LMFDB - Embedded newform 1521.4.a.u.1.3

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:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ 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)

$f(q)$	$=$	$q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} +O(q^{10})$	$q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} -48.1187 q^{10} +25.8785 q^{11} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -154.712 q^{19} -111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} +109.348 q^{28} +20.4522 q^{29} -266.424 q^{31} -256.984 q^{32} +85.5599 q^{34} -128.661 q^{35} -115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +759.390 q^{46} -467.365 q^{47} -216.341 q^{49} +23.9714 q^{50} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +609.204 q^{67} +197.505 q^{68} -541.542 q^{70} +248.038 q^{71} -852.765 q^{73} -488.181 q^{74} -1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} +541.043 q^{80} +1646.51 q^{82} -435.432 q^{83} -232.389 q^{85} +637.281 q^{86} +186.924 q^{88} +259.233 q^{89} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} -1225.17 q^{97} -910.589 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8	$ 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 - 4 * q^10 - 16 * q^11 + 176 * q^14 - 110 * q^16 + 146 * q^17 - 94 * q^19 - 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 80 * q^28 + 2 * q^29 - 302 * q^31 + 154 * q^32 - 164 * q^34 - 80 * q^35 - 374 * q^37 - 312 * q^38 - 516 * q^40 + 480 * q^41 - 260 * q^43 + 712 * q^44 + 1104 * q^46 - 24 * q^47 + 447 * q^49 + 814 * q^50 + 678 * q^53 - 1552 * q^55 - 96 * q^56 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 74 * q^67 + 460 * q^68 - 1216 * q^70 - 948 * q^71 + 222 * q^73 - 1724 * q^74 - 2392 * q^76 - 112 * q^77 - 24 * q^79 + 1100 * q^80 + 564 * q^82 - 796 * q^83 + 248 * q^85 + 1800 * q^86 + 1608 * q^88 + 1436 * q^89 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 - 3242 * q^97 - 5070 * 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 $

$2$	4.20905	1.48812	0.744062	−	0.668111i	$-0.232897\pi$
$2$	4.20905	1.48812	0.744062	+	0.668111i	$0.232897\pi$
$3$	0	0
$3$	0	0
$4$	9.71610	1.21451
$5$	−11.4322	−1.02253	−0.511264	−	0.859424i	$-0.670822\pi$
$5$	−11.4322	−1.02253	−0.511264	+	0.859424i	$0.670822\pi$
$6$	0	0
$7$	11.2543	0.607675	0.303838	−	0.952724i	$-0.401732\pi$
$7$	11.2543	0.607675	0.303838	+	0.952724i	$0.401732\pi$
$8$	7.22315	0.319221
$9$	0	0
$10$	−48.1187	−1.52165
$11$	25.8785	0.709333	0.354666	−	0.934993i	$-0.384594\pi$
$11$	25.8785	0.709333	0.354666	+	0.934993i	$0.384594\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	47.3699	0.904296
$15$	0	0
$16$	−47.3262	−0.739472
$17$	20.3276	0.290010	0.145005	−	0.989431i	$-0.453680\pi$
$17$	20.3276	0.290010	0.145005	+	0.989431i	$0.453680\pi$
$18$	0	0
$19$	−154.712	−1.86807	−0.934035	−	0.357181i	$-0.883738\pi$
$19$	−154.712	−1.86807	−0.934035	+	0.357181i	$0.883738\pi$
$20$	−111.076	−1.24187
$21$	0	0
$22$	108.924	1.05558
$23$	180.418	1.63565	0.817823	−	0.575471i	$-0.195181\pi$
$23$	180.418	1.63565	0.817823	+	0.575471i	$0.195181\pi$
$24$	0	0
$25$	5.69520	0.0455616
$26$	0	0
$27$	0	0
$28$	109.348	0.738029
$29$	20.4522	0.130961	0.0654806	−	0.997854i	$-0.479142\pi$
$29$	20.4522	0.130961	0.0654806	+	0.997854i	$0.479142\pi$
$30$	0	0
$31$	−266.424	−1.54359	−0.771794	−	0.635873i	$-0.780640\pi$
$31$	−266.424	−1.54359	−0.771794	+	0.635873i	$0.780640\pi$
$32$	−256.984	−1.41965
$33$	0	0
$34$	85.5599	0.431571
$35$	−128.661	−0.621364
$36$	0	0
$37$	−115.984	−0.515340	−0.257670	−	0.966233i	$-0.582955\pi$
$37$	−115.984	−0.515340	−0.257670	+	0.966233i	$0.582955\pi$
$38$	−651.190	−2.77992
$39$	0	0
$40$	−82.5765	−0.326412
$41$	391.184	1.49006	0.745032	−	0.667029i	$-0.232434\pi$
$41$	391.184	1.49006	0.745032	+	0.667029i	$0.232434\pi$
$42$	0	0
$43$	151.407	0.536963	0.268482	−	0.963285i	$-0.413478\pi$
$43$	151.407	0.536963	0.268482	+	0.963285i	$0.413478\pi$
$44$	251.438	0.861494
$45$	0	0
$46$	759.390	2.43404
$47$	−467.365	−1.45047	−0.725236	−	0.688500i	$-0.758269\pi$
$47$	−467.365	−1.45047	−0.725236	+	0.688500i	$0.758269\pi$
$48$	0	0
$49$	−216.341	−0.630731
$50$	23.9714	0.0678012
$51$	0	0
$52$	0	0
$53$	−79.9842	−0.207296	−0.103648	−	0.994614i	$-0.533051\pi$
$53$	−79.9842	−0.207296	−0.103648	+	0.994614i	$0.533051\pi$
$54$	0	0
$55$	−295.848	−0.725312
$56$	81.2915	0.193983
$57$	0	0
$58$	86.0843	0.194887
$59$	−873.710	−1.92792	−0.963960	−	0.266045i	$-0.914283\pi$
$59$	−873.710	−1.92792	−0.963960	+	0.266045i	$0.914283\pi$
$60$	0	0
$61$	−187.068	−0.392649	−0.196325	−	0.980539i	$-0.562901\pi$
$61$	−187.068	−0.392649	−0.196325	+	0.980539i	$0.562901\pi$
$62$	−1121.39	−2.29705
$63$	0	0
$64$	−703.047	−1.37314
$65$	0	0
$66$	0	0
$67$	609.204	1.11084	0.555418	−	0.831571i	$-0.312558\pi$
$67$	609.204	1.11084	0.555418	+	0.831571i	$0.312558\pi$
$68$	197.505	0.352221
$69$	0	0
$70$	−541.542	−0.924667
$71$	248.038	0.414601	0.207301	−	0.978277i	$-0.433532\pi$
$71$	248.038	0.414601	0.207301	+	0.978277i	$0.433532\pi$
$72$	0	0
$73$	−852.765	−1.36724	−0.683621	−	0.729838i	$-0.739596\pi$
$73$	−852.765	−1.36724	−0.683621	+	0.729838i	$0.739596\pi$
$74$	−488.181	−0.766890
$75$	0	0
$76$	−1503.20	−2.26880
$77$	291.244	0.431044
$78$	0	0
$79$	−331.221	−0.471712	−0.235856	−	0.971788i	$-0.575789\pi$
$79$	−331.221	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$80$	541.043	0.756130
$81$	0	0
$82$	1646.51	2.21740
$83$	−435.432	−0.575842	−0.287921	−	0.957654i	$-0.592964\pi$
$83$	−435.432	−0.575842	−0.287921	+	0.957654i	$0.592964\pi$
$84$	0	0
$85$	−232.389	−0.296543
$86$	637.281	0.799067
$87$	0	0
$88$	186.924	0.226434
$89$	259.233	0.308749	0.154375	−	0.988012i	$-0.450664\pi$
$89$	259.233	0.308749	0.154375	+	0.988012i	$0.450664\pi$
$90$	0	0
$91$	0	0
$92$	1752.96	1.98651
$93$	0	0
$94$	−1967.16	−2.15848
$95$	1768.70	1.91015
$96$	0	0
$97$	−1225.17	−1.28245	−0.641223	−	0.767355i	$-0.721572\pi$
$97$	−1225.17	−1.28245	−0.641223	+	0.767355i	$0.721572\pi$
$98$	−910.589	−0.938606
$99$	0	0
$100$	55.3351	0.0553351
$101$	−645.416	−0.635855	−0.317927	−	0.948115i	$-0.602987\pi$
$101$	−645.416	−0.635855	−0.317927	+	0.948115i	$0.602987\pi$
$102$	0	0
$103$	−511.137	−0.488969	−0.244484	−	0.969653i	$-0.578619\pi$
$103$	−511.137	−0.488969	−0.244484	+	0.969653i	$0.578619\pi$
$104$	0	0
$105$	0	0
$106$	−336.657	−0.308482
$107$	−608.195	−0.549499	−0.274750	−	0.961516i	$-0.588595\pi$
$107$	−608.195	−0.549499	−0.274750	+	0.961516i	$0.588595\pi$
$108$	0	0
$109$	1300.04	1.14239	0.571197	−	0.820813i	$-0.306479\pi$
$109$	1300.04	1.14239	0.571197	+	0.820813i	$0.306479\pi$
$110$	−1245.24	−1.07935
$111$	0	0
$112$	−532.623	−0.449359
$113$	−42.1953	−0.0351274	−0.0175637	−	0.999846i	$-0.505591\pi$
$113$	−42.1953	−0.0351274	−0.0175637	+	0.999846i	$0.505591\pi$
$114$	0	0
$115$	−2062.58	−1.67249
$116$	198.716	0.159054
$117$	0	0
$118$	−3677.49	−2.86899
$119$	228.773	0.176232
$120$	0	0
$121$	−661.303	−0.496847
$122$	−787.378	−0.584311
$123$	0	0
$124$	−2588.61	−1.87471
$125$	1363.92	0.975939
$126$	0	0
$127$	−311.018	−0.217310	−0.108655	−	0.994080i	$-0.534654\pi$
$127$	−311.018	−0.217310	−0.108655	+	0.994080i	$0.534654\pi$
$128$	−903.291	−0.623753
$129$	0	0
$130$	0	0
$131$	−2000.98	−1.33456	−0.667278	−	0.744809i	$-0.732541\pi$
$131$	−2000.98	−1.33456	−0.667278	+	0.744809i	$0.732541\pi$
$132$	0	0
$133$	−1741.17	−1.13518
$134$	2564.17	1.65306
$135$	0	0
$136$	146.829	0.0925773
$137$	1038.53	0.647644	0.323822	−	0.946118i	$-0.395032\pi$
$137$	1038.53	0.647644	0.323822	+	0.946118i	$0.395032\pi$
$138$	0	0
$139$	−2858.46	−1.74426	−0.872128	−	0.489277i	$-0.837261\pi$
$139$	−2858.46	−1.74426	−0.872128	+	0.489277i	$0.837261\pi$
$140$	−1250.09	−0.754655
$141$	0	0
$142$	1044.00	0.616978
$143$	0	0
$144$	0	0
$145$	−233.814	−0.133911
$146$	−3589.33	−2.03462
$147$	0	0
$148$	−1126.91	−0.625887
$149$	743.479	0.408780	0.204390	−	0.978890i	$-0.434479\pi$
$149$	743.479	0.408780	0.204390	+	0.978890i	$0.434479\pi$
$150$	0	0
$151$	−2277.24	−1.22728	−0.613640	−	0.789586i	$-0.710295\pi$
$151$	−2277.24	−1.22728	−0.613640	+	0.789586i	$0.710295\pi$
$152$	−1117.51	−0.596328
$153$	0	0
$154$	1225.86	0.641447
$155$	3045.82	1.57836
$156$	0	0
$157$	3173.51	1.61321	0.806605	−	0.591091i	$-0.201303\pi$
$157$	3173.51	1.61321	0.806605	+	0.591091i	$0.201303\pi$
$158$	−1394.12	−0.701966
$159$	0	0
$160$	2937.89	1.45163
$161$	2030.48	0.993941
$162$	0	0
$163$	2314.65	1.11225	0.556126	−	0.831098i	$-0.312287\pi$
$163$	2314.65	1.11225	0.556126	+	0.831098i	$0.312287\pi$
$164$	3800.78	1.80970
$165$	0	0
$166$	−1832.76	−0.856925
$167$	−2665.65	−1.23517	−0.617587	−	0.786502i	$-0.711890\pi$
$167$	−2665.65	−1.23517	−0.617587	+	0.786502i	$0.711890\pi$
$168$	0	0
$169$	0	0
$170$	−978.138	−0.441293
$171$	0	0
$172$	1471.09	0.652148
$173$	165.243	0.0726198	0.0363099	−	0.999341i	$-0.488440\pi$
$173$	165.243	0.0726198	0.0363099	+	0.999341i	$0.488440\pi$
$174$	0	0
$175$	64.0954	0.0276866
$176$	−1224.73	−0.524532
$177$	0	0
$178$	1091.13	0.459457
$179$	−712.339	−0.297446	−0.148723	−	0.988879i	$-0.547516\pi$
$179$	−712.339	−0.297446	−0.148723	+	0.988879i	$0.547516\pi$
$180$	0	0
$181$	2206.53	0.906133	0.453066	−	0.891477i	$-0.350330\pi$
$181$	2206.53	0.906133	0.453066	+	0.891477i	$0.350330\pi$
$182$	0	0
$183$	0	0
$184$	1303.19	0.522132
$185$	1325.95	0.526949
$186$	0	0
$187$	526.048	0.205714
$188$	−4540.96	−1.76162
$189$	0	0
$190$	7444.54	2.84254
$191$	−1470.64	−0.557129	−0.278565	−	0.960417i	$-0.589859\pi$
$191$	−1470.64	−0.557129	−0.278565	+	0.960417i	$0.589859\pi$
$192$	0	0
$193$	−369.560	−0.137832	−0.0689158	−	0.997622i	$-0.521954\pi$
$193$	−369.560	−0.137832	−0.0689158	+	0.997622i	$0.521954\pi$
$194$	−5156.80	−1.90844
$195$	0	0
$196$	−2101.99	−0.766031
$197$	−4273.41	−1.54552	−0.772761	−	0.634697i	$-0.781125\pi$
$197$	−4273.41	−1.54552	−0.772761	+	0.634697i	$0.781125\pi$
$198$	0	0
$199$	4154.31	1.47985	0.739927	−	0.672687i	$-0.234860\pi$
$199$	4154.31	1.47985	0.739927	+	0.672687i	$0.234860\pi$
$200$	41.1373	0.0145442
$201$	0	0
$202$	−2716.59	−0.946230
$203$	230.175	0.0795819
$204$	0	0
$205$	−4472.09	−1.52363
$206$	−2151.40	−0.727646
$207$	0	0
$208$	0	0
$209$	−4003.71	−1.32508
$210$	0	0
$211$	1231.59	0.401830	0.200915	−	0.979609i	$-0.435608\pi$
$211$	1231.59	0.401830	0.200915	+	0.979609i	$0.435608\pi$
$212$	−777.134	−0.251763
$213$	0	0
$214$	−2559.92	−0.817723
$215$	−1730.92	−0.549059
$216$	0	0
$217$	−2998.42	−0.938000
$218$	5471.92	1.70002
$219$	0	0
$220$	−2874.49	−0.880901
$221$	0	0
$222$	0	0
$223$	2187.24	0.656809	0.328404	−	0.944537i	$-0.393489\pi$
$223$	2187.24	0.656809	0.328404	+	0.944537i	$0.393489\pi$
$224$	−2892.17	−0.862684
$225$	0	0
$226$	−177.602	−0.0522739
$227$	4138.67	1.21010	0.605051	−	0.796187i	$-0.293153\pi$
$227$	4138.67	1.21010	0.605051	+	0.796187i	$0.293153\pi$
$228$	0	0
$229$	835.354	0.241056	0.120528	−	0.992710i	$-0.461541\pi$
$229$	835.354	0.241056	0.120528	+	0.992710i	$0.461541\pi$
$230$	−8681.50	−2.48887
$231$	0	0
$232$	147.729	0.0418056
$233$	−3685.51	−1.03625	−0.518124	−	0.855305i	$-0.673370\pi$
$233$	−3685.51	−1.03625	−0.518124	+	0.855305i	$0.673370\pi$
$234$	0	0
$235$	5343.01	1.48315
$236$	−8489.05	−2.34148
$237$	0	0
$238$	962.917	0.262255
$239$	3026.21	0.819034	0.409517	−	0.912303i	$-0.365697\pi$
$239$	3026.21	0.819034	0.409517	+	0.912303i	$0.365697\pi$
$240$	0	0
$241$	−3265.58	−0.872839	−0.436420	−	0.899743i	$-0.643754\pi$
$241$	−3265.58	−0.872839	−0.436420	+	0.899743i	$0.643754\pi$
$242$	−2783.46	−0.739370
$243$	0	0
$244$	−1817.57	−0.476877
$245$	2473.25	0.644940
$246$	0	0
$247$	0	0
$248$	−1924.42	−0.492746
$249$	0	0
$250$	5740.79	1.45232
$251$	6363.16	1.60016	0.800078	−	0.599897i	$-0.204792\pi$
$251$	6363.16	1.60016	0.800078	+	0.599897i	$0.204792\pi$
$252$	0	0
$253$	4668.96	1.16022
$254$	−1309.09	−0.323385
$255$	0	0
$256$	1822.38	0.444917
$257$	6085.36	1.47702	0.738511	−	0.674242i	$-0.235529\pi$
$257$	6085.36	1.47702	0.738511	+	0.674242i	$0.235529\pi$
$258$	0	0
$259$	−1305.31	−0.313159
$260$	0	0
$261$	0	0
$262$	−8422.24	−1.98598
$263$	−123.227	−0.0288916	−0.0144458	−	0.999896i	$-0.504598\pi$
$263$	−123.227	−0.0288916	−0.0144458	+	0.999896i	$0.504598\pi$
$264$	0	0
$265$	914.395	0.211965
$266$	−7328.69	−1.68929
$267$	0	0
$268$	5919.08	1.34913
$269$	1935.79	0.438763	0.219381	−	0.975639i	$-0.429596\pi$
$269$	1935.79	0.438763	0.219381	+	0.975639i	$0.429596\pi$
$270$	0	0
$271$	4612.69	1.03395	0.516976	−	0.856000i	$-0.327058\pi$
$271$	4612.69	1.03395	0.516976	+	0.856000i	$0.327058\pi$
$272$	−962.028	−0.214454
$273$	0	0
$274$	4371.20	0.963774
$275$	147.383	0.0323183
$276$	0	0
$277$	−5834.30	−1.26552	−0.632761	−	0.774347i	$-0.718078\pi$
$277$	−5834.30	−1.26552	−0.632761	+	0.774347i	$0.718078\pi$
$278$	−12031.4	−2.59567
$279$	0	0
$280$	−929.341	−0.198353
$281$	4691.91	0.996071	0.498036	−	0.867157i	$-0.334055\pi$
$281$	4691.91	0.996071	0.498036	+	0.867157i	$0.334055\pi$
$282$	0	0
$283$	3465.60	0.727945	0.363973	−	0.931410i	$-0.381420\pi$
$283$	3465.60	0.727945	0.363973	+	0.931410i	$0.381420\pi$
$284$	2409.96	0.503539
$285$	0	0
$286$	0	0
$287$	4402.50	0.905475
$288$	0	0
$289$	−4499.79	−0.915894
$290$	−984.133	−0.199277
$291$	0	0
$292$	−8285.55	−1.66053
$293$	2677.31	0.533822	0.266911	−	0.963721i	$-0.413997\pi$
$293$	2677.31	0.533822	0.266911	+	0.963721i	$0.413997\pi$
$294$	0	0
$295$	9988.43	1.97135
$296$	−837.767	−0.164507
$297$	0	0
$298$	3129.34	0.608315
$299$	0	0
$300$	0	0
$301$	1703.98	0.326299
$302$	−9585.02	−1.82634
$303$	0	0
$304$	7321.93	1.38139
$305$	2138.60	0.401494
$306$	0	0
$307$	−471.915	−0.0877316	−0.0438658	−	0.999037i	$-0.513967\pi$
$307$	−471.915	−0.0877316	−0.0438658	+	0.999037i	$0.513967\pi$
$308$	2829.76	0.523508
$309$	0	0
$310$	12820.0	2.34880
$311$	1518.52	0.276872	0.138436	−	0.990371i	$-0.455793\pi$
$311$	1518.52	0.276872	0.138436	+	0.990371i	$0.455793\pi$
$312$	0	0
$313$	4049.86	0.731348	0.365674	−	0.930743i	$-0.380839\pi$
$313$	4049.86	0.731348	0.365674	+	0.930743i	$0.380839\pi$
$314$	13357.5	2.40066
$315$	0	0
$316$	−3218.17	−0.572900
$317$	3253.96	0.576532	0.288266	−	0.957550i	$-0.406921\pi$
$317$	3253.96	0.576532	0.288266	+	0.957550i	$0.406921\pi$
$318$	0	0
$319$	529.272	0.0928951
$320$	8037.37	1.40407
$321$	0	0
$322$	8546.40	1.47911
$323$	−3144.92	−0.541759
$324$	0	0
$325$	0	0
$326$	9742.46	1.65517
$327$	0	0
$328$	2825.58	0.475660
$329$	−5259.86	−0.881415
$330$	0	0
$331$	3422.45	0.568322	0.284161	−	0.958777i	$-0.408285\pi$
$331$	3422.45	0.568322	0.284161	+	0.958777i	$0.408285\pi$
$332$	−4230.71	−0.699368
$333$	0	0
$334$	−11219.8	−1.83809
$335$	−6964.54	−1.13586
$336$	0	0
$337$	−9301.67	−1.50354	−0.751772	−	0.659423i	$-0.770801\pi$
$337$	−9301.67	−1.50354	−0.751772	+	0.659423i	$0.770801\pi$
$338$	0	0
$339$	0	0
$340$	−2257.92	−0.360155
$341$	−6894.66	−1.09492
$342$	0	0
$343$	−6294.99	−0.990955
$344$	1093.64	0.171410
$345$	0	0
$346$	695.518	0.108067
$347$	−216.898	−0.0335554	−0.0167777	−	0.999859i	$-0.505341\pi$
$347$	−216.898	−0.0335554	−0.0167777	+	0.999859i	$0.505341\pi$
$348$	0	0
$349$	4809.84	0.737721	0.368861	−	0.929485i	$-0.379748\pi$
$349$	4809.84	0.737721	0.368861	+	0.929485i	$0.379748\pi$
$350$	269.781	0.0412011
$351$	0	0
$352$	−6650.35	−1.00700
$353$	−2859.64	−0.431170	−0.215585	−	0.976485i	$-0.569166\pi$
$353$	−2859.64	−0.431170	−0.215585	+	0.976485i	$0.569166\pi$
$354$	0	0
$355$	−2835.62	−0.423941
$356$	2518.74	0.374980
$357$	0	0
$358$	−2998.27	−0.442636
$359$	3686.04	0.541899	0.270949	−	0.962594i	$-0.412662\pi$
$359$	3686.04	0.541899	0.270949	+	0.962594i	$0.412662\pi$
$360$	0	0
$361$	17076.8	2.48969
$362$	9287.39	1.34844
$363$	0	0
$364$	0	0
$365$	9748.98	1.39804
$366$	0	0
$367$	−3470.59	−0.493633	−0.246816	−	0.969062i	$-0.579384\pi$
$367$	−3470.59	−0.493633	−0.246816	+	0.969062i	$0.579384\pi$
$368$	−8538.52	−1.20951
$369$	0	0
$370$	5580.98	0.784166
$371$	−900.166	−0.125968
$372$	0	0
$373$	−11963.4	−1.66070	−0.830352	−	0.557240i	$-0.811860\pi$
$373$	−11963.4	−1.66070	−0.830352	+	0.557240i	$0.811860\pi$
$374$	2214.16	0.306127
$375$	0	0
$376$	−3375.85	−0.463021
$377$	0	0
$378$	0	0
$379$	−345.604	−0.0468403	−0.0234202	−	0.999726i	$-0.507456\pi$
$379$	−345.604	−0.0468403	−0.0234202	+	0.999726i	$0.507456\pi$
$380$	17184.8	2.31990
$381$	0	0
$382$	−6189.99	−0.829078
$383$	−3386.40	−0.451793	−0.225897	−	0.974151i	$-0.572531\pi$
$383$	−3386.40	−0.451793	−0.225897	+	0.974151i	$0.572531\pi$
$384$	0	0
$385$	−3329.56	−0.440754
$386$	−1555.49	−0.205110
$387$	0	0
$388$	−11903.9	−1.55755
$389$	1629.88	0.212438	0.106219	−	0.994343i	$-0.466126\pi$
$389$	1629.88	0.212438	0.106219	+	0.994343i	$0.466126\pi$
$390$	0	0
$391$	3667.47	0.474353
$392$	−1562.66	−0.201343
$393$	0	0
$394$	−17987.0	−2.29993
$395$	3786.58	0.482338
$396$	0	0
$397$	−7938.94	−1.00364	−0.501819	−	0.864973i	$-0.667336\pi$
$397$	−7938.94	−1.00364	−0.501819	+	0.864973i	$0.667336\pi$
$398$	17485.7	2.20221
$399$	0	0
$400$	−269.532	−0.0336915
$401$	214.402	0.0267001	0.0133500	−	0.999911i	$-0.495750\pi$
$401$	214.402	0.0267001	0.0133500	+	0.999911i	$0.495750\pi$
$402$	0	0
$403$	0	0
$404$	−6270.93	−0.772253
$405$	0	0
$406$	968.819	0.118428
$407$	−3001.48	−0.365548
$408$	0	0
$409$	4783.73	0.578338	0.289169	−	0.957278i	$-0.406621\pi$
$409$	4783.73	0.578338	0.289169	+	0.957278i	$0.406621\pi$
$410$	−18823.2	−2.26735
$411$	0	0
$412$	−4966.25	−0.593859
$413$	−9832.99	−1.17155
$414$	0	0
$415$	4977.95	0.588815
$416$	0	0
$417$	0	0
$418$	−16851.8	−1.97189
$419$	9903.67	1.15472	0.577358	−	0.816491i	$-0.304084\pi$
$419$	9903.67	1.15472	0.577358	+	0.816491i	$0.304084\pi$
$420$	0	0
$421$	12120.6	1.40314	0.701572	−	0.712598i	$-0.252482\pi$
$421$	12120.6	1.40314	0.701572	+	0.712598i	$0.252482\pi$
$422$	5183.82	0.597973
$423$	0	0
$424$	−577.738	−0.0661732
$425$	115.770	0.0132133
$426$	0	0
$427$	−2105.32	−0.238603
$428$	−5909.28	−0.667374
$429$	0	0
$430$	−7285.53	−0.817068
$431$	−13672.6	−1.52805	−0.764023	−	0.645189i	$-0.776779\pi$
$431$	−13672.6	−1.52805	−0.764023	+	0.645189i	$0.776779\pi$
$432$	0	0
$433$	7113.10	0.789455	0.394727	−	0.918798i	$-0.370839\pi$
$433$	7113.10	0.789455	0.394727	+	0.918798i	$0.370839\pi$
$434$	−12620.5	−1.39586
$435$	0	0
$436$	12631.3	1.38745
$437$	−27912.9	−3.05550
$438$	0	0
$439$	−6022.04	−0.654707	−0.327353	−	0.944902i	$-0.606157\pi$
$439$	−6022.04	−0.654707	−0.327353	+	0.944902i	$0.606157\pi$
$440$	−2136.96	−0.231535
$441$	0	0
$442$	0	0
$443$	12994.4	1.39364	0.696821	−	0.717245i	$-0.254597\pi$
$443$	12994.4	1.39364	0.696821	+	0.717245i	$0.254597\pi$
$444$	0	0
$445$	−2963.61	−0.315704
$446$	9206.20	0.977413
$447$	0	0
$448$	−7912.30	−0.834422
$449$	10984.3	1.15452	0.577260	−	0.816560i	$-0.304122\pi$
$449$	10984.3	1.15452	0.577260	+	0.816560i	$0.304122\pi$
$450$	0	0
$451$	10123.2	1.05695
$452$	−409.973	−0.0426627
$453$	0	0
$454$	17419.9	1.80078
$455$	0	0
$456$	0	0
$457$	−9834.10	−1.00661	−0.503304	−	0.864109i	$-0.667882\pi$
$457$	−9834.10	−1.00661	−0.503304	+	0.864109i	$0.667882\pi$
$458$	3516.05	0.358721
$459$	0	0
$460$	−20040.2	−2.03126
$461$	3401.42	0.343644	0.171822	−	0.985128i	$-0.445035\pi$
$461$	3401.42	0.343644	0.171822	+	0.985128i	$0.445035\pi$
$462$	0	0
$463$	−1739.42	−0.174596	−0.0872979	−	0.996182i	$-0.527823\pi$
$463$	−1739.42	−0.174596	−0.0872979	+	0.996182i	$0.527823\pi$
$464$	−967.925	−0.0968422
$465$	0	0
$466$	−15512.5	−1.54207
$467$	7958.82	0.788630	0.394315	−	0.918975i	$-0.370982\pi$
$467$	7958.82	0.788630	0.394315	+	0.918975i	$0.370982\pi$
$468$	0	0
$469$	6856.16	0.675028
$470$	22489.0	2.20711
$471$	0	0
$472$	−6310.94	−0.615433
$473$	3918.20	0.380886
$474$	0	0
$475$	−881.114	−0.0851122
$476$	2222.78	0.214036
$477$	0	0
$478$	12737.5	1.21882
$479$	8431.98	0.804315	0.402158	−	0.915570i	$-0.368260\pi$
$479$	8431.98	0.804315	0.402158	+	0.915570i	$0.368260\pi$
$480$	0	0
$481$	0	0
$482$	−13745.0	−1.29889
$483$	0	0
$484$	−6425.29	−0.603427
$485$	14006.4	1.31133
$486$	0	0
$487$	11684.7	1.08723	0.543617	−	0.839334i	$-0.317055\pi$
$487$	11684.7	1.08723	0.543617	+	0.839334i	$0.317055\pi$
$488$	−1351.22	−0.125342
$489$	0	0
$490$	10410.0	0.959750
$491$	3954.70	0.363489	0.181745	−	0.983346i	$-0.441826\pi$
$491$	3954.70	0.363489	0.181745	+	0.983346i	$0.441826\pi$
$492$	0	0
$493$	415.744	0.0379801
$494$	0	0
$495$	0	0
$496$	12608.8	1.14144
$497$	2791.49	0.251943
$498$	0	0
$499$	5690.37	0.510493	0.255246	−	0.966876i	$-0.417843\pi$
$499$	5690.37	0.510493	0.255246	+	0.966876i	$0.417843\pi$
$500$	13251.9	1.18529
$501$	0	0
$502$	26782.8	2.38123
$503$	−10859.1	−0.962595	−0.481298	−	0.876557i	$-0.659834\pi$
$503$	−10859.1	−0.962595	−0.481298	+	0.876557i	$0.659834\pi$
$504$	0	0
$505$	7378.53	0.650178
$506$	19651.9	1.72655
$507$	0	0
$508$	−3021.88	−0.263926
$509$	−18558.6	−1.61610	−0.808049	−	0.589115i	$-0.799476\pi$
$509$	−18558.6	−1.61610	−0.808049	+	0.589115i	$0.799476\pi$
$510$	0	0
$511$	−9597.27	−0.830838
$512$	14896.8	1.28584
$513$	0	0
$514$	25613.6	2.19799
$515$	5843.42	0.499984
$516$	0	0
$517$	−12094.7	−1.02887
$518$	−5494.13	−0.466020
$519$	0	0
$520$	0	0
$521$	−17297.5	−1.45454	−0.727271	−	0.686350i	$-0.759212\pi$
$521$	−17297.5	−1.45454	−0.727271	+	0.686350i	$0.759212\pi$
$522$	0	0
$523$	−5016.11	−0.419386	−0.209693	−	0.977767i	$-0.567247\pi$
$523$	−5016.11	−0.419386	−0.209693	+	0.977767i	$0.567247\pi$
$524$	−19441.8	−1.62084
$525$	0	0
$526$	−518.667	−0.0429942
$527$	−5415.77	−0.447656
$528$	0	0
$529$	20383.8	1.67533
$530$	3848.73	0.315431
$531$	0	0
$532$	−16917.4	−1.37869
$533$	0	0
$534$	0	0
$535$	6953.01	0.561878
$536$	4400.37	0.354603
$537$	0	0
$538$	8147.83	0.652933
$539$	−5598.57	−0.447398
$540$	0	0
$541$	−17642.3	−1.40204	−0.701018	−	0.713144i	$-0.747271\pi$
$541$	−17642.3	−1.40204	−0.701018	+	0.713144i	$0.747271\pi$
$542$	19415.0	1.53865
$543$	0	0
$544$	−5223.86	−0.411712
$545$	−14862.3	−1.16813
$546$	0	0
$547$	−18414.9	−1.43943	−0.719713	−	0.694271i	$-0.755727\pi$
$547$	−18414.9	−1.43943	−0.719713	+	0.694271i	$0.755727\pi$
$548$	10090.4	0.786571
$549$	0	0
$550$	620.343	0.0480937
$551$	−3164.20	−0.244645
$552$	0	0
$553$	−3727.66	−0.286648
$554$	−24556.9	−1.88325
$555$	0	0
$556$	−27773.1	−2.11842
$557$	8179.15	0.622193	0.311096	−	0.950378i	$-0.399304\pi$
$557$	8179.15	0.622193	0.311096	+	0.950378i	$0.399304\pi$
$558$	0	0
$559$	0	0
$560$	6089.05	0.459481
$561$	0	0
$562$	19748.5	1.48228
$563$	1880.07	0.140738	0.0703690	−	0.997521i	$-0.477582\pi$
$563$	1880.07	0.140738	0.0703690	+	0.997521i	$0.477582\pi$
$564$	0	0
$565$	482.385	0.0359187
$566$	14586.9	1.08327
$567$	0	0
$568$	1791.62	0.132350
$569$	−10118.3	−0.745485	−0.372743	−	0.927935i	$-0.621583\pi$
$569$	−10118.3	−0.745485	−0.372743	+	0.927935i	$0.621583\pi$
$570$	0	0
$571$	23428.9	1.71711	0.858555	−	0.512721i	$-0.171362\pi$
$571$	23428.9	1.71711	0.858555	+	0.512721i	$0.171362\pi$
$572$	0	0
$573$	0	0
$574$	18530.3	1.34746
$575$	1027.52	0.0745225
$576$	0	0
$577$	−20508.1	−1.47966	−0.739831	−	0.672793i	$-0.765094\pi$
$577$	−20508.1	−1.47966	−0.739831	+	0.672793i	$0.765094\pi$
$578$	−18939.8	−1.36296
$579$	0	0
$580$	−2271.76	−0.162637
$581$	−4900.49	−0.349925
$582$	0	0
$583$	−2069.87	−0.147042
$584$	−6159.65	−0.436452
$585$	0	0
$586$	11268.9	0.794394
$587$	−5968.43	−0.419665	−0.209833	−	0.977737i	$-0.567292\pi$
$587$	−5968.43	−0.419665	−0.209833	+	0.977737i	$0.567292\pi$
$588$	0	0
$589$	41219.0	2.88353
$590$	42041.8	2.93361
$591$	0	0
$592$	5489.06	0.381080
$593$	−14659.5	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$593$	−14659.5	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$594$	0	0
$595$	−2615.38	−0.180202
$596$	7223.72	0.496468
$597$	0	0
$598$	0	0
$599$	−23635.9	−1.61225	−0.806125	−	0.591746i	$-0.798439\pi$
$599$	−23635.9	−1.61225	−0.806125	+	0.591746i	$0.798439\pi$
$600$	0	0
$601$	−11527.0	−0.782356	−0.391178	−	0.920315i	$-0.627932\pi$
$601$	−11527.0	−0.782356	−0.391178	+	0.920315i	$0.627932\pi$
$602$	7172.15	0.485573
$603$	0	0
$604$	−22125.9	−1.49055
$605$	7560.15	0.508039
$606$	0	0
$607$	−5098.56	−0.340930	−0.170465	−	0.985364i	$-0.554527\pi$
$607$	−5098.56	−0.340930	−0.170465	+	0.985364i	$0.554527\pi$
$608$	39758.4	2.65200
$609$	0	0
$610$	9001.47	0.597473
$611$	0	0
$612$	0	0
$613$	−1516.39	−0.0999128	−0.0499564	−	0.998751i	$-0.515908\pi$
$613$	−1516.39	−0.0999128	−0.0499564	+	0.998751i	$0.515908\pi$
$614$	−1986.31	−0.130556
$615$	0	0
$616$	2103.70	0.137598
$617$	18539.3	1.20966	0.604832	−	0.796353i	$-0.293240\pi$
$617$	18539.3	1.20966	0.604832	+	0.796353i	$0.293240\pi$
$618$	0	0
$619$	−25684.9	−1.66779	−0.833897	−	0.551920i	$-0.813895\pi$
$619$	−25684.9	−1.66779	−0.833897	+	0.551920i	$0.813895\pi$
$620$	29593.5	1.91694
$621$	0	0
$622$	6391.51	0.412020
$623$	2917.49	0.187619
$624$	0	0
$625$	−16304.5	−1.04349
$626$	17046.1	1.08834
$627$	0	0
$628$	30834.2	1.95926
$629$	−2357.67	−0.149454
$630$	0	0
$631$	22410.9	1.41389	0.706945	−	0.707269i	$-0.250073\pi$
$631$	22410.9	1.41389	0.706945	+	0.707269i	$0.250073\pi$
$632$	−2392.46	−0.150580
$633$	0	0
$634$	13696.1	0.857950
$635$	3555.62	0.222206
$636$	0	0
$637$	0	0
$638$	2227.73	0.138239
$639$	0	0
$640$	10326.6	0.637804
$641$	−6827.81	−0.420721	−0.210361	−	0.977624i	$-0.567464\pi$
$641$	−6827.81	−0.420721	−0.210361	+	0.977624i	$0.567464\pi$
$642$	0	0
$643$	23264.3	1.42684	0.713418	−	0.700738i	$-0.247146\pi$
$643$	23264.3	1.42684	0.713418	+	0.700738i	$0.247146\pi$
$644$	19728.4	1.20715
$645$	0	0
$646$	−13237.1	−0.806204
$647$	−14745.9	−0.896014	−0.448007	−	0.894030i	$-0.647866\pi$
$647$	−14745.9	−0.896014	−0.448007	+	0.894030i	$0.647866\pi$
$648$	0	0
$649$	−22610.3	−1.36754
$650$	0	0
$651$	0	0
$652$	22489.3	1.35084
$653$	−10909.0	−0.653755	−0.326878	−	0.945067i	$-0.605997\pi$
$653$	−10909.0	−0.653755	−0.326878	+	0.945067i	$0.605997\pi$
$654$	0	0
$655$	22875.7	1.36462
$656$	−18513.2	−1.10186
$657$	0	0
$658$	−22139.0	−1.31166
$659$	4182.99	0.247263	0.123631	−	0.992328i	$-0.460546\pi$
$659$	4182.99	0.247263	0.123631	+	0.992328i	$0.460546\pi$
$660$	0	0
$661$	−2224.23	−0.130881	−0.0654406	−	0.997856i	$-0.520845\pi$
$661$	−2224.23	−0.130881	−0.0654406	+	0.997856i	$0.520845\pi$
$662$	14405.2	0.845734
$663$	0	0
$664$	−3145.19	−0.183821
$665$	19905.4	1.16075
$666$	0	0
$667$	3689.95	0.214206
$668$	−25899.7	−1.50013
$669$	0	0
$670$	−29314.1	−1.69030
$671$	−4841.04	−0.278519
$672$	0	0
$673$	−24152.5	−1.38337	−0.691687	−	0.722197i	$-0.743132\pi$
$673$	−24152.5	−1.38337	−0.691687	+	0.722197i	$0.743132\pi$
$674$	−39151.2	−2.23746
$675$	0	0
$676$	0	0
$677$	15310.7	0.869187	0.434593	−	0.900627i	$-0.356892\pi$
$677$	15310.7	0.869187	0.434593	+	0.900627i	$0.356892\pi$
$678$	0	0
$679$	−13788.4	−0.779310
$680$	−1678.58	−0.0946628
$681$	0	0
$682$	−29020.0	−1.62937
$683$	11399.6	0.638646	0.319323	−	0.947646i	$-0.396545\pi$
$683$	11399.6	0.638646	0.319323	+	0.947646i	$0.396545\pi$
$684$	0	0
$685$	−11872.6	−0.662233
$686$	−26495.9	−1.47466
$687$	0	0
$688$	−7165.54	−0.397069
$689$	0	0
$690$	0	0
$691$	3323.23	0.182955	0.0914773	−	0.995807i	$-0.470841\pi$
$691$	3323.23	0.182955	0.0914773	+	0.995807i	$0.470841\pi$
$692$	1605.52	0.0881976
$693$	0	0
$694$	−912.936	−0.0499345
$695$	32678.5	1.78355
$696$	0	0
$697$	7951.82	0.432133
$698$	20244.8	1.09782
$699$	0	0
$700$	622.758	0.0336257
$701$	12670.4	0.682673	0.341336	−	0.939941i	$-0.389120\pi$
$701$	12670.4	0.682673	0.341336	+	0.939941i	$0.389120\pi$
$702$	0	0
$703$	17944.0	0.962692
$704$	−18193.8	−0.974012
$705$	0	0
$706$	−12036.4	−0.641635
$707$	−7263.71	−0.386393
$708$	0	0
$709$	−13075.2	−0.692594	−0.346297	−	0.938125i	$-0.612561\pi$
$709$	−13075.2	−0.692594	−0.346297	+	0.938125i	$0.612561\pi$
$710$	−11935.3	−0.630877
$711$	0	0
$712$	1872.48	0.0985592
$713$	−48067.8	−2.52476
$714$	0	0
$715$	0	0
$716$	−6921.16	−0.361251
$717$	0	0
$718$	15514.7	0.806412
$719$	2988.41	0.155005	0.0775026	−	0.996992i	$-0.475305\pi$
$719$	2988.41	0.155005	0.0775026	+	0.996992i	$0.475305\pi$
$720$	0	0
$721$	−5752.48	−0.297134
$722$	71877.0	3.70496
$723$	0	0
$724$	21438.9	1.10051
$725$	116.479	0.00596680
$726$	0	0
$727$	−5507.46	−0.280963	−0.140482	−	0.990083i	$-0.544865\pi$
$727$	−5507.46	−0.280963	−0.140482	+	0.990083i	$0.544865\pi$
$728$	0	0
$729$	0	0
$730$	41033.9	2.08046
$731$	3077.75	0.155725
$732$	0	0
$733$	36585.2	1.84353	0.921764	−	0.387751i	$-0.126748\pi$
$733$	36585.2	1.84353	0.921764	+	0.387751i	$0.126748\pi$
$734$	−14607.9	−0.734587
$735$	0	0
$736$	−46364.6	−2.32204
$737$	15765.3	0.787953
$738$	0	0
$739$	−6425.89	−0.319865	−0.159933	−	0.987128i	$-0.551128\pi$
$739$	−6425.89	−0.319865	−0.159933	+	0.987128i	$0.551128\pi$
$740$	12883.0	0.639986
$741$	0	0
$742$	−3788.84	−0.187457
$743$	20411.0	1.00782	0.503908	−	0.863757i	$-0.331895\pi$
$743$	20411.0	1.00782	0.503908	+	0.863757i	$0.331895\pi$
$744$	0	0
$745$	−8499.60	−0.417988
$746$	−50354.6	−2.47133
$747$	0	0
$748$	5111.13	0.249842
$749$	−6844.81	−0.333917
$750$	0	0
$751$	−24259.5	−1.17875	−0.589375	−	0.807860i	$-0.700626\pi$
$751$	−24259.5	−1.17875	−0.589375	+	0.807860i	$0.700626\pi$
$752$	22118.6	1.07258
$753$	0	0
$754$	0	0
$755$	26033.9	1.25493
$756$	0	0
$757$	9295.39	0.446297	0.223148	−	0.974785i	$-0.428367\pi$
$757$	9295.39	0.446297	0.223148	+	0.974785i	$0.428367\pi$
$758$	−1454.66	−0.0697042
$759$	0	0
$760$	12775.6	0.609761
$761$	−21974.7	−1.04676	−0.523378	−	0.852101i	$-0.675328\pi$
$761$	−21974.7	−1.04676	−0.523378	+	0.852101i	$0.675328\pi$
$762$	0	0
$763$	14631.0	0.694204
$764$	−14288.9	−0.676641
$765$	0	0
$766$	−14253.5	−0.672324
$767$	0	0
$768$	0	0
$769$	−22987.4	−1.07795	−0.538977	−	0.842320i	$-0.681189\pi$
$769$	−22987.4	−1.07795	−0.538977	+	0.842320i	$0.681189\pi$
$770$	−14014.3	−0.655897
$771$	0	0
$772$	−3590.68	−0.167398
$773$	31970.9	1.48760	0.743799	−	0.668404i	$-0.233022\pi$
$773$	31970.9	1.48760	0.743799	+	0.668404i	$0.233022\pi$
$774$	0	0
$775$	−1517.34	−0.0703283
$776$	−8849.59	−0.409384
$777$	0	0
$778$	6860.25	0.316133
$779$	−60520.8	−2.78354
$780$	0	0
$781$	6418.85	0.294090
$782$	15436.6	0.705896
$783$	0	0
$784$	10238.6	0.466408
$785$	−36280.2	−1.64955
$786$	0	0
$787$	−6087.26	−0.275715	−0.137857	−	0.990452i	$-0.544022\pi$
$787$	−6087.26	−0.275715	−0.137857	+	0.990452i	$0.544022\pi$
$788$	−41520.9	−1.87706
$789$	0	0
$790$	15937.9	0.717779
$791$	−474.878	−0.0213460
$792$	0	0
$793$	0	0
$794$	−33415.4	−1.49354
$795$	0	0
$796$	40363.6	1.79730
$797$	−23080.0	−1.02577	−0.512883	−	0.858458i	$-0.671423\pi$
$797$	−23080.0	−1.02577	−0.512883	+	0.858458i	$0.671423\pi$
$798$	0	0
$799$	−9500.41	−0.420651
$800$	−1463.57	−0.0646813
$801$	0	0
$802$	902.429	0.0397330
$803$	−22068.3	−0.969829
$804$	0	0
$805$	−23212.9	−1.01633
$806$	0	0
$807$	0	0
$808$	−4661.94	−0.202978
$809$	32377.8	1.40710	0.703550	−	0.710646i	$-0.251597\pi$
$809$	32377.8	1.40710	0.703550	+	0.710646i	$0.251597\pi$
$810$	0	0
$811$	−26352.8	−1.14103	−0.570513	−	0.821288i	$-0.693256\pi$
$811$	−26352.8	−1.14103	−0.570513	+	0.821288i	$0.693256\pi$
$812$	2236.40	0.0966532
$813$	0	0
$814$	−12633.4	−0.543980
$815$	−26461.5	−1.13731
$816$	0	0
$817$	−23424.5	−1.00308
$818$	20135.0	0.860638
$819$	0	0
$820$	−43451.3	−1.85047
$821$	35355.3	1.50294	0.751468	−	0.659770i	$-0.229346\pi$
$821$	35355.3	1.50294	0.751468	+	0.659770i	$0.229346\pi$
$822$	0	0
$823$	−12663.3	−0.536347	−0.268173	−	0.963371i	$-0.586420\pi$
$823$	−12663.3	−0.536347	−0.268173	+	0.963371i	$0.586420\pi$
$824$	−3692.02	−0.156089
$825$	0	0
$826$	−41387.6	−1.74341
$827$	−16295.2	−0.685176	−0.342588	−	0.939486i	$-0.611303\pi$
$827$	−16295.2	−0.685176	−0.342588	+	0.939486i	$0.611303\pi$
$828$	0	0
$829$	13638.9	0.571411	0.285705	−	0.958318i	$-0.407772\pi$
$829$	13638.9	0.571411	0.285705	+	0.958318i	$0.407772\pi$
$830$	20952.4	0.876229
$831$	0	0
$832$	0	0
$833$	−4397.69	−0.182918
$834$	0	0
$835$	30474.2	1.26300
$836$	−38900.5	−1.60933
$837$	0	0
$838$	41685.0	1.71836
$839$	−1890.31	−0.0777838	−0.0388919	−	0.999243i	$-0.512383\pi$
$839$	−1890.31	−0.0777838	−0.0388919	+	0.999243i	$0.512383\pi$
$840$	0	0
$841$	−23970.7	−0.982849
$842$	51016.4	2.08805
$843$	0	0
$844$	11966.3	0.488028
$845$	0	0
$846$	0	0
$847$	−7442.50	−0.301921
$848$	3785.35	0.153289
$849$	0	0
$850$	487.280	0.0196630
$851$	−20925.6	−0.842914
$852$	0	0
$853$	−1620.21	−0.0650351	−0.0325175	−	0.999471i	$-0.510352\pi$
$853$	−1620.21	−0.0650351	−0.0325175	+	0.999471i	$0.510352\pi$
$854$	−8861.39	−0.355071
$855$	0	0
$856$	−4393.08	−0.175412
$857$	14508.4	0.578292	0.289146	−	0.957285i	$-0.406629\pi$
$857$	14508.4	0.578292	0.289146	+	0.957285i	$0.406629\pi$
$858$	0	0
$859$	29639.8	1.17730	0.588648	−	0.808389i	$-0.299660\pi$
$859$	29639.8	1.17730	0.588648	+	0.808389i	$0.299660\pi$
$860$	−16817.8	−0.666839
$861$	0	0
$862$	−57548.8	−2.27392
$863$	21528.8	0.849186	0.424593	−	0.905384i	$-0.360417\pi$
$863$	21528.8	0.849186	0.424593	+	0.905384i	$0.360417\pi$
$864$	0	0
$865$	−1889.10	−0.0742557
$866$	29939.4	1.17481
$867$	0	0
$868$	−29132.9	−1.13921
$869$	−8571.50	−0.334601
$870$	0	0
$871$	0	0
$872$	9390.36	0.364676
$873$	0	0
$874$	−117487.	−4.54696
$875$	15349.9	0.593054
$876$	0	0
$877$	−14865.3	−0.572366	−0.286183	−	0.958175i	$-0.592387\pi$
$877$	−14865.3	−0.572366	−0.286183	+	0.958175i	$0.592387\pi$
$878$	−25347.1	−0.974285
$879$	0	0
$880$	14001.4	0.536348
$881$	21336.0	0.815921	0.407961	−	0.913000i	$-0.366240\pi$
$881$	21336.0	0.815921	0.407961	+	0.913000i	$0.366240\pi$
$882$	0	0
$883$	37538.2	1.43065	0.715323	−	0.698794i	$-0.246280\pi$
$883$	37538.2	1.43065	0.715323	+	0.698794i	$0.246280\pi$
$884$	0	0
$885$	0	0
$886$	54694.1	2.07391
$887$	−34575.0	−1.30881	−0.654406	−	0.756144i	$-0.727081\pi$
$887$	−34575.0	−1.30881	−0.654406	+	0.756144i	$0.727081\pi$
$888$	0	0
$889$	−3500.29	−0.132054
$890$	−12474.0	−0.469807
$891$	0	0
$892$	21251.4	0.797703
$893$	72306.9	2.70958
$894$	0	0
$895$	8143.61	0.304146
$896$	−10165.9	−0.379039
$897$	0	0
$898$	46233.3	1.71807
$899$	−5448.96	−0.202150
$900$	0	0
$901$	−1625.89	−0.0601178
$902$	42609.2	1.57287
$903$	0	0
$904$	−304.783	−0.0112134
$905$	−25225.5	−0.926545
$906$	0	0
$907$	−10424.8	−0.381641	−0.190820	−	0.981625i	$-0.561115\pi$
$907$	−10424.8	−0.381641	−0.190820	+	0.981625i	$0.561115\pi$
$908$	40211.7	1.46968
$909$	0	0
$910$	0	0
$911$	−10961.8	−0.398661	−0.199331	−	0.979932i	$-0.563877\pi$
$911$	−10961.8	−0.398661	−0.199331	+	0.979932i	$0.563877\pi$
$912$	0	0
$913$	−11268.3	−0.408464
$914$	−41392.2	−1.49796
$915$	0	0
$916$	8116.38	0.292765
$917$	−22519.7	−0.810976
$918$	0	0
$919$	−10779.2	−0.386914	−0.193457	−	0.981109i	$-0.561970\pi$
$919$	−10779.2	−0.386914	−0.193457	+	0.981109i	$0.561970\pi$
$920$	−14898.3	−0.533895
$921$	0	0
$922$	14316.7	0.511384
$923$	0	0
$924$	0	0
$925$	−660.549	−0.0234797
$926$	−7321.32	−0.259820
$927$	0	0
$928$	−5255.88	−0.185919
$929$	5429.07	0.191735	0.0958675	−	0.995394i	$-0.469437\pi$
$929$	5429.07	0.191735	0.0958675	+	0.995394i	$0.469437\pi$
$930$	0	0
$931$	33470.5	1.17825
$932$	−35808.8	−1.25854
$933$	0	0
$934$	33499.1	1.17358
$935$	−6013.88	−0.210348
$936$	0	0
$937$	−21300.1	−0.742631	−0.371315	−	0.928507i	$-0.621093\pi$
$937$	−21300.1	−0.742631	−0.371315	+	0.928507i	$0.621093\pi$
$938$	28857.9	1.00453
$939$	0	0
$940$	51913.2	1.80130
$941$	26851.2	0.930207	0.465103	−	0.885256i	$-0.346017\pi$
$941$	26851.2	0.930207	0.465103	+	0.885256i	$0.346017\pi$
$942$	0	0
$943$	70576.7	2.43722
$944$	41349.4	1.42564
$945$	0	0
$946$	16491.9	0.566805
$947$	−8021.68	−0.275258	−0.137629	−	0.990484i	$-0.543948\pi$
$947$	−8021.68	−0.275258	−0.137629	+	0.990484i	$0.543948\pi$
$948$	0	0
$949$	0	0
$950$	−3708.65	−0.126658
$951$	0	0
$952$	1652.46	0.0562569
$953$	−35715.0	−1.21398	−0.606990	−	0.794709i	$-0.707623\pi$
$953$	−35715.0	−1.21398	−0.606990	+	0.794709i	$0.707623\pi$
$954$	0	0
$955$	16812.6	0.569680
$956$	29402.9	0.994726
$957$	0	0
$958$	35490.6	1.19692
$959$	11687.9	0.393557
$960$	0	0
$961$	41190.9	1.38266
$962$	0	0
$963$	0	0
$964$	−31728.7	−1.06007
$965$	4224.88	0.140936
$966$	0	0
$967$	−53338.8	−1.77380	−0.886898	−	0.461965i	$-0.847145\pi$
$967$	−53338.8	−1.77380	−0.886898	+	0.461965i	$0.847145\pi$
$968$	−4776.69	−0.158604
$969$	0	0
$970$	58953.6	1.95143
$971$	−23112.9	−0.763882	−0.381941	−	0.924187i	$-0.624744\pi$
$971$	−23112.9	−0.763882	−0.381941	+	0.924187i	$0.624744\pi$
$972$	0	0
$973$	−32170.0	−1.05994
$974$	49181.3	1.61794
$975$	0	0
$976$	8853.22	0.290353
$977$	−52874.6	−1.73143	−0.865715	−	0.500538i	$-0.833136\pi$
$977$	−52874.6	−1.73143	−0.865715	+	0.500538i	$0.833136\pi$
$978$	0	0
$979$	6708.57	0.219006
$980$	24030.4	0.783287
$981$	0	0
$982$	16645.5	0.540917
$983$	45173.1	1.46572	0.732858	−	0.680381i	$-0.238186\pi$
$983$	45173.1	1.46572	0.732858	+	0.680381i	$0.238186\pi$
$984$	0	0
$985$	48854.5	1.58034
$986$	1749.89	0.0565190
$987$	0	0
$988$	0	0
$989$	27316.7	0.878281
$990$	0	0
$991$	60485.6	1.93884	0.969418	−	0.245414i	$-0.0789239\pi$
$991$	60485.6	1.93884	0.969418	+	0.245414i	$0.0789239\pi$
$992$	68466.7	2.19135
$993$	0	0
$994$	11749.5	0.374922
$995$	−47492.8	−1.51319
$996$	0	0
$997$	−18108.1	−0.575214	−0.287607	−	0.957749i	$-0.592860\pi$
$997$	−18108.1	−0.575214	−0.287607	+	0.957749i	$0.592860\pi$
$998$	23951.1	0.759677
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.u.1.3		3
3.2	odd	2		507.4.a.h.1.1		3
13.12	even	2		117.4.a.f.1.1		3
39.5	even	4		507.4.b.g.337.6		6
39.8	even	4		507.4.b.g.337.1		6
39.38	odd	2		39.4.a.c.1.3	✓	3
52.51	odd	2		1872.4.a.bk.1.3		3
156.155	even	2		624.4.a.t.1.1		3
195.194	odd	2		975.4.a.l.1.1		3
273.272	even	2		1911.4.a.k.1.3		3
312.77	odd	2		2496.4.a.bl.1.3		3
312.155	even	2		2496.4.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.3	✓	3	39.38	odd	2
117.4.a.f.1.1		3	13.12	even	2
507.4.a.h.1.1		3	3.2	odd	2
507.4.b.g.337.1		6	39.8	even	4
507.4.b.g.337.6		6	39.5	even	4
624.4.a.t.1.1		3	156.155	even	2
975.4.a.l.1.1		3	195.194	odd	2
1521.4.a.u.1.3		3	1.1	even	1	trivial
1872.4.a.bk.1.3		3	52.51	odd	2
1911.4.a.k.1.3		3	273.272	even	2
2496.4.a.bl.1.3		3	312.77	odd	2
2496.4.a.bp.1.3		3	312.155	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} +O(q^{10})\)	\(q+4.20905 q^{2} +9.71610 q^{4} -11.4322 q^{5} +11.2543 q^{7} +7.22315 q^{8} -48.1187 q^{10} +25.8785 q^{11} +47.3699 q^{14} -47.3262 q^{16} +20.3276 q^{17} -154.712 q^{19} -111.076 q^{20} +108.924 q^{22} +180.418 q^{23} +5.69520 q^{25} +109.348 q^{28} +20.4522 q^{29} -266.424 q^{31} -256.984 q^{32} +85.5599 q^{34} -128.661 q^{35} -115.984 q^{37} -651.190 q^{38} -82.5765 q^{40} +391.184 q^{41} +151.407 q^{43} +251.438 q^{44} +759.390 q^{46} -467.365 q^{47} -216.341 q^{49} +23.9714 q^{50} -79.9842 q^{53} -295.848 q^{55} +81.2915 q^{56} +86.0843 q^{58} -873.710 q^{59} -187.068 q^{61} -1121.39 q^{62} -703.047 q^{64} +609.204 q^{67} +197.505 q^{68} -541.542 q^{70} +248.038 q^{71} -852.765 q^{73} -488.181 q^{74} -1503.20 q^{76} +291.244 q^{77} -331.221 q^{79} +541.043 q^{80} +1646.51 q^{82} -435.432 q^{83} -232.389 q^{85} +637.281 q^{86} +186.924 q^{88} +259.233 q^{89} +1752.96 q^{92} -1967.16 q^{94} +1768.70 q^{95} -1225.17 q^{97} -910.589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8	\( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 10 * q^4 + 4 * q^5 - 30 * q^7 - 6 * q^8 - 4 * q^10 - 16 * q^11 + 176 * q^14 - 110 * q^16 + 146 * q^17 - 94 * q^19 - 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 80 * q^28 + 2 * q^29 - 302 * q^31 + 154 * q^32 - 164 * q^34 - 80 * q^35 - 374 * q^37 - 312 * q^38 - 516 * q^40 + 480 * q^41 - 260 * q^43 + 712 * q^44 + 1104 * q^46 - 24 * q^47 + 447 * q^49 + 814 * q^50 + 678 * q^53 - 1552 * q^55 - 96 * q^56 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 74 * q^67 + 460 * q^68 - 1216 * q^70 - 948 * q^71 + 222 * q^73 - 1724 * q^74 - 2392 * q^76 - 112 * q^77 - 24 * q^79 + 1100 * q^80 + 564 * q^82 - 796 * q^83 + 248 * q^85 + 1800 * q^86 + 1608 * q^88 + 1436 * q^89 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 - 3242 * q^97 - 5070 * q^98

Introduction

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