LMFDB - Embedded newform 1089.3.b.c.485.2

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 16x^{2} + 49 $ x^4 + 16*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.2
Root			$-2.03151i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.c.485.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} -1.12702 q^{7} -7.86799i q^{8} +O(q^{10})$	$q-2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} -1.12702 q^{7} -7.86799i q^{8} -1.61895 q^{10} -9.25403 q^{13} +2.28954i q^{14} -16.4919 q^{16} -11.7514i q^{17} -17.2379 q^{19} +0.101222i q^{20} +5.11797i q^{23} +24.3649 q^{25} +18.7996i q^{26} +0.143150 q^{28} +29.0812i q^{29} -52.7298 q^{31} +2.03151i q^{32} -23.8730 q^{34} +0.898143i q^{35} +37.3649 q^{37} +35.0189i q^{38} -6.27017 q^{40} -75.0089i q^{41} -68.8407 q^{43} +10.3972 q^{46} +9.36061i q^{47} -47.7298 q^{49} -49.4975i q^{50} +1.17542 q^{52} -34.0195i q^{53} +8.86735i q^{56} +59.0786 q^{58} +69.2965i q^{59} +24.9839 q^{61} +107.121i q^{62} -61.8407 q^{64} +7.37473i q^{65} -16.8730 q^{67} +1.49262i q^{68} +1.82458 q^{70} +77.0632i q^{71} -55.2379 q^{73} -75.9071i q^{74} +2.18950 q^{76} -64.9839 q^{79} +13.1428i q^{80} -152.381 q^{82} -120.342i q^{83} -9.36492 q^{85} +139.850i q^{86} +110.926i q^{89} +10.4294 q^{91} -0.650067i q^{92} +19.0161 q^{94} +13.7372i q^{95} -14.0948 q^{97} +96.9634i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 20 q^{7}+O(q^{10}) $ 4 * q - 16 * q^4 - 20 * q^7	$ 4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100}) $ 4 * q - 16 * q^4 - 20 * q^7 + 40 * q^10 - 68 * q^13 - 4 * q^16 + 24 * q^19 + 20 * q^25 + 140 * q^28 - 56 * q^31 - 80 * q^34 + 72 * q^37 - 180 * q^40 - 12 * q^43 + 212 * q^46 - 36 * q^49 + 392 * q^52 - 120 * q^58 - 24 * q^61 + 16 * q^64 - 52 * q^67 - 380 * q^70 - 128 * q^73 - 456 * q^76 - 136 * q^79 - 656 * q^82 + 40 * q^85 + 460 * q^91 + 200 * q^94 + 176 * 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 $

$2$	− 2.03151i	− 1.01575i	−0.861430	−	0.507877i	$-0.830431\pi$
$2$	− 2.03151i	− 1.01575i	0.861430	−	0.507877i	$-0.169569\pi$
$3$	0	0
$3$	0	0
$4$	−0.127017	−0.0317542
$5$	− 0.796921i	− 0.159384i	−0.996820	−	0.0796921i	$-0.974606\pi$
$5$	− 0.796921i	− 0.159384i	0.996820	−	0.0796921i	$-0.0253937\pi$
$6$	0	0
$7$	−1.12702	−0.161002	−0.0805012	−	0.996755i	$-0.525652\pi$
$7$	−1.12702	−0.161002	−0.0805012	+	0.996755i	$0.525652\pi$
$8$	− 7.86799i	− 0.983499i
$9$	0	0
$10$	−1.61895	−0.161895
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−9.25403	−0.711849	−0.355924	−	0.934515i	$-0.615834\pi$
$13$	−9.25403	−0.711849	−0.355924	+	0.934515i	$0.615834\pi$
$14$	2.28954i	0.163539i
$15$	0	0
$16$	−16.4919	−1.03075
$17$	− 11.7514i	− 0.691257i	−0.938371	−	0.345629i	$-0.887666\pi$
$17$	− 11.7514i	− 0.691257i	0.938371	−	0.345629i	$-0.112334\pi$
$18$	0	0
$19$	−17.2379	−0.907258	−0.453629	−	0.891191i	$-0.649871\pi$
$19$	−17.2379	−0.907258	−0.453629	+	0.891191i	$0.649871\pi$
$20$	0.101222i	0.00506111i
$21$	0	0
$22$	0	0
$23$	5.11797i	0.222520i	0.993791	+	0.111260i	$0.0354887\pi$
$23$	5.11797i	0.222520i	−0.993791	+	0.111260i	$0.964511\pi$
$24$	0	0
$25$	24.3649	0.974597
$26$	18.7996i	0.723062i
$27$	0	0
$28$	0.143150	0.00511250
$29$	29.0812i	1.00280i	0.865216	+	0.501400i	$0.167181\pi$
$29$	29.0812i	1.00280i	−0.865216	+	0.501400i	$0.832819\pi$
$30$	0	0
$31$	−52.7298	−1.70096	−0.850481	−	0.526005i	$-0.823689\pi$
$31$	−52.7298	−1.70096	−0.850481	+	0.526005i	$0.823689\pi$
$32$	2.03151i	0.0634846i
$33$	0	0
$34$	−23.8730	−0.702147
$35$	0.898143i	0.0256612i
$36$	0	0
$37$	37.3649	1.00986	0.504931	−	0.863160i	$-0.331518\pi$
$37$	37.3649	1.00986	0.504931	+	0.863160i	$0.331518\pi$
$38$	35.0189i	0.921550i
$39$	0	0
$40$	−6.27017	−0.156754
$41$	− 75.0089i	− 1.82949i	−0.404037	−	0.914743i	$-0.632393\pi$
$41$	− 75.0089i	− 1.82949i	0.404037	−	0.914743i	$-0.367607\pi$
$42$	0	0
$43$	−68.8407	−1.60095	−0.800473	−	0.599368i	$-0.795419\pi$
$43$	−68.8407	−1.60095	−0.800473	+	0.599368i	$0.795419\pi$
$44$	0	0
$45$	0	0
$46$	10.3972	0.226026
$47$	9.36061i	0.199162i	0.995029	+	0.0995809i	$0.0317502\pi$
$47$	9.36061i	0.199162i	−0.995029	+	0.0995809i	$0.968250\pi$
$48$	0	0
$49$	−47.7298	−0.974078
$50$	− 49.4975i	− 0.989949i
$51$	0	0
$52$	1.17542	0.0226042
$53$	− 34.0195i	− 0.641878i	−0.947100	−	0.320939i	$-0.896002\pi$
$53$	− 34.0195i	− 0.641878i	0.947100	−	0.320939i	$-0.103998\pi$
$54$	0	0
$55$	0	0
$56$	8.86735i	0.158346i
$57$	0	0
$58$	59.0786	1.01860
$59$	69.2965i	1.17452i	0.809400	+	0.587258i	$0.199793\pi$
$59$	69.2965i	1.17452i	−0.809400	+	0.587258i	$0.800207\pi$
$60$	0	0
$61$	24.9839	0.409572	0.204786	−	0.978807i	$-0.434350\pi$
$61$	24.9839	0.409572	0.204786	+	0.978807i	$0.434350\pi$
$62$	107.121i	1.72776i
$63$	0	0
$64$	−61.8407	−0.966261
$65$	7.37473i	0.113457i
$66$	0	0
$67$	−16.8730	−0.251836	−0.125918	−	0.992041i	$-0.540188\pi$
$67$	−16.8730	−0.251836	−0.125918	+	0.992041i	$0.540188\pi$
$68$	1.49262i	0.0219503i
$69$	0	0
$70$	1.82458	0.0260655
$71$	77.0632i	1.08540i	0.839927	+	0.542699i	$0.182597\pi$
$71$	77.0632i	1.08540i	−0.839927	+	0.542699i	$0.817403\pi$
$72$	0	0
$73$	−55.2379	−0.756684	−0.378342	−	0.925666i	$-0.623506\pi$
$73$	−55.2379	−0.756684	−0.378342	+	0.925666i	$0.623506\pi$
$74$	− 75.9071i	− 1.02577i
$75$	0	0
$76$	2.18950	0.0288092
$77$	0	0
$78$	0	0
$79$	−64.9839	−0.822581	−0.411290	−	0.911504i	$-0.634922\pi$
$79$	−64.9839	−0.822581	−0.411290	+	0.911504i	$0.634922\pi$
$80$	13.1428i	0.164285i
$81$	0	0
$82$	−152.381	−1.85831
$83$	− 120.342i	− 1.44991i	−0.688799	−	0.724953i	$-0.741862\pi$
$83$	− 120.342i	− 1.44991i	0.688799	−	0.724953i	$-0.258138\pi$
$84$	0	0
$85$	−9.36492	−0.110175
$86$	139.850i	1.62617i
$87$	0	0
$88$	0	0
$89$	110.926i	1.24636i	0.782079	+	0.623179i	$0.214159\pi$
$89$	110.926i	1.24636i	−0.782079	+	0.623179i	$0.785841\pi$
$90$	0	0
$91$	10.4294	0.114609
$92$	− 0.650067i	− 0.00706595i
$93$	0	0
$94$	19.0161	0.202299
$95$	13.7372i	0.144603i
$96$	0	0
$97$	−14.0948	−0.145307	−0.0726534	−	0.997357i	$-0.523147\pi$
$97$	−14.0948	−0.145307	−0.0726534	+	0.997357i	$0.523147\pi$
$98$	96.9634i	0.989423i
$99$	0	0
$100$	−3.09475	−0.0309475
$101$	− 25.1194i	− 0.248707i	−0.992238	−	0.124353i	$-0.960314\pi$
$101$	− 25.1194i	− 0.248707i	0.992238	−	0.124353i	$-0.0396857\pi$
$102$	0	0
$103$	−56.3649	−0.547232	−0.273616	−	0.961839i	$-0.588220\pi$
$103$	−56.3649	−0.547232	−0.273616	+	0.961839i	$0.588220\pi$
$104$	72.8106i	0.700102i
$105$	0	0
$106$	−69.1109	−0.651989
$107$	− 80.7670i	− 0.754832i	−0.926044	−	0.377416i	$-0.876813\pi$
$107$	− 80.7670i	− 0.754832i	0.926044	−	0.377416i	$-0.123187\pi$
$108$	0	0
$109$	−105.698	−0.969702	−0.484851	−	0.874597i	$-0.661126\pi$
$109$	−105.698	−0.969702	−0.484851	+	0.874597i	$0.661126\pi$
$110$	0	0
$111$	0	0
$112$	18.5867	0.165953
$113$	− 182.444i	− 1.61454i	−0.590179	−	0.807272i	$-0.700943\pi$
$113$	− 182.444i	− 1.61454i	0.590179	−	0.807272i	$-0.299057\pi$
$114$	0	0
$115$	4.07862	0.0354662
$116$	− 3.69380i	− 0.0318431i
$117$	0	0
$118$	140.776	1.19302
$119$	13.2440i	0.111294i
$120$	0	0
$121$	0	0
$122$	− 50.7549i	− 0.416024i
$123$	0	0
$124$	6.69757	0.0540126
$125$	− 39.3399i	− 0.314720i
$126$	0	0
$127$	143.492	1.12986	0.564929	−	0.825140i	$-0.308904\pi$
$127$	143.492	1.12986	0.564929	+	0.825140i	$0.308904\pi$
$128$	133.756i	1.04497i
$129$	0	0
$130$	14.9818	0.115245
$131$	69.8125i	0.532920i	0.963846	+	0.266460i	$0.0858540\pi$
$131$	69.8125i	0.532920i	−0.963846	+	0.266460i	$0.914146\pi$
$132$	0	0
$133$	19.4274	0.146071
$134$	34.2776i	0.255803i
$135$	0	0
$136$	−92.4597	−0.679850
$137$	227.754i	1.66244i	0.555946	+	0.831219i	$0.312356\pi$
$137$	227.754i	1.66244i	−0.555946	+	0.831219i	$0.687644\pi$
$138$	0	0
$139$	−102.808	−0.739629	−0.369815	−	0.929106i	$-0.620579\pi$
$139$	−102.808	−0.739629	−0.369815	+	0.929106i	$0.620579\pi$
$140$	− 0.114079i	0 0.000814851i
$141$	0	0
$142$	156.554	1.10250
$143$	0	0
$144$	0	0
$145$	23.1754	0.159830
$146$	112.216i	0.768604i
$147$	0	0
$148$	−4.74597	−0.0320673
$149$	− 2.27958i	− 0.0152992i	−0.999971	−	0.00764960i	$-0.997565\pi$
$149$	− 2.27958i	− 0.0152992i	0.999971	−	0.00764960i	$-0.00243497\pi$
$150$	0	0
$151$	233.714	1.54777	0.773886	−	0.633324i	$-0.218310\pi$
$151$	233.714	1.54777	0.773886	+	0.633324i	$0.218310\pi$
$152$	135.628i	0.892287i
$153$	0	0
$154$	0	0
$155$	42.0215i	0.271107i
$156$	0	0
$157$	−170.157	−1.08380	−0.541902	−	0.840442i	$-0.682296\pi$
$157$	−170.157	−1.08380	−0.541902	+	0.840442i	$0.682296\pi$
$158$	132.015i	0.835539i
$159$	0	0
$160$	1.61895	0.0101184
$161$	− 5.76804i	− 0.0358263i
$162$	0	0
$163$	118.333	0.725967	0.362984	−	0.931796i	$-0.381758\pi$
$163$	118.333	0.725967	0.362984	+	0.931796i	$0.381758\pi$
$164$	9.52738i	0.0580938i
$165$	0	0
$166$	−244.476	−1.47275
$167$	195.857i	1.17280i	0.810023	+	0.586399i	$0.199455\pi$
$167$	195.857i	1.17280i	−0.810023	+	0.586399i	$0.800545\pi$
$168$	0	0
$169$	−83.3629	−0.493271
$170$	19.0249i	0.111911i
$171$	0	0
$172$	8.74392	0.0508367
$173$	171.185i	0.989511i	0.869032	+	0.494755i	$0.164742\pi$
$173$	171.185i	0.989511i	−0.869032	+	0.494755i	$0.835258\pi$
$174$	0	0
$175$	−27.4597	−0.156912
$176$	0	0
$177$	0	0
$178$	225.347	1.26599
$179$	− 215.228i	− 1.20239i	−0.799101	−	0.601197i	$-0.794691\pi$
$179$	− 215.228i	− 1.20239i	0.799101	−	0.601197i	$-0.205309\pi$
$180$	0	0
$181$	237.730	1.31342	0.656712	−	0.754141i	$-0.271947\pi$
$181$	237.730	1.31342	0.656712	+	0.754141i	$0.271947\pi$
$182$	− 21.1875i	− 0.116415i
$183$	0	0
$184$	40.2681	0.218848
$185$	− 29.7769i	− 0.160956i
$186$	0	0
$187$	0	0
$188$	− 1.18895i	− 0.00632422i
$189$	0	0
$190$	27.9073	0.146881
$191$	− 337.929i	− 1.76926i	−0.466294	−	0.884630i	$-0.654411\pi$
$191$	− 337.929i	− 1.76926i	0.466294	−	0.884630i	$-0.345589\pi$
$192$	0	0
$193$	252.903	1.31038	0.655190	−	0.755464i	$-0.272589\pi$
$193$	252.903	1.31038	0.655190	+	0.755464i	$0.272589\pi$
$194$	28.6336i	0.147596i
$195$	0	0
$196$	6.06248	0.0309310
$197$	94.1578i	0.477959i	0.971025	+	0.238979i	$0.0768128\pi$
$197$	94.1578i	0.477959i	−0.971025	+	0.238979i	$0.923187\pi$
$198$	0	0
$199$	179.571	0.902365	0.451182	−	0.892432i	$-0.351002\pi$
$199$	179.571	0.902365	0.451182	+	0.892432i	$0.351002\pi$
$200$	− 191.703i	− 0.958514i
$201$	0	0
$202$	−51.0302	−0.252625
$203$	− 32.7750i	− 0.161453i
$204$	0	0
$205$	−59.7762	−0.291591
$206$	114.506i	0.555853i
$207$	0	0
$208$	152.617	0.733735
$209$	0	0
$210$	0	0
$211$	−200.175	−0.948699	−0.474349	−	0.880337i	$-0.657317\pi$
$211$	−200.175	−0.948699	−0.474349	+	0.880337i	$0.657317\pi$
$212$	4.32105i	0.0203823i
$213$	0	0
$214$	−164.079	−0.766723
$215$	54.8606i	0.255166i
$216$	0	0
$217$	59.4274	0.273859
$218$	214.725i	0.984978i
$219$	0	0
$220$	0	0
$221$	108.748i	0.492071i
$222$	0	0
$223$	−182.984	−0.820555	−0.410278	−	0.911961i	$-0.634568\pi$
$223$	−182.984	−0.820555	−0.410278	+	0.911961i	$0.634568\pi$
$224$	− 2.28954i	− 0.0102212i
$225$	0	0
$226$	−370.635	−1.63998
$227$	− 321.608i	− 1.41678i	−0.705823	−	0.708388i	$-0.749423\pi$
$227$	− 321.608i	− 1.41678i	0.705823	−	0.708388i	$-0.250577\pi$
$228$	0	0
$229$	201.000	0.877729	0.438865	−	0.898553i	$-0.355381\pi$
$229$	201.000	0.877729	0.438865	+	0.898553i	$0.355381\pi$
$230$	− 8.28573i	− 0.0360249i
$231$	0	0
$232$	228.810	0.986252
$233$	39.3399i	0.168841i	0.996430	+	0.0844205i	$0.0269039\pi$
$233$	39.3399i	0.168841i	−0.996430	+	0.0844205i	$0.973096\pi$
$234$	0	0
$235$	7.45967	0.0317433
$236$	− 8.80181i	− 0.0372958i
$237$	0	0
$238$	26.9052	0.113047
$239$	− 23.7707i	− 0.0994592i	−0.998763	−	0.0497296i	$-0.984164\pi$
$239$	− 23.7707i	− 0.0994592i	0.998763	−	0.0497296i	$-0.0158359\pi$
$240$	0	0
$241$	−166.952	−0.692745	−0.346373	−	0.938097i	$-0.612587\pi$
$241$	−166.952	−0.692745	−0.346373	+	0.938097i	$0.612587\pi$
$242$	0	0
$243$	0	0
$244$	−3.17337	−0.0130056
$245$	38.0369i	0.155253i
$246$	0	0
$247$	159.520	0.645830
$248$	414.878i	1.67289i
$249$	0	0
$250$	−79.9193	−0.319677
$251$	− 247.487i	− 0.986005i	−0.870028	−	0.493003i	$-0.835899\pi$
$251$	− 247.487i	− 0.986005i	0.870028	−	0.493003i	$-0.164101\pi$
$252$	0	0
$253$	0	0
$254$	− 291.505i	− 1.14766i
$255$	0	0
$256$	24.3629	0.0951675
$257$	69.1040i	0.268887i	0.990921	+	0.134444i	$0.0429247\pi$
$257$	69.1040i	0.268887i	−0.990921	+	0.134444i	$0.957075\pi$
$258$	0	0
$259$	−42.1109	−0.162590
$260$	− 0.936714i	− 0.00360275i
$261$	0	0
$262$	141.825	0.541315
$263$	− 162.520i	− 0.617949i	−0.951070	−	0.308974i	$-0.900014\pi$
$263$	− 162.520i	− 0.617949i	0.951070	−	0.308974i	$-0.0999857\pi$
$264$	0	0
$265$	−27.1109	−0.102305
$266$	− 39.4669i	− 0.148372i
$267$	0	0
$268$	2.14315	0.00799683
$269$	− 445.060i	− 1.65450i	−0.561837	−	0.827248i	$-0.689905\pi$
$269$	− 445.060i	− 1.65450i	0.561837	−	0.827248i	$-0.310095\pi$
$270$	0	0
$271$	177.317	0.654304	0.327152	−	0.944972i	$-0.393911\pi$
$271$	177.317	0.654304	0.327152	+	0.944972i	$0.393911\pi$
$272$	193.803i	0.712510i
$273$	0	0
$274$	462.683	1.68863
$275$	0	0
$276$	0	0
$277$	−51.9193	−0.187434	−0.0937172	−	0.995599i	$-0.529875\pi$
$277$	−51.9193	−0.187434	−0.0937172	+	0.995599i	$0.529875\pi$
$278$	208.856i	0.751281i
$279$	0	0
$280$	7.06658	0.0252378
$281$	− 440.088i	− 1.56615i	−0.621927	−	0.783075i	$-0.713650\pi$
$281$	− 440.088i	− 1.56615i	0.621927	−	0.783075i	$-0.286350\pi$
$282$	0	0
$283$	−402.490	−1.42223	−0.711113	−	0.703078i	$-0.751809\pi$
$283$	−402.490	−1.42223	−0.711113	+	0.703078i	$0.751809\pi$
$284$	− 9.78831i	− 0.0344659i
$285$	0	0
$286$	0	0
$287$	84.5363i	0.294552i
$288$	0	0
$289$	150.905	0.522163
$290$	− 47.0810i	− 0.162348i
$291$	0	0
$292$	7.01613	0.0240279
$293$	− 137.303i	− 0.468610i	−0.972163	−	0.234305i	$-0.924719\pi$
$293$	− 137.303i	− 0.468610i	0.972163	−	0.234305i	$-0.0752814\pi$
$294$	0	0
$295$	55.2238	0.187199
$296$	− 293.987i	− 0.993198i
$297$	0	0
$298$	−4.63098	−0.0155402
$299$	− 47.3618i	− 0.158401i
$300$	0	0
$301$	77.5846	0.257756
$302$	− 474.791i	− 1.57215i
$303$	0	0
$304$	284.286	0.935152
$305$	− 19.9102i	− 0.0652792i
$306$	0	0
$307$	−359.413	−1.17073	−0.585364	−	0.810771i	$-0.699048\pi$
$307$	−359.413	−1.17073	−0.585364	+	0.810771i	$0.699048\pi$
$308$	0	0
$309$	0	0
$310$	85.3670	0.275377
$311$	29.2965i	0.0942009i	0.998890	+	0.0471005i	$0.0149981\pi$
$311$	29.2965i	0.0942009i	−0.998890	+	0.0471005i	$0.985002\pi$
$312$	0	0
$313$	274.681	0.877576	0.438788	−	0.898591i	$-0.355408\pi$
$313$	274.681	0.877576	0.438788	+	0.898591i	$0.355408\pi$
$314$	345.675i	1.10088i
$315$	0	0
$316$	8.25403	0.0261204
$317$	− 585.684i	− 1.84758i	−0.382895	−	0.923792i	$-0.625073\pi$
$317$	− 585.684i	− 1.84758i	0.382895	−	0.923792i	$-0.374927\pi$
$318$	0	0
$319$	0	0
$320$	49.2822i	0.154007i
$321$	0	0
$322$	−11.7178	−0.0363907
$323$	202.569i	0.627149i
$324$	0	0
$325$	−225.474	−0.693765
$326$	− 240.393i	− 0.737403i
$327$	0	0
$328$	−590.169	−1.79930
$329$	− 10.5496i	− 0.0320655i
$330$	0	0
$331$	−88.3831	−0.267018	−0.133509	−	0.991048i	$-0.542625\pi$
$331$	−88.3831	−0.267018	−0.133509	+	0.991048i	$0.542625\pi$
$332$	15.2855i	0.0460405i
$333$	0	0
$334$	397.885	1.19127
$335$	13.4464i	0.0401386i
$336$	0	0
$337$	335.429	0.995340	0.497670	−	0.867367i	$-0.334189\pi$
$337$	335.429	0.995340	0.497670	+	0.867367i	$0.334189\pi$
$338$	169.352i	0.501042i
$339$	0	0
$340$	1.18950	0.00349853
$341$	0	0
$342$	0	0
$343$	109.016	0.317831
$344$	541.638i	1.57453i
$345$	0	0
$346$	347.764	1.00510
$347$	440.046i	1.26814i	0.773274	+	0.634072i	$0.218618\pi$
$347$	440.046i	1.26814i	−0.773274	+	0.634072i	$0.781382\pi$
$348$	0	0
$349$	−470.252	−1.34743	−0.673713	−	0.738993i	$-0.735302\pi$
$349$	−470.252	−1.34743	−0.673713	+	0.738993i	$0.735302\pi$
$350$	55.7845i	0.159384i
$351$	0	0
$352$	0	0
$353$	− 459.088i	− 1.30053i	−0.759707	−	0.650266i	$-0.774658\pi$
$353$	− 459.088i	− 1.30053i	0.759707	−	0.650266i	$-0.225342\pi$
$354$	0	0
$355$	61.4133	0.172995
$356$	− 14.0894i	− 0.0395771i
$357$	0	0
$358$	−437.238	−1.22133
$359$	− 403.162i	− 1.12301i	−0.827472	−	0.561507i	$-0.810222\pi$
$359$	− 403.162i	− 1.12301i	0.827472	−	0.561507i	$-0.189778\pi$
$360$	0	0
$361$	−63.8548	−0.176883
$362$	− 482.950i	− 1.33411i
$363$	0	0
$364$	−1.32471	−0.00363932
$365$	44.0202i	0.120603i
$366$	0	0
$367$	634.458	1.72877	0.864384	−	0.502833i	$-0.167709\pi$
$367$	634.458	1.72877	0.864384	+	0.502833i	$0.167709\pi$
$368$	− 84.4052i	− 0.229362i
$369$	0	0
$370$	−60.4919	−0.163492
$371$	38.3406i	0.103344i
$372$	0	0
$373$	−43.3629	−0.116254	−0.0581272	−	0.998309i	$-0.518513\pi$
$373$	−43.3629	−0.116254	−0.0581272	+	0.998309i	$0.518513\pi$
$374$	0	0
$375$	0	0
$376$	73.6492	0.195875
$377$	− 269.118i	− 0.713842i
$378$	0	0
$379$	−230.458	−0.608068	−0.304034	−	0.952661i	$-0.598334\pi$
$379$	−230.458	−0.608068	−0.304034	+	0.952661i	$0.598334\pi$
$380$	− 1.74486i	− 0.00459173i
$381$	0	0
$382$	−686.504	−1.79713
$383$	− 715.367i	− 1.86780i	−0.357536	−	0.933899i	$-0.616383\pi$
$383$	− 715.367i	− 1.86780i	0.357536	−	0.933899i	$-0.383617\pi$
$384$	0	0
$385$	0	0
$386$	− 513.774i	− 1.33102i
$387$	0	0
$388$	1.79027	0.00461409
$389$	− 373.539i	− 0.960255i	−0.877199	−	0.480127i	$-0.840590\pi$
$389$	− 373.539i	− 0.960255i	0.877199	−	0.480127i	$-0.159410\pi$
$390$	0	0
$391$	60.1431	0.153819
$392$	375.538i	0.958005i
$393$	0	0
$394$	191.282	0.485488
$395$	51.7870i	0.131106i
$396$	0	0
$397$	265.429	0.668588	0.334294	−	0.942469i	$-0.391502\pi$
$397$	265.429	0.668588	0.334294	+	0.942469i	$0.391502\pi$
$398$	− 364.799i	− 0.916580i
$399$	0	0
$400$	−401.825	−1.00456
$401$	− 634.698i	− 1.58279i	−0.611306	−	0.791394i	$-0.709356\pi$
$401$	− 634.698i	− 1.58279i	0.611306	−	0.791394i	$-0.290644\pi$
$402$	0	0
$403$	487.964	1.21083
$404$	3.19058i	0.00789748i
$405$	0	0
$406$	−66.5826	−0.163997
$407$	0	0
$408$	0	0
$409$	−483.014	−1.18096	−0.590482	−	0.807051i	$-0.701062\pi$
$409$	−483.014	−1.18096	−0.590482	+	0.807051i	$0.701062\pi$
$410$	121.436i	0.296185i
$411$	0	0
$412$	7.15928	0.0173769
$413$	− 78.0983i	− 0.189100i
$414$	0	0
$415$	−95.9032	−0.231092
$416$	− 18.7996i	− 0.0451914i
$417$	0	0
$418$	0	0
$419$	565.911i	1.35062i	0.737533	+	0.675311i	$0.235991\pi$
$419$	565.911i	1.35062i	−0.737533	+	0.675311i	$0.764009\pi$
$420$	0	0
$421$	632.871	1.50326	0.751628	−	0.659587i	$-0.229269\pi$
$421$	632.871	1.50326	0.751628	+	0.659587i	$0.229269\pi$
$422$	406.658i	0.963644i
$423$	0	0
$424$	−267.665	−0.631286
$425$	− 286.321i	− 0.673697i
$426$	0	0
$427$	−28.1572	−0.0659420
$428$	10.2588i	0.0239690i
$429$	0	0
$430$	111.450	0.259185
$431$	325.762i	0.755829i	0.925840	+	0.377915i	$0.123359\pi$
$431$	325.762i	0.755829i	−0.925840	+	0.377915i	$0.876641\pi$
$432$	0	0
$433$	242.427	0.559879	0.279939	−	0.960018i	$-0.409686\pi$
$433$	242.427	0.559879	0.279939	+	0.960018i	$0.409686\pi$
$434$	− 120.727i	− 0.278173i
$435$	0	0
$436$	13.4254	0.0307921
$437$	− 88.2230i	− 0.201883i
$438$	0	0
$439$	43.2702	0.0985653	0.0492826	−	0.998785i	$-0.484306\pi$
$439$	43.2702	0.0985653	0.0492826	+	0.998785i	$0.484306\pi$
$440$	0	0
$441$	0	0
$442$	220.921	0.499822
$443$	650.758i	1.46898i	0.678620	+	0.734490i	$0.262578\pi$
$443$	650.758i	1.46898i	−0.678620	+	0.734490i	$0.737422\pi$
$444$	0	0
$445$	88.3992	0.198650
$446$	371.733i	0.833482i
$447$	0	0
$448$	69.6955	0.155570
$449$	− 18.7996i	− 0.0418700i	−0.999781	−	0.0209350i	$-0.993336\pi$
$449$	− 18.7996i	− 0.0418700i	0.999781	−	0.0209350i	$-0.00666430\pi$
$450$	0	0
$451$	0	0
$452$	23.1734i	0.0512685i
$453$	0	0
$454$	−653.349	−1.43909
$455$	− 8.31145i	− 0.0182669i
$456$	0	0
$457$	−20.4456	−0.0447387	−0.0223693	−	0.999750i	$-0.507121\pi$
$457$	−20.4456	−0.0447387	−0.0223693	+	0.999750i	$0.507121\pi$
$458$	− 408.333i	− 0.891556i
$459$	0	0
$460$	−0.518052	−0.00112620
$461$	− 231.268i	− 0.501666i	−0.968030	−	0.250833i	$-0.919295\pi$
$461$	− 231.268i	− 0.501666i	0.968030	−	0.250833i	$-0.0807045\pi$
$462$	0	0
$463$	−741.663	−1.60186	−0.800932	−	0.598755i	$-0.795662\pi$
$463$	−741.663	−1.60186	−0.800932	+	0.598755i	$0.795662\pi$
$464$	− 479.605i	− 1.03363i
$465$	0	0
$466$	79.9193	0.171501
$467$	− 551.632i	− 1.18122i	−0.806956	−	0.590612i	$-0.798886\pi$
$467$	− 551.632i	− 1.18122i	0.806956	−	0.590612i	$-0.201114\pi$
$468$	0	0
$469$	19.0161	0.0405461
$470$	− 15.1544i	− 0.0322433i
$471$	0	0
$472$	545.224	1.15514
$473$	0	0
$474$	0	0
$475$	−420.000	−0.884211
$476$	− 1.68221i	− 0.00353405i
$477$	0	0
$478$	−48.2904	−0.101026
$479$	− 659.380i	− 1.37658i	−0.725438	−	0.688288i	$-0.758363\pi$
$479$	− 659.380i	− 1.37658i	0.725438	−	0.688288i	$-0.241637\pi$
$480$	0	0
$481$	−345.776	−0.718869
$482$	339.163i	0.703658i
$483$	0	0
$484$	0	0
$485$	11.2324i	0.0231596i
$486$	0	0
$487$	−10.7480	−0.0220698	−0.0110349	−	0.999939i	$-0.503513\pi$
$487$	−10.7480	−0.0220698	−0.0110349	+	0.999939i	$0.503513\pi$
$488$	− 196.573i	− 0.402813i
$489$	0	0
$490$	77.2722	0.157698
$491$	186.899i	0.380649i	0.981721	+	0.190324i	$0.0609540\pi$
$491$	186.899i	0.380649i	−0.981721	+	0.190324i	$0.939046\pi$
$492$	0	0
$493$	341.744	0.693193
$494$	− 324.066i	− 0.656004i
$495$	0	0
$496$	869.617	1.75326
$497$	− 86.8515i	− 0.174752i
$498$	0	0
$499$	−433.631	−0.869000	−0.434500	−	0.900672i	$-0.643075\pi$
$499$	−433.631	−0.869000	−0.434500	+	0.900672i	$0.643075\pi$
$500$	4.99683i	0.00999366i
$501$	0	0
$502$	−502.772	−1.00154
$503$	− 72.8434i	− 0.144818i	−0.997375	−	0.0724090i	$-0.976931\pi$
$503$	− 72.8434i	− 0.144818i	0.997375	−	0.0724090i	$-0.0230687\pi$
$504$	0	0
$505$	−20.0182	−0.0396400
$506$	0	0
$507$	0	0
$508$	−18.2259	−0.0358777
$509$	− 665.644i	− 1.30775i	−0.756603	−	0.653874i	$-0.773142\pi$
$509$	− 665.644i	− 1.30775i	0.756603	−	0.653874i	$-0.226858\pi$
$510$	0	0
$511$	62.2540	0.121828
$512$	485.530i	0.948301i
$513$	0	0
$514$	140.385	0.273123
$515$	44.9184i	0.0872202i
$516$	0	0
$517$	0	0
$518$	85.5485i	0.165152i
$519$	0	0
$520$	58.0243	0.111585
$521$	30.7763i	0.0590715i	0.999564	+	0.0295358i	$0.00940289\pi$
$521$	30.7763i	0.0590715i	−0.999564	+	0.0295358i	$0.990597\pi$
$522$	0	0
$523$	−77.4274	−0.148045	−0.0740224	−	0.997257i	$-0.523584\pi$
$523$	−77.4274	−0.148045	−0.0740224	+	0.997257i	$0.523584\pi$
$524$	− 8.86735i	− 0.0169224i
$525$	0	0
$526$	−330.161	−0.627683
$527$	619.648i	1.17580i
$528$	0	0
$529$	502.806	0.950485
$530$	55.0759i	0.103917i
$531$	0	0
$532$	−2.46760	−0.00463835
$533$	694.135i	1.30232i
$534$	0	0
$535$	−64.3649	−0.120308
$536$	132.756i	0.247680i
$537$	0	0
$538$	−904.141	−1.68056
$539$	0	0
$540$	0	0
$541$	574.375	1.06169	0.530846	−	0.847469i	$-0.321874\pi$
$541$	574.375	1.06169	0.530846	+	0.847469i	$0.321874\pi$
$542$	− 360.220i	− 0.664612i
$543$	0	0
$544$	23.8730	0.0438842
$545$	84.2326i	0.154555i
$546$	0	0
$547$	1018.85	1.86262	0.931312	−	0.364224i	$-0.118666\pi$
$547$	1018.85	1.86262	0.931312	+	0.364224i	$0.118666\pi$
$548$	− 28.9285i	− 0.0527893i
$549$	0	0
$550$	0	0
$551$	− 501.299i	− 0.909798i
$552$	0	0
$553$	73.2379	0.132437
$554$	105.474i	0.190387i
$555$	0	0
$556$	13.0584	0.0234863
$557$	373.330i	0.670251i	0.942174	+	0.335125i	$0.108779\pi$
$557$	373.330i	0.670251i	−0.942174	+	0.335125i	$0.891221\pi$
$558$	0	0
$559$	637.054	1.13963
$560$	− 14.8121i	− 0.0264502i
$561$	0	0
$562$	−894.042	−1.59082
$563$	1014.70i	1.80232i	0.433491	+	0.901158i	$0.357282\pi$
$563$	1014.70i	1.80232i	−0.433491	+	0.901158i	$0.642718\pi$
$564$	0	0
$565$	−145.393	−0.257333
$566$	817.661i	1.44463i
$567$	0	0
$568$	606.333	1.06749
$569$	580.458i	1.02014i	0.860134	+	0.510068i	$0.170380\pi$
$569$	580.458i	1.02014i	−0.860134	+	0.510068i	$0.829620\pi$
$570$	0	0
$571$	604.454	1.05859	0.529294	−	0.848439i	$-0.322457\pi$
$571$	604.454	1.05859	0.529294	+	0.848439i	$0.322457\pi$
$572$	0	0
$573$	0	0
$574$	171.736	0.299192
$575$	124.699i	0.216868i
$576$	0	0
$577$	−8.90320	−0.0154302	−0.00771508	−	0.999970i	$-0.502456\pi$
$577$	−8.90320	−0.0154302	−0.00771508	+	0.999970i	$0.502456\pi$
$578$	− 306.565i	− 0.530389i
$579$	0	0
$580$	−2.94366	−0.00507528
$581$	135.628i	0.233438i
$582$	0	0
$583$	0	0
$584$	434.611i	0.744197i
$585$	0	0
$586$	−278.931	−0.475992
$587$	− 680.972i	− 1.16009i	−0.814585	−	0.580045i	$-0.803035\pi$
$587$	− 680.972i	− 1.16009i	0.814585	−	0.580045i	$-0.196965\pi$
$588$	0	0
$589$	908.952	1.54321
$590$	− 112.188i	− 0.190148i
$591$	0	0
$592$	−616.220	−1.04091
$593$	855.518i	1.44269i	0.692574	+	0.721347i	$0.256477\pi$
$593$	855.518i	1.44269i	−0.692574	+	0.721347i	$0.743523\pi$
$594$	0	0
$595$	10.5544	0.0177385
$596$	0.289545i	0 0.000485814i
$597$	0	0
$598$	−96.2159	−0.160896
$599$	186.251i	0.310937i	0.987841	+	0.155469i	$0.0496887\pi$
$599$	186.251i	0.310937i	−0.987841	+	0.155469i	$0.950311\pi$
$600$	0	0
$601$	−362.843	−0.603732	−0.301866	−	0.953350i	$-0.597609\pi$
$601$	−362.843	−0.603732	−0.301866	+	0.953350i	$0.597609\pi$
$602$	− 157.614i	− 0.261817i
$603$	0	0
$604$	−29.6855	−0.0491482
$605$	0	0
$606$	0	0
$607$	−16.0000	−0.0263591	−0.0131796	−	0.999913i	$-0.504195\pi$
$607$	−16.0000	−0.0263591	−0.0131796	+	0.999913i	$0.504195\pi$
$608$	− 35.0189i	− 0.0575969i
$609$	0	0
$610$	−40.4476	−0.0663076
$611$	− 86.6234i	− 0.141773i
$612$	0	0
$613$	1161.74	1.89517	0.947585	−	0.319502i	$-0.103516\pi$
$613$	1161.74	1.89517	0.947585	+	0.319502i	$0.103516\pi$
$614$	730.150i	1.18917i
$615$	0	0
$616$	0	0
$617$	− 163.905i	− 0.265648i	−0.991140	−	0.132824i	$-0.957595\pi$
$617$	− 163.905i	− 0.265648i	0.991140	−	0.132824i	$-0.0424045\pi$
$618$	0	0
$619$	−305.982	−0.494316	−0.247158	−	0.968975i	$-0.579497\pi$
$619$	−305.982	−0.494316	−0.247158	+	0.968975i	$0.579497\pi$
$620$	− 5.33743i	− 0.00860876i
$621$	0	0
$622$	59.5160	0.0956849
$623$	− 125.015i	− 0.200667i
$624$	0	0
$625$	577.772	0.924435
$626$	− 558.017i	− 0.891401i
$627$	0	0
$628$	21.6128	0.0344153
$629$	− 439.089i	− 0.698075i
$630$	0	0
$631$	123.764	0.196140	0.0980698	−	0.995180i	$-0.468733\pi$
$631$	123.764	0.196140	0.0980698	+	0.995180i	$0.468733\pi$
$632$	511.292i	0.809007i
$633$	0	0
$634$	−1189.82	−1.87669
$635$	− 114.352i	− 0.180081i
$636$	0	0
$637$	441.693	0.693396
$638$	0	0
$639$	0	0
$640$	106.593	0.166551
$641$	− 261.417i	− 0.407827i	−0.978989	−	0.203913i	$-0.934634\pi$
$641$	− 261.417i	− 0.407827i	0.978989	−	0.203913i	$-0.0653661\pi$
$642$	0	0
$643$	218.000	0.339036	0.169518	−	0.985527i	$-0.445779\pi$
$643$	218.000	0.339036	0.169518	+	0.985527i	$0.445779\pi$
$644$	0.732637i	0.00113763i
$645$	0	0
$646$	411.520	0.637028
$647$	858.262i	1.32653i	0.748387	+	0.663263i	$0.230829\pi$
$647$	858.262i	1.32653i	−0.748387	+	0.663263i	$0.769171\pi$
$648$	0	0
$649$	0	0
$650$	458.051i	0.704694i
$651$	0	0
$652$	−15.0302	−0.0230525
$653$	968.691i	1.48345i	0.670706	+	0.741723i	$0.265991\pi$
$653$	968.691i	1.48345i	−0.670706	+	0.741723i	$0.734009\pi$
$654$	0	0
$655$	55.6351	0.0849391
$656$	1237.04i	1.88573i
$657$	0	0
$658$	−21.4315	−0.0325707
$659$	690.156i	1.04728i	0.851940	+	0.523639i	$0.175426\pi$
$659$	690.156i	1.04728i	−0.851940	+	0.523639i	$0.824574\pi$
$660$	0	0
$661$	−1183.04	−1.78977	−0.894885	−	0.446297i	$-0.852743\pi$
$661$	−1183.04	−1.78977	−0.894885	+	0.446297i	$0.852743\pi$
$662$	179.551i	0.271225i
$663$	0	0
$664$	−946.851	−1.42598
$665$	− 15.4821i	− 0.0232814i
$666$	0	0
$667$	−148.837	−0.223143
$668$	− 24.8771i	− 0.0372412i
$669$	0	0
$670$	27.3165	0.0407709
$671$	0	0
$672$	0	0
$673$	1006.88	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$673$	1006.88	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$674$	− 681.427i	− 1.01102i
$675$	0	0
$676$	10.5885	0.0156634
$677$	− 907.102i	− 1.33989i	−0.742413	−	0.669943i	$-0.766319\pi$
$677$	− 907.102i	− 1.33989i	0.742413	−	0.669943i	$-0.233681\pi$
$678$	0	0
$679$	15.8850	0.0233947
$680$	73.6831i	0.108357i
$681$	0	0
$682$	0	0
$683$	687.644i	1.00680i	0.864054	+	0.503400i	$0.167918\pi$
$683$	687.644i	1.00680i	−0.864054	+	0.503400i	$0.832082\pi$
$684$	0	0
$685$	181.502	0.264966
$686$	− 221.467i	− 0.322838i
$687$	0	0
$688$	1135.32	1.65017
$689$	314.818i	0.456920i
$690$	0	0
$691$	199.099	0.288131	0.144066	−	0.989568i	$-0.453982\pi$
$691$	199.099	0.288131	0.144066	+	0.989568i	$0.453982\pi$
$692$	− 21.7434i	− 0.0314211i
$693$	0	0
$694$	893.955	1.28812
$695$	81.9302i	0.117885i
$696$	0	0
$697$	−881.458	−1.26465
$698$	955.320i	1.36865i
$699$	0	0
$700$	3.48784	0.00498262
$701$	81.6124i	0.116423i	0.998304	+	0.0582114i	$0.0185398\pi$
$701$	81.6124i	0.116423i	−0.998304	+	0.0582114i	$0.981460\pi$
$702$	0	0
$703$	−644.093	−0.916206
$704$	0	0
$705$	0	0
$706$	−932.639	−1.32102
$707$	28.3100i	0.0400424i
$708$	0	0
$709$	−1014.66	−1.43112	−0.715558	−	0.698553i	$-0.753827\pi$
$709$	−1014.66	−1.43112	−0.715558	+	0.698553i	$0.753827\pi$
$710$	− 124.762i	− 0.175720i
$711$	0	0
$712$	872.764	1.22579
$713$	− 269.870i	− 0.378499i
$714$	0	0
$715$	0	0
$716$	27.3376i	0.0381810i
$717$	0	0
$718$	−819.026	−1.14070
$719$	− 613.361i	− 0.853075i	−0.904470	−	0.426537i	$-0.859733\pi$
$719$	− 613.361i	− 0.853075i	0.904470	−	0.426537i	$-0.140267\pi$
$720$	0	0
$721$	63.5242	0.0881057
$722$	129.721i	0.179670i
$723$	0	0
$724$	−30.1956	−0.0417067
$725$	708.561i	0.977325i
$726$	0	0
$727$	169.163	0.232687	0.116343	−	0.993209i	$-0.462883\pi$
$727$	169.163	0.232687	0.116343	+	0.993209i	$0.462883\pi$
$728$	− 82.0588i	− 0.112718i
$729$	0	0
$730$	89.4274	0.122503
$731$	808.973i	1.10667i
$732$	0	0
$733$	−165.734	−0.226104	−0.113052	−	0.993589i	$-0.536063\pi$
$733$	−165.734	−0.226104	−0.113052	+	0.993589i	$0.536063\pi$
$734$	− 1288.90i	− 1.75600i
$735$	0	0
$736$	−10.3972	−0.0141266
$737$	0	0
$738$	0	0
$739$	−1001.21	−1.35481	−0.677406	−	0.735610i	$-0.736896\pi$
$739$	−1001.21	−1.35481	−0.677406	+	0.735610i	$0.736896\pi$
$740$	3.78216i	0.00511103i
$741$	0	0
$742$	77.8891	0.104972
$743$	110.639i	0.148909i	0.997224	+	0.0744544i	$0.0237215\pi$
$743$	110.639i	0.148909i	−0.997224	+	0.0744544i	$0.976278\pi$
$744$	0	0
$745$	−1.81665	−0.00243845
$746$	88.0919i	0.118086i
$747$	0	0
$748$	0	0
$749$	91.0257i	0.121530i
$750$	0	0
$751$	−80.0827	−0.106635	−0.0533174	−	0.998578i	$-0.516980\pi$
$751$	−80.0827	−0.106635	−0.0533174	+	0.998578i	$0.516980\pi$
$752$	− 154.375i	− 0.205285i
$753$	0	0
$754$	−546.715	−0.725087
$755$	− 186.251i	− 0.246691i
$756$	0	0
$757$	254.464	0.336148	0.168074	−	0.985774i	$-0.446245\pi$
$757$	254.464	0.336148	0.168074	+	0.985774i	$0.446245\pi$
$758$	468.176i	0.617646i
$759$	0	0
$760$	108.085	0.142216
$761$	280.475i	0.368561i	0.982874	+	0.184280i	$0.0589954\pi$
$761$	280.475i	0.368561i	−0.982874	+	0.184280i	$0.941005\pi$
$762$	0	0
$763$	119.123	0.156124
$764$	42.9226i	0.0561814i
$765$	0	0
$766$	−1453.27	−1.89722
$767$	− 641.272i	− 0.836078i
$768$	0	0
$769$	788.423	1.02526	0.512629	−	0.858610i	$-0.328672\pi$
$769$	788.423	1.02526	0.512629	+	0.858610i	$0.328672\pi$
$770$	0	0
$771$	0	0
$772$	−32.1229	−0.0416100
$773$	− 420.198i	− 0.543594i	−0.962355	−	0.271797i	$-0.912382\pi$
$773$	− 420.198i	− 0.543594i	0.962355	−	0.271797i	$-0.0876179\pi$
$774$	0	0
$775$	−1284.76	−1.65775
$776$	110.897i	0.142909i
$777$	0	0
$778$	−758.847	−0.975382
$779$	1293.00i	1.65982i
$780$	0	0
$781$	0	0
$782$	− 122.181i	− 0.156242i
$783$	0	0
$784$	787.157	1.00403
$785$	135.602i	0.172741i
$786$	0	0
$787$	246.730	0.313507	0.156753	−	0.987638i	$-0.449897\pi$
$787$	246.730	0.313507	0.156753	+	0.987638i	$0.449897\pi$
$788$	− 11.9596i	− 0.0151772i
$789$	0	0
$790$	105.206	0.133172
$791$	205.617i	0.259945i
$792$	0	0
$793$	−231.202	−0.291553
$794$	− 539.222i	− 0.679120i
$795$	0	0
$796$	−22.8085	−0.0286538
$797$	166.005i	0.208287i	0.994562	+	0.104144i	$0.0332101\pi$
$797$	166.005i	0.208287i	−0.994562	+	0.104144i	$0.966790\pi$
$798$	0	0
$799$	110.000	0.137672
$800$	49.4975i	0.0618718i
$801$	0	0
$802$	−1289.39	−1.60772
$803$	0	0
$804$	0	0
$805$	−4.59667	−0.00571015
$806$	− 991.301i	− 1.22990i
$807$	0	0
$808$	−197.639	−0.244603
$809$	− 447.222i	− 0.552809i	−0.961041	−	0.276404i	$-0.910857\pi$
$809$	− 447.222i	− 0.552809i	0.961041	−	0.276404i	$-0.0891429\pi$
$810$	0	0
$811$	885.903	1.09236	0.546180	−	0.837668i	$-0.316082\pi$
$811$	885.903	1.09236	0.546180	+	0.837668i	$0.316082\pi$
$812$	4.16297i	0.00512681i
$813$	0	0
$814$	0	0
$815$	− 94.3018i	− 0.115708i
$816$	0	0
$817$	1186.67	1.45247
$818$	981.246i	1.19957i
$819$	0	0
$820$	7.59257	0.00925923
$821$	1575.63i	1.91916i	0.281430	+	0.959582i	$0.409191\pi$
$821$	1575.63i	1.91916i	−0.281430	+	0.959582i	$0.590809\pi$
$822$	0	0
$823$	−1433.92	−1.74231	−0.871153	−	0.491012i	$-0.836627\pi$
$823$	−1433.92	−1.74231	−0.871153	+	0.491012i	$0.836627\pi$
$824$	443.479i	0.538202i
$825$	0	0
$826$	−158.657	−0.192079
$827$	9.09551i	0.0109982i	0.999985	+	0.00549910i	$0.00175043\pi$
$827$	9.09551i	0.0109982i	−0.999985	+	0.00549910i	$0.998250\pi$
$828$	0	0
$829$	206.996	0.249693	0.124847	−	0.992176i	$-0.460156\pi$
$829$	206.996	0.249693	0.124847	+	0.992176i	$0.460156\pi$
$830$	194.828i	0.234732i
$831$	0	0
$832$	572.276	0.687832
$833$	560.891i	0.673339i
$834$	0	0
$835$	156.083	0.186925
$836$	0	0
$837$	0	0
$838$	1149.65	1.37190
$839$	924.237i	1.10159i	0.834639	+	0.550797i	$0.185676\pi$
$839$	924.237i	1.10159i	−0.834639	+	0.550797i	$0.814324\pi$
$840$	0	0
$841$	−4.71575	−0.00560731
$842$	− 1285.68i	− 1.52694i
$843$	0	0
$844$	25.4256	0.0301251
$845$	66.4336i	0.0786197i
$846$	0	0
$847$	0	0
$848$	561.048i	0.661613i
$849$	0	0
$850$	−581.663	−0.684310
$851$	191.232i	0.224715i
$852$	0	0
$853$	215.111	0.252182	0.126091	−	0.992019i	$-0.459757\pi$
$853$	215.111	0.252182	0.126091	+	0.992019i	$0.459757\pi$
$854$	57.2016i	0.0669808i
$855$	0	0
$856$	−635.474	−0.742376
$857$	− 775.528i	− 0.904933i	−0.891781	−	0.452467i	$-0.850544\pi$
$857$	− 775.528i	− 0.904933i	0.891781	−	0.452467i	$-0.149456\pi$
$858$	0	0
$859$	−1070.94	−1.24673	−0.623363	−	0.781933i	$-0.714234\pi$
$859$	−1070.94	−1.24673	−0.623363	+	0.781933i	$0.714234\pi$
$860$	− 6.96821i	− 0.00810257i
$861$	0	0
$862$	661.788	0.767736
$863$	698.664i	0.809576i	0.914411	+	0.404788i	$0.132655\pi$
$863$	698.664i	0.809576i	−0.914411	+	0.404788i	$0.867345\pi$
$864$	0	0
$865$	136.421	0.157712
$866$	− 492.493i	− 0.568698i
$867$	0	0
$868$	−7.54827	−0.00869616
$869$	0	0
$870$	0	0
$871$	156.143	0.179269
$872$	831.627i	0.953701i
$873$	0	0
$874$	−179.226	−0.205064
$875$	44.3368i	0.0506706i
$876$	0	0
$877$	−1205.04	−1.37405	−0.687023	−	0.726636i	$-0.741083\pi$
$877$	−1205.04	−1.37405	−0.687023	+	0.726636i	$0.741083\pi$
$878$	− 87.9036i	− 0.100118i
$879$	0	0
$880$	0	0
$881$	37.9228i	0.0430452i	0.999768	+	0.0215226i	$0.00685139\pi$
$881$	37.9228i	0.0430452i	−0.999768	+	0.0215226i	$0.993149\pi$
$882$	0	0
$883$	444.786	0.503722	0.251861	−	0.967763i	$-0.418957\pi$
$883$	444.786	0.503722	0.251861	+	0.967763i	$0.418957\pi$
$884$	− 13.8128i	− 0.0156253i
$885$	0	0
$886$	1322.02	1.49212
$887$	− 238.917i	− 0.269354i	−0.990890	−	0.134677i	$-0.957000\pi$
$887$	− 238.917i	− 0.269354i	0.990890	−	0.134677i	$-0.0429996\pi$
$888$	0	0
$889$	−161.718	−0.181910
$890$	− 179.584i	− 0.201779i
$891$	0	0
$892$	23.2420	0.0260561
$893$	− 161.357i	− 0.180691i
$894$	0	0
$895$	−171.520	−0.191643
$896$	− 150.745i	− 0.168242i
$897$	0	0
$898$	−38.1915	−0.0425296
$899$	− 1533.45i	− 1.70572i
$900$	0	0
$901$	−399.776	−0.443703
$902$	0	0
$903$	0	0
$904$	−1435.46	−1.58790
$905$	− 189.452i	− 0.209339i
$906$	0	0
$907$	83.7360	0.0923219	0.0461610	−	0.998934i	$-0.485301\pi$
$907$	83.7360	0.0923219	0.0461610	+	0.998934i	$0.485301\pi$
$908$	40.8496i	0.0449885i
$909$	0	0
$910$	−16.8848	−0.0185547
$911$	− 682.361i	− 0.749024i	−0.927222	−	0.374512i	$-0.877810\pi$
$911$	− 682.361i	− 0.749024i	0.927222	−	0.374512i	$-0.122190\pi$
$912$	0	0
$913$	0	0
$914$	41.5353i	0.0454435i
$915$	0	0
$916$	−25.5303	−0.0278716
$917$	− 78.6799i	− 0.0858014i
$918$	0	0
$919$	−1516.74	−1.65043	−0.825214	−	0.564820i	$-0.808946\pi$
$919$	−1516.74	−1.65043	−0.825214	+	0.564820i	$0.808946\pi$
$920$	− 32.0905i	− 0.0348810i
$921$	0	0
$922$	−469.823	−0.509569
$923$	− 713.146i	− 0.772639i
$924$	0	0
$925$	910.393	0.984209
$926$	1506.69i	1.62710i
$927$	0	0
$928$	−59.0786	−0.0636623
$929$	− 901.919i	− 0.970849i	−0.874279	−	0.485425i	$-0.838665\pi$
$929$	− 901.919i	− 0.970849i	0.874279	−	0.485425i	$-0.161335\pi$
$930$	0	0
$931$	822.762	0.883740
$932$	− 4.99683i	− 0.00536140i
$933$	0	0
$934$	−1120.64	−1.19983
$935$	0	0
$936$	0	0
$937$	389.472	0.415658	0.207829	−	0.978165i	$-0.433360\pi$
$937$	389.472	0.415658	0.207829	+	0.978165i	$0.433360\pi$
$938$	− 38.6314i	− 0.0411848i
$939$	0	0
$940$	−0.947502	−0.00100798
$941$	− 372.011i	− 0.395336i	−0.980269	−	0.197668i	$-0.936663\pi$
$941$	− 372.011i	− 0.395336i	0.980269	−	0.197668i	$-0.0633367\pi$
$942$	0	0
$943$	383.893	0.407098
$944$	− 1142.83i	− 1.21063i
$945$	0	0
$946$	0	0
$947$	− 301.071i	− 0.317920i	−0.987285	−	0.158960i	$-0.949186\pi$
$947$	− 301.071i	− 0.317920i	0.987285	−	0.158960i	$-0.0508142\pi$
$948$	0	0
$949$	511.173	0.538644
$950$	853.233i	0.898139i
$951$	0	0
$952$	104.204	0.109458
$953$	1174.88i	1.23283i	0.787422	+	0.616414i	$0.211415\pi$
$953$	1174.88i	1.23283i	−0.787422	+	0.616414i	$0.788585\pi$
$954$	0	0
$955$	−269.302	−0.281992
$956$	3.01928i	0.00315824i
$957$	0	0
$958$	−1339.53	−1.39826
$959$	− 256.682i	− 0.267656i
$960$	0	0
$961$	1819.44	1.89327
$962$	702.446i	0.730194i
$963$	0	0
$964$	21.2056	0.0219975
$965$	− 201.544i	− 0.208854i
$966$	0	0
$967$	293.663	0.303685	0.151842	−	0.988405i	$-0.451479\pi$
$967$	293.663	0.303685	0.151842	+	0.988405i	$0.451479\pi$
$968$	0	0
$969$	0	0
$970$	22.8187	0.0235244
$971$	787.038i	0.810544i	0.914196	+	0.405272i	$0.132823\pi$
$971$	787.038i	0.810544i	−0.914196	+	0.405272i	$0.867177\pi$
$972$	0	0
$973$	115.867	0.119082
$974$	21.8347i	0.0224175i
$975$	0	0
$976$	−412.032	−0.422164
$977$	− 692.629i	− 0.708935i	−0.935068	−	0.354467i	$-0.884662\pi$
$977$	− 692.629i	− 0.708935i	0.935068	−	0.354467i	$-0.115338\pi$
$978$	0	0
$979$	0	0
$980$	− 4.83132i	− 0.00492992i
$981$	0	0
$982$	379.686	0.386645
$983$	1798.69i	1.82979i	0.403690	+	0.914896i	$0.367728\pi$
$983$	1798.69i	1.82979i	−0.403690	+	0.914896i	$0.632272\pi$
$984$	0	0
$985$	75.0364	0.0761791
$986$	− 694.255i	− 0.704112i
$987$	0	0
$988$	−20.2617	−0.0205078
$989$	− 352.325i	− 0.356243i
$990$	0	0
$991$	736.790	0.743482	0.371741	−	0.928337i	$-0.378761\pi$
$991$	736.790	0.743482	0.371741	+	0.928337i	$0.378761\pi$
$992$	− 107.121i	− 0.107985i
$993$	0	0
$994$	−176.439	−0.177504
$995$	− 143.104i	− 0.143823i
$996$	0	0
$997$	−1277.49	−1.28133	−0.640667	−	0.767819i	$-0.721342\pi$
$997$	−1277.49	−1.28133	−0.640667	+	0.767819i	$0.721342\pi$
$998$	880.924i	0.882689i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.c.485.2	✓	4
3.2	odd	2	inner	1089.3.b.c.485.3	yes	4
11.10	odd	2		1089.3.b.d.485.3	yes	4
33.32	even	2		1089.3.b.d.485.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.b.c.485.2	✓	4	1.1	even	1	trivial
1089.3.b.c.485.3	yes	4	3.2	odd	2	inner
1089.3.b.d.485.2	yes	4	33.32	even	2
1089.3.b.d.485.3	yes	4	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)

\(f(q)\)	\(=\)	\(q-2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} -1.12702 q^{7} -7.86799i q^{8} +O(q^{10})\)	\(q-2.03151i q^{2} -0.127017 q^{4} -0.796921i q^{5} -1.12702 q^{7} -7.86799i q^{8} -1.61895 q^{10} -9.25403 q^{13} +2.28954i q^{14} -16.4919 q^{16} -11.7514i q^{17} -17.2379 q^{19} +0.101222i q^{20} +5.11797i q^{23} +24.3649 q^{25} +18.7996i q^{26} +0.143150 q^{28} +29.0812i q^{29} -52.7298 q^{31} +2.03151i q^{32} -23.8730 q^{34} +0.898143i q^{35} +37.3649 q^{37} +35.0189i q^{38} -6.27017 q^{40} -75.0089i q^{41} -68.8407 q^{43} +10.3972 q^{46} +9.36061i q^{47} -47.7298 q^{49} -49.4975i q^{50} +1.17542 q^{52} -34.0195i q^{53} +8.86735i q^{56} +59.0786 q^{58} +69.2965i q^{59} +24.9839 q^{61} +107.121i q^{62} -61.8407 q^{64} +7.37473i q^{65} -16.8730 q^{67} +1.49262i q^{68} +1.82458 q^{70} +77.0632i q^{71} -55.2379 q^{73} -75.9071i q^{74} +2.18950 q^{76} -64.9839 q^{79} +13.1428i q^{80} -152.381 q^{82} -120.342i q^{83} -9.36492 q^{85} +139.850i q^{86} +110.926i q^{89} +10.4294 q^{91} -0.650067i q^{92} +19.0161 q^{94} +13.7372i q^{95} -14.0948 q^{97} +96.9634i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 20 q^{7}+O(q^{10}) \) 4 * q - 16 * q^4 - 20 * q^7	\( 4 q - 16 q^{4} - 20 q^{7} + 40 q^{10} - 68 q^{13} - 4 q^{16} + 24 q^{19} + 20 q^{25} + 140 q^{28} - 56 q^{31} - 80 q^{34} + 72 q^{37} - 180 q^{40} - 12 q^{43} + 212 q^{46} - 36 q^{49} + 392 q^{52} - 120 q^{58} - 24 q^{61} + 16 q^{64} - 52 q^{67} - 380 q^{70} - 128 q^{73} - 456 q^{76} - 136 q^{79} - 656 q^{82} + 40 q^{85} + 460 q^{91} + 200 q^{94} + 176 q^{97}+O(q^{100}) \) 4 * q - 16 * q^4 - 20 * q^7 + 40 * q^10 - 68 * q^13 - 4 * q^16 + 24 * q^19 + 20 * q^25 + 140 * q^28 - 56 * q^31 - 80 * q^34 + 72 * q^37 - 180 * q^40 - 12 * q^43 + 212 * q^46 - 36 * q^49 + 392 * q^52 - 120 * q^58 - 24 * q^61 + 16 * q^64 - 52 * q^67 - 380 * q^70 - 128 * q^73 - 456 * q^76 - 136 * q^79 - 656 * q^82 + 40 * q^85 + 460 * q^91 + 200 * q^94 + 176 * q^97

Introduction

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