LMFDB - Embedded newform 8112.2.a.cm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{9} -6.29590 q^{11} +2.13706 q^{15} -2.89008 q^{17} +7.20775 q^{19} +0.0489173 q^{21} -2.71379 q^{23} -0.432960 q^{25} +1.00000 q^{27} +4.91185 q^{29} -9.00969 q^{31} -6.29590 q^{33} +0.104539 q^{35} -0.176292 q^{37} -8.59179 q^{41} -6.71379 q^{43} +2.13706 q^{45} -7.20775 q^{47} -6.99761 q^{49} -2.89008 q^{51} +9.34481 q^{53} -13.4547 q^{55} +7.20775 q^{57} +4.26875 q^{59} +7.10992 q^{61} +0.0489173 q^{63} -5.38404 q^{67} -2.71379 q^{69} +8.71379 q^{71} -14.9487 q^{73} -0.432960 q^{75} -0.307979 q^{77} -13.8291 q^{79} +1.00000 q^{81} +11.1347 q^{83} -6.17629 q^{85} +4.91185 q^{87} +3.92154 q^{89} -9.00969 q^{93} +15.4034 q^{95} -2.47889 q^{97} -6.29590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9} - 5 q^{11} + q^{15} - 8 q^{17} + 4 q^{19} - 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} - 5 q^{31} - 5 q^{33} + 4 q^{35} - 8 q^{37} + 2 q^{41} - 12 q^{43} + q^{45} - 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} + 4 q^{57} + 5 q^{59} + 22 q^{61} - 9 q^{63} - 6 q^{67} + 18 q^{71} - 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} - 13 q^{83} - 26 q^{85} + 11 q^{87} - 14 q^{89} - 5 q^{93} - 8 q^{95} - 23 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9 - 5 * q^11 + q^15 - 8 * q^17 + 4 * q^19 - 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 - 5 * q^31 - 5 * q^33 + 4 * q^35 - 8 * q^37 + 2 * q^41 - 12 * q^43 + q^45 - 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 + 4 * q^57 + 5 * q^59 + 22 * q^61 - 9 * q^63 - 6 * q^67 + 18 * q^71 - 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 - 13 * q^83 - 26 * q^85 + 11 * q^87 - 14 * q^89 - 5 * q^93 - 8 * q^95 - 23 * q^97 - 5 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.13706	0.955724	0.477862	−	0.878435i	$-0.341412\pi$
$5$	2.13706	0.955724	0.477862	+	0.878435i	$0.341412\pi$
$6$	0	0
$7$	0.0489173	0.0184890	0.00924451	−	0.999957i	$-0.497057\pi$
$7$	0.0489173	0.0184890	0.00924451	+	0.999957i	$0.497057\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.29590	−1.89828	−0.949142	−	0.314848i	$-0.898047\pi$
$11$	−6.29590	−1.89828	−0.949142	+	0.314848i	$0.898047\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.13706	0.551787
$16$	0	0
$17$	−2.89008	−0.700948	−0.350474	−	0.936572i	$-0.613980\pi$
$17$	−2.89008	−0.700948	−0.350474	+	0.936572i	$0.613980\pi$
$18$	0	0
$19$	7.20775	1.65357	0.826786	−	0.562517i	$-0.190167\pi$
$19$	7.20775	1.65357	0.826786	+	0.562517i	$0.190167\pi$
$20$	0	0
$21$	0.0489173	0.0106746
$22$	0	0
$23$	−2.71379	−0.565865	−0.282932	−	0.959140i	$-0.591307\pi$
$23$	−2.71379	−0.565865	−0.282932	+	0.959140i	$0.591307\pi$
$24$	0	0
$25$	−0.432960	−0.0865921
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.91185	0.912108	0.456054	−	0.889952i	$-0.349262\pi$
$29$	4.91185	0.912108	0.456054	+	0.889952i	$0.349262\pi$
$30$	0	0
$31$	−9.00969	−1.61819	−0.809094	−	0.587679i	$-0.800042\pi$
$31$	−9.00969	−1.61819	−0.809094	+	0.587679i	$0.800042\pi$
$32$	0	0
$33$	−6.29590	−1.09597
$34$	0	0
$35$	0.104539	0.0176704
$36$	0	0
$37$	−0.176292	−0.0289822	−0.0144911	−	0.999895i	$-0.504613\pi$
$37$	−0.176292	−0.0289822	−0.0144911	+	0.999895i	$0.504613\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.59179	−1.34181	−0.670906	−	0.741542i	$-0.734095\pi$
$41$	−8.59179	−1.34181	−0.670906	+	0.741542i	$0.734095\pi$
$42$	0	0
$43$	−6.71379	−1.02384	−0.511922	−	0.859032i	$-0.671066\pi$
$43$	−6.71379	−1.02384	−0.511922	+	0.859032i	$0.671066\pi$
$44$	0	0
$45$	2.13706	0.318575
$46$	0	0
$47$	−7.20775	−1.05136	−0.525679	−	0.850683i	$-0.676189\pi$
$47$	−7.20775	−1.05136	−0.525679	+	0.850683i	$0.676189\pi$
$48$	0	0
$49$	−6.99761	−0.999658
$50$	0	0
$51$	−2.89008	−0.404693
$52$	0	0
$53$	9.34481	1.28361	0.641804	−	0.766868i	$-0.278186\pi$
$53$	9.34481	1.28361	0.641804	+	0.766868i	$0.278186\pi$
$54$	0	0
$55$	−13.4547	−1.81424
$56$	0	0
$57$	7.20775	0.954690
$58$	0	0
$59$	4.26875	0.555744	0.277872	−	0.960618i	$-0.410371\pi$
$59$	4.26875	0.555744	0.277872	+	0.960618i	$0.410371\pi$
$60$	0	0
$61$	7.10992	0.910331	0.455166	−	0.890407i	$-0.349580\pi$
$61$	7.10992	0.910331	0.455166	+	0.890407i	$0.349580\pi$
$62$	0	0
$63$	0.0489173	0.00616301
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.38404	−0.657766	−0.328883	−	0.944371i	$-0.606672\pi$
$67$	−5.38404	−0.657766	−0.328883	+	0.944371i	$0.606672\pi$
$68$	0	0
$69$	−2.71379	−0.326702
$70$	0	0
$71$	8.71379	1.03414	0.517068	−	0.855944i	$-0.327023\pi$
$71$	8.71379	1.03414	0.517068	+	0.855944i	$0.327023\pi$
$72$	0	0
$73$	−14.9487	−1.74961	−0.874806	−	0.484474i	$-0.839011\pi$
$73$	−14.9487	−1.74961	−0.874806	+	0.484474i	$0.839011\pi$
$74$	0	0
$75$	−0.432960	−0.0499939
$76$	0	0
$77$	−0.307979	−0.0350974
$78$	0	0
$79$	−13.8291	−1.55589	−0.777947	−	0.628330i	$-0.783739\pi$
$79$	−13.8291	−1.55589	−0.777947	+	0.628330i	$0.783739\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.1347	1.22219	0.611094	−	0.791558i	$-0.290730\pi$
$83$	11.1347	1.22219	0.611094	+	0.791558i	$0.290730\pi$
$84$	0	0
$85$	−6.17629	−0.669913
$86$	0	0
$87$	4.91185	0.526606
$88$	0	0
$89$	3.92154	0.415683	0.207841	−	0.978163i	$-0.433356\pi$
$89$	3.92154	0.415683	0.207841	+	0.978163i	$0.433356\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.00969	−0.934261
$94$	0	0
$95$	15.4034	1.58036
$96$	0	0
$97$	−2.47889	−0.251694	−0.125847	−	0.992050i	$-0.540165\pi$
$97$	−2.47889	−0.251694	−0.125847	+	0.992050i	$0.540165\pi$
$98$	0	0
$99$	−6.29590	−0.632761
$100$	0	0
$101$	1.65279	0.164459	0.0822295	−	0.996613i	$-0.473796\pi$
$101$	1.65279	0.164459	0.0822295	+	0.996613i	$0.473796\pi$
$102$	0	0
$103$	8.23490	0.811409	0.405704	−	0.914004i	$-0.367026\pi$
$103$	8.23490	0.811409	0.405704	+	0.914004i	$0.367026\pi$
$104$	0	0
$105$	0.104539	0.0102020
$106$	0	0
$107$	8.36658	0.808828	0.404414	−	0.914576i	$-0.367475\pi$
$107$	8.36658	0.808828	0.404414	+	0.914576i	$0.367475\pi$
$108$	0	0
$109$	−17.4276	−1.66926	−0.834630	−	0.550811i	$-0.814318\pi$
$109$	−17.4276	−1.66926	−0.834630	+	0.550811i	$0.814318\pi$
$110$	0	0
$111$	−0.176292	−0.0167329
$112$	0	0
$113$	−13.9758	−1.31474	−0.657368	−	0.753570i	$-0.728330\pi$
$113$	−13.9758	−1.31474	−0.657368	+	0.753570i	$0.728330\pi$
$114$	0	0
$115$	−5.79954	−0.540810
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.141375	−0.0129598
$120$	0	0
$121$	28.6383	2.60348
$122$	0	0
$123$	−8.59179	−0.774696
$124$	0	0
$125$	−11.6106	−1.03848
$126$	0	0
$127$	7.52111	0.667390	0.333695	−	0.942681i	$-0.391704\pi$
$127$	7.52111	0.667390	0.333695	+	0.942681i	$0.391704\pi$
$128$	0	0
$129$	−6.71379	−0.591116
$130$	0	0
$131$	−5.12498	−0.447772	−0.223886	−	0.974615i	$-0.571874\pi$
$131$	−5.12498	−0.447772	−0.223886	+	0.974615i	$0.571874\pi$
$132$	0	0
$133$	0.352584	0.0305729
$134$	0	0
$135$	2.13706	0.183929
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	−8.68963	−0.737045	−0.368522	−	0.929619i	$-0.620136\pi$
$139$	−8.68963	−0.737045	−0.368522	+	0.929619i	$0.620136\pi$
$140$	0	0
$141$	−7.20775	−0.607002
$142$	0	0
$143$	0	0
$144$	0	0
$145$	10.4969	0.871724
$146$	0	0
$147$	−6.99761	−0.577153
$148$	0	0
$149$	−4.86831	−0.398828	−0.199414	−	0.979915i	$-0.563904\pi$
$149$	−4.86831	−0.398828	−0.199414	+	0.979915i	$0.563904\pi$
$150$	0	0
$151$	−14.7463	−1.20004	−0.600019	−	0.799986i	$-0.704840\pi$
$151$	−14.7463	−1.20004	−0.600019	+	0.799986i	$0.704840\pi$
$152$	0	0
$153$	−2.89008	−0.233649
$154$	0	0
$155$	−19.2543	−1.54654
$156$	0	0
$157$	−16.7138	−1.33391	−0.666953	−	0.745100i	$-0.732402\pi$
$157$	−16.7138	−1.33391	−0.666953	+	0.745100i	$0.732402\pi$
$158$	0	0
$159$	9.34481	0.741092
$160$	0	0
$161$	−0.132751	−0.0104623
$162$	0	0
$163$	−5.54958	−0.434677	−0.217338	−	0.976096i	$-0.569738\pi$
$163$	−5.54958	−0.434677	−0.217338	+	0.976096i	$0.569738\pi$
$164$	0	0
$165$	−13.4547	−1.04745
$166$	0	0
$167$	3.92154	0.303458	0.151729	−	0.988422i	$-0.451516\pi$
$167$	3.92154	0.303458	0.151729	+	0.988422i	$0.451516\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.20775	0.551190
$172$	0	0
$173$	−3.48427	−0.264904	−0.132452	−	0.991189i	$-0.542285\pi$
$173$	−3.48427	−0.264904	−0.132452	+	0.991189i	$0.542285\pi$
$174$	0	0
$175$	−0.0211793	−0.00160100
$176$	0	0
$177$	4.26875	0.320859
$178$	0	0
$179$	−3.58881	−0.268240	−0.134120	−	0.990965i	$-0.542821\pi$
$179$	−3.58881	−0.268240	−0.134120	+	0.990965i	$0.542821\pi$
$180$	0	0
$181$	5.50604	0.409261	0.204630	−	0.978839i	$-0.434401\pi$
$181$	5.50604	0.409261	0.204630	+	0.978839i	$0.434401\pi$
$182$	0	0
$183$	7.10992	0.525580
$184$	0	0
$185$	−0.376747	−0.0276990
$186$	0	0
$187$	18.1957	1.33060
$188$	0	0
$189$	0.0489173	0.00355821
$190$	0	0
$191$	5.65817	0.409411	0.204705	−	0.978824i	$-0.434376\pi$
$191$	5.65817	0.409411	0.204705	+	0.978824i	$0.434376\pi$
$192$	0	0
$193$	−13.4034	−0.964799	−0.482400	−	0.875951i	$-0.660235\pi$
$193$	−13.4034	−0.964799	−0.482400	+	0.875951i	$0.660235\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.9812	−1.42360	−0.711801	−	0.702381i	$-0.752120\pi$
$197$	−19.9812	−1.42360	−0.711801	+	0.702381i	$0.752120\pi$
$198$	0	0
$199$	6.24160	0.442455	0.221228	−	0.975222i	$-0.428994\pi$
$199$	6.24160	0.442455	0.221228	+	0.975222i	$0.428994\pi$
$200$	0	0
$201$	−5.38404	−0.379761
$202$	0	0
$203$	0.240275	0.0168640
$204$	0	0
$205$	−18.3612	−1.28240
$206$	0	0
$207$	−2.71379	−0.188622
$208$	0	0
$209$	−45.3793	−3.13895
$210$	0	0
$211$	5.08575	0.350118	0.175059	−	0.984558i	$-0.443988\pi$
$211$	5.08575	0.350118	0.175059	+	0.984558i	$0.443988\pi$
$212$	0	0
$213$	8.71379	0.597059
$214$	0	0
$215$	−14.3478	−0.978512
$216$	0	0
$217$	−0.440730	−0.0299187
$218$	0	0
$219$	−14.9487	−1.01014
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−20.5483	−1.37601	−0.688006	−	0.725705i	$-0.741514\pi$
$223$	−20.5483	−1.37601	−0.688006	+	0.725705i	$0.741514\pi$
$224$	0	0
$225$	−0.432960	−0.0288640
$226$	0	0
$227$	4.41119	0.292781	0.146390	−	0.989227i	$-0.453234\pi$
$227$	4.41119	0.292781	0.146390	+	0.989227i	$0.453234\pi$
$228$	0	0
$229$	−0.230586	−0.0152376	−0.00761878	−	0.999971i	$-0.502425\pi$
$229$	−0.230586	−0.0152376	−0.00761878	+	0.999971i	$0.502425\pi$
$230$	0	0
$231$	−0.307979	−0.0202635
$232$	0	0
$233$	−7.82371	−0.512548	−0.256274	−	0.966604i	$-0.582495\pi$
$233$	−7.82371	−0.512548	−0.256274	+	0.966604i	$0.582495\pi$
$234$	0	0
$235$	−15.4034	−1.00481
$236$	0	0
$237$	−13.8291	−0.898296
$238$	0	0
$239$	11.1535	0.721457	0.360729	−	0.932671i	$-0.382528\pi$
$239$	11.1535	0.721457	0.360729	+	0.932671i	$0.382528\pi$
$240$	0	0
$241$	3.54527	0.228371	0.114185	−	0.993459i	$-0.463574\pi$
$241$	3.54527	0.228371	0.114185	+	0.993459i	$0.463574\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.9543	−0.955397
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.1347	0.705631
$250$	0	0
$251$	4.17092	0.263266	0.131633	−	0.991299i	$-0.457978\pi$
$251$	4.17092	0.263266	0.131633	+	0.991299i	$0.457978\pi$
$252$	0	0
$253$	17.0858	1.07417
$254$	0	0
$255$	−6.17629	−0.386774
$256$	0	0
$257$	−10.8901	−0.679305	−0.339652	−	0.940551i	$-0.610309\pi$
$257$	−10.8901	−0.679305	−0.339652	+	0.940551i	$0.610309\pi$
$258$	0	0
$259$	−0.00862374	−0.000535853 0
$260$	0	0
$261$	4.91185	0.304036
$262$	0	0
$263$	31.2271	1.92555	0.962774	−	0.270309i	$-0.0871259\pi$
$263$	31.2271	1.92555	0.962774	+	0.270309i	$0.0871259\pi$
$264$	0	0
$265$	19.9705	1.22678
$266$	0	0
$267$	3.92154	0.239995
$268$	0	0
$269$	15.9172	0.970491	0.485245	−	0.874378i	$-0.338730\pi$
$269$	15.9172	0.970491	0.485245	+	0.874378i	$0.338730\pi$
$270$	0	0
$271$	3.52111	0.213892	0.106946	−	0.994265i	$-0.465893\pi$
$271$	3.52111	0.213892	0.106946	+	0.994265i	$0.465893\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.72587	0.164376
$276$	0	0
$277$	−8.58104	−0.515585	−0.257792	−	0.966200i	$-0.582995\pi$
$277$	−8.58104	−0.515585	−0.257792	+	0.966200i	$0.582995\pi$
$278$	0	0
$279$	−9.00969	−0.539396
$280$	0	0
$281$	8.07846	0.481920	0.240960	−	0.970535i	$-0.422538\pi$
$281$	8.07846	0.481920	0.240960	+	0.970535i	$0.422538\pi$
$282$	0	0
$283$	17.4034	1.03453	0.517263	−	0.855827i	$-0.326951\pi$
$283$	17.4034	1.03453	0.517263	+	0.855827i	$0.326951\pi$
$284$	0	0
$285$	15.4034	0.912420
$286$	0	0
$287$	−0.420288	−0.0248088
$288$	0	0
$289$	−8.64742	−0.508672
$290$	0	0
$291$	−2.47889	−0.145315
$292$	0	0
$293$	19.3709	1.13166	0.565830	−	0.824522i	$-0.308556\pi$
$293$	19.3709	1.13166	0.565830	+	0.824522i	$0.308556\pi$
$294$	0	0
$295$	9.12259	0.531138
$296$	0	0
$297$	−6.29590	−0.365325
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.328421	−0.0189299
$302$	0	0
$303$	1.65279	0.0949505
$304$	0	0
$305$	15.1943	0.870025
$306$	0	0
$307$	−12.4263	−0.709204	−0.354602	−	0.935017i	$-0.615384\pi$
$307$	−12.4263	−0.709204	−0.354602	+	0.935017i	$0.615384\pi$
$308$	0	0
$309$	8.23490	0.468467
$310$	0	0
$311$	−2.71379	−0.153885	−0.0769425	−	0.997036i	$-0.524516\pi$
$311$	−2.71379	−0.153885	−0.0769425	+	0.997036i	$0.524516\pi$
$312$	0	0
$313$	15.3884	0.869801	0.434901	−	0.900478i	$-0.356783\pi$
$313$	15.3884	0.869801	0.434901	+	0.900478i	$0.356783\pi$
$314$	0	0
$315$	0.104539	0.00589013
$316$	0	0
$317$	−25.6528	−1.44080	−0.720402	−	0.693557i	$-0.756043\pi$
$317$	−25.6528	−1.44080	−0.720402	+	0.693557i	$0.756043\pi$
$318$	0	0
$319$	−30.9245	−1.73144
$320$	0	0
$321$	8.36658	0.466977
$322$	0	0
$323$	−20.8310	−1.15907
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.4276	−0.963748
$328$	0	0
$329$	−0.352584	−0.0194386
$330$	0	0
$331$	−3.82371	−0.210170	−0.105085	−	0.994463i	$-0.533511\pi$
$331$	−3.82371	−0.210170	−0.105085	+	0.994463i	$0.533511\pi$
$332$	0	0
$333$	−0.176292	−0.00966074
$334$	0	0
$335$	−11.5060	−0.628642
$336$	0	0
$337$	20.1304	1.09657	0.548285	−	0.836291i	$-0.315281\pi$
$337$	20.1304	1.09657	0.548285	+	0.836291i	$0.315281\pi$
$338$	0	0
$339$	−13.9758	−0.759063
$340$	0	0
$341$	56.7241	3.07178
$342$	0	0
$343$	−0.684726	−0.0369717
$344$	0	0
$345$	−5.79954	−0.312237
$346$	0	0
$347$	−2.93900	−0.157774	−0.0788869	−	0.996884i	$-0.525137\pi$
$347$	−2.93900	−0.157774	−0.0788869	+	0.996884i	$0.525137\pi$
$348$	0	0
$349$	−7.37329	−0.394683	−0.197342	−	0.980335i	$-0.563231\pi$
$349$	−7.37329	−0.394683	−0.197342	+	0.980335i	$0.563231\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.01075	0.107022	0.0535108	−	0.998567i	$-0.482959\pi$
$353$	2.01075	0.107022	0.0535108	+	0.998567i	$0.482959\pi$
$354$	0	0
$355$	18.6219	0.988349
$356$	0	0
$357$	−0.141375	−0.00748237
$358$	0	0
$359$	31.4577	1.66027	0.830137	−	0.557559i	$-0.188262\pi$
$359$	31.4577	1.66027	0.830137	+	0.557559i	$0.188262\pi$
$360$	0	0
$361$	32.9517	1.73430
$362$	0	0
$363$	28.6383	1.50312
$364$	0	0
$365$	−31.9463	−1.67215
$366$	0	0
$367$	4.49934	0.234863	0.117432	−	0.993081i	$-0.462534\pi$
$367$	4.49934	0.234863	0.117432	+	0.993081i	$0.462534\pi$
$368$	0	0
$369$	−8.59179	−0.447271
$370$	0	0
$371$	0.457123	0.0237327
$372$	0	0
$373$	37.8297	1.95875	0.979373	−	0.202060i	$-0.0647635\pi$
$373$	37.8297	1.95875	0.979373	+	0.202060i	$0.0647635\pi$
$374$	0	0
$375$	−11.6106	−0.599568
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8.43967	−0.433516	−0.216758	−	0.976225i	$-0.569548\pi$
$379$	−8.43967	−0.433516	−0.216758	+	0.976225i	$0.569548\pi$
$380$	0	0
$381$	7.52111	0.385318
$382$	0	0
$383$	1.28621	0.0657222	0.0328611	−	0.999460i	$-0.489538\pi$
$383$	1.28621	0.0657222	0.0328611	+	0.999460i	$0.489538\pi$
$384$	0	0
$385$	−0.658170	−0.0335434
$386$	0	0
$387$	−6.71379	−0.341281
$388$	0	0
$389$	23.3924	1.18604	0.593021	−	0.805187i	$-0.297935\pi$
$389$	23.3924	1.18604	0.593021	+	0.805187i	$0.297935\pi$
$390$	0	0
$391$	7.84309	0.396642
$392$	0	0
$393$	−5.12498	−0.258521
$394$	0	0
$395$	−29.5536	−1.48700
$396$	0	0
$397$	−37.0858	−1.86128	−0.930640	−	0.365935i	$-0.880749\pi$
$397$	−37.0858	−1.86128	−0.930640	+	0.365935i	$0.880749\pi$
$398$	0	0
$399$	0.352584	0.0176513
$400$	0	0
$401$	−4.86592	−0.242992	−0.121496	−	0.992592i	$-0.538769\pi$
$401$	−4.86592	−0.242992	−0.121496	+	0.992592i	$0.538769\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.13706	0.106192
$406$	0	0
$407$	1.10992	0.0550165
$408$	0	0
$409$	0.445042	0.0220059	0.0110030	−	0.999939i	$-0.496498\pi$
$409$	0.445042	0.0220059	0.0110030	+	0.999939i	$0.496498\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	0.208816	0.0102752
$414$	0	0
$415$	23.7955	1.16807
$416$	0	0
$417$	−8.68963	−0.425533
$418$	0	0
$419$	−17.9869	−0.878715	−0.439358	−	0.898312i	$-0.644794\pi$
$419$	−17.9869	−0.878715	−0.439358	+	0.898312i	$0.644794\pi$
$420$	0	0
$421$	21.2814	1.03719	0.518597	−	0.855019i	$-0.326455\pi$
$421$	21.2814	1.03719	0.518597	+	0.855019i	$0.326455\pi$
$422$	0	0
$423$	−7.20775	−0.350453
$424$	0	0
$425$	1.25129	0.0606966
$426$	0	0
$427$	0.347798	0.0168311
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.7138	−1.19042	−0.595211	−	0.803570i	$-0.702931\pi$
$431$	−24.7138	−1.19042	−0.595211	+	0.803570i	$0.702931\pi$
$432$	0	0
$433$	32.2174	1.54827	0.774136	−	0.633020i	$-0.218185\pi$
$433$	32.2174	1.54827	0.774136	+	0.633020i	$0.218185\pi$
$434$	0	0
$435$	10.4969	0.503290
$436$	0	0
$437$	−19.5603	−0.935698
$438$	0	0
$439$	−32.0877	−1.53146	−0.765731	−	0.643162i	$-0.777622\pi$
$439$	−32.0877	−1.53146	−0.765731	+	0.643162i	$0.777622\pi$
$440$	0	0
$441$	−6.99761	−0.333219
$442$	0	0
$443$	−20.5109	−0.974504	−0.487252	−	0.873261i	$-0.662001\pi$
$443$	−20.5109	−0.974504	−0.487252	+	0.873261i	$0.662001\pi$
$444$	0	0
$445$	8.38059	0.397278
$446$	0	0
$447$	−4.86831	−0.230263
$448$	0	0
$449$	−15.3163	−0.722823	−0.361411	−	0.932406i	$-0.617705\pi$
$449$	−15.3163	−0.722823	−0.361411	+	0.932406i	$0.617705\pi$
$450$	0	0
$451$	54.0930	2.54714
$452$	0	0
$453$	−14.7463	−0.692842
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.1860	−0.897482	−0.448741	−	0.893662i	$-0.648127\pi$
$457$	−19.1860	−0.897482	−0.448741	+	0.893662i	$0.648127\pi$
$458$	0	0
$459$	−2.89008	−0.134898
$460$	0	0
$461$	8.31229	0.387142	0.193571	−	0.981086i	$-0.437993\pi$
$461$	8.31229	0.387142	0.193571	+	0.981086i	$0.437993\pi$
$462$	0	0
$463$	6.32842	0.294107	0.147053	−	0.989129i	$-0.453021\pi$
$463$	6.32842	0.294107	0.147053	+	0.989129i	$0.453021\pi$
$464$	0	0
$465$	−19.2543	−0.892896
$466$	0	0
$467$	31.1879	1.44320	0.721602	−	0.692308i	$-0.243406\pi$
$467$	31.1879	1.44320	0.721602	+	0.692308i	$0.243406\pi$
$468$	0	0
$469$	−0.263373	−0.0121614
$470$	0	0
$471$	−16.7138	−0.770131
$472$	0	0
$473$	42.2693	1.94355
$474$	0	0
$475$	−3.12067	−0.143186
$476$	0	0
$477$	9.34481	0.427870
$478$	0	0
$479$	17.9323	0.819348	0.409674	−	0.912232i	$-0.365643\pi$
$479$	17.9323	0.819348	0.409674	+	0.912232i	$0.365643\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.132751	−0.00604040
$484$	0	0
$485$	−5.29755	−0.240549
$486$	0	0
$487$	−30.3279	−1.37429	−0.687145	−	0.726520i	$-0.741136\pi$
$487$	−30.3279	−1.37429	−0.687145	+	0.726520i	$0.741136\pi$
$488$	0	0
$489$	−5.54958	−0.250961
$490$	0	0
$491$	−30.0954	−1.35819	−0.679094	−	0.734051i	$-0.737627\pi$
$491$	−30.0954	−1.35819	−0.679094	+	0.734051i	$0.737627\pi$
$492$	0	0
$493$	−14.1957	−0.639341
$494$	0	0
$495$	−13.4547	−0.604745
$496$	0	0
$497$	0.426256	0.0191202
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	3.92154	0.175202
$502$	0	0
$503$	−33.7512	−1.50489	−0.752446	−	0.658654i	$-0.771126\pi$
$503$	−33.7512	−1.50489	−0.752446	+	0.658654i	$0.771126\pi$
$504$	0	0
$505$	3.53212	0.157177
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.7439	−0.653513	−0.326756	−	0.945109i	$-0.605956\pi$
$509$	−14.7439	−0.653513	−0.326756	+	0.945109i	$0.605956\pi$
$510$	0	0
$511$	−0.731250	−0.0323486
$512$	0	0
$513$	7.20775	0.318230
$514$	0	0
$515$	17.5985	0.775483
$516$	0	0
$517$	45.3793	1.99578
$518$	0	0
$519$	−3.48427	−0.152943
$520$	0	0
$521$	22.1086	0.968595	0.484297	−	0.874903i	$-0.339075\pi$
$521$	22.1086	0.968595	0.484297	+	0.874903i	$0.339075\pi$
$522$	0	0
$523$	17.3599	0.759095	0.379547	−	0.925172i	$-0.376080\pi$
$523$	17.3599	0.759095	0.379547	+	0.925172i	$0.376080\pi$
$524$	0	0
$525$	−0.0211793	−0.000924339 0
$526$	0	0
$527$	26.0388	1.13427
$528$	0	0
$529$	−15.6353	−0.679797
$530$	0	0
$531$	4.26875	0.185248
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.8799	0.773016
$536$	0	0
$537$	−3.58881	−0.154869
$538$	0	0
$539$	44.0562	1.89764
$540$	0	0
$541$	−8.83579	−0.379880	−0.189940	−	0.981796i	$-0.560829\pi$
$541$	−8.83579	−0.379880	−0.189940	+	0.981796i	$0.560829\pi$
$542$	0	0
$543$	5.50604	0.236287
$544$	0	0
$545$	−37.2438	−1.59535
$546$	0	0
$547$	8.10859	0.346698	0.173349	−	0.984860i	$-0.444541\pi$
$547$	8.10859	0.346698	0.173349	+	0.984860i	$0.444541\pi$
$548$	0	0
$549$	7.10992	0.303444
$550$	0	0
$551$	35.4034	1.50824
$552$	0	0
$553$	−0.676482	−0.0287669
$554$	0	0
$555$	−0.376747	−0.0159920
$556$	0	0
$557$	−9.56033	−0.405084	−0.202542	−	0.979274i	$-0.564920\pi$
$557$	−9.56033	−0.405084	−0.202542	+	0.979274i	$0.564920\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	18.1957	0.768222
$562$	0	0
$563$	45.1637	1.90342	0.951712	−	0.306991i	$-0.0993223\pi$
$563$	45.1637	1.90342	0.951712	+	0.306991i	$0.0993223\pi$
$564$	0	0
$565$	−29.8672	−1.25652
$566$	0	0
$567$	0.0489173	0.00205434
$568$	0	0
$569$	−3.20775	−0.134476	−0.0672380	−	0.997737i	$-0.521419\pi$
$569$	−3.20775	−0.134476	−0.0672380	+	0.997737i	$0.521419\pi$
$570$	0	0
$571$	−0.0241632	−0.00101120	−0.000505599	−	1.00000i	$-0.500161\pi$
$571$	−0.0241632	−0.00101120	−0.000505599		1.00000i	$0.500161\pi$
$572$	0	0
$573$	5.65817	0.236373
$574$	0	0
$575$	1.17496	0.0489994
$576$	0	0
$577$	46.0200	1.91584	0.957918	−	0.287041i	$-0.0926717\pi$
$577$	46.0200	1.91584	0.957918	+	0.287041i	$0.0926717\pi$
$578$	0	0
$579$	−13.4034	−0.557027
$580$	0	0
$581$	0.544678	0.0225971
$582$	0	0
$583$	−58.8340	−2.43665
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.9385	−1.07060	−0.535299	−	0.844663i	$-0.679801\pi$
$587$	−25.9385	−1.07060	−0.535299	+	0.844663i	$0.679801\pi$
$588$	0	0
$589$	−64.9396	−2.67579
$590$	0	0
$591$	−19.9812	−0.821917
$592$	0	0
$593$	8.30691	0.341124	0.170562	−	0.985347i	$-0.445442\pi$
$593$	8.30691	0.341124	0.170562	+	0.985347i	$0.445442\pi$
$594$	0	0
$595$	−0.302128	−0.0123860
$596$	0	0
$597$	6.24160	0.255452
$598$	0	0
$599$	−37.1702	−1.51873	−0.759366	−	0.650664i	$-0.774491\pi$
$599$	−37.1702	−1.51873	−0.759366	+	0.650664i	$0.774491\pi$
$600$	0	0
$601$	−12.5700	−0.512742	−0.256371	−	0.966578i	$-0.582527\pi$
$601$	−12.5700	−0.512742	−0.256371	+	0.966578i	$0.582527\pi$
$602$	0	0
$603$	−5.38404	−0.219255
$604$	0	0
$605$	61.2019	2.48821
$606$	0	0
$607$	−8.04892	−0.326695	−0.163348	−	0.986569i	$-0.552229\pi$
$607$	−8.04892	−0.326695	−0.163348	+	0.986569i	$0.552229\pi$
$608$	0	0
$609$	0.240275	0.00973643
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−30.4590	−1.23023	−0.615115	−	0.788438i	$-0.710890\pi$
$613$	−30.4590	−1.23023	−0.615115	+	0.788438i	$0.710890\pi$
$614$	0	0
$615$	−18.3612	−0.740395
$616$	0	0
$617$	−4.73317	−0.190550	−0.0952751	−	0.995451i	$-0.530373\pi$
$617$	−4.73317	−0.190550	−0.0952751	+	0.995451i	$0.530373\pi$
$618$	0	0
$619$	−24.9095	−1.00120	−0.500598	−	0.865680i	$-0.666886\pi$
$619$	−24.9095	−1.00120	−0.500598	+	0.865680i	$0.666886\pi$
$620$	0	0
$621$	−2.71379	−0.108901
$622$	0	0
$623$	0.191831	0.00768556
$624$	0	0
$625$	−22.6477	−0.905910
$626$	0	0
$627$	−45.3793	−1.81227
$628$	0	0
$629$	0.509499	0.0203150
$630$	0	0
$631$	24.4295	0.972523	0.486262	−	0.873813i	$-0.338360\pi$
$631$	24.4295	0.972523	0.486262	+	0.873813i	$0.338360\pi$
$632$	0	0
$633$	5.08575	0.202141
$634$	0	0
$635$	16.0731	0.637841
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.71379	0.344712
$640$	0	0
$641$	48.7982	1.92741	0.963707	−	0.266963i	$-0.0860201\pi$
$641$	48.7982	1.92741	0.963707	+	0.266963i	$0.0860201\pi$
$642$	0	0
$643$	−28.2693	−1.11483	−0.557417	−	0.830233i	$-0.688208\pi$
$643$	−28.2693	−1.11483	−0.557417	+	0.830233i	$0.688208\pi$
$644$	0	0
$645$	−14.3478	−0.564944
$646$	0	0
$647$	−34.8961	−1.37191	−0.685953	−	0.727646i	$-0.740614\pi$
$647$	−34.8961	−1.37191	−0.685953	+	0.727646i	$0.740614\pi$
$648$	0	0
$649$	−26.8756	−1.05496
$650$	0	0
$651$	−0.440730	−0.0172736
$652$	0	0
$653$	−11.7157	−0.458471	−0.229236	−	0.973371i	$-0.573623\pi$
$653$	−11.7157	−0.458471	−0.229236	+	0.973371i	$0.573623\pi$
$654$	0	0
$655$	−10.9524	−0.427946
$656$	0	0
$657$	−14.9487	−0.583204
$658$	0	0
$659$	7.13467	0.277927	0.138964	−	0.990297i	$-0.455623\pi$
$659$	7.13467	0.277927	0.138964	+	0.990297i	$0.455623\pi$
$660$	0	0
$661$	8.52888	0.331735	0.165867	−	0.986148i	$-0.446958\pi$
$661$	8.52888	0.331735	0.165867	+	0.986148i	$0.446958\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.753494	0.0292193
$666$	0	0
$667$	−13.3297	−0.516130
$668$	0	0
$669$	−20.5483	−0.794441
$670$	0	0
$671$	−44.7633	−1.72807
$672$	0	0
$673$	−31.5937	−1.21785	−0.608924	−	0.793229i	$-0.708399\pi$
$673$	−31.5937	−1.21785	−0.608924	+	0.793229i	$0.708399\pi$
$674$	0	0
$675$	−0.432960	−0.0166646
$676$	0	0
$677$	−17.3002	−0.664901	−0.332451	−	0.943121i	$-0.607875\pi$
$677$	−17.3002	−0.664901	−0.332451	+	0.943121i	$0.607875\pi$
$678$	0	0
$679$	−0.121261	−0.00465357
$680$	0	0
$681$	4.41119	0.169037
$682$	0	0
$683$	30.3957	1.16306	0.581529	−	0.813526i	$-0.302455\pi$
$683$	30.3957	1.16306	0.581529	+	0.813526i	$0.302455\pi$
$684$	0	0
$685$	−8.54825	−0.326612
$686$	0	0
$687$	−0.230586	−0.00879741
$688$	0	0
$689$	0	0
$690$	0	0
$691$	35.7318	1.35930	0.679652	−	0.733535i	$-0.262131\pi$
$691$	35.7318	1.35930	0.679652	+	0.733535i	$0.262131\pi$
$692$	0	0
$693$	−0.307979	−0.0116991
$694$	0	0
$695$	−18.5703	−0.704411
$696$	0	0
$697$	24.8310	0.940541
$698$	0	0
$699$	−7.82371	−0.295920
$700$	0	0
$701$	20.1328	0.760404	0.380202	−	0.924904i	$-0.375855\pi$
$701$	20.1328	0.760404	0.380202	+	0.924904i	$0.375855\pi$
$702$	0	0
$703$	−1.27067	−0.0479242
$704$	0	0
$705$	−15.4034	−0.580126
$706$	0	0
$707$	0.0808502	0.00304069
$708$	0	0
$709$	−16.6160	−0.624025	−0.312013	−	0.950078i	$-0.601003\pi$
$709$	−16.6160	−0.624025	−0.312013	+	0.950078i	$0.601003\pi$
$710$	0	0
$711$	−13.8291	−0.518631
$712$	0	0
$713$	24.4504	0.915675
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.1535	0.416533
$718$	0	0
$719$	−2.04833	−0.0763897	−0.0381948	−	0.999270i	$-0.512161\pi$
$719$	−2.04833	−0.0763897	−0.0381948	+	0.999270i	$0.512161\pi$
$720$	0	0
$721$	0.402829	0.0150021
$722$	0	0
$723$	3.54527	0.131850
$724$	0	0
$725$	−2.12664	−0.0789813
$726$	0	0
$727$	−29.9377	−1.11033	−0.555163	−	0.831741i	$-0.687344\pi$
$727$	−29.9377	−1.11033	−0.555163	+	0.831741i	$0.687344\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.4034	0.717661
$732$	0	0
$733$	1.46250	0.0540187	0.0270093	−	0.999635i	$-0.491402\pi$
$733$	1.46250	0.0540187	0.0270093	+	0.999635i	$0.491402\pi$
$734$	0	0
$735$	−14.9543	−0.551599
$736$	0	0
$737$	33.8974	1.24863
$738$	0	0
$739$	40.3866	1.48564	0.742822	−	0.669489i	$-0.233487\pi$
$739$	40.3866	1.48564	0.742822	+	0.669489i	$0.233487\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.77538	−0.358624	−0.179312	−	0.983792i	$-0.557387\pi$
$743$	−9.77538	−0.358624	−0.179312	+	0.983792i	$0.557387\pi$
$744$	0	0
$745$	−10.4039	−0.381169
$746$	0	0
$747$	11.1347	0.407396
$748$	0	0
$749$	0.409271	0.0149544
$750$	0	0
$751$	−34.5459	−1.26060	−0.630298	−	0.776353i	$-0.717067\pi$
$751$	−34.5459	−1.26060	−0.630298	+	0.776353i	$0.717067\pi$
$752$	0	0
$753$	4.17092	0.151997
$754$	0	0
$755$	−31.5138	−1.14690
$756$	0	0
$757$	−5.14483	−0.186992	−0.0934961	−	0.995620i	$-0.529804\pi$
$757$	−5.14483	−0.186992	−0.0934961	+	0.995620i	$0.529804\pi$
$758$	0	0
$759$	17.0858	0.620174
$760$	0	0
$761$	19.5120	0.707310	0.353655	−	0.935376i	$-0.384939\pi$
$761$	19.5120	0.707310	0.353655	+	0.935376i	$0.384939\pi$
$762$	0	0
$763$	−0.852511	−0.0308630
$764$	0	0
$765$	−6.17629	−0.223304
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.75600	−0.207567	−0.103783	−	0.994600i	$-0.533095\pi$
$769$	−5.75600	−0.207567	−0.103783	+	0.994600i	$0.533095\pi$
$770$	0	0
$771$	−10.8901	−0.392197
$772$	0	0
$773$	4.12737	0.148451	0.0742257	−	0.997241i	$-0.476351\pi$
$773$	4.12737	0.148451	0.0742257	+	0.997241i	$0.476351\pi$
$774$	0	0
$775$	3.90084	0.140122
$776$	0	0
$777$	−0.00862374	−0.000309375 0
$778$	0	0
$779$	−61.9275	−2.21878
$780$	0	0
$781$	−54.8611	−1.96309
$782$	0	0
$783$	4.91185	0.175535
$784$	0	0
$785$	−35.7184	−1.27485
$786$	0	0
$787$	−16.2392	−0.578865	−0.289433	−	0.957198i	$-0.593467\pi$
$787$	−16.2392	−0.578865	−0.289433	+	0.957198i	$0.593467\pi$
$788$	0	0
$789$	31.2271	1.11172
$790$	0	0
$791$	−0.683661	−0.0243082
$792$	0	0
$793$	0	0
$794$	0	0
$795$	19.9705	0.708279
$796$	0	0
$797$	16.1148	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$797$	16.1148	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$798$	0	0
$799$	20.8310	0.736948
$800$	0	0
$801$	3.92154	0.138561
$802$	0	0
$803$	94.1154	3.32126
$804$	0	0
$805$	−0.283698	−0.00999905
$806$	0	0
$807$	15.9172	0.560313
$808$	0	0
$809$	40.1521	1.41167	0.705837	−	0.708374i	$-0.250571\pi$
$809$	40.1521	1.41167	0.705837	+	0.708374i	$0.250571\pi$
$810$	0	0
$811$	39.8646	1.39984	0.699918	−	0.714224i	$-0.253220\pi$
$811$	39.8646	1.39984	0.699918	+	0.714224i	$0.253220\pi$
$812$	0	0
$813$	3.52111	0.123491
$814$	0	0
$815$	−11.8598	−0.415431
$816$	0	0
$817$	−48.3913	−1.69300
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.25428	0.218276	0.109138	−	0.994027i	$-0.465191\pi$
$821$	6.25428	0.218276	0.109138	+	0.994027i	$0.465191\pi$
$822$	0	0
$823$	4.74333	0.165342	0.0826711	−	0.996577i	$-0.473655\pi$
$823$	4.74333	0.165342	0.0826711	+	0.996577i	$0.473655\pi$
$824$	0	0
$825$	2.72587	0.0949027
$826$	0	0
$827$	−0.716185	−0.0249042	−0.0124521	−	0.999922i	$-0.503964\pi$
$827$	−0.716185	−0.0249042	−0.0124521	+	0.999922i	$0.503964\pi$
$828$	0	0
$829$	−36.0060	−1.25054	−0.625269	−	0.780409i	$-0.715011\pi$
$829$	−36.0060	−1.25054	−0.625269	+	0.780409i	$0.715011\pi$
$830$	0	0
$831$	−8.58104	−0.297673
$832$	0	0
$833$	20.2237	0.700709
$834$	0	0
$835$	8.38059	0.290022
$836$	0	0
$837$	−9.00969	−0.311420
$838$	0	0
$839$	−21.5555	−0.744180	−0.372090	−	0.928197i	$-0.621359\pi$
$839$	−21.5555	−0.744180	−0.372090	+	0.928197i	$0.621359\pi$
$840$	0	0
$841$	−4.87369	−0.168058
$842$	0	0
$843$	8.07846	0.278237
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.40091	0.0481358
$848$	0	0
$849$	17.4034	0.597284
$850$	0	0
$851$	0.478420	0.0164000
$852$	0	0
$853$	−37.8237	−1.29506	−0.647530	−	0.762040i	$-0.724198\pi$
$853$	−37.8237	−1.29506	−0.647530	+	0.762040i	$0.724198\pi$
$854$	0	0
$855$	15.4034	0.526786
$856$	0	0
$857$	−6.58317	−0.224877	−0.112438	−	0.993659i	$-0.535866\pi$
$857$	−6.58317	−0.224877	−0.112438	+	0.993659i	$0.535866\pi$
$858$	0	0
$859$	20.6246	0.703702	0.351851	−	0.936056i	$-0.385552\pi$
$859$	20.6246	0.703702	0.351851	+	0.936056i	$0.385552\pi$
$860$	0	0
$861$	−0.420288	−0.0143234
$862$	0	0
$863$	−15.9081	−0.541519	−0.270760	−	0.962647i	$-0.587275\pi$
$863$	−15.9081	−0.541519	−0.270760	+	0.962647i	$0.587275\pi$
$864$	0	0
$865$	−7.44611	−0.253175
$866$	0	0
$867$	−8.64742	−0.293682
$868$	0	0
$869$	87.0665	2.95353
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.47889	−0.0838978
$874$	0	0
$875$	−0.567959	−0.0192005
$876$	0	0
$877$	−15.6039	−0.526905	−0.263453	−	0.964672i	$-0.584861\pi$
$877$	−15.6039	−0.526905	−0.263453	+	0.964672i	$0.584861\pi$
$878$	0	0
$879$	19.3709	0.653364
$880$	0	0
$881$	−14.2547	−0.480255	−0.240127	−	0.970741i	$-0.577189\pi$
$881$	−14.2547	−0.480255	−0.240127	+	0.970741i	$0.577189\pi$
$882$	0	0
$883$	−1.65817	−0.0558019	−0.0279009	−	0.999611i	$-0.508882\pi$
$883$	−1.65817	−0.0558019	−0.0279009	+	0.999611i	$0.508882\pi$
$884$	0	0
$885$	9.12259	0.306652
$886$	0	0
$887$	19.2030	0.644772	0.322386	−	0.946608i	$-0.395515\pi$
$887$	19.2030	0.644772	0.322386	+	0.946608i	$0.395515\pi$
$888$	0	0
$889$	0.367913	0.0123394
$890$	0	0
$891$	−6.29590	−0.210920
$892$	0	0
$893$	−51.9517	−1.73850
$894$	0	0
$895$	−7.66951	−0.256364
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−44.2543	−1.47596
$900$	0	0
$901$	−27.0073	−0.899743
$902$	0	0
$903$	−0.328421	−0.0109292
$904$	0	0
$905$	11.7668	0.391140
$906$	0	0
$907$	32.3672	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$907$	32.3672	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$908$	0	0
$909$	1.65279	0.0548197
$910$	0	0
$911$	33.9624	1.12523	0.562613	−	0.826721i	$-0.309796\pi$
$911$	33.9624	1.12523	0.562613	+	0.826721i	$0.309796\pi$
$912$	0	0
$913$	−70.1027	−2.32006
$914$	0	0
$915$	15.1943	0.502309
$916$	0	0
$917$	−0.250700	−0.00827886
$918$	0	0
$919$	20.0780	0.662312	0.331156	−	0.943576i	$-0.392561\pi$
$919$	20.0780	0.662312	0.331156	+	0.943576i	$0.392561\pi$
$920$	0	0
$921$	−12.4263	−0.409459
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.0763274	0.00250963
$926$	0	0
$927$	8.23490	0.270470
$928$	0	0
$929$	56.1280	1.84150	0.920749	−	0.390154i	$-0.127578\pi$
$929$	56.1280	1.84150	0.920749	+	0.390154i	$0.127578\pi$
$930$	0	0
$931$	−50.4370	−1.65301
$932$	0	0
$933$	−2.71379	−0.0888456
$934$	0	0
$935$	38.8853	1.27169
$936$	0	0
$937$	27.0291	0.883001	0.441501	−	0.897261i	$-0.354446\pi$
$937$	27.0291	0.883001	0.441501	+	0.897261i	$0.354446\pi$
$938$	0	0
$939$	15.3884	0.502180
$940$	0	0
$941$	−24.0277	−0.783282	−0.391641	−	0.920118i	$-0.628092\pi$
$941$	−24.0277	−0.783282	−0.391641	+	0.920118i	$0.628092\pi$
$942$	0	0
$943$	23.3163	0.759284
$944$	0	0
$945$	0.104539	0.00340067
$946$	0	0
$947$	26.7966	0.870771	0.435386	−	0.900244i	$-0.356612\pi$
$947$	26.7966	0.870771	0.435386	+	0.900244i	$0.356612\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−25.6528	−0.831849
$952$	0	0
$953$	34.4215	1.11502	0.557510	−	0.830170i	$-0.311757\pi$
$953$	34.4215	1.11502	0.557510	+	0.830170i	$0.311757\pi$
$954$	0	0
$955$	12.0919	0.391284
$956$	0	0
$957$	−30.9245	−0.999648
$958$	0	0
$959$	−0.195669	−0.00631849
$960$	0	0
$961$	50.1745	1.61853
$962$	0	0
$963$	8.36658	0.269609
$964$	0	0
$965$	−28.6440	−0.922082
$966$	0	0
$967$	−26.2631	−0.844565	−0.422282	−	0.906464i	$-0.638771\pi$
$967$	−26.2631	−0.844565	−0.422282	+	0.906464i	$0.638771\pi$
$968$	0	0
$969$	−20.8310	−0.669188
$970$	0	0
$971$	39.9842	1.28315	0.641577	−	0.767059i	$-0.278280\pi$
$971$	39.9842	1.28315	0.641577	+	0.767059i	$0.278280\pi$
$972$	0	0
$973$	−0.425074	−0.0136272
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.58317	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$977$	4.58317	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$978$	0	0
$979$	−24.6896	−0.789084
$980$	0	0
$981$	−17.4276	−0.556420
$982$	0	0
$983$	10.6848	0.340794	0.170397	−	0.985376i	$-0.445495\pi$
$983$	10.6848	0.340794	0.170397	+	0.985376i	$0.445495\pi$
$984$	0	0
$985$	−42.7011	−1.36057
$986$	0	0
$987$	−0.352584	−0.0112229
$988$	0	0
$989$	18.2198	0.579357
$990$	0	0
$991$	−50.6021	−1.60743	−0.803714	−	0.595016i	$-0.797146\pi$
$991$	−50.6021	−1.60743	−0.803714	+	0.595016i	$0.797146\pi$
$992$	0	0
$993$	−3.82371	−0.121342
$994$	0	0
$995$	13.3387	0.422865
$996$	0	0
$997$	−35.9603	−1.13887	−0.569437	−	0.822035i	$-0.692839\pi$
$997$	−35.9603	−1.13887	−0.569437	+	0.822035i	$0.692839\pi$
$998$	0	0
$999$	−0.176292	−0.00557763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cm.1.2		3
4.3	odd	2		1014.2.a.n.1.2	yes	3
12.11	even	2		3042.2.a.ba.1.2		3
13.12	even	2		8112.2.a.cj.1.2		3
52.3	odd	6		1014.2.e.l.529.2		6
52.7	even	12		1014.2.i.h.361.5		12
52.11	even	12		1014.2.i.h.823.2		12
52.15	even	12		1014.2.i.h.823.5		12
52.19	even	12		1014.2.i.h.361.2		12
52.23	odd	6		1014.2.e.n.529.2		6
52.31	even	4		1014.2.b.f.337.2		6
52.35	odd	6		1014.2.e.l.991.2		6
52.43	odd	6		1014.2.e.n.991.2		6
52.47	even	4		1014.2.b.f.337.5		6
52.51	odd	2		1014.2.a.l.1.2	✓	3
156.47	odd	4		3042.2.b.o.1351.2		6
156.83	odd	4		3042.2.b.o.1351.5		6
156.155	even	2		3042.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.l.1.2	✓	3	52.51	odd	2
1014.2.a.n.1.2	yes	3	4.3	odd	2
1014.2.b.f.337.2		6	52.31	even	4
1014.2.b.f.337.5		6	52.47	even	4
1014.2.e.l.529.2		6	52.3	odd	6
1014.2.e.l.991.2		6	52.35	odd	6
1014.2.e.n.529.2		6	52.23	odd	6
1014.2.e.n.991.2		6	52.43	odd	6
1014.2.i.h.361.2		12	52.19	even	12
1014.2.i.h.361.5		12	52.7	even	12
1014.2.i.h.823.2		12	52.11	even	12
1014.2.i.h.823.5		12	52.15	even	12
3042.2.a.ba.1.2		3	12.11	even	2
3042.2.a.bh.1.2		3	156.155	even	2
3042.2.b.o.1351.2		6	156.47	odd	4
3042.2.b.o.1351.5		6	156.83	odd	4
8112.2.a.cj.1.2		3	13.12	even	2
8112.2.a.cm.1.2		3	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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{9} -6.29590 q^{11} +2.13706 q^{15} -2.89008 q^{17} +7.20775 q^{19} +0.0489173 q^{21} -2.71379 q^{23} -0.432960 q^{25} +1.00000 q^{27} +4.91185 q^{29} -9.00969 q^{31} -6.29590 q^{33} +0.104539 q^{35} -0.176292 q^{37} -8.59179 q^{41} -6.71379 q^{43} +2.13706 q^{45} -7.20775 q^{47} -6.99761 q^{49} -2.89008 q^{51} +9.34481 q^{53} -13.4547 q^{55} +7.20775 q^{57} +4.26875 q^{59} +7.10992 q^{61} +0.0489173 q^{63} -5.38404 q^{67} -2.71379 q^{69} +8.71379 q^{71} -14.9487 q^{73} -0.432960 q^{75} -0.307979 q^{77} -13.8291 q^{79} +1.00000 q^{81} +11.1347 q^{83} -6.17629 q^{85} +4.91185 q^{87} +3.92154 q^{89} -9.00969 q^{93} +15.4034 q^{95} -2.47889 q^{97} -6.29590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9} - 5 q^{11} + q^{15} - 8 q^{17} + 4 q^{19} - 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} - 5 q^{31} - 5 q^{33} + 4 q^{35} - 8 q^{37} + 2 q^{41} - 12 q^{43} + q^{45} - 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} + 4 q^{57} + 5 q^{59} + 22 q^{61} - 9 q^{63} - 6 q^{67} + 18 q^{71} - 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} - 13 q^{83} - 26 q^{85} + 11 q^{87} - 14 q^{89} - 5 q^{93} - 8 q^{95} - 23 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9 - 5 * q^11 + q^15 - 8 * q^17 + 4 * q^19 - 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 - 5 * q^31 - 5 * q^33 + 4 * q^35 - 8 * q^37 + 2 * q^41 - 12 * q^43 + q^45 - 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 + 4 * q^57 + 5 * q^59 + 22 * q^61 - 9 * q^63 - 6 * q^67 + 18 * q^71 - 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 - 13 * q^83 - 26 * q^85 + 11 * q^87 - 14 * q^89 - 5 * q^93 - 8 * q^95 - 23 * q^97 - 5 * q^99

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