LMFDB - Embedded newform 1089.2.d.g.1088.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1088,1089)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1089.1088");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.d (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:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1088.12
Root			$-1.90184 - 0.0324487i$ of defining polynomial
Character	$\chi$	$=$	1089.1088
Dual form			1089.2.d.g.1088.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +O(q^{10})$	$q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +0.0276065i q^{10} -2.30152i q^{13} +0.169903i q^{14} +1.38197 q^{16} +4.27621 q^{17} +6.20004i q^{19} -0.0612162i q^{20} +6.79984i q^{23} +4.99839 q^{25} -1.58412i q^{26} -0.376753i q^{28} +5.59574 q^{29} +4.79738 q^{31} +5.80538 q^{32} +2.94328 q^{34} -0.00990077 q^{35} -4.03084 q^{37} +4.26743i q^{38} -0.0973475i q^{40} -9.60293 q^{41} -1.03166i q^{43} +4.68027i q^{46} +11.1262i q^{47} +6.93907 q^{49} +3.44035 q^{50} +3.51271i q^{52} +8.96776i q^{53} -0.599123i q^{56} +3.85150 q^{58} +2.78832i q^{59} -8.48450i q^{61} +3.30199 q^{62} +1.23186 q^{64} +0.0923111 q^{65} +7.94588 q^{67} -6.52659 q^{68} -0.00681461 q^{70} +3.32850i q^{71} -11.8537i q^{73} -2.77439 q^{74} -9.46284i q^{76} -3.01100i q^{79} +0.0554289i q^{80} -6.60961 q^{82} +5.29380 q^{83} +0.171513i q^{85} -0.710085i q^{86} +8.54422i q^{89} +0.568126 q^{91} -10.3783i q^{92} +7.65807i q^{94} -0.248676 q^{95} -3.02824 q^{97} +4.77610 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4}+O(q^{10}) $ 16 * q + 16 * q^4	$ 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) $ 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.688291	0.486695	0.243348	−	0.969939i	$-0.421754\pi$
$2$	0.688291	0.486695	0.243348	+	0.969939i	$0.421754\pi$
$3$	0	0
$3$	0	0
$4$	−1.52626	−0.763128
$5$	0.0401087i	0.0179372i	0.999960	+	0.00896859i	$0.00285483\pi$
$5$	0.0401087i	0.0179372i	−0.999960	+	0.00896859i	$0.997145\pi$
$6$	0	0
$7$	0.246848i	0.0932998i	0.998911	+	0.0466499i	$0.0148545\pi$
$7$	0.246848i	0.0932998i	−0.998911	+	0.0466499i	$0.985145\pi$
$8$	−2.42709	−0.858106
$9$	0	0
$10$	0.0276065i	0.00872994i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 2.30152i	− 0.638327i	−0.947700	−	0.319164i	$-0.896598\pi$
$13$	− 2.30152i	− 0.638327i	0.947700	−	0.319164i	$-0.103402\pi$
$14$	0.169903i	0.0454086i
$15$	0	0
$16$	1.38197	0.345492
$17$	4.27621	1.03713	0.518567	−	0.855037i	$-0.326466\pi$
$17$	4.27621	1.03713	0.518567	+	0.855037i	$0.326466\pi$
$18$	0	0
$19$	6.20004i	1.42239i	0.702997	+	0.711193i	$0.251845\pi$
$19$	6.20004i	1.42239i	−0.702997	+	0.711193i	$0.748155\pi$
$20$	− 0.0612162i	− 0.0136884i
$21$	0	0
$22$	0	0
$23$	6.79984i	1.41786i	0.705277	+	0.708932i	$0.250823\pi$
$23$	6.79984i	1.41786i	−0.705277	+	0.708932i	$0.749177\pi$
$24$	0	0
$25$	4.99839	0.999678
$26$	− 1.58412i	− 0.310671i
$27$	0	0
$28$	− 0.376753i	− 0.0711997i
$29$	5.59574	1.03910	0.519552	−	0.854439i	$-0.326099\pi$
$29$	5.59574	1.03910	0.519552	+	0.854439i	$0.326099\pi$
$30$	0	0
$31$	4.79738	0.861635	0.430817	−	0.902439i	$-0.358225\pi$
$31$	4.79738	0.861635	0.430817	+	0.902439i	$0.358225\pi$
$32$	5.80538	1.02626
$33$	0	0
$34$	2.94328	0.504768
$35$	−0.00990077	−0.00167354
$36$	0	0
$37$	−4.03084	−0.662667	−0.331333	−	0.943514i	$-0.607498\pi$
$37$	−4.03084	−0.662667	−0.331333	+	0.943514i	$0.607498\pi$
$38$	4.26743i	0.692268i
$39$	0	0
$40$	− 0.0973475i	− 0.0153920i
$41$	−9.60293	−1.49973	−0.749863	−	0.661593i	$-0.769881\pi$
$41$	−9.60293	−1.49973	−0.749863	+	0.661593i	$0.769881\pi$
$42$	0	0
$43$	− 1.03166i	− 0.157327i	−0.996901	−	0.0786636i	$-0.974935\pi$
$43$	− 1.03166i	− 0.157327i	0.996901	−	0.0786636i	$-0.0250653\pi$
$44$	0	0
$45$	0	0
$46$	4.68027i	0.690068i
$47$	11.1262i	1.62292i	0.584405	+	0.811462i	$0.301328\pi$
$47$	11.1262i	1.62292i	−0.584405	+	0.811462i	$0.698672\pi$
$48$	0	0
$49$	6.93907	0.991295
$50$	3.44035	0.486539
$51$	0	0
$52$	3.51271i	0.487125i
$53$	8.96776i	1.23182i	0.787818	+	0.615908i	$0.211211\pi$
$53$	8.96776i	1.23182i	−0.787818	+	0.615908i	$0.788789\pi$
$54$	0	0
$55$	0	0
$56$	− 0.599123i	− 0.0800611i
$57$	0	0
$58$	3.85150	0.505727
$59$	2.78832i	0.363008i	0.983390	+	0.181504i	$0.0580966\pi$
$59$	2.78832i	0.363008i	−0.983390	+	0.181504i	$0.941903\pi$
$60$	0	0
$61$	− 8.48450i	− 1.08633i	−0.839627	−	0.543164i	$-0.817226\pi$
$61$	− 8.48450i	− 1.08633i	0.839627	−	0.543164i	$-0.182774\pi$
$62$	3.30199	0.419354
$63$	0	0
$64$	1.23186	0.153982
$65$	0.0923111	0.0114498
$66$	0	0
$67$	7.94588	0.970744	0.485372	−	0.874308i	$-0.338684\pi$
$67$	7.94588	0.970744	0.485372	+	0.874308i	$0.338684\pi$
$68$	−6.52659	−0.791465
$69$	0	0
$70$	−0.00681461	−0.000814502 0
$71$	3.32850i	0.395020i	0.980301	+	0.197510i	$0.0632855\pi$
$71$	3.32850i	0.395020i	−0.980301	+	0.197510i	$0.936714\pi$
$72$	0	0
$73$	− 11.8537i	− 1.38737i	−0.720278	−	0.693685i	$-0.755986\pi$
$73$	− 11.8537i	− 1.38737i	0.720278	−	0.693685i	$-0.244014\pi$
$74$	−2.77439	−0.322517
$75$	0	0
$76$	− 9.46284i	− 1.08546i
$77$	0	0
$78$	0	0
$79$	− 3.01100i	− 0.338764i	−0.985550	−	0.169382i	$-0.945823\pi$
$79$	− 3.01100i	− 0.338764i	0.985550	−	0.169382i	$-0.0541772\pi$
$80$	0.0554289i	0.00619714i
$81$	0	0
$82$	−6.60961	−0.729909
$83$	5.29380	0.581070	0.290535	−	0.956864i	$-0.406167\pi$
$83$	5.29380	0.581070	0.290535	+	0.956864i	$0.406167\pi$
$84$	0	0
$85$	0.171513i	0.0186032i
$86$	− 0.710085i	− 0.0765704i
$87$	0	0
$88$	0	0
$89$	8.54422i	0.905686i	0.891590	+	0.452843i	$0.149590\pi$
$89$	8.54422i	0.905686i	−0.891590	+	0.452843i	$0.850410\pi$
$90$	0	0
$91$	0.568126	0.0595558
$92$	− 10.3783i	− 1.08201i
$93$	0	0
$94$	7.65807i	0.789870i
$95$	−0.248676	−0.0255136
$96$	0	0
$97$	−3.02824	−0.307471	−0.153736	−	0.988112i	$-0.549130\pi$
$97$	−3.02824	−0.307471	−0.153736	+	0.988112i	$0.549130\pi$
$98$	4.77610	0.482459
$99$	0	0
$100$	−7.62882	−0.762882
$101$	6.95198	0.691747	0.345874	−	0.938281i	$-0.387583\pi$
$101$	6.95198	0.691747	0.345874	+	0.938281i	$0.387583\pi$
$102$	0	0
$103$	13.1194	1.29269	0.646347	−	0.763044i	$-0.276296\pi$
$103$	13.1194	1.29269	0.646347	+	0.763044i	$0.276296\pi$
$104$	5.58600i	0.547752i
$105$	0	0
$106$	6.17243i	0.599519i
$107$	−1.36847	−0.132295	−0.0661476	−	0.997810i	$-0.521071\pi$
$107$	−1.36847	−0.132295	−0.0661476	+	0.997810i	$0.521071\pi$
$108$	0	0
$109$	7.34454i	0.703480i	0.936098	+	0.351740i	$0.114410\pi$
$109$	7.34454i	0.703480i	−0.936098	+	0.351740i	$0.885590\pi$
$110$	0	0
$111$	0	0
$112$	0.341136i	0.0322343i
$113$	0.528542i	0.0497211i	0.999691	+	0.0248605i	$0.00791417\pi$
$113$	0.528542i	0.0497211i	−0.999691	+	0.0248605i	$0.992086\pi$
$114$	0	0
$115$	−0.272733	−0.0254325
$116$	−8.54053	−0.792968
$117$	0	0
$118$	1.91917i	0.176674i
$119$	1.05557i	0.0967644i
$120$	0	0
$121$	0	0
$122$	− 5.83980i	− 0.528711i
$123$	0	0
$124$	−7.32203	−0.657537
$125$	0.401023i	0.0358686i
$126$	0	0
$127$	− 12.7719i	− 1.13332i	−0.823952	−	0.566660i	$-0.808235\pi$
$127$	− 12.7719i	− 1.13332i	0.823952	−	0.566660i	$-0.191765\pi$
$128$	−10.7629	−0.951313
$129$	0	0
$130$	0.0635369	0.00557256
$131$	−18.1534	−1.58607	−0.793033	−	0.609178i	$-0.791499\pi$
$131$	−18.1534	−1.58607	−0.793033	+	0.609178i	$0.791499\pi$
$132$	0	0
$133$	−1.53047	−0.132708
$134$	5.46908	0.472456
$135$	0	0
$136$	−10.3787	−0.889970
$137$	− 16.0730i	− 1.37321i	−0.727032	−	0.686604i	$-0.759101\pi$
$137$	− 16.0730i	− 1.37321i	0.727032	−	0.686604i	$-0.240899\pi$
$138$	0	0
$139$	3.85730i	0.327172i	0.986529	+	0.163586i	$0.0523061\pi$
$139$	3.85730i	0.327172i	−0.986529	+	0.163586i	$0.947694\pi$
$140$	0.0151111	0.00127712
$141$	0	0
$142$	2.29098i	0.192255i
$143$	0	0
$144$	0	0
$145$	0.224438i	0.0186386i
$146$	− 8.15879i	− 0.675227i
$147$	0	0
$148$	6.15210	0.505699
$149$	−10.2700	−0.841350	−0.420675	−	0.907211i	$-0.638207\pi$
$149$	−10.2700	−0.841350	−0.420675	+	0.907211i	$0.638207\pi$
$150$	0	0
$151$	− 8.77042i	− 0.713726i	−0.934157	−	0.356863i	$-0.883846\pi$
$151$	− 8.77042i	− 0.713726i	0.934157	−	0.356863i	$-0.116154\pi$
$152$	− 15.0480i	− 1.22056i
$153$	0	0
$154$	0	0
$155$	0.192417i	0.0154553i
$156$	0	0
$157$	−13.0308	−1.03997	−0.519987	−	0.854174i	$-0.674063\pi$
$157$	−13.0308	−1.03997	−0.519987	+	0.854174i	$0.674063\pi$
$158$	− 2.07245i	− 0.164875i
$159$	0	0
$160$	0.232846i	0.0184081i
$161$	−1.67853	−0.132287
$162$	0	0
$163$	−9.17774	−0.718856	−0.359428	−	0.933173i	$-0.617028\pi$
$163$	−9.17774	−0.718856	−0.359428	+	0.933173i	$0.617028\pi$
$164$	14.6565	1.14448
$165$	0	0
$166$	3.64367	0.282804
$167$	7.60934	0.588828	0.294414	−	0.955678i	$-0.404876\pi$
$167$	7.60934	0.588828	0.294414	+	0.955678i	$0.404876\pi$
$168$	0	0
$169$	7.70300	0.592538
$170$	0.118051i	0.00905411i
$171$	0	0
$172$	1.57458i	0.120061i
$173$	16.6204	1.26363	0.631813	−	0.775121i	$-0.282311\pi$
$173$	16.6204	1.26363	0.631813	+	0.775121i	$0.282311\pi$
$174$	0	0
$175$	1.23384i	0.0932698i
$176$	0	0
$177$	0	0
$178$	5.88091i	0.440793i
$179$	− 7.04979i	− 0.526926i	−0.964669	−	0.263463i	$-0.915135\pi$
$179$	− 7.04979i	− 0.526926i	0.964669	−	0.263463i	$-0.0848647\pi$
$180$	0	0
$181$	−21.1236	−1.57011	−0.785053	−	0.619429i	$-0.787364\pi$
$181$	−21.1236	−1.57011	−0.785053	+	0.619429i	$0.787364\pi$
$182$	0.391036	0.0289855
$183$	0	0
$184$	− 16.5038i	− 1.21668i
$185$	− 0.161672i	− 0.0118864i
$186$	0	0
$187$	0	0
$188$	− 16.9814i	− 1.23850i
$189$	0	0
$190$	−0.171161	−0.0124173
$191$	13.6683i	0.989000i	0.869178	+	0.494500i	$0.164649\pi$
$191$	13.6683i	0.989000i	−0.869178	+	0.494500i	$0.835351\pi$
$192$	0	0
$193$	25.0349i	1.80205i	0.433767	+	0.901025i	$0.357184\pi$
$193$	25.0349i	1.80205i	−0.433767	+	0.901025i	$0.642816\pi$
$194$	−2.08431	−0.149645
$195$	0	0
$196$	−10.5908	−0.756485
$197$	−21.0442	−1.49934	−0.749668	−	0.661814i	$-0.769787\pi$
$197$	−21.0442	−1.49934	−0.749668	+	0.661814i	$0.769787\pi$
$198$	0	0
$199$	−10.3709	−0.735176	−0.367588	−	0.929989i	$-0.619816\pi$
$199$	−10.3709	−0.735176	−0.367588	+	0.929989i	$0.619816\pi$
$200$	−12.1315	−0.857830
$201$	0	0
$202$	4.78498	0.336670
$203$	1.38130i	0.0969481i
$204$	0	0
$205$	− 0.385161i	− 0.0269008i
$206$	9.02997	0.629148
$207$	0	0
$208$	− 3.18062i	− 0.220537i
$209$	0	0
$210$	0	0
$211$	− 18.7436i	− 1.29036i	−0.764029	−	0.645182i	$-0.776782\pi$
$211$	− 18.7436i	− 1.29036i	0.764029	−	0.645182i	$-0.223218\pi$
$212$	− 13.6871i	− 0.940033i
$213$	0	0
$214$	−0.941908	−0.0643875
$215$	0.0413787	0.00282201
$216$	0	0
$217$	1.18422i	0.0803904i
$218$	5.05518i	0.342380i
$219$	0	0
$220$	0	0
$221$	− 9.84179i	− 0.662031i
$222$	0	0
$223$	9.66894	0.647480	0.323740	−	0.946146i	$-0.395060\pi$
$223$	9.66894	0.647480	0.323740	+	0.946146i	$0.395060\pi$
$224$	1.43305i	0.0957494i
$225$	0	0
$226$	0.363791i	0.0241990i
$227$	−11.7162	−0.777630	−0.388815	−	0.921316i	$-0.627116\pi$
$227$	−11.7162	−0.777630	−0.388815	+	0.921316i	$0.627116\pi$
$228$	0	0
$229$	20.4802	1.35337	0.676684	−	0.736274i	$-0.263416\pi$
$229$	20.4802	1.35337	0.676684	+	0.736274i	$0.263416\pi$
$230$	−0.187720	−0.0123779
$231$	0	0
$232$	−13.5814	−0.891661
$233$	−16.1429	−1.05755	−0.528777	−	0.848761i	$-0.677349\pi$
$233$	−16.1429	−1.05755	−0.528777	+	0.848761i	$0.677349\pi$
$234$	0	0
$235$	−0.446258	−0.0291107
$236$	− 4.25569i	− 0.277022i
$237$	0	0
$238$	0.726543i	0.0470948i
$239$	28.3875	1.83624	0.918118	−	0.396306i	$-0.129708\pi$
$239$	28.3875	1.83624	0.918118	+	0.396306i	$0.129708\pi$
$240$	0	0
$241$	14.3654i	0.925357i	0.886526	+	0.462679i	$0.153112\pi$
$241$	14.3654i	0.925357i	−0.886526	+	0.462679i	$0.846888\pi$
$242$	0	0
$243$	0	0
$244$	12.9495i	0.829007i
$245$	0.278317i	0.0177810i
$246$	0	0
$247$	14.2695	0.907948
$248$	−11.6437	−0.739374
$249$	0	0
$250$	0.276021i	0.0174571i
$251$	− 6.94278i	− 0.438225i	−0.975700	−	0.219112i	$-0.929684\pi$
$251$	− 6.94278i	− 0.438225i	0.975700	−	0.219112i	$-0.0703161\pi$
$252$	0	0
$253$	0	0
$254$	− 8.79077i	− 0.551582i
$255$	0	0
$256$	−9.87170	−0.616981
$257$	10.4183i	0.649875i	0.945735	+	0.324938i	$0.105343\pi$
$257$	10.4183i	0.649875i	−0.945735	+	0.324938i	$0.894657\pi$
$258$	0	0
$259$	− 0.995006i	− 0.0618267i
$260$	−0.140890	−0.00873765
$261$	0	0
$262$	−12.4948	−0.771931
$263$	4.26110	0.262751	0.131375	−	0.991333i	$-0.458061\pi$
$263$	4.26110	0.262751	0.131375	+	0.991333i	$0.458061\pi$
$264$	0	0
$265$	−0.359686	−0.0220953
$266$	−1.05341	−0.0645885
$267$	0	0
$268$	−12.1274	−0.740801
$269$	− 11.0638i	− 0.674569i	−0.941403	−	0.337285i	$-0.890492\pi$
$269$	− 11.0638i	− 0.674569i	0.941403	−	0.337285i	$-0.109508\pi$
$270$	0	0
$271$	15.8957i	0.965596i	0.875732	+	0.482798i	$0.160379\pi$
$271$	15.8957i	0.965596i	−0.875732	+	0.482798i	$0.839621\pi$
$272$	5.90958	0.358321
$273$	0	0
$274$	− 11.0629i	− 0.668334i
$275$	0	0
$276$	0	0
$277$	11.3410i	0.681417i	0.940169	+	0.340709i	$0.110667\pi$
$277$	11.3410i	0.681417i	−0.940169	+	0.340709i	$0.889333\pi$
$278$	2.65495i	0.159233i
$279$	0	0
$280$	0.0240301	0.00143607
$281$	−1.56030	−0.0930794	−0.0465397	−	0.998916i	$-0.514819\pi$
$281$	−1.56030	−0.0930794	−0.0465397	+	0.998916i	$0.514819\pi$
$282$	0	0
$283$	− 26.5654i	− 1.57915i	−0.613653	−	0.789576i	$-0.710301\pi$
$283$	− 26.5654i	− 1.57915i	0.613653	−	0.789576i	$-0.289699\pi$
$284$	− 5.08014i	− 0.301451i
$285$	0	0
$286$	0	0
$287$	− 2.37046i	− 0.139924i
$288$	0	0
$289$	1.28598	0.0756456
$290$	0.154479i	0.00907131i
$291$	0	0
$292$	18.0918i	1.05874i
$293$	26.3345	1.53848	0.769240	−	0.638960i	$-0.220635\pi$
$293$	26.3345	1.53848	0.769240	+	0.638960i	$0.220635\pi$
$294$	0	0
$295$	−0.111836	−0.00651134
$296$	9.78322	0.568638
$297$	0	0
$298$	−7.06874	−0.409481
$299$	15.6500	0.905062
$300$	0	0
$301$	0.254664	0.0146786
$302$	− 6.03660i	− 0.347367i
$303$	0	0
$304$	8.56824i	0.491422i
$305$	0.340302	0.0194857
$306$	0	0
$307$	26.0083i	1.48437i	0.670195	+	0.742185i	$0.266211\pi$
$307$	26.0083i	1.48437i	−0.670195	+	0.742185i	$0.733789\pi$
$308$	0	0
$309$	0	0
$310$	0.132439i	0.00752202i
$311$	− 13.3248i	− 0.755578i	−0.925892	−	0.377789i	$-0.876684\pi$
$311$	− 13.3248i	− 0.755578i	0.925892	−	0.377789i	$-0.123316\pi$
$312$	0	0
$313$	−11.2483	−0.635792	−0.317896	−	0.948126i	$-0.602976\pi$
$313$	−11.2483	−0.635792	−0.317896	+	0.948126i	$0.602976\pi$
$314$	−8.96901	−0.506151
$315$	0	0
$316$	4.59556i	0.258520i
$317$	− 10.9526i	− 0.615161i	−0.951522	−	0.307580i	$-0.900481\pi$
$317$	− 10.9526i	− 0.615161i	0.951522	−	0.307580i	$-0.0995193\pi$
$318$	0	0
$319$	0	0
$320$	0.0494082i	0.00276200i
$321$	0	0
$322$	−1.15532	−0.0643832
$323$	26.5127i	1.47520i
$324$	0	0
$325$	− 11.5039i	− 0.638122i
$326$	−6.31695	−0.349864
$327$	0	0
$328$	23.3072	1.28692
$329$	−2.74648	−0.151419
$330$	0	0
$331$	4.23285	0.232659	0.116329	−	0.993211i	$-0.462887\pi$
$331$	4.23285	0.232659	0.116329	+	0.993211i	$0.462887\pi$
$332$	−8.07968	−0.443430
$333$	0	0
$334$	5.23744	0.286580
$335$	0.318699i	0.0174124i
$336$	0	0
$337$	− 2.66238i	− 0.145029i	−0.997367	−	0.0725146i	$-0.976898\pi$
$337$	− 2.66238i	− 0.145029i	0.997367	−	0.0725146i	$-0.0231024\pi$
$338$	5.30191	0.288386
$339$	0	0
$340$	− 0.261773i	− 0.0141966i
$341$	0	0
$342$	0	0
$343$	3.44083i	0.185787i
$344$	2.50394i	0.135003i
$345$	0	0
$346$	11.4397	0.615001
$347$	16.6032	0.891304	0.445652	−	0.895206i	$-0.352972\pi$
$347$	16.6032	0.891304	0.445652	+	0.895206i	$0.352972\pi$
$348$	0	0
$349$	13.7748i	0.737346i	0.929559	+	0.368673i	$0.120188\pi$
$349$	13.7748i	0.737346i	−0.929559	+	0.368673i	$0.879812\pi$
$350$	0.849243i	0.0453940i
$351$	0	0
$352$	0	0
$353$	− 11.0249i	− 0.586795i	−0.955990	−	0.293398i	$-0.905214\pi$
$353$	− 11.0249i	− 0.586795i	0.955990	−	0.293398i	$-0.0947860\pi$
$354$	0	0
$355$	−0.133502	−0.00708555
$356$	− 13.0407i	− 0.691154i
$357$	0	0
$358$	− 4.85231i	− 0.256452i
$359$	−19.0142	−1.00353	−0.501766	−	0.865003i	$-0.667316\pi$
$359$	−19.0142	−1.00353	−0.501766	+	0.865003i	$0.667316\pi$
$360$	0	0
$361$	−19.4404	−1.02318
$362$	−14.5392	−0.764163
$363$	0	0
$364$	−0.867106	−0.0454487
$365$	0.475437	0.0248855
$366$	0	0
$367$	−5.69718	−0.297390	−0.148695	−	0.988883i	$-0.547507\pi$
$367$	−5.69718	−0.297390	−0.148695	+	0.988883i	$0.547507\pi$
$368$	9.39715i	0.489860i
$369$	0	0
$370$	− 0.111277i	− 0.00578504i
$371$	−2.21367	−0.114928
$372$	0	0
$373$	− 22.1594i	− 1.14737i	−0.819076	−	0.573684i	$-0.805514\pi$
$373$	− 22.1594i	− 1.14737i	0.819076	−	0.573684i	$-0.194486\pi$
$374$	0	0
$375$	0	0
$376$	− 27.0043i	− 1.39264i
$377$	− 12.8787i	− 0.663288i
$378$	0	0
$379$	−0.381352	−0.0195887	−0.00979436	−	0.999952i	$-0.503118\pi$
$379$	−0.381352	−0.0195887	−0.00979436	+	0.999952i	$0.503118\pi$
$380$	0.379542	0.0194701
$381$	0	0
$382$	9.40774i	0.481342i
$383$	12.6644i	0.647122i	0.946207	+	0.323561i	$0.104880\pi$
$383$	12.6644i	0.647122i	−0.946207	+	0.323561i	$0.895120\pi$
$384$	0	0
$385$	0	0
$386$	17.2313i	0.877049i
$387$	0	0
$388$	4.62187	0.234640
$389$	− 23.7986i	− 1.20664i	−0.797500	−	0.603318i	$-0.793845\pi$
$389$	− 23.7986i	− 1.20664i	0.797500	−	0.603318i	$-0.206155\pi$
$390$	0	0
$391$	29.0775i	1.47051i
$392$	−16.8417	−0.850636
$393$	0	0
$394$	−14.4845	−0.729720
$395$	0.120768	0.00607647
$396$	0	0
$397$	5.00497	0.251192	0.125596	−	0.992081i	$-0.459916\pi$
$397$	5.00497	0.251192	0.125596	+	0.992081i	$0.459916\pi$
$398$	−7.13823	−0.357807
$399$	0	0
$400$	6.90761	0.345380
$401$	− 4.24415i	− 0.211943i	−0.994369	−	0.105971i	$-0.966205\pi$
$401$	− 4.24415i	− 0.211943i	0.994369	−	0.105971i	$-0.0337952\pi$
$402$	0	0
$403$	− 11.0413i	− 0.550005i
$404$	−10.6105	−0.527892
$405$	0	0
$406$	0.950735i	0.0471842i
$407$	0	0
$408$	0	0
$409$	− 11.8056i	− 0.583748i	−0.956457	−	0.291874i	$-0.905721\pi$
$409$	− 11.8056i	− 0.583748i	0.956457	−	0.291874i	$-0.0942788\pi$
$410$	− 0.265103i	− 0.0130925i
$411$	0	0
$412$	−20.0236	−0.986490
$413$	−0.688291	−0.0338686
$414$	0	0
$415$	0.212328i	0.0104227i
$416$	− 13.3612i	− 0.655087i
$417$	0	0
$418$	0	0
$419$	20.9795i	1.02492i	0.858712	+	0.512459i	$0.171265\pi$
$419$	20.9795i	1.02492i	−0.858712	+	0.512459i	$0.828735\pi$
$420$	0	0
$421$	−2.77978	−0.135478	−0.0677390	−	0.997703i	$-0.521579\pi$
$421$	−2.77978	−0.135478	−0.0677390	+	0.997703i	$0.521579\pi$
$422$	− 12.9011i	− 0.628014i
$423$	0	0
$424$	− 21.7656i	− 1.05703i
$425$	21.3742	1.03680
$426$	0	0
$427$	2.09438	0.101354
$428$	2.08864	0.100958
$429$	0	0
$430$	0.0284806	0.00137346
$431$	27.9292	1.34530	0.672652	−	0.739959i	$-0.265155\pi$
$431$	27.9292	1.34530	0.672652	+	0.739959i	$0.265155\pi$
$432$	0	0
$433$	−9.16809	−0.440590	−0.220295	−	0.975433i	$-0.570702\pi$
$433$	−9.16809	−0.440590	−0.220295	+	0.975433i	$0.570702\pi$
$434$	0.815091i	0.0391256i
$435$	0	0
$436$	− 11.2096i	− 0.536845i
$437$	−42.1592	−2.01675
$438$	0	0
$439$	2.93111i	0.139894i	0.997551	+	0.0699472i	$0.0222831\pi$
$439$	2.93111i	0.139894i	−0.997551	+	0.0699472i	$0.977717\pi$
$440$	0	0
$441$	0	0
$442$	− 6.77402i	− 0.322207i
$443$	35.8414i	1.70288i	0.524455	+	0.851438i	$0.324269\pi$
$443$	35.8414i	1.70288i	−0.524455	+	0.851438i	$0.675731\pi$
$444$	0	0
$445$	−0.342698	−0.0162454
$446$	6.65504	0.315125
$447$	0	0
$448$	0.304081i	0.0143665i
$449$	14.2141i	0.670806i	0.942075	+	0.335403i	$0.108872\pi$
$449$	14.2141i	0.670806i	−0.942075	+	0.335403i	$0.891128\pi$
$450$	0	0
$451$	0	0
$452$	− 0.806691i	− 0.0379435i
$453$	0	0
$454$	−8.06414	−0.378469
$455$	0.0227868i	0.00106826i
$456$	0	0
$457$	− 20.3993i	− 0.954238i	−0.878839	−	0.477119i	$-0.841681\pi$
$457$	− 20.3993i	− 0.954238i	0.878839	−	0.477119i	$-0.158319\pi$
$458$	14.0963	0.658678
$459$	0	0
$460$	0.416260	0.0194082
$461$	0.770354	0.0358790	0.0179395	−	0.999839i	$-0.494289\pi$
$461$	0.770354	0.0358790	0.0179395	+	0.999839i	$0.494289\pi$
$462$	0	0
$463$	37.7948	1.75647	0.878236	−	0.478228i	$-0.158721\pi$
$463$	37.7948	1.75647	0.878236	+	0.478228i	$0.158721\pi$
$464$	7.73312	0.359001
$465$	0	0
$466$	−11.1110	−0.514707
$467$	17.6026i	0.814551i	0.913305	+	0.407276i	$0.133521\pi$
$467$	17.6026i	0.814551i	−0.913305	+	0.407276i	$0.866479\pi$
$468$	0	0
$469$	1.96143i	0.0905702i
$470$	−0.307156	−0.0141680
$471$	0	0
$472$	− 6.76750i	− 0.311499i
$473$	0	0
$474$	0	0
$475$	30.9902i	1.42193i
$476$	− 1.61108i	− 0.0738436i
$477$	0	0
$478$	19.5389	0.893688
$479$	−2.40586	−0.109927	−0.0549633	−	0.998488i	$-0.517504\pi$
$479$	−2.40586	−0.109927	−0.0549633	+	0.998488i	$0.517504\pi$
$480$	0	0
$481$	9.27708i	0.422998i
$482$	9.88758i	0.450367i
$483$	0	0
$484$	0	0
$485$	− 0.121459i	− 0.00551517i
$486$	0	0
$487$	14.5978	0.661491	0.330745	−	0.943720i	$-0.392700\pi$
$487$	14.5978	0.661491	0.330745	+	0.943720i	$0.392700\pi$
$488$	20.5926i	0.932185i
$489$	0	0
$490$	0.191563i	0.00865395i
$491$	−25.0424	−1.13015	−0.565073	−	0.825041i	$-0.691152\pi$
$491$	−25.0424	−1.13015	−0.565073	+	0.825041i	$0.691152\pi$
$492$	0	0
$493$	23.9286	1.07769
$494$	9.82158	0.441894
$495$	0	0
$496$	6.62982	0.297688
$497$	−0.821634	−0.0368553
$498$	0	0
$499$	22.7116	1.01671	0.508355	−	0.861148i	$-0.330254\pi$
$499$	22.7116	1.01671	0.508355	+	0.861148i	$0.330254\pi$
$500$	− 0.612063i	− 0.0273723i
$501$	0	0
$502$	− 4.77866i	− 0.213282i
$503$	−10.4071	−0.464030	−0.232015	−	0.972712i	$-0.574532\pi$
$503$	−10.4071	−0.464030	−0.232015	+	0.972712i	$0.574532\pi$
$504$	0	0
$505$	0.278835i	0.0124080i
$506$	0	0
$507$	0	0
$508$	19.4931i	0.864868i
$509$	8.08550i	0.358383i	0.983814	+	0.179192i	$0.0573482\pi$
$509$	8.08550i	0.358383i	−0.983814	+	0.179192i	$0.942652\pi$
$510$	0	0
$511$	2.92606	0.129441
$512$	14.7311	0.651031
$513$	0	0
$514$	7.17082i	0.316291i
$515$	0.526203i	0.0231873i
$516$	0	0
$517$	0	0
$518$	− 0.684854i	− 0.0300908i
$519$	0	0
$520$	−0.224047	−0.00982513
$521$	− 35.0206i	− 1.53428i	−0.641479	−	0.767141i	$-0.721679\pi$
$521$	− 35.0206i	− 1.53428i	0.641479	−	0.767141i	$-0.278321\pi$
$522$	0	0
$523$	− 43.2956i	− 1.89318i	−0.322434	−	0.946592i	$-0.604501\pi$
$523$	− 43.2956i	− 1.89318i	0.322434	−	0.946592i	$-0.395499\pi$
$524$	27.7067	1.21037
$525$	0	0
$526$	2.93288	0.127880
$527$	20.5146	0.893630
$528$	0	0
$529$	−23.2378	−1.01034
$530$	−0.247568	−0.0107537
$531$	0	0
$532$	2.33588	0.101273
$533$	22.1013i	0.957316i
$534$	0	0
$535$	− 0.0548877i	− 0.00237300i
$536$	−19.2854	−0.833001
$537$	0	0
$538$	− 7.61509i	− 0.328310i
$539$	0	0
$540$	0	0
$541$	− 0.226329i	− 0.00973066i	−0.999988	−	0.00486533i	$-0.998451\pi$
$541$	− 0.226329i	− 0.00973066i	0.999988	−	0.00486533i	$-0.00154869\pi$
$542$	10.9409i	0.469951i
$543$	0	0
$544$	24.8250	1.06436
$545$	−0.294580	−0.0126184
$546$	0	0
$547$	16.4633i	0.703918i	0.936015	+	0.351959i	$0.114484\pi$
$547$	16.4633i	0.703918i	−0.936015	+	0.351959i	$0.885516\pi$
$548$	24.5315i	1.04793i
$549$	0	0
$550$	0	0
$551$	34.6938i	1.47801i
$552$	0	0
$553$	0.743260	0.0316066
$554$	7.80594i	0.331643i
$555$	0	0
$556$	− 5.88723i	− 0.249674i
$557$	26.2818	1.11360	0.556798	−	0.830648i	$-0.312030\pi$
$557$	26.2818	1.11360	0.556798	+	0.830648i	$0.312030\pi$
$558$	0	0
$559$	−2.37440	−0.100426
$560$	−0.0136825	−0.000578192 0
$561$	0	0
$562$	−1.07394	−0.0453013
$563$	−36.3658	−1.53263	−0.766317	−	0.642463i	$-0.777913\pi$
$563$	−36.3658	−1.53263	−0.766317	+	0.642463i	$0.777913\pi$
$564$	0	0
$565$	−0.0211992	−0.000891856 0
$566$	− 18.2848i	− 0.768565i
$567$	0	0
$568$	− 8.07857i	− 0.338969i
$569$	−20.2307	−0.848114	−0.424057	−	0.905636i	$-0.639394\pi$
$569$	−20.2307	−0.848114	−0.424057	+	0.905636i	$0.639394\pi$
$570$	0	0
$571$	− 24.4002i	− 1.02112i	−0.859843	−	0.510558i	$-0.829439\pi$
$571$	− 24.4002i	− 1.02112i	0.859843	−	0.510558i	$-0.170561\pi$
$572$	0	0
$573$	0	0
$574$	− 1.63157i	− 0.0681004i
$575$	33.9883i	1.41741i
$576$	0	0
$577$	−19.3030	−0.803593	−0.401796	−	0.915729i	$-0.631614\pi$
$577$	−19.3030	−0.803593	−0.401796	+	0.915729i	$0.631614\pi$
$578$	0.885126	0.0368164
$579$	0	0
$580$	− 0.342550i	− 0.0142236i
$581$	1.30676i	0.0542137i
$582$	0	0
$583$	0	0
$584$	28.7700i	1.19051i
$585$	0	0
$586$	18.1258	0.748771
$587$	− 19.1363i	− 0.789840i	−0.918715	−	0.394920i	$-0.870772\pi$
$587$	− 19.1363i	− 0.789840i	0.918715	−	0.394920i	$-0.129228\pi$
$588$	0	0
$589$	29.7439i	1.22558i
$590$	−0.0769757	−0.00316904
$591$	0	0
$592$	−5.57049	−0.228946
$593$	14.9885	0.615503	0.307751	−	0.951467i	$-0.400423\pi$
$593$	14.9885	0.615503	0.307751	+	0.951467i	$0.400423\pi$
$594$	0	0
$595$	−0.0423378	−0.00173568
$596$	15.6746	0.642058
$597$	0	0
$598$	10.7717	0.440489
$599$	33.3409i	1.36227i	0.732158	+	0.681135i	$0.238513\pi$
$599$	33.3409i	1.36227i	−0.732158	+	0.681135i	$0.761487\pi$
$600$	0	0
$601$	− 42.2730i	− 1.72435i	−0.506610	−	0.862175i	$-0.669102\pi$
$601$	− 42.2730i	− 1.72435i	0.506610	−	0.862175i	$-0.330898\pi$
$602$	0.175283	0.00714401
$603$	0	0
$604$	13.3859i	0.544664i
$605$	0	0
$606$	0	0
$607$	− 30.8859i	− 1.25362i	−0.779172	−	0.626810i	$-0.784360\pi$
$607$	− 30.8859i	− 1.25362i	0.779172	−	0.626810i	$-0.215640\pi$
$608$	35.9935i	1.45973i
$609$	0	0
$610$	0.234227	0.00948358
$611$	25.6072	1.03596
$612$	0	0
$613$	15.9745i	0.645204i	0.946535	+	0.322602i	$0.104558\pi$
$613$	15.9745i	0.645204i	−0.946535	+	0.322602i	$0.895442\pi$
$614$	17.9013i	0.722436i
$615$	0	0
$616$	0	0
$617$	2.63374i	0.106030i	0.998594	+	0.0530151i	$0.0168831\pi$
$617$	2.63374i	0.106030i	−0.998594	+	0.0530151i	$0.983117\pi$
$618$	0	0
$619$	20.2495	0.813894	0.406947	−	0.913452i	$-0.366593\pi$
$619$	20.2495	0.813894	0.406947	+	0.913452i	$0.366593\pi$
$620$	− 0.293677i	− 0.0117944i
$621$	0	0
$622$	− 9.17132i	− 0.367736i
$623$	−2.10913	−0.0845003
$624$	0	0
$625$	24.9759	0.999035
$626$	−7.74212	−0.309437
$627$	0	0
$628$	19.8884	0.793633
$629$	−17.2367	−0.687274
$630$	0	0
$631$	8.44044	0.336009	0.168004	−	0.985786i	$-0.446268\pi$
$631$	8.44044	0.336009	0.168004	+	0.985786i	$0.446268\pi$
$632$	7.30797i	0.290696i
$633$	0	0
$634$	− 7.53859i	− 0.299396i
$635$	0.512264	0.0203286
$636$	0	0
$637$	− 15.9704i	− 0.632771i
$638$	0	0
$639$	0	0
$640$	− 0.431685i	− 0.0170639i
$641$	− 32.3145i	− 1.27634i	−0.769894	−	0.638172i	$-0.779691\pi$
$641$	− 32.3145i	− 1.27634i	0.769894	−	0.638172i	$-0.220309\pi$
$642$	0	0
$643$	13.7767	0.543300	0.271650	−	0.962396i	$-0.412431\pi$
$643$	13.7767	0.543300	0.271650	+	0.962396i	$0.412431\pi$
$644$	2.56186	0.100951
$645$	0	0
$646$	18.2484i	0.717975i
$647$	− 8.52739i	− 0.335246i	−0.985851	−	0.167623i	$-0.946391\pi$
$647$	− 8.52739i	− 0.335246i	0.985851	−	0.167623i	$-0.0536092\pi$
$648$	0	0
$649$	0	0
$650$	− 7.91804i	− 0.310571i
$651$	0	0
$652$	14.0076	0.548579
$653$	− 1.20799i	− 0.0472722i	−0.999721	−	0.0236361i	$-0.992476\pi$
$653$	− 1.20799i	− 0.0472722i	0.999721	−	0.0236361i	$-0.00752430\pi$
$654$	0	0
$655$	− 0.728108i	− 0.0284496i
$656$	−13.2709	−0.518142
$657$	0	0
$658$	−1.89038	−0.0736947
$659$	9.41054	0.366583	0.183291	−	0.983059i	$-0.441325\pi$
$659$	9.41054	0.366583	0.183291	+	0.983059i	$0.441325\pi$
$660$	0	0
$661$	15.1027	0.587425	0.293713	−	0.955894i	$-0.405109\pi$
$661$	15.1027	0.587425	0.293713	+	0.955894i	$0.405109\pi$
$662$	2.91343	0.113234
$663$	0	0
$664$	−12.8485	−0.498619
$665$	− 0.0613851i	− 0.00238041i
$666$	0	0
$667$	38.0501i	1.47331i
$668$	−11.6138	−0.449351
$669$	0	0
$670$	0.219358i	0.00847453i
$671$	0	0
$672$	0	0
$673$	41.5989i	1.60352i	0.597647	+	0.801759i	$0.296102\pi$
$673$	41.5989i	1.60352i	−0.597647	+	0.801759i	$0.703898\pi$
$674$	− 1.83249i	− 0.0705850i
$675$	0	0
$676$	−11.7567	−0.452182
$677$	4.52056	0.173739	0.0868696	−	0.996220i	$-0.472314\pi$
$677$	4.52056	0.173739	0.0868696	+	0.996220i	$0.472314\pi$
$678$	0	0
$679$	− 0.747516i	− 0.0286870i
$680$	− 0.416279i	− 0.0159636i
$681$	0	0
$682$	0	0
$683$	− 34.7783i	− 1.33075i	−0.746507	−	0.665377i	$-0.768271\pi$
$683$	− 34.7783i	− 1.33075i	0.746507	−	0.665377i	$-0.231729\pi$
$684$	0	0
$685$	0.644667	0.0246315
$686$	2.36829i	0.0904219i
$687$	0	0
$688$	− 1.42572i	− 0.0543552i
$689$	20.6395	0.786302
$690$	0	0
$691$	−17.8835	−0.680320	−0.340160	−	0.940368i	$-0.610481\pi$
$691$	−17.8835	−0.680320	−0.340160	+	0.940368i	$0.610481\pi$
$692$	−25.3670	−0.964308
$693$	0	0
$694$	11.4278	0.433794
$695$	−0.154712	−0.00586854
$696$	0	0
$697$	−41.0641	−1.55542
$698$	9.48105i	0.358863i
$699$	0	0
$700$	− 1.88316i	− 0.0711768i
$701$	−17.4712	−0.659878	−0.329939	−	0.944002i	$-0.607028\pi$
$701$	−17.4712	−0.659878	−0.329939	+	0.944002i	$0.607028\pi$
$702$	0	0
$703$	− 24.9914i	− 0.942568i
$704$	0	0
$705$	0	0
$706$	− 7.58833i	− 0.285591i
$707$	1.71608i	0.0645399i
$708$	0	0
$709$	8.33849	0.313158	0.156579	−	0.987665i	$-0.449953\pi$
$709$	8.33849	0.313158	0.156579	+	0.987665i	$0.449953\pi$
$710$	−0.0918882	−0.00344850
$711$	0	0
$712$	− 20.7376i	− 0.777174i
$713$	32.6214i	1.22168i
$714$	0	0
$715$	0	0
$716$	10.7598i	0.402112i
$717$	0	0
$718$	−13.0873	−0.488415
$719$	23.6031i	0.880249i	0.897937	+	0.440124i	$0.145066\pi$
$719$	23.6031i	0.880249i	−0.897937	+	0.440124i	$0.854934\pi$
$720$	0	0
$721$	3.23850i	0.120608i
$722$	−13.3807	−0.497977
$723$	0	0
$724$	32.2400	1.19819
$725$	27.9697	1.03877
$726$	0	0
$727$	1.90462	0.0706386	0.0353193	−	0.999376i	$-0.488755\pi$
$727$	1.90462	0.0706386	0.0353193	+	0.999376i	$0.488755\pi$
$728$	−1.37889	−0.0511052
$729$	0	0
$730$	0.327239	0.0121117
$731$	− 4.41161i	− 0.163169i
$732$	0	0
$733$	− 17.6118i	− 0.650506i	−0.945627	−	0.325253i	$-0.894550\pi$
$733$	− 17.6118i	− 0.650506i	0.945627	−	0.325253i	$-0.105450\pi$
$734$	−3.92132	−0.144738
$735$	0	0
$736$	39.4756i	1.45509i
$737$	0	0
$738$	0	0
$739$	− 4.77737i	− 0.175738i	−0.996132	−	0.0878692i	$-0.971994\pi$
$739$	− 4.77737i	− 0.175738i	0.996132	−	0.0878692i	$-0.0280058\pi$
$740$	0.246753i	0.00907082i
$741$	0	0
$742$	−1.52365	−0.0559350
$743$	−1.83777	−0.0674211	−0.0337105	−	0.999432i	$-0.510732\pi$
$743$	−1.83777	−0.0674211	−0.0337105	+	0.999432i	$0.510732\pi$
$744$	0	0
$745$	− 0.411916i	− 0.0150914i
$746$	− 15.2521i	− 0.558419i
$747$	0	0
$748$	0	0
$749$	− 0.337805i	− 0.0123431i
$750$	0	0
$751$	−37.9446	−1.38462	−0.692308	−	0.721602i	$-0.743406\pi$
$751$	−37.9446	−1.38462	−0.692308	+	0.721602i	$0.743406\pi$
$752$	15.3760i	0.560707i
$753$	0	0
$754$	− 8.86431i	− 0.322819i
$755$	0.351770	0.0128022
$756$	0	0
$757$	−30.6265	−1.11314	−0.556569	−	0.830802i	$-0.687882\pi$
$757$	−30.6265	−1.11314	−0.556569	+	0.830802i	$0.687882\pi$
$758$	−0.262481	−0.00953373
$759$	0	0
$760$	0.603558	0.0218934
$761$	30.8652	1.11886	0.559432	−	0.828876i	$-0.311019\pi$
$761$	30.8652	1.11886	0.559432	+	0.828876i	$0.311019\pi$
$762$	0	0
$763$	−1.81299	−0.0656345
$764$	− 20.8613i	− 0.754734i
$765$	0	0
$766$	8.71682i	0.314951i
$767$	6.41738	0.231718
$768$	0	0
$769$	19.3154i	0.696532i	0.937396	+	0.348266i	$0.113229\pi$
$769$	19.3154i	0.696532i	−0.937396	+	0.348266i	$0.886771\pi$
$770$	0	0
$771$	0	0
$772$	− 38.2096i	− 1.37519i
$773$	− 2.94203i	− 0.105818i	−0.998599	−	0.0529088i	$-0.983151\pi$
$773$	− 2.94203i	− 0.105818i	0.998599	−	0.0529088i	$-0.0168493\pi$
$774$	0	0
$775$	23.9792	0.861358
$776$	7.34982	0.263843
$777$	0	0
$778$	− 16.3804i	− 0.587265i
$779$	− 59.5385i	− 2.13319i
$780$	0	0
$781$	0	0
$782$	20.0138i	0.715693i
$783$	0	0
$784$	9.58955	0.342484
$785$	− 0.522651i	− 0.0186542i
$786$	0	0
$787$	8.04569i	0.286798i	0.989665	+	0.143399i	$0.0458032\pi$
$787$	8.04569i	0.286798i	−0.989665	+	0.143399i	$0.954197\pi$
$788$	32.1188	1.14419
$789$	0	0
$790$	0.0831232	0.00295739
$791$	−0.130470	−0.00463897
$792$	0	0
$793$	−19.5272	−0.693433
$794$	3.44487	0.122254
$795$	0	0
$796$	15.8287	0.561033
$797$	− 54.9219i	− 1.94543i	−0.231996	−	0.972717i	$-0.574526\pi$
$797$	− 54.9219i	− 1.94543i	0.231996	−	0.972717i	$-0.425474\pi$
$798$	0	0
$799$	47.5780i	1.68319i
$800$	29.0175	1.02592
$801$	0	0
$802$	− 2.92121i	− 0.103152i
$803$	0	0
$804$	0	0
$805$	− 0.0673236i	− 0.00237285i
$806$	− 7.59961i	− 0.267685i
$807$	0	0
$808$	−16.8731	−0.593593
$809$	−24.3526	−0.856193	−0.428096	−	0.903733i	$-0.640816\pi$
$809$	−24.3526	−0.856193	−0.428096	+	0.903733i	$0.640816\pi$
$810$	0	0
$811$	20.7604i	0.728996i	0.931204	+	0.364498i	$0.118759\pi$
$811$	20.7604i	0.728996i	−0.931204	+	0.364498i	$0.881241\pi$
$812$	− 2.10821i	− 0.0739838i
$813$	0	0
$814$	0	0
$815$	− 0.368107i	− 0.0128942i
$816$	0	0
$817$	6.39635	0.223780
$818$	− 8.12567i	− 0.284107i
$819$	0	0
$820$	0.587855i	0.0205288i
$821$	−28.6373	−0.999449	−0.499725	−	0.866184i	$-0.666565\pi$
$821$	−28.6373	−0.999449	−0.499725	+	0.866184i	$0.666565\pi$
$822$	0	0
$823$	2.48784	0.0867207	0.0433603	−	0.999059i	$-0.486194\pi$
$823$	2.48784	0.0867207	0.0433603	+	0.999059i	$0.486194\pi$
$824$	−31.8420	−1.10927
$825$	0	0
$826$	−0.473745	−0.0164837
$827$	23.8333	0.828765	0.414383	−	0.910103i	$-0.363998\pi$
$827$	23.8333	0.828765	0.414383	+	0.910103i	$0.363998\pi$
$828$	0	0
$829$	−40.6340	−1.41128	−0.705639	−	0.708572i	$-0.749340\pi$
$829$	−40.6340	−1.41128	−0.705639	+	0.708572i	$0.749340\pi$
$830$	0.146143i	0.00507270i
$831$	0	0
$832$	− 2.83514i	− 0.0982909i
$833$	29.6729	1.02811
$834$	0	0
$835$	0.305201i	0.0105619i
$836$	0	0
$837$	0	0
$838$	14.4400i	0.498822i
$839$	− 57.2232i	− 1.97556i	−0.155842	−	0.987782i	$-0.549809\pi$
$839$	− 57.2232i	− 1.97556i	0.155842	−	0.987782i	$-0.450191\pi$
$840$	0	0
$841$	2.31232	0.0797352
$842$	−1.91330	−0.0659365
$843$	0	0
$844$	28.6075i	0.984712i
$845$	0.308958i	0.0106285i
$846$	0	0
$847$	0	0
$848$	12.3931i	0.425582i
$849$	0	0
$850$	14.7117	0.504606
$851$	− 27.4091i	− 0.939572i
$852$	0	0
$853$	30.5412i	1.04571i	0.852422	+	0.522855i	$0.175133\pi$
$853$	30.5412i	1.04571i	−0.852422	+	0.522855i	$0.824867\pi$
$854$	1.44154	0.0493286
$855$	0	0
$856$	3.32141	0.113523
$857$	6.14781	0.210005	0.105003	−	0.994472i	$-0.466515\pi$
$857$	6.14781	0.210005	0.105003	+	0.994472i	$0.466515\pi$
$858$	0	0
$859$	−27.5988	−0.941660	−0.470830	−	0.882224i	$-0.656045\pi$
$859$	−27.5988	−0.941660	−0.470830	+	0.882224i	$0.656045\pi$
$860$	−0.0631545	−0.00215355
$861$	0	0
$862$	19.2234	0.654753
$863$	42.8009i	1.45696i	0.685067	+	0.728481i	$0.259773\pi$
$863$	42.8009i	1.45696i	−0.685067	+	0.728481i	$0.740227\pi$
$864$	0	0
$865$	0.666624i	0.0226659i
$866$	−6.31032	−0.214433
$867$	0	0
$868$	− 1.80743i	− 0.0613481i
$869$	0	0
$870$	0	0
$871$	− 18.2876i	− 0.619652i
$872$	− 17.8259i	− 0.603660i
$873$	0	0
$874$	−29.0178	−0.981543
$875$	−0.0989917	−0.00334653
$876$	0	0
$877$	− 11.2305i	− 0.379227i	−0.981859	−	0.189613i	$-0.939277\pi$
$877$	− 11.2305i	− 0.379227i	0.981859	−	0.189613i	$-0.0607235\pi$
$878$	2.01746i	0.0680860i
$879$	0	0
$880$	0	0
$881$	− 47.3136i	− 1.59403i	−0.603957	−	0.797017i	$-0.706410\pi$
$881$	− 47.3136i	− 1.59403i	0.603957	−	0.797017i	$-0.293590\pi$
$882$	0	0
$883$	18.2617	0.614555	0.307278	−	0.951620i	$-0.400582\pi$
$883$	18.2617	0.614555	0.307278	+	0.951620i	$0.400582\pi$
$884$	15.0211i	0.505214i
$885$	0	0
$886$	24.6693i	0.828782i
$887$	−40.7104	−1.36692	−0.683460	−	0.729988i	$-0.739526\pi$
$887$	−40.7104	−1.36692	−0.683460	+	0.729988i	$0.739526\pi$
$888$	0	0
$889$	3.15271	0.105739
$890$	−0.235876	−0.00790658
$891$	0	0
$892$	−14.7573	−0.494110
$893$	−68.9829	−2.30842
$894$	0	0
$895$	0.282758	0.00945157
$896$	− 2.65680i	− 0.0887573i
$897$	0	0
$898$	9.78345i	0.326478i
$899$	26.8449	0.895327
$900$	0	0
$901$	38.3480i	1.27756i
$902$	0	0
$903$	0	0
$904$	− 1.28282i	− 0.0426660i
$905$	− 0.847242i	− 0.0281633i
$906$	0	0
$907$	1.18171	0.0392380	0.0196190	−	0.999808i	$-0.493755\pi$
$907$	1.18171	0.0392380	0.0196190	+	0.999808i	$0.493755\pi$
$908$	17.8819	0.593431
$909$	0	0
$910$	0.0156840i	0 0.000519919i
$911$	− 21.1641i	− 0.701198i	−0.936526	−	0.350599i	$-0.885978\pi$
$911$	− 21.1641i	− 0.701198i	0.936526	−	0.350599i	$-0.114022\pi$
$912$	0	0
$913$	0	0
$914$	− 14.0406i	− 0.464423i
$915$	0	0
$916$	−31.2580	−1.03279
$917$	− 4.48112i	− 0.147980i
$918$	0	0
$919$	14.0906i	0.464805i	0.972620	+	0.232402i	$0.0746586\pi$
$919$	14.0906i	0.464805i	−0.972620	+	0.232402i	$0.925341\pi$
$920$	0.661948	0.0218238
$921$	0	0
$922$	0.530228	0.0174621
$923$	7.66061	0.252152
$924$	0	0
$925$	−20.1477	−0.662454
$926$	26.0138	0.854867
$927$	0	0
$928$	32.4854	1.06638
$929$	− 34.0857i	− 1.11832i	−0.829061	−	0.559158i	$-0.811125\pi$
$929$	− 34.0857i	− 1.11832i	0.829061	−	0.559158i	$-0.188875\pi$
$930$	0	0
$931$	43.0225i	1.41000i
$932$	24.6381	0.807049
$933$	0	0
$934$	12.1157i	0.396438i
$935$	0	0
$936$	0	0
$937$	6.75011i	0.220517i	0.993903	+	0.110258i	$0.0351678\pi$
$937$	6.75011i	0.220517i	−0.993903	+	0.110258i	$0.964832\pi$
$938$	1.35003i	0.0440801i
$939$	0	0
$940$	0.681104	0.0222152
$941$	−33.2979	−1.08548	−0.542740	−	0.839901i	$-0.682613\pi$
$941$	−33.2979	−1.08548	−0.542740	+	0.839901i	$0.682613\pi$
$942$	0	0
$943$	− 65.2984i	− 2.12641i
$944$	3.85336i	0.125416i
$945$	0	0
$946$	0	0
$947$	− 0.751232i	− 0.0244117i	−0.999926	−	0.0122059i	$-0.996115\pi$
$947$	− 0.751232i	− 0.0244117i	0.999926	−	0.0122059i	$-0.00388535\pi$
$948$	0	0
$949$	−27.2815	−0.885596
$950$	21.3303i	0.692046i
$951$	0	0
$952$	− 2.56197i	− 0.0830341i
$953$	−4.65369	−0.150748	−0.0753739	−	0.997155i	$-0.524015\pi$
$953$	−4.65369	−0.150748	−0.0753739	+	0.997155i	$0.524015\pi$
$954$	0	0
$955$	−0.548217	−0.0177399
$956$	−43.3266	−1.40128
$957$	0	0
$958$	−1.65593	−0.0535007
$959$	3.96758	0.128120
$960$	0	0
$961$	−7.98515	−0.257585
$962$	6.38533i	0.205871i
$963$	0	0
$964$	− 21.9253i	− 0.706166i
$965$	−1.00412	−0.0323237
$966$	0	0
$967$	− 21.7529i	− 0.699525i	−0.936838	−	0.349763i	$-0.886262\pi$
$967$	− 21.7529i	− 0.699525i	0.936838	−	0.349763i	$-0.113738\pi$
$968$	0	0
$969$	0	0
$970$	− 0.0835991i	− 0.00268421i
$971$	5.87314i	0.188478i	0.995550	+	0.0942391i	$0.0300418\pi$
$971$	5.87314i	0.188478i	−0.995550	+	0.0942391i	$0.969958\pi$
$972$	0	0
$973$	−0.952168	−0.0305251
$974$	10.0476	0.321944
$975$	0	0
$976$	− 11.7253i	− 0.375317i
$977$	0.919993i	0.0294332i	0.999892	+	0.0147166i	$0.00468460\pi$
$977$	0.919993i	0.0294332i	−0.999892	+	0.0147166i	$0.995315\pi$
$978$	0	0
$979$	0	0
$980$	− 0.424783i	− 0.0135692i
$981$	0	0
$982$	−17.2364	−0.550037
$983$	46.6359i	1.48745i	0.668483	+	0.743727i	$0.266944\pi$
$983$	46.6359i	1.48745i	−0.668483	+	0.743727i	$0.733056\pi$
$984$	0	0
$985$	− 0.844056i	− 0.0268939i
$986$	16.4698	0.524506
$987$	0	0
$988$	−21.7789	−0.692880
$989$	7.01515	0.223069
$990$	0	0
$991$	6.21090	0.197296	0.0986478	−	0.995122i	$-0.468548\pi$
$991$	6.21090	0.197296	0.0986478	+	0.995122i	$0.468548\pi$
$992$	27.8506	0.884257
$993$	0	0
$994$	−0.565523	−0.0179373
$995$	− 0.415965i	− 0.0131870i
$996$	0	0
$997$	− 28.5668i	− 0.904719i	−0.891836	−	0.452360i	$-0.850582\pi$
$997$	− 28.5668i	− 0.904719i	0.891836	−	0.452360i	$-0.149418\pi$
$998$	15.6322	0.494828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.d.g.1088.12		16
3.2	odd	2	inner	1089.2.d.g.1088.5		16
11.3	even	5		99.2.j.a.35.3	yes	16
11.7	odd	10		99.2.j.a.17.2	✓	16
11.10	odd	2	inner	1089.2.d.g.1088.6		16
33.14	odd	10		99.2.j.a.35.2	yes	16
33.29	even	10		99.2.j.a.17.3	yes	16
33.32	even	2	inner	1089.2.d.g.1088.11		16
44.3	odd	10		1584.2.cd.c.1025.2		16
44.7	even	10		1584.2.cd.c.17.3		16
99.7	odd	30		891.2.u.c.215.2		32
99.14	odd	30		891.2.u.c.431.2		32
99.25	even	15		891.2.u.c.134.2		32
99.29	even	30		891.2.u.c.215.3		32
99.40	odd	30		891.2.u.c.512.3		32
99.47	odd	30		891.2.u.c.134.3		32
99.58	even	15		891.2.u.c.431.3		32
99.95	even	30		891.2.u.c.512.2		32
132.47	even	10		1584.2.cd.c.1025.3		16
132.95	odd	10		1584.2.cd.c.17.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.j.a.17.2	✓	16	11.7	odd	10
99.2.j.a.17.3	yes	16	33.29	even	10
99.2.j.a.35.2	yes	16	33.14	odd	10
99.2.j.a.35.3	yes	16	11.3	even	5
891.2.u.c.134.2		32	99.25	even	15
891.2.u.c.134.3		32	99.47	odd	30
891.2.u.c.215.2		32	99.7	odd	30
891.2.u.c.215.3		32	99.29	even	30
891.2.u.c.431.2		32	99.14	odd	30
891.2.u.c.431.3		32	99.58	even	15
891.2.u.c.512.2		32	99.95	even	30
891.2.u.c.512.3		32	99.40	odd	30
1089.2.d.g.1088.5		16	3.2	odd	2	inner
1089.2.d.g.1088.6		16	11.10	odd	2	inner
1089.2.d.g.1088.11		16	33.32	even	2	inner
1089.2.d.g.1088.12		16	1.1	even	1	trivial
1584.2.cd.c.17.2		16	132.95	odd	10
1584.2.cd.c.17.3		16	44.7	even	10
1584.2.cd.c.1025.2		16	44.3	odd	10
1584.2.cd.c.1025.3		16	132.47	even	10

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +O(q^{10})\)	\(q+0.688291 q^{2} -1.52626 q^{4} +0.0401087i q^{5} +0.246848i q^{7} -2.42709 q^{8} +0.0276065i q^{10} -2.30152i q^{13} +0.169903i q^{14} +1.38197 q^{16} +4.27621 q^{17} +6.20004i q^{19} -0.0612162i q^{20} +6.79984i q^{23} +4.99839 q^{25} -1.58412i q^{26} -0.376753i q^{28} +5.59574 q^{29} +4.79738 q^{31} +5.80538 q^{32} +2.94328 q^{34} -0.00990077 q^{35} -4.03084 q^{37} +4.26743i q^{38} -0.0973475i q^{40} -9.60293 q^{41} -1.03166i q^{43} +4.68027i q^{46} +11.1262i q^{47} +6.93907 q^{49} +3.44035 q^{50} +3.51271i q^{52} +8.96776i q^{53} -0.599123i q^{56} +3.85150 q^{58} +2.78832i q^{59} -8.48450i q^{61} +3.30199 q^{62} +1.23186 q^{64} +0.0923111 q^{65} +7.94588 q^{67} -6.52659 q^{68} -0.00681461 q^{70} +3.32850i q^{71} -11.8537i q^{73} -2.77439 q^{74} -9.46284i q^{76} -3.01100i q^{79} +0.0554289i q^{80} -6.60961 q^{82} +5.29380 q^{83} +0.171513i q^{85} -0.710085i q^{86} +8.54422i q^{89} +0.568126 q^{91} -10.3783i q^{92} +7.65807i q^{94} -0.248676 q^{95} -3.02824 q^{97} +4.77610 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4}+O(q^{10}) \) 16 * q + 16 * q^4	\( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) \) 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Introduction

L-functions

Modular forms

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Database

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