LMFDB - Embedded newform 1089.6.a.bk.1.1

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} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + \cdots + 5059564 $ x^10 - 3x^9 - 228x^8 + 523x^7 + 17396x^6 - 31445x^5 - 508100x^4 + 960757x^3 + 4870759x^2 - 11540360*x + 5059564
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			1.1
Root			$10.6217$ 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)

$f(q)$	$=$	$q-9.62167 q^{2} +60.5765 q^{4} +59.5053 q^{5} +30.5377 q^{7} -274.954 q^{8} +O(q^{10})$	$q-9.62167 q^{2} +60.5765 q^{4} +59.5053 q^{5} +30.5377 q^{7} -274.954 q^{8} -572.540 q^{10} +569.547 q^{13} -293.823 q^{14} +707.065 q^{16} +2001.45 q^{17} +1610.27 q^{19} +3604.62 q^{20} +1214.78 q^{23} +415.881 q^{25} -5479.99 q^{26} +1849.87 q^{28} +7599.80 q^{29} -7260.02 q^{31} +1995.37 q^{32} -19257.3 q^{34} +1817.15 q^{35} -1314.81 q^{37} -15493.5 q^{38} -16361.2 q^{40} -5840.24 q^{41} +6357.22 q^{43} -11688.2 q^{46} +30244.8 q^{47} -15874.5 q^{49} -4001.47 q^{50} +34501.2 q^{52} -7151.62 q^{53} -8396.45 q^{56} -73122.8 q^{58} +22176.0 q^{59} +24997.2 q^{61} +69853.5 q^{62} -41824.9 q^{64} +33891.1 q^{65} +15543.8 q^{67} +121241. q^{68} -17484.1 q^{70} +64626.1 q^{71} +44122.0 q^{73} +12650.7 q^{74} +97544.6 q^{76} -17446.2 q^{79} +42074.1 q^{80} +56192.8 q^{82} -118895. q^{83} +119097. q^{85} -61167.1 q^{86} +6612.00 q^{89} +17392.6 q^{91} +73587.2 q^{92} -291005. q^{94} +95819.6 q^{95} +127563. q^{97} +152739. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8}+O(q^{10}) $ 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8	$ 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8} - 212 q^{10} - 1016 q^{13} - 1566 q^{14} + 2361 q^{16} + 1669 q^{17} + 2929 q^{19} + 10189 q^{20} - 4070 q^{23} + 2425 q^{25} - 8481 q^{26} + 3272 q^{28} + 11940 q^{29} - 16085 q^{31} + 2313 q^{32} + 8270 q^{34} - 6987 q^{35} + 16136 q^{37} - 10721 q^{38} + 9332 q^{40} + 16278 q^{41} - 10844 q^{43} - 25995 q^{46} + 22411 q^{47} + 75150 q^{49} - 738 q^{50} - 8677 q^{52} + 27511 q^{53} - 84447 q^{56} + 16853 q^{58} + 39641 q^{59} - 3509 q^{61} + 227845 q^{62} - 22980 q^{64} + 67097 q^{65} + 10089 q^{67} + 273621 q^{68} - 38919 q^{70} - 60681 q^{71} - 133740 q^{73} + 317933 q^{74} + 23434 q^{76} - 12386 q^{79} + 289014 q^{80} + 385033 q^{82} + 187242 q^{83} - 191504 q^{85} + 43793 q^{86} - 102746 q^{89} - 435248 q^{91} - 30867 q^{92} - 404734 q^{94} + 648147 q^{95} - 120631 q^{97} + 1148087 q^{98}+O(q^{100}) $ 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8 - 212 * q^10 - 1016 * q^13 - 1566 * q^14 + 2361 * q^16 + 1669 * q^17 + 2929 * q^19 + 10189 * q^20 - 4070 * q^23 + 2425 * q^25 - 8481 * q^26 + 3272 * q^28 + 11940 * q^29 - 16085 * q^31 + 2313 * q^32 + 8270 * q^34 - 6987 * q^35 + 16136 * q^37 - 10721 * q^38 + 9332 * q^40 + 16278 * q^41 - 10844 * q^43 - 25995 * q^46 + 22411 * q^47 + 75150 * q^49 - 738 * q^50 - 8677 * q^52 + 27511 * q^53 - 84447 * q^56 + 16853 * q^58 + 39641 * q^59 - 3509 * q^61 + 227845 * q^62 - 22980 * q^64 + 67097 * q^65 + 10089 * q^67 + 273621 * q^68 - 38919 * q^70 - 60681 * q^71 - 133740 * q^73 + 317933 * q^74 + 23434 * q^76 - 12386 * q^79 + 289014 * q^80 + 385033 * q^82 + 187242 * q^83 - 191504 * q^85 + 43793 * q^86 - 102746 * q^89 - 435248 * q^91 - 30867 * q^92 - 404734 * q^94 + 648147 * q^95 - 120631 * q^97 + 1148087 * 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 $

$2$	−9.62167	−1.70089	−0.850443	−	0.526067i	$-0.823666\pi$
$2$	−9.62167	−1.70089	−0.850443	+	0.526067i	$0.823666\pi$
$3$	0	0
$3$	0	0
$4$	60.5765	1.89302
$5$	59.5053	1.06446	0.532232	−	0.846599i	$-0.321354\pi$
$5$	59.5053	1.06446	0.532232	+	0.846599i	$0.321354\pi$
$6$	0	0
$7$	30.5377	0.235554	0.117777	−	0.993040i	$-0.462423\pi$
$7$	30.5377	0.235554	0.117777	+	0.993040i	$0.462423\pi$
$8$	−274.954	−1.51892
$9$	0	0
$10$	−572.540	−1.81053
$11$	0	0
$11$	0	0
$12$	0	0
$13$	569.547	0.934698	0.467349	−	0.884073i	$-0.345209\pi$
$13$	569.547	0.934698	0.467349	+	0.884073i	$0.345209\pi$
$14$	−293.823	−0.400651
$15$	0	0
$16$	707.065	0.690493
$17$	2001.45	1.67966	0.839831	−	0.542849i	$-0.182654\pi$
$17$	2001.45	1.67966	0.839831	+	0.542849i	$0.182654\pi$
$18$	0	0
$19$	1610.27	1.02333	0.511664	−	0.859186i	$-0.329029\pi$
$19$	1610.27	1.02333	0.511664	+	0.859186i	$0.329029\pi$
$20$	3604.62	2.01505
$21$	0	0
$22$	0	0
$23$	1214.78	0.478827	0.239413	−	0.970918i	$-0.423045\pi$
$23$	1214.78	0.478827	0.239413	+	0.970918i	$0.423045\pi$
$24$	0	0
$25$	415.881	0.133082
$26$	−5479.99	−1.58981
$27$	0	0
$28$	1849.87	0.445908
$29$	7599.80	1.67806	0.839030	−	0.544086i	$-0.183123\pi$
$29$	7599.80	1.67806	0.839030	+	0.544086i	$0.183123\pi$
$30$	0	0
$31$	−7260.02	−1.35686	−0.678428	−	0.734667i	$-0.737338\pi$
$31$	−7260.02	−1.35686	−0.678428	+	0.734667i	$0.737338\pi$
$32$	1995.37	0.344468
$33$	0	0
$34$	−19257.3	−2.85691
$35$	1817.15	0.250739
$36$	0	0
$37$	−1314.81	−0.157892	−0.0789460	−	0.996879i	$-0.525155\pi$
$37$	−1314.81	−0.157892	−0.0789460	+	0.996879i	$0.525155\pi$
$38$	−15493.5	−1.74057
$39$	0	0
$40$	−16361.2	−1.61683
$41$	−5840.24	−0.542589	−0.271294	−	0.962496i	$-0.587452\pi$
$41$	−5840.24	−0.542589	−0.271294	+	0.962496i	$0.587452\pi$
$42$	0	0
$43$	6357.22	0.524320	0.262160	−	0.965024i	$-0.415565\pi$
$43$	6357.22	0.524320	0.262160	+	0.965024i	$0.415565\pi$
$44$	0	0
$45$	0	0
$46$	−11688.2	−0.814430
$47$	30244.8	1.99713	0.998564	−	0.0535638i	$-0.0170581\pi$
$47$	30244.8	1.99713	0.998564	+	0.0535638i	$0.0170581\pi$
$48$	0	0
$49$	−15874.5	−0.944514
$50$	−4001.47	−0.226357
$51$	0	0
$52$	34501.2	1.76940
$53$	−7151.62	−0.349716	−0.174858	−	0.984594i	$-0.555947\pi$
$53$	−7151.62	−0.349716	−0.174858	+	0.984594i	$0.555947\pi$
$54$	0	0
$55$	0	0
$56$	−8396.45	−0.357788
$57$	0	0
$58$	−73122.8	−2.85419
$59$	22176.0	0.829380	0.414690	−	0.909963i	$-0.363890\pi$
$59$	22176.0	0.829380	0.414690	+	0.909963i	$0.363890\pi$
$60$	0	0
$61$	24997.2	0.860134	0.430067	−	0.902797i	$-0.358490\pi$
$61$	24997.2	0.860134	0.430067	+	0.902797i	$0.358490\pi$
$62$	69853.5	2.30786
$63$	0	0
$64$	−41824.9	−1.27639
$65$	33891.1	0.994951
$66$	0	0
$67$	15543.8	0.423028	0.211514	−	0.977375i	$-0.432161\pi$
$67$	15543.8	0.423028	0.211514	+	0.977375i	$0.432161\pi$
$68$	121241.	3.17963
$69$	0	0
$70$	−17484.1	−0.426478
$71$	64626.1	1.52147	0.760733	−	0.649065i	$-0.224840\pi$
$71$	64626.1	1.52147	0.760733	+	0.649065i	$0.224840\pi$
$72$	0	0
$73$	44122.0	0.969054	0.484527	−	0.874776i	$-0.338992\pi$
$73$	44122.0	0.969054	0.484527	+	0.874776i	$0.338992\pi$
$74$	12650.7	0.268556
$75$	0	0
$76$	97544.6	1.93718
$77$	0	0
$78$	0	0
$79$	−17446.2	−0.314510	−0.157255	−	0.987558i	$-0.550264\pi$
$79$	−17446.2	−0.314510	−0.157255	+	0.987558i	$0.550264\pi$
$80$	42074.1	0.735005
$81$	0	0
$82$	56192.8	0.922882
$83$	−118895.	−1.89438	−0.947191	−	0.320671i	$-0.896092\pi$
$83$	−118895.	−1.89438	−0.947191	+	0.320671i	$0.896092\pi$
$84$	0	0
$85$	119097.	1.78794
$86$	−61167.1	−0.891808
$87$	0	0
$88$	0	0
$89$	6612.00	0.0884826	0.0442413	−	0.999021i	$-0.485913\pi$
$89$	6612.00	0.0884826	0.0442413	+	0.999021i	$0.485913\pi$
$90$	0	0
$91$	17392.6	0.220172
$92$	73587.2	0.906427
$93$	0	0
$94$	−291005.	−3.39689
$95$	95819.6	1.08929
$96$	0	0
$97$	127563.	1.37656	0.688282	−	0.725443i	$-0.258365\pi$
$97$	127563.	1.37656	0.688282	+	0.725443i	$0.258365\pi$
$98$	152739.	1.60651
$99$	0	0
$100$	25192.6	0.251926
$101$	74920.8	0.730801	0.365401	−	0.930850i	$-0.380932\pi$
$101$	74920.8	0.730801	0.365401	+	0.930850i	$0.380932\pi$
$102$	0	0
$103$	52734.9	0.489784	0.244892	−	0.969550i	$-0.421247\pi$
$103$	52734.9	0.489784	0.244892	+	0.969550i	$0.421247\pi$
$104$	−156599.	−1.41973
$105$	0	0
$106$	68810.6	0.594827
$107$	123265.	1.04083	0.520415	−	0.853913i	$-0.325777\pi$
$107$	123265.	1.04083	0.520415	+	0.853913i	$0.325777\pi$
$108$	0	0
$109$	−134499.	−1.08431	−0.542153	−	0.840280i	$-0.682391\pi$
$109$	−134499.	−1.08431	−0.542153	+	0.840280i	$0.682391\pi$
$110$	0	0
$111$	0	0
$112$	21592.1	0.162649
$113$	−33674.1	−0.248085	−0.124042	−	0.992277i	$-0.539586\pi$
$113$	−33674.1	−0.248085	−0.124042	+	0.992277i	$0.539586\pi$
$114$	0	0
$115$	72285.9	0.509694
$116$	460370.	3.17659
$117$	0	0
$118$	−213370.	−1.41068
$119$	61119.5	0.395651
$120$	0	0
$121$	0	0
$122$	−240514.	−1.46299
$123$	0	0
$124$	−439787.	−2.56855
$125$	−161207.	−0.922802
$126$	0	0
$127$	−234634.	−1.29087	−0.645433	−	0.763817i	$-0.723323\pi$
$127$	−234634.	−1.29087	−0.645433	+	0.763817i	$0.723323\pi$
$128$	338573.	1.82653
$129$	0	0
$130$	−326089.	−1.69230
$131$	−204823.	−1.04280	−0.521399	−	0.853313i	$-0.674590\pi$
$131$	−204823.	−1.04280	−0.521399	+	0.853313i	$0.674590\pi$
$132$	0	0
$133$	49173.9	0.241049
$134$	−149557.	−0.719523
$135$	0	0
$136$	−550305.	−2.55127
$137$	48225.3	0.219520	0.109760	−	0.993958i	$-0.464992\pi$
$137$	48225.3	0.219520	0.109760	+	0.993958i	$0.464992\pi$
$138$	0	0
$139$	117164.	0.514348	0.257174	−	0.966365i	$-0.417209\pi$
$139$	117164.	0.514348	0.257174	+	0.966365i	$0.417209\pi$
$140$	110077.	0.474653
$141$	0	0
$142$	−621811.	−2.58784
$143$	0	0
$144$	0	0
$145$	452229.	1.78623
$146$	−424527.	−1.64825
$147$	0	0
$148$	−79646.8	−0.298892
$149$	−206594.	−0.762346	−0.381173	−	0.924504i	$-0.624480\pi$
$149$	−206594.	−0.762346	−0.381173	+	0.924504i	$0.624480\pi$
$150$	0	0
$151$	100636.	0.359179	0.179590	−	0.983742i	$-0.442523\pi$
$151$	100636.	0.359179	0.179590	+	0.983742i	$0.442523\pi$
$152$	−442750.	−1.55435
$153$	0	0
$154$	0	0
$155$	−432010.	−1.44432
$156$	0	0
$157$	80258.3	0.259861	0.129930	−	0.991523i	$-0.458525\pi$
$157$	80258.3	0.259861	0.129930	+	0.991523i	$0.458525\pi$
$158$	167862.	0.534945
$159$	0	0
$160$	118735.	0.366673
$161$	37096.6	0.112790
$162$	0	0
$163$	−19082.5	−0.0562557	−0.0281278	−	0.999604i	$-0.508955\pi$
$163$	−19082.5	−0.0562557	−0.0281278	+	0.999604i	$0.508955\pi$
$164$	−353781.	−1.02713
$165$	0	0
$166$	1.14397e6	3.22213
$167$	438717.	1.21729	0.608644	−	0.793443i	$-0.291714\pi$
$167$	438717.	1.21729	0.608644	+	0.793443i	$0.291714\pi$
$168$	0	0
$169$	−46909.3	−0.126340
$170$	−1.14591e6	−3.04108
$171$	0	0
$172$	385098.	0.992545
$173$	−405525.	−1.03015	−0.515077	−	0.857144i	$-0.672237\pi$
$173$	−405525.	−1.03015	−0.515077	+	0.857144i	$0.672237\pi$
$174$	0	0
$175$	12700.0	0.0313480
$176$	0	0
$177$	0	0
$178$	−63618.5	−0.150499
$179$	−363321.	−0.847535	−0.423768	−	0.905771i	$-0.639293\pi$
$179$	−363321.	−0.847535	−0.423768	+	0.905771i	$0.639293\pi$
$180$	0	0
$181$	−626132.	−1.42059	−0.710297	−	0.703903i	$-0.751439\pi$
$181$	−626132.	−1.42059	−0.710297	+	0.703903i	$0.751439\pi$
$182$	−167346.	−0.374488
$183$	0	0
$184$	−334009.	−0.727299
$185$	−78238.4	−0.168070
$186$	0	0
$187$	0	0
$188$	1.83212e6	3.78060
$189$	0	0
$190$	−921945.	−1.85277
$191$	−488805.	−0.969510	−0.484755	−	0.874650i	$-0.661091\pi$
$191$	−488805.	−0.969510	−0.484755	+	0.874650i	$0.661091\pi$
$192$	0	0
$193$	−135003.	−0.260886	−0.130443	−	0.991456i	$-0.541640\pi$
$193$	−135003.	−0.260886	−0.130443	+	0.991456i	$0.541640\pi$
$194$	−1.22737e6	−2.34138
$195$	0	0
$196$	−961619.	−1.78798
$197$	176847.	0.324662	0.162331	−	0.986736i	$-0.448099\pi$
$197$	176847.	0.324662	0.162331	+	0.986736i	$0.448099\pi$
$198$	0	0
$199$	433048.	0.775181	0.387591	−	0.921832i	$-0.373307\pi$
$199$	433048.	0.775181	0.387591	+	0.921832i	$0.373307\pi$
$200$	−114348.	−0.202141
$201$	0	0
$202$	−720864.	−1.24301
$203$	232080.	0.395274
$204$	0	0
$205$	−347525.	−0.577566
$206$	−507397.	−0.833067
$207$	0	0
$208$	402707.	0.645403
$209$	0	0
$210$	0	0
$211$	−134526.	−0.208018	−0.104009	−	0.994576i	$-0.533167\pi$
$211$	−134526.	−0.208018	−0.104009	+	0.994576i	$0.533167\pi$
$212$	−433220.	−0.662017
$213$	0	0
$214$	−1.18601e6	−1.77033
$215$	378288.	0.558119
$216$	0	0
$217$	−221704.	−0.319613
$218$	1.29410e6	1.84428
$219$	0	0
$220$	0	0
$221$	1.13992e6	1.56998
$222$	0	0
$223$	−762824.	−1.02722	−0.513609	−	0.858025i	$-0.671692\pi$
$223$	−762824.	−1.02722	−0.513609	+	0.858025i	$0.671692\pi$
$224$	60934.0	0.0811409
$225$	0	0
$226$	324001.	0.421964
$227$	108472.	0.139718	0.0698590	−	0.997557i	$-0.477745\pi$
$227$	108472.	0.139718	0.0698590	+	0.997557i	$0.477745\pi$
$228$	0	0
$229$	−523114.	−0.659185	−0.329593	−	0.944123i	$-0.606911\pi$
$229$	−523114.	−0.659185	−0.329593	+	0.944123i	$0.606911\pi$
$230$	−695511.	−0.866931
$231$	0	0
$232$	−2.08959e6	−2.54884
$233$	−219358.	−0.264706	−0.132353	−	0.991203i	$-0.542253\pi$
$233$	−219358.	−0.264706	−0.132353	+	0.991203i	$0.542253\pi$
$234$	0	0
$235$	1.79973e6	2.12587
$236$	1.34335e6	1.57003
$237$	0	0
$238$	−588072.	−0.672958
$239$	−482553.	−0.546449	−0.273225	−	0.961950i	$-0.588090\pi$
$239$	−482553.	−0.546449	−0.273225	+	0.961950i	$0.588090\pi$
$240$	0	0
$241$	−472230.	−0.523734	−0.261867	−	0.965104i	$-0.584338\pi$
$241$	−472230.	−0.523734	−0.261867	+	0.965104i	$0.584338\pi$
$242$	0	0
$243$	0	0
$244$	1.51424e6	1.62825
$245$	−944614.	−1.00540
$246$	0	0
$247$	917125.	0.956502
$248$	1.99617e6	2.06095
$249$	0	0
$250$	1.55108e6	1.56958
$251$	−366515.	−0.367204	−0.183602	−	0.983001i	$-0.558776\pi$
$251$	−366515.	−0.367204	−0.183602	+	0.983001i	$0.558776\pi$
$252$	0	0
$253$	0	0
$254$	2.25757e6	2.19562
$255$	0	0
$256$	−1.91924e6	−1.83033
$257$	−1.68341e6	−1.58986	−0.794928	−	0.606704i	$-0.792491\pi$
$257$	−1.68341e6	−1.58986	−0.794928	+	0.606704i	$0.792491\pi$
$258$	0	0
$259$	−40151.4	−0.0371921
$260$	2.05300e6	1.88346
$261$	0	0
$262$	1.97074e6	1.77368
$263$	1.52341e6	1.35808	0.679042	−	0.734100i	$-0.262396\pi$
$263$	1.52341e6	1.35808	0.679042	+	0.734100i	$0.262396\pi$
$264$	0	0
$265$	−425560.	−0.372259
$266$	−473135.	−0.409997
$267$	0	0
$268$	941587.	0.800799
$269$	68695.3	0.0578824	0.0289412	−	0.999581i	$-0.490786\pi$
$269$	68695.3	0.0578824	0.0289412	+	0.999581i	$0.490786\pi$
$270$	0	0
$271$	1.18801e6	0.982643	0.491322	−	0.870978i	$-0.336514\pi$
$271$	1.18801e6	0.982643	0.491322	+	0.870978i	$0.336514\pi$
$272$	1.41515e6	1.15980
$273$	0	0
$274$	−464008.	−0.373378
$275$	0	0
$276$	0	0
$277$	−1.50906e6	−1.18170	−0.590848	−	0.806783i	$-0.701207\pi$
$277$	−1.50906e6	−1.18170	−0.590848	+	0.806783i	$0.701207\pi$
$278$	−1.12731e6	−0.874848
$279$	0	0
$280$	−499633.	−0.380852
$281$	−1.40761e6	−1.06345	−0.531726	−	0.846916i	$-0.678456\pi$
$281$	−1.40761e6	−1.06345	−0.531726	+	0.846916i	$0.678456\pi$
$282$	0	0
$283$	322173.	0.239124	0.119562	−	0.992827i	$-0.461851\pi$
$283$	322173.	0.239124	0.119562	+	0.992827i	$0.461851\pi$
$284$	3.91482e6	2.88016
$285$	0	0
$286$	0	0
$287$	−178347.	−0.127809
$288$	0	0
$289$	2.58593e6	1.82126
$290$	−4.35119e6	−3.03818
$291$	0	0
$292$	2.67276e6	1.83443
$293$	1.41547e6	0.963234	0.481617	−	0.876382i	$-0.340050\pi$
$293$	1.41547e6	0.963234	0.481617	+	0.876382i	$0.340050\pi$
$294$	0	0
$295$	1.31959e6	0.882844
$296$	361513.	0.239825
$297$	0	0
$298$	1.98778e6	1.29666
$299$	691875.	0.447558
$300$	0	0
$301$	194135.	0.123506
$302$	−968286.	−0.610923
$303$	0	0
$304$	1.13857e6	0.706601
$305$	1.48746e6	0.915581
$306$	0	0
$307$	637505.	0.386045	0.193022	−	0.981194i	$-0.438171\pi$
$307$	637505.	0.386045	0.193022	+	0.981194i	$0.438171\pi$
$308$	0	0
$309$	0	0
$310$	4.15666e6	2.45663
$311$	1.11700e6	0.654865	0.327433	−	0.944875i	$-0.393817\pi$
$311$	1.11700e6	0.654865	0.327433	+	0.944875i	$0.393817\pi$
$312$	0	0
$313$	−2.98340e6	−1.72128	−0.860638	−	0.509217i	$-0.829935\pi$
$313$	−2.98340e6	−1.72128	−0.860638	+	0.509217i	$0.829935\pi$
$314$	−772219.	−0.441994
$315$	0	0
$316$	−1.05683e6	−0.595372
$317$	−1.95592e6	−1.09321	−0.546605	−	0.837391i	$-0.684080\pi$
$317$	−1.95592e6	−1.09321	−0.546605	+	0.837391i	$0.684080\pi$
$318$	0	0
$319$	0	0
$320$	−2.48880e6	−1.35867
$321$	0	0
$322$	−356931.	−0.191842
$323$	3.22287e6	1.71884
$324$	0	0
$325$	236864.	0.124391
$326$	183606.	0.0956846
$327$	0	0
$328$	1.60580e6	0.824149
$329$	923606.	0.470432
$330$	0	0
$331$	−340808.	−0.170978	−0.0854888	−	0.996339i	$-0.527245\pi$
$331$	−340808.	−0.170978	−0.0854888	+	0.996339i	$0.527245\pi$
$332$	−7.20223e6	−3.58609
$333$	0	0
$334$	−4.22119e6	−2.07047
$335$	924936.	0.450298
$336$	0	0
$337$	4.02360e6	1.92992	0.964962	−	0.262388i	$-0.0845101\pi$
$337$	4.02360e6	1.92992	0.964962	+	0.262388i	$0.0845101\pi$
$338$	451346.	0.214891
$339$	0	0
$340$	7.21446e6	3.38459
$341$	0	0
$342$	0	0
$343$	−998016.	−0.458039
$344$	−1.74794e6	−0.796399
$345$	0	0
$346$	3.90182e6	1.75217
$347$	3.96933e6	1.76967	0.884837	−	0.465900i	$-0.154269\pi$
$347$	3.96933e6	1.76967	0.884837	+	0.465900i	$0.154269\pi$
$348$	0	0
$349$	−1.91022e6	−0.839500	−0.419750	−	0.907640i	$-0.637882\pi$
$349$	−1.91022e6	−0.839500	−0.419750	+	0.907640i	$0.637882\pi$
$350$	−122196.	−0.0533194
$351$	0	0
$352$	0	0
$353$	−4.22138e6	−1.80309	−0.901545	−	0.432686i	$-0.857566\pi$
$353$	−4.22138e6	−1.80309	−0.901545	+	0.432686i	$0.857566\pi$
$354$	0	0
$355$	3.84560e6	1.61954
$356$	400532.	0.167499
$357$	0	0
$358$	3.49575e6	1.44156
$359$	−1.53072e6	−0.626844	−0.313422	−	0.949614i	$-0.601476\pi$
$359$	−1.53072e6	−0.626844	−0.313422	+	0.949614i	$0.601476\pi$
$360$	0	0
$361$	116872.	0.0472001
$362$	6.02444e6	2.41627
$363$	0	0
$364$	1.05359e6	0.416789
$365$	2.62549e6	1.03152
$366$	0	0
$367$	−1.53665e6	−0.595540	−0.297770	−	0.954638i	$-0.596243\pi$
$367$	−1.53665e6	−0.595540	−0.297770	+	0.954638i	$0.596243\pi$
$368$	858930.	0.330627
$369$	0	0
$370$	752784.	0.285868
$371$	−218394.	−0.0823770
$372$	0	0
$373$	1.42900e6	0.531816	0.265908	−	0.963998i	$-0.414328\pi$
$373$	1.42900e6	0.531816	0.265908	+	0.963998i	$0.414328\pi$
$374$	0	0
$375$	0	0
$376$	−8.31592e6	−3.03348
$377$	4.32844e6	1.56848
$378$	0	0
$379$	2.86618e6	1.02496	0.512479	−	0.858700i	$-0.328727\pi$
$379$	2.86618e6	1.02496	0.512479	+	0.858700i	$0.328727\pi$
$380$	5.80442e6	2.06205
$381$	0	0
$382$	4.70312e6	1.64903
$383$	−2.84698e6	−0.991717	−0.495859	−	0.868403i	$-0.665147\pi$
$383$	−2.84698e6	−0.991717	−0.495859	+	0.868403i	$0.665147\pi$
$384$	0	0
$385$	0	0
$386$	1.29895e6	0.443737
$387$	0	0
$388$	7.72734e6	2.60586
$389$	−4.71435e6	−1.57960	−0.789801	−	0.613363i	$-0.789816\pi$
$389$	−4.71435e6	−1.57960	−0.789801	+	0.613363i	$0.789816\pi$
$390$	0	0
$391$	2.43132e6	0.804267
$392$	4.36474e6	1.43464
$393$	0	0
$394$	−1.70156e6	−0.552213
$395$	−1.03814e6	−0.334784
$396$	0	0
$397$	3.61747e6	1.15194	0.575968	−	0.817472i	$-0.304625\pi$
$397$	3.61747e6	1.15194	0.575968	+	0.817472i	$0.304625\pi$
$398$	−4.16664e6	−1.31850
$399$	0	0
$400$	294055.	0.0918922
$401$	3.48536e6	1.08240	0.541198	−	0.840895i	$-0.317971\pi$
$401$	3.48536e6	1.08240	0.541198	+	0.840895i	$0.317971\pi$
$402$	0	0
$403$	−4.13492e6	−1.26825
$404$	4.53844e6	1.38342
$405$	0	0
$406$	−2.23300e6	−0.672316
$407$	0	0
$408$	0	0
$409$	3.32176e6	0.981884	0.490942	−	0.871192i	$-0.336653\pi$
$409$	3.32176e6	0.981884	0.490942	+	0.871192i	$0.336653\pi$
$410$	3.34377e6	0.982374
$411$	0	0
$412$	3.19449e6	0.927169
$413$	677204.	0.195364
$414$	0	0
$415$	−7.07487e6	−2.01650
$416$	1.13646e6	0.321973
$417$	0	0
$418$	0	0
$419$	5.09570e6	1.41798	0.708988	−	0.705221i	$-0.249152\pi$
$419$	5.09570e6	1.41798	0.708988	+	0.705221i	$0.249152\pi$
$420$	0	0
$421$	3.19859e6	0.879534	0.439767	−	0.898112i	$-0.355061\pi$
$421$	3.19859e6	0.879534	0.439767	+	0.898112i	$0.355061\pi$
$422$	1.29437e6	0.353815
$423$	0	0
$424$	1.96637e6	0.531190
$425$	832364.	0.223533
$426$	0	0
$427$	763355.	0.202608
$428$	7.46696e6	1.97031
$429$	0	0
$430$	−3.63976e6	−0.949297
$431$	1.63848e6	0.424861	0.212430	−	0.977176i	$-0.431862\pi$
$431$	1.63848e6	0.424861	0.212430	+	0.977176i	$0.431862\pi$
$432$	0	0
$433$	−225125.	−0.0577039	−0.0288519	−	0.999584i	$-0.509185\pi$
$433$	−225125.	−0.0577039	−0.0288519	+	0.999584i	$0.509185\pi$
$434$	2.13316e6	0.543626
$435$	0	0
$436$	−8.14746e6	−2.05261
$437$	1.95613e6	0.489997
$438$	0	0
$439$	119483.	0.0295899	0.0147950	−	0.999891i	$-0.495290\pi$
$439$	119483.	0.0295899	0.0147950	+	0.999891i	$0.495290\pi$
$440$	0	0
$441$	0	0
$442$	−1.09679e7	−2.67035
$443$	930354.	0.225237	0.112618	−	0.993638i	$-0.464076\pi$
$443$	930354.	0.225237	0.112618	+	0.993638i	$0.464076\pi$
$444$	0	0
$445$	393449.	0.0941864
$446$	7.33964e6	1.74718
$447$	0	0
$448$	−1.27723e6	−0.300660
$449$	6.80769e6	1.59362	0.796808	−	0.604232i	$-0.206520\pi$
$449$	6.80769e6	1.59362	0.796808	+	0.604232i	$0.206520\pi$
$450$	0	0
$451$	0	0
$452$	−2.03986e6	−0.469628
$453$	0	0
$454$	−1.04368e6	−0.237645
$455$	1.03495e6	0.234365
$456$	0	0
$457$	−8.32853e6	−1.86543	−0.932713	−	0.360620i	$-0.882565\pi$
$457$	−8.32853e6	−1.86543	−0.932713	+	0.360620i	$0.882565\pi$
$458$	5.03323e6	1.12120
$459$	0	0
$460$	4.37883e6	0.964858
$461$	164594.	0.0360713	0.0180356	−	0.999837i	$-0.494259\pi$
$461$	164594.	0.0360713	0.0180356	+	0.999837i	$0.494259\pi$
$462$	0	0
$463$	−243247.	−0.0527345	−0.0263672	−	0.999652i	$-0.508394\pi$
$463$	−243247.	−0.0527345	−0.0263672	+	0.999652i	$0.508394\pi$
$464$	5.37356e6	1.15869
$465$	0	0
$466$	2.11059e6	0.450235
$467$	−3.26340e6	−0.692433	−0.346217	−	0.938155i	$-0.612534\pi$
$467$	−3.26340e6	−0.692433	−0.346217	+	0.938155i	$0.612534\pi$
$468$	0	0
$469$	474670.	0.0996460
$470$	−1.73164e7	−3.61586
$471$	0	0
$472$	−6.09738e6	−1.25976
$473$	0	0
$474$	0	0
$475$	669681.	0.136187
$476$	3.70241e6	0.748974
$477$	0	0
$478$	4.64296e6	0.929449
$479$	1.48977e6	0.296674	0.148337	−	0.988937i	$-0.452608\pi$
$479$	1.48977e6	0.296674	0.148337	+	0.988937i	$0.452608\pi$
$480$	0	0
$481$	−748848.	−0.147581
$482$	4.54364e6	0.890812
$483$	0	0
$484$	0	0
$485$	7.59069e6	1.46530
$486$	0	0
$487$	−5.02892e6	−0.960842	−0.480421	−	0.877038i	$-0.659516\pi$
$487$	−5.02892e6	−0.960842	−0.480421	+	0.877038i	$0.659516\pi$
$488$	−6.87306e6	−1.30647
$489$	0	0
$490$	9.08876e6	1.71007
$491$	4.45298e6	0.833579	0.416790	−	0.909003i	$-0.363155\pi$
$491$	4.45298e6	0.833579	0.416790	+	0.909003i	$0.363155\pi$
$492$	0	0
$493$	1.52106e7	2.81857
$494$	−8.82427e6	−1.62690
$495$	0	0
$496$	−5.13331e6	−0.936900
$497$	1.97353e6	0.358388
$498$	0	0
$499$	−7.77494e6	−1.39780	−0.698901	−	0.715218i	$-0.746327\pi$
$499$	−7.77494e6	−1.39780	−0.698901	+	0.715218i	$0.746327\pi$
$500$	−9.76535e6	−1.74688
$501$	0	0
$502$	3.52649e6	0.624572
$503$	3.22332e6	0.568047	0.284023	−	0.958817i	$-0.408331\pi$
$503$	3.22332e6	0.568047	0.284023	+	0.958817i	$0.408331\pi$
$504$	0	0
$505$	4.45819e6	0.777911
$506$	0	0
$507$	0	0
$508$	−1.42133e7	−2.44363
$509$	−1.24004e6	−0.212149	−0.106074	−	0.994358i	$-0.533828\pi$
$509$	−1.24004e6	−0.212149	−0.106074	+	0.994358i	$0.533828\pi$
$510$	0	0
$511$	1.34738e6	0.228265
$512$	7.63199e6	1.28666
$513$	0	0
$514$	1.61972e7	2.70416
$515$	3.13800e6	0.521357
$516$	0	0
$517$	0	0
$518$	386323.	0.0632596
$519$	0	0
$520$	−9.31847e6	−1.51125
$521$	4.50711e6	0.727450	0.363725	−	0.931506i	$-0.381505\pi$
$521$	4.50711e6	0.727450	0.363725	+	0.931506i	$0.381505\pi$
$522$	0	0
$523$	2.27029e6	0.362934	0.181467	−	0.983397i	$-0.441915\pi$
$523$	2.27029e6	0.362934	0.181467	+	0.983397i	$0.441915\pi$
$524$	−1.24075e7	−1.97403
$525$	0	0
$526$	−1.46577e7	−2.30995
$527$	−1.45305e7	−2.27906
$528$	0	0
$529$	−4.96065e6	−0.770725
$530$	4.09459e6	0.633171
$531$	0	0
$532$	2.97878e6	0.456310
$533$	−3.32629e6	−0.507157
$534$	0	0
$535$	7.33491e6	1.10793
$536$	−4.27382e6	−0.642545
$537$	0	0
$538$	−660963.	−0.0984514
$539$	0	0
$540$	0	0
$541$	−734869.	−0.107949	−0.0539743	−	0.998542i	$-0.517189\pi$
$541$	−734869.	−0.107949	−0.0539743	+	0.998542i	$0.517189\pi$
$542$	−1.14306e7	−1.67136
$543$	0	0
$544$	3.99363e6	0.578589
$545$	−8.00339e6	−1.15420
$546$	0	0
$547$	−1.00922e7	−1.44217	−0.721084	−	0.692847i	$-0.756356\pi$
$547$	−1.00922e7	−1.44217	−0.721084	+	0.692847i	$0.756356\pi$
$548$	2.92132e6	0.415554
$549$	0	0
$550$	0	0
$551$	1.22377e7	1.71721
$552$	0	0
$553$	−532768.	−0.0740841
$554$	1.45196e7	2.00993
$555$	0	0
$556$	7.09738e6	0.973669
$557$	1.71309e6	0.233960	0.116980	−	0.993134i	$-0.462679\pi$
$557$	1.71309e6	0.233960	0.116980	+	0.993134i	$0.462679\pi$
$558$	0	0
$559$	3.62073e6	0.490080
$560$	1.28485e6	0.173134
$561$	0	0
$562$	1.35436e7	1.80881
$563$	4.41357e6	0.586839	0.293419	−	0.955984i	$-0.405207\pi$
$563$	4.41357e6	0.586839	0.293419	+	0.955984i	$0.405207\pi$
$564$	0	0
$565$	−2.00379e6	−0.264077
$566$	−3.09984e6	−0.406722
$567$	0	0
$568$	−1.77692e7	−2.31098
$569$	4.24981e6	0.550287	0.275143	−	0.961403i	$-0.411275\pi$
$569$	4.24981e6	0.550287	0.275143	+	0.961403i	$0.411275\pi$
$570$	0	0
$571$	2.86179e6	0.367322	0.183661	−	0.982990i	$-0.441205\pi$
$571$	2.86179e6	0.367322	0.183661	+	0.982990i	$0.441205\pi$
$572$	0	0
$573$	0	0
$574$	1.71600e6	0.217389
$575$	505205.	0.0637232
$576$	0	0
$577$	−9.34741e6	−1.16883	−0.584415	−	0.811455i	$-0.698676\pi$
$577$	−9.34741e6	−1.16883	−0.584415	+	0.811455i	$0.698676\pi$
$578$	−2.48810e7	−3.09776
$579$	0	0
$580$	2.73944e7	3.38137
$581$	−3.63077e6	−0.446229
$582$	0	0
$583$	0	0
$584$	−1.21315e7	−1.47191
$585$	0	0
$586$	−1.36192e7	−1.63835
$587$	263999.	0.0316233	0.0158117	−	0.999875i	$-0.494967\pi$
$587$	263999.	0.0316233	0.0158117	+	0.999875i	$0.494967\pi$
$588$	0	0
$589$	−1.16906e7	−1.38851
$590$	−1.26967e7	−1.50162
$591$	0	0
$592$	−929659.	−0.109023
$593$	1.46405e6	0.170969	0.0854846	−	0.996339i	$-0.472756\pi$
$593$	1.46405e6	0.170969	0.0854846	+	0.996339i	$0.472756\pi$
$594$	0	0
$595$	3.63694e6	0.421156
$596$	−1.25147e7	−1.44313
$597$	0	0
$598$	−6.65699e6	−0.761246
$599$	−1.28854e6	−0.146733	−0.0733667	−	0.997305i	$-0.523374\pi$
$599$	−1.28854e6	−0.146733	−0.0733667	+	0.997305i	$0.523374\pi$
$600$	0	0
$601$	−1.24358e7	−1.40439	−0.702195	−	0.711985i	$-0.747796\pi$
$601$	−1.24358e7	−1.40439	−0.702195	+	0.711985i	$0.747796\pi$
$602$	−1.86790e6	−0.210069
$603$	0	0
$604$	6.09618e6	0.679932
$605$	0	0
$606$	0	0
$607$	1.41180e7	1.55526	0.777629	−	0.628724i	$-0.216422\pi$
$607$	1.41180e7	1.55526	0.777629	+	0.628724i	$0.216422\pi$
$608$	3.21309e6	0.352504
$609$	0	0
$610$	−1.43119e7	−1.55730
$611$	1.72258e7	1.86671
$612$	0	0
$613$	−1.67895e7	−1.80462	−0.902312	−	0.431085i	$-0.858131\pi$
$613$	−1.67895e7	−1.80462	−0.902312	+	0.431085i	$0.858131\pi$
$614$	−6.13386e6	−0.656618
$615$	0	0
$616$	0	0
$617$	1.49155e6	0.157734	0.0788670	−	0.996885i	$-0.474870\pi$
$617$	1.49155e6	0.157734	0.0788670	+	0.996885i	$0.474870\pi$
$618$	0	0
$619$	7.92919e6	0.831768	0.415884	−	0.909418i	$-0.363472\pi$
$619$	7.92919e6	0.831768	0.415884	+	0.909418i	$0.363472\pi$
$620$	−2.61697e7	−2.73413
$621$	0	0
$622$	−1.07474e7	−1.11385
$623$	201915.	0.0208424
$624$	0	0
$625$	−1.08923e7	−1.11537
$626$	2.87053e7	2.92770
$627$	0	0
$628$	4.86177e6	0.491921
$629$	−2.63153e6	−0.265205
$630$	0	0
$631$	1.16436e7	1.16416	0.582082	−	0.813130i	$-0.302238\pi$
$631$	1.16436e7	1.16416	0.582082	+	0.813130i	$0.302238\pi$
$632$	4.79691e6	0.477715
$633$	0	0
$634$	1.88192e7	1.85943
$635$	−1.39620e7	−1.37408
$636$	0	0
$637$	−9.04124e6	−0.882835
$638$	0	0
$639$	0	0
$640$	2.01469e7	1.94428
$641$	5.51964e6	0.530598	0.265299	−	0.964166i	$-0.414529\pi$
$641$	5.51964e6	0.530598	0.265299	+	0.964166i	$0.414529\pi$
$642$	0	0
$643$	−4.94473e6	−0.471645	−0.235822	−	0.971796i	$-0.575778\pi$
$643$	−4.94473e6	−0.471645	−0.235822	+	0.971796i	$0.575778\pi$
$644$	2.24718e6	0.213513
$645$	0	0
$646$	−3.10094e7	−2.92356
$647$	7.25445e6	0.681309	0.340654	−	0.940189i	$-0.389351\pi$
$647$	7.25445e6	0.681309	0.340654	+	0.940189i	$0.389351\pi$
$648$	0	0
$649$	0	0
$650$	−2.27903e6	−0.211576
$651$	0	0
$652$	−1.15595e6	−0.106493
$653$	1.95768e7	1.79663	0.898315	−	0.439351i	$-0.144792\pi$
$653$	1.95768e7	1.79663	0.898315	+	0.439351i	$0.144792\pi$
$654$	0	0
$655$	−1.21881e7	−1.11002
$656$	−4.12943e6	−0.374654
$657$	0	0
$658$	−8.88663e6	−0.800152
$659$	−4.24342e6	−0.380630	−0.190315	−	0.981723i	$-0.560951\pi$
$659$	−4.24342e6	−0.380630	−0.190315	+	0.981723i	$0.560951\pi$
$660$	0	0
$661$	1.86117e7	1.65685	0.828424	−	0.560101i	$-0.189238\pi$
$661$	1.86117e7	1.65685	0.828424	+	0.560101i	$0.189238\pi$
$662$	3.27914e6	0.290814
$663$	0	0
$664$	3.26905e7	2.87741
$665$	2.92611e6	0.256588
$666$	0	0
$667$	9.23210e6	0.803500
$668$	2.65760e7	2.30435
$669$	0	0
$670$	−8.89943e6	−0.765906
$671$	0	0
$672$	0	0
$673$	1.52061e7	1.29414	0.647068	−	0.762433i	$-0.275995\pi$
$673$	1.52061e7	1.29414	0.647068	+	0.762433i	$0.275995\pi$
$674$	−3.87138e7	−3.28258
$675$	0	0
$676$	−2.84160e6	−0.239164
$677$	8.17687e6	0.685670	0.342835	−	0.939396i	$-0.388613\pi$
$677$	8.17687e6	0.685670	0.342835	+	0.939396i	$0.388613\pi$
$678$	0	0
$679$	3.89549e6	0.324255
$680$	−3.27461e7	−2.71573
$681$	0	0
$682$	0	0
$683$	−7.72453e6	−0.633608	−0.316804	−	0.948491i	$-0.602610\pi$
$683$	−7.72453e6	−0.633608	−0.316804	+	0.948491i	$0.602610\pi$
$684$	0	0
$685$	2.86966e6	0.233671
$686$	9.60257e6	0.779072
$687$	0	0
$688$	4.49497e6	0.362039
$689$	−4.07319e6	−0.326878
$690$	0	0
$691$	2.01276e7	1.60360	0.801800	−	0.597592i	$-0.203876\pi$
$691$	2.01276e7	1.60360	0.801800	+	0.597592i	$0.203876\pi$
$692$	−2.45653e7	−1.95010
$693$	0	0
$694$	−3.81916e7	−3.01002
$695$	6.97188e6	0.547504
$696$	0	0
$697$	−1.16889e7	−0.911366
$698$	1.83795e7	1.42789
$699$	0	0
$700$	769324.	0.0593423
$701$	7.43586e6	0.571526	0.285763	−	0.958300i	$-0.407753\pi$
$701$	7.43586e6	0.571526	0.285763	+	0.958300i	$0.407753\pi$
$702$	0	0
$703$	−2.11721e6	−0.161575
$704$	0	0
$705$	0	0
$706$	4.06167e7	3.06685
$707$	2.28791e6	0.172143
$708$	0	0
$709$	−6.62005e6	−0.494591	−0.247295	−	0.968940i	$-0.579542\pi$
$709$	−6.62005e6	−0.494591	−0.247295	+	0.968940i	$0.579542\pi$
$710$	−3.70011e7	−2.75466
$711$	0	0
$712$	−1.81799e6	−0.134398
$713$	−8.81934e6	−0.649699
$714$	0	0
$715$	0	0
$716$	−2.20087e7	−1.60440
$717$	0	0
$718$	1.47281e7	1.06619
$719$	−1.73156e7	−1.24915	−0.624577	−	0.780963i	$-0.714729\pi$
$719$	−1.73156e7	−1.24915	−0.624577	+	0.780963i	$0.714729\pi$
$720$	0	0
$721$	1.61040e6	0.115371
$722$	−1.12450e6	−0.0802820
$723$	0	0
$724$	−3.79289e7	−2.68921
$725$	3.16061e6	0.223319
$726$	0	0
$727$	1.02515e7	0.719366	0.359683	−	0.933075i	$-0.382885\pi$
$727$	1.02515e7	0.719366	0.359683	+	0.933075i	$0.382885\pi$
$728$	−4.78217e6	−0.334423
$729$	0	0
$730$	−2.52616e7	−1.75450
$731$	1.27236e7	0.880679
$732$	0	0
$733$	−2.86700e6	−0.197091	−0.0985457	−	0.995133i	$-0.531419\pi$
$733$	−2.86700e6	−0.197091	−0.0985457	+	0.995133i	$0.531419\pi$
$734$	1.47852e7	1.01295
$735$	0	0
$736$	2.42394e6	0.164940
$737$	0	0
$738$	0	0
$739$	376546.	0.0253634	0.0126817	−	0.999920i	$-0.495963\pi$
$739$	376546.	0.0253634	0.0126817	+	0.999920i	$0.495963\pi$
$740$	−4.73941e6	−0.318159
$741$	0	0
$742$	2.10131e6	0.140114
$743$	−1.69740e7	−1.12801	−0.564005	−	0.825771i	$-0.690740\pi$
$743$	−1.69740e7	−1.12801	−0.564005	+	0.825771i	$0.690740\pi$
$744$	0	0
$745$	−1.22934e7	−0.811489
$746$	−1.37494e7	−0.904558
$747$	0	0
$748$	0	0
$749$	3.76422e6	0.245172
$750$	0	0
$751$	−2.69716e7	−1.74505	−0.872524	−	0.488571i	$-0.837518\pi$
$751$	−2.69716e7	−1.74505	−0.872524	+	0.488571i	$0.837518\pi$
$752$	2.13850e7	1.37900
$753$	0	0
$754$	−4.16469e7	−2.66780
$755$	5.98838e6	0.382333
$756$	0	0
$757$	−1.22626e7	−0.777753	−0.388877	−	0.921290i	$-0.627137\pi$
$757$	−1.22626e7	−0.777753	−0.388877	+	0.921290i	$0.627137\pi$
$758$	−2.75775e7	−1.74334
$759$	0	0
$760$	−2.63460e7	−1.65455
$761$	1.71281e7	1.07213	0.536065	−	0.844177i	$-0.319910\pi$
$761$	1.71281e7	1.07213	0.536065	+	0.844177i	$0.319910\pi$
$762$	0	0
$763$	−4.10728e6	−0.255413
$764$	−2.96101e7	−1.83530
$765$	0	0
$766$	2.73927e7	1.68680
$767$	1.26303e7	0.775219
$768$	0	0
$769$	−1.67008e7	−1.01841	−0.509203	−	0.860647i	$-0.670060\pi$
$769$	−1.67008e7	−1.01841	−0.509203	+	0.860647i	$0.670060\pi$
$770$	0	0
$771$	0	0
$772$	−8.17801e6	−0.493861
$773$	1.42948e7	0.860455	0.430227	−	0.902721i	$-0.358433\pi$
$773$	1.42948e7	0.860455	0.430227	+	0.902721i	$0.358433\pi$
$774$	0	0
$775$	−3.01931e6	−0.180573
$776$	−3.50740e7	−2.09089
$777$	0	0
$778$	4.53599e7	2.68673
$779$	−9.40436e6	−0.555246
$780$	0	0
$781$	0	0
$782$	−2.33934e7	−1.36797
$783$	0	0
$784$	−1.12243e7	−0.652181
$785$	4.77580e6	0.276612
$786$	0	0
$787$	2.81760e7	1.62159	0.810796	−	0.585328i	$-0.199034\pi$
$787$	2.81760e7	1.62159	0.810796	+	0.585328i	$0.199034\pi$
$788$	1.07127e7	0.614590
$789$	0	0
$790$	9.98867e6	0.569430
$791$	−1.02833e6	−0.0584374
$792$	0	0
$793$	1.42371e7	0.803965
$794$	−3.48061e7	−1.95931
$795$	0	0
$796$	2.62325e7	1.46743
$797$	−8.11889e6	−0.452742	−0.226371	−	0.974041i	$-0.572686\pi$
$797$	−8.11889e6	−0.452742	−0.226371	+	0.974041i	$0.572686\pi$
$798$	0	0
$799$	6.05334e7	3.35450
$800$	829837.	0.0458425
$801$	0	0
$802$	−3.35350e7	−1.84103
$803$	0	0
$804$	0	0
$805$	2.20744e6	0.120060
$806$	3.97849e7	2.15715
$807$	0	0
$808$	−2.05998e7	−1.11003
$809$	−1.76838e7	−0.949960	−0.474980	−	0.879997i	$-0.657545\pi$
$809$	−1.76838e7	−0.949960	−0.474980	+	0.879997i	$0.657545\pi$
$810$	0	0
$811$	1.03383e7	0.551945	0.275973	−	0.961165i	$-0.411000\pi$
$811$	1.03383e7	0.551945	0.275973	+	0.961165i	$0.411000\pi$
$812$	1.40586e7	0.748260
$813$	0	0
$814$	0	0
$815$	−1.13551e6	−0.0598821
$816$	0	0
$817$	1.02368e7	0.536551
$818$	−3.19609e7	−1.67007
$819$	0	0
$820$	−2.10519e7	−1.09334
$821$	1.41812e7	0.734268	0.367134	−	0.930168i	$-0.380339\pi$
$821$	1.41812e7	0.734268	0.367134	+	0.930168i	$0.380339\pi$
$822$	0	0
$823$	8.24357e6	0.424244	0.212122	−	0.977243i	$-0.431963\pi$
$823$	8.24357e6	0.424244	0.212122	+	0.977243i	$0.431963\pi$
$824$	−1.44996e7	−0.743942
$825$	0	0
$826$	−6.51583e6	−0.332292
$827$	463776.	0.0235801	0.0117900	−	0.999930i	$-0.496247\pi$
$827$	463776.	0.0235801	0.0117900	+	0.999930i	$0.496247\pi$
$828$	0	0
$829$	3.42221e7	1.72950	0.864750	−	0.502203i	$-0.167477\pi$
$829$	3.42221e7	1.72950	0.864750	+	0.502203i	$0.167477\pi$
$830$	6.80720e7	3.42984
$831$	0	0
$832$	−2.38212e7	−1.19304
$833$	−3.17719e7	−1.58646
$834$	0	0
$835$	2.61060e7	1.29576
$836$	0	0
$837$	0	0
$838$	−4.90291e7	−2.41182
$839$	7.10992e6	0.348706	0.174353	−	0.984683i	$-0.444217\pi$
$839$	7.10992e6	0.348706	0.174353	+	0.984683i	$0.444217\pi$
$840$	0	0
$841$	3.72459e7	1.81588
$842$	−3.07757e7	−1.49599
$843$	0	0
$844$	−8.14913e6	−0.393781
$845$	−2.79135e6	−0.134485
$846$	0	0
$847$	0	0
$848$	−5.05667e6	−0.241476
$849$	0	0
$850$	−8.00873e6	−0.380204
$851$	−1.59721e6	−0.0756029
$852$	0	0
$853$	−1.24081e7	−0.583891	−0.291946	−	0.956435i	$-0.594303\pi$
$853$	−1.24081e7	−0.583891	−0.291946	+	0.956435i	$0.594303\pi$
$854$	−7.34475e6	−0.344614
$855$	0	0
$856$	−3.38921e7	−1.58094
$857$	9.60549e6	0.446753	0.223376	−	0.974732i	$-0.428292\pi$
$857$	9.60549e6	0.446753	0.223376	+	0.974732i	$0.428292\pi$
$858$	0	0
$859$	1.35157e7	0.624964	0.312482	−	0.949924i	$-0.398840\pi$
$859$	1.35157e7	0.624964	0.312482	+	0.949924i	$0.398840\pi$
$860$	2.29154e7	1.05653
$861$	0	0
$862$	−1.57649e7	−0.722640
$863$	−3.40435e7	−1.55599	−0.777995	−	0.628271i	$-0.783763\pi$
$863$	−3.40435e7	−1.55599	−0.777995	+	0.628271i	$0.783763\pi$
$864$	0	0
$865$	−2.41309e7	−1.09656
$866$	2.16608e6	0.0981477
$867$	0	0
$868$	−1.34301e7	−0.605033
$869$	0	0
$870$	0	0
$871$	8.85290e6	0.395403
$872$	3.69809e7	1.64697
$873$	0	0
$874$	−1.88212e7	−0.833429
$875$	−4.92289e6	−0.217370
$876$	0	0
$877$	−3.87254e6	−0.170019	−0.0850095	−	0.996380i	$-0.527092\pi$
$877$	−3.87254e6	−0.170019	−0.0850095	+	0.996380i	$0.527092\pi$
$878$	−1.14962e6	−0.0503291
$879$	0	0
$880$	0	0
$881$	−2.91447e7	−1.26509	−0.632543	−	0.774525i	$-0.717989\pi$
$881$	−2.91447e7	−1.26509	−0.632543	+	0.774525i	$0.717989\pi$
$882$	0	0
$883$	3.82485e7	1.65087	0.825436	−	0.564496i	$-0.190930\pi$
$883$	3.82485e7	1.65087	0.825436	+	0.564496i	$0.190930\pi$
$884$	6.90522e7	2.97199
$885$	0	0
$886$	−8.95156e6	−0.383102
$887$	2.52774e7	1.07876	0.539379	−	0.842063i	$-0.318659\pi$
$887$	2.52774e7	1.07876	0.539379	+	0.842063i	$0.318659\pi$
$888$	0	0
$889$	−7.16517e6	−0.304069
$890$	−3.78564e6	−0.160200
$891$	0	0
$892$	−4.62092e7	−1.94454
$893$	4.87023e7	2.04372
$894$	0	0
$895$	−2.16195e7	−0.902170
$896$	1.03392e7	0.430248
$897$	0	0
$898$	−6.55013e7	−2.71056
$899$	−5.51747e7	−2.27689
$900$	0	0
$901$	−1.43136e7	−0.587404
$902$	0	0
$903$	0	0
$904$	9.25882e6	0.376820
$905$	−3.72582e7	−1.51217
$906$	0	0
$907$	−4.24330e7	−1.71272	−0.856359	−	0.516381i	$-0.827279\pi$
$907$	−4.24330e7	−1.71272	−0.856359	+	0.516381i	$0.827279\pi$
$908$	6.57085e6	0.264488
$909$	0	0
$910$	−9.95799e6	−0.398628
$911$	4.58167e6	0.182906	0.0914529	−	0.995809i	$-0.470849\pi$
$911$	4.58167e6	0.182906	0.0914529	+	0.995809i	$0.470849\pi$
$912$	0	0
$913$	0	0
$914$	8.01344e7	3.17288
$915$	0	0
$916$	−3.16884e7	−1.24785
$917$	−6.25482e6	−0.245636
$918$	0	0
$919$	1.85085e7	0.722905	0.361453	−	0.932390i	$-0.382281\pi$
$919$	1.85085e7	0.722905	0.361453	+	0.932390i	$0.382281\pi$
$920$	−1.98753e7	−0.774183
$921$	0	0
$922$	−1.58367e6	−0.0613532
$923$	3.68076e7	1.42211
$924$	0	0
$925$	−546806.	−0.0210126
$926$	2.34044e6	0.0896954
$927$	0	0
$928$	1.51644e7	0.578037
$929$	−4.19713e7	−1.59556	−0.797780	−	0.602948i	$-0.793993\pi$
$929$	−4.19713e7	−1.59556	−0.797780	+	0.602948i	$0.793993\pi$
$930$	0	0
$931$	−2.55622e7	−0.966548
$932$	−1.32879e7	−0.501092
$933$	0	0
$934$	3.13993e7	1.17775
$935$	0	0
$936$	0	0
$937$	7.48449e6	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$937$	7.48449e6	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$938$	−4.56712e6	−0.169487
$939$	0	0
$940$	1.09021e8	4.02431
$941$	4.14020e7	1.52422	0.762109	−	0.647449i	$-0.224164\pi$
$941$	4.14020e7	1.52422	0.762109	+	0.647449i	$0.224164\pi$
$942$	0	0
$943$	−7.09461e6	−0.259806
$944$	1.56799e7	0.572681
$945$	0	0
$946$	0	0
$947$	4.38933e7	1.59046	0.795231	−	0.606307i	$-0.207350\pi$
$947$	4.38933e7	1.59046	0.795231	+	0.606307i	$0.207350\pi$
$948$	0	0
$949$	2.51295e7	0.905772
$950$	−6.44345e6	−0.231638
$951$	0	0
$952$	−1.68050e7	−0.600962
$953$	−2.82508e6	−0.100762	−0.0503812	−	0.998730i	$-0.516044\pi$
$953$	−2.82508e6	−0.100762	−0.0503812	+	0.998730i	$0.516044\pi$
$954$	0	0
$955$	−2.90865e7	−1.03201
$956$	−2.92314e7	−1.03444
$957$	0	0
$958$	−1.43340e7	−0.504609
$959$	1.47269e6	0.0517088
$960$	0	0
$961$	2.40788e7	0.841059
$962$	7.20517e6	0.251019
$963$	0	0
$964$	−2.86060e7	−0.991437
$965$	−8.03339e6	−0.277703
$966$	0	0
$967$	−1.10642e7	−0.380499	−0.190249	−	0.981736i	$-0.560930\pi$
$967$	−1.10642e7	−0.380499	−0.190249	+	0.981736i	$0.560930\pi$
$968$	0	0
$969$	0	0
$970$	−7.30351e7	−2.49231
$971$	1.47508e7	0.502074	0.251037	−	0.967977i	$-0.419228\pi$
$971$	1.47508e7	0.502074	0.251037	+	0.967977i	$0.419228\pi$
$972$	0	0
$973$	3.57791e6	0.121157
$974$	4.83866e7	1.63428
$975$	0	0
$976$	1.76746e7	0.593917
$977$	1.86552e7	0.625263	0.312631	−	0.949875i	$-0.398790\pi$
$977$	1.86552e7	0.625263	0.312631	+	0.949875i	$0.398790\pi$
$978$	0	0
$979$	0	0
$980$	−5.72214e7	−1.90324
$981$	0	0
$982$	−4.28451e7	−1.41782
$983$	−1.43075e7	−0.472258	−0.236129	−	0.971722i	$-0.575879\pi$
$983$	−1.43075e7	−0.472258	−0.236129	+	0.971722i	$0.575879\pi$
$984$	0	0
$985$	1.05233e7	0.345590
$986$	−1.46351e8	−4.79407
$987$	0	0
$988$	5.55562e7	1.81067
$989$	7.72263e6	0.251058
$990$	0	0
$991$	3.51472e7	1.13686	0.568429	−	0.822732i	$-0.307551\pi$
$991$	3.51472e7	1.13686	0.568429	+	0.822732i	$0.307551\pi$
$992$	−1.44864e7	−0.467393
$993$	0	0
$994$	−1.89887e7	−0.609577
$995$	2.57686e7	0.825152
$996$	0	0
$997$	1.33768e7	0.426201	0.213100	−	0.977030i	$-0.431644\pi$
$997$	1.33768e7	0.426201	0.213100	+	0.977030i	$0.431644\pi$
$998$	7.48079e7	2.37750
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bk.1.1		10
3.2	odd	2		363.6.a.r.1.10		10
11.5	even	5		99.6.f.b.91.5		20
11.9	even	5		99.6.f.b.37.5		20
11.10	odd	2		1089.6.a.bi.1.10		10
33.5	odd	10		33.6.e.b.25.1	yes	20
33.20	odd	10		33.6.e.b.4.1	✓	20
33.32	even	2		363.6.a.t.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.4.1	✓	20	33.20	odd	10
33.6.e.b.25.1	yes	20	33.5	odd	10
99.6.f.b.37.5		20	11.9	even	5
99.6.f.b.91.5		20	11.5	even	5
363.6.a.r.1.10		10	3.2	odd	2
363.6.a.t.1.1		10	33.32	even	2
1089.6.a.bi.1.10		10	11.10	odd	2
1089.6.a.bk.1.1		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-9.62167 q^{2} +60.5765 q^{4} +59.5053 q^{5} +30.5377 q^{7} -274.954 q^{8} +O(q^{10})\)	\(q-9.62167 q^{2} +60.5765 q^{4} +59.5053 q^{5} +30.5377 q^{7} -274.954 q^{8} -572.540 q^{10} +569.547 q^{13} -293.823 q^{14} +707.065 q^{16} +2001.45 q^{17} +1610.27 q^{19} +3604.62 q^{20} +1214.78 q^{23} +415.881 q^{25} -5479.99 q^{26} +1849.87 q^{28} +7599.80 q^{29} -7260.02 q^{31} +1995.37 q^{32} -19257.3 q^{34} +1817.15 q^{35} -1314.81 q^{37} -15493.5 q^{38} -16361.2 q^{40} -5840.24 q^{41} +6357.22 q^{43} -11688.2 q^{46} +30244.8 q^{47} -15874.5 q^{49} -4001.47 q^{50} +34501.2 q^{52} -7151.62 q^{53} -8396.45 q^{56} -73122.8 q^{58} +22176.0 q^{59} +24997.2 q^{61} +69853.5 q^{62} -41824.9 q^{64} +33891.1 q^{65} +15543.8 q^{67} +121241. q^{68} -17484.1 q^{70} +64626.1 q^{71} +44122.0 q^{73} +12650.7 q^{74} +97544.6 q^{76} -17446.2 q^{79} +42074.1 q^{80} +56192.8 q^{82} -118895. q^{83} +119097. q^{85} -61167.1 q^{86} +6612.00 q^{89} +17392.6 q^{91} +73587.2 q^{92} -291005. q^{94} +95819.6 q^{95} +127563. q^{97} +152739. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8}+O(q^{10}) \) 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8	\( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8} - 212 q^{10} - 1016 q^{13} - 1566 q^{14} + 2361 q^{16} + 1669 q^{17} + 2929 q^{19} + 10189 q^{20} - 4070 q^{23} + 2425 q^{25} - 8481 q^{26} + 3272 q^{28} + 11940 q^{29} - 16085 q^{31} + 2313 q^{32} + 8270 q^{34} - 6987 q^{35} + 16136 q^{37} - 10721 q^{38} + 9332 q^{40} + 16278 q^{41} - 10844 q^{43} - 25995 q^{46} + 22411 q^{47} + 75150 q^{49} - 738 q^{50} - 8677 q^{52} + 27511 q^{53} - 84447 q^{56} + 16853 q^{58} + 39641 q^{59} - 3509 q^{61} + 227845 q^{62} - 22980 q^{64} + 67097 q^{65} + 10089 q^{67} + 273621 q^{68} - 38919 q^{70} - 60681 q^{71} - 133740 q^{73} + 317933 q^{74} + 23434 q^{76} - 12386 q^{79} + 289014 q^{80} + 385033 q^{82} + 187242 q^{83} - 191504 q^{85} + 43793 q^{86} - 102746 q^{89} - 435248 q^{91} - 30867 q^{92} - 404734 q^{94} + 648147 q^{95} - 120631 q^{97} + 1148087 q^{98}+O(q^{100}) \) 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8 - 212 * q^10 - 1016 * q^13 - 1566 * q^14 + 2361 * q^16 + 1669 * q^17 + 2929 * q^19 + 10189 * q^20 - 4070 * q^23 + 2425 * q^25 - 8481 * q^26 + 3272 * q^28 + 11940 * q^29 - 16085 * q^31 + 2313 * q^32 + 8270 * q^34 - 6987 * q^35 + 16136 * q^37 - 10721 * q^38 + 9332 * q^40 + 16278 * q^41 - 10844 * q^43 - 25995 * q^46 + 22411 * q^47 + 75150 * q^49 - 738 * q^50 - 8677 * q^52 + 27511 * q^53 - 84447 * q^56 + 16853 * q^58 + 39641 * q^59 - 3509 * q^61 + 227845 * q^62 - 22980 * q^64 + 67097 * q^65 + 10089 * q^67 + 273621 * q^68 - 38919 * q^70 - 60681 * q^71 - 133740 * q^73 + 317933 * q^74 + 23434 * q^76 - 12386 * q^79 + 289014 * q^80 + 385033 * q^82 + 187242 * q^83 - 191504 * q^85 + 43793 * q^86 - 102746 * q^89 - 435248 * q^91 - 30867 * q^92 - 404734 * q^94 + 648147 * q^95 - 120631 * q^97 + 1148087 * q^98

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