LMFDB - Embedded newform 1089.2.a.v.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.14896$ 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)

$f(q)$	$=$	$q+2.14896 q^{2} +2.61803 q^{4} -2.14896 q^{5} -4.23607 q^{7} +1.32813 q^{8} -4.61803 q^{10} +0.236068 q^{13} -9.10315 q^{14} -2.38197 q^{16} -6.13335 q^{17} -2.38197 q^{19} -5.62605 q^{20} +3.98439 q^{23} -0.381966 q^{25} +0.507301 q^{26} -11.0902 q^{28} +6.95418 q^{29} -4.61803 q^{31} -7.77501 q^{32} -13.1803 q^{34} +9.10315 q^{35} +1.76393 q^{37} -5.11875 q^{38} -2.85410 q^{40} -2.96979 q^{41} +2.70820 q^{43} +8.56231 q^{46} +3.47709 q^{47} +10.9443 q^{49} -0.820830 q^{50} +0.618034 q^{52} -1.83543 q^{53} -5.62605 q^{56} +14.9443 q^{58} -7.46149 q^{59} -11.3262 q^{61} -9.92398 q^{62} -11.9443 q^{64} -0.507301 q^{65} -2.85410 q^{67} -16.0573 q^{68} +19.5623 q^{70} +10.7448 q^{71} +2.47214 q^{73} +3.79062 q^{74} -6.23607 q^{76} -9.47214 q^{79} +5.11875 q^{80} -6.38197 q^{82} +8.28232 q^{83} +13.1803 q^{85} +5.81983 q^{86} +8.90937 q^{89} -1.00000 q^{91} +10.4313 q^{92} +7.47214 q^{94} +5.11875 q^{95} +11.7984 q^{97} +23.5188 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} - 8 q^{7} - 14 q^{10} - 8 q^{13} - 14 q^{16} - 14 q^{19} - 6 q^{25} - 22 q^{28} - 14 q^{31} - 8 q^{34} + 16 q^{37} + 2 q^{40} - 16 q^{43} - 6 q^{46} + 8 q^{49} - 2 q^{52} + 24 q^{58} - 14 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 - 8 * q^7 - 14 * q^10 - 8 * q^13 - 14 * q^16 - 14 * q^19 - 6 * q^25 - 22 * q^28 - 14 * q^31 - 8 * q^34 + 16 * q^37 + 2 * q^40 - 16 * q^43 - 6 * q^46 + 8 * q^49 - 2 * q^52 + 24 * q^58 - 14 * q^61 - 12 * q^64 + 2 * q^67 + 38 * q^70 - 8 * q^73 - 16 * q^76 - 20 * q^79 - 30 * q^82 + 8 * q^85 - 4 * q^91 + 12 * q^94 - 2 * q^97

Coefficient data

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

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

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

$2$	2.14896	1.51954	0.759772	−	0.650189i	$-0.225310\pi$
$2$	2.14896	1.51954	0.759772	+	0.650189i	$0.225310\pi$
$3$	0	0
$3$	0	0
$4$	2.61803	1.30902
$5$	−2.14896	−0.961045	−0.480522	−	0.876982i	$-0.659553\pi$
$5$	−2.14896	−0.961045	−0.480522	+	0.876982i	$0.659553\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	1.32813	0.469565
$9$	0	0
$10$	−4.61803	−1.46035
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	−9.10315	−2.43292
$15$	0	0
$16$	−2.38197	−0.595492
$17$	−6.13335	−1.48756	−0.743778	−	0.668426i	$-0.766968\pi$
$17$	−6.13335	−1.48756	−0.743778	+	0.668426i	$0.766968\pi$
$18$	0	0
$19$	−2.38197	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$19$	−2.38197	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$20$	−5.62605	−1.25802
$21$	0	0
$22$	0	0
$23$	3.98439	0.830803	0.415402	−	0.909638i	$-0.363641\pi$
$23$	3.98439	0.830803	0.415402	+	0.909638i	$0.363641\pi$
$24$	0	0
$25$	−0.381966	−0.0763932
$26$	0.507301	0.0994899
$27$	0	0
$28$	−11.0902	−2.09585
$29$	6.95418	1.29136	0.645680	−	0.763608i	$-0.276574\pi$
$29$	6.95418	1.29136	0.645680	+	0.763608i	$0.276574\pi$
$30$	0	0
$31$	−4.61803	−0.829423	−0.414712	−	0.909953i	$-0.636118\pi$
$31$	−4.61803	−0.829423	−0.414712	+	0.909953i	$0.636118\pi$
$32$	−7.77501	−1.37444
$33$	0	0
$34$	−13.1803	−2.26041
$35$	9.10315	1.53871
$36$	0	0
$37$	1.76393	0.289989	0.144994	−	0.989432i	$-0.453684\pi$
$37$	1.76393	0.289989	0.144994	+	0.989432i	$0.453684\pi$
$38$	−5.11875	−0.830371
$39$	0	0
$40$	−2.85410	−0.451273
$41$	−2.96979	−0.463803	−0.231902	−	0.972739i	$-0.574495\pi$
$41$	−2.96979	−0.463803	−0.231902	+	0.972739i	$0.574495\pi$
$42$	0	0
$43$	2.70820	0.412997	0.206499	−	0.978447i	$-0.433793\pi$
$43$	2.70820	0.412997	0.206499	+	0.978447i	$0.433793\pi$
$44$	0	0
$45$	0	0
$46$	8.56231	1.26244
$47$	3.47709	0.507186	0.253593	−	0.967311i	$-0.418388\pi$
$47$	3.47709	0.507186	0.253593	+	0.967311i	$0.418388\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	−0.820830	−0.116083
$51$	0	0
$52$	0.618034	0.0857059
$53$	−1.83543	−0.252116	−0.126058	−	0.992023i	$-0.540233\pi$
$53$	−1.83543	−0.252116	−0.126058	+	0.992023i	$0.540233\pi$
$54$	0	0
$55$	0	0
$56$	−5.62605	−0.751813
$57$	0	0
$58$	14.9443	1.96228
$59$	−7.46149	−0.971403	−0.485701	−	0.874125i	$-0.661436\pi$
$59$	−7.46149	−0.971403	−0.485701	+	0.874125i	$0.661436\pi$
$60$	0	0
$61$	−11.3262	−1.45018	−0.725088	−	0.688656i	$-0.758201\pi$
$61$	−11.3262	−1.45018	−0.725088	+	0.688656i	$0.758201\pi$
$62$	−9.92398	−1.26035
$63$	0	0
$64$	−11.9443	−1.49303
$65$	−0.507301	−0.0629229
$66$	0	0
$67$	−2.85410	−0.348684	−0.174342	−	0.984685i	$-0.555780\pi$
$67$	−2.85410	−0.348684	−0.174342	+	0.984685i	$0.555780\pi$
$68$	−16.0573	−1.94724
$69$	0	0
$70$	19.5623	2.33814
$71$	10.7448	1.27517	0.637587	−	0.770378i	$-0.279933\pi$
$71$	10.7448	1.27517	0.637587	+	0.770378i	$0.279933\pi$
$72$	0	0
$73$	2.47214	0.289342	0.144671	−	0.989480i	$-0.453788\pi$
$73$	2.47214	0.289342	0.144671	+	0.989480i	$0.453788\pi$
$74$	3.79062	0.440651
$75$	0	0
$76$	−6.23607	−0.715326
$77$	0	0
$78$	0	0
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	5.11875	0.572294
$81$	0	0
$82$	−6.38197	−0.704770
$83$	8.28232	0.909102	0.454551	−	0.890721i	$-0.349800\pi$
$83$	8.28232	0.909102	0.454551	+	0.890721i	$0.349800\pi$
$84$	0	0
$85$	13.1803	1.42961
$86$	5.81983	0.627568
$87$	0	0
$88$	0	0
$89$	8.90937	0.944392	0.472196	−	0.881494i	$-0.343462\pi$
$89$	8.90937	0.944392	0.472196	+	0.881494i	$0.343462\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	10.4313	1.08754
$93$	0	0
$94$	7.47214	0.770692
$95$	5.11875	0.525173
$96$	0	0
$97$	11.7984	1.19794	0.598972	−	0.800770i	$-0.295576\pi$
$97$	11.7984	1.19794	0.598972	+	0.800770i	$0.295576\pi$
$98$	23.5188	2.37576
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−0.313529	−0.0311973	−0.0155987	−	0.999878i	$-0.504965\pi$
$101$	−0.313529	−0.0311973	−0.0155987	+	0.999878i	$0.504965\pi$
$102$	0	0
$103$	−10.9443	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$103$	−10.9443	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$104$	0.313529	0.0307441
$105$	0	0
$106$	−3.94427	−0.383102
$107$	−10.9386	−1.05747	−0.528736	−	0.848786i	$-0.677334\pi$
$107$	−10.9386	−1.05747	−0.528736	+	0.848786i	$0.677334\pi$
$108$	0	0
$109$	−13.7082	−1.31301	−0.656504	−	0.754323i	$-0.727965\pi$
$109$	−13.7082	−1.31301	−0.656504	+	0.754323i	$0.727965\pi$
$110$	0	0
$111$	0	0
$112$	10.0902	0.953431
$113$	17.8928	1.68321	0.841605	−	0.540094i	$-0.181611\pi$
$113$	17.8928	1.68321	0.841605	+	0.540094i	$0.181611\pi$
$114$	0	0
$115$	−8.56231	−0.798439
$116$	18.2063	1.69041
$117$	0	0
$118$	−16.0344	−1.47609
$119$	25.9813	2.38170
$120$	0	0
$121$	0	0
$122$	−24.3396	−2.20361
$123$	0	0
$124$	−12.0902	−1.08573
$125$	11.5656	1.03446
$126$	0	0
$127$	−2.76393	−0.245259	−0.122630	−	0.992453i	$-0.539133\pi$
$127$	−2.76393	−0.245259	−0.122630	+	0.992453i	$0.539133\pi$
$128$	−10.1177	−0.894291
$129$	0	0
$130$	−1.09017	−0.0956142
$131$	2.46249	0.215149	0.107574	−	0.994197i	$-0.465692\pi$
$131$	2.46249	0.215149	0.107574	+	0.994197i	$0.465692\pi$
$132$	0	0
$133$	10.0902	0.874929
$134$	−6.13335	−0.529841
$135$	0	0
$136$	−8.14590	−0.698505
$137$	0.313529	0.0267866	0.0133933	−	0.999910i	$-0.495737\pi$
$137$	0.313529	0.0267866	0.0133933	+	0.999910i	$0.495737\pi$
$138$	0	0
$139$	−11.3820	−0.965406	−0.482703	−	0.875784i	$-0.660345\pi$
$139$	−11.3820	−0.965406	−0.482703	+	0.875784i	$0.660345\pi$
$140$	23.8323	2.01420
$141$	0	0
$142$	23.0902	1.93768
$143$	0	0
$144$	0	0
$145$	−14.9443	−1.24105
$146$	5.31252	0.439668
$147$	0	0
$148$	4.61803	0.379600
$149$	−4.61145	−0.377785	−0.188892	−	0.981998i	$-0.560490\pi$
$149$	−4.61145	−0.377785	−0.188892	+	0.981998i	$0.560490\pi$
$150$	0	0
$151$	−13.2361	−1.07714	−0.538568	−	0.842582i	$-0.681034\pi$
$151$	−13.2361	−1.07714	−0.538568	+	0.842582i	$0.681034\pi$
$152$	−3.16356	−0.256599
$153$	0	0
$154$	0	0
$155$	9.92398	0.797113
$156$	0	0
$157$	−20.6525	−1.64825	−0.824124	−	0.566410i	$-0.808332\pi$
$157$	−20.6525	−1.64825	−0.824124	+	0.566410i	$0.808332\pi$
$158$	−20.3553	−1.61938
$159$	0	0
$160$	16.7082	1.32090
$161$	−16.8782	−1.33019
$162$	0	0
$163$	1.38197	0.108244	0.0541220	−	0.998534i	$-0.482764\pi$
$163$	1.38197	0.108244	0.0541220	+	0.998534i	$0.482764\pi$
$164$	−7.77501	−0.607127
$165$	0	0
$166$	17.7984	1.38142
$167$	−10.7448	−0.831458	−0.415729	−	0.909489i	$-0.636474\pi$
$167$	−10.7448	−0.831458	−0.415729	+	0.909489i	$0.636474\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	28.3240	2.17235
$171$	0	0
$172$	7.09017	0.540620
$173$	−13.4011	−1.01886	−0.509432	−	0.860511i	$-0.670145\pi$
$173$	−13.4011	−1.01886	−0.509432	+	0.860511i	$0.670145\pi$
$174$	0	0
$175$	1.61803	0.122312
$176$	0	0
$177$	0	0
$178$	19.1459	1.43505
$179$	1.95519	0.146138	0.0730689	−	0.997327i	$-0.476721\pi$
$179$	1.95519	0.146138	0.0730689	+	0.997327i	$0.476721\pi$
$180$	0	0
$181$	12.2361	0.909500	0.454750	−	0.890619i	$-0.349729\pi$
$181$	12.2361	0.909500	0.454750	+	0.890619i	$0.349729\pi$
$182$	−2.14896	−0.159292
$183$	0	0
$184$	5.29180	0.390116
$185$	−3.79062	−0.278692
$186$	0	0
$187$	0	0
$188$	9.10315	0.663915
$189$	0	0
$190$	11.0000	0.798024
$191$	−9.92398	−0.718074	−0.359037	−	0.933323i	$-0.616895\pi$
$191$	−9.92398	−0.718074	−0.359037	+	0.933323i	$0.616895\pi$
$192$	0	0
$193$	7.38197	0.531366	0.265683	−	0.964061i	$-0.414403\pi$
$193$	7.38197	0.531366	0.265683	+	0.964061i	$0.414403\pi$
$194$	25.3542	1.82033
$195$	0	0
$196$	28.6525	2.04661
$197$	−2.46249	−0.175445	−0.0877226	−	0.996145i	$-0.527959\pi$
$197$	−2.46249	−0.175445	−0.0877226	+	0.996145i	$0.527959\pi$
$198$	0	0
$199$	0.416408	0.0295184	0.0147592	−	0.999891i	$-0.495302\pi$
$199$	0.416408	0.0295184	0.0147592	+	0.999891i	$0.495302\pi$
$200$	−0.507301	−0.0358716
$201$	0	0
$202$	−0.673762	−0.0474057
$203$	−29.4584	−2.06757
$204$	0	0
$205$	6.38197	0.445736
$206$	−23.5188	−1.63863
$207$	0	0
$208$	−0.562306	−0.0389889
$209$	0	0
$210$	0	0
$211$	−18.0344	−1.24154	−0.620771	−	0.783992i	$-0.713180\pi$
$211$	−18.0344	−1.24154	−0.620771	+	0.783992i	$0.713180\pi$
$212$	−4.80522	−0.330024
$213$	0	0
$214$	−23.5066	−1.60688
$215$	−5.81983	−0.396909
$216$	0	0
$217$	19.5623	1.32798
$218$	−29.4584	−1.99517
$219$	0	0
$220$	0	0
$221$	−1.44789	−0.0973955
$222$	0	0
$223$	3.18034	0.212971	0.106486	−	0.994314i	$-0.466040\pi$
$223$	3.18034	0.212971	0.106486	+	0.994314i	$0.466040\pi$
$224$	32.9355	2.20059
$225$	0	0
$226$	38.4508	2.55771
$227$	10.9386	0.726019	0.363009	−	0.931785i	$-0.381749\pi$
$227$	10.9386	0.726019	0.363009	+	0.931785i	$0.381749\pi$
$228$	0	0
$229$	−18.4721	−1.22067	−0.610337	−	0.792142i	$-0.708966\pi$
$229$	−18.4721	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$230$	−18.4001	−1.21326
$231$	0	0
$232$	9.23607	0.606378
$233$	23.0115	1.50753	0.753767	−	0.657142i	$-0.228235\pi$
$233$	23.0115	1.50753	0.753767	+	0.657142i	$0.228235\pi$
$234$	0	0
$235$	−7.47214	−0.487428
$236$	−19.5344	−1.27158
$237$	0	0
$238$	55.8328	3.61910
$239$	11.4459	0.740372	0.370186	−	0.928958i	$-0.379294\pi$
$239$	11.4459	0.740372	0.370186	+	0.928958i	$0.379294\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	−29.6525	−1.89831
$245$	−23.5188	−1.50256
$246$	0	0
$247$	−0.562306	−0.0357787
$248$	−6.13335	−0.389468
$249$	0	0
$250$	24.8541	1.57191
$251$	−22.6980	−1.43268	−0.716342	−	0.697749i	$-0.754185\pi$
$251$	−22.6980	−1.43268	−0.716342	+	0.697749i	$0.754185\pi$
$252$	0	0
$253$	0	0
$254$	−5.93958	−0.372683
$255$	0	0
$256$	2.14590	0.134119
$257$	−12.3865	−0.772647	−0.386323	−	0.922363i	$-0.626255\pi$
$257$	−12.3865	−0.772647	−0.386323	+	0.922363i	$0.626255\pi$
$258$	0	0
$259$	−7.47214	−0.464296
$260$	−1.32813	−0.0823672
$261$	0	0
$262$	5.29180	0.326928
$263$	−6.44688	−0.397532	−0.198766	−	0.980047i	$-0.563693\pi$
$263$	−6.44688	−0.397532	−0.198766	+	0.980047i	$0.563693\pi$
$264$	0	0
$265$	3.94427	0.242295
$266$	21.6834	1.32949
$267$	0	0
$268$	−7.47214	−0.456433
$269$	20.2355	1.23378	0.616890	−	0.787049i	$-0.288392\pi$
$269$	20.2355	1.23378	0.616890	+	0.787049i	$0.288392\pi$
$270$	0	0
$271$	−0.965558	−0.0586535	−0.0293267	−	0.999570i	$-0.509336\pi$
$271$	−0.965558	−0.0586535	−0.0293267	+	0.999570i	$0.509336\pi$
$272$	14.6094	0.885827
$273$	0	0
$274$	0.673762	0.0407035
$275$	0	0
$276$	0	0
$277$	19.3820	1.16455	0.582275	−	0.812992i	$-0.302163\pi$
$277$	19.3820	1.16455	0.582275	+	0.812992i	$0.302163\pi$
$278$	−24.4594	−1.46698
$279$	0	0
$280$	12.0902	0.722526
$281$	−16.5646	−0.988163	−0.494082	−	0.869416i	$-0.664496\pi$
$281$	−16.5646	−0.988163	−0.494082	+	0.869416i	$0.664496\pi$
$282$	0	0
$283$	−10.9443	−0.650569	−0.325285	−	0.945616i	$-0.605460\pi$
$283$	−10.9443	−0.650569	−0.325285	+	0.945616i	$0.605460\pi$
$284$	28.1303	1.66922
$285$	0	0
$286$	0	0
$287$	12.5802	0.742588
$288$	0	0
$289$	20.6180	1.21283
$290$	−32.1147	−1.88584
$291$	0	0
$292$	6.47214	0.378753
$293$	−4.99899	−0.292044	−0.146022	−	0.989281i	$-0.546647\pi$
$293$	−4.99899	−0.292044	−0.146022	+	0.989281i	$0.546647\pi$
$294$	0	0
$295$	16.0344	0.933561
$296$	2.34273	0.136169
$297$	0	0
$298$	−9.90983	−0.574061
$299$	0.940588	0.0543956
$300$	0	0
$301$	−11.4721	−0.661243
$302$	−28.4438	−1.63676
$303$	0	0
$304$	5.67376	0.325413
$305$	24.3396	1.39368
$306$	0	0
$307$	17.2705	0.985680	0.492840	−	0.870120i	$-0.335959\pi$
$307$	17.2705	0.985680	0.492840	+	0.870120i	$0.335959\pi$
$308$	0	0
$309$	0	0
$310$	21.3262	1.21125
$311$	−28.7573	−1.63068	−0.815339	−	0.578984i	$-0.803449\pi$
$311$	−28.7573	−1.63068	−0.815339	+	0.578984i	$0.803449\pi$
$312$	0	0
$313$	31.6525	1.78910	0.894552	−	0.446964i	$-0.147495\pi$
$313$	31.6525	1.78910	0.894552	+	0.446964i	$0.147495\pi$
$314$	−44.3814	−2.50459
$315$	0	0
$316$	−24.7984	−1.39502
$317$	−0.313529	−0.0176096	−0.00880478	−	0.999961i	$-0.502803\pi$
$317$	−0.313529	−0.0176096	−0.00880478	+	0.999961i	$0.502803\pi$
$318$	0	0
$319$	0	0
$320$	25.6678	1.43487
$321$	0	0
$322$	−36.2705	−2.02128
$323$	14.6094	0.812891
$324$	0	0
$325$	−0.0901699	−0.00500173
$326$	2.96979	0.164482
$327$	0	0
$328$	−3.94427	−0.217786
$329$	−14.7292	−0.812047
$330$	0	0
$331$	7.29180	0.400793	0.200397	−	0.979715i	$-0.435777\pi$
$331$	7.29180	0.400793	0.200397	+	0.979715i	$0.435777\pi$
$332$	21.6834	1.19003
$333$	0	0
$334$	−23.0902	−1.26344
$335$	6.13335	0.335101
$336$	0	0
$337$	−26.6525	−1.45185	−0.725926	−	0.687772i	$-0.758589\pi$
$337$	−26.6525	−1.45185	−0.725926	+	0.687772i	$0.758589\pi$
$338$	−27.8167	−1.51303
$339$	0	0
$340$	34.5066	1.87138
$341$	0	0
$342$	0	0
$343$	−16.7082	−0.902158
$344$	3.59685	0.193929
$345$	0	0
$346$	−28.7984	−1.54821
$347$	25.1605	1.35069	0.675343	−	0.737504i	$-0.263996\pi$
$347$	25.1605	1.35069	0.675343	+	0.737504i	$0.263996\pi$
$348$	0	0
$349$	9.18034	0.491412	0.245706	−	0.969344i	$-0.420980\pi$
$349$	9.18034	0.491412	0.245706	+	0.969344i	$0.420980\pi$
$350$	3.47709	0.185858
$351$	0	0
$352$	0	0
$353$	−33.7563	−1.79667	−0.898334	−	0.439314i	$-0.855222\pi$
$353$	−33.7563	−1.79667	−0.898334	+	0.439314i	$0.855222\pi$
$354$	0	0
$355$	−23.0902	−1.22550
$356$	23.3250	1.23622
$357$	0	0
$358$	4.20163	0.222063
$359$	2.02920	0.107097	0.0535486	−	0.998565i	$-0.482947\pi$
$359$	2.02920	0.107097	0.0535486	+	0.998565i	$0.482947\pi$
$360$	0	0
$361$	−13.3262	−0.701381
$362$	26.2948	1.38203
$363$	0	0
$364$	−2.61803	−0.137222
$365$	−5.31252	−0.278070
$366$	0	0
$367$	23.0344	1.20239	0.601194	−	0.799103i	$-0.294692\pi$
$367$	23.0344	1.20239	0.601194	+	0.799103i	$0.294692\pi$
$368$	−9.49069	−0.494736
$369$	0	0
$370$	−8.14590	−0.423485
$371$	7.77501	0.403659
$372$	0	0
$373$	−14.4164	−0.746453	−0.373227	−	0.927740i	$-0.621749\pi$
$373$	−14.4164	−0.746453	−0.373227	+	0.927740i	$0.621749\pi$
$374$	0	0
$375$	0	0
$376$	4.61803	0.238157
$377$	1.64166	0.0845498
$378$	0	0
$379$	6.23607	0.320325	0.160163	−	0.987091i	$-0.448798\pi$
$379$	6.23607	0.320325	0.160163	+	0.987091i	$0.448798\pi$
$380$	13.4011	0.687460
$381$	0	0
$382$	−21.3262	−1.09115
$383$	−34.0698	−1.74089	−0.870444	−	0.492267i	$-0.836168\pi$
$383$	−34.0698	−1.74089	−0.870444	+	0.492267i	$0.836168\pi$
$384$	0	0
$385$	0	0
$386$	15.8636	0.807434
$387$	0	0
$388$	30.8885	1.56813
$389$	22.3104	1.13118	0.565592	−	0.824685i	$-0.308648\pi$
$389$	22.3104	1.13118	0.565592	+	0.824685i	$0.308648\pi$
$390$	0	0
$391$	−24.4377	−1.23587
$392$	14.5354	0.734150
$393$	0	0
$394$	−5.29180	−0.266597
$395$	20.3553	1.02418
$396$	0	0
$397$	24.4164	1.22542	0.612712	−	0.790306i	$-0.290078\pi$
$397$	24.4164	1.22542	0.612712	+	0.790306i	$0.290078\pi$
$398$	0.894844	0.0448545
$399$	0	0
$400$	0.909830	0.0454915
$401$	−19.0271	−0.950169	−0.475085	−	0.879940i	$-0.657583\pi$
$401$	−19.0271	−0.950169	−0.475085	+	0.879940i	$0.657583\pi$
$402$	0	0
$403$	−1.09017	−0.0543052
$404$	−0.820830	−0.0408378
$405$	0	0
$406$	−63.3050	−3.14177
$407$	0	0
$408$	0	0
$409$	−5.05573	−0.249990	−0.124995	−	0.992157i	$-0.539891\pi$
$409$	−5.05573	−0.249990	−0.124995	+	0.992157i	$0.539891\pi$
$410$	13.7146	0.677316
$411$	0	0
$412$	−28.6525	−1.41161
$413$	31.6074	1.55530
$414$	0	0
$415$	−17.7984	−0.873688
$416$	−1.83543	−0.0899895
$417$	0	0
$418$	0	0
$419$	−35.2782	−1.72345	−0.861727	−	0.507372i	$-0.830617\pi$
$419$	−35.2782	−1.72345	−0.861727	+	0.507372i	$0.830617\pi$
$420$	0	0
$421$	−29.2148	−1.42384	−0.711921	−	0.702260i	$-0.752174\pi$
$421$	−29.2148	−1.42384	−0.711921	+	0.702260i	$0.752174\pi$
$422$	−38.7553	−1.88658
$423$	0	0
$424$	−2.43769	−0.118385
$425$	2.34273	0.113639
$426$	0	0
$427$	47.9787	2.32185
$428$	−28.6376	−1.38425
$429$	0	0
$430$	−12.5066	−0.603121
$431$	−35.5918	−1.71439	−0.857197	−	0.514988i	$-0.827796\pi$
$431$	−35.5918	−1.71439	−0.857197	+	0.514988i	$0.827796\pi$
$432$	0	0
$433$	−2.65248	−0.127470	−0.0637349	−	0.997967i	$-0.520301\pi$
$433$	−2.65248	−0.127470	−0.0637349	+	0.997967i	$0.520301\pi$
$434$	42.0386	2.01792
$435$	0	0
$436$	−35.8885	−1.71875
$437$	−9.49069	−0.454001
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	0	0
$442$	−3.11146	−0.147997
$443$	21.8772	1.03941	0.519707	−	0.854344i	$-0.326041\pi$
$443$	21.8772	1.03941	0.519707	+	0.854344i	$0.326041\pi$
$444$	0	0
$445$	−19.1459	−0.907603
$446$	6.83443	0.323619
$447$	0	0
$448$	50.5967	2.39047
$449$	29.4584	1.39023	0.695114	−	0.718900i	$-0.255354\pi$
$449$	29.4584	1.39023	0.695114	+	0.718900i	$0.255354\pi$
$450$	0	0
$451$	0	0
$452$	46.8439	2.20335
$453$	0	0
$454$	23.5066	1.10322
$455$	2.14896	0.100745
$456$	0	0
$457$	35.0902	1.64145	0.820724	−	0.571324i	$-0.193570\pi$
$457$	35.0902	1.64145	0.820724	+	0.571324i	$0.193570\pi$
$458$	−39.6959	−1.85487
$459$	0	0
$460$	−22.4164	−1.04517
$461$	1.52190	0.0708821	0.0354410	−	0.999372i	$-0.488716\pi$
$461$	1.52190	0.0708821	0.0354410	+	0.999372i	$0.488716\pi$
$462$	0	0
$463$	−21.5623	−1.00209	−0.501043	−	0.865423i	$-0.667050\pi$
$463$	−21.5623	−1.00209	−0.501043	+	0.865423i	$0.667050\pi$
$464$	−16.5646	−0.768994
$465$	0	0
$466$	49.4508	2.29077
$467$	21.1761	0.979912	0.489956	−	0.871747i	$-0.337013\pi$
$467$	21.1761	0.979912	0.489956	+	0.871747i	$0.337013\pi$
$468$	0	0
$469$	12.0902	0.558272
$470$	−16.0573	−0.740669
$471$	0	0
$472$	−9.90983	−0.456137
$473$	0	0
$474$	0	0
$475$	0.909830	0.0417459
$476$	68.0199	3.11769
$477$	0	0
$478$	24.5967	1.12503
$479$	−36.8459	−1.68353	−0.841765	−	0.539844i	$-0.818483\pi$
$479$	−36.8459	−1.68353	−0.841765	+	0.539844i	$0.818483\pi$
$480$	0	0
$481$	0.416408	0.0189866
$482$	30.0855	1.37035
$483$	0	0
$484$	0	0
$485$	−25.3542	−1.15128
$486$	0	0
$487$	18.8885	0.855922	0.427961	−	0.903797i	$-0.359232\pi$
$487$	18.8885	0.855922	0.427961	+	0.903797i	$0.359232\pi$
$488$	−15.0427	−0.680952
$489$	0	0
$490$	−50.5410	−2.28321
$491$	−19.4147	−0.876172	−0.438086	−	0.898933i	$-0.644343\pi$
$491$	−19.4147	−0.876172	−0.438086	+	0.898933i	$0.644343\pi$
$492$	0	0
$493$	−42.6525	−1.92097
$494$	−1.20837	−0.0543673
$495$	0	0
$496$	11.0000	0.493915
$497$	−45.5157	−2.04166
$498$	0	0
$499$	32.7426	1.46576	0.732881	−	0.680357i	$-0.238175\pi$
$499$	32.7426	1.46576	0.732881	+	0.680357i	$0.238175\pi$
$500$	30.2792	1.35413
$501$	0	0
$502$	−48.7771	−2.17703
$503$	5.93958	0.264833	0.132416	−	0.991194i	$-0.457726\pi$
$503$	5.93958	0.264833	0.132416	+	0.991194i	$0.457726\pi$
$504$	0	0
$505$	0.673762	0.0299820
$506$	0	0
$507$	0	0
$508$	−7.23607	−0.321049
$509$	16.3709	0.725626	0.362813	−	0.931862i	$-0.381816\pi$
$509$	16.3709	0.725626	0.362813	+	0.931862i	$0.381816\pi$
$510$	0	0
$511$	−10.4721	−0.463260
$512$	24.8469	1.09809
$513$	0	0
$514$	−26.6180	−1.17407
$515$	23.5188	1.03636
$516$	0	0
$517$	0	0
$518$	−16.0573	−0.705519
$519$	0	0
$520$	−0.673762	−0.0295464
$521$	−26.8021	−1.17422	−0.587111	−	0.809506i	$-0.699735\pi$
$521$	−26.8021	−1.17422	−0.587111	+	0.809506i	$0.699735\pi$
$522$	0	0
$523$	−37.4508	−1.63761	−0.818806	−	0.574071i	$-0.805363\pi$
$523$	−37.4508	−1.63761	−0.818806	+	0.574071i	$0.805363\pi$
$524$	6.44688	0.281633
$525$	0	0
$526$	−13.8541	−0.604068
$527$	28.3240	1.23381
$528$	0	0
$529$	−7.12461	−0.309766
$530$	8.47609	0.368178
$531$	0	0
$532$	26.4164	1.14530
$533$	−0.701073	−0.0303668
$534$	0	0
$535$	23.5066	1.01628
$536$	−3.79062	−0.163730
$537$	0	0
$538$	43.4853	1.87478
$539$	0	0
$540$	0	0
$541$	38.6312	1.66088	0.830442	−	0.557105i	$-0.188088\pi$
$541$	38.6312	1.66088	0.830442	+	0.557105i	$0.188088\pi$
$542$	−2.07495	−0.0891266
$543$	0	0
$544$	47.6869	2.04456
$545$	29.4584	1.26186
$546$	0	0
$547$	20.0344	0.856611	0.428305	−	0.903634i	$-0.359111\pi$
$547$	20.0344	0.856611	0.428305	+	0.903634i	$0.359111\pi$
$548$	0.820830	0.0350641
$549$	0	0
$550$	0	0
$551$	−16.5646	−0.705677
$552$	0	0
$553$	40.1246	1.70627
$554$	41.6511	1.76959
$555$	0	0
$556$	−29.7984	−1.26373
$557$	5.43228	0.230173	0.115087	−	0.993355i	$-0.463285\pi$
$557$	5.43228	0.230173	0.115087	+	0.993355i	$0.463285\pi$
$558$	0	0
$559$	0.639320	0.0270404
$560$	−21.6834	−0.916290
$561$	0	0
$562$	−35.5967	−1.50156
$563$	15.6240	0.658475	0.329237	−	0.944247i	$-0.393208\pi$
$563$	15.6240	0.658475	0.329237	+	0.944247i	$0.393208\pi$
$564$	0	0
$565$	−38.4508	−1.61764
$566$	−23.5188	−0.988570
$567$	0	0
$568$	14.2705	0.598777
$569$	31.7271	1.33007	0.665035	−	0.746812i	$-0.268417\pi$
$569$	31.7271	1.33007	0.665035	+	0.746812i	$0.268417\pi$
$570$	0	0
$571$	11.9787	0.501294	0.250647	−	0.968079i	$-0.419357\pi$
$571$	11.9787	0.501294	0.250647	+	0.968079i	$0.419357\pi$
$572$	0	0
$573$	0	0
$574$	27.0344	1.12840
$575$	−1.52190	−0.0634677
$576$	0	0
$577$	−1.34752	−0.0560982	−0.0280491	−	0.999607i	$-0.508929\pi$
$577$	−1.34752	−0.0560982	−0.0280491	+	0.999607i	$0.508929\pi$
$578$	44.3074	1.84294
$579$	0	0
$580$	−39.1246	−1.62456
$581$	−35.0845	−1.45555
$582$	0	0
$583$	0	0
$584$	3.28332	0.135865
$585$	0	0
$586$	−10.7426	−0.443775
$587$	36.7261	1.51585	0.757924	−	0.652342i	$-0.226214\pi$
$587$	36.7261	1.51585	0.757924	+	0.652342i	$0.226214\pi$
$588$	0	0
$589$	11.0000	0.453247
$590$	34.4574	1.41859
$591$	0	0
$592$	−4.20163	−0.172686
$593$	27.3094	1.12146	0.560732	−	0.827997i	$-0.310520\pi$
$593$	27.3094	1.12146	0.560732	+	0.827997i	$0.310520\pi$
$594$	0	0
$595$	−55.8328	−2.28892
$596$	−12.0729	−0.494527
$597$	0	0
$598$	2.02129	0.0826565
$599$	17.1917	0.702433	0.351217	−	0.936294i	$-0.385768\pi$
$599$	17.1917	0.702433	0.351217	+	0.936294i	$0.385768\pi$
$600$	0	0
$601$	−9.36068	−0.381830	−0.190915	−	0.981607i	$-0.561146\pi$
$601$	−9.36068	−0.381830	−0.190915	+	0.981607i	$0.561146\pi$
$602$	−24.6532	−1.00479
$603$	0	0
$604$	−34.6525	−1.40999
$605$	0	0
$606$	0	0
$607$	−28.6180	−1.16157	−0.580785	−	0.814057i	$-0.697254\pi$
$607$	−28.6180	−1.16157	−0.580785	+	0.814057i	$0.697254\pi$
$608$	18.5198	0.751078
$609$	0	0
$610$	52.3050	2.11777
$611$	0.820830	0.0332072
$612$	0	0
$613$	25.0557	1.01199	0.505996	−	0.862536i	$-0.331125\pi$
$613$	25.0557	1.01199	0.505996	+	0.862536i	$0.331125\pi$
$614$	37.1137	1.49779
$615$	0	0
$616$	0	0
$617$	16.8782	0.679489	0.339745	−	0.940518i	$-0.389659\pi$
$617$	16.8782	0.679489	0.339745	+	0.940518i	$0.389659\pi$
$618$	0	0
$619$	6.88854	0.276874	0.138437	−	0.990371i	$-0.455792\pi$
$619$	6.88854	0.276874	0.138437	+	0.990371i	$0.455792\pi$
$620$	25.9813	1.04343
$621$	0	0
$622$	−61.7984	−2.47789
$623$	−37.7407	−1.51205
$624$	0	0
$625$	−22.9443	−0.917771
$626$	68.0199	2.71862
$627$	0	0
$628$	−54.0689	−2.15758
$629$	−10.8188	−0.431375
$630$	0	0
$631$	−17.3820	−0.691965	−0.345983	−	0.938241i	$-0.612454\pi$
$631$	−17.3820	−0.691965	−0.345983	+	0.938241i	$0.612454\pi$
$632$	−12.5802	−0.500415
$633$	0	0
$634$	−0.673762	−0.0267585
$635$	5.93958	0.235705
$636$	0	0
$637$	2.58359	0.102366
$638$	0	0
$639$	0	0
$640$	21.7426	0.859454
$641$	−21.3699	−0.844058	−0.422029	−	0.906582i	$-0.638682\pi$
$641$	−21.3699	−0.844058	−0.422029	+	0.906582i	$0.638682\pi$
$642$	0	0
$643$	−41.1591	−1.62315	−0.811577	−	0.584245i	$-0.801391\pi$
$643$	−41.1591	−1.62315	−0.811577	+	0.584245i	$0.801391\pi$
$644$	−44.1876	−1.74124
$645$	0	0
$646$	31.3951	1.23522
$647$	−11.6854	−0.459400	−0.229700	−	0.973261i	$-0.573775\pi$
$647$	−11.6854	−0.459400	−0.229700	+	0.973261i	$0.573775\pi$
$648$	0	0
$649$	0	0
$650$	−0.193772	−0.00760035
$651$	0	0
$652$	3.61803	0.141693
$653$	−34.1896	−1.33794	−0.668971	−	0.743288i	$-0.733265\pi$
$653$	−34.1896	−1.33794	−0.668971	+	0.743288i	$0.733265\pi$
$654$	0	0
$655$	−5.29180	−0.206768
$656$	7.07394	0.276191
$657$	0	0
$658$	−31.6525	−1.23394
$659$	−33.7563	−1.31496	−0.657480	−	0.753472i	$-0.728377\pi$
$659$	−33.7563	−1.31496	−0.657480	+	0.753472i	$0.728377\pi$
$660$	0	0
$661$	2.43769	0.0948153	0.0474077	−	0.998876i	$-0.484904\pi$
$661$	2.43769	0.0948153	0.0474077	+	0.998876i	$0.484904\pi$
$662$	15.6698	0.609024
$663$	0	0
$664$	11.0000	0.426883
$665$	−21.6834	−0.840846
$666$	0	0
$667$	27.7082	1.07287
$668$	−28.1303	−1.08839
$669$	0	0
$670$	13.1803	0.509201
$671$	0	0
$672$	0	0
$673$	18.2361	0.702949	0.351474	−	0.936198i	$-0.385680\pi$
$673$	18.2361	0.702949	0.351474	+	0.936198i	$0.385680\pi$
$674$	−57.2751	−2.20616
$675$	0	0
$676$	−33.8885	−1.30341
$677$	−8.59584	−0.330365	−0.165183	−	0.986263i	$-0.552821\pi$
$677$	−8.59584	−0.330365	−0.165183	+	0.986263i	$0.552821\pi$
$678$	0	0
$679$	−49.9787	−1.91801
$680$	17.5052	0.671294
$681$	0	0
$682$	0	0
$683$	−0.940588	−0.0359906	−0.0179953	−	0.999838i	$-0.505728\pi$
$683$	−0.940588	−0.0359906	−0.0179953	+	0.999838i	$0.505728\pi$
$684$	0	0
$685$	−0.673762	−0.0257431
$686$	−35.9053	−1.37087
$687$	0	0
$688$	−6.45085	−0.245936
$689$	−0.433287	−0.0165069
$690$	0	0
$691$	32.3607	1.23106	0.615529	−	0.788114i	$-0.288942\pi$
$691$	32.3607	1.23106	0.615529	+	0.788114i	$0.288942\pi$
$692$	−35.0845	−1.33371
$693$	0	0
$694$	54.0689	2.05243
$695$	24.4594	0.927798
$696$	0	0
$697$	18.2148	0.689934
$698$	19.7282	0.746723
$699$	0	0
$700$	4.23607	0.160108
$701$	23.3991	0.883770	0.441885	−	0.897072i	$-0.354310\pi$
$701$	23.3991	0.883770	0.441885	+	0.897072i	$0.354310\pi$
$702$	0	0
$703$	−4.20163	−0.158467
$704$	0	0
$705$	0	0
$706$	−72.5410	−2.73012
$707$	1.32813	0.0499495
$708$	0	0
$709$	−15.2016	−0.570909	−0.285455	−	0.958392i	$-0.592145\pi$
$709$	−15.2016	−0.570909	−0.285455	+	0.958392i	$0.592145\pi$
$710$	−49.6199	−1.86220
$711$	0	0
$712$	11.8328	0.443454
$713$	−18.4001	−0.689088
$714$	0	0
$715$	0	0
$716$	5.11875	0.191297
$717$	0	0
$718$	4.36068	0.162739
$719$	−12.9678	−0.483617	−0.241808	−	0.970324i	$-0.577741\pi$
$719$	−12.9678	−0.483617	−0.241808	+	0.970324i	$0.577741\pi$
$720$	0	0
$721$	46.3607	1.72656
$722$	−28.6376	−1.06578
$723$	0	0
$724$	32.0344	1.19055
$725$	−2.65626	−0.0986511
$726$	0	0
$727$	19.5623	0.725526	0.362763	−	0.931881i	$-0.381833\pi$
$727$	19.5623	0.725526	0.362763	+	0.931881i	$0.381833\pi$
$728$	−1.32813	−0.0492238
$729$	0	0
$730$	−11.4164	−0.422540
$731$	−16.6104	−0.614357
$732$	0	0
$733$	−14.6525	−0.541202	−0.270601	−	0.962692i	$-0.587222\pi$
$733$	−14.6525	−0.541202	−0.270601	+	0.962692i	$0.587222\pi$
$734$	49.5001	1.82708
$735$	0	0
$736$	−30.9787	−1.14189
$737$	0	0
$738$	0	0
$739$	18.2361	0.670825	0.335412	−	0.942071i	$-0.391124\pi$
$739$	18.2361	0.670825	0.335412	+	0.942071i	$0.391124\pi$
$740$	−9.92398	−0.364813
$741$	0	0
$742$	16.7082	0.613377
$743$	18.1323	0.665209	0.332604	−	0.943066i	$-0.392073\pi$
$743$	18.1323	0.665209	0.332604	+	0.943066i	$0.392073\pi$
$744$	0	0
$745$	9.90983	0.363068
$746$	−30.9803	−1.13427
$747$	0	0
$748$	0	0
$749$	46.3366	1.69310
$750$	0	0
$751$	−1.50658	−0.0549758	−0.0274879	−	0.999622i	$-0.508751\pi$
$751$	−1.50658	−0.0549758	−0.0274879	+	0.999622i	$0.508751\pi$
$752$	−8.28232	−0.302025
$753$	0	0
$754$	3.52786	0.128477
$755$	28.4438	1.03518
$756$	0	0
$757$	−40.8885	−1.48612	−0.743060	−	0.669225i	$-0.766626\pi$
$757$	−40.8885	−1.48612	−0.743060	+	0.669225i	$0.766626\pi$
$758$	13.4011	0.486749
$759$	0	0
$760$	6.79837	0.246603
$761$	33.4428	1.21230	0.606150	−	0.795350i	$-0.292713\pi$
$761$	33.4428	1.21230	0.606150	+	0.795350i	$0.292713\pi$
$762$	0	0
$763$	58.0689	2.10223
$764$	−25.9813	−0.939971
$765$	0	0
$766$	−73.2148	−2.64536
$767$	−1.76142	−0.0636011
$768$	0	0
$769$	−22.2705	−0.803095	−0.401548	−	0.915838i	$-0.631528\pi$
$769$	−22.2705	−0.803095	−0.401548	+	0.915838i	$0.631528\pi$
$770$	0	0
$771$	0	0
$772$	19.3262	0.695567
$773$	−33.6823	−1.21147	−0.605734	−	0.795667i	$-0.707121\pi$
$773$	−33.6823	−1.21147	−0.605734	+	0.795667i	$0.707121\pi$
$774$	0	0
$775$	1.76393	0.0633623
$776$	15.6698	0.562513
$777$	0	0
$778$	47.9443	1.71889
$779$	7.07394	0.253450
$780$	0	0
$781$	0	0
$782$	−52.5157	−1.87796
$783$	0	0
$784$	−26.0689	−0.931032
$785$	44.3814	1.58404
$786$	0	0
$787$	27.7771	0.990146	0.495073	−	0.868851i	$-0.335141\pi$
$787$	27.7771	0.990146	0.495073	+	0.868851i	$0.335141\pi$
$788$	−6.44688	−0.229661
$789$	0	0
$790$	43.7426	1.55629
$791$	−75.7950	−2.69496
$792$	0	0
$793$	−2.67376	−0.0949481
$794$	52.4699	1.86209
$795$	0	0
$796$	1.09017	0.0386400
$797$	4.29792	0.152240	0.0761201	−	0.997099i	$-0.475747\pi$
$797$	4.29792	0.152240	0.0761201	+	0.997099i	$0.475747\pi$
$798$	0	0
$799$	−21.3262	−0.754468
$800$	2.96979	0.104998
$801$	0	0
$802$	−40.8885	−1.44382
$803$	0	0
$804$	0	0
$805$	36.2705	1.27837
$806$	−2.34273	−0.0825192
$807$	0	0
$808$	−0.416408	−0.0146492
$809$	−7.07394	−0.248707	−0.124353	−	0.992238i	$-0.539686\pi$
$809$	−7.07394	−0.248707	−0.124353	+	0.992238i	$0.539686\pi$
$810$	0	0
$811$	−44.2148	−1.55259	−0.776295	−	0.630369i	$-0.782904\pi$
$811$	−44.2148	−1.55259	−0.776295	+	0.630369i	$0.782904\pi$
$812$	−77.1231	−2.70649
$813$	0	0
$814$	0	0
$815$	−2.96979	−0.104027
$816$	0	0
$817$	−6.45085	−0.225687
$818$	−10.8646	−0.379871
$819$	0	0
$820$	16.7082	0.583476
$821$	39.7699	1.38798	0.693990	−	0.719985i	$-0.255851\pi$
$821$	39.7699	1.38798	0.693990	+	0.719985i	$0.255851\pi$
$822$	0	0
$823$	19.1115	0.666183	0.333092	−	0.942894i	$-0.391908\pi$
$823$	19.1115	0.666183	0.333092	+	0.942894i	$0.391908\pi$
$824$	−14.5354	−0.506366
$825$	0	0
$826$	67.9230	2.36334
$827$	11.6397	0.404750	0.202375	−	0.979308i	$-0.435134\pi$
$827$	11.6397	0.404750	0.202375	+	0.979308i	$0.435134\pi$
$828$	0	0
$829$	20.0344	0.695825	0.347912	−	0.937527i	$-0.386891\pi$
$829$	20.0344	0.695825	0.347912	+	0.937527i	$0.386891\pi$
$830$	−38.2480	−1.32761
$831$	0	0
$832$	−2.81966	−0.0977541
$833$	−67.1251	−2.32575
$834$	0	0
$835$	23.0902	0.799068
$836$	0	0
$837$	0	0
$838$	−75.8115	−2.61887
$839$	−13.9824	−0.482725	−0.241363	−	0.970435i	$-0.577594\pi$
$839$	−13.9824	−0.482725	−0.241363	+	0.970435i	$0.577594\pi$
$840$	0	0
$841$	19.3607	0.667610
$842$	−62.7814	−2.16359
$843$	0	0
$844$	−47.2148	−1.62520
$845$	27.8167	0.956925
$846$	0	0
$847$	0	0
$848$	4.37194	0.150133
$849$	0	0
$850$	5.03444	0.172680
$851$	7.02820	0.240924
$852$	0	0
$853$	−1.18034	−0.0404141	−0.0202070	−	0.999796i	$-0.506433\pi$
$853$	−1.18034	−0.0404141	−0.0202070	+	0.999796i	$0.506433\pi$
$854$	103.104	3.52816
$855$	0	0
$856$	−14.5279	−0.496552
$857$	13.8344	0.472573	0.236286	−	0.971683i	$-0.424070\pi$
$857$	13.8344	0.472573	0.236286	+	0.971683i	$0.424070\pi$
$858$	0	0
$859$	42.4164	1.44723	0.723615	−	0.690204i	$-0.242479\pi$
$859$	42.4164	1.44723	0.723615	+	0.690204i	$0.242479\pi$
$860$	−15.2365	−0.519560
$861$	0	0
$862$	−76.4853	−2.60510
$863$	−34.3834	−1.17042	−0.585212	−	0.810880i	$-0.698989\pi$
$863$	−34.3834	−1.17042	−0.585212	+	0.810880i	$0.698989\pi$
$864$	0	0
$865$	28.7984	0.979174
$866$	−5.70007	−0.193696
$867$	0	0
$868$	51.2148	1.73834
$869$	0	0
$870$	0	0
$871$	−0.673762	−0.0228296
$872$	−18.2063	−0.616543
$873$	0	0
$874$	−20.3951	−0.689875
$875$	−48.9928	−1.65626
$876$	0	0
$877$	11.4721	0.387387	0.193693	−	0.981062i	$-0.437953\pi$
$877$	11.4721	0.387387	0.193693	+	0.981062i	$0.437953\pi$
$878$	−53.7240	−1.81310
$879$	0	0
$880$	0	0
$881$	34.3376	1.15686	0.578432	−	0.815730i	$-0.303665\pi$
$881$	34.3376	1.15686	0.578432	+	0.815730i	$0.303665\pi$
$882$	0	0
$883$	−14.1115	−0.474888	−0.237444	−	0.971401i	$-0.576310\pi$
$883$	−14.1115	−0.474888	−0.237444	+	0.971401i	$0.576310\pi$
$884$	−3.79062	−0.127492
$885$	0	0
$886$	47.0132	1.57944
$887$	−56.5741	−1.89957	−0.949786	−	0.312902i	$-0.898699\pi$
$887$	−56.5741	−1.89957	−0.949786	+	0.312902i	$0.898699\pi$
$888$	0	0
$889$	11.7082	0.392681
$890$	−41.1438	−1.37914
$891$	0	0
$892$	8.32624	0.278783
$893$	−8.28232	−0.277157
$894$	0	0
$895$	−4.20163	−0.140445
$896$	42.8595	1.43183
$897$	0	0
$898$	63.3050	2.11251
$899$	−32.1147	−1.07108
$900$	0	0
$901$	11.2574	0.375037
$902$	0	0
$903$	0	0
$904$	23.7639	0.790377
$905$	−26.2948	−0.874070
$906$	0	0
$907$	−35.8673	−1.19095	−0.595476	−	0.803373i	$-0.703037\pi$
$907$	−35.8673	−1.19095	−0.595476	+	0.803373i	$0.703037\pi$
$908$	28.6376	0.950371
$909$	0	0
$910$	4.61803	0.153086
$911$	9.53643	0.315956	0.157978	−	0.987443i	$-0.449502\pi$
$911$	9.53643	0.315956	0.157978	+	0.987443i	$0.449502\pi$
$912$	0	0
$913$	0	0
$914$	75.4074	2.49426
$915$	0	0
$916$	−48.3607	−1.59788
$917$	−10.4313	−0.344471
$918$	0	0
$919$	−25.1803	−0.830623	−0.415311	−	0.909679i	$-0.636327\pi$
$919$	−25.1803	−0.830623	−0.415311	+	0.909679i	$0.636327\pi$
$920$	−11.3719	−0.374919
$921$	0	0
$922$	3.27051	0.107709
$923$	2.53650	0.0834901
$924$	0	0
$925$	−0.673762	−0.0221532
$926$	−46.3366	−1.52271
$927$	0	0
$928$	−54.0689	−1.77490
$929$	42.0386	1.37924	0.689621	−	0.724170i	$-0.257777\pi$
$929$	42.0386	1.37924	0.689621	+	0.724170i	$0.257777\pi$
$930$	0	0
$931$	−26.0689	−0.854373
$932$	60.2449	1.97339
$933$	0	0
$934$	45.5066	1.48902
$935$	0	0
$936$	0	0
$937$	10.8197	0.353463	0.176731	−	0.984259i	$-0.443448\pi$
$937$	10.8197	0.353463	0.176731	+	0.984259i	$0.443448\pi$
$938$	25.9813	0.848320
$939$	0	0
$940$	−19.5623	−0.638052
$941$	−9.17716	−0.299167	−0.149583	−	0.988749i	$-0.547793\pi$
$941$	−9.17716	−0.299167	−0.149583	+	0.988749i	$0.547793\pi$
$942$	0	0
$943$	−11.8328	−0.385329
$944$	17.7730	0.578462
$945$	0	0
$946$	0	0
$947$	−22.3845	−0.727397	−0.363699	−	0.931517i	$-0.618486\pi$
$947$	−22.3845	−0.727397	−0.363699	+	0.931517i	$0.618486\pi$
$948$	0	0
$949$	0.583592	0.0189442
$950$	1.95519	0.0634347
$951$	0	0
$952$	34.5066	1.11836
$953$	16.8041	0.544340	0.272170	−	0.962249i	$-0.412259\pi$
$953$	16.8041	0.544340	0.272170	+	0.962249i	$0.412259\pi$
$954$	0	0
$955$	21.3262	0.690101
$956$	29.9657	0.969160
$957$	0	0
$958$	−79.1803	−2.55820
$959$	−1.32813	−0.0428876
$960$	0	0
$961$	−9.67376	−0.312057
$962$	0.894844	0.0288509
$963$	0	0
$964$	36.6525	1.18050
$965$	−15.8636	−0.510666
$966$	0	0
$967$	−17.6869	−0.568773	−0.284386	−	0.958710i	$-0.591790\pi$
$967$	−17.6869	−0.568773	−0.284386	+	0.958710i	$0.591790\pi$
$968$	0	0
$969$	0	0
$970$	−54.4853	−1.74942
$971$	1.01460	0.0325601	0.0162801	−	0.999867i	$-0.494818\pi$
$971$	1.01460	0.0325601	0.0162801	+	0.999867i	$0.494818\pi$
$972$	0	0
$973$	48.2148	1.54569
$974$	40.5907	1.30061
$975$	0	0
$976$	26.9787	0.863568
$977$	−3.35733	−0.107411	−0.0537053	−	0.998557i	$-0.517103\pi$
$977$	−3.35733	−0.107411	−0.0537053	+	0.998557i	$0.517103\pi$
$978$	0	0
$979$	0	0
$980$	−61.5731	−1.96688
$981$	0	0
$982$	−41.7214	−1.33138
$983$	−50.7085	−1.61735	−0.808675	−	0.588256i	$-0.799815\pi$
$983$	−50.7085	−1.61735	−0.808675	+	0.588256i	$0.799815\pi$
$984$	0	0
$985$	5.29180	0.168611
$986$	−91.6585	−2.91900
$987$	0	0
$988$	−1.47214	−0.0468349
$989$	10.7905	0.343119
$990$	0	0
$991$	−58.2705	−1.85102	−0.925512	−	0.378719i	$-0.876365\pi$
$991$	−58.2705	−1.85102	−0.925512	+	0.378719i	$0.876365\pi$
$992$	35.9053	1.13999
$993$	0	0
$994$	−97.8115	−3.10239
$995$	−0.894844	−0.0283685
$996$	0	0
$997$	38.7426	1.22699	0.613496	−	0.789698i	$-0.289763\pi$
$997$	38.7426	1.22699	0.613496	+	0.789698i	$0.289763\pi$
$998$	70.3627	2.22729
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.v.1.4		4
3.2	odd	2	inner	1089.2.a.v.1.1		4
11.5	even	5		99.2.f.c.91.1	yes	8
11.9	even	5		99.2.f.c.37.1	✓	8
11.10	odd	2		1089.2.a.w.1.1		4
33.5	odd	10		99.2.f.c.91.2	yes	8
33.20	odd	10		99.2.f.c.37.2	yes	8
33.32	even	2		1089.2.a.w.1.4		4
99.5	odd	30		891.2.n.e.784.2		16
99.16	even	15		891.2.n.e.190.2		16
99.20	odd	30		891.2.n.e.433.2		16
99.31	even	15		891.2.n.e.136.2		16
99.38	odd	30		891.2.n.e.190.1		16
99.49	even	15		891.2.n.e.784.1		16
99.86	odd	30		891.2.n.e.136.1		16
99.97	even	15		891.2.n.e.433.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.f.c.37.1	✓	8	11.9	even	5
99.2.f.c.37.2	yes	8	33.20	odd	10
99.2.f.c.91.1	yes	8	11.5	even	5
99.2.f.c.91.2	yes	8	33.5	odd	10
891.2.n.e.136.1		16	99.86	odd	30
891.2.n.e.136.2		16	99.31	even	15
891.2.n.e.190.1		16	99.38	odd	30
891.2.n.e.190.2		16	99.16	even	15
891.2.n.e.433.1		16	99.97	even	15
891.2.n.e.433.2		16	99.20	odd	30
891.2.n.e.784.1		16	99.49	even	15
891.2.n.e.784.2		16	99.5	odd	30
1089.2.a.v.1.1		4	3.2	odd	2	inner
1089.2.a.v.1.4		4	1.1	even	1	trivial
1089.2.a.w.1.1		4	11.10	odd	2
1089.2.a.w.1.4		4	33.32	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)

\(f(q)\)	\(=\)	\(q+2.14896 q^{2} +2.61803 q^{4} -2.14896 q^{5} -4.23607 q^{7} +1.32813 q^{8} -4.61803 q^{10} +0.236068 q^{13} -9.10315 q^{14} -2.38197 q^{16} -6.13335 q^{17} -2.38197 q^{19} -5.62605 q^{20} +3.98439 q^{23} -0.381966 q^{25} +0.507301 q^{26} -11.0902 q^{28} +6.95418 q^{29} -4.61803 q^{31} -7.77501 q^{32} -13.1803 q^{34} +9.10315 q^{35} +1.76393 q^{37} -5.11875 q^{38} -2.85410 q^{40} -2.96979 q^{41} +2.70820 q^{43} +8.56231 q^{46} +3.47709 q^{47} +10.9443 q^{49} -0.820830 q^{50} +0.618034 q^{52} -1.83543 q^{53} -5.62605 q^{56} +14.9443 q^{58} -7.46149 q^{59} -11.3262 q^{61} -9.92398 q^{62} -11.9443 q^{64} -0.507301 q^{65} -2.85410 q^{67} -16.0573 q^{68} +19.5623 q^{70} +10.7448 q^{71} +2.47214 q^{73} +3.79062 q^{74} -6.23607 q^{76} -9.47214 q^{79} +5.11875 q^{80} -6.38197 q^{82} +8.28232 q^{83} +13.1803 q^{85} +5.81983 q^{86} +8.90937 q^{89} -1.00000 q^{91} +10.4313 q^{92} +7.47214 q^{94} +5.11875 q^{95} +11.7984 q^{97} +23.5188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 8 q^{7} - 14 q^{10} - 8 q^{13} - 14 q^{16} - 14 q^{19} - 6 q^{25} - 22 q^{28} - 14 q^{31} - 8 q^{34} + 16 q^{37} + 2 q^{40} - 16 q^{43} - 6 q^{46} + 8 q^{49} - 2 q^{52} + 24 q^{58} - 14 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 - 8 * q^7 - 14 * q^10 - 8 * q^13 - 14 * q^16 - 14 * q^19 - 6 * q^25 - 22 * q^28 - 14 * q^31 - 8 * q^34 + 16 * q^37 + 2 * q^40 - 16 * q^43 - 6 * q^46 + 8 * q^49 - 2 * q^52 + 24 * q^58 - 14 * q^61 - 12 * q^64 + 2 * q^67 + 38 * q^70 - 8 * q^73 - 16 * q^76 - 20 * q^79 - 30 * q^82 + 8 * q^85 - 4 * q^91 + 12 * q^94 - 2 * q^97

Introduction

L-functions

Modular forms

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