LMFDB - Embedded newform 1089.6.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,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, 6, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$174.657979776$
Analytic rank:	$1$
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			$-2.37228$ 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)

$f(q)$	$=$	$q+9.37228 q^{2} +55.8397 q^{4} -0.277187 q^{5} +105.081 q^{7} +223.432 q^{8} +O(q^{10})$	$q+9.37228 q^{2} +55.8397 q^{4} -0.277187 q^{5} +105.081 q^{7} +223.432 q^{8} -2.59787 q^{10} -147.549 q^{13} +984.853 q^{14} +307.198 q^{16} -1432.49 q^{17} -2033.18 q^{19} -15.4780 q^{20} -828.853 q^{23} -3124.92 q^{25} -1382.87 q^{26} +5867.71 q^{28} +4633.44 q^{29} -9835.65 q^{31} -4270.67 q^{32} -13425.7 q^{34} -29.1272 q^{35} +7134.34 q^{37} -19055.6 q^{38} -61.9324 q^{40} +18265.0 q^{41} -13822.5 q^{43} -7768.24 q^{46} -22991.2 q^{47} -5764.89 q^{49} -29287.7 q^{50} -8239.08 q^{52} -14311.9 q^{53} +23478.6 q^{56} +43425.9 q^{58} +7081.12 q^{59} +18470.2 q^{61} -92182.5 q^{62} -49856.3 q^{64} +40.8986 q^{65} +16229.5 q^{67} -79989.7 q^{68} -272.988 q^{70} -28198.9 q^{71} +39382.9 q^{73} +66865.1 q^{74} -113532. q^{76} +41243.7 q^{79} -85.1513 q^{80} +171185. q^{82} -23355.3 q^{83} +397.067 q^{85} -129549. q^{86} +103803. q^{89} -15504.6 q^{91} -46282.9 q^{92} -215480. q^{94} +563.572 q^{95} -149289. q^{97} -54030.2 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10}) $ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8	$ 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100}) $ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 - 212 * q^10 + 130 * q^13 + 74 * q^14 + 241 * q^16 - 728 * q^17 + 828 * q^19 + 1072 * q^20 + 238 * q^23 - 2918 * q^25 - 376 * q^26 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 + 1391 * q^32 - 10870 * q^34 + 14464 * q^35 - 1908 * q^37 - 8676 * q^38 + 10584 * q^40 + 36484 * q^41 - 9768 * q^43 - 3898 * q^46 - 43742 * q^47 + 40470 * q^49 - 28537 * q^50 - 13468 * q^52 + 12174 * q^53 + 69786 * q^56 + 29142 * q^58 + 2788 * q^59 + 25302 * q^61 - 94520 * q^62 - 27199 * q^64 - 15980 * q^65 - 40520 * q^67 - 93262 * q^68 + 52304 * q^70 - 31386 * q^71 + 46780 * q^73 + 34062 * q^74 - 167436 * q^76 + 16850 * q^79 + 3736 * q^80 + 237278 * q^82 + 79440 * q^83 - 40268 * q^85 - 114840 * q^86 + 54204 * q^89 - 85192 * q^91 - 66382 * q^92 - 290758 * q^94 - 164592 * q^95 - 241568 * q^97 + 113697 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	9.37228	1.65680	0.828400	−	0.560136i	$-0.189251\pi$
$2$	9.37228	1.65680	0.828400	+	0.560136i	$0.189251\pi$
$3$	0	0
$3$	0	0
$4$	55.8397	1.74499
$5$	−0.277187	−0.00495847	−0.00247923	−	0.999997i	$-0.500789\pi$
$5$	−0.277187	−0.00495847	−0.00247923	+	0.999997i	$0.500789\pi$
$6$	0	0
$7$	105.081	0.810552	0.405276	−	0.914194i	$-0.367175\pi$
$7$	105.081	0.810552	0.405276	+	0.914194i	$0.367175\pi$
$8$	223.432	1.23430
$9$	0	0
$10$	−2.59787	−0.00821519
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−147.549	−0.242146	−0.121073	−	0.992644i	$-0.538634\pi$
$13$	−147.549	−0.242146	−0.121073	+	0.992644i	$0.538634\pi$
$14$	984.853	1.34292
$15$	0	0
$16$	307.198	0.299998
$17$	−1432.49	−1.20218	−0.601089	−	0.799182i	$-0.705266\pi$
$17$	−1432.49	−1.20218	−0.601089	+	0.799182i	$0.705266\pi$
$18$	0	0
$19$	−2033.18	−1.29209	−0.646045	−	0.763300i	$-0.723578\pi$
$19$	−2033.18	−1.29209	−0.646045	+	0.763300i	$0.723578\pi$
$20$	−15.4780	−0.00865247
$21$	0	0
$22$	0	0
$23$	−828.853	−0.326707	−0.163353	−	0.986568i	$-0.552231\pi$
$23$	−828.853	−0.326707	−0.163353	+	0.986568i	$0.552231\pi$
$24$	0	0
$25$	−3124.92	−0.999975
$26$	−1382.87	−0.401188
$27$	0	0
$28$	5867.71	1.41440
$29$	4633.44	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$29$	4633.44	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$30$	0	0
$31$	−9835.65	−1.83823	−0.919113	−	0.393994i	$-0.871093\pi$
$31$	−9835.65	−1.83823	−0.919113	+	0.393994i	$0.871093\pi$
$32$	−4270.67	−0.737261
$33$	0	0
$34$	−13425.7	−1.99177
$35$	−29.1272	−0.00401910
$36$	0	0
$37$	7134.34	0.856741	0.428371	−	0.903603i	$-0.359088\pi$
$37$	7134.34	0.856741	0.428371	+	0.903603i	$0.359088\pi$
$38$	−19055.6	−2.14074
$39$	0	0
$40$	−61.9324	−0.00612023
$41$	18265.0	1.69691	0.848456	−	0.529265i	$-0.177532\pi$
$41$	18265.0	1.69691	0.848456	+	0.529265i	$0.177532\pi$
$42$	0	0
$43$	−13822.5	−1.14003	−0.570016	−	0.821634i	$-0.693063\pi$
$43$	−13822.5	−1.14003	−0.570016	+	0.821634i	$0.693063\pi$
$44$	0	0
$45$	0	0
$46$	−7768.24	−0.541288
$47$	−22991.2	−1.51816	−0.759079	−	0.650999i	$-0.774350\pi$
$47$	−22991.2	−1.51816	−0.759079	+	0.650999i	$0.774350\pi$
$48$	0	0
$49$	−5764.89	−0.343005
$50$	−29287.7	−1.65676
$51$	0	0
$52$	−8239.08	−0.422542
$53$	−14311.9	−0.699856	−0.349928	−	0.936777i	$-0.613794\pi$
$53$	−14311.9	−0.699856	−0.349928	+	0.936777i	$0.613794\pi$
$54$	0	0
$55$	0	0
$56$	23478.6	1.00046
$57$	0	0
$58$	43425.9	1.69504
$59$	7081.12	0.264833	0.132416	−	0.991194i	$-0.457726\pi$
$59$	7081.12	0.264833	0.132416	+	0.991194i	$0.457726\pi$
$60$	0	0
$61$	18470.2	0.635547	0.317774	−	0.948167i	$-0.397065\pi$
$61$	18470.2	0.635547	0.317774	+	0.948167i	$0.397065\pi$
$62$	−92182.5	−3.04557
$63$	0	0
$64$	−49856.3	−1.52149
$65$	40.8986	0.00120067
$66$	0	0
$67$	16229.5	0.441690	0.220845	−	0.975309i	$-0.429119\pi$
$67$	16229.5	0.441690	0.220845	+	0.975309i	$0.429119\pi$
$68$	−79989.7	−2.09779
$69$	0	0
$70$	−272.988	−0.00665884
$71$	−28198.9	−0.663875	−0.331938	−	0.943301i	$-0.607702\pi$
$71$	−28198.9	−0.663875	−0.331938	+	0.943301i	$0.607702\pi$
$72$	0	0
$73$	39382.9	0.864968	0.432484	−	0.901642i	$-0.357637\pi$
$73$	39382.9	0.864968	0.432484	+	0.901642i	$0.357637\pi$
$74$	66865.1	1.41945
$75$	0	0
$76$	−113532.	−2.25468
$77$	0	0
$78$	0	0
$79$	41243.7	0.743515	0.371758	−	0.928330i	$-0.378755\pi$
$79$	41243.7	0.743515	0.371758	+	0.928330i	$0.378755\pi$
$80$	−85.1513	−0.00148753
$81$	0	0
$82$	171185.	2.81145
$83$	−23355.3	−0.372126	−0.186063	−	0.982538i	$-0.559573\pi$
$83$	−23355.3	−0.372126	−0.186063	+	0.982538i	$0.559573\pi$
$84$	0	0
$85$	397.067	0.00596096
$86$	−129549.	−1.88880
$87$	0	0
$88$	0	0
$89$	103803.	1.38911	0.694555	−	0.719440i	$-0.255601\pi$
$89$	103803.	1.38911	0.694555	+	0.719440i	$0.255601\pi$
$90$	0	0
$91$	−15504.6	−0.196272
$92$	−46282.9	−0.570099
$93$	0	0
$94$	−215480.	−2.51528
$95$	563.572	0.00640678
$96$	0	0
$97$	−149289.	−1.61101	−0.805503	−	0.592592i	$-0.798105\pi$
$97$	−149289.	−1.61101	−0.805503	+	0.592592i	$0.798105\pi$
$98$	−54030.2	−0.568292
$99$	0	0
$100$	−174495.	−1.74495
$101$	62410.1	0.608768	0.304384	−	0.952549i	$-0.401549\pi$
$101$	62410.1	0.608768	0.304384	+	0.952549i	$0.401549\pi$
$102$	0	0
$103$	−90047.2	−0.836329	−0.418165	−	0.908371i	$-0.637326\pi$
$103$	−90047.2	−0.836329	−0.418165	+	0.908371i	$0.637326\pi$
$104$	−32967.1	−0.298881
$105$	0	0
$106$	−134136.	−1.15952
$107$	120458.	1.01713	0.508566	−	0.861023i	$-0.330176\pi$
$107$	120458.	1.01713	0.508566	+	0.861023i	$0.330176\pi$
$108$	0	0
$109$	91854.0	0.740511	0.370256	−	0.928930i	$-0.379270\pi$
$109$	91854.0	0.740511	0.370256	+	0.928930i	$0.379270\pi$
$110$	0	0
$111$	0	0
$112$	32280.8	0.243164
$113$	29619.6	0.218214	0.109107	−	0.994030i	$-0.465201\pi$
$113$	29619.6	0.218214	0.109107	+	0.994030i	$0.465201\pi$
$114$	0	0
$115$	229.747	0.00161996
$116$	258730.	1.78526
$117$	0	0
$118$	66366.2	0.438775
$119$	−150528.	−0.974428
$120$	0	0
$121$	0	0
$122$	173108.	1.05298
$123$	0	0
$124$	−549219.	−3.20768
$125$	1732.40	0.00991681
$126$	0	0
$127$	−271128.	−1.49165	−0.745823	−	0.666145i	$-0.767943\pi$
$127$	−271128.	−1.49165	−0.745823	+	0.666145i	$0.767943\pi$
$128$	−330606.	−1.78355
$129$	0	0
$130$	383.313	0.00198928
$131$	−223990.	−1.14038	−0.570191	−	0.821512i	$-0.693131\pi$
$131$	−223990.	−1.14038	−0.570191	+	0.821512i	$0.693131\pi$
$132$	0	0
$133$	−213650.	−1.04731
$134$	152107.	0.731792
$135$	0	0
$136$	−320064.	−1.48385
$137$	348229.	1.58513	0.792563	−	0.609789i	$-0.208746\pi$
$137$	348229.	1.58513	0.792563	+	0.609789i	$0.208746\pi$
$138$	0	0
$139$	−333802.	−1.46538	−0.732692	−	0.680560i	$-0.761736\pi$
$139$	−333802.	−1.46538	−0.732692	+	0.680560i	$0.761736\pi$
$140$	−1626.45	−0.00701328
$141$	0	0
$142$	−264288.	−1.09991
$143$	0	0
$144$	0	0
$145$	−1284.33	−0.00507290
$146$	369107.	1.43308
$147$	0	0
$148$	398379.	1.49500
$149$	310249.	1.14484	0.572419	−	0.819961i	$-0.306005\pi$
$149$	310249.	1.14484	0.572419	+	0.819961i	$0.306005\pi$
$150$	0	0
$151$	54751.5	0.195413	0.0977066	−	0.995215i	$-0.468849\pi$
$151$	54751.5	0.195413	0.0977066	+	0.995215i	$0.468849\pi$
$152$	−454278.	−1.59482
$153$	0	0
$154$	0	0
$155$	2726.31	0.00911478
$156$	0	0
$157$	221556.	0.717354	0.358677	−	0.933462i	$-0.383228\pi$
$157$	221556.	0.717354	0.358677	+	0.933462i	$0.383228\pi$
$158$	386547.	1.23186
$159$	0	0
$160$	1183.77	0.00365569
$161$	−87097.1	−0.264813
$162$	0	0
$163$	−225239.	−0.664010	−0.332005	−	0.943278i	$-0.607725\pi$
$163$	−225239.	−0.664010	−0.332005	+	0.943278i	$0.607725\pi$
$164$	1.01991e6	2.96109
$165$	0	0
$166$	−218892.	−0.616539
$167$	186796.	0.518294	0.259147	−	0.965838i	$-0.416559\pi$
$167$	186796.	0.518294	0.259147	+	0.965838i	$0.416559\pi$
$168$	0	0
$169$	−349522.	−0.941365
$170$	3721.42	0.00987613
$171$	0	0
$172$	−771846.	−1.98934
$173$	−713662.	−1.81292	−0.906458	−	0.422296i	$-0.861224\pi$
$173$	−713662.	−1.81292	−0.906458	+	0.422296i	$0.861224\pi$
$174$	0	0
$175$	−328371.	−0.810532
$176$	0	0
$177$	0	0
$178$	972875.	2.30148
$179$	−789450.	−1.84159	−0.920793	−	0.390051i	$-0.872457\pi$
$179$	−789450.	−1.84159	−0.920793	+	0.390051i	$0.872457\pi$
$180$	0	0
$181$	40330.7	0.0915039	0.0457519	−	0.998953i	$-0.485432\pi$
$181$	40330.7	0.0915039	0.0457519	+	0.998953i	$0.485432\pi$
$182$	−145314.	−0.325184
$183$	0	0
$184$	−185192.	−0.403254
$185$	−1977.55	−0.00424812
$186$	0	0
$187$	0	0
$188$	−1.28382e6	−2.64917
$189$	0	0
$190$	5281.95	0.0106148
$191$	−133736.	−0.265256	−0.132628	−	0.991166i	$-0.542342\pi$
$191$	−133736.	−0.265256	−0.132628	+	0.991166i	$0.542342\pi$
$192$	0	0
$193$	462266.	0.893303	0.446652	−	0.894708i	$-0.352616\pi$
$193$	462266.	0.893303	0.446652	+	0.894708i	$0.352616\pi$
$194$	−1.39917e6	−2.66912
$195$	0	0
$196$	−321910.	−0.598541
$197$	96432.5	0.177035	0.0885173	−	0.996075i	$-0.471787\pi$
$197$	96432.5	0.177035	0.0885173	+	0.996075i	$0.471787\pi$
$198$	0	0
$199$	−1.08264e6	−1.93800	−0.968999	−	0.247066i	$-0.920533\pi$
$199$	−1.08264e6	−1.93800	−0.968999	+	0.247066i	$0.920533\pi$
$200$	−698208.	−1.23427
$201$	0	0
$202$	584925.	1.00861
$203$	486889.	0.829258
$204$	0	0
$205$	−5062.81	−0.00841409
$206$	−843948.	−1.38563
$207$	0	0
$208$	−45326.7	−0.0726434
$209$	0	0
$210$	0	0
$211$	−875588.	−1.35392	−0.676961	−	0.736019i	$-0.736703\pi$
$211$	−875588.	−1.35392	−0.676961	+	0.736019i	$0.736703\pi$
$212$	−799174.	−1.22124
$213$	0	0
$214$	1.12897e6	1.68518
$215$	3831.43	0.00565281
$216$	0	0
$217$	−1.03354e6	−1.48998
$218$	860881.	1.22688
$219$	0	0
$220$	0	0
$221$	211362.	0.291103
$222$	0	0
$223$	988042.	1.33049	0.665247	−	0.746623i	$-0.268326\pi$
$223$	988042.	1.33049	0.665247	+	0.746623i	$0.268326\pi$
$224$	−448769.	−0.597589
$225$	0	0
$226$	277603.	0.361537
$227$	67066.0	0.0863849	0.0431925	−	0.999067i	$-0.486247\pi$
$227$	67066.0	0.0863849	0.0431925	+	0.999067i	$0.486247\pi$
$228$	0	0
$229$	−662131.	−0.834363	−0.417182	−	0.908823i	$-0.636982\pi$
$229$	−662131.	−0.834363	−0.417182	+	0.908823i	$0.636982\pi$
$230$	2153.25	0.00268396
$231$	0	0
$232$	1.03526e6	1.26278
$233$	1.37961e6	1.66482	0.832411	−	0.554159i	$-0.186960\pi$
$233$	1.37961e6	1.66482	0.832411	+	0.554159i	$0.186960\pi$
$234$	0	0
$235$	6372.85	0.00752773
$236$	395407.	0.462130
$237$	0	0
$238$	−1.41079e6	−1.61443
$239$	423791.	0.479907	0.239954	−	0.970784i	$-0.422868\pi$
$239$	423791.	0.479907	0.239954	+	0.970784i	$0.422868\pi$
$240$	0	0
$241$	1.34802e6	1.49504	0.747521	−	0.664238i	$-0.231244\pi$
$241$	1.34802e6	1.49504	0.747521	+	0.664238i	$0.231244\pi$
$242$	0	0
$243$	0	0
$244$	1.03137e6	1.10902
$245$	1597.95	0.00170078
$246$	0	0
$247$	299994.	0.312874
$248$	−2.19760e6	−2.26892
$249$	0	0
$250$	16236.5	0.0164302
$251$	1.45914e6	1.46188	0.730940	−	0.682442i	$-0.239082\pi$
$251$	1.45914e6	1.46188	0.730940	+	0.682442i	$0.239082\pi$
$252$	0	0
$253$	0	0
$254$	−2.54109e6	−2.47136
$255$	0	0
$256$	−1.50313e6	−1.43349
$257$	−548503.	−0.518020	−0.259010	−	0.965875i	$-0.583396\pi$
$257$	−548503.	−0.518020	−0.259010	+	0.965875i	$0.583396\pi$
$258$	0	0
$259$	749687.	0.694434
$260$	2283.76	0.00209516
$261$	0	0
$262$	−2.09930e6	−1.88939
$263$	905299.	0.807054	0.403527	−	0.914968i	$-0.367784\pi$
$263$	905299.	0.807054	0.403527	+	0.914968i	$0.367784\pi$
$264$	0	0
$265$	3967.08	0.00347021
$266$	−2.00239e6	−1.73518
$267$	0	0
$268$	906248.	0.770744
$269$	164041.	0.138220	0.0691100	−	0.997609i	$-0.477984\pi$
$269$	164041.	0.138220	0.0691100	+	0.997609i	$0.477984\pi$
$270$	0	0
$271$	−989636.	−0.818563	−0.409282	−	0.912408i	$-0.634221\pi$
$271$	−989636.	−0.818563	−0.409282	+	0.912408i	$0.634221\pi$
$272$	−440058.	−0.360651
$273$	0	0
$274$	3.26370e6	2.62624
$275$	0	0
$276$	0	0
$277$	−727149.	−0.569409	−0.284704	−	0.958615i	$-0.591895\pi$
$277$	−727149.	−0.569409	−0.284704	+	0.958615i	$0.591895\pi$
$278$	−3.12848e6	−2.42785
$279$	0	0
$280$	−6507.94	−0.00496077
$281$	−379100.	−0.286410	−0.143205	−	0.989693i	$-0.545741\pi$
$281$	−379100.	−0.286410	−0.143205	+	0.989693i	$0.545741\pi$
$282$	0	0
$283$	695371.	0.516120	0.258060	−	0.966129i	$-0.416917\pi$
$283$	695371.	0.516120	0.258060	+	0.966129i	$0.416917\pi$
$284$	−1.57462e6	−1.15846
$285$	0	0
$286$	0	0
$287$	1.91931e6	1.37544
$288$	0	0
$289$	632167.	0.445233
$290$	−12037.1	−0.00840479
$291$	0	0
$292$	2.19913e6	1.50936
$293$	268373.	0.182629	0.0913146	−	0.995822i	$-0.470893\pi$
$293$	268373.	0.182629	0.0913146	+	0.995822i	$0.470893\pi$
$294$	0	0
$295$	−1962.79	−0.00131316
$296$	1.59404e6	1.05747
$297$	0	0
$298$	2.90774e6	1.89677
$299$	122296.	0.0791107
$300$	0	0
$301$	−1.45249e6	−0.924055
$302$	513147.	0.323761
$303$	0	0
$304$	−624591.	−0.387625
$305$	−5119.71	−0.00315134
$306$	0	0
$307$	−1.78841e6	−1.08298	−0.541491	−	0.840706i	$-0.682140\pi$
$307$	−1.78841e6	−1.08298	−0.541491	+	0.840706i	$0.682140\pi$
$308$	0	0
$309$	0	0
$310$	25551.8	0.0151014
$311$	−362727.	−0.212657	−0.106328	−	0.994331i	$-0.533909\pi$
$311$	−362727.	−0.212657	−0.106328	+	0.994331i	$0.533909\pi$
$312$	0	0
$313$	−1.14382e6	−0.659926	−0.329963	−	0.943994i	$-0.607036\pi$
$313$	−1.14382e6	−0.659926	−0.329963	+	0.943994i	$0.607036\pi$
$314$	2.07648e6	1.18851
$315$	0	0
$316$	2.30303e6	1.29743
$317$	−3.43750e6	−1.92130	−0.960649	−	0.277766i	$-0.910406\pi$
$317$	−3.43750e6	−1.92130	−0.960649	+	0.277766i	$0.910406\pi$
$318$	0	0
$319$	0	0
$320$	13819.5	0.00754428
$321$	0	0
$322$	−816298.	−0.438742
$323$	2.91251e6	1.55332
$324$	0	0
$325$	461079.	0.242140
$326$	−2.11100e6	−1.10013
$327$	0	0
$328$	4.08098e6	2.09450
$329$	−2.41595e6	−1.23055
$330$	0	0
$331$	1.11428e6	0.559015	0.279507	−	0.960144i	$-0.409829\pi$
$331$	1.11428e6	0.559015	0.279507	+	0.960144i	$0.409829\pi$
$332$	−1.30415e6	−0.649356
$333$	0	0
$334$	1.75070e6	0.858710
$335$	−4498.59	−0.00219010
$336$	0	0
$337$	2.25804e6	1.08307	0.541535	−	0.840678i	$-0.317843\pi$
$337$	2.25804e6	1.08307	0.541535	+	0.840678i	$0.317843\pi$
$338$	−3.27582e6	−1.55965
$339$	0	0
$340$	22172.1	0.0104018
$341$	0	0
$342$	0	0
$343$	−2.37189e6	−1.08858
$344$	−3.08840e6	−1.40714
$345$	0	0
$346$	−6.68865e6	−3.00364
$347$	−1.17055e6	−0.521873	−0.260937	−	0.965356i	$-0.584031\pi$
$347$	−1.17055e6	−0.521873	−0.260937	+	0.965356i	$0.584031\pi$
$348$	0	0
$349$	−1.21704e6	−0.534863	−0.267432	−	0.963577i	$-0.586175\pi$
$349$	−1.21704e6	−0.534863	−0.267432	+	0.963577i	$0.586175\pi$
$350$	−3.07759e6	−1.34289
$351$	0	0
$352$	0	0
$353$	−286188.	−0.122240	−0.0611201	−	0.998130i	$-0.519467\pi$
$353$	−286188.	−0.122240	−0.0611201	+	0.998130i	$0.519467\pi$
$354$	0	0
$355$	7816.37	0.00329180
$356$	5.79635e6	2.42398
$357$	0	0
$358$	−7.39895e6	−3.05114
$359$	1.19066e6	0.487588	0.243794	−	0.969827i	$-0.421608\pi$
$359$	1.19066e6	0.487588	0.243794	+	0.969827i	$0.421608\pi$
$360$	0	0
$361$	1.65774e6	0.669495
$362$	377991.	0.151604
$363$	0	0
$364$	−865774.	−0.342492
$365$	−10916.4	−0.00428892
$366$	0	0
$367$	3.11120e6	1.20577	0.602884	−	0.797829i	$-0.294018\pi$
$367$	3.11120e6	1.20577	0.602884	+	0.797829i	$0.294018\pi$
$368$	−254622.	−0.0980114
$369$	0	0
$370$	−18534.1	−0.00703830
$371$	−1.50392e6	−0.567270
$372$	0	0
$373$	−3.58536e6	−1.33432	−0.667160	−	0.744914i	$-0.732490\pi$
$373$	−3.58536e6	−1.33432	−0.667160	+	0.744914i	$0.732490\pi$
$374$	0	0
$375$	0	0
$376$	−5.13697e6	−1.87386
$377$	−683659.	−0.247734
$378$	0	0
$379$	1.87822e6	0.671660	0.335830	−	0.941923i	$-0.390983\pi$
$379$	1.87822e6	0.671660	0.335830	+	0.941923i	$0.390983\pi$
$380$	31469.6	0.0111798
$381$	0	0
$382$	−1.25341e6	−0.439476
$383$	662654.	0.230829	0.115414	−	0.993317i	$-0.463180\pi$
$383$	662654.	0.230829	0.115414	+	0.993317i	$0.463180\pi$
$384$	0	0
$385$	0	0
$386$	4.33249e6	1.48003
$387$	0	0
$388$	−8.33622e6	−2.81119
$389$	5.08370e6	1.70336	0.851678	−	0.524065i	$-0.175585\pi$
$389$	5.08370e6	1.70336	0.851678	+	0.524065i	$0.175585\pi$
$390$	0	0
$391$	1.18732e6	0.392760
$392$	−1.28806e6	−0.423371
$393$	0	0
$394$	903793.	0.293311
$395$	−11432.2	−0.00368670
$396$	0	0
$397$	−518385.	−0.165073	−0.0825366	−	0.996588i	$-0.526302\pi$
$397$	−518385.	−0.165073	−0.0825366	+	0.996588i	$0.526302\pi$
$398$	−1.01468e7	−3.21088
$399$	0	0
$400$	−959971.	−0.299991
$401$	−1.12538e6	−0.349493	−0.174747	−	0.984613i	$-0.555911\pi$
$401$	−1.12538e6	−0.349493	−0.174747	+	0.984613i	$0.555911\pi$
$402$	0	0
$403$	1.45124e6	0.445119
$404$	3.48496e6	1.06229
$405$	0	0
$406$	4.56326e6	1.37392
$407$	0	0
$408$	0	0
$409$	−3.15394e6	−0.932279	−0.466139	−	0.884711i	$-0.654355\pi$
$409$	−3.15394e6	−0.932279	−0.466139	+	0.884711i	$0.654355\pi$
$410$	−47450.1	−0.0139405
$411$	0	0
$412$	−5.02821e6	−1.45939
$413$	744094.	0.214661
$414$	0	0
$415$	6473.78	0.00184518
$416$	630133.	0.178525
$417$	0	0
$418$	0	0
$419$	−801315.	−0.222981	−0.111491	−	0.993765i	$-0.535562\pi$
$419$	−801315.	−0.222981	−0.111491	+	0.993765i	$0.535562\pi$
$420$	0	0
$421$	−1.85449e6	−0.509940	−0.254970	−	0.966949i	$-0.582066\pi$
$421$	−1.85449e6	−0.509940	−0.254970	+	0.966949i	$0.582066\pi$
$422$	−8.20626e6	−2.24318
$423$	0	0
$424$	−3.19775e6	−0.863832
$425$	4.47642e6	1.20215
$426$	0	0
$427$	1.94088e6	0.515144
$428$	6.72635e6	1.77488
$429$	0	0
$430$	35909.2	0.00936558
$431$	2.46004e6	0.637895	0.318947	−	0.947772i	$-0.396671\pi$
$431$	2.46004e6	0.637895	0.318947	+	0.947772i	$0.396671\pi$
$432$	0	0
$433$	3.12223e6	0.800287	0.400144	−	0.916452i	$-0.368960\pi$
$433$	3.12223e6	0.800287	0.400144	+	0.916452i	$0.368960\pi$
$434$	−9.68667e6	−2.46860
$435$	0	0
$436$	5.12910e6	1.29218
$437$	1.68521e6	0.422134
$438$	0	0
$439$	6.09275e6	1.50887	0.754436	−	0.656374i	$-0.227911\pi$
$439$	6.09275e6	1.50887	0.754436	+	0.656374i	$0.227911\pi$
$440$	0	0
$441$	0	0
$442$	1.98094e6	0.482299
$443$	7.93457e6	1.92094	0.960470	−	0.278382i	$-0.0897982\pi$
$443$	7.93457e6	1.92094	0.960470	+	0.278382i	$0.0897982\pi$
$444$	0	0
$445$	−28772.9	−0.00688786
$446$	9.26021e6	2.20436
$447$	0	0
$448$	−5.23897e6	−1.23325
$449$	−4.61011e6	−1.07918	−0.539592	−	0.841926i	$-0.681422\pi$
$449$	−4.61011e6	−1.07918	−0.539592	+	0.841926i	$0.681422\pi$
$450$	0	0
$451$	0	0
$452$	1.65395e6	0.380781
$453$	0	0
$454$	628562.	0.143123
$455$	4297.68	0.000973208 0
$456$	0	0
$457$	8.05458e6	1.80407	0.902033	−	0.431667i	$-0.142074\pi$
$457$	8.05458e6	1.80407	0.902033	+	0.431667i	$0.142074\pi$
$458$	−6.20568e6	−1.38237
$459$	0	0
$460$	12829.0	0.00282682
$461$	3.22431e6	0.706617	0.353309	−	0.935507i	$-0.385057\pi$
$461$	3.22431e6	0.706617	0.353309	+	0.935507i	$0.385057\pi$
$462$	0	0
$463$	1.29607e6	0.280979	0.140490	−	0.990082i	$-0.455132\pi$
$463$	1.29607e6	0.280979	0.140490	+	0.990082i	$0.455132\pi$
$464$	1.42339e6	0.306922
$465$	0	0
$466$	1.29301e7	2.75828
$467$	2.13902e6	0.453860	0.226930	−	0.973911i	$-0.427131\pi$
$467$	2.13902e6	0.453860	0.226930	+	0.973911i	$0.427131\pi$
$468$	0	0
$469$	1.70542e6	0.358012
$470$	59728.2	0.0124720
$471$	0	0
$472$	1.58215e6	0.326883
$473$	0	0
$474$	0	0
$475$	6.35354e6	1.29206
$476$	−8.40543e6	−1.70037
$477$	0	0
$478$	3.97189e6	0.795111
$479$	−4.77343e6	−0.950586	−0.475293	−	0.879828i	$-0.657658\pi$
$479$	−4.77343e6	−0.950586	−0.475293	+	0.879828i	$0.657658\pi$
$480$	0	0
$481$	−1.05266e6	−0.207457
$482$	1.26340e7	2.47699
$483$	0	0
$484$	0	0
$485$	41380.8	0.00798812
$486$	0	0
$487$	7.98376e6	1.52541	0.762703	−	0.646749i	$-0.223872\pi$
$487$	7.98376e6	1.52541	0.762703	+	0.646749i	$0.223872\pi$
$488$	4.12684e6	0.784456
$489$	0	0
$490$	14976.4	0.00281786
$491$	−3.70031e6	−0.692682	−0.346341	−	0.938109i	$-0.612576\pi$
$491$	−3.70031e6	−0.692682	−0.346341	+	0.938109i	$0.612576\pi$
$492$	0	0
$493$	−6.63736e6	−1.22992
$494$	2.81163e6	0.518371
$495$	0	0
$496$	−3.02149e6	−0.551465
$497$	−2.96318e6	−0.538105
$498$	0	0
$499$	−7.05137e6	−1.26772	−0.633858	−	0.773449i	$-0.718530\pi$
$499$	−7.05137e6	−1.26772	−0.633858	+	0.773449i	$0.718530\pi$
$500$	96736.4	0.0173047
$501$	0	0
$502$	1.36754e7	2.42204
$503$	−7.77325e6	−1.36988	−0.684940	−	0.728599i	$-0.740172\pi$
$503$	−7.77325e6	−1.36988	−0.684940	+	0.728599i	$0.740172\pi$
$504$	0	0
$505$	−17299.3	−0.00301856
$506$	0	0
$507$	0	0
$508$	−1.51397e7	−2.60290
$509$	−4.11598e6	−0.704173	−0.352086	−	0.935968i	$-0.614528\pi$
$509$	−4.11598e6	−0.704173	−0.352086	+	0.935968i	$0.614528\pi$
$510$	0	0
$511$	4.13841e6	0.701102
$512$	−3.50836e6	−0.591465
$513$	0	0
$514$	−5.14073e6	−0.858256
$515$	24959.9	0.00414691
$516$	0	0
$517$	0	0
$518$	7.02628e6	1.15054
$519$	0	0
$520$	9138.05	0.00148199
$521$	2.56352e6	0.413754	0.206877	−	0.978367i	$-0.433670\pi$
$521$	2.56352e6	0.413754	0.206877	+	0.978367i	$0.433670\pi$
$522$	0	0
$523$	3.69240e6	0.590274	0.295137	−	0.955455i	$-0.404635\pi$
$523$	3.69240e6	0.590274	0.295137	+	0.955455i	$0.404635\pi$
$524$	−1.25075e7	−1.98996
$525$	0	0
$526$	8.48471e6	1.33713
$527$	1.40895e7	2.20988
$528$	0	0
$529$	−5.74935e6	−0.893263
$530$	37180.6	0.00574946
$531$	0	0
$532$	−1.19301e7	−1.82754
$533$	−2.69498e6	−0.410901
$534$	0	0
$535$	−33389.4	−0.00504341
$536$	3.62618e6	0.545177
$537$	0	0
$538$	1.53743e6	0.229003
$539$	0	0
$540$	0	0
$541$	9.33636e6	1.37146	0.685732	−	0.727854i	$-0.259482\pi$
$541$	9.33636e6	1.37146	0.685732	+	0.727854i	$0.259482\pi$
$542$	−9.27515e6	−1.35620
$543$	0	0
$544$	6.11769e6	0.886320
$545$	−25460.7	−0.00367180
$546$	0	0
$547$	−1.30814e6	−0.186933	−0.0934664	−	0.995622i	$-0.529795\pi$
$547$	−1.30814e6	−0.186933	−0.0934664	+	0.995622i	$0.529795\pi$
$548$	1.94450e7	2.76603
$549$	0	0
$550$	0	0
$551$	−9.42064e6	−1.32191
$552$	0	0
$553$	4.33395e6	0.602658
$554$	−6.81505e6	−0.943397
$555$	0	0
$556$	−1.86394e7	−2.55708
$557$	6.95945e6	0.950467	0.475234	−	0.879860i	$-0.342364\pi$
$557$	6.95945e6	0.950467	0.475234	+	0.879860i	$0.342364\pi$
$558$	0	0
$559$	2.03950e6	0.276054
$560$	−8947.82	−0.00120572
$561$	0	0
$562$	−3.55303e6	−0.474524
$563$	6.90198e6	0.917704	0.458852	−	0.888513i	$-0.348261\pi$
$563$	6.90198e6	0.917704	0.458852	+	0.888513i	$0.348261\pi$
$564$	0	0
$565$	−8210.15	−0.00108201
$566$	6.51721e6	0.855107
$567$	0	0
$568$	−6.30054e6	−0.819421
$569$	3.31261e6	0.428933	0.214466	−	0.976731i	$-0.431199\pi$
$569$	3.31261e6	0.428933	0.214466	+	0.976731i	$0.431199\pi$
$570$	0	0
$571$	−1.43930e7	−1.84741	−0.923703	−	0.383109i	$-0.874853\pi$
$571$	−1.43930e7	−1.84741	−0.923703	+	0.383109i	$0.874853\pi$
$572$	0	0
$573$	0	0
$574$	1.79883e7	2.27882
$575$	2.59010e6	0.326699
$576$	0	0
$577$	3.31761e6	0.414845	0.207423	−	0.978251i	$-0.433492\pi$
$577$	3.31761e6	0.414845	0.207423	+	0.978251i	$0.433492\pi$
$578$	5.92484e6	0.737662
$579$	0	0
$580$	−71716.5	−0.00885216
$581$	−2.45421e6	−0.301628
$582$	0	0
$583$	0	0
$584$	8.79939e6	1.06763
$585$	0	0
$586$	2.51527e6	0.302580
$587$	−2.10735e6	−0.252431	−0.126215	−	0.992003i	$-0.540283\pi$
$587$	−2.10735e6	−0.252431	−0.126215	+	0.992003i	$0.540283\pi$
$588$	0	0
$589$	1.99977e7	2.37515
$590$	−18395.8	−0.00217565
$591$	0	0
$592$	2.19166e6	0.257021
$593$	525039.	0.0613133	0.0306566	−	0.999530i	$-0.490240\pi$
$593$	525039.	0.0613133	0.0306566	+	0.999530i	$0.490240\pi$
$594$	0	0
$595$	41724.4	0.00483167
$596$	1.73242e7	1.99773
$597$	0	0
$598$	1.14619e6	0.131071
$599$	7.02964e6	0.800509	0.400254	−	0.916404i	$-0.368922\pi$
$599$	7.02964e6	0.800509	0.400254	+	0.916404i	$0.368922\pi$
$600$	0	0
$601$	1.11158e6	0.125532	0.0627660	−	0.998028i	$-0.480008\pi$
$601$	1.11158e6	0.125532	0.0627660	+	0.998028i	$0.480008\pi$
$602$	−1.36132e7	−1.53097
$603$	0	0
$604$	3.05731e6	0.340994
$605$	0	0
$606$	0	0
$607$	−1.53428e7	−1.69018	−0.845088	−	0.534627i	$-0.820452\pi$
$607$	−1.53428e7	−1.69018	−0.845088	+	0.534627i	$0.820452\pi$
$608$	8.68307e6	0.952608
$609$	0	0
$610$	−47983.3	−0.00522115
$611$	3.39232e6	0.367616
$612$	0	0
$613$	−4.78522e6	−0.514341	−0.257170	−	0.966366i	$-0.582790\pi$
$613$	−4.78522e6	−0.514341	−0.257170	+	0.966366i	$0.582790\pi$
$614$	−1.67615e7	−1.79429
$615$	0	0
$616$	0	0
$617$	−1.69116e7	−1.78843	−0.894215	−	0.447639i	$-0.852265\pi$
$617$	−1.69116e7	−1.78843	−0.894215	+	0.447639i	$0.852265\pi$
$618$	0	0
$619$	−8.65985e6	−0.908414	−0.454207	−	0.890896i	$-0.650077\pi$
$619$	−8.65985e6	−0.908414	−0.454207	+	0.890896i	$0.650077\pi$
$620$	152236.	0.0159052
$621$	0	0
$622$	−3.39958e6	−0.352330
$623$	1.09078e7	1.12595
$624$	0	0
$625$	9.76490e6	0.999926
$626$	−1.07202e7	−1.09337
$627$	0	0
$628$	1.23716e7	1.25178
$629$	−1.02199e7	−1.02996
$630$	0	0
$631$	9.04083e6	0.903930	0.451965	−	0.892036i	$-0.350723\pi$
$631$	9.04083e6	0.903930	0.451965	+	0.892036i	$0.350723\pi$
$632$	9.21516e6	0.917720
$633$	0	0
$634$	−3.22172e7	−3.18321
$635$	75153.2	0.00739627
$636$	0	0
$637$	850603.	0.0830574
$638$	0	0
$639$	0	0
$640$	91639.5	0.00884368
$641$	1.21113e7	1.16424	0.582122	−	0.813101i	$-0.302222\pi$
$641$	1.21113e7	1.16424	0.582122	+	0.813101i	$0.302222\pi$
$642$	0	0
$643$	2.24955e6	0.214570	0.107285	−	0.994228i	$-0.465784\pi$
$643$	2.24955e6	0.214570	0.107285	+	0.994228i	$0.465784\pi$
$644$	−4.86347e6	−0.462095
$645$	0	0
$646$	2.72969e7	2.57355
$647$	−8.21206e6	−0.771244	−0.385622	−	0.922657i	$-0.626013\pi$
$647$	−8.21206e6	−0.771244	−0.385622	+	0.922657i	$0.626013\pi$
$648$	0	0
$649$	0	0
$650$	4.32136e6	0.401178
$651$	0	0
$652$	−1.25773e7	−1.15869
$653$	1.15110e7	1.05641	0.528203	−	0.849118i	$-0.322866\pi$
$653$	1.15110e7	1.05641	0.528203	+	0.849118i	$0.322866\pi$
$654$	0	0
$655$	62087.1	0.00565455
$656$	5.61097e6	0.509071
$657$	0	0
$658$	−2.26429e7	−2.03877
$659$	−1.79339e7	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$659$	−1.79339e7	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$660$	0	0
$661$	−8.63046e6	−0.768299	−0.384150	−	0.923271i	$-0.625505\pi$
$661$	−8.63046e6	−0.768299	−0.384150	+	0.923271i	$0.625505\pi$
$662$	1.04433e7	0.926176
$663$	0	0
$664$	−5.21832e6	−0.459315
$665$	59220.9	0.00519303
$666$	0	0
$667$	−3.84044e6	−0.334246
$668$	1.04306e7	0.904418
$669$	0	0
$670$	−42162.1	−0.00362857
$671$	0	0
$672$	0	0
$673$	1.92878e7	1.64152	0.820758	−	0.571276i	$-0.193551\pi$
$673$	1.92878e7	1.64152	0.820758	+	0.571276i	$0.193551\pi$
$674$	2.11630e7	1.79443
$675$	0	0
$676$	−1.95172e7	−1.64267
$677$	6.67400e6	0.559647	0.279824	−	0.960051i	$-0.409724\pi$
$677$	6.67400e6	0.559647	0.279824	+	0.960051i	$0.409724\pi$
$678$	0	0
$679$	−1.56875e7	−1.30580
$680$	88717.4	0.00735761
$681$	0	0
$682$	0	0
$683$	7.83722e6	0.642851	0.321426	−	0.946935i	$-0.395838\pi$
$683$	7.83722e6	0.642851	0.321426	+	0.946935i	$0.395838\pi$
$684$	0	0
$685$	−96524.6	−0.00785980
$686$	−2.22300e7	−1.80355
$687$	0	0
$688$	−4.24626e6	−0.342007
$689$	2.11171e6	0.169467
$690$	0	0
$691$	−2.05838e7	−1.63995	−0.819974	−	0.572400i	$-0.806012\pi$
$691$	−2.05838e7	−1.63995	−0.819974	+	0.572400i	$0.806012\pi$
$692$	−3.98507e7	−3.16352
$693$	0	0
$694$	−1.09707e7	−0.864640
$695$	92525.4	0.00726606
$696$	0	0
$697$	−2.61644e7	−2.03999
$698$	−1.14065e7	−0.886162
$699$	0	0
$700$	−1.83361e7	−1.41437
$701$	−8.88356e6	−0.682797	−0.341399	−	0.939919i	$-0.610901\pi$
$701$	−8.88356e6	−0.682797	−0.341399	+	0.939919i	$0.610901\pi$
$702$	0	0
$703$	−1.45054e7	−1.10699
$704$	0	0
$705$	0	0
$706$	−2.68223e6	−0.202528
$707$	6.55815e6	0.493438
$708$	0	0
$709$	6.33167e6	0.473045	0.236523	−	0.971626i	$-0.423992\pi$
$709$	6.33167e6	0.473045	0.236523	+	0.971626i	$0.423992\pi$
$710$	73257.2	0.00545386
$711$	0	0
$712$	2.31930e7	1.71458
$713$	8.15231e6	0.600560
$714$	0	0
$715$	0	0
$716$	−4.40826e7	−3.21355
$717$	0	0
$718$	1.11592e7	0.807836
$719$	−3.88936e6	−0.280579	−0.140290	−	0.990111i	$-0.544803\pi$
$719$	−3.88936e6	−0.280579	−0.140290	+	0.990111i	$0.544803\pi$
$720$	0	0
$721$	−9.46229e6	−0.677888
$722$	1.55368e7	1.10922
$723$	0	0
$724$	2.25205e6	0.159673
$725$	−1.44792e7	−1.02305
$726$	0	0
$727$	2.34319e7	1.64426	0.822131	−	0.569299i	$-0.192785\pi$
$727$	2.34319e7	1.64426	0.822131	+	0.569299i	$0.192785\pi$
$728$	−3.46423e6	−0.242258
$729$	0	0
$730$	−102312.	−0.00710588
$731$	1.98006e7	1.37052
$732$	0	0
$733$	−1.77978e7	−1.22351	−0.611753	−	0.791049i	$-0.709535\pi$
$733$	−1.77978e7	−1.22351	−0.611753	+	0.791049i	$0.709535\pi$
$734$	2.91591e7	1.99772
$735$	0	0
$736$	3.53976e6	0.240868
$737$	0	0
$738$	0	0
$739$	−3.15950e6	−0.212817	−0.106409	−	0.994322i	$-0.533935\pi$
$739$	−3.15950e6	−0.212817	−0.106409	+	0.994322i	$0.533935\pi$
$740$	−110425.	−0.00741293
$741$	0	0
$742$	−1.40952e7	−0.939853
$743$	−4.37086e6	−0.290465	−0.145233	−	0.989398i	$-0.546393\pi$
$743$	−4.37086e6	−0.290465	−0.145233	+	0.989398i	$0.546393\pi$
$744$	0	0
$745$	−85996.8	−0.00567664
$746$	−3.36030e7	−2.21070
$747$	0	0
$748$	0	0
$749$	1.26579e7	0.824438
$750$	0	0
$751$	1.72544e7	1.11635	0.558173	−	0.829724i	$-0.311502\pi$
$751$	1.72544e7	1.11635	0.558173	+	0.829724i	$0.311502\pi$
$752$	−7.06285e6	−0.455445
$753$	0	0
$754$	−6.40745e6	−0.410447
$755$	−15176.4	−0.000968950 0
$756$	0	0
$757$	−1.57084e7	−0.996307	−0.498153	−	0.867089i	$-0.665988\pi$
$757$	−1.57084e7	−0.996307	−0.498153	+	0.867089i	$0.665988\pi$
$758$	1.76033e7	1.11281
$759$	0	0
$760$	125920.	0.00790789
$761$	5.01006e6	0.313604	0.156802	−	0.987630i	$-0.449882\pi$
$761$	5.01006e6	0.313604	0.156802	+	0.987630i	$0.449882\pi$
$762$	0	0
$763$	9.65215e6	0.600223
$764$	−7.46778e6	−0.462869
$765$	0	0
$766$	6.21058e6	0.382437
$767$	−1.04481e6	−0.0641282
$768$	0	0
$769$	−3.47489e6	−0.211897	−0.105949	−	0.994372i	$-0.533788\pi$
$769$	−3.47489e6	−0.211897	−0.105949	+	0.994372i	$0.533788\pi$
$770$	0	0
$771$	0	0
$772$	2.58128e7	1.55880
$773$	−2.29786e7	−1.38317	−0.691584	−	0.722296i	$-0.743087\pi$
$773$	−2.29786e7	−1.38317	−0.691584	+	0.722296i	$0.743087\pi$
$774$	0	0
$775$	3.07357e7	1.83818
$776$	−3.33558e7	−1.98846
$777$	0	0
$778$	4.76458e7	2.82212
$779$	−3.71361e7	−2.19256
$780$	0	0
$781$	0	0
$782$	1.11279e7	0.650724
$783$	0	0
$784$	−1.77096e6	−0.102901
$785$	−61412.3	−0.00355698
$786$	0	0
$787$	1.40930e7	0.811087	0.405544	−	0.914076i	$-0.367082\pi$
$787$	1.40930e7	0.811087	0.405544	+	0.914076i	$0.367082\pi$
$788$	5.38476e6	0.308923
$789$	0	0
$790$	−107146.	−0.00610812
$791$	3.11247e6	0.176874
$792$	0	0
$793$	−2.72526e6	−0.153895
$794$	−4.85845e6	−0.273493
$795$	0	0
$796$	−6.04545e7	−3.38178
$797$	1.08281e7	0.603819	0.301909	−	0.953337i	$-0.402376\pi$
$797$	1.08281e7	0.603819	0.301909	+	0.953337i	$0.402376\pi$
$798$	0	0
$799$	3.29346e7	1.82510
$800$	1.33455e7	0.737243
$801$	0	0
$802$	−1.05474e7	−0.579041
$803$	0	0
$804$	0	0
$805$	24142.2	0.00131307
$806$	1.36014e7	0.737474
$807$	0	0
$808$	1.39444e7	0.751402
$809$	−2.27494e7	−1.22208	−0.611040	−	0.791600i	$-0.709248\pi$
$809$	−2.27494e7	−1.22208	−0.611040	+	0.791600i	$0.709248\pi$
$810$	0	0
$811$	1.33958e7	0.715179	0.357590	−	0.933879i	$-0.383599\pi$
$811$	1.33958e7	0.715179	0.357590	+	0.933879i	$0.383599\pi$
$812$	2.71877e7	1.44705
$813$	0	0
$814$	0	0
$815$	62433.2	0.00329247
$816$	0	0
$817$	2.81038e7	1.47302
$818$	−2.95596e7	−1.54460
$819$	0	0
$820$	−282706.	−0.0146825
$821$	−1.77359e7	−0.918323	−0.459161	−	0.888353i	$-0.651850\pi$
$821$	−1.77359e7	−0.918323	−0.459161	+	0.888353i	$0.651850\pi$
$822$	0	0
$823$	−2.24560e7	−1.15567	−0.577834	−	0.816154i	$-0.696102\pi$
$823$	−2.24560e7	−1.15567	−0.577834	+	0.816154i	$0.696102\pi$
$824$	−2.01194e7	−1.03228
$825$	0	0
$826$	6.97386e6	0.355650
$827$	−1.97457e7	−1.00394	−0.501972	−	0.864884i	$-0.667392\pi$
$827$	−1.97457e7	−1.00394	−0.501972	+	0.864884i	$0.667392\pi$
$828$	0	0
$829$	2.61754e7	1.32284	0.661418	−	0.750017i	$-0.269955\pi$
$829$	2.61754e7	1.32284	0.661418	+	0.750017i	$0.269955\pi$
$830$	60674.1	0.00305709
$831$	0	0
$832$	7.35624e6	0.368424
$833$	8.25814e6	0.412354
$834$	0	0
$835$	−51777.4	−0.00256995
$836$	0	0
$837$	0	0
$838$	−7.51015e6	−0.369435
$839$	3.69992e6	0.181463	0.0907313	−	0.995875i	$-0.471080\pi$
$839$	3.69992e6	0.181463	0.0907313	+	0.995875i	$0.471080\pi$
$840$	0	0
$841$	957652.	0.0466893
$842$	−1.73808e7	−0.844869
$843$	0	0
$844$	−4.88925e7	−2.36258
$845$	96883.0	0.00466773
$846$	0	0
$847$	0	0
$848$	−4.39660e6	−0.209956
$849$	0	0
$850$	4.19542e7	1.99172
$851$	−5.91332e6	−0.279903
$852$	0	0
$853$	−2.90121e7	−1.36523	−0.682617	−	0.730776i	$-0.739158\pi$
$853$	−2.90121e7	−1.36523	−0.682617	+	0.730776i	$0.739158\pi$
$854$	1.81905e7	0.853492
$855$	0	0
$856$	2.69142e7	1.25544
$857$	−3.60120e7	−1.67493	−0.837463	−	0.546494i	$-0.815962\pi$
$857$	−3.60120e7	−1.67493	−0.837463	+	0.546494i	$0.815962\pi$
$858$	0	0
$859$	−2.14715e7	−0.992841	−0.496421	−	0.868082i	$-0.665353\pi$
$859$	−2.14715e7	−0.992841	−0.496421	+	0.868082i	$0.665353\pi$
$860$	213945.	0.00986409
$861$	0	0
$862$	2.30562e7	1.05686
$863$	−1.43154e7	−0.654298	−0.327149	−	0.944973i	$-0.606088\pi$
$863$	−1.43154e7	−0.654298	−0.327149	+	0.944973i	$0.606088\pi$
$864$	0	0
$865$	197818.	0.00898928
$866$	2.92625e7	1.32592
$867$	0	0
$868$	−5.77128e7	−2.60000
$869$	0	0
$870$	0	0
$871$	−2.39464e6	−0.106953
$872$	2.05231e7	0.914013
$873$	0	0
$874$	1.57943e7	0.699392
$875$	182043.	0.00803809
$876$	0	0
$877$	−2.25088e7	−0.988221	−0.494110	−	0.869399i	$-0.664506\pi$
$877$	−2.25088e7	−0.988221	−0.494110	+	0.869399i	$0.664506\pi$
$878$	5.71030e7	2.49990
$879$	0	0
$880$	0	0
$881$	−8.39629e6	−0.364458	−0.182229	−	0.983256i	$-0.558331\pi$
$881$	−8.39629e6	−0.364458	−0.182229	+	0.983256i	$0.558331\pi$
$882$	0	0
$883$	1.08666e7	0.469021	0.234510	−	0.972114i	$-0.424651\pi$
$883$	1.08666e7	0.469021	0.234510	+	0.972114i	$0.424651\pi$
$884$	1.18024e7	0.507971
$885$	0	0
$886$	7.43650e7	3.18262
$887$	1.53488e7	0.655038	0.327519	−	0.944845i	$-0.393788\pi$
$887$	1.53488e7	0.655038	0.327519	+	0.944845i	$0.393788\pi$
$888$	0	0
$889$	−2.84905e7	−1.20906
$890$	−269668.	−0.0114118
$891$	0	0
$892$	5.51719e7	2.32170
$893$	4.67453e7	1.96160
$894$	0	0
$895$	218825.	0.00913145
$896$	−3.47405e7	−1.44566
$897$	0	0
$898$	−4.32073e7	−1.78799
$899$	−4.55729e7	−1.88065
$900$	0	0
$901$	2.05017e7	0.841352
$902$	0	0
$903$	0	0
$904$	6.61796e6	0.269341
$905$	−11179.1	−0.000453719 0
$906$	0	0
$907$	7.93009e6	0.320081	0.160041	−	0.987110i	$-0.448838\pi$
$907$	7.93009e6	0.320081	0.160041	+	0.987110i	$0.448838\pi$
$908$	3.74494e6	0.150741
$909$	0	0
$910$	40279.1	0.00161241
$911$	4.31907e7	1.72423	0.862114	−	0.506715i	$-0.169140\pi$
$911$	4.31907e7	1.72423	0.862114	+	0.506715i	$0.169140\pi$
$912$	0	0
$913$	0	0
$914$	7.54898e7	2.98898
$915$	0	0
$916$	−3.69732e7	−1.45595
$917$	−2.35372e7	−0.924340
$918$	0	0
$919$	2.17987e7	0.851414	0.425707	−	0.904861i	$-0.360025\pi$
$919$	2.17987e7	0.851414	0.425707	+	0.904861i	$0.360025\pi$
$920$	51332.8	0.00199952
$921$	0	0
$922$	3.02191e7	1.17072
$923$	4.16072e6	0.160755
$924$	0	0
$925$	−2.22943e7	−0.856720
$926$	1.21471e7	0.465527
$927$	0	0
$928$	−1.97879e7	−0.754276
$929$	1.33390e7	0.507089	0.253545	−	0.967324i	$-0.418404\pi$
$929$	1.33390e7	0.507089	0.253545	+	0.967324i	$0.418404\pi$
$930$	0	0
$931$	1.17211e7	0.443194
$932$	7.70371e7	2.90510
$933$	0	0
$934$	2.00475e7	0.751956
$935$	0	0
$936$	0	0
$937$	−1.33697e7	−0.497476	−0.248738	−	0.968571i	$-0.580016\pi$
$937$	−1.33697e7	−0.497476	−0.248738	+	0.968571i	$0.580016\pi$
$938$	1.59836e7	0.593155
$939$	0	0
$940$	355858.	0.0131358
$941$	2.28926e7	0.842793	0.421397	−	0.906876i	$-0.361540\pi$
$941$	2.28926e7	0.842793	0.421397	+	0.906876i	$0.361540\pi$
$942$	0	0
$943$	−1.51390e7	−0.554393
$944$	2.17531e6	0.0794494
$945$	0	0
$946$	0	0
$947$	−3.33079e6	−0.120690	−0.0603451	−	0.998178i	$-0.519220\pi$
$947$	−3.33079e6	−0.120690	−0.0603451	+	0.998178i	$0.519220\pi$
$948$	0	0
$949$	−5.81089e6	−0.209449
$950$	5.95472e7	2.14068
$951$	0	0
$952$	−3.36328e7	−1.20274
$953$	4.68461e7	1.67087	0.835433	−	0.549593i	$-0.185217\pi$
$953$	4.68461e7	1.67087	0.835433	+	0.549593i	$0.185217\pi$
$954$	0	0
$955$	37069.9	0.00131526
$956$	2.36644e7	0.837433
$957$	0	0
$958$	−4.47379e7	−1.57493
$959$	3.65924e7	1.28483
$960$	0	0
$961$	6.81109e7	2.37907
$962$	−9.86586e6	−0.343714
$963$	0	0
$964$	7.52730e7	2.60883
$965$	−128134.	−0.00442942
$966$	0	0
$967$	3.65887e7	1.25829	0.629146	−	0.777287i	$-0.283405\pi$
$967$	3.65887e7	1.25829	0.629146	+	0.777287i	$0.283405\pi$
$968$	0	0
$969$	0	0
$970$	387833.	0.0132347
$971$	−1.62147e7	−0.551901	−0.275951	−	0.961172i	$-0.588993\pi$
$971$	−1.62147e7	−0.551901	−0.275951	+	0.961172i	$0.588993\pi$
$972$	0	0
$973$	−3.50764e7	−1.18777
$974$	7.48261e7	2.52729
$975$	0	0
$976$	5.67403e6	0.190663
$977$	1.50181e6	0.0503360	0.0251680	−	0.999683i	$-0.491988\pi$
$977$	1.50181e6	0.0503360	0.0251680	+	0.999683i	$0.491988\pi$
$978$	0	0
$979$	0	0
$980$	89229.1	0.00296784
$981$	0	0
$982$	−3.46803e7	−1.14764
$983$	−3.72690e7	−1.23017	−0.615083	−	0.788462i	$-0.710878\pi$
$983$	−3.72690e7	−1.23017	−0.615083	+	0.788462i	$0.710878\pi$
$984$	0	0
$985$	−26729.8	−0.000877820 0
$986$	−6.22072e7	−2.03774
$987$	0	0
$988$	1.67516e7	0.545962
$989$	1.14569e7	0.372456
$990$	0	0
$991$	−3.20015e7	−1.03511	−0.517554	−	0.855651i	$-0.673157\pi$
$991$	−3.20015e7	−1.03511	−0.517554	+	0.855651i	$0.673157\pi$
$992$	4.20049e7	1.35525
$993$	0	0
$994$	−2.77718e7	−0.891534
$995$	300095.	0.00960950
$996$	0	0
$997$	5.59172e7	1.78159	0.890795	−	0.454406i	$-0.150148\pi$
$997$	5.59172e7	1.78159	0.890795	+	0.454406i	$0.150148\pi$
$998$	−6.60874e7	−2.10035
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.p.1.2		2
3.2	odd	2		363.6.a.f.1.1		2
11.10	odd	2		99.6.a.d.1.1		2
33.32	even	2		33.6.a.e.1.2	✓	2
132.131	odd	2		528.6.a.o.1.1		2
165.164	even	2		825.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.e.1.2	✓	2	33.32	even	2
99.6.a.d.1.1		2	11.10	odd	2
363.6.a.f.1.1		2	3.2	odd	2
528.6.a.o.1.1		2	132.131	odd	2
825.6.a.c.1.1		2	165.164	even	2
1089.6.a.p.1.2		2	1.1	even	1	trivial

Overview	Random
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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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+9.37228 q^{2} +55.8397 q^{4} -0.277187 q^{5} +105.081 q^{7} +223.432 q^{8} +O(q^{10})\)	\(q+9.37228 q^{2} +55.8397 q^{4} -0.277187 q^{5} +105.081 q^{7} +223.432 q^{8} -2.59787 q^{10} -147.549 q^{13} +984.853 q^{14} +307.198 q^{16} -1432.49 q^{17} -2033.18 q^{19} -15.4780 q^{20} -828.853 q^{23} -3124.92 q^{25} -1382.87 q^{26} +5867.71 q^{28} +4633.44 q^{29} -9835.65 q^{31} -4270.67 q^{32} -13425.7 q^{34} -29.1272 q^{35} +7134.34 q^{37} -19055.6 q^{38} -61.9324 q^{40} +18265.0 q^{41} -13822.5 q^{43} -7768.24 q^{46} -22991.2 q^{47} -5764.89 q^{49} -29287.7 q^{50} -8239.08 q^{52} -14311.9 q^{53} +23478.6 q^{56} +43425.9 q^{58} +7081.12 q^{59} +18470.2 q^{61} -92182.5 q^{62} -49856.3 q^{64} +40.8986 q^{65} +16229.5 q^{67} -79989.7 q^{68} -272.988 q^{70} -28198.9 q^{71} +39382.9 q^{73} +66865.1 q^{74} -113532. q^{76} +41243.7 q^{79} -85.1513 q^{80} +171185. q^{82} -23355.3 q^{83} +397.067 q^{85} -129549. q^{86} +103803. q^{89} -15504.6 q^{91} -46282.9 q^{92} -215480. q^{94} +563.572 q^{95} -149289. q^{97} -54030.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10}) \) 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8	\( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100}) \) 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 - 212 * q^10 + 130 * q^13 + 74 * q^14 + 241 * q^16 - 728 * q^17 + 828 * q^19 + 1072 * q^20 + 238 * q^23 - 2918 * q^25 - 376 * q^26 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 + 1391 * q^32 - 10870 * q^34 + 14464 * q^35 - 1908 * q^37 - 8676 * q^38 + 10584 * q^40 + 36484 * q^41 - 9768 * q^43 - 3898 * q^46 - 43742 * q^47 + 40470 * q^49 - 28537 * q^50 - 13468 * q^52 + 12174 * q^53 + 69786 * q^56 + 29142 * q^58 + 2788 * q^59 + 25302 * q^61 - 94520 * q^62 - 27199 * q^64 - 15980 * q^65 - 40520 * q^67 - 93262 * q^68 + 52304 * q^70 - 31386 * q^71 + 46780 * q^73 + 34062 * q^74 - 167436 * q^76 + 16850 * q^79 + 3736 * q^80 + 237278 * q^82 + 79440 * q^83 - 40268 * q^85 - 114840 * q^86 + 54204 * q^89 - 85192 * q^91 - 66382 * q^92 - 290758 * q^94 - 164592 * q^95 - 241568 * q^97 + 113697 * q^98

Introduction

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