.220287	−	0.975435i	$-0.429301\pi$
$401$	3537.82	0.440574	0.220287	+	0.975435i	$0.429301\pi$
$402$	0	0
$403$	−3752.08	−0.463782
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6497.14	0.791281
$408$	0	0
$409$	−11611.9	−1.40384	−0.701920	−	0.712255i	$-0.747674\pi$
$409$	−11611.9	−1.40384	−0.701920	+	0.712255i	$0.747674\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13853.6	−1.65058
$414$	0	0
$415$	1845.07	0.218243
$416$	0	0
$417$	0	0
$418$	0	0
$419$	710.256	0.0828122	0.0414061	−	0.999142i	$-0.486816\pi$
$419$	710.256	0.0828122	0.0414061	+	0.999142i	$0.486816\pi$
$420$	0	0
$421$	−8869.78	−1.02681	−0.513405	−	0.858147i	$-0.671616\pi$
$421$	−8869.78	−1.02681	−0.513405	+	0.858147i	$0.671616\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13893.2	−1.58570
$426$	0	0
$427$	9984.79	1.13161
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1105.67	0.123569	0.0617847	−	0.998090i	$-0.480321\pi$
$431$	1105.67	0.123569	0.0617847	+	0.998090i	$0.480321\pi$
$432$	0	0
$433$	11029.1	1.22407	0.612037	−	0.790829i	$-0.290350\pi$
$433$	11029.1	1.22407	0.612037	+	0.790829i	$0.290350\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5761.09	−0.630642
$438$	0	0
$439$	−16036.8	−1.74349	−0.871746	−	0.489958i	$-0.837012\pi$
$439$	−16036.8	−1.74349	−0.871746	+	0.489958i	$0.837012\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15576.2	−1.67054	−0.835268	−	0.549842i	$-0.814688\pi$
$443$	−15576.2	−1.67054	−0.835268	+	0.549842i	$0.814688\pi$
$444$	0	0
$445$	−1797.99	−0.191534
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1293.39	0.135944	0.0679722	−	0.997687i	$-0.478347\pi$
$449$	1293.39	0.135944	0.0679722	+	0.997687i	$0.478347\pi$
$450$	0	0
$451$	−9511.34	−0.993063
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3135.96	0.323112
$456$	0	0
$457$	12202.4	1.24902	0.624510	−	0.781016i	$-0.285299\pi$
$457$	12202.4	1.24902	0.624510	+	0.781016i	$0.285299\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6755.98	−0.682554	−0.341277	−	0.939963i	$-0.610859\pi$
$461$	−6755.98	−0.682554	−0.341277	+	0.939963i	$0.610859\pi$
$462$	0	0
$463$	−6938.69	−0.696475	−0.348238	−	0.937406i	$-0.613220\pi$
$463$	−6938.69	−0.696475	−0.348238	+	0.937406i	$0.613220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8682.20	−0.860309	−0.430155	−	0.902755i	$-0.641541\pi$
$467$	−8682.20	−0.860309	−0.430155	+	0.902755i	$0.641541\pi$
$468$	0	0
$469$	385.674	0.0379718
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17120.3	1.66426
$474$	0	0
$475$	11630.3	1.12344
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2803.30	0.267403	0.133702	−	0.991022i	$-0.457314\pi$
$479$	2803.30	0.267403	0.133702	+	0.991022i	$0.457314\pi$
$480$	0	0
$481$	−8622.91	−0.817403
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−823.275	−0.0770783
$486$	0	0
$487$	6475.32	0.602515	0.301257	−	0.953543i	$-0.402594\pi$
$487$	6475.32	0.602515	0.301257	+	0.953543i	$0.402594\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12485.7	−1.14760	−0.573802	−	0.818994i	$-0.694532\pi$
$491$	−12485.7	−1.14760	−0.573802	+	0.818994i	$0.694532\pi$
$492$	0	0
$493$	3248.41	0.296757
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3743.77	0.337889
$498$	0	0
$499$	−16093.5	−1.44378	−0.721889	−	0.692009i	$-0.756726\pi$
$499$	−16093.5	−1.44378	−0.721889	+	0.692009i	$0.756726\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10158.6	−0.900498	−0.450249	−	0.892903i	$-0.648665\pi$
$503$	−10158.6	−0.900498	−0.450249	+	0.892903i	$0.648665\pi$
$504$	0	0
$505$	1438.05	0.126717
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12230.7	−1.06506	−0.532529	−	0.846412i	$-0.678758\pi$
$509$	−12230.7	−1.06506	−0.532529	+	0.846412i	$0.678758\pi$
$510$	0	0
$511$	−4675.11	−0.404725
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2134.18	−0.182608
$516$	0	0
$517$	−27693.1	−2.35579
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14621.3	−1.22950	−0.614752	−	0.788721i	$-0.710744\pi$
$521$	−14621.3	−1.22950	−0.614752	+	0.788721i	$0.710744\pi$
$522$	0	0
$523$	−7672.76	−0.641504	−0.320752	−	0.947163i	$-0.603936\pi$
$523$	−7672.76	−0.641504	−0.320752	+	0.947163i	$0.603936\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6894.24	0.569863
$528$	0	0
$529$	−8810.69	−0.724146
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12623.3	1.02585
$534$	0	0
$535$	4142.76	0.334780
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2461.74	−0.196725
$540$	0	0
$541$	4538.20	0.360651	0.180326	−	0.983607i	$-0.442285\pi$
$541$	4538.20	0.360651	0.180326	+	0.983607i	$0.442285\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2887.42	0.226942
$546$	0	0
$547$	−9075.58	−0.709404	−0.354702	−	0.934979i	$-0.615418\pi$
$547$	−9075.58	−0.709404	−0.354702	+	0.934979i	$0.615418\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2719.32	−0.210248
$552$	0	0
$553$	−14199.7	−1.09192
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24020.8	1.82728	0.913640	−	0.406524i	$-0.133259\pi$
$557$	24020.8	1.82728	0.913640	+	0.406524i	$0.133259\pi$
$558$	0	0
$559$	−22721.9	−1.71920
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11321.7	0.847516	0.423758	−	0.905775i	$-0.360711\pi$
$563$	11321.7	0.847516	0.423758	+	0.905775i	$0.360711\pi$
$564$	0	0
$565$	5875.99	0.437530
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15713.0	−1.15769	−0.578844	−	0.815438i	$-0.696496\pi$
$569$	−15713.0	−1.15769	−0.578844	+	0.815438i	$0.696496\pi$
$570$	0	0
$571$	1895.19	0.138899	0.0694495	−	0.997585i	$-0.477876\pi$
$571$	1895.19	0.138899	0.0694495	+	0.997585i	$0.477876\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6775.63	−0.491415
$576$	0	0
$577$	−9282.29	−0.669717	−0.334858	−	0.942268i	$-0.608688\pi$
$577$	−9282.29	−0.669717	−0.334858	+	0.942268i	$0.608688\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11124.6	−0.794366
$582$	0	0
$583$	10141.9	0.720473
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12013.6	0.844723	0.422362	−	0.906427i	$-0.361201\pi$
$587$	12013.6	0.844723	0.422362	+	0.906427i	$0.361201\pi$
$588$	0	0
$589$	−5771.32	−0.403741
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15698.8	1.08714	0.543569	−	0.839365i	$-0.317073\pi$
$593$	15698.8	1.08714	0.543569	+	0.839365i	$0.317073\pi$
$594$	0	0
$595$	−5762.16	−0.397018
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15896.0	−1.08429	−0.542147	−	0.840284i	$-0.682388\pi$
$599$	−15896.0	−1.08429	−0.542147	+	0.840284i	$0.682388\pi$
$600$	0	0
$601$	−15127.0	−1.02670	−0.513348	−	0.858180i	$-0.671595\pi$
$601$	−15127.0	−1.02670	−0.513348	+	0.858180i	$0.671595\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2955.19	0.198587
$606$	0	0
$607$	−25776.1	−1.72359	−0.861796	−	0.507255i	$-0.830660\pi$
$607$	−25776.1	−1.72359	−0.861796	+	0.507255i	$0.830660\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36753.9	2.43356
$612$	0	0
$613$	16964.7	1.11778	0.558889	−	0.829243i	$-0.311228\pi$
$613$	16964.7	1.11778	0.558889	+	0.829243i	$0.311228\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5966.47	0.389305	0.194652	−	0.980872i	$-0.437642\pi$
$617$	5966.47	0.389305	0.194652	+	0.980872i	$0.437642\pi$
$618$	0	0
$619$	11911.5	0.773448	0.386724	−	0.922195i	$-0.373607\pi$
$619$	11911.5	0.773448	0.386724	+	0.922195i	$0.373607\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10840.7	0.697150
$624$	0	0
$625$	12672.8	0.811061
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15844.1	1.00437
$630$	0	0
$631$	2161.74	0.136383	0.0681915	−	0.997672i	$-0.478277\pi$
$631$	2161.74	0.136383	0.0681915	+	0.997672i	$0.478277\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2552.58	0.159521
$636$	0	0
$637$	3267.18	0.203219
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5300.77	0.326627	0.163313	−	0.986574i	$-0.447782\pi$
$641$	5300.77	0.326627	0.163313	+	0.986574i	$0.447782\pi$
$642$	0	0
$643$	−19007.1	−1.16574	−0.582868	−	0.812567i	$-0.698069\pi$
$643$	−19007.1	−1.16574	−0.582868	+	0.812567i	$0.698069\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7966.88	0.484096	0.242048	−	0.970264i	$-0.422181\pi$
$647$	7966.88	0.484096	0.242048	+	0.970264i	$0.422181\pi$
$648$	0	0
$649$	39460.7	2.38670
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−721.559	−0.0432416	−0.0216208	−	0.999766i	$-0.506883\pi$
$653$	−721.559	−0.0432416	−0.0216208	+	0.999766i	$0.506883\pi$
$654$	0	0
$655$	3255.13	0.194181
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3550.81	0.209894	0.104947	−	0.994478i	$-0.466533\pi$
$659$	3550.81	0.209894	0.104947	+	0.994478i	$0.466533\pi$
$660$	0	0
$661$	4651.72	0.273723	0.136861	−	0.990590i	$-0.456298\pi$
$661$	4651.72	0.273723	0.136861	+	0.990590i	$0.456298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4823.63	0.281282
$666$	0	0
$667$	1584.23	0.0919663
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−28440.8	−1.63628
$672$	0	0
$673$	24984.3	1.43101	0.715507	−	0.698605i	$-0.246196\pi$
$673$	24984.3	1.43101	0.715507	+	0.698605i	$0.246196\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9777.60	0.555072	0.277536	−	0.960715i	$-0.410482\pi$
$677$	9777.60	0.555072	0.277536	+	0.960715i	$0.410482\pi$
$678$	0	0
$679$	4963.82	0.280551
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13516.6	−0.757248	−0.378624	−	0.925551i	$-0.623603\pi$
$683$	−13516.6	−0.757248	−0.378624	+	0.925551i	$0.623603\pi$
$684$	0	0
$685$	438.493	0.0244583
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13460.2	−0.744257
$690$	0	0
$691$	−26368.1	−1.45165	−0.725826	−	0.687879i	$-0.758542\pi$
$691$	−26368.1	−1.45165	−0.725826	+	0.687879i	$0.758542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2028.08	−0.110690
$696$	0	0
$697$	−23194.6	−1.26049
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17932.3	−0.966183	−0.483091	−	0.875570i	$-0.660486\pi$
$701$	−17932.3	−0.966183	−0.483091	+	0.875570i	$0.660486\pi$
$702$	0	0
$703$	−13263.5	−0.711581
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8670.51	−0.461228
$708$	0	0
$709$	−10294.4	−0.545295	−0.272648	−	0.962114i	$-0.587899\pi$
$709$	−10294.4	−0.545295	−0.272648	+	0.962114i	$0.587899\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3362.27	0.176603
$714$	0	0
$715$	−8932.50	−0.467212
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3732.23	0.193587	0.0967934	−	0.995304i	$-0.469141\pi$
$719$	3732.23	0.193587	0.0967934	+	0.995304i	$0.469141\pi$
$720$	0	0
$721$	12867.8	0.664660
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3198.19	−0.163832
$726$	0	0
$727$	17161.7	0.875507	0.437753	−	0.899095i	$-0.355774\pi$
$727$	17161.7	0.875507	0.437753	+	0.899095i	$0.355774\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	41750.2	2.11243
$732$	0	0
$733$	−11006.9	−0.554638	−0.277319	−	0.960778i	$-0.589446\pi$
$733$	−11006.9	−0.554638	−0.277319	+	0.960778i	$0.589446\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1098.56	−0.0549063
$738$	0	0
$739$	−3124.07	−0.155509	−0.0777543	−	0.996973i	$-0.524775\pi$
$739$	−3124.07	−0.155509	−0.0777543	+	0.996973i	$0.524775\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13015.8	−0.642670	−0.321335	−	0.946966i	$-0.604132\pi$
$743$	−13015.8	−0.642670	−0.321335	+	0.946966i	$0.604132\pi$
$744$	0	0
$745$	3307.61	0.162660
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24978.2	−1.21854
$750$	0	0
$751$	−33593.7	−1.63229	−0.816147	−	0.577844i	$-0.803894\pi$
$751$	−33593.7	−1.63229	−0.816147	+	0.577844i	$0.803894\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7258.55	0.349888
$756$	0	0
$757$	39640.0	1.90322	0.951612	−	0.307303i	$-0.0994265\pi$
$757$	39640.0	1.90322	0.951612	+	0.307303i	$0.0994265\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14778.4	−0.703965	−0.351983	−	0.936007i	$-0.614492\pi$
$761$	−14778.4	−0.703965	−0.351983	+	0.936007i	$0.614492\pi$
$762$	0	0
$763$	−17409.3	−0.826027
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52371.7	−2.46549
$768$	0	0
$769$	−34311.7	−1.60899	−0.804494	−	0.593960i	$-0.797564\pi$
$769$	−34311.7	−1.60899	−0.804494	+	0.593960i	$0.797564\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29124.5	−1.35516	−0.677578	−	0.735451i	$-0.736971\pi$
$773$	−29124.5	−1.35516	−0.677578	+	0.735451i	$0.736971\pi$
$774$	0	0
$775$	−6787.66	−0.314607
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19416.8	0.893039
$780$	0	0
$781$	−10663.8	−0.488579
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6334.35	−0.288003
$786$	0	0
$787$	−12726.1	−0.576414	−0.288207	−	0.957568i	$-0.593059\pi$
$787$	−12726.1	−0.576414	−0.288207	+	0.957568i	$0.593059\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−35428.5	−1.59253
$792$	0	0
$793$	37746.2	1.69030
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19129.5	0.850189	0.425095	−	0.905149i	$-0.360241\pi$
$797$	19129.5	0.850189	0.425095	+	0.905149i	$0.360241\pi$
$798$	0	0
$799$	−67533.3	−2.99019
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13316.6	0.585222
$804$	0	0
$805$	−2810.16	−0.123038
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13785.9	0.599120	0.299560	−	0.954077i	$-0.403160\pi$
$809$	13785.9	0.599120	0.299560	+	0.954077i	$0.403160\pi$
$810$	0	0
$811$	−1552.80	−0.0672335	−0.0336167	−	0.999435i	$-0.510703\pi$
$811$	−1552.80	−0.0672335	−0.0336167	+	0.999435i	$0.510703\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5352.12	0.230033
$816$	0	0
$817$	−34950.0	−1.49663
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29137.6	−1.23862	−0.619311	−	0.785146i	$-0.712588\pi$
$821$	−29137.6	−1.23862	−0.619311	+	0.785146i	$0.712588\pi$
$822$	0	0
$823$	−20421.3	−0.864936	−0.432468	−	0.901649i	$-0.642357\pi$
$823$	−20421.3	−0.864936	−0.432468	+	0.901649i	$0.642357\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7589.68	0.319128	0.159564	−	0.987188i	$-0.448991\pi$
$827$	7589.68	0.319128	0.159564	+	0.987188i	$0.448991\pi$
$828$	0	0
$829$	16739.2	0.701297	0.350649	−	0.936507i	$-0.385961\pi$
$829$	16739.2	0.701297	0.350649	+	0.936507i	$0.385961\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6003.27	−0.249701
$834$	0	0
$835$	−10341.0	−0.428581
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8775.89	0.361117	0.180559	−	0.983564i	$-0.442209\pi$
$839$	8775.89	0.361117	0.180559	+	0.983564i	$0.442209\pi$
$840$	0	0
$841$	−23641.2	−0.969340
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5623.56	0.228942
$846$	0	0
$847$	−17817.9	−0.722822
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7727.07	0.311258
$852$	0	0
$853$	−6097.89	−0.244769	−0.122384	−	0.992483i	$-0.539054\pi$
$853$	−6097.89	−0.244769	−0.122384	+	0.992483i	$0.539054\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33948.7	1.35317	0.676584	−	0.736366i	$-0.263460\pi$
$857$	33948.7	1.35317	0.676584	+	0.736366i	$0.263460\pi$
$858$	0	0
$859$	−41205.9	−1.63670	−0.818351	−	0.574719i	$-0.805111\pi$
$859$	−41205.9	−1.63670	−0.818351	+	0.574719i	$0.805111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7042.40	0.277782	0.138891	−	0.990308i	$-0.455646\pi$
$863$	7042.40	0.277782	0.138891	+	0.990308i	$0.455646\pi$
$864$	0	0
$865$	10613.1	0.417174
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40446.6	1.57889
$870$	0	0
$871$	1457.99	0.0567189
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11736.4	0.453443
$876$	0	0
$877$	−43330.1	−1.66836	−0.834182	−	0.551490i	$-0.814060\pi$
$877$	−43330.1	−1.66836	−0.834182	+	0.551490i	$0.814060\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41153.3	−1.57377	−0.786884	−	0.617100i	$-0.788307\pi$
$881$	−41153.3	−1.57377	−0.786884	+	0.617100i	$0.788307\pi$
$882$	0	0
$883$	−37967.0	−1.44699	−0.723494	−	0.690330i	$-0.757465\pi$
$883$	−37967.0	−1.44699	−0.723494	+	0.690330i	$0.757465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27747.3	1.05035	0.525177	−	0.850993i	$-0.323999\pi$
$887$	27747.3	1.05035	0.525177	+	0.850993i	$0.323999\pi$
$888$	0	0
$889$	−15390.4	−0.580629
$890$	0	0
$891$	0	0
$892$	0	0
$893$	56533.7	2.11851
$894$	0	0
$895$	4595.18	0.171620
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1587.04	0.0588774
$900$	0	0
$901$	24732.4	0.914491
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7084.63	−0.260222
$906$	0	0
$907$	38020.0	1.39188	0.695939	−	0.718101i	$-0.254989\pi$
$907$	38020.0	1.39188	0.695939	+	0.718101i	$0.254989\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20911.5	−0.760515	−0.380258	−	0.924881i	$-0.624165\pi$
$911$	−20911.5	−0.760515	−0.380258	+	0.924881i	$0.624165\pi$
$912$	0	0
$913$	31687.5	1.14863
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19626.3	−0.706782
$918$	0	0
$919$	47740.8	1.71363	0.856815	−	0.515625i	$-0.172440\pi$
$919$	47740.8	1.71363	0.856815	+	0.515625i	$0.172440\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14152.8	0.504708
$924$	0	0
$925$	−15599.2	−0.554485
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10533.3	0.371999	0.185999	−	0.982550i	$-0.440448\pi$
$929$	10533.3	0.371999	0.185999	+	0.982550i	$0.440448\pi$
$930$	0	0
$931$	5025.47	0.176910
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16413.0	0.574077
$936$	0	0
$937$	28024.6	0.977080	0.488540	−	0.872542i	$-0.337530\pi$
$937$	28024.6	0.977080	0.488540	+	0.872542i	$0.337530\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11186.5	0.387533	0.193767	−	0.981048i	$-0.437930\pi$
$941$	11186.5	0.387533	0.193767	+	0.981048i	$0.437930\pi$
$942$	0	0
$943$	−11311.9	−0.390631
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30894.7	−1.06013	−0.530064	−	0.847957i	$-0.677832\pi$
$947$	−30894.7	−1.06013	−0.530064	+	0.847957i	$0.677832\pi$
$948$	0	0
$949$	−17673.6	−0.604542
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14163.7	−0.481434	−0.240717	−	0.970595i	$-0.577383\pi$
$953$	−14163.7	−0.481434	−0.240717	+	0.970595i	$0.577383\pi$
$954$	0	0
$955$	13237.7	0.448548
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2643.83	−0.0890238
$960$	0	0
$961$	−26422.8	−0.886938
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14820.6	−0.494395
$966$	0	0
$967$	22586.0	0.751105	0.375552	−	0.926801i	$-0.377453\pi$
$967$	22586.0	0.751105	0.375552	+	0.926801i	$0.377453\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−362.454	−0.0119791	−0.00598955	−	0.999982i	$-0.501907\pi$
$971$	−362.454	−0.0119791	−0.00598955	+	0.999982i	$0.501907\pi$
$972$	0	0
$973$	12228.0	0.402890
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11084.6	0.362975	0.181488	−	0.983393i	$-0.441909\pi$
$977$	11084.6	0.362975	0.181488	+	0.983393i	$0.441909\pi$
$978$	0	0
$979$	−30878.8	−1.00806
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25263.3	0.819710	0.409855	−	0.912151i	$-0.365579\pi$
$983$	25263.3	0.819710	0.409855	+	0.912151i	$0.365579\pi$
$984$	0	0
$985$	−5029.65	−0.162699
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20361.3	0.654653
$990$	0	0
$991$	−47472.7	−1.52172	−0.760858	−	0.648918i	$-0.775222\pi$
$991$	−47472.7	−1.52172	−0.760858	+	0.648918i	$0.775222\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13175.0	0.419774
$996$	0	0
$997$	−7171.61	−0.227811	−0.113905	−	0.993492i	$-0.536336\pi$
$997$	−7171.61	−0.227811	−0.113905	+	0.993492i	$0.536336\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.cd.1.2		3
3.2	odd	2		1728.4.a.bx.1.2		3
4.3	odd	2		1728.4.a.cc.1.2		3
8.3	odd	2		864.4.a.k.1.2	✓	3
8.5	even	2		864.4.a.l.1.2	yes	3
12.11	even	2		1728.4.a.bw.1.2		3
24.5	odd	2		864.4.a.r.1.2	yes	3
24.11	even	2		864.4.a.q.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.4.a.k.1.2	✓	3	8.3	odd	2
864.4.a.l.1.2	yes	3	8.5	even	2
864.4.a.q.1.2	yes	3	24.11	even	2
864.4.a.r.1.2	yes	3	24.5	odd	2
1728.4.a.bw.1.2		3	12.11	even	2
1728.4.a.bx.1.2		3	3.2	odd	2
1728.4.a.cc.1.2		3	4.3	odd	2
1728.4.a.cd.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q+2.83638 q^{5} -17.1016 q^{7} +48.7123 q^{11} -64.6502 q^{13} +118.791 q^{17} -99.4428 q^{19} +57.9337 q^{23} -116.955 q^{25} +27.3455 q^{29} +58.0366 q^{31} -48.5066 q^{35} +133.378 q^{37} -195.255 q^{41} +351.459 q^{43} -568.504 q^{47} -50.5363 q^{49} +208.201 q^{53} +138.167 q^{55} +810.077 q^{59} -583.852 q^{61} -183.373 q^{65} -22.5520 q^{67} -218.914 q^{71} +273.373 q^{73} -833.057 q^{77} +830.315 q^{79} +650.502 q^{83} +336.937 q^{85} -633.902 q^{89} +1105.62 q^{91} -282.058 q^{95} -290.255 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 9 q^{7} + 18 q^{11} - 3 q^{13} - 18 q^{17} - 27 q^{19} - 90 q^{23} + 21 q^{25} + 72 q^{29} + 144 q^{31} + 450 q^{35} + 159 q^{37} + 168 q^{41} + 180 q^{43} - 522 q^{47} + 150 q^{49} - 84 q^{53}+ \cdots - 117 q^{97}+O(q^{100}) \) 3 * q + 6 * q^5 + 9 * q^7 + 18 * q^11 - 3 * q^13 - 18 * q^17 - 27 * q^19 - 90 * q^23 + 21 * q^25 + 72 * q^29 + 144 * q^31 + 450 * q^35 + 159 * q^37 + 168 * q^41 + 180 * q^43 - 522 * q^47 + 150 * q^49 - 84 * q^53 + 324 * q^55 + 1458 * q^59 - 33 * q^61 - 246 * q^65 + 837 * q^67 - 1908 * q^71 - 165 * q^73 - 954 * q^77 + 531 * q^79 + 2268 * q^83 - 708 * q^85 - 618 * q^89 + 1719 * q^91 - 2934 * q^95 - 117 * q^97

Introduction

L-functions

Modular forms

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Database

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