LMFDB - Embedded newform 5733.2.a.br.1.2

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			1.2
Root			$-0.146243$ 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)

$f(q)$	$=$	$q-1.83237 q^{2} +1.35758 q^{4} -2.62555 q^{5} +1.17715 q^{8} +O(q^{10})$	$q-1.83237 q^{2} +1.35758 q^{4} -2.62555 q^{5} +1.17715 q^{8} +4.81098 q^{10} -3.26469 q^{11} +1.00000 q^{13} -4.87214 q^{16} -4.53021 q^{17} +4.06615 q^{19} -3.56440 q^{20} +5.98212 q^{22} +4.53266 q^{23} +1.89352 q^{25} -1.83237 q^{26} +1.42268 q^{29} -2.80328 q^{31} +6.57326 q^{32} +8.30102 q^{34} -10.0503 q^{37} -7.45070 q^{38} -3.09067 q^{40} +2.84271 q^{41} +9.72632 q^{43} -4.43208 q^{44} -8.30551 q^{46} +9.44956 q^{47} -3.46963 q^{50} +1.35758 q^{52} -5.26439 q^{53} +8.57161 q^{55} -2.60687 q^{58} +2.56791 q^{59} -11.1830 q^{61} +5.13664 q^{62} -2.30037 q^{64} -2.62555 q^{65} -1.98172 q^{67} -6.15013 q^{68} +11.7544 q^{71} +12.1391 q^{73} +18.4159 q^{74} +5.52013 q^{76} +11.9089 q^{79} +12.7920 q^{80} -5.20889 q^{82} -13.2233 q^{83} +11.8943 q^{85} -17.8222 q^{86} -3.84303 q^{88} -10.6666 q^{89} +6.15345 q^{92} -17.3151 q^{94} -10.6759 q^{95} -13.7422 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.83237	−1.29568	−0.647841	−	0.761776i	$-0.724328\pi$
$2$	−1.83237	−1.29568	−0.647841	+	0.761776i	$0.724328\pi$
$3$	0	0
$3$	0	0
$4$	1.35758	0.678791
$5$	−2.62555	−1.17418	−0.587091	−	0.809521i	$-0.699727\pi$
$5$	−2.62555	−1.17418	−0.587091	+	0.809521i	$0.699727\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.17715	0.416185
$9$	0	0
$10$	4.81098	1.52137
$11$	−3.26469	−0.984341	−0.492170	−	0.870499i	$-0.663796\pi$
$11$	−3.26469	−0.984341	−0.492170	+	0.870499i	$0.663796\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−4.87214	−1.21803
$17$	−4.53021	−1.09874	−0.549369	−	0.835580i	$-0.685132\pi$
$17$	−4.53021	−1.09874	−0.549369	+	0.835580i	$0.685132\pi$
$18$	0	0
$19$	4.06615	0.932839	0.466420	−	0.884564i	$-0.345544\pi$
$19$	4.06615	0.932839	0.466420	+	0.884564i	$0.345544\pi$
$20$	−3.56440	−0.797024
$21$	0	0
$22$	5.98212	1.27539
$23$	4.53266	0.945125	0.472562	−	0.881297i	$-0.343329\pi$
$23$	4.53266	0.945125	0.472562	+	0.881297i	$0.343329\pi$
$24$	0	0
$25$	1.89352	0.378705
$26$	−1.83237	−0.359357
$27$	0	0
$28$	0	0
$29$	1.42268	0.264184	0.132092	−	0.991237i	$-0.457831\pi$
$29$	1.42268	0.264184	0.132092	+	0.991237i	$0.457831\pi$
$30$	0	0
$31$	−2.80328	−0.503484	−0.251742	−	0.967794i	$-0.581003\pi$
$31$	−2.80328	−0.503484	−0.251742	+	0.967794i	$0.581003\pi$
$32$	6.57326	1.16200
$33$	0	0
$34$	8.30102	1.42361
$35$	0	0
$36$	0	0
$37$	−10.0503	−1.65227	−0.826133	−	0.563475i	$-0.809464\pi$
$37$	−10.0503	−1.65227	−0.826133	+	0.563475i	$0.809464\pi$
$38$	−7.45070	−1.20866
$39$	0	0
$40$	−3.09067	−0.488677
$41$	2.84271	0.443956	0.221978	−	0.975052i	$-0.428749\pi$
$41$	2.84271	0.443956	0.221978	+	0.975052i	$0.428749\pi$
$42$	0	0
$43$	9.72632	1.48325	0.741625	−	0.670815i	$-0.234056\pi$
$43$	9.72632	1.48325	0.741625	+	0.670815i	$0.234056\pi$
$44$	−4.43208	−0.668161
$45$	0	0
$46$	−8.30551	−1.22458
$47$	9.44956	1.37836	0.689180	−	0.724590i	$-0.257971\pi$
$47$	9.44956	1.37836	0.689180	+	0.724590i	$0.257971\pi$
$48$	0	0
$49$	0	0
$50$	−3.46963	−0.490680
$51$	0	0
$52$	1.35758	0.188263
$53$	−5.26439	−0.723119	−0.361559	−	0.932349i	$-0.617756\pi$
$53$	−5.26439	−0.723119	−0.361559	+	0.932349i	$0.617756\pi$
$54$	0	0
$55$	8.57161	1.15580
$56$	0	0
$57$	0	0
$58$	−2.60687	−0.342299
$59$	2.56791	0.334313	0.167156	−	0.985930i	$-0.446542\pi$
$59$	2.56791	0.334313	0.167156	+	0.985930i	$0.446542\pi$
$60$	0	0
$61$	−11.1830	−1.43183	−0.715917	−	0.698185i	$-0.753991\pi$
$61$	−11.1830	−1.43183	−0.715917	+	0.698185i	$0.753991\pi$
$62$	5.13664	0.652354
$63$	0	0
$64$	−2.30037	−0.287546
$65$	−2.62555	−0.325660
$66$	0	0
$67$	−1.98172	−0.242106	−0.121053	−	0.992646i	$-0.538627\pi$
$67$	−1.98172	−0.242106	−0.121053	+	0.992646i	$0.538627\pi$
$68$	−6.15013	−0.745813
$69$	0	0
$70$	0	0
$71$	11.7544	1.39499	0.697495	−	0.716590i	$-0.254298\pi$
$71$	11.7544	1.39499	0.697495	+	0.716590i	$0.254298\pi$
$72$	0	0
$73$	12.1391	1.42078	0.710388	−	0.703810i	$-0.248519\pi$
$73$	12.1391	1.42078	0.710388	+	0.703810i	$0.248519\pi$
$74$	18.4159	2.14081
$75$	0	0
$76$	5.52013	0.633202
$77$	0	0
$78$	0	0
$79$	11.9089	1.33986	0.669928	−	0.742426i	$-0.266325\pi$
$79$	11.9089	1.33986	0.669928	+	0.742426i	$0.266325\pi$
$80$	12.7920	1.43019
$81$	0	0
$82$	−5.20889	−0.575226
$83$	−13.2233	−1.45145	−0.725723	−	0.687987i	$-0.758495\pi$
$83$	−13.2233	−1.45145	−0.725723	+	0.687987i	$0.758495\pi$
$84$	0	0
$85$	11.8943	1.29012
$86$	−17.8222	−1.92182
$87$	0	0
$88$	−3.84303	−0.409668
$89$	−10.6666	−1.13065	−0.565326	−	0.824867i	$-0.691250\pi$
$89$	−10.6666	−1.13065	−0.565326	+	0.824867i	$0.691250\pi$
$90$	0	0
$91$	0	0
$92$	6.15345	0.641542
$93$	0	0
$94$	−17.3151	−1.78592
$95$	−10.6759	−1.09532
$96$	0	0
$97$	−13.7422	−1.39531	−0.697655	−	0.716433i	$-0.745773\pi$
$97$	−13.7422	−1.39531	−0.697655	+	0.716433i	$0.745773\pi$
$98$	0	0
$99$	0	0
$100$	2.57061	0.257061
$101$	−5.89458	−0.586533	−0.293266	−	0.956031i	$-0.594742\pi$
$101$	−5.89458	−0.586533	−0.293266	+	0.956031i	$0.594742\pi$
$102$	0	0
$103$	2.78737	0.274647	0.137324	−	0.990526i	$-0.456150\pi$
$103$	2.78737	0.274647	0.137324	+	0.990526i	$0.456150\pi$
$104$	1.17715	0.115429
$105$	0	0
$106$	9.64630	0.936932
$107$	17.6647	1.70771	0.853856	−	0.520509i	$-0.174258\pi$
$107$	17.6647	1.70771	0.853856	+	0.520509i	$0.174258\pi$
$108$	0	0
$109$	−9.56450	−0.916113	−0.458057	−	0.888923i	$-0.651454\pi$
$109$	−9.56450	−0.916113	−0.458057	+	0.888923i	$0.651454\pi$
$110$	−15.7064	−1.49754
$111$	0	0
$112$	0	0
$113$	17.6017	1.65583	0.827916	−	0.560852i	$-0.189526\pi$
$113$	17.6017	1.65583	0.827916	+	0.560852i	$0.189526\pi$
$114$	0	0
$115$	−11.9007	−1.10975
$116$	1.93140	0.179326
$117$	0	0
$118$	−4.70535	−0.433163
$119$	0	0
$120$	0	0
$121$	−0.341808	−0.0310735
$122$	20.4914	1.85520
$123$	0	0
$124$	−3.80568	−0.341760
$125$	8.15622	0.729514
$126$	0	0
$127$	−1.59482	−0.141517	−0.0707586	−	0.997493i	$-0.522542\pi$
$127$	−1.59482	−0.141517	−0.0707586	+	0.997493i	$0.522542\pi$
$128$	−8.93138	−0.789430
$129$	0	0
$130$	4.81098	0.421951
$131$	−4.30279	−0.375937	−0.187968	−	0.982175i	$-0.560190\pi$
$131$	−4.30279	−0.375937	−0.187968	+	0.982175i	$0.560190\pi$
$132$	0	0
$133$	0	0
$134$	3.63125	0.313692
$135$	0	0
$136$	−5.33273	−0.457278
$137$	−9.82234	−0.839179	−0.419590	−	0.907714i	$-0.637826\pi$
$137$	−9.82234	−0.839179	−0.419590	+	0.907714i	$0.637826\pi$
$138$	0	0
$139$	10.0811	0.855070	0.427535	−	0.903999i	$-0.359382\pi$
$139$	10.0811	0.855070	0.427535	+	0.903999i	$0.359382\pi$
$140$	0	0
$141$	0	0
$142$	−21.5384	−1.80746
$143$	−3.26469	−0.273007
$144$	0	0
$145$	−3.73531	−0.310201
$146$	−22.2434	−1.84087
$147$	0	0
$148$	−13.6442	−1.12154
$149$	13.8124	1.13155	0.565777	−	0.824558i	$-0.308576\pi$
$149$	13.8124	1.13155	0.565777	+	0.824558i	$0.308576\pi$
$150$	0	0
$151$	21.2123	1.72623	0.863117	−	0.505004i	$-0.168509\pi$
$151$	21.2123	1.72623	0.863117	+	0.505004i	$0.168509\pi$
$152$	4.78647	0.388234
$153$	0	0
$154$	0	0
$155$	7.36015	0.591182
$156$	0	0
$157$	23.5155	1.87674	0.938372	−	0.345627i	$-0.112334\pi$
$157$	23.5155	1.87674	0.938372	+	0.345627i	$0.112334\pi$
$158$	−21.8215	−1.73603
$159$	0	0
$160$	−17.2584	−1.36440
$161$	0	0
$162$	0	0
$163$	7.83062	0.613341	0.306671	−	0.951816i	$-0.400785\pi$
$163$	7.83062	0.613341	0.306671	+	0.951816i	$0.400785\pi$
$164$	3.85920	0.301353
$165$	0	0
$166$	24.2300	1.88061
$167$	−12.7116	−0.983654	−0.491827	−	0.870693i	$-0.663671\pi$
$167$	−12.7116	−0.983654	−0.491827	+	0.870693i	$0.663671\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−21.7948	−1.67158
$171$	0	0
$172$	13.2043	1.00682
$173$	−5.24907	−0.399079	−0.199540	−	0.979890i	$-0.563945\pi$
$173$	−5.24907	−0.399079	−0.199540	+	0.979890i	$0.563945\pi$
$174$	0	0
$175$	0	0
$176$	15.9060	1.19896
$177$	0	0
$178$	19.5451	1.46497
$179$	10.5255	0.786714	0.393357	−	0.919386i	$-0.371314\pi$
$179$	10.5255	0.786714	0.393357	+	0.919386i	$0.371314\pi$
$180$	0	0
$181$	−18.8177	−1.39871	−0.699356	−	0.714774i	$-0.746530\pi$
$181$	−18.8177	−1.39871	−0.699356	+	0.714774i	$0.746530\pi$
$182$	0	0
$183$	0	0
$184$	5.33562	0.393347
$185$	26.3877	1.94006
$186$	0	0
$187$	14.7897	1.08153
$188$	12.8285	0.935618
$189$	0	0
$190$	19.5622	1.41919
$191$	−14.9144	−1.07917	−0.539585	−	0.841931i	$-0.681419\pi$
$191$	−14.9144	−1.07917	−0.539585	+	0.841931i	$0.681419\pi$
$192$	0	0
$193$	−0.0531356	−0.00382479	−0.00191239	−	0.999998i	$-0.500609\pi$
$193$	−0.0531356	−0.00382479	−0.00191239	+	0.999998i	$0.500609\pi$
$194$	25.1808	1.80788
$195$	0	0
$196$	0	0
$197$	−10.0478	−0.715875	−0.357938	−	0.933745i	$-0.616520\pi$
$197$	−10.0478	−0.715875	−0.357938	+	0.933745i	$0.616520\pi$
$198$	0	0
$199$	23.2914	1.65109	0.825543	−	0.564340i	$-0.190869\pi$
$199$	23.2914	1.65109	0.825543	+	0.564340i	$0.190869\pi$
$200$	2.22896	0.157611
$201$	0	0
$202$	10.8011	0.759959
$203$	0	0
$204$	0	0
$205$	−7.46367	−0.521285
$206$	−5.10749	−0.355855
$207$	0	0
$208$	−4.87214	−0.337822
$209$	−13.2747	−0.918232
$210$	0	0
$211$	−2.47457	−0.170356	−0.0851780	−	0.996366i	$-0.527146\pi$
$211$	−2.47457	−0.170356	−0.0851780	+	0.996366i	$0.527146\pi$
$212$	−7.14683	−0.490846
$213$	0	0
$214$	−32.3683	−2.21265
$215$	−25.5369	−1.74161
$216$	0	0
$217$	0	0
$218$	17.5257	1.18699
$219$	0	0
$220$	11.6367	0.784543
$221$	−4.53021	−0.304735
$222$	0	0
$223$	6.17027	0.413192	0.206596	−	0.978426i	$-0.433761\pi$
$223$	6.17027	0.413192	0.206596	+	0.978426i	$0.433761\pi$
$224$	0	0
$225$	0	0
$226$	−32.2529	−2.14543
$227$	−11.5939	−0.769513	−0.384757	−	0.923018i	$-0.625715\pi$
$227$	−11.5939	−0.769513	−0.384757	+	0.923018i	$0.625715\pi$
$228$	0	0
$229$	−10.2382	−0.676558	−0.338279	−	0.941046i	$-0.609845\pi$
$229$	−10.2382	−0.676558	−0.338279	+	0.941046i	$0.609845\pi$
$230$	21.8065	1.43788
$231$	0	0
$232$	1.67470	0.109950
$233$	−21.4822	−1.40735	−0.703673	−	0.710524i	$-0.748458\pi$
$233$	−21.4822	−1.40735	−0.703673	+	0.710524i	$0.748458\pi$
$234$	0	0
$235$	−24.8103	−1.61845
$236$	3.48614	0.226928
$237$	0	0
$238$	0	0
$239$	−1.08591	−0.0702414	−0.0351207	−	0.999383i	$-0.511182\pi$
$239$	−1.08591	−0.0702414	−0.0351207	+	0.999383i	$0.511182\pi$
$240$	0	0
$241$	−19.9830	−1.28722	−0.643608	−	0.765355i	$-0.722563\pi$
$241$	−19.9830	−1.28722	−0.643608	+	0.765355i	$0.722563\pi$
$242$	0.626319	0.0402613
$243$	0	0
$244$	−15.1818	−0.971915
$245$	0	0
$246$	0	0
$247$	4.06615	0.258723
$248$	−3.29988	−0.209542
$249$	0	0
$250$	−14.9452	−0.945218
$251$	−20.1497	−1.27184	−0.635920	−	0.771755i	$-0.719379\pi$
$251$	−20.1497	−1.27184	−0.635920	+	0.771755i	$0.719379\pi$
$252$	0	0
$253$	−14.7977	−0.930325
$254$	2.92230	0.183361
$255$	0	0
$256$	20.9663	1.31040
$257$	15.0174	0.936759	0.468379	−	0.883527i	$-0.344838\pi$
$257$	15.0174	0.936759	0.468379	+	0.883527i	$0.344838\pi$
$258$	0	0
$259$	0	0
$260$	−3.56440	−0.221055
$261$	0	0
$262$	7.88431	0.487094
$263$	21.1662	1.30516	0.652582	−	0.757718i	$-0.273686\pi$
$263$	21.1662	1.30516	0.652582	+	0.757718i	$0.273686\pi$
$264$	0	0
$265$	13.8219	0.849074
$266$	0	0
$267$	0	0
$268$	−2.69035	−0.164339
$269$	4.10480	0.250274	0.125137	−	0.992139i	$-0.460063\pi$
$269$	4.10480	0.250274	0.125137	+	0.992139i	$0.460063\pi$
$270$	0	0
$271$	3.43244	0.208506	0.104253	−	0.994551i	$-0.466755\pi$
$271$	3.43244	0.208506	0.104253	+	0.994551i	$0.466755\pi$
$272$	22.0718	1.33830
$273$	0	0
$274$	17.9982	1.08731
$275$	−6.18176	−0.372774
$276$	0	0
$277$	−0.361012	−0.0216911	−0.0108455	−	0.999941i	$-0.503452\pi$
$277$	−0.361012	−0.0216911	−0.0108455	+	0.999941i	$0.503452\pi$
$278$	−18.4724	−1.10790
$279$	0	0
$280$	0	0
$281$	−18.5213	−1.10489	−0.552445	−	0.833550i	$-0.686305\pi$
$281$	−18.5213	−1.10489	−0.552445	+	0.833550i	$0.686305\pi$
$282$	0	0
$283$	−1.61158	−0.0957983	−0.0478991	−	0.998852i	$-0.515253\pi$
$283$	−1.61158	−0.0957983	−0.0478991	+	0.998852i	$0.515253\pi$
$284$	15.9575	0.946906
$285$	0	0
$286$	5.98212	0.353730
$287$	0	0
$288$	0	0
$289$	3.52281	0.207224
$290$	6.84447	0.401921
$291$	0	0
$292$	16.4798	0.964410
$293$	4.41671	0.258027	0.129013	−	0.991643i	$-0.458819\pi$
$293$	4.41671	0.258027	0.129013	+	0.991643i	$0.458819\pi$
$294$	0	0
$295$	−6.74217	−0.392544
$296$	−11.8308	−0.687649
$297$	0	0
$298$	−25.3094	−1.46613
$299$	4.53266	0.262130
$300$	0	0
$301$	0	0
$302$	−38.8688	−2.23665
$303$	0	0
$304$	−19.8108	−1.13623
$305$	29.3615	1.68123
$306$	0	0
$307$	−5.78353	−0.330083	−0.165042	−	0.986287i	$-0.552776\pi$
$307$	−5.78353	−0.330083	−0.165042	+	0.986287i	$0.552776\pi$
$308$	0	0
$309$	0	0
$310$	−13.4865	−0.765983
$311$	−14.2895	−0.810284	−0.405142	−	0.914254i	$-0.632778\pi$
$311$	−14.2895	−0.810284	−0.405142	+	0.914254i	$0.632778\pi$
$312$	0	0
$313$	2.73725	0.154718	0.0773592	−	0.997003i	$-0.475351\pi$
$313$	2.73725	0.154718	0.0773592	+	0.997003i	$0.475351\pi$
$314$	−43.0892	−2.43166
$315$	0	0
$316$	16.1673	0.909481
$317$	−10.6512	−0.598231	−0.299116	−	0.954217i	$-0.596692\pi$
$317$	−10.6512	−0.598231	−0.299116	+	0.954217i	$0.596692\pi$
$318$	0	0
$319$	−4.64460	−0.260047
$320$	6.03975	0.337632
$321$	0	0
$322$	0	0
$323$	−18.4205	−1.02495
$324$	0	0
$325$	1.89352	0.105034
$326$	−14.3486	−0.794695
$327$	0	0
$328$	3.34629	0.184768
$329$	0	0
$330$	0	0
$331$	−3.41626	−0.187774	−0.0938872	−	0.995583i	$-0.529929\pi$
$331$	−3.41626	−0.187774	−0.0938872	+	0.995583i	$0.529929\pi$
$332$	−17.9517	−0.985228
$333$	0	0
$334$	23.2924	1.27450
$335$	5.20312	0.284277
$336$	0	0
$337$	24.9606	1.35969	0.679844	−	0.733357i	$-0.262047\pi$
$337$	24.9606	1.35969	0.679844	+	0.733357i	$0.262047\pi$
$338$	−1.83237	−0.0996678
$339$	0	0
$340$	16.1475	0.875720
$341$	9.15183	0.495599
$342$	0	0
$343$	0	0
$344$	11.4493	0.617306
$345$	0	0
$346$	9.61824	0.517080
$347$	19.1833	1.02981	0.514907	−	0.857246i	$-0.327827\pi$
$347$	19.1833	1.02981	0.514907	+	0.857246i	$0.327827\pi$
$348$	0	0
$349$	−3.94421	−0.211129	−0.105564	−	0.994412i	$-0.533665\pi$
$349$	−3.94421	−0.211129	−0.105564	+	0.994412i	$0.533665\pi$
$350$	0	0
$351$	0	0
$352$	−21.4596	−1.14380
$353$	−28.0232	−1.49152	−0.745762	−	0.666212i	$-0.767914\pi$
$353$	−28.0232	−1.49152	−0.745762	+	0.666212i	$0.767914\pi$
$354$	0	0
$355$	−30.8618	−1.63797
$356$	−14.4807	−0.767476
$357$	0	0
$358$	−19.2866	−1.01933
$359$	−33.2232	−1.75345	−0.876726	−	0.480989i	$-0.840278\pi$
$359$	−33.2232	−1.75345	−0.876726	+	0.480989i	$0.840278\pi$
$360$	0	0
$361$	−2.46641	−0.129811
$362$	34.4811	1.81228
$363$	0	0
$364$	0	0
$365$	−31.8719	−1.66825
$366$	0	0
$367$	−28.3091	−1.47772	−0.738862	−	0.673857i	$-0.764637\pi$
$367$	−28.3091	−1.47772	−0.738862	+	0.673857i	$0.764637\pi$
$368$	−22.0837	−1.15119
$369$	0	0
$370$	−48.3520	−2.51370
$371$	0	0
$372$	0	0
$373$	−12.9515	−0.670602	−0.335301	−	0.942111i	$-0.608838\pi$
$373$	−12.9515	−0.670602	−0.335301	+	0.942111i	$0.608838\pi$
$374$	−27.1003	−1.40132
$375$	0	0
$376$	11.1235	0.573653
$377$	1.42268	0.0732716
$378$	0	0
$379$	0.168981	0.00867995	0.00433997	−	0.999991i	$-0.498619\pi$
$379$	0.168981	0.00867995	0.00433997	+	0.999991i	$0.498619\pi$
$380$	−14.4934	−0.743495
$381$	0	0
$382$	27.3288	1.39826
$383$	−10.1933	−0.520854	−0.260427	−	0.965494i	$-0.583863\pi$
$383$	−10.1933	−0.520854	−0.260427	+	0.965494i	$0.583863\pi$
$384$	0	0
$385$	0	0
$386$	0.0973641	0.00495570
$387$	0	0
$388$	−18.6562	−0.947124
$389$	28.6665	1.45345	0.726723	−	0.686930i	$-0.241042\pi$
$389$	28.6665	1.45345	0.726723	+	0.686930i	$0.241042\pi$
$390$	0	0
$391$	−20.5339	−1.03844
$392$	0	0
$393$	0	0
$394$	18.4113	0.927547
$395$	−31.2674	−1.57323
$396$	0	0
$397$	−28.6411	−1.43746	−0.718728	−	0.695292i	$-0.755275\pi$
$397$	−28.6411	−1.43746	−0.718728	+	0.695292i	$0.755275\pi$
$398$	−42.6785	−2.13928
$399$	0	0
$400$	−9.22550	−0.461275
$401$	14.2963	0.713922	0.356961	−	0.934119i	$-0.383813\pi$
$401$	14.2963	0.713922	0.356961	+	0.934119i	$0.383813\pi$
$402$	0	0
$403$	−2.80328	−0.139641
$404$	−8.00237	−0.398133
$405$	0	0
$406$	0	0
$407$	32.8112	1.62639
$408$	0	0
$409$	25.6988	1.27072	0.635362	−	0.772215i	$-0.280851\pi$
$409$	25.6988	1.27072	0.635362	+	0.772215i	$0.280851\pi$
$410$	13.6762	0.675420
$411$	0	0
$412$	3.78408	0.186428
$413$	0	0
$414$	0	0
$415$	34.7185	1.70426
$416$	6.57326	0.322281
$417$	0	0
$418$	24.3242	1.18974
$419$	−18.7999	−0.918433	−0.459216	−	0.888324i	$-0.651870\pi$
$419$	−18.7999	−0.918433	−0.459216	+	0.888324i	$0.651870\pi$
$420$	0	0
$421$	−18.0283	−0.878645	−0.439322	−	0.898329i	$-0.644781\pi$
$421$	−18.0283	−0.878645	−0.439322	+	0.898329i	$0.644781\pi$
$422$	4.53432	0.220727
$423$	0	0
$424$	−6.19697	−0.300951
$425$	−8.57806	−0.416097
$426$	0	0
$427$	0	0
$428$	23.9813	1.15918
$429$	0	0
$430$	46.7931	2.25657
$431$	−20.5583	−0.990256	−0.495128	−	0.868820i	$-0.664879\pi$
$431$	−20.5583	−0.990256	−0.495128	+	0.868820i	$0.664879\pi$
$432$	0	0
$433$	18.9235	0.909404	0.454702	−	0.890644i	$-0.349746\pi$
$433$	18.9235	0.909404	0.454702	+	0.890644i	$0.349746\pi$
$434$	0	0
$435$	0	0
$436$	−12.9846	−0.621849
$437$	18.4305	0.881650
$438$	0	0
$439$	−14.6550	−0.699445	−0.349723	−	0.936853i	$-0.613724\pi$
$439$	−14.6550	−0.699445	−0.349723	+	0.936853i	$0.613724\pi$
$440$	10.0901	0.481025
$441$	0	0
$442$	8.30102	0.394839
$443$	11.9592	0.568201	0.284100	−	0.958795i	$-0.408305\pi$
$443$	11.9592	0.568201	0.284100	+	0.958795i	$0.408305\pi$
$444$	0	0
$445$	28.0056	1.32759
$446$	−11.3062	−0.535365
$447$	0	0
$448$	0	0
$449$	2.32245	0.109603	0.0548015	−	0.998497i	$-0.482547\pi$
$449$	2.32245	0.109603	0.0548015	+	0.998497i	$0.482547\pi$
$450$	0	0
$451$	−9.28055	−0.437004
$452$	23.8958	1.12396
$453$	0	0
$454$	21.2443	0.997044
$455$	0	0
$456$	0	0
$457$	−18.5805	−0.869157	−0.434579	−	0.900634i	$-0.643103\pi$
$457$	−18.5805	−0.869157	−0.434579	+	0.900634i	$0.643103\pi$
$458$	18.7601	0.876604
$459$	0	0
$460$	−16.1562	−0.753287
$461$	−27.8926	−1.29909	−0.649543	−	0.760325i	$-0.725040\pi$
$461$	−27.8926	−1.29909	−0.649543	+	0.760325i	$0.725040\pi$
$462$	0	0
$463$	−3.66462	−0.170309	−0.0851547	−	0.996368i	$-0.527138\pi$
$463$	−3.66462	−0.170309	−0.0851547	+	0.996368i	$0.527138\pi$
$464$	−6.93147	−0.321786
$465$	0	0
$466$	39.3634	1.82347
$467$	−38.7532	−1.79328	−0.896641	−	0.442757i	$-0.854000\pi$
$467$	−38.7532	−1.79328	−0.896641	+	0.442757i	$0.854000\pi$
$468$	0	0
$469$	0	0
$470$	45.4617	2.09699
$471$	0	0
$472$	3.02281	0.139136
$473$	−31.7534	−1.46002
$474$	0	0
$475$	7.69935	0.353270
$476$	0	0
$477$	0	0
$478$	1.98978	0.0910105
$479$	6.03430	0.275714	0.137857	−	0.990452i	$-0.455979\pi$
$479$	6.03430	0.275714	0.137857	+	0.990452i	$0.455979\pi$
$480$	0	0
$481$	−10.0503	−0.458256
$482$	36.6162	1.66782
$483$	0	0
$484$	−0.464032	−0.0210924
$485$	36.0809	1.63835
$486$	0	0
$487$	−3.80249	−0.172307	−0.0861537	−	0.996282i	$-0.527458\pi$
$487$	−3.80249	−0.172307	−0.0861537	+	0.996282i	$0.527458\pi$
$488$	−13.1640	−0.595908
$489$	0	0
$490$	0	0
$491$	0.381464	0.0172152	0.00860761	−	0.999963i	$-0.497260\pi$
$491$	0.381464	0.0172152	0.00860761	+	0.999963i	$0.497260\pi$
$492$	0	0
$493$	−6.44502	−0.290269
$494$	−7.45070	−0.335223
$495$	0	0
$496$	13.6580	0.613260
$497$	0	0
$498$	0	0
$499$	34.5739	1.54774	0.773871	−	0.633343i	$-0.218318\pi$
$499$	34.5739	1.54774	0.773871	+	0.633343i	$0.218318\pi$
$500$	11.0727	0.495187
$501$	0	0
$502$	36.9218	1.64790
$503$	2.41090	0.107497	0.0537485	−	0.998555i	$-0.482883\pi$
$503$	2.41090	0.107497	0.0537485	+	0.998555i	$0.482883\pi$
$504$	0	0
$505$	15.4765	0.688696
$506$	27.1149	1.20540
$507$	0	0
$508$	−2.16509	−0.0960605
$509$	21.9153	0.971380	0.485690	−	0.874131i	$-0.338568\pi$
$509$	21.9153	0.971380	0.485690	+	0.874131i	$0.338568\pi$
$510$	0	0
$511$	0	0
$512$	−20.5553	−0.908426
$513$	0	0
$514$	−27.5174	−1.21374
$515$	−7.31837	−0.322486
$516$	0	0
$517$	−30.8499	−1.35678
$518$	0	0
$519$	0	0
$520$	−3.09067	−0.135535
$521$	30.1450	1.32068	0.660338	−	0.750968i	$-0.270413\pi$
$521$	30.1450	1.32068	0.660338	+	0.750968i	$0.270413\pi$
$522$	0	0
$523$	−18.0993	−0.791428	−0.395714	−	0.918374i	$-0.629503\pi$
$523$	−18.0993	−0.791428	−0.395714	+	0.918374i	$0.629503\pi$
$524$	−5.84139	−0.255182
$525$	0	0
$526$	−38.7843	−1.69108
$527$	12.6994	0.553196
$528$	0	0
$529$	−2.45500	−0.106739
$530$	−25.3269	−1.10013
$531$	0	0
$532$	0	0
$533$	2.84271	0.123131
$534$	0	0
$535$	−46.3796	−2.00517
$536$	−2.33278	−0.100761
$537$	0	0
$538$	−7.52151	−0.324275
$539$	0	0
$540$	0	0
$541$	4.08890	0.175795	0.0878977	−	0.996130i	$-0.471985\pi$
$541$	4.08890	0.175795	0.0878977	+	0.996130i	$0.471985\pi$
$542$	−6.28950	−0.270157
$543$	0	0
$544$	−29.7782	−1.27673
$545$	25.1121	1.07568
$546$	0	0
$547$	−40.4264	−1.72851	−0.864255	−	0.503055i	$-0.832209\pi$
$547$	−40.4264	−1.72851	−0.864255	+	0.503055i	$0.832209\pi$
$548$	−13.3346	−0.569627
$549$	0	0
$550$	11.3273	0.482997
$551$	5.78482	0.246442
$552$	0	0
$553$	0	0
$554$	0.661507	0.0281047
$555$	0	0
$556$	13.6859	0.580413
$557$	−19.7690	−0.837640	−0.418820	−	0.908069i	$-0.637556\pi$
$557$	−19.7690	−0.837640	−0.418820	+	0.908069i	$0.637556\pi$
$558$	0	0
$559$	9.72632	0.411379
$560$	0	0
$561$	0	0
$562$	33.9379	1.43158
$563$	−8.22392	−0.346597	−0.173298	−	0.984869i	$-0.555443\pi$
$563$	−8.22392	−0.346597	−0.173298	+	0.984869i	$0.555443\pi$
$564$	0	0
$565$	−46.2143	−1.94425
$566$	2.95300	0.124124
$567$	0	0
$568$	13.8367	0.580574
$569$	−14.5770	−0.611099	−0.305550	−	0.952176i	$-0.598840\pi$
$569$	−14.5770	−0.611099	−0.305550	+	0.952176i	$0.598840\pi$
$570$	0	0
$571$	−2.58822	−0.108314	−0.0541568	−	0.998532i	$-0.517247\pi$
$571$	−2.58822	−0.108314	−0.0541568	+	0.998532i	$0.517247\pi$
$572$	−4.43208	−0.185315
$573$	0	0
$574$	0	0
$575$	8.58269	0.357923
$576$	0	0
$577$	−23.8937	−0.994707	−0.497353	−	0.867548i	$-0.665695\pi$
$577$	−23.8937	−0.994707	−0.497353	+	0.867548i	$0.665695\pi$
$578$	−6.45509	−0.268496
$579$	0	0
$580$	−5.07099	−0.210561
$581$	0	0
$582$	0	0
$583$	17.1866	0.711795
$584$	14.2896	0.591306
$585$	0	0
$586$	−8.09304	−0.334320
$587$	−28.5759	−1.17945	−0.589726	−	0.807604i	$-0.700764\pi$
$587$	−28.5759	−1.17945	−0.589726	+	0.807604i	$0.700764\pi$
$588$	0	0
$589$	−11.3986	−0.469669
$590$	12.3541	0.508612
$591$	0	0
$592$	48.9666	2.01252
$593$	19.9575	0.819556	0.409778	−	0.912185i	$-0.365606\pi$
$593$	19.9575	0.819556	0.409778	+	0.912185i	$0.365606\pi$
$594$	0	0
$595$	0	0
$596$	18.7514	0.768088
$597$	0	0
$598$	−8.30551	−0.339638
$599$	−1.17716	−0.0480973	−0.0240486	−	0.999711i	$-0.507656\pi$
$599$	−1.17716	−0.0480973	−0.0240486	+	0.999711i	$0.507656\pi$
$600$	0	0
$601$	−30.9250	−1.26146	−0.630729	−	0.776003i	$-0.717244\pi$
$601$	−30.9250	−1.26146	−0.630729	+	0.776003i	$0.717244\pi$
$602$	0	0
$603$	0	0
$604$	28.7974	1.17175
$605$	0.897435	0.0364859
$606$	0	0
$607$	2.40708	0.0977005	0.0488503	−	0.998806i	$-0.484444\pi$
$607$	2.40708	0.0977005	0.0488503	+	0.998806i	$0.484444\pi$
$608$	26.7279	1.08396
$609$	0	0
$610$	−53.8011	−2.17834
$611$	9.44956	0.382288
$612$	0	0
$613$	−27.7742	−1.12179	−0.560895	−	0.827887i	$-0.689543\pi$
$613$	−27.7742	−1.12179	−0.560895	+	0.827887i	$0.689543\pi$
$614$	10.5976	0.427683
$615$	0	0
$616$	0	0
$617$	−26.2125	−1.05527	−0.527637	−	0.849470i	$-0.676922\pi$
$617$	−26.2125	−1.05527	−0.527637	+	0.849470i	$0.676922\pi$
$618$	0	0
$619$	−17.2357	−0.692763	−0.346381	−	0.938094i	$-0.612590\pi$
$619$	−17.2357	−0.692763	−0.346381	+	0.938094i	$0.612590\pi$
$620$	9.99200	0.401288
$621$	0	0
$622$	26.1837	1.04987
$623$	0	0
$624$	0	0
$625$	−30.8822	−1.23529
$626$	−5.01565	−0.200466
$627$	0	0
$628$	31.9242	1.27392
$629$	45.5302	1.81541
$630$	0	0
$631$	39.2125	1.56103	0.780513	−	0.625140i	$-0.214958\pi$
$631$	39.2125	1.56103	0.780513	+	0.625140i	$0.214958\pi$
$632$	14.0185	0.557628
$633$	0	0
$634$	19.5170	0.775117
$635$	4.18728	0.166167
$636$	0	0
$637$	0	0
$638$	8.51062	0.336939
$639$	0	0
$640$	23.4498	0.926935
$641$	22.8735	0.903451	0.451725	−	0.892157i	$-0.350809\pi$
$641$	22.8735	0.903451	0.451725	+	0.892157i	$0.350809\pi$
$642$	0	0
$643$	−46.7072	−1.84195	−0.920976	−	0.389620i	$-0.872606\pi$
$643$	−46.7072	−1.84195	−0.920976	+	0.389620i	$0.872606\pi$
$644$	0	0
$645$	0	0
$646$	33.7532	1.32800
$647$	−5.93075	−0.233162	−0.116581	−	0.993181i	$-0.537193\pi$
$647$	−5.93075	−0.233162	−0.116581	+	0.993181i	$0.537193\pi$
$648$	0	0
$649$	−8.38341	−0.329078
$650$	−3.46963	−0.136090
$651$	0	0
$652$	10.6307	0.416330
$653$	18.8046	0.735881	0.367940	−	0.929849i	$-0.380063\pi$
$653$	18.8046	0.735881	0.367940	+	0.929849i	$0.380063\pi$
$654$	0	0
$655$	11.2972	0.441419
$656$	−13.8500	−0.540754
$657$	0	0
$658$	0	0
$659$	−23.1086	−0.900184	−0.450092	−	0.892982i	$-0.648609\pi$
$659$	−23.1086	−0.900184	−0.450092	+	0.892982i	$0.648609\pi$
$660$	0	0
$661$	−33.9087	−1.31889	−0.659447	−	0.751751i	$-0.729210\pi$
$661$	−33.9087	−1.31889	−0.659447	+	0.751751i	$0.729210\pi$
$662$	6.25985	0.243296
$663$	0	0
$664$	−15.5658	−0.604071
$665$	0	0
$666$	0	0
$667$	6.44851	0.249687
$668$	−17.2570	−0.667695
$669$	0	0
$670$	−9.53404	−0.368332
$671$	36.5090	1.40941
$672$	0	0
$673$	6.00430	0.231449	0.115724	−	0.993281i	$-0.463081\pi$
$673$	6.00430	0.231449	0.115724	+	0.993281i	$0.463081\pi$
$674$	−45.7370	−1.76172
$675$	0	0
$676$	1.35758	0.0522147
$677$	−20.2075	−0.776637	−0.388318	−	0.921525i	$-0.626944\pi$
$677$	−20.2075	−0.776637	−0.388318	+	0.921525i	$0.626944\pi$
$678$	0	0
$679$	0	0
$680$	14.0014	0.536928
$681$	0	0
$682$	−16.7695	−0.642139
$683$	31.6508	1.21108	0.605541	−	0.795814i	$-0.292957\pi$
$683$	31.6508	1.21108	0.605541	+	0.795814i	$0.292957\pi$
$684$	0	0
$685$	25.7891	0.985350
$686$	0	0
$687$	0	0
$688$	−47.3879	−1.80665
$689$	−5.26439	−0.200557
$690$	0	0
$691$	9.69199	0.368701	0.184350	−	0.982861i	$-0.440982\pi$
$691$	9.69199	0.368701	0.184350	+	0.982861i	$0.440982\pi$
$692$	−7.12604	−0.270891
$693$	0	0
$694$	−35.1509	−1.33431
$695$	−26.4685	−1.00401
$696$	0	0
$697$	−12.8781	−0.487791
$698$	7.22725	0.273556
$699$	0	0
$700$	0	0
$701$	5.10365	0.192762	0.0963811	−	0.995345i	$-0.469273\pi$
$701$	5.10365	0.192762	0.0963811	+	0.995345i	$0.469273\pi$
$702$	0	0
$703$	−40.8662	−1.54130
$704$	7.51000	0.283044
$705$	0	0
$706$	51.3489	1.93254
$707$	0	0
$708$	0	0
$709$	26.1445	0.981878	0.490939	−	0.871194i	$-0.336654\pi$
$709$	26.1445	0.981878	0.490939	+	0.871194i	$0.336654\pi$
$710$	56.5502	2.12229
$711$	0	0
$712$	−12.5561	−0.470561
$713$	−12.7063	−0.475855
$714$	0	0
$715$	8.57161	0.320560
$716$	14.2892	0.534014
$717$	0	0
$718$	60.8772	2.27192
$719$	8.72884	0.325531	0.162765	−	0.986665i	$-0.447959\pi$
$719$	8.72884	0.325531	0.162765	+	0.986665i	$0.447959\pi$
$720$	0	0
$721$	0	0
$722$	4.51937	0.168193
$723$	0	0
$724$	−25.5466	−0.949432
$725$	2.69387	0.100048
$726$	0	0
$727$	−21.8712	−0.811158	−0.405579	−	0.914060i	$-0.632930\pi$
$727$	−21.8712	−0.811158	−0.405579	+	0.914060i	$0.632930\pi$
$728$	0	0
$729$	0	0
$730$	58.4011	2.16152
$731$	−44.0623	−1.62970
$732$	0	0
$733$	−35.5640	−1.31359	−0.656793	−	0.754071i	$-0.728088\pi$
$733$	−35.5640	−1.31359	−0.656793	+	0.754071i	$0.728088\pi$
$734$	51.8728	1.91466
$735$	0	0
$736$	29.7943	1.09823
$737$	6.46971	0.238315
$738$	0	0
$739$	−36.4894	−1.34228	−0.671142	−	0.741329i	$-0.734196\pi$
$739$	−36.4894	−1.34228	−0.671142	+	0.741329i	$0.734196\pi$
$740$	35.8234	1.31690
$741$	0	0
$742$	0	0
$743$	−12.4588	−0.457071	−0.228535	−	0.973536i	$-0.573394\pi$
$743$	−12.4588	−0.457071	−0.228535	+	0.973536i	$0.573394\pi$
$744$	0	0
$745$	−36.2651	−1.32865
$746$	23.7319	0.868886
$747$	0	0
$748$	20.0783	0.734134
$749$	0	0
$750$	0	0
$751$	−18.9424	−0.691217	−0.345608	−	0.938379i	$-0.612327\pi$
$751$	−18.9424	−0.691217	−0.345608	+	0.938379i	$0.612327\pi$
$752$	−46.0395	−1.67889
$753$	0	0
$754$	−2.60687	−0.0949366
$755$	−55.6940	−2.02691
$756$	0	0
$757$	−25.0956	−0.912114	−0.456057	−	0.889951i	$-0.650739\pi$
$757$	−25.0956	−0.912114	−0.456057	+	0.889951i	$0.650739\pi$
$758$	−0.309635	−0.0112464
$759$	0	0
$760$	−12.5671	−0.455857
$761$	−28.0617	−1.01723	−0.508617	−	0.860993i	$-0.669843\pi$
$761$	−28.0617	−1.01723	−0.508617	+	0.860993i	$0.669843\pi$
$762$	0	0
$763$	0	0
$764$	−20.2476	−0.732531
$765$	0	0
$766$	18.6779	0.674861
$767$	2.56791	0.0927217
$768$	0	0
$769$	−23.2636	−0.838907	−0.419454	−	0.907777i	$-0.637778\pi$
$769$	−23.2636	−0.838907	−0.419454	+	0.907777i	$0.637778\pi$
$770$	0	0
$771$	0	0
$772$	−0.0721359	−0.00259623
$773$	−45.8118	−1.64774	−0.823869	−	0.566781i	$-0.808189\pi$
$773$	−45.8118	−1.64774	−0.823869	+	0.566781i	$0.808189\pi$
$774$	0	0
$775$	−5.30807	−0.190672
$776$	−16.1766	−0.580708
$777$	0	0
$778$	−52.5276	−1.88320
$779$	11.5589	0.414140
$780$	0	0
$781$	−38.3744	−1.37315
$782$	37.6257	1.34549
$783$	0	0
$784$	0	0
$785$	−61.7413	−2.20364
$786$	0	0
$787$	31.2715	1.11471	0.557355	−	0.830274i	$-0.311816\pi$
$787$	31.2715	1.11471	0.557355	+	0.830274i	$0.311816\pi$
$788$	−13.6407	−0.485929
$789$	0	0
$790$	57.2935	2.03841
$791$	0	0
$792$	0	0
$793$	−11.1830	−0.397119
$794$	52.4811	1.86248
$795$	0	0
$796$	31.6200	1.12074
$797$	11.6121	0.411323	0.205661	−	0.978623i	$-0.434065\pi$
$797$	11.6121	0.411323	0.205661	+	0.978623i	$0.434065\pi$
$798$	0	0
$799$	−42.8085	−1.51446
$800$	12.4466	0.440054
$801$	0	0
$802$	−26.1961	−0.925016
$803$	−39.6304	−1.39853
$804$	0	0
$805$	0	0
$806$	5.13664	0.180931
$807$	0	0
$808$	−6.93880	−0.244106
$809$	−7.30665	−0.256888	−0.128444	−	0.991717i	$-0.540998\pi$
$809$	−7.30665	−0.256888	−0.128444	+	0.991717i	$0.540998\pi$
$810$	0	0
$811$	−15.9662	−0.560651	−0.280325	−	0.959905i	$-0.590442\pi$
$811$	−15.9662	−0.560651	−0.280325	+	0.959905i	$0.590442\pi$
$812$	0	0
$813$	0	0
$814$	−60.1223	−2.10729
$815$	−20.5597	−0.720175
$816$	0	0
$817$	39.5487	1.38363
$818$	−47.0897	−1.64645
$819$	0	0
$820$	−10.1325	−0.353844
$821$	−1.12788	−0.0393634	−0.0196817	−	0.999806i	$-0.506265\pi$
$821$	−1.12788	−0.0393634	−0.0196817	+	0.999806i	$0.506265\pi$
$822$	0	0
$823$	22.4044	0.780968	0.390484	−	0.920610i	$-0.372308\pi$
$823$	22.4044	0.780968	0.390484	+	0.920610i	$0.372308\pi$
$824$	3.28115	0.114304
$825$	0	0
$826$	0	0
$827$	6.32296	0.219871	0.109935	−	0.993939i	$-0.464936\pi$
$827$	6.32296	0.219871	0.109935	+	0.993939i	$0.464936\pi$
$828$	0	0
$829$	−44.3704	−1.54105	−0.770523	−	0.637412i	$-0.780005\pi$
$829$	−44.3704	−1.54105	−0.770523	+	0.637412i	$0.780005\pi$
$830$	−63.6171	−2.20818
$831$	0	0
$832$	−2.30037	−0.0797510
$833$	0	0
$834$	0	0
$835$	33.3750	1.15499
$836$	−18.0215	−0.623287
$837$	0	0
$838$	34.4483	1.19000
$839$	9.89476	0.341605	0.170803	−	0.985305i	$-0.445364\pi$
$839$	9.89476	0.341605	0.170803	+	0.985305i	$0.445364\pi$
$840$	0	0
$841$	−26.9760	−0.930207
$842$	33.0345	1.13844
$843$	0	0
$844$	−3.35942	−0.115636
$845$	−2.62555	−0.0903217
$846$	0	0
$847$	0	0
$848$	25.6488	0.880783
$849$	0	0
$850$	15.7182	0.539129
$851$	−45.5548	−1.56160
$852$	0	0
$853$	25.9407	0.888193	0.444097	−	0.895979i	$-0.353525\pi$
$853$	25.9407	0.888193	0.444097	+	0.895979i	$0.353525\pi$
$854$	0	0
$855$	0	0
$856$	20.7940	0.710725
$857$	−10.0244	−0.342426	−0.171213	−	0.985234i	$-0.554769\pi$
$857$	−10.0244	−0.342426	−0.171213	+	0.985234i	$0.554769\pi$
$858$	0	0
$859$	37.4378	1.27736	0.638680	−	0.769472i	$-0.279481\pi$
$859$	37.4378	1.27736	0.638680	+	0.769472i	$0.279481\pi$
$860$	−34.6685	−1.18219
$861$	0	0
$862$	37.6703	1.28306
$863$	−21.0026	−0.714938	−0.357469	−	0.933925i	$-0.616360\pi$
$863$	−21.0026	−0.714938	−0.357469	+	0.933925i	$0.616360\pi$
$864$	0	0
$865$	13.7817	0.468592
$866$	−34.6748	−1.17830
$867$	0	0
$868$	0	0
$869$	−38.8788	−1.31887
$870$	0	0
$871$	−1.98172	−0.0671481
$872$	−11.2588	−0.381273
$873$	0	0
$874$	−33.7715	−1.14234
$875$	0	0
$876$	0	0
$877$	7.56302	0.255385	0.127692	−	0.991814i	$-0.459243\pi$
$877$	7.56302	0.255385	0.127692	+	0.991814i	$0.459243\pi$
$878$	26.8534	0.906258
$879$	0	0
$880$	−41.7620	−1.40780
$881$	34.3550	1.15745	0.578725	−	0.815522i	$-0.303550\pi$
$881$	34.3550	1.15745	0.578725	+	0.815522i	$0.303550\pi$
$882$	0	0
$883$	49.6697	1.67152	0.835759	−	0.549096i	$-0.185028\pi$
$883$	49.6697	1.67152	0.835759	+	0.549096i	$0.185028\pi$
$884$	−6.15013	−0.206851
$885$	0	0
$886$	−21.9138	−0.736207
$887$	48.0325	1.61277	0.806386	−	0.591389i	$-0.201420\pi$
$887$	48.0325	1.61277	0.806386	+	0.591389i	$0.201420\pi$
$888$	0	0
$889$	0	0
$890$	−51.3166	−1.72014
$891$	0	0
$892$	8.37665	0.280471
$893$	38.4234	1.28579
$894$	0	0
$895$	−27.6353	−0.923745
$896$	0	0
$897$	0	0
$898$	−4.25558	−0.142011
$899$	−3.98816	−0.133013
$900$	0	0
$901$	23.8488	0.794518
$902$	17.0054	0.566218
$903$	0	0
$904$	20.7199	0.689133
$905$	49.4069	1.64234
$906$	0	0
$907$	2.40195	0.0797556	0.0398778	−	0.999205i	$-0.487303\pi$
$907$	2.40195	0.0797556	0.0398778	+	0.999205i	$0.487303\pi$
$908$	−15.7396	−0.522338
$909$	0	0
$910$	0	0
$911$	33.9555	1.12500	0.562499	−	0.826798i	$-0.309840\pi$
$911$	33.9555	1.12500	0.562499	+	0.826798i	$0.309840\pi$
$912$	0	0
$913$	43.1700	1.42872
$914$	34.0463	1.12615
$915$	0	0
$916$	−13.8992	−0.459241
$917$	0	0
$918$	0	0
$919$	−9.37134	−0.309132	−0.154566	−	0.987982i	$-0.549398\pi$
$919$	−9.37134	−0.309132	−0.154566	+	0.987982i	$0.549398\pi$
$920$	−14.0089	−0.461861
$921$	0	0
$922$	51.1095	1.68320
$923$	11.7544	0.386901
$924$	0	0
$925$	−19.0306	−0.625721
$926$	6.71494	0.220667
$927$	0	0
$928$	9.35162	0.306982
$929$	43.4273	1.42480	0.712402	−	0.701771i	$-0.247607\pi$
$929$	43.4273	1.42480	0.712402	+	0.701771i	$0.247607\pi$
$930$	0	0
$931$	0	0
$932$	−29.1639	−0.955294
$933$	0	0
$934$	71.0102	2.32352
$935$	−38.8312	−1.26992
$936$	0	0
$937$	2.59416	0.0847476	0.0423738	−	0.999102i	$-0.486508\pi$
$937$	2.59416	0.0847476	0.0423738	+	0.999102i	$0.486508\pi$
$938$	0	0
$939$	0	0
$940$	−33.6820	−1.09859
$941$	−43.4060	−1.41499	−0.707497	−	0.706716i	$-0.750176\pi$
$941$	−43.4060	−1.41499	−0.707497	+	0.706716i	$0.750176\pi$
$942$	0	0
$943$	12.8850	0.419594
$944$	−12.5112	−0.407204
$945$	0	0
$946$	58.1840	1.89172
$947$	7.50058	0.243736	0.121868	−	0.992546i	$-0.461112\pi$
$947$	7.50058	0.243736	0.121868	+	0.992546i	$0.461112\pi$
$948$	0	0
$949$	12.1391	0.394052
$950$	−14.1081	−0.457726
$951$	0	0
$952$	0	0
$953$	0.708439	0.0229486	0.0114743	−	0.999934i	$-0.496348\pi$
$953$	0.708439	0.0229486	0.0114743	+	0.999934i	$0.496348\pi$
$954$	0	0
$955$	39.1586	1.26714
$956$	−1.47421	−0.0476792
$957$	0	0
$958$	−11.0571	−0.357238
$959$	0	0
$960$	0	0
$961$	−23.1416	−0.746504
$962$	18.4159	0.593754
$963$	0	0
$964$	−27.1285	−0.873750
$965$	0.139510	0.00449100
$966$	0	0
$967$	−16.9761	−0.545915	−0.272957	−	0.962026i	$-0.588002\pi$
$967$	−16.9761	−0.545915	−0.272957	+	0.962026i	$0.588002\pi$
$968$	−0.402359	−0.0129323
$969$	0	0
$970$	−66.1136	−2.12278
$971$	6.86359	0.220263	0.110132	−	0.993917i	$-0.464873\pi$
$971$	6.86359	0.220263	0.110132	+	0.993917i	$0.464873\pi$
$972$	0	0
$973$	0	0
$974$	6.96757	0.223255
$975$	0	0
$976$	54.4850	1.74402
$977$	−0.581913	−0.0186171	−0.00930853	−	0.999957i	$-0.502963\pi$
$977$	−0.581913	−0.0186171	−0.00930853	+	0.999957i	$0.502963\pi$
$978$	0	0
$979$	34.8230	1.11295
$980$	0	0
$981$	0	0
$982$	−0.698983	−0.0223054
$983$	−34.7195	−1.10738	−0.553689	−	0.832723i	$-0.686780\pi$
$983$	−34.7195	−1.10738	−0.553689	+	0.832723i	$0.686780\pi$
$984$	0	0
$985$	26.3810	0.840568
$986$	11.8097	0.376097
$987$	0	0
$988$	5.52013	0.175619
$989$	44.0861	1.40186
$990$	0	0
$991$	24.1688	0.767748	0.383874	−	0.923385i	$-0.374590\pi$
$991$	24.1688	0.767748	0.383874	+	0.923385i	$0.374590\pi$
$992$	−18.4267	−0.585047
$993$	0	0
$994$	0	0
$995$	−61.1528	−1.93868
$996$	0	0
$997$	−35.1852	−1.11433	−0.557163	−	0.830403i	$-0.688110\pi$
$997$	−35.1852	−1.11433	−0.557163	+	0.830403i	$0.688110\pi$
$998$	−63.3523	−2.00538
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.br.1.2		6
3.2	odd	2		637.2.a.n.1.5	yes	6
7.6	odd	2		5733.2.a.bu.1.2		6
21.2	odd	6		637.2.e.n.508.2		12
21.5	even	6		637.2.e.o.508.2		12
21.11	odd	6		637.2.e.n.79.2		12
21.17	even	6		637.2.e.o.79.2		12
21.20	even	2		637.2.a.m.1.5	✓	6
39.38	odd	2		8281.2.a.cd.1.2		6
273.272	even	2		8281.2.a.cc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.5	✓	6	21.20	even	2
637.2.a.n.1.5	yes	6	3.2	odd	2
637.2.e.n.79.2		12	21.11	odd	6
637.2.e.n.508.2		12	21.2	odd	6
637.2.e.o.79.2		12	21.17	even	6
637.2.e.o.508.2		12	21.5	even	6
5733.2.a.br.1.2		6	1.1	even	1	trivial
5733.2.a.bu.1.2		6	7.6	odd	2
8281.2.a.cc.1.2		6	273.272	even	2
8281.2.a.cd.1.2		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)

\(f(q)\)	\(=\)	\(q-1.83237 q^{2} +1.35758 q^{4} -2.62555 q^{5} +1.17715 q^{8} +O(q^{10})\)	\(q-1.83237 q^{2} +1.35758 q^{4} -2.62555 q^{5} +1.17715 q^{8} +4.81098 q^{10} -3.26469 q^{11} +1.00000 q^{13} -4.87214 q^{16} -4.53021 q^{17} +4.06615 q^{19} -3.56440 q^{20} +5.98212 q^{22} +4.53266 q^{23} +1.89352 q^{25} -1.83237 q^{26} +1.42268 q^{29} -2.80328 q^{31} +6.57326 q^{32} +8.30102 q^{34} -10.0503 q^{37} -7.45070 q^{38} -3.09067 q^{40} +2.84271 q^{41} +9.72632 q^{43} -4.43208 q^{44} -8.30551 q^{46} +9.44956 q^{47} -3.46963 q^{50} +1.35758 q^{52} -5.26439 q^{53} +8.57161 q^{55} -2.60687 q^{58} +2.56791 q^{59} -11.1830 q^{61} +5.13664 q^{62} -2.30037 q^{64} -2.62555 q^{65} -1.98172 q^{67} -6.15013 q^{68} +11.7544 q^{71} +12.1391 q^{73} +18.4159 q^{74} +5.52013 q^{76} +11.9089 q^{79} +12.7920 q^{80} -5.20889 q^{82} -13.2233 q^{83} +11.8943 q^{85} -17.8222 q^{86} -3.84303 q^{88} -10.6666 q^{89} +6.15345 q^{92} -17.3151 q^{94} -10.6759 q^{95} -13.7422 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{5}+O(q^{10}) \) 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

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Modular forms

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