LMFDB - Embedded newform 8450.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, 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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{12} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +8.00000 q^{21} +2.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} +4.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -10.0000 q^{41} +8.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} -2.00000 q^{48} +9.00000 q^{49} +4.00000 q^{51} -2.00000 q^{53} +4.00000 q^{54} -4.00000 q^{56} +12.0000 q^{57} +2.00000 q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +12.0000 q^{69} -10.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} -4.00000 q^{79} -11.0000 q^{81} -10.0000 q^{82} +8.00000 q^{84} +10.0000 q^{86} -4.00000 q^{87} +2.00000 q^{88} +14.0000 q^{89} -6.00000 q^{92} -12.0000 q^{93} -12.0000 q^{94} -2.00000 q^{96} +14.0000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	2.00000	0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	8.00000	1.23443
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−2.00000	−0.288675
$49$	9.00000	1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−4.00000	−0.534522
$57$	12.0000	1.58944
$58$	2.00000	0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000	0.762001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	12.0000	1.44463
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	10.0000	1.07833
$87$	−4.00000	−0.428845
$88$	2.00000	0.213201
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−12.0000	−1.24434
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−2.00000	−0.204124
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	9.00000	0.909137
$99$	2.00000	0.201008
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	4.00000	0.396059
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	4.00000	0.384900
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−4.00000	−0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	12.0000	1.12390
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	−10.0000	−0.920575
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	20.0000	1.80334
$124$	6.00000	0.538816
$125$	0	0
$126$	−4.00000	−0.356348
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	−20.0000	−1.76090
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−4.00000	−0.348155
$133$	24.0000	2.08106
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	12.0000	1.02151
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	−10.0000	−0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	−18.0000	−1.48461
$148$	−2.00000	−0.164399
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	−6.00000	−0.486664
$153$	−2.00000	−0.161690
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.00000	−0.318223
$159$	4.00000	0.317221
$160$	0	0
$161$	24.0000	1.89146
$162$	−11.0000	−0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	0	0
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	8.00000	0.617213
$169$	0	0
$170$	0	0
$171$	−6.00000	−0.458831
$172$	10.0000	0.762493
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	2.00000	0.150756
$177$	20.0000	1.50329
$178$	14.0000	1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−12.0000	−0.879883
$187$	−4.00000	−0.292509
$188$	−12.0000	−0.875190
$189$	−16.0000	−1.16383
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	2.00000	0.142134
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	14.0000	0.985037
$203$	−8.00000	−0.561490
$204$	4.00000	0.280056
$205$	0	0
$206$	18.0000	1.25412
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−2.00000	−0.137361
$213$	20.0000	1.37038
$214$	−6.00000	−0.410152
$215$	0	0
$216$	4.00000	0.272166
$217$	−24.0000	−1.62923
$218$	6.00000	0.406371
$219$	−20.0000	−1.35147
$220$	0	0
$221$	0	0
$222$	4.00000	0.268462
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	12.0000	0.794719
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	2.00000	0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	8.00000	0.519656
$238$	8.00000	0.518563
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	2.00000	0.128037
$245$	0	0
$246$	20.0000	1.27515
$247$	0	0
$248$	6.00000	0.381000
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.00000	−0.251976
$253$	−12.0000	−0.754434
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−20.0000	−1.24515
$259$	8.00000	0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	24.0000	1.47153
$267$	−28.0000	−1.71357
$268$	−12.0000	−0.733017
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	12.0000	0.722315
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−8.00000	−0.479808
$279$	6.00000	0.359211
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	24.0000	1.42918
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	0	0
$287$	40.0000	2.36113
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−28.0000	−1.64139
$292$	10.0000	0.585206
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−2.00000	−0.116248
$297$	8.00000	0.464207
$298$	−2.00000	−0.115857
$299$	0	0
$300$	0	0
$301$	−40.0000	−2.30556
$302$	−6.00000	−0.345261
$303$	−28.0000	−1.60856
$304$	−6.00000	−0.344124
$305$	0	0
$306$	−2.00000	−0.114332
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	−8.00000	−0.455842
$309$	−36.0000	−2.04797
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	4.00000	0.224309
$319$	4.00000	0.223957
$320$	0	0
$321$	12.0000	0.669775
$322$	24.0000	1.33747
$323$	12.0000	0.667698
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−12.0000	−0.663602
$328$	−10.0000	−0.552158
$329$	48.0000	2.64633
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	20.0000	1.09435
$335$	0	0
$336$	8.00000	0.436436
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	12.0000	0.649836
$342$	−6.00000	−0.324443
$343$	−8.00000	−0.431959
$344$	10.0000	0.539164
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−4.00000	−0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	14.0000	0.741999
$357$	−16.0000	−0.846810
$358$	−4.00000	−0.211407
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	10.0000	0.525588
$363$	14.0000	0.734809
$364$	0	0
$365$	0	0
$366$	−4.00000	−0.209083
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	−6.00000	−0.312772
$369$	−10.0000	−0.520579
$370$	0	0
$371$	8.00000	0.415339
$372$	−12.0000	−0.622171
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	−16.0000	−0.822951
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−28.0000	−1.43448
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	10.0000	0.508329
$388$	14.0000	0.710742
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	9.00000	0.454569
$393$	−8.00000	−0.403547
$394$	−6.00000	−0.302276
$395$	0	0
$396$	2.00000	0.100504
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	0	0
$399$	−48.0000	−2.40301
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	24.0000	1.19701
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	−4.00000	−0.198273
$408$	4.00000	0.198030
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	18.0000	0.886796
$413$	40.0000	1.96827
$414$	−6.00000	−0.294884
$415$	0	0
$416$	0	0
$417$	16.0000	0.783523
$418$	−12.0000	−0.586939
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	28.0000	1.36302
$423$	−12.0000	−0.583460
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	20.0000	0.969003
$427$	−8.00000	−0.387147
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	4.00000	0.192450
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	−24.0000	−1.15204
$435$	0	0
$436$	6.00000	0.287348
$437$	36.0000	1.72211
$438$	−20.0000	−0.955637
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	4.00000	0.189405
$447$	4.00000	0.189194
$448$	−4.00000	−0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	−2.00000	−0.0940721
$453$	12.0000	0.563809
$454$	−4.00000	−0.187729
$455$	0	0
$456$	12.0000	0.561951
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−10.0000	−0.467269
$459$	−8.00000	−0.373408
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	16.0000	0.744387
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	48.0000	2.21643
$470$	0	0
$471$	20.0000	0.921551
$472$	−10.0000	−0.460287
$473$	20.0000	0.919601
$474$	8.00000	0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	−2.00000	−0.0915737
$478$	26.0000	1.18921
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	0	0
$482$	22.0000	1.00207
$483$	−48.0000	−2.18408
$484$	−7.00000	−0.318182
$485$	0	0
$486$	10.0000	0.453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	2.00000	0.0905357
$489$	8.00000	0.361773
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	20.0000	0.901670
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	6.00000	0.269408
$497$	40.0000	1.79425
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	−40.0000	−1.78707
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	14.0000	0.621150
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	1.00000	0.0441942
$513$	−24.0000	−1.05963
$514$	30.0000	1.32324
$515$	0	0
$516$	−20.0000	−0.880451
$517$	−24.0000	−1.05552
$518$	8.00000	0.351500
$519$	20.0000	0.877903
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	2.00000	0.0875376
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−2.00000	−0.0872041
$527$	−12.0000	−0.522728
$528$	−4.00000	−0.174078
$529$	13.0000	0.565217
$530$	0	0
$531$	−10.0000	−0.433963
$532$	24.0000	1.04053
$533$	0	0
$534$	−28.0000	−1.21168
$535$	0	0
$536$	−12.0000	−0.518321
$537$	8.00000	0.345225
$538$	6.00000	0.258678
$539$	18.0000	0.775315
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	2.00000	0.0859074
$543$	−20.0000	−0.858282
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−18.0000	−0.768922
$549$	2.00000	0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	12.0000	0.510754
$553$	16.0000	0.680389
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−8.00000	−0.339276
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	6.00000	0.254000
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	6.00000	0.253095
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	−14.0000	−0.588464
$567$	44.0000	1.84783
$568$	−10.0000	−0.419591
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	0	0
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−13.0000	−0.540729
$579$	−28.0000	−1.16364
$580$	0	0
$581$	0	0
$582$	−28.0000	−1.16064
$583$	−4.00000	−0.165663
$584$	10.0000	0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−18.0000	−0.742307
$589$	−36.0000	−1.48335
$590$	0	0
$591$	12.0000	0.493614
$592$	−2.00000	−0.0821995
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	8.00000	0.328244
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−40.0000	−1.63028
$603$	−12.0000	−0.488678
$604$	−6.00000	−0.244137
$605$	0	0
$606$	−28.0000	−1.13742
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−6.00000	−0.243332
$609$	16.0000	0.648353
$610$	0	0
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	24.0000	0.968561
$615$	0	0
$616$	−8.00000	−0.322329
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−36.0000	−1.44813
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	−12.0000	−0.481156
$623$	−56.0000	−2.24359
$624$	0	0
$625$	0	0
$626$	6.00000	0.239808
$627$	24.0000	0.958468
$628$	−10.0000	−0.399043
$629$	4.00000	0.159490
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	−4.00000	−0.159111
$633$	−56.0000	−2.22580
$634$	−18.0000	−0.714871
$635$	0	0
$636$	4.00000	0.158610
$637$	0	0
$638$	4.00000	0.158362
$639$	−10.0000	−0.395594
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	12.0000	0.472134
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	−11.0000	−0.432121
$649$	−20.0000	−0.785069
$650$	0	0
$651$	48.0000	1.88127
$652$	−4.00000	−0.156652
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	−10.0000	−0.390434
$657$	10.0000	0.390137
$658$	48.0000	1.87123
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−12.0000	−0.464642
$668$	20.0000	0.773823
$669$	−8.00000	−0.309298
$670$	0	0
$671$	4.00000	0.154418
$672$	8.00000	0.308607
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	4.00000	0.153619
$679$	−56.0000	−2.14908
$680$	0	0
$681$	8.00000	0.306561
$682$	12.0000	0.459504
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−8.00000	−0.305441
$687$	20.0000	0.763048
$688$	10.0000	0.381246
$689$	0	0
$690$	0	0
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	−10.0000	−0.380143
$693$	−8.00000	−0.303895
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−4.00000	−0.151620
$697$	20.0000	0.757554
$698$	−2.00000	−0.0757011
$699$	−12.0000	−0.453882
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	2.00000	0.0753778
$705$	0	0
$706$	34.0000	1.27961
$707$	−56.0000	−2.10610
$708$	20.0000	0.751646
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	14.0000	0.524672
$713$	−36.0000	−1.34821
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−52.0000	−1.94198
$718$	6.00000	0.223918
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−72.0000	−2.68142
$722$	17.0000	0.632674
$723$	−44.0000	−1.63638
$724$	10.0000	0.371647
$725$	0	0
$726$	14.0000	0.519589
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−20.0000	−0.739727
$732$	−4.00000	−0.147844
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−30.0000	−1.10732
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−24.0000	−0.884051
$738$	−10.0000	−0.368105
$739$	−42.0000	−1.54499	−0.772497	−	0.635018i	$-0.780993\pi$
$739$	−42.0000	−1.54499	−0.772497	+	0.635018i	$0.780993\pi$
$740$	0	0
$741$	0	0
$742$	8.00000	0.293689
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	−4.00000	−0.146254
$749$	24.0000	0.876941
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−16.0000	−0.581914
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−6.00000	−0.217930
$759$	24.0000	0.871145
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	−28.0000	−1.01433
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−60.0000	−2.16085
$772$	14.0000	0.503871
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	14.0000	0.502571
$777$	−16.0000	−0.573997
$778$	−10.0000	−0.358517
$779$	60.0000	2.14972
$780$	0	0
$781$	−20.0000	−0.715656
$782$	12.0000	0.429119
$783$	8.00000	0.285897
$784$	9.00000	0.321429
$785$	0	0
$786$	−8.00000	−0.285351
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−6.00000	−0.213741
$789$	4.00000	0.142404
$790$	0	0
$791$	8.00000	0.284447
$792$	2.00000	0.0710669
$793$	0	0
$794$	38.0000	1.34857
$795$	0	0
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	−48.0000	−1.69918
$799$	24.0000	0.849059
$800$	0	0
$801$	14.0000	0.494666
$802$	6.00000	0.211867
$803$	20.0000	0.705785
$804$	24.0000	0.846415
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	14.0000	0.492518
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	−8.00000	−0.280745
$813$	−4.00000	−0.140286
$814$	−4.00000	−0.140200
$815$	0	0
$816$	4.00000	0.140028
$817$	−60.0000	−2.09913
$818$	−34.0000	−1.18878
$819$	0	0
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	36.0000	1.25564
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	40.0000	1.39178
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	−6.00000	−0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	−18.0000	−0.623663
$834$	16.0000	0.554035
$835$	0	0
$836$	−12.0000	−0.415029
$837$	24.0000	0.829561
$838$	16.0000	0.552711
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	−12.0000	−0.413302
$844$	28.0000	0.963800
$845$	0	0
$846$	−12.0000	−0.412568
$847$	28.0000	0.962091
$848$	−2.00000	−0.0686803
$849$	28.0000	0.960958
$850$	0	0
$851$	12.0000	0.411355
$852$	20.0000	0.685189
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−6.00000	−0.205076
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−80.0000	−2.72639
$862$	−18.0000	−0.613082
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	38.0000	1.29129
$867$	26.0000	0.883006
$868$	−24.0000	−0.814613
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	36.0000	1.21772
$875$	0	0
$876$	−20.0000	−0.675737
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−32.0000	−1.07995
$879$	−44.0000	−1.48408
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	9.00000	0.303046
$883$	−38.0000	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$884$	0	0
$885$	0	0
$886$	14.0000	0.470339
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	4.00000	0.134231
$889$	−56.0000	−1.87818
$890$	0	0
$891$	−22.0000	−0.737028
$892$	4.00000	0.133930
$893$	72.0000	2.40939
$894$	4.00000	0.133780
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	6.00000	0.200223
$899$	12.0000	0.400222
$900$	0	0
$901$	4.00000	0.133259
$902$	−20.0000	−0.665927
$903$	80.0000	2.66223
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	12.0000	0.398673
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	−4.00000	−0.132745
$909$	14.0000	0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	12.0000	0.397360
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−16.0000	−0.528367
$918$	−8.00000	−0.264039
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−48.0000	−1.58165
$922$	6.00000	0.197599
$923$	0	0
$924$	16.0000	0.526361
$925$	0	0
$926$	16.0000	0.525793
$927$	18.0000	0.591198
$928$	2.00000	0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	6.00000	0.196537
$933$	24.0000	0.785725
$934$	−10.0000	−0.327210
$935$	0	0
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	48.0000	1.56726
$939$	−12.0000	−0.391605
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	20.0000	0.651635
$943$	60.0000	1.95387
$944$	−10.0000	−0.325472
$945$	0	0
$946$	20.0000	0.650256
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	0	0
$951$	36.0000	1.16738
$952$	8.00000	0.259281
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	26.0000	0.840900
$957$	−8.00000	−0.258603
$958$	−2.00000	−0.0646171
$959$	72.0000	2.32500
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−6.00000	−0.193347
$964$	22.0000	0.708572
$965$	0	0
$966$	−48.0000	−1.54437
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−7.00000	−0.224989
$969$	−24.0000	−0.770991
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	10.0000	0.320750
$973$	32.0000	1.02587
$974$	−16.0000	−0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	8.00000	0.255812
$979$	28.0000	0.894884
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	−4.00000	−0.127386
$987$	−96.0000	−3.05571
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	6.00000	0.190500
$993$	−28.0000	−0.888553
$994$	40.0000	1.26872
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−38.0000	−1.20287
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.n.1.1		1
5.4	even	2		1690.2.a.e.1.1		1
13.12	even	2		650.2.a.c.1.1		1
39.38	odd	2		5850.2.a.cb.1.1		1
52.51	odd	2		5200.2.a.bd.1.1		1
65.4	even	6		1690.2.e.a.991.1		2
65.9	even	6		1690.2.e.g.991.1		2
65.12	odd	4		650.2.b.g.599.1		2
65.19	odd	12		1690.2.l.a.361.2		4
65.24	odd	12		1690.2.l.a.1161.2		4
65.29	even	6		1690.2.e.g.191.1		2
65.34	odd	4		1690.2.d.e.1351.1		2
65.38	odd	4		650.2.b.g.599.2		2
65.44	odd	4		1690.2.d.e.1351.2		2
65.49	even	6		1690.2.e.a.191.1		2
65.54	odd	12		1690.2.l.a.1161.1		4
65.59	odd	12		1690.2.l.a.361.1		4
65.64	even	2		130.2.a.c.1.1	✓	1
195.38	even	4		5850.2.e.u.5149.1		2
195.77	even	4		5850.2.e.u.5149.2		2
195.194	odd	2		1170.2.a.d.1.1		1
260.259	odd	2		1040.2.a.b.1.1		1
455.454	odd	2		6370.2.a.l.1.1		1
520.259	odd	2		4160.2.a.t.1.1		1
520.389	even	2		4160.2.a.c.1.1		1
780.779	even	2		9360.2.a.by.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.c.1.1	✓	1	65.64	even	2
650.2.a.c.1.1		1	13.12	even	2
650.2.b.g.599.1		2	65.12	odd	4
650.2.b.g.599.2		2	65.38	odd	4
1040.2.a.b.1.1		1	260.259	odd	2
1170.2.a.d.1.1		1	195.194	odd	2
1690.2.a.e.1.1		1	5.4	even	2
1690.2.d.e.1351.1		2	65.34	odd	4
1690.2.d.e.1351.2		2	65.44	odd	4
1690.2.e.a.191.1		2	65.49	even	6
1690.2.e.a.991.1		2	65.4	even	6
1690.2.e.g.191.1		2	65.29	even	6
1690.2.e.g.991.1		2	65.9	even	6
1690.2.l.a.361.1		4	65.59	odd	12
1690.2.l.a.361.2		4	65.19	odd	12
1690.2.l.a.1161.1		4	65.54	odd	12
1690.2.l.a.1161.2		4	65.24	odd	12
4160.2.a.c.1.1		1	520.389	even	2
4160.2.a.t.1.1		1	520.259	odd	2
5200.2.a.bd.1.1		1	52.51	odd	2
5850.2.a.cb.1.1		1	39.38	odd	2
5850.2.e.u.5149.1		2	195.38	even	4
5850.2.e.u.5149.2		2	195.77	even	4
6370.2.a.l.1.1		1	455.454	odd	2
8450.2.a.n.1.1		1	1.1	even	1	trivial
9360.2.a.by.1.1		1	780.779	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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