LMFDB - Embedded newform 8649.2.a.bs.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8649,2,Mod(1,8649)] 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("8649.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8649, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,8,0,8,16,0,-16,0,0,8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 24x^{14} + 220x^{12} - 992x^{10} + 2366x^{8} - 2944x^{6} + 1688x^{4} - 288x^{2} + 4 $ x^16 - 24x^14 + 220x^12 - 992x^10 + 2366x^8 - 2944x^6 + 1688x^4 - 288*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 961)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.86943$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.124175 q^{2} -1.98458 q^{4} +1.22944 q^{5} +2.66703 q^{7} +0.494784 q^{8} -0.152666 q^{10} +5.63192 q^{11} -0.0134654 q^{13} -0.331177 q^{14} +3.90772 q^{16} -3.54485 q^{17} -5.96385 q^{19} -2.43993 q^{20} -0.699342 q^{22} +4.08969 q^{23} -3.48847 q^{25} +0.00167206 q^{26} -5.29293 q^{28} +2.96497 q^{29} -1.47481 q^{32} +0.440181 q^{34} +3.27896 q^{35} +10.6352 q^{37} +0.740559 q^{38} +0.608309 q^{40} +7.14473 q^{41} -3.17563 q^{43} -11.1770 q^{44} -0.507836 q^{46} +9.81029 q^{47} +0.113032 q^{49} +0.433179 q^{50} +0.0267232 q^{52} -7.90828 q^{53} +6.92413 q^{55} +1.31960 q^{56} -0.368174 q^{58} +10.3909 q^{59} +3.70951 q^{61} -7.63231 q^{64} -0.0165550 q^{65} +3.56973 q^{67} +7.03504 q^{68} -0.407164 q^{70} -3.76169 q^{71} -3.55021 q^{73} -1.32063 q^{74} +11.8357 q^{76} +15.0205 q^{77} -3.25987 q^{79} +4.80433 q^{80} -0.887194 q^{82} -2.07024 q^{83} -4.35820 q^{85} +0.394333 q^{86} +2.78658 q^{88} -8.80431 q^{89} -0.0359126 q^{91} -8.11632 q^{92} -1.21819 q^{94} -7.33223 q^{95} +3.38749 q^{97} -0.0140358 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56}+ \cdots - 16 q^{98}+O(q^{100}) $ 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * 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 $

$2$	−0.124175	−0.0878047	−0.0439024	−	0.999036i	$-0.513979\pi$
$2$	−0.124175	−0.0878047	−0.0439024	+	0.999036i	$0.513979\pi$
$3$	0	0
$3$	0	0
$4$	−1.98458	−0.992290
$5$	1.22944	0.549824	0.274912	−	0.961469i	$-0.411351\pi$
$5$	1.22944	0.549824	0.274912	+	0.961469i	$0.411351\pi$
$6$	0	0
$7$	2.66703	1.00804	0.504021	−	0.863692i	$-0.331854\pi$
$7$	2.66703	1.00804	0.504021	+	0.863692i	$0.331854\pi$
$8$	0.494784	0.174933
$9$	0	0
$10$	−0.152666	−0.0482772
$11$	5.63192	1.69809	0.849044	−	0.528322i	$-0.177179\pi$
$11$	5.63192	1.69809	0.849044	+	0.528322i	$0.177179\pi$
$12$	0	0
$13$	−0.0134654	−0.00373463	−0.00186731	−	0.999998i	$-0.500594\pi$
$13$	−0.0134654	−0.00373463	−0.00186731	+	0.999998i	$0.500594\pi$
$14$	−0.331177	−0.0885108
$15$	0	0
$16$	3.90772	0.976930
$17$	−3.54485	−0.859753	−0.429876	−	0.902888i	$-0.641443\pi$
$17$	−3.54485	−0.859753	−0.429876	+	0.902888i	$0.641443\pi$
$18$	0	0
$19$	−5.96385	−1.36820	−0.684101	−	0.729388i	$-0.739805\pi$
$19$	−5.96385	−1.36820	−0.684101	+	0.729388i	$0.739805\pi$
$20$	−2.43993	−0.545585
$21$	0	0
$22$	−0.699342	−0.149100
$23$	4.08969	0.852759	0.426380	−	0.904544i	$-0.359789\pi$
$23$	4.08969	0.852759	0.426380	+	0.904544i	$0.359789\pi$
$24$	0	0
$25$	−3.48847	−0.697693
$26$	0.00167206	0.000327918 0
$27$	0	0
$28$	−5.29293	−1.00027
$29$	2.96497	0.550581	0.275290	−	0.961361i	$-0.411226\pi$
$29$	2.96497	0.550581	0.275290	+	0.961361i	$0.411226\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−1.47481	−0.260712
$33$	0	0
$34$	0.440181	0.0754903
$35$	3.27896	0.554246
$36$	0	0
$37$	10.6352	1.74842	0.874211	−	0.485546i	$-0.161379\pi$
$37$	10.6352	1.74842	0.874211	+	0.485546i	$0.161379\pi$
$38$	0.740559	0.120135
$39$	0	0
$40$	0.608309	0.0961822
$41$	7.14473	1.11582	0.557910	−	0.829902i	$-0.311604\pi$
$41$	7.14473	1.11582	0.557910	+	0.829902i	$0.311604\pi$
$42$	0	0
$43$	−3.17563	−0.484279	−0.242139	−	0.970241i	$-0.577849\pi$
$43$	−3.17563	−0.484279	−0.242139	+	0.970241i	$0.577849\pi$
$44$	−11.1770	−1.68500
$45$	0	0
$46$	−0.507836	−0.0748763
$47$	9.81029	1.43098	0.715489	−	0.698624i	$-0.246204\pi$
$47$	9.81029	1.43098	0.715489	+	0.698624i	$0.246204\pi$
$48$	0	0
$49$	0.113032	0.0161475
$50$	0.433179	0.0612608
$51$	0	0
$52$	0.0267232	0.00370584
$53$	−7.90828	−1.08629	−0.543143	−	0.839640i	$-0.682766\pi$
$53$	−7.90828	−1.08629	−0.543143	+	0.839640i	$0.682766\pi$
$54$	0	0
$55$	6.92413	0.933650
$56$	1.31960	0.176339
$57$	0	0
$58$	−0.368174	−0.0483436
$59$	10.3909	1.35278	0.676392	−	0.736542i	$-0.263543\pi$
$59$	10.3909	1.35278	0.676392	+	0.736542i	$0.263543\pi$
$60$	0	0
$61$	3.70951	0.474954	0.237477	−	0.971393i	$-0.423680\pi$
$61$	3.70951	0.474954	0.237477	+	0.971393i	$0.423680\pi$
$62$	0	0
$63$	0	0
$64$	−7.63231	−0.954039
$65$	−0.0165550	−0.00205339
$66$	0	0
$67$	3.56973	0.436112	0.218056	−	0.975936i	$-0.430028\pi$
$67$	3.56973	0.436112	0.218056	+	0.975936i	$0.430028\pi$
$68$	7.03504	0.853124
$69$	0	0
$70$	−0.407164	−0.0486654
$71$	−3.76169	−0.446430	−0.223215	−	0.974769i	$-0.571655\pi$
$71$	−3.76169	−0.446430	−0.223215	+	0.974769i	$0.571655\pi$
$72$	0	0
$73$	−3.55021	−0.415520	−0.207760	−	0.978180i	$-0.566617\pi$
$73$	−3.55021	−0.415520	−0.207760	+	0.978180i	$0.566617\pi$
$74$	−1.32063	−0.153520
$75$	0	0
$76$	11.8357	1.35765
$77$	15.0205	1.71174
$78$	0	0
$79$	−3.25987	−0.366765	−0.183382	−	0.983042i	$-0.558705\pi$
$79$	−3.25987	−0.366765	−0.183382	+	0.983042i	$0.558705\pi$
$80$	4.80433	0.537140
$81$	0	0
$82$	−0.887194	−0.0979742
$83$	−2.07024	−0.227238	−0.113619	−	0.993524i	$-0.536244\pi$
$83$	−2.07024	−0.227238	−0.113619	+	0.993524i	$0.536244\pi$
$84$	0	0
$85$	−4.35820	−0.472713
$86$	0.394333	0.0425220
$87$	0	0
$88$	2.78658	0.297051
$89$	−8.80431	−0.933255	−0.466627	−	0.884454i	$-0.654531\pi$
$89$	−8.80431	−0.933255	−0.466627	+	0.884454i	$0.654531\pi$
$90$	0	0
$91$	−0.0359126	−0.00376466
$92$	−8.11632	−0.846185
$93$	0	0
$94$	−1.21819	−0.125647
$95$	−7.33223	−0.752271
$96$	0	0
$97$	3.38749	0.343948	0.171974	−	0.985101i	$-0.444986\pi$
$97$	3.38749	0.343948	0.171974	+	0.985101i	$0.444986\pi$
$98$	−0.0140358	−0.00141783
$99$	0	0
$100$	6.92314	0.692314
$101$	8.66837	0.862535	0.431267	−	0.902224i	$-0.358067\pi$
$101$	8.66837	0.862535	0.431267	+	0.902224i	$0.358067\pi$
$102$	0	0
$103$	2.68096	0.264163	0.132081	−	0.991239i	$-0.457834\pi$
$103$	2.68096	0.264163	0.132081	+	0.991239i	$0.457834\pi$
$104$	−0.00666246	−0.000653308 0
$105$	0	0
$106$	0.982008	0.0953811
$107$	7.97570	0.771040	0.385520	−	0.922700i	$-0.374022\pi$
$107$	7.97570	0.771040	0.385520	+	0.922700i	$0.374022\pi$
$108$	0	0
$109$	6.29895	0.603330	0.301665	−	0.953414i	$-0.402458\pi$
$109$	6.29895	0.603330	0.301665	+	0.953414i	$0.402458\pi$
$110$	−0.859802	−0.0819789
$111$	0	0
$112$	10.4220	0.984786
$113$	2.74728	0.258442	0.129221	−	0.991616i	$-0.458752\pi$
$113$	2.74728	0.258442	0.129221	+	0.991616i	$0.458752\pi$
$114$	0	0
$115$	5.02805	0.468868
$116$	−5.88422	−0.546336
$117$	0	0
$118$	−1.29029	−0.118781
$119$	−9.45421	−0.866666
$120$	0	0
$121$	20.7185	1.88350
$122$	−0.460627	−0.0417032
$123$	0	0
$124$	0	0
$125$	−10.4361	−0.933433
$126$	0	0
$127$	1.73534	0.153986	0.0769931	−	0.997032i	$-0.475468\pi$
$127$	1.73534	0.153986	0.0769931	+	0.997032i	$0.475468\pi$
$128$	3.89735	0.344481
$129$	0	0
$130$	0.00205571	0.000180297 0
$131$	−7.06939	−0.617656	−0.308828	−	0.951118i	$-0.599937\pi$
$131$	−7.06939	−0.617656	−0.308828	+	0.951118i	$0.599937\pi$
$132$	0	0
$133$	−15.9058	−1.37920
$134$	−0.443270	−0.0382927
$135$	0	0
$136$	−1.75394	−0.150399
$137$	9.53710	0.814809	0.407405	−	0.913248i	$-0.366434\pi$
$137$	9.53710	0.814809	0.407405	+	0.913248i	$0.366434\pi$
$138$	0	0
$139$	13.3484	1.13220	0.566098	−	0.824338i	$-0.308453\pi$
$139$	13.3484	1.13220	0.566098	+	0.824338i	$0.308453\pi$
$140$	−6.50737	−0.549973
$141$	0	0
$142$	0.467106	0.0391987
$143$	−0.0758360	−0.00634173
$144$	0	0
$145$	3.64526	0.302723
$146$	0.440846	0.0364846
$147$	0	0
$148$	−21.1065	−1.73494
$149$	−18.1985	−1.49088	−0.745440	−	0.666573i	$-0.767761\pi$
$149$	−18.1985	−1.49088	−0.745440	+	0.666573i	$0.767761\pi$
$150$	0	0
$151$	−6.59561	−0.536743	−0.268371	−	0.963316i	$-0.586485\pi$
$151$	−6.59561	−0.536743	−0.268371	+	0.963316i	$0.586485\pi$
$152$	−2.95082	−0.239343
$153$	0	0
$154$	−1.86516	−0.150299
$155$	0	0
$156$	0	0
$157$	15.7251	1.25500	0.627502	−	0.778615i	$-0.284078\pi$
$157$	15.7251	1.25500	0.627502	+	0.778615i	$0.284078\pi$
$158$	0.404794	0.0322037
$159$	0	0
$160$	−1.81319	−0.143346
$161$	10.9073	0.859617
$162$	0	0
$163$	−7.19636	−0.563663	−0.281831	−	0.959464i	$-0.590942\pi$
$163$	−7.19636	−0.563663	−0.281831	+	0.959464i	$0.590942\pi$
$164$	−14.1793	−1.10722
$165$	0	0
$166$	0.257071	0.0199526
$167$	−20.4703	−1.58404	−0.792018	−	0.610498i	$-0.790969\pi$
$167$	−20.4703	−1.58404	−0.792018	+	0.610498i	$0.790969\pi$
$168$	0	0
$169$	−12.9998	−0.999986
$170$	0.541178	0.0415064
$171$	0	0
$172$	6.30229	0.480545
$173$	21.0674	1.60173	0.800863	−	0.598848i	$-0.204375\pi$
$173$	21.0674	1.60173	0.800863	+	0.598848i	$0.204375\pi$
$174$	0	0
$175$	−9.30383	−0.703304
$176$	22.0080	1.65891
$177$	0	0
$178$	1.09327	0.0819442
$179$	−10.4789	−0.783231	−0.391615	−	0.920129i	$-0.628084\pi$
$179$	−10.4789	−0.783231	−0.391615	+	0.920129i	$0.628084\pi$
$180$	0	0
$181$	9.53710	0.708887	0.354443	−	0.935077i	$-0.384670\pi$
$181$	9.53710	0.708887	0.354443	+	0.935077i	$0.384670\pi$
$182$	0.00445943	0.000330555 0
$183$	0	0
$184$	2.02351	0.149175
$185$	13.0754	0.961325
$186$	0	0
$187$	−19.9643	−1.45994
$188$	−19.4693	−1.41994
$189$	0	0
$190$	0.910477	0.0660529
$191$	10.0046	0.723905	0.361952	−	0.932197i	$-0.382110\pi$
$191$	10.0046	0.723905	0.361952	+	0.932197i	$0.382110\pi$
$192$	0	0
$193$	−20.6806	−1.48862	−0.744310	−	0.667835i	$-0.767221\pi$
$193$	−20.6806	−1.48862	−0.744310	+	0.667835i	$0.767221\pi$
$194$	−0.420641	−0.0302003
$195$	0	0
$196$	−0.224322	−0.0160230
$197$	−0.753285	−0.0536693	−0.0268347	−	0.999640i	$-0.508543\pi$
$197$	−0.753285	−0.0536693	−0.0268347	+	0.999640i	$0.508543\pi$
$198$	0	0
$199$	−9.89063	−0.701128	−0.350564	−	0.936539i	$-0.614010\pi$
$199$	−9.89063	−0.701128	−0.350564	+	0.936539i	$0.614010\pi$
$200$	−1.72604	−0.122049
$201$	0	0
$202$	−1.07639	−0.0757346
$203$	7.90765	0.555008
$204$	0	0
$205$	8.78405	0.613505
$206$	−0.332907	−0.0231947
$207$	0	0
$208$	−0.0526190	−0.00364847
$209$	−33.5879	−2.32333
$210$	0	0
$211$	14.0629	0.968128	0.484064	−	0.875033i	$-0.339160\pi$
$211$	14.0629	0.968128	0.484064	+	0.875033i	$0.339160\pi$
$212$	15.6946	1.07791
$213$	0	0
$214$	−0.990379	−0.0677009
$215$	−3.90426	−0.266268
$216$	0	0
$217$	0	0
$218$	−0.782170	−0.0529753
$219$	0	0
$220$	−13.7415	−0.926452
$221$	0.0477328	0.00321086
$222$	0	0
$223$	−9.89557	−0.662656	−0.331328	−	0.943516i	$-0.607497\pi$
$223$	−9.89557	−0.662656	−0.331328	+	0.943516i	$0.607497\pi$
$224$	−3.93335	−0.262808
$225$	0	0
$226$	−0.341142	−0.0226924
$227$	−9.40457	−0.624204	−0.312102	−	0.950049i	$-0.601033\pi$
$227$	−9.40457	−0.624204	−0.312102	+	0.950049i	$0.601033\pi$
$228$	0	0
$229$	14.7546	0.975011	0.487506	−	0.873120i	$-0.337907\pi$
$229$	14.7546	0.975011	0.487506	+	0.873120i	$0.337907\pi$
$230$	−0.624356	−0.0411688
$231$	0	0
$232$	1.46702	0.0963145
$233$	28.5860	1.87273	0.936365	−	0.351028i	$-0.114168\pi$
$233$	28.5860	1.87273	0.936365	+	0.351028i	$0.114168\pi$
$234$	0	0
$235$	12.0612	0.786786
$236$	−20.6216	−1.34235
$237$	0	0
$238$	1.17397	0.0760974
$239$	−9.00876	−0.582729	−0.291364	−	0.956612i	$-0.594109\pi$
$239$	−9.00876	−0.582729	−0.291364	+	0.956612i	$0.594109\pi$
$240$	0	0
$241$	−0.788163	−0.0507700	−0.0253850	−	0.999678i	$-0.508081\pi$
$241$	−0.788163	−0.0507700	−0.0253850	+	0.999678i	$0.508081\pi$
$242$	−2.57271	−0.165380
$243$	0	0
$244$	−7.36182	−0.471292
$245$	0.138967	0.00887829
$246$	0	0
$247$	0.0803056	0.00510972
$248$	0	0
$249$	0	0
$250$	1.29590	0.0819598
$251$	−19.7718	−1.24798	−0.623992	−	0.781430i	$-0.714490\pi$
$251$	−19.7718	−1.24798	−0.623992	+	0.781430i	$0.714490\pi$
$252$	0	0
$253$	23.0328	1.44806
$254$	−0.215485	−0.0135207
$255$	0	0
$256$	14.7807	0.923792
$257$	9.02462	0.562940	0.281470	−	0.959570i	$-0.409178\pi$
$257$	9.02462	0.562940	0.281470	+	0.959570i	$0.409178\pi$
$258$	0	0
$259$	28.3645	1.76248
$260$	0.0328546	0.00203756
$261$	0	0
$262$	0.877840	0.0542331
$263$	−3.12800	−0.192881	−0.0964403	−	0.995339i	$-0.530746\pi$
$263$	−3.12800	−0.192881	−0.0964403	+	0.995339i	$0.530746\pi$
$264$	0	0
$265$	−9.72280	−0.597267
$266$	1.97509	0.121101
$267$	0	0
$268$	−7.08441	−0.432749
$269$	13.9776	0.852231	0.426115	−	0.904669i	$-0.359882\pi$
$269$	13.9776	0.852231	0.426115	+	0.904669i	$0.359882\pi$
$270$	0	0
$271$	−14.0587	−0.854008	−0.427004	−	0.904250i	$-0.640431\pi$
$271$	−14.0587	−0.854008	−0.427004	+	0.904250i	$0.640431\pi$
$272$	−13.8523	−0.839918
$273$	0	0
$274$	−1.18427	−0.0715441
$275$	−19.6468	−1.18474
$276$	0	0
$277$	−22.0218	−1.32316	−0.661581	−	0.749874i	$-0.730114\pi$
$277$	−22.0218	−1.32316	−0.661581	+	0.749874i	$0.730114\pi$
$278$	−1.65753	−0.0994121
$279$	0	0
$280$	1.62238	0.0969556
$281$	−2.32625	−0.138772	−0.0693862	−	0.997590i	$-0.522104\pi$
$281$	−2.32625	−0.138772	−0.0693862	+	0.997590i	$0.522104\pi$
$282$	0	0
$283$	13.9295	0.828021	0.414010	−	0.910272i	$-0.364128\pi$
$283$	13.9295	0.828021	0.414010	+	0.910272i	$0.364128\pi$
$284$	7.46537	0.442988
$285$	0	0
$286$	0.00941691	0.000556834 0
$287$	19.0552	1.12479
$288$	0	0
$289$	−4.43403	−0.260825
$290$	−0.452649	−0.0265805
$291$	0	0
$292$	7.04567	0.412317
$293$	−25.8604	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$293$	−25.8604	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$294$	0	0
$295$	12.7751	0.743794
$296$	5.26214	0.305856
$297$	0	0
$298$	2.25979	0.130906
$299$	−0.0550693	−0.00318474
$300$	0	0
$301$	−8.46949	−0.488173
$302$	0.819007	0.0471286
$303$	0	0
$304$	−23.3051	−1.33664
$305$	4.56064	0.261141
$306$	0	0
$307$	−0.655663	−0.0374206	−0.0187103	−	0.999825i	$-0.505956\pi$
$307$	−0.655663	−0.0374206	−0.0187103	+	0.999825i	$0.505956\pi$
$308$	−29.8094	−1.69855
$309$	0	0
$310$	0	0
$311$	22.0394	1.24974	0.624869	−	0.780730i	$-0.285152\pi$
$311$	22.0394	1.24974	0.624869	+	0.780730i	$0.285152\pi$
$312$	0	0
$313$	−9.51336	−0.537727	−0.268863	−	0.963178i	$-0.586648\pi$
$313$	−9.51336	−0.537727	−0.268863	+	0.963178i	$0.586648\pi$
$314$	−1.95266	−0.110195
$315$	0	0
$316$	6.46948	0.363937
$317$	2.09503	0.117669	0.0588343	−	0.998268i	$-0.481262\pi$
$317$	2.09503	0.117669	0.0588343	+	0.998268i	$0.481262\pi$
$318$	0	0
$319$	16.6985	0.934934
$320$	−9.38350	−0.524554
$321$	0	0
$322$	−1.35441	−0.0754784
$323$	21.1410	1.17631
$324$	0	0
$325$	0.0469736	0.00260562
$326$	0.893606	0.0494923
$327$	0	0
$328$	3.53510	0.195193
$329$	26.1643	1.44248
$330$	0	0
$331$	−2.50739	−0.137819	−0.0689093	−	0.997623i	$-0.521952\pi$
$331$	−2.50739	−0.137819	−0.0689093	+	0.997623i	$0.521952\pi$
$332$	4.10855	0.225486
$333$	0	0
$334$	2.54189	0.139086
$335$	4.38878	0.239785
$336$	0	0
$337$	27.2137	1.48242	0.741212	−	0.671271i	$-0.234251\pi$
$337$	27.2137	1.48242	0.741212	+	0.671271i	$0.234251\pi$
$338$	1.61425	0.0878035
$339$	0	0
$340$	8.64920	0.469069
$341$	0	0
$342$	0	0
$343$	−18.3677	−0.991764
$344$	−1.57125	−0.0847161
$345$	0	0
$346$	−2.61604	−0.140639
$347$	24.3284	1.30602	0.653010	−	0.757350i	$-0.273506\pi$
$347$	24.3284	1.30602	0.653010	+	0.757350i	$0.273506\pi$
$348$	0	0
$349$	12.1656	0.651211	0.325605	−	0.945506i	$-0.394432\pi$
$349$	12.1656	0.651211	0.325605	+	0.945506i	$0.394432\pi$
$350$	1.15530	0.0617534
$351$	0	0
$352$	−8.30600	−0.442711
$353$	28.0154	1.49111	0.745556	−	0.666443i	$-0.232184\pi$
$353$	28.0154	1.49111	0.745556	+	0.666443i	$0.232184\pi$
$354$	0	0
$355$	−4.62479	−0.245458
$356$	17.4729	0.926060
$357$	0	0
$358$	1.30122	0.0687714
$359$	24.3366	1.28444	0.642219	−	0.766521i	$-0.278014\pi$
$359$	24.3366	1.28444	0.642219	+	0.766521i	$0.278014\pi$
$360$	0	0
$361$	16.5675	0.871975
$362$	−1.18427	−0.0622436
$363$	0	0
$364$	0.0712714	0.00373564
$365$	−4.36478	−0.228463
$366$	0	0
$367$	1.68920	0.0881753	0.0440876	−	0.999028i	$-0.485962\pi$
$367$	1.68920	0.0881753	0.0440876	+	0.999028i	$0.485962\pi$
$368$	15.9814	0.833087
$369$	0	0
$370$	−1.62364	−0.0844089
$371$	−21.0916	−1.09502
$372$	0	0
$373$	24.2842	1.25739	0.628695	−	0.777652i	$-0.283589\pi$
$373$	24.2842	1.25739	0.628695	+	0.777652i	$0.283589\pi$
$374$	2.47906	0.128189
$375$	0	0
$376$	4.85397	0.250324
$377$	−0.0399245	−0.00205621
$378$	0	0
$379$	13.3445	0.685462	0.342731	−	0.939433i	$-0.388648\pi$
$379$	13.3445	0.685462	0.342731	+	0.939433i	$0.388648\pi$
$380$	14.5514	0.746471
$381$	0	0
$382$	−1.24231	−0.0635622
$383$	25.0445	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$383$	25.0445	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$384$	0	0
$385$	18.4669	0.941158
$386$	2.56800	0.130708
$387$	0	0
$388$	−6.72276	−0.341296
$389$	21.9366	1.11223	0.556115	−	0.831105i	$-0.312291\pi$
$389$	21.9366	1.11223	0.556115	+	0.831105i	$0.312291\pi$
$390$	0	0
$391$	−14.4973	−0.733162
$392$	0.0559267	0.00282472
$393$	0	0
$394$	0.0935389	0.00471242
$395$	−4.00784	−0.201656
$396$	0	0
$397$	2.21630	0.111233	0.0556165	−	0.998452i	$-0.482288\pi$
$397$	2.21630	0.111233	0.0556165	+	0.998452i	$0.482288\pi$
$398$	1.22817	0.0615624
$399$	0	0
$400$	−13.6320	−0.681598
$401$	−26.6696	−1.33181	−0.665907	−	0.746035i	$-0.731955\pi$
$401$	−26.6696	−1.33181	−0.665907	+	0.746035i	$0.731955\pi$
$402$	0	0
$403$	0	0
$404$	−17.2031	−0.855885
$405$	0	0
$406$	−0.981930	−0.0487323
$407$	59.8968	2.96897
$408$	0	0
$409$	37.8777	1.87293	0.936465	−	0.350760i	$-0.114077\pi$
$409$	37.8777	1.87293	0.936465	+	0.350760i	$0.114077\pi$
$410$	−1.09076	−0.0538686
$411$	0	0
$412$	−5.32058	−0.262126
$413$	27.7129	1.36366
$414$	0	0
$415$	−2.54524	−0.124941
$416$	0.0198589	0.000973661 0
$417$	0	0
$418$	4.17077	0.203999
$419$	9.16702	0.447838	0.223919	−	0.974608i	$-0.428115\pi$
$419$	9.16702	0.447838	0.223919	+	0.974608i	$0.428115\pi$
$420$	0	0
$421$	−3.26808	−0.159277	−0.0796383	−	0.996824i	$-0.525377\pi$
$421$	−3.26808	−0.159277	−0.0796383	+	0.996824i	$0.525377\pi$
$422$	−1.74625	−0.0850062
$423$	0	0
$424$	−3.91289	−0.190027
$425$	12.3661	0.599843
$426$	0	0
$427$	9.89336	0.478773
$428$	−15.8284	−0.765095
$429$	0	0
$430$	0.484810	0.0233796
$431$	23.4706	1.13054	0.565269	−	0.824907i	$-0.308772\pi$
$431$	23.4706	1.13054	0.565269	+	0.824907i	$0.308772\pi$
$432$	0	0
$433$	−27.3421	−1.31398	−0.656990	−	0.753900i	$-0.728171\pi$
$433$	−27.3421	−1.31398	−0.656990	+	0.753900i	$0.728171\pi$
$434$	0	0
$435$	0	0
$436$	−12.5008	−0.598679
$437$	−24.3903	−1.16675
$438$	0	0
$439$	−33.0439	−1.57710	−0.788551	−	0.614970i	$-0.789168\pi$
$439$	−33.0439	−1.57710	−0.788551	+	0.614970i	$0.789168\pi$
$440$	3.42595	0.163326
$441$	0	0
$442$	−0.00592720	−0.000281928 0
$443$	13.3265	0.633162	0.316581	−	0.948566i	$-0.397465\pi$
$443$	13.3265	0.633162	0.316581	+	0.948566i	$0.397465\pi$
$444$	0	0
$445$	−10.8244	−0.513126
$446$	1.22878	0.0581844
$447$	0	0
$448$	−20.3556	−0.961711
$449$	11.2943	0.533012	0.266506	−	0.963833i	$-0.414131\pi$
$449$	11.2943	0.533012	0.266506	+	0.963833i	$0.414131\pi$
$450$	0	0
$451$	40.2385	1.89476
$452$	−5.45219	−0.256449
$453$	0	0
$454$	1.16781	0.0548080
$455$	−0.0441525	−0.00206990
$456$	0	0
$457$	−4.55068	−0.212872	−0.106436	−	0.994320i	$-0.533944\pi$
$457$	−4.55068	−0.212872	−0.106436	+	0.994320i	$0.533944\pi$
$458$	−1.83215	−0.0856106
$459$	0	0
$460$	−9.97857	−0.465253
$461$	28.7961	1.34117	0.670585	−	0.741833i	$-0.266043\pi$
$461$	28.7961	1.34117	0.670585	+	0.741833i	$0.266043\pi$
$462$	0	0
$463$	−12.0515	−0.560082	−0.280041	−	0.959988i	$-0.590348\pi$
$463$	−12.0515	−0.560082	−0.280041	+	0.959988i	$0.590348\pi$
$464$	11.5863	0.537879
$465$	0	0
$466$	−3.54965	−0.164435
$467$	19.1300	0.885230	0.442615	−	0.896712i	$-0.354051\pi$
$467$	19.1300	0.885230	0.442615	+	0.896712i	$0.354051\pi$
$468$	0	0
$469$	9.52056	0.439619
$470$	−1.49770	−0.0690836
$471$	0	0
$472$	5.14126	0.236646
$473$	−17.8849	−0.822348
$474$	0	0
$475$	20.8047	0.954585
$476$	18.7626	0.859985
$477$	0	0
$478$	1.11866	0.0511663
$479$	12.2911	0.561593	0.280797	−	0.959767i	$-0.409401\pi$
$479$	12.2911	0.561593	0.280797	+	0.959767i	$0.409401\pi$
$480$	0	0
$481$	−0.143208	−0.00652971
$482$	0.0978698	0.00445785
$483$	0	0
$484$	−41.1176	−1.86898
$485$	4.16474	0.189111
$486$	0	0
$487$	−15.1267	−0.685456	−0.342728	−	0.939435i	$-0.611351\pi$
$487$	−15.1267	−0.685456	−0.342728	+	0.939435i	$0.611351\pi$
$488$	1.83540	0.0830849
$489$	0	0
$490$	−0.0172562	−0.000779556 0
$491$	30.5340	1.37798	0.688990	−	0.724771i	$-0.258054\pi$
$491$	30.5340	1.37798	0.688990	+	0.724771i	$0.258054\pi$
$492$	0	0
$493$	−10.5104	−0.473363
$494$	−0.00997192	−0.000448658 0
$495$	0	0
$496$	0	0
$497$	−10.0325	−0.450020
$498$	0	0
$499$	−28.8272	−1.29048	−0.645240	−	0.763980i	$-0.723243\pi$
$499$	−28.8272	−1.29048	−0.645240	+	0.763980i	$0.723243\pi$
$500$	20.7113	0.926237
$501$	0	0
$502$	2.45516	0.109579
$503$	3.11786	0.139018	0.0695092	−	0.997581i	$-0.477857\pi$
$503$	3.11786	0.139018	0.0695092	+	0.997581i	$0.477857\pi$
$504$	0	0
$505$	10.6573	0.474243
$506$	−2.86009	−0.127147
$507$	0	0
$508$	−3.44392	−0.152799
$509$	−13.2957	−0.589323	−0.294661	−	0.955602i	$-0.595207\pi$
$509$	−13.2957	−0.589323	−0.294661	+	0.955602i	$0.595207\pi$
$510$	0	0
$511$	−9.46849	−0.418862
$512$	−9.63009	−0.425594
$513$	0	0
$514$	−1.12063	−0.0494288
$515$	3.29609	0.145243
$516$	0	0
$517$	55.2507	2.42992
$518$	−3.52215	−0.154754
$519$	0	0
$520$	−0.00819113	−0.000359205 0
$521$	−8.44298	−0.369894	−0.184947	−	0.982749i	$-0.559211\pi$
$521$	−8.44298	−0.369894	−0.184947	+	0.982749i	$0.559211\pi$
$522$	0	0
$523$	−39.6642	−1.73440	−0.867198	−	0.497964i	$-0.834081\pi$
$523$	−39.6642	−1.73440	−0.867198	+	0.497964i	$0.834081\pi$
$524$	14.0298	0.612894
$525$	0	0
$526$	0.388418	0.0169358
$527$	0	0
$528$	0	0
$529$	−6.27443	−0.272801
$530$	1.20732	0.0524428
$531$	0	0
$532$	31.5662	1.36857
$533$	−0.0962066	−0.00416717
$534$	0	0
$535$	9.80568	0.423937
$536$	1.76624	0.0762901
$537$	0	0
$538$	−1.73567	−0.0748299
$539$	0.636590	0.0274199
$540$	0	0
$541$	21.8292	0.938509	0.469255	−	0.883063i	$-0.344523\pi$
$541$	21.8292	0.938509	0.469255	+	0.883063i	$0.344523\pi$
$542$	1.74574	0.0749859
$543$	0	0
$544$	5.22797	0.224148
$545$	7.74422	0.331726
$546$	0	0
$547$	27.3401	1.16898	0.584490	−	0.811401i	$-0.301295\pi$
$547$	27.3401	1.16898	0.584490	+	0.811401i	$0.301295\pi$
$548$	−18.9271	−0.808527
$549$	0	0
$550$	2.43963	0.104026
$551$	−17.6826	−0.753305
$552$	0	0
$553$	−8.69417	−0.369714
$554$	2.73455	0.116180
$555$	0	0
$556$	−26.4909	−1.12347
$557$	−27.9171	−1.18289	−0.591443	−	0.806347i	$-0.701441\pi$
$557$	−27.9171	−1.18289	−0.591443	+	0.806347i	$0.701441\pi$
$558$	0	0
$559$	0.0427611	0.00180860
$560$	12.8133	0.541460
$561$	0	0
$562$	0.288861	0.0121849
$563$	25.7662	1.08591	0.542957	−	0.839760i	$-0.317305\pi$
$563$	25.7662	1.08591	0.542957	+	0.839760i	$0.317305\pi$
$564$	0	0
$565$	3.37762	0.142098
$566$	−1.72969	−0.0727042
$567$	0	0
$568$	−1.86122	−0.0780952
$569$	16.2134	0.679701	0.339851	−	0.940479i	$-0.389623\pi$
$569$	16.2134	0.679701	0.339851	+	0.940479i	$0.389623\pi$
$570$	0	0
$571$	9.37945	0.392518	0.196259	−	0.980552i	$-0.437121\pi$
$571$	9.37945	0.392518	0.196259	+	0.980552i	$0.437121\pi$
$572$	0.150503	0.00629283
$573$	0	0
$574$	−2.36617	−0.0987620
$575$	−14.2667	−0.594964
$576$	0	0
$577$	31.6764	1.31870	0.659352	−	0.751834i	$-0.270831\pi$
$577$	31.6764	1.31870	0.659352	+	0.751834i	$0.270831\pi$
$578$	0.550594	0.0229017
$579$	0	0
$580$	−7.23432	−0.300389
$581$	−5.52138	−0.229065
$582$	0	0
$583$	−44.5388	−1.84461
$584$	−1.75658	−0.0726880
$585$	0	0
$586$	3.21120	0.132654
$587$	−0.186802	−0.00771014	−0.00385507	−	0.999993i	$-0.501227\pi$
$587$	−0.186802	−0.00771014	−0.00385507	+	0.999993i	$0.501227\pi$
$588$	0	0
$589$	0	0
$590$	−1.58634	−0.0653086
$591$	0	0
$592$	41.5596	1.70809
$593$	13.4668	0.553015	0.276508	−	0.961012i	$-0.410823\pi$
$593$	13.4668	0.553015	0.276508	+	0.961012i	$0.410823\pi$
$594$	0	0
$595$	−11.6234	−0.476514
$596$	36.1164	1.47939
$597$	0	0
$598$	0.00683821	0.000279635 0
$599$	18.4914	0.755540	0.377770	−	0.925899i	$-0.376691\pi$
$599$	18.4914	0.755540	0.377770	+	0.925899i	$0.376691\pi$
$600$	0	0
$601$	−17.8606	−0.728548	−0.364274	−	0.931292i	$-0.618683\pi$
$601$	−17.8606	−0.728548	−0.364274	+	0.931292i	$0.618683\pi$
$602$	1.05170	0.0428639
$603$	0	0
$604$	13.0895	0.532605
$605$	25.4723	1.03560
$606$	0	0
$607$	28.9604	1.17547	0.587734	−	0.809054i	$-0.300020\pi$
$607$	28.9604	1.17547	0.587734	+	0.809054i	$0.300020\pi$
$608$	8.79553	0.356706
$609$	0	0
$610$	−0.566315	−0.0229294
$611$	−0.132099	−0.00534417
$612$	0	0
$613$	−20.6444	−0.833821	−0.416910	−	0.908948i	$-0.636887\pi$
$613$	−20.6444	−0.833821	−0.416910	+	0.908948i	$0.636887\pi$
$614$	0.0814167	0.00328571
$615$	0	0
$616$	7.43189	0.299439
$617$	−0.697562	−0.0280828	−0.0140414	−	0.999901i	$-0.504470\pi$
$617$	−0.697562	−0.0280828	−0.0140414	+	0.999901i	$0.504470\pi$
$618$	0	0
$619$	−2.41091	−0.0969026	−0.0484513	−	0.998826i	$-0.515429\pi$
$619$	−2.41091	−0.0969026	−0.0484513	+	0.998826i	$0.515429\pi$
$620$	0	0
$621$	0	0
$622$	−2.73673	−0.109733
$623$	−23.4813	−0.940760
$624$	0	0
$625$	4.61172	0.184469
$626$	1.18132	0.0472150
$627$	0	0
$628$	−31.2078	−1.24533
$629$	−37.7003	−1.50321
$630$	0	0
$631$	36.4136	1.44960	0.724801	−	0.688959i	$-0.241932\pi$
$631$	36.4136	1.44960	0.724801	+	0.688959i	$0.241932\pi$
$632$	−1.61293	−0.0641590
$633$	0	0
$634$	−0.260150	−0.0103319
$635$	2.13350	0.0846654
$636$	0	0
$637$	−0.00152203	−6.03049e−5 0
$638$	−2.07353	−0.0820917
$639$	0	0
$640$	4.79158	0.189404
$641$	−2.14971	−0.0849085	−0.0424542	−	0.999098i	$-0.513518\pi$
$641$	−2.14971	−0.0849085	−0.0424542	+	0.999098i	$0.513518\pi$
$642$	0	0
$643$	−17.7453	−0.699806	−0.349903	−	0.936786i	$-0.613785\pi$
$643$	−17.7453	−0.699806	−0.349903	+	0.936786i	$0.613785\pi$
$644$	−21.6464	−0.852989
$645$	0	0
$646$	−2.62517	−0.103286
$647$	−40.1703	−1.57926	−0.789629	−	0.613584i	$-0.789727\pi$
$647$	−40.1703	−1.57926	−0.789629	+	0.613584i	$0.789727\pi$
$648$	0	0
$649$	58.5209	2.29715
$650$	−0.00583292	−0.000228786 0
$651$	0	0
$652$	14.2818	0.559317
$653$	2.41626	0.0945556	0.0472778	−	0.998882i	$-0.484945\pi$
$653$	2.41626	0.0945556	0.0472778	+	0.998882i	$0.484945\pi$
$654$	0	0
$655$	−8.69143	−0.339602
$656$	27.9196	1.09008
$657$	0	0
$658$	−3.24894	−0.126657
$659$	−26.3680	−1.02715	−0.513576	−	0.858044i	$-0.671679\pi$
$659$	−26.3680	−1.02715	−0.513576	+	0.858044i	$0.671679\pi$
$660$	0	0
$661$	−37.8137	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$661$	−37.8137	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$662$	0.311354	0.0121011
$663$	0	0
$664$	−1.02432	−0.0397513
$665$	−19.5552	−0.758320
$666$	0	0
$667$	12.1258	0.469513
$668$	40.6249	1.57182
$669$	0	0
$670$	−0.544976	−0.0210542
$671$	20.8916	0.806513
$672$	0	0
$673$	25.8793	0.997575	0.498788	−	0.866724i	$-0.333779\pi$
$673$	25.8793	0.997575	0.498788	+	0.866724i	$0.333779\pi$
$674$	−3.37925	−0.130164
$675$	0	0
$676$	25.7992	0.992276
$677$	19.9464	0.766602	0.383301	−	0.923623i	$-0.374787\pi$
$677$	19.9464	0.766602	0.383301	+	0.923623i	$0.374787\pi$
$678$	0	0
$679$	9.03454	0.346714
$680$	−2.15637	−0.0826929
$681$	0	0
$682$	0	0
$683$	10.3960	0.397793	0.198897	−	0.980020i	$-0.436264\pi$
$683$	10.3960	0.397793	0.198897	+	0.980020i	$0.436264\pi$
$684$	0	0
$685$	11.7253	0.448002
$686$	2.28081	0.0870816
$687$	0	0
$688$	−12.4095	−0.473107
$689$	0.106488	0.00405687
$690$	0	0
$691$	−13.4503	−0.511675	−0.255837	−	0.966720i	$-0.582351\pi$
$691$	−13.4503	−0.511675	−0.255837	+	0.966720i	$0.582351\pi$
$692$	−41.8100	−1.58938
$693$	0	0
$694$	−3.02098	−0.114675
$695$	16.4111	0.622509
$696$	0	0
$697$	−25.3270	−0.959328
$698$	−1.51066	−0.0571794
$699$	0	0
$700$	18.4642	0.697881
$701$	1.52309	0.0575264	0.0287632	−	0.999586i	$-0.490843\pi$
$701$	1.52309	0.0575264	0.0287632	+	0.999586i	$0.490843\pi$
$702$	0	0
$703$	−63.4270	−2.39219
$704$	−42.9846	−1.62004
$705$	0	0
$706$	−3.47881	−0.130927
$707$	23.1188	0.869471
$708$	0	0
$709$	1.54507	0.0580263	0.0290131	−	0.999579i	$-0.490764\pi$
$709$	1.54507	0.0580263	0.0290131	+	0.999579i	$0.490764\pi$
$710$	0.574281	0.0215524
$711$	0	0
$712$	−4.35623	−0.163257
$713$	0	0
$714$	0	0
$715$	−0.0932362	−0.00348684
$716$	20.7962	0.777192
$717$	0	0
$718$	−3.02199	−0.112780
$719$	36.4715	1.36016	0.680079	−	0.733139i	$-0.261946\pi$
$719$	36.4715	1.36016	0.680079	+	0.733139i	$0.261946\pi$
$720$	0	0
$721$	7.15019	0.266287
$722$	−2.05727	−0.0765635
$723$	0	0
$724$	−18.9271	−0.703422
$725$	−10.3432	−0.384136
$726$	0	0
$727$	11.3252	0.420027	0.210013	−	0.977698i	$-0.432649\pi$
$727$	11.3252	0.420027	0.210013	+	0.977698i	$0.432649\pi$
$728$	−0.0177690	−0.000658561 0
$729$	0	0
$730$	0.541995	0.0200601
$731$	11.2571	0.416360
$732$	0	0
$733$	1.82359	0.0673559	0.0336779	−	0.999433i	$-0.489278\pi$
$733$	1.82359	0.0673559	0.0336779	+	0.999433i	$0.489278\pi$
$734$	−0.209755	−0.00774221
$735$	0	0
$736$	−6.03151	−0.222324
$737$	20.1044	0.740556
$738$	0	0
$739$	35.2840	1.29794	0.648971	−	0.760813i	$-0.275200\pi$
$739$	35.2840	1.29794	0.648971	+	0.760813i	$0.275200\pi$
$740$	−25.9493	−0.953914
$741$	0	0
$742$	2.61904	0.0961481
$743$	−26.9859	−0.990018	−0.495009	−	0.868888i	$-0.664835\pi$
$743$	−26.9859	−0.990018	−0.495009	+	0.868888i	$0.664835\pi$
$744$	0	0
$745$	−22.3741	−0.819722
$746$	−3.01549	−0.110405
$747$	0	0
$748$	39.6208	1.44868
$749$	21.2714	0.777240
$750$	0	0
$751$	−21.1606	−0.772161	−0.386080	−	0.922465i	$-0.626171\pi$
$751$	−21.1606	−0.772161	−0.386080	+	0.922465i	$0.626171\pi$
$752$	38.3359	1.39797
$753$	0	0
$754$	0.00495761	0.000180545 0
$755$	−8.10894	−0.295114
$756$	0	0
$757$	−53.6180	−1.94878	−0.974389	−	0.224869i	$-0.927804\pi$
$757$	−53.6180	−1.94878	−0.974389	+	0.224869i	$0.927804\pi$
$758$	−1.65705	−0.0601868
$759$	0	0
$760$	−3.62787	−0.131597
$761$	−8.24779	−0.298982	−0.149491	−	0.988763i	$-0.547764\pi$
$761$	−8.24779	−0.298982	−0.149491	+	0.988763i	$0.547764\pi$
$762$	0	0
$763$	16.7995	0.608182
$764$	−19.8549	−0.718323
$765$	0	0
$766$	−3.10989	−0.112365
$767$	−0.139918	−0.00505214
$768$	0	0
$769$	26.3326	0.949577	0.474788	−	0.880100i	$-0.342525\pi$
$769$	26.3326	0.949577	0.474788	+	0.880100i	$0.342525\pi$
$770$	−2.29311	−0.0826381
$771$	0	0
$772$	41.0422	1.47714
$773$	−11.8440	−0.426000	−0.213000	−	0.977052i	$-0.568323\pi$
$773$	−11.8440	−0.426000	−0.213000	+	0.977052i	$0.568323\pi$
$774$	0	0
$775$	0	0
$776$	1.67608	0.0601677
$777$	0	0
$778$	−2.72397	−0.0976591
$779$	−42.6101	−1.52667
$780$	0	0
$781$	−21.1855	−0.758078
$782$	1.80020	0.0643751
$783$	0	0
$784$	0.441700	0.0157750
$785$	19.3332	0.690032
$786$	0	0
$787$	−38.8768	−1.38581	−0.692904	−	0.721030i	$-0.743669\pi$
$787$	−38.8768	−1.38581	−0.692904	+	0.721030i	$0.743669\pi$
$788$	1.49496	0.0532556
$789$	0	0
$790$	0.497672	0.0177064
$791$	7.32706	0.260520
$792$	0	0
$793$	−0.0499500	−0.00177378
$794$	−0.275208	−0.00976678
$795$	0	0
$796$	19.6288	0.695723
$797$	52.0663	1.84428	0.922141	−	0.386854i	$-0.126438\pi$
$797$	52.0663	1.84428	0.922141	+	0.386854i	$0.126438\pi$
$798$	0	0
$799$	−34.7760	−1.23029
$800$	5.14482	0.181897
$801$	0	0
$802$	3.31168	0.116940
$803$	−19.9945	−0.705590
$804$	0	0
$805$	13.4099	0.472638
$806$	0	0
$807$	0	0
$808$	4.28897	0.150885
$809$	31.2082	1.09722	0.548612	−	0.836077i	$-0.315157\pi$
$809$	31.2082	1.09722	0.548612	+	0.836077i	$0.315157\pi$
$810$	0	0
$811$	16.4241	0.576729	0.288364	−	0.957521i	$-0.406889\pi$
$811$	16.4241	0.576729	0.288364	+	0.957521i	$0.406889\pi$
$812$	−15.6934	−0.550729
$813$	0	0
$814$	−7.43766	−0.260690
$815$	−8.84753	−0.309916
$816$	0	0
$817$	18.9390	0.662591
$818$	−4.70345	−0.164452
$819$	0	0
$820$	−17.4327	−0.608775
$821$	−40.2675	−1.40534	−0.702672	−	0.711514i	$-0.748010\pi$
$821$	−40.2675	−1.40534	−0.702672	+	0.711514i	$0.748010\pi$
$822$	0	0
$823$	−23.3005	−0.812205	−0.406102	−	0.913828i	$-0.633112\pi$
$823$	−23.3005	−0.812205	−0.406102	+	0.913828i	$0.633112\pi$
$824$	1.32650	0.0462107
$825$	0	0
$826$	−3.44124	−0.119736
$827$	24.1368	0.839319	0.419660	−	0.907682i	$-0.362149\pi$
$827$	24.1368	0.839319	0.419660	+	0.907682i	$0.362149\pi$
$828$	0	0
$829$	22.7194	0.789076	0.394538	−	0.918880i	$-0.370905\pi$
$829$	22.7194	0.789076	0.394538	+	0.918880i	$0.370905\pi$
$830$	0.316055	0.0109704
$831$	0	0
$832$	0.102772	0.00356298
$833$	−0.400683	−0.0138829
$834$	0	0
$835$	−25.1670	−0.870941
$836$	66.6580	2.30541
$837$	0	0
$838$	−1.13831	−0.0393223
$839$	6.01460	0.207647	0.103823	−	0.994596i	$-0.466892\pi$
$839$	6.01460	0.207647	0.103823	+	0.994596i	$0.466892\pi$
$840$	0	0
$841$	−20.2090	−0.696861
$842$	0.405813	0.0139852
$843$	0	0
$844$	−27.9089	−0.960664
$845$	−15.9826	−0.549817
$846$	0	0
$847$	55.2568	1.89865
$848$	−30.9034	−1.06123
$849$	0	0
$850$	−1.53555	−0.0526691
$851$	43.4948	1.49098
$852$	0	0
$853$	10.8138	0.370258	0.185129	−	0.982714i	$-0.440730\pi$
$853$	10.8138	0.370258	0.185129	+	0.982714i	$0.440730\pi$
$854$	−1.22850	−0.0420385
$855$	0	0
$856$	3.94625	0.134880
$857$	34.0840	1.16429	0.582144	−	0.813086i	$-0.302214\pi$
$857$	34.0840	1.16429	0.582144	+	0.813086i	$0.302214\pi$
$858$	0	0
$859$	7.51793	0.256508	0.128254	−	0.991741i	$-0.459063\pi$
$859$	7.51793	0.256508	0.128254	+	0.991741i	$0.459063\pi$
$860$	7.74832	0.264215
$861$	0	0
$862$	−2.91445	−0.0992666
$863$	−22.2748	−0.758243	−0.379121	−	0.925347i	$-0.623774\pi$
$863$	−22.2748	−0.758243	−0.379121	+	0.925347i	$0.623774\pi$
$864$	0	0
$865$	25.9012	0.880668
$866$	3.39520	0.115374
$867$	0	0
$868$	0	0
$869$	−18.3594	−0.622798
$870$	0	0
$871$	−0.0480678	−0.00162872
$872$	3.11662	0.105542
$873$	0	0
$874$	3.02866	0.102446
$875$	−27.8334	−0.940939
$876$	0	0
$877$	−27.1655	−0.917313	−0.458656	−	0.888614i	$-0.651669\pi$
$877$	−27.1655	−0.917313	−0.458656	+	0.888614i	$0.651669\pi$
$878$	4.10322	0.138477
$879$	0	0
$880$	27.0576	0.912111
$881$	−22.7696	−0.767126	−0.383563	−	0.923515i	$-0.625303\pi$
$881$	−22.7696	−0.767126	−0.383563	+	0.923515i	$0.625303\pi$
$882$	0	0
$883$	20.5333	0.691000	0.345500	−	0.938419i	$-0.387709\pi$
$883$	20.5333	0.691000	0.345500	+	0.938419i	$0.387709\pi$
$884$	−0.0947296	−0.00318610
$885$	0	0
$886$	−1.65481	−0.0555946
$887$	−53.7079	−1.80333	−0.901667	−	0.432431i	$-0.857656\pi$
$887$	−53.7079	−1.80333	−0.901667	+	0.432431i	$0.857656\pi$
$888$	0	0
$889$	4.62819	0.155224
$890$	1.34412	0.0450549
$891$	0	0
$892$	19.6386	0.657547
$893$	−58.5071	−1.95787
$894$	0	0
$895$	−12.8832	−0.430639
$896$	10.3943	0.347251
$897$	0	0
$898$	−1.40247	−0.0468010
$899$	0	0
$900$	0	0
$901$	28.0337	0.933937
$902$	−4.99661	−0.166369
$903$	0	0
$904$	1.35931	0.0452099
$905$	11.7253	0.389763
$906$	0	0
$907$	53.3488	1.77142	0.885709	−	0.464241i	$-0.153673\pi$
$907$	53.3488	1.77142	0.885709	+	0.464241i	$0.153673\pi$
$908$	18.6641	0.619391
$909$	0	0
$910$	0.00548262	0.000181747 0
$911$	−44.5464	−1.47589	−0.737944	−	0.674861i	$-0.764203\pi$
$911$	−44.5464	−1.47589	−0.737944	+	0.674861i	$0.764203\pi$
$912$	0	0
$913$	−11.6594	−0.385870
$914$	0.565079	0.0186912
$915$	0	0
$916$	−29.2817	−0.967494
$917$	−18.8543	−0.622623
$918$	0	0
$919$	−11.6589	−0.384593	−0.192297	−	0.981337i	$-0.561594\pi$
$919$	−11.6589	−0.384593	−0.192297	+	0.981337i	$0.561594\pi$
$920$	2.48780	0.0820203
$921$	0	0
$922$	−3.57575	−0.117761
$923$	0.0506526	0.00166725
$924$	0	0
$925$	−37.1007	−1.21986
$926$	1.49649	0.0491779
$927$	0	0
$928$	−4.37276	−0.143543
$929$	16.1137	0.528675	0.264337	−	0.964430i	$-0.414847\pi$
$929$	16.1137	0.528675	0.264337	+	0.964430i	$0.414847\pi$
$930$	0	0
$931$	−0.674109	−0.0220930
$932$	−56.7312	−1.85829
$933$	0	0
$934$	−2.37546	−0.0777274
$935$	−24.5450	−0.802708
$936$	0	0
$937$	45.2884	1.47951	0.739754	−	0.672877i	$-0.234942\pi$
$937$	45.2884	1.47951	0.739754	+	0.672877i	$0.234942\pi$
$938$	−1.18221	−0.0386006
$939$	0	0
$940$	−23.9364	−0.780720
$941$	−24.5088	−0.798964	−0.399482	−	0.916741i	$-0.630810\pi$
$941$	−24.5088	−0.798964	−0.399482	+	0.916741i	$0.630810\pi$
$942$	0	0
$943$	29.2197	0.951525
$944$	40.6049	1.32158
$945$	0	0
$946$	2.22085	0.0722060
$947$	−44.7537	−1.45430	−0.727150	−	0.686478i	$-0.759156\pi$
$947$	−44.7537	−1.45430	−0.727150	+	0.686478i	$0.759156\pi$
$948$	0	0
$949$	0.0478049	0.00155181
$950$	−2.58341	−0.0838170
$951$	0	0
$952$	−4.67779	−0.151608
$953$	−30.8112	−0.998071	−0.499036	−	0.866581i	$-0.666312\pi$
$953$	−30.8112	−0.998071	−0.499036	+	0.866581i	$0.666312\pi$
$954$	0	0
$955$	12.3001	0.398020
$956$	17.8786	0.578236
$957$	0	0
$958$	−1.52624	−0.0493105
$959$	25.4357	0.821361
$960$	0	0
$961$	0	0
$962$	0.0177828	0.000573339 0
$963$	0	0
$964$	1.56417	0.0503786
$965$	−25.4256	−0.818479
$966$	0	0
$967$	−44.5447	−1.43246	−0.716230	−	0.697864i	$-0.754134\pi$
$967$	−44.5447	−1.43246	−0.716230	+	0.697864i	$0.754134\pi$
$968$	10.2512	0.329486
$969$	0	0
$970$	−0.517155	−0.0166048
$971$	−6.35732	−0.204016	−0.102008	−	0.994784i	$-0.532527\pi$
$971$	−6.35732	−0.204016	−0.102008	+	0.994784i	$0.532527\pi$
$972$	0	0
$973$	35.6005	1.14130
$974$	1.87835	0.0601863
$975$	0	0
$976$	14.4957	0.463997
$977$	13.4329	0.429758	0.214879	−	0.976641i	$-0.431064\pi$
$977$	13.4329	0.429758	0.214879	+	0.976641i	$0.431064\pi$
$978$	0	0
$979$	−49.5852	−1.58475
$980$	−0.275792	−0.00880984
$981$	0	0
$982$	−3.79155	−0.120993
$983$	42.1777	1.34526	0.672629	−	0.739980i	$-0.265165\pi$
$983$	42.1777	1.34526	0.672629	+	0.739980i	$0.265165\pi$
$984$	0	0
$985$	−0.926123	−0.0295087
$986$	1.30512	0.0415635
$987$	0	0
$988$	−0.159373	−0.00507033
$989$	−12.9873	−0.412973
$990$	0	0
$991$	−40.1209	−1.27448	−0.637241	−	0.770665i	$-0.719924\pi$
$991$	−40.1209	−1.27448	−0.637241	+	0.770665i	$0.719924\pi$
$992$	0	0
$993$	0	0
$994$	1.24579	0.0395139
$995$	−12.1600	−0.385497
$996$	0	0
$997$	−40.6053	−1.28598	−0.642991	−	0.765874i	$-0.722307\pi$
$997$	−40.6053	−1.28598	−0.642991	+	0.765874i	$0.722307\pi$
$998$	3.57960	0.113310
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bs.1.6		16
3.2	odd	2		961.2.a.l.1.12	yes	16
31.30	odd	2	inner	8649.2.a.bs.1.5		16
93.2	odd	10		961.2.d.s.531.12		64
93.5	odd	6		961.2.c.l.521.11		32
93.8	odd	10		961.2.d.s.374.5		64
93.11	even	30		961.2.g.w.338.6		128
93.14	odd	30		961.2.g.w.816.5		128
93.17	even	30		961.2.g.w.816.6		128
93.20	odd	30		961.2.g.w.338.5		128
93.23	even	10		961.2.d.s.374.6		64
93.26	even	6		961.2.c.l.521.12		32
93.29	even	10		961.2.d.s.531.11		64
93.35	odd	10		961.2.d.s.388.5		64
93.38	odd	30		961.2.g.w.235.6		128
93.41	odd	30		961.2.g.w.844.11		128
93.44	even	30		961.2.g.w.448.11		128
93.47	odd	10		961.2.d.s.628.12		64
93.50	odd	30		961.2.g.w.547.12		128
93.53	even	30		961.2.g.w.732.5		128
93.56	odd	6		961.2.c.l.439.11		32
93.59	odd	30		961.2.g.w.846.11		128
93.65	even	30		961.2.g.w.846.12		128
93.68	even	6		961.2.c.l.439.12		32
93.71	odd	30		961.2.g.w.732.6		128
93.74	even	30		961.2.g.w.547.11		128
93.77	even	10		961.2.d.s.628.11		64
93.80	odd	30		961.2.g.w.448.12		128
93.83	even	30		961.2.g.w.844.12		128
93.86	even	30		961.2.g.w.235.5		128
93.89	even	10		961.2.d.s.388.6		64
93.92	even	2		961.2.a.l.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.l.1.11	✓	16	93.92	even	2
961.2.a.l.1.12	yes	16	3.2	odd	2
961.2.c.l.439.11		32	93.56	odd	6
961.2.c.l.439.12		32	93.68	even	6
961.2.c.l.521.11		32	93.5	odd	6
961.2.c.l.521.12		32	93.26	even	6
961.2.d.s.374.5		64	93.8	odd	10
961.2.d.s.374.6		64	93.23	even	10
961.2.d.s.388.5		64	93.35	odd	10
961.2.d.s.388.6		64	93.89	even	10
961.2.d.s.531.11		64	93.29	even	10
961.2.d.s.531.12		64	93.2	odd	10
961.2.d.s.628.11		64	93.77	even	10
961.2.d.s.628.12		64	93.47	odd	10
961.2.g.w.235.5		128	93.86	even	30
961.2.g.w.235.6		128	93.38	odd	30
961.2.g.w.338.5		128	93.20	odd	30
961.2.g.w.338.6		128	93.11	even	30
961.2.g.w.448.11		128	93.44	even	30
961.2.g.w.448.12		128	93.80	odd	30
961.2.g.w.547.11		128	93.74	even	30
961.2.g.w.547.12		128	93.50	odd	30
961.2.g.w.732.5		128	93.53	even	30
961.2.g.w.732.6		128	93.71	odd	30
961.2.g.w.816.5		128	93.14	odd	30
961.2.g.w.816.6		128	93.17	even	30
961.2.g.w.844.11		128	93.41	odd	30
961.2.g.w.844.12		128	93.83	even	30
961.2.g.w.846.11		128	93.59	odd	30
961.2.g.w.846.12		128	93.65	even	30
8649.2.a.bs.1.5		16	31.30	odd	2	inner
8649.2.a.bs.1.6		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.124175 q^{2} -1.98458 q^{4} +1.22944 q^{5} +2.66703 q^{7} +0.494784 q^{8} -0.152666 q^{10} +5.63192 q^{11} -0.0134654 q^{13} -0.331177 q^{14} +3.90772 q^{16} -3.54485 q^{17} -5.96385 q^{19} -2.43993 q^{20} -0.699342 q^{22} +4.08969 q^{23} -3.48847 q^{25} +0.00167206 q^{26} -5.29293 q^{28} +2.96497 q^{29} -1.47481 q^{32} +0.440181 q^{34} +3.27896 q^{35} +10.6352 q^{37} +0.740559 q^{38} +0.608309 q^{40} +7.14473 q^{41} -3.17563 q^{43} -11.1770 q^{44} -0.507836 q^{46} +9.81029 q^{47} +0.113032 q^{49} +0.433179 q^{50} +0.0267232 q^{52} -7.90828 q^{53} +6.92413 q^{55} +1.31960 q^{56} -0.368174 q^{58} +10.3909 q^{59} +3.70951 q^{61} -7.63231 q^{64} -0.0165550 q^{65} +3.56973 q^{67} +7.03504 q^{68} -0.407164 q^{70} -3.76169 q^{71} -3.55021 q^{73} -1.32063 q^{74} +11.8357 q^{76} +15.0205 q^{77} -3.25987 q^{79} +4.80433 q^{80} -0.887194 q^{82} -2.07024 q^{83} -4.35820 q^{85} +0.394333 q^{86} +2.78658 q^{88} -8.80431 q^{89} -0.0359126 q^{91} -8.11632 q^{92} -1.21819 q^{94} -7.33223 q^{95} +3.38749 q^{97} -0.0140358 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{2} + 8 q^{4} + 16 q^{5} - 16 q^{7} + 8 q^{10} + 8 q^{14} - 8 q^{16} - 32 q^{19} + 24 q^{20} - 8 q^{28} + 8 q^{32} + 16 q^{35} + 24 q^{38} + 32 q^{41} + 32 q^{47} - 16 q^{49} + 32 q^{50} + 48 q^{56}+ \cdots - 16 q^{98}+O(q^{100}) \) 16 * q + 8 * q^2 + 8 * q^4 + 16 * q^5 - 16 * q^7 + 8 * q^10 + 8 * q^14 - 8 * q^16 - 32 * q^19 + 24 * q^20 - 8 * q^28 + 8 * q^32 + 16 * q^35 + 24 * q^38 + 32 * q^41 + 32 * q^47 - 16 * q^49 + 32 * q^50 + 48 * q^56 + 64 * q^59 - 16 * q^64 + 16 * q^67 + 88 * q^70 + 48 * q^71 + 40 * q^76 + 40 * q^80 + 88 * q^82 - 32 * q^94 + 48 * q^95 - 16 * q^98

Introduction

L-functions

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