LMFDB - Embedded newform 1089.3.c.j.604.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$8$
Coefficient field:	8.0.41108373504.15
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 $ x^8 - 4x^7 - 4x^6 + 20x^5 + 20x^4 - 28x^3 + 4x^2 + 12*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 363)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.3
Root			$-0.468648 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.j.604.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07698i q^{2} -0.313852 q^{4} +1.66935 q^{5} +3.02335i q^{7} -7.65606i q^{8} +O(q^{10})$	$q-2.07698i q^{2} -0.313852 q^{4} +1.66935 q^{5} +3.02335i q^{7} -7.65606i q^{8} -3.46720i q^{10} +10.7987i q^{13} +6.27944 q^{14} -17.1569 q^{16} +29.7619i q^{17} +36.3356i q^{19} -0.523928 q^{20} +20.1967 q^{23} -22.2133 q^{25} +22.4288 q^{26} -0.948884i q^{28} +1.27301i q^{29} -41.1127 q^{31} +5.01033i q^{32} +61.8149 q^{34} +5.04701i q^{35} -45.8791 q^{37} +75.4683 q^{38} -12.7806i q^{40} +13.5353i q^{41} -13.7677i q^{43} -41.9482i q^{46} -59.4079 q^{47} +39.8594 q^{49} +46.1366i q^{50} -3.38921i q^{52} -3.33524 q^{53} +23.1469 q^{56} +2.64402 q^{58} +82.4159 q^{59} +45.2608i q^{61} +85.3903i q^{62} -58.2213 q^{64} +18.0268i q^{65} +47.1972 q^{67} -9.34084i q^{68} +10.4826 q^{70} -80.8174 q^{71} +78.3935i q^{73} +95.2901i q^{74} -11.4040i q^{76} +65.1628i q^{79} -28.6408 q^{80} +28.1125 q^{82} -12.9986i q^{83} +49.6829i q^{85} -28.5953 q^{86} +159.166 q^{89} -32.6483 q^{91} -6.33878 q^{92} +123.389i q^{94} +60.6567i q^{95} +83.2933 q^{97} -82.7872i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} - 16 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 - 16 * q^5	$ 8 q - 24 q^{4} - 16 q^{5} - 136 q^{14} + 72 q^{16} + 16 q^{20} - 88 q^{25} + 120 q^{26} - 128 q^{31} + 96 q^{34} + 80 q^{37} + 216 q^{38} + 32 q^{47} - 152 q^{49} - 80 q^{53} + 696 q^{56} - 176 q^{58} + 64 q^{59} + 8 q^{64} + 464 q^{67} + 304 q^{70} + 128 q^{71} - 80 q^{80} + 528 q^{82} - 24 q^{86} + 720 q^{89} + 80 q^{91} + 1248 q^{92} + 416 q^{97}+O(q^{100}) $ 8 * q - 24 * q^4 - 16 * q^5 - 136 * q^14 + 72 * q^16 + 16 * q^20 - 88 * q^25 + 120 * q^26 - 128 * q^31 + 96 * q^34 + 80 * q^37 + 216 * q^38 + 32 * q^47 - 152 * q^49 - 80 * q^53 + 696 * q^56 - 176 * q^58 + 64 * q^59 + 8 * q^64 + 464 * q^67 + 304 * q^70 + 128 * q^71 - 80 * q^80 + 528 * q^82 - 24 * q^86 + 720 * q^89 + 80 * q^91 + 1248 * q^92 + 416 * 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 $

$2$	− 2.07698i	− 1.03849i	−0.854625	−	0.519245i	$-0.826213\pi$
$2$	− 2.07698i	− 1.03849i	0.854625	−	0.519245i	$-0.173787\pi$
$3$	0	0
$3$	0	0
$4$	−0.313852	−0.0784630
$5$	1.66935	0.333869	0.166935	−	0.985968i	$-0.446613\pi$
$5$	1.66935	0.333869	0.166935	+	0.985968i	$0.446613\pi$
$6$	0	0
$7$	3.02335i	0.431907i	0.976404	+	0.215953i	$0.0692859\pi$
$7$	3.02335i	0.431907i	−0.976404	+	0.215953i	$0.930714\pi$
$8$	− 7.65606i	− 0.957008i
$9$	0	0
$10$	− 3.46720i	− 0.346720i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	10.7987i	0.830672i	0.909668	+	0.415336i	$0.136336\pi$
$13$	10.7987i	0.830672i	−0.909668	+	0.415336i	$0.863664\pi$
$14$	6.27944	0.448531
$15$	0	0
$16$	−17.1569	−1.07231
$17$	29.7619i	1.75070i	0.483489	+	0.875350i	$0.339369\pi$
$17$	29.7619i	1.75070i	−0.483489	+	0.875350i	$0.660631\pi$
$18$	0	0
$19$	36.3356i	1.91240i	0.292716	+	0.956199i	$0.405441\pi$
$19$	36.3356i	1.91240i	−0.292716	+	0.956199i	$0.594559\pi$
$20$	−0.523928	−0.0261964
$21$	0	0
$22$	0	0
$23$	20.1967	0.878118	0.439059	−	0.898458i	$-0.355312\pi$
$23$	20.1967	0.878118	0.439059	+	0.898458i	$0.355312\pi$
$24$	0	0
$25$	−22.2133	−0.888531
$26$	22.4288	0.862646
$27$	0	0
$28$	− 0.948884i	− 0.0338887i
$29$	1.27301i	0.0438970i	0.999759	+	0.0219485i	$0.00698698\pi$
$29$	1.27301i	0.0438970i	−0.999759	+	0.0219485i	$0.993013\pi$
$30$	0	0
$31$	−41.1127	−1.32622	−0.663108	−	0.748523i	$-0.730763\pi$
$31$	−41.1127	−1.32622	−0.663108	+	0.748523i	$0.730763\pi$
$32$	5.01033i	0.156573i
$33$	0	0
$34$	61.8149	1.81809
$35$	5.04701i	0.144200i
$36$	0	0
$37$	−45.8791	−1.23998	−0.619988	−	0.784611i	$-0.712863\pi$
$37$	−45.8791	−1.23998	−0.619988	+	0.784611i	$0.712863\pi$
$38$	75.4683	1.98601
$39$	0	0
$40$	− 12.7806i	− 0.319515i
$41$	13.5353i	0.330129i	0.986283	+	0.165064i	$0.0527832\pi$
$41$	13.5353i	0.330129i	−0.986283	+	0.165064i	$0.947217\pi$
$42$	0	0
$43$	− 13.7677i	− 0.320180i	−0.987102	−	0.160090i	$-0.948822\pi$
$43$	− 13.7677i	− 0.320180i	0.987102	−	0.160090i	$-0.0511784\pi$
$44$	0	0
$45$	0	0
$46$	− 41.9482i	− 0.911918i
$47$	−59.4079	−1.26400	−0.631999	−	0.774969i	$-0.717765\pi$
$47$	−59.4079	−1.26400	−0.631999	+	0.774969i	$0.717765\pi$
$48$	0	0
$49$	39.8594	0.813457
$50$	46.1366i	0.922732i
$51$	0	0
$52$	− 3.38921i	− 0.0651771i
$53$	−3.33524	−0.0629291	−0.0314646	−	0.999505i	$-0.510017\pi$
$53$	−3.33524	−0.0629291	−0.0314646	+	0.999505i	$0.510017\pi$
$54$	0	0
$55$	0	0
$56$	23.1469	0.413338
$57$	0	0
$58$	2.64402	0.0455866
$59$	82.4159	1.39688	0.698440	−	0.715668i	$-0.253878\pi$
$59$	82.4159	1.39688	0.698440	+	0.715668i	$0.253878\pi$
$60$	0	0
$61$	45.2608i	0.741981i	0.928637	+	0.370990i	$0.120982\pi$
$61$	45.2608i	0.741981i	−0.928637	+	0.370990i	$0.879018\pi$
$62$	85.3903i	1.37726i
$63$	0	0
$64$	−58.2213	−0.909707
$65$	18.0268i	0.277336i
$66$	0	0
$67$	47.1972	0.704436	0.352218	−	0.935918i	$-0.385428\pi$
$67$	47.1972	0.704436	0.352218	+	0.935918i	$0.385428\pi$
$68$	− 9.34084i	− 0.137365i
$69$	0	0
$70$	10.4826	0.149751
$71$	−80.8174	−1.13827	−0.569137	−	0.822243i	$-0.692723\pi$
$71$	−80.8174	−1.13827	−0.569137	+	0.822243i	$0.692723\pi$
$72$	0	0
$73$	78.3935i	1.07388i	0.843619	+	0.536942i	$0.180420\pi$
$73$	78.3935i	1.07388i	−0.843619	+	0.536942i	$0.819580\pi$
$74$	95.2901i	1.28770i
$75$	0	0
$76$	− 11.4040i	− 0.150053i
$77$	0	0
$78$	0	0
$79$	65.1628i	0.824845i	0.910993	+	0.412423i	$0.135317\pi$
$79$	65.1628i	0.824845i	−0.910993	+	0.412423i	$0.864683\pi$
$80$	−28.6408	−0.358010
$81$	0	0
$82$	28.1125	0.342836
$83$	− 12.9986i	− 0.156610i	−0.996929	−	0.0783049i	$-0.975049\pi$
$83$	− 12.9986i	− 0.156610i	0.996929	−	0.0783049i	$-0.0249508\pi$
$84$	0	0
$85$	49.6829i	0.584505i
$86$	−28.5953	−0.332504
$87$	0	0
$88$	0	0
$89$	159.166	1.78838	0.894189	−	0.447690i	$-0.147753\pi$
$89$	159.166	1.78838	0.894189	+	0.447690i	$0.147753\pi$
$90$	0	0
$91$	−32.6483	−0.358773
$92$	−6.33878	−0.0688998
$93$	0	0
$94$	123.389i	1.31265i
$95$	60.6567i	0.638491i
$96$	0	0
$97$	83.2933	0.858694	0.429347	−	0.903140i	$-0.358744\pi$
$97$	83.2933	0.858694	0.429347	+	0.903140i	$0.358744\pi$
$98$	− 82.7872i	− 0.844767i
$99$	0	0
$100$	6.97169	0.0697169
$101$	− 180.370i	− 1.78584i	−0.450217	−	0.892919i	$-0.648653\pi$
$101$	− 180.370i	− 1.78584i	0.450217	−	0.892919i	$-0.351347\pi$
$102$	0	0
$103$	139.259	1.35203	0.676017	−	0.736886i	$-0.263705\pi$
$103$	139.259	1.35203	0.676017	+	0.736886i	$0.263705\pi$
$104$	82.6758	0.794960
$105$	0	0
$106$	6.92724i	0.0653513i
$107$	24.3108i	0.227203i	0.993526	+	0.113602i	$0.0362388\pi$
$107$	24.3108i	0.227203i	−0.993526	+	0.113602i	$0.963761\pi$
$108$	0	0
$109$	− 0.466514i	− 0.00427994i	−0.999998	−	0.00213997i	$-0.999319\pi$
$109$	− 0.466514i	− 0.00427994i	0.999998	−	0.00213997i	$-0.000681174\pi$
$110$	0	0
$111$	0	0
$112$	− 51.8713i	− 0.463136i
$113$	−5.85786	−0.0518394	−0.0259197	−	0.999664i	$-0.508251\pi$
$113$	−5.85786	−0.0518394	−0.0259197	+	0.999664i	$0.508251\pi$
$114$	0	0
$115$	33.7153	0.293177
$116$	− 0.399538i	− 0.00344429i
$117$	0	0
$118$	− 171.176i	− 1.45065i
$119$	−89.9806	−0.756139
$120$	0	0
$121$	0	0
$122$	94.0059	0.770540
$123$	0	0
$124$	12.9033	0.104059
$125$	−78.8153	−0.630523
$126$	0	0
$127$	− 74.7220i	− 0.588362i	−0.955750	−	0.294181i	$-0.904953\pi$
$127$	− 74.7220i	− 0.588362i	0.955750	−	0.294181i	$-0.0950468\pi$
$128$	140.966i	1.10130i
$129$	0	0
$130$	37.4414	0.288011
$131$	− 138.278i	− 1.05556i	−0.849381	−	0.527779i	$-0.823025\pi$
$131$	− 138.278i	− 1.05556i	0.849381	−	0.527779i	$-0.176975\pi$
$132$	0	0
$133$	−109.855	−0.825978
$134$	− 98.0277i	− 0.731550i
$135$	0	0
$136$	227.859	1.67543
$137$	−95.6934	−0.698492	−0.349246	−	0.937031i	$-0.613562\pi$
$137$	−95.6934	−0.698492	−0.349246	+	0.937031i	$0.613562\pi$
$138$	0	0
$139$	25.8790i	0.186180i	0.995658	+	0.0930901i	$0.0296745\pi$
$139$	25.8790i	0.186180i	−0.995658	+	0.0930901i	$0.970326\pi$
$140$	− 1.58402i	− 0.0113144i
$141$	0	0
$142$	167.856i	1.18209i
$143$	0	0
$144$	0	0
$145$	2.12510i	0.0146558i
$146$	162.822	1.11522
$147$	0	0
$148$	14.3993	0.0972923
$149$	− 159.060i	− 1.06752i	−0.845637	−	0.533758i	$-0.820779\pi$
$149$	− 159.060i	− 1.06752i	0.845637	−	0.533758i	$-0.179221\pi$
$150$	0	0
$151$	200.393i	1.32710i	0.748131	+	0.663552i	$0.230952\pi$
$151$	200.393i	1.32710i	−0.748131	+	0.663552i	$0.769048\pi$
$152$	278.187	1.83018
$153$	0	0
$154$	0	0
$155$	−68.6314	−0.442783
$156$	0	0
$157$	−90.1406	−0.574144	−0.287072	−	0.957909i	$-0.592682\pi$
$157$	−90.1406	−0.574144	−0.287072	+	0.957909i	$0.592682\pi$
$158$	135.342	0.856594
$159$	0	0
$160$	8.36398i	0.0522749i
$161$	61.0617i	0.379265i
$162$	0	0
$163$	263.559	1.61693	0.808463	−	0.588547i	$-0.200300\pi$
$163$	263.559	1.61693	0.808463	+	0.588547i	$0.200300\pi$
$164$	− 4.24807i	− 0.0259029i
$165$	0	0
$166$	−26.9979	−0.162638
$167$	− 126.384i	− 0.756792i	−0.925644	−	0.378396i	$-0.876476\pi$
$167$	− 126.384i	− 0.756792i	0.925644	−	0.378396i	$-0.123524\pi$
$168$	0	0
$169$	52.3872	0.309983
$170$	103.191	0.607003
$171$	0	0
$172$	4.32103i	0.0251223i
$173$	214.578i	1.24033i	0.784470	+	0.620166i	$0.212935\pi$
$173$	214.578i	1.24033i	−0.784470	+	0.620166i	$0.787065\pi$
$174$	0	0
$175$	− 67.1585i	− 0.383763i
$176$	0	0
$177$	0	0
$178$	− 330.584i	− 1.85721i
$179$	−230.957	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$179$	−230.957	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$180$	0	0
$181$	−15.8757	−0.0877109	−0.0438555	−	0.999038i	$-0.513964\pi$
$181$	−15.8757	−0.0877109	−0.0438555	+	0.999038i	$0.513964\pi$
$182$	67.8100i	0.372582i
$183$	0	0
$184$	− 154.627i	− 0.840366i
$185$	−76.5882	−0.413990
$186$	0	0
$187$	0	0
$188$	18.6453	0.0991771
$189$	0	0
$190$	125.983	0.663067
$191$	83.7129	0.438287	0.219144	−	0.975693i	$-0.429674\pi$
$191$	83.7129	0.438287	0.219144	+	0.975693i	$0.429674\pi$
$192$	0	0
$193$	− 291.132i	− 1.50845i	−0.656613	−	0.754227i	$-0.728012\pi$
$193$	− 291.132i	− 1.50845i	0.656613	−	0.754227i	$-0.271988\pi$
$194$	− 172.999i	− 0.891746i
$195$	0	0
$196$	−12.5099	−0.0638263
$197$	346.358i	1.75816i	0.476674	+	0.879080i	$0.341842\pi$
$197$	346.358i	1.75816i	−0.476674	+	0.879080i	$0.658158\pi$
$198$	0	0
$199$	−26.8349	−0.134849	−0.0674244	−	0.997724i	$-0.521478\pi$
$199$	−26.8349	−0.134849	−0.0674244	+	0.997724i	$0.521478\pi$
$200$	170.066i	0.850331i
$201$	0	0
$202$	−374.625	−1.85458
$203$	−3.84876	−0.0189594
$204$	0	0
$205$	22.5951i	0.110220i
$206$	− 289.239i	− 1.40407i
$207$	0	0
$208$	− 185.273i	− 0.890736i
$209$	0	0
$210$	0	0
$211$	− 3.97783i	− 0.0188523i	−0.999956	−	0.00942613i	$-0.997000\pi$
$211$	− 3.97783i	− 0.0188523i	0.999956	−	0.00942613i	$-0.00300047\pi$
$212$	1.04677	0.00493761
$213$	0	0
$214$	50.4930	0.235949
$215$	− 22.9831i	− 0.106898i
$216$	0	0
$217$	− 124.298i	− 0.572802i
$218$	−0.968940	−0.00444468
$219$	0	0
$220$	0	0
$221$	−321.391	−1.45426
$222$	0	0
$223$	304.173	1.36400	0.682002	−	0.731350i	$-0.261109\pi$
$223$	304.173	1.36400	0.682002	+	0.731350i	$0.261109\pi$
$224$	−15.1480	−0.0676249
$225$	0	0
$226$	12.1667i	0.0538348i
$227$	70.8729i	0.312215i	0.987740	+	0.156108i	$0.0498947\pi$
$227$	70.8729i	0.312215i	−0.987740	+	0.156108i	$0.950105\pi$
$228$	0	0
$229$	−97.0927	−0.423985	−0.211993	−	0.977271i	$-0.567995\pi$
$229$	−97.0927	−0.423985	−0.211993	+	0.977271i	$0.567995\pi$
$230$	− 70.0261i	− 0.304461i
$231$	0	0
$232$	9.74626	0.0420097
$233$	124.688i	0.535143i	0.963538	+	0.267572i	$0.0862212\pi$
$233$	124.688i	0.535143i	−0.963538	+	0.267572i	$0.913779\pi$
$234$	0	0
$235$	−99.1724	−0.422010
$236$	−25.8664	−0.109603
$237$	0	0
$238$	186.888i	0.785244i
$239$	24.5500i	0.102720i	0.998680	+	0.0513599i	$0.0163555\pi$
$239$	24.5500i	0.102720i	−0.998680	+	0.0513599i	$0.983644\pi$
$240$	0	0
$241$	− 221.322i	− 0.918347i	−0.888347	−	0.459174i	$-0.848146\pi$
$241$	− 221.322i	− 0.918347i	0.888347	−	0.459174i	$-0.151854\pi$
$242$	0	0
$243$	0	0
$244$	− 14.2052i	− 0.0582180i
$245$	66.5391	0.271588
$246$	0	0
$247$	−392.379	−1.58858
$248$	314.761i	1.26920i
$249$	0	0
$250$	163.698i	0.654792i
$251$	95.2322	0.379411	0.189706	−	0.981841i	$-0.439247\pi$
$251$	95.2322	0.379411	0.189706	+	0.981841i	$0.439247\pi$
$252$	0	0
$253$	0	0
$254$	−155.196	−0.611008
$255$	0	0
$256$	59.8983	0.233978
$257$	279.641	1.08810	0.544048	−	0.839054i	$-0.316891\pi$
$257$	279.641	1.08810	0.544048	+	0.839054i	$0.316891\pi$
$258$	0	0
$259$	− 138.709i	− 0.535554i
$260$	− 5.65776i	− 0.0217606i
$261$	0	0
$262$	−287.201	−1.09619
$263$	− 167.625i	− 0.637357i	−0.947863	−	0.318678i	$-0.896761\pi$
$263$	− 167.625i	− 0.637357i	0.947863	−	0.318678i	$-0.103239\pi$
$264$	0	0
$265$	−5.56768	−0.0210101
$266$	228.167i	0.857770i
$267$	0	0
$268$	−14.8129	−0.0552722
$269$	−199.168	−0.740403	−0.370201	−	0.928952i	$-0.620711\pi$
$269$	−199.168	−0.740403	−0.370201	+	0.928952i	$0.620711\pi$
$270$	0	0
$271$	102.977i	0.379987i	0.981785	+	0.189994i	$0.0608467\pi$
$271$	102.977i	0.379987i	−0.981785	+	0.189994i	$0.939153\pi$
$272$	− 510.622i	− 1.87729i
$273$	0	0
$274$	198.753i	0.725377i
$275$	0	0
$276$	0	0
$277$	235.576i	0.850453i	0.905087	+	0.425227i	$0.139806\pi$
$277$	235.576i	0.850453i	−0.905087	+	0.425227i	$0.860194\pi$
$278$	53.7503	0.193346
$279$	0	0
$280$	38.6402	0.138001
$281$	− 235.579i	− 0.838360i	−0.907903	−	0.419180i	$-0.862318\pi$
$281$	− 235.579i	− 0.838360i	0.907903	−	0.419180i	$-0.137682\pi$
$282$	0	0
$283$	154.316i	0.545285i	0.962115	+	0.272642i	$0.0878975\pi$
$283$	154.316i	0.545285i	−0.962115	+	0.272642i	$0.912102\pi$
$284$	25.3647	0.0893124
$285$	0	0
$286$	0	0
$287$	−40.9218	−0.142585
$288$	0	0
$289$	−596.771	−2.06495
$290$	4.41379	0.0152200
$291$	0	0
$292$	− 24.6040i	− 0.0842601i
$293$	− 528.461i	− 1.80362i	−0.432132	−	0.901811i	$-0.642238\pi$
$293$	− 528.461i	− 1.80362i	0.432132	−	0.901811i	$-0.357762\pi$
$294$	0	0
$295$	137.581	0.466375
$296$	351.253i	1.18667i
$297$	0	0
$298$	−330.365	−1.10861
$299$	218.099i	0.729429i
$300$	0	0
$301$	41.6246	0.138288
$302$	416.212	1.37818
$303$	0	0
$304$	− 623.406i	− 2.05068i
$305$	75.5560i	0.247724i
$306$	0	0
$307$	149.102i	0.485674i	0.970067	+	0.242837i	$0.0780779\pi$
$307$	149.102i	0.485674i	−0.970067	+	0.242837i	$0.921922\pi$
$308$	0	0
$309$	0	0
$310$	142.546i	0.459826i
$311$	447.770	1.43978	0.719888	−	0.694090i	$-0.244193\pi$
$311$	447.770	1.43978	0.719888	+	0.694090i	$0.244193\pi$
$312$	0	0
$313$	−377.905	−1.20737	−0.603683	−	0.797225i	$-0.706301\pi$
$313$	−377.905	−1.20737	−0.603683	+	0.797225i	$0.706301\pi$
$314$	187.220i	0.596243i
$315$	0	0
$316$	− 20.4515i	− 0.0647199i
$317$	−8.12673	−0.0256364	−0.0128182	−	0.999918i	$-0.504080\pi$
$317$	−8.12673	−0.0256364	−0.0128182	+	0.999918i	$0.504080\pi$
$318$	0	0
$319$	0	0
$320$	−97.1914	−0.303723
$321$	0	0
$322$	126.824	0.393863
$323$	−1081.42	−3.34804
$324$	0	0
$325$	− 239.876i	− 0.738078i
$326$	− 547.407i	− 1.67916i
$327$	0	0
$328$	103.627	0.315936
$329$	− 179.611i	− 0.545929i
$330$	0	0
$331$	524.134	1.58349	0.791743	−	0.610855i	$-0.209174\pi$
$331$	524.134	1.58349	0.791743	+	0.610855i	$0.209174\pi$
$332$	4.07964i	0.0122881i
$333$	0	0
$334$	−262.498	−0.785922
$335$	78.7885	0.235189
$336$	0	0
$337$	− 274.599i	− 0.814833i	−0.913242	−	0.407417i	$-0.866430\pi$
$337$	− 274.599i	− 0.814833i	0.913242	−	0.407417i	$-0.133570\pi$
$338$	− 108.807i	− 0.321915i
$339$	0	0
$340$	− 15.5931i	− 0.0458620i
$341$	0	0
$342$	0	0
$343$	268.653i	0.783244i
$344$	−105.407	−0.306415
$345$	0	0
$346$	445.674	1.28807
$347$	− 335.062i	− 0.965596i	−0.875732	−	0.482798i	$-0.839621\pi$
$347$	− 335.062i	− 0.965596i	0.875732	−	0.482798i	$-0.160379\pi$
$348$	0	0
$349$	59.1720i	0.169547i	0.996400	+	0.0847737i	$0.0270167\pi$
$349$	59.1720i	0.169547i	−0.996400	+	0.0847737i	$0.972983\pi$
$350$	−139.487	−0.398534
$351$	0	0
$352$	0	0
$353$	392.466	1.11180	0.555900	−	0.831249i	$-0.312374\pi$
$353$	392.466	1.11180	0.555900	+	0.831249i	$0.312374\pi$
$354$	0	0
$355$	−134.912	−0.380035
$356$	−49.9545	−0.140322
$357$	0	0
$358$	479.693i	1.33993i
$359$	512.245i	1.42687i	0.700724	+	0.713433i	$0.252860\pi$
$359$	512.245i	1.42687i	−0.700724	+	0.713433i	$0.747140\pi$
$360$	0	0
$361$	−959.274	−2.65727
$362$	32.9735i	0.0910870i
$363$	0	0
$364$	10.2468	0.0281504
$365$	130.866i	0.358537i
$366$	0	0
$367$	−272.693	−0.743032	−0.371516	−	0.928427i	$-0.621162\pi$
$367$	−272.693	−0.743032	−0.371516	+	0.928427i	$0.621162\pi$
$368$	−346.513	−0.941612
$369$	0	0
$370$	159.072i	0.429925i
$371$	− 10.0836i	− 0.0271795i
$372$	0	0
$373$	226.096i	0.606156i	0.952966	+	0.303078i	$0.0980143\pi$
$373$	226.096i	0.606156i	−0.952966	+	0.303078i	$0.901986\pi$
$374$	0	0
$375$	0	0
$376$	454.830i	1.20966i
$377$	−13.7469	−0.0364640
$378$	0	0
$379$	316.284	0.834524	0.417262	−	0.908786i	$-0.362990\pi$
$379$	316.284	0.834524	0.417262	+	0.908786i	$0.362990\pi$
$380$	− 19.0372i	− 0.0500980i
$381$	0	0
$382$	− 173.870i	− 0.455157i
$383$	−689.107	−1.79923	−0.899617	−	0.436680i	$-0.856154\pi$
$383$	−689.107	−1.79923	−0.899617	+	0.436680i	$0.856154\pi$
$384$	0	0
$385$	0	0
$386$	−604.675	−1.56652
$387$	0	0
$388$	−26.1418	−0.0673757
$389$	−430.083	−1.10561	−0.552806	−	0.833310i	$-0.686443\pi$
$389$	−430.083	−1.10561	−0.552806	+	0.833310i	$0.686443\pi$
$390$	0	0
$391$	601.093i	1.53732i
$392$	− 305.166i	− 0.778484i
$393$	0	0
$394$	719.378	1.82583
$395$	108.779i	0.275390i
$396$	0	0
$397$	−239.773	−0.603962	−0.301981	−	0.953314i	$-0.597648\pi$
$397$	−239.773	−0.603962	−0.301981	+	0.953314i	$0.597648\pi$
$398$	55.7356i	0.140039i
$399$	0	0
$400$	381.111	0.952778
$401$	−548.277	−1.36727	−0.683637	−	0.729822i	$-0.739603\pi$
$401$	−548.277	−1.36727	−0.683637	+	0.729822i	$0.739603\pi$
$402$	0	0
$403$	− 443.966i	− 1.10165i
$404$	56.6094i	0.140122i
$405$	0	0
$406$	7.99380i	0.0196892i
$407$	0	0
$408$	0	0
$409$	58.3814i	0.142742i	0.997450	+	0.0713709i	$0.0227374\pi$
$409$	58.3814i	0.142742i	−0.997450	+	0.0713709i	$0.977263\pi$
$410$	46.9295	0.114462
$411$	0	0
$412$	−43.7069	−0.106085
$413$	249.172i	0.603322i
$414$	0	0
$415$	− 21.6992i	− 0.0522872i
$416$	−54.1053	−0.130061
$417$	0	0
$418$	0	0
$419$	−29.3426	−0.0700300	−0.0350150	−	0.999387i	$-0.511148\pi$
$419$	−29.3426	−0.0700300	−0.0350150	+	0.999387i	$0.511148\pi$
$420$	0	0
$421$	432.667	1.02771	0.513856	−	0.857876i	$-0.328216\pi$
$421$	432.667	1.02771	0.513856	+	0.857876i	$0.328216\pi$
$422$	−8.26187	−0.0195779
$423$	0	0
$424$	25.5348i	0.0602237i
$425$	− 661.110i	− 1.55555i
$426$	0	0
$427$	−136.839	−0.320466
$428$	− 7.62999i	− 0.0178271i
$429$	0	0
$430$	−47.7355	−0.111013
$431$	533.656i	1.23818i	0.785319	+	0.619091i	$0.212499\pi$
$431$	533.656i	1.23818i	−0.785319	+	0.619091i	$0.787501\pi$
$432$	0	0
$433$	−271.237	−0.626414	−0.313207	−	0.949685i	$-0.601403\pi$
$433$	−271.237	−0.626414	−0.313207	+	0.949685i	$0.601403\pi$
$434$	−258.165	−0.594849
$435$	0	0
$436$	0.146416i	0 0.000335817i
$437$	733.860i	1.67931i
$438$	0	0
$439$	508.788i	1.15897i	0.814982	+	0.579486i	$0.196747\pi$
$439$	508.788i	1.15897i	−0.814982	+	0.579486i	$0.803253\pi$
$440$	0	0
$441$	0	0
$442$	667.524i	1.51023i
$443$	−404.718	−0.913585	−0.456793	−	0.889573i	$-0.651002\pi$
$443$	−404.718	−0.913585	−0.456793	+	0.889573i	$0.651002\pi$
$444$	0	0
$445$	265.703	0.597084
$446$	− 631.762i	− 1.41651i
$447$	0	0
$448$	− 176.023i	− 0.392909i
$449$	629.680	1.40241	0.701203	−	0.712962i	$-0.252647\pi$
$449$	629.680	1.40241	0.701203	+	0.712962i	$0.252647\pi$
$450$	0	0
$451$	0	0
$452$	1.83850	0.00406748
$453$	0	0
$454$	147.202	0.324233
$455$	−54.5014	−0.119783
$456$	0	0
$457$	490.340i	1.07295i	0.843915	+	0.536477i	$0.180245\pi$
$457$	490.340i	1.07295i	−0.843915	+	0.536477i	$0.819755\pi$
$458$	201.660i	0.440305i
$459$	0	0
$460$	−10.5816	−0.0230035
$461$	− 264.802i	− 0.574407i	−0.957870	−	0.287203i	$-0.907274\pi$
$461$	− 264.802i	− 0.574407i	0.957870	−	0.287203i	$-0.0927256\pi$
$462$	0	0
$463$	−430.645	−0.930118	−0.465059	−	0.885280i	$-0.653967\pi$
$463$	−430.645	−0.930118	−0.465059	+	0.885280i	$0.653967\pi$
$464$	− 21.8409i	− 0.0470710i
$465$	0	0
$466$	258.976	0.555741
$467$	−38.5052	−0.0824522	−0.0412261	−	0.999150i	$-0.513126\pi$
$467$	−38.5052	−0.0824522	−0.0412261	+	0.999150i	$0.513126\pi$
$468$	0	0
$469$	142.694i	0.304251i
$470$	205.979i	0.438253i
$471$	0	0
$472$	− 630.981i	− 1.33683i
$473$	0	0
$474$	0	0
$475$	− 807.132i	− 1.69923i
$476$	28.2406	0.0593290
$477$	0	0
$478$	50.9899	0.106673
$479$	− 690.603i	− 1.44176i	−0.693060	−	0.720880i	$-0.743738\pi$
$479$	− 690.603i	− 1.44176i	0.693060	−	0.720880i	$-0.256262\pi$
$480$	0	0
$481$	− 495.437i	− 1.03001i
$482$	−459.681	−0.953695
$483$	0	0
$484$	0	0
$485$	139.045	0.286692
$486$	0	0
$487$	144.315	0.296335	0.148168	−	0.988962i	$-0.452662\pi$
$487$	144.315	0.296335	0.148168	+	0.988962i	$0.452662\pi$
$488$	346.520	0.710081
$489$	0	0
$490$	− 138.200i	− 0.282042i
$491$	244.479i	0.497921i	0.968514	+	0.248961i	$0.0800890\pi$
$491$	244.479i	0.497921i	−0.968514	+	0.248961i	$0.919911\pi$
$492$	0	0
$493$	−37.8873	−0.0768505
$494$	814.963i	1.64972i
$495$	0	0
$496$	705.367	1.42211
$497$	− 244.339i	− 0.491628i
$498$	0	0
$499$	−166.165	−0.332997	−0.166498	−	0.986042i	$-0.553246\pi$
$499$	−166.165	−0.332997	−0.166498	+	0.986042i	$0.553246\pi$
$500$	24.7364	0.0494727
$501$	0	0
$502$	− 197.796i	− 0.394015i
$503$	− 240.497i	− 0.478126i	−0.971004	−	0.239063i	$-0.923160\pi$
$503$	− 240.497i	− 0.478126i	0.971004	−	0.239063i	$-0.0768402\pi$
$504$	0	0
$505$	− 301.099i	− 0.596237i
$506$	0	0
$507$	0	0
$508$	23.4516i	0.0461647i
$509$	238.912	0.469375	0.234687	−	0.972071i	$-0.424593\pi$
$509$	238.912	0.469375	0.234687	+	0.972071i	$0.424593\pi$
$510$	0	0
$511$	−237.011	−0.463817
$512$	439.455i	0.858311i
$513$	0	0
$514$	− 580.808i	− 1.12998i
$515$	232.472	0.451402
$516$	0	0
$517$	0	0
$518$	−288.095	−0.556168
$519$	0	0
$520$	138.015	0.265413
$521$	−608.850	−1.16862	−0.584309	−	0.811531i	$-0.698635\pi$
$521$	−608.850	−1.16862	−0.584309	+	0.811531i	$0.698635\pi$
$522$	0	0
$523$	− 43.0443i	− 0.0823027i	−0.999153	−	0.0411514i	$-0.986897\pi$
$523$	− 43.0443i	− 0.0823027i	0.999153	−	0.0411514i	$-0.0131026\pi$
$524$	43.3989i	0.0828224i
$525$	0	0
$526$	−348.154	−0.661889
$527$	− 1223.59i	− 2.32181i
$528$	0	0
$529$	−121.092	−0.228908
$530$	11.5640i	0.0218188i
$531$	0	0
$532$	34.4782	0.0648087
$533$	−146.164	−0.274229
$534$	0	0
$535$	40.5831i	0.0758562i
$536$	− 361.345i	− 0.674151i
$537$	0	0
$538$	413.669i	0.768901i
$539$	0	0
$540$	0	0
$541$	− 631.174i	− 1.16668i	−0.812228	−	0.583340i	$-0.801745\pi$
$541$	− 631.174i	− 1.16668i	0.812228	−	0.583340i	$-0.198255\pi$
$542$	213.880	0.394613
$543$	0	0
$544$	−149.117	−0.274112
$545$	− 0.778773i	− 0.00142894i
$546$	0	0
$547$	1001.13i	1.83021i	0.403214	+	0.915106i	$0.367893\pi$
$547$	1001.13i	1.83021i	−0.403214	+	0.915106i	$0.632107\pi$
$548$	30.0336	0.0548058
$549$	0	0
$550$	0	0
$551$	−46.2556	−0.0839485
$552$	0	0
$553$	−197.010	−0.356256
$554$	489.286	0.883188
$555$	0	0
$556$	− 8.12219i	− 0.0146083i
$557$	536.341i	0.962910i	0.876471	+	0.481455i	$0.159892\pi$
$557$	536.341i	0.962910i	−0.876471	+	0.481455i	$0.840108\pi$
$558$	0	0
$559$	148.674	0.265965
$560$	− 86.5911i	− 0.154627i
$561$	0	0
$562$	−489.293	−0.870629
$563$	113.718i	0.201986i	0.994887	+	0.100993i	$0.0322020\pi$
$563$	113.718i	0.201986i	−0.994887	+	0.100993i	$0.967798\pi$
$564$	0	0
$565$	−9.77879	−0.0173076
$566$	320.510	0.566273
$567$	0	0
$568$	618.743i	1.08934i
$569$	− 434.581i	− 0.763762i	−0.924211	−	0.381881i	$-0.875276\pi$
$569$	− 434.581i	− 0.763762i	0.924211	−	0.381881i	$-0.124724\pi$
$570$	0	0
$571$	− 91.3599i	− 0.160000i	−0.996795	−	0.0800000i	$-0.974508\pi$
$571$	− 91.3599i	− 0.160000i	0.996795	−	0.0800000i	$-0.0254920\pi$
$572$	0	0
$573$	0	0
$574$	84.9939i	0.148073i
$575$	−448.636	−0.780236
$576$	0	0
$577$	−244.639	−0.423985	−0.211992	−	0.977271i	$-0.567995\pi$
$577$	−244.639	−0.423985	−0.211992	+	0.977271i	$0.567995\pi$
$578$	1239.48i	2.14443i
$579$	0	0
$580$	− 0.666967i	− 0.00114994i
$581$	39.2993	0.0676408
$582$	0	0
$583$	0	0
$584$	600.185	1.02771
$585$	0	0
$586$	−1097.60	−1.87304
$587$	604.382	1.02961	0.514806	−	0.857307i	$-0.327864\pi$
$587$	604.382	1.02961	0.514806	+	0.857307i	$0.327864\pi$
$588$	0	0
$589$	− 1493.85i	− 2.53626i
$590$	− 285.753i	− 0.484327i
$591$	0	0
$592$	787.144	1.32964
$593$	168.161i	0.283577i	0.989897	+	0.141789i	$0.0452854\pi$
$593$	168.161i	0.283577i	−0.989897	+	0.141789i	$0.954715\pi$
$594$	0	0
$595$	−150.209	−0.252452
$596$	49.9213i	0.0837606i
$597$	0	0
$598$	452.988	0.757505
$599$	954.528	1.59354	0.796768	−	0.604285i	$-0.206541\pi$
$599$	954.528	1.59354	0.796768	+	0.604285i	$0.206541\pi$
$600$	0	0
$601$	330.613i	0.550105i	0.961429	+	0.275052i	$0.0886952\pi$
$601$	330.613i	0.550105i	−0.961429	+	0.275052i	$0.911305\pi$
$602$	− 86.4536i	− 0.143611i
$603$	0	0
$604$	− 62.8936i	− 0.104129i
$605$	0	0
$606$	0	0
$607$	− 50.4233i	− 0.0830697i	−0.999137	−	0.0415349i	$-0.986775\pi$
$607$	− 50.4233i	− 0.0830697i	0.999137	−	0.0415349i	$-0.0132248\pi$
$608$	−182.053	−0.299430
$609$	0	0
$610$	156.928	0.257260
$611$	− 641.531i	− 1.04997i
$612$	0	0
$613$	− 736.938i	− 1.20218i	−0.799180	−	0.601091i	$-0.794733\pi$
$613$	− 736.938i	− 1.20218i	0.799180	−	0.601091i	$-0.205267\pi$
$614$	309.682	0.504367
$615$	0	0
$616$	0	0
$617$	412.246	0.668147	0.334073	−	0.942547i	$-0.391577\pi$
$617$	412.246	0.668147	0.334073	+	0.942547i	$0.391577\pi$
$618$	0	0
$619$	298.754	0.482639	0.241320	−	0.970446i	$-0.422420\pi$
$619$	298.754	0.482639	0.241320	+	0.970446i	$0.422420\pi$
$620$	21.5401	0.0347421
$621$	0	0
$622$	− 930.011i	− 1.49519i
$623$	481.213i	0.772412i
$624$	0	0
$625$	423.762	0.678019
$626$	784.902i	1.25384i
$627$	0	0
$628$	28.2908	0.0450491
$629$	− 1365.45i	− 2.17083i
$630$	0	0
$631$	406.980	0.644977	0.322488	−	0.946573i	$-0.395481\pi$
$631$	406.980	0.644977	0.322488	+	0.946573i	$0.395481\pi$
$632$	498.890	0.789383
$633$	0	0
$634$	16.8791i	0.0266231i
$635$	− 124.737i	− 0.196436i
$636$	0	0
$637$	430.431i	0.675716i
$638$	0	0
$639$	0	0
$640$	235.321i	0.367689i
$641$	850.412	1.32670	0.663348	−	0.748311i	$-0.269135\pi$
$641$	850.412	1.32670	0.663348	+	0.748311i	$0.269135\pi$
$642$	0	0
$643$	−498.009	−0.774509	−0.387254	−	0.921973i	$-0.626576\pi$
$643$	−498.009	−0.774509	−0.387254	+	0.921973i	$0.626576\pi$
$644$	− 19.1643i	− 0.0297583i
$645$	0	0
$646$	2246.08i	3.47691i
$647$	705.900	1.09104	0.545518	−	0.838099i	$-0.316333\pi$
$647$	705.900	1.09104	0.545518	+	0.838099i	$0.316333\pi$
$648$	0	0
$649$	0	0
$650$	−498.217	−0.766488
$651$	0	0
$652$	−82.7186	−0.126869
$653$	641.754	0.982778	0.491389	−	0.870940i	$-0.336489\pi$
$653$	641.754	0.982778	0.491389	+	0.870940i	$0.336489\pi$
$654$	0	0
$655$	− 230.834i	− 0.352419i
$656$	− 232.223i	− 0.353999i
$657$	0	0
$658$	−373.048	−0.566942
$659$	198.456i	0.301147i	0.988599	+	0.150574i	$0.0481120\pi$
$659$	198.456i	0.301147i	−0.988599	+	0.150574i	$0.951888\pi$
$660$	0	0
$661$	663.285	1.00346	0.501728	−	0.865025i	$-0.332698\pi$
$661$	663.285	1.00346	0.501728	+	0.865025i	$0.332698\pi$
$662$	− 1088.62i	− 1.64443i
$663$	0	0
$664$	−99.5182	−0.149877
$665$	−183.386	−0.275769
$666$	0	0
$667$	25.7107i	0.0385467i
$668$	39.6660i	0.0593802i
$669$	0	0
$670$	− 163.642i	− 0.244242i
$671$	0	0
$672$	0	0
$673$	− 942.510i	− 1.40046i	−0.713917	−	0.700231i	$-0.753080\pi$
$673$	− 942.510i	− 1.40046i	0.713917	−	0.700231i	$-0.246920\pi$
$674$	−570.337	−0.846197
$675$	0	0
$676$	−16.4418	−0.0243222
$677$	645.489i	0.953455i	0.879051	+	0.476728i	$0.158177\pi$
$677$	645.489i	0.953455i	−0.879051	+	0.476728i	$0.841823\pi$
$678$	0	0
$679$	251.825i	0.370876i
$680$	380.376	0.559376
$681$	0	0
$682$	0	0
$683$	−6.47577	−0.00948135	−0.00474068	−	0.999989i	$-0.501509\pi$
$683$	−6.47577	−0.00948135	−0.00474068	+	0.999989i	$0.501509\pi$
$684$	0	0
$685$	−159.745	−0.233205
$686$	557.987	0.813392
$687$	0	0
$688$	236.212i	0.343331i
$689$	− 36.0164i	− 0.0522735i
$690$	0	0
$691$	1037.55	1.50152	0.750759	−	0.660576i	$-0.229688\pi$
$691$	1037.55	1.50152	0.750759	+	0.660576i	$0.229688\pi$
$692$	− 67.3456i	− 0.0973203i
$693$	0	0
$694$	−695.917	−1.00276
$695$	43.2011i	0.0621598i
$696$	0	0
$697$	−402.836	−0.577956
$698$	122.899	0.176073
$699$	0	0
$700$	21.0778i	0.0301112i
$701$	− 324.727i	− 0.463234i	−0.972807	−	0.231617i	$-0.925598\pi$
$701$	− 324.727i	− 0.463234i	0.972807	−	0.231617i	$-0.0744016\pi$
$702$	0	0
$703$	− 1667.05i	− 2.37133i
$704$	0	0
$705$	0	0
$706$	− 815.144i	− 1.15459i
$707$	545.320	0.771316
$708$	0	0
$709$	−276.053	−0.389356	−0.194678	−	0.980867i	$-0.562366\pi$
$709$	−276.053	−0.389356	−0.194678	+	0.980867i	$0.562366\pi$
$710$	280.210i	0.394662i
$711$	0	0
$712$	− 1218.58i	− 1.71149i
$713$	−830.342	−1.16458
$714$	0	0
$715$	0	0
$716$	72.4863	0.101238
$717$	0	0
$718$	1063.92	1.48179
$719$	605.393	0.841994	0.420997	−	0.907062i	$-0.361680\pi$
$719$	605.393	0.841994	0.420997	+	0.907062i	$0.361680\pi$
$720$	0	0
$721$	421.030i	0.583952i
$722$	1992.39i	2.75955i
$723$	0	0
$724$	4.98262	0.00688207
$725$	− 28.2778i	− 0.0390038i
$726$	0	0
$727$	401.322	0.552025	0.276012	−	0.961154i	$-0.410987\pi$
$727$	401.322	0.552025	0.276012	+	0.961154i	$0.410987\pi$
$728$	249.958i	0.343349i
$729$	0	0
$730$	271.806	0.372337
$731$	409.754	0.560539
$732$	0	0
$733$	1262.41i	1.72225i	0.508392	+	0.861126i	$0.330240\pi$
$733$	1262.41i	1.72225i	−0.508392	+	0.861126i	$0.669760\pi$
$734$	566.378i	0.771632i
$735$	0	0
$736$	101.192i	0.137490i
$737$	0	0
$738$	0	0
$739$	− 532.115i	− 0.720048i	−0.932943	−	0.360024i	$-0.882769\pi$
$739$	− 532.115i	− 0.720048i	0.932943	−	0.360024i	$-0.117231\pi$
$740$	24.0374	0.0324829
$741$	0	0
$742$	−20.9435	−0.0282257
$743$	593.543i	0.798847i	0.916767	+	0.399423i	$0.130790\pi$
$743$	593.543i	0.798847i	−0.916767	+	0.399423i	$0.869210\pi$
$744$	0	0
$745$	− 265.526i	− 0.356411i
$746$	469.598	0.629488
$747$	0	0
$748$	0	0
$749$	−73.4999	−0.0981307
$750$	0	0
$751$	916.296	1.22010	0.610051	−	0.792362i	$-0.291149\pi$
$751$	916.296	1.22010	0.610051	+	0.792362i	$0.291149\pi$
$752$	1019.26	1.35539
$753$	0	0
$754$	28.5521i	0.0378675i
$755$	334.525i	0.443079i
$756$	0	0
$757$	−1263.61	−1.66924	−0.834619	−	0.550828i	$-0.814312\pi$
$757$	−1263.61	−1.66924	−0.834619	+	0.550828i	$0.814312\pi$
$758$	− 656.917i	− 0.866645i
$759$	0	0
$760$	464.391	0.611041
$761$	67.8695i	0.0891846i	0.999005	+	0.0445923i	$0.0141989\pi$
$761$	67.8695i	0.0891846i	−0.999005	+	0.0445923i	$0.985801\pi$
$762$	0	0
$763$	1.41043	0.00184854
$764$	−26.2735	−0.0343894
$765$	0	0
$766$	1431.26i	1.86849i
$767$	889.988i	1.16035i
$768$	0	0
$769$	213.139i	0.277163i	0.990351	+	0.138582i	$0.0442543\pi$
$769$	213.139i	0.277163i	−0.990351	+	0.138582i	$0.955746\pi$
$770$	0	0
$771$	0	0
$772$	91.3723i	0.118358i
$773$	−628.593	−0.813186	−0.406593	−	0.913609i	$-0.633283\pi$
$773$	−628.593	−0.813186	−0.406593	+	0.913609i	$0.633283\pi$
$774$	0	0
$775$	913.248	1.17838
$776$	− 637.699i	− 0.821777i
$777$	0	0
$778$	893.274i	1.14817i
$779$	−491.812	−0.631338
$780$	0	0
$781$	0	0
$782$	1248.46	1.59650
$783$	0	0
$784$	−683.863	−0.872275
$785$	−150.476	−0.191689
$786$	0	0
$787$	366.569i	0.465780i	0.972503	+	0.232890i	$0.0748182\pi$
$787$	366.569i	0.465780i	−0.972503	+	0.232890i	$0.925182\pi$
$788$	− 108.705i	− 0.137951i
$789$	0	0
$790$	225.932	0.285990
$791$	− 17.7103i	− 0.0223898i
$792$	0	0
$793$	−488.760	−0.616343
$794$	498.004i	0.627209i
$795$	0	0
$796$	8.42220	0.0105807
$797$	−877.739	−1.10130	−0.550652	−	0.834735i	$-0.685621\pi$
$797$	−877.739	−1.10130	−0.550652	+	0.834735i	$0.685621\pi$
$798$	0	0
$799$	− 1768.09i	− 2.21288i
$800$	− 111.296i	− 0.139120i
$801$	0	0
$802$	1138.76i	1.41990i
$803$	0	0
$804$	0	0
$805$	101.933i	0.126625i
$806$	−922.108	−1.14405
$807$	0	0
$808$	−1380.92	−1.70906
$809$	380.698i	0.470579i	0.971925	+	0.235289i	$0.0756038\pi$
$809$	380.698i	0.470579i	−0.971925	+	0.235289i	$0.924396\pi$
$810$	0	0
$811$	− 501.430i	− 0.618286i	−0.951016	−	0.309143i	$-0.899958\pi$
$811$	− 501.430i	− 0.618286i	0.951016	−	0.309143i	$-0.100042\pi$
$812$	1.20794	0.00148761
$813$	0	0
$814$	0	0
$815$	439.971	0.539842
$816$	0	0
$817$	500.259	0.612312
$818$	121.257	0.148236
$819$	0	0
$820$	− 7.09151i	− 0.00864818i
$821$	− 507.368i	− 0.617988i	−0.951064	−	0.308994i	$-0.900008\pi$
$821$	− 507.368i	− 0.617988i	0.951064	−	0.308994i	$-0.0999922\pi$
$822$	0	0
$823$	−219.897	−0.267189	−0.133595	−	0.991036i	$-0.542652\pi$
$823$	−219.897	−0.267189	−0.133595	+	0.991036i	$0.542652\pi$
$824$	− 1066.18i	− 1.29391i
$825$	0	0
$826$	517.526	0.626544
$827$	132.603i	0.160342i	0.996781	+	0.0801711i	$0.0255467\pi$
$827$	132.603i	0.160342i	−0.996781	+	0.0801711i	$0.974453\pi$
$828$	0	0
$829$	90.0297	0.108600	0.0543002	−	0.998525i	$-0.482707\pi$
$829$	90.0297	0.108600	0.0543002	+	0.998525i	$0.482707\pi$
$830$	−45.0688	−0.0542998
$831$	0	0
$832$	− 628.716i	− 0.755669i
$833$	1186.29i	1.42412i
$834$	0	0
$835$	− 210.979i	− 0.252670i
$836$	0	0
$837$	0	0
$838$	60.9439i	0.0727255i
$839$	−775.856	−0.924738	−0.462369	−	0.886687i	$-0.653001\pi$
$839$	−775.856	−0.924738	−0.462369	+	0.886687i	$0.653001\pi$
$840$	0	0
$841$	839.379	0.998073
$842$	− 898.641i	− 1.06727i
$843$	0	0
$844$	1.24845i	0.00147920i
$845$	87.4523	0.103494
$846$	0	0
$847$	0	0
$848$	57.2225	0.0674793
$849$	0	0
$850$	−1373.11	−1.61543
$851$	−926.608	−1.08885
$852$	0	0
$853$	1512.33i	1.77295i	0.462772	+	0.886477i	$0.346855\pi$
$853$	1512.33i	1.77295i	−0.462772	+	0.886477i	$0.653145\pi$
$854$	284.212i	0.332801i
$855$	0	0
$856$	186.125	0.217435
$857$	1555.00i	1.81447i	0.420624	+	0.907235i	$0.361811\pi$
$857$	1555.00i	1.81447i	−0.420624	+	0.907235i	$0.638189\pi$
$858$	0	0
$859$	1048.08	1.22011	0.610057	−	0.792358i	$-0.291147\pi$
$859$	1048.08	1.22011	0.610057	+	0.792358i	$0.291147\pi$
$860$	7.21330i	0.00838756i
$861$	0	0
$862$	1108.39	1.28584
$863$	507.805	0.588418	0.294209	−	0.955741i	$-0.404944\pi$
$863$	507.805	0.588418	0.294209	+	0.955741i	$0.404944\pi$
$864$	0	0
$865$	358.204i	0.414109i
$866$	563.355i	0.650525i
$867$	0	0
$868$	39.0112i	0.0449438i
$869$	0	0
$870$	0	0
$871$	509.670i	0.585155i
$872$	−3.57166	−0.00409594
$873$	0	0
$874$	1524.21	1.74395
$875$	− 238.286i	− 0.272327i
$876$	0	0
$877$	− 923.870i	− 1.05344i	−0.850038	−	0.526722i	$-0.823421\pi$
$877$	− 923.870i	− 1.05344i	0.850038	−	0.526722i	$-0.176579\pi$
$878$	1056.74	1.20358
$879$	0	0
$880$	0	0
$881$	173.500	0.196935	0.0984677	−	0.995140i	$-0.468606\pi$
$881$	173.500	0.196935	0.0984677	+	0.995140i	$0.468606\pi$
$882$	0	0
$883$	−251.549	−0.284880	−0.142440	−	0.989803i	$-0.545495\pi$
$883$	−251.549	−0.284880	−0.142440	+	0.989803i	$0.545495\pi$
$884$	100.869	0.114106
$885$	0	0
$886$	840.592i	0.948750i
$887$	897.015i	1.01129i	0.862741	+	0.505646i	$0.168746\pi$
$887$	897.015i	1.01129i	−0.862741	+	0.505646i	$0.831254\pi$
$888$	0	0
$889$	225.910	0.254117
$890$	− 551.859i	− 0.620067i
$891$	0	0
$892$	−95.4653	−0.107024
$893$	− 2158.62i	− 2.41727i
$894$	0	0
$895$	−385.547	−0.430779
$896$	−426.189	−0.475657
$897$	0	0
$898$	− 1307.83i	− 1.45639i
$899$	− 52.3370i	− 0.0582169i
$900$	0	0
$901$	− 99.2632i	− 0.110170i
$902$	0	0
$903$	0	0
$904$	44.8481i	0.0496107i
$905$	−26.5020	−0.0292840
$906$	0	0
$907$	807.083	0.889838	0.444919	−	0.895571i	$-0.353232\pi$
$907$	807.083	0.889838	0.444919	+	0.895571i	$0.353232\pi$
$908$	− 22.2436i	− 0.0244974i
$909$	0	0
$910$	113.198i	0.124394i
$911$	−282.911	−0.310550	−0.155275	−	0.987871i	$-0.549626\pi$
$911$	−282.911	−0.310550	−0.155275	+	0.987871i	$0.549626\pi$
$912$	0	0
$913$	0	0
$914$	1018.43	1.11425
$915$	0	0
$916$	30.4727	0.0332672
$917$	418.063	0.455903
$918$	0	0
$919$	− 451.445i	− 0.491235i	−0.969367	−	0.245617i	$-0.921009\pi$
$919$	− 451.445i	− 0.491235i	0.969367	−	0.245617i	$-0.0789907\pi$
$920$	− 258.127i	− 0.280572i
$921$	0	0
$922$	−549.988	−0.596516
$923$	− 872.727i	− 0.945533i
$924$	0	0
$925$	1019.13	1.10176
$926$	894.441i	0.965919i
$927$	0	0
$928$	−6.37821	−0.00687307
$929$	633.711	0.682143	0.341072	−	0.940037i	$-0.389210\pi$
$929$	633.711	0.682143	0.341072	+	0.940037i	$0.389210\pi$
$930$	0	0
$931$	1448.31i	1.55565i
$932$	− 39.1337i	− 0.0419890i
$933$	0	0
$934$	79.9745i	0.0856259i
$935$	0	0
$936$	0	0
$937$	837.661i	0.893982i	0.894538	+	0.446991i	$0.147504\pi$
$937$	837.661i	0.893982i	−0.894538	+	0.446991i	$0.852496\pi$
$938$	296.372	0.315961
$939$	0	0
$940$	31.1255	0.0331122
$941$	725.601i	0.771095i	0.922688	+	0.385548i	$0.125987\pi$
$941$	725.601i	0.771095i	−0.922688	+	0.385548i	$0.874013\pi$
$942$	0	0
$943$	273.368i	0.289892i
$944$	−1414.00	−1.49788
$945$	0	0
$946$	0	0
$947$	699.606	0.738761	0.369380	−	0.929278i	$-0.379570\pi$
$947$	699.606	0.738761	0.369380	+	0.929278i	$0.379570\pi$
$948$	0	0
$949$	−846.551	−0.892045
$950$	−1676.40	−1.76463
$951$	0	0
$952$	688.897i	0.723631i
$953$	32.8437i	0.0344634i	0.999852	+	0.0172317i	$0.00548530\pi$
$953$	32.8437i	0.0344634i	−0.999852	+	0.0172317i	$0.994515\pi$
$954$	0	0
$955$	139.746	0.146331
$956$	− 7.70507i	− 0.00805970i
$957$	0	0
$958$	−1434.37	−1.49725
$959$	− 289.314i	− 0.301683i
$960$	0	0
$961$	729.255	0.758850
$962$	−1029.01	−1.06966
$963$	0	0
$964$	69.4623i	0.0720563i
$965$	− 486.000i	− 0.503627i
$966$	0	0
$967$	140.387i	0.145178i	0.997362	+	0.0725888i	$0.0231261\pi$
$967$	140.387i	0.145178i	−0.997362	+	0.0725888i	$0.976874\pi$
$968$	0	0
$969$	0	0
$970$	− 288.795i	− 0.297726i
$971$	1664.59	1.71430	0.857151	−	0.515066i	$-0.172232\pi$
$971$	1664.59	1.71430	0.857151	+	0.515066i	$0.172232\pi$
$972$	0	0
$973$	−78.2413	−0.0804125
$974$	− 299.740i	− 0.307741i
$975$	0	0
$976$	− 776.535i	− 0.795631i
$977$	−1471.35	−1.50599	−0.752994	−	0.658028i	$-0.771391\pi$
$977$	−1471.35	−1.50599	−0.752994	+	0.658028i	$0.771391\pi$
$978$	0	0
$979$	0	0
$980$	−20.8834	−0.0213096
$981$	0	0
$982$	507.779	0.517087
$983$	−874.915	−0.890046	−0.445023	−	0.895519i	$-0.646805\pi$
$983$	−874.915	−0.890046	−0.445023	+	0.895519i	$0.646805\pi$
$984$	0	0
$985$	578.191i	0.586996i
$986$	78.6912i	0.0798085i
$987$	0	0
$988$	123.149	0.124645
$989$	− 278.063i	− 0.281156i
$990$	0	0
$991$	−470.488	−0.474761	−0.237380	−	0.971417i	$-0.576289\pi$
$991$	−470.488	−0.474761	−0.237380	+	0.971417i	$0.576289\pi$
$992$	− 205.988i	− 0.207650i
$993$	0	0
$994$	−507.488	−0.510551
$995$	−44.7968	−0.0450219
$996$	0	0
$997$	− 1045.67i	− 1.04882i	−0.851466	−	0.524410i	$-0.824286\pi$
$997$	− 1045.67i	− 1.04882i	0.851466	−	0.524410i	$-0.175714\pi$
$998$	345.122i	0.345814i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.j.604.3		8
3.2	odd	2		363.3.c.d.241.6	yes	8
11.10	odd	2	inner	1089.3.c.j.604.6		8
33.2	even	10		363.3.g.h.40.3		32
33.5	odd	10		363.3.g.h.118.3		32
33.8	even	10		363.3.g.h.112.3		32
33.14	odd	10		363.3.g.h.112.6		32
33.17	even	10		363.3.g.h.118.6		32
33.20	odd	10		363.3.g.h.40.6		32
33.26	odd	10		363.3.g.h.94.3		32
33.29	even	10		363.3.g.h.94.6		32
33.32	even	2		363.3.c.d.241.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.d.241.3	✓	8	33.32	even	2
363.3.c.d.241.6	yes	8	3.2	odd	2
363.3.g.h.40.3		32	33.2	even	10
363.3.g.h.40.6		32	33.20	odd	10
363.3.g.h.94.3		32	33.26	odd	10
363.3.g.h.94.6		32	33.29	even	10
363.3.g.h.112.3		32	33.8	even	10
363.3.g.h.112.6		32	33.14	odd	10
363.3.g.h.118.3		32	33.5	odd	10
363.3.g.h.118.6		32	33.17	even	10
1089.3.c.j.604.3		8	1.1	even	1	trivial
1089.3.c.j.604.6		8	11.10	odd	2	inner

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.07698i q^{2} -0.313852 q^{4} +1.66935 q^{5} +3.02335i q^{7} -7.65606i q^{8} +O(q^{10})\)	\(q-2.07698i q^{2} -0.313852 q^{4} +1.66935 q^{5} +3.02335i q^{7} -7.65606i q^{8} -3.46720i q^{10} +10.7987i q^{13} +6.27944 q^{14} -17.1569 q^{16} +29.7619i q^{17} +36.3356i q^{19} -0.523928 q^{20} +20.1967 q^{23} -22.2133 q^{25} +22.4288 q^{26} -0.948884i q^{28} +1.27301i q^{29} -41.1127 q^{31} +5.01033i q^{32} +61.8149 q^{34} +5.04701i q^{35} -45.8791 q^{37} +75.4683 q^{38} -12.7806i q^{40} +13.5353i q^{41} -13.7677i q^{43} -41.9482i q^{46} -59.4079 q^{47} +39.8594 q^{49} +46.1366i q^{50} -3.38921i q^{52} -3.33524 q^{53} +23.1469 q^{56} +2.64402 q^{58} +82.4159 q^{59} +45.2608i q^{61} +85.3903i q^{62} -58.2213 q^{64} +18.0268i q^{65} +47.1972 q^{67} -9.34084i q^{68} +10.4826 q^{70} -80.8174 q^{71} +78.3935i q^{73} +95.2901i q^{74} -11.4040i q^{76} +65.1628i q^{79} -28.6408 q^{80} +28.1125 q^{82} -12.9986i q^{83} +49.6829i q^{85} -28.5953 q^{86} +159.166 q^{89} -32.6483 q^{91} -6.33878 q^{92} +123.389i q^{94} +60.6567i q^{95} +83.2933 q^{97} -82.7872i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} - 16 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 - 16 * q^5	\( 8 q - 24 q^{4} - 16 q^{5} - 136 q^{14} + 72 q^{16} + 16 q^{20} - 88 q^{25} + 120 q^{26} - 128 q^{31} + 96 q^{34} + 80 q^{37} + 216 q^{38} + 32 q^{47} - 152 q^{49} - 80 q^{53} + 696 q^{56} - 176 q^{58} + 64 q^{59} + 8 q^{64} + 464 q^{67} + 304 q^{70} + 128 q^{71} - 80 q^{80} + 528 q^{82} - 24 q^{86} + 720 q^{89} + 80 q^{91} + 1248 q^{92} + 416 q^{97}+O(q^{100}) \) 8 * q - 24 * q^4 - 16 * q^5 - 136 * q^14 + 72 * q^16 + 16 * q^20 - 88 * q^25 + 120 * q^26 - 128 * q^31 + 96 * q^34 + 80 * q^37 + 216 * q^38 + 32 * q^47 - 152 * q^49 - 80 * q^53 + 696 * q^56 - 176 * q^58 + 64 * q^59 + 8 * q^64 + 464 * q^67 + 304 * q^70 + 128 * q^71 - 80 * q^80 + 528 * q^82 - 24 * q^86 + 720 * q^89 + 80 * q^91 + 1248 * q^92 + 416 * q^97

Introduction

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