LMFDB - Embedded newform 5733.2.a.bi.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:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			$0.792287$ 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)

$f(q)$	$=$	$q+0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} -2.67181 q^{8} +O(q^{10})$	$q+0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} -2.67181 q^{8} -2.00000 q^{10} +3.46410 q^{11} +1.00000 q^{13} +0.627719 q^{16} -1.58457 q^{17} -1.62772 q^{19} +3.46410 q^{20} +2.74456 q^{22} +4.40387 q^{23} +1.37228 q^{25} +0.792287 q^{26} -5.98844 q^{29} -6.37228 q^{31} +5.84096 q^{32} -1.25544 q^{34} +6.74456 q^{37} -1.28962 q^{38} +6.74456 q^{40} +1.58457 q^{41} +11.1168 q^{43} -4.75372 q^{44} +3.48913 q^{46} +9.15759 q^{47} +1.08724 q^{50} -1.37228 q^{52} -0.939764 q^{53} -8.74456 q^{55} -4.74456 q^{58} +5.04868 q^{59} -6.00000 q^{61} -5.04868 q^{62} +3.37228 q^{64} -2.52434 q^{65} -8.74456 q^{67} +2.17448 q^{68} +1.58457 q^{71} -8.37228 q^{73} +5.34363 q^{74} +2.23369 q^{76} -11.8614 q^{79} -1.58457 q^{80} +1.25544 q^{82} +12.3267 q^{83} +4.00000 q^{85} +8.80773 q^{86} -9.25544 q^{88} -2.52434 q^{89} -6.04334 q^{92} +7.25544 q^{94} +4.10891 q^{95} -3.62772 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 4 q + 6 q^{4} - 8 q^{10} + 4 q^{13} + 14 q^{16} - 18 q^{19} - 12 q^{22} - 6 q^{25} - 14 q^{31} - 28 q^{34} + 4 q^{37} + 4 q^{40} + 10 q^{43} - 32 q^{46} + 6 q^{52} - 12 q^{55} + 4 q^{58} - 24 q^{61} + 2 q^{64} - 12 q^{67} - 22 q^{73} - 60 q^{76} + 10 q^{79} + 28 q^{82} + 16 q^{85} - 60 q^{88} + 52 q^{94} - 26 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 - 8 * q^10 + 4 * q^13 + 14 * q^16 - 18 * q^19 - 12 * q^22 - 6 * q^25 - 14 * q^31 - 28 * q^34 + 4 * q^37 + 4 * q^40 + 10 * q^43 - 32 * q^46 + 6 * q^52 - 12 * q^55 + 4 * q^58 - 24 * q^61 + 2 * q^64 - 12 * q^67 - 22 * q^73 - 60 * q^76 + 10 * q^79 + 28 * q^82 + 16 * q^85 - 60 * q^88 + 52 * q^94 - 26 * 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 $

$2$	0.792287	0.560232	0.280116	−	0.959966i	$-0.409627\pi$
$2$	0.792287	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$3$	0	0
$3$	0	0
$4$	−1.37228	−0.686141
$5$	−2.52434	−1.12892	−0.564459	−	0.825461i	$-0.690915\pi$
$5$	−2.52434	−1.12892	−0.564459	+	0.825461i	$0.690915\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.67181	−0.944629
$9$	0	0
$10$	−2.00000	−0.632456
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0.627719	0.156930
$17$	−1.58457	−0.384316	−0.192158	−	0.981364i	$-0.561549\pi$
$17$	−1.58457	−0.384316	−0.192158	+	0.981364i	$0.561549\pi$
$18$	0	0
$19$	−1.62772	−0.373424	−0.186712	−	0.982415i	$-0.559783\pi$
$19$	−1.62772	−0.373424	−0.186712	+	0.982415i	$0.559783\pi$
$20$	3.46410	0.774597
$21$	0	0
$22$	2.74456	0.585143
$23$	4.40387	0.918269	0.459135	−	0.888367i	$-0.348160\pi$
$23$	4.40387	0.918269	0.459135	+	0.888367i	$0.348160\pi$
$24$	0	0
$25$	1.37228	0.274456
$26$	0.792287	0.155380
$27$	0	0
$28$	0	0
$29$	−5.98844	−1.11203	−0.556013	−	0.831174i	$-0.687669\pi$
$29$	−5.98844	−1.11203	−0.556013	+	0.831174i	$0.687669\pi$
$30$	0	0
$31$	−6.37228	−1.14450	−0.572248	−	0.820081i	$-0.693928\pi$
$31$	−6.37228	−1.14450	−0.572248	+	0.820081i	$0.693928\pi$
$32$	5.84096	1.03255
$33$	0	0
$34$	−1.25544	−0.215306
$35$	0	0
$36$	0	0
$37$	6.74456	1.10880	0.554400	−	0.832251i	$-0.312948\pi$
$37$	6.74456	1.10880	0.554400	+	0.832251i	$0.312948\pi$
$38$	−1.28962	−0.209204
$39$	0	0
$40$	6.74456	1.06641
$41$	1.58457	0.247469	0.123734	−	0.992315i	$-0.460513\pi$
$41$	1.58457	0.247469	0.123734	+	0.992315i	$0.460513\pi$
$42$	0	0
$43$	11.1168	1.69530	0.847651	−	0.530554i	$-0.178016\pi$
$43$	11.1168	1.69530	0.847651	+	0.530554i	$0.178016\pi$
$44$	−4.75372	−0.716651
$45$	0	0
$46$	3.48913	0.514443
$47$	9.15759	1.33577	0.667886	−	0.744264i	$-0.267199\pi$
$47$	9.15759	1.33577	0.667886	+	0.744264i	$0.267199\pi$
$48$	0	0
$49$	0	0
$50$	1.08724	0.153759
$51$	0	0
$52$	−1.37228	−0.190301
$53$	−0.939764	−0.129086	−0.0645432	−	0.997915i	$-0.520559\pi$
$53$	−0.939764	−0.129086	−0.0645432	+	0.997915i	$0.520559\pi$
$54$	0	0
$55$	−8.74456	−1.17912
$56$	0	0
$57$	0	0
$58$	−4.74456	−0.622992
$59$	5.04868	0.657282	0.328641	−	0.944455i	$-0.393409\pi$
$59$	5.04868	0.657282	0.328641	+	0.944455i	$0.393409\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−5.04868	−0.641182
$63$	0	0
$64$	3.37228	0.421535
$65$	−2.52434	−0.313106
$66$	0	0
$67$	−8.74456	−1.06832	−0.534159	−	0.845384i	$-0.679372\pi$
$67$	−8.74456	−1.06832	−0.534159	+	0.845384i	$0.679372\pi$
$68$	2.17448	0.263695
$69$	0	0
$70$	0	0
$71$	1.58457	0.188054	0.0940272	−	0.995570i	$-0.470026\pi$
$71$	1.58457	0.188054	0.0940272	+	0.995570i	$0.470026\pi$
$72$	0	0
$73$	−8.37228	−0.979901	−0.489951	−	0.871750i	$-0.662985\pi$
$73$	−8.37228	−0.979901	−0.489951	+	0.871750i	$0.662985\pi$
$74$	5.34363	0.621184
$75$	0	0
$76$	2.23369	0.256222
$77$	0	0
$78$	0	0
$79$	−11.8614	−1.33451	−0.667256	−	0.744828i	$-0.732531\pi$
$79$	−11.8614	−1.33451	−0.667256	+	0.744828i	$0.732531\pi$
$80$	−1.58457	−0.177161
$81$	0	0
$82$	1.25544	0.138640
$83$	12.3267	1.35303	0.676517	−	0.736427i	$-0.263488\pi$
$83$	12.3267	1.35303	0.676517	+	0.736427i	$0.263488\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	8.80773	0.949762
$87$	0	0
$88$	−9.25544	−0.986633
$89$	−2.52434	−0.267579	−0.133790	−	0.991010i	$-0.542715\pi$
$89$	−2.52434	−0.267579	−0.133790	+	0.991010i	$0.542715\pi$
$90$	0	0
$91$	0	0
$92$	−6.04334	−0.630062
$93$	0	0
$94$	7.25544	0.748341
$95$	4.10891	0.421565
$96$	0	0
$97$	−3.62772	−0.368339	−0.184170	−	0.982894i	$-0.558960\pi$
$97$	−3.62772	−0.368339	−0.184170	+	0.982894i	$0.558960\pi$
$98$	0	0
$99$	0	0
$100$	−1.88316	−0.188316
$101$	−17.3205	−1.72345	−0.861727	−	0.507371i	$-0.830617\pi$
$101$	−17.3205	−1.72345	−0.861727	+	0.507371i	$0.830617\pi$
$102$	0	0
$103$	13.4891	1.32912	0.664562	−	0.747234i	$-0.268618\pi$
$103$	13.4891	1.32912	0.664562	+	0.747234i	$0.268618\pi$
$104$	−2.67181	−0.261993
$105$	0	0
$106$	−0.744563	−0.0723183
$107$	−16.7306	−1.61741	−0.808704	−	0.588216i	$-0.799831\pi$
$107$	−16.7306	−1.61741	−0.808704	+	0.588216i	$0.799831\pi$
$108$	0	0
$109$	−7.48913	−0.717328	−0.358664	−	0.933467i	$-0.616768\pi$
$109$	−7.48913	−0.717328	−0.358664	+	0.933467i	$0.616768\pi$
$110$	−6.92820	−0.660578
$111$	0	0
$112$	0	0
$113$	5.98844	0.563345	0.281672	−	0.959511i	$-0.409111\pi$
$113$	5.98844	0.563345	0.281672	+	0.959511i	$0.409111\pi$
$114$	0	0
$115$	−11.1168	−1.03665
$116$	8.21782	0.763006
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.75372	−0.430382
$123$	0	0
$124$	8.74456	0.785285
$125$	9.15759	0.819080
$126$	0	0
$127$	1.48913	0.132139	0.0660693	−	0.997815i	$-0.478954\pi$
$127$	1.48913	0.132139	0.0660693	+	0.997815i	$0.478954\pi$
$128$	−9.01011	−0.796389
$129$	0	0
$130$	−2.00000	−0.175412
$131$	10.0974	0.882210	0.441105	−	0.897456i	$-0.354587\pi$
$131$	10.0974	0.882210	0.441105	+	0.897456i	$0.354587\pi$
$132$	0	0
$133$	0	0
$134$	−6.92820	−0.598506
$135$	0	0
$136$	4.23369	0.363036
$137$	−17.0256	−1.45459	−0.727296	−	0.686324i	$-0.759223\pi$
$137$	−17.0256	−1.45459	−0.727296	+	0.686324i	$0.759223\pi$
$138$	0	0
$139$	−4.74456	−0.402429	−0.201214	−	0.979547i	$-0.564489\pi$
$139$	−4.74456	−0.402429	−0.201214	+	0.979547i	$0.564489\pi$
$140$	0	0
$141$	0	0
$142$	1.25544	0.105354
$143$	3.46410	0.289683
$144$	0	0
$145$	15.1168	1.25539
$146$	−6.63325	−0.548972
$147$	0	0
$148$	−9.25544	−0.760792
$149$	−15.1460	−1.24081	−0.620405	−	0.784281i	$-0.713032\pi$
$149$	−15.1460	−1.24081	−0.620405	+	0.784281i	$0.713032\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.34896	0.352747
$153$	0	0
$154$	0	0
$155$	16.0858	1.29204
$156$	0	0
$157$	−1.25544	−0.100195	−0.0500974	−	0.998744i	$-0.515953\pi$
$157$	−1.25544	−0.100195	−0.0500974	+	0.998744i	$0.515953\pi$
$158$	−9.39764	−0.747636
$159$	0	0
$160$	−14.7446	−1.16566
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.17448	−0.169798
$165$	0	0
$166$	9.76631	0.758013
$167$	12.9166	0.999520	0.499760	−	0.866164i	$-0.333422\pi$
$167$	12.9166	0.999520	0.499760	+	0.866164i	$0.333422\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.16915	0.243063
$171$	0	0
$172$	−15.2554	−1.16322
$173$	−11.6819	−0.888160	−0.444080	−	0.895987i	$-0.646469\pi$
$173$	−11.6819	−0.888160	−0.444080	+	0.895987i	$0.646469\pi$
$174$	0	0
$175$	0	0
$176$	2.17448	0.163908
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−13.2116	−0.987481	−0.493741	−	0.869609i	$-0.664371\pi$
$179$	−13.2116	−0.987481	−0.493741	+	0.869609i	$0.664371\pi$
$180$	0	0
$181$	−11.4891	−0.853980	−0.426990	−	0.904256i	$-0.640426\pi$
$181$	−11.4891	−0.853980	−0.426990	+	0.904256i	$0.640426\pi$
$182$	0	0
$183$	0	0
$184$	−11.7663	−0.867424
$185$	−17.0256	−1.25174
$186$	0	0
$187$	−5.48913	−0.401405
$188$	−12.5668	−0.916527
$189$	0	0
$190$	3.25544	0.236174
$191$	−6.63325	−0.479965	−0.239983	−	0.970777i	$-0.577142\pi$
$191$	−6.63325	−0.479965	−0.239983	+	0.970777i	$0.577142\pi$
$192$	0	0
$193$	−11.4891	−0.827005	−0.413503	−	0.910503i	$-0.635695\pi$
$193$	−11.4891	−0.827005	−0.413503	+	0.910503i	$0.635695\pi$
$194$	−2.87419	−0.206355
$195$	0	0
$196$	0	0
$197$	11.3870	0.811288	0.405644	−	0.914031i	$-0.367047\pi$
$197$	11.3870	0.811288	0.405644	+	0.914031i	$0.367047\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−3.66648	−0.259259
$201$	0	0
$202$	−13.7228	−0.965534
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	10.6873	0.744617
$207$	0	0
$208$	0.627719	0.0435245
$209$	−5.63858	−0.390029
$210$	0	0
$211$	−20.6060	−1.41857	−0.709287	−	0.704920i	$-0.750983\pi$
$211$	−20.6060	−1.41857	−0.709287	+	0.704920i	$0.750983\pi$
$212$	1.28962	0.0885715
$213$	0	0
$214$	−13.2554	−0.906123
$215$	−28.0627	−1.91386
$216$	0	0
$217$	0	0
$218$	−5.93354	−0.401870
$219$	0	0
$220$	12.0000	0.809040
$221$	−1.58457	−0.106590
$222$	0	0
$223$	16.6060	1.11202	0.556009	−	0.831176i	$-0.312332\pi$
$223$	16.6060	1.11202	0.556009	+	0.831176i	$0.312332\pi$
$224$	0	0
$225$	0	0
$226$	4.74456	0.315604
$227$	−11.3870	−0.755780	−0.377890	−	0.925851i	$-0.623350\pi$
$227$	−11.3870	−0.755780	−0.377890	+	0.925851i	$0.623350\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.80773	−0.580765
$231$	0	0
$232$	16.0000	1.05045
$233$	21.1345	1.38456	0.692282	−	0.721627i	$-0.256605\pi$
$233$	21.1345	1.38456	0.692282	+	0.721627i	$0.256605\pi$
$234$	0	0
$235$	−23.1168	−1.50798
$236$	−6.92820	−0.450988
$237$	0	0
$238$	0	0
$239$	2.17448	0.140656	0.0703278	−	0.997524i	$-0.477595\pi$
$239$	2.17448	0.140656	0.0703278	+	0.997524i	$0.477595\pi$
$240$	0	0
$241$	−25.8614	−1.66588	−0.832940	−	0.553364i	$-0.813344\pi$
$241$	−25.8614	−1.66588	−0.832940	+	0.553364i	$0.813344\pi$
$242$	0.792287	0.0509301
$243$	0	0
$244$	8.23369	0.527108
$245$	0	0
$246$	0	0
$247$	−1.62772	−0.103569
$248$	17.0256	1.08112
$249$	0	0
$250$	7.25544	0.458874
$251$	−25.2434	−1.59335	−0.796674	−	0.604409i	$-0.793409\pi$
$251$	−25.2434	−1.59335	−0.796674	+	0.604409i	$0.793409\pi$
$252$	0	0
$253$	15.2554	0.959101
$254$	1.17981	0.0740282
$255$	0	0
$256$	−13.8832	−0.867697
$257$	−6.63325	−0.413771	−0.206885	−	0.978365i	$-0.566333\pi$
$257$	−6.63325	−0.413771	−0.206885	+	0.978365i	$0.566333\pi$
$258$	0	0
$259$	0	0
$260$	3.46410	0.214834
$261$	0	0
$262$	8.00000	0.494242
$263$	14.5012	0.894183	0.447092	−	0.894488i	$-0.352460\pi$
$263$	14.5012	0.894183	0.447092	+	0.894488i	$0.352460\pi$
$264$	0	0
$265$	2.37228	0.145728
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	19.2000	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$269$	19.2000	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$270$	0	0
$271$	−13.4891	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$271$	−13.4891	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$272$	−0.994667	−0.0603105
$273$	0	0
$274$	−13.4891	−0.814908
$275$	4.75372	0.286660
$276$	0	0
$277$	25.8614	1.55386	0.776931	−	0.629586i	$-0.216776\pi$
$277$	25.8614	1.55386	0.776931	+	0.629586i	$0.216776\pi$
$278$	−3.75906	−0.225453
$279$	0	0
$280$	0	0
$281$	−18.9051	−1.12778	−0.563891	−	0.825849i	$-0.690696\pi$
$281$	−18.9051	−1.12778	−0.563891	+	0.825849i	$0.690696\pi$
$282$	0	0
$283$	22.9783	1.36592	0.682958	−	0.730458i	$-0.260693\pi$
$283$	22.9783	1.36592	0.682958	+	0.730458i	$0.260693\pi$
$284$	−2.17448	−0.129032
$285$	0	0
$286$	2.74456	0.162289
$287$	0	0
$288$	0	0
$289$	−14.4891	−0.852301
$290$	11.9769	0.703307
$291$	0	0
$292$	11.4891	0.672350
$293$	16.3807	0.956973	0.478487	−	0.878095i	$-0.341186\pi$
$293$	16.3807	0.956973	0.478487	+	0.878095i	$0.341186\pi$
$294$	0	0
$295$	−12.7446	−0.742017
$296$	−18.0202	−1.04740
$297$	0	0
$298$	−12.0000	−0.695141
$299$	4.40387	0.254682
$300$	0	0
$301$	0	0
$302$	−6.33830	−0.364728
$303$	0	0
$304$	−1.02175	−0.0586013
$305$	15.1460	0.867259
$306$	0	0
$307$	−19.1168	−1.09106	−0.545528	−	0.838093i	$-0.683671\pi$
$307$	−19.1168	−1.09106	−0.545528	+	0.838093i	$0.683671\pi$
$308$	0	0
$309$	0	0
$310$	12.7446	0.723843
$311$	32.7615	1.85773	0.928867	−	0.370414i	$-0.120784\pi$
$311$	32.7615	1.85773	0.928867	+	0.370414i	$0.120784\pi$
$312$	0	0
$313$	4.51087	0.254970	0.127485	−	0.991841i	$-0.459310\pi$
$313$	4.51087	0.254970	0.127485	+	0.991841i	$0.459310\pi$
$314$	−0.994667	−0.0561323
$315$	0	0
$316$	16.2772	0.915663
$317$	−30.8820	−1.73450	−0.867252	−	0.497870i	$-0.834116\pi$
$317$	−30.8820	−1.73450	−0.867252	+	0.497870i	$0.834116\pi$
$318$	0	0
$319$	−20.7446	−1.16147
$320$	−8.51278	−0.475879
$321$	0	0
$322$	0	0
$323$	2.57924	0.143513
$324$	0	0
$325$	1.37228	0.0761205
$326$	−9.50744	−0.526569
$327$	0	0
$328$	−4.23369	−0.233766
$329$	0	0
$330$	0	0
$331$	8.74456	0.480645	0.240322	−	0.970693i	$-0.422747\pi$
$331$	8.74456	0.480645	0.240322	+	0.970693i	$0.422747\pi$
$332$	−16.9157	−0.928372
$333$	0	0
$334$	10.2337	0.559962
$335$	22.0742	1.20604
$336$	0	0
$337$	−1.11684	−0.0608384	−0.0304192	−	0.999537i	$-0.509684\pi$
$337$	−1.11684	−0.0608384	−0.0304192	+	0.999537i	$0.509684\pi$
$338$	0.792287	0.0430947
$339$	0	0
$340$	−5.48913	−0.297690
$341$	−22.0742	−1.19539
$342$	0	0
$343$	0	0
$344$	−29.7021	−1.60143
$345$	0	0
$346$	−9.25544	−0.497575
$347$	4.05401	0.217631	0.108815	−	0.994062i	$-0.465294\pi$
$347$	4.05401	0.217631	0.108815	+	0.994062i	$0.465294\pi$
$348$	0	0
$349$	−24.3723	−1.30462	−0.652309	−	0.757953i	$-0.726200\pi$
$349$	−24.3723	−1.30462	−0.652309	+	0.757953i	$0.726200\pi$
$350$	0	0
$351$	0	0
$352$	20.2337	1.07846
$353$	−28.1176	−1.49655	−0.748274	−	0.663390i	$-0.769117\pi$
$353$	−28.1176	−1.49655	−0.748274	+	0.663390i	$0.769117\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	3.46410	0.183597
$357$	0	0
$358$	−10.4674	−0.553218
$359$	19.2000	1.01334	0.506670	−	0.862140i	$-0.330876\pi$
$359$	19.2000	1.01334	0.506670	+	0.862140i	$0.330876\pi$
$360$	0	0
$361$	−16.3505	−0.860554
$362$	−9.10268	−0.478426
$363$	0	0
$364$	0	0
$365$	21.1345	1.10623
$366$	0	0
$367$	−14.2337	−0.742992	−0.371496	−	0.928434i	$-0.621155\pi$
$367$	−14.2337	−0.742992	−0.371496	+	0.928434i	$0.621155\pi$
$368$	2.76439	0.144104
$369$	0	0
$370$	−13.4891	−0.701266
$371$	0	0
$372$	0	0
$373$	−0.510875	−0.0264521	−0.0132260	−	0.999913i	$-0.504210\pi$
$373$	−0.510875	−0.0264521	−0.0132260	+	0.999913i	$0.504210\pi$
$374$	−4.34896	−0.224880
$375$	0	0
$376$	−24.4674	−1.26181
$377$	−5.98844	−0.308420
$378$	0	0
$379$	18.2337	0.936602	0.468301	−	0.883569i	$-0.344866\pi$
$379$	18.2337	0.936602	0.468301	+	0.883569i	$0.344866\pi$
$380$	−5.63858	−0.289253
$381$	0	0
$382$	−5.25544	−0.268892
$383$	24.6535	1.25973	0.629867	−	0.776703i	$-0.283109\pi$
$383$	24.6535	1.25973	0.629867	+	0.776703i	$0.283109\pi$
$384$	0	0
$385$	0	0
$386$	−9.10268	−0.463314
$387$	0	0
$388$	4.97825	0.252732
$389$	−26.5330	−1.34528	−0.672638	−	0.739972i	$-0.734839\pi$
$389$	−26.5330	−1.34528	−0.672638	+	0.739972i	$0.734839\pi$
$390$	0	0
$391$	−6.97825	−0.352905
$392$	0	0
$393$	0	0
$394$	9.02175	0.454509
$395$	29.9422	1.50656
$396$	0	0
$397$	−2.13859	−0.107333	−0.0536665	−	0.998559i	$-0.517091\pi$
$397$	−2.13859	−0.107333	−0.0536665	+	0.998559i	$0.517091\pi$
$398$	−12.6766	−0.635420
$399$	0	0
$400$	0.861407	0.0430703
$401$	17.0256	0.850216	0.425108	−	0.905143i	$-0.360236\pi$
$401$	17.0256	0.850216	0.425108	+	0.905143i	$0.360236\pi$
$402$	0	0
$403$	−6.37228	−0.317426
$404$	23.7686	1.18253
$405$	0	0
$406$	0	0
$407$	23.3639	1.15810
$408$	0	0
$409$	−32.0951	−1.58700	−0.793500	−	0.608570i	$-0.791743\pi$
$409$	−32.0951	−1.58700	−0.793500	+	0.608570i	$0.791743\pi$
$410$	−3.16915	−0.156513
$411$	0	0
$412$	−18.5109	−0.911965
$413$	0	0
$414$	0	0
$415$	−31.1168	−1.52747
$416$	5.84096	0.286377
$417$	0	0
$418$	−4.46738	−0.218506
$419$	7.62792	0.372648	0.186324	−	0.982488i	$-0.440343\pi$
$419$	7.62792	0.372648	0.186324	+	0.982488i	$0.440343\pi$
$420$	0	0
$421$	26.7446	1.30345	0.651725	−	0.758455i	$-0.274046\pi$
$421$	26.7446	1.30345	0.651725	+	0.758455i	$0.274046\pi$
$422$	−16.3258	−0.794730
$423$	0	0
$424$	2.51087	0.121939
$425$	−2.17448	−0.105478
$426$	0	0
$427$	0	0
$428$	22.9591	1.10977
$429$	0	0
$430$	−22.2337	−1.07220
$431$	23.6588	1.13960	0.569802	−	0.821782i	$-0.307020\pi$
$431$	23.6588	1.13960	0.569802	+	0.821782i	$0.307020\pi$
$432$	0	0
$433$	−17.7228	−0.851704	−0.425852	−	0.904793i	$-0.640026\pi$
$433$	−17.7228	−0.851704	−0.425852	+	0.904793i	$0.640026\pi$
$434$	0	0
$435$	0	0
$436$	10.2772	0.492188
$437$	−7.16825	−0.342904
$438$	0	0
$439$	4.74456	0.226446	0.113223	−	0.993570i	$-0.463883\pi$
$439$	4.74456	0.226446	0.113223	+	0.993570i	$0.463883\pi$
$440$	23.3639	1.11383
$441$	0	0
$442$	−1.25544	−0.0597151
$443$	32.8164	1.55915	0.779577	−	0.626307i	$-0.215434\pi$
$443$	32.8164	1.55915	0.779577	+	0.626307i	$0.215434\pi$
$444$	0	0
$445$	6.37228	0.302075
$446$	13.1567	0.622987
$447$	0	0
$448$	0	0
$449$	−7.51811	−0.354802	−0.177401	−	0.984139i	$-0.556769\pi$
$449$	−7.51811	−0.354802	−0.177401	+	0.984139i	$0.556769\pi$
$450$	0	0
$451$	5.48913	0.258473
$452$	−8.21782	−0.386534
$453$	0	0
$454$	−9.02175	−0.423412
$455$	0	0
$456$	0	0
$457$	−30.7446	−1.43817	−0.719085	−	0.694922i	$-0.755439\pi$
$457$	−30.7446	−1.43817	−0.719085	+	0.694922i	$0.755439\pi$
$458$	−7.92287	−0.370211
$459$	0	0
$460$	15.2554	0.711288
$461$	36.2256	1.68719	0.843597	−	0.536977i	$-0.180434\pi$
$461$	36.2256	1.68719	0.843597	+	0.536977i	$0.180434\pi$
$462$	0	0
$463$	−27.2554	−1.26667	−0.633334	−	0.773879i	$-0.718314\pi$
$463$	−27.2554	−1.26667	−0.633334	+	0.773879i	$0.718314\pi$
$464$	−3.75906	−0.174510
$465$	0	0
$466$	16.7446	0.775677
$467$	5.04868	0.233625	0.116812	−	0.993154i	$-0.462732\pi$
$467$	5.04868	0.233625	0.116812	+	0.993154i	$0.462732\pi$
$468$	0	0
$469$	0	0
$470$	−18.3152	−0.844816
$471$	0	0
$472$	−13.4891	−0.620887
$473$	38.5099	1.77069
$474$	0	0
$475$	−2.23369	−0.102489
$476$	0	0
$477$	0	0
$478$	1.72281	0.0787996
$479$	−33.1113	−1.51290	−0.756448	−	0.654054i	$-0.773067\pi$
$479$	−33.1113	−1.51290	−0.756448	+	0.654054i	$0.773067\pi$
$480$	0	0
$481$	6.74456	0.307526
$482$	−20.4897	−0.933278
$483$	0	0
$484$	−1.37228	−0.0623764
$485$	9.15759	0.415825
$486$	0	0
$487$	28.7446	1.30254	0.651270	−	0.758846i	$-0.274236\pi$
$487$	28.7446	1.30254	0.651270	+	0.758846i	$0.274236\pi$
$488$	16.0309	0.725684
$489$	0	0
$490$	0	0
$491$	−10.9822	−0.495620	−0.247810	−	0.968809i	$-0.579711\pi$
$491$	−10.9822	−0.495620	−0.247810	+	0.968809i	$0.579711\pi$
$492$	0	0
$493$	9.48913	0.427369
$494$	−1.28962	−0.0580228
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	14.9783	0.670519	0.335259	−	0.942126i	$-0.391176\pi$
$499$	14.9783	0.670519	0.335259	+	0.942126i	$0.391176\pi$
$500$	−12.5668	−0.562004
$501$	0	0
$502$	−20.0000	−0.892644
$503$	−32.1716	−1.43446	−0.717230	−	0.696837i	$-0.754590\pi$
$503$	−32.1716	−1.43446	−0.717230	+	0.696837i	$0.754590\pi$
$504$	0	0
$505$	43.7228	1.94564
$506$	12.0867	0.537319
$507$	0	0
$508$	−2.04350	−0.0906656
$509$	25.8882	1.14747	0.573737	−	0.819040i	$-0.305493\pi$
$509$	25.8882	1.14747	0.573737	+	0.819040i	$0.305493\pi$
$510$	0	0
$511$	0	0
$512$	7.02078	0.310277
$513$	0	0
$514$	−5.25544	−0.231807
$515$	−34.0511	−1.50047
$516$	0	0
$517$	31.7228	1.39517
$518$	0	0
$519$	0	0
$520$	6.74456	0.295769
$521$	26.1282	1.14470	0.572349	−	0.820010i	$-0.306032\pi$
$521$	26.1282	1.14470	0.572349	+	0.820010i	$0.306032\pi$
$522$	0	0
$523$	23.7228	1.03733	0.518663	−	0.854979i	$-0.326430\pi$
$523$	23.7228	1.03733	0.518663	+	0.854979i	$0.326430\pi$
$524$	−13.8564	−0.605320
$525$	0	0
$526$	11.4891	0.500950
$527$	10.0974	0.439848
$528$	0	0
$529$	−3.60597	−0.156781
$530$	1.87953	0.0816415
$531$	0	0
$532$	0	0
$533$	1.58457	0.0686355
$534$	0	0
$535$	42.2337	1.82592
$536$	23.3639	1.00916
$537$	0	0
$538$	15.2119	0.655833
$539$	0	0
$540$	0	0
$541$	28.2337	1.21386	0.606931	−	0.794755i	$-0.292401\pi$
$541$	28.2337	1.21386	0.606931	+	0.794755i	$0.292401\pi$
$542$	−10.6873	−0.459057
$543$	0	0
$544$	−9.25544	−0.396824
$545$	18.9051	0.809805
$546$	0	0
$547$	−39.8614	−1.70435	−0.852175	−	0.523256i	$-0.824717\pi$
$547$	−39.8614	−1.70435	−0.852175	+	0.523256i	$0.824717\pi$
$548$	23.3639	0.998054
$549$	0	0
$550$	3.76631	0.160596
$551$	9.74749	0.415257
$552$	0	0
$553$	0	0
$554$	20.4897	0.870522
$555$	0	0
$556$	6.51087	0.276123
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	11.1168	0.470192
$560$	0	0
$561$	0	0
$562$	−14.9783	−0.631819
$563$	17.0256	0.717542	0.358771	−	0.933426i	$-0.383196\pi$
$563$	17.0256	0.717542	0.358771	+	0.933426i	$0.383196\pi$
$564$	0	0
$565$	−15.1168	−0.635970
$566$	18.2054	0.765229
$567$	0	0
$568$	−4.23369	−0.177642
$569$	28.6526	1.20118	0.600589	−	0.799558i	$-0.294933\pi$
$569$	28.6526	1.20118	0.600589	+	0.799558i	$0.294933\pi$
$570$	0	0
$571$	3.11684	0.130436	0.0652179	−	0.997871i	$-0.479226\pi$
$571$	3.11684	0.130436	0.0652179	+	0.997871i	$0.479226\pi$
$572$	−4.75372	−0.198763
$573$	0	0
$574$	0	0
$575$	6.04334	0.252025
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−11.4795	−0.477486
$579$	0	0
$580$	−20.7446	−0.861371
$581$	0	0
$582$	0	0
$583$	−3.25544	−0.134826
$584$	22.3692	0.925643
$585$	0	0
$586$	12.9783	0.536127
$587$	−23.0140	−0.949889	−0.474945	−	0.880016i	$-0.657532\pi$
$587$	−23.0140	−0.949889	−0.474945	+	0.880016i	$0.657532\pi$
$588$	0	0
$589$	10.3723	0.427382
$590$	−10.0974	−0.415701
$591$	0	0
$592$	4.23369	0.174004
$593$	35.3956	1.45352	0.726762	−	0.686889i	$-0.241024\pi$
$593$	35.3956	1.45352	0.726762	+	0.686889i	$0.241024\pi$
$594$	0	0
$595$	0	0
$596$	20.7846	0.851371
$597$	0	0
$598$	3.48913	0.142681
$599$	22.1291	0.904172	0.452086	−	0.891974i	$-0.350680\pi$
$599$	22.1291	0.904172	0.452086	+	0.891974i	$0.350680\pi$
$600$	0	0
$601$	−12.9783	−0.529394	−0.264697	−	0.964332i	$-0.585272\pi$
$601$	−12.9783	−0.529394	−0.264697	+	0.964332i	$0.585272\pi$
$602$	0	0
$603$	0	0
$604$	10.9783	0.446699
$605$	−2.52434	−0.102629
$606$	0	0
$607$	−6.23369	−0.253018	−0.126509	−	0.991965i	$-0.540377\pi$
$607$	−6.23369	−0.253018	−0.126509	+	0.991965i	$0.540377\pi$
$608$	−9.50744	−0.385578
$609$	0	0
$610$	12.0000	0.485866
$611$	9.15759	0.370476
$612$	0	0
$613$	4.23369	0.170997	0.0854985	−	0.996338i	$-0.472752\pi$
$613$	4.23369	0.170997	0.0854985	+	0.996338i	$0.472752\pi$
$614$	−15.1460	−0.611244
$615$	0	0
$616$	0	0
$617$	−18.9051	−0.761090	−0.380545	−	0.924762i	$-0.624264\pi$
$617$	−18.9051	−0.761090	−0.380545	+	0.924762i	$0.624264\pi$
$618$	0	0
$619$	−1.48913	−0.0598530	−0.0299265	−	0.999552i	$-0.509527\pi$
$619$	−1.48913	−0.0598530	−0.0299265	+	0.999552i	$0.509527\pi$
$620$	−22.0742	−0.886522
$621$	0	0
$622$	25.9565	1.04076
$623$	0	0
$624$	0	0
$625$	−29.9783	−1.19913
$626$	3.57391	0.142842
$627$	0	0
$628$	1.72281	0.0687477
$629$	−10.6873	−0.426129
$630$	0	0
$631$	−12.7446	−0.507353	−0.253677	−	0.967289i	$-0.581640\pi$
$631$	−12.7446	−0.507353	−0.253677	+	0.967289i	$0.581640\pi$
$632$	31.6915	1.26062
$633$	0	0
$634$	−24.4674	−0.971724
$635$	−3.75906	−0.149174
$636$	0	0
$637$	0	0
$638$	−16.4356	−0.650694
$639$	0	0
$640$	22.7446	0.899058
$641$	−6.57835	−0.259829	−0.129915	−	0.991525i	$-0.541470\pi$
$641$	−6.57835	−0.259829	−0.129915	+	0.991525i	$0.541470\pi$
$642$	0	0
$643$	21.4891	0.847448	0.423724	−	0.905791i	$-0.360723\pi$
$643$	21.4891	0.847448	0.423724	+	0.905791i	$0.360723\pi$
$644$	0	0
$645$	0	0
$646$	2.04350	0.0804004
$647$	7.62792	0.299884	0.149942	−	0.988695i	$-0.452091\pi$
$647$	7.62792	0.299884	0.149942	+	0.988695i	$0.452091\pi$
$648$	0	0
$649$	17.4891	0.686508
$650$	1.08724	0.0426451
$651$	0	0
$652$	16.4674	0.644912
$653$	−47.9075	−1.87477	−0.937383	−	0.348300i	$-0.886759\pi$
$653$	−47.9075	−1.87477	−0.937383	+	0.348300i	$0.886759\pi$
$654$	0	0
$655$	−25.4891	−0.995943
$656$	0.994667	0.0388352
$657$	0	0
$658$	0	0
$659$	33.4063	1.30132	0.650662	−	0.759367i	$-0.274491\pi$
$659$	33.4063	1.30132	0.650662	+	0.759367i	$0.274491\pi$
$660$	0	0
$661$	−22.8832	−0.890052	−0.445026	−	0.895518i	$-0.646805\pi$
$661$	−22.8832	−0.890052	−0.445026	+	0.895518i	$0.646805\pi$
$662$	6.92820	0.269272
$663$	0	0
$664$	−32.9348	−1.27812
$665$	0	0
$666$	0	0
$667$	−26.3723	−1.02114
$668$	−17.7253	−0.685811
$669$	0	0
$670$	17.4891	0.675664
$671$	−20.7846	−0.802381
$672$	0	0
$673$	−36.3723	−1.40205	−0.701024	−	0.713137i	$-0.747274\pi$
$673$	−36.3723	−1.40205	−0.701024	+	0.713137i	$0.747274\pi$
$674$	−0.884861	−0.0340836
$675$	0	0
$676$	−1.37228	−0.0527801
$677$	47.0227	1.80723	0.903614	−	0.428348i	$-0.140904\pi$
$677$	47.0227	1.80723	0.903614	+	0.428348i	$0.140904\pi$
$678$	0	0
$679$	0	0
$680$	−10.6873	−0.409838
$681$	0	0
$682$	−17.4891	−0.669693
$683$	−15.4410	−0.590833	−0.295416	−	0.955369i	$-0.595458\pi$
$683$	−15.4410	−0.590833	−0.295416	+	0.955369i	$0.595458\pi$
$684$	0	0
$685$	42.9783	1.64211
$686$	0	0
$687$	0	0
$688$	6.97825	0.266043
$689$	−0.939764	−0.0358022
$690$	0	0
$691$	23.1168	0.879406	0.439703	−	0.898143i	$-0.355084\pi$
$691$	23.1168	0.879406	0.439703	+	0.898143i	$0.355084\pi$
$692$	16.0309	0.609403
$693$	0	0
$694$	3.21194	0.121924
$695$	11.9769	0.454309
$696$	0	0
$697$	−2.51087	−0.0951062
$698$	−19.3098	−0.730888
$699$	0	0
$700$	0	0
$701$	−14.2063	−0.536563	−0.268282	−	0.963341i	$-0.586456\pi$
$701$	−14.2063	−0.536563	−0.268282	+	0.963341i	$0.586456\pi$
$702$	0	0
$703$	−10.9783	−0.414053
$704$	11.6819	0.440279
$705$	0	0
$706$	−22.2772	−0.838413
$707$	0	0
$708$	0	0
$709$	−34.4674	−1.29445	−0.647225	−	0.762299i	$-0.724070\pi$
$709$	−34.4674	−1.29445	−0.647225	+	0.762299i	$0.724070\pi$
$710$	−3.16915	−0.118936
$711$	0	0
$712$	6.74456	0.252763
$713$	−28.0627	−1.05096
$714$	0	0
$715$	−8.74456	−0.327028
$716$	18.1300	0.677551
$717$	0	0
$718$	15.2119	0.567705
$719$	52.3663	1.95293	0.976466	−	0.215669i	$-0.0691934\pi$
$719$	52.3663	1.95293	0.976466	+	0.215669i	$0.0691934\pi$
$720$	0	0
$721$	0	0
$722$	−12.9543	−0.482110
$723$	0	0
$724$	15.7663	0.585950
$725$	−8.21782	−0.305202
$726$	0	0
$727$	15.2554	0.565793	0.282896	−	0.959150i	$-0.408705\pi$
$727$	15.2554	0.565793	0.282896	+	0.959150i	$0.408705\pi$
$728$	0	0
$729$	0	0
$730$	16.7446	0.619744
$731$	−17.6155	−0.651531
$732$	0	0
$733$	−26.8832	−0.992952	−0.496476	−	0.868050i	$-0.665373\pi$
$733$	−26.8832	−0.992952	−0.496476	+	0.868050i	$0.665373\pi$
$734$	−11.2772	−0.416248
$735$	0	0
$736$	25.7228	0.948155
$737$	−30.2921	−1.11582
$738$	0	0
$739$	15.2554	0.561180	0.280590	−	0.959828i	$-0.409470\pi$
$739$	15.2554	0.561180	0.280590	+	0.959828i	$0.409470\pi$
$740$	23.3639	0.858872
$741$	0	0
$742$	0	0
$743$	−22.3692	−0.820646	−0.410323	−	0.911940i	$-0.634584\pi$
$743$	−22.3692	−0.820646	−0.410323	+	0.911940i	$0.634584\pi$
$744$	0	0
$745$	38.2337	1.40077
$746$	−0.404759	−0.0148193
$747$	0	0
$748$	7.53262	0.275420
$749$	0	0
$750$	0	0
$751$	−18.3723	−0.670414	−0.335207	−	0.942144i	$-0.608806\pi$
$751$	−18.3723	−0.670414	−0.335207	+	0.942144i	$0.608806\pi$
$752$	5.74839	0.209622
$753$	0	0
$754$	−4.74456	−0.172787
$755$	20.1947	0.734960
$756$	0	0
$757$	20.3723	0.740443	0.370222	−	0.928943i	$-0.379282\pi$
$757$	20.3723	0.740443	0.370222	+	0.928943i	$0.379282\pi$
$758$	14.4463	0.524714
$759$	0	0
$760$	−10.9783	−0.398223
$761$	−25.2983	−0.917062	−0.458531	−	0.888678i	$-0.651624\pi$
$761$	−25.2983	−0.917062	−0.458531	+	0.888678i	$0.651624\pi$
$762$	0	0
$763$	0	0
$764$	9.10268	0.329324
$765$	0	0
$766$	19.5326	0.705742
$767$	5.04868	0.182297
$768$	0	0
$769$	0.372281	0.0134248	0.00671240	−	0.999977i	$-0.497863\pi$
$769$	0.372281	0.0134248	0.00671240	+	0.999977i	$0.497863\pi$
$770$	0	0
$771$	0	0
$772$	15.7663	0.567442
$773$	−45.7330	−1.64490	−0.822451	−	0.568835i	$-0.807394\pi$
$773$	−45.7330	−1.64490	−0.822451	+	0.568835i	$0.807394\pi$
$774$	0	0
$775$	−8.74456	−0.314114
$776$	9.69259	0.347944
$777$	0	0
$778$	−21.0217	−0.753666
$779$	−2.57924	−0.0924109
$780$	0	0
$781$	5.48913	0.196416
$782$	−5.52878	−0.197709
$783$	0	0
$784$	0	0
$785$	3.16915	0.113112
$786$	0	0
$787$	0.883156	0.0314811	0.0157406	−	0.999876i	$-0.494989\pi$
$787$	0.883156	0.0314811	0.0157406	+	0.999876i	$0.494989\pi$
$788$	−15.6261	−0.556658
$789$	0	0
$790$	23.7228	0.844020
$791$	0	0
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−1.69438	−0.0601313
$795$	0	0
$796$	21.9565	0.778228
$797$	−9.80240	−0.347219	−0.173609	−	0.984815i	$-0.555543\pi$
$797$	−9.80240	−0.347219	−0.173609	+	0.984815i	$0.555543\pi$
$798$	0	0
$799$	−14.5109	−0.513358
$800$	8.01544	0.283389
$801$	0	0
$802$	13.4891	0.476318
$803$	−29.0024	−1.02347
$804$	0	0
$805$	0	0
$806$	−5.04868	−0.177832
$807$	0	0
$808$	46.2772	1.62803
$809$	−42.6188	−1.49840	−0.749198	−	0.662346i	$-0.769561\pi$
$809$	−42.6188	−1.49840	−0.749198	+	0.662346i	$0.769561\pi$
$810$	0	0
$811$	28.4674	0.999625	0.499812	−	0.866134i	$-0.333402\pi$
$811$	28.4674	0.999625	0.499812	+	0.866134i	$0.333402\pi$
$812$	0	0
$813$	0	0
$814$	18.5109	0.648806
$815$	30.2921	1.06108
$816$	0	0
$817$	−18.0951	−0.633067
$818$	−25.4285	−0.889088
$819$	0	0
$820$	5.48913	0.191689
$821$	25.2434	0.881000	0.440500	−	0.897753i	$-0.354801\pi$
$821$	25.2434	0.881000	0.440500	+	0.897753i	$0.354801\pi$
$822$	0	0
$823$	18.9783	0.661540	0.330770	−	0.943711i	$-0.392692\pi$
$823$	18.9783	0.661540	0.330770	+	0.943711i	$0.392692\pi$
$824$	−36.0404	−1.25553
$825$	0	0
$826$	0	0
$827$	9.10268	0.316531	0.158266	−	0.987397i	$-0.449410\pi$
$827$	9.10268	0.316531	0.158266	+	0.987397i	$0.449410\pi$
$828$	0	0
$829$	−30.4674	−1.05818	−0.529088	−	0.848567i	$-0.677466\pi$
$829$	−30.4674	−1.05818	−0.529088	+	0.848567i	$0.677466\pi$
$830$	−24.6535	−0.855734
$831$	0	0
$832$	3.37228	0.116913
$833$	0	0
$834$	0	0
$835$	−32.6060	−1.12838
$836$	7.73772	0.267615
$837$	0	0
$838$	6.04350	0.208769
$839$	−39.0998	−1.34987	−0.674937	−	0.737875i	$-0.735829\pi$
$839$	−39.0998	−1.34987	−0.674937	+	0.737875i	$0.735829\pi$
$840$	0	0
$841$	6.86141	0.236600
$842$	21.1894	0.730234
$843$	0	0
$844$	28.2772	0.973341
$845$	−2.52434	−0.0868399
$846$	0	0
$847$	0	0
$848$	−0.589907	−0.0202575
$849$	0	0
$850$	−1.72281	−0.0590920
$851$	29.7021	1.01818
$852$	0	0
$853$	53.5842	1.83469	0.917344	−	0.398095i	$-0.130328\pi$
$853$	53.5842	1.83469	0.917344	+	0.398095i	$0.130328\pi$
$854$	0	0
$855$	0	0
$856$	44.7011	1.52785
$857$	−0.294954	−0.0100754	−0.00503771	−	0.999987i	$-0.501604\pi$
$857$	−0.294954	−0.0100754	−0.00503771	+	0.999987i	$0.501604\pi$
$858$	0	0
$859$	−16.7446	−0.571317	−0.285659	−	0.958331i	$-0.592212\pi$
$859$	−16.7446	−0.571317	−0.285659	+	0.958331i	$0.592212\pi$
$860$	38.5099	1.31318
$861$	0	0
$862$	18.7446	0.638442
$863$	−45.7330	−1.55677	−0.778385	−	0.627787i	$-0.783961\pi$
$863$	−45.7330	−1.55677	−0.778385	+	0.627787i	$0.783961\pi$
$864$	0	0
$865$	29.4891	1.00266
$866$	−14.0416	−0.477151
$867$	0	0
$868$	0	0
$869$	−41.0891	−1.39385
$870$	0	0
$871$	−8.74456	−0.296298
$872$	20.0096	0.677609
$873$	0	0
$874$	−5.67931	−0.192106
$875$	0	0
$876$	0	0
$877$	2.74456	0.0926773	0.0463386	−	0.998926i	$-0.485245\pi$
$877$	2.74456	0.0926773	0.0463386	+	0.998926i	$0.485245\pi$
$878$	3.75906	0.126862
$879$	0	0
$880$	−5.48913	−0.185038
$881$	−9.80240	−0.330251	−0.165126	−	0.986273i	$-0.552803\pi$
$881$	−9.80240	−0.330251	−0.165126	+	0.986273i	$0.552803\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	2.17448	0.0731357
$885$	0	0
$886$	26.0000	0.873487
$887$	−10.6873	−0.358843	−0.179422	−	0.983772i	$-0.557423\pi$
$887$	−10.6873	−0.358843	−0.179422	+	0.983772i	$0.557423\pi$
$888$	0	0
$889$	0	0
$890$	5.04868	0.169232
$891$	0	0
$892$	−22.7881	−0.763001
$893$	−14.9060	−0.498809
$894$	0	0
$895$	33.3505	1.11479
$896$	0	0
$897$	0	0
$898$	−5.95650	−0.198771
$899$	38.1600	1.27271
$900$	0	0
$901$	1.48913	0.0496100
$902$	4.34896	0.144805
$903$	0	0
$904$	−16.0000	−0.532152
$905$	29.0024	0.964074
$906$	0	0
$907$	−8.13859	−0.270238	−0.135119	−	0.990829i	$-0.543142\pi$
$907$	−8.13859	−0.270238	−0.135119	+	0.990829i	$0.543142\pi$
$908$	15.6261	0.518571
$909$	0	0
$910$	0	0
$911$	−8.86263	−0.293632	−0.146816	−	0.989164i	$-0.546903\pi$
$911$	−8.86263	−0.293632	−0.146816	+	0.989164i	$0.546903\pi$
$912$	0	0
$913$	42.7011	1.41320
$914$	−24.3585	−0.805708
$915$	0	0
$916$	13.7228	0.453415
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	29.7021	0.979251
$921$	0	0
$922$	28.7011	0.945219
$923$	1.58457	0.0521569
$924$	0	0
$925$	9.25544	0.304317
$926$	−21.5941	−0.709627
$927$	0	0
$928$	−34.9783	−1.14822
$929$	3.81396	0.125132	0.0625660	−	0.998041i	$-0.480072\pi$
$929$	3.81396	0.125132	0.0625660	+	0.998041i	$0.480072\pi$
$930$	0	0
$931$	0	0
$932$	−29.0024	−0.950006
$933$	0	0
$934$	4.00000	0.130884
$935$	13.8564	0.453153
$936$	0	0
$937$	50.7446	1.65775	0.828876	−	0.559432i	$-0.188981\pi$
$937$	50.7446	1.65775	0.828876	+	0.559432i	$0.188981\pi$
$938$	0	0
$939$	0	0
$940$	31.7228	1.03468
$941$	−32.8164	−1.06978	−0.534892	−	0.844921i	$-0.679648\pi$
$941$	−32.8164	−1.06978	−0.534892	+	0.844921i	$0.679648\pi$
$942$	0	0
$943$	6.97825	0.227243
$944$	3.16915	0.103147
$945$	0	0
$946$	30.5109	0.991994
$947$	−36.9253	−1.19991	−0.599956	−	0.800033i	$-0.704815\pi$
$947$	−36.9253	−1.19991	−0.599956	+	0.800033i	$0.704815\pi$
$948$	0	0
$949$	−8.37228	−0.271776
$950$	−1.76972	−0.0574174
$951$	0	0
$952$	0	0
$953$	24.3036	0.787271	0.393636	−	0.919267i	$-0.371217\pi$
$953$	24.3036	0.787271	0.393636	+	0.919267i	$0.371217\pi$
$954$	0	0
$955$	16.7446	0.541841
$956$	−2.98400	−0.0965095
$957$	0	0
$958$	−26.2337	−0.847572
$959$	0	0
$960$	0	0
$961$	9.60597	0.309870
$962$	5.34363	0.172286
$963$	0	0
$964$	35.4891	1.14303
$965$	29.0024	0.933621
$966$	0	0
$967$	−55.7228	−1.79192	−0.895962	−	0.444130i	$-0.853513\pi$
$967$	−55.7228	−1.79192	−0.895962	+	0.444130i	$0.853513\pi$
$968$	−2.67181	−0.0858754
$969$	0	0
$970$	7.25544	0.232958
$971$	−24.6535	−0.791168	−0.395584	−	0.918430i	$-0.629458\pi$
$971$	−24.6535	−0.791168	−0.395584	+	0.918430i	$0.629458\pi$
$972$	0	0
$973$	0	0
$974$	22.7739	0.729724
$975$	0	0
$976$	−3.76631	−0.120557
$977$	−7.62792	−0.244039	−0.122019	−	0.992528i	$-0.538937\pi$
$977$	−7.62792	−0.244039	−0.122019	+	0.992528i	$0.538937\pi$
$978$	0	0
$979$	−8.74456	−0.279477
$980$	0	0
$981$	0	0
$982$	−8.70106	−0.277662
$983$	33.7013	1.07490	0.537452	−	0.843295i	$-0.319387\pi$
$983$	33.7013	1.07490	0.537452	+	0.843295i	$0.319387\pi$
$984$	0	0
$985$	−28.7446	−0.915878
$986$	7.51811	0.239425
$987$	0	0
$988$	2.23369	0.0710631
$989$	48.9571	1.55674
$990$	0	0
$991$	52.4674	1.66668	0.833341	−	0.552760i	$-0.186425\pi$
$991$	52.4674	1.66668	0.833341	+	0.552760i	$0.186425\pi$
$992$	−37.2203	−1.18174
$993$	0	0
$994$	0	0
$995$	40.3894	1.28043
$996$	0	0
$997$	−1.25544	−0.0397601	−0.0198800	−	0.999802i	$-0.506328\pi$
$997$	−1.25544	−0.0397601	−0.0198800	+	0.999802i	$0.506328\pi$
$998$	11.8671	0.375646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bi.1.3		4
3.2	odd	2	inner	5733.2.a.bi.1.2		4
7.6	odd	2		819.2.a.l.1.3	yes	4
21.20	even	2		819.2.a.l.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.a.l.1.2	✓	4	21.20	even	2
819.2.a.l.1.3	yes	4	7.6	odd	2
5733.2.a.bi.1.2		4	3.2	odd	2	inner
5733.2.a.bi.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} -2.67181 q^{8} +O(q^{10})\)	\(q+0.792287 q^{2} -1.37228 q^{4} -2.52434 q^{5} -2.67181 q^{8} -2.00000 q^{10} +3.46410 q^{11} +1.00000 q^{13} +0.627719 q^{16} -1.58457 q^{17} -1.62772 q^{19} +3.46410 q^{20} +2.74456 q^{22} +4.40387 q^{23} +1.37228 q^{25} +0.792287 q^{26} -5.98844 q^{29} -6.37228 q^{31} +5.84096 q^{32} -1.25544 q^{34} +6.74456 q^{37} -1.28962 q^{38} +6.74456 q^{40} +1.58457 q^{41} +11.1168 q^{43} -4.75372 q^{44} +3.48913 q^{46} +9.15759 q^{47} +1.08724 q^{50} -1.37228 q^{52} -0.939764 q^{53} -8.74456 q^{55} -4.74456 q^{58} +5.04868 q^{59} -6.00000 q^{61} -5.04868 q^{62} +3.37228 q^{64} -2.52434 q^{65} -8.74456 q^{67} +2.17448 q^{68} +1.58457 q^{71} -8.37228 q^{73} +5.34363 q^{74} +2.23369 q^{76} -11.8614 q^{79} -1.58457 q^{80} +1.25544 q^{82} +12.3267 q^{83} +4.00000 q^{85} +8.80773 q^{86} -9.25544 q^{88} -2.52434 q^{89} -6.04334 q^{92} +7.25544 q^{94} +4.10891 q^{95} -3.62772 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} - 8 q^{10} + 4 q^{13} + 14 q^{16} - 18 q^{19} - 12 q^{22} - 6 q^{25} - 14 q^{31} - 28 q^{34} + 4 q^{37} + 4 q^{40} + 10 q^{43} - 32 q^{46} + 6 q^{52} - 12 q^{55} + 4 q^{58} - 24 q^{61} + 2 q^{64} - 12 q^{67} - 22 q^{73} - 60 q^{76} + 10 q^{79} + 28 q^{82} + 16 q^{85} - 60 q^{88} + 52 q^{94} - 26 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 - 8 * q^10 + 4 * q^13 + 14 * q^16 - 18 * q^19 - 12 * q^22 - 6 * q^25 - 14 * q^31 - 28 * q^34 + 4 * q^37 + 4 * q^40 + 10 * q^43 - 32 * q^46 + 6 * q^52 - 12 * q^55 + 4 * q^58 - 24 * q^61 + 2 * q^64 - 12 * q^67 - 22 * q^73 - 60 * q^76 + 10 * q^79 + 28 * q^82 + 16 * q^85 - 60 * q^88 + 52 * q^94 - 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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