LMFDB - Embedded newform 5733.2.a.cb.1.6

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 92x^{8} - 228x^{6} + 225x^{4} - 60x^{2} + 4 $ x^12 - 16x^10 + 92x^8 - 228x^6 + 225x^4 - 60*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.318530$ 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)

$f(q)$	$=$	$q-0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} +1.24180 q^{8} +O(q^{10})$	$q-0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} +1.24180 q^{8} +0.0887749 q^{10} -4.26805 q^{11} +1.00000 q^{13} +3.40153 q^{16} -6.27885 q^{17} -0.935311 q^{19} +0.529127 q^{20} +1.35950 q^{22} +0.554198 q^{23} -4.92233 q^{25} -0.318530 q^{26} -4.49614 q^{29} -0.861767 q^{31} -3.56709 q^{32} +2.00000 q^{34} +4.40123 q^{37} +0.297924 q^{38} -0.346092 q^{40} +0.979596 q^{41} -1.71659 q^{43} +8.10306 q^{44} -0.176529 q^{46} -4.42810 q^{47} +1.56791 q^{50} -1.89854 q^{52} +9.30603 q^{53} +1.18951 q^{55} +1.43215 q^{58} +12.5106 q^{59} -1.97463 q^{61} +0.274498 q^{62} -5.66683 q^{64} -0.278702 q^{65} -3.97361 q^{67} +11.9206 q^{68} -2.43794 q^{71} -4.83639 q^{73} -1.40192 q^{74} +1.77572 q^{76} +12.5478 q^{79} -0.948013 q^{80} -0.312030 q^{82} -11.1808 q^{83} +1.74993 q^{85} +0.546784 q^{86} -5.30007 q^{88} -12.1828 q^{89} -1.05217 q^{92} +1.41048 q^{94} +0.260673 q^{95} +12.9099 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4}+O(q^{10}) $ 12 * q + 8 * q^4	$ 12 q + 8 q^{4} + 8 q^{10} + 12 q^{13} + 28 q^{19} + 16 q^{25} + 12 q^{31} + 24 q^{34} + 32 q^{40} - 20 q^{43} - 8 q^{46} + 8 q^{52} + 56 q^{55} + 8 q^{58} + 24 q^{61} - 40 q^{64} + 8 q^{67} + 12 q^{73} + 64 q^{76} - 20 q^{79} + 40 q^{82} + 32 q^{85} + 8 q^{88} + 8 q^{94} + 68 q^{97}+O(q^{100}) $ 12 * q + 8 * q^4 + 8 * q^10 + 12 * q^13 + 28 * q^19 + 16 * q^25 + 12 * q^31 + 24 * q^34 + 32 * q^40 - 20 * q^43 - 8 * q^46 + 8 * q^52 + 56 * q^55 + 8 * q^58 + 24 * q^61 - 40 * q^64 + 8 * q^67 + 12 * q^73 + 64 * q^76 - 20 * q^79 + 40 * q^82 + 32 * q^85 + 8 * q^88 + 8 * q^94 + 68 * 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 $

$2$	−0.318530	−0.225235	−0.112617	−	0.993638i	$-0.535923\pi$
$2$	−0.318530	−0.225235	−0.112617	+	0.993638i	$0.535923\pi$
$3$	0	0
$3$	0	0
$4$	−1.89854	−0.949269
$5$	−0.278702	−0.124639	−0.0623197	−	0.998056i	$-0.519850\pi$
$5$	−0.278702	−0.124639	−0.0623197	+	0.998056i	$0.519850\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.24180	0.439043
$9$	0	0
$10$	0.0887749	0.0280731
$11$	−4.26805	−1.28687	−0.643433	−	0.765503i	$-0.722490\pi$
$11$	−4.26805	−1.28687	−0.643433	+	0.765503i	$0.722490\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	3.40153	0.850382
$17$	−6.27885	−1.52284	−0.761422	−	0.648256i	$-0.775499\pi$
$17$	−6.27885	−1.52284	−0.761422	+	0.648256i	$0.775499\pi$
$18$	0	0
$19$	−0.935311	−0.214575	−0.107287	−	0.994228i	$-0.534217\pi$
$19$	−0.935311	−0.214575	−0.107287	+	0.994228i	$0.534217\pi$
$20$	0.529127	0.118316
$21$	0	0
$22$	1.35950	0.289847
$23$	0.554198	0.115558	0.0577792	−	0.998329i	$-0.481598\pi$
$23$	0.554198	0.115558	0.0577792	+	0.998329i	$0.481598\pi$
$24$	0	0
$25$	−4.92233	−0.984465
$26$	−0.318530	−0.0624688
$27$	0	0
$28$	0	0
$29$	−4.49614	−0.834913	−0.417456	−	0.908697i	$-0.637078\pi$
$29$	−4.49614	−0.834913	−0.417456	+	0.908697i	$0.637078\pi$
$30$	0	0
$31$	−0.861767	−0.154778	−0.0773890	−	0.997001i	$-0.524658\pi$
$31$	−0.861767	−0.154778	−0.0773890	+	0.997001i	$0.524658\pi$
$32$	−3.56709	−0.630578
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	4.40123	0.723557	0.361779	−	0.932264i	$-0.382170\pi$
$37$	4.40123	0.723557	0.361779	+	0.932264i	$0.382170\pi$
$38$	0.297924	0.0483297
$39$	0	0
$40$	−0.346092	−0.0547220
$41$	0.979596	0.152987	0.0764936	−	0.997070i	$-0.475628\pi$
$41$	0.979596	0.152987	0.0764936	+	0.997070i	$0.475628\pi$
$42$	0	0
$43$	−1.71659	−0.261777	−0.130889	−	0.991397i	$-0.541783\pi$
$43$	−1.71659	−0.261777	−0.130889	+	0.991397i	$0.541783\pi$
$44$	8.10306	1.22158
$45$	0	0
$46$	−0.176529	−0.0260277
$47$	−4.42810	−0.645905	−0.322953	−	0.946415i	$-0.604675\pi$
$47$	−4.42810	−0.645905	−0.322953	+	0.946415i	$0.604675\pi$
$48$	0	0
$49$	0	0
$50$	1.56791	0.221736
$51$	0	0
$52$	−1.89854	−0.263280
$53$	9.30603	1.27828	0.639141	−	0.769090i	$-0.279290\pi$
$53$	9.30603	1.27828	0.639141	+	0.769090i	$0.279290\pi$
$54$	0	0
$55$	1.18951	0.160394
$56$	0	0
$57$	0	0
$58$	1.43215	0.188051
$59$	12.5106	1.62874	0.814372	−	0.580343i	$-0.197081\pi$
$59$	12.5106	1.62874	0.814372	+	0.580343i	$0.197081\pi$
$60$	0	0
$61$	−1.97463	−0.252825	−0.126413	−	0.991978i	$-0.540346\pi$
$61$	−1.97463	−0.252825	−0.126413	+	0.991978i	$0.540346\pi$
$62$	0.274498	0.0348613
$63$	0	0
$64$	−5.66683	−0.708354
$65$	−0.278702	−0.0345687
$66$	0	0
$67$	−3.97361	−0.485453	−0.242727	−	0.970095i	$-0.578042\pi$
$67$	−3.97361	−0.485453	−0.242727	+	0.970095i	$0.578042\pi$
$68$	11.9206	1.44559
$69$	0	0
$70$	0	0
$71$	−2.43794	−0.289330	−0.144665	−	0.989481i	$-0.546210\pi$
$71$	−2.43794	−0.289330	−0.144665	+	0.989481i	$0.546210\pi$
$72$	0	0
$73$	−4.83639	−0.566057	−0.283029	−	0.959111i	$-0.591339\pi$
$73$	−4.83639	−0.566057	−0.283029	+	0.959111i	$0.591339\pi$
$74$	−1.40192	−0.162970
$75$	0	0
$76$	1.77572	0.203689
$77$	0	0
$78$	0	0
$79$	12.5478	1.41174	0.705871	−	0.708341i	$-0.250556\pi$
$79$	12.5478	1.41174	0.705871	+	0.708341i	$0.250556\pi$
$80$	−0.948013	−0.105991
$81$	0	0
$82$	−0.312030	−0.0344580
$83$	−11.1808	−1.22725	−0.613623	−	0.789599i	$-0.710289\pi$
$83$	−11.1808	−1.22725	−0.613623	+	0.789599i	$0.710289\pi$
$84$	0	0
$85$	1.74993	0.189806
$86$	0.546784	0.0589613
$87$	0	0
$88$	−5.30007	−0.564989
$89$	−12.1828	−1.29137	−0.645686	−	0.763603i	$-0.723428\pi$
$89$	−12.1828	−1.29137	−0.645686	+	0.763603i	$0.723428\pi$
$90$	0	0
$91$	0	0
$92$	−1.05217	−0.109696
$93$	0	0
$94$	1.41048	0.145480
$95$	0.260673	0.0267445
$96$	0	0
$97$	12.9099	1.31081	0.655403	−	0.755280i	$-0.272499\pi$
$97$	12.9099	1.31081	0.655403	+	0.755280i	$0.272499\pi$
$98$	0	0
$99$	0	0
$100$	9.34523	0.934523
$101$	−5.10341	−0.507809	−0.253904	−	0.967229i	$-0.581715\pi$
$101$	−5.10341	−0.507809	−0.253904	+	0.967229i	$0.581715\pi$
$102$	0	0
$103$	0.361087	0.0355789	0.0177895	−	0.999842i	$-0.494337\pi$
$103$	0.361087	0.0355789	0.0177895	+	0.999842i	$0.494337\pi$
$104$	1.24180	0.121769
$105$	0	0
$106$	−2.96425	−0.287913
$107$	−13.2663	−1.28250	−0.641252	−	0.767330i	$-0.721585\pi$
$107$	−13.2663	−1.28250	−0.641252	+	0.767330i	$0.721585\pi$
$108$	0	0
$109$	−13.7826	−1.32014	−0.660069	−	0.751205i	$-0.729473\pi$
$109$	−13.7826	−1.32014	−0.660069	+	0.751205i	$0.729473\pi$
$110$	−0.378896	−0.0361263
$111$	0	0
$112$	0	0
$113$	−2.99313	−0.281570	−0.140785	−	0.990040i	$-0.544963\pi$
$113$	−2.99313	−0.281570	−0.140785	+	0.990040i	$0.544963\pi$
$114$	0	0
$115$	−0.154456	−0.0144031
$116$	8.53610	0.792557
$117$	0	0
$118$	−3.98501	−0.366849
$119$	0	0
$120$	0	0
$121$	7.21625	0.656022
$122$	0.628978	0.0569449
$123$	0	0
$124$	1.63610	0.146926
$125$	2.76537	0.247342
$126$	0	0
$127$	−6.71157	−0.595556	−0.297778	−	0.954635i	$-0.596245\pi$
$127$	−6.71157	−0.595556	−0.297778	+	0.954635i	$0.596245\pi$
$128$	8.93923	0.790124
$129$	0	0
$130$	0.0887749	0.00778607
$131$	9.52731	0.832405	0.416203	−	0.909272i	$-0.363361\pi$
$131$	9.52731	0.832405	0.416203	+	0.909272i	$0.363361\pi$
$132$	0	0
$133$	0	0
$134$	1.26571	0.109341
$135$	0	0
$136$	−7.79708	−0.668594
$137$	21.1133	1.80383	0.901915	−	0.431913i	$-0.142161\pi$
$137$	21.1133	1.80383	0.901915	+	0.431913i	$0.142161\pi$
$138$	0	0
$139$	14.6811	1.24524	0.622618	−	0.782526i	$-0.286069\pi$
$139$	14.6811	1.24524	0.622618	+	0.782526i	$0.286069\pi$
$140$	0	0
$141$	0	0
$142$	0.776556	0.0651671
$143$	−4.26805	−0.356912
$144$	0	0
$145$	1.25308	0.104063
$146$	1.54054	0.127496
$147$	0	0
$148$	−8.35590	−0.686851
$149$	−13.5524	−1.11026	−0.555130	−	0.831764i	$-0.687331\pi$
$149$	−13.5524	−1.11026	−0.555130	+	0.831764i	$0.687331\pi$
$150$	0	0
$151$	−8.52294	−0.693587	−0.346794	−	0.937941i	$-0.612730\pi$
$151$	−8.52294	−0.693587	−0.346794	+	0.937941i	$0.612730\pi$
$152$	−1.16147	−0.0942076
$153$	0	0
$154$	0	0
$155$	0.240176	0.0192914
$156$	0	0
$157$	13.9303	1.11176	0.555879	−	0.831263i	$-0.312382\pi$
$157$	13.9303	1.11176	0.555879	+	0.831263i	$0.312382\pi$
$158$	−3.99686	−0.317973
$159$	0	0
$160$	0.994155	0.0785948
$161$	0	0
$162$	0	0
$163$	17.6518	1.38260	0.691298	−	0.722569i	$-0.257039\pi$
$163$	17.6518	1.38260	0.691298	+	0.722569i	$0.257039\pi$
$164$	−1.85980	−0.145226
$165$	0	0
$166$	3.56140	0.276418
$167$	−7.45254	−0.576695	−0.288348	−	0.957526i	$-0.593106\pi$
$167$	−7.45254	−0.576695	−0.288348	+	0.957526i	$0.593106\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.557404	−0.0427509
$171$	0	0
$172$	3.25901	0.248497
$173$	19.1846	1.45858	0.729290	−	0.684205i	$-0.239851\pi$
$173$	19.1846	1.45858	0.729290	+	0.684205i	$0.239851\pi$
$174$	0	0
$175$	0	0
$176$	−14.5179	−1.09433
$177$	0	0
$178$	3.88058	0.290861
$179$	−19.6322	−1.46738	−0.733692	−	0.679483i	$-0.762204\pi$
$179$	−19.6322	−1.46738	−0.733692	+	0.679483i	$0.762204\pi$
$180$	0	0
$181$	13.5654	1.00831	0.504153	−	0.863614i	$-0.331805\pi$
$181$	13.5654	1.00831	0.504153	+	0.863614i	$0.331805\pi$
$182$	0	0
$183$	0	0
$184$	0.688204	0.0507351
$185$	−1.22663	−0.0901837
$186$	0	0
$187$	26.7984	1.95970
$188$	8.40693	0.613138
$189$	0	0
$190$	−0.0830321	−0.00602378
$191$	−2.86026	−0.206961	−0.103481	−	0.994631i	$-0.532998\pi$
$191$	−2.86026	−0.206961	−0.103481	+	0.994631i	$0.532998\pi$
$192$	0	0
$193$	0.318196	0.0229043	0.0114521	−	0.999934i	$-0.496355\pi$
$193$	0.318196	0.0229043	0.0114521	+	0.999934i	$0.496355\pi$
$194$	−4.11220	−0.295239
$195$	0	0
$196$	0	0
$197$	2.15763	0.153725	0.0768624	−	0.997042i	$-0.475510\pi$
$197$	2.15763	0.153725	0.0768624	+	0.997042i	$0.475510\pi$
$198$	0	0
$199$	13.7721	0.976280	0.488140	−	0.872765i	$-0.337675\pi$
$199$	13.7721	0.976280	0.488140	+	0.872765i	$0.337675\pi$
$200$	−6.11255	−0.432222
$201$	0	0
$202$	1.62559	0.114376
$203$	0	0
$204$	0	0
$205$	−0.273015	−0.0190682
$206$	−0.115017	−0.00801361
$207$	0	0
$208$	3.40153	0.235853
$209$	3.99195	0.276129
$210$	0	0
$211$	−5.05624	−0.348086	−0.174043	−	0.984738i	$-0.555683\pi$
$211$	−5.05624	−0.348086	−0.174043	+	0.984738i	$0.555683\pi$
$212$	−17.6679	−1.21343
$213$	0	0
$214$	4.22572	0.288864
$215$	0.478417	0.0326277
$216$	0	0
$217$	0	0
$218$	4.39018	0.297340
$219$	0	0
$220$	−2.25834	−0.152257
$221$	−6.27885	−0.422361
$222$	0	0
$223$	23.9620	1.60462	0.802309	−	0.596910i	$-0.203605\pi$
$223$	23.9620	1.60462	0.802309	+	0.596910i	$0.203605\pi$
$224$	0	0
$225$	0	0
$226$	0.953402	0.0634194
$227$	−18.8419	−1.25058	−0.625292	−	0.780391i	$-0.715020\pi$
$227$	−18.8419	−1.25058	−0.625292	+	0.780391i	$0.715020\pi$
$228$	0	0
$229$	18.9695	1.25354	0.626768	−	0.779206i	$-0.284377\pi$
$229$	18.9695	1.25354	0.626768	+	0.779206i	$0.284377\pi$
$230$	0.0491989	0.00324408
$231$	0	0
$232$	−5.58331	−0.366562
$233$	27.3238	1.79004	0.895022	−	0.446022i	$-0.147160\pi$
$233$	27.3238	1.79004	0.895022	+	0.446022i	$0.147160\pi$
$234$	0	0
$235$	1.23412	0.0805052
$236$	−23.7519	−1.54612
$237$	0	0
$238$	0	0
$239$	23.8749	1.54434	0.772170	−	0.635416i	$-0.219171\pi$
$239$	23.8749	1.54434	0.772170	+	0.635416i	$0.219171\pi$
$240$	0	0
$241$	16.3941	1.05603	0.528017	−	0.849234i	$-0.322936\pi$
$241$	16.3941	1.05603	0.528017	+	0.849234i	$0.322936\pi$
$242$	−2.29859	−0.147759
$243$	0	0
$244$	3.74891	0.239999
$245$	0	0
$246$	0	0
$247$	−0.935311	−0.0595124
$248$	−1.07014	−0.0679541
$249$	0	0
$250$	−0.880853	−0.0557100
$251$	7.88947	0.497979	0.248989	−	0.968506i	$-0.419902\pi$
$251$	7.88947	0.497979	0.248989	+	0.968506i	$0.419902\pi$
$252$	0	0
$253$	−2.36535	−0.148708
$254$	2.13783	0.134140
$255$	0	0
$256$	8.48625	0.530391
$257$	−21.0559	−1.31343	−0.656715	−	0.754139i	$-0.728054\pi$
$257$	−21.0559	−1.31343	−0.656715	+	0.754139i	$0.728054\pi$
$258$	0	0
$259$	0	0
$260$	0.529127	0.0328150
$261$	0	0
$262$	−3.03473	−0.187486
$263$	9.45167	0.582815	0.291408	−	0.956599i	$-0.405876\pi$
$263$	9.45167	0.582815	0.291408	+	0.956599i	$0.405876\pi$
$264$	0	0
$265$	−2.59361	−0.159324
$266$	0	0
$267$	0	0
$268$	7.54405	0.460826
$269$	−13.4282	−0.818731	−0.409365	−	0.912371i	$-0.634250\pi$
$269$	−13.4282	−0.818731	−0.409365	+	0.912371i	$0.634250\pi$
$270$	0	0
$271$	8.94268	0.543229	0.271615	−	0.962406i	$-0.412442\pi$
$271$	8.94268	0.543229	0.271615	+	0.962406i	$0.412442\pi$
$272$	−21.3577	−1.29500
$273$	0	0
$274$	−6.72521	−0.406285
$275$	21.0087	1.26687
$276$	0	0
$277$	−6.08015	−0.365321	−0.182661	−	0.983176i	$-0.558471\pi$
$277$	−6.08015	−0.365321	−0.182661	+	0.983176i	$0.558471\pi$
$278$	−4.67637	−0.280470
$279$	0	0
$280$	0	0
$281$	−16.7806	−1.00105	−0.500523	−	0.865723i	$-0.666859\pi$
$281$	−16.7806	−1.00105	−0.500523	+	0.865723i	$0.666859\pi$
$282$	0	0
$283$	−4.49011	−0.266909	−0.133455	−	0.991055i	$-0.542607\pi$
$283$	−4.49011	−0.266909	−0.133455	+	0.991055i	$0.542607\pi$
$284$	4.62852	0.274652
$285$	0	0
$286$	1.35950	0.0803890
$287$	0	0
$288$	0	0
$289$	22.4239	1.31906
$290$	−0.399144	−0.0234386
$291$	0	0
$292$	9.18208	0.537341
$293$	−7.74333	−0.452370	−0.226185	−	0.974084i	$-0.572625\pi$
$293$	−7.74333	−0.452370	−0.226185	+	0.974084i	$0.572625\pi$
$294$	0	0
$295$	−3.48674	−0.203006
$296$	5.46545	0.317673
$297$	0	0
$298$	4.31686	0.250069
$299$	0.554198	0.0320501
$300$	0	0
$301$	0	0
$302$	2.71481	0.156220
$303$	0	0
$304$	−3.18148	−0.182471
$305$	0.550333	0.0315120
$306$	0	0
$307$	7.02736	0.401073	0.200536	−	0.979686i	$-0.435732\pi$
$307$	7.02736	0.401073	0.200536	+	0.979686i	$0.435732\pi$
$308$	0	0
$309$	0	0
$310$	−0.0765033	−0.00434509
$311$	33.4422	1.89633	0.948165	−	0.317777i	$-0.102936\pi$
$311$	33.4422	1.89633	0.948165	+	0.317777i	$0.102936\pi$
$312$	0	0
$313$	12.8389	0.725700	0.362850	−	0.931848i	$-0.381804\pi$
$313$	12.8389	0.725700	0.362850	+	0.931848i	$0.381804\pi$
$314$	−4.43721	−0.250406
$315$	0	0
$316$	−23.8225	−1.34012
$317$	3.14685	0.176745	0.0883723	−	0.996088i	$-0.471833\pi$
$317$	3.14685	0.176745	0.0883723	+	0.996088i	$0.471833\pi$
$318$	0	0
$319$	19.1898	1.07442
$320$	1.57936	0.0882888
$321$	0	0
$322$	0	0
$323$	5.87267	0.326764
$324$	0	0
$325$	−4.92233	−0.273041
$326$	−5.62263	−0.311409
$327$	0	0
$328$	1.21646	0.0671679
$329$	0	0
$330$	0	0
$331$	6.26160	0.344169	0.172084	−	0.985082i	$-0.444950\pi$
$331$	6.26160	0.344169	0.172084	+	0.985082i	$0.444950\pi$
$332$	21.2271	1.16499
$333$	0	0
$334$	2.37386	0.129892
$335$	1.10745	0.0605066
$336$	0	0
$337$	−15.8596	−0.863929	−0.431964	−	0.901891i	$-0.642179\pi$
$337$	−15.8596	−0.863929	−0.431964	+	0.901891i	$0.642179\pi$
$338$	−0.318530	−0.0173257
$339$	0	0
$340$	−3.32231	−0.180177
$341$	3.67806	0.199178
$342$	0	0
$343$	0	0
$344$	−2.13166	−0.114931
$345$	0	0
$346$	−6.11087	−0.328522
$347$	−13.1466	−0.705747	−0.352873	−	0.935671i	$-0.614795\pi$
$347$	−13.1466	−0.705747	−0.352873	+	0.935671i	$0.614795\pi$
$348$	0	0
$349$	29.8740	1.59912	0.799559	−	0.600588i	$-0.205067\pi$
$349$	29.8740	1.59912	0.799559	+	0.600588i	$0.205067\pi$
$350$	0	0
$351$	0	0
$352$	15.2245	0.811469
$353$	−0.661559	−0.0352112	−0.0176056	−	0.999845i	$-0.505604\pi$
$353$	−0.661559	−0.0352112	−0.0176056	+	0.999845i	$0.505604\pi$
$354$	0	0
$355$	0.679458	0.0360619
$356$	23.1295	1.22586
$357$	0	0
$358$	6.25345	0.330505
$359$	−13.2792	−0.700851	−0.350425	−	0.936591i	$-0.613963\pi$
$359$	−13.2792	−0.700851	−0.350425	+	0.936591i	$0.613963\pi$
$360$	0	0
$361$	−18.1252	−0.953958
$362$	−4.32097	−0.227105
$363$	0	0
$364$	0	0
$365$	1.34791	0.0705530
$366$	0	0
$367$	36.0453	1.88155	0.940776	−	0.339029i	$-0.110098\pi$
$367$	36.0453	1.88155	0.940776	+	0.339029i	$0.110098\pi$
$368$	1.88512	0.0982687
$369$	0	0
$370$	0.390718	0.0203125
$371$	0	0
$372$	0	0
$373$	−5.95464	−0.308320	−0.154160	−	0.988046i	$-0.549267\pi$
$373$	−5.95464	−0.308320	−0.154160	+	0.988046i	$0.549267\pi$
$374$	−8.53610	−0.441391
$375$	0	0
$376$	−5.49882	−0.283580
$377$	−4.49614	−0.231563
$378$	0	0
$379$	−18.7760	−0.964457	−0.482229	−	0.876045i	$-0.660173\pi$
$379$	−18.7760	−0.964457	−0.482229	+	0.876045i	$0.660173\pi$
$380$	−0.494898	−0.0253877
$381$	0	0
$382$	0.911079	0.0466149
$383$	11.2426	0.574473	0.287236	−	0.957860i	$-0.407264\pi$
$383$	11.2426	0.574473	0.287236	+	0.957860i	$0.407264\pi$
$384$	0	0
$385$	0	0
$386$	−0.101355	−0.00515883
$387$	0	0
$388$	−24.5100	−1.24431
$389$	10.2488	0.519632	0.259816	−	0.965658i	$-0.416338\pi$
$389$	10.2488	0.519632	0.259816	+	0.965658i	$0.416338\pi$
$390$	0	0
$391$	−3.47973	−0.175977
$392$	0	0
$393$	0	0
$394$	−0.687269	−0.0346241
$395$	−3.49711	−0.175958
$396$	0	0
$397$	−4.24651	−0.213126	−0.106563	−	0.994306i	$-0.533985\pi$
$397$	−4.24651	−0.213126	−0.106563	+	0.994306i	$0.533985\pi$
$398$	−4.38683	−0.219892
$399$	0	0
$400$	−16.7434	−0.837171
$401$	16.0419	0.801092	0.400546	−	0.916277i	$-0.368820\pi$
$401$	16.0419	0.801092	0.400546	+	0.916277i	$0.368820\pi$
$402$	0	0
$403$	−0.861767	−0.0429277
$404$	9.68903	0.482047
$405$	0	0
$406$	0	0
$407$	−18.7847	−0.931121
$408$	0	0
$409$	10.4352	0.515986	0.257993	−	0.966147i	$-0.416939\pi$
$409$	10.4352	0.515986	0.257993	+	0.966147i	$0.416939\pi$
$410$	0.0869635	0.00429482
$411$	0	0
$412$	−0.685537	−0.0337740
$413$	0	0
$414$	0	0
$415$	3.11610	0.152963
$416$	−3.56709	−0.174891
$417$	0	0
$418$	−1.27156	−0.0621938
$419$	26.0460	1.27243	0.636215	−	0.771512i	$-0.280499\pi$
$419$	26.0460	1.27243	0.636215	+	0.771512i	$0.280499\pi$
$420$	0	0
$421$	19.4906	0.949912	0.474956	−	0.880009i	$-0.342464\pi$
$421$	19.4906	0.949912	0.474956	+	0.880009i	$0.342464\pi$
$422$	1.61056	0.0784009
$423$	0	0
$424$	11.5562	0.561220
$425$	30.9065	1.49919
$426$	0	0
$427$	0	0
$428$	25.1866	1.21744
$429$	0	0
$430$	−0.152390	−0.00734889
$431$	−3.84680	−0.185294	−0.0926468	−	0.995699i	$-0.529533\pi$
$431$	−3.84680	−0.185294	−0.0926468	+	0.995699i	$0.529533\pi$
$432$	0	0
$433$	1.34412	0.0645943	0.0322972	−	0.999478i	$-0.489718\pi$
$433$	1.34412	0.0645943	0.0322972	+	0.999478i	$0.489718\pi$
$434$	0	0
$435$	0	0
$436$	26.1669	1.25317
$437$	−0.518348	−0.0247959
$438$	0	0
$439$	36.1985	1.72766	0.863831	−	0.503781i	$-0.168058\pi$
$439$	36.1985	1.72766	0.863831	+	0.503781i	$0.168058\pi$
$440$	1.47714	0.0704198
$441$	0	0
$442$	2.00000	0.0951303
$443$	−11.1890	−0.531608	−0.265804	−	0.964027i	$-0.585637\pi$
$443$	−11.1890	−0.531608	−0.265804	+	0.964027i	$0.585637\pi$
$444$	0	0
$445$	3.39536	0.160956
$446$	−7.63262	−0.361415
$447$	0	0
$448$	0	0
$449$	10.3847	0.490082	0.245041	−	0.969513i	$-0.421199\pi$
$449$	10.3847	0.490082	0.245041	+	0.969513i	$0.421199\pi$
$450$	0	0
$451$	−4.18096	−0.196874
$452$	5.68258	0.267286
$453$	0	0
$454$	6.00172	0.281675
$455$	0	0
$456$	0	0
$457$	−27.0132	−1.26363	−0.631813	−	0.775121i	$-0.717689\pi$
$457$	−27.0132	−1.26363	−0.631813	+	0.775121i	$0.717689\pi$
$458$	−6.04233	−0.282340
$459$	0	0
$460$	0.293241	0.0136724
$461$	17.3981	0.810309	0.405154	−	0.914248i	$-0.367218\pi$
$461$	17.3981	0.810309	0.405154	+	0.914248i	$0.367218\pi$
$462$	0	0
$463$	5.57924	0.259289	0.129645	−	0.991561i	$-0.458616\pi$
$463$	5.57924	0.259289	0.129645	+	0.991561i	$0.458616\pi$
$464$	−15.2937	−0.709994
$465$	0	0
$466$	−8.70345	−0.403180
$467$	−1.10964	−0.0513479	−0.0256740	−	0.999670i	$-0.508173\pi$
$467$	−1.10964	−0.0513479	−0.0256740	+	0.999670i	$0.508173\pi$
$468$	0	0
$469$	0	0
$470$	−0.393104	−0.0181326
$471$	0	0
$472$	15.5357	0.715088
$473$	7.32648	0.336872
$474$	0	0
$475$	4.60390	0.211242
$476$	0	0
$477$	0	0
$478$	−7.60487	−0.347839
$479$	28.7672	1.31441	0.657204	−	0.753713i	$-0.271739\pi$
$479$	28.7672	1.31441	0.657204	+	0.753713i	$0.271739\pi$
$480$	0	0
$481$	4.40123	0.200679
$482$	−5.22200	−0.237856
$483$	0	0
$484$	−13.7003	−0.622742
$485$	−3.59803	−0.163378
$486$	0	0
$487$	−8.05808	−0.365147	−0.182573	−	0.983192i	$-0.558443\pi$
$487$	−8.05808	−0.365147	−0.182573	+	0.983192i	$0.558443\pi$
$488$	−2.45209	−0.111001
$489$	0	0
$490$	0	0
$491$	−33.6810	−1.52000	−0.760001	−	0.649922i	$-0.774802\pi$
$491$	−33.6810	−1.52000	−0.760001	+	0.649922i	$0.774802\pi$
$492$	0	0
$493$	28.2306	1.27144
$494$	0.297924	0.0134042
$495$	0	0
$496$	−2.93132	−0.131620
$497$	0	0
$498$	0	0
$499$	4.98394	0.223112	0.111556	−	0.993758i	$-0.464417\pi$
$499$	4.98394	0.223112	0.111556	+	0.993758i	$0.464417\pi$
$500$	−5.25017	−0.234795
$501$	0	0
$502$	−2.51303	−0.112162
$503$	−10.4861	−0.467554	−0.233777	−	0.972290i	$-0.575109\pi$
$503$	−10.4861	−0.467554	−0.233777	+	0.972290i	$0.575109\pi$
$504$	0	0
$505$	1.42233	0.0632929
$506$	0.753433	0.0334942
$507$	0	0
$508$	12.7422	0.565343
$509$	27.7170	1.22854	0.614268	−	0.789097i	$-0.289451\pi$
$509$	27.7170	1.22854	0.614268	+	0.789097i	$0.289451\pi$
$510$	0	0
$511$	0	0
$512$	−20.5816	−0.909586
$513$	0	0
$514$	6.70692	0.295830
$515$	−0.100636	−0.00443454
$516$	0	0
$517$	18.8994	0.831193
$518$	0	0
$519$	0	0
$520$	−0.346092	−0.0151772
$521$	−4.84437	−0.212236	−0.106118	−	0.994354i	$-0.533842\pi$
$521$	−4.84437	−0.212236	−0.106118	+	0.994354i	$0.533842\pi$
$522$	0	0
$523$	−26.8799	−1.17538	−0.587688	−	0.809088i	$-0.699962\pi$
$523$	−26.8799	−1.17538	−0.587688	+	0.809088i	$0.699962\pi$
$524$	−18.0880	−0.790177
$525$	0	0
$526$	−3.01064	−0.131270
$527$	5.41091	0.235703
$528$	0	0
$529$	−22.6929	−0.986646
$530$	0.826142	0.0358853
$531$	0	0
$532$	0	0
$533$	0.979596	0.0424310
$534$	0	0
$535$	3.69735	0.159851
$536$	−4.93443	−0.213135
$537$	0	0
$538$	4.27727	0.184406
$539$	0	0
$540$	0	0
$541$	13.9541	0.599933	0.299966	−	0.953950i	$-0.403025\pi$
$541$	13.9541	0.599933	0.299966	+	0.953950i	$0.403025\pi$
$542$	−2.84851	−0.122354
$543$	0	0
$544$	22.3972	0.960272
$545$	3.84125	0.164541
$546$	0	0
$547$	39.2066	1.67635	0.838177	−	0.545399i	$-0.183622\pi$
$547$	39.2066	1.67635	0.838177	+	0.545399i	$0.183622\pi$
$548$	−40.0844	−1.71232
$549$	0	0
$550$	−6.69190	−0.285344
$551$	4.20529	0.179151
$552$	0	0
$553$	0	0
$554$	1.93671	0.0822829
$555$	0	0
$556$	−27.8727	−1.18206
$557$	7.47685	0.316804	0.158402	−	0.987375i	$-0.449366\pi$
$557$	7.47685	0.316804	0.158402	+	0.987375i	$0.449366\pi$
$558$	0	0
$559$	−1.71659	−0.0726039
$560$	0	0
$561$	0	0
$562$	5.34512	0.225470
$563$	−18.2725	−0.770096	−0.385048	−	0.922897i	$-0.625815\pi$
$563$	−18.2725	−0.770096	−0.385048	+	0.922897i	$0.625815\pi$
$564$	0	0
$565$	0.834192	0.0350947
$566$	1.43023	0.0601172
$567$	0	0
$568$	−3.02743	−0.127028
$569$	−26.0547	−1.09227	−0.546136	−	0.837697i	$-0.683902\pi$
$569$	−26.0547	−1.09227	−0.546136	+	0.837697i	$0.683902\pi$
$570$	0	0
$571$	−2.26065	−0.0946052	−0.0473026	−	0.998881i	$-0.515063\pi$
$571$	−2.26065	−0.0946052	−0.0473026	+	0.998881i	$0.515063\pi$
$572$	8.10306	0.338806
$573$	0	0
$574$	0	0
$575$	−2.72794	−0.113763
$576$	0	0
$577$	−35.8836	−1.49385	−0.746927	−	0.664906i	$-0.768471\pi$
$577$	−35.8836	−1.49385	−0.746927	+	0.664906i	$0.768471\pi$
$578$	−7.14269	−0.297097
$579$	0	0
$580$	−2.37903	−0.0987838
$581$	0	0
$582$	0	0
$583$	−39.7186	−1.64498
$584$	−6.00584	−0.248523
$585$	0	0
$586$	2.46648	0.101889
$587$	21.2452	0.876885	0.438443	−	0.898759i	$-0.355530\pi$
$587$	21.2452	0.876885	0.438443	+	0.898759i	$0.355530\pi$
$588$	0	0
$589$	0.806020	0.0332115
$590$	1.11063	0.0457239
$591$	0	0
$592$	14.9709	0.615300
$593$	−19.0481	−0.782214	−0.391107	−	0.920345i	$-0.627908\pi$
$593$	−19.0481	−0.782214	−0.391107	+	0.920345i	$0.627908\pi$
$594$	0	0
$595$	0	0
$596$	25.7298	1.05394
$597$	0	0
$598$	−0.176529	−0.00721879
$599$	21.9481	0.896776	0.448388	−	0.893839i	$-0.351998\pi$
$599$	21.9481	0.896776	0.448388	+	0.893839i	$0.351998\pi$
$600$	0	0
$601$	−11.1249	−0.453794	−0.226897	−	0.973919i	$-0.572858\pi$
$601$	−11.1249	−0.453794	−0.226897	+	0.973919i	$0.572858\pi$
$602$	0	0
$603$	0	0
$604$	16.1811	0.658401
$605$	−2.01118	−0.0817662
$606$	0	0
$607$	8.29441	0.336660	0.168330	−	0.985731i	$-0.446163\pi$
$607$	8.29441	0.336660	0.168330	+	0.985731i	$0.446163\pi$
$608$	3.33634	0.135306
$609$	0	0
$610$	−0.175297	−0.00709758
$611$	−4.42810	−0.179142
$612$	0	0
$613$	−24.9909	−1.00937	−0.504686	−	0.863303i	$-0.668392\pi$
$613$	−24.9909	−1.00937	−0.504686	+	0.863303i	$0.668392\pi$
$614$	−2.23842	−0.0903354
$615$	0	0
$616$	0	0
$617$	29.5367	1.18910	0.594551	−	0.804058i	$-0.297330\pi$
$617$	29.5367	1.18910	0.594551	+	0.804058i	$0.297330\pi$
$618$	0	0
$619$	24.2203	0.973494	0.486747	−	0.873543i	$-0.338183\pi$
$619$	24.2203	0.973494	0.486747	+	0.873543i	$0.338183\pi$
$620$	−0.455984	−0.0183128
$621$	0	0
$622$	−10.6523	−0.427119
$623$	0	0
$624$	0	0
$625$	23.8409	0.953636
$626$	−4.08958	−0.163453
$627$	0	0
$628$	−26.4472	−1.05536
$629$	−27.6346	−1.10187
$630$	0	0
$631$	22.9592	0.913992	0.456996	−	0.889469i	$-0.348925\pi$
$631$	22.9592	0.913992	0.456996	+	0.889469i	$0.348925\pi$
$632$	15.5819	0.619815
$633$	0	0
$634$	−1.00236	−0.0398090
$635$	1.87053	0.0742296
$636$	0	0
$637$	0	0
$638$	−6.11251	−0.241996
$639$	0	0
$640$	−2.49138	−0.0984805
$641$	−40.5768	−1.60269	−0.801344	−	0.598204i	$-0.795881\pi$
$641$	−40.5768	−1.60269	−0.801344	+	0.598204i	$0.795881\pi$
$642$	0	0
$643$	−9.87685	−0.389505	−0.194752	−	0.980852i	$-0.562390\pi$
$643$	−9.87685	−0.389505	−0.194752	+	0.980852i	$0.562390\pi$
$644$	0	0
$645$	0	0
$646$	−1.87062	−0.0735986
$647$	44.6596	1.75575	0.877876	−	0.478888i	$-0.158960\pi$
$647$	44.6596	1.75575	0.877876	+	0.478888i	$0.158960\pi$
$648$	0	0
$649$	−53.3960	−2.09597
$650$	1.56791	0.0614984
$651$	0	0
$652$	−33.5127	−1.31246
$653$	0.169205	0.00662149	0.00331074	−	0.999995i	$-0.498946\pi$
$653$	0.169205	0.00662149	0.00331074	+	0.999995i	$0.498946\pi$
$654$	0	0
$655$	−2.65528	−0.103750
$656$	3.33212	0.130097
$657$	0	0
$658$	0	0
$659$	33.3418	1.29881	0.649405	−	0.760443i	$-0.275018\pi$
$659$	33.3418	1.29881	0.649405	+	0.760443i	$0.275018\pi$
$660$	0	0
$661$	−37.0200	−1.43991	−0.719956	−	0.694019i	$-0.755838\pi$
$661$	−37.0200	−1.43991	−0.719956	+	0.694019i	$0.755838\pi$
$662$	−1.99451	−0.0775187
$663$	0	0
$664$	−13.8843	−0.538814
$665$	0	0
$666$	0	0
$667$	−2.49175	−0.0964811
$668$	14.1489	0.547439
$669$	0	0
$670$	−0.352756	−0.0136282
$671$	8.42781	0.325352
$672$	0	0
$673$	32.6406	1.25820	0.629102	−	0.777323i	$-0.283423\pi$
$673$	32.6406	1.25820	0.629102	+	0.777323i	$0.283423\pi$
$674$	5.05176	0.194587
$675$	0	0
$676$	−1.89854	−0.0730207
$677$	−34.7106	−1.33404	−0.667018	−	0.745042i	$-0.732429\pi$
$677$	−34.7106	−1.33404	−0.667018	+	0.745042i	$0.732429\pi$
$678$	0	0
$679$	0	0
$680$	2.17306	0.0833331
$681$	0	0
$682$	−1.17157	−0.0448618
$683$	−25.9340	−0.992337	−0.496169	−	0.868226i	$-0.665260\pi$
$683$	−25.9340	−0.992337	−0.496169	+	0.868226i	$0.665260\pi$
$684$	0	0
$685$	−5.88432	−0.224828
$686$	0	0
$687$	0	0
$688$	−5.83902	−0.222611
$689$	9.30603	0.354531
$690$	0	0
$691$	−38.3626	−1.45938	−0.729691	−	0.683777i	$-0.760336\pi$
$691$	−38.3626	−1.45938	−0.729691	+	0.683777i	$0.760336\pi$
$692$	−36.4227	−1.38458
$693$	0	0
$694$	4.18759	0.158959
$695$	−4.09165	−0.155205
$696$	0	0
$697$	−6.15073	−0.232976
$698$	−9.51575	−0.360176
$699$	0	0
$700$	0	0
$701$	−26.2477	−0.991363	−0.495682	−	0.868504i	$-0.665082\pi$
$701$	−26.2477	−0.991363	−0.495682	+	0.868504i	$0.665082\pi$
$702$	0	0
$703$	−4.11652	−0.155257
$704$	24.1863	0.911556
$705$	0	0
$706$	0.210726	0.00793078
$707$	0	0
$708$	0	0
$709$	34.9709	1.31336	0.656679	−	0.754170i	$-0.271961\pi$
$709$	34.9709	1.31336	0.656679	+	0.754170i	$0.271961\pi$
$710$	−0.216428	−0.00812238
$711$	0	0
$712$	−15.1286	−0.566967
$713$	−0.477590	−0.0178859
$714$	0	0
$715$	1.18951	0.0444853
$716$	37.2726	1.39294
$717$	0	0
$718$	4.22983	0.157856
$719$	−14.2166	−0.530190	−0.265095	−	0.964222i	$-0.585403\pi$
$719$	−14.2166	−0.530190	−0.265095	+	0.964222i	$0.585403\pi$
$720$	0	0
$721$	0	0
$722$	5.77341	0.214864
$723$	0	0
$724$	−25.7544	−0.957154
$725$	22.1315	0.821942
$726$	0	0
$727$	0.566912	0.0210256	0.0105128	−	0.999945i	$-0.496654\pi$
$727$	0.566912	0.0210256	0.0105128	+	0.999945i	$0.496654\pi$
$728$	0	0
$729$	0	0
$730$	−0.429350	−0.0158910
$731$	10.7782	0.398646
$732$	0	0
$733$	28.2007	1.04162	0.520809	−	0.853673i	$-0.325630\pi$
$733$	28.2007	1.04162	0.520809	+	0.853673i	$0.325630\pi$
$734$	−11.4815	−0.423790
$735$	0	0
$736$	−1.97687	−0.0728686
$737$	16.9595	0.624713
$738$	0	0
$739$	−44.6598	−1.64284	−0.821419	−	0.570325i	$-0.806817\pi$
$739$	−44.6598	−1.64284	−0.821419	+	0.570325i	$0.806817\pi$
$740$	2.32881	0.0856086
$741$	0	0
$742$	0	0
$743$	−3.31876	−0.121754	−0.0608768	−	0.998145i	$-0.519390\pi$
$743$	−3.31876	−0.121754	−0.0608768	+	0.998145i	$0.519390\pi$
$744$	0	0
$745$	3.77709	0.138382
$746$	1.89673	0.0694443
$747$	0	0
$748$	−50.8779	−1.86028
$749$	0	0
$750$	0	0
$751$	−33.1595	−1.21001	−0.605004	−	0.796223i	$-0.706828\pi$
$751$	−33.1595	−1.21001	−0.605004	+	0.796223i	$0.706828\pi$
$752$	−15.0623	−0.549266
$753$	0	0
$754$	1.43215	0.0521560
$755$	2.37536	0.0864482
$756$	0	0
$757$	18.8780	0.686133	0.343067	−	0.939311i	$-0.388534\pi$
$757$	18.8780	0.686133	0.343067	+	0.939311i	$0.388534\pi$
$758$	5.98071	0.217229
$759$	0	0
$760$	0.323704	0.0117420
$761$	−28.3937	−1.02927	−0.514635	−	0.857409i	$-0.672073\pi$
$761$	−28.3937	−1.02927	−0.514635	+	0.857409i	$0.672073\pi$
$762$	0	0
$763$	0	0
$764$	5.43032	0.196462
$765$	0	0
$766$	−3.58112	−0.129391
$767$	12.5106	0.451732
$768$	0	0
$769$	36.8171	1.32766	0.663830	−	0.747884i	$-0.268930\pi$
$769$	36.8171	1.32766	0.663830	+	0.747884i	$0.268930\pi$
$770$	0	0
$771$	0	0
$772$	−0.604108	−0.0217423
$773$	14.3835	0.517337	0.258669	−	0.965966i	$-0.416716\pi$
$773$	14.3835	0.517337	0.258669	+	0.965966i	$0.416716\pi$
$774$	0	0
$775$	4.24190	0.152373
$776$	16.0316	0.575500
$777$	0	0
$778$	−3.26453	−0.117039
$779$	−0.916226	−0.0328272
$780$	0	0
$781$	10.4052	0.372329
$782$	1.10840	0.0396362
$783$	0	0
$784$	0	0
$785$	−3.88240	−0.138569
$786$	0	0
$787$	37.1507	1.32428	0.662140	−	0.749380i	$-0.269648\pi$
$787$	37.1507	1.32428	0.662140	+	0.749380i	$0.269648\pi$
$788$	−4.09634	−0.145926
$789$	0	0
$790$	1.11393	0.0396319
$791$	0	0
$792$	0	0
$793$	−1.97463	−0.0701211
$794$	1.35264	0.0480034
$795$	0	0
$796$	−26.1469	−0.926753
$797$	23.7554	0.841460	0.420730	−	0.907186i	$-0.361774\pi$
$797$	23.7554	0.841460	0.420730	+	0.907186i	$0.361774\pi$
$798$	0	0
$799$	27.8034	0.983613
$800$	17.5584	0.620782
$801$	0	0
$802$	−5.10981	−0.180434
$803$	20.6420	0.728439
$804$	0	0
$805$	0	0
$806$	0.274498	0.00966879
$807$	0	0
$808$	−6.33742	−0.222950
$809$	−45.9182	−1.61440	−0.807200	−	0.590279i	$-0.799018\pi$
$809$	−45.9182	−1.61440	−0.807200	+	0.590279i	$0.799018\pi$
$810$	0	0
$811$	−9.21250	−0.323495	−0.161747	−	0.986832i	$-0.551713\pi$
$811$	−9.21250	−0.323495	−0.161747	+	0.986832i	$0.551713\pi$
$812$	0	0
$813$	0	0
$814$	5.98347	0.209721
$815$	−4.91960	−0.172326
$816$	0	0
$817$	1.60554	0.0561708
$818$	−3.32391	−0.116218
$819$	0	0
$820$	0.518330	0.0181009
$821$	9.22432	0.321931	0.160965	−	0.986960i	$-0.448539\pi$
$821$	9.22432	0.321931	0.160965	+	0.986960i	$0.448539\pi$
$822$	0	0
$823$	32.3510	1.12768	0.563842	−	0.825883i	$-0.309323\pi$
$823$	32.3510	1.12768	0.563842	+	0.825883i	$0.309323\pi$
$824$	0.448398	0.0156207
$825$	0	0
$826$	0	0
$827$	25.5118	0.887133	0.443566	−	0.896242i	$-0.353713\pi$
$827$	25.5118	0.887133	0.443566	+	0.896242i	$0.353713\pi$
$828$	0	0
$829$	−37.2508	−1.29377	−0.646887	−	0.762586i	$-0.723929\pi$
$829$	−37.2508	−1.29377	−0.646887	+	0.762586i	$0.723929\pi$
$830$	−0.992570	−0.0344526
$831$	0	0
$832$	−5.66683	−0.196462
$833$	0	0
$834$	0	0
$835$	2.07704	0.0718789
$836$	−7.57888	−0.262121
$837$	0	0
$838$	−8.29642	−0.286595
$839$	−24.1576	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$839$	−24.1576	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$840$	0	0
$841$	−8.78471	−0.302921
$842$	−6.20833	−0.213953
$843$	0	0
$844$	9.59946	0.330427
$845$	−0.278702	−0.00958764
$846$	0	0
$847$	0	0
$848$	31.6547	1.08703
$849$	0	0
$850$	−9.84465	−0.337669
$851$	2.43915	0.0836131
$852$	0	0
$853$	−39.3757	−1.34820	−0.674100	−	0.738640i	$-0.735468\pi$
$853$	−39.3757	−1.34820	−0.674100	+	0.738640i	$0.735468\pi$
$854$	0	0
$855$	0	0
$856$	−16.4741	−0.563074
$857$	−50.6221	−1.72922	−0.864610	−	0.502444i	$-0.832434\pi$
$857$	−50.6221	−1.72922	−0.864610	+	0.502444i	$0.832434\pi$
$858$	0	0
$859$	16.7350	0.570991	0.285496	−	0.958380i	$-0.407842\pi$
$859$	16.7350	0.570991	0.285496	+	0.958380i	$0.407842\pi$
$860$	−0.908292	−0.0309725
$861$	0	0
$862$	1.22532	0.0417345
$863$	−19.8012	−0.674040	−0.337020	−	0.941497i	$-0.609419\pi$
$863$	−19.8012	−0.674040	−0.337020	+	0.941497i	$0.609419\pi$
$864$	0	0
$865$	−5.34679	−0.181796
$866$	−0.428142	−0.0145489
$867$	0	0
$868$	0	0
$869$	−53.5548	−1.81672
$870$	0	0
$871$	−3.97361	−0.134641
$872$	−17.1153	−0.579597
$873$	0	0
$874$	0.165109	0.00558490
$875$	0	0
$876$	0	0
$877$	−20.7398	−0.700333	−0.350167	−	0.936687i	$-0.613875\pi$
$877$	−20.7398	−0.700333	−0.350167	+	0.936687i	$0.613875\pi$
$878$	−11.5303	−0.389129
$879$	0	0
$880$	4.04616	0.136396
$881$	1.62325	0.0546888	0.0273444	−	0.999626i	$-0.491295\pi$
$881$	1.62325	0.0546888	0.0273444	+	0.999626i	$0.491295\pi$
$882$	0	0
$883$	−11.8833	−0.399904	−0.199952	−	0.979806i	$-0.564079\pi$
$883$	−11.8833	−0.399904	−0.199952	+	0.979806i	$0.564079\pi$
$884$	11.9206	0.400934
$885$	0	0
$886$	3.56404	0.119736
$887$	−31.7030	−1.06448	−0.532241	−	0.846593i	$-0.678650\pi$
$887$	−31.7030	−1.06448	−0.532241	+	0.846593i	$0.678650\pi$
$888$	0	0
$889$	0	0
$890$	−1.08152	−0.0362528
$891$	0	0
$892$	−45.4929	−1.52321
$893$	4.14165	0.138595
$894$	0	0
$895$	5.47155	0.182894
$896$	0	0
$897$	0	0
$898$	−3.30782	−0.110383
$899$	3.87463	0.129226
$900$	0	0
$901$	−58.4311	−1.94662
$902$	1.33176	0.0443428
$903$	0	0
$904$	−3.71688	−0.123621
$905$	−3.78070	−0.125675
$906$	0	0
$907$	−4.45928	−0.148068	−0.0740341	−	0.997256i	$-0.523587\pi$
$907$	−4.45928	−0.148068	−0.0740341	+	0.997256i	$0.523587\pi$
$908$	35.7722	1.18714
$909$	0	0
$910$	0	0
$911$	46.4395	1.53861	0.769305	−	0.638882i	$-0.220603\pi$
$911$	46.4395	1.53861	0.769305	+	0.638882i	$0.220603\pi$
$912$	0	0
$913$	47.7200	1.57930
$914$	8.60452	0.284612
$915$	0	0
$916$	−36.0142	−1.18994
$917$	0	0
$918$	0	0
$919$	−19.5628	−0.645317	−0.322658	−	0.946515i	$-0.604577\pi$
$919$	−19.5628	−0.645317	−0.322658	+	0.946515i	$0.604577\pi$
$920$	−0.191804	−0.00632358
$921$	0	0
$922$	−5.54180	−0.182509
$923$	−2.43794	−0.0802457
$924$	0	0
$925$	−21.6643	−0.712317
$926$	−1.77715	−0.0584009
$927$	0	0
$928$	16.0381	0.526478
$929$	−18.1377	−0.595078	−0.297539	−	0.954710i	$-0.596166\pi$
$929$	−18.1377	−0.595078	−0.297539	+	0.954710i	$0.596166\pi$
$930$	0	0
$931$	0	0
$932$	−51.8754	−1.69923
$933$	0	0
$934$	0.353453	0.0115653
$935$	−7.46878	−0.244255
$936$	0	0
$937$	−40.3697	−1.31882	−0.659410	−	0.751784i	$-0.729194\pi$
$937$	−40.3697	−1.31882	−0.659410	+	0.751784i	$0.729194\pi$
$938$	0	0
$939$	0	0
$940$	−2.34303	−0.0764211
$941$	21.5312	0.701898	0.350949	−	0.936395i	$-0.385859\pi$
$941$	21.5312	0.701898	0.350949	+	0.936395i	$0.385859\pi$
$942$	0	0
$943$	0.542890	0.0176789
$944$	42.5552	1.38505
$945$	0	0
$946$	−2.33370	−0.0758752
$947$	−55.9123	−1.81691	−0.908454	−	0.417986i	$-0.862736\pi$
$947$	−55.9123	−1.81691	−0.908454	+	0.417986i	$0.862736\pi$
$948$	0	0
$949$	−4.83639	−0.156996
$950$	−1.46648	−0.0475789
$951$	0	0
$952$	0	0
$953$	21.2497	0.688346	0.344173	−	0.938906i	$-0.388159\pi$
$953$	21.2497	0.688346	0.344173	+	0.938906i	$0.388159\pi$
$954$	0	0
$955$	0.797161	0.0257955
$956$	−45.3274	−1.46599
$957$	0	0
$958$	−9.16321	−0.296050
$959$	0	0
$960$	0	0
$961$	−30.2574	−0.976044
$962$	−1.40192	−0.0451998
$963$	0	0
$964$	−31.1248	−1.00246
$965$	−0.0886819	−0.00285477
$966$	0	0
$967$	52.4550	1.68684	0.843419	−	0.537257i	$-0.180539\pi$
$967$	52.4550	1.68684	0.843419	+	0.537257i	$0.180539\pi$
$968$	8.96114	0.288022
$969$	0	0
$970$	1.14608	0.0367984
$971$	16.7283	0.536836	0.268418	−	0.963302i	$-0.413499\pi$
$971$	16.7283	0.536836	0.268418	+	0.963302i	$0.413499\pi$
$972$	0	0
$973$	0	0
$974$	2.56674	0.0822436
$975$	0	0
$976$	−6.71675	−0.214998
$977$	10.9472	0.350232	0.175116	−	0.984548i	$-0.443970\pi$
$977$	10.9472	0.350232	0.175116	+	0.984548i	$0.443970\pi$
$978$	0	0
$979$	51.9967	1.66182
$980$	0	0
$981$	0	0
$982$	10.7284	0.342357
$983$	2.17536	0.0693832	0.0346916	−	0.999398i	$-0.488955\pi$
$983$	2.17536	0.0693832	0.0346916	+	0.999398i	$0.488955\pi$
$984$	0	0
$985$	−0.601336	−0.0191601
$986$	−8.99228	−0.286373
$987$	0	0
$988$	1.77572	0.0564933
$989$	−0.951330	−0.0302505
$990$	0	0
$991$	24.0099	0.762700	0.381350	−	0.924431i	$-0.375459\pi$
$991$	24.0099	0.762700	0.381350	+	0.924431i	$0.375459\pi$
$992$	3.07400	0.0975996
$993$	0	0
$994$	0	0
$995$	−3.83832	−0.121683
$996$	0	0
$997$	26.5083	0.839527	0.419763	−	0.907634i	$-0.362113\pi$
$997$	26.5083	0.839527	0.419763	+	0.907634i	$0.362113\pi$
$998$	−1.58753	−0.0502524
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.cb.1.6	yes	12
3.2	odd	2	inner	5733.2.a.cb.1.7	yes	12
7.6	odd	2		5733.2.a.ca.1.6	✓	12
21.20	even	2		5733.2.a.ca.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5733.2.a.ca.1.6	✓	12	7.6	odd	2
5733.2.a.ca.1.7	yes	12	21.20	even	2
5733.2.a.cb.1.6	yes	12	1.1	even	1	trivial
5733.2.a.cb.1.7	yes	12	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)

\(f(q)\)	\(=\)	\(q-0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} +1.24180 q^{8} +O(q^{10})\)	\(q-0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} +1.24180 q^{8} +0.0887749 q^{10} -4.26805 q^{11} +1.00000 q^{13} +3.40153 q^{16} -6.27885 q^{17} -0.935311 q^{19} +0.529127 q^{20} +1.35950 q^{22} +0.554198 q^{23} -4.92233 q^{25} -0.318530 q^{26} -4.49614 q^{29} -0.861767 q^{31} -3.56709 q^{32} +2.00000 q^{34} +4.40123 q^{37} +0.297924 q^{38} -0.346092 q^{40} +0.979596 q^{41} -1.71659 q^{43} +8.10306 q^{44} -0.176529 q^{46} -4.42810 q^{47} +1.56791 q^{50} -1.89854 q^{52} +9.30603 q^{53} +1.18951 q^{55} +1.43215 q^{58} +12.5106 q^{59} -1.97463 q^{61} +0.274498 q^{62} -5.66683 q^{64} -0.278702 q^{65} -3.97361 q^{67} +11.9206 q^{68} -2.43794 q^{71} -4.83639 q^{73} -1.40192 q^{74} +1.77572 q^{76} +12.5478 q^{79} -0.948013 q^{80} -0.312030 q^{82} -11.1808 q^{83} +1.74993 q^{85} +0.546784 q^{86} -5.30007 q^{88} -12.1828 q^{89} -1.05217 q^{92} +1.41048 q^{94} +0.260673 q^{95} +12.9099 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4}+O(q^{10}) \) 12 * q + 8 * q^4	\( 12 q + 8 q^{4} + 8 q^{10} + 12 q^{13} + 28 q^{19} + 16 q^{25} + 12 q^{31} + 24 q^{34} + 32 q^{40} - 20 q^{43} - 8 q^{46} + 8 q^{52} + 56 q^{55} + 8 q^{58} + 24 q^{61} - 40 q^{64} + 8 q^{67} + 12 q^{73} + 64 q^{76} - 20 q^{79} + 40 q^{82} + 32 q^{85} + 8 q^{88} + 8 q^{94} + 68 q^{97}+O(q^{100}) \) 12 * q + 8 * q^4 + 8 * q^10 + 12 * q^13 + 28 * q^19 + 16 * q^25 + 12 * q^31 + 24 * q^34 + 32 * q^40 - 20 * q^43 - 8 * q^46 + 8 * q^52 + 56 * q^55 + 8 * q^58 + 24 * q^61 - 40 * q^64 + 8 * q^67 + 12 * q^73 + 64 * q^76 - 20 * q^79 + 40 * q^82 + 32 * q^85 + 8 * q^88 + 8 * q^94 + 68 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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