LMFDB - Embedded newform 1089.4.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.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:	$64.2530799963$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 $ x^6 - x^5 - 30x^4 + 3x^3 + 211x^2 + 208x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-4.12458$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.12458 q^{2} +18.2613 q^{4} -14.9367 q^{5} +12.3403 q^{7} +52.5849 q^{8} -76.5442 q^{10} -30.5222 q^{13} +63.2387 q^{14} +123.385 q^{16} +132.065 q^{17} +22.3068 q^{19} -272.763 q^{20} +158.801 q^{23} +98.1044 q^{25} -156.413 q^{26} +225.349 q^{28} -47.7165 q^{29} +68.0343 q^{31} +211.617 q^{32} +676.778 q^{34} -184.323 q^{35} +158.409 q^{37} +114.313 q^{38} -785.444 q^{40} +286.159 q^{41} -103.549 q^{43} +813.790 q^{46} +464.238 q^{47} -190.718 q^{49} +502.744 q^{50} -557.375 q^{52} +6.04835 q^{53} +648.912 q^{56} -244.527 q^{58} +221.619 q^{59} +274.338 q^{61} +348.647 q^{62} +97.3683 q^{64} +455.900 q^{65} +187.178 q^{67} +2411.68 q^{68} -944.576 q^{70} +457.714 q^{71} +270.332 q^{73} +811.780 q^{74} +407.352 q^{76} -493.647 q^{79} -1842.96 q^{80} +1466.45 q^{82} -271.033 q^{83} -1972.61 q^{85} -530.646 q^{86} +77.4891 q^{89} -376.652 q^{91} +2899.92 q^{92} +2379.02 q^{94} -333.190 q^{95} -1269.39 q^{97} -977.348 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} + 17 q^{4} - 9 q^{5} - q^{7} + 24 q^{8} - 50 q^{10} - 66 q^{13} + 42 q^{14} - 71 q^{16} + 80 q^{17} + 90 q^{19} - 455 q^{20} + 196 q^{23} + 351 q^{25} - 93 q^{26} + 52 q^{28} + 579 q^{29}+ \cdots + 1405 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 + 17 * q^4 - 9 * q^5 - q^7 + 24 * q^8 - 50 * q^10 - 66 * q^13 + 42 * q^14 - 71 * q^16 + 80 * q^17 + 90 * q^19 - 455 * q^20 + 196 * q^23 + 351 * q^25 - 93 * q^26 + 52 * q^28 + 579 * q^29 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 192 * q^37 + 769 * q^38 - 1476 * q^40 + 990 * q^41 - 208 * q^43 - 493 * q^46 - 680 * q^47 - 963 * q^49 + 1656 * q^50 - 887 * q^52 - 197 * q^53 + 429 * q^56 + 395 * q^58 + 1229 * q^59 - 928 * q^61 + 2117 * q^62 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 665 * q^70 + 786 * q^71 - 593 * q^73 + 1543 * q^74 + 434 * q^76 + 421 * q^79 - 18 * q^80 - 739 * q^82 + 2043 * q^83 - 1900 * q^85 - 2683 * q^86 - 158 * q^89 + 698 * q^91 + 3915 * q^92 + 4934 * q^94 + 1038 * q^95 - 1711 * q^97 + 1405 * q^98

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 $

$2$	5.12458	1.81181	0.905906	−	0.423478i	$-0.139191\pi$
$2$	5.12458	1.81181	0.905906	+	0.423478i	$0.139191\pi$
$3$	0	0
$3$	0	0
$4$	18.2613	2.28266
$5$	−14.9367	−1.33598	−0.667989	−	0.744172i	$-0.732845\pi$
$5$	−14.9367	−1.33598	−0.667989	+	0.744172i	$0.732845\pi$
$6$	0	0
$7$	12.3403	0.666312	0.333156	−	0.942872i	$-0.391886\pi$
$7$	12.3403	0.666312	0.333156	+	0.942872i	$0.391886\pi$
$8$	52.5849	2.32395
$9$	0	0
$10$	−76.5442	−2.42054
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−30.5222	−0.651179	−0.325590	−	0.945511i	$-0.605563\pi$
$13$	−30.5222	−0.651179	−0.325590	+	0.945511i	$0.605563\pi$
$14$	63.2387	1.20723
$15$	0	0
$16$	123.385	1.92789
$17$	132.065	1.88415	0.942073	−	0.335409i	$-0.108874\pi$
$17$	132.065	1.88415	0.942073	+	0.335409i	$0.108874\pi$
$18$	0	0
$19$	22.3068	0.269344	0.134672	−	0.990890i	$-0.457002\pi$
$19$	22.3068	0.269344	0.134672	+	0.990890i	$0.457002\pi$
$20$	−272.763	−3.04959
$21$	0	0
$22$	0	0
$23$	158.801	1.43967	0.719834	−	0.694147i	$-0.244218\pi$
$23$	158.801	1.43967	0.719834	+	0.694147i	$0.244218\pi$
$24$	0	0
$25$	98.1044	0.784835
$26$	−156.413	−1.17982
$27$	0	0
$28$	225.349	1.52097
$29$	−47.7165	−0.305543	−0.152771	−	0.988262i	$-0.548820\pi$
$29$	−47.7165	−0.305543	−0.152771	+	0.988262i	$0.548820\pi$
$30$	0	0
$31$	68.0343	0.394172	0.197086	−	0.980386i	$-0.436852\pi$
$31$	68.0343	0.394172	0.197086	+	0.980386i	$0.436852\pi$
$32$	211.617	1.16903
$33$	0	0
$34$	676.778	3.41372
$35$	−184.323	−0.890177
$36$	0	0
$37$	158.409	0.703846	0.351923	−	0.936029i	$-0.385528\pi$
$37$	158.409	0.703846	0.351923	+	0.936029i	$0.385528\pi$
$38$	114.313	0.488001
$39$	0	0
$40$	−785.444	−3.10474
$41$	286.159	1.09001	0.545007	−	0.838432i	$-0.316527\pi$
$41$	286.159	1.09001	0.545007	+	0.838432i	$0.316527\pi$
$42$	0	0
$43$	−103.549	−0.367235	−0.183617	−	0.982998i	$-0.558781\pi$
$43$	−103.549	−0.367235	−0.183617	+	0.982998i	$0.558781\pi$
$44$	0	0
$45$	0	0
$46$	813.790	2.60841
$47$	464.238	1.44077	0.720383	−	0.693576i	$-0.243966\pi$
$47$	464.238	1.44077	0.720383	+	0.693576i	$0.243966\pi$
$48$	0	0
$49$	−190.718	−0.556029
$50$	502.744	1.42197
$51$	0	0
$52$	−557.375	−1.48642
$53$	6.04835	0.0156756	0.00783778	−	0.999969i	$-0.497505\pi$
$53$	6.04835	0.0156756	0.00783778	+	0.999969i	$0.497505\pi$
$54$	0	0
$55$	0	0
$56$	648.912	1.54847
$57$	0	0
$58$	−244.527	−0.553586
$59$	221.619	0.489023	0.244511	−	0.969646i	$-0.421372\pi$
$59$	221.619	0.489023	0.244511	+	0.969646i	$0.421372\pi$
$60$	0	0
$61$	274.338	0.575825	0.287913	−	0.957657i	$-0.407039\pi$
$61$	274.338	0.575825	0.287913	+	0.957657i	$0.407039\pi$
$62$	348.647	0.714165
$63$	0	0
$64$	97.3683	0.190172
$65$	455.900	0.869961
$66$	0	0
$67$	187.178	0.341304	0.170652	−	0.985331i	$-0.445413\pi$
$67$	187.178	0.341304	0.170652	+	0.985331i	$0.445413\pi$
$68$	2411.68	4.30087
$69$	0	0
$70$	−944.576	−1.61283
$71$	457.714	0.765080	0.382540	−	0.923939i	$-0.375049\pi$
$71$	457.714	0.765080	0.382540	+	0.923939i	$0.375049\pi$
$72$	0	0
$73$	270.332	0.433424	0.216712	−	0.976236i	$-0.430467\pi$
$73$	270.332	0.433424	0.216712	+	0.976236i	$0.430467\pi$
$74$	811.780	1.27524
$75$	0	0
$76$	407.352	0.614822
$77$	0	0
$78$	0	0
$79$	−493.647	−0.703033	−0.351517	−	0.936182i	$-0.614334\pi$
$79$	−493.647	−0.703033	−0.351517	+	0.936182i	$0.614334\pi$
$80$	−1842.96	−2.57562
$81$	0	0
$82$	1466.45	1.97490
$83$	−271.033	−0.358431	−0.179216	−	0.983810i	$-0.557356\pi$
$83$	−271.033	−0.358431	−0.179216	+	0.983810i	$0.557356\pi$
$84$	0	0
$85$	−1972.61	−2.51718
$86$	−530.646	−0.665361
$87$	0	0
$88$	0	0
$89$	77.4891	0.0922902	0.0461451	−	0.998935i	$-0.485306\pi$
$89$	77.4891	0.0922902	0.0461451	+	0.998935i	$0.485306\pi$
$90$	0	0
$91$	−376.652	−0.433889
$92$	2899.92	3.28628
$93$	0	0
$94$	2379.02	2.61040
$95$	−333.190	−0.359837
$96$	0	0
$97$	−1269.39	−1.32873	−0.664364	−	0.747409i	$-0.731298\pi$
$97$	−1269.39	−1.32873	−0.664364	+	0.747409i	$0.731298\pi$
$98$	−977.348	−1.00742
$99$	0	0
$100$	1791.51	1.79151
$101$	−1238.06	−1.21972	−0.609859	−	0.792510i	$-0.708774\pi$
$101$	−1238.06	−1.21972	−0.609859	+	0.792510i	$0.708774\pi$
$102$	0	0
$103$	911.965	0.872413	0.436206	−	0.899847i	$-0.356322\pi$
$103$	911.965	0.872413	0.436206	+	0.899847i	$0.356322\pi$
$104$	−1605.01	−1.51331
$105$	0	0
$106$	30.9953	0.0284012
$107$	−1583.34	−1.43054	−0.715269	−	0.698849i	$-0.753696\pi$
$107$	−1583.34	−1.43054	−0.715269	+	0.698849i	$0.753696\pi$
$108$	0	0
$109$	−296.898	−0.260896	−0.130448	−	0.991455i	$-0.541642\pi$
$109$	−296.898	−0.260896	−0.130448	+	0.991455i	$0.541642\pi$
$110$	0	0
$111$	0	0
$112$	1522.60	1.28458
$113$	109.404	0.0910787	0.0455393	−	0.998963i	$-0.485499\pi$
$113$	109.404	0.0910787	0.0455393	+	0.998963i	$0.485499\pi$
$114$	0	0
$115$	−2371.96	−1.92336
$116$	−871.367	−0.697451
$117$	0	0
$118$	1135.70	0.886017
$119$	1629.72	1.25543
$120$	0	0
$121$	0	0
$122$	1405.86	1.04329
$123$	0	0
$124$	1242.40	0.899761
$125$	401.731	0.287455
$126$	0	0
$127$	−1387.37	−0.969366	−0.484683	−	0.874690i	$-0.661065\pi$
$127$	−1387.37	−0.969366	−0.484683	+	0.874690i	$0.661065\pi$
$128$	−1193.97	−0.824474
$129$	0	0
$130$	2336.30	1.57621
$131$	1515.23	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$131$	1515.23	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$132$	0	0
$133$	275.272	0.179467
$134$	959.207	0.618379
$135$	0	0
$136$	6944.63	4.37865
$137$	2970.09	1.85220	0.926101	−	0.377275i	$-0.123139\pi$
$137$	2970.09	1.85220	0.926101	+	0.377275i	$0.123139\pi$
$138$	0	0
$139$	2152.08	1.31322	0.656608	−	0.754232i	$-0.271990\pi$
$139$	2152.08	1.31322	0.656608	+	0.754232i	$0.271990\pi$
$140$	−3365.97	−2.03198
$141$	0	0
$142$	2345.59	1.38618
$143$	0	0
$144$	0	0
$145$	712.727	0.408198
$146$	1385.34	0.785283
$147$	0	0
$148$	2892.76	1.60664
$149$	−132.876	−0.0730579	−0.0365290	−	0.999333i	$-0.511630\pi$
$149$	−132.876	−0.0730579	−0.0365290	+	0.999333i	$0.511630\pi$
$150$	0	0
$151$	−212.982	−0.114783	−0.0573915	−	0.998352i	$-0.518278\pi$
$151$	−212.982	−0.114783	−0.0573915	+	0.998352i	$0.518278\pi$
$152$	1173.00	0.625941
$153$	0	0
$154$	0	0
$155$	−1016.21	−0.526604
$156$	0	0
$157$	−2925.09	−1.48693	−0.743464	−	0.668776i	$-0.766818\pi$
$157$	−2925.09	−1.48693	−0.743464	+	0.668776i	$0.766818\pi$
$158$	−2529.73	−1.27376
$159$	0	0
$160$	−3160.86	−1.56180
$161$	1959.65	0.959267
$162$	0	0
$163$	−633.267	−0.304302	−0.152151	−	0.988357i	$-0.548620\pi$
$163$	−633.267	−0.304302	−0.152151	+	0.988357i	$0.548620\pi$
$164$	5225.64	2.48813
$165$	0	0
$166$	−1388.93	−0.649410
$167$	−1533.33	−0.710496	−0.355248	−	0.934772i	$-0.615604\pi$
$167$	−1533.33	−0.710496	−0.355248	+	0.934772i	$0.615604\pi$
$168$	0	0
$169$	−1265.40	−0.575965
$170$	−10108.8	−4.56065
$171$	0	0
$172$	−1890.94	−0.838274
$173$	3326.68	1.46198	0.730991	−	0.682388i	$-0.239058\pi$
$173$	3326.68	1.46198	0.730991	+	0.682388i	$0.239058\pi$
$174$	0	0
$175$	1210.63	0.522945
$176$	0	0
$177$	0	0
$178$	397.099	0.167212
$179$	−4261.40	−1.77940	−0.889698	−	0.456549i	$-0.849085\pi$
$179$	−4261.40	−1.77940	−0.889698	+	0.456549i	$0.849085\pi$
$180$	0	0
$181$	741.930	0.304681	0.152340	−	0.988328i	$-0.451319\pi$
$181$	741.930	0.304681	0.152340	+	0.988328i	$0.451319\pi$
$182$	−1930.18	−0.786125
$183$	0	0
$184$	8350.55	3.34571
$185$	−2366.11	−0.940323
$186$	0	0
$187$	0	0
$188$	8477.59	3.28879
$189$	0	0
$190$	−1707.46	−0.651958
$191$	−4132.16	−1.56541	−0.782704	−	0.622394i	$-0.786160\pi$
$191$	−4132.16	−1.56541	−0.782704	+	0.622394i	$0.786160\pi$
$192$	0	0
$193$	−3081.78	−1.14938	−0.574692	−	0.818370i	$-0.694878\pi$
$193$	−3081.78	−1.14938	−0.574692	+	0.818370i	$0.694878\pi$
$194$	−6505.07	−2.40741
$195$	0	0
$196$	−3482.76	−1.26923
$197$	3059.84	1.10662	0.553310	−	0.832975i	$-0.313364\pi$
$197$	3059.84	1.10662	0.553310	+	0.832975i	$0.313364\pi$
$198$	0	0
$199$	−971.928	−0.346222	−0.173111	−	0.984902i	$-0.555382\pi$
$199$	−971.928	−0.346222	−0.173111	+	0.984902i	$0.555382\pi$
$200$	5158.81	1.82391
$201$	0	0
$202$	−6344.54	−2.20990
$203$	−588.835	−0.203587
$204$	0	0
$205$	−4274.27	−1.45623
$206$	4673.43	1.58065
$207$	0	0
$208$	−3765.98	−1.25540
$209$	0	0
$210$	0	0
$211$	−3327.16	−1.08555	−0.542775	−	0.839878i	$-0.682626\pi$
$211$	−3327.16	−1.08555	−0.542775	+	0.839878i	$0.682626\pi$
$212$	110.451	0.0357821
$213$	0	0
$214$	−8113.97	−2.59187
$215$	1546.68	0.490617
$216$	0	0
$217$	839.562	0.262641
$218$	−1521.48	−0.472694
$219$	0	0
$220$	0	0
$221$	−4030.91	−1.22692
$222$	0	0
$223$	4915.02	1.47594	0.737969	−	0.674834i	$-0.235785\pi$
$223$	4915.02	1.47594	0.737969	+	0.674834i	$0.235785\pi$
$224$	2611.41	0.778939
$225$	0	0
$226$	560.651	0.165017
$227$	−1062.64	−0.310706	−0.155353	−	0.987859i	$-0.549651\pi$
$227$	−1062.64	−0.310706	−0.155353	+	0.987859i	$0.549651\pi$
$228$	0	0
$229$	−4312.88	−1.24456	−0.622278	−	0.782796i	$-0.713793\pi$
$229$	−4312.88	−1.24456	−0.622278	+	0.782796i	$0.713793\pi$
$230$	−12155.3	−3.48477
$231$	0	0
$232$	−2509.17	−0.710065
$233$	5154.61	1.44931	0.724655	−	0.689111i	$-0.241999\pi$
$233$	5154.61	1.44931	0.724655	+	0.689111i	$0.241999\pi$
$234$	0	0
$235$	−6934.17	−1.92483
$236$	4047.05	1.11627
$237$	0	0
$238$	8351.62	2.27460
$239$	−124.293	−0.0336395	−0.0168197	−	0.999859i	$-0.505354\pi$
$239$	−124.293	−0.0336395	−0.0168197	+	0.999859i	$0.505354\pi$
$240$	0	0
$241$	445.539	0.119086	0.0595429	−	0.998226i	$-0.481036\pi$
$241$	445.539	0.119086	0.0595429	+	0.998226i	$0.481036\pi$
$242$	0	0
$243$	0	0
$244$	5009.76	1.31442
$245$	2848.69	0.742841
$246$	0	0
$247$	−680.853	−0.175391
$248$	3577.58	0.916034
$249$	0	0
$250$	2058.70	0.520815
$251$	−1471.10	−0.369940	−0.184970	−	0.982744i	$-0.559219\pi$
$251$	−1471.10	−0.369940	−0.184970	+	0.982744i	$0.559219\pi$
$252$	0	0
$253$	0	0
$254$	−7109.70	−1.75631
$255$	0	0
$256$	−6897.52	−1.68396
$257$	−1463.01	−0.355097	−0.177549	−	0.984112i	$-0.556817\pi$
$257$	−1463.01	−0.355097	−0.177549	+	0.984112i	$0.556817\pi$
$258$	0	0
$259$	1954.81	0.468981
$260$	8325.34	1.98583
$261$	0	0
$262$	7764.91	1.83098
$263$	−734.717	−0.172261	−0.0861304	−	0.996284i	$-0.527450\pi$
$263$	−734.717	−0.172261	−0.0861304	+	0.996284i	$0.527450\pi$
$264$	0	0
$265$	−90.3423	−0.0209422
$266$	1410.65	0.325161
$267$	0	0
$268$	3418.11	0.779083
$269$	−3427.29	−0.776823	−0.388411	−	0.921486i	$-0.626976\pi$
$269$	−3427.29	−0.776823	−0.388411	+	0.921486i	$0.626976\pi$
$270$	0	0
$271$	−1719.34	−0.385398	−0.192699	−	0.981258i	$-0.561724\pi$
$271$	−1719.34	−0.385398	−0.192699	+	0.981258i	$0.561724\pi$
$272$	16294.8	3.63243
$273$	0	0
$274$	15220.5	3.35584
$275$	0	0
$276$	0	0
$277$	−5156.07	−1.11841	−0.559203	−	0.829031i	$-0.688893\pi$
$277$	−5156.07	−1.11841	−0.559203	+	0.829031i	$0.688893\pi$
$278$	11028.5	2.37930
$279$	0	0
$280$	−9692.59	−2.06872
$281$	5020.69	1.06587	0.532934	−	0.846157i	$-0.321089\pi$
$281$	5020.69	1.06587	0.532934	+	0.846157i	$0.321089\pi$
$282$	0	0
$283$	1878.02	0.394477	0.197238	−	0.980356i	$-0.436803\pi$
$283$	1878.02	0.394477	0.197238	+	0.980356i	$0.436803\pi$
$284$	8358.46	1.74642
$285$	0	0
$286$	0	0
$287$	3531.28	0.726289
$288$	0	0
$289$	12528.2	2.55000
$290$	3652.42	0.739578
$291$	0	0
$292$	4936.61	0.989361
$293$	4811.63	0.959379	0.479690	−	0.877438i	$-0.340749\pi$
$293$	4811.63	0.959379	0.479690	+	0.877438i	$0.340749\pi$
$294$	0	0
$295$	−3310.25	−0.653323
$296$	8329.93	1.63570
$297$	0	0
$298$	−680.934	−0.132367
$299$	−4846.96	−0.937482
$300$	0	0
$301$	−1277.82	−0.244693
$302$	−1091.44	−0.207965
$303$	0	0
$304$	2752.33	0.519266
$305$	−4097.69	−0.769289
$306$	0	0
$307$	598.905	0.111340	0.0556699	−	0.998449i	$-0.482271\pi$
$307$	598.905	0.111340	0.0556699	+	0.998449i	$0.482271\pi$
$308$	0	0
$309$	0	0
$310$	−5207.63	−0.954108
$311$	−5131.15	−0.935566	−0.467783	−	0.883843i	$-0.654947\pi$
$311$	−5131.15	−0.935566	−0.467783	+	0.883843i	$0.654947\pi$
$312$	0	0
$313$	−9541.45	−1.72305	−0.861525	−	0.507716i	$-0.830490\pi$
$313$	−9541.45	−1.72305	−0.861525	+	0.507716i	$0.830490\pi$
$314$	−14989.9	−2.69403
$315$	0	0
$316$	−9014.64	−1.60479
$317$	−6603.42	−1.16998	−0.584992	−	0.811039i	$-0.698902\pi$
$317$	−6603.42	−1.16998	−0.584992	+	0.811039i	$0.698902\pi$
$318$	0	0
$319$	0	0
$320$	−1454.36	−0.254066
$321$	0	0
$322$	10042.4	1.73801
$323$	2945.95	0.507483
$324$	0	0
$325$	−2994.36	−0.511068
$326$	−3245.23	−0.551339
$327$	0	0
$328$	15047.7	2.53313
$329$	5728.82	0.960000
$330$	0	0
$331$	−6549.79	−1.08764	−0.543820	−	0.839202i	$-0.683023\pi$
$331$	−6549.79	−1.08764	−0.543820	+	0.839202i	$0.683023\pi$
$332$	−4949.43	−0.818178
$333$	0	0
$334$	−7857.69	−1.28729
$335$	−2795.81	−0.455975
$336$	0	0
$337$	2690.38	0.434879	0.217439	−	0.976074i	$-0.430230\pi$
$337$	2690.38	0.434879	0.217439	+	0.976074i	$0.430230\pi$
$338$	−6484.62	−1.04354
$339$	0	0
$340$	−36022.5	−5.74587
$341$	0	0
$342$	0	0
$343$	−6586.22	−1.03680
$344$	−5445.12	−0.853434
$345$	0	0
$346$	17047.8	2.64884
$347$	−778.334	−0.120413	−0.0602063	−	0.998186i	$-0.519176\pi$
$347$	−778.334	−0.120413	−0.0602063	+	0.998186i	$0.519176\pi$
$348$	0	0
$349$	5416.06	0.830702	0.415351	−	0.909661i	$-0.363659\pi$
$349$	5416.06	0.830702	0.415351	+	0.909661i	$0.363659\pi$
$350$	6203.99	0.947478
$351$	0	0
$352$	0	0
$353$	4504.51	0.679181	0.339591	−	0.940573i	$-0.389711\pi$
$353$	4504.51	0.679181	0.339591	+	0.940573i	$0.389711\pi$
$354$	0	0
$355$	−6836.73	−1.02213
$356$	1415.05	0.210667
$357$	0	0
$358$	−21837.9	−3.22393
$359$	8588.42	1.26262	0.631308	−	0.775532i	$-0.282518\pi$
$359$	8588.42	1.26262	0.631308	+	0.775532i	$0.282518\pi$
$360$	0	0
$361$	−6361.41	−0.927454
$362$	3802.08	0.552025
$363$	0	0
$364$	−6878.16	−0.990422
$365$	−4037.86	−0.579045
$366$	0	0
$367$	2178.08	0.309795	0.154897	−	0.987931i	$-0.450495\pi$
$367$	2178.08	0.309795	0.154897	+	0.987931i	$0.450495\pi$
$368$	19593.7	2.77552
$369$	0	0
$370$	−12125.3	−1.70369
$371$	74.6383	0.0104448
$372$	0	0
$373$	12814.7	1.77888	0.889440	−	0.457052i	$-0.151095\pi$
$373$	12814.7	1.77888	0.889440	+	0.457052i	$0.151095\pi$
$374$	0	0
$375$	0	0
$376$	24411.9	3.34826
$377$	1456.41	0.198963
$378$	0	0
$379$	7302.82	0.989764	0.494882	−	0.868960i	$-0.335211\pi$
$379$	7302.82	0.989764	0.494882	+	0.868960i	$0.335211\pi$
$380$	−6084.48	−0.821388
$381$	0	0
$382$	−21175.6	−2.83622
$383$	−11314.0	−1.50945	−0.754726	−	0.656040i	$-0.772230\pi$
$383$	−11314.0	−1.50945	−0.754726	+	0.656040i	$0.772230\pi$
$384$	0	0
$385$	0	0
$386$	−15792.8	−2.08247
$387$	0	0
$388$	−23180.7	−3.03304
$389$	2955.81	0.385259	0.192629	−	0.981272i	$-0.438299\pi$
$389$	2955.81	0.385259	0.192629	+	0.981272i	$0.438299\pi$
$390$	0	0
$391$	20972.1	2.71254
$392$	−10028.9	−1.29218
$393$	0	0
$394$	15680.4	2.00499
$395$	7373.45	0.939236
$396$	0	0
$397$	12391.9	1.56658	0.783288	−	0.621659i	$-0.213541\pi$
$397$	12391.9	1.56658	0.783288	+	0.621659i	$0.213541\pi$
$398$	−4980.72	−0.627289
$399$	0	0
$400$	12104.6	1.51308
$401$	−7592.65	−0.945533	−0.472766	−	0.881188i	$-0.656745\pi$
$401$	−7592.65	−0.945533	−0.472766	+	0.881188i	$0.656745\pi$
$402$	0	0
$403$	−2076.56	−0.256677
$404$	−22608.6	−2.78421
$405$	0	0
$406$	−3017.53	−0.368861
$407$	0	0
$408$	0	0
$409$	7275.77	0.879618	0.439809	−	0.898091i	$-0.355046\pi$
$409$	7275.77	0.879618	0.439809	+	0.898091i	$0.355046\pi$
$410$	−21903.8	−2.63842
$411$	0	0
$412$	16653.7	1.99143
$413$	2734.84	0.325842
$414$	0	0
$415$	4048.34	0.478856
$416$	−6459.02	−0.761248
$417$	0	0
$418$	0	0
$419$	−11095.7	−1.29370	−0.646851	−	0.762616i	$-0.723915\pi$
$419$	−11095.7	−1.29370	−0.646851	+	0.762616i	$0.723915\pi$
$420$	0	0
$421$	−5103.54	−0.590811	−0.295406	−	0.955372i	$-0.595455\pi$
$421$	−5103.54	−0.590811	−0.295406	+	0.955372i	$0.595455\pi$
$422$	−17050.3	−1.96681
$423$	0	0
$424$	318.052	0.0364292
$425$	12956.2	1.47874
$426$	0	0
$427$	3385.40	0.383679
$428$	−28913.9	−3.26544
$429$	0	0
$430$	7926.09	0.888906
$431$	−14437.7	−1.61355	−0.806774	−	0.590861i	$-0.798788\pi$
$431$	−14437.7	−1.61355	−0.806774	+	0.590861i	$0.798788\pi$
$432$	0	0
$433$	4236.83	0.470229	0.235114	−	0.971968i	$-0.424454\pi$
$433$	4236.83	0.470229	0.235114	+	0.971968i	$0.424454\pi$
$434$	4302.40	0.475857
$435$	0	0
$436$	−5421.74	−0.595538
$437$	3542.35	0.387766
$438$	0	0
$439$	−1809.83	−0.196761	−0.0983807	−	0.995149i	$-0.531366\pi$
$439$	−1809.83	−0.196761	−0.0983807	+	0.995149i	$0.531366\pi$
$440$	0	0
$441$	0	0
$442$	−20656.7	−2.22294
$443$	−4797.37	−0.514514	−0.257257	−	0.966343i	$-0.582819\pi$
$443$	−4797.37	−0.514514	−0.257257	+	0.966343i	$0.582819\pi$
$444$	0	0
$445$	−1157.43	−0.123298
$446$	25187.4	2.67412
$447$	0	0
$448$	1201.55	0.126714
$449$	9177.20	0.964585	0.482293	−	0.876010i	$-0.339804\pi$
$449$	9177.20	0.964585	0.482293	+	0.876010i	$0.339804\pi$
$450$	0	0
$451$	0	0
$452$	1997.87	0.207902
$453$	0	0
$454$	−5445.61	−0.562940
$455$	5625.93	0.579665
$456$	0	0
$457$	−6179.47	−0.632524	−0.316262	−	0.948672i	$-0.602428\pi$
$457$	−6179.47	−0.632524	−0.316262	+	0.948672i	$0.602428\pi$
$458$	−22101.7	−2.25490
$459$	0	0
$460$	−43315.2	−4.39039
$461$	−16827.2	−1.70004	−0.850022	−	0.526747i	$-0.823411\pi$
$461$	−16827.2	−1.70004	−0.850022	+	0.526747i	$0.823411\pi$
$462$	0	0
$463$	−818.694	−0.0821770	−0.0410885	−	0.999156i	$-0.513083\pi$
$463$	−818.694	−0.0821770	−0.0410885	+	0.999156i	$0.513083\pi$
$464$	−5887.51	−0.589053
$465$	0	0
$466$	26415.2	2.62588
$467$	16789.7	1.66367	0.831836	−	0.555021i	$-0.187290\pi$
$467$	16789.7	1.66367	0.831836	+	0.555021i	$0.187290\pi$
$468$	0	0
$469$	2309.82	0.227415
$470$	−35534.7	−3.48743
$471$	0	0
$472$	11653.8	1.13646
$473$	0	0
$474$	0	0
$475$	2188.40	0.211391
$476$	29760.8	2.86572
$477$	0	0
$478$	−636.949	−0.0609485
$479$	−4656.56	−0.444183	−0.222092	−	0.975026i	$-0.571288\pi$
$479$	−4656.56	−0.444183	−0.222092	+	0.975026i	$0.571288\pi$
$480$	0	0
$481$	−4835.00	−0.458330
$482$	2283.20	0.215761
$483$	0	0
$484$	0	0
$485$	18960.4	1.77515
$486$	0	0
$487$	17627.2	1.64018	0.820089	−	0.572236i	$-0.193924\pi$
$487$	17627.2	1.64018	0.820089	+	0.572236i	$0.193924\pi$
$488$	14426.0	1.33819
$489$	0	0
$490$	14598.3	1.34589
$491$	−14954.8	−1.37454	−0.687269	−	0.726403i	$-0.741191\pi$
$491$	−14954.8	−1.37454	−0.687269	+	0.726403i	$0.741191\pi$
$492$	0	0
$493$	−6301.69	−0.575687
$494$	−3489.09	−0.317776
$495$	0	0
$496$	8394.41	0.759920
$497$	5648.32	0.509782
$498$	0	0
$499$	−21474.7	−1.92653	−0.963265	−	0.268554i	$-0.913454\pi$
$499$	−21474.7	−1.92653	−0.963265	+	0.268554i	$0.913454\pi$
$500$	7336.14	0.656164
$501$	0	0
$502$	−7538.77	−0.670263
$503$	11832.4	1.04887	0.524435	−	0.851450i	$-0.324277\pi$
$503$	11832.4	1.04887	0.524435	+	0.851450i	$0.324277\pi$
$504$	0	0
$505$	18492.5	1.62952
$506$	0	0
$507$	0	0
$508$	−25335.2	−2.21274
$509$	13571.6	1.18183	0.590914	−	0.806735i	$-0.298767\pi$
$509$	13571.6	1.18183	0.590914	+	0.806735i	$0.298767\pi$
$510$	0	0
$511$	3335.97	0.288796
$512$	−25795.1	−2.22655
$513$	0	0
$514$	−7497.31	−0.643370
$515$	−13621.7	−1.16552
$516$	0	0
$517$	0	0
$518$	10017.6	0.849706
$519$	0	0
$520$	23973.5	2.02174
$521$	2124.44	0.178644	0.0893221	−	0.996003i	$-0.471530\pi$
$521$	2124.44	0.178644	0.0893221	+	0.996003i	$0.471530\pi$
$522$	0	0
$523$	5558.58	0.464741	0.232371	−	0.972627i	$-0.425352\pi$
$523$	5558.58	0.464741	0.232371	+	0.972627i	$0.425352\pi$
$524$	27670.1	2.30682
$525$	0	0
$526$	−3765.11	−0.312104
$527$	8984.95	0.742677
$528$	0	0
$529$	13050.8	1.07264
$530$	−462.966	−0.0379433
$531$	0	0
$532$	5026.83	0.409663
$533$	−8734.21	−0.709794
$534$	0	0
$535$	23649.9	1.91117
$536$	9842.72	0.793173
$537$	0	0
$538$	−17563.4	−1.40746
$539$	0	0
$540$	0	0
$541$	−13186.0	−1.04789	−0.523946	−	0.851751i	$-0.675541\pi$
$541$	−13186.0	−1.04789	−0.523946	+	0.851751i	$0.675541\pi$
$542$	−8810.92	−0.698268
$543$	0	0
$544$	27947.2	2.20262
$545$	4434.67	0.348551
$546$	0	0
$547$	−20380.7	−1.59308	−0.796539	−	0.604587i	$-0.793338\pi$
$547$	−20380.7	−1.59308	−0.796539	+	0.604587i	$0.793338\pi$
$548$	54237.7	4.22796
$549$	0	0
$550$	0	0
$551$	−1064.40	−0.0822961
$552$	0	0
$553$	−6091.74	−0.468439
$554$	−26422.7	−2.02634
$555$	0	0
$556$	39299.8	2.99763
$557$	−8523.55	−0.648392	−0.324196	−	0.945990i	$-0.605094\pi$
$557$	−8523.55	−0.648392	−0.324196	+	0.945990i	$0.605094\pi$
$558$	0	0
$559$	3160.55	0.239136
$560$	−22742.7	−1.71616
$561$	0	0
$562$	25728.9	1.93115
$563$	24660.1	1.84600	0.923000	−	0.384801i	$-0.125730\pi$
$563$	24660.1	1.84600	0.923000	+	0.384801i	$0.125730\pi$
$564$	0	0
$565$	−1634.14	−0.121679
$566$	9624.08	0.714718
$567$	0	0
$568$	24068.9	1.77801
$569$	6276.91	0.462464	0.231232	−	0.972899i	$-0.425724\pi$
$569$	6276.91	0.462464	0.231232	+	0.972899i	$0.425724\pi$
$570$	0	0
$571$	−13032.1	−0.955127	−0.477563	−	0.878597i	$-0.658480\pi$
$571$	−13032.1	−0.955127	−0.477563	+	0.878597i	$0.658480\pi$
$572$	0	0
$573$	0	0
$574$	18096.3	1.31590
$575$	15579.1	1.12990
$576$	0	0
$577$	7724.04	0.557289	0.278645	−	0.960394i	$-0.410115\pi$
$577$	7724.04	0.557289	0.278645	+	0.960394i	$0.410115\pi$
$578$	64201.6	4.62013
$579$	0	0
$580$	13015.3	0.931779
$581$	−3344.63	−0.238827
$582$	0	0
$583$	0	0
$584$	14215.4	1.00725
$585$	0	0
$586$	24657.6	1.73822
$587$	−3858.98	−0.271341	−0.135670	−	0.990754i	$-0.543319\pi$
$587$	−3858.98	−0.271341	−0.135670	+	0.990754i	$0.543319\pi$
$588$	0	0
$589$	1517.63	0.106168
$590$	−16963.7	−1.18370
$591$	0	0
$592$	19545.3	1.35694
$593$	26982.6	1.86854	0.934270	−	0.356567i	$-0.116053\pi$
$593$	26982.6	1.86854	0.934270	+	0.356567i	$0.116053\pi$
$594$	0	0
$595$	−24342.6	−1.67722
$596$	−2426.49	−0.166767
$597$	0	0
$598$	−24838.6	−1.69854
$599$	22912.4	1.56289	0.781447	−	0.623971i	$-0.214482\pi$
$599$	22912.4	1.56289	0.781447	+	0.623971i	$0.214482\pi$
$600$	0	0
$601$	−3091.89	−0.209851	−0.104926	−	0.994480i	$-0.533460\pi$
$601$	−3091.89	−0.209851	−0.104926	+	0.994480i	$0.533460\pi$
$602$	−6548.31	−0.443338
$603$	0	0
$604$	−3889.33	−0.262011
$605$	0	0
$606$	0	0
$607$	13059.0	0.873225	0.436612	−	0.899650i	$-0.356178\pi$
$607$	13059.0	0.873225	0.436612	+	0.899650i	$0.356178\pi$
$608$	4720.50	0.314871
$609$	0	0
$610$	−20999.0	−1.39381
$611$	−14169.5	−0.938197
$612$	0	0
$613$	27025.2	1.78065	0.890325	−	0.455326i	$-0.150477\pi$
$613$	27025.2	1.78065	0.890325	+	0.455326i	$0.150477\pi$
$614$	3069.13	0.201727
$615$	0	0
$616$	0	0
$617$	−23885.0	−1.55847	−0.779235	−	0.626732i	$-0.784392\pi$
$617$	−23885.0	−1.55847	−0.779235	+	0.626732i	$0.784392\pi$
$618$	0	0
$619$	−14347.6	−0.931631	−0.465815	−	0.884882i	$-0.654239\pi$
$619$	−14347.6	−0.931631	−0.465815	+	0.884882i	$0.654239\pi$
$620$	−18557.3	−1.20206
$621$	0	0
$622$	−26295.0	−1.69507
$623$	956.236	0.0614940
$624$	0	0
$625$	−18263.6	−1.16887
$626$	−48895.9	−3.12184
$627$	0	0
$628$	−53416.0	−3.39415
$629$	20920.3	1.32615
$630$	0	0
$631$	4672.42	0.294780	0.147390	−	0.989078i	$-0.452913\pi$
$631$	4672.42	0.294780	0.147390	+	0.989078i	$0.452913\pi$
$632$	−25958.4	−1.63381
$633$	0	0
$634$	−33839.7	−2.11979
$635$	20722.7	1.29505
$636$	0	0
$637$	5821.13	0.362074
$638$	0	0
$639$	0	0
$640$	17833.9	1.10148
$641$	−1007.49	−0.0620802	−0.0310401	−	0.999518i	$-0.509882\pi$
$641$	−1007.49	−0.0620802	−0.0310401	+	0.999518i	$0.509882\pi$
$642$	0	0
$643$	−238.620	−0.0146349	−0.00731747	−	0.999973i	$-0.502329\pi$
$643$	−238.620	−0.0146349	−0.00731747	+	0.999973i	$0.502329\pi$
$644$	35785.8	2.18969
$645$	0	0
$646$	15096.8	0.919464
$647$	−8030.33	−0.487952	−0.243976	−	0.969781i	$-0.578452\pi$
$647$	−8030.33	−0.487952	−0.243976	+	0.969781i	$0.578452\pi$
$648$	0	0
$649$	0	0
$650$	−15344.8	−0.925960
$651$	0	0
$652$	−11564.3	−0.694620
$653$	−292.050	−0.0175020	−0.00875101	−	0.999962i	$-0.502786\pi$
$653$	−292.050	−0.0175020	−0.00875101	+	0.999962i	$0.502786\pi$
$654$	0	0
$655$	−22632.5	−1.35011
$656$	35307.8	2.10143
$657$	0	0
$658$	29357.8	1.73934
$659$	−9034.21	−0.534025	−0.267013	−	0.963693i	$-0.586037\pi$
$659$	−9034.21	−0.534025	−0.267013	+	0.963693i	$0.586037\pi$
$660$	0	0
$661$	11884.6	0.699332	0.349666	−	0.936874i	$-0.386295\pi$
$661$	11884.6	0.699332	0.349666	+	0.936874i	$0.386295\pi$
$662$	−33564.9	−1.97060
$663$	0	0
$664$	−14252.3	−0.832975
$665$	−4111.65	−0.239764
$666$	0	0
$667$	−7577.45	−0.439880
$668$	−28000.7	−1.62182
$669$	0	0
$670$	−14327.4	−0.826141
$671$	0	0
$672$	0	0
$673$	7483.49	0.428629	0.214314	−	0.976765i	$-0.431248\pi$
$673$	7483.49	0.428629	0.214314	+	0.976765i	$0.431248\pi$
$674$	13787.0	0.787919
$675$	0	0
$676$	−23107.8	−1.31474
$677$	−31698.6	−1.79952	−0.899761	−	0.436384i	$-0.856259\pi$
$677$	−31698.6	−1.79952	−0.899761	+	0.436384i	$0.856259\pi$
$678$	0	0
$679$	−15664.6	−0.885347
$680$	−103730.	−5.84978
$681$	0	0
$682$	0	0
$683$	7905.89	0.442914	0.221457	−	0.975170i	$-0.428919\pi$
$683$	7905.89	0.442914	0.221457	+	0.975170i	$0.428919\pi$
$684$	0	0
$685$	−44363.3	−2.47450
$686$	−33751.6	−1.87849
$687$	0	0
$688$	−12776.4	−0.707989
$689$	−184.609	−0.0102076
$690$	0	0
$691$	−6903.03	−0.380034	−0.190017	−	0.981781i	$-0.560854\pi$
$691$	−6903.03	−0.380034	−0.190017	+	0.981781i	$0.560854\pi$
$692$	60749.5	3.33721
$693$	0	0
$694$	−3988.64	−0.218165
$695$	−32144.9	−1.75443
$696$	0	0
$697$	37791.6	2.05374
$698$	27755.0	1.50508
$699$	0	0
$700$	22107.8	1.19371
$701$	20702.1	1.11542	0.557709	−	0.830036i	$-0.311680\pi$
$701$	20702.1	1.11542	0.557709	+	0.830036i	$0.311680\pi$
$702$	0	0
$703$	3533.61	0.189577
$704$	0	0
$705$	0	0
$706$	23083.7	1.23055
$707$	−15278.0	−0.812713
$708$	0	0
$709$	−6302.17	−0.333826	−0.166913	−	0.985972i	$-0.553380\pi$
$709$	−6302.17	−0.333826	−0.166913	+	0.985972i	$0.553380\pi$
$710$	−35035.4	−1.85191
$711$	0	0
$712$	4074.76	0.214477
$713$	10803.9	0.567476
$714$	0	0
$715$	0	0
$716$	−77818.7	−4.06176
$717$	0	0
$718$	44012.0	2.28762
$719$	−1385.72	−0.0718756	−0.0359378	−	0.999354i	$-0.511442\pi$
$719$	−1385.72	−0.0718756	−0.0359378	+	0.999354i	$0.511442\pi$
$720$	0	0
$721$	11253.9	0.581299
$722$	−32599.5	−1.68037
$723$	0	0
$724$	13548.6	0.695484
$725$	−4681.20	−0.239801
$726$	0	0
$727$	−11818.1	−0.602900	−0.301450	−	0.953482i	$-0.597471\pi$
$727$	−11818.1	−0.602900	−0.301450	+	0.953482i	$0.597471\pi$
$728$	−19806.2	−1.00833
$729$	0	0
$730$	−20692.3	−1.04912
$731$	−13675.2	−0.691924
$732$	0	0
$733$	9537.65	0.480602	0.240301	−	0.970698i	$-0.422754\pi$
$733$	9537.65	0.480602	0.240301	+	0.970698i	$0.422754\pi$
$734$	11161.7	0.561290
$735$	0	0
$736$	33605.1	1.68301
$737$	0	0
$738$	0	0
$739$	3411.51	0.169817	0.0849084	−	0.996389i	$-0.472940\pi$
$739$	3411.51	0.169817	0.0849084	+	0.996389i	$0.472940\pi$
$740$	−43208.2	−2.14644
$741$	0	0
$742$	382.490	0.0189240
$743$	−2452.91	−0.121115	−0.0605576	−	0.998165i	$-0.519288\pi$
$743$	−2452.91	−0.121115	−0.0605576	+	0.998165i	$0.519288\pi$
$744$	0	0
$745$	1984.73	0.0976037
$746$	65670.2	3.22300
$747$	0	0
$748$	0	0
$749$	−19538.9	−0.953185
$750$	0	0
$751$	−11556.1	−0.561500	−0.280750	−	0.959781i	$-0.590583\pi$
$751$	−11556.1	−0.561500	−0.280750	+	0.959781i	$0.590583\pi$
$752$	57280.0	2.77764
$753$	0	0
$754$	7463.51	0.360484
$755$	3181.24	0.153347
$756$	0	0
$757$	25642.8	1.23118	0.615589	−	0.788067i	$-0.288918\pi$
$757$	25642.8	1.23118	0.615589	+	0.788067i	$0.288918\pi$
$758$	37423.9	1.79327
$759$	0	0
$760$	−17520.8	−0.836243
$761$	−6054.02	−0.288381	−0.144191	−	0.989550i	$-0.546058\pi$
$761$	−6054.02	−0.288381	−0.144191	+	0.989550i	$0.546058\pi$
$762$	0	0
$763$	−3663.80	−0.173838
$764$	−75458.7	−3.57330
$765$	0	0
$766$	−57979.7	−2.73484
$767$	−6764.30	−0.318442
$768$	0	0
$769$	−23509.7	−1.10245	−0.551224	−	0.834357i	$-0.685839\pi$
$769$	−23509.7	−1.10245	−0.551224	+	0.834357i	$0.685839\pi$
$770$	0	0
$771$	0	0
$772$	−56277.3	−2.62366
$773$	−4985.36	−0.231967	−0.115984	−	0.993251i	$-0.537002\pi$
$773$	−4985.36	−0.231967	−0.115984	+	0.993251i	$0.537002\pi$
$774$	0	0
$775$	6674.46	0.309360
$776$	−66750.6	−3.08789
$777$	0	0
$778$	15147.3	0.698016
$779$	6383.30	0.293589
$780$	0	0
$781$	0	0
$782$	107473.	4.91462
$783$	0	0
$784$	−23531.7	−1.07196
$785$	43691.1	1.98650
$786$	0	0
$787$	−23093.6	−1.04599	−0.522996	−	0.852335i	$-0.675186\pi$
$787$	−23093.6	−1.04599	−0.522996	+	0.852335i	$0.675186\pi$
$788$	55876.6	2.52604
$789$	0	0
$790$	37785.8	1.70172
$791$	1350.08	0.0606868
$792$	0	0
$793$	−8373.39	−0.374965
$794$	63503.2	2.83834
$795$	0	0
$796$	−17748.7	−0.790308
$797$	−17376.5	−0.772281	−0.386141	−	0.922440i	$-0.626192\pi$
$797$	−17376.5	−0.772281	−0.386141	+	0.922440i	$0.626192\pi$
$798$	0	0
$799$	61309.6	2.71461
$800$	20760.6	0.917496
$801$	0	0
$802$	−38909.1	−1.71313
$803$	0	0
$804$	0	0
$805$	−29270.7	−1.28156
$806$	−10641.5	−0.465050
$807$	0	0
$808$	−65103.3	−2.83456
$809$	3335.36	0.144951	0.0724753	−	0.997370i	$-0.476910\pi$
$809$	3335.36	0.144951	0.0724753	+	0.997370i	$0.476910\pi$
$810$	0	0
$811$	−4873.73	−0.211023	−0.105512	−	0.994418i	$-0.533648\pi$
$811$	−4873.73	−0.211023	−0.105512	+	0.994418i	$0.533648\pi$
$812$	−10752.9	−0.464720
$813$	0	0
$814$	0	0
$815$	9458.91	0.406541
$816$	0	0
$817$	−2309.85	−0.0989125
$818$	37285.3	1.59370
$819$	0	0
$820$	−78053.7	−3.32409
$821$	18640.4	0.792392	0.396196	−	0.918166i	$-0.370330\pi$
$821$	18640.4	0.792392	0.396196	+	0.918166i	$0.370330\pi$
$822$	0	0
$823$	18819.8	0.797106	0.398553	−	0.917145i	$-0.369512\pi$
$823$	18819.8	0.797106	0.398553	+	0.917145i	$0.369512\pi$
$824$	47955.6	2.02744
$825$	0	0
$826$	14014.9	0.590364
$827$	−34989.3	−1.47122	−0.735610	−	0.677406i	$-0.763104\pi$
$827$	−34989.3	−1.47122	−0.735610	+	0.677406i	$0.763104\pi$
$828$	0	0
$829$	14231.8	0.596250	0.298125	−	0.954527i	$-0.403639\pi$
$829$	14231.8	0.596250	0.298125	+	0.954527i	$0.403639\pi$
$830$	20746.0	0.867597
$831$	0	0
$832$	−2971.89	−0.123836
$833$	−25187.2	−1.04764
$834$	0	0
$835$	22902.9	0.949207
$836$	0	0
$837$	0	0
$838$	−56860.9	−2.34395
$839$	25354.5	1.04331	0.521654	−	0.853157i	$-0.325315\pi$
$839$	25354.5	1.04331	0.521654	+	0.853157i	$0.325315\pi$
$840$	0	0
$841$	−22112.1	−0.906644
$842$	−26153.5	−1.07044
$843$	0	0
$844$	−60758.3	−2.47795
$845$	18900.8	0.769476
$846$	0	0
$847$	0	0
$848$	746.276	0.0302208
$849$	0	0
$850$	66394.9	2.67921
$851$	25155.6	1.01330
$852$	0	0
$853$	9664.40	0.387928	0.193964	−	0.981009i	$-0.437865\pi$
$853$	9664.40	0.387928	0.193964	+	0.981009i	$0.437865\pi$
$854$	17348.7	0.695154
$855$	0	0
$856$	−83260.0	−3.32449
$857$	−38619.7	−1.53935	−0.769676	−	0.638435i	$-0.779582\pi$
$857$	−38619.7	−1.53935	−0.769676	+	0.638435i	$0.779582\pi$
$858$	0	0
$859$	−30501.9	−1.21154	−0.605769	−	0.795641i	$-0.707134\pi$
$859$	−30501.9	−1.21154	−0.605769	+	0.795641i	$0.707134\pi$
$860$	28244.4	1.11991
$861$	0	0
$862$	−73987.1	−2.92344
$863$	29563.6	1.16611	0.583056	−	0.812432i	$-0.301857\pi$
$863$	29563.6	1.16611	0.583056	+	0.812432i	$0.301857\pi$
$864$	0	0
$865$	−49689.5	−1.95317
$866$	21712.0	0.851966
$867$	0	0
$868$	15331.5	0.599522
$869$	0	0
$870$	0	0
$871$	−5713.07	−0.222250
$872$	−15612.3	−0.606308
$873$	0	0
$874$	18153.1	0.702559
$875$	4957.47	0.191535
$876$	0	0
$877$	23007.5	0.885872	0.442936	−	0.896553i	$-0.353937\pi$
$877$	23007.5	0.885872	0.442936	+	0.896553i	$0.353937\pi$
$878$	−9274.60	−0.356495
$879$	0	0
$880$	0	0
$881$	−34057.4	−1.30241	−0.651205	−	0.758902i	$-0.725736\pi$
$881$	−34057.4	−1.30241	−0.651205	+	0.758902i	$0.725736\pi$
$882$	0	0
$883$	34653.7	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$883$	34653.7	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$884$	−73609.8	−2.80064
$885$	0	0
$886$	−24584.5	−0.932203
$887$	−5147.89	−0.194870	−0.0974348	−	0.995242i	$-0.531064\pi$
$887$	−5147.89	−0.194870	−0.0974348	+	0.995242i	$0.531064\pi$
$888$	0	0
$889$	−17120.6	−0.645900
$890$	−5931.34	−0.223392
$891$	0	0
$892$	89754.8	3.36907
$893$	10355.7	0.388062
$894$	0	0
$895$	63651.2	2.37723
$896$	−14733.8	−0.549356
$897$	0	0
$898$	47029.3	1.74765
$899$	−3246.36	−0.120436
$900$	0	0
$901$	798.776	0.0295351
$902$	0	0
$903$	0	0
$904$	5753.02	0.211662
$905$	−11082.0	−0.407047
$906$	0	0
$907$	40684.2	1.48941	0.744706	−	0.667392i	$-0.232589\pi$
$907$	40684.2	1.48941	0.744706	+	0.667392i	$0.232589\pi$
$908$	−19405.3	−0.709237
$909$	0	0
$910$	28830.5	1.05024
$911$	−25223.0	−0.917315	−0.458657	−	0.888613i	$-0.651669\pi$
$911$	−25223.0	−0.917315	−0.458657	+	0.888613i	$0.651669\pi$
$912$	0	0
$913$	0	0
$914$	−31667.2	−1.14601
$915$	0	0
$916$	−78758.9	−2.84090
$917$	18698.3	0.673363
$918$	0	0
$919$	9198.23	0.330165	0.165083	−	0.986280i	$-0.447211\pi$
$919$	9198.23	0.330165	0.165083	+	0.986280i	$0.447211\pi$
$920$	−124729.	−4.46979
$921$	0	0
$922$	−86232.2	−3.08016
$923$	−13970.4	−0.498205
$924$	0	0
$925$	15540.6	0.552403
$926$	−4195.46	−0.148889
$927$	0	0
$928$	−10097.6	−0.357189
$929$	33459.9	1.18168	0.590841	−	0.806788i	$-0.298796\pi$
$929$	33459.9	1.18168	0.590841	+	0.806788i	$0.298796\pi$
$930$	0	0
$931$	−4254.31	−0.149763
$932$	94129.9	3.30829
$933$	0	0
$934$	86040.2	3.01426
$935$	0	0
$936$	0	0
$937$	31048.7	1.08252	0.541258	−	0.840857i	$-0.317948\pi$
$937$	31048.7	1.08252	0.541258	+	0.840857i	$0.317948\pi$
$938$	11836.9	0.412034
$939$	0	0
$940$	−126627.	−4.39374
$941$	21640.9	0.749704	0.374852	−	0.927085i	$-0.377693\pi$
$941$	21640.9	0.749704	0.374852	+	0.927085i	$0.377693\pi$
$942$	0	0
$943$	45442.4	1.56926
$944$	27344.5	0.942783
$945$	0	0
$946$	0	0
$947$	31725.1	1.08862	0.544311	−	0.838883i	$-0.316791\pi$
$947$	31725.1	1.08862	0.544311	+	0.838883i	$0.316791\pi$
$948$	0	0
$949$	−8251.12	−0.282237
$950$	11214.6	0.383000
$951$	0	0
$952$	85698.6	2.91755
$953$	52072.7	1.76999	0.884995	−	0.465601i	$-0.154162\pi$
$953$	52072.7	1.76999	0.884995	+	0.465601i	$0.154162\pi$
$954$	0	0
$955$	61720.8	2.09135
$956$	−2269.75	−0.0767877
$957$	0	0
$958$	−23862.9	−0.804777
$959$	36651.7	1.23414
$960$	0	0
$961$	−25162.3	−0.844629
$962$	−24777.3	−0.830408
$963$	0	0
$964$	8136.13	0.271833
$965$	46031.5	1.53555
$966$	0	0
$967$	4519.31	0.150291	0.0751454	−	0.997173i	$-0.476058\pi$
$967$	4519.31	0.150291	0.0751454	+	0.997173i	$0.476058\pi$
$968$	0	0
$969$	0	0
$970$	97164.2	3.21624
$971$	−24955.7	−0.824785	−0.412392	−	0.911006i	$-0.635307\pi$
$971$	−24955.7	−0.824785	−0.412392	+	0.911006i	$0.635307\pi$
$972$	0	0
$973$	26557.3	0.875012
$974$	90332.2	2.97169
$975$	0	0
$976$	33849.2	1.11013
$977$	−56789.9	−1.85964	−0.929820	−	0.368014i	$-0.880038\pi$
$977$	−56789.9	−1.85964	−0.929820	+	0.368014i	$0.880038\pi$
$978$	0	0
$979$	0	0
$980$	52020.8	1.69566
$981$	0	0
$982$	−76636.8	−2.49041
$983$	31484.6	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$983$	31484.6	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$984$	0	0
$985$	−45703.8	−1.47842
$986$	−32293.5	−1.04304
$987$	0	0
$988$	−12433.3	−0.400359
$989$	−16443.7	−0.528696
$990$	0	0
$991$	9490.27	0.304206	0.152103	−	0.988365i	$-0.451395\pi$
$991$	9490.27	0.304206	0.152103	+	0.988365i	$0.451395\pi$
$992$	14397.2	0.460799
$993$	0	0
$994$	28945.2	0.923629
$995$	14517.4	0.462545
$996$	0	0
$997$	−40887.6	−1.29882	−0.649409	−	0.760439i	$-0.724984\pi$
$997$	−40887.6	−1.29882	−0.649409	+	0.760439i	$0.724984\pi$
$998$	−110049.	−3.49051
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.bk.1.6		6
3.2	odd	2		363.4.a.u.1.1		6
11.7	odd	10		99.4.f.d.82.1		12
11.8	odd	10		99.4.f.d.64.1		12
11.10	odd	2		1089.4.a.bi.1.1		6
33.8	even	10		33.4.e.c.31.3	yes	12
33.29	even	10		33.4.e.c.16.3	✓	12
33.32	even	2		363.4.a.v.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.16.3	✓	12	33.29	even	10
33.4.e.c.31.3	yes	12	33.8	even	10
99.4.f.d.64.1		12	11.8	odd	10
99.4.f.d.82.1		12	11.7	odd	10
363.4.a.u.1.1		6	3.2	odd	2
363.4.a.v.1.6		6	33.32	even	2
1089.4.a.bi.1.1		6	11.10	odd	2
1089.4.a.bk.1.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+5.12458 q^{2} +18.2613 q^{4} -14.9367 q^{5} +12.3403 q^{7} +52.5849 q^{8} -76.5442 q^{10} -30.5222 q^{13} +63.2387 q^{14} +123.385 q^{16} +132.065 q^{17} +22.3068 q^{19} -272.763 q^{20} +158.801 q^{23} +98.1044 q^{25} -156.413 q^{26} +225.349 q^{28} -47.7165 q^{29} +68.0343 q^{31} +211.617 q^{32} +676.778 q^{34} -184.323 q^{35} +158.409 q^{37} +114.313 q^{38} -785.444 q^{40} +286.159 q^{41} -103.549 q^{43} +813.790 q^{46} +464.238 q^{47} -190.718 q^{49} +502.744 q^{50} -557.375 q^{52} +6.04835 q^{53} +648.912 q^{56} -244.527 q^{58} +221.619 q^{59} +274.338 q^{61} +348.647 q^{62} +97.3683 q^{64} +455.900 q^{65} +187.178 q^{67} +2411.68 q^{68} -944.576 q^{70} +457.714 q^{71} +270.332 q^{73} +811.780 q^{74} +407.352 q^{76} -493.647 q^{79} -1842.96 q^{80} +1466.45 q^{82} -271.033 q^{83} -1972.61 q^{85} -530.646 q^{86} +77.4891 q^{89} -376.652 q^{91} +2899.92 q^{92} +2379.02 q^{94} -333.190 q^{95} -1269.39 q^{97} -977.348 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} + 17 q^{4} - 9 q^{5} - q^{7} + 24 q^{8} - 50 q^{10} - 66 q^{13} + 42 q^{14} - 71 q^{16} + 80 q^{17} + 90 q^{19} - 455 q^{20} + 196 q^{23} + 351 q^{25} - 93 q^{26} + 52 q^{28} + 579 q^{29}+ \cdots + 1405 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 + 17 * q^4 - 9 * q^5 - q^7 + 24 * q^8 - 50 * q^10 - 66 * q^13 + 42 * q^14 - 71 * q^16 + 80 * q^17 + 90 * q^19 - 455 * q^20 + 196 * q^23 + 351 * q^25 - 93 * q^26 + 52 * q^28 + 579 * q^29 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 192 * q^37 + 769 * q^38 - 1476 * q^40 + 990 * q^41 - 208 * q^43 - 493 * q^46 - 680 * q^47 - 963 * q^49 + 1656 * q^50 - 887 * q^52 - 197 * q^53 + 429 * q^56 + 395 * q^58 + 1229 * q^59 - 928 * q^61 + 2117 * q^62 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 665 * q^70 + 786 * q^71 - 593 * q^73 + 1543 * q^74 + 434 * q^76 + 421 * q^79 - 18 * q^80 - 739 * q^82 + 2043 * q^83 - 1900 * q^85 - 2683 * q^86 - 158 * q^89 + 698 * q^91 + 3915 * q^92 + 4934 * q^94 + 1038 * q^95 - 1711 * q^97 + 1405 * q^98

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