LMFDB - Embedded newform 882.4.a.bg.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:	no (minimal twist has level 98)
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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} -39.5980 q^{10} +14.0000 q^{11} -50.9117 q^{13} +16.0000 q^{16} +1.41421 q^{17} +1.41421 q^{19} -79.1960 q^{20} +28.0000 q^{22} -140.000 q^{23} +267.000 q^{25} -101.823 q^{26} +286.000 q^{29} +93.3381 q^{31} +32.0000 q^{32} +2.82843 q^{34} -38.0000 q^{37} +2.82843 q^{38} -158.392 q^{40} -125.865 q^{41} -34.0000 q^{43} +56.0000 q^{44} -280.000 q^{46} +523.259 q^{47} +534.000 q^{50} -203.647 q^{52} +74.0000 q^{53} -277.186 q^{55} +572.000 q^{58} +434.164 q^{59} -14.1421 q^{61} +186.676 q^{62} +64.0000 q^{64} +1008.00 q^{65} +684.000 q^{67} +5.65685 q^{68} -588.000 q^{71} +270.115 q^{73} -76.0000 q^{74} +5.65685 q^{76} +1220.00 q^{79} -316.784 q^{80} -251.730 q^{82} +422.850 q^{83} -28.0000 q^{85} -68.0000 q^{86} +112.000 q^{88} +618.011 q^{89} -560.000 q^{92} +1046.52 q^{94} -28.0000 q^{95} -1483.51 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 28 q^{11} + 32 q^{16} + 56 q^{22} - 280 q^{23} + 534 q^{25} + 572 q^{29} + 64 q^{32} - 76 q^{37} - 68 q^{43} + 112 q^{44} - 560 q^{46} + 1068 q^{50} + 148 q^{53} + 1144 q^{58} + 128 q^{64} + 2016 q^{65} + 1368 q^{67} - 1176 q^{71} - 152 q^{74} + 2440 q^{79} - 56 q^{85} - 136 q^{86} + 224 q^{88} - 1120 q^{92} - 56 q^{95}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 28 * q^11 + 32 * q^16 + 56 * q^22 - 280 * q^23 + 534 * q^25 + 572 * q^29 + 64 * q^32 - 76 * q^37 - 68 * q^43 + 112 * q^44 - 560 * q^46 + 1068 * q^50 + 148 * q^53 + 1144 * q^58 + 128 * q^64 + 2016 * q^65 + 1368 * q^67 - 1176 * q^71 - 152 * q^74 + 2440 * q^79 - 56 * q^85 - 136 * q^86 + 224 * q^88 - 1120 * q^92 - 56 * q^95

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−19.7990	−1.77088	−0.885438	−	0.464758i	$-0.846141\pi$
$5$	−19.7990	−1.77088	−0.885438	+	0.464758i	$0.846141\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−39.5980	−1.25220
$11$	14.0000	0.383742	0.191871	−	0.981420i	$-0.438545\pi$
$11$	14.0000	0.383742	0.191871	+	0.981420i	$0.438545\pi$
$12$	0	0
$13$	−50.9117	−1.08618	−0.543091	−	0.839674i	$-0.682746\pi$
$13$	−50.9117	−1.08618	−0.543091	+	0.839674i	$0.682746\pi$
$14$	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
$19$	1.41421	0.0170759	0.00853797	−	0.999964i	$-0.497282\pi$
$19$	1.41421	0.0170759	0.00853797	+	0.999964i	$0.497282\pi$
$20$	−79.1960	−0.885438
$21$	0	0
$22$	28.0000	0.271346
$23$	−140.000	−1.26922	−0.634609	−	0.772833i	$-0.718839\pi$
$23$	−140.000	−1.26922	−0.634609	+	0.772833i	$0.718839\pi$
$24$	0	0
$25$	267.000	2.13600
$26$	−101.823	−0.768046
$27$	0	0
$28$	0	0
$29$	286.000	1.83134	0.915670	−	0.401931i	$-0.131661\pi$
$29$	286.000	1.83134	0.915670	+	0.401931i	$0.131661\pi$
$30$	0	0
$31$	93.3381	0.540775	0.270387	−	0.962752i	$-0.412848\pi$
$31$	93.3381	0.540775	0.270387	+	0.962752i	$0.412848\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	2.82843	0.0142668
$35$	0	0
$36$	0	0
$37$	−38.0000	−0.168842	−0.0844211	−	0.996430i	$-0.526904\pi$
$37$	−38.0000	−0.168842	−0.0844211	+	0.996430i	$0.526904\pi$
$38$	2.82843	0.0120745
$39$	0	0
$40$	−158.392	−0.626099
$41$	−125.865	−0.479434	−0.239717	−	0.970843i	$-0.577055\pi$
$41$	−125.865	−0.479434	−0.239717	+	0.970843i	$0.577055\pi$
$42$	0	0
$43$	−34.0000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$43$	−34.0000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$44$	56.0000	0.191871
$45$	0	0
$46$	−280.000	−0.897473
$47$	523.259	1.62394	0.811970	−	0.583699i	$-0.198395\pi$
$47$	523.259	1.62394	0.811970	+	0.583699i	$0.198395\pi$
$48$	0	0
$49$	0	0
$50$	534.000	1.51038
$51$	0	0
$52$	−203.647	−0.543091
$53$	74.0000	0.191786	0.0958932	−	0.995392i	$-0.469429\pi$
$53$	74.0000	0.191786	0.0958932	+	0.995392i	$0.469429\pi$
$54$	0	0
$55$	−277.186	−0.679559
$56$	0	0
$57$	0	0
$58$	572.000	1.29495
$59$	434.164	0.958022	0.479011	−	0.877809i	$-0.340995\pi$
$59$	434.164	0.958022	0.479011	+	0.877809i	$0.340995\pi$
$60$	0	0
$61$	−14.1421	−0.0296839	−0.0148419	−	0.999890i	$-0.504725\pi$
$61$	−14.1421	−0.0296839	−0.0148419	+	0.999890i	$0.504725\pi$
$62$	186.676	0.382385
$63$	0	0
$64$	64.0000	0.125000
$65$	1008.00	1.92349
$66$	0	0
$67$	684.000	1.24722	0.623611	−	0.781735i	$-0.285665\pi$
$67$	684.000	1.24722	0.623611	+	0.781735i	$0.285665\pi$
$68$	5.65685	0.0100882
$69$	0	0
$70$	0	0
$71$	−588.000	−0.982856	−0.491428	−	0.870918i	$-0.663525\pi$
$71$	−588.000	−0.982856	−0.491428	+	0.870918i	$0.663525\pi$
$72$	0	0
$73$	270.115	0.433076	0.216538	−	0.976274i	$-0.430523\pi$
$73$	270.115	0.433076	0.216538	+	0.976274i	$0.430523\pi$
$74$	−76.0000	−0.119389
$75$	0	0
$76$	5.65685	0.00853797
$77$	0	0
$78$	0	0
$79$	1220.00	1.73748	0.868739	−	0.495271i	$-0.164931\pi$
$79$	1220.00	1.73748	0.868739	+	0.495271i	$0.164931\pi$
$80$	−316.784	−0.442719
$81$	0	0
$82$	−251.730	−0.339011
$83$	422.850	0.559202	0.279601	−	0.960116i	$-0.409798\pi$
$83$	422.850	0.559202	0.279601	+	0.960116i	$0.409798\pi$
$84$	0	0
$85$	−28.0000	−0.0357297
$86$	−68.0000	−0.0852631
$87$	0	0
$88$	112.000	0.135673
$89$	618.011	0.736057	0.368028	−	0.929815i	$-0.380033\pi$
$89$	618.011	0.736057	0.368028	+	0.929815i	$0.380033\pi$
$90$	0	0
$91$	0	0
$92$	−560.000	−0.634609
$93$	0	0
$94$	1046.52	1.14830
$95$	−28.0000	−0.0302394
$96$	0	0
$97$	−1483.51	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$97$	−1483.51	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$98$	0	0
$99$	0	0
$100$	1068.00	1.06800
$101$	1128.54	1.11182	0.555912	−	0.831241i	$-0.312369\pi$
$101$	1128.54	1.11182	0.555912	+	0.831241i	$0.312369\pi$
$102$	0	0
$103$	868.327	0.830668	0.415334	−	0.909669i	$-0.363665\pi$
$103$	868.327	0.830668	0.415334	+	0.909669i	$0.363665\pi$
$104$	−407.294	−0.384023
$105$	0	0
$106$	148.000	0.135613
$107$	1684.00	1.52148	0.760740	−	0.649056i	$-0.224836\pi$
$107$	1684.00	1.52148	0.760740	+	0.649056i	$0.224836\pi$
$108$	0	0
$109$	−818.000	−0.718809	−0.359405	−	0.933182i	$-0.617020\pi$
$109$	−818.000	−0.718809	−0.359405	+	0.933182i	$0.617020\pi$
$110$	−554.372	−0.480521
$111$	0	0
$112$	0	0
$113$	540.000	0.449548	0.224774	−	0.974411i	$-0.427836\pi$
$113$	540.000	0.449548	0.224774	+	0.974411i	$0.427836\pi$
$114$	0	0
$115$	2771.86	2.24763
$116$	1144.00	0.915670
$117$	0	0
$118$	868.327	0.677424
$119$	0	0
$120$	0	0
$121$	−1135.00	−0.852742
$122$	−28.2843	−0.0209897
$123$	0	0
$124$	373.352	0.270387
$125$	−2811.46	−2.01171
$126$	0	0
$127$	1720.00	1.20177	0.600887	−	0.799334i	$-0.294814\pi$
$127$	1720.00	1.20177	0.600887	+	0.799334i	$0.294814\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	2016.00	1.36011
$131$	−1735.24	−1.15732	−0.578659	−	0.815570i	$-0.696424\pi$
$131$	−1735.24	−1.15732	−0.578659	+	0.815570i	$0.696424\pi$
$132$	0	0
$133$	0	0
$134$	1368.00	0.881919
$135$	0	0
$136$	11.3137	0.00713340
$137$	−828.000	−0.516356	−0.258178	−	0.966097i	$-0.583122\pi$
$137$	−828.000	−0.516356	−0.258178	+	0.966097i	$0.583122\pi$
$138$	0	0
$139$	425.678	0.259752	0.129876	−	0.991530i	$-0.458542\pi$
$139$	425.678	0.259752	0.129876	+	0.991530i	$0.458542\pi$
$140$	0	0
$141$	0	0
$142$	−1176.00	−0.694984
$143$	−712.764	−0.416813
$144$	0	0
$145$	−5662.51	−3.24308
$146$	540.230	0.306231
$147$	0	0
$148$	−152.000	−0.0844211
$149$	−2050.00	−1.12713	−0.563566	−	0.826071i	$-0.690571\pi$
$149$	−2050.00	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$150$	0	0
$151$	−472.000	−0.254376	−0.127188	−	0.991879i	$-0.540595\pi$
$151$	−472.000	−0.254376	−0.127188	+	0.991879i	$0.540595\pi$
$152$	11.3137	0.00603726
$153$	0	0
$154$	0	0
$155$	−1848.00	−0.957645
$156$	0	0
$157$	2211.83	1.12435	0.562176	−	0.827018i	$-0.309964\pi$
$157$	2211.83	1.12435	0.562176	+	0.827018i	$0.309964\pi$
$158$	2440.00	1.22858
$159$	0	0
$160$	−633.568	−0.313050
$161$	0	0
$162$	0	0
$163$	3286.00	1.57901	0.789507	−	0.613741i	$-0.210336\pi$
$163$	3286.00	1.57901	0.789507	+	0.613741i	$0.210336\pi$
$164$	−503.460	−0.239717
$165$	0	0
$166$	845.700	0.395416
$167$	−1490.58	−0.690686	−0.345343	−	0.938476i	$-0.612237\pi$
$167$	−1490.58	−0.690686	−0.345343	+	0.938476i	$0.612237\pi$
$168$	0	0
$169$	395.000	0.179791
$170$	−56.0000	−0.0252647
$171$	0	0
$172$	−136.000	−0.0602901
$173$	−2070.41	−0.909886	−0.454943	−	0.890521i	$-0.650340\pi$
$173$	−2070.41	−0.909886	−0.454943	+	0.890521i	$0.650340\pi$
$174$	0	0
$175$	0	0
$176$	224.000	0.0959354
$177$	0	0
$178$	1236.02	0.520471
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	0	0
$181$	−3784.44	−1.55412	−0.777058	−	0.629429i	$-0.783289\pi$
$181$	−3784.44	−1.55412	−0.777058	+	0.629429i	$0.783289\pi$
$182$	0	0
$183$	0	0
$184$	−1120.00	−0.448736
$185$	752.362	0.298999
$186$	0	0
$187$	19.7990	0.00774249
$188$	2093.04	0.811970
$189$	0	0
$190$	−56.0000	−0.0213825
$191$	−1028.00	−0.389442	−0.194721	−	0.980859i	$-0.562380\pi$
$191$	−1028.00	−0.389442	−0.194721	+	0.980859i	$0.562380\pi$
$192$	0	0
$193$	4592.00	1.71264	0.856320	−	0.516446i	$-0.172745\pi$
$193$	4592.00	1.71264	0.856320	+	0.516446i	$0.172745\pi$
$194$	−2967.02	−1.09804
$195$	0	0
$196$	0	0
$197$	−794.000	−0.287158	−0.143579	−	0.989639i	$-0.545861\pi$
$197$	−794.000	−0.287158	−0.143579	+	0.989639i	$0.545861\pi$
$198$	0	0
$199$	−2486.19	−0.885634	−0.442817	−	0.896612i	$-0.646021\pi$
$199$	−2486.19	−0.885634	−0.442817	+	0.896612i	$0.646021\pi$
$200$	2136.00	0.755190
$201$	0	0
$202$	2257.08	0.786178
$203$	0	0
$204$	0	0
$205$	2492.00	0.849019
$206$	1736.65	0.587371
$207$	0	0
$208$	−814.587	−0.271545
$209$	19.7990	0.00655275
$210$	0	0
$211$	−2748.00	−0.896588	−0.448294	−	0.893886i	$-0.647968\pi$
$211$	−2748.00	−0.896588	−0.448294	+	0.893886i	$0.647968\pi$
$212$	296.000	0.0958932
$213$	0	0
$214$	3368.00	1.07585
$215$	673.166	0.213533
$216$	0	0
$217$	0	0
$218$	−1636.00	−0.508275
$219$	0	0
$220$	−1108.74	−0.339779
$221$	−72.0000	−0.0219151
$222$	0	0
$223$	3428.05	1.02941	0.514707	−	0.857366i	$-0.327901\pi$
$223$	3428.05	1.02941	0.514707	+	0.857366i	$0.327901\pi$
$224$	0	0
$225$	0	0
$226$	1080.00	0.317878
$227$	5290.57	1.54691	0.773453	−	0.633854i	$-0.218528\pi$
$227$	5290.57	1.54691	0.773453	+	0.633854i	$0.218528\pi$
$228$	0	0
$229$	2749.23	0.793338	0.396669	−	0.917962i	$-0.370166\pi$
$229$	2749.23	0.793338	0.396669	+	0.917962i	$0.370166\pi$
$230$	5543.72	1.58931
$231$	0	0
$232$	2288.00	0.647477
$233$	−72.0000	−0.0202441	−0.0101221	−	0.999949i	$-0.503222\pi$
$233$	−72.0000	−0.0202441	−0.0101221	+	0.999949i	$0.503222\pi$
$234$	0	0
$235$	−10360.0	−2.87580
$236$	1736.65	0.479011
$237$	0	0
$238$	0	0
$239$	−4308.00	−1.16595	−0.582974	−	0.812491i	$-0.698111\pi$
$239$	−4308.00	−1.16595	−0.582974	+	0.812491i	$0.698111\pi$
$240$	0	0
$241$	1540.08	0.411640	0.205820	−	0.978590i	$-0.434014\pi$
$241$	1540.08	0.411640	0.205820	+	0.978590i	$0.434014\pi$
$242$	−2270.00	−0.602980
$243$	0	0
$244$	−56.5685	−0.0148419
$245$	0	0
$246$	0	0
$247$	−72.0000	−0.0185476
$248$	746.705	0.191193
$249$	0	0
$250$	−5622.91	−1.42250
$251$	931.967	0.234363	0.117182	−	0.993110i	$-0.462614\pi$
$251$	931.967	0.234363	0.117182	+	0.993110i	$0.462614\pi$
$252$	0	0
$253$	−1960.00	−0.487052
$254$	3440.00	0.849783
$255$	0	0
$256$	256.000	0.0625000
$257$	937.624	0.227577	0.113789	−	0.993505i	$-0.463701\pi$
$257$	937.624	0.227577	0.113789	+	0.993505i	$0.463701\pi$
$258$	0	0
$259$	0	0
$260$	4032.00	0.961746
$261$	0	0
$262$	−3470.48	−0.818347
$263$	−7140.00	−1.67404	−0.837018	−	0.547176i	$-0.815703\pi$
$263$	−7140.00	−1.67404	−0.837018	+	0.547176i	$0.815703\pi$
$264$	0	0
$265$	−1465.13	−0.339630
$266$	0	0
$267$	0	0
$268$	2736.00	0.623611
$269$	4610.34	1.04497	0.522485	−	0.852648i	$-0.325005\pi$
$269$	4610.34	1.04497	0.522485	+	0.852648i	$0.325005\pi$
$270$	0	0
$271$	2364.57	0.530026	0.265013	−	0.964245i	$-0.414624\pi$
$271$	2364.57	0.530026	0.265013	+	0.964245i	$0.414624\pi$
$272$	22.6274	0.00504408
$273$	0	0
$274$	−1656.00	−0.365119
$275$	3738.00	0.819672
$276$	0	0
$277$	4006.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$277$	4006.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$278$	851.357	0.183673
$279$	0	0
$280$	0	0
$281$	5984.00	1.27038	0.635188	−	0.772358i	$-0.280923\pi$
$281$	5984.00	1.27038	0.635188	+	0.772358i	$0.280923\pi$
$282$	0	0
$283$	4928.53	1.03523	0.517617	−	0.855613i	$-0.326819\pi$
$283$	4928.53	1.03523	0.517617	+	0.855613i	$0.326819\pi$
$284$	−2352.00	−0.491428
$285$	0	0
$286$	−1425.53	−0.294731
$287$	0	0
$288$	0	0
$289$	−4911.00	−0.999593
$290$	−11325.0	−2.29320
$291$	0	0
$292$	1080.46	0.216538
$293$	1971.41	0.393076	0.196538	−	0.980496i	$-0.437030\pi$
$293$	1971.41	0.393076	0.196538	+	0.980496i	$0.437030\pi$
$294$	0	0
$295$	−8596.00	−1.69654
$296$	−304.000	−0.0596947
$297$	0	0
$298$	−4100.00	−0.797002
$299$	7127.64	1.37860
$300$	0	0
$301$	0	0
$302$	−944.000	−0.179871
$303$	0	0
$304$	22.6274	0.00426898
$305$	280.000	0.0525664
$306$	0	0
$307$	4767.31	0.886270	0.443135	−	0.896455i	$-0.353866\pi$
$307$	4767.31	0.886270	0.443135	+	0.896455i	$0.353866\pi$
$308$	0	0
$309$	0	0
$310$	−3696.00	−0.677157
$311$	−6776.91	−1.23564	−0.617819	−	0.786320i	$-0.711984\pi$
$311$	−6776.91	−1.23564	−0.617819	+	0.786320i	$0.711984\pi$
$312$	0	0
$313$	6190.01	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$313$	6190.01	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$314$	4423.66	0.795037
$315$	0	0
$316$	4880.00	0.868739
$317$	−9826.00	−1.74096	−0.870478	−	0.492207i	$-0.836190\pi$
$317$	−9826.00	−1.74096	−0.870478	+	0.492207i	$0.836190\pi$
$318$	0	0
$319$	4004.00	0.702762
$320$	−1267.14	−0.221359
$321$	0	0
$322$	0	0
$323$	2.00000	0.000344529 0
$324$	0	0
$325$	−13593.4	−2.32008
$326$	6572.00	1.11653
$327$	0	0
$328$	−1006.92	−0.169506
$329$	0	0
$330$	0	0
$331$	−5738.00	−0.952837	−0.476418	−	0.879219i	$-0.658065\pi$
$331$	−5738.00	−0.952837	−0.476418	+	0.879219i	$0.658065\pi$
$332$	1691.40	0.279601
$333$	0	0
$334$	−2981.16	−0.488389
$335$	−13542.5	−2.20868
$336$	0	0
$337$	−2254.00	−0.364342	−0.182171	−	0.983267i	$-0.558312\pi$
$337$	−2254.00	−0.364342	−0.182171	+	0.983267i	$0.558312\pi$
$338$	790.000	0.127131
$339$	0	0
$340$	−112.000	−0.0178649
$341$	1306.73	0.207518
$342$	0	0
$343$	0	0
$344$	−272.000	−0.0426316
$345$	0	0
$346$	−4140.82	−0.643386
$347$	1986.00	0.307245	0.153623	−	0.988130i	$-0.450906\pi$
$347$	1986.00	0.307245	0.153623	+	0.988130i	$0.450906\pi$
$348$	0	0
$349$	−6771.25	−1.03856	−0.519279	−	0.854605i	$-0.673800\pi$
$349$	−6771.25	−1.03856	−0.519279	+	0.854605i	$0.673800\pi$
$350$	0	0
$351$	0	0
$352$	448.000	0.0678366
$353$	6993.29	1.05443	0.527217	−	0.849731i	$-0.323236\pi$
$353$	6993.29	1.05443	0.527217	+	0.849731i	$0.323236\pi$
$354$	0	0
$355$	11641.8	1.74052
$356$	2472.05	0.368028
$357$	0	0
$358$	−1080.00	−0.159441
$359$	−5944.00	−0.873850	−0.436925	−	0.899498i	$-0.643933\pi$
$359$	−5944.00	−0.873850	−0.436925	+	0.899498i	$0.643933\pi$
$360$	0	0
$361$	−6857.00	−0.999708
$362$	−7568.87	−1.09893
$363$	0	0
$364$	0	0
$365$	−5348.00	−0.766924
$366$	0	0
$367$	842.871	0.119884	0.0599421	−	0.998202i	$-0.480908\pi$
$367$	842.871	0.119884	0.0599421	+	0.998202i	$0.480908\pi$
$368$	−2240.00	−0.317305
$369$	0	0
$370$	1504.72	0.211424
$371$	0	0
$372$	0	0
$373$	−5726.00	−0.794855	−0.397428	−	0.917634i	$-0.630097\pi$
$373$	−5726.00	−0.794855	−0.397428	+	0.917634i	$0.630097\pi$
$374$	39.5980	0.00547477
$375$	0	0
$376$	4186.07	0.574149
$377$	−14560.7	−1.98917
$378$	0	0
$379$	10330.0	1.40004	0.700022	−	0.714122i	$-0.253174\pi$
$379$	10330.0	1.40004	0.700022	+	0.714122i	$0.253174\pi$
$380$	−112.000	−0.0151197
$381$	0	0
$382$	−2056.00	−0.275377
$383$	1004.09	0.133960	0.0669800	−	0.997754i	$-0.478664\pi$
$383$	1004.09	0.133960	0.0669800	+	0.997754i	$0.478664\pi$
$384$	0	0
$385$	0	0
$386$	9184.00	1.21102
$387$	0	0
$388$	−5934.04	−0.776431
$389$	−5210.00	−0.679068	−0.339534	−	0.940594i	$-0.610269\pi$
$389$	−5210.00	−0.679068	−0.339534	+	0.940594i	$0.610269\pi$
$390$	0	0
$391$	−197.990	−0.0256081
$392$	0	0
$393$	0	0
$394$	−1588.00	−0.203051
$395$	−24154.8	−3.07686
$396$	0	0
$397$	73.5391	0.00929678	0.00464839	−	0.999989i	$-0.498520\pi$
$397$	73.5391	0.00929678	0.00464839	+	0.999989i	$0.498520\pi$
$398$	−4972.37	−0.626238
$399$	0	0
$400$	4272.00	0.534000
$401$	498.000	0.0620173	0.0310086	−	0.999519i	$-0.490128\pi$
$401$	498.000	0.0620173	0.0310086	+	0.999519i	$0.490128\pi$
$402$	0	0
$403$	−4752.00	−0.587380
$404$	4514.17	0.555912
$405$	0	0
$406$	0	0
$407$	−532.000	−0.0647918
$408$	0	0
$409$	−3355.93	−0.405721	−0.202861	−	0.979208i	$-0.565024\pi$
$409$	−3355.93	−0.405721	−0.202861	+	0.979208i	$0.565024\pi$
$410$	4984.00	0.600347
$411$	0	0
$412$	3473.31	0.415334
$413$	0	0
$414$	0	0
$415$	−8372.00	−0.990278
$416$	−1629.17	−0.192012
$417$	0	0
$418$	39.5980	0.00463349
$419$	14545.2	1.69589	0.847946	−	0.530082i	$-0.177839\pi$
$419$	14545.2	1.69589	0.847946	+	0.530082i	$0.177839\pi$
$420$	0	0
$421$	10854.0	1.25651	0.628256	−	0.778007i	$-0.283769\pi$
$421$	10854.0	1.25651	0.628256	+	0.778007i	$0.283769\pi$
$422$	−5496.00	−0.633984
$423$	0	0
$424$	592.000	0.0678067
$425$	377.595	0.0430966
$426$	0	0
$427$	0	0
$428$	6736.00	0.760740
$429$	0	0
$430$	1346.33	0.150990
$431$	5364.00	0.599477	0.299739	−	0.954021i	$-0.403100\pi$
$431$	5364.00	0.599477	0.299739	+	0.954021i	$0.403100\pi$
$432$	0	0
$433$	6487.00	0.719966	0.359983	−	0.932959i	$-0.382783\pi$
$433$	6487.00	0.719966	0.359983	+	0.932959i	$0.382783\pi$
$434$	0	0
$435$	0	0
$436$	−3272.00	−0.359405
$437$	−197.990	−0.0216731
$438$	0	0
$439$	−13932.8	−1.51476	−0.757378	−	0.652977i	$-0.773520\pi$
$439$	−13932.8	−1.51476	−0.757378	+	0.652977i	$0.773520\pi$
$440$	−2217.49	−0.240260
$441$	0	0
$442$	−144.000	−0.0154963
$443$	−5996.00	−0.643067	−0.321533	−	0.946898i	$-0.604198\pi$
$443$	−5996.00	−0.643067	−0.321533	+	0.946898i	$0.604198\pi$
$444$	0	0
$445$	−12236.0	−1.30347
$446$	6856.11	0.727906
$447$	0	0
$448$	0	0
$449$	−2622.00	−0.275590	−0.137795	−	0.990461i	$-0.544001\pi$
$449$	−2622.00	−0.275590	−0.137795	+	0.990461i	$0.544001\pi$
$450$	0	0
$451$	−1762.11	−0.183979
$452$	2160.00	0.224774
$453$	0	0
$454$	10581.1	1.09383
$455$	0	0
$456$	0	0
$457$	11208.0	1.14724	0.573619	−	0.819122i	$-0.305539\pi$
$457$	11208.0	1.14724	0.573619	+	0.819122i	$0.305539\pi$
$458$	5498.46	0.560974
$459$	0	0
$460$	11087.4	1.12381
$461$	9786.36	0.988712	0.494356	−	0.869260i	$-0.335404\pi$
$461$	9786.36	0.988712	0.494356	+	0.869260i	$0.335404\pi$
$462$	0	0
$463$	3952.00	0.396685	0.198342	−	0.980133i	$-0.436444\pi$
$463$	3952.00	0.396685	0.198342	+	0.980133i	$0.436444\pi$
$464$	4576.00	0.457835
$465$	0	0
$466$	−144.000	−0.0143147
$467$	17506.5	1.73470	0.867352	−	0.497696i	$-0.165820\pi$
$467$	17506.5	1.73470	0.867352	+	0.497696i	$0.165820\pi$
$468$	0	0
$469$	0	0
$470$	−20720.0	−2.03349
$471$	0	0
$472$	3473.31	0.338712
$473$	−476.000	−0.0462717
$474$	0	0
$475$	377.595	0.0364742
$476$	0	0
$477$	0	0
$478$	−8616.00	−0.824449
$479$	2288.20	0.218268	0.109134	−	0.994027i	$-0.465192\pi$
$479$	2288.20	0.218268	0.109134	+	0.994027i	$0.465192\pi$
$480$	0	0
$481$	1934.64	0.183393
$482$	3080.16	0.291073
$483$	0	0
$484$	−4540.00	−0.426371
$485$	29372.0	2.74993
$486$	0	0
$487$	972.000	0.0904426	0.0452213	−	0.998977i	$-0.485601\pi$
$487$	972.000	0.0904426	0.0452213	+	0.998977i	$0.485601\pi$
$488$	−113.137	−0.0104948
$489$	0	0
$490$	0	0
$491$	7404.00	0.680525	0.340263	−	0.940330i	$-0.389484\pi$
$491$	7404.00	0.680525	0.340263	+	0.940330i	$0.389484\pi$
$492$	0	0
$493$	404.465	0.0369497
$494$	−144.000	−0.0131151
$495$	0	0
$496$	1493.41	0.135194
$497$	0	0
$498$	0	0
$499$	−12244.0	−1.09843	−0.549215	−	0.835681i	$-0.685073\pi$
$499$	−12244.0	−1.09843	−0.549215	+	0.835681i	$0.685073\pi$
$500$	−11245.8	−1.00586
$501$	0	0
$502$	1863.93	0.165720
$503$	−2415.48	−0.214117	−0.107058	−	0.994253i	$-0.534143\pi$
$503$	−2415.48	−0.214117	−0.107058	+	0.994253i	$0.534143\pi$
$504$	0	0
$505$	−22344.0	−1.96890
$506$	−3920.00	−0.344398
$507$	0	0
$508$	6880.00	0.600887
$509$	−5707.77	−0.497038	−0.248519	−	0.968627i	$-0.579944\pi$
$509$	−5707.77	−0.497038	−0.248519	+	0.968627i	$0.579944\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	1875.25	0.160921
$515$	−17192.0	−1.47101
$516$	0	0
$517$	7325.63	0.623173
$518$	0	0
$519$	0	0
$520$	8064.00	0.680057
$521$	1.41421	0.000118921 0	5.94605e−5	−	1.00000i	$-0.499981\pi$
$521$	1.41421	0.000118921 0	5.94605e−5		1.00000i	$0.499981\pi$
$522$	0	0
$523$	12257.0	1.02478	0.512391	−	0.858752i	$-0.328760\pi$
$523$	12257.0	1.02478	0.512391	+	0.858752i	$0.328760\pi$
$524$	−6940.96	−0.578659
$525$	0	0
$526$	−14280.0	−1.18372
$527$	132.000	0.0109108
$528$	0	0
$529$	7433.00	0.610915
$530$	−2930.25	−0.240155
$531$	0	0
$532$	0	0
$533$	6408.00	0.520753
$534$	0	0
$535$	−33341.5	−2.69435
$536$	5472.00	0.440960
$537$	0	0
$538$	9220.67	0.738906
$539$	0	0
$540$	0	0
$541$	2050.00	0.162914	0.0814569	−	0.996677i	$-0.474043\pi$
$541$	2050.00	0.162914	0.0814569	+	0.996677i	$0.474043\pi$
$542$	4729.13	0.374785
$543$	0	0
$544$	45.2548	0.00356670
$545$	16195.6	1.27292
$546$	0	0
$547$	14554.0	1.13763	0.568815	−	0.822465i	$-0.307402\pi$
$547$	14554.0	1.13763	0.568815	+	0.822465i	$0.307402\pi$
$548$	−3312.00	−0.258178
$549$	0	0
$550$	7476.00	0.579596
$551$	404.465	0.0312719
$552$	0	0
$553$	0	0
$554$	8012.00	0.614435
$555$	0	0
$556$	1702.71	0.129876
$557$	−6954.00	−0.528995	−0.264498	−	0.964386i	$-0.585206\pi$
$557$	−6954.00	−0.528995	−0.264498	+	0.964386i	$0.585206\pi$
$558$	0	0
$559$	1731.00	0.130972
$560$	0	0
$561$	0	0
$562$	11968.0	0.898291
$563$	−1636.25	−0.122486	−0.0612429	−	0.998123i	$-0.519506\pi$
$563$	−1636.25	−0.122486	−0.0612429	+	0.998123i	$0.519506\pi$
$564$	0	0
$565$	−10691.5	−0.796094
$566$	9857.07	0.732020
$567$	0	0
$568$	−4704.00	−0.347492
$569$	7142.00	0.526201	0.263100	−	0.964768i	$-0.415255\pi$
$569$	7142.00	0.526201	0.263100	+	0.964768i	$0.415255\pi$
$570$	0	0
$571$	−20606.0	−1.51022	−0.755109	−	0.655599i	$-0.772416\pi$
$571$	−20606.0	−1.51022	−0.755109	+	0.655599i	$0.772416\pi$
$572$	−2851.05	−0.208407
$573$	0	0
$574$	0	0
$575$	−37380.0	−2.71105
$576$	0	0
$577$	8803.48	0.635171	0.317585	−	0.948230i	$-0.397128\pi$
$577$	8803.48	0.635171	0.317585	+	0.948230i	$0.397128\pi$
$578$	−9822.00	−0.706819
$579$	0	0
$580$	−22650.0	−1.62154
$581$	0	0
$582$	0	0
$583$	1036.00	0.0735965
$584$	2160.92	0.153115
$585$	0	0
$586$	3942.83	0.277947
$587$	6503.97	0.457321	0.228661	−	0.973506i	$-0.426565\pi$
$587$	6503.97	0.457321	0.228661	+	0.973506i	$0.426565\pi$
$588$	0	0
$589$	132.000	0.00923424
$590$	−17192.0	−1.19963
$591$	0	0
$592$	−608.000	−0.0422106
$593$	−23140.8	−1.60249	−0.801246	−	0.598335i	$-0.795829\pi$
$593$	−23140.8	−1.60249	−0.801246	+	0.598335i	$0.795829\pi$
$594$	0	0
$595$	0	0
$596$	−8200.00	−0.563566
$597$	0	0
$598$	14255.3	0.974818
$599$	11296.0	0.770521	0.385260	−	0.922808i	$-0.374112\pi$
$599$	11296.0	0.770521	0.385260	+	0.922808i	$0.374112\pi$
$600$	0	0
$601$	−8727.11	−0.592323	−0.296162	−	0.955138i	$-0.595707\pi$
$601$	−8727.11	−0.592323	−0.296162	+	0.955138i	$0.595707\pi$
$602$	0	0
$603$	0	0
$604$	−1888.00	−0.127188
$605$	22471.9	1.51010
$606$	0	0
$607$	−19736.8	−1.31975	−0.659877	−	0.751374i	$-0.729392\pi$
$607$	−19736.8	−1.31975	−0.659877	+	0.751374i	$0.729392\pi$
$608$	45.2548	0.00301863
$609$	0	0
$610$	560.000	0.0371701
$611$	−26640.0	−1.76389
$612$	0	0
$613$	16962.0	1.11760	0.558800	−	0.829302i	$-0.311262\pi$
$613$	16962.0	1.11760	0.558800	+	0.829302i	$0.311262\pi$
$614$	9534.63	0.626688
$615$	0	0
$616$	0	0
$617$	19034.0	1.24194	0.620972	−	0.783832i	$-0.286738\pi$
$617$	19034.0	1.24194	0.620972	+	0.783832i	$0.286738\pi$
$618$	0	0
$619$	18677.5	1.21278	0.606392	−	0.795166i	$-0.292616\pi$
$619$	18677.5	1.21278	0.606392	+	0.795166i	$0.292616\pi$
$620$	−7392.00	−0.478822
$621$	0	0
$622$	−13553.8	−0.873728
$623$	0	0
$624$	0	0
$625$	22289.0	1.42650
$626$	12380.0	0.790424
$627$	0	0
$628$	8847.32	0.562176
$629$	−53.7401	−0.00340661
$630$	0	0
$631$	−14716.0	−0.928423	−0.464211	−	0.885724i	$-0.653662\pi$
$631$	−14716.0	−0.928423	−0.464211	+	0.885724i	$0.653662\pi$
$632$	9760.00	0.614291
$633$	0	0
$634$	−19652.0	−1.23104
$635$	−34054.3	−2.12819
$636$	0	0
$637$	0	0
$638$	8008.00	0.496928
$639$	0	0
$640$	−2534.27	−0.156525
$641$	−4730.00	−0.291457	−0.145728	−	0.989325i	$-0.546553\pi$
$641$	−4730.00	−0.291457	−0.145728	+	0.989325i	$0.546553\pi$
$642$	0	0
$643$	−19056.5	−1.16877	−0.584383	−	0.811478i	$-0.698663\pi$
$643$	−19056.5	−1.16877	−0.584383	+	0.811478i	$0.698663\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.000243619 0
$647$	9342.29	0.567672	0.283836	−	0.958873i	$-0.408393\pi$
$647$	9342.29	0.567672	0.283836	+	0.958873i	$0.408393\pi$
$648$	0	0
$649$	6078.29	0.367633
$650$	−27186.8	−1.64055
$651$	0	0
$652$	13144.0	0.789507
$653$	−3774.00	−0.226169	−0.113084	−	0.993585i	$-0.536073\pi$
$653$	−3774.00	−0.226169	−0.113084	+	0.993585i	$0.536073\pi$
$654$	0	0
$655$	34356.0	2.04947
$656$	−2013.84	−0.119859
$657$	0	0
$658$	0	0
$659$	21150.0	1.25021	0.625104	−	0.780541i	$-0.285057\pi$
$659$	21150.0	1.25021	0.625104	+	0.780541i	$0.285057\pi$
$660$	0	0
$661$	10377.5	0.610647	0.305324	−	0.952249i	$-0.401235\pi$
$661$	10377.5	0.610647	0.305324	+	0.952249i	$0.401235\pi$
$662$	−11476.0	−0.673757
$663$	0	0
$664$	3382.80	0.197708
$665$	0	0
$666$	0	0
$667$	−40040.0	−2.32437
$668$	−5962.32	−0.345343
$669$	0	0
$670$	−27085.0	−1.56177
$671$	−197.990	−0.0113909
$672$	0	0
$673$	−1164.00	−0.0666700	−0.0333350	−	0.999444i	$-0.510613\pi$
$673$	−1164.00	−0.0666700	−0.0333350	+	0.999444i	$0.510613\pi$
$674$	−4508.00	−0.257629
$675$	0	0
$676$	1580.00	0.0898953
$677$	27152.9	1.54146	0.770732	−	0.637160i	$-0.219891\pi$
$677$	27152.9	1.54146	0.770732	+	0.637160i	$0.219891\pi$
$678$	0	0
$679$	0	0
$680$	−224.000	−0.0126324
$681$	0	0
$682$	2613.47	0.146737
$683$	16596.0	0.929763	0.464882	−	0.885373i	$-0.346097\pi$
$683$	16596.0	0.929763	0.464882	+	0.885373i	$0.346097\pi$
$684$	0	0
$685$	16393.6	0.914403
$686$	0	0
$687$	0	0
$688$	−544.000	−0.0301451
$689$	−3767.46	−0.208315
$690$	0	0
$691$	−11298.2	−0.622000	−0.311000	−	0.950410i	$-0.600664\pi$
$691$	−11298.2	−0.622000	−0.311000	+	0.950410i	$0.600664\pi$
$692$	−8281.63	−0.454943
$693$	0	0
$694$	3972.00	0.217255
$695$	−8428.00	−0.459989
$696$	0	0
$697$	−178.000	−0.00967321
$698$	−13542.5	−0.734372
$699$	0	0
$700$	0	0
$701$	2754.00	0.148384	0.0741920	−	0.997244i	$-0.476362\pi$
$701$	2754.00	0.148384	0.0741920	+	0.997244i	$0.476362\pi$
$702$	0	0
$703$	−53.7401	−0.00288314
$704$	896.000	0.0479677
$705$	0	0
$706$	13986.6	0.745597
$707$	0	0
$708$	0	0
$709$	29434.0	1.55912	0.779561	−	0.626327i	$-0.215442\pi$
$709$	29434.0	1.55912	0.779561	+	0.626327i	$0.215442\pi$
$710$	23283.6	1.23073
$711$	0	0
$712$	4944.09	0.260235
$713$	−13067.3	−0.686361
$714$	0	0
$715$	14112.0	0.738124
$716$	−2160.00	−0.112742
$717$	0	0
$718$	−11888.0	−0.617906
$719$	−17669.2	−0.916480	−0.458240	−	0.888828i	$-0.651520\pi$
$719$	−17669.2	−0.916480	−0.458240	+	0.888828i	$0.651520\pi$
$720$	0	0
$721$	0	0
$722$	−13714.0	−0.706901
$723$	0	0
$724$	−15137.7	−0.777058
$725$	76362.0	3.91174
$726$	0	0
$727$	−28445.5	−1.45115	−0.725574	−	0.688144i	$-0.758426\pi$
$727$	−28445.5	−1.45115	−0.725574	+	0.688144i	$0.758426\pi$
$728$	0	0
$729$	0	0
$730$	−10696.0	−0.542297
$731$	−48.0833	−0.00243286
$732$	0	0
$733$	−22341.7	−1.12580	−0.562900	−	0.826525i	$-0.690314\pi$
$733$	−22341.7	−1.12580	−0.562900	+	0.826525i	$0.690314\pi$
$734$	1685.74	0.0847710
$735$	0	0
$736$	−4480.00	−0.224368
$737$	9576.00	0.478611
$738$	0	0
$739$	20670.0	1.02890	0.514451	−	0.857520i	$-0.327996\pi$
$739$	20670.0	1.02890	0.514451	+	0.857520i	$0.327996\pi$
$740$	3009.45	0.149499
$741$	0	0
$742$	0	0
$743$	25400.0	1.25415	0.627076	−	0.778958i	$-0.284251\pi$
$743$	25400.0	1.25415	0.627076	+	0.778958i	$0.284251\pi$
$744$	0	0
$745$	40587.9	1.99601
$746$	−11452.0	−0.562048
$747$	0	0
$748$	79.1960	0.00387124
$749$	0	0
$750$	0	0
$751$	29180.0	1.41783	0.708917	−	0.705292i	$-0.249184\pi$
$751$	29180.0	1.41783	0.708917	+	0.705292i	$0.249184\pi$
$752$	8372.14	0.405985
$753$	0	0
$754$	−29121.5	−1.40655
$755$	9345.12	0.450469
$756$	0	0
$757$	−26206.0	−1.25822	−0.629110	−	0.777316i	$-0.716581\pi$
$757$	−26206.0	−1.25822	−0.629110	+	0.777316i	$0.716581\pi$
$758$	20660.0	0.989980
$759$	0	0
$760$	−224.000	−0.0106912
$761$	6863.18	0.326925	0.163463	−	0.986550i	$-0.447734\pi$
$761$	6863.18	0.326925	0.163463	+	0.986550i	$0.447734\pi$
$762$	0	0
$763$	0	0
$764$	−4112.00	−0.194721
$765$	0	0
$766$	2008.18	0.0947240
$767$	−22104.0	−1.04059
$768$	0	0
$769$	9058.04	0.424761	0.212380	−	0.977187i	$-0.431878\pi$
$769$	9058.04	0.424761	0.212380	+	0.977187i	$0.431878\pi$
$770$	0	0
$771$	0	0
$772$	18368.0	0.856320
$773$	−132.936	−0.00618548	−0.00309274	−	0.999995i	$-0.500984\pi$
$773$	−132.936	−0.00618548	−0.00309274	+	0.999995i	$0.500984\pi$
$774$	0	0
$775$	24921.3	1.15509
$776$	−11868.1	−0.549020
$777$	0	0
$778$	−10420.0	−0.480174
$779$	−178.000	−0.00818679
$780$	0	0
$781$	−8232.00	−0.377163
$782$	−395.980	−0.0181077
$783$	0	0
$784$	0	0
$785$	−43792.0	−1.99109
$786$	0	0
$787$	8729.94	0.395411	0.197706	−	0.980261i	$-0.436651\pi$
$787$	8729.94	0.395411	0.197706	+	0.980261i	$0.436651\pi$
$788$	−3176.00	−0.143579
$789$	0	0
$790$	−48309.5	−2.17567
$791$	0	0
$792$	0	0
$793$	720.000	0.0322421
$794$	147.078	0.00657382
$795$	0	0
$796$	−9944.75	−0.442817
$797$	7517.96	0.334128	0.167064	−	0.985946i	$-0.446571\pi$
$797$	7517.96	0.334128	0.167064	+	0.985946i	$0.446571\pi$
$798$	0	0
$799$	740.000	0.0327651
$800$	8544.00	0.377595
$801$	0	0
$802$	996.000	0.0438528
$803$	3781.61	0.166189
$804$	0	0
$805$	0	0
$806$	−9504.00	−0.415340
$807$	0	0
$808$	9028.34	0.393089
$809$	−3776.00	−0.164100	−0.0820501	−	0.996628i	$-0.526147\pi$
$809$	−3776.00	−0.164100	−0.0820501	+	0.996628i	$0.526147\pi$
$810$	0	0
$811$	36227.9	1.56860	0.784300	−	0.620382i	$-0.213023\pi$
$811$	36227.9	1.56860	0.784300	+	0.620382i	$0.213023\pi$
$812$	0	0
$813$	0	0
$814$	−1064.00	−0.0458147
$815$	−65059.5	−2.79624
$816$	0	0
$817$	−48.0833	−0.00205902
$818$	−6711.86	−0.286888
$819$	0	0
$820$	9968.00	0.424509
$821$	16410.0	0.697580	0.348790	−	0.937201i	$-0.386593\pi$
$821$	16410.0	0.697580	0.348790	+	0.937201i	$0.386593\pi$
$822$	0	0
$823$	22072.0	0.934850	0.467425	−	0.884033i	$-0.345182\pi$
$823$	22072.0	0.934850	0.467425	+	0.884033i	$0.345182\pi$
$824$	6946.62	0.293686
$825$	0	0
$826$	0	0
$827$	11628.0	0.488930	0.244465	−	0.969658i	$-0.421388\pi$
$827$	11628.0	0.488930	0.244465	+	0.969658i	$0.421388\pi$
$828$	0	0
$829$	30906.2	1.29483	0.647417	−	0.762136i	$-0.275849\pi$
$829$	30906.2	1.29483	0.647417	+	0.762136i	$0.275849\pi$
$830$	−16744.0	−0.700232
$831$	0	0
$832$	−3258.35	−0.135773
$833$	0	0
$834$	0	0
$835$	29512.0	1.22312
$836$	79.1960	0.00327638
$837$	0	0
$838$	29090.4	1.19918
$839$	17884.1	0.735911	0.367955	−	0.929843i	$-0.380058\pi$
$839$	17884.1	0.735911	0.367955	+	0.929843i	$0.380058\pi$
$840$	0	0
$841$	57407.0	2.35381
$842$	21708.0	0.888488
$843$	0	0
$844$	−10992.0	−0.448294
$845$	−7820.60	−0.318387
$846$	0	0
$847$	0	0
$848$	1184.00	0.0479466
$849$	0	0
$850$	755.190	0.0304739
$851$	5320.00	0.214298
$852$	0	0
$853$	20755.0	0.833104	0.416552	−	0.909112i	$-0.363238\pi$
$853$	20755.0	0.833104	0.416552	+	0.909112i	$0.363238\pi$
$854$	0	0
$855$	0	0
$856$	13472.0	0.537925
$857$	44919.7	1.79046	0.895231	−	0.445602i	$-0.147010\pi$
$857$	44919.7	1.79046	0.895231	+	0.445602i	$0.147010\pi$
$858$	0	0
$859$	−69.2965	−0.00275246	−0.00137623	−	0.999999i	$-0.500438\pi$
$859$	−69.2965	−0.00275246	−0.00137623	+	0.999999i	$0.500438\pi$
$860$	2692.66	0.106766
$861$	0	0
$862$	10728.0	0.423895
$863$	5452.00	0.215050	0.107525	−	0.994202i	$-0.465707\pi$
$863$	5452.00	0.215050	0.107525	+	0.994202i	$0.465707\pi$
$864$	0	0
$865$	40992.0	1.61129
$866$	12974.0	0.509093
$867$	0	0
$868$	0	0
$869$	17080.0	0.666743
$870$	0	0
$871$	−34823.6	−1.35471
$872$	−6544.00	−0.254137
$873$	0	0
$874$	−395.980	−0.0153252
$875$	0	0
$876$	0	0
$877$	31106.0	1.19769	0.598845	−	0.800865i	$-0.295626\pi$
$877$	31106.0	1.19769	0.598845	+	0.800865i	$0.295626\pi$
$878$	−27865.7	−1.07109
$879$	0	0
$880$	−4434.97	−0.169890
$881$	5943.94	0.227306	0.113653	−	0.993521i	$-0.463745\pi$
$881$	5943.94	0.227306	0.113653	+	0.993521i	$0.463745\pi$
$882$	0	0
$883$	34796.0	1.32614	0.663068	−	0.748559i	$-0.269254\pi$
$883$	34796.0	1.32614	0.663068	+	0.748559i	$0.269254\pi$
$884$	−288.000	−0.0109576
$885$	0	0
$886$	−11992.0	−0.454717
$887$	9964.55	0.377200	0.188600	−	0.982054i	$-0.439605\pi$
$887$	9964.55	0.377200	0.188600	+	0.982054i	$0.439605\pi$
$888$	0	0
$889$	0	0
$890$	−24472.0	−0.921689
$891$	0	0
$892$	13712.2	0.514707
$893$	740.000	0.0277303
$894$	0	0
$895$	10691.5	0.399303
$896$	0	0
$897$	0	0
$898$	−5244.00	−0.194871
$899$	26694.7	0.990343
$900$	0	0
$901$	104.652	0.00386954
$902$	−3524.22	−0.130093
$903$	0	0
$904$	4320.00	0.158939
$905$	74928.0	2.75214
$906$	0	0
$907$	−29756.0	−1.08934	−0.544670	−	0.838650i	$-0.683345\pi$
$907$	−29756.0	−1.08934	−0.544670	+	0.838650i	$0.683345\pi$
$908$	21162.3	0.773453
$909$	0	0
$910$	0	0
$911$	−21440.0	−0.779735	−0.389868	−	0.920871i	$-0.627479\pi$
$911$	−21440.0	−0.779735	−0.389868	+	0.920871i	$0.627479\pi$
$912$	0	0
$913$	5919.90	0.214589
$914$	22416.0	0.811220
$915$	0	0
$916$	10996.9	0.396669
$917$	0	0
$918$	0	0
$919$	−8288.00	−0.297493	−0.148746	−	0.988875i	$-0.547524\pi$
$919$	−8288.00	−0.297493	−0.148746	+	0.988875i	$0.547524\pi$
$920$	22174.9	0.794656
$921$	0	0
$922$	19572.7	0.699125
$923$	29936.1	1.06756
$924$	0	0
$925$	−10146.0	−0.360647
$926$	7904.00	0.280498
$927$	0	0
$928$	9152.00	0.323738
$929$	−45581.5	−1.60978	−0.804888	−	0.593427i	$-0.797774\pi$
$929$	−45581.5	−1.60978	−0.804888	+	0.593427i	$0.797774\pi$
$930$	0	0
$931$	0	0
$932$	−288.000	−0.0101221
$933$	0	0
$934$	35013.1	1.22662
$935$	−392.000	−0.0137110
$936$	0	0
$937$	−11665.8	−0.406731	−0.203365	−	0.979103i	$-0.565188\pi$
$937$	−11665.8	−0.406731	−0.203365	+	0.979103i	$0.565188\pi$
$938$	0	0
$939$	0	0
$940$	−41440.0	−1.43790
$941$	14.1421	0.000489926 0	0.000244963	−	1.00000i	$-0.499922\pi$
$941$	14.1421	0.000489926 0	0.000244963		1.00000i	$0.499922\pi$
$942$	0	0
$943$	17621.1	0.608507
$944$	6946.62	0.239505
$945$	0	0
$946$	−952.000	−0.0327190
$947$	−14034.0	−0.481567	−0.240783	−	0.970579i	$-0.577404\pi$
$947$	−14034.0	−0.481567	−0.240783	+	0.970579i	$0.577404\pi$
$948$	0	0
$949$	−13752.0	−0.470399
$950$	755.190	0.0257912
$951$	0	0
$952$	0	0
$953$	42698.0	1.45134	0.725668	−	0.688045i	$-0.241531\pi$
$953$	42698.0	1.45134	0.725668	+	0.688045i	$0.241531\pi$
$954$	0	0
$955$	20353.4	0.689654
$956$	−17232.0	−0.582974
$957$	0	0
$958$	4576.40	0.154339
$959$	0	0
$960$	0	0
$961$	−21079.0	−0.707563
$962$	3869.29	0.129679
$963$	0	0
$964$	6160.31	0.205820
$965$	−90917.0	−3.03287
$966$	0	0
$967$	−48492.0	−1.61261	−0.806307	−	0.591497i	$-0.798537\pi$
$967$	−48492.0	−1.61261	−0.806307	+	0.591497i	$0.798537\pi$
$968$	−9080.00	−0.301490
$969$	0	0
$970$	58744.0	1.94449
$971$	52669.6	1.74073	0.870364	−	0.492409i	$-0.163884\pi$
$971$	52669.6	1.74073	0.870364	+	0.492409i	$0.163884\pi$
$972$	0	0
$973$	0	0
$974$	1944.00	0.0639525
$975$	0	0
$976$	−226.274	−0.00742096
$977$	55380.0	1.81347	0.906737	−	0.421698i	$-0.138566\pi$
$977$	55380.0	1.81347	0.906737	+	0.421698i	$0.138566\pi$
$978$	0	0
$979$	8652.16	0.282456
$980$	0	0
$981$	0	0
$982$	14808.0	0.481204
$983$	−50535.5	−1.63971	−0.819854	−	0.572573i	$-0.805945\pi$
$983$	−50535.5	−1.63971	−0.819854	+	0.572573i	$0.805945\pi$
$984$	0	0
$985$	15720.4	0.508521
$986$	808.930	0.0261274
$987$	0	0
$988$	−288.000	−0.00927379
$989$	4760.00	0.153043
$990$	0	0
$991$	−39712.0	−1.27295	−0.636475	−	0.771297i	$-0.719608\pi$
$991$	−39712.0	−1.27295	−0.636475	+	0.771297i	$0.719608\pi$
$992$	2986.82	0.0955964
$993$	0	0
$994$	0	0
$995$	49224.0	1.56835
$996$	0	0
$997$	−2186.37	−0.0694515	−0.0347258	−	0.999397i	$-0.511056\pi$
$997$	−2186.37	−0.0694515	−0.0347258	+	0.999397i	$0.511056\pi$
$998$	−24488.0	−0.776708
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.bg.1.1		2
3.2	odd	2		98.4.a.g.1.2	yes	2
7.2	even	3		882.4.g.ba.361.2		4
7.3	odd	6		882.4.g.ba.667.1		4
7.4	even	3		882.4.g.ba.667.2		4
7.5	odd	6		882.4.g.ba.361.1		4
7.6	odd	2	inner	882.4.a.bg.1.2		2
12.11	even	2		784.4.a.y.1.1		2
15.14	odd	2		2450.4.a.bx.1.1		2
21.2	odd	6		98.4.c.h.67.1		4
21.5	even	6		98.4.c.h.67.2		4
21.11	odd	6		98.4.c.h.79.1		4
21.17	even	6		98.4.c.h.79.2		4
21.20	even	2		98.4.a.g.1.1	✓	2
84.83	odd	2		784.4.a.y.1.2		2
105.104	even	2		2450.4.a.bx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.a.g.1.1	✓	2	21.20	even	2
98.4.a.g.1.2	yes	2	3.2	odd	2
98.4.c.h.67.1		4	21.2	odd	6
98.4.c.h.67.2		4	21.5	even	6
98.4.c.h.79.1		4	21.11	odd	6
98.4.c.h.79.2		4	21.17	even	6
784.4.a.y.1.1		2	12.11	even	2
784.4.a.y.1.2		2	84.83	odd	2
882.4.a.bg.1.1		2	1.1	even	1	trivial
882.4.a.bg.1.2		2	7.6	odd	2	inner
882.4.g.ba.361.1		4	7.5	odd	6
882.4.g.ba.361.2		4	7.2	even	3
882.4.g.ba.667.1		4	7.3	odd	6
882.4.g.ba.667.2		4	7.4	even	3
2450.4.a.bx.1.1		2	15.14	odd	2
2450.4.a.bx.1.2		2	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -19.7990 q^{5} +8.00000 q^{8} -39.5980 q^{10} +14.0000 q^{11} -50.9117 q^{13} +16.0000 q^{16} +1.41421 q^{17} +1.41421 q^{19} -79.1960 q^{20} +28.0000 q^{22} -140.000 q^{23} +267.000 q^{25} -101.823 q^{26} +286.000 q^{29} +93.3381 q^{31} +32.0000 q^{32} +2.82843 q^{34} -38.0000 q^{37} +2.82843 q^{38} -158.392 q^{40} -125.865 q^{41} -34.0000 q^{43} +56.0000 q^{44} -280.000 q^{46} +523.259 q^{47} +534.000 q^{50} -203.647 q^{52} +74.0000 q^{53} -277.186 q^{55} +572.000 q^{58} +434.164 q^{59} -14.1421 q^{61} +186.676 q^{62} +64.0000 q^{64} +1008.00 q^{65} +684.000 q^{67} +5.65685 q^{68} -588.000 q^{71} +270.115 q^{73} -76.0000 q^{74} +5.65685 q^{76} +1220.00 q^{79} -316.784 q^{80} -251.730 q^{82} +422.850 q^{83} -28.0000 q^{85} -68.0000 q^{86} +112.000 q^{88} +618.011 q^{89} -560.000 q^{92} +1046.52 q^{94} -28.0000 q^{95} -1483.51 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 28 q^{11} + 32 q^{16} + 56 q^{22} - 280 q^{23} + 534 q^{25} + 572 q^{29} + 64 q^{32} - 76 q^{37} - 68 q^{43} + 112 q^{44} - 560 q^{46} + 1068 q^{50} + 148 q^{53} + 1144 q^{58} + 128 q^{64} + 2016 q^{65} + 1368 q^{67} - 1176 q^{71} - 152 q^{74} + 2440 q^{79} - 56 q^{85} - 136 q^{86} + 224 q^{88} - 1120 q^{92} - 56 q^{95}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 28 * q^11 + 32 * q^16 + 56 * q^22 - 280 * q^23 + 534 * q^25 + 572 * q^29 + 64 * q^32 - 76 * q^37 - 68 * q^43 + 112 * q^44 - 560 * q^46 + 1068 * q^50 + 148 * q^53 + 1144 * q^58 + 128 * q^64 + 2016 * q^65 + 1368 * q^67 - 1176 * q^71 - 152 * q^74 + 2440 * q^79 - 56 * q^85 - 136 * q^86 + 224 * q^88 - 1120 * q^92 - 56 * q^95

Introduction

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