LMFDB - Embedded newform 1053.2.a.k.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1053,2,Mod(1,1053)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1053.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,2,0,4,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.403137.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 3 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.50814$ of defining polynomial
Character	$\chi$	$=$	1053.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.50814 q^{2} +0.274488 q^{4} -2.64315 q^{5} +0.343091 q^{7} -2.60231 q^{8} -3.98624 q^{10} +5.88760 q^{11} +1.00000 q^{13} +0.517430 q^{14} -4.47363 q^{16} +7.72803 q^{17} +1.60678 q^{19} -0.725512 q^{20} +8.87933 q^{22} +5.23365 q^{23} +1.98624 q^{25} +1.50814 q^{26} +0.0941744 q^{28} -0.409497 q^{29} -2.62837 q^{31} -1.54224 q^{32} +11.6550 q^{34} -0.906842 q^{35} +5.94321 q^{37} +2.42326 q^{38} +6.87831 q^{40} +6.61311 q^{41} +1.31085 q^{43} +1.61607 q^{44} +7.89308 q^{46} -3.60086 q^{47} -6.88229 q^{49} +2.99553 q^{50} +0.274488 q^{52} -3.87831 q^{53} -15.5618 q^{55} -0.892832 q^{56} -0.617579 q^{58} +0.452530 q^{59} +3.74029 q^{61} -3.96396 q^{62} +6.62136 q^{64} -2.64315 q^{65} -2.80970 q^{67} +2.12125 q^{68} -1.36765 q^{70} +16.5889 q^{71} -5.15974 q^{73} +8.96319 q^{74} +0.441042 q^{76} +2.01998 q^{77} -9.29811 q^{79} +11.8245 q^{80} +9.97350 q^{82} -1.13904 q^{83} -20.4264 q^{85} +1.97695 q^{86} -15.3214 q^{88} -9.01137 q^{89} +0.343091 q^{91} +1.43657 q^{92} -5.43060 q^{94} -4.24697 q^{95} +4.32267 q^{97} -10.3795 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 4 q^{4} + q^{5} - 2 q^{7} + 12 q^{8} - 2 q^{10} + 11 q^{11} + 5 q^{13} - 5 q^{14} + 10 q^{16} + 7 q^{17} + 3 q^{19} - q^{20} - 7 q^{22} + 18 q^{23} - 8 q^{25} + 2 q^{26} - 19 q^{28} + 4 q^{29}+ \cdots - 39 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 4 * q^4 + q^5 - 2 * q^7 + 12 * q^8 - 2 * q^10 + 11 * q^11 + 5 * q^13 - 5 * q^14 + 10 * q^16 + 7 * q^17 + 3 * q^19 - q^20 - 7 * q^22 + 18 * q^23 - 8 * q^25 + 2 * q^26 - 19 * q^28 + 4 * q^29 - 16 * q^31 + 31 * q^32 + 13 * q^34 + 11 * q^35 + 5 * q^37 + 24 * q^38 + 18 * q^40 + 12 * q^41 + q^44 + 6 * q^46 + 15 * q^47 + 3 * q^49 - 5 * q^50 + 4 * q^52 - 3 * q^53 - 17 * q^55 - 24 * q^56 - 2 * q^58 + 3 * q^59 - q^61 - 4 * q^62 + 62 * q^64 + q^65 - 4 * q^67 - 16 * q^68 - 25 * q^70 + 15 * q^71 - 5 * q^73 + 11 * q^74 + 9 * q^76 + q^77 + 13 * q^79 + 32 * q^80 - 21 * q^82 + 26 * q^83 - 7 * q^85 - 6 * q^86 + 3 * q^88 + 29 * q^89 - 2 * q^91 - 6 * q^92 + 12 * q^94 + 12 * q^95 - 16 * q^97 - 39 * q^98

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 $

$2$	1.50814	1.06642	0.533208	−	0.845984i	$-0.320986\pi$
$2$	1.50814	1.06642	0.533208	+	0.845984i	$0.320986\pi$
$3$	0	0
$3$	0	0
$4$	0.274488	0.137244
$5$	−2.64315	−1.18205	−0.591026	−	0.806652i	$-0.701277\pi$
$5$	−2.64315	−1.18205	−0.591026	+	0.806652i	$0.701277\pi$
$6$	0	0
$7$	0.343091	0.129676	0.0648382	−	0.997896i	$-0.479347\pi$
$7$	0.343091	0.129676	0.0648382	+	0.997896i	$0.479347\pi$
$8$	−2.60231	−0.920057
$9$	0	0
$10$	−3.98624	−1.26056
$11$	5.88760	1.77518	0.887589	−	0.460637i	$-0.152379\pi$
$11$	5.88760	1.77518	0.887589	+	0.460637i	$0.152379\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.517430	0.138289
$15$	0	0
$16$	−4.47363	−1.11841
$17$	7.72803	1.87432	0.937162	−	0.348895i	$-0.113443\pi$
$17$	7.72803	1.87432	0.937162	+	0.348895i	$0.113443\pi$
$18$	0	0
$19$	1.60678	0.368621	0.184311	−	0.982868i	$-0.440995\pi$
$19$	1.60678	0.368621	0.184311	+	0.982868i	$0.440995\pi$
$20$	−0.725512	−0.162229
$21$	0	0
$22$	8.87933	1.89308
$23$	5.23365	1.09129	0.545646	−	0.838016i	$-0.316284\pi$
$23$	5.23365	1.09129	0.545646	+	0.838016i	$0.316284\pi$
$24$	0	0
$25$	1.98624	0.397248
$26$	1.50814	0.295771
$27$	0	0
$28$	0.0941744	0.0177973
$29$	−0.409497	−0.0760417	−0.0380209	−	0.999277i	$-0.512105\pi$
$29$	−0.409497	−0.0760417	−0.0380209	+	0.999277i	$0.512105\pi$
$30$	0	0
$31$	−2.62837	−0.472070	−0.236035	−	0.971745i	$-0.575848\pi$
$31$	−2.62837	−0.472070	−0.236035	+	0.971745i	$0.575848\pi$
$32$	−1.54224	−0.272631
$33$	0	0
$34$	11.6550	1.99881
$35$	−0.906842	−0.153284
$36$	0	0
$37$	5.94321	0.977057	0.488529	−	0.872548i	$-0.337534\pi$
$37$	5.94321	0.977057	0.488529	+	0.872548i	$0.337534\pi$
$38$	2.42326	0.393104
$39$	0	0
$40$	6.87831	1.08756
$41$	6.61311	1.03279	0.516397	−	0.856349i	$-0.327273\pi$
$41$	6.61311	1.03279	0.516397	+	0.856349i	$0.327273\pi$
$42$	0	0
$43$	1.31085	0.199903	0.0999517	−	0.994992i	$-0.468131\pi$
$43$	1.31085	0.199903	0.0999517	+	0.994992i	$0.468131\pi$
$44$	1.61607	0.243632
$45$	0	0
$46$	7.89308	1.16377
$47$	−3.60086	−0.525239	−0.262620	−	0.964899i	$-0.584586\pi$
$47$	−3.60086	−0.525239	−0.262620	+	0.964899i	$0.584586\pi$
$48$	0	0
$49$	−6.88229	−0.983184
$50$	2.99553	0.423632
$51$	0	0
$52$	0.274488	0.0380646
$53$	−3.87831	−0.532727	−0.266363	−	0.963873i	$-0.585822\pi$
$53$	−3.87831	−0.532727	−0.266363	+	0.963873i	$0.585822\pi$
$54$	0	0
$55$	−15.5618	−2.09835
$56$	−0.892832	−0.119310
$57$	0	0
$58$	−0.617579	−0.0810921
$59$	0.452530	0.0589144	0.0294572	−	0.999566i	$-0.490622\pi$
$59$	0.452530	0.0589144	0.0294572	+	0.999566i	$0.490622\pi$
$60$	0	0
$61$	3.74029	0.478895	0.239447	−	0.970909i	$-0.423034\pi$
$61$	3.74029	0.478895	0.239447	+	0.970909i	$0.423034\pi$
$62$	−3.96396	−0.503423
$63$	0	0
$64$	6.62136	0.827669
$65$	−2.64315	−0.327842
$66$	0	0
$67$	−2.80970	−0.343260	−0.171630	−	0.985161i	$-0.554903\pi$
$67$	−2.80970	−0.343260	−0.171630	+	0.985161i	$0.554903\pi$
$68$	2.12125	0.257239
$69$	0	0
$70$	−1.36765	−0.163465
$71$	16.5889	1.96874	0.984369	−	0.176119i	$-0.0563542\pi$
$71$	16.5889	1.96874	0.984369	+	0.176119i	$0.0563542\pi$
$72$	0	0
$73$	−5.15974	−0.603902	−0.301951	−	0.953323i	$-0.597638\pi$
$73$	−5.15974	−0.603902	−0.301951	+	0.953323i	$0.597638\pi$
$74$	8.96319	1.04195
$75$	0	0
$76$	0.441042	0.0505910
$77$	2.01998	0.230199
$78$	0	0
$79$	−9.29811	−1.04612	−0.523060	−	0.852296i	$-0.675210\pi$
$79$	−9.29811	−1.04612	−0.523060	+	0.852296i	$0.675210\pi$
$80$	11.8245	1.32202
$81$	0	0
$82$	9.97350	1.10139
$83$	−1.13904	−0.125026	−0.0625128	−	0.998044i	$-0.519911\pi$
$83$	−1.13904	−0.125026	−0.0625128	+	0.998044i	$0.519911\pi$
$84$	0	0
$85$	−20.4264	−2.21555
$86$	1.97695	0.213180
$87$	0	0
$88$	−15.3214	−1.63327
$89$	−9.01137	−0.955203	−0.477602	−	0.878576i	$-0.658494\pi$
$89$	−9.01137	−0.955203	−0.477602	+	0.878576i	$0.658494\pi$
$90$	0	0
$91$	0.343091	0.0359658
$92$	1.43657	0.149773
$93$	0	0
$94$	−5.43060	−0.560124
$95$	−4.24697	−0.435730
$96$	0	0
$97$	4.32267	0.438900	0.219450	−	0.975624i	$-0.429574\pi$
$97$	4.32267	0.438900	0.219450	+	0.975624i	$0.429574\pi$
$98$	−10.3795	−1.04848
$99$	0	0
$100$	0.545199	0.0545199
$101$	6.63159	0.659868	0.329934	−	0.944004i	$-0.392974\pi$
$101$	6.63159	0.659868	0.329934	+	0.944004i	$0.392974\pi$
$102$	0	0
$103$	−14.2302	−1.40215	−0.701074	−	0.713088i	$-0.747296\pi$
$103$	−14.2302	−1.40215	−0.701074	+	0.713088i	$0.747296\pi$
$104$	−2.60231	−0.255178
$105$	0	0
$106$	−5.84903	−0.568108
$107$	−0.957659	−0.0925804	−0.0462902	−	0.998928i	$-0.514740\pi$
$107$	−0.957659	−0.0925804	−0.0462902	+	0.998928i	$0.514740\pi$
$108$	0	0
$109$	5.77436	0.553083	0.276541	−	0.961002i	$-0.410812\pi$
$109$	5.77436	0.553083	0.276541	+	0.961002i	$0.410812\pi$
$110$	−23.4694	−2.23772
$111$	0	0
$112$	−1.53487	−0.145031
$113$	11.8771	1.11731	0.558653	−	0.829401i	$-0.311318\pi$
$113$	11.8771	1.11731	0.558653	+	0.829401i	$0.311318\pi$
$114$	0	0
$115$	−13.8333	−1.28996
$116$	−0.112402	−0.0104363
$117$	0	0
$118$	0.682479	0.0628273
$119$	2.65142	0.243056
$120$	0	0
$121$	23.6638	2.15126
$122$	5.64088	0.510701
$123$	0	0
$124$	−0.721457	−0.0647887
$125$	7.96582	0.712484
$126$	0	0
$127$	−11.9783	−1.06290	−0.531451	−	0.847089i	$-0.678353\pi$
$127$	−11.9783	−1.06290	−0.531451	+	0.847089i	$0.678353\pi$
$128$	13.0704	1.15527
$129$	0	0
$130$	−3.98624	−0.349616
$131$	−12.0381	−1.05178	−0.525888	−	0.850554i	$-0.676267\pi$
$131$	−12.0381	−1.05178	−0.525888	+	0.850554i	$0.676267\pi$
$132$	0	0
$133$	0.551274	0.0478015
$134$	−4.23743	−0.366058
$135$	0	0
$136$	−20.1108	−1.72449
$137$	−0.0307320	−0.00262561	−0.00131280	−	0.999999i	$-0.500418\pi$
$137$	−0.0307320	−0.00262561	−0.00131280	+	0.999999i	$0.500418\pi$
$138$	0	0
$139$	11.2989	0.958356	0.479178	−	0.877718i	$-0.340935\pi$
$139$	11.2989	0.958356	0.479178	+	0.877718i	$0.340935\pi$
$140$	−0.248917	−0.0210373
$141$	0	0
$142$	25.0184	2.09949
$143$	5.88760	0.492346
$144$	0	0
$145$	1.08236	0.0898853
$146$	−7.78161	−0.644011
$147$	0	0
$148$	1.63134	0.134095
$149$	−1.51879	−0.124424	−0.0622120	−	0.998063i	$-0.519815\pi$
$149$	−1.51879	−0.124424	−0.0622120	+	0.998063i	$0.519815\pi$
$150$	0	0
$151$	−9.15198	−0.744778	−0.372389	−	0.928077i	$-0.621461\pi$
$151$	−9.15198	−0.744778	−0.372389	+	0.928077i	$0.621461\pi$
$152$	−4.18136	−0.339153
$153$	0	0
$154$	3.04642	0.245488
$155$	6.94719	0.558012
$156$	0	0
$157$	2.02031	0.161238	0.0806191	−	0.996745i	$-0.474310\pi$
$157$	2.02031	0.161238	0.0806191	+	0.996745i	$0.474310\pi$
$158$	−14.0229	−1.11560
$159$	0	0
$160$	4.07636	0.322265
$161$	1.79562	0.141515
$162$	0	0
$163$	−13.4360	−1.05239	−0.526194	−	0.850365i	$-0.676381\pi$
$163$	−13.4360	−1.05239	−0.526194	+	0.850365i	$0.676381\pi$
$164$	1.81522	0.141745
$165$	0	0
$166$	−1.71783	−0.133329
$167$	11.8367	0.915954	0.457977	−	0.888964i	$-0.348574\pi$
$167$	11.8367	0.915954	0.457977	+	0.888964i	$0.348574\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−30.8058	−2.36270
$171$	0	0
$172$	0.359813	0.0274355
$173$	−18.5170	−1.40782	−0.703910	−	0.710289i	$-0.748564\pi$
$173$	−18.5170	−1.40782	−0.703910	+	0.710289i	$0.748564\pi$
$174$	0	0
$175$	0.681463	0.0515137
$176$	−26.3389	−1.98537
$177$	0	0
$178$	−13.5904	−1.01864
$179$	18.6972	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$179$	18.6972	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$180$	0	0
$181$	5.93713	0.441304	0.220652	−	0.975353i	$-0.429182\pi$
$181$	5.93713	0.441304	0.220652	+	0.975353i	$0.429182\pi$
$182$	0.517430	0.0383545
$183$	0	0
$184$	−13.6196	−1.00405
$185$	−15.7088	−1.15493
$186$	0	0
$187$	45.4996	3.32726
$188$	−0.988391	−0.0720858
$189$	0	0
$190$	−6.40503	−0.464670
$191$	−5.16424	−0.373671	−0.186835	−	0.982391i	$-0.559823\pi$
$191$	−5.16424	−0.373671	−0.186835	+	0.982391i	$0.559823\pi$
$192$	0	0
$193$	22.9193	1.64977	0.824883	−	0.565304i	$-0.191241\pi$
$193$	22.9193	1.64977	0.824883	+	0.565304i	$0.191241\pi$
$194$	6.51919	0.468050
$195$	0	0
$196$	−1.88910	−0.134936
$197$	−14.5085	−1.03369	−0.516843	−	0.856080i	$-0.672893\pi$
$197$	−14.5085	−1.03369	−0.516843	+	0.856080i	$0.672893\pi$
$198$	0	0
$199$	−9.40273	−0.666542	−0.333271	−	0.942831i	$-0.608152\pi$
$199$	−9.40273	−0.666542	−0.333271	+	0.942831i	$0.608152\pi$
$200$	−5.16883	−0.365491
$201$	0	0
$202$	10.0014	0.703694
$203$	−0.140495	−0.00986082
$204$	0	0
$205$	−17.4794	−1.22082
$206$	−21.4612	−1.49527
$207$	0	0
$208$	−4.47363	−0.310191
$209$	9.46010	0.654369
$210$	0	0
$211$	15.5659	1.07160	0.535801	−	0.844344i	$-0.320010\pi$
$211$	15.5659	1.07160	0.535801	+	0.844344i	$0.320010\pi$
$212$	−1.06455	−0.0731135
$213$	0	0
$214$	−1.44428	−0.0987293
$215$	−3.46478	−0.236296
$216$	0	0
$217$	−0.901773	−0.0612163
$218$	8.70854	0.589817
$219$	0	0
$220$	−4.27152	−0.287986
$221$	7.72803	0.519844
$222$	0	0
$223$	−3.84296	−0.257344	−0.128672	−	0.991687i	$-0.541071\pi$
$223$	−3.84296	−0.257344	−0.128672	+	0.991687i	$0.541071\pi$
$224$	−0.529128	−0.0353538
$225$	0	0
$226$	17.9124	1.19151
$227$	24.0584	1.59681	0.798406	−	0.602120i	$-0.205677\pi$
$227$	24.0584	1.59681	0.798406	+	0.602120i	$0.205677\pi$
$228$	0	0
$229$	−20.1459	−1.33128	−0.665641	−	0.746272i	$-0.731842\pi$
$229$	−20.1459	−1.33128	−0.665641	+	0.746272i	$0.731842\pi$
$230$	−20.8626	−1.37564
$231$	0	0
$232$	1.06564	0.0699627
$233$	5.86085	0.383957	0.191978	−	0.981399i	$-0.438510\pi$
$233$	5.86085	0.383957	0.191978	+	0.981399i	$0.438510\pi$
$234$	0	0
$235$	9.51761	0.620860
$236$	0.124214	0.00808564
$237$	0	0
$238$	3.99872	0.259198
$239$	9.16945	0.593122	0.296561	−	0.955014i	$-0.404160\pi$
$239$	9.16945	0.593122	0.296561	+	0.955014i	$0.404160\pi$
$240$	0	0
$241$	−24.3733	−1.57002	−0.785010	−	0.619483i	$-0.787342\pi$
$241$	−24.3733	−1.57002	−0.785010	+	0.619483i	$0.787342\pi$
$242$	35.6884	2.29413
$243$	0	0
$244$	1.02666	0.0657254
$245$	18.1909	1.16218
$246$	0	0
$247$	1.60678	0.102237
$248$	6.83986	0.434331
$249$	0	0
$250$	12.0136	0.759805
$251$	−11.2925	−0.712775	−0.356387	−	0.934338i	$-0.615992\pi$
$251$	−11.2925	−0.712775	−0.356387	+	0.934338i	$0.615992\pi$
$252$	0	0
$253$	30.8136	1.93724
$254$	−18.0649	−1.13350
$255$	0	0
$256$	6.46930	0.404331
$257$	−24.6348	−1.53667	−0.768337	−	0.640045i	$-0.778916\pi$
$257$	−24.6348	−1.53667	−0.768337	+	0.640045i	$0.778916\pi$
$258$	0	0
$259$	2.03906	0.126701
$260$	−0.725512	−0.0449944
$261$	0	0
$262$	−18.1552	−1.12163
$263$	−22.3752	−1.37971	−0.689856	−	0.723947i	$-0.742326\pi$
$263$	−22.3752	−1.37971	−0.689856	+	0.723947i	$0.742326\pi$
$264$	0	0
$265$	10.2510	0.629711
$266$	0.831398	0.0509763
$267$	0	0
$268$	−0.771229	−0.0471103
$269$	−24.7265	−1.50760	−0.753799	−	0.657105i	$-0.771781\pi$
$269$	−24.7265	−1.50760	−0.753799	+	0.657105i	$0.771781\pi$
$270$	0	0
$271$	10.1047	0.613817	0.306909	−	0.951739i	$-0.400705\pi$
$271$	10.1047	0.613817	0.306909	+	0.951739i	$0.400705\pi$
$272$	−34.5724	−2.09626
$273$	0	0
$274$	−0.0463481	−0.00279999
$275$	11.6942	0.705186
$276$	0	0
$277$	10.5940	0.636534	0.318267	−	0.948001i	$-0.396899\pi$
$277$	10.5940	0.636534	0.318267	+	0.948001i	$0.396899\pi$
$278$	17.0403	1.02201
$279$	0	0
$280$	2.35989	0.141030
$281$	−21.2107	−1.26532	−0.632662	−	0.774428i	$-0.718038\pi$
$281$	−21.2107	−1.26532	−0.632662	+	0.774428i	$0.718038\pi$
$282$	0	0
$283$	0.210870	0.0125349	0.00626745	−	0.999980i	$-0.498005\pi$
$283$	0.210870	0.0125349	0.00626745	+	0.999980i	$0.498005\pi$
$284$	4.55344	0.270197
$285$	0	0
$286$	8.87933	0.525045
$287$	2.26890	0.133929
$288$	0	0
$289$	42.7225	2.51309
$290$	1.63235	0.0958552
$291$	0	0
$292$	−1.41629	−0.0828818
$293$	26.4926	1.54772	0.773858	−	0.633359i	$-0.218324\pi$
$293$	26.4926	1.54772	0.773858	+	0.633359i	$0.218324\pi$
$294$	0	0
$295$	−1.19611	−0.0696399
$296$	−15.4661	−0.898949
$297$	0	0
$298$	−2.29054	−0.132688
$299$	5.23365	0.302670
$300$	0	0
$301$	0.449743	0.0259227
$302$	−13.8025	−0.794243
$303$	0	0
$304$	−7.18816	−0.412269
$305$	−9.88614	−0.566079
$306$	0	0
$307$	−6.00218	−0.342563	−0.171281	−	0.985222i	$-0.554791\pi$
$307$	−6.00218	−0.342563	−0.171281	+	0.985222i	$0.554791\pi$
$308$	0.554461	0.0315934
$309$	0	0
$310$	10.4773	0.595073
$311$	−29.0153	−1.64530	−0.822652	−	0.568545i	$-0.807506\pi$
$311$	−29.0153	−1.64530	−0.822652	+	0.568545i	$0.807506\pi$
$312$	0	0
$313$	−6.81984	−0.385480	−0.192740	−	0.981250i	$-0.561737\pi$
$313$	−6.81984	−0.385480	−0.192740	+	0.981250i	$0.561737\pi$
$314$	3.04691	0.171947
$315$	0	0
$316$	−2.55222	−0.143573
$317$	17.2718	0.970081	0.485040	−	0.874492i	$-0.338805\pi$
$317$	17.2718	0.970081	0.485040	+	0.874492i	$0.338805\pi$
$318$	0	0
$319$	−2.41095	−0.134988
$320$	−17.5012	−0.978349
$321$	0	0
$322$	2.70805	0.150914
$323$	12.4173	0.690916
$324$	0	0
$325$	1.98624	0.110177
$326$	−20.2634	−1.12228
$327$	0	0
$328$	−17.2094	−0.950230
$329$	−1.23542	−0.0681111
$330$	0	0
$331$	30.2097	1.66047	0.830237	−	0.557411i	$-0.188205\pi$
$331$	30.2097	1.66047	0.830237	+	0.557411i	$0.188205\pi$
$332$	−0.312652	−0.0171590
$333$	0	0
$334$	17.8515	0.976788
$335$	7.42647	0.405751
$336$	0	0
$337$	9.40866	0.512522	0.256261	−	0.966608i	$-0.417509\pi$
$337$	9.40866	0.512522	0.256261	+	0.966608i	$0.417509\pi$
$338$	1.50814	0.0820320
$339$	0	0
$340$	−5.60678	−0.304071
$341$	−15.4748	−0.838008
$342$	0	0
$343$	−4.76289	−0.257172
$344$	−3.41125	−0.183923
$345$	0	0
$346$	−27.9262	−1.50132
$347$	11.6833	0.627193	0.313597	−	0.949556i	$-0.398466\pi$
$347$	11.6833	0.627193	0.313597	+	0.949556i	$0.398466\pi$
$348$	0	0
$349$	−4.74967	−0.254244	−0.127122	−	0.991887i	$-0.540574\pi$
$349$	−4.74967	−0.254244	−0.127122	+	0.991887i	$0.540574\pi$
$350$	1.02774	0.0549351
$351$	0	0
$352$	−9.08006	−0.483969
$353$	0.273647	0.0145648	0.00728238	−	0.999973i	$-0.497682\pi$
$353$	0.273647	0.0145648	0.00728238	+	0.999973i	$0.497682\pi$
$354$	0	0
$355$	−43.8469	−2.32715
$356$	−2.47351	−0.131096
$357$	0	0
$358$	28.1979	1.49031
$359$	−11.3893	−0.601105	−0.300553	−	0.953765i	$-0.597171\pi$
$359$	−11.3893	−0.601105	−0.300553	+	0.953765i	$0.597171\pi$
$360$	0	0
$361$	−16.4182	−0.864118
$362$	8.95403	0.470613
$363$	0	0
$364$	0.0941744	0.00493608
$365$	13.6380	0.713844
$366$	0	0
$367$	−7.60908	−0.397191	−0.198595	−	0.980082i	$-0.563638\pi$
$367$	−7.60908	−0.397191	−0.198595	+	0.980082i	$0.563638\pi$
$368$	−23.4134	−1.22051
$369$	0	0
$370$	−23.6911	−1.23164
$371$	−1.33061	−0.0690821
$372$	0	0
$373$	−28.1189	−1.45594	−0.727970	−	0.685609i	$-0.759536\pi$
$373$	−28.1189	−1.45594	−0.727970	+	0.685609i	$0.759536\pi$
$374$	68.6197	3.54824
$375$	0	0
$376$	9.37056	0.483250
$377$	−0.409497	−0.0210902
$378$	0	0
$379$	−16.8213	−0.864052	−0.432026	−	0.901861i	$-0.642201\pi$
$379$	−16.8213	−0.864052	−0.432026	+	0.901861i	$0.642201\pi$
$380$	−1.16574	−0.0598013
$381$	0	0
$382$	−7.78839	−0.398489
$383$	4.84715	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$383$	4.84715	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$384$	0	0
$385$	−5.33912	−0.272107
$386$	34.5655	1.75934
$387$	0	0
$388$	1.18652	0.0602364
$389$	−36.3669	−1.84387	−0.921937	−	0.387340i	$-0.873394\pi$
$389$	−36.3669	−1.84387	−0.921937	+	0.387340i	$0.873394\pi$
$390$	0	0
$391$	40.4458	2.04543
$392$	17.9099	0.904586
$393$	0	0
$394$	−21.8808	−1.10234
$395$	24.5763	1.23657
$396$	0	0
$397$	−29.0866	−1.45981	−0.729907	−	0.683547i	$-0.760437\pi$
$397$	−29.0866	−1.45981	−0.729907	+	0.683547i	$0.760437\pi$
$398$	−14.1806	−0.710811
$399$	0	0
$400$	−8.88571	−0.444286
$401$	4.38902	0.219177	0.109589	−	0.993977i	$-0.465047\pi$
$401$	4.38902	0.219177	0.109589	+	0.993977i	$0.465047\pi$
$402$	0	0
$403$	−2.62837	−0.130929
$404$	1.82029	0.0905628
$405$	0	0
$406$	−0.211886	−0.0105157
$407$	34.9912	1.73445
$408$	0	0
$409$	0.557473	0.0275652	0.0137826	−	0.999905i	$-0.495613\pi$
$409$	0.557473	0.0275652	0.0137826	+	0.999905i	$0.495613\pi$
$410$	−26.3615	−1.30190
$411$	0	0
$412$	−3.90603	−0.192436
$413$	0.155259	0.00763981
$414$	0	0
$415$	3.01065	0.147787
$416$	−1.54224	−0.0756143
$417$	0	0
$418$	14.2672	0.697829
$419$	−7.98303	−0.389996	−0.194998	−	0.980804i	$-0.562470\pi$
$419$	−7.98303	−0.389996	−0.194998	+	0.980804i	$0.562470\pi$
$420$	0	0
$421$	−13.0285	−0.634968	−0.317484	−	0.948264i	$-0.602838\pi$
$421$	−13.0285	−0.634968	−0.317484	+	0.948264i	$0.602838\pi$
$422$	23.4756	1.14277
$423$	0	0
$424$	10.0926	0.490139
$425$	15.3497	0.744572
$426$	0	0
$427$	1.28326	0.0621013
$428$	−0.262866	−0.0127061
$429$	0	0
$430$	−5.22538	−0.251990
$431$	13.8846	0.668796	0.334398	−	0.942432i	$-0.391467\pi$
$431$	13.8846	0.668796	0.334398	+	0.942432i	$0.391467\pi$
$432$	0	0
$433$	−8.20792	−0.394447	−0.197224	−	0.980359i	$-0.563193\pi$
$433$	−8.20792	−0.394447	−0.197224	+	0.980359i	$0.563193\pi$
$434$	−1.36000	−0.0652821
$435$	0	0
$436$	1.58499	0.0759072
$437$	8.40935	0.402274
$438$	0	0
$439$	−10.6950	−0.510442	−0.255221	−	0.966883i	$-0.582148\pi$
$439$	−10.6950	−0.510442	−0.255221	+	0.966883i	$0.582148\pi$
$440$	40.4967	1.93061
$441$	0	0
$442$	11.6550	0.554370
$443$	19.0992	0.907430	0.453715	−	0.891147i	$-0.350098\pi$
$443$	19.0992	0.907430	0.453715	+	0.891147i	$0.350098\pi$
$444$	0	0
$445$	23.8184	1.12910
$446$	−5.79572	−0.274435
$447$	0	0
$448$	2.27173	0.107329
$449$	−28.5873	−1.34912	−0.674558	−	0.738221i	$-0.735666\pi$
$449$	−28.5873	−1.34912	−0.674558	+	0.738221i	$0.735666\pi$
$450$	0	0
$451$	38.9353	1.83339
$452$	3.26013	0.153343
$453$	0	0
$454$	36.2834	1.70287
$455$	−0.906842	−0.0425134
$456$	0	0
$457$	7.09382	0.331835	0.165918	−	0.986140i	$-0.446941\pi$
$457$	7.09382	0.331835	0.165918	+	0.986140i	$0.446941\pi$
$458$	−30.3829	−1.41970
$459$	0	0
$460$	−3.79708	−0.177040
$461$	−20.8133	−0.969371	−0.484685	−	0.874689i	$-0.661066\pi$
$461$	−20.8133	−0.969371	−0.484685	+	0.874689i	$0.661066\pi$
$462$	0	0
$463$	−19.5743	−0.909694	−0.454847	−	0.890570i	$-0.650306\pi$
$463$	−19.5743	−0.909694	−0.454847	+	0.890570i	$0.650306\pi$
$464$	1.83194	0.0850457
$465$	0	0
$466$	8.83898	0.409458
$467$	32.2324	1.49154	0.745768	−	0.666206i	$-0.232083\pi$
$467$	32.2324	1.49154	0.745768	+	0.666206i	$0.232083\pi$
$468$	0	0
$469$	−0.963986	−0.0445127
$470$	14.3539	0.662095
$471$	0	0
$472$	−1.17763	−0.0542046
$473$	7.71778	0.354864
$474$	0	0
$475$	3.19146	0.146434
$476$	0.727783	0.0333579
$477$	0	0
$478$	13.8288	0.632515
$479$	−1.05077	−0.0480111	−0.0240055	−	0.999712i	$-0.507642\pi$
$479$	−1.05077	−0.0480111	−0.0240055	+	0.999712i	$0.507642\pi$
$480$	0	0
$481$	5.94321	0.270987
$482$	−36.7583	−1.67430
$483$	0	0
$484$	6.49543	0.295247
$485$	−11.4255	−0.518803
$486$	0	0
$487$	18.2019	0.824808	0.412404	−	0.911001i	$-0.364689\pi$
$487$	18.2019	0.824808	0.412404	+	0.911001i	$0.364689\pi$
$488$	−9.73341	−0.440611
$489$	0	0
$490$	27.4345	1.23936
$491$	1.37475	0.0620418	0.0310209	−	0.999519i	$-0.490124\pi$
$491$	1.37475	0.0620418	0.0310209	+	0.999519i	$0.490124\pi$
$492$	0	0
$493$	−3.16461	−0.142527
$494$	2.42326	0.109027
$495$	0	0
$496$	11.7584	0.527967
$497$	5.69150	0.255299
$498$	0	0
$499$	23.6934	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$499$	23.6934	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$500$	2.18652	0.0977841
$501$	0	0
$502$	−17.0306	−0.760115
$503$	−8.31698	−0.370836	−0.185418	−	0.982660i	$-0.559364\pi$
$503$	−8.31698	−0.370836	−0.185418	+	0.982660i	$0.559364\pi$
$504$	0	0
$505$	−17.5283	−0.779998
$506$	46.4713	2.06590
$507$	0	0
$508$	−3.28789	−0.145877
$509$	9.67977	0.429048	0.214524	−	0.976719i	$-0.431180\pi$
$509$	9.67977	0.429048	0.214524	+	0.976719i	$0.431180\pi$
$510$	0	0
$511$	−1.77026	−0.0783118
$512$	−16.3842	−0.724086
$513$	0	0
$514$	−37.1527	−1.63873
$515$	37.6127	1.65741
$516$	0	0
$517$	−21.2004	−0.932393
$518$	3.07520	0.135116
$519$	0	0
$520$	6.87831	0.301634
$521$	−13.7581	−0.602753	−0.301377	−	0.953505i	$-0.597446\pi$
$521$	−13.7581	−0.602753	−0.301377	+	0.953505i	$0.597446\pi$
$522$	0	0
$523$	−11.2006	−0.489769	−0.244885	−	0.969552i	$-0.578750\pi$
$523$	−11.2006	−0.489769	−0.244885	+	0.969552i	$0.578750\pi$
$524$	−3.30432	−0.144350
$525$	0	0
$526$	−33.7449	−1.47135
$527$	−20.3122	−0.884812
$528$	0	0
$529$	4.39112	0.190918
$530$	15.4599	0.671534
$531$	0	0
$532$	0.151318	0.00656046
$533$	6.61311	0.286446
$534$	0	0
$535$	2.53124	0.109435
$536$	7.31174	0.315819
$537$	0	0
$538$	−37.2910	−1.60773
$539$	−40.5201	−1.74533
$540$	0	0
$541$	−24.0677	−1.03475	−0.517375	−	0.855759i	$-0.673091\pi$
$541$	−24.0677	−1.03475	−0.517375	+	0.855759i	$0.673091\pi$
$542$	15.2393	0.654585
$543$	0	0
$544$	−11.9185	−0.510999
$545$	−15.2625	−0.653773
$546$	0	0
$547$	21.6378	0.925164	0.462582	−	0.886576i	$-0.346923\pi$
$547$	21.6378	0.925164	0.462582	+	0.886576i	$0.346923\pi$
$548$	−0.00843555	−0.000360349 0
$549$	0	0
$550$	17.6365	0.752022
$551$	−0.657973	−0.0280306
$552$	0	0
$553$	−3.19010	−0.135657
$554$	15.9773	0.678811
$555$	0	0
$556$	3.10140	0.131529
$557$	23.1251	0.979840	0.489920	−	0.871767i	$-0.337026\pi$
$557$	23.1251	0.979840	0.489920	+	0.871767i	$0.337026\pi$
$558$	0	0
$559$	1.31085	0.0554432
$560$	4.05688	0.171434
$561$	0	0
$562$	−31.9887	−1.34936
$563$	−25.2335	−1.06346	−0.531732	−	0.846913i	$-0.678459\pi$
$563$	−25.2335	−1.06346	−0.531732	+	0.846913i	$0.678459\pi$
$564$	0	0
$565$	−31.3930	−1.32071
$566$	0.318021	0.0133674
$567$	0	0
$568$	−43.1695	−1.81135
$569$	33.4048	1.40040	0.700200	−	0.713946i	$-0.253094\pi$
$569$	33.4048	1.40040	0.700200	+	0.713946i	$0.253094\pi$
$570$	0	0
$571$	38.4889	1.61071	0.805355	−	0.592792i	$-0.201974\pi$
$571$	38.4889	1.61071	0.805355	+	0.592792i	$0.201974\pi$
$572$	1.61607	0.0675714
$573$	0	0
$574$	3.42182	0.142824
$575$	10.3953	0.433514
$576$	0	0
$577$	−40.5935	−1.68993	−0.844965	−	0.534821i	$-0.820379\pi$
$577$	−40.5935	−1.68993	−0.844965	+	0.534821i	$0.820379\pi$
$578$	64.4316	2.68000
$579$	0	0
$580$	0.297095	0.0123362
$581$	−0.390794	−0.0162129
$582$	0	0
$583$	−22.8339	−0.945684
$584$	13.4273	0.555624
$585$	0	0
$586$	39.9546	1.65051
$587$	−9.72421	−0.401361	−0.200681	−	0.979657i	$-0.564315\pi$
$587$	−9.72421	−0.401361	−0.200681	+	0.979657i	$0.564315\pi$
$588$	0	0
$589$	−4.22323	−0.174015
$590$	−1.80389	−0.0742652
$591$	0	0
$592$	−26.5877	−1.09275
$593$	12.0667	0.495518	0.247759	−	0.968822i	$-0.420306\pi$
$593$	12.0667	0.495518	0.247759	+	0.968822i	$0.420306\pi$
$594$	0	0
$595$	−7.00811	−0.287304
$596$	−0.416889	−0.0170764
$597$	0	0
$598$	7.89308	0.322772
$599$	−4.67924	−0.191189	−0.0955943	−	0.995420i	$-0.530475\pi$
$599$	−4.67924	−0.191189	−0.0955943	+	0.995420i	$0.530475\pi$
$600$	0	0
$601$	19.4861	0.794855	0.397427	−	0.917634i	$-0.369903\pi$
$601$	19.4861	0.794855	0.397427	+	0.917634i	$0.369903\pi$
$602$	0.678275	0.0276444
$603$	0	0
$604$	−2.51211	−0.102216
$605$	−62.5470	−2.54290
$606$	0	0
$607$	−11.6192	−0.471610	−0.235805	−	0.971800i	$-0.575773\pi$
$607$	−11.6192	−0.471610	−0.235805	+	0.971800i	$0.575773\pi$
$608$	−2.47804	−0.100498
$609$	0	0
$610$	−14.9097	−0.603676
$611$	−3.60086	−0.145675
$612$	0	0
$613$	24.0957	0.973218	0.486609	−	0.873620i	$-0.338234\pi$
$613$	24.0957	0.973218	0.486609	+	0.873620i	$0.338234\pi$
$614$	−9.05213	−0.365314
$615$	0	0
$616$	−5.25664	−0.211796
$617$	6.70091	0.269768	0.134884	−	0.990861i	$-0.456934\pi$
$617$	6.70091	0.269768	0.134884	+	0.990861i	$0.456934\pi$
$618$	0	0
$619$	10.5804	0.425262	0.212631	−	0.977133i	$-0.431797\pi$
$619$	10.5804	0.425262	0.212631	+	0.977133i	$0.431797\pi$
$620$	1.90692	0.0765837
$621$	0	0
$622$	−43.7591	−1.75458
$623$	−3.09172	−0.123867
$624$	0	0
$625$	−30.9861	−1.23944
$626$	−10.2853	−0.411082
$627$	0	0
$628$	0.554550	0.0221290
$629$	45.9293	1.83132
$630$	0	0
$631$	31.1707	1.24089	0.620443	−	0.784252i	$-0.286953\pi$
$631$	31.1707	1.24089	0.620443	+	0.784252i	$0.286953\pi$
$632$	24.1966	0.962490
$633$	0	0
$634$	26.0483	1.03451
$635$	31.6604	1.25641
$636$	0	0
$637$	−6.88229	−0.272686
$638$	−3.63606	−0.143953
$639$	0	0
$640$	−34.5470	−1.36559
$641$	−15.9019	−0.628086	−0.314043	−	0.949409i	$-0.601684\pi$
$641$	−15.9019	−0.628086	−0.314043	+	0.949409i	$0.601684\pi$
$642$	0	0
$643$	−23.9100	−0.942920	−0.471460	−	0.881887i	$-0.656273\pi$
$643$	−23.9100	−0.942920	−0.471460	+	0.881887i	$0.656273\pi$
$644$	0.492876	0.0194220
$645$	0	0
$646$	18.7270	0.736804
$647$	41.8996	1.64725	0.823623	−	0.567138i	$-0.191950\pi$
$647$	41.8996	1.64725	0.823623	+	0.567138i	$0.191950\pi$
$648$	0	0
$649$	2.66432	0.104584
$650$	2.99553	0.117494
$651$	0	0
$652$	−3.68801	−0.144434
$653$	−3.18545	−0.124656	−0.0623282	−	0.998056i	$-0.519853\pi$
$653$	−3.18545	−0.124656	−0.0623282	+	0.998056i	$0.519853\pi$
$654$	0	0
$655$	31.8186	1.24325
$656$	−29.5846	−1.15509
$657$	0	0
$658$	−1.86319	−0.0726348
$659$	−14.4257	−0.561947	−0.280973	−	0.959716i	$-0.590657\pi$
$659$	−14.4257	−0.561947	−0.280973	+	0.959716i	$0.590657\pi$
$660$	0	0
$661$	12.2706	0.477272	0.238636	−	0.971109i	$-0.423300\pi$
$661$	12.2706	0.477272	0.238636	+	0.971109i	$0.423300\pi$
$662$	45.5604	1.77076
$663$	0	0
$664$	2.96413	0.115031
$665$	−1.45710	−0.0565039
$666$	0	0
$667$	−2.14317	−0.0829837
$668$	3.24904	0.125709
$669$	0	0
$670$	11.2002	0.432700
$671$	22.0213	0.850123
$672$	0	0
$673$	43.0030	1.65764	0.828821	−	0.559514i	$-0.189012\pi$
$673$	43.0030	1.65764	0.828821	+	0.559514i	$0.189012\pi$
$674$	14.1896	0.546562
$675$	0	0
$676$	0.274488	0.0105572
$677$	−39.0920	−1.50243	−0.751213	−	0.660059i	$-0.770531\pi$
$677$	−39.0920	−1.50243	−0.751213	+	0.660059i	$0.770531\pi$
$678$	0	0
$679$	1.48307	0.0569150
$680$	53.1558	2.03843
$681$	0	0
$682$	−23.3382	−0.893666
$683$	12.2513	0.468783	0.234391	−	0.972142i	$-0.424690\pi$
$683$	12.2513	0.468783	0.234391	+	0.972142i	$0.424690\pi$
$684$	0	0
$685$	0.0812292	0.00310361
$686$	−7.18311	−0.274253
$687$	0	0
$688$	−5.86428	−0.223573
$689$	−3.87831	−0.147752
$690$	0	0
$691$	23.1225	0.879620	0.439810	−	0.898091i	$-0.355046\pi$
$691$	23.1225	0.879620	0.439810	+	0.898091i	$0.355046\pi$
$692$	−5.08269	−0.193215
$693$	0	0
$694$	17.6201	0.668849
$695$	−29.8646	−1.13283
$696$	0	0
$697$	51.1063	1.93579
$698$	−7.16318	−0.271130
$699$	0	0
$700$	0.187053	0.00706994
$701$	−16.6318	−0.628173	−0.314086	−	0.949394i	$-0.601698\pi$
$701$	−16.6318	−0.628173	−0.314086	+	0.949394i	$0.601698\pi$
$702$	0	0
$703$	9.54945	0.360164
$704$	38.9839	1.46926
$705$	0	0
$706$	0.412698	0.0155321
$707$	2.27524	0.0855693
$708$	0	0
$709$	−26.5936	−0.998743	−0.499371	−	0.866388i	$-0.666436\pi$
$709$	−26.5936	−0.998743	−0.499371	+	0.866388i	$0.666436\pi$
$710$	−66.1273	−2.48171
$711$	0	0
$712$	23.4504	0.878842
$713$	−13.7560	−0.515166
$714$	0	0
$715$	−15.5618	−0.581978
$716$	5.13214	0.191797
$717$	0	0
$718$	−17.1767	−0.641029
$719$	27.2922	1.01783	0.508914	−	0.860817i	$-0.330047\pi$
$719$	27.2922	1.01783	0.508914	+	0.860817i	$0.330047\pi$
$720$	0	0
$721$	−4.88228	−0.181826
$722$	−24.7610	−0.921510
$723$	0	0
$724$	1.62967	0.0605662
$725$	−0.813360	−0.0302074
$726$	0	0
$727$	6.67878	0.247702	0.123851	−	0.992301i	$-0.460476\pi$
$727$	6.67878	0.247702	0.123851	+	0.992301i	$0.460476\pi$
$728$	−0.892832	−0.0330906
$729$	0	0
$730$	20.5680	0.761255
$731$	10.1303	0.374684
$732$	0	0
$733$	−18.2908	−0.675587	−0.337793	−	0.941220i	$-0.609681\pi$
$733$	−18.2908	−0.675587	−0.337793	+	0.941220i	$0.609681\pi$
$734$	−11.4756	−0.423571
$735$	0	0
$736$	−8.07153	−0.297520
$737$	−16.5424	−0.609348
$738$	0	0
$739$	1.08400	0.0398757	0.0199378	−	0.999801i	$-0.493653\pi$
$739$	1.08400	0.0398757	0.0199378	+	0.999801i	$0.493653\pi$
$740$	−4.31187	−0.158508
$741$	0	0
$742$	−2.00675	−0.0736702
$743$	12.8618	0.471852	0.235926	−	0.971771i	$-0.424188\pi$
$743$	12.8618	0.471852	0.235926	+	0.971771i	$0.424188\pi$
$744$	0	0
$745$	4.01438	0.147076
$746$	−42.4072	−1.55264
$747$	0	0
$748$	12.4891	0.456646
$749$	−0.328565	−0.0120055
$750$	0	0
$751$	−9.14971	−0.333877	−0.166939	−	0.985967i	$-0.553388\pi$
$751$	−9.14971	−0.333877	−0.166939	+	0.985967i	$0.553388\pi$
$752$	16.1089	0.587432
$753$	0	0
$754$	−0.617579	−0.0224909
$755$	24.1901	0.880367
$756$	0	0
$757$	31.3956	1.14109	0.570546	−	0.821266i	$-0.306732\pi$
$757$	31.3956	1.14109	0.570546	+	0.821266i	$0.306732\pi$
$758$	−25.3689	−0.921439
$759$	0	0
$760$	11.0520	0.400897
$761$	−42.6877	−1.54743	−0.773714	−	0.633534i	$-0.781603\pi$
$761$	−42.6877	−1.54743	−0.773714	+	0.633534i	$0.781603\pi$
$762$	0	0
$763$	1.98113	0.0717218
$764$	−1.41752	−0.0512841
$765$	0	0
$766$	7.31019	0.264128
$767$	0.452530	0.0163399
$768$	0	0
$769$	−9.00487	−0.324724	−0.162362	−	0.986731i	$-0.551911\pi$
$769$	−9.00487	−0.324724	−0.162362	+	0.986731i	$0.551911\pi$
$770$	−8.05215	−0.290179
$771$	0	0
$772$	6.29106	0.226420
$773$	18.7179	0.673235	0.336617	−	0.941641i	$-0.390717\pi$
$773$	18.7179	0.673235	0.336617	+	0.941641i	$0.390717\pi$
$774$	0	0
$775$	−5.22059	−0.187529
$776$	−11.2489	−0.403813
$777$	0	0
$778$	−54.8464	−1.96634
$779$	10.6258	0.380710
$780$	0	0
$781$	97.6687	3.49486
$782$	60.9980	2.18128
$783$	0	0
$784$	30.7888	1.09960
$785$	−5.33998	−0.190592
$786$	0	0
$787$	30.4325	1.08480	0.542401	−	0.840120i	$-0.317515\pi$
$787$	30.4325	1.08480	0.542401	+	0.840120i	$0.317515\pi$
$788$	−3.98240	−0.141867
$789$	0	0
$790$	37.0645	1.31870
$791$	4.07494	0.144888
$792$	0	0
$793$	3.74029	0.132822
$794$	−43.8666	−1.55677
$795$	0	0
$796$	−2.58093	−0.0914788
$797$	−27.7866	−0.984252	−0.492126	−	0.870524i	$-0.663780\pi$
$797$	−27.7866	−0.984252	−0.492126	+	0.870524i	$0.663780\pi$
$798$	0	0
$799$	−27.8276	−0.984468
$800$	−3.06325	−0.108302
$801$	0	0
$802$	6.61926	0.233734
$803$	−30.3785	−1.07203
$804$	0	0
$805$	−4.74610	−0.167278
$806$	−3.96396	−0.139624
$807$	0	0
$808$	−17.2575	−0.607116
$809$	−25.1521	−0.884299	−0.442150	−	0.896941i	$-0.645784\pi$
$809$	−25.1521	−0.884299	−0.442150	+	0.896941i	$0.645784\pi$
$810$	0	0
$811$	46.6061	1.63656	0.818281	−	0.574819i	$-0.194928\pi$
$811$	46.6061	1.63656	0.818281	+	0.574819i	$0.194928\pi$
$812$	−0.0385642	−0.00135334
$813$	0	0
$814$	52.7717	1.84965
$815$	35.5133	1.24398
$816$	0	0
$817$	2.10626	0.0736887
$818$	0.840747	0.0293960
$819$	0	0
$820$	−4.79789	−0.167550
$821$	10.3955	0.362807	0.181403	−	0.983409i	$-0.441936\pi$
$821$	10.3955	0.362807	0.181403	+	0.983409i	$0.441936\pi$
$822$	0	0
$823$	7.33931	0.255832	0.127916	−	0.991785i	$-0.459171\pi$
$823$	7.33931	0.255832	0.127916	+	0.991785i	$0.459171\pi$
$824$	37.0316	1.29006
$825$	0	0
$826$	0.234153	0.00814722
$827$	−11.6262	−0.404281	−0.202141	−	0.979357i	$-0.564790\pi$
$827$	−11.6262	−0.404281	−0.202141	+	0.979357i	$0.564790\pi$
$828$	0	0
$829$	5.35678	0.186049	0.0930243	−	0.995664i	$-0.470347\pi$
$829$	5.35678	0.186049	0.0930243	+	0.995664i	$0.470347\pi$
$830$	4.54048	0.157602
$831$	0	0
$832$	6.62136	0.229554
$833$	−53.1866	−1.84281
$834$	0	0
$835$	−31.2863	−1.08271
$836$	2.59668	0.0898081
$837$	0	0
$838$	−12.0395	−0.415899
$839$	48.0656	1.65941	0.829705	−	0.558202i	$-0.188509\pi$
$839$	48.0656	1.65941	0.829705	+	0.558202i	$0.188509\pi$
$840$	0	0
$841$	−28.8323	−0.994218
$842$	−19.6487	−0.677141
$843$	0	0
$844$	4.27266	0.147071
$845$	−2.64315	−0.0909271
$846$	0	0
$847$	8.11885	0.278967
$848$	17.3501	0.595806
$849$	0	0
$850$	23.1496	0.794024
$851$	31.1047	1.06625
$852$	0	0
$853$	−24.6097	−0.842620	−0.421310	−	0.906917i	$-0.638430\pi$
$853$	−24.6097	−0.842620	−0.421310	+	0.906917i	$0.638430\pi$
$854$	1.93534	0.0662259
$855$	0	0
$856$	2.49213	0.0851793
$857$	33.2803	1.13683	0.568417	−	0.822740i	$-0.307556\pi$
$857$	33.2803	1.13683	0.568417	+	0.822740i	$0.307556\pi$
$858$	0	0
$859$	13.8267	0.471761	0.235881	−	0.971782i	$-0.424203\pi$
$859$	13.8267	0.471761	0.235881	+	0.971782i	$0.424203\pi$
$860$	−0.951041	−0.0324302
$861$	0	0
$862$	20.9399	0.713215
$863$	−23.1145	−0.786827	−0.393413	−	0.919362i	$-0.628706\pi$
$863$	−23.1145	−0.786827	−0.393413	+	0.919362i	$0.628706\pi$
$864$	0	0
$865$	48.9432	1.66412
$866$	−12.3787	−0.420645
$867$	0	0
$868$	−0.247526	−0.00840157
$869$	−54.7435	−1.85705
$870$	0	0
$871$	−2.80970	−0.0952032
$872$	−15.0267	−0.508868
$873$	0	0
$874$	12.6825	0.428991
$875$	2.73300	0.0923924
$876$	0	0
$877$	46.9409	1.58508	0.792540	−	0.609819i	$-0.208758\pi$
$877$	46.9409	1.58508	0.792540	+	0.609819i	$0.208758\pi$
$878$	−16.1295	−0.544344
$879$	0	0
$880$	69.6178	2.34682
$881$	−10.1247	−0.341109	−0.170555	−	0.985348i	$-0.554556\pi$
$881$	−10.1247	−0.341109	−0.170555	+	0.985348i	$0.554556\pi$
$882$	0	0
$883$	−34.3946	−1.15747	−0.578736	−	0.815515i	$-0.696454\pi$
$883$	−34.3946	−1.15747	−0.578736	+	0.815515i	$0.696454\pi$
$884$	2.12125	0.0713454
$885$	0	0
$886$	28.8043	0.967698
$887$	−28.9212	−0.971078	−0.485539	−	0.874215i	$-0.661377\pi$
$887$	−28.9212	−0.971078	−0.485539	+	0.874215i	$0.661377\pi$
$888$	0	0
$889$	−4.10965	−0.137833
$890$	35.9215	1.20409
$891$	0	0
$892$	−1.05485	−0.0353188
$893$	−5.78580	−0.193614
$894$	0	0
$895$	−49.4194	−1.65191
$896$	4.48435	0.149811
$897$	0	0
$898$	−43.1136	−1.43872
$899$	1.07631	0.0358970
$900$	0	0
$901$	−29.9717	−0.998502
$902$	58.7200	1.95516
$903$	0	0
$904$	−30.9080	−1.02799
$905$	−15.6927	−0.521644
$906$	0	0
$907$	−40.0696	−1.33049	−0.665244	−	0.746626i	$-0.731673\pi$
$907$	−40.0696	−1.33049	−0.665244	+	0.746626i	$0.731673\pi$
$908$	6.60373	0.219153
$909$	0	0
$910$	−1.36765	−0.0453370
$911$	−56.6924	−1.87830	−0.939151	−	0.343505i	$-0.888386\pi$
$911$	−56.6924	−1.87830	−0.939151	+	0.343505i	$0.888386\pi$
$912$	0	0
$913$	−6.70620	−0.221943
$914$	10.6985	0.353874
$915$	0	0
$916$	−5.52981	−0.182710
$917$	−4.13018	−0.136390
$918$	0	0
$919$	−26.3183	−0.868162	−0.434081	−	0.900874i	$-0.642927\pi$
$919$	−26.3183	−0.868162	−0.434081	+	0.900874i	$0.642927\pi$
$920$	35.9987	1.18684
$921$	0	0
$922$	−31.3893	−1.03375
$923$	16.5889	0.546030
$924$	0	0
$925$	11.8046	0.388134
$926$	−29.5208	−0.970113
$927$	0	0
$928$	0.631541	0.0207314
$929$	−6.25536	−0.205232	−0.102616	−	0.994721i	$-0.532721\pi$
$929$	−6.25536	−0.205232	−0.102616	+	0.994721i	$0.532721\pi$
$930$	0	0
$931$	−11.0583	−0.362423
$932$	1.60873	0.0526957
$933$	0	0
$934$	48.6109	1.59060
$935$	−120.262	−3.93299
$936$	0	0
$937$	−29.4349	−0.961598	−0.480799	−	0.876831i	$-0.659653\pi$
$937$	−29.4349	−0.961598	−0.480799	+	0.876831i	$0.659653\pi$
$938$	−1.45383	−0.0474691
$939$	0	0
$940$	2.61247	0.0852093
$941$	−6.77088	−0.220724	−0.110362	−	0.993891i	$-0.535201\pi$
$941$	−6.77088	−0.220724	−0.110362	+	0.993891i	$0.535201\pi$
$942$	0	0
$943$	34.6107	1.12708
$944$	−2.02445	−0.0658904
$945$	0	0
$946$	11.6395	0.378433
$947$	−43.6091	−1.41710	−0.708552	−	0.705658i	$-0.750651\pi$
$947$	−43.6091	−1.41710	−0.708552	+	0.705658i	$0.750651\pi$
$948$	0	0
$949$	−5.15974	−0.167492
$950$	4.81317	0.156160
$951$	0	0
$952$	−6.89984	−0.223625
$953$	11.8773	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$953$	11.8773	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$954$	0	0
$955$	13.6498	0.441699
$956$	2.51690	0.0814024
$957$	0	0
$958$	−1.58471	−0.0511998
$959$	−0.0105439	−0.000340479 0
$960$	0	0
$961$	−24.0916	−0.777150
$962$	8.96319	0.288985
$963$	0	0
$964$	−6.69016	−0.215476
$965$	−60.5791	−1.95011
$966$	0	0
$967$	20.5197	0.659870	0.329935	−	0.944004i	$-0.392973\pi$
$967$	20.5197	0.659870	0.329935	+	0.944004i	$0.392973\pi$
$968$	−61.5807	−1.97928
$969$	0	0
$970$	−17.2312	−0.553260
$971$	−17.5599	−0.563526	−0.281763	−	0.959484i	$-0.590919\pi$
$971$	−17.5599	−0.563526	−0.281763	+	0.959484i	$0.590919\pi$
$972$	0	0
$973$	3.87654	0.124276
$974$	27.4510	0.879588
$975$	0	0
$976$	−16.7327	−0.535600
$977$	−14.3945	−0.460519	−0.230260	−	0.973129i	$-0.573958\pi$
$977$	−14.3945	−0.460519	−0.230260	+	0.973129i	$0.573958\pi$
$978$	0	0
$979$	−53.0553	−1.69566
$980$	4.99318	0.159501
$981$	0	0
$982$	2.07332	0.0661623
$983$	21.7891	0.694964	0.347482	−	0.937687i	$-0.387037\pi$
$983$	21.7891	0.694964	0.347482	+	0.937687i	$0.387037\pi$
$984$	0	0
$985$	38.3480	1.22187
$986$	−4.77267	−0.151993
$987$	0	0
$988$	0.441042	0.0140314
$989$	6.86055	0.218153
$990$	0	0
$991$	−57.2494	−1.81859	−0.909294	−	0.416154i	$-0.863378\pi$
$991$	−57.2494	−1.81859	−0.909294	+	0.416154i	$0.863378\pi$
$992$	4.05357	0.128701
$993$	0	0
$994$	8.58359	0.272255
$995$	24.8528	0.787888
$996$	0	0
$997$	−23.4890	−0.743903	−0.371951	−	0.928252i	$-0.621311\pi$
$997$	−23.4890	−0.743903	−0.371951	+	0.928252i	$0.621311\pi$
$998$	35.7329	1.13111
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.a.k.1.4		5
3.2	odd	2		1053.2.a.j.1.2		5
9.2	odd	6		351.2.e.b.118.4		10
9.4	even	3		117.2.e.b.79.2	yes	10
9.5	odd	6		351.2.e.b.235.4		10
9.7	even	3		117.2.e.b.40.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.e.b.40.2	✓	10	9.7	even	3
117.2.e.b.79.2	yes	10	9.4	even	3
351.2.e.b.118.4		10	9.2	odd	6
351.2.e.b.235.4		10	9.5	odd	6
1053.2.a.j.1.2		5	3.2	odd	2
1053.2.a.k.1.4		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50814 q^{2} +0.274488 q^{4} -2.64315 q^{5} +0.343091 q^{7} -2.60231 q^{8} -3.98624 q^{10} +5.88760 q^{11} +1.00000 q^{13} +0.517430 q^{14} -4.47363 q^{16} +7.72803 q^{17} +1.60678 q^{19} -0.725512 q^{20} +8.87933 q^{22} +5.23365 q^{23} +1.98624 q^{25} +1.50814 q^{26} +0.0941744 q^{28} -0.409497 q^{29} -2.62837 q^{31} -1.54224 q^{32} +11.6550 q^{34} -0.906842 q^{35} +5.94321 q^{37} +2.42326 q^{38} +6.87831 q^{40} +6.61311 q^{41} +1.31085 q^{43} +1.61607 q^{44} +7.89308 q^{46} -3.60086 q^{47} -6.88229 q^{49} +2.99553 q^{50} +0.274488 q^{52} -3.87831 q^{53} -15.5618 q^{55} -0.892832 q^{56} -0.617579 q^{58} +0.452530 q^{59} +3.74029 q^{61} -3.96396 q^{62} +6.62136 q^{64} -2.64315 q^{65} -2.80970 q^{67} +2.12125 q^{68} -1.36765 q^{70} +16.5889 q^{71} -5.15974 q^{73} +8.96319 q^{74} +0.441042 q^{76} +2.01998 q^{77} -9.29811 q^{79} +11.8245 q^{80} +9.97350 q^{82} -1.13904 q^{83} -20.4264 q^{85} +1.97695 q^{86} -15.3214 q^{88} -9.01137 q^{89} +0.343091 q^{91} +1.43657 q^{92} -5.43060 q^{94} -4.24697 q^{95} +4.32267 q^{97} -10.3795 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} + q^{5} - 2 q^{7} + 12 q^{8} - 2 q^{10} + 11 q^{11} + 5 q^{13} - 5 q^{14} + 10 q^{16} + 7 q^{17} + 3 q^{19} - q^{20} - 7 q^{22} + 18 q^{23} - 8 q^{25} + 2 q^{26} - 19 q^{28} + 4 q^{29}+ \cdots - 39 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 4 * q^4 + q^5 - 2 * q^7 + 12 * q^8 - 2 * q^10 + 11 * q^11 + 5 * q^13 - 5 * q^14 + 10 * q^16 + 7 * q^17 + 3 * q^19 - q^20 - 7 * q^22 + 18 * q^23 - 8 * q^25 + 2 * q^26 - 19 * q^28 + 4 * q^29 - 16 * q^31 + 31 * q^32 + 13 * q^34 + 11 * q^35 + 5 * q^37 + 24 * q^38 + 18 * q^40 + 12 * q^41 + q^44 + 6 * q^46 + 15 * q^47 + 3 * q^49 - 5 * q^50 + 4 * q^52 - 3 * q^53 - 17 * q^55 - 24 * q^56 - 2 * q^58 + 3 * q^59 - q^61 - 4 * q^62 + 62 * q^64 + q^65 - 4 * q^67 - 16 * q^68 - 25 * q^70 + 15 * q^71 - 5 * q^73 + 11 * q^74 + 9 * q^76 + q^77 + 13 * q^79 + 32 * q^80 - 21 * q^82 + 26 * q^83 - 7 * q^85 - 6 * q^86 + 3 * q^88 + 29 * q^89 - 2 * q^91 - 6 * q^92 + 12 * q^94 + 12 * q^95 - 16 * q^97 - 39 * q^98

Introduction

L-functions

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Database

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