LMFDB - Embedded newform 1089.3.b.j.485.10

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

$q$-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	$0.0654503\pi$
$2$	0.816689i	0.408345i	−0.978935	+	0.204172i	$0.934550\pi$
$3$	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	$-0.662228\pi$
$7$	−6.83027	−0.975753	−0.487877	+	0.872913i	$0.662228\pi$
$8$	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	$-0.404237\pi$
$13$	7.70460	0.592661	0.296331	+	0.955085i	$0.404237\pi$
$14$	− 5.57821i	− 0.398444i
$15$	0	0
$16$	8.44109	0.527568
$17$	9.39296i	0.552527i	0.961082	+	0.276263i	$0.0890962\pi$
$17$	9.39296i	0.552527i	−0.961082	+	0.276263i	$0.910904\pi$
$18$	0	0
$19$	−9.42028	−0.495804	−0.247902	−	0.968785i	$-0.579741\pi$
$19$	−9.42028	−0.495804	−0.247902	+	0.968785i	$0.579741\pi$
$20$	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	$0.298445\pi$
$29$	46.7559i	1.61227i	−0.591730	+	0.806137i	$0.701555\pi$
$30$	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	$0.0295471\pi$
$41$	7.60072i	0.185383i	−0.995695	+	0.0926917i	$0.970453\pi$
$42$	0	0
$43$	−50.6851	−1.17872	−0.589362	−	0.807869i	$-0.700621\pi$
$43$	−50.6851	−1.17872	−0.589362	+	0.807869i	$0.700621\pi$
$44$	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	$-0.732616\pi$
$61$	−81.4295	−1.33491	−0.667455	+	0.744650i	$0.732616\pi$
$62$	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	$-0.601807\pi$
$73$	−45.9042	−0.628825	−0.314412	+	0.949287i	$0.601807\pi$
$74$	− 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	$-0.317328\pi$
$79$	85.7775	1.08579	0.542896	+	0.839800i	$0.317328\pi$
$80$	8.83751i	0.110469i
$81$	0	0
$82$	−6.20742	−0.0757003
$83$	159.135i	1.91729i	0.284610	+	0.958643i	$0.408136\pi$
$83$	159.135i	1.91729i	−0.284610	+	0.958643i	$0.591864\pi$
$84$	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	$-0.716928\pi$
$101$	− 156.879i	− 1.55326i	0.629958	−	0.776629i	$-0.283072\pi$
$102$	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	$0.0773744\pi$
$107$	51.5081i	0.481384i	−0.970601	+	0.240692i	$0.922626\pi$
$108$	0	0
$109$	63.0507	0.578446	0.289223	−	0.957262i	$-0.406603\pi$
$109$	63.0507	0.578446	0.289223	+	0.957262i	$0.406603\pi$
$110$	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	$-0.596268\pi$
$127$	−75.6526	−0.595690	−0.297845	+	0.954614i	$0.596268\pi$
$128$	130.395i	1.01871i
$129$	0	0
$130$	−6.58776	−0.0506751
$131$	124.201i	0.948097i	0.880499	+	0.474049i	$0.157208\pi$
$131$	124.201i	0.948097i	−0.880499	+	0.474049i	$0.842792\pi$
$132$	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	$-0.591787\pi$
$139$	−79.0567	−0.568753	−0.284377	+	0.958713i	$0.591787\pi$
$140$	− 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	$-0.976773\pi$
$149$	− 21.7256i	− 0.145809i	0.997339	−	0.0729046i	$-0.0232269\pi$
$150$	0	0
$151$	124.089	0.821784	0.410892	−	0.911684i	$-0.365217\pi$
$151$	124.089	0.821784	0.410892	+	0.911684i	$0.365217\pi$
$152$	− 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	$-0.947790\pi$
$167$	− 54.5378i	− 0.326574i	0.986579	−	0.163287i	$-0.0522096\pi$
$168$	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	$0.0940319\pi$
$173$	100.732i	0.582264i	−0.956683	+	0.291132i	$0.905968\pi$
$174$	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	$-0.298856\pi$
$193$	228.006	1.18138	0.590689	+	0.806899i	$0.298856\pi$
$194$	125.328i	0.646020i
$195$	0	0
$196$	−7.82387	−0.0399177
$197$	170.096i	0.863429i	0.902010	+	0.431715i	$0.142091\pi$
$197$	170.096i	0.863429i	−0.902010	+	0.431715i	$0.857909\pi$
$198$	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	$-0.592055\pi$
$211$	−120.349	−0.570372	−0.285186	+	0.958472i	$0.592055\pi$
$212$	− 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	$-0.845433\pi$
$227$	− 211.895i	− 0.933456i	0.884401	−	0.466728i	$-0.154567\pi$
$228$	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	$-0.867004\pi$
$233$	− 189.088i	− 0.811537i	0.913976	−	0.405769i	$-0.132996\pi$
$234$	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	$-0.940522\pi$
$239$	− 88.7986i	− 0.371542i	0.982593	−	0.185771i	$-0.0594783\pi$
$240$	0	0
$241$	155.144	0.643750	0.321875	−	0.946782i	$-0.395687\pi$
$241$	155.144	0.643750	0.321875	+	0.946782i	$0.395687\pi$
$242$	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	$-0.766701\pi$
$263$	− 351.921i	− 1.33810i	0.743217	−	0.669050i	$-0.233299\pi$
$264$	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	$-0.265322\pi$
$271$	364.367	1.34453	0.672264	+	0.740312i	$0.265322\pi$
$272$	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	$-0.376914\pi$
$277$	208.925	0.754244	0.377122	+	0.926164i	$0.376914\pi$
$278$	− 64.5648i	− 0.232247i
$279$	0	0
$280$	42.8261	0.152950
$281$	− 474.459i	− 1.68847i	−0.535976	−	0.844233i	$-0.680056\pi$
$281$	− 474.459i	− 1.68847i	0.535976	−	0.844233i	$-0.319944\pi$
$282$	0	0
$283$	−172.881	−0.610888	−0.305444	−	0.952210i	$-0.598805\pi$
$283$	−172.881	−0.610888	−0.305444	+	0.952210i	$0.598805\pi$
$284$	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	$-0.775202\pi$
$293$	− 380.294i	− 1.29793i	0.760817	−	0.648966i	$-0.224798\pi$
$294$	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	$-0.311040\pi$
$307$	343.458	1.11876	0.559378	+	0.828912i	$0.311040\pi$
$308$	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	$-0.273652\pi$
$337$	439.894	1.30532	0.652662	+	0.757649i	$0.273652\pi$
$338$	− 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	$-0.800013\pi$
$347$	− 407.900i	− 1.17550i	0.809041	−	0.587752i	$-0.199987\pi$
$348$	0	0
$349$	478.460	1.37094	0.685472	−	0.728099i	$-0.259596\pi$
$349$	478.460	1.37094	0.685472	+	0.728099i	$0.259596\pi$
$350$	− 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	$-0.833855\pi$
$359$	− 357.980i	− 0.997159i	0.866844	−	0.498579i	$-0.166145\pi$
$360$	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	$-0.724237\pi$
$373$	−483.127	−1.29525	−0.647623	+	0.761961i	$0.724237\pi$
$374$	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	$-0.453248\pi$
$409$	119.713	0.292697	0.146349	+	0.989233i	$0.453248\pi$
$410$	− 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	$-0.942596\pi$
$431$	− 154.611i	− 0.358726i	0.983783	−	0.179363i	$-0.0574037\pi$
$432$	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	$-0.196226\pi$
$439$	716.385	1.63186	0.815928	+	0.578153i	$0.196226\pi$
$440$	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	$-0.344892\pi$
$457$	427.961	0.936457	0.468228	+	0.883607i	$0.344892\pi$
$458$	227.364i	0.496427i
$459$	0	0
$460$	16.8339	0.0365954
$461$	− 313.392i	− 0.679809i	−0.940460	−	0.339904i	$-0.889605\pi$
$461$	− 313.392i	− 0.679809i	0.940460	−	0.339904i	$-0.110395\pi$
$462$	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	$-0.993075\pi$
$479$	− 20.8402i	− 0.0435078i	0.999763	−	0.0217539i	$-0.00692503\pi$
$480$	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	$-0.968407\pi$
$491$	− 97.3072i	− 0.198182i	0.995078	−	0.0990908i	$-0.0315934\pi$
$492$	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	$0.249495\pi$
$503$	710.221i	1.41197i	−0.708227	+	0.705985i	$0.750505\pi$
$504$	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	$-0.593570\pi$
$523$	−303.071	−0.579485	−0.289743	+	0.957105i	$0.593570\pi$
$524$	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	$-0.0903776\pi$
$541$	1038.68	1.91992	0.959962	+	0.280130i	$0.0903776\pi$
$542$	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	$-0.587555\pi$
$547$	−297.137	−0.543212	−0.271606	+	0.962409i	$0.587555\pi$
$548$	− 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	$-0.840993\pi$
$557$	− 533.625i	− 0.958034i	0.877806	−	0.479017i	$-0.159007\pi$
$558$	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	$-0.939853\pi$
$563$	− 211.503i	− 0.375672i	0.982200	−	0.187836i	$-0.0601473\pi$
$564$	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	$-0.781325\pi$
$569$	− 721.732i	− 1.26842i	0.773160	−	0.634211i	$-0.218675\pi$
$570$	0	0
$571$	−798.387	−1.39823	−0.699113	−	0.715011i	$-0.746422\pi$
$571$	−798.387	−1.39823	−0.699113	+	0.715011i	$0.746422\pi$
$572$	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	$0.123069\pi$
$593$	447.206i	0.754141i	−0.926185	+	0.377071i	$0.876931\pi$
$594$	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	$-0.446635\pi$
$601$	200.574	0.333734	0.166867	+	0.985979i	$0.446635\pi$
$602$	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	$-0.768858\pi$
$607$	−907.748	−1.49547	−0.747733	+	0.664000i	$0.768858\pi$
$608$	− 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	$-0.281549\pi$
$613$	776.875	1.26733	0.633666	+	0.773606i	$0.281549\pi$
$614$	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	$-0.995353\pi$
$659$	− 19.2417i	− 0.0291983i	0.999893	−	0.0145992i	$-0.00464722\pi$
$660$	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	$-0.500342\pi$
$673$	−1.44765	−0.00215105	−0.00107552	+	0.999999i	$0.500342\pi$
$674$	359.257i	0.533022i
$675$	0	0
$676$	−365.430	−0.540576
$677$	1049.30i	1.54993i	0.632004	+	0.774965i	$0.282233\pi$
$677$	1049.30i	1.54993i	−0.632004	+	0.774965i	$0.717767\pi$
$678$	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	$-0.716913\pi$
$701$	− 1088.88i	− 1.55332i	0.629921	−	0.776659i	$-0.283087\pi$
$702$	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	$-0.550968\pi$
$733$	−233.736	−0.318876	−0.159438	+	0.987208i	$0.550968\pi$
$734$	− 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	$-0.330606\pi$
$739$	749.941	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$740$	− 164.647i	− 0.222496i
$741$	0	0
$742$	−214.264	−0.288765
$743$	534.169i	0.718936i	0.933157	+	0.359468i	$0.117042\pi$
$743$	534.169i	0.718936i	−0.933157	+	0.359468i	$0.882958\pi$
$744$	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	$0.0413894\pi$
$761$	197.346i	0.259325i	−0.991558	+	0.129662i	$0.958611\pi$
$762$	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	$-0.776735\pi$
$769$	−1174.93	−1.52787	−0.763934	+	0.645294i	$0.776735\pi$
$770$	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	$-0.636538\pi$
$787$	−654.650	−0.831829	−0.415915	+	0.909404i	$0.636538\pi$
$788$	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	$0.201196\pi$
$809$	955.949i	1.18164i	−0.806802	+	0.590821i	$0.798804\pi$
$810$	0	0
$811$	301.766	0.372091	0.186045	−	0.982541i	$-0.440433\pi$
$811$	301.766	0.372091	0.186045	+	0.982541i	$0.440433\pi$
$812$	− 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	$-0.955136\pi$
$821$	− 230.665i	− 0.280956i	0.990084	−	0.140478i	$-0.0448640\pi$
$822$	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	$0.130538\pi$
$827$	659.445i	0.797395i	−0.917083	+	0.398697i	$0.869462\pi$
$828$	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	$-0.830383\pi$
$853$	−1469.47	−1.72271	−0.861354	+	0.508005i	$0.830383\pi$
$854$	454.231i	0.531886i
$855$	0	0
$856$	−308.472	−0.360364
$857$	297.274i	0.346877i	0.984845	+	0.173439i	$0.0554878\pi$
$857$	297.274i	0.346877i	−0.984845	+	0.173439i	$0.944512\pi$
$858$	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	$-0.596434\pi$
$877$	−523.294	−0.596687	−0.298343	+	0.954459i	$0.596434\pi$
$878$	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	$0.212542\pi$
$887$	1098.46i	1.23839i	−0.785235	+	0.619197i	$0.787458\pi$
$888$	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	$-0.488897\pi$
$919$	64.0998	0.0697495	0.0348747	+	0.999392i	$0.488897\pi$
$920$	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	$-0.533355\pi$
$937$	−196.014	−0.209193	−0.104597	+	0.994515i	$0.533355\pi$
$938$	− 391.305i	− 0.417169i
$939$	0	0
$940$	−230.821	−0.245555
$941$	− 477.070i	− 0.506982i	−0.967338	−	0.253491i	$-0.918421\pi$
$941$	− 477.070i	− 0.506982i	0.967338	−	0.253491i	$-0.0815788\pi$
$942$	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	$0.402488\pi$
$953$	1817.26i	1.90689i	−0.301575	+	0.953443i	$0.597512\pi$
$954$	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	$-0.771626\pi$
$967$	−1457.23	−1.50696	−0.753480	+	0.657471i	$0.771626\pi$
$968$	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	$-0.295445\pi$
$997$	1195.01	1.19860	0.599301	+	0.800524i	$0.295445\pi$
$998$	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.j.485.10		16
3.2	odd	2	inner	1089.3.b.j.485.7		16
11.2	odd	10		99.3.l.a.26.4	✓	32
11.6	odd	10		99.3.l.a.80.5	yes	32
11.10	odd	2		1089.3.b.i.485.7		16
33.2	even	10		99.3.l.a.26.5	yes	32
33.17	even	10		99.3.l.a.80.4	yes	32
33.32	even	2		1089.3.b.i.485.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.26.4	✓	32	11.2	odd	10
99.3.l.a.26.5	yes	32	33.2	even	10
99.3.l.a.80.4	yes	32	33.17	even	10
99.3.l.a.80.5	yes	32	11.6	odd	10
1089.3.b.i.485.7		16	11.10	odd	2
1089.3.b.i.485.10		16	33.32	even	2
1089.3.b.j.485.7		16	3.2	odd	2	inner
1089.3.b.j.485.10		16	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:	\( k \)	\(=\)	\( 3 \)
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

Modular forms

Varieties

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Database

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