LMFDB - Embedded newform 3276.2.a.s.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3276,2,Mod(1,3276)] 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("3276.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3276.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-4,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.04547$ of defining polynomial
Character	$\chi$	$=$	3276.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04547 q^{5} -1.00000 q^{7} +6.09095 q^{11} -1.00000 q^{13} +6.92820 q^{17} +5.27492 q^{19} +3.04547 q^{23} +4.27492 q^{25} -3.88273 q^{29} -1.27492 q^{31} -3.04547 q^{35} -4.54983 q^{37} -6.92820 q^{41} -11.8248 q^{43} -3.04547 q^{47} +1.00000 q^{49} +2.20822 q^{53} +18.5498 q^{55} +0.837253 q^{59} -2.00000 q^{61} -3.04547 q^{65} +14.5498 q^{67} -12.1819 q^{71} +7.27492 q^{73} -6.09095 q^{77} +13.2749 q^{79} +9.13642 q^{83} +21.0997 q^{85} +4.71998 q^{89} +1.00000 q^{91} +16.0646 q^{95} -3.27492 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{13} + 6 q^{19} + 2 q^{25} + 10 q^{31} + 12 q^{37} - 2 q^{43} + 4 q^{49} + 44 q^{55} - 8 q^{61} + 28 q^{67} + 14 q^{73} + 38 q^{79} + 24 q^{85} + 4 q^{91} + 2 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^13 + 6 * q^19 + 2 * q^25 + 10 * q^31 + 12 * q^37 - 2 * q^43 + 4 * q^49 + 44 * q^55 - 8 * q^61 + 28 * q^67 + 14 * q^73 + 38 * q^79 + 24 * q^85 + 4 * q^91 + 2 * 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
$5$	3.04547	1.36198	0.680989	−	0.732294i	$-0.261550\pi$
$5$	3.04547	1.36198	0.680989	+	0.732294i	$0.261550\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.09095	1.83649	0.918245	−	0.396012i	$-0.129606\pi$
$11$	6.09095	1.83649	0.918245	+	0.396012i	$0.129606\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	5.27492	1.21015	0.605075	−	0.796169i	$-0.293143\pi$
$19$	5.27492	1.21015	0.605075	+	0.796169i	$0.293143\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.04547	0.635025	0.317513	−	0.948254i	$-0.397152\pi$
$23$	3.04547	0.635025	0.317513	+	0.948254i	$0.397152\pi$
$24$	0	0
$25$	4.27492	0.854983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.88273	−0.721005	−0.360502	−	0.932758i	$-0.617395\pi$
$29$	−3.88273	−0.721005	−0.360502	+	0.932758i	$0.617395\pi$
$30$	0	0
$31$	−1.27492	−0.228982	−0.114491	−	0.993424i	$-0.536524\pi$
$31$	−1.27492	−0.228982	−0.114491	+	0.993424i	$0.536524\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.04547	−0.514779
$36$	0	0
$37$	−4.54983	−0.747988	−0.373994	−	0.927431i	$-0.622012\pi$
$37$	−4.54983	−0.747988	−0.373994	+	0.927431i	$0.622012\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	−11.8248	−1.80326	−0.901629	−	0.432511i	$-0.857628\pi$
$43$	−11.8248	−1.80326	−0.901629	+	0.432511i	$0.857628\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.04547	−0.444228	−0.222114	−	0.975021i	$-0.571296\pi$
$47$	−3.04547	−0.444228	−0.222114	+	0.975021i	$0.571296\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.20822	0.303323	0.151661	−	0.988433i	$-0.451538\pi$
$53$	2.20822	0.303323	0.151661	+	0.988433i	$0.451538\pi$
$54$	0	0
$55$	18.5498	2.50126
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.837253	0.109001	0.0545006	−	0.998514i	$-0.482643\pi$
$59$	0.837253	0.109001	0.0545006	+	0.998514i	$0.482643\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.04547	−0.377745
$66$	0	0
$67$	14.5498	1.77755	0.888773	−	0.458348i	$-0.151559\pi$
$67$	14.5498	1.77755	0.888773	+	0.458348i	$0.151559\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.1819	−1.44573	−0.722863	−	0.690992i	$-0.757174\pi$
$71$	−12.1819	−1.44573	−0.722863	+	0.690992i	$0.757174\pi$
$72$	0	0
$73$	7.27492	0.851465	0.425732	−	0.904849i	$-0.360017\pi$
$73$	7.27492	0.851465	0.425732	+	0.904849i	$0.360017\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.09095	−0.694128
$78$	0	0
$79$	13.2749	1.49354	0.746772	−	0.665080i	$-0.231602\pi$
$79$	13.2749	1.49354	0.746772	+	0.665080i	$0.231602\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.13642	1.00285	0.501426	−	0.865200i	$-0.332809\pi$
$83$	9.13642	1.00285	0.501426	+	0.865200i	$0.332809\pi$
$84$	0	0
$85$	21.0997	2.28858
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.71998	0.500317	0.250159	−	0.968205i	$-0.419517\pi$
$89$	4.71998	0.500317	0.250159	+	0.968205i	$0.419517\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.0646	1.64820
$96$	0	0
$97$	−3.27492	−0.332517	−0.166259	−	0.986082i	$-0.553169\pi$
$97$	−3.27492	−0.332517	−0.166259	+	0.986082i	$0.553169\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.837253	−0.0833098	−0.0416549	−	0.999132i	$-0.513263\pi$
$101$	−0.837253	−0.0833098	−0.0416549	+	0.999132i	$0.513263\pi$
$102$	0	0
$103$	−17.0997	−1.68488	−0.842440	−	0.538790i	$-0.818882\pi$
$103$	−17.0997	−1.68488	−0.842440	+	0.538790i	$0.818882\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.837253	0.0809403	0.0404702	−	0.999181i	$-0.487114\pi$
$107$	0.837253	0.0809403	0.0404702	+	0.999181i	$0.487114\pi$
$108$	0	0
$109$	−11.0997	−1.06316	−0.531578	−	0.847010i	$-0.678401\pi$
$109$	−11.0997	−1.06316	−0.531578	+	0.847010i	$0.678401\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.29917	−0.780720	−0.390360	−	0.920662i	$-0.627650\pi$
$113$	−8.29917	−0.780720	−0.390360	+	0.920662i	$0.627650\pi$
$114$	0	0
$115$	9.27492	0.864890
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	26.0997	2.37270
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.20822	−0.197509
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.67451	−0.146302	−0.0731512	−	0.997321i	$-0.523306\pi$
$131$	−1.67451	−0.146302	−0.0731512	+	0.997321i	$0.523306\pi$
$132$	0	0
$133$	−5.27492	−0.457393
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	5.45017	0.462277	0.231139	−	0.972921i	$-0.425755\pi$
$139$	5.45017	0.462277	0.231139	+	0.972921i	$0.425755\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.09095	−0.509351
$144$	0	0
$145$	−11.8248	−0.981992
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.2728	−1.49697	−0.748485	−	0.663152i	$-0.769218\pi$
$149$	−18.2728	−1.49697	−0.748485	+	0.663152i	$0.769218\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.88273	−0.311868
$156$	0	0
$157$	−20.5498	−1.64006	−0.820028	−	0.572324i	$-0.806042\pi$
$157$	−20.5498	−1.64006	−0.820028	+	0.572324i	$0.806042\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.04547	−0.240017
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.13642	0.706998	0.353499	−	0.935435i	$-0.384992\pi$
$167$	9.13642	0.706998	0.353499	+	0.935435i	$0.384992\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.1101	1.45291	0.726457	−	0.687212i	$-0.241166\pi$
$173$	19.1101	1.45291	0.726457	+	0.687212i	$0.241166\pi$
$174$	0	0
$175$	−4.27492	−0.323153
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.5529	1.01299	0.506494	−	0.862243i	$-0.330941\pi$
$179$	13.5529	1.01299	0.506494	+	0.862243i	$0.330941\pi$
$180$	0	0
$181$	15.0997	1.12235	0.561175	−	0.827697i	$-0.310350\pi$
$181$	15.0997	1.12235	0.561175	+	0.827697i	$0.310350\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−13.8564	−1.01874
$186$	0	0
$187$	42.1993	3.08592
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.837253	−0.0605815	−0.0302908	−	0.999541i	$-0.509643\pi$
$191$	−0.837253	−0.0605815	−0.0302908	+	0.999541i	$0.509643\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.2728	−1.30189	−0.650943	−	0.759126i	$-0.725626\pi$
$197$	−18.2728	−1.30189	−0.650943	+	0.759126i	$0.725626\pi$
$198$	0	0
$199$	17.0997	1.21216	0.606082	−	0.795402i	$-0.292740\pi$
$199$	17.0997	1.21216	0.606082	+	0.795402i	$0.292740\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.88273	0.272514
$204$	0	0
$205$	−21.0997	−1.47366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.1293	2.22243
$210$	0	0
$211$	14.7251	1.01372	0.506858	−	0.862029i	$-0.330807\pi$
$211$	14.7251	1.01372	0.506858	+	0.862029i	$0.330807\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−36.0120	−2.45600
$216$	0	0
$217$	1.27492	0.0865470
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−14.3746	−0.962593	−0.481297	−	0.876558i	$-0.659834\pi$
$223$	−14.3746	−0.962593	−0.481297	+	0.876558i	$0.659834\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.3446	−0.752971	−0.376485	−	0.926423i	$-0.622867\pi$
$227$	−11.3446	−0.752971	−0.376485	+	0.926423i	$0.622867\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.3901	−0.942728	−0.471364	−	0.881939i	$-0.656238\pi$
$233$	−14.3901	−0.942728	−0.471364	+	0.881939i	$0.656238\pi$
$234$	0	0
$235$	−9.27492	−0.605029
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	−21.8248	−1.40586	−0.702928	−	0.711261i	$-0.748124\pi$
$241$	−21.8248	−1.40586	−0.702928	+	0.711261i	$0.748124\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.04547	0.194568
$246$	0	0
$247$	−5.27492	−0.335635
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.41644	0.278763	0.139382	−	0.990239i	$-0.455489\pi$
$251$	4.41644	0.278763	0.139382	+	0.990239i	$0.455489\pi$
$252$	0	0
$253$	18.5498	1.16622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.3446	−0.707660	−0.353830	−	0.935310i	$-0.615121\pi$
$257$	−11.3446	−0.707660	−0.353830	+	0.935310i	$0.615121\pi$
$258$	0	0
$259$	4.54983	0.282713
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.5764	1.14547	0.572735	−	0.819741i	$-0.305883\pi$
$263$	18.5764	1.14547	0.572735	+	0.819741i	$0.305883\pi$
$264$	0	0
$265$	6.72508	0.413119
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.6175	1.80581	0.902905	−	0.429840i	$-0.141430\pi$
$269$	29.6175	1.80581	0.902905	+	0.429840i	$0.141430\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.0383	1.57017
$276$	0	0
$277$	−12.7251	−0.764576	−0.382288	−	0.924043i	$-0.624864\pi$
$277$	−12.7251	−0.764576	−0.382288	+	0.924043i	$0.624864\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.5983	−0.990174	−0.495087	−	0.868843i	$-0.664864\pi$
$281$	−16.5983	−0.990174	−0.495087	+	0.868843i	$0.664864\pi$
$282$	0	0
$283$	13.0997	0.778694	0.389347	−	0.921091i	$-0.372701\pi$
$283$	13.0997	0.778694	0.389347	+	0.921091i	$0.372701\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.92820	0.408959
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.04547	−0.177919	−0.0889593	−	0.996035i	$-0.528354\pi$
$293$	−3.04547	−0.177919	−0.0889593	+	0.996035i	$0.528354\pi$
$294$	0	0
$295$	2.54983	0.148457
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.04547	−0.176124
$300$	0	0
$301$	11.8248	0.681567
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.09095	−0.348767
$306$	0	0
$307$	−7.82475	−0.446582	−0.223291	−	0.974752i	$-0.571680\pi$
$307$	−7.82475	−0.446582	−0.223291	+	0.974752i	$0.571680\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.1293	−1.82188	−0.910941	−	0.412536i	$-0.864643\pi$
$311$	−32.1293	−1.82188	−0.910941	+	0.412536i	$0.864643\pi$
$312$	0	0
$313$	−0.900331	−0.0508897	−0.0254449	−	0.999676i	$-0.508100\pi$
$313$	−0.900331	−0.0508897	−0.0254449	+	0.999676i	$0.508100\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.8564	−0.778253	−0.389127	−	0.921184i	$-0.627223\pi$
$317$	−13.8564	−0.778253	−0.389127	+	0.921184i	$0.627223\pi$
$318$	0	0
$319$	−23.6495	−1.32412
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36.5457	2.03346
$324$	0	0
$325$	−4.27492	−0.237130
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.04547	0.167902
$330$	0	0
$331$	9.45017	0.519428	0.259714	−	0.965686i	$-0.416372\pi$
$331$	9.45017	0.519428	0.259714	+	0.965686i	$0.416372\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	44.3112	2.42098
$336$	0	0
$337$	−34.9244	−1.90245	−0.951227	−	0.308491i	$-0.900176\pi$
$337$	−34.9244	−1.90245	−0.951227	+	0.308491i	$0.900176\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.76546	−0.420523
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.3446	−0.609013	−0.304506	−	0.952510i	$-0.598491\pi$
$347$	−11.3446	−0.609013	−0.304506	+	0.952510i	$0.598491\pi$
$348$	0	0
$349$	12.3746	0.662396	0.331198	−	0.943561i	$-0.392547\pi$
$349$	12.3746	0.662396	0.331198	+	0.943561i	$0.392547\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.57919	0.190501	0.0952505	−	0.995453i	$-0.469635\pi$
$353$	3.57919	0.190501	0.0952505	+	0.995453i	$0.469635\pi$
$354$	0	0
$355$	−37.0997	−1.96905
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.7128	−1.46263	−0.731313	−	0.682042i	$-0.761092\pi$
$359$	−27.7128	−1.46263	−0.731313	+	0.682042i	$0.761092\pi$
$360$	0	0
$361$	8.82475	0.464461
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.1556	1.15968
$366$	0	0
$367$	−14.5498	−0.759495	−0.379748	−	0.925090i	$-0.623989\pi$
$367$	−14.5498	−0.759495	−0.379748	+	0.925090i	$0.623989\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.20822	−0.114645
$372$	0	0
$373$	11.0997	0.574719	0.287360	−	0.957823i	$-0.407223\pi$
$373$	11.0997	0.574719	0.287360	+	0.957823i	$0.407223\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.88273	0.199971
$378$	0	0
$379$	14.5498	0.747375	0.373687	−	0.927555i	$-0.378093\pi$
$379$	14.5498	0.747375	0.373687	+	0.927555i	$0.378093\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.3446	−0.579684	−0.289842	−	0.957074i	$-0.593603\pi$
$383$	−11.3446	−0.579684	−0.289842	+	0.957074i	$0.593603\pi$
$384$	0	0
$385$	−18.5498	−0.945387
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7128	1.40510	0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128	1.40510	0.702548	+	0.711637i	$0.252046\pi$
$390$	0	0
$391$	21.0997	1.06706
$392$	0	0
$393$	0	0
$394$	0	0
$395$	40.4284	2.03417
$396$	0	0
$397$	−27.2749	−1.36889	−0.684444	−	0.729065i	$-0.739955\pi$
$397$	−27.2749	−1.36889	−0.684444	+	0.729065i	$0.739955\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.7128	−1.38391	−0.691956	−	0.721940i	$-0.743251\pi$
$401$	−27.7128	−1.38391	−0.691956	+	0.721940i	$0.743251\pi$
$402$	0	0
$403$	1.27492	0.0635082
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.7128	−1.37367
$408$	0	0
$409$	−16.3746	−0.809671	−0.404836	−	0.914389i	$-0.632671\pi$
$409$	−16.3746	−0.809671	−0.404836	+	0.914389i	$0.632671\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.837253	−0.0411986
$414$	0	0
$415$	27.8248	1.36586
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.41644	0.215757	0.107879	−	0.994164i	$-0.465594\pi$
$419$	4.41644	0.215757	0.107879	+	0.994164i	$0.465594\pi$
$420$	0	0
$421$	−0.549834	−0.0267973	−0.0133986	−	0.999910i	$-0.504265\pi$
$421$	−0.549834	−0.0267973	−0.0133986	+	0.999910i	$0.504265\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.6175	1.43666
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.43996	0.454707	0.227354	−	0.973812i	$-0.426993\pi$
$431$	9.43996	0.454707	0.227354	+	0.973812i	$0.426993\pi$
$432$	0	0
$433$	−21.6495	−1.04041	−0.520204	−	0.854042i	$-0.674144\pi$
$433$	−21.6495	−1.04041	−0.520204	+	0.854042i	$0.674144\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.0646	0.768475
$438$	0	0
$439$	−14.5498	−0.694426	−0.347213	−	0.937786i	$-0.612872\pi$
$439$	−14.5498	−0.694426	−0.347213	+	0.937786i	$0.612872\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.9928	1.09242	0.546211	−	0.837648i	$-0.316070\pi$
$443$	22.9928	1.09242	0.546211	+	0.837648i	$0.316070\pi$
$444$	0	0
$445$	14.3746	0.681421
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.5074	0.495875	0.247937	−	0.968776i	$-0.420247\pi$
$449$	10.5074	0.495875	0.247937	+	0.968776i	$0.420247\pi$
$450$	0	0
$451$	−42.1993	−1.98709
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.04547	0.142774
$456$	0	0
$457$	29.6495	1.38695	0.693473	−	0.720483i	$-0.256080\pi$
$457$	29.6495	1.38695	0.693473	+	0.720483i	$0.256080\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.6175	1.37942	0.689712	−	0.724084i	$-0.257737\pi$
$461$	29.6175	1.37942	0.689712	+	0.724084i	$0.257737\pi$
$462$	0	0
$463$	23.6495	1.09909	0.549543	−	0.835466i	$-0.314802\pi$
$463$	23.6495	1.09909	0.549543	+	0.835466i	$0.314802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.7802	−1.33179	−0.665895	−	0.746045i	$-0.731950\pi$
$467$	−28.7802	−1.33179	−0.665895	+	0.746045i	$0.731950\pi$
$468$	0	0
$469$	−14.5498	−0.671849
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−72.0240	−3.31167
$474$	0	0
$475$	22.5498	1.03466
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.13642	−0.417454	−0.208727	−	0.977974i	$-0.566932\pi$
$479$	−9.13642	−0.417454	−0.208727	+	0.977974i	$0.566932\pi$
$480$	0	0
$481$	4.54983	0.207455
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.97368	−0.452881
$486$	0	0
$487$	15.6495	0.709147	0.354573	−	0.935028i	$-0.384626\pi$
$487$	15.6495	0.709147	0.354573	+	0.935028i	$0.384626\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.8756	1.21288	0.606439	−	0.795130i	$-0.292598\pi$
$491$	26.8756	1.21288	0.606439	+	0.795130i	$0.292598\pi$
$492$	0	0
$493$	−26.9003	−1.21153
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.1819	0.546433
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.09095	0.271582	0.135791	−	0.990738i	$-0.456642\pi$
$503$	6.09095	0.271582	0.135791	+	0.990738i	$0.456642\pi$
$504$	0	0
$505$	−2.54983	−0.113466
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.5237	1.70753	0.853767	−	0.520654i	$-0.174312\pi$
$509$	38.5237	1.70753	0.853767	+	0.520654i	$0.174312\pi$
$510$	0	0
$511$	−7.27492	−0.321823
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−52.0766	−2.29477
$516$	0	0
$517$	−18.5498	−0.815821
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.6410	1.51765	0.758825	−	0.651294i	$-0.225774\pi$
$521$	34.6410	1.51765	0.758825	+	0.651294i	$0.225774\pi$
$522$	0	0
$523$	−7.64950	−0.334489	−0.167245	−	0.985915i	$-0.553487\pi$
$523$	−7.64950	−0.334489	−0.167245	+	0.985915i	$0.553487\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.83289	−0.384767
$528$	0	0
$529$	−13.7251	−0.596743
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	2.54983	0.110239
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.09095	0.262356
$540$	0	0
$541$	−29.6495	−1.27473	−0.637366	−	0.770561i	$-0.719976\pi$
$541$	−29.6495	−1.27473	−0.637366	+	0.770561i	$0.719976\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−33.8038	−1.44799
$546$	0	0
$547$	−35.8248	−1.53176	−0.765878	−	0.642986i	$-0.777695\pi$
$547$	−35.8248	−1.53176	−0.765878	+	0.642986i	$0.777695\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.4811	−0.872523
$552$	0	0
$553$	−13.2749	−0.564507
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.6893	0.961376	0.480688	−	0.876892i	$-0.340387\pi$
$557$	22.6893	0.961376	0.480688	+	0.876892i	$0.340387\pi$
$558$	0	0
$559$	11.8248	0.500134
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.2202	−1.61079	−0.805395	−	0.592739i	$-0.798047\pi$
$563$	−38.2202	−1.61079	−0.805395	+	0.592739i	$0.798047\pi$
$564$	0	0
$565$	−25.2749	−1.06332
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.3901	−0.603265	−0.301633	−	0.953424i	$-0.597532\pi$
$569$	−14.3901	−0.603265	−0.301633	+	0.953424i	$0.597532\pi$
$570$	0	0
$571$	−27.8248	−1.16443	−0.582215	−	0.813035i	$-0.697814\pi$
$571$	−27.8248	−1.16443	−0.582215	+	0.813035i	$0.697814\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.0192	0.542936
$576$	0	0
$577$	36.1993	1.50700	0.753499	−	0.657449i	$-0.228364\pi$
$577$	36.1993	1.50700	0.753499	+	0.657449i	$0.228364\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.13642	−0.379043
$582$	0	0
$583$	13.4502	0.557049
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.5002	−1.38270	−0.691351	−	0.722519i	$-0.742984\pi$
$587$	−33.5002	−1.38270	−0.691351	+	0.722519i	$0.742984\pi$
$588$	0	0
$589$	−6.72508	−0.277102
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.0838	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$593$	−29.0838	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$594$	0	0
$595$	−21.0997	−0.865002
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.7583	−1.25675	−0.628375	−	0.777911i	$-0.716280\pi$
$599$	−30.7583	−1.25675	−0.628375	+	0.777911i	$0.716280\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	79.4859	3.23156
$606$	0	0
$607$	19.6495	0.797549	0.398774	−	0.917049i	$-0.369436\pi$
$607$	19.6495	0.797549	0.398774	+	0.917049i	$0.369436\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.04547	0.123207
$612$	0	0
$613$	−20.5498	−0.830000	−0.415000	−	0.909821i	$-0.636218\pi$
$613$	−20.5498	−0.830000	−0.415000	+	0.909821i	$0.636218\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−44.3112	−1.78390	−0.891950	−	0.452134i	$-0.850663\pi$
$617$	−44.3112	−1.78390	−0.891950	+	0.452134i	$0.850663\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.71998	−0.189102
$624$	0	0
$625$	−28.0997	−1.12399
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−31.5222	−1.25687
$630$	0	0
$631$	2.54983	0.101507	0.0507537	−	0.998711i	$-0.483838\pi$
$631$	2.54983	0.101507	0.0507537	+	0.998711i	$0.483838\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.1819	0.483424
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.3227	0.526215	0.263107	−	0.964767i	$-0.415253\pi$
$641$	13.3227	0.526215	0.263107	+	0.964767i	$0.415253\pi$
$642$	0	0
$643$	41.0997	1.62081	0.810406	−	0.585868i	$-0.199246\pi$
$643$	41.0997	1.62081	0.810406	+	0.585868i	$0.199246\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.4547	−1.19730	−0.598650	−	0.801011i	$-0.704296\pi$
$647$	−30.4547	−1.19730	−0.598650	+	0.801011i	$0.704296\pi$
$648$	0	0
$649$	5.09967	0.200180
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.7128	1.08449	0.542243	−	0.840222i	$-0.317575\pi$
$653$	27.7128	1.08449	0.542243	+	0.840222i	$0.317575\pi$
$654$	0	0
$655$	−5.09967	−0.199261
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.2657	−1.60748	−0.803741	−	0.594979i	$-0.797160\pi$
$659$	−41.2657	−1.60748	−0.803741	+	0.594979i	$0.797160\pi$
$660$	0	0
$661$	−16.7251	−0.650530	−0.325265	−	0.945623i	$-0.605454\pi$
$661$	−16.7251	−0.650530	−0.325265	+	0.945623i	$0.605454\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0646	−0.622960
$666$	0	0
$667$	−11.8248	−0.457856
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.1819	−0.470277
$672$	0	0
$673$	34.9244	1.34624	0.673119	−	0.739534i	$-0.264954\pi$
$673$	34.9244	1.34624	0.673119	+	0.739534i	$0.264954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.3682	−0.629080	−0.314540	−	0.949244i	$-0.601850\pi$
$677$	−16.3682	−0.629080	−0.314540	+	0.949244i	$0.601850\pi$
$678$	0	0
$679$	3.27492	0.125680
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−39.8947	−1.52653	−0.763264	−	0.646086i	$-0.776405\pi$
$683$	−39.8947	−1.52653	−0.763264	+	0.646086i	$0.776405\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.20822	−0.0841265
$690$	0	0
$691$	−37.2749	−1.41800	−0.709002	−	0.705206i	$-0.750854\pi$
$691$	−37.2749	−1.41800	−0.709002	+	0.705206i	$0.750854\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.5983	0.629611
$696$	0	0
$697$	−48.0000	−1.81813
$698$	0	0
$699$	0	0
$700$	0	0
$701$	52.6103	1.98706	0.993532	−	0.113550i	$-0.0362222\pi$
$701$	52.6103	1.98706	0.993532	+	0.113550i	$0.0362222\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.837253	0.0314882
$708$	0	0
$709$	−19.0997	−0.717303	−0.358652	−	0.933472i	$-0.616763\pi$
$709$	−19.0997	−0.717303	−0.358652	+	0.933472i	$0.616763\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.88273	−0.145409
$714$	0	0
$715$	−18.5498	−0.693724
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.09095	−0.227154	−0.113577	−	0.993529i	$-0.536231\pi$
$719$	−6.09095	−0.227154	−0.113577	+	0.993529i	$0.536231\pi$
$720$	0	0
$721$	17.0997	0.636825
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−16.5983	−0.616447
$726$	0	0
$727$	−30.5498	−1.13303	−0.566515	−	0.824051i	$-0.691709\pi$
$727$	−30.5498	−1.13303	−0.566515	+	0.824051i	$0.691709\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−81.9243	−3.03008
$732$	0	0
$733$	−46.9244	−1.73319	−0.866597	−	0.499010i	$-0.833697\pi$
$733$	−46.9244	−1.73319	−0.866597	+	0.499010i	$0.833697\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	88.6223	3.26444
$738$	0	0
$739$	22.5498	0.829509	0.414755	−	0.909933i	$-0.363867\pi$
$739$	22.5498	0.829509	0.414755	+	0.909933i	$0.363867\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.34901	−0.122863	−0.0614317	−	0.998111i	$-0.519567\pi$
$743$	−3.34901	−0.122863	−0.0614317	+	0.998111i	$0.519567\pi$
$744$	0	0
$745$	−55.6495	−2.03884
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.837253	−0.0305926
$750$	0	0
$751$	50.3746	1.83819	0.919097	−	0.394030i	$-0.128920\pi$
$751$	50.3746	1.83819	0.919097	+	0.394030i	$0.128920\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.92442	−0.251672	−0.125836	−	0.992051i	$-0.540161\pi$
$757$	−6.92442	−0.251672	−0.125836	+	0.992051i	$0.540161\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.5529	0.491291	0.245646	−	0.969360i	$-0.421000\pi$
$761$	13.5529	0.491291	0.245646	+	0.969360i	$0.421000\pi$
$762$	0	0
$763$	11.0997	0.401835
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.837253	−0.0302315
$768$	0	0
$769$	0.725083	0.0261472	0.0130736	−	0.999915i	$-0.495838\pi$
$769$	0.725083	0.0261472	0.0130736	+	0.999915i	$0.495838\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.25370	0.188962	0.0944812	−	0.995527i	$-0.469881\pi$
$773$	5.25370	0.188962	0.0944812	+	0.995527i	$0.469881\pi$
$774$	0	0
$775$	−5.45017	−0.195776
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.5457	−1.30939
$780$	0	0
$781$	−74.1993	−2.65506
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−62.5840	−2.23372
$786$	0	0
$787$	31.4743	1.12194	0.560968	−	0.827837i	$-0.310429\pi$
$787$	31.4743	1.12194	0.560968	+	0.827837i	$0.310429\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.29917	0.295085
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.9139	−1.87431	−0.937153	−	0.348920i	$-0.886549\pi$
$797$	−52.9139	−1.87431	−0.937153	+	0.348920i	$0.886549\pi$
$798$	0	0
$799$	−21.0997	−0.746453
$800$	0	0
$801$	0	0
$802$	0	0
$803$	44.3112	1.56371
$804$	0	0
$805$	−9.27492	−0.326898
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.4811	0.720076	0.360038	−	0.932938i	$-0.382764\pi$
$809$	20.4811	0.720076	0.360038	+	0.932938i	$0.382764\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	60.9095	2.13357
$816$	0	0
$817$	−62.3746	−2.18221
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.4547	1.06288	0.531439	−	0.847097i	$-0.321651\pi$
$821$	30.4547	1.06288	0.531439	+	0.847097i	$0.321651\pi$
$822$	0	0
$823$	9.09967	0.317195	0.158597	−	0.987343i	$-0.449303\pi$
$823$	9.09967	0.317195	0.158597	+	0.987343i	$0.449303\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.6893	0.788984	0.394492	−	0.918899i	$-0.370921\pi$
$827$	22.6893	0.788984	0.394492	+	0.918899i	$0.370921\pi$
$828$	0	0
$829$	7.09967	0.246582	0.123291	−	0.992371i	$-0.460655\pi$
$829$	7.09967	0.246582	0.123291	+	0.992371i	$0.460655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.92820	0.240048
$834$	0	0
$835$	27.8248	0.962915
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.8520	−0.754416	−0.377208	−	0.926129i	$-0.623116\pi$
$839$	−21.8520	−0.754416	−0.377208	+	0.926129i	$0.623116\pi$
$840$	0	0
$841$	−13.9244	−0.480152
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.04547	0.104768
$846$	0	0
$847$	−26.0997	−0.896795
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.8564	−0.474991
$852$	0	0
$853$	−11.6254	−0.398047	−0.199023	−	0.979995i	$-0.563777\pi$
$853$	−11.6254	−0.398047	−0.199023	+	0.979995i	$0.563777\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.6937	−0.501926	−0.250963	−	0.967997i	$-0.580747\pi$
$857$	−14.6937	−0.501926	−0.250963	+	0.967997i	$0.580747\pi$
$858$	0	0
$859$	−5.45017	−0.185957	−0.0929786	−	0.995668i	$-0.529639\pi$
$859$	−5.45017	−0.185957	−0.0929786	+	0.995668i	$0.529639\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.3638	0.829353	0.414677	−	0.909969i	$-0.363895\pi$
$863$	24.3638	0.829353	0.414677	+	0.909969i	$0.363895\pi$
$864$	0	0
$865$	58.1993	1.97884
$866$	0	0
$867$	0	0
$868$	0	0
$869$	80.8569	2.74288
$870$	0	0
$871$	−14.5498	−0.493002
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.20822	0.0746515
$876$	0	0
$877$	40.5498	1.36927	0.684635	−	0.728886i	$-0.259961\pi$
$877$	40.5498	1.36927	0.684635	+	0.728886i	$0.259961\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.2011	−0.849045	−0.424523	−	0.905417i	$-0.639558\pi$
$881$	−25.2011	−0.849045	−0.424523	+	0.905417i	$0.639558\pi$
$882$	0	0
$883$	13.0997	0.440839	0.220419	−	0.975405i	$-0.429257\pi$
$883$	13.0997	0.440839	0.220419	+	0.975405i	$0.429257\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.5074	0.352804	0.176402	−	0.984318i	$-0.443554\pi$
$887$	10.5074	0.352804	0.176402	+	0.984318i	$0.443554\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16.0646	−0.537582
$894$	0	0
$895$	41.2749	1.37967
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.95016	0.165097
$900$	0	0
$901$	15.2990	0.509684
$902$	0	0
$903$	0	0
$904$	0	0
$905$	45.9857	1.52862
$906$	0	0
$907$	−38.3746	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$907$	−38.3746	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.13642	−0.302703	−0.151352	−	0.988480i	$-0.548363\pi$
$911$	−9.13642	−0.302703	−0.151352	+	0.988480i	$0.548363\pi$
$912$	0	0
$913$	55.6495	1.84173
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.67451	0.0552971
$918$	0	0
$919$	14.9003	0.491517	0.245758	−	0.969331i	$-0.420963\pi$
$919$	14.9003	0.491517	0.245758	+	0.969331i	$0.420963\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.1819	0.400972
$924$	0	0
$925$	−19.4502	−0.639518
$926$	0	0
$927$	0	0
$928$	0	0
$929$	50.0986	1.64368	0.821840	−	0.569718i	$-0.192948\pi$
$929$	50.0986	1.64368	0.821840	+	0.569718i	$0.192948\pi$
$930$	0	0
$931$	5.27492	0.172878
$932$	0	0
$933$	0	0
$934$	0	0
$935$	128.517	4.20296
$936$	0	0
$937$	−40.5498	−1.32471	−0.662353	−	0.749192i	$-0.730442\pi$
$937$	−40.5498	−1.32471	−0.662353	+	0.749192i	$0.730442\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.9402	1.39981	0.699905	−	0.714236i	$-0.253226\pi$
$941$	42.9402	1.39981	0.699905	+	0.714236i	$0.253226\pi$
$942$	0	0
$943$	−21.0997	−0.687100
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.41644	−0.143515	−0.0717576	−	0.997422i	$-0.522861\pi$
$947$	−4.41644	−0.143515	−0.0717576	+	0.997422i	$0.522861\pi$
$948$	0	0
$949$	−7.27492	−0.236154
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−49.2613	−1.59573	−0.797865	−	0.602836i	$-0.794037\pi$
$953$	−49.2613	−1.59573	−0.797865	+	0.602836i	$0.794037\pi$
$954$	0	0
$955$	−2.54983	−0.0825107
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.3746	−0.947567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.09095	0.196075
$966$	0	0
$967$	−23.6495	−0.760517	−0.380258	−	0.924880i	$-0.624165\pi$
$967$	−23.6495	−0.760517	−0.380258	+	0.924880i	$0.624165\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.2964	0.747616	0.373808	−	0.927506i	$-0.378052\pi$
$971$	23.2964	0.747616	0.373808	+	0.927506i	$0.378052\pi$
$972$	0	0
$973$	−5.45017	−0.174724
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.2964	0.745317	0.372658	−	0.927969i	$-0.378446\pi$
$977$	23.2964	0.745317	0.372658	+	0.927969i	$0.378446\pi$
$978$	0	0
$979$	28.7492	0.918828
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.0603	−0.767403	−0.383702	−	0.923457i	$-0.625351\pi$
$983$	−24.0603	−0.767403	−0.383702	+	0.923457i	$0.625351\pi$
$984$	0	0
$985$	−55.6495	−1.77314
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−36.0120	−1.14511
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	52.0766	1.65094
$996$	0	0
$997$	−44.5498	−1.41091	−0.705454	−	0.708756i	$-0.749257\pi$
$997$	−44.5498	−1.41091	−0.705454	+	0.708756i	$0.749257\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3276.2.a.s.1.4	yes	4
3.2	odd	2	inner	3276.2.a.s.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3276.2.a.s.1.1	✓	4	3.2	odd	2	inner
3276.2.a.s.1.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3276.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04547 q^{5} -1.00000 q^{7} +6.09095 q^{11} -1.00000 q^{13} +6.92820 q^{17} +5.27492 q^{19} +3.04547 q^{23} +4.27492 q^{25} -3.88273 q^{29} -1.27492 q^{31} -3.04547 q^{35} -4.54983 q^{37} -6.92820 q^{41} -11.8248 q^{43} -3.04547 q^{47} +1.00000 q^{49} +2.20822 q^{53} +18.5498 q^{55} +0.837253 q^{59} -2.00000 q^{61} -3.04547 q^{65} +14.5498 q^{67} -12.1819 q^{71} +7.27492 q^{73} -6.09095 q^{77} +13.2749 q^{79} +9.13642 q^{83} +21.0997 q^{85} +4.71998 q^{89} +1.00000 q^{91} +16.0646 q^{95} -3.27492 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{13} + 6 q^{19} + 2 q^{25} + 10 q^{31} + 12 q^{37} - 2 q^{43} + 4 q^{49} + 44 q^{55} - 8 q^{61} + 28 q^{67} + 14 q^{73} + 38 q^{79} + 24 q^{85} + 4 q^{91} + 2 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^13 + 6 * q^19 + 2 * q^25 + 10 * q^31 + 12 * q^37 - 2 * q^43 + 4 * q^49 + 44 * q^55 - 8 * q^61 + 28 * q^67 + 14 * q^73 + 38 * q^79 + 24 * q^85 + 4 * q^91 + 2 * q^97

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