LMFDB - Embedded newform 2205.4.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 $ x^5 - 2x^4 - 30x^3 + 22x^2 + 153x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.10571$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.10571 q^{2} +18.0682 q^{4} +5.00000 q^{5} -51.4055 q^{8} +O(q^{10})$	$q-5.10571 q^{2} +18.0682 q^{4} +5.00000 q^{5} -51.4055 q^{8} -25.5285 q^{10} +40.7354 q^{11} -70.9263 q^{13} +117.915 q^{16} +22.6701 q^{17} -77.6588 q^{19} +90.3412 q^{20} -207.983 q^{22} +90.0622 q^{23} +25.0000 q^{25} +362.129 q^{26} -213.332 q^{29} +175.729 q^{31} -190.798 q^{32} -115.747 q^{34} -354.933 q^{37} +396.503 q^{38} -257.028 q^{40} +249.001 q^{41} +297.920 q^{43} +736.018 q^{44} -459.831 q^{46} +168.771 q^{47} -127.643 q^{50} -1281.51 q^{52} +102.473 q^{53} +203.677 q^{55} +1089.21 q^{58} -23.4797 q^{59} +874.628 q^{61} -897.220 q^{62} +30.8343 q^{64} -354.631 q^{65} -810.771 q^{67} +409.609 q^{68} +632.487 q^{71} -914.872 q^{73} +1812.18 q^{74} -1403.16 q^{76} -520.886 q^{79} +589.577 q^{80} -1271.33 q^{82} -1027.67 q^{83} +113.350 q^{85} -1521.09 q^{86} -2094.02 q^{88} +1003.75 q^{89} +1627.27 q^{92} -861.694 q^{94} -388.294 q^{95} +639.620 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} + 25 q^{4} + 25 q^{5} - 21 q^{8}+O(q^{10}) $ 5 * q - 3 * q^2 + 25 * q^4 + 25 * q^5 - 21 * q^8	$ 5 q - 3 q^{2} + 25 q^{4} + 25 q^{5} - 21 q^{8} - 15 q^{10} - 43 q^{11} - 123 q^{13} + 161 q^{16} + 124 q^{17} - 37 q^{19} + 125 q^{20} - 221 q^{22} - 77 q^{23} + 125 q^{25} - 79 q^{26} - 360 q^{29} - 314 q^{31} + 59 q^{32} - 176 q^{34} + 225 q^{37} + 759 q^{38} - 105 q^{40} + 341 q^{41} + 32 q^{43} - 679 q^{44} - 331 q^{46} + 25 q^{47} - 75 q^{50} - 2299 q^{52} + 317 q^{53} - 215 q^{55} + 8 q^{58} + 676 q^{59} + 188 q^{61} - 348 q^{62} - 1103 q^{64} - 615 q^{65} - 1776 q^{67} + 1280 q^{68} + 6 q^{71} - 2006 q^{73} + 2729 q^{74} - 1417 q^{76} + 200 q^{79} + 805 q^{80} + 539 q^{82} - 332 q^{83} + 620 q^{85} - 4262 q^{86} - 4529 q^{88} + 894 q^{89} + 3687 q^{92} - 4233 q^{94} - 185 q^{95} + 576 q^{97}+O(q^{100}) $ 5 * q - 3 * q^2 + 25 * q^4 + 25 * q^5 - 21 * q^8 - 15 * q^10 - 43 * q^11 - 123 * q^13 + 161 * q^16 + 124 * q^17 - 37 * q^19 + 125 * q^20 - 221 * q^22 - 77 * q^23 + 125 * q^25 - 79 * q^26 - 360 * q^29 - 314 * q^31 + 59 * q^32 - 176 * q^34 + 225 * q^37 + 759 * q^38 - 105 * q^40 + 341 * q^41 + 32 * q^43 - 679 * q^44 - 331 * q^46 + 25 * q^47 - 75 * q^50 - 2299 * q^52 + 317 * q^53 - 215 * q^55 + 8 * q^58 + 676 * q^59 + 188 * q^61 - 348 * q^62 - 1103 * q^64 - 615 * q^65 - 1776 * q^67 + 1280 * q^68 + 6 * q^71 - 2006 * q^73 + 2729 * q^74 - 1417 * q^76 + 200 * q^79 + 805 * q^80 + 539 * q^82 - 332 * q^83 + 620 * q^85 - 4262 * q^86 - 4529 * q^88 + 894 * q^89 + 3687 * q^92 - 4233 * q^94 - 185 * q^95 + 576 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−5.10571	−1.80514	−0.902570	−	0.430543i	$-0.858322\pi$
$2$	−5.10571	−1.80514	−0.902570	+	0.430543i	$0.858322\pi$
$3$	0	0
$3$	0	0
$4$	18.0682	2.25853
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−51.4055	−2.27182
$9$	0	0
$10$	−25.5285	−0.807283
$11$	40.7354	1.11656	0.558281	−	0.829652i	$-0.311461\pi$
$11$	40.7354	1.11656	0.558281	+	0.829652i	$0.311461\pi$
$12$	0	0
$13$	−70.9263	−1.51319	−0.756593	−	0.653887i	$-0.773137\pi$
$13$	−70.9263	−1.51319	−0.756593	+	0.653887i	$0.773137\pi$
$14$	0	0
$15$	0	0
$16$	117.915	1.84243
$17$	22.6701	0.323430	0.161715	−	0.986838i	$-0.448298\pi$
$17$	22.6701	0.323430	0.161715	+	0.986838i	$0.448298\pi$
$18$	0	0
$19$	−77.6588	−0.937692	−0.468846	−	0.883280i	$-0.655330\pi$
$19$	−77.6588	−0.937692	−0.468846	+	0.883280i	$0.655330\pi$
$20$	90.3412	1.01005
$21$	0	0
$22$	−207.983	−2.01555
$23$	90.0622	0.816490	0.408245	−	0.912872i	$-0.366141\pi$
$23$	90.0622	0.816490	0.408245	+	0.912872i	$0.366141\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	362.129	2.73151
$27$	0	0
$28$	0	0
$29$	−213.332	−1.36603	−0.683013	−	0.730406i	$-0.739331\pi$
$29$	−213.332	−1.36603	−0.683013	+	0.730406i	$0.739331\pi$
$30$	0	0
$31$	175.729	1.01812	0.509062	−	0.860730i	$-0.329992\pi$
$31$	175.729	1.01812	0.509062	+	0.860730i	$0.329992\pi$
$32$	−190.798	−1.05402
$33$	0	0
$34$	−115.747	−0.583836
$35$	0	0
$36$	0	0
$37$	−354.933	−1.57704	−0.788521	−	0.615007i	$-0.789153\pi$
$37$	−354.933	−1.57704	−0.788521	+	0.615007i	$0.789153\pi$
$38$	396.503	1.69267
$39$	0	0
$40$	−257.028	−1.01599
$41$	249.001	0.948475	0.474238	−	0.880397i	$-0.342724\pi$
$41$	249.001	0.948475	0.474238	+	0.880397i	$0.342724\pi$
$42$	0	0
$43$	297.920	1.05657	0.528284	−	0.849068i	$-0.322836\pi$
$43$	297.920	1.05657	0.528284	+	0.849068i	$0.322836\pi$
$44$	736.018	2.52179
$45$	0	0
$46$	−459.831	−1.47388
$47$	168.771	0.523782	0.261891	−	0.965097i	$-0.415654\pi$
$47$	168.771	0.523782	0.261891	+	0.965097i	$0.415654\pi$
$48$	0	0
$49$	0	0
$50$	−127.643	−0.361028
$51$	0	0
$52$	−1281.51	−3.41757
$53$	102.473	0.265581	0.132790	−	0.991144i	$-0.457606\pi$
$53$	102.473	0.265581	0.132790	+	0.991144i	$0.457606\pi$
$54$	0	0
$55$	203.677	0.499342
$56$	0	0
$57$	0	0
$58$	1089.21	2.46587
$59$	−23.4797	−0.0518102	−0.0259051	−	0.999664i	$-0.508247\pi$
$59$	−23.4797	−0.0518102	−0.0259051	+	0.999664i	$0.508247\pi$
$60$	0	0
$61$	874.628	1.83581	0.917907	−	0.396796i	$-0.129878\pi$
$61$	874.628	1.83581	0.917907	+	0.396796i	$0.129878\pi$
$62$	−897.220	−1.83786
$63$	0	0
$64$	30.8343	0.0602232
$65$	−354.631	−0.676717
$66$	0	0
$67$	−810.771	−1.47838	−0.739190	−	0.673497i	$-0.764792\pi$
$67$	−810.771	−1.47838	−0.739190	+	0.673497i	$0.764792\pi$
$68$	409.609	0.730476
$69$	0	0
$70$	0	0
$71$	632.487	1.05722	0.528608	−	0.848866i	$-0.322714\pi$
$71$	632.487	1.05722	0.528608	+	0.848866i	$0.322714\pi$
$72$	0	0
$73$	−914.872	−1.46682	−0.733409	−	0.679788i	$-0.762072\pi$
$73$	−914.872	−1.46682	−0.733409	+	0.679788i	$0.762072\pi$
$74$	1812.18	2.84678
$75$	0	0
$76$	−1403.16	−2.11781
$77$	0	0
$78$	0	0
$79$	−520.886	−0.741827	−0.370913	−	0.928668i	$-0.620955\pi$
$79$	−520.886	−0.741827	−0.370913	+	0.928668i	$0.620955\pi$
$80$	589.577	0.823960
$81$	0	0
$82$	−1271.33	−1.71213
$83$	−1027.67	−1.35905	−0.679527	−	0.733651i	$-0.737815\pi$
$83$	−1027.67	−1.35905	−0.679527	+	0.733651i	$0.737815\pi$
$84$	0	0
$85$	113.350	0.144642
$86$	−1521.09	−1.90725
$87$	0	0
$88$	−2094.02	−2.53663
$89$	1003.75	1.19548	0.597738	−	0.801692i	$-0.296066\pi$
$89$	1003.75	1.19548	0.597738	+	0.801692i	$0.296066\pi$
$90$	0	0
$91$	0	0
$92$	1627.27	1.84407
$93$	0	0
$94$	−861.694	−0.945500
$95$	−388.294	−0.419349
$96$	0	0
$97$	639.620	0.669522	0.334761	−	0.942303i	$-0.391344\pi$
$97$	639.620	0.669522	0.334761	+	0.942303i	$0.391344\pi$
$98$	0	0
$99$	0	0
$100$	451.706	0.451706
$101$	−633.869	−0.624479	−0.312239	−	0.950003i	$-0.601079\pi$
$101$	−633.869	−0.624479	−0.312239	+	0.950003i	$0.601079\pi$
$102$	0	0
$103$	387.578	0.370769	0.185385	−	0.982666i	$-0.440647\pi$
$103$	387.578	0.370769	0.185385	+	0.982666i	$0.440647\pi$
$104$	3646.00	3.43769
$105$	0	0
$106$	−523.198	−0.479410
$107$	−138.730	−0.125341	−0.0626707	−	0.998034i	$-0.519962\pi$
$107$	−138.730	−0.125341	−0.0626707	+	0.998034i	$0.519962\pi$
$108$	0	0
$109$	−1712.97	−1.50526	−0.752628	−	0.658447i	$-0.771214\pi$
$109$	−1712.97	−1.50526	−0.752628	+	0.658447i	$0.771214\pi$
$110$	−1039.92	−0.901382
$111$	0	0
$112$	0	0
$113$	−2083.35	−1.73438	−0.867192	−	0.497975i	$-0.834077\pi$
$113$	−2083.35	−1.73438	−0.867192	+	0.497975i	$0.834077\pi$
$114$	0	0
$115$	450.311	0.365145
$116$	−3854.54	−3.08521
$117$	0	0
$118$	119.881	0.0935246
$119$	0	0
$120$	0	0
$121$	328.375	0.246713
$122$	−4465.59	−3.31390
$123$	0	0
$124$	3175.11	2.29946
$125$	125.000	0.0894427
$126$	0	0
$127$	432.085	0.301900	0.150950	−	0.988541i	$-0.451767\pi$
$127$	432.085	0.301900	0.150950	+	0.988541i	$0.451767\pi$
$128$	1368.95	0.945308
$129$	0	0
$130$	1810.64	1.22157
$131$	−657.499	−0.438519	−0.219259	−	0.975667i	$-0.570364\pi$
$131$	−657.499	−0.438519	−0.219259	+	0.975667i	$0.570364\pi$
$132$	0	0
$133$	0	0
$134$	4139.56	2.66868
$135$	0	0
$136$	−1165.37	−0.734775
$137$	−292.515	−0.182418	−0.0912090	−	0.995832i	$-0.529073\pi$
$137$	−292.515	−0.182418	−0.0912090	+	0.995832i	$0.529073\pi$
$138$	0	0
$139$	1673.02	1.02089	0.510445	−	0.859910i	$-0.329481\pi$
$139$	1673.02	1.02089	0.510445	+	0.859910i	$0.329481\pi$
$140$	0	0
$141$	0	0
$142$	−3229.29	−1.90842
$143$	−2889.21	−1.68957
$144$	0	0
$145$	−1066.66	−0.610906
$146$	4671.07	2.64781
$147$	0	0
$148$	−6413.01	−3.56180
$149$	−2521.01	−1.38610	−0.693050	−	0.720890i	$-0.743734\pi$
$149$	−2521.01	−1.38610	−0.693050	+	0.720890i	$0.743734\pi$
$150$	0	0
$151$	1379.32	0.743361	0.371680	−	0.928361i	$-0.378782\pi$
$151$	1379.32	0.743361	0.371680	+	0.928361i	$0.378782\pi$
$152$	3992.09	2.13027
$153$	0	0
$154$	0	0
$155$	878.644	0.455319
$156$	0	0
$157$	−1992.73	−1.01298	−0.506488	−	0.862247i	$-0.669057\pi$
$157$	−1992.73	−1.01298	−0.506488	+	0.862247i	$0.669057\pi$
$158$	2659.49	1.33910
$159$	0	0
$160$	−953.990	−0.471372
$161$	0	0
$162$	0	0
$163$	1902.94	0.914418	0.457209	−	0.889359i	$-0.348849\pi$
$163$	1902.94	0.914418	0.457209	+	0.889359i	$0.348849\pi$
$164$	4499.02	2.14216
$165$	0	0
$166$	5246.98	2.45328
$167$	−378.184	−0.175238	−0.0876189	−	0.996154i	$-0.527926\pi$
$167$	−378.184	−0.175238	−0.0876189	+	0.996154i	$0.527926\pi$
$168$	0	0
$169$	2833.54	1.28973
$170$	−578.734	−0.261099
$171$	0	0
$172$	5382.89	2.38629
$173$	−664.037	−0.291825	−0.145913	−	0.989297i	$-0.546612\pi$
$173$	−664.037	−0.291825	−0.145913	+	0.989297i	$0.546612\pi$
$174$	0	0
$175$	0	0
$176$	4803.34	2.05719
$177$	0	0
$178$	−5124.86	−2.15800
$179$	334.999	0.139883	0.0699414	−	0.997551i	$-0.477719\pi$
$179$	334.999	0.139883	0.0699414	+	0.997551i	$0.477719\pi$
$180$	0	0
$181$	−1225.58	−0.503298	−0.251649	−	0.967819i	$-0.580973\pi$
$181$	−1225.58	−0.503298	−0.251649	+	0.967819i	$0.580973\pi$
$182$	0	0
$183$	0	0
$184$	−4629.69	−1.85492
$185$	−1774.66	−0.705275
$186$	0	0
$187$	923.475	0.361129
$188$	3049.39	1.18298
$189$	0	0
$190$	1982.51	0.756983
$191$	1581.40	0.599088	0.299544	−	0.954082i	$-0.403165\pi$
$191$	1581.40	0.599088	0.299544	+	0.954082i	$0.403165\pi$
$192$	0	0
$193$	−3894.63	−1.45255	−0.726273	−	0.687406i	$-0.758749\pi$
$193$	−3894.63	−1.45255	−0.726273	+	0.687406i	$0.758749\pi$
$194$	−3265.71	−1.20858
$195$	0	0
$196$	0	0
$197$	−993.858	−0.359439	−0.179719	−	0.983718i	$-0.557519\pi$
$197$	−993.858	−0.359439	−0.179719	+	0.983718i	$0.557519\pi$
$198$	0	0
$199$	3079.88	1.09712	0.548560	−	0.836111i	$-0.315176\pi$
$199$	3079.88	1.09712	0.548560	+	0.836111i	$0.315176\pi$
$200$	−1285.14	−0.454365
$201$	0	0
$202$	3236.35	1.12727
$203$	0	0
$204$	0	0
$205$	1245.01	0.424171
$206$	−1978.86	−0.669290
$207$	0	0
$208$	−8363.31	−2.78794
$209$	−3163.46	−1.04699
$210$	0	0
$211$	2114.72	0.689968	0.344984	−	0.938609i	$-0.387884\pi$
$211$	2114.72	0.689968	0.344984	+	0.938609i	$0.387884\pi$
$212$	1851.51	0.599822
$213$	0	0
$214$	708.315	0.226259
$215$	1489.60	0.472511
$216$	0	0
$217$	0	0
$218$	8745.92	2.71720
$219$	0	0
$220$	3680.09	1.12778
$221$	−1607.90	−0.489409
$222$	0	0
$223$	−2578.44	−0.774282	−0.387141	−	0.922021i	$-0.626537\pi$
$223$	−2578.44	−0.774282	−0.387141	+	0.922021i	$0.626537\pi$
$224$	0	0
$225$	0	0
$226$	10637.0	3.13080
$227$	2925.92	0.855509	0.427754	−	0.903895i	$-0.359305\pi$
$227$	2925.92	0.855509	0.427754	+	0.903895i	$0.359305\pi$
$228$	0	0
$229$	1047.88	0.302383	0.151191	−	0.988505i	$-0.451689\pi$
$229$	1047.88	0.302383	0.151191	+	0.988505i	$0.451689\pi$
$230$	−2299.16	−0.659139
$231$	0	0
$232$	10966.4	3.10337
$233$	3869.99	1.08812	0.544058	−	0.839047i	$-0.316887\pi$
$233$	3869.99	1.08812	0.544058	+	0.839047i	$0.316887\pi$
$234$	0	0
$235$	843.854	0.234242
$236$	−424.237	−0.117015
$237$	0	0
$238$	0	0
$239$	6060.77	1.64033	0.820164	−	0.572128i	$-0.193882\pi$
$239$	6060.77	1.64033	0.820164	+	0.572128i	$0.193882\pi$
$240$	0	0
$241$	4184.12	1.11835	0.559176	−	0.829049i	$-0.311118\pi$
$241$	4184.12	1.11835	0.559176	+	0.829049i	$0.311118\pi$
$242$	−1676.58	−0.445351
$243$	0	0
$244$	15803.0	4.14624
$245$	0	0
$246$	0	0
$247$	5508.05	1.41890
$248$	−9033.43	−2.31300
$249$	0	0
$250$	−638.213	−0.161457
$251$	−2631.65	−0.661786	−0.330893	−	0.943668i	$-0.607350\pi$
$251$	−2631.65	−0.661786	−0.330893	+	0.943668i	$0.607350\pi$
$252$	0	0
$253$	3668.72	0.911663
$254$	−2206.10	−0.544972
$255$	0	0
$256$	−7236.14	−1.76664
$257$	3288.77	0.798240	0.399120	−	0.916899i	$-0.369316\pi$
$257$	3288.77	0.798240	0.399120	+	0.916899i	$0.369316\pi$
$258$	0	0
$259$	0	0
$260$	−6407.57	−1.52839
$261$	0	0
$262$	3357.00	0.791588
$263$	5408.90	1.26816	0.634081	−	0.773266i	$-0.281378\pi$
$263$	5408.90	1.26816	0.634081	+	0.773266i	$0.281378\pi$
$264$	0	0
$265$	512.366	0.118771
$266$	0	0
$267$	0	0
$268$	−14649.2	−3.33897
$269$	539.788	0.122348	0.0611738	−	0.998127i	$-0.480516\pi$
$269$	539.788	0.122348	0.0611738	+	0.998127i	$0.480516\pi$
$270$	0	0
$271$	−4091.14	−0.917045	−0.458522	−	0.888683i	$-0.651621\pi$
$271$	−4091.14	−0.917045	−0.458522	+	0.888683i	$0.651621\pi$
$272$	2673.15	0.595896
$273$	0	0
$274$	1493.50	0.329290
$275$	1018.39	0.223313
$276$	0	0
$277$	3091.34	0.670543	0.335271	−	0.942122i	$-0.391172\pi$
$277$	3091.34	0.670543	0.335271	+	0.942122i	$0.391172\pi$
$278$	−8541.95	−1.84285
$279$	0	0
$280$	0	0
$281$	−7215.73	−1.53187	−0.765933	−	0.642920i	$-0.777723\pi$
$281$	−7215.73	−1.53187	−0.765933	+	0.642920i	$0.777723\pi$
$282$	0	0
$283$	−2964.72	−0.622735	−0.311368	−	0.950290i	$-0.600787\pi$
$283$	−2964.72	−0.622735	−0.311368	+	0.950290i	$0.600787\pi$
$284$	11427.9	2.38776
$285$	0	0
$286$	14751.5	3.04990
$287$	0	0
$288$	0	0
$289$	−4399.07	−0.895393
$290$	5446.05	1.10277
$291$	0	0
$292$	−16530.1	−3.31285
$293$	−7924.10	−1.57997	−0.789984	−	0.613127i	$-0.789911\pi$
$293$	−7924.10	−1.57997	−0.789984	+	0.613127i	$0.789911\pi$
$294$	0	0
$295$	−117.399	−0.0231702
$296$	18245.5	3.58276
$297$	0	0
$298$	12871.5	2.50210
$299$	−6387.78	−1.23550
$300$	0	0
$301$	0	0
$302$	−7042.41	−1.34187
$303$	0	0
$304$	−9157.17	−1.72763
$305$	4373.14	0.821001
$306$	0	0
$307$	4067.00	0.756079	0.378039	−	0.925790i	$-0.376598\pi$
$307$	4067.00	0.756079	0.378039	+	0.925790i	$0.376598\pi$
$308$	0	0
$309$	0	0
$310$	−4486.10	−0.821914
$311$	7652.58	1.39530	0.697650	−	0.716439i	$-0.254229\pi$
$311$	7652.58	1.39530	0.697650	+	0.716439i	$0.254229\pi$
$312$	0	0
$313$	−1822.12	−0.329049	−0.164525	−	0.986373i	$-0.552609\pi$
$313$	−1822.12	−0.329049	−0.164525	+	0.986373i	$0.552609\pi$
$314$	10174.3	1.82856
$315$	0	0
$316$	−9411.50	−1.67544
$317$	−3148.86	−0.557910	−0.278955	−	0.960304i	$-0.589988\pi$
$317$	−3148.86	−0.557910	−0.278955	+	0.960304i	$0.589988\pi$
$318$	0	0
$319$	−8690.17	−1.52525
$320$	154.172	0.0269327
$321$	0	0
$322$	0	0
$323$	−1760.53	−0.303277
$324$	0	0
$325$	−1773.16	−0.302637
$326$	−9715.87	−1.65065
$327$	0	0
$328$	−12800.0	−2.15477
$329$	0	0
$330$	0	0
$331$	−5856.12	−0.972452	−0.486226	−	0.873833i	$-0.661627\pi$
$331$	−5856.12	−0.972452	−0.486226	+	0.873833i	$0.661627\pi$
$332$	−18568.2	−3.06946
$333$	0	0
$334$	1930.89	0.316329
$335$	−4053.86	−0.661152
$336$	0	0
$337$	−7065.69	−1.14212	−0.571058	−	0.820910i	$-0.693467\pi$
$337$	−7065.69	−1.14212	−0.571058	+	0.820910i	$0.693467\pi$
$338$	−14467.2	−2.32814
$339$	0	0
$340$	2048.04	0.326679
$341$	7158.39	1.13680
$342$	0	0
$343$	0	0
$344$	−15314.7	−2.40033
$345$	0	0
$346$	3390.38	0.526785
$347$	−9897.45	−1.53119	−0.765595	−	0.643323i	$-0.777555\pi$
$347$	−9897.45	−1.53119	−0.765595	+	0.643323i	$0.777555\pi$
$348$	0	0
$349$	−11832.6	−1.81486	−0.907431	−	0.420201i	$-0.861960\pi$
$349$	−11832.6	−1.81486	−0.907431	+	0.420201i	$0.861960\pi$
$350$	0	0
$351$	0	0
$352$	−7772.23	−1.17688
$353$	7007.38	1.05656	0.528279	−	0.849071i	$-0.322837\pi$
$353$	7007.38	1.05656	0.528279	+	0.849071i	$0.322837\pi$
$354$	0	0
$355$	3162.43	0.472802
$356$	18136.0	2.70002
$357$	0	0
$358$	−1710.41	−0.252508
$359$	−2814.39	−0.413754	−0.206877	−	0.978367i	$-0.566330\pi$
$359$	−2814.39	−0.413754	−0.206877	+	0.978367i	$0.566330\pi$
$360$	0	0
$361$	−828.115	−0.120734
$362$	6257.47	0.908523
$363$	0	0
$364$	0	0
$365$	−4574.36	−0.655981
$366$	0	0
$367$	−2349.86	−0.334227	−0.167114	−	0.985938i	$-0.553445\pi$
$367$	−2349.86	−0.334227	−0.167114	+	0.985938i	$0.553445\pi$
$368$	10619.7	1.50433
$369$	0	0
$370$	9060.91	1.27312
$371$	0	0
$372$	0	0
$373$	−1659.30	−0.230336	−0.115168	−	0.993346i	$-0.536741\pi$
$373$	−1659.30	−0.230336	−0.115168	+	0.993346i	$0.536741\pi$
$374$	−4714.99	−0.651889
$375$	0	0
$376$	−8675.75	−1.18994
$377$	15130.8	2.06705
$378$	0	0
$379$	5327.80	0.722087	0.361043	−	0.932549i	$-0.382421\pi$
$379$	5327.80	0.722087	0.361043	+	0.932549i	$0.382421\pi$
$380$	−7015.79	−0.947111
$381$	0	0
$382$	−8074.15	−1.08144
$383$	6239.75	0.832471	0.416235	−	0.909257i	$-0.363349\pi$
$383$	6239.75	0.832471	0.416235	+	0.909257i	$0.363349\pi$
$384$	0	0
$385$	0	0
$386$	19884.8	2.62205
$387$	0	0
$388$	11556.8	1.51213
$389$	−9485.21	−1.23630	−0.618148	−	0.786062i	$-0.712117\pi$
$389$	−9485.21	−1.23630	−0.618148	+	0.786062i	$0.712117\pi$
$390$	0	0
$391$	2041.72	0.264077
$392$	0	0
$393$	0	0
$394$	5074.35	0.648837
$395$	−2604.43	−0.331755
$396$	0	0
$397$	−7892.74	−0.997796	−0.498898	−	0.866661i	$-0.666262\pi$
$397$	−7892.74	−0.997796	−0.498898	+	0.866661i	$0.666262\pi$
$398$	−15725.0	−1.98046
$399$	0	0
$400$	2947.89	0.368486
$401$	12459.7	1.55163	0.775817	−	0.630958i	$-0.217338\pi$
$401$	12459.7	1.55163	0.775817	+	0.630958i	$0.217338\pi$
$402$	0	0
$403$	−12463.8	−1.54061
$404$	−11452.9	−1.41040
$405$	0	0
$406$	0	0
$407$	−14458.3	−1.76087
$408$	0	0
$409$	−3403.62	−0.411487	−0.205744	−	0.978606i	$-0.565961\pi$
$409$	−3403.62	−0.411487	−0.205744	+	0.978606i	$0.565961\pi$
$410$	−6356.64	−0.765688
$411$	0	0
$412$	7002.86	0.837393
$413$	0	0
$414$	0	0
$415$	−5138.35	−0.607787
$416$	13532.6	1.59493
$417$	0	0
$418$	16151.7	1.88997
$419$	−1376.83	−0.160532	−0.0802658	−	0.996773i	$-0.525577\pi$
$419$	−1376.83	−0.160532	−0.0802658	+	0.996773i	$0.525577\pi$
$420$	0	0
$421$	−15757.3	−1.82414	−0.912069	−	0.410036i	$-0.865516\pi$
$421$	−15757.3	−1.82414	−0.912069	+	0.410036i	$0.865516\pi$
$422$	−10797.1	−1.24549
$423$	0	0
$424$	−5267.69	−0.603353
$425$	566.752	0.0646859
$426$	0	0
$427$	0	0
$428$	−2506.61	−0.283088
$429$	0	0
$430$	−7605.46	−0.852949
$431$	−5083.70	−0.568152	−0.284076	−	0.958802i	$-0.591687\pi$
$431$	−5083.70	−0.568152	−0.284076	+	0.958802i	$0.591687\pi$
$432$	0	0
$433$	−4631.44	−0.514025	−0.257013	−	0.966408i	$-0.582738\pi$
$433$	−4631.44	−0.514025	−0.257013	+	0.966408i	$0.582738\pi$
$434$	0	0
$435$	0	0
$436$	−30950.4	−3.39966
$437$	−6994.12	−0.765616
$438$	0	0
$439$	2338.87	0.254279	0.127139	−	0.991885i	$-0.459420\pi$
$439$	2338.87	0.254279	0.127139	+	0.991885i	$0.459420\pi$
$440$	−10470.1	−1.13442
$441$	0	0
$442$	8209.49	0.883451
$443$	−11811.5	−1.26677	−0.633387	−	0.773835i	$-0.718336\pi$
$443$	−11811.5	−1.26677	−0.633387	+	0.773835i	$0.718336\pi$
$444$	0	0
$445$	5018.75	0.534633
$446$	13164.7	1.39769
$447$	0	0
$448$	0	0
$449$	−9582.29	−1.00716	−0.503582	−	0.863948i	$-0.667985\pi$
$449$	−9582.29	−1.00716	−0.503582	+	0.863948i	$0.667985\pi$
$450$	0	0
$451$	10143.2	1.05903
$452$	−37642.5	−3.91716
$453$	0	0
$454$	−14938.9	−1.54431
$455$	0	0
$456$	0	0
$457$	−6571.95	−0.672697	−0.336349	−	0.941738i	$-0.609192\pi$
$457$	−6571.95	−0.672697	−0.336349	+	0.941738i	$0.609192\pi$
$458$	−5350.15	−0.545843
$459$	0	0
$460$	8136.33	0.824692
$461$	−4197.59	−0.424081	−0.212041	−	0.977261i	$-0.568011\pi$
$461$	−4197.59	−0.424081	−0.212041	+	0.977261i	$0.568011\pi$
$462$	0	0
$463$	−6125.75	−0.614876	−0.307438	−	0.951568i	$-0.599472\pi$
$463$	−6125.75	−0.614876	−0.307438	+	0.951568i	$0.599472\pi$
$464$	−25155.2	−2.51681
$465$	0	0
$466$	−19759.0	−1.96420
$467$	3228.08	0.319867	0.159934	−	0.987128i	$-0.448872\pi$
$467$	3228.08	0.319867	0.159934	+	0.987128i	$0.448872\pi$
$468$	0	0
$469$	0	0
$470$	−4308.47	−0.422840
$471$	0	0
$472$	1206.99	0.117704
$473$	12135.9	1.17972
$474$	0	0
$475$	−1941.47	−0.187538
$476$	0	0
$477$	0	0
$478$	−30944.5	−2.96102
$479$	3126.31	0.298214	0.149107	−	0.988821i	$-0.452360\pi$
$479$	3126.31	0.298214	0.149107	+	0.988821i	$0.452360\pi$
$480$	0	0
$481$	25174.1	2.38636
$482$	−21362.9	−2.01878
$483$	0	0
$484$	5933.15	0.557208
$485$	3198.10	0.299419
$486$	0	0
$487$	−869.038	−0.0808621	−0.0404311	−	0.999182i	$-0.512873\pi$
$487$	−869.038	−0.0808621	−0.0404311	+	0.999182i	$0.512873\pi$
$488$	−44960.7	−4.17065
$489$	0	0
$490$	0	0
$491$	−6617.71	−0.608254	−0.304127	−	0.952631i	$-0.598365\pi$
$491$	−6617.71	−0.608254	−0.304127	+	0.952631i	$0.598365\pi$
$492$	0	0
$493$	−4836.25	−0.441813
$494$	−28122.5	−2.56132
$495$	0	0
$496$	20721.2	1.87582
$497$	0	0
$498$	0	0
$499$	−888.154	−0.0796778	−0.0398389	−	0.999206i	$-0.512684\pi$
$499$	−888.154	−0.0796778	−0.0398389	+	0.999206i	$0.512684\pi$
$500$	2258.53	0.202009
$501$	0	0
$502$	13436.4	1.19462
$503$	−10757.4	−0.953572	−0.476786	−	0.879019i	$-0.658198\pi$
$503$	−10757.4	−0.953572	−0.476786	+	0.879019i	$0.658198\pi$
$504$	0	0
$505$	−3169.35	−0.279275
$506$	−18731.4	−1.64568
$507$	0	0
$508$	7807.01	0.681850
$509$	8860.03	0.771540	0.385770	−	0.922595i	$-0.373936\pi$
$509$	8860.03	0.771540	0.385770	+	0.922595i	$0.373936\pi$
$510$	0	0
$511$	0	0
$512$	25994.0	2.24372
$513$	0	0
$514$	−16791.5	−1.44093
$515$	1937.89	0.165813
$516$	0	0
$517$	6874.95	0.584835
$518$	0	0
$519$	0	0
$520$	18230.0	1.53738
$521$	−75.3875	−0.00633932	−0.00316966	−	0.999995i	$-0.501009\pi$
$521$	−75.3875	−0.00633932	−0.00316966	+	0.999995i	$0.501009\pi$
$522$	0	0
$523$	−8666.59	−0.724596	−0.362298	−	0.932062i	$-0.618008\pi$
$523$	−8666.59	−0.724596	−0.362298	+	0.932062i	$0.618008\pi$
$524$	−11879.9	−0.990408
$525$	0	0
$526$	−27616.2	−2.28921
$527$	3983.79	0.329291
$528$	0	0
$529$	−4055.80	−0.333344
$530$	−2615.99	−0.214399
$531$	0	0
$532$	0	0
$533$	−17660.7	−1.43522
$534$	0	0
$535$	−693.650	−0.0560544
$536$	41678.1	3.35862
$537$	0	0
$538$	−2756.00	−0.220854
$539$	0	0
$540$	0	0
$541$	12284.8	0.976272	0.488136	−	0.872768i	$-0.337677\pi$
$541$	12284.8	0.976272	0.488136	+	0.872768i	$0.337677\pi$
$542$	20888.2	1.65539
$543$	0	0
$544$	−4325.40	−0.340901
$545$	−8564.85	−0.673171
$546$	0	0
$547$	−2906.85	−0.227218	−0.113609	−	0.993526i	$-0.536241\pi$
$547$	−2906.85	−0.227218	−0.113609	+	0.993526i	$0.536241\pi$
$548$	−5285.24	−0.411997
$549$	0	0
$550$	−5199.58	−0.403110
$551$	16567.1	1.28091
$552$	0	0
$553$	0	0
$554$	−15783.5	−1.21042
$555$	0	0
$556$	30228.5	2.30571
$557$	−3873.82	−0.294684	−0.147342	−	0.989086i	$-0.547072\pi$
$557$	−3873.82	−0.294684	−0.147342	+	0.989086i	$0.547072\pi$
$558$	0	0
$559$	−21130.4	−1.59878
$560$	0	0
$561$	0	0
$562$	36841.4	2.76523
$563$	−5461.09	−0.408805	−0.204403	−	0.978887i	$-0.565525\pi$
$563$	−5461.09	−0.408805	−0.204403	+	0.978887i	$0.565525\pi$
$564$	0	0
$565$	−10416.8	−0.775640
$566$	15137.0	1.12412
$567$	0	0
$568$	−32513.3	−2.40181
$569$	−13105.0	−0.965534	−0.482767	−	0.875749i	$-0.660368\pi$
$569$	−13105.0	−0.965534	−0.482767	+	0.875749i	$0.660368\pi$
$570$	0	0
$571$	3439.26	0.252064	0.126032	−	0.992026i	$-0.459776\pi$
$571$	3439.26	0.252064	0.126032	+	0.992026i	$0.459776\pi$
$572$	−52203.0	−3.81594
$573$	0	0
$574$	0	0
$575$	2251.56	0.163298
$576$	0	0
$577$	−17792.2	−1.28371	−0.641853	−	0.766828i	$-0.721834\pi$
$577$	−17792.2	−1.28371	−0.641853	+	0.766828i	$0.721834\pi$
$578$	22460.3	1.61631
$579$	0	0
$580$	−19272.7	−1.37975
$581$	0	0
$582$	0	0
$583$	4174.29	0.296538
$584$	47029.4	3.33235
$585$	0	0
$586$	40458.1	2.85206
$587$	14416.6	1.01369	0.506846	−	0.862037i	$-0.330811\pi$
$587$	14416.6	1.01369	0.506846	+	0.862037i	$0.330811\pi$
$588$	0	0
$589$	−13646.9	−0.954686
$590$	599.403	0.0418255
$591$	0	0
$592$	−41852.1	−2.90559
$593$	8172.03	0.565911	0.282955	−	0.959133i	$-0.408685\pi$
$593$	8172.03	0.565911	0.282955	+	0.959133i	$0.408685\pi$
$594$	0	0
$595$	0	0
$596$	−45550.2	−3.13055
$597$	0	0
$598$	32614.1	2.23025
$599$	2492.18	0.169996	0.0849980	−	0.996381i	$-0.472912\pi$
$599$	2492.18	0.169996	0.0849980	+	0.996381i	$0.472912\pi$
$600$	0	0
$601$	13835.4	0.939031	0.469516	−	0.882924i	$-0.344429\pi$
$601$	13835.4	0.939031	0.469516	+	0.882924i	$0.344429\pi$
$602$	0	0
$603$	0	0
$604$	24921.9	1.67890
$605$	1641.87	0.110333
$606$	0	0
$607$	−7335.90	−0.490536	−0.245268	−	0.969455i	$-0.578876\pi$
$607$	−7335.90	−0.490536	−0.245268	+	0.969455i	$0.578876\pi$
$608$	14817.1	0.988345
$609$	0	0
$610$	−22328.0	−1.48202
$611$	−11970.3	−0.792579
$612$	0	0
$613$	6012.04	0.396124	0.198062	−	0.980189i	$-0.436535\pi$
$613$	6012.04	0.396124	0.198062	+	0.980189i	$0.436535\pi$
$614$	−20764.9	−1.36483
$615$	0	0
$616$	0	0
$617$	22927.4	1.49598	0.747991	−	0.663708i	$-0.231018\pi$
$617$	22927.4	1.49598	0.747991	+	0.663708i	$0.231018\pi$
$618$	0	0
$619$	−17921.5	−1.16369	−0.581847	−	0.813298i	$-0.697670\pi$
$619$	−17921.5	−1.16369	−0.581847	+	0.813298i	$0.697670\pi$
$620$	15875.6	1.02835
$621$	0	0
$622$	−39071.8	−2.51871
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	9303.22	0.593980
$627$	0	0
$628$	−36005.1	−2.28784
$629$	−8046.35	−0.510062
$630$	0	0
$631$	19270.7	1.21578	0.607888	−	0.794023i	$-0.292017\pi$
$631$	19270.7	1.21578	0.607888	+	0.794023i	$0.292017\pi$
$632$	26776.4	1.68530
$633$	0	0
$634$	16077.1	1.00711
$635$	2160.42	0.135014
$636$	0	0
$637$	0	0
$638$	44369.5	2.75330
$639$	0	0
$640$	6844.76	0.422755
$641$	−28238.6	−1.74003	−0.870013	−	0.493029i	$-0.835890\pi$
$641$	−28238.6	−1.74003	−0.870013	+	0.493029i	$0.835890\pi$
$642$	0	0
$643$	19055.3	1.16869	0.584346	−	0.811505i	$-0.301351\pi$
$643$	19055.3	1.16869	0.584346	+	0.811505i	$0.301351\pi$
$644$	0	0
$645$	0	0
$646$	8988.75	0.547458
$647$	−16211.3	−0.985057	−0.492529	−	0.870296i	$-0.663927\pi$
$647$	−16211.3	−0.985057	−0.492529	+	0.870296i	$0.663927\pi$
$648$	0	0
$649$	−956.457	−0.0578493
$650$	9053.22	0.546302
$651$	0	0
$652$	34382.9	2.06524
$653$	26532.1	1.59001	0.795007	−	0.606600i	$-0.207467\pi$
$653$	26532.1	1.59001	0.795007	+	0.606600i	$0.207467\pi$
$654$	0	0
$655$	−3287.50	−0.196112
$656$	29361.1	1.74750
$657$	0	0
$658$	0	0
$659$	−1883.19	−0.111318	−0.0556592	−	0.998450i	$-0.517726\pi$
$659$	−1883.19	−0.111318	−0.0556592	+	0.998450i	$0.517726\pi$
$660$	0	0
$661$	−4674.27	−0.275050	−0.137525	−	0.990498i	$-0.543915\pi$
$661$	−4674.27	−0.275050	−0.137525	+	0.990498i	$0.543915\pi$
$662$	29899.6	1.75541
$663$	0	0
$664$	52827.9	3.08753
$665$	0	0
$666$	0	0
$667$	−19213.2	−1.11535
$668$	−6833.11	−0.395780
$669$	0	0
$670$	20697.8	1.19347
$671$	35628.3	2.04980
$672$	0	0
$673$	−23344.3	−1.33709	−0.668543	−	0.743674i	$-0.733082\pi$
$673$	−23344.3	−1.33709	−0.668543	+	0.743674i	$0.733082\pi$
$674$	36075.4	2.06168
$675$	0	0
$676$	51197.0	2.91289
$677$	1399.49	0.0794490	0.0397245	−	0.999211i	$-0.487352\pi$
$677$	1399.49	0.0794490	0.0397245	+	0.999211i	$0.487352\pi$
$678$	0	0
$679$	0	0
$680$	−5826.83	−0.328601
$681$	0	0
$682$	−36548.6	−2.05208
$683$	−33020.7	−1.84993	−0.924964	−	0.380054i	$-0.875905\pi$
$683$	−33020.7	−1.84993	−0.924964	+	0.380054i	$0.875905\pi$
$684$	0	0
$685$	−1462.58	−0.0815798
$686$	0	0
$687$	0	0
$688$	35129.4	1.94665
$689$	−7268.04	−0.401873
$690$	0	0
$691$	−720.176	−0.0396480	−0.0198240	−	0.999803i	$-0.506311\pi$
$691$	−720.176	−0.0396480	−0.0198240	+	0.999803i	$0.506311\pi$
$692$	−11998.0	−0.659096
$693$	0	0
$694$	50533.5	2.76401
$695$	8365.10	0.456556
$696$	0	0
$697$	5644.88	0.306765
$698$	60414.0	3.27608
$699$	0	0
$700$	0	0
$701$	−11675.1	−0.629048	−0.314524	−	0.949249i	$-0.601845\pi$
$701$	−11675.1	−0.629048	−0.314524	+	0.949249i	$0.601845\pi$
$702$	0	0
$703$	27563.6	1.47878
$704$	1256.05	0.0672430
$705$	0	0
$706$	−35777.6	−1.90724
$707$	0	0
$708$	0	0
$709$	−26771.7	−1.41810	−0.709050	−	0.705158i	$-0.750876\pi$
$709$	−26771.7	−1.41810	−0.709050	+	0.705158i	$0.750876\pi$
$710$	−16146.5	−0.853473
$711$	0	0
$712$	−51598.3	−2.71591
$713$	15826.5	0.831288
$714$	0	0
$715$	−14446.1	−0.755597
$716$	6052.85	0.315930
$717$	0	0
$718$	14369.4	0.746884
$719$	34783.5	1.80418	0.902089	−	0.431550i	$-0.142033\pi$
$719$	34783.5	1.80418	0.902089	+	0.431550i	$0.142033\pi$
$720$	0	0
$721$	0	0
$722$	4228.11	0.217942
$723$	0	0
$724$	−22144.1	−1.13671
$725$	−5333.30	−0.273205
$726$	0	0
$727$	−25368.1	−1.29415	−0.647077	−	0.762425i	$-0.724009\pi$
$727$	−25368.1	−1.29415	−0.647077	+	0.762425i	$0.724009\pi$
$728$	0	0
$729$	0	0
$730$	23355.3	1.18414
$731$	6753.87	0.341725
$732$	0	0
$733$	−31691.5	−1.59694	−0.798468	−	0.602038i	$-0.794356\pi$
$733$	−31691.5	−1.59694	−0.798468	+	0.602038i	$0.794356\pi$
$734$	11997.7	0.603327
$735$	0	0
$736$	−17183.7	−0.860596
$737$	−33027.1	−1.65070
$738$	0	0
$739$	−13196.0	−0.656862	−0.328431	−	0.944528i	$-0.606520\pi$
$739$	−13196.0	−0.656862	−0.328431	+	0.944528i	$0.606520\pi$
$740$	−32065.1	−1.59289
$741$	0	0
$742$	0	0
$743$	−8244.34	−0.407073	−0.203537	−	0.979067i	$-0.565244\pi$
$743$	−8244.34	−0.407073	−0.203537	+	0.979067i	$0.565244\pi$
$744$	0	0
$745$	−12605.0	−0.619883
$746$	8471.90	0.415788
$747$	0	0
$748$	16685.6	0.815622
$749$	0	0
$750$	0	0
$751$	31769.8	1.54367	0.771834	−	0.635824i	$-0.219339\pi$
$751$	31769.8	1.54367	0.771834	+	0.635824i	$0.219339\pi$
$752$	19900.7	0.965031
$753$	0	0
$754$	−77253.7	−3.73132
$755$	6896.60	0.332441
$756$	0	0
$757$	−3019.09	−0.144954	−0.0724772	−	0.997370i	$-0.523090\pi$
$757$	−3019.09	−0.144954	−0.0724772	+	0.997370i	$0.523090\pi$
$758$	−27202.2	−1.30347
$759$	0	0
$760$	19960.4	0.952686
$761$	24096.3	1.14782	0.573910	−	0.818918i	$-0.305426\pi$
$761$	24096.3	1.14782	0.573910	+	0.818918i	$0.305426\pi$
$762$	0	0
$763$	0	0
$764$	28573.1	1.35306
$765$	0	0
$766$	−31858.3	−1.50273
$767$	1665.33	0.0783984
$768$	0	0
$769$	−7954.16	−0.372996	−0.186498	−	0.982455i	$-0.559714\pi$
$769$	−7954.16	−0.372996	−0.186498	+	0.982455i	$0.559714\pi$
$770$	0	0
$771$	0	0
$772$	−70369.1	−3.28062
$773$	36239.4	1.68621	0.843105	−	0.537749i	$-0.180725\pi$
$773$	36239.4	1.68621	0.843105	+	0.537749i	$0.180725\pi$
$774$	0	0
$775$	4393.22	0.203625
$776$	−32880.0	−1.52104
$777$	0	0
$778$	48428.7	2.23169
$779$	−19337.1	−0.889377
$780$	0	0
$781$	25764.6	1.18045
$782$	−10424.4	−0.476696
$783$	0	0
$784$	0	0
$785$	−9963.65	−0.453017
$786$	0	0
$787$	9684.13	0.438630	0.219315	−	0.975654i	$-0.429618\pi$
$787$	9684.13	0.438630	0.219315	+	0.975654i	$0.429618\pi$
$788$	−17957.3	−0.811803
$789$	0	0
$790$	13297.5	0.598864
$791$	0	0
$792$	0	0
$793$	−62034.1	−2.77793
$794$	40298.0	1.80116
$795$	0	0
$796$	55648.1	2.47788
$797$	6717.98	0.298573	0.149287	−	0.988794i	$-0.452302\pi$
$797$	6717.98	0.298573	0.149287	+	0.988794i	$0.452302\pi$
$798$	0	0
$799$	3826.05	0.169407
$800$	−4769.95	−0.210804
$801$	0	0
$802$	−63615.3	−2.80092
$803$	−37267.7	−1.63779
$804$	0	0
$805$	0	0
$806$	63636.5	2.78102
$807$	0	0
$808$	32584.4	1.41871
$809$	1210.73	0.0526167	0.0263083	−	0.999654i	$-0.491625\pi$
$809$	1210.73	0.0526167	0.0263083	+	0.999654i	$0.491625\pi$
$810$	0	0
$811$	−34825.1	−1.50786	−0.753930	−	0.656955i	$-0.771844\pi$
$811$	−34825.1	−1.50786	−0.753930	+	0.656955i	$0.771844\pi$
$812$	0	0
$813$	0	0
$814$	73820.0	3.17861
$815$	9514.72	0.408940
$816$	0	0
$817$	−23136.1	−0.990734
$818$	17377.9	0.742792
$819$	0	0
$820$	22495.1	0.958003
$821$	−30169.6	−1.28249	−0.641247	−	0.767334i	$-0.721583\pi$
$821$	−30169.6	−1.28249	−0.641247	+	0.767334i	$0.721583\pi$
$822$	0	0
$823$	−26016.4	−1.10191	−0.550956	−	0.834534i	$-0.685737\pi$
$823$	−26016.4	−1.10191	−0.550956	+	0.834534i	$0.685737\pi$
$824$	−19923.7	−0.842322
$825$	0	0
$826$	0	0
$827$	−15602.4	−0.656045	−0.328022	−	0.944670i	$-0.606382\pi$
$827$	−15602.4	−0.656045	−0.328022	+	0.944670i	$0.606382\pi$
$828$	0	0
$829$	19506.0	0.817215	0.408607	−	0.912710i	$-0.366015\pi$
$829$	19506.0	0.817215	0.408607	+	0.912710i	$0.366015\pi$
$830$	26234.9	1.09714
$831$	0	0
$832$	−2186.96	−0.0911289
$833$	0	0
$834$	0	0
$835$	−1890.92	−0.0783687
$836$	−57158.2	−2.36466
$837$	0	0
$838$	7029.71	0.289782
$839$	−12231.2	−0.503300	−0.251650	−	0.967818i	$-0.580973\pi$
$839$	−12231.2	−0.503300	−0.251650	+	0.967818i	$0.580973\pi$
$840$	0	0
$841$	21121.6	0.866028
$842$	80452.0	3.29283
$843$	0	0
$844$	38209.3	1.55831
$845$	14167.7	0.576784
$846$	0	0
$847$	0	0
$848$	12083.2	0.489314
$849$	0	0
$850$	−2893.67	−0.116767
$851$	−31966.0	−1.28764
$852$	0	0
$853$	−1234.83	−0.0495660	−0.0247830	−	0.999693i	$-0.507889\pi$
$853$	−1234.83	−0.0495660	−0.0247830	+	0.999693i	$0.507889\pi$
$854$	0	0
$855$	0	0
$856$	7131.49	0.284754
$857$	24101.4	0.960662	0.480331	−	0.877087i	$-0.340517\pi$
$857$	24101.4	0.960662	0.480331	+	0.877087i	$0.340517\pi$
$858$	0	0
$859$	−13362.9	−0.530777	−0.265389	−	0.964142i	$-0.585500\pi$
$859$	−13362.9	−0.530777	−0.265389	+	0.964142i	$0.585500\pi$
$860$	26914.5	1.06718
$861$	0	0
$862$	25955.9	1.02559
$863$	21934.1	0.865174	0.432587	−	0.901592i	$-0.357601\pi$
$863$	21934.1	0.865174	0.432587	+	0.901592i	$0.357601\pi$
$864$	0	0
$865$	−3320.18	−0.130508
$866$	23646.8	0.927888
$867$	0	0
$868$	0	0
$869$	−21218.5	−0.828296
$870$	0	0
$871$	57505.0	2.23706
$872$	88056.1	3.41967
$873$	0	0
$874$	35709.9	1.38204
$875$	0	0
$876$	0	0
$877$	−770.318	−0.0296600	−0.0148300	−	0.999890i	$-0.504721\pi$
$877$	−770.318	−0.0296600	−0.0148300	+	0.999890i	$0.504721\pi$
$878$	−11941.6	−0.459009
$879$	0	0
$880$	24016.7	0.920003
$881$	17887.9	0.684064	0.342032	−	0.939688i	$-0.388885\pi$
$881$	17887.9	0.684064	0.342032	+	0.939688i	$0.388885\pi$
$882$	0	0
$883$	583.389	0.0222340	0.0111170	−	0.999938i	$-0.496461\pi$
$883$	583.389	0.0222340	0.0111170	+	0.999938i	$0.496461\pi$
$884$	−29052.0	−1.10534
$885$	0	0
$886$	60306.0	2.28670
$887$	−42443.8	−1.60668	−0.803340	−	0.595521i	$-0.796946\pi$
$887$	−42443.8	−1.60668	−0.803340	+	0.595521i	$0.796946\pi$
$888$	0	0
$889$	0	0
$890$	−25624.3	−0.965087
$891$	0	0
$892$	−46587.8	−1.74874
$893$	−13106.5	−0.491146
$894$	0	0
$895$	1675.00	0.0625575
$896$	0	0
$897$	0	0
$898$	48924.4	1.81807
$899$	−37488.6	−1.39078
$900$	0	0
$901$	2323.08	0.0858967
$902$	−51788.1	−1.91170
$903$	0	0
$904$	107096.	3.94021
$905$	−6127.92	−0.225082
$906$	0	0
$907$	−27717.8	−1.01472	−0.507362	−	0.861733i	$-0.669379\pi$
$907$	−27717.8	−1.01472	−0.507362	+	0.861733i	$0.669379\pi$
$908$	52866.3	1.93219
$909$	0	0
$910$	0	0
$911$	−20446.7	−0.743609	−0.371805	−	0.928311i	$-0.621261\pi$
$911$	−20446.7	−0.743609	−0.371805	+	0.928311i	$0.621261\pi$
$912$	0	0
$913$	−41862.6	−1.51747
$914$	33554.4	1.21431
$915$	0	0
$916$	18933.3	0.682941
$917$	0	0
$918$	0	0
$919$	17455.5	0.626556	0.313278	−	0.949662i	$-0.398573\pi$
$919$	17455.5	0.626556	0.313278	+	0.949662i	$0.398573\pi$
$920$	−23148.5	−0.829546
$921$	0	0
$922$	21431.7	0.765526
$923$	−44859.9	−1.59976
$924$	0	0
$925$	−8873.32	−0.315409
$926$	31276.3	1.10994
$927$	0	0
$928$	40703.3	1.43982
$929$	20748.8	0.732773	0.366387	−	0.930463i	$-0.380595\pi$
$929$	20748.8	0.732773	0.366387	+	0.930463i	$0.380595\pi$
$930$	0	0
$931$	0	0
$932$	69923.8	2.45754
$933$	0	0
$934$	−16481.7	−0.577405
$935$	4617.38	0.161502
$936$	0	0
$937$	−12135.5	−0.423106	−0.211553	−	0.977366i	$-0.567852\pi$
$937$	−12135.5	−0.423106	−0.211553	+	0.977366i	$0.567852\pi$
$938$	0	0
$939$	0	0
$940$	15247.0	0.529044
$941$	−25266.0	−0.875292	−0.437646	−	0.899147i	$-0.644188\pi$
$941$	−25266.0	−0.875292	−0.437646	+	0.899147i	$0.644188\pi$
$942$	0	0
$943$	22425.6	0.774421
$944$	−2768.62	−0.0954566
$945$	0	0
$946$	−61962.3	−2.12957
$947$	42768.7	1.46758	0.733789	−	0.679377i	$-0.237750\pi$
$947$	42768.7	1.46758	0.733789	+	0.679377i	$0.237750\pi$
$948$	0	0
$949$	64888.4	2.21957
$950$	9912.57	0.338533
$951$	0	0
$952$	0	0
$953$	25855.5	0.878846	0.439423	−	0.898280i	$-0.355183\pi$
$953$	25855.5	0.878846	0.439423	+	0.898280i	$0.355183\pi$
$954$	0	0
$955$	7906.98	0.267920
$956$	109507.	3.70473
$957$	0	0
$958$	−15962.0	−0.538318
$959$	0	0
$960$	0	0
$961$	1089.64	0.0365761
$962$	−128531.	−4.30771
$963$	0	0
$964$	75599.6	2.52583
$965$	−19473.1	−0.649598
$966$	0	0
$967$	−20336.0	−0.676278	−0.338139	−	0.941096i	$-0.609797\pi$
$967$	−20336.0	−0.676278	−0.338139	+	0.941096i	$0.609797\pi$
$968$	−16880.3	−0.560488
$969$	0	0
$970$	−16328.6	−0.540493
$971$	−31123.7	−1.02864	−0.514319	−	0.857599i	$-0.671955\pi$
$971$	−31123.7	−1.02864	−0.514319	+	0.857599i	$0.671955\pi$
$972$	0	0
$973$	0	0
$974$	4437.05	0.145967
$975$	0	0
$976$	103132.	3.38236
$977$	−31317.3	−1.02552	−0.512758	−	0.858533i	$-0.671376\pi$
$977$	−31317.3	−1.02552	−0.512758	+	0.858533i	$0.671376\pi$
$978$	0	0
$979$	40888.2	1.33482
$980$	0	0
$981$	0	0
$982$	33788.1	1.09798
$983$	−4002.12	−0.129855	−0.0649277	−	0.997890i	$-0.520682\pi$
$983$	−4002.12	−0.129855	−0.0649277	+	0.997890i	$0.520682\pi$
$984$	0	0
$985$	−4969.29	−0.160746
$986$	24692.5	0.797535
$987$	0	0
$988$	99520.7	3.20463
$989$	26831.3	0.862677
$990$	0	0
$991$	10042.5	0.321906	0.160953	−	0.986962i	$-0.448543\pi$
$991$	10042.5	0.321906	0.160953	+	0.986962i	$0.448543\pi$
$992$	−33528.7	−1.07312
$993$	0	0
$994$	0	0
$995$	15399.4	0.490647
$996$	0	0
$997$	9407.44	0.298833	0.149417	−	0.988774i	$-0.452260\pi$
$997$	9407.44	0.298833	0.149417	+	0.988774i	$0.452260\pi$
$998$	4534.65	0.143830
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bs.1.1		5
3.2	odd	2		735.4.a.z.1.5		5
7.3	odd	6		315.4.j.h.226.5		10
7.5	odd	6		315.4.j.h.46.5		10
7.6	odd	2		2205.4.a.br.1.1		5
21.5	even	6		105.4.i.d.46.1	yes	10
21.17	even	6		105.4.i.d.16.1	✓	10
21.20	even	2		735.4.a.ba.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.1	✓	10	21.17	even	6
105.4.i.d.46.1	yes	10	21.5	even	6
315.4.j.h.46.5		10	7.5	odd	6
315.4.j.h.226.5		10	7.3	odd	6
735.4.a.z.1.5		5	3.2	odd	2
735.4.a.ba.1.5		5	21.20	even	2
2205.4.a.br.1.1		5	7.6	odd	2
2205.4.a.bs.1.1		5	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-5.10571 q^{2} +18.0682 q^{4} +5.00000 q^{5} -51.4055 q^{8} +O(q^{10})\)	\(q-5.10571 q^{2} +18.0682 q^{4} +5.00000 q^{5} -51.4055 q^{8} -25.5285 q^{10} +40.7354 q^{11} -70.9263 q^{13} +117.915 q^{16} +22.6701 q^{17} -77.6588 q^{19} +90.3412 q^{20} -207.983 q^{22} +90.0622 q^{23} +25.0000 q^{25} +362.129 q^{26} -213.332 q^{29} +175.729 q^{31} -190.798 q^{32} -115.747 q^{34} -354.933 q^{37} +396.503 q^{38} -257.028 q^{40} +249.001 q^{41} +297.920 q^{43} +736.018 q^{44} -459.831 q^{46} +168.771 q^{47} -127.643 q^{50} -1281.51 q^{52} +102.473 q^{53} +203.677 q^{55} +1089.21 q^{58} -23.4797 q^{59} +874.628 q^{61} -897.220 q^{62} +30.8343 q^{64} -354.631 q^{65} -810.771 q^{67} +409.609 q^{68} +632.487 q^{71} -914.872 q^{73} +1812.18 q^{74} -1403.16 q^{76} -520.886 q^{79} +589.577 q^{80} -1271.33 q^{82} -1027.67 q^{83} +113.350 q^{85} -1521.09 q^{86} -2094.02 q^{88} +1003.75 q^{89} +1627.27 q^{92} -861.694 q^{94} -388.294 q^{95} +639.620 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} + 25 q^{4} + 25 q^{5} - 21 q^{8}+O(q^{10}) \) 5 * q - 3 * q^2 + 25 * q^4 + 25 * q^5 - 21 * q^8	\( 5 q - 3 q^{2} + 25 q^{4} + 25 q^{5} - 21 q^{8} - 15 q^{10} - 43 q^{11} - 123 q^{13} + 161 q^{16} + 124 q^{17} - 37 q^{19} + 125 q^{20} - 221 q^{22} - 77 q^{23} + 125 q^{25} - 79 q^{26} - 360 q^{29} - 314 q^{31} + 59 q^{32} - 176 q^{34} + 225 q^{37} + 759 q^{38} - 105 q^{40} + 341 q^{41} + 32 q^{43} - 679 q^{44} - 331 q^{46} + 25 q^{47} - 75 q^{50} - 2299 q^{52} + 317 q^{53} - 215 q^{55} + 8 q^{58} + 676 q^{59} + 188 q^{61} - 348 q^{62} - 1103 q^{64} - 615 q^{65} - 1776 q^{67} + 1280 q^{68} + 6 q^{71} - 2006 q^{73} + 2729 q^{74} - 1417 q^{76} + 200 q^{79} + 805 q^{80} + 539 q^{82} - 332 q^{83} + 620 q^{85} - 4262 q^{86} - 4529 q^{88} + 894 q^{89} + 3687 q^{92} - 4233 q^{94} - 185 q^{95} + 576 q^{97}+O(q^{100}) \) 5 * q - 3 * q^2 + 25 * q^4 + 25 * q^5 - 21 * q^8 - 15 * q^10 - 43 * q^11 - 123 * q^13 + 161 * q^16 + 124 * q^17 - 37 * q^19 + 125 * q^20 - 221 * q^22 - 77 * q^23 + 125 * q^25 - 79 * q^26 - 360 * q^29 - 314 * q^31 + 59 * q^32 - 176 * q^34 + 225 * q^37 + 759 * q^38 - 105 * q^40 + 341 * q^41 + 32 * q^43 - 679 * q^44 - 331 * q^46 + 25 * q^47 - 75 * q^50 - 2299 * q^52 + 317 * q^53 - 215 * q^55 + 8 * q^58 + 676 * q^59 + 188 * q^61 - 348 * q^62 - 1103 * q^64 - 615 * q^65 - 1776 * q^67 + 1280 * q^68 + 6 * q^71 - 2006 * q^73 + 2729 * q^74 - 1417 * q^76 + 200 * q^79 + 805 * q^80 + 539 * q^82 - 332 * q^83 + 620 * q^85 - 4262 * q^86 - 4529 * q^88 + 894 * q^89 + 3687 * q^92 - 4233 * q^94 - 185 * q^95 + 576 * q^97

Introduction

L-functions

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