LMFDB - Embedded newform 5733.2.a.bc.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.529317 q^{2} -1.71982 q^{4} +1.77846 q^{5} -1.96896 q^{8} +O(q^{10})$	$q+0.529317 q^{2} -1.71982 q^{4} +1.77846 q^{5} -1.96896 q^{8} +0.941367 q^{10} +6.49828 q^{11} +1.00000 q^{13} +2.39744 q^{16} -2.94137 q^{17} +4.83709 q^{19} -3.05863 q^{20} +3.43965 q^{22} +5.77846 q^{23} -1.83709 q^{25} +0.529317 q^{26} +2.83709 q^{29} -6.27674 q^{31} +5.20693 q^{32} -1.55691 q^{34} +9.55691 q^{37} +2.56035 q^{38} -3.50172 q^{40} -3.05863 q^{41} +2.71982 q^{43} -11.1759 q^{44} +3.05863 q^{46} -8.71982 q^{47} -0.972402 q^{50} -1.71982 q^{52} -6.39400 q^{53} +11.5569 q^{55} +1.50172 q^{58} +1.55691 q^{59} -3.88273 q^{61} -3.32238 q^{62} -2.03877 q^{64} +1.77846 q^{65} +5.67418 q^{67} +5.05863 q^{68} -10.0552 q^{71} +15.8337 q^{73} +5.05863 q^{74} -8.31894 q^{76} -1.28018 q^{79} +4.26375 q^{80} -1.61899 q^{82} -2.83709 q^{83} -5.23109 q^{85} +1.43965 q^{86} -12.7949 q^{88} -7.66119 q^{89} -9.93793 q^{92} -4.61555 q^{94} +8.60256 q^{95} +17.7164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 12 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 12 * q^8	$ 3 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 12 q^{8} + 2 q^{10} + 2 q^{11} + 3 q^{13} + 18 q^{16} - 8 q^{17} + 7 q^{19} - 10 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} + 2 q^{26} + q^{29} + 7 q^{31} + 36 q^{32} + 12 q^{34} + 12 q^{37} + 26 q^{38} - 28 q^{40} - 10 q^{41} - q^{43} - 36 q^{44} + 10 q^{46} - 17 q^{47} - 20 q^{50} + 4 q^{52} + 5 q^{53} + 18 q^{55} + 22 q^{58} - 12 q^{59} - 10 q^{61} + 10 q^{62} + 58 q^{64} - 3 q^{65} + 2 q^{67} + 16 q^{68} + 4 q^{71} + 5 q^{73} + 16 q^{74} + 30 q^{76} - 13 q^{79} - 8 q^{80} - 24 q^{82} - q^{83} + 16 q^{85} - 14 q^{86} - 60 q^{88} - 13 q^{89} + 6 q^{92} + 2 q^{94} + 15 q^{95} + 9 q^{97}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 12 * q^8 + 2 * q^10 + 2 * q^11 + 3 * q^13 + 18 * q^16 - 8 * q^17 + 7 * q^19 - 10 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 + 2 * q^26 + q^29 + 7 * q^31 + 36 * q^32 + 12 * q^34 + 12 * q^37 + 26 * q^38 - 28 * q^40 - 10 * q^41 - q^43 - 36 * q^44 + 10 * q^46 - 17 * q^47 - 20 * q^50 + 4 * q^52 + 5 * q^53 + 18 * q^55 + 22 * q^58 - 12 * q^59 - 10 * q^61 + 10 * q^62 + 58 * q^64 - 3 * q^65 + 2 * q^67 + 16 * q^68 + 4 * q^71 + 5 * q^73 + 16 * q^74 + 30 * q^76 - 13 * q^79 - 8 * q^80 - 24 * q^82 - q^83 + 16 * q^85 - 14 * q^86 - 60 * q^88 - 13 * q^89 + 6 * q^92 + 2 * q^94 + 15 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.529317	0.374283	0.187142	−	0.982333i	$-0.440078\pi$
$2$	0.529317	0.374283	0.187142	+	0.982333i	$0.440078\pi$
$3$	0	0
$3$	0	0
$4$	−1.71982	−0.859912
$5$	1.77846	0.795350	0.397675	−	0.917526i	$-0.369817\pi$
$5$	1.77846	0.795350	0.397675	+	0.917526i	$0.369817\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.96896	−0.696134
$9$	0	0
$10$	0.941367	0.297686
$11$	6.49828	1.95931	0.979653	−	0.200700i	$-0.0643217\pi$
$11$	6.49828	1.95931	0.979653	+	0.200700i	$0.0643217\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	2.39744	0.599361
$17$	−2.94137	−0.713386	−0.356693	−	0.934222i	$-0.616096\pi$
$17$	−2.94137	−0.713386	−0.356693	+	0.934222i	$0.616096\pi$
$18$	0	0
$19$	4.83709	1.10970	0.554852	−	0.831949i	$-0.312775\pi$
$19$	4.83709	1.10970	0.554852	+	0.831949i	$0.312775\pi$
$20$	−3.05863	−0.683931
$21$	0	0
$22$	3.43965	0.733335
$23$	5.77846	1.20489	0.602446	−	0.798160i	$-0.294193\pi$
$23$	5.77846	1.20489	0.602446	+	0.798160i	$0.294193\pi$
$24$	0	0
$25$	−1.83709	−0.367418
$26$	0.529317	0.103808
$27$	0	0
$28$	0	0
$29$	2.83709	0.526834	0.263417	−	0.964682i	$-0.415150\pi$
$29$	2.83709	0.526834	0.263417	+	0.964682i	$0.415150\pi$
$30$	0	0
$31$	−6.27674	−1.12734	−0.563668	−	0.826002i	$-0.690610\pi$
$31$	−6.27674	−1.12734	−0.563668	+	0.826002i	$0.690610\pi$
$32$	5.20693	0.920465
$33$	0	0
$34$	−1.55691	−0.267009
$35$	0	0
$36$	0	0
$37$	9.55691	1.57115	0.785574	−	0.618768i	$-0.212368\pi$
$37$	9.55691	1.57115	0.785574	+	0.618768i	$0.212368\pi$
$38$	2.56035	0.415344
$39$	0	0
$40$	−3.50172	−0.553670
$41$	−3.05863	−0.477678	−0.238839	−	0.971059i	$-0.576767\pi$
$41$	−3.05863	−0.477678	−0.238839	+	0.971059i	$0.576767\pi$
$42$	0	0
$43$	2.71982	0.414769	0.207385	−	0.978259i	$-0.433505\pi$
$43$	2.71982	0.414769	0.207385	+	0.978259i	$0.433505\pi$
$44$	−11.1759	−1.68483
$45$	0	0
$46$	3.05863	0.450971
$47$	−8.71982	−1.27192	−0.635959	−	0.771723i	$-0.719395\pi$
$47$	−8.71982	−1.27192	−0.635959	+	0.771723i	$0.719395\pi$
$48$	0	0
$49$	0	0
$50$	−0.972402	−0.137518
$51$	0	0
$52$	−1.71982	−0.238497
$53$	−6.39400	−0.878284	−0.439142	−	0.898418i	$-0.644718\pi$
$53$	−6.39400	−0.878284	−0.439142	+	0.898418i	$0.644718\pi$
$54$	0	0
$55$	11.5569	1.55833
$56$	0	0
$57$	0	0
$58$	1.50172	0.197185
$59$	1.55691	0.202693	0.101346	−	0.994851i	$-0.467685\pi$
$59$	1.55691	0.202693	0.101346	+	0.994851i	$0.467685\pi$
$60$	0	0
$61$	−3.88273	−0.497133	−0.248567	−	0.968615i	$-0.579959\pi$
$61$	−3.88273	−0.497133	−0.248567	+	0.968615i	$0.579959\pi$
$62$	−3.32238	−0.421943
$63$	0	0
$64$	−2.03877	−0.254846
$65$	1.77846	0.220590
$66$	0	0
$67$	5.67418	0.693211	0.346606	−	0.938011i	$-0.387334\pi$
$67$	5.67418	0.693211	0.346606	+	0.938011i	$0.387334\pi$
$68$	5.05863	0.613449
$69$	0	0
$70$	0	0
$71$	−10.0552	−1.19333	−0.596666	−	0.802490i	$-0.703508\pi$
$71$	−10.0552	−1.19333	−0.596666	+	0.802490i	$0.703508\pi$
$72$	0	0
$73$	15.8337	1.85319	0.926594	−	0.376062i	$-0.122722\pi$
$73$	15.8337	1.85319	0.926594	+	0.376062i	$0.122722\pi$
$74$	5.05863	0.588054
$75$	0	0
$76$	−8.31894	−0.954248
$77$	0	0
$78$	0	0
$79$	−1.28018	−0.144031	−0.0720155	−	0.997404i	$-0.522943\pi$
$79$	−1.28018	−0.144031	−0.0720155	+	0.997404i	$0.522943\pi$
$80$	4.26375	0.476702
$81$	0	0
$82$	−1.61899	−0.178787
$83$	−2.83709	−0.311411	−0.155706	−	0.987804i	$-0.549765\pi$
$83$	−2.83709	−0.311411	−0.155706	+	0.987804i	$0.549765\pi$
$84$	0	0
$85$	−5.23109	−0.567392
$86$	1.43965	0.155241
$87$	0	0
$88$	−12.7949	−1.36394
$89$	−7.66119	−0.812085	−0.406042	−	0.913854i	$-0.633091\pi$
$89$	−7.66119	−0.812085	−0.406042	+	0.913854i	$0.633091\pi$
$90$	0	0
$91$	0	0
$92$	−9.93793	−1.03610
$93$	0	0
$94$	−4.61555	−0.476057
$95$	8.60256	0.882604
$96$	0	0
$97$	17.7164	1.79883	0.899413	−	0.437099i	$-0.143994\pi$
$97$	17.7164	1.79883	0.899413	+	0.437099i	$0.143994\pi$
$98$	0	0
$99$	0	0
$100$	3.15947	0.315947
$101$	6.73281	0.669940	0.334970	−	0.942229i	$-0.391274\pi$
$101$	6.73281	0.669940	0.334970	+	0.942229i	$0.391274\pi$
$102$	0	0
$103$	−10.8793	−1.07197	−0.535984	−	0.844228i	$-0.680059\pi$
$103$	−10.8793	−1.07197	−0.535984	+	0.844228i	$0.680059\pi$
$104$	−1.96896	−0.193073
$105$	0	0
$106$	−3.38445	−0.328727
$107$	8.49828	0.821560	0.410780	−	0.911735i	$-0.365256\pi$
$107$	8.49828	0.821560	0.410780	+	0.911735i	$0.365256\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	6.11727	0.583258
$111$	0	0
$112$	0	0
$113$	−14.6026	−1.37369	−0.686847	−	0.726802i	$-0.741006\pi$
$113$	−14.6026	−1.37369	−0.686847	+	0.726802i	$0.741006\pi$
$114$	0	0
$115$	10.2767	0.958311
$116$	−4.87930	−0.453031
$117$	0	0
$118$	0.824101	0.0758646
$119$	0	0
$120$	0	0
$121$	31.2277	2.83888
$122$	−2.05520	−0.186069
$123$	0	0
$124$	10.7949	0.969409
$125$	−12.1595	−1.08758
$126$	0	0
$127$	−9.88273	−0.876951	−0.438475	−	0.898743i	$-0.644481\pi$
$127$	−9.88273	−0.876951	−0.438475	+	0.898743i	$0.644481\pi$
$128$	−11.4930	−1.01585
$129$	0	0
$130$	0.941367	0.0825633
$131$	3.76547	0.328990	0.164495	−	0.986378i	$-0.447400\pi$
$131$	3.76547	0.328990	0.164495	+	0.986378i	$0.447400\pi$
$132$	0	0
$133$	0	0
$134$	3.00344	0.259458
$135$	0	0
$136$	5.79145	0.496612
$137$	16.9966	1.45211	0.726057	−	0.687634i	$-0.241351\pi$
$137$	16.9966	1.45211	0.726057	+	0.687634i	$0.241351\pi$
$138$	0	0
$139$	8.55348	0.725496	0.362748	−	0.931887i	$-0.381839\pi$
$139$	8.55348	0.725496	0.362748	+	0.931887i	$0.381839\pi$
$140$	0	0
$141$	0	0
$142$	−5.32238	−0.446644
$143$	6.49828	0.543414
$144$	0	0
$145$	5.04564	0.419018
$146$	8.38101	0.693618
$147$	0	0
$148$	−16.4362	−1.35105
$149$	15.5569	1.27447	0.637236	−	0.770669i	$-0.280078\pi$
$149$	15.5569	1.27447	0.637236	+	0.770669i	$0.280078\pi$
$150$	0	0
$151$	4.99656	0.406614	0.203307	−	0.979115i	$-0.434831\pi$
$151$	4.99656	0.406614	0.203307	+	0.979115i	$0.434831\pi$
$152$	−9.52406	−0.772503
$153$	0	0
$154$	0	0
$155$	−11.1629	−0.896626
$156$	0	0
$157$	18.7880	1.49945	0.749723	−	0.661752i	$-0.230187\pi$
$157$	18.7880	1.49945	0.749723	+	0.661752i	$0.230187\pi$
$158$	−0.677618	−0.0539084
$159$	0	0
$160$	9.26031	0.732092
$161$	0	0
$162$	0	0
$163$	−9.88273	−0.774075	−0.387038	−	0.922064i	$-0.626502\pi$
$163$	−9.88273	−0.774075	−0.387038	+	0.922064i	$0.626502\pi$
$164$	5.26031	0.410761
$165$	0	0
$166$	−1.50172	−0.116556
$167$	−7.04564	−0.545208	−0.272604	−	0.962126i	$-0.587885\pi$
$167$	−7.04564	−0.545208	−0.272604	+	0.962126i	$0.587885\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.76891	−0.212365
$171$	0	0
$172$	−4.67762	−0.356665
$173$	−14.2897	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$173$	−14.2897	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$174$	0	0
$175$	0	0
$176$	15.5793	1.17433
$177$	0	0
$178$	−4.05520	−0.303950
$179$	8.65775	0.647111	0.323555	−	0.946209i	$-0.395122\pi$
$179$	8.65775	0.647111	0.323555	+	0.946209i	$0.395122\pi$
$180$	0	0
$181$	−26.3449	−1.95820	−0.979101	−	0.203373i	$-0.934810\pi$
$181$	−26.3449	−1.95820	−0.979101	+	0.203373i	$0.934810\pi$
$182$	0	0
$183$	0	0
$184$	−11.3776	−0.838766
$185$	16.9966	1.24961
$186$	0	0
$187$	−19.1138	−1.39774
$188$	14.9966	1.09374
$189$	0	0
$190$	4.55348	0.330344
$191$	−6.61555	−0.478684	−0.239342	−	0.970935i	$-0.576932\pi$
$191$	−6.61555	−0.478684	−0.239342	+	0.970935i	$0.576932\pi$
$192$	0	0
$193$	11.8827	0.855338	0.427669	−	0.903935i	$-0.359335\pi$
$193$	11.8827	0.855338	0.427669	+	0.903935i	$0.359335\pi$
$194$	9.37758	0.673271
$195$	0	0
$196$	0	0
$197$	−11.5569	−0.823396	−0.411698	−	0.911320i	$-0.635064\pi$
$197$	−11.5569	−0.823396	−0.411698	+	0.911320i	$0.635064\pi$
$198$	0	0
$199$	−0.996562	−0.0706444	−0.0353222	−	0.999376i	$-0.511246\pi$
$199$	−0.996562	−0.0706444	−0.0353222	+	0.999376i	$0.511246\pi$
$200$	3.61717	0.255772
$201$	0	0
$202$	3.56379	0.250747
$203$	0	0
$204$	0	0
$205$	−5.43965	−0.379921
$206$	−5.75859	−0.401220
$207$	0	0
$208$	2.39744	0.166233
$209$	31.4328	2.17425
$210$	0	0
$211$	26.8302	1.84707	0.923534	−	0.383516i	$-0.125287\pi$
$211$	26.8302	1.84707	0.923534	+	0.383516i	$0.125287\pi$
$212$	10.9966	0.755247
$213$	0	0
$214$	4.49828	0.307496
$215$	4.83709	0.329887
$216$	0	0
$217$	0	0
$218$	5.29317	0.358498
$219$	0	0
$220$	−19.8759	−1.34003
$221$	−2.94137	−0.197858
$222$	0	0
$223$	2.92838	0.196099	0.0980493	−	0.995182i	$-0.468740\pi$
$223$	2.92838	0.196099	0.0980493	+	0.995182i	$0.468740\pi$
$224$	0	0
$225$	0	0
$226$	−7.72938	−0.514150
$227$	−9.79145	−0.649881	−0.324941	−	0.945734i	$-0.605344\pi$
$227$	−9.79145	−0.649881	−0.324941	+	0.945734i	$0.605344\pi$
$228$	0	0
$229$	−3.88273	−0.256578	−0.128289	−	0.991737i	$-0.540949\pi$
$229$	−3.88273	−0.256578	−0.128289	+	0.991737i	$0.540949\pi$
$230$	5.43965	0.358680
$231$	0	0
$232$	−5.58613	−0.366747
$233$	15.8337	1.03730	0.518649	−	0.854988i	$-0.326435\pi$
$233$	15.8337	1.03730	0.518649	+	0.854988i	$0.326435\pi$
$234$	0	0
$235$	−15.5078	−1.01162
$236$	−2.67762	−0.174298
$237$	0	0
$238$	0	0
$239$	6.94137	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$239$	6.94137	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$240$	0	0
$241$	−7.28018	−0.468957	−0.234479	−	0.972121i	$-0.575338\pi$
$241$	−7.28018	−0.468957	−0.234479	+	0.972121i	$0.575338\pi$
$242$	16.5293	1.06254
$243$	0	0
$244$	6.67762	0.427491
$245$	0	0
$246$	0	0
$247$	4.83709	0.307777
$248$	12.3587	0.784777
$249$	0	0
$250$	−6.43621	−0.407062
$251$	23.3224	1.47210	0.736048	−	0.676930i	$-0.236690\pi$
$251$	23.3224	1.47210	0.736048	+	0.676930i	$0.236690\pi$
$252$	0	0
$253$	37.5500	2.36075
$254$	−5.23109	−0.328228
$255$	0	0
$256$	−2.00591	−0.125370
$257$	15.4948	0.966542	0.483271	−	0.875471i	$-0.339449\pi$
$257$	15.4948	0.966542	0.483271	+	0.875471i	$0.339449\pi$
$258$	0	0
$259$	0	0
$260$	−3.05863	−0.189688
$261$	0	0
$262$	1.99312	0.123136
$263$	−3.42666	−0.211297	−0.105648	−	0.994404i	$-0.533692\pi$
$263$	−3.42666	−0.211297	−0.105648	+	0.994404i	$0.533692\pi$
$264$	0	0
$265$	−11.3715	−0.698543
$266$	0	0
$267$	0	0
$268$	−9.75859	−0.596101
$269$	−0.172462	−0.0105152	−0.00525759	−	0.999986i	$-0.501674\pi$
$269$	−0.172462	−0.0105152	−0.00525759	+	0.999986i	$0.501674\pi$
$270$	0	0
$271$	25.9931	1.57897	0.789485	−	0.613770i	$-0.210348\pi$
$271$	25.9931	1.57897	0.789485	+	0.613770i	$0.210348\pi$
$272$	−7.05176	−0.427576
$273$	0	0
$274$	8.99656	0.543502
$275$	−11.9379	−0.719884
$276$	0	0
$277$	3.72326	0.223709	0.111855	−	0.993725i	$-0.464321\pi$
$277$	3.72326	0.223709	0.111855	+	0.993725i	$0.464321\pi$
$278$	4.52750	0.271541
$279$	0	0
$280$	0	0
$281$	21.5500	1.28557	0.642784	−	0.766048i	$-0.277779\pi$
$281$	21.5500	1.28557	0.642784	+	0.766048i	$0.277779\pi$
$282$	0	0
$283$	31.8759	1.89482	0.947412	−	0.320018i	$-0.103689\pi$
$283$	31.8759	1.89482	0.947412	+	0.320018i	$0.103689\pi$
$284$	17.2932	1.02616
$285$	0	0
$286$	3.43965	0.203391
$287$	0	0
$288$	0	0
$289$	−8.34836	−0.491080
$290$	2.67074	0.156831
$291$	0	0
$292$	−27.2311	−1.59358
$293$	3.42666	0.200188	0.100094	−	0.994978i	$-0.468086\pi$
$293$	3.42666	0.200188	0.100094	+	0.994978i	$0.468086\pi$
$294$	0	0
$295$	2.76891	0.161212
$296$	−18.8172	−1.09373
$297$	0	0
$298$	8.23453	0.477014
$299$	5.77846	0.334177
$300$	0	0
$301$	0	0
$302$	2.64476	0.152189
$303$	0	0
$304$	11.5966	0.665113
$305$	−6.90528	−0.395395
$306$	0	0
$307$	3.39744	0.193902	0.0969511	−	0.995289i	$-0.469091\pi$
$307$	3.39744	0.193902	0.0969511	+	0.995289i	$0.469091\pi$
$308$	0	0
$309$	0	0
$310$	−5.90871	−0.335592
$311$	−0.443086	−0.0251251	−0.0125625	−	0.999921i	$-0.503999\pi$
$311$	−0.443086	−0.0251251	−0.0125625	+	0.999921i	$0.503999\pi$
$312$	0	0
$313$	19.8827	1.12384	0.561919	−	0.827192i	$-0.310063\pi$
$313$	19.8827	1.12384	0.561919	+	0.827192i	$0.310063\pi$
$314$	9.94480	0.561218
$315$	0	0
$316$	2.20168	0.123854
$317$	−9.46563	−0.531643	−0.265821	−	0.964022i	$-0.585643\pi$
$317$	−9.46563	−0.531643	−0.265821	+	0.964022i	$0.585643\pi$
$318$	0	0
$319$	18.4362	1.03223
$320$	−3.62586	−0.202692
$321$	0	0
$322$	0	0
$323$	−14.2277	−0.791648
$324$	0	0
$325$	−1.83709	−0.101903
$326$	−5.23109	−0.289724
$327$	0	0
$328$	6.02234	0.332528
$329$	0	0
$330$	0	0
$331$	−27.4328	−1.50784	−0.753921	−	0.656965i	$-0.771840\pi$
$331$	−27.4328	−1.50784	−0.753921	+	0.656965i	$0.771840\pi$
$332$	4.87930	0.267786
$333$	0	0
$334$	−3.72938	−0.204062
$335$	10.0913	0.551346
$336$	0	0
$337$	−20.2767	−1.10454	−0.552272	−	0.833664i	$-0.686239\pi$
$337$	−20.2767	−1.10454	−0.552272	+	0.833664i	$0.686239\pi$
$338$	0.529317	0.0287910
$339$	0	0
$340$	8.99656	0.487907
$341$	−40.7880	−2.20879
$342$	0	0
$343$	0	0
$344$	−5.35524	−0.288735
$345$	0	0
$346$	−7.56379	−0.406632
$347$	−16.7328	−0.898265	−0.449132	−	0.893465i	$-0.648267\pi$
$347$	−16.7328	−0.898265	−0.449132	+	0.893465i	$0.648267\pi$
$348$	0	0
$349$	−22.3940	−1.19872	−0.599362	−	0.800478i	$-0.704579\pi$
$349$	−22.3940	−1.19872	−0.599362	+	0.800478i	$0.704579\pi$
$350$	0	0
$351$	0	0
$352$	33.8361	1.80347
$353$	−27.1690	−1.44606	−0.723031	−	0.690816i	$-0.757251\pi$
$353$	−27.1690	−1.44606	−0.723031	+	0.690816i	$0.757251\pi$
$354$	0	0
$355$	−17.8827	−0.949117
$356$	13.1759	0.698321
$357$	0	0
$358$	4.58269	0.242203
$359$	29.0586	1.53366	0.766828	−	0.641853i	$-0.221834\pi$
$359$	29.0586	1.53366	0.766828	+	0.641853i	$0.221834\pi$
$360$	0	0
$361$	4.39744	0.231444
$362$	−13.9448	−0.732923
$363$	0	0
$364$	0	0
$365$	28.1595	1.47393
$366$	0	0
$367$	4.44309	0.231927	0.115964	−	0.993253i	$-0.463004\pi$
$367$	4.44309	0.231927	0.115964	+	0.993253i	$0.463004\pi$
$368$	13.8535	0.722165
$369$	0	0
$370$	8.99656	0.467709
$371$	0	0
$372$	0	0
$373$	31.3415	1.62280	0.811400	−	0.584491i	$-0.198706\pi$
$373$	31.3415	1.62280	0.811400	+	0.584491i	$0.198706\pi$
$374$	−10.1173	−0.523151
$375$	0	0
$376$	17.1690	0.885425
$377$	2.83709	0.146118
$378$	0	0
$379$	−11.5569	−0.593639	−0.296819	−	0.954934i	$-0.595926\pi$
$379$	−11.5569	−0.593639	−0.296819	+	0.954934i	$0.595926\pi$
$380$	−14.7949	−0.758962
$381$	0	0
$382$	−3.50172	−0.179164
$383$	−25.6673	−1.31154	−0.655769	−	0.754962i	$-0.727655\pi$
$383$	−25.6673	−1.31154	−0.655769	+	0.754962i	$0.727655\pi$
$384$	0	0
$385$	0	0
$386$	6.28973	0.320139
$387$	0	0
$388$	−30.4691	−1.54683
$389$	24.2277	1.22839	0.614195	−	0.789154i	$-0.289481\pi$
$389$	24.2277	1.22839	0.614195	+	0.789154i	$0.289481\pi$
$390$	0	0
$391$	−16.9966	−0.859553
$392$	0	0
$393$	0	0
$394$	−6.11727	−0.308183
$395$	−2.27674	−0.114555
$396$	0	0
$397$	14.3680	0.721111	0.360555	−	0.932738i	$-0.382587\pi$
$397$	14.3680	0.721111	0.360555	+	0.932738i	$0.382587\pi$
$398$	−0.527497	−0.0264410
$399$	0	0
$400$	−4.40432	−0.220216
$401$	21.8827	1.09277	0.546386	−	0.837534i	$-0.316003\pi$
$401$	21.8827	1.09277	0.546386	+	0.837534i	$0.316003\pi$
$402$	0	0
$403$	−6.27674	−0.312667
$404$	−11.5793	−0.576089
$405$	0	0
$406$	0	0
$407$	62.1035	3.07836
$408$	0	0
$409$	−7.39057	−0.365440	−0.182720	−	0.983165i	$-0.558490\pi$
$409$	−7.39057	−0.365440	−0.182720	+	0.983165i	$0.558490\pi$
$410$	−2.87930	−0.142198
$411$	0	0
$412$	18.7105	0.921799
$413$	0	0
$414$	0	0
$415$	−5.04564	−0.247681
$416$	5.20693	0.255291
$417$	0	0
$418$	16.6379	0.813786
$419$	33.7846	1.65048	0.825242	−	0.564779i	$-0.191039\pi$
$419$	33.7846	1.65048	0.825242	+	0.564779i	$0.191039\pi$
$420$	0	0
$421$	−35.5500	−1.73260	−0.866301	−	0.499522i	$-0.833509\pi$
$421$	−35.5500	−1.73260	−0.866301	+	0.499522i	$0.833509\pi$
$422$	14.2017	0.691327
$423$	0	0
$424$	12.5896	0.611403
$425$	5.40356	0.262111
$426$	0	0
$427$	0	0
$428$	−14.6155	−0.706469
$429$	0	0
$430$	2.56035	0.123471
$431$	−23.7294	−1.14300	−0.571502	−	0.820601i	$-0.693639\pi$
$431$	−23.7294	−1.14300	−0.571502	+	0.820601i	$0.693639\pi$
$432$	0	0
$433$	5.55691	0.267048	0.133524	−	0.991046i	$-0.457371\pi$
$433$	5.55691	0.267048	0.133524	+	0.991046i	$0.457371\pi$
$434$	0	0
$435$	0	0
$436$	−17.1982	−0.823646
$437$	27.9509	1.33707
$438$	0	0
$439$	−5.32926	−0.254352	−0.127176	−	0.991880i	$-0.540591\pi$
$439$	−5.32926	−0.254352	−0.127176	+	0.991880i	$0.540591\pi$
$440$	−22.7552	−1.08481
$441$	0	0
$442$	−1.55691	−0.0740549
$443$	−5.33537	−0.253491	−0.126746	−	0.991935i	$-0.540453\pi$
$443$	−5.33537	−0.253491	−0.126746	+	0.991935i	$0.540453\pi$
$444$	0	0
$445$	−13.6251	−0.645892
$446$	1.55004	0.0733965
$447$	0	0
$448$	0	0
$449$	−26.2277	−1.23776	−0.618880	−	0.785486i	$-0.712413\pi$
$449$	−26.2277	−1.23776	−0.618880	+	0.785486i	$0.712413\pi$
$450$	0	0
$451$	−19.8759	−0.935918
$452$	25.1138	1.18126
$453$	0	0
$454$	−5.18278	−0.243240
$455$	0	0
$456$	0	0
$457$	11.4396	0.535124	0.267562	−	0.963541i	$-0.413782\pi$
$457$	11.4396	0.535124	0.267562	+	0.963541i	$0.413782\pi$
$458$	−2.05520	−0.0960330
$459$	0	0
$460$	−17.6742	−0.824063
$461$	−28.0552	−1.30666	−0.653330	−	0.757073i	$-0.726629\pi$
$461$	−28.0552	−1.30666	−0.653330	+	0.757073i	$0.726629\pi$
$462$	0	0
$463$	−10.5604	−0.490781	−0.245391	−	0.969424i	$-0.578916\pi$
$463$	−10.5604	−0.490781	−0.245391	+	0.969424i	$0.578916\pi$
$464$	6.80176	0.315764
$465$	0	0
$466$	8.38101	0.388243
$467$	16.6776	0.771748	0.385874	−	0.922551i	$-0.373900\pi$
$467$	16.6776	0.771748	0.385874	+	0.922551i	$0.373900\pi$
$468$	0	0
$469$	0	0
$470$	−8.20855	−0.378632
$471$	0	0
$472$	−3.06551	−0.141101
$473$	17.6742	0.812660
$474$	0	0
$475$	−8.88617	−0.407726
$476$	0	0
$477$	0	0
$478$	3.67418	0.168053
$479$	4.06819	0.185880	0.0929401	−	0.995672i	$-0.470374\pi$
$479$	4.06819	0.185880	0.0929401	+	0.995672i	$0.470374\pi$
$480$	0	0
$481$	9.55691	0.435758
$482$	−3.85352	−0.175523
$483$	0	0
$484$	−53.7061	−2.44119
$485$	31.5078	1.43070
$486$	0	0
$487$	−0.443086	−0.0200781	−0.0100391	−	0.999950i	$-0.503196\pi$
$487$	−0.443086	−0.0200781	−0.0100391	+	0.999950i	$0.503196\pi$
$488$	7.64496	0.346071
$489$	0	0
$490$	0	0
$491$	−20.7328	−0.935659	−0.467829	−	0.883819i	$-0.654964\pi$
$491$	−20.7328	−0.935659	−0.467829	+	0.883819i	$0.654964\pi$
$492$	0	0
$493$	−8.34492	−0.375836
$494$	2.56035	0.115196
$495$	0	0
$496$	−15.0481	−0.675680
$497$	0	0
$498$	0	0
$499$	−13.9931	−0.626418	−0.313209	−	0.949684i	$-0.601404\pi$
$499$	−13.9931	−0.626418	−0.313209	+	0.949684i	$0.601404\pi$
$500$	20.9122	0.935220
$501$	0	0
$502$	12.3449	0.550981
$503$	5.67418	0.252999	0.126500	−	0.991967i	$-0.459626\pi$
$503$	5.67418	0.252999	0.126500	+	0.991967i	$0.459626\pi$
$504$	0	0
$505$	11.9740	0.532837
$506$	19.8759	0.883590
$507$	0	0
$508$	16.9966	0.754101
$509$	−6.22154	−0.275765	−0.137883	−	0.990449i	$-0.544030\pi$
$509$	−6.22154	−0.275765	−0.137883	+	0.990449i	$0.544030\pi$
$510$	0	0
$511$	0	0
$512$	21.9243	0.968926
$513$	0	0
$514$	8.20168	0.361760
$515$	−19.3484	−0.852591
$516$	0	0
$517$	−56.6639	−2.49207
$518$	0	0
$519$	0	0
$520$	−3.50172	−0.153561
$521$	−10.2897	−0.450801	−0.225401	−	0.974266i	$-0.572369\pi$
$521$	−10.2897	−0.450801	−0.225401	+	0.974266i	$0.572369\pi$
$522$	0	0
$523$	2.32582	0.101701	0.0508505	−	0.998706i	$-0.483807\pi$
$523$	2.32582	0.101701	0.0508505	+	0.998706i	$0.483807\pi$
$524$	−6.47594	−0.282903
$525$	0	0
$526$	−1.81379	−0.0790849
$527$	18.4622	0.804226
$528$	0	0
$529$	10.3906	0.451764
$530$	−6.01910	−0.261453
$531$	0	0
$532$	0	0
$533$	−3.05863	−0.132484
$534$	0	0
$535$	15.1138	0.653428
$536$	−11.1723	−0.482568
$537$	0	0
$538$	−0.0912868	−0.00393565
$539$	0	0
$540$	0	0
$541$	5.55691	0.238910	0.119455	−	0.992840i	$-0.461885\pi$
$541$	5.55691	0.238910	0.119455	+	0.992840i	$0.461885\pi$
$542$	13.7586	0.590982
$543$	0	0
$544$	−15.3155	−0.656647
$545$	17.7846	0.761807
$546$	0	0
$547$	−5.48873	−0.234681	−0.117341	−	0.993092i	$-0.537437\pi$
$547$	−5.48873	−0.234681	−0.117341	+	0.993092i	$0.537437\pi$
$548$	−29.2311	−1.24869
$549$	0	0
$550$	−6.31894	−0.269441
$551$	13.7233	0.584631
$552$	0	0
$553$	0	0
$554$	1.97078	0.0837306
$555$	0	0
$556$	−14.7105	−0.623863
$557$	22.9897	0.974104	0.487052	−	0.873373i	$-0.338072\pi$
$557$	22.9897	0.974104	0.487052	+	0.873373i	$0.338072\pi$
$558$	0	0
$559$	2.71982	0.115036
$560$	0	0
$561$	0	0
$562$	11.4068	0.481167
$563$	−32.7620	−1.38075	−0.690377	−	0.723449i	$-0.742556\pi$
$563$	−32.7620	−1.38075	−0.690377	+	0.723449i	$0.742556\pi$
$564$	0	0
$565$	−25.9700	−1.09257
$566$	16.8724	0.709201
$567$	0	0
$568$	19.7983	0.830719
$569$	−36.1526	−1.51560	−0.757798	−	0.652489i	$-0.773725\pi$
$569$	−36.1526	−1.51560	−0.757798	+	0.652489i	$0.773725\pi$
$570$	0	0
$571$	6.48529	0.271401	0.135700	−	0.990750i	$-0.456672\pi$
$571$	6.48529	0.271401	0.135700	+	0.990750i	$0.456672\pi$
$572$	−11.1759	−0.467288
$573$	0	0
$574$	0	0
$575$	−10.6155	−0.442699
$576$	0	0
$577$	−3.99312	−0.166236	−0.0831180	−	0.996540i	$-0.526488\pi$
$577$	−3.99312	−0.166236	−0.0831180	+	0.996540i	$0.526488\pi$
$578$	−4.41893	−0.183803
$579$	0	0
$580$	−8.67762	−0.360318
$581$	0	0
$582$	0	0
$583$	−41.5500	−1.72083
$584$	−31.1759	−1.29007
$585$	0	0
$586$	1.81379	0.0749268
$587$	4.16635	0.171964	0.0859818	−	0.996297i	$-0.472597\pi$
$587$	4.16635	0.171964	0.0859818	+	0.996297i	$0.472597\pi$
$588$	0	0
$589$	−30.3611	−1.25101
$590$	1.46563	0.0603389
$591$	0	0
$592$	22.9122	0.941684
$593$	16.1303	0.662390	0.331195	−	0.943562i	$-0.392548\pi$
$593$	16.1303	0.662390	0.331195	+	0.943562i	$0.392548\pi$
$594$	0	0
$595$	0	0
$596$	−26.7552	−1.09593
$597$	0	0
$598$	3.05863	0.125077
$599$	33.2372	1.35804	0.679018	−	0.734122i	$-0.262406\pi$
$599$	33.2372	1.35804	0.679018	+	0.734122i	$0.262406\pi$
$600$	0	0
$601$	28.9897	1.18251	0.591257	−	0.806483i	$-0.298632\pi$
$601$	28.9897	1.18251	0.591257	+	0.806483i	$0.298632\pi$
$602$	0	0
$603$	0	0
$604$	−8.59321	−0.349653
$605$	55.5370	2.25790
$606$	0	0
$607$	25.7846	1.04656	0.523282	−	0.852160i	$-0.324708\pi$
$607$	25.7846	1.04656	0.523282	+	0.852160i	$0.324708\pi$
$608$	25.1864	1.02144
$609$	0	0
$610$	−3.65508	−0.147990
$611$	−8.71982	−0.352766
$612$	0	0
$613$	−7.43965	−0.300485	−0.150242	−	0.988649i	$-0.548005\pi$
$613$	−7.43965	−0.300485	−0.150242	+	0.988649i	$0.548005\pi$
$614$	1.79832	0.0725744
$615$	0	0
$616$	0	0
$617$	2.67074	0.107520	0.0537600	−	0.998554i	$-0.482879\pi$
$617$	2.67074	0.107520	0.0537600	+	0.998554i	$0.482879\pi$
$618$	0	0
$619$	4.46907	0.179627	0.0898134	−	0.995959i	$-0.471373\pi$
$619$	4.46907	0.179627	0.0898134	+	0.995959i	$0.471373\pi$
$620$	19.1982	0.771020
$621$	0	0
$622$	−0.234533	−0.00940390
$623$	0	0
$624$	0	0
$625$	−12.4396	−0.497586
$626$	10.5243	0.420634
$627$	0	0
$628$	−32.3121	−1.28939
$629$	−28.1104	−1.12083
$630$	0	0
$631$	23.0878	0.919113	0.459556	−	0.888149i	$-0.348008\pi$
$631$	23.0878	0.919113	0.459556	+	0.888149i	$0.348008\pi$
$632$	2.52062	0.100265
$633$	0	0
$634$	−5.01031	−0.198985
$635$	−17.5760	−0.697483
$636$	0	0
$637$	0	0
$638$	9.75859	0.386346
$639$	0	0
$640$	−20.4398	−0.807956
$641$	−41.8268	−1.65206	−0.826029	−	0.563627i	$-0.809405\pi$
$641$	−41.8268	−1.65206	−0.826029	+	0.563627i	$0.809405\pi$
$642$	0	0
$643$	−3.11383	−0.122797	−0.0613987	−	0.998113i	$-0.519556\pi$
$643$	−3.11383	−0.122797	−0.0613987	+	0.998113i	$0.519556\pi$
$644$	0	0
$645$	0	0
$646$	−7.53093	−0.296301
$647$	30.4362	1.19657	0.598285	−	0.801283i	$-0.295849\pi$
$647$	30.4362	1.19657	0.598285	+	0.801283i	$0.295849\pi$
$648$	0	0
$649$	10.1173	0.397137
$650$	−0.972402	−0.0381408
$651$	0	0
$652$	16.9966	0.665637
$653$	−3.99312	−0.156263	−0.0781315	−	0.996943i	$-0.524895\pi$
$653$	−3.99312	−0.156263	−0.0781315	+	0.996943i	$0.524895\pi$
$654$	0	0
$655$	6.69672	0.261663
$656$	−7.33290	−0.286302
$657$	0	0
$658$	0	0
$659$	−32.6578	−1.27217	−0.636083	−	0.771621i	$-0.719446\pi$
$659$	−32.6578	−1.27217	−0.636083	+	0.771621i	$0.719446\pi$
$660$	0	0
$661$	−43.7355	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$661$	−43.7355	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$662$	−14.5206	−0.564360
$663$	0	0
$664$	5.58613	0.216784
$665$	0	0
$666$	0	0
$667$	16.3940	0.634778
$668$	12.1173	0.468831
$669$	0	0
$670$	5.34149	0.206360
$671$	−25.2311	−0.974036
$672$	0	0
$673$	−9.63198	−0.371285	−0.185643	−	0.982617i	$-0.559437\pi$
$673$	−9.63198	−0.371285	−0.185643	+	0.982617i	$0.559437\pi$
$674$	−10.7328	−0.413413
$675$	0	0
$676$	−1.71982	−0.0661471
$677$	16.4914	0.633816	0.316908	−	0.948456i	$-0.397355\pi$
$677$	16.4914	0.633816	0.316908	+	0.948456i	$0.397355\pi$
$678$	0	0
$679$	0	0
$680$	10.2998	0.394981
$681$	0	0
$682$	−21.5898	−0.826715
$683$	−41.9311	−1.60445	−0.802224	−	0.597024i	$-0.796350\pi$
$683$	−41.9311	−1.60445	−0.802224	+	0.597024i	$0.796350\pi$
$684$	0	0
$685$	30.2277	1.15494
$686$	0	0
$687$	0	0
$688$	6.52062	0.248596
$689$	−6.39400	−0.243592
$690$	0	0
$691$	−38.1786	−1.45238	−0.726191	−	0.687493i	$-0.758711\pi$
$691$	−38.1786	−1.45238	−0.726191	+	0.687493i	$0.758711\pi$
$692$	24.5758	0.934232
$693$	0	0
$694$	−8.85696	−0.336205
$695$	15.2120	0.577024
$696$	0	0
$697$	8.99656	0.340769
$698$	−11.8535	−0.448662
$699$	0	0
$700$	0	0
$701$	−16.5957	−0.626810	−0.313405	−	0.949620i	$-0.601470\pi$
$701$	−16.5957	−0.626810	−0.313405	+	0.949620i	$0.601470\pi$
$702$	0	0
$703$	46.2277	1.74351
$704$	−13.2485	−0.499321
$705$	0	0
$706$	−14.3810	−0.541237
$707$	0	0
$708$	0	0
$709$	21.7655	0.817419	0.408710	−	0.912664i	$-0.365979\pi$
$709$	21.7655	0.817419	0.408710	+	0.912664i	$0.365979\pi$
$710$	−9.46563	−0.355239
$711$	0	0
$712$	15.0846	0.565320
$713$	−36.2699	−1.35832
$714$	0	0
$715$	11.5569	0.432204
$716$	−14.8898	−0.556458
$717$	0	0
$718$	15.3812	0.574022
$719$	36.7880	1.37196	0.685981	−	0.727620i	$-0.259373\pi$
$719$	36.7880	1.37196	0.685981	+	0.727620i	$0.259373\pi$
$720$	0	0
$721$	0	0
$722$	2.32764	0.0866258
$723$	0	0
$724$	45.3086	1.68388
$725$	−5.21199	−0.193568
$726$	0	0
$727$	32.3189	1.19864	0.599322	−	0.800508i	$-0.295437\pi$
$727$	32.3189	1.19864	0.599322	+	0.800508i	$0.295437\pi$
$728$	0	0
$729$	0	0
$730$	14.9053	0.551669
$731$	−8.00000	−0.295891
$732$	0	0
$733$	12.7198	0.469817	0.234909	−	0.972017i	$-0.424521\pi$
$733$	12.7198	0.469817	0.234909	+	0.972017i	$0.424521\pi$
$734$	2.35180	0.0868065
$735$	0	0
$736$	30.0881	1.10906
$737$	36.8724	1.35821
$738$	0	0
$739$	23.1982	0.853361	0.426681	−	0.904402i	$-0.359683\pi$
$739$	23.1982	0.853361	0.426681	+	0.904402i	$0.359683\pi$
$740$	−29.2311	−1.07456
$741$	0	0
$742$	0	0
$743$	−31.8138	−1.16713	−0.583567	−	0.812065i	$-0.698344\pi$
$743$	−31.8138	−1.16713	−0.583567	+	0.812065i	$0.698344\pi$
$744$	0	0
$745$	27.6673	1.01365
$746$	16.5896	0.607387
$747$	0	0
$748$	32.8724	1.20193
$749$	0	0
$750$	0	0
$751$	−0.863070	−0.0314939	−0.0157469	−	0.999876i	$-0.505013\pi$
$751$	−0.863070	−0.0314939	−0.0157469	+	0.999876i	$0.505013\pi$
$752$	−20.9053	−0.762337
$753$	0	0
$754$	1.50172	0.0546894
$755$	8.88617	0.323401
$756$	0	0
$757$	−13.3974	−0.486938	−0.243469	−	0.969909i	$-0.578285\pi$
$757$	−13.3974	−0.486938	−0.243469	+	0.969909i	$0.578285\pi$
$758$	−6.11727	−0.222189
$759$	0	0
$760$	−16.9381	−0.614411
$761$	−35.5630	−1.28916	−0.644579	−	0.764537i	$-0.722967\pi$
$761$	−35.5630	−1.28916	−0.644579	+	0.764537i	$0.722967\pi$
$762$	0	0
$763$	0	0
$764$	11.3776	0.411626
$765$	0	0
$766$	−13.5861	−0.490887
$767$	1.55691	0.0562169
$768$	0	0
$769$	48.3871	1.74488	0.872442	−	0.488717i	$-0.162535\pi$
$769$	48.3871	1.74488	0.872442	+	0.488717i	$0.162535\pi$
$770$	0	0
$771$	0	0
$772$	−20.4362	−0.735515
$773$	17.9379	0.645182	0.322591	−	0.946538i	$-0.395446\pi$
$773$	17.9379	0.645182	0.322591	+	0.946538i	$0.395446\pi$
$774$	0	0
$775$	11.5309	0.414203
$776$	−34.8829	−1.25222
$777$	0	0
$778$	12.8241	0.459766
$779$	−14.7949	−0.530082
$780$	0	0
$781$	−65.3415	−2.33810
$782$	−8.99656	−0.321716
$783$	0	0
$784$	0	0
$785$	33.4137	1.19258
$786$	0	0
$787$	3.71639	0.132475	0.0662374	−	0.997804i	$-0.478901\pi$
$787$	3.71639	0.132475	0.0662374	+	0.997804i	$0.478901\pi$
$788$	19.8759	0.708048
$789$	0	0
$790$	−1.20512	−0.0428761
$791$	0	0
$792$	0	0
$793$	−3.88273	−0.137880
$794$	7.60523	0.269900
$795$	0	0
$796$	1.71391	0.0607480
$797$	14.9673	0.530171	0.265085	−	0.964225i	$-0.414600\pi$
$797$	14.9673	0.530171	0.265085	+	0.964225i	$0.414600\pi$
$798$	0	0
$799$	25.6482	0.907368
$800$	−9.56561	−0.338195
$801$	0	0
$802$	11.5829	0.409006
$803$	102.892	3.63096
$804$	0	0
$805$	0	0
$806$	−3.32238	−0.117026
$807$	0	0
$808$	−13.2567	−0.466368
$809$	−11.7233	−0.412168	−0.206084	−	0.978534i	$-0.566072\pi$
$809$	−11.7233	−0.412168	−0.206084	+	0.978534i	$0.566072\pi$
$810$	0	0
$811$	0.886172	0.0311177	0.0155588	−	0.999879i	$-0.495047\pi$
$811$	0.886172	0.0311177	0.0155588	+	0.999879i	$0.495047\pi$
$812$	0	0
$813$	0	0
$814$	32.8724	1.15218
$815$	−17.5760	−0.615661
$816$	0	0
$817$	13.1560	0.460271
$818$	−3.91195	−0.136778
$819$	0	0
$820$	9.35524	0.326699
$821$	53.7846	1.87709	0.938547	−	0.345151i	$-0.112172\pi$
$821$	53.7846	1.87709	0.938547	+	0.345151i	$0.112172\pi$
$822$	0	0
$823$	−35.7655	−1.24671	−0.623353	−	0.781941i	$-0.714230\pi$
$823$	−35.7655	−1.24671	−0.623353	+	0.781941i	$0.714230\pi$
$824$	21.4209	0.746234
$825$	0	0
$826$	0	0
$827$	−7.41043	−0.257686	−0.128843	−	0.991665i	$-0.541126\pi$
$827$	−7.41043	−0.257686	−0.128843	+	0.991665i	$0.541126\pi$
$828$	0	0
$829$	31.9931	1.11117	0.555584	−	0.831461i	$-0.312495\pi$
$829$	31.9931	1.11117	0.555584	+	0.831461i	$0.312495\pi$
$830$	−2.67074	−0.0927028
$831$	0	0
$832$	−2.03877	−0.0706816
$833$	0	0
$834$	0	0
$835$	−12.5304	−0.433631
$836$	−54.0588	−1.86966
$837$	0	0
$838$	17.8827	0.617749
$839$	−1.55691	−0.0537506	−0.0268753	−	0.999639i	$-0.508556\pi$
$839$	−1.55691	−0.0537506	−0.0268753	+	0.999639i	$0.508556\pi$
$840$	0	0
$841$	−20.9509	−0.722445
$842$	−18.8172	−0.648484
$843$	0	0
$844$	−46.1432	−1.58832
$845$	1.77846	0.0611808
$846$	0	0
$847$	0	0
$848$	−15.3293	−0.526409
$849$	0	0
$850$	2.86019	0.0981038
$851$	55.2242	1.89306
$852$	0	0
$853$	−35.4750	−1.21464	−0.607320	−	0.794457i	$-0.707755\pi$
$853$	−35.4750	−1.21464	−0.607320	+	0.794457i	$0.707755\pi$
$854$	0	0
$855$	0	0
$856$	−16.7328	−0.571916
$857$	−2.49828	−0.0853397	−0.0426698	−	0.999089i	$-0.513586\pi$
$857$	−2.49828	−0.0853397	−0.0426698	+	0.999089i	$0.513586\pi$
$858$	0	0
$859$	−52.0122	−1.77463	−0.887317	−	0.461160i	$-0.847434\pi$
$859$	−52.0122	−1.77463	−0.887317	+	0.461160i	$0.847434\pi$
$860$	−8.31894	−0.283674
$861$	0	0
$862$	−12.5604	−0.427807
$863$	−34.8172	−1.18519	−0.592596	−	0.805500i	$-0.701897\pi$
$863$	−34.8172	−1.18519	−0.592596	+	0.805500i	$0.701897\pi$
$864$	0	0
$865$	−25.4137	−0.864091
$866$	2.94137	0.0999517
$867$	0	0
$868$	0	0
$869$	−8.31894	−0.282201
$870$	0	0
$871$	5.67418	0.192262
$872$	−19.6896	−0.666776
$873$	0	0
$874$	14.7949	0.500444
$875$	0	0
$876$	0	0
$877$	−11.2571	−0.380124	−0.190062	−	0.981772i	$-0.560869\pi$
$877$	−11.2571	−0.380124	−0.190062	+	0.981772i	$0.560869\pi$
$878$	−2.82086	−0.0951996
$879$	0	0
$880$	27.7070	0.934004
$881$	−9.50172	−0.320121	−0.160061	−	0.987107i	$-0.551169\pi$
$881$	−9.50172	−0.320121	−0.160061	+	0.987107i	$0.551169\pi$
$882$	0	0
$883$	23.7655	0.799772	0.399886	−	0.916565i	$-0.369050\pi$
$883$	23.7655	0.799772	0.399886	+	0.916565i	$0.369050\pi$
$884$	5.05863	0.170140
$885$	0	0
$886$	−2.82410	−0.0948775
$887$	18.3518	0.616193	0.308097	−	0.951355i	$-0.400308\pi$
$887$	18.3518	0.616193	0.308097	+	0.951355i	$0.400308\pi$
$888$	0	0
$889$	0	0
$890$	−7.21199	−0.241746
$891$	0	0
$892$	−5.03629	−0.168628
$893$	−42.1786	−1.41145
$894$	0	0
$895$	15.3974	0.514680
$896$	0	0
$897$	0	0
$898$	−13.8827	−0.463273
$899$	−17.8077	−0.593919
$900$	0	0
$901$	18.8071	0.626556
$902$	−10.5206	−0.350298
$903$	0	0
$904$	28.7519	0.956275
$905$	−46.8533	−1.55746
$906$	0	0
$907$	35.7164	1.18594	0.592972	−	0.805223i	$-0.297955\pi$
$907$	35.7164	1.18594	0.592972	+	0.805223i	$0.297955\pi$
$908$	16.8396	0.558841
$909$	0	0
$910$	0	0
$911$	16.2147	0.537216	0.268608	−	0.963250i	$-0.413436\pi$
$911$	16.2147	0.537216	0.268608	+	0.963250i	$0.413436\pi$
$912$	0	0
$913$	−18.4362	−0.610149
$914$	6.05520	0.200288
$915$	0	0
$916$	6.67762	0.220635
$917$	0	0
$918$	0	0
$919$	−34.4622	−1.13680	−0.568401	−	0.822751i	$-0.692438\pi$
$919$	−34.4622	−1.13680	−0.568401	+	0.822751i	$0.692438\pi$
$920$	−20.2345	−0.667113
$921$	0	0
$922$	−14.8501	−0.489061
$923$	−10.0552	−0.330971
$924$	0	0
$925$	−17.5569	−0.577268
$926$	−5.58977	−0.183691
$927$	0	0
$928$	14.7725	0.484933
$929$	−28.4492	−0.933388	−0.466694	−	0.884419i	$-0.654555\pi$
$929$	−28.4492	−0.933388	−0.466694	+	0.884419i	$0.654555\pi$
$930$	0	0
$931$	0	0
$932$	−27.2311	−0.891984
$933$	0	0
$934$	8.82774	0.288852
$935$	−33.9931	−1.11169
$936$	0	0
$937$	−2.91215	−0.0951358	−0.0475679	−	0.998868i	$-0.515147\pi$
$937$	−2.91215	−0.0951358	−0.0475679	+	0.998868i	$0.515147\pi$
$938$	0	0
$939$	0	0
$940$	26.6707	0.869904
$941$	−59.0096	−1.92366	−0.961828	−	0.273654i	$-0.911768\pi$
$941$	−59.0096	−1.92366	−0.961828	+	0.273654i	$0.911768\pi$
$942$	0	0
$943$	−17.6742	−0.575551
$944$	3.73261	0.121486
$945$	0	0
$946$	9.35524	0.304165
$947$	−34.8432	−1.13225	−0.566126	−	0.824319i	$-0.691558\pi$
$947$	−34.8432	−1.13225	−0.566126	+	0.824319i	$0.691558\pi$
$948$	0	0
$949$	15.8337	0.513982
$950$	−4.70360	−0.152605
$951$	0	0
$952$	0	0
$953$	−23.9578	−0.776069	−0.388035	−	0.921645i	$-0.626846\pi$
$953$	−23.9578	−0.776069	−0.388035	+	0.921645i	$0.626846\pi$
$954$	0	0
$955$	−11.7655	−0.380722
$956$	−11.9379	−0.386100
$957$	0	0
$958$	2.15336	0.0695718
$959$	0	0
$960$	0	0
$961$	8.39744	0.270885
$962$	5.05863	0.163097
$963$	0	0
$964$	12.5206	0.403262
$965$	21.1329	0.680293
$966$	0	0
$967$	−37.3155	−1.19999	−0.599993	−	0.800005i	$-0.704830\pi$
$967$	−37.3155	−1.19999	−0.599993	+	0.800005i	$0.704830\pi$
$968$	−61.4861	−1.97624
$969$	0	0
$970$	16.6776	0.535486
$971$	20.7880	0.667119	0.333559	−	0.942729i	$-0.391750\pi$
$971$	20.7880	0.667119	0.333559	+	0.942729i	$0.391750\pi$
$972$	0	0
$973$	0	0
$974$	−0.234533	−0.00751491
$975$	0	0
$976$	−9.30863	−0.297962
$977$	−49.0810	−1.57024	−0.785120	−	0.619344i	$-0.787399\pi$
$977$	−49.0810	−1.57024	−0.785120	+	0.619344i	$0.787399\pi$
$978$	0	0
$979$	−49.7846	−1.59112
$980$	0	0
$981$	0	0
$982$	−10.9742	−0.350201
$983$	10.1855	0.324865	0.162433	−	0.986720i	$-0.448066\pi$
$983$	10.1855	0.324865	0.162433	+	0.986720i	$0.448066\pi$
$984$	0	0
$985$	−20.5535	−0.654888
$986$	−4.41711	−0.140669
$987$	0	0
$988$	−8.31894	−0.264661
$989$	15.7164	0.499752
$990$	0	0
$991$	4.34492	0.138021	0.0690105	−	0.997616i	$-0.478016\pi$
$991$	4.34492	0.138021	0.0690105	+	0.997616i	$0.478016\pi$
$992$	−32.6826	−1.03767
$993$	0	0
$994$	0	0
$995$	−1.77234	−0.0561871
$996$	0	0
$997$	−44.9637	−1.42401	−0.712007	−	0.702172i	$-0.752214\pi$
$997$	−44.9637	−1.42401	−0.712007	+	0.702172i	$0.752214\pi$
$998$	−7.40679	−0.234458
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bc.1.2		3
3.2	odd	2		1911.2.a.n.1.2		3
7.6	odd	2		819.2.a.j.1.2		3
21.20	even	2		273.2.a.d.1.2	✓	3
84.83	odd	2		4368.2.a.bq.1.3		3
105.104	even	2		6825.2.a.bd.1.2		3
273.272	even	2		3549.2.a.t.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.d.1.2	✓	3	21.20	even	2
819.2.a.j.1.2		3	7.6	odd	2
1911.2.a.n.1.2		3	3.2	odd	2
3549.2.a.t.1.2		3	273.272	even	2
4368.2.a.bq.1.3		3	84.83	odd	2
5733.2.a.bc.1.2		3	1.1	even	1	trivial
6825.2.a.bd.1.2		3	105.104	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)

\(f(q)\)	\(=\)	\(q+0.529317 q^{2} -1.71982 q^{4} +1.77846 q^{5} -1.96896 q^{8} +O(q^{10})\)	\(q+0.529317 q^{2} -1.71982 q^{4} +1.77846 q^{5} -1.96896 q^{8} +0.941367 q^{10} +6.49828 q^{11} +1.00000 q^{13} +2.39744 q^{16} -2.94137 q^{17} +4.83709 q^{19} -3.05863 q^{20} +3.43965 q^{22} +5.77846 q^{23} -1.83709 q^{25} +0.529317 q^{26} +2.83709 q^{29} -6.27674 q^{31} +5.20693 q^{32} -1.55691 q^{34} +9.55691 q^{37} +2.56035 q^{38} -3.50172 q^{40} -3.05863 q^{41} +2.71982 q^{43} -11.1759 q^{44} +3.05863 q^{46} -8.71982 q^{47} -0.972402 q^{50} -1.71982 q^{52} -6.39400 q^{53} +11.5569 q^{55} +1.50172 q^{58} +1.55691 q^{59} -3.88273 q^{61} -3.32238 q^{62} -2.03877 q^{64} +1.77846 q^{65} +5.67418 q^{67} +5.05863 q^{68} -10.0552 q^{71} +15.8337 q^{73} +5.05863 q^{74} -8.31894 q^{76} -1.28018 q^{79} +4.26375 q^{80} -1.61899 q^{82} -2.83709 q^{83} -5.23109 q^{85} +1.43965 q^{86} -12.7949 q^{88} -7.66119 q^{89} -9.93793 q^{92} -4.61555 q^{94} +8.60256 q^{95} +17.7164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 12 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 12 * q^8	\( 3 q + 2 q^{2} + 4 q^{4} - 3 q^{5} + 12 q^{8} + 2 q^{10} + 2 q^{11} + 3 q^{13} + 18 q^{16} - 8 q^{17} + 7 q^{19} - 10 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} + 2 q^{26} + q^{29} + 7 q^{31} + 36 q^{32} + 12 q^{34} + 12 q^{37} + 26 q^{38} - 28 q^{40} - 10 q^{41} - q^{43} - 36 q^{44} + 10 q^{46} - 17 q^{47} - 20 q^{50} + 4 q^{52} + 5 q^{53} + 18 q^{55} + 22 q^{58} - 12 q^{59} - 10 q^{61} + 10 q^{62} + 58 q^{64} - 3 q^{65} + 2 q^{67} + 16 q^{68} + 4 q^{71} + 5 q^{73} + 16 q^{74} + 30 q^{76} - 13 q^{79} - 8 q^{80} - 24 q^{82} - q^{83} + 16 q^{85} - 14 q^{86} - 60 q^{88} - 13 q^{89} + 6 q^{92} + 2 q^{94} + 15 q^{95} + 9 q^{97}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^4 - 3 * q^5 + 12 * q^8 + 2 * q^10 + 2 * q^11 + 3 * q^13 + 18 * q^16 - 8 * q^17 + 7 * q^19 - 10 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 + 2 * q^26 + q^29 + 7 * q^31 + 36 * q^32 + 12 * q^34 + 12 * q^37 + 26 * q^38 - 28 * q^40 - 10 * q^41 - q^43 - 36 * q^44 + 10 * q^46 - 17 * q^47 - 20 * q^50 + 4 * q^52 + 5 * q^53 + 18 * q^55 + 22 * q^58 - 12 * q^59 - 10 * q^61 + 10 * q^62 + 58 * q^64 - 3 * q^65 + 2 * q^67 + 16 * q^68 + 4 * q^71 + 5 * q^73 + 16 * q^74 + 30 * q^76 - 13 * q^79 - 8 * q^80 - 24 * q^82 - q^83 + 16 * q^85 - 14 * q^86 - 60 * q^88 - 13 * q^89 + 6 * q^92 + 2 * q^94 + 15 * q^95 + 9 * q^97

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