LMFDB - Embedded newform 513.2.a.h.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 = [4,0,0,4] 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:	$4$
Coefficient field:	4.4.27648.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 3 $ x^4 - 6*x^2 + 3
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			$-2.33441$ 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)

$f(q)$	$=$	\(q-2.33441 q^{2} +3.44949 q^{4} -1.04930 q^{5} -0.449490 q^{7} -3.38371 q^{8} +2.44949 q^{10} -1.28512 q^{11} +2.00000 q^{13} +1.04930 q^{14} +1.00000 q^{16} +2.09859 q^{17} +1.00000 q^{19} -3.61953 q^{20} +3.00000 q^{22} +3.61953 q^{23} -3.89898 q^{25} -4.66883 q^{26} -1.55051 q^{28} -3.38371 q^{29} +2.55051 q^{31} +4.43300 q^{32} -4.89898 q^{34} +0.471647 q^{35} -0.449490 q^{37} -2.33441 q^{38} +3.55051 q^{40} +11.6721 q^{41} +9.34847 q^{43} -4.43300 q^{44} -8.44949 q^{46} +2.33441 q^{47} -6.79796 q^{49} +9.10183 q^{50} +6.89898 q^{52} +11.2004 q^{53} +1.34847 q^{55} +1.52094 q^{56} +7.89898 q^{58} +1.52094 q^{59} +6.34847 q^{61} -5.95395 q^{62} -12.3485 q^{64} -2.09859 q^{65} +12.3485 q^{67} +7.23907 q^{68} -1.10102 q^{70} +13.5348 q^{71} +6.34847 q^{73} +1.04930 q^{74} +3.44949 q^{76} +0.577648 q^{77} +5.00000 q^{79} -1.04930 q^{80} -27.2474 q^{82} -4.43300 q^{83} -2.20204 q^{85} -21.8232 q^{86} +4.34847 q^{88} -5.95395 q^{89} -0.898979 q^{91} +12.4855 q^{92} -5.44949 q^{94} -1.04930 q^{95} +10.4495 q^{97} +15.8693 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 4 q^{19} + 12 q^{22} + 4 q^{25} - 16 q^{28} + 20 q^{31} + 8 q^{37} + 24 q^{40} + 8 q^{43} - 24 q^{46} + 12 q^{49} + 8 q^{52} - 24 q^{55} + 12 q^{58} - 4 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^13 + 4 * q^16 + 4 * q^19 + 12 * q^22 + 4 * q^25 - 16 * q^28 + 20 * q^31 + 8 * q^37 + 24 * q^40 + 8 * q^43 - 24 * q^46 + 12 * q^49 + 8 * q^52 - 24 * q^55 + 12 * q^58 - 4 * q^61 - 20 * q^64 + 20 * q^67 - 24 * q^70 - 4 * q^73 + 4 * q^76 + 20 * q^79 - 60 * q^82 - 48 * q^85 - 12 * q^88 + 16 * q^91 - 12 * q^94 + 32 * q^97

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 $

$2$	−2.33441	−1.65068	−0.825340	−	0.564636i	$-0.809017\pi$
$2$	−2.33441	−1.65068	−0.825340	+	0.564636i	$0.809017\pi$
$3$	0	0
$3$	0	0
$4$	3.44949	1.72474
$5$	−1.04930	−0.469259	−0.234630	−	0.972085i	$-0.575388\pi$
$5$	−1.04930	−0.469259	−0.234630	+	0.972085i	$0.575388\pi$
$6$	0	0
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	−3.38371	−1.19632
$9$	0	0
$10$	2.44949	0.774597
$11$	−1.28512	−0.387478	−0.193739	−	0.981053i	$-0.562061\pi$
$11$	−1.28512	−0.387478	−0.193739	+	0.981053i	$0.562061\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.04930	0.280436
$15$	0	0
$16$	1.00000	0.250000
$17$	2.09859	0.508983	0.254491	−	0.967075i	$-0.418092\pi$
$17$	2.09859	0.508983	0.254491	+	0.967075i	$0.418092\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.61953	−0.809352
$21$	0	0
$22$	3.00000	0.639602
$23$	3.61953	0.754725	0.377362	−	0.926066i	$-0.376831\pi$
$23$	3.61953	0.754725	0.377362	+	0.926066i	$0.376831\pi$
$24$	0	0
$25$	−3.89898	−0.779796
$26$	−4.66883	−0.915633
$27$	0	0
$28$	−1.55051	−0.293019
$29$	−3.38371	−0.628339	−0.314170	−	0.949367i	$-0.601726\pi$
$29$	−3.38371	−0.628339	−0.314170	+	0.949367i	$0.601726\pi$
$30$	0	0
$31$	2.55051	0.458085	0.229043	−	0.973416i	$-0.426440\pi$
$31$	2.55051	0.458085	0.229043	+	0.973416i	$0.426440\pi$
$32$	4.43300	0.783652
$33$	0	0
$34$	−4.89898	−0.840168
$35$	0.471647	0.0797230
$36$	0	0
$37$	−0.449490	−0.0738957	−0.0369478	−	0.999317i	$-0.511764\pi$
$37$	−0.449490	−0.0738957	−0.0369478	+	0.999317i	$0.511764\pi$
$38$	−2.33441	−0.378692
$39$	0	0
$40$	3.55051	0.561385
$41$	11.6721	1.82287	0.911436	−	0.411443i	$-0.134975\pi$
$41$	11.6721	1.82287	0.911436	+	0.411443i	$0.134975\pi$
$42$	0	0
$43$	9.34847	1.42563	0.712814	−	0.701353i	$-0.247420\pi$
$43$	9.34847	1.42563	0.712814	+	0.701353i	$0.247420\pi$
$44$	−4.43300	−0.668301
$45$	0	0
$46$	−8.44949	−1.24581
$47$	2.33441	0.340509	0.170255	−	0.985400i	$-0.445541\pi$
$47$	2.33441	0.340509	0.170255	+	0.985400i	$0.445541\pi$
$48$	0	0
$49$	−6.79796	−0.971137
$50$	9.10183	1.28719
$51$	0	0
$52$	6.89898	0.956716
$53$	11.2004	1.53850	0.769248	−	0.638950i	$-0.220631\pi$
$53$	11.2004	1.53850	0.769248	+	0.638950i	$0.220631\pi$
$54$	0	0
$55$	1.34847	0.181828
$56$	1.52094	0.203245
$57$	0	0
$58$	7.89898	1.03719
$59$	1.52094	0.198010	0.0990049	−	0.995087i	$-0.468434\pi$
$59$	1.52094	0.198010	0.0990049	+	0.995087i	$0.468434\pi$
$60$	0	0
$61$	6.34847	0.812838	0.406419	−	0.913687i	$-0.366777\pi$
$61$	6.34847	0.812838	0.406419	+	0.913687i	$0.366777\pi$
$62$	−5.95395	−0.756152
$63$	0	0
$64$	−12.3485	−1.54356
$65$	−2.09859	−0.260298
$66$	0	0
$67$	12.3485	1.50861	0.754303	−	0.656527i	$-0.227975\pi$
$67$	12.3485	1.50861	0.754303	+	0.656527i	$0.227975\pi$
$68$	7.23907	0.877866
$69$	0	0
$70$	−1.10102	−0.131597
$71$	13.5348	1.60629	0.803145	−	0.595784i	$-0.203158\pi$
$71$	13.5348	1.60629	0.803145	+	0.595784i	$0.203158\pi$
$72$	0	0
$73$	6.34847	0.743032	0.371516	−	0.928427i	$-0.378838\pi$
$73$	6.34847	0.743032	0.371516	+	0.928427i	$0.378838\pi$
$74$	1.04930	0.121978
$75$	0	0
$76$	3.44949	0.395684
$77$	0.577648	0.0658291
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−1.04930	−0.117315
$81$	0	0
$82$	−27.2474	−3.00898
$83$	−4.43300	−0.486585	−0.243293	−	0.969953i	$-0.578228\pi$
$83$	−4.43300	−0.486585	−0.243293	+	0.969953i	$0.578228\pi$
$84$	0	0
$85$	−2.20204	−0.238845
$86$	−21.8232	−2.35326
$87$	0	0
$88$	4.34847	0.463548
$89$	−5.95395	−0.631117	−0.315559	−	0.948906i	$-0.602192\pi$
$89$	−5.95395	−0.631117	−0.315559	+	0.948906i	$0.602192\pi$
$90$	0	0
$91$	−0.898979	−0.0942387
$92$	12.4855	1.30171
$93$	0	0
$94$	−5.44949	−0.562072
$95$	−1.04930	−0.107655
$96$	0	0
$97$	10.4495	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$97$	10.4495	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$98$	15.8693	1.60304
$99$	0	0
$100$	−13.4495	−1.34495
$101$	−12.9572	−1.28929	−0.644644	−	0.764483i	$-0.722995\pi$
$101$	−12.9572	−1.28929	−0.644644	+	0.764483i	$0.722995\pi$
$102$	0	0
$103$	−7.79796	−0.768356	−0.384178	−	0.923259i	$-0.625515\pi$
$103$	−7.79796	−0.768356	−0.384178	+	0.923259i	$0.625515\pi$
$104$	−6.76742	−0.663600
$105$	0	0
$106$	−26.1464	−2.53957
$107$	−0.577648	−0.0558433	−0.0279217	−	0.999610i	$-0.508889\pi$
$107$	−0.577648	−0.0558433	−0.0279217	+	0.999610i	$0.508889\pi$
$108$	0	0
$109$	−16.2474	−1.55622	−0.778112	−	0.628126i	$-0.783822\pi$
$109$	−16.2474	−1.55622	−0.778112	+	0.628126i	$0.783822\pi$
$110$	−3.14789	−0.300139
$111$	0	0
$112$	−0.449490	−0.0424728
$113$	−1.28512	−0.120894	−0.0604469	−	0.998171i	$-0.519253\pi$
$113$	−1.28512	−0.120894	−0.0604469	+	0.998171i	$0.519253\pi$
$114$	0	0
$115$	−3.79796	−0.354162
$116$	−11.6721	−1.08372
$117$	0	0
$118$	−3.55051	−0.326851
$119$	−0.943295	−0.0864717
$120$	0	0
$121$	−9.34847	−0.849861
$122$	−14.8200	−1.34174
$123$	0	0
$124$	8.79796	0.790080
$125$	9.33766	0.835185
$126$	0	0
$127$	5.79796	0.514486	0.257243	−	0.966347i	$-0.417186\pi$
$127$	5.79796	0.514486	0.257243	+	0.966347i	$0.417186\pi$
$128$	19.9604	1.76427
$129$	0	0
$130$	4.89898	0.429669
$131$	−20.1963	−1.76456	−0.882278	−	0.470730i	$-0.843991\pi$
$131$	−20.1963	−1.76456	−0.882278	+	0.470730i	$0.843991\pi$
$132$	0	0
$133$	−0.449490	−0.0389757
$134$	−28.8264	−2.49023
$135$	0	0
$136$	−7.10102	−0.608907
$137$	9.33766	0.797770	0.398885	−	0.917001i	$-0.369397\pi$
$137$	9.33766	0.797770	0.398885	+	0.917001i	$0.369397\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	1.62694	0.137502
$141$	0	0
$142$	−31.5959	−2.65147
$143$	−2.57024	−0.214934
$144$	0	0
$145$	3.55051	0.294854
$146$	−14.8200	−1.22651
$147$	0	0
$148$	−1.55051	−0.127451
$149$	4.66883	0.382485	0.191243	−	0.981543i	$-0.438748\pi$
$149$	4.66883	0.382485	0.191243	+	0.981543i	$0.438748\pi$
$150$	0	0
$151$	19.6969	1.60291	0.801457	−	0.598052i	$-0.204059\pi$
$151$	19.6969	1.60291	0.801457	+	0.598052i	$0.204059\pi$
$152$	−3.38371	−0.274455
$153$	0	0
$154$	−1.34847	−0.108663
$155$	−2.67624	−0.214961
$156$	0	0
$157$	0.348469	0.0278109	0.0139054	−	0.999903i	$-0.495574\pi$
$157$	0.348469	0.0278109	0.0139054	+	0.999903i	$0.495574\pi$
$158$	−11.6721	−0.928580
$159$	0	0
$160$	−4.65153	−0.367736
$161$	−1.62694	−0.128221
$162$	0	0
$163$	8.24745	0.645990	0.322995	−	0.946401i	$-0.395310\pi$
$163$	8.24745	0.645990	0.322995	+	0.946401i	$0.395310\pi$
$164$	40.2627	3.14399
$165$	0	0
$166$	10.3485	0.803197
$167$	−16.1051	−1.24625	−0.623124	−	0.782123i	$-0.714137\pi$
$167$	−16.1051	−1.24625	−0.623124	+	0.782123i	$0.714137\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	5.14048	0.394257
$171$	0	0
$172$	32.2474	2.45884
$173$	−7.47489	−0.568305	−0.284153	−	0.958779i	$-0.591712\pi$
$173$	−7.47489	−0.568305	−0.284153	+	0.958779i	$0.591712\pi$
$174$	0	0
$175$	1.75255	0.132480
$176$	−1.28512	−0.0968695
$177$	0	0
$178$	13.8990	1.04177
$179$	8.28836	0.619501	0.309751	−	0.950818i	$-0.399754\pi$
$179$	8.28836	0.619501	0.309751	+	0.950818i	$0.399754\pi$
$180$	0	0
$181$	19.1464	1.42314	0.711571	−	0.702614i	$-0.247984\pi$
$181$	19.1464	1.42314	0.711571	+	0.702614i	$0.247984\pi$
$182$	2.09859	0.155558
$183$	0	0
$184$	−12.2474	−0.902894
$185$	0.471647	0.0346762
$186$	0	0
$187$	−2.69694	−0.197220
$188$	8.05254	0.587292
$189$	0	0
$190$	2.44949	0.177705
$191$	−15.2916	−1.10646	−0.553231	−	0.833028i	$-0.686605\pi$
$191$	−15.2916	−1.10646	−0.553231	+	0.833028i	$0.686605\pi$
$192$	0	0
$193$	4.69694	0.338093	0.169047	−	0.985608i	$-0.445931\pi$
$193$	4.69694	0.338093	0.169047	+	0.985608i	$0.445931\pi$
$194$	−24.3934	−1.75135
$195$	0	0
$196$	−23.4495	−1.67496
$197$	−22.8725	−1.62960	−0.814799	−	0.579744i	$-0.803153\pi$
$197$	−22.8725	−1.62960	−0.814799	+	0.579744i	$0.803153\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	13.1930	0.932887
$201$	0	0
$202$	30.2474	2.12820
$203$	1.52094	0.106749
$204$	0	0
$205$	−12.2474	−0.855399
$206$	18.2037	1.26831
$207$	0	0
$208$	2.00000	0.138675
$209$	−1.28512	−0.0888935
$210$	0	0
$211$	−2.34847	−0.161675	−0.0808376	−	0.996727i	$-0.525760\pi$
$211$	−2.34847	−0.161675	−0.0808376	+	0.996727i	$0.525760\pi$
$212$	38.6357	2.65351
$213$	0	0
$214$	1.34847	0.0921795
$215$	−9.80930	−0.668989
$216$	0	0
$217$	−1.14643	−0.0778246
$218$	37.9283	2.56883
$219$	0	0
$220$	4.65153	0.313606
$221$	4.19718	0.282333
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−1.99259	−0.133136
$225$	0	0
$226$	3.00000	0.199557
$227$	−7.71071	−0.511778	−0.255889	−	0.966706i	$-0.582368\pi$
$227$	−7.71071	−0.511778	−0.255889	+	0.966706i	$0.582368\pi$
$228$	0	0
$229$	−2.10102	−0.138839	−0.0694197	−	0.997588i	$-0.522115\pi$
$229$	−2.10102	−0.138839	−0.0694197	+	0.997588i	$0.522115\pi$
$230$	8.86601	0.584607
$231$	0	0
$232$	11.4495	0.751696
$233$	18.0977	1.18562	0.592809	−	0.805343i	$-0.298019\pi$
$233$	18.0977	1.18562	0.592809	+	0.805343i	$0.298019\pi$
$234$	0	0
$235$	−2.44949	−0.159787
$236$	5.24648	0.341517
$237$	0	0
$238$	2.20204	0.142737
$239$	21.0097	1.35901	0.679503	−	0.733673i	$-0.262195\pi$
$239$	21.0097	1.35901	0.679503	+	0.733673i	$0.262195\pi$
$240$	0	0
$241$	−8.65153	−0.557294	−0.278647	−	0.960394i	$-0.589886\pi$
$241$	−8.65153	−0.557294	−0.278647	+	0.960394i	$0.589886\pi$
$242$	21.8232	1.40285
$243$	0	0
$244$	21.8990	1.40194
$245$	7.13307	0.455715
$246$	0	0
$247$	2.00000	0.127257
$248$	−8.63019	−0.548017
$249$	0	0
$250$	−21.7980	−1.37862
$251$	9.91530	0.625848	0.312924	−	0.949778i	$-0.398691\pi$
$251$	9.91530	0.625848	0.312924	+	0.949778i	$0.398691\pi$
$252$	0	0
$253$	−4.65153	−0.292439
$254$	−13.5348	−0.849251
$255$	0	0
$256$	−21.8990	−1.36869
$257$	−29.5339	−1.84227	−0.921137	−	0.389237i	$-0.872739\pi$
$257$	−29.5339	−1.84227	−0.921137	+	0.389237i	$0.872739\pi$
$258$	0	0
$259$	0.202041	0.0125542
$260$	−7.23907	−0.448948
$261$	0	0
$262$	47.1464	2.91272
$263$	19.9604	1.23081	0.615407	−	0.788210i	$-0.288992\pi$
$263$	19.9604	1.23081	0.615407	+	0.788210i	$0.288992\pi$
$264$	0	0
$265$	−11.7526	−0.721953
$266$	1.04930	0.0643364
$267$	0	0
$268$	42.5959	2.60196
$269$	1.52094	0.0927335	0.0463668	−	0.998924i	$-0.485236\pi$
$269$	1.52094	0.0927335	0.0463668	+	0.998924i	$0.485236\pi$
$270$	0	0
$271$	−22.2474	−1.35144	−0.675718	−	0.737160i	$-0.736166\pi$
$271$	−22.2474	−1.35144	−0.675718	+	0.737160i	$0.736166\pi$
$272$	2.09859	0.127246
$273$	0	0
$274$	−21.7980	−1.31686
$275$	5.01065	0.302154
$276$	0	0
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	23.3441	1.40009
$279$	0	0
$280$	−1.59592	−0.0953743
$281$	−28.1190	−1.67744	−0.838719	−	0.544565i	$-0.816695\pi$
$281$	−28.1190	−1.67744	−0.838719	+	0.544565i	$0.816695\pi$
$282$	0	0
$283$	−2.89898	−0.172326	−0.0861632	−	0.996281i	$-0.527461\pi$
$283$	−2.89898	−0.172326	−0.0861632	+	0.996281i	$0.527461\pi$
$284$	46.6883	2.77044
$285$	0	0
$286$	6.00000	0.354787
$287$	−5.24648	−0.309690
$288$	0	0
$289$	−12.5959	−0.740936
$290$	−8.28836	−0.486709
$291$	0	0
$292$	21.8990	1.28154
$293$	−1.28512	−0.0750775	−0.0375387	−	0.999295i	$-0.511952\pi$
$293$	−1.28512	−0.0750775	−0.0375387	+	0.999295i	$0.511952\pi$
$294$	0	0
$295$	−1.59592	−0.0929179
$296$	1.52094	0.0884030
$297$	0	0
$298$	−10.8990	−0.631361
$299$	7.23907	0.418646
$300$	0	0
$301$	−4.20204	−0.242202
$302$	−45.9808	−2.64590
$303$	0	0
$304$	1.00000	0.0573539
$305$	−6.66142	−0.381432
$306$	0	0
$307$	8.79796	0.502126	0.251063	−	0.967971i	$-0.419220\pi$
$307$	8.79796	0.502126	0.251063	+	0.967971i	$0.419220\pi$
$308$	1.99259	0.113538
$309$	0	0
$310$	6.24745	0.354831
$311$	−10.1511	−0.575618	−0.287809	−	0.957688i	$-0.592927\pi$
$311$	−10.1511	−0.575618	−0.287809	+	0.957688i	$0.592927\pi$
$312$	0	0
$313$	−13.2474	−0.748790	−0.374395	−	0.927269i	$-0.622150\pi$
$313$	−13.2474	−0.748790	−0.374395	+	0.927269i	$0.622150\pi$
$314$	−0.813472	−0.0459069
$315$	0	0
$316$	17.2474	0.970245
$317$	−16.6827	−0.936995	−0.468498	−	0.883465i	$-0.655205\pi$
$317$	−16.6827	−0.936995	−0.468498	+	0.883465i	$0.655205\pi$
$318$	0	0
$319$	4.34847	0.243468
$320$	12.9572	0.724329
$321$	0	0
$322$	3.79796	0.211652
$323$	2.09859	0.116769
$324$	0	0
$325$	−7.79796	−0.432553
$326$	−19.2530	−1.06632
$327$	0	0
$328$	−39.4949	−2.18074
$329$	−1.04930	−0.0578495
$330$	0	0
$331$	21.5959	1.18702	0.593510	−	0.804827i	$-0.297742\pi$
$331$	21.5959	1.18702	0.593510	+	0.804827i	$0.297742\pi$
$332$	−15.2916	−0.839236
$333$	0	0
$334$	37.5959	2.05716
$335$	−12.9572	−0.707927
$336$	0	0
$337$	−11.3485	−0.618191	−0.309095	−	0.951031i	$-0.600026\pi$
$337$	−11.3485	−0.618191	−0.309095	+	0.951031i	$0.600026\pi$
$338$	21.0097	1.14278
$339$	0	0
$340$	−7.59592	−0.411946
$341$	−3.27771	−0.177498
$342$	0	0
$343$	6.20204	0.334879
$344$	−31.6325	−1.70551
$345$	0	0
$346$	17.4495	0.938090
$347$	6.66142	0.357604	0.178802	−	0.983885i	$-0.442778\pi$
$347$	6.66142	0.357604	0.178802	+	0.983885i	$0.442778\pi$
$348$	0	0
$349$	24.5959	1.31659	0.658295	−	0.752760i	$-0.271278\pi$
$349$	24.5959	1.31659	0.658295	+	0.752760i	$0.271278\pi$
$350$	−4.09118	−0.218683
$351$	0	0
$352$	−5.69694	−0.303648
$353$	17.7320	0.943780	0.471890	−	0.881657i	$-0.343572\pi$
$353$	17.7320	0.943780	0.471890	+	0.881657i	$0.343572\pi$
$354$	0	0
$355$	−14.2020	−0.753766
$356$	−20.5381	−1.08852
$357$	0	0
$358$	−19.3485	−1.02260
$359$	−11.3302	−0.597988	−0.298994	−	0.954255i	$-0.596651\pi$
$359$	−11.3302	−0.597988	−0.298994	+	0.954255i	$0.596651\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−44.6957	−2.34915
$363$	0	0
$364$	−3.10102	−0.162538
$365$	−6.66142	−0.348675
$366$	0	0
$367$	22.6969	1.18477	0.592385	−	0.805655i	$-0.298186\pi$
$367$	22.6969	1.18477	0.592385	+	0.805655i	$0.298186\pi$
$368$	3.61953	0.188681
$369$	0	0
$370$	−1.10102	−0.0572393
$371$	−5.03448	−0.261377
$372$	0	0
$373$	−20.0454	−1.03791	−0.518956	−	0.854801i	$-0.673679\pi$
$373$	−20.0454	−1.03791	−0.518956	+	0.854801i	$0.673679\pi$
$374$	6.29577	0.325547
$375$	0	0
$376$	−7.89898	−0.407359
$377$	−6.76742	−0.348540
$378$	0	0
$379$	−13.4949	−0.693186	−0.346593	−	0.938016i	$-0.612662\pi$
$379$	−13.4949	−0.693186	−0.346593	+	0.938016i	$0.612662\pi$
$380$	−3.61953	−0.185678
$381$	0	0
$382$	35.6969	1.82641
$383$	30.6892	1.56815	0.784073	−	0.620669i	$-0.213139\pi$
$383$	30.6892	1.56815	0.784073	+	0.620669i	$0.213139\pi$
$384$	0	0
$385$	−0.606123	−0.0308909
$386$	−10.9646	−0.558083
$387$	0	0
$388$	36.0454	1.82993
$389$	26.4920	1.34320	0.671600	−	0.740914i	$-0.265608\pi$
$389$	26.4920	1.34320	0.671600	+	0.740914i	$0.265608\pi$
$390$	0	0
$391$	7.59592	0.384142
$392$	23.0023	1.16179
$393$	0	0
$394$	53.3939	2.68994
$395$	−5.24648	−0.263979
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	−32.6818	−1.63819
$399$	0	0
$400$	−3.89898	−0.194949
$401$	33.2594	1.66090	0.830449	−	0.557095i	$-0.188084\pi$
$401$	33.2594	1.66090	0.830449	+	0.557095i	$0.188084\pi$
$402$	0	0
$403$	5.10102	0.254100
$404$	−44.6957	−2.22369
$405$	0	0
$406$	−3.55051	−0.176209
$407$	0.577648	0.0286329
$408$	0	0
$409$	−22.2474	−1.10007	−0.550033	−	0.835143i	$-0.685385\pi$
$409$	−22.2474	−1.10007	−0.550033	+	0.835143i	$0.685385\pi$
$410$	28.5906	1.41199
$411$	0	0
$412$	−26.8990	−1.32522
$413$	−0.683648	−0.0336401
$414$	0	0
$415$	4.65153	0.228335
$416$	8.86601	0.434692
$417$	0	0
$418$	3.00000	0.146735
$419$	12.1437	0.593259	0.296630	−	0.954993i	$-0.404137\pi$
$419$	12.1437	0.593259	0.296630	+	0.954993i	$0.404137\pi$
$420$	0	0
$421$	−32.8990	−1.60340	−0.801699	−	0.597728i	$-0.796070\pi$
$421$	−32.8990	−1.60340	−0.801699	+	0.597728i	$0.796070\pi$
$422$	5.48230	0.266874
$423$	0	0
$424$	−37.8990	−1.84054
$425$	−8.18236	−0.396903
$426$	0	0
$427$	−2.85357	−0.138094
$428$	−1.99259	−0.0963155
$429$	0	0
$430$	22.8990	1.10429
$431$	−1.52094	−0.0732612	−0.0366306	−	0.999329i	$-0.511662\pi$
$431$	−1.52094	−0.0732612	−0.0366306	+	0.999329i	$0.511662\pi$
$432$	0	0
$433$	21.3485	1.02594	0.512971	−	0.858406i	$-0.328545\pi$
$433$	21.3485	1.02594	0.512971	+	0.858406i	$0.328545\pi$
$434$	2.67624	0.128464
$435$	0	0
$436$	−56.0454	−2.68409
$437$	3.61953	0.173146
$438$	0	0
$439$	31.6969	1.51281	0.756406	−	0.654102i	$-0.226953\pi$
$439$	31.6969	1.51281	0.756406	+	0.654102i	$0.226953\pi$
$440$	−4.56283	−0.217524
$441$	0	0
$442$	−9.79796	−0.466041
$443$	−20.5381	−0.975794	−0.487897	−	0.872901i	$-0.662236\pi$
$443$	−20.5381	−0.975794	−0.487897	+	0.872901i	$0.662236\pi$
$444$	0	0
$445$	6.24745	0.296157
$446$	−53.6915	−2.54237
$447$	0	0
$448$	5.55051	0.262237
$449$	−13.2990	−0.627619	−0.313810	−	0.949486i	$-0.601605\pi$
$449$	−13.2990	−0.627619	−0.313810	+	0.949486i	$0.601605\pi$
$450$	0	0
$451$	−15.0000	−0.706322
$452$	−4.43300	−0.208511
$453$	0	0
$454$	18.0000	0.844782
$455$	0.943295	0.0442223
$456$	0	0
$457$	−30.1464	−1.41019	−0.705095	−	0.709113i	$-0.749096\pi$
$457$	−30.1464	−1.41019	−0.705095	+	0.709113i	$0.749096\pi$
$458$	4.90465	0.229179
$459$	0	0
$460$	−13.1010	−0.610838
$461$	7.23907	0.337157	0.168578	−	0.985688i	$-0.446082\pi$
$461$	7.23907	0.337157	0.168578	+	0.985688i	$0.446082\pi$
$462$	0	0
$463$	6.89898	0.320623	0.160311	−	0.987066i	$-0.448750\pi$
$463$	6.89898	0.320623	0.160311	+	0.987066i	$0.448750\pi$
$464$	−3.38371	−0.157085
$465$	0	0
$466$	−42.2474	−1.95708
$467$	−15.8693	−0.734341	−0.367171	−	0.930154i	$-0.619674\pi$
$467$	−15.8693	−0.734341	−0.367171	+	0.930154i	$0.619674\pi$
$468$	0	0
$469$	−5.55051	−0.256299
$470$	5.71812	0.263757
$471$	0	0
$472$	−5.14643	−0.236884
$473$	−12.0139	−0.552399
$474$	0	0
$475$	−3.89898	−0.178897
$476$	−3.25389	−0.149142
$477$	0	0
$478$	−49.0454	−2.24328
$479$	39.3432	1.79764	0.898819	−	0.438320i	$-0.144427\pi$
$479$	39.3432	1.79764	0.898819	+	0.438320i	$0.144427\pi$
$480$	0	0
$481$	−0.898979	−0.0409899
$482$	20.1963	0.919914
$483$	0	0
$484$	−32.2474	−1.46579
$485$	−10.9646	−0.497877
$486$	0	0
$487$	−8.34847	−0.378305	−0.189153	−	0.981948i	$-0.560574\pi$
$487$	−8.34847	−0.378305	−0.189153	+	0.981948i	$0.560574\pi$
$488$	−21.4814	−0.972416
$489$	0	0
$490$	−16.6515	−0.752239
$491$	−13.7707	−0.621461	−0.310731	−	0.950498i	$-0.600574\pi$
$491$	−13.7707	−0.621461	−0.310731	+	0.950498i	$0.600574\pi$
$492$	0	0
$493$	−7.10102	−0.319814
$494$	−4.66883	−0.210061
$495$	0	0
$496$	2.55051	0.114521
$497$	−6.08377	−0.272894
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	32.2102	1.44048
$501$	0	0
$502$	−23.1464	−1.03308
$503$	38.6357	1.72268	0.861341	−	0.508027i	$-0.169625\pi$
$503$	38.6357	1.72268	0.861341	+	0.508027i	$0.169625\pi$
$504$	0	0
$505$	13.5959	0.605010
$506$	10.8586	0.482724
$507$	0	0
$508$	20.0000	0.887357
$509$	30.3474	1.34512	0.672562	−	0.740041i	$-0.265194\pi$
$509$	30.3474	1.34512	0.672562	+	0.740041i	$0.265194\pi$
$510$	0	0
$511$	−2.85357	−0.126235
$512$	11.2004	0.494993
$513$	0	0
$514$	68.9444	3.04101
$515$	8.18236	0.360558
$516$	0	0
$517$	−3.00000	−0.131940
$518$	−0.471647	−0.0207230
$519$	0	0
$520$	7.10102	0.311400
$521$	3.14789	0.137911	0.0689557	−	0.997620i	$-0.478033\pi$
$521$	3.14789	0.137911	0.0689557	+	0.997620i	$0.478033\pi$
$522$	0	0
$523$	−23.0454	−1.00771	−0.503853	−	0.863790i	$-0.668085\pi$
$523$	−23.0454	−1.00771	−0.503853	+	0.863790i	$0.668085\pi$
$524$	−69.6668	−3.04341
$525$	0	0
$526$	−46.5959	−2.03168
$527$	5.35248	0.233158
$528$	0	0
$529$	−9.89898	−0.430390
$530$	27.4353	1.19171
$531$	0	0
$532$	−1.55051	−0.0672231
$533$	23.3441	1.01115
$534$	0	0
$535$	0.606123	0.0262050
$536$	−41.7836	−1.80478
$537$	0	0
$538$	−3.55051	−0.153073
$539$	8.73619	0.376294
$540$	0	0
$541$	21.0454	0.904813	0.452406	−	0.891812i	$-0.350566\pi$
$541$	21.0454	0.904813	0.452406	+	0.891812i	$0.350566\pi$
$542$	51.9348	2.23079
$543$	0	0
$544$	9.30306	0.398865
$545$	17.0484	0.730272
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	32.2102	1.37595
$549$	0	0
$550$	−11.6969	−0.498759
$551$	−3.38371	−0.144151
$552$	0	0
$553$	−2.24745	−0.0955712
$554$	30.3474	1.28934
$555$	0	0
$556$	−34.4949	−1.46291
$557$	18.7813	0.795790	0.397895	−	0.917431i	$-0.369741\pi$
$557$	18.7813	0.795790	0.397895	+	0.917431i	$0.369741\pi$
$558$	0	0
$559$	18.6969	0.790796
$560$	0.471647	0.0199307
$561$	0	0
$562$	65.6413	2.76891
$563$	−23.4501	−0.988306	−0.494153	−	0.869375i	$-0.664522\pi$
$563$	−23.4501	−0.988306	−0.494153	+	0.869375i	$0.664522\pi$
$564$	0	0
$565$	1.34847	0.0567305
$566$	6.76742	0.284456
$567$	0	0
$568$	−45.7980	−1.92164
$569$	−3.14789	−0.131966	−0.0659831	−	0.997821i	$-0.521018\pi$
$569$	−3.14789	−0.131966	−0.0659831	+	0.997821i	$0.521018\pi$
$570$	0	0
$571$	−18.4495	−0.772087	−0.386044	−	0.922481i	$-0.626159\pi$
$571$	−18.4495	−0.772087	−0.386044	+	0.922481i	$0.626159\pi$
$572$	−8.86601	−0.370706
$573$	0	0
$574$	12.2474	0.511199
$575$	−14.1125	−0.588531
$576$	0	0
$577$	11.3031	0.470553	0.235276	−	0.971929i	$-0.424400\pi$
$577$	11.3031	0.470553	0.235276	+	0.971929i	$0.424400\pi$
$578$	29.4041	1.22305
$579$	0	0
$580$	12.2474	0.508548
$581$	1.99259	0.0826666
$582$	0	0
$583$	−14.3939	−0.596133
$584$	−21.4814	−0.888906
$585$	0	0
$586$	3.00000	0.123929
$587$	43.5404	1.79710	0.898552	−	0.438866i	$-0.144620\pi$
$587$	43.5404	1.79710	0.898552	+	0.438866i	$0.144620\pi$
$588$	0	0
$589$	2.55051	0.105092
$590$	3.72553	0.153378
$591$	0	0
$592$	−0.449490	−0.0184739
$593$	44.2240	1.81606	0.908032	−	0.418901i	$-0.137585\pi$
$593$	44.2240	1.81606	0.908032	+	0.418901i	$0.137585\pi$
$594$	0	0
$595$	0.989795	0.0405776
$596$	16.1051	0.659690
$597$	0	0
$598$	−16.8990	−0.691051
$599$	−19.2530	−0.786655	−0.393327	−	0.919399i	$-0.628676\pi$
$599$	−19.2530	−0.786655	−0.393327	+	0.919399i	$0.628676\pi$
$600$	0	0
$601$	−33.1464	−1.35207	−0.676036	−	0.736869i	$-0.736303\pi$
$601$	−33.1464	−1.35207	−0.676036	+	0.736869i	$0.736303\pi$
$602$	9.80930	0.399797
$603$	0	0
$604$	67.9444	2.76462
$605$	9.80930	0.398805
$606$	0	0
$607$	23.2474	0.943585	0.471792	−	0.881710i	$-0.343607\pi$
$607$	23.2474	0.943585	0.471792	+	0.881710i	$0.343607\pi$
$608$	4.43300	0.179782
$609$	0	0
$610$	15.5505	0.629622
$611$	4.66883	0.188881
$612$	0	0
$613$	−32.8990	−1.32878	−0.664389	−	0.747387i	$-0.731308\pi$
$613$	−32.8990	−1.32878	−0.664389	+	0.747387i	$0.731308\pi$
$614$	−20.5381	−0.828849
$615$	0	0
$616$	−1.95459	−0.0787528
$617$	−47.2659	−1.90285	−0.951427	−	0.307873i	$-0.900383\pi$
$617$	−47.2659	−1.90285	−0.951427	+	0.307873i	$0.900383\pi$
$618$	0	0
$619$	16.2020	0.651215	0.325608	−	0.945505i	$-0.394431\pi$
$619$	16.2020	0.651215	0.325608	+	0.945505i	$0.394431\pi$
$620$	−9.23166	−0.370752
$621$	0	0
$622$	23.6969	0.950161
$623$	2.67624	0.107221
$624$	0	0
$625$	9.69694	0.387878
$626$	30.9250	1.23601
$627$	0	0
$628$	1.20204	0.0479667
$629$	−0.943295	−0.0376116
$630$	0	0
$631$	31.3939	1.24977	0.624885	−	0.780717i	$-0.285146\pi$
$631$	31.3939	1.24977	0.624885	+	0.780717i	$0.285146\pi$
$632$	−16.9185	−0.672984
$633$	0	0
$634$	38.9444	1.54668
$635$	−6.08377	−0.241427
$636$	0	0
$637$	−13.5959	−0.538690
$638$	−10.1511	−0.401887
$639$	0	0
$640$	−20.9444	−0.827900
$641$	36.5372	1.44313	0.721565	−	0.692346i	$-0.243423\pi$
$641$	36.5372	1.44313	0.721565	+	0.692346i	$0.243423\pi$
$642$	0	0
$643$	3.34847	0.132051	0.0660254	−	0.997818i	$-0.478968\pi$
$643$	3.34847	0.132051	0.0660254	+	0.997818i	$0.478968\pi$
$644$	−5.61212	−0.221149
$645$	0	0
$646$	−4.89898	−0.192748
$647$	24.7353	0.972443	0.486222	−	0.873835i	$-0.338375\pi$
$647$	24.7353	0.972443	0.486222	+	0.873835i	$0.338375\pi$
$648$	0	0
$649$	−1.95459	−0.0767245
$650$	18.2037	0.714007
$651$	0	0
$652$	28.4495	1.11417
$653$	−25.4427	−0.995651	−0.497826	−	0.867277i	$-0.665868\pi$
$653$	−25.4427	−0.995651	−0.497826	+	0.867277i	$0.665868\pi$
$654$	0	0
$655$	21.1918	0.828034
$656$	11.6721	0.455718
$657$	0	0
$658$	2.44949	0.0954911
$659$	31.6325	1.23223	0.616114	−	0.787657i	$-0.288706\pi$
$659$	31.6325	1.23223	0.616114	+	0.787657i	$0.288706\pi$
$660$	0	0
$661$	0.404082	0.0157170	0.00785849	−	0.999969i	$-0.497499\pi$
$661$	0.404082	0.0157170	0.00785849	+	0.999969i	$0.497499\pi$
$662$	−50.4138	−1.95939
$663$	0	0
$664$	15.0000	0.582113
$665$	0.471647	0.0182897
$666$	0	0
$667$	−12.2474	−0.474223
$668$	−55.5543	−2.14946
$669$	0	0
$670$	30.2474	1.16856
$671$	−8.15854	−0.314957
$672$	0	0
$673$	24.8990	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$673$	24.8990	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$674$	26.4920	1.02043
$675$	0	0
$676$	−31.0454	−1.19405
$677$	−47.7376	−1.83470	−0.917352	−	0.398077i	$-0.869678\pi$
$677$	−47.7376	−1.83470	−0.917352	+	0.398077i	$0.869678\pi$
$678$	0	0
$679$	−4.69694	−0.180252
$680$	7.45107	0.285735
$681$	0	0
$682$	7.65153	0.292992
$683$	−40.0269	−1.53159	−0.765793	−	0.643087i	$-0.777653\pi$
$683$	−40.0269	−1.53159	−0.765793	+	0.643087i	$0.777653\pi$
$684$	0	0
$685$	−9.79796	−0.374361
$686$	−14.4781	−0.552778
$687$	0	0
$688$	9.34847	0.356407
$689$	22.4008	0.853404
$690$	0	0
$691$	18.0454	0.686480	0.343240	−	0.939248i	$-0.388476\pi$
$691$	18.0454	0.686480	0.343240	+	0.939248i	$0.388476\pi$
$692$	−25.7846	−0.980182
$693$	0	0
$694$	−15.5505	−0.590289
$695$	10.4930	0.398020
$696$	0	0
$697$	24.4949	0.927810
$698$	−57.4171	−2.17327
$699$	0	0
$700$	6.04541	0.228495
$701$	−44.2240	−1.67032	−0.835160	−	0.550008i	$-0.814625\pi$
$701$	−44.2240	−1.67032	−0.835160	+	0.550008i	$0.814625\pi$
$702$	0	0
$703$	−0.449490	−0.0169528
$704$	15.8693	0.598095
$705$	0	0
$706$	−41.3939	−1.55788
$707$	5.82412	0.219039
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	33.1534	1.24423
$711$	0	0
$712$	20.1464	0.755019
$713$	9.23166	0.345728
$714$	0	0
$715$	2.69694	0.100860
$716$	28.5906	1.06848
$717$	0	0
$718$	26.4495	0.987086
$719$	−53.2199	−1.98477	−0.992383	−	0.123188i	$-0.960688\pi$
$719$	−53.2199	−1.98477	−0.992383	+	0.123188i	$0.960688\pi$
$720$	0	0
$721$	3.50510	0.130537
$722$	−2.33441	−0.0868779
$723$	0	0
$724$	66.0454	2.45456
$725$	13.1930	0.489976
$726$	0	0
$727$	−9.39388	−0.348400	−0.174200	−	0.984710i	$-0.555734\pi$
$727$	−9.39388	−0.348400	−0.174200	+	0.984710i	$0.555734\pi$
$728$	3.04189	0.112740
$729$	0	0
$730$	15.5505	0.575550
$731$	19.6186	0.725620
$732$	0	0
$733$	−11.0454	−0.407971	−0.203986	−	0.978974i	$-0.565390\pi$
$733$	−11.0454	−0.407971	−0.203986	+	0.978974i	$0.565390\pi$
$734$	−52.9841	−1.95568
$735$	0	0
$736$	16.0454	0.591442
$737$	−15.8693	−0.584551
$738$	0	0
$739$	−29.3485	−1.07960	−0.539800	−	0.841793i	$-0.681500\pi$
$739$	−29.3485	−1.07960	−0.539800	+	0.841793i	$0.681500\pi$
$740$	1.62694	0.0598076
$741$	0	0
$742$	11.7526	0.431450
$743$	−6.76742	−0.248273	−0.124136	−	0.992265i	$-0.539616\pi$
$743$	−6.76742	−0.248273	−0.124136	+	0.992265i	$0.539616\pi$
$744$	0	0
$745$	−4.89898	−0.179485
$746$	46.7943	1.71326
$747$	0	0
$748$	−9.30306	−0.340154
$749$	0.259647	0.00948729
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	2.33441	0.0851273
$753$	0	0
$754$	15.7980	0.575328
$755$	−20.6679	−0.752182
$756$	0	0
$757$	2.30306	0.0837062	0.0418531	−	0.999124i	$-0.486674\pi$
$757$	2.30306	0.0837062	0.0418531	+	0.999124i	$0.486674\pi$
$758$	31.5027	1.14423
$759$	0	0
$760$	3.55051	0.128791
$761$	−10.0213	−0.363272	−0.181636	−	0.983366i	$-0.558139\pi$
$761$	−10.0213	−0.363272	−0.181636	+	0.983366i	$0.558139\pi$
$762$	0	0
$763$	7.30306	0.264389
$764$	−52.7482	−1.90836
$765$	0	0
$766$	−71.6413	−2.58851
$767$	3.04189	0.109836
$768$	0	0
$769$	−27.3939	−0.987848	−0.493924	−	0.869505i	$-0.664438\pi$
$769$	−27.3939	−0.987848	−0.493924	+	0.869505i	$0.664438\pi$
$770$	1.41494	0.0509910
$771$	0	0
$772$	16.2020	0.583124
$773$	28.9324	1.04063	0.520314	−	0.853975i	$-0.325815\pi$
$773$	28.9324	1.04063	0.520314	+	0.853975i	$0.325815\pi$
$774$	0	0
$775$	−9.94439	−0.357213
$776$	−35.3580	−1.26928
$777$	0	0
$778$	−61.8434	−2.21719
$779$	11.6721	0.418195
$780$	0	0
$781$	−17.3939	−0.622402
$782$	−17.7320	−0.634096
$783$	0	0
$784$	−6.79796	−0.242784
$785$	−0.365647	−0.0130505
$786$	0	0
$787$	5.24745	0.187051	0.0935257	−	0.995617i	$-0.470186\pi$
$787$	5.24745	0.187051	0.0935257	+	0.995617i	$0.470186\pi$
$788$	−78.8984	−2.81064
$789$	0	0
$790$	12.2474	0.435745
$791$	0.577648	0.0205388
$792$	0	0
$793$	12.6969	0.450882
$794$	30.3474	1.07699
$795$	0	0
$796$	48.2929	1.71169
$797$	−17.2842	−0.612238	−0.306119	−	0.951993i	$-0.599030\pi$
$797$	−17.2842	−0.612238	−0.306119	+	0.951993i	$0.599030\pi$
$798$	0	0
$799$	4.89898	0.173313
$800$	−17.2842	−0.611089
$801$	0	0
$802$	−77.6413	−2.74161
$803$	−8.15854	−0.287909
$804$	0	0
$805$	1.70714	0.0601689
$806$	−11.9079	−0.419438
$807$	0	0
$808$	43.8434	1.54240
$809$	−31.1609	−1.09556	−0.547779	−	0.836623i	$-0.684526\pi$
$809$	−31.1609	−1.09556	−0.547779	+	0.836623i	$0.684526\pi$
$810$	0	0
$811$	−23.8990	−0.839207	−0.419603	−	0.907708i	$-0.637831\pi$
$811$	−23.8990	−0.839207	−0.419603	+	0.907708i	$0.637831\pi$
$812$	5.24648	0.184115
$813$	0	0
$814$	−1.34847	−0.0472638
$815$	−8.65401	−0.303137
$816$	0	0
$817$	9.34847	0.327062
$818$	51.9348	1.81586
$819$	0	0
$820$	−42.2474	−1.47534
$821$	21.3516	0.745174	0.372587	−	0.927997i	$-0.378471\pi$
$821$	21.3516	0.745174	0.372587	+	0.927997i	$0.378471\pi$
$822$	0	0
$823$	19.7526	0.688531	0.344265	−	0.938872i	$-0.388128\pi$
$823$	19.7526	0.688531	0.344265	+	0.938872i	$0.388128\pi$
$824$	26.3860	0.919201
$825$	0	0
$826$	1.59592	0.0555291
$827$	−11.8019	−0.410392	−0.205196	−	0.978721i	$-0.565783\pi$
$827$	−11.8019	−0.410392	−0.205196	+	0.978721i	$0.565783\pi$
$828$	0	0
$829$	24.2929	0.843726	0.421863	−	0.906660i	$-0.361376\pi$
$829$	24.2929	0.843726	0.421863	+	0.906660i	$0.361376\pi$
$830$	−10.8586	−0.376907
$831$	0	0
$832$	−24.6969	−0.856212
$833$	−14.2661	−0.494292
$834$	0	0
$835$	16.8990	0.584813
$836$	−4.43300	−0.153319
$837$	0	0
$838$	−28.3485	−0.979282
$839$	−45.0613	−1.55569	−0.777845	−	0.628456i	$-0.783687\pi$
$839$	−45.0613	−1.55569	−0.777845	+	0.628456i	$0.783687\pi$
$840$	0	0
$841$	−17.5505	−0.605190
$842$	76.7998	2.64670
$843$	0	0
$844$	−8.10102	−0.278849
$845$	9.44366	0.324872
$846$	0	0
$847$	4.20204	0.144384
$848$	11.2004	0.384624
$849$	0	0
$850$	19.1010	0.655160
$851$	−1.62694	−0.0557709
$852$	0	0
$853$	1.69694	0.0581021	0.0290510	−	0.999578i	$-0.490751\pi$
$853$	1.69694	0.0581021	0.0290510	+	0.999578i	$0.490751\pi$
$854$	6.66142	0.227949
$855$	0	0
$856$	1.95459	0.0668066
$857$	40.2627	1.37535	0.687674	−	0.726020i	$-0.258632\pi$
$857$	40.2627	1.37535	0.687674	+	0.726020i	$0.258632\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−33.8371	−1.15384
$861$	0	0
$862$	3.55051	0.120931
$863$	17.9917	0.612443	0.306222	−	0.951960i	$-0.400935\pi$
$863$	17.9917	0.612443	0.306222	+	0.951960i	$0.400935\pi$
$864$	0	0
$865$	7.84337	0.266682
$866$	−49.8362	−1.69350
$867$	0	0
$868$	−3.95459	−0.134228
$869$	−6.42559	−0.217973
$870$	0	0
$871$	24.6969	0.836824
$872$	54.9766	1.86174
$873$	0	0
$874$	−8.44949	−0.285808
$875$	−4.19718	−0.141891
$876$	0	0
$877$	42.6515	1.44024	0.720120	−	0.693850i	$-0.244087\pi$
$877$	42.6515	1.44024	0.720120	+	0.693850i	$0.244087\pi$
$878$	−73.9938	−2.49717
$879$	0	0
$880$	1.34847	0.0454569
$881$	5.14048	0.173187	0.0865935	−	0.996244i	$-0.472402\pi$
$881$	5.14048	0.173187	0.0865935	+	0.996244i	$0.472402\pi$
$882$	0	0
$883$	−32.0454	−1.07841	−0.539207	−	0.842173i	$-0.681276\pi$
$883$	−32.0454	−1.07841	−0.539207	+	0.842173i	$0.681276\pi$
$884$	14.4781	0.486952
$885$	0	0
$886$	47.9444	1.61072
$887$	−28.1190	−0.944143	−0.472071	−	0.881560i	$-0.656494\pi$
$887$	−28.1190	−0.944143	−0.472071	+	0.881560i	$0.656494\pi$
$888$	0	0
$889$	−2.60612	−0.0874066
$890$	−14.5841	−0.488861
$891$	0	0
$892$	79.3383	2.65644
$893$	2.33441	0.0781182
$894$	0	0
$895$	−8.69694	−0.290707
$896$	−8.97201	−0.299734
$897$	0	0
$898$	31.0454	1.03600
$899$	−8.63019	−0.287833
$900$	0	0
$901$	23.5051	0.783069
$902$	35.0162	1.16591
$903$	0	0
$904$	4.34847	0.144628
$905$	−20.0903	−0.667823
$906$	0	0
$907$	−41.5959	−1.38117	−0.690585	−	0.723251i	$-0.742647\pi$
$907$	−41.5959	−1.38117	−0.690585	+	0.723251i	$0.742647\pi$
$908$	−26.5980	−0.882687
$909$	0	0
$910$	−2.20204	−0.0729969
$911$	−19.7246	−0.653505	−0.326753	−	0.945110i	$-0.605954\pi$
$911$	−19.7246	−0.653505	−0.326753	+	0.945110i	$0.605954\pi$
$912$	0	0
$913$	5.69694	0.188541
$914$	70.3743	2.32777
$915$	0	0
$916$	−7.24745	−0.239462
$917$	9.07801	0.299782
$918$	0	0
$919$	−21.7526	−0.717550	−0.358775	−	0.933424i	$-0.616806\pi$
$919$	−21.7526	−0.717550	−0.358775	+	0.933424i	$0.616806\pi$
$920$	12.8512	0.423691
$921$	0	0
$922$	−16.8990	−0.556538
$923$	27.0697	0.891009
$924$	0	0
$925$	1.75255	0.0576235
$926$	−16.1051	−0.529246
$927$	0	0
$928$	−15.0000	−0.492399
$929$	20.4083	0.669573	0.334787	−	0.942294i	$-0.391336\pi$
$929$	20.4083	0.669573	0.334787	+	0.942294i	$0.391336\pi$
$930$	0	0
$931$	−6.79796	−0.222794
$932$	62.4277	2.04489
$933$	0	0
$934$	37.0454	1.21216
$935$	2.82988	0.0925471
$936$	0	0
$937$	10.3939	0.339553	0.169777	−	0.985483i	$-0.445695\pi$
$937$	10.3939	0.339553	0.169777	+	0.985483i	$0.445695\pi$
$938$	12.9572	0.423067
$939$	0	0
$940$	−8.44949	−0.275592
$941$	−7.81671	−0.254818	−0.127409	−	0.991850i	$-0.540666\pi$
$941$	−7.81671	−0.254818	−0.127409	+	0.991850i	$0.540666\pi$
$942$	0	0
$943$	42.2474	1.37577
$944$	1.52094	0.0495025
$945$	0	0
$946$	28.0454	0.911835
$947$	−8.05254	−0.261672	−0.130836	−	0.991404i	$-0.541766\pi$
$947$	−8.05254	−0.261672	−0.130836	+	0.991404i	$0.541766\pi$
$948$	0	0
$949$	12.6969	0.412160
$950$	9.10183	0.295302
$951$	0	0
$952$	3.19184	0.103448
$953$	−39.2134	−1.27025	−0.635123	−	0.772411i	$-0.719051\pi$
$953$	−39.2134	−1.27025	−0.635123	+	0.772411i	$0.719051\pi$
$954$	0	0
$955$	16.0454	0.519217
$956$	72.4728	2.34394
$957$	0	0
$958$	−91.8434	−2.96732
$959$	−4.19718	−0.135534
$960$	0	0
$961$	−24.4949	−0.790158
$962$	2.09859	0.0676613
$963$	0	0
$964$	−29.8434	−0.961190
$965$	−4.92848	−0.158653
$966$	0	0
$967$	−14.0454	−0.451670	−0.225835	−	0.974166i	$-0.572511\pi$
$967$	−14.0454	−0.451670	−0.225835	+	0.974166i	$0.572511\pi$
$968$	31.6325	1.01671
$969$	0	0
$970$	25.5959	0.821835
$971$	37.8223	1.21377	0.606887	−	0.794788i	$-0.292418\pi$
$971$	37.8223	1.21377	0.606887	+	0.794788i	$0.292418\pi$
$972$	0	0
$973$	4.49490	0.144100
$974$	19.4888	0.624461
$975$	0	0
$976$	6.34847	0.203210
$977$	13.5587	0.433780	0.216890	−	0.976196i	$-0.430409\pi$
$977$	13.5587	0.433780	0.216890	+	0.976196i	$0.430409\pi$
$978$	0	0
$979$	7.65153	0.244544
$980$	24.6054	0.785992
$981$	0	0
$982$	32.1464	1.02583
$983$	−29.0623	−0.926942	−0.463471	−	0.886112i	$-0.653396\pi$
$983$	−29.0623	−0.926942	−0.463471	+	0.886112i	$0.653396\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	16.5767	0.527910
$987$	0	0
$988$	6.89898	0.219486
$989$	33.8371	1.07596
$990$	0	0
$991$	−18.6969	−0.593928	−0.296964	−	0.954889i	$-0.595974\pi$
$991$	−18.6969	−0.593928	−0.296964	+	0.954889i	$0.595974\pi$
$992$	11.3064	0.358979
$993$	0	0
$994$	14.2020	0.450461
$995$	−14.6901	−0.465709
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	79.3701	2.51242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.h.1.1	✓	4
3.2	odd	2	inner	513.2.a.h.1.4	yes	4
4.3	odd	2		8208.2.a.bu.1.2		4
12.11	even	2		8208.2.a.bu.1.3		4
19.18	odd	2		9747.2.a.be.1.4		4
57.56	even	2		9747.2.a.be.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.h.1.1	✓	4	1.1	even	1	trivial
513.2.a.h.1.4	yes	4	3.2	odd	2	inner
8208.2.a.bu.1.2		4	4.3	odd	2
8208.2.a.bu.1.3		4	12.11	even	2
9747.2.a.be.1.1		4	57.56	even	2
9747.2.a.be.1.4		4	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)

\(f(q)\)	\(=\)	\(q-2.33441 q^{2} +3.44949 q^{4} -1.04930 q^{5} -0.449490 q^{7} -3.38371 q^{8} +2.44949 q^{10} -1.28512 q^{11} +2.00000 q^{13} +1.04930 q^{14} +1.00000 q^{16} +2.09859 q^{17} +1.00000 q^{19} -3.61953 q^{20} +3.00000 q^{22} +3.61953 q^{23} -3.89898 q^{25} -4.66883 q^{26} -1.55051 q^{28} -3.38371 q^{29} +2.55051 q^{31} +4.43300 q^{32} -4.89898 q^{34} +0.471647 q^{35} -0.449490 q^{37} -2.33441 q^{38} +3.55051 q^{40} +11.6721 q^{41} +9.34847 q^{43} -4.43300 q^{44} -8.44949 q^{46} +2.33441 q^{47} -6.79796 q^{49} +9.10183 q^{50} +6.89898 q^{52} +11.2004 q^{53} +1.34847 q^{55} +1.52094 q^{56} +7.89898 q^{58} +1.52094 q^{59} +6.34847 q^{61} -5.95395 q^{62} -12.3485 q^{64} -2.09859 q^{65} +12.3485 q^{67} +7.23907 q^{68} -1.10102 q^{70} +13.5348 q^{71} +6.34847 q^{73} +1.04930 q^{74} +3.44949 q^{76} +0.577648 q^{77} +5.00000 q^{79} -1.04930 q^{80} -27.2474 q^{82} -4.43300 q^{83} -2.20204 q^{85} -21.8232 q^{86} +4.34847 q^{88} -5.95395 q^{89} -0.898979 q^{91} +12.4855 q^{92} -5.44949 q^{94} -1.04930 q^{95} +10.4495 q^{97} +15.8693 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 4 q^{19} + 12 q^{22} + 4 q^{25} - 16 q^{28} + 20 q^{31} + 8 q^{37} + 24 q^{40} + 8 q^{43} - 24 q^{46} + 12 q^{49} + 8 q^{52} - 24 q^{55} + 12 q^{58} - 4 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^13 + 4 * q^16 + 4 * q^19 + 12 * q^22 + 4 * q^25 - 16 * q^28 + 20 * q^31 + 8 * q^37 + 24 * q^40 + 8 * q^43 - 24 * q^46 + 12 * q^49 + 8 * q^52 - 24 * q^55 + 12 * q^58 - 4 * q^61 - 20 * q^64 + 20 * q^67 - 24 * q^70 - 4 * q^73 + 4 * q^76 + 20 * q^79 - 60 * q^82 - 48 * q^85 - 12 * q^88 + 16 * q^91 - 12 * q^94 + 32 * q^97

Introduction

L-functions

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Varieties

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Database

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