LMFDB - Embedded newform 2205.4.a.bs.1.5

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			1.5
Root			$5.80647$ 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)

$f(q)$	$=$	$q+4.80647 q^{2} +15.1022 q^{4} +5.00000 q^{5} +34.1365 q^{8} +24.0324 q^{10} -53.2563 q^{11} -60.9548 q^{13} +43.2586 q^{16} +31.1198 q^{17} +17.9133 q^{19} +75.5109 q^{20} -255.975 q^{22} +34.8223 q^{23} +25.0000 q^{25} -292.978 q^{26} -141.423 q^{29} -117.739 q^{31} -65.1706 q^{32} +149.576 q^{34} +175.319 q^{37} +86.0996 q^{38} +170.682 q^{40} +411.157 q^{41} -498.509 q^{43} -804.286 q^{44} +167.373 q^{46} -290.262 q^{47} +120.162 q^{50} -920.551 q^{52} -582.633 q^{53} -266.281 q^{55} -679.745 q^{58} -657.710 q^{59} +417.167 q^{61} -565.911 q^{62} -659.310 q^{64} -304.774 q^{65} -567.911 q^{67} +469.977 q^{68} -887.933 q^{71} +1092.51 q^{73} +842.666 q^{74} +270.529 q^{76} +135.425 q^{79} +216.293 q^{80} +1976.22 q^{82} -464.523 q^{83} +155.599 q^{85} -2396.07 q^{86} -1817.98 q^{88} +31.8799 q^{89} +525.894 q^{92} -1395.14 q^{94} +89.5663 q^{95} +254.781 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	4.80647	1.69935	0.849673	−	0.527311i	$-0.176800\pi$
$2$	4.80647	1.69935	0.849673	+	0.527311i	$0.176800\pi$
$3$	0	0
$3$	0	0
$4$	15.1022	1.88777
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	34.1365	1.50863
$9$	0	0
$10$	24.0324	0.759970
$11$	−53.2563	−1.45976	−0.729881	−	0.683575i	$-0.760424\pi$
$11$	−53.2563	−1.45976	−0.729881	+	0.683575i	$0.760424\pi$
$12$	0	0
$13$	−60.9548	−1.30045	−0.650224	−	0.759743i	$-0.725325\pi$
$13$	−60.9548	−1.30045	−0.650224	+	0.759743i	$0.725325\pi$
$14$	0	0
$15$	0	0
$16$	43.2586	0.675915
$17$	31.1198	0.443980	0.221990	−	0.975049i	$-0.428745\pi$
$17$	31.1198	0.443980	0.221990	+	0.975049i	$0.428745\pi$
$18$	0	0
$19$	17.9133	0.216294	0.108147	−	0.994135i	$-0.465508\pi$
$19$	17.9133	0.216294	0.108147	+	0.994135i	$0.465508\pi$
$20$	75.5109	0.844238
$21$	0	0
$22$	−255.975	−2.48064
$23$	34.8223	0.315694	0.157847	−	0.987464i	$-0.449545\pi$
$23$	34.8223	0.315694	0.157847	+	0.987464i	$0.449545\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−292.978	−2.20991
$27$	0	0
$28$	0	0
$29$	−141.423	−0.905571	−0.452786	−	0.891619i	$-0.649570\pi$
$29$	−141.423	−0.905571	−0.452786	+	0.891619i	$0.649570\pi$
$30$	0	0
$31$	−117.739	−0.682149	−0.341075	−	0.940036i	$-0.610791\pi$
$31$	−117.739	−0.682149	−0.341075	+	0.940036i	$0.610791\pi$
$32$	−65.1706	−0.360020
$33$	0	0
$34$	149.576	0.754475
$35$	0	0
$36$	0	0
$37$	175.319	0.778980	0.389490	−	0.921031i	$-0.372651\pi$
$37$	175.319	0.778980	0.389490	+	0.921031i	$0.372651\pi$
$38$	86.0996	0.367558
$39$	0	0
$40$	170.682	0.674681
$41$	411.157	1.56615	0.783073	−	0.621930i	$-0.213651\pi$
$41$	411.157	1.56615	0.783073	+	0.621930i	$0.213651\pi$
$42$	0	0
$43$	−498.509	−1.76795	−0.883976	−	0.467532i	$-0.845143\pi$
$43$	−498.509	−1.76795	−0.883976	+	0.467532i	$0.845143\pi$
$44$	−804.286	−2.75570
$45$	0	0
$46$	167.373	0.536473
$47$	−290.262	−0.900832	−0.450416	−	0.892819i	$-0.648724\pi$
$47$	−290.262	−0.900832	−0.450416	+	0.892819i	$0.648724\pi$
$48$	0	0
$49$	0	0
$50$	120.162	0.339869
$51$	0	0
$52$	−920.551	−2.45495
$53$	−582.633	−1.51001	−0.755007	−	0.655717i	$-0.772367\pi$
$53$	−582.633	−1.51001	−0.755007	+	0.655717i	$0.772367\pi$
$54$	0	0
$55$	−266.281	−0.652825
$56$	0	0
$57$	0	0
$58$	−679.745	−1.53888
$59$	−657.710	−1.45130	−0.725649	−	0.688065i	$-0.758460\pi$
$59$	−657.710	−1.45130	−0.725649	+	0.688065i	$0.758460\pi$
$60$	0	0
$61$	417.167	0.875618	0.437809	−	0.899068i	$-0.355755\pi$
$61$	417.167	0.875618	0.437809	+	0.899068i	$0.355755\pi$
$62$	−565.911	−1.15921
$63$	0	0
$64$	−659.310	−1.28771
$65$	−304.774	−0.581578
$66$	0	0
$67$	−567.911	−1.03554	−0.517771	−	0.855519i	$-0.673238\pi$
$67$	−567.911	−1.03554	−0.517771	+	0.855519i	$0.673238\pi$
$68$	469.977	0.838134
$69$	0	0
$70$	0	0
$71$	−887.933	−1.48420	−0.742100	−	0.670289i	$-0.766170\pi$
$71$	−887.933	−1.48420	−0.742100	+	0.670289i	$0.766170\pi$
$72$	0	0
$73$	1092.51	1.75162	0.875809	−	0.482657i	$-0.160328\pi$
$73$	1092.51	1.75162	0.875809	+	0.482657i	$0.160328\pi$
$74$	842.666	1.32376
$75$	0	0
$76$	270.529	0.408314
$77$	0	0
$78$	0	0
$79$	135.425	0.192867	0.0964337	−	0.995339i	$-0.469256\pi$
$79$	135.425	0.192867	0.0964337	+	0.995339i	$0.469256\pi$
$80$	216.293	0.302278
$81$	0	0
$82$	1976.22	2.66142
$83$	−464.523	−0.614313	−0.307157	−	0.951659i	$-0.599378\pi$
$83$	−464.523	−0.614313	−0.307157	+	0.951659i	$0.599378\pi$
$84$	0	0
$85$	155.599	0.198554
$86$	−2396.07	−3.00436
$87$	0	0
$88$	−1817.98	−2.20224
$89$	31.8799	0.0379692	0.0189846	−	0.999820i	$-0.493957\pi$
$89$	31.8799	0.0379692	0.0189846	+	0.999820i	$0.493957\pi$
$90$	0	0
$91$	0	0
$92$	525.894	0.595959
$93$	0	0
$94$	−1395.14	−1.53082
$95$	89.5663	0.0967295
$96$	0	0
$97$	254.781	0.266691	0.133346	−	0.991070i	$-0.457428\pi$
$97$	254.781	0.266691	0.133346	+	0.991070i	$0.457428\pi$
$98$	0	0
$99$	0	0
$100$	377.555	0.377555
$101$	1359.53	1.33939	0.669694	−	0.742637i	$-0.266425\pi$
$101$	1359.53	1.33939	0.669694	+	0.742637i	$0.266425\pi$
$102$	0	0
$103$	−1448.06	−1.38526	−0.692629	−	0.721294i	$-0.743547\pi$
$103$	−1448.06	−1.38526	−0.692629	+	0.721294i	$0.743547\pi$
$104$	−2080.78	−1.96190
$105$	0	0
$106$	−2800.41	−2.56603
$107$	1666.45	1.50563	0.752814	−	0.658234i	$-0.228696\pi$
$107$	1666.45	1.50563	0.752814	+	0.658234i	$0.228696\pi$
$108$	0	0
$109$	−1118.40	−0.982780	−0.491390	−	0.870940i	$-0.663511\pi$
$109$	−1118.40	−0.982780	−0.491390	+	0.870940i	$0.663511\pi$
$110$	−1279.87	−1.10937
$111$	0	0
$112$	0	0
$113$	−806.732	−0.671602	−0.335801	−	0.941933i	$-0.609007\pi$
$113$	−806.732	−0.671602	−0.335801	+	0.941933i	$0.609007\pi$
$114$	0	0
$115$	174.112	0.141183
$116$	−2135.80	−1.70951
$117$	0	0
$118$	−3161.27	−2.46625
$119$	0	0
$120$	0	0
$121$	1505.23	1.13090
$122$	2005.10	1.48798
$123$	0	0
$124$	−1778.12	−1.28774
$125$	125.000	0.0894427
$126$	0	0
$127$	−1433.92	−1.00189	−0.500945	−	0.865479i	$-0.667014\pi$
$127$	−1433.92	−1.00189	−0.500945	+	0.865479i	$0.667014\pi$
$128$	−2647.59	−1.82825
$129$	0	0
$130$	−1464.89	−0.988301
$131$	650.933	0.434140	0.217070	−	0.976156i	$-0.430350\pi$
$131$	650.933	0.434140	0.217070	+	0.976156i	$0.430350\pi$
$132$	0	0
$133$	0	0
$134$	−2729.65	−1.75974
$135$	0	0
$136$	1062.32	0.669803
$137$	617.782	0.385260	0.192630	−	0.981271i	$-0.438298\pi$
$137$	617.782	0.385260	0.192630	+	0.981271i	$0.438298\pi$
$138$	0	0
$139$	−1525.08	−0.930614	−0.465307	−	0.885149i	$-0.654056\pi$
$139$	−1525.08	−0.930614	−0.465307	+	0.885149i	$0.654056\pi$
$140$	0	0
$141$	0	0
$142$	−4267.82	−2.52217
$143$	3246.23	1.89834
$144$	0	0
$145$	−707.114	−0.404984
$146$	5251.10	2.97660
$147$	0	0
$148$	2647.70	1.47054
$149$	1387.82	0.763051	0.381525	−	0.924358i	$-0.375399\pi$
$149$	1387.82	0.763051	0.381525	+	0.924358i	$0.375399\pi$
$150$	0	0
$151$	2284.34	1.23111	0.615554	−	0.788095i	$-0.288932\pi$
$151$	2284.34	1.23111	0.615554	+	0.788095i	$0.288932\pi$
$152$	611.495	0.326308
$153$	0	0
$154$	0	0
$155$	−588.697	−0.305066
$156$	0	0
$157$	−268.308	−0.136391	−0.0681953	−	0.997672i	$-0.521724\pi$
$157$	−268.308	−0.136391	−0.0681953	+	0.997672i	$0.521724\pi$
$158$	650.918	0.327748
$159$	0	0
$160$	−325.853	−0.161006
$161$	0	0
$162$	0	0
$163$	−2727.58	−1.31068	−0.655340	−	0.755334i	$-0.727475\pi$
$163$	−2727.58	−1.31068	−0.655340	+	0.755334i	$0.727475\pi$
$164$	6209.37	2.95653
$165$	0	0
$166$	−2232.72	−1.04393
$167$	2234.41	1.03535	0.517675	−	0.855577i	$-0.326798\pi$
$167$	2234.41	1.03535	0.517675	+	0.855577i	$0.326798\pi$
$168$	0	0
$169$	1518.49	0.691164
$170$	747.882	0.337412
$171$	0	0
$172$	−7528.58	−3.33749
$173$	2339.66	1.02822	0.514108	−	0.857725i	$-0.328123\pi$
$173$	2339.66	1.02822	0.514108	+	0.857725i	$0.328123\pi$
$174$	0	0
$175$	0	0
$176$	−2303.79	−0.986675
$177$	0	0
$178$	153.230	0.0645228
$179$	−2445.75	−1.02125	−0.510626	−	0.859803i	$-0.670586\pi$
$179$	−2445.75	−1.02125	−0.510626	+	0.859803i	$0.670586\pi$
$180$	0	0
$181$	−1178.45	−0.483944	−0.241972	−	0.970283i	$-0.577794\pi$
$181$	−1178.45	−0.483944	−0.241972	+	0.970283i	$0.577794\pi$
$182$	0	0
$183$	0	0
$184$	1188.71	0.476266
$185$	876.595	0.348371
$186$	0	0
$187$	−1657.32	−0.648105
$188$	−4383.60	−1.70057
$189$	0	0
$190$	430.498	0.164377
$191$	2275.29	0.861958	0.430979	−	0.902362i	$-0.358168\pi$
$191$	2275.29	0.861958	0.430979	+	0.902362i	$0.358168\pi$
$192$	0	0
$193$	4411.04	1.64515	0.822574	−	0.568658i	$-0.192537\pi$
$193$	4411.04	1.64515	0.822574	+	0.568658i	$0.192537\pi$
$194$	1224.60	0.453200
$195$	0	0
$196$	0	0
$197$	2011.41	0.727447	0.363723	−	0.931507i	$-0.381505\pi$
$197$	2011.41	0.727447	0.363723	+	0.931507i	$0.381505\pi$
$198$	0	0
$199$	−1111.24	−0.395846	−0.197923	−	0.980218i	$-0.563420\pi$
$199$	−1111.24	−0.395846	−0.197923	+	0.980218i	$0.563420\pi$
$200$	853.412	0.301727
$201$	0	0
$202$	6534.54	2.27608
$203$	0	0
$204$	0	0
$205$	2055.79	0.700402
$206$	−6960.06	−2.35403
$207$	0	0
$208$	−2636.82	−0.878992
$209$	−953.993	−0.315737
$210$	0	0
$211$	3031.04	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$211$	3031.04	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$212$	−8799.03	−2.85056
$213$	0	0
$214$	8009.77	2.55858
$215$	−2492.55	−0.790652
$216$	0	0
$217$	0	0
$218$	−5375.55	−1.67008
$219$	0	0
$220$	−4021.43	−1.23239
$221$	−1896.90	−0.577373
$222$	0	0
$223$	3789.93	1.13808	0.569042	−	0.822309i	$-0.307314\pi$
$223$	3789.93	1.13808	0.569042	+	0.822309i	$0.307314\pi$
$224$	0	0
$225$	0	0
$226$	−3877.54	−1.14128
$227$	3876.31	1.13339	0.566696	−	0.823927i	$-0.308222\pi$
$227$	3876.31	1.13339	0.566696	+	0.823927i	$0.308222\pi$
$228$	0	0
$229$	−5926.44	−1.71018	−0.855088	−	0.518483i	$-0.826497\pi$
$229$	−5926.44	−1.71018	−0.855088	+	0.518483i	$0.826497\pi$
$230$	836.863	0.239918
$231$	0	0
$232$	−4827.68	−1.36618
$233$	−4643.03	−1.30547	−0.652737	−	0.757585i	$-0.726379\pi$
$233$	−4643.03	−1.30547	−0.652737	+	0.757585i	$0.726379\pi$
$234$	0	0
$235$	−1451.31	−0.402864
$236$	−9932.86	−2.73972
$237$	0	0
$238$	0	0
$239$	−944.025	−0.255497	−0.127749	−	0.991807i	$-0.540775\pi$
$239$	−944.025	−0.255497	−0.127749	+	0.991807i	$0.540775\pi$
$240$	0	0
$241$	−1515.01	−0.404939	−0.202470	−	0.979289i	$-0.564897\pi$
$241$	−1515.01	−0.404939	−0.202470	+	0.979289i	$0.564897\pi$
$242$	7234.85	1.92179
$243$	0	0
$244$	6300.13	1.65297
$245$	0	0
$246$	0	0
$247$	−1091.90	−0.281279
$248$	−4019.21	−1.02911
$249$	0	0
$250$	600.809	0.151994
$251$	3219.82	0.809695	0.404847	−	0.914384i	$-0.367325\pi$
$251$	3219.82	0.809695	0.404847	+	0.914384i	$0.367325\pi$
$252$	0	0
$253$	−1854.51	−0.460838
$254$	−6892.11	−1.70256
$255$	0	0
$256$	−7451.09	−1.81911
$257$	2214.55	0.537509	0.268754	−	0.963209i	$-0.413388\pi$
$257$	2214.55	0.537509	0.268754	+	0.963209i	$0.413388\pi$
$258$	0	0
$259$	0	0
$260$	−4602.75	−1.09789
$261$	0	0
$262$	3128.69	0.737753
$263$	5457.02	1.27945	0.639723	−	0.768605i	$-0.279049\pi$
$263$	5457.02	1.27945	0.639723	+	0.768605i	$0.279049\pi$
$264$	0	0
$265$	−2913.16	−0.675299
$266$	0	0
$267$	0	0
$268$	−8576.69	−1.95487
$269$	−5820.76	−1.31932	−0.659661	−	0.751563i	$-0.729300\pi$
$269$	−5820.76	−1.31932	−0.659661	+	0.751563i	$0.729300\pi$
$270$	0	0
$271$	−3300.74	−0.739874	−0.369937	−	0.929057i	$-0.620621\pi$
$271$	−3300.74	−0.739874	−0.369937	+	0.929057i	$0.620621\pi$
$272$	1346.20	0.300093
$273$	0	0
$274$	2969.35	0.654690
$275$	−1331.41	−0.291952
$276$	0	0
$277$	−3871.25	−0.839714	−0.419857	−	0.907590i	$-0.637920\pi$
$277$	−3871.25	−0.839714	−0.419857	+	0.907590i	$0.637920\pi$
$278$	−7330.24	−1.58143
$279$	0	0
$280$	0	0
$281$	1411.40	0.299634	0.149817	−	0.988714i	$-0.452131\pi$
$281$	1411.40	0.299634	0.149817	+	0.988714i	$0.452131\pi$
$282$	0	0
$283$	4752.51	0.998259	0.499129	−	0.866528i	$-0.333653\pi$
$283$	4752.51	0.998259	0.499129	+	0.866528i	$0.333653\pi$
$284$	−13409.7	−2.80183
$285$	0	0
$286$	15602.9	3.22594
$287$	0	0
$288$	0	0
$289$	−3944.56	−0.802882
$290$	−3398.73	−0.688207
$291$	0	0
$292$	16499.2	3.30666
$293$	−1301.32	−0.259468	−0.129734	−	0.991549i	$-0.541412\pi$
$293$	−1301.32	−0.259468	−0.129734	+	0.991549i	$0.541412\pi$
$294$	0	0
$295$	−3288.55	−0.649040
$296$	5984.77	1.17520
$297$	0	0
$298$	6670.51	1.29669
$299$	−2122.59	−0.410543
$300$	0	0
$301$	0	0
$302$	10979.6	2.09208
$303$	0	0
$304$	774.902	0.146196
$305$	2085.83	0.391588
$306$	0	0
$307$	−6138.12	−1.14111	−0.570555	−	0.821259i	$-0.693272\pi$
$307$	−6138.12	−1.14111	−0.570555	+	0.821259i	$0.693272\pi$
$308$	0	0
$309$	0	0
$310$	−2829.56	−0.518413
$311$	4082.04	0.744281	0.372140	−	0.928176i	$-0.378624\pi$
$311$	4082.04	0.744281	0.372140	+	0.928176i	$0.378624\pi$
$312$	0	0
$313$	−7143.19	−1.28996	−0.644979	−	0.764200i	$-0.723134\pi$
$313$	−7143.19	−1.28996	−0.644979	+	0.764200i	$0.723134\pi$
$314$	−1289.62	−0.231775
$315$	0	0
$316$	2045.22	0.364090
$317$	7481.62	1.32558	0.662791	−	0.748804i	$-0.269372\pi$
$317$	7481.62	1.32558	0.662791	+	0.748804i	$0.269372\pi$
$318$	0	0
$319$	7531.66	1.32192
$320$	−3296.55	−0.575883
$321$	0	0
$322$	0	0
$323$	557.457	0.0960301
$324$	0	0
$325$	−1523.87	−0.260090
$326$	−13110.1	−2.22730
$327$	0	0
$328$	14035.5	2.36274
$329$	0	0
$330$	0	0
$331$	−1944.12	−0.322835	−0.161417	−	0.986886i	$-0.551607\pi$
$331$	−1944.12	−0.322835	−0.161417	+	0.986886i	$0.551607\pi$
$332$	−7015.31	−1.15968
$333$	0	0
$334$	10739.6	1.75942
$335$	−2839.55	−0.463109
$336$	0	0
$337$	−1920.36	−0.310411	−0.155205	−	0.987882i	$-0.549604\pi$
$337$	−1920.36	−0.310411	−0.155205	+	0.987882i	$0.549604\pi$
$338$	7298.57	1.17453
$339$	0	0
$340$	2349.89	0.374825
$341$	6270.36	0.995775
$342$	0	0
$343$	0	0
$344$	−17017.3	−2.66719
$345$	0	0
$346$	11245.5	1.74729
$347$	−2093.45	−0.323868	−0.161934	−	0.986802i	$-0.551773\pi$
$347$	−2093.45	−0.323868	−0.161934	+	0.986802i	$0.551773\pi$
$348$	0	0
$349$	−2199.00	−0.337277	−0.168638	−	0.985678i	$-0.553937\pi$
$349$	−2199.00	−0.337277	−0.168638	+	0.985678i	$0.553937\pi$
$350$	0	0
$351$	0	0
$352$	3470.75	0.525544
$353$	−5861.17	−0.883735	−0.441867	−	0.897080i	$-0.645684\pi$
$353$	−5861.17	−0.883735	−0.441867	+	0.897080i	$0.645684\pi$
$354$	0	0
$355$	−4439.66	−0.663755
$356$	481.456	0.0716772
$357$	0	0
$358$	−11755.4	−1.73546
$359$	−7271.44	−1.06900	−0.534502	−	0.845167i	$-0.679501\pi$
$359$	−7271.44	−1.06900	−0.534502	+	0.845167i	$0.679501\pi$
$360$	0	0
$361$	−6538.12	−0.953217
$362$	−5664.21	−0.822387
$363$	0	0
$364$	0	0
$365$	5462.53	0.783348
$366$	0	0
$367$	12638.0	1.79754	0.898768	−	0.438425i	$-0.144463\pi$
$367$	12638.0	1.79754	0.898768	+	0.438425i	$0.144463\pi$
$368$	1506.36	0.213382
$369$	0	0
$370$	4213.33	0.592002
$371$	0	0
$372$	0	0
$373$	−6212.81	−0.862431	−0.431216	−	0.902249i	$-0.641915\pi$
$373$	−6212.81	−0.862431	−0.431216	+	0.902249i	$0.641915\pi$
$374$	−7965.89	−1.10135
$375$	0	0
$376$	−9908.53	−1.35903
$377$	8620.40	1.17765
$378$	0	0
$379$	5075.53	0.687896	0.343948	−	0.938989i	$-0.388236\pi$
$379$	5075.53	0.687896	0.343948	+	0.938989i	$0.388236\pi$
$380$	1352.65	0.182603
$381$	0	0
$382$	10936.1	1.46476
$383$	−8413.71	−1.12251	−0.561254	−	0.827644i	$-0.689681\pi$
$383$	−8413.71	−1.12251	−0.561254	+	0.827644i	$0.689681\pi$
$384$	0	0
$385$	0	0
$386$	21201.5	2.79567
$387$	0	0
$388$	3847.74	0.503453
$389$	−3708.69	−0.483389	−0.241694	−	0.970352i	$-0.577703\pi$
$389$	−3708.69	−0.483389	−0.241694	+	0.970352i	$0.577703\pi$
$390$	0	0
$391$	1083.66	0.140162
$392$	0	0
$393$	0	0
$394$	9667.79	1.23618
$395$	677.126	0.0862529
$396$	0	0
$397$	4520.05	0.571422	0.285711	−	0.958316i	$-0.407770\pi$
$397$	4520.05	0.571422	0.285711	+	0.958316i	$0.407770\pi$
$398$	−5341.12	−0.672679
$399$	0	0
$400$	1081.46	0.135183
$401$	12591.8	1.56809	0.784043	−	0.620707i	$-0.213154\pi$
$401$	12591.8	1.56809	0.784043	+	0.620707i	$0.213154\pi$
$402$	0	0
$403$	7176.78	0.887099
$404$	20531.9	2.52846
$405$	0	0
$406$	0	0
$407$	−9336.84	−1.13713
$408$	0	0
$409$	−9017.62	−1.09020	−0.545101	−	0.838370i	$-0.683509\pi$
$409$	−9017.62	−1.09020	−0.545101	+	0.838370i	$0.683509\pi$
$410$	9881.08	1.19022
$411$	0	0
$412$	−21868.9	−2.61505
$413$	0	0
$414$	0	0
$415$	−2322.61	−0.274729
$416$	3972.46	0.468188
$417$	0	0
$418$	−4585.34	−0.536547
$419$	3.80670	0.000443842 0	0.000221921	−	1.00000i	$-0.499929\pi$
$419$	3.80670	0.000443842 0	0.000221921		1.00000i	$0.499929\pi$
$420$	0	0
$421$	12076.0	1.39798	0.698989	−	0.715132i	$-0.253633\pi$
$421$	12076.0	1.39798	0.698989	+	0.715132i	$0.253633\pi$
$422$	14568.6	1.68054
$423$	0	0
$424$	−19889.0	−2.27806
$425$	777.995	0.0887960
$426$	0	0
$427$	0	0
$428$	25167.1	2.84228
$429$	0	0
$430$	−11980.4	−1.34359
$431$	10317.7	1.15310	0.576551	−	0.817061i	$-0.304398\pi$
$431$	10317.7	1.15310	0.576551	+	0.817061i	$0.304398\pi$
$432$	0	0
$433$	−12728.4	−1.41267	−0.706334	−	0.707878i	$-0.749653\pi$
$433$	−12728.4	−1.41267	−0.706334	+	0.707878i	$0.749653\pi$
$434$	0	0
$435$	0	0
$436$	−16890.2	−1.85527
$437$	623.781	0.0682826
$438$	0	0
$439$	9972.58	1.08420	0.542102	−	0.840313i	$-0.317629\pi$
$439$	9972.58	1.08420	0.542102	+	0.840313i	$0.317629\pi$
$440$	−9089.91	−0.984874
$441$	0	0
$442$	−9117.40	−0.981155
$443$	−10462.8	−1.12213	−0.561063	−	0.827773i	$-0.689607\pi$
$443$	−10462.8	−1.12213	−0.561063	+	0.827773i	$0.689607\pi$
$444$	0	0
$445$	159.399	0.0169803
$446$	18216.2	1.93400
$447$	0	0
$448$	0	0
$449$	11664.7	1.22604	0.613019	−	0.790068i	$-0.289955\pi$
$449$	11664.7	1.22604	0.613019	+	0.790068i	$0.289955\pi$
$450$	0	0
$451$	−21896.7	−2.28620
$452$	−12183.4	−1.26783
$453$	0	0
$454$	18631.4	1.92602
$455$	0	0
$456$	0	0
$457$	4542.53	0.464968	0.232484	−	0.972600i	$-0.425315\pi$
$457$	4542.53	0.464968	0.232484	+	0.972600i	$0.425315\pi$
$458$	−28485.3	−2.90618
$459$	0	0
$460$	2629.47	0.266521
$461$	7887.54	0.796875	0.398437	−	0.917195i	$-0.369553\pi$
$461$	7887.54	0.796875	0.398437	+	0.917195i	$0.369553\pi$
$462$	0	0
$463$	−5444.15	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$463$	−5444.15	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$464$	−6117.75	−0.612089
$465$	0	0
$466$	−22316.6	−2.21845
$467$	−10578.0	−1.04816	−0.524082	−	0.851668i	$-0.675591\pi$
$467$	−10578.0	−1.04816	−0.524082	+	0.851668i	$0.675591\pi$
$468$	0	0
$469$	0	0
$470$	−6975.69	−0.684606
$471$	0	0
$472$	−22451.9	−2.18948
$473$	26548.7	2.58079
$474$	0	0
$475$	447.831	0.0432588
$476$	0	0
$477$	0	0
$478$	−4537.43	−0.434178
$479$	11644.7	1.11077	0.555386	−	0.831592i	$-0.312570\pi$
$479$	11644.7	1.11077	0.555386	+	0.831592i	$0.312570\pi$
$480$	0	0
$481$	−10686.5	−1.01302
$482$	−7281.85	−0.688131
$483$	0	0
$484$	22732.3	2.13489
$485$	1273.90	0.119268
$486$	0	0
$487$	−9237.23	−0.859504	−0.429752	−	0.902947i	$-0.641399\pi$
$487$	−9237.23	−0.859504	−0.429752	+	0.902947i	$0.641399\pi$
$488$	14240.6	1.32099
$489$	0	0
$490$	0	0
$491$	2704.47	0.248576	0.124288	−	0.992246i	$-0.460335\pi$
$491$	2704.47	0.248576	0.124288	+	0.992246i	$0.460335\pi$
$492$	0	0
$493$	−4401.05	−0.402056
$494$	−5248.18	−0.477990
$495$	0	0
$496$	−5093.24	−0.461075
$497$	0	0
$498$	0	0
$499$	−763.351	−0.0684815	−0.0342408	−	0.999414i	$-0.510901\pi$
$499$	−763.351	−0.0684815	−0.0342408	+	0.999414i	$0.510901\pi$
$500$	1887.77	0.168848
$501$	0	0
$502$	15476.0	1.37595
$503$	5902.36	0.523207	0.261604	−	0.965175i	$-0.415749\pi$
$503$	5902.36	0.523207	0.261604	+	0.965175i	$0.415749\pi$
$504$	0	0
$505$	6797.64	0.598993
$506$	−8913.64	−0.783122
$507$	0	0
$508$	−21655.4	−1.89134
$509$	2200.30	0.191604	0.0958020	−	0.995400i	$-0.469458\pi$
$509$	2200.30	0.191604	0.0958020	+	0.995400i	$0.469458\pi$
$510$	0	0
$511$	0	0
$512$	−14632.8	−1.26305
$513$	0	0
$514$	10644.2	0.913413
$515$	−7240.29	−0.619506
$516$	0	0
$517$	15458.3	1.31500
$518$	0	0
$519$	0	0
$520$	−10403.9	−0.877388
$521$	−1213.69	−0.102059	−0.0510296	−	0.998697i	$-0.516250\pi$
$521$	−1213.69	−0.102059	−0.0510296	+	0.998697i	$0.516250\pi$
$522$	0	0
$523$	8864.18	0.741116	0.370558	−	0.928809i	$-0.379167\pi$
$523$	8864.18	0.741116	0.370558	+	0.928809i	$0.379167\pi$
$524$	9830.52	0.819558
$525$	0	0
$526$	26229.0	2.17422
$527$	−3664.03	−0.302861
$528$	0	0
$529$	−10954.4	−0.900337
$530$	−14002.0	−1.14757
$531$	0	0
$532$	0	0
$533$	−25062.0	−2.03669
$534$	0	0
$535$	8332.27	0.673337
$536$	−19386.5	−1.56225
$537$	0	0
$538$	−27977.3	−2.24198
$539$	0	0
$540$	0	0
$541$	−11585.7	−0.920714	−0.460357	−	0.887734i	$-0.652279\pi$
$541$	−11585.7	−0.920714	−0.460357	+	0.887734i	$0.652279\pi$
$542$	−15864.9	−1.25730
$543$	0	0
$544$	−2028.10	−0.159842
$545$	−5591.98	−0.439513
$546$	0	0
$547$	4233.45	0.330913	0.165456	−	0.986217i	$-0.447090\pi$
$547$	4233.45	0.330913	0.165456	+	0.986217i	$0.447090\pi$
$548$	9329.86	0.727284
$549$	0	0
$550$	−6399.37	−0.496128
$551$	−2533.34	−0.195869
$552$	0	0
$553$	0	0
$554$	−18607.1	−1.42696
$555$	0	0
$556$	−23032.0	−1.75679
$557$	−2198.23	−0.167221	−0.0836106	−	0.996499i	$-0.526645\pi$
$557$	−2198.23	−0.167221	−0.0836106	+	0.996499i	$0.526645\pi$
$558$	0	0
$559$	30386.5	2.29913
$560$	0	0
$561$	0	0
$562$	6783.87	0.509182
$563$	−5918.67	−0.443059	−0.221529	−	0.975154i	$-0.571105\pi$
$563$	−5918.67	−0.443059	−0.221529	+	0.975154i	$0.571105\pi$
$564$	0	0
$565$	−4033.66	−0.300349
$566$	22842.8	1.69639
$567$	0	0
$568$	−30310.9	−2.23911
$569$	2560.97	0.188684	0.0943422	−	0.995540i	$-0.469925\pi$
$569$	2560.97	0.188684	0.0943422	+	0.995540i	$0.469925\pi$
$570$	0	0
$571$	−24683.8	−1.80908	−0.904540	−	0.426389i	$-0.859786\pi$
$571$	−24683.8	−1.80908	−0.904540	+	0.426389i	$0.859786\pi$
$572$	49025.1	3.58364
$573$	0	0
$574$	0	0
$575$	870.559	0.0631388
$576$	0	0
$577$	−10144.6	−0.731932	−0.365966	−	0.930628i	$-0.619261\pi$
$577$	−10144.6	−0.731932	−0.365966	+	0.930628i	$0.619261\pi$
$578$	−18959.4	−1.36437
$579$	0	0
$580$	−10679.0	−0.764518
$581$	0	0
$582$	0	0
$583$	31028.8	2.20426
$584$	37294.3	2.64255
$585$	0	0
$586$	−6254.78	−0.440926
$587$	12204.6	0.858158	0.429079	−	0.903267i	$-0.358838\pi$
$587$	12204.6	0.858158	0.429079	+	0.903267i	$0.358838\pi$
$588$	0	0
$589$	−2109.10	−0.147545
$590$	−15806.3	−1.10294
$591$	0	0
$592$	7584.05	0.526525
$593$	10372.0	0.718255	0.359128	−	0.933288i	$-0.383074\pi$
$593$	10372.0	0.718255	0.359128	+	0.933288i	$0.383074\pi$
$594$	0	0
$595$	0	0
$596$	20959.1	1.44047
$597$	0	0
$598$	−10202.2	−0.697655
$599$	20908.1	1.42618	0.713091	−	0.701072i	$-0.247295\pi$
$599$	20908.1	1.42618	0.713091	+	0.701072i	$0.247295\pi$
$600$	0	0
$601$	758.785	0.0515000	0.0257500	−	0.999668i	$-0.491803\pi$
$601$	758.785	0.0515000	0.0257500	+	0.999668i	$0.491803\pi$
$602$	0	0
$603$	0	0
$604$	34498.6	2.32405
$605$	7526.15	0.505755
$606$	0	0
$607$	−18780.8	−1.25583	−0.627917	−	0.778281i	$-0.716092\pi$
$607$	−18780.8	−1.25583	−0.627917	+	0.778281i	$0.716092\pi$
$608$	−1167.42	−0.0778702
$609$	0	0
$610$	10025.5	0.665444
$611$	17692.9	1.17149
$612$	0	0
$613$	27346.0	1.80179	0.900893	−	0.434040i	$-0.142912\pi$
$613$	27346.0	1.80179	0.900893	+	0.434040i	$0.142912\pi$
$614$	−29502.7	−1.93914
$615$	0	0
$616$	0	0
$617$	16384.9	1.06910	0.534548	−	0.845138i	$-0.320482\pi$
$617$	16384.9	1.06910	0.534548	+	0.845138i	$0.320482\pi$
$618$	0	0
$619$	9536.46	0.619229	0.309614	−	0.950862i	$-0.399800\pi$
$619$	9536.46	0.619229	0.309614	+	0.950862i	$0.399800\pi$
$620$	−8890.62	−0.575896
$621$	0	0
$622$	19620.2	1.26479
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−34333.6	−2.19209
$627$	0	0
$628$	−4052.04	−0.257475
$629$	5455.89	0.345852
$630$	0	0
$631$	−20656.2	−1.30318	−0.651592	−	0.758570i	$-0.725899\pi$
$631$	−20656.2	−1.30318	−0.651592	+	0.758570i	$0.725899\pi$
$632$	4622.94	0.290966
$633$	0	0
$634$	35960.2	2.25262
$635$	−7169.61	−0.448059
$636$	0	0
$637$	0	0
$638$	36200.7	2.24639
$639$	0	0
$640$	−13237.9	−0.817618
$641$	17904.5	1.10326	0.551628	−	0.834091i	$-0.314007\pi$
$641$	17904.5	1.10326	0.551628	+	0.834091i	$0.314007\pi$
$642$	0	0
$643$	4497.56	0.275842	0.137921	−	0.990443i	$-0.455958\pi$
$643$	4497.56	0.275842	0.137921	+	0.990443i	$0.455958\pi$
$644$	0	0
$645$	0	0
$646$	2679.40	0.163188
$647$	261.315	0.0158784	0.00793922	−	0.999968i	$-0.497473\pi$
$647$	261.315	0.0158784	0.00793922	+	0.999968i	$0.497473\pi$
$648$	0	0
$649$	35027.2	2.11855
$650$	−7324.44	−0.441982
$651$	0	0
$652$	−41192.5	−2.47427
$653$	−3791.14	−0.227196	−0.113598	−	0.993527i	$-0.536238\pi$
$653$	−3791.14	−0.227196	−0.113598	+	0.993527i	$0.536238\pi$
$654$	0	0
$655$	3254.67	0.194153
$656$	17786.1	1.05858
$657$	0	0
$658$	0	0
$659$	−1149.18	−0.0679298	−0.0339649	−	0.999423i	$-0.510813\pi$
$659$	−1149.18	−0.0679298	−0.0339649	+	0.999423i	$0.510813\pi$
$660$	0	0
$661$	−1943.10	−0.114338	−0.0571692	−	0.998365i	$-0.518207\pi$
$661$	−1943.10	−0.114338	−0.0571692	+	0.998365i	$0.518207\pi$
$662$	−9344.35	−0.548608
$663$	0	0
$664$	−15857.2	−0.926773
$665$	0	0
$666$	0	0
$667$	−4924.68	−0.285883
$668$	33744.4	1.95451
$669$	0	0
$670$	−13648.2	−0.786981
$671$	−22216.7	−1.27819
$672$	0	0
$673$	31546.2	1.80686	0.903431	−	0.428734i	$-0.141040\pi$
$673$	31546.2	1.80686	0.903431	+	0.428734i	$0.141040\pi$
$674$	−9230.14	−0.527495
$675$	0	0
$676$	22932.5	1.30476
$677$	−3409.55	−0.193560	−0.0967798	−	0.995306i	$-0.530854\pi$
$677$	−3409.55	−0.193560	−0.0967798	+	0.995306i	$0.530854\pi$
$678$	0	0
$679$	0	0
$680$	5311.60	0.299545
$681$	0	0
$682$	30138.3	1.69217
$683$	−2632.86	−0.147502	−0.0737508	−	0.997277i	$-0.523497\pi$
$683$	−2632.86	−0.147502	−0.0737508	+	0.997277i	$0.523497\pi$
$684$	0	0
$685$	3088.91	0.172294
$686$	0	0
$687$	0	0
$688$	−21564.8	−1.19499
$689$	35514.2	1.96369
$690$	0	0
$691$	−25251.1	−1.39015	−0.695076	−	0.718936i	$-0.744629\pi$
$691$	−25251.1	−1.39015	−0.695076	+	0.718936i	$0.744629\pi$
$692$	35334.1	1.94104
$693$	0	0
$694$	−10062.1	−0.550363
$695$	−7625.39	−0.416183
$696$	0	0
$697$	12795.1	0.695337
$698$	−10569.4	−0.573149
$699$	0	0
$700$	0	0
$701$	18046.3	0.972325	0.486162	−	0.873869i	$-0.338396\pi$
$701$	18046.3	0.972325	0.486162	+	0.873869i	$0.338396\pi$
$702$	0	0
$703$	3140.53	0.168489
$704$	35112.4	1.87975
$705$	0	0
$706$	−28171.5	−1.50177
$707$	0	0
$708$	0	0
$709$	−21736.6	−1.15139	−0.575693	−	0.817666i	$-0.695268\pi$
$709$	−21736.6	−1.15139	−0.575693	+	0.817666i	$0.695268\pi$
$710$	−21339.1	−1.12795
$711$	0	0
$712$	1088.27	0.0572816
$713$	−4099.96	−0.215350
$714$	0	0
$715$	16231.1	0.848965
$716$	−36936.2	−1.92789
$717$	0	0
$718$	−34950.0	−1.81661
$719$	10921.1	0.566467	0.283233	−	0.959051i	$-0.408593\pi$
$719$	10921.1	0.566467	0.283233	+	0.959051i	$0.408593\pi$
$720$	0	0
$721$	0	0
$722$	−31425.3	−1.61984
$723$	0	0
$724$	−17797.2	−0.913576
$725$	−3535.57	−0.181114
$726$	0	0
$727$	15670.3	0.799420	0.399710	−	0.916642i	$-0.369111\pi$
$727$	15670.3	0.799420	0.399710	+	0.916642i	$0.369111\pi$
$728$	0	0
$729$	0	0
$730$	26255.5	1.33118
$731$	−15513.5	−0.784935
$732$	0	0
$733$	8032.15	0.404740	0.202370	−	0.979309i	$-0.435136\pi$
$733$	8032.15	0.404740	0.202370	+	0.979309i	$0.435136\pi$
$734$	60744.0	3.05463
$735$	0	0
$736$	−2269.39	−0.113656
$737$	30244.8	1.51164
$738$	0	0
$739$	9602.50	0.477989	0.238994	−	0.971021i	$-0.423182\pi$
$739$	9602.50	0.477989	0.238994	+	0.971021i	$0.423182\pi$
$740$	13238.5	0.657645
$741$	0	0
$742$	0	0
$743$	17641.7	0.871078	0.435539	−	0.900170i	$-0.356558\pi$
$743$	17641.7	0.871078	0.435539	+	0.900170i	$0.356558\pi$
$744$	0	0
$745$	6939.09	0.341247
$746$	−29861.7	−1.46557
$747$	0	0
$748$	−25029.2	−1.22347
$749$	0	0
$750$	0	0
$751$	5688.94	0.276421	0.138211	−	0.990403i	$-0.455865\pi$
$751$	5688.94	0.276421	0.138211	+	0.990403i	$0.455865\pi$
$752$	−12556.3	−0.608886
$753$	0	0
$754$	41433.7	2.00123
$755$	11421.7	0.550568
$756$	0	0
$757$	−18939.2	−0.909324	−0.454662	−	0.890664i	$-0.650240\pi$
$757$	−18939.2	−0.909324	−0.454662	+	0.890664i	$0.650240\pi$
$758$	24395.4	1.16897
$759$	0	0
$760$	3057.48	0.145929
$761$	−16101.6	−0.766992	−0.383496	−	0.923542i	$-0.625280\pi$
$761$	−16101.6	−0.766992	−0.383496	+	0.923542i	$0.625280\pi$
$762$	0	0
$763$	0	0
$764$	34361.8	1.62718
$765$	0	0
$766$	−40440.3	−1.90753
$767$	40090.6	1.88734
$768$	0	0
$769$	−10781.2	−0.505568	−0.252784	−	0.967523i	$-0.581346\pi$
$769$	−10781.2	−0.505568	−0.252784	+	0.967523i	$0.581346\pi$
$770$	0	0
$771$	0	0
$772$	66616.3	3.10567
$773$	−10224.8	−0.475755	−0.237878	−	0.971295i	$-0.576452\pi$
$773$	−10224.8	−0.475755	−0.237878	+	0.971295i	$0.576452\pi$
$774$	0	0
$775$	−2943.49	−0.136430
$776$	8697.31	0.402339
$777$	0	0
$778$	−17825.7	−0.821444
$779$	7365.16	0.338748
$780$	0	0
$781$	47288.0	2.16658
$782$	5208.60	0.238183
$783$	0	0
$784$	0	0
$785$	−1341.54	−0.0609958
$786$	0	0
$787$	12823.4	0.580817	0.290409	−	0.956903i	$-0.406209\pi$
$787$	12823.4	0.580817	0.290409	+	0.956903i	$0.406209\pi$
$788$	30376.7	1.37326
$789$	0	0
$790$	3254.59	0.146573
$791$	0	0
$792$	0	0
$793$	−25428.3	−1.13870
$794$	21725.5	0.971044
$795$	0	0
$796$	−16782.1	−0.747268
$797$	−28745.8	−1.27758	−0.638789	−	0.769382i	$-0.720564\pi$
$797$	−28745.8	−1.27758	−0.638789	+	0.769382i	$0.720564\pi$
$798$	0	0
$799$	−9032.91	−0.399951
$800$	−1629.27	−0.0720041
$801$	0	0
$802$	60522.0	2.66472
$803$	−58182.8	−2.55694
$804$	0	0
$805$	0	0
$806$	34495.0	1.50749
$807$	0	0
$808$	46409.5	2.02065
$809$	31758.8	1.38020	0.690099	−	0.723715i	$-0.257567\pi$
$809$	31758.8	1.38020	0.690099	+	0.723715i	$0.257567\pi$
$810$	0	0
$811$	3269.28	0.141554	0.0707769	−	0.997492i	$-0.477452\pi$
$811$	3269.28	0.141554	0.0707769	+	0.997492i	$0.477452\pi$
$812$	0	0
$813$	0	0
$814$	−44877.3	−1.93237
$815$	−13637.9	−0.586154
$816$	0	0
$817$	−8929.92	−0.382397
$818$	−43343.0	−1.85263
$819$	0	0
$820$	31046.9	1.32220
$821$	−17959.3	−0.763441	−0.381720	−	0.924278i	$-0.624668\pi$
$821$	−17959.3	−0.763441	−0.381720	+	0.924278i	$0.624668\pi$
$822$	0	0
$823$	−22816.3	−0.966373	−0.483187	−	0.875517i	$-0.660521\pi$
$823$	−22816.3	−0.966373	−0.483187	+	0.875517i	$0.660521\pi$
$824$	−49431.6	−2.08985
$825$	0	0
$826$	0	0
$827$	−38776.6	−1.63047	−0.815233	−	0.579133i	$-0.803391\pi$
$827$	−38776.6	−1.63047	−0.815233	+	0.579133i	$0.803391\pi$
$828$	0	0
$829$	43239.0	1.81152	0.905761	−	0.423789i	$-0.139300\pi$
$829$	43239.0	1.81152	0.905761	+	0.423789i	$0.139300\pi$
$830$	−11163.6	−0.466860
$831$	0	0
$832$	40188.1	1.67460
$833$	0	0
$834$	0	0
$835$	11172.0	0.463023
$836$	−14407.4	−0.596040
$837$	0	0
$838$	18.2968	0.000754240 0
$839$	−38639.2	−1.58996	−0.794978	−	0.606638i	$-0.792518\pi$
$839$	−38639.2	−1.58996	−0.794978	+	0.606638i	$0.792518\pi$
$840$	0	0
$841$	−4388.57	−0.179940
$842$	58043.1	2.37565
$843$	0	0
$844$	45775.3	1.86689
$845$	7592.44	0.309098
$846$	0	0
$847$	0	0
$848$	−25203.9	−1.02064
$849$	0	0
$850$	3739.41	0.150895
$851$	6105.02	0.245919
$852$	0	0
$853$	−21024.3	−0.843914	−0.421957	−	0.906616i	$-0.638657\pi$
$853$	−21024.3	−0.843914	−0.421957	+	0.906616i	$0.638657\pi$
$854$	0	0
$855$	0	0
$856$	56886.9	2.27144
$857$	40140.9	1.59998	0.799991	−	0.600011i	$-0.204837\pi$
$857$	40140.9	1.59998	0.799991	+	0.600011i	$0.204837\pi$
$858$	0	0
$859$	−16970.0	−0.674050	−0.337025	−	0.941496i	$-0.609421\pi$
$859$	−16970.0	−0.674050	−0.337025	+	0.941496i	$0.609421\pi$
$860$	−37642.9	−1.49257
$861$	0	0
$862$	49591.8	1.95952
$863$	−28638.9	−1.12964	−0.564819	−	0.825215i	$-0.691054\pi$
$863$	−28638.9	−1.12964	−0.564819	+	0.825215i	$0.691054\pi$
$864$	0	0
$865$	11698.3	0.459832
$866$	−61178.5	−2.40061
$867$	0	0
$868$	0	0
$869$	−7212.24	−0.281540
$870$	0	0
$871$	34616.9	1.34667
$872$	−38178.1	−1.48265
$873$	0	0
$874$	2998.19	0.116036
$875$	0	0
$876$	0	0
$877$	1459.67	0.0562026	0.0281013	−	0.999605i	$-0.491054\pi$
$877$	1459.67	0.0562026	0.0281013	+	0.999605i	$0.491054\pi$
$878$	47933.0	1.84244
$879$	0	0
$880$	−11519.0	−0.441254
$881$	−34268.7	−1.31049	−0.655246	−	0.755416i	$-0.727435\pi$
$881$	−34268.7	−1.31049	−0.655246	+	0.755416i	$0.727435\pi$
$882$	0	0
$883$	−23106.9	−0.880645	−0.440322	−	0.897840i	$-0.645136\pi$
$883$	−23106.9	−0.880645	−0.440322	+	0.897840i	$0.645136\pi$
$884$	−28647.4	−1.08995
$885$	0	0
$886$	−50289.1	−1.90688
$887$	22858.2	0.865280	0.432640	−	0.901567i	$-0.357582\pi$
$887$	22858.2	0.865280	0.432640	+	0.901567i	$0.357582\pi$
$888$	0	0
$889$	0	0
$890$	766.148	0.0288555
$891$	0	0
$892$	57236.3	2.14844
$893$	−5199.54	−0.194844
$894$	0	0
$895$	−12228.8	−0.456718
$896$	0	0
$897$	0	0
$898$	56066.0	2.08346
$899$	16651.1	0.617735
$900$	0	0
$901$	−18131.4	−0.670416
$902$	−105246.	−3.88504
$903$	0	0
$904$	−27539.0	−1.01320
$905$	−5892.27	−0.216426
$906$	0	0
$907$	−4420.76	−0.161840	−0.0809199	−	0.996721i	$-0.525786\pi$
$907$	−4420.76	−0.161840	−0.0809199	+	0.996721i	$0.525786\pi$
$908$	58540.8	2.13959
$909$	0	0
$910$	0	0
$911$	−31813.3	−1.15699	−0.578496	−	0.815685i	$-0.696360\pi$
$911$	−31813.3	−1.15699	−0.578496	+	0.815685i	$0.696360\pi$
$912$	0	0
$913$	24738.7	0.896750
$914$	21833.5	0.790142
$915$	0	0
$916$	−89502.2	−3.22842
$917$	0	0
$918$	0	0
$919$	22036.5	0.790988	0.395494	−	0.918469i	$-0.370573\pi$
$919$	22036.5	0.790988	0.395494	+	0.918469i	$0.370573\pi$
$920$	5943.56	0.212993
$921$	0	0
$922$	37911.3	1.35417
$923$	54123.8	1.93012
$924$	0	0
$925$	4382.98	0.155796
$926$	−26167.1	−0.928624
$927$	0	0
$928$	9216.62	0.326024
$929$	−27015.4	−0.954086	−0.477043	−	0.878880i	$-0.658291\pi$
$929$	−27015.4	−0.954086	−0.477043	+	0.878880i	$0.658291\pi$
$930$	0	0
$931$	0	0
$932$	−70120.0	−2.46444
$933$	0	0
$934$	−50843.0	−1.78119
$935$	−8286.62	−0.289841
$936$	0	0
$937$	−33711.1	−1.17534	−0.587670	−	0.809101i	$-0.699954\pi$
$937$	−33711.1	−1.17534	−0.587670	+	0.809101i	$0.699954\pi$
$938$	0	0
$939$	0	0
$940$	−21918.0	−0.760517
$941$	23168.8	0.802638	0.401319	−	0.915938i	$-0.368552\pi$
$941$	23168.8	0.802638	0.401319	+	0.915938i	$0.368552\pi$
$942$	0	0
$943$	14317.5	0.494423
$944$	−28451.6	−0.980954
$945$	0	0
$946$	127606.	4.38565
$947$	−39932.2	−1.37024	−0.685122	−	0.728428i	$-0.740251\pi$
$947$	−39932.2	−1.37024	−0.685122	+	0.728428i	$0.740251\pi$
$948$	0	0
$949$	−66593.5	−2.27789
$950$	2152.49	0.0735116
$951$	0	0
$952$	0	0
$953$	31125.8	1.05799	0.528995	−	0.848625i	$-0.322569\pi$
$953$	31125.8	1.05799	0.528995	+	0.848625i	$0.322569\pi$
$954$	0	0
$955$	11376.4	0.385479
$956$	−14256.8	−0.482321
$957$	0	0
$958$	55970.0	1.88759
$959$	0	0
$960$	0	0
$961$	−15928.4	−0.534672
$962$	−51364.6	−1.72148
$963$	0	0
$964$	−22880.0	−0.764433
$965$	22055.2	0.735732
$966$	0	0
$967$	−35679.1	−1.18652	−0.593259	−	0.805011i	$-0.702159\pi$
$967$	−35679.1	−1.18652	−0.593259	+	0.805011i	$0.702159\pi$
$968$	51383.3	1.70612
$969$	0	0
$970$	6122.98	0.202677
$971$	−4137.64	−0.136749	−0.0683744	−	0.997660i	$-0.521781\pi$
$971$	−4137.64	−0.136749	−0.0683744	+	0.997660i	$0.521781\pi$
$972$	0	0
$973$	0	0
$974$	−44398.5	−1.46059
$975$	0	0
$976$	18046.0	0.591844
$977$	−30405.7	−0.995664	−0.497832	−	0.867273i	$-0.665870\pi$
$977$	−30405.7	−0.995664	−0.497832	+	0.867273i	$0.665870\pi$
$978$	0	0
$979$	−1697.80	−0.0554259
$980$	0	0
$981$	0	0
$982$	12998.9	0.422416
$983$	10011.3	0.324832	0.162416	−	0.986722i	$-0.448071\pi$
$983$	10011.3	0.324832	0.162416	+	0.986722i	$0.448071\pi$
$984$	0	0
$985$	10057.1	0.325324
$986$	−21153.5	−0.683231
$987$	0	0
$988$	−16490.1	−0.530991
$989$	−17359.3	−0.558132
$990$	0	0
$991$	−2889.51	−0.0926219	−0.0463110	−	0.998927i	$-0.514747\pi$
$991$	−2889.51	−0.0926219	−0.0463110	+	0.998927i	$0.514747\pi$
$992$	7673.15	0.245588
$993$	0	0
$994$	0	0
$995$	−5556.18	−0.177028
$996$	0	0
$997$	6005.63	0.190772	0.0953862	−	0.995440i	$-0.469591\pi$
$997$	6005.63	0.190772	0.0953862	+	0.995440i	$0.469591\pi$
$998$	−3669.03	−0.116374
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bs.1.5		5
3.2	odd	2		735.4.a.z.1.1		5
7.3	odd	6		315.4.j.h.226.1		10
7.5	odd	6		315.4.j.h.46.1		10
7.6	odd	2		2205.4.a.br.1.5		5
21.5	even	6		105.4.i.d.46.5	yes	10
21.17	even	6		105.4.i.d.16.5	✓	10
21.20	even	2		735.4.a.ba.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.5	✓	10	21.17	even	6
105.4.i.d.46.5	yes	10	21.5	even	6
315.4.j.h.46.1		10	7.5	odd	6
315.4.j.h.226.1		10	7.3	odd	6
735.4.a.z.1.1		5	3.2	odd	2
735.4.a.ba.1.1		5	21.20	even	2
2205.4.a.br.1.5		5	7.6	odd	2
2205.4.a.bs.1.5		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)

\(f(q)\)	\(=\)	\(q+4.80647 q^{2} +15.1022 q^{4} +5.00000 q^{5} +34.1365 q^{8} +24.0324 q^{10} -53.2563 q^{11} -60.9548 q^{13} +43.2586 q^{16} +31.1198 q^{17} +17.9133 q^{19} +75.5109 q^{20} -255.975 q^{22} +34.8223 q^{23} +25.0000 q^{25} -292.978 q^{26} -141.423 q^{29} -117.739 q^{31} -65.1706 q^{32} +149.576 q^{34} +175.319 q^{37} +86.0996 q^{38} +170.682 q^{40} +411.157 q^{41} -498.509 q^{43} -804.286 q^{44} +167.373 q^{46} -290.262 q^{47} +120.162 q^{50} -920.551 q^{52} -582.633 q^{53} -266.281 q^{55} -679.745 q^{58} -657.710 q^{59} +417.167 q^{61} -565.911 q^{62} -659.310 q^{64} -304.774 q^{65} -567.911 q^{67} +469.977 q^{68} -887.933 q^{71} +1092.51 q^{73} +842.666 q^{74} +270.529 q^{76} +135.425 q^{79} +216.293 q^{80} +1976.22 q^{82} -464.523 q^{83} +155.599 q^{85} -2396.07 q^{86} -1817.98 q^{88} +31.8799 q^{89} +525.894 q^{92} -1395.14 q^{94} +89.5663 q^{95} +254.781 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} + 25 q^{4} + 25 q^{5} - 21 q^{8} - 15 q^{10} - 43 q^{11} - 123 q^{13} + 161 q^{16} + 124 q^{17} - 37 q^{19} + 125 q^{20} - 221 q^{22} - 77 q^{23} + 125 q^{25} - 79 q^{26} - 360 q^{29} - 314 q^{31}+ \cdots + 576 q^{97}+O(q^{100}) \) 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

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