LMFDB - Embedded newform 819.2.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [819,2,Mod(1,819)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("819.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,7,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.52616$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.670843 q^{2} -1.54997 q^{4} -2.54997 q^{5} +1.00000 q^{7} -2.38147 q^{8} -1.71063 q^{10} +3.05232 q^{11} +1.00000 q^{13} +0.670843 q^{14} +1.50235 q^{16} -1.34169 q^{17} +7.60228 q^{19} +3.95238 q^{20} +2.04762 q^{22} -1.84403 q^{23} +1.50235 q^{25} +0.670843 q^{26} -1.54997 q^{28} +5.60228 q^{29} +10.2857 q^{31} +5.77078 q^{32} -0.900061 q^{34} -2.54997 q^{35} -9.20457 q^{37} +5.09994 q^{38} +6.07268 q^{40} +8.04762 q^{41} +1.49765 q^{43} -4.73100 q^{44} -1.23706 q^{46} +3.89166 q^{47} +1.00000 q^{49} +1.00784 q^{50} -1.54997 q^{52} -0.502345 q^{53} -7.78331 q^{55} -2.38147 q^{56} +3.75825 q^{58} +4.28937 q^{59} -0.683372 q^{61} +6.90006 q^{62} +0.866598 q^{64} -2.54997 q^{65} -7.68806 q^{67} +2.07957 q^{68} -1.71063 q^{70} +14.2569 q^{71} -12.0189 q^{73} -6.17482 q^{74} -11.7833 q^{76} +3.05232 q^{77} +4.91891 q^{79} -3.83094 q^{80} +5.39869 q^{82} +1.20828 q^{83} +3.42126 q^{85} +1.00469 q^{86} -7.26900 q^{88} -13.7545 q^{89} +1.00000 q^{91} +2.85819 q^{92} +2.61069 q^{94} -19.3856 q^{95} -7.18572 q^{97} +0.670843 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29}+ \cdots - q^{98}+O(q^{100}) $ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - q^98

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 $

$2$	0.670843	0.474358	0.237179	−	0.971466i	$-0.423777\pi$
$2$	0.670843	0.474358	0.237179	+	0.971466i	$0.423777\pi$
$3$	0	0
$3$	0	0
$4$	−1.54997	−0.774985
$5$	−2.54997	−1.14038	−0.570191	−	0.821512i	$-0.693131\pi$
$5$	−2.54997	−1.14038	−0.570191	+	0.821512i	$0.693131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.38147	−0.841978
$9$	0	0
$10$	−1.71063	−0.540948
$11$	3.05232	0.920308	0.460154	−	0.887839i	$-0.347794\pi$
$11$	3.05232	0.920308	0.460154	+	0.887839i	$0.347794\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.670843	0.179290
$15$	0	0
$16$	1.50235	0.375586
$17$	−1.34169	−0.325407	−0.162703	−	0.986675i	$-0.552021\pi$
$17$	−1.34169	−0.325407	−0.162703	+	0.986675i	$0.552021\pi$
$18$	0	0
$19$	7.60228	1.74408	0.872042	−	0.489431i	$-0.162796\pi$
$19$	7.60228	1.74408	0.872042	+	0.489431i	$0.162796\pi$
$20$	3.95238	0.883778
$21$	0	0
$22$	2.04762	0.436555
$23$	−1.84403	−0.384507	−0.192254	−	0.981345i	$-0.561580\pi$
$23$	−1.84403	−0.384507	−0.192254	+	0.981345i	$0.561580\pi$
$24$	0	0
$25$	1.50235	0.300469
$26$	0.670843	0.131563
$27$	0	0
$28$	−1.54997	−0.292917
$29$	5.60228	1.04032	0.520159	−	0.854069i	$-0.325873\pi$
$29$	5.60228	1.04032	0.520159	+	0.854069i	$0.325873\pi$
$30$	0	0
$31$	10.2857	1.84736	0.923679	−	0.383167i	$-0.125166\pi$
$31$	10.2857	1.84736	0.923679	+	0.383167i	$0.125166\pi$
$32$	5.77078	1.02014
$33$	0	0
$34$	−0.900061	−0.154359
$35$	−2.54997	−0.431024
$36$	0	0
$37$	−9.20457	−1.51322	−0.756611	−	0.653865i	$-0.773146\pi$
$37$	−9.20457	−1.51322	−0.756611	+	0.653865i	$0.773146\pi$
$38$	5.09994	0.827319
$39$	0	0
$40$	6.07268	0.960175
$41$	8.04762	1.25683	0.628414	−	0.777879i	$-0.283704\pi$
$41$	8.04762	1.25683	0.628414	+	0.777879i	$0.283704\pi$
$42$	0	0
$43$	1.49765	0.228390	0.114195	−	0.993458i	$-0.463571\pi$
$43$	1.49765	0.228390	0.114195	+	0.993458i	$0.463571\pi$
$44$	−4.73100	−0.713224
$45$	0	0
$46$	−1.23706	−0.182394
$47$	3.89166	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$47$	3.89166	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00784	0.142530
$51$	0	0
$52$	−1.54997	−0.214942
$53$	−0.502345	−0.0690025	−0.0345012	−	0.999405i	$-0.510984\pi$
$53$	−0.502345	−0.0690025	−0.0345012	+	0.999405i	$0.510984\pi$
$54$	0	0
$55$	−7.78331	−1.04950
$56$	−2.38147	−0.318238
$57$	0	0
$58$	3.75825	0.493483
$59$	4.28937	0.558429	0.279214	−	0.960229i	$-0.409926\pi$
$59$	4.28937	0.558429	0.279214	+	0.960229i	$0.409926\pi$
$60$	0	0
$61$	−0.683372	−0.0874968	−0.0437484	−	0.999043i	$-0.513930\pi$
$61$	−0.683372	−0.0874968	−0.0437484	+	0.999043i	$0.513930\pi$
$62$	6.90006	0.876309
$63$	0	0
$64$	0.866598	0.108325
$65$	−2.54997	−0.316285
$66$	0	0
$67$	−7.68806	−0.939246	−0.469623	−	0.882867i	$-0.655610\pi$
$67$	−7.68806	−0.939246	−0.469623	+	0.882867i	$0.655610\pi$
$68$	2.07957	0.252185
$69$	0	0
$70$	−1.71063	−0.204459
$71$	14.2569	1.69198	0.845990	−	0.533198i	$-0.179010\pi$
$71$	14.2569	1.69198	0.845990	+	0.533198i	$0.179010\pi$
$72$	0	0
$73$	−12.0189	−1.40670	−0.703350	−	0.710844i	$-0.748313\pi$
$73$	−12.0189	−1.40670	−0.703350	+	0.710844i	$0.748313\pi$
$74$	−6.17482	−0.717808
$75$	0	0
$76$	−11.7833	−1.35164
$77$	3.05232	0.347844
$78$	0	0
$79$	4.91891	0.553421	0.276710	−	0.960953i	$-0.410756\pi$
$79$	4.91891	0.553421	0.276710	+	0.960953i	$0.410756\pi$
$80$	−3.83094	−0.428312
$81$	0	0
$82$	5.39869	0.596186
$83$	1.20828	0.132626	0.0663132	−	0.997799i	$-0.478876\pi$
$83$	1.20828	0.132626	0.0663132	+	0.997799i	$0.478876\pi$
$84$	0	0
$85$	3.42126	0.371088
$86$	1.00469	0.108339
$87$	0	0
$88$	−7.26900	−0.774878
$89$	−13.7545	−1.45798	−0.728989	−	0.684525i	$-0.760010\pi$
$89$	−13.7545	−1.45798	−0.728989	+	0.684525i	$0.760010\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	2.85819	0.297987
$93$	0	0
$94$	2.61069	0.269272
$95$	−19.3856	−1.98892
$96$	0	0
$97$	−7.18572	−0.729599	−0.364800	−	0.931086i	$-0.618862\pi$
$97$	−7.18572	−0.729599	−0.364800	+	0.931086i	$0.618862\pi$
$98$	0.670843	0.0677654
$99$	0	0
$100$	−2.32859	−0.232859
$101$	18.4416	1.83501	0.917505	−	0.397724i	$-0.130200\pi$
$101$	18.4416	1.83501	0.917505	+	0.397724i	$0.130200\pi$
$102$	0	0
$103$	1.89537	0.186756	0.0933782	−	0.995631i	$-0.470233\pi$
$103$	1.89537	0.186756	0.0933782	+	0.995631i	$0.470233\pi$
$104$	−2.38147	−0.233523
$105$	0	0
$106$	−0.336995	−0.0327318
$107$	−18.5463	−1.79293	−0.896467	−	0.443110i	$-0.853875\pi$
$107$	−18.5463	−1.79293	−0.896467	+	0.443110i	$0.853875\pi$
$108$	0	0
$109$	−1.42126	−0.136132	−0.0680659	−	0.997681i	$-0.521683\pi$
$109$	−1.42126	−0.136132	−0.0680659	+	0.997681i	$0.521683\pi$
$110$	−5.22138	−0.497839
$111$	0	0
$112$	1.50235	0.141958
$113$	−5.60228	−0.527019	−0.263509	−	0.964657i	$-0.584880\pi$
$113$	−5.60228	−0.527019	−0.263509	+	0.964657i	$0.584880\pi$
$114$	0	0
$115$	4.70222	0.438485
$116$	−8.68337	−0.806231
$117$	0	0
$118$	2.87749	0.264895
$119$	−1.34169	−0.122992
$120$	0	0
$121$	−1.68337	−0.153034
$122$	−0.458435	−0.0415048
$123$	0	0
$124$	−15.9425	−1.43167
$125$	8.91891	0.797732
$126$	0	0
$127$	14.1046	1.25158	0.625792	−	0.779990i	$-0.284776\pi$
$127$	14.1046	1.25158	0.625792	+	0.779990i	$0.284776\pi$
$128$	−10.9602	−0.968755
$129$	0	0
$130$	−1.71063	−0.150032
$131$	8.78800	0.767811	0.383906	−	0.923372i	$-0.374579\pi$
$131$	8.78800	0.767811	0.383906	+	0.923372i	$0.374579\pi$
$132$	0	0
$133$	7.60228	0.659202
$134$	−5.15748	−0.445539
$135$	0	0
$136$	3.19519	0.273985
$137$	1.97743	0.168944	0.0844718	−	0.996426i	$-0.473080\pi$
$137$	1.97743	0.168944	0.0844718	+	0.996426i	$0.473080\pi$
$138$	0	0
$139$	13.0999	1.11112	0.555561	−	0.831476i	$-0.312503\pi$
$139$	13.0999	1.11112	0.555561	+	0.831476i	$0.312503\pi$
$140$	3.95238	0.334037
$141$	0	0
$142$	9.56413	0.802604
$143$	3.05232	0.255247
$144$	0	0
$145$	−14.2857	−1.18636
$146$	−8.06276	−0.667279
$147$	0	0
$148$	14.2668	1.17272
$149$	−14.5939	−1.19558	−0.597789	−	0.801654i	$-0.703954\pi$
$149$	−14.5939	−1.19558	−0.597789	+	0.801654i	$0.703954\pi$
$150$	0	0
$151$	−5.41188	−0.440412	−0.220206	−	0.975453i	$-0.570673\pi$
$151$	−5.41188	−0.440412	−0.220206	+	0.975453i	$0.570673\pi$
$152$	−18.1046	−1.46848
$153$	0	0
$154$	2.04762	0.165002
$155$	−26.2281	−2.10669
$156$	0	0
$157$	23.4044	1.86788	0.933939	−	0.357432i	$-0.116348\pi$
$157$	23.4044	1.86788	0.933939	+	0.357432i	$0.116348\pi$
$158$	3.29982	0.262519
$159$	0	0
$160$	−14.7153	−1.16335
$161$	−1.84403	−0.145330
$162$	0	0
$163$	−10.1046	−0.791456	−0.395728	−	0.918368i	$-0.629508\pi$
$163$	−10.1046	−0.791456	−0.395728	+	0.918368i	$0.629508\pi$
$164$	−12.4736	−0.974022
$165$	0	0
$166$	0.810569	0.0629123
$167$	3.09623	0.239593	0.119797	−	0.992798i	$-0.461776\pi$
$167$	3.09623	0.239593	0.119797	+	0.992798i	$0.461776\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.29513	0.176028
$171$	0	0
$172$	−2.32132	−0.176999
$173$	−2.75356	−0.209349	−0.104675	−	0.994507i	$-0.533380\pi$
$173$	−2.75356	−0.209349	−0.104675	+	0.994507i	$0.533380\pi$
$174$	0	0
$175$	1.50235	0.113567
$176$	4.58563	0.345655
$177$	0	0
$178$	−9.22714	−0.691603
$179$	1.57723	0.117887	0.0589437	−	0.998261i	$-0.481227\pi$
$179$	1.57723	0.117887	0.0589437	+	0.998261i	$0.481227\pi$
$180$	0	0
$181$	9.51651	0.707356	0.353678	−	0.935367i	$-0.384931\pi$
$181$	9.51651	0.707356	0.353678	+	0.935367i	$0.384931\pi$
$182$	0.670843	0.0497262
$183$	0	0
$184$	4.39151	0.323746
$185$	23.4714	1.72565
$186$	0	0
$187$	−4.09525	−0.299474
$188$	−6.03195	−0.439925
$189$	0	0
$190$	−13.0047	−0.943459
$191$	22.8508	1.65342	0.826712	−	0.562626i	$-0.190209\pi$
$191$	22.8508	1.65342	0.826712	+	0.562626i	$0.190209\pi$
$192$	0	0
$193$	19.5760	1.40911	0.704556	−	0.709649i	$-0.251146\pi$
$193$	19.5760	1.40911	0.704556	+	0.709649i	$0.251146\pi$
$194$	−4.82049	−0.346091
$195$	0	0
$196$	−1.54997	−0.110712
$197$	21.2271	1.51237	0.756185	−	0.654357i	$-0.227061\pi$
$197$	21.2271	1.51237	0.756185	+	0.654357i	$0.227061\pi$
$198$	0	0
$199$	−6.68337	−0.473772	−0.236886	−	0.971537i	$-0.576127\pi$
$199$	−6.68337	−0.473772	−0.236886	+	0.971537i	$0.576127\pi$
$200$	−3.57779	−0.252988
$201$	0	0
$202$	12.3714	0.870451
$203$	5.60228	0.393203
$204$	0	0
$205$	−20.5212	−1.43326
$206$	1.27150	0.0885893
$207$	0	0
$208$	1.50235	0.104169
$209$	23.2046	1.60509
$210$	0	0
$211$	−4.60698	−0.317157	−0.158579	−	0.987346i	$-0.550691\pi$
$211$	−4.60698	−0.317157	−0.158579	+	0.987346i	$0.550691\pi$
$212$	0.778620	0.0534759
$213$	0	0
$214$	−12.4416	−0.850492
$215$	−3.81897	−0.260452
$216$	0	0
$217$	10.2857	0.698236
$218$	−0.953441	−0.0645752
$219$	0	0
$220$	12.0639	0.813348
$221$	−1.34169	−0.0902516
$222$	0	0
$223$	8.60698	0.576366	0.288183	−	0.957575i	$-0.406949\pi$
$223$	8.60698	0.576366	0.288183	+	0.957575i	$0.406949\pi$
$224$	5.77078	0.385577
$225$	0	0
$226$	−3.75825	−0.249995
$227$	14.4892	0.961685	0.480843	−	0.876807i	$-0.340331\pi$
$227$	14.4892	0.961685	0.480843	+	0.876807i	$0.340331\pi$
$228$	0	0
$229$	−20.1999	−1.33485	−0.667423	−	0.744679i	$-0.732603\pi$
$229$	−20.1999	−1.33485	−0.667423	+	0.744679i	$0.732603\pi$
$230$	3.15445	0.207999
$231$	0	0
$232$	−13.3417	−0.875925
$233$	−7.28097	−0.476992	−0.238496	−	0.971143i	$-0.576654\pi$
$233$	−7.28097	−0.476992	−0.238496	+	0.971143i	$0.576654\pi$
$234$	0	0
$235$	−9.92360	−0.647345
$236$	−6.64839	−0.432774
$237$	0	0
$238$	−0.900061	−0.0583423
$239$	−27.6688	−1.78974	−0.894872	−	0.446323i	$-0.852733\pi$
$239$	−27.6688	−1.78974	−0.894872	+	0.446323i	$0.852733\pi$
$240$	0	0
$241$	−22.2187	−1.43123	−0.715617	−	0.698493i	$-0.753854\pi$
$241$	−22.2187	−1.43123	−0.715617	+	0.698493i	$0.753854\pi$
$242$	−1.12928	−0.0725928
$243$	0	0
$244$	1.05921	0.0678087
$245$	−2.54997	−0.162912
$246$	0	0
$247$	7.60228	0.483722
$248$	−24.4950	−1.55543
$249$	0	0
$250$	5.98319	0.378410
$251$	−17.3092	−1.09255	−0.546274	−	0.837607i	$-0.683954\pi$
$251$	−17.3092	−1.09255	−0.546274	+	0.837607i	$0.683954\pi$
$252$	0	0
$253$	−5.62857	−0.353865
$254$	9.46199	0.593698
$255$	0	0
$256$	−9.08578	−0.567861
$257$	5.55837	0.346722	0.173361	−	0.984858i	$-0.444537\pi$
$257$	5.55837	0.346722	0.173361	+	0.984858i	$0.444537\pi$
$258$	0	0
$259$	−9.20457	−0.571944
$260$	3.95238	0.245116
$261$	0	0
$262$	5.89537	0.364217
$263$	−25.0486	−1.54456	−0.772281	−	0.635281i	$-0.780884\pi$
$263$	−25.0486	−1.54456	−0.772281	+	0.635281i	$0.780884\pi$
$264$	0	0
$265$	1.28097	0.0786891
$266$	5.09994	0.312697
$267$	0	0
$268$	11.9163	0.727902
$269$	4.55368	0.277643	0.138821	−	0.990317i	$-0.455669\pi$
$269$	4.55368	0.277643	0.138821	+	0.990317i	$0.455669\pi$
$270$	0	0
$271$	8.88325	0.539619	0.269810	−	0.962914i	$-0.413039\pi$
$271$	8.88325	0.539619	0.269810	+	0.962914i	$0.413039\pi$
$272$	−2.01568	−0.122218
$273$	0	0
$274$	1.32655	0.0801397
$275$	4.58563	0.276524
$276$	0	0
$277$	−5.38560	−0.323589	−0.161795	−	0.986824i	$-0.551728\pi$
$277$	−5.38560	−0.323589	−0.161795	+	0.986824i	$0.551728\pi$
$278$	8.78800	0.527069
$279$	0	0
$280$	6.07268	0.362912
$281$	12.9727	0.773889	0.386944	−	0.922103i	$-0.373531\pi$
$281$	12.9727	0.773889	0.386944	+	0.922103i	$0.373531\pi$
$282$	0	0
$283$	−1.52589	−0.0907047	−0.0453523	−	0.998971i	$-0.514441\pi$
$283$	−1.52589	−0.0907047	−0.0453523	+	0.998971i	$0.514441\pi$
$284$	−22.0977	−1.31126
$285$	0	0
$286$	2.04762	0.121079
$287$	8.04762	0.475036
$288$	0	0
$289$	−15.1999	−0.894111
$290$	−9.58343	−0.562759
$291$	0	0
$292$	18.6289	1.09017
$293$	−18.8545	−1.10149	−0.550745	−	0.834673i	$-0.685656\pi$
$293$	−18.8545	−1.10149	−0.550745	+	0.834673i	$0.685656\pi$
$294$	0	0
$295$	−10.9378	−0.636821
$296$	21.9204	1.27410
$297$	0	0
$298$	−9.79020	−0.567131
$299$	−1.84403	−0.106643
$300$	0	0
$301$	1.49765	0.0863234
$302$	−3.63052	−0.208913
$303$	0	0
$304$	11.4213	0.655054
$305$	1.74258	0.0997797
$306$	0	0
$307$	15.3856	0.878102	0.439051	−	0.898462i	$-0.355315\pi$
$307$	15.3856	0.878102	0.439051	+	0.898462i	$0.355315\pi$
$308$	−4.73100	−0.269574
$309$	0	0
$310$	−17.5949	−0.999326
$311$	17.0999	0.969649	0.484824	−	0.874612i	$-0.338884\pi$
$311$	17.0999	0.969649	0.484824	+	0.874612i	$0.338884\pi$
$312$	0	0
$313$	4.89263	0.276548	0.138274	−	0.990394i	$-0.455845\pi$
$313$	4.89263	0.276548	0.138274	+	0.990394i	$0.455845\pi$
$314$	15.7007	0.886042
$315$	0	0
$316$	−7.62417	−0.428893
$317$	−1.44382	−0.0810933	−0.0405466	−	0.999178i	$-0.512910\pi$
$317$	−1.44382	−0.0810933	−0.0405466	+	0.999178i	$0.512910\pi$
$318$	0	0
$319$	17.0999	0.957413
$320$	−2.20980	−0.123531
$321$	0	0
$322$	−1.23706	−0.0689384
$323$	−10.1999	−0.567536
$324$	0	0
$325$	1.50235	0.0833351
$326$	−6.77862	−0.375433
$327$	0	0
$328$	−19.1652	−1.05822
$329$	3.89166	0.214554
$330$	0	0
$331$	−14.9953	−0.824217	−0.412108	−	0.911135i	$-0.635207\pi$
$331$	−14.9953	−0.824217	−0.412108	+	0.911135i	$0.635207\pi$
$332$	−1.87280	−0.102783
$333$	0	0
$334$	2.07708	0.113653
$335$	19.6043	1.07110
$336$	0	0
$337$	−14.9115	−0.812280	−0.406140	−	0.913811i	$-0.633126\pi$
$337$	−14.9115	−0.812280	−0.406140	+	0.913811i	$0.633126\pi$
$338$	0.670843	0.0364890
$339$	0	0
$340$	−5.30285	−0.287587
$341$	31.3951	1.70014
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.56662	−0.192299
$345$	0	0
$346$	−1.84721	−0.0993065
$347$	−7.07488	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$347$	−7.07488	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$348$	0	0
$349$	21.1188	1.13046	0.565232	−	0.824932i	$-0.308787\pi$
$349$	21.1188	1.13046	0.565232	+	0.824932i	$0.308787\pi$
$350$	1.00784	0.0538712
$351$	0	0
$352$	17.6142	0.938843
$353$	17.3643	0.924206	0.462103	−	0.886826i	$-0.347095\pi$
$353$	17.3643	0.924206	0.462103	+	0.886826i	$0.347095\pi$
$354$	0	0
$355$	−36.3546	−1.92950
$356$	21.3191	1.12991
$357$	0	0
$358$	1.05807	0.0559208
$359$	−18.3521	−0.968589	−0.484294	−	0.874905i	$-0.660924\pi$
$359$	−18.3521	−0.968589	−0.484294	+	0.874905i	$0.660924\pi$
$360$	0	0
$361$	38.7947	2.04183
$362$	6.38408	0.335540
$363$	0	0
$364$	−1.54997	−0.0812405
$365$	30.6477	1.60417
$366$	0	0
$367$	8.40719	0.438852	0.219426	−	0.975629i	$-0.429582\pi$
$367$	8.40719	0.438852	0.219426	+	0.975629i	$0.429582\pi$
$368$	−2.77037	−0.144416
$369$	0	0
$370$	15.7456	0.818575
$371$	−0.502345	−0.0260805
$372$	0	0
$373$	27.0925	1.40280	0.701399	−	0.712769i	$-0.252559\pi$
$373$	27.0925	1.40280	0.701399	+	0.712769i	$0.252559\pi$
$374$	−2.74727	−0.142058
$375$	0	0
$376$	−9.26787	−0.477954
$377$	5.60228	0.288532
$378$	0	0
$379$	−6.41657	−0.329597	−0.164798	−	0.986327i	$-0.552697\pi$
$379$	−6.41657	−0.329597	−0.164798	+	0.986327i	$0.552697\pi$
$380$	30.0471	1.54138
$381$	0	0
$382$	15.3293	0.784314
$383$	−32.5939	−1.66547	−0.832735	−	0.553672i	$-0.813226\pi$
$383$	−32.5939	−1.66547	−0.832735	+	0.553672i	$0.813226\pi$
$384$	0	0
$385$	−7.78331	−0.396674
$386$	13.1324	0.668423
$387$	0	0
$388$	11.1376	0.565428
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	2.47411	0.125121
$392$	−2.38147	−0.120283
$393$	0	0
$394$	14.2401	0.717405
$395$	−12.5431	−0.631111
$396$	0	0
$397$	15.7520	0.790573	0.395286	−	0.918558i	$-0.370645\pi$
$397$	15.7520	0.790573	0.395286	+	0.918558i	$0.370645\pi$
$398$	−4.48349	−0.224737
$399$	0	0
$400$	2.25704	0.112852
$401$	−26.1867	−1.30770	−0.653851	−	0.756624i	$-0.726848\pi$
$401$	−26.1867	−1.30770	−0.653851	+	0.756624i	$0.726848\pi$
$402$	0	0
$403$	10.2857	0.512365
$404$	−28.5840	−1.42211
$405$	0	0
$406$	3.75825	0.186519
$407$	−28.0952	−1.39263
$408$	0	0
$409$	−17.3856	−0.859662	−0.429831	−	0.902909i	$-0.641427\pi$
$409$	−17.3856	−0.859662	−0.429831	+	0.902909i	$0.641427\pi$
$410$	−13.7665	−0.679879
$411$	0	0
$412$	−2.93777	−0.144733
$413$	4.28937	0.211066
$414$	0	0
$415$	−3.08109	−0.151245
$416$	5.77078	0.282936
$417$	0	0
$418$	15.5666	0.761388
$419$	12.4761	0.609496	0.304748	−	0.952433i	$-0.401428\pi$
$419$	12.4761	0.609496	0.304748	+	0.952433i	$0.401428\pi$
$420$	0	0
$421$	9.57405	0.466611	0.233305	−	0.972404i	$-0.425046\pi$
$421$	9.57405	0.466611	0.233305	+	0.972404i	$0.425046\pi$
$422$	−3.09056	−0.150446
$423$	0	0
$424$	1.19632	0.0580985
$425$	−2.01568	−0.0977746
$426$	0	0
$427$	−0.683372	−0.0330707
$428$	28.7461	1.38950
$429$	0	0
$430$	−2.56193	−0.123547
$431$	−5.20208	−0.250575	−0.125288	−	0.992120i	$-0.539985\pi$
$431$	−5.20208	−0.250575	−0.125288	+	0.992120i	$0.539985\pi$
$432$	0	0
$433$	−17.2547	−0.829207	−0.414604	−	0.910002i	$-0.636080\pi$
$433$	−17.2547	−0.829207	−0.414604	+	0.910002i	$0.636080\pi$
$434$	6.90006	0.331214
$435$	0	0
$436$	2.20291	0.105500
$437$	−14.0189	−0.670613
$438$	0	0
$439$	−22.1925	−1.05919	−0.529594	−	0.848251i	$-0.677656\pi$
$439$	−22.1925	−1.05919	−0.529594	+	0.848251i	$0.677656\pi$
$440$	18.5357	0.883657
$441$	0	0
$442$	−0.900061	−0.0428115
$443$	37.9319	1.80220	0.901098	−	0.433615i	$-0.142762\pi$
$443$	37.9319	1.80220	0.901098	+	0.433615i	$0.142762\pi$
$444$	0	0
$445$	35.0737	1.66265
$446$	5.77393	0.273403
$447$	0	0
$448$	0.866598	0.0409429
$449$	33.3150	1.57223	0.786115	−	0.618080i	$-0.212089\pi$
$449$	33.3150	1.57223	0.786115	+	0.618080i	$0.212089\pi$
$450$	0	0
$451$	24.5639	1.15667
$452$	8.68337	0.408431
$453$	0	0
$454$	9.72001	0.456183
$455$	−2.54997	−0.119544
$456$	0	0
$457$	−7.79269	−0.364527	−0.182263	−	0.983250i	$-0.558342\pi$
$457$	−7.79269	−0.364527	−0.182263	+	0.983250i	$0.558342\pi$
$458$	−13.5509	−0.633194
$459$	0	0
$460$	−7.28831	−0.339819
$461$	−30.4568	−1.41851	−0.709256	−	0.704951i	$-0.750969\pi$
$461$	−30.4568	−1.41851	−0.709256	+	0.704951i	$0.750969\pi$
$462$	0	0
$463$	39.4639	1.83405	0.917023	−	0.398835i	$-0.130585\pi$
$463$	39.4639	1.83405	0.917023	+	0.398835i	$0.130585\pi$
$464$	8.41657	0.390729
$465$	0	0
$466$	−4.88438	−0.226265
$467$	9.93307	0.459648	0.229824	−	0.973232i	$-0.426185\pi$
$467$	9.93307	0.459648	0.229824	+	0.973232i	$0.426185\pi$
$468$	0	0
$469$	−7.68806	−0.355002
$470$	−6.65718	−0.307073
$471$	0	0
$472$	−10.2150	−0.470184
$473$	4.57131	0.210189
$474$	0	0
$475$	11.4213	0.524043
$476$	2.07957	0.0953171
$477$	0	0
$478$	−18.5614	−0.848978
$479$	0.941479	0.0430173	0.0215086	−	0.999769i	$-0.493153\pi$
$479$	0.941479	0.0430173	0.0215086	+	0.999769i	$0.493153\pi$
$480$	0	0
$481$	−9.20457	−0.419692
$482$	−14.9053	−0.678917
$483$	0	0
$484$	2.60918	0.118599
$485$	18.3234	0.832021
$486$	0	0
$487$	−39.3499	−1.78312	−0.891558	−	0.452907i	$-0.850387\pi$
$487$	−39.3499	−1.78312	−0.891558	+	0.452907i	$0.850387\pi$
$488$	1.62743	0.0736703
$489$	0	0
$490$	−1.71063	−0.0772784
$491$	−5.45374	−0.246124	−0.123062	−	0.992399i	$-0.539271\pi$
$491$	−5.45374	−0.246124	−0.123062	+	0.992399i	$0.539271\pi$
$492$	0	0
$493$	−7.51651	−0.338526
$494$	5.09994	0.229457
$495$	0	0
$496$	15.4526	0.693843
$497$	14.2569	0.639509
$498$	0	0
$499$	−31.3574	−1.40375	−0.701874	−	0.712301i	$-0.747653\pi$
$499$	−31.3574	−1.40375	−0.701874	+	0.712301i	$0.747653\pi$
$500$	−13.8240	−0.618230
$501$	0	0
$502$	−11.6118	−0.518258
$503$	−43.9926	−1.96153	−0.980766	−	0.195188i	$-0.937468\pi$
$503$	−43.9926	−1.96153	−0.980766	+	0.195188i	$0.937468\pi$
$504$	0	0
$505$	−47.0256	−2.09261
$506$	−3.77588	−0.167858
$507$	0	0
$508$	−21.8617	−0.969958
$509$	41.8498	1.85496	0.927480	−	0.373874i	$-0.121971\pi$
$509$	41.8498	1.85496	0.927480	+	0.373874i	$0.121971\pi$
$510$	0	0
$511$	−12.0189	−0.531683
$512$	15.8253	0.699386
$513$	0	0
$514$	3.72880	0.164470
$515$	−4.83314	−0.212973
$516$	0	0
$517$	11.8786	0.522418
$518$	−6.17482	−0.271306
$519$	0	0
$520$	6.07268	0.266305
$521$	−8.32957	−0.364925	−0.182462	−	0.983213i	$-0.558407\pi$
$521$	−8.32957	−0.364925	−0.182462	+	0.983213i	$0.558407\pi$
$522$	0	0
$523$	25.3092	1.10669	0.553347	−	0.832951i	$-0.313350\pi$
$523$	25.3092	1.10669	0.553347	+	0.832951i	$0.313350\pi$
$524$	−13.6211	−0.595042
$525$	0	0
$526$	−16.8037	−0.732675
$527$	−13.8001	−0.601143
$528$	0	0
$529$	−19.5995	−0.852154
$530$	0.859327	0.0373268
$531$	0	0
$532$	−11.7833	−0.510871
$533$	8.04762	0.348581
$534$	0	0
$535$	47.2924	2.04463
$536$	18.3089	0.790824
$537$	0	0
$538$	3.05481	0.131702
$539$	3.05232	0.131473
$540$	0	0
$541$	27.1501	1.16727	0.583636	−	0.812015i	$-0.301630\pi$
$541$	27.1501	1.16727	0.583636	+	0.812015i	$0.301630\pi$
$542$	5.95927	0.255972
$543$	0	0
$544$	−7.74258	−0.331960
$545$	3.62417	0.155242
$546$	0	0
$547$	27.5949	1.17987	0.589935	−	0.807450i	$-0.299153\pi$
$547$	27.5949	1.17987	0.589935	+	0.807450i	$0.299153\pi$
$548$	−3.06496	−0.130929
$549$	0	0
$550$	3.07624	0.131171
$551$	42.5902	1.81440
$552$	0	0
$553$	4.91891	0.209173
$554$	−3.61289	−0.153497
$555$	0	0
$556$	−20.3045	−0.861103
$557$	−7.19881	−0.305024	−0.152512	−	0.988302i	$-0.548736\pi$
$557$	−7.19881	−0.305024	−0.152512	+	0.988302i	$0.548736\pi$
$558$	0	0
$559$	1.49765	0.0633440
$560$	−3.83094	−0.161887
$561$	0	0
$562$	8.70267	0.367100
$563$	20.2594	0.853831	0.426915	−	0.904292i	$-0.359600\pi$
$563$	20.2594	0.853831	0.426915	+	0.904292i	$0.359600\pi$
$564$	0	0
$565$	14.2857	0.601002
$566$	−1.02363	−0.0430265
$567$	0	0
$568$	−33.9524	−1.42461
$569$	−7.28097	−0.305234	−0.152617	−	0.988285i	$-0.548770\pi$
$569$	−7.28097	−0.305234	−0.152617	+	0.988285i	$0.548770\pi$
$570$	0	0
$571$	−22.1141	−0.925446	−0.462723	−	0.886503i	$-0.653128\pi$
$571$	−22.1141	−0.925446	−0.462723	+	0.886503i	$0.653128\pi$
$572$	−4.73100	−0.197813
$573$	0	0
$574$	5.39869	0.225337
$575$	−2.77037	−0.115533
$576$	0	0
$577$	20.3139	0.845678	0.422839	−	0.906205i	$-0.361034\pi$
$577$	20.3139	0.845678	0.422839	+	0.906205i	$0.361034\pi$
$578$	−10.1967	−0.424128
$579$	0	0
$580$	22.1423	0.919410
$581$	1.20828	0.0501281
$582$	0	0
$583$	−1.53332	−0.0635035
$584$	28.6226	1.18441
$585$	0	0
$586$	−12.6484	−0.522500
$587$	−35.8842	−1.48110	−0.740550	−	0.672001i	$-0.765435\pi$
$587$	−35.8842	−1.48110	−0.740550	+	0.672001i	$0.765435\pi$
$588$	0	0
$589$	78.1945	3.22195
$590$	−7.33752	−0.302081
$591$	0	0
$592$	−13.8284	−0.568346
$593$	−20.2664	−0.832239	−0.416120	−	0.909310i	$-0.636610\pi$
$593$	−20.2664	−0.832239	−0.416120	+	0.909310i	$0.636610\pi$
$594$	0	0
$595$	3.42126	0.140258
$596$	22.6201	0.926554
$597$	0	0
$598$	−1.23706	−0.0505870
$599$	−14.9365	−0.610291	−0.305145	−	0.952306i	$-0.598705\pi$
$599$	−14.9365	−0.610291	−0.305145	+	0.952306i	$0.598705\pi$
$600$	0	0
$601$	4.89263	0.199575	0.0997873	−	0.995009i	$-0.468184\pi$
$601$	4.89263	0.199575	0.0997873	+	0.995009i	$0.468184\pi$
$602$	1.00469	0.0409481
$603$	0	0
$604$	8.38824	0.341313
$605$	4.29255	0.174517
$606$	0	0
$607$	20.6164	0.836796	0.418398	−	0.908264i	$-0.362592\pi$
$607$	20.6164	0.836796	0.418398	+	0.908264i	$0.362592\pi$
$608$	43.8711	1.77921
$609$	0	0
$610$	1.16900	0.0473313
$611$	3.89166	0.157440
$612$	0	0
$613$	−9.73818	−0.393321	−0.196661	−	0.980472i	$-0.563010\pi$
$613$	−9.73818	−0.393321	−0.196661	+	0.980472i	$0.563010\pi$
$614$	10.3213	0.416535
$615$	0	0
$616$	−7.26900	−0.292877
$617$	0.763482	0.0307366	0.0153683	−	0.999882i	$-0.495108\pi$
$617$	0.763482	0.0307366	0.0153683	+	0.999882i	$0.495108\pi$
$618$	0	0
$619$	−41.0925	−1.65165	−0.825824	−	0.563928i	$-0.809289\pi$
$619$	−41.0925	−1.65165	−0.825824	+	0.563928i	$0.809289\pi$
$620$	40.6528	1.63265
$621$	0	0
$622$	11.4714	0.459960
$623$	−13.7545	−0.551064
$624$	0	0
$625$	−30.2547	−1.21019
$626$	3.28219	0.131183
$627$	0	0
$628$	−36.2762	−1.44758
$629$	12.3496	0.492412
$630$	0	0
$631$	−14.4667	−0.575910	−0.287955	−	0.957644i	$-0.592975\pi$
$631$	−14.4667	−0.575910	−0.287955	+	0.957644i	$0.592975\pi$
$632$	−11.7143	−0.465968
$633$	0	0
$634$	−0.968580	−0.0384672
$635$	−35.9664	−1.42728
$636$	0	0
$637$	1.00000	0.0396214
$638$	11.4714	0.454156
$639$	0	0
$640$	27.9482	1.10475
$641$	−35.7069	−1.41034	−0.705169	−	0.709039i	$-0.749129\pi$
$641$	−35.7069	−1.41034	−0.705169	+	0.709039i	$0.749129\pi$
$642$	0	0
$643$	−26.4091	−1.04147	−0.520737	−	0.853717i	$-0.674343\pi$
$643$	−26.4091	−1.04147	−0.520737	+	0.853717i	$0.674343\pi$
$644$	2.85819	0.112629
$645$	0	0
$646$	−6.84252	−0.269215
$647$	−37.3543	−1.46855	−0.734275	−	0.678852i	$-0.762478\pi$
$647$	−37.3543	−1.46855	−0.734275	+	0.678852i	$0.762478\pi$
$648$	0	0
$649$	13.0925	0.513926
$650$	1.00784	0.0395307
$651$	0	0
$652$	15.6619	0.613366
$653$	0.987881	0.0386588	0.0193294	−	0.999813i	$-0.493847\pi$
$653$	0.987881	0.0386588	0.0193294	+	0.999813i	$0.493847\pi$
$654$	0	0
$655$	−22.4091	−0.875598
$656$	12.0903	0.472047
$657$	0	0
$658$	2.61069	0.101775
$659$	1.24653	0.0485578	0.0242789	−	0.999705i	$-0.492271\pi$
$659$	1.24653	0.0485578	0.0242789	+	0.999705i	$0.492271\pi$
$660$	0	0
$661$	−33.8994	−1.31853	−0.659266	−	0.751910i	$-0.729133\pi$
$661$	−33.8994	−1.31853	−0.659266	+	0.751910i	$0.729133\pi$
$662$	−10.0595	−0.390973
$663$	0	0
$664$	−2.87749	−0.111668
$665$	−19.3856	−0.751741
$666$	0	0
$667$	−10.3308	−0.400010
$668$	−4.79906	−0.185681
$669$	0	0
$670$	13.1514	0.508084
$671$	−2.08587	−0.0805240
$672$	0	0
$673$	42.5808	1.64137	0.820684	−	0.571382i	$-0.193592\pi$
$673$	42.5808	1.64137	0.820684	+	0.571382i	$0.193592\pi$
$674$	−10.0033	−0.385311
$675$	0	0
$676$	−1.54997	−0.0596142
$677$	33.8629	1.30146	0.650728	−	0.759311i	$-0.274464\pi$
$677$	33.8629	1.30146	0.650728	+	0.759311i	$0.274464\pi$
$678$	0	0
$679$	−7.18572	−0.275763
$680$	−8.14763	−0.312447
$681$	0	0
$682$	21.0612	0.806473
$683$	34.5163	1.32073	0.660364	−	0.750946i	$-0.270402\pi$
$683$	34.5163	1.32073	0.660364	+	0.750946i	$0.270402\pi$
$684$	0	0
$685$	−5.04240	−0.192660
$686$	0.670843	0.0256129
$687$	0	0
$688$	2.24999	0.0857802
$689$	−0.502345	−0.0191378
$690$	0	0
$691$	−7.90679	−0.300789	−0.150394	−	0.988626i	$-0.548054\pi$
$691$	−7.90679	−0.300789	−0.150394	+	0.988626i	$0.548054\pi$
$692$	4.26794	0.162243
$693$	0	0
$694$	−4.74613	−0.180161
$695$	−33.4044	−1.26710
$696$	0	0
$697$	−10.7974	−0.408980
$698$	14.1674	0.536244
$699$	0	0
$700$	−2.32859	−0.0880124
$701$	5.19510	0.196216	0.0981081	−	0.995176i	$-0.468721\pi$
$701$	5.19510	0.196216	0.0981081	+	0.995176i	$0.468721\pi$
$702$	0	0
$703$	−69.9758	−2.63919
$704$	2.64513	0.0996921
$705$	0	0
$706$	11.6487	0.438404
$707$	18.4416	0.693569
$708$	0	0
$709$	33.1971	1.24674	0.623372	−	0.781925i	$-0.285762\pi$
$709$	33.1971	1.24674	0.623372	+	0.781925i	$0.285762\pi$
$710$	−24.3882	−0.915275
$711$	0	0
$712$	32.7561	1.22759
$713$	−18.9671	−0.710323
$714$	0	0
$715$	−7.78331	−0.291079
$716$	−2.44465	−0.0913610
$717$	0	0
$718$	−12.3114	−0.459457
$719$	−30.7261	−1.14589	−0.572944	−	0.819594i	$-0.694199\pi$
$719$	−30.7261	−1.14589	−0.572944	+	0.819594i	$0.694199\pi$
$720$	0	0
$721$	1.89537	0.0705873
$722$	26.0252	0.968557
$723$	0	0
$724$	−14.7503	−0.548190
$725$	8.41657	0.312583
$726$	0	0
$727$	1.83783	0.0681612	0.0340806	−	0.999419i	$-0.489150\pi$
$727$	1.83783	0.0681612	0.0340806	+	0.999419i	$0.489150\pi$
$728$	−2.38147	−0.0882632
$729$	0	0
$730$	20.5598	0.760952
$731$	−2.00938	−0.0743197
$732$	0	0
$733$	−19.1857	−0.708641	−0.354320	−	0.935124i	$-0.615288\pi$
$733$	−19.1857	−0.708641	−0.354320	+	0.935124i	$0.615288\pi$
$734$	5.63990	0.208173
$735$	0	0
$736$	−10.6415	−0.392251
$737$	−23.4664	−0.864396
$738$	0	0
$739$	−21.4882	−0.790456	−0.395228	−	0.918583i	$-0.629334\pi$
$739$	−21.4882	−0.790456	−0.395228	+	0.918583i	$0.629334\pi$
$740$	−36.3799	−1.33735
$741$	0	0
$742$	−0.336995	−0.0123715
$743$	−19.9430	−0.731637	−0.365819	−	0.930686i	$-0.619211\pi$
$743$	−19.9430	−0.731637	−0.365819	+	0.930686i	$0.619211\pi$
$744$	0	0
$745$	37.2140	1.36341
$746$	18.1748	0.665427
$747$	0	0
$748$	6.34751	0.232088
$749$	−18.5463	−0.677665
$750$	0	0
$751$	38.4498	1.40305	0.701526	−	0.712644i	$-0.252502\pi$
$751$	38.4498	1.40305	0.701526	+	0.712644i	$0.252502\pi$
$752$	5.84661	0.213204
$753$	0	0
$754$	3.75825	0.136868
$755$	13.8001	0.502238
$756$	0	0
$757$	−16.2931	−0.592182	−0.296091	−	0.955160i	$-0.595683\pi$
$757$	−16.2931	−0.592182	−0.296091	+	0.955160i	$0.595683\pi$
$758$	−4.30451	−0.156347
$759$	0	0
$760$	46.1663	1.67463
$761$	−35.3380	−1.28100	−0.640500	−	0.767958i	$-0.721273\pi$
$761$	−35.3380	−1.28100	−0.640500	+	0.767958i	$0.721273\pi$
$762$	0	0
$763$	−1.42126	−0.0514530
$764$	−35.4180	−1.28138
$765$	0	0
$766$	−21.8654	−0.790028
$767$	4.28937	0.154880
$768$	0	0
$769$	−11.5428	−0.416244	−0.208122	−	0.978103i	$-0.566735\pi$
$769$	−11.5428	−0.416244	−0.208122	+	0.978103i	$0.566735\pi$
$770$	−5.22138	−0.188165
$771$	0	0
$772$	−30.3422	−1.09204
$773$	5.89786	0.212131	0.106066	−	0.994359i	$-0.466175\pi$
$773$	5.89786	0.212131	0.106066	+	0.994359i	$0.466175\pi$
$774$	0	0
$775$	15.4526	0.555074
$776$	17.1126	0.614306
$777$	0	0
$778$	−4.02506	−0.144305
$779$	61.1803	2.19201
$780$	0	0
$781$	43.5165	1.55714
$782$	1.65974	0.0593522
$783$	0	0
$784$	1.50235	0.0536552
$785$	−59.6806	−2.13009
$786$	0	0
$787$	6.33079	0.225668	0.112834	−	0.993614i	$-0.464007\pi$
$787$	6.33079	0.225668	0.112834	+	0.993614i	$0.464007\pi$
$788$	−32.9014	−1.17206
$789$	0	0
$790$	−8.41444	−0.299372
$791$	−5.60228	−0.199194
$792$	0	0
$793$	−0.683372	−0.0242672
$794$	10.5672	0.375014
$795$	0	0
$796$	10.3590	0.367166
$797$	22.0721	0.781835	0.390918	−	0.920426i	$-0.372158\pi$
$797$	22.0721	0.781835	0.390918	+	0.920426i	$0.372158\pi$
$798$	0	0
$799$	−5.22138	−0.184719
$800$	8.66971	0.306520
$801$	0	0
$802$	−17.5672	−0.620318
$803$	−36.6853	−1.29460
$804$	0	0
$805$	4.70222	0.165732
$806$	6.90006	0.243044
$807$	0	0
$808$	−43.9182	−1.54504
$809$	−36.2019	−1.27279	−0.636396	−	0.771363i	$-0.719576\pi$
$809$	−36.2019	−1.27279	−0.636396	+	0.771363i	$0.719576\pi$
$810$	0	0
$811$	−52.4950	−1.84335	−0.921674	−	0.387964i	$-0.873178\pi$
$811$	−52.4950	−1.84335	−0.921674	+	0.387964i	$0.873178\pi$
$812$	−8.68337	−0.304727
$813$	0	0
$814$	−18.8475	−0.660605
$815$	25.7665	0.902561
$816$	0	0
$817$	11.3856	0.398332
$818$	−11.6630	−0.407787
$819$	0	0
$820$	31.8072	1.11076
$821$	0.972743	0.0339490	0.0169745	−	0.999856i	$-0.494597\pi$
$821$	0.972743	0.0339490	0.0169745	+	0.999856i	$0.494597\pi$
$822$	0	0
$823$	−29.5572	−1.03030	−0.515150	−	0.857100i	$-0.672264\pi$
$823$	−29.5572	−1.03030	−0.515150	+	0.857100i	$0.672264\pi$
$824$	−4.51377	−0.157245
$825$	0	0
$826$	2.87749	0.100121
$827$	−15.3191	−0.532698	−0.266349	−	0.963877i	$-0.585817\pi$
$827$	−15.3191	−0.532698	−0.266349	+	0.963877i	$0.585817\pi$
$828$	0	0
$829$	27.0236	0.938570	0.469285	−	0.883047i	$-0.344512\pi$
$829$	27.0236	0.938570	0.469285	+	0.883047i	$0.344512\pi$
$830$	−2.06693	−0.0717440
$831$	0	0
$832$	0.866598	0.0300439
$833$	−1.34169	−0.0464867
$834$	0	0
$835$	−7.89528	−0.273227
$836$	−35.9664	−1.24392
$837$	0	0
$838$	8.36948	0.289119
$839$	−46.2200	−1.59569	−0.797846	−	0.602862i	$-0.794027\pi$
$839$	−46.2200	−1.59569	−0.797846	+	0.602862i	$0.794027\pi$
$840$	0	0
$841$	2.38560	0.0822619
$842$	6.42268	0.221340
$843$	0	0
$844$	7.14067	0.245792
$845$	−2.54997	−0.0877216
$846$	0	0
$847$	−1.68337	−0.0578413
$848$	−0.754696	−0.0259164
$849$	0	0
$850$	−1.35220	−0.0463801
$851$	16.9735	0.581845
$852$	0	0
$853$	−15.1094	−0.517336	−0.258668	−	0.965966i	$-0.583284\pi$
$853$	−15.1094	−0.517336	−0.258668	+	0.965966i	$0.583284\pi$
$854$	−0.458435	−0.0156873
$855$	0	0
$856$	44.1674	1.50961
$857$	0.191631	0.00654599	0.00327299	−	0.999995i	$-0.498958\pi$
$857$	0.191631	0.00654599	0.00327299	+	0.999995i	$0.498958\pi$
$858$	0	0
$859$	7.06419	0.241027	0.120514	−	0.992712i	$-0.461546\pi$
$859$	7.06419	0.241027	0.120514	+	0.992712i	$0.461546\pi$
$860$	5.91929	0.201846
$861$	0	0
$862$	−3.48978	−0.118862
$863$	−41.6142	−1.41657	−0.708283	−	0.705929i	$-0.750530\pi$
$863$	−41.6142	−1.41657	−0.708283	+	0.705929i	$0.750530\pi$
$864$	0	0
$865$	7.02150	0.238738
$866$	−11.5752	−0.393341
$867$	0	0
$868$	−15.9425	−0.541122
$869$	15.0141	0.509318
$870$	0	0
$871$	−7.68806	−0.260500
$872$	3.38469	0.114620
$873$	0	0
$874$	−9.40445	−0.318110
$875$	8.91891	0.301514
$876$	0	0
$877$	−52.5471	−1.77439	−0.887194	−	0.461396i	$-0.847349\pi$
$877$	−52.5471	−1.77439	−0.887194	+	0.461396i	$0.847349\pi$
$878$	−14.8876	−0.502434
$879$	0	0
$880$	−11.6932	−0.394178
$881$	−0.934500	−0.0314841	−0.0157421	−	0.999876i	$-0.505011\pi$
$881$	−0.934500	−0.0314841	−0.0157421	+	0.999876i	$0.505011\pi$
$882$	0	0
$883$	−42.6448	−1.43511	−0.717555	−	0.696501i	$-0.754739\pi$
$883$	−42.6448	−1.43511	−0.717555	+	0.696501i	$0.754739\pi$
$884$	2.07957	0.0699436
$885$	0	0
$886$	25.4463	0.854886
$887$	−24.8833	−0.835498	−0.417749	−	0.908563i	$-0.637181\pi$
$887$	−24.8833	−0.835498	−0.417749	+	0.908563i	$0.637181\pi$
$888$	0	0
$889$	14.1046	0.473054
$890$	23.5289	0.788691
$891$	0	0
$892$	−13.3406	−0.446675
$893$	29.5855	0.990040
$894$	0	0
$895$	−4.02188	−0.134437
$896$	−10.9602	−0.366155
$897$	0	0
$898$	22.3491	0.745799
$899$	57.6232	1.92184
$900$	0	0
$901$	0.673990	0.0224539
$902$	16.4785	0.548674
$903$	0	0
$904$	13.3417	0.443738
$905$	−24.2668	−0.806656
$906$	0	0
$907$	42.2207	1.40191	0.700957	−	0.713203i	$-0.252756\pi$
$907$	42.2207	1.40191	0.700957	+	0.713203i	$0.252756\pi$
$908$	−22.4579	−0.745292
$909$	0	0
$910$	−1.71063	−0.0567068
$911$	−14.1108	−0.467513	−0.233756	−	0.972295i	$-0.575102\pi$
$911$	−14.1108	−0.467513	−0.233756	+	0.972295i	$0.575102\pi$
$912$	0	0
$913$	3.68806	0.122057
$914$	−5.22767	−0.172916
$915$	0	0
$916$	31.3092	1.03449
$917$	8.78800	0.290205
$918$	0	0
$919$	−18.1547	−0.598870	−0.299435	−	0.954117i	$-0.596798\pi$
$919$	−18.1547	−0.598870	−0.299435	+	0.954117i	$0.596798\pi$
$920$	−11.1982	−0.369194
$921$	0	0
$922$	−20.4317	−0.672882
$923$	14.2569	0.469271
$924$	0	0
$925$	−13.8284	−0.454676
$926$	26.4741	0.869993
$927$	0	0
$928$	32.3296	1.06127
$929$	8.24741	0.270589	0.135294	−	0.990805i	$-0.456802\pi$
$929$	8.24741	0.270589	0.135294	+	0.990805i	$0.456802\pi$
$930$	0	0
$931$	7.60228	0.249155
$932$	11.2853	0.369662
$933$	0	0
$934$	6.66353	0.218037
$935$	10.4428	0.341515
$936$	0	0
$937$	−21.6693	−0.707905	−0.353953	−	0.935263i	$-0.615163\pi$
$937$	−21.6693	−0.707905	−0.353953	+	0.935263i	$0.615163\pi$
$938$	−5.15748	−0.168398
$939$	0	0
$940$	15.3813	0.501682
$941$	−42.5238	−1.38624	−0.693118	−	0.720824i	$-0.743763\pi$
$941$	−42.5238	−1.38624	−0.693118	+	0.720824i	$0.743763\pi$
$942$	0	0
$943$	−14.8401	−0.483259
$944$	6.44412	0.209738
$945$	0	0
$946$	3.06663	0.0997049
$947$	6.79792	0.220903	0.110451	−	0.993882i	$-0.464770\pi$
$947$	6.79792	0.220903	0.110451	+	0.993882i	$0.464770\pi$
$948$	0	0
$949$	−12.0189	−0.390148
$950$	7.66187	0.248584
$951$	0	0
$952$	3.19519	0.103557
$953$	−57.3782	−1.85866	−0.929331	−	0.369249i	$-0.879615\pi$
$953$	−57.3782	−1.85866	−0.929331	+	0.369249i	$0.879615\pi$
$954$	0	0
$955$	−58.2688	−1.88553
$956$	42.8857	1.38702
$957$	0	0
$958$	0.631585	0.0204056
$959$	1.97743	0.0638547
$960$	0	0
$961$	74.7947	2.41273
$962$	−6.17482	−0.199084
$963$	0	0
$964$	34.4384	1.10918
$965$	−49.9182	−1.60692
$966$	0	0
$967$	7.52393	0.241953	0.120977	−	0.992655i	$-0.461397\pi$
$967$	7.52393	0.241953	0.120977	+	0.992655i	$0.461397\pi$
$968$	4.00890	0.128851
$969$	0	0
$970$	12.2921	0.394675
$971$	18.5807	0.596283	0.298141	−	0.954522i	$-0.403633\pi$
$971$	18.5807	0.596283	0.298141	+	0.954522i	$0.403633\pi$
$972$	0	0
$973$	13.0999	0.419965
$974$	−26.3976	−0.845835
$975$	0	0
$976$	−1.02666	−0.0328626
$977$	38.5939	1.23473	0.617364	−	0.786678i	$-0.288201\pi$
$977$	38.5939	1.23473	0.617364	+	0.786678i	$0.288201\pi$
$978$	0	0
$979$	−41.9832	−1.34179
$980$	3.95238	0.126254
$981$	0	0
$982$	−3.65861	−0.116751
$983$	0.591837	0.0188767	0.00943834	−	0.999955i	$-0.496996\pi$
$983$	0.591837	0.0188767	0.00943834	+	0.999955i	$0.496996\pi$
$984$	0	0
$985$	−54.1286	−1.72468
$986$	−5.04240	−0.160583
$987$	0	0
$988$	−11.7833	−0.374877
$989$	−2.76172	−0.0878177
$990$	0	0
$991$	9.41686	0.299136	0.149568	−	0.988751i	$-0.452212\pi$
$991$	9.41686	0.299136	0.149568	+	0.988751i	$0.452212\pi$
$992$	59.3563	1.88456
$993$	0	0
$994$	9.56413	0.303356
$995$	17.0424	0.540280
$996$	0	0
$997$	22.1716	0.702180	0.351090	−	0.936342i	$-0.385811\pi$
$997$	22.1716	0.702180	0.351090	+	0.936342i	$0.385811\pi$
$998$	−21.0359	−0.665879
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.k.1.3		4
3.2	odd	2		273.2.a.e.1.2	✓	4
7.6	odd	2		5733.2.a.bf.1.3		4
12.11	even	2		4368.2.a.br.1.4		4
15.14	odd	2		6825.2.a.bg.1.3		4
21.20	even	2		1911.2.a.s.1.2		4
39.38	odd	2		3549.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.2	✓	4	3.2	odd	2
819.2.a.k.1.3		4	1.1	even	1	trivial
1911.2.a.s.1.2		4	21.20	even	2
3549.2.a.w.1.3		4	39.38	odd	2
4368.2.a.br.1.4		4	12.11	even	2
5733.2.a.bf.1.3		4	7.6	odd	2
6825.2.a.bg.1.3		4	15.14	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+0.670843 q^{2} -1.54997 q^{4} -2.54997 q^{5} +1.00000 q^{7} -2.38147 q^{8} -1.71063 q^{10} +3.05232 q^{11} +1.00000 q^{13} +0.670843 q^{14} +1.50235 q^{16} -1.34169 q^{17} +7.60228 q^{19} +3.95238 q^{20} +2.04762 q^{22} -1.84403 q^{23} +1.50235 q^{25} +0.670843 q^{26} -1.54997 q^{28} +5.60228 q^{29} +10.2857 q^{31} +5.77078 q^{32} -0.900061 q^{34} -2.54997 q^{35} -9.20457 q^{37} +5.09994 q^{38} +6.07268 q^{40} +8.04762 q^{41} +1.49765 q^{43} -4.73100 q^{44} -1.23706 q^{46} +3.89166 q^{47} +1.00000 q^{49} +1.00784 q^{50} -1.54997 q^{52} -0.502345 q^{53} -7.78331 q^{55} -2.38147 q^{56} +3.75825 q^{58} +4.28937 q^{59} -0.683372 q^{61} +6.90006 q^{62} +0.866598 q^{64} -2.54997 q^{65} -7.68806 q^{67} +2.07957 q^{68} -1.71063 q^{70} +14.2569 q^{71} -12.0189 q^{73} -6.17482 q^{74} -11.7833 q^{76} +3.05232 q^{77} +4.91891 q^{79} -3.83094 q^{80} +5.39869 q^{82} +1.20828 q^{83} +3.42126 q^{85} +1.00469 q^{86} -7.26900 q^{88} -13.7545 q^{89} +1.00000 q^{91} +2.85819 q^{92} +2.61069 q^{94} -19.3856 q^{95} -7.18572 q^{97} +0.670843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29}+ \cdots - q^{98}+O(q^{100}) \) 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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