LMFDB - Embedded newform 2205.4.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,4,Mod(1,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,-2,0,14,15,0,0,-66,0,-10,20,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 14 $ x^3 - x^2 - 18*x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.37989$ 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)

$f(q)$	$=$	$q+3.37989 q^{2} +3.42368 q^{4} +5.00000 q^{5} -15.4675 q^{8} +16.8995 q^{10} -45.0132 q^{11} +35.4937 q^{13} -79.6679 q^{16} +29.4413 q^{17} -3.18839 q^{19} +17.1184 q^{20} -152.140 q^{22} +23.6429 q^{23} +25.0000 q^{25} +119.965 q^{26} -9.22109 q^{29} +80.2011 q^{31} -145.529 q^{32} +99.5084 q^{34} -61.1957 q^{37} -10.7764 q^{38} -77.3373 q^{40} +282.085 q^{41} -58.8504 q^{43} -154.111 q^{44} +79.9105 q^{46} +371.773 q^{47} +84.4973 q^{50} +121.519 q^{52} +256.184 q^{53} -225.066 q^{55} -31.1663 q^{58} +571.131 q^{59} -835.587 q^{61} +271.071 q^{62} +145.470 q^{64} +177.468 q^{65} +933.953 q^{67} +100.798 q^{68} -378.339 q^{71} +494.430 q^{73} -206.835 q^{74} -10.9160 q^{76} +1078.92 q^{79} -398.339 q^{80} +953.418 q^{82} -722.570 q^{83} +147.206 q^{85} -198.908 q^{86} +696.241 q^{88} +89.5596 q^{89} +80.9458 q^{92} +1256.55 q^{94} -15.9420 q^{95} +101.540 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 14 q^{4} + 15 q^{5} - 66 q^{8} - 10 q^{10} + 20 q^{11} + 114 q^{16} + 234 q^{17} + 82 q^{19} + 70 q^{20} - 236 q^{22} - 30 q^{23} + 75 q^{25} + 76 q^{26} + 32 q^{29} + 362 q^{31} - 430 q^{32}+ \cdots + 3052 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 14 * q^4 + 15 * q^5 - 66 * q^8 - 10 * q^10 + 20 * q^11 + 114 * q^16 + 234 * q^17 + 82 * q^19 + 70 * q^20 - 236 * q^22 - 30 * q^23 + 75 * q^25 + 76 * q^26 + 32 * q^29 + 362 * q^31 - 430 * q^32 - 596 * q^34 + 302 * q^37 + 160 * q^38 - 330 * q^40 + 626 * q^41 - 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 - 50 * q^50 + 684 * q^52 - 298 * q^53 + 100 * q^55 - 1184 * q^58 + 428 * q^59 - 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 - 1712 * q^76 - 488 * q^79 + 570 * q^80 + 1476 * q^82 + 948 * q^83 + 1170 * q^85 + 1928 * q^86 + 124 * q^88 - 142 * q^89 + 3220 * q^92 - 328 * q^94 + 410 * q^95 + 3052 * q^97

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 $

$2$	3.37989	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$2$	3.37989	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$3$	0	0
$3$	0	0
$4$	3.42368	0.427960
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−15.4675	−0.683572
$9$	0	0
$10$	16.8995	0.534408
$11$	−45.0132	−1.23382	−0.616909	−	0.787034i	$-0.711615\pi$
$11$	−45.0132	−1.23382	−0.616909	+	0.787034i	$0.711615\pi$
$12$	0	0
$13$	35.4937	0.757244	0.378622	−	0.925551i	$-0.376398\pi$
$13$	35.4937	0.757244	0.378622	+	0.925551i	$0.376398\pi$
$14$	0	0
$15$	0	0
$16$	−79.6679	−1.24481
$17$	29.4413	0.420033	0.210016	−	0.977698i	$-0.432648\pi$
$17$	29.4413	0.420033	0.210016	+	0.977698i	$0.432648\pi$
$18$	0	0
$19$	−3.18839	−0.0384983	−0.0192491	−	0.999815i	$-0.506128\pi$
$19$	−3.18839	−0.0384983	−0.0192491	+	0.999815i	$0.506128\pi$
$20$	17.1184	0.191390
$21$	0	0
$22$	−152.140	−1.47438
$23$	23.6429	0.214343	0.107171	−	0.994241i	$-0.465821\pi$
$23$	23.6429	0.214343	0.107171	+	0.994241i	$0.465821\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	119.965	0.904886
$27$	0	0
$28$	0	0
$29$	−9.22109	−0.0590453	−0.0295227	−	0.999564i	$-0.509399\pi$
$29$	−9.22109	−0.0590453	−0.0295227	+	0.999564i	$0.509399\pi$
$30$	0	0
$31$	80.2011	0.464662	0.232331	−	0.972637i	$-0.425365\pi$
$31$	80.2011	0.464662	0.232331	+	0.972637i	$0.425365\pi$
$32$	−145.529	−0.803942
$33$	0	0
$34$	99.5084	0.501928
$35$	0	0
$36$	0	0
$37$	−61.1957	−0.271906	−0.135953	−	0.990715i	$-0.543410\pi$
$37$	−61.1957	−0.271906	−0.135953	+	0.990715i	$0.543410\pi$
$38$	−10.7764	−0.0460044
$39$	0	0
$40$	−77.3373	−0.305703
$41$	282.085	1.07449	0.537247	−	0.843425i	$-0.319464\pi$
$41$	282.085	1.07449	0.537247	+	0.843425i	$0.319464\pi$
$42$	0	0
$43$	−58.8504	−0.208712	−0.104356	−	0.994540i	$-0.533278\pi$
$43$	−58.8504	−0.208712	−0.104356	+	0.994540i	$0.533278\pi$
$44$	−154.111	−0.528025
$45$	0	0
$46$	79.9105	0.256134
$47$	371.773	1.15380	0.576901	−	0.816814i	$-0.304262\pi$
$47$	371.773	1.15380	0.576901	+	0.816814i	$0.304262\pi$
$48$	0	0
$49$	0	0
$50$	84.4973	0.238995
$51$	0	0
$52$	121.519	0.324070
$53$	256.184	0.663953	0.331977	−	0.943288i	$-0.392284\pi$
$53$	256.184	0.663953	0.331977	+	0.943288i	$0.392284\pi$
$54$	0	0
$55$	−225.066	−0.551780
$56$	0	0
$57$	0	0
$58$	−31.1663	−0.0705575
$59$	571.131	1.26025	0.630127	−	0.776492i	$-0.283003\pi$
$59$	571.131	1.26025	0.630127	+	0.776492i	$0.283003\pi$
$60$	0	0
$61$	−835.587	−1.75387	−0.876934	−	0.480610i	$-0.840415\pi$
$61$	−835.587	−1.75387	−0.876934	+	0.480610i	$0.840415\pi$
$62$	271.071	0.555259
$63$	0	0
$64$	145.470	0.284121
$65$	177.468	0.338650
$66$	0	0
$67$	933.953	1.70299	0.851497	−	0.524360i	$-0.175695\pi$
$67$	933.953	1.70299	0.851497	+	0.524360i	$0.175695\pi$
$68$	100.798	0.179757
$69$	0	0
$70$	0	0
$71$	−378.339	−0.632403	−0.316201	−	0.948692i	$-0.602408\pi$
$71$	−378.339	−0.632403	−0.316201	+	0.948692i	$0.602408\pi$
$72$	0	0
$73$	494.430	0.792722	0.396361	−	0.918095i	$-0.370273\pi$
$73$	494.430	0.792722	0.396361	+	0.918095i	$0.370273\pi$
$74$	−206.835	−0.324920
$75$	0	0
$76$	−10.9160	−0.0164757
$77$	0	0
$78$	0	0
$79$	1078.92	1.53656	0.768279	−	0.640115i	$-0.221113\pi$
$79$	1078.92	1.53656	0.768279	+	0.640115i	$0.221113\pi$
$80$	−398.339	−0.556696
$81$	0	0
$82$	953.418	1.28399
$83$	−722.570	−0.955571	−0.477785	−	0.878477i	$-0.658560\pi$
$83$	−722.570	−0.955571	−0.477785	+	0.878477i	$0.658560\pi$
$84$	0	0
$85$	147.206	0.187844
$86$	−198.908	−0.249405
$87$	0	0
$88$	696.241	0.843404
$89$	89.5596	0.106666	0.0533332	−	0.998577i	$-0.483015\pi$
$89$	89.5596	0.106666	0.0533332	+	0.998577i	$0.483015\pi$
$90$	0	0
$91$	0	0
$92$	80.9458	0.0917302
$93$	0	0
$94$	1256.55	1.37876
$95$	−15.9420	−0.0172169
$96$	0	0
$97$	101.540	0.106287	0.0531436	−	0.998587i	$-0.483076\pi$
$97$	101.540	0.106287	0.0531436	+	0.998587i	$0.483076\pi$
$98$	0	0
$99$	0	0
$100$	85.5920	0.0855920
$101$	1024.92	1.00974	0.504868	−	0.863197i	$-0.331541\pi$
$101$	1024.92	1.00974	0.504868	+	0.863197i	$0.331541\pi$
$102$	0	0
$103$	−607.877	−0.581514	−0.290757	−	0.956797i	$-0.593907\pi$
$103$	−607.877	−0.581514	−0.290757	+	0.956797i	$0.593907\pi$
$104$	−548.997	−0.517631
$105$	0	0
$106$	865.873	0.793406
$107$	−169.320	−0.152980	−0.0764898	−	0.997070i	$-0.524371\pi$
$107$	−169.320	−0.152980	−0.0764898	+	0.997070i	$0.524371\pi$
$108$	0	0
$109$	455.114	0.399926	0.199963	−	0.979803i	$-0.435918\pi$
$109$	455.114	0.399926	0.199963	+	0.979803i	$0.435918\pi$
$110$	−760.700	−0.659363
$111$	0	0
$112$	0	0
$113$	2039.49	1.69787	0.848934	−	0.528498i	$-0.177245\pi$
$113$	2039.49	1.69787	0.848934	+	0.528498i	$0.177245\pi$
$114$	0	0
$115$	118.215	0.0958571
$116$	−31.5701	−0.0252690
$117$	0	0
$118$	1930.36	1.50597
$119$	0	0
$120$	0	0
$121$	695.192	0.522308
$122$	−2824.20	−2.09583
$123$	0	0
$124$	274.583	0.198857
$125$	125.000	0.0894427
$126$	0	0
$127$	−2506.64	−1.75141	−0.875703	−	0.482849i	$-0.839602\pi$
$127$	−2506.64	−1.75141	−0.875703	+	0.482849i	$0.839602\pi$
$128$	1655.91	1.14346
$129$	0	0
$130$	599.824	0.404677
$131$	1485.93	0.991039	0.495519	−	0.868597i	$-0.334978\pi$
$131$	1485.93	0.991039	0.495519	+	0.868597i	$0.334978\pi$
$132$	0	0
$133$	0	0
$134$	3156.66	2.03503
$135$	0	0
$136$	−455.382	−0.287123
$137$	−1238.20	−0.772165	−0.386082	−	0.922464i	$-0.626172\pi$
$137$	−1238.20	−0.772165	−0.386082	+	0.922464i	$0.626172\pi$
$138$	0	0
$139$	2861.84	1.74632	0.873158	−	0.487437i	$-0.162068\pi$
$139$	2861.84	1.74632	0.873158	+	0.487437i	$0.162068\pi$
$140$	0	0
$141$	0	0
$142$	−1278.75	−0.755704
$143$	−1597.68	−0.934301
$144$	0	0
$145$	−46.1055	−0.0264059
$146$	1671.12	0.947281
$147$	0	0
$148$	−209.515	−0.116365
$149$	−1536.08	−0.844569	−0.422284	−	0.906463i	$-0.638772\pi$
$149$	−1536.08	−0.844569	−0.422284	+	0.906463i	$0.638772\pi$
$150$	0	0
$151$	3520.06	1.89707	0.948537	−	0.316665i	$-0.102563\pi$
$151$	3520.06	1.89707	0.948537	+	0.316665i	$0.102563\pi$
$152$	49.3163	0.0263163
$153$	0	0
$154$	0	0
$155$	401.005	0.207803
$156$	0	0
$157$	962.671	0.489360	0.244680	−	0.969604i	$-0.421317\pi$
$157$	962.671	0.489360	0.244680	+	0.969604i	$0.421317\pi$
$158$	3646.64	1.83615
$159$	0	0
$160$	−727.646	−0.359534
$161$	0	0
$162$	0	0
$163$	−1558.10	−0.748709	−0.374354	−	0.927286i	$-0.622136\pi$
$163$	−1558.10	−0.748709	−0.374354	+	0.927286i	$0.622136\pi$
$164$	965.770	0.459841
$165$	0	0
$166$	−2442.21	−1.14188
$167$	1145.95	0.530996	0.265498	−	0.964111i	$-0.414464\pi$
$167$	1145.95	0.530996	0.265498	+	0.964111i	$0.414464\pi$
$168$	0	0
$169$	−937.200	−0.426582
$170$	497.542	0.224469
$171$	0	0
$172$	−201.485	−0.0893203
$173$	1277.72	0.561522	0.280761	−	0.959778i	$-0.409413\pi$
$173$	1277.72	0.561522	0.280761	+	0.959778i	$0.409413\pi$
$174$	0	0
$175$	0	0
$176$	3586.11	1.53587
$177$	0	0
$178$	302.702	0.127463
$179$	1888.13	0.788412	0.394206	−	0.919022i	$-0.371020\pi$
$179$	1888.13	0.788412	0.394206	+	0.919022i	$0.371020\pi$
$180$	0	0
$181$	1193.79	0.490242	0.245121	−	0.969493i	$-0.421172\pi$
$181$	1193.79	0.490242	0.245121	+	0.969493i	$0.421172\pi$
$182$	0	0
$183$	0	0
$184$	−365.696	−0.146519
$185$	−305.979	−0.121600
$186$	0	0
$187$	−1325.25	−0.518244
$188$	1272.83	0.493781
$189$	0	0
$190$	−53.8821	−0.0205738
$191$	−401.072	−0.151940	−0.0759700	−	0.997110i	$-0.524205\pi$
$191$	−401.072	−0.151940	−0.0759700	+	0.997110i	$0.524205\pi$
$192$	0	0
$193$	3984.55	1.48608	0.743042	−	0.669245i	$-0.233382\pi$
$193$	3984.55	1.48608	0.743042	+	0.669245i	$0.233382\pi$
$194$	343.195	0.127010
$195$	0	0
$196$	0	0
$197$	−3478.55	−1.25805	−0.629027	−	0.777383i	$-0.716547\pi$
$197$	−3478.55	−1.25805	−0.629027	+	0.777383i	$0.716547\pi$
$198$	0	0
$199$	2082.09	0.741684	0.370842	−	0.928696i	$-0.379069\pi$
$199$	2082.09	0.741684	0.370842	+	0.928696i	$0.379069\pi$
$200$	−386.687	−0.136714
$201$	0	0
$202$	3464.12	1.20661
$203$	0	0
$204$	0	0
$205$	1410.43	0.480529
$206$	−2054.56	−0.694893
$207$	0	0
$208$	−2827.70	−0.942625
$209$	143.520	0.0474999
$210$	0	0
$211$	−2423.61	−0.790751	−0.395375	−	0.918520i	$-0.629385\pi$
$211$	−2423.61	−0.790751	−0.395375	+	0.918520i	$0.629385\pi$
$212$	877.091	0.284146
$213$	0	0
$214$	−572.285	−0.182807
$215$	−294.252	−0.0933387
$216$	0	0
$217$	0	0
$218$	1538.24	0.477901
$219$	0	0
$220$	−770.555	−0.236140
$221$	1044.98	0.318067
$222$	0	0
$223$	−959.968	−0.288270	−0.144135	−	0.989558i	$-0.546040\pi$
$223$	−959.968	−0.288270	−0.144135	+	0.989558i	$0.546040\pi$
$224$	0	0
$225$	0	0
$226$	6893.26	2.02891
$227$	−1018.79	−0.297883	−0.148942	−	0.988846i	$-0.547587\pi$
$227$	−1018.79	−0.297883	−0.148942	+	0.988846i	$0.547587\pi$
$228$	0	0
$229$	−3281.03	−0.946799	−0.473399	−	0.880848i	$-0.656973\pi$
$229$	−3281.03	−0.946799	−0.473399	+	0.880848i	$0.656973\pi$
$230$	399.553	0.114547
$231$	0	0
$232$	142.627	0.0403617
$233$	−4682.68	−1.31662	−0.658311	−	0.752746i	$-0.728729\pi$
$233$	−4682.68	−1.31662	−0.658311	+	0.752746i	$0.728729\pi$
$234$	0	0
$235$	1858.87	0.515996
$236$	1955.37	0.539339
$237$	0	0
$238$	0	0
$239$	3146.54	0.851602	0.425801	−	0.904817i	$-0.359992\pi$
$239$	3146.54	0.851602	0.425801	+	0.904817i	$0.359992\pi$
$240$	0	0
$241$	−7082.09	−1.89294	−0.946468	−	0.322798i	$-0.895376\pi$
$241$	−7082.09	−1.89294	−0.946468	+	0.322798i	$0.895376\pi$
$242$	2349.67	0.624144
$243$	0	0
$244$	−2860.78	−0.750586
$245$	0	0
$246$	0	0
$247$	−113.168	−0.0291526
$248$	−1240.51	−0.317630
$249$	0	0
$250$	422.487	0.106882
$251$	7196.53	1.80972	0.904862	−	0.425705i	$-0.139974\pi$
$251$	7196.53	1.80972	0.904862	+	0.425705i	$0.139974\pi$
$252$	0	0
$253$	−1064.24	−0.264460
$254$	−8472.19	−2.09288
$255$	0	0
$256$	4433.03	1.08228
$257$	3886.10	0.943222	0.471611	−	0.881807i	$-0.343673\pi$
$257$	3886.10	0.943222	0.471611	+	0.881807i	$0.343673\pi$
$258$	0	0
$259$	0	0
$260$	607.595	0.144929
$261$	0	0
$262$	5022.28	1.18426
$263$	−2066.88	−0.484597	−0.242299	−	0.970202i	$-0.577901\pi$
$263$	−2066.88	−0.484597	−0.242299	+	0.970202i	$0.577901\pi$
$264$	0	0
$265$	1280.92	0.296929
$266$	0	0
$267$	0	0
$268$	3197.56	0.728813
$269$	−2230.84	−0.505638	−0.252819	−	0.967514i	$-0.581358\pi$
$269$	−2230.84	−0.505638	−0.252819	+	0.967514i	$0.581358\pi$
$270$	0	0
$271$	836.594	0.187526	0.0937629	−	0.995595i	$-0.470110\pi$
$271$	836.594	0.187526	0.0937629	+	0.995595i	$0.470110\pi$
$272$	−2345.52	−0.522861
$273$	0	0
$274$	−4184.98	−0.922716
$275$	−1125.33	−0.246764
$276$	0	0
$277$	−3674.18	−0.796968	−0.398484	−	0.917175i	$-0.630464\pi$
$277$	−3674.18	−0.796968	−0.398484	+	0.917175i	$0.630464\pi$
$278$	9672.71	2.08680
$279$	0	0
$280$	0	0
$281$	4363.28	0.926305	0.463152	−	0.886279i	$-0.346718\pi$
$281$	4363.28	0.926305	0.463152	+	0.886279i	$0.346718\pi$
$282$	0	0
$283$	7166.82	1.50538	0.752691	−	0.658374i	$-0.228755\pi$
$283$	7166.82	1.50538	0.752691	+	0.658374i	$0.228755\pi$
$284$	−1295.31	−0.270643
$285$	0	0
$286$	−5400.00	−1.11646
$287$	0	0
$288$	0	0
$289$	−4046.21	−0.823572
$290$	−155.832	−0.0315543
$291$	0	0
$292$	1692.77	0.339253
$293$	5529.75	1.10257	0.551283	−	0.834319i	$-0.314139\pi$
$293$	5529.75	1.10257	0.551283	+	0.834319i	$0.314139\pi$
$294$	0	0
$295$	2855.66	0.563603
$296$	946.543	0.185867
$297$	0	0
$298$	−5191.79	−1.00924
$299$	839.173	0.162310
$300$	0	0
$301$	0	0
$302$	11897.4	2.26695
$303$	0	0
$304$	254.012	0.0479230
$305$	−4177.94	−0.784354
$306$	0	0
$307$	8431.95	1.56755	0.783774	−	0.621047i	$-0.213292\pi$
$307$	8431.95	1.56755	0.783774	+	0.621047i	$0.213292\pi$
$308$	0	0
$309$	0	0
$310$	1355.36	0.248319
$311$	−3579.03	−0.652566	−0.326283	−	0.945272i	$-0.605796\pi$
$311$	−3579.03	−0.652566	−0.326283	+	0.945272i	$0.605796\pi$
$312$	0	0
$313$	7766.04	1.40244	0.701218	−	0.712947i	$-0.252640\pi$
$313$	7766.04	1.40244	0.701218	+	0.712947i	$0.252640\pi$
$314$	3253.72	0.584772
$315$	0	0
$316$	3693.88	0.657586
$317$	−10110.0	−1.79128	−0.895639	−	0.444781i	$-0.853281\pi$
$317$	−10110.0	−1.79128	−0.895639	+	0.444781i	$0.853281\pi$
$318$	0	0
$319$	415.071	0.0728512
$320$	727.349	0.127063
$321$	0	0
$322$	0	0
$323$	−93.8703	−0.0161705
$324$	0	0
$325$	887.342	0.151449
$326$	−5266.20	−0.894686
$327$	0	0
$328$	−4363.14	−0.734495
$329$	0	0
$330$	0	0
$331$	−3124.35	−0.518821	−0.259411	−	0.965767i	$-0.583528\pi$
$331$	−3124.35	−0.518821	−0.259411	+	0.965767i	$0.583528\pi$
$332$	−2473.85	−0.408946
$333$	0	0
$334$	3873.19	0.634525
$335$	4669.77	0.761602
$336$	0	0
$337$	−4502.27	−0.727757	−0.363879	−	0.931446i	$-0.618548\pi$
$337$	−4502.27	−0.727757	−0.363879	+	0.931446i	$0.618548\pi$
$338$	−3167.64	−0.509753
$339$	0	0
$340$	503.988	0.0803899
$341$	−3610.11	−0.573309
$342$	0	0
$343$	0	0
$344$	910.267	0.142670
$345$	0	0
$346$	4318.56	0.671003
$347$	11504.3	1.77978	0.889888	−	0.456179i	$-0.150782\pi$
$347$	11504.3	1.77978	0.889888	+	0.456179i	$0.150782\pi$
$348$	0	0
$349$	−917.499	−0.140724	−0.0703619	−	0.997522i	$-0.522415\pi$
$349$	−917.499	−0.140724	−0.0703619	+	0.997522i	$0.522415\pi$
$350$	0	0
$351$	0	0
$352$	6550.74	0.991919
$353$	3548.38	0.535017	0.267509	−	0.963555i	$-0.413800\pi$
$353$	3548.38	0.535017	0.267509	+	0.963555i	$0.413800\pi$
$354$	0	0
$355$	−1891.70	−0.282819
$356$	306.624	0.0456489
$357$	0	0
$358$	6381.69	0.942130
$359$	7875.13	1.15775	0.578876	−	0.815415i	$-0.303491\pi$
$359$	7875.13	1.15775	0.578876	+	0.815415i	$0.303491\pi$
$360$	0	0
$361$	−6848.83	−0.998518
$362$	4034.89	0.585826
$363$	0	0
$364$	0	0
$365$	2472.15	0.354516
$366$	0	0
$367$	−6090.90	−0.866327	−0.433164	−	0.901315i	$-0.642603\pi$
$367$	−6090.90	−0.866327	−0.433164	+	0.901315i	$0.642603\pi$
$368$	−1883.58	−0.266816
$369$	0	0
$370$	−1034.18	−0.145309
$371$	0	0
$372$	0	0
$373$	−11782.2	−1.63555	−0.817777	−	0.575535i	$-0.804794\pi$
$373$	−11782.2	−1.63555	−0.817777	+	0.575535i	$0.804794\pi$
$374$	−4479.19	−0.619288
$375$	0	0
$376$	−5750.39	−0.788706
$377$	−327.290	−0.0447117
$378$	0	0
$379$	6408.54	0.868561	0.434280	−	0.900778i	$-0.357003\pi$
$379$	6408.54	0.868561	0.434280	+	0.900778i	$0.357003\pi$
$380$	−54.5802	−0.00736817
$381$	0	0
$382$	−1355.58	−0.181564
$383$	−3769.38	−0.502889	−0.251444	−	0.967872i	$-0.580906\pi$
$383$	−3769.38	−0.502889	−0.251444	+	0.967872i	$0.580906\pi$
$384$	0	0
$385$	0	0
$386$	13467.4	1.77583
$387$	0	0
$388$	347.642	0.0454867
$389$	3205.72	0.417831	0.208915	−	0.977934i	$-0.433007\pi$
$389$	3205.72	0.417831	0.208915	+	0.977934i	$0.433007\pi$
$390$	0	0
$391$	696.077	0.0900311
$392$	0	0
$393$	0	0
$394$	−11757.1	−1.50334
$395$	5394.61	0.687170
$396$	0	0
$397$	8989.91	1.13650	0.568250	−	0.822856i	$-0.307620\pi$
$397$	8989.91	1.13650	0.568250	+	0.822856i	$0.307620\pi$
$398$	7037.23	0.886293
$399$	0	0
$400$	−1991.70	−0.248962
$401$	15270.8	1.90172	0.950859	−	0.309625i	$-0.100203\pi$
$401$	15270.8	1.90172	0.950859	+	0.309625i	$0.100203\pi$
$402$	0	0
$403$	2846.63	0.351863
$404$	3509.00	0.432127
$405$	0	0
$406$	0	0
$407$	2754.62	0.335483
$408$	0	0
$409$	−13450.1	−1.62608	−0.813040	−	0.582208i	$-0.802189\pi$
$409$	−13450.1	−1.62608	−0.813040	+	0.582208i	$0.802189\pi$
$410$	4767.09	0.574219
$411$	0	0
$412$	−2081.18	−0.248865
$413$	0	0
$414$	0	0
$415$	−3612.85	−0.427344
$416$	−5165.36	−0.608781
$417$	0	0
$418$	485.082	0.0567611
$419$	−7525.80	−0.877468	−0.438734	−	0.898617i	$-0.644573\pi$
$419$	−7525.80	−0.877468	−0.438734	+	0.898617i	$0.644573\pi$
$420$	0	0
$421$	6677.45	0.773014	0.386507	−	0.922286i	$-0.373681\pi$
$421$	6677.45	0.773014	0.386507	+	0.922286i	$0.373681\pi$
$422$	−8191.55	−0.944926
$423$	0	0
$424$	−3962.51	−0.453860
$425$	736.032	0.0840066
$426$	0	0
$427$	0	0
$428$	−579.699	−0.0654692
$429$	0	0
$430$	−994.541	−0.111537
$431$	7676.15	0.857882	0.428941	−	0.903332i	$-0.358887\pi$
$431$	7676.15	0.857882	0.428941	+	0.903332i	$0.358887\pi$
$432$	0	0
$433$	−4853.67	−0.538690	−0.269345	−	0.963044i	$-0.586807\pi$
$433$	−4853.67	−0.538690	−0.269345	+	0.963044i	$0.586807\pi$
$434$	0	0
$435$	0	0
$436$	1558.16	0.171153
$437$	−75.3828	−0.00825183
$438$	0	0
$439$	−5301.90	−0.576414	−0.288207	−	0.957568i	$-0.593059\pi$
$439$	−5301.90	−0.576414	−0.288207	+	0.957568i	$0.593059\pi$
$440$	3481.20	0.377182
$441$	0	0
$442$	3531.92	0.380082
$443$	1994.30	0.213887	0.106944	−	0.994265i	$-0.465894\pi$
$443$	1994.30	0.213887	0.106944	+	0.994265i	$0.465894\pi$
$444$	0	0
$445$	447.798	0.0477026
$446$	−3244.59	−0.344475
$447$	0	0
$448$	0	0
$449$	432.941	0.0455051	0.0227525	−	0.999741i	$-0.492757\pi$
$449$	432.941	0.0455051	0.0227525	+	0.999741i	$0.492757\pi$
$450$	0	0
$451$	−12697.6	−1.32573
$452$	6982.57	0.726620
$453$	0	0
$454$	−3443.40	−0.355962
$455$	0	0
$456$	0	0
$457$	−4192.20	−0.429109	−0.214555	−	0.976712i	$-0.568830\pi$
$457$	−4192.20	−0.429109	−0.214555	+	0.976712i	$0.568830\pi$
$458$	−11089.5	−1.13140
$459$	0	0
$460$	404.729	0.0410230
$461$	16787.0	1.69598	0.847992	−	0.530009i	$-0.177811\pi$
$461$	16787.0	1.69598	0.847992	+	0.530009i	$0.177811\pi$
$462$	0	0
$463$	5604.79	0.562585	0.281292	−	0.959622i	$-0.409237\pi$
$463$	5604.79	0.562585	0.281292	+	0.959622i	$0.409237\pi$
$464$	734.625	0.0735002
$465$	0	0
$466$	−15827.0	−1.57333
$467$	13011.1	1.28926	0.644630	−	0.764495i	$-0.277012\pi$
$467$	13011.1	1.28926	0.644630	+	0.764495i	$0.277012\pi$
$468$	0	0
$469$	0	0
$470$	6282.77	0.616601
$471$	0	0
$472$	−8833.96	−0.861474
$473$	2649.05	0.257512
$474$	0	0
$475$	−79.7098	−0.00769965
$476$	0	0
$477$	0	0
$478$	10635.0	1.01764
$479$	−16424.3	−1.56670	−0.783348	−	0.621583i	$-0.786490\pi$
$479$	−16424.3	−1.56670	−0.783348	+	0.621583i	$0.786490\pi$
$480$	0	0
$481$	−2172.06	−0.205899
$482$	−23936.7	−2.26201
$483$	0	0
$484$	2380.12	0.223527
$485$	507.701	0.0475330
$486$	0	0
$487$	3448.18	0.320846	0.160423	−	0.987048i	$-0.448714\pi$
$487$	3448.18	0.320846	0.160423	+	0.987048i	$0.448714\pi$
$488$	12924.4	1.19890
$489$	0	0
$490$	0	0
$491$	19657.5	1.80679	0.903393	−	0.428814i	$-0.141068\pi$
$491$	19657.5	1.80679	0.903393	+	0.428814i	$0.141068\pi$
$492$	0	0
$493$	−271.481	−0.0248010
$494$	−382.495	−0.0348365
$495$	0	0
$496$	−6389.45	−0.578417
$497$	0	0
$498$	0	0
$499$	4259.62	0.382138	0.191069	−	0.981577i	$-0.438805\pi$
$499$	4259.62	0.382138	0.191069	+	0.981577i	$0.438805\pi$
$500$	427.960	0.0382779
$501$	0	0
$502$	24323.5	2.16257
$503$	−14981.4	−1.32801	−0.664005	−	0.747728i	$-0.731145\pi$
$503$	−14981.4	−1.32801	−0.664005	+	0.747728i	$0.731145\pi$
$504$	0	0
$505$	5124.60	0.451567
$506$	−3597.03	−0.316023
$507$	0	0
$508$	−8581.95	−0.749532
$509$	3433.20	0.298966	0.149483	−	0.988764i	$-0.452239\pi$
$509$	3433.20	0.298966	0.149483	+	0.988764i	$0.452239\pi$
$510$	0	0
$511$	0	0
$512$	1735.91	0.149838
$513$	0	0
$514$	13134.6	1.12713
$515$	−3039.39	−0.260061
$516$	0	0
$517$	−16734.7	−1.42358
$518$	0	0
$519$	0	0
$520$	−2744.99	−0.231492
$521$	10205.7	0.858194	0.429097	−	0.903258i	$-0.358832\pi$
$521$	10205.7	0.858194	0.429097	+	0.903258i	$0.358832\pi$
$522$	0	0
$523$	−6563.35	−0.548748	−0.274374	−	0.961623i	$-0.588471\pi$
$523$	−6563.35	−0.548748	−0.274374	+	0.961623i	$0.588471\pi$
$524$	5087.34	0.424125
$525$	0	0
$526$	−6985.82	−0.579081
$527$	2361.22	0.195173
$528$	0	0
$529$	−11608.0	−0.954057
$530$	4329.37	0.354822
$531$	0	0
$532$	0	0
$533$	10012.2	0.813655
$534$	0	0
$535$	−846.602	−0.0684146
$536$	−14445.9	−1.16412
$537$	0	0
$538$	−7540.00	−0.604224
$539$	0	0
$540$	0	0
$541$	−8321.50	−0.661311	−0.330655	−	0.943752i	$-0.607270\pi$
$541$	−8321.50	−0.661311	−0.330655	+	0.943752i	$0.607270\pi$
$542$	2827.60	0.224088
$543$	0	0
$544$	−4284.56	−0.337682
$545$	2275.57	0.178853
$546$	0	0
$547$	−6107.98	−0.477438	−0.238719	−	0.971089i	$-0.576727\pi$
$547$	−6107.98	−0.477438	−0.238719	+	0.971089i	$0.576727\pi$
$548$	−4239.20	−0.330456
$549$	0	0
$550$	−3803.50	−0.294876
$551$	29.4004	0.00227314
$552$	0	0
$553$	0	0
$554$	−12418.3	−0.952355
$555$	0	0
$556$	9798.02	0.747354
$557$	2474.65	0.188248	0.0941240	−	0.995560i	$-0.469995\pi$
$557$	2474.65	0.188248	0.0941240	+	0.995560i	$0.469995\pi$
$558$	0	0
$559$	−2088.82	−0.158046
$560$	0	0
$561$	0	0
$562$	14747.4	1.10691
$563$	−2322.75	−0.173876	−0.0869381	−	0.996214i	$-0.527708\pi$
$563$	−2322.75	−0.173876	−0.0869381	+	0.996214i	$0.527708\pi$
$564$	0	0
$565$	10197.5	0.759310
$566$	24223.1	1.79889
$567$	0	0
$568$	5851.95	0.432293
$569$	−3303.12	−0.243364	−0.121682	−	0.992569i	$-0.538829\pi$
$569$	−3303.12	−0.243364	−0.121682	+	0.992569i	$0.538829\pi$
$570$	0	0
$571$	−13356.7	−0.978914	−0.489457	−	0.872028i	$-0.662805\pi$
$571$	−13356.7	−0.978914	−0.489457	+	0.872028i	$0.662805\pi$
$572$	−5469.96	−0.399844
$573$	0	0
$574$	0	0
$575$	591.073	0.0428686
$576$	0	0
$577$	−19598.7	−1.41405	−0.707023	−	0.707191i	$-0.749962\pi$
$577$	−19598.7	−1.41405	−0.707023	+	0.707191i	$0.749962\pi$
$578$	−13675.8	−0.984147
$579$	0	0
$580$	−157.850	−0.0113007
$581$	0	0
$582$	0	0
$583$	−11531.7	−0.819198
$584$	−7647.59	−0.541882
$585$	0	0
$586$	18690.0	1.31754
$587$	−20293.7	−1.42693	−0.713467	−	0.700689i	$-0.752876\pi$
$587$	−20293.7	−1.42693	−0.713467	+	0.700689i	$0.752876\pi$
$588$	0	0
$589$	−255.712	−0.0178887
$590$	9651.82	0.673490
$591$	0	0
$592$	4875.33	0.338471
$593$	−26600.4	−1.84207	−0.921035	−	0.389479i	$-0.872655\pi$
$593$	−26600.4	−1.84207	−0.921035	+	0.389479i	$0.872655\pi$
$594$	0	0
$595$	0	0
$596$	−5259.06	−0.361442
$597$	0	0
$598$	2836.32	0.193956
$599$	10224.5	0.697430	0.348715	−	0.937229i	$-0.386618\pi$
$599$	10224.5	0.697430	0.348715	+	0.937229i	$0.386618\pi$
$600$	0	0
$601$	9417.06	0.639152	0.319576	−	0.947561i	$-0.396460\pi$
$601$	9417.06	0.639152	0.319576	+	0.947561i	$0.396460\pi$
$602$	0	0
$603$	0	0
$604$	12051.6	0.811873
$605$	3475.96	0.233583
$606$	0	0
$607$	−9840.73	−0.658028	−0.329014	−	0.944325i	$-0.606716\pi$
$607$	−9840.73	−0.658028	−0.329014	+	0.944325i	$0.606716\pi$
$608$	464.004	0.0309504
$609$	0	0
$610$	−14121.0	−0.937282
$611$	13195.6	0.873709
$612$	0	0
$613$	−20819.3	−1.37175	−0.685876	−	0.727718i	$-0.740581\pi$
$613$	−20819.3	−1.37175	−0.685876	+	0.727718i	$0.740581\pi$
$614$	28499.1	1.87318
$615$	0	0
$616$	0	0
$617$	−21750.5	−1.41920	−0.709598	−	0.704607i	$-0.751123\pi$
$617$	−21750.5	−1.41920	−0.709598	+	0.704607i	$0.751123\pi$
$618$	0	0
$619$	−4123.18	−0.267730	−0.133865	−	0.991000i	$-0.542739\pi$
$619$	−4123.18	−0.267730	−0.133865	+	0.991000i	$0.542739\pi$
$620$	1372.91	0.0889316
$621$	0	0
$622$	−12096.7	−0.779798
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	26248.4	1.67587
$627$	0	0
$628$	3295.88	0.209427
$629$	−1801.68	−0.114209
$630$	0	0
$631$	−15561.6	−0.981773	−0.490887	−	0.871224i	$-0.663327\pi$
$631$	−15561.6	−0.981773	−0.490887	+	0.871224i	$0.663327\pi$
$632$	−16688.2	−1.05035
$633$	0	0
$634$	−34170.8	−2.14053
$635$	−12533.2	−0.783253
$636$	0	0
$637$	0	0
$638$	1402.90	0.0870552
$639$	0	0
$640$	8279.53	0.511370
$641$	−15463.9	−0.952866	−0.476433	−	0.879211i	$-0.658070\pi$
$641$	−15463.9	−0.952866	−0.476433	+	0.879211i	$0.658070\pi$
$642$	0	0
$643$	6231.42	0.382182	0.191091	−	0.981572i	$-0.438797\pi$
$643$	6231.42	0.382182	0.191091	+	0.981572i	$0.438797\pi$
$644$	0	0
$645$	0	0
$646$	−317.272	−0.0193234
$647$	15001.4	0.911538	0.455769	−	0.890098i	$-0.349364\pi$
$647$	15001.4	0.911538	0.455769	+	0.890098i	$0.349364\pi$
$648$	0	0
$649$	−25708.5	−1.55492
$650$	2999.12	0.180977
$651$	0	0
$652$	−5334.42	−0.320417
$653$	−6309.85	−0.378137	−0.189069	−	0.981964i	$-0.560547\pi$
$653$	−6309.85	−0.378137	−0.189069	+	0.981964i	$0.560547\pi$
$654$	0	0
$655$	7429.64	0.443206
$656$	−22473.1	−1.33754
$657$	0	0
$658$	0	0
$659$	−6287.74	−0.371678	−0.185839	−	0.982580i	$-0.559500\pi$
$659$	−6287.74	−0.371678	−0.185839	+	0.982580i	$0.559500\pi$
$660$	0	0
$661$	−131.962	−0.00776511	−0.00388256	−	0.999992i	$-0.501236\pi$
$661$	−131.962	−0.00776511	−0.00388256	+	0.999992i	$0.501236\pi$
$662$	−10560.0	−0.619977
$663$	0	0
$664$	11176.3	0.653201
$665$	0	0
$666$	0	0
$667$	−218.013	−0.0126559
$668$	3923.37	0.227245
$669$	0	0
$670$	15783.3	0.910094
$671$	37612.5	2.16396
$672$	0	0
$673$	1586.31	0.0908583	0.0454291	−	0.998968i	$-0.485534\pi$
$673$	1586.31	0.0908583	0.0454291	+	0.998968i	$0.485534\pi$
$674$	−15217.2	−0.869650
$675$	0	0
$676$	−3208.67	−0.182560
$677$	17191.5	0.975958	0.487979	−	0.872855i	$-0.337734\pi$
$677$	17191.5	0.975958	0.487979	+	0.872855i	$0.337734\pi$
$678$	0	0
$679$	0	0
$680$	−2276.91	−0.128405
$681$	0	0
$682$	−12201.8	−0.685089
$683$	28445.7	1.59362	0.796812	−	0.604227i	$-0.206518\pi$
$683$	28445.7	1.59362	0.796812	+	0.604227i	$0.206518\pi$
$684$	0	0
$685$	−6191.00	−0.345323
$686$	0	0
$687$	0	0
$688$	4688.49	0.259806
$689$	9092.90	0.502775
$690$	0	0
$691$	5503.89	0.303007	0.151503	−	0.988457i	$-0.451589\pi$
$691$	5503.89	0.303007	0.151503	+	0.988457i	$0.451589\pi$
$692$	4374.51	0.240309
$693$	0	0
$694$	38883.3	2.12678
$695$	14309.2	0.780976
$696$	0	0
$697$	8304.94	0.451323
$698$	−3101.05	−0.168161
$699$	0	0
$700$	0	0
$701$	−9737.40	−0.524646	−0.262323	−	0.964980i	$-0.584489\pi$
$701$	−9737.40	−0.524646	−0.262323	+	0.964980i	$0.584489\pi$
$702$	0	0
$703$	195.116	0.0104679
$704$	−6548.07	−0.350553
$705$	0	0
$706$	11993.1	0.639331
$707$	0	0
$708$	0	0
$709$	−11496.7	−0.608981	−0.304490	−	0.952515i	$-0.598486\pi$
$709$	−11496.7	−0.608981	−0.304490	+	0.952515i	$0.598486\pi$
$710$	−6393.73	−0.337961
$711$	0	0
$712$	−1385.26	−0.0729141
$713$	1896.19	0.0995971
$714$	0	0
$715$	−7988.42	−0.417832
$716$	6464.37	0.337409
$717$	0	0
$718$	26617.1	1.38348
$719$	−9513.43	−0.493451	−0.246725	−	0.969085i	$-0.579355\pi$
$719$	−9513.43	−0.493451	−0.246725	+	0.969085i	$0.579355\pi$
$720$	0	0
$721$	0	0
$722$	−23148.3	−1.19320
$723$	0	0
$724$	4087.16	0.209804
$725$	−230.527	−0.0118091
$726$	0	0
$727$	3298.37	0.168266	0.0841332	−	0.996455i	$-0.473188\pi$
$727$	3298.37	0.168266	0.0841332	+	0.996455i	$0.473188\pi$
$728$	0	0
$729$	0	0
$730$	8355.61	0.423637
$731$	−1732.63	−0.0876658
$732$	0	0
$733$	−22811.5	−1.14947	−0.574735	−	0.818339i	$-0.694895\pi$
$733$	−22811.5	−1.14947	−0.574735	+	0.818339i	$0.694895\pi$
$734$	−20586.6	−1.03524
$735$	0	0
$736$	−3440.73	−0.172319
$737$	−42040.3	−2.10118
$738$	0	0
$739$	24860.4	1.23749	0.618746	−	0.785591i	$-0.287641\pi$
$739$	24860.4	1.23749	0.618746	+	0.785591i	$0.287641\pi$
$740$	−1047.57	−0.0520400
$741$	0	0
$742$	0	0
$743$	6616.66	0.326705	0.163352	−	0.986568i	$-0.447769\pi$
$743$	6616.66	0.326705	0.163352	+	0.986568i	$0.447769\pi$
$744$	0	0
$745$	−7680.41	−0.377703
$746$	−39822.7	−1.95444
$747$	0	0
$748$	−4537.22	−0.221788
$749$	0	0
$750$	0	0
$751$	−18976.4	−0.922051	−0.461025	−	0.887387i	$-0.652518\pi$
$751$	−18976.4	−0.922051	−0.461025	+	0.887387i	$0.652518\pi$
$752$	−29618.4	−1.43626
$753$	0	0
$754$	−1106.21	−0.0534293
$755$	17600.3	0.848398
$756$	0	0
$757$	−7069.89	−0.339444	−0.169722	−	0.985492i	$-0.554287\pi$
$757$	−7069.89	−0.339444	−0.169722	+	0.985492i	$0.554287\pi$
$758$	21660.2	1.03791
$759$	0	0
$760$	246.582	0.0117690
$761$	1345.22	0.0640793	0.0320397	−	0.999487i	$-0.489800\pi$
$761$	1345.22	0.0640793	0.0320397	+	0.999487i	$0.489800\pi$
$762$	0	0
$763$	0	0
$764$	−1373.14	−0.0650242
$765$	0	0
$766$	−12740.1	−0.600939
$767$	20271.5	0.954320
$768$	0	0
$769$	−36811.4	−1.72621	−0.863103	−	0.505027i	$-0.831482\pi$
$769$	−36811.4	−1.72621	−0.863103	+	0.505027i	$0.831482\pi$
$770$	0	0
$771$	0	0
$772$	13641.8	0.635985
$773$	11439.2	0.532265	0.266133	−	0.963936i	$-0.414254\pi$
$773$	11439.2	0.532265	0.266133	+	0.963936i	$0.414254\pi$
$774$	0	0
$775$	2005.03	0.0929325
$776$	−1570.57	−0.0726549
$777$	0	0
$778$	10835.0	0.499297
$779$	−899.398	−0.0413662
$780$	0	0
$781$	17030.3	0.780270
$782$	2352.67	0.107585
$783$	0	0
$784$	0	0
$785$	4813.35	0.218848
$786$	0	0
$787$	1200.30	0.0543662	0.0271831	−	0.999630i	$-0.491346\pi$
$787$	1200.30	0.0543662	0.0271831	+	0.999630i	$0.491346\pi$
$788$	−11909.5	−0.538397
$789$	0	0
$790$	18233.2	0.821150
$791$	0	0
$792$	0	0
$793$	−29658.1	−1.32811
$794$	30385.0	1.35809
$795$	0	0
$796$	7128.40	0.317411
$797$	−28945.9	−1.28647	−0.643236	−	0.765668i	$-0.722409\pi$
$797$	−28945.9	−1.28647	−0.643236	+	0.765668i	$0.722409\pi$
$798$	0	0
$799$	10945.5	0.484634
$800$	−3638.23	−0.160788
$801$	0	0
$802$	51613.8	2.27250
$803$	−22255.9	−0.978075
$804$	0	0
$805$	0	0
$806$	9621.31	0.420467
$807$	0	0
$808$	−15852.9	−0.690227
$809$	43557.5	1.89296	0.946478	−	0.322770i	$-0.104614\pi$
$809$	43557.5	1.89296	0.946478	+	0.322770i	$0.104614\pi$
$810$	0	0
$811$	36631.5	1.58608	0.793038	−	0.609172i	$-0.208498\pi$
$811$	36631.5	1.58608	0.793038	+	0.609172i	$0.208498\pi$
$812$	0	0
$813$	0	0
$814$	9310.32	0.400893
$815$	−7790.48	−0.334833
$816$	0	0
$817$	187.638	0.00803504
$818$	−45460.0	−1.94312
$819$	0	0
$820$	4828.85	0.205647
$821$	17226.4	0.732287	0.366143	−	0.930558i	$-0.380678\pi$
$821$	17226.4	0.732287	0.366143	+	0.930558i	$0.380678\pi$
$822$	0	0
$823$	−7294.54	−0.308957	−0.154479	−	0.987996i	$-0.549370\pi$
$823$	−7294.54	−0.308957	−0.154479	+	0.987996i	$0.549370\pi$
$824$	9402.32	0.397507
$825$	0	0
$826$	0	0
$827$	−46248.5	−1.94464	−0.972321	−	0.233649i	$-0.924933\pi$
$827$	−46248.5	−1.94464	−0.972321	+	0.233649i	$0.924933\pi$
$828$	0	0
$829$	8189.74	0.343114	0.171557	−	0.985174i	$-0.445120\pi$
$829$	8189.74	0.343114	0.171557	+	0.985174i	$0.445120\pi$
$830$	−12211.1	−0.510665
$831$	0	0
$832$	5163.26	0.215149
$833$	0	0
$834$	0	0
$835$	5729.75	0.237468
$836$	491.366	0.0203281
$837$	0	0
$838$	−25436.4	−1.04855
$839$	−34877.9	−1.43518	−0.717592	−	0.696464i	$-0.754756\pi$
$839$	−34877.9	−1.43518	−0.717592	+	0.696464i	$0.754756\pi$
$840$	0	0
$841$	−24304.0	−0.996514
$842$	22569.1	0.923731
$843$	0	0
$844$	−8297.68	−0.338410
$845$	−4686.00	−0.190773
$846$	0	0
$847$	0	0
$848$	−20409.6	−0.826496
$849$	0	0
$850$	2487.71	0.100386
$851$	−1446.85	−0.0582811
$852$	0	0
$853$	36422.3	1.46199	0.730995	−	0.682383i	$-0.239056\pi$
$853$	36422.3	1.46199	0.730995	+	0.682383i	$0.239056\pi$
$854$	0	0
$855$	0	0
$856$	2618.96	0.104573
$857$	−27699.1	−1.10407	−0.552033	−	0.833823i	$-0.686148\pi$
$857$	−27699.1	−1.10407	−0.552033	+	0.833823i	$0.686148\pi$
$858$	0	0
$859$	−45673.0	−1.81413	−0.907067	−	0.420986i	$-0.861684\pi$
$859$	−45673.0	−1.81413	−0.907067	+	0.420986i	$0.861684\pi$
$860$	−1007.43	−0.0399453
$861$	0	0
$862$	25944.6	1.02515
$863$	−7734.49	−0.305081	−0.152541	−	0.988297i	$-0.548746\pi$
$863$	−7734.49	−0.305081	−0.152541	+	0.988297i	$0.548746\pi$
$864$	0	0
$865$	6388.60	0.251120
$866$	−16404.9	−0.643720
$867$	0	0
$868$	0	0
$869$	−48565.8	−1.89583
$870$	0	0
$871$	33149.4	1.28958
$872$	−7039.46	−0.273379
$873$	0	0
$874$	−254.786	−0.00986071
$875$	0	0
$876$	0	0
$877$	16893.9	0.650475	0.325237	−	0.945632i	$-0.394556\pi$
$877$	16893.9	0.650475	0.325237	+	0.945632i	$0.394556\pi$
$878$	−17919.9	−0.688799
$879$	0	0
$880$	17930.5	0.686862
$881$	44345.0	1.69582	0.847912	−	0.530136i	$-0.177859\pi$
$881$	44345.0	1.69582	0.847912	+	0.530136i	$0.177859\pi$
$882$	0	0
$883$	−17457.4	−0.665332	−0.332666	−	0.943045i	$-0.607948\pi$
$883$	−17457.4	−0.665332	−0.332666	+	0.943045i	$0.607948\pi$
$884$	3577.67	0.136120
$885$	0	0
$886$	6740.52	0.255589
$887$	−28981.4	−1.09707	−0.548535	−	0.836128i	$-0.684814\pi$
$887$	−28981.4	−1.09707	−0.548535	+	0.836128i	$0.684814\pi$
$888$	0	0
$889$	0	0
$890$	1513.51	0.0570033
$891$	0	0
$892$	−3286.63	−0.123368
$893$	−1185.36	−0.0444194
$894$	0	0
$895$	9440.66	0.352588
$896$	0	0
$897$	0	0
$898$	1463.30	0.0543773
$899$	−739.541	−0.0274361
$900$	0	0
$901$	7542.37	0.278882
$902$	−42916.4	−1.58421
$903$	0	0
$904$	−31545.8	−1.16062
$905$	5968.95	0.219243
$906$	0	0
$907$	31108.2	1.13884	0.569422	−	0.822046i	$-0.307167\pi$
$907$	31108.2	1.13884	0.569422	+	0.822046i	$0.307167\pi$
$908$	−3488.01	−0.127482
$909$	0	0
$910$	0	0
$911$	4115.73	0.149682	0.0748409	−	0.997195i	$-0.476155\pi$
$911$	4115.73	0.149682	0.0748409	+	0.997195i	$0.476155\pi$
$912$	0	0
$913$	32525.2	1.17900
$914$	−14169.2	−0.512774
$915$	0	0
$916$	−11233.2	−0.405192
$917$	0	0
$918$	0	0
$919$	−13909.3	−0.499266	−0.249633	−	0.968341i	$-0.580310\pi$
$919$	−13909.3	−0.499266	−0.249633	+	0.968341i	$0.580310\pi$
$920$	−1828.48	−0.0655252
$921$	0	0
$922$	56738.3	2.02666
$923$	−13428.6	−0.478883
$924$	0	0
$925$	−1529.89	−0.0543812
$926$	18943.6	0.672273
$927$	0	0
$928$	1341.94	0.0474690
$929$	−22380.1	−0.790383	−0.395192	−	0.918599i	$-0.629322\pi$
$929$	−22380.1	−0.790383	−0.395192	+	0.918599i	$0.629322\pi$
$930$	0	0
$931$	0	0
$932$	−16032.0	−0.563461
$933$	0	0
$934$	43976.3	1.54063
$935$	−6626.24	−0.231766
$936$	0	0
$937$	52911.1	1.84475	0.922374	−	0.386297i	$-0.126246\pi$
$937$	52911.1	1.84475	0.922374	+	0.386297i	$0.126246\pi$
$938$	0	0
$939$	0	0
$940$	6364.16	0.220826
$941$	42984.9	1.48913	0.744564	−	0.667551i	$-0.232658\pi$
$941$	42984.9	1.48913	0.744564	+	0.667551i	$0.232658\pi$
$942$	0	0
$943$	6669.31	0.230310
$944$	−45500.8	−1.56878
$945$	0	0
$946$	8953.50	0.307720
$947$	25210.3	0.865075	0.432537	−	0.901616i	$-0.357618\pi$
$947$	25210.3	0.865075	0.432537	+	0.901616i	$0.357618\pi$
$948$	0	0
$949$	17549.1	0.600284
$950$	−269.411	−0.00920088
$951$	0	0
$952$	0	0
$953$	−32792.0	−1.11462	−0.557312	−	0.830303i	$-0.688167\pi$
$953$	−32792.0	−1.11462	−0.557312	+	0.830303i	$0.688167\pi$
$954$	0	0
$955$	−2005.36	−0.0679496
$956$	10772.8	0.364452
$957$	0	0
$958$	−55512.5	−1.87216
$959$	0	0
$960$	0	0
$961$	−23358.8	−0.784089
$962$	−7341.34	−0.246044
$963$	0	0
$964$	−24246.8	−0.810101
$965$	19922.8	0.664597
$966$	0	0
$967$	−22158.8	−0.736897	−0.368449	−	0.929648i	$-0.620111\pi$
$967$	−22158.8	−0.736897	−0.368449	+	0.929648i	$0.620111\pi$
$968$	−10752.9	−0.357035
$969$	0	0
$970$	1715.98	0.0568007
$971$	−45086.4	−1.49010	−0.745052	−	0.667007i	$-0.767575\pi$
$971$	−45086.4	−1.49010	−0.745052	+	0.667007i	$0.767575\pi$
$972$	0	0
$973$	0	0
$974$	11654.5	0.383402
$975$	0	0
$976$	66569.4	2.18323
$977$	31577.0	1.03402	0.517010	−	0.855980i	$-0.327045\pi$
$977$	31577.0	1.03402	0.517010	+	0.855980i	$0.327045\pi$
$978$	0	0
$979$	−4031.37	−0.131607
$980$	0	0
$981$	0	0
$982$	66440.3	2.15906
$983$	47938.2	1.55543	0.777716	−	0.628615i	$-0.216378\pi$
$983$	47938.2	1.55543	0.777716	+	0.628615i	$0.216378\pi$
$984$	0	0
$985$	−17392.8	−0.562619
$986$	−917.576	−0.0296365
$987$	0	0
$988$	−387.450	−0.0124761
$989$	−1391.39	−0.0447359
$990$	0	0
$991$	−10906.5	−0.349603	−0.174801	−	0.984604i	$-0.555928\pi$
$991$	−10906.5	−0.349603	−0.174801	+	0.984604i	$0.555928\pi$
$992$	−11671.6	−0.373562
$993$	0	0
$994$	0	0
$995$	10410.4	0.331691
$996$	0	0
$997$	12617.3	0.400796	0.200398	−	0.979715i	$-0.435776\pi$
$997$	12617.3	0.400796	0.200398	+	0.979715i	$0.435776\pi$
$998$	14397.1	0.456644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bk.1.3		3
3.2	odd	2		2205.4.a.bl.1.1		3
7.6	odd	2		315.4.a.n.1.3	✓	3
21.20	even	2		315.4.a.o.1.1	yes	3
35.34	odd	2		1575.4.a.be.1.1		3
105.104	even	2		1575.4.a.bb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.3	✓	3	7.6	odd	2
315.4.a.o.1.1	yes	3	21.20	even	2
1575.4.a.bb.1.3		3	105.104	even	2
1575.4.a.be.1.1		3	35.34	odd	2
2205.4.a.bk.1.3		3	1.1	even	1	trivial
2205.4.a.bl.1.1		3	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37989 q^{2} +3.42368 q^{4} +5.00000 q^{5} -15.4675 q^{8} +16.8995 q^{10} -45.0132 q^{11} +35.4937 q^{13} -79.6679 q^{16} +29.4413 q^{17} -3.18839 q^{19} +17.1184 q^{20} -152.140 q^{22} +23.6429 q^{23} +25.0000 q^{25} +119.965 q^{26} -9.22109 q^{29} +80.2011 q^{31} -145.529 q^{32} +99.5084 q^{34} -61.1957 q^{37} -10.7764 q^{38} -77.3373 q^{40} +282.085 q^{41} -58.8504 q^{43} -154.111 q^{44} +79.9105 q^{46} +371.773 q^{47} +84.4973 q^{50} +121.519 q^{52} +256.184 q^{53} -225.066 q^{55} -31.1663 q^{58} +571.131 q^{59} -835.587 q^{61} +271.071 q^{62} +145.470 q^{64} +177.468 q^{65} +933.953 q^{67} +100.798 q^{68} -378.339 q^{71} +494.430 q^{73} -206.835 q^{74} -10.9160 q^{76} +1078.92 q^{79} -398.339 q^{80} +953.418 q^{82} -722.570 q^{83} +147.206 q^{85} -198.908 q^{86} +696.241 q^{88} +89.5596 q^{89} +80.9458 q^{92} +1256.55 q^{94} -15.9420 q^{95} +101.540 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 14 q^{4} + 15 q^{5} - 66 q^{8} - 10 q^{10} + 20 q^{11} + 114 q^{16} + 234 q^{17} + 82 q^{19} + 70 q^{20} - 236 q^{22} - 30 q^{23} + 75 q^{25} + 76 q^{26} + 32 q^{29} + 362 q^{31} - 430 q^{32}+ \cdots + 3052 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 14 * q^4 + 15 * q^5 - 66 * q^8 - 10 * q^10 + 20 * q^11 + 114 * q^16 + 234 * q^17 + 82 * q^19 + 70 * q^20 - 236 * q^22 - 30 * q^23 + 75 * q^25 + 76 * q^26 + 32 * q^29 + 362 * q^31 - 430 * q^32 - 596 * q^34 + 302 * q^37 + 160 * q^38 - 330 * q^40 + 626 * q^41 - 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 - 50 * q^50 + 684 * q^52 - 298 * q^53 + 100 * q^55 - 1184 * q^58 + 428 * q^59 - 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 - 1712 * q^76 - 488 * q^79 + 570 * q^80 + 1476 * q^82 + 948 * q^83 + 1170 * q^85 + 1928 * q^86 + 124 * q^88 - 142 * q^89 + 3220 * q^92 - 328 * q^94 + 410 * q^95 + 3052 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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