LMFDB - Embedded newform 8112.2.a.bp.1.1

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:	$2$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{9} -4.73205 q^{11} -0.267949 q^{15} -2.26795 q^{17} -1.26795 q^{19} +0.732051 q^{21} +6.19615 q^{23} -4.92820 q^{25} -1.00000 q^{27} +2.46410 q^{29} +5.46410 q^{31} +4.73205 q^{33} -0.196152 q^{35} +10.4641 q^{37} +11.3923 q^{41} -7.66025 q^{43} +0.267949 q^{45} +8.19615 q^{47} -6.46410 q^{49} +2.26795 q^{51} +0.464102 q^{53} -1.26795 q^{55} +1.26795 q^{57} -8.00000 q^{59} +1.19615 q^{61} -0.732051 q^{63} +11.1244 q^{67} -6.19615 q^{69} -1.26795 q^{71} -9.73205 q^{73} +4.92820 q^{75} +3.46410 q^{77} +9.46410 q^{79} +1.00000 q^{81} -10.1962 q^{83} -0.607695 q^{85} -2.46410 q^{87} -2.53590 q^{89} -5.46410 q^{93} -0.339746 q^{95} -6.00000 q^{97} -4.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 8 q^{17} - 6 q^{19} - 2 q^{21} + 2 q^{23} + 4 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 6 q^{33} + 10 q^{35} + 14 q^{37} + 2 q^{41} + 2 q^{43} + 4 q^{45} + 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} - 6 q^{55} + 6 q^{57} - 16 q^{59} - 8 q^{61} + 2 q^{63} - 2 q^{67} - 2 q^{69} - 6 q^{71} - 16 q^{73} - 4 q^{75} + 12 q^{79} + 2 q^{81} - 10 q^{83} - 22 q^{85} + 2 q^{87} - 12 q^{89} - 4 q^{93} - 18 q^{95} - 12 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 8 * q^17 - 6 * q^19 - 2 * q^21 + 2 * q^23 + 4 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 6 * q^33 + 10 * q^35 + 14 * q^37 + 2 * q^41 + 2 * q^43 + 4 * q^45 + 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 - 6 * q^55 + 6 * q^57 - 16 * q^59 - 8 * q^61 + 2 * q^63 - 2 * q^67 - 2 * q^69 - 6 * q^71 - 16 * q^73 - 4 * q^75 + 12 * q^79 + 2 * q^81 - 10 * q^83 - 22 * q^85 + 2 * q^87 - 12 * q^89 - 4 * q^93 - 18 * q^95 - 12 * q^97 - 6 * 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
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.267949	−0.0691842
$16$	0	0
$17$	−2.26795	−0.550058	−0.275029	−	0.961436i	$-0.588688\pi$
$17$	−2.26795	−0.550058	−0.275029	+	0.961436i	$0.588688\pi$
$18$	0	0
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	6.19615	1.29199	0.645994	−	0.763343i	$-0.276443\pi$
$23$	6.19615	1.29199	0.645994	+	0.763343i	$0.276443\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.46410	0.457572	0.228786	−	0.973477i	$-0.426524\pi$
$29$	2.46410	0.457572	0.228786	+	0.973477i	$0.426524\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	4.73205	0.823744
$34$	0	0
$35$	−0.196152	−0.0331558
$36$	0	0
$37$	10.4641	1.72029	0.860144	−	0.510052i	$-0.170374\pi$
$37$	10.4641	1.72029	0.860144	+	0.510052i	$0.170374\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.3923	1.77918	0.889590	−	0.456761i	$-0.150990\pi$
$41$	11.3923	1.77918	0.889590	+	0.456761i	$0.150990\pi$
$42$	0	0
$43$	−7.66025	−1.16818	−0.584089	−	0.811690i	$-0.698548\pi$
$43$	−7.66025	−1.16818	−0.584089	+	0.811690i	$0.698548\pi$
$44$	0	0
$45$	0.267949	0.0399435
$46$	0	0
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	2.26795	0.317576
$52$	0	0
$53$	0.464102	0.0637493	0.0318746	−	0.999492i	$-0.489852\pi$
$53$	0.464102	0.0637493	0.0318746	+	0.999492i	$0.489852\pi$
$54$	0	0
$55$	−1.26795	−0.170970
$56$	0	0
$57$	1.26795	0.167944
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	1.19615	0.153152	0.0765758	−	0.997064i	$-0.475601\pi$
$61$	1.19615	0.153152	0.0765758	+	0.997064i	$0.475601\pi$
$62$	0	0
$63$	−0.732051	−0.0922297
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.1244	1.35906	0.679528	−	0.733649i	$-0.262185\pi$
$67$	11.1244	1.35906	0.679528	+	0.733649i	$0.262185\pi$
$68$	0	0
$69$	−6.19615	−0.745929
$70$	0	0
$71$	−1.26795	−0.150478	−0.0752389	−	0.997166i	$-0.523972\pi$
$71$	−1.26795	−0.150478	−0.0752389	+	0.997166i	$0.523972\pi$
$72$	0	0
$73$	−9.73205	−1.13905	−0.569525	−	0.821974i	$-0.692873\pi$
$73$	−9.73205	−1.13905	−0.569525	+	0.821974i	$0.692873\pi$
$74$	0	0
$75$	4.92820	0.569060
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	9.46410	1.06479	0.532397	−	0.846495i	$-0.321291\pi$
$79$	9.46410	1.06479	0.532397	+	0.846495i	$0.321291\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.1962	−1.11917	−0.559587	−	0.828772i	$-0.689040\pi$
$83$	−10.1962	−1.11917	−0.559587	+	0.828772i	$0.689040\pi$
$84$	0	0
$85$	−0.607695	−0.0659138
$86$	0	0
$87$	−2.46410	−0.264179
$88$	0	0
$89$	−2.53590	−0.268805	−0.134402	−	0.990927i	$-0.542911\pi$
$89$	−2.53590	−0.268805	−0.134402	+	0.990927i	$0.542911\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.46410	−0.566601
$94$	0	0
$95$	−0.339746	−0.0348572
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−4.73205	−0.475589
$100$	0	0
$101$	−11.9282	−1.18690	−0.593450	−	0.804871i	$-0.702235\pi$
$101$	−11.9282	−1.18690	−0.593450	+	0.804871i	$0.702235\pi$
$102$	0	0
$103$	18.7321	1.84572	0.922862	−	0.385131i	$-0.125844\pi$
$103$	18.7321	1.84572	0.922862	+	0.385131i	$0.125844\pi$
$104$	0	0
$105$	0.196152	0.0191425
$106$	0	0
$107$	0.196152	0.0189628	0.00948139	−	0.999955i	$-0.496982\pi$
$107$	0.196152	0.0189628	0.00948139	+	0.999955i	$0.496982\pi$
$108$	0	0
$109$	−5.46410	−0.523366	−0.261683	−	0.965154i	$-0.584277\pi$
$109$	−5.46410	−0.523366	−0.261683	+	0.965154i	$0.584277\pi$
$110$	0	0
$111$	−10.4641	−0.993209
$112$	0	0
$113$	−18.6603	−1.75541	−0.877705	−	0.479202i	$-0.840926\pi$
$113$	−18.6603	−1.75541	−0.877705	+	0.479202i	$0.840926\pi$
$114$	0	0
$115$	1.66025	0.154819
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.66025	0.152195
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	−11.3923	−1.02721
$124$	0	0
$125$	−2.66025	−0.237940
$126$	0	0
$127$	−17.8564	−1.58450	−0.792250	−	0.610197i	$-0.791090\pi$
$127$	−17.8564	−1.58450	−0.792250	+	0.610197i	$0.791090\pi$
$128$	0	0
$129$	7.66025	0.674448
$130$	0	0
$131$	−13.4641	−1.17636	−0.588182	−	0.808729i	$-0.700156\pi$
$131$	−13.4641	−1.17636	−0.588182	+	0.808729i	$0.700156\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	0	0
$135$	−0.267949	−0.0230614
$136$	0	0
$137$	−1.92820	−0.164738	−0.0823688	−	0.996602i	$-0.526249\pi$
$137$	−1.92820	−0.164738	−0.0823688	+	0.996602i	$0.526249\pi$
$138$	0	0
$139$	9.85641	0.836009	0.418005	−	0.908445i	$-0.362730\pi$
$139$	9.85641	0.836009	0.418005	+	0.908445i	$0.362730\pi$
$140$	0	0
$141$	−8.19615	−0.690241
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.660254	0.0548311
$146$	0	0
$147$	6.46410	0.533150
$148$	0	0
$149$	2.80385	0.229700	0.114850	−	0.993383i	$-0.463361\pi$
$149$	2.80385	0.229700	0.114850	+	0.993383i	$0.463361\pi$
$150$	0	0
$151$	−3.26795	−0.265942	−0.132971	−	0.991120i	$-0.542452\pi$
$151$	−3.26795	−0.265942	−0.132971	+	0.991120i	$0.542452\pi$
$152$	0	0
$153$	−2.26795	−0.183353
$154$	0	0
$155$	1.46410	0.117599
$156$	0	0
$157$	−23.5885	−1.88256	−0.941282	−	0.337622i	$-0.890378\pi$
$157$	−23.5885	−1.88256	−0.941282	+	0.337622i	$0.890378\pi$
$158$	0	0
$159$	−0.464102	−0.0368057
$160$	0	0
$161$	−4.53590	−0.357479
$162$	0	0
$163$	6.53590	0.511931	0.255966	−	0.966686i	$-0.417607\pi$
$163$	6.53590	0.511931	0.255966	+	0.966686i	$0.417607\pi$
$164$	0	0
$165$	1.26795	0.0987097
$166$	0	0
$167$	2.53590	0.196234	0.0981169	−	0.995175i	$-0.468718\pi$
$167$	2.53590	0.196234	0.0981169	+	0.995175i	$0.468718\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1.26795	−0.0969625
$172$	0	0
$173$	−16.3923	−1.24628	−0.623142	−	0.782109i	$-0.714144\pi$
$173$	−16.3923	−1.24628	−0.623142	+	0.782109i	$0.714144\pi$
$174$	0	0
$175$	3.60770	0.272716
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−22.0526	−1.64829	−0.824143	−	0.566382i	$-0.808343\pi$
$179$	−22.0526	−1.64829	−0.824143	+	0.566382i	$0.808343\pi$
$180$	0	0
$181$	8.80385	0.654385	0.327192	−	0.944958i	$-0.393897\pi$
$181$	8.80385	0.654385	0.327192	+	0.944958i	$0.393897\pi$
$182$	0	0
$183$	−1.19615	−0.0884221
$184$	0	0
$185$	2.80385	0.206143
$186$	0	0
$187$	10.7321	0.784805
$188$	0	0
$189$	0.732051	0.0532489
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	−8.26795	−0.595140	−0.297570	−	0.954700i	$-0.596176\pi$
$193$	−8.26795	−0.595140	−0.297570	+	0.954700i	$0.596176\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.85641	0.702240	0.351120	−	0.936330i	$-0.385801\pi$
$197$	9.85641	0.702240	0.351120	+	0.936330i	$0.385801\pi$
$198$	0	0
$199$	3.80385	0.269648	0.134824	−	0.990870i	$-0.456953\pi$
$199$	3.80385	0.269648	0.134824	+	0.990870i	$0.456953\pi$
$200$	0	0
$201$	−11.1244	−0.784652
$202$	0	0
$203$	−1.80385	−0.126605
$204$	0	0
$205$	3.05256	0.213200
$206$	0	0
$207$	6.19615	0.430662
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	4.39230	0.302379	0.151189	−	0.988505i	$-0.451690\pi$
$211$	4.39230	0.302379	0.151189	+	0.988505i	$0.451690\pi$
$212$	0	0
$213$	1.26795	0.0868784
$214$	0	0
$215$	−2.05256	−0.139983
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	9.73205	0.657631
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.0718	0.875352	0.437676	−	0.899133i	$-0.355802\pi$
$223$	13.0718	0.875352	0.437676	+	0.899133i	$0.355802\pi$
$224$	0	0
$225$	−4.92820	−0.328547
$226$	0	0
$227$	1.80385	0.119726	0.0598628	−	0.998207i	$-0.480934\pi$
$227$	1.80385	0.119726	0.0598628	+	0.998207i	$0.480934\pi$
$228$	0	0
$229$	15.8564	1.04782	0.523910	−	0.851773i	$-0.324473\pi$
$229$	15.8564	1.04782	0.523910	+	0.851773i	$0.324473\pi$
$230$	0	0
$231$	−3.46410	−0.227921
$232$	0	0
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	2.19615	0.143261
$236$	0	0
$237$	−9.46410	−0.614759
$238$	0	0
$239$	9.66025	0.624870	0.312435	−	0.949939i	$-0.398855\pi$
$239$	9.66025	0.624870	0.312435	+	0.949939i	$0.398855\pi$
$240$	0	0
$241$	−17.5885	−1.13297	−0.566486	−	0.824071i	$-0.691698\pi$
$241$	−17.5885	−1.13297	−0.566486	+	0.824071i	$0.691698\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.73205	−0.110657
$246$	0	0
$247$	0	0
$248$	0	0
$249$	10.1962	0.646155
$250$	0	0
$251$	−6.53590	−0.412542	−0.206271	−	0.978495i	$-0.566133\pi$
$251$	−6.53590	−0.412542	−0.206271	+	0.978495i	$0.566133\pi$
$252$	0	0
$253$	−29.3205	−1.84336
$254$	0	0
$255$	0.607695	0.0380553
$256$	0	0
$257$	26.6603	1.66302	0.831510	−	0.555509i	$-0.187477\pi$
$257$	26.6603	1.66302	0.831510	+	0.555509i	$0.187477\pi$
$258$	0	0
$259$	−7.66025	−0.475985
$260$	0	0
$261$	2.46410	0.152524
$262$	0	0
$263$	−28.0526	−1.72979	−0.864897	−	0.501949i	$-0.832617\pi$
$263$	−28.0526	−1.72979	−0.864897	+	0.501949i	$0.832617\pi$
$264$	0	0
$265$	0.124356	0.00763911
$266$	0	0
$267$	2.53590	0.155194
$268$	0	0
$269$	−1.46410	−0.0892679	−0.0446339	−	0.999003i	$-0.514212\pi$
$269$	−1.46410	−0.0892679	−0.0446339	+	0.999003i	$0.514212\pi$
$270$	0	0
$271$	−5.85641	−0.355751	−0.177876	−	0.984053i	$-0.556922\pi$
$271$	−5.85641	−0.355751	−0.177876	+	0.984053i	$0.556922\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.3205	1.40628
$276$	0	0
$277$	−2.26795	−0.136268	−0.0681339	−	0.997676i	$-0.521705\pi$
$277$	−2.26795	−0.136268	−0.0681339	+	0.997676i	$0.521705\pi$
$278$	0	0
$279$	5.46410	0.327127
$280$	0	0
$281$	−22.3205	−1.33153	−0.665765	−	0.746162i	$-0.731895\pi$
$281$	−22.3205	−1.33153	−0.665765	+	0.746162i	$0.731895\pi$
$282$	0	0
$283$	−8.33975	−0.495746	−0.247873	−	0.968792i	$-0.579732\pi$
$283$	−8.33975	−0.495746	−0.247873	+	0.968792i	$0.579732\pi$
$284$	0	0
$285$	0.339746	0.0201248
$286$	0	0
$287$	−8.33975	−0.492280
$288$	0	0
$289$	−11.8564	−0.697436
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−14.5167	−0.848072	−0.424036	−	0.905645i	$-0.639387\pi$
$293$	−14.5167	−0.848072	−0.424036	+	0.905645i	$0.639387\pi$
$294$	0	0
$295$	−2.14359	−0.124805
$296$	0	0
$297$	4.73205	0.274581
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.60770	0.323222
$302$	0	0
$303$	11.9282	0.685257
$304$	0	0
$305$	0.320508	0.0183522
$306$	0	0
$307$	−8.58846	−0.490169	−0.245085	−	0.969502i	$-0.578816\pi$
$307$	−8.58846	−0.490169	−0.245085	+	0.969502i	$0.578816\pi$
$308$	0	0
$309$	−18.7321	−1.06563
$310$	0	0
$311$	−15.6603	−0.888012	−0.444006	−	0.896024i	$-0.646443\pi$
$311$	−15.6603	−0.888012	−0.444006	+	0.896024i	$0.646443\pi$
$312$	0	0
$313$	13.4641	0.761036	0.380518	−	0.924774i	$-0.375746\pi$
$313$	13.4641	0.761036	0.380518	+	0.924774i	$0.375746\pi$
$314$	0	0
$315$	−0.196152	−0.0110519
$316$	0	0
$317$	3.33975	0.187579	0.0937894	−	0.995592i	$-0.470102\pi$
$317$	3.33975	0.187579	0.0937894	+	0.995592i	$0.470102\pi$
$318$	0	0
$319$	−11.6603	−0.652849
$320$	0	0
$321$	−0.196152	−0.0109482
$322$	0	0
$323$	2.87564	0.160005
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.46410	0.302166
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	10.4641	0.573429
$334$	0	0
$335$	2.98076	0.162856
$336$	0	0
$337$	6.85641	0.373492	0.186746	−	0.982408i	$-0.440206\pi$
$337$	6.85641	0.373492	0.186746	+	0.982408i	$0.440206\pi$
$338$	0	0
$339$	18.6603	1.01349
$340$	0	0
$341$	−25.8564	−1.40020
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	−1.66025	−0.0893851
$346$	0	0
$347$	−8.87564	−0.476470	−0.238235	−	0.971208i	$-0.576569\pi$
$347$	−8.87564	−0.476470	−0.238235	+	0.971208i	$0.576569\pi$
$348$	0	0
$349$	−19.3205	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$349$	−19.3205	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.7846	1.05303	0.526514	−	0.850166i	$-0.323499\pi$
$353$	19.7846	1.05303	0.526514	+	0.850166i	$0.323499\pi$
$354$	0	0
$355$	−0.339746	−0.0180318
$356$	0	0
$357$	−1.66025	−0.0878700
$358$	0	0
$359$	−23.1244	−1.22046	−0.610228	−	0.792226i	$-0.708922\pi$
$359$	−23.1244	−1.22046	−0.610228	+	0.792226i	$0.708922\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	0	0
$363$	−11.3923	−0.597941
$364$	0	0
$365$	−2.60770	−0.136493
$366$	0	0
$367$	14.7321	0.769007	0.384503	−	0.923124i	$-0.374373\pi$
$367$	14.7321	0.769007	0.384503	+	0.923124i	$0.374373\pi$
$368$	0	0
$369$	11.3923	0.593060
$370$	0	0
$371$	−0.339746	−0.0176387
$372$	0	0
$373$	−10.2679	−0.531654	−0.265827	−	0.964021i	$-0.585645\pi$
$373$	−10.2679	−0.531654	−0.265827	+	0.964021i	$0.585645\pi$
$374$	0	0
$375$	2.66025	0.137375
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.46410	−0.0752058	−0.0376029	−	0.999293i	$-0.511972\pi$
$379$	−1.46410	−0.0752058	−0.0376029	+	0.999293i	$0.511972\pi$
$380$	0	0
$381$	17.8564	0.914811
$382$	0	0
$383$	−5.46410	−0.279203	−0.139601	−	0.990208i	$-0.544582\pi$
$383$	−5.46410	−0.279203	−0.139601	+	0.990208i	$0.544582\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	0	0
$387$	−7.66025	−0.389393
$388$	0	0
$389$	−29.7846	−1.51014	−0.755070	−	0.655644i	$-0.772397\pi$
$389$	−29.7846	−1.51014	−0.755070	+	0.655644i	$0.772397\pi$
$390$	0	0
$391$	−14.0526	−0.710668
$392$	0	0
$393$	13.4641	0.679174
$394$	0	0
$395$	2.53590	0.127595
$396$	0	0
$397$	−0.392305	−0.0196892	−0.00984461	−	0.999952i	$-0.503134\pi$
$397$	−0.392305	−0.0196892	−0.00984461	+	0.999952i	$0.503134\pi$
$398$	0	0
$399$	−0.928203	−0.0464683
$400$	0	0
$401$	21.9282	1.09504	0.547521	−	0.836792i	$-0.315572\pi$
$401$	21.9282	1.09504	0.547521	+	0.836792i	$0.315572\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.267949	0.0133145
$406$	0	0
$407$	−49.5167	−2.45445
$408$	0	0
$409$	14.2679	0.705505	0.352752	−	0.935717i	$-0.385246\pi$
$409$	14.2679	0.705505	0.352752	+	0.935717i	$0.385246\pi$
$410$	0	0
$411$	1.92820	0.0951113
$412$	0	0
$413$	5.85641	0.288175
$414$	0	0
$415$	−2.73205	−0.134111
$416$	0	0
$417$	−9.85641	−0.482670
$418$	0	0
$419$	10.5359	0.514712	0.257356	−	0.966317i	$-0.417149\pi$
$419$	10.5359	0.514712	0.257356	+	0.966317i	$0.417149\pi$
$420$	0	0
$421$	32.7128	1.59432	0.797162	−	0.603765i	$-0.206333\pi$
$421$	32.7128	1.59432	0.797162	+	0.603765i	$0.206333\pi$
$422$	0	0
$423$	8.19615	0.398511
$424$	0	0
$425$	11.1769	0.542160
$426$	0	0
$427$	−0.875644	−0.0423754
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.1244	0.535841	0.267921	−	0.963441i	$-0.413663\pi$
$431$	11.1244	0.535841	0.267921	+	0.963441i	$0.413663\pi$
$432$	0	0
$433$	−14.8564	−0.713953	−0.356977	−	0.934113i	$-0.616192\pi$
$433$	−14.8564	−0.713953	−0.356977	+	0.934113i	$0.616192\pi$
$434$	0	0
$435$	−0.660254	−0.0316568
$436$	0	0
$437$	−7.85641	−0.375823
$438$	0	0
$439$	17.6603	0.842878	0.421439	−	0.906857i	$-0.361525\pi$
$439$	17.6603	0.842878	0.421439	+	0.906857i	$0.361525\pi$
$440$	0	0
$441$	−6.46410	−0.307814
$442$	0	0
$443$	−36.3923	−1.72905	−0.864525	−	0.502589i	$-0.832381\pi$
$443$	−36.3923	−1.72905	−0.864525	+	0.502589i	$0.832381\pi$
$444$	0	0
$445$	−0.679492	−0.0322110
$446$	0	0
$447$	−2.80385	−0.132617
$448$	0	0
$449$	−23.3205	−1.10056	−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205	−1.10056	−0.550281	+	0.834979i	$0.685480\pi$
$450$	0	0
$451$	−53.9090	−2.53847
$452$	0	0
$453$	3.26795	0.153542
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.6603	0.872890	0.436445	−	0.899731i	$-0.356237\pi$
$457$	18.6603	0.872890	0.436445	+	0.899731i	$0.356237\pi$
$458$	0	0
$459$	2.26795	0.105859
$460$	0	0
$461$	25.7321	1.19846	0.599231	−	0.800577i	$-0.295473\pi$
$461$	25.7321	1.19846	0.599231	+	0.800577i	$0.295473\pi$
$462$	0	0
$463$	28.0526	1.30371	0.651856	−	0.758342i	$-0.273990\pi$
$463$	28.0526	1.30371	0.651856	+	0.758342i	$0.273990\pi$
$464$	0	0
$465$	−1.46410	−0.0678961
$466$	0	0
$467$	12.5885	0.582524	0.291262	−	0.956643i	$-0.405925\pi$
$467$	12.5885	0.582524	0.291262	+	0.956643i	$0.405925\pi$
$468$	0	0
$469$	−8.14359	−0.376036
$470$	0	0
$471$	23.5885	1.08690
$472$	0	0
$473$	36.2487	1.66672
$474$	0	0
$475$	6.24871	0.286711
$476$	0	0
$477$	0.464102	0.0212498
$478$	0	0
$479$	−26.5359	−1.21246	−0.606228	−	0.795291i	$-0.707318\pi$
$479$	−26.5359	−1.21246	−0.606228	+	0.795291i	$0.707318\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.53590	0.206391
$484$	0	0
$485$	−1.60770	−0.0730017
$486$	0	0
$487$	−21.1244	−0.957236	−0.478618	−	0.878023i	$-0.658862\pi$
$487$	−21.1244	−0.957236	−0.478618	+	0.878023i	$0.658862\pi$
$488$	0	0
$489$	−6.53590	−0.295564
$490$	0	0
$491$	−5.26795	−0.237739	−0.118870	−	0.992910i	$-0.537927\pi$
$491$	−5.26795	−0.237739	−0.118870	+	0.992910i	$0.537927\pi$
$492$	0	0
$493$	−5.58846	−0.251691
$494$	0	0
$495$	−1.26795	−0.0569901
$496$	0	0
$497$	0.928203	0.0416356
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	−2.53590	−0.113296
$502$	0	0
$503$	10.9808	0.489608	0.244804	−	0.969573i	$-0.421276\pi$
$503$	10.9808	0.489608	0.244804	+	0.969573i	$0.421276\pi$
$504$	0	0
$505$	−3.19615	−0.142227
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.2679	0.455119	0.227559	−	0.973764i	$-0.426925\pi$
$509$	10.2679	0.455119	0.227559	+	0.973764i	$0.426925\pi$
$510$	0	0
$511$	7.12436	0.315163
$512$	0	0
$513$	1.26795	0.0559813
$514$	0	0
$515$	5.01924	0.221174
$516$	0	0
$517$	−38.7846	−1.70575
$518$	0	0
$519$	16.3923	0.719542
$520$	0	0
$521$	−17.4449	−0.764273	−0.382137	−	0.924106i	$-0.624812\pi$
$521$	−17.4449	−0.764273	−0.382137	+	0.924106i	$0.624812\pi$
$522$	0	0
$523$	−36.4449	−1.59362	−0.796811	−	0.604228i	$-0.793482\pi$
$523$	−36.4449	−1.59362	−0.796811	+	0.604228i	$0.793482\pi$
$524$	0	0
$525$	−3.60770	−0.157453
$526$	0	0
$527$	−12.3923	−0.539817
$528$	0	0
$529$	15.3923	0.669231
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.0525589	0.00227232
$536$	0	0
$537$	22.0526	0.951638
$538$	0	0
$539$	30.5885	1.31754
$540$	0	0
$541$	40.3205	1.73351	0.866757	−	0.498731i	$-0.166200\pi$
$541$	40.3205	1.73351	0.866757	+	0.498731i	$0.166200\pi$
$542$	0	0
$543$	−8.80385	−0.377809
$544$	0	0
$545$	−1.46410	−0.0627152
$546$	0	0
$547$	−6.19615	−0.264928	−0.132464	−	0.991188i	$-0.542289\pi$
$547$	−6.19615	−0.264928	−0.132464	+	0.991188i	$0.542289\pi$
$548$	0	0
$549$	1.19615	0.0510505
$550$	0	0
$551$	−3.12436	−0.133102
$552$	0	0
$553$	−6.92820	−0.294617
$554$	0	0
$555$	−2.80385	−0.119017
$556$	0	0
$557$	−30.3731	−1.28695	−0.643474	−	0.765468i	$-0.722508\pi$
$557$	−30.3731	−1.28695	−0.643474	+	0.765468i	$0.722508\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−10.7321	−0.453108
$562$	0	0
$563$	−21.0718	−0.888070	−0.444035	−	0.896009i	$-0.646453\pi$
$563$	−21.0718	−0.888070	−0.444035	+	0.896009i	$0.646453\pi$
$564$	0	0
$565$	−5.00000	−0.210352
$566$	0	0
$567$	−0.732051	−0.0307432
$568$	0	0
$569$	−38.6410	−1.61992	−0.809958	−	0.586488i	$-0.800510\pi$
$569$	−38.6410	−1.61992	−0.809958	+	0.586488i	$0.800510\pi$
$570$	0	0
$571$	24.0526	1.00657	0.503284	−	0.864121i	$-0.332125\pi$
$571$	24.0526	1.00657	0.503284	+	0.864121i	$0.332125\pi$
$572$	0	0
$573$	6.92820	0.289430
$574$	0	0
$575$	−30.5359	−1.27343
$576$	0	0
$577$	0.267949	0.0111549	0.00557744	−	0.999984i	$-0.498225\pi$
$577$	0.267949	0.0111549	0.00557744	+	0.999984i	$0.498225\pi$
$578$	0	0
$579$	8.26795	0.343604
$580$	0	0
$581$	7.46410	0.309663
$582$	0	0
$583$	−2.19615	−0.0909553
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	0	0
$591$	−9.85641	−0.405438
$592$	0	0
$593$	−36.8564	−1.51351	−0.756756	−	0.653698i	$-0.773217\pi$
$593$	−36.8564	−1.51351	−0.756756	+	0.653698i	$0.773217\pi$
$594$	0	0
$595$	0.444864	0.0182376
$596$	0	0
$597$	−3.80385	−0.155681
$598$	0	0
$599$	9.46410	0.386693	0.193346	−	0.981131i	$-0.438066\pi$
$599$	9.46410	0.386693	0.193346	+	0.981131i	$0.438066\pi$
$600$	0	0
$601$	−5.92820	−0.241816	−0.120908	−	0.992664i	$-0.538581\pi$
$601$	−5.92820	−0.241816	−0.120908	+	0.992664i	$0.538581\pi$
$602$	0	0
$603$	11.1244	0.453019
$604$	0	0
$605$	3.05256	0.124104
$606$	0	0
$607$	−0.784610	−0.0318463	−0.0159232	−	0.999873i	$-0.505069\pi$
$607$	−0.784610	−0.0318463	−0.0159232	+	0.999873i	$0.505069\pi$
$608$	0	0
$609$	1.80385	0.0730956
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.3923	0.460131	0.230065	−	0.973175i	$-0.426106\pi$
$613$	11.3923	0.460131	0.230065	+	0.973175i	$0.426106\pi$
$614$	0	0
$615$	−3.05256	−0.123091
$616$	0	0
$617$	35.2487	1.41906	0.709530	−	0.704675i	$-0.248907\pi$
$617$	35.2487	1.41906	0.709530	+	0.704675i	$0.248907\pi$
$618$	0	0
$619$	−10.5359	−0.423474	−0.211737	−	0.977327i	$-0.567912\pi$
$619$	−10.5359	−0.423474	−0.211737	+	0.977327i	$0.567912\pi$
$620$	0	0
$621$	−6.19615	−0.248643
$622$	0	0
$623$	1.85641	0.0743754
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	−23.7321	−0.946259
$630$	0	0
$631$	−47.7128	−1.89942	−0.949709	−	0.313135i	$-0.898621\pi$
$631$	−47.7128	−1.89942	−0.949709	+	0.313135i	$0.898621\pi$
$632$	0	0
$633$	−4.39230	−0.174578
$634$	0	0
$635$	−4.78461	−0.189871
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.26795	−0.0501593
$640$	0	0
$641$	25.9808	1.02618	0.513089	−	0.858335i	$-0.328501\pi$
$641$	25.9808	1.02618	0.513089	+	0.858335i	$0.328501\pi$
$642$	0	0
$643$	−13.8564	−0.546443	−0.273222	−	0.961951i	$-0.588089\pi$
$643$	−13.8564	−0.546443	−0.273222	+	0.961951i	$0.588089\pi$
$644$	0	0
$645$	2.05256	0.0808194
$646$	0	0
$647$	−26.2487	−1.03194	−0.515972	−	0.856606i	$-0.672569\pi$
$647$	−26.2487	−1.03194	−0.515972	+	0.856606i	$0.672569\pi$
$648$	0	0
$649$	37.8564	1.48599
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	−10.5359	−0.412302	−0.206151	−	0.978520i	$-0.566094\pi$
$653$	−10.5359	−0.412302	−0.206151	+	0.978520i	$0.566094\pi$
$654$	0	0
$655$	−3.60770	−0.140964
$656$	0	0
$657$	−9.73205	−0.379683
$658$	0	0
$659$	−38.2487	−1.48996	−0.744979	−	0.667088i	$-0.767541\pi$
$659$	−38.2487	−1.48996	−0.744979	+	0.667088i	$0.767541\pi$
$660$	0	0
$661$	−9.39230	−0.365318	−0.182659	−	0.983176i	$-0.558470\pi$
$661$	−9.39230	−0.365318	−0.182659	+	0.983176i	$0.558470\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.248711	0.00964461
$666$	0	0
$667$	15.2679	0.591177
$668$	0	0
$669$	−13.0718	−0.505385
$670$	0	0
$671$	−5.66025	−0.218512
$672$	0	0
$673$	14.0718	0.542428	0.271214	−	0.962519i	$-0.412575\pi$
$673$	14.0718	0.542428	0.271214	+	0.962519i	$0.412575\pi$
$674$	0	0
$675$	4.92820	0.189687
$676$	0	0
$677$	−38.5359	−1.48105	−0.740527	−	0.672026i	$-0.765424\pi$
$677$	−38.5359	−1.48105	−0.740527	+	0.672026i	$0.765424\pi$
$678$	0	0
$679$	4.39230	0.168561
$680$	0	0
$681$	−1.80385	−0.0691236
$682$	0	0
$683$	37.8564	1.44854	0.724268	−	0.689519i	$-0.242178\pi$
$683$	37.8564	1.44854	0.724268	+	0.689519i	$0.242178\pi$
$684$	0	0
$685$	−0.516660	−0.0197406
$686$	0	0
$687$	−15.8564	−0.604960
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−26.3397	−1.00201	−0.501006	−	0.865444i	$-0.667036\pi$
$691$	−26.3397	−1.00201	−0.501006	+	0.865444i	$0.667036\pi$
$692$	0	0
$693$	3.46410	0.131590
$694$	0	0
$695$	2.64102	0.100179
$696$	0	0
$697$	−25.8372	−0.978653
$698$	0	0
$699$	−19.8564	−0.751038
$700$	0	0
$701$	31.3205	1.18296	0.591480	−	0.806320i	$-0.298544\pi$
$701$	31.3205	1.18296	0.591480	+	0.806320i	$0.298544\pi$
$702$	0	0
$703$	−13.2679	−0.500410
$704$	0	0
$705$	−2.19615	−0.0827119
$706$	0	0
$707$	8.73205	0.328403
$708$	0	0
$709$	40.8564	1.53439	0.767197	−	0.641411i	$-0.221651\pi$
$709$	40.8564	1.53439	0.767197	+	0.641411i	$0.221651\pi$
$710$	0	0
$711$	9.46410	0.354932
$712$	0	0
$713$	33.8564	1.26793
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.66025	−0.360769
$718$	0	0
$719$	22.5359	0.840447	0.420224	−	0.907421i	$-0.361952\pi$
$719$	22.5359	0.840447	0.420224	+	0.907421i	$0.361952\pi$
$720$	0	0
$721$	−13.7128	−0.510692
$722$	0	0
$723$	17.5885	0.654122
$724$	0	0
$725$	−12.1436	−0.451002
$726$	0	0
$727$	−20.9808	−0.778133	−0.389067	−	0.921210i	$-0.627202\pi$
$727$	−20.9808	−0.778133	−0.389067	+	0.921210i	$0.627202\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.3731	0.642566
$732$	0	0
$733$	19.0000	0.701781	0.350891	−	0.936416i	$-0.385879\pi$
$733$	19.0000	0.701781	0.350891	+	0.936416i	$0.385879\pi$
$734$	0	0
$735$	1.73205	0.0638877
$736$	0	0
$737$	−52.6410	−1.93906
$738$	0	0
$739$	10.9282	0.402000	0.201000	−	0.979591i	$-0.435581\pi$
$739$	10.9282	0.402000	0.201000	+	0.979591i	$0.435581\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.6077	1.01283	0.506414	−	0.862290i	$-0.330971\pi$
$743$	27.6077	1.01283	0.506414	+	0.862290i	$0.330971\pi$
$744$	0	0
$745$	0.751289	0.0275251
$746$	0	0
$747$	−10.1962	−0.373058
$748$	0	0
$749$	−0.143594	−0.00524679
$750$	0	0
$751$	−15.9090	−0.580526	−0.290263	−	0.956947i	$-0.593743\pi$
$751$	−15.9090	−0.580526	−0.290263	+	0.956947i	$0.593743\pi$
$752$	0	0
$753$	6.53590	0.238181
$754$	0	0
$755$	−0.875644	−0.0318680
$756$	0	0
$757$	7.07180	0.257029	0.128514	−	0.991708i	$-0.458979\pi$
$757$	7.07180	0.257029	0.128514	+	0.991708i	$0.458979\pi$
$758$	0	0
$759$	29.3205	1.06427
$760$	0	0
$761$	−23.3205	−0.845368	−0.422684	−	0.906277i	$-0.638912\pi$
$761$	−23.3205	−0.845368	−0.422684	+	0.906277i	$0.638912\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	−0.607695	−0.0219713
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−16.1436	−0.582153	−0.291076	−	0.956700i	$-0.594013\pi$
$769$	−16.1436	−0.582153	−0.291076	+	0.956700i	$0.594013\pi$
$770$	0	0
$771$	−26.6603	−0.960146
$772$	0	0
$773$	−35.0718	−1.26144	−0.630722	−	0.776009i	$-0.717241\pi$
$773$	−35.0718	−1.26144	−0.630722	+	0.776009i	$0.717241\pi$
$774$	0	0
$775$	−26.9282	−0.967290
$776$	0	0
$777$	7.66025	0.274810
$778$	0	0
$779$	−14.4449	−0.517541
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	−2.46410	−0.0880598
$784$	0	0
$785$	−6.32051	−0.225589
$786$	0	0
$787$	−39.3205	−1.40162	−0.700812	−	0.713346i	$-0.747179\pi$
$787$	−39.3205	−1.40162	−0.700812	+	0.713346i	$0.747179\pi$
$788$	0	0
$789$	28.0526	0.998698
$790$	0	0
$791$	13.6603	0.485703
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.124356	−0.00441044
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−18.5885	−0.657612
$800$	0	0
$801$	−2.53590	−0.0896016
$802$	0	0
$803$	46.0526	1.62516
$804$	0	0
$805$	−1.21539	−0.0428369
$806$	0	0
$807$	1.46410	0.0515388
$808$	0	0
$809$	−22.4115	−0.787948	−0.393974	−	0.919122i	$-0.628900\pi$
$809$	−22.4115	−0.787948	−0.393974	+	0.919122i	$0.628900\pi$
$810$	0	0
$811$	45.1769	1.58638	0.793188	−	0.608977i	$-0.208420\pi$
$811$	45.1769	1.58638	0.793188	+	0.608977i	$0.208420\pi$
$812$	0	0
$813$	5.85641	0.205393
$814$	0	0
$815$	1.75129	0.0613450
$816$	0	0
$817$	9.71281	0.339808
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.9282	0.451197	0.225599	−	0.974220i	$-0.427566\pi$
$821$	12.9282	0.451197	0.225599	+	0.974220i	$0.427566\pi$
$822$	0	0
$823$	41.5692	1.44901	0.724506	−	0.689269i	$-0.242068\pi$
$823$	41.5692	1.44901	0.724506	+	0.689269i	$0.242068\pi$
$824$	0	0
$825$	−23.3205	−0.811916
$826$	0	0
$827$	−33.4641	−1.16366	−0.581830	−	0.813310i	$-0.697663\pi$
$827$	−33.4641	−1.16366	−0.581830	+	0.813310i	$0.697663\pi$
$828$	0	0
$829$	12.1244	0.421096	0.210548	−	0.977583i	$-0.432475\pi$
$829$	12.1244	0.421096	0.210548	+	0.977583i	$0.432475\pi$
$830$	0	0
$831$	2.26795	0.0786743
$832$	0	0
$833$	14.6603	0.507948
$834$	0	0
$835$	0.679492	0.0235148
$836$	0	0
$837$	−5.46410	−0.188867
$838$	0	0
$839$	−14.1436	−0.488291	−0.244146	−	0.969739i	$-0.578507\pi$
$839$	−14.1436	−0.488291	−0.244146	+	0.969739i	$0.578507\pi$
$840$	0	0
$841$	−22.9282	−0.790628
$842$	0	0
$843$	22.3205	0.768759
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.33975	−0.286557
$848$	0	0
$849$	8.33975	0.286219
$850$	0	0
$851$	64.8372	2.22259
$852$	0	0
$853$	8.17691	0.279972	0.139986	−	0.990153i	$-0.455294\pi$
$853$	8.17691	0.279972	0.139986	+	0.990153i	$0.455294\pi$
$854$	0	0
$855$	−0.339746	−0.0116191
$856$	0	0
$857$	−19.4449	−0.664224	−0.332112	−	0.943240i	$-0.607761\pi$
$857$	−19.4449	−0.664224	−0.332112	+	0.943240i	$0.607761\pi$
$858$	0	0
$859$	22.8756	0.780507	0.390253	−	0.920707i	$-0.372387\pi$
$859$	22.8756	0.780507	0.390253	+	0.920707i	$0.372387\pi$
$860$	0	0
$861$	8.33975	0.284218
$862$	0	0
$863$	−7.12436	−0.242516	−0.121258	−	0.992621i	$-0.538693\pi$
$863$	−7.12436	−0.242516	−0.121258	+	0.992621i	$0.538693\pi$
$864$	0	0
$865$	−4.39230	−0.149343
$866$	0	0
$867$	11.8564	0.402665
$868$	0	0
$869$	−44.7846	−1.51921
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	1.94744	0.0658355
$876$	0	0
$877$	10.0718	0.340100	0.170050	−	0.985435i	$-0.445607\pi$
$877$	10.0718	0.340100	0.170050	+	0.985435i	$0.445607\pi$
$878$	0	0
$879$	14.5167	0.489635
$880$	0	0
$881$	51.8372	1.74644	0.873219	−	0.487327i	$-0.162028\pi$
$881$	51.8372	1.74644	0.873219	+	0.487327i	$0.162028\pi$
$882$	0	0
$883$	−29.0718	−0.978344	−0.489172	−	0.872187i	$-0.662701\pi$
$883$	−29.0718	−0.978344	−0.489172	+	0.872187i	$0.662701\pi$
$884$	0	0
$885$	2.14359	0.0720561
$886$	0	0
$887$	−10.1436	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$887$	−10.1436	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$888$	0	0
$889$	13.0718	0.438414
$890$	0	0
$891$	−4.73205	−0.158530
$892$	0	0
$893$	−10.3923	−0.347765
$894$	0	0
$895$	−5.90897	−0.197515
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.4641	0.449053
$900$	0	0
$901$	−1.05256	−0.0350658
$902$	0	0
$903$	−5.60770	−0.186612
$904$	0	0
$905$	2.35898	0.0784153
$906$	0	0
$907$	−15.6077	−0.518245	−0.259123	−	0.965844i	$-0.583433\pi$
$907$	−15.6077	−0.518245	−0.259123	+	0.965844i	$0.583433\pi$
$908$	0	0
$909$	−11.9282	−0.395634
$910$	0	0
$911$	9.46410	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$911$	9.46410	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$912$	0	0
$913$	48.2487	1.59680
$914$	0	0
$915$	−0.320508	−0.0105957
$916$	0	0
$917$	9.85641	0.325487
$918$	0	0
$919$	57.9615	1.91197	0.955987	−	0.293409i	$-0.0947896\pi$
$919$	57.9615	1.91197	0.955987	+	0.293409i	$0.0947896\pi$
$920$	0	0
$921$	8.58846	0.282999
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−51.5692	−1.69559
$926$	0	0
$927$	18.7321	0.615241
$928$	0	0
$929$	9.24871	0.303440	0.151720	−	0.988423i	$-0.451519\pi$
$929$	9.24871	0.303440	0.151720	+	0.988423i	$0.451519\pi$
$930$	0	0
$931$	8.19615	0.268618
$932$	0	0
$933$	15.6603	0.512694
$934$	0	0
$935$	2.87564	0.0940436
$936$	0	0
$937$	43.2487	1.41287	0.706437	−	0.707776i	$-0.250301\pi$
$937$	43.2487	1.41287	0.706437	+	0.707776i	$0.250301\pi$
$938$	0	0
$939$	−13.4641	−0.439384
$940$	0	0
$941$	−56.6410	−1.84644	−0.923222	−	0.384267i	$-0.874454\pi$
$941$	−56.6410	−1.84644	−0.923222	+	0.384267i	$0.874454\pi$
$942$	0	0
$943$	70.5885	2.29868
$944$	0	0
$945$	0.196152	0.00638084
$946$	0	0
$947$	−34.9282	−1.13501	−0.567507	−	0.823369i	$-0.692092\pi$
$947$	−34.9282	−1.13501	−0.567507	+	0.823369i	$0.692092\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−3.33975	−0.108299
$952$	0	0
$953$	−41.5692	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$953$	−41.5692	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$954$	0	0
$955$	−1.85641	−0.0600719
$956$	0	0
$957$	11.6603	0.376922
$958$	0	0
$959$	1.41154	0.0455811
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0.196152	0.00632092
$964$	0	0
$965$	−2.21539	−0.0713159
$966$	0	0
$967$	18.8756	0.607000	0.303500	−	0.952831i	$-0.401845\pi$
$967$	18.8756	0.607000	0.303500	+	0.952831i	$0.401845\pi$
$968$	0	0
$969$	−2.87564	−0.0923790
$970$	0	0
$971$	18.2487	0.585629	0.292815	−	0.956169i	$-0.405408\pi$
$971$	18.2487	0.585629	0.292815	+	0.956169i	$0.405408\pi$
$972$	0	0
$973$	−7.21539	−0.231315
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.0718	1.02607	0.513034	−	0.858368i	$-0.328522\pi$
$977$	32.0718	1.02607	0.513034	+	0.858368i	$0.328522\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−5.46410	−0.174455
$982$	0	0
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	2.64102	0.0841498
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	−47.4641	−1.50927
$990$	0	0
$991$	8.58846	0.272821	0.136411	−	0.990652i	$-0.456443\pi$
$991$	8.58846	0.272821	0.136411	+	0.990652i	$0.456443\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	1.01924	0.0323120
$996$	0	0
$997$	38.6603	1.22438	0.612191	−	0.790710i	$-0.290288\pi$
$997$	38.6603	1.22438	0.612191	+	0.790710i	$0.290288\pi$
$998$	0	0
$999$	−10.4641	−0.331070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bp.1.1		2
4.3	odd	2		1014.2.a.k.1.1		2
12.11	even	2		3042.2.a.p.1.2		2
13.2	odd	12		624.2.bv.e.433.1		4
13.7	odd	12		624.2.bv.e.49.2		4
13.12	even	2		8112.2.a.bj.1.2		2
39.2	even	12		1872.2.by.h.433.2		4
39.20	even	12		1872.2.by.h.1297.1		4
52.3	odd	6		1014.2.e.g.529.1		4
52.7	even	12		78.2.i.a.49.2	yes	4
52.11	even	12		1014.2.i.a.823.1		4
52.15	even	12		78.2.i.a.43.2	✓	4
52.19	even	12		1014.2.i.a.361.1		4
52.23	odd	6		1014.2.e.i.529.2		4
52.31	even	4		1014.2.b.e.337.2		4
52.35	odd	6		1014.2.e.g.991.1		4
52.43	odd	6		1014.2.e.i.991.2		4
52.47	even	4		1014.2.b.e.337.3		4
52.51	odd	2		1014.2.a.i.1.2		2
156.47	odd	4		3042.2.b.i.1351.2		4
156.59	odd	12		234.2.l.c.127.1		4
156.83	odd	4		3042.2.b.i.1351.3		4
156.119	odd	12		234.2.l.c.199.1		4
156.155	even	2		3042.2.a.y.1.1		2
260.7	odd	12		1950.2.y.g.49.1		4
260.59	even	12		1950.2.bc.d.751.1		4
260.67	odd	12		1950.2.y.b.199.2		4
260.119	even	12		1950.2.bc.d.901.1		4
260.163	odd	12		1950.2.y.b.49.2		4
260.223	odd	12		1950.2.y.g.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.2	✓	4	52.15	even	12
78.2.i.a.49.2	yes	4	52.7	even	12
234.2.l.c.127.1		4	156.59	odd	12
234.2.l.c.199.1		4	156.119	odd	12
624.2.bv.e.49.2		4	13.7	odd	12
624.2.bv.e.433.1		4	13.2	odd	12
1014.2.a.i.1.2		2	52.51	odd	2
1014.2.a.k.1.1		2	4.3	odd	2
1014.2.b.e.337.2		4	52.31	even	4
1014.2.b.e.337.3		4	52.47	even	4
1014.2.e.g.529.1		4	52.3	odd	6
1014.2.e.g.991.1		4	52.35	odd	6
1014.2.e.i.529.2		4	52.23	odd	6
1014.2.e.i.991.2		4	52.43	odd	6
1014.2.i.a.361.1		4	52.19	even	12
1014.2.i.a.823.1		4	52.11	even	12
1872.2.by.h.433.2		4	39.2	even	12
1872.2.by.h.1297.1		4	39.20	even	12
1950.2.y.b.49.2		4	260.163	odd	12
1950.2.y.b.199.2		4	260.67	odd	12
1950.2.y.g.49.1		4	260.7	odd	12
1950.2.y.g.199.1		4	260.223	odd	12
1950.2.bc.d.751.1		4	260.59	even	12
1950.2.bc.d.901.1		4	260.119	even	12
3042.2.a.p.1.2		2	12.11	even	2
3042.2.a.y.1.1		2	156.155	even	2
3042.2.b.i.1351.2		4	156.47	odd	4
3042.2.b.i.1351.3		4	156.83	odd	4
8112.2.a.bj.1.2		2	13.12	even	2
8112.2.a.bp.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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{9} -4.73205 q^{11} -0.267949 q^{15} -2.26795 q^{17} -1.26795 q^{19} +0.732051 q^{21} +6.19615 q^{23} -4.92820 q^{25} -1.00000 q^{27} +2.46410 q^{29} +5.46410 q^{31} +4.73205 q^{33} -0.196152 q^{35} +10.4641 q^{37} +11.3923 q^{41} -7.66025 q^{43} +0.267949 q^{45} +8.19615 q^{47} -6.46410 q^{49} +2.26795 q^{51} +0.464102 q^{53} -1.26795 q^{55} +1.26795 q^{57} -8.00000 q^{59} +1.19615 q^{61} -0.732051 q^{63} +11.1244 q^{67} -6.19615 q^{69} -1.26795 q^{71} -9.73205 q^{73} +4.92820 q^{75} +3.46410 q^{77} +9.46410 q^{79} +1.00000 q^{81} -10.1962 q^{83} -0.607695 q^{85} -2.46410 q^{87} -2.53590 q^{89} -5.46410 q^{93} -0.339746 q^{95} -6.00000 q^{97} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 8 q^{17} - 6 q^{19} - 2 q^{21} + 2 q^{23} + 4 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 6 q^{33} + 10 q^{35} + 14 q^{37} + 2 q^{41} + 2 q^{43} + 4 q^{45} + 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} - 6 q^{55} + 6 q^{57} - 16 q^{59} - 8 q^{61} + 2 q^{63} - 2 q^{67} - 2 q^{69} - 6 q^{71} - 16 q^{73} - 4 q^{75} + 12 q^{79} + 2 q^{81} - 10 q^{83} - 22 q^{85} + 2 q^{87} - 12 q^{89} - 4 q^{93} - 18 q^{95} - 12 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 8 * q^17 - 6 * q^19 - 2 * q^21 + 2 * q^23 + 4 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 6 * q^33 + 10 * q^35 + 14 * q^37 + 2 * q^41 + 2 * q^43 + 4 * q^45 + 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 - 6 * q^55 + 6 * q^57 - 16 * q^59 - 8 * q^61 + 2 * q^63 - 2 * q^67 - 2 * q^69 - 6 * q^71 - 16 * q^73 - 4 * q^75 + 12 * q^79 + 2 * q^81 - 10 * q^83 - 22 * q^85 + 2 * q^87 - 12 * q^89 - 4 * q^93 - 18 * q^95 - 12 * q^97 - 6 * q^99

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