LMFDB - Embedded newform 441.4.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.6257832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 19x^{2} + 42 $ x^4 - 19*x^2 + 42
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.05539$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} +1.80961 q^{8} +O(q^{10})$	$q+4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} +1.80961 q^{8} +40.2311 q^{10} +13.5396 q^{11} +18.5538 q^{13} -60.2311 q^{16} +93.7102 q^{17} +131.908 q^{19} +83.7899 q^{20} +54.9084 q^{22} +198.278 q^{23} -26.5858 q^{25} +75.2429 q^{26} -188.358 q^{29} -83.9244 q^{31} -258.738 q^{32} +380.032 q^{34} +80.1555 q^{37} +534.941 q^{38} +17.9520 q^{40} +385.828 q^{41} -397.048 q^{43} +114.359 q^{44} +804.096 q^{46} -272.277 q^{47} -107.816 q^{50} +156.709 q^{52} +36.9996 q^{53} +134.318 q^{55} -763.865 q^{58} -395.749 q^{59} +13.4738 q^{61} -340.347 q^{62} -567.435 q^{64} +184.061 q^{65} +340.522 q^{67} +791.498 q^{68} +211.140 q^{71} +486.124 q^{73} +325.062 q^{74} +1114.13 q^{76} +293.741 q^{79} -597.516 q^{80} +1564.69 q^{82} -889.635 q^{83} +929.643 q^{85} -1610.19 q^{86} +24.5014 q^{88} -1144.36 q^{89} +1674.70 q^{92} -1104.19 q^{94} +1308.58 q^{95} +1384.61 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 4 q + 6 q^{4} + 22 q^{10} + 102 q^{13} - 102 q^{16} + 222 q^{19} - 86 q^{22} + 366 q^{25} + 220 q^{31} + 1020 q^{34} - 374 q^{37} + 822 q^{40} - 838 q^{43} + 1716 q^{46} - 40 q^{52} + 2510 q^{55} - 1694 q^{58} + 1332 q^{61} - 686 q^{64} + 1890 q^{67} + 1750 q^{73} + 2456 q^{76} + 8 q^{79} + 2480 q^{82} - 1116 q^{85} + 2682 q^{88} - 1416 q^{94} + 3010 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 + 22 * q^10 + 102 * q^13 - 102 * q^16 + 222 * q^19 - 86 * q^22 + 366 * q^25 + 220 * q^31 + 1020 * q^34 - 374 * q^37 + 822 * q^40 - 838 * q^43 + 1716 * q^46 - 40 * q^52 + 2510 * q^55 - 1694 * q^58 + 1332 * q^61 - 686 * q^64 + 1890 * q^67 + 1750 * q^73 + 2456 * q^76 + 8 * q^79 + 2480 * q^82 - 1116 * q^85 + 2682 * q^88 - 1416 * q^94 + 3010 * 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 $

$2$	4.05539	1.43380	0.716899	−	0.697177i	$-0.245561\pi$
$2$	4.05539	1.43380	0.716899	+	0.697177i	$0.245561\pi$
$3$	0	0
$3$	0	0
$4$	8.44622	1.05578
$5$	9.92039	0.887307	0.443654	−	0.896198i	$-0.353682\pi$
$5$	9.92039	0.887307	0.443654	+	0.896198i	$0.353682\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.80961	0.0799740
$9$	0	0
$10$	40.2311	1.27222
$11$	13.5396	0.371122	0.185561	−	0.982633i	$-0.440590\pi$
$11$	13.5396	0.371122	0.185561	+	0.982633i	$0.440590\pi$
$12$	0	0
$13$	18.5538	0.395838	0.197919	−	0.980218i	$-0.436582\pi$
$13$	18.5538	0.395838	0.197919	+	0.980218i	$0.436582\pi$
$14$	0	0
$15$	0	0
$16$	−60.2311	−0.941111
$17$	93.7102	1.33695	0.668473	−	0.743737i	$-0.266948\pi$
$17$	93.7102	1.33695	0.668473	+	0.743737i	$0.266948\pi$
$18$	0	0
$19$	131.908	1.59273	0.796365	−	0.604816i	$-0.206753\pi$
$19$	131.908	1.59273	0.796365	+	0.604816i	$0.206753\pi$
$20$	83.7899	0.936799
$21$	0	0
$22$	54.9084	0.532115
$23$	198.278	1.79756	0.898779	−	0.438401i	$-0.144455\pi$
$23$	198.278	1.79756	0.898779	+	0.438401i	$0.144455\pi$
$24$	0	0
$25$	−26.5858	−0.212686
$26$	75.2429	0.567552
$27$	0	0
$28$	0	0
$29$	−188.358	−1.20611	−0.603054	−	0.797700i	$-0.706050\pi$
$29$	−188.358	−1.20611	−0.603054	+	0.797700i	$0.706050\pi$
$30$	0	0
$31$	−83.9244	−0.486235	−0.243117	−	0.969997i	$-0.578170\pi$
$31$	−83.9244	−0.486235	−0.243117	+	0.969997i	$0.578170\pi$
$32$	−258.738	−1.42934
$33$	0	0
$34$	380.032	1.91691
$35$	0	0
$36$	0	0
$37$	80.1555	0.356148	0.178074	−	0.984017i	$-0.443013\pi$
$37$	80.1555	0.356148	0.178074	+	0.984017i	$0.443013\pi$
$38$	534.941	2.28365
$39$	0	0
$40$	17.9520	0.0709615
$41$	385.828	1.46967	0.734833	−	0.678249i	$-0.237261\pi$
$41$	385.828	1.46967	0.734833	+	0.678249i	$0.237261\pi$
$42$	0	0
$43$	−397.048	−1.40812	−0.704061	−	0.710139i	$-0.748632\pi$
$43$	−397.048	−1.40812	−0.704061	+	0.710139i	$0.748632\pi$
$44$	114.359	0.391823
$45$	0	0
$46$	804.096	2.57734
$47$	−272.277	−0.845016	−0.422508	−	0.906359i	$-0.638850\pi$
$47$	−272.277	−0.845016	−0.422508	+	0.906359i	$0.638850\pi$
$48$	0	0
$49$	0	0
$50$	−107.816	−0.304949
$51$	0	0
$52$	156.709	0.417917
$53$	36.9996	0.0958922	0.0479461	−	0.998850i	$-0.484732\pi$
$53$	36.9996	0.0958922	0.0479461	+	0.998850i	$0.484732\pi$
$54$	0	0
$55$	134.318	0.329299
$56$	0	0
$57$	0	0
$58$	−763.865	−1.72932
$59$	−395.749	−0.873256	−0.436628	−	0.899642i	$-0.643827\pi$
$59$	−395.749	−0.873256	−0.436628	+	0.899642i	$0.643827\pi$
$60$	0	0
$61$	13.4738	0.0282810	0.0141405	−	0.999900i	$-0.495499\pi$
$61$	13.4738	0.0282810	0.0141405	+	0.999900i	$0.495499\pi$
$62$	−340.347	−0.697162
$63$	0	0
$64$	−567.435	−1.10827
$65$	184.061	0.351230
$66$	0	0
$67$	340.522	0.620916	0.310458	−	0.950587i	$-0.399518\pi$
$67$	340.522	0.620916	0.310458	+	0.950587i	$0.399518\pi$
$68$	791.498	1.41152
$69$	0	0
$70$	0	0
$71$	211.140	0.352925	0.176463	−	0.984307i	$-0.443534\pi$
$71$	211.140	0.352925	0.176463	+	0.984307i	$0.443534\pi$
$72$	0	0
$73$	486.124	0.779404	0.389702	−	0.920941i	$-0.372578\pi$
$73$	486.124	0.779404	0.389702	+	0.920941i	$0.372578\pi$
$74$	325.062	0.510645
$75$	0	0
$76$	1114.13	1.68157
$77$	0	0
$78$	0	0
$79$	293.741	0.418335	0.209168	−	0.977880i	$-0.432925\pi$
$79$	293.741	0.418335	0.209168	+	0.977880i	$0.432925\pi$
$80$	−597.516	−0.835055
$81$	0	0
$82$	1564.69	2.10720
$83$	−889.635	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$83$	−889.635	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$84$	0	0
$85$	929.643	1.18628
$86$	−1610.19	−2.01896
$87$	0	0
$88$	24.5014	0.0296801
$89$	−1144.36	−1.36295	−0.681474	−	0.731843i	$-0.738661\pi$
$89$	−1144.36	−1.36295	−0.681474	+	0.731843i	$0.738661\pi$
$90$	0	0
$91$	0	0
$92$	1674.70	1.89782
$93$	0	0
$94$	−1104.19	−1.21158
$95$	1308.58	1.41324
$96$	0	0
$97$	1384.61	1.44933	0.724667	−	0.689099i	$-0.241993\pi$
$97$	1384.61	1.44933	0.724667	+	0.689099i	$0.241993\pi$
$98$	0	0
$99$	0	0
$100$	−224.549	−0.224549
$101$	−1785.99	−1.75953	−0.879765	−	0.475409i	$-0.842300\pi$
$101$	−1785.99	−1.75953	−0.879765	+	0.475409i	$0.842300\pi$
$102$	0	0
$103$	489.661	0.468425	0.234212	−	0.972185i	$-0.424749\pi$
$103$	489.661	0.468425	0.234212	+	0.972185i	$0.424749\pi$
$104$	33.5750	0.0316568
$105$	0	0
$106$	150.048	0.137490
$107$	−282.068	−0.254846	−0.127423	−	0.991848i	$-0.540671\pi$
$107$	−282.068	−0.254846	−0.127423	+	0.991848i	$0.540671\pi$
$108$	0	0
$109$	−287.231	−0.252401	−0.126201	−	0.992005i	$-0.540278\pi$
$109$	−287.231	−0.252401	−0.126201	+	0.992005i	$0.540278\pi$
$110$	544.713	0.472149
$111$	0	0
$112$	0	0
$113$	−1895.21	−1.57776	−0.788879	−	0.614548i	$-0.789338\pi$
$113$	−1895.21	−1.57776	−0.788879	+	0.614548i	$0.789338\pi$
$114$	0	0
$115$	1967.00	1.59499
$116$	−1590.91	−1.27338
$117$	0	0
$118$	−1604.92	−1.25207
$119$	0	0
$120$	0	0
$121$	−1147.68	−0.862268
$122$	54.6415	0.0405493
$123$	0	0
$124$	−708.844	−0.513356
$125$	−1503.79	−1.07603
$126$	0	0
$127$	−1222.92	−0.854463	−0.427231	−	0.904142i	$-0.640511\pi$
$127$	−1222.92	−0.854463	−0.427231	+	0.904142i	$0.640511\pi$
$128$	−231.269	−0.159699
$129$	0	0
$130$	746.439	0.503593
$131$	−1190.61	−0.794074	−0.397037	−	0.917803i	$-0.629962\pi$
$131$	−1190.61	−0.794074	−0.397037	+	0.917803i	$0.629962\pi$
$132$	0	0
$133$	0	0
$134$	1380.95	0.890268
$135$	0	0
$136$	169.579	0.106921
$137$	−106.961	−0.0667031	−0.0333516	−	0.999444i	$-0.510618\pi$
$137$	−106.961	−0.0667031	−0.0333516	+	0.999444i	$0.510618\pi$
$138$	0	0
$139$	−1096.78	−0.669262	−0.334631	−	0.942349i	$-0.608612\pi$
$139$	−1096.78	−0.669262	−0.334631	+	0.942349i	$0.608612\pi$
$140$	0	0
$141$	0	0
$142$	856.256	0.506024
$143$	251.211	0.146904
$144$	0	0
$145$	−1868.58	−1.07019
$146$	1971.42	1.11751
$147$	0	0
$148$	677.012	0.376014
$149$	−716.662	−0.394035	−0.197018	−	0.980400i	$-0.563126\pi$
$149$	−716.662	−0.394035	−0.197018	+	0.980400i	$0.563126\pi$
$150$	0	0
$151$	154.368	0.0831939	0.0415970	−	0.999134i	$-0.486755\pi$
$151$	154.368	0.0831939	0.0415970	+	0.999134i	$0.486755\pi$
$152$	238.702	0.127377
$153$	0	0
$154$	0	0
$155$	−832.564	−0.431439
$156$	0	0
$157$	−2093.75	−1.06433	−0.532163	−	0.846642i	$-0.678621\pi$
$157$	−2093.75	−1.06433	−0.532163	+	0.846642i	$0.678621\pi$
$158$	1191.24	0.599808
$159$	0	0
$160$	−2566.78	−1.26826
$161$	0	0
$162$	0	0
$163$	3007.86	1.44536	0.722680	−	0.691183i	$-0.242910\pi$
$163$	3007.86	1.44536	0.722680	+	0.691183i	$0.242910\pi$
$164$	3258.79	1.55164
$165$	0	0
$166$	−3607.82	−1.68687
$167$	2230.43	1.03351	0.516754	−	0.856134i	$-0.327140\pi$
$167$	2230.43	1.03351	0.516754	+	0.856134i	$0.327140\pi$
$168$	0	0
$169$	−1852.76	−0.843312
$170$	3770.07	1.70089
$171$	0	0
$172$	−3353.56	−1.48666
$173$	563.170	0.247497	0.123749	−	0.992314i	$-0.460508\pi$
$173$	563.170	0.247497	0.123749	+	0.992314i	$0.460508\pi$
$174$	0	0
$175$	0	0
$176$	−815.506	−0.349267
$177$	0	0
$178$	−4640.85	−1.95419
$179$	−1839.50	−0.768104	−0.384052	−	0.923312i	$-0.625472\pi$
$179$	−1839.50	−0.768104	−0.384052	+	0.923312i	$0.625472\pi$
$180$	0	0
$181$	2324.71	0.954664	0.477332	−	0.878723i	$-0.341604\pi$
$181$	2324.71	0.954664	0.477332	+	0.878723i	$0.341604\pi$
$182$	0	0
$183$	0	0
$184$	358.805	0.143758
$185$	795.175	0.316013
$186$	0	0
$187$	1268.80	0.496170
$188$	−2299.71	−0.892149
$189$	0	0
$190$	5306.82	2.02630
$191$	3127.27	1.18472	0.592360	−	0.805673i	$-0.298196\pi$
$191$	3127.27	1.18472	0.592360	+	0.805673i	$0.298196\pi$
$192$	0	0
$193$	−3709.29	−1.38342	−0.691711	−	0.722175i	$-0.743143\pi$
$193$	−3709.29	−1.38342	−0.691711	+	0.722175i	$0.743143\pi$
$194$	5615.12	2.07805
$195$	0	0
$196$	0	0
$197$	−851.150	−0.307827	−0.153913	−	0.988084i	$-0.549188\pi$
$197$	−851.150	−0.307827	−0.153913	+	0.988084i	$0.549188\pi$
$198$	0	0
$199$	3397.78	1.21036	0.605182	−	0.796087i	$-0.293100\pi$
$199$	3397.78	1.21036	0.605182	+	0.796087i	$0.293100\pi$
$200$	−48.1098	−0.0170094
$201$	0	0
$202$	−7242.89	−2.52281
$203$	0	0
$204$	0	0
$205$	3827.57	1.30404
$206$	1985.77	0.671627
$207$	0	0
$208$	−1117.51	−0.372527
$209$	1785.99	0.591098
$210$	0	0
$211$	216.732	0.0707132	0.0353566	−	0.999375i	$-0.488743\pi$
$211$	216.732	0.0707132	0.0353566	+	0.999375i	$0.488743\pi$
$212$	312.507	0.101241
$213$	0	0
$214$	−1143.90	−0.365398
$215$	−3938.87	−1.24944
$216$	0	0
$217$	0	0
$218$	−1164.84	−0.361893
$219$	0	0
$220$	1134.48	0.347667
$221$	1738.68	0.529214
$222$	0	0
$223$	−2254.86	−0.677115	−0.338558	−	0.940946i	$-0.609939\pi$
$223$	−2254.86	−0.677115	−0.338558	+	0.940946i	$0.609939\pi$
$224$	0	0
$225$	0	0
$226$	−7685.84	−2.26219
$227$	3390.80	0.991433	0.495716	−	0.868484i	$-0.334906\pi$
$227$	3390.80	0.991433	0.495716	+	0.868484i	$0.334906\pi$
$228$	0	0
$229$	2587.85	0.746768	0.373384	−	0.927677i	$-0.378197\pi$
$229$	2587.85	0.746768	0.373384	+	0.927677i	$0.378197\pi$
$230$	7976.95	2.28689
$231$	0	0
$232$	−340.853	−0.0964574
$233$	954.420	0.268353	0.134176	−	0.990957i	$-0.457161\pi$
$233$	954.420	0.268353	0.134176	+	0.990957i	$0.457161\pi$
$234$	0	0
$235$	−2701.10	−0.749788
$236$	−3342.58	−0.921964
$237$	0	0
$238$	0	0
$239$	−199.504	−0.0539951	−0.0269976	−	0.999635i	$-0.508595\pi$
$239$	−199.504	−0.0539951	−0.0269976	+	0.999635i	$0.508595\pi$
$240$	0	0
$241$	4794.43	1.28148	0.640739	−	0.767759i	$-0.278628\pi$
$241$	4794.43	1.28148	0.640739	+	0.767759i	$0.278628\pi$
$242$	−4654.29	−1.23632
$243$	0	0
$244$	113.803	0.0298585
$245$	0	0
$246$	0	0
$247$	2447.40	0.630463
$248$	−151.870	−0.0388862
$249$	0	0
$250$	−6098.46	−1.54280
$251$	−6249.73	−1.57163	−0.785816	−	0.618460i	$-0.787757\pi$
$251$	−6249.73	−1.57163	−0.785816	+	0.618460i	$0.787757\pi$
$252$	0	0
$253$	2684.61	0.667114
$254$	−4959.43	−1.22513
$255$	0	0
$256$	3601.59	0.879294
$257$	3837.85	0.931512	0.465756	−	0.884913i	$-0.345782\pi$
$257$	3837.85	0.931512	0.465756	+	0.884913i	$0.345782\pi$
$258$	0	0
$259$	0	0
$260$	1554.62	0.370821
$261$	0	0
$262$	−4828.38	−1.13854
$263$	−207.203	−0.0485806	−0.0242903	−	0.999705i	$-0.507733\pi$
$263$	−207.203	−0.0485806	−0.0242903	+	0.999705i	$0.507733\pi$
$264$	0	0
$265$	367.051	0.0850858
$266$	0	0
$267$	0	0
$268$	2876.12	0.655549
$269$	−4804.08	−1.08888	−0.544442	−	0.838798i	$-0.683259\pi$
$269$	−4804.08	−1.08888	−0.544442	+	0.838798i	$0.683259\pi$
$270$	0	0
$271$	3215.75	0.720823	0.360411	−	0.932793i	$-0.382636\pi$
$271$	3215.75	0.720823	0.360411	+	0.932793i	$0.382636\pi$
$272$	−5644.27	−1.25821
$273$	0	0
$274$	−433.771	−0.0956388
$275$	−359.961	−0.0789326
$276$	0	0
$277$	2054.39	0.445619	0.222810	−	0.974862i	$-0.428477\pi$
$277$	2054.39	0.445619	0.222810	+	0.974862i	$0.428477\pi$
$278$	−4447.87	−0.959587
$279$	0	0
$280$	0	0
$281$	−1768.61	−0.375468	−0.187734	−	0.982220i	$-0.560114\pi$
$281$	−1768.61	−0.375468	−0.187734	+	0.982220i	$0.560114\pi$
$282$	0	0
$283$	2340.53	0.491625	0.245813	−	0.969317i	$-0.420945\pi$
$283$	2340.53	0.491625	0.245813	+	0.969317i	$0.420945\pi$
$284$	1783.34	0.372611
$285$	0	0
$286$	1018.76	0.210631
$287$	0	0
$288$	0	0
$289$	3868.61	0.787423
$290$	−7577.84	−1.53443
$291$	0	0
$292$	4105.91	0.822877
$293$	3633.47	0.724470	0.362235	−	0.932087i	$-0.382014\pi$
$293$	3633.47	0.724470	0.362235	+	0.932087i	$0.382014\pi$
$294$	0	0
$295$	−3925.98	−0.774846
$296$	145.050	0.0284826
$297$	0	0
$298$	−2906.35	−0.564967
$299$	3678.81	0.711542
$300$	0	0
$301$	0	0
$302$	626.023	0.119283
$303$	0	0
$304$	−7944.99	−1.49894
$305$	133.665	0.0250939
$306$	0	0
$307$	5954.32	1.10694	0.553471	−	0.832868i	$-0.313303\pi$
$307$	5954.32	1.10694	0.553471	+	0.832868i	$0.313303\pi$
$308$	0	0
$309$	0	0
$310$	−3376.37	−0.618597
$311$	−1180.09	−0.215167	−0.107584	−	0.994196i	$-0.534311\pi$
$311$	−1180.09	−0.215167	−0.107584	+	0.994196i	$0.534311\pi$
$312$	0	0
$313$	−9746.25	−1.76003	−0.880017	−	0.474943i	$-0.842469\pi$
$313$	−9746.25	−1.76003	−0.880017	+	0.474943i	$0.842469\pi$
$314$	−8490.97	−1.52603
$315$	0	0
$316$	2481.00	0.441669
$317$	−8591.91	−1.52230	−0.761151	−	0.648574i	$-0.775366\pi$
$317$	−8591.91	−1.52230	−0.761151	+	0.648574i	$0.775366\pi$
$318$	0	0
$319$	−2550.29	−0.447614
$320$	−5629.18	−0.983377
$321$	0	0
$322$	0	0
$323$	12361.2	2.12939
$324$	0	0
$325$	−493.267	−0.0841892
$326$	12198.0	2.07235
$327$	0	0
$328$	698.197	0.117535
$329$	0	0
$330$	0	0
$331$	−5251.20	−0.872000	−0.436000	−	0.899947i	$-0.643605\pi$
$331$	−5251.20	−0.872000	−0.436000	+	0.899947i	$0.643605\pi$
$332$	−7514.05	−1.24213
$333$	0	0
$334$	9045.28	1.48184
$335$	3378.11	0.550943
$336$	0	0
$337$	−8496.45	−1.37339	−0.686693	−	0.726947i	$-0.740938\pi$
$337$	−8496.45	−1.37339	−0.686693	+	0.726947i	$0.740938\pi$
$338$	−7513.66	−1.20914
$339$	0	0
$340$	7851.97	1.25245
$341$	−1136.30	−0.180453
$342$	0	0
$343$	0	0
$344$	−718.500	−0.112613
$345$	0	0
$346$	2283.88	0.354861
$347$	5830.15	0.901956	0.450978	−	0.892535i	$-0.351075\pi$
$347$	5830.15	0.901956	0.450978	+	0.892535i	$0.351075\pi$
$348$	0	0
$349$	1811.13	0.277786	0.138893	−	0.990307i	$-0.455646\pi$
$349$	1811.13	0.277786	0.138893	+	0.990307i	$0.455646\pi$
$350$	0	0
$351$	0	0
$352$	−3503.21	−0.530459
$353$	−3327.76	−0.501753	−0.250876	−	0.968019i	$-0.580719\pi$
$353$	−3327.76	−0.501753	−0.250876	+	0.968019i	$0.580719\pi$
$354$	0	0
$355$	2094.59	0.313153
$356$	−9665.55	−1.43897
$357$	0	0
$358$	−7459.89	−1.10131
$359$	870.861	0.128029	0.0640143	−	0.997949i	$-0.479610\pi$
$359$	870.861	0.128029	0.0640143	+	0.997949i	$0.479610\pi$
$360$	0	0
$361$	10540.8	1.53679
$362$	9427.61	1.36880
$363$	0	0
$364$	0	0
$365$	4822.54	0.691570
$366$	0	0
$367$	1174.72	0.167084	0.0835418	−	0.996504i	$-0.473377\pi$
$367$	1174.72	0.167084	0.0835418	+	0.996504i	$0.473377\pi$
$368$	−11942.5	−1.69170
$369$	0	0
$370$	3224.75	0.453099
$371$	0	0
$372$	0	0
$373$	3628.33	0.503667	0.251834	−	0.967771i	$-0.418966\pi$
$373$	3628.33	0.503667	0.251834	+	0.967771i	$0.418966\pi$
$374$	5145.48	0.711408
$375$	0	0
$376$	−492.715	−0.0675793
$377$	−3494.75	−0.477424
$378$	0	0
$379$	7321.99	0.992362	0.496181	−	0.868219i	$-0.334735\pi$
$379$	7321.99	0.992362	0.496181	+	0.868219i	$0.334735\pi$
$380$	11052.6	1.49207
$381$	0	0
$382$	12682.3	1.69865
$383$	7354.89	0.981247	0.490623	−	0.871372i	$-0.336769\pi$
$383$	7354.89	0.981247	0.490623	+	0.871372i	$0.336769\pi$
$384$	0	0
$385$	0	0
$386$	−15042.6	−1.98355
$387$	0	0
$388$	11694.7	1.53018
$389$	−9069.62	−1.18213	−0.591064	−	0.806624i	$-0.701292\pi$
$389$	−9069.62	−1.18213	−0.591064	+	0.806624i	$0.701292\pi$
$390$	0	0
$391$	18580.7	2.40324
$392$	0	0
$393$	0	0
$394$	−3451.75	−0.441362
$395$	2914.03	0.371192
$396$	0	0
$397$	7376.83	0.932575	0.466288	−	0.884633i	$-0.345591\pi$
$397$	7376.83	0.932575	0.466288	+	0.884633i	$0.345591\pi$
$398$	13779.4	1.73542
$399$	0	0
$400$	1601.29	0.200161
$401$	2853.51	0.355356	0.177678	−	0.984089i	$-0.443142\pi$
$401$	2853.51	0.355356	0.177678	+	0.984089i	$0.443142\pi$
$402$	0	0
$403$	−1557.12	−0.192470
$404$	−15084.9	−1.85767
$405$	0	0
$406$	0	0
$407$	1085.27	0.132175
$408$	0	0
$409$	11260.1	1.36130	0.680652	−	0.732607i	$-0.261696\pi$
$409$	11260.1	1.36130	0.680652	+	0.732607i	$0.261696\pi$
$410$	15522.3	1.86974
$411$	0	0
$412$	4135.79	0.494553
$413$	0	0
$414$	0	0
$415$	−8825.53	−1.04392
$416$	−4800.56	−0.565786
$417$	0	0
$418$	7242.89	0.847515
$419$	9221.47	1.07517	0.537587	−	0.843208i	$-0.319336\pi$
$419$	9221.47	1.07517	0.537587	+	0.843208i	$0.319336\pi$
$420$	0	0
$421$	−8520.28	−0.986349	−0.493175	−	0.869930i	$-0.664164\pi$
$421$	−8520.28	−0.986349	−0.493175	+	0.869930i	$0.664164\pi$
$422$	878.936	0.101388
$423$	0	0
$424$	66.9547	0.00766889
$425$	−2491.36	−0.284350
$426$	0	0
$427$	0	0
$428$	−2382.41	−0.269061
$429$	0	0
$430$	−15973.7	−1.79144
$431$	−9162.10	−1.02395	−0.511976	−	0.859000i	$-0.671086\pi$
$431$	−9162.10	−1.02395	−0.511976	+	0.859000i	$0.671086\pi$
$432$	0	0
$433$	−10976.2	−1.21820	−0.609100	−	0.793093i	$-0.708469\pi$
$433$	−10976.2	−1.21820	−0.609100	+	0.793093i	$0.708469\pi$
$434$	0	0
$435$	0	0
$436$	−2426.02	−0.266480
$437$	26154.6	2.86303
$438$	0	0
$439$	3983.18	0.433045	0.216523	−	0.976278i	$-0.430528\pi$
$439$	3983.18	0.433045	0.216523	+	0.976278i	$0.430528\pi$
$440$	243.063	0.0263354
$441$	0	0
$442$	7051.03	0.758786
$443$	−4524.45	−0.485244	−0.242622	−	0.970121i	$-0.578007\pi$
$443$	−4524.45	−0.485244	−0.242622	+	0.970121i	$0.578007\pi$
$444$	0	0
$445$	−11352.5	−1.20935
$446$	−9144.35	−0.970847
$447$	0	0
$448$	0	0
$449$	2076.49	0.218253	0.109127	−	0.994028i	$-0.465195\pi$
$449$	2076.49	0.218253	0.109127	+	0.994028i	$0.465195\pi$
$450$	0	0
$451$	5223.96	0.545425
$452$	−16007.4	−1.66576
$453$	0	0
$454$	13751.0	1.42151
$455$	0	0
$456$	0	0
$457$	−1847.59	−0.189117	−0.0945587	−	0.995519i	$-0.530144\pi$
$457$	−1847.59	−0.189117	−0.0945587	+	0.995519i	$0.530144\pi$
$458$	10494.7	1.07071
$459$	0	0
$460$	16613.7	1.68395
$461$	−876.945	−0.0885974	−0.0442987	−	0.999018i	$-0.514105\pi$
$461$	−876.945	−0.0885974	−0.0442987	+	0.999018i	$0.514105\pi$
$462$	0	0
$463$	16245.2	1.63062	0.815310	−	0.579025i	$-0.196566\pi$
$463$	16245.2	1.63062	0.815310	+	0.579025i	$0.196566\pi$
$464$	11345.0	1.13508
$465$	0	0
$466$	3870.55	0.384763
$467$	18961.8	1.87890	0.939449	−	0.342689i	$-0.111338\pi$
$467$	18961.8	1.87890	0.939449	+	0.342689i	$0.111338\pi$
$468$	0	0
$469$	0	0
$470$	−10954.0	−1.07505
$471$	0	0
$472$	−716.149	−0.0698378
$473$	−5375.87	−0.522585
$474$	0	0
$475$	−3506.89	−0.338752
$476$	0	0
$477$	0	0
$478$	−809.067	−0.0774181
$479$	8377.76	0.799143	0.399572	−	0.916702i	$-0.369159\pi$
$479$	8377.76	0.799143	0.399572	+	0.916702i	$0.369159\pi$
$480$	0	0
$481$	1487.19	0.140977
$482$	19443.3	1.83738
$483$	0	0
$484$	−9693.55	−0.910364
$485$	13735.8	1.28600
$486$	0	0
$487$	−4558.85	−0.424191	−0.212096	−	0.977249i	$-0.568029\pi$
$487$	−4558.85	−0.424191	−0.212096	+	0.977249i	$0.568029\pi$
$488$	24.3822	0.00226175
$489$	0	0
$490$	0	0
$491$	−15809.9	−1.45314	−0.726570	−	0.687092i	$-0.758887\pi$
$491$	−15809.9	−1.45314	−0.726570	+	0.687092i	$0.758887\pi$
$492$	0	0
$493$	−17651.1	−1.61250
$494$	9925.17	0.903957
$495$	0	0
$496$	5054.86	0.457601
$497$	0	0
$498$	0	0
$499$	13386.1	1.20089	0.600444	−	0.799667i	$-0.294990\pi$
$499$	13386.1	1.20089	0.600444	+	0.799667i	$0.294990\pi$
$500$	−12701.3	−1.13604
$501$	0	0
$502$	−25345.1	−2.25340
$503$	−5720.55	−0.507091	−0.253545	−	0.967323i	$-0.581597\pi$
$503$	−5720.55	−0.507091	−0.253545	+	0.967323i	$0.581597\pi$
$504$	0	0
$505$	−17717.7	−1.56124
$506$	10887.1	0.956507
$507$	0	0
$508$	−10329.1	−0.902123
$509$	15293.2	1.33175	0.665873	−	0.746065i	$-0.268059\pi$
$509$	15293.2	1.33175	0.665873	+	0.746065i	$0.268059\pi$
$510$	0	0
$511$	0	0
$512$	16456.0	1.42043
$513$	0	0
$514$	15564.0	1.33560
$515$	4857.63	0.415637
$516$	0	0
$517$	−3686.53	−0.313604
$518$	0	0
$519$	0	0
$520$	333.078	0.0280893
$521$	−4368.69	−0.367363	−0.183681	−	0.982986i	$-0.558801\pi$
$521$	−4368.69	−0.367363	−0.183681	+	0.982986i	$0.558801\pi$
$522$	0	0
$523$	−2422.17	−0.202512	−0.101256	−	0.994860i	$-0.532286\pi$
$523$	−2422.17	−0.202512	−0.101256	+	0.994860i	$0.532286\pi$
$524$	−10056.1	−0.838366
$525$	0	0
$526$	−840.291	−0.0696548
$527$	−7864.58	−0.650069
$528$	0	0
$529$	27147.2	2.23122
$530$	1488.54	0.121996
$531$	0	0
$532$	0	0
$533$	7158.57	0.581749
$534$	0	0
$535$	−2798.23	−0.226127
$536$	616.210	0.0496571
$537$	0	0
$538$	−19482.4	−1.56124
$539$	0	0
$540$	0	0
$541$	13162.5	1.04603	0.523014	−	0.852324i	$-0.324807\pi$
$541$	13162.5	1.04603	0.523014	+	0.852324i	$0.324807\pi$
$542$	13041.1	1.03351
$543$	0	0
$544$	−24246.4	−1.91095
$545$	−2849.45	−0.223958
$546$	0	0
$547$	12112.4	0.946778	0.473389	−	0.880853i	$-0.343031\pi$
$547$	12112.4	0.946778	0.473389	+	0.880853i	$0.343031\pi$
$548$	−903.420	−0.0704237
$549$	0	0
$550$	−1459.78	−0.113173
$551$	−24846.0	−1.92101
$552$	0	0
$553$	0	0
$554$	8331.38	0.638928
$555$	0	0
$556$	−9263.63	−0.706592
$557$	8359.65	0.635924	0.317962	−	0.948103i	$-0.397002\pi$
$557$	8359.65	0.635924	0.317962	+	0.948103i	$0.397002\pi$
$558$	0	0
$559$	−7366.74	−0.557388
$560$	0	0
$561$	0	0
$562$	−7172.42	−0.538346
$563$	−13638.4	−1.02094	−0.510471	−	0.859895i	$-0.670529\pi$
$563$	−13638.4	−1.02094	−0.510471	+	0.859895i	$0.670529\pi$
$564$	0	0
$565$	−18801.3	−1.39996
$566$	9491.76	0.704891
$567$	0	0
$568$	382.080	0.0282249
$569$	−15491.3	−1.14135	−0.570677	−	0.821175i	$-0.693319\pi$
$569$	−15491.3	−1.14135	−0.570677	+	0.821175i	$0.693319\pi$
$570$	0	0
$571$	−4648.15	−0.340664	−0.170332	−	0.985387i	$-0.554484\pi$
$571$	−4648.15	−0.340664	−0.170332	+	0.985387i	$0.554484\pi$
$572$	2121.78	0.155098
$573$	0	0
$574$	0	0
$575$	−5271.38	−0.382316
$576$	0	0
$577$	−5479.06	−0.395314	−0.197657	−	0.980271i	$-0.563333\pi$
$577$	−5479.06	−0.395314	−0.197657	+	0.980271i	$0.563333\pi$
$578$	15688.7	1.12901
$579$	0	0
$580$	−15782.5	−1.12988
$581$	0	0
$582$	0	0
$583$	500.960	0.0355877
$584$	879.692	0.0623321
$585$	0	0
$586$	14735.2	1.03874
$587$	−4408.22	−0.309960	−0.154980	−	0.987918i	$-0.549531\pi$
$587$	−4408.22	−0.309960	−0.154980	+	0.987918i	$0.549531\pi$
$588$	0	0
$589$	−11070.3	−0.774441
$590$	−15921.4	−1.11097
$591$	0	0
$592$	−4827.86	−0.335175
$593$	−2815.26	−0.194956	−0.0974779	−	0.995238i	$-0.531078\pi$
$593$	−2815.26	−0.194956	−0.0974779	+	0.995238i	$0.531078\pi$
$594$	0	0
$595$	0	0
$596$	−6053.09	−0.416014
$597$	0	0
$598$	14919.0	1.02021
$599$	−19719.1	−1.34507	−0.672537	−	0.740064i	$-0.734795\pi$
$599$	−19719.1	−1.34507	−0.672537	+	0.740064i	$0.734795\pi$
$600$	0	0
$601$	−13982.8	−0.949033	−0.474517	−	0.880247i	$-0.657377\pi$
$601$	−13982.8	−0.949033	−0.474517	+	0.880247i	$0.657377\pi$
$602$	0	0
$603$	0	0
$604$	1303.83	0.0878343
$605$	−11385.4	−0.765097
$606$	0	0
$607$	13388.3	0.895245	0.447622	−	0.894223i	$-0.352271\pi$
$607$	13388.3	0.895245	0.447622	+	0.894223i	$0.352271\pi$
$608$	−34129.7	−2.27655
$609$	0	0
$610$	542.065	0.0359797
$611$	−5051.77	−0.334489
$612$	0	0
$613$	−27791.9	−1.83116	−0.915582	−	0.402131i	$-0.868269\pi$
$613$	−27791.9	−1.83116	−0.915582	+	0.402131i	$0.868269\pi$
$614$	24147.1	1.58713
$615$	0	0
$616$	0	0
$617$	19107.2	1.24672	0.623361	−	0.781935i	$-0.285767\pi$
$617$	19107.2	1.24672	0.623361	+	0.781935i	$0.285767\pi$
$618$	0	0
$619$	1092.94	0.0709675	0.0354837	−	0.999370i	$-0.488703\pi$
$619$	1092.94	0.0709675	0.0354837	+	0.999370i	$0.488703\pi$
$620$	−7032.02	−0.455504
$621$	0	0
$622$	−4785.75	−0.308507
$623$	0	0
$624$	0	0
$625$	−11595.0	−0.742078
$626$	−39524.9	−2.52353
$627$	0	0
$628$	−17684.3	−1.12369
$629$	7511.40	0.476151
$630$	0	0
$631$	19235.2	1.21353	0.606767	−	0.794879i	$-0.292466\pi$
$631$	19235.2	1.21353	0.606767	+	0.794879i	$0.292466\pi$
$632$	531.556	0.0334560
$633$	0	0
$634$	−34843.6	−2.18267
$635$	−12131.9	−0.758171
$636$	0	0
$637$	0	0
$638$	−10342.4	−0.641788
$639$	0	0
$640$	−2294.28	−0.141702
$641$	19950.7	1.22933	0.614667	−	0.788787i	$-0.289290\pi$
$641$	19950.7	1.22933	0.614667	+	0.788787i	$0.289290\pi$
$642$	0	0
$643$	688.125	0.0422037	0.0211019	−	0.999777i	$-0.493283\pi$
$643$	688.125	0.0422037	0.0211019	+	0.999777i	$0.493283\pi$
$644$	0	0
$645$	0	0
$646$	50129.4	3.05312
$647$	−10966.2	−0.666345	−0.333173	−	0.942866i	$-0.608119\pi$
$647$	−10966.2	−0.666345	−0.333173	+	0.942866i	$0.608119\pi$
$648$	0	0
$649$	−5358.28	−0.324085
$650$	−2000.39	−0.120710
$651$	0	0
$652$	25405.0	1.52598
$653$	12925.1	0.774575	0.387287	−	0.921959i	$-0.373412\pi$
$653$	12925.1	0.774575	0.387287	+	0.921959i	$0.373412\pi$
$654$	0	0
$655$	−11811.3	−0.704588
$656$	−23238.9	−1.38312
$657$	0	0
$658$	0	0
$659$	11779.0	0.696273	0.348137	−	0.937444i	$-0.386815\pi$
$659$	11779.0	0.696273	0.348137	+	0.937444i	$0.386815\pi$
$660$	0	0
$661$	−25040.1	−1.47344	−0.736721	−	0.676196i	$-0.763627\pi$
$661$	−25040.1	−1.47344	−0.736721	+	0.676196i	$0.763627\pi$
$662$	−21295.7	−1.25027
$663$	0	0
$664$	−1609.89	−0.0940900
$665$	0	0
$666$	0	0
$667$	−37347.2	−2.16805
$668$	18838.7	1.09116
$669$	0	0
$670$	13699.6	0.789941
$671$	182.430	0.0104957
$672$	0	0
$673$	4104.64	0.235100	0.117550	−	0.993067i	$-0.462496\pi$
$673$	4104.64	0.235100	0.117550	+	0.993067i	$0.462496\pi$
$674$	−34456.5	−1.96916
$675$	0	0
$676$	−15648.8	−0.890350
$677$	−12153.4	−0.689945	−0.344973	−	0.938613i	$-0.612112\pi$
$677$	−12153.4	−0.689945	−0.344973	+	0.938613i	$0.612112\pi$
$678$	0	0
$679$	0	0
$680$	1682.29	0.0948717
$681$	0	0
$682$	−4608.16	−0.258733
$683$	20414.0	1.14366	0.571830	−	0.820372i	$-0.306234\pi$
$683$	20414.0	1.14366	0.571830	+	0.820372i	$0.306234\pi$
$684$	0	0
$685$	−1061.10	−0.0591862
$686$	0	0
$687$	0	0
$688$	23914.6	1.32520
$689$	686.483	0.0379578
$690$	0	0
$691$	16093.5	0.886001	0.443000	−	0.896521i	$-0.353914\pi$
$691$	16093.5	0.886001	0.443000	+	0.896521i	$0.353914\pi$
$692$	4756.66	0.261302
$693$	0	0
$694$	23643.6	1.29322
$695$	−10880.5	−0.593841
$696$	0	0
$697$	36156.1	1.96486
$698$	7344.83	0.398290
$699$	0	0
$700$	0	0
$701$	−20803.0	−1.12085	−0.560426	−	0.828204i	$-0.689363\pi$
$701$	−20803.0	−1.12085	−0.560426	+	0.828204i	$0.689363\pi$
$702$	0	0
$703$	10573.2	0.567248
$704$	−7682.84	−0.411304
$705$	0	0
$706$	−13495.4	−0.719412
$707$	0	0
$708$	0	0
$709$	141.492	0.00749484	0.00374742	−	0.999993i	$-0.498807\pi$
$709$	141.492	0.00749484	0.00374742	+	0.999993i	$0.498807\pi$
$710$	8494.40	0.448999
$711$	0	0
$712$	−2070.85	−0.109000
$713$	−16640.4	−0.874035
$714$	0	0
$715$	2492.11	0.130349
$716$	−15536.8	−0.810947
$717$	0	0
$718$	3531.68	0.183567
$719$	6664.46	0.345678	0.172839	−	0.984950i	$-0.444706\pi$
$719$	6664.46	0.345678	0.172839	+	0.984950i	$0.444706\pi$
$720$	0	0
$721$	0	0
$722$	42747.2	2.20345
$723$	0	0
$724$	19635.0	1.00791
$725$	5007.64	0.256523
$726$	0	0
$727$	−4837.23	−0.246772	−0.123386	−	0.992359i	$-0.539375\pi$
$727$	−4837.23	−0.246772	−0.123386	+	0.992359i	$0.539375\pi$
$728$	0	0
$729$	0	0
$730$	19557.3	0.991572
$731$	−37207.5	−1.88258
$732$	0	0
$733$	−31602.5	−1.59245	−0.796225	−	0.605001i	$-0.793173\pi$
$733$	−31602.5	−1.59245	−0.796225	+	0.605001i	$0.793173\pi$
$734$	4763.93	0.239564
$735$	0	0
$736$	−51302.0	−2.56932
$737$	4610.53	0.230436
$738$	0	0
$739$	−8227.58	−0.409549	−0.204774	−	0.978809i	$-0.565646\pi$
$739$	−8227.58	−0.409549	−0.204774	+	0.978809i	$0.565646\pi$
$740$	6716.22	0.333639
$741$	0	0
$742$	0	0
$743$	−37020.3	−1.82792	−0.913959	−	0.405805i	$-0.866991\pi$
$743$	−37020.3	−1.82792	−0.913959	+	0.405805i	$0.866991\pi$
$744$	0	0
$745$	−7109.57	−0.349630
$746$	14714.3	0.722158
$747$	0	0
$748$	10716.6	0.523846
$749$	0	0
$750$	0	0
$751$	−25149.9	−1.22202	−0.611008	−	0.791625i	$-0.709236\pi$
$751$	−25149.9	−1.22202	−0.611008	+	0.791625i	$0.709236\pi$
$752$	16399.6	0.795254
$753$	0	0
$754$	−14172.6	−0.684529
$755$	1531.39	0.0738186
$756$	0	0
$757$	20460.8	0.982377	0.491189	−	0.871053i	$-0.336563\pi$
$757$	20460.8	0.982377	0.491189	+	0.871053i	$0.336563\pi$
$758$	29693.6	1.42285
$759$	0	0
$760$	2368.02	0.113023
$761$	−32659.1	−1.55571	−0.777853	−	0.628447i	$-0.783691\pi$
$761$	−32659.1	−1.55571	−0.777853	+	0.628447i	$0.783691\pi$
$762$	0	0
$763$	0	0
$764$	26413.7	1.25080
$765$	0	0
$766$	29827.0	1.40691
$767$	−7342.64	−0.345668
$768$	0	0
$769$	−11005.3	−0.516075	−0.258037	−	0.966135i	$-0.583076\pi$
$769$	−11005.3	−0.516075	−0.258037	+	0.966135i	$0.583076\pi$
$770$	0	0
$771$	0	0
$772$	−31329.5	−1.46059
$773$	3038.18	0.141366	0.0706829	−	0.997499i	$-0.477482\pi$
$773$	3038.18	0.141366	0.0706829	+	0.997499i	$0.477482\pi$
$774$	0	0
$775$	2231.20	0.103415
$776$	2505.59	0.115909
$777$	0	0
$778$	−36780.9	−1.69493
$779$	50894.0	2.34078
$780$	0	0
$781$	2858.75	0.130979
$782$	75352.0	3.44576
$783$	0	0
$784$	0	0
$785$	−20770.8	−0.944384
$786$	0	0
$787$	−12306.8	−0.557419	−0.278710	−	0.960375i	$-0.589907\pi$
$787$	−12306.8	−0.557419	−0.278710	+	0.960375i	$0.589907\pi$
$788$	−7189.00	−0.324997
$789$	0	0
$790$	11817.5	0.532214
$791$	0	0
$792$	0	0
$793$	249.990	0.0111947
$794$	29915.9	1.33712
$795$	0	0
$796$	28698.4	1.27788
$797$	−3007.06	−0.133646	−0.0668228	−	0.997765i	$-0.521286\pi$
$797$	−3007.06	−0.133646	−0.0668228	+	0.997765i	$0.521286\pi$
$798$	0	0
$799$	−25515.2	−1.12974
$800$	6878.74	0.304000
$801$	0	0
$802$	11572.1	0.509508
$803$	6581.92	0.289254
$804$	0	0
$805$	0	0
$806$	−6314.72	−0.275963
$807$	0	0
$808$	−3231.94	−0.140717
$809$	10585.0	0.460009	0.230005	−	0.973190i	$-0.426126\pi$
$809$	10585.0	0.460009	0.230005	+	0.973190i	$0.426126\pi$
$810$	0	0
$811$	−18217.8	−0.788796	−0.394398	−	0.918940i	$-0.629047\pi$
$811$	−18217.8	−0.788796	−0.394398	+	0.918940i	$0.629047\pi$
$812$	0	0
$813$	0	0
$814$	4401.22	0.189512
$815$	29839.1	1.28248
$816$	0	0
$817$	−52374.0	−2.24276
$818$	45663.9	1.95184
$819$	0	0
$820$	32328.5	1.37678
$821$	25596.5	1.08809	0.544047	−	0.839055i	$-0.316891\pi$
$821$	25596.5	1.08809	0.544047	+	0.839055i	$0.316891\pi$
$822$	0	0
$823$	43778.9	1.85424	0.927118	−	0.374769i	$-0.122278\pi$
$823$	43778.9	1.85424	0.927118	+	0.374769i	$0.122278\pi$
$824$	886.094	0.0374618
$825$	0	0
$826$	0	0
$827$	−2735.78	−0.115033	−0.0575166	−	0.998345i	$-0.518318\pi$
$827$	−2735.78	−0.115033	−0.0575166	+	0.998345i	$0.518318\pi$
$828$	0	0
$829$	−31144.2	−1.30480	−0.652402	−	0.757873i	$-0.726239\pi$
$829$	−31144.2	−1.30480	−0.652402	+	0.757873i	$0.726239\pi$
$830$	−35791.0	−1.49677
$831$	0	0
$832$	−10528.1	−0.438696
$833$	0	0
$834$	0	0
$835$	22126.8	0.917039
$836$	15084.9	0.624068
$837$	0	0
$838$	37396.7	1.54158
$839$	14977.3	0.616300	0.308150	−	0.951338i	$-0.400290\pi$
$839$	14977.3	0.616300	0.308150	+	0.951338i	$0.400290\pi$
$840$	0	0
$841$	11089.6	0.454698
$842$	−34553.1	−1.41423
$843$	0	0
$844$	1830.57	0.0746574
$845$	−18380.1	−0.748277
$846$	0	0
$847$	0	0
$848$	−2228.53	−0.0902452
$849$	0	0
$850$	−10103.4	−0.407700
$851$	15893.1	0.640198
$852$	0	0
$853$	42861.7	1.72047	0.860233	−	0.509901i	$-0.170318\pi$
$853$	42861.7	1.72047	0.860233	+	0.509901i	$0.170318\pi$
$854$	0	0
$855$	0	0
$856$	−510.432	−0.0203811
$857$	9390.00	0.374278	0.187139	−	0.982333i	$-0.440079\pi$
$857$	9390.00	0.374278	0.187139	+	0.982333i	$0.440079\pi$
$858$	0	0
$859$	33561.2	1.33305	0.666526	−	0.745482i	$-0.267781\pi$
$859$	33561.2	1.33305	0.666526	+	0.745482i	$0.267781\pi$
$860$	−33268.6	−1.31913
$861$	0	0
$862$	−37155.9	−1.46814
$863$	−25691.6	−1.01339	−0.506693	−	0.862127i	$-0.669132\pi$
$863$	−25691.6	−1.01339	−0.506693	+	0.862127i	$0.669132\pi$
$864$	0	0
$865$	5586.87	0.219606
$866$	−44512.7	−1.74665
$867$	0	0
$868$	0	0
$869$	3977.14	0.155254
$870$	0	0
$871$	6317.97	0.245782
$872$	−519.775	−0.0201856
$873$	0	0
$874$	106067.	4.10500
$875$	0	0
$876$	0	0
$877$	−5351.52	−0.206052	−0.103026	−	0.994679i	$-0.532853\pi$
$877$	−5351.52	−0.206052	−0.103026	+	0.994679i	$0.532853\pi$
$878$	16153.4	0.620900
$879$	0	0
$880$	−8090.14	−0.309907
$881$	34212.7	1.30835	0.654174	−	0.756344i	$-0.273016\pi$
$881$	34212.7	1.30835	0.654174	+	0.756344i	$0.273016\pi$
$882$	0	0
$883$	17149.2	0.653587	0.326794	−	0.945096i	$-0.394032\pi$
$883$	17149.2	0.653587	0.326794	+	0.945096i	$0.394032\pi$
$884$	14685.3	0.558732
$885$	0	0
$886$	−18348.4	−0.695742
$887$	−4020.87	−0.152207	−0.0761035	−	0.997100i	$-0.524248\pi$
$887$	−4020.87	−0.152207	−0.0761035	+	0.997100i	$0.524248\pi$
$888$	0	0
$889$	0	0
$890$	−46039.0	−1.73397
$891$	0	0
$892$	−19045.1	−0.714883
$893$	−35915.7	−1.34588
$894$	0	0
$895$	−18248.5	−0.681544
$896$	0	0
$897$	0	0
$898$	8420.99	0.312931
$899$	15807.8	0.586452
$900$	0	0
$901$	3467.24	0.128203
$902$	21185.2	0.782030
$903$	0	0
$904$	−3429.59	−0.126180
$905$	23062.0	0.847080
$906$	0	0
$907$	−22967.2	−0.840810	−0.420405	−	0.907337i	$-0.638112\pi$
$907$	−22967.2	−0.840810	−0.420405	+	0.907337i	$0.638112\pi$
$908$	28639.4	1.04673
$909$	0	0
$910$	0	0
$911$	−9860.77	−0.358619	−0.179309	−	0.983793i	$-0.557386\pi$
$911$	−9860.77	−0.358619	−0.179309	+	0.983793i	$0.557386\pi$
$912$	0	0
$913$	−12045.3	−0.436628
$914$	−7492.71	−0.271156
$915$	0	0
$916$	21857.5	0.788421
$917$	0	0
$918$	0	0
$919$	−5271.55	−0.189219	−0.0946096	−	0.995514i	$-0.530160\pi$
$919$	−5271.55	−0.189219	−0.0946096	+	0.995514i	$0.530160\pi$
$920$	3559.49	0.127558
$921$	0	0
$922$	−3556.36	−0.127031
$923$	3917.44	0.139701
$924$	0	0
$925$	−2131.00	−0.0757478
$926$	65880.6	2.33798
$927$	0	0
$928$	48735.3	1.72394
$929$	14902.6	0.526307	0.263153	−	0.964754i	$-0.415238\pi$
$929$	14902.6	0.526307	0.263153	+	0.964754i	$0.415238\pi$
$930$	0	0
$931$	0	0
$932$	8061.24	0.283321
$933$	0	0
$934$	76897.4	2.69396
$935$	12587.0	0.440255
$936$	0	0
$937$	−21934.8	−0.764757	−0.382378	−	0.924006i	$-0.624895\pi$
$937$	−21934.8	−0.764757	−0.382378	+	0.924006i	$0.624895\pi$
$938$	0	0
$939$	0	0
$940$	−22814.1	−0.791610
$941$	−14521.5	−0.503069	−0.251535	−	0.967848i	$-0.580935\pi$
$941$	−14521.5	−0.503069	−0.251535	+	0.967848i	$0.580935\pi$
$942$	0	0
$943$	76501.3	2.64181
$944$	23836.4	0.821831
$945$	0	0
$946$	−21801.3	−0.749282
$947$	−1943.73	−0.0666979	−0.0333489	−	0.999444i	$-0.510617\pi$
$947$	−1943.73	−0.0666979	−0.0333489	+	0.999444i	$0.510617\pi$
$948$	0	0
$949$	9019.43	0.308517
$950$	−14221.8	−0.485702
$951$	0	0
$952$	0	0
$953$	16904.1	0.574583	0.287292	−	0.957843i	$-0.407245\pi$
$953$	16904.1	0.574583	0.287292	+	0.957843i	$0.407245\pi$
$954$	0	0
$955$	31023.8	1.05121
$956$	−1685.05	−0.0570068
$957$	0	0
$958$	33975.1	1.14581
$959$	0	0
$960$	0	0
$961$	−22747.7	−0.763576
$962$	6031.13	0.202133
$963$	0	0
$964$	40494.8	1.35296
$965$	−36797.6	−1.22752
$966$	0	0
$967$	26699.3	0.887891	0.443946	−	0.896054i	$-0.353578\pi$
$967$	26699.3	0.887891	0.443946	+	0.896054i	$0.353578\pi$
$968$	−2076.85	−0.0689591
$969$	0	0
$970$	55704.2	1.84387
$971$	−11089.5	−0.366507	−0.183253	−	0.983066i	$-0.558663\pi$
$971$	−11089.5	−0.366507	−0.183253	+	0.983066i	$0.558663\pi$
$972$	0	0
$973$	0	0
$974$	−18487.9	−0.608205
$975$	0	0
$976$	−811.541	−0.0266156
$977$	−51891.9	−1.69925	−0.849625	−	0.527387i	$-0.823172\pi$
$977$	−51891.9	−1.69925	−0.849625	+	0.527387i	$0.823172\pi$
$978$	0	0
$979$	−15494.2	−0.505820
$980$	0	0
$981$	0	0
$982$	−64115.5	−2.08351
$983$	−24560.7	−0.796911	−0.398456	−	0.917188i	$-0.630454\pi$
$983$	−24560.7	−0.796911	−0.398456	+	0.917188i	$0.630454\pi$
$984$	0	0
$985$	−8443.74	−0.273137
$986$	−71582.0	−2.31200
$987$	0	0
$988$	20671.3	0.665629
$989$	−78725.9	−2.53118
$990$	0	0
$991$	−18507.5	−0.593250	−0.296625	−	0.954994i	$-0.595861\pi$
$991$	−18507.5	−0.593250	−0.296625	+	0.954994i	$0.595861\pi$
$992$	21714.4	0.694993
$993$	0	0
$994$	0	0
$995$	33707.4	1.07397
$996$	0	0
$997$	−42966.2	−1.36485	−0.682424	−	0.730957i	$-0.739074\pi$
$997$	−42966.2	−1.36485	−0.682424	+	0.730957i	$0.739074\pi$
$998$	54285.8	1.72183
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.w.1.4		4
3.2	odd	2	inner	441.4.a.w.1.1		4
7.2	even	3		63.4.e.d.46.1	yes	8
7.3	odd	6		441.4.e.x.226.1		8
7.4	even	3		63.4.e.d.37.1	✓	8
7.5	odd	6		441.4.e.x.361.1		8
7.6	odd	2		441.4.a.v.1.4		4
21.2	odd	6		63.4.e.d.46.4	yes	8
21.5	even	6		441.4.e.x.361.4		8
21.11	odd	6		63.4.e.d.37.4	yes	8
21.17	even	6		441.4.e.x.226.4		8
21.20	even	2		441.4.a.v.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.e.d.37.1	✓	8	7.4	even	3
63.4.e.d.37.4	yes	8	21.11	odd	6
63.4.e.d.46.1	yes	8	7.2	even	3
63.4.e.d.46.4	yes	8	21.2	odd	6
441.4.a.v.1.1		4	21.20	even	2
441.4.a.v.1.4		4	7.6	odd	2
441.4.a.w.1.1		4	3.2	odd	2	inner
441.4.a.w.1.4		4	1.1	even	1	trivial
441.4.e.x.226.1		8	7.3	odd	6
441.4.e.x.226.4		8	21.17	even	6
441.4.e.x.361.1		8	7.5	odd	6
441.4.e.x.361.4		8	21.5	even	6

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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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} +1.80961 q^{8} +O(q^{10})\)	\(q+4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} +1.80961 q^{8} +40.2311 q^{10} +13.5396 q^{11} +18.5538 q^{13} -60.2311 q^{16} +93.7102 q^{17} +131.908 q^{19} +83.7899 q^{20} +54.9084 q^{22} +198.278 q^{23} -26.5858 q^{25} +75.2429 q^{26} -188.358 q^{29} -83.9244 q^{31} -258.738 q^{32} +380.032 q^{34} +80.1555 q^{37} +534.941 q^{38} +17.9520 q^{40} +385.828 q^{41} -397.048 q^{43} +114.359 q^{44} +804.096 q^{46} -272.277 q^{47} -107.816 q^{50} +156.709 q^{52} +36.9996 q^{53} +134.318 q^{55} -763.865 q^{58} -395.749 q^{59} +13.4738 q^{61} -340.347 q^{62} -567.435 q^{64} +184.061 q^{65} +340.522 q^{67} +791.498 q^{68} +211.140 q^{71} +486.124 q^{73} +325.062 q^{74} +1114.13 q^{76} +293.741 q^{79} -597.516 q^{80} +1564.69 q^{82} -889.635 q^{83} +929.643 q^{85} -1610.19 q^{86} +24.5014 q^{88} -1144.36 q^{89} +1674.70 q^{92} -1104.19 q^{94} +1308.58 q^{95} +1384.61 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} + 22 q^{10} + 102 q^{13} - 102 q^{16} + 222 q^{19} - 86 q^{22} + 366 q^{25} + 220 q^{31} + 1020 q^{34} - 374 q^{37} + 822 q^{40} - 838 q^{43} + 1716 q^{46} - 40 q^{52} + 2510 q^{55} - 1694 q^{58} + 1332 q^{61} - 686 q^{64} + 1890 q^{67} + 1750 q^{73} + 2456 q^{76} + 8 q^{79} + 2480 q^{82} - 1116 q^{85} + 2682 q^{88} - 1416 q^{94} + 3010 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 + 22 * q^10 + 102 * q^13 - 102 * q^16 + 222 * q^19 - 86 * q^22 + 366 * q^25 + 220 * q^31 + 1020 * q^34 - 374 * q^37 + 822 * q^40 - 838 * q^43 + 1716 * q^46 - 40 * q^52 + 2510 * q^55 - 1694 * q^58 + 1332 * q^61 - 686 * q^64 + 1890 * q^67 + 1750 * q^73 + 2456 * q^76 + 8 * q^79 + 2480 * q^82 - 1116 * q^85 + 2682 * q^88 - 1416 * q^94 + 3010 * q^97

Introduction

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