LMFDB - Embedded newform 1521.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
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:	$\Z[a_1, a_2]$
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			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{2} -1.80194 q^{4} -0.246980 q^{5} -1.75302 q^{7} -1.69202 q^{8} +O(q^{10})$	$q+0.445042 q^{2} -1.80194 q^{4} -0.246980 q^{5} -1.75302 q^{7} -1.69202 q^{8} -0.109916 q^{10} +5.65279 q^{11} -0.780167 q^{14} +2.85086 q^{16} +3.80194 q^{17} -5.58211 q^{19} +0.445042 q^{20} +2.51573 q^{22} -8.34481 q^{23} -4.93900 q^{25} +3.15883 q^{28} +5.93900 q^{29} -5.26875 q^{31} +4.65279 q^{32} +1.69202 q^{34} +0.432960 q^{35} -3.19806 q^{37} -2.48427 q^{38} +0.417895 q^{40} +0.445042 q^{41} +1.71379 q^{43} -10.1860 q^{44} -3.71379 q^{46} -6.73556 q^{47} -3.92692 q^{49} -2.19806 q^{50} +1.06100 q^{53} -1.39612 q^{55} +2.96615 q^{56} +2.64310 q^{58} -13.7017 q^{59} -8.51573 q^{61} -2.34481 q^{62} -3.63102 q^{64} -5.96077 q^{67} -6.85086 q^{68} +0.192685 q^{70} -5.71917 q^{71} -7.35690 q^{73} -1.42327 q^{74} +10.0586 q^{76} -9.90946 q^{77} +4.45473 q^{79} -0.704103 q^{80} +0.198062 q^{82} -10.1860 q^{83} -0.939001 q^{85} +0.762709 q^{86} -9.56465 q^{88} +0.137063 q^{89} +15.0368 q^{92} -2.99761 q^{94} +1.37867 q^{95} +13.6896 q^{97} -1.74764 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7}+O(q^{10}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7} - q^{10} - q^{11} - q^{14} - 5 q^{16} + 7 q^{17} - 11 q^{19} + q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} + q^{28} + 8 q^{29} - 8 q^{31} - 4 q^{32} - 18 q^{35} - 14 q^{37} - 20 q^{38} + 7 q^{40} + q^{41} - 3 q^{43} - 16 q^{44} - 3 q^{46} - 9 q^{47} + 17 q^{49} - 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} + 12 q^{58} - 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} - 5 q^{67} - 7 q^{68} + 8 q^{70} - 6 q^{71} - 18 q^{73} - 7 q^{74} - q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 5 q^{82} - 16 q^{83} + 7 q^{85} - 15 q^{86} - 7 q^{88} - 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} - 5 q^{97} - 13 q^{98}+O(q^{100}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7 - q^10 - q^11 - q^14 - 5 * q^16 + 7 * q^17 - 11 * q^19 + q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 + q^28 + 8 * q^29 - 8 * q^31 - 4 * q^32 - 18 * q^35 - 14 * q^37 - 20 * q^38 + 7 * q^40 + q^41 - 3 * q^43 - 16 * q^44 - 3 * q^46 - 9 * q^47 + 17 * q^49 - 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 + 12 * q^58 - 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 - 5 * q^67 - 7 * q^68 + 8 * q^70 - 6 * q^71 - 18 * q^73 - 7 * q^74 - q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 5 * q^82 - 16 * q^83 + 7 * q^85 - 15 * q^86 - 7 * q^88 - 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 - 5 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.445042	0.314692	0.157346	−	0.987544i	$-0.449706\pi$
$2$	0.445042	0.314692	0.157346	+	0.987544i	$0.449706\pi$
$3$	0	0
$3$	0	0
$4$	−1.80194	−0.900969
$5$	−0.246980	−0.110453	−0.0552263	−	0.998474i	$-0.517588\pi$
$5$	−0.246980	−0.110453	−0.0552263	+	0.998474i	$0.517588\pi$
$6$	0	0
$7$	−1.75302	−0.662579	−0.331290	−	0.943529i	$-0.607484\pi$
$7$	−1.75302	−0.662579	−0.331290	+	0.943529i	$0.607484\pi$
$8$	−1.69202	−0.598220
$9$	0	0
$10$	−0.109916	−0.0347586
$11$	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
$14$	−0.780167	−0.208509
$15$	0	0
$16$	2.85086	0.712714
$17$	3.80194	0.922105	0.461053	−	0.887373i	$-0.347472\pi$
$17$	3.80194	0.922105	0.461053	+	0.887373i	$0.347472\pi$
$18$	0	0
$19$	−5.58211	−1.28062	−0.640311	−	0.768115i	$-0.721195\pi$
$19$	−5.58211	−1.28062	−0.640311	+	0.768115i	$0.721195\pi$
$20$	0.445042	0.0995144
$21$	0	0
$22$	2.51573	0.536355
$23$	−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
$27$	0	0
$28$	3.15883	0.596963
$29$	5.93900	1.10284	0.551422	−	0.834226i	$-0.314085\pi$
$29$	5.93900	1.10284	0.551422	+	0.834226i	$0.314085\pi$
$30$	0	0
$31$	−5.26875	−0.946295	−0.473148	−	0.880983i	$-0.656882\pi$
$31$	−5.26875	−0.946295	−0.473148	+	0.880983i	$0.656882\pi$
$32$	4.65279	0.822505
$33$	0	0
$34$	1.69202	0.290179
$35$	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	$0.584672\pi$
$38$	−2.48427	−0.403002
$39$	0	0
$40$	0.417895	0.0660750
$41$	0.445042	0.0695039	0.0347519	−	0.999396i	$-0.488936\pi$
$41$	0.445042	0.0695039	0.0347519	+	0.999396i	$0.488936\pi$
$42$	0	0
$43$	1.71379	0.261351	0.130675	−	0.991425i	$-0.458285\pi$
$43$	1.71379	0.261351	0.130675	+	0.991425i	$0.458285\pi$
$44$	−10.1860	−1.53559
$45$	0	0
$46$	−3.71379	−0.547569
$47$	−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
$50$	−2.19806	−0.310853
$51$	0	0
$52$	0	0
$53$	1.06100	0.145739	0.0728697	−	0.997341i	$-0.476784\pi$
$53$	1.06100	0.145739	0.0728697	+	0.997341i	$0.476784\pi$
$54$	0	0
$55$	−1.39612	−0.188253
$56$	2.96615	0.396368
$57$	0	0
$58$	2.64310	0.347057
$59$	−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	$0.683533\pi$
$62$	−2.34481	−0.297792
$63$	0	0
$64$	−3.63102	−0.453878
$65$	0	0
$66$	0	0
$67$	−5.96077	−0.728224	−0.364112	−	0.931355i	$-0.618627\pi$
$67$	−5.96077	−0.728224	−0.364112	+	0.931355i	$0.618627\pi$
$68$	−6.85086	−0.830788
$69$	0	0
$70$	0.192685	0.0230303
$71$	−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	$0.641673\pi$
$74$	−1.42327	−0.165452
$75$	0	0
$76$	10.0586	1.15380
$77$	−9.90946	−1.12929
$78$	0	0
$79$	4.45473	0.501196	0.250598	−	0.968091i	$-0.419373\pi$
$79$	4.45473	0.501196	0.250598	+	0.968091i	$0.419373\pi$
$80$	−0.704103	−0.0787211
$81$	0	0
$82$	0.198062	0.0218723
$83$	−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
$86$	0.762709	0.0822450
$87$	0	0
$88$	−9.56465	−1.01959
$89$	0.137063	0.0145287	0.00726434	−	0.999974i	$-0.497688\pi$
$89$	0.137063	0.0145287	0.00726434	+	0.999974i	$0.497688\pi$
$90$	0	0
$91$	0	0
$92$	15.0368	1.56770
$93$	0	0
$94$	−2.99761	−0.309180
$95$	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	$0.255411\pi$
$98$	−1.74764	−0.176539
$99$	0	0
$100$	8.89977	0.889977
$101$	5.41119	0.538434	0.269217	−	0.963080i	$-0.413235\pi$
$101$	5.41119	0.538434	0.269217	+	0.963080i	$0.413235\pi$
$102$	0	0
$103$	13.7560	1.35542	0.677710	−	0.735330i	$-0.262973\pi$
$103$	13.7560	1.35542	0.677710	+	0.735330i	$0.262973\pi$
$104$	0	0
$105$	0	0
$106$	0.472189	0.0458630
$107$	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	$0.697622\pi$
$110$	−0.621334	−0.0592419
$111$	0	0
$112$	−4.99761	−0.472229
$113$	1.63773	0.154064	0.0770322	−	0.997029i	$-0.475456\pi$
$113$	1.63773	0.154064	0.0770322	+	0.997029i	$0.475456\pi$
$114$	0	0
$115$	2.06100	0.192189
$116$	−10.7017	−0.993629
$117$	0	0
$118$	−6.09783	−0.561351
$119$	−6.66487	−0.610968
$120$	0	0
$121$	20.9541	1.90492
$122$	−3.78986	−0.343117
$123$	0	0
$124$	9.49396	0.852583
$125$	2.45473	0.219558
$126$	0	0
$127$	10.7995	0.958305	0.479152	−	0.877732i	$-0.340944\pi$
$127$	10.7995	0.958305	0.479152	+	0.877732i	$0.340944\pi$
$128$	−10.9215	−0.965337
$129$	0	0
$130$	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
$134$	−2.65279	−0.229166
$135$	0	0
$136$	−6.43296	−0.551622
$137$	−9.54825	−0.815762	−0.407881	−	0.913035i	$-0.633732\pi$
$137$	−9.54825	−0.815762	−0.407881	+	0.913035i	$0.633732\pi$
$138$	0	0
$139$	−4.09246	−0.347118	−0.173559	−	0.984823i	$-0.555527\pi$
$139$	−4.09246	−0.347118	−0.173559	+	0.984823i	$0.555527\pi$
$140$	−0.780167	−0.0659362
$141$	0	0
$142$	−2.54527	−0.213594
$143$	0	0
$144$	0	0
$145$	−1.46681	−0.121812
$146$	−3.27413	−0.270969
$147$	0	0
$148$	5.76271	0.473692
$149$	−15.3884	−1.26066	−0.630332	−	0.776326i	$-0.717081\pi$
$149$	−15.3884	−1.26066	−0.630332	+	0.776326i	$0.717081\pi$
$150$	0	0
$151$	3.67456	0.299032	0.149516	−	0.988759i	$-0.452228\pi$
$151$	3.67456	0.299032	0.149516	+	0.988759i	$0.452228\pi$
$152$	9.44504	0.766094
$153$	0	0
$154$	−4.41013	−0.355378
$155$	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	$0.562358\pi$
$158$	1.98254	0.157723
$159$	0	0
$160$	−1.14914	−0.0908479
$161$	14.6286	1.15290
$162$	0	0
$163$	−8.63102	−0.676034	−0.338017	−	0.941140i	$-0.609756\pi$
$163$	−8.63102	−0.676034	−0.338017	+	0.941140i	$0.609756\pi$
$164$	−0.801938	−0.0626208
$165$	0	0
$166$	−4.53319	−0.351844
$167$	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
$170$	−0.417895	−0.0320511
$171$	0	0
$172$	−3.08815	−0.235469
$173$	−4.77479	−0.363021	−0.181510	−	0.983389i	$-0.558099\pi$
$173$	−4.77479	−0.363021	−0.181510	+	0.983389i	$0.558099\pi$
$174$	0	0
$175$	8.65817	0.654496
$176$	16.1153	1.21474
$177$	0	0
$178$	0.0609989	0.00457206
$179$	−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
$183$	0	0
$184$	14.1196	1.04091
$185$	0.789856	0.0580714
$186$	0	0
$187$	21.4916	1.57162
$188$	12.1371	0.885186
$189$	0	0
$190$	0.613564	0.0445126
$191$	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	$0.392628\pi$
$194$	6.09246	0.437413
$195$	0	0
$196$	7.07606	0.505433
$197$	4.11231	0.292990	0.146495	−	0.989211i	$-0.453201\pi$
$197$	4.11231	0.292990	0.146495	+	0.989211i	$0.453201\pi$
$198$	0	0
$199$	−24.7724	−1.75607	−0.878034	−	0.478598i	$-0.841145\pi$
$199$	−24.7724	−1.75607	−0.878034	+	0.478598i	$0.841145\pi$
$200$	8.35690	0.590922
$201$	0	0
$202$	2.40821	0.169441
$203$	−10.4112	−0.730722
$204$	0	0
$205$	−0.109916	−0.00767688
$206$	6.12200	0.426540
$207$	0	0
$208$	0	0
$209$	−31.5545	−2.18267
$210$	0	0
$211$	−5.93900	−0.408858	−0.204429	−	0.978881i	$-0.565534\pi$
$211$	−5.93900	−0.408858	−0.204429	+	0.978881i	$0.565534\pi$
$212$	−1.91185	−0.131307
$213$	0	0
$214$	5.70410	0.389924
$215$	−0.423272	−0.0288669
$216$	0	0
$217$	9.23623	0.626996
$218$	−5.40581	−0.366128
$219$	0	0
$220$	2.51573	0.169610
$221$	0	0
$222$	0	0
$223$	−14.2010	−0.950972	−0.475486	−	0.879723i	$-0.657728\pi$
$223$	−14.2010	−0.950972	−0.475486	+	0.879723i	$0.657728\pi$
$224$	−8.15644	−0.544975
$225$	0	0
$226$	0.728857	0.0484829
$227$	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	$0.519376\pi$
$230$	0.917231	0.0604804
$231$	0	0
$232$	−10.0489	−0.659744
$233$	23.4252	1.53464	0.767318	−	0.641267i	$-0.221591\pi$
$233$	23.4252	1.53464	0.767318	+	0.641267i	$0.221591\pi$
$234$	0	0
$235$	1.66355	0.108518
$236$	24.6896	1.60716
$237$	0	0
$238$	−2.96615	−0.192267
$239$	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	$0.589667\pi$
$242$	9.32544	0.599462
$243$	0	0
$244$	15.3448	0.982351
$245$	0.969869	0.0619627
$246$	0	0
$247$	0	0
$248$	8.91484	0.566093
$249$	0	0
$250$	1.09246	0.0690931
$251$	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
$254$	4.80625	0.301571
$255$	0	0
$256$	2.40150	0.150094
$257$	20.5961	1.28475	0.642375	−	0.766391i	$-0.277949\pi$
$257$	20.5961	1.28475	0.642375	+	0.766391i	$0.277949\pi$
$258$	0	0
$259$	5.60627	0.348357
$260$	0	0
$261$	0	0
$262$	−0.403894	−0.0249527
$263$	−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
$266$	4.35498	0.267021
$267$	0	0
$268$	10.7409	0.656107
$269$	−27.3032	−1.66471	−0.832353	−	0.554247i	$-0.813006\pi$
$269$	−27.3032	−1.66471	−0.832353	+	0.554247i	$0.813006\pi$
$270$	0	0
$271$	−27.9855	−1.70000	−0.850000	−	0.526783i	$-0.823398\pi$
$271$	−27.9855	−1.70000	−0.850000	+	0.526783i	$0.823398\pi$
$272$	10.8388	0.657197
$273$	0	0
$274$	−4.24937	−0.256714
$275$	−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	$0.520126\pi$
$278$	−1.82132	−0.109235
$279$	0	0
$280$	−0.732578	−0.0437799
$281$	27.2349	1.62470	0.812349	−	0.583172i	$-0.198189\pi$
$281$	27.2349	1.62470	0.812349	+	0.583172i	$0.198189\pi$
$282$	0	0
$283$	5.28382	0.314090	0.157045	−	0.987591i	$-0.449803\pi$
$283$	5.28382	0.314090	0.157045	+	0.987591i	$0.449803\pi$
$284$	10.3056	0.611524
$285$	0	0
$286$	0	0
$287$	−0.780167	−0.0460518
$288$	0	0
$289$	−2.54527	−0.149722
$290$	−0.652793	−0.0383333
$291$	0	0
$292$	13.2567	0.775788
$293$	32.6625	1.90816	0.954081	−	0.299548i	$-0.0968358\pi$
$293$	32.6625	1.90816	0.954081	+	0.299548i	$0.0968358\pi$
$294$	0	0
$295$	3.38404	0.197027
$296$	5.41119	0.314519
$297$	0	0
$298$	−6.84846	−0.396721
$299$	0	0
$300$	0	0
$301$	−3.00431	−0.173166
$302$	1.63533	0.0941029
$303$	0	0
$304$	−15.9138	−0.912717
$305$	2.10321	0.120430
$306$	0	0
$307$	20.7614	1.18491	0.592457	−	0.805602i	$-0.298158\pi$
$307$	20.7614	1.18491	0.592457	+	0.805602i	$0.298158\pi$
$308$	17.8562	1.01745
$309$	0	0
$310$	0.579121	0.0328919
$311$	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	$0.538522\pi$
$314$	−2.17092	−0.122512
$315$	0	0
$316$	−8.02715	−0.451562
$317$	15.4776	0.869307	0.434653	−	0.900598i	$-0.356871\pi$
$317$	15.4776	0.869307	0.434653	+	0.900598i	$0.356871\pi$
$318$	0	0
$319$	33.5719	1.87967
$320$	0.896789	0.0501320
$321$	0	0
$322$	6.51035	0.362808
$323$	−21.2228	−1.18087
$324$	0	0
$325$	0	0
$326$	−3.84117	−0.212743
$327$	0	0
$328$	−0.753020	−0.0415786
$329$	11.8076	0.650973
$330$	0	0
$331$	−6.06829	−0.333544	−0.166772	−	0.985996i	$-0.553334\pi$
$331$	−6.06829	−0.333544	−0.166772	+	0.985996i	$0.553334\pi$
$332$	18.3545	1.00733
$333$	0	0
$334$	4.21446	0.230605
$335$	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
$339$	0	0
$340$	1.69202	0.0917627
$341$	−29.7832	−1.61285
$342$	0	0
$343$	19.1551	1.03428
$344$	−2.89977	−0.156345
$345$	0	0
$346$	−2.12498	−0.114240
$347$	−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	$0.297527\pi$
$350$	3.85325	0.205965
$351$	0	0
$352$	26.3013	1.40186
$353$	−5.07069	−0.269885	−0.134943	−	0.990853i	$-0.543085\pi$
$353$	−5.07069	−0.269885	−0.134943	+	0.990853i	$0.543085\pi$
$354$	0	0
$355$	1.41252	0.0749687
$356$	−0.246980	−0.0130899
$357$	0	0
$358$	−1.52888	−0.0808036
$359$	−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
$362$	−6.00192	−0.315454
$363$	0	0
$364$	0	0
$365$	1.81700	0.0951063
$366$	0	0
$367$	−1.17928	−0.0615577	−0.0307789	−	0.999526i	$-0.509799\pi$
$367$	−1.17928	−0.0615577	−0.0307789	+	0.999526i	$0.509799\pi$
$368$	−23.7899	−1.24013
$369$	0	0
$370$	0.351519	0.0182746
$371$	−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	$0.784305\pi$
$374$	9.56465	0.494576
$375$	0	0
$376$	11.3967	0.587741
$377$	0	0
$378$	0	0
$379$	19.1631	0.984345	0.492172	−	0.870498i	$-0.336203\pi$
$379$	19.1631	0.984345	0.492172	+	0.870498i	$0.336203\pi$
$380$	−2.48427	−0.127440
$381$	0	0
$382$	0.579121	0.0296304
$383$	−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
$386$	4.09246	0.208301
$387$	0	0
$388$	−24.6679	−1.25232
$389$	24.0315	1.21844	0.609222	−	0.793000i	$-0.291482\pi$
$389$	24.0315	1.21844	0.609222	+	0.793000i	$0.291482\pi$
$390$	0	0
$391$	−31.7265	−1.60448
$392$	6.64443	0.335594
$393$	0	0
$394$	1.83015	0.0922016
$395$	−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	$0.766516\pi$
$398$	−11.0248	−0.552621
$399$	0	0
$400$	−14.0804	−0.704019
$401$	−21.1032	−1.05384	−0.526922	−	0.849914i	$-0.676654\pi$
$401$	−21.1032	−1.05384	−0.526922	+	0.849914i	$0.676654\pi$
$402$	0	0
$403$	0	0
$404$	−9.75063	−0.485112
$405$	0	0
$406$	−4.63342	−0.229953
$407$	−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	$0.150288\pi$
$410$	−0.0489173	−0.00241586
$411$	0	0
$412$	−24.7875	−1.22119
$413$	24.0194	1.18192
$414$	0	0
$415$	2.51573	0.123492
$416$	0	0
$417$	0	0
$418$	−14.0431	−0.686869
$419$	−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	$0.483739\pi$
$422$	−2.64310	−0.128664
$423$	0	0
$424$	−1.79523	−0.0871842
$425$	−18.7778	−0.910856
$426$	0	0
$427$	14.9282	0.722429
$428$	−23.0954	−1.11636
$429$	0	0
$430$	−0.188374	−0.00908418
$431$	−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	$0.598297\pi$
$434$	4.11051	0.197311
$435$	0	0
$436$	21.8877	1.04823
$437$	46.5816	2.22830
$438$	0	0
$439$	−10.9269	−0.521513	−0.260757	−	0.965405i	$-0.583972\pi$
$439$	−10.9269	−0.521513	−0.260757	+	0.965405i	$0.583972\pi$
$440$	2.36227	0.112617
$441$	0	0
$442$	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
$446$	−6.32006	−0.299264
$447$	0	0
$448$	6.36526	0.300730
$449$	−28.8200	−1.36010	−0.680050	−	0.733166i	$-0.738042\pi$
$449$	−28.8200	−1.36010	−0.680050	+	0.733166i	$0.738042\pi$
$450$	0	0
$451$	2.51573	0.118461
$452$	−2.95108	−0.138807
$453$	0	0
$454$	7.12392	0.334342
$455$	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	$0.361098\pi$
$458$	−0.819396	−0.0382879
$459$	0	0
$460$	−3.71379	−0.173156
$461$	−7.56763	−0.352460	−0.176230	−	0.984349i	$-0.556390\pi$
$461$	−7.56763	−0.352460	−0.176230	+	0.984349i	$0.556390\pi$
$462$	0	0
$463$	−35.3551	−1.64309	−0.821545	−	0.570143i	$-0.806888\pi$
$463$	−35.3551	−1.64309	−0.821545	+	0.570143i	$0.806888\pi$
$464$	16.9312	0.786013
$465$	0	0
$466$	10.4252	0.482938
$467$	−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
$470$	0.740348	0.0341497
$471$	0	0
$472$	23.1836	1.06711
$473$	9.68771	0.445441
$474$	0	0
$475$	27.5700	1.26500
$476$	12.0097	0.550463
$477$	0	0
$478$	6.49827	0.297224
$479$	−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
$482$	−3.84117	−0.174960
$483$	0	0
$484$	−37.7579	−1.71627
$485$	−3.38106	−0.153526
$486$	0	0
$487$	−16.0073	−0.725360	−0.362680	−	0.931914i	$-0.618138\pi$
$487$	−16.0073	−0.725360	−0.362680	+	0.931914i	$0.618138\pi$
$488$	14.4088	0.652256
$489$	0	0
$490$	0.431632	0.0194992
$491$	−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
$495$	0	0
$496$	−15.0204	−0.674438
$497$	10.0258	0.449719
$498$	0	0
$499$	8.06770	0.361160	0.180580	−	0.983560i	$-0.442203\pi$
$499$	8.06770	0.361160	0.180580	+	0.983560i	$0.442203\pi$
$500$	−4.42327	−0.197815
$501$	0	0
$502$	1.69202	0.0755186
$503$	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
$506$	−20.9933	−0.933266
$507$	0	0
$508$	−19.4601	−0.863403
$509$	16.0495	0.711382	0.355691	−	0.934604i	$-0.384246\pi$
$509$	16.0495	0.711382	0.355691	+	0.934604i	$0.384246\pi$
$510$	0	0
$511$	12.8968	0.570520
$512$	22.9119	1.01257
$513$	0	0
$514$	9.16613	0.404301
$515$	−3.39745	−0.149710
$516$	0	0
$517$	−38.0747	−1.67452
$518$	2.49502	0.109625
$519$	0	0
$520$	0	0
$521$	2.69309	0.117986	0.0589931	−	0.998258i	$-0.481211\pi$
$521$	2.69309	0.117986	0.0589931	+	0.998258i	$0.481211\pi$
$522$	0	0
$523$	35.3957	1.54774	0.773872	−	0.633342i	$-0.218317\pi$
$523$	35.3957	1.54774	0.773872	+	0.633342i	$0.218317\pi$
$524$	1.63533	0.0714399
$525$	0	0
$526$	−0.148080	−0.00645659
$527$	−20.0315	−0.872584
$528$	0	0
$529$	46.6359	2.02765
$530$	−0.116621	−0.00506569
$531$	0	0
$532$	−17.6329	−0.764485
$533$	0	0
$534$	0	0
$535$	−3.16554	−0.136858
$536$	10.0858	0.435638
$537$	0	0
$538$	−12.1511	−0.523870
$539$	−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	$0.231656\pi$
$542$	−12.4547	−0.534976
$543$	0	0
$544$	17.6896	0.758437
$545$	3.00000	0.128506
$546$	0	0
$547$	−26.1183	−1.11674	−0.558368	−	0.829593i	$-0.688572\pi$
$547$	−26.1183	−1.11674	−0.558368	+	0.829593i	$0.688572\pi$
$548$	17.2054	0.734976
$549$	0	0
$550$	−12.4252	−0.529812
$551$	−33.1521	−1.41233
$552$	0	0
$553$	−7.80923	−0.332082
$554$	−0.936017	−0.0397676
$555$	0	0
$556$	7.37435	0.312742
$557$	24.7748	1.04974	0.524871	−	0.851182i	$-0.324114\pi$
$557$	24.7748	1.04974	0.524871	+	0.851182i	$0.324114\pi$
$558$	0	0
$559$	0	0
$560$	1.23431	0.0521590
$561$	0	0
$562$	12.1207	0.511280
$563$	−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
$566$	2.35152	0.0988417
$567$	0	0
$568$	9.67696	0.406036
$569$	33.7458	1.41470	0.707350	−	0.706864i	$-0.249891\pi$
$569$	33.7458	1.41470	0.707350	+	0.706864i	$0.249891\pi$
$570$	0	0
$571$	23.0887	0.966234	0.483117	−	0.875556i	$-0.339505\pi$
$571$	23.0887	0.966234	0.483117	+	0.875556i	$0.339505\pi$
$572$	0	0
$573$	0	0
$574$	−0.347207	−0.0144921
$575$	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	$0.523676\pi$
$578$	−1.13275	−0.0471162
$579$	0	0
$580$	2.64310	0.109749
$581$	17.8562	0.740801
$582$	0	0
$583$	5.99761	0.248396
$584$	12.4480	0.515103
$585$	0	0
$586$	14.5362	0.600484
$587$	−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
$590$	1.50604	0.0620027
$591$	0	0
$592$	−9.11721	−0.374715
$593$	−21.8538	−0.897430	−0.448715	−	0.893675i	$-0.648118\pi$
$593$	−21.8538	−0.897430	−0.448715	+	0.893675i	$0.648118\pi$
$594$	0	0
$595$	1.64609	0.0674830
$596$	27.7289	1.13582
$597$	0	0
$598$	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	$0.428787\pi$
$602$	−1.33704	−0.0544939
$603$	0	0
$604$	−6.62133	−0.269418
$605$	−5.17523	−0.210403
$606$	0	0
$607$	−29.6359	−1.20289	−0.601443	−	0.798916i	$-0.705407\pi$
$607$	−29.6359	−1.20289	−0.601443	+	0.798916i	$0.705407\pi$
$608$	−25.9724	−1.05332
$609$	0	0
$610$	0.936017	0.0378982
$611$	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	$0.566266\pi$
$614$	9.23968	0.372883
$615$	0	0
$616$	16.7670	0.675563
$617$	26.2828	1.05810	0.529052	−	0.848590i	$-0.322548\pi$
$617$	26.2828	1.05810	0.529052	+	0.848590i	$0.322548\pi$
$618$	0	0
$619$	−29.0834	−1.16896	−0.584479	−	0.811408i	$-0.698701\pi$
$619$	−29.0834	−1.16896	−0.584479	+	0.811408i	$0.698701\pi$
$620$	−2.34481	−0.0941700
$621$	0	0
$622$	5.02954	0.201666
$623$	−0.240275	−0.00962641
$624$	0	0
$625$	24.0887	0.963549
$626$	−1.90110	−0.0759833
$627$	0	0
$628$	8.78986	0.350753
$629$	−12.1588	−0.484804
$630$	0	0
$631$	−25.4480	−1.01307	−0.506535	−	0.862219i	$-0.669074\pi$
$631$	−25.4480	−1.01307	−0.506535	+	0.862219i	$0.669074\pi$
$632$	−7.53750	−0.299826
$633$	0	0
$634$	6.88816	0.273564
$635$	−2.66727	−0.105847
$636$	0	0
$637$	0	0
$638$	14.9409	0.591517
$639$	0	0
$640$	2.69740	0.106624
$641$	26.7409	1.05620	0.528102	−	0.849181i	$-0.322904\pi$
$641$	26.7409	1.05620	0.528102	+	0.849181i	$0.322904\pi$
$642$	0	0
$643$	32.9614	1.29987	0.649935	−	0.759990i	$-0.274796\pi$
$643$	32.9614	1.29987	0.649935	+	0.759990i	$0.274796\pi$
$644$	−26.3599	−1.03872
$645$	0	0
$646$	−9.44504	−0.371610
$647$	−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
$651$	0	0
$652$	15.5526	0.609085
$653$	−36.1517	−1.41472	−0.707362	−	0.706852i	$-0.750115\pi$
$653$	−36.1517	−1.41472	−0.707362	+	0.706852i	$0.750115\pi$
$654$	0	0
$655$	0.224144	0.00875805
$656$	1.26875	0.0495364
$657$	0	0
$658$	5.25487	0.204856
$659$	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	$0.432044\pi$
$662$	−2.70065	−0.104964
$663$	0	0
$664$	17.2349	0.668844
$665$	−2.41683	−0.0937206
$666$	0	0
$667$	−49.5599	−1.91897
$668$	−17.0640	−0.660225
$669$	0	0
$670$	0.655186	0.0253120
$671$	−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	$0.369112\pi$
$674$	5.39565	0.207833
$675$	0	0
$676$	0	0
$677$	25.5786	0.983067	0.491534	−	0.870859i	$-0.336436\pi$
$677$	25.5786	0.983067	0.491534	+	0.870859i	$0.336436\pi$
$678$	0	0
$679$	−23.9982	−0.920966
$680$	1.58881	0.0609281
$681$	0	0
$682$	−13.2547	−0.507551
$683$	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
$686$	8.52483	0.325479
$687$	0	0
$688$	4.88577	0.186268
$689$	0	0
$690$	0	0
$691$	−2.62996	−0.100048	−0.0500242	−	0.998748i	$-0.515930\pi$
$691$	−2.62996	−0.100048	−0.0500242	+	0.998748i	$0.515930\pi$
$692$	8.60388	0.327070
$693$	0	0
$694$	−10.3026	−0.391081
$695$	1.01075	0.0383401
$696$	0	0
$697$	1.69202	0.0640899
$698$	9.87800	0.373888
$699$	0	0
$700$	−15.6015	−0.589681
$701$	−40.0925	−1.51427	−0.757136	−	0.653258i	$-0.773402\pi$
$701$	−40.0925	−1.51427	−0.757136	+	0.653258i	$0.773402\pi$
$702$	0	0
$703$	17.8519	0.673298
$704$	−20.5254	−0.773581
$705$	0	0
$706$	−2.25667	−0.0849308
$707$	−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	$0.643545\pi$
$710$	0.628630	0.0235921
$711$	0	0
$712$	−0.231914	−0.00869135
$713$	43.9667	1.64657
$714$	0	0
$715$	0	0
$716$	6.19029	0.231342
$717$	0	0
$718$	−7.42088	−0.276945
$719$	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
$722$	5.41166	0.201401
$723$	0	0
$724$	24.3013	0.903150
$725$	−29.3327	−1.08939
$726$	0	0
$727$	−16.5472	−0.613701	−0.306851	−	0.951758i	$-0.599275\pi$
$727$	−16.5472	−0.613701	−0.306851	+	0.951758i	$0.599275\pi$
$728$	0	0
$729$	0	0
$730$	0.808643	0.0299292
$731$	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	$0.613337\pi$
$734$	−0.524827	−0.0193717
$735$	0	0
$736$	−38.8267	−1.43117
$737$	−33.6950	−1.24117
$738$	0	0
$739$	−47.3239	−1.74084	−0.870419	−	0.492312i	$-0.836152\pi$
$739$	−47.3239	−1.74084	−0.870419	+	0.492312i	$0.836152\pi$
$740$	−1.42327	−0.0523205
$741$	0	0
$742$	−0.827757	−0.0303879
$743$	−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
$746$	−13.3924	−0.490331
$747$	0	0
$748$	−38.7265	−1.41598
$749$	−22.4685	−0.820980
$750$	0	0
$751$	0.710808	0.0259377	0.0129689	−	0.999916i	$-0.495872\pi$
$751$	0.710808	0.0259377	0.0129689	+	0.999916i	$0.495872\pi$
$752$	−19.2021	−0.700229
$753$	0	0
$754$	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	$0.443092\pi$
$758$	8.52840	0.309766
$759$	0	0
$760$	−2.33273	−0.0846171
$761$	−18.8810	−0.684435	−0.342218	−	0.939621i	$-0.611178\pi$
$761$	−18.8810	−0.684435	−0.342218	+	0.939621i	$0.611178\pi$
$762$	0	0
$763$	21.2935	0.770877
$764$	−2.34481	−0.0848324
$765$	0	0
$766$	−6.84846	−0.247445
$767$	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	$0.428021\pi$
$770$	1.08921	0.0392524
$771$	0	0
$772$	−16.5700	−0.596368
$773$	45.6746	1.64280	0.821400	−	0.570353i	$-0.193193\pi$
$773$	45.6746	1.64280	0.821400	+	0.570353i	$0.193193\pi$
$774$	0	0
$775$	26.0224	0.934751
$776$	−23.1631	−0.831508
$777$	0	0
$778$	10.6950	0.383435
$779$	−2.48427	−0.0890082
$780$	0	0
$781$	−32.3293	−1.15683
$782$	−14.1196	−0.504916
$783$	0	0
$784$	−11.1951	−0.399824
$785$	1.20477	0.0430000
$786$	0	0
$787$	4.51871	0.161075	0.0805374	−	0.996752i	$-0.474336\pi$
$787$	4.51871	0.161075	0.0805374	+	0.996752i	$0.474336\pi$
$788$	−7.41013	−0.263975
$789$	0	0
$790$	−0.489647	−0.0174209
$791$	−2.87097	−0.102080
$792$	0	0
$793$	0	0
$794$	−13.1739	−0.467524
$795$	0	0
$796$	44.6383	1.58216
$797$	28.7391	1.01799	0.508996	−	0.860769i	$-0.330017\pi$
$797$	28.7391	1.01799	0.508996	+	0.860769i	$0.330017\pi$
$798$	0	0
$799$	−25.6082	−0.905953
$800$	−22.9801	−0.812471
$801$	0	0
$802$	−9.39181	−0.331636
$803$	−41.5870	−1.46757
$804$	0	0
$805$	−3.61297	−0.127341
$806$	0	0
$807$	0	0
$808$	−9.15585	−0.322102
$809$	−5.42891	−0.190870	−0.0954352	−	0.995436i	$-0.530424\pi$
$809$	−5.42891	−0.190870	−0.0954352	+	0.995436i	$0.530424\pi$
$810$	0	0
$811$	−0.629104	−0.0220908	−0.0110454	−	0.999939i	$-0.503516\pi$
$811$	−0.629104	−0.0220908	−0.0110454	+	0.999939i	$0.503516\pi$
$812$	18.7603	0.658358
$813$	0	0
$814$	−8.04546	−0.281993
$815$	2.13169	0.0746697
$816$	0	0
$817$	−9.56657	−0.334692
$818$	16.0315	0.560527
$819$	0	0
$820$	0.198062	0.00691663
$821$	36.2640	1.26562	0.632811	−	0.774307i	$-0.281901\pi$
$821$	36.2640	1.26562	0.632811	+	0.774307i	$0.281901\pi$
$822$	0	0
$823$	41.7396	1.45495	0.727476	−	0.686133i	$-0.240693\pi$
$823$	41.7396	1.45495	0.727476	+	0.686133i	$0.240693\pi$
$824$	−23.2755	−0.810839
$825$	0	0
$826$	10.6896	0.371940
$827$	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	$0.410500\pi$
$830$	1.11960	0.0388621
$831$	0	0
$832$	0	0
$833$	−14.9299	−0.517290
$834$	0	0
$835$	−2.33885	−0.0809391
$836$	56.8592	1.96652
$837$	0	0
$838$	−2.65625	−0.0917587
$839$	−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
$842$	0.932559	0.0321381
$843$	0	0
$844$	10.7017	0.368368
$845$	0	0
$846$	0	0
$847$	−36.7329	−1.26216
$848$	3.02475	0.103870
$849$	0	0
$850$	−8.35690	−0.286639
$851$	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	$0.601938\pi$
$854$	6.64370	0.227343
$855$	0	0
$856$	−21.6866	−0.741234
$857$	−28.1849	−0.962778	−0.481389	−	0.876507i	$-0.659868\pi$
$857$	−28.1849	−0.962778	−0.481389	+	0.876507i	$0.659868\pi$
$858$	0	0
$859$	33.3957	1.13944	0.569722	−	0.821837i	$-0.307051\pi$
$859$	33.3957	1.13944	0.569722	+	0.821837i	$0.307051\pi$
$860$	0.762709	0.0260082
$861$	0	0
$862$	−1.28429	−0.0437431
$863$	−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
$866$	−5.62910	−0.191285
$867$	0	0
$868$	−16.6431	−0.564904
$869$	25.1817	0.854230
$870$	0	0
$871$	0	0
$872$	20.5526	0.695998
$873$	0	0
$874$	20.7308	0.701229
$875$	−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	$0.329872\pi$
$878$	−4.86294	−0.164116
$879$	0	0
$880$	−3.98015	−0.134171
$881$	3.56273	0.120031	0.0600157	−	0.998197i	$-0.480885\pi$
$881$	3.56273	0.120031	0.0600157	+	0.998197i	$0.480885\pi$
$882$	0	0
$883$	−10.2088	−0.343554	−0.171777	−	0.985136i	$-0.554951\pi$
$883$	−10.2088	−0.343554	−0.171777	+	0.985136i	$0.554951\pi$
$884$	0	0
$885$	0	0
$886$	8.56273	0.287670
$887$	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
$890$	−0.0150655	−0.000504996 0
$891$	0	0
$892$	25.5894	0.856797
$893$	37.5986	1.25819
$894$	0	0
$895$	0.848462	0.0283610
$896$	19.1457	0.639613
$897$	0	0
$898$	−12.8261	−0.428013
$899$	−31.2911	−1.04362
$900$	0	0
$901$	4.03385	0.134387
$902$	1.11960	0.0372788
$903$	0	0
$904$	−2.77107	−0.0921644
$905$	3.33081	0.110720
$906$	0	0
$907$	−41.0804	−1.36405	−0.682026	−	0.731328i	$-0.738901\pi$
$907$	−41.0804	−1.36405	−0.682026	+	0.731328i	$0.738901\pi$
$908$	−28.8442	−0.957227
$909$	0	0
$910$	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
$914$	8.04221	0.266013
$915$	0	0
$916$	3.31767	0.109619
$917$	1.59094	0.0525375
$918$	0	0
$919$	29.0019	0.956685	0.478343	−	0.878173i	$-0.341238\pi$
$919$	29.0019	0.956685	0.478343	+	0.878173i	$0.341238\pi$
$920$	−3.48725	−0.114971
$921$	0	0
$922$	−3.36791	−0.110916
$923$	0	0
$924$	0	0
$925$	15.7952	0.519344
$926$	−15.7345	−0.517068
$927$	0	0
$928$	27.6329	0.907096
$929$	−50.7211	−1.66410	−0.832052	−	0.554697i	$-0.812834\pi$
$929$	−50.7211	−1.66410	−0.832052	+	0.554697i	$0.812834\pi$
$930$	0	0
$931$	21.9205	0.718415
$932$	−42.2107	−1.38266
$933$	0	0
$934$	−5.78554	−0.189309
$935$	−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	$0.183706\pi$
$938$	4.65040	0.151841
$939$	0	0
$940$	−2.99761	−0.0977712
$941$	−34.7036	−1.13131	−0.565653	−	0.824643i	$-0.691376\pi$
$941$	−34.7036	−1.13131	−0.565653	+	0.824643i	$0.691376\pi$
$942$	0	0
$943$	−3.71379	−0.120938
$944$	−39.0616	−1.27135
$945$	0	0
$946$	4.31144	0.140177
$947$	−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
$950$	12.2698	0.398085
$951$	0	0
$952$	11.2771	0.365493
$953$	−26.0151	−0.842711	−0.421355	−	0.906896i	$-0.638445\pi$
$953$	−26.0151	−0.842711	−0.421355	+	0.906896i	$0.638445\pi$
$954$	0	0
$955$	−0.321388	−0.0103999
$956$	−26.3110	−0.850957
$957$	0	0
$958$	−11.3604	−0.367036
$959$	16.7383	0.540507
$960$	0	0
$961$	−3.24027	−0.104525
$962$	0	0
$963$	0	0
$964$	15.5526	0.500914
$965$	−2.27114	−0.0731107
$966$	0	0
$967$	−36.6644	−1.17905	−0.589524	−	0.807751i	$-0.700685\pi$
$967$	−36.6644	−1.17905	−0.589524	+	0.807751i	$0.700685\pi$
$968$	−35.4547	−1.13956
$969$	0	0
$970$	−1.50471	−0.0483134
$971$	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
$974$	−7.12392	−0.228265
$975$	0	0
$976$	−24.2771	−0.777091
$977$	28.8998	0.924586	0.462293	−	0.886727i	$-0.347027\pi$
$977$	28.8998	0.924586	0.462293	+	0.886727i	$0.347027\pi$
$978$	0	0
$979$	0.774791	0.0247624
$980$	−1.74764	−0.0558264
$981$	0	0
$982$	−9.22952	−0.294526
$983$	−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
$986$	10.0489	0.320023
$987$	0	0
$988$	0	0
$989$	−14.3013	−0.454754
$990$	0	0
$991$	−5.18300	−0.164643	−0.0823217	−	0.996606i	$-0.526233\pi$
$991$	−5.18300	−0.164643	−0.0823217	+	0.996606i	$0.526233\pi$
$992$	−24.5144	−0.778333
$993$	0	0
$994$	4.46191	0.141523
$995$	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	$0.785642\pi$
$998$	3.59047	0.113654
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.q.1.2		3
3.2	odd	2		507.2.a.j.1.2	✓	3
12.11	even	2		8112.2.a.by.1.3		3
13.5	odd	4		1521.2.b.m.1351.3		6
13.8	odd	4		1521.2.b.m.1351.4		6
13.12	even	2		1521.2.a.p.1.2		3
39.2	even	12		507.2.j.h.316.3		12
39.5	even	4		507.2.b.g.337.4		6
39.8	even	4		507.2.b.g.337.3		6
39.11	even	12		507.2.j.h.316.4		12
39.17	odd	6		507.2.e.j.484.2		6
39.20	even	12		507.2.j.h.361.3		12
39.23	odd	6		507.2.e.j.22.2		6
39.29	odd	6		507.2.e.k.22.2		6
39.32	even	12		507.2.j.h.361.4		12
39.35	odd	6		507.2.e.k.484.2		6
39.38	odd	2		507.2.a.k.1.2	yes	3
156.155	even	2		8112.2.a.cf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.2	✓	3	3.2	odd	2
507.2.a.k.1.2	yes	3	39.38	odd	2
507.2.b.g.337.3		6	39.8	even	4
507.2.b.g.337.4		6	39.5	even	4
507.2.e.j.22.2		6	39.23	odd	6
507.2.e.j.484.2		6	39.17	odd	6
507.2.e.k.22.2		6	39.29	odd	6
507.2.e.k.484.2		6	39.35	odd	6
507.2.j.h.316.3		12	39.2	even	12
507.2.j.h.316.4		12	39.11	even	12
507.2.j.h.361.3		12	39.20	even	12
507.2.j.h.361.4		12	39.32	even	12
1521.2.a.p.1.2		3	13.12	even	2
1521.2.a.q.1.2		3	1.1	even	1	trivial
1521.2.b.m.1351.3		6	13.5	odd	4
1521.2.b.m.1351.4		6	13.8	odd	4
8112.2.a.by.1.3		3	12.11	even	2
8112.2.a.cf.1.1		3	156.155	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} -1.80194 q^{4} -0.246980 q^{5} -1.75302 q^{7} -1.69202 q^{8} +O(q^{10})\)	\(q+0.445042 q^{2} -1.80194 q^{4} -0.246980 q^{5} -1.75302 q^{7} -1.69202 q^{8} -0.109916 q^{10} +5.65279 q^{11} -0.780167 q^{14} +2.85086 q^{16} +3.80194 q^{17} -5.58211 q^{19} +0.445042 q^{20} +2.51573 q^{22} -8.34481 q^{23} -4.93900 q^{25} +3.15883 q^{28} +5.93900 q^{29} -5.26875 q^{31} +4.65279 q^{32} +1.69202 q^{34} +0.432960 q^{35} -3.19806 q^{37} -2.48427 q^{38} +0.417895 q^{40} +0.445042 q^{41} +1.71379 q^{43} -10.1860 q^{44} -3.71379 q^{46} -6.73556 q^{47} -3.92692 q^{49} -2.19806 q^{50} +1.06100 q^{53} -1.39612 q^{55} +2.96615 q^{56} +2.64310 q^{58} -13.7017 q^{59} -8.51573 q^{61} -2.34481 q^{62} -3.63102 q^{64} -5.96077 q^{67} -6.85086 q^{68} +0.192685 q^{70} -5.71917 q^{71} -7.35690 q^{73} -1.42327 q^{74} +10.0586 q^{76} -9.90946 q^{77} +4.45473 q^{79} -0.704103 q^{80} +0.198062 q^{82} -10.1860 q^{83} -0.939001 q^{85} +0.762709 q^{86} -9.56465 q^{88} +0.137063 q^{89} +15.0368 q^{92} -2.99761 q^{94} +1.37867 q^{95} +13.6896 q^{97} -1.74764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7}+O(q^{10}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7} - q^{10} - q^{11} - q^{14} - 5 q^{16} + 7 q^{17} - 11 q^{19} + q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} + q^{28} + 8 q^{29} - 8 q^{31} - 4 q^{32} - 18 q^{35} - 14 q^{37} - 20 q^{38} + 7 q^{40} + q^{41} - 3 q^{43} - 16 q^{44} - 3 q^{46} - 9 q^{47} + 17 q^{49} - 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} + 12 q^{58} - 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} - 5 q^{67} - 7 q^{68} + 8 q^{70} - 6 q^{71} - 18 q^{73} - 7 q^{74} - q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 5 q^{82} - 16 q^{83} + 7 q^{85} - 15 q^{86} - 7 q^{88} - 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} - 5 q^{97} - 13 q^{98}+O(q^{100}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7 - q^10 - q^11 - q^14 - 5 * q^16 + 7 * q^17 - 11 * q^19 + q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 + q^28 + 8 * q^29 - 8 * q^31 - 4 * q^32 - 18 * q^35 - 14 * q^37 - 20 * q^38 + 7 * q^40 + q^41 - 3 * q^43 - 16 * q^44 - 3 * q^46 - 9 * q^47 + 17 * q^49 - 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 + 12 * q^58 - 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 - 5 * q^67 - 7 * q^68 + 8 * q^70 - 6 * q^71 - 18 * q^73 - 7 * q^74 - q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 5 * q^82 - 16 * q^83 + 7 * q^85 - 15 * q^86 - 7 * q^88 - 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 - 5 * q^97 - 13 * q^98

Introduction

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