LMFDB - Embedded newform 8208.2.a.bs.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-1,0,-1,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.88980.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 12 $ x^4 - x^3 - 10*x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.33812$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.33812 q^{5} +2.33812 q^{7} +0.771960 q^{11} -6.14306 q^{13} -2.00000 q^{17} -1.00000 q^{19} -8.02510 q^{23} +0.466816 q^{25} +6.38184 q^{29} -2.88204 q^{31} +5.46682 q^{35} +7.49192 q^{37} -10.2639 q^{41} +8.02510 q^{43} -7.25314 q^{47} -1.53318 q^{49} -9.70922 q^{53} +1.80494 q^{55} +3.80494 q^{59} -8.01437 q^{61} -14.3632 q^{65} -12.9480 q^{67} +11.6870 q^{71} -13.5027 q^{73} +1.80494 q^{77} +14.2969 q^{79} +11.1760 q^{83} -4.67624 q^{85} -0.151348 q^{89} -14.3632 q^{91} -2.33812 q^{95} -5.34886 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} - q^{7} + 3 q^{11} - 3 q^{13} - 8 q^{17} - 4 q^{19} + q^{25} + 3 q^{29} - q^{31} + 21 q^{35} - 3 q^{37} - 8 q^{41} + 3 q^{47} - 7 q^{49} - 7 q^{53} - 4 q^{55} + 4 q^{59} - q^{61} - 15 q^{65}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q - q^5 - q^7 + 3 * q^11 - 3 * q^13 - 8 * q^17 - 4 * q^19 + q^25 + 3 * q^29 - q^31 + 21 * q^35 - 3 * q^37 - 8 * q^41 + 3 * q^47 - 7 * q^49 - 7 * q^53 - 4 * q^55 + 4 * q^59 - q^61 - 15 * q^65 - 19 * q^67 + 25 * q^71 - 20 * q^73 - 4 * q^77 + 13 * q^79 + 12 * q^83 + 2 * q^85 - 24 * q^89 - 15 * q^91 + q^95 - 10 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.33812	1.04564	0.522820	−	0.852443i	$-0.324880\pi$
$5$	2.33812	1.04564	0.522820	+	0.852443i	$0.324880\pi$
$6$	0	0
$7$	2.33812	0.883727	0.441864	−	0.897082i	$-0.354318\pi$
$7$	2.33812	0.883727	0.441864	+	0.897082i	$0.354318\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.771960	0.232755	0.116377	−	0.993205i	$-0.462872\pi$
$11$	0.771960	0.232755	0.116377	+	0.993205i	$0.462872\pi$
$12$	0	0
$13$	−6.14306	−1.70378	−0.851889	−	0.523722i	$-0.824543\pi$
$13$	−6.14306	−1.70378	−0.851889	+	0.523722i	$0.824543\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.02510	−1.67335	−0.836675	−	0.547700i	$-0.815504\pi$
$23$	−8.02510	−1.67335	−0.836675	+	0.547700i	$0.815504\pi$
$24$	0	0
$25$	0.466816	0.0933631
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.38184	1.18508	0.592539	−	0.805542i	$-0.298126\pi$
$29$	6.38184	1.18508	0.592539	+	0.805542i	$0.298126\pi$
$30$	0	0
$31$	−2.88204	−0.517630	−0.258815	−	0.965927i	$-0.583332\pi$
$31$	−2.88204	−0.517630	−0.258815	+	0.965927i	$0.583332\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.46682	0.924060
$36$	0	0
$37$	7.49192	1.23166	0.615832	−	0.787878i	$-0.288820\pi$
$37$	7.49192	1.23166	0.615832	+	0.787878i	$0.288820\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.2639	−1.60295	−0.801474	−	0.598029i	$-0.795951\pi$
$41$	−10.2639	−1.60295	−0.801474	+	0.598029i	$0.795951\pi$
$42$	0	0
$43$	8.02510	1.22382	0.611908	−	0.790929i	$-0.290402\pi$
$43$	8.02510	1.22382	0.611908	+	0.790929i	$0.290402\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.25314	−1.05798	−0.528990	−	0.848628i	$-0.677429\pi$
$47$	−7.25314	−1.05798	−0.528990	+	0.848628i	$0.677429\pi$
$48$	0	0
$49$	−1.53318	−0.219026
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.70922	−1.33366	−0.666832	−	0.745208i	$-0.732350\pi$
$53$	−9.70922	−1.33366	−0.666832	+	0.745208i	$0.732350\pi$
$54$	0	0
$55$	1.80494	0.243378
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.80494	0.495361	0.247680	−	0.968842i	$-0.420332\pi$
$59$	3.80494	0.495361	0.247680	+	0.968842i	$0.420332\pi$
$60$	0	0
$61$	−8.01437	−1.02613	−0.513067	−	0.858348i	$-0.671491\pi$
$61$	−8.01437	−1.02613	−0.513067	+	0.858348i	$0.671491\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.3632	−1.78154
$66$	0	0
$67$	−12.9480	−1.58185	−0.790925	−	0.611913i	$-0.790400\pi$
$67$	−12.9480	−1.58185	−0.790925	+	0.611913i	$0.790400\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.6870	1.38699	0.693495	−	0.720462i	$-0.256070\pi$
$71$	11.6870	1.38699	0.693495	+	0.720462i	$0.256070\pi$
$72$	0	0
$73$	−13.5027	−1.58037	−0.790183	−	0.612871i	$-0.790015\pi$
$73$	−13.5027	−1.58037	−0.790183	+	0.612871i	$0.790015\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.80494	0.205692
$78$	0	0
$79$	14.2969	1.60852	0.804261	−	0.594276i	$-0.202561\pi$
$79$	14.2969	1.60852	0.804261	+	0.594276i	$0.202561\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.1760	1.22673	0.613365	−	0.789800i	$-0.289816\pi$
$83$	11.1760	1.22673	0.613365	+	0.789800i	$0.289816\pi$
$84$	0	0
$85$	−4.67624	−0.507210
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.151348	−0.0160429	−0.00802144	−	0.999968i	$-0.502553\pi$
$89$	−0.151348	−0.0160429	−0.00802144	+	0.999968i	$0.502553\pi$
$90$	0	0
$91$	−14.3632	−1.50568
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.33812	−0.239886
$96$	0	0
$97$	−5.34886	−0.543094	−0.271547	−	0.962425i	$-0.587535\pi$
$97$	−5.34886	−0.543094	−0.271547	+	0.962425i	$0.587535\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.4041	−1.43326	−0.716630	−	0.697454i	$-0.754316\pi$
$101$	−14.4041	−1.43326	−0.716630	+	0.697454i	$0.754316\pi$
$102$	0	0
$103$	−5.13232	−0.505703	−0.252851	−	0.967505i	$-0.581368\pi$
$103$	−5.13232	−0.505703	−0.252851	+	0.967505i	$0.581368\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.19506	0.405552	0.202776	−	0.979225i	$-0.435004\pi$
$107$	4.19506	0.405552	0.202776	+	0.979225i	$0.435004\pi$
$108$	0	0
$109$	−5.20580	−0.498625	−0.249313	−	0.968423i	$-0.580205\pi$
$109$	−5.20580	−0.498625	−0.249313	+	0.968423i	$0.580205\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.23878	−0.210606	−0.105303	−	0.994440i	$-0.533581\pi$
$113$	−2.23878	−0.210606	−0.105303	+	0.994440i	$0.533581\pi$
$114$	0	0
$115$	−18.7637	−1.74972
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.67624	−0.428671
$120$	0	0
$121$	−10.4041	−0.945825
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.5991	−0.948016
$126$	0	0
$127$	3.76409	0.334009	0.167004	−	0.985956i	$-0.446591\pi$
$127$	3.76409	0.334009	0.167004	+	0.985956i	$0.446591\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.40449	0.297452	0.148726	−	0.988878i	$-0.452483\pi$
$131$	3.40449	0.297452	0.148726	+	0.988878i	$0.452483\pi$
$132$	0	0
$133$	−2.33812	−0.202741
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.74261	−0.661496	−0.330748	−	0.943719i	$-0.607301\pi$
$137$	−7.74261	−0.661496	−0.330748	+	0.943719i	$0.607301\pi$
$138$	0	0
$139$	−15.3525	−1.30218	−0.651091	−	0.759000i	$-0.725688\pi$
$139$	−15.3525	−1.30218	−0.651091	+	0.759000i	$0.725688\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.74220	−0.396563
$144$	0	0
$145$	14.9215	1.23916
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.5170	1.27120	0.635602	−	0.772017i	$-0.280752\pi$
$149$	15.5170	1.27120	0.635602	+	0.772017i	$0.280752\pi$
$150$	0	0
$151$	−22.2811	−1.81321	−0.906605	−	0.421979i	$-0.861335\pi$
$151$	−22.2811	−1.81321	−0.906605	+	0.421979i	$0.861335\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.73857	−0.541255
$156$	0	0
$157$	−12.9480	−1.03336	−0.516681	−	0.856178i	$-0.672833\pi$
$157$	−12.9480	−1.03336	−0.516681	+	0.856178i	$0.672833\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−18.7637	−1.47878
$162$	0	0
$163$	−1.97490	−0.154686	−0.0773429	−	0.997005i	$-0.524644\pi$
$163$	−1.97490	−0.154686	−0.0773429	+	0.997005i	$0.524644\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.46682	0.113506	0.0567528	−	0.998388i	$-0.481925\pi$
$167$	1.46682	0.113506	0.0567528	+	0.998388i	$0.481925\pi$
$168$	0	0
$169$	24.7372	1.90286
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.4069	−1.09534	−0.547670	−	0.836695i	$-0.684485\pi$
$173$	−14.4069	−1.09534	−0.547670	+	0.836695i	$0.684485\pi$
$174$	0	0
$175$	1.09147	0.0825075
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.04127	0.0778278	0.0389139	−	0.999243i	$-0.487610\pi$
$179$	1.04127	0.0778278	0.0389139	+	0.999243i	$0.487610\pi$
$180$	0	0
$181$	8.16816	0.607135	0.303567	−	0.952810i	$-0.401822\pi$
$181$	8.16816	0.607135	0.303567	+	0.952810i	$0.401822\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.5170	1.28788
$186$	0	0
$187$	−1.54392	−0.112903
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.9258	1.15235	0.576174	−	0.817327i	$-0.304545\pi$
$191$	15.9258	1.15235	0.576174	+	0.817327i	$0.304545\pi$
$192$	0	0
$193$	25.4687	1.83327	0.916637	−	0.399721i	$-0.130893\pi$
$193$	25.4687	1.83327	0.916637	+	0.399721i	$0.130893\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7641	−0.838156	−0.419078	−	0.907950i	$-0.637647\pi$
$197$	−11.7641	−0.838156	−0.419078	+	0.907950i	$0.637647\pi$
$198$	0	0
$199$	−0.610290	−0.0432623	−0.0216311	−	0.999766i	$-0.506886\pi$
$199$	−0.610290	−0.0432623	−0.0216311	+	0.999766i	$0.506886\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.9215	1.04728
$204$	0	0
$205$	−23.9982	−1.67611
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.771960	−0.0533976
$210$	0	0
$211$	−1.39375	−0.0959500	−0.0479750	−	0.998849i	$-0.515277\pi$
$211$	−1.39375	−0.0959500	−0.0479750	+	0.998849i	$0.515277\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.7637	1.27967
$216$	0	0
$217$	−6.73857	−0.457444
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.2861	0.826454
$222$	0	0
$223$	27.1933	1.82100	0.910498	−	0.413514i	$-0.135699\pi$
$223$	27.1933	1.82100	0.910498	+	0.413514i	$0.135699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.77983	0.118132	0.0590659	−	0.998254i	$-0.481188\pi$
$227$	1.77983	0.118132	0.0590659	+	0.998254i	$0.481188\pi$
$228$	0	0
$229$	−14.7085	−0.971962	−0.485981	−	0.873969i	$-0.661538\pi$
$229$	−14.7085	−0.971962	−0.485981	+	0.873969i	$0.661538\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.8175	1.88790	0.943949	−	0.330092i	$-0.107080\pi$
$233$	28.8175	1.88790	0.943949	+	0.330092i	$0.107080\pi$
$234$	0	0
$235$	−16.9587	−1.10627
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.6931	1.72663	0.863315	−	0.504665i	$-0.168384\pi$
$239$	26.6931	1.72663	0.863315	+	0.504665i	$0.168384\pi$
$240$	0	0
$241$	6.93726	0.446868	0.223434	−	0.974719i	$-0.428273\pi$
$241$	6.93726	0.446868	0.223434	+	0.974719i	$0.428273\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.58477	−0.229023
$246$	0	0
$247$	6.14306	0.390874
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.5740	0.667427	0.333714	−	0.942675i	$-0.391698\pi$
$251$	10.5740	0.667427	0.333714	+	0.942675i	$0.391698\pi$
$252$	0	0
$253$	−6.19506	−0.389480
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.3669	1.89423	0.947116	−	0.320891i	$-0.103982\pi$
$257$	30.3669	1.89423	0.947116	+	0.320891i	$0.103982\pi$
$258$	0	0
$259$	17.5170	1.08845
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.0868	−1.30027	−0.650134	−	0.759819i	$-0.725287\pi$
$263$	−21.0868	−1.30027	−0.650134	+	0.759819i	$0.725287\pi$
$264$	0	0
$265$	−22.7013	−1.39453
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.3274	−0.690643	−0.345321	−	0.938484i	$-0.612230\pi$
$269$	−11.3274	−0.690643	−0.345321	+	0.938484i	$0.612230\pi$
$270$	0	0
$271$	−3.15017	−0.191359	−0.0956795	−	0.995412i	$-0.530502\pi$
$271$	−3.15017	−0.191359	−0.0956795	+	0.995412i	$0.530502\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.360363	0.0217307
$276$	0	0
$277$	1.55466	0.0934103	0.0467052	−	0.998909i	$-0.485128\pi$
$277$	1.55466	0.0934103	0.0467052	+	0.998909i	$0.485128\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.8229	−0.645642	−0.322821	−	0.946460i	$-0.604631\pi$
$281$	−10.8229	−0.645642	−0.322821	+	0.946460i	$0.604631\pi$
$282$	0	0
$283$	−20.8964	−1.24216	−0.621081	−	0.783746i	$-0.713306\pi$
$283$	−20.8964	−1.24216	−0.621081	+	0.783746i	$0.713306\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.9982	−1.41657
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.4693	−0.962145	−0.481072	−	0.876681i	$-0.659753\pi$
$293$	−16.4693	−0.962145	−0.481072	+	0.876681i	$0.659753\pi$
$294$	0	0
$295$	8.89641	0.517969
$296$	0	0
$297$	0	0
$298$	0	0
$299$	49.2987	2.85102
$300$	0	0
$301$	18.7637	1.08152
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.7386	−1.07297
$306$	0	0
$307$	−23.6977	−1.35250	−0.676250	−	0.736672i	$-0.736396\pi$
$307$	−23.6977	−1.35250	−0.676250	+	0.736672i	$0.736396\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.8666	1.18324	0.591620	−	0.806217i	$-0.298489\pi$
$311$	20.8666	1.18324	0.591620	+	0.806217i	$0.298489\pi$
$312$	0	0
$313$	5.77273	0.326294	0.163147	−	0.986602i	$-0.447836\pi$
$313$	5.77273	0.326294	0.163147	+	0.986602i	$0.447836\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.1574	0.626664	0.313332	−	0.949644i	$-0.398555\pi$
$317$	11.1574	0.626664	0.313332	+	0.949644i	$0.398555\pi$
$318$	0	0
$319$	4.92652	0.275832
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−2.86768	−0.159070
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−16.9587	−0.934965
$330$	0	0
$331$	−10.8193	−0.594683	−0.297341	−	0.954771i	$-0.596100\pi$
$331$	−10.8193	−0.594683	−0.297341	+	0.954771i	$0.596100\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−30.2740	−1.65405
$336$	0	0
$337$	0.128279	0.00698783	0.00349391	−	0.999994i	$-0.498888\pi$
$337$	0.128279	0.00698783	0.00349391	+	0.999994i	$0.498888\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.22482	−0.120481
$342$	0	0
$343$	−19.9516	−1.07729
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.7831	−1.11569	−0.557846	−	0.829944i	$-0.688372\pi$
$347$	−20.7831	−1.11569	−0.557846	+	0.829944i	$0.688372\pi$
$348$	0	0
$349$	−10.8642	−0.581547	−0.290774	−	0.956792i	$-0.593913\pi$
$349$	−10.8642	−0.581547	−0.290774	+	0.956792i	$0.593913\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.9624	0.796366	0.398183	−	0.917306i	$-0.369641\pi$
$353$	14.9624	0.796366	0.398183	+	0.917306i	$0.369641\pi$
$354$	0	0
$355$	27.3256	1.45029
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.2453	1.70184	0.850920	−	0.525295i	$-0.176045\pi$
$359$	32.2453	1.70184	0.850920	+	0.525295i	$0.176045\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−31.5709	−1.65249
$366$	0	0
$367$	−21.3041	−1.11207	−0.556033	−	0.831160i	$-0.687677\pi$
$367$	−21.3041	−1.11207	−0.556033	+	0.831160i	$0.687677\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−22.7013	−1.17860
$372$	0	0
$373$	30.8547	1.59760	0.798799	−	0.601598i	$-0.205469\pi$
$373$	30.8547	1.59760	0.798799	+	0.601598i	$0.205469\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−39.2040	−2.01911
$378$	0	0
$379$	29.8426	1.53291	0.766456	−	0.642297i	$-0.222018\pi$
$379$	29.8426	1.53291	0.766456	+	0.642297i	$0.222018\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.2004	−1.59426	−0.797132	−	0.603805i	$-0.793651\pi$
$383$	−31.2004	−1.59426	−0.797132	+	0.603805i	$0.793651\pi$
$384$	0	0
$385$	4.22017	0.215080
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.3557	1.84331	0.921654	−	0.388013i	$-0.126839\pi$
$389$	36.3557	1.84331	0.921654	+	0.388013i	$0.126839\pi$
$390$	0	0
$391$	16.0502	0.811694
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.4278	1.68194
$396$	0	0
$397$	−29.9067	−1.50098	−0.750488	−	0.660884i	$-0.770181\pi$
$397$	−29.9067	−1.50098	−0.750488	+	0.660884i	$0.770181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.7888	1.63739	0.818697	−	0.574226i	$-0.194697\pi$
$401$	32.7888	1.63739	0.818697	+	0.574226i	$0.194697\pi$
$402$	0	0
$403$	17.7046	0.881927
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.78346	0.286676
$408$	0	0
$409$	−31.7224	−1.56857	−0.784286	−	0.620399i	$-0.786971\pi$
$409$	−31.7224	−1.56857	−0.784286	+	0.620399i	$0.786971\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.89641	0.437764
$414$	0	0
$415$	26.1309	1.28272
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.0759	−0.638801	−0.319400	−	0.947620i	$-0.603481\pi$
$419$	−13.0759	−0.638801	−0.319400	+	0.947620i	$0.603481\pi$
$420$	0	0
$421$	23.7713	1.15854	0.579272	−	0.815134i	$-0.303337\pi$
$421$	23.7713	1.15854	0.579272	+	0.815134i	$0.303337\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.933631	−0.0452878
$426$	0	0
$427$	−18.7386	−0.906823
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.42024	0.0684105	0.0342053	−	0.999415i	$-0.489110\pi$
$431$	1.42024	0.0684105	0.0342053	+	0.999415i	$0.489110\pi$
$432$	0	0
$433$	0.303963	0.0146075	0.00730376	−	0.999973i	$-0.497675\pi$
$433$	0.303963	0.0146075	0.00730376	+	0.999973i	$0.497675\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.02510	0.383893
$438$	0	0
$439$	−24.3915	−1.16414	−0.582072	−	0.813137i	$-0.697758\pi$
$439$	−24.3915	−1.16414	−0.582072	+	0.813137i	$0.697758\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.2262	−1.29356	−0.646779	−	0.762678i	$-0.723884\pi$
$443$	−27.2262	−1.29356	−0.646779	+	0.762678i	$0.723884\pi$
$444$	0	0
$445$	−0.353871	−0.0167751
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.30151	−0.155808	−0.0779041	−	0.996961i	$-0.524823\pi$
$449$	−3.30151	−0.155808	−0.0779041	+	0.996961i	$0.524823\pi$
$450$	0	0
$451$	−7.92331	−0.373094
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−33.5830	−1.57439
$456$	0	0
$457$	−17.7831	−0.831856	−0.415928	−	0.909398i	$-0.636543\pi$
$457$	−17.7831	−0.831856	−0.415928	+	0.909398i	$0.636543\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.2305	−0.476481	−0.238241	−	0.971206i	$-0.576571\pi$
$461$	−10.2305	−0.476481	−0.238241	+	0.971206i	$0.576571\pi$
$462$	0	0
$463$	3.97853	0.184898	0.0924489	−	0.995717i	$-0.470531\pi$
$463$	3.97853	0.184898	0.0924489	+	0.995717i	$0.470531\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.42269	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$467$	4.42269	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$468$	0	0
$469$	−30.2740	−1.39792
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.19506	0.284849
$474$	0	0
$475$	−0.466816	−0.0214190
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.7071	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$479$	22.7071	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$480$	0	0
$481$	−46.0233	−2.09848
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.5063	−0.567881
$486$	0	0
$487$	−3.72825	−0.168943	−0.0844715	−	0.996426i	$-0.526920\pi$
$487$	−3.72825	−0.168943	−0.0844715	+	0.996426i	$0.526920\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2854	0.825206	0.412603	−	0.910911i	$-0.364620\pi$
$491$	18.2854	0.825206	0.412603	+	0.910911i	$0.364620\pi$
$492$	0	0
$493$	−12.7637	−0.574847
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.3256	1.22572
$498$	0	0
$499$	−17.1753	−0.768871	−0.384435	−	0.923152i	$-0.625604\pi$
$499$	−17.1753	−0.768871	−0.384435	+	0.923152i	$0.625604\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.12404	0.406821	0.203410	−	0.979094i	$-0.434797\pi$
$503$	9.12404	0.406821	0.203410	+	0.979094i	$0.434797\pi$
$504$	0	0
$505$	−33.6785	−1.49867
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.70196	0.430032	0.215016	−	0.976611i	$-0.431020\pi$
$509$	9.70196	0.430032	0.215016	+	0.976611i	$0.431020\pi$
$510$	0	0
$511$	−31.5709	−1.39661
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−5.59914	−0.246250
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.32237	0.189367	0.0946833	−	0.995507i	$-0.469816\pi$
$521$	4.32237	0.189367	0.0946833	+	0.995507i	$0.469816\pi$
$522$	0	0
$523$	−1.50808	−0.0659438	−0.0329719	−	0.999456i	$-0.510497\pi$
$523$	−1.50808	−0.0659438	−0.0329719	+	0.999456i	$0.510497\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.76409	0.251088
$528$	0	0
$529$	41.4023	1.80010
$530$	0	0
$531$	0	0
$532$	0	0
$533$	63.0516	2.73107
$534$	0	0
$535$	9.80857	0.424061
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.18356	−0.0509794
$540$	0	0
$541$	−12.9933	−0.558626	−0.279313	−	0.960200i	$-0.590107\pi$
$541$	−12.9933	−0.558626	−0.279313	+	0.960200i	$0.590107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.1718	−0.521382
$546$	0	0
$547$	10.5174	0.449693	0.224847	−	0.974394i	$-0.427812\pi$
$547$	10.5174	0.449693	0.224847	+	0.974394i	$0.427812\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.38184	−0.271875
$552$	0	0
$553$	33.4278	1.42149
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.14710	−0.133347	−0.0666735	−	0.997775i	$-0.521239\pi$
$557$	−3.14710	−0.133347	−0.0666735	+	0.997775i	$0.521239\pi$
$558$	0	0
$559$	−49.2987	−2.08511
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.8175	0.961643	0.480822	−	0.876818i	$-0.340338\pi$
$563$	22.8175	0.961643	0.480822	+	0.876818i	$0.340338\pi$
$564$	0	0
$565$	−5.23453	−0.220218
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.7638	−1.08008	−0.540038	−	0.841641i	$-0.681590\pi$
$569$	−25.7638	−1.08008	−0.540038	+	0.841641i	$0.681590\pi$
$570$	0	0
$571$	−17.2687	−0.722672	−0.361336	−	0.932436i	$-0.617679\pi$
$571$	−17.2687	−0.722672	−0.361336	+	0.932436i	$0.617679\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.74624	−0.156229
$576$	0	0
$577$	−42.2880	−1.76047	−0.880235	−	0.474537i	$-0.842615\pi$
$577$	−42.2880	−1.76047	−0.880235	+	0.474537i	$0.842615\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26.1309	1.08409
$582$	0	0
$583$	−7.49514	−0.310417
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.9108	0.491610	0.245805	−	0.969319i	$-0.420948\pi$
$587$	11.9108	0.491610	0.245805	+	0.969319i	$0.420948\pi$
$588$	0	0
$589$	2.88204	0.118753
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.3484	0.507090	0.253545	−	0.967324i	$-0.418403\pi$
$593$	12.3484	0.507090	0.253545	+	0.967324i	$0.418403\pi$
$594$	0	0
$595$	−10.9336	−0.448235
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.7942	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$599$	−28.7942	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$600$	0	0
$601$	−23.9054	−0.975120	−0.487560	−	0.873090i	$-0.662113\pi$
$601$	−23.9054	−0.975120	−0.487560	+	0.873090i	$0.662113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−24.3260	−0.988993
$606$	0	0
$607$	16.3220	0.662488	0.331244	−	0.943545i	$-0.392532\pi$
$607$	16.3220	0.662488	0.331244	+	0.943545i	$0.392532\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44.5565	1.80256
$612$	0	0
$613$	−18.7422	−0.756990	−0.378495	−	0.925603i	$-0.623558\pi$
$613$	−18.7422	−0.756990	−0.378495	+	0.925603i	$0.623558\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.85514	−0.0746853	−0.0373426	−	0.999303i	$-0.511889\pi$
$617$	−1.85514	−0.0746853	−0.0373426	+	0.999303i	$0.511889\pi$
$618$	0	0
$619$	−13.0506	−0.524549	−0.262274	−	0.964993i	$-0.584473\pi$
$619$	−13.0506	−0.524549	−0.262274	+	0.964993i	$0.584473\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.353871	−0.0141775
$624$	0	0
$625$	−27.1162	−1.08465
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.9838	−0.597445
$630$	0	0
$631$	−3.16495	−0.125995	−0.0629973	−	0.998014i	$-0.520066\pi$
$631$	−3.16495	−0.125995	−0.0629973	+	0.998014i	$0.520066\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.80089	0.349253
$636$	0	0
$637$	9.41844	0.373172
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.8268	−0.427632	−0.213816	−	0.976874i	$-0.568589\pi$
$641$	−10.8268	−0.427632	−0.213816	+	0.976874i	$0.568589\pi$
$642$	0	0
$643$	−5.54029	−0.218488	−0.109244	−	0.994015i	$-0.534843\pi$
$643$	−5.54029	−0.218488	−0.109244	+	0.994015i	$0.534843\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.6056	1.20323	0.601616	−	0.798786i	$-0.294524\pi$
$647$	30.6056	1.20323	0.601616	+	0.798786i	$0.294524\pi$
$648$	0	0
$649$	2.93726	0.115298
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.56192	−0.335054	−0.167527	−	0.985868i	$-0.553578\pi$
$653$	−8.56192	−0.335054	−0.167527	+	0.985868i	$0.553578\pi$
$654$	0	0
$655$	7.96012	0.311027
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.8337	0.460974	0.230487	−	0.973075i	$-0.425968\pi$
$659$	11.8337	0.460974	0.230487	+	0.973075i	$0.425968\pi$
$660$	0	0
$661$	24.9735	0.971357	0.485679	−	0.874137i	$-0.338573\pi$
$661$	24.9735	0.971357	0.485679	+	0.874137i	$0.338573\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.46682	−0.211994
$666$	0	0
$667$	−51.2149	−1.98305
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.18677	−0.238838
$672$	0	0
$673$	20.3306	0.783687	0.391844	−	0.920032i	$-0.371837\pi$
$673$	20.3306	0.783687	0.391844	+	0.920032i	$0.371837\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.7600	−0.413542	−0.206771	−	0.978389i	$-0.566296\pi$
$677$	−10.7600	−0.413542	−0.206771	+	0.978389i	$0.566296\pi$
$678$	0	0
$679$	−12.5063	−0.479947
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.3557	−0.778889	−0.389445	−	0.921050i	$-0.627333\pi$
$683$	−20.3557	−0.778889	−0.389445	+	0.921050i	$0.627333\pi$
$684$	0	0
$685$	−18.1032	−0.691687
$686$	0	0
$687$	0	0
$688$	0	0
$689$	59.6443	2.27227
$690$	0	0
$691$	6.02510	0.229206	0.114603	−	0.993411i	$-0.463440\pi$
$691$	6.02510	0.229206	0.114603	+	0.993411i	$0.463440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−35.8960	−1.36161
$696$	0	0
$697$	20.5278	0.777544
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.68489	−0.139176	−0.0695881	−	0.997576i	$-0.522169\pi$
$701$	−3.68489	−0.139176	−0.0695881	+	0.997576i	$0.522169\pi$
$702$	0	0
$703$	−7.49192	−0.282563
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.6785	−1.26661
$708$	0	0
$709$	−7.24122	−0.271950	−0.135975	−	0.990712i	$-0.543417\pi$
$709$	−7.24122	−0.271950	−0.135975	+	0.990712i	$0.543417\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23.1287	0.866176
$714$	0	0
$715$	−11.0878	−0.414662
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.7433	−0.997356	−0.498678	−	0.866787i	$-0.666181\pi$
$719$	−26.7433	−0.997356	−0.498678	+	0.866787i	$0.666181\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.97914	0.110643
$726$	0	0
$727$	−2.79783	−0.103766	−0.0518829	−	0.998653i	$-0.516522\pi$
$727$	−2.79783	−0.103766	−0.0518829	+	0.998653i	$0.516522\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.0502	−0.593638
$732$	0	0
$733$	−11.2936	−0.417140	−0.208570	−	0.978007i	$-0.566881\pi$
$733$	−11.2936	−0.417140	−0.208570	+	0.978007i	$0.566881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.99534	−0.368183
$738$	0	0
$739$	−2.95301	−0.108628	−0.0543141	−	0.998524i	$-0.517297\pi$
$739$	−2.95301	−0.108628	−0.0543141	+	0.998524i	$0.517297\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.9520	−1.57576	−0.787879	−	0.615830i	$-0.788821\pi$
$743$	−42.9520	−1.57576	−0.787879	+	0.615830i	$0.788821\pi$
$744$	0	0
$745$	36.2807	1.32922
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.80857	0.358397
$750$	0	0
$751$	25.4910	0.930178	0.465089	−	0.885264i	$-0.346022\pi$
$751$	25.4910	0.930178	0.465089	+	0.885264i	$0.346022\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−52.0960	−1.89597
$756$	0	0
$757$	1.81930	0.0661237	0.0330619	−	0.999453i	$-0.489474\pi$
$757$	1.81930	0.0661237	0.0330619	+	0.999453i	$0.489474\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.3869	−1.64528	−0.822638	−	0.568566i	$-0.807498\pi$
$761$	−45.3869	−1.64528	−0.822638	+	0.568566i	$0.807498\pi$
$762$	0	0
$763$	−12.1718	−0.440649
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.3740	−0.843985
$768$	0	0
$769$	8.28152	0.298639	0.149320	−	0.988789i	$-0.452292\pi$
$769$	8.28152	0.298639	0.149320	+	0.988789i	$0.452292\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36.0383	−1.29621	−0.648104	−	0.761552i	$-0.724438\pi$
$773$	−36.0383	−1.29621	−0.648104	+	0.761552i	$0.724438\pi$
$774$	0	0
$775$	−1.34538	−0.0483276
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.2639	0.367742
$780$	0	0
$781$	9.02189	0.322828
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−30.2740	−1.08053
$786$	0	0
$787$	−35.4727	−1.26446	−0.632232	−	0.774779i	$-0.717861\pi$
$787$	−35.4727	−1.26446	−0.632232	+	0.774779i	$0.717861\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.23453	−0.186119
$792$	0	0
$793$	49.2327	1.74831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.23208	0.291595	0.145798	−	0.989314i	$-0.453425\pi$
$797$	8.23208	0.291595	0.145798	+	0.989314i	$0.453425\pi$
$798$	0	0
$799$	14.5063	0.513195
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.4235	−0.367838
$804$	0	0
$805$	−43.8718	−1.54628
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.34355	0.152711	0.0763555	−	0.997081i	$-0.475672\pi$
$809$	4.34355	0.152711	0.0763555	+	0.997081i	$0.475672\pi$
$810$	0	0
$811$	2.98579	0.104845	0.0524226	−	0.998625i	$-0.483306\pi$
$811$	2.98579	0.104845	0.0524226	+	0.998625i	$0.483306\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.61755	−0.161746
$816$	0	0
$817$	−8.02510	−0.280763
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.57362	−0.194521	−0.0972604	−	0.995259i	$-0.531008\pi$
$821$	−5.57362	−0.194521	−0.0972604	+	0.995259i	$0.531008\pi$
$822$	0	0
$823$	7.37033	0.256914	0.128457	−	0.991715i	$-0.458998\pi$
$823$	7.37033	0.256914	0.128457	+	0.991715i	$0.458998\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.30550	0.288810	0.144405	−	0.989519i	$-0.453873\pi$
$827$	8.30550	0.288810	0.144405	+	0.989519i	$0.453873\pi$
$828$	0	0
$829$	27.6777	0.961285	0.480643	−	0.876917i	$-0.340403\pi$
$829$	27.6777	0.961285	0.480643	+	0.876917i	$0.340403\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.06637	0.106243
$834$	0	0
$835$	3.42959	0.118686
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.1073	1.35013	0.675066	−	0.737757i	$-0.264115\pi$
$839$	39.1073	1.35013	0.675066	+	0.737757i	$0.264115\pi$
$840$	0	0
$841$	11.7278	0.404408
$842$	0	0
$843$	0	0
$844$	0	0
$845$	57.8386	1.98971
$846$	0	0
$847$	−24.3260	−0.835851
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−60.1234	−2.06100
$852$	0	0
$853$	48.1127	1.64735	0.823673	−	0.567064i	$-0.191921\pi$
$853$	48.1127	1.64735	0.823673	+	0.567064i	$0.191921\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.2406	−0.623087	−0.311544	−	0.950232i	$-0.600846\pi$
$857$	−18.2406	−0.623087	−0.311544	+	0.950232i	$0.600846\pi$
$858$	0	0
$859$	−19.8903	−0.678647	−0.339324	−	0.940670i	$-0.610198\pi$
$859$	−19.8903	−0.678647	−0.339324	+	0.940670i	$0.610198\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.69271	0.329944	0.164972	−	0.986298i	$-0.447247\pi$
$863$	9.69271	0.329944	0.164972	+	0.986298i	$0.447247\pi$
$864$	0	0
$865$	−33.6852	−1.14533
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.0366	0.374391
$870$	0	0
$871$	79.5403	2.69512
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.7821	−0.837787
$876$	0	0
$877$	3.59203	0.121294	0.0606472	−	0.998159i	$-0.480684\pi$
$877$	3.59203	0.121294	0.0606472	+	0.998159i	$0.480684\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.2557	1.72685	0.863425	−	0.504477i	$-0.168315\pi$
$881$	51.2557	1.72685	0.863425	+	0.504477i	$0.168315\pi$
$882$	0	0
$883$	8.78878	0.295766	0.147883	−	0.989005i	$-0.452754\pi$
$883$	8.78878	0.295766	0.147883	+	0.989005i	$0.452754\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.8501	−1.30446	−0.652230	−	0.758021i	$-0.726166\pi$
$887$	−38.8501	−1.30446	−0.652230	+	0.758021i	$0.726166\pi$
$888$	0	0
$889$	8.80089	0.295173
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.25314	0.242717
$894$	0	0
$895$	2.43461	0.0813799
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.3927	−0.613432
$900$	0	0
$901$	19.4184	0.646922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.0982	0.634844
$906$	0	0
$907$	8.12159	0.269673	0.134836	−	0.990868i	$-0.456949\pi$
$907$	8.12159	0.269673	0.134836	+	0.990868i	$0.456949\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	57.5255	1.90591	0.952953	−	0.303117i	$-0.0980274\pi$
$911$	57.5255	1.90591	0.952953	+	0.303117i	$0.0980274\pi$
$912$	0	0
$913$	8.62746	0.285527
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.96012	0.262866
$918$	0	0
$919$	36.3614	1.19945	0.599726	−	0.800205i	$-0.295276\pi$
$919$	36.3614	1.19945	0.599726	+	0.800205i	$0.295276\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−71.7938	−2.36312
$924$	0	0
$925$	3.49734	0.114992
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.61392	−0.151378	−0.0756889	−	0.997131i	$-0.524116\pi$
$929$	−4.61392	−0.151378	−0.0756889	+	0.997131i	$0.524116\pi$
$930$	0	0
$931$	1.53318	0.0502481
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.60988	−0.118056
$936$	0	0
$937$	−37.5024	−1.22515	−0.612575	−	0.790413i	$-0.709866\pi$
$937$	−37.5024	−1.22515	−0.612575	+	0.790413i	$0.709866\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.5103	1.35320	0.676599	−	0.736352i	$-0.263453\pi$
$941$	41.5103	1.35320	0.676599	+	0.736352i	$0.263453\pi$
$942$	0	0
$943$	82.3687	2.68229
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.7289	−0.381138	−0.190569	−	0.981674i	$-0.561033\pi$
$947$	−11.7289	−0.381138	−0.190569	+	0.981674i	$0.561033\pi$
$948$	0	0
$949$	82.9476	2.69259
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20.7971	0.673683	0.336841	−	0.941561i	$-0.390641\pi$
$953$	20.7971	0.673683	0.336841	+	0.941561i	$0.390641\pi$
$954$	0	0
$955$	37.2364	1.20494
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.1032	−0.584582
$960$	0	0
$961$	−22.6938	−0.732059
$962$	0	0
$963$	0	0
$964$	0	0
$965$	59.5488	1.91694
$966$	0	0
$967$	−10.0753	−0.324000	−0.162000	−	0.986791i	$-0.551794\pi$
$967$	−10.0753	−0.324000	−0.162000	+	0.986791i	$0.551794\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.8526	−1.37521	−0.687603	−	0.726087i	$-0.741337\pi$
$971$	−42.8526	−1.37521	−0.687603	+	0.726087i	$0.741337\pi$
$972$	0	0
$973$	−35.8960	−1.15077
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.9254	1.88519	0.942595	−	0.333938i	$-0.108378\pi$
$977$	58.9254	1.88519	0.942595	+	0.333938i	$0.108378\pi$
$978$	0	0
$979$	−0.116835	−0.00373406
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29.6979	0.947215	0.473608	−	0.880736i	$-0.342952\pi$
$983$	29.6979	0.947215	0.473608	+	0.880736i	$0.342952\pi$
$984$	0	0
$985$	−27.5059	−0.876410
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−64.4023	−2.04787
$990$	0	0
$991$	11.8887	0.377658	0.188829	−	0.982010i	$-0.439531\pi$
$991$	11.8887	0.377658	0.188829	+	0.982010i	$0.439531\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.42693	−0.0452368
$996$	0	0
$997$	61.9964	1.96345	0.981723	−	0.190315i	$-0.0609511\pi$
$997$	61.9964	1.96345	0.981723	+	0.190315i	$0.0609511\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bs.1.4		4
3.2	odd	2		8208.2.a.bx.1.1		4
4.3	odd	2		4104.2.a.k.1.4	✓	4
12.11	even	2		4104.2.a.l.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.k.1.4	✓	4	4.3	odd	2
4104.2.a.l.1.1	yes	4	12.11	even	2
8208.2.a.bs.1.4		4	1.1	even	1	trivial
8208.2.a.bx.1.1		4	3.2	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:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+2.33812 q^{5} +2.33812 q^{7} +0.771960 q^{11} -6.14306 q^{13} -2.00000 q^{17} -1.00000 q^{19} -8.02510 q^{23} +0.466816 q^{25} +6.38184 q^{29} -2.88204 q^{31} +5.46682 q^{35} +7.49192 q^{37} -10.2639 q^{41} +8.02510 q^{43} -7.25314 q^{47} -1.53318 q^{49} -9.70922 q^{53} +1.80494 q^{55} +3.80494 q^{59} -8.01437 q^{61} -14.3632 q^{65} -12.9480 q^{67} +11.6870 q^{71} -13.5027 q^{73} +1.80494 q^{77} +14.2969 q^{79} +11.1760 q^{83} -4.67624 q^{85} -0.151348 q^{89} -14.3632 q^{91} -2.33812 q^{95} -5.34886 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} - q^{7} + 3 q^{11} - 3 q^{13} - 8 q^{17} - 4 q^{19} + q^{25} + 3 q^{29} - q^{31} + 21 q^{35} - 3 q^{37} - 8 q^{41} + 3 q^{47} - 7 q^{49} - 7 q^{53} - 4 q^{55} + 4 q^{59} - q^{61} - 15 q^{65}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q - q^5 - q^7 + 3 * q^11 - 3 * q^13 - 8 * q^17 - 4 * q^19 + q^25 + 3 * q^29 - q^31 + 21 * q^35 - 3 * q^37 - 8 * q^41 + 3 * q^47 - 7 * q^49 - 7 * q^53 - 4 * q^55 + 4 * q^59 - q^61 - 15 * q^65 - 19 * q^67 + 25 * q^71 - 20 * q^73 - 4 * q^77 + 13 * q^79 + 12 * q^83 + 2 * q^85 - 24 * q^89 - 15 * q^91 + q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more