LMFDB - Embedded newform 513.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, 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("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.879385 q^{2} -1.22668 q^{4} +0.879385 q^{5} -4.41147 q^{7} -2.83750 q^{8} +0.773318 q^{10} -1.04189 q^{11} +0.0418891 q^{13} -3.87939 q^{14} -0.0418891 q^{16} -4.75877 q^{17} +1.00000 q^{19} -1.07873 q^{20} -0.916222 q^{22} -6.87939 q^{23} -4.22668 q^{25} +0.0368366 q^{26} +5.41147 q^{28} -3.87939 q^{29} +7.78106 q^{31} +5.63816 q^{32} -4.18479 q^{34} -3.87939 q^{35} +1.22668 q^{37} +0.879385 q^{38} -2.49525 q^{40} +12.1557 q^{41} +2.55438 q^{43} +1.27807 q^{44} -6.04963 q^{46} -10.4338 q^{47} +12.4611 q^{49} -3.71688 q^{50} -0.0513845 q^{52} -2.83750 q^{53} -0.916222 q^{55} +12.5175 q^{56} -3.41147 q^{58} -6.16250 q^{59} -5.95811 q^{61} +6.84255 q^{62} +5.04189 q^{64} +0.0368366 q^{65} -2.59627 q^{67} +5.83750 q^{68} -3.41147 q^{70} -6.71688 q^{71} +8.19253 q^{73} +1.07873 q^{74} -1.22668 q^{76} +4.59627 q^{77} -2.40373 q^{79} -0.0368366 q^{80} +10.6895 q^{82} -12.1925 q^{83} -4.18479 q^{85} +2.24628 q^{86} +2.95636 q^{88} +8.80066 q^{89} -0.184793 q^{91} +8.43882 q^{92} -9.17530 q^{94} +0.879385 q^{95} -1.14290 q^{97} +10.9581 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 9 q^{10} - 3 q^{13} - 6 q^{14} + 3 q^{16} - 3 q^{17} + 3 q^{19} - 12 q^{20} - 9 q^{22} - 15 q^{23} - 6 q^{25} + 12 q^{26} + 6 q^{28} - 6 q^{29} + 6 q^{31}+ \cdots + 36 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 6 * q^8 + 9 * q^10 - 3 * q^13 - 6 * q^14 + 3 * q^16 - 3 * q^17 + 3 * q^19 - 12 * q^20 - 9 * q^22 - 15 * q^23 - 6 * q^25 + 12 * q^26 + 6 * q^28 - 6 * q^29 + 6 * q^31 - 9 * q^34 - 6 * q^35 - 3 * q^37 - 3 * q^38 + 9 * q^40 - 3 * q^41 - 3 * q^43 + 27 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 - 30 * q^52 - 6 * q^53 - 9 * q^55 + 15 * q^56 - 21 * q^59 - 21 * q^61 + 3 * q^62 + 12 * q^64 + 12 * q^65 + 6 * q^67 + 15 * q^68 - 12 * q^71 - 3 * q^73 + 12 * q^74 + 3 * q^76 - 21 * q^79 - 12 * q^80 + 45 * q^82 - 9 * q^83 - 9 * q^85 + 3 * q^86 - 45 * q^88 + 12 * q^89 + 3 * q^91 - 6 * q^92 + 9 * q^94 - 3 * q^95 - 3 * q^97 + 36 * 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 $

$2$	0.879385	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$2$	0.879385	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	0.879385	0.393273	0.196637	−	0.980476i	$-0.436998\pi$
$5$	0.879385	0.393273	0.196637	+	0.980476i	$0.436998\pi$
$6$	0	0
$7$	−4.41147	−1.66738	−0.833690	−	0.552232i	$-0.813776\pi$
$7$	−4.41147	−1.66738	−0.833690	+	0.552232i	$0.813776\pi$
$8$	−2.83750	−1.00321
$9$	0	0
$10$	0.773318	0.244545
$11$	−1.04189	−0.314141	−0.157071	−	0.987587i	$-0.550205\pi$
$11$	−1.04189	−0.314141	−0.157071	+	0.987587i	$0.550205\pi$
$12$	0	0
$13$	0.0418891	0.0116179	0.00580897	−	0.999983i	$-0.498151\pi$
$13$	0.0418891	0.0116179	0.00580897	+	0.999983i	$0.498151\pi$
$14$	−3.87939	−1.03681
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	−4.75877	−1.15417	−0.577086	−	0.816684i	$-0.695810\pi$
$17$	−4.75877	−1.15417	−0.577086	+	0.816684i	$0.695810\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.07873	−0.241210
$21$	0	0
$22$	−0.916222	−0.195339
$23$	−6.87939	−1.43445	−0.717225	−	0.696841i	$-0.754588\pi$
$23$	−6.87939	−1.43445	−0.717225	+	0.696841i	$0.754588\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	0.0368366	0.00722426
$27$	0	0
$28$	5.41147	1.02267
$29$	−3.87939	−0.720384	−0.360192	−	0.932878i	$-0.617289\pi$
$29$	−3.87939	−0.720384	−0.360192	+	0.932878i	$0.617289\pi$
$30$	0	0
$31$	7.78106	1.39752	0.698760	−	0.715356i	$-0.253736\pi$
$31$	7.78106	1.39752	0.698760	+	0.715356i	$0.253736\pi$
$32$	5.63816	0.996695
$33$	0	0
$34$	−4.18479	−0.717686
$35$	−3.87939	−0.655736
$36$	0	0
$37$	1.22668	0.201665	0.100833	−	0.994903i	$-0.467849\pi$
$37$	1.22668	0.201665	0.100833	+	0.994903i	$0.467849\pi$
$38$	0.879385	0.142655
$39$	0	0
$40$	−2.49525	−0.394534
$41$	12.1557	1.89840	0.949200	−	0.314672i	$-0.101895\pi$
$41$	12.1557	1.89840	0.949200	+	0.314672i	$0.101895\pi$
$42$	0	0
$43$	2.55438	0.389539	0.194769	−	0.980849i	$-0.437604\pi$
$43$	2.55438	0.389539	0.194769	+	0.980849i	$0.437604\pi$
$44$	1.27807	0.192676
$45$	0	0
$46$	−6.04963	−0.891969
$47$	−10.4338	−1.52192	−0.760960	−	0.648798i	$-0.775272\pi$
$47$	−10.4338	−1.52192	−0.760960	+	0.648798i	$0.775272\pi$
$48$	0	0
$49$	12.4611	1.78016
$50$	−3.71688	−0.525646
$51$	0	0
$52$	−0.0513845	−0.00712575
$53$	−2.83750	−0.389760	−0.194880	−	0.980827i	$-0.562432\pi$
$53$	−2.83750	−0.389760	−0.194880	+	0.980827i	$0.562432\pi$
$54$	0	0
$55$	−0.916222	−0.123543
$56$	12.5175	1.67273
$57$	0	0
$58$	−3.41147	−0.447948
$59$	−6.16250	−0.802290	−0.401145	−	0.916015i	$-0.631388\pi$
$59$	−6.16250	−0.802290	−0.401145	+	0.916015i	$0.631388\pi$
$60$	0	0
$61$	−5.95811	−0.762858	−0.381429	−	0.924398i	$-0.624568\pi$
$61$	−5.95811	−0.762858	−0.381429	+	0.924398i	$0.624568\pi$
$62$	6.84255	0.869005
$63$	0	0
$64$	5.04189	0.630236
$65$	0.0368366	0.00456902
$66$	0	0
$67$	−2.59627	−0.317184	−0.158592	−	0.987344i	$-0.550696\pi$
$67$	−2.59627	−0.317184	−0.158592	+	0.987344i	$0.550696\pi$
$68$	5.83750	0.707900
$69$	0	0
$70$	−3.41147	−0.407749
$71$	−6.71688	−0.797147	−0.398574	−	0.917136i	$-0.630495\pi$
$71$	−6.71688	−0.797147	−0.398574	+	0.917136i	$0.630495\pi$
$72$	0	0
$73$	8.19253	0.958863	0.479432	−	0.877579i	$-0.340843\pi$
$73$	8.19253	0.958863	0.479432	+	0.877579i	$0.340843\pi$
$74$	1.07873	0.125399
$75$	0	0
$76$	−1.22668	−0.140710
$77$	4.59627	0.523793
$78$	0	0
$79$	−2.40373	−0.270441	−0.135221	−	0.990816i	$-0.543174\pi$
$79$	−2.40373	−0.270441	−0.135221	+	0.990816i	$0.543174\pi$
$80$	−0.0368366	−0.00411846
$81$	0	0
$82$	10.6895	1.18046
$83$	−12.1925	−1.33830	−0.669152	−	0.743125i	$-0.733343\pi$
$83$	−12.1925	−1.33830	−0.669152	+	0.743125i	$0.733343\pi$
$84$	0	0
$85$	−4.18479	−0.453904
$86$	2.24628	0.242223
$87$	0	0
$88$	2.95636	0.315149
$89$	8.80066	0.932868	0.466434	−	0.884556i	$-0.345539\pi$
$89$	8.80066	0.932868	0.466434	+	0.884556i	$0.345539\pi$
$90$	0	0
$91$	−0.184793	−0.0193715
$92$	8.43882	0.879807
$93$	0	0
$94$	−9.17530	−0.946360
$95$	0.879385	0.0902230
$96$	0	0
$97$	−1.14290	−0.116044	−0.0580221	−	0.998315i	$-0.518479\pi$
$97$	−1.14290	−0.116044	−0.0580221	+	0.998315i	$0.518479\pi$
$98$	10.9581	1.10694
$99$	0	0
$100$	5.18479	0.518479
$101$	15.7169	1.56389	0.781944	−	0.623349i	$-0.214228\pi$
$101$	15.7169	1.56389	0.781944	+	0.623349i	$0.214228\pi$
$102$	0	0
$103$	9.18479	0.905004	0.452502	−	0.891763i	$-0.350531\pi$
$103$	9.18479	0.905004	0.452502	+	0.891763i	$0.350531\pi$
$104$	−0.118860	−0.0116552
$105$	0	0
$106$	−2.49525	−0.242360
$107$	1.47060	0.142168	0.0710841	−	0.997470i	$-0.477354\pi$
$107$	1.47060	0.142168	0.0710841	+	0.997470i	$0.477354\pi$
$108$	0	0
$109$	18.2841	1.75129	0.875647	−	0.482951i	$-0.160435\pi$
$109$	18.2841	1.75129	0.875647	+	0.482951i	$0.160435\pi$
$110$	−0.805712	−0.0768216
$111$	0	0
$112$	0.184793	0.0174613
$113$	6.68004	0.628406	0.314203	−	0.949356i	$-0.398263\pi$
$113$	6.68004	0.628406	0.314203	+	0.949356i	$0.398263\pi$
$114$	0	0
$115$	−6.04963	−0.564131
$116$	4.75877	0.441841
$117$	0	0
$118$	−5.41921	−0.498879
$119$	20.9932	1.92444
$120$	0	0
$121$	−9.91447	−0.901315
$122$	−5.23947	−0.474360
$123$	0	0
$124$	−9.54488	−0.857156
$125$	−8.11381	−0.725721
$126$	0	0
$127$	−19.0993	−1.69479	−0.847393	−	0.530967i	$-0.821829\pi$
$127$	−19.0993	−1.69479	−0.847393	+	0.530967i	$0.821829\pi$
$128$	−6.84255	−0.604802
$129$	0	0
$130$	0.0323936	0.00284111
$131$	−4.27126	−0.373182	−0.186591	−	0.982438i	$-0.559744\pi$
$131$	−4.27126	−0.373182	−0.186591	+	0.982438i	$0.559744\pi$
$132$	0	0
$133$	−4.41147	−0.382523
$134$	−2.28312	−0.197231
$135$	0	0
$136$	13.5030	1.15787
$137$	−5.47565	−0.467817	−0.233908	−	0.972259i	$-0.575152\pi$
$137$	−5.47565	−0.467817	−0.233908	+	0.972259i	$0.575152\pi$
$138$	0	0
$139$	−15.4688	−1.31205	−0.656025	−	0.754739i	$-0.727763\pi$
$139$	−15.4688	−1.31205	−0.656025	+	0.754739i	$0.727763\pi$
$140$	4.75877	0.402190
$141$	0	0
$142$	−5.90673	−0.495681
$143$	−0.0436438	−0.00364967
$144$	0	0
$145$	−3.41147	−0.283308
$146$	7.20439	0.596240
$147$	0	0
$148$	−1.50475	−0.123690
$149$	−14.9932	−1.22829	−0.614145	−	0.789193i	$-0.710499\pi$
$149$	−14.9932	−1.22829	−0.614145	+	0.789193i	$0.710499\pi$
$150$	0	0
$151$	13.7638	1.12008	0.560042	−	0.828464i	$-0.310785\pi$
$151$	13.7638	1.12008	0.560042	+	0.828464i	$0.310785\pi$
$152$	−2.83750	−0.230151
$153$	0	0
$154$	4.04189	0.325705
$155$	6.84255	0.549607
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−2.11381	−0.168166
$159$	0	0
$160$	4.95811	0.391973
$161$	30.3482	2.39178
$162$	0	0
$163$	−12.7314	−0.997203	−0.498601	−	0.866831i	$-0.666153\pi$
$163$	−12.7314	−0.997203	−0.498601	+	0.866831i	$0.666153\pi$
$164$	−14.9112	−1.16437
$165$	0	0
$166$	−10.7219	−0.832183
$167$	−2.64496	−0.204673	−0.102337	−	0.994750i	$-0.532632\pi$
$167$	−2.64496	−0.204673	−0.102337	+	0.994750i	$0.532632\pi$
$168$	0	0
$169$	−12.9982	−0.999865
$170$	−3.68004	−0.282247
$171$	0	0
$172$	−3.13341	−0.238920
$173$	−21.3550	−1.62359	−0.811797	−	0.583940i	$-0.801510\pi$
$173$	−21.3550	−1.62359	−0.811797	+	0.583940i	$0.801510\pi$
$174$	0	0
$175$	18.6459	1.40950
$176$	0.0436438	0.00328977
$177$	0	0
$178$	7.73917	0.580075
$179$	−1.04189	−0.0778744	−0.0389372	−	0.999242i	$-0.512397\pi$
$179$	−1.04189	−0.0778744	−0.0389372	+	0.999242i	$0.512397\pi$
$180$	0	0
$181$	−14.1411	−1.05110	−0.525552	−	0.850762i	$-0.676141\pi$
$181$	−14.1411	−1.05110	−0.525552	+	0.850762i	$0.676141\pi$
$182$	−0.162504	−0.0120456
$183$	0	0
$184$	19.5202	1.43905
$185$	1.07873	0.0793095
$186$	0	0
$187$	4.95811	0.362573
$188$	12.7989	0.933456
$189$	0	0
$190$	0.773318	0.0561024
$191$	−16.2344	−1.17468	−0.587341	−	0.809340i	$-0.699825\pi$
$191$	−16.2344	−1.17468	−0.587341	+	0.809340i	$0.699825\pi$
$192$	0	0
$193$	12.4270	0.894512	0.447256	−	0.894406i	$-0.352401\pi$
$193$	12.4270	0.894512	0.447256	+	0.894406i	$0.352401\pi$
$194$	−1.00505	−0.0721586
$195$	0	0
$196$	−15.2858	−1.09184
$197$	−1.52940	−0.108965	−0.0544826	−	0.998515i	$-0.517351\pi$
$197$	−1.52940	−0.108965	−0.0544826	+	0.998515i	$0.517351\pi$
$198$	0	0
$199$	10.8307	0.767767	0.383884	−	0.923381i	$-0.374586\pi$
$199$	10.8307	0.767767	0.383884	+	0.923381i	$0.374586\pi$
$200$	11.9932	0.848047
$201$	0	0
$202$	13.8212	0.972456
$203$	17.1138	1.20115
$204$	0	0
$205$	10.6895	0.746590
$206$	8.07697	0.562749
$207$	0	0
$208$	−0.00175469	−0.000121666 0
$209$	−1.04189	−0.0720690
$210$	0	0
$211$	−7.87258	−0.541971	−0.270985	−	0.962583i	$-0.587349\pi$
$211$	−7.87258	−0.541971	−0.270985	+	0.962583i	$0.587349\pi$
$212$	3.48070	0.239056
$213$	0	0
$214$	1.29322	0.0884029
$215$	2.24628	0.153195
$216$	0	0
$217$	−34.3259	−2.33020
$218$	16.0787	1.08899
$219$	0	0
$220$	1.12391	0.0757742
$221$	−0.199340	−0.0134091
$222$	0	0
$223$	18.0651	1.20973	0.604865	−	0.796328i	$-0.293227\pi$
$223$	18.0651	1.20973	0.604865	+	0.796328i	$0.293227\pi$
$224$	−24.8726	−1.66187
$225$	0	0
$226$	5.87433	0.390755
$227$	−18.6800	−1.23984	−0.619919	−	0.784666i	$-0.712835\pi$
$227$	−18.6800	−1.23984	−0.619919	+	0.784666i	$0.712835\pi$
$228$	0	0
$229$	−6.92396	−0.457548	−0.228774	−	0.973480i	$-0.573472\pi$
$229$	−6.92396	−0.457548	−0.228774	+	0.973480i	$0.573472\pi$
$230$	−5.31996	−0.350787
$231$	0	0
$232$	11.0077	0.722694
$233$	11.2831	0.739182	0.369591	−	0.929195i	$-0.379498\pi$
$233$	11.2831	0.739182	0.369591	+	0.929195i	$0.379498\pi$
$234$	0	0
$235$	−9.17530	−0.598530
$236$	7.55943	0.492077
$237$	0	0
$238$	18.4611	1.19666
$239$	−26.9564	−1.74366	−0.871831	−	0.489807i	$-0.837067\pi$
$239$	−26.9564	−1.74366	−0.871831	+	0.489807i	$0.837067\pi$
$240$	0	0
$241$	19.6382	1.26500	0.632502	−	0.774558i	$-0.282028\pi$
$241$	19.6382	1.26500	0.632502	+	0.774558i	$0.282028\pi$
$242$	−8.71864	−0.560455
$243$	0	0
$244$	7.30871	0.467892
$245$	10.9581	0.700088
$246$	0	0
$247$	0.0418891	0.00266534
$248$	−22.0787	−1.40200
$249$	0	0
$250$	−7.13516	−0.451267
$251$	18.1857	1.14787	0.573936	−	0.818900i	$-0.305416\pi$
$251$	18.1857	1.14787	0.573936	+	0.818900i	$0.305416\pi$
$252$	0	0
$253$	7.16756	0.450620
$254$	−16.7956	−1.05385
$255$	0	0
$256$	−16.1010	−1.00631
$257$	−19.1976	−1.19751	−0.598756	−	0.800931i	$-0.704338\pi$
$257$	−19.1976	−1.19751	−0.598756	+	0.800931i	$0.704338\pi$
$258$	0	0
$259$	−5.41147	−0.336253
$260$	−0.0451868	−0.00280237
$261$	0	0
$262$	−3.75608	−0.232052
$263$	−3.28817	−0.202757	−0.101379	−	0.994848i	$-0.532325\pi$
$263$	−3.28817	−0.202757	−0.101379	+	0.994848i	$0.532325\pi$
$264$	0	0
$265$	−2.49525	−0.153282
$266$	−3.87939	−0.237860
$267$	0	0
$268$	3.18479	0.194542
$269$	−30.3783	−1.85220	−0.926098	−	0.377284i	$-0.876858\pi$
$269$	−30.3783	−1.85220	−0.926098	+	0.377284i	$0.876858\pi$
$270$	0	0
$271$	−29.9317	−1.81822	−0.909111	−	0.416554i	$-0.863238\pi$
$271$	−29.9317	−1.81822	−0.909111	+	0.416554i	$0.863238\pi$
$272$	0.199340	0.0120868
$273$	0	0
$274$	−4.81521	−0.290897
$275$	4.40373	0.265555
$276$	0	0
$277$	7.51249	0.451382	0.225691	−	0.974199i	$-0.427536\pi$
$277$	7.51249	0.451382	0.225691	+	0.974199i	$0.427536\pi$
$278$	−13.6031	−0.815858
$279$	0	0
$280$	11.0077	0.657838
$281$	4.75877	0.283884	0.141942	−	0.989875i	$-0.454665\pi$
$281$	4.75877	0.283884	0.141942	+	0.989875i	$0.454665\pi$
$282$	0	0
$283$	12.4534	0.740276	0.370138	−	0.928977i	$-0.379310\pi$
$283$	12.4534	0.740276	0.370138	+	0.928977i	$0.379310\pi$
$284$	8.23947	0.488923
$285$	0	0
$286$	−0.0383797	−0.00226944
$287$	−53.6245	−3.16536
$288$	0	0
$289$	5.64590	0.332112
$290$	−3.00000	−0.176166
$291$	0	0
$292$	−10.0496	−0.588110
$293$	5.76382	0.336726	0.168363	−	0.985725i	$-0.446152\pi$
$293$	5.76382	0.336726	0.168363	+	0.985725i	$0.446152\pi$
$294$	0	0
$295$	−5.41921	−0.315519
$296$	−3.48070	−0.202312
$297$	0	0
$298$	−13.1848	−0.763775
$299$	−0.288171	−0.0166654
$300$	0	0
$301$	−11.2686	−0.649510
$302$	12.1037	0.696490
$303$	0	0
$304$	−0.0418891	−0.00240250
$305$	−5.23947	−0.300011
$306$	0	0
$307$	31.7547	1.81233	0.906167	−	0.422920i	$-0.138995\pi$
$307$	31.7547	1.81233	0.906167	+	0.422920i	$0.138995\pi$
$308$	−5.63816	−0.321264
$309$	0	0
$310$	6.01724	0.341756
$311$	−17.6382	−1.00017	−0.500084	−	0.865977i	$-0.666698\pi$
$311$	−17.6382	−1.00017	−0.500084	+	0.865977i	$0.666698\pi$
$312$	0	0
$313$	13.6209	0.769900	0.384950	−	0.922937i	$-0.374219\pi$
$313$	13.6209	0.769900	0.384950	+	0.922937i	$0.374219\pi$
$314$	4.39693	0.248133
$315$	0	0
$316$	2.94862	0.165873
$317$	−13.6263	−0.765329	−0.382665	−	0.923887i	$-0.624993\pi$
$317$	−13.6263	−0.765329	−0.382665	+	0.923887i	$0.624993\pi$
$318$	0	0
$319$	4.04189	0.226302
$320$	4.43376	0.247855
$321$	0	0
$322$	26.6878	1.48725
$323$	−4.75877	−0.264785
$324$	0	0
$325$	−0.177052	−0.00982106
$326$	−11.1958	−0.620080
$327$	0	0
$328$	−34.4917	−1.90449
$329$	46.0283	2.53762
$330$	0	0
$331$	24.6287	1.35371	0.676857	−	0.736115i	$-0.263342\pi$
$331$	24.6287	1.35371	0.676857	+	0.736115i	$0.263342\pi$
$332$	14.9564	0.820837
$333$	0	0
$334$	−2.32594	−0.127270
$335$	−2.28312	−0.124740
$336$	0	0
$337$	−23.5526	−1.28299	−0.641497	−	0.767126i	$-0.721686\pi$
$337$	−23.5526	−1.28299	−0.641497	+	0.767126i	$0.721686\pi$
$338$	−11.4305	−0.621735
$339$	0	0
$340$	5.13341	0.278398
$341$	−8.10700	−0.439019
$342$	0	0
$343$	−24.0915	−1.30082
$344$	−7.24804	−0.390788
$345$	0	0
$346$	−18.7793	−1.00958
$347$	32.2395	1.73071	0.865353	−	0.501163i	$-0.167094\pi$
$347$	32.2395	1.73071	0.865353	+	0.501163i	$0.167094\pi$
$348$	0	0
$349$	−14.1584	−0.757881	−0.378940	−	0.925421i	$-0.623711\pi$
$349$	−14.1584	−0.757881	−0.378940	+	0.925421i	$0.623711\pi$
$350$	16.3969	0.876453
$351$	0	0
$352$	−5.87433	−0.313103
$353$	15.9982	0.851501	0.425750	−	0.904841i	$-0.360010\pi$
$353$	15.9982	0.851501	0.425750	+	0.904841i	$0.360010\pi$
$354$	0	0
$355$	−5.90673	−0.313496
$356$	−10.7956	−0.572166
$357$	0	0
$358$	−0.916222	−0.0484238
$359$	22.2645	1.17507	0.587536	−	0.809198i	$-0.300098\pi$
$359$	22.2645	1.17507	0.587536	+	0.809198i	$0.300098\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.4355	−0.653596
$363$	0	0
$364$	0.226682	0.0118813
$365$	7.20439	0.377095
$366$	0	0
$367$	−10.3182	−0.538606	−0.269303	−	0.963056i	$-0.586793\pi$
$367$	−10.3182	−0.538606	−0.269303	+	0.963056i	$0.586793\pi$
$368$	0.288171	0.0150220
$369$	0	0
$370$	0.948615	0.0493162
$371$	12.5175	0.649878
$372$	0	0
$373$	10.7638	0.557330	0.278665	−	0.960388i	$-0.410108\pi$
$373$	10.7638	0.557330	0.278665	+	0.960388i	$0.410108\pi$
$374$	4.36009	0.225455
$375$	0	0
$376$	29.6058	1.52680
$377$	−0.162504	−0.00836937
$378$	0	0
$379$	−24.2591	−1.24610	−0.623052	−	0.782180i	$-0.714108\pi$
$379$	−24.2591	−1.24610	−0.623052	+	0.782180i	$0.714108\pi$
$380$	−1.07873	−0.0553375
$381$	0	0
$382$	−14.2763	−0.730440
$383$	19.8188	1.01269	0.506347	−	0.862330i	$-0.330995\pi$
$383$	19.8188	1.01269	0.506347	+	0.862330i	$0.330995\pi$
$384$	0	0
$385$	4.04189	0.205994
$386$	10.9281	0.556225
$387$	0	0
$388$	1.40198	0.0711747
$389$	34.7151	1.76013	0.880063	−	0.474856i	$-0.157500\pi$
$389$	34.7151	1.76013	0.880063	+	0.474856i	$0.157500\pi$
$390$	0	0
$391$	32.7374	1.65560
$392$	−35.3583	−1.78587
$393$	0	0
$394$	−1.34493	−0.0677567
$395$	−2.11381	−0.106357
$396$	0	0
$397$	−8.55262	−0.429244	−0.214622	−	0.976697i	$-0.568852\pi$
$397$	−8.55262	−0.429244	−0.214622	+	0.976697i	$0.568852\pi$
$398$	9.52435	0.477412
$399$	0	0
$400$	0.177052	0.00885259
$401$	−12.9682	−0.647602	−0.323801	−	0.946125i	$-0.604961\pi$
$401$	−12.9682	−0.647602	−0.323801	+	0.946125i	$0.604961\pi$
$402$	0	0
$403$	0.325941	0.0162363
$404$	−19.2796	−0.959196
$405$	0	0
$406$	15.0496	0.746901
$407$	−1.27807	−0.0633514
$408$	0	0
$409$	−22.4611	−1.11063	−0.555315	−	0.831640i	$-0.687402\pi$
$409$	−22.4611	−1.11063	−0.555315	+	0.831640i	$0.687402\pi$
$410$	9.40022	0.464244
$411$	0	0
$412$	−11.2668	−0.555076
$413$	27.1857	1.33772
$414$	0	0
$415$	−10.7219	−0.526319
$416$	0.236177	0.0115795
$417$	0	0
$418$	−0.916222	−0.0448139
$419$	3.68004	0.179782	0.0898910	−	0.995952i	$-0.471348\pi$
$419$	3.68004	0.179782	0.0898910	+	0.995952i	$0.471348\pi$
$420$	0	0
$421$	2.41147	0.117528	0.0587640	−	0.998272i	$-0.481284\pi$
$421$	2.41147	0.117528	0.0587640	+	0.998272i	$0.481284\pi$
$422$	−6.92303	−0.337008
$423$	0	0
$424$	8.05138	0.391010
$425$	20.1138	0.975663
$426$	0	0
$427$	26.2841	1.27197
$428$	−1.80396	−0.0871976
$429$	0	0
$430$	1.97535	0.0952597
$431$	13.6919	0.659516	0.329758	−	0.944066i	$-0.393033\pi$
$431$	13.6919	0.659516	0.329758	+	0.944066i	$0.393033\pi$
$432$	0	0
$433$	−27.6382	−1.32821	−0.664103	−	0.747642i	$-0.731186\pi$
$433$	−27.6382	−1.32821	−0.664103	+	0.747642i	$0.731186\pi$
$434$	−30.1857	−1.44896
$435$	0	0
$436$	−22.4287	−1.07414
$437$	−6.87939	−0.329086
$438$	0	0
$439$	−3.80747	−0.181720	−0.0908602	−	0.995864i	$-0.528962\pi$
$439$	−3.80747	−0.181720	−0.0908602	+	0.995864i	$0.528962\pi$
$440$	2.59978	0.123939
$441$	0	0
$442$	−0.175297	−0.00833803
$443$	9.16250	0.435324	0.217662	−	0.976024i	$-0.430157\pi$
$443$	9.16250	0.435324	0.217662	+	0.976024i	$0.430157\pi$
$444$	0	0
$445$	7.73917	0.366872
$446$	15.8862	0.752233
$447$	0	0
$448$	−22.2422	−1.05084
$449$	5.93313	0.280002	0.140001	−	0.990151i	$-0.455289\pi$
$449$	5.93313	0.280002	0.140001	+	0.990151i	$0.455289\pi$
$450$	0	0
$451$	−12.6649	−0.596366
$452$	−8.19429	−0.385427
$453$	0	0
$454$	−16.4270	−0.770955
$455$	−0.162504	−0.00761830
$456$	0	0
$457$	28.6705	1.34115	0.670576	−	0.741841i	$-0.266047\pi$
$457$	28.6705	1.34115	0.670576	+	0.741841i	$0.266047\pi$
$458$	−6.08883	−0.284512
$459$	0	0
$460$	7.42097	0.346004
$461$	25.3669	1.18145	0.590727	−	0.806871i	$-0.298841\pi$
$461$	25.3669	1.18145	0.590727	+	0.806871i	$0.298841\pi$
$462$	0	0
$463$	15.0591	0.699857	0.349928	−	0.936776i	$-0.386206\pi$
$463$	15.0591	0.699857	0.349928	+	0.936776i	$0.386206\pi$
$464$	0.162504	0.00754405
$465$	0	0
$466$	9.92221	0.459637
$467$	−20.7050	−0.958114	−0.479057	−	0.877784i	$-0.659021\pi$
$467$	−20.7050	−0.958114	−0.479057	+	0.877784i	$0.659021\pi$
$468$	0	0
$469$	11.4534	0.528867
$470$	−8.06862	−0.372178
$471$	0	0
$472$	17.4861	0.804862
$473$	−2.66138	−0.122370
$474$	0	0
$475$	−4.22668	−0.193933
$476$	−25.7520	−1.18034
$477$	0	0
$478$	−23.7050	−1.08424
$479$	−23.6013	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$479$	−23.6013	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$480$	0	0
$481$	0.0513845	0.00234293
$482$	17.2695	0.786604
$483$	0	0
$484$	12.1619	0.552813
$485$	−1.00505	−0.0456371
$486$	0	0
$487$	−15.4020	−0.697930	−0.348965	−	0.937136i	$-0.613467\pi$
$487$	−15.4020	−0.697930	−0.348965	+	0.937136i	$0.613467\pi$
$488$	16.9061	0.765304
$489$	0	0
$490$	9.63640	0.435328
$491$	3.13247	0.141367	0.0706833	−	0.997499i	$-0.477482\pi$
$491$	3.13247	0.141367	0.0706833	+	0.997499i	$0.477482\pi$
$492$	0	0
$493$	18.4611	0.831446
$494$	0.0368366	0.00165736
$495$	0	0
$496$	−0.325941	−0.0146352
$497$	29.6313	1.32915
$498$	0	0
$499$	21.9145	0.981026	0.490513	−	0.871434i	$-0.336809\pi$
$499$	21.9145	0.981026	0.490513	+	0.871434i	$0.336809\pi$
$500$	9.95306	0.445114
$501$	0	0
$502$	15.9923	0.713769
$503$	25.7820	1.14956	0.574781	−	0.818307i	$-0.305087\pi$
$503$	25.7820	1.14956	0.574781	+	0.818307i	$0.305087\pi$
$504$	0	0
$505$	13.8212	0.615035
$506$	6.30304	0.280204
$507$	0	0
$508$	23.4287	1.03948
$509$	29.0182	1.28621	0.643104	−	0.765779i	$-0.277646\pi$
$509$	29.0182	1.28621	0.643104	+	0.765779i	$0.277646\pi$
$510$	0	0
$511$	−36.1411	−1.59879
$512$	−0.473897	−0.0209435
$513$	0	0
$514$	−16.8821	−0.744636
$515$	8.07697	0.355914
$516$	0	0
$517$	10.8708	0.478098
$518$	−4.75877	−0.209088
$519$	0	0
$520$	−0.104524	−0.00458367
$521$	10.5526	0.462319	0.231159	−	0.972916i	$-0.425748\pi$
$521$	10.5526	0.462319	0.231159	+	0.972916i	$0.425748\pi$
$522$	0	0
$523$	29.3182	1.28200	0.640998	−	0.767543i	$-0.278521\pi$
$523$	29.3182	1.28200	0.640998	+	0.767543i	$0.278521\pi$
$524$	5.23947	0.228888
$525$	0	0
$526$	−2.89157	−0.126078
$527$	−37.0283	−1.61298
$528$	0	0
$529$	24.3259	1.05765
$530$	−2.19429	−0.0953138
$531$	0	0
$532$	5.41147	0.234617
$533$	0.509191	0.0220555
$534$	0	0
$535$	1.29322	0.0559109
$536$	7.36690	0.318201
$537$	0	0
$538$	−26.7142	−1.15173
$539$	−12.9831	−0.559221
$540$	0	0
$541$	−40.0820	−1.72326	−0.861630	−	0.507536i	$-0.830556\pi$
$541$	−40.0820	−1.72326	−0.861630	+	0.507536i	$0.830556\pi$
$542$	−26.3215	−1.13061
$543$	0	0
$544$	−26.8307	−1.15036
$545$	16.0787	0.688737
$546$	0	0
$547$	−26.2576	−1.12270	−0.561348	−	0.827580i	$-0.689717\pi$
$547$	−26.2576	−1.12270	−0.561348	+	0.827580i	$0.689717\pi$
$548$	6.71688	0.286931
$549$	0	0
$550$	3.87258	0.165127
$551$	−3.87939	−0.165267
$552$	0	0
$553$	10.6040	0.450928
$554$	6.60637	0.280678
$555$	0	0
$556$	18.9753	0.804734
$557$	43.5895	1.84694	0.923472	−	0.383665i	$-0.125338\pi$
$557$	43.5895	1.84694	0.923472	+	0.383665i	$0.125338\pi$
$558$	0	0
$559$	0.107000	0.00452564
$560$	0.162504	0.00686704
$561$	0	0
$562$	4.18479	0.176525
$563$	7.47060	0.314848	0.157424	−	0.987531i	$-0.449681\pi$
$563$	7.47060	0.314848	0.157424	+	0.987531i	$0.449681\pi$
$564$	0	0
$565$	5.87433	0.247135
$566$	10.9513	0.460318
$567$	0	0
$568$	19.0591	0.799703
$569$	20.5945	0.863367	0.431683	−	0.902025i	$-0.357920\pi$
$569$	20.5945	0.863367	0.431683	+	0.902025i	$0.357920\pi$
$570$	0	0
$571$	−4.02641	−0.168500	−0.0842500	−	0.996445i	$-0.526849\pi$
$571$	−4.02641	−0.168500	−0.0842500	+	0.996445i	$0.526849\pi$
$572$	0.0535370	0.00223849
$573$	0	0
$574$	−47.1566	−1.96828
$575$	29.0770	1.21259
$576$	0	0
$577$	19.8895	0.828010	0.414005	−	0.910275i	$-0.364130\pi$
$577$	19.8895	0.828010	0.414005	+	0.910275i	$0.364130\pi$
$578$	4.96492	0.206513
$579$	0	0
$580$	4.18479	0.173764
$581$	53.7870	2.23146
$582$	0	0
$583$	2.95636	0.122440
$584$	−23.2463	−0.961938
$585$	0	0
$586$	5.06862	0.209383
$587$	−16.4638	−0.679533	−0.339767	−	0.940510i	$-0.610348\pi$
$587$	−16.4638	−0.679533	−0.339767	+	0.940510i	$0.610348\pi$
$588$	0	0
$589$	7.78106	0.320613
$590$	−4.76558	−0.196196
$591$	0	0
$592$	−0.0513845	−0.00211189
$593$	−23.4088	−0.961284	−0.480642	−	0.876917i	$-0.659596\pi$
$593$	−23.4088	−0.961284	−0.480642	+	0.876917i	$0.659596\pi$
$594$	0	0
$595$	18.4611	0.756831
$596$	18.3919	0.753361
$597$	0	0
$598$	−0.253413	−0.0103628
$599$	−40.3465	−1.64851	−0.824256	−	0.566217i	$-0.808406\pi$
$599$	−40.3465	−1.64851	−0.824256	+	0.566217i	$0.808406\pi$
$600$	0	0
$601$	0.228436	0.00931811	0.00465906	−	0.999989i	$-0.498517\pi$
$601$	0.228436	0.00931811	0.00465906	+	0.999989i	$0.498517\pi$
$602$	−9.90941	−0.403878
$603$	0	0
$604$	−16.8838	−0.686993
$605$	−8.71864	−0.354463
$606$	0	0
$607$	−2.28405	−0.0927068	−0.0463534	−	0.998925i	$-0.514760\pi$
$607$	−2.28405	−0.0927068	−0.0463534	+	0.998925i	$0.514760\pi$
$608$	5.63816	0.228657
$609$	0	0
$610$	−4.60752	−0.186553
$611$	−0.437061	−0.0176816
$612$	0	0
$613$	43.2235	1.74578	0.872890	−	0.487917i	$-0.162243\pi$
$613$	43.2235	1.74578	0.872890	+	0.487917i	$0.162243\pi$
$614$	27.9246	1.12694
$615$	0	0
$616$	−13.0419	−0.525473
$617$	−42.0096	−1.69124	−0.845622	−	0.533783i	$-0.820770\pi$
$617$	−42.0096	−1.69124	−0.845622	+	0.533783i	$0.820770\pi$
$618$	0	0
$619$	−33.1334	−1.33174	−0.665872	−	0.746066i	$-0.731940\pi$
$619$	−33.1334	−1.33174	−0.665872	+	0.746066i	$0.731940\pi$
$620$	−8.39363	−0.337096
$621$	0	0
$622$	−15.5107	−0.621924
$623$	−38.8239	−1.55545
$624$	0	0
$625$	13.9982	0.559930
$626$	11.9780	0.478739
$627$	0	0
$628$	−6.13341	−0.244750
$629$	−5.83750	−0.232756
$630$	0	0
$631$	33.3601	1.32804	0.664022	−	0.747713i	$-0.268848\pi$
$631$	33.3601	1.32804	0.664022	+	0.747713i	$0.268848\pi$
$632$	6.82058	0.271308
$633$	0	0
$634$	−11.9828	−0.475896
$635$	−16.7956	−0.666513
$636$	0	0
$637$	0.521984	0.0206818
$638$	3.55438	0.140719
$639$	0	0
$640$	−6.01724	−0.237852
$641$	−2.18748	−0.0864003	−0.0432002	−	0.999066i	$-0.513755\pi$
$641$	−2.18748	−0.0864003	−0.0432002	+	0.999066i	$0.513755\pi$
$642$	0	0
$643$	14.4875	0.571332	0.285666	−	0.958329i	$-0.407785\pi$
$643$	14.4875	0.571332	0.285666	+	0.958329i	$0.407785\pi$
$644$	−37.2276	−1.46697
$645$	0	0
$646$	−4.18479	−0.164648
$647$	−10.6399	−0.418298	−0.209149	−	0.977884i	$-0.567069\pi$
$647$	−10.6399	−0.418298	−0.209149	+	0.977884i	$0.567069\pi$
$648$	0	0
$649$	6.42065	0.252032
$650$	−0.155697	−0.00610693
$651$	0	0
$652$	15.6174	0.611625
$653$	−14.7638	−0.577753	−0.288877	−	0.957366i	$-0.593282\pi$
$653$	−14.7638	−0.577753	−0.288877	+	0.957366i	$0.593282\pi$
$654$	0	0
$655$	−3.75608	−0.146762
$656$	−0.509191	−0.0198806
$657$	0	0
$658$	40.4766	1.57794
$659$	49.2945	1.92024	0.960120	−	0.279588i	$-0.0901977\pi$
$659$	49.2945	1.92024	0.960120	+	0.279588i	$0.0901977\pi$
$660$	0	0
$661$	−26.9905	−1.04981	−0.524904	−	0.851161i	$-0.675899\pi$
$661$	−26.9905	−1.04981	−0.524904	+	0.851161i	$0.675899\pi$
$662$	21.6581	0.841765
$663$	0	0
$664$	34.5963	1.34260
$665$	−3.87939	−0.150436
$666$	0	0
$667$	26.6878	1.03336
$668$	3.24453	0.125535
$669$	0	0
$670$	−2.00774	−0.0775658
$671$	6.20769	0.239645
$672$	0	0
$673$	17.1429	0.660810	0.330405	−	0.943839i	$-0.392815\pi$
$673$	17.1429	0.660810	0.330405	+	0.943839i	$0.392815\pi$
$674$	−20.7118	−0.797790
$675$	0	0
$676$	15.9447	0.613258
$677$	−31.6783	−1.21750	−0.608748	−	0.793364i	$-0.708328\pi$
$677$	−31.6783	−1.21750	−0.608748	+	0.793364i	$0.708328\pi$
$678$	0	0
$679$	5.04189	0.193490
$680$	11.8743	0.455360
$681$	0	0
$682$	−7.12918	−0.272990
$683$	7.36009	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$683$	7.36009	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$684$	0	0
$685$	−4.81521	−0.183980
$686$	−21.1857	−0.808875
$687$	0	0
$688$	−0.107000	−0.00407936
$689$	−0.118860	−0.00452821
$690$	0	0
$691$	−24.2763	−0.923514	−0.461757	−	0.887006i	$-0.652781\pi$
$691$	−24.2763	−0.923514	−0.461757	+	0.887006i	$0.652781\pi$
$692$	26.1958	0.995816
$693$	0	0
$694$	28.3509	1.07619
$695$	−13.6031	−0.515994
$696$	0	0
$697$	−57.8462	−2.19108
$698$	−12.4507	−0.471265
$699$	0	0
$700$	−22.8726	−0.864502
$701$	43.5390	1.64445	0.822223	−	0.569166i	$-0.192734\pi$
$701$	43.5390	1.64445	0.822223	+	0.569166i	$0.192734\pi$
$702$	0	0
$703$	1.22668	0.0462652
$704$	−5.25309	−0.197983
$705$	0	0
$706$	14.0686	0.529480
$707$	−69.3346	−2.60760
$708$	0	0
$709$	−13.6132	−0.511254	−0.255627	−	0.966776i	$-0.582282\pi$
$709$	−13.6132	−0.511254	−0.255627	+	0.966776i	$0.582282\pi$
$710$	−5.19429	−0.194938
$711$	0	0
$712$	−24.9718	−0.935859
$713$	−53.5289	−2.00467
$714$	0	0
$715$	−0.0383797	−0.00143532
$716$	1.27807	0.0477636
$717$	0	0
$718$	19.5790	0.730683
$719$	−33.4807	−1.24862	−0.624310	−	0.781177i	$-0.714620\pi$
$719$	−33.4807	−1.24862	−0.624310	+	0.781177i	$0.714620\pi$
$720$	0	0
$721$	−40.5185	−1.50899
$722$	0.879385	0.0327273
$723$	0	0
$724$	17.3467	0.644685
$725$	16.3969	0.608967
$726$	0	0
$727$	−6.87433	−0.254955	−0.127477	−	0.991841i	$-0.540688\pi$
$727$	−6.87433	−0.254955	−0.127477	+	0.991841i	$0.540688\pi$
$728$	0.524348	0.0194336
$729$	0	0
$730$	6.33544	0.234485
$731$	−12.1557	−0.449595
$732$	0	0
$733$	−48.9796	−1.80910	−0.904551	−	0.426365i	$-0.859794\pi$
$733$	−48.9796	−1.80910	−0.904551	+	0.426365i	$0.859794\pi$
$734$	−9.07367	−0.334915
$735$	0	0
$736$	−38.7870	−1.42971
$737$	2.70502	0.0996408
$738$	0	0
$739$	40.5276	1.49083	0.745417	−	0.666599i	$-0.232251\pi$
$739$	40.5276	1.49083	0.745417	+	0.666599i	$0.232251\pi$
$740$	−1.32325	−0.0486437
$741$	0	0
$742$	11.0077	0.404107
$743$	3.39187	0.124436	0.0622179	−	0.998063i	$-0.480183\pi$
$743$	3.39187	0.124436	0.0622179	+	0.998063i	$0.480183\pi$
$744$	0	0
$745$	−13.1848	−0.483053
$746$	9.46555	0.346558
$747$	0	0
$748$	−6.08202	−0.222381
$749$	−6.48751	−0.237049
$750$	0	0
$751$	11.6800	0.426211	0.213105	−	0.977029i	$-0.431642\pi$
$751$	11.6800	0.426211	0.213105	+	0.977029i	$0.431642\pi$
$752$	0.437061	0.0159380
$753$	0	0
$754$	−0.142903	−0.00520424
$755$	12.1037	0.440499
$756$	0	0
$757$	−43.4083	−1.57770	−0.788851	−	0.614585i	$-0.789323\pi$
$757$	−43.4083	−1.57770	−0.788851	+	0.614585i	$0.789323\pi$
$758$	−21.3331	−0.774852
$759$	0	0
$760$	−2.49525	−0.0905123
$761$	15.6800	0.568401	0.284201	−	0.958765i	$-0.408272\pi$
$761$	15.6800	0.568401	0.284201	+	0.958765i	$0.408272\pi$
$762$	0	0
$763$	−80.6596	−2.92007
$764$	19.9145	0.720480
$765$	0	0
$766$	17.4284	0.629713
$767$	−0.258142	−0.00932095
$768$	0	0
$769$	18.1908	0.655976	0.327988	−	0.944682i	$-0.393629\pi$
$769$	18.1908	0.655976	0.327988	+	0.944682i	$0.393629\pi$
$770$	3.55438	0.128091
$771$	0	0
$772$	−15.2439	−0.548641
$773$	−19.1307	−0.688084	−0.344042	−	0.938954i	$-0.611796\pi$
$773$	−19.1307	−0.688084	−0.344042	+	0.938954i	$0.611796\pi$
$774$	0	0
$775$	−32.8881	−1.18137
$776$	3.24298	0.116416
$777$	0	0
$778$	30.5280	1.09448
$779$	12.1557	0.435523
$780$	0	0
$781$	6.99825	0.250417
$782$	28.7888	1.02949
$783$	0	0
$784$	−0.521984	−0.0186423
$785$	4.39693	0.156933
$786$	0	0
$787$	−43.0729	−1.53538	−0.767691	−	0.640821i	$-0.778594\pi$
$787$	−43.0729	−1.53538	−0.767691	+	0.640821i	$0.778594\pi$
$788$	1.87609	0.0668328
$789$	0	0
$790$	−1.85885	−0.0661350
$791$	−29.4688	−1.04779
$792$	0	0
$793$	−0.249580	−0.00886284
$794$	−7.52105	−0.266912
$795$	0	0
$796$	−13.2858	−0.470903
$797$	−4.89124	−0.173257	−0.0866284	−	0.996241i	$-0.527609\pi$
$797$	−4.89124	−0.173257	−0.0866284	+	0.996241i	$0.527609\pi$
$798$	0	0
$799$	49.6519	1.75656
$800$	−23.8307	−0.842542
$801$	0	0
$802$	−11.4041	−0.402691
$803$	−8.53571	−0.301219
$804$	0	0
$805$	26.6878	0.940621
$806$	0.286628	0.0100960
$807$	0	0
$808$	−44.5966	−1.56890
$809$	39.0669	1.37352	0.686759	−	0.726885i	$-0.259033\pi$
$809$	39.0669	1.37352	0.686759	+	0.726885i	$0.259033\pi$
$810$	0	0
$811$	51.4178	1.80552	0.902761	−	0.430142i	$-0.141536\pi$
$811$	51.4178	1.80552	0.902761	+	0.430142i	$0.141536\pi$
$812$	−20.9932	−0.736717
$813$	0	0
$814$	−1.12391	−0.0393931
$815$	−11.1958	−0.392173
$816$	0	0
$817$	2.55438	0.0893664
$818$	−19.7520	−0.690611
$819$	0	0
$820$	−13.1127	−0.457914
$821$	−8.47565	−0.295802	−0.147901	−	0.989002i	$-0.547252\pi$
$821$	−8.47565	−0.295802	−0.147901	+	0.989002i	$0.547252\pi$
$822$	0	0
$823$	22.8571	0.796748	0.398374	−	0.917223i	$-0.369575\pi$
$823$	22.8571	0.796748	0.398374	+	0.917223i	$0.369575\pi$
$824$	−26.0618	−0.907906
$825$	0	0
$826$	23.9067	0.831821
$827$	30.5708	1.06305	0.531525	−	0.847042i	$-0.321619\pi$
$827$	30.5708	1.06305	0.531525	+	0.847042i	$0.321619\pi$
$828$	0	0
$829$	22.9504	0.797099	0.398550	−	0.917147i	$-0.369514\pi$
$829$	22.9504	0.797099	0.398550	+	0.917147i	$0.369514\pi$
$830$	−9.42871	−0.327275
$831$	0	0
$832$	0.211200	0.00732204
$833$	−59.2995	−2.05461
$834$	0	0
$835$	−2.32594	−0.0804925
$836$	1.27807	0.0442028
$837$	0	0
$838$	3.23618	0.111792
$839$	18.5408	0.640098	0.320049	−	0.947401i	$-0.396301\pi$
$839$	18.5408	0.640098	0.320049	+	0.947401i	$0.396301\pi$
$840$	0	0
$841$	−13.9504	−0.481047
$842$	2.12061	0.0730812
$843$	0	0
$844$	9.65715	0.332413
$845$	−11.4305	−0.393220
$846$	0	0
$847$	43.7374	1.50284
$848$	0.118860	0.00408167
$849$	0	0
$850$	17.6878	0.606686
$851$	−8.43882	−0.289279
$852$	0	0
$853$	−46.3182	−1.58590	−0.792952	−	0.609283i	$-0.791457\pi$
$853$	−46.3182	−1.58590	−0.792952	+	0.609283i	$0.791457\pi$
$854$	23.1138	0.790938
$855$	0	0
$856$	−4.17282	−0.142624
$857$	2.41685	0.0825581	0.0412790	−	0.999148i	$-0.486857\pi$
$857$	2.41685	0.0825581	0.0412790	+	0.999148i	$0.486857\pi$
$858$	0	0
$859$	−31.1489	−1.06279	−0.531393	−	0.847125i	$-0.678331\pi$
$859$	−31.1489	−1.06279	−0.531393	+	0.847125i	$0.678331\pi$
$860$	−2.75547	−0.0939608
$861$	0	0
$862$	12.0405	0.410100
$863$	−57.6364	−1.96197	−0.980983	−	0.194094i	$-0.937823\pi$
$863$	−57.6364	−1.96197	−0.980983	+	0.194094i	$0.937823\pi$
$864$	0	0
$865$	−18.7793	−0.638516
$866$	−24.3046	−0.825903
$867$	0	0
$868$	42.1070	1.42920
$869$	2.50442	0.0849567
$870$	0	0
$871$	−0.108755	−0.00368503
$872$	−51.8809	−1.75691
$873$	0	0
$874$	−6.04963	−0.204632
$875$	35.7939	1.21005
$876$	0	0
$877$	−23.5526	−0.795316	−0.397658	−	0.917534i	$-0.630177\pi$
$877$	−23.5526	−0.795316	−0.397658	+	0.917534i	$0.630177\pi$
$878$	−3.34823	−0.112997
$879$	0	0
$880$	0.0383797	0.00129378
$881$	−7.83244	−0.263882	−0.131941	−	0.991258i	$-0.542121\pi$
$881$	−7.83244	−0.263882	−0.131941	+	0.991258i	$0.542121\pi$
$882$	0	0
$883$	−20.4706	−0.688891	−0.344445	−	0.938806i	$-0.611933\pi$
$883$	−20.4706	−0.688891	−0.344445	+	0.938806i	$0.611933\pi$
$884$	0.244527	0.00822434
$885$	0	0
$886$	8.05737	0.270693
$887$	2.56448	0.0861069	0.0430534	−	0.999073i	$-0.486291\pi$
$887$	2.56448	0.0861069	0.0430534	+	0.999073i	$0.486291\pi$
$888$	0	0
$889$	84.2559	2.82585
$890$	6.80571	0.228128
$891$	0	0
$892$	−22.1601	−0.741976
$893$	−10.4338	−0.349153
$894$	0	0
$895$	−0.916222	−0.0306259
$896$	30.1857	1.00843
$897$	0	0
$898$	5.21751	0.174111
$899$	−30.1857	−1.00675
$900$	0	0
$901$	13.5030	0.449850
$902$	−11.1373	−0.370832
$903$	0	0
$904$	−18.9546	−0.630421
$905$	−12.4355	−0.413371
$906$	0	0
$907$	28.6441	0.951113	0.475557	−	0.879685i	$-0.342247\pi$
$907$	28.6441	0.951113	0.475557	+	0.879685i	$0.342247\pi$
$908$	22.9145	0.760443
$909$	0	0
$910$	−0.142903	−0.00473720
$911$	9.14889	0.303116	0.151558	−	0.988448i	$-0.451571\pi$
$911$	9.14889	0.303116	0.151558	+	0.988448i	$0.451571\pi$
$912$	0	0
$913$	12.7033	0.420417
$914$	25.2125	0.833954
$915$	0	0
$916$	8.49350	0.280633
$917$	18.8425	0.622236
$918$	0	0
$919$	1.22668	0.0404645	0.0202322	−	0.999795i	$-0.493559\pi$
$919$	1.22668	0.0404645	0.0202322	+	0.999795i	$0.493559\pi$
$920$	17.1658	0.565940
$921$	0	0
$922$	22.3073	0.734651
$923$	−0.281364	−0.00926121
$924$	0	0
$925$	−5.18479	−0.170475
$926$	13.2428	0.435185
$927$	0	0
$928$	−21.8726	−0.718003
$929$	−6.45748	−0.211863	−0.105932	−	0.994373i	$-0.533782\pi$
$929$	−6.45748	−0.211863	−0.105932	+	0.994373i	$0.533782\pi$
$930$	0	0
$931$	12.4611	0.408396
$932$	−13.8408	−0.453370
$933$	0	0
$934$	−18.2077	−0.595774
$935$	4.36009	0.142590
$936$	0	0
$937$	27.4270	0.896000	0.448000	−	0.894034i	$-0.352137\pi$
$937$	27.4270	0.896000	0.448000	+	0.894034i	$0.352137\pi$
$938$	10.0719	0.328860
$939$	0	0
$940$	11.2552	0.367103
$941$	−12.5175	−0.408060	−0.204030	−	0.978965i	$-0.565404\pi$
$941$	−12.5175	−0.408060	−0.204030	+	0.978965i	$0.565404\pi$
$942$	0	0
$943$	−83.6237	−2.72316
$944$	0.258142	0.00840179
$945$	0	0
$946$	−2.34038	−0.0760922
$947$	54.1353	1.75916	0.879580	−	0.475751i	$-0.157824\pi$
$947$	54.1353	1.75916	0.879580	+	0.475751i	$0.157824\pi$
$948$	0	0
$949$	0.343178	0.0111400
$950$	−3.71688	−0.120592
$951$	0	0
$952$	−59.5681	−1.93061
$953$	−45.8070	−1.48383	−0.741917	−	0.670492i	$-0.766083\pi$
$953$	−45.8070	−1.48383	−0.741917	+	0.670492i	$0.766083\pi$
$954$	0	0
$955$	−14.2763	−0.461971
$956$	33.0669	1.06946
$957$	0	0
$958$	−20.7547	−0.670552
$959$	24.1557	0.780028
$960$	0	0
$961$	29.5449	0.953061
$962$	0.0451868	0.00145688
$963$	0	0
$964$	−24.0898	−0.775879
$965$	10.9281	0.351787
$966$	0	0
$967$	31.2844	1.00604	0.503019	−	0.864275i	$-0.332223\pi$
$967$	31.2844	1.00604	0.503019	+	0.864275i	$0.332223\pi$
$968$	28.1323	0.904205
$969$	0	0
$970$	−0.883828	−0.0283780
$971$	31.6699	1.01634	0.508168	−	0.861258i	$-0.330323\pi$
$971$	31.6699	1.01634	0.508168	+	0.861258i	$0.330323\pi$
$972$	0	0
$973$	68.2404	2.18769
$974$	−13.5443	−0.433986
$975$	0	0
$976$	0.249580	0.00798885
$977$	6.74691	0.215853	0.107926	−	0.994159i	$-0.465579\pi$
$977$	6.74691	0.215853	0.107926	+	0.994159i	$0.465579\pi$
$978$	0	0
$979$	−9.16931	−0.293052
$980$	−13.4421	−0.429393
$981$	0	0
$982$	2.75465	0.0879045
$983$	−30.5476	−0.974316	−0.487158	−	0.873314i	$-0.661966\pi$
$983$	−30.5476	−0.974316	−0.487158	+	0.873314i	$0.661966\pi$
$984$	0	0
$985$	−1.34493	−0.0428531
$986$	16.2344	0.517009
$987$	0	0
$988$	−0.0513845	−0.00163476
$989$	−17.5725	−0.558775
$990$	0	0
$991$	12.5730	0.399396	0.199698	−	0.979858i	$-0.436004\pi$
$991$	12.5730	0.399396	0.199698	+	0.979858i	$0.436004\pi$
$992$	43.8708	1.39290
$993$	0	0
$994$	26.0574	0.826490
$995$	9.52435	0.301942
$996$	0	0
$997$	34.8895	1.10496	0.552481	−	0.833526i	$-0.313681\pi$
$997$	34.8895	1.10496	0.552481	+	0.833526i	$0.313681\pi$
$998$	19.2713	0.610021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.d.1.3	✓	3
3.2	odd	2		513.2.a.g.1.1	yes	3
4.3	odd	2		8208.2.a.bh.1.3		3
12.11	even	2		8208.2.a.bn.1.1		3
19.18	odd	2		9747.2.a.bc.1.1		3
57.56	even	2		9747.2.a.w.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.d.1.3	✓	3	1.1	even	1	trivial
513.2.a.g.1.1	yes	3	3.2	odd	2
8208.2.a.bh.1.3		3	4.3	odd	2
8208.2.a.bn.1.1		3	12.11	even	2
9747.2.a.w.1.3		3	57.56	even	2
9747.2.a.bc.1.1		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q+0.879385 q^{2} -1.22668 q^{4} +0.879385 q^{5} -4.41147 q^{7} -2.83750 q^{8} +0.773318 q^{10} -1.04189 q^{11} +0.0418891 q^{13} -3.87939 q^{14} -0.0418891 q^{16} -4.75877 q^{17} +1.00000 q^{19} -1.07873 q^{20} -0.916222 q^{22} -6.87939 q^{23} -4.22668 q^{25} +0.0368366 q^{26} +5.41147 q^{28} -3.87939 q^{29} +7.78106 q^{31} +5.63816 q^{32} -4.18479 q^{34} -3.87939 q^{35} +1.22668 q^{37} +0.879385 q^{38} -2.49525 q^{40} +12.1557 q^{41} +2.55438 q^{43} +1.27807 q^{44} -6.04963 q^{46} -10.4338 q^{47} +12.4611 q^{49} -3.71688 q^{50} -0.0513845 q^{52} -2.83750 q^{53} -0.916222 q^{55} +12.5175 q^{56} -3.41147 q^{58} -6.16250 q^{59} -5.95811 q^{61} +6.84255 q^{62} +5.04189 q^{64} +0.0368366 q^{65} -2.59627 q^{67} +5.83750 q^{68} -3.41147 q^{70} -6.71688 q^{71} +8.19253 q^{73} +1.07873 q^{74} -1.22668 q^{76} +4.59627 q^{77} -2.40373 q^{79} -0.0368366 q^{80} +10.6895 q^{82} -12.1925 q^{83} -4.18479 q^{85} +2.24628 q^{86} +2.95636 q^{88} +8.80066 q^{89} -0.184793 q^{91} +8.43882 q^{92} -9.17530 q^{94} +0.879385 q^{95} -1.14290 q^{97} +10.9581 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 9 q^{10} - 3 q^{13} - 6 q^{14} + 3 q^{16} - 3 q^{17} + 3 q^{19} - 12 q^{20} - 9 q^{22} - 15 q^{23} - 6 q^{25} + 12 q^{26} + 6 q^{28} - 6 q^{29} + 6 q^{31}+ \cdots + 36 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 6 * q^8 + 9 * q^10 - 3 * q^13 - 6 * q^14 + 3 * q^16 - 3 * q^17 + 3 * q^19 - 12 * q^20 - 9 * q^22 - 15 * q^23 - 6 * q^25 + 12 * q^26 + 6 * q^28 - 6 * q^29 + 6 * q^31 - 9 * q^34 - 6 * q^35 - 3 * q^37 - 3 * q^38 + 9 * q^40 - 3 * q^41 - 3 * q^43 + 27 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 - 30 * q^52 - 6 * q^53 - 9 * q^55 + 15 * q^56 - 21 * q^59 - 21 * q^61 + 3 * q^62 + 12 * q^64 + 12 * q^65 + 6 * q^67 + 15 * q^68 - 12 * q^71 - 3 * q^73 + 12 * q^74 + 3 * q^76 - 21 * q^79 - 12 * q^80 + 45 * q^82 - 9 * q^83 - 9 * q^85 + 3 * q^86 - 45 * q^88 + 12 * q^89 + 3 * q^91 - 6 * q^92 + 9 * q^94 - 3 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

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