LMFDB - Embedded newform 648.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 $ x^5 - x^4 - 19x^3 + 4x^2 + 81*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.80998$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.3969 q^{5} +30.6325 q^{7} +O(q^{10})$	$q+12.3969 q^{5} +30.6325 q^{7} +0.944238 q^{11} +90.8738 q^{13} +53.4025 q^{17} -70.4876 q^{19} +97.1887 q^{23} +28.6823 q^{25} -156.610 q^{29} -43.0772 q^{31} +379.748 q^{35} -64.1933 q^{37} -85.0644 q^{41} +167.284 q^{43} -558.822 q^{47} +595.353 q^{49} -503.158 q^{53} +11.7056 q^{55} -61.7789 q^{59} +269.008 q^{61} +1126.55 q^{65} -126.506 q^{67} -854.343 q^{71} +404.568 q^{73} +28.9244 q^{77} +646.318 q^{79} -822.859 q^{83} +662.023 q^{85} +1185.67 q^{89} +2783.70 q^{91} -873.826 q^{95} +130.566 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 + 3 * q^7	$ 5 q + 5 q^{5} + 3 q^{7} - 25 q^{11} + 29 q^{13} + 28 q^{17} + 64 q^{19} + 89 q^{23} + 322 q^{25} + 129 q^{29} + 241 q^{31} - 243 q^{35} + 366 q^{37} + 171 q^{41} + 803 q^{43} - 477 q^{47} + 1072 q^{49} + 374 q^{53} + 1469 q^{55} - 607 q^{59} + 1349 q^{61} + 527 q^{65} + 1549 q^{67} - 812 q^{71} + 1928 q^{73} + 903 q^{77} + 1727 q^{79} - 1025 q^{83} + 2902 q^{85} + 2310 q^{89} + 2985 q^{91} - 2012 q^{95} + 2875 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 + 3 * q^7 - 25 * q^11 + 29 * q^13 + 28 * q^17 + 64 * q^19 + 89 * q^23 + 322 * q^25 + 129 * q^29 + 241 * q^31 - 243 * q^35 + 366 * q^37 + 171 * q^41 + 803 * q^43 - 477 * q^47 + 1072 * q^49 + 374 * q^53 + 1469 * q^55 - 607 * q^59 + 1349 * q^61 + 527 * q^65 + 1549 * q^67 - 812 * q^71 + 1928 * q^73 + 903 * q^77 + 1727 * q^79 - 1025 * q^83 + 2902 * q^85 + 2310 * q^89 + 2985 * q^91 - 2012 * q^95 + 2875 * 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
$5$	12.3969	1.10881	0.554405	−	0.832247i	$-0.312946\pi$
$5$	12.3969	1.10881	0.554405	+	0.832247i	$0.312946\pi$
$6$	0	0
$7$	30.6325	1.65400	0.827001	−	0.562200i	$-0.190045\pi$
$7$	30.6325	1.65400	0.827001	+	0.562200i	$0.190045\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.944238	0.0258817	0.0129408	−	0.999916i	$-0.495881\pi$
$11$	0.944238	0.0258817	0.0129408	+	0.999916i	$0.495881\pi$
$12$	0	0
$13$	90.8738	1.93876	0.969379	−	0.245568i	$-0.0789745\pi$
$13$	90.8738	1.93876	0.969379	+	0.245568i	$0.0789745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	53.4025	0.761882	0.380941	−	0.924599i	$-0.375600\pi$
$17$	53.4025	0.761882	0.380941	+	0.924599i	$0.375600\pi$
$18$	0	0
$19$	−70.4876	−0.851104	−0.425552	−	0.904934i	$-0.639920\pi$
$19$	−70.4876	−0.851104	−0.425552	+	0.904934i	$0.639920\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	97.1887	0.881098	0.440549	−	0.897729i	$-0.354784\pi$
$23$	97.1887	0.881098	0.440549	+	0.897729i	$0.354784\pi$
$24$	0	0
$25$	28.6823	0.229459
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−156.610	−1.00282	−0.501410	−	0.865210i	$-0.667185\pi$
$29$	−156.610	−1.00282	−0.501410	+	0.865210i	$0.667185\pi$
$30$	0	0
$31$	−43.0772	−0.249577	−0.124789	−	0.992183i	$-0.539825\pi$
$31$	−43.0772	−0.249577	−0.124789	+	0.992183i	$0.539825\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	379.748	1.83397
$36$	0	0
$37$	−64.1933	−0.285225	−0.142612	−	0.989779i	$-0.545550\pi$
$37$	−64.1933	−0.285225	−0.142612	+	0.989779i	$0.545550\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−85.0644	−0.324020	−0.162010	−	0.986789i	$-0.551798\pi$
$41$	−85.0644	−0.324020	−0.162010	+	0.986789i	$0.551798\pi$
$42$	0	0
$43$	167.284	0.593271	0.296635	−	0.954991i	$-0.404135\pi$
$43$	167.284	0.593271	0.296635	+	0.954991i	$0.404135\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−558.822	−1.73431	−0.867156	−	0.498037i	$-0.834054\pi$
$47$	−558.822	−1.73431	−0.867156	+	0.498037i	$0.834054\pi$
$48$	0	0
$49$	595.353	1.73572
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−503.158	−1.30404	−0.652020	−	0.758202i	$-0.726078\pi$
$53$	−503.158	−1.30404	−0.652020	+	0.758202i	$0.726078\pi$
$54$	0	0
$55$	11.7056	0.0286979
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−61.7789	−0.136321	−0.0681604	−	0.997674i	$-0.521713\pi$
$59$	−61.7789	−0.136321	−0.0681604	+	0.997674i	$0.521713\pi$
$60$	0	0
$61$	269.008	0.564638	0.282319	−	0.959321i	$-0.408896\pi$
$61$	269.008	0.564638	0.282319	+	0.959321i	$0.408896\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1126.55	2.14971
$66$	0	0
$67$	−126.506	−0.230673	−0.115337	−	0.993326i	$-0.536795\pi$
$67$	−126.506	−0.230673	−0.115337	+	0.993326i	$0.536795\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−854.343	−1.42805	−0.714027	−	0.700118i	$-0.753131\pi$
$71$	−854.343	−1.42805	−0.714027	+	0.700118i	$0.753131\pi$
$72$	0	0
$73$	404.568	0.648645	0.324323	−	0.945947i	$-0.394864\pi$
$73$	404.568	0.648645	0.324323	+	0.945947i	$0.394864\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	28.9244	0.0428084
$78$	0	0
$79$	646.318	0.920462	0.460231	−	0.887799i	$-0.347767\pi$
$79$	646.318	0.920462	0.460231	+	0.887799i	$0.347767\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−822.859	−1.08820	−0.544099	−	0.839021i	$-0.683128\pi$
$83$	−822.859	−1.08820	−0.544099	+	0.839021i	$0.683128\pi$
$84$	0	0
$85$	662.023	0.844782
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1185.67	1.41214	0.706070	−	0.708142i	$-0.250467\pi$
$89$	1185.67	1.41214	0.706070	+	0.708142i	$0.250467\pi$
$90$	0	0
$91$	2783.70	3.20671
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−873.826	−0.943712
$96$	0	0
$97$	130.566	0.136670	0.0683350	−	0.997662i	$-0.478231\pi$
$97$	130.566	0.136670	0.0683350	+	0.997662i	$0.478231\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1553.31	1.53030	0.765148	−	0.643855i	$-0.222666\pi$
$101$	1553.31	1.53030	0.765148	+	0.643855i	$0.222666\pi$
$102$	0	0
$103$	189.382	0.181168	0.0905842	−	0.995889i	$-0.471127\pi$
$103$	189.382	0.181168	0.0905842	+	0.995889i	$0.471127\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−179.800	−0.162448	−0.0812239	−	0.996696i	$-0.525883\pi$
$107$	−179.800	−0.162448	−0.0812239	+	0.996696i	$0.525883\pi$
$108$	0	0
$109$	1720.46	1.51184	0.755918	−	0.654666i	$-0.227191\pi$
$109$	1720.46	1.51184	0.755918	+	0.654666i	$0.227191\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1279.48	1.06517	0.532583	−	0.846378i	$-0.321221\pi$
$113$	1279.48	1.06517	0.532583	+	0.846378i	$0.321221\pi$
$114$	0	0
$115$	1204.84	0.976969
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1635.85	1.26015
$120$	0	0
$121$	−1330.11	−0.999330
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1194.04	−0.854384
$126$	0	0
$127$	−1583.06	−1.10609	−0.553047	−	0.833150i	$-0.686535\pi$
$127$	−1583.06	−1.10609	−0.553047	+	0.833150i	$0.686535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−763.730	−0.509370	−0.254685	−	0.967024i	$-0.581972\pi$
$131$	−763.730	−0.509370	−0.254685	+	0.967024i	$0.581972\pi$
$132$	0	0
$133$	−2159.22	−1.40773
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−965.233	−0.601937	−0.300969	−	0.953634i	$-0.597310\pi$
$137$	−965.233	−0.601937	−0.300969	+	0.953634i	$0.597310\pi$
$138$	0	0
$139$	1405.25	0.857496	0.428748	−	0.903424i	$-0.358955\pi$
$139$	1405.25	0.857496	0.428748	+	0.903424i	$0.358955\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	85.8066	0.0501784
$144$	0	0
$145$	−1941.48	−1.11194
$146$	0	0
$147$	0	0
$148$	0	0
$149$	358.567	0.197148	0.0985738	−	0.995130i	$-0.468572\pi$
$149$	358.567	0.197148	0.0985738	+	0.995130i	$0.468572\pi$
$150$	0	0
$151$	−762.546	−0.410961	−0.205481	−	0.978661i	$-0.565876\pi$
$151$	−762.546	−0.410961	−0.205481	+	0.978661i	$0.565876\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−534.022	−0.276733
$156$	0	0
$157$	939.097	0.477376	0.238688	−	0.971096i	$-0.423283\pi$
$157$	939.097	0.477376	0.238688	+	0.971096i	$0.423283\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2977.14	1.45734
$162$	0	0
$163$	−988.378	−0.474943	−0.237472	−	0.971394i	$-0.576319\pi$
$163$	−988.378	−0.474943	−0.237472	+	0.971394i	$0.576319\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1729.58	0.801432	0.400716	−	0.916202i	$-0.368761\pi$
$167$	1729.58	0.801432	0.400716	+	0.916202i	$0.368761\pi$
$168$	0	0
$169$	6061.05	2.75879
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3557.68	−1.56350	−0.781750	−	0.623593i	$-0.785672\pi$
$173$	−3557.68	−1.56350	−0.781750	+	0.623593i	$0.785672\pi$
$174$	0	0
$175$	878.613	0.379525
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3718.32	−1.55263	−0.776313	−	0.630348i	$-0.782912\pi$
$179$	−3718.32	−1.55263	−0.776313	+	0.630348i	$0.782912\pi$
$180$	0	0
$181$	849.530	0.348868	0.174434	−	0.984669i	$-0.444190\pi$
$181$	849.530	0.348868	0.174434	+	0.984669i	$0.444190\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−795.796	−0.316260
$186$	0	0
$187$	50.4247	0.0197188
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2344.71	−0.888259	−0.444130	−	0.895963i	$-0.646487\pi$
$191$	−2344.71	−0.888259	−0.444130	+	0.895963i	$0.646487\pi$
$192$	0	0
$193$	644.601	0.240411	0.120206	−	0.992749i	$-0.461645\pi$
$193$	644.601	0.240411	0.120206	+	0.992749i	$0.461645\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1697.78	0.614018	0.307009	−	0.951707i	$-0.400672\pi$
$197$	1697.78	0.614018	0.307009	+	0.951707i	$0.400672\pi$
$198$	0	0
$199$	1085.70	0.386748	0.193374	−	0.981125i	$-0.438057\pi$
$199$	1085.70	0.386748	0.193374	+	0.981125i	$0.438057\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4797.37	−1.65867
$204$	0	0
$205$	−1054.53	−0.359277
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−66.5571	−0.0220280
$210$	0	0
$211$	4665.49	1.52221	0.761104	−	0.648630i	$-0.224658\pi$
$211$	4665.49	1.52221	0.761104	+	0.648630i	$0.224658\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2073.80	0.657824
$216$	0	0
$217$	−1319.56	−0.412801
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4852.89	1.47711
$222$	0	0
$223$	−5353.59	−1.60764	−0.803818	−	0.594876i	$-0.797201\pi$
$223$	−5353.59	−1.60764	−0.803818	+	0.594876i	$0.797201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3710.51	−1.08491	−0.542456	−	0.840084i	$-0.682506\pi$
$227$	−3710.51	−1.08491	−0.542456	+	0.840084i	$0.682506\pi$
$228$	0	0
$229$	−3587.12	−1.03512	−0.517562	−	0.855646i	$-0.673160\pi$
$229$	−3587.12	−1.03512	−0.517562	+	0.855646i	$0.673160\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5513.12	1.55011	0.775057	−	0.631892i	$-0.217721\pi$
$233$	5513.12	1.55011	0.775057	+	0.631892i	$0.217721\pi$
$234$	0	0
$235$	−6927.65	−1.92302
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1160.40	−0.314058	−0.157029	−	0.987594i	$-0.550192\pi$
$239$	−1160.40	−0.314058	−0.157029	+	0.987594i	$0.550192\pi$
$240$	0	0
$241$	108.331	0.0289552	0.0144776	−	0.999895i	$-0.495391\pi$
$241$	108.331	0.0289552	0.0144776	+	0.999895i	$0.495391\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7380.51	1.92459
$246$	0	0
$247$	−6405.48	−1.65009
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3819.74	−0.960558	−0.480279	−	0.877116i	$-0.659465\pi$
$251$	−3819.74	−0.960558	−0.480279	+	0.877116i	$0.659465\pi$
$252$	0	0
$253$	91.7693	0.0228043
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1840.57	0.446737	0.223368	−	0.974734i	$-0.428295\pi$
$257$	1840.57	0.446737	0.223368	+	0.974734i	$0.428295\pi$
$258$	0	0
$259$	−1966.40	−0.471762
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1840.16	0.431442	0.215721	−	0.976455i	$-0.430790\pi$
$263$	1840.16	0.431442	0.215721	+	0.976455i	$0.430790\pi$
$264$	0	0
$265$	−6237.59	−1.44593
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5336.35	−1.20953	−0.604764	−	0.796405i	$-0.706732\pi$
$269$	−5336.35	−1.20953	−0.604764	+	0.796405i	$0.706732\pi$
$270$	0	0
$271$	−1649.56	−0.369755	−0.184878	−	0.982762i	$-0.559189\pi$
$271$	−1649.56	−0.369755	−0.184878	+	0.982762i	$0.559189\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.0830	0.00593878
$276$	0	0
$277$	−3627.20	−0.786778	−0.393389	−	0.919372i	$-0.628697\pi$
$277$	−3627.20	−0.786778	−0.393389	+	0.919372i	$0.628697\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1905.22	−0.404468	−0.202234	−	0.979337i	$-0.564820\pi$
$281$	−1905.22	−0.404468	−0.202234	+	0.979337i	$0.564820\pi$
$282$	0	0
$283$	−6015.01	−1.26345	−0.631723	−	0.775194i	$-0.717652\pi$
$283$	−6015.01	−1.26345	−0.631723	+	0.775194i	$0.717652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2605.74	−0.535930
$288$	0	0
$289$	−2061.18	−0.419535
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6129.45	1.22214	0.611068	−	0.791578i	$-0.290740\pi$
$293$	6129.45	1.22214	0.611068	+	0.791578i	$0.290740\pi$
$294$	0	0
$295$	−765.865	−0.151154
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8831.91	1.70824
$300$	0	0
$301$	5124.35	0.981271
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3334.85	0.626076
$306$	0	0
$307$	3453.04	0.641939	0.320970	−	0.947090i	$-0.395991\pi$
$307$	3453.04	0.641939	0.320970	+	0.947090i	$0.395991\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3391.61	−0.618395	−0.309197	−	0.950998i	$-0.600060\pi$
$311$	−3391.61	−0.618395	−0.309197	+	0.950998i	$0.600060\pi$
$312$	0	0
$313$	−6189.64	−1.11776	−0.558880	−	0.829248i	$-0.688769\pi$
$313$	−6189.64	−1.11776	−0.558880	+	0.829248i	$0.688769\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6057.64	1.07328	0.536642	−	0.843810i	$-0.319693\pi$
$317$	6057.64	1.07328	0.536642	+	0.843810i	$0.319693\pi$
$318$	0	0
$319$	−147.877	−0.0259547
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3764.21	−0.648441
$324$	0	0
$325$	2606.47	0.444865
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17118.2	−2.86855
$330$	0	0
$331$	1080.21	0.179376	0.0896881	−	0.995970i	$-0.471413\pi$
$331$	1080.21	0.179376	0.0896881	+	0.995970i	$0.471413\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1568.27	−0.255773
$336$	0	0
$337$	304.332	0.0491929	0.0245965	−	0.999697i	$-0.492170\pi$
$337$	304.332	0.0491929	0.0245965	+	0.999697i	$0.492170\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−40.6751	−0.00645948
$342$	0	0
$343$	7730.22	1.21689
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7765.41	1.20135	0.600676	−	0.799493i	$-0.294898\pi$
$347$	7765.41	1.20135	0.600676	+	0.799493i	$0.294898\pi$
$348$	0	0
$349$	−8139.66	−1.24844	−0.624221	−	0.781248i	$-0.714583\pi$
$349$	−8139.66	−1.24844	−0.624221	+	0.781248i	$0.714583\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1467.59	0.221280	0.110640	−	0.993861i	$-0.464710\pi$
$353$	1467.59	0.221280	0.110640	+	0.993861i	$0.464710\pi$
$354$	0	0
$355$	−10591.2	−1.58344
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7409.40	1.08928	0.544642	−	0.838669i	$-0.316665\pi$
$359$	7409.40	1.08928	0.544642	+	0.838669i	$0.316665\pi$
$360$	0	0
$361$	−1890.49	−0.275622
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5015.38	0.719224
$366$	0	0
$367$	10865.4	1.54543	0.772713	−	0.634756i	$-0.218899\pi$
$367$	10865.4	1.54543	0.772713	+	0.634756i	$0.218899\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15413.0	−2.15688
$372$	0	0
$373$	−7075.08	−0.982128	−0.491064	−	0.871124i	$-0.663392\pi$
$373$	−7075.08	−0.982128	−0.491064	+	0.871124i	$0.663392\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14231.8	−1.94423
$378$	0	0
$379$	1585.21	0.214847	0.107423	−	0.994213i	$-0.465740\pi$
$379$	1585.21	0.214847	0.107423	+	0.994213i	$0.465740\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4616.39	−0.615891	−0.307946	−	0.951404i	$-0.599641\pi$
$383$	−4616.39	−0.615891	−0.307946	+	0.951404i	$0.599641\pi$
$384$	0	0
$385$	358.572	0.0474663
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3829.85	−0.499180	−0.249590	−	0.968352i	$-0.580296\pi$
$389$	−3829.85	−0.499180	−0.249590	+	0.968352i	$0.580296\pi$
$390$	0	0
$391$	5190.12	0.671293
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8012.32	1.02062
$396$	0	0
$397$	−3476.78	−0.439532	−0.219766	−	0.975553i	$-0.570529\pi$
$397$	−3476.78	−0.439532	−0.219766	+	0.975553i	$0.570529\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	231.262	0.0287996	0.0143998	−	0.999896i	$-0.495416\pi$
$401$	231.262	0.0287996	0.0143998	+	0.999896i	$0.495416\pi$
$402$	0	0
$403$	−3914.59	−0.483870
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−60.6138	−0.00738209
$408$	0	0
$409$	6459.67	0.780954	0.390477	−	0.920613i	$-0.372310\pi$
$409$	6459.67	0.780954	0.390477	+	0.920613i	$0.372310\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1892.45	−0.225475
$414$	0	0
$415$	−10200.9	−1.20660
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5235.99	−0.610489	−0.305244	−	0.952274i	$-0.598738\pi$
$419$	−5235.99	−0.610489	−0.305244	+	0.952274i	$0.598738\pi$
$420$	0	0
$421$	1934.90	0.223994	0.111997	−	0.993709i	$-0.464275\pi$
$421$	1934.90	0.223994	0.111997	+	0.993709i	$0.464275\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1531.71	0.174820
$426$	0	0
$427$	8240.39	0.933912
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4607.87	0.514973	0.257487	−	0.966282i	$-0.417106\pi$
$431$	4607.87	0.514973	0.257487	+	0.966282i	$0.417106\pi$
$432$	0	0
$433$	−15913.6	−1.76619	−0.883094	−	0.469196i	$-0.844544\pi$
$433$	−15913.6	−1.76619	−0.883094	+	0.469196i	$0.844544\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6850.60	−0.749906
$438$	0	0
$439$	−3143.43	−0.341749	−0.170874	−	0.985293i	$-0.554659\pi$
$439$	−3143.43	−0.341749	−0.170874	+	0.985293i	$0.554659\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5716.29	−0.613068	−0.306534	−	0.951860i	$-0.599169\pi$
$443$	−5716.29	−0.613068	−0.306534	+	0.951860i	$0.599169\pi$
$444$	0	0
$445$	14698.6	1.56579
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1282.73	−0.134823	−0.0674116	−	0.997725i	$-0.521474\pi$
$449$	−1282.73	−0.134823	−0.0674116	+	0.997725i	$0.521474\pi$
$450$	0	0
$451$	−80.3211	−0.00838619
$452$	0	0
$453$	0	0
$454$	0	0
$455$	34509.1	3.55563
$456$	0	0
$457$	18913.3	1.93595	0.967973	−	0.251053i	$-0.0807769\pi$
$457$	18913.3	1.93595	0.967973	+	0.251053i	$0.0807769\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4721.62	0.477023	0.238512	−	0.971140i	$-0.423340\pi$
$461$	4721.62	0.477023	0.238512	+	0.971140i	$0.423340\pi$
$462$	0	0
$463$	4841.35	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$463$	4841.35	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5794.52	0.574172	0.287086	−	0.957905i	$-0.407313\pi$
$467$	5794.52	0.574172	0.287086	+	0.957905i	$0.407313\pi$
$468$	0	0
$469$	−3875.19	−0.381534
$470$	0	0
$471$	0	0
$472$	0	0
$473$	157.956	0.0153548
$474$	0	0
$475$	−2021.75	−0.195293
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11908.1	1.13590	0.567949	−	0.823064i	$-0.307737\pi$
$479$	11908.1	1.13590	0.567949	+	0.823064i	$0.307737\pi$
$480$	0	0
$481$	−5833.49	−0.552982
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1618.61	0.151541
$486$	0	0
$487$	1937.87	0.180315	0.0901574	−	0.995928i	$-0.471263\pi$
$487$	1937.87	0.180315	0.0901574	+	0.995928i	$0.471263\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6917.13	−0.635776	−0.317888	−	0.948128i	$-0.602974\pi$
$491$	−6917.13	−0.635776	−0.317888	+	0.948128i	$0.602974\pi$
$492$	0	0
$493$	−8363.37	−0.764031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−26170.7	−2.36201
$498$	0	0
$499$	−6456.01	−0.579179	−0.289590	−	0.957151i	$-0.593519\pi$
$499$	−6456.01	−0.579179	−0.289590	+	0.957151i	$0.593519\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3324.68	0.294712	0.147356	−	0.989084i	$-0.452924\pi$
$503$	3324.68	0.294712	0.147356	+	0.989084i	$0.452924\pi$
$504$	0	0
$505$	19256.1	1.69681
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5507.96	−0.479638	−0.239819	−	0.970818i	$-0.577088\pi$
$509$	−5507.96	−0.479638	−0.239819	+	0.970818i	$0.577088\pi$
$510$	0	0
$511$	12392.9	1.07286
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2347.74	0.200881
$516$	0	0
$517$	−527.662	−0.0448869
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18594.3	1.56359	0.781794	−	0.623536i	$-0.214305\pi$
$521$	18594.3	1.56359	0.781794	+	0.623536i	$0.214305\pi$
$522$	0	0
$523$	−9587.79	−0.801615	−0.400807	−	0.916162i	$-0.631270\pi$
$523$	−9587.79	−0.801615	−0.400807	+	0.916162i	$0.631270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2300.43	−0.190148
$528$	0	0
$529$	−2721.36	−0.223667
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7730.13	−0.628197
$534$	0	0
$535$	−2228.96	−0.180124
$536$	0	0
$537$	0	0
$538$	0	0
$539$	562.155	0.0449235
$540$	0	0
$541$	16195.6	1.28707	0.643535	−	0.765416i	$-0.277467\pi$
$541$	16195.6	1.28707	0.643535	+	0.765416i	$0.277467\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	21328.3	1.67634
$546$	0	0
$547$	8954.74	0.699958	0.349979	−	0.936758i	$-0.386189\pi$
$547$	8954.74	0.699958	0.349979	+	0.936758i	$0.386189\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11039.1	0.853505
$552$	0	0
$553$	19798.4	1.52245
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2729.48	−0.207633	−0.103817	−	0.994596i	$-0.533106\pi$
$557$	−2729.48	−0.207633	−0.103817	+	0.994596i	$0.533106\pi$
$558$	0	0
$559$	15201.8	1.15021
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23175.6	−1.73487	−0.867437	−	0.497546i	$-0.834234\pi$
$563$	−23175.6	−1.73487	−0.867437	+	0.497546i	$0.834234\pi$
$564$	0	0
$565$	15861.6	1.18107
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15079.5	1.11101	0.555507	−	0.831512i	$-0.312524\pi$
$569$	15079.5	1.11101	0.555507	+	0.831512i	$0.312524\pi$
$570$	0	0
$571$	12295.4	0.901134	0.450567	−	0.892743i	$-0.351222\pi$
$571$	12295.4	0.901134	0.450567	+	0.892743i	$0.351222\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2787.60	0.202175
$576$	0	0
$577$	17778.8	1.28274	0.641370	−	0.767232i	$-0.278366\pi$
$577$	17778.8	1.28274	0.641370	+	0.767232i	$0.278366\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−25206.3	−1.79988
$582$	0	0
$583$	−475.102	−0.0337508
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6848.86	0.481572	0.240786	−	0.970578i	$-0.422595\pi$
$587$	6848.86	0.481572	0.240786	+	0.970578i	$0.422595\pi$
$588$	0	0
$589$	3036.41	0.212416
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12568.2	−0.870342	−0.435171	−	0.900348i	$-0.643312\pi$
$593$	−12568.2	−0.870342	−0.435171	+	0.900348i	$0.643312\pi$
$594$	0	0
$595$	20279.5	1.39727
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20486.0	1.39739	0.698693	−	0.715422i	$-0.253765\pi$
$599$	20486.0	1.39739	0.698693	+	0.715422i	$0.253765\pi$
$600$	0	0
$601$	−4563.59	−0.309738	−0.154869	−	0.987935i	$-0.549496\pi$
$601$	−4563.59	−0.309738	−0.154869	+	0.987935i	$0.549496\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16489.2	−1.10807
$606$	0	0
$607$	24197.6	1.61804	0.809019	−	0.587782i	$-0.199999\pi$
$607$	24197.6	1.61804	0.809019	+	0.587782i	$0.199999\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−50782.3	−3.36241
$612$	0	0
$613$	8861.98	0.583902	0.291951	−	0.956433i	$-0.405696\pi$
$613$	8861.98	0.583902	0.291951	+	0.956433i	$0.405696\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16360.7	1.06752	0.533759	−	0.845637i	$-0.320779\pi$
$617$	16360.7	1.06752	0.533759	+	0.845637i	$0.320779\pi$
$618$	0	0
$619$	−10312.4	−0.669612	−0.334806	−	0.942287i	$-0.608671\pi$
$619$	−10312.4	−0.669612	−0.334806	+	0.942287i	$0.608671\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	36320.0	2.33568
$624$	0	0
$625$	−18387.6	−1.17681
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3428.08	−0.217308
$630$	0	0
$631$	−4335.08	−0.273497	−0.136749	−	0.990606i	$-0.543665\pi$
$631$	−4335.08	−0.273497	−0.136749	+	0.990606i	$0.543665\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19625.0	−1.22645
$636$	0	0
$637$	54102.0	3.36515
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18368.9	1.13187	0.565935	−	0.824450i	$-0.308515\pi$
$641$	18368.9	1.13187	0.565935	+	0.824450i	$0.308515\pi$
$642$	0	0
$643$	−8048.87	−0.493649	−0.246824	−	0.969060i	$-0.579387\pi$
$643$	−8048.87	−0.493649	−0.246824	+	0.969060i	$0.579387\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4001.44	0.243142	0.121571	−	0.992583i	$-0.461207\pi$
$647$	4001.44	0.243142	0.121571	+	0.992583i	$0.461207\pi$
$648$	0	0
$649$	−58.3340	−0.00352821
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3093.01	−0.185358	−0.0926792	−	0.995696i	$-0.529543\pi$
$653$	−3093.01	−0.185358	−0.0926792	+	0.995696i	$0.529543\pi$
$654$	0	0
$655$	−9467.86	−0.564794
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18334.9	−1.08380	−0.541902	−	0.840442i	$-0.682296\pi$
$659$	−18334.9	−1.08380	−0.541902	+	0.840442i	$0.682296\pi$
$660$	0	0
$661$	−32425.5	−1.90803	−0.954013	−	0.299764i	$-0.903092\pi$
$661$	−32425.5	−1.90803	−0.954013	+	0.299764i	$0.903092\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26767.5	−1.56090
$666$	0	0
$667$	−15220.7	−0.883583
$668$	0	0
$669$	0	0
$670$	0	0
$671$	254.007	0.0146138
$672$	0	0
$673$	−27780.7	−1.59118	−0.795591	−	0.605834i	$-0.792840\pi$
$673$	−27780.7	−1.59118	−0.795591	+	0.605834i	$0.792840\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33485.5	−1.90096	−0.950482	−	0.310780i	$-0.899410\pi$
$677$	−33485.5	−1.90096	−0.950482	+	0.310780i	$0.899410\pi$
$678$	0	0
$679$	3999.58	0.226053
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32940.7	1.84545	0.922724	−	0.385461i	$-0.125958\pi$
$683$	32940.7	1.84545	0.922724	+	0.385461i	$0.125958\pi$
$684$	0	0
$685$	−11965.9	−0.667434
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−45723.9	−2.52822
$690$	0	0
$691$	−33655.5	−1.85285	−0.926423	−	0.376484i	$-0.877133\pi$
$691$	−33655.5	−1.85285	−0.926423	+	0.376484i	$0.877133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17420.7	0.950800
$696$	0	0
$697$	−4542.65	−0.246865
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4650.86	0.250585	0.125293	−	0.992120i	$-0.460013\pi$
$701$	4650.86	0.250585	0.125293	+	0.992120i	$0.460013\pi$
$702$	0	0
$703$	4524.83	0.242756
$704$	0	0
$705$	0	0
$706$	0	0
$707$	47581.8	2.53111
$708$	0	0
$709$	−25747.6	−1.36385	−0.681926	−	0.731421i	$-0.738857\pi$
$709$	−25747.6	−1.36385	−0.681926	+	0.731421i	$0.738857\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4186.61	−0.219902
$714$	0	0
$715$	1063.73	0.0556382
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29032.1	1.50586	0.752931	−	0.658100i	$-0.228639\pi$
$719$	29032.1	1.50586	0.752931	+	0.658100i	$0.228639\pi$
$720$	0	0
$721$	5801.25	0.299653
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4491.95	−0.230106
$726$	0	0
$727$	20219.0	1.03147	0.515736	−	0.856748i	$-0.327519\pi$
$727$	20219.0	1.03147	0.515736	+	0.856748i	$0.327519\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8933.40	0.452002
$732$	0	0
$733$	−4972.66	−0.250572	−0.125286	−	0.992121i	$-0.539985\pi$
$733$	−4972.66	−0.250572	−0.125286	+	0.992121i	$0.539985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−119.451	−0.00597022
$738$	0	0
$739$	29562.3	1.47154	0.735769	−	0.677233i	$-0.236821\pi$
$739$	29562.3	1.47154	0.735769	+	0.677233i	$0.236821\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2935.04	−0.144921	−0.0724603	−	0.997371i	$-0.523085\pi$
$743$	−2935.04	−0.144921	−0.0724603	+	0.997371i	$0.523085\pi$
$744$	0	0
$745$	4445.11	0.218599
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5507.73	−0.268689
$750$	0	0
$751$	−10344.6	−0.502636	−0.251318	−	0.967905i	$-0.580864\pi$
$751$	−10344.6	−0.502636	−0.251318	+	0.967905i	$0.580864\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9453.18	−0.455678
$756$	0	0
$757$	−39052.5	−1.87502	−0.937508	−	0.347962i	$-0.886874\pi$
$757$	−39052.5	−1.87502	−0.937508	+	0.347962i	$0.886874\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21790.8	1.03800	0.518998	−	0.854776i	$-0.326305\pi$
$761$	21790.8	1.03800	0.518998	+	0.854776i	$0.326305\pi$
$762$	0	0
$763$	52702.0	2.50058
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5614.09	−0.264293
$768$	0	0
$769$	24968.5	1.17085	0.585426	−	0.810726i	$-0.300927\pi$
$769$	24968.5	1.17085	0.585426	+	0.810726i	$0.300927\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13776.8	−0.641033	−0.320516	−	0.947243i	$-0.603856\pi$
$773$	−13776.8	−0.641033	−0.320516	+	0.947243i	$0.603856\pi$
$774$	0	0
$775$	−1235.55	−0.0572676
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5995.99	0.275775
$780$	0	0
$781$	−806.704	−0.0369605
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11641.9	0.529320
$786$	0	0
$787$	15211.2	0.688971	0.344485	−	0.938792i	$-0.388053\pi$
$787$	15211.2	0.688971	0.344485	+	0.938792i	$0.388053\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	39193.9	1.76179
$792$	0	0
$793$	24445.7	1.09470
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6556.81	−0.291410	−0.145705	−	0.989328i	$-0.546545\pi$
$797$	−6556.81	−0.291410	−0.145705	+	0.989328i	$0.546545\pi$
$798$	0	0
$799$	−29842.5	−1.32134
$800$	0	0
$801$	0	0
$802$	0	0
$803$	382.009	0.0167880
$804$	0	0
$805$	36907.2	1.61591
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16020.5	0.696231	0.348115	−	0.937452i	$-0.386822\pi$
$809$	16020.5	0.696231	0.348115	+	0.937452i	$0.386822\pi$
$810$	0	0
$811$	−18637.9	−0.806984	−0.403492	−	0.914983i	$-0.632204\pi$
$811$	−18637.9	−0.806984	−0.403492	+	0.914983i	$0.632204\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12252.8	−0.526622
$816$	0	0
$817$	−11791.5	−0.504935
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4934.86	0.209778	0.104889	−	0.994484i	$-0.466551\pi$
$821$	4934.86	0.209778	0.104889	+	0.994484i	$0.466551\pi$
$822$	0	0
$823$	−14397.3	−0.609791	−0.304895	−	0.952386i	$-0.598621\pi$
$823$	−14397.3	−0.609791	−0.304895	+	0.952386i	$0.598621\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30026.5	−1.26254	−0.631271	−	0.775562i	$-0.717466\pi$
$827$	−30026.5	−1.26254	−0.631271	+	0.775562i	$0.717466\pi$
$828$	0	0
$829$	−16117.7	−0.675262	−0.337631	−	0.941279i	$-0.609626\pi$
$829$	−16117.7	−0.675262	−0.337631	+	0.941279i	$0.609626\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31793.3	1.32242
$834$	0	0
$835$	21441.4	0.888636
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46793.8	−1.92551	−0.962753	−	0.270381i	$-0.912850\pi$
$839$	−46793.8	−1.92551	−0.962753	+	0.270381i	$0.912850\pi$
$840$	0	0
$841$	137.780	0.00564927
$842$	0	0
$843$	0	0
$844$	0	0
$845$	75138.1	3.05897
$846$	0	0
$847$	−40744.6	−1.65289
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6238.86	−0.251311
$852$	0	0
$853$	20288.3	0.814371	0.407186	−	0.913345i	$-0.366510\pi$
$853$	20288.3	0.814371	0.407186	+	0.913345i	$0.366510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22542.0	−0.898507	−0.449253	−	0.893404i	$-0.648310\pi$
$857$	−22542.0	−0.898507	−0.449253	+	0.893404i	$0.648310\pi$
$858$	0	0
$859$	26490.5	1.05220	0.526102	−	0.850422i	$-0.323653\pi$
$859$	26490.5	1.05220	0.526102	+	0.850422i	$0.323653\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24555.8	−0.968585	−0.484293	−	0.874906i	$-0.660923\pi$
$863$	−24555.8	−0.968585	−0.484293	+	0.874906i	$0.660923\pi$
$864$	0	0
$865$	−44104.1	−1.73362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	610.279	0.0238231
$870$	0	0
$871$	−11496.1	−0.447220
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36576.4	−1.41315
$876$	0	0
$877$	−42047.4	−1.61897	−0.809487	−	0.587138i	$-0.800255\pi$
$877$	−42047.4	−1.61897	−0.809487	+	0.587138i	$0.800255\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35487.4	−1.35709	−0.678547	−	0.734557i	$-0.737390\pi$
$881$	−35487.4	−1.35709	−0.678547	+	0.734557i	$0.737390\pi$
$882$	0	0
$883$	−32872.6	−1.25283	−0.626416	−	0.779489i	$-0.715479\pi$
$883$	−32872.6	−1.25283	−0.626416	+	0.779489i	$0.715479\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41109.7	1.55618	0.778088	−	0.628156i	$-0.216190\pi$
$887$	41109.7	1.55618	0.778088	+	0.628156i	$0.216190\pi$
$888$	0	0
$889$	−48493.2	−1.82948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	39390.1	1.47608
$894$	0	0
$895$	−46095.5	−1.72157
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6746.33	0.250281
$900$	0	0
$901$	−26869.9	−0.993525
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10531.5	0.386828
$906$	0	0
$907$	−3948.63	−0.144556	−0.0722779	−	0.997385i	$-0.523027\pi$
$907$	−3948.63	−0.144556	−0.0722779	+	0.997385i	$0.523027\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2354.34	−0.0856232	−0.0428116	−	0.999083i	$-0.513632\pi$
$911$	−2354.34	−0.0856232	−0.0428116	+	0.999083i	$0.513632\pi$
$912$	0	0
$913$	−776.975	−0.0281644
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23395.0	−0.842499
$918$	0	0
$919$	46120.8	1.65548	0.827739	−	0.561114i	$-0.189627\pi$
$919$	46120.8	1.65548	0.827739	+	0.561114i	$0.189627\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−77637.4	−2.76865
$924$	0	0
$925$	−1841.21	−0.0654472
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31332.7	1.10656	0.553278	−	0.832996i	$-0.313377\pi$
$929$	31332.7	1.10656	0.553278	+	0.832996i	$0.313377\pi$
$930$	0	0
$931$	−41965.0	−1.47728
$932$	0	0
$933$	0	0
$934$	0	0
$935$	625.108	0.0218644
$936$	0	0
$937$	−1934.65	−0.0674516	−0.0337258	−	0.999431i	$-0.510737\pi$
$937$	−1934.65	−0.0674516	−0.0337258	+	0.999431i	$0.510737\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20684.1	0.716559	0.358280	−	0.933614i	$-0.383363\pi$
$941$	20684.1	0.716559	0.358280	+	0.933614i	$0.383363\pi$
$942$	0	0
$943$	−8267.30	−0.285493
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4841.49	−0.166132	−0.0830661	−	0.996544i	$-0.526471\pi$
$947$	−4841.49	−0.166132	−0.0830661	+	0.996544i	$0.526471\pi$
$948$	0	0
$949$	36764.6	1.25757
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24802.0	−0.843038	−0.421519	−	0.906820i	$-0.638503\pi$
$953$	−24802.0	−0.843038	−0.421519	+	0.906820i	$0.638503\pi$
$954$	0	0
$955$	−29067.1	−0.984910
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−29567.5	−0.995606
$960$	0	0
$961$	−27935.4	−0.937711
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7991.03	0.266570
$966$	0	0
$967$	−48565.1	−1.61504	−0.807522	−	0.589837i	$-0.799192\pi$
$967$	−48565.1	−1.61504	−0.807522	+	0.589837i	$0.799192\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20272.7	−0.670012	−0.335006	−	0.942216i	$-0.608738\pi$
$971$	−20272.7	−0.670012	−0.335006	+	0.942216i	$0.608738\pi$
$972$	0	0
$973$	43046.5	1.41830
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12109.5	−0.396539	−0.198270	−	0.980148i	$-0.563532\pi$
$977$	−12109.5	−0.396539	−0.198270	+	0.980148i	$0.563532\pi$
$978$	0	0
$979$	1119.55	0.0365486
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25330.4	−0.821885	−0.410942	−	0.911661i	$-0.634800\pi$
$983$	−25330.4	−0.821885	−0.410942	+	0.911661i	$0.634800\pi$
$984$	0	0
$985$	21047.1	0.680829
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16258.2	0.522729
$990$	0	0
$991$	−54229.7	−1.73831	−0.869154	−	0.494541i	$-0.835336\pi$
$991$	−54229.7	−1.73831	−0.869154	+	0.494541i	$0.835336\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13459.2	0.428830
$996$	0	0
$997$	−33647.4	−1.06883	−0.534415	−	0.845222i	$-0.679468\pi$
$997$	−33647.4	−1.06883	−0.534415	+	0.845222i	$0.679468\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.l.1.4		5
3.2	odd	2		648.4.a.k.1.2		5
4.3	odd	2		1296.4.a.bd.1.4		5
9.2	odd	6		216.4.i.b.145.4		10
9.4	even	3		72.4.i.b.25.2	✓	10
9.5	odd	6		216.4.i.b.73.4		10
9.7	even	3		72.4.i.b.49.2	yes	10
12.11	even	2		1296.4.a.bc.1.2		5
36.7	odd	6		144.4.i.f.49.4		10
36.11	even	6		432.4.i.f.145.4		10
36.23	even	6		432.4.i.f.289.4		10
36.31	odd	6		144.4.i.f.97.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.b.25.2	✓	10	9.4	even	3
72.4.i.b.49.2	yes	10	9.7	even	3
144.4.i.f.49.4		10	36.7	odd	6
144.4.i.f.97.4		10	36.31	odd	6
216.4.i.b.73.4		10	9.5	odd	6
216.4.i.b.145.4		10	9.2	odd	6
432.4.i.f.145.4		10	36.11	even	6
432.4.i.f.289.4		10	36.23	even	6
648.4.a.k.1.2		5	3.2	odd	2
648.4.a.l.1.4		5	1.1	even	1	trivial
1296.4.a.bc.1.2		5	12.11	even	2
1296.4.a.bd.1.4		5	4.3	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+12.3969 q^{5} +30.6325 q^{7} +O(q^{10})\)	\(q+12.3969 q^{5} +30.6325 q^{7} +0.944238 q^{11} +90.8738 q^{13} +53.4025 q^{17} -70.4876 q^{19} +97.1887 q^{23} +28.6823 q^{25} -156.610 q^{29} -43.0772 q^{31} +379.748 q^{35} -64.1933 q^{37} -85.0644 q^{41} +167.284 q^{43} -558.822 q^{47} +595.353 q^{49} -503.158 q^{53} +11.7056 q^{55} -61.7789 q^{59} +269.008 q^{61} +1126.55 q^{65} -126.506 q^{67} -854.343 q^{71} +404.568 q^{73} +28.9244 q^{77} +646.318 q^{79} -822.859 q^{83} +662.023 q^{85} +1185.67 q^{89} +2783.70 q^{91} -873.826 q^{95} +130.566 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 + 3 * q^7	\( 5 q + 5 q^{5} + 3 q^{7} - 25 q^{11} + 29 q^{13} + 28 q^{17} + 64 q^{19} + 89 q^{23} + 322 q^{25} + 129 q^{29} + 241 q^{31} - 243 q^{35} + 366 q^{37} + 171 q^{41} + 803 q^{43} - 477 q^{47} + 1072 q^{49} + 374 q^{53} + 1469 q^{55} - 607 q^{59} + 1349 q^{61} + 527 q^{65} + 1549 q^{67} - 812 q^{71} + 1928 q^{73} + 903 q^{77} + 1727 q^{79} - 1025 q^{83} + 2902 q^{85} + 2310 q^{89} + 2985 q^{91} - 2012 q^{95} + 2875 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 + 3 * q^7 - 25 * q^11 + 29 * q^13 + 28 * q^17 + 64 * q^19 + 89 * q^23 + 322 * q^25 + 129 * q^29 + 241 * q^31 - 243 * q^35 + 366 * q^37 + 171 * q^41 + 803 * q^43 - 477 * q^47 + 1072 * q^49 + 374 * q^53 + 1469 * q^55 - 607 * q^59 + 1349 * q^61 + 527 * q^65 + 1549 * q^67 - 812 * q^71 + 1928 * q^73 + 903 * q^77 + 1727 * q^79 - 1025 * q^83 + 2902 * q^85 + 2310 * q^89 + 2985 * q^91 - 2012 * q^95 + 2875 * q^97

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