LMFDB - Embedded newform 882.4.a.y.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -7.07107 q^{5} -8.00000 q^{8} +14.1421 q^{10} +40.0000 q^{11} -63.6396 q^{13} +16.0000 q^{16} +1.41421 q^{17} +11.3137 q^{19} -28.2843 q^{20} -80.0000 q^{22} +68.0000 q^{23} -75.0000 q^{25} +127.279 q^{26} +110.000 q^{29} -118.794 q^{31} -32.0000 q^{32} -2.82843 q^{34} -20.0000 q^{37} -22.6274 q^{38} +56.5685 q^{40} -49.4975 q^{41} -340.000 q^{43} +160.000 q^{44} -136.000 q^{46} +90.5097 q^{47} +150.000 q^{50} -254.558 q^{52} +628.000 q^{53} -282.843 q^{55} -220.000 q^{58} -876.812 q^{59} +917.825 q^{61} +237.588 q^{62} +64.0000 q^{64} +450.000 q^{65} +540.000 q^{67} +5.65685 q^{68} -420.000 q^{71} +289.914 q^{73} +40.0000 q^{74} +45.2548 q^{76} -760.000 q^{79} -113.137 q^{80} +98.9949 q^{82} +944.695 q^{83} -10.0000 q^{85} +680.000 q^{86} -320.000 q^{88} +1152.58 q^{89} +272.000 q^{92} -181.019 q^{94} -80.0000 q^{95} -502.046 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 80 q^{11} + 32 q^{16} - 160 q^{22} + 136 q^{23} - 150 q^{25} + 220 q^{29} - 64 q^{32} - 40 q^{37} - 680 q^{43} + 320 q^{44} - 272 q^{46} + 300 q^{50} + 1256 q^{53} - 440 q^{58}+ \cdots - 160 q^{95}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 80 * q^11 + 32 * q^16 - 160 * q^22 + 136 * q^23 - 150 * q^25 + 220 * q^29 - 64 * q^32 - 40 * q^37 - 680 * q^43 + 320 * q^44 - 272 * q^46 + 300 * q^50 + 1256 * q^53 - 440 * q^58 + 128 * q^64 + 900 * q^65 + 1080 * q^67 - 840 * q^71 + 80 * q^74 - 1520 * q^79 - 20 * q^85 + 1360 * q^86 - 640 * q^88 + 544 * q^92 - 160 * 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
$5$	−7.07107	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−7.07107	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	14.1421	0.447214
$11$	40.0000	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$11$	40.0000	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$12$	0	0
$13$	−63.6396	−1.35773	−0.678864	−	0.734264i	$-0.737527\pi$
$13$	−63.6396	−1.35773	−0.678864	+	0.734264i	$0.737527\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	1.41421	0.0201763	0.0100882	−	0.999949i	$-0.496789\pi$
$17$	1.41421	0.0201763	0.0100882	+	0.999949i	$0.496789\pi$
$18$	0	0
$19$	11.3137	0.136608	0.0683038	−	0.997665i	$-0.478241\pi$
$19$	11.3137	0.136608	0.0683038	+	0.997665i	$0.478241\pi$
$20$	−28.2843	−0.316228
$21$	0	0
$22$	−80.0000	−0.775275
$23$	68.0000	0.616477	0.308239	−	0.951309i	$-0.400260\pi$
$23$	68.0000	0.616477	0.308239	+	0.951309i	$0.400260\pi$
$24$	0	0
$25$	−75.0000	−0.600000
$26$	127.279	0.960058
$27$	0	0
$28$	0	0
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	0	0
$31$	−118.794	−0.688259	−0.344129	−	0.938922i	$-0.611826\pi$
$31$	−118.794	−0.688259	−0.344129	+	0.938922i	$0.611826\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−2.82843	−0.0142668
$35$	0	0
$36$	0	0
$37$	−20.0000	−0.0888643	−0.0444322	−	0.999012i	$-0.514148\pi$
$37$	−20.0000	−0.0888643	−0.0444322	+	0.999012i	$0.514148\pi$
$38$	−22.6274	−0.0965961
$39$	0	0
$40$	56.5685	0.223607
$41$	−49.4975	−0.188542	−0.0942708	−	0.995547i	$-0.530052\pi$
$41$	−49.4975	−0.188542	−0.0942708	+	0.995547i	$0.530052\pi$
$42$	0	0
$43$	−340.000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$43$	−340.000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$44$	160.000	0.548202
$45$	0	0
$46$	−136.000	−0.435915
$47$	90.5097	0.280898	0.140449	−	0.990088i	$-0.455145\pi$
$47$	90.5097	0.280898	0.140449	+	0.990088i	$0.455145\pi$
$48$	0	0
$49$	0	0
$50$	150.000	0.424264
$51$	0	0
$52$	−254.558	−0.678864
$53$	628.000	1.62759	0.813797	−	0.581150i	$-0.197397\pi$
$53$	628.000	1.62759	0.813797	+	0.581150i	$0.197397\pi$
$54$	0	0
$55$	−282.843	−0.693427
$56$	0	0
$57$	0	0
$58$	−220.000	−0.498059
$59$	−876.812	−1.93477	−0.967383	−	0.253316i	$-0.918479\pi$
$59$	−876.812	−1.93477	−0.967383	+	0.253316i	$0.918479\pi$
$60$	0	0
$61$	917.825	1.92648	0.963241	−	0.268639i	$-0.0865738\pi$
$61$	917.825	1.92648	0.963241	+	0.268639i	$0.0865738\pi$
$62$	237.588	0.486672
$63$	0	0
$64$	64.0000	0.125000
$65$	450.000	0.858702
$66$	0	0
$67$	540.000	0.984649	0.492325	−	0.870412i	$-0.336147\pi$
$67$	540.000	0.984649	0.492325	+	0.870412i	$0.336147\pi$
$68$	5.65685	0.0100882
$69$	0	0
$70$	0	0
$71$	−420.000	−0.702040	−0.351020	−	0.936368i	$-0.614165\pi$
$71$	−420.000	−0.702040	−0.351020	+	0.936368i	$0.614165\pi$
$72$	0	0
$73$	289.914	0.464820	0.232410	−	0.972618i	$-0.425339\pi$
$73$	289.914	0.464820	0.232410	+	0.972618i	$0.425339\pi$
$74$	40.0000	0.0628366
$75$	0	0
$76$	45.2548	0.0683038
$77$	0	0
$78$	0	0
$79$	−760.000	−1.08236	−0.541182	−	0.840906i	$-0.682023\pi$
$79$	−760.000	−1.08236	−0.541182	+	0.840906i	$0.682023\pi$
$80$	−113.137	−0.158114
$81$	0	0
$82$	98.9949	0.133319
$83$	944.695	1.24932	0.624661	−	0.780896i	$-0.285237\pi$
$83$	944.695	1.24932	0.624661	+	0.780896i	$0.285237\pi$
$84$	0	0
$85$	−10.0000	−0.0127606
$86$	680.000	0.852631
$87$	0	0
$88$	−320.000	−0.387638
$89$	1152.58	1.37274	0.686369	−	0.727254i	$-0.259204\pi$
$89$	1152.58	1.37274	0.686369	+	0.727254i	$0.259204\pi$
$90$	0	0
$91$	0	0
$92$	272.000	0.308239
$93$	0	0
$94$	−181.019	−0.198625
$95$	−80.0000	−0.0863982
$96$	0	0
$97$	−502.046	−0.525516	−0.262758	−	0.964862i	$-0.584632\pi$
$97$	−502.046	−0.525516	−0.262758	+	0.964862i	$0.584632\pi$
$98$	0	0
$99$	0	0
$100$	−300.000	−0.300000
$101$	−1760.70	−1.73461	−0.867306	−	0.497776i	$-0.834150\pi$
$101$	−1760.70	−1.73461	−0.867306	+	0.497776i	$0.834150\pi$
$102$	0	0
$103$	226.274	0.216461	0.108230	−	0.994126i	$-0.465482\pi$
$103$	226.274	0.216461	0.108230	+	0.994126i	$0.465482\pi$
$104$	509.117	0.480029
$105$	0	0
$106$	−1256.00	−1.15088
$107$	2024.00	1.82867	0.914334	−	0.404961i	$-0.132715\pi$
$107$	2024.00	1.82867	0.914334	+	0.404961i	$0.132715\pi$
$108$	0	0
$109$	−404.000	−0.355011	−0.177505	−	0.984120i	$-0.556803\pi$
$109$	−404.000	−0.355011	−0.177505	+	0.984120i	$0.556803\pi$
$110$	565.685	0.490327
$111$	0	0
$112$	0	0
$113$	1008.00	0.839156	0.419578	−	0.907719i	$-0.362178\pi$
$113$	1008.00	0.839156	0.419578	+	0.907719i	$0.362178\pi$
$114$	0	0
$115$	−480.833	−0.389895
$116$	440.000	0.352181
$117$	0	0
$118$	1753.62	1.36809
$119$	0	0
$120$	0	0
$121$	269.000	0.202104
$122$	−1835.65	−1.36223
$123$	0	0
$124$	−475.176	−0.344129
$125$	1414.21	1.01193
$126$	0	0
$127$	1000.00	0.698706	0.349353	−	0.936991i	$-0.386401\pi$
$127$	1000.00	0.698706	0.349353	+	0.936991i	$0.386401\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−900.000	−0.607194
$131$	84.8528	0.0565926	0.0282963	−	0.999600i	$-0.490992\pi$
$131$	84.8528	0.0565926	0.0282963	+	0.999600i	$0.490992\pi$
$132$	0	0
$133$	0	0
$134$	−1080.00	−0.696252
$135$	0	0
$136$	−11.3137	−0.00713340
$137$	2034.00	1.26844	0.634220	−	0.773152i	$-0.281321\pi$
$137$	2034.00	1.26844	0.634220	+	0.773152i	$0.281321\pi$
$138$	0	0
$139$	−1736.65	−1.05972	−0.529860	−	0.848085i	$-0.677756\pi$
$139$	−1736.65	−1.05972	−0.529860	+	0.848085i	$0.677756\pi$
$140$	0	0
$141$	0	0
$142$	840.000	0.496417
$143$	−2545.58	−1.48862
$144$	0	0
$145$	−777.817	−0.445477
$146$	−579.828	−0.328677
$147$	0	0
$148$	−80.0000	−0.0444322
$149$	2140.00	1.17661	0.588307	−	0.808637i	$-0.299794\pi$
$149$	2140.00	1.17661	0.588307	+	0.808637i	$0.299794\pi$
$150$	0	0
$151$	2120.00	1.14254	0.571269	−	0.820763i	$-0.306451\pi$
$151$	2120.00	1.14254	0.571269	+	0.820763i	$0.306451\pi$
$152$	−90.5097	−0.0482980
$153$	0	0
$154$	0	0
$155$	840.000	0.435293
$156$	0	0
$157$	1746.55	0.887835	0.443918	−	0.896068i	$-0.353588\pi$
$157$	1746.55	0.887835	0.443918	+	0.896068i	$0.353588\pi$
$158$	1520.00	0.765346
$159$	0	0
$160$	226.274	0.111803
$161$	0	0
$162$	0	0
$163$	3340.00	1.60496	0.802482	−	0.596677i	$-0.203513\pi$
$163$	3340.00	1.60496	0.802482	+	0.596677i	$0.203513\pi$
$164$	−197.990	−0.0942708
$165$	0	0
$166$	−1889.39	−0.883404
$167$	367.696	0.170378	0.0851890	−	0.996365i	$-0.472851\pi$
$167$	367.696	0.170378	0.0851890	+	0.996365i	$0.472851\pi$
$168$	0	0
$169$	1853.00	0.843423
$170$	20.0000	0.00902312
$171$	0	0
$172$	−1360.00	−0.602901
$173$	3389.87	1.48975	0.744876	−	0.667203i	$-0.232509\pi$
$173$	3389.87	1.48975	0.744876	+	0.667203i	$0.232509\pi$
$174$	0	0
$175$	0	0
$176$	640.000	0.274101
$177$	0	0
$178$	−2305.17	−0.970672
$179$	720.000	0.300644	0.150322	−	0.988637i	$-0.451969\pi$
$179$	720.000	0.300644	0.150322	+	0.988637i	$0.451969\pi$
$180$	0	0
$181$	−1854.03	−0.761377	−0.380689	−	0.924703i	$-0.624313\pi$
$181$	−1854.03	−0.761377	−0.380689	+	0.924703i	$0.624313\pi$
$182$	0	0
$183$	0	0
$184$	−544.000	−0.217958
$185$	141.421	0.0562027
$186$	0	0
$187$	56.5685	0.0221214
$188$	362.039	0.140449
$189$	0	0
$190$	160.000	0.0610927
$191$	3980.00	1.50776	0.753881	−	0.657011i	$-0.228179\pi$
$191$	3980.00	1.50776	0.753881	+	0.657011i	$0.228179\pi$
$192$	0	0
$193$	3710.00	1.38369	0.691844	−	0.722047i	$-0.256799\pi$
$193$	3710.00	1.38369	0.691844	+	0.722047i	$0.256799\pi$
$194$	1004.09	0.371596
$195$	0	0
$196$	0	0
$197$	956.000	0.345747	0.172874	−	0.984944i	$-0.444695\pi$
$197$	956.000	0.345747	0.172874	+	0.984944i	$0.444695\pi$
$198$	0	0
$199$	4089.91	1.45691	0.728457	−	0.685092i	$-0.240238\pi$
$199$	4089.91	1.45691	0.728457	+	0.685092i	$0.240238\pi$
$200$	600.000	0.212132
$201$	0	0
$202$	3521.39	1.22656
$203$	0	0
$204$	0	0
$205$	350.000	0.119244
$206$	−452.548	−0.153061
$207$	0	0
$208$	−1018.23	−0.339432
$209$	452.548	0.149777
$210$	0	0
$211$	2868.00	0.935741	0.467870	−	0.883797i	$-0.345021\pi$
$211$	2868.00	0.935741	0.467870	+	0.883797i	$0.345021\pi$
$212$	2512.00	0.813797
$213$	0	0
$214$	−4048.00	−1.29306
$215$	2404.16	0.762617
$216$	0	0
$217$	0	0
$218$	808.000	0.251031
$219$	0	0
$220$	−1131.37	−0.346714
$221$	−90.0000	−0.0273939
$222$	0	0
$223$	2630.44	0.789897	0.394949	−	0.918703i	$-0.370762\pi$
$223$	2630.44	0.789897	0.394949	+	0.918703i	$0.370762\pi$
$224$	0	0
$225$	0	0
$226$	−2016.00	−0.593373
$227$	−169.706	−0.0496201	−0.0248100	−	0.999692i	$-0.507898\pi$
$227$	−169.706	−0.0496201	−0.0248100	+	0.999692i	$0.507898\pi$
$228$	0	0
$229$	−3143.80	−0.907196	−0.453598	−	0.891206i	$-0.649860\pi$
$229$	−3143.80	−0.907196	−0.453598	+	0.891206i	$0.649860\pi$
$230$	961.665	0.275697
$231$	0	0
$232$	−880.000	−0.249029
$233$	−4482.00	−1.26020	−0.630098	−	0.776516i	$-0.716985\pi$
$233$	−4482.00	−1.26020	−0.630098	+	0.776516i	$0.716985\pi$
$234$	0	0
$235$	−640.000	−0.177655
$236$	−3507.25	−0.967383
$237$	0	0
$238$	0	0
$239$	−1740.00	−0.470926	−0.235463	−	0.971883i	$-0.575661\pi$
$239$	−1740.00	−0.470926	−0.235463	+	0.971883i	$0.575661\pi$
$240$	0	0
$241$	1260.06	0.336796	0.168398	−	0.985719i	$-0.446141\pi$
$241$	1260.06	0.336796	0.168398	+	0.985719i	$0.446141\pi$
$242$	−538.000	−0.142909
$243$	0	0
$244$	3671.30	0.963241
$245$	0	0
$246$	0	0
$247$	−720.000	−0.185476
$248$	950.352	0.243336
$249$	0	0
$250$	−2828.43	−0.715542
$251$	−5826.56	−1.46522	−0.732608	−	0.680651i	$-0.761697\pi$
$251$	−5826.56	−1.46522	−0.732608	+	0.680651i	$0.761697\pi$
$252$	0	0
$253$	2720.00	0.675909
$254$	−2000.00	−0.494060
$255$	0	0
$256$	256.000	0.0625000
$257$	−4688.12	−1.13789	−0.568943	−	0.822377i	$-0.692648\pi$
$257$	−4688.12	−1.13789	−0.568943	+	0.822377i	$0.692648\pi$
$258$	0	0
$259$	0	0
$260$	1800.00	0.429351
$261$	0	0
$262$	−169.706	−0.0400170
$263$	2172.00	0.509244	0.254622	−	0.967041i	$-0.418049\pi$
$263$	2172.00	0.509244	0.254622	+	0.967041i	$0.418049\pi$
$264$	0	0
$265$	−4440.63	−1.02938
$266$	0	0
$267$	0	0
$268$	2160.00	0.492325
$269$	−2708.22	−0.613840	−0.306920	−	0.951735i	$-0.599298\pi$
$269$	−2708.22	−0.613840	−0.306920	+	0.951735i	$0.599298\pi$
$270$	0	0
$271$	6188.60	1.38720	0.693599	−	0.720361i	$-0.256024\pi$
$271$	6188.60	1.38720	0.693599	+	0.720361i	$0.256024\pi$
$272$	22.6274	0.00504408
$273$	0	0
$274$	−4068.00	−0.896923
$275$	−3000.00	−0.657843
$276$	0	0
$277$	6130.00	1.32966	0.664830	−	0.746994i	$-0.268504\pi$
$277$	6130.00	1.32966	0.664830	+	0.746994i	$0.268504\pi$
$278$	3473.31	0.749335
$279$	0	0
$280$	0	0
$281$	−1970.00	−0.418222	−0.209111	−	0.977892i	$-0.567057\pi$
$281$	−1970.00	−0.418222	−0.209111	+	0.977892i	$0.567057\pi$
$282$	0	0
$283$	−1555.63	−0.326759	−0.163380	−	0.986563i	$-0.552240\pi$
$283$	−1555.63	−0.326759	−0.163380	+	0.986563i	$0.552240\pi$
$284$	−1680.00	−0.351020
$285$	0	0
$286$	5091.17	1.05261
$287$	0	0
$288$	0	0
$289$	−4911.00	−0.999593
$290$	1555.63	0.315000
$291$	0	0
$292$	1159.66	0.232410
$293$	7686.25	1.53254	0.766272	−	0.642516i	$-0.222109\pi$
$293$	7686.25	1.53254	0.766272	+	0.642516i	$0.222109\pi$
$294$	0	0
$295$	6200.00	1.22365
$296$	160.000	0.0314183
$297$	0	0
$298$	−4280.00	−0.831992
$299$	−4327.49	−0.837008
$300$	0	0
$301$	0	0
$302$	−4240.00	−0.807896
$303$	0	0
$304$	181.019	0.0341519
$305$	−6490.00	−1.21841
$306$	0	0
$307$	−8598.42	−1.59849	−0.799247	−	0.601003i	$-0.794768\pi$
$307$	−8598.42	−1.59849	−0.799247	+	0.601003i	$0.794768\pi$
$308$	0	0
$309$	0	0
$310$	−1680.00	−0.307799
$311$	7580.18	1.38210	0.691050	−	0.722807i	$-0.257149\pi$
$311$	7580.18	1.38210	0.691050	+	0.722807i	$0.257149\pi$
$312$	0	0
$313$	3075.91	0.555466	0.277733	−	0.960658i	$-0.410417\pi$
$313$	3075.91	0.555466	0.277733	+	0.960658i	$0.410417\pi$
$314$	−3493.11	−0.627794
$315$	0	0
$316$	−3040.00	−0.541182
$317$	−2324.00	−0.411763	−0.205881	−	0.978577i	$-0.566006\pi$
$317$	−2324.00	−0.411763	−0.205881	+	0.978577i	$0.566006\pi$
$318$	0	0
$319$	4400.00	0.772266
$320$	−452.548	−0.0790569
$321$	0	0
$322$	0	0
$323$	16.0000	0.00275623
$324$	0	0
$325$	4772.97	0.814636
$326$	−6680.00	−1.13488
$327$	0	0
$328$	395.980	0.0666595
$329$	0	0
$330$	0	0
$331$	9508.00	1.57887	0.789436	−	0.613832i	$-0.210373\pi$
$331$	9508.00	1.57887	0.789436	+	0.613832i	$0.210373\pi$
$332$	3778.78	0.624661
$333$	0	0
$334$	−735.391	−0.120475
$335$	−3818.38	−0.622747
$336$	0	0
$337$	−4720.00	−0.762952	−0.381476	−	0.924379i	$-0.624584\pi$
$337$	−4720.00	−0.762952	−0.381476	+	0.924379i	$0.624584\pi$
$338$	−3706.00	−0.596390
$339$	0	0
$340$	−40.0000	−0.00638031
$341$	−4751.76	−0.754610
$342$	0	0
$343$	0	0
$344$	2720.00	0.426316
$345$	0	0
$346$	−6779.74	−1.05341
$347$	−6504.00	−1.00620	−0.503102	−	0.864227i	$-0.667808\pi$
$347$	−6504.00	−1.00620	−0.503102	+	0.864227i	$0.667808\pi$
$348$	0	0
$349$	−5256.63	−0.806249	−0.403125	−	0.915145i	$-0.632076\pi$
$349$	−5256.63	−0.806249	−0.403125	+	0.915145i	$0.632076\pi$
$350$	0	0
$351$	0	0
$352$	−1280.00	−0.193819
$353$	12211.7	1.84126	0.920630	−	0.390435i	$-0.127676\pi$
$353$	12211.7	1.84126	0.920630	+	0.390435i	$0.127676\pi$
$354$	0	0
$355$	2969.85	0.444009
$356$	4610.34	0.686369
$357$	0	0
$358$	−1440.00	−0.212588
$359$	−7340.00	−1.07908	−0.539541	−	0.841959i	$-0.681402\pi$
$359$	−7340.00	−1.07908	−0.539541	+	0.841959i	$0.681402\pi$
$360$	0	0
$361$	−6731.00	−0.981338
$362$	3708.07	0.538375
$363$	0	0
$364$	0	0
$365$	−2050.00	−0.293978
$366$	0	0
$367$	−7665.04	−1.09022	−0.545111	−	0.838364i	$-0.683513\pi$
$367$	−7665.04	−1.09022	−0.545111	+	0.838364i	$0.683513\pi$
$368$	1088.00	0.154119
$369$	0	0
$370$	−282.843	−0.0397413
$371$	0	0
$372$	0	0
$373$	−2990.00	−0.415057	−0.207529	−	0.978229i	$-0.566542\pi$
$373$	−2990.00	−0.415057	−0.207529	+	0.978229i	$0.566542\pi$
$374$	−113.137	−0.0156422
$375$	0	0
$376$	−724.077	−0.0993123
$377$	−7000.36	−0.956331
$378$	0	0
$379$	−11900.0	−1.61283	−0.806414	−	0.591351i	$-0.798595\pi$
$379$	−11900.0	−1.61283	−0.806414	+	0.591351i	$0.798595\pi$
$380$	−320.000	−0.0431991
$381$	0	0
$382$	−7960.00	−1.06615
$383$	−9712.82	−1.29583	−0.647914	−	0.761714i	$-0.724358\pi$
$383$	−9712.82	−1.29583	−0.647914	+	0.761714i	$0.724358\pi$
$384$	0	0
$385$	0	0
$386$	−7420.00	−0.978415
$387$	0	0
$388$	−2008.18	−0.262758
$389$	2150.00	0.280230	0.140115	−	0.990135i	$-0.455253\pi$
$389$	2150.00	0.280230	0.140115	+	0.990135i	$0.455253\pi$
$390$	0	0
$391$	96.1665	0.0124382
$392$	0	0
$393$	0	0
$394$	−1912.00	−0.244480
$395$	5374.01	0.684546
$396$	0	0
$397$	3401.18	0.429976	0.214988	−	0.976617i	$-0.431029\pi$
$397$	3401.18	0.429976	0.214988	+	0.976617i	$0.431029\pi$
$398$	−8179.81	−1.03019
$399$	0	0
$400$	−1200.00	−0.150000
$401$	−12090.0	−1.50560	−0.752800	−	0.658249i	$-0.771297\pi$
$401$	−12090.0	−1.50560	−0.752800	+	0.658249i	$0.771297\pi$
$402$	0	0
$403$	7560.00	0.934468
$404$	−7042.78	−0.867306
$405$	0	0
$406$	0	0
$407$	−800.000	−0.0974313
$408$	0	0
$409$	8192.54	0.990452	0.495226	−	0.868764i	$-0.335085\pi$
$409$	8192.54	0.990452	0.495226	+	0.868764i	$0.335085\pi$
$410$	−700.000	−0.0843184
$411$	0	0
$412$	905.097	0.108230
$413$	0	0
$414$	0	0
$415$	−6680.00	−0.790140
$416$	2036.47	0.240015
$417$	0	0
$418$	−905.097	−0.105908
$419$	−1046.52	−0.122019	−0.0610093	−	0.998137i	$-0.519432\pi$
$419$	−1046.52	−0.122019	−0.0610093	+	0.998137i	$0.519432\pi$
$420$	0	0
$421$	−3870.00	−0.448010	−0.224005	−	0.974588i	$-0.571913\pi$
$421$	−3870.00	−0.448010	−0.224005	+	0.974588i	$0.571913\pi$
$422$	−5736.00	−0.661669
$423$	0	0
$424$	−5024.00	−0.575441
$425$	−106.066	−0.0121058
$426$	0	0
$427$	0	0
$428$	8096.00	0.914334
$429$	0	0
$430$	−4808.33	−0.539251
$431$	−2700.00	−0.301750	−0.150875	−	0.988553i	$-0.548209\pi$
$431$	−2700.00	−0.301750	−0.150875	+	0.988553i	$0.548209\pi$
$432$	0	0
$433$	−5876.06	−0.652160	−0.326080	−	0.945342i	$-0.605728\pi$
$433$	−5876.06	−0.652160	−0.326080	+	0.945342i	$0.605728\pi$
$434$	0	0
$435$	0	0
$436$	−1616.00	−0.177505
$437$	769.332	0.0842154
$438$	0	0
$439$	6346.99	0.690035	0.345017	−	0.938596i	$-0.387873\pi$
$439$	6346.99	0.690035	0.345017	+	0.938596i	$0.387873\pi$
$440$	2262.74	0.245164
$441$	0	0
$442$	180.000	0.0193704
$443$	−6928.00	−0.743023	−0.371512	−	0.928428i	$-0.621160\pi$
$443$	−6928.00	−0.743023	−0.371512	+	0.928428i	$0.621160\pi$
$444$	0	0
$445$	−8150.00	−0.868196
$446$	−5260.87	−0.558542
$447$	0	0
$448$	0	0
$449$	−1320.00	−0.138741	−0.0693704	−	0.997591i	$-0.522099\pi$
$449$	−1320.00	−0.138741	−0.0693704	+	0.997591i	$0.522099\pi$
$450$	0	0
$451$	−1979.90	−0.206718
$452$	4032.00	0.419578
$453$	0	0
$454$	339.411	0.0350867
$455$	0	0
$456$	0	0
$457$	1290.00	0.132043	0.0660215	−	0.997818i	$-0.478969\pi$
$457$	1290.00	0.132043	0.0660215	+	0.997818i	$0.478969\pi$
$458$	6287.59	0.641485
$459$	0	0
$460$	−1923.33	−0.194947
$461$	−17642.3	−1.78240	−0.891198	−	0.453615i	$-0.850134\pi$
$461$	−17642.3	−1.78240	−0.891198	+	0.453615i	$0.850134\pi$
$462$	0	0
$463$	5680.00	0.570134	0.285067	−	0.958508i	$-0.407984\pi$
$463$	5680.00	0.570134	0.285067	+	0.958508i	$0.407984\pi$
$464$	1760.00	0.176090
$465$	0	0
$466$	8964.00	0.891093
$467$	7693.32	0.762322	0.381161	−	0.924509i	$-0.375524\pi$
$467$	7693.32	0.762322	0.381161	+	0.924509i	$0.375524\pi$
$468$	0	0
$469$	0	0
$470$	1280.00	0.125621
$471$	0	0
$472$	7014.50	0.684043
$473$	−13600.0	−1.32205
$474$	0	0
$475$	−848.528	−0.0819645
$476$	0	0
$477$	0	0
$478$	3480.00	0.332995
$479$	16518.0	1.57563	0.787816	−	0.615911i	$-0.211212\pi$
$479$	16518.0	1.57563	0.787816	+	0.615911i	$0.211212\pi$
$480$	0	0
$481$	1272.79	0.120653
$482$	−2520.13	−0.238151
$483$	0	0
$484$	1076.00	0.101052
$485$	3550.00	0.332365
$486$	0	0
$487$	13680.0	1.27290	0.636448	−	0.771320i	$-0.280403\pi$
$487$	13680.0	1.27290	0.636448	+	0.771320i	$0.280403\pi$
$488$	−7342.60	−0.681114
$489$	0	0
$490$	0	0
$491$	2280.00	0.209562	0.104781	−	0.994495i	$-0.466586\pi$
$491$	2280.00	0.209562	0.104781	+	0.994495i	$0.466586\pi$
$492$	0	0
$493$	155.563	0.0142114
$494$	1440.00	0.131151
$495$	0	0
$496$	−1900.70	−0.172065
$497$	0	0
$498$	0	0
$499$	860.000	0.0771521	0.0385760	−	0.999256i	$-0.487718\pi$
$499$	860.000	0.0771521	0.0385760	+	0.999256i	$0.487718\pi$
$500$	5656.85	0.505964
$501$	0	0
$502$	11653.1	1.03606
$503$	5730.39	0.507963	0.253982	−	0.967209i	$-0.418260\pi$
$503$	5730.39	0.507963	0.253982	+	0.967209i	$0.418260\pi$
$504$	0	0
$505$	12450.0	1.09706
$506$	−5440.00	−0.477940
$507$	0	0
$508$	4000.00	0.349353
$509$	4589.12	0.399625	0.199813	−	0.979834i	$-0.435967\pi$
$509$	4589.12	0.399625	0.199813	+	0.979834i	$0.435967\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	9376.24	0.804607
$515$	−1600.00	−0.136902
$516$	0	0
$517$	3620.39	0.307978
$518$	0	0
$519$	0	0
$520$	−3600.00	−0.303597
$521$	14587.6	1.22667	0.613335	−	0.789823i	$-0.289828\pi$
$521$	14587.6	1.22667	0.613335	+	0.789823i	$0.289828\pi$
$522$	0	0
$523$	−6109.40	−0.510794	−0.255397	−	0.966836i	$-0.582206\pi$
$523$	−6109.40	−0.510794	−0.255397	+	0.966836i	$0.582206\pi$
$524$	339.411	0.0282963
$525$	0	0
$526$	−4344.00	−0.360090
$527$	−168.000	−0.0138865
$528$	0	0
$529$	−7543.00	−0.619956
$530$	8881.26	0.727882
$531$	0	0
$532$	0	0
$533$	3150.00	0.255988
$534$	0	0
$535$	−14311.8	−1.15655
$536$	−4320.00	−0.348126
$537$	0	0
$538$	5416.44	0.434051
$539$	0	0
$540$	0	0
$541$	−17210.0	−1.36768	−0.683841	−	0.729631i	$-0.739692\pi$
$541$	−17210.0	−1.36768	−0.683841	+	0.729631i	$0.739692\pi$
$542$	−12377.2	−0.980897
$543$	0	0
$544$	−45.2548	−0.00356670
$545$	2856.71	0.224529
$546$	0	0
$547$	4060.00	0.317355	0.158677	−	0.987330i	$-0.449277\pi$
$547$	4060.00	0.317355	0.158677	+	0.987330i	$0.449277\pi$
$548$	8136.00	0.634220
$549$	0	0
$550$	6000.00	0.465165
$551$	1244.51	0.0962211
$552$	0	0
$553$	0	0
$554$	−12260.0	−0.940212
$555$	0	0
$556$	−6946.62	−0.529860
$557$	10356.0	0.787788	0.393894	−	0.919156i	$-0.371128\pi$
$557$	10356.0	0.787788	0.393894	+	0.919156i	$0.371128\pi$
$558$	0	0
$559$	21637.5	1.63715
$560$	0	0
$561$	0	0
$562$	3940.00	0.295728
$563$	−23210.1	−1.73746	−0.868728	−	0.495289i	$-0.835062\pi$
$563$	−23210.1	−1.73746	−0.868728	+	0.495289i	$0.835062\pi$
$564$	0	0
$565$	−7127.64	−0.530729
$566$	3111.27	0.231054
$567$	0	0
$568$	3360.00	0.248209
$569$	5890.00	0.433957	0.216979	−	0.976176i	$-0.430380\pi$
$569$	5890.00	0.433957	0.216979	+	0.976176i	$0.430380\pi$
$570$	0	0
$571$	−5612.00	−0.411305	−0.205652	−	0.978625i	$-0.565932\pi$
$571$	−5612.00	−0.411305	−0.205652	+	0.978625i	$0.565932\pi$
$572$	−10182.3	−0.744309
$573$	0	0
$574$	0	0
$575$	−5100.00	−0.369886
$576$	0	0
$577$	17797.9	1.28412	0.642058	−	0.766656i	$-0.278081\pi$
$577$	17797.9	1.28412	0.642058	+	0.766656i	$0.278081\pi$
$578$	9822.00	0.706819
$579$	0	0
$580$	−3111.27	−0.222739
$581$	0	0
$582$	0	0
$583$	25120.0	1.78450
$584$	−2319.31	−0.164339
$585$	0	0
$586$	−15372.5	−1.08367
$587$	7942.22	0.558451	0.279225	−	0.960226i	$-0.409922\pi$
$587$	7942.22	0.558451	0.279225	+	0.960226i	$0.409922\pi$
$588$	0	0
$589$	−1344.00	−0.0940213
$590$	−12400.0	−0.865254
$591$	0	0
$592$	−320.000	−0.0222161
$593$	−14078.5	−0.974932	−0.487466	−	0.873142i	$-0.662079\pi$
$593$	−14078.5	−0.974932	−0.487466	+	0.873142i	$0.662079\pi$
$594$	0	0
$595$	0	0
$596$	8560.00	0.588307
$597$	0	0
$598$	8654.99	0.591854
$599$	−7300.00	−0.497946	−0.248973	−	0.968510i	$-0.580093\pi$
$599$	−7300.00	−0.497946	−0.248973	+	0.968510i	$0.580093\pi$
$600$	0	0
$601$	8727.11	0.592323	0.296162	−	0.955138i	$-0.404293\pi$
$601$	8727.11	0.592323	0.296162	+	0.955138i	$0.404293\pi$
$602$	0	0
$603$	0	0
$604$	8480.00	0.571269
$605$	−1902.12	−0.127822
$606$	0	0
$607$	−10606.6	−0.709240	−0.354620	−	0.935011i	$-0.615390\pi$
$607$	−10606.6	−0.709240	−0.354620	+	0.935011i	$0.615390\pi$
$608$	−362.039	−0.0241490
$609$	0	0
$610$	12980.0	0.861549
$611$	−5760.00	−0.381382
$612$	0	0
$613$	−13980.0	−0.921121	−0.460560	−	0.887628i	$-0.652351\pi$
$613$	−13980.0	−0.921121	−0.460560	+	0.887628i	$0.652351\pi$
$614$	17196.8	1.13031
$615$	0	0
$616$	0	0
$617$	−2654.00	−0.173170	−0.0865851	−	0.996244i	$-0.527595\pi$
$617$	−2654.00	−0.173170	−0.0865851	+	0.996244i	$0.527595\pi$
$618$	0	0
$619$	−23883.2	−1.55081	−0.775403	−	0.631467i	$-0.782453\pi$
$619$	−23883.2	−1.55081	−0.775403	+	0.631467i	$0.782453\pi$
$620$	3360.00	0.217647
$621$	0	0
$622$	−15160.4	−0.977292
$623$	0	0
$624$	0	0
$625$	−625.000	−0.0400000
$626$	−6151.83	−0.392774
$627$	0	0
$628$	6986.21	0.443918
$629$	−28.2843	−0.00179295
$630$	0	0
$631$	−6400.00	−0.403772	−0.201886	−	0.979409i	$-0.564707\pi$
$631$	−6400.00	−0.403772	−0.201886	+	0.979409i	$0.564707\pi$
$632$	6080.00	0.382673
$633$	0	0
$634$	4648.00	0.291160
$635$	−7071.07	−0.441900
$636$	0	0
$637$	0	0
$638$	−8800.00	−0.546074
$639$	0	0
$640$	905.097	0.0559017
$641$	15350.0	0.945848	0.472924	−	0.881103i	$-0.343199\pi$
$641$	15350.0	0.945848	0.472924	+	0.881103i	$0.343199\pi$
$642$	0	0
$643$	17847.4	1.09461	0.547303	−	0.836934i	$-0.315655\pi$
$643$	17847.4	1.09461	0.547303	+	0.836934i	$0.315655\pi$
$644$	0	0
$645$	0	0
$646$	−32.0000	−0.00194895
$647$	−14000.7	−0.850734	−0.425367	−	0.905021i	$-0.639855\pi$
$647$	−14000.7	−0.850734	−0.425367	+	0.905021i	$0.639855\pi$
$648$	0	0
$649$	−35072.5	−2.12129
$650$	−9545.94	−0.576035
$651$	0	0
$652$	13360.0	0.802482
$653$	26382.0	1.58102	0.790511	−	0.612448i	$-0.209815\pi$
$653$	26382.0	1.58102	0.790511	+	0.612448i	$0.209815\pi$
$654$	0	0
$655$	−600.000	−0.0357923
$656$	−791.960	−0.0471354
$657$	0	0
$658$	0	0
$659$	14400.0	0.851205	0.425603	−	0.904910i	$-0.360062\pi$
$659$	14400.0	0.851205	0.425603	+	0.904910i	$0.360062\pi$
$660$	0	0
$661$	28582.7	1.68190	0.840951	−	0.541112i	$-0.181996\pi$
$661$	28582.7	1.68190	0.840951	+	0.541112i	$0.181996\pi$
$662$	−19016.0	−1.11643
$663$	0	0
$664$	−7557.56	−0.441702
$665$	0	0
$666$	0	0
$667$	7480.00	0.434223
$668$	1470.78	0.0851890
$669$	0	0
$670$	7636.75	0.440349
$671$	36713.0	2.11220
$672$	0	0
$673$	−18120.0	−1.03785	−0.518926	−	0.854819i	$-0.673668\pi$
$673$	−18120.0	−1.03785	−0.518926	+	0.854819i	$0.673668\pi$
$674$	9440.00	0.539488
$675$	0	0
$676$	7412.00	0.421711
$677$	17797.9	1.01038	0.505191	−	0.863008i	$-0.331422\pi$
$677$	17797.9	1.01038	0.505191	+	0.863008i	$0.331422\pi$
$678$	0	0
$679$	0	0
$680$	80.0000	0.00451156
$681$	0	0
$682$	9503.52	0.533590
$683$	1728.00	0.0968083	0.0484042	−	0.998828i	$-0.484586\pi$
$683$	1728.00	0.0968083	0.0484042	+	0.998828i	$0.484586\pi$
$684$	0	0
$685$	−14382.6	−0.802232
$686$	0	0
$687$	0	0
$688$	−5440.00	−0.301451
$689$	−39965.7	−2.20983
$690$	0	0
$691$	17089.4	0.940825	0.470412	−	0.882447i	$-0.344105\pi$
$691$	17089.4	0.940825	0.470412	+	0.882447i	$0.344105\pi$
$692$	13559.5	0.744876
$693$	0	0
$694$	13008.0	0.711494
$695$	12280.0	0.670226
$696$	0	0
$697$	−70.0000	−0.00380407
$698$	10513.3	0.570104
$699$	0	0
$700$	0	0
$701$	13410.0	0.722523	0.361262	−	0.932465i	$-0.382346\pi$
$701$	13410.0	0.722523	0.361262	+	0.932465i	$0.382346\pi$
$702$	0	0
$703$	−226.274	−0.0121395
$704$	2560.00	0.137051
$705$	0	0
$706$	−24423.5	−1.30197
$707$	0	0
$708$	0	0
$709$	−140.000	−0.00741581	−0.00370791	−	0.999993i	$-0.501180\pi$
$709$	−140.000	−0.00741581	−0.00370791	+	0.999993i	$0.501180\pi$
$710$	−5939.70	−0.313962
$711$	0	0
$712$	−9220.67	−0.485336
$713$	−8077.99	−0.424296
$714$	0	0
$715$	18000.0	0.941485
$716$	2880.00	0.150322
$717$	0	0
$718$	14680.0	0.763026
$719$	25936.7	1.34531	0.672653	−	0.739958i	$-0.265155\pi$
$719$	25936.7	1.34531	0.672653	+	0.739958i	$0.265155\pi$
$720$	0	0
$721$	0	0
$722$	13462.0	0.693911
$723$	0	0
$724$	−7416.14	−0.380689
$725$	−8250.00	−0.422617
$726$	0	0
$727$	9277.24	0.473279	0.236639	−	0.971598i	$-0.423954\pi$
$727$	9277.24	0.473279	0.236639	+	0.971598i	$0.423954\pi$
$728$	0	0
$729$	0	0
$730$	4100.00	0.207874
$731$	−480.833	−0.0243286
$732$	0	0
$733$	−25477.1	−1.28379	−0.641894	−	0.766793i	$-0.721851\pi$
$733$	−25477.1	−1.28379	−0.641894	+	0.766793i	$0.721851\pi$
$734$	15330.1	0.770904
$735$	0	0
$736$	−2176.00	−0.108979
$737$	21600.0	1.07957
$738$	0	0
$739$	−6924.00	−0.344660	−0.172330	−	0.985039i	$-0.555129\pi$
$739$	−6924.00	−0.344660	−0.172330	+	0.985039i	$0.555129\pi$
$740$	565.685	0.0281014
$741$	0	0
$742$	0	0
$743$	−29108.0	−1.43724	−0.718620	−	0.695403i	$-0.755226\pi$
$743$	−29108.0	−1.43724	−0.718620	+	0.695403i	$0.755226\pi$
$744$	0	0
$745$	−15132.1	−0.744157
$746$	5980.00	0.293490
$747$	0	0
$748$	226.274	0.0110607
$749$	0	0
$750$	0	0
$751$	31448.0	1.52803	0.764017	−	0.645196i	$-0.223224\pi$
$751$	31448.0	1.52803	0.764017	+	0.645196i	$0.223224\pi$
$752$	1448.15	0.0702244
$753$	0	0
$754$	14000.7	0.676228
$755$	−14990.7	−0.722604
$756$	0	0
$757$	−13300.0	−0.638569	−0.319284	−	0.947659i	$-0.603443\pi$
$757$	−13300.0	−0.638569	−0.319284	+	0.947659i	$0.603443\pi$
$758$	23800.0	1.14044
$759$	0	0
$760$	640.000	0.0305464
$761$	4801.26	0.228706	0.114353	−	0.993440i	$-0.463520\pi$
$761$	4801.26	0.228706	0.114353	+	0.993440i	$0.463520\pi$
$762$	0	0
$763$	0	0
$764$	15920.0	0.753881
$765$	0	0
$766$	19425.6	0.916288
$767$	55800.0	2.62689
$768$	0	0
$769$	16932.4	0.794015	0.397007	−	0.917815i	$-0.370049\pi$
$769$	16932.4	0.794015	0.397007	+	0.917815i	$0.370049\pi$
$770$	0	0
$771$	0	0
$772$	14840.0	0.691844
$773$	−19441.2	−0.904594	−0.452297	−	0.891867i	$-0.649395\pi$
$773$	−19441.2	−0.904594	−0.452297	+	0.891867i	$0.649395\pi$
$774$	0	0
$775$	8909.55	0.412955
$776$	4016.37	0.185798
$777$	0	0
$778$	−4300.00	−0.198152
$779$	−560.000	−0.0257562
$780$	0	0
$781$	−16800.0	−0.769720
$782$	−192.333	−0.00879516
$783$	0	0
$784$	0	0
$785$	−12350.0	−0.561516
$786$	0	0
$787$	−20732.4	−0.939046	−0.469523	−	0.882920i	$-0.655574\pi$
$787$	−20732.4	−0.939046	−0.469523	+	0.882920i	$0.655574\pi$
$788$	3824.00	0.172874
$789$	0	0
$790$	−10748.0	−0.484047
$791$	0	0
$792$	0	0
$793$	−58410.0	−2.61564
$794$	−6802.37	−0.304039
$795$	0	0
$796$	16359.6	0.728457
$797$	28582.7	1.27033	0.635163	−	0.772378i	$-0.280933\pi$
$797$	28582.7	1.27033	0.635163	+	0.772378i	$0.280933\pi$
$798$	0	0
$799$	128.000	0.00566748
$800$	2400.00	0.106066
$801$	0	0
$802$	24180.0	1.06462
$803$	11596.6	0.509631
$804$	0	0
$805$	0	0
$806$	−15120.0	−0.660768
$807$	0	0
$808$	14085.6	0.613278
$809$	22280.0	0.968261	0.484130	−	0.874996i	$-0.339136\pi$
$809$	22280.0	0.968261	0.484130	+	0.874996i	$0.339136\pi$
$810$	0	0
$811$	26100.7	1.13011	0.565056	−	0.825053i	$-0.308855\pi$
$811$	26100.7	1.13011	0.565056	+	0.825053i	$0.308855\pi$
$812$	0	0
$813$	0	0
$814$	1600.00	0.0688943
$815$	−23617.4	−1.01507
$816$	0	0
$817$	−3846.66	−0.164722
$818$	−16385.1	−0.700356
$819$	0	0
$820$	1400.00	0.0596221
$821$	−3540.00	−0.150483	−0.0752417	−	0.997165i	$-0.523973\pi$
$821$	−3540.00	−0.150483	−0.0752417	+	0.997165i	$0.523973\pi$
$822$	0	0
$823$	−28400.0	−1.20287	−0.601435	−	0.798922i	$-0.705404\pi$
$823$	−28400.0	−1.20287	−0.601435	+	0.798922i	$0.705404\pi$
$824$	−1810.19	−0.0765304
$825$	0	0
$826$	0	0
$827$	38736.0	1.62876	0.814379	−	0.580334i	$-0.197078\pi$
$827$	38736.0	1.62876	0.814379	+	0.580334i	$0.197078\pi$
$828$	0	0
$829$	−23409.5	−0.980754	−0.490377	−	0.871511i	$-0.663141\pi$
$829$	−23409.5	−0.980754	−0.490377	+	0.871511i	$0.663141\pi$
$830$	13360.0	0.558714
$831$	0	0
$832$	−4072.94	−0.169716
$833$	0	0
$834$	0	0
$835$	−2600.00	−0.107757
$836$	1810.19	0.0748886
$837$	0	0
$838$	2093.04	0.0862801
$839$	−29387.4	−1.20925	−0.604627	−	0.796509i	$-0.706678\pi$
$839$	−29387.4	−1.20925	−0.604627	+	0.796509i	$0.706678\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	7740.00	0.316791
$843$	0	0
$844$	11472.0	0.467870
$845$	−13102.7	−0.533427
$846$	0	0
$847$	0	0
$848$	10048.0	0.406898
$849$	0	0
$850$	212.132	0.00856008
$851$	−1360.00	−0.0547828
$852$	0	0
$853$	22252.7	0.893220	0.446610	−	0.894729i	$-0.352631\pi$
$853$	22252.7	0.893220	0.446610	+	0.894729i	$0.352631\pi$
$854$	0	0
$855$	0	0
$856$	−16192.0	−0.646532
$857$	−13832.4	−0.551350	−0.275675	−	0.961251i	$-0.588901\pi$
$857$	−13832.4	−0.551350	−0.275675	+	0.961251i	$0.588901\pi$
$858$	0	0
$859$	39665.9	1.57553	0.787766	−	0.615975i	$-0.211238\pi$
$859$	39665.9	1.57553	0.787766	+	0.615975i	$0.211238\pi$
$860$	9616.65	0.381308
$861$	0	0
$862$	5400.00	0.213370
$863$	31988.0	1.26174	0.630871	−	0.775887i	$-0.282698\pi$
$863$	31988.0	1.26174	0.630871	+	0.775887i	$0.282698\pi$
$864$	0	0
$865$	−23970.0	−0.942202
$866$	11752.1	0.461147
$867$	0	0
$868$	0	0
$869$	−30400.0	−1.18671
$870$	0	0
$871$	−34365.4	−1.33688
$872$	3232.00	0.125515
$873$	0	0
$874$	−1538.66	−0.0595493
$875$	0	0
$876$	0	0
$877$	−33460.0	−1.28833	−0.644164	−	0.764887i	$-0.722795\pi$
$877$	−33460.0	−1.28833	−0.644164	+	0.764887i	$0.722795\pi$
$878$	−12694.0	−0.487928
$879$	0	0
$880$	−4525.48	−0.173357
$881$	12791.6	0.489170	0.244585	−	0.969628i	$-0.421348\pi$
$881$	12791.6	0.489170	0.244585	+	0.969628i	$0.421348\pi$
$882$	0	0
$883$	−35260.0	−1.34382	−0.671910	−	0.740633i	$-0.734526\pi$
$883$	−35260.0	−1.34382	−0.671910	+	0.740633i	$0.734526\pi$
$884$	−360.000	−0.0136970
$885$	0	0
$886$	13856.0	0.525397
$887$	15437.6	0.584377	0.292188	−	0.956361i	$-0.405617\pi$
$887$	15437.6	0.584377	0.292188	+	0.956361i	$0.405617\pi$
$888$	0	0
$889$	0	0
$890$	16300.0	0.613907
$891$	0	0
$892$	10521.7	0.394949
$893$	1024.00	0.0383727
$894$	0	0
$895$	−5091.17	−0.190144
$896$	0	0
$897$	0	0
$898$	2640.00	0.0981046
$899$	−13067.3	−0.484783
$900$	0	0
$901$	888.126	0.0328388
$902$	3959.80	0.146172
$903$	0	0
$904$	−8064.00	−0.296687
$905$	13110.0	0.481537
$906$	0	0
$907$	33100.0	1.21176	0.605881	−	0.795556i	$-0.292821\pi$
$907$	33100.0	1.21176	0.605881	+	0.795556i	$0.292821\pi$
$908$	−678.823	−0.0248100
$909$	0	0
$910$	0	0
$911$	39620.0	1.44091	0.720455	−	0.693502i	$-0.243933\pi$
$911$	39620.0	1.44091	0.720455	+	0.693502i	$0.243933\pi$
$912$	0	0
$913$	37787.8	1.36976
$914$	−2580.00	−0.0933685
$915$	0	0
$916$	−12575.2	−0.453598
$917$	0	0
$918$	0	0
$919$	20944.0	0.751772	0.375886	−	0.926666i	$-0.377338\pi$
$919$	20944.0	0.751772	0.375886	+	0.926666i	$0.377338\pi$
$920$	3846.66	0.137849
$921$	0	0
$922$	35284.6	1.26034
$923$	26728.6	0.953179
$924$	0	0
$925$	1500.00	0.0533186
$926$	−11360.0	−0.403146
$927$	0	0
$928$	−3520.00	−0.124515
$929$	46237.7	1.63295	0.816475	−	0.577381i	$-0.195925\pi$
$929$	46237.7	1.63295	0.816475	+	0.577381i	$0.195925\pi$
$930$	0	0
$931$	0	0
$932$	−17928.0	−0.630098
$933$	0	0
$934$	−15386.6	−0.539043
$935$	−400.000	−0.0139908
$936$	0	0
$937$	−50522.8	−1.76148	−0.880740	−	0.473600i	$-0.842954\pi$
$937$	−50522.8	−1.76148	−0.880740	+	0.473600i	$0.842954\pi$
$938$	0	0
$939$	0	0
$940$	−2560.00	−0.0888277
$941$	9878.28	0.342213	0.171107	−	0.985253i	$-0.445266\pi$
$941$	9878.28	0.342213	0.171107	+	0.985253i	$0.445266\pi$
$942$	0	0
$943$	−3365.83	−0.116232
$944$	−14029.0	−0.483692
$945$	0	0
$946$	27200.0	0.934829
$947$	30216.0	1.03684	0.518420	−	0.855126i	$-0.326520\pi$
$947$	30216.0	1.03684	0.518420	+	0.855126i	$0.326520\pi$
$948$	0	0
$949$	−18450.0	−0.631098
$950$	1697.06	0.0579577
$951$	0	0
$952$	0	0
$953$	35512.0	1.20708	0.603540	−	0.797333i	$-0.293757\pi$
$953$	35512.0	1.20708	0.603540	+	0.797333i	$0.293757\pi$
$954$	0	0
$955$	−28142.8	−0.953593
$956$	−6960.00	−0.235463
$957$	0	0
$958$	−33036.0	−1.11414
$959$	0	0
$960$	0	0
$961$	−15679.0	−0.526300
$962$	−2545.58	−0.0853149
$963$	0	0
$964$	5040.26	0.168398
$965$	−26233.7	−0.875121
$966$	0	0
$967$	−720.000	−0.0239438	−0.0119719	−	0.999928i	$-0.503811\pi$
$967$	−720.000	−0.0239438	−0.0119719	+	0.999928i	$0.503811\pi$
$968$	−2152.00	−0.0714544
$969$	0	0
$970$	−7100.00	−0.235018
$971$	−15386.6	−0.508528	−0.254264	−	0.967135i	$-0.581833\pi$
$971$	−15386.6	−0.508528	−0.254264	+	0.967135i	$0.581833\pi$
$972$	0	0
$973$	0	0
$974$	−27360.0	−0.900073
$975$	0	0
$976$	14685.2	0.481620
$977$	−41574.0	−1.36138	−0.680691	−	0.732571i	$-0.738320\pi$
$977$	−41574.0	−1.36138	−0.680691	+	0.732571i	$0.738320\pi$
$978$	0	0
$979$	46103.4	1.50508
$980$	0	0
$981$	0	0
$982$	−4560.00	−0.148183
$983$	44236.6	1.43533	0.717665	−	0.696389i	$-0.245211\pi$
$983$	44236.6	1.43533	0.717665	+	0.696389i	$0.245211\pi$
$984$	0	0
$985$	−6759.94	−0.218670
$986$	−311.127	−0.0100490
$987$	0	0
$988$	−2880.00	−0.0927379
$989$	−23120.0	−0.743350
$990$	0	0
$991$	12272.0	0.393373	0.196687	−	0.980466i	$-0.436982\pi$
$991$	12272.0	0.393373	0.196687	+	0.980466i	$0.436982\pi$
$992$	3801.41	0.121668
$993$	0	0
$994$	0	0
$995$	−28920.0	−0.921433
$996$	0	0
$997$	57692.8	1.83265	0.916324	−	0.400437i	$-0.131142\pi$
$997$	57692.8	1.83265	0.916324	+	0.400437i	$0.131142\pi$
$998$	−1720.00	−0.0545548
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -7.07107 q^{5} -8.00000 q^{8} +14.1421 q^{10} +40.0000 q^{11} -63.6396 q^{13} +16.0000 q^{16} +1.41421 q^{17} +11.3137 q^{19} -28.2843 q^{20} -80.0000 q^{22} +68.0000 q^{23} -75.0000 q^{25} +127.279 q^{26} +110.000 q^{29} -118.794 q^{31} -32.0000 q^{32} -2.82843 q^{34} -20.0000 q^{37} -22.6274 q^{38} +56.5685 q^{40} -49.4975 q^{41} -340.000 q^{43} +160.000 q^{44} -136.000 q^{46} +90.5097 q^{47} +150.000 q^{50} -254.558 q^{52} +628.000 q^{53} -282.843 q^{55} -220.000 q^{58} -876.812 q^{59} +917.825 q^{61} +237.588 q^{62} +64.0000 q^{64} +450.000 q^{65} +540.000 q^{67} +5.65685 q^{68} -420.000 q^{71} +289.914 q^{73} +40.0000 q^{74} +45.2548 q^{76} -760.000 q^{79} -113.137 q^{80} +98.9949 q^{82} +944.695 q^{83} -10.0000 q^{85} +680.000 q^{86} -320.000 q^{88} +1152.58 q^{89} +272.000 q^{92} -181.019 q^{94} -80.0000 q^{95} -502.046 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 80 q^{11} + 32 q^{16} - 160 q^{22} + 136 q^{23} - 150 q^{25} + 220 q^{29} - 64 q^{32} - 40 q^{37} - 680 q^{43} + 320 q^{44} - 272 q^{46} + 300 q^{50} + 1256 q^{53} - 440 q^{58}+ \cdots - 160 q^{95}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 80 * q^11 + 32 * q^16 - 160 * q^22 + 136 * q^23 - 150 * q^25 + 220 * q^29 - 64 * q^32 - 40 * q^37 - 680 * q^43 + 320 * q^44 - 272 * q^46 + 300 * q^50 + 1256 * q^53 - 440 * q^58 + 128 * q^64 + 900 * q^65 + 1080 * q^67 - 840 * q^71 + 80 * q^74 - 1520 * q^79 - 20 * q^85 + 1360 * q^86 - 640 * q^88 + 544 * q^92 - 160 * q^95

Introduction

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