LMFDB - Embedded newform 1521.4.a.t.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +O(q^{10})$	$q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +7.80776 q^{10} +22.4233 q^{11} +2.38447 q^{14} +59.4233 q^{16} -67.9848 q^{17} -80.8078 q^{19} -139.039 q^{20} +9.83143 q^{22} -140.531 q^{23} +192.116 q^{25} -42.4621 q^{28} +106.693 q^{29} -276.155 q^{31} +81.5009 q^{32} -29.8078 q^{34} +96.8466 q^{35} -4.29168 q^{37} -35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +27.5294 q^{43} -175.076 q^{44} -61.6155 q^{46} -318.617 q^{47} -313.423 q^{49} +84.2329 q^{50} +67.6562 q^{53} +399.309 q^{55} -37.6932 q^{56} +46.7793 q^{58} +291.115 q^{59} +663.311 q^{61} -121.080 q^{62} -439.652 q^{64} -425.101 q^{67} +530.810 q^{68} +42.4621 q^{70} +152.963 q^{71} +117.268 q^{73} -1.88167 q^{74} +630.928 q^{76} +121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -99.8647 q^{82} -336.155 q^{83} -1210.66 q^{85} +12.0702 q^{86} -155.413 q^{88} -718.194 q^{89} +1097.23 q^{92} -139.697 q^{94} -1439.01 q^{95} +759.368 q^{97} -137.420 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10}) $ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8	$ 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100}) $ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 - 5 * q^10 - 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 - 141 * q^19 - 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 + 80 * q^28 - 34 * q^29 - 140 * q^31 - 105 * q^32 - 39 * q^34 + 70 * q^35 - 190 * q^37 - 310 * q^38 - 185 * q^40 - 538 * q^41 + 455 * q^43 - 680 * q^44 - 82 * q^46 - 60 * q^47 - 565 * q^49 - 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 - 595 * q^58 + 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 - 475 * q^67 + 505 * q^68 - 80 * q^70 - 127 * q^71 + 585 * q^73 - 849 * q^74 - 140 * q^76 - 255 * q^77 + 240 * q^79 + 1065 * q^80 - 1515 * q^82 - 260 * q^83 - 1205 * q^85 + 1962 * q^86 - 1020 * q^88 - 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 - 415 * q^97 - 1285 * 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 $

$2$	0.438447	0.155014	0.0775072	−	0.996992i	$-0.475304\pi$
$2$	0.438447	0.155014	0.0775072	+	0.996992i	$0.475304\pi$
$3$	0	0
$3$	0	0
$4$	−7.80776	−0.975971
$5$	17.8078	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	17.8078	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	5.43845	0.293649	0.146824	−	0.989163i	$-0.453095\pi$
$7$	5.43845	0.293649	0.146824	+	0.989163i	$0.453095\pi$
$8$	−6.93087	−0.306304
$9$	0	0
$10$	7.80776	0.246903
$11$	22.4233	0.614625	0.307313	−	0.951609i	$-0.400570\pi$
$11$	22.4233	0.614625	0.307313	+	0.951609i	$0.400570\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.38447	0.0455198
$15$	0	0
$16$	59.4233	0.928489
$17$	−67.9848	−0.969926	−0.484963	−	0.874535i	$-0.661167\pi$
$17$	−67.9848	−0.969926	−0.484963	+	0.874535i	$0.661167\pi$
$18$	0	0
$19$	−80.8078	−0.975714	−0.487857	−	0.872923i	$-0.662221\pi$
$19$	−80.8078	−0.975714	−0.487857	+	0.872923i	$0.662221\pi$
$20$	−139.039	−1.55450
$21$	0	0
$22$	9.83143	0.0952758
$23$	−140.531	−1.27403	−0.637017	−	0.770850i	$-0.719832\pi$
$23$	−140.531	−1.27403	−0.637017	+	0.770850i	$0.719832\pi$
$24$	0	0
$25$	192.116	1.53693
$26$	0	0
$27$	0	0
$28$	−42.4621	−0.286592
$29$	106.693	0.683187	0.341594	−	0.939848i	$-0.389033\pi$
$29$	106.693	0.683187	0.341594	+	0.939848i	$0.389033\pi$
$30$	0	0
$31$	−276.155	−1.59997	−0.799983	−	0.600023i	$-0.795158\pi$
$31$	−276.155	−1.59997	−0.799983	+	0.600023i	$0.795158\pi$
$32$	81.5009	0.450233
$33$	0	0
$34$	−29.8078	−0.150353
$35$	96.8466	0.467716
$36$	0	0
$37$	−4.29168	−0.0190688	−0.00953442	−	0.999955i	$-0.503035\pi$
$37$	−4.29168	−0.0190688	−0.00953442	+	0.999955i	$0.503035\pi$
$38$	−35.4299	−0.151250
$39$	0	0
$40$	−123.423	−0.487873
$41$	−227.769	−0.867598	−0.433799	−	0.901010i	$-0.642827\pi$
$41$	−227.769	−0.867598	−0.433799	+	0.901010i	$0.642827\pi$
$42$	0	0
$43$	27.5294	0.0976323	0.0488162	−	0.998808i	$-0.484455\pi$
$43$	27.5294	0.0976323	0.0488162	+	0.998808i	$0.484455\pi$
$44$	−175.076	−0.599856
$45$	0	0
$46$	−61.6155	−0.197494
$47$	−318.617	−0.988832	−0.494416	−	0.869225i	$-0.664618\pi$
$47$	−318.617	−0.988832	−0.494416	+	0.869225i	$0.664618\pi$
$48$	0	0
$49$	−313.423	−0.913771
$50$	84.2329	0.238247
$51$	0	0
$52$	0	0
$53$	67.6562	0.175345	0.0876726	−	0.996149i	$-0.472057\pi$
$53$	67.6562	0.175345	0.0876726	+	0.996149i	$0.472057\pi$
$54$	0	0
$55$	399.309	0.978960
$56$	−37.6932	−0.0899457
$57$	0	0
$58$	46.7793	0.105904
$59$	291.115	0.642371	0.321186	−	0.947016i	$-0.395919\pi$
$59$	291.115	0.642371	0.321186	+	0.947016i	$0.395919\pi$
$60$	0	0
$61$	663.311	1.39227	0.696133	−	0.717913i	$-0.254902\pi$
$61$	663.311	1.39227	0.696133	+	0.717913i	$0.254902\pi$
$62$	−121.080	−0.248018
$63$	0	0
$64$	−439.652	−0.858696
$65$	0	0
$66$	0	0
$67$	−425.101	−0.775140	−0.387570	−	0.921840i	$-0.626685\pi$
$67$	−425.101	−0.775140	−0.387570	+	0.921840i	$0.626685\pi$
$68$	530.810	0.946619
$69$	0	0
$70$	42.4621	0.0725028
$71$	152.963	0.255681	0.127841	−	0.991795i	$-0.459195\pi$
$71$	152.963	0.255681	0.127841	+	0.991795i	$0.459195\pi$
$72$	0	0
$73$	117.268	0.188016	0.0940081	−	0.995571i	$-0.470032\pi$
$73$	117.268	0.188016	0.0940081	+	0.995571i	$0.470032\pi$
$74$	−1.88167	−0.00295595
$75$	0	0
$76$	630.928	0.952268
$77$	121.948	0.180484
$78$	0	0
$79$	202.462	0.288339	0.144169	−	0.989553i	$-0.453949\pi$
$79$	202.462	0.288339	0.144169	+	0.989553i	$0.453949\pi$
$80$	1058.20	1.47887
$81$	0	0
$82$	−99.8647	−0.134490
$83$	−336.155	−0.444552	−0.222276	−	0.974984i	$-0.571349\pi$
$83$	−336.155	−0.444552	−0.222276	+	0.974984i	$0.571349\pi$
$84$	0	0
$85$	−1210.66	−1.54487
$86$	12.0702	0.0151344
$87$	0	0
$88$	−155.413	−0.188262
$89$	−718.194	−0.855376	−0.427688	−	0.903927i	$-0.640672\pi$
$89$	−718.194	−0.855376	−0.427688	+	0.903927i	$0.640672\pi$
$90$	0	0
$91$	0	0
$92$	1097.23	1.24342
$93$	0	0
$94$	−139.697	−0.153283
$95$	−1439.01	−1.55409
$96$	0	0
$97$	759.368	0.794868	0.397434	−	0.917631i	$-0.369901\pi$
$97$	759.368	0.794868	0.397434	+	0.917631i	$0.369901\pi$
$98$	−137.420	−0.141648
$99$	0	0
$100$	−1500.00	−1.50000
$101$	348.697	0.343531	0.171766	−	0.985138i	$-0.445053\pi$
$101$	348.697	0.343531	0.171766	+	0.985138i	$0.445053\pi$
$102$	0	0
$103$	−580.303	−0.555136	−0.277568	−	0.960706i	$-0.589528\pi$
$103$	−580.303	−0.555136	−0.277568	+	0.960706i	$0.589528\pi$
$104$	0	0
$105$	0	0
$106$	29.6637	0.0271810
$107$	−571.493	−0.516340	−0.258170	−	0.966100i	$-0.583119\pi$
$107$	−571.493	−0.516340	−0.258170	+	0.966100i	$0.583119\pi$
$108$	0	0
$109$	176.004	0.154661	0.0773307	−	0.997005i	$-0.475360\pi$
$109$	176.004	0.154661	0.0773307	+	0.997005i	$0.475360\pi$
$110$	175.076	0.151753
$111$	0	0
$112$	323.170	0.272649
$113$	−1264.88	−1.05301	−0.526505	−	0.850172i	$-0.676498\pi$
$113$	−1264.88	−1.05301	−0.526505	+	0.850172i	$0.676498\pi$
$114$	0	0
$115$	−2502.55	−2.02925
$116$	−833.035	−0.666770
$117$	0	0
$118$	127.638	0.0995768
$119$	−369.732	−0.284817
$120$	0	0
$121$	−828.196	−0.622236
$122$	290.827	0.215821
$123$	0	0
$124$	2156.16	1.56152
$125$	1195.19	0.855211
$126$	0	0
$127$	2604.11	1.81950	0.909752	−	0.415151i	$-0.136271\pi$
$127$	2604.11	1.81950	0.909752	+	0.415151i	$0.136271\pi$
$128$	−844.772	−0.583344
$129$	0	0
$130$	0	0
$131$	−2131.70	−1.42174	−0.710870	−	0.703324i	$-0.751698\pi$
$131$	−2131.70	−1.42174	−0.710870	+	0.703324i	$0.751698\pi$
$132$	0	0
$133$	−439.469	−0.286517
$134$	−186.384	−0.120158
$135$	0	0
$136$	471.194	0.297092
$137$	−687.985	−0.429040	−0.214520	−	0.976720i	$-0.568819\pi$
$137$	−687.985	−0.429040	−0.214520	+	0.976720i	$0.568819\pi$
$138$	0	0
$139$	−679.580	−0.414685	−0.207343	−	0.978268i	$-0.566482\pi$
$139$	−679.580	−0.414685	−0.207343	+	0.978268i	$0.566482\pi$
$140$	−756.155	−0.456477
$141$	0	0
$142$	67.0662	0.0396343
$143$	0	0
$144$	0	0
$145$	1899.97	1.08816
$146$	51.4158	0.0291452
$147$	0	0
$148$	33.5084	0.0186106
$149$	1975.46	1.08615	0.543074	−	0.839685i	$-0.317260\pi$
$149$	1975.46	1.08615	0.543074	+	0.839685i	$0.317260\pi$
$150$	0	0
$151$	1803.24	0.971824	0.485912	−	0.874008i	$-0.338487\pi$
$151$	1803.24	0.971824	0.485912	+	0.874008i	$0.338487\pi$
$152$	560.068	0.298865
$153$	0	0
$154$	53.4677	0.0279776
$155$	−4917.71	−2.54839
$156$	0	0
$157$	−397.168	−0.201894	−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168	−0.201894	−0.100947	+	0.994892i	$0.532187\pi$
$158$	88.7689	0.0446967
$159$	0	0
$160$	1451.35	0.717120
$161$	−764.272	−0.374118
$162$	0	0
$163$	941.393	0.452365	0.226183	−	0.974085i	$-0.427375\pi$
$163$	941.393	0.452365	0.226183	+	0.974085i	$0.427375\pi$
$164$	1778.37	0.846750
$165$	0	0
$166$	−147.386	−0.0689120
$167$	−3680.43	−1.70539	−0.852696	−	0.522408i	$-0.825034\pi$
$167$	−3680.43	−1.70539	−0.852696	+	0.522408i	$0.825034\pi$
$168$	0	0
$169$	0	0
$170$	−530.810	−0.239478
$171$	0	0
$172$	−214.943	−0.0952863
$173$	−1422.77	−0.625269	−0.312634	−	0.949874i	$-0.601211\pi$
$173$	−1422.77	−0.625269	−0.312634	+	0.949874i	$0.601211\pi$
$174$	0	0
$175$	1044.82	0.451318
$176$	1332.47	0.570673
$177$	0	0
$178$	−314.890	−0.132596
$179$	−1167.89	−0.487666	−0.243833	−	0.969817i	$-0.578405\pi$
$179$	−1167.89	−0.487666	−0.243833	+	0.969817i	$0.578405\pi$
$180$	0	0
$181$	−1133.96	−0.465673	−0.232836	−	0.972516i	$-0.574801\pi$
$181$	−1133.96	−0.465673	−0.232836	+	0.972516i	$0.574801\pi$
$182$	0	0
$183$	0	0
$184$	974.004	0.390242
$185$	−76.4252	−0.0303724
$186$	0	0
$187$	−1524.44	−0.596141
$188$	2487.69	0.965071
$189$	0	0
$190$	−630.928	−0.240907
$191$	−2682.12	−1.01608	−0.508040	−	0.861333i	$-0.669630\pi$
$191$	−2682.12	−1.01608	−0.508040	+	0.861333i	$0.669630\pi$
$192$	0	0
$193$	−1970.67	−0.734983	−0.367491	−	0.930027i	$-0.619783\pi$
$193$	−1970.67	−0.734983	−0.367491	+	0.930027i	$0.619783\pi$
$194$	332.943	0.123216
$195$	0	0
$196$	2447.14	0.891813
$197$	4016.05	1.45244	0.726222	−	0.687460i	$-0.241274\pi$
$197$	4016.05	1.45244	0.726222	+	0.687460i	$0.241274\pi$
$198$	0	0
$199$	−4226.06	−1.50541	−0.752707	−	0.658356i	$-0.771252\pi$
$199$	−4226.06	−1.50541	−0.752707	+	0.658356i	$0.771252\pi$
$200$	−1331.53	−0.470768
$201$	0	0
$202$	152.885	0.0532523
$203$	580.245	0.200617
$204$	0	0
$205$	−4056.06	−1.38189
$206$	−254.432	−0.0860541
$207$	0	0
$208$	0	0
$209$	−1811.98	−0.599699
$210$	0	0
$211$	1364.67	0.445249	0.222625	−	0.974904i	$-0.428538\pi$
$211$	1364.67	0.445249	0.222625	+	0.974904i	$0.428538\pi$
$212$	−528.244	−0.171132
$213$	0	0
$214$	−250.570	−0.0800401
$215$	490.237	0.155506
$216$	0	0
$217$	−1501.86	−0.469828
$218$	77.1683	0.0239748
$219$	0	0
$220$	−3117.71	−0.955436
$221$	0	0
$222$	0	0
$223$	−1059.47	−0.318149	−0.159075	−	0.987267i	$-0.550851\pi$
$223$	−1059.47	−0.318149	−0.159075	+	0.987267i	$0.550851\pi$
$224$	443.239	0.132210
$225$	0	0
$226$	−554.584	−0.163232
$227$	−3464.19	−1.01289	−0.506446	−	0.862272i	$-0.669041\pi$
$227$	−3464.19	−1.01289	−0.506446	+	0.862272i	$0.669041\pi$
$228$	0	0
$229$	−2324.64	−0.670815	−0.335407	−	0.942073i	$-0.608874\pi$
$229$	−2324.64	−0.670815	−0.335407	+	0.942073i	$0.608874\pi$
$230$	−1097.23	−0.314563
$231$	0	0
$232$	−739.476	−0.209263
$233$	3731.01	1.04904	0.524521	−	0.851398i	$-0.324245\pi$
$233$	3731.01	1.04904	0.524521	+	0.851398i	$0.324245\pi$
$234$	0	0
$235$	−5673.86	−1.57499
$236$	−2272.95	−0.626935
$237$	0	0
$238$	−162.108	−0.0441508
$239$	−6044.47	−1.63592	−0.817958	−	0.575278i	$-0.804894\pi$
$239$	−6044.47	−1.63592	−0.817958	+	0.575278i	$0.804894\pi$
$240$	0	0
$241$	−5173.96	−1.38292	−0.691461	−	0.722414i	$-0.743033\pi$
$241$	−5173.96	−1.38292	−0.691461	+	0.722414i	$0.743033\pi$
$242$	−363.120	−0.0964556
$243$	0	0
$244$	−5178.97	−1.35881
$245$	−5581.37	−1.45543
$246$	0	0
$247$	0	0
$248$	1914.00	0.490076
$249$	0	0
$250$	524.029	0.132570
$251$	−5620.73	−1.41346	−0.706728	−	0.707486i	$-0.749829\pi$
$251$	−5620.73	−1.41346	−0.706728	+	0.707486i	$0.749829\pi$
$252$	0	0
$253$	−3151.17	−0.783054
$254$	1141.76	0.282050
$255$	0	0
$256$	3146.83	0.768270
$257$	1674.14	0.406342	0.203171	−	0.979143i	$-0.434875\pi$
$257$	1674.14	0.406342	0.203171	+	0.979143i	$0.434875\pi$
$258$	0	0
$259$	−23.3401	−0.00559954
$260$	0	0
$261$	0	0
$262$	−934.640	−0.220390
$263$	6309.18	1.47924	0.739622	−	0.673023i	$-0.235004\pi$
$263$	6309.18	1.47924	0.739622	+	0.673023i	$0.235004\pi$
$264$	0	0
$265$	1204.81	0.279285
$266$	−192.684	−0.0444143
$267$	0	0
$268$	3319.09	0.756514
$269$	2482.73	0.562731	0.281366	−	0.959601i	$-0.409213\pi$
$269$	2482.73	0.562731	0.281366	+	0.959601i	$0.409213\pi$
$270$	0	0
$271$	2835.72	0.635638	0.317819	−	0.948151i	$-0.397050\pi$
$271$	2835.72	0.635638	0.317819	+	0.948151i	$0.397050\pi$
$272$	−4039.88	−0.900566
$273$	0	0
$274$	−301.645	−0.0665075
$275$	4307.88	0.944637
$276$	0	0
$277$	−3837.51	−0.832396	−0.416198	−	0.909274i	$-0.636638\pi$
$277$	−3837.51	−0.832396	−0.416198	+	0.909274i	$0.636638\pi$
$278$	−297.960	−0.0642822
$279$	0	0
$280$	−671.231	−0.143263
$281$	9122.13	1.93659	0.968293	−	0.249819i	$-0.0803712\pi$
$281$	9122.13	1.93659	0.968293	+	0.249819i	$0.0803712\pi$
$282$	0	0
$283$	2127.85	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$283$	2127.85	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$284$	−1194.30	−0.249537
$285$	0	0
$286$	0	0
$287$	−1238.71	−0.254769
$288$	0	0
$289$	−291.061	−0.0592430
$290$	833.035	0.168681
$291$	0	0
$292$	−915.601	−0.183498
$293$	−8274.77	−1.64989	−0.824944	−	0.565215i	$-0.808793\pi$
$293$	−8274.77	−1.64989	−0.824944	+	0.565215i	$0.808793\pi$
$294$	0	0
$295$	5184.10	1.02315
$296$	29.7450	0.00584086
$297$	0	0
$298$	866.136	0.168369
$299$	0	0
$300$	0	0
$301$	149.717	0.0286696
$302$	790.625	0.150647
$303$	0	0
$304$	−4801.86	−0.905940
$305$	11812.1	2.21757
$306$	0	0
$307$	−3610.49	−0.671211	−0.335605	−	0.942003i	$-0.608941\pi$
$307$	−3610.49	−0.671211	−0.335605	+	0.942003i	$0.608941\pi$
$308$	−952.140	−0.176147
$309$	0	0
$310$	−2156.16	−0.395037
$311$	−3331.06	−0.607354	−0.303677	−	0.952775i	$-0.598214\pi$
$311$	−3331.06	−0.607354	−0.303677	+	0.952775i	$0.598214\pi$
$312$	0	0
$313$	−358.125	−0.0646724	−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125	−0.0646724	−0.0323362	+	0.999477i	$0.510295\pi$
$314$	−174.137	−0.0312966
$315$	0	0
$316$	−1580.78	−0.281410
$317$	−3047.46	−0.539944	−0.269972	−	0.962868i	$-0.587014\pi$
$317$	−3047.46	−0.539944	−0.269972	+	0.962868i	$0.587014\pi$
$318$	0	0
$319$	2392.41	0.419904
$320$	−7829.23	−1.36771
$321$	0	0
$322$	−335.093	−0.0579938
$323$	5493.70	0.946371
$324$	0	0
$325$	0	0
$326$	412.751	0.0701232
$327$	0	0
$328$	1578.64	0.265749
$329$	−1732.78	−0.290369
$330$	0	0
$331$	7694.77	1.27777	0.638887	−	0.769301i	$-0.279395\pi$
$331$	7694.77	1.27777	0.638887	+	0.769301i	$0.279395\pi$
$332$	2624.62	0.433870
$333$	0	0
$334$	−1613.68	−0.264360
$335$	−7570.10	−1.23462
$336$	0	0
$337$	4712.21	0.761693	0.380846	−	0.924638i	$-0.375633\pi$
$337$	4712.21	0.761693	0.380846	+	0.924638i	$0.375633\pi$
$338$	0	0
$339$	0	0
$340$	9452.53	1.50775
$341$	−6192.31	−0.983380
$342$	0	0
$343$	−3569.92	−0.561976
$344$	−190.803	−0.0299052
$345$	0	0
$346$	−623.811	−0.0969257
$347$	5261.98	0.814058	0.407029	−	0.913415i	$-0.366565\pi$
$347$	5261.98	0.814058	0.407029	+	0.913415i	$0.366565\pi$
$348$	0	0
$349$	50.3345	0.00772018	0.00386009	−	0.999993i	$-0.498771\pi$
$349$	50.3345	0.00772018	0.00386009	+	0.999993i	$0.498771\pi$
$350$	458.096	0.0699608
$351$	0	0
$352$	1827.52	0.276725
$353$	−9057.64	−1.36569	−0.682846	−	0.730562i	$-0.739258\pi$
$353$	−9057.64	−1.36569	−0.682846	+	0.730562i	$0.739258\pi$
$354$	0	0
$355$	2723.93	0.407243
$356$	5607.49	0.834821
$357$	0	0
$358$	−512.059	−0.0755953
$359$	−7177.86	−1.05525	−0.527623	−	0.849479i	$-0.676917\pi$
$359$	−7177.86	−1.05525	−0.527623	+	0.849479i	$0.676917\pi$
$360$	0	0
$361$	−329.105	−0.0479815
$362$	−497.183	−0.0721861
$363$	0	0
$364$	0	0
$365$	2088.28	0.299467
$366$	0	0
$367$	4004.14	0.569522	0.284761	−	0.958599i	$-0.408086\pi$
$367$	4004.14	0.569522	0.284761	+	0.958599i	$0.408086\pi$
$368$	−8350.83	−1.18293
$369$	0	0
$370$	−33.5084	−0.00470816
$371$	367.945	0.0514899
$372$	0	0
$373$	−10014.2	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$373$	−10014.2	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$374$	−668.388	−0.0924105
$375$	0	0
$376$	2208.30	0.302883
$377$	0	0
$378$	0	0
$379$	−8169.12	−1.10717	−0.553587	−	0.832791i	$-0.686742\pi$
$379$	−8169.12	−1.10717	−0.553587	+	0.832791i	$0.686742\pi$
$380$	11235.4	1.51675
$381$	0	0
$382$	−1175.97	−0.157507
$383$	7310.25	0.975290	0.487645	−	0.873042i	$-0.337856\pi$
$383$	7310.25	0.975290	0.487645	+	0.873042i	$0.337856\pi$
$384$	0	0
$385$	2171.62	0.287470
$386$	−864.033	−0.113933
$387$	0	0
$388$	−5928.97	−0.775767
$389$	−8785.47	−1.14509	−0.572546	−	0.819872i	$-0.694044\pi$
$389$	−8785.47	−1.14509	−0.572546	+	0.819872i	$0.694044\pi$
$390$	0	0
$391$	9553.99	1.23572
$392$	2172.30	0.279892
$393$	0	0
$394$	1760.82	0.225150
$395$	3605.40	0.459259
$396$	0	0
$397$	11266.8	1.42434	0.712171	−	0.702006i	$-0.247712\pi$
$397$	11266.8	1.42434	0.712171	+	0.702006i	$0.247712\pi$
$398$	−1852.90	−0.233361
$399$	0	0
$400$	11416.2	1.42702
$401$	−1576.23	−0.196293	−0.0981464	−	0.995172i	$-0.531291\pi$
$401$	−1576.23	−0.196293	−0.0981464	+	0.995172i	$0.531291\pi$
$402$	0	0
$403$	0	0
$404$	−2722.54	−0.335276
$405$	0	0
$406$	254.407	0.0310985
$407$	−96.2335	−0.0117202
$408$	0	0
$409$	6755.78	0.816753	0.408377	−	0.912814i	$-0.366095\pi$
$409$	6755.78	0.816753	0.408377	+	0.912814i	$0.366095\pi$
$410$	−1778.37	−0.214213
$411$	0	0
$412$	4530.87	0.541796
$413$	1583.21	0.188631
$414$	0	0
$415$	−5986.17	−0.708072
$416$	0	0
$417$	0	0
$418$	−794.456	−0.0929620
$419$	−10756.2	−1.25411	−0.627057	−	0.778973i	$-0.715741\pi$
$419$	−10756.2	−1.25411	−0.627057	+	0.778973i	$0.715741\pi$
$420$	0	0
$421$	7886.03	0.912925	0.456463	−	0.889743i	$-0.349116\pi$
$421$	7886.03	0.912925	0.456463	+	0.889743i	$0.349116\pi$
$422$	598.335	0.0690201
$423$	0	0
$424$	−468.916	−0.0537089
$425$	−13061.0	−1.49071
$426$	0	0
$427$	3607.38	0.408837
$428$	4462.08	0.503932
$429$	0	0
$430$	214.943	0.0241057
$431$	14084.6	1.57409	0.787044	−	0.616897i	$-0.211610\pi$
$431$	14084.6	1.57409	0.787044	+	0.616897i	$0.211610\pi$
$432$	0	0
$433$	1864.14	0.206894	0.103447	−	0.994635i	$-0.467013\pi$
$433$	1864.14	0.206894	0.103447	+	0.994635i	$0.467013\pi$
$434$	−658.485	−0.0728301
$435$	0	0
$436$	−1374.20	−0.150945
$437$	11356.0	1.24309
$438$	0	0
$439$	6154.49	0.669106	0.334553	−	0.942377i	$-0.391415\pi$
$439$	6154.49	0.669106	0.334553	+	0.942377i	$0.391415\pi$
$440$	−2767.56	−0.299859
$441$	0	0
$442$	0	0
$443$	14539.3	1.55933	0.779663	−	0.626200i	$-0.215391\pi$
$443$	14539.3	1.55933	0.779663	+	0.626200i	$0.215391\pi$
$444$	0	0
$445$	−12789.4	−1.36242
$446$	−464.521	−0.0493177
$447$	0	0
$448$	−2391.03	−0.252155
$449$	7043.87	0.740358	0.370179	−	0.928960i	$-0.379296\pi$
$449$	7043.87	0.740358	0.370179	+	0.928960i	$0.379296\pi$
$450$	0	0
$451$	−5107.33	−0.533248
$452$	9875.90	1.02771
$453$	0	0
$454$	−1518.87	−0.157013
$455$	0	0
$456$	0	0
$457$	14098.9	1.44314	0.721572	−	0.692340i	$-0.243420\pi$
$457$	14098.9	1.44314	0.721572	+	0.692340i	$0.243420\pi$
$458$	−1019.23	−0.103986
$459$	0	0
$460$	19539.3	1.98049
$461$	−14449.7	−1.45985	−0.729924	−	0.683529i	$-0.760444\pi$
$461$	−14449.7	−1.45985	−0.729924	+	0.683529i	$0.760444\pi$
$462$	0	0
$463$	15806.5	1.58659	0.793293	−	0.608840i	$-0.208365\pi$
$463$	15806.5	1.58659	0.793293	+	0.608840i	$0.208365\pi$
$464$	6340.06	0.634332
$465$	0	0
$466$	1635.85	0.162617
$467$	15071.3	1.49340	0.746699	−	0.665162i	$-0.231638\pi$
$467$	15071.3	1.49340	0.746699	+	0.665162i	$0.231638\pi$
$468$	0	0
$469$	−2311.89	−0.227619
$470$	−2487.69	−0.244146
$471$	0	0
$472$	−2017.68	−0.196761
$473$	617.299	0.0600073
$474$	0	0
$475$	−15524.5	−1.49961
$476$	2886.78	0.277973
$477$	0	0
$478$	−2650.18	−0.253591
$479$	392.545	0.0374443	0.0187222	−	0.999825i	$-0.494040\pi$
$479$	392.545	0.0374443	0.0187222	+	0.999825i	$0.494040\pi$
$480$	0	0
$481$	0	0
$482$	−2268.51	−0.214373
$483$	0	0
$484$	6466.36	0.607284
$485$	13522.7	1.26605
$486$	0	0
$487$	9497.89	0.883758	0.441879	−	0.897075i	$-0.354312\pi$
$487$	9497.89	0.883758	0.441879	+	0.897075i	$0.354312\pi$
$488$	−4597.32	−0.426457
$489$	0	0
$490$	−2447.14	−0.225613
$491$	1893.82	0.174067	0.0870337	−	0.996205i	$-0.472261\pi$
$491$	1893.82	0.174067	0.0870337	+	0.996205i	$0.472261\pi$
$492$	0	0
$493$	−7253.52	−0.662641
$494$	0	0
$495$	0	0
$496$	−16410.1	−1.48555
$497$	831.881	0.0750804
$498$	0	0
$499$	−13370.1	−1.19945	−0.599727	−	0.800205i	$-0.704724\pi$
$499$	−13370.1	−1.19945	−0.599727	+	0.800205i	$0.704724\pi$
$500$	−9331.79	−0.834661
$501$	0	0
$502$	−2464.39	−0.219106
$503$	−5554.71	−0.492391	−0.246195	−	0.969220i	$-0.579180\pi$
$503$	−5554.71	−0.492391	−0.246195	+	0.969220i	$0.579180\pi$
$504$	0	0
$505$	6209.51	0.547168
$506$	−1381.62	−0.121385
$507$	0	0
$508$	−20332.3	−1.77578
$509$	−2197.55	−0.191365	−0.0956824	−	0.995412i	$-0.530503\pi$
$509$	−2197.55	−0.191365	−0.0956824	+	0.995412i	$0.530503\pi$
$510$	0	0
$511$	637.756	0.0552107
$512$	8137.89	0.702437
$513$	0	0
$514$	734.022	0.0629890
$515$	−10333.9	−0.884206
$516$	0	0
$517$	−7144.45	−0.607761
$518$	−10.2334	−0.000868010 0
$519$	0	0
$520$	0	0
$521$	−17005.2	−1.42997	−0.714983	−	0.699142i	$-0.753565\pi$
$521$	−17005.2	−1.42997	−0.714983	+	0.699142i	$0.753565\pi$
$522$	0	0
$523$	−14486.2	−1.21116	−0.605581	−	0.795783i	$-0.707059\pi$
$523$	−14486.2	−1.21116	−0.605581	+	0.795783i	$0.707059\pi$
$524$	16643.8	1.38758
$525$	0	0
$526$	2766.24	0.229304
$527$	18774.4	1.55185
$528$	0	0
$529$	7582.03	0.623163
$530$	528.244	0.0432933
$531$	0	0
$532$	3431.27	0.279632
$533$	0	0
$534$	0	0
$535$	−10177.0	−0.822413
$536$	2946.32	0.237429
$537$	0	0
$538$	1088.55	0.0872315
$539$	−7027.98	−0.561626
$540$	0	0
$541$	−15266.7	−1.21325	−0.606623	−	0.794990i	$-0.707476\pi$
$541$	−15266.7	−1.21325	−0.606623	+	0.794990i	$0.707476\pi$
$542$	1243.31	0.0985330
$543$	0	0
$544$	−5540.83	−0.436693
$545$	3134.23	0.246341
$546$	0	0
$547$	15260.5	1.19286	0.596430	−	0.802665i	$-0.296586\pi$
$547$	15260.5	1.19286	0.596430	+	0.802665i	$0.296586\pi$
$548$	5371.62	0.418731
$549$	0	0
$550$	1888.78	0.146432
$551$	−8621.64	−0.666595
$552$	0	0
$553$	1101.08	0.0846703
$554$	−1682.55	−0.129033
$555$	0	0
$556$	5306.00	0.404721
$557$	10442.1	0.794337	0.397169	−	0.917746i	$-0.369993\pi$
$557$	10442.1	0.794337	0.397169	+	0.917746i	$0.369993\pi$
$558$	0	0
$559$	0	0
$560$	5754.94	0.434269
$561$	0	0
$562$	3999.57	0.300199
$563$	−7145.26	−0.534879	−0.267440	−	0.963575i	$-0.586178\pi$
$563$	−7145.26	−0.534879	−0.267440	+	0.963575i	$0.586178\pi$
$564$	0	0
$565$	−22524.7	−1.67721
$566$	932.950	0.0692841
$567$	0	0
$568$	−1060.17	−0.0783162
$569$	−4438.86	−0.327042	−0.163521	−	0.986540i	$-0.552285\pi$
$569$	−4438.86	−0.327042	−0.163521	+	0.986540i	$0.552285\pi$
$570$	0	0
$571$	10117.3	0.741497	0.370748	−	0.928733i	$-0.379101\pi$
$571$	10117.3	0.741497	0.370748	+	0.928733i	$0.379101\pi$
$572$	0	0
$573$	0	0
$574$	−543.109	−0.0394929
$575$	−26998.4	−1.95810
$576$	0	0
$577$	3105.60	0.224069	0.112035	−	0.993704i	$-0.464263\pi$
$577$	3105.60	0.224069	0.112035	+	0.993704i	$0.464263\pi$
$578$	−127.615	−0.00918352
$579$	0	0
$580$	−14834.5	−1.06202
$581$	−1828.16	−0.130542
$582$	0	0
$583$	1517.08	0.107772
$584$	−812.769	−0.0575901
$585$	0	0
$586$	−3628.05	−0.255757
$587$	−19662.3	−1.38254	−0.691270	−	0.722597i	$-0.742948\pi$
$587$	−19662.3	−1.38254	−0.691270	+	0.722597i	$0.742948\pi$
$588$	0	0
$589$	22315.5	1.56111
$590$	2272.95	0.158603
$591$	0	0
$592$	−255.026	−0.0177052
$593$	−6395.51	−0.442888	−0.221444	−	0.975173i	$-0.571077\pi$
$593$	−6395.51	−0.442888	−0.221444	+	0.975173i	$0.571077\pi$
$594$	0	0
$595$	−6584.10	−0.453650
$596$	−15423.9	−1.06005
$597$	0	0
$598$	0	0
$599$	−8878.48	−0.605618	−0.302809	−	0.953051i	$-0.597924\pi$
$599$	−8878.48	−0.605618	−0.302809	+	0.953051i	$0.597924\pi$
$600$	0	0
$601$	19100.6	1.29639	0.648194	−	0.761475i	$-0.275525\pi$
$601$	19100.6	1.29639	0.648194	+	0.761475i	$0.275525\pi$
$602$	65.6430	0.00444420
$603$	0	0
$604$	−14079.3	−0.948472
$605$	−14748.3	−0.991082
$606$	0	0
$607$	16595.8	1.10972	0.554861	−	0.831943i	$-0.312771\pi$
$607$	16595.8	1.10972	0.554861	+	0.831943i	$0.312771\pi$
$608$	−6585.91	−0.439299
$609$	0	0
$610$	5178.97	0.343755
$611$	0	0
$612$	0	0
$613$	16469.2	1.08513	0.542564	−	0.840015i	$-0.317454\pi$
$613$	16469.2	1.08513	0.542564	+	0.840015i	$0.317454\pi$
$614$	−1583.01	−0.104047
$615$	0	0
$616$	−845.205	−0.0552829
$617$	−10116.0	−0.660055	−0.330027	−	0.943971i	$-0.607058\pi$
$617$	−10116.0	−0.660055	−0.330027	+	0.943971i	$0.607058\pi$
$618$	0	0
$619$	18854.8	1.22430	0.612148	−	0.790743i	$-0.290306\pi$
$619$	18854.8	1.22430	0.612148	+	0.790743i	$0.290306\pi$
$620$	38396.3	2.48715
$621$	0	0
$622$	−1460.49	−0.0941487
$623$	−3905.86	−0.251180
$624$	0	0
$625$	−2730.82	−0.174773
$626$	−157.019	−0.0100252
$627$	0	0
$628$	3100.99	0.197043
$629$	291.769	0.0184954
$630$	0	0
$631$	18946.2	1.19531	0.597653	−	0.801755i	$-0.296100\pi$
$631$	18946.2	1.19531	0.597653	+	0.801755i	$0.296100\pi$
$632$	−1403.24	−0.0883194
$633$	0	0
$634$	−1336.15	−0.0836991
$635$	46373.3	2.89806
$636$	0	0
$637$	0	0
$638$	1048.95	0.0650912
$639$	0	0
$640$	−15043.5	−0.929135
$641$	23586.9	1.45340	0.726698	−	0.686957i	$-0.241054\pi$
$641$	23586.9	1.45340	0.726698	+	0.686957i	$0.241054\pi$
$642$	0	0
$643$	−27153.0	−1.66534	−0.832669	−	0.553772i	$-0.813188\pi$
$643$	−27153.0	−1.66534	−0.832669	+	0.553772i	$0.813188\pi$
$644$	5967.25	0.365128
$645$	0	0
$646$	2408.70	0.146701
$647$	−6856.72	−0.416639	−0.208319	−	0.978061i	$-0.566799\pi$
$647$	−6856.72	−0.416639	−0.208319	+	0.978061i	$0.566799\pi$
$648$	0	0
$649$	6527.75	0.394817
$650$	0	0
$651$	0	0
$652$	−7350.17	−0.441495
$653$	8073.89	0.483853	0.241926	−	0.970295i	$-0.422221\pi$
$653$	8073.89	0.483853	0.241926	+	0.970295i	$0.422221\pi$
$654$	0	0
$655$	−37960.9	−2.26451
$656$	−13534.8	−0.805555
$657$	0	0
$658$	−759.734	−0.0450114
$659$	−5305.73	−0.313629	−0.156815	−	0.987628i	$-0.550123\pi$
$659$	−5305.73	−0.313629	−0.156815	+	0.987628i	$0.550123\pi$
$660$	0	0
$661$	25848.3	1.52100	0.760502	−	0.649336i	$-0.224953\pi$
$661$	25848.3	1.52100	0.760502	+	0.649336i	$0.224953\pi$
$662$	3373.75	0.198073
$663$	0	0
$664$	2329.85	0.136168
$665$	−7825.96	−0.456357
$666$	0	0
$667$	−14993.7	−0.870404
$668$	28735.9	1.66441
$669$	0	0
$670$	−3319.09	−0.191385
$671$	14873.6	0.855722
$672$	0	0
$673$	−14529.1	−0.832177	−0.416089	−	0.909324i	$-0.636599\pi$
$673$	−14529.1	−0.832177	−0.416089	+	0.909324i	$0.636599\pi$
$674$	2066.06	0.118073
$675$	0	0
$676$	0	0
$677$	−12058.1	−0.684535	−0.342267	−	0.939603i	$-0.611195\pi$
$677$	−12058.1	−0.684535	−0.342267	+	0.939603i	$0.611195\pi$
$678$	0	0
$679$	4129.78	0.233412
$680$	8390.91	0.473201
$681$	0	0
$682$	−2715.00	−0.152438
$683$	−30028.8	−1.68231	−0.841156	−	0.540792i	$-0.818125\pi$
$683$	−30028.8	−1.68231	−0.841156	+	0.540792i	$0.818125\pi$
$684$	0	0
$685$	−12251.5	−0.683364
$686$	−1565.22	−0.0871144
$687$	0	0
$688$	1635.89	0.0906505
$689$	0	0
$690$	0	0
$691$	449.696	0.0247572	0.0123786	−	0.999923i	$-0.496060\pi$
$691$	449.696	0.0247572	0.0123786	+	0.999923i	$0.496060\pi$
$692$	11108.7	0.610244
$693$	0	0
$694$	2307.10	0.126191
$695$	−12101.8	−0.660500
$696$	0	0
$697$	15484.8	0.841506
$698$	22.0690	0.00119674
$699$	0	0
$700$	−8157.67	−0.440473
$701$	26986.0	1.45399	0.726994	−	0.686644i	$-0.240917\pi$
$701$	26986.0	1.45399	0.726994	+	0.686644i	$0.240917\pi$
$702$	0	0
$703$	346.801	0.0186057
$704$	−9858.46	−0.527776
$705$	0	0
$706$	−3971.29	−0.211702
$707$	1896.37	0.100877
$708$	0	0
$709$	−9098.87	−0.481968	−0.240984	−	0.970529i	$-0.577470\pi$
$709$	−9098.87	−0.481968	−0.240984	+	0.970529i	$0.577470\pi$
$710$	1194.30	0.0631285
$711$	0	0
$712$	4977.71	0.262005
$713$	38808.4	2.03841
$714$	0	0
$715$	0	0
$716$	9118.62	0.475948
$717$	0	0
$718$	−3147.11	−0.163578
$719$	6293.55	0.326439	0.163220	−	0.986590i	$-0.447812\pi$
$719$	6293.55	0.326439	0.163220	+	0.986590i	$0.447812\pi$
$720$	0	0
$721$	−3155.95	−0.163015
$722$	−144.295	−0.00743783
$723$	0	0
$724$	8853.72	0.454483
$725$	20497.5	1.05001
$726$	0	0
$727$	18070.7	0.921878	0.460939	−	0.887432i	$-0.347513\pi$
$727$	18070.7	0.921878	0.460939	+	0.887432i	$0.347513\pi$
$728$	0	0
$729$	0	0
$730$	915.601	0.0464218
$731$	−1871.58	−0.0946962
$732$	0	0
$733$	34771.5	1.75214	0.876068	−	0.482188i	$-0.160158\pi$
$733$	34771.5	1.75214	0.876068	+	0.482188i	$0.160158\pi$
$734$	1755.61	0.0882842
$735$	0	0
$736$	−11453.4	−0.573613
$737$	−9532.17	−0.476421
$738$	0	0
$739$	23631.5	1.17632	0.588158	−	0.808746i	$-0.299853\pi$
$739$	23631.5	1.17632	0.588158	+	0.808746i	$0.299853\pi$
$740$	596.710	0.0296425
$741$	0	0
$742$	161.324	0.00798167
$743$	−32502.8	−1.60486	−0.802431	−	0.596745i	$-0.796460\pi$
$743$	−32502.8	−1.60486	−0.802431	+	0.596745i	$0.796460\pi$
$744$	0	0
$745$	35178.6	1.72999
$746$	−4390.69	−0.215489
$747$	0	0
$748$	11902.5	0.581816
$749$	−3108.04	−0.151622
$750$	0	0
$751$	2020.86	0.0981920	0.0490960	−	0.998794i	$-0.484366\pi$
$751$	2020.86	0.0981920	0.0490960	+	0.998794i	$0.484366\pi$
$752$	−18933.3	−0.918120
$753$	0	0
$754$	0	0
$755$	32111.6	1.54790
$756$	0	0
$757$	12568.2	0.603434	0.301717	−	0.953398i	$-0.402440\pi$
$757$	12568.2	0.603434	0.301717	+	0.953398i	$0.402440\pi$
$758$	−3581.73	−0.171628
$759$	0	0
$760$	9973.56	0.476025
$761$	8704.81	0.414651	0.207325	−	0.978272i	$-0.433524\pi$
$761$	8704.81	0.414651	0.207325	+	0.978272i	$0.433524\pi$
$762$	0	0
$763$	957.187	0.0454161
$764$	20941.4	0.991665
$765$	0	0
$766$	3205.16	0.151184
$767$	0	0
$768$	0	0
$769$	−21915.9	−1.02771	−0.513853	−	0.857878i	$-0.671782\pi$
$769$	−21915.9	−1.02771	−0.513853	+	0.857878i	$0.671782\pi$
$770$	952.140	0.0445620
$771$	0	0
$772$	15386.5	0.717322
$773$	23077.5	1.07379	0.536896	−	0.843649i	$-0.319597\pi$
$773$	23077.5	1.07379	0.536896	+	0.843649i	$0.319597\pi$
$774$	0	0
$775$	−53054.0	−2.45904
$776$	−5263.08	−0.243471
$777$	0	0
$778$	−3851.96	−0.177506
$779$	18405.5	0.846528
$780$	0	0
$781$	3429.94	0.157148
$782$	4188.92	0.191554
$783$	0	0
$784$	−18624.6	−0.848426
$785$	−7072.67	−0.321572
$786$	0	0
$787$	−16522.4	−0.748362	−0.374181	−	0.927356i	$-0.622076\pi$
$787$	−16522.4	−0.748362	−0.374181	+	0.927356i	$0.622076\pi$
$788$	−31356.4	−1.41754
$789$	0	0
$790$	1580.78	0.0711918
$791$	−6879.00	−0.309215
$792$	0	0
$793$	0	0
$794$	4939.89	0.220794
$795$	0	0
$796$	32996.1	1.46924
$797$	11719.4	0.520855	0.260427	−	0.965493i	$-0.416137\pi$
$797$	11719.4	0.520855	0.260427	+	0.965493i	$0.416137\pi$
$798$	0	0
$799$	21661.2	0.959095
$800$	15657.7	0.691978
$801$	0	0
$802$	−691.096	−0.0304282
$803$	2629.53	0.115559
$804$	0	0
$805$	−13610.0	−0.595886
$806$	0	0
$807$	0	0
$808$	−2416.77	−0.105225
$809$	24096.0	1.04718	0.523592	−	0.851969i	$-0.324592\pi$
$809$	24096.0	1.04718	0.523592	+	0.851969i	$0.324592\pi$
$810$	0	0
$811$	16622.6	0.719729	0.359864	−	0.933005i	$-0.382823\pi$
$811$	16622.6	0.719729	0.359864	+	0.933005i	$0.382823\pi$
$812$	−4530.42	−0.195796
$813$	0	0
$814$	−42.1933	−0.00181680
$815$	16764.1	0.720516
$816$	0	0
$817$	−2224.59	−0.0952613
$818$	2962.05	0.126609
$819$	0	0
$820$	31668.7	1.34868
$821$	38005.5	1.61559	0.807797	−	0.589461i	$-0.200660\pi$
$821$	38005.5	1.61559	0.807797	+	0.589461i	$0.200660\pi$
$822$	0	0
$823$	−15859.5	−0.671722	−0.335861	−	0.941912i	$-0.609027\pi$
$823$	−15859.5	−0.671722	−0.335861	+	0.941912i	$0.609027\pi$
$824$	4022.01	0.170040
$825$	0	0
$826$	694.155	0.0292406
$827$	−12201.0	−0.513023	−0.256512	−	0.966541i	$-0.582573\pi$
$827$	−12201.0	−0.513023	−0.256512	+	0.966541i	$0.582573\pi$
$828$	0	0
$829$	−5431.41	−0.227552	−0.113776	−	0.993506i	$-0.536295\pi$
$829$	−5431.41	−0.227552	−0.113776	+	0.993506i	$0.536295\pi$
$830$	−2624.62	−0.109761
$831$	0	0
$832$	0	0
$833$	21308.0	0.886290
$834$	0	0
$835$	−65540.3	−2.71630
$836$	14147.5	0.585288
$837$	0	0
$838$	−4716.02	−0.194406
$839$	7960.90	0.327582	0.163791	−	0.986495i	$-0.447628\pi$
$839$	7960.90	0.327582	0.163791	+	0.986495i	$0.447628\pi$
$840$	0	0
$841$	−13005.6	−0.533255
$842$	3457.61	0.141517
$843$	0	0
$844$	−10655.0	−0.434550
$845$	0	0
$846$	0	0
$847$	−4504.10	−0.182719
$848$	4020.35	0.162806
$849$	0	0
$850$	−5726.56	−0.231082
$851$	603.115	0.0242944
$852$	0	0
$853$	13576.7	0.544969	0.272485	−	0.962160i	$-0.412155\pi$
$853$	13576.7	0.544969	0.272485	+	0.962160i	$0.412155\pi$
$854$	1581.65	0.0633756
$855$	0	0
$856$	3960.95	0.158157
$857$	31223.9	1.24456	0.622281	−	0.782794i	$-0.286206\pi$
$857$	31223.9	1.24456	0.622281	+	0.782794i	$0.286206\pi$
$858$	0	0
$859$	−11815.8	−0.469323	−0.234661	−	0.972077i	$-0.575398\pi$
$859$	−11815.8	−0.469323	−0.234661	+	0.972077i	$0.575398\pi$
$860$	−3827.65	−0.151770
$861$	0	0
$862$	6175.36	0.244006
$863$	1790.84	0.0706384	0.0353192	−	0.999376i	$-0.488755\pi$
$863$	1790.84	0.0706384	0.0353192	+	0.999376i	$0.488755\pi$
$864$	0	0
$865$	−25336.4	−0.995912
$866$	817.328	0.0320715
$867$	0	0
$868$	11726.1	0.458538
$869$	4539.87	0.177220
$870$	0	0
$871$	0	0
$872$	−1219.86	−0.0473734
$873$	0	0
$874$	4979.01	0.192698
$875$	6500.00	0.251132
$876$	0	0
$877$	43542.5	1.67654	0.838270	−	0.545255i	$-0.183567\pi$
$877$	43542.5	1.67654	0.838270	+	0.545255i	$0.183567\pi$
$878$	2698.42	0.103721
$879$	0	0
$880$	23728.2	0.908953
$881$	1020.04	0.0390080	0.0195040	−	0.999810i	$-0.493791\pi$
$881$	1020.04	0.0390080	0.0195040	+	0.999810i	$0.493791\pi$
$882$	0	0
$883$	−34781.9	−1.32560	−0.662800	−	0.748797i	$-0.730632\pi$
$883$	−34781.9	−1.32560	−0.662800	+	0.748797i	$0.730632\pi$
$884$	0	0
$885$	0	0
$886$	6374.70	0.241718
$887$	−49785.1	−1.88458	−0.942288	−	0.334802i	$-0.891330\pi$
$887$	−49785.1	−1.88458	−0.942288	+	0.334802i	$0.891330\pi$
$888$	0	0
$889$	14162.3	0.534295
$890$	−5607.49	−0.211195
$891$	0	0
$892$	8272.08	0.310504
$893$	25746.8	0.964818
$894$	0	0
$895$	−20797.5	−0.776743
$896$	−4594.25	−0.171298
$897$	0	0
$898$	3088.36	0.114766
$899$	−29463.9	−1.09308
$900$	0	0
$901$	−4599.60	−0.170072
$902$	−2239.29	−0.0826611
$903$	0	0
$904$	8766.74	0.322541
$905$	−20193.3	−0.741712
$906$	0	0
$907$	17389.9	0.636627	0.318314	−	0.947985i	$-0.396884\pi$
$907$	17389.9	0.636627	0.318314	+	0.947985i	$0.396884\pi$
$908$	27047.6	0.988553
$909$	0	0
$910$	0	0
$911$	−20419.5	−0.742621	−0.371311	−	0.928509i	$-0.621091\pi$
$911$	−20419.5	−0.742621	−0.371311	+	0.928509i	$0.621091\pi$
$912$	0	0
$913$	−7537.71	−0.273233
$914$	6181.60	0.223708
$915$	0	0
$916$	18150.2	0.654695
$917$	−11593.2	−0.417492
$918$	0	0
$919$	−33231.8	−1.19283	−0.596417	−	0.802674i	$-0.703410\pi$
$919$	−33231.8	−1.19283	−0.596417	+	0.802674i	$0.703410\pi$
$920$	17344.8	0.621567
$921$	0	0
$922$	−6335.43	−0.226297
$923$	0	0
$924$	0	0
$925$	−824.502	−0.0293075
$926$	6930.31	0.245944
$927$	0	0
$928$	8695.59	0.307594
$929$	25222.8	0.890780	0.445390	−	0.895337i	$-0.353065\pi$
$929$	25222.8	0.890780	0.445390	+	0.895337i	$0.353065\pi$
$930$	0	0
$931$	25327.0	0.891579
$932$	−29130.9	−1.02383
$933$	0	0
$934$	6607.97	0.231498
$935$	−27146.9	−0.949519
$936$	0	0
$937$	−26979.4	−0.940639	−0.470319	−	0.882496i	$-0.655861\pi$
$937$	−26979.4	−0.940639	−0.470319	+	0.882496i	$0.655861\pi$
$938$	−1013.64	−0.0352842
$939$	0	0
$940$	44300.2	1.53714
$941$	−7641.67	−0.264730	−0.132365	−	0.991201i	$-0.542257\pi$
$941$	−7641.67	−0.264730	−0.132365	+	0.991201i	$0.542257\pi$
$942$	0	0
$943$	32008.7	1.10535
$944$	17299.0	0.596434
$945$	0	0
$946$	270.653	0.00930200
$947$	2869.32	0.0984587	0.0492293	−	0.998788i	$-0.484323\pi$
$947$	2869.32	0.0984587	0.0492293	+	0.998788i	$0.484323\pi$
$948$	0	0
$949$	0	0
$950$	−6806.67	−0.232461
$951$	0	0
$952$	2562.56	0.0872407
$953$	−12313.6	−0.418548	−0.209274	−	0.977857i	$-0.567110\pi$
$953$	−12313.6	−0.418548	−0.209274	+	0.977857i	$0.567110\pi$
$954$	0	0
$955$	−47762.6	−1.61839
$956$	47193.8	1.59661
$957$	0	0
$958$	172.110	0.00580442
$959$	−3741.57	−0.125987
$960$	0	0
$961$	46470.7	1.55989
$962$	0	0
$963$	0	0
$964$	40397.1	1.34969
$965$	−35093.2	−1.17066
$966$	0	0
$967$	−17838.0	−0.593207	−0.296603	−	0.955001i	$-0.595854\pi$
$967$	−17838.0	−0.593207	−0.296603	+	0.955001i	$0.595854\pi$
$968$	5740.12	0.190593
$969$	0	0
$970$	5928.97	0.196255
$971$	−41525.3	−1.37241	−0.686206	−	0.727408i	$-0.740725\pi$
$971$	−41525.3	−1.37241	−0.686206	+	0.727408i	$0.740725\pi$
$972$	0	0
$973$	−3695.86	−0.121772
$974$	4164.32	0.136995
$975$	0	0
$976$	39416.1	1.29270
$977$	31654.4	1.03655	0.518277	−	0.855213i	$-0.326574\pi$
$977$	31654.4	1.03655	0.518277	+	0.855213i	$0.326574\pi$
$978$	0	0
$979$	−16104.3	−0.525735
$980$	43578.0	1.42046
$981$	0	0
$982$	830.342	0.0269830
$983$	39913.2	1.29505	0.647525	−	0.762045i	$-0.275804\pi$
$983$	39913.2	1.29505	0.647525	+	0.762045i	$0.275804\pi$
$984$	0	0
$985$	71516.8	2.31342
$986$	−3180.28	−0.102719
$987$	0	0
$988$	0	0
$989$	−3868.74	−0.124387
$990$	0	0
$991$	−2700.94	−0.0865773	−0.0432887	−	0.999063i	$-0.513784\pi$
$991$	−2700.94	−0.0865773	−0.0432887	+	0.999063i	$0.513784\pi$
$992$	−22506.9	−0.720358
$993$	0	0
$994$	364.736	0.0116386
$995$	−75256.6	−2.39778
$996$	0	0
$997$	−9729.08	−0.309050	−0.154525	−	0.987989i	$-0.549385\pi$
$997$	−9729.08	−0.309050	−0.154525	+	0.987989i	$0.549385\pi$
$998$	−5862.08	−0.185933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.t.1.1		2
3.2	odd	2		169.4.a.f.1.2		2
13.3	even	3		117.4.g.d.100.2		4
13.9	even	3		117.4.g.d.55.2		4
13.12	even	2		1521.4.a.l.1.2		2
39.2	even	12		169.4.e.g.147.2		8
39.5	even	4		169.4.b.e.168.3		4
39.8	even	4		169.4.b.e.168.2		4
39.11	even	12		169.4.e.g.147.3		8
39.17	odd	6		169.4.c.f.146.2		4
39.20	even	12		169.4.e.g.23.2		8
39.23	odd	6		169.4.c.f.22.2		4
39.29	odd	6		13.4.c.b.9.1	yes	4
39.32	even	12		169.4.e.g.23.3		8
39.35	odd	6		13.4.c.b.3.1	✓	4
39.38	odd	2		169.4.a.j.1.1		2
156.35	even	6		208.4.i.e.81.1		4
156.107	even	6		208.4.i.e.113.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.1	✓	4	39.35	odd	6
13.4.c.b.9.1	yes	4	39.29	odd	6
117.4.g.d.55.2		4	13.9	even	3
117.4.g.d.100.2		4	13.3	even	3
169.4.a.f.1.2		2	3.2	odd	2
169.4.a.j.1.1		2	39.38	odd	2
169.4.b.e.168.2		4	39.8	even	4
169.4.b.e.168.3		4	39.5	even	4
169.4.c.f.22.2		4	39.23	odd	6
169.4.c.f.146.2		4	39.17	odd	6
169.4.e.g.23.2		8	39.20	even	12
169.4.e.g.23.3		8	39.32	even	12
169.4.e.g.147.2		8	39.2	even	12
169.4.e.g.147.3		8	39.11	even	12
208.4.i.e.81.1		4	156.35	even	6
208.4.i.e.113.1		4	156.107	even	6
1521.4.a.l.1.2		2	13.12	even	2
1521.4.a.t.1.1		2	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)

\(f(q)\)	\(=\)	\(q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +O(q^{10})\)	\(q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +7.80776 q^{10} +22.4233 q^{11} +2.38447 q^{14} +59.4233 q^{16} -67.9848 q^{17} -80.8078 q^{19} -139.039 q^{20} +9.83143 q^{22} -140.531 q^{23} +192.116 q^{25} -42.4621 q^{28} +106.693 q^{29} -276.155 q^{31} +81.5009 q^{32} -29.8078 q^{34} +96.8466 q^{35} -4.29168 q^{37} -35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +27.5294 q^{43} -175.076 q^{44} -61.6155 q^{46} -318.617 q^{47} -313.423 q^{49} +84.2329 q^{50} +67.6562 q^{53} +399.309 q^{55} -37.6932 q^{56} +46.7793 q^{58} +291.115 q^{59} +663.311 q^{61} -121.080 q^{62} -439.652 q^{64} -425.101 q^{67} +530.810 q^{68} +42.4621 q^{70} +152.963 q^{71} +117.268 q^{73} -1.88167 q^{74} +630.928 q^{76} +121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -99.8647 q^{82} -336.155 q^{83} -1210.66 q^{85} +12.0702 q^{86} -155.413 q^{88} -718.194 q^{89} +1097.23 q^{92} -139.697 q^{94} -1439.01 q^{95} +759.368 q^{97} -137.420 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10}) \) 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8	\( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100}) \) 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 - 5 * q^10 - 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 - 141 * q^19 - 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 + 80 * q^28 - 34 * q^29 - 140 * q^31 - 105 * q^32 - 39 * q^34 + 70 * q^35 - 190 * q^37 - 310 * q^38 - 185 * q^40 - 538 * q^41 + 455 * q^43 - 680 * q^44 - 82 * q^46 - 60 * q^47 - 565 * q^49 - 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 - 595 * q^58 + 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 - 475 * q^67 + 505 * q^68 - 80 * q^70 - 127 * q^71 + 585 * q^73 - 849 * q^74 - 140 * q^76 - 255 * q^77 + 240 * q^79 + 1065 * q^80 - 1515 * q^82 - 260 * q^83 - 1205 * q^85 + 1962 * q^86 - 1020 * q^88 - 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 - 415 * q^97 - 1285 * q^98

Introduction

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