LMFDB - Embedded newform 1089.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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, 0]))

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

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

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:	yes
Analytic conductor:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.46410 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.46410 q^{7} +1.73205 q^{8} -5.19615 q^{10} -1.73205 q^{13} +6.00000 q^{14} -5.00000 q^{16} -1.73205 q^{17} +6.92820 q^{19} +3.00000 q^{20} +6.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} -3.46410 q^{28} +1.73205 q^{29} +4.00000 q^{31} +5.19615 q^{32} +3.00000 q^{34} -10.3923 q^{35} -11.0000 q^{37} -12.0000 q^{38} +5.19615 q^{40} +1.73205 q^{41} +3.46410 q^{43} -10.3923 q^{46} +5.00000 q^{49} -6.92820 q^{50} -1.73205 q^{52} +9.00000 q^{53} -6.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -6.92820 q^{62} +1.00000 q^{64} -5.19615 q^{65} -2.00000 q^{67} -1.73205 q^{68} +18.0000 q^{70} +6.00000 q^{71} +6.92820 q^{73} +19.0526 q^{74} +6.92820 q^{76} -15.0000 q^{80} -3.00000 q^{82} -5.19615 q^{85} -6.00000 q^{86} -9.00000 q^{89} +6.00000 q^{91} +6.00000 q^{92} +20.7846 q^{95} -7.00000 q^{97} -8.66025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{5}+O(q^{10}) $ 2 * q + 2 * q^4 + 6 * q^5	$ 2 q + 2 q^{4} + 6 q^{5} + 12 q^{14} - 10 q^{16} + 6 q^{20} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 8 q^{31} + 6 q^{34} - 22 q^{37} - 24 q^{38} + 10 q^{49} + 18 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} + 2 q^{64} - 4 q^{67} + 36 q^{70} + 12 q^{71} - 30 q^{80} - 6 q^{82} - 12 q^{86} - 18 q^{89} + 12 q^{91} + 12 q^{92} - 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^5 + 12 * q^14 - 10 * q^16 + 6 * q^20 + 12 * q^23 + 8 * q^25 + 6 * q^26 + 8 * q^31 + 6 * q^34 - 22 * q^37 - 24 * q^38 + 10 * q^49 + 18 * q^53 - 12 * q^56 - 6 * q^58 + 12 * q^59 + 2 * q^64 - 4 * q^67 + 36 * q^70 + 12 * q^71 - 30 * q^80 - 6 * q^82 - 12 * q^86 - 18 * q^89 + 12 * q^91 + 12 * q^92 - 14 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−5.19615	−1.64317
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.73205	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	−1.73205	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−1.73205	−0.420084	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205	−0.420084	−0.210042	+	0.977692i	$0.567360\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	3.00000	0.588348
$27$	0	0
$28$	−3.46410	−0.654654
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	3.00000	0.514496
$35$	−10.3923	−1.75662
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	−12.0000	−1.94666
$39$	0	0
$40$	5.19615	0.821584
$41$	1.73205	0.270501	0.135250	−	0.990811i	$-0.456816\pi$
$41$	1.73205	0.270501	0.135250	+	0.990811i	$0.456816\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	0	0
$46$	−10.3923	−1.53226
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	−6.92820	−0.979796
$51$	0	0
$52$	−1.73205	−0.240192
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−6.00000	−0.801784
$57$	0	0
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−6.92820	−0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.19615	−0.644503
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−1.73205	−0.210042
$69$	0	0
$70$	18.0000	2.15141
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	19.0526	2.21481
$75$	0	0
$76$	6.92820	0.794719
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−15.0000	−1.67705
$81$	0	0
$82$	−3.00000	−0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−5.19615	−0.563602
$86$	−6.00000	−0.646997
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	20.7846	2.13246
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−8.66025	−0.874818
$99$	0	0
$100$	4.00000	0.400000
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−15.5885	−1.51408
$107$	−3.46410	−0.334887	−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	0	0
$109$	15.5885	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$109$	15.5885	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$110$	0	0
$111$	0	0
$112$	17.3205	1.63663
$113$	21.0000	1.97551	0.987757	−	0.156001i	$-0.0498603\pi$
$113$	21.0000	1.97551	0.987757	+	0.156001i	$0.0498603\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	1.73205	0.160817
$117$	0	0
$118$	−10.3923	−0.956689
$119$	6.00000	0.550019
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	9.00000	0.789352
$131$	17.3205	1.51330	0.756650	−	0.653820i	$-0.226835\pi$
$131$	17.3205	1.51330	0.756650	+	0.653820i	$0.226835\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	3.46410	0.299253
$135$	0	0
$136$	−3.00000	−0.257248
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	10.3923	0.881464	0.440732	−	0.897639i	$-0.354719\pi$
$139$	10.3923	0.881464	0.440732	+	0.897639i	$0.354719\pi$
$140$	−10.3923	−0.878310
$141$	0	0
$142$	−10.3923	−0.872103
$143$	0	0
$144$	0	0
$145$	5.19615	0.431517
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−11.0000	−0.904194
$149$	12.1244	0.993266	0.496633	−	0.867961i	$-0.334570\pi$
$149$	12.1244	0.993266	0.496633	+	0.867961i	$0.334570\pi$
$150$	0	0
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	15.5885	1.23238
$161$	−20.7846	−1.63806
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	1.73205	0.135250
$165$	0	0
$166$	0	0
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−10.0000	−0.769231
$170$	9.00000	0.690268
$171$	0	0
$172$	3.46410	0.264135
$173$	−20.7846	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	−13.8564	−1.04745
$176$	0	0
$177$	0	0
$178$	15.5885	1.16840
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−10.3923	−0.770329
$183$	0	0
$184$	10.3923	0.766131
$185$	−33.0000	−2.42621
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−36.0000	−2.61171
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−5.19615	−0.374027	−0.187014	−	0.982357i	$-0.559881\pi$
$193$	−5.19615	−0.374027	−0.187014	+	0.982357i	$0.559881\pi$
$194$	12.1244	0.870478
$195$	0	0
$196$	5.00000	0.357143
$197$	19.0526	1.35744	0.678719	−	0.734398i	$-0.262535\pi$
$197$	19.0526	1.35744	0.678719	+	0.734398i	$0.262535\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	6.92820	0.489898
$201$	0	0
$202$	−24.0000	−1.68863
$203$	−6.00000	−0.421117
$204$	0	0
$205$	5.19615	0.362915
$206$	−24.2487	−1.68949
$207$	0	0
$208$	8.66025	0.600481
$209$	0	0
$210$	0	0
$211$	−17.3205	−1.19239	−0.596196	−	0.802839i	$-0.703322\pi$
$211$	−17.3205	−1.19239	−0.596196	+	0.802839i	$0.703322\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	6.00000	0.410152
$215$	10.3923	0.708749
$216$	0	0
$217$	−13.8564	−0.940634
$218$	−27.0000	−1.82867
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	−18.0000	−1.20268
$225$	0	0
$226$	−36.3731	−2.41950
$227$	24.2487	1.60944	0.804722	−	0.593652i	$-0.202314\pi$
$227$	24.2487	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$228$	0	0
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	−31.1769	−2.05574
$231$	0	0
$232$	3.00000	0.196960
$233$	−29.4449	−1.92900	−0.964499	−	0.264088i	$-0.914929\pi$
$233$	−29.4449	−1.92900	−0.964499	+	0.264088i	$0.914929\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	−10.3923	−0.673633
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	−20.7846	−1.33885	−0.669427	−	0.742878i	$-0.733460\pi$
$241$	−20.7846	−1.33885	−0.669427	+	0.742878i	$0.733460\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.0000	0.958315
$246$	0	0
$247$	−12.0000	−0.763542
$248$	6.92820	0.439941
$249$	0	0
$250$	5.19615	0.328634
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	19.0000	1.18750
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	0	0
$259$	38.1051	2.36774
$260$	−5.19615	−0.322252
$261$	0	0
$262$	−30.0000	−1.85341
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	41.5692	2.54877
$267$	0	0
$268$	−2.00000	−0.122169
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−3.46410	−0.210429	−0.105215	−	0.994450i	$-0.533553\pi$
$271$	−3.46410	−0.210429	−0.105215	+	0.994450i	$0.533553\pi$
$272$	8.66025	0.525105
$273$	0	0
$274$	−10.3923	−0.627822
$275$	0	0
$276$	0	0
$277$	−5.19615	−0.312207	−0.156103	−	0.987741i	$-0.549893\pi$
$277$	−5.19615	−0.312207	−0.156103	+	0.987741i	$0.549893\pi$
$278$	−18.0000	−1.07957
$279$	0	0
$280$	−18.0000	−1.07571
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	−31.1769	−1.85328	−0.926638	−	0.375956i	$-0.877314\pi$
$283$	−31.1769	−1.85328	−0.926638	+	0.375956i	$0.877314\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−14.0000	−0.823529
$290$	−9.00000	−0.528498
$291$	0	0
$292$	6.92820	0.405442
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	−19.0526	−1.10741
$297$	0	0
$298$	−21.0000	−1.21650
$299$	−10.3923	−0.601003
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−24.0000	−1.38104
$303$	0	0
$304$	−34.6410	−1.98680
$305$	0	0
$306$	0	0
$307$	3.46410	0.197707	0.0988534	−	0.995102i	$-0.468483\pi$
$307$	3.46410	0.197707	0.0988534	+	0.995102i	$0.468483\pi$
$308$	0	0
$309$	0	0
$310$	−20.7846	−1.18049
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	7.00000	0.395663	0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000	0.395663	0.197832	+	0.980236i	$0.436610\pi$
$314$	24.2487	1.36843
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	36.0000	2.00620
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−6.92820	−0.384308
$326$	−3.46410	−0.191859
$327$	0	0
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	0	0
$334$	−6.00000	−0.328305
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−12.1244	−0.660456	−0.330228	−	0.943901i	$-0.607126\pi$
$337$	−12.1244	−0.660456	−0.330228	+	0.943901i	$0.607126\pi$
$338$	17.3205	0.942111
$339$	0	0
$340$	−5.19615	−0.281801
$341$	0	0
$342$	0	0
$343$	6.92820	0.374088
$344$	6.00000	0.323498
$345$	0	0
$346$	36.0000	1.93537
$347$	−27.7128	−1.48770	−0.743851	−	0.668346i	$-0.767003\pi$
$347$	−27.7128	−1.48770	−0.743851	+	0.668346i	$0.767003\pi$
$348$	0	0
$349$	−1.73205	−0.0927146	−0.0463573	−	0.998925i	$-0.514761\pi$
$349$	−1.73205	−0.0927146	−0.0463573	+	0.998925i	$0.514761\pi$
$350$	24.0000	1.28285
$351$	0	0
$352$	0	0
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	−9.00000	−0.476999
$357$	0	0
$358$	20.7846	1.09850
$359$	−31.1769	−1.64545	−0.822727	−	0.568436i	$-0.807549\pi$
$359$	−31.1769	−1.64545	−0.822727	+	0.568436i	$0.807549\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	12.1244	0.637242
$363$	0	0
$364$	6.00000	0.314485
$365$	20.7846	1.08792
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	−30.0000	−1.56386
$369$	0	0
$370$	57.1577	2.97149
$371$	−31.1769	−1.61862
$372$	0	0
$373$	20.7846	1.07619	0.538093	−	0.842885i	$-0.319145\pi$
$373$	20.7846	1.07619	0.538093	+	0.842885i	$0.319145\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	20.7846	1.06623
$381$	0	0
$382$	20.7846	1.06343
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	9.00000	0.458088
$387$	0	0
$388$	−7.00000	−0.355371
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	0	0
$391$	−10.3923	−0.525561
$392$	8.66025	0.437409
$393$	0	0
$394$	−33.0000	−1.66252
$395$	0	0
$396$	0	0
$397$	11.0000	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$397$	11.0000	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$398$	−17.3205	−0.868199
$399$	0	0
$400$	−20.0000	−1.00000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	−6.92820	−0.345118
$404$	13.8564	0.689382
$405$	0	0
$406$	10.3923	0.515761
$407$	0	0
$408$	0	0
$409$	1.73205	0.0856444	0.0428222	−	0.999083i	$-0.486365\pi$
$409$	1.73205	0.0856444	0.0428222	+	0.999083i	$0.486365\pi$
$410$	−9.00000	−0.444478
$411$	0	0
$412$	14.0000	0.689730
$413$	−20.7846	−1.02274
$414$	0	0
$415$	0	0
$416$	−9.00000	−0.441261
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	30.0000	1.46038
$423$	0	0
$424$	15.5885	0.757042
$425$	−6.92820	−0.336067
$426$	0	0
$427$	0	0
$428$	−3.46410	−0.167444
$429$	0	0
$430$	−18.0000	−0.868037
$431$	17.3205	0.834300	0.417150	−	0.908838i	$-0.363029\pi$
$431$	17.3205	0.834300	0.417150	+	0.908838i	$0.363029\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	24.0000	1.15204
$435$	0	0
$436$	15.5885	0.746552
$437$	41.5692	1.98853
$438$	0	0
$439$	−6.92820	−0.330665	−0.165333	−	0.986238i	$-0.552870\pi$
$439$	−6.92820	−0.330665	−0.165333	+	0.986238i	$0.552870\pi$
$440$	0	0
$441$	0	0
$442$	−5.19615	−0.247156
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	−27.0000	−1.27992
$446$	−34.6410	−1.64030
$447$	0	0
$448$	−3.46410	−0.163663
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	0	0
$452$	21.0000	0.987757
$453$	0	0
$454$	−42.0000	−1.97116
$455$	18.0000	0.843853
$456$	0	0
$457$	−29.4449	−1.37737	−0.688686	−	0.725059i	$-0.741812\pi$
$457$	−29.4449	−1.37737	−0.688686	+	0.725059i	$0.741812\pi$
$458$	39.8372	1.86147
$459$	0	0
$460$	18.0000	0.839254
$461$	15.5885	0.726027	0.363013	−	0.931784i	$-0.381748\pi$
$461$	15.5885	0.726027	0.363013	+	0.931784i	$0.381748\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−8.66025	−0.402042
$465$	0	0
$466$	51.0000	2.36253
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	6.92820	0.319915
$470$	0	0
$471$	0	0
$472$	10.3923	0.478345
$473$	0	0
$474$	0	0
$475$	27.7128	1.27155
$476$	6.00000	0.275010
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−24.2487	−1.10795	−0.553976	−	0.832533i	$-0.686890\pi$
$479$	−24.2487	−1.10795	−0.553976	+	0.832533i	$0.686890\pi$
$480$	0	0
$481$	19.0526	0.868722
$482$	36.0000	1.63976
$483$	0	0
$484$	0	0
$485$	−21.0000	−0.953561
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	0	0
$489$	0	0
$490$	−25.9808	−1.17369
$491$	20.7846	0.937996	0.468998	−	0.883199i	$-0.344615\pi$
$491$	20.7846	0.937996	0.468998	+	0.883199i	$0.344615\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	20.7846	0.935144
$495$	0	0
$496$	−20.0000	−0.898027
$497$	−20.7846	−0.932317
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−10.3923	−0.463831
$503$	−31.1769	−1.39011	−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$504$	0	0
$505$	41.5692	1.84981
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−8.66025	−0.382733
$513$	0	0
$514$	15.5885	0.687577
$515$	42.0000	1.85074
$516$	0	0
$517$	0	0
$518$	−66.0000	−2.89987
$519$	0	0
$520$	−9.00000	−0.394676
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−41.5692	−1.81770	−0.908848	−	0.417128i	$-0.863037\pi$
$523$	−41.5692	−1.81770	−0.908848	+	0.417128i	$0.863037\pi$
$524$	17.3205	0.756650
$525$	0	0
$526$	24.0000	1.04645
$527$	−6.92820	−0.301797
$528$	0	0
$529$	13.0000	0.565217
$530$	−46.7654	−2.03136
$531$	0	0
$532$	−24.0000	−1.04053
$533$	−3.00000	−0.129944
$534$	0	0
$535$	−10.3923	−0.449299
$536$	−3.46410	−0.149626
$537$	0	0
$538$	−36.3731	−1.56815
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	6.00000	0.257722
$543$	0	0
$544$	−9.00000	−0.385872
$545$	46.7654	2.00321
$546$	0	0
$547$	−34.6410	−1.48114	−0.740571	−	0.671978i	$-0.765445\pi$
$547$	−34.6410	−1.48114	−0.740571	+	0.671978i	$0.765445\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	9.00000	0.382373
$555$	0	0
$556$	10.3923	0.440732
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	51.9615	2.19578
$561$	0	0
$562$	−12.0000	−0.506189
$563$	−17.3205	−0.729972	−0.364986	−	0.931013i	$-0.618926\pi$
$563$	−17.3205	−0.729972	−0.364986	+	0.931013i	$0.618926\pi$
$564$	0	0
$565$	63.0000	2.65043
$566$	54.0000	2.26979
$567$	0	0
$568$	10.3923	0.436051
$569$	27.7128	1.16178	0.580891	−	0.813982i	$-0.302704\pi$
$569$	27.7128	1.16178	0.580891	+	0.813982i	$0.302704\pi$
$570$	0	0
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	0	0
$574$	10.3923	0.433766
$575$	24.0000	1.00087
$576$	0	0
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	24.2487	1.00861
$579$	0	0
$580$	5.19615	0.215758
$581$	0	0
$582$	0	0
$583$	0	0
$584$	12.0000	0.496564
$585$	0	0
$586$	33.0000	1.36322
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	27.7128	1.14189
$590$	−31.1769	−1.28353
$591$	0	0
$592$	55.0000	2.26049
$593$	22.5167	0.924648	0.462324	−	0.886711i	$-0.347016\pi$
$593$	22.5167	0.924648	0.462324	+	0.886711i	$0.347016\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	12.1244	0.496633
$597$	0	0
$598$	18.0000	0.736075
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−22.5167	−0.918474	−0.459237	−	0.888314i	$-0.651877\pi$
$601$	−22.5167	−0.918474	−0.459237	+	0.888314i	$0.651877\pi$
$602$	20.7846	0.847117
$603$	0	0
$604$	13.8564	0.563809
$605$	0	0
$606$	0	0
$607$	20.7846	0.843621	0.421811	−	0.906684i	$-0.361395\pi$
$607$	20.7846	0.843621	0.421811	+	0.906684i	$0.361395\pi$
$608$	36.0000	1.45999
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.0526	0.769526	0.384763	−	0.923015i	$-0.374283\pi$
$613$	19.0526	0.769526	0.384763	+	0.923015i	$0.374283\pi$
$614$	−6.00000	−0.242140
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	20.7846	0.833387
$623$	31.1769	1.24908
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−12.1244	−0.484587
$627$	0	0
$628$	−14.0000	−0.558661
$629$	19.0526	0.759675
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	0	0
$634$	10.3923	0.412731
$635$	0	0
$636$	0	0
$637$	−8.66025	−0.343132
$638$	0	0
$639$	0	0
$640$	−36.3731	−1.43777
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−20.7846	−0.819028
$645$	0	0
$646$	20.7846	0.817760
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	12.0000	0.470679
$651$	0	0
$652$	2.00000	0.0783260
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	51.9615	2.03030
$656$	−8.66025	−0.338126
$657$	0	0
$658$	0	0
$659$	−13.8564	−0.539769	−0.269884	−	0.962893i	$-0.586986\pi$
$659$	−13.8564	−0.539769	−0.269884	+	0.962893i	$0.586986\pi$
$660$	0	0
$661$	−41.0000	−1.59472	−0.797358	−	0.603507i	$-0.793769\pi$
$661$	−41.0000	−1.59472	−0.797358	+	0.603507i	$0.793769\pi$
$662$	−34.6410	−1.34636
$663$	0	0
$664$	0	0
$665$	−72.0000	−2.79204
$666$	0	0
$667$	10.3923	0.402392
$668$	3.46410	0.134030
$669$	0	0
$670$	10.3923	0.401490
$671$	0	0
$672$	0	0
$673$	−20.7846	−0.801188	−0.400594	−	0.916256i	$-0.631196\pi$
$673$	−20.7846	−0.801188	−0.400594	+	0.916256i	$0.631196\pi$
$674$	21.0000	0.808890
$675$	0	0
$676$	−10.0000	−0.384615
$677$	−32.9090	−1.26479	−0.632397	−	0.774644i	$-0.717929\pi$
$677$	−32.9090	−1.26479	−0.632397	+	0.774644i	$0.717929\pi$
$678$	0	0
$679$	24.2487	0.930580
$680$	−9.00000	−0.345134
$681$	0	0
$682$	0	0
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	−12.0000	−0.458162
$687$	0	0
$688$	−17.3205	−0.660338
$689$	−15.5885	−0.593873
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	48.0000	1.82206
$695$	31.1769	1.18261
$696$	0	0
$697$	−3.00000	−0.113633
$698$	3.00000	0.113552
$699$	0	0
$700$	−13.8564	−0.523723
$701$	39.8372	1.50463	0.752315	−	0.658804i	$-0.228937\pi$
$701$	39.8372	1.50463	0.752315	+	0.658804i	$0.228937\pi$
$702$	0	0
$703$	−76.2102	−2.87432
$704$	0	0
$705$	0	0
$706$	36.3731	1.36892
$707$	−48.0000	−1.80523
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−31.1769	−1.17005
$711$	0	0
$712$	−15.5885	−0.584202
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	54.0000	2.01526
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−48.4974	−1.80614
$722$	−50.2295	−1.86935
$723$	0	0
$724$	−7.00000	−0.260153
$725$	6.92820	0.257307
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	10.3923	0.385164
$729$	0	0
$730$	−36.0000	−1.33242
$731$	−6.00000	−0.221918
$732$	0	0
$733$	8.66025	0.319874	0.159937	−	0.987127i	$-0.448871\pi$
$733$	8.66025	0.319874	0.159937	+	0.987127i	$0.448871\pi$
$734$	−45.0333	−1.66221
$735$	0	0
$736$	31.1769	1.14920
$737$	0	0
$738$	0	0
$739$	−27.7128	−1.01943	−0.509716	−	0.860343i	$-0.670250\pi$
$739$	−27.7128	−1.01943	−0.509716	+	0.860343i	$0.670250\pi$
$740$	−33.0000	−1.21310
$741$	0	0
$742$	54.0000	1.98240
$743$	51.9615	1.90628	0.953142	−	0.302524i	$-0.0978293\pi$
$743$	51.9615	1.90628	0.953142	+	0.302524i	$0.0978293\pi$
$744$	0	0
$745$	36.3731	1.33261
$746$	−36.0000	−1.31805
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	0	0
$754$	5.19615	0.189233
$755$	41.5692	1.51286
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	−24.2487	−0.880753
$759$	0	0
$760$	36.0000	1.30586
$761$	−32.9090	−1.19295	−0.596475	−	0.802632i	$-0.703432\pi$
$761$	−32.9090	−1.19295	−0.596475	+	0.802632i	$0.703432\pi$
$762$	0	0
$763$	−54.0000	−1.95493
$764$	−12.0000	−0.434145
$765$	0	0
$766$	10.3923	0.375489
$767$	−10.3923	−0.375244
$768$	0	0
$769$	25.9808	0.936890	0.468445	−	0.883493i	$-0.344814\pi$
$769$	25.9808	0.936890	0.468445	+	0.883493i	$0.344814\pi$
$770$	0	0
$771$	0	0
$772$	−5.19615	−0.187014
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	−12.1244	−0.435239
$777$	0	0
$778$	−46.7654	−1.67662
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	18.0000	0.643679
$783$	0	0
$784$	−25.0000	−0.892857
$785$	−42.0000	−1.49904
$786$	0	0
$787$	−6.92820	−0.246964	−0.123482	−	0.992347i	$-0.539406\pi$
$787$	−6.92820	−0.246964	−0.123482	+	0.992347i	$0.539406\pi$
$788$	19.0526	0.678719
$789$	0	0
$790$	0	0
$791$	−72.7461	−2.58655
$792$	0	0
$793$	0	0
$794$	−19.0526	−0.676150
$795$	0	0
$796$	10.0000	0.354441
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	0	0
$800$	20.7846	0.734847
$801$	0	0
$802$	5.19615	0.183483
$803$	0	0
$804$	0	0
$805$	−62.3538	−2.19768
$806$	12.0000	0.422682
$807$	0	0
$808$	24.0000	0.844317
$809$	−20.7846	−0.730748	−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$810$	0	0
$811$	−27.7128	−0.973128	−0.486564	−	0.873645i	$-0.661750\pi$
$811$	−27.7128	−0.973128	−0.486564	+	0.873645i	$0.661750\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	24.0000	0.839654
$818$	−3.00000	−0.104893
$819$	0	0
$820$	5.19615	0.181458
$821$	−55.4256	−1.93437	−0.967184	−	0.254078i	$-0.918228\pi$
$821$	−55.4256	−1.93437	−0.967184	+	0.254078i	$0.918228\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	24.2487	0.844744
$825$	0	0
$826$	36.0000	1.25260
$827$	38.1051	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$827$	38.1051	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	0	0
$832$	−1.73205	−0.0600481
$833$	−8.66025	−0.300060
$834$	0	0
$835$	10.3923	0.359641
$836$	0	0
$837$	0	0
$838$	−51.9615	−1.79498
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−12.1244	−0.417833
$843$	0	0
$844$	−17.3205	−0.596196
$845$	−30.0000	−1.03203
$846$	0	0
$847$	0	0
$848$	−45.0000	−1.54531
$849$	0	0
$850$	12.0000	0.411597
$851$	−66.0000	−2.26245
$852$	0	0
$853$	8.66025	0.296521	0.148261	−	0.988948i	$-0.452633\pi$
$853$	8.66025	0.296521	0.148261	+	0.988948i	$0.452633\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−41.5692	−1.41998	−0.709989	−	0.704213i	$-0.751300\pi$
$857$	−41.5692	−1.41998	−0.709989	+	0.704213i	$0.751300\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	10.3923	0.354375
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−62.3538	−2.12009
$866$	32.9090	1.11829
$867$	0	0
$868$	−13.8564	−0.470317
$869$	0	0
$870$	0	0
$871$	3.46410	0.117377
$872$	27.0000	0.914335
$873$	0	0
$874$	−72.0000	−2.43544
$875$	10.3923	0.351324
$876$	0	0
$877$	−19.0526	−0.643359	−0.321680	−	0.946849i	$-0.604247\pi$
$877$	−19.0526	−0.643359	−0.321680	+	0.946849i	$0.604247\pi$
$878$	12.0000	0.404980
$879$	0	0
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−31.1769	−1.04741
$887$	−31.1769	−1.04682	−0.523409	−	0.852081i	$-0.675340\pi$
$887$	−31.1769	−1.04682	−0.523409	+	0.852081i	$0.675340\pi$
$888$	0	0
$889$	0	0
$890$	46.7654	1.56758
$891$	0	0
$892$	20.0000	0.669650
$893$	0	0
$894$	0	0
$895$	−36.0000	−1.20335
$896$	42.0000	1.40312
$897$	0	0
$898$	36.3731	1.21378
$899$	6.92820	0.231069
$900$	0	0
$901$	−15.5885	−0.519327
$902$	0	0
$903$	0	0
$904$	36.3731	1.20975
$905$	−21.0000	−0.698064
$906$	0	0
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	24.2487	0.804722
$909$	0	0
$910$	−31.1769	−1.03350
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	0	0
$914$	51.0000	1.68693
$915$	0	0
$916$	−23.0000	−0.759941
$917$	−60.0000	−1.98137
$918$	0	0
$919$	−24.2487	−0.799891	−0.399946	−	0.916539i	$-0.630971\pi$
$919$	−24.2487	−0.799891	−0.399946	+	0.916539i	$0.630971\pi$
$920$	31.1769	1.02787
$921$	0	0
$922$	−27.0000	−0.889198
$923$	−10.3923	−0.342067
$924$	0	0
$925$	−44.0000	−1.44671
$926$	58.8897	1.93524
$927$	0	0
$928$	9.00000	0.295439
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	34.6410	1.13531
$932$	−29.4449	−0.964499
$933$	0	0
$934$	−31.1769	−1.02014
$935$	0	0
$936$	0	0
$937$	50.2295	1.64093	0.820463	−	0.571700i	$-0.193716\pi$
$937$	50.2295	1.64093	0.820463	+	0.571700i	$0.193716\pi$
$938$	−12.0000	−0.391814
$939$	0	0
$940$	0	0
$941$	−29.4449	−0.959875	−0.479938	−	0.877303i	$-0.659341\pi$
$941$	−29.4449	−0.959875	−0.479938	+	0.877303i	$0.659341\pi$
$942$	0	0
$943$	10.3923	0.338420
$944$	−30.0000	−0.976417
$945$	0	0
$946$	0	0
$947$	−54.0000	−1.75476	−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000	−1.75476	−0.877382	+	0.479792i	$0.840712\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	−48.0000	−1.55733
$951$	0	0
$952$	10.3923	0.336817
$953$	57.1577	1.85152	0.925759	−	0.378113i	$-0.123427\pi$
$953$	57.1577	1.85152	0.925759	+	0.378113i	$0.123427\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	6.92820	0.224074
$957$	0	0
$958$	42.0000	1.35696
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−33.0000	−1.06396
$963$	0	0
$964$	−20.7846	−0.669427
$965$	−15.5885	−0.501810
$966$	0	0
$967$	10.3923	0.334194	0.167097	−	0.985940i	$-0.446561\pi$
$967$	10.3923	0.334194	0.167097	+	0.985940i	$0.446561\pi$
$968$	0	0
$969$	0	0
$970$	36.3731	1.16787
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	−36.0000	−1.15411
$974$	65.8179	2.10894
$975$	0	0
$976$	0	0
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	0	0
$979$	0	0
$980$	15.0000	0.479157
$981$	0	0
$982$	−36.0000	−1.14881
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	57.1577	1.82120
$986$	5.19615	0.165479
$987$	0	0
$988$	−12.0000	−0.381771
$989$	20.7846	0.660912
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	20.7846	0.659912
$993$	0	0
$994$	36.0000	1.14185
$995$	30.0000	0.951064
$996$	0	0
$997$	−1.73205	−0.0548546	−0.0274273	−	0.999624i	$-0.508731\pi$
$997$	−1.73205	−0.0548546	−0.0274273	+	0.999624i	$0.508731\pi$
$998$	−38.1051	−1.20620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.o.1.1		2
3.2	odd	2		363.2.a.f.1.2	yes	2
11.10	odd	2	inner	1089.2.a.o.1.2		2
12.11	even	2		5808.2.a.ca.1.2		2
15.14	odd	2		9075.2.a.bo.1.1		2
33.2	even	10		363.2.e.m.202.2		8
33.5	odd	10		363.2.e.m.124.1		8
33.8	even	10		363.2.e.m.130.1		8
33.14	odd	10		363.2.e.m.130.2		8
33.17	even	10		363.2.e.m.124.2		8
33.20	odd	10		363.2.e.m.202.1		8
33.26	odd	10		363.2.e.m.148.2		8
33.29	even	10		363.2.e.m.148.1		8
33.32	even	2		363.2.a.f.1.1	✓	2
132.131	odd	2		5808.2.a.ca.1.1		2
165.164	even	2		9075.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.f.1.1	✓	2	33.32	even	2
363.2.a.f.1.2	yes	2	3.2	odd	2
363.2.e.m.124.1		8	33.5	odd	10
363.2.e.m.124.2		8	33.17	even	10
363.2.e.m.130.1		8	33.8	even	10
363.2.e.m.130.2		8	33.14	odd	10
363.2.e.m.148.1		8	33.29	even	10
363.2.e.m.148.2		8	33.26	odd	10
363.2.e.m.202.1		8	33.20	odd	10
363.2.e.m.202.2		8	33.2	even	10
1089.2.a.o.1.1		2	1.1	even	1	trivial
1089.2.a.o.1.2		2	11.10	odd	2	inner
5808.2.a.ca.1.1		2	132.131	odd	2
5808.2.a.ca.1.2		2	12.11	even	2
9075.2.a.bo.1.1		2	15.14	odd	2
9075.2.a.bo.1.2		2	165.164	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.46410 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.46410 q^{7} +1.73205 q^{8} -5.19615 q^{10} -1.73205 q^{13} +6.00000 q^{14} -5.00000 q^{16} -1.73205 q^{17} +6.92820 q^{19} +3.00000 q^{20} +6.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} -3.46410 q^{28} +1.73205 q^{29} +4.00000 q^{31} +5.19615 q^{32} +3.00000 q^{34} -10.3923 q^{35} -11.0000 q^{37} -12.0000 q^{38} +5.19615 q^{40} +1.73205 q^{41} +3.46410 q^{43} -10.3923 q^{46} +5.00000 q^{49} -6.92820 q^{50} -1.73205 q^{52} +9.00000 q^{53} -6.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -6.92820 q^{62} +1.00000 q^{64} -5.19615 q^{65} -2.00000 q^{67} -1.73205 q^{68} +18.0000 q^{70} +6.00000 q^{71} +6.92820 q^{73} +19.0526 q^{74} +6.92820 q^{76} -15.0000 q^{80} -3.00000 q^{82} -5.19615 q^{85} -6.00000 q^{86} -9.00000 q^{89} +6.00000 q^{91} +6.00000 q^{92} +20.7846 q^{95} -7.00000 q^{97} -8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{5}+O(q^{10}) \) 2 * q + 2 * q^4 + 6 * q^5	\( 2 q + 2 q^{4} + 6 q^{5} + 12 q^{14} - 10 q^{16} + 6 q^{20} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 8 q^{31} + 6 q^{34} - 22 q^{37} - 24 q^{38} + 10 q^{49} + 18 q^{53} - 12 q^{56} - 6 q^{58} + 12 q^{59} + 2 q^{64} - 4 q^{67} + 36 q^{70} + 12 q^{71} - 30 q^{80} - 6 q^{82} - 12 q^{86} - 18 q^{89} + 12 q^{91} + 12 q^{92} - 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^5 + 12 * q^14 - 10 * q^16 + 6 * q^20 + 12 * q^23 + 8 * q^25 + 6 * q^26 + 8 * q^31 + 6 * q^34 - 22 * q^37 - 24 * q^38 + 10 * q^49 + 18 * q^53 - 12 * q^56 - 6 * q^58 + 12 * q^59 + 2 * q^64 - 4 * q^67 + 36 * q^70 + 12 * q^71 - 30 * q^80 - 6 * q^82 - 12 * q^86 - 18 * q^89 + 12 * q^91 + 12 * q^92 - 14 * q^97

Introduction

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