LMFDB - Embedded newform 5733.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.46162368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 17x^{2} - 7 $ x^6 - 8x^4 + 17x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 819)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.734503$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.734503 q^{2} -1.46050 q^{4} -2.54175 q^{5} +2.54175 q^{8} +O(q^{10})$	$q-0.734503 q^{2} -1.46050 q^{4} -2.54175 q^{5} +2.54175 q^{8} +1.86693 q^{10} +3.71224 q^{11} +1.00000 q^{13} +1.05408 q^{16} +2.10577 q^{17} -5.92101 q^{19} +3.71224 q^{20} -2.72665 q^{22} -6.25399 q^{23} +1.46050 q^{25} -0.734503 q^{26} -8.83547 q^{29} +5.70175 q^{31} -5.85773 q^{32} -1.54669 q^{34} +6.56867 q^{37} +4.34900 q^{38} -6.46050 q^{40} +10.3045 q^{41} -10.6477 q^{43} -5.42175 q^{44} +4.59358 q^{46} -1.46901 q^{47} -1.07275 q^{50} -1.46050 q^{52} +6.45477 q^{53} -9.43560 q^{55} +6.48968 q^{58} +11.4352 q^{59} +9.11537 q^{61} -4.18795 q^{62} +2.19436 q^{64} -2.54175 q^{65} -14.0541 q^{67} -3.07548 q^{68} +14.4527 q^{71} +6.14027 q^{73} -4.82471 q^{74} +8.64766 q^{76} -7.96790 q^{79} -2.67922 q^{80} -7.56867 q^{82} -12.1515 q^{83} -5.35234 q^{85} +7.82074 q^{86} +9.43560 q^{88} +11.8132 q^{89} +9.13399 q^{92} +1.07899 q^{94} +15.0497 q^{95} -7.10097 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4}+O(q^{10}) $ 6 * q + 4 * q^4	$ 6 q + 4 q^{4} + 4 q^{10} + 6 q^{13} - 12 q^{16} - 10 q^{19} - 18 q^{22} - 4 q^{25} - 8 q^{31} - 6 q^{34} - 10 q^{37} - 26 q^{40} - 40 q^{43} + 22 q^{46} + 4 q^{52} - 36 q^{58} + 2 q^{61} - 14 q^{64} - 66 q^{67} + 28 q^{73} + 28 q^{76} - 20 q^{79} + 4 q^{82} - 56 q^{85} + 32 q^{94} - 22 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 + 4 * q^10 + 6 * q^13 - 12 * q^16 - 10 * q^19 - 18 * q^22 - 4 * q^25 - 8 * q^31 - 6 * q^34 - 10 * q^37 - 26 * q^40 - 40 * q^43 + 22 * q^46 + 4 * q^52 - 36 * q^58 + 2 * q^61 - 14 * q^64 - 66 * q^67 + 28 * q^73 + 28 * q^76 - 20 * q^79 + 4 * q^82 - 56 * q^85 + 32 * q^94 - 22 * 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 $

$2$	−0.734503	−0.519372	−0.259686	−	0.965693i	$-0.583619\pi$
$2$	−0.734503	−0.519372	−0.259686	+	0.965693i	$0.583619\pi$
$3$	0	0
$3$	0	0
$4$	−1.46050	−0.730252
$5$	−2.54175	−1.13671	−0.568353	−	0.822785i	$-0.692419\pi$
$5$	−2.54175	−1.13671	−0.568353	+	0.822785i	$0.692419\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.54175	0.898645
$9$	0	0
$10$	1.86693	0.590374
$11$	3.71224	1.11928	0.559641	−	0.828735i	$-0.310939\pi$
$11$	3.71224	1.11928	0.559641	+	0.828735i	$0.310939\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	1.05408	0.263521
$17$	2.10577	0.510723	0.255362	−	0.966846i	$-0.417805\pi$
$17$	2.10577	0.510723	0.255362	+	0.966846i	$0.417805\pi$
$18$	0	0
$19$	−5.92101	−1.35837	−0.679186	−	0.733966i	$-0.737667\pi$
$19$	−5.92101	−1.35837	−0.679186	+	0.733966i	$0.737667\pi$
$20$	3.71224	0.830082
$21$	0	0
$22$	−2.72665	−0.581325
$23$	−6.25399	−1.30405	−0.652024	−	0.758198i	$-0.726080\pi$
$23$	−6.25399	−1.30405	−0.652024	+	0.758198i	$0.726080\pi$
$24$	0	0
$25$	1.46050	0.292101
$26$	−0.734503	−0.144048
$27$	0	0
$28$	0	0
$29$	−8.83547	−1.64071	−0.820353	−	0.571858i	$-0.806223\pi$
$29$	−8.83547	−1.64071	−0.820353	+	0.571858i	$0.806223\pi$
$30$	0	0
$31$	5.70175	1.02406	0.512032	−	0.858966i	$-0.328893\pi$
$31$	5.70175	1.02406	0.512032	+	0.858966i	$0.328893\pi$
$32$	−5.85773	−1.03551
$33$	0	0
$34$	−1.54669	−0.265256
$35$	0	0
$36$	0	0
$37$	6.56867	1.07988	0.539942	−	0.841702i	$-0.318446\pi$
$37$	6.56867	1.07988	0.539942	+	0.841702i	$0.318446\pi$
$38$	4.34900	0.705501
$39$	0	0
$40$	−6.46050	−1.02150
$41$	10.3045	1.60929	0.804645	−	0.593757i	$-0.202356\pi$
$41$	10.3045	1.60929	0.804645	+	0.593757i	$0.202356\pi$
$42$	0	0
$43$	−10.6477	−1.62375	−0.811877	−	0.583829i	$-0.801554\pi$
$43$	−10.6477	−1.62375	−0.811877	+	0.583829i	$0.801554\pi$
$44$	−5.42175	−0.817359
$45$	0	0
$46$	4.59358	0.677286
$47$	−1.46901	−0.214277	−0.107138	−	0.994244i	$-0.534169\pi$
$47$	−1.46901	−0.214277	−0.107138	+	0.994244i	$0.534169\pi$
$48$	0	0
$49$	0	0
$50$	−1.07275	−0.151709
$51$	0	0
$52$	−1.46050	−0.202536
$53$	6.45477	0.886631	0.443315	−	0.896366i	$-0.353802\pi$
$53$	6.45477	0.886631	0.443315	+	0.896366i	$0.353802\pi$
$54$	0	0
$55$	−9.43560	−1.27230
$56$	0	0
$57$	0	0
$58$	6.48968	0.852137
$59$	11.4352	1.48874	0.744371	−	0.667766i	$-0.232750\pi$
$59$	11.4352	1.48874	0.744371	+	0.667766i	$0.232750\pi$
$60$	0	0
$61$	9.11537	1.16710	0.583551	−	0.812076i	$-0.301663\pi$
$61$	9.11537	1.16710	0.583551	+	0.812076i	$0.301663\pi$
$62$	−4.18795	−0.531871
$63$	0	0
$64$	2.19436	0.274294
$65$	−2.54175	−0.315266
$66$	0	0
$67$	−14.0541	−1.71698	−0.858490	−	0.512831i	$-0.828597\pi$
$67$	−14.0541	−1.71698	−0.858490	+	0.512831i	$0.828597\pi$
$68$	−3.07548	−0.372957
$69$	0	0
$70$	0	0
$71$	14.4527	1.71522	0.857610	−	0.514300i	$-0.171948\pi$
$71$	14.4527	1.71522	0.857610	+	0.514300i	$0.171948\pi$
$72$	0	0
$73$	6.14027	0.718664	0.359332	−	0.933210i	$-0.383004\pi$
$73$	6.14027	0.718664	0.359332	+	0.933210i	$0.383004\pi$
$74$	−4.82471	−0.560861
$75$	0	0
$76$	8.64766	0.991955
$77$	0	0
$78$	0	0
$79$	−7.96790	−0.896458	−0.448229	−	0.893919i	$-0.647945\pi$
$79$	−7.96790	−0.896458	−0.448229	+	0.893919i	$0.647945\pi$
$80$	−2.67922	−0.299546
$81$	0	0
$82$	−7.56867	−0.835820
$83$	−12.1515	−1.33380	−0.666898	−	0.745149i	$-0.732378\pi$
$83$	−12.1515	−1.33380	−0.666898	+	0.745149i	$0.732378\pi$
$84$	0	0
$85$	−5.35234	−0.580542
$86$	7.82074	0.843333
$87$	0	0
$88$	9.43560	1.00584
$89$	11.8132	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$89$	11.8132	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$90$	0	0
$91$	0	0
$92$	9.13399	0.952284
$93$	0	0
$94$	1.07899	0.111289
$95$	15.0497	1.54407
$96$	0	0
$97$	−7.10097	−0.720994	−0.360497	−	0.932760i	$-0.617393\pi$
$97$	−7.10097	−0.720994	−0.360497	+	0.932760i	$0.617393\pi$
$98$	0	0
$99$	0	0
$100$	−2.13307	−0.213307
$101$	8.49723	0.845506	0.422753	−	0.906245i	$-0.361064\pi$
$101$	8.49723	0.845506	0.422753	+	0.906245i	$0.361064\pi$
$102$	0	0
$103$	−2.18716	−0.215507	−0.107754	−	0.994178i	$-0.534366\pi$
$103$	−2.18716	−0.215507	−0.107754	+	0.994178i	$0.534366\pi$
$104$	2.54175	0.249239
$105$	0	0
$106$	−4.74105	−0.460491
$107$	7.52751	0.727712	0.363856	−	0.931455i	$-0.381460\pi$
$107$	7.52751	0.727712	0.363856	+	0.931455i	$0.381460\pi$
$108$	0	0
$109$	0.0540842	0.00518033	0.00259016	−	0.999997i	$-0.499176\pi$
$109$	0.0540842	0.00518033	0.00259016	+	0.999997i	$0.499176\pi$
$110$	6.93048	0.660795
$111$	0	0
$112$	0	0
$113$	6.25399	0.588326	0.294163	−	0.955755i	$-0.404959\pi$
$113$	6.25399	0.588326	0.294163	+	0.955755i	$0.404959\pi$
$114$	0	0
$115$	15.8961	1.48232
$116$	12.9042	1.19813
$117$	0	0
$118$	−8.39922	−0.773211
$119$	0	0
$120$	0	0
$121$	2.78074	0.252794
$122$	−6.69527	−0.606161
$123$	0	0
$124$	−8.32743	−0.747825
$125$	8.99652	0.804673
$126$	0	0
$127$	−2.49261	−0.221183	−0.110592	−	0.993866i	$-0.535275\pi$
$127$	−2.49261	−0.221183	−0.110592	+	0.993866i	$0.535275\pi$
$128$	10.1037	0.893050
$129$	0	0
$130$	1.86693	0.163740
$131$	−8.85376	−0.773557	−0.386779	−	0.922173i	$-0.626412\pi$
$131$	−8.85376	−0.773557	−0.386779	+	0.922173i	$0.626412\pi$
$132$	0	0
$133$	0	0
$134$	10.3228	0.891752
$135$	0	0
$136$	5.35234	0.458959
$137$	−2.71895	−0.232295	−0.116148	−	0.993232i	$-0.537055\pi$
$137$	−2.71895	−0.232295	−0.116148	+	0.993232i	$0.537055\pi$
$138$	0	0
$139$	−13.1694	−1.11702	−0.558509	−	0.829498i	$-0.688626\pi$
$139$	−13.1694	−1.11702	−0.558509	+	0.829498i	$0.688626\pi$
$140$	0	0
$141$	0	0
$142$	−10.6156	−0.890838
$143$	3.71224	0.310433
$144$	0	0
$145$	22.4576	1.86500
$146$	−4.51005	−0.373254
$147$	0	0
$148$	−9.59358	−0.788587
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−7.29533	−0.593685	−0.296843	−	0.954926i	$-0.595934\pi$
$151$	−7.29533	−0.593685	−0.296843	+	0.954926i	$0.595934\pi$
$152$	−15.0497	−1.22070
$153$	0	0
$154$	0	0
$155$	−14.4924	−1.16406
$156$	0	0
$157$	5.67257	0.452720	0.226360	−	0.974044i	$-0.427317\pi$
$157$	5.67257	0.452720	0.226360	+	0.974044i	$0.427317\pi$
$158$	5.85245	0.465596
$159$	0	0
$160$	14.8889	1.17707
$161$	0	0
$162$	0	0
$163$	−20.7161	−1.62261	−0.811307	−	0.584621i	$-0.801243\pi$
$163$	−20.7161	−1.62261	−0.811307	+	0.584621i	$0.801243\pi$
$164$	−15.0497	−1.17519
$165$	0	0
$166$	8.92528	0.692736
$167$	14.3152	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$167$	14.3152	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.93131	0.301518
$171$	0	0
$172$	15.5510	1.18575
$173$	−22.4162	−1.70427	−0.852136	−	0.523320i	$-0.824693\pi$
$173$	−22.4162	−1.70427	−0.852136	+	0.523320i	$0.824693\pi$
$174$	0	0
$175$	0	0
$176$	3.91302	0.294955
$177$	0	0
$178$	−8.67684	−0.650357
$179$	−8.26202	−0.617532	−0.308766	−	0.951138i	$-0.599916\pi$
$179$	−8.26202	−0.617532	−0.308766	+	0.951138i	$0.599916\pi$
$180$	0	0
$181$	0.787935	0.0585668	0.0292834	−	0.999571i	$-0.490677\pi$
$181$	0.787935	0.0585668	0.0292834	+	0.999571i	$0.490677\pi$
$182$	0	0
$183$	0	0
$184$	−15.8961	−1.17188
$185$	−16.6959	−1.22751
$186$	0	0
$187$	7.81711	0.571644
$188$	2.14549	0.156476
$189$	0	0
$190$	−11.0541	−0.801948
$191$	−0.396261	−0.0286724	−0.0143362	−	0.999897i	$-0.504564\pi$
$191$	−0.396261	−0.0286724	−0.0143362	+	0.999897i	$0.504564\pi$
$192$	0	0
$193$	3.78794	0.272662	0.136331	−	0.990663i	$-0.456469\pi$
$193$	3.78794	0.272662	0.136331	+	0.990663i	$0.456469\pi$
$194$	5.21569	0.374464
$195$	0	0
$196$	0	0
$197$	−12.0484	−0.858415	−0.429207	−	0.903206i	$-0.641207\pi$
$197$	−12.0484	−0.858415	−0.429207	+	0.903206i	$0.641207\pi$
$198$	0	0
$199$	5.16225	0.365942	0.182971	−	0.983118i	$-0.441428\pi$
$199$	5.16225	0.365942	0.182971	+	0.983118i	$0.441428\pi$
$200$	3.71224	0.262495
$201$	0	0
$202$	−6.24124	−0.439132
$203$	0	0
$204$	0	0
$205$	−26.1914	−1.82929
$206$	1.60648	0.111928
$207$	0	0
$208$	1.05408	0.0730876
$209$	−21.9802	−1.52040
$210$	0	0
$211$	17.9794	1.23775	0.618875	−	0.785489i	$-0.287589\pi$
$211$	17.9794	1.23775	0.618875	+	0.785489i	$0.287589\pi$
$212$	−9.42722	−0.647464
$213$	0	0
$214$	−5.52898	−0.377954
$215$	27.0637	1.84573
$216$	0	0
$217$	0	0
$218$	−0.0397250	−0.00269052
$219$	0	0
$220$	13.7807	0.929097
$221$	2.10577	0.141649
$222$	0	0
$223$	−1.96790	−0.131780	−0.0658901	−	0.997827i	$-0.520989\pi$
$223$	−1.96790	−0.131780	−0.0658901	+	0.997827i	$0.520989\pi$
$224$	0	0
$225$	0	0
$226$	−4.59358	−0.305560
$227$	−12.4285	−0.824911	−0.412456	−	0.910978i	$-0.635329\pi$
$227$	−12.4285	−0.824911	−0.412456	+	0.910978i	$0.635329\pi$
$228$	0	0
$229$	−21.0584	−1.39158	−0.695788	−	0.718248i	$-0.744945\pi$
$229$	−21.0584	−1.39158	−0.695788	+	0.718248i	$0.744945\pi$
$230$	−11.6757	−0.769876
$231$	0	0
$232$	−22.4576	−1.47441
$233$	7.26872	0.476190	0.238095	−	0.971242i	$-0.423477\pi$
$233$	7.26872	0.476190	0.238095	+	0.971242i	$0.423477\pi$
$234$	0	0
$235$	3.73385	0.243570
$236$	−16.7012	−1.08716
$237$	0	0
$238$	0	0
$239$	−12.8698	−0.832479	−0.416239	−	0.909255i	$-0.636652\pi$
$239$	−12.8698	−0.832479	−0.416239	+	0.909255i	$0.636652\pi$
$240$	0	0
$241$	−13.1986	−0.850198	−0.425099	−	0.905147i	$-0.639761\pi$
$241$	−13.1986	−0.850198	−0.425099	+	0.905147i	$0.639761\pi$
$242$	−2.04246	−0.131294
$243$	0	0
$244$	−13.3130	−0.852280
$245$	0	0
$246$	0	0
$247$	−5.92101	−0.376745
$248$	14.4924	0.920270
$249$	0	0
$250$	−6.60797	−0.417925
$251$	10.1617	0.641402	0.320701	−	0.947180i	$-0.396081\pi$
$251$	10.1617	0.641402	0.320701	+	0.947180i	$0.396081\pi$
$252$	0	0
$253$	−23.2163	−1.45960
$254$	1.83083	0.114876
$255$	0	0
$256$	−11.8099	−0.738120
$257$	0.240499	0.0150019	0.00750095	−	0.999972i	$-0.497612\pi$
$257$	0.240499	0.0150019	0.00750095	+	0.999972i	$0.497612\pi$
$258$	0	0
$259$	0	0
$260$	3.71224	0.230223
$261$	0	0
$262$	6.50312	0.401764
$263$	−16.7592	−1.03342	−0.516710	−	0.856161i	$-0.672843\pi$
$263$	−16.7592	−1.03342	−0.516710	+	0.856161i	$0.672843\pi$
$264$	0	0
$265$	−16.4064	−1.00784
$266$	0	0
$267$	0	0
$268$	20.5261	1.25383
$269$	−14.8307	−0.904242	−0.452121	−	0.891957i	$-0.649332\pi$
$269$	−14.8307	−0.904242	−0.452121	+	0.891957i	$0.649332\pi$
$270$	0	0
$271$	−27.2704	−1.65656	−0.828280	−	0.560314i	$-0.810680\pi$
$271$	−27.2704	−1.65656	−0.828280	+	0.560314i	$0.810680\pi$
$272$	2.21966	0.134586
$273$	0	0
$274$	1.99707	0.120648
$275$	5.42175	0.326944
$276$	0	0
$277$	−27.6912	−1.66381	−0.831903	−	0.554922i	$-0.812748\pi$
$277$	−27.6912	−1.66381	−0.831903	+	0.554922i	$0.812748\pi$
$278$	9.67300	0.580148
$279$	0	0
$280$	0	0
$281$	16.6959	0.995996	0.497998	−	0.867178i	$-0.334069\pi$
$281$	16.6959	0.995996	0.497998	+	0.867178i	$0.334069\pi$
$282$	0	0
$283$	0.320233	0.0190359	0.00951794	−	0.999955i	$-0.496970\pi$
$283$	0.320233	0.0190359	0.00951794	+	0.999955i	$0.496970\pi$
$284$	−21.1082	−1.25254
$285$	0	0
$286$	−2.72665	−0.161230
$287$	0	0
$288$	0	0
$289$	−12.5657	−0.739162
$290$	−16.4952	−0.968630
$291$	0	0
$292$	−8.96790	−0.524806
$293$	8.07954	0.472012	0.236006	−	0.971752i	$-0.424162\pi$
$293$	8.07954	0.472012	0.236006	+	0.971752i	$0.424162\pi$
$294$	0	0
$295$	−29.0656	−1.69226
$296$	16.6959	0.970432
$297$	0	0
$298$	0	0
$299$	−6.25399	−0.361678
$300$	0	0
$301$	0	0
$302$	5.35844	0.308344
$303$	0	0
$304$	−6.24124	−0.357960
$305$	−23.1690	−1.32665
$306$	0	0
$307$	4.94299	0.282111	0.141056	−	0.990002i	$-0.454950\pi$
$307$	4.94299	0.282111	0.141056	+	0.990002i	$0.454950\pi$
$308$	0	0
$309$	0	0
$310$	10.6447	0.604581
$311$	−4.05048	−0.229682	−0.114841	−	0.993384i	$-0.536636\pi$
$311$	−4.05048	−0.229682	−0.114841	+	0.993384i	$0.536636\pi$
$312$	0	0
$313$	10.3671	0.585984	0.292992	−	0.956115i	$-0.405349\pi$
$313$	10.3671	0.585984	0.292992	+	0.956115i	$0.405349\pi$
$314$	−4.16652	−0.235130
$315$	0	0
$316$	11.6372	0.654641
$317$	−12.0773	−0.678328	−0.339164	−	0.940727i	$-0.610144\pi$
$317$	−12.0773	−0.678328	−0.339164	+	0.940727i	$0.610144\pi$
$318$	0	0
$319$	−32.7994	−1.83641
$320$	−5.57751	−0.311792
$321$	0	0
$322$	0	0
$323$	−12.4683	−0.693753
$324$	0	0
$325$	1.46050	0.0810142
$326$	15.2161	0.842740
$327$	0	0
$328$	26.1914	1.44618
$329$	0	0
$330$	0	0
$331$	−22.1766	−1.21894	−0.609469	−	0.792810i	$-0.708617\pi$
$331$	−22.1766	−1.21894	−0.609469	+	0.792810i	$0.708617\pi$
$332$	17.7473	0.974007
$333$	0	0
$334$	−10.5146	−0.575333
$335$	35.7220	1.95170
$336$	0	0
$337$	−5.95311	−0.324287	−0.162143	−	0.986767i	$-0.551841\pi$
$337$	−5.95311	−0.324287	−0.162143	+	0.986767i	$0.551841\pi$
$338$	−0.734503	−0.0399517
$339$	0	0
$340$	7.81711	0.423942
$341$	21.1663	1.14622
$342$	0	0
$343$	0	0
$344$	−27.0637	−1.45918
$345$	0	0
$346$	16.4648	0.885152
$347$	19.0422	1.02224	0.511120	−	0.859510i	$-0.329231\pi$
$347$	19.0422	1.02224	0.511120	+	0.859510i	$0.329231\pi$
$348$	0	0
$349$	3.61556	0.193537	0.0967683	−	0.995307i	$-0.469149\pi$
$349$	3.61556	0.193537	0.0967683	+	0.995307i	$0.469149\pi$
$350$	0	0
$351$	0	0
$352$	−21.7453	−1.15903
$353$	5.97691	0.318119	0.159059	−	0.987269i	$-0.449154\pi$
$353$	5.97691	0.318119	0.159059	+	0.987269i	$0.449154\pi$
$354$	0	0
$355$	−36.7352	−1.94970
$356$	−17.2533	−0.914420
$357$	0	0
$358$	6.06848	0.320729
$359$	15.1528	0.799733	0.399866	−	0.916573i	$-0.369057\pi$
$359$	15.1528	0.799733	0.399866	+	0.916573i	$0.369057\pi$
$360$	0	0
$361$	16.0584	0.845177
$362$	−0.578741	−0.0304180
$363$	0	0
$364$	0	0
$365$	−15.6070	−0.816910
$366$	0	0
$367$	6.75156	0.352429	0.176214	−	0.984352i	$-0.443615\pi$
$367$	6.75156	0.352429	0.176214	+	0.984352i	$0.443615\pi$
$368$	−6.59224	−0.343644
$369$	0	0
$370$	12.2632	0.637535
$371$	0	0
$372$	0	0
$373$	6.30972	0.326705	0.163352	−	0.986568i	$-0.447769\pi$
$373$	6.30972	0.326705	0.163352	+	0.986568i	$0.447769\pi$
$374$	−5.74170	−0.296896
$375$	0	0
$376$	−3.73385	−0.192559
$377$	−8.83547	−0.455050
$378$	0	0
$379$	0.0861875	0.00442715	0.00221358	−	0.999998i	$-0.499295\pi$
$379$	0.0861875	0.00442715	0.00221358	+	0.999998i	$0.499295\pi$
$380$	−21.9802	−1.12756
$381$	0	0
$382$	0.291055	0.0148917
$383$	−21.3779	−1.09236	−0.546180	−	0.837668i	$-0.683918\pi$
$383$	−21.3779	−1.09236	−0.546180	+	0.837668i	$0.683918\pi$
$384$	0	0
$385$	0	0
$386$	−2.78225	−0.141613
$387$	0	0
$388$	10.3710	0.526508
$389$	32.0839	1.62672	0.813360	−	0.581761i	$-0.197636\pi$
$389$	32.0839	1.62672	0.813360	+	0.581761i	$0.197636\pi$
$390$	0	0
$391$	−13.1694	−0.666008
$392$	0	0
$393$	0	0
$394$	8.84961	0.445837
$395$	20.2524	1.01901
$396$	0	0
$397$	−12.1196	−0.608267	−0.304134	−	0.952629i	$-0.598367\pi$
$397$	−12.1196	−0.608267	−0.304134	+	0.952629i	$0.598367\pi$
$398$	−3.79169	−0.190060
$399$	0	0
$400$	1.53950	0.0769748
$401$	−26.6277	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$401$	−26.6277	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$402$	0	0
$403$	5.70175	0.284024
$404$	−12.4102	−0.617433
$405$	0	0
$406$	0	0
$407$	24.3845	1.20869
$408$	0	0
$409$	6.48968	0.320894	0.160447	−	0.987044i	$-0.448706\pi$
$409$	6.48968	0.320894	0.160447	+	0.987044i	$0.448706\pi$
$410$	19.2377	0.950082
$411$	0	0
$412$	3.19436	0.157375
$413$	0	0
$414$	0	0
$415$	30.8860	1.51613
$416$	−5.85773	−0.287199
$417$	0	0
$418$	16.1445	0.789655
$419$	−31.3097	−1.52958	−0.764789	−	0.644280i	$-0.777157\pi$
$419$	−31.3097	−1.52958	−0.764789	+	0.644280i	$0.777157\pi$
$420$	0	0
$421$	−24.3097	−1.18478	−0.592392	−	0.805650i	$-0.701816\pi$
$421$	−24.3097	−1.18478	−0.592392	+	0.805650i	$0.701816\pi$
$422$	−13.2059	−0.642853
$423$	0	0
$424$	16.4064	0.796766
$425$	3.07548	0.149183
$426$	0	0
$427$	0	0
$428$	−10.9940	−0.531414
$429$	0	0
$430$	−19.8784	−0.958621
$431$	−19.7950	−0.953492	−0.476746	−	0.879041i	$-0.658184\pi$
$431$	−19.7950	−0.953492	−0.476746	+	0.879041i	$0.658184\pi$
$432$	0	0
$433$	31.8932	1.53269	0.766344	−	0.642430i	$-0.222074\pi$
$433$	31.8932	1.53269	0.766344	+	0.642430i	$0.222074\pi$
$434$	0	0
$435$	0	0
$436$	−0.0789903	−0.00378295
$437$	37.0300	1.77138
$438$	0	0
$439$	28.8463	1.37676	0.688379	−	0.725351i	$-0.258323\pi$
$439$	28.8463	1.37676	0.688379	+	0.725351i	$0.258323\pi$
$440$	−23.9830	−1.14334
$441$	0	0
$442$	−1.54669	−0.0735687
$443$	−3.25796	−0.154790	−0.0773952	−	0.997000i	$-0.524660\pi$
$443$	−3.25796	−0.154790	−0.0773952	+	0.997000i	$0.524660\pi$
$444$	0	0
$445$	−30.0263	−1.42338
$446$	1.44543	0.0684429
$447$	0	0
$448$	0	0
$449$	−1.03302	−0.0487513	−0.0243756	−	0.999703i	$-0.507760\pi$
$449$	−1.03302	−0.0487513	−0.0243756	+	0.999703i	$0.507760\pi$
$450$	0	0
$451$	38.2527	1.80125
$452$	−9.13399	−0.429627
$453$	0	0
$454$	9.12880	0.428436
$455$	0	0
$456$	0	0
$457$	−17.4648	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$457$	−17.4648	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$458$	15.4674	0.722746
$459$	0	0
$460$	−23.2163	−1.08247
$461$	−32.4060	−1.50930	−0.754649	−	0.656128i	$-0.772193\pi$
$461$	−32.4060	−1.50930	−0.754649	+	0.656128i	$0.772193\pi$
$462$	0	0
$463$	−34.8788	−1.62095	−0.810477	−	0.585770i	$-0.800792\pi$
$463$	−34.8788	−1.62095	−0.810477	+	0.585770i	$0.800792\pi$
$464$	−9.31333	−0.432361
$465$	0	0
$466$	−5.33890	−0.247320
$467$	−32.5252	−1.50509	−0.752543	−	0.658543i	$-0.771173\pi$
$467$	−32.5252	−1.50509	−0.752543	+	0.658543i	$0.771173\pi$
$468$	0	0
$469$	0	0
$470$	−2.74253	−0.126503
$471$	0	0
$472$	29.0656	1.33785
$473$	−39.5267	−1.81744
$474$	0	0
$475$	−8.64766	−0.396782
$476$	0	0
$477$	0	0
$478$	9.45292	0.432366
$479$	−5.04378	−0.230456	−0.115228	−	0.993339i	$-0.536760\pi$
$479$	−5.04378	−0.230456	−0.115228	+	0.993339i	$0.536760\pi$
$480$	0	0
$481$	6.56867	0.299506
$482$	9.69444	0.441569
$483$	0	0
$484$	−4.06128	−0.184604
$485$	18.0489	0.819559
$486$	0	0
$487$	−11.2019	−0.507608	−0.253804	−	0.967256i	$-0.581682\pi$
$487$	−11.2019	−0.507608	−0.253804	+	0.967256i	$0.581682\pi$
$488$	23.1690	1.04881
$489$	0	0
$490$	0	0
$491$	21.6420	0.976689	0.488344	−	0.872651i	$-0.337601\pi$
$491$	21.6420	0.976689	0.488344	+	0.872651i	$0.337601\pi$
$492$	0	0
$493$	−18.6054	−0.837947
$494$	4.34900	0.195671
$495$	0	0
$496$	6.01012	0.269862
$497$	0	0
$498$	0	0
$499$	−12.8128	−0.573582	−0.286791	−	0.957993i	$-0.592588\pi$
$499$	−12.8128	−0.573582	−0.286791	+	0.957993i	$0.592588\pi$
$500$	−13.1395	−0.587615
$501$	0	0
$502$	−7.46382	−0.333127
$503$	24.9763	1.11364	0.556818	−	0.830635i	$-0.312022\pi$
$503$	24.9763	1.11364	0.556818	+	0.830635i	$0.312022\pi$
$504$	0	0
$505$	−21.5979	−0.961092
$506$	17.0525	0.758075
$507$	0	0
$508$	3.64047	0.161520
$509$	35.6007	1.57797	0.788986	−	0.614411i	$-0.210606\pi$
$509$	35.6007	1.57797	0.788986	+	0.614411i	$0.210606\pi$
$510$	0	0
$511$	0	0
$512$	−11.5330	−0.509691
$513$	0	0
$514$	−0.176647	−0.00779157
$515$	5.55922	0.244968
$516$	0	0
$517$	−5.45331	−0.239836
$518$	0	0
$519$	0	0
$520$	−6.46050	−0.283312
$521$	−36.8078	−1.61258	−0.806288	−	0.591523i	$-0.798527\pi$
$521$	−36.8078	−1.61258	−0.806288	+	0.591523i	$0.798527\pi$
$522$	0	0
$523$	15.5175	0.678534	0.339267	−	0.940690i	$-0.389821\pi$
$523$	15.5175	0.678534	0.339267	+	0.940690i	$0.389821\pi$
$524$	12.9310	0.564892
$525$	0	0
$526$	12.3097	0.536729
$527$	12.0065	0.523013
$528$	0	0
$529$	16.1124	0.700541
$530$	12.0506	0.523443
$531$	0	0
$532$	0	0
$533$	10.3045	0.446336
$534$	0	0
$535$	−19.1331	−0.827195
$536$	−35.7220	−1.54296
$537$	0	0
$538$	10.8932	0.469638
$539$	0	0
$540$	0	0
$541$	22.5261	0.968471	0.484235	−	0.874938i	$-0.339098\pi$
$541$	22.5261	0.968471	0.484235	+	0.874938i	$0.339098\pi$
$542$	20.0302	0.860371
$543$	0	0
$544$	−12.3350	−0.528859
$545$	−0.137469	−0.00588851
$546$	0	0
$547$	1.73812	0.0743168	0.0371584	−	0.999309i	$-0.488169\pi$
$547$	1.73812	0.0743168	0.0371584	+	0.999309i	$0.488169\pi$
$548$	3.97103	0.169634
$549$	0	0
$550$	−3.98229	−0.169805
$551$	52.3149	2.22869
$552$	0	0
$553$	0	0
$554$	20.3393	0.864134
$555$	0	0
$556$	19.2340	0.815705
$557$	26.3056	1.11461	0.557303	−	0.830309i	$-0.311836\pi$
$557$	26.3056	1.11461	0.557303	+	0.830309i	$0.311836\pi$
$558$	0	0
$559$	−10.6477	−0.450348
$560$	0	0
$561$	0	0
$562$	−12.2632	−0.517293
$563$	13.7365	0.578924	0.289462	−	0.957189i	$-0.406524\pi$
$563$	13.7365	0.578924	0.289462	+	0.957189i	$0.406524\pi$
$564$	0	0
$565$	−15.8961	−0.668754
$566$	−0.235212	−0.00988671
$567$	0	0
$568$	36.7352	1.54137
$569$	−19.6522	−0.823865	−0.411933	−	0.911214i	$-0.635146\pi$
$569$	−19.6522	−0.823865	−0.411933	+	0.911214i	$0.635146\pi$
$570$	0	0
$571$	−16.7516	−0.701031	−0.350515	−	0.936557i	$-0.613994\pi$
$571$	−16.7516	−0.701031	−0.350515	+	0.936557i	$0.613994\pi$
$572$	−5.42175	−0.226695
$573$	0	0
$574$	0	0
$575$	−9.13399	−0.380914
$576$	0	0
$577$	23.3930	0.973863	0.486931	−	0.873440i	$-0.338116\pi$
$577$	23.3930	0.973863	0.486931	+	0.873440i	$0.338116\pi$
$578$	9.22958	0.383900
$579$	0	0
$580$	−32.7994	−1.36192
$581$	0	0
$582$	0	0
$583$	23.9617	0.992390
$584$	15.6070	0.645824
$585$	0	0
$586$	−5.93445	−0.245150
$587$	6.05322	0.249843	0.124922	−	0.992167i	$-0.460132\pi$
$587$	6.05322	0.249843	0.124922	+	0.992167i	$0.460132\pi$
$588$	0	0
$589$	−33.7601	−1.39106
$590$	21.3487	0.878914
$591$	0	0
$592$	6.92393	0.284572
$593$	6.54722	0.268862	0.134431	−	0.990923i	$-0.457079\pi$
$593$	6.54722	0.268862	0.134431	+	0.990923i	$0.457079\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	4.59358	0.187845
$599$	2.36241	0.0965255	0.0482627	−	0.998835i	$-0.484632\pi$
$599$	2.36241	0.0965255	0.0482627	+	0.998835i	$0.484632\pi$
$600$	0	0
$601$	−40.5408	−1.65370	−0.826848	−	0.562426i	$-0.809868\pi$
$601$	−40.5408	−1.65370	−0.826848	+	0.562426i	$0.809868\pi$
$602$	0	0
$603$	0	0
$604$	10.6549	0.433540
$605$	−7.06795	−0.287353
$606$	0	0
$607$	11.4461	0.464583	0.232292	−	0.972646i	$-0.425378\pi$
$607$	11.4461	0.464583	0.232292	+	0.972646i	$0.425378\pi$
$608$	34.6837	1.40661
$609$	0	0
$610$	17.0177	0.689027
$611$	−1.46901	−0.0594296
$612$	0	0
$613$	12.7060	0.513191	0.256596	−	0.966519i	$-0.417399\pi$
$613$	12.7060	0.513191	0.256596	+	0.966519i	$0.417399\pi$
$614$	−3.63064	−0.146521
$615$	0	0
$616$	0	0
$617$	−30.9500	−1.24600	−0.623000	−	0.782221i	$-0.714087\pi$
$617$	−30.9500	−1.24600	−0.623000	+	0.782221i	$0.714087\pi$
$618$	0	0
$619$	−4.56440	−0.183459	−0.0917294	−	0.995784i	$-0.529239\pi$
$619$	−4.56440	−0.183459	−0.0917294	+	0.995784i	$0.529239\pi$
$620$	21.1663	0.850058
$621$	0	0
$622$	2.97509	0.119290
$623$	0	0
$624$	0	0
$625$	−30.1694	−1.20678
$626$	−7.61468	−0.304344
$627$	0	0
$628$	−8.28482	−0.330600
$629$	13.8321	0.551522
$630$	0	0
$631$	15.0803	0.600339	0.300169	−	0.953886i	$-0.402957\pi$
$631$	15.0803	0.600339	0.300169	+	0.953886i	$0.402957\pi$
$632$	−20.2524	−0.805598
$633$	0	0
$634$	8.87081	0.352305
$635$	6.33559	0.251420
$636$	0	0
$637$	0	0
$638$	24.0913	0.953783
$639$	0	0
$640$	−25.6811	−1.01514
$641$	6.04236	0.238659	0.119329	−	0.992855i	$-0.461926\pi$
$641$	6.04236	0.238659	0.119329	+	0.992855i	$0.461926\pi$
$642$	0	0
$643$	16.2881	0.642341	0.321171	−	0.947021i	$-0.395924\pi$
$643$	16.2881	0.642341	0.321171	+	0.947021i	$0.395924\pi$
$644$	0	0
$645$	0	0
$646$	9.15798	0.360316
$647$	−7.16040	−0.281505	−0.140752	−	0.990045i	$-0.544952\pi$
$647$	−7.16040	−0.281505	−0.140752	+	0.990045i	$0.544952\pi$
$648$	0	0
$649$	42.4504	1.66632
$650$	−1.07275	−0.0420765
$651$	0	0
$652$	30.2560	1.18492
$653$	−23.5255	−0.920625	−0.460313	−	0.887757i	$-0.652263\pi$
$653$	−23.5255	−0.920625	−0.460313	+	0.887757i	$0.652263\pi$
$654$	0	0
$655$	22.5041	0.879307
$656$	10.8618	0.424082
$657$	0	0
$658$	0	0
$659$	8.73459	0.340251	0.170126	−	0.985422i	$-0.445583\pi$
$659$	8.73459	0.340251	0.170126	+	0.985422i	$0.445583\pi$
$660$	0	0
$661$	−11.2016	−0.435690	−0.217845	−	0.975983i	$-0.569903\pi$
$661$	−11.2016	−0.435690	−0.217845	+	0.975983i	$0.569903\pi$
$662$	16.2888	0.633083
$663$	0	0
$664$	−30.8860	−1.19861
$665$	0	0
$666$	0	0
$667$	55.2570	2.13956
$668$	−20.9075	−0.808934
$669$	0	0
$670$	−26.2379	−1.01366
$671$	33.8384	1.30632
$672$	0	0
$673$	−11.0728	−0.426823	−0.213412	−	0.976962i	$-0.568458\pi$
$673$	−11.0728	−0.426823	−0.213412	+	0.976962i	$0.568458\pi$
$674$	4.37258	0.168426
$675$	0	0
$676$	−1.46050	−0.0561733
$677$	−36.0838	−1.38681	−0.693407	−	0.720546i	$-0.743891\pi$
$677$	−36.0838	−1.38681	−0.693407	+	0.720546i	$0.743891\pi$
$678$	0	0
$679$	0	0
$680$	−13.6043	−0.521702
$681$	0	0
$682$	−15.5467	−0.595314
$683$	−48.9248	−1.87205	−0.936027	−	0.351928i	$-0.885526\pi$
$683$	−48.9248	−1.87205	−0.936027	+	0.351928i	$0.885526\pi$
$684$	0	0
$685$	6.91089	0.264051
$686$	0	0
$687$	0	0
$688$	−11.2235	−0.427893
$689$	6.45477	0.245907
$690$	0	0
$691$	37.6706	1.43306	0.716529	−	0.697558i	$-0.245730\pi$
$691$	37.6706	1.43306	0.716529	+	0.697558i	$0.245730\pi$
$692$	32.7390	1.24455
$693$	0	0
$694$	−13.9866	−0.530923
$695$	33.4735	1.26972
$696$	0	0
$697$	21.6988	0.821902
$698$	−2.65564	−0.100517
$699$	0	0
$700$	0	0
$701$	19.9325	0.752839	0.376420	−	0.926449i	$-0.377155\pi$
$701$	19.9325	0.752839	0.376420	+	0.926449i	$0.377155\pi$
$702$	0	0
$703$	−38.8932	−1.46688
$704$	8.14598	0.307013
$705$	0	0
$706$	−4.39006	−0.165222
$707$	0	0
$708$	0	0
$709$	−4.07607	−0.153080	−0.0765399	−	0.997067i	$-0.524387\pi$
$709$	−4.07607	−0.153080	−0.0765399	+	0.997067i	$0.524387\pi$
$710$	26.9821	1.01262
$711$	0	0
$712$	30.0263	1.12528
$713$	−35.6587	−1.33543
$714$	0	0
$715$	−9.43560	−0.352871
$716$	12.0667	0.450954
$717$	0	0
$718$	−11.1298	−0.415359
$719$	−4.12465	−0.153823	−0.0769117	−	0.997038i	$-0.524506\pi$
$719$	−4.12465	−0.153823	−0.0769117	+	0.997038i	$0.524506\pi$
$720$	0	0
$721$	0	0
$722$	−11.7949	−0.438961
$723$	0	0
$724$	−1.15078	−0.0427685
$725$	−12.9042	−0.479252
$726$	0	0
$727$	7.39203	0.274155	0.137078	−	0.990560i	$-0.456229\pi$
$727$	7.39203	0.274155	0.137078	+	0.990560i	$0.456229\pi$
$728$	0	0
$729$	0	0
$730$	11.4634	0.424281
$731$	−22.4215	−0.829289
$732$	0	0
$733$	7.05836	0.260706	0.130353	−	0.991468i	$-0.458389\pi$
$733$	7.05836	0.260706	0.130353	+	0.991468i	$0.458389\pi$
$734$	−4.95904	−0.183042
$735$	0	0
$736$	36.6342	1.35036
$737$	−52.1722	−1.92179
$738$	0	0
$739$	−33.9033	−1.24715	−0.623577	−	0.781762i	$-0.714321\pi$
$739$	−33.9033	−1.24715	−0.623577	+	0.781762i	$0.714321\pi$
$740$	24.3845	0.896392
$741$	0	0
$742$	0	0
$743$	16.3211	0.598763	0.299382	−	0.954133i	$-0.403220\pi$
$743$	16.3211	0.598763	0.299382	+	0.954133i	$0.403220\pi$
$744$	0	0
$745$	0	0
$746$	−4.63451	−0.169682
$747$	0	0
$748$	−11.4169	−0.417444
$749$	0	0
$750$	0	0
$751$	−23.5467	−0.859231	−0.429616	−	0.903012i	$-0.641351\pi$
$751$	−23.5467	−0.859231	−0.429616	+	0.903012i	$0.641351\pi$
$752$	−1.54846	−0.0564664
$753$	0	0
$754$	6.48968	0.236340
$755$	18.5429	0.674846
$756$	0	0
$757$	35.1019	1.27580	0.637901	−	0.770119i	$-0.279803\pi$
$757$	35.1019	1.27580	0.637901	+	0.770119i	$0.279803\pi$
$758$	−0.0633050	−0.00229934
$759$	0	0
$760$	38.2527	1.38757
$761$	−45.4692	−1.64826	−0.824128	−	0.566404i	$-0.808334\pi$
$761$	−45.4692	−1.64826	−0.824128	+	0.566404i	$0.808334\pi$
$762$	0	0
$763$	0	0
$764$	0.578741	0.0209381
$765$	0	0
$766$	15.7021	0.567341
$767$	11.4352	0.412903
$768$	0	0
$769$	42.4543	1.53094	0.765470	−	0.643472i	$-0.222507\pi$
$769$	42.4543	1.53094	0.765470	+	0.643472i	$0.222507\pi$
$770$	0	0
$771$	0	0
$772$	−5.53230	−0.199112
$773$	−46.6021	−1.67616	−0.838080	−	0.545547i	$-0.816322\pi$
$773$	−46.6021	−1.67616	−0.838080	+	0.545547i	$0.816322\pi$
$774$	0	0
$775$	8.32743	0.299130
$776$	−18.0489	−0.647918
$777$	0	0
$778$	−23.5657	−0.844873
$779$	−61.0129	−2.18601
$780$	0	0
$781$	53.6519	1.91982
$782$	9.67300	0.345906
$783$	0	0
$784$	0	0
$785$	−14.4183	−0.514610
$786$	0	0
$787$	−26.3815	−0.940399	−0.470200	−	0.882560i	$-0.655818\pi$
$787$	−26.3815	−0.940399	−0.470200	+	0.882560i	$0.655818\pi$
$788$	17.5968	0.626859
$789$	0	0
$790$	−14.8755	−0.529245
$791$	0	0
$792$	0	0
$793$	9.11537	0.323696
$794$	8.90191	0.315917
$795$	0	0
$796$	−7.53950	−0.267230
$797$	−17.6474	−0.625102	−0.312551	−	0.949901i	$-0.601183\pi$
$797$	−17.6474	−0.625102	−0.312551	+	0.949901i	$0.601183\pi$
$798$	0	0
$799$	−3.09338	−0.109436
$800$	−8.55525	−0.302474
$801$	0	0
$802$	19.5582	0.690623
$803$	22.7942	0.804389
$804$	0	0
$805$	0	0
$806$	−4.18795	−0.147514
$807$	0	0
$808$	21.5979	0.759810
$809$	−17.9716	−0.631848	−0.315924	−	0.948784i	$-0.602314\pi$
$809$	−17.9716	−0.631848	−0.315924	+	0.948784i	$0.602314\pi$
$810$	0	0
$811$	−43.5159	−1.52805	−0.764026	−	0.645186i	$-0.776780\pi$
$811$	−43.5159	−1.52805	−0.764026	+	0.645186i	$0.776780\pi$
$812$	0	0
$813$	0	0
$814$	−17.9105	−0.627763
$815$	52.6553	1.84443
$816$	0	0
$817$	63.0449	2.20566
$818$	−4.76669	−0.166664
$819$	0	0
$820$	38.2527	1.33584
$821$	36.7283	1.28183	0.640913	−	0.767613i	$-0.278556\pi$
$821$	36.7283	1.28183	0.640913	+	0.767613i	$0.278556\pi$
$822$	0	0
$823$	−41.2235	−1.43696	−0.718481	−	0.695547i	$-0.755162\pi$
$823$	−41.2235	−1.43696	−0.718481	+	0.695547i	$0.755162\pi$
$824$	−5.55922	−0.193664
$825$	0	0
$826$	0	0
$827$	−10.0060	−0.347941	−0.173971	−	0.984751i	$-0.555660\pi$
$827$	−10.0060	−0.347941	−0.173971	+	0.984751i	$0.555660\pi$
$828$	0	0
$829$	−14.9607	−0.519607	−0.259803	−	0.965662i	$-0.583658\pi$
$829$	−14.9607	−0.519607	−0.259803	+	0.965662i	$0.583658\pi$
$830$	−22.6859	−0.787438
$831$	0	0
$832$	2.19436	0.0760756
$833$	0	0
$834$	0	0
$835$	−36.3858	−1.25918
$836$	32.1022	1.11028
$837$	0	0
$838$	22.9971	0.794421
$839$	27.0026	0.932232	0.466116	−	0.884724i	$-0.345653\pi$
$839$	27.0026	0.932232	0.466116	+	0.884724i	$0.345653\pi$
$840$	0	0
$841$	49.0656	1.69192
$842$	17.8556	0.615344
$843$	0	0
$844$	−26.2590	−0.903870
$845$	−2.54175	−0.0874389
$846$	0	0
$847$	0	0
$848$	6.80387	0.233646
$849$	0	0
$850$	−2.25895	−0.0774814
$851$	−41.0804	−1.40822
$852$	0	0
$853$	10.5438	0.361012	0.180506	−	0.983574i	$-0.442227\pi$
$853$	10.5438	0.361012	0.180506	+	0.983574i	$0.442227\pi$
$854$	0	0
$855$	0	0
$856$	19.1331	0.653955
$857$	7.48035	0.255524	0.127762	−	0.991805i	$-0.459221\pi$
$857$	7.48035	0.255524	0.127762	+	0.991805i	$0.459221\pi$
$858$	0	0
$859$	−43.3724	−1.47985	−0.739923	−	0.672692i	$-0.765138\pi$
$859$	−43.3724	−1.47985	−0.739923	+	0.672692i	$0.765138\pi$
$860$	−39.5267	−1.34785
$861$	0	0
$862$	14.5395	0.495217
$863$	−32.0228	−1.09007	−0.545034	−	0.838414i	$-0.683483\pi$
$863$	−32.0228	−1.09007	−0.545034	+	0.838414i	$0.683483\pi$
$864$	0	0
$865$	56.9764	1.93726
$866$	−23.4256	−0.796036
$867$	0	0
$868$	0	0
$869$	−29.5788	−1.00339
$870$	0	0
$871$	−14.0541	−0.476204
$872$	0.137469	0.00465528
$873$	0	0
$874$	−27.1986	−0.920007
$875$	0	0
$876$	0	0
$877$	32.6156	1.10135	0.550675	−	0.834720i	$-0.314371\pi$
$877$	32.6156	1.10135	0.550675	+	0.834720i	$0.314371\pi$
$878$	−21.1877	−0.715050
$879$	0	0
$880$	−9.94592	−0.335277
$881$	−34.6890	−1.16870	−0.584351	−	0.811501i	$-0.698651\pi$
$881$	−34.6890	−1.16870	−0.584351	+	0.811501i	$0.698651\pi$
$882$	0	0
$883$	7.36381	0.247812	0.123906	−	0.992294i	$-0.460458\pi$
$883$	7.36381	0.247812	0.123906	+	0.992294i	$0.460458\pi$
$884$	−3.07548	−0.103440
$885$	0	0
$886$	2.39298	0.0803939
$887$	31.2517	1.04933	0.524664	−	0.851309i	$-0.324191\pi$
$887$	31.2517	1.04933	0.524664	+	0.851309i	$0.324191\pi$
$888$	0	0
$889$	0	0
$890$	22.0544	0.739265
$891$	0	0
$892$	2.87412	0.0962327
$893$	8.69800	0.291068
$894$	0	0
$895$	21.0000	0.701953
$896$	0	0
$897$	0	0
$898$	0.758757	0.0253201
$899$	−50.3776	−1.68019
$900$	0	0
$901$	13.5922	0.452823
$902$	−28.0967	−0.935519
$903$	0	0
$904$	15.8961	0.528697
$905$	−2.00274	−0.0665732
$906$	0	0
$907$	−56.7204	−1.88337	−0.941685	−	0.336495i	$-0.890758\pi$
$907$	−56.7204	−1.88337	−0.941685	+	0.336495i	$0.890758\pi$
$908$	18.1519	0.602393
$909$	0	0
$910$	0	0
$911$	22.4742	0.744604	0.372302	−	0.928112i	$-0.378569\pi$
$911$	22.4742	0.744604	0.372302	+	0.928112i	$0.378569\pi$
$912$	0	0
$913$	−45.1091	−1.49289
$914$	12.8279	0.424310
$915$	0	0
$916$	30.7558	1.01620
$917$	0	0
$918$	0	0
$919$	2.48541	0.0819861	0.0409931	−	0.999159i	$-0.486948\pi$
$919$	2.48541	0.0819861	0.0409931	+	0.999159i	$0.486948\pi$
$920$	40.4040	1.33208
$921$	0	0
$922$	23.8023	0.783888
$923$	14.4527	0.475717
$924$	0	0
$925$	9.59358	0.315435
$926$	25.6186	0.841879
$927$	0	0
$928$	51.7558	1.69897
$929$	35.1991	1.15485	0.577423	−	0.816445i	$-0.304058\pi$
$929$	35.1991	1.15485	0.577423	+	0.816445i	$0.304058\pi$
$930$	0	0
$931$	0	0
$932$	−10.6160	−0.347739
$933$	0	0
$934$	23.8899	0.781700
$935$	−19.8692	−0.649791
$936$	0	0
$937$	−40.0613	−1.30875	−0.654373	−	0.756172i	$-0.727067\pi$
$937$	−40.0613	−1.30875	−0.654373	+	0.756172i	$0.727067\pi$
$938$	0	0
$939$	0	0
$940$	−5.45331	−0.177867
$941$	−32.0076	−1.04342	−0.521709	−	0.853123i	$-0.674705\pi$
$941$	−32.0076	−1.04342	−0.521709	+	0.853123i	$0.674705\pi$
$942$	0	0
$943$	−64.4441	−2.09859
$944$	12.0537	0.392315
$945$	0	0
$946$	29.0325	0.943928
$947$	22.7383	0.738896	0.369448	−	0.929251i	$-0.379547\pi$
$947$	22.7383	0.738896	0.369448	+	0.929251i	$0.379547\pi$
$948$	0	0
$949$	6.14027	0.199322
$950$	6.35174	0.206078
$951$	0	0
$952$	0	0
$953$	−16.7884	−0.543830	−0.271915	−	0.962321i	$-0.587657\pi$
$953$	−16.7884	−0.543830	−0.271915	+	0.962321i	$0.587657\pi$
$954$	0	0
$955$	1.00720	0.0325921
$956$	18.7964	0.607920
$957$	0	0
$958$	3.70467	0.119693
$959$	0	0
$960$	0	0
$961$	1.50993	0.0487073
$962$	−4.82471	−0.155555
$963$	0	0
$964$	19.2767	0.620859
$965$	−9.62799	−0.309936
$966$	0	0
$967$	44.3533	1.42631	0.713153	−	0.701009i	$-0.247267\pi$
$967$	44.3533	1.42631	0.713153	+	0.701009i	$0.247267\pi$
$968$	7.06795	0.227172
$969$	0	0
$970$	−13.2570	−0.425656
$971$	−4.07935	−0.130913	−0.0654563	−	0.997855i	$-0.520850\pi$
$971$	−4.07935	−0.130913	−0.0654563	+	0.997855i	$0.520850\pi$
$972$	0	0
$973$	0	0
$974$	8.22786	0.263638
$975$	0	0
$976$	9.60836	0.307556
$977$	−9.76018	−0.312256	−0.156128	−	0.987737i	$-0.549901\pi$
$977$	−9.76018	−0.312256	−0.156128	+	0.987737i	$0.549901\pi$
$978$	0	0
$979$	43.8535	1.40156
$980$	0	0
$981$	0	0
$982$	−15.8961	−0.507265
$983$	−51.3163	−1.63674	−0.818368	−	0.574694i	$-0.805121\pi$
$983$	−51.3163	−1.63674	−0.818368	+	0.574694i	$0.805121\pi$
$984$	0	0
$985$	30.6241	0.975765
$986$	13.6658	0.435206
$987$	0	0
$988$	8.64766	0.275119
$989$	66.5904	2.11745
$990$	0	0
$991$	−0.704673	−0.0223847	−0.0111923	−	0.999937i	$-0.503563\pi$
$991$	−0.704673	−0.0223847	−0.0111923	+	0.999937i	$0.503563\pi$
$992$	−33.3993	−1.06043
$993$	0	0
$994$	0	0
$995$	−13.1212	−0.415969
$996$	0	0
$997$	−34.0259	−1.07761	−0.538805	−	0.842431i	$-0.681124\pi$
$997$	−34.0259	−1.07761	−0.538805	+	0.842431i	$0.681124\pi$
$998$	9.41107	0.297902
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bt.1.3		6
3.2	odd	2	inner	5733.2.a.bt.1.4		6
7.3	odd	6		819.2.j.i.352.4	yes	12
7.5	odd	6		819.2.j.i.235.4	yes	12
7.6	odd	2		5733.2.a.bs.1.3		6
21.5	even	6		819.2.j.i.235.3	✓	12
21.17	even	6		819.2.j.i.352.3	yes	12
21.20	even	2		5733.2.a.bs.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.j.i.235.3	✓	12	21.5	even	6
819.2.j.i.235.4	yes	12	7.5	odd	6
819.2.j.i.352.3	yes	12	21.17	even	6
819.2.j.i.352.4	yes	12	7.3	odd	6
5733.2.a.bs.1.3		6	7.6	odd	2
5733.2.a.bs.1.4		6	21.20	even	2
5733.2.a.bt.1.3		6	1.1	even	1	trivial
5733.2.a.bt.1.4		6	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-0.734503 q^{2} -1.46050 q^{4} -2.54175 q^{5} +2.54175 q^{8} +O(q^{10})\)	\(q-0.734503 q^{2} -1.46050 q^{4} -2.54175 q^{5} +2.54175 q^{8} +1.86693 q^{10} +3.71224 q^{11} +1.00000 q^{13} +1.05408 q^{16} +2.10577 q^{17} -5.92101 q^{19} +3.71224 q^{20} -2.72665 q^{22} -6.25399 q^{23} +1.46050 q^{25} -0.734503 q^{26} -8.83547 q^{29} +5.70175 q^{31} -5.85773 q^{32} -1.54669 q^{34} +6.56867 q^{37} +4.34900 q^{38} -6.46050 q^{40} +10.3045 q^{41} -10.6477 q^{43} -5.42175 q^{44} +4.59358 q^{46} -1.46901 q^{47} -1.07275 q^{50} -1.46050 q^{52} +6.45477 q^{53} -9.43560 q^{55} +6.48968 q^{58} +11.4352 q^{59} +9.11537 q^{61} -4.18795 q^{62} +2.19436 q^{64} -2.54175 q^{65} -14.0541 q^{67} -3.07548 q^{68} +14.4527 q^{71} +6.14027 q^{73} -4.82471 q^{74} +8.64766 q^{76} -7.96790 q^{79} -2.67922 q^{80} -7.56867 q^{82} -12.1515 q^{83} -5.35234 q^{85} +7.82074 q^{86} +9.43560 q^{88} +11.8132 q^{89} +9.13399 q^{92} +1.07899 q^{94} +15.0497 q^{95} -7.10097 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4}+O(q^{10}) \) 6 * q + 4 * q^4	\( 6 q + 4 q^{4} + 4 q^{10} + 6 q^{13} - 12 q^{16} - 10 q^{19} - 18 q^{22} - 4 q^{25} - 8 q^{31} - 6 q^{34} - 10 q^{37} - 26 q^{40} - 40 q^{43} + 22 q^{46} + 4 q^{52} - 36 q^{58} + 2 q^{61} - 14 q^{64} - 66 q^{67} + 28 q^{73} + 28 q^{76} - 20 q^{79} + 4 q^{82} - 56 q^{85} + 32 q^{94} - 22 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 + 4 * q^10 + 6 * q^13 - 12 * q^16 - 10 * q^19 - 18 * q^22 - 4 * q^25 - 8 * q^31 - 6 * q^34 - 10 * q^37 - 26 * q^40 - 40 * q^43 + 22 * q^46 + 4 * q^52 - 36 * q^58 + 2 * q^61 - 14 * q^64 - 66 * q^67 + 28 * q^73 + 28 * q^76 - 20 * q^79 + 4 * q^82 - 56 * q^85 + 32 * q^94 - 22 * q^97

Introduction

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