LMFDB - Embedded newform 1089.3.b.i.485.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 $ x^16 + 48x^14 + 921x^12 + 8986x^10 + 46812x^8 + 125072x^6 + 152129x^4 + 65614*x^2 + 5041
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.7
Root			$-0.816689i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.i.485.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.816689i q^{2} +3.33302 q^{4} +1.04696i q^{5} +6.83027 q^{7} -5.98880i q^{8} +O(q^{10})$	$q-0.816689i q^{2} +3.33302 q^{4} +1.04696i q^{5} +6.83027 q^{7} -5.98880i q^{8} +0.855043 q^{10} -7.70460 q^{13} -5.57821i q^{14} +8.44109 q^{16} -9.39296i q^{17} +9.42028 q^{19} +3.48955i q^{20} -4.82409i q^{23} +23.9039 q^{25} +6.29226i q^{26} +22.7654 q^{28} -46.7559i q^{29} +18.9914 q^{31} -30.8489i q^{32} -7.67113 q^{34} +7.15104i q^{35} -47.1829 q^{37} -7.69344i q^{38} +6.27005 q^{40} -7.60072i q^{41} +50.6851 q^{43} -3.93978 q^{46} +66.1465i q^{47} -2.34738 q^{49} -19.5220i q^{50} -25.6796 q^{52} -38.4109i q^{53} -40.9051i q^{56} -38.1851 q^{58} +117.266i q^{59} +81.4295 q^{61} -15.5100i q^{62} +8.57036 q^{64} -8.06643i q^{65} +70.1488 q^{67} -31.3069i q^{68} +5.84018 q^{70} +43.2633i q^{71} +45.9042 q^{73} +38.5338i q^{74} +31.3980 q^{76} -85.7775 q^{79} +8.83751i q^{80} -6.20742 q^{82} -159.135i q^{83} +9.83408 q^{85} -41.3940i q^{86} -154.015i q^{89} -52.6245 q^{91} -16.0788i q^{92} +54.0211 q^{94} +9.86269i q^{95} +153.459 q^{97} +1.91708i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 - 8 * q^7	$ 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 - 8 * q^7 + 24 * q^10 + 4 * q^13 + 28 * q^16 - 20 * q^19 - 44 * q^25 + 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 + 224 * q^40 + 272 * q^43 + 208 * q^46 + 348 * q^49 + 520 * q^52 - 44 * q^58 + 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 - 4 * q^73 + 1052 * q^76 + 216 * q^79 + 348 * q^82 + 416 * q^85 - 168 * q^91 + 1140 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.816689i	− 0.408345i	−0.978935	−	0.204172i	$-0.934550\pi$
$2$	− 0.816689i	− 0.408345i	0.978935	−	0.204172i	$-0.0654503\pi$
$3$	0	0
$3$	0	0
$4$	3.33302	0.833255
$5$	1.04696i	0.209393i	0.994504	+	0.104696i	$0.0333870\pi$
$5$	1.04696i	0.209393i	−0.994504	+	0.104696i	$0.966613\pi$
$6$	0	0
$7$	6.83027	0.975753	0.487877	−	0.872913i	$-0.337772\pi$
$7$	6.83027	0.975753	0.487877	+	0.872913i	$0.337772\pi$
$8$	− 5.98880i	− 0.748600i
$9$	0	0
$10$	0.855043	0.0855043
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−7.70460	−0.592661	−0.296331	−	0.955085i	$-0.595763\pi$
$13$	−7.70460	−0.592661	−0.296331	+	0.955085i	$0.595763\pi$
$14$	− 5.57821i	− 0.398444i
$15$	0	0
$16$	8.44109	0.527568
$17$	− 9.39296i	− 0.552527i	−0.961082	−	0.276263i	$-0.910904\pi$
$17$	− 9.39296i	− 0.552527i	0.961082	−	0.276263i	$-0.0890962\pi$
$18$	0	0
$19$	9.42028	0.495804	0.247902	−	0.968785i	$-0.420259\pi$
$19$	9.42028	0.495804	0.247902	+	0.968785i	$0.420259\pi$
$20$	3.48955i	0.174477i
$21$	0	0
$22$	0	0
$23$	− 4.82409i	− 0.209743i	−0.994486	−	0.104872i	$-0.966557\pi$
$23$	− 4.82409i	− 0.209743i	0.994486	−	0.104872i	$-0.0334431\pi$
$24$	0	0
$25$	23.9039	0.956155
$26$	6.29226i	0.242010i
$27$	0	0
$28$	22.7654	0.813051
$29$	− 46.7559i	− 1.61227i	−0.591730	−	0.806137i	$-0.701555\pi$
$29$	− 46.7559i	− 1.61227i	0.591730	−	0.806137i	$-0.298445\pi$
$30$	0	0
$31$	18.9914	0.612625	0.306312	−	0.951931i	$-0.400905\pi$
$31$	18.9914	0.612625	0.306312	+	0.951931i	$0.400905\pi$
$32$	− 30.8489i	− 0.964029i
$33$	0	0
$34$	−7.67113	−0.225621
$35$	7.15104i	0.204315i
$36$	0	0
$37$	−47.1829	−1.27521	−0.637607	−	0.770361i	$-0.720076\pi$
$37$	−47.1829	−1.27521	−0.637607	+	0.770361i	$0.720076\pi$
$38$	− 7.69344i	− 0.202459i
$39$	0	0
$40$	6.27005	0.156751
$41$	− 7.60072i	− 0.185383i	−0.995695	−	0.0926917i	$-0.970453\pi$
$41$	− 7.60072i	− 0.185383i	0.995695	−	0.0926917i	$-0.0295471\pi$
$42$	0	0
$43$	50.6851	1.17872	0.589362	−	0.807869i	$-0.299379\pi$
$43$	50.6851	1.17872	0.589362	+	0.807869i	$0.299379\pi$
$44$	0	0
$45$	0	0
$46$	−3.93978	−0.0856475
$47$	66.1465i	1.40737i	0.710511	+	0.703686i	$0.248464\pi$
$47$	66.1465i	1.40737i	−0.710511	+	0.703686i	$0.751536\pi$
$48$	0	0
$49$	−2.34738	−0.0479058
$50$	− 19.5220i	− 0.390441i
$51$	0	0
$52$	−25.6796	−0.493838
$53$	− 38.4109i	− 0.724734i	−0.932036	−	0.362367i	$-0.881969\pi$
$53$	− 38.4109i	− 0.724734i	0.932036	−	0.362367i	$-0.118031\pi$
$54$	0	0
$55$	0	0
$56$	− 40.9051i	− 0.730448i
$57$	0	0
$58$	−38.1851	−0.658363
$59$	117.266i	1.98756i	0.111382	+	0.993778i	$0.464472\pi$
$59$	117.266i	1.98756i	−0.111382	+	0.993778i	$0.535528\pi$
$60$	0	0
$61$	81.4295	1.33491	0.667455	−	0.744650i	$-0.267384\pi$
$61$	81.4295	1.33491	0.667455	+	0.744650i	$0.267384\pi$
$62$	− 15.5100i	− 0.250162i
$63$	0	0
$64$	8.57036	0.133912
$65$	− 8.06643i	− 0.124099i
$66$	0	0
$67$	70.1488	1.04700	0.523499	−	0.852027i	$-0.324627\pi$
$67$	70.1488	1.04700	0.523499	+	0.852027i	$0.324627\pi$
$68$	− 31.3069i	− 0.460396i
$69$	0	0
$70$	5.84018	0.0834311
$71$	43.2633i	0.609342i	0.952458	+	0.304671i	$0.0985465\pi$
$71$	43.2633i	0.609342i	−0.952458	+	0.304671i	$0.901453\pi$
$72$	0	0
$73$	45.9042	0.628825	0.314412	−	0.949287i	$-0.398193\pi$
$73$	45.9042	0.628825	0.314412	+	0.949287i	$0.398193\pi$
$74$	38.5338i	0.520727i
$75$	0	0
$76$	31.3980	0.413131
$77$	0	0
$78$	0	0
$79$	−85.7775	−1.08579	−0.542896	−	0.839800i	$-0.682672\pi$
$79$	−85.7775	−1.08579	−0.542896	+	0.839800i	$0.682672\pi$
$80$	8.83751i	0.110469i
$81$	0	0
$82$	−6.20742	−0.0757003
$83$	− 159.135i	− 1.91729i	−0.284610	−	0.958643i	$-0.591864\pi$
$83$	− 159.135i	− 1.91729i	0.284610	−	0.958643i	$-0.408136\pi$
$84$	0	0
$85$	9.83408	0.115695
$86$	− 41.3940i	− 0.481325i
$87$	0	0
$88$	0	0
$89$	− 154.015i	− 1.73051i	−0.501335	−	0.865253i	$-0.667158\pi$
$89$	− 154.015i	− 1.73051i	0.501335	−	0.865253i	$-0.332842\pi$
$90$	0	0
$91$	−52.6245	−0.578291
$92$	− 16.0788i	− 0.174769i
$93$	0	0
$94$	54.0211	0.574693
$95$	9.86269i	0.103818i
$96$	0	0
$97$	153.459	1.58205	0.791023	−	0.611786i	$-0.209549\pi$
$97$	153.459	1.58205	0.791023	+	0.611786i	$0.209549\pi$
$98$	1.91708i	0.0195621i
$99$	0	0
$100$	79.6720	0.796720
$101$	156.879i	1.55326i	0.629958	+	0.776629i	$0.283072\pi$
$101$	156.879i	1.55326i	−0.629958	+	0.776629i	$0.716928\pi$
$102$	0	0
$103$	−63.3825	−0.615364	−0.307682	−	0.951489i	$-0.599553\pi$
$103$	−63.3825	−0.615364	−0.307682	+	0.951489i	$0.599553\pi$
$104$	46.1413i	0.443666i
$105$	0	0
$106$	−31.3698	−0.295941
$107$	− 51.5081i	− 0.481384i	−0.970601	−	0.240692i	$-0.922626\pi$
$107$	− 51.5081i	− 0.481384i	0.970601	−	0.240692i	$-0.0773744\pi$
$108$	0	0
$109$	−63.0507	−0.578446	−0.289223	−	0.957262i	$-0.593397\pi$
$109$	−63.0507	−0.578446	−0.289223	+	0.957262i	$0.593397\pi$
$110$	0	0
$111$	0	0
$112$	57.6549	0.514776
$113$	86.7762i	0.767931i	0.923347	+	0.383966i	$0.125442\pi$
$113$	86.7762i	0.767931i	−0.923347	+	0.383966i	$0.874558\pi$
$114$	0	0
$115$	5.05064	0.0439187
$116$	− 155.838i	− 1.34343i
$117$	0	0
$118$	95.7697	0.811607
$119$	− 64.1565i	− 0.539130i
$120$	0	0
$121$	0	0
$122$	− 66.5026i	− 0.545103i
$123$	0	0
$124$	63.2986	0.510473
$125$	51.2005i	0.409604i
$126$	0	0
$127$	75.6526	0.595690	0.297845	−	0.954614i	$-0.403732\pi$
$127$	75.6526	0.595690	0.297845	+	0.954614i	$0.403732\pi$
$128$	− 130.395i	− 1.01871i
$129$	0	0
$130$	−6.58776	−0.0506751
$131$	− 124.201i	− 0.948097i	−0.880499	−	0.474049i	$-0.842792\pi$
$131$	− 124.201i	− 0.948097i	0.880499	−	0.474049i	$-0.157208\pi$
$132$	0	0
$133$	64.3431	0.483783
$134$	− 57.2898i	− 0.427536i
$135$	0	0
$136$	−56.2525	−0.413621
$137$	− 53.1596i	− 0.388026i	−0.980999	−	0.194013i	$-0.937850\pi$
$137$	− 53.1596i	− 0.388026i	0.980999	−	0.194013i	$-0.0621504\pi$
$138$	0	0
$139$	79.0567	0.568753	0.284377	−	0.958713i	$-0.408213\pi$
$139$	79.0567	0.568753	0.284377	+	0.958713i	$0.408213\pi$
$140$	23.8346i	0.170247i
$141$	0	0
$142$	35.3327	0.248822
$143$	0	0
$144$	0	0
$145$	48.9517	0.337598
$146$	− 37.4895i	− 0.256777i
$147$	0	0
$148$	−157.262	−1.06258
$149$	21.7256i	0.145809i	0.997339	+	0.0729046i	$0.0232269\pi$
$149$	21.7256i	0.145809i	−0.997339	+	0.0729046i	$0.976773\pi$
$150$	0	0
$151$	−124.089	−0.821784	−0.410892	−	0.911684i	$-0.634783\pi$
$151$	−124.089	−0.821784	−0.410892	+	0.911684i	$0.634783\pi$
$152$	− 56.4162i	− 0.371159i
$153$	0	0
$154$	0	0
$155$	19.8833i	0.128279i
$156$	0	0
$157$	−181.214	−1.15423	−0.577114	−	0.816664i	$-0.695821\pi$
$157$	−181.214	−1.15423	−0.577114	+	0.816664i	$0.695821\pi$
$158$	70.0536i	0.443377i
$159$	0	0
$160$	32.2977	0.201861
$161$	− 32.9499i	− 0.204658i
$162$	0	0
$163$	−276.772	−1.69799	−0.848995	−	0.528402i	$-0.822792\pi$
$163$	−276.772	−1.69799	−0.848995	+	0.528402i	$0.822792\pi$
$164$	− 25.3333i	− 0.154472i
$165$	0	0
$166$	−129.964	−0.782914
$167$	54.5378i	0.326574i	0.986579	+	0.163287i	$0.0522096\pi$
$167$	54.5378i	0.326574i	−0.986579	+	0.163287i	$0.947790\pi$
$168$	0	0
$169$	−109.639	−0.648753
$170$	− 8.03139i	− 0.0472434i
$171$	0	0
$172$	168.934	0.982177
$173$	− 100.732i	− 0.582264i	−0.956683	−	0.291132i	$-0.905968\pi$
$173$	− 100.732i	− 0.582264i	0.956683	−	0.291132i	$-0.0940319\pi$
$174$	0	0
$175$	163.270	0.932971
$176$	0	0
$177$	0	0
$178$	−125.782	−0.706643
$179$	− 81.8331i	− 0.457168i	−0.973524	−	0.228584i	$-0.926590\pi$
$179$	− 81.8331i	− 0.457168i	0.973524	−	0.228584i	$-0.0734096\pi$
$180$	0	0
$181$	203.797	1.12595	0.562976	−	0.826473i	$-0.309656\pi$
$181$	203.797	1.12595	0.562976	+	0.826473i	$0.309656\pi$
$182$	42.9778i	0.236142i
$183$	0	0
$184$	−28.8905	−0.157014
$185$	− 49.3988i	− 0.267021i
$186$	0	0
$187$	0	0
$188$	220.468i	1.17270i
$189$	0	0
$190$	8.05475	0.0423934
$191$	201.738i	1.05622i	0.849175	+	0.528111i	$0.177099\pi$
$191$	201.738i	1.05622i	−0.849175	+	0.528111i	$0.822901\pi$
$192$	0	0
$193$	−228.006	−1.18138	−0.590689	−	0.806899i	$-0.701144\pi$
$193$	−228.006	−1.18138	−0.590689	+	0.806899i	$0.701144\pi$
$194$	− 125.328i	− 0.646020i
$195$	0	0
$196$	−7.82387	−0.0399177
$197$	− 170.096i	− 0.863429i	−0.902010	−	0.431715i	$-0.857909\pi$
$197$	− 170.096i	− 0.863429i	0.902010	−	0.431715i	$-0.142091\pi$
$198$	0	0
$199$	143.997	0.723602	0.361801	−	0.932255i	$-0.382162\pi$
$199$	143.997	0.723602	0.361801	+	0.932255i	$0.382162\pi$
$200$	− 143.155i	− 0.715777i
$201$	0	0
$202$	128.121	0.634265
$203$	− 319.356i	− 1.57318i
$204$	0	0
$205$	7.95767	0.0388179
$206$	51.7638i	0.251281i
$207$	0	0
$208$	−65.0352	−0.312669
$209$	0	0
$210$	0	0
$211$	120.349	0.570372	0.285186	−	0.958472i	$-0.407945\pi$
$211$	120.349	0.570372	0.285186	+	0.958472i	$0.407945\pi$
$212$	− 128.024i	− 0.603888i
$213$	0	0
$214$	−42.0661	−0.196571
$215$	53.0654i	0.246816i
$216$	0	0
$217$	129.716	0.597771
$218$	51.4928i	0.236205i
$219$	0	0
$220$	0	0
$221$	72.3689i	0.327461i
$222$	0	0
$223$	108.662	0.487273	0.243636	−	0.969867i	$-0.421660\pi$
$223$	108.662	0.487273	0.243636	+	0.969867i	$0.421660\pi$
$224$	− 210.707i	− 0.940655i
$225$	0	0
$226$	70.8692	0.313581
$227$	211.895i	0.933456i	0.884401	+	0.466728i	$0.154567\pi$
$227$	211.895i	0.933456i	−0.884401	+	0.466728i	$0.845433\pi$
$228$	0	0
$229$	278.397	1.21571	0.607853	−	0.794049i	$-0.292031\pi$
$229$	278.397	1.21571	0.607853	+	0.794049i	$0.292031\pi$
$230$	− 4.12481i	− 0.0179339i
$231$	0	0
$232$	−280.012	−1.20695
$233$	189.088i	0.811537i	0.913976	+	0.405769i	$0.132996\pi$
$233$	189.088i	0.811537i	−0.913976	+	0.405769i	$0.867004\pi$
$234$	0	0
$235$	−69.2529	−0.294693
$236$	390.849i	1.65614i
$237$	0	0
$238$	−52.3959	−0.220151
$239$	88.7986i	0.371542i	0.982593	+	0.185771i	$0.0594783\pi$
$239$	88.7986i	0.371542i	−0.982593	+	0.185771i	$0.940522\pi$
$240$	0	0
$241$	−155.144	−0.643750	−0.321875	−	0.946782i	$-0.604313\pi$
$241$	−155.144	−0.643750	−0.321875	+	0.946782i	$0.604313\pi$
$242$	0	0
$243$	0	0
$244$	271.406	1.11232
$245$	− 2.45762i	− 0.0100311i
$246$	0	0
$247$	−72.5795	−0.293844
$248$	− 113.735i	− 0.458611i
$249$	0	0
$250$	41.8149	0.167260
$251$	299.708i	1.19406i	0.802221	+	0.597028i	$0.203652\pi$
$251$	299.708i	1.19406i	−0.802221	+	0.597028i	$0.796348\pi$
$252$	0	0
$253$	0	0
$254$	− 61.7847i	− 0.243247i
$255$	0	0
$256$	−72.2108	−0.282073
$257$	− 370.370i	− 1.44113i	−0.693387	−	0.720565i	$-0.743882\pi$
$257$	− 370.370i	− 1.44113i	0.693387	−	0.720565i	$-0.256118\pi$
$258$	0	0
$259$	−322.272	−1.24429
$260$	− 26.8855i	− 0.103406i
$261$	0	0
$262$	−101.433	−0.387150
$263$	351.921i	1.33810i	0.743217	+	0.669050i	$0.233299\pi$
$263$	351.921i	1.33810i	−0.743217	+	0.669050i	$0.766701\pi$
$264$	0	0
$265$	40.2148	0.151754
$266$	− 52.5483i	− 0.197550i
$267$	0	0
$268$	233.807	0.872415
$269$	− 153.234i	− 0.569644i	−0.958580	−	0.284822i	$-0.908065\pi$
$269$	− 153.234i	− 0.569644i	0.958580	−	0.284822i	$-0.0919345\pi$
$270$	0	0
$271$	−364.367	−1.34453	−0.672264	−	0.740312i	$-0.734678\pi$
$271$	−364.367	−1.34453	−0.672264	+	0.740312i	$0.734678\pi$
$272$	− 79.2868i	− 0.291496i
$273$	0	0
$274$	−43.4148	−0.158448
$275$	0	0
$276$	0	0
$277$	−208.925	−0.754244	−0.377122	−	0.926164i	$-0.623086\pi$
$277$	−208.925	−0.754244	−0.377122	+	0.926164i	$0.623086\pi$
$278$	− 64.5648i	− 0.232247i
$279$	0	0
$280$	42.8261	0.152950
$281$	474.459i	1.68847i	0.535976	+	0.844233i	$0.319944\pi$
$281$	474.459i	1.68847i	−0.535976	+	0.844233i	$0.680056\pi$
$282$	0	0
$283$	172.881	0.610888	0.305444	−	0.952210i	$-0.401195\pi$
$283$	172.881	0.610888	0.305444	+	0.952210i	$0.401195\pi$
$284$	144.197i	0.507737i
$285$	0	0
$286$	0	0
$287$	− 51.9150i	− 0.180888i
$288$	0	0
$289$	200.772	0.694714
$290$	− 39.9783i	− 0.137856i
$291$	0	0
$292$	153.000	0.523971
$293$	380.294i	1.29793i	0.760817	+	0.648966i	$0.224798\pi$
$293$	380.294i	1.29793i	−0.760817	+	0.648966i	$0.775202\pi$
$294$	0	0
$295$	−122.773	−0.416179
$296$	282.569i	0.954625i
$297$	0	0
$298$	17.7430	0.0595404
$299$	37.1677i	0.124307i
$300$	0	0
$301$	346.193	1.15014
$302$	101.342i	0.335571i
$303$	0	0
$304$	79.5175	0.261571
$305$	85.2537i	0.279520i
$306$	0	0
$307$	−343.458	−1.11876	−0.559378	−	0.828912i	$-0.688960\pi$
$307$	−343.458	−1.11876	−0.559378	+	0.828912i	$0.688960\pi$
$308$	0	0
$309$	0	0
$310$	16.2384	0.0523821
$311$	182.587i	0.587096i	0.955944	+	0.293548i	$0.0948361\pi$
$311$	182.587i	0.587096i	−0.955944	+	0.293548i	$0.905164\pi$
$312$	0	0
$313$	−62.5252	−0.199761	−0.0998806	−	0.994999i	$-0.531846\pi$
$313$	−62.5252	−0.199761	−0.0998806	+	0.994999i	$0.531846\pi$
$314$	147.995i	0.471322i
$315$	0	0
$316$	−285.898	−0.904741
$317$	− 194.548i	− 0.613717i	−0.951755	−	0.306859i	$-0.900722\pi$
$317$	− 194.548i	− 0.613717i	0.951755	−	0.306859i	$-0.0992779\pi$
$318$	0	0
$319$	0	0
$320$	8.97285i	0.0280402i
$321$	0	0
$322$	−26.9098	−0.0835708
$323$	− 88.4843i	− 0.273945i
$324$	0	0
$325$	−184.170	−0.566676
$326$	226.037i	0.693365i
$327$	0	0
$328$	−45.5191	−0.138778
$329$	451.799i	1.37325i
$330$	0	0
$331$	−17.6498	−0.0533226	−0.0266613	−	0.999645i	$-0.508488\pi$
$331$	−17.6498	−0.0533226	−0.0266613	+	0.999645i	$0.508488\pi$
$332$	− 530.399i	− 1.59759i
$333$	0	0
$334$	44.5404	0.133355
$335$	73.4432i	0.219233i
$336$	0	0
$337$	−439.894	−1.30532	−0.652662	−	0.757649i	$-0.726348\pi$
$337$	−439.894	−1.30532	−0.652662	+	0.757649i	$0.726348\pi$
$338$	89.5412i	0.264915i
$339$	0	0
$340$	32.7772	0.0964034
$341$	0	0
$342$	0	0
$343$	−350.717	−1.02250
$344$	− 303.543i	− 0.882392i
$345$	0	0
$346$	−82.2664	−0.237764
$347$	407.900i	1.17550i	0.809041	+	0.587752i	$0.199987\pi$
$347$	407.900i	1.17550i	−0.809041	+	0.587752i	$0.800013\pi$
$348$	0	0
$349$	−478.460	−1.37094	−0.685472	−	0.728099i	$-0.740404\pi$
$349$	−478.460	−1.37094	−0.685472	+	0.728099i	$0.740404\pi$
$350$	− 133.341i	− 0.380974i
$351$	0	0
$352$	0	0
$353$	551.674i	1.56282i	0.624020	+	0.781409i	$0.285498\pi$
$353$	551.674i	1.56282i	−0.624020	+	0.781409i	$0.714502\pi$
$354$	0	0
$355$	−45.2951	−0.127592
$356$	− 513.335i	− 1.44195i
$357$	0	0
$358$	−66.8322	−0.186682
$359$	357.980i	0.997159i	0.866844	+	0.498579i	$0.166145\pi$
$359$	357.980i	0.997159i	−0.866844	+	0.498579i	$0.833855\pi$
$360$	0	0
$361$	−272.258	−0.754178
$362$	− 166.439i	− 0.459777i
$363$	0	0
$364$	−175.398	−0.481864
$365$	48.0600i	0.131671i
$366$	0	0
$367$	−105.660	−0.287902	−0.143951	−	0.989585i	$-0.545981\pi$
$367$	−105.660	−0.287902	−0.143951	+	0.989585i	$0.545981\pi$
$368$	− 40.7206i	− 0.110654i
$369$	0	0
$370$	−40.3435	−0.109036
$371$	− 262.357i	− 0.707161i
$372$	0	0
$373$	483.127	1.29525	0.647623	−	0.761961i	$-0.275763\pi$
$373$	483.127	1.29525	0.647623	+	0.761961i	$0.275763\pi$
$374$	0	0
$375$	0	0
$376$	396.138	1.05356
$377$	360.235i	0.955532i
$378$	0	0
$379$	448.279	1.18279	0.591397	−	0.806380i	$-0.298577\pi$
$379$	448.279	1.18279	0.591397	+	0.806380i	$0.298577\pi$
$380$	32.8725i	0.0865066i
$381$	0	0
$382$	164.758	0.431302
$383$	479.726i	1.25255i	0.779603	+	0.626274i	$0.215421\pi$
$383$	479.726i	1.25255i	−0.779603	+	0.626274i	$0.784579\pi$
$384$	0	0
$385$	0	0
$386$	186.210i	0.482409i
$387$	0	0
$388$	511.480	1.31825
$389$	304.721i	0.783344i	0.920105	+	0.391672i	$0.128103\pi$
$389$	304.721i	0.783344i	−0.920105	+	0.391672i	$0.871897\pi$
$390$	0	0
$391$	−45.3125	−0.115889
$392$	14.0580i	0.0358622i
$393$	0	0
$394$	−138.915	−0.352577
$395$	− 89.8059i	− 0.227357i
$396$	0	0
$397$	−433.795	−1.09268	−0.546342	−	0.837562i	$-0.683980\pi$
$397$	−433.795	−1.09268	−0.546342	+	0.837562i	$0.683980\pi$
$398$	− 117.601i	− 0.295479i
$399$	0	0
$400$	201.775	0.504437
$401$	− 53.2849i	− 0.132880i	−0.997790	−	0.0664401i	$-0.978836\pi$
$401$	− 53.2849i	− 0.132880i	0.997790	−	0.0664401i	$-0.0211641\pi$
$402$	0	0
$403$	−146.321	−0.363079
$404$	522.881i	1.29426i
$405$	0	0
$406$	−260.814	−0.642400
$407$	0	0
$408$	0	0
$409$	−119.713	−0.292697	−0.146349	−	0.989233i	$-0.546752\pi$
$409$	−119.713	−0.292697	−0.146349	+	0.989233i	$0.546752\pi$
$410$	− 6.49894i	− 0.0158511i
$411$	0	0
$412$	−211.255	−0.512755
$413$	800.957i	1.93936i
$414$	0	0
$415$	166.608	0.401466
$416$	237.679i	0.571343i
$417$	0	0
$418$	0	0
$419$	− 165.561i	− 0.395133i	−0.980289	−	0.197566i	$-0.936696\pi$
$419$	− 165.561i	− 0.395133i	0.980289	−	0.197566i	$-0.0633038\pi$
$420$	0	0
$421$	−3.79320	−0.00900999	−0.00450499	−	0.999990i	$-0.501434\pi$
$421$	−3.79320	−0.00900999	−0.00450499	+	0.999990i	$0.501434\pi$
$422$	− 98.2873i	− 0.232908i
$423$	0	0
$424$	−230.035	−0.542535
$425$	− 224.528i	− 0.528301i
$426$	0	0
$427$	556.186	1.30254
$428$	− 171.678i	− 0.401116i
$429$	0	0
$430$	43.3380	0.100786
$431$	154.611i	0.358726i	0.983783	+	0.179363i	$0.0574037\pi$
$431$	154.611i	0.358726i	−0.983783	+	0.179363i	$0.942596\pi$
$432$	0	0
$433$	−265.483	−0.613124	−0.306562	−	0.951851i	$-0.599179\pi$
$433$	−265.483	−0.613124	−0.306562	+	0.951851i	$0.599179\pi$
$434$	− 105.938i	− 0.244096i
$435$	0	0
$436$	−210.149	−0.481993
$437$	− 45.4443i	− 0.103992i
$438$	0	0
$439$	−716.385	−1.63186	−0.815928	−	0.578153i	$-0.803774\pi$
$439$	−716.385	−1.63186	−0.815928	+	0.578153i	$0.803774\pi$
$440$	0	0
$441$	0	0
$442$	59.1029	0.133717
$443$	− 41.7661i	− 0.0942801i	−0.998888	−	0.0471400i	$-0.984989\pi$
$443$	− 41.7661i	− 0.0942801i	0.998888	−	0.0471400i	$-0.0150107\pi$
$444$	0	0
$445$	161.248	0.362355
$446$	− 88.7429i	− 0.198975i
$447$	0	0
$448$	58.5379	0.130665
$449$	265.740i	0.591848i	0.955211	+	0.295924i	$0.0956275\pi$
$449$	265.740i	0.591848i	−0.955211	+	0.295924i	$0.904372\pi$
$450$	0	0
$451$	0	0
$452$	289.227i	0.639882i
$453$	0	0
$454$	173.052	0.381172
$455$	− 55.0959i	− 0.121090i
$456$	0	0
$457$	−427.961	−0.936457	−0.468228	−	0.883607i	$-0.655108\pi$
$457$	−427.961	−0.936457	−0.468228	+	0.883607i	$0.655108\pi$
$458$	− 227.364i	− 0.496427i
$459$	0	0
$460$	16.8339	0.0365954
$461$	313.392i	0.679809i	0.940460	+	0.339904i	$0.110395\pi$
$461$	313.392i	0.679809i	−0.940460	+	0.339904i	$0.889605\pi$
$462$	0	0
$463$	−529.774	−1.14422	−0.572110	−	0.820177i	$-0.693875\pi$
$463$	−529.774	−1.14422	−0.572110	+	0.820177i	$0.693875\pi$
$464$	− 394.671i	− 0.850584i
$465$	0	0
$466$	154.426	0.331387
$467$	− 69.3305i	− 0.148459i	−0.997241	−	0.0742297i	$-0.976350\pi$
$467$	− 69.3305i	− 0.148459i	0.997241	−	0.0742297i	$-0.0236498\pi$
$468$	0	0
$469$	479.135	1.02161
$470$	56.5581i	0.120336i
$471$	0	0
$472$	702.281	1.48788
$473$	0	0
$474$	0	0
$475$	225.181	0.474066
$476$	− 213.835i	− 0.449233i
$477$	0	0
$478$	72.5209	0.151717
$479$	20.8402i	0.0435078i	0.999763	+	0.0217539i	$0.00692503\pi$
$479$	20.8402i	0.0435078i	−0.999763	+	0.0217539i	$0.993075\pi$
$480$	0	0
$481$	363.526	0.755770
$482$	126.704i	0.262872i
$483$	0	0
$484$	0	0
$485$	160.665i	0.331269i
$486$	0	0
$487$	83.2183	0.170879	0.0854397	−	0.996343i	$-0.472770\pi$
$487$	83.2183	0.170879	0.0854397	+	0.996343i	$0.472770\pi$
$488$	− 487.665i	− 0.999313i
$489$	0	0
$490$	−2.00711	−0.00409615
$491$	97.3072i	0.198182i	0.995078	+	0.0990908i	$0.0315934\pi$
$491$	97.3072i	0.198182i	−0.995078	+	0.0990908i	$0.968407\pi$
$492$	0	0
$493$	−439.176	−0.890824
$494$	59.2749i	0.119990i
$495$	0	0
$496$	160.308	0.323201
$497$	295.500i	0.594567i
$498$	0	0
$499$	304.369	0.609959	0.304979	−	0.952359i	$-0.401350\pi$
$499$	304.369	0.609959	0.304979	+	0.952359i	$0.401350\pi$
$500$	170.652i	0.341305i
$501$	0	0
$502$	244.768	0.487586
$503$	− 710.221i	− 1.41197i	−0.708227	−	0.705985i	$-0.750505\pi$
$503$	− 710.221i	− 1.41197i	0.708227	−	0.705985i	$-0.249495\pi$
$504$	0	0
$505$	−164.247	−0.325241
$506$	0	0
$507$	0	0
$508$	252.152	0.496361
$509$	− 758.041i	− 1.48927i	−0.667470	−	0.744637i	$-0.732623\pi$
$509$	− 758.041i	− 1.48927i	0.667470	−	0.744637i	$-0.267377\pi$
$510$	0	0
$511$	313.538	0.613578
$512$	− 462.606i	− 0.903528i
$513$	0	0
$514$	−302.478	−0.588478
$515$	− 66.3591i	− 0.128853i
$516$	0	0
$517$	0	0
$518$	263.196i	0.508101i
$519$	0	0
$520$	−48.3082	−0.0929004
$521$	270.903i	0.519968i	0.965613	+	0.259984i	$0.0837173\pi$
$521$	270.903i	0.519968i	−0.965613	+	0.259984i	$0.916283\pi$
$522$	0	0
$523$	303.071	0.579485	0.289743	−	0.957105i	$-0.406430\pi$
$523$	303.071	0.579485	0.289743	+	0.957105i	$0.406430\pi$
$524$	− 413.963i	− 0.790006i
$525$	0	0
$526$	287.410	0.546406
$527$	− 178.385i	− 0.338492i
$528$	0	0
$529$	505.728	0.956008
$530$	− 32.8430i	− 0.0619679i
$531$	0	0
$532$	214.457	0.403114
$533$	58.5604i	0.109870i
$534$	0	0
$535$	53.9271	0.100798
$536$	− 420.107i	− 0.783782i
$537$	0	0
$538$	−125.145	−0.232611
$539$	0	0
$540$	0	0
$541$	−1038.68	−1.91992	−0.959962	−	0.280130i	$-0.909622\pi$
$541$	−1038.68	−1.91992	−0.959962	+	0.280130i	$0.909622\pi$
$542$	297.575i	0.549031i
$543$	0	0
$544$	−289.763	−0.532652
$545$	− 66.0117i	− 0.121122i
$546$	0	0
$547$	297.137	0.543212	0.271606	−	0.962409i	$-0.412445\pi$
$547$	297.137	0.543212	0.271606	+	0.962409i	$0.412445\pi$
$548$	− 177.182i	− 0.323324i
$549$	0	0
$550$	0	0
$551$	− 440.454i	− 0.799372i
$552$	0	0
$553$	−585.884	−1.05946
$554$	170.627i	0.307991i
$555$	0	0
$556$	263.498	0.473916
$557$	533.625i	0.958034i	0.877806	+	0.479017i	$0.159007\pi$
$557$	533.625i	0.958034i	−0.877806	+	0.479017i	$0.840993\pi$
$558$	0	0
$559$	−390.508	−0.698584
$560$	60.3626i	0.107790i
$561$	0	0
$562$	387.486	0.689476
$563$	211.503i	0.375672i	0.982200	+	0.187836i	$0.0601473\pi$
$563$	211.503i	0.375672i	−0.982200	+	0.187836i	$0.939853\pi$
$564$	0	0
$565$	−90.8515	−0.160799
$566$	− 141.190i	− 0.249453i
$567$	0	0
$568$	259.095	0.456153
$569$	721.732i	1.26842i	0.773160	+	0.634211i	$0.218675\pi$
$569$	721.732i	1.26842i	−0.773160	+	0.634211i	$0.781325\pi$
$570$	0	0
$571$	798.387	1.39823	0.699113	−	0.715011i	$-0.253578\pi$
$571$	798.387	1.39823	0.699113	+	0.715011i	$0.253578\pi$
$572$	0	0
$573$	0	0
$574$	−42.3984	−0.0738648
$575$	− 115.314i	− 0.200547i
$576$	0	0
$577$	−589.480	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$577$	−589.480	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$578$	− 163.969i	− 0.283683i
$579$	0	0
$580$	163.157	0.281305
$581$	− 1086.93i	− 1.87080i
$582$	0	0
$583$	0	0
$584$	− 274.911i	− 0.470738i
$585$	0	0
$586$	310.582	0.530004
$587$	126.283i	0.215133i	0.994198	+	0.107567i	$0.0343059\pi$
$587$	126.283i	0.215133i	−0.994198	+	0.107567i	$0.965694\pi$
$588$	0	0
$589$	178.904	0.303742
$590$	100.267i	0.169945i
$591$	0	0
$592$	−398.275	−0.672763
$593$	− 447.206i	− 0.754141i	−0.926185	−	0.377071i	$-0.876931\pi$
$593$	− 447.206i	− 0.754141i	0.926185	−	0.377071i	$-0.123069\pi$
$594$	0	0
$595$	67.1694	0.112890
$596$	72.4118i	0.121496i
$597$	0	0
$598$	30.3544	0.0507599
$599$	− 656.278i	− 1.09562i	−0.836602	−	0.547811i	$-0.815461\pi$
$599$	− 656.278i	− 1.09562i	0.836602	−	0.547811i	$-0.184539\pi$
$600$	0	0
$601$	−200.574	−0.333734	−0.166867	−	0.985979i	$-0.553365\pi$
$601$	−200.574	−0.333734	−0.166867	+	0.985979i	$0.553365\pi$
$602$	− 282.732i	− 0.469655i
$603$	0	0
$604$	−413.592	−0.684755
$605$	0	0
$606$	0	0
$607$	907.748	1.49547	0.747733	−	0.664000i	$-0.231142\pi$
$607$	907.748	1.49547	0.747733	+	0.664000i	$0.231142\pi$
$608$	− 290.606i	− 0.477970i
$609$	0	0
$610$	69.6258	0.114141
$611$	− 509.632i	− 0.834095i
$612$	0	0
$613$	−776.875	−1.26733	−0.633666	−	0.773606i	$-0.718451\pi$
$613$	−776.875	−1.26733	−0.633666	+	0.773606i	$0.718451\pi$
$614$	280.499i	0.456838i
$615$	0	0
$616$	0	0
$617$	− 490.222i	− 0.794525i	−0.917705	−	0.397263i	$-0.869960\pi$
$617$	− 490.222i	− 0.794525i	0.917705	−	0.397263i	$-0.130040\pi$
$618$	0	0
$619$	−299.087	−0.483178	−0.241589	−	0.970379i	$-0.577669\pi$
$619$	−299.087	−0.483178	−0.241589	+	0.970379i	$0.577669\pi$
$620$	66.2713i	0.106889i
$621$	0	0
$622$	149.117	0.239738
$623$	− 1051.97i	− 1.68855i
$624$	0	0
$625$	543.992	0.870387
$626$	51.0637i	0.0815714i
$627$	0	0
$628$	−603.989	−0.961765
$629$	443.187i	0.704591i
$630$	0	0
$631$	−844.049	−1.33764	−0.668819	−	0.743426i	$-0.733200\pi$
$631$	−844.049	−1.33764	−0.668819	+	0.743426i	$0.733200\pi$
$632$	513.704i	0.812823i
$633$	0	0
$634$	−158.886	−0.250608
$635$	79.2055i	0.124733i
$636$	0	0
$637$	18.0856	0.0283919
$638$	0	0
$639$	0	0
$640$	136.519	0.213311
$641$	962.296i	1.50124i	0.660733	+	0.750621i	$0.270246\pi$
$641$	962.296i	1.50124i	−0.660733	+	0.750621i	$0.729754\pi$
$642$	0	0
$643$	26.2872	0.0408821	0.0204411	−	0.999791i	$-0.493493\pi$
$643$	26.2872	0.0408821	0.0204411	+	0.999791i	$0.493493\pi$
$644$	− 109.823i	− 0.170532i
$645$	0	0
$646$	−72.2642	−0.111864
$647$	− 293.755i	− 0.454027i	−0.973892	−	0.227013i	$-0.927104\pi$
$647$	− 293.755i	− 0.454027i	0.973892	−	0.227013i	$-0.0728961\pi$
$648$	0	0
$649$	0	0
$650$	150.409i	0.231399i
$651$	0	0
$652$	−922.487	−1.41486
$653$	600.075i	0.918952i	0.888190	+	0.459476i	$0.151963\pi$
$653$	600.075i	0.918952i	−0.888190	+	0.459476i	$0.848037\pi$
$654$	0	0
$655$	130.034	0.198524
$656$	− 64.1583i	− 0.0978023i
$657$	0	0
$658$	368.979	0.560759
$659$	19.2417i	0.0291983i	0.999893	+	0.0145992i	$0.00464722\pi$
$659$	19.2417i	0.0291983i	−0.999893	+	0.0145992i	$0.995353\pi$
$660$	0	0
$661$	543.444	0.822154	0.411077	−	0.911601i	$-0.365153\pi$
$661$	543.444	0.822154	0.411077	+	0.911601i	$0.365153\pi$
$662$	14.4144i	0.0217740i
$663$	0	0
$664$	−953.026	−1.43528
$665$	67.3648i	0.101301i
$666$	0	0
$667$	−225.555	−0.338163
$668$	181.775i	0.272119i
$669$	0	0
$670$	59.9803	0.0895228
$671$	0	0
$672$	0	0
$673$	1.44765	0.00215105	0.00107552	−	0.999999i	$-0.499658\pi$
$673$	1.44765	0.00215105	0.00107552	+	0.999999i	$0.499658\pi$
$674$	359.257i	0.533022i
$675$	0	0
$676$	−365.430	−0.540576
$677$	− 1049.30i	− 1.54993i	−0.632004	−	0.774965i	$-0.717767\pi$
$677$	− 1049.30i	− 1.54993i	0.632004	−	0.774965i	$-0.282233\pi$
$678$	0	0
$679$	1048.16	1.54369
$680$	− 58.8943i	− 0.0866093i
$681$	0	0
$682$	0	0
$683$	− 810.057i	− 1.18603i	−0.805192	−	0.593014i	$-0.797938\pi$
$683$	− 810.057i	− 1.18603i	0.805192	−	0.593014i	$-0.202062\pi$
$684$	0	0
$685$	55.6561	0.0812498
$686$	286.426i	0.417531i
$687$	0	0
$688$	427.837	0.621857
$689$	295.940i	0.429522i
$690$	0	0
$691$	−106.488	−0.154107	−0.0770534	−	0.997027i	$-0.524551\pi$
$691$	−106.488	−0.154107	−0.0770534	+	0.997027i	$0.524551\pi$
$692$	− 335.740i	− 0.485174i
$693$	0	0
$694$	333.127	0.480011
$695$	82.7694i	0.119093i
$696$	0	0
$697$	−71.3932	−0.102429
$698$	390.753i	0.559818i
$699$	0	0
$700$	544.182	0.777402
$701$	1088.88i	1.55332i	0.629921	+	0.776659i	$0.283087\pi$
$701$	1088.88i	1.55332i	−0.629921	+	0.776659i	$0.716913\pi$
$702$	0	0
$703$	−444.477	−0.632257
$704$	0	0
$705$	0	0
$706$	450.547	0.638168
$707$	1071.53i	1.51560i
$708$	0	0
$709$	58.7724	0.0828948	0.0414474	−	0.999141i	$-0.486803\pi$
$709$	58.7724	0.0828948	0.0414474	+	0.999141i	$0.486803\pi$
$710$	36.9920i	0.0521014i
$711$	0	0
$712$	−922.365	−1.29546
$713$	− 91.6161i	− 0.128494i
$714$	0	0
$715$	0	0
$716$	− 272.751i	− 0.380938i
$717$	0	0
$718$	292.358	0.407184
$719$	728.639i	1.01341i	0.862121	+	0.506703i	$0.169136\pi$
$719$	728.639i	1.01341i	−0.862121	+	0.506703i	$0.830864\pi$
$720$	0	0
$721$	−432.920	−0.600444
$722$	222.350i	0.307965i
$723$	0	0
$724$	679.261	0.938205
$725$	− 1117.65i	− 1.54158i
$726$	0	0
$727$	1161.64	1.59785	0.798925	−	0.601431i	$-0.205403\pi$
$727$	1161.64	1.59785	0.798925	+	0.601431i	$0.205403\pi$
$728$	315.157i	0.432908i
$729$	0	0
$730$	39.2501	0.0537672
$731$	− 476.083i	− 0.651276i
$732$	0	0
$733$	233.736	0.318876	0.159438	−	0.987208i	$-0.449032\pi$
$733$	233.736	0.318876	0.159438	+	0.987208i	$0.449032\pi$
$734$	86.2914i	0.117563i
$735$	0	0
$736$	−148.818	−0.202198
$737$	0	0
$738$	0	0
$739$	−749.941	−1.01481	−0.507403	−	0.861709i	$-0.669394\pi$
$739$	−749.941	−1.01481	−0.507403	+	0.861709i	$0.669394\pi$
$740$	− 164.647i	− 0.222496i
$741$	0	0
$742$	−214.264	−0.288765
$743$	− 534.169i	− 0.718936i	−0.933157	−	0.359468i	$-0.882958\pi$
$743$	− 534.169i	− 0.718936i	0.933157	−	0.359468i	$-0.117042\pi$
$744$	0	0
$745$	−22.7459	−0.0305314
$746$	− 394.565i	− 0.528907i
$747$	0	0
$748$	0	0
$749$	− 351.815i	− 0.469712i
$750$	0	0
$751$	−1188.62	−1.58272	−0.791358	−	0.611354i	$-0.790625\pi$
$751$	−1188.62	−1.58272	−0.791358	+	0.611354i	$0.790625\pi$
$752$	558.349i	0.742485i
$753$	0	0
$754$	294.200	0.390186
$755$	− 129.917i	− 0.172075i
$756$	0	0
$757$	148.360	0.195984	0.0979919	−	0.995187i	$-0.468758\pi$
$757$	148.360	0.195984	0.0979919	+	0.995187i	$0.468758\pi$
$758$	− 366.105i	− 0.482988i
$759$	0	0
$760$	59.0656	0.0777179
$761$	− 197.346i	− 0.259325i	−0.991558	−	0.129662i	$-0.958611\pi$
$761$	− 197.346i	− 0.259325i	0.991558	−	0.129662i	$-0.0413894\pi$
$762$	0	0
$763$	−430.653	−0.564421
$764$	672.398i	0.880102i
$765$	0	0
$766$	391.787	0.511471
$767$	− 903.485i	− 1.17795i
$768$	0	0
$769$	1174.93	1.52787	0.763934	−	0.645294i	$-0.223265\pi$
$769$	1174.93	1.52787	0.763934	+	0.645294i	$0.223265\pi$
$770$	0	0
$771$	0	0
$772$	−759.948	−0.984389
$773$	− 716.104i	− 0.926396i	−0.886255	−	0.463198i	$-0.846702\pi$
$773$	− 716.104i	− 0.926396i	0.886255	−	0.463198i	$-0.153298\pi$
$774$	0	0
$775$	453.967	0.585764
$776$	− 919.032i	− 1.18432i
$777$	0	0
$778$	248.862	0.319874
$779$	− 71.6009i	− 0.0919139i
$780$	0	0
$781$	0	0
$782$	37.0062i	0.0473225i
$783$	0	0
$784$	−19.8145	−0.0252735
$785$	− 189.724i	− 0.241687i
$786$	0	0
$787$	654.650	0.831829	0.415915	−	0.909404i	$-0.363462\pi$
$787$	654.650	0.831829	0.415915	+	0.909404i	$0.363462\pi$
$788$	− 566.932i	− 0.719457i
$789$	0	0
$790$	−73.3435	−0.0928399
$791$	592.705i	0.749311i
$792$	0	0
$793$	−627.382	−0.791149
$794$	354.276i	0.446191i
$795$	0	0
$796$	479.944	0.602945
$797$	519.505i	0.651826i	0.945400	+	0.325913i	$0.105672\pi$
$797$	519.505i	0.651826i	−0.945400	+	0.325913i	$0.894328\pi$
$798$	0	0
$799$	621.311	0.777611
$800$	− 737.409i	− 0.921761i
$801$	0	0
$802$	−43.5172	−0.0542609
$803$	0	0
$804$	0	0
$805$	34.4973	0.0428538
$806$	119.499i	0.148261i
$807$	0	0
$808$	939.517	1.16277
$809$	− 955.949i	− 1.18164i	−0.806802	−	0.590821i	$-0.798804\pi$
$809$	− 955.949i	− 1.18164i	0.806802	−	0.590821i	$-0.201196\pi$
$810$	0	0
$811$	−301.766	−0.372091	−0.186045	−	0.982541i	$-0.559567\pi$
$811$	−301.766	−0.372091	−0.186045	+	0.982541i	$0.559567\pi$
$812$	− 1064.42i	− 1.31086i
$813$	0	0
$814$	0	0
$815$	− 289.770i	− 0.355546i
$816$	0	0
$817$	477.468	0.584416
$818$	97.7685i	0.119521i
$819$	0	0
$820$	26.5231	0.0323452
$821$	230.665i	0.280956i	0.990084	+	0.140478i	$0.0448640\pi$
$821$	230.665i	0.280956i	−0.990084	+	0.140478i	$0.955136\pi$
$822$	0	0
$823$	−128.308	−0.155903	−0.0779516	−	0.996957i	$-0.524838\pi$
$823$	−128.308	−0.155903	−0.0779516	+	0.996957i	$0.524838\pi$
$824$	379.585i	0.460661i
$825$	0	0
$826$	654.133	0.791929
$827$	− 659.445i	− 0.797395i	−0.917083	−	0.398697i	$-0.869462\pi$
$827$	− 659.445i	− 0.797395i	0.917083	−	0.398697i	$-0.130538\pi$
$828$	0	0
$829$	404.300	0.487696	0.243848	−	0.969813i	$-0.421590\pi$
$829$	404.300	0.487696	0.243848	+	0.969813i	$0.421590\pi$
$830$	− 136.067i	− 0.163936i
$831$	0	0
$832$	−66.0312	−0.0793644
$833$	22.0489i	0.0264692i
$834$	0	0
$835$	−57.0990	−0.0683821
$836$	0	0
$837$	0	0
$838$	−135.212	−0.161350
$839$	− 465.824i	− 0.555213i	−0.960695	−	0.277607i	$-0.910459\pi$
$839$	− 465.824i	− 0.555213i	0.960695	−	0.277607i	$-0.0895412\pi$
$840$	0	0
$841$	−1345.12	−1.59942
$842$	3.09787i	0.00367918i
$843$	0	0
$844$	401.124	0.475265
$845$	− 114.788i	− 0.135844i
$846$	0	0
$847$	0	0
$848$	− 324.230i	− 0.382346i
$849$	0	0
$850$	−183.370	−0.215729
$851$	227.615i	0.267468i
$852$	0	0
$853$	1469.47	1.72271	0.861354	−	0.508005i	$-0.169617\pi$
$853$	1469.47	1.72271	0.861354	+	0.508005i	$0.169617\pi$
$854$	− 454.231i	− 0.531886i
$855$	0	0
$856$	−308.472	−0.360364
$857$	− 297.274i	− 0.346877i	−0.984845	−	0.173439i	$-0.944512\pi$
$857$	− 297.274i	− 0.346877i	0.984845	−	0.173439i	$-0.0554878\pi$
$858$	0	0
$859$	38.4367	0.0447459	0.0223729	−	0.999750i	$-0.492878\pi$
$859$	38.4367	0.0447459	0.0223729	+	0.999750i	$0.492878\pi$
$860$	176.868i	0.205661i
$861$	0	0
$862$	126.269	0.146484
$863$	− 996.462i	− 1.15465i	−0.816515	−	0.577324i	$-0.804097\pi$
$863$	− 996.462i	− 1.15465i	0.816515	−	0.577324i	$-0.195903\pi$
$864$	0	0
$865$	105.462	0.121922
$866$	216.817i	0.250366i
$867$	0	0
$868$	432.347	0.498095
$869$	0	0
$870$	0	0
$871$	−540.468	−0.620515
$872$	377.598i	0.433025i
$873$	0	0
$874$	−37.1139	−0.0424644
$875$	349.714i	0.399673i
$876$	0	0
$877$	523.294	0.596687	0.298343	−	0.954459i	$-0.403566\pi$
$877$	523.294	0.596687	0.298343	+	0.954459i	$0.403566\pi$
$878$	585.064i	0.666360i
$879$	0	0
$880$	0	0
$881$	− 360.136i	− 0.408780i	−0.978889	−	0.204390i	$-0.934479\pi$
$881$	− 360.136i	− 0.408780i	0.978889	−	0.204390i	$-0.0655211\pi$
$882$	0	0
$883$	1072.08	1.21413	0.607065	−	0.794652i	$-0.292347\pi$
$883$	1072.08	1.21413	0.607065	+	0.794652i	$0.292347\pi$
$884$	241.207i	0.272859i
$885$	0	0
$886$	−34.1099	−0.0384988
$887$	− 1098.46i	− 1.23839i	−0.785235	−	0.619197i	$-0.787458\pi$
$887$	− 1098.46i	− 1.23839i	0.785235	−	0.619197i	$-0.212542\pi$
$888$	0	0
$889$	516.728	0.581246
$890$	− 131.690i	− 0.147966i
$891$	0	0
$892$	362.172	0.406022
$893$	623.119i	0.697782i
$894$	0	0
$895$	85.6762	0.0957276
$896$	− 890.634i	− 0.994011i
$897$	0	0
$898$	217.027	0.241678
$899$	− 887.959i	− 0.987719i
$900$	0	0
$901$	−360.792	−0.400435
$902$	0	0
$903$	0	0
$904$	519.685	0.574873
$905$	213.368i	0.235766i
$906$	0	0
$907$	679.995	0.749719	0.374860	−	0.927082i	$-0.377691\pi$
$907$	679.995	0.749719	0.374860	+	0.927082i	$0.377691\pi$
$908$	706.249i	0.777807i
$909$	0	0
$910$	−44.9962	−0.0494464
$911$	31.1262i	0.0341671i	0.999854	+	0.0170836i	$0.00543813\pi$
$911$	31.1262i	0.0341671i	−0.999854	+	0.0170836i	$0.994562\pi$
$912$	0	0
$913$	0	0
$914$	349.511i	0.382397i
$915$	0	0
$916$	927.902	1.01299
$917$	− 848.325i	− 0.925109i
$918$	0	0
$919$	−64.0998	−0.0697495	−0.0348747	−	0.999392i	$-0.511103\pi$
$919$	−64.0998	−0.0697495	−0.0348747	+	0.999392i	$0.511103\pi$
$920$	− 30.2473i	− 0.0328775i
$921$	0	0
$922$	255.944	0.277596
$923$	− 333.326i	− 0.361133i
$924$	0	0
$925$	−1127.86	−1.21930
$926$	432.661i	0.467236i
$927$	0	0
$928$	−1442.37	−1.55428
$929$	1572.05i	1.69219i	0.533030	+	0.846096i	$0.321053\pi$
$929$	1572.05i	1.69219i	−0.533030	+	0.846096i	$0.678947\pi$
$930$	0	0
$931$	−22.1130	−0.0237519
$932$	630.235i	0.676217i
$933$	0	0
$934$	−56.6215	−0.0606226
$935$	0	0
$936$	0	0
$937$	196.014	0.209193	0.104597	−	0.994515i	$-0.466645\pi$
$937$	196.014	0.209193	0.104597	+	0.994515i	$0.466645\pi$
$938$	− 391.305i	− 0.417169i
$939$	0	0
$940$	−230.821	−0.245555
$941$	477.070i	0.506982i	0.967338	+	0.253491i	$0.0815788\pi$
$941$	477.070i	0.506982i	−0.967338	+	0.253491i	$0.918421\pi$
$942$	0	0
$943$	−36.6666	−0.0388829
$944$	989.851i	1.04857i
$945$	0	0
$946$	0	0
$947$	− 151.161i	− 0.159621i	−0.996810	−	0.0798106i	$-0.974568\pi$
$947$	− 151.161i	− 0.159621i	0.996810	−	0.0798106i	$-0.0254316\pi$
$948$	0	0
$949$	−353.673	−0.372680
$950$	− 183.903i	− 0.193582i
$951$	0	0
$952$	−384.220	−0.403592
$953$	− 1817.26i	− 1.90689i	−0.301575	−	0.953443i	$-0.597512\pi$
$953$	− 1817.26i	− 1.90689i	0.301575	−	0.953443i	$-0.402488\pi$
$954$	0	0
$955$	−211.213	−0.221165
$956$	295.967i	0.309589i
$957$	0	0
$958$	17.0200	0.0177662
$959$	− 363.094i	− 0.378618i
$960$	0	0
$961$	−600.328	−0.624691
$962$	− 296.887i	− 0.308615i
$963$	0	0
$964$	−517.097	−0.536408
$965$	− 238.714i	− 0.247372i
$966$	0	0
$967$	1457.23	1.50696	0.753480	−	0.657471i	$-0.228374\pi$
$967$	1457.23	1.50696	0.753480	+	0.657471i	$0.228374\pi$
$968$	0	0
$969$	0	0
$970$	131.214	0.135272
$971$	500.940i	0.515901i	0.966158	+	0.257950i	$0.0830471\pi$
$971$	500.940i	0.515901i	−0.966158	+	0.257950i	$0.916953\pi$
$972$	0	0
$973$	539.979	0.554963
$974$	− 67.9635i	− 0.0697777i
$975$	0	0
$976$	687.354	0.704256
$977$	253.634i	0.259605i	0.991540	+	0.129802i	$0.0414343\pi$
$977$	253.634i	0.259605i	−0.991540	+	0.129802i	$0.958566\pi$
$978$	0	0
$979$	0	0
$980$	− 8.19130i	− 0.00835847i
$981$	0	0
$982$	79.4697	0.0809264
$983$	660.173i	0.671590i	0.941935	+	0.335795i	$0.109005\pi$
$983$	660.173i	0.671590i	−0.941935	+	0.335795i	$0.890995\pi$
$984$	0	0
$985$	178.084	0.180796
$986$	358.671i	0.363763i
$987$	0	0
$988$	−241.909	−0.244847
$989$	− 244.510i	− 0.247229i
$990$	0	0
$991$	33.5854	0.0338905	0.0169452	−	0.999856i	$-0.494606\pi$
$991$	33.5854	0.0338905	0.0169452	+	0.999856i	$0.494606\pi$
$992$	− 585.864i	− 0.590588i
$993$	0	0
$994$	241.332	0.242788
$995$	150.759i	0.151517i
$996$	0	0
$997$	−1195.01	−1.19860	−0.599301	−	0.800524i	$-0.704555\pi$
$997$	−1195.01	−1.19860	−0.599301	+	0.800524i	$0.704555\pi$
$998$	− 248.575i	− 0.249073i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.i.485.7		16
3.2	odd	2	inner	1089.3.b.i.485.10		16
11.5	even	5		99.3.l.a.80.5	yes	32
11.9	even	5		99.3.l.a.26.4	✓	32
11.10	odd	2		1089.3.b.j.485.10		16
33.5	odd	10		99.3.l.a.80.4	yes	32
33.20	odd	10		99.3.l.a.26.5	yes	32
33.32	even	2		1089.3.b.j.485.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.26.4	✓	32	11.9	even	5
99.3.l.a.26.5	yes	32	33.20	odd	10
99.3.l.a.80.4	yes	32	33.5	odd	10
99.3.l.a.80.5	yes	32	11.5	even	5
1089.3.b.i.485.7		16	1.1	even	1	trivial
1089.3.b.i.485.10		16	3.2	odd	2	inner
1089.3.b.j.485.7		16	33.32	even	2
1089.3.b.j.485.10		16	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.816689i q^{2} +3.33302 q^{4} +1.04696i q^{5} +6.83027 q^{7} -5.98880i q^{8} +O(q^{10})\)	\(q-0.816689i q^{2} +3.33302 q^{4} +1.04696i q^{5} +6.83027 q^{7} -5.98880i q^{8} +0.855043 q^{10} -7.70460 q^{13} -5.57821i q^{14} +8.44109 q^{16} -9.39296i q^{17} +9.42028 q^{19} +3.48955i q^{20} -4.82409i q^{23} +23.9039 q^{25} +6.29226i q^{26} +22.7654 q^{28} -46.7559i q^{29} +18.9914 q^{31} -30.8489i q^{32} -7.67113 q^{34} +7.15104i q^{35} -47.1829 q^{37} -7.69344i q^{38} +6.27005 q^{40} -7.60072i q^{41} +50.6851 q^{43} -3.93978 q^{46} +66.1465i q^{47} -2.34738 q^{49} -19.5220i q^{50} -25.6796 q^{52} -38.4109i q^{53} -40.9051i q^{56} -38.1851 q^{58} +117.266i q^{59} +81.4295 q^{61} -15.5100i q^{62} +8.57036 q^{64} -8.06643i q^{65} +70.1488 q^{67} -31.3069i q^{68} +5.84018 q^{70} +43.2633i q^{71} +45.9042 q^{73} +38.5338i q^{74} +31.3980 q^{76} -85.7775 q^{79} +8.83751i q^{80} -6.20742 q^{82} -159.135i q^{83} +9.83408 q^{85} -41.3940i q^{86} -154.015i q^{89} -52.6245 q^{91} -16.0788i q^{92} +54.0211 q^{94} +9.86269i q^{95} +153.459 q^{97} +1.91708i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 - 8 * q^7	\( 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 - 8 * q^7 + 24 * q^10 + 4 * q^13 + 28 * q^16 - 20 * q^19 - 44 * q^25 + 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 + 224 * q^40 + 272 * q^43 + 208 * q^46 + 348 * q^49 + 520 * q^52 - 44 * q^58 + 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 - 4 * q^73 + 1052 * q^76 + 216 * q^79 + 348 * q^82 + 416 * q^85 - 168 * q^91 + 1140 * q^94 - 44 * q^97

Introduction

L-functions

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