LMFDB - Embedded newform 8379.2.a.cx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,0,20,0,0,0,0,0,24,0,0,24,0,0,28,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 30 x^{18} + 373 x^{16} - 2492 x^{14} + 9710 x^{12} - 22456 x^{10} + 30154 x^{8} - 22292 x^{6} + \cdots + 49 $ x^20 - 30x^18 + 373x^16 - 2492x^14 + 9710x^12 - 22456x^10 + 30154x^8 - 22292x^6 + 8401x^4 - 1354*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.39042$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39042 q^{2} +3.71410 q^{4} +2.19330 q^{5} -4.09743 q^{8} -5.24290 q^{10} +6.36271 q^{11} +4.92923 q^{13} +2.36636 q^{16} -3.58339 q^{17} +1.00000 q^{19} +8.14614 q^{20} -15.2095 q^{22} +3.94863 q^{23} -0.189447 q^{25} -11.7829 q^{26} -9.74344 q^{29} +0.500669 q^{31} +2.53826 q^{32} +8.56581 q^{34} -4.22117 q^{37} -2.39042 q^{38} -8.98688 q^{40} -10.7537 q^{41} +6.83768 q^{43} +23.6318 q^{44} -9.43889 q^{46} -8.65488 q^{47} +0.452859 q^{50} +18.3077 q^{52} -6.92482 q^{53} +13.9553 q^{55} +23.2909 q^{58} +15.3086 q^{59} +9.25254 q^{61} -1.19681 q^{62} -10.8002 q^{64} +10.8113 q^{65} +13.8432 q^{67} -13.3091 q^{68} +2.00326 q^{71} +2.90535 q^{73} +10.0904 q^{74} +3.71410 q^{76} -5.47327 q^{79} +5.19014 q^{80} +25.7058 q^{82} -2.75270 q^{83} -7.85945 q^{85} -16.3449 q^{86} -26.0707 q^{88} +1.72013 q^{89} +14.6656 q^{92} +20.6888 q^{94} +2.19330 q^{95} +4.29271 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} + 24 q^{10} + 24 q^{13} + 28 q^{16} + 20 q^{19} - 16 q^{22} + 28 q^{25} + 40 q^{31} + 32 q^{34} - 16 q^{37} + 56 q^{40} + 8 q^{43} - 16 q^{46} + 64 q^{52} + 32 q^{55} + 8 q^{58} + 80 q^{61}+ \cdots + 64 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 24 * q^10 + 24 * q^13 + 28 * q^16 + 20 * q^19 - 16 * q^22 + 28 * q^25 + 40 * q^31 + 32 * q^34 - 16 * q^37 + 56 * q^40 + 8 * q^43 - 16 * q^46 + 64 * q^52 + 32 * q^55 + 8 * q^58 + 80 * q^61 + 12 * q^64 + 16 * q^67 + 48 * q^73 + 20 * q^76 - 16 * q^79 + 32 * q^82 - 16 * q^85 - 32 * q^88 + 96 * q^94 + 64 * 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 $

$2$	−2.39042	−1.69028	−0.845141	−	0.534544i	$-0.820484\pi$
$2$	−2.39042	−1.69028	−0.845141	+	0.534544i	$0.820484\pi$
$3$	0	0
$3$	0	0
$4$	3.71410	1.85705
$5$	2.19330	0.980872	0.490436	−	0.871477i	$-0.336837\pi$
$5$	2.19330	0.980872	0.490436	+	0.871477i	$0.336837\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−4.09743	−1.44866
$9$	0	0
$10$	−5.24290	−1.65795
$11$	6.36271	1.91843	0.959214	−	0.282680i	$-0.0912233\pi$
$11$	6.36271	1.91843	0.959214	+	0.282680i	$0.0912233\pi$
$12$	0	0
$13$	4.92923	1.36712	0.683562	−	0.729893i	$-0.260430\pi$
$13$	4.92923	1.36712	0.683562	+	0.729893i	$0.260430\pi$
$14$	0	0
$15$	0	0
$16$	2.36636	0.591591
$17$	−3.58339	−0.869101	−0.434550	−	0.900648i	$-0.643093\pi$
$17$	−3.58339	−0.869101	−0.434550	+	0.900648i	$0.643093\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	8.14614	1.82153
$21$	0	0
$22$	−15.2095	−3.24269
$23$	3.94863	0.823347	0.411673	−	0.911331i	$-0.364944\pi$
$23$	3.94863	0.823347	0.411673	+	0.911331i	$0.364944\pi$
$24$	0	0
$25$	−0.189447	−0.0378895
$26$	−11.7829	−2.31082
$27$	0	0
$28$	0	0
$29$	−9.74344	−1.80931	−0.904656	−	0.426143i	$-0.859872\pi$
$29$	−9.74344	−1.80931	−0.904656	+	0.426143i	$0.859872\pi$
$30$	0	0
$31$	0.500669	0.0899228	0.0449614	−	0.998989i	$-0.485684\pi$
$31$	0.500669	0.0899228	0.0449614	+	0.998989i	$0.485684\pi$
$32$	2.53826	0.448705
$33$	0	0
$34$	8.56581	1.46903
$35$	0	0
$36$	0	0
$37$	−4.22117	−0.693956	−0.346978	−	0.937873i	$-0.612792\pi$
$37$	−4.22117	−0.693956	−0.346978	+	0.937873i	$0.612792\pi$
$38$	−2.39042	−0.387777
$39$	0	0
$40$	−8.98688	−1.42095
$41$	−10.7537	−1.67944	−0.839722	−	0.543016i	$-0.817282\pi$
$41$	−10.7537	−1.67944	−0.839722	+	0.543016i	$0.817282\pi$
$42$	0	0
$43$	6.83768	1.04274	0.521368	−	0.853332i	$-0.325422\pi$
$43$	6.83768	1.04274	0.521368	+	0.853332i	$0.325422\pi$
$44$	23.6318	3.56262
$45$	0	0
$46$	−9.43889	−1.39169
$47$	−8.65488	−1.26244	−0.631222	−	0.775602i	$-0.717446\pi$
$47$	−8.65488	−1.26244	−0.631222	+	0.775602i	$0.717446\pi$
$48$	0	0
$49$	0	0
$50$	0.452859	0.0640439
$51$	0	0
$52$	18.3077	2.53882
$53$	−6.92482	−0.951197	−0.475599	−	0.879662i	$-0.657768\pi$
$53$	−6.92482	−0.951197	−0.475599	+	0.879662i	$0.657768\pi$
$54$	0	0
$55$	13.9553	1.88173
$56$	0	0
$57$	0	0
$58$	23.2909	3.05825
$59$	15.3086	1.99301	0.996507	−	0.0835042i	$-0.0266112\pi$
$59$	15.3086	1.99301	0.996507	+	0.0835042i	$0.0266112\pi$
$60$	0	0
$61$	9.25254	1.18467	0.592333	−	0.805693i	$-0.298207\pi$
$61$	9.25254	1.18467	0.592333	+	0.805693i	$0.298207\pi$
$62$	−1.19681	−0.151995
$63$	0	0
$64$	−10.8002	−1.35003
$65$	10.8113	1.34097
$66$	0	0
$67$	13.8432	1.69122	0.845609	−	0.533802i	$-0.179237\pi$
$67$	13.8432	1.69122	0.845609	+	0.533802i	$0.179237\pi$
$68$	−13.3091	−1.61397
$69$	0	0
$70$	0	0
$71$	2.00326	0.237744	0.118872	−	0.992910i	$-0.462072\pi$
$71$	2.00326	0.237744	0.118872	+	0.992910i	$0.462072\pi$
$72$	0	0
$73$	2.90535	0.340045	0.170023	−	0.985440i	$-0.445616\pi$
$73$	2.90535	0.340045	0.170023	+	0.985440i	$0.445616\pi$
$74$	10.0904	1.17298
$75$	0	0
$76$	3.71410	0.426037
$77$	0	0
$78$	0	0
$79$	−5.47327	−0.615791	−0.307895	−	0.951420i	$-0.599625\pi$
$79$	−5.47327	−0.615791	−0.307895	+	0.951420i	$0.599625\pi$
$80$	5.19014	0.580275
$81$	0	0
$82$	25.7058	2.83873
$83$	−2.75270	−0.302148	−0.151074	−	0.988522i	$-0.548273\pi$
$83$	−2.75270	−0.302148	−0.151074	+	0.988522i	$0.548273\pi$
$84$	0	0
$85$	−7.85945	−0.852477
$86$	−16.3449	−1.76252
$87$	0	0
$88$	−26.0707	−2.77915
$89$	1.72013	0.182333	0.0911666	−	0.995836i	$-0.470940\pi$
$89$	1.72013	0.182333	0.0911666	+	0.995836i	$0.470940\pi$
$90$	0	0
$91$	0	0
$92$	14.6656	1.52900
$93$	0	0
$94$	20.6888	2.13389
$95$	2.19330	0.225028
$96$	0	0
$97$	4.29271	0.435858	0.217929	−	0.975965i	$-0.430070\pi$
$97$	4.29271	0.435858	0.217929	+	0.975965i	$0.430070\pi$
$98$	0	0
$99$	0	0
$100$	−0.703627	−0.0703627
$101$	−0.00555371	−0.000552615 0	−0.000276307	−	1.00000i	$-0.500088\pi$
$101$	−0.00555371	−0.000552615 0	−0.000276307		1.00000i	$0.500088\pi$
$102$	0	0
$103$	16.6252	1.63812	0.819062	−	0.573704i	$-0.194494\pi$
$103$	16.6252	1.63812	0.819062	+	0.573704i	$0.194494\pi$
$104$	−20.1972	−1.98050
$105$	0	0
$106$	16.5532	1.60779
$107$	−7.97389	−0.770865	−0.385432	−	0.922736i	$-0.625948\pi$
$107$	−7.97389	−0.770865	−0.385432	+	0.922736i	$0.625948\pi$
$108$	0	0
$109$	2.75836	0.264203	0.132102	−	0.991236i	$-0.457828\pi$
$109$	2.75836	0.264203	0.132102	+	0.991236i	$0.457828\pi$
$110$	−33.3590	−3.18066
$111$	0	0
$112$	0	0
$113$	4.26115	0.400856	0.200428	−	0.979708i	$-0.435767\pi$
$113$	4.26115	0.400856	0.200428	+	0.979708i	$0.435767\pi$
$114$	0	0
$115$	8.66053	0.807598
$116$	−36.1882	−3.35999
$117$	0	0
$118$	−36.5941	−3.36876
$119$	0	0
$120$	0	0
$121$	29.4841	2.68037
$122$	−22.1174	−2.00242
$123$	0	0
$124$	1.85954	0.166991
$125$	−11.3820	−1.01804
$126$	0	0
$127$	−16.9218	−1.50157	−0.750785	−	0.660546i	$-0.770325\pi$
$127$	−16.9218	−1.50157	−0.750785	+	0.660546i	$0.770325\pi$
$128$	20.7406	1.83322
$129$	0	0
$130$	−25.8435	−2.26662
$131$	8.87465	0.775382	0.387691	−	0.921789i	$-0.373273\pi$
$131$	8.87465	0.775382	0.387691	+	0.921789i	$0.373273\pi$
$132$	0	0
$133$	0	0
$134$	−33.0911	−2.85864
$135$	0	0
$136$	14.6827	1.25903
$137$	−5.92450	−0.506165	−0.253082	−	0.967445i	$-0.581444\pi$
$137$	−5.92450	−0.506165	−0.253082	+	0.967445i	$0.581444\pi$
$138$	0	0
$139$	11.2138	0.951141	0.475570	−	0.879678i	$-0.342242\pi$
$139$	11.2138	0.951141	0.475570	+	0.879678i	$0.342242\pi$
$140$	0	0
$141$	0	0
$142$	−4.78864	−0.401854
$143$	31.3633	2.62273
$144$	0	0
$145$	−21.3703	−1.77470
$146$	−6.94500	−0.574772
$147$	0	0
$148$	−15.6779	−1.28871
$149$	17.7866	1.45713	0.728566	−	0.684975i	$-0.240187\pi$
$149$	17.7866	1.45713	0.728566	+	0.684975i	$0.240187\pi$
$150$	0	0
$151$	13.6969	1.11464	0.557321	−	0.830297i	$-0.311829\pi$
$151$	13.6969	1.11464	0.557321	+	0.830297i	$0.311829\pi$
$152$	−4.09743	−0.332345
$153$	0	0
$154$	0	0
$155$	1.09812	0.0882027
$156$	0	0
$157$	2.51975	0.201098	0.100549	−	0.994932i	$-0.467940\pi$
$157$	2.51975	0.201098	0.100549	+	0.994932i	$0.467940\pi$
$158$	13.0834	1.04086
$159$	0	0
$160$	5.56715	0.440122
$161$	0	0
$162$	0	0
$163$	13.6563	1.06965	0.534823	−	0.844964i	$-0.320378\pi$
$163$	13.6563	1.06965	0.534823	+	0.844964i	$0.320378\pi$
$164$	−39.9403	−3.11882
$165$	0	0
$166$	6.58010	0.510715
$167$	−7.42644	−0.574675	−0.287337	−	0.957829i	$-0.592770\pi$
$167$	−7.42644	−0.574675	−0.287337	+	0.957829i	$0.592770\pi$
$168$	0	0
$169$	11.2973	0.869027
$170$	18.7874	1.44093
$171$	0	0
$172$	25.3959	1.93642
$173$	11.9783	0.910696	0.455348	−	0.890313i	$-0.349515\pi$
$173$	11.9783	0.910696	0.455348	+	0.890313i	$0.349515\pi$
$174$	0	0
$175$	0	0
$176$	15.0565	1.13492
$177$	0	0
$178$	−4.11183	−0.308195
$179$	−6.47186	−0.483730	−0.241865	−	0.970310i	$-0.577759\pi$
$179$	−6.47186	−0.483730	−0.241865	+	0.970310i	$0.577759\pi$
$180$	0	0
$181$	−2.55804	−0.190138	−0.0950688	−	0.995471i	$-0.530307\pi$
$181$	−2.55804	−0.190138	−0.0950688	+	0.995471i	$0.530307\pi$
$182$	0	0
$183$	0	0
$184$	−16.1792	−1.19275
$185$	−9.25828	−0.680682
$186$	0	0
$187$	−22.8001	−1.66731
$188$	−32.1451	−2.34443
$189$	0	0
$190$	−5.24290	−0.380360
$191$	23.7476	1.71832	0.859158	−	0.511710i	$-0.170988\pi$
$191$	23.7476	1.71832	0.859158	+	0.511710i	$0.170988\pi$
$192$	0	0
$193$	19.5170	1.40486	0.702432	−	0.711751i	$-0.252098\pi$
$193$	19.5170	1.40486	0.702432	+	0.711751i	$0.252098\pi$
$194$	−10.2614	−0.736723
$195$	0	0
$196$	0	0
$197$	−5.10766	−0.363906	−0.181953	−	0.983307i	$-0.558242\pi$
$197$	−5.10766	−0.363906	−0.181953	+	0.983307i	$0.558242\pi$
$198$	0	0
$199$	12.4766	0.884442	0.442221	−	0.896906i	$-0.354191\pi$
$199$	12.4766	0.884442	0.442221	+	0.896906i	$0.354191\pi$
$200$	0.776247	0.0548889
$201$	0	0
$202$	0.0132757	0.000934074 0
$203$	0	0
$204$	0	0
$205$	−23.5860	−1.64732
$206$	−39.7411	−2.76889
$207$	0	0
$208$	11.6644	0.808778
$209$	6.36271	0.440118
$210$	0	0
$211$	−6.77802	−0.466618	−0.233309	−	0.972403i	$-0.574955\pi$
$211$	−6.77802	−0.466618	−0.233309	+	0.972403i	$0.574955\pi$
$212$	−25.7195	−1.76642
$213$	0	0
$214$	19.0609	1.30298
$215$	14.9971	1.02279
$216$	0	0
$217$	0	0
$218$	−6.59364	−0.446578
$219$	0	0
$220$	51.8315	3.49448
$221$	−17.6634	−1.18817
$222$	0	0
$223$	−7.01204	−0.469561	−0.234781	−	0.972048i	$-0.575437\pi$
$223$	−7.01204	−0.469561	−0.234781	+	0.972048i	$0.575437\pi$
$224$	0	0
$225$	0	0
$226$	−10.1859	−0.677559
$227$	16.6020	1.10192	0.550958	−	0.834533i	$-0.314263\pi$
$227$	16.6020	1.10192	0.550958	+	0.834533i	$0.314263\pi$
$228$	0	0
$229$	16.9021	1.11692	0.558461	−	0.829530i	$-0.311392\pi$
$229$	16.9021	1.11692	0.558461	+	0.829530i	$0.311392\pi$
$230$	−20.7023	−1.36507
$231$	0	0
$232$	39.9231	2.62108
$233$	12.2730	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$233$	12.2730	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$234$	0	0
$235$	−18.9827	−1.23830
$236$	56.8579	3.70113
$237$	0	0
$238$	0	0
$239$	1.65508	0.107059	0.0535293	−	0.998566i	$-0.482953\pi$
$239$	1.65508	0.107059	0.0535293	+	0.998566i	$0.482953\pi$
$240$	0	0
$241$	−22.5871	−1.45496	−0.727482	−	0.686127i	$-0.759309\pi$
$241$	−22.5871	−1.45496	−0.727482	+	0.686127i	$0.759309\pi$
$242$	−70.4793	−4.53058
$243$	0	0
$244$	34.3649	2.19999
$245$	0	0
$246$	0	0
$247$	4.92923	0.313640
$248$	−2.05145	−0.130267
$249$	0	0
$250$	27.2078	1.72077
$251$	27.3643	1.72722	0.863609	−	0.504163i	$-0.168199\pi$
$251$	27.3643	1.72722	0.863609	+	0.504163i	$0.168199\pi$
$252$	0	0
$253$	25.1240	1.57953
$254$	40.4503	2.53808
$255$	0	0
$256$	−27.9782	−1.74864
$257$	−25.9466	−1.61851	−0.809254	−	0.587460i	$-0.800128\pi$
$257$	−25.9466	−1.61851	−0.809254	+	0.587460i	$0.800128\pi$
$258$	0	0
$259$	0	0
$260$	40.1542	2.49026
$261$	0	0
$262$	−21.2141	−1.31061
$263$	14.0020	0.863399	0.431699	−	0.902018i	$-0.357914\pi$
$263$	14.0020	0.863399	0.431699	+	0.902018i	$0.357914\pi$
$264$	0	0
$265$	−15.1882	−0.933003
$266$	0	0
$267$	0	0
$268$	51.4152	3.14068
$269$	−4.50634	−0.274756	−0.137378	−	0.990519i	$-0.543868\pi$
$269$	−4.50634	−0.274756	−0.137378	+	0.990519i	$0.543868\pi$
$270$	0	0
$271$	11.8141	0.717656	0.358828	−	0.933404i	$-0.383177\pi$
$271$	11.8141	0.717656	0.358828	+	0.933404i	$0.383177\pi$
$272$	−8.47961	−0.514152
$273$	0	0
$274$	14.1621	0.855561
$275$	−1.20540	−0.0726882
$276$	0	0
$277$	13.0003	0.781114	0.390557	−	0.920579i	$-0.372282\pi$
$277$	13.0003	0.781114	0.390557	+	0.920579i	$0.372282\pi$
$278$	−26.8056	−1.60770
$279$	0	0
$280$	0	0
$281$	−23.6212	−1.40912	−0.704560	−	0.709644i	$-0.748856\pi$
$281$	−23.6212	−1.40912	−0.704560	+	0.709644i	$0.748856\pi$
$282$	0	0
$283$	−28.1543	−1.67360	−0.836801	−	0.547507i	$-0.815577\pi$
$283$	−28.1543	−1.67360	−0.836801	+	0.547507i	$0.815577\pi$
$284$	7.44033	0.441502
$285$	0	0
$286$	−74.9714	−4.43315
$287$	0	0
$288$	0	0
$289$	−4.15929	−0.244664
$290$	51.0839	2.99975
$291$	0	0
$292$	10.7908	0.631482
$293$	8.39057	0.490182	0.245091	−	0.969500i	$-0.421182\pi$
$293$	8.39057	0.490182	0.245091	+	0.969500i	$0.421182\pi$
$294$	0	0
$295$	33.5764	1.95489
$296$	17.2959	1.00531
$297$	0	0
$298$	−42.5174	−2.46296
$299$	19.4637	1.12562
$300$	0	0
$301$	0	0
$302$	−32.7414	−1.88406
$303$	0	0
$304$	2.36636	0.135720
$305$	20.2936	1.16201
$306$	0	0
$307$	9.48925	0.541580	0.270790	−	0.962638i	$-0.412715\pi$
$307$	9.48925	0.541580	0.270790	+	0.962638i	$0.412715\pi$
$308$	0	0
$309$	0	0
$310$	−2.62496	−0.149087
$311$	15.3798	0.872109	0.436054	−	0.899920i	$-0.356376\pi$
$311$	15.3798	0.872109	0.436054	+	0.899920i	$0.356376\pi$
$312$	0	0
$313$	−2.15660	−0.121898	−0.0609490	−	0.998141i	$-0.519413\pi$
$313$	−2.15660	−0.121898	−0.0609490	+	0.998141i	$0.519413\pi$
$314$	−6.02326	−0.339912
$315$	0	0
$316$	−20.3283	−1.14356
$317$	2.11686	0.118895	0.0594475	−	0.998231i	$-0.481066\pi$
$317$	2.11686	0.118895	0.0594475	+	0.998231i	$0.481066\pi$
$318$	0	0
$319$	−61.9947	−3.47104
$320$	−23.6881	−1.32421
$321$	0	0
$322$	0	0
$323$	−3.58339	−0.199385
$324$	0	0
$325$	−0.933830	−0.0517996
$326$	−32.6443	−1.80800
$327$	0	0
$328$	44.0625	2.43294
$329$	0	0
$330$	0	0
$331$	0.891779	0.0490166	0.0245083	−	0.999700i	$-0.492198\pi$
$331$	0.891779	0.0490166	0.0245083	+	0.999700i	$0.492198\pi$
$332$	−10.2238	−0.561104
$333$	0	0
$334$	17.7523	0.971362
$335$	30.3623	1.65887
$336$	0	0
$337$	−18.4091	−1.00281	−0.501405	−	0.865213i	$-0.667183\pi$
$337$	−18.4091	−1.00281	−0.501405	+	0.865213i	$0.667183\pi$
$338$	−27.0054	−1.46890
$339$	0	0
$340$	−29.1908	−1.58309
$341$	3.18561	0.172510
$342$	0	0
$343$	0	0
$344$	−28.0169	−1.51057
$345$	0	0
$346$	−28.6333	−1.53933
$347$	−18.8503	−1.01194	−0.505968	−	0.862552i	$-0.668865\pi$
$347$	−18.8503	−1.01194	−0.505968	+	0.862552i	$0.668865\pi$
$348$	0	0
$349$	24.5754	1.31549	0.657746	−	0.753240i	$-0.271510\pi$
$349$	24.5754	1.31549	0.657746	+	0.753240i	$0.271510\pi$
$350$	0	0
$351$	0	0
$352$	16.1502	0.860808
$353$	−23.0527	−1.22697	−0.613486	−	0.789705i	$-0.710233\pi$
$353$	−23.0527	−1.22697	−0.613486	+	0.789705i	$0.710233\pi$
$354$	0	0
$355$	4.39375	0.233196
$356$	6.38874	0.338602
$357$	0	0
$358$	15.4705	0.817640
$359$	−8.54108	−0.450781	−0.225390	−	0.974269i	$-0.572366\pi$
$359$	−8.54108	−0.450781	−0.225390	+	0.974269i	$0.572366\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.11479	0.321386
$363$	0	0
$364$	0	0
$365$	6.37229	0.333541
$366$	0	0
$367$	13.9542	0.728403	0.364202	−	0.931320i	$-0.381342\pi$
$367$	13.9542	0.728403	0.364202	+	0.931320i	$0.381342\pi$
$368$	9.34390	0.487084
$369$	0	0
$370$	22.1312	1.15054
$371$	0	0
$372$	0	0
$373$	23.9949	1.24241	0.621206	−	0.783648i	$-0.286643\pi$
$373$	23.9949	1.24241	0.621206	+	0.783648i	$0.286643\pi$
$374$	54.5018	2.81822
$375$	0	0
$376$	35.4628	1.82885
$377$	−48.0277	−2.47355
$378$	0	0
$379$	18.1983	0.934783	0.467392	−	0.884050i	$-0.345194\pi$
$379$	18.1983	0.934783	0.467392	+	0.884050i	$0.345194\pi$
$380$	8.14614	0.417888
$381$	0	0
$382$	−56.7667	−2.90444
$383$	−13.3959	−0.684497	−0.342249	−	0.939609i	$-0.611189\pi$
$383$	−13.3959	−0.684497	−0.342249	+	0.939609i	$0.611189\pi$
$384$	0	0
$385$	0	0
$386$	−46.6538	−2.37461
$387$	0	0
$388$	15.9436	0.809412
$389$	6.37101	0.323023	0.161511	−	0.986871i	$-0.448363\pi$
$389$	6.37101	0.323023	0.161511	+	0.986871i	$0.448363\pi$
$390$	0	0
$391$	−14.1495	−0.715571
$392$	0	0
$393$	0	0
$394$	12.2094	0.615103
$395$	−12.0045	−0.604012
$396$	0	0
$397$	23.3452	1.17166	0.585832	−	0.810433i	$-0.300768\pi$
$397$	23.3452	1.17166	0.585832	+	0.810433i	$0.300768\pi$
$398$	−29.8243	−1.49496
$399$	0	0
$400$	−0.448301	−0.0224151
$401$	12.1079	0.604642	0.302321	−	0.953206i	$-0.402239\pi$
$401$	12.1079	0.604642	0.302321	+	0.953206i	$0.402239\pi$
$402$	0	0
$403$	2.46791	0.122936
$404$	−0.0206271	−0.00102623
$405$	0	0
$406$	0	0
$407$	−26.8581	−1.33130
$408$	0	0
$409$	22.1376	1.09464	0.547318	−	0.836925i	$-0.315649\pi$
$409$	22.1376	1.09464	0.547318	+	0.836925i	$0.315649\pi$
$410$	56.3805	2.78444
$411$	0	0
$412$	61.7476	3.04208
$413$	0	0
$414$	0	0
$415$	−6.03748	−0.296368
$416$	12.5117	0.613435
$417$	0	0
$418$	−15.2095	−0.743923
$419$	5.72444	0.279657	0.139829	−	0.990176i	$-0.455345\pi$
$419$	5.72444	0.279657	0.139829	+	0.990176i	$0.455345\pi$
$420$	0	0
$421$	−0.894050	−0.0435733	−0.0217867	−	0.999763i	$-0.506935\pi$
$421$	−0.894050	−0.0435733	−0.0217867	+	0.999763i	$0.506935\pi$
$422$	16.2023	0.788716
$423$	0	0
$424$	28.3740	1.37796
$425$	0.678864	0.0329298
$426$	0	0
$427$	0	0
$428$	−29.6159	−1.43154
$429$	0	0
$430$	−35.8493	−1.72881
$431$	6.91659	0.333161	0.166580	−	0.986028i	$-0.446728\pi$
$431$	6.91659	0.333161	0.166580	+	0.986028i	$0.446728\pi$
$432$	0	0
$433$	−27.1219	−1.30340	−0.651698	−	0.758478i	$-0.725943\pi$
$433$	−27.1219	−1.30340	−0.651698	+	0.758478i	$0.725943\pi$
$434$	0	0
$435$	0	0
$436$	10.2448	0.490639
$437$	3.94863	0.188889
$438$	0	0
$439$	−10.8706	−0.518825	−0.259413	−	0.965767i	$-0.583529\pi$
$439$	−10.8706	−0.518825	−0.259413	+	0.965767i	$0.583529\pi$
$440$	−57.1809	−2.72599
$441$	0	0
$442$	42.2229	2.00834
$443$	−12.7648	−0.606475	−0.303237	−	0.952915i	$-0.598068\pi$
$443$	−12.7648	−0.606475	−0.303237	+	0.952915i	$0.598068\pi$
$444$	0	0
$445$	3.77275	0.178846
$446$	16.7617	0.793691
$447$	0	0
$448$	0	0
$449$	−18.2883	−0.863077	−0.431539	−	0.902094i	$-0.642029\pi$
$449$	−18.2883	−0.863077	−0.431539	+	0.902094i	$0.642029\pi$
$450$	0	0
$451$	−68.4226	−3.22190
$452$	15.8264	0.744410
$453$	0	0
$454$	−39.6858	−1.86255
$455$	0	0
$456$	0	0
$457$	−25.1547	−1.17669	−0.588343	−	0.808612i	$-0.700219\pi$
$457$	−25.1547	−1.17669	−0.588343	+	0.808612i	$0.700219\pi$
$458$	−40.4031	−1.88791
$459$	0	0
$460$	32.1661	1.49975
$461$	2.04065	0.0950424	0.0475212	−	0.998870i	$-0.484868\pi$
$461$	2.04065	0.0950424	0.0475212	+	0.998870i	$0.484868\pi$
$462$	0	0
$463$	0.789344	0.0366839	0.0183420	−	0.999832i	$-0.494161\pi$
$463$	0.789344	0.0366839	0.0183420	+	0.999832i	$0.494161\pi$
$464$	−23.0565	−1.07037
$465$	0	0
$466$	−29.3377	−1.35904
$467$	12.3150	0.569869	0.284935	−	0.958547i	$-0.408028\pi$
$467$	12.3150	0.569869	0.284935	+	0.958547i	$0.408028\pi$
$468$	0	0
$469$	0	0
$470$	45.3767	2.09307
$471$	0	0
$472$	−62.7260	−2.88720
$473$	43.5062	2.00042
$474$	0	0
$475$	−0.189447	−0.00869244
$476$	0	0
$477$	0	0
$478$	−3.95635	−0.180959
$479$	32.6218	1.49053	0.745265	−	0.666769i	$-0.232323\pi$
$479$	32.6218	1.49053	0.745265	+	0.666769i	$0.232323\pi$
$480$	0	0
$481$	−20.8071	−0.948723
$482$	53.9927	2.45930
$483$	0	0
$484$	109.507	4.97759
$485$	9.41518	0.427521
$486$	0	0
$487$	−5.51446	−0.249884	−0.124942	−	0.992164i	$-0.539874\pi$
$487$	−5.51446	−0.249884	−0.124942	+	0.992164i	$0.539874\pi$
$488$	−37.9116	−1.71618
$489$	0	0
$490$	0	0
$491$	13.9087	0.627692	0.313846	−	0.949474i	$-0.398382\pi$
$491$	13.9087	0.627692	0.313846	+	0.949474i	$0.398382\pi$
$492$	0	0
$493$	34.9146	1.57247
$494$	−11.7829	−0.530139
$495$	0	0
$496$	1.18476	0.0531975
$497$	0	0
$498$	0	0
$499$	−17.0006	−0.761051	−0.380526	−	0.924770i	$-0.624257\pi$
$499$	−17.0006	−0.761051	−0.380526	+	0.924770i	$0.624257\pi$
$500$	−42.2739	−1.89055
$501$	0	0
$502$	−65.4121	−2.91948
$503$	−38.2113	−1.70376	−0.851879	−	0.523739i	$-0.824537\pi$
$503$	−38.2113	−1.70376	−0.851879	+	0.523739i	$0.824537\pi$
$504$	0	0
$505$	−0.0121809	−0.000542044 0
$506$	−60.0569	−2.66985
$507$	0	0
$508$	−62.8495	−2.78850
$509$	−5.30397	−0.235094	−0.117547	−	0.993067i	$-0.537503\pi$
$509$	−5.30397	−0.235094	−0.117547	+	0.993067i	$0.537503\pi$
$510$	0	0
$511$	0	0
$512$	25.3984	1.12246
$513$	0	0
$514$	62.0234	2.73573
$515$	36.4639	1.60679
$516$	0	0
$517$	−55.0685	−2.42191
$518$	0	0
$519$	0	0
$520$	−44.2984	−1.94261
$521$	−3.18660	−0.139608	−0.0698038	−	0.997561i	$-0.522237\pi$
$521$	−3.18660	−0.139608	−0.0698038	+	0.997561i	$0.522237\pi$
$522$	0	0
$523$	−5.47204	−0.239276	−0.119638	−	0.992818i	$-0.538173\pi$
$523$	−5.47204	−0.239276	−0.119638	+	0.992818i	$0.538173\pi$
$524$	32.9614	1.43992
$525$	0	0
$526$	−33.4706	−1.45939
$527$	−1.79409	−0.0781519
$528$	0	0
$529$	−7.40830	−0.322100
$530$	36.3061	1.57704
$531$	0	0
$532$	0	0
$533$	−53.0075	−2.29601
$534$	0	0
$535$	−17.4891	−0.756120
$536$	−56.7216	−2.45000
$537$	0	0
$538$	10.7720	0.464416
$539$	0	0
$540$	0	0
$541$	−36.6714	−1.57663	−0.788314	−	0.615274i	$-0.789046\pi$
$541$	−36.6714	−1.57663	−0.788314	+	0.615274i	$0.789046\pi$
$542$	−28.2407	−1.21304
$543$	0	0
$544$	−9.09557	−0.389970
$545$	6.04991	0.259150
$546$	0	0
$547$	−37.6225	−1.60862	−0.804311	−	0.594208i	$-0.797466\pi$
$547$	−37.6225	−1.60862	−0.804311	+	0.594208i	$0.797466\pi$
$548$	−22.0042	−0.939974
$549$	0	0
$550$	2.88141	0.122864
$551$	−9.74344	−0.415085
$552$	0	0
$553$	0	0
$554$	−31.0762	−1.32030
$555$	0	0
$556$	41.6492	1.76632
$557$	4.72745	0.200308	0.100154	−	0.994972i	$-0.468066\pi$
$557$	4.72745	0.200308	0.100154	+	0.994972i	$0.468066\pi$
$558$	0	0
$559$	33.7045	1.42555
$560$	0	0
$561$	0	0
$562$	56.4645	2.38181
$563$	−29.3726	−1.23791	−0.618953	−	0.785428i	$-0.712443\pi$
$563$	−29.3726	−1.23791	−0.618953	+	0.785428i	$0.712443\pi$
$564$	0	0
$565$	9.34597	0.393188
$566$	67.3007	2.82886
$567$	0	0
$568$	−8.20823	−0.344410
$569$	−4.92439	−0.206441	−0.103220	−	0.994658i	$-0.532915\pi$
$569$	−4.92439	−0.206441	−0.103220	+	0.994658i	$0.532915\pi$
$570$	0	0
$571$	−2.99018	−0.125135	−0.0625676	−	0.998041i	$-0.519929\pi$
$571$	−2.99018	−0.125135	−0.0625676	+	0.998041i	$0.519929\pi$
$572$	116.486	4.87054
$573$	0	0
$574$	0	0
$575$	−0.748058	−0.0311962
$576$	0	0
$577$	1.49749	0.0623412	0.0311706	−	0.999514i	$-0.490076\pi$
$577$	1.49749	0.0623412	0.0311706	+	0.999514i	$0.490076\pi$
$578$	9.94244	0.413551
$579$	0	0
$580$	−79.3714	−3.29572
$581$	0	0
$582$	0	0
$583$	−44.0606	−1.82480
$584$	−11.9045	−0.492610
$585$	0	0
$586$	−20.0570	−0.828546
$587$	−12.3454	−0.509548	−0.254774	−	0.967001i	$-0.582001\pi$
$587$	−12.3454	−0.509548	−0.254774	+	0.967001i	$0.582001\pi$
$588$	0	0
$589$	0.500669	0.0206297
$590$	−80.2617	−3.30432
$591$	0	0
$592$	−9.98882	−0.410538
$593$	5.13280	0.210779	0.105389	−	0.994431i	$-0.466391\pi$
$593$	5.13280	0.210779	0.105389	+	0.994431i	$0.466391\pi$
$594$	0	0
$595$	0	0
$596$	66.0612	2.70597
$597$	0	0
$598$	−46.5265	−1.90261
$599$	35.9941	1.47068	0.735340	−	0.677699i	$-0.237023\pi$
$599$	35.9941	1.47068	0.735340	+	0.677699i	$0.237023\pi$
$600$	0	0
$601$	38.8330	1.58403	0.792016	−	0.610500i	$-0.209031\pi$
$601$	38.8330	1.58403	0.792016	+	0.610500i	$0.209031\pi$
$602$	0	0
$603$	0	0
$604$	50.8719	2.06995
$605$	64.6673	2.62910
$606$	0	0
$607$	−21.8458	−0.886693	−0.443346	−	0.896350i	$-0.646209\pi$
$607$	−21.8458	−0.886693	−0.443346	+	0.896350i	$0.646209\pi$
$608$	2.53826	0.102940
$609$	0	0
$610$	−48.5101	−1.96412
$611$	−42.6619	−1.72592
$612$	0	0
$613$	−36.5678	−1.47696	−0.738480	−	0.674276i	$-0.764456\pi$
$613$	−36.5678	−1.47696	−0.738480	+	0.674276i	$0.764456\pi$
$614$	−22.6833	−0.915423
$615$	0	0
$616$	0	0
$617$	−6.29559	−0.253451	−0.126725	−	0.991938i	$-0.540447\pi$
$617$	−6.29559	−0.253451	−0.126725	+	0.991938i	$0.540447\pi$
$618$	0	0
$619$	−38.0518	−1.52943	−0.764716	−	0.644368i	$-0.777121\pi$
$619$	−38.0518	−1.52943	−0.764716	+	0.644368i	$0.777121\pi$
$620$	4.07851	0.163797
$621$	0	0
$622$	−36.7642	−1.47411
$623$	0	0
$624$	0	0
$625$	−24.0169	−0.960675
$626$	5.15517	0.206042
$627$	0	0
$628$	9.35862	0.373449
$629$	15.1261	0.603117
$630$	0	0
$631$	−42.5926	−1.69558	−0.847792	−	0.530328i	$-0.822069\pi$
$631$	−42.5926	−1.69558	−0.847792	+	0.530328i	$0.822069\pi$
$632$	22.4263	0.892071
$633$	0	0
$634$	−5.06019	−0.200966
$635$	−37.1146	−1.47285
$636$	0	0
$637$	0	0
$638$	148.193	5.86703
$639$	0	0
$640$	45.4902	1.79816
$641$	46.8912	1.85209	0.926045	−	0.377414i	$-0.123187\pi$
$641$	46.8912	1.85209	0.926045	+	0.377414i	$0.123187\pi$
$642$	0	0
$643$	−10.9693	−0.432587	−0.216293	−	0.976328i	$-0.569397\pi$
$643$	−10.9693	−0.432587	−0.216293	+	0.976328i	$0.569397\pi$
$644$	0	0
$645$	0	0
$646$	8.56581	0.337017
$647$	28.3982	1.11645	0.558225	−	0.829690i	$-0.311483\pi$
$647$	28.3982	1.11645	0.558225	+	0.829690i	$0.311483\pi$
$648$	0	0
$649$	97.4044	3.82346
$650$	2.23225	0.0875559
$651$	0	0
$652$	50.7210	1.98639
$653$	9.44253	0.369515	0.184757	−	0.982784i	$-0.440850\pi$
$653$	9.44253	0.369515	0.184757	+	0.982784i	$0.440850\pi$
$654$	0	0
$655$	19.4647	0.760550
$656$	−25.4471	−0.993544
$657$	0	0
$658$	0	0
$659$	−30.8658	−1.20236	−0.601181	−	0.799113i	$-0.705303\pi$
$659$	−30.8658	−1.20236	−0.601181	+	0.799113i	$0.705303\pi$
$660$	0	0
$661$	31.0096	1.20613	0.603066	−	0.797691i	$-0.293945\pi$
$661$	31.0096	1.20613	0.603066	+	0.797691i	$0.293945\pi$
$662$	−2.13173	−0.0828519
$663$	0	0
$664$	11.2790	0.437709
$665$	0	0
$666$	0	0
$667$	−38.4733	−1.48969
$668$	−27.5826	−1.06720
$669$	0	0
$670$	−72.5786	−2.80396
$671$	58.8712	2.27270
$672$	0	0
$673$	−0.411588	−0.0158656	−0.00793278	−	0.999969i	$-0.502525\pi$
$673$	−0.411588	−0.0158656	−0.00793278	+	0.999969i	$0.502525\pi$
$674$	44.0056	1.69503
$675$	0	0
$676$	41.9595	1.61383
$677$	−13.8372	−0.531806	−0.265903	−	0.964000i	$-0.585670\pi$
$677$	−13.8372	−0.531806	−0.265903	+	0.964000i	$0.585670\pi$
$678$	0	0
$679$	0	0
$680$	32.2035	1.23495
$681$	0	0
$682$	−7.61494	−0.291591
$683$	−17.4041	−0.665949	−0.332975	−	0.942936i	$-0.608052\pi$
$683$	−17.4041	−0.665949	−0.332975	+	0.942936i	$0.608052\pi$
$684$	0	0
$685$	−12.9942	−0.496483
$686$	0	0
$687$	0	0
$688$	16.1804	0.616873
$689$	−34.1341	−1.30040
$690$	0	0
$691$	18.8299	0.716324	0.358162	−	0.933659i	$-0.383403\pi$
$691$	18.8299	0.716324	0.358162	+	0.933659i	$0.383403\pi$
$692$	44.4888	1.69121
$693$	0	0
$694$	45.0601	1.71046
$695$	24.5952	0.932947
$696$	0	0
$697$	38.5347	1.45961
$698$	−58.7456	−2.22355
$699$	0	0
$700$	0	0
$701$	25.8485	0.976286	0.488143	−	0.872764i	$-0.337675\pi$
$701$	25.8485	0.976286	0.488143	+	0.872764i	$0.337675\pi$
$702$	0	0
$703$	−4.22117	−0.159204
$704$	−68.7187	−2.58993
$705$	0	0
$706$	55.1057	2.07393
$707$	0	0
$708$	0	0
$709$	−44.7265	−1.67974	−0.839870	−	0.542788i	$-0.817369\pi$
$709$	−44.7265	−1.67974	−0.839870	+	0.542788i	$0.817369\pi$
$710$	−10.5029	−0.394167
$711$	0	0
$712$	−7.04810	−0.264139
$713$	1.97696	0.0740376
$714$	0	0
$715$	68.7890	2.57256
$716$	−24.0372	−0.898311
$717$	0	0
$718$	20.4168	0.761947
$719$	30.1687	1.12510	0.562551	−	0.826763i	$-0.309820\pi$
$719$	30.1687	1.12510	0.562551	+	0.826763i	$0.309820\pi$
$720$	0	0
$721$	0	0
$722$	−2.39042	−0.0889622
$723$	0	0
$724$	−9.50082	−0.353095
$725$	1.84587	0.0685539
$726$	0	0
$727$	35.2752	1.30828	0.654142	−	0.756372i	$-0.273030\pi$
$727$	35.2752	1.30828	0.654142	+	0.756372i	$0.273030\pi$
$728$	0	0
$729$	0	0
$730$	−15.2325	−0.563778
$731$	−24.5021	−0.906243
$732$	0	0
$733$	−41.4144	−1.52968	−0.764838	−	0.644223i	$-0.777181\pi$
$733$	−41.4144	−1.52968	−0.764838	+	0.644223i	$0.777181\pi$
$734$	−33.3564	−1.23121
$735$	0	0
$736$	10.0226	0.369440
$737$	88.0804	3.24448
$738$	0	0
$739$	−1.44475	−0.0531458	−0.0265729	−	0.999647i	$-0.508459\pi$
$739$	−1.44475	−0.0531458	−0.0265729	+	0.999647i	$0.508459\pi$
$740$	−34.3862	−1.26406
$741$	0	0
$742$	0	0
$743$	0.788993	0.0289453	0.0144727	−	0.999895i	$-0.495393\pi$
$743$	0.788993	0.0289453	0.0144727	+	0.999895i	$0.495393\pi$
$744$	0	0
$745$	39.0112	1.42926
$746$	−57.3580	−2.10002
$747$	0	0
$748$	−84.6819	−3.09628
$749$	0	0
$750$	0	0
$751$	17.5000	0.638585	0.319292	−	0.947656i	$-0.396555\pi$
$751$	17.5000	0.638585	0.319292	+	0.947656i	$0.396555\pi$
$752$	−20.4806	−0.746851
$753$	0	0
$754$	114.806	4.18100
$755$	30.0415	1.09332
$756$	0	0
$757$	3.17391	0.115358	0.0576788	−	0.998335i	$-0.481630\pi$
$757$	3.17391	0.115358	0.0576788	+	0.998335i	$0.481630\pi$
$758$	−43.5015	−1.58005
$759$	0	0
$760$	−8.98688	−0.325988
$761$	−22.5797	−0.818515	−0.409258	−	0.912419i	$-0.634212\pi$
$761$	−22.5797	−0.818515	−0.409258	+	0.912419i	$0.634212\pi$
$762$	0	0
$763$	0	0
$764$	88.2011	3.19100
$765$	0	0
$766$	32.0218	1.15699
$767$	75.4599	2.72470
$768$	0	0
$769$	45.6701	1.64691	0.823453	−	0.567385i	$-0.192045\pi$
$769$	45.6701	1.64691	0.823453	+	0.567385i	$0.192045\pi$
$770$	0	0
$771$	0	0
$772$	72.4881	2.60890
$773$	37.9098	1.36352	0.681761	−	0.731575i	$-0.261214\pi$
$773$	37.9098	1.36352	0.681761	+	0.731575i	$0.261214\pi$
$774$	0	0
$775$	−0.0948504	−0.00340713
$776$	−17.5891	−0.631410
$777$	0	0
$778$	−15.2294	−0.546000
$779$	−10.7537	−0.385291
$780$	0	0
$781$	12.7462	0.456094
$782$	33.8233	1.20952
$783$	0	0
$784$	0	0
$785$	5.52656	0.197251
$786$	0	0
$787$	7.80181	0.278105	0.139052	−	0.990285i	$-0.455594\pi$
$787$	7.80181	0.278105	0.139052	+	0.990285i	$0.455594\pi$
$788$	−18.9704	−0.675792
$789$	0	0
$790$	28.6958	1.02095
$791$	0	0
$792$	0	0
$793$	45.6079	1.61958
$794$	−55.8049	−1.98044
$795$	0	0
$796$	46.3394	1.64246
$797$	−38.6404	−1.36871	−0.684357	−	0.729147i	$-0.739917\pi$
$797$	−38.6404	−1.36871	−0.684357	+	0.729147i	$0.739917\pi$
$798$	0	0
$799$	31.0139	1.09719
$800$	−0.480866	−0.0170012
$801$	0	0
$802$	−28.9431	−1.02202
$803$	18.4859	0.652353
$804$	0	0
$805$	0	0
$806$	−5.89935	−0.207796
$807$	0	0
$808$	0.0227559	0.000800551 0
$809$	42.6750	1.50037	0.750187	−	0.661226i	$-0.229963\pi$
$809$	42.6750	1.50037	0.750187	+	0.661226i	$0.229963\pi$
$810$	0	0
$811$	1.08816	0.0382103	0.0191052	−	0.999817i	$-0.493918\pi$
$811$	1.08816	0.0382103	0.0191052	+	0.999817i	$0.493918\pi$
$812$	0	0
$813$	0	0
$814$	64.2020	2.25028
$815$	29.9524	1.04919
$816$	0	0
$817$	6.83768	0.239220
$818$	−52.9182	−1.85024
$819$	0	0
$820$	−87.6010	−3.05916
$821$	−48.5646	−1.69492	−0.847458	−	0.530863i	$-0.821868\pi$
$821$	−48.5646	−1.69492	−0.847458	+	0.530863i	$0.821868\pi$
$822$	0	0
$823$	23.9237	0.833926	0.416963	−	0.908923i	$-0.363094\pi$
$823$	23.9237	0.833926	0.416963	+	0.908923i	$0.363094\pi$
$824$	−68.1204	−2.37309
$825$	0	0
$826$	0	0
$827$	6.63171	0.230607	0.115304	−	0.993330i	$-0.463216\pi$
$827$	6.63171	0.230607	0.115304	+	0.993330i	$0.463216\pi$
$828$	0	0
$829$	47.8849	1.66311	0.831555	−	0.555442i	$-0.187451\pi$
$829$	47.8849	1.66311	0.831555	+	0.555442i	$0.187451\pi$
$830$	14.4321	0.500946
$831$	0	0
$832$	−53.2368	−1.84566
$833$	0	0
$834$	0	0
$835$	−16.2884	−0.563683
$836$	23.6318	0.817322
$837$	0	0
$838$	−13.6838	−0.472700
$839$	17.4432	0.602208	0.301104	−	0.953591i	$-0.402645\pi$
$839$	17.4432	0.602208	0.301104	+	0.953591i	$0.402645\pi$
$840$	0	0
$841$	65.9347	2.27361
$842$	2.13715	0.0736512
$843$	0	0
$844$	−25.1743	−0.866534
$845$	24.7784	0.852404
$846$	0	0
$847$	0	0
$848$	−16.3866	−0.562719
$849$	0	0
$850$	−1.62277	−0.0556606
$851$	−16.6678	−0.571366
$852$	0	0
$853$	−14.3567	−0.491564	−0.245782	−	0.969325i	$-0.579045\pi$
$853$	−14.3567	−0.491564	−0.245782	+	0.969325i	$0.579045\pi$
$854$	0	0
$855$	0	0
$856$	32.6724	1.11672
$857$	44.1973	1.50975	0.754875	−	0.655868i	$-0.227697\pi$
$857$	44.1973	1.50975	0.754875	+	0.655868i	$0.227697\pi$
$858$	0	0
$859$	56.5625	1.92989	0.964944	−	0.262455i	$-0.0845322\pi$
$859$	56.5625	1.92989	0.964944	+	0.262455i	$0.0845322\pi$
$860$	55.7007	1.89938
$861$	0	0
$862$	−16.5336	−0.563135
$863$	−54.6050	−1.85877	−0.929387	−	0.369106i	$-0.879664\pi$
$863$	−54.6050	−1.85877	−0.929387	+	0.369106i	$0.879664\pi$
$864$	0	0
$865$	26.2721	0.893277
$866$	64.8328	2.20311
$867$	0	0
$868$	0	0
$869$	−34.8248	−1.18135
$870$	0	0
$871$	68.2365	2.31210
$872$	−11.3022	−0.382740
$873$	0	0
$874$	−9.43889	−0.319275
$875$	0	0
$876$	0	0
$877$	23.1578	0.781985	0.390992	−	0.920394i	$-0.372132\pi$
$877$	23.1578	0.781985	0.390992	+	0.920394i	$0.372132\pi$
$878$	25.9853	0.876961
$879$	0	0
$880$	33.0233	1.11322
$881$	20.5800	0.693358	0.346679	−	0.937984i	$-0.387309\pi$
$881$	20.5800	0.693358	0.346679	+	0.937984i	$0.387309\pi$
$882$	0	0
$883$	27.0434	0.910084	0.455042	−	0.890470i	$-0.349624\pi$
$883$	27.0434	0.910084	0.455042	+	0.890470i	$0.349624\pi$
$884$	−65.6037	−2.20649
$885$	0	0
$886$	30.5133	1.02511
$887$	−20.7239	−0.695840	−0.347920	−	0.937524i	$-0.613112\pi$
$887$	−20.7239	−0.695840	−0.347920	+	0.937524i	$0.613112\pi$
$888$	0	0
$889$	0	0
$890$	−9.01846	−0.302300
$891$	0	0
$892$	−26.0435	−0.872000
$893$	−8.65488	−0.289625
$894$	0	0
$895$	−14.1947	−0.474477
$896$	0	0
$897$	0	0
$898$	43.7167	1.45884
$899$	−4.87824	−0.162698
$900$	0	0
$901$	24.8144	0.826686
$902$	163.559	5.44591
$903$	0	0
$904$	−17.4598	−0.580703
$905$	−5.61054	−0.186501
$906$	0	0
$907$	4.53996	0.150747	0.0753735	−	0.997155i	$-0.475985\pi$
$907$	4.53996	0.150747	0.0753735	+	0.997155i	$0.475985\pi$
$908$	61.6617	2.04631
$909$	0	0
$910$	0	0
$911$	5.37973	0.178238	0.0891192	−	0.996021i	$-0.471595\pi$
$911$	5.37973	0.178238	0.0891192	+	0.996021i	$0.471595\pi$
$912$	0	0
$913$	−17.5146	−0.579649
$914$	60.1302	1.98893
$915$	0	0
$916$	62.7762	2.07418
$917$	0	0
$918$	0	0
$919$	−15.2344	−0.502536	−0.251268	−	0.967918i	$-0.580847\pi$
$919$	−15.2344	−0.502536	−0.251268	+	0.967918i	$0.580847\pi$
$920$	−35.4859	−1.16993
$921$	0	0
$922$	−4.87800	−0.160648
$923$	9.87455	0.325025
$924$	0	0
$925$	0.799689	0.0262936
$926$	−1.88686	−0.0620061
$927$	0	0
$928$	−24.7314	−0.811847
$929$	29.0939	0.954540	0.477270	−	0.878757i	$-0.341626\pi$
$929$	29.0939	0.954540	0.477270	+	0.878757i	$0.341626\pi$
$930$	0	0
$931$	0	0
$932$	45.5833	1.49313
$933$	0	0
$934$	−29.4380	−0.963240
$935$	−50.0074	−1.63542
$936$	0	0
$937$	43.9646	1.43626	0.718131	−	0.695908i	$-0.244998\pi$
$937$	43.9646	1.43626	0.718131	+	0.695908i	$0.244998\pi$
$938$	0	0
$939$	0	0
$940$	−70.5039	−2.29958
$941$	13.1347	0.428177	0.214089	−	0.976814i	$-0.431322\pi$
$941$	13.1347	0.428177	0.214089	+	0.976814i	$0.431322\pi$
$942$	0	0
$943$	−42.4624	−1.38277
$944$	36.2258	1.17905
$945$	0	0
$946$	−103.998	−3.38127
$947$	−46.8502	−1.52243	−0.761213	−	0.648502i	$-0.775396\pi$
$947$	−46.8502	−1.52243	−0.761213	+	0.648502i	$0.775396\pi$
$948$	0	0
$949$	14.3211	0.464884
$950$	0.452859	0.0146927
$951$	0	0
$952$	0	0
$953$	−0.621467	−0.0201313	−0.0100656	−	0.999949i	$-0.503204\pi$
$953$	−0.621467	−0.0201313	−0.0100656	+	0.999949i	$0.503204\pi$
$954$	0	0
$955$	52.0856	1.68545
$956$	6.14716	0.198813
$957$	0	0
$958$	−77.9799	−2.51941
$959$	0	0
$960$	0	0
$961$	−30.7493	−0.991914
$962$	49.7378	1.60361
$963$	0	0
$964$	−83.8909	−2.70194
$965$	42.8065	1.37799
$966$	0	0
$967$	−15.3631	−0.494045	−0.247022	−	0.969010i	$-0.579452\pi$
$967$	−15.3631	−0.494045	−0.247022	+	0.969010i	$0.579452\pi$
$968$	−120.809	−3.88294
$969$	0	0
$970$	−22.5062	−0.722632
$971$	15.8647	0.509123	0.254562	−	0.967057i	$-0.418069\pi$
$971$	15.8647	0.509123	0.254562	+	0.967057i	$0.418069\pi$
$972$	0	0
$973$	0	0
$974$	13.1819	0.422374
$975$	0	0
$976$	21.8949	0.700837
$977$	−50.0609	−1.60159	−0.800795	−	0.598938i	$-0.795589\pi$
$977$	−50.0609	−1.60159	−0.800795	+	0.598938i	$0.795589\pi$
$978$	0	0
$979$	10.9447	0.349793
$980$	0	0
$981$	0	0
$982$	−33.2477	−1.06098
$983$	21.5881	0.688553	0.344277	−	0.938868i	$-0.388124\pi$
$983$	21.5881	0.688553	0.344277	+	0.938868i	$0.388124\pi$
$984$	0	0
$985$	−11.2026	−0.356945
$986$	−83.4605	−2.65792
$987$	0	0
$988$	18.3077	0.582445
$989$	26.9995	0.858534
$990$	0	0
$991$	−38.9528	−1.23738	−0.618688	−	0.785637i	$-0.712336\pi$
$991$	−38.9528	−1.23738	−0.618688	+	0.785637i	$0.712336\pi$
$992$	1.27083	0.0403488
$993$	0	0
$994$	0	0
$995$	27.3649	0.867525
$996$	0	0
$997$	−13.1724	−0.417174	−0.208587	−	0.978004i	$-0.566886\pi$
$997$	−13.1724	−0.417174	−0.208587	+	0.978004i	$0.566886\pi$
$998$	40.6386	1.28639
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.cx.1.3	yes	20
3.2	odd	2	inner	8379.2.a.cx.1.18	yes	20
7.6	odd	2		8379.2.a.cw.1.3	✓	20
21.20	even	2		8379.2.a.cw.1.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8379.2.a.cw.1.3	✓	20	7.6	odd	2
8379.2.a.cw.1.18	yes	20	21.20	even	2
8379.2.a.cx.1.3	yes	20	1.1	even	1	trivial
8379.2.a.cx.1.18	yes	20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39042 q^{2} +3.71410 q^{4} +2.19330 q^{5} -4.09743 q^{8} -5.24290 q^{10} +6.36271 q^{11} +4.92923 q^{13} +2.36636 q^{16} -3.58339 q^{17} +1.00000 q^{19} +8.14614 q^{20} -15.2095 q^{22} +3.94863 q^{23} -0.189447 q^{25} -11.7829 q^{26} -9.74344 q^{29} +0.500669 q^{31} +2.53826 q^{32} +8.56581 q^{34} -4.22117 q^{37} -2.39042 q^{38} -8.98688 q^{40} -10.7537 q^{41} +6.83768 q^{43} +23.6318 q^{44} -9.43889 q^{46} -8.65488 q^{47} +0.452859 q^{50} +18.3077 q^{52} -6.92482 q^{53} +13.9553 q^{55} +23.2909 q^{58} +15.3086 q^{59} +9.25254 q^{61} -1.19681 q^{62} -10.8002 q^{64} +10.8113 q^{65} +13.8432 q^{67} -13.3091 q^{68} +2.00326 q^{71} +2.90535 q^{73} +10.0904 q^{74} +3.71410 q^{76} -5.47327 q^{79} +5.19014 q^{80} +25.7058 q^{82} -2.75270 q^{83} -7.85945 q^{85} -16.3449 q^{86} -26.0707 q^{88} +1.72013 q^{89} +14.6656 q^{92} +20.6888 q^{94} +2.19330 q^{95} +4.29271 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} + 24 q^{10} + 24 q^{13} + 28 q^{16} + 20 q^{19} - 16 q^{22} + 28 q^{25} + 40 q^{31} + 32 q^{34} - 16 q^{37} + 56 q^{40} + 8 q^{43} - 16 q^{46} + 64 q^{52} + 32 q^{55} + 8 q^{58} + 80 q^{61}+ \cdots + 64 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 24 * q^10 + 24 * q^13 + 28 * q^16 + 20 * q^19 - 16 * q^22 + 28 * q^25 + 40 * q^31 + 32 * q^34 - 16 * q^37 + 56 * q^40 + 8 * q^43 - 16 * q^46 + 64 * q^52 + 32 * q^55 + 8 * q^58 + 80 * q^61 + 12 * q^64 + 16 * q^67 + 48 * q^73 + 20 * q^76 - 16 * q^79 + 32 * q^82 - 16 * q^85 - 32 * q^88 + 96 * q^94 + 64 * q^97

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