LMFDB - Embedded newform 1089.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,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, 2, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$8.69570878012$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.54336$ 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)

$f(q)$	$=$	$q-1.54336 q^{2} +0.381966 q^{4} +1.54336 q^{5} +0.236068 q^{7} +2.49721 q^{8} -2.38197 q^{10} -4.23607 q^{13} -0.364338 q^{14} -4.61803 q^{16} -5.94827 q^{17} -4.61803 q^{19} +0.589512 q^{20} +7.49164 q^{23} -2.61803 q^{25} +6.53779 q^{26} +0.0901699 q^{28} +1.90770 q^{29} -2.38197 q^{31} +2.13287 q^{32} +9.18034 q^{34} +0.364338 q^{35} +6.23607 q^{37} +7.12730 q^{38} +3.85410 q^{40} +5.58394 q^{41} -10.7082 q^{43} -11.5623 q^{46} +0.953850 q^{47} -6.94427 q^{49} +4.04057 q^{50} -1.61803 q^{52} -9.03500 q^{53} +0.589512 q^{56} -2.94427 q^{58} -8.44549 q^{59} +4.32624 q^{61} +3.67624 q^{62} +5.94427 q^{64} -6.53779 q^{65} +3.85410 q^{67} -2.27204 q^{68} -0.562306 q^{70} -7.71681 q^{71} -6.47214 q^{73} -9.62451 q^{74} -1.76393 q^{76} -0.527864 q^{79} -7.12730 q^{80} -8.61803 q^{82} +4.40491 q^{83} -9.18034 q^{85} +16.5266 q^{86} -16.7518 q^{89} -1.00000 q^{91} +2.86155 q^{92} -1.47214 q^{94} -7.12730 q^{95} -12.7984 q^{97} +10.7175 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} - 8 q^{7} - 14 q^{10} - 8 q^{13} - 14 q^{16} - 14 q^{19} - 6 q^{25} - 22 q^{28} - 14 q^{31} - 8 q^{34} + 16 q^{37} + 2 q^{40} - 16 q^{43} - 6 q^{46} + 8 q^{49} - 2 q^{52} + 24 q^{58} - 14 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 - 8 * q^7 - 14 * q^10 - 8 * q^13 - 14 * q^16 - 14 * q^19 - 6 * q^25 - 22 * q^28 - 14 * q^31 - 8 * q^34 + 16 * q^37 + 2 * q^40 - 16 * q^43 - 6 * q^46 + 8 * q^49 - 2 * q^52 + 24 * q^58 - 14 * q^61 - 12 * q^64 + 2 * q^67 + 38 * q^70 - 8 * q^73 - 16 * q^76 - 20 * q^79 - 30 * q^82 + 8 * q^85 - 4 * q^91 + 12 * q^94 - 2 * q^97

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 $

$2$	−1.54336	−1.09132	−0.545661	−	0.838006i	$-0.683721\pi$
$2$	−1.54336	−1.09132	−0.545661	+	0.838006i	$0.683721\pi$
$3$	0	0
$3$	0	0
$4$	0.381966	0.190983
$5$	1.54336	0.690212	0.345106	−	0.938564i	$-0.387843\pi$
$5$	1.54336	0.690212	0.345106	+	0.938564i	$0.387843\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	2.49721	0.882898
$9$	0	0
$10$	−2.38197	−0.753244
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	−0.364338	−0.0973735
$15$	0	0
$16$	−4.61803	−1.15451
$17$	−5.94827	−1.44267	−0.721334	−	0.692587i	$-0.756471\pi$
$17$	−5.94827	−1.44267	−0.721334	+	0.692587i	$0.756471\pi$
$18$	0	0
$19$	−4.61803	−1.05945	−0.529725	−	0.848170i	$-0.677705\pi$
$19$	−4.61803	−1.05945	−0.529725	+	0.848170i	$0.677705\pi$
$20$	0.589512	0.131819
$21$	0	0
$22$	0	0
$23$	7.49164	1.56211	0.781057	−	0.624460i	$-0.214681\pi$
$23$	7.49164	1.56211	0.781057	+	0.624460i	$0.214681\pi$
$24$	0	0
$25$	−2.61803	−0.523607
$26$	6.53779	1.28217
$27$	0	0
$28$	0.0901699	0.0170405
$29$	1.90770	0.354251	0.177126	−	0.984188i	$-0.443320\pi$
$29$	1.90770	0.354251	0.177126	+	0.984188i	$0.443320\pi$
$30$	0	0
$31$	−2.38197	−0.427814	−0.213907	−	0.976854i	$-0.568619\pi$
$31$	−2.38197	−0.427814	−0.213907	+	0.976854i	$0.568619\pi$
$32$	2.13287	0.377042
$33$	0	0
$34$	9.18034	1.57442
$35$	0.364338	0.0615844
$36$	0	0
$37$	6.23607	1.02520	0.512602	−	0.858627i	$-0.328682\pi$
$37$	6.23607	1.02520	0.512602	+	0.858627i	$0.328682\pi$
$38$	7.12730	1.15620
$39$	0	0
$40$	3.85410	0.609387
$41$	5.58394	0.872064	0.436032	−	0.899931i	$-0.356383\pi$
$41$	5.58394	0.872064	0.436032	+	0.899931i	$0.356383\pi$
$42$	0	0
$43$	−10.7082	−1.63299	−0.816493	−	0.577355i	$-0.804085\pi$
$43$	−10.7082	−1.63299	−0.816493	+	0.577355i	$0.804085\pi$
$44$	0	0
$45$	0	0
$46$	−11.5623	−1.70477
$47$	0.953850	0.139133	0.0695667	−	0.997577i	$-0.477838\pi$
$47$	0.953850	0.139133	0.0695667	+	0.997577i	$0.477838\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	4.04057	0.571423
$51$	0	0
$52$	−1.61803	−0.224381
$53$	−9.03500	−1.24105	−0.620526	−	0.784186i	$-0.713081\pi$
$53$	−9.03500	−1.24105	−0.620526	+	0.784186i	$0.713081\pi$
$54$	0	0
$55$	0	0
$56$	0.589512	0.0787768
$57$	0	0
$58$	−2.94427	−0.386602
$59$	−8.44549	−1.09951	−0.549754	−	0.835326i	$-0.685279\pi$
$59$	−8.44549	−1.09951	−0.549754	+	0.835326i	$0.685279\pi$
$60$	0	0
$61$	4.32624	0.553918	0.276959	−	0.960882i	$-0.410673\pi$
$61$	4.32624	0.553918	0.276959	+	0.960882i	$0.410673\pi$
$62$	3.67624	0.466882
$63$	0	0
$64$	5.94427	0.743034
$65$	−6.53779	−0.810913
$66$	0	0
$67$	3.85410	0.470853	0.235427	−	0.971892i	$-0.424351\pi$
$67$	3.85410	0.470853	0.235427	+	0.971892i	$0.424351\pi$
$68$	−2.27204	−0.275525
$69$	0	0
$70$	−0.562306	−0.0672084
$71$	−7.71681	−0.915817	−0.457908	−	0.888999i	$-0.651401\pi$
$71$	−7.71681	−0.915817	−0.457908	+	0.888999i	$0.651401\pi$
$72$	0	0
$73$	−6.47214	−0.757506	−0.378753	−	0.925498i	$-0.623647\pi$
$73$	−6.47214	−0.757506	−0.378753	+	0.925498i	$0.623647\pi$
$74$	−9.62451	−1.11883
$75$	0	0
$76$	−1.76393	−0.202337
$77$	0	0
$78$	0	0
$79$	−0.527864	−0.0593893	−0.0296947	−	0.999559i	$-0.509453\pi$
$79$	−0.527864	−0.0593893	−0.0296947	+	0.999559i	$0.509453\pi$
$80$	−7.12730	−0.796856
$81$	0	0
$82$	−8.61803	−0.951703
$83$	4.40491	0.483502	0.241751	−	0.970338i	$-0.422278\pi$
$83$	4.40491	0.483502	0.241751	+	0.970338i	$0.422278\pi$
$84$	0	0
$85$	−9.18034	−0.995748
$86$	16.5266	1.78211
$87$	0	0
$88$	0	0
$89$	−16.7518	−1.77569	−0.887844	−	0.460145i	$-0.847798\pi$
$89$	−16.7518	−1.77569	−0.887844	+	0.460145i	$0.847798\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	2.86155	0.298337
$93$	0	0
$94$	−1.47214	−0.151839
$95$	−7.12730	−0.731245
$96$	0	0
$97$	−12.7984	−1.29948	−0.649739	−	0.760157i	$-0.725122\pi$
$97$	−12.7984	−1.29948	−0.649739	+	0.760157i	$0.725122\pi$
$98$	10.7175	1.08263
$99$	0	0
$100$	−1.00000	−0.100000
$101$	10.5784	1.05259	0.526293	−	0.850303i	$-0.323581\pi$
$101$	10.5784	1.05259	0.526293	+	0.850303i	$0.323581\pi$
$102$	0	0
$103$	6.94427	0.684239	0.342120	−	0.939656i	$-0.388855\pi$
$103$	6.94427	0.684239	0.342120	+	0.939656i	$0.388855\pi$
$104$	−10.5784	−1.03729
$105$	0	0
$106$	13.9443	1.35439
$107$	−9.39934	−0.908668	−0.454334	−	0.890831i	$-0.650123\pi$
$107$	−9.39934	−0.908668	−0.454334	+	0.890831i	$0.650123\pi$
$108$	0	0
$109$	−0.291796	−0.0279490	−0.0139745	−	0.999902i	$-0.504448\pi$
$109$	−0.291796	−0.0279490	−0.0139745	+	0.999902i	$0.504448\pi$
$110$	0	0
$111$	0	0
$112$	−1.09017	−0.103011
$113$	11.3070	1.06368	0.531838	−	0.846846i	$-0.321501\pi$
$113$	11.3070	1.06368	0.531838	+	0.846846i	$0.321501\pi$
$114$	0	0
$115$	11.5623	1.07819
$116$	0.728677	0.0676559
$117$	0	0
$118$	13.0344	1.19992
$119$	−1.40420	−0.128723
$120$	0	0
$121$	0	0
$122$	−6.67695	−0.604503
$123$	0	0
$124$	−0.909830	−0.0817052
$125$	−11.7574	−1.05161
$126$	0	0
$127$	−7.23607	−0.642097	−0.321049	−	0.947063i	$-0.604035\pi$
$127$	−7.23607	−0.642097	−0.321049	+	0.947063i	$0.604035\pi$
$128$	−13.4399	−1.18793
$129$	0	0
$130$	10.0902	0.884966
$131$	−12.1217	−1.05908	−0.529540	−	0.848285i	$-0.677635\pi$
$131$	−12.1217	−1.05908	−0.529540	+	0.848285i	$0.677635\pi$
$132$	0	0
$133$	−1.09017	−0.0945297
$134$	−5.94827	−0.513853
$135$	0	0
$136$	−14.8541	−1.27373
$137$	−10.5784	−0.903770	−0.451885	−	0.892076i	$-0.649248\pi$
$137$	−10.5784	−0.903770	−0.451885	+	0.892076i	$0.649248\pi$
$138$	0	0
$139$	−13.6180	−1.15507	−0.577533	−	0.816367i	$-0.695985\pi$
$139$	−13.6180	−1.15507	−0.577533	+	0.816367i	$0.695985\pi$
$140$	0.139165	0.0117616
$141$	0	0
$142$	11.9098	0.999451
$143$	0	0
$144$	0	0
$145$	2.94427	0.244508
$146$	9.98885	0.826683
$147$	0	0
$148$	2.38197	0.195796
$149$	13.6651	1.11949	0.559744	−	0.828666i	$-0.310900\pi$
$149$	13.6651	1.11949	0.559744	+	0.828666i	$0.310900\pi$
$150$	0	0
$151$	−8.76393	−0.713199	−0.356599	−	0.934257i	$-0.616064\pi$
$151$	−8.76393	−0.713199	−0.356599	+	0.934257i	$0.616064\pi$
$152$	−11.5322	−0.935386
$153$	0	0
$154$	0	0
$155$	−3.67624	−0.295282
$156$	0	0
$157$	10.6525	0.850160	0.425080	−	0.905156i	$-0.360246\pi$
$157$	10.6525	0.850160	0.425080	+	0.905156i	$0.360246\pi$
$158$	0.814685	0.0648129
$159$	0	0
$160$	3.29180	0.260239
$161$	1.76854	0.139380
$162$	0	0
$163$	3.61803	0.283386	0.141693	−	0.989911i	$-0.454745\pi$
$163$	3.61803	0.283386	0.141693	+	0.989911i	$0.454745\pi$
$164$	2.13287	0.166549
$165$	0	0
$166$	−6.79837	−0.527656
$167$	7.71681	0.597145	0.298572	−	0.954387i	$-0.403490\pi$
$167$	7.71681	0.597145	0.298572	+	0.954387i	$0.403490\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	14.1686	1.08668
$171$	0	0
$172$	−4.09017	−0.311873
$173$	2.72239	0.206979	0.103490	−	0.994631i	$-0.466999\pi$
$173$	2.72239	0.206979	0.103490	+	0.994631i	$0.466999\pi$
$174$	0	0
$175$	−0.618034	−0.0467190
$176$	0	0
$177$	0	0
$178$	25.8541	1.93785
$179$	−18.6595	−1.39468	−0.697339	−	0.716742i	$-0.745633\pi$
$179$	−18.6595	−1.39468	−0.697339	+	0.716742i	$0.745633\pi$
$180$	0	0
$181$	7.76393	0.577089	0.288544	−	0.957467i	$-0.406829\pi$
$181$	7.76393	0.577089	0.288544	+	0.957467i	$0.406829\pi$
$182$	1.54336	0.114402
$183$	0	0
$184$	18.7082	1.37919
$185$	9.62451	0.707608
$186$	0	0
$187$	0	0
$188$	0.364338	0.0265721
$189$	0	0
$190$	11.0000	0.798024
$191$	3.67624	0.266003	0.133002	−	0.991116i	$-0.457538\pi$
$191$	3.67624	0.266003	0.133002	+	0.991116i	$0.457538\pi$
$192$	0	0
$193$	9.61803	0.692321	0.346161	−	0.938175i	$-0.387485\pi$
$193$	9.61803	0.692321	0.346161	+	0.938175i	$0.387485\pi$
$194$	19.7525	1.41815
$195$	0	0
$196$	−2.65248	−0.189463
$197$	12.1217	0.863637	0.431818	−	0.901961i	$-0.357872\pi$
$197$	12.1217	0.863637	0.431818	+	0.901961i	$0.357872\pi$
$198$	0	0
$199$	−26.4164	−1.87261	−0.936305	−	0.351189i	$-0.885778\pi$
$199$	−26.4164	−1.87261	−0.936305	+	0.351189i	$0.885778\pi$
$200$	−6.53779	−0.462291
$201$	0	0
$202$	−16.3262	−1.14871
$203$	0.450347	0.0316082
$204$	0	0
$205$	8.61803	0.601910
$206$	−10.7175	−0.746725
$207$	0	0
$208$	19.5623	1.35640
$209$	0	0
$210$	0	0
$211$	11.0344	0.759642	0.379821	−	0.925060i	$-0.375986\pi$
$211$	11.0344	0.759642	0.379821	+	0.925060i	$0.375986\pi$
$212$	−3.45106	−0.237020
$213$	0	0
$214$	14.5066	0.991649
$215$	−16.5266	−1.12711
$216$	0	0
$217$	−0.562306	−0.0381718
$218$	0.450347	0.0305013
$219$	0	0
$220$	0	0
$221$	25.1973	1.69495
$222$	0	0
$223$	−19.1803	−1.28441	−0.642205	−	0.766533i	$-0.721980\pi$
$223$	−19.1803	−1.28441	−0.642205	+	0.766533i	$0.721980\pi$
$224$	0.503503	0.0336417
$225$	0	0
$226$	−17.4508	−1.16081
$227$	9.39934	0.623856	0.311928	−	0.950106i	$-0.399025\pi$
$227$	9.39934	0.623856	0.311928	+	0.950106i	$0.399025\pi$
$228$	0	0
$229$	−9.52786	−0.629619	−0.314809	−	0.949155i	$-0.601941\pi$
$229$	−9.52786	−0.629619	−0.314809	+	0.949155i	$0.601941\pi$
$230$	−17.8448	−1.17665
$231$	0	0
$232$	4.76393	0.312767
$233$	4.17974	0.273824	0.136912	−	0.990583i	$-0.456282\pi$
$233$	4.17974	0.273824	0.136912	+	0.990583i	$0.456282\pi$
$234$	0	0
$235$	1.47214	0.0960316
$236$	−3.22589	−0.209987
$237$	0	0
$238$	2.16718	0.140478
$239$	15.9371	1.03089	0.515443	−	0.856924i	$-0.327627\pi$
$239$	15.9371	1.03089	0.515443	+	0.856924i	$0.327627\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	1.65248	0.105789
$245$	−10.7175	−0.684718
$246$	0	0
$247$	19.5623	1.24472
$248$	−5.94827	−0.377716
$249$	0	0
$250$	18.1459	1.14765
$251$	−14.7581	−0.931523	−0.465761	−	0.884910i	$-0.654220\pi$
$251$	−14.7581	−0.931523	−0.465761	+	0.884910i	$0.654220\pi$
$252$	0	0
$253$	0	0
$254$	11.1679	0.700735
$255$	0	0
$256$	8.85410	0.553381
$257$	15.7980	0.985450	0.492725	−	0.870185i	$-0.336001\pi$
$257$	15.7980	0.985450	0.492725	+	0.870185i	$0.336001\pi$
$258$	0	0
$259$	1.47214	0.0914741
$260$	−2.49721	−0.154871
$261$	0	0
$262$	18.7082	1.15580
$263$	4.63009	0.285503	0.142752	−	0.989759i	$-0.454405\pi$
$263$	4.63009	0.285503	0.142752	+	0.989759i	$0.454405\pi$
$264$	0	0
$265$	−13.9443	−0.856590
$266$	1.68253	0.103162
$267$	0	0
$268$	1.47214	0.0899250
$269$	26.8798	1.63889	0.819446	−	0.573157i	$-0.194281\pi$
$269$	26.8798	1.63889	0.819446	+	0.573157i	$0.194281\pi$
$270$	0	0
$271$	−30.0344	−1.82446	−0.912231	−	0.409676i	$-0.865642\pi$
$271$	−30.0344	−1.82446	−0.912231	+	0.409676i	$0.865642\pi$
$272$	27.4693	1.66557
$273$	0	0
$274$	16.3262	0.986304
$275$	0	0
$276$	0	0
$277$	21.6180	1.29890	0.649451	−	0.760404i	$-0.274999\pi$
$277$	21.6180	1.29890	0.649451	+	0.760404i	$0.274999\pi$
$278$	21.0176	1.26055
$279$	0	0
$280$	0.909830	0.0543727
$281$	−8.80982	−0.525550	−0.262775	−	0.964857i	$-0.584638\pi$
$281$	−8.80982	−0.525550	−0.262775	+	0.964857i	$0.584638\pi$
$282$	0	0
$283$	6.94427	0.412794	0.206397	−	0.978468i	$-0.433826\pi$
$283$	6.94427	0.412794	0.206397	+	0.978468i	$0.433826\pi$
$284$	−2.94756	−0.174905
$285$	0	0
$286$	0	0
$287$	1.31819	0.0778102
$288$	0	0
$289$	18.3820	1.08129
$290$	−4.54408	−0.266837
$291$	0	0
$292$	−2.47214	−0.144671
$293$	−20.5672	−1.20155	−0.600775	−	0.799418i	$-0.705141\pi$
$293$	−20.5672	−1.20155	−0.600775	+	0.799418i	$0.705141\pi$
$294$	0	0
$295$	−13.0344	−0.758895
$296$	15.5728	0.905150
$297$	0	0
$298$	−21.0902	−1.22172
$299$	−31.7351	−1.83529
$300$	0	0
$301$	−2.52786	−0.145704
$302$	13.5259	0.778329
$303$	0	0
$304$	21.3262	1.22314
$305$	6.67695	0.382321
$306$	0	0
$307$	−16.2705	−0.928607	−0.464304	−	0.885676i	$-0.653695\pi$
$307$	−16.2705	−0.928607	−0.464304	+	0.885676i	$0.653695\pi$
$308$	0	0
$309$	0	0
$310$	5.67376	0.322248
$311$	24.1043	1.36683	0.683414	−	0.730031i	$-0.260494\pi$
$311$	24.1043	1.36683	0.683414	+	0.730031i	$0.260494\pi$
$312$	0	0
$313$	0.347524	0.0196432	0.00982161	−	0.999952i	$-0.496874\pi$
$313$	0.347524	0.0196432	0.00982161	+	0.999952i	$0.496874\pi$
$314$	−16.4406	−0.927798
$315$	0	0
$316$	−0.201626	−0.0113424
$317$	10.5784	0.594140	0.297070	−	0.954856i	$-0.403991\pi$
$317$	10.5784	0.594140	0.297070	+	0.954856i	$0.403991\pi$
$318$	0	0
$319$	0	0
$320$	9.17416	0.512851
$321$	0	0
$322$	−2.72949	−0.152109
$323$	27.4693	1.52843
$324$	0	0
$325$	11.0902	0.615172
$326$	−5.58394	−0.309266
$327$	0	0
$328$	13.9443	0.769944
$329$	0.225173	0.0124142
$330$	0	0
$331$	20.7082	1.13823	0.569113	−	0.822259i	$-0.307287\pi$
$331$	20.7082	1.13823	0.569113	+	0.822259i	$0.307287\pi$
$332$	1.68253	0.0923407
$333$	0	0
$334$	−11.9098	−0.651677
$335$	5.94827	0.324989
$336$	0	0
$337$	4.65248	0.253437	0.126718	−	0.991939i	$-0.459556\pi$
$337$	4.65248	0.253437	0.126718	+	0.991939i	$0.459556\pi$
$338$	−7.63080	−0.415061
$339$	0	0
$340$	−3.50658	−0.190171
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	−26.7407	−1.44176
$345$	0	0
$346$	−4.20163	−0.225881
$347$	2.63638	0.141528	0.0707641	−	0.997493i	$-0.477456\pi$
$347$	2.63638	0.141528	0.0707641	+	0.997493i	$0.477456\pi$
$348$	0	0
$349$	−13.1803	−0.705527	−0.352764	−	0.935712i	$-0.614758\pi$
$349$	−13.1803	−0.705527	−0.352764	+	0.935712i	$0.614758\pi$
$350$	0.953850	0.0509854
$351$	0	0
$352$	0	0
$353$	3.53707	0.188259	0.0941296	−	0.995560i	$-0.469993\pi$
$353$	3.53707	0.188259	0.0941296	+	0.995560i	$0.469993\pi$
$354$	0	0
$355$	−11.9098	−0.632108
$356$	−6.39862	−0.339126
$357$	0	0
$358$	28.7984	1.52204
$359$	26.1511	1.38020	0.690102	−	0.723712i	$-0.257566\pi$
$359$	26.1511	1.38020	0.690102	+	0.723712i	$0.257566\pi$
$360$	0	0
$361$	2.32624	0.122434
$362$	−11.9826	−0.629789
$363$	0	0
$364$	−0.381966	−0.0200205
$365$	−9.98885	−0.522840
$366$	0	0
$367$	−6.03444	−0.314995	−0.157498	−	0.987519i	$-0.550343\pi$
$367$	−6.03444	−0.314995	−0.157498	+	0.987519i	$0.550343\pi$
$368$	−34.5966	−1.80347
$369$	0	0
$370$	−14.8541	−0.772228
$371$	−2.13287	−0.110733
$372$	0	0
$373$	12.4164	0.642897	0.321449	−	0.946927i	$-0.395830\pi$
$373$	12.4164	0.642897	0.321449	+	0.946927i	$0.395830\pi$
$374$	0	0
$375$	0	0
$376$	2.38197	0.122841
$377$	−8.08115	−0.416200
$378$	0	0
$379$	1.76393	0.0906071	0.0453036	−	0.998973i	$-0.485574\pi$
$379$	1.76393	0.0906071	0.0453036	+	0.998973i	$0.485574\pi$
$380$	−2.72239	−0.139655
$381$	0	0
$382$	−5.67376	−0.290295
$383$	14.1154	0.721265	0.360632	−	0.932708i	$-0.382561\pi$
$383$	14.1154	0.721265	0.360632	+	0.932708i	$0.382561\pi$
$384$	0	0
$385$	0	0
$386$	−14.8441	−0.755545
$387$	0	0
$388$	−4.88854	−0.248178
$389$	−19.4742	−0.987381	−0.493690	−	0.869638i	$-0.664352\pi$
$389$	−19.4742	−0.987381	−0.493690	+	0.869638i	$0.664352\pi$
$390$	0	0
$391$	−44.5623	−2.25361
$392$	−17.3413	−0.875869
$393$	0	0
$394$	−18.7082	−0.942506
$395$	−0.814685	−0.0409913
$396$	0	0
$397$	−2.41641	−0.121276	−0.0606380	−	0.998160i	$-0.519314\pi$
$397$	−2.41641	−0.121276	−0.0606380	+	0.998160i	$0.519314\pi$
$398$	40.7701	2.04362
$399$	0	0
$400$	12.0902	0.604508
$401$	3.31190	0.165388	0.0826941	−	0.996575i	$-0.473648\pi$
$401$	3.31190	0.165388	0.0826941	+	0.996575i	$0.473648\pi$
$402$	0	0
$403$	10.0902	0.502627
$404$	4.04057	0.201026
$405$	0	0
$406$	−0.695048	−0.0344947
$407$	0	0
$408$	0	0
$409$	−22.9443	−1.13452	−0.567261	−	0.823538i	$-0.691997\pi$
$409$	−22.9443	−1.13452	−0.567261	+	0.823538i	$0.691997\pi$
$410$	−13.3007	−0.656877
$411$	0	0
$412$	2.65248	0.130678
$413$	−1.99371	−0.0981040
$414$	0	0
$415$	6.79837	0.333719
$416$	−9.03500	−0.442977
$417$	0	0
$418$	0	0
$419$	−16.0763	−0.785378	−0.392689	−	0.919671i	$-0.628455\pi$
$419$	−16.0763	−0.785378	−0.392689	+	0.919671i	$0.628455\pi$
$420$	0	0
$421$	22.2148	1.08268	0.541341	−	0.840803i	$-0.317917\pi$
$421$	22.2148	1.08268	0.541341	+	0.840803i	$0.317917\pi$
$422$	−17.0301	−0.829014
$423$	0	0
$424$	−22.5623	−1.09572
$425$	15.5728	0.755391
$426$	0	0
$427$	1.02129	0.0494235
$428$	−3.59023	−0.173540
$429$	0	0
$430$	25.5066	1.23004
$431$	−5.49793	−0.264826	−0.132413	−	0.991195i	$-0.542272\pi$
$431$	−5.49793	−0.264826	−0.132413	+	0.991195i	$0.542272\pi$
$432$	0	0
$433$	28.6525	1.37695	0.688475	−	0.725260i	$-0.258280\pi$
$433$	28.6525	1.37695	0.688475	+	0.725260i	$0.258280\pi$
$434$	0.867842	0.0416577
$435$	0	0
$436$	−0.111456	−0.00533778
$437$	−34.5966	−1.65498
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	0	0
$442$	−38.8885	−1.84974
$443$	18.7987	0.893152	0.446576	−	0.894746i	$-0.352643\pi$
$443$	18.7987	0.893152	0.446576	+	0.894746i	$0.352643\pi$
$444$	0	0
$445$	−25.8541	−1.22560
$446$	29.6022	1.40171
$447$	0	0
$448$	1.40325	0.0662974
$449$	−0.450347	−0.0212532	−0.0106266	−	0.999944i	$-0.503383\pi$
$449$	−0.450347	−0.0212532	−0.0106266	+	0.999944i	$0.503383\pi$
$450$	0	0
$451$	0	0
$452$	4.31890	0.203144
$453$	0	0
$454$	−14.5066	−0.680827
$455$	−1.54336	−0.0723539
$456$	0	0
$457$	23.9098	1.11845	0.559227	−	0.829014i	$-0.311098\pi$
$457$	23.9098	1.11845	0.559227	+	0.829014i	$0.311098\pi$
$458$	14.7049	0.687117
$459$	0	0
$460$	4.41641	0.205916
$461$	19.6134	0.913485	0.456743	−	0.889599i	$-0.349016\pi$
$461$	19.6134	0.913485	0.456743	+	0.889599i	$0.349016\pi$
$462$	0	0
$463$	−1.43769	−0.0668153	−0.0334077	−	0.999442i	$-0.510636\pi$
$463$	−1.43769	−0.0668153	−0.0334077	+	0.999442i	$0.510636\pi$
$464$	−8.80982	−0.408986
$465$	0	0
$466$	−6.45085	−0.298830
$467$	−4.85526	−0.224675	−0.112337	−	0.993670i	$-0.535834\pi$
$467$	−4.85526	−0.224675	−0.112337	+	0.993670i	$0.535834\pi$
$468$	0	0
$469$	0.909830	0.0420120
$470$	−2.27204	−0.104801
$471$	0	0
$472$	−21.0902	−0.970754
$473$	0	0
$474$	0	0
$475$	12.0902	0.554735
$476$	−0.536356	−0.0245838
$477$	0	0
$478$	−24.5967	−1.12503
$479$	36.8155	1.68214	0.841072	−	0.540923i	$-0.181925\pi$
$479$	36.8155	1.68214	0.841072	+	0.540923i	$0.181925\pi$
$480$	0	0
$481$	−26.4164	−1.20448
$482$	−21.6071	−0.984175
$483$	0	0
$484$	0	0
$485$	−19.7525	−0.896916
$486$	0	0
$487$	−16.8885	−0.765293	−0.382646	−	0.923895i	$-0.624987\pi$
$487$	−16.8885	−0.765293	−0.382646	+	0.923895i	$0.624987\pi$
$488$	10.8035	0.489053
$489$	0	0
$490$	16.5410	0.747247
$491$	−30.9204	−1.39542	−0.697709	−	0.716381i	$-0.745797\pi$
$491$	−30.9204	−1.39542	−0.697709	+	0.716381i	$0.745797\pi$
$492$	0	0
$493$	−11.3475	−0.511067
$494$	−30.1917	−1.35839
$495$	0	0
$496$	11.0000	0.493915
$497$	−1.82169	−0.0817140
$498$	0	0
$499$	−9.74265	−0.436141	−0.218070	−	0.975933i	$-0.569976\pi$
$499$	−9.74265	−0.436141	−0.218070	+	0.975933i	$0.569976\pi$
$500$	−4.49092	−0.200840
$501$	0	0
$502$	22.7771	1.01659
$503$	−11.1679	−0.497951	−0.248975	−	0.968510i	$-0.580094\pi$
$503$	−11.1679	−0.497951	−0.248975	+	0.968510i	$0.580094\pi$
$504$	0	0
$505$	16.3262	0.726508
$506$	0	0
$507$	0	0
$508$	−2.76393	−0.122630
$509$	−8.30632	−0.368171	−0.184086	−	0.982910i	$-0.558932\pi$
$509$	−8.30632	−0.368171	−0.184086	+	0.982910i	$0.558932\pi$
$510$	0	0
$511$	−1.52786	−0.0675887
$512$	13.2147	0.584014
$513$	0	0
$514$	−24.3820	−1.07544
$515$	10.7175	0.472271
$516$	0	0
$517$	0	0
$518$	−2.27204	−0.0998276
$519$	0	0
$520$	−16.3262	−0.715953
$521$	5.44477	0.238540	0.119270	−	0.992862i	$-0.461945\pi$
$521$	5.44477	0.238540	0.119270	+	0.992862i	$0.461945\pi$
$522$	0	0
$523$	18.4508	0.806799	0.403400	−	0.915024i	$-0.367828\pi$
$523$	18.4508	0.806799	0.403400	+	0.915024i	$0.367828\pi$
$524$	−4.63009	−0.202266
$525$	0	0
$526$	−7.14590	−0.311576
$527$	14.1686	0.617193
$528$	0	0
$529$	33.1246	1.44020
$530$	21.5211	0.934815
$531$	0	0
$532$	−0.416408	−0.0180536
$533$	−23.6539	−1.02457
$534$	0	0
$535$	−14.5066	−0.627174
$536$	9.62451	0.415716
$537$	0	0
$538$	−41.4853	−1.78856
$539$	0	0
$540$	0	0
$541$	−39.6312	−1.70388	−0.851939	−	0.523641i	$-0.824573\pi$
$541$	−39.6312	−1.70388	−0.851939	+	0.523641i	$0.824573\pi$
$542$	46.3540	1.99108
$543$	0	0
$544$	−12.6869	−0.543947
$545$	−0.450347	−0.0192907
$546$	0	0
$547$	−9.03444	−0.386285	−0.193142	−	0.981171i	$-0.561868\pi$
$547$	−9.03444	−0.386285	−0.193142	+	0.981171i	$0.561868\pi$
$548$	−4.04057	−0.172605
$549$	0	0
$550$	0	0
$551$	−8.80982	−0.375311
$552$	0	0
$553$	−0.124612	−0.00529903
$554$	−33.3645	−1.41752
$555$	0	0
$556$	−5.20163	−0.220598
$557$	−17.7057	−0.750213	−0.375106	−	0.926982i	$-0.622394\pi$
$557$	−17.7057	−0.750213	−0.375106	+	0.926982i	$0.622394\pi$
$558$	0	0
$559$	45.3607	1.91855
$560$	−1.68253	−0.0710997
$561$	0	0
$562$	13.5967	0.573544
$563$	40.5449	1.70876	0.854382	−	0.519645i	$-0.173936\pi$
$563$	40.5449	1.70876	0.854382	+	0.519645i	$0.173936\pi$
$564$	0	0
$565$	17.4508	0.734163
$566$	−10.7175	−0.450491
$567$	0	0
$568$	−19.2705	−0.808573
$569$	−29.6882	−1.24459	−0.622297	−	0.782781i	$-0.713801\pi$
$569$	−29.6882	−1.24459	−0.622297	+	0.782781i	$0.713801\pi$
$570$	0	0
$571$	−34.9787	−1.46381	−0.731907	−	0.681405i	$-0.761369\pi$
$571$	−34.9787	−1.46381	−0.731907	+	0.681405i	$0.761369\pi$
$572$	0	0
$573$	0	0
$574$	−2.03444	−0.0849160
$575$	−19.6134	−0.817934
$576$	0	0
$577$	−32.6525	−1.35934	−0.679670	−	0.733518i	$-0.737877\pi$
$577$	−32.6525	−1.35934	−0.679670	+	0.733518i	$0.737877\pi$
$578$	−28.3700	−1.18004
$579$	0	0
$580$	1.12461	0.0466970
$581$	1.03986	0.0431406
$582$	0	0
$583$	0	0
$584$	−16.1623	−0.668801
$585$	0	0
$586$	31.7426	1.31128
$587$	−9.12101	−0.376464	−0.188232	−	0.982125i	$-0.560276\pi$
$587$	−9.12101	−0.376464	−0.188232	+	0.982125i	$0.560276\pi$
$588$	0	0
$589$	11.0000	0.453247
$590$	20.1169	0.828198
$591$	0	0
$592$	−28.7984	−1.18361
$593$	1.09301	0.0448847	0.0224424	−	0.999748i	$-0.492856\pi$
$593$	1.09301	0.0448847	0.0224424	+	0.999748i	$0.492856\pi$
$594$	0	0
$595$	−2.16718	−0.0888459
$596$	5.21960	0.213803
$597$	0	0
$598$	48.9787	2.00289
$599$	−12.3469	−0.504480	−0.252240	−	0.967665i	$-0.581167\pi$
$599$	−12.3469	−0.504480	−0.252240	+	0.967665i	$0.581167\pi$
$600$	0	0
$601$	35.3607	1.44239	0.721196	−	0.692731i	$-0.243593\pi$
$601$	35.3607	1.44239	0.721196	+	0.692731i	$0.243593\pi$
$602$	3.90141	0.159010
$603$	0	0
$604$	−3.34752	−0.136209
$605$	0	0
$606$	0	0
$607$	−26.3820	−1.07081	−0.535405	−	0.844595i	$-0.679841\pi$
$607$	−26.3820	−1.07081	−0.535405	+	0.844595i	$0.679841\pi$
$608$	−9.84968	−0.399457
$609$	0	0
$610$	−10.3050	−0.417235
$611$	−4.04057	−0.163464
$612$	0	0
$613$	42.9443	1.73450	0.867251	−	0.497870i	$-0.165885\pi$
$613$	42.9443	1.73450	0.867251	+	0.497870i	$0.165885\pi$
$614$	25.1113	1.01341
$615$	0	0
$616$	0	0
$617$	−1.76854	−0.0711986	−0.0355993	−	0.999366i	$-0.511334\pi$
$617$	−1.76854	−0.0711986	−0.0355993	+	0.999366i	$0.511334\pi$
$618$	0	0
$619$	−28.8885	−1.16113	−0.580564	−	0.814214i	$-0.697168\pi$
$619$	−28.8885	−1.16113	−0.580564	+	0.814214i	$0.697168\pi$
$620$	−1.40420	−0.0563939
$621$	0	0
$622$	−37.2016	−1.49165
$623$	−3.95457	−0.158436
$624$	0	0
$625$	−5.05573	−0.202229
$626$	−0.536356	−0.0214371
$627$	0	0
$628$	4.06888	0.162366
$629$	−37.0938	−1.47903
$630$	0	0
$631$	−19.6180	−0.780982	−0.390491	−	0.920607i	$-0.627695\pi$
$631$	−19.6180	−0.780982	−0.390491	+	0.920607i	$0.627695\pi$
$632$	−1.31819	−0.0524347
$633$	0	0
$634$	−16.3262	−0.648398
$635$	−11.1679	−0.443183
$636$	0	0
$637$	29.4164	1.16552
$638$	0	0
$639$	0	0
$640$	−20.7426	−0.819925
$641$	−12.2609	−0.484276	−0.242138	−	0.970242i	$-0.577849\pi$
$641$	−12.2609	−0.484276	−0.242138	+	0.970242i	$0.577849\pi$
$642$	0	0
$643$	28.1591	1.11048	0.555242	−	0.831689i	$-0.312626\pi$
$643$	28.1591	1.11048	0.555242	+	0.831689i	$0.312626\pi$
$644$	0.675520	0.0266192
$645$	0	0
$646$	−42.3951	−1.66801
$647$	39.4519	1.55101	0.775507	−	0.631339i	$-0.217494\pi$
$647$	39.4519	1.55101	0.775507	+	0.631339i	$0.217494\pi$
$648$	0	0
$649$	0	0
$650$	−17.1161	−0.671350
$651$	0	0
$652$	1.38197	0.0541220
$653$	41.8099	1.63615	0.818075	−	0.575112i	$-0.195042\pi$
$653$	41.8099	1.63615	0.818075	+	0.575112i	$0.195042\pi$
$654$	0	0
$655$	−18.7082	−0.730990
$656$	−25.7868	−1.00681
$657$	0	0
$658$	−0.347524	−0.0135479
$659$	3.53707	0.137785	0.0688924	−	0.997624i	$-0.478053\pi$
$659$	3.53707	0.137785	0.0688924	+	0.997624i	$0.478053\pi$
$660$	0	0
$661$	22.5623	0.877572	0.438786	−	0.898592i	$-0.355409\pi$
$661$	22.5623	0.877572	0.438786	+	0.898592i	$0.355409\pi$
$662$	−31.9603	−1.24217
$663$	0	0
$664$	11.0000	0.426883
$665$	−1.68253	−0.0652456
$666$	0	0
$667$	14.2918	0.553381
$668$	2.94756	0.114044
$669$	0	0
$670$	−9.18034	−0.354667
$671$	0	0
$672$	0	0
$673$	13.7639	0.530561	0.265280	−	0.964171i	$-0.414536\pi$
$673$	13.7639	0.530561	0.265280	+	0.964171i	$0.414536\pi$
$674$	−7.18045	−0.276581
$675$	0	0
$676$	1.88854	0.0726363
$677$	6.17345	0.237265	0.118632	−	0.992938i	$-0.462149\pi$
$677$	6.17345	0.237265	0.118632	+	0.992938i	$0.462149\pi$
$678$	0	0
$679$	−3.02129	−0.115946
$680$	−22.9253	−0.879143
$681$	0	0
$682$	0	0
$683$	31.7351	1.21431	0.607155	−	0.794584i	$-0.292311\pi$
$683$	31.7351	1.21431	0.607155	+	0.794584i	$0.292311\pi$
$684$	0	0
$685$	−16.3262	−0.623793
$686$	5.08043	0.193972
$687$	0	0
$688$	49.4508	1.88530
$689$	38.2729	1.45808
$690$	0	0
$691$	−12.3607	−0.470222	−0.235111	−	0.971968i	$-0.575545\pi$
$691$	−12.3607	−0.470222	−0.235111	+	0.971968i	$0.575545\pi$
$692$	1.03986	0.0395295
$693$	0	0
$694$	−4.06888	−0.154453
$695$	−21.0176	−0.797241
$696$	0	0
$697$	−33.2148	−1.25810
$698$	20.3420	0.769957
$699$	0	0
$700$	−0.236068	−0.00892253
$701$	38.4120	1.45080	0.725401	−	0.688326i	$-0.241654\pi$
$701$	38.4120	1.45080	0.725401	+	0.688326i	$0.241654\pi$
$702$	0	0
$703$	−28.7984	−1.08615
$704$	0	0
$705$	0	0
$706$	−5.45898	−0.205451
$707$	2.49721	0.0939173
$708$	0	0
$709$	−39.7984	−1.49466	−0.747330	−	0.664453i	$-0.768664\pi$
$709$	−39.7984	−1.49466	−0.747330	+	0.664453i	$0.768664\pi$
$710$	18.3812	0.689833
$711$	0	0
$712$	−41.8328	−1.56775
$713$	−17.8448	−0.668294
$714$	0	0
$715$	0	0
$716$	−7.12730	−0.266360
$717$	0	0
$718$	−40.3607	−1.50625
$719$	−35.5505	−1.32581	−0.662905	−	0.748704i	$-0.730677\pi$
$719$	−35.5505	−1.32581	−0.662905	+	0.748704i	$0.730677\pi$
$720$	0	0
$721$	1.63932	0.0610515
$722$	−3.59023	−0.133614
$723$	0	0
$724$	2.96556	0.110214
$725$	−4.99442	−0.185488
$726$	0	0
$727$	−0.562306	−0.0208548	−0.0104274	−	0.999946i	$-0.503319\pi$
$727$	−0.562306	−0.0208548	−0.0104274	+	0.999946i	$0.503319\pi$
$728$	−2.49721	−0.0925528
$729$	0	0
$730$	15.4164	0.570587
$731$	63.6953	2.35586
$732$	0	0
$733$	16.6525	0.615073	0.307537	−	0.951536i	$-0.400495\pi$
$733$	16.6525	0.615073	0.307537	+	0.951536i	$0.400495\pi$
$734$	9.31333	0.343761
$735$	0	0
$736$	15.9787	0.588983
$737$	0	0
$738$	0	0
$739$	13.7639	0.506314	0.253157	−	0.967425i	$-0.418531\pi$
$739$	13.7639	0.506314	0.253157	+	0.967425i	$0.418531\pi$
$740$	3.67624	0.135141
$741$	0	0
$742$	3.29180	0.120846
$743$	−44.0820	−1.61721	−0.808605	−	0.588351i	$-0.799777\pi$
$743$	−44.0820	−1.61721	−0.808605	+	0.588351i	$0.799777\pi$
$744$	0	0
$745$	21.0902	0.772684
$746$	−19.1630	−0.701608
$747$	0	0
$748$	0	0
$749$	−2.21888	−0.0810762
$750$	0	0
$751$	36.5066	1.33214	0.666072	−	0.745887i	$-0.267974\pi$
$751$	36.5066	1.33214	0.666072	+	0.745887i	$0.267974\pi$
$752$	−4.40491	−0.160631
$753$	0	0
$754$	12.4721	0.454208
$755$	−13.5259	−0.492259
$756$	0	0
$757$	−5.11146	−0.185779	−0.0928895	−	0.995676i	$-0.529610\pi$
$757$	−5.11146	−0.185779	−0.0928895	+	0.995676i	$0.529610\pi$
$758$	−2.72239	−0.0988815
$759$	0	0
$760$	−17.7984	−0.645615
$761$	7.04129	0.255247	0.127623	−	0.991823i	$-0.459265\pi$
$761$	7.04129	0.255247	0.127623	+	0.991823i	$0.459265\pi$
$762$	0	0
$763$	−0.0688837	−0.00249376
$764$	1.40420	0.0508021
$765$	0	0
$766$	−21.7852	−0.787132
$767$	35.7757	1.29178
$768$	0	0
$769$	11.2705	0.406425	0.203212	−	0.979135i	$-0.434862\pi$
$769$	11.2705	0.406425	0.203212	+	0.979135i	$0.434862\pi$
$770$	0	0
$771$	0	0
$772$	3.67376	0.132222
$773$	48.3477	1.73895	0.869473	−	0.493980i	$-0.164458\pi$
$773$	48.3477	1.73895	0.869473	+	0.493980i	$0.164458\pi$
$774$	0	0
$775$	6.23607	0.224006
$776$	−31.9603	−1.14731
$777$	0	0
$778$	30.0557	1.07755
$779$	−25.7868	−0.923908
$780$	0	0
$781$	0	0
$782$	68.7758	2.45942
$783$	0	0
$784$	32.0689	1.14532
$785$	16.4406	0.586791
$786$	0	0
$787$	−43.7771	−1.56048	−0.780242	−	0.625477i	$-0.784904\pi$
$787$	−43.7771	−1.56048	−0.780242	+	0.625477i	$0.784904\pi$
$788$	4.63009	0.164940
$789$	0	0
$790$	1.25735	0.0447347
$791$	2.66923	0.0949069
$792$	0	0
$793$	−18.3262	−0.650784
$794$	3.72939	0.132351
$795$	0	0
$796$	−10.0902	−0.357637
$797$	−3.08672	−0.109337	−0.0546687	−	0.998505i	$-0.517410\pi$
$797$	−3.08672	−0.109337	−0.0546687	+	0.998505i	$0.517410\pi$
$798$	0	0
$799$	−5.67376	−0.200723
$800$	−5.58394	−0.197422
$801$	0	0
$802$	−5.11146	−0.180492
$803$	0	0
$804$	0	0
$805$	2.72949	0.0962019
$806$	−15.5728	−0.548528
$807$	0	0
$808$	26.4164	0.929326
$809$	25.7868	0.906616	0.453308	−	0.891354i	$-0.350244\pi$
$809$	25.7868	0.906616	0.453308	+	0.891354i	$0.350244\pi$
$810$	0	0
$811$	7.21478	0.253345	0.126673	−	0.991945i	$-0.459570\pi$
$811$	7.21478	0.253345	0.126673	+	0.991945i	$0.459570\pi$
$812$	0.172017	0.00603662
$813$	0	0
$814$	0	0
$815$	5.58394	0.195597
$816$	0	0
$817$	49.4508	1.73007
$818$	35.4113	1.23813
$819$	0	0
$820$	3.29180	0.114955
$821$	30.1057	1.05070	0.525348	−	0.850887i	$-0.323935\pi$
$821$	30.1057	1.05070	0.525348	+	0.850887i	$0.323935\pi$
$822$	0	0
$823$	54.8885	1.91329	0.956647	−	0.291249i	$-0.0940709\pi$
$823$	54.8885	1.91329	0.956647	+	0.291249i	$0.0940709\pi$
$824$	17.3413	0.604113
$825$	0	0
$826$	3.07701	0.107063
$827$	33.0533	1.14937	0.574687	−	0.818373i	$-0.305124\pi$
$827$	33.0533	1.14937	0.574687	+	0.818373i	$0.305124\pi$
$828$	0	0
$829$	−9.03444	−0.313779	−0.156890	−	0.987616i	$-0.550147\pi$
$829$	−9.03444	−0.313779	−0.156890	+	0.987616i	$0.550147\pi$
$830$	−10.4924	−0.364195
$831$	0	0
$832$	−25.1803	−0.872971
$833$	41.3064	1.43118
$834$	0	0
$835$	11.9098	0.412157
$836$	0	0
$837$	0	0
$838$	24.8115	0.857100
$839$	−48.6261	−1.67876	−0.839379	−	0.543547i	$-0.817081\pi$
$839$	−48.6261	−1.67876	−0.839379	+	0.543547i	$0.817081\pi$
$840$	0	0
$841$	−25.3607	−0.874506
$842$	−34.2854	−1.18155
$843$	0	0
$844$	4.21478	0.145079
$845$	7.63080	0.262508
$846$	0	0
$847$	0	0
$848$	41.7239	1.43281
$849$	0	0
$850$	−24.0344	−0.824375
$851$	46.7184	1.60148
$852$	0	0
$853$	21.1803	0.725201	0.362601	−	0.931945i	$-0.381889\pi$
$853$	21.1803	0.725201	0.362601	+	0.931945i	$0.381889\pi$
$854$	−1.57621	−0.0539369
$855$	0	0
$856$	−23.4721	−0.802261
$857$	−40.9953	−1.40037	−0.700186	−	0.713961i	$-0.746899\pi$
$857$	−40.9953	−1.40037	−0.700186	+	0.713961i	$0.746899\pi$
$858$	0	0
$859$	15.5836	0.531705	0.265853	−	0.964014i	$-0.414347\pi$
$859$	15.5836	0.531705	0.265853	+	0.964014i	$0.414347\pi$
$860$	−6.31261	−0.215258
$861$	0	0
$862$	8.48529	0.289010
$863$	24.6938	0.840586	0.420293	−	0.907388i	$-0.361927\pi$
$863$	24.6938	0.840586	0.420293	+	0.907388i	$0.361927\pi$
$864$	0	0
$865$	4.20163	0.142860
$866$	−44.2211	−1.50270
$867$	0	0
$868$	−0.214782	−0.00729017
$869$	0	0
$870$	0	0
$871$	−16.3262	−0.553193
$872$	−0.728677	−0.0246761
$873$	0	0
$874$	53.3951	1.80612
$875$	−2.77554	−0.0938304
$876$	0	0
$877$	2.52786	0.0853599	0.0426800	−	0.999089i	$-0.486410\pi$
$877$	2.52786	0.0853599	0.0426800	+	0.999089i	$0.486410\pi$
$878$	38.5840	1.30215
$879$	0	0
$880$	0	0
$881$	47.8114	1.61081	0.805403	−	0.592728i	$-0.201949\pi$
$881$	47.8114	1.61081	0.805403	+	0.592728i	$0.201949\pi$
$882$	0	0
$883$	−49.8885	−1.67888	−0.839442	−	0.543450i	$-0.817118\pi$
$883$	−49.8885	−1.67888	−0.839442	+	0.543450i	$0.817118\pi$
$884$	9.62451	0.323707
$885$	0	0
$886$	−29.0132	−0.974716
$887$	16.4735	0.553125	0.276563	−	0.960996i	$-0.410805\pi$
$887$	16.4735	0.553125	0.276563	+	0.960996i	$0.410805\pi$
$888$	0	0
$889$	−1.70820	−0.0572913
$890$	39.9022	1.33753
$891$	0	0
$892$	−7.32624	−0.245301
$893$	−4.40491	−0.147405
$894$	0	0
$895$	−28.7984	−0.962623
$896$	−3.17273	−0.105994
$897$	0	0
$898$	0.695048	0.0231941
$899$	−4.54408	−0.151553
$900$	0	0
$901$	53.7426	1.79043
$902$	0	0
$903$	0	0
$904$	28.2361	0.939118
$905$	11.9826	0.398314
$906$	0	0
$907$	46.8673	1.55620	0.778101	−	0.628139i	$-0.216183\pi$
$907$	46.8673	1.55620	0.778101	+	0.628139i	$0.216183\pi$
$908$	3.59023	0.119146
$909$	0	0
$910$	2.38197	0.0789614
$911$	−37.9085	−1.25597	−0.627983	−	0.778227i	$-0.716119\pi$
$911$	−37.9085	−1.25597	−0.627983	+	0.778227i	$0.716119\pi$
$912$	0	0
$913$	0	0
$914$	−36.9015	−1.22059
$915$	0	0
$916$	−3.63932	−0.120247
$917$	−2.86155	−0.0944967
$918$	0	0
$919$	−2.81966	−0.0930120	−0.0465060	−	0.998918i	$-0.514809\pi$
$919$	−2.81966	−0.0930120	−0.0465060	+	0.998918i	$0.514809\pi$
$920$	28.8735	0.951932
$921$	0	0
$922$	−30.2705	−0.996906
$923$	32.6889	1.07597
$924$	0	0
$925$	−16.3262	−0.536803
$926$	2.21888	0.0729170
$927$	0	0
$928$	4.06888	0.133568
$929$	0.867842	0.0284730	0.0142365	−	0.999899i	$-0.495468\pi$
$929$	0.867842	0.0284730	0.0142365	+	0.999899i	$0.495468\pi$
$930$	0	0
$931$	32.0689	1.05102
$932$	1.59652	0.0522957
$933$	0	0
$934$	7.49342	0.245192
$935$	0	0
$936$	0	0
$937$	33.1803	1.08395	0.541977	−	0.840393i	$-0.317676\pi$
$937$	33.1803	1.08395	0.541977	+	0.840393i	$0.317676\pi$
$938$	−1.40420	−0.0458487
$939$	0	0
$940$	0.562306	0.0183404
$941$	−45.1750	−1.47266	−0.736331	−	0.676621i	$-0.763443\pi$
$941$	−45.1750	−1.47266	−0.736331	+	0.676621i	$0.763443\pi$
$942$	0	0
$943$	41.8328	1.36226
$944$	39.0015	1.26939
$945$	0	0
$946$	0	0
$947$	−25.3365	−0.823324	−0.411662	−	0.911337i	$-0.635052\pi$
$947$	−25.3365	−0.823324	−0.411662	+	0.911337i	$0.635052\pi$
$948$	0	0
$949$	27.4164	0.889974
$950$	−18.6595	−0.605394
$951$	0	0
$952$	−3.50658	−0.113649
$953$	−46.5792	−1.50885	−0.754424	−	0.656387i	$-0.772084\pi$
$953$	−46.5792	−1.50885	−0.754424	+	0.656387i	$0.772084\pi$
$954$	0	0
$955$	5.67376	0.183599
$956$	6.08744	0.196882
$957$	0	0
$958$	−56.8197	−1.83576
$959$	−2.49721	−0.0806392
$960$	0	0
$961$	−25.3262	−0.816975
$962$	40.7701	1.31448
$963$	0	0
$964$	5.34752	0.172232
$965$	14.8441	0.477849
$966$	0	0
$967$	42.6869	1.37272	0.686359	−	0.727263i	$-0.259208\pi$
$967$	42.6869	1.37272	0.686359	+	0.727263i	$0.259208\pi$
$968$	0	0
$969$	0	0
$970$	30.4853	0.978824
$971$	13.0756	0.419615	0.209808	−	0.977743i	$-0.432716\pi$
$971$	13.0756	0.419615	0.209808	+	0.977743i	$0.432716\pi$
$972$	0	0
$973$	−3.21478	−0.103061
$974$	26.0651	0.835181
$975$	0	0
$976$	−19.9787	−0.639503
$977$	−28.6484	−0.916542	−0.458271	−	0.888812i	$-0.651531\pi$
$977$	−28.6484	−0.916542	−0.458271	+	0.888812i	$0.651531\pi$
$978$	0	0
$979$	0	0
$980$	−4.09373	−0.130769
$981$	0	0
$982$	47.7214	1.52285
$983$	−39.5050	−1.26002	−0.630008	−	0.776589i	$-0.716948\pi$
$983$	−39.5050	−1.26002	−0.630008	+	0.776589i	$0.716948\pi$
$984$	0	0
$985$	18.7082	0.596093
$986$	17.5133	0.557738
$987$	0	0
$988$	7.47214	0.237720
$989$	−80.2220	−2.55091
$990$	0	0
$991$	−24.7295	−0.785558	−0.392779	−	0.919633i	$-0.628486\pi$
$991$	−24.7295	−0.785558	−0.392779	+	0.919633i	$0.628486\pi$
$992$	−5.08043	−0.161304
$993$	0	0
$994$	2.81153	0.0891763
$995$	−40.7701	−1.29250
$996$	0	0
$997$	−3.74265	−0.118531	−0.0592654	−	0.998242i	$-0.518876\pi$
$997$	−3.74265	−0.118531	−0.0592654	+	0.998242i	$0.518876\pi$
$998$	15.0364	0.475970
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.v.1.2		4
3.2	odd	2	inner	1089.2.a.v.1.3		4
11.3	even	5		99.2.f.c.64.1	✓	8
11.4	even	5		99.2.f.c.82.1	yes	8
11.10	odd	2		1089.2.a.w.1.3		4
33.14	odd	10		99.2.f.c.64.2	yes	8
33.26	odd	10		99.2.f.c.82.2	yes	8
33.32	even	2		1089.2.a.w.1.2		4
99.4	even	15		891.2.n.e.379.2		16
99.14	odd	30		891.2.n.e.460.2		16
99.25	even	15		891.2.n.e.757.2		16
99.47	odd	30		891.2.n.e.757.1		16
99.58	even	15		891.2.n.e.460.1		16
99.59	odd	30		891.2.n.e.379.1		16
99.70	even	15		891.2.n.e.676.1		16
99.92	odd	30		891.2.n.e.676.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.f.c.64.1	✓	8	11.3	even	5
99.2.f.c.64.2	yes	8	33.14	odd	10
99.2.f.c.82.1	yes	8	11.4	even	5
99.2.f.c.82.2	yes	8	33.26	odd	10
891.2.n.e.379.1		16	99.59	odd	30
891.2.n.e.379.2		16	99.4	even	15
891.2.n.e.460.1		16	99.58	even	15
891.2.n.e.460.2		16	99.14	odd	30
891.2.n.e.676.1		16	99.70	even	15
891.2.n.e.676.2		16	99.92	odd	30
891.2.n.e.757.1		16	99.47	odd	30
891.2.n.e.757.2		16	99.25	even	15
1089.2.a.v.1.2		4	1.1	even	1	trivial
1089.2.a.v.1.3		4	3.2	odd	2	inner
1089.2.a.w.1.2		4	33.32	even	2
1089.2.a.w.1.3		4	11.10	odd	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-1.54336 q^{2} +0.381966 q^{4} +1.54336 q^{5} +0.236068 q^{7} +2.49721 q^{8} -2.38197 q^{10} -4.23607 q^{13} -0.364338 q^{14} -4.61803 q^{16} -5.94827 q^{17} -4.61803 q^{19} +0.589512 q^{20} +7.49164 q^{23} -2.61803 q^{25} +6.53779 q^{26} +0.0901699 q^{28} +1.90770 q^{29} -2.38197 q^{31} +2.13287 q^{32} +9.18034 q^{34} +0.364338 q^{35} +6.23607 q^{37} +7.12730 q^{38} +3.85410 q^{40} +5.58394 q^{41} -10.7082 q^{43} -11.5623 q^{46} +0.953850 q^{47} -6.94427 q^{49} +4.04057 q^{50} -1.61803 q^{52} -9.03500 q^{53} +0.589512 q^{56} -2.94427 q^{58} -8.44549 q^{59} +4.32624 q^{61} +3.67624 q^{62} +5.94427 q^{64} -6.53779 q^{65} +3.85410 q^{67} -2.27204 q^{68} -0.562306 q^{70} -7.71681 q^{71} -6.47214 q^{73} -9.62451 q^{74} -1.76393 q^{76} -0.527864 q^{79} -7.12730 q^{80} -8.61803 q^{82} +4.40491 q^{83} -9.18034 q^{85} +16.5266 q^{86} -16.7518 q^{89} -1.00000 q^{91} +2.86155 q^{92} -1.47214 q^{94} -7.12730 q^{95} -12.7984 q^{97} +10.7175 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 8 q^{7} - 14 q^{10} - 8 q^{13} - 14 q^{16} - 14 q^{19} - 6 q^{25} - 22 q^{28} - 14 q^{31} - 8 q^{34} + 16 q^{37} + 2 q^{40} - 16 q^{43} - 6 q^{46} + 8 q^{49} - 2 q^{52} + 24 q^{58} - 14 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 - 8 * q^7 - 14 * q^10 - 8 * q^13 - 14 * q^16 - 14 * q^19 - 6 * q^25 - 22 * q^28 - 14 * q^31 - 8 * q^34 + 16 * q^37 + 2 * q^40 - 16 * q^43 - 6 * q^46 + 8 * q^49 - 2 * q^52 + 24 * q^58 - 14 * q^61 - 12 * q^64 + 2 * q^67 + 38 * q^70 - 8 * q^73 - 16 * q^76 - 20 * q^79 - 30 * q^82 + 8 * q^85 - 4 * q^91 + 12 * q^94 - 2 * q^97

Introduction

L-functions

Modular forms

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