LMFDB - Embedded newform 882.4.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.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:	$52.0396846251$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{58}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 58 $ x^2 - 58
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.61577$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +15.2315 q^{5} -8.00000 q^{8} -30.4631 q^{10} -2.00000 q^{11} +30.4631 q^{13} +16.0000 q^{16} -45.6946 q^{17} -152.315 q^{19} +60.9262 q^{20} +4.00000 q^{22} -30.0000 q^{23} +107.000 q^{25} -60.9262 q^{26} -212.000 q^{29} -213.242 q^{31} -32.0000 q^{32} +91.3893 q^{34} +246.000 q^{37} +304.631 q^{38} -121.852 q^{40} -319.862 q^{41} -284.000 q^{43} -8.00000 q^{44} +60.0000 q^{46} +60.9262 q^{47} -214.000 q^{50} +121.852 q^{52} -548.000 q^{53} -30.4631 q^{55} +424.000 q^{58} -670.188 q^{59} +517.873 q^{61} +426.483 q^{62} +64.0000 q^{64} +464.000 q^{65} +652.000 q^{67} -182.779 q^{68} -770.000 q^{71} +974.819 q^{73} -492.000 q^{74} -609.262 q^{76} +472.000 q^{79} +243.705 q^{80} +639.725 q^{82} +182.779 q^{83} -696.000 q^{85} +568.000 q^{86} +16.0000 q^{88} +715.883 q^{89} -120.000 q^{92} -121.852 q^{94} -2320.00 q^{95} -304.631 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} - 4 q^{11} + 32 q^{16} + 8 q^{22} - 60 q^{23} + 214 q^{25} - 424 q^{29} - 64 q^{32} + 492 q^{37} - 568 q^{43} - 16 q^{44} + 120 q^{46} - 428 q^{50} - 1096 q^{53} + 848 q^{58}+ \cdots - 4640 q^{95}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 - 4 * q^11 + 32 * q^16 + 8 * q^22 - 60 * q^23 + 214 * q^25 - 424 * q^29 - 64 * q^32 + 492 * q^37 - 568 * q^43 - 16 * q^44 + 120 * q^46 - 428 * q^50 - 1096 * q^53 + 848 * q^58 + 128 * q^64 + 928 * q^65 + 1304 * q^67 - 1540 * q^71 - 984 * q^74 + 944 * q^79 - 1392 * q^85 + 1136 * q^86 + 32 * q^88 - 240 * q^92 - 4640 * q^95

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	15.2315	1.36235	0.681175	−	0.732120i	$-0.261469\pi$
$5$	15.2315	1.36235	0.681175	+	0.732120i	$0.261469\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−30.4631	−0.963328
$11$	−2.00000	−0.0548202	−0.0274101	−	0.999624i	$-0.508726\pi$
$11$	−2.00000	−0.0548202	−0.0274101	+	0.999624i	$0.508726\pi$
$12$	0	0
$13$	30.4631	0.649919	0.324959	−	0.945728i	$-0.394649\pi$
$13$	30.4631	0.649919	0.324959	+	0.945728i	$0.394649\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−45.6946	−0.651916	−0.325958	−	0.945384i	$-0.605687\pi$
$17$	−45.6946	−0.651916	−0.325958	+	0.945384i	$0.605687\pi$
$18$	0	0
$19$	−152.315	−1.83913	−0.919567	−	0.392932i	$-0.871461\pi$
$19$	−152.315	−1.83913	−0.919567	+	0.392932i	$0.871461\pi$
$20$	60.9262	0.681175
$21$	0	0
$22$	4.00000	0.0387638
$23$	−30.0000	−0.271975	−0.135988	−	0.990711i	$-0.543421\pi$
$23$	−30.0000	−0.271975	−0.135988	+	0.990711i	$0.543421\pi$
$24$	0	0
$25$	107.000	0.856000
$26$	−60.9262	−0.459562
$27$	0	0
$28$	0	0
$29$	−212.000	−1.35750	−0.678748	−	0.734371i	$-0.737477\pi$
$29$	−212.000	−1.35750	−0.678748	+	0.734371i	$0.737477\pi$
$30$	0	0
$31$	−213.242	−1.23546	−0.617731	−	0.786389i	$-0.711948\pi$
$31$	−213.242	−1.23546	−0.617731	+	0.786389i	$0.711948\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	91.3893	0.460974
$35$	0	0
$36$	0	0
$37$	246.000	1.09303	0.546516	−	0.837449i	$-0.315954\pi$
$37$	246.000	1.09303	0.546516	+	0.837449i	$0.315954\pi$
$38$	304.631	1.30046
$39$	0	0
$40$	−121.852	−0.481664
$41$	−319.862	−1.21839	−0.609197	−	0.793019i	$-0.708508\pi$
$41$	−319.862	−1.21839	−0.609197	+	0.793019i	$0.708508\pi$
$42$	0	0
$43$	−284.000	−1.00720	−0.503600	−	0.863937i	$-0.667991\pi$
$43$	−284.000	−1.00720	−0.503600	+	0.863937i	$0.667991\pi$
$44$	−8.00000	−0.0274101
$45$	0	0
$46$	60.0000	0.192316
$47$	60.9262	0.189085	0.0945425	−	0.995521i	$-0.469861\pi$
$47$	60.9262	0.189085	0.0945425	+	0.995521i	$0.469861\pi$
$48$	0	0
$49$	0	0
$50$	−214.000	−0.605283
$51$	0	0
$52$	121.852	0.324959
$53$	−548.000	−1.42026	−0.710128	−	0.704072i	$-0.751363\pi$
$53$	−548.000	−1.42026	−0.710128	+	0.704072i	$0.751363\pi$
$54$	0	0
$55$	−30.4631	−0.0746844
$56$	0	0
$57$	0	0
$58$	424.000	0.959895
$59$	−670.188	−1.47883	−0.739416	−	0.673249i	$-0.764898\pi$
$59$	−670.188	−1.47883	−0.739416	+	0.673249i	$0.764898\pi$
$60$	0	0
$61$	517.873	1.08700	0.543498	−	0.839410i	$-0.317099\pi$
$61$	517.873	1.08700	0.543498	+	0.839410i	$0.317099\pi$
$62$	426.483	0.873604
$63$	0	0
$64$	64.0000	0.125000
$65$	464.000	0.885417
$66$	0	0
$67$	652.000	1.18887	0.594436	−	0.804143i	$-0.297375\pi$
$67$	652.000	1.18887	0.594436	+	0.804143i	$0.297375\pi$
$68$	−182.779	−0.325958
$69$	0	0
$70$	0	0
$71$	−770.000	−1.28707	−0.643537	−	0.765415i	$-0.722534\pi$
$71$	−770.000	−1.28707	−0.643537	+	0.765415i	$0.722534\pi$
$72$	0	0
$73$	974.819	1.56293	0.781465	−	0.623949i	$-0.214473\pi$
$73$	974.819	1.56293	0.781465	+	0.623949i	$0.214473\pi$
$74$	−492.000	−0.772890
$75$	0	0
$76$	−609.262	−0.919567
$77$	0	0
$78$	0	0
$79$	472.000	0.672204	0.336102	−	0.941826i	$-0.390891\pi$
$79$	472.000	0.672204	0.336102	+	0.941826i	$0.390891\pi$
$80$	243.705	0.340588
$81$	0	0
$82$	639.725	0.861534
$83$	182.779	0.241718	0.120859	−	0.992670i	$-0.461435\pi$
$83$	182.779	0.241718	0.120859	+	0.992670i	$0.461435\pi$
$84$	0	0
$85$	−696.000	−0.888139
$86$	568.000	0.712198
$87$	0	0
$88$	16.0000	0.0193819
$89$	715.883	0.852623	0.426311	−	0.904577i	$-0.359813\pi$
$89$	715.883	0.852623	0.426311	+	0.904577i	$0.359813\pi$
$90$	0	0
$91$	0	0
$92$	−120.000	−0.135988
$93$	0	0
$94$	−121.852	−0.133703
$95$	−2320.00	−2.50555
$96$	0	0
$97$	−304.631	−0.318872	−0.159436	−	0.987208i	$-0.550968\pi$
$97$	−304.631	−0.318872	−0.159436	+	0.987208i	$0.550968\pi$
$98$	0	0
$99$	0	0
$100$	428.000	0.428000
$101$	1020.51	1.00540	0.502698	−	0.864462i	$-0.332341\pi$
$101$	1020.51	1.00540	0.502698	+	0.864462i	$0.332341\pi$
$102$	0	0
$103$	578.799	0.553696	0.276848	−	0.960914i	$-0.410710\pi$
$103$	578.799	0.553696	0.276848	+	0.960914i	$0.410710\pi$
$104$	−243.705	−0.229781
$105$	0	0
$106$	1096.00	1.00427
$107$	1646.00	1.48715	0.743574	−	0.668654i	$-0.233129\pi$
$107$	1646.00	1.48715	0.743574	+	0.668654i	$0.233129\pi$
$108$	0	0
$109$	982.000	0.862922	0.431461	−	0.902131i	$-0.357998\pi$
$109$	982.000	0.862922	0.431461	+	0.902131i	$0.357998\pi$
$110$	60.9262	0.0528099
$111$	0	0
$112$	0	0
$113$	−1288.00	−1.07226	−0.536128	−	0.844137i	$-0.680113\pi$
$113$	−1288.00	−1.07226	−0.536128	+	0.844137i	$0.680113\pi$
$114$	0	0
$115$	−456.946	−0.370526
$116$	−848.000	−0.678748
$117$	0	0
$118$	1340.38	1.04569
$119$	0	0
$120$	0	0
$121$	−1327.00	−0.996995
$122$	−1035.75	−0.768623
$123$	0	0
$124$	−852.967	−0.617731
$125$	−274.168	−0.196179
$126$	0	0
$127$	−1072.00	−0.749013	−0.374506	−	0.927224i	$-0.622188\pi$
$127$	−1072.00	−0.749013	−0.374506	+	0.927224i	$0.622188\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−928.000	−0.626084
$131$	−2741.68	−1.82856	−0.914281	−	0.405081i	$-0.867243\pi$
$131$	−2741.68	−1.82856	−0.914281	+	0.405081i	$0.867243\pi$
$132$	0	0
$133$	0	0
$134$	−1304.00	−0.840660
$135$	0	0
$136$	365.557	0.230487
$137$	−2936.00	−1.83094	−0.915472	−	0.402381i	$-0.868183\pi$
$137$	−2936.00	−1.83094	−0.915472	+	0.402381i	$0.868183\pi$
$138$	0	0
$139$	−182.779	−0.111533	−0.0557665	−	0.998444i	$-0.517760\pi$
$139$	−182.779	−0.111533	−0.0557665	+	0.998444i	$0.517760\pi$
$140$	0	0
$141$	0	0
$142$	1540.00	0.910098
$143$	−60.9262	−0.0356287
$144$	0	0
$145$	−3229.09	−1.84939
$146$	−1949.64	−1.10516
$147$	0	0
$148$	984.000	0.546516
$149$	964.000	0.530027	0.265013	−	0.964245i	$-0.414624\pi$
$149$	964.000	0.530027	0.265013	+	0.964245i	$0.414624\pi$
$150$	0	0
$151$	−1688.00	−0.909718	−0.454859	−	0.890563i	$-0.650310\pi$
$151$	−1688.00	−0.909718	−0.454859	+	0.890563i	$0.650310\pi$
$152$	1218.52	0.650232
$153$	0	0
$154$	0	0
$155$	−3248.00	−1.68313
$156$	0	0
$157$	−3442.33	−1.74986	−0.874929	−	0.484251i	$-0.839092\pi$
$157$	−3442.33	−1.74986	−0.874929	+	0.484251i	$0.839092\pi$
$158$	−944.000	−0.475320
$159$	0	0
$160$	−487.409	−0.240832
$161$	0	0
$162$	0	0
$163$	−3324.00	−1.59727	−0.798637	−	0.601813i	$-0.794445\pi$
$163$	−3324.00	−1.59727	−0.798637	+	0.601813i	$0.794445\pi$
$164$	−1279.45	−0.609197
$165$	0	0
$166$	−365.557	−0.170920
$167$	3046.31	1.41156	0.705780	−	0.708431i	$-0.250597\pi$
$167$	3046.31	1.41156	0.705780	+	0.708431i	$0.250597\pi$
$168$	0	0
$169$	−1269.00	−0.577606
$170$	1392.00	0.628009
$171$	0	0
$172$	−1136.00	−0.503600
$173$	3274.78	1.43917	0.719587	−	0.694402i	$-0.244331\pi$
$173$	3274.78	1.43917	0.719587	+	0.694402i	$0.244331\pi$
$174$	0	0
$175$	0	0
$176$	−32.0000	−0.0137051
$177$	0	0
$178$	−1431.77	−0.602895
$179$	−1254.00	−0.523622	−0.261811	−	0.965119i	$-0.584320\pi$
$179$	−1254.00	−0.523622	−0.261811	+	0.965119i	$0.584320\pi$
$180$	0	0
$181$	3076.77	1.26351	0.631753	−	0.775170i	$-0.282336\pi$
$181$	3076.77	1.26351	0.631753	+	0.775170i	$0.282336\pi$
$182$	0	0
$183$	0	0
$184$	240.000	0.0961578
$185$	3746.96	1.48909
$186$	0	0
$187$	91.3893	0.0357382
$188$	243.705	0.0945425
$189$	0	0
$190$	4640.00	1.77169
$191$	−906.000	−0.343224	−0.171612	−	0.985165i	$-0.554898\pi$
$191$	−906.000	−0.343224	−0.171612	+	0.985165i	$0.554898\pi$
$192$	0	0
$193$	182.000	0.0678790	0.0339395	−	0.999424i	$-0.489195\pi$
$193$	182.000	0.0678790	0.0339395	+	0.999424i	$0.489195\pi$
$194$	609.262	0.225477
$195$	0	0
$196$	0	0
$197$	−3468.00	−1.25424	−0.627119	−	0.778924i	$-0.715766\pi$
$197$	−3468.00	−1.25424	−0.627119	+	0.778924i	$0.715766\pi$
$198$	0	0
$199$	3899.28	1.38901	0.694503	−	0.719489i	$-0.255624\pi$
$199$	3899.28	1.38901	0.694503	+	0.719489i	$0.255624\pi$
$200$	−856.000	−0.302642
$201$	0	0
$202$	−2041.03	−0.710922
$203$	0	0
$204$	0	0
$205$	−4872.00	−1.65988
$206$	−1157.60	−0.391523
$207$	0	0
$208$	487.409	0.162480
$209$	304.631	0.100822
$210$	0	0
$211$	−2620.00	−0.854826	−0.427413	−	0.904057i	$-0.640575\pi$
$211$	−2620.00	−0.854826	−0.427413	+	0.904057i	$0.640575\pi$
$212$	−2192.00	−0.710128
$213$	0	0
$214$	−3292.00	−1.05157
$215$	−4325.76	−1.37216
$216$	0	0
$217$	0	0
$218$	−1964.00	−0.610178
$219$	0	0
$220$	−121.852	−0.0373422
$221$	−1392.00	−0.423693
$222$	0	0
$223$	−1401.30	−0.420799	−0.210399	−	0.977616i	$-0.567476\pi$
$223$	−1401.30	−0.420799	−0.210399	+	0.977616i	$0.567476\pi$
$224$	0	0
$225$	0	0
$226$	2576.00	0.758199
$227$	−3899.28	−1.14011	−0.570053	−	0.821608i	$-0.693077\pi$
$227$	−3899.28	−1.14011	−0.570053	+	0.821608i	$0.693077\pi$
$228$	0	0
$229$	5513.82	1.59111	0.795553	−	0.605883i	$-0.207180\pi$
$229$	5513.82	1.59111	0.795553	+	0.605883i	$0.207180\pi$
$230$	913.893	0.262001
$231$	0	0
$232$	1696.00	0.479948
$233$	712.000	0.200192	0.100096	−	0.994978i	$-0.468085\pi$
$233$	712.000	0.200192	0.100096	+	0.994978i	$0.468085\pi$
$234$	0	0
$235$	928.000	0.257600
$236$	−2680.75	−0.739416
$237$	0	0
$238$	0	0
$239$	2586.00	0.699893	0.349947	−	0.936770i	$-0.386200\pi$
$239$	2586.00	0.699893	0.349947	+	0.936770i	$0.386200\pi$
$240$	0	0
$241$	2254.27	0.602532	0.301266	−	0.953540i	$-0.402591\pi$
$241$	2254.27	0.602532	0.301266	+	0.953540i	$0.402591\pi$
$242$	2654.00	0.704982
$243$	0	0
$244$	2071.49	0.543498
$245$	0	0
$246$	0	0
$247$	−4640.00	−1.19529
$248$	1705.93	0.436802
$249$	0	0
$250$	548.336	0.138719
$251$	−4508.54	−1.13377	−0.566885	−	0.823797i	$-0.691852\pi$
$251$	−4508.54	−1.13377	−0.566885	+	0.823797i	$0.691852\pi$
$252$	0	0
$253$	60.0000	0.0149098
$254$	2144.00	0.529632
$255$	0	0
$256$	256.000	0.0625000
$257$	4554.23	1.10539	0.552695	−	0.833384i	$-0.313599\pi$
$257$	4554.23	1.10539	0.552695	+	0.833384i	$0.313599\pi$
$258$	0	0
$259$	0	0
$260$	1856.00	0.442709
$261$	0	0
$262$	5483.36	1.29299
$263$	2578.00	0.604435	0.302217	−	0.953239i	$-0.402273\pi$
$263$	2578.00	0.604435	0.302217	+	0.953239i	$0.402273\pi$
$264$	0	0
$265$	−8346.89	−1.93489
$266$	0	0
$267$	0	0
$268$	2608.00	0.594436
$269$	929.124	0.210594	0.105297	−	0.994441i	$-0.466421\pi$
$269$	929.124	0.210594	0.105297	+	0.994441i	$0.466421\pi$
$270$	0	0
$271$	1127.13	0.252651	0.126326	−	0.991989i	$-0.459682\pi$
$271$	1127.13	0.252651	0.126326	+	0.991989i	$0.459682\pi$
$272$	−731.114	−0.162979
$273$	0	0
$274$	5872.00	1.29467
$275$	−214.000	−0.0469261
$276$	0	0
$277$	1510.00	0.327535	0.163767	−	0.986499i	$-0.447635\pi$
$277$	1510.00	0.327535	0.163767	+	0.986499i	$0.447635\pi$
$278$	365.557	0.0788657
$279$	0	0
$280$	0	0
$281$	4008.00	0.850880	0.425440	−	0.904987i	$-0.360119\pi$
$281$	4008.00	0.850880	0.425440	+	0.904987i	$0.360119\pi$
$282$	0	0
$283$	−2406.58	−0.505500	−0.252750	−	0.967532i	$-0.581335\pi$
$283$	−2406.58	−0.505500	−0.252750	+	0.967532i	$0.581335\pi$
$284$	−3080.00	−0.643537
$285$	0	0
$286$	121.852	0.0251933
$287$	0	0
$288$	0	0
$289$	−2825.00	−0.575005
$290$	6458.18	1.30771
$291$	0	0
$292$	3899.28	0.781465
$293$	−5254.88	−1.04776	−0.523880	−	0.851792i	$-0.675516\pi$
$293$	−5254.88	−1.04776	−0.523880	+	0.851792i	$0.675516\pi$
$294$	0	0
$295$	−10208.0	−2.01469
$296$	−1968.00	−0.386445
$297$	0	0
$298$	−1928.00	−0.374785
$299$	−913.893	−0.176762
$300$	0	0
$301$	0	0
$302$	3376.00	0.643268
$303$	0	0
$304$	−2437.05	−0.459784
$305$	7888.00	1.48087
$306$	0	0
$307$	6366.79	1.18362	0.591811	−	0.806077i	$-0.298413\pi$
$307$	6366.79	1.18362	0.591811	+	0.806077i	$0.298413\pi$
$308$	0	0
$309$	0	0
$310$	6496.00	1.19015
$311$	−7372.07	−1.34415	−0.672077	−	0.740482i	$-0.734597\pi$
$311$	−7372.07	−1.34415	−0.672077	+	0.740482i	$0.734597\pi$
$312$	0	0
$313$	−548.336	−0.0990216	−0.0495108	−	0.998774i	$-0.515766\pi$
$313$	−548.336	−0.0990216	−0.0495108	+	0.998774i	$0.515766\pi$
$314$	6884.66	1.23734
$315$	0	0
$316$	1888.00	0.336102
$317$	−3780.00	−0.669735	−0.334867	−	0.942265i	$-0.608692\pi$
$317$	−3780.00	−0.669735	−0.334867	+	0.942265i	$0.608692\pi$
$318$	0	0
$319$	424.000	0.0744183
$320$	974.819	0.170294
$321$	0	0
$322$	0	0
$323$	6960.00	1.19896
$324$	0	0
$325$	3259.55	0.556330
$326$	6648.00	1.12944
$327$	0	0
$328$	2558.90	0.430767
$329$	0	0
$330$	0	0
$331$	6260.00	1.03952	0.519759	−	0.854313i	$-0.326022\pi$
$331$	6260.00	1.03952	0.519759	+	0.854313i	$0.326022\pi$
$332$	731.114	0.120859
$333$	0	0
$334$	−6092.62	−0.998124
$335$	9930.97	1.61966
$336$	0	0
$337$	−3166.00	−0.511760	−0.255880	−	0.966709i	$-0.582365\pi$
$337$	−3166.00	−0.511760	−0.255880	+	0.966709i	$0.582365\pi$
$338$	2538.00	0.408429
$339$	0	0
$340$	−2784.00	−0.444069
$341$	426.483	0.0677283
$342$	0	0
$343$	0	0
$344$	2272.00	0.356099
$345$	0	0
$346$	−6549.56	−1.01765
$347$	3618.00	0.559725	0.279862	−	0.960040i	$-0.409711\pi$
$347$	3618.00	0.559725	0.279862	+	0.960040i	$0.409711\pi$
$348$	0	0
$349$	4478.07	0.686836	0.343418	−	0.939183i	$-0.388415\pi$
$349$	4478.07	0.686836	0.343418	+	0.939183i	$0.388415\pi$
$350$	0	0
$351$	0	0
$352$	64.0000	0.00969094
$353$	−3701.27	−0.558069	−0.279035	−	0.960281i	$-0.590014\pi$
$353$	−3701.27	−0.558069	−0.279035	+	0.960281i	$0.590014\pi$
$354$	0	0
$355$	−11728.3	−1.75345
$356$	2863.53	0.426311
$357$	0	0
$358$	2508.00	0.370257
$359$	−130.000	−0.0191118	−0.00955590	−	0.999954i	$-0.503042\pi$
$359$	−130.000	−0.0191118	−0.00955590	+	0.999954i	$0.503042\pi$
$360$	0	0
$361$	16341.0	2.38242
$362$	−6153.54	−0.893434
$363$	0	0
$364$	0	0
$365$	14848.0	2.12926
$366$	0	0
$367$	−7798.55	−1.10921	−0.554606	−	0.832113i	$-0.687131\pi$
$367$	−7798.55	−1.10921	−0.554606	+	0.832113i	$0.687131\pi$
$368$	−480.000	−0.0679938
$369$	0	0
$370$	−7493.92	−1.05295
$371$	0	0
$372$	0	0
$373$	−50.0000	−0.00694076	−0.00347038	−	0.999994i	$-0.501105\pi$
$373$	−50.0000	−0.00694076	−0.00347038	+	0.999994i	$0.501105\pi$
$374$	−182.779	−0.0252707
$375$	0	0
$376$	−487.409	−0.0668517
$377$	−6458.18	−0.882263
$378$	0	0
$379$	−4956.00	−0.671696	−0.335848	−	0.941916i	$-0.609023\pi$
$379$	−4956.00	−0.671696	−0.335848	+	0.941916i	$0.609023\pi$
$380$	−9280.00	−1.25277
$381$	0	0
$382$	1812.00	0.242696
$383$	6762.81	0.902254	0.451127	−	0.892460i	$-0.351022\pi$
$383$	6762.81	0.902254	0.451127	+	0.892460i	$0.351022\pi$
$384$	0	0
$385$	0	0
$386$	−364.000	−0.0479977
$387$	0	0
$388$	−1218.52	−0.159436
$389$	13140.0	1.71266	0.856330	−	0.516430i	$-0.172739\pi$
$389$	13140.0	1.71266	0.856330	+	0.516430i	$0.172739\pi$
$390$	0	0
$391$	1370.84	0.177305
$392$	0	0
$393$	0	0
$394$	6936.00	0.886880
$395$	7189.29	0.915778
$396$	0	0
$397$	−3807.89	−0.481391	−0.240696	−	0.970601i	$-0.577376\pi$
$397$	−3807.89	−0.481391	−0.240696	+	0.970601i	$0.577376\pi$
$398$	−7798.55	−0.982176
$399$	0	0
$400$	1712.00	0.214000
$401$	−4824.00	−0.600746	−0.300373	−	0.953822i	$-0.597111\pi$
$401$	−4824.00	−0.600746	−0.300373	+	0.953822i	$0.597111\pi$
$402$	0	0
$403$	−6496.00	−0.802950
$404$	4082.05	0.502698
$405$	0	0
$406$	0	0
$407$	−492.000	−0.0599202
$408$	0	0
$409$	−5300.58	−0.640823	−0.320412	−	0.947278i	$-0.603821\pi$
$409$	−5300.58	−0.640823	−0.320412	+	0.947278i	$0.603821\pi$
$410$	9744.00	1.17371
$411$	0	0
$412$	2315.20	0.276848
$413$	0	0
$414$	0	0
$415$	2784.00	0.329304
$416$	−974.819	−0.114890
$417$	0	0
$418$	−609.262	−0.0712918
$419$	−10540.2	−1.22894	−0.614468	−	0.788942i	$-0.710629\pi$
$419$	−10540.2	−1.22894	−0.614468	+	0.788942i	$0.710629\pi$
$420$	0	0
$421$	−4458.00	−0.516080	−0.258040	−	0.966134i	$-0.583077\pi$
$421$	−4458.00	−0.516080	−0.258040	+	0.966134i	$0.583077\pi$
$422$	5240.00	0.604453
$423$	0	0
$424$	4384.00	0.502136
$425$	−4889.33	−0.558040
$426$	0	0
$427$	0	0
$428$	6584.00	0.743574
$429$	0	0
$430$	8651.52	0.970263
$431$	16214.0	1.81207	0.906034	−	0.423206i	$-0.139095\pi$
$431$	16214.0	1.81207	0.906034	+	0.423206i	$0.139095\pi$
$432$	0	0
$433$	3594.64	0.398955	0.199478	−	0.979902i	$-0.436075\pi$
$433$	3594.64	0.398955	0.199478	+	0.979902i	$0.436075\pi$
$434$	0	0
$435$	0	0
$436$	3928.00	0.431461
$437$	4569.46	0.500199
$438$	0	0
$439$	8590.59	0.933956	0.466978	−	0.884269i	$-0.345343\pi$
$439$	8590.59	0.933956	0.466978	+	0.884269i	$0.345343\pi$
$440$	243.705	0.0264049
$441$	0	0
$442$	2784.00	0.299596
$443$	−15006.0	−1.60938	−0.804691	−	0.593693i	$-0.797669\pi$
$443$	−15006.0	−1.60938	−0.804691	+	0.593693i	$0.797669\pi$
$444$	0	0
$445$	10904.0	1.16157
$446$	2802.60	0.297550
$447$	0	0
$448$	0	0
$449$	−1824.00	−0.191715	−0.0958573	−	0.995395i	$-0.530559\pi$
$449$	−1824.00	−0.191715	−0.0958573	+	0.995395i	$0.530559\pi$
$450$	0	0
$451$	639.725	0.0667926
$452$	−5152.00	−0.536128
$453$	0	0
$454$	7798.55	0.806177
$455$	0	0
$456$	0	0
$457$	−586.000	−0.0599823	−0.0299912	−	0.999550i	$-0.509548\pi$
$457$	−586.000	−0.0599823	−0.0299912	+	0.999550i	$0.509548\pi$
$458$	−11027.6	−1.12508
$459$	0	0
$460$	−1827.79	−0.185263
$461$	15399.1	1.55576	0.777882	−	0.628410i	$-0.216294\pi$
$461$	15399.1	1.55576	0.777882	+	0.628410i	$0.216294\pi$
$462$	0	0
$463$	−88.0000	−0.00883306	−0.00441653	−	0.999990i	$-0.501406\pi$
$463$	−88.0000	−0.00883306	−0.00441653	+	0.999990i	$0.501406\pi$
$464$	−3392.00	−0.339374
$465$	0	0
$466$	−1424.00	−0.141557
$467$	−9748.19	−0.965937	−0.482968	−	0.875638i	$-0.660441\pi$
$467$	−9748.19	−0.965937	−0.482968	+	0.875638i	$0.660441\pi$
$468$	0	0
$469$	0	0
$470$	−1856.00	−0.182151
$471$	0	0
$472$	5361.50	0.522846
$473$	568.000	0.0552149
$474$	0	0
$475$	−16297.8	−1.57430
$476$	0	0
$477$	0	0
$478$	−5172.00	−0.494899
$479$	4935.02	0.470745	0.235373	−	0.971905i	$-0.424369\pi$
$479$	4935.02	0.470745	0.235373	+	0.971905i	$0.424369\pi$
$480$	0	0
$481$	7493.92	0.710381
$482$	−4508.54	−0.426054
$483$	0	0
$484$	−5308.00	−0.498497
$485$	−4640.00	−0.434416
$486$	0	0
$487$	−824.000	−0.0766715	−0.0383357	−	0.999265i	$-0.512206\pi$
$487$	−824.000	−0.0766715	−0.0383357	+	0.999265i	$0.512206\pi$
$488$	−4142.98	−0.384311
$489$	0	0
$490$	0	0
$491$	15426.0	1.41785	0.708926	−	0.705283i	$-0.249180\pi$
$491$	15426.0	1.41785	0.708926	+	0.705283i	$0.249180\pi$
$492$	0	0
$493$	9687.26	0.884974
$494$	9280.00	0.845196
$495$	0	0
$496$	−3411.87	−0.308866
$497$	0	0
$498$	0	0
$499$	5844.00	0.524275	0.262138	−	0.965030i	$-0.415573\pi$
$499$	5844.00	0.524275	0.262138	+	0.965030i	$0.415573\pi$
$500$	−1096.67	−0.0980893
$501$	0	0
$502$	9017.08	0.801697
$503$	10174.7	0.901921	0.450960	−	0.892544i	$-0.351082\pi$
$503$	10174.7	0.901921	0.450960	+	0.892544i	$0.351082\pi$
$504$	0	0
$505$	15544.0	1.36970
$506$	−120.000	−0.0105428
$507$	0	0
$508$	−4288.00	−0.374506
$509$	−11804.4	−1.02794	−0.513971	−	0.857807i	$-0.671826\pi$
$509$	−11804.4	−1.02794	−0.513971	+	0.857807i	$0.671826\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−9108.46	−0.781629
$515$	8816.00	0.754329
$516$	0	0
$517$	−121.852	−0.0103657
$518$	0	0
$519$	0	0
$520$	−3712.00	−0.313042
$521$	−1568.85	−0.131924	−0.0659621	−	0.997822i	$-0.521012\pi$
$521$	−1568.85	−0.131924	−0.0659621	+	0.997822i	$0.521012\pi$
$522$	0	0
$523$	19130.8	1.59949	0.799744	−	0.600341i	$-0.204968\pi$
$523$	19130.8	1.59949	0.799744	+	0.600341i	$0.204968\pi$
$524$	−10966.7	−0.914281
$525$	0	0
$526$	−5156.00	−0.427400
$527$	9744.00	0.805418
$528$	0	0
$529$	−11267.0	−0.926029
$530$	16693.8	1.36817
$531$	0	0
$532$	0	0
$533$	−9744.00	−0.791856
$534$	0	0
$535$	25071.1	2.02602
$536$	−5216.00	−0.420330
$537$	0	0
$538$	−1858.25	−0.148912
$539$	0	0
$540$	0	0
$541$	−6626.00	−0.526569	−0.263285	−	0.964718i	$-0.584806\pi$
$541$	−6626.00	−0.526569	−0.263285	+	0.964718i	$0.584806\pi$
$542$	−2254.27	−0.178652
$543$	0	0
$544$	1462.23	0.115244
$545$	14957.4	1.17560
$546$	0	0
$547$	−19964.0	−1.56051	−0.780255	−	0.625462i	$-0.784911\pi$
$547$	−19964.0	−1.56051	−0.780255	+	0.625462i	$0.784911\pi$
$548$	−11744.0	−0.915472
$549$	0	0
$550$	428.000	0.0331818
$551$	32290.9	2.49662
$552$	0	0
$553$	0	0
$554$	−3020.00	−0.231602
$555$	0	0
$556$	−731.114	−0.0557665
$557$	−6780.00	−0.515759	−0.257880	−	0.966177i	$-0.583024\pi$
$557$	−6780.00	−0.515759	−0.257880	+	0.966177i	$0.583024\pi$
$558$	0	0
$559$	−8651.52	−0.654598
$560$	0	0
$561$	0	0
$562$	−8016.00	−0.601663
$563$	18948.0	1.41841	0.709205	−	0.705002i	$-0.249054\pi$
$563$	18948.0	1.41841	0.709205	+	0.705002i	$0.249054\pi$
$564$	0	0
$565$	−19618.2	−1.46079
$566$	4813.17	0.357443
$567$	0	0
$568$	6160.00	0.455049
$569$	192.000	0.0141460	0.00707299	−	0.999975i	$-0.497749\pi$
$569$	192.000	0.0141460	0.00707299	+	0.999975i	$0.497749\pi$
$570$	0	0
$571$	−23028.0	−1.68773	−0.843863	−	0.536558i	$-0.819724\pi$
$571$	−23028.0	−1.68773	−0.843863	+	0.536558i	$0.819724\pi$
$572$	−243.705	−0.0178143
$573$	0	0
$574$	0	0
$575$	−3210.00	−0.232811
$576$	0	0
$577$	−10723.0	−0.773665	−0.386832	−	0.922150i	$-0.626431\pi$
$577$	−10723.0	−0.773665	−0.386832	+	0.922150i	$0.626431\pi$
$578$	5650.00	0.406590
$579$	0	0
$580$	−12916.4	−0.924694
$581$	0	0
$582$	0	0
$583$	1096.00	0.0778588
$584$	−7798.55	−0.552579
$585$	0	0
$586$	10509.8	0.740878
$587$	−19496.4	−1.37087	−0.685436	−	0.728133i	$-0.740388\pi$
$587$	−19496.4	−1.37087	−0.685436	+	0.728133i	$0.740388\pi$
$588$	0	0
$589$	32480.0	2.27218
$590$	20416.0	1.42460
$591$	0	0
$592$	3936.00	0.273258
$593$	−13388.5	−0.927152	−0.463576	−	0.886057i	$-0.653434\pi$
$593$	−13388.5	−0.927152	−0.463576	+	0.886057i	$0.653434\pi$
$594$	0	0
$595$	0	0
$596$	3856.00	0.265013
$597$	0	0
$598$	1827.79	0.124989
$599$	−18066.0	−1.23232	−0.616158	−	0.787623i	$-0.711312\pi$
$599$	−18066.0	−1.23232	−0.616158	+	0.787623i	$0.711312\pi$
$600$	0	0
$601$	19861.9	1.34806	0.674031	−	0.738703i	$-0.264561\pi$
$601$	19861.9	1.34806	0.674031	+	0.738703i	$0.264561\pi$
$602$	0	0
$603$	0	0
$604$	−6752.00	−0.454859
$605$	−20212.3	−1.35826
$606$	0	0
$607$	−5300.58	−0.354438	−0.177219	−	0.984171i	$-0.556710\pi$
$607$	−5300.58	−0.354438	−0.177219	+	0.984171i	$0.556710\pi$
$608$	4874.09	0.325116
$609$	0	0
$610$	−15776.0	−1.04713
$611$	1856.00	0.122890
$612$	0	0
$613$	15938.0	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	15938.0	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−12733.6	−0.836947
$615$	0	0
$616$	0	0
$617$	11696.0	0.763149	0.381575	−	0.924338i	$-0.375382\pi$
$617$	11696.0	0.763149	0.381575	+	0.924338i	$0.375382\pi$
$618$	0	0
$619$	−13464.7	−0.874300	−0.437150	−	0.899389i	$-0.644012\pi$
$619$	−13464.7	−0.874300	−0.437150	+	0.899389i	$0.644012\pi$
$620$	−12992.0	−0.841567
$621$	0	0
$622$	14744.1	0.950460
$623$	0	0
$624$	0	0
$625$	−17551.0	−1.12326
$626$	1096.67	0.0700189
$627$	0	0
$628$	−13769.3	−0.874929
$629$	−11240.9	−0.712565
$630$	0	0
$631$	11856.0	0.747987	0.373994	−	0.927431i	$-0.377988\pi$
$631$	11856.0	0.747987	0.373994	+	0.927431i	$0.377988\pi$
$632$	−3776.00	−0.237660
$633$	0	0
$634$	7560.00	0.473574
$635$	−16328.2	−1.02042
$636$	0	0
$637$	0	0
$638$	−848.000	−0.0526217
$639$	0	0
$640$	−1949.64	−0.120416
$641$	−5608.00	−0.345558	−0.172779	−	0.984961i	$-0.555275\pi$
$641$	−5608.00	−0.345558	−0.172779	+	0.984961i	$0.555275\pi$
$642$	0	0
$643$	25314.8	1.55260	0.776298	−	0.630366i	$-0.217095\pi$
$643$	25314.8	1.55260	0.776298	+	0.630366i	$0.217095\pi$
$644$	0	0
$645$	0	0
$646$	−13920.0	−0.847794
$647$	5848.91	0.355401	0.177701	−	0.984085i	$-0.443134\pi$
$647$	5848.91	0.355401	0.177701	+	0.984085i	$0.443134\pi$
$648$	0	0
$649$	1340.38	0.0810699
$650$	−6519.10	−0.393385
$651$	0	0
$652$	−13296.0	−0.798637
$653$	28636.0	1.71610	0.858050	−	0.513566i	$-0.171676\pi$
$653$	28636.0	1.71610	0.858050	+	0.513566i	$0.171676\pi$
$654$	0	0
$655$	−41760.0	−2.49114
$656$	−5117.80	−0.304598
$657$	0	0
$658$	0	0
$659$	−31786.0	−1.87892	−0.939459	−	0.342662i	$-0.888672\pi$
$659$	−31786.0	−1.87892	−0.939459	+	0.342662i	$0.888672\pi$
$660$	0	0
$661$	−9474.02	−0.557484	−0.278742	−	0.960366i	$-0.589917\pi$
$661$	−9474.02	−0.557484	−0.278742	+	0.960366i	$0.589917\pi$
$662$	−12520.0	−0.735051
$663$	0	0
$664$	−1462.23	−0.0854600
$665$	0	0
$666$	0	0
$667$	6360.00	0.369206
$668$	12185.2	0.705780
$669$	0	0
$670$	−19861.9	−1.14527
$671$	−1035.75	−0.0595894
$672$	0	0
$673$	24986.0	1.43111	0.715557	−	0.698555i	$-0.246173\pi$
$673$	24986.0	1.43111	0.715557	+	0.698555i	$0.246173\pi$
$674$	6332.00	0.361869
$675$	0	0
$676$	−5076.00	−0.288803
$677$	45.6946	0.00259407	0.00129704	−	0.999999i	$-0.499587\pi$
$677$	45.6946	0.00259407	0.00129704	+	0.999999i	$0.499587\pi$
$678$	0	0
$679$	0	0
$680$	5568.00	0.314004
$681$	0	0
$682$	−852.967	−0.0478912
$683$	−30094.0	−1.68597	−0.842983	−	0.537940i	$-0.819203\pi$
$683$	−30094.0	−1.68597	−0.842983	+	0.537940i	$0.819203\pi$
$684$	0	0
$685$	−44719.8	−2.49439
$686$	0	0
$687$	0	0
$688$	−4544.00	−0.251800
$689$	−16693.8	−0.923051
$690$	0	0
$691$	−16937.5	−0.932463	−0.466232	−	0.884663i	$-0.654389\pi$
$691$	−16937.5	−0.932463	−0.466232	+	0.884663i	$0.654389\pi$
$692$	13099.1	0.719587
$693$	0	0
$694$	−7236.00	−0.395785
$695$	−2784.00	−0.151947
$696$	0	0
$697$	14616.0	0.794290
$698$	−8956.15	−0.485667
$699$	0	0
$700$	0	0
$701$	−7660.00	−0.412716	−0.206358	−	0.978477i	$-0.566161\pi$
$701$	−7660.00	−0.412716	−0.206358	+	0.978477i	$0.566161\pi$
$702$	0	0
$703$	−37469.6	−2.01023
$704$	−128.000	−0.00685253
$705$	0	0
$706$	7402.53	0.394615
$707$	0	0
$708$	0	0
$709$	10654.0	0.564343	0.282172	−	0.959364i	$-0.408945\pi$
$709$	10654.0	0.564343	0.282172	+	0.959364i	$0.408945\pi$
$710$	23456.6	1.23987
$711$	0	0
$712$	−5727.06	−0.301448
$713$	6397.25	0.336015
$714$	0	0
$715$	−928.000	−0.0485388
$716$	−5016.00	−0.261811
$717$	0	0
$718$	260.000	0.0135141
$719$	37104.0	1.92454	0.962272	−	0.272089i	$-0.0877144\pi$
$719$	37104.0	1.92454	0.962272	+	0.272089i	$0.0877144\pi$
$720$	0	0
$721$	0	0
$722$	−32682.0	−1.68462
$723$	0	0
$724$	12307.1	0.631753
$725$	−22684.0	−1.16202
$726$	0	0
$727$	17760.0	0.906027	0.453013	−	0.891504i	$-0.350349\pi$
$727$	17760.0	0.906027	0.453013	+	0.891504i	$0.350349\pi$
$728$	0	0
$729$	0	0
$730$	−29696.0	−1.50561
$731$	12977.3	0.656610
$732$	0	0
$733$	2528.44	0.127408	0.0637039	−	0.997969i	$-0.479709\pi$
$733$	2528.44	0.127408	0.0637039	+	0.997969i	$0.479709\pi$
$734$	15597.1	0.784332
$735$	0	0
$736$	960.000	0.0480789
$737$	−1304.00	−0.0651743
$738$	0	0
$739$	−8156.00	−0.405986	−0.202993	−	0.979180i	$-0.565067\pi$
$739$	−8156.00	−0.405986	−0.202993	+	0.979180i	$0.565067\pi$
$740$	14987.8	0.744546
$741$	0	0
$742$	0	0
$743$	2910.00	0.143684	0.0718422	−	0.997416i	$-0.477112\pi$
$743$	2910.00	0.143684	0.0718422	+	0.997416i	$0.477112\pi$
$744$	0	0
$745$	14683.2	0.722082
$746$	100.000	0.00490786
$747$	0	0
$748$	365.557	0.0178691
$749$	0	0
$750$	0	0
$751$	13584.0	0.660036	0.330018	−	0.943975i	$-0.392945\pi$
$751$	13584.0	0.660036	0.330018	+	0.943975i	$0.392945\pi$
$752$	974.819	0.0472713
$753$	0	0
$754$	12916.4	0.623854
$755$	−25710.9	−1.23936
$756$	0	0
$757$	19054.0	0.914834	0.457417	−	0.889252i	$-0.348775\pi$
$757$	19054.0	0.914834	0.457417	+	0.889252i	$0.348775\pi$
$758$	9912.00	0.474960
$759$	0	0
$760$	18560.0	0.885845
$761$	−5011.18	−0.238706	−0.119353	−	0.992852i	$-0.538082\pi$
$761$	−5011.18	−0.238706	−0.119353	+	0.992852i	$0.538082\pi$
$762$	0	0
$763$	0	0
$764$	−3624.00	−0.171612
$765$	0	0
$766$	−13525.6	−0.637990
$767$	−20416.0	−0.961120
$768$	0	0
$769$	21506.9	1.00853	0.504265	−	0.863549i	$-0.331763\pi$
$769$	21506.9	1.00853	0.504265	+	0.863549i	$0.331763\pi$
$770$	0	0
$771$	0	0
$772$	728.000	0.0339395
$773$	−23715.5	−1.10348	−0.551739	−	0.834017i	$-0.686035\pi$
$773$	−23715.5	−1.10348	−0.551739	+	0.834017i	$0.686035\pi$
$774$	0	0
$775$	−22816.9	−1.05756
$776$	2437.05	0.112738
$777$	0	0
$778$	−26280.0	−1.21103
$779$	48720.0	2.24079
$780$	0	0
$781$	1540.00	0.0705577
$782$	−2741.68	−0.125374
$783$	0	0
$784$	0	0
$785$	−52432.0	−2.38392
$786$	0	0
$787$	36982.2	1.67506	0.837530	−	0.546391i	$-0.183999\pi$
$787$	36982.2	1.67506	0.837530	+	0.546391i	$0.183999\pi$
$788$	−13872.0	−0.627119
$789$	0	0
$790$	−14378.6	−0.647553
$791$	0	0
$792$	0	0
$793$	15776.0	0.706459
$794$	7615.77	0.340395
$795$	0	0
$796$	15597.1	0.694503
$797$	5346.27	0.237609	0.118805	−	0.992918i	$-0.462094\pi$
$797$	5346.27	0.237609	0.118805	+	0.992918i	$0.462094\pi$
$798$	0	0
$799$	−2784.00	−0.123268
$800$	−3424.00	−0.151321
$801$	0	0
$802$	9648.00	0.424791
$803$	−1949.64	−0.0856802
$804$	0	0
$805$	0	0
$806$	12992.0	0.567771
$807$	0	0
$808$	−8164.11	−0.355461
$809$	−19776.0	−0.859440	−0.429720	−	0.902962i	$-0.641388\pi$
$809$	−19776.0	−0.859440	−0.429720	+	0.902962i	$0.641388\pi$
$810$	0	0
$811$	−15962.7	−0.691153	−0.345576	−	0.938391i	$-0.612317\pi$
$811$	−15962.7	−0.691153	−0.345576	+	0.938391i	$0.612317\pi$
$812$	0	0
$813$	0	0
$814$	984.000	0.0423700
$815$	−50629.7	−2.17605
$816$	0	0
$817$	43257.6	1.85238
$818$	10601.2	0.453130
$819$	0	0
$820$	−19488.0	−0.829940
$821$	−29972.0	−1.27409	−0.637046	−	0.770826i	$-0.719844\pi$
$821$	−29972.0	−1.27409	−0.637046	+	0.770826i	$0.719844\pi$
$822$	0	0
$823$	−38816.0	−1.64403	−0.822017	−	0.569462i	$-0.807151\pi$
$823$	−38816.0	−1.64403	−0.822017	+	0.569462i	$0.807151\pi$
$824$	−4630.39	−0.195761
$825$	0	0
$826$	0	0
$827$	−34386.0	−1.44585	−0.722925	−	0.690926i	$-0.757203\pi$
$827$	−34386.0	−1.44585	−0.722925	+	0.690926i	$0.757203\pi$
$828$	0	0
$829$	13129.6	0.550072	0.275036	−	0.961434i	$-0.411310\pi$
$829$	13129.6	0.550072	0.275036	+	0.961434i	$0.411310\pi$
$830$	−5568.00	−0.232853
$831$	0	0
$832$	1949.64	0.0812398
$833$	0	0
$834$	0	0
$835$	46400.0	1.92304
$836$	1218.52	0.0504109
$837$	0	0
$838$	21080.5	0.868989
$839$	27051.2	1.11313	0.556563	−	0.830806i	$-0.312120\pi$
$839$	27051.2	1.11313	0.556563	+	0.830806i	$0.312120\pi$
$840$	0	0
$841$	20555.0	0.842798
$842$	8916.00	0.364924
$843$	0	0
$844$	−10480.0	−0.427413
$845$	−19328.8	−0.786902
$846$	0	0
$847$	0	0
$848$	−8768.00	−0.355064
$849$	0	0
$850$	9778.65	0.394594
$851$	−7380.00	−0.297277
$852$	0	0
$853$	−14774.6	−0.593051	−0.296526	−	0.955025i	$-0.595828\pi$
$853$	−14774.6	−0.593051	−0.296526	+	0.955025i	$0.595828\pi$
$854$	0	0
$855$	0	0
$856$	−13168.0	−0.525786
$857$	19603.0	0.781360	0.390680	−	0.920527i	$-0.372240\pi$
$857$	19603.0	0.781360	0.390680	+	0.920527i	$0.372240\pi$
$858$	0	0
$859$	12520.3	0.497309	0.248654	−	0.968592i	$-0.420012\pi$
$859$	12520.3	0.497309	0.248654	+	0.968592i	$0.420012\pi$
$860$	−17303.0	−0.686080
$861$	0	0
$862$	−32428.0	−1.28132
$863$	−30774.0	−1.21386	−0.606929	−	0.794756i	$-0.707599\pi$
$863$	−30774.0	−1.21386	−0.606929	+	0.794756i	$0.707599\pi$
$864$	0	0
$865$	49880.0	1.96066
$866$	−7189.29	−0.282104
$867$	0	0
$868$	0	0
$869$	−944.000	−0.0368504
$870$	0	0
$871$	19861.9	0.772671
$872$	−7856.00	−0.305089
$873$	0	0
$874$	−9138.93	−0.353694
$875$	0	0
$876$	0	0
$877$	−16758.0	−0.645242	−0.322621	−	0.946528i	$-0.604564\pi$
$877$	−16758.0	−0.645242	−0.322621	+	0.946528i	$0.604564\pi$
$878$	−17181.2	−0.660406
$879$	0	0
$880$	−487.409	−0.0186711
$881$	−25208.2	−0.964002	−0.482001	−	0.876171i	$-0.660090\pi$
$881$	−25208.2	−0.964002	−0.482001	+	0.876171i	$0.660090\pi$
$882$	0	0
$883$	−5468.00	−0.208395	−0.104198	−	0.994557i	$-0.533227\pi$
$883$	−5468.00	−0.208395	−0.104198	+	0.994557i	$0.533227\pi$
$884$	−5568.00	−0.211846
$885$	0	0
$886$	30012.0	1.13801
$887$	27051.2	1.02400	0.512002	−	0.858984i	$-0.328904\pi$
$887$	27051.2	1.02400	0.512002	+	0.858984i	$0.328904\pi$
$888$	0	0
$889$	0	0
$890$	−21808.0	−0.821355
$891$	0	0
$892$	−5605.21	−0.210399
$893$	−9280.00	−0.347753
$894$	0	0
$895$	−19100.4	−0.713357
$896$	0	0
$897$	0	0
$898$	3648.00	0.135563
$899$	45207.2	1.67714
$900$	0	0
$901$	25040.7	0.925888
$902$	−1279.45	−0.0472295
$903$	0	0
$904$	10304.0	0.379099
$905$	46864.0	1.72134
$906$	0	0
$907$	−5876.00	−0.215115	−0.107558	−	0.994199i	$-0.534303\pi$
$907$	−5876.00	−0.215115	−0.107558	+	0.994199i	$0.534303\pi$
$908$	−15597.1	−0.570053
$909$	0	0
$910$	0	0
$911$	−17962.0	−0.653247	−0.326623	−	0.945155i	$-0.605911\pi$
$911$	−17962.0	−0.653247	−0.326623	+	0.945155i	$0.605911\pi$
$912$	0	0
$913$	−365.557	−0.0132510
$914$	1172.00	0.0424139
$915$	0	0
$916$	22055.3	0.795553
$917$	0	0
$918$	0	0
$919$	53424.0	1.91762	0.958811	−	0.284044i	$-0.0916761\pi$
$919$	53424.0	1.91762	0.958811	+	0.284044i	$0.0916761\pi$
$920$	3655.57	0.131001
$921$	0	0
$922$	−30798.2	−1.10009
$923$	−23456.6	−0.836493
$924$	0	0
$925$	26322.0	0.935635
$926$	176.000	0.00624592
$927$	0	0
$928$	6784.00	0.239974
$929$	−5986.00	−0.211404	−0.105702	−	0.994398i	$-0.533709\pi$
$929$	−5986.00	−0.211404	−0.105702	+	0.994398i	$0.533709\pi$
$930$	0	0
$931$	0	0
$932$	2848.00	0.100096
$933$	0	0
$934$	19496.4	0.683020
$935$	1392.00	0.0486880
$936$	0	0
$937$	−3655.57	−0.127452	−0.0637259	−	0.997967i	$-0.520298\pi$
$937$	−3655.57	−0.127452	−0.0637259	+	0.997967i	$0.520298\pi$
$938$	0	0
$939$	0	0
$940$	3712.00	0.128800
$941$	53142.9	1.84103	0.920514	−	0.390709i	$-0.127770\pi$
$941$	53142.9	1.84103	0.920514	+	0.390709i	$0.127770\pi$
$942$	0	0
$943$	9595.87	0.331373
$944$	−10723.0	−0.369708
$945$	0	0
$946$	−1136.00	−0.0390429
$947$	−10202.0	−0.350074	−0.175037	−	0.984562i	$-0.556005\pi$
$947$	−10202.0	−0.350074	−0.175037	+	0.984562i	$0.556005\pi$
$948$	0	0
$949$	29696.0	1.01578
$950$	32595.5	1.11320
$951$	0	0
$952$	0	0
$953$	8856.00	0.301022	0.150511	−	0.988608i	$-0.451908\pi$
$953$	8856.00	0.301022	0.150511	+	0.988608i	$0.451908\pi$
$954$	0	0
$955$	−13799.8	−0.467592
$956$	10344.0	0.349947
$957$	0	0
$958$	−9870.04	−0.332867
$959$	0	0
$960$	0	0
$961$	15681.0	0.526367
$962$	−14987.8	−0.502315
$963$	0	0
$964$	9017.08	0.301266
$965$	2772.14	0.0924750
$966$	0	0
$967$	−40760.0	−1.35548	−0.677742	−	0.735300i	$-0.737041\pi$
$967$	−40760.0	−1.35548	−0.677742	+	0.735300i	$0.737041\pi$
$968$	10616.0	0.352491
$969$	0	0
$970$	9280.00	0.307178
$971$	−46182.0	−1.52632	−0.763158	−	0.646212i	$-0.776352\pi$
$971$	−46182.0	−1.52632	−0.763158	+	0.646212i	$0.776352\pi$
$972$	0	0
$973$	0	0
$974$	1648.00	0.0542149
$975$	0	0
$976$	8285.96	0.271749
$977$	−23360.0	−0.764946	−0.382473	−	0.923967i	$-0.624928\pi$
$977$	−23360.0	−0.764946	−0.382473	+	0.923967i	$0.624928\pi$
$978$	0	0
$979$	−1431.77	−0.0467410
$980$	0	0
$981$	0	0
$982$	−30852.0	−1.00257
$983$	−12550.8	−0.407231	−0.203616	−	0.979051i	$-0.565269\pi$
$983$	−12550.8	−0.407231	−0.203616	+	0.979051i	$0.565269\pi$
$984$	0	0
$985$	−52823.0	−1.70871
$986$	−19374.5	−0.625771
$987$	0	0
$988$	−18560.0	−0.597644
$989$	8520.00	0.273934
$990$	0	0
$991$	−19480.0	−0.624422	−0.312211	−	0.950013i	$-0.601070\pi$
$991$	−19480.0	−0.624422	−0.312211	+	0.950013i	$0.601070\pi$
$992$	6823.73	0.218401
$993$	0	0
$994$	0	0
$995$	59392.0	1.89231
$996$	0	0
$997$	−36525.2	−1.16025	−0.580123	−	0.814529i	$-0.696996\pi$
$997$	−36525.2	−1.16025	−0.580123	+	0.814529i	$0.696996\pi$
$998$	−11688.0	−0.370719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.x.1.2	yes	2
3.2	odd	2		882.4.a.bf.1.1	yes	2
7.2	even	3		882.4.g.bh.361.1		4
7.3	odd	6		882.4.g.bh.667.2		4
7.4	even	3		882.4.g.bh.667.1		4
7.5	odd	6		882.4.g.bh.361.2		4
7.6	odd	2	inner	882.4.a.x.1.1	✓	2
21.2	odd	6		882.4.g.bb.361.2		4
21.5	even	6		882.4.g.bb.361.1		4
21.11	odd	6		882.4.g.bb.667.2		4
21.17	even	6		882.4.g.bb.667.1		4
21.20	even	2		882.4.a.bf.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.4.a.x.1.1	✓	2	7.6	odd	2	inner
882.4.a.x.1.2	yes	2	1.1	even	1	trivial
882.4.a.bf.1.1	yes	2	3.2	odd	2
882.4.a.bf.1.2	yes	2	21.20	even	2
882.4.g.bb.361.1		4	21.5	even	6
882.4.g.bb.361.2		4	21.2	odd	6
882.4.g.bb.667.1		4	21.17	even	6
882.4.g.bb.667.2		4	21.11	odd	6
882.4.g.bh.361.1		4	7.2	even	3
882.4.g.bh.361.2		4	7.5	odd	6
882.4.g.bh.667.1		4	7.4	even	3
882.4.g.bh.667.2		4	7.3	odd	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +15.2315 q^{5} -8.00000 q^{8} -30.4631 q^{10} -2.00000 q^{11} +30.4631 q^{13} +16.0000 q^{16} -45.6946 q^{17} -152.315 q^{19} +60.9262 q^{20} +4.00000 q^{22} -30.0000 q^{23} +107.000 q^{25} -60.9262 q^{26} -212.000 q^{29} -213.242 q^{31} -32.0000 q^{32} +91.3893 q^{34} +246.000 q^{37} +304.631 q^{38} -121.852 q^{40} -319.862 q^{41} -284.000 q^{43} -8.00000 q^{44} +60.0000 q^{46} +60.9262 q^{47} -214.000 q^{50} +121.852 q^{52} -548.000 q^{53} -30.4631 q^{55} +424.000 q^{58} -670.188 q^{59} +517.873 q^{61} +426.483 q^{62} +64.0000 q^{64} +464.000 q^{65} +652.000 q^{67} -182.779 q^{68} -770.000 q^{71} +974.819 q^{73} -492.000 q^{74} -609.262 q^{76} +472.000 q^{79} +243.705 q^{80} +639.725 q^{82} +182.779 q^{83} -696.000 q^{85} +568.000 q^{86} +16.0000 q^{88} +715.883 q^{89} -120.000 q^{92} -121.852 q^{94} -2320.00 q^{95} -304.631 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} - 4 q^{11} + 32 q^{16} + 8 q^{22} - 60 q^{23} + 214 q^{25} - 424 q^{29} - 64 q^{32} + 492 q^{37} - 568 q^{43} - 16 q^{44} + 120 q^{46} - 428 q^{50} - 1096 q^{53} + 848 q^{58}+ \cdots - 4640 q^{95}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 - 4 * q^11 + 32 * q^16 + 8 * q^22 - 60 * q^23 + 214 * q^25 - 424 * q^29 - 64 * q^32 + 492 * q^37 - 568 * q^43 - 16 * q^44 + 120 * q^46 - 428 * q^50 - 1096 * q^53 + 848 * q^58 + 128 * q^64 + 928 * q^65 + 1304 * q^67 - 1540 * q^71 - 984 * q^74 + 944 * q^79 - 1392 * q^85 + 1136 * q^86 + 32 * q^88 - 240 * q^92 - 4640 * q^95

Introduction

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