LMFDB - Embedded newform 5733.2.a.br.1.5

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:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.35100$ 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)

$f(q)$	$=$	$q+1.17619 q^{2} -0.616586 q^{4} -3.14862 q^{5} -3.07759 q^{8} +O(q^{10})$	$q+1.17619 q^{2} -0.616586 q^{4} -3.14862 q^{5} -3.07759 q^{8} -3.70337 q^{10} +0.773390 q^{11} +1.00000 q^{13} -2.38665 q^{16} +5.75340 q^{17} -1.22298 q^{19} +1.94140 q^{20} +0.909650 q^{22} +2.99182 q^{23} +4.91383 q^{25} +1.17619 q^{26} +2.46882 q^{29} -6.13487 q^{31} +3.34804 q^{32} +6.76707 q^{34} +4.99933 q^{37} -1.43845 q^{38} +9.69018 q^{40} +2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} +3.51894 q^{46} -5.37169 q^{47} +5.77958 q^{50} -0.616586 q^{52} +9.79015 q^{53} -2.43511 q^{55} +2.90379 q^{58} -2.50456 q^{59} -10.9167 q^{61} -7.21575 q^{62} +8.71122 q^{64} -3.14862 q^{65} +4.32518 q^{67} -3.54746 q^{68} -10.6649 q^{71} -5.17450 q^{73} +5.88014 q^{74} +0.754072 q^{76} -0.542038 q^{79} +7.51467 q^{80} +3.01081 q^{82} -15.2259 q^{83} -18.1153 q^{85} -3.21275 q^{86} -2.38018 q^{88} -9.23208 q^{89} -1.84471 q^{92} -6.31811 q^{94} +3.85070 q^{95} -1.26291 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 4 * q^4 - 6 * q^5	$ 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.17619	0.831689	0.415845	−	0.909436i	$-0.363486\pi$
$2$	1.17619	0.831689	0.415845	+	0.909436i	$0.363486\pi$
$3$	0	0
$3$	0	0
$4$	−0.616586	−0.308293
$5$	−3.14862	−1.40811	−0.704054	−	0.710147i	$-0.748629\pi$
$5$	−3.14862	−1.40811	−0.704054	+	0.710147i	$0.748629\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.07759	−1.08809
$9$	0	0
$10$	−3.70337	−1.17111
$11$	0.773390	0.233186	0.116593	−	0.993180i	$-0.462803\pi$
$11$	0.773390	0.233186	0.116593	+	0.993180i	$0.462803\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−2.38665	−0.596663
$17$	5.75340	1.39540	0.697702	−	0.716388i	$-0.254206\pi$
$17$	5.75340	1.39540	0.697702	+	0.716388i	$0.254206\pi$
$18$	0	0
$19$	−1.22298	−0.280571	−0.140285	−	0.990111i	$-0.544802\pi$
$19$	−1.22298	−0.280571	−0.140285	+	0.990111i	$0.544802\pi$
$20$	1.94140	0.434109
$21$	0	0
$22$	0.909650	0.193938
$23$	2.99182	0.623838	0.311919	−	0.950109i	$-0.399028\pi$
$23$	2.99182	0.623838	0.311919	+	0.950109i	$0.399028\pi$
$24$	0	0
$25$	4.91383	0.982767
$26$	1.17619	0.230669
$27$	0	0
$28$	0	0
$29$	2.46882	0.458449	0.229224	−	0.973374i	$-0.426381\pi$
$29$	2.46882	0.458449	0.229224	+	0.973374i	$0.426381\pi$
$30$	0	0
$31$	−6.13487	−1.10185	−0.550927	−	0.834553i	$-0.685726\pi$
$31$	−6.13487	−1.10185	−0.550927	+	0.834553i	$0.685726\pi$
$32$	3.34804	0.591855
$33$	0	0
$34$	6.76707	1.16054
$35$	0	0
$36$	0	0
$37$	4.99933	0.821884	0.410942	−	0.911661i	$-0.365200\pi$
$37$	4.99933	0.821884	0.410942	+	0.911661i	$0.365200\pi$
$38$	−1.43845	−0.233348
$39$	0	0
$40$	9.69018	1.53215
$41$	2.55981	0.399774	0.199887	−	0.979819i	$-0.435942\pi$
$41$	2.55981	0.399774	0.199887	+	0.979819i	$0.435942\pi$
$42$	0	0
$43$	−2.73150	−0.416550	−0.208275	−	0.978070i	$-0.566785\pi$
$43$	−2.73150	−0.416550	−0.208275	+	0.978070i	$0.566785\pi$
$44$	−0.476861	−0.0718895
$45$	0	0
$46$	3.51894	0.518839
$47$	−5.37169	−0.783542	−0.391771	−	0.920063i	$-0.628137\pi$
$47$	−5.37169	−0.783542	−0.391771	+	0.920063i	$0.628137\pi$
$48$	0	0
$49$	0	0
$50$	5.77958	0.817357
$51$	0	0
$52$	−0.616586	−0.0855050
$53$	9.79015	1.34478	0.672390	−	0.740197i	$-0.265268\pi$
$53$	9.79015	1.34478	0.672390	+	0.740197i	$0.265268\pi$
$54$	0	0
$55$	−2.43511	−0.328351
$56$	0	0
$57$	0	0
$58$	2.90379	0.381287
$59$	−2.50456	−0.326066	−0.163033	−	0.986621i	$-0.552128\pi$
$59$	−2.50456	−0.326066	−0.163033	+	0.986621i	$0.552128\pi$
$60$	0	0
$61$	−10.9167	−1.39774	−0.698870	−	0.715248i	$-0.746314\pi$
$61$	−10.9167	−1.39774	−0.698870	+	0.715248i	$0.746314\pi$
$62$	−7.21575	−0.916401
$63$	0	0
$64$	8.71122	1.08890
$65$	−3.14862	−0.390539
$66$	0	0
$67$	4.32518	0.528404	0.264202	−	0.964467i	$-0.414891\pi$
$67$	4.32518	0.528404	0.264202	+	0.964467i	$0.414891\pi$
$68$	−3.54746	−0.430193
$69$	0	0
$70$	0	0
$71$	−10.6649	−1.26570	−0.632848	−	0.774276i	$-0.718114\pi$
$71$	−10.6649	−1.26570	−0.632848	+	0.774276i	$0.718114\pi$
$72$	0	0
$73$	−5.17450	−0.605630	−0.302815	−	0.953049i	$-0.597926\pi$
$73$	−5.17450	−0.605630	−0.302815	+	0.953049i	$0.597926\pi$
$74$	5.88014	0.683552
$75$	0	0
$76$	0.754072	0.0864979
$77$	0	0
$78$	0	0
$79$	−0.542038	−0.0609841	−0.0304920	−	0.999535i	$-0.509707\pi$
$79$	−0.542038	−0.0609841	−0.0304920	+	0.999535i	$0.509707\pi$
$80$	7.51467	0.840165
$81$	0	0
$82$	3.01081	0.332488
$83$	−15.2259	−1.67125	−0.835627	−	0.549297i	$-0.814896\pi$
$83$	−15.2259	−1.67125	−0.835627	+	0.549297i	$0.814896\pi$
$84$	0	0
$85$	−18.1153	−1.96488
$86$	−3.21275	−0.346440
$87$	0	0
$88$	−2.38018	−0.253728
$89$	−9.23208	−0.978599	−0.489299	−	0.872116i	$-0.662747\pi$
$89$	−9.23208	−0.978599	−0.489299	+	0.872116i	$0.662747\pi$
$90$	0	0
$91$	0	0
$92$	−1.84471	−0.192325
$93$	0	0
$94$	−6.31811	−0.651663
$95$	3.85070	0.395074
$96$	0	0
$97$	−1.26291	−0.128229	−0.0641145	−	0.997943i	$-0.520422\pi$
$97$	−1.26291	−0.128229	−0.0641145	+	0.997943i	$0.520422\pi$
$98$	0	0
$99$	0	0
$100$	−3.02980	−0.302980
$101$	−0.605447	−0.0602443	−0.0301221	−	0.999546i	$-0.509590\pi$
$101$	−0.605447	−0.0602443	−0.0301221	+	0.999546i	$0.509590\pi$
$102$	0	0
$103$	7.64804	0.753584	0.376792	−	0.926298i	$-0.377027\pi$
$103$	7.64804	0.753584	0.376792	+	0.926298i	$0.377027\pi$
$104$	−3.07759	−0.301783
$105$	0	0
$106$	11.5150	1.11844
$107$	−4.82965	−0.466900	−0.233450	−	0.972369i	$-0.575001\pi$
$107$	−4.82965	−0.466900	−0.233450	+	0.972369i	$0.575001\pi$
$108$	0	0
$109$	7.23092	0.692596	0.346298	−	0.938125i	$-0.387439\pi$
$109$	7.23092	0.692596	0.346298	+	0.938125i	$0.387439\pi$
$110$	−2.86415	−0.273086
$111$	0	0
$112$	0	0
$113$	9.19375	0.864875	0.432438	−	0.901664i	$-0.357654\pi$
$113$	9.19375	0.864875	0.432438	+	0.901664i	$0.357654\pi$
$114$	0	0
$115$	−9.42012	−0.878430
$116$	−1.52224	−0.141336
$117$	0	0
$118$	−2.94583	−0.271186
$119$	0	0
$120$	0	0
$121$	−10.4019	−0.945624
$122$	−12.8401	−1.16249
$123$	0	0
$124$	3.78267	0.339694
$125$	0.271305	0.0242663
$126$	0	0
$127$	−11.2118	−0.994888	−0.497444	−	0.867496i	$-0.665728\pi$
$127$	−11.2118	−0.994888	−0.497444	+	0.867496i	$0.665728\pi$
$128$	3.54994	0.313773
$129$	0	0
$130$	−3.70337	−0.324807
$131$	−15.7380	−1.37503	−0.687517	−	0.726169i	$-0.741299\pi$
$131$	−15.7380	−1.37503	−0.687517	+	0.726169i	$0.741299\pi$
$132$	0	0
$133$	0	0
$134$	5.08721	0.439468
$135$	0	0
$136$	−17.7066	−1.51833
$137$	18.6210	1.59090	0.795449	−	0.606020i	$-0.207235\pi$
$137$	18.6210	1.59090	0.795449	+	0.606020i	$0.207235\pi$
$138$	0	0
$139$	−11.9137	−1.01050	−0.505252	−	0.862972i	$-0.668601\pi$
$139$	−11.9137	−1.01050	−0.505252	+	0.862972i	$0.668601\pi$
$140$	0	0
$141$	0	0
$142$	−12.5440	−1.05267
$143$	0.773390	0.0646741
$144$	0	0
$145$	−7.77339	−0.645545
$146$	−6.08618	−0.503696
$147$	0	0
$148$	−3.08251	−0.253381
$149$	−11.3753	−0.931904	−0.465952	−	0.884810i	$-0.654288\pi$
$149$	−11.3753	−0.931904	−0.465952	+	0.884810i	$0.654288\pi$
$150$	0	0
$151$	−3.69250	−0.300492	−0.150246	−	0.988649i	$-0.548007\pi$
$151$	−3.69250	−0.300492	−0.150246	+	0.988649i	$0.548007\pi$
$152$	3.76383	0.305287
$153$	0	0
$154$	0	0
$155$	19.3164	1.55153
$156$	0	0
$157$	1.12065	0.0894374	0.0447187	−	0.999000i	$-0.485761\pi$
$157$	1.12065	0.0894374	0.0447187	+	0.999000i	$0.485761\pi$
$158$	−0.637538	−0.0507198
$159$	0	0
$160$	−10.5417	−0.833396
$161$	0	0
$162$	0	0
$163$	−20.3435	−1.59342	−0.796712	−	0.604359i	$-0.793429\pi$
$163$	−20.3435	−1.59342	−0.796712	+	0.604359i	$0.793429\pi$
$164$	−1.57834	−0.123248
$165$	0	0
$166$	−17.9084	−1.38996
$167$	−13.0063	−1.00646	−0.503228	−	0.864154i	$-0.667854\pi$
$167$	−13.0063	−1.00646	−0.503228	+	0.864154i	$0.667854\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−21.3070	−1.63417
$171$	0	0
$172$	1.68420	0.128419
$173$	25.9110	1.96998	0.984989	−	0.172616i	$-0.0552221\pi$
$173$	25.9110	1.96998	0.984989	+	0.172616i	$0.0552221\pi$
$174$	0	0
$175$	0	0
$176$	−1.84581	−0.139133
$177$	0	0
$178$	−10.8587	−0.813890
$179$	−9.64163	−0.720649	−0.360325	−	0.932827i	$-0.617334\pi$
$179$	−9.64163	−0.720649	−0.360325	+	0.932827i	$0.617334\pi$
$180$	0	0
$181$	−2.92683	−0.217550	−0.108775	−	0.994066i	$-0.534693\pi$
$181$	−2.92683	−0.217550	−0.108775	+	0.994066i	$0.534693\pi$
$182$	0	0
$183$	0	0
$184$	−9.20760	−0.678793
$185$	−15.7410	−1.15730
$186$	0	0
$187$	4.44962	0.325388
$188$	3.31211	0.241560
$189$	0	0
$190$	4.52914	0.328579
$191$	16.1364	1.16759	0.583796	−	0.811901i	$-0.301567\pi$
$191$	16.1364	1.16759	0.583796	+	0.811901i	$0.301567\pi$
$192$	0	0
$193$	−0.0863817	−0.00621789	−0.00310894	−	0.999995i	$-0.500990\pi$
$193$	−0.0863817	−0.00621789	−0.00310894	+	0.999995i	$0.500990\pi$
$194$	−1.48542	−0.106647
$195$	0	0
$196$	0	0
$197$	−15.0589	−1.07290	−0.536451	−	0.843932i	$-0.680235\pi$
$197$	−15.0589	−1.07290	−0.536451	+	0.843932i	$0.680235\pi$
$198$	0	0
$199$	15.4466	1.09498	0.547492	−	0.836811i	$-0.315583\pi$
$199$	15.4466	1.09498	0.547492	+	0.836811i	$0.315583\pi$
$200$	−15.1228	−1.06934
$201$	0	0
$202$	−0.712119	−0.0501045
$203$	0	0
$204$	0	0
$205$	−8.05987	−0.562925
$206$	8.99552	0.626748
$207$	0	0
$208$	−2.38665	−0.165484
$209$	−0.945840	−0.0654251
$210$	0	0
$211$	−25.0561	−1.72493	−0.862466	−	0.506115i	$-0.831081\pi$
$211$	−25.0561	−1.72493	−0.862466	+	0.506115i	$0.831081\pi$
$212$	−6.03647	−0.414586
$213$	0	0
$214$	−5.68057	−0.388316
$215$	8.60047	0.586547
$216$	0	0
$217$	0	0
$218$	8.50491	0.576025
$219$	0	0
$220$	1.50146	0.101228
$221$	5.75340	0.387015
$222$	0	0
$223$	−20.6640	−1.38376	−0.691881	−	0.722012i	$-0.743218\pi$
$223$	−20.6640	−1.38376	−0.691881	+	0.722012i	$0.743218\pi$
$224$	0	0
$225$	0	0
$226$	10.8136	0.719307
$227$	−0.982378	−0.0652027	−0.0326013	−	0.999468i	$-0.510379\pi$
$227$	−0.982378	−0.0652027	−0.0326013	+	0.999468i	$0.510379\pi$
$228$	0	0
$229$	−14.2742	−0.943269	−0.471634	−	0.881794i	$-0.656336\pi$
$229$	−14.2742	−0.943269	−0.471634	+	0.881794i	$0.656336\pi$
$230$	−11.0798	−0.730581
$231$	0	0
$232$	−7.59802	−0.498835
$233$	−1.39534	−0.0914119	−0.0457060	−	0.998955i	$-0.514554\pi$
$233$	−1.39534	−0.0914119	−0.0457060	+	0.998955i	$0.514554\pi$
$234$	0	0
$235$	16.9134	1.10331
$236$	1.54428	0.100524
$237$	0	0
$238$	0	0
$239$	−2.54875	−0.164865	−0.0824326	−	0.996597i	$-0.526269\pi$
$239$	−2.54875	−0.164865	−0.0824326	+	0.996597i	$0.526269\pi$
$240$	0	0
$241$	29.6746	1.91151	0.955755	−	0.294164i	$-0.0950413\pi$
$241$	29.6746	1.91151	0.955755	+	0.294164i	$0.0950413\pi$
$242$	−12.2345	−0.786466
$243$	0	0
$244$	6.73108	0.430913
$245$	0	0
$246$	0	0
$247$	−1.22298	−0.0778163
$248$	18.8806	1.19892
$249$	0	0
$250$	0.319106	0.0201820
$251$	−18.0858	−1.14157	−0.570784	−	0.821100i	$-0.693361\pi$
$251$	−18.0858	−1.14157	−0.570784	+	0.821100i	$0.693361\pi$
$252$	0	0
$253$	2.31384	0.145470
$254$	−13.1872	−0.827437
$255$	0	0
$256$	−13.2470	−0.827940
$257$	−14.7403	−0.919473	−0.459737	−	0.888055i	$-0.652056\pi$
$257$	−14.7403	−0.919473	−0.459737	+	0.888055i	$0.652056\pi$
$258$	0	0
$259$	0	0
$260$	1.94140	0.120400
$261$	0	0
$262$	−18.5108	−1.14360
$263$	8.15922	0.503119	0.251560	−	0.967842i	$-0.419057\pi$
$263$	8.15922	0.503119	0.251560	+	0.967842i	$0.419057\pi$
$264$	0	0
$265$	−30.8255	−1.89360
$266$	0	0
$267$	0	0
$268$	−2.66684	−0.162903
$269$	−0.442582	−0.0269847	−0.0134924	−	0.999909i	$-0.504295\pi$
$269$	−0.442582	−0.0269847	−0.0134924	+	0.999909i	$0.504295\pi$
$270$	0	0
$271$	−21.0831	−1.28071	−0.640354	−	0.768080i	$-0.721212\pi$
$271$	−21.0831	−1.28071	−0.640354	+	0.768080i	$0.721212\pi$
$272$	−13.7314	−0.832586
$273$	0	0
$274$	21.9018	1.32313
$275$	3.80031	0.229167
$276$	0	0
$277$	3.59421	0.215955	0.107977	−	0.994153i	$-0.465563\pi$
$277$	3.59421	0.215955	0.107977	+	0.994153i	$0.465563\pi$
$278$	−14.0127	−0.840426
$279$	0	0
$280$	0	0
$281$	9.01252	0.537642	0.268821	−	0.963190i	$-0.413366\pi$
$281$	9.01252	0.537642	0.268821	+	0.963190i	$0.413366\pi$
$282$	0	0
$283$	27.7411	1.64904	0.824520	−	0.565833i	$-0.191445\pi$
$283$	27.7411	1.64904	0.824520	+	0.565833i	$0.191445\pi$
$284$	6.57585	0.390205
$285$	0	0
$286$	0.909650	0.0537887
$287$	0	0
$288$	0	0
$289$	16.1016	0.947153
$290$	−9.14296	−0.536893
$291$	0	0
$292$	3.19052	0.186711
$293$	−12.7409	−0.744333	−0.372167	−	0.928166i	$-0.621385\pi$
$293$	−12.7409	−0.744333	−0.372167	+	0.928166i	$0.621385\pi$
$294$	0	0
$295$	7.88593	0.459137
$296$	−15.3859	−0.894287
$297$	0	0
$298$	−13.3795	−0.775055
$299$	2.99182	0.173021
$300$	0	0
$301$	0	0
$302$	−4.34307	−0.249916
$303$	0	0
$304$	2.91883	0.167406
$305$	34.3726	1.96817
$306$	0	0
$307$	10.1384	0.578626	0.289313	−	0.957235i	$-0.406573\pi$
$307$	10.1384	0.578626	0.289313	+	0.957235i	$0.406573\pi$
$308$	0	0
$309$	0	0
$310$	22.7197	1.29039
$311$	−20.6013	−1.16819	−0.584096	−	0.811685i	$-0.698551\pi$
$311$	−20.6013	−1.16819	−0.584096	+	0.811685i	$0.698551\pi$
$312$	0	0
$313$	16.5456	0.935213	0.467606	−	0.883937i	$-0.345117\pi$
$313$	16.5456	0.935213	0.467606	+	0.883937i	$0.345117\pi$
$314$	1.31809	0.0743841
$315$	0	0
$316$	0.334213	0.0188010
$317$	21.2340	1.19262	0.596310	−	0.802754i	$-0.296633\pi$
$317$	21.2340	1.19262	0.596310	+	0.802754i	$0.296633\pi$
$318$	0	0
$319$	1.90936	0.106904
$320$	−27.4284	−1.53329
$321$	0	0
$322$	0	0
$323$	−7.03629	−0.391510
$324$	0	0
$325$	4.91383	0.272570
$326$	−23.9277	−1.32523
$327$	0	0
$328$	−7.87804	−0.434992
$329$	0	0
$330$	0	0
$331$	16.2415	0.892712	0.446356	−	0.894855i	$-0.352721\pi$
$331$	16.2415	0.892712	0.446356	+	0.894855i	$0.352721\pi$
$332$	9.38804	0.515236
$333$	0	0
$334$	−15.2978	−0.837058
$335$	−13.6184	−0.744050
$336$	0	0
$337$	6.42141	0.349797	0.174898	−	0.984586i	$-0.444040\pi$
$337$	6.42141	0.349797	0.174898	+	0.984586i	$0.444040\pi$
$338$	1.17619	0.0639761
$339$	0	0
$340$	11.1696	0.605758
$341$	−4.74464	−0.256937
$342$	0	0
$343$	0	0
$344$	8.40645	0.453245
$345$	0	0
$346$	30.4762	1.63841
$347$	−14.4809	−0.777376	−0.388688	−	0.921369i	$-0.627072\pi$
$347$	−14.4809	−0.777376	−0.388688	+	0.921369i	$0.627072\pi$
$348$	0	0
$349$	1.74500	0.0934079	0.0467040	−	0.998909i	$-0.485128\pi$
$349$	1.74500	0.0934079	0.0467040	+	0.998909i	$0.485128\pi$
$350$	0	0
$351$	0	0
$352$	2.58934	0.138012
$353$	−15.9580	−0.849358	−0.424679	−	0.905344i	$-0.639613\pi$
$353$	−15.9580	−0.849358	−0.424679	+	0.905344i	$0.639613\pi$
$354$	0	0
$355$	33.5799	1.78224
$356$	5.69237	0.301695
$357$	0	0
$358$	−11.3404	−0.599356
$359$	−8.81400	−0.465185	−0.232593	−	0.972574i	$-0.574721\pi$
$359$	−8.81400	−0.465185	−0.232593	+	0.972574i	$0.574721\pi$
$360$	0	0
$361$	−17.5043	−0.921280
$362$	−3.44250	−0.180934
$363$	0	0
$364$	0	0
$365$	16.2926	0.852792
$366$	0	0
$367$	−19.7080	−1.02875	−0.514374	−	0.857566i	$-0.671976\pi$
$367$	−19.7080	−1.02875	−0.514374	+	0.857566i	$0.671976\pi$
$368$	−7.14043	−0.372221
$369$	0	0
$370$	−18.5144	−0.962515
$371$	0	0
$372$	0	0
$373$	−0.365792	−0.0189400	−0.00947000	−	0.999955i	$-0.503014\pi$
$373$	−0.365792	−0.0189400	−0.00947000	+	0.999955i	$0.503014\pi$
$374$	5.23358	0.270622
$375$	0	0
$376$	16.5319	0.852566
$377$	2.46882	0.127151
$378$	0	0
$379$	7.39215	0.379709	0.189855	−	0.981812i	$-0.439198\pi$
$379$	7.39215	0.379709	0.189855	+	0.981812i	$0.439198\pi$
$380$	−2.37429	−0.121798
$381$	0	0
$382$	18.9795	0.971073
$383$	3.35364	0.171363	0.0856814	−	0.996323i	$-0.472693\pi$
$383$	3.35364	0.171363	0.0856814	+	0.996323i	$0.472693\pi$
$384$	0	0
$385$	0	0
$386$	−0.101601	−0.00517135
$387$	0	0
$388$	0.778692	0.0395321
$389$	2.59344	0.131492	0.0657462	−	0.997836i	$-0.479057\pi$
$389$	2.59344	0.131492	0.0657462	+	0.997836i	$0.479057\pi$
$390$	0	0
$391$	17.2131	0.870506
$392$	0	0
$393$	0	0
$394$	−17.7121	−0.892321
$395$	1.70668	0.0858721
$396$	0	0
$397$	−33.4914	−1.68088	−0.840441	−	0.541902i	$-0.817704\pi$
$397$	−33.4914	−1.68088	−0.840441	+	0.541902i	$0.817704\pi$
$398$	18.1681	0.910686
$399$	0	0
$400$	−11.7276	−0.586380
$401$	−39.5156	−1.97331	−0.986657	−	0.162811i	$-0.947944\pi$
$401$	−39.5156	−1.97331	−0.986657	+	0.162811i	$0.947944\pi$
$402$	0	0
$403$	−6.13487	−0.305599
$404$	0.373310	0.0185729
$405$	0	0
$406$	0	0
$407$	3.86643	0.191652
$408$	0	0
$409$	−10.9799	−0.542922	−0.271461	−	0.962449i	$-0.587507\pi$
$409$	−10.9799	−0.542922	−0.271461	+	0.962449i	$0.587507\pi$
$410$	−9.47990	−0.468179
$411$	0	0
$412$	−4.71567	−0.232324
$413$	0	0
$414$	0	0
$415$	47.9405	2.35331
$416$	3.34804	0.164151
$417$	0	0
$418$	−1.11248	−0.0544134
$419$	−31.5621	−1.54191	−0.770954	−	0.636891i	$-0.780220\pi$
$419$	−31.5621	−1.54191	−0.770954	+	0.636891i	$0.780220\pi$
$420$	0	0
$421$	−17.7055	−0.862914	−0.431457	−	0.902134i	$-0.642000\pi$
$421$	−17.7055	−0.862914	−0.431457	+	0.902134i	$0.642000\pi$
$422$	−29.4706	−1.43461
$423$	0	0
$424$	−30.1301	−1.46325
$425$	28.2712	1.37136
$426$	0	0
$427$	0	0
$428$	2.97789	0.143942
$429$	0	0
$430$	10.1158	0.487825
$431$	21.4816	1.03473	0.517367	−	0.855764i	$-0.326912\pi$
$431$	21.4816	1.03473	0.517367	+	0.855764i	$0.326912\pi$
$432$	0	0
$433$	34.7200	1.66854	0.834269	−	0.551358i	$-0.185890\pi$
$433$	34.7200	1.66854	0.834269	+	0.551358i	$0.185890\pi$
$434$	0	0
$435$	0	0
$436$	−4.45848	−0.213522
$437$	−3.65894	−0.175031
$438$	0	0
$439$	0.728505	0.0347697	0.0173848	−	0.999849i	$-0.494466\pi$
$439$	0.728505	0.0347697	0.0173848	+	0.999849i	$0.494466\pi$
$440$	7.49428	0.357276
$441$	0	0
$442$	6.76707	0.321877
$443$	−0.837291	−0.0397809	−0.0198904	−	0.999802i	$-0.506332\pi$
$443$	−0.837291	−0.0397809	−0.0198904	+	0.999802i	$0.506332\pi$
$444$	0	0
$445$	29.0684	1.37797
$446$	−24.3047	−1.15086
$447$	0	0
$448$	0	0
$449$	−26.4312	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$449$	−26.4312	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$450$	0	0
$451$	1.97973	0.0932217
$452$	−5.66873	−0.266635
$453$	0	0
$454$	−1.15546	−0.0542284
$455$	0	0
$456$	0	0
$457$	−2.40475	−0.112489	−0.0562447	−	0.998417i	$-0.517913\pi$
$457$	−2.40475	−0.112489	−0.0562447	+	0.998417i	$0.517913\pi$
$458$	−16.7892	−0.784507
$459$	0	0
$460$	5.80831	0.270814
$461$	22.2702	1.03722	0.518612	−	0.855010i	$-0.326449\pi$
$461$	22.2702	1.03722	0.518612	+	0.855010i	$0.326449\pi$
$462$	0	0
$463$	32.3085	1.50151	0.750753	−	0.660583i	$-0.229691\pi$
$463$	32.3085	1.50151	0.750753	+	0.660583i	$0.229691\pi$
$464$	−5.89221	−0.273539
$465$	0	0
$466$	−1.64118	−0.0760263
$467$	12.8744	0.595756	0.297878	−	0.954604i	$-0.403721\pi$
$467$	12.8744	0.595756	0.297878	+	0.954604i	$0.403721\pi$
$468$	0	0
$469$	0	0
$470$	19.8934	0.917612
$471$	0	0
$472$	7.70802	0.354791
$473$	−2.11251	−0.0971335
$474$	0	0
$475$	−6.00952	−0.275736
$476$	0	0
$477$	0	0
$478$	−2.99781	−0.137117
$479$	−10.5419	−0.481670	−0.240835	−	0.970566i	$-0.577421\pi$
$479$	−10.5419	−0.481670	−0.240835	+	0.970566i	$0.577421\pi$
$480$	0	0
$481$	4.99933	0.227950
$482$	34.9029	1.58978
$483$	0	0
$484$	6.41364	0.291529
$485$	3.97643	0.180560
$486$	0	0
$487$	38.2416	1.73289	0.866446	−	0.499271i	$-0.166399\pi$
$487$	38.2416	1.73289	0.866446	+	0.499271i	$0.166399\pi$
$488$	33.5972	1.52087
$489$	0	0
$490$	0	0
$491$	15.4291	0.696306	0.348153	−	0.937438i	$-0.386809\pi$
$491$	15.4291	0.696306	0.348153	+	0.937438i	$0.386809\pi$
$492$	0	0
$493$	14.2041	0.639721
$494$	−1.43845	−0.0647190
$495$	0	0
$496$	14.6418	0.657436
$497$	0	0
$498$	0	0
$499$	−38.8212	−1.73788	−0.868938	−	0.494921i	$-0.835197\pi$
$499$	−38.8212	−1.73788	−0.868938	+	0.494921i	$0.835197\pi$
$500$	−0.167283	−0.00748112
$501$	0	0
$502$	−21.2723	−0.949430
$503$	27.0935	1.20804	0.604020	−	0.796969i	$-0.293565\pi$
$503$	27.0935	1.20804	0.604020	+	0.796969i	$0.293565\pi$
$504$	0	0
$505$	1.90633	0.0848304
$506$	2.72151	0.120986
$507$	0	0
$508$	6.91304	0.306717
$509$	15.7693	0.698961	0.349480	−	0.936944i	$-0.386358\pi$
$509$	15.7693	0.698961	0.349480	+	0.936944i	$0.386358\pi$
$510$	0	0
$511$	0	0
$512$	−22.6809	−1.00236
$513$	0	0
$514$	−17.3373	−0.764716
$515$	−24.0808	−1.06113
$516$	0	0
$517$	−4.15441	−0.182711
$518$	0	0
$519$	0	0
$520$	9.69018	0.424943
$521$	−7.30737	−0.320142	−0.160071	−	0.987106i	$-0.551172\pi$
$521$	−7.30737	−0.320142	−0.160071	+	0.987106i	$0.551172\pi$
$522$	0	0
$523$	14.0826	0.615788	0.307894	−	0.951421i	$-0.400376\pi$
$523$	14.0826	0.615788	0.307894	+	0.951421i	$0.400376\pi$
$524$	9.70381	0.423913
$525$	0	0
$526$	9.59677	0.418439
$527$	−35.2963	−1.53753
$528$	0	0
$529$	−14.0490	−0.610827
$530$	−36.2565	−1.57488
$531$	0	0
$532$	0	0
$533$	2.55981	0.110877
$534$	0	0
$535$	15.2067	0.657445
$536$	−13.3111	−0.574953
$537$	0	0
$538$	−0.520559	−0.0224429
$539$	0	0
$540$	0	0
$541$	−39.1750	−1.68426	−0.842132	−	0.539271i	$-0.818700\pi$
$541$	−39.1750	−1.68426	−0.842132	+	0.539271i	$0.818700\pi$
$542$	−24.7977	−1.06515
$543$	0	0
$544$	19.2626	0.825877
$545$	−22.7674	−0.975250
$546$	0	0
$547$	38.7917	1.65862	0.829308	−	0.558792i	$-0.188735\pi$
$547$	38.7917	1.65862	0.829308	+	0.558792i	$0.188735\pi$
$548$	−11.4814	−0.490463
$549$	0	0
$550$	4.46987	0.190596
$551$	−3.01932	−0.128627
$552$	0	0
$553$	0	0
$554$	4.22746	0.179607
$555$	0	0
$556$	7.34580	0.311531
$557$	27.4693	1.16391	0.581956	−	0.813220i	$-0.302287\pi$
$557$	27.4693	1.16391	0.581956	+	0.813220i	$0.302287\pi$
$558$	0	0
$559$	−2.73150	−0.115530
$560$	0	0
$561$	0	0
$562$	10.6004	0.447151
$563$	−14.1210	−0.595129	−0.297565	−	0.954702i	$-0.596174\pi$
$563$	−14.1210	−0.595129	−0.297565	+	0.954702i	$0.596174\pi$
$564$	0	0
$565$	−28.9477	−1.21784
$566$	32.6288	1.37149
$567$	0	0
$568$	32.8223	1.37720
$569$	8.28649	0.347388	0.173694	−	0.984800i	$-0.444430\pi$
$569$	8.28649	0.347388	0.173694	+	0.984800i	$0.444430\pi$
$570$	0	0
$571$	−23.5274	−0.984591	−0.492295	−	0.870428i	$-0.663842\pi$
$571$	−23.5274	−0.984591	−0.492295	+	0.870428i	$0.663842\pi$
$572$	−0.476861	−0.0199386
$573$	0	0
$574$	0	0
$575$	14.7013	0.613087
$576$	0	0
$577$	−17.7732	−0.739906	−0.369953	−	0.929050i	$-0.620626\pi$
$577$	−17.7732	−0.739906	−0.369953	+	0.929050i	$0.620626\pi$
$578$	18.9385	0.787737
$579$	0	0
$580$	4.79296	0.199017
$581$	0	0
$582$	0	0
$583$	7.57160	0.313584
$584$	15.9250	0.658982
$585$	0	0
$586$	−14.9857	−0.619054
$587$	6.64096	0.274102	0.137051	−	0.990564i	$-0.456238\pi$
$587$	6.64096	0.274102	0.137051	+	0.990564i	$0.456238\pi$
$588$	0	0
$589$	7.50282	0.309148
$590$	9.27532	0.381859
$591$	0	0
$592$	−11.9317	−0.490388
$593$	−34.0504	−1.39828	−0.699141	−	0.714984i	$-0.746434\pi$
$593$	−34.0504	−1.39828	−0.699141	+	0.714984i	$0.746434\pi$
$594$	0	0
$595$	0	0
$596$	7.01387	0.287299
$597$	0	0
$598$	3.51894	0.143900
$599$	−9.38441	−0.383437	−0.191718	−	0.981450i	$-0.561406\pi$
$599$	−9.38441	−0.383437	−0.191718	+	0.981450i	$0.561406\pi$
$600$	0	0
$601$	8.80294	0.359079	0.179540	−	0.983751i	$-0.442539\pi$
$601$	8.80294	0.359079	0.179540	+	0.983751i	$0.442539\pi$
$602$	0	0
$603$	0	0
$604$	2.27675	0.0926394
$605$	32.7516	1.33154
$606$	0	0
$607$	17.7720	0.721342	0.360671	−	0.932693i	$-0.382548\pi$
$607$	17.7720	0.721342	0.360671	+	0.932693i	$0.382548\pi$
$608$	−4.09458	−0.166057
$609$	0	0
$610$	40.4286	1.63691
$611$	−5.37169	−0.217315
$612$	0	0
$613$	10.6081	0.428457	0.214228	−	0.976784i	$-0.431276\pi$
$613$	10.6081	0.428457	0.214228	+	0.976784i	$0.431276\pi$
$614$	11.9246	0.481237
$615$	0	0
$616$	0	0
$617$	−49.3483	−1.98669	−0.993344	−	0.115188i	$-0.963253\pi$
$617$	−49.3483	−1.98669	−0.993344	+	0.115188i	$0.963253\pi$
$618$	0	0
$619$	7.42203	0.298317	0.149158	−	0.988813i	$-0.452344\pi$
$619$	7.42203	0.298317	0.149158	+	0.988813i	$0.452344\pi$
$620$	−11.9102	−0.478325
$621$	0	0
$622$	−24.2309	−0.971573
$623$	0	0
$624$	0	0
$625$	−25.4234	−1.01694
$626$	19.4607	0.777806
$627$	0	0
$628$	−0.690975	−0.0275729
$629$	28.7631	1.14686
$630$	0	0
$631$	35.5184	1.41396	0.706982	−	0.707231i	$-0.250056\pi$
$631$	35.5184	1.41396	0.706982	+	0.707231i	$0.250056\pi$
$632$	1.66817	0.0663564
$633$	0	0
$634$	24.9752	0.991890
$635$	35.3018	1.40091
$636$	0	0
$637$	0	0
$638$	2.24576	0.0889106
$639$	0	0
$640$	−11.1774	−0.441827
$641$	−37.1554	−1.46755	−0.733775	−	0.679393i	$-0.762243\pi$
$641$	−37.1554	−1.46755	−0.733775	+	0.679393i	$0.762243\pi$
$642$	0	0
$643$	39.7694	1.56835	0.784176	−	0.620538i	$-0.213086\pi$
$643$	39.7694	1.56835	0.784176	+	0.620538i	$0.213086\pi$
$644$	0	0
$645$	0	0
$646$	−8.27599	−0.325614
$647$	−22.9754	−0.903256	−0.451628	−	0.892206i	$-0.649157\pi$
$647$	−22.9754	−0.903256	−0.451628	+	0.892206i	$0.649157\pi$
$648$	0	0
$649$	−1.93700	−0.0760340
$650$	5.77958	0.226694
$651$	0	0
$652$	12.5435	0.491241
$653$	−18.1757	−0.711268	−0.355634	−	0.934625i	$-0.615735\pi$
$653$	−18.1757	−0.711268	−0.355634	+	0.934625i	$0.615735\pi$
$654$	0	0
$655$	49.5530	1.93619
$656$	−6.10936	−0.238531
$657$	0	0
$658$	0	0
$659$	−14.1044	−0.549431	−0.274716	−	0.961526i	$-0.588584\pi$
$659$	−14.1044	−0.549431	−0.274716	+	0.961526i	$0.588584\pi$
$660$	0	0
$661$	−12.8557	−0.500027	−0.250013	−	0.968242i	$-0.580435\pi$
$661$	−12.8557	−0.500027	−0.250013	+	0.968242i	$0.580435\pi$
$662$	19.1030	0.742459
$663$	0	0
$664$	46.8590	1.81848
$665$	0	0
$666$	0	0
$667$	7.38627	0.285997
$668$	8.01948	0.310283
$669$	0	0
$670$	−16.0177	−0.618819
$671$	−8.44287	−0.325933
$672$	0	0
$673$	−45.6138	−1.75828	−0.879141	−	0.476561i	$-0.841883\pi$
$673$	−45.6138	−1.75828	−0.879141	+	0.476561i	$0.841883\pi$
$674$	7.55278	0.290922
$675$	0	0
$676$	−0.616586	−0.0237148
$677$	−10.2469	−0.393821	−0.196910	−	0.980421i	$-0.563091\pi$
$677$	−10.2469	−0.393821	−0.196910	+	0.980421i	$0.563091\pi$
$678$	0	0
$679$	0	0
$680$	55.7515	2.13797
$681$	0	0
$682$	−5.58058	−0.213692
$683$	−2.90040	−0.110981	−0.0554904	−	0.998459i	$-0.517672\pi$
$683$	−2.90040	−0.110981	−0.0554904	+	0.998459i	$0.517672\pi$
$684$	0	0
$685$	−58.6305	−2.24016
$686$	0	0
$687$	0	0
$688$	6.51914	0.248540
$689$	9.79015	0.372975
$690$	0	0
$691$	6.91470	0.263048	0.131524	−	0.991313i	$-0.458013\pi$
$691$	6.91470	0.263048	0.131524	+	0.991313i	$0.458013\pi$
$692$	−15.9764	−0.607330
$693$	0	0
$694$	−17.0323	−0.646536
$695$	37.5117	1.42290
$696$	0	0
$697$	14.7276	0.557847
$698$	2.05245	0.0776864
$699$	0	0
$700$	0	0
$701$	26.0973	0.985682	0.492841	−	0.870119i	$-0.335958\pi$
$701$	26.0973	0.985682	0.492841	+	0.870119i	$0.335958\pi$
$702$	0	0
$703$	−6.11408	−0.230597
$704$	6.73717	0.253916
$705$	0	0
$706$	−18.7696	−0.706402
$707$	0	0
$708$	0	0
$709$	−3.38472	−0.127116	−0.0635580	−	0.997978i	$-0.520245\pi$
$709$	−3.38472	−0.127116	−0.0635580	+	0.997978i	$0.520245\pi$
$710$	39.4962	1.48227
$711$	0	0
$712$	28.4126	1.06481
$713$	−18.3544	−0.687378
$714$	0	0
$715$	−2.43511	−0.0910681
$716$	5.94489	0.222171
$717$	0	0
$718$	−10.3669	−0.386890
$719$	2.41574	0.0900918	0.0450459	−	0.998985i	$-0.485657\pi$
$719$	2.41574	0.0900918	0.0450459	+	0.998985i	$0.485657\pi$
$720$	0	0
$721$	0	0
$722$	−20.5883	−0.766219
$723$	0	0
$724$	1.80464	0.0670691
$725$	12.1314	0.450548
$726$	0	0
$727$	17.0150	0.631050	0.315525	−	0.948917i	$-0.397819\pi$
$727$	17.0150	0.631050	0.315525	+	0.948917i	$0.397819\pi$
$728$	0	0
$729$	0	0
$730$	19.1631	0.709258
$731$	−15.7154	−0.581256
$732$	0	0
$733$	4.18453	0.154559	0.0772795	−	0.997009i	$-0.475377\pi$
$733$	4.18453	0.154559	0.0772795	+	0.997009i	$0.475377\pi$
$734$	−23.1803	−0.855599
$735$	0	0
$736$	10.0167	0.369221
$737$	3.34505	0.123216
$738$	0	0
$739$	−28.0794	−1.03292	−0.516458	−	0.856312i	$-0.672750\pi$
$739$	−28.0794	−1.03292	−0.516458	+	0.856312i	$0.672750\pi$
$740$	9.70567	0.356788
$741$	0	0
$742$	0	0
$743$	26.5210	0.972961	0.486481	−	0.873691i	$-0.338280\pi$
$743$	26.5210	0.972961	0.486481	+	0.873691i	$0.338280\pi$
$744$	0	0
$745$	35.8167	1.31222
$746$	−0.430240	−0.0157522
$747$	0	0
$748$	−2.74357	−0.100315
$749$	0	0
$750$	0	0
$751$	−23.7162	−0.865418	−0.432709	−	0.901534i	$-0.642442\pi$
$751$	−23.7162	−0.865418	−0.432709	+	0.901534i	$0.642442\pi$
$752$	12.8204	0.467510
$753$	0	0
$754$	2.90379	0.105750
$755$	11.6263	0.423125
$756$	0	0
$757$	−52.3661	−1.90328	−0.951639	−	0.307219i	$-0.900602\pi$
$757$	−52.3661	−1.90328	−0.951639	+	0.307219i	$0.900602\pi$
$758$	8.69454	0.315800
$759$	0	0
$760$	−11.8509	−0.429877
$761$	−23.8695	−0.865268	−0.432634	−	0.901570i	$-0.642416\pi$
$761$	−23.8695	−0.865268	−0.432634	+	0.901570i	$0.642416\pi$
$762$	0	0
$763$	0	0
$764$	−9.94949	−0.359960
$765$	0	0
$766$	3.94450	0.142521
$767$	−2.50456	−0.0904345
$768$	0	0
$769$	41.7599	1.50590	0.752950	−	0.658077i	$-0.228630\pi$
$769$	41.7599	1.50590	0.752950	+	0.658077i	$0.228630\pi$
$770$	0	0
$771$	0	0
$772$	0.0532617	0.00191693
$773$	−2.97178	−0.106888	−0.0534438	−	0.998571i	$-0.517020\pi$
$773$	−2.97178	−0.106888	−0.0534438	+	0.998571i	$0.517020\pi$
$774$	0	0
$775$	−30.1457	−1.08287
$776$	3.88672	0.139525
$777$	0	0
$778$	3.05037	0.109361
$779$	−3.13059	−0.112165
$780$	0	0
$781$	−8.24815	−0.295142
$782$	20.2459	0.723990
$783$	0	0
$784$	0	0
$785$	−3.52850	−0.125937
$786$	0	0
$787$	−48.6142	−1.73291	−0.866454	−	0.499256i	$-0.833607\pi$
$787$	−48.6142	−1.73291	−0.866454	+	0.499256i	$0.833607\pi$
$788$	9.28509	0.330768
$789$	0	0
$790$	2.00737	0.0714189
$791$	0	0
$792$	0	0
$793$	−10.9167	−0.387664
$794$	−39.3921	−1.39797
$795$	0	0
$796$	−9.52418	−0.337576
$797$	−25.6470	−0.908463	−0.454232	−	0.890884i	$-0.650086\pi$
$797$	−25.6470	−0.908463	−0.454232	+	0.890884i	$0.650086\pi$
$798$	0	0
$799$	−30.9055	−1.09336
$800$	16.4517	0.581655
$801$	0	0
$802$	−46.4777	−1.64118
$803$	−4.00191	−0.141224
$804$	0	0
$805$	0	0
$806$	−7.21575	−0.254164
$807$	0	0
$808$	1.86332	0.0655514
$809$	11.9205	0.419101	0.209550	−	0.977798i	$-0.432800\pi$
$809$	11.9205	0.419101	0.209550	+	0.977798i	$0.432800\pi$
$810$	0	0
$811$	−10.5564	−0.370685	−0.185343	−	0.982674i	$-0.559339\pi$
$811$	−10.5564	−0.370685	−0.185343	+	0.982674i	$0.559339\pi$
$812$	0	0
$813$	0	0
$814$	4.54764	0.159395
$815$	64.0540	2.24371
$816$	0	0
$817$	3.34057	0.116872
$818$	−12.9144	−0.451543
$819$	0	0
$820$	4.96960	0.173546
$821$	46.2139	1.61288	0.806438	−	0.591319i	$-0.201393\pi$
$821$	46.2139	1.61288	0.806438	+	0.591319i	$0.201393\pi$
$822$	0	0
$823$	41.2171	1.43674	0.718368	−	0.695663i	$-0.244889\pi$
$823$	41.2171	1.43674	0.718368	+	0.695663i	$0.244889\pi$
$824$	−23.5375	−0.819969
$825$	0	0
$826$	0	0
$827$	−5.94430	−0.206704	−0.103352	−	0.994645i	$-0.532957\pi$
$827$	−5.94430	−0.206704	−0.103352	+	0.994645i	$0.532957\pi$
$828$	0	0
$829$	34.8106	1.20902	0.604511	−	0.796596i	$-0.293368\pi$
$829$	34.8106	1.20902	0.604511	+	0.796596i	$0.293368\pi$
$830$	56.3869	1.95722
$831$	0	0
$832$	8.71122	0.302007
$833$	0	0
$834$	0	0
$835$	40.9519	1.41720
$836$	0.583191	0.0201701
$837$	0	0
$838$	−37.1229	−1.28239
$839$	−10.7896	−0.372500	−0.186250	−	0.982502i	$-0.559633\pi$
$839$	−10.7896	−0.372500	−0.186250	+	0.982502i	$0.559633\pi$
$840$	0	0
$841$	−22.9049	−0.789825
$842$	−20.8250	−0.717676
$843$	0	0
$844$	15.4492	0.531784
$845$	−3.14862	−0.108316
$846$	0	0
$847$	0	0
$848$	−23.3657	−0.802381
$849$	0	0
$850$	33.2523	1.14054
$851$	14.9571	0.512722
$852$	0	0
$853$	39.1053	1.33894	0.669470	−	0.742839i	$-0.266521\pi$
$853$	39.1053	1.33894	0.669470	+	0.742839i	$0.266521\pi$
$854$	0	0
$855$	0	0
$856$	14.8637	0.508030
$857$	33.3654	1.13974	0.569870	−	0.821735i	$-0.306994\pi$
$857$	33.3654	1.13974	0.569870	+	0.821735i	$0.306994\pi$
$858$	0	0
$859$	−29.4914	−1.00623	−0.503116	−	0.864219i	$-0.667813\pi$
$859$	−29.4914	−1.00623	−0.503116	+	0.864219i	$0.667813\pi$
$860$	−5.30293	−0.180828
$861$	0	0
$862$	25.2664	0.860578
$863$	24.6224	0.838157	0.419079	−	0.907950i	$-0.362353\pi$
$863$	24.6224	0.838157	0.419079	+	0.907950i	$0.362353\pi$
$864$	0	0
$865$	−81.5841	−2.77394
$866$	40.8372	1.38771
$867$	0	0
$868$	0	0
$869$	−0.419207	−0.0142206
$870$	0	0
$871$	4.32518	0.146553
$872$	−22.2538	−0.753609
$873$	0	0
$874$	−4.30359	−0.145571
$875$	0	0
$876$	0	0
$877$	−4.62556	−0.156194	−0.0780971	−	0.996946i	$-0.524884\pi$
$877$	−4.62556	−0.156194	−0.0780971	+	0.996946i	$0.524884\pi$
$878$	0.856858	0.0289176
$879$	0	0
$880$	5.81177	0.195915
$881$	42.1720	1.42081	0.710405	−	0.703793i	$-0.248512\pi$
$881$	42.1720	1.42081	0.710405	+	0.703793i	$0.248512\pi$
$882$	0	0
$883$	20.7992	0.699950	0.349975	−	0.936759i	$-0.386190\pi$
$883$	20.7992	0.699950	0.349975	+	0.936759i	$0.386190\pi$
$884$	−3.54746	−0.119314
$885$	0	0
$886$	−0.984810	−0.0330853
$887$	−34.4889	−1.15802	−0.579012	−	0.815319i	$-0.696562\pi$
$887$	−34.4889	−1.15802	−0.579012	+	0.815319i	$0.696562\pi$
$888$	0	0
$889$	0	0
$890$	34.1898	1.14605
$891$	0	0
$892$	12.7411	0.426604
$893$	6.56947	0.219839
$894$	0	0
$895$	30.3579	1.01475
$896$	0	0
$897$	0	0
$898$	−31.0880	−1.03742
$899$	−15.1459	−0.505144
$900$	0	0
$901$	56.3267	1.87651
$902$	2.32853	0.0775315
$903$	0	0
$904$	−28.2946	−0.941065
$905$	9.21550	0.306334
$906$	0	0
$907$	13.3619	0.443675	0.221838	−	0.975084i	$-0.428795\pi$
$907$	13.3619	0.443675	0.221838	+	0.975084i	$0.428795\pi$
$908$	0.605720	0.0201015
$909$	0	0
$910$	0	0
$911$	−5.29058	−0.175285	−0.0876424	−	0.996152i	$-0.527933\pi$
$911$	−5.29058	−0.175285	−0.0876424	+	0.996152i	$0.527933\pi$
$912$	0	0
$913$	−11.7755	−0.389713
$914$	−2.82843	−0.0935562
$915$	0	0
$916$	8.80129	0.290803
$917$	0	0
$918$	0	0
$919$	−18.8306	−0.621164	−0.310582	−	0.950547i	$-0.600524\pi$
$919$	−18.8306	−0.621164	−0.310582	+	0.950547i	$0.600524\pi$
$920$	28.9913	0.955814
$921$	0	0
$922$	26.1939	0.862649
$923$	−10.6649	−0.351041
$924$	0	0
$925$	24.5659	0.807721
$926$	38.0009	1.24879
$927$	0	0
$928$	8.26571	0.271335
$929$	44.9537	1.47488	0.737442	−	0.675411i	$-0.236034\pi$
$929$	44.9537	1.47488	0.737442	+	0.675411i	$0.236034\pi$
$930$	0	0
$931$	0	0
$932$	0.860348	0.0281816
$933$	0	0
$934$	15.1427	0.495484
$935$	−14.0102	−0.458182
$936$	0	0
$937$	37.9272	1.23903	0.619514	−	0.784986i	$-0.287330\pi$
$937$	37.9272	1.23903	0.619514	+	0.784986i	$0.287330\pi$
$938$	0	0
$939$	0	0
$940$	−10.4286	−0.340143
$941$	−35.7869	−1.16662	−0.583310	−	0.812250i	$-0.698243\pi$
$941$	−35.7869	−1.16662	−0.583310	+	0.812250i	$0.698243\pi$
$942$	0	0
$943$	7.65848	0.249394
$944$	5.97752	0.194552
$945$	0	0
$946$	−2.48471	−0.0807849
$947$	14.7443	0.479125	0.239563	−	0.970881i	$-0.422996\pi$
$947$	14.7443	0.479125	0.239563	+	0.970881i	$0.422996\pi$
$948$	0	0
$949$	−5.17450	−0.167971
$950$	−7.06831	−0.229326
$951$	0	0
$952$	0	0
$953$	−0.649669	−0.0210448	−0.0105224	−	0.999945i	$-0.503349\pi$
$953$	−0.649669	−0.0210448	−0.0105224	+	0.999945i	$0.503349\pi$
$954$	0	0
$955$	−50.8076	−1.64409
$956$	1.57153	0.0508268
$957$	0	0
$958$	−12.3992	−0.400600
$959$	0	0
$960$	0	0
$961$	6.63659	0.214083
$962$	5.88014	0.189583
$963$	0	0
$964$	−18.2969	−0.589305
$965$	0.271983	0.00875545
$966$	0	0
$967$	−13.0802	−0.420632	−0.210316	−	0.977633i	$-0.567449\pi$
$967$	−13.0802	−0.420632	−0.210316	+	0.977633i	$0.567449\pi$
$968$	32.0127	1.02893
$969$	0	0
$970$	4.67702	0.150170
$971$	29.1203	0.934515	0.467258	−	0.884121i	$-0.345242\pi$
$971$	29.1203	0.934515	0.467258	+	0.884121i	$0.345242\pi$
$972$	0	0
$973$	0	0
$974$	44.9792	1.44123
$975$	0	0
$976$	26.0544	0.833980
$977$	18.5000	0.591868	0.295934	−	0.955208i	$-0.404369\pi$
$977$	18.5000	0.591868	0.295934	+	0.955208i	$0.404369\pi$
$978$	0	0
$979$	−7.14000	−0.228195
$980$	0	0
$981$	0	0
$982$	18.1475	0.579111
$983$	11.0462	0.352318	0.176159	−	0.984362i	$-0.443633\pi$
$983$	11.0462	0.352318	0.176159	+	0.984362i	$0.443633\pi$
$984$	0	0
$985$	47.4148	1.51076
$986$	16.7067	0.532049
$987$	0	0
$988$	0.754072	0.0239902
$989$	−8.17216	−0.259860
$990$	0	0
$991$	−40.4757	−1.28575	−0.642877	−	0.765969i	$-0.722259\pi$
$991$	−40.4757	−1.28575	−0.642877	+	0.765969i	$0.722259\pi$
$992$	−20.5398	−0.652138
$993$	0	0
$994$	0	0
$995$	−48.6357	−1.54185
$996$	0	0
$997$	−12.0623	−0.382018	−0.191009	−	0.981588i	$-0.561176\pi$
$997$	−12.0623	−0.382018	−0.191009	+	0.981588i	$0.561176\pi$
$998$	−45.6610	−1.44537
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.br.1.5		6
3.2	odd	2		637.2.a.n.1.2	yes	6
7.6	odd	2		5733.2.a.bu.1.5		6
21.2	odd	6		637.2.e.n.508.5		12
21.5	even	6		637.2.e.o.508.5		12
21.11	odd	6		637.2.e.n.79.5		12
21.17	even	6		637.2.e.o.79.5		12
21.20	even	2		637.2.a.m.1.2	✓	6
39.38	odd	2		8281.2.a.cd.1.5		6
273.272	even	2		8281.2.a.cc.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.2	✓	6	21.20	even	2
637.2.a.n.1.2	yes	6	3.2	odd	2
637.2.e.n.79.5		12	21.11	odd	6
637.2.e.n.508.5		12	21.2	odd	6
637.2.e.o.79.5		12	21.17	even	6
637.2.e.o.508.5		12	21.5	even	6
5733.2.a.br.1.5		6	1.1	even	1	trivial
5733.2.a.bu.1.5		6	7.6	odd	2
8281.2.a.cc.1.5		6	273.272	even	2
8281.2.a.cd.1.5		6	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.17619 q^{2} -0.616586 q^{4} -3.14862 q^{5} -3.07759 q^{8} +O(q^{10})\)	\(q+1.17619 q^{2} -0.616586 q^{4} -3.14862 q^{5} -3.07759 q^{8} -3.70337 q^{10} +0.773390 q^{11} +1.00000 q^{13} -2.38665 q^{16} +5.75340 q^{17} -1.22298 q^{19} +1.94140 q^{20} +0.909650 q^{22} +2.99182 q^{23} +4.91383 q^{25} +1.17619 q^{26} +2.46882 q^{29} -6.13487 q^{31} +3.34804 q^{32} +6.76707 q^{34} +4.99933 q^{37} -1.43845 q^{38} +9.69018 q^{40} +2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} +3.51894 q^{46} -5.37169 q^{47} +5.77958 q^{50} -0.616586 q^{52} +9.79015 q^{53} -2.43511 q^{55} +2.90379 q^{58} -2.50456 q^{59} -10.9167 q^{61} -7.21575 q^{62} +8.71122 q^{64} -3.14862 q^{65} +4.32518 q^{67} -3.54746 q^{68} -10.6649 q^{71} -5.17450 q^{73} +5.88014 q^{74} +0.754072 q^{76} -0.542038 q^{79} +7.51467 q^{80} +3.01081 q^{82} -15.2259 q^{83} -18.1153 q^{85} -3.21275 q^{86} -2.38018 q^{88} -9.23208 q^{89} -1.84471 q^{92} -6.31811 q^{94} +3.85070 q^{95} -1.26291 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 4 * q^4 - 6 * q^5	\( 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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