LMFDB - Embedded newform 1089.4.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1089,4,Mod(1,1089)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,1,0,1,-16,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2530799963$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +11.7663 q^{10} +15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684 q^{17} +78.7011 q^{19} +11.7663 q^{20} -112.000 q^{23} -112.826 q^{25} +50.6576 q^{26} +16.0000 q^{28} +243.125 q^{29} +278.717 q^{31} -143.606 q^{32} +246.745 q^{34} +16.5544 q^{35} +102.380 q^{37} +265.402 q^{38} -54.4512 q^{40} -241.255 q^{41} +280.016 q^{43} -377.696 q^{46} +169.870 q^{47} -320.489 q^{49} -380.481 q^{50} +50.6576 q^{52} +409.652 q^{53} -74.0435 q^{56} +819.886 q^{58} -196.000 q^{59} +701.359 q^{61} +939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} +900.587 q^{67} +246.745 q^{68} +55.8260 q^{70} -756.500 q^{71} +1019.81 q^{73} +345.255 q^{74} +265.402 q^{76} +327.549 q^{79} -277.755 q^{80} -813.581 q^{82} -756.619 q^{83} +255.294 q^{85} +944.293 q^{86} -508.978 q^{89} +71.2716 q^{91} -377.696 q^{92} +572.848 q^{94} +274.598 q^{95} +614.358 q^{97} -1080.78 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8} + 58 q^{10} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} + 54 q^{19} + 58 q^{20} - 224 q^{23} + 142 q^{25} - 94 q^{26} + 32 q^{28} + 222 q^{29} - 40 q^{31}+ \cdots - 375 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 + 58 * q^10 + 76 * q^13 + 32 * q^14 - 119 * q^16 - 26 * q^17 + 54 * q^19 + 58 * q^20 - 224 * q^23 + 142 * q^25 - 94 * q^26 + 32 * q^28 + 222 * q^29 - 40 * q^31 - 247 * q^32 + 482 * q^34 + 148 * q^35 - 48 * q^37 + 324 * q^38 - 534 * q^40 - 494 * q^41 + 66 * q^43 - 112 * q^46 + 64 * q^47 - 618 * q^49 - 985 * q^50 - 94 * q^52 + 84 * q^53 - 240 * q^56 + 870 * q^58 - 392 * q^59 + 1104 * q^61 + 1696 * q^62 + 713 * q^64 - 1136 * q^65 + 928 * q^67 + 482 * q^68 - 256 * q^70 - 456 * q^71 + 592 * q^73 + 702 * q^74 + 324 * q^76 + 230 * q^79 + 490 * q^80 - 214 * q^82 + 348 * q^83 + 2188 * q^85 + 1452 * q^86 - 972 * q^89 - 340 * q^91 - 112 * q^92 + 824 * q^94 + 756 * q^95 - 1184 * q^97 - 375 * 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 $

$2$	3.37228	1.19228	0.596141	−	0.802880i	$-0.296700\pi$
$2$	3.37228	1.19228	0.596141	+	0.802880i	$0.296700\pi$
$3$	0	0
$3$	0	0
$4$	3.37228	0.421535
$5$	3.48913	0.312077	0.156038	−	0.987751i	$-0.450128\pi$
$5$	3.48913	0.312077	0.156038	+	0.987751i	$0.450128\pi$
$6$	0	0
$7$	4.74456	0.256182	0.128091	−	0.991762i	$-0.459115\pi$
$7$	4.74456	0.256182	0.128091	+	0.991762i	$0.459115\pi$
$8$	−15.6060	−0.689693
$9$	0	0
$10$	11.7663	0.372083
$11$	0	0
$11$	0	0
$12$	0	0
$13$	15.0217	0.320483	0.160242	−	0.987078i	$-0.448773\pi$
$13$	15.0217	0.320483	0.160242	+	0.987078i	$0.448773\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	−79.6060	−1.24384
$17$	73.1684	1.04388	0.521940	−	0.852982i	$-0.325209\pi$
$17$	73.1684	1.04388	0.521940	+	0.852982i	$0.325209\pi$
$18$	0	0
$19$	78.7011	0.950277	0.475138	−	0.879911i	$-0.342398\pi$
$19$	78.7011	0.950277	0.475138	+	0.879911i	$0.342398\pi$
$20$	11.7663	0.131551
$21$	0	0
$22$	0	0
$23$	−112.000	−1.01537	−0.507687	−	0.861541i	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	0	0
$25$	−112.826	−0.902608
$26$	50.6576	0.382106
$27$	0	0
$28$	16.0000	0.107990
$29$	243.125	1.55680	0.778399	−	0.627769i	$-0.216032\pi$
$29$	243.125	1.55680	0.778399	+	0.627769i	$0.216032\pi$
$30$	0	0
$31$	278.717	1.61481	0.807405	−	0.589998i	$-0.200871\pi$
$31$	278.717	1.61481	0.807405	+	0.589998i	$0.200871\pi$
$32$	−143.606	−0.793318
$33$	0	0
$34$	246.745	1.24460
$35$	16.5544	0.0799486
$36$	0	0
$37$	102.380	0.454898	0.227449	−	0.973790i	$-0.426961\pi$
$37$	102.380	0.454898	0.227449	+	0.973790i	$0.426961\pi$
$38$	265.402	1.13300
$39$	0	0
$40$	−54.4512	−0.215237
$41$	−241.255	−0.918970	−0.459485	−	0.888186i	$-0.651966\pi$
$41$	−241.255	−0.918970	−0.459485	+	0.888186i	$0.651966\pi$
$42$	0	0
$43$	280.016	0.993071	0.496536	−	0.868016i	$-0.334605\pi$
$43$	280.016	0.993071	0.496536	+	0.868016i	$0.334605\pi$
$44$	0	0
$45$	0	0
$46$	−377.696	−1.21061
$47$	169.870	0.527192	0.263596	−	0.964633i	$-0.415091\pi$
$47$	169.870	0.527192	0.263596	+	0.964633i	$0.415091\pi$
$48$	0	0
$49$	−320.489	−0.934371
$50$	−380.481	−1.07616
$51$	0	0
$52$	50.6576	0.135095
$53$	409.652	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$53$	409.652	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$54$	0	0
$55$	0	0
$56$	−74.0435	−0.176687
$57$	0	0
$58$	819.886	1.85614
$59$	−196.000	−0.432492	−0.216246	−	0.976339i	$-0.569381\pi$
$59$	−196.000	−0.432492	−0.216246	+	0.976339i	$0.569381\pi$
$60$	0	0
$61$	701.359	1.47213	0.736064	−	0.676912i	$-0.236682\pi$
$61$	701.359	1.47213	0.736064	+	0.676912i	$0.236682\pi$
$62$	939.913	1.92531
$63$	0	0
$64$	152.568	0.297984
$65$	52.4128	0.100015
$66$	0	0
$67$	900.587	1.64215	0.821076	−	0.570819i	$-0.193374\pi$
$67$	900.587	1.64215	0.821076	+	0.570819i	$0.193374\pi$
$68$	246.745	0.440032
$69$	0	0
$70$	55.8260	0.0953212
$71$	−756.500	−1.26451	−0.632254	−	0.774762i	$-0.717870\pi$
$71$	−756.500	−1.26451	−0.632254	+	0.774762i	$0.717870\pi$
$72$	0	0
$73$	1019.81	1.63507	0.817536	−	0.575877i	$-0.195339\pi$
$73$	1019.81	1.63507	0.817536	+	0.575877i	$0.195339\pi$
$74$	345.255	0.542367
$75$	0	0
$76$	265.402	0.400575
$77$	0	0
$78$	0	0
$79$	327.549	0.466483	0.233241	−	0.972419i	$-0.425067\pi$
$79$	327.549	0.466483	0.233241	+	0.972419i	$0.425067\pi$
$80$	−277.755	−0.388175
$81$	0	0
$82$	−813.581	−1.09567
$83$	−756.619	−1.00060	−0.500300	−	0.865852i	$-0.666777\pi$
$83$	−756.619	−1.00060	−0.500300	+	0.865852i	$0.666777\pi$
$84$	0	0
$85$	255.294	0.325771
$86$	944.293	1.18402
$87$	0	0
$88$	0	0
$89$	−508.978	−0.606198	−0.303099	−	0.952959i	$-0.598021\pi$
$89$	−508.978	−0.606198	−0.303099	+	0.952959i	$0.598021\pi$
$90$	0	0
$91$	71.2716	0.0821022
$92$	−377.696	−0.428016
$93$	0	0
$94$	572.848	0.628561
$95$	274.598	0.296559
$96$	0	0
$97$	614.358	0.643079	0.321539	−	0.946896i	$-0.395800\pi$
$97$	614.358	0.643079	0.321539	+	0.946896i	$0.395800\pi$
$98$	−1080.78	−1.11403
$99$	0	0
$100$	−380.481	−0.380481
$101$	−1015.92	−1.00087	−0.500434	−	0.865775i	$-0.666826\pi$
$101$	−1015.92	−1.00087	−0.500434	+	0.865775i	$0.666826\pi$
$102$	0	0
$103$	1102.16	1.05436	0.527181	−	0.849753i	$-0.323249\pi$
$103$	1102.16	1.05436	0.527181	+	0.849753i	$0.323249\pi$
$104$	−234.429	−0.221035
$105$	0	0
$106$	1381.46	1.26584
$107$	1377.58	1.24463	0.622315	−	0.782767i	$-0.286192\pi$
$107$	1377.58	1.24463	0.622315	+	0.782767i	$0.286192\pi$
$108$	0	0
$109$	−320.217	−0.281388	−0.140694	−	0.990053i	$-0.544933\pi$
$109$	−320.217	−0.281388	−0.140694	+	0.990053i	$0.544933\pi$
$110$	0	0
$111$	0	0
$112$	−377.696	−0.318651
$113$	1629.45	1.35651	0.678254	−	0.734828i	$-0.262737\pi$
$113$	1629.45	1.35651	0.678254	+	0.734828i	$0.262737\pi$
$114$	0	0
$115$	−390.782	−0.316875
$116$	819.886	0.656245
$117$	0	0
$118$	−660.967	−0.515652
$119$	347.152	0.267423
$120$	0	0
$121$	0	0
$122$	2365.18	1.75519
$123$	0	0
$124$	939.913	0.680699
$125$	−829.805	−0.593760
$126$	0	0
$127$	−2291.26	−1.60091	−0.800457	−	0.599390i	$-0.795410\pi$
$127$	−2291.26	−1.60091	−0.800457	+	0.599390i	$0.795410\pi$
$128$	1663.35	1.14860
$129$	0	0
$130$	176.751	0.119247
$131$	−1147.41	−0.765267	−0.382633	−	0.923900i	$-0.624983\pi$
$131$	−1147.41	−0.765267	−0.382633	+	0.923900i	$0.624983\pi$
$132$	0	0
$133$	373.402	0.243444
$134$	3037.03	1.95791
$135$	0	0
$136$	−1141.86	−0.719956
$137$	−1268.60	−0.791121	−0.395561	−	0.918440i	$-0.629450\pi$
$137$	−1268.60	−0.791121	−0.395561	+	0.918440i	$0.629450\pi$
$138$	0	0
$139$	486.288	0.296737	0.148368	−	0.988932i	$-0.452598\pi$
$139$	486.288	0.296737	0.148368	+	0.988932i	$0.452598\pi$
$140$	55.8260	0.0337011
$141$	0	0
$142$	−2551.13	−1.50765
$143$	0	0
$144$	0	0
$145$	848.293	0.485841
$146$	3439.10	1.94947
$147$	0	0
$148$	345.255	0.191756
$149$	2354.11	1.29434	0.647169	−	0.762346i	$-0.275953\pi$
$149$	2354.11	1.29434	0.647169	+	0.762346i	$0.275953\pi$
$150$	0	0
$151$	570.070	0.307229	0.153615	−	0.988131i	$-0.450909\pi$
$151$	570.070	0.307229	0.153615	+	0.988131i	$0.450909\pi$
$152$	−1228.21	−0.655399
$153$	0	0
$154$	0	0
$155$	972.479	0.503945
$156$	0	0
$157$	−2072.67	−1.05361	−0.526807	−	0.849985i	$-0.676611\pi$
$157$	−2072.67	−1.05361	−0.526807	+	0.849985i	$0.676611\pi$
$158$	1104.59	0.556179
$159$	0	0
$160$	−501.059	−0.247576
$161$	−531.391	−0.260121
$162$	0	0
$163$	2676.51	1.28614	0.643069	−	0.765808i	$-0.277661\pi$
$163$	2676.51	1.28614	0.643069	+	0.765808i	$0.277661\pi$
$164$	−813.581	−0.387378
$165$	0	0
$166$	−2551.53	−1.19300
$167$	−1188.12	−0.550536	−0.275268	−	0.961368i	$-0.588767\pi$
$167$	−1188.12	−0.550536	−0.275268	+	0.961368i	$0.588767\pi$
$168$	0	0
$169$	−1971.35	−0.897290
$170$	860.923	0.388410
$171$	0	0
$172$	944.293	0.418615
$173$	807.147	0.354718	0.177359	−	0.984146i	$-0.443245\pi$
$173$	807.147	0.354718	0.177359	+	0.984146i	$0.443245\pi$
$174$	0	0
$175$	−535.310	−0.231232
$176$	0	0
$177$	0	0
$178$	−1716.42	−0.722758
$179$	1950.39	0.814408	0.407204	−	0.913337i	$-0.366504\pi$
$179$	1950.39	0.814408	0.407204	+	0.913337i	$0.366504\pi$
$180$	0	0
$181$	1061.61	0.435959	0.217980	−	0.975953i	$-0.430053\pi$
$181$	1061.61	0.435959	0.217980	+	0.975953i	$0.430053\pi$
$182$	240.348	0.0978889
$183$	0	0
$184$	1747.87	0.700297
$185$	357.218	0.141963
$186$	0	0
$187$	0	0
$188$	572.848	0.222230
$189$	0	0
$190$	926.021	0.353582
$191$	−2136.41	−0.809348	−0.404674	−	0.914461i	$-0.632615\pi$
$191$	−2136.41	−0.809348	−0.404674	+	0.914461i	$0.632615\pi$
$192$	0	0
$193$	−3947.76	−1.47236	−0.736181	−	0.676784i	$-0.763373\pi$
$193$	−3947.76	−1.47236	−0.736181	+	0.676784i	$0.763373\pi$
$194$	2071.79	0.766731
$195$	0	0
$196$	−1080.78	−0.393870
$197$	923.886	0.334133	0.167066	−	0.985946i	$-0.446571\pi$
$197$	923.886	0.334133	0.167066	+	0.985946i	$0.446571\pi$
$198$	0	0
$199$	−476.152	−0.169616	−0.0848078	−	0.996397i	$-0.527028\pi$
$199$	−476.152	−0.169616	−0.0848078	+	0.996397i	$0.527028\pi$
$200$	1760.76	0.622522
$201$	0	0
$202$	−3425.96	−1.19332
$203$	1153.52	0.398824
$204$	0	0
$205$	−841.770	−0.286789
$206$	3716.80	1.25710
$207$	0	0
$208$	−1195.82	−0.398631
$209$	0	0
$210$	0	0
$211$	4918.24	1.60467	0.802336	−	0.596872i	$-0.203590\pi$
$211$	4918.24	1.60467	0.802336	+	0.596872i	$0.203590\pi$
$212$	1381.46	0.447543
$213$	0	0
$214$	4645.57	1.48395
$215$	977.012	0.309915
$216$	0	0
$217$	1322.39	0.413686
$218$	−1079.86	−0.335494
$219$	0	0
$220$	0	0
$221$	1099.12	0.334546
$222$	0	0
$223$	2100.29	0.630700	0.315350	−	0.948975i	$-0.397878\pi$
$223$	2100.29	0.630700	0.315350	+	0.948975i	$0.397878\pi$
$224$	−681.348	−0.203234
$225$	0	0
$226$	5494.95	1.61734
$227$	−2257.16	−0.659970	−0.329985	−	0.943986i	$-0.607044\pi$
$227$	−2257.16	−0.659970	−0.329985	+	0.943986i	$0.607044\pi$
$228$	0	0
$229$	−5311.07	−1.53260	−0.766301	−	0.642482i	$-0.777905\pi$
$229$	−5311.07	−1.53260	−0.766301	+	0.642482i	$0.777905\pi$
$230$	−1317.83	−0.377804
$231$	0	0
$232$	−3794.20	−1.07371
$233$	2466.27	0.693435	0.346718	−	0.937970i	$-0.387296\pi$
$233$	2466.27	0.693435	0.346718	+	0.937970i	$0.387296\pi$
$234$	0	0
$235$	592.696	0.164524
$236$	−660.967	−0.182311
$237$	0	0
$238$	1170.70	0.318844
$239$	1429.40	0.386863	0.193432	−	0.981114i	$-0.438038\pi$
$239$	1429.40	0.386863	0.193432	+	0.981114i	$0.438038\pi$
$240$	0	0
$241$	978.989	0.261669	0.130835	−	0.991404i	$-0.458234\pi$
$241$	978.989	0.261669	0.130835	+	0.991404i	$0.458234\pi$
$242$	0	0
$243$	0	0
$244$	2365.18	0.620553
$245$	−1118.23	−0.291595
$246$	0	0
$247$	1182.23	0.304548
$248$	−4349.65	−1.11372
$249$	0	0
$250$	−2798.33	−0.707929
$251$	6530.63	1.64227	0.821135	−	0.570734i	$-0.193341\pi$
$251$	6530.63	1.64227	0.821135	+	0.570734i	$0.193341\pi$
$252$	0	0
$253$	0	0
$254$	−7726.76	−1.90874
$255$	0	0
$256$	4388.74	1.07147
$257$	−8130.26	−1.97335	−0.986676	−	0.162696i	$-0.947981\pi$
$257$	−8130.26	−1.97335	−0.986676	+	0.162696i	$0.947981\pi$
$258$	0	0
$259$	485.750	0.116537
$260$	176.751	0.0421600
$261$	0	0
$262$	−3869.40	−0.912414
$263$	−4549.42	−1.06665	−0.533326	−	0.845910i	$-0.679058\pi$
$263$	−4549.42	−1.06665	−0.533326	+	0.845910i	$0.679058\pi$
$264$	0	0
$265$	1429.33	0.331332
$266$	1259.22	0.290254
$267$	0	0
$268$	3037.03	0.692225
$269$	29.1522	0.00660760	0.00330380	−	0.999995i	$-0.498948\pi$
$269$	29.1522	0.00660760	0.00330380	+	0.999995i	$0.498948\pi$
$270$	0	0
$271$	−7711.22	−1.72850	−0.864250	−	0.503063i	$-0.832206\pi$
$271$	−7711.22	−1.72850	−0.864250	+	0.503063i	$0.832206\pi$
$272$	−5824.64	−1.29842
$273$	0	0
$274$	−4278.07	−0.943239
$275$	0	0
$276$	0	0
$277$	−1127.52	−0.244571	−0.122286	−	0.992495i	$-0.539022\pi$
$277$	−1127.52	−0.244571	−0.122286	+	0.992495i	$0.539022\pi$
$278$	1639.90	0.353794
$279$	0	0
$280$	−258.347	−0.0551400
$281$	−1872.47	−0.397517	−0.198758	−	0.980049i	$-0.563691\pi$
$281$	−1872.47	−0.397517	−0.198758	+	0.980049i	$0.563691\pi$
$282$	0	0
$283$	−2124.48	−0.446245	−0.223123	−	0.974790i	$-0.571625\pi$
$283$	−2124.48	−0.446245	−0.223123	+	0.974790i	$0.571625\pi$
$284$	−2551.13	−0.533034
$285$	0	0
$286$	0	0
$287$	−1144.65	−0.235424
$288$	0	0
$289$	440.621	0.0896846
$290$	2860.68	0.579259
$291$	0	0
$292$	3439.10	0.689241
$293$	−3324.19	−0.662802	−0.331401	−	0.943490i	$-0.607521\pi$
$293$	−3324.19	−0.662802	−0.331401	+	0.943490i	$0.607521\pi$
$294$	0	0
$295$	−683.869	−0.134971
$296$	−1597.75	−0.313740
$297$	0	0
$298$	7938.73	1.54322
$299$	−1682.44	−0.325411
$300$	0	0
$301$	1328.55	0.254407
$302$	1922.44	0.366304
$303$	0	0
$304$	−6265.07	−1.18200
$305$	2447.13	0.459417
$306$	0	0
$307$	1698.94	0.315843	0.157921	−	0.987452i	$-0.449521\pi$
$307$	1698.94	0.315843	0.157921	+	0.987452i	$0.449521\pi$
$308$	0	0
$309$	0	0
$310$	3279.47	0.600844
$311$	−6928.83	−1.26334	−0.631668	−	0.775239i	$-0.717630\pi$
$311$	−6928.83	−1.26334	−0.631668	+	0.775239i	$0.717630\pi$
$312$	0	0
$313$	−3560.75	−0.643020	−0.321510	−	0.946906i	$-0.604190\pi$
$313$	−3560.75	−0.643020	−0.321510	+	0.946906i	$0.604190\pi$
$314$	−6989.64	−1.25620
$315$	0	0
$316$	1104.59	0.196639
$317$	−332.750	−0.0589561	−0.0294780	−	0.999565i	$-0.509385\pi$
$317$	−332.750	−0.0589561	−0.0294780	+	0.999565i	$0.509385\pi$
$318$	0	0
$319$	0	0
$320$	532.329	0.0929940
$321$	0	0
$322$	−1792.00	−0.310137
$323$	5758.43	0.991975
$324$	0	0
$325$	−1694.84	−0.289271
$326$	9025.94	1.53344
$327$	0	0
$328$	3765.02	0.633807
$329$	805.957	0.135057
$330$	0	0
$331$	−541.445	−0.0899108	−0.0449554	−	0.998989i	$-0.514315\pi$
$331$	−541.445	−0.0899108	−0.0449554	+	0.998989i	$0.514315\pi$
$332$	−2551.53	−0.421788
$333$	0	0
$334$	−4006.67	−0.656393
$335$	3142.26	0.512478
$336$	0	0
$337$	−816.531	−0.131986	−0.0659930	−	0.997820i	$-0.521022\pi$
$337$	−816.531	−0.131986	−0.0659930	+	0.997820i	$0.521022\pi$
$338$	−6647.94	−1.06982
$339$	0	0
$340$	860.923	0.137324
$341$	0	0
$342$	0	0
$343$	−3147.97	−0.495552
$344$	−4369.92	−0.684914
$345$	0	0
$346$	2721.93	0.422924
$347$	6260.53	0.968539	0.484269	−	0.874919i	$-0.339086\pi$
$347$	6260.53	0.968539	0.484269	+	0.874919i	$0.339086\pi$
$348$	0	0
$349$	12768.5	1.95840	0.979198	−	0.202906i	$-0.0650386\pi$
$349$	12768.5	1.95840	0.979198	+	0.202906i	$0.0650386\pi$
$350$	−1805.22	−0.275694
$351$	0	0
$352$	0	0
$353$	2649.28	0.399453	0.199727	−	0.979852i	$-0.435995\pi$
$353$	2649.28	0.399453	0.199727	+	0.979852i	$0.435995\pi$
$354$	0	0
$355$	−2639.52	−0.394623
$356$	−1716.42	−0.255534
$357$	0	0
$358$	6577.27	0.971004
$359$	−3203.91	−0.471020	−0.235510	−	0.971872i	$-0.575676\pi$
$359$	−3203.91	−0.471020	−0.235510	+	0.971872i	$0.575676\pi$
$360$	0	0
$361$	−665.143	−0.0969737
$362$	3580.04	0.519786
$363$	0	0
$364$	240.348	0.0346089
$365$	3558.26	0.510268
$366$	0	0
$367$	−8429.40	−1.19894	−0.599470	−	0.800397i	$-0.704622\pi$
$367$	−8429.40	−1.19894	−0.599470	+	0.800397i	$0.704622\pi$
$368$	8915.87	1.26297
$369$	0	0
$370$	1204.64	0.169260
$371$	1943.62	0.271988
$372$	0	0
$373$	9388.53	1.30327	0.651635	−	0.758533i	$-0.274083\pi$
$373$	9388.53	1.30327	0.651635	+	0.758533i	$0.274083\pi$
$374$	0	0
$375$	0	0
$376$	−2650.98	−0.363600
$377$	3652.16	0.498928
$378$	0	0
$379$	−14264.5	−1.93329	−0.966647	−	0.256112i	$-0.917558\pi$
$379$	−14264.5	−1.93329	−0.966647	+	0.256112i	$0.917558\pi$
$380$	926.021	0.125010
$381$	0	0
$382$	−7204.58	−0.964970
$383$	−13462.2	−1.79605	−0.898026	−	0.439942i	$-0.854999\pi$
$383$	−13462.2	−1.79605	−0.898026	+	0.439942i	$0.854999\pi$
$384$	0	0
$385$	0	0
$386$	−13313.0	−1.75547
$387$	0	0
$388$	2071.79	0.271080
$389$	941.881	0.122764	0.0613821	−	0.998114i	$-0.480449\pi$
$389$	941.881	0.122764	0.0613821	+	0.998114i	$0.480449\pi$
$390$	0	0
$391$	−8194.87	−1.05993
$392$	5001.54	0.644429
$393$	0	0
$394$	3115.60	0.398380
$395$	1142.86	0.145578
$396$	0	0
$397$	−847.839	−0.107183	−0.0535917	−	0.998563i	$-0.517067\pi$
$397$	−847.839	−0.107183	−0.0535917	+	0.998563i	$0.517067\pi$
$398$	−1605.72	−0.202230
$399$	0	0
$400$	8981.62	1.12270
$401$	−12203.6	−1.51975	−0.759875	−	0.650069i	$-0.774740\pi$
$401$	−12203.6	−1.51975	−0.759875	+	0.650069i	$0.774740\pi$
$402$	0	0
$403$	4186.82	0.517520
$404$	−3425.96	−0.421901
$405$	0	0
$406$	3890.00	0.475511
$407$	0	0
$408$	0	0
$409$	−8759.53	−1.05900	−0.529500	−	0.848310i	$-0.677620\pi$
$409$	−8759.53	−1.05900	−0.529500	+	0.848310i	$0.677620\pi$
$410$	−2838.69	−0.341934
$411$	0	0
$412$	3716.80	0.444451
$413$	−929.934	−0.110797
$414$	0	0
$415$	−2639.94	−0.312264
$416$	−2157.21	−0.254245
$417$	0	0
$418$	0	0
$419$	11188.4	1.30451	0.652256	−	0.757999i	$-0.273823\pi$
$419$	11188.4	1.30451	0.652256	+	0.757999i	$0.273823\pi$
$420$	0	0
$421$	−14082.3	−1.63023	−0.815116	−	0.579298i	$-0.803327\pi$
$421$	−14082.3	−1.63023	−0.815116	+	0.579298i	$0.803327\pi$
$422$	16585.7	1.91322
$423$	0	0
$424$	−6393.02	−0.732246
$425$	−8255.30	−0.942214
$426$	0	0
$427$	3327.64	0.377133
$428$	4645.57	0.524655
$429$	0	0
$430$	3294.76	0.369505
$431$	−5616.05	−0.627647	−0.313823	−	0.949481i	$-0.601610\pi$
$431$	−5616.05	−0.627647	−0.313823	+	0.949481i	$0.601610\pi$
$432$	0	0
$433$	7195.75	0.798627	0.399314	−	0.916814i	$-0.369248\pi$
$433$	7195.75	0.798627	0.399314	+	0.916814i	$0.369248\pi$
$434$	4459.48	0.493230
$435$	0	0
$436$	−1079.86	−0.118615
$437$	−8814.52	−0.964887
$438$	0	0
$439$	−101.959	−0.0110848	−0.00554240	−	0.999985i	$-0.501764\pi$
$439$	−101.959	−0.0110848	−0.00554240	+	0.999985i	$0.501764\pi$
$440$	0	0
$441$	0	0
$442$	3706.53	0.398873
$443$	−4953.74	−0.531285	−0.265642	−	0.964072i	$-0.585584\pi$
$443$	−4953.74	−0.531285	−0.265642	+	0.964072i	$0.585584\pi$
$444$	0	0
$445$	−1775.89	−0.189180
$446$	7082.78	0.751972
$447$	0	0
$448$	723.869	0.0763383
$449$	11602.0	1.21945	0.609723	−	0.792615i	$-0.291281\pi$
$449$	11602.0	1.21945	0.609723	+	0.792615i	$0.291281\pi$
$450$	0	0
$451$	0	0
$452$	5494.95	0.571816
$453$	0	0
$454$	−7611.79	−0.786870
$455$	248.676	0.0256222
$456$	0	0
$457$	3530.68	0.361397	0.180698	−	0.983539i	$-0.442164\pi$
$457$	3530.68	0.361397	0.180698	+	0.983539i	$0.442164\pi$
$458$	−17910.4	−1.82729
$459$	0	0
$460$	−1317.83	−0.133574
$461$	11566.3	1.16854	0.584271	−	0.811559i	$-0.301381\pi$
$461$	11566.3	1.16854	0.584271	+	0.811559i	$0.301381\pi$
$462$	0	0
$463$	10888.5	1.09294	0.546470	−	0.837479i	$-0.315971\pi$
$463$	10888.5	1.09294	0.546470	+	0.837479i	$0.315971\pi$
$464$	−19354.2	−1.93641
$465$	0	0
$466$	8316.94	0.826770
$467$	−10688.0	−1.05906	−0.529529	−	0.848292i	$-0.677631\pi$
$467$	−10688.0	−1.05906	−0.529529	+	0.848292i	$0.677631\pi$
$468$	0	0
$469$	4272.89	0.420690
$470$	1998.74	0.196159
$471$	0	0
$472$	3058.77	0.298287
$473$	0	0
$474$	0	0
$475$	−8879.53	−0.857728
$476$	1170.70	0.112728
$477$	0	0
$478$	4820.35	0.461250
$479$	2341.90	0.223391	0.111696	−	0.993742i	$-0.464372\pi$
$479$	2341.90	0.223391	0.111696	+	0.993742i	$0.464372\pi$
$480$	0	0
$481$	1537.93	0.145787
$482$	3301.43	0.311983
$483$	0	0
$484$	0	0
$485$	2143.57	0.200690
$486$	0	0
$487$	6748.91	0.627972	0.313986	−	0.949428i	$-0.398335\pi$
$487$	6748.91	0.627972	0.313986	+	0.949428i	$0.398335\pi$
$488$	−10945.4	−1.01532
$489$	0	0
$490$	−3770.98	−0.347664
$491$	7361.40	0.676609	0.338305	−	0.941037i	$-0.390147\pi$
$491$	7361.40	0.676609	0.338305	+	0.941037i	$0.390147\pi$
$492$	0	0
$493$	17789.1	1.62511
$494$	3986.80	0.363107
$495$	0	0
$496$	−22187.6	−2.00857
$497$	−3589.26	−0.323944
$498$	0	0
$499$	10381.7	0.931359	0.465680	−	0.884953i	$-0.345810\pi$
$499$	10381.7	0.931359	0.465680	+	0.884953i	$0.345810\pi$
$500$	−2798.33	−0.250291
$501$	0	0
$502$	22023.1	1.95805
$503$	19149.0	1.69744	0.848721	−	0.528840i	$-0.177373\pi$
$503$	19149.0	1.69744	0.848721	+	0.528840i	$0.177373\pi$
$504$	0	0
$505$	−3544.67	−0.312348
$506$	0	0
$507$	0	0
$508$	−7726.76	−0.674841
$509$	−16073.2	−1.39967	−0.699836	−	0.714303i	$-0.746744\pi$
$509$	−16073.2	−1.39967	−0.699836	+	0.714303i	$0.746744\pi$
$510$	0	0
$511$	4838.58	0.418877
$512$	1493.27	0.128894
$513$	0	0
$514$	−27417.5	−2.35279
$515$	3845.58	0.329042
$516$	0	0
$517$	0	0
$518$	1638.09	0.138945
$519$	0	0
$520$	−817.952	−0.0689799
$521$	18955.3	1.59395	0.796975	−	0.604012i	$-0.206432\pi$
$521$	18955.3	1.59395	0.796975	+	0.604012i	$0.206432\pi$
$522$	0	0
$523$	4442.19	0.371402	0.185701	−	0.982606i	$-0.440544\pi$
$523$	4442.19	0.371402	0.185701	+	0.982606i	$0.440544\pi$
$524$	−3869.40	−0.322587
$525$	0	0
$526$	−15341.9	−1.27175
$527$	20393.3	1.68567
$528$	0	0
$529$	377.000	0.0309855
$530$	4820.09	0.395041
$531$	0	0
$532$	1259.22	0.102620
$533$	−3624.08	−0.294515
$534$	0	0
$535$	4806.54	0.388420
$536$	−14054.5	−1.13258
$537$	0	0
$538$	98.3096	0.00787812
$539$	0	0
$540$	0	0
$541$	−2180.90	−0.173316	−0.0866580	−	0.996238i	$-0.527619\pi$
$541$	−2180.90	−0.173316	−0.0866580	+	0.996238i	$0.527619\pi$
$542$	−26004.4	−2.06086
$543$	0	0
$544$	−10507.4	−0.828129
$545$	−1117.28	−0.0878146
$546$	0	0
$547$	−8225.04	−0.642920	−0.321460	−	0.946923i	$-0.604174\pi$
$547$	−8225.04	−0.642920	−0.321460	+	0.946923i	$0.604174\pi$
$548$	−4278.07	−0.333485
$549$	0	0
$550$	0	0
$551$	19134.2	1.47939
$552$	0	0
$553$	1554.08	0.119505
$554$	−3802.32	−0.291598
$555$	0	0
$556$	1639.90	0.125085
$557$	−25181.9	−1.91561	−0.957804	−	0.287423i	$-0.907201\pi$
$557$	−25181.9	−1.91561	−0.957804	+	0.287423i	$0.907201\pi$
$558$	0	0
$559$	4206.33	0.318263
$560$	−1317.83	−0.0994435
$561$	0	0
$562$	−6314.50	−0.473952
$563$	−4504.50	−0.337197	−0.168599	−	0.985685i	$-0.553924\pi$
$563$	−4504.50	−0.337197	−0.168599	+	0.985685i	$0.553924\pi$
$564$	0	0
$565$	5685.34	0.423335
$566$	−7164.36	−0.532050
$567$	0	0
$568$	11805.9	0.872122
$569$	−13447.0	−0.990732	−0.495366	−	0.868684i	$-0.664966\pi$
$569$	−13447.0	−0.990732	−0.495366	+	0.868684i	$0.664966\pi$
$570$	0	0
$571$	2605.52	0.190959	0.0954795	−	0.995431i	$-0.469562\pi$
$571$	2605.52	0.190959	0.0954795	+	0.995431i	$0.469562\pi$
$572$	0	0
$573$	0	0
$574$	−3860.09	−0.280691
$575$	12636.5	0.916485
$576$	0	0
$577$	6339.65	0.457406	0.228703	−	0.973496i	$-0.426552\pi$
$577$	6339.65	0.457406	0.228703	+	0.973496i	$0.426552\pi$
$578$	1485.90	0.106929
$579$	0	0
$580$	2860.68	0.204799
$581$	−3589.83	−0.256336
$582$	0	0
$583$	0	0
$584$	−15915.2	−1.12770
$585$	0	0
$586$	−11210.1	−0.790247
$587$	13370.6	0.940140	0.470070	−	0.882629i	$-0.344229\pi$
$587$	13370.6	0.940140	0.470070	+	0.882629i	$0.344229\pi$
$588$	0	0
$589$	21935.3	1.53452
$590$	−2306.20	−0.160923
$591$	0	0
$592$	−8150.09	−0.565822
$593$	14319.3	0.991608	0.495804	−	0.868434i	$-0.334873\pi$
$593$	14319.3	0.991608	0.495804	+	0.868434i	$0.334873\pi$
$594$	0	0
$595$	1211.26	0.0834567
$596$	7938.73	0.545609
$597$	0	0
$598$	−5673.65	−0.387981
$599$	5788.63	0.394853	0.197427	−	0.980318i	$-0.436742\pi$
$599$	5788.63	0.394853	0.197427	+	0.980318i	$0.436742\pi$
$600$	0	0
$601$	−23968.1	−1.62675	−0.813375	−	0.581739i	$-0.802372\pi$
$601$	−23968.1	−1.62675	−0.813375	+	0.581739i	$0.802372\pi$
$602$	4480.26	0.303325
$603$	0	0
$604$	1922.44	0.129508
$605$	0	0
$606$	0	0
$607$	23526.6	1.57317	0.786585	−	0.617482i	$-0.211847\pi$
$607$	23526.6	1.57317	0.786585	+	0.617482i	$0.211847\pi$
$608$	−11301.9	−0.753872
$609$	0	0
$610$	8252.40	0.547754
$611$	2551.74	0.168956
$612$	0	0
$613$	−1228.07	−0.0809159	−0.0404579	−	0.999181i	$-0.512882\pi$
$613$	−1228.07	−0.0809159	−0.0404579	+	0.999181i	$0.512882\pi$
$614$	5729.31	0.376573
$615$	0	0
$616$	0	0
$617$	9844.90	0.642368	0.321184	−	0.947017i	$-0.395919\pi$
$617$	9844.90	0.642368	0.321184	+	0.947017i	$0.395919\pi$
$618$	0	0
$619$	−6551.68	−0.425419	−0.212709	−	0.977115i	$-0.568229\pi$
$619$	−6551.68	−0.425419	−0.212709	+	0.977115i	$0.568229\pi$
$620$	3279.47	0.212430
$621$	0	0
$622$	−23365.9	−1.50625
$623$	−2414.88	−0.155297
$624$	0	0
$625$	11208.0	0.717309
$626$	−12007.8	−0.766661
$627$	0	0
$628$	−6989.64	−0.444135
$629$	7491.01	0.474859
$630$	0	0
$631$	−26440.5	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$631$	−26440.5	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$632$	−5111.72	−0.321730
$633$	0	0
$634$	−1122.13	−0.0702923
$635$	−7994.48	−0.499608
$636$	0	0
$637$	−4814.31	−0.299450
$638$	0	0
$639$	0	0
$640$	5803.64	0.358451
$641$	27927.2	1.72084	0.860421	−	0.509584i	$-0.170201\pi$
$641$	27927.2	1.72084	0.860421	+	0.509584i	$0.170201\pi$
$642$	0	0
$643$	−16737.7	−1.02655	−0.513274	−	0.858225i	$-0.671568\pi$
$643$	−16737.7	−1.02655	−0.513274	+	0.858225i	$0.671568\pi$
$644$	−1792.00	−0.109650
$645$	0	0
$646$	19419.1	1.18271
$647$	−7818.70	−0.475092	−0.237546	−	0.971376i	$-0.576343\pi$
$647$	−7818.70	−0.475092	−0.237546	+	0.971376i	$0.576343\pi$
$648$	0	0
$649$	0	0
$650$	−5715.49	−0.344892
$651$	0	0
$652$	9025.94	0.542152
$653$	−19747.6	−1.18344	−0.591719	−	0.806144i	$-0.701550\pi$
$653$	−19747.6	−1.18344	−0.591719	+	0.806144i	$0.701550\pi$
$654$	0	0
$655$	−4003.47	−0.238822
$656$	19205.4	1.14305
$657$	0	0
$658$	2717.91	0.161026
$659$	7867.72	0.465072	0.232536	−	0.972588i	$-0.425298\pi$
$659$	7867.72	0.465072	0.232536	+	0.972588i	$0.425298\pi$
$660$	0	0
$661$	4227.41	0.248755	0.124378	−	0.992235i	$-0.460307\pi$
$661$	4227.41	0.248755	0.124378	+	0.992235i	$0.460307\pi$
$662$	−1825.90	−0.107199
$663$	0	0
$664$	11807.8	0.690106
$665$	1302.85	0.0759733
$666$	0	0
$667$	−27230.0	−1.58073
$668$	−4006.67	−0.232070
$669$	0	0
$670$	10596.6	0.611018
$671$	0	0
$672$	0	0
$673$	−29397.6	−1.68379	−0.841897	−	0.539638i	$-0.818561\pi$
$673$	−29397.6	−1.68379	−0.841897	+	0.539638i	$0.818561\pi$
$674$	−2753.57	−0.157364
$675$	0	0
$676$	−6647.94	−0.378239
$677$	5737.14	0.325696	0.162848	−	0.986651i	$-0.447932\pi$
$677$	5737.14	0.325696	0.162848	+	0.986651i	$0.447932\pi$
$678$	0	0
$679$	2914.86	0.164745
$680$	−3984.11	−0.224682
$681$	0	0
$682$	0	0
$683$	−32097.6	−1.79821	−0.899107	−	0.437729i	$-0.855783\pi$
$683$	−32097.6	−1.79821	−0.899107	+	0.437729i	$0.855783\pi$
$684$	0	0
$685$	−4426.30	−0.246891
$686$	−10615.8	−0.590837
$687$	0	0
$688$	−22291.0	−1.23523
$689$	6153.69	0.340257
$690$	0	0
$691$	−16456.2	−0.905965	−0.452983	−	0.891519i	$-0.649640\pi$
$691$	−16456.2	−0.905965	−0.452983	+	0.891519i	$0.649640\pi$
$692$	2721.93	0.149526
$693$	0	0
$694$	21112.3	1.15477
$695$	1696.72	0.0926047
$696$	0	0
$697$	−17652.3	−0.959294
$698$	43058.9	2.33496
$699$	0	0
$700$	−1805.22	−0.0974725
$701$	27238.1	1.46758	0.733788	−	0.679379i	$-0.237751\pi$
$701$	27238.1	1.46758	0.733788	+	0.679379i	$0.237751\pi$
$702$	0	0
$703$	8057.44	0.432279
$704$	0	0
$705$	0	0
$706$	8934.12	0.476261
$707$	−4820.09	−0.256405
$708$	0	0
$709$	28761.4	1.52349	0.761747	−	0.647875i	$-0.224342\pi$
$709$	28761.4	1.52349	0.761747	+	0.647875i	$0.224342\pi$
$710$	−8901.21	−0.470502
$711$	0	0
$712$	7943.10	0.418090
$713$	−31216.3	−1.63964
$714$	0	0
$715$	0	0
$716$	6577.27	0.343302
$717$	0	0
$718$	−10804.5	−0.561588
$719$	27272.0	1.41456	0.707282	−	0.706931i	$-0.249921\pi$
$719$	27272.0	1.41456	0.707282	+	0.706931i	$0.249921\pi$
$720$	0	0
$721$	5229.28	0.270109
$722$	−2243.05	−0.115620
$723$	0	0
$724$	3580.04	0.183772
$725$	−27430.8	−1.40518
$726$	0	0
$727$	3979.75	0.203027	0.101514	−	0.994834i	$-0.467631\pi$
$727$	3979.75	0.203027	0.101514	+	0.994834i	$0.467631\pi$
$728$	−1112.26	−0.0566253
$729$	0	0
$730$	11999.5	0.608383
$731$	20488.3	1.03665
$732$	0	0
$733$	−9342.48	−0.470767	−0.235384	−	0.971903i	$-0.575635\pi$
$733$	−9342.48	−0.470767	−0.235384	+	0.971903i	$0.575635\pi$
$734$	−28426.3	−1.42947
$735$	0	0
$736$	16083.9	0.805515
$737$	0	0
$738$	0	0
$739$	28928.0	1.43997	0.719983	−	0.693992i	$-0.244150\pi$
$739$	28928.0	1.43997	0.719983	+	0.693992i	$0.244150\pi$
$740$	1204.64	0.0598425
$741$	0	0
$742$	6554.43	0.324287
$743$	4857.04	0.239822	0.119911	−	0.992785i	$-0.461739\pi$
$743$	4857.04	0.239822	0.119911	+	0.992785i	$0.461739\pi$
$744$	0	0
$745$	8213.80	0.403933
$746$	31660.8	1.55386
$747$	0	0
$748$	0	0
$749$	6536.00	0.318852
$750$	0	0
$751$	14355.4	0.697517	0.348759	−	0.937213i	$-0.386603\pi$
$751$	14355.4	0.697517	0.348759	+	0.937213i	$0.386603\pi$
$752$	−13522.6	−0.655744
$753$	0	0
$754$	12316.1	0.594863
$755$	1989.05	0.0958792
$756$	0	0
$757$	−17714.9	−0.850538	−0.425269	−	0.905067i	$-0.639821\pi$
$757$	−17714.9	−0.850538	−0.425269	+	0.905067i	$0.639821\pi$
$758$	−48103.9	−2.30503
$759$	0	0
$760$	−4285.37	−0.204535
$761$	−7945.82	−0.378497	−0.189248	−	0.981929i	$-0.560605\pi$
$761$	−7945.82	−0.378497	−0.189248	+	0.981929i	$0.560605\pi$
$762$	0	0
$763$	−1519.29	−0.0720866
$764$	−7204.58	−0.341168
$765$	0	0
$766$	−45398.4	−2.14140
$767$	−2944.26	−0.138606
$768$	0	0
$769$	−27308.1	−1.28057	−0.640284	−	0.768139i	$-0.721183\pi$
$769$	−27308.1	−1.28057	−0.640284	+	0.768139i	$0.721183\pi$
$770$	0	0
$771$	0	0
$772$	−13313.0	−0.620653
$773$	18872.6	0.878136	0.439068	−	0.898454i	$-0.355309\pi$
$773$	18872.6	0.878136	0.439068	+	0.898454i	$0.355309\pi$
$774$	0	0
$775$	−31446.6	−1.45754
$776$	−9587.65	−0.443527
$777$	0	0
$778$	3176.29	0.146369
$779$	−18987.1	−0.873276
$780$	0	0
$781$	0	0
$782$	−27635.4	−1.26373
$783$	0	0
$784$	25512.8	1.16221
$785$	−7231.82	−0.328808
$786$	0	0
$787$	14512.1	0.657307	0.328654	−	0.944450i	$-0.393405\pi$
$787$	14512.1	0.657307	0.328654	+	0.944450i	$0.393405\pi$
$788$	3115.60	0.140849
$789$	0	0
$790$	3854.04	0.173570
$791$	7731.01	0.347513
$792$	0	0
$793$	10535.6	0.471792
$794$	−2859.15	−0.127793
$795$	0	0
$796$	−1605.72	−0.0714989
$797$	−29108.9	−1.29371	−0.646856	−	0.762612i	$-0.723917\pi$
$797$	−29108.9	−1.29371	−0.646856	+	0.762612i	$0.723917\pi$
$798$	0	0
$799$	12429.1	0.550325
$800$	16202.5	0.716056
$801$	0	0
$802$	−41154.0	−1.81197
$803$	0	0
$804$	0	0
$805$	−1854.09	−0.0811777
$806$	14119.1	0.617029
$807$	0	0
$808$	15854.4	0.690291
$809$	−3000.83	−0.130413	−0.0652063	−	0.997872i	$-0.520771\pi$
$809$	−3000.83	−0.130413	−0.0652063	+	0.997872i	$0.520771\pi$
$810$	0	0
$811$	−6239.39	−0.270154	−0.135077	−	0.990835i	$-0.543128\pi$
$811$	−6239.39	−0.270154	−0.135077	+	0.990835i	$0.543128\pi$
$812$	3890.00	0.168118
$813$	0	0
$814$	0	0
$815$	9338.68	0.401374
$816$	0	0
$817$	22037.6	0.943693
$818$	−29539.6	−1.26263
$819$	0	0
$820$	−2838.69	−0.120892
$821$	14922.4	0.634342	0.317171	−	0.948368i	$-0.397267\pi$
$821$	14922.4	0.634342	0.317171	+	0.948368i	$0.397267\pi$
$822$	0	0
$823$	−25737.8	−1.09011	−0.545057	−	0.838399i	$-0.683492\pi$
$823$	−25737.8	−1.09011	−0.545057	+	0.838399i	$0.683492\pi$
$824$	−17200.3	−0.727186
$825$	0	0
$826$	−3136.00	−0.132101
$827$	27043.4	1.13711	0.568555	−	0.822645i	$-0.307503\pi$
$827$	27043.4	1.13711	0.568555	+	0.822645i	$0.307503\pi$
$828$	0	0
$829$	−9795.41	−0.410384	−0.205192	−	0.978722i	$-0.565782\pi$
$829$	−9795.41	−0.410384	−0.205192	+	0.978722i	$0.565782\pi$
$830$	−8902.62	−0.372306
$831$	0	0
$832$	2291.84	0.0954990
$833$	−23449.7	−0.975370
$834$	0	0
$835$	−4145.50	−0.171809
$836$	0	0
$837$	0	0
$838$	37730.5	1.55535
$839$	−28875.5	−1.18819	−0.594095	−	0.804395i	$-0.702490\pi$
$839$	−28875.5	−1.18819	−0.594095	+	0.804395i	$0.702490\pi$
$840$	0	0
$841$	34720.7	1.42362
$842$	−47489.3	−1.94369
$843$	0	0
$844$	16585.7	0.676426
$845$	−6878.28	−0.280024
$846$	0	0
$847$	0	0
$848$	−32610.7	−1.32059
$849$	0	0
$850$	−27839.2	−1.12338
$851$	−11466.6	−0.461892
$852$	0	0
$853$	47157.1	1.89288	0.946441	−	0.322878i	$-0.104650\pi$
$853$	47157.1	1.89288	0.946441	+	0.322878i	$0.104650\pi$
$854$	11221.7	0.449649
$855$	0	0
$856$	−21498.4	−0.858412
$857$	5021.31	0.200145	0.100073	−	0.994980i	$-0.468092\pi$
$857$	5021.31	0.200145	0.100073	+	0.994980i	$0.468092\pi$
$858$	0	0
$859$	−22921.1	−0.910428	−0.455214	−	0.890382i	$-0.650437\pi$
$859$	−22921.1	−0.910428	−0.455214	+	0.890382i	$0.650437\pi$
$860$	3294.76	0.130640
$861$	0	0
$862$	−18938.9	−0.748332
$863$	19488.1	0.768693	0.384347	−	0.923189i	$-0.374427\pi$
$863$	19488.1	0.768693	0.384347	+	0.923189i	$0.374427\pi$
$864$	0	0
$865$	2816.24	0.110699
$866$	24266.1	0.952188
$867$	0	0
$868$	4459.48	0.174383
$869$	0	0
$870$	0	0
$871$	13528.4	0.526282
$872$	4997.30	0.194071
$873$	0	0
$874$	−29725.0	−1.15042
$875$	−3937.06	−0.152111
$876$	0	0
$877$	8455.67	0.325573	0.162787	−	0.986661i	$-0.447952\pi$
$877$	8455.67	0.325573	0.162787	+	0.986661i	$0.447952\pi$
$878$	−343.834	−0.0132162
$879$	0	0
$880$	0	0
$881$	11291.2	0.431794	0.215897	−	0.976416i	$-0.430732\pi$
$881$	11291.2	0.431794	0.215897	+	0.976416i	$0.430732\pi$
$882$	0	0
$883$	31818.1	1.21264	0.606322	−	0.795219i	$-0.292644\pi$
$883$	31818.1	1.21264	0.606322	+	0.795219i	$0.292644\pi$
$884$	3706.53	0.141023
$885$	0	0
$886$	−16705.4	−0.633441
$887$	17481.1	0.661732	0.330866	−	0.943678i	$-0.392659\pi$
$887$	17481.1	0.661732	0.330866	+	0.943678i	$0.392659\pi$
$888$	0	0
$889$	−10871.0	−0.410126
$890$	−5988.80	−0.225556
$891$	0	0
$892$	7082.78	0.265862
$893$	13368.9	0.500978
$894$	0	0
$895$	6805.16	0.254158
$896$	7891.87	0.294251
$897$	0	0
$898$	39125.1	1.45392
$899$	67763.1	2.51393
$900$	0	0
$901$	29973.6	1.10829
$902$	0	0
$903$	0	0
$904$	−25429.1	−0.935574
$905$	3704.08	0.136053
$906$	0	0
$907$	10607.4	0.388326	0.194163	−	0.980969i	$-0.437801\pi$
$907$	10607.4	0.388326	0.194163	+	0.980969i	$0.437801\pi$
$908$	−7611.79	−0.278201
$909$	0	0
$910$	838.604	0.0305489
$911$	41249.2	1.50016	0.750080	−	0.661347i	$-0.230015\pi$
$911$	41249.2	1.50016	0.750080	+	0.661347i	$0.230015\pi$
$912$	0	0
$913$	0	0
$914$	11906.5	0.430887
$915$	0	0
$916$	−17910.4	−0.646045
$917$	−5443.97	−0.196048
$918$	0	0
$919$	13858.1	0.497429	0.248714	−	0.968577i	$-0.419992\pi$
$919$	13858.1	0.497429	0.248714	+	0.968577i	$0.419992\pi$
$920$	6098.53	0.218546
$921$	0	0
$922$	39004.9	1.39323
$923$	−11363.9	−0.405253
$924$	0	0
$925$	−11551.2	−0.410595
$926$	36719.0	1.30309
$927$	0	0
$928$	−34914.2	−1.23504
$929$	−20893.7	−0.737890	−0.368945	−	0.929451i	$-0.620281\pi$
$929$	−20893.7	−0.737890	−0.368945	+	0.929451i	$0.620281\pi$
$930$	0	0
$931$	−25222.8	−0.887911
$932$	8316.94	0.292307
$933$	0	0
$934$	−36042.9	−1.26270
$935$	0	0
$936$	0	0
$937$	−3203.52	−0.111691	−0.0558454	−	0.998439i	$-0.517785\pi$
$937$	−3203.52	−0.111691	−0.0558454	+	0.998439i	$0.517785\pi$
$938$	14409.4	0.501581
$939$	0	0
$940$	1998.74	0.0693528
$941$	19951.6	0.691182	0.345591	−	0.938385i	$-0.387678\pi$
$941$	19951.6	0.691182	0.345591	+	0.938385i	$0.387678\pi$
$942$	0	0
$943$	27020.6	0.933099
$944$	15602.8	0.537952
$945$	0	0
$946$	0	0
$947$	38216.7	1.31138	0.655689	−	0.755031i	$-0.272378\pi$
$947$	38216.7	1.31138	0.655689	+	0.755031i	$0.272378\pi$
$948$	0	0
$949$	15319.4	0.524014
$950$	−29944.3	−1.02265
$951$	0	0
$952$	−5417.65	−0.184440
$953$	−47661.4	−1.62004	−0.810022	−	0.586399i	$-0.800545\pi$
$953$	−47661.4	−1.62004	−0.810022	+	0.586399i	$0.800545\pi$
$954$	0	0
$955$	−7454.21	−0.252579
$956$	4820.35	0.163077
$957$	0	0
$958$	7897.56	0.266345
$959$	−6018.94	−0.202671
$960$	0	0
$961$	47892.3	1.60761
$962$	5186.34	0.173819
$963$	0	0
$964$	3301.43	0.110303
$965$	−13774.2	−0.459490
$966$	0	0
$967$	−18933.2	−0.629628	−0.314814	−	0.949153i	$-0.601942\pi$
$967$	−18933.2	−0.629628	−0.314814	+	0.949153i	$0.601942\pi$
$968$	0	0
$969$	0	0
$970$	7228.73	0.239279
$971$	40660.3	1.34382	0.671911	−	0.740632i	$-0.265474\pi$
$971$	40660.3	1.34382	0.671911	+	0.740632i	$0.265474\pi$
$972$	0	0
$973$	2307.23	0.0760188
$974$	22759.2	0.748720
$975$	0	0
$976$	−55832.3	−1.83110
$977$	22502.8	0.736876	0.368438	−	0.929652i	$-0.379893\pi$
$977$	22502.8	0.736876	0.368438	+	0.929652i	$0.379893\pi$
$978$	0	0
$979$	0	0
$980$	−3770.98	−0.122918
$981$	0	0
$982$	24824.7	0.806709
$983$	4435.20	0.143907	0.0719536	−	0.997408i	$-0.477077\pi$
$983$	4435.20	0.143907	0.0719536	+	0.997408i	$0.477077\pi$
$984$	0	0
$985$	3223.55	0.104275
$986$	59989.8	1.93759
$987$	0	0
$988$	3986.80	0.128378
$989$	−31361.8	−1.00834
$990$	0	0
$991$	7362.76	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$991$	7362.76	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$992$	−40025.5	−1.28106
$993$	0	0
$994$	−12104.0	−0.386233
$995$	−1661.35	−0.0529331
$996$	0	0
$997$	53480.1	1.69883	0.849413	−	0.527728i	$-0.176956\pi$
$997$	53480.1	1.69883	0.849413	+	0.527728i	$0.176956\pi$
$998$	35010.0	1.11044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.t.1.2		2
3.2	odd	2		363.4.a.j.1.1		2
11.10	odd	2		99.4.a.e.1.1		2
33.32	even	2		33.4.a.d.1.2	✓	2
44.43	even	2		1584.4.a.x.1.2		2
55.54	odd	2		2475.4.a.o.1.2		2
132.131	odd	2		528.4.a.o.1.1		2
165.32	odd	4		825.4.c.i.199.4		4
165.98	odd	4		825.4.c.i.199.1		4
165.164	even	2		825.4.a.k.1.1		2
231.230	odd	2		1617.4.a.j.1.2		2
264.131	odd	2		2112.4.a.bh.1.2		2
264.197	even	2		2112.4.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.d.1.2	✓	2	33.32	even	2
99.4.a.e.1.1		2	11.10	odd	2
363.4.a.j.1.1		2	3.2	odd	2
528.4.a.o.1.1		2	132.131	odd	2
825.4.a.k.1.1		2	165.164	even	2
825.4.c.i.199.1		4	165.98	odd	4
825.4.c.i.199.4		4	165.32	odd	4
1089.4.a.t.1.2		2	1.1	even	1	trivial
1584.4.a.x.1.2		2	44.43	even	2
1617.4.a.j.1.2		2	231.230	odd	2
2112.4.a.ba.1.2		2	264.197	even	2
2112.4.a.bh.1.2		2	264.131	odd	2
2475.4.a.o.1.2		2	55.54	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.37228 q^{2} +3.37228 q^{4} +3.48913 q^{5} +4.74456 q^{7} -15.6060 q^{8} +11.7663 q^{10} +15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684 q^{17} +78.7011 q^{19} +11.7663 q^{20} -112.000 q^{23} -112.826 q^{25} +50.6576 q^{26} +16.0000 q^{28} +243.125 q^{29} +278.717 q^{31} -143.606 q^{32} +246.745 q^{34} +16.5544 q^{35} +102.380 q^{37} +265.402 q^{38} -54.4512 q^{40} -241.255 q^{41} +280.016 q^{43} -377.696 q^{46} +169.870 q^{47} -320.489 q^{49} -380.481 q^{50} +50.6576 q^{52} +409.652 q^{53} -74.0435 q^{56} +819.886 q^{58} -196.000 q^{59} +701.359 q^{61} +939.913 q^{62} +152.568 q^{64} +52.4128 q^{65} +900.587 q^{67} +246.745 q^{68} +55.8260 q^{70} -756.500 q^{71} +1019.81 q^{73} +345.255 q^{74} +265.402 q^{76} +327.549 q^{79} -277.755 q^{80} -813.581 q^{82} -756.619 q^{83} +255.294 q^{85} +944.293 q^{86} -508.978 q^{89} +71.2716 q^{91} -377.696 q^{92} +572.848 q^{94} +274.598 q^{95} +614.358 q^{97} -1080.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8} + 58 q^{10} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} + 54 q^{19} + 58 q^{20} - 224 q^{23} + 142 q^{25} - 94 q^{26} + 32 q^{28} + 222 q^{29} - 40 q^{31}+ \cdots - 375 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 + 58 * q^10 + 76 * q^13 + 32 * q^14 - 119 * q^16 - 26 * q^17 + 54 * q^19 + 58 * q^20 - 224 * q^23 + 142 * q^25 - 94 * q^26 + 32 * q^28 + 222 * q^29 - 40 * q^31 - 247 * q^32 + 482 * q^34 + 148 * q^35 - 48 * q^37 + 324 * q^38 - 534 * q^40 - 494 * q^41 + 66 * q^43 - 112 * q^46 + 64 * q^47 - 618 * q^49 - 985 * q^50 - 94 * q^52 + 84 * q^53 - 240 * q^56 + 870 * q^58 - 392 * q^59 + 1104 * q^61 + 1696 * q^62 + 713 * q^64 - 1136 * q^65 + 928 * q^67 + 482 * q^68 - 256 * q^70 - 456 * q^71 + 592 * q^73 + 702 * q^74 + 324 * q^76 + 230 * q^79 + 490 * q^80 - 214 * q^82 + 348 * q^83 + 2188 * q^85 + 1452 * q^86 - 972 * q^89 - 340 * q^91 - 112 * q^92 + 824 * q^94 + 756 * q^95 - 1184 * q^97 - 375 * q^98

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