LMFDB - Embedded newform 1089.6.a.bh.1.8

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 252 x^{8} + 45 x^{7} + 21644 x^{6} + 14121 x^{5} - 727612 x^{4} - 1049829 x^{3} + \cdots - 5072980 $ x^10 - x^9 - 252x^8 + 45x^7 + 21644x^6 + 14121x^5 - 727612x^4 - 1049829x^3 + 8255839x^2 + 16664560x - 5072980
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$6.64018$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.64018 q^{2} -0.188416 q^{4} -68.1325 q^{5} +131.688 q^{7} -181.548 q^{8} +O(q^{10})$	$q+5.64018 q^{2} -0.188416 q^{4} -68.1325 q^{5} +131.688 q^{7} -181.548 q^{8} -384.279 q^{10} -13.1678 q^{13} +742.742 q^{14} -1017.94 q^{16} -2001.86 q^{17} +2781.55 q^{19} +12.8373 q^{20} -2936.26 q^{23} +1517.03 q^{25} -74.2688 q^{26} -24.8121 q^{28} +2382.77 q^{29} -1953.88 q^{31} +68.2131 q^{32} -11290.8 q^{34} -8972.21 q^{35} -9142.17 q^{37} +15688.4 q^{38} +12369.3 q^{40} -9449.20 q^{41} -1942.32 q^{43} -16561.0 q^{46} -11525.6 q^{47} +534.655 q^{49} +8556.33 q^{50} +2.48103 q^{52} +18706.8 q^{53} -23907.7 q^{56} +13439.3 q^{58} +45451.5 q^{59} -34349.1 q^{61} -11020.2 q^{62} +32958.7 q^{64} +897.155 q^{65} +37935.2 q^{67} +377.183 q^{68} -50604.8 q^{70} +24243.2 q^{71} -59045.4 q^{73} -51563.4 q^{74} -524.090 q^{76} +42086.7 q^{79} +69354.4 q^{80} -53295.1 q^{82} -18630.7 q^{83} +136392. q^{85} -10955.0 q^{86} +82154.9 q^{89} -1734.04 q^{91} +553.240 q^{92} -65006.5 q^{94} -189514. q^{95} +117153. q^{97} +3015.55 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8}+O(q^{10}) $ 10 * q - 9 * q^2 + 193 * q^4 - 11 * q^5 + 470 * q^7 - 324 * q^8	$ 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8} + 976 q^{10} + 2308 q^{13} - 540 q^{14} + 2801 q^{16} - 3093 q^{17} + 5305 q^{19} + 2229 q^{20} - 700 q^{23} + 8381 q^{25} + 15559 q^{26} + 24656 q^{28} - 10392 q^{29} - 101 q^{31} - 34557 q^{32} - 4542 q^{34} - 19867 q^{35} - 1284 q^{37} + 35769 q^{38} + 66596 q^{40} - 17944 q^{41} + 31812 q^{43} + 36417 q^{46} - 8787 q^{47} - 23810 q^{49} + 910 q^{50} + 51663 q^{52} + 3261 q^{53} - 84819 q^{56} + 53125 q^{58} - 49375 q^{59} + 63175 q^{61} - 28399 q^{62} + 124764 q^{64} + 14105 q^{65} + 5365 q^{67} + 53313 q^{68} + 40297 q^{70} + 236675 q^{71} + 200912 q^{73} + 180329 q^{74} + 39606 q^{76} + 210802 q^{79} - 270298 q^{80} - 369223 q^{82} + 178968 q^{83} + 107352 q^{85} - 465999 q^{86} - 90816 q^{89} + 7500 q^{91} + 136407 q^{92} + 71890 q^{94} - 335807 q^{95} + 271521 q^{97} + 75285 q^{98}+O(q^{100}) $ 10 * q - 9 * q^2 + 193 * q^4 - 11 * q^5 + 470 * q^7 - 324 * q^8 + 976 * q^10 + 2308 * q^13 - 540 * q^14 + 2801 * q^16 - 3093 * q^17 + 5305 * q^19 + 2229 * q^20 - 700 * q^23 + 8381 * q^25 + 15559 * q^26 + 24656 * q^28 - 10392 * q^29 - 101 * q^31 - 34557 * q^32 - 4542 * q^34 - 19867 * q^35 - 1284 * q^37 + 35769 * q^38 + 66596 * q^40 - 17944 * q^41 + 31812 * q^43 + 36417 * q^46 - 8787 * q^47 - 23810 * q^49 + 910 * q^50 + 51663 * q^52 + 3261 * q^53 - 84819 * q^56 + 53125 * q^58 - 49375 * q^59 + 63175 * q^61 - 28399 * q^62 + 124764 * q^64 + 14105 * q^65 + 5365 * q^67 + 53313 * q^68 + 40297 * q^70 + 236675 * q^71 + 200912 * q^73 + 180329 * q^74 + 39606 * q^76 + 210802 * q^79 - 270298 * q^80 - 369223 * q^82 + 178968 * q^83 + 107352 * q^85 - 465999 * q^86 - 90816 * q^89 + 7500 * q^91 + 136407 * q^92 + 71890 * q^94 - 335807 * q^95 + 271521 * q^97 + 75285 * 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 $

$2$	5.64018	0.997052	0.498526	−	0.866875i	$-0.333875\pi$
$2$	5.64018	0.997052	0.498526	+	0.866875i	$0.333875\pi$
$3$	0	0
$3$	0	0
$4$	−0.188416	−0.00588801
$5$	−68.1325	−1.21879	−0.609395	−	0.792867i	$-0.708588\pi$
$5$	−68.1325	−1.21879	−0.609395	+	0.792867i	$0.708588\pi$
$6$	0	0
$7$	131.688	1.01578	0.507891	−	0.861422i	$-0.330425\pi$
$7$	131.688	1.01578	0.507891	+	0.861422i	$0.330425\pi$
$8$	−181.548	−1.00292
$9$	0	0
$10$	−384.279	−1.21520
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−13.1678	−0.0216100	−0.0108050	−	0.999942i	$-0.503439\pi$
$13$	−13.1678	−0.0216100	−0.0108050	+	0.999942i	$0.503439\pi$
$14$	742.742	1.01279
$15$	0	0
$16$	−1017.94	−0.994077
$17$	−2001.86	−1.68001	−0.840004	−	0.542581i	$-0.817447\pi$
$17$	−2001.86	−1.68001	−0.840004	+	0.542581i	$0.817447\pi$
$18$	0	0
$19$	2781.55	1.76768	0.883839	−	0.467791i	$-0.154950\pi$
$19$	2781.55	1.76768	0.883839	+	0.467791i	$0.154950\pi$
$20$	12.8373	0.00717625
$21$	0	0
$22$	0	0
$23$	−2936.26	−1.15738	−0.578689	−	0.815548i	$-0.696436\pi$
$23$	−2936.26	−1.15738	−0.578689	+	0.815548i	$0.696436\pi$
$24$	0	0
$25$	1517.03	0.485450
$26$	−74.2688	−0.0215463
$27$	0	0
$28$	−24.8121	−0.00598093
$29$	2382.77	0.526124	0.263062	−	0.964779i	$-0.415268\pi$
$29$	2382.77	0.526124	0.263062	+	0.964779i	$0.415268\pi$
$30$	0	0
$31$	−1953.88	−0.365169	−0.182585	−	0.983190i	$-0.558446\pi$
$31$	−1953.88	−0.365169	−0.182585	+	0.983190i	$0.558446\pi$
$32$	68.2131	0.0117759
$33$	0	0
$34$	−11290.8	−1.67505
$35$	−8972.21	−1.23802
$36$	0	0
$37$	−9142.17	−1.09785	−0.548927	−	0.835870i	$-0.684964\pi$
$37$	−9142.17	−1.09785	−0.548927	+	0.835870i	$0.684964\pi$
$38$	15688.4	1.76247
$39$	0	0
$40$	12369.3	1.22235
$41$	−9449.20	−0.877880	−0.438940	−	0.898516i	$-0.644646\pi$
$41$	−9449.20	−0.877880	−0.438940	+	0.898516i	$0.644646\pi$
$42$	0	0
$43$	−1942.32	−0.160195	−0.0800977	−	0.996787i	$-0.525523\pi$
$43$	−1942.32	−0.160195	−0.0800977	+	0.996787i	$0.525523\pi$
$44$	0	0
$45$	0	0
$46$	−16561.0	−1.15397
$47$	−11525.6	−0.761062	−0.380531	−	0.924768i	$-0.624259\pi$
$47$	−11525.6	−0.761062	−0.380531	+	0.924768i	$0.624259\pi$
$48$	0	0
$49$	534.655	0.0318114
$50$	8556.33	0.484019
$51$	0	0
$52$	2.48103	0.000127240 0
$53$	18706.8	0.914765	0.457382	−	0.889270i	$-0.348787\pi$
$53$	18706.8	0.914765	0.457382	+	0.889270i	$0.348787\pi$
$54$	0	0
$55$	0	0
$56$	−23907.7	−1.01875
$57$	0	0
$58$	13439.3	0.524572
$59$	45451.5	1.69988	0.849941	−	0.526879i	$-0.176638\pi$
$59$	45451.5	1.69988	0.849941	+	0.526879i	$0.176638\pi$
$60$	0	0
$61$	−34349.1	−1.18193	−0.590964	−	0.806698i	$-0.701252\pi$
$61$	−34349.1	−1.18193	−0.590964	+	0.806698i	$0.701252\pi$
$62$	−11020.2	−0.364093
$63$	0	0
$64$	32958.7	1.00582
$65$	897.155	0.0263381
$66$	0	0
$67$	37935.2	1.03242	0.516208	−	0.856463i	$-0.327343\pi$
$67$	37935.2	1.03242	0.516208	+	0.856463i	$0.327343\pi$
$68$	377.183	0.00989190
$69$	0	0
$70$	−50604.8	−1.23437
$71$	24243.2	0.570748	0.285374	−	0.958416i	$-0.407882\pi$
$71$	24243.2	0.570748	0.285374	+	0.958416i	$0.407882\pi$
$72$	0	0
$73$	−59045.4	−1.29682	−0.648409	−	0.761292i	$-0.724565\pi$
$73$	−59045.4	−1.29682	−0.648409	+	0.761292i	$0.724565\pi$
$74$	−51563.4	−1.09462
$75$	0	0
$76$	−524.090	−0.0104081
$77$	0	0
$78$	0	0
$79$	42086.7	0.758712	0.379356	−	0.925251i	$-0.376145\pi$
$79$	42086.7	0.758712	0.379356	+	0.925251i	$0.376145\pi$
$80$	69354.4	1.21157
$81$	0	0
$82$	−53295.1	−0.875292
$83$	−18630.7	−0.296848	−0.148424	−	0.988924i	$-0.547420\pi$
$83$	−18630.7	−0.296848	−0.148424	+	0.988924i	$0.547420\pi$
$84$	0	0
$85$	136392.	2.04758
$86$	−10955.0	−0.159723
$87$	0	0
$88$	0	0
$89$	82154.9	1.09941	0.549704	−	0.835360i	$-0.314741\pi$
$89$	82154.9	1.09941	0.549704	+	0.835360i	$0.314741\pi$
$90$	0	0
$91$	−1734.04	−0.0219511
$92$	553.240	0.00681465
$93$	0	0
$94$	−65006.5	−0.758818
$95$	−189514.	−2.15443
$96$	0	0
$97$	117153.	1.26423	0.632114	−	0.774876i	$-0.282188\pi$
$97$	117153.	1.26423	0.632114	+	0.774876i	$0.282188\pi$
$98$	3015.55	0.0317176
$99$	0	0
$100$	−285.833	−0.00285833
$101$	−30554.8	−0.298041	−0.149020	−	0.988834i	$-0.547612\pi$
$101$	−30554.8	−0.298041	−0.149020	+	0.988834i	$0.547612\pi$
$102$	0	0
$103$	−4791.65	−0.0445033	−0.0222517	−	0.999752i	$-0.507084\pi$
$103$	−4791.65	−0.0445033	−0.0222517	+	0.999752i	$0.507084\pi$
$104$	2390.59	0.0216732
$105$	0	0
$106$	105510.	0.912068
$107$	79013.0	0.667174	0.333587	−	0.942719i	$-0.391741\pi$
$107$	79013.0	0.667174	0.333587	+	0.942719i	$0.391741\pi$
$108$	0	0
$109$	173349.	1.39751	0.698756	−	0.715360i	$-0.253737\pi$
$109$	173349.	1.39751	0.698756	+	0.715360i	$0.253737\pi$
$110$	0	0
$111$	0	0
$112$	−134050.	−1.00977
$113$	−48167.4	−0.354860	−0.177430	−	0.984133i	$-0.556778\pi$
$113$	−48167.4	−0.354860	−0.177430	+	0.984133i	$0.556778\pi$
$114$	0	0
$115$	200055.	1.41060
$116$	−448.953	−0.00309782
$117$	0	0
$118$	256355.	1.69487
$119$	−263620.	−1.70652
$120$	0	0
$121$	0	0
$122$	−193735.	−1.17844
$123$	0	0
$124$	368.143	0.00215012
$125$	109555.	0.627128
$126$	0	0
$127$	−110728.	−0.609186	−0.304593	−	0.952483i	$-0.598520\pi$
$127$	−110728.	−0.609186	−0.304593	+	0.952483i	$0.598520\pi$
$128$	183710.	0.991077
$129$	0	0
$130$	5060.11	0.0262604
$131$	−106644.	−0.542949	−0.271474	−	0.962446i	$-0.587511\pi$
$131$	−106644.	−0.542949	−0.271474	+	0.962446i	$0.587511\pi$
$132$	0	0
$133$	366296.	1.79557
$134$	213961.	1.02937
$135$	0	0
$136$	363434.	1.68492
$137$	52032.8	0.236851	0.118426	−	0.992963i	$-0.462215\pi$
$137$	52032.8	0.236851	0.118426	+	0.992963i	$0.462215\pi$
$138$	0	0
$139$	430171.	1.88845	0.944223	−	0.329308i	$-0.106815\pi$
$139$	430171.	1.88845	0.944223	+	0.329308i	$0.106815\pi$
$140$	1690.51	0.00728950
$141$	0	0
$142$	136736.	0.569065
$143$	0	0
$144$	0	0
$145$	−162344.	−0.641234
$146$	−333027.	−1.29299
$147$	0	0
$148$	1722.53	0.00646418
$149$	−242307.	−0.894130	−0.447065	−	0.894502i	$-0.647531\pi$
$149$	−242307.	−0.894130	−0.447065	+	0.894502i	$0.647531\pi$
$150$	0	0
$151$	−37456.2	−0.133685	−0.0668423	−	0.997764i	$-0.521292\pi$
$151$	−37456.2	−0.133685	−0.0668423	+	0.997764i	$0.521292\pi$
$152$	−504986.	−1.77284
$153$	0	0
$154$	0	0
$155$	133123.	0.445065
$156$	0	0
$157$	168218.	0.544658	0.272329	−	0.962204i	$-0.412206\pi$
$157$	168218.	0.544658	0.272329	+	0.962204i	$0.412206\pi$
$158$	237376.	0.756476
$159$	0	0
$160$	−4647.52	−0.0143523
$161$	−386670.	−1.17564
$162$	0	0
$163$	−278854.	−0.822068	−0.411034	−	0.911620i	$-0.634832\pi$
$163$	−278854.	−0.822068	−0.411034	+	0.911620i	$0.634832\pi$
$164$	1780.38	0.00516897
$165$	0	0
$166$	−105080.	−0.295973
$167$	467221.	1.29638	0.648188	−	0.761480i	$-0.275527\pi$
$167$	467221.	1.29638	0.648188	+	0.761480i	$0.275527\pi$
$168$	0	0
$169$	−371120.	−0.999533
$170$	769273.	2.04154
$171$	0	0
$172$	365.965	0.000943231 0
$173$	195317.	0.496164	0.248082	−	0.968739i	$-0.420200\pi$
$173$	195317.	0.496164	0.248082	+	0.968739i	$0.420200\pi$
$174$	0	0
$175$	199774.	0.493111
$176$	0	0
$177$	0	0
$178$	463368.	1.09617
$179$	582378.	1.35854	0.679270	−	0.733889i	$-0.262297\pi$
$179$	582378.	1.35854	0.679270	+	0.733889i	$0.262297\pi$
$180$	0	0
$181$	699478.	1.58700	0.793502	−	0.608568i	$-0.208256\pi$
$181$	699478.	1.58700	0.793502	+	0.608568i	$0.208256\pi$
$182$	−9780.28	−0.0218863
$183$	0	0
$184$	533074.	1.16076
$185$	622878.	1.33806
$186$	0	0
$187$	0	0
$188$	2171.61	0.00448114
$189$	0	0
$190$	−1.06889e6	−2.14808
$191$	−937877.	−1.86021	−0.930106	−	0.367290i	$-0.880286\pi$
$191$	−937877.	−1.86021	−0.930106	+	0.367290i	$0.880286\pi$
$192$	0	0
$193$	−839.294	−0.00162189	−0.000810944	−	1.00000i	$-0.500258\pi$
$193$	−839.294	−0.00162189	−0.000810944		1.00000i	$0.500258\pi$
$194$	660765.	1.26050
$195$	0	0
$196$	−100.738	−0.000187306 0
$197$	286777.	0.526476	0.263238	−	0.964731i	$-0.415210\pi$
$197$	286777.	0.526476	0.263238	+	0.964731i	$0.415210\pi$
$198$	0	0
$199$	855459.	1.53132	0.765661	−	0.643245i	$-0.222412\pi$
$199$	855459.	1.53132	0.765661	+	0.643245i	$0.222412\pi$
$200$	−275415.	−0.486869
$201$	0	0
$202$	−172334.	−0.297162
$203$	313782.	0.534426
$204$	0	0
$205$	643797.	1.06995
$206$	−27025.8	−0.0443721
$207$	0	0
$208$	13404.0	0.0214820
$209$	0	0
$210$	0	0
$211$	251108.	0.388289	0.194144	−	0.980973i	$-0.437807\pi$
$211$	251108.	0.388289	0.194144	+	0.980973i	$0.437807\pi$
$212$	−3524.66	−0.00538614
$213$	0	0
$214$	445647.	0.665207
$215$	132335.	0.195245
$216$	0	0
$217$	−257302.	−0.370932
$218$	977720.	1.39339
$219$	0	0
$220$	0	0
$221$	26360.1	0.0363050
$222$	0	0
$223$	1.12120e6	1.50981	0.754903	−	0.655837i	$-0.227684\pi$
$223$	1.12120e6	1.50981	0.754903	+	0.655837i	$0.227684\pi$
$224$	8982.82	0.0119617
$225$	0	0
$226$	−271673.	−0.353814
$227$	430136.	0.554040	0.277020	−	0.960864i	$-0.410653\pi$
$227$	430136.	0.554040	0.277020	+	0.960864i	$0.410653\pi$
$228$	0	0
$229$	301590.	0.380039	0.190019	−	0.981780i	$-0.439145\pi$
$229$	301590.	0.380039	0.190019	+	0.981780i	$0.439145\pi$
$230$	1.12834e6	1.40644
$231$	0	0
$232$	−432589.	−0.527661
$233$	−162885.	−0.196558	−0.0982792	−	0.995159i	$-0.531334\pi$
$233$	−162885.	−0.196558	−0.0982792	+	0.995159i	$0.531334\pi$
$234$	0	0
$235$	785269.	0.927575
$236$	−8563.81	−0.0100089
$237$	0	0
$238$	−1.48686e6	−1.70149
$239$	860256.	0.974166	0.487083	−	0.873356i	$-0.338061\pi$
$239$	860256.	0.974166	0.487083	+	0.873356i	$0.338061\pi$
$240$	0	0
$241$	−1.20676e6	−1.33838	−0.669190	−	0.743091i	$-0.733359\pi$
$241$	−1.20676e6	−1.33838	−0.669190	+	0.743091i	$0.733359\pi$
$242$	0	0
$243$	0	0
$244$	6471.93	0.00695920
$245$	−36427.3	−0.0387715
$246$	0	0
$247$	−36626.9	−0.0381996
$248$	354724.	0.366236
$249$	0	0
$250$	617908.	0.625279
$251$	455597.	0.456454	0.228227	−	0.973608i	$-0.426707\pi$
$251$	455597.	0.456454	0.228227	+	0.973608i	$0.426707\pi$
$252$	0	0
$253$	0	0
$254$	−624528.	−0.607390
$255$	0	0
$256$	−18521.4	−0.0176634
$257$	1.26128e6	1.19119	0.595593	−	0.803286i	$-0.296917\pi$
$257$	1.26128e6	1.19119	0.595593	+	0.803286i	$0.296917\pi$
$258$	0	0
$259$	−1.20391e6	−1.11518
$260$	−169.039	−0.000155079 0
$261$	0	0
$262$	−601492.	−0.541348
$263$	1.72764e6	1.54015	0.770077	−	0.637951i	$-0.220218\pi$
$263$	1.72764e6	1.54015	0.770077	+	0.637951i	$0.220218\pi$
$264$	0	0
$265$	−1.27454e6	−1.11491
$266$	2.06598e6	1.79028
$267$	0	0
$268$	−7147.60	−0.00607888
$269$	1.43854e6	1.21211	0.606055	−	0.795423i	$-0.292751\pi$
$269$	1.43854e6	1.21211	0.606055	+	0.795423i	$0.292751\pi$
$270$	0	0
$271$	−1.12657e6	−0.931824	−0.465912	−	0.884831i	$-0.654274\pi$
$271$	−1.12657e6	−0.931824	−0.465912	+	0.884831i	$0.654274\pi$
$272$	2.03776e6	1.67006
$273$	0	0
$274$	293474.	0.236153
$275$	0	0
$276$	0	0
$277$	749776.	0.587127	0.293563	−	0.955940i	$-0.405159\pi$
$277$	749776.	0.587127	0.293563	+	0.955940i	$0.405159\pi$
$278$	2.42624e6	1.88288
$279$	0	0
$280$	1.62889e6	1.24164
$281$	−1.78795e6	−1.35079	−0.675396	−	0.737455i	$-0.736027\pi$
$281$	−1.78795e6	−1.35079	−0.675396	+	0.737455i	$0.736027\pi$
$282$	0	0
$283$	−1.06643e6	−0.791527	−0.395764	−	0.918352i	$-0.629520\pi$
$283$	−1.06643e6	−0.791527	−0.395764	+	0.918352i	$0.629520\pi$
$284$	−4567.82	−0.00336057
$285$	0	0
$286$	0	0
$287$	−1.24434e6	−0.891734
$288$	0	0
$289$	2.58758e6	1.82243
$290$	−915650.	−0.639344
$291$	0	0
$292$	11125.1	0.00763568
$293$	−760365.	−0.517431	−0.258716	−	0.965954i	$-0.583299\pi$
$293$	−760365.	−0.517431	−0.258716	+	0.965954i	$0.583299\pi$
$294$	0	0
$295$	−3.09672e6	−2.07180
$296$	1.65975e6	1.10106
$297$	0	0
$298$	−1.36665e6	−0.891493
$299$	38664.2	0.0250110
$300$	0	0
$301$	−255780.	−0.162723
$302$	−211260.	−0.133290
$303$	0	0
$304$	−2.83144e6	−1.75721
$305$	2.34029e6	1.44052
$306$	0	0
$307$	1.51732e6	0.918821	0.459410	−	0.888224i	$-0.348061\pi$
$307$	1.51732e6	0.918821	0.459410	+	0.888224i	$0.348061\pi$
$308$	0	0
$309$	0	0
$310$	750836.	0.443752
$311$	−2.46463e6	−1.44494	−0.722472	−	0.691400i	$-0.756994\pi$
$311$	−2.46463e6	−1.44494	−0.722472	+	0.691400i	$0.756994\pi$
$312$	0	0
$313$	−2.63328e6	−1.51928	−0.759638	−	0.650347i	$-0.774624\pi$
$313$	−2.63328e6	−1.51928	−0.759638	+	0.650347i	$0.774624\pi$
$314$	948781.	0.543052
$315$	0	0
$316$	−7929.82	−0.00446730
$317$	−211170.	−0.118028	−0.0590139	−	0.998257i	$-0.518796\pi$
$317$	−211170.	−0.118028	−0.0590139	+	0.998257i	$0.518796\pi$
$318$	0	0
$319$	0	0
$320$	−2.24555e6	−1.22588
$321$	0	0
$322$	−2.18089e6	−1.17218
$323$	−5.56828e6	−2.96971
$324$	0	0
$325$	−19976.0	−0.0104906
$326$	−1.57278e6	−0.819644
$327$	0	0
$328$	1.71549e6	0.880446
$329$	−1.51778e6	−0.773072
$330$	0	0
$331$	−2.93493e6	−1.47240	−0.736202	−	0.676762i	$-0.763383\pi$
$331$	−2.93493e6	−1.47240	−0.736202	+	0.676762i	$0.763383\pi$
$332$	3510.33	0.00174784
$333$	0	0
$334$	2.63521e6	1.29255
$335$	−2.58462e6	−1.25830
$336$	0	0
$337$	618188.	0.296514	0.148257	−	0.988949i	$-0.452634\pi$
$337$	618188.	0.296514	0.148257	+	0.988949i	$0.452634\pi$
$338$	−2.09318e6	−0.996586
$339$	0	0
$340$	−25698.4	−0.0120562
$341$	0	0
$342$	0	0
$343$	−2.14287e6	−0.983468
$344$	352625.	0.160663
$345$	0	0
$346$	1.10162e6	0.494701
$347$	−2.52463e6	−1.12558	−0.562788	−	0.826601i	$-0.690271\pi$
$347$	−2.52463e6	−1.12558	−0.562788	+	0.826601i	$0.690271\pi$
$348$	0	0
$349$	665578.	0.292506	0.146253	−	0.989247i	$-0.453279\pi$
$349$	665578.	0.292506	0.146253	+	0.989247i	$0.453279\pi$
$350$	1.12676e6	0.491657
$351$	0	0
$352$	0	0
$353$	−1.17460e6	−0.501712	−0.250856	−	0.968024i	$-0.580712\pi$
$353$	−1.17460e6	−0.501712	−0.250856	+	0.968024i	$0.580712\pi$
$354$	0	0
$355$	−1.65175e6	−0.695622
$356$	−15479.3	−0.00647332
$357$	0	0
$358$	3.28471e6	1.35453
$359$	1.73622e6	0.711000	0.355500	−	0.934676i	$-0.384311\pi$
$359$	1.73622e6	0.711000	0.355500	+	0.934676i	$0.384311\pi$
$360$	0	0
$361$	5.26093e6	2.12469
$362$	3.94518e6	1.58232
$363$	0	0
$364$	326.721	0.000129248 0
$365$	4.02291e6	1.58055
$366$	0	0
$367$	−3.99095e6	−1.54672	−0.773359	−	0.633968i	$-0.781425\pi$
$367$	−3.99095e6	−1.54672	−0.773359	+	0.633968i	$0.781425\pi$
$368$	2.98893e6	1.15052
$369$	0	0
$370$	3.51314e6	1.33411
$371$	2.46345e6	0.929201
$372$	0	0
$373$	3.60332e6	1.34101	0.670504	−	0.741906i	$-0.266078\pi$
$373$	3.60332e6	1.34101	0.670504	+	0.741906i	$0.266078\pi$
$374$	0	0
$375$	0	0
$376$	2.09246e6	0.763286
$377$	−31375.9	−0.0113695
$378$	0	0
$379$	2.13144e6	0.762209	0.381105	−	0.924532i	$-0.375544\pi$
$379$	2.13144e6	0.762209	0.381105	+	0.924532i	$0.375544\pi$
$380$	35707.5	0.0126853
$381$	0	0
$382$	−5.28979e6	−1.85473
$383$	−1.96998e6	−0.686223	−0.343112	−	0.939295i	$-0.611481\pi$
$383$	−1.96998e6	−0.686223	−0.343112	+	0.939295i	$0.611481\pi$
$384$	0	0
$385$	0	0
$386$	−4733.77	−0.00161711
$387$	0	0
$388$	−22073.6	−0.00744378
$389$	−993539.	−0.332898	−0.166449	−	0.986050i	$-0.553230\pi$
$389$	−993539.	−0.332898	−0.166449	+	0.986050i	$0.553230\pi$
$390$	0	0
$391$	5.87798e6	1.94440
$392$	−97065.7	−0.0319044
$393$	0	0
$394$	1.61747e6	0.524923
$395$	−2.86747e6	−0.924712
$396$	0	0
$397$	32927.1	0.0104852	0.00524260	−	0.999986i	$-0.498331\pi$
$397$	32927.1	0.0104852	0.00524260	+	0.999986i	$0.498331\pi$
$398$	4.82494e6	1.52681
$399$	0	0
$400$	−1.54424e6	−0.482575
$401$	4.32766e6	1.34398	0.671989	−	0.740561i	$-0.265440\pi$
$401$	4.32766e6	1.34398	0.671989	+	0.740561i	$0.265440\pi$
$402$	0	0
$403$	25728.3	0.00789131
$404$	5757.02	0.00175487
$405$	0	0
$406$	1.76979e6	0.532851
$407$	0	0
$408$	0	0
$409$	5.16462e6	1.52662	0.763308	−	0.646034i	$-0.223574\pi$
$409$	5.16462e6	1.52662	0.763308	+	0.646034i	$0.223574\pi$
$410$	3.63113e6	1.06680
$411$	0	0
$412$	902.826	0.000262036 0
$413$	5.98541e6	1.72671
$414$	0	0
$415$	1.26936e6	0.361796
$416$	−898.217	−0.000254477 0
$417$	0	0
$418$	0	0
$419$	−2.23401e6	−0.621656	−0.310828	−	0.950466i	$-0.600606\pi$
$419$	−2.23401e6	−0.621656	−0.310828	+	0.950466i	$0.600606\pi$
$420$	0	0
$421$	2.07619e6	0.570903	0.285451	−	0.958393i	$-0.407856\pi$
$421$	2.07619e6	0.570903	0.285451	+	0.958393i	$0.407856\pi$
$422$	1.41630e6	0.387144
$423$	0	0
$424$	−3.39619e6	−0.917438
$425$	−3.03688e6	−0.815560
$426$	0	0
$427$	−4.52336e6	−1.20058
$428$	−14887.3	−0.00392832
$429$	0	0
$430$	746393.	0.194669
$431$	3.74643e6	0.971459	0.485729	−	0.874109i	$-0.338554\pi$
$431$	3.74643e6	0.971459	0.485729	+	0.874109i	$0.338554\pi$
$432$	0	0
$433$	−4.51785e6	−1.15801	−0.579005	−	0.815324i	$-0.696559\pi$
$433$	−4.51785e6	−1.15801	−0.579005	+	0.815324i	$0.696559\pi$
$434$	−1.45123e6	−0.369838
$435$	0	0
$436$	−32661.8	−0.00822856
$437$	−8.16737e6	−2.04587
$438$	0	0
$439$	−1.07202e6	−0.265486	−0.132743	−	0.991150i	$-0.542379\pi$
$439$	−1.07202e6	−0.265486	−0.132743	+	0.991150i	$0.542379\pi$
$440$	0	0
$441$	0	0
$442$	148676.	0.0361980
$443$	1.13234e6	0.274136	0.137068	−	0.990562i	$-0.456232\pi$
$443$	1.13234e6	0.274136	0.137068	+	0.990562i	$0.456232\pi$
$444$	0	0
$445$	−5.59742e6	−1.33995
$446$	6.32377e6	1.50535
$447$	0	0
$448$	4.34025e6	1.02169
$449$	1.91657e6	0.448652	0.224326	−	0.974514i	$-0.427982\pi$
$449$	1.91657e6	0.448652	0.224326	+	0.974514i	$0.427982\pi$
$450$	0	0
$451$	0	0
$452$	9075.52	0.00208942
$453$	0	0
$454$	2.42604e6	0.552407
$455$	118144.	0.0267537
$456$	0	0
$457$	−2.22872e6	−0.499189	−0.249594	−	0.968350i	$-0.580297\pi$
$457$	−2.22872e6	−0.499189	−0.249594	+	0.968350i	$0.580297\pi$
$458$	1.70102e6	0.378918
$459$	0	0
$460$	−37693.6	−0.00830563
$461$	571870.	0.125327	0.0626635	−	0.998035i	$-0.480041\pi$
$461$	571870.	0.125327	0.0626635	+	0.998035i	$0.480041\pi$
$462$	0	0
$463$	−5.75256e6	−1.24712	−0.623561	−	0.781775i	$-0.714315\pi$
$463$	−5.75256e6	−1.24712	−0.623561	+	0.781775i	$0.714315\pi$
$464$	−2.42551e6	−0.523007
$465$	0	0
$466$	−918701.	−0.195979
$467$	5.79604e6	1.22981	0.614907	−	0.788600i	$-0.289194\pi$
$467$	5.79604e6	1.22981	0.614907	+	0.788600i	$0.289194\pi$
$468$	0	0
$469$	4.99560e6	1.04871
$470$	4.42906e6	0.924840
$471$	0	0
$472$	−8.25165e6	−1.70485
$473$	0	0
$474$	0	0
$475$	4.21970e6	0.858120
$476$	49670.3	0.0100480
$477$	0	0
$478$	4.85200e6	0.971294
$479$	−2.45226e6	−0.488345	−0.244173	−	0.969732i	$-0.578516\pi$
$479$	−2.45226e6	−0.488345	−0.244173	+	0.969732i	$0.578516\pi$
$480$	0	0
$481$	120382.	0.0237247
$482$	−6.80636e6	−1.33443
$483$	0	0
$484$	0	0
$485$	−7.98194e6	−1.54083
$486$	0	0
$487$	−385125.	−0.0735834	−0.0367917	−	0.999323i	$-0.511714\pi$
$487$	−385125.	−0.0735834	−0.0367917	+	0.999323i	$0.511714\pi$
$488$	6.23602e6	1.18538
$489$	0	0
$490$	−205457.	−0.0386572
$491$	−2.89916e6	−0.542711	−0.271356	−	0.962479i	$-0.587472\pi$
$491$	−2.89916e6	−0.542711	−0.271356	+	0.962479i	$0.587472\pi$
$492$	0	0
$493$	−4.76998e6	−0.883892
$494$	−206582.	−0.0380869
$495$	0	0
$496$	1.98893e6	0.363006
$497$	3.19253e6	0.579755
$498$	0	0
$499$	−4.47558e6	−0.804634	−0.402317	−	0.915501i	$-0.631795\pi$
$499$	−4.47558e6	−0.804634	−0.402317	+	0.915501i	$0.631795\pi$
$500$	−20641.9	−0.00369254
$501$	0	0
$502$	2.56965e6	0.455108
$503$	−7.47742e6	−1.31775	−0.658873	−	0.752254i	$-0.728967\pi$
$503$	−7.47742e6	−1.31775	−0.658873	+	0.752254i	$0.728967\pi$
$504$	0	0
$505$	2.08177e6	0.363249
$506$	0	0
$507$	0	0
$508$	20863.0	0.00358689
$509$	1.72733e6	0.295515	0.147758	−	0.989024i	$-0.452794\pi$
$509$	1.72733e6	0.295515	0.147758	+	0.989024i	$0.452794\pi$
$510$	0	0
$511$	−7.77556e6	−1.31728
$512$	−5.98318e6	−1.00869
$513$	0	0
$514$	7.11386e6	1.18767
$515$	326467.	0.0542402
$516$	0	0
$517$	0	0
$518$	−6.79027e6	−1.11189
$519$	0	0
$520$	−162877.	−0.0264151
$521$	2.84137e6	0.458599	0.229299	−	0.973356i	$-0.426357\pi$
$521$	2.84137e6	0.458599	0.229299	+	0.973356i	$0.426357\pi$
$522$	0	0
$523$	4.12365e6	0.659216	0.329608	−	0.944118i	$-0.393083\pi$
$523$	4.12365e6	0.659216	0.329608	+	0.944118i	$0.393083\pi$
$524$	20093.5	0.00319689
$525$	0	0
$526$	9.74420e6	1.53561
$527$	3.91140e6	0.613487
$528$	0	0
$529$	2.18530e6	0.339524
$530$	−7.18863e6	−1.11162
$531$	0	0
$532$	−69016.2	−0.0105724
$533$	124425.	0.0189710
$534$	0	0
$535$	−5.38335e6	−0.813145
$536$	−6.88707e6	−1.03543
$537$	0	0
$538$	8.11363e6	1.20854
$539$	0	0
$540$	0	0
$541$	7.00979e6	1.02970	0.514851	−	0.857279i	$-0.327847\pi$
$541$	7.00979e6	1.02970	0.514851	+	0.857279i	$0.327847\pi$
$542$	−6.35404e6	−0.929077
$543$	0	0
$544$	−136553.	−0.0197835
$545$	−1.18107e7	−1.70327
$546$	0	0
$547$	−1.88423e6	−0.269256	−0.134628	−	0.990896i	$-0.542984\pi$
$547$	−1.88423e6	−0.269256	−0.134628	+	0.990896i	$0.542984\pi$
$548$	−9803.82	−0.00139458
$549$	0	0
$550$	0	0
$551$	6.62781e6	0.930017
$552$	0	0
$553$	5.54230e6	0.770686
$554$	4.22887e6	0.585396
$555$	0	0
$556$	−81051.3	−0.0111192
$557$	−1.42698e6	−0.194885	−0.0974426	−	0.995241i	$-0.531066\pi$
$557$	−1.42698e6	−0.194885	−0.0974426	+	0.995241i	$0.531066\pi$
$558$	0	0
$559$	25576.1	0.00346182
$560$	9.13313e6	1.23069
$561$	0	0
$562$	−1.00843e7	−1.34681
$563$	−5.78916e6	−0.769741	−0.384871	−	0.922970i	$-0.625754\pi$
$563$	−5.78916e6	−0.769741	−0.384871	+	0.922970i	$0.625754\pi$
$564$	0	0
$565$	3.28176e6	0.432500
$566$	−6.01485e6	−0.789194
$567$	0	0
$568$	−4.40131e6	−0.572416
$569$	1.34088e7	1.73624	0.868119	−	0.496356i	$-0.165329\pi$
$569$	1.34088e7	1.73624	0.868119	+	0.496356i	$0.165329\pi$
$570$	0	0
$571$	7.12645e6	0.914709	0.457354	−	0.889285i	$-0.348797\pi$
$571$	7.12645e6	0.914709	0.457354	+	0.889285i	$0.348797\pi$
$572$	0	0
$573$	0	0
$574$	−7.01831e6	−0.889105
$575$	−4.45440e6	−0.561850
$576$	0	0
$577$	−7.93579e6	−0.992318	−0.496159	−	0.868232i	$-0.665257\pi$
$577$	−7.93579e6	−0.992318	−0.496159	+	0.868232i	$0.665257\pi$
$578$	1.45944e7	1.81705
$579$	0	0
$580$	30588.3	0.00377559
$581$	−2.45344e6	−0.301533
$582$	0	0
$583$	0	0
$584$	1.07196e7	1.30061
$585$	0	0
$586$	−4.28859e6	−0.515906
$587$	−9.35402e6	−1.12048	−0.560239	−	0.828331i	$-0.689291\pi$
$587$	−9.35402e6	−1.12048	−0.560239	+	0.828331i	$0.689291\pi$
$588$	0	0
$589$	−5.43482e6	−0.645501
$590$	−1.74661e7	−2.06569
$591$	0	0
$592$	9.30614e6	1.09135
$593$	−2.97324e6	−0.347210	−0.173605	−	0.984815i	$-0.555542\pi$
$593$	−2.97324e6	−0.347210	−0.173605	+	0.984815i	$0.555542\pi$
$594$	0	0
$595$	1.79611e7	2.07989
$596$	45654.6	0.00526464
$597$	0	0
$598$	218073.	0.0249372
$599$	9.24308e6	1.05257	0.526283	−	0.850309i	$-0.323585\pi$
$599$	9.24308e6	1.05257	0.526283	+	0.850309i	$0.323585\pi$
$600$	0	0
$601$	45710.0	0.00516208	0.00258104	−	0.999997i	$-0.499178\pi$
$601$	45710.0	0.00516208	0.00258104	+	0.999997i	$0.499178\pi$
$602$	−1.44264e6	−0.162244
$603$	0	0
$604$	7057.36	0.000787136 0
$605$	0	0
$606$	0	0
$607$	−1.22465e7	−1.34908	−0.674542	−	0.738237i	$-0.735659\pi$
$607$	−1.22465e7	−1.34908	−0.674542	+	0.738237i	$0.735659\pi$
$608$	189738.	0.0208159
$609$	0	0
$610$	1.31996e7	1.43628
$611$	151767.	0.0164466
$612$	0	0
$613$	1.24917e7	1.34268	0.671339	−	0.741150i	$-0.265719\pi$
$613$	1.24917e7	1.34268	0.671339	+	0.741150i	$0.265719\pi$
$614$	8.55794e6	0.916112
$615$	0	0
$616$	0	0
$617$	2.08647e6	0.220648	0.110324	−	0.993896i	$-0.464811\pi$
$617$	2.08647e6	0.220648	0.110324	+	0.993896i	$0.464811\pi$
$618$	0	0
$619$	−6.67616e6	−0.700325	−0.350163	−	0.936689i	$-0.613874\pi$
$619$	−6.67616e6	−0.700325	−0.350163	+	0.936689i	$0.613874\pi$
$620$	−25082.5	−0.00262054
$621$	0	0
$622$	−1.39010e7	−1.44068
$623$	1.08188e7	1.11676
$624$	0	0
$625$	−1.22050e7	−1.24979
$626$	−1.48522e7	−1.51480
$627$	0	0
$628$	−31695.1	−0.00320695
$629$	1.83013e7	1.84440
$630$	0	0
$631$	688276.	0.0688160	0.0344080	−	0.999408i	$-0.489045\pi$
$631$	688276.	0.0688160	0.0344080	+	0.999408i	$0.489045\pi$
$632$	−7.64077e6	−0.760930
$633$	0	0
$634$	−1.19104e6	−0.117680
$635$	7.54420e6	0.742470
$636$	0	0
$637$	−7040.23	−0.000687446 0
$638$	0	0
$639$	0	0
$640$	−1.25166e7	−1.20792
$641$	1.81701e6	0.174668	0.0873339	−	0.996179i	$-0.472165\pi$
$641$	1.81701e6	0.174668	0.0873339	+	0.996179i	$0.472165\pi$
$642$	0	0
$643$	−2.54263e6	−0.242525	−0.121262	−	0.992621i	$-0.538694\pi$
$643$	−2.54263e6	−0.242525	−0.121262	+	0.992621i	$0.538694\pi$
$644$	72854.9	0.00692220
$645$	0	0
$646$	−3.14061e7	−2.96096
$647$	1.60308e7	1.50555	0.752774	−	0.658279i	$-0.228715\pi$
$647$	1.60308e7	1.50555	0.752774	+	0.658279i	$0.228715\pi$
$648$	0	0
$649$	0	0
$650$	−112668.	−0.0104597
$651$	0	0
$652$	52540.6	0.00484034
$653$	1.17255e6	0.107609	0.0538045	−	0.998551i	$-0.482865\pi$
$653$	1.17255e6	0.107609	0.0538045	+	0.998551i	$0.482865\pi$
$654$	0	0
$655$	7.26593e6	0.661741
$656$	9.61867e6	0.872681
$657$	0	0
$658$	−8.56056e6	−0.770793
$659$	−1.52616e7	−1.36895	−0.684474	−	0.729037i	$-0.739968\pi$
$659$	−1.52616e7	−1.36895	−0.684474	+	0.729037i	$0.739968\pi$
$660$	0	0
$661$	3.21107e6	0.285855	0.142928	−	0.989733i	$-0.454348\pi$
$661$	3.21107e6	0.285855	0.142928	+	0.989733i	$0.454348\pi$
$662$	−1.65535e7	−1.46806
$663$	0	0
$664$	3.38237e6	0.297716
$665$	−2.49567e7	−2.18843
$666$	0	0
$667$	−6.99645e6	−0.608924
$668$	−88032.0	−0.00763307
$669$	0	0
$670$	−1.45777e7	−1.25459
$671$	0	0
$672$	0	0
$673$	−6.22661e6	−0.529925	−0.264962	−	0.964259i	$-0.585360\pi$
$673$	−6.22661e6	−0.529925	−0.264962	+	0.964259i	$0.585360\pi$
$674$	3.48669e6	0.295640
$675$	0	0
$676$	69925.0	0.00588526
$677$	−8.35757e6	−0.700823	−0.350412	−	0.936596i	$-0.613958\pi$
$677$	−8.35757e6	−0.700823	−0.350412	+	0.936596i	$0.613958\pi$
$678$	0	0
$679$	1.54276e7	1.28418
$680$	−2.47617e7	−2.05356
$681$	0	0
$682$	0	0
$683$	1.01680e7	0.834037	0.417018	−	0.908898i	$-0.363075\pi$
$683$	1.01680e7	0.834037	0.417018	+	0.908898i	$0.363075\pi$
$684$	0	0
$685$	−3.54512e6	−0.288672
$686$	−1.20862e7	−0.980568
$687$	0	0
$688$	1.97716e6	0.159247
$689$	−246327.	−0.0197681
$690$	0	0
$691$	1.74812e7	1.39276	0.696380	−	0.717673i	$-0.254793\pi$
$691$	1.74812e7	1.39276	0.696380	+	0.717673i	$0.254793\pi$
$692$	−36800.9	−0.00292142
$693$	0	0
$694$	−1.42394e7	−1.12226
$695$	−2.93086e7	−2.30162
$696$	0	0
$697$	1.89160e7	1.47485
$698$	3.75398e6	0.291644
$699$	0	0
$700$	−37640.8	−0.00290344
$701$	2.19442e7	1.68665	0.843323	−	0.537406i	$-0.180596\pi$
$701$	2.19442e7	1.68665	0.843323	+	0.537406i	$0.180596\pi$
$702$	0	0
$703$	−2.54294e7	−1.94065
$704$	0	0
$705$	0	0
$706$	−6.62498e6	−0.500233
$707$	−4.02369e6	−0.302744
$708$	0	0
$709$	2.29284e7	1.71301	0.856503	−	0.516142i	$-0.172632\pi$
$709$	2.29284e7	1.71301	0.856503	+	0.516142i	$0.172632\pi$
$710$	−9.31616e6	−0.693571
$711$	0	0
$712$	−1.49151e7	−1.10262
$713$	5.73711e6	0.422639
$714$	0	0
$715$	0	0
$716$	−109729.	−0.00799909
$717$	0	0
$718$	9.79261e6	0.708904
$719$	−2.22562e6	−0.160557	−0.0802783	−	0.996772i	$-0.525581\pi$
$719$	−2.22562e6	−0.160557	−0.0802783	+	0.996772i	$0.525581\pi$
$720$	0	0
$721$	−631002.	−0.0452056
$722$	2.96726e7	2.11842
$723$	0	0
$724$	−131793.	−0.00934429
$725$	3.61474e6	0.255407
$726$	0	0
$727$	197853.	0.0138837	0.00694187	−	0.999976i	$-0.497790\pi$
$727$	197853.	0.0138837	0.00694187	+	0.999976i	$0.497790\pi$
$728$	314812.	0.0220152
$729$	0	0
$730$	2.26899e7	1.57589
$731$	3.88825e6	0.269129
$732$	0	0
$733$	1.33968e7	0.920963	0.460481	−	0.887669i	$-0.347677\pi$
$733$	1.33968e7	0.920963	0.460481	+	0.887669i	$0.347677\pi$
$734$	−2.25097e7	−1.54216
$735$	0	0
$736$	−200292.	−0.0136291
$737$	0	0
$738$	0	0
$739$	714300.	0.0481138	0.0240569	−	0.999711i	$-0.492342\pi$
$739$	714300.	0.0481138	0.0240569	+	0.999711i	$0.492342\pi$
$740$	−117360.	−0.00787848
$741$	0	0
$742$	1.38943e7	0.926461
$743$	1.16716e7	0.775636	0.387818	−	0.921736i	$-0.373229\pi$
$743$	1.16716e7	0.775636	0.387818	+	0.921736i	$0.373229\pi$
$744$	0	0
$745$	1.65090e7	1.08976
$746$	2.03234e7	1.33705
$747$	0	0
$748$	0	0
$749$	1.04050e7	0.677703
$750$	0	0
$751$	−9.72938e6	−0.629485	−0.314743	−	0.949177i	$-0.601918\pi$
$751$	−9.72938e6	−0.629485	−0.314743	+	0.949177i	$0.601918\pi$
$752$	1.17323e7	0.756554
$753$	0	0
$754$	−176966.	−0.0113360
$755$	2.55198e6	0.162933
$756$	0	0
$757$	−2.44576e7	−1.55122	−0.775612	−	0.631210i	$-0.782559\pi$
$757$	−2.44576e7	−1.55122	−0.775612	+	0.631210i	$0.782559\pi$
$758$	1.20217e7	0.759962
$759$	0	0
$760$	3.44059e7	2.16072
$761$	1.64368e6	0.102886	0.0514430	−	0.998676i	$-0.483618\pi$
$761$	1.64368e6	0.102886	0.0514430	+	0.998676i	$0.483618\pi$
$762$	0	0
$763$	2.28280e7	1.41957
$764$	176711.	0.0109529
$765$	0	0
$766$	−1.11110e7	−0.684200
$767$	−598497.	−0.0367345
$768$	0	0
$769$	9.18833e6	0.560300	0.280150	−	0.959956i	$-0.409616\pi$
$769$	9.18833e6	0.560300	0.280150	+	0.959956i	$0.409616\pi$
$770$	0	0
$771$	0	0
$772$	158.137	9.54969e−6 0
$773$	1.90739e7	1.14813	0.574065	−	0.818810i	$-0.305366\pi$
$773$	1.90739e7	1.14813	0.574065	+	0.818810i	$0.305366\pi$
$774$	0	0
$775$	−2.96410e6	−0.177271
$776$	−2.12690e7	−1.26792
$777$	0	0
$778$	−5.60373e6	−0.331916
$779$	−2.62834e7	−1.55181
$780$	0	0
$781$	0	0
$782$	3.31529e7	1.93867
$783$	0	0
$784$	−544244.	−0.0316230
$785$	−1.14611e7	−0.663824
$786$	0	0
$787$	−6.23318e6	−0.358734	−0.179367	−	0.983782i	$-0.557405\pi$
$787$	−6.23318e6	−0.358734	−0.179367	+	0.983782i	$0.557405\pi$
$788$	−54033.4	−0.00309989
$789$	0	0
$790$	−1.61730e7	−0.921985
$791$	−6.34305e6	−0.360460
$792$	0	0
$793$	452303.	0.0255415
$794$	185714.	0.0104543
$795$	0	0
$796$	−161182.	−0.00901643
$797$	−7.79047e6	−0.434428	−0.217214	−	0.976124i	$-0.569697\pi$
$797$	−7.79047e6	−0.434428	−0.217214	+	0.976124i	$0.569697\pi$
$798$	0	0
$799$	2.30727e7	1.27859
$800$	103481.	0.00571659
$801$	0	0
$802$	2.44088e7	1.34001
$803$	0	0
$804$	0	0
$805$	2.63448e7	1.43286
$806$	145112.	0.00786805
$807$	0	0
$808$	5.54717e6	0.298912
$809$	7.94586e6	0.426844	0.213422	−	0.976960i	$-0.431539\pi$
$809$	7.94586e6	0.426844	0.213422	+	0.976960i	$0.431539\pi$
$810$	0	0
$811$	−9.91732e6	−0.529471	−0.264736	−	0.964321i	$-0.585285\pi$
$811$	−9.91732e6	−0.529471	−0.264736	+	0.964321i	$0.585285\pi$
$812$	−59121.6	−0.00314671
$813$	0	0
$814$	0	0
$815$	1.89990e7	1.00193
$816$	0	0
$817$	−5.40267e6	−0.283174
$818$	2.91294e7	1.52212
$819$	0	0
$820$	−121302.	−0.00629989
$821$	2.26042e7	1.17039	0.585196	−	0.810892i	$-0.301018\pi$
$821$	2.26042e7	1.17039	0.585196	+	0.810892i	$0.301018\pi$
$822$	0	0
$823$	−2.57251e6	−0.132391	−0.0661954	−	0.997807i	$-0.521086\pi$
$823$	−2.57251e6	−0.132391	−0.0661954	+	0.997807i	$0.521086\pi$
$824$	869917.	0.0446334
$825$	0	0
$826$	3.37588e7	1.72162
$827$	−9.04532e6	−0.459897	−0.229948	−	0.973203i	$-0.573856\pi$
$827$	−9.04532e6	−0.459897	−0.229948	+	0.973203i	$0.573856\pi$
$828$	0	0
$829$	−3.76134e7	−1.90089	−0.950443	−	0.310899i	$-0.899370\pi$
$829$	−3.76134e7	−1.90089	−0.950443	+	0.310899i	$0.899370\pi$
$830$	7.15939e6	0.360729
$831$	0	0
$832$	−433993.	−0.0217358
$833$	−1.07030e6	−0.0534435
$834$	0	0
$835$	−3.18329e7	−1.58001
$836$	0	0
$837$	0	0
$838$	−1.26002e7	−0.619824
$839$	−2.14562e7	−1.05232	−0.526160	−	0.850386i	$-0.676369\pi$
$839$	−2.14562e7	−1.05232	−0.526160	+	0.850386i	$0.676369\pi$
$840$	0	0
$841$	−1.48335e7	−0.723194
$842$	1.17101e7	0.569220
$843$	0	0
$844$	−47312.9	−0.00228625
$845$	2.52853e7	1.21822
$846$	0	0
$847$	0	0
$848$	−1.90423e7	−0.909347
$849$	0	0
$850$	−1.71286e7	−0.813156
$851$	2.68438e7	1.27063
$852$	0	0
$853$	−1.78088e7	−0.838036	−0.419018	−	0.907978i	$-0.637626\pi$
$853$	−1.78088e7	−0.838036	−0.419018	+	0.907978i	$0.637626\pi$
$854$	−2.55125e7	−1.19704
$855$	0	0
$856$	−1.43447e7	−0.669124
$857$	−1.19745e7	−0.556935	−0.278467	−	0.960446i	$-0.589826\pi$
$857$	−1.19745e7	−0.556935	−0.278467	+	0.960446i	$0.589826\pi$
$858$	0	0
$859$	1.62906e7	0.753277	0.376639	−	0.926360i	$-0.377080\pi$
$859$	1.62906e7	0.753277	0.376639	+	0.926360i	$0.377080\pi$
$860$	−24934.1	−0.00114960
$861$	0	0
$862$	2.11305e7	0.968595
$863$	−3.84892e7	−1.75919	−0.879594	−	0.475725i	$-0.842186\pi$
$863$	−3.84892e7	−1.75919	−0.879594	+	0.475725i	$0.842186\pi$
$864$	0	0
$865$	−1.33074e7	−0.604720
$866$	−2.54815e7	−1.15460
$867$	0	0
$868$	48479.9	0.00218405
$869$	0	0
$870$	0	0
$871$	−499523.	−0.0223105
$872$	−3.14713e7	−1.40160
$873$	0	0
$874$	−4.60654e7	−2.03984
$875$	1.44270e7	0.637025
$876$	0	0
$877$	−1.88587e7	−0.827968	−0.413984	−	0.910284i	$-0.635863\pi$
$877$	−1.88587e7	−0.827968	−0.413984	+	0.910284i	$0.635863\pi$
$878$	−6.04639e6	−0.264703
$879$	0	0
$880$	0	0
$881$	1.97073e7	0.855436	0.427718	−	0.903912i	$-0.359318\pi$
$881$	1.97073e7	0.855436	0.427718	+	0.903912i	$0.359318\pi$
$882$	0	0
$883$	2.57448e7	1.11119	0.555594	−	0.831454i	$-0.312491\pi$
$883$	2.57448e7	1.11119	0.555594	+	0.831454i	$0.312491\pi$
$884$	−4966.67	−0.000213764 0
$885$	0	0
$886$	6.38659e6	0.273328
$887$	2.86268e7	1.22170	0.610849	−	0.791747i	$-0.290828\pi$
$887$	2.86268e7	1.22170	0.610849	+	0.791747i	$0.290828\pi$
$888$	0	0
$889$	−1.45816e7	−0.618799
$890$	−3.15704e7	−1.33600
$891$	0	0
$892$	−211252.	−0.00888975
$893$	−3.20591e7	−1.34531
$894$	0	0
$895$	−3.96788e7	−1.65577
$896$	2.41923e7	1.00672
$897$	0	0
$898$	1.08098e7	0.447330
$899$	−4.65566e6	−0.192124
$900$	0	0
$901$	−3.74484e7	−1.53681
$902$	0	0
$903$	0	0
$904$	8.74471e6	0.355897
$905$	−4.76572e7	−1.93422
$906$	0	0
$907$	−2.70774e7	−1.09292	−0.546460	−	0.837485i	$-0.684025\pi$
$907$	−2.70774e7	−1.09292	−0.546460	+	0.837485i	$0.684025\pi$
$908$	−81044.7	−0.00326219
$909$	0	0
$910$	666355.	0.0266749
$911$	1.34155e7	0.535563	0.267781	−	0.963480i	$-0.413709\pi$
$911$	1.34155e7	0.535563	0.267781	+	0.963480i	$0.413709\pi$
$912$	0	0
$913$	0	0
$914$	−1.25704e7	−0.497717
$915$	0	0
$916$	−56824.4	−0.00223767
$917$	−1.40437e7	−0.551517
$918$	0	0
$919$	9.04796e6	0.353396	0.176698	−	0.984265i	$-0.443458\pi$
$919$	9.04796e6	0.353396	0.176698	+	0.984265i	$0.443458\pi$
$920$	−3.63196e7	−1.41472
$921$	0	0
$922$	3.22545e6	0.124958
$923$	−319230.	−0.0123339
$924$	0	0
$925$	−1.38690e7	−0.532954
$926$	−3.24455e7	−1.24345
$927$	0	0
$928$	162536.	0.00619556
$929$	−1.14651e7	−0.435850	−0.217925	−	0.975966i	$-0.569929\pi$
$929$	−1.14651e7	−0.435850	−0.217925	+	0.975966i	$0.569929\pi$
$930$	0	0
$931$	1.48717e6	0.0562324
$932$	30690.2	0.00115734
$933$	0	0
$934$	3.26907e7	1.22619
$935$	0	0
$936$	0	0
$937$	3.07135e7	1.14283	0.571413	−	0.820662i	$-0.306395\pi$
$937$	3.07135e7	1.14283	0.571413	+	0.820662i	$0.306395\pi$
$938$	2.81760e7	1.04562
$939$	0	0
$940$	−147957.	−0.00546157
$941$	−1.70045e7	−0.626023	−0.313012	−	0.949749i	$-0.601338\pi$
$941$	−1.70045e7	−0.626023	−0.313012	+	0.949749i	$0.601338\pi$
$942$	0	0
$943$	2.77453e7	1.01604
$944$	−4.62667e7	−1.68981
$945$	0	0
$946$	0	0
$947$	1.03184e7	0.373885	0.186943	−	0.982371i	$-0.440142\pi$
$947$	1.03184e7	0.373885	0.186943	+	0.982371i	$0.440142\pi$
$948$	0	0
$949$	777499.	0.0280243
$950$	2.37999e7	0.855590
$951$	0	0
$952$	4.78598e7	1.71151
$953$	−4.67887e7	−1.66882	−0.834408	−	0.551147i	$-0.814190\pi$
$953$	−4.67887e7	−1.66882	−0.834408	+	0.551147i	$0.814190\pi$
$954$	0	0
$955$	6.38999e7	2.26721
$956$	−162086.	−0.00573590
$957$	0	0
$958$	−1.38312e7	−0.486906
$959$	6.85208e6	0.240589
$960$	0	0
$961$	−2.48115e7	−0.866651
$962$	678978.	0.0236547
$963$	0	0
$964$	227374.	0.00788039
$965$	57183.2	0.00197674
$966$	0	0
$967$	5.92997e6	0.203932	0.101966	−	0.994788i	$-0.467487\pi$
$967$	5.92997e6	0.203932	0.101966	+	0.994788i	$0.467487\pi$
$968$	0	0
$969$	0	0
$970$	−4.50195e7	−1.53628
$971$	2.93211e7	0.998004	0.499002	−	0.866601i	$-0.333700\pi$
$971$	2.93211e7	0.998004	0.499002	+	0.866601i	$0.333700\pi$
$972$	0	0
$973$	5.66483e7	1.91825
$974$	−2.17217e6	−0.0733664
$975$	0	0
$976$	3.49652e7	1.17493
$977$	5.55590e6	0.186216	0.0931082	−	0.995656i	$-0.470320\pi$
$977$	5.55590e6	0.186216	0.0931082	+	0.995656i	$0.470320\pi$
$978$	0	0
$979$	0	0
$980$	6863.50	0.000228287 0
$981$	0	0
$982$	−1.63518e7	−0.541111
$983$	−8.06213e6	−0.266113	−0.133057	−	0.991108i	$-0.542479\pi$
$983$	−8.06213e6	−0.266113	−0.133057	+	0.991108i	$0.542479\pi$
$984$	0	0
$985$	−1.95388e7	−0.641663
$986$	−2.69035e7	−0.881286
$987$	0	0
$988$	6901.11	0.000224919 0
$989$	5.70316e6	0.185407
$990$	0	0
$991$	−2.89898e7	−0.937693	−0.468846	−	0.883280i	$-0.655330\pi$
$991$	−2.89898e7	−0.937693	−0.468846	+	0.883280i	$0.655330\pi$
$992$	−133280.	−0.00430018
$993$	0	0
$994$	1.80064e7	0.578046
$995$	−5.82845e7	−1.86636
$996$	0	0
$997$	−1.02351e7	−0.326103	−0.163052	−	0.986618i	$-0.552134\pi$
$997$	−1.02351e7	−0.326103	−0.163052	+	0.986618i	$0.552134\pi$
$998$	−2.52431e7	−0.802261
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bh.1.8		10
3.2	odd	2		363.6.a.u.1.3		10
11.3	even	5		99.6.f.c.64.4		20
11.4	even	5		99.6.f.c.82.4		20
11.10	odd	2		1089.6.a.bl.1.3		10
33.14	odd	10		33.6.e.a.31.2	yes	20
33.26	odd	10		33.6.e.a.16.2	✓	20
33.32	even	2		363.6.a.q.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.2	✓	20	33.26	odd	10
33.6.e.a.31.2	yes	20	33.14	odd	10
99.6.f.c.64.4		20	11.3	even	5
99.6.f.c.82.4		20	11.4	even	5
363.6.a.q.1.8		10	33.32	even	2
363.6.a.u.1.3		10	3.2	odd	2
1089.6.a.bh.1.8		10	1.1	even	1	trivial
1089.6.a.bl.1.3		10	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+5.64018 q^{2} -0.188416 q^{4} -68.1325 q^{5} +131.688 q^{7} -181.548 q^{8} +O(q^{10})\)	\(q+5.64018 q^{2} -0.188416 q^{4} -68.1325 q^{5} +131.688 q^{7} -181.548 q^{8} -384.279 q^{10} -13.1678 q^{13} +742.742 q^{14} -1017.94 q^{16} -2001.86 q^{17} +2781.55 q^{19} +12.8373 q^{20} -2936.26 q^{23} +1517.03 q^{25} -74.2688 q^{26} -24.8121 q^{28} +2382.77 q^{29} -1953.88 q^{31} +68.2131 q^{32} -11290.8 q^{34} -8972.21 q^{35} -9142.17 q^{37} +15688.4 q^{38} +12369.3 q^{40} -9449.20 q^{41} -1942.32 q^{43} -16561.0 q^{46} -11525.6 q^{47} +534.655 q^{49} +8556.33 q^{50} +2.48103 q^{52} +18706.8 q^{53} -23907.7 q^{56} +13439.3 q^{58} +45451.5 q^{59} -34349.1 q^{61} -11020.2 q^{62} +32958.7 q^{64} +897.155 q^{65} +37935.2 q^{67} +377.183 q^{68} -50604.8 q^{70} +24243.2 q^{71} -59045.4 q^{73} -51563.4 q^{74} -524.090 q^{76} +42086.7 q^{79} +69354.4 q^{80} -53295.1 q^{82} -18630.7 q^{83} +136392. q^{85} -10955.0 q^{86} +82154.9 q^{89} -1734.04 q^{91} +553.240 q^{92} -65006.5 q^{94} -189514. q^{95} +117153. q^{97} +3015.55 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8}+O(q^{10}) \) 10 * q - 9 * q^2 + 193 * q^4 - 11 * q^5 + 470 * q^7 - 324 * q^8	\( 10 q - 9 q^{2} + 193 q^{4} - 11 q^{5} + 470 q^{7} - 324 q^{8} + 976 q^{10} + 2308 q^{13} - 540 q^{14} + 2801 q^{16} - 3093 q^{17} + 5305 q^{19} + 2229 q^{20} - 700 q^{23} + 8381 q^{25} + 15559 q^{26} + 24656 q^{28} - 10392 q^{29} - 101 q^{31} - 34557 q^{32} - 4542 q^{34} - 19867 q^{35} - 1284 q^{37} + 35769 q^{38} + 66596 q^{40} - 17944 q^{41} + 31812 q^{43} + 36417 q^{46} - 8787 q^{47} - 23810 q^{49} + 910 q^{50} + 51663 q^{52} + 3261 q^{53} - 84819 q^{56} + 53125 q^{58} - 49375 q^{59} + 63175 q^{61} - 28399 q^{62} + 124764 q^{64} + 14105 q^{65} + 5365 q^{67} + 53313 q^{68} + 40297 q^{70} + 236675 q^{71} + 200912 q^{73} + 180329 q^{74} + 39606 q^{76} + 210802 q^{79} - 270298 q^{80} - 369223 q^{82} + 178968 q^{83} + 107352 q^{85} - 465999 q^{86} - 90816 q^{89} + 7500 q^{91} + 136407 q^{92} + 71890 q^{94} - 335807 q^{95} + 271521 q^{97} + 75285 q^{98}+O(q^{100}) \) 10 * q - 9 * q^2 + 193 * q^4 - 11 * q^5 + 470 * q^7 - 324 * q^8 + 976 * q^10 + 2308 * q^13 - 540 * q^14 + 2801 * q^16 - 3093 * q^17 + 5305 * q^19 + 2229 * q^20 - 700 * q^23 + 8381 * q^25 + 15559 * q^26 + 24656 * q^28 - 10392 * q^29 - 101 * q^31 - 34557 * q^32 - 4542 * q^34 - 19867 * q^35 - 1284 * q^37 + 35769 * q^38 + 66596 * q^40 - 17944 * q^41 + 31812 * q^43 + 36417 * q^46 - 8787 * q^47 - 23810 * q^49 + 910 * q^50 + 51663 * q^52 + 3261 * q^53 - 84819 * q^56 + 53125 * q^58 - 49375 * q^59 + 63175 * q^61 - 28399 * q^62 + 124764 * q^64 + 14105 * q^65 + 5365 * q^67 + 53313 * q^68 + 40297 * q^70 + 236675 * q^71 + 200912 * q^73 + 180329 * q^74 + 39606 * q^76 + 210802 * q^79 - 270298 * q^80 - 369223 * q^82 + 178968 * q^83 + 107352 * q^85 - 465999 * q^86 - 90816 * q^89 + 7500 * q^91 + 136407 * q^92 + 71890 * q^94 - 335807 * q^95 + 271521 * q^97 + 75285 * q^98

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