LMFDB - Embedded newform 1089.3.c.b.604.2

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.2
Root			$0.500000 + 3.07253i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.b.604.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} -9.89949i q^{7} +9.94987i q^{8} +O(q^{10})$	$q-3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} -9.89949i q^{7} +9.94987i q^{8} -31.1127i q^{10} -12.7279i q^{13} -32.8329 q^{14} +5.00000 q^{16} +4.24264i q^{19} -65.6658 q^{20} +28.1425 q^{23} +63.0000 q^{25} -42.2137 q^{26} +69.2965i q^{28} -13.2665i q^{29} -44.0000 q^{31} +23.2164i q^{32} -92.8655i q^{35} +44.0000 q^{37} +14.0712 q^{38} +93.3381i q^{40} -39.7995i q^{41} +29.6985i q^{43} -93.3381i q^{46} +9.38083 q^{47} -49.0000 q^{49} -208.947i q^{50} +89.0955i q^{52} -65.6658 q^{53} +98.4987 q^{56} -44.0000 q^{58} +18.7617 q^{59} -69.2965i q^{61} +145.931i q^{62} +97.0000 q^{64} -119.398i q^{65} -88.0000 q^{67} -308.000 q^{70} -46.9042 q^{71} +41.0122i q^{73} -145.931i q^{74} -29.6985i q^{76} +12.7279i q^{79} +46.9042 q^{80} -132.000 q^{82} +92.8655i q^{83} +98.4987 q^{86} +112.570 q^{89} -126.000 q^{91} -196.997 q^{92} -31.1127i q^{94} +39.7995i q^{95} +70.0000 q^{97} +162.515i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 28 q^{4}+O(q^{10}) $ 4 * q - 28 * q^4	$ 4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97}+O(q^{100}) $ 4 * q - 28 * q^4 + 20 * q^16 + 252 * q^25 - 176 * q^31 + 176 * q^37 - 196 * q^49 - 176 * q^58 + 388 * q^64 - 352 * q^67 - 1232 * q^70 - 528 * q^82 - 504 * q^91 + 280 * 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 $

$2$	− 3.31662i	− 1.65831i	−0.559017	−	0.829156i	$-0.688821\pi$
$2$	− 3.31662i	− 1.65831i	0.559017	−	0.829156i	$-0.311179\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−1.75000
$5$	9.38083	1.87617	0.938083	−	0.346410i	$-0.112599\pi$
$5$	9.38083	1.87617	0.938083	+	0.346410i	$0.112599\pi$
$6$	0	0
$7$	− 9.89949i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$7$	− 9.89949i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$8$	9.94987i	1.24373i
$9$	0	0
$10$	− 31.1127i	− 3.11127i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 12.7279i	− 0.979071i	−0.871983	−	0.489535i	$-0.837166\pi$
$13$	− 12.7279i	− 0.979071i	0.871983	−	0.489535i	$-0.162834\pi$
$14$	−32.8329	−2.34521
$15$	0	0
$16$	5.00000	0.312500
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	4.24264i	0.223297i	0.993748	+	0.111648i	$0.0356131\pi$
$19$	4.24264i	0.223297i	−0.993748	+	0.111648i	$0.964387\pi$
$20$	−65.6658	−3.28329
$21$	0	0
$22$	0	0
$23$	28.1425	1.22359	0.611793	−	0.791018i	$-0.290448\pi$
$23$	28.1425	1.22359	0.611793	+	0.791018i	$0.290448\pi$
$24$	0	0
$25$	63.0000	2.52000
$26$	−42.2137	−1.62361
$27$	0	0
$28$	69.2965i	2.47487i
$29$	− 13.2665i	− 0.457465i	−0.973489	−	0.228733i	$-0.926542\pi$
$29$	− 13.2665i	− 0.457465i	0.973489	−	0.228733i	$-0.0734582\pi$
$30$	0	0
$31$	−44.0000	−1.41935	−0.709677	−	0.704527i	$-0.751159\pi$
$31$	−44.0000	−1.41935	−0.709677	+	0.704527i	$0.751159\pi$
$32$	23.2164i	0.725512i
$33$	0	0
$34$	0	0
$35$	− 92.8655i	− 2.65330i
$36$	0	0
$37$	44.0000	1.18919	0.594595	−	0.804026i	$-0.297313\pi$
$37$	44.0000	1.18919	0.594595	+	0.804026i	$0.297313\pi$
$38$	14.0712	0.370296
$39$	0	0
$40$	93.3381i	2.33345i
$41$	− 39.7995i	− 0.970719i	−0.874315	−	0.485360i	$-0.838689\pi$
$41$	− 39.7995i	− 0.970719i	0.874315	−	0.485360i	$-0.161311\pi$
$42$	0	0
$43$	29.6985i	0.690662i	0.938481	+	0.345331i	$0.112233\pi$
$43$	29.6985i	0.690662i	−0.938481	+	0.345331i	$0.887767\pi$
$44$	0	0
$45$	0	0
$46$	− 93.3381i	− 2.02909i
$47$	9.38083	0.199592	0.0997961	−	0.995008i	$-0.468181\pi$
$47$	9.38083	0.199592	0.0997961	+	0.995008i	$0.468181\pi$
$48$	0	0
$49$	−49.0000	−1.00000
$50$	− 208.947i	− 4.17895i
$51$	0	0
$52$	89.0955i	1.71337i
$53$	−65.6658	−1.23898	−0.619489	−	0.785005i	$-0.712660\pi$
$53$	−65.6658	−1.23898	−0.619489	+	0.785005i	$0.712660\pi$
$54$	0	0
$55$	0	0
$56$	98.4987	1.75891
$57$	0	0
$58$	−44.0000	−0.758621
$59$	18.7617	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$59$	18.7617	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$60$	0	0
$61$	− 69.2965i	− 1.13601i	−0.823026	−	0.568004i	$-0.807716\pi$
$61$	− 69.2965i	− 1.13601i	0.823026	−	0.568004i	$-0.192284\pi$
$62$	145.931i	2.35373i
$63$	0	0
$64$	97.0000	1.51562
$65$	− 119.398i	− 1.83690i
$66$	0	0
$67$	−88.0000	−1.31343	−0.656716	−	0.754138i	$-0.728055\pi$
$67$	−88.0000	−1.31343	−0.656716	+	0.754138i	$0.728055\pi$
$68$	0	0
$69$	0	0
$70$	−308.000	−4.40000
$71$	−46.9042	−0.660622	−0.330311	−	0.943872i	$-0.607154\pi$
$71$	−46.9042	−0.660622	−0.330311	+	0.943872i	$0.607154\pi$
$72$	0	0
$73$	41.0122i	0.561811i	0.959735	+	0.280905i	$0.0906347\pi$
$73$	41.0122i	0.561811i	−0.959735	+	0.280905i	$0.909365\pi$
$74$	− 145.931i	− 1.97205i
$75$	0	0
$76$	− 29.6985i	− 0.390770i
$77$	0	0
$78$	0	0
$79$	12.7279i	0.161113i	0.996750	+	0.0805565i	$0.0256697\pi$
$79$	12.7279i	0.161113i	−0.996750	+	0.0805565i	$0.974330\pi$
$80$	46.9042	0.586302
$81$	0	0
$82$	−132.000	−1.60976
$83$	92.8655i	1.11886i	0.828877	+	0.559431i	$0.188980\pi$
$83$	92.8655i	1.11886i	−0.828877	+	0.559431i	$0.811020\pi$
$84$	0	0
$85$	0	0
$86$	98.4987	1.14533
$87$	0	0
$88$	0	0
$89$	112.570	1.26483	0.632416	−	0.774629i	$-0.282064\pi$
$89$	112.570	1.26483	0.632416	+	0.774629i	$0.282064\pi$
$90$	0	0
$91$	−126.000	−1.38462
$92$	−196.997	−2.14128
$93$	0	0
$94$	− 31.1127i	− 0.330986i
$95$	39.7995i	0.418942i
$96$	0	0
$97$	70.0000	0.721649	0.360825	−	0.932634i	$-0.382495\pi$
$97$	70.0000	0.721649	0.360825	+	0.932634i	$0.382495\pi$
$98$	162.515i	1.65831i
$99$	0	0
$100$	−441.000	−4.41000
$101$	− 26.5330i	− 0.262703i	−0.991336	−	0.131351i	$-0.958068\pi$
$101$	− 26.5330i	− 0.262703i	0.991336	−	0.131351i	$-0.0419316\pi$
$102$	0	0
$103$	14.0000	0.135922	0.0679612	−	0.997688i	$-0.478351\pi$
$103$	14.0000	0.135922	0.0679612	+	0.997688i	$0.478351\pi$
$104$	126.641	1.21770
$105$	0	0
$106$	217.789i	2.05461i
$107$	− 26.5330i	− 0.247972i	−0.992284	−	0.123986i	$-0.960432\pi$
$107$	− 26.5330i	− 0.247972i	0.992284	−	0.123986i	$-0.0395678\pi$
$108$	0	0
$109$	63.6396i	0.583850i	0.956441	+	0.291925i	$0.0942957\pi$
$109$	63.6396i	0.583850i	−0.956441	+	0.291925i	$0.905704\pi$
$110$	0	0
$111$	0	0
$112$	− 49.4975i	− 0.441942i
$113$	−131.332	−1.16223	−0.581113	−	0.813823i	$-0.697383\pi$
$113$	−131.332	−1.16223	−0.581113	+	0.813823i	$0.697383\pi$
$114$	0	0
$115$	264.000	2.29565
$116$	92.8655i	0.800565i
$117$	0	0
$118$	− 62.2254i	− 0.527334i
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−229.830	−1.88386
$123$	0	0
$124$	308.000	2.48387
$125$	356.472	2.85177
$126$	0	0
$127$	117.380i	0.924250i	0.886815	+	0.462125i	$0.152913\pi$
$127$	117.380i	0.924250i	−0.886815	+	0.462125i	$0.847087\pi$
$128$	− 228.847i	− 1.78787i
$129$	0	0
$130$	−396.000	−3.04615
$131$	− 198.997i	− 1.51906i	−0.650469	−	0.759532i	$-0.725428\pi$
$131$	− 198.997i	− 1.51906i	0.650469	−	0.759532i	$-0.274572\pi$
$132$	0	0
$133$	42.0000	0.315789
$134$	291.863i	2.17808i
$135$	0	0
$136$	0	0
$137$	−131.332	−0.958625	−0.479313	−	0.877644i	$-0.659114\pi$
$137$	−131.332	−0.958625	−0.479313	+	0.877644i	$0.659114\pi$
$138$	0	0
$139$	227.688i	1.63805i	0.573761	+	0.819023i	$0.305484\pi$
$139$	227.688i	1.63805i	−0.573761	+	0.819023i	$0.694516\pi$
$140$	650.058i	4.64327i
$141$	0	0
$142$	155.563i	1.09552i
$143$	0	0
$144$	0	0
$145$	− 124.451i	− 0.858281i
$146$	136.022	0.931658
$147$	0	0
$148$	−308.000	−2.08108
$149$	119.398i	0.801332i	0.916224	+	0.400666i	$0.131221\pi$
$149$	119.398i	0.801332i	−0.916224	+	0.400666i	$0.868779\pi$
$150$	0	0
$151$	89.0955i	0.590036i	0.955492	+	0.295018i	$0.0953257\pi$
$151$	89.0955i	0.590036i	−0.955492	+	0.295018i	$0.904674\pi$
$152$	−42.2137	−0.277722
$153$	0	0
$154$	0	0
$155$	−412.757	−2.66295
$156$	0	0
$157$	−44.0000	−0.280255	−0.140127	−	0.990133i	$-0.544751\pi$
$157$	−44.0000	−0.280255	−0.140127	+	0.990133i	$0.544751\pi$
$158$	42.2137	0.267176
$159$	0	0
$160$	217.789i	1.36118i
$161$	− 278.596i	− 1.73041i
$162$	0	0
$163$	194.000	1.19018	0.595092	−	0.803658i	$-0.297116\pi$
$163$	194.000	1.19018	0.595092	+	0.803658i	$0.297116\pi$
$164$	278.596i	1.69876i
$165$	0	0
$166$	308.000	1.85542
$167$	− 238.797i	− 1.42992i	−0.699164	−	0.714961i	$-0.746444\pi$
$167$	− 238.797i	− 1.42992i	0.699164	−	0.714961i	$-0.253556\pi$
$168$	0	0
$169$	7.00000	0.0414201
$170$	0	0
$171$	0	0
$172$	− 207.889i	− 1.20866i
$173$	− 198.997i	− 1.15027i	−0.818057	−	0.575137i	$-0.804949\pi$
$173$	− 198.997i	− 1.15027i	0.818057	−	0.575137i	$-0.195051\pi$
$174$	0	0
$175$	− 623.668i	− 3.56382i
$176$	0	0
$177$	0	0
$178$	− 373.352i	− 2.09749i
$179$	−75.0467	−0.419255	−0.209628	−	0.977781i	$-0.567225\pi$
$179$	−75.0467	−0.419255	−0.209628	+	0.977781i	$0.567225\pi$
$180$	0	0
$181$	−10.0000	−0.0552486	−0.0276243	−	0.999618i	$-0.508794\pi$
$181$	−10.0000	−0.0552486	−0.0276243	+	0.999618i	$0.508794\pi$
$182$	417.895i	2.29612i
$183$	0	0
$184$	280.014i	1.52182i
$185$	412.757	2.23112
$186$	0	0
$187$	0	0
$188$	−65.6658	−0.349286
$189$	0	0
$190$	132.000	0.694737
$191$	−103.189	−0.540257	−0.270129	−	0.962824i	$-0.587066\pi$
$191$	−103.189	−0.540257	−0.270129	+	0.962824i	$0.587066\pi$
$192$	0	0
$193$	− 148.492i	− 0.769391i	−0.923044	−	0.384695i	$-0.874306\pi$
$193$	− 148.492i	− 0.769391i	0.923044	−	0.384695i	$-0.125694\pi$
$194$	− 232.164i	− 1.19672i
$195$	0	0
$196$	343.000	1.75000
$197$	344.929i	1.75091i	0.483301	+	0.875454i	$0.339438\pi$
$197$	344.929i	1.75091i	−0.483301	+	0.875454i	$0.660562\pi$
$198$	0	0
$199$	308.000	1.54774	0.773869	−	0.633345i	$-0.218319\pi$
$199$	308.000	1.54774	0.773869	+	0.633345i	$0.218319\pi$
$200$	626.842i	3.13421i
$201$	0	0
$202$	−88.0000	−0.435644
$203$	−131.332	−0.646954
$204$	0	0
$205$	− 373.352i	− 1.82123i
$206$	− 46.4327i	− 0.225402i
$207$	0	0
$208$	− 63.6396i	− 0.305960i
$209$	0	0
$210$	0	0
$211$	168.291i	0.797590i	0.917040	+	0.398795i	$0.130571\pi$
$211$	168.291i	0.797590i	−0.917040	+	0.398795i	$0.869429\pi$
$212$	459.661	2.16821
$213$	0	0
$214$	−88.0000	−0.411215
$215$	278.596i	1.29580i
$216$	0	0
$217$	435.578i	2.00727i
$218$	211.069	0.968205
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	126.000	0.565022	0.282511	−	0.959264i	$-0.408833\pi$
$223$	126.000	0.565022	0.282511	+	0.959264i	$0.408833\pi$
$224$	229.830	1.02603
$225$	0	0
$226$	435.578i	1.92734i
$227$	397.995i	1.75328i	0.481145	+	0.876641i	$0.340221\pi$
$227$	397.995i	1.75328i	−0.481145	+	0.876641i	$0.659779\pi$
$228$	0	0
$229$	−132.000	−0.576419	−0.288210	−	0.957567i	$-0.593060\pi$
$229$	−132.000	−0.576419	−0.288210	+	0.957567i	$0.593060\pi$
$230$	− 875.589i	− 3.80691i
$231$	0	0
$232$	132.000	0.568966
$233$	92.8655i	0.398564i	0.979942	+	0.199282i	$0.0638610\pi$
$233$	92.8655i	0.398564i	−0.979942	+	0.199282i	$0.936139\pi$
$234$	0	0
$235$	88.0000	0.374468
$236$	−131.332	−0.556490
$237$	0	0
$238$	0	0
$239$	− 106.132i	− 0.444067i	−0.975039	−	0.222033i	$-0.928731\pi$
$239$	− 106.132i	− 0.444067i	0.975039	−	0.222033i	$-0.0712694\pi$
$240$	0	0
$241$	156.978i	0.651360i	0.945480	+	0.325680i	$0.105593\pi$
$241$	156.978i	0.651360i	−0.945480	+	0.325680i	$0.894407\pi$
$242$	0	0
$243$	0	0
$244$	485.075i	1.98801i
$245$	−459.661	−1.87617
$246$	0	0
$247$	54.0000	0.218623
$248$	− 437.794i	− 1.76530i
$249$	0	0
$250$	− 1182.28i	− 4.72913i
$251$	168.855	0.672729	0.336364	−	0.941732i	$-0.390803\pi$
$251$	168.855	0.672729	0.336364	+	0.941732i	$0.390803\pi$
$252$	0	0
$253$	0	0
$254$	389.305	1.53269
$255$	0	0
$256$	−371.000	−1.44922
$257$	187.617	0.730026	0.365013	−	0.931002i	$-0.381065\pi$
$257$	187.617	0.730026	0.365013	+	0.931002i	$0.381065\pi$
$258$	0	0
$259$	− 435.578i	− 1.68177i
$260$	835.789i	3.21457i
$261$	0	0
$262$	−660.000	−2.51908
$263$	39.7995i	0.151329i	0.997133	+	0.0756644i	$0.0241078\pi$
$263$	39.7995i	0.151329i	−0.997133	+	0.0756644i	$0.975892\pi$
$264$	0	0
$265$	−616.000	−2.32453
$266$	− 139.298i	− 0.523678i
$267$	0	0
$268$	616.000	2.29851
$269$	328.329	1.22055	0.610277	−	0.792188i	$-0.291058\pi$
$269$	328.329	1.22055	0.610277	+	0.792188i	$0.291058\pi$
$270$	0	0
$271$	114.551i	0.422699i	0.977411	+	0.211349i	$0.0677857\pi$
$271$	114.551i	0.422699i	−0.977411	+	0.211349i	$0.932214\pi$
$272$	0	0
$273$	0	0
$274$	435.578i	1.58970i
$275$	0	0
$276$	0	0
$277$	− 386.080i	− 1.39379i	−0.717172	−	0.696896i	$-0.754564\pi$
$277$	− 386.080i	− 1.39379i	0.717172	−	0.696896i	$-0.245436\pi$
$278$	755.157	2.71639
$279$	0	0
$280$	924.000	3.30000
$281$	− 477.594i	− 1.69962i	−0.527087	−	0.849811i	$-0.676716\pi$
$281$	− 477.594i	− 1.69962i	0.527087	−	0.849811i	$-0.323284\pi$
$282$	0	0
$283$	− 57.9828i	− 0.204886i	−0.994739	−	0.102443i	$-0.967334\pi$
$283$	− 57.9828i	− 0.204886i	0.994739	−	0.102443i	$-0.0326659\pi$
$284$	328.329	1.15609
$285$	0	0
$286$	0	0
$287$	−393.995	−1.37280
$288$	0	0
$289$	289.000	1.00000
$290$	−412.757	−1.42330
$291$	0	0
$292$	− 287.085i	− 0.983169i
$293$	− 371.462i	− 1.26779i	−0.773420	−	0.633894i	$-0.781455\pi$
$293$	− 371.462i	− 1.26779i	0.773420	−	0.633894i	$-0.218545\pi$
$294$	0	0
$295$	176.000	0.596610
$296$	437.794i	1.47904i
$297$	0	0
$298$	396.000	1.32886
$299$	− 358.195i	− 1.19798i
$300$	0	0
$301$	294.000	0.976744
$302$	295.496	0.978464
$303$	0	0
$304$	21.2132i	0.0697803i
$305$	− 650.058i	− 2.13134i
$306$	0	0
$307$	530.330i	1.72746i	0.503955	+	0.863730i	$0.331878\pi$
$307$	530.330i	1.72746i	−0.503955	+	0.863730i	$0.668122\pi$
$308$	0	0
$309$	0	0
$310$	1368.96i	4.41600i
$311$	−459.661	−1.47801	−0.739004	−	0.673701i	$-0.764704\pi$
$311$	−459.661	−1.47801	−0.739004	+	0.673701i	$0.764704\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	145.931i	0.464750i
$315$	0	0
$316$	− 89.0955i	− 0.281948i
$317$	−365.852	−1.15411	−0.577054	−	0.816706i	$-0.695798\pi$
$317$	−365.852	−1.15411	−0.577054	+	0.816706i	$0.695798\pi$
$318$	0	0
$319$	0	0
$320$	909.941	2.84356
$321$	0	0
$322$	−924.000	−2.86957
$323$	0	0
$324$	0	0
$325$	− 801.859i	− 2.46726i
$326$	− 643.425i	− 1.97370i
$327$	0	0
$328$	396.000	1.20732
$329$	− 92.8655i	− 0.282266i
$330$	0	0
$331$	−350.000	−1.05740	−0.528701	−	0.848808i	$-0.677321\pi$
$331$	−350.000	−1.05740	−0.528701	+	0.848808i	$0.677321\pi$
$332$	− 650.058i	− 1.95801i
$333$	0	0
$334$	−792.000	−2.37126
$335$	−825.513	−2.46422
$336$	0	0
$337$	− 414.365i	− 1.22957i	−0.788695	−	0.614784i	$-0.789243\pi$
$337$	− 414.365i	− 1.22957i	0.788695	−	0.614784i	$-0.210757\pi$
$338$	− 23.2164i	− 0.0686875i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−295.496	−0.859001
$345$	0	0
$346$	−660.000	−1.90751
$347$	− 464.327i	− 1.33812i	−0.743209	−	0.669060i	$-0.766697\pi$
$347$	− 464.327i	− 1.33812i	0.743209	−	0.669060i	$-0.233303\pi$
$348$	0	0
$349$	− 284.257i	− 0.814490i	−0.913319	−	0.407245i	$-0.866490\pi$
$349$	− 284.257i	− 0.814490i	0.913319	−	0.407245i	$-0.133510\pi$
$350$	−2068.47	−5.90992
$351$	0	0
$352$	0	0
$353$	281.425	0.797238	0.398619	−	0.917117i	$-0.369490\pi$
$353$	281.425	0.797238	0.398619	+	0.917117i	$0.369490\pi$
$354$	0	0
$355$	−440.000	−1.23944
$356$	−787.990	−2.21345
$357$	0	0
$358$	248.902i	0.695256i
$359$	650.058i	1.81075i	0.424615	+	0.905374i	$0.360409\pi$
$359$	650.058i	1.81075i	−0.424615	+	0.905374i	$0.639591\pi$
$360$	0	0
$361$	343.000	0.950139
$362$	33.1662i	0.0916195i
$363$	0	0
$364$	882.000	2.42308
$365$	384.728i	1.05405i
$366$	0	0
$367$	338.000	0.920981	0.460490	−	0.887665i	$-0.347674\pi$
$367$	338.000	0.920981	0.460490	+	0.887665i	$0.347674\pi$
$368$	140.712	0.382371
$369$	0	0
$370$	− 1368.96i	− 3.69989i
$371$	650.058i	1.75218i
$372$	0	0
$373$	− 250.316i	− 0.671088i	−0.942025	−	0.335544i	$-0.891080\pi$
$373$	− 250.316i	− 0.671088i	0.942025	−	0.335544i	$-0.108920\pi$
$374$	0	0
$375$	0	0
$376$	93.3381i	0.248240i
$377$	−168.855	−0.447891
$378$	0	0
$379$	−78.0000	−0.205805	−0.102902	−	0.994691i	$-0.532813\pi$
$379$	−78.0000	−0.205805	−0.102902	+	0.994691i	$0.532813\pi$
$380$	− 278.596i	− 0.733149i
$381$	0	0
$382$	342.240i	0.895915i
$383$	−347.091	−0.906242	−0.453121	−	0.891449i	$-0.649689\pi$
$383$	−347.091	−0.906242	−0.453121	+	0.891449i	$0.649689\pi$
$384$	0	0
$385$	0	0
$386$	−492.494	−1.27589
$387$	0	0
$388$	−490.000	−1.26289
$389$	121.951	0.313498	0.156749	−	0.987638i	$-0.449899\pi$
$389$	121.951	0.313498	0.156749	+	0.987638i	$0.449899\pi$
$390$	0	0
$391$	0	0
$392$	− 487.544i	− 1.24373i
$393$	0	0
$394$	1144.00	2.90355
$395$	119.398i	0.302275i
$396$	0	0
$397$	266.000	0.670025	0.335013	−	0.942214i	$-0.391259\pi$
$397$	266.000	0.670025	0.335013	+	0.942214i	$0.391259\pi$
$398$	− 1021.52i	− 2.56663i
$399$	0	0
$400$	315.000	0.787500
$401$	262.663	0.655021	0.327510	−	0.944848i	$-0.393790\pi$
$401$	262.663	0.655021	0.327510	+	0.944848i	$0.393790\pi$
$402$	0	0
$403$	560.029i	1.38965i
$404$	185.731i	0.459730i
$405$	0	0
$406$	435.578i	1.07285i
$407$	0	0
$408$	0	0
$409$	− 106.066i	− 0.259330i	−0.991558	−	0.129665i	$-0.958610\pi$
$409$	− 106.066i	− 0.259330i	0.991558	−	0.129665i	$-0.0413902\pi$
$410$	−1238.27	−3.02017
$411$	0	0
$412$	−98.0000	−0.237864
$413$	− 185.731i	− 0.449712i
$414$	0	0
$415$	871.156i	2.09917i
$416$	295.496	0.710327
$417$	0	0
$418$	0	0
$419$	37.5233	0.0895545	0.0447772	−	0.998997i	$-0.485742\pi$
$419$	37.5233	0.0895545	0.0447772	+	0.998997i	$0.485742\pi$
$420$	0	0
$421$	484.000	1.14964	0.574822	−	0.818279i	$-0.305071\pi$
$421$	484.000	1.14964	0.574822	+	0.818279i	$0.305071\pi$
$422$	558.159	1.32265
$423$	0	0
$424$	− 653.367i	− 1.54096i
$425$	0	0
$426$	0	0
$427$	−686.000	−1.60656
$428$	185.731i	0.433951i
$429$	0	0
$430$	924.000	2.14884
$431$	− 411.261i	− 0.954203i	−0.878848	−	0.477101i	$-0.841687\pi$
$431$	− 411.261i	− 0.954203i	0.878848	−	0.477101i	$-0.158313\pi$
$432$	0	0
$433$	690.000	1.59353	0.796767	−	0.604287i	$-0.206542\pi$
$433$	690.000	1.59353	0.796767	+	0.604287i	$0.206542\pi$
$434$	1444.65	3.32868
$435$	0	0
$436$	− 445.477i	− 1.02174i
$437$	119.398i	0.273223i
$438$	0	0
$439$	− 499.217i	− 1.13717i	−0.822625	−	0.568585i	$-0.807491\pi$
$439$	− 499.217i	− 1.13717i	0.822625	−	0.568585i	$-0.192509\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	525.327	1.18584	0.592919	−	0.805262i	$-0.297975\pi$
$443$	525.327	1.18584	0.592919	+	0.805262i	$0.297975\pi$
$444$	0	0
$445$	1056.00	2.37303
$446$	− 417.895i	− 0.936984i
$447$	0	0
$448$	− 960.251i	− 2.14342i
$449$	−262.663	−0.584996	−0.292498	−	0.956266i	$-0.594487\pi$
$449$	−262.663	−0.584996	−0.292498	+	0.956266i	$0.594487\pi$
$450$	0	0
$451$	0	0
$452$	919.321	2.03390
$453$	0	0
$454$	1320.00	2.90749
$455$	−1181.98	−2.59777
$456$	0	0
$457$	660.438i	1.44516i	0.691288	+	0.722580i	$0.257044\pi$
$457$	660.438i	1.44516i	−0.691288	+	0.722580i	$0.742956\pi$
$458$	437.794i	0.955883i
$459$	0	0
$460$	−1848.00	−4.01739
$461$	− 371.462i	− 0.805774i	−0.915250	−	0.402887i	$-0.868007\pi$
$461$	− 371.462i	− 0.805774i	0.915250	−	0.402887i	$-0.131993\pi$
$462$	0	0
$463$	174.000	0.375810	0.187905	−	0.982187i	$-0.439830\pi$
$463$	174.000	0.375810	0.187905	+	0.982187i	$0.439830\pi$
$464$	− 66.3325i	− 0.142958i
$465$	0	0
$466$	308.000	0.660944
$467$	863.036	1.84804	0.924022	−	0.382339i	$-0.124881\pi$
$467$	863.036	1.84804	0.924022	+	0.382339i	$0.124881\pi$
$468$	0	0
$469$	871.156i	1.85747i
$470$	− 291.863i	− 0.620985i
$471$	0	0
$472$	186.676i	0.395500i
$473$	0	0
$474$	0	0
$475$	267.286i	0.562708i
$476$	0	0
$477$	0	0
$478$	−352.000	−0.736402
$479$	623.525i	1.30172i	0.759196	+	0.650862i	$0.225592\pi$
$479$	623.525i	1.30172i	−0.759196	+	0.650862i	$0.774408\pi$
$480$	0	0
$481$	− 560.029i	− 1.16430i
$482$	520.636	1.08016
$483$	0	0
$484$	0	0
$485$	656.658	1.35393
$486$	0	0
$487$	−484.000	−0.993840	−0.496920	−	0.867796i	$-0.665536\pi$
$487$	−484.000	−0.993840	−0.496920	+	0.867796i	$0.665536\pi$
$488$	689.491	1.41289
$489$	0	0
$490$	1524.52i	3.11127i
$491$	− 106.132i	− 0.216155i	−0.994142	−	0.108077i	$-0.965531\pi$
$491$	− 106.132i	− 0.216155i	0.994142	−	0.108077i	$-0.0344694\pi$
$492$	0	0
$493$	0	0
$494$	− 179.098i	− 0.362546i
$495$	0	0
$496$	−220.000	−0.443548
$497$	464.327i	0.934261i
$498$	0	0
$499$	880.000	1.76353	0.881764	−	0.471692i	$-0.156356\pi$
$499$	880.000	1.76353	0.881764	+	0.471692i	$0.156356\pi$
$500$	−2495.30	−4.99060
$501$	0	0
$502$	− 560.029i	− 1.11559i
$503$	− 504.127i	− 1.00224i	−0.865378	−	0.501120i	$-0.832921\pi$
$503$	− 504.127i	− 1.00224i	0.865378	−	0.501120i	$-0.167079\pi$
$504$	0	0
$505$	− 248.902i	− 0.492874i
$506$	0	0
$507$	0	0
$508$	− 821.658i	− 1.61744i
$509$	196.997	0.387028	0.193514	−	0.981097i	$-0.438011\pi$
$509$	196.997	0.387028	0.193514	+	0.981097i	$0.438011\pi$
$510$	0	0
$511$	406.000	0.794521
$512$	315.079i	0.615389i
$513$	0	0
$514$	− 622.254i	− 1.21061i
$515$	131.332	0.255013
$516$	0	0
$517$	0	0
$518$	−1444.65	−2.78890
$519$	0	0
$520$	1188.00	2.28462
$521$	544.088	1.04432	0.522158	−	0.852849i	$-0.325127\pi$
$521$	544.088	1.04432	0.522158	+	0.852849i	$0.325127\pi$
$522$	0	0
$523$	411.536i	0.786876i	0.919351	+	0.393438i	$0.128714\pi$
$523$	411.536i	0.786876i	−0.919351	+	0.393438i	$0.871286\pi$
$524$	1392.98i	2.65836i
$525$	0	0
$526$	132.000	0.250951
$527$	0	0
$528$	0	0
$529$	263.000	0.497164
$530$	2043.04i	3.85479i
$531$	0	0
$532$	−294.000	−0.552632
$533$	−506.565	−0.950403
$534$	0	0
$535$	− 248.902i	− 0.465237i
$536$	− 875.589i	− 1.63356i
$537$	0	0
$538$	− 1088.94i	− 2.02406i
$539$	0	0
$540$	0	0
$541$	− 422.850i	− 0.781608i	−0.920474	−	0.390804i	$-0.872197\pi$
$541$	− 422.850i	− 0.781608i	0.920474	−	0.390804i	$-0.127803\pi$
$542$	379.924	0.700966
$543$	0	0
$544$	0	0
$545$	596.992i	1.09540i
$546$	0	0
$547$	999.849i	1.82788i	0.405852	+	0.913939i	$0.366975\pi$
$547$	999.849i	1.82788i	−0.405852	+	0.913939i	$0.633025\pi$
$548$	919.321	1.67759
$549$	0	0
$550$	0	0
$551$	56.2850	0.102151
$552$	0	0
$553$	126.000	0.227848
$554$	−1280.48	−2.31134
$555$	0	0
$556$	− 1593.82i	− 2.86658i
$557$	− 610.259i	− 1.09562i	−0.836604	−	0.547809i	$-0.815462\pi$
$557$	− 610.259i	− 1.09562i	0.836604	−	0.547809i	$-0.184538\pi$
$558$	0	0
$559$	378.000	0.676208
$560$	− 464.327i	− 0.829156i
$561$	0	0
$562$	−1584.00	−2.81851
$563$	− 159.198i	− 0.282767i	−0.989955	−	0.141384i	$-0.954845\pi$
$563$	− 159.198i	− 0.282767i	0.989955	−	0.141384i	$-0.0451551\pi$
$564$	0	0
$565$	−1232.00	−2.18053
$566$	−192.307	−0.339765
$567$	0	0
$568$	− 466.690i	− 0.821638i
$569$	742.924i	1.30567i	0.757502	+	0.652833i	$0.226420\pi$
$569$	742.924i	1.30567i	−0.757502	+	0.652833i	$0.773580\pi$
$570$	0	0
$571$	− 465.276i	− 0.814845i	−0.913240	−	0.407422i	$-0.866428\pi$
$571$	− 465.276i	− 0.814845i	0.913240	−	0.407422i	$-0.133572\pi$
$572$	0	0
$573$	0	0
$574$	1306.73i	2.27654i
$575$	1772.98	3.08344
$576$	0	0
$577$	880.000	1.52513	0.762565	−	0.646912i	$-0.223940\pi$
$577$	880.000	1.52513	0.762565	+	0.646912i	$0.223940\pi$
$578$	− 958.505i	− 1.65831i
$579$	0	0
$580$	871.156i	1.50199i
$581$	919.321	1.58231
$582$	0	0
$583$	0	0
$584$	−408.066	−0.698743
$585$	0	0
$586$	−1232.00	−2.10239
$587$	694.182	1.18259	0.591296	−	0.806455i	$-0.298616\pi$
$587$	694.182	1.18259	0.591296	+	0.806455i	$0.298616\pi$
$588$	0	0
$589$	− 186.676i	− 0.316938i
$590$	− 583.726i	− 0.989366i
$591$	0	0
$592$	220.000	0.371622
$593$	928.655i	1.56603i	0.622004	+	0.783014i	$0.286319\pi$
$593$	928.655i	1.56603i	−0.622004	+	0.783014i	$0.713681\pi$
$594$	0	0
$595$	0	0
$596$	− 835.789i	− 1.40233i
$597$	0	0
$598$	−1188.00	−1.98662
$599$	647.277	1.08060	0.540298	−	0.841474i	$-0.318311\pi$
$599$	647.277	1.08060	0.540298	+	0.841474i	$0.318311\pi$
$600$	0	0
$601$	165.463i	0.275313i	0.990480	+	0.137656i	$0.0439570\pi$
$601$	165.463i	0.275313i	−0.990480	+	0.137656i	$0.956043\pi$
$602$	− 975.088i	− 1.61975i
$603$	0	0
$604$	− 623.668i	− 1.03256i
$605$	0	0
$606$	0	0
$607$	− 173.948i	− 0.286570i	−0.989681	−	0.143285i	$-0.954233\pi$
$607$	− 173.948i	− 0.286570i	0.989681	−	0.143285i	$-0.0457666\pi$
$608$	−98.4987	−0.162004
$609$	0	0
$610$	−2156.00	−3.53443
$611$	− 119.398i	− 0.195415i
$612$	0	0
$613$	89.0955i	0.145343i	0.997356	+	0.0726717i	$0.0231525\pi$
$613$	89.0955i	0.145343i	−0.997356	+	0.0726717i	$0.976847\pi$
$614$	1758.91	2.86467
$615$	0	0
$616$	0	0
$617$	375.233	0.608158	0.304079	−	0.952647i	$-0.401651\pi$
$617$	375.233	0.608158	0.304079	+	0.952647i	$0.401651\pi$
$618$	0	0
$619$	490.000	0.791599	0.395800	−	0.918337i	$-0.370467\pi$
$619$	490.000	0.791599	0.395800	+	0.918337i	$0.370467\pi$
$620$	2889.30	4.66016
$621$	0	0
$622$	1524.52i	2.45100i
$623$	− 1114.39i	− 1.78874i
$624$	0	0
$625$	1769.00	2.83040
$626$	0	0
$627$	0	0
$628$	308.000	0.490446
$629$	0	0
$630$	0	0
$631$	−308.000	−0.488114	−0.244057	−	0.969761i	$-0.578478\pi$
$631$	−308.000	−0.488114	−0.244057	+	0.969761i	$0.578478\pi$
$632$	−126.641	−0.200382
$633$	0	0
$634$	1213.40i	1.91387i
$635$	1101.12i	1.73405i
$636$	0	0
$637$	623.668i	0.979071i
$638$	0	0
$639$	0	0
$640$	− 2146.78i	− 3.35434i
$641$	1257.03	1.96105	0.980524	−	0.196401i	$-0.0629253\pi$
$641$	1257.03	1.96105	0.980524	+	0.196401i	$0.0629253\pi$
$642$	0	0
$643$	−714.000	−1.11042	−0.555210	−	0.831710i	$-0.687362\pi$
$643$	−714.000	−1.11042	−0.555210	+	0.831710i	$0.687362\pi$
$644$	1950.18i	3.02822i
$645$	0	0
$646$	0	0
$647$	46.9042	0.0724948	0.0362474	−	0.999343i	$-0.488460\pi$
$647$	46.9042	0.0724948	0.0362474	+	0.999343i	$0.488460\pi$
$648$	0	0
$649$	0	0
$650$	−2659.47	−4.09149
$651$	0	0
$652$	−1358.00	−2.08282
$653$	−853.656	−1.30728	−0.653641	−	0.756804i	$-0.726760\pi$
$653$	−853.656	−1.30728	−0.653641	+	0.756804i	$0.726760\pi$
$654$	0	0
$655$	− 1866.76i	− 2.85002i
$656$	− 198.997i	− 0.303350i
$657$	0	0
$658$	−308.000	−0.468085
$659$	− 1034.79i	− 1.57024i	−0.619345	−	0.785119i	$-0.712602\pi$
$659$	− 1034.79i	− 1.57024i	0.619345	−	0.785119i	$-0.287398\pi$
$660$	0	0
$661$	−308.000	−0.465961	−0.232980	−	0.972481i	$-0.574848\pi$
$661$	−308.000	−0.465961	−0.232980	+	0.972481i	$0.574848\pi$
$662$	1160.82i	1.75350i
$663$	0	0
$664$	−924.000	−1.39157
$665$	393.995	0.592474
$666$	0	0
$667$	− 373.352i	− 0.559749i
$668$	1671.58i	2.50236i
$669$	0	0
$670$	2737.92i	4.08644i
$671$	0	0
$672$	0	0
$673$	852.771i	1.26712i	0.773694	+	0.633559i	$0.218407\pi$
$673$	852.771i	1.26712i	−0.773694	+	0.633559i	$0.781593\pi$
$674$	−1374.29	−2.03901
$675$	0	0
$676$	−49.0000	−0.0724852
$677$	− 371.462i	− 0.548688i	−0.961632	−	0.274344i	$-0.911539\pi$
$677$	− 371.462i	− 0.548688i	0.961632	−	0.274344i	$-0.0884607\pi$
$678$	0	0
$679$	− 692.965i	− 1.02057i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−262.663	−0.384573	−0.192286	−	0.981339i	$-0.561590\pi$
$683$	−262.663	−0.384573	−0.192286	+	0.981339i	$0.561590\pi$
$684$	0	0
$685$	−1232.00	−1.79854
$686$	0	0
$687$	0	0
$688$	148.492i	0.215832i
$689$	835.789i	1.21305i
$690$	0	0
$691$	−1126.00	−1.62952	−0.814761	−	0.579797i	$-0.803132\pi$
$691$	−1126.00	−1.62952	−0.814761	+	0.579797i	$0.803132\pi$
$692$	1392.98i	2.01298i
$693$	0	0
$694$	−1540.00	−2.21902
$695$	2135.91i	3.07325i
$696$	0	0
$697$	0	0
$698$	−942.774	−1.35068
$699$	0	0
$700$	4365.68i	6.23668i
$701$	1114.39i	1.58971i	0.606800	+	0.794854i	$0.292453\pi$
$701$	1114.39i	1.58971i	−0.606800	+	0.794854i	$0.707547\pi$
$702$	0	0
$703$	186.676i	0.265542i
$704$	0	0
$705$	0	0
$706$	− 933.381i	− 1.32207i
$707$	−262.663	−0.371518
$708$	0	0
$709$	−44.0000	−0.0620592	−0.0310296	−	0.999518i	$-0.509879\pi$
$709$	−44.0000	−0.0620592	−0.0310296	+	0.999518i	$0.509879\pi$
$710$	1459.31i	2.05537i
$711$	0	0
$712$	1120.06i	1.57311i
$713$	−1238.27	−1.73670
$714$	0	0
$715$	0	0
$716$	525.327	0.733696
$717$	0	0
$718$	2156.00	3.00279
$719$	−778.609	−1.08291	−0.541453	−	0.840731i	$-0.682125\pi$
$719$	−778.609	−1.08291	−0.541453	+	0.840731i	$0.682125\pi$
$720$	0	0
$721$	− 138.593i	− 0.192223i
$722$	− 1137.60i	− 1.57563i
$723$	0	0
$724$	70.0000	0.0966851
$725$	− 835.789i	− 1.15281i
$726$	0	0
$727$	−1012.00	−1.39202	−0.696011	−	0.718031i	$-0.745044\pi$
$727$	−1012.00	−1.39202	−0.696011	+	0.718031i	$0.745044\pi$
$728$	− 1253.68i	− 1.72209i
$729$	0	0
$730$	1276.00	1.74795
$731$	0	0
$732$	0	0
$733$	− 425.678i	− 0.580734i	−0.956915	−	0.290367i	$-0.906223\pi$
$733$	− 425.678i	− 0.580734i	0.956915	−	0.290367i	$-0.0937774\pi$
$734$	− 1121.02i	− 1.52727i
$735$	0	0
$736$	653.367i	0.887726i
$737$	0	0
$738$	0	0
$739$	292.742i	0.396133i	0.980189	+	0.198066i	$0.0634662\pi$
$739$	292.742i	0.396133i	−0.980189	+	0.198066i	$0.936534\pi$
$740$	−2889.30	−3.90445
$741$	0	0
$742$	2156.00	2.90566
$743$	1286.85i	1.73197i	0.500074	+	0.865983i	$0.333306\pi$
$743$	1286.85i	1.73197i	−0.500074	+	0.865983i	$0.666694\pi$
$744$	0	0
$745$	1120.06i	1.50343i
$746$	−830.204	−1.11287
$747$	0	0
$748$	0	0
$749$	−262.663	−0.350685
$750$	0	0
$751$	572.000	0.761651	0.380826	−	0.924647i	$-0.375640\pi$
$751$	572.000	0.761651	0.380826	+	0.924647i	$0.375640\pi$
$752$	46.9042	0.0623725
$753$	0	0
$754$	560.029i	0.742743i
$755$	835.789i	1.10701i
$756$	0	0
$757$	−748.000	−0.988111	−0.494055	−	0.869430i	$-0.664486\pi$
$757$	−748.000	−0.988111	−0.494055	+	0.869430i	$0.664486\pi$
$758$	258.697i	0.341289i
$759$	0	0
$760$	−396.000	−0.521053
$761$	451.061i	0.592721i	0.955076	+	0.296361i	$0.0957730\pi$
$761$	451.061i	0.592721i	−0.955076	+	0.296361i	$0.904227\pi$
$762$	0	0
$763$	630.000	0.825688
$764$	722.324	0.945450
$765$	0	0
$766$	1151.17i	1.50283i
$767$	− 238.797i	− 0.311339i
$768$	0	0
$769$	779.232i	1.01331i	0.862150	+	0.506653i	$0.169117\pi$
$769$	779.232i	1.01331i	−0.862150	+	0.506653i	$0.830883\pi$
$770$	0	0
$771$	0	0
$772$	1039.45i	1.34643i
$773$	872.417	1.12861	0.564306	−	0.825566i	$-0.309144\pi$
$773$	872.417	1.12861	0.564306	+	0.825566i	$0.309144\pi$
$774$	0	0
$775$	−2772.00	−3.57677
$776$	696.491i	0.897540i
$777$	0	0
$778$	− 404.465i	− 0.519878i
$779$	168.855	0.216759
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−245.000	−0.312500
$785$	−412.757	−0.525805
$786$	0	0
$787$	− 335.169i	− 0.425881i	−0.977065	−	0.212941i	$-0.931696\pi$
$787$	− 335.169i	− 0.425881i	0.977065	−	0.212941i	$-0.0683041\pi$
$788$	− 2414.50i	− 3.06409i
$789$	0	0
$790$	396.000	0.501266
$791$	1300.12i	1.64364i
$792$	0	0
$793$	−882.000	−1.11223
$794$	− 882.222i	− 1.11111i
$795$	0	0
$796$	−2156.00	−2.70854
$797$	290.806	0.364876	0.182438	−	0.983217i	$-0.441601\pi$
$797$	290.806	0.364876	0.182438	+	0.983217i	$0.441601\pi$
$798$	0	0
$799$	0	0
$800$	1462.63i	1.82829i
$801$	0	0
$802$	− 871.156i	− 1.08623i
$803$	0	0
$804$	0	0
$805$	− 2613.47i	− 3.24654i
$806$	1857.40	2.30447
$807$	0	0
$808$	264.000	0.326733
$809$	331.662i	0.409966i	0.978766	+	0.204983i	$0.0657139\pi$
$809$	331.662i	0.409966i	−0.978766	+	0.204983i	$0.934286\pi$
$810$	0	0
$811$	− 626.497i	− 0.772499i	−0.922394	−	0.386249i	$-0.873770\pi$
$811$	− 626.497i	− 0.772499i	0.922394	−	0.386249i	$-0.126230\pi$
$812$	919.321	1.13217
$813$	0	0
$814$	0	0
$815$	1819.88	2.23298
$816$	0	0
$817$	−126.000	−0.154223
$818$	−351.781	−0.430050
$819$	0	0
$820$	2613.47i	3.18715i
$821$	159.198i	0.193907i	0.995289	+	0.0969537i	$0.0309099\pi$
$821$	159.198i	0.193907i	−0.995289	+	0.0969537i	$0.969090\pi$
$822$	0	0
$823$	−18.0000	−0.0218712	−0.0109356	−	0.999940i	$-0.503481\pi$
$823$	−18.0000	−0.0218712	−0.0109356	+	0.999940i	$0.503481\pi$
$824$	139.298i	0.169051i
$825$	0	0
$826$	−616.000	−0.745763
$827$	− 198.997i	− 0.240626i	−0.992736	−	0.120313i	$-0.961610\pi$
$827$	− 198.997i	− 0.240626i	0.992736	−	0.120313i	$-0.0383898\pi$
$828$	0	0
$829$	−602.000	−0.726176	−0.363088	−	0.931755i	$-0.618278\pi$
$829$	−602.000	−0.726176	−0.363088	+	0.931755i	$0.618278\pi$
$830$	2889.30	3.48108
$831$	0	0
$832$	− 1234.61i	− 1.48390i
$833$	0	0
$834$	0	0
$835$	− 2240.11i	− 2.68277i
$836$	0	0
$837$	0	0
$838$	− 124.451i	− 0.148509i
$839$	−1153.84	−1.37526	−0.687629	−	0.726062i	$-0.741349\pi$
$839$	−1153.84	−1.37526	−0.687629	+	0.726062i	$0.741349\pi$
$840$	0	0
$841$	665.000	0.790725
$842$	− 1605.25i	− 1.90647i
$843$	0	0
$844$	− 1178.04i	− 1.39578i
$845$	65.6658	0.0777110
$846$	0	0
$847$	0	0
$848$	−328.329	−0.387181
$849$	0	0
$850$	0	0
$851$	1238.27	1.45508
$852$	0	0
$853$	629.325i	0.737778i	0.929473	+	0.368889i	$0.120262\pi$
$853$	629.325i	0.737778i	−0.929473	+	0.368889i	$0.879738\pi$
$854$	2275.20i	2.66417i
$855$	0	0
$856$	264.000	0.308411
$857$	92.8655i	0.108361i	0.998531	+	0.0541806i	$0.0172547\pi$
$857$	92.8655i	0.108361i	−0.998531	+	0.0541806i	$0.982745\pi$
$858$	0	0
$859$	526.000	0.612340	0.306170	−	0.951977i	$-0.400952\pi$
$859$	526.000	0.612340	0.306170	+	0.951977i	$0.400952\pi$
$860$	− 1950.18i	− 2.26765i
$861$	0	0
$862$	−1364.00	−1.58237
$863$	−1604.12	−1.85877	−0.929387	−	0.369106i	$-0.879664\pi$
$863$	−1604.12	−1.85877	−0.929387	+	0.369106i	$0.879664\pi$
$864$	0	0
$865$	− 1866.76i	− 2.15811i
$866$	− 2288.47i	− 2.64258i
$867$	0	0
$868$	− 3049.04i	− 3.51272i
$869$	0	0
$870$	0	0
$871$	1120.06i	1.28594i
$872$	−633.206	−0.726154
$873$	0	0
$874$	396.000	0.453089
$875$	− 3528.89i	− 4.03302i
$876$	0	0
$877$	346.482i	0.395077i	0.980295	+	0.197538i	$0.0632947\pi$
$877$	346.482i	0.395077i	−0.980295	+	0.197538i	$0.936705\pi$
$878$	−1655.72	−1.88578
$879$	0	0
$880$	0	0
$881$	−1538.46	−1.74626	−0.873131	−	0.487486i	$-0.837914\pi$
$881$	−1538.46	−1.74626	−0.873131	+	0.487486i	$0.837914\pi$
$882$	0	0
$883$	1408.00	1.59456	0.797282	−	0.603607i	$-0.206270\pi$
$883$	1408.00	1.59456	0.797282	+	0.603607i	$0.206270\pi$
$884$	0	0
$885$	0	0
$886$	− 1742.31i	− 1.96649i
$887$	− 278.596i	− 0.314088i	−0.987592	−	0.157044i	$-0.949803\pi$
$887$	− 278.596i	− 0.314088i	0.987592	−	0.157044i	$-0.0501965\pi$
$888$	0	0
$889$	1162.00	1.30709
$890$	− 3502.36i	− 3.93523i
$891$	0	0
$892$	−882.000	−0.988789
$893$	39.7995i	0.0445683i
$894$	0	0
$895$	−704.000	−0.786592
$896$	−2265.47	−2.52843
$897$	0	0
$898$	871.156i	0.970106i
$899$	583.726i	0.649306i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	− 1306.73i	− 1.44550i
$905$	−93.8083	−0.103656
$906$	0	0
$907$	−714.000	−0.787211	−0.393605	−	0.919280i	$-0.628772\pi$
$907$	−714.000	−0.787211	−0.393605	+	0.919280i	$0.628772\pi$
$908$	− 2785.96i	− 3.06824i
$909$	0	0
$910$	3920.20i	4.30791i
$911$	−1510.31	−1.65786	−0.828932	−	0.559350i	$-0.811051\pi$
$911$	−1510.31	−1.65786	−0.828932	+	0.559350i	$0.811051\pi$
$912$	0	0
$913$	0	0
$914$	2190.42	2.39653
$915$	0	0
$916$	924.000	1.00873
$917$	−1969.97	−2.14828
$918$	0	0
$919$	− 504.874i	− 0.549373i	−0.961534	−	0.274687i	$-0.911426\pi$
$919$	− 504.874i	− 0.549373i	0.961534	−	0.274687i	$-0.0885742\pi$
$920$	2626.77i	2.85518i
$921$	0	0
$922$	−1232.00	−1.33623
$923$	596.992i	0.646796i
$924$	0	0
$925$	2772.00	2.99676
$926$	− 577.093i	− 0.623210i
$927$	0	0
$928$	308.000	0.331897
$929$	206.378	0.222151	0.111076	−	0.993812i	$-0.464570\pi$
$929$	206.378	0.222151	0.111076	+	0.993812i	$0.464570\pi$
$930$	0	0
$931$	− 207.889i	− 0.223297i
$932$	− 650.058i	− 0.697488i
$933$	0	0
$934$	− 2862.37i	− 3.06463i
$935$	0	0
$936$	0	0
$937$	− 168.291i	− 0.179607i	−0.995960	−	0.0898033i	$-0.971376\pi$
$937$	− 168.291i	− 0.179607i	0.995960	−	0.0898033i	$-0.0286238\pi$
$938$	2889.30	3.08027
$939$	0	0
$940$	−616.000	−0.655319
$941$	− 530.660i	− 0.563932i	−0.959424	−	0.281966i	$-0.909013\pi$
$941$	− 530.660i	− 0.563932i	0.959424	−	0.281966i	$-0.0909865\pi$
$942$	0	0
$943$	− 1120.06i	− 1.18776i
$944$	93.8083	0.0993732
$945$	0	0
$946$	0	0
$947$	694.182	0.733032	0.366516	−	0.930412i	$-0.380550\pi$
$947$	694.182	0.733032	0.366516	+	0.930412i	$0.380550\pi$
$948$	0	0
$949$	522.000	0.550053
$950$	886.489	0.933146
$951$	0	0
$952$	0	0
$953$	− 172.464i	− 0.180970i	−0.995898	−	0.0904850i	$-0.971158\pi$
$953$	− 172.464i	− 0.180970i	0.995898	−	0.0904850i	$-0.0288417\pi$
$954$	0	0
$955$	−968.000	−1.01361
$956$	742.924i	0.777117i
$957$	0	0
$958$	2068.00	2.15866
$959$	1300.12i	1.35570i
$960$	0	0
$961$	975.000	1.01457
$962$	−1857.40	−1.93077
$963$	0	0
$964$	− 1098.84i	− 1.13988i
$965$	− 1392.98i	− 1.44351i
$966$	0	0
$967$	683.065i	0.706376i	0.935552	+	0.353188i	$0.114902\pi$
$967$	683.065i	0.706376i	−0.935552	+	0.353188i	$0.885098\pi$
$968$	0	0
$969$	0	0
$970$	− 2177.89i	− 2.24525i
$971$	56.2850	0.0579660	0.0289830	−	0.999580i	$-0.490773\pi$
$971$	56.2850	0.0579660	0.0289830	+	0.999580i	$0.490773\pi$
$972$	0	0
$973$	2254.00	2.31655
$974$	1605.25i	1.64810i
$975$	0	0
$976$	− 346.482i	− 0.355002i
$977$	337.710	0.345660	0.172830	−	0.984952i	$-0.444709\pi$
$977$	337.710	0.345660	0.172830	+	0.984952i	$0.444709\pi$
$978$	0	0
$979$	0	0
$980$	3217.63	3.28329
$981$	0	0
$982$	−352.000	−0.358452
$983$	−459.661	−0.467610	−0.233805	−	0.972283i	$-0.575118\pi$
$983$	−459.661	−0.467610	−0.233805	+	0.972283i	$0.575118\pi$
$984$	0	0
$985$	3235.72i	3.28500i
$986$	0	0
$987$	0	0
$988$	−378.000	−0.382591
$989$	835.789i	0.845085i
$990$	0	0
$991$	−1452.00	−1.46519	−0.732593	−	0.680667i	$-0.761690\pi$
$991$	−1452.00	−1.46519	−0.732593	+	0.680667i	$0.761690\pi$
$992$	− 1021.52i	− 1.02976i
$993$	0	0
$994$	1540.00	1.54930
$995$	2889.30	2.90382
$996$	0	0
$997$	1752.21i	1.75748i	0.477298	+	0.878742i	$0.341616\pi$
$997$	1752.21i	1.75748i	−0.477298	+	0.878742i	$0.658384\pi$
$998$	− 2918.63i	− 2.92448i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.b.604.2	yes	4
3.2	odd	2	inner	1089.3.c.b.604.3	yes	4
11.10	odd	2	inner	1089.3.c.b.604.4	yes	4
33.32	even	2	inner	1089.3.c.b.604.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.c.b.604.1	✓	4	33.32	even	2	inner
1089.3.c.b.604.2	yes	4	1.1	even	1	trivial
1089.3.c.b.604.3	yes	4	3.2	odd	2	inner
1089.3.c.b.604.4	yes	4	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)

\(f(q)\)	\(=\)	\(q-3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} -9.89949i q^{7} +9.94987i q^{8} +O(q^{10})\)	\(q-3.31662i q^{2} -7.00000 q^{4} +9.38083 q^{5} -9.89949i q^{7} +9.94987i q^{8} -31.1127i q^{10} -12.7279i q^{13} -32.8329 q^{14} +5.00000 q^{16} +4.24264i q^{19} -65.6658 q^{20} +28.1425 q^{23} +63.0000 q^{25} -42.2137 q^{26} +69.2965i q^{28} -13.2665i q^{29} -44.0000 q^{31} +23.2164i q^{32} -92.8655i q^{35} +44.0000 q^{37} +14.0712 q^{38} +93.3381i q^{40} -39.7995i q^{41} +29.6985i q^{43} -93.3381i q^{46} +9.38083 q^{47} -49.0000 q^{49} -208.947i q^{50} +89.0955i q^{52} -65.6658 q^{53} +98.4987 q^{56} -44.0000 q^{58} +18.7617 q^{59} -69.2965i q^{61} +145.931i q^{62} +97.0000 q^{64} -119.398i q^{65} -88.0000 q^{67} -308.000 q^{70} -46.9042 q^{71} +41.0122i q^{73} -145.931i q^{74} -29.6985i q^{76} +12.7279i q^{79} +46.9042 q^{80} -132.000 q^{82} +92.8655i q^{83} +98.4987 q^{86} +112.570 q^{89} -126.000 q^{91} -196.997 q^{92} -31.1127i q^{94} +39.7995i q^{95} +70.0000 q^{97} +162.515i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{4}+O(q^{10}) \) 4 * q - 28 * q^4	\( 4 q - 28 q^{4} + 20 q^{16} + 252 q^{25} - 176 q^{31} + 176 q^{37} - 196 q^{49} - 176 q^{58} + 388 q^{64} - 352 q^{67} - 1232 q^{70} - 528 q^{82} - 504 q^{91} + 280 q^{97}+O(q^{100}) \) 4 * q - 28 * q^4 + 20 * q^16 + 252 * q^25 - 176 * q^31 + 176 * q^37 - 196 * q^49 - 176 * q^58 + 388 * q^64 - 352 * q^67 - 1232 * q^70 - 528 * q^82 - 504 * q^91 + 280 * q^97

Introduction

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