LMFDB - Embedded newform 7128.2.a.w.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7128,2,Mod(1,7128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7128.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7128 = 2^{3} \cdot 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-4,0,5,0,0,0,-6] 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:	$56.9173665608$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.30796308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 10x^{2} + 2x - 2 $ x^6 - 7x^4 - x^3 + 10x^2 + 2*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.35114$ of defining polynomial
Character	$\chi$	$=$	7128.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52788 q^{5} -0.259373 q^{7} -1.00000 q^{11} -1.32579 q^{13} -3.05274 q^{17} -1.26851 q^{19} +2.37853 q^{23} +1.39018 q^{25} -5.21676 q^{29} -0.0664179 q^{31} -0.655663 q^{35} -5.84709 q^{37} +7.16375 q^{41} +3.64955 q^{43} -4.30658 q^{47} -6.93273 q^{49} -5.63237 q^{53} -2.52788 q^{55} +1.87571 q^{59} +10.4781 q^{61} -3.35144 q^{65} -8.76056 q^{67} -0.266153 q^{71} -11.9822 q^{73} +0.259373 q^{77} -2.90010 q^{79} +3.14646 q^{83} -7.71696 q^{85} -14.4329 q^{89} +0.343874 q^{91} -3.20663 q^{95} +1.61043 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 16 * 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
$5$	2.52788	1.13050	0.565251	−	0.824919i	$-0.308779\pi$
$5$	2.52788	1.13050	0.565251	+	0.824919i	$0.308779\pi$
$6$	0	0
$7$	−0.259373	−0.0980337	−0.0490168	−	0.998798i	$-0.515609\pi$
$7$	−0.259373	−0.0980337	−0.0490168	+	0.998798i	$0.515609\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.32579	−0.367708	−0.183854	−	0.982954i	$-0.558857\pi$
$13$	−1.32579	−0.367708	−0.183854	+	0.982954i	$0.558857\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.05274	−0.740398	−0.370199	−	0.928952i	$-0.620710\pi$
$17$	−3.05274	−0.740398	−0.370199	+	0.928952i	$0.620710\pi$
$18$	0	0
$19$	−1.26851	−0.291016	−0.145508	−	0.989357i	$-0.546482\pi$
$19$	−1.26851	−0.291016	−0.145508	+	0.989357i	$0.546482\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.37853	0.495958	0.247979	−	0.968765i	$-0.420234\pi$
$23$	2.37853	0.495958	0.247979	+	0.968765i	$0.420234\pi$
$24$	0	0
$25$	1.39018	0.278035
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.21676	−0.968728	−0.484364	−	0.874867i	$-0.660949\pi$
$29$	−5.21676	−0.968728	−0.484364	+	0.874867i	$0.660949\pi$
$30$	0	0
$31$	−0.0664179	−0.0119290	−0.00596450	−	0.999982i	$-0.501899\pi$
$31$	−0.0664179	−0.0119290	−0.00596450	+	0.999982i	$0.501899\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.655663	−0.110827
$36$	0	0
$37$	−5.84709	−0.961256	−0.480628	−	0.876924i	$-0.659591\pi$
$37$	−5.84709	−0.961256	−0.480628	+	0.876924i	$0.659591\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.16375	1.11879	0.559395	−	0.828901i	$-0.311034\pi$
$41$	7.16375	1.11879	0.559395	+	0.828901i	$0.311034\pi$
$42$	0	0
$43$	3.64955	0.556551	0.278276	−	0.960501i	$-0.410237\pi$
$43$	3.64955	0.556551	0.278276	+	0.960501i	$0.410237\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.30658	−0.628179	−0.314089	−	0.949393i	$-0.601699\pi$
$47$	−4.30658	−0.628179	−0.314089	+	0.949393i	$0.601699\pi$
$48$	0	0
$49$	−6.93273	−0.990389
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.63237	−0.773665	−0.386833	−	0.922150i	$-0.626431\pi$
$53$	−5.63237	−0.773665	−0.386833	+	0.922150i	$0.626431\pi$
$54$	0	0
$55$	−2.52788	−0.340859
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.87571	0.244196	0.122098	−	0.992518i	$-0.461038\pi$
$59$	1.87571	0.244196	0.122098	+	0.992518i	$0.461038\pi$
$60$	0	0
$61$	10.4781	1.34158	0.670792	−	0.741646i	$-0.265955\pi$
$61$	10.4781	1.34158	0.670792	+	0.741646i	$0.265955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.35144	−0.415695
$66$	0	0
$67$	−8.76056	−1.07027	−0.535136	−	0.844766i	$-0.679740\pi$
$67$	−8.76056	−1.07027	−0.535136	+	0.844766i	$0.679740\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.266153	−0.0315866	−0.0157933	−	0.999875i	$-0.505027\pi$
$71$	−0.266153	−0.0315866	−0.0157933	+	0.999875i	$0.505027\pi$
$72$	0	0
$73$	−11.9822	−1.40241	−0.701204	−	0.712961i	$-0.747354\pi$
$73$	−11.9822	−1.40241	−0.701204	+	0.712961i	$0.747354\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.259373	0.0295583
$78$	0	0
$79$	−2.90010	−0.326287	−0.163143	−	0.986602i	$-0.552163\pi$
$79$	−2.90010	−0.326287	−0.163143	+	0.986602i	$0.552163\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.14646	0.345369	0.172685	−	0.984977i	$-0.444756\pi$
$83$	3.14646	0.345369	0.172685	+	0.984977i	$0.444756\pi$
$84$	0	0
$85$	−7.71696	−0.837022
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.4329	−1.52989	−0.764944	−	0.644097i	$-0.777233\pi$
$89$	−14.4329	−1.52989	−0.764944	+	0.644097i	$0.777233\pi$
$90$	0	0
$91$	0.343874	0.0360478
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.20663	−0.328994
$96$	0	0
$97$	1.61043	0.163514	0.0817570	−	0.996652i	$-0.473947\pi$
$97$	1.61043	0.163514	0.0817570	+	0.996652i	$0.473947\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.04480	0.402473	0.201236	−	0.979543i	$-0.435504\pi$
$101$	4.04480	0.402473	0.201236	+	0.979543i	$0.435504\pi$
$102$	0	0
$103$	10.9819	1.08208	0.541038	−	0.840998i	$-0.318032\pi$
$103$	10.9819	1.08208	0.541038	+	0.840998i	$0.318032\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.1271	−0.979028	−0.489514	−	0.871996i	$-0.662826\pi$
$107$	−10.1271	−0.979028	−0.489514	+	0.871996i	$0.662826\pi$
$108$	0	0
$109$	−11.9561	−1.14518	−0.572592	−	0.819840i	$-0.694062\pi$
$109$	−11.9561	−1.14518	−0.572592	+	0.819840i	$0.694062\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.6474	−1.75420	−0.877102	−	0.480304i	$-0.840526\pi$
$113$	−18.6474	−1.75420	−0.877102	+	0.480304i	$0.840526\pi$
$114$	0	0
$115$	6.01264	0.560681
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.791797	0.0725839
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.12520	−0.816183
$126$	0	0
$127$	−4.53148	−0.402104	−0.201052	−	0.979581i	$-0.564436\pi$
$127$	−4.53148	−0.402104	−0.201052	+	0.979581i	$0.564436\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.6797	1.63205	0.816024	−	0.578017i	$-0.196173\pi$
$131$	18.6797	1.63205	0.816024	+	0.578017i	$0.196173\pi$
$132$	0	0
$133$	0.329016	0.0285293
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.7450	0.918010	0.459005	−	0.888434i	$-0.348206\pi$
$137$	10.7450	0.918010	0.459005	+	0.888434i	$0.348206\pi$
$138$	0	0
$139$	7.16887	0.608056	0.304028	−	0.952663i	$-0.401668\pi$
$139$	7.16887	0.608056	0.304028	+	0.952663i	$0.401668\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.32579	0.110868
$144$	0	0
$145$	−13.1873	−1.09515
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.8460	−1.70777	−0.853884	−	0.520463i	$-0.825759\pi$
$149$	−20.8460	−1.70777	−0.853884	+	0.520463i	$0.825759\pi$
$150$	0	0
$151$	−6.50797	−0.529611	−0.264806	−	0.964302i	$-0.585308\pi$
$151$	−6.50797	−0.529611	−0.264806	+	0.964302i	$0.585308\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.167896	−0.0134858
$156$	0	0
$157$	9.15085	0.730317	0.365159	−	0.930945i	$-0.381015\pi$
$157$	9.15085	0.730317	0.365159	+	0.930945i	$0.381015\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.616926	−0.0486206
$162$	0	0
$163$	−7.23853	−0.566965	−0.283483	−	0.958977i	$-0.591490\pi$
$163$	−7.23853	−0.566965	−0.283483	+	0.958977i	$0.591490\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.42898	−0.265342	−0.132671	−	0.991160i	$-0.542355\pi$
$167$	−3.42898	−0.265342	−0.132671	+	0.991160i	$0.542355\pi$
$168$	0	0
$169$	−11.2423	−0.864791
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.42465	−0.488457	−0.244228	−	0.969718i	$-0.578535\pi$
$173$	−6.42465	−0.488457	−0.244228	+	0.969718i	$0.578535\pi$
$174$	0	0
$175$	−0.360574	−0.0272568
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.80213	−0.508415	−0.254207	−	0.967150i	$-0.581815\pi$
$179$	−6.80213	−0.508415	−0.254207	+	0.967150i	$0.581815\pi$
$180$	0	0
$181$	17.1173	1.27232	0.636158	−	0.771559i	$-0.280523\pi$
$181$	17.1173	1.27232	0.636158	+	0.771559i	$0.280523\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−14.7808	−1.08670
$186$	0	0
$187$	3.05274	0.223238
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.3386	−1.61636	−0.808181	−	0.588935i	$-0.799548\pi$
$191$	−22.3386	−1.61636	−0.808181	+	0.588935i	$0.799548\pi$
$192$	0	0
$193$	7.72114	0.555780	0.277890	−	0.960613i	$-0.410365\pi$
$193$	7.72114	0.555780	0.277890	+	0.960613i	$0.410365\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.6608	−1.25828	−0.629139	−	0.777293i	$-0.716592\pi$
$197$	−17.6608	−1.25828	−0.629139	+	0.777293i	$0.716592\pi$
$198$	0	0
$199$	26.4171	1.87266	0.936330	−	0.351120i	$-0.114199\pi$
$199$	26.4171	1.87266	0.936330	+	0.351120i	$0.114199\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.35309	0.0949680
$204$	0	0
$205$	18.1091	1.26479
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.26851	0.0877445
$210$	0	0
$211$	9.89679	0.681323	0.340662	−	0.940186i	$-0.389349\pi$
$211$	9.89679	0.681323	0.340662	+	0.940186i	$0.389349\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.22562	0.629182
$216$	0	0
$217$	0.0172270	0.00116944
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.04729	0.272250
$222$	0	0
$223$	−11.5335	−0.772339	−0.386170	−	0.922428i	$-0.626202\pi$
$223$	−11.5335	−0.772339	−0.386170	+	0.922428i	$0.626202\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.04859	0.600576	0.300288	−	0.953849i	$-0.402917\pi$
$227$	9.04859	0.600576	0.300288	+	0.953849i	$0.402917\pi$
$228$	0	0
$229$	−17.3028	−1.14340	−0.571700	−	0.820463i	$-0.693716\pi$
$229$	−17.3028	−1.14340	−0.571700	+	0.820463i	$0.693716\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.93329	0.454215	0.227107	−	0.973870i	$-0.427073\pi$
$233$	6.93329	0.454215	0.227107	+	0.973870i	$0.427073\pi$
$234$	0	0
$235$	−10.8865	−0.710158
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.1719	1.56355	0.781775	−	0.623561i	$-0.214315\pi$
$239$	24.1719	1.56355	0.781775	+	0.623561i	$0.214315\pi$
$240$	0	0
$241$	0.282703	0.0182105	0.00910526	−	0.999959i	$-0.497102\pi$
$241$	0.282703	0.0182105	0.00910526	+	0.999959i	$0.497102\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−17.5251	−1.11964
$246$	0	0
$247$	1.68178	0.107009
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.4910	−1.48274	−0.741371	−	0.671096i	$-0.765824\pi$
$251$	−23.4910	−1.48274	−0.741371	+	0.671096i	$0.765824\pi$
$252$	0	0
$253$	−2.37853	−0.149537
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.588237	−0.0366932	−0.0183466	−	0.999832i	$-0.505840\pi$
$257$	−0.588237	−0.0366932	−0.0183466	+	0.999832i	$0.505840\pi$
$258$	0	0
$259$	1.51658	0.0942355
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−29.1557	−1.79782	−0.898908	−	0.438137i	$-0.855638\pi$
$263$	−29.1557	−1.79782	−0.898908	+	0.438137i	$0.855638\pi$
$264$	0	0
$265$	−14.2380	−0.874630
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.87374	−0.419099	−0.209550	−	0.977798i	$-0.567200\pi$
$269$	−6.87374	−0.419099	−0.209550	+	0.977798i	$0.567200\pi$
$270$	0	0
$271$	25.1479	1.52763	0.763815	−	0.645436i	$-0.223324\pi$
$271$	25.1479	1.52763	0.763815	+	0.645436i	$0.223324\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.39018	−0.0838308
$276$	0	0
$277$	10.7243	0.644361	0.322181	−	0.946678i	$-0.395584\pi$
$277$	10.7243	0.644361	0.322181	+	0.946678i	$0.395584\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.7227	1.77311	0.886554	−	0.462625i	$-0.153092\pi$
$281$	29.7227	1.77311	0.886554	+	0.462625i	$0.153092\pi$
$282$	0	0
$283$	19.8530	1.18014	0.590068	−	0.807353i	$-0.299101\pi$
$283$	19.8530	1.18014	0.590068	+	0.807353i	$0.299101\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.85808	−0.109679
$288$	0	0
$289$	−7.68079	−0.451811
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.93524	−0.229899	−0.114950	−	0.993371i	$-0.536671\pi$
$293$	−3.93524	−0.229899	−0.114950	+	0.993371i	$0.536671\pi$
$294$	0	0
$295$	4.74156	0.276065
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.15343	−0.182368
$300$	0	0
$301$	−0.946594	−0.0545608
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.4874	1.51666
$306$	0	0
$307$	26.9063	1.53563	0.767813	−	0.640674i	$-0.221345\pi$
$307$	26.9063	1.53563	0.767813	+	0.640674i	$0.221345\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.32622	−0.472137	−0.236068	−	0.971736i	$-0.575859\pi$
$311$	−8.32622	−0.472137	−0.236068	+	0.971736i	$0.575859\pi$
$312$	0	0
$313$	−24.7058	−1.39645	−0.698226	−	0.715878i	$-0.746027\pi$
$313$	−24.7058	−1.39645	−0.698226	+	0.715878i	$0.746027\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.4793	−0.981738	−0.490869	−	0.871233i	$-0.663321\pi$
$317$	−17.4793	−0.981738	−0.490869	+	0.871233i	$0.663321\pi$
$318$	0	0
$319$	5.21676	0.292082
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.87242	0.215467
$324$	0	0
$325$	−1.84308	−0.102236
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.11701	0.0615827
$330$	0	0
$331$	10.1116	0.555783	0.277892	−	0.960612i	$-0.410364\pi$
$331$	10.1116	0.555783	0.277892	+	0.960612i	$0.410364\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.1456	−1.20995
$336$	0	0
$337$	7.42830	0.404646	0.202323	−	0.979319i	$-0.435151\pi$
$337$	7.42830	0.404646	0.202323	+	0.979319i	$0.435151\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.0664179	0.00359673
$342$	0	0
$343$	3.61377	0.195125
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.6580	−1.16266	−0.581331	−	0.813667i	$-0.697468\pi$
$347$	−21.6580	−1.16266	−0.581331	+	0.813667i	$0.697468\pi$
$348$	0	0
$349$	6.49158	0.347487	0.173743	−	0.984791i	$-0.444414\pi$
$349$	6.49158	0.347487	0.173743	+	0.984791i	$0.444414\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.3562	0.764101	0.382050	−	0.924141i	$-0.375218\pi$
$353$	14.3562	0.764101	0.382050	+	0.924141i	$0.375218\pi$
$354$	0	0
$355$	−0.672804	−0.0357087
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.1280	−1.32621	−0.663103	−	0.748528i	$-0.730761\pi$
$359$	−25.1280	−1.32621	−0.663103	+	0.748528i	$0.730761\pi$
$360$	0	0
$361$	−17.3909	−0.915310
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.2895	−1.58542
$366$	0	0
$367$	13.8865	0.724871	0.362435	−	0.932009i	$-0.381945\pi$
$367$	13.8865	0.724871	0.362435	+	0.932009i	$0.381945\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.46088	0.0758453
$372$	0	0
$373$	18.1306	0.938765	0.469383	−	0.882995i	$-0.344476\pi$
$373$	18.1306	0.938765	0.469383	+	0.882995i	$0.344476\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.91633	0.356209
$378$	0	0
$379$	33.7566	1.73396	0.866981	−	0.498341i	$-0.166057\pi$
$379$	33.7566	1.73396	0.866981	+	0.498341i	$0.166057\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.1854	−0.775936	−0.387968	−	0.921673i	$-0.626823\pi$
$383$	−15.1854	−0.775936	−0.387968	+	0.921673i	$0.626823\pi$
$384$	0	0
$385$	0.655663	0.0334157
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.0810	−1.17025	−0.585126	−	0.810942i	$-0.698955\pi$
$389$	−23.0810	−1.17025	−0.585126	+	0.810942i	$0.698955\pi$
$390$	0	0
$391$	−7.26103	−0.367206
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.33111	−0.368868
$396$	0	0
$397$	−11.7765	−0.591046	−0.295523	−	0.955336i	$-0.595494\pi$
$397$	−11.7765	−0.591046	−0.295523	+	0.955336i	$0.595494\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.04565	−0.301906	−0.150953	−	0.988541i	$-0.548234\pi$
$401$	−6.04565	−0.301906	−0.150953	+	0.988541i	$0.548234\pi$
$402$	0	0
$403$	0.0880562	0.00438639
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.84709	0.289830
$408$	0	0
$409$	13.8329	0.683991	0.341995	−	0.939702i	$-0.388897\pi$
$409$	13.8329	0.683991	0.341995	+	0.939702i	$0.388897\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.486508	−0.0239395
$414$	0	0
$415$	7.95388	0.390441
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.52062	0.220847	0.110423	−	0.993885i	$-0.464779\pi$
$419$	4.52062	0.220847	0.110423	+	0.993885i	$0.464779\pi$
$420$	0	0
$421$	−19.6123	−0.955847	−0.477924	−	0.878401i	$-0.658610\pi$
$421$	−19.6123	−0.955847	−0.477924	+	0.878401i	$0.658610\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.24385	−0.205857
$426$	0	0
$427$	−2.71773	−0.131520
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0517	1.73655	0.868274	−	0.496085i	$-0.165230\pi$
$431$	36.0517	1.73655	0.868274	+	0.496085i	$0.165230\pi$
$432$	0	0
$433$	3.07491	0.147771	0.0738854	−	0.997267i	$-0.476460\pi$
$433$	3.07491	0.147771	0.0738854	+	0.997267i	$0.476460\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.01718	−0.144331
$438$	0	0
$439$	−3.86096	−0.184273	−0.0921367	−	0.995746i	$-0.529370\pi$
$439$	−3.86096	−0.184273	−0.0921367	+	0.995746i	$0.529370\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.22060	−0.105504	−0.0527521	−	0.998608i	$-0.516799\pi$
$443$	−2.22060	−0.105504	−0.0527521	+	0.998608i	$0.516799\pi$
$444$	0	0
$445$	−36.4847	−1.72954
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.2230	1.04877	0.524383	−	0.851482i	$-0.324296\pi$
$449$	22.2230	1.04877	0.524383	+	0.851482i	$0.324296\pi$
$450$	0	0
$451$	−7.16375	−0.337328
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.869272	0.0407521
$456$	0	0
$457$	11.4845	0.537220	0.268610	−	0.963249i	$-0.413436\pi$
$457$	11.4845	0.537220	0.268610	+	0.963249i	$0.413436\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.26633	0.0589787	0.0294893	−	0.999565i	$-0.490612\pi$
$461$	1.26633	0.0589787	0.0294893	+	0.999565i	$0.490612\pi$
$462$	0	0
$463$	−28.7947	−1.33820	−0.669102	−	0.743170i	$-0.733321\pi$
$463$	−28.7947	−1.33820	−0.669102	+	0.743170i	$0.733321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.0031	−1.43465	−0.717326	−	0.696738i	$-0.754634\pi$
$467$	−31.0031	−1.43465	−0.717326	+	0.696738i	$0.754634\pi$
$468$	0	0
$469$	2.27225	0.104923
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.64955	−0.167806
$474$	0	0
$475$	−1.76345	−0.0809126
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.4856	−1.16447	−0.582233	−	0.813022i	$-0.697821\pi$
$479$	−25.4856	−1.16447	−0.582233	+	0.813022i	$0.697821\pi$
$480$	0	0
$481$	7.75202	0.353462
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.07096	0.184853
$486$	0	0
$487$	32.8068	1.48662	0.743309	−	0.668949i	$-0.233255\pi$
$487$	32.8068	1.48662	0.743309	+	0.668949i	$0.233255\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.7606	−1.34308	−0.671540	−	0.740969i	$-0.734367\pi$
$491$	−29.7606	−1.34308	−0.671540	+	0.740969i	$0.734367\pi$
$492$	0	0
$493$	15.9254	0.717244
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.0690329	0.00309655
$498$	0	0
$499$	−5.70263	−0.255285	−0.127642	−	0.991820i	$-0.540741\pi$
$499$	−5.70263	−0.255285	−0.127642	+	0.991820i	$0.540741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.3856	0.730598	0.365299	−	0.930890i	$-0.380967\pi$
$503$	16.3856	0.730598	0.365299	+	0.930890i	$0.380967\pi$
$504$	0	0
$505$	10.2248	0.454996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1731	0.628212	0.314106	−	0.949388i	$-0.398295\pi$
$509$	14.1731	0.628212	0.314106	+	0.949388i	$0.398295\pi$
$510$	0	0
$511$	3.10785	0.137483
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.7608	1.22329
$516$	0	0
$517$	4.30658	0.189403
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.4457	−1.46528	−0.732642	−	0.680614i	$-0.761713\pi$
$521$	−33.4457	−1.46528	−0.732642	+	0.680614i	$0.761713\pi$
$522$	0	0
$523$	−33.5212	−1.46578	−0.732890	−	0.680347i	$-0.761829\pi$
$523$	−33.5212	−1.46578	−0.732890	+	0.680347i	$0.761829\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.202757	0.00883221
$528$	0	0
$529$	−17.3426	−0.754026
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.49763	−0.411388
$534$	0	0
$535$	−25.6002	−1.10679
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.93273	0.298614
$540$	0	0
$541$	−41.2381	−1.77297	−0.886483	−	0.462762i	$-0.846858\pi$
$541$	−41.2381	−1.77297	−0.886483	+	0.462762i	$0.846858\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−30.2235	−1.29463
$546$	0	0
$547$	−21.8642	−0.934844	−0.467422	−	0.884034i	$-0.654817\pi$
$547$	−21.8642	−0.934844	−0.467422	+	0.884034i	$0.654817\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.61749	0.281915
$552$	0	0
$553$	0.752207	0.0319871
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.1892	0.770702	0.385351	−	0.922770i	$-0.374080\pi$
$557$	18.1892	0.770702	0.385351	+	0.922770i	$0.374080\pi$
$558$	0	0
$559$	−4.83854	−0.204648
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.6284	−0.911530	−0.455765	−	0.890100i	$-0.650634\pi$
$563$	−21.6284	−0.911530	−0.455765	+	0.890100i	$0.650634\pi$
$564$	0	0
$565$	−47.1385	−1.98313
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.01029	−0.0423536	−0.0211768	−	0.999776i	$-0.506741\pi$
$569$	−1.01029	−0.0423536	−0.0211768	+	0.999776i	$0.506741\pi$
$570$	0	0
$571$	23.8078	0.996326	0.498163	−	0.867083i	$-0.334008\pi$
$571$	23.8078	0.996326	0.498163	+	0.867083i	$0.334008\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.30658	0.137894
$576$	0	0
$577$	18.1176	0.754247	0.377124	−	0.926163i	$-0.376913\pi$
$577$	18.1176	0.754247	0.377124	+	0.926163i	$0.376913\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.816107	−0.0338578
$582$	0	0
$583$	5.63237	0.233269
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.8619	−1.23253	−0.616266	−	0.787538i	$-0.711356\pi$
$587$	−29.8619	−1.23253	−0.616266	+	0.787538i	$0.711356\pi$
$588$	0	0
$589$	0.0842516	0.00347153
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.46598	0.224461	0.112230	−	0.993682i	$-0.464201\pi$
$593$	5.46598	0.224461	0.112230	+	0.993682i	$0.464201\pi$
$594$	0	0
$595$	2.00157	0.0820563
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.13549	0.209830	0.104915	−	0.994481i	$-0.466543\pi$
$599$	5.13549	0.209830	0.104915	+	0.994481i	$0.466543\pi$
$600$	0	0
$601$	1.74595	0.0712187	0.0356094	−	0.999366i	$-0.488663\pi$
$601$	1.74595	0.0712187	0.0356094	+	0.999366i	$0.488663\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.52788	0.102773
$606$	0	0
$607$	1.17090	0.0475253	0.0237627	−	0.999718i	$-0.492435\pi$
$607$	1.17090	0.0475253	0.0237627	+	0.999718i	$0.492435\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.70962	0.230987
$612$	0	0
$613$	20.8168	0.840783	0.420392	−	0.907343i	$-0.361893\pi$
$613$	20.8168	0.840783	0.420392	+	0.907343i	$0.361893\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.2447	1.25786	0.628932	−	0.777461i	$-0.283493\pi$
$617$	31.2447	1.25786	0.628932	+	0.777461i	$0.283493\pi$
$618$	0	0
$619$	−28.0767	−1.12850	−0.564250	−	0.825604i	$-0.690835\pi$
$619$	−28.0767	−1.12850	−0.564250	+	0.825604i	$0.690835\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.74351	0.149980
$624$	0	0
$625$	−30.0183	−1.20073
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.8497	0.711712
$630$	0	0
$631$	31.2790	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$631$	31.2790	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.4550	−0.454579
$636$	0	0
$637$	9.19134	0.364174
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.9866	−1.73736	−0.868682	−	0.495370i	$-0.835033\pi$
$641$	−43.9866	−1.73736	−0.868682	+	0.495370i	$0.835033\pi$
$642$	0	0
$643$	49.3147	1.94478	0.972390	−	0.233361i	$-0.0749722\pi$
$643$	49.3147	1.94478	0.972390	+	0.233361i	$0.0749722\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.2796	−0.954528	−0.477264	−	0.878760i	$-0.658371\pi$
$647$	−24.2796	−0.954528	−0.477264	+	0.878760i	$0.658371\pi$
$648$	0	0
$649$	−1.87571	−0.0736280
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.02174	−0.157383	−0.0786914	−	0.996899i	$-0.525074\pi$
$653$	−4.02174	−0.157383	−0.0786914	+	0.996899i	$0.525074\pi$
$654$	0	0
$655$	47.2199	1.84504
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.9173	0.659003	0.329501	−	0.944155i	$-0.393119\pi$
$659$	16.9173	0.659003	0.329501	+	0.944155i	$0.393119\pi$
$660$	0	0
$661$	22.2545	0.865601	0.432800	−	0.901490i	$-0.357526\pi$
$661$	22.2545	0.865601	0.432800	+	0.901490i	$0.357526\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.831714	0.0322525
$666$	0	0
$667$	−12.4082	−0.480448
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.4781	−0.404503
$672$	0	0
$673$	−39.0710	−1.50608	−0.753038	−	0.657977i	$-0.771412\pi$
$673$	−39.0710	−1.50608	−0.753038	+	0.657977i	$0.771412\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.0748	1.42490	0.712450	−	0.701723i	$-0.247585\pi$
$677$	37.0748	1.42490	0.712450	+	0.701723i	$0.247585\pi$
$678$	0	0
$679$	−0.417701	−0.0160299
$680$	0	0
$681$	0	0
$682$	0	0
$683$	37.6043	1.43889	0.719443	−	0.694551i	$-0.244397\pi$
$683$	37.6043	1.43889	0.719443	+	0.694551i	$0.244397\pi$
$684$	0	0
$685$	27.1622	1.03781
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.46734	0.284483
$690$	0	0
$691$	−16.2879	−0.619621	−0.309810	−	0.950798i	$-0.600266\pi$
$691$	−16.2879	−0.619621	−0.309810	+	0.950798i	$0.600266\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.1221	0.687409
$696$	0	0
$697$	−21.8691	−0.828350
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.3983	0.506047	0.253023	−	0.967460i	$-0.418575\pi$
$701$	13.3983	0.506047	0.253023	+	0.967460i	$0.418575\pi$
$702$	0	0
$703$	7.41708	0.279741
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.04911	−0.0394559
$708$	0	0
$709$	−18.8671	−0.708568	−0.354284	−	0.935138i	$-0.615275\pi$
$709$	−18.8671	−0.708568	−0.354284	+	0.935138i	$0.615275\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.157977	−0.00591628
$714$	0	0
$715$	3.35144	0.125337
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.49032	0.242048	0.121024	−	0.992650i	$-0.461382\pi$
$719$	6.49032	0.242048	0.121024	+	0.992650i	$0.461382\pi$
$720$	0	0
$721$	−2.84840	−0.106080
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.25222	−0.269341
$726$	0	0
$727$	−32.6534	−1.21105	−0.605524	−	0.795827i	$-0.707037\pi$
$727$	−32.6534	−1.21105	−0.605524	+	0.795827i	$0.707037\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.1411	−0.412069
$732$	0	0
$733$	18.4922	0.683027	0.341513	−	0.939877i	$-0.389061\pi$
$733$	18.4922	0.683027	0.341513	+	0.939877i	$0.389061\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.76056	0.322699
$738$	0	0
$739$	11.5633	0.425364	0.212682	−	0.977122i	$-0.431780\pi$
$739$	11.5633	0.425364	0.212682	+	0.977122i	$0.431780\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.3689	−1.04075	−0.520377	−	0.853937i	$-0.674209\pi$
$743$	−28.3689	−1.04075	−0.520377	+	0.853937i	$0.674209\pi$
$744$	0	0
$745$	−52.6961	−1.93064
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.62670	0.0959777
$750$	0	0
$751$	15.8079	0.576839	0.288420	−	0.957504i	$-0.406870\pi$
$751$	15.8079	0.576839	0.288420	+	0.957504i	$0.406870\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.4514	−0.598727
$756$	0	0
$757$	11.6372	0.422960	0.211480	−	0.977382i	$-0.432172\pi$
$757$	11.6372	0.422960	0.211480	+	0.977382i	$0.432172\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.1271	0.802105	0.401053	−	0.916055i	$-0.368644\pi$
$761$	22.1271	0.802105	0.401053	+	0.916055i	$0.368644\pi$
$762$	0	0
$763$	3.10108	0.112267
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.48680	−0.0897930
$768$	0	0
$769$	−52.0277	−1.87617	−0.938084	−	0.346408i	$-0.887401\pi$
$769$	−52.0277	−1.87617	−0.938084	+	0.346408i	$0.887401\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.0327	−0.900365	−0.450182	−	0.892937i	$-0.648641\pi$
$773$	−25.0327	−0.900365	−0.450182	+	0.892937i	$0.648641\pi$
$774$	0	0
$775$	−0.0923326	−0.00331669
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.08727	−0.325585
$780$	0	0
$781$	0.266153	0.00952372
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23.1322	0.825625
$786$	0	0
$787$	32.3104	1.15174	0.575870	−	0.817541i	$-0.304663\pi$
$787$	32.3104	1.15174	0.575870	+	0.817541i	$0.304663\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.83664	0.171971
$792$	0	0
$793$	−13.8918	−0.493311
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.9047	0.775905	0.387953	−	0.921679i	$-0.373182\pi$
$797$	21.9047	0.775905	0.387953	+	0.921679i	$0.373182\pi$
$798$	0	0
$799$	13.1469	0.465102
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.9822	0.422842
$804$	0	0
$805$	−1.55951	−0.0549657
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.7750	−0.800726	−0.400363	−	0.916357i	$-0.631116\pi$
$809$	−22.7750	−0.800726	−0.400363	+	0.916357i	$0.631116\pi$
$810$	0	0
$811$	−40.8286	−1.43369	−0.716843	−	0.697235i	$-0.754413\pi$
$811$	−40.8286	−1.43369	−0.716843	+	0.697235i	$0.754413\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.2981	−0.640955
$816$	0	0
$817$	−4.62948	−0.161965
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.4439	−0.364493	−0.182246	−	0.983253i	$-0.558337\pi$
$821$	−10.4439	−0.364493	−0.182246	+	0.983253i	$0.558337\pi$
$822$	0	0
$823$	3.70212	0.129048	0.0645239	−	0.997916i	$-0.479447\pi$
$823$	3.70212	0.129048	0.0645239	+	0.997916i	$0.479447\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.1184	−1.18641	−0.593207	−	0.805050i	$-0.702138\pi$
$827$	−34.1184	−1.18641	−0.593207	+	0.805050i	$0.702138\pi$
$828$	0	0
$829$	35.6910	1.23960	0.619800	−	0.784760i	$-0.287214\pi$
$829$	35.6910	1.23960	0.619800	+	0.784760i	$0.287214\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21.1638	0.733282
$834$	0	0
$835$	−8.66805	−0.299970
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.3082	−0.355877	−0.177939	−	0.984042i	$-0.556943\pi$
$839$	−10.3082	−0.355877	−0.177939	+	0.984042i	$0.556943\pi$
$840$	0	0
$841$	−1.78544	−0.0615668
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−28.4191	−0.977648
$846$	0	0
$847$	−0.259373	−0.00891215
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.9075	−0.476743
$852$	0	0
$853$	−22.5393	−0.771730	−0.385865	−	0.922555i	$-0.626097\pi$
$853$	−22.5393	−0.771730	−0.385865	+	0.922555i	$0.626097\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.5365	0.428238	0.214119	−	0.976808i	$-0.431312\pi$
$857$	12.5365	0.428238	0.214119	+	0.976808i	$0.431312\pi$
$858$	0	0
$859$	14.4875	0.494307	0.247154	−	0.968976i	$-0.420505\pi$
$859$	14.4875	0.494307	0.247154	+	0.968976i	$0.420505\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0520	1.36338	0.681692	−	0.731639i	$-0.261244\pi$
$863$	40.0520	1.36338	0.681692	+	0.731639i	$0.261244\pi$
$864$	0	0
$865$	−16.2407	−0.552202
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.90010	0.0983792
$870$	0	0
$871$	11.6147	0.393548
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.36683	0.0800134
$876$	0	0
$877$	−31.1005	−1.05019	−0.525095	−	0.851044i	$-0.675970\pi$
$877$	−31.1005	−1.05019	−0.525095	+	0.851044i	$0.675970\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.6899	0.393844	0.196922	−	0.980419i	$-0.436906\pi$
$881$	11.6899	0.393844	0.196922	+	0.980419i	$0.436906\pi$
$882$	0	0
$883$	−21.5218	−0.724266	−0.362133	−	0.932126i	$-0.617951\pi$
$883$	−21.5218	−0.724266	−0.362133	+	0.932126i	$0.617951\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.4671	−1.02298	−0.511492	−	0.859288i	$-0.670907\pi$
$887$	−30.4671	−1.02298	−0.511492	+	0.859288i	$0.670907\pi$
$888$	0	0
$889$	1.17534	0.0394197
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.46292	0.182810
$894$	0	0
$895$	−17.1950	−0.574764
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.346486	0.0115560
$900$	0	0
$901$	17.1941	0.572820
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43.2704	1.43836
$906$	0	0
$907$	40.7022	1.35150	0.675748	−	0.737133i	$-0.263821\pi$
$907$	40.7022	1.35150	0.675748	+	0.737133i	$0.263821\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.5755	1.14554	0.572768	−	0.819717i	$-0.305869\pi$
$911$	34.5755	1.14554	0.572768	+	0.819717i	$0.305869\pi$
$912$	0	0
$913$	−3.14646	−0.104133
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.84499	−0.159996
$918$	0	0
$919$	−57.8136	−1.90710	−0.953548	−	0.301242i	$-0.902599\pi$
$919$	−57.8136	−1.90710	−0.953548	+	0.301242i	$0.902599\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.352864	0.0116146
$924$	0	0
$925$	−8.12850	−0.267263
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.5854	−1.00347	−0.501737	−	0.865020i	$-0.667305\pi$
$929$	−30.5854	−1.00347	−0.501737	+	0.865020i	$0.667305\pi$
$930$	0	0
$931$	8.79421	0.288219
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.71696	0.252371
$936$	0	0
$937$	−47.5246	−1.55256	−0.776280	−	0.630388i	$-0.782896\pi$
$937$	−47.5246	−1.55256	−0.776280	+	0.630388i	$0.782896\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.0155	−0.945879	−0.472940	−	0.881095i	$-0.656807\pi$
$941$	−29.0155	−0.945879	−0.472940	+	0.881095i	$0.656807\pi$
$942$	0	0
$943$	17.0392	0.554872
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.0502	1.20397	0.601985	−	0.798508i	$-0.294377\pi$
$947$	37.0502	1.20397	0.601985	+	0.798508i	$0.294377\pi$
$948$	0	0
$949$	15.8859	0.515677
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.6665	−0.377915	−0.188958	−	0.981985i	$-0.560511\pi$
$953$	−11.6665	−0.377915	−0.188958	+	0.981985i	$0.560511\pi$
$954$	0	0
$955$	−56.4692	−1.82730
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.78697	−0.0899960
$960$	0	0
$961$	−30.9956	−0.999858
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.5181	0.628310
$966$	0	0
$967$	−35.4370	−1.13958	−0.569789	−	0.821791i	$-0.692975\pi$
$967$	−35.4370	−1.13958	−0.569789	+	0.821791i	$0.692975\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.9907	0.769898	0.384949	−	0.922938i	$-0.374219\pi$
$971$	23.9907	0.769898	0.384949	+	0.922938i	$0.374219\pi$
$972$	0	0
$973$	−1.85941	−0.0596100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.6605	−1.39682	−0.698412	−	0.715696i	$-0.746109\pi$
$977$	−43.6605	−1.39682	−0.698412	+	0.715696i	$0.746109\pi$
$978$	0	0
$979$	14.4329	0.461278
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.7661	−0.534755	−0.267378	−	0.963592i	$-0.586157\pi$
$983$	−16.7661	−0.534755	−0.267378	+	0.963592i	$0.586157\pi$
$984$	0	0
$985$	−44.6443	−1.42249
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.68056	0.276026
$990$	0	0
$991$	−23.3381	−0.741358	−0.370679	−	0.928761i	$-0.620875\pi$
$991$	−23.3381	−0.741358	−0.370679	+	0.928761i	$0.620875\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	66.7793	2.11705
$996$	0	0
$997$	55.8914	1.77010	0.885049	−	0.465498i	$-0.154125\pi$
$997$	55.8914	1.77010	0.885049	+	0.465498i	$0.154125\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7128.2.a.w.1.6		6
3.2	odd	2		7128.2.a.ba.1.1		6
9.2	odd	6		792.2.q.f.265.1	✓	12
9.4	even	3		2376.2.q.f.1585.1		12
9.5	odd	6		792.2.q.f.529.1	yes	12
9.7	even	3		2376.2.q.f.793.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.q.f.265.1	✓	12	9.2	odd	6
792.2.q.f.529.1	yes	12	9.5	odd	6
2376.2.q.f.793.1		12	9.7	even	3
2376.2.q.f.1585.1		12	9.4	even	3
7128.2.a.w.1.6		6	1.1	even	1	trivial
7128.2.a.ba.1.1		6	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:	\( N \)	\(=\)	\( 7128 = 2^{3} \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7128.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52788 q^{5} -0.259373 q^{7} -1.00000 q^{11} -1.32579 q^{13} -3.05274 q^{17} -1.26851 q^{19} +2.37853 q^{23} +1.39018 q^{25} -5.21676 q^{29} -0.0664179 q^{31} -0.655663 q^{35} -5.84709 q^{37} +7.16375 q^{41} +3.64955 q^{43} -4.30658 q^{47} -6.93273 q^{49} -5.63237 q^{53} -2.52788 q^{55} +1.87571 q^{59} +10.4781 q^{61} -3.35144 q^{65} -8.76056 q^{67} -0.266153 q^{71} -11.9822 q^{73} +0.259373 q^{77} -2.90010 q^{79} +3.14646 q^{83} -7.71696 q^{85} -14.4329 q^{89} +0.343874 q^{91} -3.20663 q^{95} +1.61043 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 16 * q^97

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