LMFDB - Embedded newform 1089.3.c.c.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:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.c.604.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03528i q^{2} +2.92820 q^{4} -1.46410 q^{5} -0.517638i q^{7} -7.17260i q^{8} +O(q^{10})$	$q-1.03528i q^{2} +2.92820 q^{4} -1.46410 q^{5} -0.517638i q^{7} -7.17260i q^{8} +1.51575i q^{10} +18.9396i q^{13} -0.535898 q^{14} +4.28719 q^{16} +18.2832i q^{17} +17.3867i q^{19} -4.28719 q^{20} -40.6410 q^{23} -22.8564 q^{25} +19.6077 q^{26} -1.51575i q^{28} +43.0555i q^{29} +29.7321 q^{31} -33.1288i q^{32} +18.9282 q^{34} +0.757875i q^{35} -30.2295 q^{37} +18.0000 q^{38} +10.5014i q^{40} +0.480473i q^{41} -30.0502i q^{43} +42.0747i q^{46} +71.2820 q^{47} +48.7321 q^{49} +23.6627i q^{50} +55.4589i q^{52} -45.4256 q^{53} -3.71281 q^{56} +44.5744 q^{58} +33.3205 q^{59} +70.4704i q^{61} -30.7809i q^{62} -17.1487 q^{64} -27.7295i q^{65} -79.8372 q^{67} +53.5370i q^{68} +0.784610 q^{70} +74.2487 q^{71} +2.06059i q^{73} +31.2959i q^{74} +50.9117i q^{76} -17.5154i q^{79} -6.27688 q^{80} +0.497423 q^{82} +48.1032i q^{83} -26.7685i q^{85} -31.1103 q^{86} -56.7846 q^{89} +9.80385 q^{91} -119.005 q^{92} -73.7966i q^{94} -25.4558i q^{95} -138.923 q^{97} -50.4511i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 8 q^{5}+O(q^{10}) $ 4 * q - 16 * q^4 + 8 * q^5	$ 4 q - 16 q^{4} + 8 q^{5} - 16 q^{14} + 128 q^{16} - 128 q^{20} - 24 q^{23} - 36 q^{25} + 120 q^{26} + 112 q^{31} + 48 q^{34} + 80 q^{37} + 72 q^{38} + 8 q^{47} + 188 q^{49} + 40 q^{53} + 96 q^{56} + 400 q^{58} + 64 q^{59} - 512 q^{64} - 160 q^{67} - 80 q^{70} + 200 q^{71} + 640 q^{80} - 192 q^{82} - 360 q^{86} - 144 q^{89} + 60 q^{91} - 864 q^{92} - 140 q^{97}+O(q^{100}) $ 4 * q - 16 * q^4 + 8 * q^5 - 16 * q^14 + 128 * q^16 - 128 * q^20 - 24 * q^23 - 36 * q^25 + 120 * q^26 + 112 * q^31 + 48 * q^34 + 80 * q^37 + 72 * q^38 + 8 * q^47 + 188 * q^49 + 40 * q^53 + 96 * q^56 + 400 * q^58 + 64 * q^59 - 512 * q^64 - 160 * q^67 - 80 * q^70 + 200 * q^71 + 640 * q^80 - 192 * q^82 - 360 * q^86 - 144 * q^89 + 60 * q^91 - 864 * q^92 - 140 * 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 $

$2$	− 1.03528i	− 0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	− 1.03528i	− 0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$	0	0
$3$	0	0
$4$	2.92820	0.732051
$5$	−1.46410	−0.292820	−0.146410	−	0.989224i	$-0.546772\pi$
$5$	−1.46410	−0.292820	−0.146410	+	0.989224i	$0.546772\pi$
$6$	0	0
$7$	− 0.517638i	− 0.0739483i	−0.999316	−	0.0369741i	$-0.988228\pi$
$7$	− 0.517638i	− 0.0739483i	0.999316	−	0.0369741i	$-0.0117719\pi$
$8$	− 7.17260i	− 0.896575i
$9$	0	0
$10$	1.51575i	0.151575i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	18.9396i	1.45689i	0.685104	+	0.728445i	$0.259757\pi$
$13$	18.9396i	1.45689i	−0.685104	+	0.728445i	$0.740243\pi$
$14$	−0.535898	−0.0382785
$15$	0	0
$16$	4.28719	0.267949
$17$	18.2832i	1.07548i	0.843109	+	0.537742i	$0.180723\pi$
$17$	18.2832i	1.07548i	−0.843109	+	0.537742i	$0.819277\pi$
$18$	0	0
$19$	17.3867i	0.915088i	0.889187	+	0.457544i	$0.151271\pi$
$19$	17.3867i	0.915088i	−0.889187	+	0.457544i	$0.848729\pi$
$20$	−4.28719	−0.214359
$21$	0	0
$22$	0	0
$23$	−40.6410	−1.76700	−0.883500	−	0.468430i	$-0.844820\pi$
$23$	−40.6410	−1.76700	−0.883500	+	0.468430i	$0.844820\pi$
$24$	0	0
$25$	−22.8564	−0.914256
$26$	19.6077	0.754142
$27$	0	0
$28$	− 1.51575i	− 0.0541339i
$29$	43.0555i	1.48467i	0.670027	+	0.742337i	$0.266283\pi$
$29$	43.0555i	1.48467i	−0.670027	+	0.742337i	$0.733717\pi$
$30$	0	0
$31$	29.7321	0.959098	0.479549	−	0.877515i	$-0.340800\pi$
$31$	29.7321	0.959098	0.479549	+	0.877515i	$0.340800\pi$
$32$	− 33.1288i	− 1.03528i
$33$	0	0
$34$	18.9282	0.556712
$35$	0.757875i	0.0216536i
$36$	0	0
$37$	−30.2295	−0.817013	−0.408506	−	0.912755i	$-0.633950\pi$
$37$	−30.2295	−0.817013	−0.408506	+	0.912755i	$0.633950\pi$
$38$	18.0000	0.473684
$39$	0	0
$40$	10.5014i	0.262536i
$41$	0.480473i	0.0117189i	0.999983	+	0.00585943i	$0.00186513\pi$
$41$	0.480473i	0.0117189i	−0.999983	+	0.00585943i	$0.998135\pi$
$42$	0	0
$43$	− 30.0502i	− 0.698842i	−0.936966	−	0.349421i	$-0.886378\pi$
$43$	− 30.0502i	− 0.698842i	0.936966	−	0.349421i	$-0.113622\pi$
$44$	0	0
$45$	0	0
$46$	42.0747i	0.914667i
$47$	71.2820	1.51664	0.758319	−	0.651883i	$-0.226021\pi$
$47$	71.2820	1.51664	0.758319	+	0.651883i	$0.226021\pi$
$48$	0	0
$49$	48.7321	0.994532
$50$	23.6627i	0.473254i
$51$	0	0
$52$	55.4589i	1.06652i
$53$	−45.4256	−0.857087	−0.428544	−	0.903521i	$-0.640973\pi$
$53$	−45.4256	−0.857087	−0.428544	+	0.903521i	$0.640973\pi$
$54$	0	0
$55$	0	0
$56$	−3.71281	−0.0663002
$57$	0	0
$58$	44.5744	0.768524
$59$	33.3205	0.564754	0.282377	−	0.959303i	$-0.408877\pi$
$59$	33.3205	0.564754	0.282377	+	0.959303i	$0.408877\pi$
$60$	0	0
$61$	70.4704i	1.15525i	0.816301	+	0.577627i	$0.196021\pi$
$61$	70.4704i	1.15525i	−0.816301	+	0.577627i	$0.803979\pi$
$62$	− 30.7809i	− 0.496466i
$63$	0	0
$64$	−17.1487	−0.267949
$65$	− 27.7295i	− 0.426607i
$66$	0	0
$67$	−79.8372	−1.19160	−0.595800	−	0.803133i	$-0.703165\pi$
$67$	−79.8372	−1.19160	−0.595800	+	0.803133i	$0.703165\pi$
$68$	53.5370i	0.787309i
$69$	0	0
$70$	0.784610	0.0112087
$71$	74.2487	1.04576	0.522878	−	0.852407i	$-0.324858\pi$
$71$	74.2487	1.04576	0.522878	+	0.852407i	$0.324858\pi$
$72$	0	0
$73$	2.06059i	0.0282273i	0.999900	+	0.0141137i	$0.00449267\pi$
$73$	2.06059i	0.0282273i	−0.999900	+	0.0141137i	$0.995507\pi$
$74$	31.2959i	0.422917i
$75$	0	0
$76$	50.9117i	0.669891i
$77$	0	0
$78$	0	0
$79$	− 17.5154i	− 0.221714i	−0.993836	−	0.110857i	$-0.964640\pi$
$79$	− 17.5154i	− 0.221714i	0.993836	−	0.110857i	$-0.0353596\pi$
$80$	−6.27688	−0.0784610
$81$	0	0
$82$	0.497423	0.00606613
$83$	48.1032i	0.579556i	0.957094	+	0.289778i	$0.0935815\pi$
$83$	48.1032i	0.579556i	−0.957094	+	0.289778i	$0.906418\pi$
$84$	0	0
$85$	− 26.7685i	− 0.314924i
$86$	−31.1103	−0.361747
$87$	0	0
$88$	0	0
$89$	−56.7846	−0.638029	−0.319015	−	0.947750i	$-0.603352\pi$
$89$	−56.7846	−0.638029	−0.319015	+	0.947750i	$0.603352\pi$
$90$	0	0
$91$	9.80385	0.107735
$92$	−119.005	−1.29353
$93$	0	0
$94$	− 73.7966i	− 0.785070i
$95$	− 25.4558i	− 0.267956i
$96$	0	0
$97$	−138.923	−1.43220	−0.716098	−	0.698000i	$-0.754074\pi$
$97$	−138.923	−1.43220	−0.716098	+	0.698000i	$0.754074\pi$
$98$	− 50.4511i	− 0.514807i
$99$	0	0
$100$	−66.9282	−0.669282
$101$	191.040i	1.89149i	0.324913	+	0.945744i	$0.394665\pi$
$101$	191.040i	1.89149i	−0.324913	+	0.945744i	$0.605335\pi$
$102$	0	0
$103$	64.1384	0.622703	0.311352	−	0.950295i	$-0.399218\pi$
$103$	64.1384	0.622703	0.311352	+	0.950295i	$0.399218\pi$
$104$	135.846	1.30621
$105$	0	0
$106$	47.0281i	0.443661i
$107$	188.767i	1.76417i	0.471086	+	0.882087i	$0.343862\pi$
$107$	188.767i	1.76417i	−0.471086	+	0.882087i	$0.656138\pi$
$108$	0	0
$109$	− 120.570i	− 1.10615i	−0.833133	−	0.553073i	$-0.813455\pi$
$109$	− 120.570i	− 1.10615i	0.833133	−	0.553073i	$-0.186545\pi$
$110$	0	0
$111$	0	0
$112$	− 2.21921i	− 0.0198144i
$113$	160.354	1.41906	0.709530	−	0.704675i	$-0.248907\pi$
$113$	160.354	1.41906	0.709530	+	0.704675i	$0.248907\pi$
$114$	0	0
$115$	59.5026	0.517414
$116$	126.075i	1.08686i
$117$	0	0
$118$	− 34.4959i	− 0.292338i
$119$	9.46410	0.0795303
$120$	0	0
$121$	0	0
$122$	72.9564	0.598003
$123$	0	0
$124$	87.0615	0.702109
$125$	70.0666	0.560533
$126$	0	0
$127$	112.721i	0.887567i	0.896134	+	0.443783i	$0.146364\pi$
$127$	112.721i	0.887567i	−0.896134	+	0.443783i	$0.853636\pi$
$128$	− 114.762i	− 0.896575i
$129$	0	0
$130$	−28.7077	−0.220828
$131$	71.3398i	0.544579i	0.962215	+	0.272289i	$0.0877808\pi$
$131$	71.3398i	0.544579i	−0.962215	+	0.272289i	$0.912219\pi$
$132$	0	0
$133$	9.00000	0.0676692
$134$	82.6535i	0.616817i
$135$	0	0
$136$	131.138	0.964253
$137$	−82.7461	−0.603986	−0.301993	−	0.953310i	$-0.597652\pi$
$137$	−82.7461	−0.603986	−0.301993	+	0.953310i	$0.597652\pi$
$138$	0	0
$139$	− 19.6975i	− 0.141708i	−0.997487	−	0.0708542i	$-0.977428\pi$
$139$	− 19.6975i	− 0.141708i	0.997487	−	0.0708542i	$-0.0225725\pi$
$140$	2.21921i	0.0158515i
$141$	0	0
$142$	− 76.8679i	− 0.541323i
$143$	0	0
$144$	0	0
$145$	− 63.0377i	− 0.434743i
$146$	2.13328	0.0146115
$147$	0	0
$148$	−88.5180	−0.598095
$149$	81.9700i	0.550134i	0.961425	+	0.275067i	$0.0887000\pi$
$149$	81.9700i	0.550134i	−0.961425	+	0.275067i	$0.911300\pi$
$150$	0	0
$151$	− 252.778i	− 1.67403i	−0.547183	−	0.837013i	$-0.684300\pi$
$151$	− 252.778i	− 1.67403i	0.547183	−	0.837013i	$-0.315700\pi$
$152$	124.708	0.820445
$153$	0	0
$154$	0	0
$155$	−43.5307	−0.280844
$156$	0	0
$157$	177.329	1.12949	0.564743	−	0.825267i	$-0.308975\pi$
$157$	177.329	1.12949	0.564743	+	0.825267i	$0.308975\pi$
$158$	−18.1333	−0.114768
$159$	0	0
$160$	48.5040i	0.303150i
$161$	21.0373i	0.130667i
$162$	0	0
$163$	70.5026	0.432531	0.216266	−	0.976335i	$-0.430612\pi$
$163$	70.5026	0.432531	0.216266	+	0.976335i	$0.430612\pi$
$164$	1.40692i	0.00857880i
$165$	0	0
$166$	49.8001	0.300000
$167$	− 100.288i	− 0.600525i	−0.953857	−	0.300263i	$-0.902926\pi$
$167$	− 100.288i	− 0.600525i	0.953857	−	0.300263i	$-0.0970743\pi$
$168$	0	0
$169$	−189.708	−1.12253
$170$	−27.7128	−0.163017
$171$	0	0
$172$	− 87.9931i	− 0.511588i
$173$	− 101.472i	− 0.586541i	−0.956029	−	0.293271i	$-0.905256\pi$
$173$	− 101.472i	− 0.586541i	0.956029	−	0.293271i	$-0.0947437\pi$
$174$	0	0
$175$	11.8313i	0.0676077i
$176$	0	0
$177$	0	0
$178$	58.7878i	0.330268i
$179$	71.6462	0.400258	0.200129	−	0.979770i	$-0.435864\pi$
$179$	71.6462	0.400258	0.200129	+	0.979770i	$0.435864\pi$
$180$	0	0
$181$	−66.6359	−0.368154	−0.184077	−	0.982912i	$-0.558930\pi$
$181$	−66.6359	−0.368154	−0.184077	+	0.982912i	$0.558930\pi$
$182$	− 10.1497i	− 0.0557675i
$183$	0	0
$184$	291.502i	1.58425i
$185$	44.2590	0.239238
$186$	0	0
$187$	0	0
$188$	208.728	1.11026
$189$	0	0
$190$	−26.3538	−0.138704
$191$	173.779	0.909840	0.454920	−	0.890532i	$-0.349668\pi$
$191$	173.779	0.909840	0.454920	+	0.890532i	$0.349668\pi$
$192$	0	0
$193$	117.734i	0.610021i	0.952349	+	0.305011i	$0.0986600\pi$
$193$	117.734i	0.610021i	−0.952349	+	0.305011i	$0.901340\pi$
$194$	143.824i	0.741359i
$195$	0	0
$196$	142.697	0.728048
$197$	− 166.768i	− 0.846540i	−0.906004	−	0.423270i	$-0.860882\pi$
$197$	− 166.768i	− 0.846540i	0.906004	−	0.423270i	$-0.139118\pi$
$198$	0	0
$199$	−29.7321	−0.149407	−0.0747036	−	0.997206i	$-0.523801\pi$
$199$	−29.7321	−0.149407	−0.0747036	+	0.997206i	$0.523801\pi$
$200$	163.940i	0.819700i
$201$	0	0
$202$	197.779	0.979106
$203$	22.2872	0.109789
$204$	0	0
$205$	− 0.703462i	− 0.00343152i
$206$	− 66.4010i	− 0.322335i
$207$	0	0
$208$	81.1975i	0.390373i
$209$	0	0
$210$	0	0
$211$	24.5865i	0.116524i	0.998301	+	0.0582618i	$0.0185558\pi$
$211$	24.5865i	0.116524i	−0.998301	+	0.0582618i	$0.981444\pi$
$212$	−133.015	−0.627431
$213$	0	0
$214$	195.426	0.913204
$215$	43.9966i	0.204635i
$216$	0	0
$217$	− 15.3904i	− 0.0709237i
$218$	−124.823	−0.572583
$219$	0	0
$220$	0	0
$221$	−346.277	−1.56686
$222$	0	0
$223$	−159.862	−0.716868	−0.358434	−	0.933555i	$-0.616689\pi$
$223$	−159.862	−0.716868	−0.358434	+	0.933555i	$0.616689\pi$
$224$	−17.1487	−0.0765569
$225$	0	0
$226$	− 166.010i	− 0.734560i
$227$	160.963i	0.709087i	0.935039	+	0.354544i	$0.115364\pi$
$227$	160.963i	0.709087i	−0.935039	+	0.354544i	$0.884636\pi$
$228$	0	0
$229$	19.7898	0.0864182	0.0432091	−	0.999066i	$-0.486242\pi$
$229$	19.7898	0.0864182	0.0432091	+	0.999066i	$0.486242\pi$
$230$	− 61.6016i	− 0.267833i
$231$	0	0
$232$	308.820	1.33112
$233$	− 398.922i	− 1.71211i	−0.516882	−	0.856057i	$-0.672907\pi$
$233$	− 398.922i	− 1.71211i	0.516882	−	0.856057i	$-0.327093\pi$
$234$	0	0
$235$	−104.364	−0.444103
$236$	97.5692	0.413429
$237$	0	0
$238$	− 9.79796i	− 0.0411679i
$239$	− 363.411i	− 1.52055i	−0.649602	−	0.760274i	$-0.725065\pi$
$239$	− 363.411i	− 1.52055i	0.649602	−	0.760274i	$-0.274935\pi$
$240$	0	0
$241$	433.688i	1.79954i	0.436368	+	0.899769i	$0.356265\pi$
$241$	433.688i	1.79954i	−0.436368	+	0.899769i	$0.643735\pi$
$242$	0	0
$243$	0	0
$244$	206.352i	0.845704i
$245$	−71.3487	−0.291219
$246$	0	0
$247$	−329.296	−1.33318
$248$	− 213.256i	− 0.859904i
$249$	0	0
$250$	− 72.5383i	− 0.290153i
$251$	−359.703	−1.43308	−0.716539	−	0.697547i	$-0.754275\pi$
$251$	−359.703	−1.43308	−0.716539	+	0.697547i	$0.754275\pi$
$252$	0	0
$253$	0	0
$254$	116.697	0.459438
$255$	0	0
$256$	−187.405	−0.732051
$257$	95.1769	0.370338	0.185169	−	0.982707i	$-0.440717\pi$
$257$	95.1769	0.370338	0.185169	+	0.982707i	$0.440717\pi$
$258$	0	0
$259$	15.6479i	0.0604167i
$260$	− 81.1975i	− 0.312298i
$261$	0	0
$262$	73.8564	0.281895
$263$	260.641i	0.991032i	0.868599	+	0.495516i	$0.165021\pi$
$263$	260.641i	0.991032i	−0.868599	+	0.495516i	$0.834979\pi$
$264$	0	0
$265$	66.5077	0.250973
$266$	− 9.31749i	− 0.0350281i
$267$	0	0
$268$	−233.779	−0.872311
$269$	−386.631	−1.43729	−0.718644	−	0.695378i	$-0.755237\pi$
$269$	−386.631	−1.43729	−0.718644	+	0.695378i	$0.755237\pi$
$270$	0	0
$271$	− 292.579i	− 1.07963i	−0.841785	−	0.539814i	$-0.818495\pi$
$271$	− 292.579i	− 1.07963i	0.841785	−	0.539814i	$-0.181505\pi$
$272$	78.3837i	0.288175i
$273$	0	0
$274$	85.6651i	0.312646i
$275$	0	0
$276$	0	0
$277$	− 146.940i	− 0.530468i	−0.964184	−	0.265234i	$-0.914551\pi$
$277$	− 146.940i	− 0.530468i	0.964184	−	0.265234i	$-0.0854491\pi$
$278$	−20.3923	−0.0733536
$279$	0	0
$280$	5.43594	0.0194141
$281$	− 180.911i	− 0.643810i	−0.946772	−	0.321905i	$-0.895677\pi$
$281$	− 180.911i	− 0.643810i	0.946772	−	0.321905i	$-0.104323\pi$
$282$	0	0
$283$	254.150i	0.898055i	0.893518	+	0.449028i	$0.148230\pi$
$283$	254.150i	0.898055i	−0.893518	+	0.449028i	$0.851770\pi$
$284$	217.415	0.765547
$285$	0	0
$286$	0	0
$287$	0.248711	0.000866590 0
$288$	0	0
$289$	−45.2769	−0.156667
$290$	−65.2614	−0.225039
$291$	0	0
$292$	6.03384i	0.0206638i
$293$	− 174.104i	− 0.594212i	−0.954844	−	0.297106i	$-0.903978\pi$
$293$	− 174.104i	− 0.594212i	0.954844	−	0.297106i	$-0.0960215\pi$
$294$	0	0
$295$	−48.7846	−0.165372
$296$	216.824i	0.732514i
$297$	0	0
$298$	84.8616	0.284770
$299$	− 769.724i	− 2.57433i
$300$	0	0
$301$	−15.5551	−0.0516782
$302$	−261.695	−0.866540
$303$	0	0
$304$	74.5399i	0.245197i
$305$	− 103.176i	− 0.338282i
$306$	0	0
$307$	− 485.896i	− 1.58272i	−0.611349	−	0.791361i	$-0.709373\pi$
$307$	− 485.896i	− 1.58272i	0.611349	−	0.791361i	$-0.290627\pi$
$308$	0	0
$309$	0	0
$310$	45.0663i	0.145375i
$311$	−43.6359	−0.140308	−0.0701541	−	0.997536i	$-0.522349\pi$
$311$	−43.6359	−0.140308	−0.0701541	+	0.997536i	$0.522349\pi$
$312$	0	0
$313$	496.344	1.58576	0.792881	−	0.609376i	$-0.208580\pi$
$313$	496.344	1.58576	0.792881	+	0.609376i	$0.208580\pi$
$314$	− 183.585i	− 0.584665i
$315$	0	0
$316$	− 51.2887i	− 0.162306i
$317$	193.723	0.611114	0.305557	−	0.952174i	$-0.401157\pi$
$317$	193.723	0.611114	0.305557	+	0.952174i	$0.401157\pi$
$318$	0	0
$319$	0	0
$320$	25.1075	0.0784610
$321$	0	0
$322$	21.7795	0.0676381
$323$	−317.885	−0.984163
$324$	0	0
$325$	− 432.891i	− 1.33197i
$326$	− 72.9896i	− 0.223895i
$327$	0	0
$328$	3.44624	0.0105068
$329$	− 36.8983i	− 0.112153i
$330$	0	0
$331$	−28.1384	−0.0850104	−0.0425052	−	0.999096i	$-0.513534\pi$
$331$	−28.1384	−0.0850104	−0.0425052	+	0.999096i	$0.513534\pi$
$332$	140.856i	0.424265i
$333$	0	0
$334$	−103.825	−0.310855
$335$	116.890	0.348925
$336$	0	0
$337$	337.282i	1.00084i	0.865784	+	0.500418i	$0.166820\pi$
$337$	337.282i	1.00084i	−0.865784	+	0.500418i	$0.833180\pi$
$338$	196.400i	0.581065i
$339$	0	0
$340$	− 78.3837i	− 0.230540i
$341$	0	0
$342$	0	0
$343$	− 50.5898i	− 0.147492i
$344$	−215.538	−0.626565
$345$	0	0
$346$	−105.051	−0.303616
$347$	− 585.178i	− 1.68639i	−0.537607	−	0.843196i	$-0.680671\pi$
$347$	− 585.178i	− 1.68639i	0.537607	−	0.843196i	$-0.319329\pi$
$348$	0	0
$349$	− 28.3341i	− 0.0811864i	−0.999176	−	0.0405932i	$-0.987075\pi$
$349$	− 28.3341i	− 0.0811864i	0.999176	−	0.0405932i	$-0.0129248\pi$
$350$	12.2487	0.0349963
$351$	0	0
$352$	0	0
$353$	−25.0436	−0.0709451	−0.0354726	−	0.999371i	$-0.511294\pi$
$353$	−25.0436	−0.0709451	−0.0354726	+	0.999371i	$0.511294\pi$
$354$	0	0
$355$	−108.708	−0.306219
$356$	−166.277	−0.467070
$357$	0	0
$358$	− 74.1736i	− 0.207189i
$359$	452.737i	1.26111i	0.776146	+	0.630553i	$0.217172\pi$
$359$	452.737i	1.26111i	−0.776146	+	0.630553i	$0.782828\pi$
$360$	0	0
$361$	58.7039	0.162615
$362$	68.9865i	0.190570i
$363$	0	0
$364$	28.7077	0.0788672
$365$	− 3.01692i	− 0.00826553i
$366$	0	0
$367$	188.067	0.512443	0.256222	−	0.966618i	$-0.417522\pi$
$367$	188.067	0.512443	0.256222	+	0.966618i	$0.417522\pi$
$368$	−174.236	−0.473466
$369$	0	0
$370$	− 45.8203i	− 0.123839i
$371$	23.5140i	0.0633801i
$372$	0	0
$373$	87.2506i	0.233916i	0.993137	+	0.116958i	$0.0373142\pi$
$373$	87.2506i	0.233916i	−0.993137	+	0.116958i	$0.962686\pi$
$374$	0	0
$375$	0	0
$376$	− 511.278i	− 1.35978i
$377$	−815.454	−2.16301
$378$	0	0
$379$	−716.123	−1.88951	−0.944753	−	0.327782i	$-0.893699\pi$
$379$	−716.123	−1.88951	−0.944753	+	0.327782i	$0.893699\pi$
$380$	− 74.5399i	− 0.196158i
$381$	0	0
$382$	− 179.910i	− 0.470968i
$383$	−29.0821	−0.0759324	−0.0379662	−	0.999279i	$-0.512088\pi$
$383$	−29.0821	−0.0759324	−0.0379662	+	0.999279i	$0.512088\pi$
$384$	0	0
$385$	0	0
$386$	121.887	0.315770
$387$	0	0
$388$	−406.795	−1.04844
$389$	−191.541	−0.492393	−0.246197	−	0.969220i	$-0.579181\pi$
$389$	−191.541	−0.492393	−0.246197	+	0.969220i	$0.579181\pi$
$390$	0	0
$391$	− 743.049i	− 1.90038i
$392$	− 349.536i	− 0.891673i
$393$	0	0
$394$	−172.651	−0.438201
$395$	25.6443i	0.0649224i
$396$	0	0
$397$	−669.420	−1.68620	−0.843099	−	0.537759i	$-0.819271\pi$
$397$	−669.420	−1.68620	−0.843099	+	0.537759i	$0.819271\pi$
$398$	30.7809i	0.0773389i
$399$	0	0
$400$	−97.9897	−0.244974
$401$	286.851	0.715340	0.357670	−	0.933848i	$-0.383571\pi$
$401$	286.851	0.715340	0.357670	+	0.933848i	$0.383571\pi$
$402$	0	0
$403$	563.113i	1.39730i
$404$	559.405i	1.38467i
$405$	0	0
$406$	− 23.0734i	− 0.0568310i
$407$	0	0
$408$	0	0
$409$	− 460.462i	− 1.12582i	−0.826517	−	0.562911i	$-0.809681\pi$
$409$	− 460.462i	− 1.12582i	0.826517	−	0.562911i	$-0.190319\pi$
$410$	−0.728277	−0.00177629
$411$	0	0
$412$	187.810	0.455850
$413$	− 17.2480i	− 0.0417626i
$414$	0	0
$415$	− 70.4279i	− 0.169706i
$416$	627.446	1.50828
$417$	0	0
$418$	0	0
$419$	487.520	1.16353	0.581767	−	0.813356i	$-0.302362\pi$
$419$	487.520	1.16353	0.581767	+	0.813356i	$0.302362\pi$
$420$	0	0
$421$	−119.979	−0.284987	−0.142493	−	0.989796i	$-0.545512\pi$
$421$	−119.979	−0.284987	−0.142493	+	0.989796i	$0.545512\pi$
$422$	25.4538	0.0603170
$423$	0	0
$424$	325.820i	0.768443i
$425$	− 417.889i	− 0.983269i
$426$	0	0
$427$	36.4782	0.0854290
$428$	552.747i	1.29147i
$429$	0	0
$430$	45.5486	0.105927
$431$	285.216i	0.661754i	0.943674	+	0.330877i	$0.107345\pi$
$431$	285.216i	0.661754i	−0.943674	+	0.330877i	$0.892655\pi$
$432$	0	0
$433$	405.928	0.937479	0.468739	−	0.883337i	$-0.344708\pi$
$433$	405.928	0.937479	0.468739	+	0.883337i	$0.344708\pi$
$434$	−15.9334	−0.0367128
$435$	0	0
$436$	− 353.053i	− 0.809755i
$437$	− 706.612i	− 1.61696i
$438$	0	0
$439$	473.244i	1.07800i	0.842304	+	0.539002i	$0.181199\pi$
$439$	473.244i	1.07800i	−0.842304	+	0.539002i	$0.818801\pi$
$440$	0	0
$441$	0	0
$442$	358.492i	0.811068i
$443$	−110.382	−0.249169	−0.124585	−	0.992209i	$-0.539760\pi$
$443$	−110.382	−0.249169	−0.124585	+	0.992209i	$0.539760\pi$
$444$	0	0
$445$	83.1384	0.186828
$446$	165.501i	0.371078i
$447$	0	0
$448$	8.87685i	0.0198144i
$449$	−186.192	−0.414682	−0.207341	−	0.978269i	$-0.566481\pi$
$449$	−186.192	−0.414682	−0.207341	+	0.978269i	$0.566481\pi$
$450$	0	0
$451$	0	0
$452$	469.549	1.03882
$453$	0	0
$454$	166.641	0.367051
$455$	−14.3538	−0.0315469
$456$	0	0
$457$	679.450i	1.48676i	0.668869	+	0.743380i	$0.266779\pi$
$457$	679.450i	1.48676i	−0.668869	+	0.743380i	$0.733221\pi$
$458$	− 20.4879i	− 0.0447333i
$459$	0	0
$460$	174.236	0.378773
$461$	− 664.834i	− 1.44216i	−0.692853	−	0.721078i	$-0.743647\pi$
$461$	− 664.834i	− 1.44216i	0.692853	−	0.721078i	$-0.256353\pi$
$462$	0	0
$463$	−736.774	−1.59131	−0.795653	−	0.605753i	$-0.792872\pi$
$463$	−736.774	−1.59131	−0.795653	+	0.605753i	$0.792872\pi$
$464$	184.587i	0.397817i
$465$	0	0
$466$	−412.995	−0.886255
$467$	−717.031	−1.53540	−0.767699	−	0.640811i	$-0.778598\pi$
$467$	−717.031	−1.53540	−0.767699	+	0.640811i	$0.778598\pi$
$468$	0	0
$469$	41.3268i	0.0881168i
$470$	108.046i	0.229884i
$471$	0	0
$472$	− 238.995i	− 0.506345i
$473$	0	0
$474$	0	0
$475$	− 397.397i	− 0.836625i
$476$	27.7128	0.0582202
$477$	0	0
$478$	−376.231	−0.787094
$479$	− 40.0386i	− 0.0835879i	−0.999126	−	0.0417940i	$-0.986693\pi$
$479$	− 40.0386i	− 0.0835879i	0.999126	−	0.0417940i	$-0.0133073\pi$
$480$	0	0
$481$	− 572.534i	− 1.19030i
$482$	448.987	0.931509
$483$	0	0
$484$	0	0
$485$	203.397	0.419376
$486$	0	0
$487$	220.785	0.453356	0.226678	−	0.973970i	$-0.427213\pi$
$487$	220.785	0.453356	0.226678	+	0.973970i	$0.427213\pi$
$488$	505.457	1.03577
$489$	0	0
$490$	73.8656i	0.150746i
$491$	697.557i	1.42069i	0.703855	+	0.710343i	$0.251460\pi$
$491$	697.557i	1.42069i	−0.703855	+	0.710343i	$0.748540\pi$
$492$	0	0
$493$	−787.195	−1.59674
$494$	340.912i	0.690106i
$495$	0	0
$496$	127.467	0.256990
$497$	− 38.4340i	− 0.0773319i
$498$	0	0
$499$	453.347	0.908512	0.454256	−	0.890871i	$-0.349905\pi$
$499$	453.347	0.908512	0.454256	+	0.890871i	$0.349905\pi$
$500$	205.169	0.410339
$501$	0	0
$502$	372.391i	0.741816i
$503$	− 118.428i	− 0.235443i	−0.993047	−	0.117721i	$-0.962441\pi$
$503$	− 118.428i	− 0.235443i	0.993047	−	0.117721i	$-0.0375589\pi$
$504$	0	0
$505$	− 279.702i	− 0.553866i
$506$	0	0
$507$	0	0
$508$	330.070i	0.649744i
$509$	−221.551	−0.435268	−0.217634	−	0.976030i	$-0.569834\pi$
$509$	−221.551	−0.435268	−0.217634	+	0.976030i	$0.569834\pi$
$510$	0	0
$511$	1.06664	0.00208736
$512$	− 265.031i	− 0.517638i
$513$	0	0
$514$	− 98.5344i	− 0.191701i
$515$	−93.9052	−0.182340
$516$	0	0
$517$	0	0
$518$	16.1999	0.0312740
$519$	0	0
$520$	−198.892	−0.382486
$521$	20.0179	0.0384220	0.0192110	−	0.999815i	$-0.493885\pi$
$521$	20.0179	0.0384220	0.0192110	+	0.999815i	$0.493885\pi$
$522$	0	0
$523$	− 365.707i	− 0.699249i	−0.936890	−	0.349625i	$-0.886309\pi$
$523$	− 365.707i	− 0.699249i	0.936890	−	0.349625i	$-0.113691\pi$
$524$	208.897i	0.398659i
$525$	0	0
$526$	269.836	0.512996
$527$	543.598i	1.03150i
$528$	0	0
$529$	1122.69	2.12229
$530$	− 68.8539i	− 0.129913i
$531$	0	0
$532$	26.3538	0.0495373
$533$	−9.09996	−0.0170731
$534$	0	0
$535$	− 276.374i	− 0.516586i
$536$	572.640i	1.06836i
$537$	0	0
$538$	400.270i	0.743995i
$539$	0	0
$540$	0	0
$541$	− 69.4451i	− 0.128364i	−0.997938	−	0.0641822i	$-0.979556\pi$
$541$	− 69.4451i	− 0.128364i	0.997938	−	0.0641822i	$-0.0204439\pi$
$542$	−302.900	−0.558856
$543$	0	0
$544$	605.703	1.11342
$545$	176.527i	0.323902i
$546$	0	0
$547$	518.324i	0.947575i	0.880639	+	0.473788i	$0.157114\pi$
$547$	518.324i	0.947575i	−0.880639	+	0.473788i	$0.842886\pi$
$548$	−242.297	−0.442149
$549$	0	0
$550$	0	0
$551$	−748.592	−1.35861
$552$	0	0
$553$	−9.06664	−0.0163954
$554$	−152.123	−0.274590
$555$	0	0
$556$	− 57.6781i	− 0.103738i
$557$	− 258.605i	− 0.464282i	−0.972682	−	0.232141i	$-0.925427\pi$
$557$	− 258.605i	− 0.464282i	0.972682	−	0.232141i	$-0.0745731\pi$
$558$	0	0
$559$	569.138	1.01814
$560$	3.24915i	0.00580206i
$561$	0	0
$562$	−187.292	−0.333260
$563$	− 111.527i	− 0.198094i	−0.995083	−	0.0990471i	$-0.968421\pi$
$563$	− 111.527i	− 0.198094i	0.995083	−	0.0990471i	$-0.0315795\pi$
$564$	0	0
$565$	−234.774	−0.415530
$566$	263.115	0.464868
$567$	0	0
$568$	− 532.557i	− 0.937600i
$569$	− 429.624i	− 0.755050i	−0.925999	−	0.377525i	$-0.876775\pi$
$569$	− 429.624i	− 0.755050i	0.925999	−	0.377525i	$-0.123225\pi$
$570$	0	0
$571$	− 51.0504i	− 0.0894052i	−0.999000	−	0.0447026i	$-0.985766\pi$
$571$	− 51.0504i	− 0.0894052i	0.999000	−	0.0447026i	$-0.0142340\pi$
$572$	0	0
$573$	0	0
$574$	− 0.257485i	0 0.000448580i
$575$	928.908	1.61549
$576$	0	0
$577$	78.9756	0.136873	0.0684364	−	0.997655i	$-0.478199\pi$
$577$	78.9756	0.136873	0.0684364	+	0.997655i	$0.478199\pi$
$578$	46.8741i	0.0810970i
$579$	0	0
$580$	− 184.587i	− 0.318254i
$581$	24.9000	0.0428572
$582$	0	0
$583$	0	0
$584$	14.7798	0.0253079
$585$	0	0
$586$	−180.246	−0.307587
$587$	215.195	0.366601	0.183300	−	0.983057i	$-0.441322\pi$
$587$	215.195	0.366601	0.183300	+	0.983057i	$0.441322\pi$
$588$	0	0
$589$	516.941i	0.877659i
$590$	50.5055i	0.0856026i
$591$	0	0
$592$	−129.599	−0.218918
$593$	− 1035.06i	− 1.74547i	−0.488198	−	0.872733i	$-0.662346\pi$
$593$	− 1035.06i	− 1.74547i	0.488198	−	0.872733i	$-0.337654\pi$
$594$	0	0
$595$	−13.8564	−0.0232881
$596$	240.025i	0.402726i
$597$	0	0
$598$	−796.877	−1.33257
$599$	494.936	0.826270	0.413135	−	0.910670i	$-0.364434\pi$
$599$	494.936	0.826270	0.413135	+	0.910670i	$0.364434\pi$
$600$	0	0
$601$	841.844i	1.40074i	0.713781	+	0.700369i	$0.246981\pi$
$601$	841.844i	1.40074i	−0.713781	+	0.700369i	$0.753019\pi$
$602$	16.1039i	0.0267506i
$603$	0	0
$604$	− 740.185i	− 1.22547i
$605$	0	0
$606$	0	0
$607$	758.377i	1.24939i	0.780870	+	0.624693i	$0.214776\pi$
$607$	758.377i	1.24939i	−0.780870	+	0.624693i	$0.785224\pi$
$608$	576.000	0.947368
$609$	0	0
$610$	−106.816	−0.175107
$611$	1350.05i	2.20958i
$612$	0	0
$613$	436.987i	0.712867i	0.934321	+	0.356433i	$0.116007\pi$
$613$	436.987i	0.712867i	−0.934321	+	0.356433i	$0.883993\pi$
$614$	−503.036	−0.819277
$615$	0	0
$616$	0	0
$617$	−927.769	−1.50368	−0.751839	−	0.659347i	$-0.770833\pi$
$617$	−927.769	−1.50368	−0.751839	+	0.659347i	$0.770833\pi$
$618$	0	0
$619$	1130.97	1.82710	0.913549	−	0.406728i	$-0.133330\pi$
$619$	1130.97	1.82710	0.913549	+	0.406728i	$0.133330\pi$
$620$	−127.467	−0.205592
$621$	0	0
$622$	45.1752i	0.0726289i
$623$	29.3939i	0.0471812i
$624$	0	0
$625$	468.825	0.750121
$626$	− 513.853i	− 0.820851i
$627$	0	0
$628$	519.257	0.826842
$629$	− 552.693i	− 0.878685i
$630$	0	0
$631$	615.138	0.974863	0.487431	−	0.873161i	$-0.337934\pi$
$631$	615.138	0.974863	0.487431	+	0.873161i	$0.337934\pi$
$632$	−125.631	−0.198783
$633$	0	0
$634$	− 200.557i	− 0.316336i
$635$	− 165.035i	− 0.259898i
$636$	0	0
$637$	922.965i	1.44892i
$638$	0	0
$639$	0	0
$640$	168.023i	0.262536i
$641$	982.200	1.53229	0.766147	−	0.642666i	$-0.222172\pi$
$641$	982.200	1.53229	0.766147	+	0.642666i	$0.222172\pi$
$642$	0	0
$643$	−14.6359	−0.0227618	−0.0113809	−	0.999935i	$-0.503623\pi$
$643$	−14.6359	−0.0227618	−0.0113809	+	0.999935i	$0.503623\pi$
$644$	61.6016i	0.0956547i
$645$	0	0
$646$	329.098i	0.509440i
$647$	−272.000	−0.420402	−0.210201	−	0.977658i	$-0.567412\pi$
$647$	−272.000	−0.420402	−0.210201	+	0.977658i	$0.567412\pi$
$648$	0	0
$649$	0	0
$650$	−448.161	−0.689479
$651$	0	0
$652$	206.446	0.316635
$653$	−173.797	−0.266152	−0.133076	−	0.991106i	$-0.542485\pi$
$653$	−173.797	−0.266152	−0.133076	+	0.991106i	$0.542485\pi$
$654$	0	0
$655$	− 104.449i	− 0.159464i
$656$	2.05988i	0.00314006i
$657$	0	0
$658$	−38.1999	−0.0580546
$659$	506.577i	0.768705i	0.923186	+	0.384352i	$0.125575\pi$
$659$	506.577i	0.768705i	−0.923186	+	0.384352i	$0.874425\pi$
$660$	0	0
$661$	1203.30	1.82042	0.910211	−	0.414146i	$-0.135920\pi$
$661$	1203.30	1.82042	0.910211	+	0.414146i	$0.135920\pi$
$662$	29.1311i	0.0440046i
$663$	0	0
$664$	345.025	0.519616
$665$	−13.1769	−0.0198149
$666$	0	0
$667$	− 1749.82i	− 2.62342i
$668$	− 293.663i	− 0.439615i
$669$	0	0
$670$	− 121.013i	− 0.180617i
$671$	0	0
$672$	0	0
$673$	95.9462i	0.142565i	0.997456	+	0.0712825i	$0.0227092\pi$
$673$	95.9462i	0.142565i	−0.997456	+	0.0712825i	$0.977291\pi$
$674$	349.180	0.518071
$675$	0	0
$676$	−555.503	−0.821749
$677$	30.7862i	0.0454745i	0.999741	+	0.0227372i	$0.00723811\pi$
$677$	30.7862i	0.0454745i	−0.999741	+	0.0227372i	$0.992762\pi$
$678$	0	0
$679$	71.9119i	0.105908i
$680$	−192.000	−0.282353
$681$	0	0
$682$	0	0
$683$	63.6537	0.0931972	0.0465986	−	0.998914i	$-0.485162\pi$
$683$	63.6537	0.0931972	0.0465986	+	0.998914i	$0.485162\pi$
$684$	0	0
$685$	121.149	0.176859
$686$	−52.3744	−0.0763476
$687$	0	0
$688$	− 128.831i	− 0.187254i
$689$	− 860.342i	− 1.24868i
$690$	0	0
$691$	−217.000	−0.314038	−0.157019	−	0.987596i	$-0.550188\pi$
$691$	−217.000	−0.314038	−0.157019	+	0.987596i	$0.550188\pi$
$692$	− 297.130i	− 0.429378i
$693$	0	0
$694$	−605.821	−0.872940
$695$	28.8391i	0.0414951i
$696$	0	0
$697$	−8.78461	−0.0126035
$698$	−29.3336	−0.0420252
$699$	0	0
$700$	34.6446i	0.0494923i
$701$	− 155.816i	− 0.222276i	−0.993805	−	0.111138i	$-0.964550\pi$
$701$	− 155.816i	− 0.222276i	0.993805	−	0.111138i	$-0.0354496\pi$
$702$	0	0
$703$	− 525.590i	− 0.747638i
$704$	0	0
$705$	0	0
$706$	25.9271i	0.0367239i
$707$	98.8897	0.139872
$708$	0	0
$709$	1164.09	1.64188	0.820939	−	0.571015i	$-0.193450\pi$
$709$	1164.09	1.64188	0.820939	+	0.571015i	$0.193450\pi$
$710$	112.542i	0.158510i
$711$	0	0
$712$	407.294i	0.572041i
$713$	−1208.34	−1.69473
$714$	0	0
$715$	0	0
$716$	209.795	0.293009
$717$	0	0
$718$	468.708	0.652796
$719$	872.795	1.21390	0.606951	−	0.794740i	$-0.292393\pi$
$719$	872.795	1.21390	0.606951	+	0.794740i	$0.292393\pi$
$720$	0	0
$721$	− 33.2005i	− 0.0460478i
$722$	− 60.7747i	− 0.0841755i
$723$	0	0
$724$	−195.123	−0.269507
$725$	− 984.095i	− 1.35737i
$726$	0	0
$727$	−155.292	−0.213607	−0.106804	−	0.994280i	$-0.534062\pi$
$727$	−155.292	−0.213607	−0.106804	+	0.994280i	$0.534062\pi$
$728$	− 70.3191i	− 0.0965922i
$729$	0	0
$730$	−3.12334	−0.00427855
$731$	549.415	0.751594
$732$	0	0
$733$	− 338.483i	− 0.461777i	−0.972980	−	0.230889i	$-0.925837\pi$
$733$	− 338.483i	− 0.461777i	0.972980	−	0.230889i	$-0.0741633\pi$
$734$	− 194.701i	− 0.265260i
$735$	0	0
$736$	1346.39i	1.82933i
$737$	0	0
$738$	0	0
$739$	1138.70i	1.54087i	0.637521	+	0.770433i	$0.279960\pi$
$739$	1138.70i	1.54087i	−0.637521	+	0.770433i	$0.720040\pi$
$740$	129.599	0.175134
$741$	0	0
$742$	24.3435	0.0328080
$743$	− 307.566i	− 0.413951i	−0.978346	−	0.206976i	$-0.933638\pi$
$743$	− 307.566i	− 0.413951i	0.978346	−	0.206976i	$-0.0663621\pi$
$744$	0	0
$745$	− 120.012i	− 0.161090i
$746$	90.3284	0.121084
$747$	0	0
$748$	0	0
$749$	97.7128	0.130458
$750$	0	0
$751$	−760.240	−1.01230	−0.506152	−	0.862444i	$-0.668932\pi$
$751$	−760.240	−1.01230	−0.506152	+	0.862444i	$0.668932\pi$
$752$	305.599	0.406382
$753$	0	0
$754$	844.220i	1.11966i
$755$	370.093i	0.490189i
$756$	0	0
$757$	−436.471	−0.576579	−0.288290	−	0.957543i	$-0.593087\pi$
$757$	−436.471	−0.576579	−0.288290	+	0.957543i	$0.593087\pi$
$758$	741.385i	0.978081i
$759$	0	0
$760$	−182.585	−0.240243
$761$	1273.44i	1.67338i	0.547678	+	0.836689i	$0.315512\pi$
$761$	1273.44i	1.67338i	−0.547678	+	0.836689i	$0.684488\pi$
$762$	0	0
$763$	−62.4115	−0.0817976
$764$	508.862	0.666049
$765$	0	0
$766$	30.1080i	0.0393055i
$767$	631.076i	0.822785i
$768$	0	0
$769$	− 1228.99i	− 1.59816i	−0.601223	−	0.799081i	$-0.705320\pi$
$769$	− 1228.99i	− 1.59816i	0.601223	−	0.799081i	$-0.294680\pi$
$770$	0	0
$771$	0	0
$772$	344.749i	0.446567i
$773$	−180.218	−0.233141	−0.116570	−	0.993182i	$-0.537190\pi$
$773$	−180.218	−0.233141	−0.116570	+	0.993182i	$0.537190\pi$
$774$	0	0
$775$	−679.568	−0.876862
$776$	996.440i	1.28407i
$777$	0	0
$778$	198.298i	0.254882i
$779$	−8.35383	−0.0107238
$780$	0	0
$781$	0	0
$782$	−769.261	−0.983710
$783$	0	0
$784$	208.923	0.266484
$785$	−259.628	−0.330737
$786$	0	0
$787$	1029.91i	1.30865i	0.756214	+	0.654324i	$0.227047\pi$
$787$	1029.91i	1.30865i	−0.756214	+	0.654324i	$0.772953\pi$
$788$	− 488.332i	− 0.619710i
$789$	0	0
$790$	26.5490	0.0336063
$791$	− 83.0052i	− 0.104937i
$792$	0	0
$793$	−1334.68	−1.68308
$794$	693.035i	0.872840i
$795$	0	0
$796$	−87.0615	−0.109374
$797$	82.2872	0.103246	0.0516231	−	0.998667i	$-0.483561\pi$
$797$	82.2872	0.103246	0.0516231	+	0.998667i	$0.483561\pi$
$798$	0	0
$799$	1303.27i	1.63112i
$800$	757.206i	0.946508i
$801$	0	0
$802$	− 296.970i	− 0.370287i
$803$	0	0
$804$	0	0
$805$	− 30.8008i	− 0.0382619i
$806$	582.977	0.723297
$807$	0	0
$808$	1370.26	1.69586
$809$	− 152.720i	− 0.188777i	−0.995535	−	0.0943884i	$-0.969910\pi$
$809$	− 152.720i	− 0.188777i	0.995535	−	0.0943884i	$-0.0300896\pi$
$810$	0	0
$811$	− 633.536i	− 0.781179i	−0.920565	−	0.390589i	$-0.872271\pi$
$811$	− 633.536i	− 0.781179i	0.920565	−	0.390589i	$-0.127729\pi$
$812$	65.2614	0.0803712
$813$	0	0
$814$	0	0
$815$	−103.223	−0.126654
$816$	0	0
$817$	522.473	0.639502
$818$	−476.705	−0.582769
$819$	0	0
$820$	− 2.05988i	− 0.00251205i
$821$	1213.67i	1.47828i	0.673551	+	0.739141i	$0.264768\pi$
$821$	1213.67i	1.47828i	−0.673551	+	0.739141i	$0.735232\pi$
$822$	0	0
$823$	−775.077	−0.941770	−0.470885	−	0.882195i	$-0.656065\pi$
$823$	−775.077	−0.941770	−0.470885	+	0.882195i	$0.656065\pi$
$824$	− 460.040i	− 0.558300i
$825$	0	0
$826$	−17.8564	−0.0216179
$827$	132.470i	0.160182i	0.996788	+	0.0800908i	$0.0255210\pi$
$827$	132.470i	0.160182i	−0.996788	+	0.0800908i	$0.974479\pi$
$828$	0	0
$829$	−623.133	−0.751669	−0.375834	−	0.926687i	$-0.622644\pi$
$829$	−623.133	−0.751669	−0.375834	+	0.926687i	$0.622644\pi$
$830$	−72.9124	−0.0878462
$831$	0	0
$832$	− 324.790i	− 0.390373i
$833$	890.980i	1.06960i
$834$	0	0
$835$	146.831i	0.175846i
$836$	0	0
$837$	0	0
$838$	− 504.718i	− 0.602289i
$839$	789.051	0.940466	0.470233	−	0.882542i	$-0.344170\pi$
$839$	789.051	0.940466	0.470233	+	0.882542i	$0.344170\pi$
$840$	0	0
$841$	−1012.78	−1.20426
$842$	124.212i	0.147520i
$843$	0	0
$844$	71.9942i	0.0853012i
$845$	277.751	0.328700
$846$	0	0
$847$	0	0
$848$	−194.748	−0.229656
$849$	0	0
$850$	−432.631	−0.508977
$851$	1228.56	1.44366
$852$	0	0
$853$	− 281.101i	− 0.329544i	−0.986332	−	0.164772i	$-0.947311\pi$
$853$	− 281.101i	− 0.329544i	0.986332	−	0.164772i	$-0.0526889\pi$
$854$	− 37.7650i	− 0.0442213i
$855$	0	0
$856$	1353.95	1.58172
$857$	− 581.651i	− 0.678706i	−0.940659	−	0.339353i	$-0.889792\pi$
$857$	− 581.651i	− 0.678706i	0.940659	−	0.339353i	$-0.110208\pi$
$858$	0	0
$859$	−466.733	−0.543345	−0.271673	−	0.962390i	$-0.587577\pi$
$859$	−466.733	−0.543345	−0.271673	+	0.962390i	$0.587577\pi$
$860$	128.831i	0.149803i
$861$	0	0
$862$	295.277	0.342549
$863$	1206.35	1.39786	0.698929	−	0.715191i	$-0.253660\pi$
$863$	1206.35	1.39786	0.698929	+	0.715191i	$0.253660\pi$
$864$	0	0
$865$	148.565i	0.171751i
$866$	− 420.248i	− 0.485275i
$867$	0	0
$868$	− 45.0663i	− 0.0519197i
$869$	0	0
$870$	0	0
$871$	− 1512.08i	− 1.73603i
$872$	−864.800	−0.991743
$873$	0	0
$874$	−731.538	−0.837000
$875$	− 36.2692i	− 0.0414505i
$876$	0	0
$877$	− 702.830i	− 0.801403i	−0.916209	−	0.400701i	$-0.868766\pi$
$877$	− 702.830i	− 0.801403i	0.916209	−	0.400701i	$-0.131234\pi$
$878$	489.938	0.558016
$879$	0	0
$880$	0	0
$881$	704.831	0.800035	0.400017	−	0.916508i	$-0.369004\pi$
$881$	704.831	0.800035	0.400017	+	0.916508i	$0.369004\pi$
$882$	0	0
$883$	117.481	0.133047	0.0665235	−	0.997785i	$-0.478809\pi$
$883$	117.481	0.133047	0.0665235	+	0.997785i	$0.478809\pi$
$884$	−1013.97	−1.14702
$885$	0	0
$886$	114.276i	0.128980i
$887$	192.313i	0.216813i	0.994107	+	0.108406i	$0.0345748\pi$
$887$	192.313i	0.216813i	−0.994107	+	0.108406i	$0.965425\pi$
$888$	0	0
$889$	58.3487	0.0656341
$890$	− 86.0712i	− 0.0967093i
$891$	0	0
$892$	−468.107	−0.524784
$893$	1239.36i	1.38786i
$894$	0	0
$895$	−104.897	−0.117204
$896$	−59.4050	−0.0663002
$897$	0	0
$898$	192.761i	0.214655i
$899$	1280.13i	1.42395i
$900$	0	0
$901$	− 830.528i	− 0.921784i
$902$	0	0
$903$	0	0
$904$	− 1150.15i	− 1.27229i
$905$	97.5617	0.107803
$906$	0	0
$907$	1508.77	1.66347	0.831736	−	0.555171i	$-0.187347\pi$
$907$	1508.77	1.66347	0.831736	+	0.555171i	$0.187347\pi$
$908$	471.332i	0.519088i
$909$	0	0
$910$	14.8602i	0.0163299i
$911$	1157.04	1.27007	0.635037	−	0.772481i	$-0.280985\pi$
$911$	1157.04	1.27007	0.635037	+	0.772481i	$0.280985\pi$
$912$	0	0
$913$	0	0
$914$	703.418	0.769604
$915$	0	0
$916$	57.9485	0.0632625
$917$	36.9282	0.0402707
$918$	0	0
$919$	− 1126.79i	− 1.22611i	−0.790042	−	0.613053i	$-0.789941\pi$
$919$	− 1126.79i	− 1.22611i	0.790042	−	0.613053i	$-0.210059\pi$
$920$	− 426.788i	− 0.463900i
$921$	0	0
$922$	−688.287	−0.746515
$923$	1406.24i	1.52355i
$924$	0	0
$925$	690.937	0.746959
$926$	762.765i	0.823720i
$927$	0	0
$928$	1426.38	1.53705
$929$	−394.462	−0.424609	−0.212304	−	0.977204i	$-0.568097\pi$
$929$	−394.462	−0.424609	−0.212304	+	0.977204i	$0.568097\pi$
$930$	0	0
$931$	847.288i	0.910084i
$932$	− 1168.13i	− 1.25335i
$933$	0	0
$934$	742.325i	0.794780i
$935$	0	0
$936$	0	0
$937$	1373.13i	1.46545i	0.680523	+	0.732727i	$0.261753\pi$
$937$	1373.13i	1.46545i	−0.680523	+	0.732727i	$0.738247\pi$
$938$	42.7846	0.0456126
$939$	0	0
$940$	−305.599	−0.325106
$941$	1662.55i	1.76679i	0.468627	+	0.883396i	$0.344749\pi$
$941$	1662.55i	1.76679i	−0.468627	+	0.883396i	$0.655251\pi$
$942$	0	0
$943$	− 19.5269i	− 0.0207072i
$944$	142.851	0.151325
$945$	0	0
$946$	0	0
$947$	1058.43	1.11766	0.558832	−	0.829281i	$-0.311250\pi$
$947$	1058.43	1.11766	0.558832	+	0.829281i	$0.311250\pi$
$948$	0	0
$949$	−39.0268	−0.0411241
$950$	−411.415	−0.433069
$951$	0	0
$952$	− 67.8823i	− 0.0713049i
$953$	− 1108.44i	− 1.16310i	−0.813510	−	0.581551i	$-0.802446\pi$
$953$	− 1108.44i	− 1.16310i	0.813510	−	0.581551i	$-0.197554\pi$
$954$	0	0
$955$	−254.431	−0.266420
$956$	− 1064.14i	− 1.11312i
$957$	0	0
$958$	−41.4510	−0.0432683
$959$	42.8326i	0.0446638i
$960$	0	0
$961$	−77.0052	−0.0801302
$962$	−592.730	−0.616144
$963$	0	0
$964$	1269.93i	1.31735i
$965$	− 172.375i	− 0.178627i
$966$	0	0
$967$	− 1243.66i	− 1.28611i	−0.765822	−	0.643053i	$-0.777668\pi$
$967$	− 1243.66i	− 1.28611i	0.765822	−	0.643053i	$-0.222332\pi$
$968$	0	0
$969$	0	0
$970$	− 210.573i	− 0.217085i
$971$	108.466	0.111706	0.0558530	−	0.998439i	$-0.482212\pi$
$971$	108.466	0.111706	0.0558530	+	0.998439i	$0.482212\pi$
$972$	0	0
$973$	−10.1962	−0.0104791
$974$	− 228.573i	− 0.234675i
$975$	0	0
$976$	302.120i	0.309549i
$977$	496.108	0.507787	0.253893	−	0.967232i	$-0.418289\pi$
$977$	496.108	0.507787	0.253893	+	0.967232i	$0.418289\pi$
$978$	0	0
$979$	0	0
$980$	−208.923	−0.213187
$981$	0	0
$982$	722.164	0.735401
$983$	−145.733	−0.148254	−0.0741269	−	0.997249i	$-0.523617\pi$
$983$	−145.733	−0.148254	−0.0741269	+	0.997249i	$0.523617\pi$
$984$	0	0
$985$	244.166i	0.247884i
$986$	814.964i	0.826535i
$987$	0	0
$988$	−964.246	−0.975957
$989$	1221.27i	1.23485i
$990$	0	0
$991$	−345.149	−0.348283	−0.174142	−	0.984721i	$-0.555715\pi$
$991$	−345.149	−0.348283	−0.174142	+	0.984721i	$0.555715\pi$
$992$	− 984.988i	− 0.992932i
$993$	0	0
$994$	−39.7898	−0.0400299
$995$	43.5307	0.0437495
$996$	0	0
$997$	918.997i	0.921762i	0.887462	+	0.460881i	$0.152466\pi$
$997$	918.997i	0.921762i	−0.887462	+	0.460881i	$0.847534\pi$
$998$	− 469.340i	− 0.470280i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.c.604.2		4
3.2	odd	2		363.3.c.a.241.3	yes	4
11.10	odd	2	inner	1089.3.c.c.604.3		4
33.2	even	10		363.3.g.d.40.2		16
33.5	odd	10		363.3.g.d.118.2		16
33.8	even	10		363.3.g.d.112.2		16
33.14	odd	10		363.3.g.d.112.3		16
33.17	even	10		363.3.g.d.118.3		16
33.20	odd	10		363.3.g.d.40.3		16
33.26	odd	10		363.3.g.d.94.2		16
33.29	even	10		363.3.g.d.94.3		16
33.32	even	2		363.3.c.a.241.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.a.241.2	✓	4	33.32	even	2
363.3.c.a.241.3	yes	4	3.2	odd	2
363.3.g.d.40.2		16	33.2	even	10
363.3.g.d.40.3		16	33.20	odd	10
363.3.g.d.94.2		16	33.26	odd	10
363.3.g.d.94.3		16	33.29	even	10
363.3.g.d.112.2		16	33.8	even	10
363.3.g.d.112.3		16	33.14	odd	10
363.3.g.d.118.2		16	33.5	odd	10
363.3.g.d.118.3		16	33.17	even	10
1089.3.c.c.604.2		4	1.1	even	1	trivial
1089.3.c.c.604.3		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)

\(f(q)\)	\(=\)	\(q-1.03528i q^{2} +2.92820 q^{4} -1.46410 q^{5} -0.517638i q^{7} -7.17260i q^{8} +O(q^{10})\)	\(q-1.03528i q^{2} +2.92820 q^{4} -1.46410 q^{5} -0.517638i q^{7} -7.17260i q^{8} +1.51575i q^{10} +18.9396i q^{13} -0.535898 q^{14} +4.28719 q^{16} +18.2832i q^{17} +17.3867i q^{19} -4.28719 q^{20} -40.6410 q^{23} -22.8564 q^{25} +19.6077 q^{26} -1.51575i q^{28} +43.0555i q^{29} +29.7321 q^{31} -33.1288i q^{32} +18.9282 q^{34} +0.757875i q^{35} -30.2295 q^{37} +18.0000 q^{38} +10.5014i q^{40} +0.480473i q^{41} -30.0502i q^{43} +42.0747i q^{46} +71.2820 q^{47} +48.7321 q^{49} +23.6627i q^{50} +55.4589i q^{52} -45.4256 q^{53} -3.71281 q^{56} +44.5744 q^{58} +33.3205 q^{59} +70.4704i q^{61} -30.7809i q^{62} -17.1487 q^{64} -27.7295i q^{65} -79.8372 q^{67} +53.5370i q^{68} +0.784610 q^{70} +74.2487 q^{71} +2.06059i q^{73} +31.2959i q^{74} +50.9117i q^{76} -17.5154i q^{79} -6.27688 q^{80} +0.497423 q^{82} +48.1032i q^{83} -26.7685i q^{85} -31.1103 q^{86} -56.7846 q^{89} +9.80385 q^{91} -119.005 q^{92} -73.7966i q^{94} -25.4558i q^{95} -138.923 q^{97} -50.4511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 8 q^{5}+O(q^{10}) \) 4 * q - 16 * q^4 + 8 * q^5	\( 4 q - 16 q^{4} + 8 q^{5} - 16 q^{14} + 128 q^{16} - 128 q^{20} - 24 q^{23} - 36 q^{25} + 120 q^{26} + 112 q^{31} + 48 q^{34} + 80 q^{37} + 72 q^{38} + 8 q^{47} + 188 q^{49} + 40 q^{53} + 96 q^{56} + 400 q^{58} + 64 q^{59} - 512 q^{64} - 160 q^{67} - 80 q^{70} + 200 q^{71} + 640 q^{80} - 192 q^{82} - 360 q^{86} - 144 q^{89} + 60 q^{91} - 864 q^{92} - 140 q^{97}+O(q^{100}) \) 4 * q - 16 * q^4 + 8 * q^5 - 16 * q^14 + 128 * q^16 - 128 * q^20 - 24 * q^23 - 36 * q^25 + 120 * q^26 + 112 * q^31 + 48 * q^34 + 80 * q^37 + 72 * q^38 + 8 * q^47 + 188 * q^49 + 40 * q^53 + 96 * q^56 + 400 * q^58 + 64 * q^59 - 512 * q^64 - 160 * q^67 - 80 * q^70 + 200 * q^71 + 640 * q^80 - 192 * q^82 - 360 * q^86 - 144 * q^89 + 60 * q^91 - 864 * q^92 - 140 * q^97

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