LMFDB - Embedded newform 5733.2.a.ca.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 92x^{8} - 228x^{6} + 225x^{4} - 60x^{2} + 4 $ x^12 - 16x^10 + 92x^8 - 228x^6 + 225x^4 - 60*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.80357$ 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)

$f(q)$	$=$	$q-1.80357 q^{2} +1.25286 q^{4} -2.82630 q^{5} +1.34752 q^{8} +O(q^{10})$	$q-1.80357 q^{2} +1.25286 q^{4} -2.82630 q^{5} +1.34752 q^{8} +5.09742 q^{10} +4.59281 q^{11} -1.00000 q^{13} -4.93606 q^{16} +1.10891 q^{17} -4.36428 q^{19} -3.54095 q^{20} -8.28344 q^{22} +5.18803 q^{23} +2.98796 q^{25} +1.80357 q^{26} -7.33174 q^{29} -0.141436 q^{31} +6.20748 q^{32} -2.00000 q^{34} +1.79053 q^{37} +7.87127 q^{38} -3.80851 q^{40} -6.09158 q^{41} +2.16426 q^{43} +5.75413 q^{44} -9.35696 q^{46} +4.59221 q^{47} -5.38900 q^{50} -1.25286 q^{52} +10.7412 q^{53} -12.9806 q^{55} +13.2233 q^{58} +6.63170 q^{59} -14.7006 q^{61} +0.255089 q^{62} -1.32348 q^{64} +2.82630 q^{65} -6.85125 q^{67} +1.38931 q^{68} -5.54024 q^{71} -12.8420 q^{73} -3.22934 q^{74} -5.46781 q^{76} -1.66525 q^{79} +13.9508 q^{80} +10.9866 q^{82} +10.9639 q^{83} -3.13412 q^{85} -3.90338 q^{86} +6.18892 q^{88} +1.16985 q^{89} +6.49986 q^{92} -8.28236 q^{94} +12.3348 q^{95} +9.06483 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4}+O(q^{10}) $ 12 * q + 8 * q^4	$ 12 q + 8 q^{4} - 8 q^{10} - 12 q^{13} - 28 q^{19} + 16 q^{25} - 12 q^{31} - 24 q^{34} - 32 q^{40} - 20 q^{43} - 8 q^{46} - 8 q^{52} - 56 q^{55} + 8 q^{58} - 24 q^{61} - 40 q^{64} + 8 q^{67} - 12 q^{73} - 64 q^{76} - 20 q^{79} - 40 q^{82} + 32 q^{85} + 8 q^{88} - 8 q^{94} - 68 q^{97}+O(q^{100}) $ 12 * q + 8 * q^4 - 8 * q^10 - 12 * q^13 - 28 * q^19 + 16 * q^25 - 12 * q^31 - 24 * q^34 - 32 * q^40 - 20 * q^43 - 8 * q^46 - 8 * q^52 - 56 * q^55 + 8 * q^58 - 24 * q^61 - 40 * q^64 + 8 * q^67 - 12 * q^73 - 64 * q^76 - 20 * q^79 - 40 * q^82 + 32 * q^85 + 8 * q^88 - 8 * q^94 - 68 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.80357	−1.27531	−0.637657	−	0.770320i	$-0.720096\pi$
$2$	−1.80357	−1.27531	−0.637657	+	0.770320i	$0.720096\pi$
$3$	0	0
$3$	0	0
$4$	1.25286	0.626428
$5$	−2.82630	−1.26396	−0.631980	−	0.774985i	$-0.717757\pi$
$5$	−2.82630	−1.26396	−0.631980	+	0.774985i	$0.717757\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.34752	0.476422
$9$	0	0
$10$	5.09742	1.61195
$11$	4.59281	1.38478	0.692392	−	0.721522i	$-0.256557\pi$
$11$	4.59281	1.38478	0.692392	+	0.721522i	$0.256557\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	−4.93606	−1.23402
$17$	1.10891	0.268951	0.134475	−	0.990917i	$-0.457065\pi$
$17$	1.10891	0.268951	0.134475	+	0.990917i	$0.457065\pi$
$18$	0	0
$19$	−4.36428	−1.00123	−0.500617	−	0.865669i	$-0.666893\pi$
$19$	−4.36428	−1.00123	−0.500617	+	0.865669i	$0.666893\pi$
$20$	−3.54095	−0.791780
$21$	0	0
$22$	−8.28344	−1.76604
$23$	5.18803	1.08178	0.540889	−	0.841094i	$-0.318088\pi$
$23$	5.18803	1.08178	0.540889	+	0.841094i	$0.318088\pi$
$24$	0	0
$25$	2.98796	0.597593
$26$	1.80357	0.353709
$27$	0	0
$28$	0	0
$29$	−7.33174	−1.36147	−0.680735	−	0.732530i	$-0.738339\pi$
$29$	−7.33174	−1.36147	−0.680735	+	0.732530i	$0.738339\pi$
$30$	0	0
$31$	−0.141436	−0.0254026	−0.0127013	−	0.999919i	$-0.504043\pi$
$31$	−0.141436	−0.0254026	−0.0127013	+	0.999919i	$0.504043\pi$
$32$	6.20748	1.09734
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	1.79053	0.294361	0.147180	−	0.989110i	$-0.452980\pi$
$37$	1.79053	0.294361	0.147180	+	0.989110i	$0.452980\pi$
$38$	7.87127	1.27689
$39$	0	0
$40$	−3.80851	−0.602178
$41$	−6.09158	−0.951344	−0.475672	−	0.879623i	$-0.657795\pi$
$41$	−6.09158	−0.951344	−0.475672	+	0.879623i	$0.657795\pi$
$42$	0	0
$43$	2.16426	0.330046	0.165023	−	0.986290i	$-0.447230\pi$
$43$	2.16426	0.330046	0.165023	+	0.986290i	$0.447230\pi$
$44$	5.75413	0.867468
$45$	0	0
$46$	−9.35696	−1.37961
$47$	4.59221	0.669843	0.334921	−	0.942246i	$-0.391290\pi$
$47$	4.59221	0.669843	0.334921	+	0.942246i	$0.391290\pi$
$48$	0	0
$49$	0	0
$50$	−5.38900	−0.762119
$51$	0	0
$52$	−1.25286	−0.173740
$53$	10.7412	1.47542	0.737709	−	0.675119i	$-0.235908\pi$
$53$	10.7412	1.47542	0.737709	+	0.675119i	$0.235908\pi$
$54$	0	0
$55$	−12.9806	−1.75031
$56$	0	0
$57$	0	0
$58$	13.2233	1.73630
$59$	6.63170	0.863374	0.431687	−	0.902023i	$-0.357918\pi$
$59$	6.63170	0.863374	0.431687	+	0.902023i	$0.357918\pi$
$60$	0	0
$61$	−14.7006	−1.88221	−0.941107	−	0.338110i	$-0.890212\pi$
$61$	−14.7006	−1.88221	−0.941107	+	0.338110i	$0.890212\pi$
$62$	0.255089	0.0323963
$63$	0	0
$64$	−1.32348	−0.165435
$65$	2.82630	0.350559
$66$	0	0
$67$	−6.85125	−0.837013	−0.418507	−	0.908214i	$-0.637446\pi$
$67$	−6.85125	−0.837013	−0.418507	+	0.908214i	$0.637446\pi$
$68$	1.38931	0.168478
$69$	0	0
$70$	0	0
$71$	−5.54024	−0.657505	−0.328753	−	0.944416i	$-0.606628\pi$
$71$	−5.54024	−0.657505	−0.328753	+	0.944416i	$0.606628\pi$
$72$	0	0
$73$	−12.8420	−1.50304	−0.751521	−	0.659710i	$-0.770679\pi$
$73$	−12.8420	−1.50304	−0.751521	+	0.659710i	$0.770679\pi$
$74$	−3.22934	−0.375403
$75$	0	0
$76$	−5.46781	−0.627201
$77$	0	0
$78$	0	0
$79$	−1.66525	−0.187355	−0.0936776	−	0.995603i	$-0.529862\pi$
$79$	−1.66525	−0.187355	−0.0936776	+	0.995603i	$0.529862\pi$
$80$	13.9508	1.55975
$81$	0	0
$82$	10.9866	1.21326
$83$	10.9639	1.20344	0.601720	−	0.798707i	$-0.294482\pi$
$83$	10.9639	1.20344	0.601720	+	0.798707i	$0.294482\pi$
$84$	0	0
$85$	−3.13412	−0.339943
$86$	−3.90338	−0.420913
$87$	0	0
$88$	6.18892	0.659741
$89$	1.16985	0.124003	0.0620017	−	0.998076i	$-0.480252\pi$
$89$	1.16985	0.124003	0.0620017	+	0.998076i	$0.480252\pi$
$90$	0	0
$91$	0	0
$92$	6.49986	0.677657
$93$	0	0
$94$	−8.28236	−0.854260
$95$	12.3348	1.26552
$96$	0	0
$97$	9.06483	0.920394	0.460197	−	0.887817i	$-0.347779\pi$
$97$	9.06483	0.920394	0.460197	+	0.887817i	$0.347779\pi$
$98$	0	0
$99$	0	0
$100$	3.74349	0.374349
$101$	−3.73726	−0.371871	−0.185936	−	0.982562i	$-0.559532\pi$
$101$	−3.73726	−0.371871	−0.185936	+	0.982562i	$0.559532\pi$
$102$	0	0
$103$	−12.1522	−1.19739	−0.598697	−	0.800976i	$-0.704315\pi$
$103$	−12.1522	−1.19739	−0.598697	+	0.800976i	$0.704315\pi$
$104$	−1.34752	−0.132136
$105$	0	0
$106$	−19.3725	−1.88162
$107$	15.0359	1.45357	0.726787	−	0.686863i	$-0.241013\pi$
$107$	15.0359	1.45357	0.726787	+	0.686863i	$0.241013\pi$
$108$	0	0
$109$	10.3873	0.994922	0.497461	−	0.867486i	$-0.334266\pi$
$109$	10.3873	0.994922	0.497461	+	0.867486i	$0.334266\pi$
$110$	23.4115	2.23220
$111$	0	0
$112$	0	0
$113$	−18.9977	−1.78715	−0.893576	−	0.448912i	$-0.851812\pi$
$113$	−18.9977	−1.78715	−0.893576	+	0.448912i	$0.851812\pi$
$114$	0	0
$115$	−14.6629	−1.36732
$116$	−9.18562	−0.852863
$117$	0	0
$118$	−11.9607	−1.10107
$119$	0	0
$120$	0	0
$121$	10.0939	0.917626
$122$	26.5134	2.40041
$123$	0	0
$124$	−0.177198	−0.0159129
$125$	5.68661	0.508626
$126$	0	0
$127$	17.4584	1.54918	0.774590	−	0.632464i	$-0.217956\pi$
$127$	17.4584	1.54918	0.774590	+	0.632464i	$0.217956\pi$
$128$	−10.0280	−0.886356
$129$	0	0
$130$	−5.09742	−0.447073
$131$	−3.42616	−0.299345	−0.149673	−	0.988736i	$-0.547822\pi$
$131$	−3.42616	−0.299345	−0.149673	+	0.988736i	$0.547822\pi$
$132$	0	0
$133$	0	0
$134$	12.3567	1.06746
$135$	0	0
$136$	1.49429	0.128134
$137$	−3.30220	−0.282126	−0.141063	−	0.990001i	$-0.545052\pi$
$137$	−3.30220	−0.282126	−0.141063	+	0.990001i	$0.545052\pi$
$138$	0	0
$139$	7.70920	0.653886	0.326943	−	0.945044i	$-0.393981\pi$
$139$	7.70920	0.653886	0.326943	+	0.945044i	$0.393981\pi$
$140$	0	0
$141$	0	0
$142$	9.99220	0.838526
$143$	−4.59281	−0.384070
$144$	0	0
$145$	20.7217	1.72084
$146$	23.1614	1.91685
$147$	0	0
$148$	2.24327	0.184396
$149$	8.77077	0.718530	0.359265	−	0.933236i	$-0.383027\pi$
$149$	8.77077	0.718530	0.359265	+	0.933236i	$0.383027\pi$
$150$	0	0
$151$	19.3557	1.57515	0.787573	−	0.616222i	$-0.211337\pi$
$151$	19.3557	1.57515	0.787573	+	0.616222i	$0.211337\pi$
$152$	−5.88097	−0.477010
$153$	0	0
$154$	0	0
$155$	0.399739	0.0321078
$156$	0	0
$157$	11.6584	0.930440	0.465220	−	0.885195i	$-0.345975\pi$
$157$	11.6584	0.930440	0.465220	+	0.885195i	$0.345975\pi$
$158$	3.00339	0.238937
$159$	0	0
$160$	−17.5442	−1.38699
$161$	0	0
$162$	0	0
$163$	10.5346	0.825132	0.412566	−	0.910928i	$-0.364633\pi$
$163$	10.5346	0.825132	0.412566	+	0.910928i	$0.364633\pi$
$164$	−7.63187	−0.595949
$165$	0	0
$166$	−19.7741	−1.53477
$167$	7.02011	0.543233	0.271616	−	0.962406i	$-0.412442\pi$
$167$	7.02011	0.543233	0.271616	+	0.962406i	$0.412442\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.65260	0.433534
$171$	0	0
$172$	2.71150	0.206750
$173$	1.25036	0.0950634	0.0475317	−	0.998870i	$-0.484864\pi$
$173$	1.25036	0.0950634	0.0475317	+	0.998870i	$0.484864\pi$
$174$	0	0
$175$	0	0
$176$	−22.6704	−1.70885
$177$	0	0
$178$	−2.10990	−0.158143
$179$	−4.42417	−0.330678	−0.165339	−	0.986237i	$-0.552872\pi$
$179$	−4.42417	−0.330678	−0.165339	+	0.986237i	$0.552872\pi$
$180$	0	0
$181$	−6.07062	−0.451226	−0.225613	−	0.974217i	$-0.572438\pi$
$181$	−6.07062	−0.451226	−0.225613	+	0.974217i	$0.572438\pi$
$182$	0	0
$183$	0	0
$184$	6.99099	0.515383
$185$	−5.06056	−0.372060
$186$	0	0
$187$	5.09303	0.372439
$188$	5.75338	0.419608
$189$	0	0
$190$	−22.2466	−1.61393
$191$	−2.67800	−0.193773	−0.0968866	−	0.995295i	$-0.530888\pi$
$191$	−2.67800	−0.193773	−0.0968866	+	0.995295i	$0.530888\pi$
$192$	0	0
$193$	−24.4560	−1.76038	−0.880192	−	0.474618i	$-0.842586\pi$
$193$	−24.4560	−1.76038	−0.880192	+	0.474618i	$0.842586\pi$
$194$	−16.3490	−1.17379
$195$	0	0
$196$	0	0
$197$	−5.17485	−0.368692	−0.184346	−	0.982861i	$-0.559017\pi$
$197$	−5.17485	−0.368692	−0.184346	+	0.982861i	$0.559017\pi$
$198$	0	0
$199$	17.3049	1.22671	0.613356	−	0.789807i	$-0.289819\pi$
$199$	17.3049	1.22671	0.613356	+	0.789807i	$0.289819\pi$
$200$	4.02635	0.284706
$201$	0	0
$202$	6.74040	0.474253
$203$	0	0
$204$	0	0
$205$	17.2166	1.20246
$206$	21.9174	1.52705
$207$	0	0
$208$	4.93606	0.342254
$209$	−20.0443	−1.38649
$210$	0	0
$211$	−25.0246	−1.72276	−0.861382	−	0.507957i	$-0.830401\pi$
$211$	−25.0246	−1.72276	−0.861382	+	0.507957i	$0.830401\pi$
$212$	13.4572	0.924243
$213$	0	0
$214$	−27.1182	−1.85376
$215$	−6.11684	−0.417165
$216$	0	0
$217$	0	0
$218$	−18.7342	−1.26884
$219$	0	0
$220$	−16.2629	−1.09644
$221$	−1.10891	−0.0745936
$222$	0	0
$223$	−9.74896	−0.652839	−0.326419	−	0.945225i	$-0.605842\pi$
$223$	−9.74896	−0.652839	−0.326419	+	0.945225i	$0.605842\pi$
$224$	0	0
$225$	0	0
$226$	34.2636	2.27918
$227$	10.8716	0.721574	0.360787	−	0.932648i	$-0.382508\pi$
$227$	10.8716	0.721574	0.360787	+	0.932648i	$0.382508\pi$
$228$	0	0
$229$	−22.8123	−1.50748	−0.753739	−	0.657174i	$-0.771752\pi$
$229$	−22.8123	−1.50748	−0.753739	+	0.657174i	$0.771752\pi$
$230$	26.4456	1.74377
$231$	0	0
$232$	−9.87970	−0.648634
$233$	−13.6434	−0.893807	−0.446904	−	0.894582i	$-0.647473\pi$
$233$	−13.6434	−0.893807	−0.446904	+	0.894582i	$0.647473\pi$
$234$	0	0
$235$	−12.9790	−0.846654
$236$	8.30857	0.540842
$237$	0	0
$238$	0	0
$239$	−15.5693	−1.00710	−0.503548	−	0.863967i	$-0.667972\pi$
$239$	−15.5693	−1.00710	−0.503548	+	0.863967i	$0.667972\pi$
$240$	0	0
$241$	26.0184	1.67599	0.837996	−	0.545677i	$-0.183727\pi$
$241$	26.0184	1.67599	0.837996	+	0.545677i	$0.183727\pi$
$242$	−18.2050	−1.17026
$243$	0	0
$244$	−18.4177	−1.17907
$245$	0	0
$246$	0	0
$247$	4.36428	0.277692
$248$	−0.190588	−0.0121023
$249$	0	0
$250$	−10.2562	−0.648659
$251$	24.2444	1.53030	0.765148	−	0.643855i	$-0.222666\pi$
$251$	24.2444	1.53030	0.765148	+	0.643855i	$0.222666\pi$
$252$	0	0
$253$	23.8276	1.49803
$254$	−31.4873	−1.97569
$255$	0	0
$256$	20.7331	1.29582
$257$	−7.79094	−0.485985	−0.242993	−	0.970028i	$-0.578129\pi$
$257$	−7.79094	−0.485985	−0.242993	+	0.970028i	$0.578129\pi$
$258$	0	0
$259$	0	0
$260$	3.54095	0.219600
$261$	0	0
$262$	6.17932	0.381759
$263$	27.6719	1.70633	0.853163	−	0.521645i	$-0.174681\pi$
$263$	27.6719	1.70633	0.853163	+	0.521645i	$0.174681\pi$
$264$	0	0
$265$	−30.3578	−1.86487
$266$	0	0
$267$	0	0
$268$	−8.58363	−0.524329
$269$	−28.3862	−1.73074	−0.865369	−	0.501135i	$-0.832916\pi$
$269$	−28.3862	−1.73074	−0.865369	+	0.501135i	$0.832916\pi$
$270$	0	0
$271$	2.58151	0.156816	0.0784079	−	0.996921i	$-0.475016\pi$
$271$	2.58151	0.156816	0.0784079	+	0.996921i	$0.475016\pi$
$272$	−5.47367	−0.331890
$273$	0	0
$274$	5.95573	0.359799
$275$	13.7231	0.827537
$276$	0	0
$277$	−27.4673	−1.65035	−0.825174	−	0.564879i	$-0.808923\pi$
$277$	−27.4673	−1.65035	−0.825174	+	0.564879i	$0.808923\pi$
$278$	−13.9041	−0.833911
$279$	0	0
$280$	0	0
$281$	1.54958	0.0924404	0.0462202	−	0.998931i	$-0.485282\pi$
$281$	1.54958	0.0924404	0.0462202	+	0.998931i	$0.485282\pi$
$282$	0	0
$283$	−26.3200	−1.56456	−0.782280	−	0.622927i	$-0.785943\pi$
$283$	−26.3200	−1.56456	−0.782280	+	0.622927i	$0.785943\pi$
$284$	−6.94112	−0.411880
$285$	0	0
$286$	8.28344	0.489810
$287$	0	0
$288$	0	0
$289$	−15.7703	−0.927665
$290$	−37.3730	−2.19462
$291$	0	0
$292$	−16.0892	−0.941547
$293$	−1.23200	−0.0719743	−0.0359872	−	0.999352i	$-0.511458\pi$
$293$	−1.23200	−0.0719743	−0.0359872	+	0.999352i	$0.511458\pi$
$294$	0	0
$295$	−18.7432	−1.09127
$296$	2.41278	0.140240
$297$	0	0
$298$	−15.8187	−0.916351
$299$	−5.18803	−0.300031
$300$	0	0
$301$	0	0
$302$	−34.9093	−2.00881
$303$	0	0
$304$	21.5424	1.23554
$305$	41.5482	2.37904
$306$	0	0
$307$	−16.3966	−0.935801	−0.467900	−	0.883781i	$-0.654989\pi$
$307$	−16.3966	−0.935801	−0.467900	+	0.883781i	$0.654989\pi$
$308$	0	0
$309$	0	0
$310$	−0.720956	−0.0409476
$311$	3.67447	0.208360	0.104180	−	0.994558i	$-0.466778\pi$
$311$	3.67447	0.208360	0.104180	+	0.994558i	$0.466778\pi$
$312$	0	0
$313$	−19.7460	−1.11611	−0.558056	−	0.829804i	$-0.688452\pi$
$313$	−19.7460	−1.11611	−0.558056	+	0.829804i	$0.688452\pi$
$314$	−21.0267	−1.18660
$315$	0	0
$316$	−2.08632	−0.117365
$317$	−9.10190	−0.511214	−0.255607	−	0.966781i	$-0.582275\pi$
$317$	−9.10190	−0.511214	−0.255607	+	0.966781i	$0.582275\pi$
$318$	0	0
$319$	−33.6733	−1.88534
$320$	3.74054	0.209103
$321$	0	0
$322$	0	0
$323$	−4.83960	−0.269283
$324$	0	0
$325$	−2.98796	−0.165742
$326$	−18.9998	−1.05230
$327$	0	0
$328$	−8.20855	−0.453241
$329$	0	0
$330$	0	0
$331$	−1.50009	−0.0824523	−0.0412261	−	0.999150i	$-0.513126\pi$
$331$	−1.50009	−0.0824523	−0.0412261	+	0.999150i	$0.513126\pi$
$332$	13.7361	0.753869
$333$	0	0
$334$	−12.6613	−0.692793
$335$	19.3637	1.05795
$336$	0	0
$337$	4.65624	0.253642	0.126821	−	0.991926i	$-0.459523\pi$
$337$	4.65624	0.253642	0.126821	+	0.991926i	$0.459523\pi$
$338$	−1.80357	−0.0981011
$339$	0	0
$340$	−3.92660	−0.212950
$341$	−0.649586	−0.0351771
$342$	0	0
$343$	0	0
$344$	2.91639	0.157241
$345$	0	0
$346$	−2.25512	−0.121236
$347$	−19.0696	−1.02371	−0.511856	−	0.859071i	$-0.671042\pi$
$347$	−19.0696	−1.02371	−0.511856	+	0.859071i	$0.671042\pi$
$348$	0	0
$349$	13.6199	0.729055	0.364527	−	0.931193i	$-0.381231\pi$
$349$	13.6199	0.729055	0.364527	+	0.931193i	$0.381231\pi$
$350$	0	0
$351$	0	0
$352$	28.5098	1.51957
$353$	−26.1032	−1.38934	−0.694668	−	0.719331i	$-0.744449\pi$
$353$	−26.1032	−1.38934	−0.694668	+	0.719331i	$0.744449\pi$
$354$	0	0
$355$	15.6584	0.831060
$356$	1.46565	0.0776792
$357$	0	0
$358$	7.97930	0.421719
$359$	−17.7855	−0.938680	−0.469340	−	0.883017i	$-0.655508\pi$
$359$	−17.7855	−0.938680	−0.469340	+	0.883017i	$0.655508\pi$
$360$	0	0
$361$	0.0469157	0.00246925
$362$	10.9488	0.575455
$363$	0	0
$364$	0	0
$365$	36.2953	1.89978
$366$	0	0
$367$	−9.07809	−0.473872	−0.236936	−	0.971525i	$-0.576143\pi$
$367$	−9.07809	−0.473872	−0.236936	+	0.971525i	$0.576143\pi$
$368$	−25.6084	−1.33493
$369$	0	0
$370$	9.12707	0.474494
$371$	0	0
$372$	0	0
$373$	−16.5940	−0.859204	−0.429602	−	0.903018i	$-0.641346\pi$
$373$	−16.5940	−0.859204	−0.429602	+	0.903018i	$0.641346\pi$
$374$	−9.18562	−0.474977
$375$	0	0
$376$	6.18811	0.319128
$377$	7.33174	0.377604
$378$	0	0
$379$	−0.722207	−0.0370973	−0.0185486	−	0.999828i	$-0.505905\pi$
$379$	−0.722207	−0.0370973	−0.0185486	+	0.999828i	$0.505905\pi$
$380$	15.4537	0.792757
$381$	0	0
$382$	4.82995	0.247122
$383$	−15.4930	−0.791654	−0.395827	−	0.918325i	$-0.629542\pi$
$383$	−15.4930	−0.791654	−0.395827	+	0.918325i	$0.629542\pi$
$384$	0	0
$385$	0	0
$386$	44.1081	2.24504
$387$	0	0
$388$	11.3569	0.576561
$389$	−19.6945	−0.998552	−0.499276	−	0.866443i	$-0.666401\pi$
$389$	−19.6945	−0.998552	−0.499276	+	0.866443i	$0.666401\pi$
$390$	0	0
$391$	5.75307	0.290945
$392$	0	0
$393$	0	0
$394$	9.33318	0.470199
$395$	4.70649	0.236809
$396$	0	0
$397$	4.09791	0.205668	0.102834	−	0.994699i	$-0.467209\pi$
$397$	4.09791	0.205668	0.102834	+	0.994699i	$0.467209\pi$
$398$	−31.2105	−1.56444
$399$	0	0
$400$	−14.7488	−0.737439
$401$	4.89445	0.244417	0.122209	−	0.992504i	$-0.461002\pi$
$401$	4.89445	0.244417	0.122209	+	0.992504i	$0.461002\pi$
$402$	0	0
$403$	0.141436	0.00704541
$404$	−4.68225	−0.232951
$405$	0	0
$406$	0	0
$407$	8.22355	0.407626
$408$	0	0
$409$	4.63252	0.229063	0.114532	−	0.993420i	$-0.463463\pi$
$409$	4.63252	0.229063	0.114532	+	0.993420i	$0.463463\pi$
$410$	−31.0513	−1.53352
$411$	0	0
$412$	−15.2250	−0.750081
$413$	0	0
$414$	0	0
$415$	−30.9872	−1.52110
$416$	−6.20748	−0.304347
$417$	0	0
$418$	36.1512	1.76821
$419$	−0.896067	−0.0437757	−0.0218879	−	0.999760i	$-0.506968\pi$
$419$	−0.896067	−0.0437757	−0.0218879	+	0.999760i	$0.506968\pi$
$420$	0	0
$421$	4.86195	0.236957	0.118479	−	0.992957i	$-0.462198\pi$
$421$	4.86195	0.236957	0.118479	+	0.992957i	$0.462198\pi$
$422$	45.1336	2.19707
$423$	0	0
$424$	14.4740	0.702921
$425$	3.31339	0.160723
$426$	0	0
$427$	0	0
$428$	18.8378	0.910560
$429$	0	0
$430$	11.0321	0.532017
$431$	23.8248	1.14760	0.573799	−	0.818996i	$-0.305469\pi$
$431$	23.8248	1.14760	0.573799	+	0.818996i	$0.305469\pi$
$432$	0	0
$433$	−25.1365	−1.20798	−0.603992	−	0.796990i	$-0.706424\pi$
$433$	−25.1365	−1.20798	−0.603992	+	0.796990i	$0.706424\pi$
$434$	0	0
$435$	0	0
$436$	13.0138	0.623247
$437$	−22.6420	−1.08311
$438$	0	0
$439$	37.3691	1.78353	0.891765	−	0.452499i	$-0.149467\pi$
$439$	37.3691	1.78353	0.891765	+	0.452499i	$0.149467\pi$
$440$	−17.4917	−0.833886
$441$	0	0
$442$	2.00000	0.0951303
$443$	−11.5346	−0.548024	−0.274012	−	0.961726i	$-0.588351\pi$
$443$	−11.5346	−0.548024	−0.274012	+	0.961726i	$0.588351\pi$
$444$	0	0
$445$	−3.30633	−0.156735
$446$	17.5829	0.832575
$447$	0	0
$448$	0	0
$449$	−2.45217	−0.115725	−0.0578625	−	0.998325i	$-0.518428\pi$
$449$	−2.45217	−0.115725	−0.0578625	+	0.998325i	$0.518428\pi$
$450$	0	0
$451$	−27.9774	−1.31741
$452$	−23.8014	−1.11952
$453$	0	0
$454$	−19.6077	−0.920234
$455$	0	0
$456$	0	0
$457$	35.0302	1.63865	0.819323	−	0.573333i	$-0.194350\pi$
$457$	35.0302	1.63865	0.819323	+	0.573333i	$0.194350\pi$
$458$	41.1435	1.92251
$459$	0	0
$460$	−18.3705	−0.856530
$461$	−32.4393	−1.51085	−0.755424	−	0.655236i	$-0.772569\pi$
$461$	−32.4393	−1.51085	−0.755424	+	0.655236i	$0.772569\pi$
$462$	0	0
$463$	8.77761	0.407930	0.203965	−	0.978978i	$-0.434617\pi$
$463$	8.77761	0.407930	0.203965	+	0.978978i	$0.434617\pi$
$464$	36.1899	1.68008
$465$	0	0
$466$	24.6068	1.13989
$467$	−23.4814	−1.08659	−0.543295	−	0.839542i	$-0.682824\pi$
$467$	−23.4814	−1.08659	−0.543295	+	0.839542i	$0.682824\pi$
$468$	0	0
$469$	0	0
$470$	23.4084	1.07975
$471$	0	0
$472$	8.93638	0.411330
$473$	9.94002	0.457043
$474$	0	0
$475$	−13.0403	−0.598330
$476$	0	0
$477$	0	0
$478$	28.0803	1.28436
$479$	33.1913	1.51655	0.758274	−	0.651936i	$-0.226043\pi$
$479$	33.1913	1.51655	0.758274	+	0.651936i	$0.226043\pi$
$480$	0	0
$481$	−1.79053	−0.0816410
$482$	−46.9259	−2.13742
$483$	0	0
$484$	12.6462	0.574827
$485$	−25.6199	−1.16334
$486$	0	0
$487$	−5.44738	−0.246844	−0.123422	−	0.992354i	$-0.539387\pi$
$487$	−5.44738	−0.246844	−0.123422	+	0.992354i	$0.539387\pi$
$488$	−19.8094	−0.896727
$489$	0	0
$490$	0	0
$491$	−1.21560	−0.0548594	−0.0274297	−	0.999624i	$-0.508732\pi$
$491$	−1.21560	−0.0548594	−0.0274297	+	0.999624i	$0.508732\pi$
$492$	0	0
$493$	−8.13026	−0.366169
$494$	−7.87127	−0.354145
$495$	0	0
$496$	0.698135	0.0313472
$497$	0	0
$498$	0	0
$499$	0.350196	0.0156769	0.00783847	−	0.999969i	$-0.497505\pi$
$499$	0.350196	0.0156769	0.00783847	+	0.999969i	$0.497505\pi$
$500$	7.12451	0.318618
$501$	0	0
$502$	−43.7265	−1.95161
$503$	23.5734	1.05109	0.525543	−	0.850767i	$-0.323862\pi$
$503$	23.5734	1.05109	0.525543	+	0.850767i	$0.323862\pi$
$504$	0	0
$505$	10.5626	0.470030
$506$	−42.9747	−1.91046
$507$	0	0
$508$	21.8728	0.970449
$509$	10.1526	0.450005	0.225003	−	0.974358i	$-0.427761\pi$
$509$	10.1526	0.450005	0.225003	+	0.974358i	$0.427761\pi$
$510$	0	0
$511$	0	0
$512$	−17.3376	−0.766220
$513$	0	0
$514$	14.0515	0.619784
$515$	34.3458	1.51346
$516$	0	0
$517$	21.0911	0.927587
$518$	0	0
$519$	0	0
$520$	3.80851	0.167014
$521$	−11.4239	−0.500491	−0.250246	−	0.968182i	$-0.580511\pi$
$521$	−11.4239	−0.500491	−0.250246	+	0.968182i	$0.580511\pi$
$522$	0	0
$523$	−38.0950	−1.66578	−0.832889	−	0.553440i	$-0.813315\pi$
$523$	−38.0950	−1.66578	−0.832889	+	0.553440i	$0.813315\pi$
$524$	−4.29249	−0.187518
$525$	0	0
$526$	−49.9082	−2.17610
$527$	−0.156840	−0.00683205
$528$	0	0
$529$	3.91565	0.170245
$530$	54.7524	2.37829
$531$	0	0
$532$	0	0
$533$	6.09158	0.263855
$534$	0	0
$535$	−42.4959	−1.83726
$536$	−9.23222	−0.398771
$537$	0	0
$538$	51.1965	2.20724
$539$	0	0
$540$	0	0
$541$	17.4194	0.748918	0.374459	−	0.927244i	$-0.377828\pi$
$541$	17.4194	0.748918	0.374459	+	0.927244i	$0.377828\pi$
$542$	−4.65593	−0.199989
$543$	0	0
$544$	6.88355	0.295130
$545$	−29.3576	−1.25754
$546$	0	0
$547$	6.57889	0.281293	0.140646	−	0.990060i	$-0.455082\pi$
$547$	6.57889	0.281293	0.140646	+	0.990060i	$0.455082\pi$
$548$	−4.13718	−0.176731
$549$	0	0
$550$	−24.7506	−1.05537
$551$	31.9977	1.36315
$552$	0	0
$553$	0	0
$554$	49.5391	2.10471
$555$	0	0
$556$	9.65852	0.409613
$557$	−24.5369	−1.03966	−0.519831	−	0.854269i	$-0.674005\pi$
$557$	−24.5369	−1.03966	−0.519831	+	0.854269i	$0.674005\pi$
$558$	0	0
$559$	−2.16426	−0.0915383
$560$	0	0
$561$	0	0
$562$	−2.79478	−0.117891
$563$	−39.8101	−1.67780	−0.838899	−	0.544288i	$-0.816800\pi$
$563$	−39.8101	−1.67780	−0.838899	+	0.544288i	$0.816800\pi$
$564$	0	0
$565$	53.6931	2.25889
$566$	47.4698	1.99531
$567$	0	0
$568$	−7.46561	−0.313250
$569$	−43.4205	−1.82028	−0.910142	−	0.414297i	$-0.864028\pi$
$569$	−43.4205	−1.82028	−0.910142	+	0.414297i	$0.864028\pi$
$570$	0	0
$571$	42.4653	1.77712	0.888559	−	0.458763i	$-0.151707\pi$
$571$	42.4653	1.77712	0.888559	+	0.458763i	$0.151707\pi$
$572$	−5.75413	−0.240592
$573$	0	0
$574$	0	0
$575$	15.5016	0.646463
$576$	0	0
$577$	−31.3411	−1.30475	−0.652374	−	0.757897i	$-0.726227\pi$
$577$	−31.3411	−1.30475	−0.652374	+	0.757897i	$0.726227\pi$
$578$	28.4428	1.18307
$579$	0	0
$580$	25.9613	1.07798
$581$	0	0
$582$	0	0
$583$	49.3323	2.04313
$584$	−17.3049	−0.716081
$585$	0	0
$586$	2.22200	0.0917899
$587$	−11.6120	−0.479278	−0.239639	−	0.970862i	$-0.577029\pi$
$587$	−11.6120	−0.479278	−0.239639	+	0.970862i	$0.577029\pi$
$588$	0	0
$589$	0.617264	0.0254339
$590$	33.8046	1.39171
$591$	0	0
$592$	−8.83815	−0.363246
$593$	−9.72687	−0.399435	−0.199717	−	0.979854i	$-0.564002\pi$
$593$	−9.72687	−0.399435	−0.199717	+	0.979854i	$0.564002\pi$
$594$	0	0
$595$	0	0
$596$	10.9885	0.450107
$597$	0	0
$598$	9.35696	0.382635
$599$	39.8630	1.62876	0.814379	−	0.580334i	$-0.197078\pi$
$599$	39.8630	1.62876	0.814379	+	0.580334i	$0.197078\pi$
$600$	0	0
$601$	−21.1274	−0.861806	−0.430903	−	0.902398i	$-0.641805\pi$
$601$	−21.1274	−0.861806	−0.430903	+	0.902398i	$0.641805\pi$
$602$	0	0
$603$	0	0
$604$	24.2499	0.986715
$605$	−28.5283	−1.15984
$606$	0	0
$607$	−22.0919	−0.896684	−0.448342	−	0.893862i	$-0.647985\pi$
$607$	−22.0919	−0.896684	−0.448342	+	0.893862i	$0.647985\pi$
$608$	−27.0911	−1.09869
$609$	0	0
$610$	−74.9349	−3.03403
$611$	−4.59221	−0.185781
$612$	0	0
$613$	−34.5488	−1.39541	−0.697707	−	0.716383i	$-0.745796\pi$
$613$	−34.5488	−1.39541	−0.697707	+	0.716383i	$0.745796\pi$
$614$	29.5723	1.19344
$615$	0	0
$616$	0	0
$617$	4.78056	0.192458	0.0962291	−	0.995359i	$-0.469322\pi$
$617$	4.78056	0.192458	0.0962291	+	0.995359i	$0.469322\pi$
$618$	0	0
$619$	6.14189	0.246863	0.123432	−	0.992353i	$-0.460610\pi$
$619$	6.14189	0.246863	0.123432	+	0.992353i	$0.460610\pi$
$620$	0.500816	0.0201132
$621$	0	0
$622$	−6.62716	−0.265725
$623$	0	0
$624$	0	0
$625$	−31.0119	−1.24048
$626$	35.6133	1.42339
$627$	0	0
$628$	14.6063	0.582854
$629$	1.98554	0.0791686
$630$	0	0
$631$	32.2502	1.28386	0.641930	−	0.766763i	$-0.278134\pi$
$631$	32.2502	1.28386	0.641930	+	0.766763i	$0.278134\pi$
$632$	−2.24396	−0.0892601
$633$	0	0
$634$	16.4159	0.651958
$635$	−49.3425	−1.95810
$636$	0	0
$637$	0	0
$638$	60.7320	2.40440
$639$	0	0
$640$	28.3420	1.12032
$641$	−25.5963	−1.01099	−0.505497	−	0.862828i	$-0.668691\pi$
$641$	−25.5963	−1.01099	−0.505497	+	0.862828i	$0.668691\pi$
$642$	0	0
$643$	−32.8406	−1.29511	−0.647553	−	0.762020i	$-0.724208\pi$
$643$	−32.8406	−1.29511	−0.647553	+	0.762020i	$0.724208\pi$
$644$	0	0
$645$	0	0
$646$	8.72855	0.343420
$647$	−26.5987	−1.04570	−0.522851	−	0.852424i	$-0.675132\pi$
$647$	−26.5987	−1.04570	−0.522851	+	0.852424i	$0.675132\pi$
$648$	0	0
$649$	30.4581	1.19559
$650$	5.38900	0.211374
$651$	0	0
$652$	13.1983	0.516886
$653$	−5.29185	−0.207086	−0.103543	−	0.994625i	$-0.533018\pi$
$653$	−5.29185	−0.207086	−0.103543	+	0.994625i	$0.533018\pi$
$654$	0	0
$655$	9.68336	0.378360
$656$	30.0684	1.17397
$657$	0	0
$658$	0	0
$659$	−13.8708	−0.540331	−0.270166	−	0.962814i	$-0.587078\pi$
$659$	−13.8708	−0.540331	−0.270166	+	0.962814i	$0.587078\pi$
$660$	0	0
$661$	−45.3663	−1.76454	−0.882272	−	0.470740i	$-0.843987\pi$
$661$	−45.3663	−1.76454	−0.882272	+	0.470740i	$0.843987\pi$
$662$	2.70551	0.105153
$663$	0	0
$664$	14.7741	0.573345
$665$	0	0
$666$	0	0
$667$	−38.0373	−1.47281
$668$	8.79520	0.340296
$669$	0	0
$670$	−34.9237	−1.34922
$671$	−67.5168	−2.60646
$672$	0	0
$673$	−23.9469	−0.923084	−0.461542	−	0.887118i	$-0.652704\pi$
$673$	−23.9469	−0.923084	−0.461542	+	0.887118i	$0.652704\pi$
$674$	−8.39785	−0.323473
$675$	0	0
$676$	1.25286	0.0481868
$677$	46.6781	1.79398	0.896992	−	0.442047i	$-0.145747\pi$
$677$	46.6781	1.79398	0.896992	+	0.442047i	$0.145747\pi$
$678$	0	0
$679$	0	0
$680$	−4.22330	−0.161956
$681$	0	0
$682$	1.17157	0.0448618
$683$	−31.3332	−1.19893	−0.599467	−	0.800400i	$-0.704621\pi$
$683$	−31.3332	−1.19893	−0.599467	+	0.800400i	$0.704621\pi$
$684$	0	0
$685$	9.33299	0.356595
$686$	0	0
$687$	0	0
$688$	−10.6829	−0.407282
$689$	−10.7412	−0.409207
$690$	0	0
$691$	−26.4803	−1.00736	−0.503679	−	0.863891i	$-0.668021\pi$
$691$	−26.4803	−1.00736	−0.503679	+	0.863891i	$0.668021\pi$
$692$	1.56653	0.0595504
$693$	0	0
$694$	34.3934	1.30555
$695$	−21.7885	−0.826485
$696$	0	0
$697$	−6.75503	−0.255865
$698$	−24.5644	−0.929775
$699$	0	0
$700$	0	0
$701$	41.6410	1.57276	0.786378	−	0.617745i	$-0.211954\pi$
$701$	41.6410	1.57276	0.786378	+	0.617745i	$0.211954\pi$
$702$	0	0
$703$	−7.81436	−0.294724
$704$	−6.07848	−0.229091
$705$	0	0
$706$	47.0790	1.77184
$707$	0	0
$708$	0	0
$709$	−30.1202	−1.13119	−0.565595	−	0.824683i	$-0.691353\pi$
$709$	−30.1202	−1.13119	−0.565595	+	0.824683i	$0.691353\pi$
$710$	−28.2409	−1.05986
$711$	0	0
$712$	1.57639	0.0590779
$713$	−0.733772	−0.0274800
$714$	0	0
$715$	12.9806	0.485449
$716$	−5.54285	−0.207146
$717$	0	0
$718$	32.0773	1.19711
$719$	−50.9866	−1.90148	−0.950740	−	0.309990i	$-0.899674\pi$
$719$	−50.9866	−1.90148	−0.950740	+	0.309990i	$0.899674\pi$
$720$	0	0
$721$	0	0
$722$	−0.0846156	−0.00314907
$723$	0	0
$724$	−7.60562	−0.282660
$725$	−21.9070	−0.813605
$726$	0	0
$727$	−24.0386	−0.891543	−0.445771	−	0.895147i	$-0.647071\pi$
$727$	−24.0386	−0.891543	−0.445771	+	0.895147i	$0.647071\pi$
$728$	0	0
$729$	0	0
$730$	−65.4610	−2.42282
$731$	2.39997	0.0887662
$732$	0	0
$733$	−21.6554	−0.799860	−0.399930	−	0.916546i	$-0.630966\pi$
$733$	−21.6554	−0.799860	−0.399930	+	0.916546i	$0.630966\pi$
$734$	16.3729	0.604337
$735$	0	0
$736$	32.2046	1.18708
$737$	−31.4665	−1.15908
$738$	0	0
$739$	28.2634	1.03969	0.519843	−	0.854262i	$-0.325991\pi$
$739$	28.2634	1.03969	0.519843	+	0.854262i	$0.325991\pi$
$740$	−6.34016	−0.233069
$741$	0	0
$742$	0	0
$743$	29.4246	1.07948	0.539742	−	0.841831i	$-0.318522\pi$
$743$	29.4246	1.07948	0.539742	+	0.841831i	$0.318522\pi$
$744$	0	0
$745$	−24.7888	−0.908192
$746$	29.9284	1.09576
$747$	0	0
$748$	6.38083	0.233306
$749$	0	0
$750$	0	0
$751$	−44.0042	−1.60574	−0.802869	−	0.596156i	$-0.796694\pi$
$751$	−44.0042	−1.60574	−0.802869	+	0.596156i	$0.796694\pi$
$752$	−22.6674	−0.826597
$753$	0	0
$754$	−13.2233	−0.481564
$755$	−54.7050	−1.99092
$756$	0	0
$757$	1.19964	0.0436017	0.0218008	−	0.999762i	$-0.493060\pi$
$757$	1.19964	0.0436017	0.0218008	+	0.999762i	$0.493060\pi$
$758$	1.30255	0.0473107
$759$	0	0
$760$	16.6214	0.602921
$761$	−0.0412816	−0.00149646	−0.000748228	−	1.00000i	$-0.500238\pi$
$761$	−0.0412816	−0.00149646	−0.000748228		1.00000i	$0.500238\pi$
$762$	0	0
$763$	0	0
$764$	−3.35515	−0.121385
$765$	0	0
$766$	27.9426	1.00961
$767$	−6.63170	−0.239457
$768$	0	0
$769$	−15.7537	−0.568095	−0.284047	−	0.958810i	$-0.591677\pi$
$769$	−15.7537	−0.568095	−0.284047	+	0.958810i	$0.591677\pi$
$770$	0	0
$771$	0	0
$772$	−30.6399	−1.10275
$773$	28.6610	1.03086	0.515432	−	0.856930i	$-0.327631\pi$
$773$	28.6610	1.03086	0.515432	+	0.856930i	$0.327631\pi$
$774$	0	0
$775$	−0.422604	−0.0151804
$776$	12.2151	0.438496
$777$	0	0
$778$	35.5204	1.27347
$779$	26.5853	0.952518
$780$	0	0
$781$	−25.4453	−0.910503
$782$	−10.3761	−0.371047
$783$	0	0
$784$	0	0
$785$	−32.9501	−1.17604
$786$	0	0
$787$	52.4263	1.86879	0.934397	−	0.356233i	$-0.115939\pi$
$787$	52.4263	1.86879	0.934397	+	0.356233i	$0.115939\pi$
$788$	−6.48334	−0.230959
$789$	0	0
$790$	−8.48848	−0.302006
$791$	0	0
$792$	0	0
$793$	14.7006	0.522032
$794$	−7.39086	−0.262292
$795$	0	0
$796$	21.6805	0.768446
$797$	−42.6181	−1.50961	−0.754805	−	0.655949i	$-0.772269\pi$
$797$	−42.6181	−1.50961	−0.754805	+	0.655949i	$0.772269\pi$
$798$	0	0
$799$	5.09236	0.180155
$800$	18.5477	0.655761
$801$	0	0
$802$	−8.82747	−0.311709
$803$	−58.9808	−2.08139
$804$	0	0
$805$	0	0
$806$	−0.255089	−0.00898511
$807$	0	0
$808$	−5.03605	−0.177167
$809$	33.7332	1.18600	0.592999	−	0.805203i	$-0.297944\pi$
$809$	33.7332	1.18600	0.592999	+	0.805203i	$0.297944\pi$
$810$	0	0
$811$	40.6837	1.42860	0.714298	−	0.699841i	$-0.246746\pi$
$811$	40.6837	1.42860	0.714298	+	0.699841i	$0.246746\pi$
$812$	0	0
$813$	0	0
$814$	−14.8317	−0.519852
$815$	−29.7739	−1.04293
$816$	0	0
$817$	−9.44542	−0.330453
$818$	−8.35506	−0.292128
$819$	0	0
$820$	21.5699	0.753255
$821$	−31.5104	−1.09972	−0.549861	−	0.835256i	$-0.685319\pi$
$821$	−31.5104	−1.09972	−0.549861	+	0.835256i	$0.685319\pi$
$822$	0	0
$823$	30.3445	1.05774	0.528872	−	0.848701i	$-0.322615\pi$
$823$	30.3445	1.05774	0.528872	+	0.848701i	$0.322615\pi$
$824$	−16.3754	−0.570464
$825$	0	0
$826$	0	0
$827$	35.0092	1.21739	0.608695	−	0.793404i	$-0.291693\pi$
$827$	35.0092	1.21739	0.608695	+	0.793404i	$0.291693\pi$
$828$	0	0
$829$	−13.2821	−0.461306	−0.230653	−	0.973036i	$-0.574086\pi$
$829$	−13.2821	−0.461306	−0.230653	+	0.973036i	$0.574086\pi$
$830$	55.8874	1.93988
$831$	0	0
$832$	1.32348	0.0458833
$833$	0	0
$834$	0	0
$835$	−19.8409	−0.686624
$836$	−25.1126	−0.868538
$837$	0	0
$838$	1.61612	0.0558278
$839$	36.5802	1.26289	0.631445	−	0.775421i	$-0.282462\pi$
$839$	36.5802	1.26289	0.631445	+	0.775421i	$0.282462\pi$
$840$	0	0
$841$	24.7544	0.853600
$842$	−8.76886	−0.302195
$843$	0	0
$844$	−31.3522	−1.07919
$845$	−2.82630	−0.0972276
$846$	0	0
$847$	0	0
$848$	−53.0193	−1.82069
$849$	0	0
$850$	−5.97593	−0.204973
$851$	9.28931	0.318433
$852$	0	0
$853$	−7.58341	−0.259651	−0.129825	−	0.991537i	$-0.541442\pi$
$853$	−7.58341	−0.259651	−0.129825	+	0.991537i	$0.541442\pi$
$854$	0	0
$855$	0	0
$856$	20.2612	0.692514
$857$	41.6435	1.42251	0.711257	−	0.702932i	$-0.248126\pi$
$857$	41.6435	1.42251	0.711257	+	0.702932i	$0.248126\pi$
$858$	0	0
$859$	−45.4281	−1.54999	−0.774994	−	0.631969i	$-0.782247\pi$
$859$	−45.4281	−1.54999	−0.774994	+	0.631969i	$0.782247\pi$
$860$	−7.66352	−0.261324
$861$	0	0
$862$	−42.9696	−1.46355
$863$	−19.0240	−0.647585	−0.323792	−	0.946128i	$-0.604958\pi$
$863$	−19.0240	−0.647585	−0.323792	+	0.946128i	$0.604958\pi$
$864$	0	0
$865$	−3.53390	−0.120156
$866$	45.3355	1.54056
$867$	0	0
$868$	0	0
$869$	−7.64817	−0.259446
$870$	0	0
$871$	6.85125	0.232146
$872$	13.9971	0.474003
$873$	0	0
$874$	40.8364	1.38131
$875$	0	0
$876$	0	0
$877$	−44.9665	−1.51841	−0.759206	−	0.650851i	$-0.774412\pi$
$877$	−44.9665	−1.51841	−0.759206	+	0.650851i	$0.774412\pi$
$878$	−67.3977	−2.27456
$879$	0	0
$880$	64.0733	2.15991
$881$	48.0593	1.61916	0.809579	−	0.587011i	$-0.199695\pi$
$881$	48.0593	1.61916	0.809579	+	0.587011i	$0.199695\pi$
$882$	0	0
$883$	−9.93064	−0.334193	−0.167096	−	0.985941i	$-0.553439\pi$
$883$	−9.93064	−0.334193	−0.167096	+	0.985941i	$0.553439\pi$
$884$	−1.38931	−0.0467275
$885$	0	0
$886$	20.8034	0.698903
$887$	−24.7325	−0.830436	−0.415218	−	0.909722i	$-0.636295\pi$
$887$	−24.7325	−0.830436	−0.415218	+	0.909722i	$0.636295\pi$
$888$	0	0
$889$	0	0
$890$	5.96319	0.199887
$891$	0	0
$892$	−12.2141	−0.408957
$893$	−20.0417	−0.670669
$894$	0	0
$895$	12.5040	0.417964
$896$	0	0
$897$	0	0
$898$	4.42265	0.147586
$899$	1.03697	0.0345848
$900$	0	0
$901$	11.9111	0.396815
$902$	50.4592	1.68011
$903$	0	0
$904$	−25.5998	−0.851438
$905$	17.1574	0.570331
$906$	0	0
$907$	5.36347	0.178091	0.0890456	−	0.996028i	$-0.471618\pi$
$907$	5.36347	0.178091	0.0890456	+	0.996028i	$0.471618\pi$
$908$	13.6206	0.452014
$909$	0	0
$910$	0	0
$911$	50.0131	1.65701	0.828505	−	0.559982i	$-0.189192\pi$
$911$	50.0131	1.65701	0.828505	+	0.559982i	$0.189192\pi$
$912$	0	0
$913$	50.3549	1.66651
$914$	−63.1794	−2.08979
$915$	0	0
$916$	−28.5805	−0.944326
$917$	0	0
$918$	0	0
$919$	−25.3719	−0.836940	−0.418470	−	0.908231i	$-0.637434\pi$
$919$	−25.3719	−0.836940	−0.418470	+	0.908231i	$0.637434\pi$
$920$	−19.7586	−0.651423
$921$	0	0
$922$	58.5065	1.92681
$923$	5.54024	0.182359
$924$	0	0
$925$	5.35003	0.175908
$926$	−15.8310	−0.520239
$927$	0	0
$928$	−45.5116	−1.49399
$929$	2.05440	0.0674027	0.0337014	−	0.999432i	$-0.489270\pi$
$929$	2.05440	0.0674027	0.0337014	+	0.999432i	$0.489270\pi$
$930$	0	0
$931$	0	0
$932$	−17.0932	−0.559906
$933$	0	0
$934$	42.3503	1.38575
$935$	−14.3944	−0.470748
$936$	0	0
$937$	11.3021	0.369222	0.184611	−	0.982812i	$-0.440897\pi$
$937$	11.3021	0.369222	0.184611	+	0.982812i	$0.440897\pi$
$938$	0	0
$939$	0	0
$940$	−16.2608	−0.530368
$941$	−3.88126	−0.126526	−0.0632628	−	0.997997i	$-0.520151\pi$
$941$	−3.88126	−0.126526	−0.0632628	+	0.997997i	$0.520151\pi$
$942$	0	0
$943$	−31.6033	−1.02914
$944$	−32.7345	−1.06542
$945$	0	0
$946$	−17.9275	−0.582873
$947$	30.7104	0.997955	0.498978	−	0.866615i	$-0.333709\pi$
$947$	30.7104	0.997955	0.498978	+	0.866615i	$0.333709\pi$
$948$	0	0
$949$	12.8420	0.416869
$950$	23.5191	0.763059
$951$	0	0
$952$	0	0
$953$	−48.0784	−1.55741	−0.778706	−	0.627388i	$-0.784124\pi$
$953$	−48.0784	−1.55741	−0.778706	+	0.627388i	$0.784124\pi$
$954$	0	0
$955$	7.56882	0.244921
$956$	−19.5061	−0.630873
$957$	0	0
$958$	−59.8627	−1.93408
$959$	0	0
$960$	0	0
$961$	−30.9800	−0.999355
$962$	3.22934	0.104118
$963$	0	0
$964$	32.5973	1.04989
$965$	69.1200	2.22505
$966$	0	0
$967$	44.3725	1.42692	0.713462	−	0.700693i	$-0.247126\pi$
$967$	44.3725	1.42692	0.713462	+	0.700693i	$0.247126\pi$
$968$	13.6018	0.437177
$969$	0	0
$970$	46.2073	1.48363
$971$	−43.0284	−1.38085	−0.690423	−	0.723406i	$-0.742576\pi$
$971$	−43.0284	−1.38085	−0.690423	+	0.723406i	$0.742576\pi$
$972$	0	0
$973$	0	0
$974$	9.82472	0.314804
$975$	0	0
$976$	72.5629	2.32268
$977$	21.8581	0.699301	0.349651	−	0.936880i	$-0.386300\pi$
$977$	21.8581	0.699301	0.349651	+	0.936880i	$0.386300\pi$
$978$	0	0
$979$	5.37288	0.171718
$980$	0	0
$981$	0	0
$982$	2.19242	0.0699630
$983$	28.7784	0.917890	0.458945	−	0.888465i	$-0.348228\pi$
$983$	28.7784	0.917890	0.458945	+	0.888465i	$0.348228\pi$
$984$	0	0
$985$	14.6257	0.466012
$986$	14.6635	0.466980
$987$	0	0
$988$	5.46781	0.173954
$989$	11.2282	0.357037
$990$	0	0
$991$	−53.2665	−1.69206	−0.846032	−	0.533131i	$-0.821015\pi$
$991$	−53.2665	−1.69206	−0.846032	+	0.533131i	$0.821015\pi$
$992$	−0.877958	−0.0278752
$993$	0	0
$994$	0	0
$995$	−48.9088	−1.55051
$996$	0	0
$997$	−19.8446	−0.628486	−0.314243	−	0.949343i	$-0.601751\pi$
$997$	−19.8446	−0.628486	−0.314243	+	0.949343i	$0.601751\pi$
$998$	−0.631603	−0.0199930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.ca.1.3	✓	12
3.2	odd	2	inner	5733.2.a.ca.1.10	yes	12
7.6	odd	2		5733.2.a.cb.1.3	yes	12
21.20	even	2		5733.2.a.cb.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5733.2.a.ca.1.3	✓	12	1.1	even	1	trivial
5733.2.a.ca.1.10	yes	12	3.2	odd	2	inner
5733.2.a.cb.1.3	yes	12	7.6	odd	2
5733.2.a.cb.1.10	yes	12	21.20	even	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)

\(f(q)\)	\(=\)	\(q-1.80357 q^{2} +1.25286 q^{4} -2.82630 q^{5} +1.34752 q^{8} +O(q^{10})\)	\(q-1.80357 q^{2} +1.25286 q^{4} -2.82630 q^{5} +1.34752 q^{8} +5.09742 q^{10} +4.59281 q^{11} -1.00000 q^{13} -4.93606 q^{16} +1.10891 q^{17} -4.36428 q^{19} -3.54095 q^{20} -8.28344 q^{22} +5.18803 q^{23} +2.98796 q^{25} +1.80357 q^{26} -7.33174 q^{29} -0.141436 q^{31} +6.20748 q^{32} -2.00000 q^{34} +1.79053 q^{37} +7.87127 q^{38} -3.80851 q^{40} -6.09158 q^{41} +2.16426 q^{43} +5.75413 q^{44} -9.35696 q^{46} +4.59221 q^{47} -5.38900 q^{50} -1.25286 q^{52} +10.7412 q^{53} -12.9806 q^{55} +13.2233 q^{58} +6.63170 q^{59} -14.7006 q^{61} +0.255089 q^{62} -1.32348 q^{64} +2.82630 q^{65} -6.85125 q^{67} +1.38931 q^{68} -5.54024 q^{71} -12.8420 q^{73} -3.22934 q^{74} -5.46781 q^{76} -1.66525 q^{79} +13.9508 q^{80} +10.9866 q^{82} +10.9639 q^{83} -3.13412 q^{85} -3.90338 q^{86} +6.18892 q^{88} +1.16985 q^{89} +6.49986 q^{92} -8.28236 q^{94} +12.3348 q^{95} +9.06483 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4}+O(q^{10}) \) 12 * q + 8 * q^4	\( 12 q + 8 q^{4} - 8 q^{10} - 12 q^{13} - 28 q^{19} + 16 q^{25} - 12 q^{31} - 24 q^{34} - 32 q^{40} - 20 q^{43} - 8 q^{46} - 8 q^{52} - 56 q^{55} + 8 q^{58} - 24 q^{61} - 40 q^{64} + 8 q^{67} - 12 q^{73} - 64 q^{76} - 20 q^{79} - 40 q^{82} + 32 q^{85} + 8 q^{88} - 8 q^{94} - 68 q^{97}+O(q^{100}) \) 12 * q + 8 * q^4 - 8 * q^10 - 12 * q^13 - 28 * q^19 + 16 * q^25 - 12 * q^31 - 24 * q^34 - 32 * q^40 - 20 * q^43 - 8 * q^46 - 8 * q^52 - 56 * q^55 + 8 * q^58 - 24 * q^61 - 40 * q^64 + 8 * q^67 - 12 * q^73 - 64 * q^76 - 20 * q^79 - 40 * q^82 + 32 * q^85 + 8 * q^88 - 8 * q^94 - 68 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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