LMFDB - Embedded newform 1521.4.a.bj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.05129$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05129 q^{2} -6.89480 q^{4} +17.8886 q^{5} +30.1975 q^{7} +15.6587 q^{8} +O(q^{10})$	$q-1.05129 q^{2} -6.89480 q^{4} +17.8886 q^{5} +30.1975 q^{7} +15.6587 q^{8} -18.8061 q^{10} -50.8457 q^{11} -31.7462 q^{14} +38.6966 q^{16} +2.99137 q^{17} -72.7016 q^{19} -123.339 q^{20} +53.4533 q^{22} +41.9071 q^{23} +195.003 q^{25} -208.206 q^{28} +135.233 q^{29} +316.820 q^{31} -165.951 q^{32} -3.14479 q^{34} +540.192 q^{35} -261.777 q^{37} +76.4301 q^{38} +280.113 q^{40} +198.911 q^{41} -201.351 q^{43} +350.571 q^{44} -44.0563 q^{46} +97.3687 q^{47} +568.890 q^{49} -205.004 q^{50} +150.458 q^{53} -909.560 q^{55} +472.853 q^{56} -142.168 q^{58} +497.812 q^{59} +525.066 q^{61} -333.068 q^{62} -135.112 q^{64} -777.584 q^{67} -20.6249 q^{68} -567.896 q^{70} +1012.16 q^{71} -612.910 q^{73} +275.202 q^{74} +501.263 q^{76} -1535.41 q^{77} +718.804 q^{79} +692.230 q^{80} -209.112 q^{82} +397.730 q^{83} +53.5116 q^{85} +211.677 q^{86} -796.176 q^{88} -648.413 q^{89} -288.941 q^{92} -102.362 q^{94} -1300.53 q^{95} -272.412 q^{97} -598.066 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * q^98

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 $

$2$	−1.05129	−0.371686	−0.185843	−	0.982579i	$-0.559502\pi$
$2$	−1.05129	−0.371686	−0.185843	+	0.982579i	$0.559502\pi$
$3$	0	0
$3$	0	0
$4$	−6.89480	−0.861850
$5$	17.8886	1.60001	0.800004	−	0.599994i	$-0.204831\pi$
$5$	17.8886	1.60001	0.800004	+	0.599994i	$0.204831\pi$
$6$	0	0
$7$	30.1975	1.63051	0.815256	−	0.579100i	$-0.196596\pi$
$7$	30.1975	1.63051	0.815256	+	0.579100i	$0.196596\pi$
$8$	15.6587	0.692023
$9$	0	0
$10$	−18.8061	−0.594700
$11$	−50.8457	−1.39369	−0.696843	−	0.717224i	$-0.745413\pi$
$11$	−50.8457	−1.39369	−0.696843	+	0.717224i	$0.745413\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−31.7462	−0.606038
$15$	0	0
$16$	38.6966	0.604635
$17$	2.99137	0.0426773	0.0213387	−	0.999772i	$-0.493207\pi$
$17$	2.99137	0.0426773	0.0213387	+	0.999772i	$0.493207\pi$
$18$	0	0
$19$	−72.7016	−0.877836	−0.438918	−	0.898527i	$-0.644638\pi$
$19$	−72.7016	−0.877836	−0.438918	+	0.898527i	$0.644638\pi$
$20$	−123.339	−1.37897
$21$	0	0
$22$	53.4533	0.518013
$23$	41.9071	0.379923	0.189961	−	0.981792i	$-0.439164\pi$
$23$	41.9071	0.379923	0.189961	+	0.981792i	$0.439164\pi$
$24$	0	0
$25$	195.003	1.56003
$26$	0	0
$27$	0	0
$28$	−208.206	−1.40526
$29$	135.233	0.865935	0.432967	−	0.901410i	$-0.357466\pi$
$29$	135.233	0.865935	0.432967	+	0.901410i	$0.357466\pi$
$30$	0	0
$31$	316.820	1.83557	0.917783	−	0.397082i	$-0.129977\pi$
$31$	316.820	1.83557	0.917783	+	0.397082i	$0.129977\pi$
$32$	−165.951	−0.916757
$33$	0	0
$34$	−3.14479	−0.0158625
$35$	540.192	2.60883
$36$	0	0
$37$	−261.777	−1.16313	−0.581565	−	0.813500i	$-0.697559\pi$
$37$	−261.777	−1.16313	−0.581565	+	0.813500i	$0.697559\pi$
$38$	76.4301	0.326279
$39$	0	0
$40$	280.113	1.10724
$41$	198.911	0.757674	0.378837	−	0.925463i	$-0.376324\pi$
$41$	198.911	0.757674	0.378837	+	0.925463i	$0.376324\pi$
$42$	0	0
$43$	−201.351	−0.714085	−0.357043	−	0.934088i	$-0.616215\pi$
$43$	−201.351	−0.714085	−0.357043	+	0.934088i	$0.616215\pi$
$44$	350.571	1.20115
$45$	0	0
$46$	−44.0563	−0.141212
$47$	97.3687	0.302185	0.151092	−	0.988520i	$-0.451721\pi$
$47$	97.3687	0.302185	0.151092	+	0.988520i	$0.451721\pi$
$48$	0	0
$49$	568.890	1.65857
$50$	−205.004	−0.579839
$51$	0	0
$52$	0	0
$53$	150.458	0.389943	0.194972	−	0.980809i	$-0.437539\pi$
$53$	150.458	0.389943	0.194972	+	0.980809i	$0.437539\pi$
$54$	0	0
$55$	−909.560	−2.22991
$56$	472.853	1.12835
$57$	0	0
$58$	−142.168	−0.321855
$59$	497.812	1.09847	0.549234	−	0.835668i	$-0.314919\pi$
$59$	497.812	1.09847	0.549234	+	0.835668i	$0.314919\pi$
$60$	0	0
$61$	525.066	1.10209	0.551047	−	0.834474i	$-0.314228\pi$
$61$	525.066	1.10209	0.551047	+	0.834474i	$0.314228\pi$
$62$	−333.068	−0.682253
$63$	0	0
$64$	−135.112	−0.263890
$65$	0	0
$66$	0	0
$67$	−777.584	−1.41787	−0.708933	−	0.705276i	$-0.750823\pi$
$67$	−777.584	−1.41787	−0.708933	+	0.705276i	$0.750823\pi$
$68$	−20.6249	−0.0367814
$69$	0	0
$70$	−567.896	−0.969666
$71$	1012.16	1.69185	0.845924	−	0.533303i	$-0.179049\pi$
$71$	1012.16	1.69185	0.845924	+	0.533303i	$0.179049\pi$
$72$	0	0
$73$	−612.910	−0.982680	−0.491340	−	0.870968i	$-0.663493\pi$
$73$	−612.910	−0.982680	−0.491340	+	0.870968i	$0.663493\pi$
$74$	275.202	0.432318
$75$	0	0
$76$	501.263	0.756563
$77$	−1535.41	−2.27242
$78$	0	0
$79$	718.804	1.02369	0.511846	−	0.859077i	$-0.328962\pi$
$79$	718.804	1.02369	0.511846	+	0.859077i	$0.328962\pi$
$80$	692.230	0.967421
$81$	0	0
$82$	−209.112	−0.281616
$83$	397.730	0.525982	0.262991	−	0.964798i	$-0.415291\pi$
$83$	397.730	0.525982	0.262991	+	0.964798i	$0.415291\pi$
$84$	0	0
$85$	53.5116	0.0682841
$86$	211.677	0.265415
$87$	0	0
$88$	−796.176	−0.964462
$89$	−648.413	−0.772265	−0.386133	−	0.922443i	$-0.626189\pi$
$89$	−648.413	−0.772265	−0.386133	+	0.922443i	$0.626189\pi$
$90$	0	0
$91$	0	0
$92$	−288.941	−0.327436
$93$	0	0
$94$	−102.362	−0.112318
$95$	−1300.53	−1.40454
$96$	0	0
$97$	−272.412	−0.285147	−0.142574	−	0.989784i	$-0.545538\pi$
$97$	−272.412	−0.285147	−0.142574	+	0.989784i	$0.545538\pi$
$98$	−598.066	−0.616467
$99$	0	0
$100$	−1344.51	−1.34451
$101$	−416.057	−0.409893	−0.204947	−	0.978773i	$-0.565702\pi$
$101$	−416.057	−0.409893	−0.204947	+	0.978773i	$0.565702\pi$
$102$	0	0
$103$	261.509	0.250168	0.125084	−	0.992146i	$-0.460080\pi$
$103$	261.509	0.250168	0.125084	+	0.992146i	$0.460080\pi$
$104$	0	0
$105$	0	0
$106$	−158.174	−0.144936
$107$	−1041.57	−0.941052	−0.470526	−	0.882386i	$-0.655936\pi$
$107$	−1041.57	−0.941052	−0.470526	+	0.882386i	$0.655936\pi$
$108$	0	0
$109$	609.644	0.535718	0.267859	−	0.963458i	$-0.413684\pi$
$109$	609.644	0.535718	0.267859	+	0.963458i	$0.413684\pi$
$110$	956.207	0.828825
$111$	0	0
$112$	1168.54	0.985865
$113$	333.997	0.278051	0.139025	−	0.990289i	$-0.455603\pi$
$113$	333.997	0.278051	0.139025	+	0.990289i	$0.455603\pi$
$114$	0	0
$115$	749.660	0.607880
$116$	−932.403	−0.746306
$117$	0	0
$118$	−523.343	−0.408285
$119$	90.3320	0.0695859
$120$	0	0
$121$	1254.28	0.942361
$122$	−551.994	−0.409633
$123$	0	0
$124$	−2184.41	−1.58198
$125$	1252.26	0.896048
$126$	0	0
$127$	−1110.44	−0.775872	−0.387936	−	0.921686i	$-0.626812\pi$
$127$	−1110.44	−0.775872	−0.387936	+	0.921686i	$0.626812\pi$
$128$	1469.65	1.01484
$129$	0	0
$130$	0	0
$131$	557.489	0.371817	0.185909	−	0.982567i	$-0.440477\pi$
$131$	557.489	0.371817	0.185909	+	0.982567i	$0.440477\pi$
$132$	0	0
$133$	−2195.41	−1.43132
$134$	817.463	0.527000
$135$	0	0
$136$	46.8410	0.0295337
$137$	−1364.89	−0.851170	−0.425585	−	0.904918i	$-0.639932\pi$
$137$	−1364.89	−0.851170	−0.425585	+	0.904918i	$0.639932\pi$
$138$	0	0
$139$	−1936.99	−1.18197	−0.590985	−	0.806683i	$-0.701261\pi$
$139$	−1936.99	−1.18197	−0.590985	+	0.806683i	$0.701261\pi$
$140$	−3724.52	−2.24842
$141$	0	0
$142$	−1064.07	−0.628836
$143$	0	0
$144$	0	0
$145$	2419.13	1.38550
$146$	644.343	0.365248
$147$	0	0
$148$	1804.90	1.00244
$149$	3499.94	1.92434	0.962169	−	0.272452i	$-0.0878347\pi$
$149$	3499.94	1.92434	0.962169	+	0.272452i	$0.0878347\pi$
$150$	0	0
$151$	−672.434	−0.362397	−0.181198	−	0.983447i	$-0.557998\pi$
$151$	−672.434	−0.362397	−0.181198	+	0.983447i	$0.557998\pi$
$152$	−1138.41	−0.607482
$153$	0	0
$154$	1614.16	0.844627
$155$	5667.48	2.93692
$156$	0	0
$157$	1272.02	0.646612	0.323306	−	0.946295i	$-0.395206\pi$
$157$	1272.02	0.646612	0.323306	+	0.946295i	$0.395206\pi$
$158$	−755.668	−0.380492
$159$	0	0
$160$	−2968.63	−1.46682
$161$	1265.49	0.619469
$162$	0	0
$163$	−1476.22	−0.709366	−0.354683	−	0.934987i	$-0.615411\pi$
$163$	−1476.22	−0.709366	−0.354683	+	0.934987i	$0.615411\pi$
$164$	−1371.45	−0.653001
$165$	0	0
$166$	−418.127	−0.195500
$167$	3234.78	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$167$	3234.78	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$168$	0	0
$169$	0	0
$170$	−56.2559	−0.0253802
$171$	0	0
$172$	1388.27	0.615434
$173$	−424.597	−0.186598	−0.0932991	−	0.995638i	$-0.529741\pi$
$173$	−424.597	−0.186598	−0.0932991	+	0.995638i	$0.529741\pi$
$174$	0	0
$175$	5888.62	2.54364
$176$	−1967.56	−0.842671
$177$	0	0
$178$	681.667	0.287040
$179$	4400.04	1.83729	0.918644	−	0.395086i	$-0.129285\pi$
$179$	4400.04	1.83729	0.918644	+	0.395086i	$0.129285\pi$
$180$	0	0
$181$	−345.151	−0.141740	−0.0708698	−	0.997486i	$-0.522577\pi$
$181$	−345.151	−0.141740	−0.0708698	+	0.997486i	$0.522577\pi$
$182$	0	0
$183$	0	0
$184$	656.209	0.262915
$185$	−4682.83	−1.86102
$186$	0	0
$187$	−152.098	−0.0594788
$188$	−671.338	−0.260438
$189$	0	0
$190$	1367.23	0.522049
$191$	4076.61	1.54436	0.772181	−	0.635402i	$-0.219166\pi$
$191$	4076.61	1.54436	0.772181	+	0.635402i	$0.219166\pi$
$192$	0	0
$193$	−3221.08	−1.20134	−0.600670	−	0.799497i	$-0.705099\pi$
$193$	−3221.08	−1.20134	−0.600670	+	0.799497i	$0.705099\pi$
$194$	286.383	0.105985
$195$	0	0
$196$	−3922.38	−1.42944
$197$	3068.80	1.10986	0.554931	−	0.831897i	$-0.312745\pi$
$197$	3068.80	1.10986	0.554931	+	0.831897i	$0.312745\pi$
$198$	0	0
$199$	3992.05	1.42206	0.711028	−	0.703164i	$-0.248230\pi$
$199$	3992.05	1.42206	0.711028	+	0.703164i	$0.248230\pi$
$200$	3053.50	1.07957
$201$	0	0
$202$	437.395	0.152351
$203$	4083.70	1.41192
$204$	0	0
$205$	3558.24	1.21228
$206$	−274.921	−0.0929838
$207$	0	0
$208$	0	0
$209$	3696.56	1.22343
$210$	0	0
$211$	2868.29	0.935837	0.467918	−	0.883772i	$-0.345004\pi$
$211$	2868.29	0.935837	0.467918	+	0.883772i	$0.345004\pi$
$212$	−1037.38	−0.336072
$213$	0	0
$214$	1094.99	0.349775
$215$	−3601.89	−1.14254
$216$	0	0
$217$	9567.18	2.99291
$218$	−640.909	−0.199119
$219$	0	0
$220$	6271.23	1.92185
$221$	0	0
$222$	0	0
$223$	3099.47	0.930745	0.465372	−	0.885115i	$-0.345920\pi$
$223$	3099.47	0.930745	0.465372	+	0.885115i	$0.345920\pi$
$224$	−5011.30	−1.49478
$225$	0	0
$226$	−351.126	−0.103347
$227$	−258.860	−0.0756879	−0.0378440	−	0.999284i	$-0.512049\pi$
$227$	−258.860	−0.0756879	−0.0378440	+	0.999284i	$0.512049\pi$
$228$	0	0
$229$	804.986	0.232292	0.116146	−	0.993232i	$-0.462946\pi$
$229$	804.986	0.232292	0.116146	+	0.993232i	$0.462946\pi$
$230$	−788.107	−0.225940
$231$	0	0
$232$	2117.57	0.599247
$233$	1717.65	0.482947	0.241474	−	0.970407i	$-0.422369\pi$
$233$	1717.65	0.482947	0.241474	+	0.970407i	$0.422369\pi$
$234$	0	0
$235$	1741.79	0.483498
$236$	−3432.32	−0.946715
$237$	0	0
$238$	−94.9647	−0.0258641
$239$	2355.26	0.637445	0.318722	−	0.947848i	$-0.396746\pi$
$239$	2355.26	0.637445	0.318722	+	0.947848i	$0.396746\pi$
$240$	0	0
$241$	2170.07	0.580027	0.290014	−	0.957023i	$-0.406340\pi$
$241$	2170.07	0.580027	0.290014	+	0.957023i	$0.406340\pi$
$242$	−1318.61	−0.350262
$243$	0	0
$244$	−3620.22	−0.949840
$245$	10176.7	2.65373
$246$	0	0
$247$	0	0
$248$	4960.98	1.27025
$249$	0	0
$250$	−1316.49	−0.333048
$251$	−2860.82	−0.719416	−0.359708	−	0.933065i	$-0.617124\pi$
$251$	−2860.82	−0.719416	−0.359708	+	0.933065i	$0.617124\pi$
$252$	0	0
$253$	−2130.79	−0.529493
$254$	1167.39	0.288381
$255$	0	0
$256$	−464.125	−0.113312
$257$	−2276.69	−0.552593	−0.276296	−	0.961073i	$-0.589107\pi$
$257$	−2276.69	−0.552593	−0.276296	+	0.961073i	$0.589107\pi$
$258$	0	0
$259$	−7905.00	−1.89650
$260$	0	0
$261$	0	0
$262$	−586.080	−0.138199
$263$	3933.20	0.922172	0.461086	−	0.887355i	$-0.347460\pi$
$263$	3933.20	0.922172	0.461086	+	0.887355i	$0.347460\pi$
$264$	0	0
$265$	2691.49	0.623912
$266$	2308.00	0.532002
$267$	0	0
$268$	5361.29	1.22199
$269$	1622.92	0.367848	0.183924	−	0.982940i	$-0.441120\pi$
$269$	1622.92	0.367848	0.183924	+	0.982940i	$0.441120\pi$
$270$	0	0
$271$	−1554.77	−0.348508	−0.174254	−	0.984701i	$-0.555751\pi$
$271$	−1554.77	−0.348508	−0.174254	+	0.984701i	$0.555751\pi$
$272$	115.756	0.0258042
$273$	0	0
$274$	1434.89	0.316368
$275$	−9915.08	−2.17419
$276$	0	0
$277$	−8258.33	−1.79132	−0.895658	−	0.444743i	$-0.853295\pi$
$277$	−8258.33	−1.79132	−0.895658	+	0.444743i	$0.853295\pi$
$278$	2036.33	0.439321
$279$	0	0
$280$	8458.70	1.80537
$281$	−5023.16	−1.06639	−0.533197	−	0.845991i	$-0.679009\pi$
$281$	−5023.16	−1.06639	−0.533197	+	0.845991i	$0.679009\pi$
$282$	0	0
$283$	−4804.71	−1.00922	−0.504612	−	0.863346i	$-0.668364\pi$
$283$	−4804.71	−1.00922	−0.504612	+	0.863346i	$0.668364\pi$
$284$	−6978.64	−1.45812
$285$	0	0
$286$	0	0
$287$	6006.61	1.23540
$288$	0	0
$289$	−4904.05	−0.998179
$290$	−2543.20	−0.514971
$291$	0	0
$292$	4225.89	0.846923
$293$	2496.53	0.497778	0.248889	−	0.968532i	$-0.419935\pi$
$293$	2496.53	0.497778	0.248889	+	0.968532i	$0.419935\pi$
$294$	0	0
$295$	8905.18	1.75756
$296$	−4099.08	−0.804912
$297$	0	0
$298$	−3679.44	−0.715249
$299$	0	0
$300$	0	0
$301$	−6080.29	−1.16433
$302$	706.920	0.134698
$303$	0	0
$304$	−2813.31	−0.530770
$305$	9392.71	1.76336
$306$	0	0
$307$	1914.68	0.355950	0.177975	−	0.984035i	$-0.443045\pi$
$307$	1914.68	0.355950	0.177975	+	0.984035i	$0.443045\pi$
$308$	10586.4	1.95849
$309$	0	0
$310$	−5958.14	−1.09161
$311$	−3171.99	−0.578350	−0.289175	−	0.957276i	$-0.593381\pi$
$311$	−3171.99	−0.578350	−0.289175	+	0.957276i	$0.593381\pi$
$312$	0	0
$313$	1240.50	0.224017	0.112009	−	0.993707i	$-0.464272\pi$
$313$	1240.50	0.224017	0.112009	+	0.993707i	$0.464272\pi$
$314$	−1337.25	−0.240336
$315$	0	0
$316$	−4956.01	−0.882270
$317$	−9521.41	−1.68699	−0.843495	−	0.537137i	$-0.819506\pi$
$317$	−9521.41	−1.68699	−0.843495	+	0.537137i	$0.819506\pi$
$318$	0	0
$319$	−6876.01	−1.20684
$320$	−2416.96	−0.422226
$321$	0	0
$322$	−1330.39	−0.230248
$323$	−217.477	−0.0374637
$324$	0	0
$325$	0	0
$326$	1551.93	0.263661
$327$	0	0
$328$	3114.68	0.524327
$329$	2940.29	0.492716
$330$	0	0
$331$	−1997.12	−0.331637	−0.165818	−	0.986156i	$-0.553027\pi$
$331$	−1997.12	−0.331637	−0.165818	+	0.986156i	$0.553027\pi$
$332$	−2742.27	−0.453317
$333$	0	0
$334$	−3400.68	−0.557116
$335$	−13909.9	−2.26860
$336$	0	0
$337$	5200.25	0.840581	0.420290	−	0.907390i	$-0.361928\pi$
$337$	5200.25	0.840581	0.420290	+	0.907390i	$0.361928\pi$
$338$	0	0
$339$	0	0
$340$	−368.952	−0.0588506
$341$	−16108.9	−2.55820
$342$	0	0
$343$	6821.32	1.07381
$344$	−3152.88	−0.494163
$345$	0	0
$346$	446.372	0.0693558
$347$	−4622.17	−0.715075	−0.357538	−	0.933899i	$-0.616384\pi$
$347$	−4622.17	−0.715075	−0.357538	+	0.933899i	$0.616384\pi$
$348$	0	0
$349$	9870.06	1.51385	0.756923	−	0.653504i	$-0.226702\pi$
$349$	9870.06	1.51385	0.756923	+	0.653504i	$0.226702\pi$
$350$	−6190.62	−0.945436
$351$	0	0
$352$	8437.87	1.27767
$353$	8962.46	1.35134	0.675671	−	0.737203i	$-0.263854\pi$
$353$	8962.46	1.35134	0.675671	+	0.737203i	$0.263854\pi$
$354$	0	0
$355$	18106.2	2.70697
$356$	4470.68	0.665577
$357$	0	0
$358$	−4625.70	−0.682893
$359$	9174.16	1.34873	0.674365	−	0.738399i	$-0.264418\pi$
$359$	9174.16	1.34873	0.674365	+	0.738399i	$0.264418\pi$
$360$	0	0
$361$	−1573.48	−0.229404
$362$	362.852	0.0526826
$363$	0	0
$364$	0	0
$365$	−10964.1	−1.57230
$366$	0	0
$367$	10360.7	1.47364	0.736818	−	0.676092i	$-0.236328\pi$
$367$	10360.7	1.47364	0.736818	+	0.676092i	$0.236328\pi$
$368$	1621.66	0.229715
$369$	0	0
$370$	4922.99	0.691713
$371$	4543.46	0.635807
$372$	0	0
$373$	−5480.60	−0.760790	−0.380395	−	0.924824i	$-0.624212\pi$
$373$	−5480.60	−0.760790	−0.380395	+	0.924824i	$0.624212\pi$
$374$	159.899	0.0221074
$375$	0	0
$376$	1524.67	0.209119
$377$	0	0
$378$	0	0
$379$	4262.83	0.577748	0.288874	−	0.957367i	$-0.406719\pi$
$379$	4262.83	0.577748	0.288874	+	0.957367i	$0.406719\pi$
$380$	8966.91	1.21051
$381$	0	0
$382$	−4285.68	−0.574017
$383$	901.981	0.120337	0.0601685	−	0.998188i	$-0.480836\pi$
$383$	901.981	0.120337	0.0601685	+	0.998188i	$0.480836\pi$
$384$	0	0
$385$	−27466.4	−3.63590
$386$	3386.28	0.446520
$387$	0	0
$388$	1878.23	0.245754
$389$	1093.46	0.142521	0.0712606	−	0.997458i	$-0.477298\pi$
$389$	1093.46	0.142521	0.0712606	+	0.997458i	$0.477298\pi$
$390$	0	0
$391$	125.360	0.0162141
$392$	8908.07	1.14777
$393$	0	0
$394$	−3226.18	−0.412519
$395$	12858.4	1.63792
$396$	0	0
$397$	2587.62	0.327126	0.163563	−	0.986533i	$-0.447701\pi$
$397$	2587.62	0.327126	0.163563	+	0.986533i	$0.447701\pi$
$398$	−4196.79	−0.528558
$399$	0	0
$400$	7545.98	0.943247
$401$	−422.775	−0.0526494	−0.0263247	−	0.999653i	$-0.508380\pi$
$401$	−422.775	−0.0526494	−0.0263247	+	0.999653i	$0.508380\pi$
$402$	0	0
$403$	0	0
$404$	2868.63	0.353267
$405$	0	0
$406$	−4293.13	−0.524789
$407$	13310.2	1.62104
$408$	0	0
$409$	−3028.09	−0.366087	−0.183044	−	0.983105i	$-0.558595\pi$
$409$	−3028.09	−0.366087	−0.183044	+	0.983105i	$0.558595\pi$
$410$	−3740.73	−0.450588
$411$	0	0
$412$	−1803.05	−0.215607
$413$	15032.7	1.79107
$414$	0	0
$415$	7114.84	0.841576
$416$	0	0
$417$	0	0
$418$	−3886.14	−0.454730
$419$	9629.51	1.12275	0.561375	−	0.827562i	$-0.310273\pi$
$419$	9629.51	1.12275	0.561375	+	0.827562i	$0.310273\pi$
$420$	0	0
$421$	−8288.54	−0.959521	−0.479761	−	0.877399i	$-0.659277\pi$
$421$	−8288.54	−0.959521	−0.479761	+	0.877399i	$0.659277\pi$
$422$	−3015.40	−0.347837
$423$	0	0
$424$	2355.97	0.269850
$425$	583.328	0.0665778
$426$	0	0
$427$	15855.7	1.79698
$428$	7181.43	0.811046
$429$	0	0
$430$	3786.61	0.424667
$431$	3873.84	0.432938	0.216469	−	0.976289i	$-0.430546\pi$
$431$	3873.84	0.432938	0.216469	+	0.976289i	$0.430546\pi$
$432$	0	0
$433$	3016.87	0.334830	0.167415	−	0.985886i	$-0.446458\pi$
$433$	3016.87	0.334830	0.167415	+	0.985886i	$0.446458\pi$
$434$	−10057.8	−1.11242
$435$	0	0
$436$	−4203.37	−0.461709
$437$	−3046.71	−0.333510
$438$	0	0
$439$	−2862.77	−0.311235	−0.155618	−	0.987817i	$-0.549737\pi$
$439$	−2862.77	−0.311235	−0.155618	+	0.987817i	$0.549737\pi$
$440$	−14242.5	−1.54315
$441$	0	0
$442$	0	0
$443$	−13067.0	−1.40143	−0.700713	−	0.713443i	$-0.747135\pi$
$443$	−13067.0	−1.40143	−0.700713	+	0.713443i	$0.747135\pi$
$444$	0	0
$445$	−11599.2	−1.23563
$446$	−3258.43	−0.345944
$447$	0	0
$448$	−4080.04	−0.430276
$449$	−9527.11	−1.00136	−0.500682	−	0.865631i	$-0.666917\pi$
$449$	−9527.11	−1.00136	−0.500682	+	0.865631i	$0.666917\pi$
$450$	0	0
$451$	−10113.7	−1.05596
$452$	−2302.84	−0.239638
$453$	0	0
$454$	272.136	0.0281321
$455$	0	0
$456$	0	0
$457$	−6223.04	−0.636983	−0.318492	−	0.947926i	$-0.603176\pi$
$457$	−6223.04	−0.636983	−0.318492	+	0.947926i	$0.603176\pi$
$458$	−846.270	−0.0863397
$459$	0	0
$460$	−5168.76	−0.523901
$461$	17482.6	1.76626	0.883128	−	0.469132i	$-0.155433\pi$
$461$	17482.6	1.76626	0.883128	+	0.469132i	$0.155433\pi$
$462$	0	0
$463$	3280.46	0.329278	0.164639	−	0.986354i	$-0.447354\pi$
$463$	3280.46	0.329278	0.164639	+	0.986354i	$0.447354\pi$
$464$	5233.06	0.523575
$465$	0	0
$466$	−1805.74	−0.179504
$467$	−8459.42	−0.838234	−0.419117	−	0.907932i	$-0.637660\pi$
$467$	−8459.42	−0.838234	−0.419117	+	0.907932i	$0.637660\pi$
$468$	0	0
$469$	−23481.1	−2.31185
$470$	−1831.12	−0.179709
$471$	0	0
$472$	7795.09	0.760165
$473$	10237.8	0.995211
$474$	0	0
$475$	−14177.0	−1.36945
$476$	−622.821	−0.0599726
$477$	0	0
$478$	−2476.05	−0.236929
$479$	17593.9	1.67826	0.839129	−	0.543933i	$-0.183065\pi$
$479$	17593.9	1.67826	0.839129	+	0.543933i	$0.183065\pi$
$480$	0	0
$481$	0	0
$482$	−2281.36	−0.215588
$483$	0	0
$484$	−8648.02	−0.812174
$485$	−4873.08	−0.456238
$486$	0	0
$487$	−12736.9	−1.18515	−0.592573	−	0.805517i	$-0.701888\pi$
$487$	−12736.9	−1.18515	−0.592573	+	0.805517i	$0.701888\pi$
$488$	8221.84	0.762675
$489$	0	0
$490$	−10698.6	−0.986352
$491$	5458.09	0.501671	0.250835	−	0.968030i	$-0.419295\pi$
$491$	5458.09	0.501671	0.250835	+	0.968030i	$0.419295\pi$
$492$	0	0
$493$	404.532	0.0369558
$494$	0	0
$495$	0	0
$496$	12259.9	1.10985
$497$	30564.7	2.75858
$498$	0	0
$499$	18109.7	1.62466	0.812328	−	0.583201i	$-0.198200\pi$
$499$	18109.7	1.62466	0.812328	+	0.583201i	$0.198200\pi$
$500$	−8634.12	−0.772259
$501$	0	0
$502$	3007.54	0.267396
$503$	−19414.1	−1.72094	−0.860469	−	0.509503i	$-0.829829\pi$
$503$	−19414.1	−1.72094	−0.860469	+	0.509503i	$0.829829\pi$
$504$	0	0
$505$	−7442.70	−0.655833
$506$	2240.07	0.196805
$507$	0	0
$508$	7656.27	0.668686
$509$	16124.1	1.40410	0.702051	−	0.712126i	$-0.252268\pi$
$509$	16124.1	1.40410	0.702051	+	0.712126i	$0.252268\pi$
$510$	0	0
$511$	−18508.3	−1.60227
$512$	−11269.2	−0.972724
$513$	0	0
$514$	2393.46	0.205391
$515$	4678.05	0.400271
$516$	0	0
$517$	−4950.78	−0.421151
$518$	8310.41	0.704901
$519$	0	0
$520$	0	0
$521$	−18662.7	−1.56934	−0.784672	−	0.619911i	$-0.787169\pi$
$521$	−18662.7	−1.56934	−0.784672	+	0.619911i	$0.787169\pi$
$522$	0	0
$523$	−2290.01	−0.191463	−0.0957314	−	0.995407i	$-0.530519\pi$
$523$	−2290.01	−0.191463	−0.0957314	+	0.995407i	$0.530519\pi$
$524$	−3843.77	−0.320450
$525$	0	0
$526$	−4134.91	−0.342758
$527$	947.727	0.0783370
$528$	0	0
$529$	−10410.8	−0.855659
$530$	−2829.52	−0.231899
$531$	0	0
$532$	15136.9	1.23359
$533$	0	0
$534$	0	0
$535$	−18632.3	−1.50569
$536$	−12175.9	−0.981195
$537$	0	0
$538$	−1706.15	−0.136724
$539$	−28925.6	−2.31153
$540$	0	0
$541$	11740.6	0.933029	0.466514	−	0.884514i	$-0.345510\pi$
$541$	11740.6	0.933029	0.466514	+	0.884514i	$0.345510\pi$
$542$	1634.51	0.129535
$543$	0	0
$544$	−496.420	−0.0391247
$545$	10905.7	0.857153
$546$	0	0
$547$	312.599	0.0244347	0.0122174	−	0.999925i	$-0.496111\pi$
$547$	312.599	0.0244347	0.0122174	+	0.999925i	$0.496111\pi$
$548$	9410.63	0.733581
$549$	0	0
$550$	10423.6	0.808114
$551$	−9831.64	−0.760149
$552$	0	0
$553$	21706.1	1.66914
$554$	8681.86	0.665806
$555$	0	0
$556$	13355.2	1.01868
$557$	−7567.62	−0.575674	−0.287837	−	0.957679i	$-0.592936\pi$
$557$	−7567.62	−0.575674	−0.287837	+	0.957679i	$0.592936\pi$
$558$	0	0
$559$	0	0
$560$	20903.6	1.57739
$561$	0	0
$562$	5280.78	0.396363
$563$	−13068.3	−0.978261	−0.489131	−	0.872211i	$-0.662686\pi$
$563$	−13068.3	−0.978261	−0.489131	+	0.872211i	$0.662686\pi$
$564$	0	0
$565$	5974.74	0.444884
$566$	5051.12	0.375114
$567$	0	0
$568$	15849.1	1.17080
$569$	5959.19	0.439055	0.219528	−	0.975606i	$-0.429548\pi$
$569$	5959.19	0.439055	0.219528	+	0.975606i	$0.429548\pi$
$570$	0	0
$571$	−5460.53	−0.400203	−0.200102	−	0.979775i	$-0.564127\pi$
$571$	−5460.53	−0.400203	−0.200102	+	0.979775i	$0.564127\pi$
$572$	0	0
$573$	0	0
$574$	−6314.66	−0.459179
$575$	8172.02	0.592690
$576$	0	0
$577$	−8560.26	−0.617623	−0.308811	−	0.951123i	$-0.599931\pi$
$577$	−8560.26	−0.617623	−0.308811	+	0.951123i	$0.599931\pi$
$578$	5155.56	0.371009
$579$	0	0
$580$	−16679.4	−1.19410
$581$	12010.4	0.857620
$582$	0	0
$583$	−7650.14	−0.543458
$584$	−9597.36	−0.680037
$585$	0	0
$586$	−2624.57	−0.185017
$587$	−25603.9	−1.80032	−0.900158	−	0.435564i	$-0.856549\pi$
$587$	−25603.9	−1.80032	−0.900158	+	0.435564i	$0.856549\pi$
$588$	0	0
$589$	−23033.3	−1.61133
$590$	−9361.89	−0.653259
$591$	0	0
$592$	−10129.9	−0.703269
$593$	16311.5	1.12956	0.564782	−	0.825240i	$-0.308960\pi$
$593$	16311.5	1.12956	0.564782	+	0.825240i	$0.308960\pi$
$594$	0	0
$595$	1615.92	0.111338
$596$	−24131.4	−1.65849
$597$	0	0
$598$	0	0
$599$	−6549.91	−0.446782	−0.223391	−	0.974729i	$-0.571713\pi$
$599$	−6549.91	−0.446782	−0.223391	+	0.974729i	$0.571713\pi$
$600$	0	0
$601$	11467.3	0.778306	0.389153	−	0.921173i	$-0.372768\pi$
$601$	11467.3	0.778306	0.389153	+	0.921173i	$0.372768\pi$
$602$	6392.12	0.432763
$603$	0	0
$604$	4636.30	0.312332
$605$	22437.4	1.50779
$606$	0	0
$607$	17594.3	1.17649	0.588245	−	0.808682i	$-0.299819\pi$
$607$	17594.3	1.17649	0.588245	+	0.808682i	$0.299819\pi$
$608$	12064.9	0.804762
$609$	0	0
$610$	−9874.42	−0.655416
$611$	0	0
$612$	0	0
$613$	12783.2	0.842265	0.421132	−	0.906999i	$-0.361633\pi$
$613$	12783.2	0.842265	0.421132	+	0.906999i	$0.361633\pi$
$614$	−2012.87	−0.132301
$615$	0	0
$616$	−24042.5	−1.57257
$617$	−9767.34	−0.637307	−0.318653	−	0.947871i	$-0.603231\pi$
$617$	−9767.34	−0.637307	−0.318653	+	0.947871i	$0.603231\pi$
$618$	0	0
$619$	24677.0	1.60235	0.801173	−	0.598433i	$-0.204210\pi$
$619$	24677.0	1.60235	0.801173	+	0.598433i	$0.204210\pi$
$620$	−39076.1	−2.53119
$621$	0	0
$622$	3334.66	0.214964
$623$	−19580.5	−1.25919
$624$	0	0
$625$	−1974.11	−0.126343
$626$	−1304.12	−0.0832639
$627$	0	0
$628$	−8770.30	−0.557282
$629$	−783.071	−0.0496393
$630$	0	0
$631$	22666.9	1.43004	0.715021	−	0.699103i	$-0.246417\pi$
$631$	22666.9	1.43004	0.715021	+	0.699103i	$0.246417\pi$
$632$	11255.5	0.708419
$633$	0	0
$634$	10009.7	0.627030
$635$	−19864.3	−1.24140
$636$	0	0
$637$	0	0
$638$	7228.64	0.448565
$639$	0	0
$640$	26290.0	1.62375
$641$	10770.3	0.663653	0.331826	−	0.943341i	$-0.392335\pi$
$641$	10770.3	0.663653	0.331826	+	0.943341i	$0.392335\pi$
$642$	0	0
$643$	−21220.8	−1.30151	−0.650753	−	0.759289i	$-0.725547\pi$
$643$	−21220.8	−1.30151	−0.650753	+	0.759289i	$0.725547\pi$
$644$	−8725.29	−0.533889
$645$	0	0
$646$	228.631	0.0139247
$647$	19430.5	1.18067	0.590334	−	0.807159i	$-0.298996\pi$
$647$	19430.5	1.18067	0.590334	+	0.807159i	$0.298996\pi$
$648$	0	0
$649$	−25311.6	−1.53092
$650$	0	0
$651$	0	0
$652$	10178.3	0.611367
$653$	−28307.3	−1.69640	−0.848201	−	0.529674i	$-0.822314\pi$
$653$	−28307.3	−1.69640	−0.848201	+	0.529674i	$0.822314\pi$
$654$	0	0
$655$	9972.72	0.594910
$656$	7697.17	0.458116
$657$	0	0
$658$	−3091.09	−0.183136
$659$	−25525.5	−1.50885	−0.754425	−	0.656387i	$-0.772084\pi$
$659$	−25525.5	−1.50885	−0.754425	+	0.656387i	$0.772084\pi$
$660$	0	0
$661$	−26065.8	−1.53380	−0.766901	−	0.641765i	$-0.778202\pi$
$661$	−26065.8	−1.53380	−0.766901	+	0.641765i	$0.778202\pi$
$662$	2099.55	0.123265
$663$	0	0
$664$	6227.92	0.363991
$665$	−39272.8	−2.29013
$666$	0	0
$667$	5667.21	0.328988
$668$	−22303.2	−1.29182
$669$	0	0
$670$	14623.3	0.843205
$671$	−26697.3	−1.53597
$672$	0	0
$673$	27351.3	1.56659	0.783295	−	0.621650i	$-0.213537\pi$
$673$	27351.3	1.56659	0.783295	+	0.621650i	$0.213537\pi$
$674$	−5466.95	−0.312432
$675$	0	0
$676$	0	0
$677$	20998.8	1.19209	0.596047	−	0.802949i	$-0.296737\pi$
$677$	20998.8	1.19209	0.596047	+	0.802949i	$0.296737\pi$
$678$	0	0
$679$	−8226.17	−0.464936
$680$	837.921	0.0472541
$681$	0	0
$682$	16935.1	0.950847
$683$	21227.9	1.18926	0.594629	−	0.804000i	$-0.297299\pi$
$683$	21227.9	1.18926	0.594629	+	0.804000i	$0.297299\pi$
$684$	0	0
$685$	−24416.0	−1.36188
$686$	−7171.15	−0.399119
$687$	0	0
$688$	−7791.59	−0.431761
$689$	0	0
$690$	0	0
$691$	−11371.3	−0.626025	−0.313012	−	0.949749i	$-0.601338\pi$
$691$	−11371.3	−0.626025	−0.313012	+	0.949749i	$0.601338\pi$
$692$	2927.51	0.160820
$693$	0	0
$694$	4859.22	0.265783
$695$	−34650.2	−1.89116
$696$	0	0
$697$	595.016	0.0323355
$698$	−10376.3	−0.562675
$699$	0	0
$700$	−40600.8	−2.19224
$701$	8695.98	0.468534	0.234267	−	0.972172i	$-0.424731\pi$
$701$	8695.98	0.468534	0.234267	+	0.972172i	$0.424731\pi$
$702$	0	0
$703$	19031.6	1.02104
$704$	6869.84	0.367780
$705$	0	0
$706$	−9422.11	−0.502274
$707$	−12563.9	−0.668336
$708$	0	0
$709$	9428.79	0.499444	0.249722	−	0.968318i	$-0.419661\pi$
$709$	9428.79	0.499444	0.249722	+	0.968318i	$0.419661\pi$
$710$	−19034.7	−1.00614
$711$	0	0
$712$	−10153.3	−0.534425
$713$	13277.0	0.697374
$714$	0	0
$715$	0	0
$716$	−30337.4	−1.58347
$717$	0	0
$718$	−9644.66	−0.501303
$719$	−790.174	−0.0409854	−0.0204927	−	0.999790i	$-0.506523\pi$
$719$	−790.174	−0.0409854	−0.0204927	+	0.999790i	$0.506523\pi$
$720$	0	0
$721$	7896.93	0.407902
$722$	1654.18	0.0852663
$723$	0	0
$724$	2379.75	0.122158
$725$	26370.9	1.35088
$726$	0	0
$727$	−1591.88	−0.0812101	−0.0406050	−	0.999175i	$-0.512929\pi$
$727$	−1591.88	−0.0812101	−0.0406050	+	0.999175i	$0.512929\pi$
$728$	0	0
$729$	0	0
$730$	11526.4	0.584400
$731$	−602.314	−0.0304752
$732$	0	0
$733$	2695.20	0.135811	0.0679056	−	0.997692i	$-0.478368\pi$
$733$	2695.20	0.135811	0.0679056	+	0.997692i	$0.478368\pi$
$734$	−10892.1	−0.547729
$735$	0	0
$736$	−6954.50	−0.348297
$737$	39536.8	1.97606
$738$	0	0
$739$	1979.17	0.0985181	0.0492590	−	0.998786i	$-0.484314\pi$
$739$	1979.17	0.0985181	0.0492590	+	0.998786i	$0.484314\pi$
$740$	32287.1	1.60392
$741$	0	0
$742$	−4776.47	−0.236320
$743$	−19324.2	−0.954156	−0.477078	−	0.878861i	$-0.658304\pi$
$743$	−19324.2	−0.954156	−0.477078	+	0.878861i	$0.658304\pi$
$744$	0	0
$745$	62609.2	3.07896
$746$	5761.67	0.282775
$747$	0	0
$748$	1048.69	0.0512618
$749$	−31452.9	−1.53440
$750$	0	0
$751$	13898.2	0.675303	0.337652	−	0.941271i	$-0.390367\pi$
$751$	13898.2	0.675303	0.337652	+	0.941271i	$0.390367\pi$
$752$	3767.84	0.182712
$753$	0	0
$754$	0	0
$755$	−12028.9	−0.579838
$756$	0	0
$757$	16195.7	0.777601	0.388800	−	0.921322i	$-0.372890\pi$
$757$	16195.7	0.777601	0.388800	+	0.921322i	$0.372890\pi$
$758$	−4481.45	−0.214741
$759$	0	0
$760$	−20364.6	−0.971977
$761$	22912.0	1.09141	0.545704	−	0.837978i	$-0.316262\pi$
$761$	22912.0	1.09141	0.545704	+	0.837978i	$0.316262\pi$
$762$	0	0
$763$	18409.7	0.873495
$764$	−28107.4	−1.33101
$765$	0	0
$766$	−948.240	−0.0447275
$767$	0	0
$768$	0	0
$769$	17213.9	0.807215	0.403607	−	0.914932i	$-0.367756\pi$
$769$	17213.9	0.807215	0.403607	+	0.914932i	$0.367756\pi$
$770$	28875.1	1.35141
$771$	0	0
$772$	22208.7	1.03537
$773$	−19732.2	−0.918136	−0.459068	−	0.888401i	$-0.651817\pi$
$773$	−19732.2	−0.918136	−0.459068	+	0.888401i	$0.651817\pi$
$774$	0	0
$775$	61781.0	2.86353
$776$	−4265.62	−0.197328
$777$	0	0
$778$	−1149.54	−0.0529731
$779$	−14461.1	−0.665113
$780$	0	0
$781$	−51463.9	−2.35791
$782$	−131.789	−0.00602654
$783$	0	0
$784$	22014.1	1.00283
$785$	22754.7	1.03458
$786$	0	0
$787$	−34771.8	−1.57494	−0.787472	−	0.616350i	$-0.788611\pi$
$787$	−34771.8	−1.57494	−0.787472	+	0.616350i	$0.788611\pi$
$788$	−21158.7	−0.956534
$789$	0	0
$790$	−13517.9	−0.608790
$791$	10085.9	0.453365
$792$	0	0
$793$	0	0
$794$	−2720.33	−0.121588
$795$	0	0
$796$	−27524.4	−1.22560
$797$	16429.2	0.730178	0.365089	−	0.930973i	$-0.381039\pi$
$797$	16429.2	0.730178	0.365089	+	0.930973i	$0.381039\pi$
$798$	0	0
$799$	291.266	0.0128964
$800$	−32360.9	−1.43017
$801$	0	0
$802$	444.458	0.0195690
$803$	31163.8	1.36955
$804$	0	0
$805$	22637.9	0.991156
$806$	0	0
$807$	0	0
$808$	−6514.91	−0.283656
$809$	1805.75	0.0784755	0.0392378	−	0.999230i	$-0.487507\pi$
$809$	1805.75	0.0784755	0.0392378	+	0.999230i	$0.487507\pi$
$810$	0	0
$811$	8758.70	0.379235	0.189618	−	0.981858i	$-0.439275\pi$
$811$	8758.70	0.379235	0.189618	+	0.981858i	$0.439275\pi$
$812$	−28156.3	−1.21686
$813$	0	0
$814$	−13992.8	−0.602516
$815$	−26407.6	−1.13499
$816$	0	0
$817$	14638.5	0.626850
$818$	3183.39	0.136069
$819$	0	0
$820$	−24533.4	−1.04481
$821$	−10480.4	−0.445518	−0.222759	−	0.974874i	$-0.571506\pi$
$821$	−10480.4	−0.445518	−0.222759	+	0.974874i	$0.571506\pi$
$822$	0	0
$823$	−38187.3	−1.61741	−0.808703	−	0.588217i	$-0.799830\pi$
$823$	−38187.3	−1.61741	−0.808703	+	0.588217i	$0.799830\pi$
$824$	4094.89	0.173122
$825$	0	0
$826$	−15803.7	−0.665714
$827$	10529.0	0.442720	0.221360	−	0.975192i	$-0.428950\pi$
$827$	10529.0	0.442720	0.221360	+	0.975192i	$0.428950\pi$
$828$	0	0
$829$	9131.19	0.382557	0.191278	−	0.981536i	$-0.438737\pi$
$829$	9131.19	0.382557	0.191278	+	0.981536i	$0.438737\pi$
$830$	−7479.73	−0.312801
$831$	0	0
$832$	0	0
$833$	1701.76	0.0707834
$834$	0	0
$835$	57865.8	2.39824
$836$	−25487.0	−1.05441
$837$	0	0
$838$	−10123.4	−0.417310
$839$	−40623.1	−1.67159	−0.835796	−	0.549041i	$-0.814993\pi$
$839$	−40623.1	−1.67159	−0.835796	+	0.549041i	$0.814993\pi$
$840$	0	0
$841$	−6101.07	−0.250157
$842$	8713.62	0.356640
$843$	0	0
$844$	−19776.3	−0.806551
$845$	0	0
$846$	0	0
$847$	37876.2	1.53653
$848$	5822.22	0.235773
$849$	0	0
$850$	−613.244	−0.0247460
$851$	−10970.3	−0.441900
$852$	0	0
$853$	−40704.0	−1.63385	−0.816927	−	0.576742i	$-0.804324\pi$
$853$	−40704.0	−1.63385	−0.816927	+	0.576742i	$0.804324\pi$
$854$	−16668.8	−0.667911
$855$	0	0
$856$	−16309.7	−0.651229
$857$	−45984.1	−1.83289	−0.916445	−	0.400160i	$-0.868955\pi$
$857$	−45984.1	−1.83289	−0.916445	+	0.400160i	$0.868955\pi$
$858$	0	0
$859$	7787.52	0.309321	0.154660	−	0.987968i	$-0.450572\pi$
$859$	7787.52	0.309321	0.154660	+	0.987968i	$0.450572\pi$
$860$	24834.3	0.984700
$861$	0	0
$862$	−4072.51	−0.160917
$863$	34878.2	1.37575	0.687873	−	0.725831i	$-0.258545\pi$
$863$	34878.2	1.37575	0.687873	+	0.725831i	$0.258545\pi$
$864$	0	0
$865$	−7595.46	−0.298559
$866$	−3171.59	−0.124452
$867$	0	0
$868$	−65963.8	−2.57944
$869$	−36548.1	−1.42671
$870$	0	0
$871$	0	0
$872$	9546.22	0.370729
$873$	0	0
$874$	3202.96	0.123961
$875$	37815.3	1.46102
$876$	0	0
$877$	−35625.1	−1.37169	−0.685846	−	0.727747i	$-0.740568\pi$
$877$	−35625.1	−1.37169	−0.685846	+	0.727747i	$0.740568\pi$
$878$	3009.58	0.115682
$879$	0	0
$880$	−35196.9	−1.34828
$881$	−15602.6	−0.596669	−0.298335	−	0.954461i	$-0.596431\pi$
$881$	−15602.6	−0.596669	−0.298335	+	0.954461i	$0.596431\pi$
$882$	0	0
$883$	−25341.6	−0.965812	−0.482906	−	0.875672i	$-0.660419\pi$
$883$	−25341.6	−0.965812	−0.482906	+	0.875672i	$0.660419\pi$
$884$	0	0
$885$	0	0
$886$	13737.1	0.520890
$887$	5091.34	0.192729	0.0963644	−	0.995346i	$-0.469279\pi$
$887$	5091.34	0.192729	0.0963644	+	0.995346i	$0.469279\pi$
$888$	0	0
$889$	−33532.6	−1.26507
$890$	12194.1	0.459266
$891$	0	0
$892$	−21370.2	−0.802162
$893$	−7078.86	−0.265269
$894$	0	0
$895$	78710.8	2.93968
$896$	44379.7	1.65471
$897$	0	0
$898$	10015.7	0.372192
$899$	42844.5	1.58948
$900$	0	0
$901$	450.076	0.0166417
$902$	10632.4	0.392485
$903$	0	0
$904$	5229.95	0.192418
$905$	−6174.28	−0.226785
$906$	0	0
$907$	−551.828	−0.0202019	−0.0101010	−	0.999949i	$-0.503215\pi$
$907$	−551.828	−0.0202019	−0.0101010	+	0.999949i	$0.503215\pi$
$908$	1784.79	0.0652316
$909$	0	0
$910$	0	0
$911$	−37661.8	−1.36969	−0.684847	−	0.728687i	$-0.740131\pi$
$911$	−37661.8	−1.36969	−0.684847	+	0.728687i	$0.740131\pi$
$912$	0	0
$913$	−20222.8	−0.733054
$914$	6542.19	0.236757
$915$	0	0
$916$	−5550.22	−0.200201
$917$	16834.8	0.606252
$918$	0	0
$919$	16265.4	0.583836	0.291918	−	0.956443i	$-0.405707\pi$
$919$	16265.4	0.583836	0.291918	+	0.956443i	$0.405707\pi$
$920$	11738.7	0.420667
$921$	0	0
$922$	−18379.2	−0.656492
$923$	0	0
$924$	0	0
$925$	−51047.3	−1.81451
$926$	−3448.70	−0.122388
$927$	0	0
$928$	−22442.0	−0.793852
$929$	−10133.6	−0.357881	−0.178940	−	0.983860i	$-0.557267\pi$
$929$	−10133.6	−0.357881	−0.178940	+	0.983860i	$0.557267\pi$
$930$	0	0
$931$	−41359.2	−1.45595
$932$	−11842.8	−0.416228
$933$	0	0
$934$	8893.26	0.311559
$935$	−2720.83	−0.0951666
$936$	0	0
$937$	−45619.6	−1.59053	−0.795265	−	0.606261i	$-0.792669\pi$
$937$	−45619.6	−1.59053	−0.795265	+	0.606261i	$0.792669\pi$
$938$	24685.3	0.859280
$939$	0	0
$940$	−12009.3	−0.416703
$941$	−26998.8	−0.935321	−0.467660	−	0.883908i	$-0.654903\pi$
$941$	−26998.8	−0.935321	−0.467660	+	0.883908i	$0.654903\pi$
$942$	0	0
$943$	8335.76	0.287858
$944$	19263.7	0.664173
$945$	0	0
$946$	−10762.9	−0.369905
$947$	16343.4	0.560812	0.280406	−	0.959882i	$-0.409531\pi$
$947$	16343.4	0.560812	0.280406	+	0.959882i	$0.409531\pi$
$948$	0	0
$949$	0	0
$950$	14904.1	0.509004
$951$	0	0
$952$	1414.48	0.0481550
$953$	19617.5	0.666812	0.333406	−	0.942783i	$-0.391802\pi$
$953$	19617.5	0.666812	0.333406	+	0.942783i	$0.391802\pi$
$954$	0	0
$955$	72925.0	2.47099
$956$	−16239.1	−0.549382
$957$	0	0
$958$	−18496.2	−0.623784
$959$	−41216.2	−1.38784
$960$	0	0
$961$	70583.9	2.36930
$962$	0	0
$963$	0	0
$964$	−14962.2	−0.499896
$965$	−57620.8	−1.92215
$966$	0	0
$967$	−42185.0	−1.40287	−0.701436	−	0.712732i	$-0.747457\pi$
$967$	−42185.0	−1.40287	−0.701436	+	0.712732i	$0.747457\pi$
$968$	19640.4	0.652135
$969$	0	0
$970$	5123.00	0.169577
$971$	42711.7	1.41162	0.705811	−	0.708400i	$-0.250583\pi$
$971$	42711.7	1.41162	0.705811	+	0.708400i	$0.250583\pi$
$972$	0	0
$973$	−58492.4	−1.92722
$974$	13390.2	0.440502
$975$	0	0
$976$	20318.3	0.666365
$977$	−30169.0	−0.987915	−0.493957	−	0.869486i	$-0.664450\pi$
$977$	−30169.0	−0.987915	−0.493957	+	0.869486i	$0.664450\pi$
$978$	0	0
$979$	32969.0	1.07630
$980$	−70166.1	−2.28712
$981$	0	0
$982$	−5738.01	−0.186464
$983$	23434.7	0.760378	0.380189	−	0.924909i	$-0.375859\pi$
$983$	23434.7	0.760378	0.380189	+	0.924909i	$0.375859\pi$
$984$	0	0
$985$	54896.6	1.77579
$986$	−425.278	−0.0137359
$987$	0	0
$988$	0	0
$989$	−8438.01	−0.271297
$990$	0	0
$991$	−41688.8	−1.33632	−0.668158	−	0.744019i	$-0.732917\pi$
$991$	−41688.8	−1.33632	−0.668158	+	0.744019i	$0.732917\pi$
$992$	−52576.5	−1.68277
$993$	0	0
$994$	−32132.2	−1.02532
$995$	71412.4	2.27530
$996$	0	0
$997$	−46249.7	−1.46915	−0.734574	−	0.678528i	$-0.762618\pi$
$997$	−46249.7	−1.46915	−0.734574	+	0.678528i	$0.762618\pi$
$998$	−19038.5	−0.603861
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bj.1.4		9
3.2	odd	2		507.4.a.n.1.6	✓	9
13.12	even	2		1521.4.a.be.1.6		9
39.5	even	4		507.4.b.j.337.9		18
39.8	even	4		507.4.b.j.337.10		18
39.38	odd	2		507.4.a.q.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.6	✓	9	3.2	odd	2
507.4.a.q.1.4	yes	9	39.38	odd	2
507.4.b.j.337.9		18	39.5	even	4
507.4.b.j.337.10		18	39.8	even	4
1521.4.a.be.1.6		9	13.12	even	2
1521.4.a.bj.1.4		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05129 q^{2} -6.89480 q^{4} +17.8886 q^{5} +30.1975 q^{7} +15.6587 q^{8} +O(q^{10})\)	\(q-1.05129 q^{2} -6.89480 q^{4} +17.8886 q^{5} +30.1975 q^{7} +15.6587 q^{8} -18.8061 q^{10} -50.8457 q^{11} -31.7462 q^{14} +38.6966 q^{16} +2.99137 q^{17} -72.7016 q^{19} -123.339 q^{20} +53.4533 q^{22} +41.9071 q^{23} +195.003 q^{25} -208.206 q^{28} +135.233 q^{29} +316.820 q^{31} -165.951 q^{32} -3.14479 q^{34} +540.192 q^{35} -261.777 q^{37} +76.4301 q^{38} +280.113 q^{40} +198.911 q^{41} -201.351 q^{43} +350.571 q^{44} -44.0563 q^{46} +97.3687 q^{47} +568.890 q^{49} -205.004 q^{50} +150.458 q^{53} -909.560 q^{55} +472.853 q^{56} -142.168 q^{58} +497.812 q^{59} +525.066 q^{61} -333.068 q^{62} -135.112 q^{64} -777.584 q^{67} -20.6249 q^{68} -567.896 q^{70} +1012.16 q^{71} -612.910 q^{73} +275.202 q^{74} +501.263 q^{76} -1535.41 q^{77} +718.804 q^{79} +692.230 q^{80} -209.112 q^{82} +397.730 q^{83} +53.5116 q^{85} +211.677 q^{86} -796.176 q^{88} -648.413 q^{89} -288.941 q^{92} -102.362 q^{94} -1300.53 q^{95} -272.412 q^{97} -598.066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * q^98

Introduction

L-functions

Modular forms

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Database

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