LMFDB - Embedded newform 513.2.a.f.1.1

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [3,1,0,3] 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:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*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			1.1
Root			$-1.69963$ 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)

$f(q)$	$=$	\(q-1.69963 q^{2} +0.888736 q^{4} +3.69963 q^{5} -1.58836 q^{7} +1.88874 q^{8} -6.28799 q^{10} +4.58836 q^{11} +1.58836 q^{13} +2.69963 q^{14} -4.98762 q^{16} -5.57598 q^{17} -1.00000 q^{19} +3.28799 q^{20} -7.79851 q^{22} +9.09888 q^{23} +8.68725 q^{25} -2.69963 q^{26} -1.41164 q^{28} -7.65383 q^{29} -2.58836 q^{31} +4.69963 q^{32} +9.47710 q^{34} -5.87636 q^{35} +0.287992 q^{37} +1.69963 q^{38} +6.98762 q^{40} +3.69963 q^{41} +9.32141 q^{43} +4.07784 q^{44} -15.4647 q^{46} +9.00000 q^{47} -4.47710 q^{49} -14.7651 q^{50} +1.41164 q^{52} +8.24219 q^{53} +16.9752 q^{55} -3.00000 q^{56} +13.0087 q^{58} +7.46472 q^{59} +0.189108 q^{61} +4.39926 q^{62} +1.98762 q^{64} +5.87636 q^{65} +8.08650 q^{67} -4.95558 q^{68} +9.98762 q^{70} -6.21015 q^{71} -1.22253 q^{73} -0.489480 q^{74} -0.888736 q^{76} -7.28799 q^{77} -5.90978 q^{79} -18.4523 q^{80} -6.28799 q^{82} -2.84431 q^{83} -20.6291 q^{85} -15.8429 q^{86} +8.66621 q^{88} +1.03342 q^{89} -2.52290 q^{91} +8.08650 q^{92} -15.2967 q^{94} -3.69963 q^{95} -15.4647 q^{97} +7.60940 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} - 7 q^{10} + 8 q^{11} - q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 10 q^{28} - 6 q^{29}+ \cdots - 8 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 + 5 * q^5 + q^7 + 6 * q^8 - 7 * q^10 + 8 * q^11 - q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - 3 * q^19 - 2 * q^20 + q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 10 * q^28 - 6 * q^29 - 2 * q^31 + 8 * q^32 + 23 * q^34 - 11 * q^37 - q^38 + 3 * q^40 + 5 * q^41 + 9 * q^43 + 19 * q^44 - 23 * q^46 + 27 * q^47 - 8 * q^49 - 27 * q^50 + 10 * q^52 + 2 * q^53 + 15 * q^55 - 9 * q^56 - 4 * q^58 - q^59 + 7 * q^61 + q^62 - 12 * q^64 - 12 * q^67 - q^68 + 12 * q^70 - 3 * q^73 - 14 * q^74 - 3 * q^76 - 10 * q^77 + 7 * q^79 - 14 * q^80 - 7 * q^82 + 5 * q^83 - 9 * q^85 - 37 * q^86 + 27 * q^88 - 4 * q^89 - 13 * q^91 - 12 * q^92 + 9 * q^94 - 5 * q^95 - 23 * q^97 - 8 * 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 $

$2$	−1.69963	−1.20182	−0.600909	−	0.799317i	$-0.705195\pi$
$2$	−1.69963	−1.20182	−0.600909	+	0.799317i	$0.705195\pi$
$3$	0	0
$3$	0	0
$4$	0.888736	0.444368
$5$	3.69963	1.65452	0.827262	−	0.561816i	$-0.189897\pi$
$5$	3.69963	1.65452	0.827262	+	0.561816i	$0.189897\pi$
$6$	0	0
$7$	−1.58836	−0.600345	−0.300173	−	0.953885i	$-0.597044\pi$
$7$	−1.58836	−0.600345	−0.300173	+	0.953885i	$0.597044\pi$
$8$	1.88874	0.667769
$9$	0	0
$10$	−6.28799	−1.98844
$11$	4.58836	1.38344	0.691722	−	0.722164i	$-0.256852\pi$
$11$	4.58836	1.38344	0.691722	+	0.722164i	$0.256852\pi$
$12$	0	0
$13$	1.58836	0.440533	0.220266	−	0.975440i	$-0.429307\pi$
$13$	1.58836	0.440533	0.220266	+	0.975440i	$0.429307\pi$
$14$	2.69963	0.721506
$15$	0	0
$16$	−4.98762	−1.24691
$17$	−5.57598	−1.35237	−0.676187	−	0.736730i	$-0.736369\pi$
$17$	−5.57598	−1.35237	−0.676187	+	0.736730i	$0.736369\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.28799	0.735217
$21$	0	0
$22$	−7.79851	−1.66265
$23$	9.09888	1.89725	0.948624	−	0.316405i	$-0.102476\pi$
$23$	9.09888	1.89725	0.948624	+	0.316405i	$0.102476\pi$
$24$	0	0
$25$	8.68725	1.73745
$26$	−2.69963	−0.529441
$27$	0	0
$28$	−1.41164	−0.266774
$29$	−7.65383	−1.42128	−0.710640	−	0.703556i	$-0.751595\pi$
$29$	−7.65383	−1.42128	−0.710640	+	0.703556i	$0.751595\pi$
$30$	0	0
$31$	−2.58836	−0.464884	−0.232442	−	0.972610i	$-0.574672\pi$
$31$	−2.58836	−0.464884	−0.232442	+	0.972610i	$0.574672\pi$
$32$	4.69963	0.830785
$33$	0	0
$34$	9.47710	1.62531
$35$	−5.87636	−0.993285
$36$	0	0
$37$	0.287992	0.0473456	0.0236728	−	0.999720i	$-0.492464\pi$
$37$	0.287992	0.0473456	0.0236728	+	0.999720i	$0.492464\pi$
$38$	1.69963	0.275716
$39$	0	0
$40$	6.98762	1.10484
$41$	3.69963	0.577785	0.288892	−	0.957362i	$-0.406713\pi$
$41$	3.69963	0.577785	0.288892	+	0.957362i	$0.406713\pi$
$42$	0	0
$43$	9.32141	1.42150	0.710751	−	0.703444i	$-0.248355\pi$
$43$	9.32141	1.42150	0.710751	+	0.703444i	$0.248355\pi$
$44$	4.07784	0.614758
$45$	0	0
$46$	−15.4647	−2.28015
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	−4.47710	−0.639586
$50$	−14.7651	−2.08810
$51$	0	0
$52$	1.41164	0.195759
$53$	8.24219	1.13215	0.566076	−	0.824353i	$-0.308461\pi$
$53$	8.24219	1.13215	0.566076	+	0.824353i	$0.308461\pi$
$54$	0	0
$55$	16.9752	2.28894
$56$	−3.00000	−0.400892
$57$	0	0
$58$	13.0087	1.70812
$59$	7.46472	0.971824	0.485912	−	0.874008i	$-0.338488\pi$
$59$	7.46472	0.971824	0.485912	+	0.874008i	$0.338488\pi$
$60$	0	0
$61$	0.189108	0.0242128	0.0121064	−	0.999927i	$-0.496146\pi$
$61$	0.189108	0.0242128	0.0121064	+	0.999927i	$0.496146\pi$
$62$	4.39926	0.558706
$63$	0	0
$64$	1.98762	0.248453
$65$	5.87636	0.728872
$66$	0	0
$67$	8.08650	0.987924	0.493962	−	0.869484i	$-0.335548\pi$
$67$	8.08650	0.987924	0.493962	+	0.869484i	$0.335548\pi$
$68$	−4.95558	−0.600952
$69$	0	0
$70$	9.98762	1.19375
$71$	−6.21015	−0.737009	−0.368505	−	0.929626i	$-0.620130\pi$
$71$	−6.21015	−0.737009	−0.368505	+	0.929626i	$0.620130\pi$
$72$	0	0
$73$	−1.22253	−0.143086	−0.0715431	−	0.997438i	$-0.522792\pi$
$73$	−1.22253	−0.143086	−0.0715431	+	0.997438i	$0.522792\pi$
$74$	−0.489480	−0.0569008
$75$	0	0
$76$	−0.888736	−0.101945
$77$	−7.28799	−0.830544
$78$	0	0
$79$	−5.90978	−0.664902	−0.332451	−	0.943121i	$-0.607876\pi$
$79$	−5.90978	−0.664902	−0.332451	+	0.943121i	$0.607876\pi$
$80$	−18.4523	−2.06303
$81$	0	0
$82$	−6.28799	−0.694393
$83$	−2.84431	−0.312204	−0.156102	−	0.987741i	$-0.549893\pi$
$83$	−2.84431	−0.312204	−0.156102	+	0.987741i	$0.549893\pi$
$84$	0	0
$85$	−20.6291	−2.23754
$86$	−15.8429	−1.70839
$87$	0	0
$88$	8.66621	0.923821
$89$	1.03342	0.109542	0.0547712	−	0.998499i	$-0.482557\pi$
$89$	1.03342	0.109542	0.0547712	+	0.998499i	$0.482557\pi$
$90$	0	0
$91$	−2.52290	−0.264472
$92$	8.08650	0.843076
$93$	0	0
$94$	−15.2967	−1.57773
$95$	−3.69963	−0.379574
$96$	0	0
$97$	−15.4647	−1.57020	−0.785102	−	0.619366i	$-0.787390\pi$
$97$	−15.4647	−1.57020	−0.785102	+	0.619366i	$0.787390\pi$
$98$	7.60940	0.768666
$99$	0	0
$100$	7.72067	0.772067
$101$	7.36584	0.732928	0.366464	−	0.930432i	$-0.380568\pi$
$101$	7.36584	0.732928	0.366464	+	0.930432i	$0.380568\pi$
$102$	0	0
$103$	−5.74543	−0.566114	−0.283057	−	0.959103i	$-0.591349\pi$
$103$	−5.74543	−0.566114	−0.283057	+	0.959103i	$0.591349\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−14.0087	−1.36064
$107$	0.699628	0.0676356	0.0338178	−	0.999428i	$-0.489233\pi$
$107$	0.699628	0.0676356	0.0338178	+	0.999428i	$0.489233\pi$
$108$	0	0
$109$	−8.49814	−0.813974	−0.406987	−	0.913434i	$-0.633421\pi$
$109$	−8.49814	−0.813974	−0.406987	+	0.913434i	$0.633421\pi$
$110$	−28.8516	−2.75089
$111$	0	0
$112$	7.92216	0.748573
$113$	−10.4647	−0.984438	−0.492219	−	0.870471i	$-0.663814\pi$
$113$	−10.4647	−0.984438	−0.492219	+	0.870471i	$0.663814\pi$
$114$	0	0
$115$	33.6625	3.13904
$116$	−6.80223	−0.631571
$117$	0	0
$118$	−12.6872	−1.16796
$119$	8.85669	0.811892
$120$	0	0
$121$	10.0531	0.913917
$122$	−0.321413	−0.0290993
$123$	0	0
$124$	−2.30037	−0.206579
$125$	13.6414	1.22013
$126$	0	0
$127$	−10.7527	−0.954149	−0.477074	−	0.878863i	$-0.658303\pi$
$127$	−10.7527	−0.954149	−0.477074	+	0.878863i	$0.658303\pi$
$128$	−12.7775	−1.12938
$129$	0	0
$130$	−9.98762	−0.875972
$131$	16.3090	1.42493	0.712463	−	0.701709i	$-0.247579\pi$
$131$	16.3090	1.42493	0.712463	+	0.701709i	$0.247579\pi$
$132$	0	0
$133$	1.58836	0.137729
$134$	−13.7441	−1.18731
$135$	0	0
$136$	−10.5316	−0.903074
$137$	−15.2101	−1.29949	−0.649745	−	0.760152i	$-0.725124\pi$
$137$	−15.2101	−1.29949	−0.649745	+	0.760152i	$0.725124\pi$
$138$	0	0
$139$	2.62178	0.222377	0.111188	−	0.993799i	$-0.464534\pi$
$139$	2.62178	0.222377	0.111188	+	0.993799i	$0.464534\pi$
$140$	−5.22253	−0.441384
$141$	0	0
$142$	10.5549	0.885751
$143$	7.28799	0.609453
$144$	0	0
$145$	−28.3163	−2.35154
$146$	2.07784	0.171964
$147$	0	0
$148$	0.255949	0.0210389
$149$	−16.7651	−1.37345	−0.686725	−	0.726917i	$-0.740952\pi$
$149$	−16.7651	−1.37345	−0.686725	+	0.726917i	$0.740952\pi$
$150$	0	0
$151$	−15.8182	−1.28726	−0.643632	−	0.765335i	$-0.722573\pi$
$151$	−15.8182	−1.28726	−0.643632	+	0.765335i	$0.722573\pi$
$152$	−1.88874	−0.153197
$153$	0	0
$154$	12.3869	0.998163
$155$	−9.57598	−0.769162
$156$	0	0
$157$	14.9084	1.18982	0.594910	−	0.803792i	$-0.297188\pi$
$157$	14.9084	1.18982	0.594910	+	0.803792i	$0.297188\pi$
$158$	10.0444	0.799091
$159$	0	0
$160$	17.3869	1.37455
$161$	−14.4523	−1.13900
$162$	0	0
$163$	11.3004	0.885113	0.442557	−	0.896741i	$-0.354072\pi$
$163$	11.3004	0.885113	0.442557	+	0.896741i	$0.354072\pi$
$164$	3.28799	0.256749
$165$	0	0
$166$	4.83427	0.375212
$167$	19.4647	1.50623	0.753113	−	0.657892i	$-0.228552\pi$
$167$	19.4647	1.50623	0.753113	+	0.657892i	$0.228552\pi$
$168$	0	0
$169$	−10.4771	−0.805931
$170$	35.0617	2.68911
$171$	0	0
$172$	8.28427	0.631670
$173$	−15.9098	−1.20960	−0.604799	−	0.796378i	$-0.706747\pi$
$173$	−15.9098	−1.20960	−0.604799	+	0.796378i	$0.706747\pi$
$174$	0	0
$175$	−13.7985	−1.04307
$176$	−22.8850	−1.72502
$177$	0	0
$178$	−1.75643	−0.131650
$179$	−12.7861	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$179$	−12.7861	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$180$	0	0
$181$	19.7861	1.47069	0.735346	−	0.677692i	$-0.237020\pi$
$181$	19.7861	1.47069	0.735346	+	0.677692i	$0.237020\pi$
$182$	4.28799	0.317847
$183$	0	0
$184$	17.1854	1.26692
$185$	1.06546	0.0783345
$186$	0	0
$187$	−25.5846	−1.87093
$188$	7.99862	0.583360
$189$	0	0
$190$	6.28799	0.456179
$191$	−8.43268	−0.610167	−0.305084	−	0.952326i	$-0.598684\pi$
$191$	−8.43268	−0.610167	−0.305084	+	0.952326i	$0.598684\pi$
$192$	0	0
$193$	−6.76509	−0.486962	−0.243481	−	0.969906i	$-0.578289\pi$
$193$	−6.76509	−0.486962	−0.243481	+	0.969906i	$0.578289\pi$
$194$	26.2843	1.88710
$195$	0	0
$196$	−3.97896	−0.284211
$197$	20.2509	1.44281	0.721407	−	0.692512i	$-0.243496\pi$
$197$	20.2509	1.44281	0.721407	+	0.692512i	$0.243496\pi$
$198$	0	0
$199$	0.0320432	0.00227148	0.00113574	−	0.999999i	$-0.499638\pi$
$199$	0.0320432	0.00227148	0.00113574	+	0.999999i	$0.499638\pi$
$200$	16.4079	1.16021
$201$	0	0
$202$	−12.5192	−0.880847
$203$	12.1571	0.853259
$204$	0	0
$205$	13.6872	0.955959
$206$	9.76509	0.680366
$207$	0	0
$208$	−7.92216	−0.549303
$209$	−4.58836	−0.317384
$210$	0	0
$211$	−5.35483	−0.368642	−0.184321	−	0.982866i	$-0.559009\pi$
$211$	−5.35483	−0.368642	−0.184321	+	0.982866i	$0.559009\pi$
$212$	7.32513	0.503092
$213$	0	0
$214$	−1.18911	−0.0812857
$215$	34.4858	2.35191
$216$	0	0
$217$	4.11126	0.279091
$218$	14.4437	0.978249
$219$	0	0
$220$	15.0865	1.01713
$221$	−8.85669	−0.595766
$222$	0	0
$223$	−7.84294	−0.525202	−0.262601	−	0.964905i	$-0.584580\pi$
$223$	−7.84294	−0.525202	−0.262601	+	0.964905i	$0.584580\pi$
$224$	−7.46472	−0.498758
$225$	0	0
$226$	17.7861	1.18312
$227$	8.55632	0.567903	0.283951	−	0.958839i	$-0.408355\pi$
$227$	8.55632	0.567903	0.283951	+	0.958839i	$0.408355\pi$
$228$	0	0
$229$	−5.03204	−0.332527	−0.166263	−	0.986081i	$-0.553170\pi$
$229$	−5.03204	−0.332527	−0.166263	+	0.986081i	$0.553170\pi$
$230$	−57.2137	−3.77256
$231$	0	0
$232$	−14.4561	−0.949087
$233$	15.2559	0.999450	0.499725	−	0.866184i	$-0.333434\pi$
$233$	15.2559	0.999450	0.499725	+	0.866184i	$0.333434\pi$
$234$	0	0
$235$	33.2967	2.17203
$236$	6.63416	0.431847
$237$	0	0
$238$	−15.0531	−0.975747
$239$	−24.6167	−1.59232	−0.796161	−	0.605085i	$-0.793139\pi$
$239$	−24.6167	−1.59232	−0.796161	+	0.605085i	$0.793139\pi$
$240$	0	0
$241$	−11.5229	−0.742255	−0.371128	−	0.928582i	$-0.621029\pi$
$241$	−11.5229	−0.742255	−0.371128	+	0.928582i	$0.621029\pi$
$242$	−17.0865	−1.09836
$243$	0	0
$244$	0.168067	0.0107594
$245$	−16.5636	−1.05821
$246$	0	0
$247$	−1.58836	−0.101065
$248$	−4.88874	−0.310435
$249$	0	0
$250$	−23.1854	−1.46637
$251$	−26.6094	−1.67957	−0.839785	−	0.542919i	$-0.817319\pi$
$251$	−26.6094	−1.67957	−0.839785	+	0.542919i	$0.817319\pi$
$252$	0	0
$253$	41.7490	2.62474
$254$	18.2756	1.14671
$255$	0	0
$256$	17.7417	1.10886
$257$	26.6167	1.66030	0.830152	−	0.557538i	$-0.188254\pi$
$257$	26.6167	1.66030	0.830152	+	0.557538i	$0.188254\pi$
$258$	0	0
$259$	−0.457436	−0.0284237
$260$	5.22253	0.323887
$261$	0	0
$262$	−27.7193	−1.71250
$263$	−5.45234	−0.336206	−0.168103	−	0.985769i	$-0.553764\pi$
$263$	−5.45234	−0.336206	−0.168103	+	0.985769i	$0.553764\pi$
$264$	0	0
$265$	30.4930	1.87317
$266$	−2.69963	−0.165525
$267$	0	0
$268$	7.18677	0.439002
$269$	19.2312	1.17255	0.586273	−	0.810113i	$-0.300595\pi$
$269$	19.2312	1.17255	0.586273	+	0.810113i	$0.300595\pi$
$270$	0	0
$271$	−24.1730	−1.46841	−0.734203	−	0.678930i	$-0.762444\pi$
$271$	−24.1730	−1.46841	−0.734203	+	0.678930i	$0.762444\pi$
$272$	27.8109	1.68628
$273$	0	0
$274$	25.8516	1.56175
$275$	39.8603	2.40366
$276$	0	0
$277$	−0.00137742	−8.27610e−5 0	−4.13805e−5	−	1.00000i	$-0.500013\pi$
$277$	−0.00137742	−8.27610e−5 0	−4.13805e−5		1.00000i	$0.500013\pi$
$278$	−4.45606	−0.267257
$279$	0	0
$280$	−11.0989	−0.663285
$281$	−7.24729	−0.432337	−0.216168	−	0.976356i	$-0.569356\pi$
$281$	−7.24729	−0.432337	−0.216168	+	0.976356i	$0.569356\pi$
$282$	0	0
$283$	5.37450	0.319481	0.159740	−	0.987159i	$-0.448934\pi$
$283$	5.37450	0.319481	0.159740	+	0.987159i	$0.448934\pi$
$284$	−5.51918	−0.327503
$285$	0	0
$286$	−12.3869	−0.732451
$287$	−5.87636	−0.346870
$288$	0	0
$289$	14.0916	0.828918
$290$	48.1272	2.82613
$291$	0	0
$292$	−1.08650	−0.0635829
$293$	−7.13602	−0.416891	−0.208445	−	0.978034i	$-0.566840\pi$
$293$	−7.13602	−0.416891	−0.208445	+	0.978034i	$0.566840\pi$
$294$	0	0
$295$	27.6167	1.60791
$296$	0.543941	0.0316159
$297$	0	0
$298$	28.4944	1.65064
$299$	14.4523	0.835800
$300$	0	0
$301$	−14.8058	−0.853392
$302$	26.8850	1.54706
$303$	0	0
$304$	4.98762	0.286060
$305$	0.699628	0.0400606
$306$	0	0
$307$	−10.7280	−0.612277	−0.306138	−	0.951987i	$-0.599037\pi$
$307$	−10.7280	−0.612277	−0.306138	+	0.951987i	$0.599037\pi$
$308$	−6.47710	−0.369067
$309$	0	0
$310$	16.2756	0.924393
$311$	−8.83056	−0.500735	−0.250367	−	0.968151i	$-0.580551\pi$
$311$	−8.83056	−0.500735	−0.250367	+	0.968151i	$0.580551\pi$
$312$	0	0
$313$	−12.8196	−0.724604	−0.362302	−	0.932061i	$-0.618009\pi$
$313$	−12.8196	−0.724604	−0.362302	+	0.932061i	$0.618009\pi$
$314$	−25.3387	−1.42995
$315$	0	0
$316$	−5.25223	−0.295461
$317$	−13.5512	−0.761113	−0.380556	−	0.924758i	$-0.624267\pi$
$317$	−13.5512	−0.761113	−0.380556	+	0.924758i	$0.624267\pi$
$318$	0	0
$319$	−35.1185	−1.96626
$320$	7.35346	0.411071
$321$	0	0
$322$	24.5636	1.36888
$323$	5.57598	0.310256
$324$	0	0
$325$	13.7985	0.765404
$326$	−19.2064	−1.06375
$327$	0	0
$328$	6.98762	0.385827
$329$	−14.2953	−0.788124
$330$	0	0
$331$	2.94692	0.161977	0.0809886	−	0.996715i	$-0.474192\pi$
$331$	2.94692	0.161977	0.0809886	+	0.996715i	$0.474192\pi$
$332$	−2.52784	−0.138733
$333$	0	0
$334$	−33.0828	−1.81021
$335$	29.9171	1.63454
$336$	0	0
$337$	−29.0173	−1.58067	−0.790337	−	0.612672i	$-0.790095\pi$
$337$	−29.0173	−1.58067	−0.790337	+	0.612672i	$0.790095\pi$
$338$	17.8072	0.968583
$339$	0	0
$340$	−18.3338	−0.994289
$341$	−11.8764	−0.643141
$342$	0	0
$343$	18.2298	0.984317
$344$	17.6057	0.949235
$345$	0	0
$346$	27.0407	1.45372
$347$	−0.119925	−0.00643793	−0.00321897	−	0.999995i	$-0.501025\pi$
$347$	−0.119925	−0.00643793	−0.00321897	+	0.999995i	$0.501025\pi$
$348$	0	0
$349$	−11.0902	−0.593646	−0.296823	−	0.954933i	$-0.595927\pi$
$349$	−11.0902	−0.593646	−0.296823	+	0.954933i	$0.595927\pi$
$350$	23.4523	1.25358
$351$	0	0
$352$	21.5636	1.14934
$353$	−12.6291	−0.672177	−0.336089	−	0.941830i	$-0.609104\pi$
$353$	−12.6291	−0.672177	−0.336089	+	0.941830i	$0.609104\pi$
$354$	0	0
$355$	−22.9752	−1.21940
$356$	0.918438	0.0486771
$357$	0	0
$358$	21.7317	1.14855
$359$	−16.8726	−0.890504	−0.445252	−	0.895405i	$-0.646886\pi$
$359$	−16.8726	−0.890504	−0.445252	+	0.895405i	$0.646886\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−33.6291	−1.76750
$363$	0	0
$364$	−2.24219	−0.117523
$365$	−4.52290	−0.236739
$366$	0	0
$367$	−29.8319	−1.55721	−0.778607	−	0.627512i	$-0.784073\pi$
$367$	−29.8319	−1.55721	−0.778607	+	0.627512i	$0.784073\pi$
$368$	−45.3818	−2.36569
$369$	0	0
$370$	−1.81089	−0.0941438
$371$	−13.0916	−0.679682
$372$	0	0
$373$	10.6625	0.552083	0.276041	−	0.961146i	$-0.410977\pi$
$373$	10.6625	0.552083	0.276041	+	0.961146i	$0.410977\pi$
$374$	43.4844	2.24852
$375$	0	0
$376$	16.9986	0.876637
$377$	−12.1571	−0.626121
$378$	0	0
$379$	−4.25457	−0.218543	−0.109271	−	0.994012i	$-0.534852\pi$
$379$	−4.25457	−0.218543	−0.109271	+	0.994012i	$0.534852\pi$
$380$	−3.28799	−0.168670
$381$	0	0
$382$	14.3324	0.733310
$383$	10.8233	0.553043	0.276522	−	0.961008i	$-0.410818\pi$
$383$	10.8233	0.553043	0.276522	+	0.961008i	$0.410818\pi$
$384$	0	0
$385$	−26.9629	−1.37415
$386$	11.4981	0.585240
$387$	0	0
$388$	−13.7441	−0.697748
$389$	33.2595	1.68632	0.843162	−	0.537659i	$-0.180691\pi$
$389$	33.2595	1.68632	0.843162	+	0.537659i	$0.180691\pi$
$390$	0	0
$391$	−50.7352	−2.56579
$392$	−8.45606	−0.427095
$393$	0	0
$394$	−34.4189	−1.73400
$395$	−21.8640	−1.10010
$396$	0	0
$397$	−31.1024	−1.56099	−0.780494	−	0.625164i	$-0.785032\pi$
$397$	−31.1024	−1.56099	−0.780494	+	0.625164i	$0.785032\pi$
$398$	−0.0544615	−0.00272991
$399$	0	0
$400$	−43.3287	−2.16643
$401$	−25.0334	−1.25011	−0.625055	−	0.780581i	$-0.714923\pi$
$401$	−25.0334	−1.25011	−0.625055	+	0.780581i	$0.714923\pi$
$402$	0	0
$403$	−4.11126	−0.204797
$404$	6.54628	0.325690
$405$	0	0
$406$	−20.6625	−1.02546
$407$	1.32141	0.0655000
$408$	0	0
$409$	38.7417	1.91565	0.957827	−	0.287345i	$-0.0927726\pi$
$409$	38.7417	1.91565	0.957827	+	0.287345i	$0.0927726\pi$
$410$	−23.2632	−1.14889
$411$	0	0
$412$	−5.10617	−0.251563
$413$	−11.8567	−0.583430
$414$	0	0
$415$	−10.5229	−0.516549
$416$	7.46472	0.365988
$417$	0	0
$418$	7.79851	0.381438
$419$	4.41892	0.215878	0.107939	−	0.994157i	$-0.465575\pi$
$419$	4.41892	0.215878	0.107939	+	0.994157i	$0.465575\pi$
$420$	0	0
$421$	−13.4574	−0.655875	−0.327938	−	0.944699i	$-0.606354\pi$
$421$	−13.4574	−0.655875	−0.327938	+	0.944699i	$0.606354\pi$
$422$	9.10123	0.443041
$423$	0	0
$424$	15.5673	0.756016
$425$	−48.4400	−2.34968
$426$	0	0
$427$	−0.300372	−0.0145360
$428$	0.621785	0.0300551
$429$	0	0
$430$	−58.6130	−2.82657
$431$	34.6945	1.67118	0.835588	−	0.549356i	$-0.185127\pi$
$431$	34.6945	1.67118	0.835588	+	0.549356i	$0.185127\pi$
$432$	0	0
$433$	23.7848	1.14302	0.571511	−	0.820594i	$-0.306357\pi$
$433$	23.7848	1.14302	0.571511	+	0.820594i	$0.306357\pi$
$434$	−6.98762	−0.335417
$435$	0	0
$436$	−7.55260	−0.361704
$437$	−9.09888	−0.435259
$438$	0	0
$439$	1.95420	0.0932689	0.0466344	−	0.998912i	$-0.485150\pi$
$439$	1.95420	0.0932689	0.0466344	+	0.998912i	$0.485150\pi$
$440$	32.0617	1.52848
$441$	0	0
$442$	15.0531	0.716002
$443$	−19.9281	−0.946811	−0.473405	−	0.880845i	$-0.656975\pi$
$443$	−19.9281	−0.946811	−0.473405	+	0.880845i	$0.656975\pi$
$444$	0	0
$445$	3.82327	0.181240
$446$	13.3301	0.631197
$447$	0	0
$448$	−3.15706	−0.149157
$449$	−40.2522	−1.89962	−0.949810	−	0.312827i	$-0.898724\pi$
$449$	−40.2522	−1.89962	−0.949810	+	0.312827i	$0.898724\pi$
$450$	0	0
$451$	16.9752	0.799333
$452$	−9.30037	−0.437453
$453$	0	0
$454$	−14.5426	−0.682516
$455$	−9.33379	−0.437575
$456$	0	0
$457$	24.0159	1.12342	0.561709	−	0.827335i	$-0.310144\pi$
$457$	24.0159	1.12342	0.561709	+	0.827335i	$0.310144\pi$
$458$	8.55260	0.399637
$459$	0	0
$460$	29.9171	1.39489
$461$	15.9876	0.744618	0.372309	−	0.928109i	$-0.378566\pi$
$461$	15.9876	0.744618	0.372309	+	0.928109i	$0.378566\pi$
$462$	0	0
$463$	13.0617	0.607031	0.303515	−	0.952827i	$-0.401840\pi$
$463$	13.0617	0.607031	0.303515	+	0.952827i	$0.401840\pi$
$464$	38.1744	1.77220
$465$	0	0
$466$	−25.9294	−1.20116
$467$	1.40063	0.0648136	0.0324068	−	0.999475i	$-0.489683\pi$
$467$	1.40063	0.0648136	0.0324068	+	0.999475i	$0.489683\pi$
$468$	0	0
$469$	−12.8443	−0.593095
$470$	−56.5919	−2.61039
$471$	0	0
$472$	14.0989	0.648954
$473$	42.7700	1.96657
$474$	0	0
$475$	−8.68725	−0.398598
$476$	7.87126	0.360779
$477$	0	0
$478$	41.8392	1.91368
$479$	16.3287	0.746077	0.373039	−	0.927816i	$-0.378316\pi$
$479$	16.3287	0.746077	0.373039	+	0.927816i	$0.378316\pi$
$480$	0	0
$481$	0.457436	0.0208573
$482$	19.5846	0.892056
$483$	0	0
$484$	8.93454	0.406115
$485$	−57.2137	−2.59794
$486$	0	0
$487$	35.3832	1.60336	0.801682	−	0.597751i	$-0.203939\pi$
$487$	35.3832	1.60336	0.801682	+	0.597751i	$0.203939\pi$
$488$	0.357174	0.0161685
$489$	0	0
$490$	28.1520	1.27178
$491$	−7.64654	−0.345084	−0.172542	−	0.985002i	$-0.555198\pi$
$491$	−7.64654	−0.345084	−0.172542	+	0.985002i	$0.555198\pi$
$492$	0	0
$493$	42.6776	1.92210
$494$	2.69963	0.121462
$495$	0	0
$496$	12.9098	0.579666
$497$	9.86398	0.442460
$498$	0	0
$499$	−12.1447	−0.543671	−0.271835	−	0.962344i	$-0.587631\pi$
$499$	−12.1447	−0.543671	−0.271835	+	0.962344i	$0.587631\pi$
$500$	12.1236	0.542186
$501$	0	0
$502$	45.2261	2.01854
$503$	−27.2261	−1.21395	−0.606976	−	0.794720i	$-0.707618\pi$
$503$	−27.2261	−1.21395	−0.606976	+	0.794720i	$0.707618\pi$
$504$	0	0
$505$	27.2509	1.21265
$506$	−70.9578	−3.15446
$507$	0	0
$508$	−9.55632	−0.423993
$509$	−38.5141	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$509$	−38.5141	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$510$	0	0
$511$	1.94182	0.0859011
$512$	−4.59937	−0.203265
$513$	0	0
$514$	−45.2385	−1.99538
$515$	−21.2559	−0.936649
$516$	0	0
$517$	41.2953	1.81616
$518$	0.777472	0.0341601
$519$	0	0
$520$	11.0989	0.486718
$521$	−22.3535	−0.979323	−0.489661	−	0.871913i	$-0.662880\pi$
$521$	−22.3535	−0.979323	−0.489661	+	0.871913i	$0.662880\pi$
$522$	0	0
$523$	−6.12227	−0.267708	−0.133854	−	0.991001i	$-0.542735\pi$
$523$	−6.12227	−0.267708	−0.133854	+	0.991001i	$0.542735\pi$
$524$	14.4944	0.633192
$525$	0	0
$526$	9.26695	0.404058
$527$	14.4327	0.628697
$528$	0	0
$529$	59.7897	2.59955
$530$	−51.8268	−2.25121
$531$	0	0
$532$	1.41164	0.0612022
$533$	5.87636	0.254533
$534$	0	0
$535$	2.58836	0.111905
$536$	15.2733	0.659705
$537$	0	0
$538$	−32.6859	−1.40919
$539$	−20.5426	−0.884831
$540$	0	0
$541$	−23.0073	−0.989160	−0.494580	−	0.869132i	$-0.664678\pi$
$541$	−23.0073	−0.989160	−0.494580	+	0.869132i	$0.664678\pi$
$542$	41.0851	1.76476
$543$	0	0
$544$	−26.2051	−1.12353
$545$	−31.4400	−1.34674
$546$	0	0
$547$	−13.1716	−0.563178	−0.281589	−	0.959535i	$-0.590862\pi$
$547$	−13.1716	−0.563178	−0.281589	+	0.959535i	$0.590862\pi$
$548$	−13.5178	−0.577452
$549$	0	0
$550$	−67.7476	−2.88877
$551$	7.65383	0.326064
$552$	0	0
$553$	9.38688	0.399171
$554$	0.00234110	9.94638e−5 0
$555$	0	0
$556$	2.33007	0.0988171
$557$	−1.18773	−0.0503257	−0.0251629	−	0.999683i	$-0.508010\pi$
$557$	−1.18773	−0.0503257	−0.0251629	+	0.999683i	$0.508010\pi$
$558$	0	0
$559$	14.8058	0.626218
$560$	29.3090	1.23853
$561$	0	0
$562$	12.3177	0.519591
$563$	−43.2929	−1.82458	−0.912290	−	0.409545i	$-0.865687\pi$
$563$	−43.2929	−1.82458	−0.912290	+	0.409545i	$0.865687\pi$
$564$	0	0
$565$	−38.7156	−1.62878
$566$	−9.13465	−0.383958
$567$	0	0
$568$	−11.7293	−0.492152
$569$	2.54256	0.106590	0.0532949	−	0.998579i	$-0.483028\pi$
$569$	2.54256	0.106590	0.0532949	+	0.998579i	$0.483028\pi$
$570$	0	0
$571$	28.3621	1.18692	0.593459	−	0.804864i	$-0.297762\pi$
$571$	28.3621	1.18692	0.593459	+	0.804864i	$0.297762\pi$
$572$	6.47710	0.270821
$573$	0	0
$574$	9.98762	0.416875
$575$	79.0443	3.29637
$576$	0	0
$577$	−1.67859	−0.0698805	−0.0349403	−	0.999389i	$-0.511124\pi$
$577$	−1.67859	−0.0698805	−0.0349403	+	0.999389i	$0.511124\pi$
$578$	−23.9505	−0.996209
$579$	0	0
$580$	−25.1657	−1.04495
$581$	4.51780	0.187430
$582$	0	0
$583$	37.8182	1.56627
$584$	−2.30903	−0.0955485
$585$	0	0
$586$	12.1286	0.501027
$587$	22.6662	0.935535	0.467767	−	0.883852i	$-0.345059\pi$
$587$	22.6662	0.935535	0.467767	+	0.883852i	$0.345059\pi$
$588$	0	0
$589$	2.58836	0.106652
$590$	−46.9381	−1.93241
$591$	0	0
$592$	−1.43640	−0.0590355
$593$	6.77747	0.278317	0.139159	−	0.990270i	$-0.455560\pi$
$593$	6.77747	0.278317	0.139159	+	0.990270i	$0.455560\pi$
$594$	0	0
$595$	32.7665	1.34329
$596$	−14.8997	−0.610317
$597$	0	0
$598$	−24.5636	−1.00448
$599$	4.30766	0.176006	0.0880030	−	0.996120i	$-0.471951\pi$
$599$	4.30766	0.176006	0.0880030	+	0.996120i	$0.471951\pi$
$600$	0	0
$601$	−15.5215	−0.633136	−0.316568	−	0.948570i	$-0.602531\pi$
$601$	−15.5215	−0.633136	−0.316568	+	0.948570i	$0.602531\pi$
$602$	25.1643	1.02562
$603$	0	0
$604$	−14.0582	−0.572019
$605$	37.1927	1.51210
$606$	0	0
$607$	24.7600	1.00498	0.502489	−	0.864584i	$-0.332418\pi$
$607$	24.7600	1.00498	0.502489	+	0.864584i	$0.332418\pi$
$608$	−4.69963	−0.190595
$609$	0	0
$610$	−1.18911	−0.0481456
$611$	14.2953	0.578325
$612$	0	0
$613$	−21.7775	−0.879584	−0.439792	−	0.898100i	$-0.644948\pi$
$613$	−21.7775	−0.879584	−0.439792	+	0.898100i	$0.644948\pi$
$614$	18.2335	0.735845
$615$	0	0
$616$	−13.7651	−0.554611
$617$	−0.452340	−0.0182105	−0.00910527	−	0.999959i	$-0.502898\pi$
$617$	−0.452340	−0.0182105	−0.00910527	+	0.999959i	$0.502898\pi$
$618$	0	0
$619$	25.1323	1.01015	0.505076	−	0.863075i	$-0.331464\pi$
$619$	25.1323	1.01015	0.505076	+	0.863075i	$0.331464\pi$
$620$	−8.51052	−0.341791
$621$	0	0
$622$	15.0087	0.601792
$623$	−1.64145	−0.0657632
$624$	0	0
$625$	7.03204	0.281282
$626$	21.7885	0.870843
$627$	0	0
$628$	13.2496	0.528718
$629$	−1.60584	−0.0640290
$630$	0	0
$631$	−6.26461	−0.249390	−0.124695	−	0.992195i	$-0.539795\pi$
$631$	−6.26461	−0.249390	−0.124695	+	0.992195i	$0.539795\pi$
$632$	−11.1620	−0.444001
$633$	0	0
$634$	23.0320	0.914719
$635$	−39.7810	−1.57866
$636$	0	0
$637$	−7.11126	−0.281759
$638$	59.6885	2.36309
$639$	0	0
$640$	−47.2719	−1.86859
$641$	19.4858	0.769641	0.384821	−	0.922991i	$-0.374263\pi$
$641$	19.4858	0.769641	0.384821	+	0.922991i	$0.374263\pi$
$642$	0	0
$643$	−26.8887	−1.06039	−0.530194	−	0.847876i	$-0.677881\pi$
$643$	−26.8887	−1.06039	−0.530194	+	0.847876i	$0.677881\pi$
$644$	−12.8443	−0.506137
$645$	0	0
$646$	−9.47710	−0.372872
$647$	43.7280	1.71912	0.859562	−	0.511032i	$-0.170737\pi$
$647$	43.7280	1.71912	0.859562	+	0.511032i	$0.170737\pi$
$648$	0	0
$649$	34.2509	1.34446
$650$	−23.4523	−0.919876
$651$	0	0
$652$	10.0430	0.393316
$653$	4.84294	0.189519	0.0947594	−	0.995500i	$-0.469792\pi$
$653$	4.84294	0.189519	0.0947594	+	0.995500i	$0.469792\pi$
$654$	0	0
$655$	60.3374	2.35758
$656$	−18.4523	−0.720443
$657$	0	0
$658$	24.2967	0.947182
$659$	26.5846	1.03559	0.517795	−	0.855504i	$-0.326753\pi$
$659$	26.5846	1.03559	0.517795	+	0.855504i	$0.326753\pi$
$660$	0	0
$661$	7.97524	0.310201	0.155100	−	0.987899i	$-0.450430\pi$
$661$	7.97524	0.310201	0.155100	+	0.987899i	$0.450430\pi$
$662$	−5.00866	−0.194667
$663$	0	0
$664$	−5.37216	−0.208480
$665$	5.87636	0.227875
$666$	0	0
$667$	−69.6413	−2.69652
$668$	17.2990	0.669318
$669$	0	0
$670$	−50.8479	−1.96442
$671$	0.867695	0.0334970
$672$	0	0
$673$	−32.7476	−1.26233	−0.631164	−	0.775649i	$-0.717422\pi$
$673$	−32.7476	−1.26233	−0.631164	+	0.775649i	$0.717422\pi$
$674$	49.3187	1.89968
$675$	0	0
$676$	−9.31137	−0.358130
$677$	36.8282	1.41542	0.707712	−	0.706501i	$-0.249728\pi$
$677$	36.8282	1.41542	0.707712	+	0.706501i	$0.249728\pi$
$678$	0	0
$679$	24.5636	0.942665
$680$	−38.9629	−1.49416
$681$	0	0
$682$	20.1854	0.772938
$683$	41.6784	1.59478	0.797390	−	0.603464i	$-0.206213\pi$
$683$	41.6784	1.59478	0.797390	+	0.603464i	$0.206213\pi$
$684$	0	0
$685$	−56.2719	−2.15004
$686$	−30.9839	−1.18297
$687$	0	0
$688$	−46.4917	−1.77248
$689$	13.0916	0.498750
$690$	0	0
$691$	18.3077	0.696456	0.348228	−	0.937410i	$-0.386783\pi$
$691$	18.3077	0.696456	0.348228	+	0.937410i	$0.386783\pi$
$692$	−14.1396	−0.537507
$693$	0	0
$694$	0.203829	0.00773722
$695$	9.69963	0.367928
$696$	0	0
$697$	−20.6291	−0.781382
$698$	18.8493	0.713455
$699$	0	0
$700$	−12.2632	−0.463507
$701$	−14.8589	−0.561212	−0.280606	−	0.959823i	$-0.590535\pi$
$701$	−14.8589	−0.561212	−0.280606	+	0.959823i	$0.590535\pi$
$702$	0	0
$703$	−0.287992	−0.0108618
$704$	9.11993	0.343720
$705$	0	0
$706$	21.4647	0.807835
$707$	−11.6996	−0.440010
$708$	0	0
$709$	−1.97168	−0.0740478	−0.0370239	−	0.999314i	$-0.511788\pi$
$709$	−1.97168	−0.0740478	−0.0370239	+	0.999314i	$0.511788\pi$
$710$	39.0494	1.46550
$711$	0	0
$712$	1.95186	0.0731490
$713$	−23.5512	−0.882000
$714$	0	0
$715$	26.9629	1.00835
$716$	−11.3635	−0.424674
$717$	0	0
$718$	28.6772	1.07022
$719$	−3.49814	−0.130459	−0.0652293	−	0.997870i	$-0.520778\pi$
$719$	−3.49814	−0.130459	−0.0652293	+	0.997870i	$0.520778\pi$
$720$	0	0
$721$	9.12583	0.339864
$722$	−1.69963	−0.0632536
$723$	0	0
$724$	17.5846	0.653528
$725$	−66.4907	−2.46940
$726$	0	0
$727$	−8.69453	−0.322462	−0.161231	−	0.986917i	$-0.551546\pi$
$727$	−8.69453	−0.322462	−0.161231	+	0.986917i	$0.551546\pi$
$728$	−4.76509	−0.176606
$729$	0	0
$730$	7.68725	0.284518
$731$	−51.9761	−1.92240
$732$	0	0
$733$	15.0124	0.554495	0.277247	−	0.960799i	$-0.410578\pi$
$733$	15.0124	0.554495	0.277247	+	0.960799i	$0.410578\pi$
$734$	50.7032	1.87149
$735$	0	0
$736$	42.7614	1.57621
$737$	37.1038	1.36674
$738$	0	0
$739$	8.81955	0.324433	0.162216	−	0.986755i	$-0.448136\pi$
$739$	8.81955	0.324433	0.162216	+	0.986755i	$0.448136\pi$
$740$	0.946916	0.0348093
$741$	0	0
$742$	22.2509	0.816855
$743$	−8.60940	−0.315848	−0.157924	−	0.987451i	$-0.550480\pi$
$743$	−8.60940	−0.315848	−0.157924	+	0.987451i	$0.550480\pi$
$744$	0	0
$745$	−62.0246	−2.27241
$746$	−18.1223	−0.663503
$747$	0	0
$748$	−22.7380	−0.831383
$749$	−1.11126	−0.0406047
$750$	0	0
$751$	−3.39788	−0.123990	−0.0619952	−	0.998076i	$-0.519746\pi$
$751$	−3.39788	−0.123990	−0.0619952	+	0.998076i	$0.519746\pi$
$752$	−44.8886	−1.63692
$753$	0	0
$754$	20.6625	0.752483
$755$	−58.5214	−2.12981
$756$	0	0
$757$	−39.5585	−1.43778	−0.718889	−	0.695125i	$-0.755349\pi$
$757$	−39.5585	−1.43778	−0.718889	+	0.695125i	$0.755349\pi$
$758$	7.23119	0.262649
$759$	0	0
$760$	−6.98762	−0.253468
$761$	−21.8355	−0.791536	−0.395768	−	0.918350i	$-0.629522\pi$
$761$	−21.8355	−0.791536	−0.395768	+	0.918350i	$0.629522\pi$
$762$	0	0
$763$	13.4981	0.488666
$764$	−7.49442	−0.271139
$765$	0	0
$766$	−18.3955	−0.664658
$767$	11.8567	0.428120
$768$	0	0
$769$	−6.19049	−0.223235	−0.111617	−	0.993751i	$-0.535603\pi$
$769$	−6.19049	−0.223235	−0.111617	+	0.993751i	$0.535603\pi$
$770$	45.8268	1.65148
$771$	0	0
$772$	−6.01238	−0.216390
$773$	40.9184	1.47173	0.735867	−	0.677126i	$-0.236775\pi$
$773$	40.9184	1.47173	0.735867	+	0.677126i	$0.236775\pi$
$774$	0	0
$775$	−22.4858	−0.807712
$776$	−29.2088	−1.04853
$777$	0	0
$778$	−56.5288	−2.02666
$779$	−3.69963	−0.132553
$780$	0	0
$781$	−28.4944	−1.01961
$782$	86.2310	3.08362
$783$	0	0
$784$	22.3301	0.797503
$785$	55.1555	1.96859
$786$	0	0
$787$	−28.9171	−1.03078	−0.515391	−	0.856955i	$-0.672353\pi$
$787$	−28.9171	−1.03078	−0.515391	+	0.856955i	$0.672353\pi$
$788$	17.9977	0.641140
$789$	0	0
$790$	37.1606	1.32212
$791$	16.6218	0.591003
$792$	0	0
$793$	0.300372	0.0106665
$794$	52.8626	1.87602
$795$	0	0
$796$	0.0284779	0.00100937
$797$	−12.4596	−0.441343	−0.220671	−	0.975348i	$-0.570825\pi$
$797$	−12.4596	−0.441343	−0.220671	+	0.975348i	$0.570825\pi$
$798$	0	0
$799$	−50.1839	−1.77538
$800$	40.8268	1.44345
$801$	0	0
$802$	42.5475	1.50240
$803$	−5.60940	−0.197952
$804$	0	0
$805$	−53.4683	−1.88451
$806$	6.98762	0.246128
$807$	0	0
$808$	13.9121	0.489427
$809$	−9.07922	−0.319208	−0.159604	−	0.987181i	$-0.551022\pi$
$809$	−9.07922	−0.319208	−0.159604	+	0.987181i	$0.551022\pi$
$810$	0	0
$811$	25.5622	0.897611	0.448806	−	0.893629i	$-0.351850\pi$
$811$	25.5622	0.897611	0.448806	+	0.893629i	$0.351850\pi$
$812$	10.8044	0.379161
$813$	0	0
$814$	−2.24591	−0.0787191
$815$	41.8072	1.46444
$816$	0	0
$817$	−9.32141	−0.326115
$818$	−65.8465	−2.30227
$819$	0	0
$820$	12.1643	0.424797
$821$	9.10026	0.317601	0.158801	−	0.987311i	$-0.449237\pi$
$821$	9.10026	0.317601	0.158801	+	0.987311i	$0.449237\pi$
$822$	0	0
$823$	−29.0865	−1.01389	−0.506946	−	0.861978i	$-0.669226\pi$
$823$	−29.0865	−1.01389	−0.506946	+	0.861978i	$0.669226\pi$
$824$	−10.8516	−0.378033
$825$	0	0
$826$	20.1520	0.701177
$827$	3.94182	0.137071	0.0685353	−	0.997649i	$-0.478167\pi$
$827$	3.94182	0.137071	0.0685353	+	0.997649i	$0.478167\pi$
$828$	0	0
$829$	56.1184	1.94907	0.974536	−	0.224230i	$-0.0719868\pi$
$829$	56.1184	1.94907	0.974536	+	0.224230i	$0.0719868\pi$
$830$	17.8850	0.620798
$831$	0	0
$832$	3.15706	0.109452
$833$	24.9642	0.864960
$834$	0	0
$835$	72.0122	2.49209
$836$	−4.07784	−0.141035
$837$	0	0
$838$	−7.51052	−0.259447
$839$	30.1868	1.04216	0.521081	−	0.853507i	$-0.325529\pi$
$839$	30.1868	1.04216	0.521081	+	0.853507i	$0.325529\pi$
$840$	0	0
$841$	29.5811	1.02004
$842$	22.8726	0.788243
$843$	0	0
$844$	−4.75903	−0.163813
$845$	−38.7614	−1.33343
$846$	0	0
$847$	−15.9680	−0.548665
$848$	−41.1089	−1.41169
$849$	0	0
$850$	82.3299	2.82389
$851$	2.62041	0.0898264
$852$	0	0
$853$	19.3200	0.661505	0.330753	−	0.943717i	$-0.392697\pi$
$853$	19.3200	0.661505	0.330753	+	0.943717i	$0.392697\pi$
$854$	0.510520	0.0174697
$855$	0	0
$856$	1.32141	0.0451650
$857$	−49.4472	−1.68909	−0.844543	−	0.535488i	$-0.820128\pi$
$857$	−49.4472	−1.68909	−0.844543	+	0.535488i	$0.820128\pi$
$858$	0	0
$859$	−35.1075	−1.19785	−0.598927	−	0.800804i	$-0.704406\pi$
$859$	−35.1075	−1.19785	−0.598927	+	0.800804i	$0.704406\pi$
$860$	30.6487	1.04511
$861$	0	0
$862$	−58.9678	−2.00845
$863$	43.8158	1.49151	0.745754	−	0.666221i	$-0.232089\pi$
$863$	43.8158	1.49151	0.745754	+	0.666221i	$0.232089\pi$
$864$	0	0
$865$	−58.8603	−2.00131
$866$	−40.4252	−1.37371
$867$	0	0
$868$	3.65383	0.124019
$869$	−27.1162	−0.919854
$870$	0	0
$871$	12.8443	0.435213
$872$	−16.0507	−0.543547
$873$	0	0
$874$	15.4647	0.523102
$875$	−21.6676	−0.732498
$876$	0	0
$877$	−9.82327	−0.331708	−0.165854	−	0.986150i	$-0.553038\pi$
$877$	−9.82327	−0.331708	−0.165854	+	0.986150i	$0.553038\pi$
$878$	−3.32141	−0.112092
$879$	0	0
$880$	−84.6661	−2.85409
$881$	14.6648	0.494071	0.247035	−	0.969006i	$-0.420544\pi$
$881$	14.6648	0.494071	0.247035	+	0.969006i	$0.420544\pi$
$882$	0	0
$883$	−14.9432	−0.502879	−0.251439	−	0.967873i	$-0.580904\pi$
$883$	−14.9432	−0.502879	−0.251439	+	0.967873i	$0.580904\pi$
$884$	−7.87126	−0.264739
$885$	0	0
$886$	33.8703	1.13789
$887$	20.3004	0.681620	0.340810	−	0.940132i	$-0.389299\pi$
$887$	20.3004	0.681620	0.340810	+	0.940132i	$0.389299\pi$
$888$	0	0
$889$	17.0792	0.572819
$890$	−6.49814	−0.217818
$891$	0	0
$892$	−6.97030	−0.233383
$893$	−9.00000	−0.301174
$894$	0	0
$895$	−47.3039	−1.58120
$896$	20.2953	0.678018
$897$	0	0
$898$	68.4138	2.28300
$899$	19.8109	0.660730
$900$	0	0
$901$	−45.9583	−1.53109
$902$	−28.8516	−0.960653
$903$	0	0
$904$	−19.7651	−0.657377
$905$	73.2013	2.43329
$906$	0	0
$907$	54.9120	1.82332	0.911661	−	0.410943i	$-0.134801\pi$
$907$	54.9120	1.82332	0.911661	+	0.410943i	$0.134801\pi$
$908$	7.60431	0.252358
$909$	0	0
$910$	15.8640	0.525886
$911$	−29.7751	−0.986494	−0.493247	−	0.869889i	$-0.664190\pi$
$911$	−29.7751	−0.986494	−0.493247	+	0.869889i	$0.664190\pi$
$912$	0	0
$913$	−13.0507	−0.431917
$914$	−40.8182	−1.35015
$915$	0	0
$916$	−4.47216	−0.147764
$917$	−25.9047	−0.855448
$918$	0	0
$919$	19.5563	0.645103	0.322552	−	0.946552i	$-0.395459\pi$
$919$	19.5563	0.645103	0.322552	+	0.946552i	$0.395459\pi$
$920$	63.5795	2.09616
$921$	0	0
$922$	−27.1730	−0.894895
$923$	−9.86398	−0.324677
$924$	0	0
$925$	2.50186	0.0822606
$926$	−22.2001	−0.729541
$927$	0	0
$928$	−35.9701	−1.18078
$929$	23.0641	0.756708	0.378354	−	0.925661i	$-0.376490\pi$
$929$	23.0641	0.756708	0.378354	+	0.925661i	$0.376490\pi$
$930$	0	0
$931$	4.47710	0.146731
$932$	13.5585	0.444124
$933$	0	0
$934$	−2.38056	−0.0778942
$935$	−94.6537	−3.09551
$936$	0	0
$937$	−4.31632	−0.141008	−0.0705040	−	0.997511i	$-0.522461\pi$
$937$	−4.31632	−0.141008	−0.0705040	+	0.997511i	$0.522461\pi$
$938$	21.8306	0.712793
$939$	0	0
$940$	29.5919	0.965182
$941$	21.9322	0.714969	0.357485	−	0.933919i	$-0.383634\pi$
$941$	21.9322	0.714969	0.357485	+	0.933919i	$0.383634\pi$
$942$	0	0
$943$	33.6625	1.09620
$944$	−37.2312	−1.21177
$945$	0	0
$946$	−72.6932	−2.36346
$947$	21.5288	0.699592	0.349796	−	0.936826i	$-0.386251\pi$
$947$	21.5288	0.699592	0.349796	+	0.936826i	$0.386251\pi$
$948$	0	0
$949$	−1.94182	−0.0630341
$950$	14.7651	0.479043
$951$	0	0
$952$	16.7280	0.542156
$953$	−26.4313	−0.856194	−0.428097	−	0.903733i	$-0.640816\pi$
$953$	−26.4313	−0.856194	−0.428097	+	0.903733i	$0.640816\pi$
$954$	0	0
$955$	−31.1978	−1.00954
$956$	−21.8777	−0.707576
$957$	0	0
$958$	−27.7527	−0.896649
$959$	24.1593	0.780143
$960$	0	0
$961$	−24.3004	−0.783883
$962$	−0.777472	−0.0250667
$963$	0	0
$964$	−10.2408	−0.329834
$965$	−25.0283	−0.805690
$966$	0	0
$967$	−22.2953	−0.716968	−0.358484	−	0.933536i	$-0.616706\pi$
$967$	−22.2953	−0.716968	−0.358484	+	0.933536i	$0.616706\pi$
$968$	18.9876	0.610285
$969$	0	0
$970$	97.2420	3.12225
$971$	30.6167	0.982536	0.491268	−	0.871008i	$-0.336534\pi$
$971$	30.6167	0.982536	0.491268	+	0.871008i	$0.336534\pi$
$972$	0	0
$973$	−4.16435	−0.133503
$974$	−60.1382	−1.92695
$975$	0	0
$976$	−0.943197	−0.0301910
$977$	27.1593	0.868901	0.434451	−	0.900696i	$-0.356943\pi$
$977$	27.1593	0.868901	0.434451	+	0.900696i	$0.356943\pi$
$978$	0	0
$979$	4.74171	0.151546
$980$	−14.7207	−0.470235
$981$	0	0
$982$	12.9963	0.414728
$983$	36.3128	1.15820	0.579098	−	0.815258i	$-0.303405\pi$
$983$	36.3128	1.15820	0.579098	+	0.815258i	$0.303405\pi$
$984$	0	0
$985$	74.9206	2.38717
$986$	−72.5361	−2.31002
$987$	0	0
$988$	−1.41164	−0.0449101
$989$	84.8145	2.69694
$990$	0	0
$991$	25.1671	0.799459	0.399730	−	0.916633i	$-0.369104\pi$
$991$	25.1671	0.799459	0.399730	+	0.916633i	$0.369104\pi$
$992$	−12.1643	−0.386218
$993$	0	0
$994$	−16.7651	−0.531756
$995$	0.118548	0.00375822
$996$	0	0
$997$	−31.3360	−0.992420	−0.496210	−	0.868202i	$-0.665275\pi$
$997$	−31.3360	−0.992420	−0.496210	+	0.868202i	$0.665275\pi$
$998$	20.6414	0.653394
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.f.1.1	yes	3
3.2	odd	2		513.2.a.e.1.3	✓	3
4.3	odd	2		8208.2.a.bp.1.3		3
12.11	even	2		8208.2.a.bf.1.1		3
19.18	odd	2		9747.2.a.x.1.3		3
57.56	even	2		9747.2.a.ba.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.e.1.3	✓	3	3.2	odd	2
513.2.a.f.1.1	yes	3	1.1	even	1	trivial
8208.2.a.bf.1.1		3	12.11	even	2
8208.2.a.bp.1.3		3	4.3	odd	2
9747.2.a.x.1.3		3	19.18	odd	2
9747.2.a.ba.1.1		3	57.56	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.69963 q^{2} +0.888736 q^{4} +3.69963 q^{5} -1.58836 q^{7} +1.88874 q^{8} -6.28799 q^{10} +4.58836 q^{11} +1.58836 q^{13} +2.69963 q^{14} -4.98762 q^{16} -5.57598 q^{17} -1.00000 q^{19} +3.28799 q^{20} -7.79851 q^{22} +9.09888 q^{23} +8.68725 q^{25} -2.69963 q^{26} -1.41164 q^{28} -7.65383 q^{29} -2.58836 q^{31} +4.69963 q^{32} +9.47710 q^{34} -5.87636 q^{35} +0.287992 q^{37} +1.69963 q^{38} +6.98762 q^{40} +3.69963 q^{41} +9.32141 q^{43} +4.07784 q^{44} -15.4647 q^{46} +9.00000 q^{47} -4.47710 q^{49} -14.7651 q^{50} +1.41164 q^{52} +8.24219 q^{53} +16.9752 q^{55} -3.00000 q^{56} +13.0087 q^{58} +7.46472 q^{59} +0.189108 q^{61} +4.39926 q^{62} +1.98762 q^{64} +5.87636 q^{65} +8.08650 q^{67} -4.95558 q^{68} +9.98762 q^{70} -6.21015 q^{71} -1.22253 q^{73} -0.489480 q^{74} -0.888736 q^{76} -7.28799 q^{77} -5.90978 q^{79} -18.4523 q^{80} -6.28799 q^{82} -2.84431 q^{83} -20.6291 q^{85} -15.8429 q^{86} +8.66621 q^{88} +1.03342 q^{89} -2.52290 q^{91} +8.08650 q^{92} -15.2967 q^{94} -3.69963 q^{95} -15.4647 q^{97} +7.60940 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} - 7 q^{10} + 8 q^{11} - q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 10 q^{28} - 6 q^{29}+ \cdots - 8 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 + 5 * q^5 + q^7 + 6 * q^8 - 7 * q^10 + 8 * q^11 - q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - 3 * q^19 - 2 * q^20 + q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 10 * q^28 - 6 * q^29 - 2 * q^31 + 8 * q^32 + 23 * q^34 - 11 * q^37 - q^38 + 3 * q^40 + 5 * q^41 + 9 * q^43 + 19 * q^44 - 23 * q^46 + 27 * q^47 - 8 * q^49 - 27 * q^50 + 10 * q^52 + 2 * q^53 + 15 * q^55 - 9 * q^56 - 4 * q^58 - q^59 + 7 * q^61 + q^62 - 12 * q^64 - 12 * q^67 - q^68 + 12 * q^70 - 3 * q^73 - 14 * q^74 - 3 * q^76 - 10 * q^77 + 7 * q^79 - 14 * q^80 - 7 * q^82 + 5 * q^83 - 9 * q^85 - 37 * q^86 + 27 * q^88 - 4 * q^89 - 13 * q^91 - 12 * q^92 + 9 * q^94 - 5 * q^95 - 23 * q^97 - 8 * q^98

Introduction

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