LMFDB - Embedded newform 1323.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.760877 q^{2} -1.42107 q^{4} -3.18194 q^{5} +2.60301 q^{8} +O(q^{10})$	$q-0.760877 q^{2} -1.42107 q^{4} -3.18194 q^{5} +2.60301 q^{8} +2.42107 q^{10} -2.23912 q^{11} +3.70370 q^{13} +0.861564 q^{16} +5.60301 q^{17} +4.42107 q^{19} +4.52175 q^{20} +1.70370 q^{22} +0.942820 q^{23} +5.12476 q^{25} -2.81806 q^{26} -10.1248 q^{29} -5.70370 q^{31} -5.86156 q^{32} -4.26320 q^{34} +3.12476 q^{37} -3.36389 q^{38} -8.28263 q^{40} -3.98633 q^{41} -3.28263 q^{43} +3.18194 q^{44} -0.717370 q^{46} +0.225450 q^{47} -3.89931 q^{50} -5.26320 q^{52} -10.6602 q^{53} +7.12476 q^{55} +7.70370 q^{58} +2.05718 q^{59} -5.84213 q^{61} +4.33981 q^{62} +2.73680 q^{64} -11.7850 q^{65} +7.42107 q^{67} -7.96225 q^{68} -7.26320 q^{71} +7.55950 q^{73} -2.37756 q^{74} -6.28263 q^{76} -6.82846 q^{79} -2.74145 q^{80} +3.03310 q^{82} +8.11109 q^{83} -17.8285 q^{85} +2.49768 q^{86} -5.82846 q^{88} -9.72777 q^{89} -1.33981 q^{92} -0.171540 q^{94} -14.0676 q^{95} +0.842133 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8	$ 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34} - 8 q^{37} + 7 q^{38} - 24 q^{40} - 2 q^{41} - 9 q^{43} + q^{44} - 3 q^{46} - 9 q^{47} - 4 q^{50} + 9 q^{52} - 24 q^{53} + 4 q^{55} + 14 q^{58} + 15 q^{59} - q^{61} + 21 q^{62} + 33 q^{64} - 10 q^{65} + 14 q^{67} - 39 q^{68} + 3 q^{71} + 7 q^{73} - 18 q^{76} + 6 q^{79} + 16 q^{80} + 43 q^{82} - 3 q^{83} - 27 q^{85} + 32 q^{86} + 9 q^{88} + 5 q^{89} - 12 q^{92} - 27 q^{94} - 16 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 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 $

$2$	−0.760877	−0.538021	−0.269011	−	0.963137i	$-0.586697\pi$
$2$	−0.760877	−0.538021	−0.269011	+	0.963137i	$0.586697\pi$
$3$	0	0
$3$	0	0
$4$	−1.42107	−0.710533
$5$	−3.18194	−1.42301	−0.711504	−	0.702682i	$-0.751986\pi$
$5$	−3.18194	−1.42301	−0.711504	+	0.702682i	$0.751986\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.60301	0.920303
$9$	0	0
$10$	2.42107	0.765608
$11$	−2.23912	−0.675121	−0.337561	−	0.941304i	$-0.609602\pi$
$11$	−2.23912	−0.675121	−0.337561	+	0.941304i	$0.609602\pi$
$12$	0	0
$13$	3.70370	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$13$	3.70370	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$14$	0	0
$15$	0	0
$16$	0.861564	0.215391
$17$	5.60301	1.35893	0.679465	−	0.733708i	$-0.262212\pi$
$17$	5.60301	1.35893	0.679465	+	0.733708i	$0.262212\pi$
$18$	0	0
$19$	4.42107	1.01426	0.507131	−	0.861869i	$-0.330706\pi$
$19$	4.42107	1.01426	0.507131	+	0.861869i	$0.330706\pi$
$20$	4.52175	1.01109
$21$	0	0
$22$	1.70370	0.363229
$23$	0.942820	0.196592	0.0982958	−	0.995157i	$-0.468661\pi$
$23$	0.942820	0.196592	0.0982958	+	0.995157i	$0.468661\pi$
$24$	0	0
$25$	5.12476	1.02495
$26$	−2.81806	−0.552666
$27$	0	0
$28$	0	0
$29$	−10.1248	−1.88012	−0.940061	−	0.341007i	$-0.889232\pi$
$29$	−10.1248	−1.88012	−0.940061	+	0.341007i	$0.889232\pi$
$30$	0	0
$31$	−5.70370	−1.02441	−0.512207	−	0.858862i	$-0.671172\pi$
$31$	−5.70370	−1.02441	−0.512207	+	0.858862i	$0.671172\pi$
$32$	−5.86156	−1.03619
$33$	0	0
$34$	−4.26320	−0.731133
$35$	0	0
$36$	0	0
$37$	3.12476	0.513708	0.256854	−	0.966450i	$-0.417314\pi$
$37$	3.12476	0.513708	0.256854	+	0.966450i	$0.417314\pi$
$38$	−3.36389	−0.545694
$39$	0	0
$40$	−8.28263	−1.30960
$41$	−3.98633	−0.622560	−0.311280	−	0.950318i	$-0.600758\pi$
$41$	−3.98633	−0.622560	−0.311280	+	0.950318i	$0.600758\pi$
$42$	0	0
$43$	−3.28263	−0.500596	−0.250298	−	0.968169i	$-0.580529\pi$
$43$	−3.28263	−0.500596	−0.250298	+	0.968169i	$0.580529\pi$
$44$	3.18194	0.479696
$45$	0	0
$46$	−0.717370	−0.105770
$47$	0.225450	0.0328853	0.0164426	−	0.999865i	$-0.494766\pi$
$47$	0.225450	0.0328853	0.0164426	+	0.999865i	$0.494766\pi$
$48$	0	0
$49$	0	0
$50$	−3.89931	−0.551446
$51$	0	0
$52$	−5.26320	−0.729874
$53$	−10.6602	−1.46429	−0.732145	−	0.681149i	$-0.761481\pi$
$53$	−10.6602	−1.46429	−0.732145	+	0.681149i	$0.761481\pi$
$54$	0	0
$55$	7.12476	0.960703
$56$	0	0
$57$	0	0
$58$	7.70370	1.01154
$59$	2.05718	0.267822	0.133911	−	0.990993i	$-0.457246\pi$
$59$	2.05718	0.267822	0.133911	+	0.990993i	$0.457246\pi$
$60$	0	0
$61$	−5.84213	−0.748009	−0.374004	−	0.927427i	$-0.622015\pi$
$61$	−5.84213	−0.748009	−0.374004	+	0.927427i	$0.622015\pi$
$62$	4.33981	0.551156
$63$	0	0
$64$	2.73680	0.342100
$65$	−11.7850	−1.46174
$66$	0	0
$67$	7.42107	0.906628	0.453314	−	0.891351i	$-0.350242\pi$
$67$	7.42107	0.906628	0.453314	+	0.891351i	$0.350242\pi$
$68$	−7.96225	−0.965565
$69$	0	0
$70$	0	0
$71$	−7.26320	−0.861983	−0.430992	−	0.902356i	$-0.641836\pi$
$71$	−7.26320	−0.861983	−0.430992	+	0.902356i	$0.641836\pi$
$72$	0	0
$73$	7.55950	0.884773	0.442386	−	0.896825i	$-0.354132\pi$
$73$	7.55950	0.884773	0.442386	+	0.896825i	$0.354132\pi$
$74$	−2.37756	−0.276386
$75$	0	0
$76$	−6.28263	−0.720667
$77$	0	0
$78$	0	0
$79$	−6.82846	−0.768262	−0.384131	−	0.923279i	$-0.625499\pi$
$79$	−6.82846	−0.768262	−0.384131	+	0.923279i	$0.625499\pi$
$80$	−2.74145	−0.306503
$81$	0	0
$82$	3.03310	0.334950
$83$	8.11109	0.890308	0.445154	−	0.895454i	$-0.353149\pi$
$83$	8.11109	0.890308	0.445154	+	0.895454i	$0.353149\pi$
$84$	0	0
$85$	−17.8285	−1.93377
$86$	2.49768	0.269331
$87$	0	0
$88$	−5.82846	−0.621316
$89$	−9.72777	−1.03114	−0.515571	−	0.856847i	$-0.672420\pi$
$89$	−9.72777	−1.03114	−0.515571	+	0.856847i	$0.672420\pi$
$90$	0	0
$91$	0	0
$92$	−1.33981	−0.139685
$93$	0	0
$94$	−0.171540	−0.0176930
$95$	−14.0676	−1.44330
$96$	0	0
$97$	0.842133	0.0855057	0.0427528	−	0.999086i	$-0.486387\pi$
$97$	0.842133	0.0855057	0.0427528	+	0.999086i	$0.486387\pi$
$98$	0	0
$99$	0	0
$100$	−7.28263	−0.728263
$101$	7.74720	0.770876	0.385438	−	0.922734i	$-0.374050\pi$
$101$	7.74720	0.770876	0.385438	+	0.922734i	$0.374050\pi$
$102$	0	0
$103$	−2.43474	−0.239902	−0.119951	−	0.992780i	$-0.538274\pi$
$103$	−2.43474	−0.239902	−0.119951	+	0.992780i	$0.538274\pi$
$104$	9.64076	0.945354
$105$	0	0
$106$	8.11109	0.787819
$107$	−11.4646	−1.10832	−0.554161	−	0.832409i	$-0.686961\pi$
$107$	−11.4646	−1.10832	−0.554161	+	0.832409i	$0.686961\pi$
$108$	0	0
$109$	−18.2495	−1.74799	−0.873994	−	0.485937i	$-0.838478\pi$
$109$	−18.2495	−1.74799	−0.873994	+	0.485937i	$0.838478\pi$
$110$	−5.42107	−0.516878
$111$	0	0
$112$	0	0
$113$	5.24953	0.493834	0.246917	−	0.969037i	$-0.420583\pi$
$113$	5.24953	0.493834	0.246917	+	0.969037i	$0.420583\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	14.3880	1.33589
$117$	0	0
$118$	−1.56526	−0.144094
$119$	0	0
$120$	0	0
$121$	−5.98633	−0.544212
$122$	4.44514	0.402444
$123$	0	0
$124$	8.10533	0.727880
$125$	−0.396990	−0.0355079
$126$	0	0
$127$	20.1053	1.78406	0.892030	−	0.451976i	$-0.149281\pi$
$127$	20.1053	1.78406	0.892030	+	0.451976i	$0.149281\pi$
$128$	9.64076	0.852131
$129$	0	0
$130$	8.96690	0.786449
$131$	4.08126	0.356581	0.178291	−	0.983978i	$-0.442943\pi$
$131$	4.08126	0.356581	0.178291	+	0.983978i	$0.442943\pi$
$132$	0	0
$133$	0	0
$134$	−5.64652	−0.487785
$135$	0	0
$136$	14.5847	1.25063
$137$	1.88564	0.161101	0.0805506	−	0.996751i	$-0.474332\pi$
$137$	1.88564	0.161101	0.0805506	+	0.996751i	$0.474332\pi$
$138$	0	0
$139$	−12.7954	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$139$	−12.7954	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$140$	0	0
$141$	0	0
$142$	5.52640	0.463765
$143$	−8.29303	−0.693498
$144$	0	0
$145$	32.2164	2.67543
$146$	−5.75185	−0.476026
$147$	0	0
$148$	−4.44050	−0.365007
$149$	−8.05718	−0.660070	−0.330035	−	0.943969i	$-0.607061\pi$
$149$	−8.05718	−0.660070	−0.330035	+	0.943969i	$0.607061\pi$
$150$	0	0
$151$	−6.28263	−0.511273	−0.255637	−	0.966773i	$-0.582285\pi$
$151$	−6.28263	−0.511273	−0.255637	+	0.966773i	$0.582285\pi$
$152$	11.5081	0.933429
$153$	0	0
$154$	0	0
$155$	18.1488	1.45775
$156$	0	0
$157$	0.703697	0.0561611	0.0280806	−	0.999606i	$-0.491061\pi$
$157$	0.703697	0.0561611	0.0280806	+	0.999606i	$0.491061\pi$
$158$	5.19562	0.413341
$159$	0	0
$160$	18.6512	1.47450
$161$	0	0
$162$	0	0
$163$	−19.2359	−1.50667	−0.753334	−	0.657638i	$-0.771556\pi$
$163$	−19.2359	−1.50667	−0.753334	+	0.657638i	$0.771556\pi$
$164$	5.66484	0.442349
$165$	0	0
$166$	−6.17154	−0.479004
$167$	−23.3880	−1.80981	−0.904907	−	0.425608i	$-0.860060\pi$
$167$	−23.3880	−1.80981	−0.904907	+	0.425608i	$0.860060\pi$
$168$	0	0
$169$	0.717370	0.0551823
$170$	13.5653	1.04041
$171$	0	0
$172$	4.66484	0.355690
$173$	8.23912	0.626409	0.313204	−	0.949686i	$-0.398597\pi$
$173$	8.23912	0.626409	0.313204	+	0.949686i	$0.398597\pi$
$174$	0	0
$175$	0	0
$176$	−1.92915	−0.145415
$177$	0	0
$178$	7.40164	0.554776
$179$	−9.90972	−0.740687	−0.370344	−	0.928895i	$-0.620760\pi$
$179$	−9.90972	−0.740687	−0.370344	+	0.928895i	$0.620760\pi$
$180$	0	0
$181$	−9.38796	−0.697802	−0.348901	−	0.937160i	$-0.613445\pi$
$181$	−9.38796	−0.697802	−0.348901	+	0.937160i	$0.613445\pi$
$182$	0	0
$183$	0	0
$184$	2.45417	0.180924
$185$	−9.94282	−0.731011
$186$	0	0
$187$	−12.5458	−0.917442
$188$	−0.320380	−0.0233661
$189$	0	0
$190$	10.7037	0.776528
$191$	24.7382	1.78999	0.894996	−	0.446075i	$-0.147178\pi$
$191$	24.7382	1.78999	0.894996	+	0.446075i	$0.147178\pi$
$192$	0	0
$193$	−0.828460	−0.0596339	−0.0298169	−	0.999555i	$-0.509492\pi$
$193$	−0.828460	−0.0596339	−0.0298169	+	0.999555i	$0.509492\pi$
$194$	−0.640760	−0.0460039
$195$	0	0
$196$	0	0
$197$	−5.86156	−0.417619	−0.208810	−	0.977956i	$-0.566959\pi$
$197$	−5.86156	−0.417619	−0.208810	+	0.977956i	$0.566959\pi$
$198$	0	0
$199$	−9.24953	−0.655682	−0.327841	−	0.944733i	$-0.606321\pi$
$199$	−9.24953	−0.655682	−0.327841	+	0.944733i	$0.606321\pi$
$200$	13.3398	0.943267
$201$	0	0
$202$	−5.89467	−0.414747
$203$	0	0
$204$	0	0
$205$	12.6843	0.885908
$206$	1.85254	0.129072
$207$	0	0
$208$	3.19097	0.221254
$209$	−9.89931	−0.684750
$210$	0	0
$211$	−12.5595	−0.864632	−0.432316	−	0.901722i	$-0.642303\pi$
$211$	−12.5595	−0.864632	−0.432316	+	0.901722i	$0.642303\pi$
$212$	15.1488	1.04043
$213$	0	0
$214$	8.72313	0.596301
$215$	10.4451	0.712353
$216$	0	0
$217$	0	0
$218$	13.8856	0.940454
$219$	0	0
$220$	−10.1248	−0.682611
$221$	20.7518	1.39592
$222$	0	0
$223$	−21.3880	−1.43224	−0.716122	−	0.697975i	$-0.754085\pi$
$223$	−21.3880	−1.43224	−0.716122	+	0.697975i	$0.754085\pi$
$224$	0	0
$225$	0	0
$226$	−3.99424	−0.265693
$227$	12.6224	0.837781	0.418890	−	0.908037i	$-0.362419\pi$
$227$	12.6224	0.837781	0.418890	+	0.908037i	$0.362419\pi$
$228$	0	0
$229$	−28.9201	−1.91110	−0.955548	−	0.294837i	$-0.904735\pi$
$229$	−28.9201	−1.91110	−0.955548	+	0.294837i	$0.904735\pi$
$230$	2.28263	0.150512
$231$	0	0
$232$	−26.3549	−1.73028
$233$	−21.4509	−1.40530	−0.702648	−	0.711538i	$-0.747999\pi$
$233$	−21.4509	−1.40530	−0.702648	+	0.711538i	$0.747999\pi$
$234$	0	0
$235$	−0.717370	−0.0467960
$236$	−2.92339	−0.190296
$237$	0	0
$238$	0	0
$239$	7.73680	0.500452	0.250226	−	0.968187i	$-0.419495\pi$
$239$	7.73680	0.500452	0.250226	+	0.968187i	$0.419495\pi$
$240$	0	0
$241$	6.09166	0.392398	0.196199	−	0.980564i	$-0.437140\pi$
$241$	6.09166	0.392398	0.196199	+	0.980564i	$0.437140\pi$
$242$	4.55486	0.292797
$243$	0	0
$244$	8.30206	0.531485
$245$	0	0
$246$	0	0
$247$	16.3743	1.04187
$248$	−14.8468	−0.942771
$249$	0	0
$250$	0.302060	0.0191040
$251$	1.40164	0.0884705	0.0442352	−	0.999021i	$-0.485915\pi$
$251$	1.40164	0.0884705	0.0442352	+	0.999021i	$0.485915\pi$
$252$	0	0
$253$	−2.11109	−0.132723
$254$	−15.2977	−0.959862
$255$	0	0
$256$	−12.8090	−0.800564
$257$	17.9565	1.12010	0.560048	−	0.828460i	$-0.310783\pi$
$257$	17.9565	1.12010	0.560048	+	0.828460i	$0.310783\pi$
$258$	0	0
$259$	0	0
$260$	16.7472	1.03862
$261$	0	0
$262$	−3.10533	−0.191848
$263$	7.17619	0.442503	0.221251	−	0.975217i	$-0.428986\pi$
$263$	7.17619	0.442503	0.221251	+	0.975217i	$0.428986\pi$
$264$	0	0
$265$	33.9201	2.08370
$266$	0	0
$267$	0	0
$268$	−10.5458	−0.644189
$269$	−3.39699	−0.207118	−0.103559	−	0.994623i	$-0.533023\pi$
$269$	−3.39699	−0.207118	−0.103559	+	0.994623i	$0.533023\pi$
$270$	0	0
$271$	−10.2359	−0.621784	−0.310892	−	0.950445i	$-0.600628\pi$
$271$	−10.2359	−0.621784	−0.310892	+	0.950445i	$0.600628\pi$
$272$	4.82735	0.292701
$273$	0	0
$274$	−1.43474	−0.0866758
$275$	−11.4750	−0.691967
$276$	0	0
$277$	−3.55950	−0.213870	−0.106935	−	0.994266i	$-0.534104\pi$
$277$	−3.55950	−0.213870	−0.106935	+	0.994266i	$0.534104\pi$
$278$	9.73569	0.583908
$279$	0	0
$280$	0	0
$281$	−6.98633	−0.416769	−0.208385	−	0.978047i	$-0.566821\pi$
$281$	−6.98633	−0.416769	−0.208385	+	0.978047i	$0.566821\pi$
$282$	0	0
$283$	−30.2164	−1.79618	−0.898090	−	0.439812i	$-0.855045\pi$
$283$	−30.2164	−1.79618	−0.898090	+	0.439812i	$0.855045\pi$
$284$	10.3215	0.612468
$285$	0	0
$286$	6.30998	0.373117
$287$	0	0
$288$	0	0
$289$	14.3937	0.846689
$290$	−24.5127	−1.43944
$291$	0	0
$292$	−10.7426	−0.628661
$293$	15.2359	0.890088	0.445044	−	0.895509i	$-0.353188\pi$
$293$	15.2359	0.890088	0.445044	+	0.895509i	$0.353188\pi$
$294$	0	0
$295$	−6.54583	−0.381113
$296$	8.13379	0.472767
$297$	0	0
$298$	6.13052	0.355132
$299$	3.49192	0.201943
$300$	0	0
$301$	0	0
$302$	4.78031	0.275076
$303$	0	0
$304$	3.80903	0.218463
$305$	18.5893	1.06442
$306$	0	0
$307$	−1.03310	−0.0589623	−0.0294812	−	0.999565i	$-0.509386\pi$
$307$	−1.03310	−0.0589623	−0.0294812	+	0.999565i	$0.509386\pi$
$308$	0	0
$309$	0	0
$310$	−13.8090	−0.784300
$311$	9.32038	0.528510	0.264255	−	0.964453i	$-0.414874\pi$
$311$	9.32038	0.528510	0.264255	+	0.964453i	$0.414874\pi$
$312$	0	0
$313$	6.09166	0.344321	0.172160	−	0.985069i	$-0.444925\pi$
$313$	6.09166	0.344321	0.172160	+	0.985069i	$0.444925\pi$
$314$	−0.535426	−0.0302159
$315$	0	0
$316$	9.70370	0.545876
$317$	−23.3009	−1.30871	−0.654356	−	0.756187i	$-0.727060\pi$
$317$	−23.3009	−1.30871	−0.654356	+	0.756187i	$0.727060\pi$
$318$	0	0
$319$	22.6706	1.26931
$320$	−8.70834	−0.486811
$321$	0	0
$322$	0	0
$323$	24.7713	1.37831
$324$	0	0
$325$	18.9806	1.05285
$326$	14.6361	0.810619
$327$	0	0
$328$	−10.3764	−0.572944
$329$	0	0
$330$	0	0
$331$	14.6764	0.806685	0.403343	−	0.915049i	$-0.367848\pi$
$331$	14.6764	0.806685	0.403343	+	0.915049i	$0.367848\pi$
$332$	−11.5264	−0.632593
$333$	0	0
$334$	17.7954	0.973719
$335$	−23.6134	−1.29014
$336$	0	0
$337$	25.6238	1.39582	0.697909	−	0.716186i	$-0.254114\pi$
$337$	25.6238	1.39582	0.697909	+	0.716186i	$0.254114\pi$
$338$	−0.545830	−0.0296892
$339$	0	0
$340$	25.3354	1.37401
$341$	12.7713	0.691604
$342$	0	0
$343$	0	0
$344$	−8.54472	−0.460700
$345$	0	0
$346$	−6.26896	−0.337021
$347$	11.2930	0.606242	0.303121	−	0.952952i	$-0.401971\pi$
$347$	11.2930	0.606242	0.303121	+	0.952952i	$0.401971\pi$
$348$	0	0
$349$	−11.2963	−0.604677	−0.302339	−	0.953201i	$-0.597767\pi$
$349$	−11.2963	−0.604677	−0.302339	+	0.953201i	$0.597767\pi$
$350$	0	0
$351$	0	0
$352$	13.1248	0.699552
$353$	−18.5023	−0.984779	−0.492390	−	0.870375i	$-0.663876\pi$
$353$	−18.5023	−0.984779	−0.492390	+	0.870375i	$0.663876\pi$
$354$	0	0
$355$	23.1111	1.22661
$356$	13.8238	0.732661
$357$	0	0
$358$	7.54007	0.398505
$359$	19.8960	1.05007	0.525037	−	0.851080i	$-0.324052\pi$
$359$	19.8960	1.05007	0.525037	+	0.851080i	$0.324052\pi$
$360$	0	0
$361$	0.545830	0.0287279
$362$	7.14308	0.375432
$363$	0	0
$364$	0	0
$365$	−24.0539	−1.25904
$366$	0	0
$367$	5.75047	0.300172	0.150086	−	0.988673i	$-0.452045\pi$
$367$	5.75047	0.300172	0.150086	+	0.988673i	$0.452045\pi$
$368$	0.812299	0.0423440
$369$	0	0
$370$	7.56526	0.393299
$371$	0	0
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	9.54583	0.493603
$375$	0	0
$376$	0.586849	0.0302644
$377$	−37.4991	−1.93130
$378$	0	0
$379$	8.95322	0.459896	0.229948	−	0.973203i	$-0.426144\pi$
$379$	8.95322	0.459896	0.229948	+	0.973203i	$0.426144\pi$
$380$	19.9910	1.02552
$381$	0	0
$382$	−18.8227	−0.963053
$383$	20.1650	1.03038	0.515192	−	0.857075i	$-0.327721\pi$
$383$	20.1650	1.03038	0.515192	+	0.857075i	$0.327721\pi$
$384$	0	0
$385$	0	0
$386$	0.630356	0.0320843
$387$	0	0
$388$	−1.19673	−0.0607546
$389$	−33.5997	−1.70357	−0.851787	−	0.523888i	$-0.824481\pi$
$389$	−33.5997	−1.70357	−0.851787	+	0.523888i	$0.824481\pi$
$390$	0	0
$391$	5.28263	0.267154
$392$	0	0
$393$	0	0
$394$	4.45993	0.224688
$395$	21.7278	1.09324
$396$	0	0
$397$	−4.13844	−0.207702	−0.103851	−	0.994593i	$-0.533117\pi$
$397$	−4.13844	−0.207702	−0.103851	+	0.994593i	$0.533117\pi$
$398$	7.03775	0.352771
$399$	0	0
$400$	4.41531	0.220765
$401$	15.4692	0.772496	0.386248	−	0.922395i	$-0.373771\pi$
$401$	15.4692	0.772496	0.386248	+	0.922395i	$0.373771\pi$
$402$	0	0
$403$	−21.1248	−1.05230
$404$	−11.0093	−0.547733
$405$	0	0
$406$	0	0
$407$	−6.99673	−0.346815
$408$	0	0
$409$	27.3937	1.35453	0.677266	−	0.735738i	$-0.263165\pi$
$409$	27.3937	1.35453	0.677266	+	0.735738i	$0.263165\pi$
$410$	−9.65116	−0.476637
$411$	0	0
$412$	3.45993	0.170458
$413$	0	0
$414$	0	0
$415$	−25.8090	−1.26692
$416$	−21.7095	−1.06439
$417$	0	0
$418$	7.53216	0.368410
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	19.9590	0.972741	0.486371	−	0.873753i	$-0.338321\pi$
$421$	19.9590	0.972741	0.486371	+	0.873753i	$0.338321\pi$
$422$	9.55623	0.465190
$423$	0	0
$424$	−27.7486	−1.34759
$425$	28.7141	1.39284
$426$	0	0
$427$	0	0
$428$	16.2919	0.787500
$429$	0	0
$430$	−7.94747	−0.383261
$431$	20.1294	0.969600	0.484800	−	0.874625i	$-0.338892\pi$
$431$	20.1294	0.969600	0.484800	+	0.874625i	$0.338892\pi$
$432$	0	0
$433$	11.8558	0.569754	0.284877	−	0.958564i	$-0.408047\pi$
$433$	11.8558	0.569754	0.284877	+	0.958564i	$0.408047\pi$
$434$	0	0
$435$	0	0
$436$	25.9338	1.24200
$437$	4.16827	0.199395
$438$	0	0
$439$	−2.15787	−0.102989	−0.0514947	−	0.998673i	$-0.516399\pi$
$439$	−2.15787	−0.102989	−0.0514947	+	0.998673i	$0.516399\pi$
$440$	18.5458	0.884138
$441$	0	0
$442$	−15.7896	−0.751035
$443$	1.96225	0.0932293	0.0466147	−	0.998913i	$-0.485157\pi$
$443$	1.96225	0.0932293	0.0466147	+	0.998913i	$0.485157\pi$
$444$	0	0
$445$	30.9532	1.46732
$446$	16.2736	0.770577
$447$	0	0
$448$	0	0
$449$	−2.15211	−0.101564	−0.0507822	−	0.998710i	$-0.516171\pi$
$449$	−2.15211	−0.101564	−0.0507822	+	0.998710i	$0.516171\pi$
$450$	0	0
$451$	8.92588	0.420303
$452$	−7.45993	−0.350885
$453$	0	0
$454$	−9.60412	−0.450744
$455$	0	0
$456$	0	0
$457$	−13.1053	−0.613042	−0.306521	−	0.951864i	$-0.599165\pi$
$457$	−13.1053	−0.613042	−0.306521	+	0.951864i	$0.599165\pi$
$458$	22.0046	1.02821
$459$	0	0
$460$	4.26320	0.198773
$461$	5.48727	0.255568	0.127784	−	0.991802i	$-0.459214\pi$
$461$	5.48727	0.255568	0.127784	+	0.991802i	$0.459214\pi$
$462$	0	0
$463$	−20.4991	−0.952672	−0.476336	−	0.879263i	$-0.658035\pi$
$463$	−20.4991	−0.952672	−0.476336	+	0.879263i	$0.658035\pi$
$464$	−8.72313	−0.404961
$465$	0	0
$466$	16.3215	0.756078
$467$	−39.3516	−1.82097	−0.910487	−	0.413537i	$-0.864293\pi$
$467$	−39.3516	−1.82097	−0.910487	+	0.413537i	$0.864293\pi$
$468$	0	0
$469$	0	0
$470$	0.545830	0.0251773
$471$	0	0
$472$	5.35486	0.246477
$473$	7.35021	0.337963
$474$	0	0
$475$	22.6569	1.03957
$476$	0	0
$477$	0	0
$478$	−5.88675	−0.269254
$479$	16.7472	0.765199	0.382600	−	0.923914i	$-0.375029\pi$
$479$	16.7472	0.765199	0.382600	+	0.923914i	$0.375029\pi$
$480$	0	0
$481$	11.5732	0.527691
$482$	−4.63500	−0.211119
$483$	0	0
$484$	8.50697	0.386680
$485$	−2.67962	−0.121675
$486$	0	0
$487$	4.00576	0.181518	0.0907591	−	0.995873i	$-0.471071\pi$
$487$	4.00576	0.181518	0.0907591	+	0.995873i	$0.471071\pi$
$488$	−15.2071	−0.688394
$489$	0	0
$490$	0	0
$491$	24.6512	1.11249	0.556246	−	0.831018i	$-0.312241\pi$
$491$	24.6512	1.11249	0.556246	+	0.831018i	$0.312241\pi$
$492$	0	0
$493$	−56.7292	−2.55495
$494$	−12.4588	−0.560549
$495$	0	0
$496$	−4.91410	−0.220649
$497$	0	0
$498$	0	0
$499$	−24.5595	−1.09943	−0.549717	−	0.835351i	$-0.685265\pi$
$499$	−24.5595	−1.09943	−0.549717	+	0.835351i	$0.685265\pi$
$500$	0.564149	0.0252295
$501$	0	0
$502$	−1.06647	−0.0475990
$503$	−12.3743	−0.551742	−0.275871	−	0.961195i	$-0.588966\pi$
$503$	−12.3743	−0.551742	−0.275871	+	0.961195i	$0.588966\pi$
$504$	0	0
$505$	−24.6512	−1.09696
$506$	1.60628	0.0714078
$507$	0	0
$508$	−28.5710	−1.26763
$509$	−11.1956	−0.496237	−0.248118	−	0.968730i	$-0.579812\pi$
$509$	−11.1956	−0.496237	−0.248118	+	0.968730i	$0.579812\pi$
$510$	0	0
$511$	0	0
$512$	−9.53543	−0.421410
$513$	0	0
$514$	−13.6627	−0.602635
$515$	7.74720	0.341383
$516$	0	0
$517$	−0.504811	−0.0222016
$518$	0	0
$519$	0	0
$520$	−30.6764	−1.34525
$521$	−31.6192	−1.38526	−0.692631	−	0.721293i	$-0.743548\pi$
$521$	−31.6192	−1.38526	−0.692631	+	0.721293i	$0.743548\pi$
$522$	0	0
$523$	−28.2359	−1.23467	−0.617334	−	0.786701i	$-0.711787\pi$
$523$	−28.2359	−1.23467	−0.617334	+	0.786701i	$0.711787\pi$
$524$	−5.79974	−0.253363
$525$	0	0
$526$	−5.46019	−0.238076
$527$	−31.9579	−1.39211
$528$	0	0
$529$	−22.1111	−0.961352
$530$	−25.8090	−1.12107
$531$	0	0
$532$	0	0
$533$	−14.7641	−0.639506
$534$	0	0
$535$	36.4796	1.57715
$536$	19.3171	0.834372
$537$	0	0
$538$	2.58469	0.111434
$539$	0	0
$540$	0	0
$541$	42.8090	1.84050	0.920252	−	0.391326i	$-0.127984\pi$
$541$	42.8090	1.84050	0.920252	+	0.391326i	$0.127984\pi$
$542$	7.78822	0.334533
$543$	0	0
$544$	−32.8424	−1.40811
$545$	58.0690	2.48740
$546$	0	0
$547$	13.5516	0.579424	0.289712	−	0.957114i	$-0.406440\pi$
$547$	13.5516	0.579424	0.289712	+	0.957114i	$0.406440\pi$
$548$	−2.67962	−0.114468
$549$	0	0
$550$	8.73104	0.372293
$551$	−44.7623	−1.90694
$552$	0	0
$553$	0	0
$554$	2.70834	0.115066
$555$	0	0
$556$	18.1831	0.771133
$557$	−32.7850	−1.38914	−0.694572	−	0.719424i	$-0.744406\pi$
$557$	−32.7850	−1.38914	−0.694572	+	0.719424i	$0.744406\pi$
$558$	0	0
$559$	−12.1579	−0.514223
$560$	0	0
$561$	0	0
$562$	5.31573	0.224231
$563$	17.1546	0.722980	0.361490	−	0.932376i	$-0.382268\pi$
$563$	17.1546	0.722980	0.361490	+	0.932376i	$0.382268\pi$
$564$	0	0
$565$	−16.7037	−0.702730
$566$	22.9910	0.966383
$567$	0	0
$568$	−18.9062	−0.793286
$569$	−12.8993	−0.540767	−0.270384	−	0.962753i	$-0.587151\pi$
$569$	−12.8993	−0.540767	−0.270384	+	0.962753i	$0.587151\pi$
$570$	0	0
$571$	−0.282630	−0.0118277	−0.00591385	−	0.999983i	$-0.501882\pi$
$571$	−0.282630	−0.0118277	−0.00591385	+	0.999983i	$0.501882\pi$
$572$	11.7850	0.492754
$573$	0	0
$574$	0	0
$575$	4.83173	0.201497
$576$	0	0
$577$	−2.16578	−0.0901627	−0.0450814	−	0.998983i	$-0.514355\pi$
$577$	−2.16578	−0.0901627	−0.0450814	+	0.998983i	$0.514355\pi$
$578$	−10.9518	−0.455537
$579$	0	0
$580$	−45.7817	−1.90098
$581$	0	0
$582$	0	0
$583$	23.8695	0.988573
$584$	19.6775	0.814259
$585$	0	0
$586$	−11.5926	−0.478886
$587$	16.7759	0.692417	0.346208	−	0.938158i	$-0.387469\pi$
$587$	16.7759	0.692417	0.346208	+	0.938158i	$0.387469\pi$
$588$	0	0
$589$	−25.2164	−1.03902
$590$	4.98057	0.205047
$591$	0	0
$592$	2.69218	0.110648
$593$	25.1866	1.03429	0.517145	−	0.855898i	$-0.326995\pi$
$593$	25.1866	1.03429	0.517145	+	0.855898i	$0.326995\pi$
$594$	0	0
$595$	0	0
$596$	11.4498	0.469002
$597$	0	0
$598$	−2.65692	−0.108650
$599$	−28.7324	−1.17397	−0.586987	−	0.809596i	$-0.699686\pi$
$599$	−28.7324	−1.17397	−0.586987	+	0.809596i	$0.699686\pi$
$600$	0	0
$601$	−36.7954	−1.50091	−0.750457	−	0.660919i	$-0.770167\pi$
$601$	−36.7954	−1.50091	−0.750457	+	0.660919i	$0.770167\pi$
$602$	0	0
$603$	0	0
$604$	8.92804	0.363277
$605$	19.0482	0.774418
$606$	0	0
$607$	37.9007	1.53834	0.769171	−	0.639043i	$-0.220670\pi$
$607$	37.9007	1.53834	0.769171	+	0.639043i	$0.220670\pi$
$608$	−25.9144	−1.05097
$609$	0	0
$610$	−14.1442	−0.572682
$611$	0.834999	0.0337805
$612$	0	0
$613$	38.0391	1.53639	0.768193	−	0.640218i	$-0.221156\pi$
$613$	38.0391	1.53639	0.768193	+	0.640218i	$0.221156\pi$
$614$	0.786064	0.0317230
$615$	0	0
$616$	0	0
$617$	−26.5264	−1.06791	−0.533956	−	0.845512i	$-0.679295\pi$
$617$	−26.5264	−1.06791	−0.533956	+	0.845512i	$0.679295\pi$
$618$	0	0
$619$	12.7037	0.510605	0.255302	−	0.966861i	$-0.417825\pi$
$619$	12.7037	0.510605	0.255302	+	0.966861i	$0.417825\pi$
$620$	−25.7907	−1.03578
$621$	0	0
$622$	−7.09166	−0.284350
$623$	0	0
$624$	0	0
$625$	−24.3606	−0.974425
$626$	−4.63500	−0.185252
$627$	0	0
$628$	−1.00000	−0.0399043
$629$	17.5081	0.698093
$630$	0	0
$631$	20.6764	0.823113	0.411556	−	0.911384i	$-0.364985\pi$
$631$	20.6764	0.823113	0.411556	+	0.911384i	$0.364985\pi$
$632$	−17.7745	−0.707034
$633$	0	0
$634$	17.7292	0.704114
$635$	−63.9740	−2.53873
$636$	0	0
$637$	0	0
$638$	−17.2495	−0.682915
$639$	0	0
$640$	−30.6764	−1.21259
$641$	−27.9740	−1.10491	−0.552454	−	0.833543i	$-0.686308\pi$
$641$	−27.9740	−1.10491	−0.552454	+	0.833543i	$0.686308\pi$
$642$	0	0
$643$	27.9806	1.10345	0.551723	−	0.834027i	$-0.313971\pi$
$643$	27.9806	1.10345	0.551723	+	0.834027i	$0.313971\pi$
$644$	0	0
$645$	0	0
$646$	−18.8479	−0.741560
$647$	4.61668	0.181501	0.0907503	−	0.995874i	$-0.471073\pi$
$647$	4.61668	0.181501	0.0907503	+	0.995874i	$0.471073\pi$
$648$	0	0
$649$	−4.60628	−0.180812
$650$	−14.4419	−0.566457
$651$	0	0
$652$	27.3354	1.07054
$653$	11.5803	0.453173	0.226586	−	0.973991i	$-0.427243\pi$
$653$	11.5803	0.453173	0.226586	+	0.973991i	$0.427243\pi$
$654$	0	0
$655$	−12.9863	−0.507418
$656$	−3.43447	−0.134094
$657$	0	0
$658$	0	0
$659$	4.73680	0.184520	0.0922598	−	0.995735i	$-0.470591\pi$
$659$	4.73680	0.184520	0.0922598	+	0.995735i	$0.470591\pi$
$660$	0	0
$661$	−13.8227	−0.537641	−0.268820	−	0.963190i	$-0.586634\pi$
$661$	−13.8227	−0.537641	−0.268820	+	0.963190i	$0.586634\pi$
$662$	−11.1669	−0.434014
$663$	0	0
$664$	21.1132	0.819353
$665$	0	0
$666$	0	0
$667$	−9.54583	−0.369616
$668$	33.2359	1.28593
$669$	0	0
$670$	17.9669	0.694122
$671$	13.0813	0.504996
$672$	0	0
$673$	6.02735	0.232337	0.116169	−	0.993230i	$-0.462939\pi$
$673$	6.02735	0.232337	0.116169	+	0.993230i	$0.462939\pi$
$674$	−19.4966	−0.750980
$675$	0	0
$676$	−1.01943	−0.0392089
$677$	−22.8856	−0.879567	−0.439783	−	0.898104i	$-0.644945\pi$
$677$	−22.8856	−0.879567	−0.439783	+	0.898104i	$0.644945\pi$
$678$	0	0
$679$	0	0
$680$	−46.4077	−1.77965
$681$	0	0
$682$	−9.71737	−0.372097
$683$	10.2988	0.394072	0.197036	−	0.980396i	$-0.436868\pi$
$683$	10.2988	0.394072	0.197036	+	0.980396i	$0.436868\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	−2.82819	−0.107824
$689$	−39.4821	−1.50415
$690$	0	0
$691$	29.0722	1.10596	0.552980	−	0.833195i	$-0.313491\pi$
$691$	29.0722	1.10596	0.552980	+	0.833195i	$0.313491\pi$
$692$	−11.7083	−0.445084
$693$	0	0
$694$	−8.59261	−0.326171
$695$	40.7141	1.54437
$696$	0	0
$697$	−22.3354	−0.846015
$698$	8.59509	0.325329
$699$	0	0
$700$	0	0
$701$	−27.4153	−1.03546	−0.517731	−	0.855543i	$-0.673223\pi$
$701$	−27.4153	−1.03546	−0.517731	+	0.855543i	$0.673223\pi$
$702$	0	0
$703$	13.8148	0.521035
$704$	−6.12803	−0.230959
$705$	0	0
$706$	14.0780	0.529832
$707$	0	0
$708$	0	0
$709$	−36.9669	−1.38832	−0.694160	−	0.719820i	$-0.744224\pi$
$709$	−36.9669	−1.38832	−0.694160	+	0.719820i	$0.744224\pi$
$710$	−17.5847	−0.659942
$711$	0	0
$712$	−25.3215	−0.948963
$713$	−5.37756	−0.201391
$714$	0	0
$715$	26.3880	0.986854
$716$	14.0824	0.526283
$717$	0	0
$718$	−15.1384	−0.564961
$719$	40.5199	1.51114	0.755568	−	0.655070i	$-0.227361\pi$
$719$	40.5199	1.51114	0.755568	+	0.655070i	$0.227361\pi$
$720$	0	0
$721$	0	0
$722$	−0.415309	−0.0154562
$723$	0	0
$724$	13.3409	0.495811
$725$	−51.8870	−1.92704
$726$	0	0
$727$	−15.2416	−0.565280	−0.282640	−	0.959226i	$-0.591210\pi$
$727$	−15.2416	−0.565280	−0.282640	+	0.959226i	$0.591210\pi$
$728$	0	0
$729$	0	0
$730$	18.3021	0.677390
$731$	−18.3926	−0.680275
$732$	0	0
$733$	32.5789	1.20333	0.601665	−	0.798748i	$-0.294504\pi$
$733$	32.5789	1.20333	0.601665	+	0.798748i	$0.294504\pi$
$734$	−4.37540	−0.161499
$735$	0	0
$736$	−5.52640	−0.203706
$737$	−16.6167	−0.612083
$738$	0	0
$739$	7.01367	0.258002	0.129001	−	0.991644i	$-0.458823\pi$
$739$	7.01367	0.258002	0.129001	+	0.991644i	$0.458823\pi$
$740$	14.1294	0.519407
$741$	0	0
$742$	0	0
$743$	−35.0118	−1.28446	−0.642229	−	0.766513i	$-0.721990\pi$
$743$	−35.0118	−1.28446	−0.642229	+	0.766513i	$0.721990\pi$
$744$	0	0
$745$	25.6375	0.939285
$746$	−1.52175	−0.0557154
$747$	0	0
$748$	17.8285	0.651873
$749$	0	0
$750$	0	0
$751$	−4.27687	−0.156065	−0.0780327	−	0.996951i	$-0.524864\pi$
$751$	−4.27687	−0.156065	−0.0780327	+	0.996951i	$0.524864\pi$
$752$	0.194240	0.00708319
$753$	0	0
$754$	28.5322	1.03908
$755$	19.9910	0.727546
$756$	0	0
$757$	−34.9611	−1.27068	−0.635342	−	0.772231i	$-0.719141\pi$
$757$	−34.9611	−1.27068	−0.635342	+	0.772231i	$0.719141\pi$
$758$	−6.81230	−0.247434
$759$	0	0
$760$	−36.6181	−1.32828
$761$	−4.58358	−0.166155	−0.0830773	−	0.996543i	$-0.526475\pi$
$761$	−4.58358	−0.166155	−0.0830773	+	0.996543i	$0.526475\pi$
$762$	0	0
$763$	0	0
$764$	−35.1546	−1.27185
$765$	0	0
$766$	−15.3431	−0.554368
$767$	7.61917	0.275112
$768$	0	0
$769$	16.9590	0.611556	0.305778	−	0.952103i	$-0.401083\pi$
$769$	16.9590	0.611556	0.305778	+	0.952103i	$0.401083\pi$
$770$	0	0
$771$	0	0
$772$	1.17730	0.0423718
$773$	−40.2840	−1.44892	−0.724458	−	0.689319i	$-0.757910\pi$
$773$	−40.2840	−1.44892	−0.724458	+	0.689319i	$0.757910\pi$
$774$	0	0
$775$	−29.2301	−1.04998
$776$	2.19208	0.0786911
$777$	0	0
$778$	25.5653	0.916559
$779$	−17.6238	−0.631439
$780$	0	0
$781$	16.2632	0.581943
$782$	−4.01943	−0.143735
$783$	0	0
$784$	0	0
$785$	−2.23912	−0.0799177
$786$	0	0
$787$	47.2107	1.68288	0.841439	−	0.540352i	$-0.181709\pi$
$787$	47.2107	1.68288	0.841439	+	0.540352i	$0.181709\pi$
$788$	8.32967	0.296732
$789$	0	0
$790$	−16.5322	−0.588188
$791$	0	0
$792$	0	0
$793$	−21.6375	−0.768370
$794$	3.14884	0.111748
$795$	0	0
$796$	13.1442	0.465884
$797$	4.02735	0.142656	0.0713280	−	0.997453i	$-0.477276\pi$
$797$	4.02735	0.142656	0.0713280	+	0.997453i	$0.477276\pi$
$798$	0	0
$799$	1.26320	0.0446888
$800$	−30.0391	−1.06204
$801$	0	0
$802$	−11.7702	−0.415619
$803$	−16.9267	−0.597329
$804$	0	0
$805$	0	0
$806$	16.0733	0.566159
$807$	0	0
$808$	20.1660	0.709439
$809$	−11.8824	−0.417762	−0.208881	−	0.977941i	$-0.566982\pi$
$809$	−11.8824	−0.417762	−0.208881	+	0.977941i	$0.566982\pi$
$810$	0	0
$811$	−21.1111	−0.741311	−0.370655	−	0.928770i	$-0.620867\pi$
$811$	−21.1111	−0.741311	−0.370655	+	0.928770i	$0.620867\pi$
$812$	0	0
$813$	0	0
$814$	5.32365	0.186594
$815$	61.2074	2.14400
$816$	0	0
$817$	−14.5127	−0.507736
$818$	−20.8432	−0.728767
$819$	0	0
$820$	−18.0252	−0.629467
$821$	37.2919	1.30150	0.650749	−	0.759293i	$-0.274455\pi$
$821$	37.2919	1.30150	0.650749	+	0.759293i	$0.274455\pi$
$822$	0	0
$823$	−21.4523	−0.747779	−0.373890	−	0.927473i	$-0.621976\pi$
$823$	−21.4523	−0.747779	−0.373890	+	0.927473i	$0.621976\pi$
$824$	−6.33765	−0.220783
$825$	0	0
$826$	0	0
$827$	28.6375	0.995823	0.497912	−	0.867228i	$-0.334100\pi$
$827$	28.6375	0.995823	0.497912	+	0.867228i	$0.334100\pi$
$828$	0	0
$829$	−39.7292	−1.37985	−0.689925	−	0.723881i	$-0.742357\pi$
$829$	−39.7292	−1.37985	−0.689925	+	0.723881i	$0.742357\pi$
$830$	19.6375	0.681627
$831$	0	0
$832$	10.1363	0.351412
$833$	0	0
$834$	0	0
$835$	74.4192	2.57538
$836$	14.0676	0.486538
$837$	0	0
$838$	15.9784	0.551965
$839$	34.3606	1.18626	0.593130	−	0.805107i	$-0.297892\pi$
$839$	34.3606	1.18626	0.593130	+	0.805107i	$0.297892\pi$
$840$	0	0
$841$	73.5108	2.53486
$842$	−15.1863	−0.523355
$843$	0	0
$844$	17.8479	0.614350
$845$	−2.28263	−0.0785249
$846$	0	0
$847$	0	0
$848$	−9.18443	−0.315395
$849$	0	0
$850$	−21.8479	−0.749376
$851$	2.94609	0.100991
$852$	0	0
$853$	−1.51462	−0.0518596	−0.0259298	−	0.999664i	$-0.508255\pi$
$853$	−1.51462	−0.0518596	−0.0259298	+	0.999664i	$0.508255\pi$
$854$	0	0
$855$	0	0
$856$	−29.8424	−1.01999
$857$	4.50946	0.154040	0.0770201	−	0.997030i	$-0.475459\pi$
$857$	4.50946	0.154040	0.0770201	+	0.997030i	$0.475459\pi$
$858$	0	0
$859$	12.6063	0.430121	0.215060	−	0.976601i	$-0.431005\pi$
$859$	12.6063	0.430121	0.215060	+	0.976601i	$0.431005\pi$
$860$	−14.8432	−0.506150
$861$	0	0
$862$	−15.3160	−0.521665
$863$	10.6602	0.362877	0.181439	−	0.983402i	$-0.441925\pi$
$863$	10.6602	0.362877	0.181439	+	0.983402i	$0.441925\pi$
$864$	0	0
$865$	−26.2164	−0.891385
$866$	−9.02081	−0.306540
$867$	0	0
$868$	0	0
$869$	15.2898	0.518670
$870$	0	0
$871$	27.4854	0.931307
$872$	−47.5037	−1.60868
$873$	0	0
$874$	−3.17154	−0.107279
$875$	0	0
$876$	0	0
$877$	−32.2380	−1.08860	−0.544300	−	0.838891i	$-0.683205\pi$
$877$	−32.2380	−1.08860	−0.544300	+	0.838891i	$0.683205\pi$
$878$	1.64187	0.0554104
$879$	0	0
$880$	6.13844	0.206927
$881$	55.6375	1.87447	0.937237	−	0.348692i	$-0.113374\pi$
$881$	55.6375	1.87447	0.937237	+	0.348692i	$0.113374\pi$
$882$	0	0
$883$	−42.4854	−1.42975	−0.714873	−	0.699254i	$-0.753516\pi$
$883$	−42.4854	−1.42975	−0.714873	+	0.699254i	$0.753516\pi$
$884$	−29.4898	−0.991848
$885$	0	0
$886$	−1.49303	−0.0501593
$887$	8.95649	0.300730	0.150365	−	0.988631i	$-0.451955\pi$
$887$	8.95649	0.300730	0.150365	+	0.988631i	$0.451955\pi$
$888$	0	0
$889$	0	0
$890$	−23.5516	−0.789451
$891$	0	0
$892$	30.3937	1.01766
$893$	0.996730	0.0333543
$894$	0	0
$895$	31.5322	1.05400
$896$	0	0
$897$	0	0
$898$	1.63749	0.0546437
$899$	57.7486	1.92602
$900$	0	0
$901$	−59.7292	−1.98987
$902$	−6.79149	−0.226132
$903$	0	0
$904$	13.6646	0.454477
$905$	29.8720	0.992978
$906$	0	0
$907$	10.7874	0.358191	0.179096	−	0.983832i	$-0.442683\pi$
$907$	10.7874	0.358191	0.179096	+	0.983832i	$0.442683\pi$
$908$	−17.9373	−0.595271
$909$	0	0
$910$	0	0
$911$	47.4854	1.57326	0.786630	−	0.617424i	$-0.211824\pi$
$911$	47.4854	1.57326	0.786630	+	0.617424i	$0.211824\pi$
$912$	0	0
$913$	−18.1617	−0.601066
$914$	9.97154	0.329829
$915$	0	0
$916$	41.0974	1.35790
$917$	0	0
$918$	0	0
$919$	56.2750	1.85634	0.928170	−	0.372156i	$-0.121381\pi$
$919$	56.2750	1.85634	0.928170	+	0.372156i	$0.121381\pi$
$920$	−7.80903	−0.257456
$921$	0	0
$922$	−4.17514	−0.137501
$923$	−26.9007	−0.885447
$924$	0	0
$925$	16.0137	0.526526
$926$	15.5973	0.512558
$927$	0	0
$928$	59.3469	1.94816
$929$	−0.760877	−0.0249636	−0.0124818	−	0.999922i	$-0.503973\pi$
$929$	−0.760877	−0.0249636	−0.0124818	+	0.999922i	$0.503973\pi$
$930$	0	0
$931$	0	0
$932$	30.4832	0.998509
$933$	0	0
$934$	29.9417	0.979723
$935$	39.9201	1.30553
$936$	0	0
$937$	−30.3218	−0.990569	−0.495284	−	0.868731i	$-0.664936\pi$
$937$	−30.3218	−0.990569	−0.495284	+	0.868731i	$0.664936\pi$
$938$	0	0
$939$	0	0
$940$	1.01943	0.0332502
$941$	−55.0813	−1.79560	−0.897799	−	0.440406i	$-0.854835\pi$
$941$	−55.0813	−1.79560	−0.897799	+	0.440406i	$0.854835\pi$
$942$	0	0
$943$	−3.75839	−0.122390
$944$	1.77239	0.0576864
$945$	0	0
$946$	−5.59261	−0.181831
$947$	−30.6205	−0.995034	−0.497517	−	0.867454i	$-0.665755\pi$
$947$	−30.6205	−0.995034	−0.497517	+	0.867454i	$0.665755\pi$
$948$	0	0
$949$	27.9981	0.908857
$950$	−17.2391	−0.559311
$951$	0	0
$952$	0	0
$953$	−7.83422	−0.253775	−0.126888	−	0.991917i	$-0.540499\pi$
$953$	−7.83422	−0.253775	−0.126888	+	0.991917i	$0.540499\pi$
$954$	0	0
$955$	−78.7155	−2.54717
$956$	−10.9945	−0.355588
$957$	0	0
$958$	−12.7426	−0.411693
$959$	0	0
$960$	0	0
$961$	1.53216	0.0494244
$962$	−8.80576	−0.283909
$963$	0	0
$964$	−8.65665	−0.278812
$965$	2.63611	0.0848595
$966$	0	0
$967$	−5.29630	−0.170318	−0.0851588	−	0.996367i	$-0.527140\pi$
$967$	−5.29630	−0.170318	−0.0851588	+	0.996367i	$0.527140\pi$
$968$	−15.5825	−0.500840
$969$	0	0
$970$	2.03886	0.0654639
$971$	−52.2405	−1.67648	−0.838239	−	0.545303i	$-0.816414\pi$
$971$	−52.2405	−1.67648	−0.838239	+	0.545303i	$0.816414\pi$
$972$	0	0
$973$	0	0
$974$	−3.04789	−0.0976606
$975$	0	0
$976$	−5.03337	−0.161114
$977$	5.94609	0.190232	0.0951161	−	0.995466i	$-0.469678\pi$
$977$	5.94609	0.190232	0.0951161	+	0.995466i	$0.469678\pi$
$978$	0	0
$979$	21.7817	0.696146
$980$	0	0
$981$	0	0
$982$	−18.7565	−0.598544
$983$	18.5641	0.592104	0.296052	−	0.955172i	$-0.404330\pi$
$983$	18.5641	0.592104	0.296052	+	0.955172i	$0.404330\pi$
$984$	0	0
$985$	18.6512	0.594275
$986$	43.1639	1.37462
$987$	0	0
$988$	−23.2690	−0.740284
$989$	−3.09493	−0.0984130
$990$	0	0
$991$	18.6375	0.592039	0.296020	−	0.955182i	$-0.404341\pi$
$991$	18.6375	0.592039	0.296020	+	0.955182i	$0.404341\pi$
$992$	33.4326	1.06149
$993$	0	0
$994$	0	0
$995$	29.4315	0.933040
$996$	0	0
$997$	−30.5595	−0.967829	−0.483915	−	0.875115i	$-0.660785\pi$
$997$	−30.5595	−0.967829	−0.483915	+	0.875115i	$0.660785\pi$
$998$	18.6868	0.591519
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.x.1.2		3
3.2	odd	2		1323.2.a.ba.1.2		3
7.2	even	3		189.2.e.f.109.2	yes	6
7.4	even	3		189.2.e.f.163.2	yes	6
7.6	odd	2		1323.2.a.y.1.2		3
21.2	odd	6		189.2.e.e.109.2	✓	6
21.11	odd	6		189.2.e.e.163.2	yes	6
21.20	even	2		1323.2.a.z.1.2		3
63.2	odd	6		567.2.g.h.109.2		6
63.4	even	3		567.2.g.i.541.2		6
63.11	odd	6		567.2.h.i.352.2		6
63.16	even	3		567.2.g.i.109.2		6
63.23	odd	6		567.2.h.i.298.2		6
63.25	even	3		567.2.h.h.352.2		6
63.32	odd	6		567.2.g.h.541.2		6
63.58	even	3		567.2.h.h.298.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.e.109.2	✓	6	21.2	odd	6
189.2.e.e.163.2	yes	6	21.11	odd	6
189.2.e.f.109.2	yes	6	7.2	even	3
189.2.e.f.163.2	yes	6	7.4	even	3
567.2.g.h.109.2		6	63.2	odd	6
567.2.g.h.541.2		6	63.32	odd	6
567.2.g.i.109.2		6	63.16	even	3
567.2.g.i.541.2		6	63.4	even	3
567.2.h.h.298.2		6	63.58	even	3
567.2.h.h.352.2		6	63.25	even	3
567.2.h.i.298.2		6	63.23	odd	6
567.2.h.i.352.2		6	63.11	odd	6
1323.2.a.x.1.2		3	1.1	even	1	trivial
1323.2.a.y.1.2		3	7.6	odd	2
1323.2.a.z.1.2		3	21.20	even	2
1323.2.a.ba.1.2		3	3.2	odd	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-0.760877 q^{2} -1.42107 q^{4} -3.18194 q^{5} +2.60301 q^{8} +O(q^{10})\)	\(q-0.760877 q^{2} -1.42107 q^{4} -3.18194 q^{5} +2.60301 q^{8} +2.42107 q^{10} -2.23912 q^{11} +3.70370 q^{13} +0.861564 q^{16} +5.60301 q^{17} +4.42107 q^{19} +4.52175 q^{20} +1.70370 q^{22} +0.942820 q^{23} +5.12476 q^{25} -2.81806 q^{26} -10.1248 q^{29} -5.70370 q^{31} -5.86156 q^{32} -4.26320 q^{34} +3.12476 q^{37} -3.36389 q^{38} -8.28263 q^{40} -3.98633 q^{41} -3.28263 q^{43} +3.18194 q^{44} -0.717370 q^{46} +0.225450 q^{47} -3.89931 q^{50} -5.26320 q^{52} -10.6602 q^{53} +7.12476 q^{55} +7.70370 q^{58} +2.05718 q^{59} -5.84213 q^{61} +4.33981 q^{62} +2.73680 q^{64} -11.7850 q^{65} +7.42107 q^{67} -7.96225 q^{68} -7.26320 q^{71} +7.55950 q^{73} -2.37756 q^{74} -6.28263 q^{76} -6.82846 q^{79} -2.74145 q^{80} +3.03310 q^{82} +8.11109 q^{83} -17.8285 q^{85} +2.49768 q^{86} -5.82846 q^{88} -9.72777 q^{89} -1.33981 q^{92} -0.171540 q^{94} -14.0676 q^{95} +0.842133 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8	\( 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34} - 8 q^{37} + 7 q^{38} - 24 q^{40} - 2 q^{41} - 9 q^{43} + q^{44} - 3 q^{46} - 9 q^{47} - 4 q^{50} + 9 q^{52} - 24 q^{53} + 4 q^{55} + 14 q^{58} + 15 q^{59} - q^{61} + 21 q^{62} + 33 q^{64} - 10 q^{65} + 14 q^{67} - 39 q^{68} + 3 q^{71} + 7 q^{73} - 18 q^{76} + 6 q^{79} + 16 q^{80} + 43 q^{82} - 3 q^{83} - 27 q^{85} + 32 q^{86} + 9 q^{88} + 5 q^{89} - 12 q^{92} - 27 q^{94} - 16 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more