LMFDB - Embedded newform 648.4.a.k.1.2

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			1.2
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	$-0.687054\pi$
$5$	−12.3969	−1.10881	−0.554405	+	0.832247i	$0.687054\pi$
$6$	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	$-0.504119\pi$
$11$	−0.944238	−0.0258817	−0.0129408	+	0.999916i	$0.504119\pi$
$12$	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	$-0.624400\pi$
$17$	−53.4025	−0.761882	−0.380941	+	0.924599i	$0.624400\pi$
$18$	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	$-0.645216\pi$
$23$	−97.1887	−0.881098	−0.440549	+	0.897729i	$0.645216\pi$
$24$	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	$-0.332815\pi$
$29$	156.610	1.00282	0.501410	+	0.865210i	$0.332815\pi$
$30$	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	$-0.448202\pi$
$41$	85.0644	0.324020	0.162010	+	0.986789i	$0.448202\pi$
$42$	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	$-0.165946\pi$
$47$	558.822	1.73431	0.867156	+	0.498037i	$0.165946\pi$
$48$	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	$-0.273922\pi$
$53$	503.158	1.30404	0.652020	+	0.758202i	$0.273922\pi$
$54$	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	$-0.478287\pi$
$59$	61.7789	0.136321	0.0681604	+	0.997674i	$0.478287\pi$
$60$	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	$-0.246869\pi$
$71$	854.343	1.42805	0.714027	+	0.700118i	$0.246869\pi$
$72$	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	$-0.316872\pi$
$83$	822.859	1.08820	0.544099	+	0.839021i	$0.316872\pi$
$84$	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	$-0.749533\pi$
$89$	−1185.67	−1.41214	−0.706070	+	0.708142i	$0.749533\pi$
$90$	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	$-0.777334\pi$
$101$	−1553.31	−1.53030	−0.765148	+	0.643855i	$0.777334\pi$
$102$	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	$-0.474117\pi$
$107$	179.800	0.162448	0.0812239	+	0.996696i	$0.474117\pi$
$108$	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	$-0.678779\pi$
$113$	−1279.48	−1.06517	−0.532583	+	0.846378i	$0.678779\pi$
$114$	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	$-0.418028\pi$
$131$	763.730	0.509370	0.254685	+	0.967024i	$0.418028\pi$
$132$	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	$-0.402690\pi$
$137$	965.233	0.601937	0.300969	+	0.953634i	$0.402690\pi$
$138$	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	$-0.531428\pi$
$149$	−358.567	−0.197148	−0.0985738	+	0.995130i	$0.531428\pi$
$150$	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	$-0.631239\pi$
$167$	−1729.58	−0.801432	−0.400716	+	0.916202i	$0.631239\pi$
$168$	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	$-0.214328\pi$
$173$	3557.68	1.56350	0.781750	+	0.623593i	$0.214328\pi$
$174$	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	$-0.217088\pi$
$179$	3718.32	1.55263	0.776313	+	0.630348i	$0.217088\pi$
$180$	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	$-0.353513\pi$
$191$	2344.71	0.888259	0.444130	+	0.895963i	$0.353513\pi$
$192$	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	$-0.599328\pi$
$197$	−1697.78	−0.614018	−0.307009	+	0.951707i	$0.599328\pi$
$198$	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	$-0.317494\pi$
$227$	3710.51	1.08491	0.542456	+	0.840084i	$0.317494\pi$
$228$	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	$-0.782279\pi$
$233$	−5513.12	−1.55011	−0.775057	+	0.631892i	$0.782279\pi$
$234$	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	$-0.449808\pi$
$239$	1160.40	0.314058	0.157029	+	0.987594i	$0.449808\pi$
$240$	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	$-0.340535\pi$
$251$	3819.74	0.960558	0.480279	+	0.877116i	$0.340535\pi$
$252$	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	$-0.571705\pi$
$257$	−1840.57	−0.446737	−0.223368	+	0.974734i	$0.571705\pi$
$258$	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	$-0.569210\pi$
$263$	−1840.16	−0.431442	−0.215721	+	0.976455i	$0.569210\pi$
$264$	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	$-0.293268\pi$
$269$	5336.35	1.20953	0.604764	+	0.796405i	$0.293268\pi$
$270$	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	$-0.435180\pi$
$281$	1905.22	0.404468	0.202234	+	0.979337i	$0.435180\pi$
$282$	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	$-0.709260\pi$
$293$	−6129.45	−1.22214	−0.611068	+	0.791578i	$0.709260\pi$
$294$	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	$-0.399940\pi$
$311$	3391.61	0.618395	0.309197	+	0.950998i	$0.399940\pi$
$312$	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	$-0.680307\pi$
$317$	−6057.64	−1.07328	−0.536642	+	0.843810i	$0.680307\pi$
$318$	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	$-0.705102\pi$
$347$	−7765.41	−1.20135	−0.600676	+	0.799493i	$0.705102\pi$
$348$	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	$-0.535290\pi$
$353$	−1467.59	−0.221280	−0.110640	+	0.993861i	$0.535290\pi$
$354$	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	$-0.683335\pi$
$359$	−7409.40	−1.08928	−0.544642	+	0.838669i	$0.683335\pi$
$360$	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	$-0.400359\pi$
$383$	4616.39	0.615891	0.307946	+	0.951404i	$0.400359\pi$
$384$	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	$-0.419704\pi$
$389$	3829.85	0.499180	0.249590	+	0.968352i	$0.419704\pi$
$390$	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	$-0.504584\pi$
$401$	−231.262	−0.0287996	−0.0143998	+	0.999896i	$0.504584\pi$
$402$	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	$-0.401262\pi$
$419$	5235.99	0.610489	0.305244	+	0.952274i	$0.401262\pi$
$420$	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	$-0.582894\pi$
$431$	−4607.87	−0.514973	−0.257487	+	0.966282i	$0.582894\pi$
$432$	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	$-0.400831\pi$
$443$	5716.29	0.613068	0.306534	+	0.951860i	$0.400831\pi$
$444$	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	$-0.478526\pi$
$449$	1282.73	0.134823	0.0674116	+	0.997725i	$0.478526\pi$
$450$	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	$-0.576660\pi$
$461$	−4721.62	−0.477023	−0.238512	+	0.971140i	$0.576660\pi$
$462$	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	$-0.592687\pi$
$467$	−5794.52	−0.574172	−0.287086	+	0.957905i	$0.592687\pi$
$468$	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	$-0.692263\pi$
$479$	−11908.1	−1.13590	−0.567949	+	0.823064i	$0.692263\pi$
$480$	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	$-0.397026\pi$
$491$	6917.13	0.635776	0.317888	+	0.948128i	$0.397026\pi$
$492$	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	$-0.547076\pi$
$503$	−3324.68	−0.294712	−0.147356	+	0.989084i	$0.547076\pi$
$504$	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	$-0.422912\pi$
$509$	5507.96	0.479638	0.239819	+	0.970818i	$0.422912\pi$
$510$	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	$-0.785695\pi$
$521$	−18594.3	−1.56359	−0.781794	+	0.623536i	$0.785695\pi$
$522$	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	$-0.466894\pi$
$557$	2729.48	0.207633	0.103817	+	0.994596i	$0.466894\pi$
$558$	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	$-0.165766\pi$
$563$	23175.6	1.73487	0.867437	+	0.497546i	$0.165766\pi$
$564$	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	$-0.687476\pi$
$569$	−15079.5	−1.11101	−0.555507	+	0.831512i	$0.687476\pi$
$570$	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	$-0.577405\pi$
$587$	−6848.86	−0.481572	−0.240786	+	0.970578i	$0.577405\pi$
$588$	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	$-0.356688\pi$
$593$	12568.2	0.870342	0.435171	+	0.900348i	$0.356688\pi$
$594$	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	$-0.746235\pi$
$599$	−20486.0	−1.39739	−0.698693	+	0.715422i	$0.746235\pi$
$600$	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	$-0.679221\pi$
$617$	−16360.7	−1.06752	−0.533759	+	0.845637i	$0.679221\pi$
$618$	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	$-0.691485\pi$
$641$	−18368.9	−1.13187	−0.565935	+	0.824450i	$0.691485\pi$
$642$	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	$-0.538793\pi$
$647$	−4001.44	−0.243142	−0.121571	+	0.992583i	$0.538793\pi$
$648$	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	$-0.470457\pi$
$653$	3093.01	0.185358	0.0926792	+	0.995696i	$0.470457\pi$
$654$	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	$-0.317704\pi$
$659$	18334.9	1.08380	0.541902	+	0.840442i	$0.317704\pi$
$660$	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	$-0.100590\pi$
$677$	33485.5	1.90096	0.950482	+	0.310780i	$0.100590\pi$
$678$	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	$-0.874042\pi$
$683$	−32940.7	−1.84545	−0.922724	+	0.385461i	$0.874042\pi$
$684$	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	$-0.539987\pi$
$701$	−4650.86	−0.250585	−0.125293	+	0.992120i	$0.539987\pi$
$702$	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	$-0.771361\pi$
$719$	−29032.1	−1.50586	−0.752931	+	0.658100i	$0.771361\pi$
$720$	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	$-0.476915\pi$
$743$	2935.04	0.144921	0.0724603	+	0.997371i	$0.476915\pi$
$744$	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	$-0.673695\pi$
$761$	−21790.8	−1.03800	−0.518998	+	0.854776i	$0.673695\pi$
$762$	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	$-0.396144\pi$
$773$	13776.8	0.641033	0.320516	+	0.947243i	$0.396144\pi$
$774$	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	$-0.453455\pi$
$797$	6556.81	0.291410	0.145705	+	0.989328i	$0.453455\pi$
$798$	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	$-0.613178\pi$
$809$	−16020.5	−0.696231	−0.348115	+	0.937452i	$0.613178\pi$
$810$	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	$-0.533449\pi$
$821$	−4934.86	−0.209778	−0.104889	+	0.994484i	$0.533449\pi$
$822$	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	$-0.282534\pi$
$827$	30026.5	1.26254	0.631271	+	0.775562i	$0.282534\pi$
$828$	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	$-0.0871497\pi$
$839$	46793.8	1.92551	0.962753	+	0.270381i	$0.0871497\pi$
$840$	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	$-0.351690\pi$
$857$	22542.0	0.898507	0.449253	+	0.893404i	$0.351690\pi$
$858$	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	$-0.339077\pi$
$863$	24555.8	0.968585	0.484293	+	0.874906i	$0.339077\pi$
$864$	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	$-0.262610\pi$
$881$	35487.4	1.35709	0.678547	+	0.734557i	$0.262610\pi$
$882$	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	$-0.783810\pi$
$887$	−41109.7	−1.55618	−0.778088	+	0.628156i	$0.783810\pi$
$888$	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	$-0.486368\pi$
$911$	2354.34	0.0856232	0.0428116	+	0.999083i	$0.486368\pi$
$912$	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	$-0.686623\pi$
$929$	−31332.7	−1.10656	−0.553278	+	0.832996i	$0.686623\pi$
$930$	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	$-0.616637\pi$
$941$	−20684.1	−0.716559	−0.358280	+	0.933614i	$0.616637\pi$
$942$	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	$-0.473529\pi$
$947$	4841.49	0.166132	0.0830661	+	0.996544i	$0.473529\pi$
$948$	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	$-0.361497\pi$
$953$	24802.0	0.843038	0.421519	+	0.906820i	$0.361497\pi$
$954$	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	$-0.391262\pi$
$971$	20272.7	0.670012	0.335006	+	0.942216i	$0.391262\pi$
$972$	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	$-0.436468\pi$
$977$	12109.5	0.396539	0.198270	+	0.980148i	$0.436468\pi$
$978$	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	$-0.365200\pi$
$983$	25330.4	0.821885	0.410942	+	0.911661i	$0.365200\pi$
$984$	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.k.1.2		5
3.2	odd	2		648.4.a.l.1.4		5
4.3	odd	2		1296.4.a.bc.1.2		5
9.2	odd	6		72.4.i.b.49.2	yes	10
9.4	even	3		216.4.i.b.73.4		10
9.5	odd	6		72.4.i.b.25.2	✓	10
9.7	even	3		216.4.i.b.145.4		10
12.11	even	2		1296.4.a.bd.1.4		5
36.7	odd	6		432.4.i.f.145.4		10
36.11	even	6		144.4.i.f.49.4		10
36.23	even	6		144.4.i.f.97.4		10
36.31	odd	6		432.4.i.f.289.4		10

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more