LMFDB - Embedded newform 8550.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2850)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} -0.267949 q^{11} +0.732051 q^{13} -2.73205 q^{14} +1.00000 q^{16} -4.19615 q^{17} -1.00000 q^{19} -0.267949 q^{22} +7.92820 q^{23} +0.732051 q^{26} -2.73205 q^{28} -1.73205 q^{29} +4.46410 q^{31} +1.00000 q^{32} -4.19615 q^{34} -2.00000 q^{37} -1.00000 q^{38} -10.9282 q^{41} +2.19615 q^{43} -0.267949 q^{44} +7.92820 q^{46} +3.46410 q^{47} +0.464102 q^{49} +0.732051 q^{52} -1.73205 q^{53} -2.73205 q^{56} -1.73205 q^{58} -2.19615 q^{59} -6.66025 q^{61} +4.46410 q^{62} +1.00000 q^{64} -3.73205 q^{67} -4.19615 q^{68} -1.80385 q^{71} +4.46410 q^{73} -2.00000 q^{74} -1.00000 q^{76} +0.732051 q^{77} -12.4641 q^{79} -10.9282 q^{82} -0.267949 q^{83} +2.19615 q^{86} -0.267949 q^{88} +16.8564 q^{89} -2.00000 q^{91} +7.92820 q^{92} +3.46410 q^{94} +9.12436 q^{97} +0.464102 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{26} - 2 q^{28} + 2 q^{31} + 2 q^{32} + 2 q^{34} - 4 q^{37} - 2 q^{38} - 8 q^{41} - 6 q^{43} - 4 q^{44} + 2 q^{46} - 6 q^{49} - 2 q^{52} - 2 q^{56} + 6 q^{59} + 4 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{68} - 14 q^{71} + 2 q^{73} - 4 q^{74} - 2 q^{76} - 2 q^{77} - 18 q^{79} - 8 q^{82} - 4 q^{83} - 6 q^{86} - 4 q^{88} + 6 q^{89} - 4 q^{91} + 2 q^{92} - 6 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^22 + 2 * q^23 - 2 * q^26 - 2 * q^28 + 2 * q^31 + 2 * q^32 + 2 * q^34 - 4 * q^37 - 2 * q^38 - 8 * q^41 - 6 * q^43 - 4 * q^44 + 2 * q^46 - 6 * q^49 - 2 * q^52 - 2 * q^56 + 6 * q^59 + 4 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^68 - 14 * q^71 + 2 * q^73 - 4 * q^74 - 2 * q^76 - 2 * q^77 - 18 * q^79 - 8 * q^82 - 4 * q^83 - 6 * q^86 - 4 * q^88 + 6 * q^89 - 4 * q^91 + 2 * q^92 - 6 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−0.267949	−0.0807897	−0.0403949	−	0.999184i	$-0.512862\pi$
$11$	−0.267949	−0.0807897	−0.0403949	+	0.999184i	$0.512862\pi$
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	−2.73205	−0.730171
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.19615	−1.01772	−0.508858	−	0.860850i	$-0.669932\pi$
$17$	−4.19615	−1.01772	−0.508858	+	0.860850i	$0.669932\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−0.267949	−0.0571270
$23$	7.92820	1.65314	0.826572	−	0.562831i	$-0.190288\pi$
$23$	7.92820	1.65314	0.826572	+	0.562831i	$0.190288\pi$
$24$	0	0
$25$	0	0
$26$	0.732051	0.143567
$27$	0	0
$28$	−2.73205	−0.516309
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	4.46410	0.801776	0.400888	−	0.916127i	$-0.368702\pi$
$31$	4.46410	0.801776	0.400888	+	0.916127i	$0.368702\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.19615	−0.719634
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−10.9282	−1.70670	−0.853349	−	0.521340i	$-0.825432\pi$
$41$	−10.9282	−1.70670	−0.853349	+	0.521340i	$0.825432\pi$
$42$	0	0
$43$	2.19615	0.334910	0.167455	−	0.985880i	$-0.446445\pi$
$43$	2.19615	0.334910	0.167455	+	0.985880i	$0.446445\pi$
$44$	−0.267949	−0.0403949
$45$	0	0
$46$	7.92820	1.16895
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	0.732051	0.101517
$53$	−1.73205	−0.237915	−0.118958	−	0.992899i	$-0.537955\pi$
$53$	−1.73205	−0.237915	−0.118958	+	0.992899i	$0.537955\pi$
$54$	0	0
$55$	0	0
$56$	−2.73205	−0.365086
$57$	0	0
$58$	−1.73205	−0.227429
$59$	−2.19615	−0.285915	−0.142957	−	0.989729i	$-0.545661\pi$
$59$	−2.19615	−0.285915	−0.142957	+	0.989729i	$0.545661\pi$
$60$	0	0
$61$	−6.66025	−0.852758	−0.426379	−	0.904545i	$-0.640211\pi$
$61$	−6.66025	−0.852758	−0.426379	+	0.904545i	$0.640211\pi$
$62$	4.46410	0.566941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−3.73205	−0.455943	−0.227971	−	0.973668i	$-0.573209\pi$
$67$	−3.73205	−0.455943	−0.227971	+	0.973668i	$0.573209\pi$
$68$	−4.19615	−0.508858
$69$	0	0
$70$	0	0
$71$	−1.80385	−0.214077	−0.107039	−	0.994255i	$-0.534137\pi$
$71$	−1.80385	−0.214077	−0.107039	+	0.994255i	$0.534137\pi$
$72$	0	0
$73$	4.46410	0.522484	0.261242	−	0.965273i	$-0.415868\pi$
$73$	4.46410	0.522484	0.261242	+	0.965273i	$0.415868\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0.732051	0.0834249
$78$	0	0
$79$	−12.4641	−1.40232	−0.701160	−	0.713003i	$-0.747334\pi$
$79$	−12.4641	−1.40232	−0.701160	+	0.713003i	$0.747334\pi$
$80$	0	0
$81$	0	0
$82$	−10.9282	−1.20682
$83$	−0.267949	−0.0294112	−0.0147056	−	0.999892i	$-0.504681\pi$
$83$	−0.267949	−0.0294112	−0.0147056	+	0.999892i	$0.504681\pi$
$84$	0	0
$85$	0	0
$86$	2.19615	0.236817
$87$	0	0
$88$	−0.267949	−0.0285635
$89$	16.8564	1.78678	0.893388	−	0.449286i	$-0.148322\pi$
$89$	16.8564	1.78678	0.893388	+	0.449286i	$0.148322\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	7.92820	0.826572
$93$	0	0
$94$	3.46410	0.357295
$95$	0	0
$96$	0	0
$97$	9.12436	0.926438	0.463219	−	0.886244i	$-0.346694\pi$
$97$	9.12436	0.926438	0.463219	+	0.886244i	$0.346694\pi$
$98$	0.464102	0.0468813
$99$	0	0
$100$	0	0
$101$	−12.7321	−1.26689	−0.633443	−	0.773789i	$-0.718359\pi$
$101$	−12.7321	−1.26689	−0.633443	+	0.773789i	$0.718359\pi$
$102$	0	0
$103$	−10.4641	−1.03106	−0.515529	−	0.856872i	$-0.672405\pi$
$103$	−10.4641	−1.03106	−0.515529	+	0.856872i	$0.672405\pi$
$104$	0.732051	0.0717835
$105$	0	0
$106$	−1.73205	−0.168232
$107$	8.19615	0.792352	0.396176	−	0.918175i	$-0.370337\pi$
$107$	8.19615	0.792352	0.396176	+	0.918175i	$0.370337\pi$
$108$	0	0
$109$	−18.3923	−1.76166	−0.880832	−	0.473430i	$-0.843016\pi$
$109$	−18.3923	−1.76166	−0.880832	+	0.473430i	$0.843016\pi$
$110$	0	0
$111$	0	0
$112$	−2.73205	−0.258155
$113$	−7.39230	−0.695410	−0.347705	−	0.937604i	$-0.613039\pi$
$113$	−7.39230	−0.695410	−0.347705	+	0.937604i	$0.613039\pi$
$114$	0	0
$115$	0	0
$116$	−1.73205	−0.160817
$117$	0	0
$118$	−2.19615	−0.202172
$119$	11.4641	1.05091
$120$	0	0
$121$	−10.9282	−0.993473
$122$	−6.66025	−0.602991
$123$	0	0
$124$	4.46410	0.400888
$125$	0	0
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−13.7321	−1.19977	−0.599887	−	0.800084i	$-0.704788\pi$
$131$	−13.7321	−1.19977	−0.599887	+	0.800084i	$0.704788\pi$
$132$	0	0
$133$	2.73205	0.236899
$134$	−3.73205	−0.322400
$135$	0	0
$136$	−4.19615	−0.359817
$137$	−8.39230	−0.717003	−0.358501	−	0.933529i	$-0.616712\pi$
$137$	−8.39230	−0.717003	−0.358501	+	0.933529i	$0.616712\pi$
$138$	0	0
$139$	3.12436	0.265004	0.132502	−	0.991183i	$-0.457699\pi$
$139$	3.12436	0.265004	0.132502	+	0.991183i	$0.457699\pi$
$140$	0	0
$141$	0	0
$142$	−1.80385	−0.151376
$143$	−0.196152	−0.0164031
$144$	0	0
$145$	0	0
$146$	4.46410	0.369452
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−5.85641	−0.479776	−0.239888	−	0.970801i	$-0.577111\pi$
$149$	−5.85641	−0.479776	−0.239888	+	0.970801i	$0.577111\pi$
$150$	0	0
$151$	−4.53590	−0.369126	−0.184563	−	0.982821i	$-0.559087\pi$
$151$	−4.53590	−0.369126	−0.184563	+	0.982821i	$0.559087\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0.732051	0.0589903
$155$	0	0
$156$	0	0
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	−12.4641	−0.991591
$159$	0	0
$160$	0	0
$161$	−21.6603	−1.70707
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−10.9282	−0.853349
$165$	0	0
$166$	−0.267949	−0.0207969
$167$	−4.92820	−0.381356	−0.190678	−	0.981653i	$-0.561069\pi$
$167$	−4.92820	−0.381356	−0.190678	+	0.981653i	$0.561069\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	0	0
$172$	2.19615	0.167455
$173$	17.5885	1.33723	0.668613	−	0.743611i	$-0.266888\pi$
$173$	17.5885	1.33723	0.668613	+	0.743611i	$0.266888\pi$
$174$	0	0
$175$	0	0
$176$	−0.267949	−0.0201974
$177$	0	0
$178$	16.8564	1.26344
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.5885	1.08435	0.542176	−	0.840265i	$-0.317601\pi$
$181$	14.5885	1.08435	0.542176	+	0.840265i	$0.317601\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	7.92820	0.584475
$185$	0	0
$186$	0	0
$187$	1.12436	0.0822210
$188$	3.46410	0.252646
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−16.0526	−1.15549	−0.577744	−	0.816218i	$-0.696067\pi$
$193$	−16.0526	−1.15549	−0.577744	+	0.816218i	$0.696067\pi$
$194$	9.12436	0.655091
$195$	0	0
$196$	0.464102	0.0331501
$197$	6.58846	0.469408	0.234704	−	0.972067i	$-0.424588\pi$
$197$	6.58846	0.469408	0.234704	+	0.972067i	$0.424588\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	−12.7321	−0.895824
$203$	4.73205	0.332125
$204$	0	0
$205$	0	0
$206$	−10.4641	−0.729069
$207$	0	0
$208$	0.732051	0.0507586
$209$	0.267949	0.0185344
$210$	0	0
$211$	−13.7321	−0.945353	−0.472677	−	0.881236i	$-0.656712\pi$
$211$	−13.7321	−0.945353	−0.472677	+	0.881236i	$0.656712\pi$
$212$	−1.73205	−0.118958
$213$	0	0
$214$	8.19615	0.560277
$215$	0	0
$216$	0	0
$217$	−12.1962	−0.827929
$218$	−18.3923	−1.24568
$219$	0	0
$220$	0	0
$221$	−3.07180	−0.206631
$222$	0	0
$223$	−10.4641	−0.700728	−0.350364	−	0.936614i	$-0.613942\pi$
$223$	−10.4641	−0.700728	−0.350364	+	0.936614i	$0.613942\pi$
$224$	−2.73205	−0.182543
$225$	0	0
$226$	−7.39230	−0.491729
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−9.73205	−0.643112	−0.321556	−	0.946891i	$-0.604206\pi$
$229$	−9.73205	−0.643112	−0.321556	+	0.946891i	$0.604206\pi$
$230$	0	0
$231$	0	0
$232$	−1.73205	−0.113715
$233$	−3.12436	−0.204683	−0.102342	−	0.994749i	$-0.532634\pi$
$233$	−3.12436	−0.204683	−0.102342	+	0.994749i	$0.532634\pi$
$234$	0	0
$235$	0	0
$236$	−2.19615	−0.142957
$237$	0	0
$238$	11.4641	0.743107
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−19.8038	−1.27568	−0.637839	−	0.770170i	$-0.720171\pi$
$241$	−19.8038	−1.27568	−0.637839	+	0.770170i	$0.720171\pi$
$242$	−10.9282	−0.702492
$243$	0	0
$244$	−6.66025	−0.426379
$245$	0	0
$246$	0	0
$247$	−0.732051	−0.0465793
$248$	4.46410	0.283471
$249$	0	0
$250$	0	0
$251$	−4.53590	−0.286303	−0.143152	−	0.989701i	$-0.545724\pi$
$251$	−4.53590	−0.286303	−0.143152	+	0.989701i	$0.545724\pi$
$252$	0	0
$253$	−2.12436	−0.133557
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	0	0
$259$	5.46410	0.339523
$260$	0	0
$261$	0	0
$262$	−13.7321	−0.848369
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	0	0
$266$	2.73205	0.167513
$267$	0	0
$268$	−3.73205	−0.227971
$269$	27.4641	1.67452	0.837258	−	0.546808i	$-0.184157\pi$
$269$	27.4641	1.67452	0.837258	+	0.546808i	$0.184157\pi$
$270$	0	0
$271$	−26.1962	−1.59130	−0.795651	−	0.605755i	$-0.792871\pi$
$271$	−26.1962	−1.59130	−0.795651	+	0.605755i	$0.792871\pi$
$272$	−4.19615	−0.254429
$273$	0	0
$274$	−8.39230	−0.506998
$275$	0	0
$276$	0	0
$277$	0.803848	0.0482985	0.0241493	−	0.999708i	$-0.492312\pi$
$277$	0.803848	0.0482985	0.0241493	+	0.999708i	$0.492312\pi$
$278$	3.12436	0.187386
$279$	0	0
$280$	0	0
$281$	23.7846	1.41887	0.709435	−	0.704770i	$-0.248950\pi$
$281$	23.7846	1.41887	0.709435	+	0.704770i	$0.248950\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−1.80385	−0.107039
$285$	0	0
$286$	−0.196152	−0.0115987
$287$	29.8564	1.76237
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	0	0
$292$	4.46410	0.261242
$293$	24.2679	1.41775	0.708874	−	0.705335i	$-0.249203\pi$
$293$	24.2679	1.41775	0.708874	+	0.705335i	$0.249203\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−5.85641	−0.339253
$299$	5.80385	0.335645
$300$	0	0
$301$	−6.00000	−0.345834
$302$	−4.53590	−0.261012
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−23.7321	−1.35446	−0.677230	−	0.735772i	$-0.736820\pi$
$307$	−23.7321	−1.35446	−0.677230	+	0.735772i	$0.736820\pi$
$308$	0.732051	0.0417125
$309$	0	0
$310$	0	0
$311$	−5.32051	−0.301698	−0.150849	−	0.988557i	$-0.548201\pi$
$311$	−5.32051	−0.301698	−0.150849	+	0.988557i	$0.548201\pi$
$312$	0	0
$313$	9.92820	0.561175	0.280588	−	0.959828i	$-0.409471\pi$
$313$	9.92820	0.561175	0.280588	+	0.959828i	$0.409471\pi$
$314$	−13.8564	−0.781962
$315$	0	0
$316$	−12.4641	−0.701160
$317$	−1.73205	−0.0972817	−0.0486408	−	0.998816i	$-0.515489\pi$
$317$	−1.73205	−0.0972817	−0.0486408	+	0.998816i	$0.515489\pi$
$318$	0	0
$319$	0.464102	0.0259847
$320$	0	0
$321$	0	0
$322$	−21.6603	−1.20708
$323$	4.19615	0.233480
$324$	0	0
$325$	0	0
$326$	−2.00000	−0.110770
$327$	0	0
$328$	−10.9282	−0.603409
$329$	−9.46410	−0.521773
$330$	0	0
$331$	−25.7321	−1.41436	−0.707181	−	0.707033i	$-0.750033\pi$
$331$	−25.7321	−1.41436	−0.707181	+	0.707033i	$0.750033\pi$
$332$	−0.267949	−0.0147056
$333$	0	0
$334$	−4.92820	−0.269659
$335$	0	0
$336$	0	0
$337$	17.4641	0.951330	0.475665	−	0.879626i	$-0.342207\pi$
$337$	17.4641	0.951330	0.475665	+	0.879626i	$0.342207\pi$
$338$	−12.4641	−0.677958
$339$	0	0
$340$	0	0
$341$	−1.19615	−0.0647753
$342$	0	0
$343$	17.8564	0.964155
$344$	2.19615	0.118409
$345$	0	0
$346$	17.5885	0.945561
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	0	0
$349$	21.0526	1.12692	0.563459	−	0.826144i	$-0.309470\pi$
$349$	21.0526	1.12692	0.563459	+	0.826144i	$0.309470\pi$
$350$	0	0
$351$	0	0
$352$	−0.267949	−0.0142817
$353$	22.0526	1.17374	0.586870	−	0.809681i	$-0.300360\pi$
$353$	22.0526	1.17374	0.586870	+	0.809681i	$0.300360\pi$
$354$	0	0
$355$	0	0
$356$	16.8564	0.893388
$357$	0	0
$358$	12.0000	0.634220
$359$	3.07180	0.162123	0.0810616	−	0.996709i	$-0.474169\pi$
$359$	3.07180	0.162123	0.0810616	+	0.996709i	$0.474169\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.5885	0.766752
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	−16.7321	−0.873406	−0.436703	−	0.899606i	$-0.643854\pi$
$367$	−16.7321	−0.873406	−0.436703	+	0.899606i	$0.643854\pi$
$368$	7.92820	0.413286
$369$	0	0
$370$	0	0
$371$	4.73205	0.245676
$372$	0	0
$373$	13.5167	0.699866	0.349933	−	0.936775i	$-0.386204\pi$
$373$	13.5167	0.699866	0.349933	+	0.936775i	$0.386204\pi$
$374$	1.12436	0.0581390
$375$	0	0
$376$	3.46410	0.178647
$377$	−1.26795	−0.0653027
$378$	0	0
$379$	20.3923	1.04748	0.523741	−	0.851877i	$-0.324536\pi$
$379$	20.3923	1.04748	0.523741	+	0.851877i	$0.324536\pi$
$380$	0	0
$381$	0	0
$382$	−3.00000	−0.153493
$383$	−25.5167	−1.30384	−0.651920	−	0.758288i	$-0.726036\pi$
$383$	−25.5167	−1.30384	−0.651920	+	0.758288i	$0.726036\pi$
$384$	0	0
$385$	0	0
$386$	−16.0526	−0.817054
$387$	0	0
$388$	9.12436	0.463219
$389$	33.7128	1.70931	0.854654	−	0.519198i	$-0.173769\pi$
$389$	33.7128	1.70931	0.854654	+	0.519198i	$0.173769\pi$
$390$	0	0
$391$	−33.2679	−1.68243
$392$	0.464102	0.0234407
$393$	0	0
$394$	6.58846	0.331922
$395$	0	0
$396$	0	0
$397$	8.26795	0.414956	0.207478	−	0.978240i	$-0.433474\pi$
$397$	8.26795	0.414956	0.207478	+	0.978240i	$0.433474\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	0	0
$401$	10.8564	0.542143	0.271072	−	0.962559i	$-0.412622\pi$
$401$	10.8564	0.542143	0.271072	+	0.962559i	$0.412622\pi$
$402$	0	0
$403$	3.26795	0.162788
$404$	−12.7321	−0.633443
$405$	0	0
$406$	4.73205	0.234848
$407$	0.535898	0.0265635
$408$	0	0
$409$	22.3923	1.10723	0.553614	−	0.832773i	$-0.313248\pi$
$409$	22.3923	1.10723	0.553614	+	0.832773i	$0.313248\pi$
$410$	0	0
$411$	0	0
$412$	−10.4641	−0.515529
$413$	6.00000	0.295241
$414$	0	0
$415$	0	0
$416$	0.732051	0.0358917
$417$	0	0
$418$	0.267949	0.0131058
$419$	13.8564	0.676930	0.338465	−	0.940979i	$-0.390092\pi$
$419$	13.8564	0.676930	0.338465	+	0.940979i	$0.390092\pi$
$420$	0	0
$421$	−34.7846	−1.69530	−0.847649	−	0.530557i	$-0.821983\pi$
$421$	−34.7846	−1.69530	−0.847649	+	0.530557i	$0.821983\pi$
$422$	−13.7321	−0.668466
$423$	0	0
$424$	−1.73205	−0.0841158
$425$	0	0
$426$	0	0
$427$	18.1962	0.880574
$428$	8.19615	0.396176
$429$	0	0
$430$	0	0
$431$	−5.32051	−0.256280	−0.128140	−	0.991756i	$-0.540901\pi$
$431$	−5.32051	−0.256280	−0.128140	+	0.991756i	$0.540901\pi$
$432$	0	0
$433$	13.8038	0.663371	0.331685	−	0.943390i	$-0.392383\pi$
$433$	13.8038	0.663371	0.331685	+	0.943390i	$0.392383\pi$
$434$	−12.1962	−0.585434
$435$	0	0
$436$	−18.3923	−0.880832
$437$	−7.92820	−0.379257
$438$	0	0
$439$	−15.7846	−0.753358	−0.376679	−	0.926344i	$-0.622934\pi$
$439$	−15.7846	−0.753358	−0.376679	+	0.926344i	$0.622934\pi$
$440$	0	0
$441$	0	0
$442$	−3.07180	−0.146110
$443$	−20.6603	−0.981598	−0.490799	−	0.871273i	$-0.663295\pi$
$443$	−20.6603	−0.981598	−0.490799	+	0.871273i	$0.663295\pi$
$444$	0	0
$445$	0	0
$446$	−10.4641	−0.495490
$447$	0	0
$448$	−2.73205	−0.129077
$449$	−17.5359	−0.827570	−0.413785	−	0.910375i	$-0.635794\pi$
$449$	−17.5359	−0.827570	−0.413785	+	0.910375i	$0.635794\pi$
$450$	0	0
$451$	2.92820	0.137884
$452$	−7.39230	−0.347705
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	18.9282	0.885424	0.442712	−	0.896664i	$-0.354016\pi$
$457$	18.9282	0.885424	0.442712	+	0.896664i	$0.354016\pi$
$458$	−9.73205	−0.454749
$459$	0	0
$460$	0	0
$461$	−9.85641	−0.459059	−0.229529	−	0.973302i	$-0.573719\pi$
$461$	−9.85641	−0.459059	−0.229529	+	0.973302i	$0.573719\pi$
$462$	0	0
$463$	0.679492	0.0315787	0.0157893	−	0.999875i	$-0.494974\pi$
$463$	0.679492	0.0315787	0.0157893	+	0.999875i	$0.494974\pi$
$464$	−1.73205	−0.0804084
$465$	0	0
$466$	−3.12436	−0.144733
$467$	−3.19615	−0.147900	−0.0739501	−	0.997262i	$-0.523561\pi$
$467$	−3.19615	−0.147900	−0.0739501	+	0.997262i	$0.523561\pi$
$468$	0	0
$469$	10.1962	0.470815
$470$	0	0
$471$	0	0
$472$	−2.19615	−0.101086
$473$	−0.588457	−0.0270573
$474$	0	0
$475$	0	0
$476$	11.4641	0.525456
$477$	0	0
$478$	14.0000	0.640345
$479$	−30.8564	−1.40987	−0.704933	−	0.709274i	$-0.749023\pi$
$479$	−30.8564	−1.40987	−0.704933	+	0.709274i	$0.749023\pi$
$480$	0	0
$481$	−1.46410	−0.0667573
$482$	−19.8038	−0.902041
$483$	0	0
$484$	−10.9282	−0.496737
$485$	0	0
$486$	0	0
$487$	−29.8564	−1.35292	−0.676461	−	0.736478i	$-0.736487\pi$
$487$	−29.8564	−1.35292	−0.676461	+	0.736478i	$0.736487\pi$
$488$	−6.66025	−0.301496
$489$	0	0
$490$	0	0
$491$	−10.9282	−0.493183	−0.246591	−	0.969120i	$-0.579311\pi$
$491$	−10.9282	−0.493183	−0.246591	+	0.969120i	$0.579311\pi$
$492$	0	0
$493$	7.26795	0.327332
$494$	−0.732051	−0.0329365
$495$	0	0
$496$	4.46410	0.200444
$497$	4.92820	0.221060
$498$	0	0
$499$	−31.3731	−1.40445	−0.702226	−	0.711954i	$-0.747810\pi$
$499$	−31.3731	−1.40445	−0.702226	+	0.711954i	$0.747810\pi$
$500$	0	0
$501$	0	0
$502$	−4.53590	−0.202447
$503$	9.85641	0.439475	0.219738	−	0.975559i	$-0.429480\pi$
$503$	9.85641	0.439475	0.219738	+	0.975559i	$0.429480\pi$
$504$	0	0
$505$	0	0
$506$	−2.12436	−0.0944391
$507$	0	0
$508$	−5.00000	−0.221839
$509$	−38.1244	−1.68983	−0.844916	−	0.534899i	$-0.820350\pi$
$509$	−38.1244	−1.68983	−0.844916	+	0.534899i	$0.820350\pi$
$510$	0	0
$511$	−12.1962	−0.539526
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.00000	−0.396973
$515$	0	0
$516$	0	0
$517$	−0.928203	−0.0408223
$518$	5.46410	0.240079
$519$	0	0
$520$	0	0
$521$	38.1769	1.67256	0.836280	−	0.548302i	$-0.184726\pi$
$521$	38.1769	1.67256	0.836280	+	0.548302i	$0.184726\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−13.7321	−0.599887
$525$	0	0
$526$	−15.0000	−0.654031
$527$	−18.7321	−0.815981
$528$	0	0
$529$	39.8564	1.73289
$530$	0	0
$531$	0	0
$532$	2.73205	0.118449
$533$	−8.00000	−0.346518
$534$	0	0
$535$	0	0
$536$	−3.73205	−0.161200
$537$	0	0
$538$	27.4641	1.18406
$539$	−0.124356	−0.00535638
$540$	0	0
$541$	6.80385	0.292520	0.146260	−	0.989246i	$-0.453276\pi$
$541$	6.80385	0.292520	0.146260	+	0.989246i	$0.453276\pi$
$542$	−26.1962	−1.12522
$543$	0	0
$544$	−4.19615	−0.179909
$545$	0	0
$546$	0	0
$547$	−15.4449	−0.660375	−0.330187	−	0.943915i	$-0.607112\pi$
$547$	−15.4449	−0.660375	−0.330187	+	0.943915i	$0.607112\pi$
$548$	−8.39230	−0.358501
$549$	0	0
$550$	0	0
$551$	1.73205	0.0737878
$552$	0	0
$553$	34.0526	1.44806
$554$	0.803848	0.0341522
$555$	0	0
$556$	3.12436	0.132502
$557$	−14.1962	−0.601510	−0.300755	−	0.953701i	$-0.597239\pi$
$557$	−14.1962	−0.601510	−0.300755	+	0.953701i	$0.597239\pi$
$558$	0	0
$559$	1.60770	0.0679983
$560$	0	0
$561$	0	0
$562$	23.7846	1.00329
$563$	9.66025	0.407131	0.203566	−	0.979061i	$-0.434747\pi$
$563$	9.66025	0.407131	0.203566	+	0.979061i	$0.434747\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	−1.80385	−0.0756878
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	4.19615	0.175604	0.0878018	−	0.996138i	$-0.472016\pi$
$571$	4.19615	0.175604	0.0878018	+	0.996138i	$0.472016\pi$
$572$	−0.196152	−0.00820154
$573$	0	0
$574$	29.8564	1.24618
$575$	0	0
$576$	0	0
$577$	31.2487	1.30090	0.650450	−	0.759549i	$-0.274580\pi$
$577$	31.2487	1.30090	0.650450	+	0.759549i	$0.274580\pi$
$578$	0.607695	0.0252768
$579$	0	0
$580$	0	0
$581$	0.732051	0.0303706
$582$	0	0
$583$	0.464102	0.0192211
$584$	4.46410	0.184726
$585$	0	0
$586$	24.2679	1.00250
$587$	15.7321	0.649331	0.324666	−	0.945829i	$-0.394748\pi$
$587$	15.7321	0.649331	0.324666	+	0.945829i	$0.394748\pi$
$588$	0	0
$589$	−4.46410	−0.183940
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	48.3923	1.98723	0.993617	−	0.112807i	$-0.0359843\pi$
$593$	48.3923	1.98723	0.993617	+	0.112807i	$0.0359843\pi$
$594$	0	0
$595$	0	0
$596$	−5.85641	−0.239888
$597$	0	0
$598$	5.80385	0.237337
$599$	−13.6603	−0.558143	−0.279071	−	0.960270i	$-0.590027\pi$
$599$	−13.6603	−0.558143	−0.279071	+	0.960270i	$0.590027\pi$
$600$	0	0
$601$	28.0526	1.14429	0.572144	−	0.820153i	$-0.306112\pi$
$601$	28.0526	1.14429	0.572144	+	0.820153i	$0.306112\pi$
$602$	−6.00000	−0.244542
$603$	0	0
$604$	−4.53590	−0.184563
$605$	0	0
$606$	0	0
$607$	29.7846	1.20892	0.604460	−	0.796635i	$-0.293389\pi$
$607$	29.7846	1.20892	0.604460	+	0.796635i	$0.293389\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	2.53590	0.102591
$612$	0	0
$613$	−2.78461	−0.112469	−0.0562347	−	0.998418i	$-0.517909\pi$
$613$	−2.78461	−0.112469	−0.0562347	+	0.998418i	$0.517909\pi$
$614$	−23.7321	−0.957748
$615$	0	0
$616$	0.732051	0.0294952
$617$	−25.6077	−1.03093	−0.515463	−	0.856912i	$-0.672380\pi$
$617$	−25.6077	−1.03093	−0.515463	+	0.856912i	$0.672380\pi$
$618$	0	0
$619$	10.1962	0.409818	0.204909	−	0.978781i	$-0.434310\pi$
$619$	10.1962	0.409818	0.204909	+	0.978781i	$0.434310\pi$
$620$	0	0
$621$	0	0
$622$	−5.32051	−0.213333
$623$	−46.0526	−1.84506
$624$	0	0
$625$	0	0
$626$	9.92820	0.396811
$627$	0	0
$628$	−13.8564	−0.552931
$629$	8.39230	0.334623
$630$	0	0
$631$	−10.9282	−0.435045	−0.217522	−	0.976055i	$-0.569798\pi$
$631$	−10.9282	−0.435045	−0.217522	+	0.976055i	$0.569798\pi$
$632$	−12.4641	−0.495795
$633$	0	0
$634$	−1.73205	−0.0687885
$635$	0	0
$636$	0	0
$637$	0.339746	0.0134612
$638$	0.464102	0.0183740
$639$	0	0
$640$	0	0
$641$	−17.4641	−0.689791	−0.344895	−	0.938641i	$-0.612086\pi$
$641$	−17.4641	−0.689791	−0.344895	+	0.938641i	$0.612086\pi$
$642$	0	0
$643$	40.9282	1.61405	0.807025	−	0.590517i	$-0.201076\pi$
$643$	40.9282	1.61405	0.807025	+	0.590517i	$0.201076\pi$
$644$	−21.6603	−0.853534
$645$	0	0
$646$	4.19615	0.165095
$647$	11.3923	0.447878	0.223939	−	0.974603i	$-0.428108\pi$
$647$	11.3923	0.447878	0.223939	+	0.974603i	$0.428108\pi$
$648$	0	0
$649$	0.588457	0.0230990
$650$	0	0
$651$	0	0
$652$	−2.00000	−0.0783260
$653$	10.5359	0.412302	0.206151	−	0.978520i	$-0.433906\pi$
$653$	10.5359	0.412302	0.206151	+	0.978520i	$0.433906\pi$
$654$	0	0
$655$	0	0
$656$	−10.9282	−0.426675
$657$	0	0
$658$	−9.46410	−0.368949
$659$	−38.5885	−1.50319	−0.751596	−	0.659623i	$-0.770716\pi$
$659$	−38.5885	−1.50319	−0.751596	+	0.659623i	$0.770716\pi$
$660$	0	0
$661$	46.6410	1.81413	0.907063	−	0.420996i	$-0.138319\pi$
$661$	46.6410	1.81413	0.907063	+	0.420996i	$0.138319\pi$
$662$	−25.7321	−1.00010
$663$	0	0
$664$	−0.267949	−0.0103984
$665$	0	0
$666$	0	0
$667$	−13.7321	−0.531707
$668$	−4.92820	−0.190678
$669$	0	0
$670$	0	0
$671$	1.78461	0.0688941
$672$	0	0
$673$	−44.9808	−1.73388	−0.866940	−	0.498412i	$-0.833917\pi$
$673$	−44.9808	−1.73388	−0.866940	+	0.498412i	$0.833917\pi$
$674$	17.4641	0.672692
$675$	0	0
$676$	−12.4641	−0.479389
$677$	−8.66025	−0.332841	−0.166420	−	0.986055i	$-0.553221\pi$
$677$	−8.66025	−0.332841	−0.166420	+	0.986055i	$0.553221\pi$
$678$	0	0
$679$	−24.9282	−0.956657
$680$	0	0
$681$	0	0
$682$	−1.19615	−0.0458030
$683$	4.98076	0.190584	0.0952918	−	0.995449i	$-0.469622\pi$
$683$	4.98076	0.190584	0.0952918	+	0.995449i	$0.469622\pi$
$684$	0	0
$685$	0	0
$686$	17.8564	0.681761
$687$	0	0
$688$	2.19615	0.0837275
$689$	−1.26795	−0.0483050
$690$	0	0
$691$	−11.8038	−0.449040	−0.224520	−	0.974470i	$-0.572081\pi$
$691$	−11.8038	−0.449040	−0.224520	+	0.974470i	$0.572081\pi$
$692$	17.5885	0.668613
$693$	0	0
$694$	−20.7846	−0.788973
$695$	0	0
$696$	0	0
$697$	45.8564	1.73694
$698$	21.0526	0.796851
$699$	0	0
$700$	0	0
$701$	−24.3923	−0.921285	−0.460642	−	0.887586i	$-0.652381\pi$
$701$	−24.3923	−0.921285	−0.460642	+	0.887586i	$0.652381\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−0.267949	−0.0100987
$705$	0	0
$706$	22.0526	0.829959
$707$	34.7846	1.30821
$708$	0	0
$709$	−39.3013	−1.47599	−0.737995	−	0.674806i	$-0.764227\pi$
$709$	−39.3013	−1.47599	−0.737995	+	0.674806i	$0.764227\pi$
$710$	0	0
$711$	0	0
$712$	16.8564	0.631721
$713$	35.3923	1.32545
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	3.07180	0.114638
$719$	50.1769	1.87128	0.935642	−	0.352952i	$-0.114822\pi$
$719$	50.1769	1.87128	0.935642	+	0.352952i	$0.114822\pi$
$720$	0	0
$721$	28.5885	1.06469
$722$	1.00000	0.0372161
$723$	0	0
$724$	14.5885	0.542176
$725$	0	0
$726$	0	0
$727$	38.1051	1.41324	0.706620	−	0.707593i	$-0.250219\pi$
$727$	38.1051	1.41324	0.706620	+	0.707593i	$0.250219\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	0	0
$731$	−9.21539	−0.340844
$732$	0	0
$733$	24.5167	0.905544	0.452772	−	0.891626i	$-0.350435\pi$
$733$	24.5167	0.905544	0.452772	+	0.891626i	$0.350435\pi$
$734$	−16.7321	−0.617591
$735$	0	0
$736$	7.92820	0.292237
$737$	1.00000	0.0368355
$738$	0	0
$739$	9.17691	0.337578	0.168789	−	0.985652i	$-0.446014\pi$
$739$	9.17691	0.337578	0.168789	+	0.985652i	$0.446014\pi$
$740$	0	0
$741$	0	0
$742$	4.73205	0.173719
$743$	18.9282	0.694408	0.347204	−	0.937790i	$-0.387131\pi$
$743$	18.9282	0.694408	0.347204	+	0.937790i	$0.387131\pi$
$744$	0	0
$745$	0	0
$746$	13.5167	0.494880
$747$	0	0
$748$	1.12436	0.0411105
$749$	−22.3923	−0.818197
$750$	0	0
$751$	0.535898	0.0195552	0.00977760	−	0.999952i	$-0.496888\pi$
$751$	0.535898	0.0195552	0.00977760	+	0.999952i	$0.496888\pi$
$752$	3.46410	0.126323
$753$	0	0
$754$	−1.26795	−0.0461760
$755$	0	0
$756$	0	0
$757$	17.5885	0.639263	0.319632	−	0.947542i	$-0.396441\pi$
$757$	17.5885	0.639263	0.319632	+	0.947542i	$0.396441\pi$
$758$	20.3923	0.740682
$759$	0	0
$760$	0	0
$761$	−4.73205	−0.171537	−0.0857684	−	0.996315i	$-0.527334\pi$
$761$	−4.73205	−0.171537	−0.0857684	+	0.996315i	$0.527334\pi$
$762$	0	0
$763$	50.2487	1.81913
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−25.5167	−0.921954
$767$	−1.60770	−0.0580505
$768$	0	0
$769$	20.7128	0.746923	0.373462	−	0.927646i	$-0.378171\pi$
$769$	20.7128	0.746923	0.373462	+	0.927646i	$0.378171\pi$
$770$	0	0
$771$	0	0
$772$	−16.0526	−0.577744
$773$	10.6795	0.384115	0.192057	−	0.981384i	$-0.438484\pi$
$773$	10.6795	0.384115	0.192057	+	0.981384i	$0.438484\pi$
$774$	0	0
$775$	0	0
$776$	9.12436	0.327545
$777$	0	0
$778$	33.7128	1.20866
$779$	10.9282	0.391544
$780$	0	0
$781$	0.483340	0.0172952
$782$	−33.2679	−1.18966
$783$	0	0
$784$	0.464102	0.0165751
$785$	0	0
$786$	0	0
$787$	−22.1244	−0.788648	−0.394324	−	0.918971i	$-0.629021\pi$
$787$	−22.1244	−0.788648	−0.394324	+	0.918971i	$0.629021\pi$
$788$	6.58846	0.234704
$789$	0	0
$790$	0	0
$791$	20.1962	0.718093
$792$	0	0
$793$	−4.87564	−0.173139
$794$	8.26795	0.293419
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	0	0
$799$	−14.5359	−0.514243
$800$	0	0
$801$	0	0
$802$	10.8564	0.383353
$803$	−1.19615	−0.0422113
$804$	0	0
$805$	0	0
$806$	3.26795	0.115109
$807$	0	0
$808$	−12.7321	−0.447912
$809$	7.51666	0.264272	0.132136	−	0.991232i	$-0.457816\pi$
$809$	7.51666	0.264272	0.132136	+	0.991232i	$0.457816\pi$
$810$	0	0
$811$	−54.3731	−1.90930	−0.954648	−	0.297736i	$-0.903769\pi$
$811$	−54.3731	−1.90930	−0.954648	+	0.297736i	$0.903769\pi$
$812$	4.73205	0.166062
$813$	0	0
$814$	0.535898	0.0187832
$815$	0	0
$816$	0	0
$817$	−2.19615	−0.0768336
$818$	22.3923	0.782929
$819$	0	0
$820$	0	0
$821$	−47.5167	−1.65834	−0.829171	−	0.558994i	$-0.811187\pi$
$821$	−47.5167	−1.65834	−0.829171	+	0.558994i	$0.811187\pi$
$822$	0	0
$823$	34.7846	1.21252	0.606258	−	0.795268i	$-0.292670\pi$
$823$	34.7846	1.21252	0.606258	+	0.795268i	$0.292670\pi$
$824$	−10.4641	−0.364534
$825$	0	0
$826$	6.00000	0.208767
$827$	−3.26795	−0.113638	−0.0568189	−	0.998385i	$-0.518096\pi$
$827$	−3.26795	−0.113638	−0.0568189	+	0.998385i	$0.518096\pi$
$828$	0	0
$829$	−4.19615	−0.145738	−0.0728692	−	0.997342i	$-0.523216\pi$
$829$	−4.19615	−0.145738	−0.0728692	+	0.997342i	$0.523216\pi$
$830$	0	0
$831$	0	0
$832$	0.732051	0.0253793
$833$	−1.94744	−0.0674748
$834$	0	0
$835$	0	0
$836$	0.267949	0.00926722
$837$	0	0
$838$	13.8564	0.478662
$839$	15.4115	0.532066	0.266033	−	0.963964i	$-0.414287\pi$
$839$	15.4115	0.532066	0.266033	+	0.963964i	$0.414287\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−34.7846	−1.19876
$843$	0	0
$844$	−13.7321	−0.472677
$845$	0	0
$846$	0	0
$847$	29.8564	1.02588
$848$	−1.73205	−0.0594789
$849$	0	0
$850$	0	0
$851$	−15.8564	−0.543551
$852$	0	0
$853$	−11.4641	−0.392523	−0.196262	−	0.980552i	$-0.562880\pi$
$853$	−11.4641	−0.392523	−0.196262	+	0.980552i	$0.562880\pi$
$854$	18.1962	0.622660
$855$	0	0
$856$	8.19615	0.280139
$857$	29.0718	0.993074	0.496537	−	0.868016i	$-0.334605\pi$
$857$	29.0718	0.993074	0.496537	+	0.868016i	$0.334605\pi$
$858$	0	0
$859$	43.0718	1.46959	0.734795	−	0.678289i	$-0.237278\pi$
$859$	43.0718	1.46959	0.734795	+	0.678289i	$0.237278\pi$
$860$	0	0
$861$	0	0
$862$	−5.32051	−0.181217
$863$	20.6795	0.703938	0.351969	−	0.936012i	$-0.385512\pi$
$863$	20.6795	0.703938	0.351969	+	0.936012i	$0.385512\pi$
$864$	0	0
$865$	0	0
$866$	13.8038	0.469074
$867$	0	0
$868$	−12.1962	−0.413964
$869$	3.33975	0.113293
$870$	0	0
$871$	−2.73205	−0.0925720
$872$	−18.3923	−0.622842
$873$	0	0
$874$	−7.92820	−0.268175
$875$	0	0
$876$	0	0
$877$	−31.1769	−1.05277	−0.526385	−	0.850246i	$-0.676453\pi$
$877$	−31.1769	−1.05277	−0.526385	+	0.850246i	$0.676453\pi$
$878$	−15.7846	−0.532705
$879$	0	0
$880$	0	0
$881$	33.4641	1.12743	0.563717	−	0.825968i	$-0.309371\pi$
$881$	33.4641	1.12743	0.563717	+	0.825968i	$0.309371\pi$
$882$	0	0
$883$	30.7321	1.03422	0.517108	−	0.855920i	$-0.327009\pi$
$883$	30.7321	1.03422	0.517108	+	0.855920i	$0.327009\pi$
$884$	−3.07180	−0.103316
$885$	0	0
$886$	−20.6603	−0.694095
$887$	28.3923	0.953320	0.476660	−	0.879088i	$-0.341847\pi$
$887$	28.3923	0.953320	0.476660	+	0.879088i	$0.341847\pi$
$888$	0	0
$889$	13.6603	0.458150
$890$	0	0
$891$	0	0
$892$	−10.4641	−0.350364
$893$	−3.46410	−0.115922
$894$	0	0
$895$	0	0
$896$	−2.73205	−0.0912714
$897$	0	0
$898$	−17.5359	−0.585181
$899$	−7.73205	−0.257878
$900$	0	0
$901$	7.26795	0.242130
$902$	2.92820	0.0974985
$903$	0	0
$904$	−7.39230	−0.245864
$905$	0	0
$906$	0	0
$907$	14.6795	0.487425	0.243712	−	0.969848i	$-0.421635\pi$
$907$	14.6795	0.487425	0.243712	+	0.969848i	$0.421635\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	0	0
$911$	42.1051	1.39500	0.697502	−	0.716582i	$-0.254295\pi$
$911$	42.1051	1.39500	0.697502	+	0.716582i	$0.254295\pi$
$912$	0	0
$913$	0.0717968	0.00237613
$914$	18.9282	0.626089
$915$	0	0
$916$	−9.73205	−0.321556
$917$	37.5167	1.23891
$918$	0	0
$919$	−49.5167	−1.63340	−0.816702	−	0.577060i	$-0.804200\pi$
$919$	−49.5167	−1.63340	−0.816702	+	0.577060i	$0.804200\pi$
$920$	0	0
$921$	0	0
$922$	−9.85641	−0.324603
$923$	−1.32051	−0.0434651
$924$	0	0
$925$	0	0
$926$	0.679492	0.0223295
$927$	0	0
$928$	−1.73205	−0.0568574
$929$	−29.0718	−0.953815	−0.476907	−	0.878954i	$-0.658242\pi$
$929$	−29.0718	−0.953815	−0.476907	+	0.878954i	$0.658242\pi$
$930$	0	0
$931$	−0.464102	−0.0152103
$932$	−3.12436	−0.102342
$933$	0	0
$934$	−3.19615	−0.104581
$935$	0	0
$936$	0	0
$937$	−54.7846	−1.78974	−0.894868	−	0.446332i	$-0.852730\pi$
$937$	−54.7846	−1.78974	−0.894868	+	0.446332i	$0.852730\pi$
$938$	10.1962	0.332916
$939$	0	0
$940$	0	0
$941$	15.7321	0.512850	0.256425	−	0.966564i	$-0.417455\pi$
$941$	15.7321	0.512850	0.256425	+	0.966564i	$0.417455\pi$
$942$	0	0
$943$	−86.6410	−2.82142
$944$	−2.19615	−0.0714787
$945$	0	0
$946$	−0.588457	−0.0191324
$947$	21.8564	0.710238	0.355119	−	0.934821i	$-0.384440\pi$
$947$	21.8564	0.710238	0.355119	+	0.934821i	$0.384440\pi$
$948$	0	0
$949$	3.26795	0.106082
$950$	0	0
$951$	0	0
$952$	11.4641	0.371554
$953$	43.2487	1.40096	0.700482	−	0.713670i	$-0.252969\pi$
$953$	43.2487	1.40096	0.700482	+	0.713670i	$0.252969\pi$
$954$	0	0
$955$	0	0
$956$	14.0000	0.452792
$957$	0	0
$958$	−30.8564	−0.996925
$959$	22.9282	0.740390
$960$	0	0
$961$	−11.0718	−0.357155
$962$	−1.46410	−0.0472045
$963$	0	0
$964$	−19.8038	−0.637839
$965$	0	0
$966$	0	0
$967$	58.1051	1.86853	0.934267	−	0.356573i	$-0.116055\pi$
$967$	58.1051	1.86853	0.934267	+	0.356573i	$0.116055\pi$
$968$	−10.9282	−0.351246
$969$	0	0
$970$	0	0
$971$	−55.7654	−1.78960	−0.894798	−	0.446471i	$-0.852681\pi$
$971$	−55.7654	−1.78960	−0.894798	+	0.446471i	$0.852681\pi$
$972$	0	0
$973$	−8.53590	−0.273648
$974$	−29.8564	−0.956661
$975$	0	0
$976$	−6.66025	−0.213190
$977$	5.46410	0.174812	0.0874060	−	0.996173i	$-0.472142\pi$
$977$	5.46410	0.174812	0.0874060	+	0.996173i	$0.472142\pi$
$978$	0	0
$979$	−4.51666	−0.144353
$980$	0	0
$981$	0	0
$982$	−10.9282	−0.348733
$983$	−47.6603	−1.52013	−0.760063	−	0.649849i	$-0.774832\pi$
$983$	−47.6603	−1.52013	−0.760063	+	0.649849i	$0.774832\pi$
$984$	0	0
$985$	0	0
$986$	7.26795	0.231459
$987$	0	0
$988$	−0.732051	−0.0232896
$989$	17.4115	0.553655
$990$	0	0
$991$	−5.67949	−0.180415	−0.0902075	−	0.995923i	$-0.528753\pi$
$991$	−5.67949	−0.180415	−0.0902075	+	0.995923i	$0.528753\pi$
$992$	4.46410	0.141735
$993$	0	0
$994$	4.92820	0.156313
$995$	0	0
$996$	0	0
$997$	24.9474	0.790093	0.395047	−	0.918661i	$-0.370728\pi$
$997$	24.9474	0.790093	0.395047	+	0.918661i	$0.370728\pi$
$998$	−31.3731	−0.993097
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.by.1.1		2
3.2	odd	2		2850.2.a.bd.1.1	✓	2
5.4	even	2		8550.2.a.bs.1.2		2
15.2	even	4		2850.2.d.x.799.1		4
15.8	even	4		2850.2.d.x.799.4		4
15.14	odd	2		2850.2.a.bi.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.bd.1.1	✓	2	3.2	odd	2
2850.2.a.bi.1.2	yes	2	15.14	odd	2
2850.2.d.x.799.1		4	15.2	even	4
2850.2.d.x.799.4		4	15.8	even	4
8550.2.a.bs.1.2		2	5.4	even	2
8550.2.a.by.1.1		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.73205 q^{7} +1.00000 q^{8} -0.267949 q^{11} +0.732051 q^{13} -2.73205 q^{14} +1.00000 q^{16} -4.19615 q^{17} -1.00000 q^{19} -0.267949 q^{22} +7.92820 q^{23} +0.732051 q^{26} -2.73205 q^{28} -1.73205 q^{29} +4.46410 q^{31} +1.00000 q^{32} -4.19615 q^{34} -2.00000 q^{37} -1.00000 q^{38} -10.9282 q^{41} +2.19615 q^{43} -0.267949 q^{44} +7.92820 q^{46} +3.46410 q^{47} +0.464102 q^{49} +0.732051 q^{52} -1.73205 q^{53} -2.73205 q^{56} -1.73205 q^{58} -2.19615 q^{59} -6.66025 q^{61} +4.46410 q^{62} +1.00000 q^{64} -3.73205 q^{67} -4.19615 q^{68} -1.80385 q^{71} +4.46410 q^{73} -2.00000 q^{74} -1.00000 q^{76} +0.732051 q^{77} -12.4641 q^{79} -10.9282 q^{82} -0.267949 q^{83} +2.19615 q^{86} -0.267949 q^{88} +16.8564 q^{89} -2.00000 q^{91} +7.92820 q^{92} +3.46410 q^{94} +9.12436 q^{97} +0.464102 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{26} - 2 q^{28} + 2 q^{31} + 2 q^{32} + 2 q^{34} - 4 q^{37} - 2 q^{38} - 8 q^{41} - 6 q^{43} - 4 q^{44} + 2 q^{46} - 6 q^{49} - 2 q^{52} - 2 q^{56} + 6 q^{59} + 4 q^{61} + 2 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{68} - 14 q^{71} + 2 q^{73} - 4 q^{74} - 2 q^{76} - 2 q^{77} - 18 q^{79} - 8 q^{82} - 4 q^{83} - 6 q^{86} - 4 q^{88} + 6 q^{89} - 4 q^{91} + 2 q^{92} - 6 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 4 * q^22 + 2 * q^23 - 2 * q^26 - 2 * q^28 + 2 * q^31 + 2 * q^32 + 2 * q^34 - 4 * q^37 - 2 * q^38 - 8 * q^41 - 6 * q^43 - 4 * q^44 + 2 * q^46 - 6 * q^49 - 2 * q^52 - 2 * q^56 + 6 * q^59 + 4 * q^61 + 2 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^68 - 14 * q^71 + 2 * q^73 - 4 * q^74 - 2 * q^76 - 2 * q^77 - 18 * q^79 - 8 * q^82 - 4 * q^83 - 6 * q^86 - 4 * q^88 + 6 * q^89 - 4 * q^91 + 2 * q^92 - 6 * q^97 - 6 * q^98

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