LMFDB - Embedded newform 8112.2.a.by.1.3

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +5.65279 q^{11} -0.246980 q^{15} -3.80194 q^{17} +5.58211 q^{19} -1.75302 q^{21} -8.34481 q^{23} -4.93900 q^{25} -1.00000 q^{27} -5.93900 q^{29} +5.26875 q^{31} -5.65279 q^{33} +0.432960 q^{35} -3.19806 q^{37} -0.445042 q^{41} -1.71379 q^{43} +0.246980 q^{45} -6.73556 q^{47} -3.92692 q^{49} +3.80194 q^{51} -1.06100 q^{53} +1.39612 q^{55} -5.58211 q^{57} -13.7017 q^{59} -8.51573 q^{61} +1.75302 q^{63} +5.96077 q^{67} +8.34481 q^{69} -5.71917 q^{71} -7.35690 q^{73} +4.93900 q^{75} +9.90946 q^{77} -4.45473 q^{79} +1.00000 q^{81} -10.1860 q^{83} -0.939001 q^{85} +5.93900 q^{87} -0.137063 q^{89} -5.26875 q^{93} +1.37867 q^{95} +13.6896 q^{97} +5.65279 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - 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
$5$	0.246980	0.110453	0.0552263	−	0.998474i	$-0.482412\pi$
$5$	0.246980	0.110453	0.0552263	+	0.998474i	$0.482412\pi$
$6$	0	0
$7$	1.75302	0.662579	0.331290	−	0.943529i	$-0.392516\pi$
$7$	1.75302	0.662579	0.331290	+	0.943529i	$0.392516\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.65279	1.70438	0.852191	−	0.523232i	$-0.175274\pi$
$11$	5.65279	1.70438	0.852191	+	0.523232i	$0.175274\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.246980	−0.0637699
$16$	0	0
$17$	−3.80194	−0.922105	−0.461053	−	0.887373i	$-0.652528\pi$
$17$	−3.80194	−0.922105	−0.461053	+	0.887373i	$0.652528\pi$
$18$	0	0
$19$	5.58211	1.28062	0.640311	−	0.768115i	$-0.278805\pi$
$19$	5.58211	1.28062	0.640311	+	0.768115i	$0.278805\pi$
$20$	0	0
$21$	−1.75302	−0.382540
$22$	0	0
$23$	−8.34481	−1.74001	−0.870007	−	0.493039i	$-0.835886\pi$
$23$	−8.34481	−1.74001	−0.870007	+	0.493039i	$0.835886\pi$
$24$	0	0
$25$	−4.93900	−0.987800
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.93900	−1.10284	−0.551422	−	0.834226i	$-0.685915\pi$
$29$	−5.93900	−1.10284	−0.551422	+	0.834226i	$0.685915\pi$
$30$	0	0
$31$	5.26875	0.946295	0.473148	−	0.880983i	$-0.343118\pi$
$31$	5.26875	0.946295	0.473148	+	0.880983i	$0.343118\pi$
$32$	0	0
$33$	−5.65279	−0.984025
$34$	0	0
$35$	0.432960	0.0731836
$36$	0	0
$37$	−3.19806	−0.525758	−0.262879	−	0.964829i	$-0.584672\pi$
$37$	−3.19806	−0.525758	−0.262879	+	0.964829i	$0.584672\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.445042	−0.0695039	−0.0347519	−	0.999396i	$-0.511064\pi$
$41$	−0.445042	−0.0695039	−0.0347519	+	0.999396i	$0.511064\pi$
$42$	0	0
$43$	−1.71379	−0.261351	−0.130675	−	0.991425i	$-0.541715\pi$
$43$	−1.71379	−0.261351	−0.130675	+	0.991425i	$0.541715\pi$
$44$	0	0
$45$	0.246980	0.0368175
$46$	0	0
$47$	−6.73556	−0.982483	−0.491241	−	0.871024i	$-0.663457\pi$
$47$	−6.73556	−0.982483	−0.491241	+	0.871024i	$0.663457\pi$
$48$	0	0
$49$	−3.92692	−0.560988
$50$	0	0
$51$	3.80194	0.532378
$52$	0	0
$53$	−1.06100	−0.145739	−0.0728697	−	0.997341i	$-0.523216\pi$
$53$	−1.06100	−0.145739	−0.0728697	+	0.997341i	$0.523216\pi$
$54$	0	0
$55$	1.39612	0.188253
$56$	0	0
$57$	−5.58211	−0.739368
$58$	0	0
$59$	−13.7017	−1.78381	−0.891905	−	0.452222i	$-0.850631\pi$
$59$	−13.7017	−1.78381	−0.891905	+	0.452222i	$0.850631\pi$
$60$	0	0
$61$	−8.51573	−1.09033	−0.545164	−	0.838330i	$-0.683533\pi$
$61$	−8.51573	−1.09033	−0.545164	+	0.838330i	$0.683533\pi$
$62$	0	0
$63$	1.75302	0.220860
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.96077	0.728224	0.364112	−	0.931355i	$-0.381373\pi$
$67$	5.96077	0.728224	0.364112	+	0.931355i	$0.381373\pi$
$68$	0	0
$69$	8.34481	1.00460
$70$	0	0
$71$	−5.71917	−0.678740	−0.339370	−	0.940653i	$-0.610214\pi$
$71$	−5.71917	−0.678740	−0.339370	+	0.940653i	$0.610214\pi$
$72$	0	0
$73$	−7.35690	−0.861060	−0.430530	−	0.902576i	$-0.641673\pi$
$73$	−7.35690	−0.861060	−0.430530	+	0.902576i	$0.641673\pi$
$74$	0	0
$75$	4.93900	0.570307
$76$	0	0
$77$	9.90946	1.12929
$78$	0	0
$79$	−4.45473	−0.501196	−0.250598	−	0.968091i	$-0.580627\pi$
$79$	−4.45473	−0.501196	−0.250598	+	0.968091i	$0.580627\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.1860	−1.11806	−0.559028	−	0.829149i	$-0.688826\pi$
$83$	−10.1860	−1.11806	−0.559028	+	0.829149i	$0.688826\pi$
$84$	0	0
$85$	−0.939001	−0.101849
$86$	0	0
$87$	5.93900	0.636728
$88$	0	0
$89$	−0.137063	−0.0145287	−0.00726434	−	0.999974i	$-0.502312\pi$
$89$	−0.137063	−0.0145287	−0.00726434	+	0.999974i	$0.502312\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.26875	−0.546344
$94$	0	0
$95$	1.37867	0.141448
$96$	0	0
$97$	13.6896	1.38997	0.694986	−	0.719024i	$-0.255411\pi$
$97$	13.6896	1.38997	0.694986	+	0.719024i	$0.255411\pi$
$98$	0	0
$99$	5.65279	0.568127
$100$	0	0
$101$	−5.41119	−0.538434	−0.269217	−	0.963080i	$-0.586765\pi$
$101$	−5.41119	−0.538434	−0.269217	+	0.963080i	$0.586765\pi$
$102$	0	0
$103$	−13.7560	−1.35542	−0.677710	−	0.735330i	$-0.737027\pi$
$103$	−13.7560	−1.35542	−0.677710	+	0.735330i	$0.737027\pi$
$104$	0	0
$105$	−0.432960	−0.0422526
$106$	0	0
$107$	12.8170	1.23907	0.619533	−	0.784970i	$-0.287322\pi$
$107$	12.8170	1.23907	0.619533	+	0.784970i	$0.287322\pi$
$108$	0	0
$109$	−12.1468	−1.16345	−0.581724	−	0.813386i	$-0.697622\pi$
$109$	−12.1468	−1.16345	−0.581724	+	0.813386i	$0.697622\pi$
$110$	0	0
$111$	3.19806	0.303547
$112$	0	0
$113$	−1.63773	−0.154064	−0.0770322	−	0.997029i	$-0.524544\pi$
$113$	−1.63773	−0.154064	−0.0770322	+	0.997029i	$0.524544\pi$
$114$	0	0
$115$	−2.06100	−0.192189
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.66487	−0.610968
$120$	0	0
$121$	20.9541	1.90492
$122$	0	0
$123$	0.445042	0.0401281
$124$	0	0
$125$	−2.45473	−0.219558
$126$	0	0
$127$	−10.7995	−0.958305	−0.479152	−	0.877732i	$-0.659056\pi$
$127$	−10.7995	−0.958305	−0.479152	+	0.877732i	$0.659056\pi$
$128$	0	0
$129$	1.71379	0.150891
$130$	0	0
$131$	−0.907542	−0.0792923	−0.0396462	−	0.999214i	$-0.512623\pi$
$131$	−0.907542	−0.0792923	−0.0396462	+	0.999214i	$0.512623\pi$
$132$	0	0
$133$	9.78554	0.848514
$134$	0	0
$135$	−0.246980	−0.0212566
$136$	0	0
$137$	9.54825	0.815762	0.407881	−	0.913035i	$-0.366268\pi$
$137$	9.54825	0.815762	0.407881	+	0.913035i	$0.366268\pi$
$138$	0	0
$139$	4.09246	0.347118	0.173559	−	0.984823i	$-0.444473\pi$
$139$	4.09246	0.347118	0.173559	+	0.984823i	$0.444473\pi$
$140$	0	0
$141$	6.73556	0.567237
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.46681	−0.121812
$146$	0	0
$147$	3.92692	0.323887
$148$	0	0
$149$	15.3884	1.26066	0.630332	−	0.776326i	$-0.282919\pi$
$149$	15.3884	1.26066	0.630332	+	0.776326i	$0.282919\pi$
$150$	0	0
$151$	−3.67456	−0.299032	−0.149516	−	0.988759i	$-0.547772\pi$
$151$	−3.67456	−0.299032	−0.149516	+	0.988759i	$0.547772\pi$
$152$	0	0
$153$	−3.80194	−0.307368
$154$	0	0
$155$	1.30127	0.104521
$156$	0	0
$157$	−4.87800	−0.389307	−0.194653	−	0.980872i	$-0.562358\pi$
$157$	−4.87800	−0.389307	−0.194653	+	0.980872i	$0.562358\pi$
$158$	0	0
$159$	1.06100	0.0841427
$160$	0	0
$161$	−14.6286	−1.15290
$162$	0	0
$163$	8.63102	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$163$	8.63102	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$164$	0	0
$165$	−1.39612	−0.108688
$166$	0	0
$167$	9.46980	0.732795	0.366397	−	0.930458i	$-0.380591\pi$
$167$	9.46980	0.732795	0.366397	+	0.930458i	$0.380591\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.58211	0.426874
$172$	0	0
$173$	4.77479	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$173$	4.77479	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$174$	0	0
$175$	−8.65817	−0.654496
$176$	0	0
$177$	13.7017	1.02988
$178$	0	0
$179$	−3.43535	−0.256770	−0.128385	−	0.991724i	$-0.540979\pi$
$179$	−3.43535	−0.256770	−0.128385	+	0.991724i	$0.540979\pi$
$180$	0	0
$181$	−13.4862	−1.00242	−0.501210	−	0.865326i	$-0.667112\pi$
$181$	−13.4862	−1.00242	−0.501210	+	0.865326i	$0.667112\pi$
$182$	0	0
$183$	8.51573	0.629501
$184$	0	0
$185$	−0.789856	−0.0580714
$186$	0	0
$187$	−21.4916	−1.57162
$188$	0	0
$189$	−1.75302	−0.127513
$190$	0	0
$191$	1.30127	0.0941569	0.0470784	−	0.998891i	$-0.485009\pi$
$191$	1.30127	0.0941569	0.0470784	+	0.998891i	$0.485009\pi$
$192$	0	0
$193$	9.19567	0.661919	0.330959	−	0.943645i	$-0.392628\pi$
$193$	9.19567	0.661919	0.330959	+	0.943645i	$0.392628\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.11231	−0.292990	−0.146495	−	0.989211i	$-0.546799\pi$
$197$	−4.11231	−0.292990	−0.146495	+	0.989211i	$0.546799\pi$
$198$	0	0
$199$	24.7724	1.75607	0.878034	−	0.478598i	$-0.158855\pi$
$199$	24.7724	1.75607	0.878034	+	0.478598i	$0.158855\pi$
$200$	0	0
$201$	−5.96077	−0.420440
$202$	0	0
$203$	−10.4112	−0.730722
$204$	0	0
$205$	−0.109916	−0.00767688
$206$	0	0
$207$	−8.34481	−0.580005
$208$	0	0
$209$	31.5545	2.18267
$210$	0	0
$211$	5.93900	0.408858	0.204429	−	0.978881i	$-0.434466\pi$
$211$	5.93900	0.408858	0.204429	+	0.978881i	$0.434466\pi$
$212$	0	0
$213$	5.71917	0.391871
$214$	0	0
$215$	−0.423272	−0.0288669
$216$	0	0
$217$	9.23623	0.626996
$218$	0	0
$219$	7.35690	0.497133
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.2010	0.950972	0.475486	−	0.879723i	$-0.342272\pi$
$223$	14.2010	0.950972	0.475486	+	0.879723i	$0.342272\pi$
$224$	0	0
$225$	−4.93900	−0.329267
$226$	0	0
$227$	16.0073	1.06244	0.531221	−	0.847233i	$-0.321733\pi$
$227$	16.0073	1.06244	0.531221	+	0.847233i	$0.321733\pi$
$228$	0	0
$229$	−1.84117	−0.121668	−0.0608339	−	0.998148i	$-0.519376\pi$
$229$	−1.84117	−0.121668	−0.0608339	+	0.998148i	$0.519376\pi$
$230$	0	0
$231$	−9.90946	−0.651995
$232$	0	0
$233$	−23.4252	−1.53464	−0.767318	−	0.641267i	$-0.778409\pi$
$233$	−23.4252	−1.53464	−0.767318	+	0.641267i	$0.778409\pi$
$234$	0	0
$235$	−1.66355	−0.108518
$236$	0	0
$237$	4.45473	0.289366
$238$	0	0
$239$	14.6015	0.944491	0.472246	−	0.881467i	$-0.343444\pi$
$239$	14.6015	0.944491	0.472246	+	0.881467i	$0.343444\pi$
$240$	0	0
$241$	−8.63102	−0.555973	−0.277987	−	0.960585i	$-0.589667\pi$
$241$	−8.63102	−0.555973	−0.277987	+	0.960585i	$0.589667\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.969869	−0.0619627
$246$	0	0
$247$	0	0
$248$	0	0
$249$	10.1860	0.645510
$250$	0	0
$251$	3.80194	0.239976	0.119988	−	0.992775i	$-0.461714\pi$
$251$	3.80194	0.239976	0.119988	+	0.992775i	$0.461714\pi$
$252$	0	0
$253$	−47.1715	−2.96565
$254$	0	0
$255$	0.939001	0.0588025
$256$	0	0
$257$	−20.5961	−1.28475	−0.642375	−	0.766391i	$-0.722051\pi$
$257$	−20.5961	−1.28475	−0.642375	+	0.766391i	$0.722051\pi$
$258$	0	0
$259$	−5.60627	−0.348357
$260$	0	0
$261$	−5.93900	−0.367615
$262$	0	0
$263$	−0.332733	−0.0205172	−0.0102586	−	0.999947i	$-0.503265\pi$
$263$	−0.332733	−0.0205172	−0.0102586	+	0.999947i	$0.503265\pi$
$264$	0	0
$265$	−0.262045	−0.0160973
$266$	0	0
$267$	0.137063	0.00838814
$268$	0	0
$269$	27.3032	1.66471	0.832353	−	0.554247i	$-0.186994\pi$
$269$	27.3032	1.66471	0.832353	+	0.554247i	$0.186994\pi$
$270$	0	0
$271$	27.9855	1.70000	0.850000	−	0.526783i	$-0.176602\pi$
$271$	27.9855	1.70000	0.850000	+	0.526783i	$0.176602\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−27.9191	−1.68359
$276$	0	0
$277$	−2.10321	−0.126370	−0.0631849	−	0.998002i	$-0.520126\pi$
$277$	−2.10321	−0.126370	−0.0631849	+	0.998002i	$0.520126\pi$
$278$	0	0
$279$	5.26875	0.315432
$280$	0	0
$281$	−27.2349	−1.62470	−0.812349	−	0.583172i	$-0.801811\pi$
$281$	−27.2349	−1.62470	−0.812349	+	0.583172i	$0.801811\pi$
$282$	0	0
$283$	−5.28382	−0.314090	−0.157045	−	0.987591i	$-0.550197\pi$
$283$	−5.28382	−0.314090	−0.157045	+	0.987591i	$0.550197\pi$
$284$	0	0
$285$	−1.37867	−0.0816651
$286$	0	0
$287$	−0.780167	−0.0460518
$288$	0	0
$289$	−2.54527	−0.149722
$290$	0	0
$291$	−13.6896	−0.802500
$292$	0	0
$293$	−32.6625	−1.90816	−0.954081	−	0.299548i	$-0.903164\pi$
$293$	−32.6625	−1.90816	−0.954081	+	0.299548i	$0.903164\pi$
$294$	0	0
$295$	−3.38404	−0.197027
$296$	0	0
$297$	−5.65279	−0.328008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.00431	−0.173166
$302$	0	0
$303$	5.41119	0.310865
$304$	0	0
$305$	−2.10321	−0.120430
$306$	0	0
$307$	−20.7614	−1.18491	−0.592457	−	0.805602i	$-0.701842\pi$
$307$	−20.7614	−1.18491	−0.592457	+	0.805602i	$0.701842\pi$
$308$	0	0
$309$	13.7560	0.782552
$310$	0	0
$311$	11.3013	0.640836	0.320418	−	0.947276i	$-0.396177\pi$
$311$	11.3013	0.640836	0.320418	+	0.947276i	$0.396177\pi$
$312$	0	0
$313$	−4.27173	−0.241453	−0.120726	−	0.992686i	$-0.538522\pi$
$313$	−4.27173	−0.241453	−0.120726	+	0.992686i	$0.538522\pi$
$314$	0	0
$315$	0.432960	0.0243945
$316$	0	0
$317$	−15.4776	−0.869307	−0.434653	−	0.900598i	$-0.643129\pi$
$317$	−15.4776	−0.869307	−0.434653	+	0.900598i	$0.643129\pi$
$318$	0	0
$319$	−33.5719	−1.87967
$320$	0	0
$321$	−12.8170	−0.715375
$322$	0	0
$323$	−21.2228	−1.18087
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.1468	0.671717
$328$	0	0
$329$	−11.8076	−0.650973
$330$	0	0
$331$	6.06829	0.333544	0.166772	−	0.985996i	$-0.446666\pi$
$331$	6.06829	0.333544	0.166772	+	0.985996i	$0.446666\pi$
$332$	0	0
$333$	−3.19806	−0.175253
$334$	0	0
$335$	1.47219	0.0804343
$336$	0	0
$337$	12.1239	0.660432	0.330216	−	0.943905i	$-0.392878\pi$
$337$	12.1239	0.660432	0.330216	+	0.943905i	$0.392878\pi$
$338$	0	0
$339$	1.63773	0.0889491
$340$	0	0
$341$	29.7832	1.61285
$342$	0	0
$343$	−19.1551	−1.03428
$344$	0	0
$345$	2.06100	0.110960
$346$	0	0
$347$	−23.1497	−1.24274	−0.621371	−	0.783516i	$-0.713424\pi$
$347$	−23.1497	−1.24274	−0.621371	+	0.783516i	$0.713424\pi$
$348$	0	0
$349$	22.1957	1.18811	0.594053	−	0.804426i	$-0.297527\pi$
$349$	22.1957	1.18811	0.594053	+	0.804426i	$0.297527\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.07069	0.269885	0.134943	−	0.990853i	$-0.456915\pi$
$353$	5.07069	0.269885	0.134943	+	0.990853i	$0.456915\pi$
$354$	0	0
$355$	−1.41252	−0.0749687
$356$	0	0
$357$	6.66487	0.352743
$358$	0	0
$359$	−16.6746	−0.880050	−0.440025	−	0.897986i	$-0.645030\pi$
$359$	−16.6746	−0.880050	−0.440025	+	0.897986i	$0.645030\pi$
$360$	0	0
$361$	12.1599	0.639995
$362$	0	0
$363$	−20.9541	−1.09980
$364$	0	0
$365$	−1.81700	−0.0951063
$366$	0	0
$367$	1.17928	0.0615577	0.0307789	−	0.999526i	$-0.490201\pi$
$367$	1.17928	0.0615577	0.0307789	+	0.999526i	$0.490201\pi$
$368$	0	0
$369$	−0.445042	−0.0231680
$370$	0	0
$371$	−1.85995	−0.0965639
$372$	0	0
$373$	−30.0925	−1.55813	−0.779064	−	0.626944i	$-0.784305\pi$
$373$	−30.0925	−1.55813	−0.779064	+	0.626944i	$0.784305\pi$
$374$	0	0
$375$	2.45473	0.126762
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−19.1631	−0.984345	−0.492172	−	0.870498i	$-0.663797\pi$
$379$	−19.1631	−0.984345	−0.492172	+	0.870498i	$0.663797\pi$
$380$	0	0
$381$	10.7995	0.553277
$382$	0	0
$383$	−15.3884	−0.786308	−0.393154	−	0.919473i	$-0.628616\pi$
$383$	−15.3884	−0.786308	−0.393154	+	0.919473i	$0.628616\pi$
$384$	0	0
$385$	2.44743	0.124733
$386$	0	0
$387$	−1.71379	−0.0871169
$388$	0	0
$389$	−24.0315	−1.21844	−0.609222	−	0.793000i	$-0.708518\pi$
$389$	−24.0315	−1.21844	−0.609222	+	0.793000i	$0.708518\pi$
$390$	0	0
$391$	31.7265	1.60448
$392$	0	0
$393$	0.907542	0.0457794
$394$	0	0
$395$	−1.10023	−0.0553585
$396$	0	0
$397$	−29.6015	−1.48566	−0.742828	−	0.669482i	$-0.766516\pi$
$397$	−29.6015	−1.48566	−0.742828	+	0.669482i	$0.766516\pi$
$398$	0	0
$399$	−9.78554	−0.489890
$400$	0	0
$401$	21.1032	1.05384	0.526922	−	0.849914i	$-0.323346\pi$
$401$	21.1032	1.05384	0.526922	+	0.849914i	$0.323346\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.246980	0.0122725
$406$	0	0
$407$	−18.0780	−0.896092
$408$	0	0
$409$	36.0224	1.78119	0.890596	−	0.454796i	$-0.150288\pi$
$409$	36.0224	1.78119	0.890596	+	0.454796i	$0.150288\pi$
$410$	0	0
$411$	−9.54825	−0.470981
$412$	0	0
$413$	−24.0194	−1.18192
$414$	0	0
$415$	−2.51573	−0.123492
$416$	0	0
$417$	−4.09246	−0.200409
$418$	0	0
$419$	−5.96854	−0.291582	−0.145791	−	0.989315i	$-0.546573\pi$
$419$	−5.96854	−0.291582	−0.145791	+	0.989315i	$0.546573\pi$
$420$	0	0
$421$	2.09544	0.102126	0.0510628	−	0.998695i	$-0.483739\pi$
$421$	2.09544	0.102126	0.0510628	+	0.998695i	$0.483739\pi$
$422$	0	0
$423$	−6.73556	−0.327494
$424$	0	0
$425$	18.7778	0.910856
$426$	0	0
$427$	−14.9282	−0.722429
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.88577	−0.139003	−0.0695014	−	0.997582i	$-0.522141\pi$
$431$	−2.88577	−0.139003	−0.0695014	+	0.997582i	$0.522141\pi$
$432$	0	0
$433$	−12.6485	−0.607847	−0.303924	−	0.952696i	$-0.598297\pi$
$433$	−12.6485	−0.607847	−0.303924	+	0.952696i	$0.598297\pi$
$434$	0	0
$435$	1.46681	0.0703283
$436$	0	0
$437$	−46.5816	−2.22830
$438$	0	0
$439$	10.9269	0.521513	0.260757	−	0.965405i	$-0.416028\pi$
$439$	10.9269	0.521513	0.260757	+	0.965405i	$0.416028\pi$
$440$	0	0
$441$	−3.92692	−0.186996
$442$	0	0
$443$	19.2403	0.914133	0.457067	−	0.889433i	$-0.348900\pi$
$443$	19.2403	0.914133	0.457067	+	0.889433i	$0.348900\pi$
$444$	0	0
$445$	−0.0338518	−0.00160473
$446$	0	0
$447$	−15.3884	−0.727844
$448$	0	0
$449$	28.8200	1.36010	0.680050	−	0.733166i	$-0.261958\pi$
$449$	28.8200	1.36010	0.680050	+	0.733166i	$0.261958\pi$
$450$	0	0
$451$	−2.51573	−0.118461
$452$	0	0
$453$	3.67456	0.172646
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0707	0.845311	0.422656	−	0.906290i	$-0.361098\pi$
$457$	18.0707	0.845311	0.422656	+	0.906290i	$0.361098\pi$
$458$	0	0
$459$	3.80194	0.177459
$460$	0	0
$461$	7.56763	0.352460	0.176230	−	0.984349i	$-0.443610\pi$
$461$	7.56763	0.352460	0.176230	+	0.984349i	$0.443610\pi$
$462$	0	0
$463$	35.3551	1.64309	0.821545	−	0.570143i	$-0.193112\pi$
$463$	35.3551	1.64309	0.821545	+	0.570143i	$0.193112\pi$
$464$	0	0
$465$	−1.30127	−0.0603451
$466$	0	0
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	10.4494	0.482506
$470$	0	0
$471$	4.87800	0.224766
$472$	0	0
$473$	−9.68771	−0.445441
$474$	0	0
$475$	−27.5700	−1.26500
$476$	0	0
$477$	−1.06100	−0.0485798
$478$	0	0
$479$	−25.5265	−1.16633	−0.583167	−	0.812352i	$-0.698187\pi$
$479$	−25.5265	−1.16633	−0.583167	+	0.812352i	$0.698187\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.6286	0.665626
$484$	0	0
$485$	3.38106	0.153526
$486$	0	0
$487$	16.0073	0.725360	0.362680	−	0.931914i	$-0.381862\pi$
$487$	16.0073	0.725360	0.362680	+	0.931914i	$0.381862\pi$
$488$	0	0
$489$	−8.63102	−0.390308
$490$	0	0
$491$	−20.7385	−0.935917	−0.467959	−	0.883750i	$-0.655010\pi$
$491$	−20.7385	−0.935917	−0.467959	+	0.883750i	$0.655010\pi$
$492$	0	0
$493$	22.5797	1.01694
$494$	0	0
$495$	1.39612	0.0627511
$496$	0	0
$497$	−10.0258	−0.449719
$498$	0	0
$499$	−8.06770	−0.361160	−0.180580	−	0.983560i	$-0.557797\pi$
$499$	−8.06770	−0.361160	−0.180580	+	0.983560i	$0.557797\pi$
$500$	0	0
$501$	−9.46980	−0.423079
$502$	0	0
$503$	30.2422	1.34843	0.674216	−	0.738534i	$-0.264482\pi$
$503$	30.2422	1.34843	0.674216	+	0.738534i	$0.264482\pi$
$504$	0	0
$505$	−1.33645	−0.0594714
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.0495	−0.711382	−0.355691	−	0.934604i	$-0.615754\pi$
$509$	−16.0495	−0.711382	−0.355691	+	0.934604i	$0.615754\pi$
$510$	0	0
$511$	−12.8968	−0.570520
$512$	0	0
$513$	−5.58211	−0.246456
$514$	0	0
$515$	−3.39745	−0.149710
$516$	0	0
$517$	−38.0747	−1.67452
$518$	0	0
$519$	−4.77479	−0.209590
$520$	0	0
$521$	−2.69309	−0.117986	−0.0589931	−	0.998258i	$-0.518789\pi$
$521$	−2.69309	−0.117986	−0.0589931	+	0.998258i	$0.518789\pi$
$522$	0	0
$523$	−35.3957	−1.54774	−0.773872	−	0.633342i	$-0.781683\pi$
$523$	−35.3957	−1.54774	−0.773872	+	0.633342i	$0.781683\pi$
$524$	0	0
$525$	8.65817	0.377874
$526$	0	0
$527$	−20.0315	−0.872584
$528$	0	0
$529$	46.6359	2.02765
$530$	0	0
$531$	−13.7017	−0.594604
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3.16554	0.136858
$536$	0	0
$537$	3.43535	0.148246
$538$	0	0
$539$	−22.1981	−0.956138
$540$	0	0
$541$	34.7338	1.49332	0.746660	−	0.665205i	$-0.231656\pi$
$541$	34.7338	1.49332	0.746660	+	0.665205i	$0.231656\pi$
$542$	0	0
$543$	13.4862	0.578748
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	26.1183	1.11674	0.558368	−	0.829593i	$-0.311428\pi$
$547$	26.1183	1.11674	0.558368	+	0.829593i	$0.311428\pi$
$548$	0	0
$549$	−8.51573	−0.363442
$550$	0	0
$551$	−33.1521	−1.41233
$552$	0	0
$553$	−7.80923	−0.332082
$554$	0	0
$555$	0.789856	0.0335275
$556$	0	0
$557$	−24.7748	−1.04974	−0.524871	−	0.851182i	$-0.675886\pi$
$557$	−24.7748	−1.04974	−0.524871	+	0.851182i	$0.675886\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	21.4916	0.907375
$562$	0	0
$563$	−5.26098	−0.221724	−0.110862	−	0.993836i	$-0.535361\pi$
$563$	−5.26098	−0.221724	−0.110862	+	0.993836i	$0.535361\pi$
$564$	0	0
$565$	−0.404485	−0.0170168
$566$	0	0
$567$	1.75302	0.0736199
$568$	0	0
$569$	−33.7458	−1.41470	−0.707350	−	0.706864i	$-0.750109\pi$
$569$	−33.7458	−1.41470	−0.707350	+	0.706864i	$0.750109\pi$
$570$	0	0
$571$	−23.0887	−0.966234	−0.483117	−	0.875556i	$-0.660495\pi$
$571$	−23.0887	−0.966234	−0.483117	+	0.875556i	$0.660495\pi$
$572$	0	0
$573$	−1.30127	−0.0543615
$574$	0	0
$575$	41.2150	1.71879
$576$	0	0
$577$	−3.57002	−0.148622	−0.0743110	−	0.997235i	$-0.523676\pi$
$577$	−3.57002	−0.148622	−0.0743110	+	0.997235i	$0.523676\pi$
$578$	0	0
$579$	−9.19567	−0.382159
$580$	0	0
$581$	−17.8562	−0.740801
$582$	0	0
$583$	−5.99761	−0.248396
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.4625	−0.473108	−0.236554	−	0.971618i	$-0.576018\pi$
$587$	−11.4625	−0.473108	−0.236554	+	0.971618i	$0.576018\pi$
$588$	0	0
$589$	29.4107	1.21185
$590$	0	0
$591$	4.11231	0.169158
$592$	0	0
$593$	21.8538	0.897430	0.448715	−	0.893675i	$-0.351882\pi$
$593$	21.8538	0.897430	0.448715	+	0.893675i	$0.351882\pi$
$594$	0	0
$595$	−1.64609	−0.0674830
$596$	0	0
$597$	−24.7724	−1.01387
$598$	0	0
$599$	−27.0573	−1.10553	−0.552765	−	0.833337i	$-0.686427\pi$
$599$	−27.0573	−1.10553	−0.552765	+	0.833337i	$0.686427\pi$
$600$	0	0
$601$	10.8780	0.443723	0.221861	−	0.975078i	$-0.428787\pi$
$601$	10.8780	0.443723	0.221861	+	0.975078i	$0.428787\pi$
$602$	0	0
$603$	5.96077	0.242741
$604$	0	0
$605$	5.17523	0.210403
$606$	0	0
$607$	29.6359	1.20289	0.601443	−	0.798916i	$-0.294593\pi$
$607$	29.6359	1.20289	0.601443	+	0.798916i	$0.294593\pi$
$608$	0	0
$609$	10.4112	0.421883
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.2343	−0.413360	−0.206680	−	0.978409i	$-0.566266\pi$
$613$	−10.2343	−0.413360	−0.206680	+	0.978409i	$0.566266\pi$
$614$	0	0
$615$	0.109916	0.00443225
$616$	0	0
$617$	−26.2828	−1.05810	−0.529052	−	0.848590i	$-0.677452\pi$
$617$	−26.2828	−1.05810	−0.529052	+	0.848590i	$0.677452\pi$
$618$	0	0
$619$	29.0834	1.16896	0.584479	−	0.811408i	$-0.301299\pi$
$619$	29.0834	1.16896	0.584479	+	0.811408i	$0.301299\pi$
$620$	0	0
$621$	8.34481	0.334866
$622$	0	0
$623$	−0.240275	−0.00962641
$624$	0	0
$625$	24.0887	0.963549
$626$	0	0
$627$	−31.5545	−1.26016
$628$	0	0
$629$	12.1588	0.484804
$630$	0	0
$631$	25.4480	1.01307	0.506535	−	0.862219i	$-0.330926\pi$
$631$	25.4480	1.01307	0.506535	+	0.862219i	$0.330926\pi$
$632$	0	0
$633$	−5.93900	−0.236054
$634$	0	0
$635$	−2.66727	−0.105847
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.71917	−0.226247
$640$	0	0
$641$	−26.7409	−1.05620	−0.528102	−	0.849181i	$-0.677096\pi$
$641$	−26.7409	−1.05620	−0.528102	+	0.849181i	$0.677096\pi$
$642$	0	0
$643$	−32.9614	−1.29987	−0.649935	−	0.759990i	$-0.725204\pi$
$643$	−32.9614	−1.29987	−0.649935	+	0.759990i	$0.725204\pi$
$644$	0	0
$645$	0.423272	0.0166663
$646$	0	0
$647$	−34.4946	−1.35612	−0.678060	−	0.735006i	$-0.737179\pi$
$647$	−34.4946	−1.35612	−0.678060	+	0.735006i	$0.737179\pi$
$648$	0	0
$649$	−77.4529	−3.04029
$650$	0	0
$651$	−9.23623	−0.361996
$652$	0	0
$653$	36.1517	1.41472	0.707362	−	0.706852i	$-0.249885\pi$
$653$	36.1517	1.41472	0.707362	+	0.706852i	$0.249885\pi$
$654$	0	0
$655$	−0.224144	−0.00875805
$656$	0	0
$657$	−7.35690	−0.287020
$658$	0	0
$659$	6.81700	0.265553	0.132776	−	0.991146i	$-0.457611\pi$
$659$	6.81700	0.265553	0.132776	+	0.991146i	$0.457611\pi$
$660$	0	0
$661$	10.8944	0.423743	0.211871	−	0.977298i	$-0.432044\pi$
$661$	10.8944	0.423743	0.211871	+	0.977298i	$0.432044\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.41683	0.0937206
$666$	0	0
$667$	49.5599	1.91897
$668$	0	0
$669$	−14.2010	−0.549044
$670$	0	0
$671$	−48.1377	−1.85833
$672$	0	0
$673$	20.7385	0.799412	0.399706	−	0.916643i	$-0.369112\pi$
$673$	20.7385	0.799412	0.399706	+	0.916643i	$0.369112\pi$
$674$	0	0
$675$	4.93900	0.190102
$676$	0	0
$677$	−25.5786	−0.983067	−0.491534	−	0.870859i	$-0.663564\pi$
$677$	−25.5786	−0.983067	−0.491534	+	0.870859i	$0.663564\pi$
$678$	0	0
$679$	23.9982	0.920966
$680$	0	0
$681$	−16.0073	−0.613401
$682$	0	0
$683$	21.6310	0.827688	0.413844	−	0.910348i	$-0.364186\pi$
$683$	21.6310	0.827688	0.413844	+	0.910348i	$0.364186\pi$
$684$	0	0
$685$	2.35822	0.0901031
$686$	0	0
$687$	1.84117	0.0702449
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.62996	0.100048	0.0500242	−	0.998748i	$-0.484070\pi$
$691$	2.62996	0.100048	0.0500242	+	0.998748i	$0.484070\pi$
$692$	0	0
$693$	9.90946	0.376429
$694$	0	0
$695$	1.01075	0.0383401
$696$	0	0
$697$	1.69202	0.0640899
$698$	0	0
$699$	23.4252	0.886022
$700$	0	0
$701$	40.0925	1.51427	0.757136	−	0.653258i	$-0.226598\pi$
$701$	40.0925	1.51427	0.757136	+	0.653258i	$0.226598\pi$
$702$	0	0
$703$	−17.8519	−0.673298
$704$	0	0
$705$	1.66355	0.0626528
$706$	0	0
$707$	−9.48593	−0.356755
$708$	0	0
$709$	−23.2097	−0.871657	−0.435829	−	0.900030i	$-0.643545\pi$
$709$	−23.2097	−0.871657	−0.435829	+	0.900030i	$0.643545\pi$
$710$	0	0
$711$	−4.45473	−0.167065
$712$	0	0
$713$	−43.9667	−1.64657
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.6015	−0.545302
$718$	0	0
$719$	26.0146	0.970181	0.485090	−	0.874464i	$-0.338787\pi$
$719$	26.0146	0.970181	0.485090	+	0.874464i	$0.338787\pi$
$720$	0	0
$721$	−24.1146	−0.898073
$722$	0	0
$723$	8.63102	0.320991
$724$	0	0
$725$	29.3327	1.08939
$726$	0	0
$727$	16.5472	0.613701	0.306851	−	0.951758i	$-0.400725\pi$
$727$	16.5472	0.613701	0.306851	+	0.951758i	$0.400725\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.51573	0.240993
$732$	0	0
$733$	−18.8750	−0.697165	−0.348582	−	0.937278i	$-0.613337\pi$
$733$	−18.8750	−0.697165	−0.348582	+	0.937278i	$0.613337\pi$
$734$	0	0
$735$	0.969869	0.0357742
$736$	0	0
$737$	33.6950	1.24117
$738$	0	0
$739$	47.3239	1.74084	0.870419	−	0.492312i	$-0.163848\pi$
$739$	47.3239	1.74084	0.870419	+	0.492312i	$0.163848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.88769	−0.326058	−0.163029	−	0.986621i	$-0.552126\pi$
$743$	−8.88769	−0.326058	−0.163029	+	0.986621i	$0.552126\pi$
$744$	0	0
$745$	3.80061	0.139244
$746$	0	0
$747$	−10.1860	−0.372686
$748$	0	0
$749$	22.4685	0.820980
$750$	0	0
$751$	−0.710808	−0.0259377	−0.0129689	−	0.999916i	$-0.504128\pi$
$751$	−0.710808	−0.0259377	−0.0129689	+	0.999916i	$0.504128\pi$
$752$	0	0
$753$	−3.80194	−0.138550
$754$	0	0
$755$	−0.907542	−0.0330288
$756$	0	0
$757$	9.78554	0.355662	0.177831	−	0.984061i	$-0.443092\pi$
$757$	9.78554	0.355662	0.177831	+	0.984061i	$0.443092\pi$
$758$	0	0
$759$	47.1715	1.71222
$760$	0	0
$761$	18.8810	0.684435	0.342218	−	0.939621i	$-0.388822\pi$
$761$	18.8810	0.684435	0.342218	+	0.939621i	$0.388822\pi$
$762$	0	0
$763$	−21.2935	−0.770877
$764$	0	0
$765$	−0.939001	−0.0339497
$766$	0	0
$767$	0	0
$768$	0	0
$769$	12.4349	0.448413	0.224207	−	0.974542i	$-0.428021\pi$
$769$	12.4349	0.448413	0.224207	+	0.974542i	$0.428021\pi$
$770$	0	0
$771$	20.5961	0.741751
$772$	0	0
$773$	−45.6746	−1.64280	−0.821400	−	0.570353i	$-0.806807\pi$
$773$	−45.6746	−1.64280	−0.821400	+	0.570353i	$0.806807\pi$
$774$	0	0
$775$	−26.0224	−0.934751
$776$	0	0
$777$	5.60627	0.201124
$778$	0	0
$779$	−2.48427	−0.0890082
$780$	0	0
$781$	−32.3293	−1.15683
$782$	0	0
$783$	5.93900	0.212243
$784$	0	0
$785$	−1.20477	−0.0430000
$786$	0	0
$787$	−4.51871	−0.161075	−0.0805374	−	0.996752i	$-0.525664\pi$
$787$	−4.51871	−0.161075	−0.0805374	+	0.996752i	$0.525664\pi$
$788$	0	0
$789$	0.332733	0.0118456
$790$	0	0
$791$	−2.87097	−0.102080
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.262045	0.00929378
$796$	0	0
$797$	−28.7391	−1.01799	−0.508996	−	0.860769i	$-0.669983\pi$
$797$	−28.7391	−1.01799	−0.508996	+	0.860769i	$0.669983\pi$
$798$	0	0
$799$	25.6082	0.905953
$800$	0	0
$801$	−0.137063	−0.00484289
$802$	0	0
$803$	−41.5870	−1.46757
$804$	0	0
$805$	−3.61297	−0.127341
$806$	0	0
$807$	−27.3032	−0.961118
$808$	0	0
$809$	5.42891	0.190870	0.0954352	−	0.995436i	$-0.469576\pi$
$809$	5.42891	0.190870	0.0954352	+	0.995436i	$0.469576\pi$
$810$	0	0
$811$	0.629104	0.0220908	0.0110454	−	0.999939i	$-0.496484\pi$
$811$	0.629104	0.0220908	0.0110454	+	0.999939i	$0.496484\pi$
$812$	0	0
$813$	−27.9855	−0.981495
$814$	0	0
$815$	2.13169	0.0746697
$816$	0	0
$817$	−9.56657	−0.334692
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.2640	−1.26562	−0.632811	−	0.774307i	$-0.718099\pi$
$821$	−36.2640	−1.26562	−0.632811	+	0.774307i	$0.718099\pi$
$822$	0	0
$823$	−41.7396	−1.45495	−0.727476	−	0.686133i	$-0.759307\pi$
$823$	−41.7396	−1.45495	−0.727476	+	0.686133i	$0.759307\pi$
$824$	0	0
$825$	27.9191	0.972020
$826$	0	0
$827$	38.1997	1.32833	0.664167	−	0.747584i	$-0.268786\pi$
$827$	38.1997	1.32833	0.664167	+	0.747584i	$0.268786\pi$
$828$	0	0
$829$	15.9788	0.554967	0.277484	−	0.960730i	$-0.410500\pi$
$829$	15.9788	0.554967	0.277484	+	0.960730i	$0.410500\pi$
$830$	0	0
$831$	2.10321	0.0729596
$832$	0	0
$833$	14.9299	0.517290
$834$	0	0
$835$	2.33885	0.0809391
$836$	0	0
$837$	−5.26875	−0.182115
$838$	0	0
$839$	−4.63879	−0.160149	−0.0800744	−	0.996789i	$-0.525516\pi$
$839$	−4.63879	−0.160149	−0.0800744	+	0.996789i	$0.525516\pi$
$840$	0	0
$841$	6.27173	0.216267
$842$	0	0
$843$	27.2349	0.938020
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36.7329	1.26216
$848$	0	0
$849$	5.28382	0.181340
$850$	0	0
$851$	26.6872	0.914827
$852$	0	0
$853$	−18.3884	−0.629605	−0.314803	−	0.949157i	$-0.601938\pi$
$853$	−18.3884	−0.629605	−0.314803	+	0.949157i	$0.601938\pi$
$854$	0	0
$855$	1.37867	0.0471494
$856$	0	0
$857$	28.1849	0.962778	0.481389	−	0.876507i	$-0.340132\pi$
$857$	28.1849	0.962778	0.481389	+	0.876507i	$0.340132\pi$
$858$	0	0
$859$	−33.3957	−1.13944	−0.569722	−	0.821837i	$-0.692949\pi$
$859$	−33.3957	−1.13944	−0.569722	+	0.821837i	$0.692949\pi$
$860$	0	0
$861$	0.780167	0.0265880
$862$	0	0
$863$	−43.0640	−1.46592	−0.732958	−	0.680274i	$-0.761861\pi$
$863$	−43.0640	−1.46592	−0.732958	+	0.680274i	$0.761861\pi$
$864$	0	0
$865$	1.17928	0.0400966
$866$	0	0
$867$	2.54527	0.0864419
$868$	0	0
$869$	−25.1817	−0.854230
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.6896	0.463324
$874$	0	0
$875$	−4.30319	−0.145474
$876$	0	0
$877$	30.1702	1.01877	0.509387	−	0.860537i	$-0.329872\pi$
$877$	30.1702	1.01877	0.509387	+	0.860537i	$0.329872\pi$
$878$	0	0
$879$	32.6625	1.10168
$880$	0	0
$881$	−3.56273	−0.120031	−0.0600157	−	0.998197i	$-0.519115\pi$
$881$	−3.56273	−0.120031	−0.0600157	+	0.998197i	$0.519115\pi$
$882$	0	0
$883$	10.2088	0.343554	0.171777	−	0.985136i	$-0.445049\pi$
$883$	10.2088	0.343554	0.171777	+	0.985136i	$0.445049\pi$
$884$	0	0
$885$	3.38404	0.113753
$886$	0	0
$887$	9.33645	0.313487	0.156744	−	0.987639i	$-0.449900\pi$
$887$	9.33645	0.313487	0.156744	+	0.987639i	$0.449900\pi$
$888$	0	0
$889$	−18.9318	−0.634953
$890$	0	0
$891$	5.65279	0.189376
$892$	0	0
$893$	−37.5986	−1.25819
$894$	0	0
$895$	−0.848462	−0.0283610
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31.2911	−1.04362
$900$	0	0
$901$	4.03385	0.134387
$902$	0	0
$903$	3.00431	0.0999772
$904$	0	0
$905$	−3.33081	−0.110720
$906$	0	0
$907$	41.0804	1.36405	0.682026	−	0.731328i	$-0.261099\pi$
$907$	41.0804	1.36405	0.682026	+	0.731328i	$0.261099\pi$
$908$	0	0
$909$	−5.41119	−0.179478
$910$	0	0
$911$	18.9705	0.628519	0.314260	−	0.949337i	$-0.398244\pi$
$911$	18.9705	0.628519	0.314260	+	0.949337i	$0.398244\pi$
$912$	0	0
$913$	−57.5792	−1.90559
$914$	0	0
$915$	2.10321	0.0695300
$916$	0	0
$917$	−1.59094	−0.0525375
$918$	0	0
$919$	−29.0019	−0.956685	−0.478343	−	0.878173i	$-0.658762\pi$
$919$	−29.0019	−0.956685	−0.478343	+	0.878173i	$0.658762\pi$
$920$	0	0
$921$	20.7614	0.684111
$922$	0	0
$923$	0	0
$924$	0	0
$925$	15.7952	0.519344
$926$	0	0
$927$	−13.7560	−0.451806
$928$	0	0
$929$	50.7211	1.66410	0.832052	−	0.554697i	$-0.187166\pi$
$929$	50.7211	1.66410	0.832052	+	0.554697i	$0.187166\pi$
$930$	0	0
$931$	−21.9205	−0.718415
$932$	0	0
$933$	−11.3013	−0.369987
$934$	0	0
$935$	−5.30798	−0.173589
$936$	0	0
$937$	51.3051	1.67606	0.838032	−	0.545620i	$-0.183706\pi$
$937$	51.3051	1.67606	0.838032	+	0.545620i	$0.183706\pi$
$938$	0	0
$939$	4.27173	0.139403
$940$	0	0
$941$	34.7036	1.13131	0.565653	−	0.824643i	$-0.308624\pi$
$941$	34.7036	1.13131	0.565653	+	0.824643i	$0.308624\pi$
$942$	0	0
$943$	3.71379	0.120938
$944$	0	0
$945$	−0.432960	−0.0140842
$946$	0	0
$947$	−13.0127	−0.422855	−0.211428	−	0.977394i	$-0.567811\pi$
$947$	−13.0127	−0.422855	−0.211428	+	0.977394i	$0.567811\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	15.4776	0.501894
$952$	0	0
$953$	26.0151	0.842711	0.421355	−	0.906896i	$-0.361555\pi$
$953$	26.0151	0.842711	0.421355	+	0.906896i	$0.361555\pi$
$954$	0	0
$955$	0.321388	0.0103999
$956$	0	0
$957$	33.5719	1.08523
$958$	0	0
$959$	16.7383	0.540507
$960$	0	0
$961$	−3.24027	−0.104525
$962$	0	0
$963$	12.8170	0.413022
$964$	0	0
$965$	2.27114	0.0731107
$966$	0	0
$967$	36.6644	1.17905	0.589524	−	0.807751i	$-0.299315\pi$
$967$	36.6644	1.17905	0.589524	+	0.807751i	$0.299315\pi$
$968$	0	0
$969$	21.2228	0.681775
$970$	0	0
$971$	37.8465	1.21455	0.607277	−	0.794490i	$-0.292262\pi$
$971$	37.8465	1.21455	0.607277	+	0.794490i	$0.292262\pi$
$972$	0	0
$973$	7.17416	0.229993
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.8998	−0.924586	−0.462293	−	0.886727i	$-0.652973\pi$
$977$	−28.8998	−0.924586	−0.462293	+	0.886727i	$0.652973\pi$
$978$	0	0
$979$	−0.774791	−0.0247624
$980$	0	0
$981$	−12.1468	−0.387816
$982$	0	0
$983$	−19.3991	−0.618735	−0.309368	−	0.950942i	$-0.600117\pi$
$983$	−19.3991	−0.618735	−0.309368	+	0.950942i	$0.600117\pi$
$984$	0	0
$985$	−1.01566	−0.0323615
$986$	0	0
$987$	11.8076	0.375839
$988$	0	0
$989$	14.3013	0.454754
$990$	0	0
$991$	5.18300	0.164643	0.0823217	−	0.996606i	$-0.473767\pi$
$991$	5.18300	0.164643	0.0823217	+	0.996606i	$0.473767\pi$
$992$	0	0
$993$	−6.06829	−0.192572
$994$	0	0
$995$	6.11828	0.193962
$996$	0	0
$997$	−49.3642	−1.56338	−0.781690	−	0.623667i	$-0.785642\pi$
$997$	−49.3642	−1.56338	−0.781690	+	0.623667i	$0.785642\pi$
$998$	0	0
$999$	3.19806	0.101182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.by.1.3		3
4.3	odd	2		507.2.a.j.1.2	✓	3
12.11	even	2		1521.2.a.q.1.2		3
13.12	even	2		8112.2.a.cf.1.1		3
52.3	odd	6		507.2.e.k.22.2		6
52.7	even	12		507.2.j.h.361.3		12
52.11	even	12		507.2.j.h.316.4		12
52.15	even	12		507.2.j.h.316.3		12
52.19	even	12		507.2.j.h.361.4		12
52.23	odd	6		507.2.e.j.22.2		6
52.31	even	4		507.2.b.g.337.4		6
52.35	odd	6		507.2.e.k.484.2		6
52.43	odd	6		507.2.e.j.484.2		6
52.47	even	4		507.2.b.g.337.3		6
52.51	odd	2		507.2.a.k.1.2	yes	3
156.47	odd	4		1521.2.b.m.1351.4		6
156.83	odd	4		1521.2.b.m.1351.3		6
156.155	even	2		1521.2.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.2	✓	3	4.3	odd	2
507.2.a.k.1.2	yes	3	52.51	odd	2
507.2.b.g.337.3		6	52.47	even	4
507.2.b.g.337.4		6	52.31	even	4
507.2.e.j.22.2		6	52.23	odd	6
507.2.e.j.484.2		6	52.43	odd	6
507.2.e.k.22.2		6	52.3	odd	6
507.2.e.k.484.2		6	52.35	odd	6
507.2.j.h.316.3		12	52.15	even	12
507.2.j.h.316.4		12	52.11	even	12
507.2.j.h.361.3		12	52.7	even	12
507.2.j.h.361.4		12	52.19	even	12
1521.2.a.p.1.2		3	156.155	even	2
1521.2.a.q.1.2		3	12.11	even	2
1521.2.b.m.1351.3		6	156.83	odd	4
1521.2.b.m.1351.4		6	156.47	odd	4
8112.2.a.by.1.3		3	1.1	even	1	trivial
8112.2.a.cf.1.1		3	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.246980 q^{5} +1.75302 q^{7} +1.00000 q^{9} +5.65279 q^{11} -0.246980 q^{15} -3.80194 q^{17} +5.58211 q^{19} -1.75302 q^{21} -8.34481 q^{23} -4.93900 q^{25} -1.00000 q^{27} -5.93900 q^{29} +5.26875 q^{31} -5.65279 q^{33} +0.432960 q^{35} -3.19806 q^{37} -0.445042 q^{41} -1.71379 q^{43} +0.246980 q^{45} -6.73556 q^{47} -3.92692 q^{49} +3.80194 q^{51} -1.06100 q^{53} +1.39612 q^{55} -5.58211 q^{57} -13.7017 q^{59} -8.51573 q^{61} +1.75302 q^{63} +5.96077 q^{67} +8.34481 q^{69} -5.71917 q^{71} -7.35690 q^{73} +4.93900 q^{75} +9.90946 q^{77} -4.45473 q^{79} +1.00000 q^{81} -10.1860 q^{83} -0.939001 q^{85} +5.93900 q^{87} -0.137063 q^{89} -5.26875 q^{93} +1.37867 q^{95} +13.6896 q^{97} +5.65279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - q^99

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