LMFDB - Embedded newform 8624.2.a.cr.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.89289.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - x + 15 $ x^4 - 9*x^2 - x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.61001$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40788 q^{3} +2.61001 q^{5} -1.01788 q^{9} +O(q^{10})$	$q+1.40788 q^{3} +2.61001 q^{5} -1.01788 q^{9} +1.00000 q^{11} -1.40788 q^{13} +3.67457 q^{15} -7.09671 q^{17} -0.123301 q^{19} -2.20213 q^{23} +1.81214 q^{25} -5.65668 q^{27} -4.42576 q^{29} -1.46882 q^{31} +1.40788 q^{33} +1.22001 q^{37} -1.98212 q^{39} +5.70672 q^{41} +1.59212 q^{43} -2.65668 q^{45} -10.2882 q^{47} -9.99130 q^{51} +1.26669 q^{53} +2.61001 q^{55} -0.173593 q^{57} +0.302459 q^{59} +6.27031 q^{61} -3.67457 q^{65} -12.2988 q^{67} -3.10033 q^{69} -7.09671 q^{71} +4.94126 q^{73} +2.55127 q^{75} -5.98573 q^{79} -4.91026 q^{81} -8.31673 q^{83} -18.5225 q^{85} -6.23092 q^{87} +10.7246 q^{89} -2.06792 q^{93} -0.321817 q^{95} -13.1613 q^{97} -1.01788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 10 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 10 q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 3 q^{17} - 13 q^{19} - 10 q^{23} + 2 q^{25} - 23 q^{27} + 4 q^{29} - q^{31} - 2 q^{33} - 8 q^{37} - 22 q^{39} - 9 q^{41} + 14 q^{43} - 11 q^{45} - 11 q^{47} + 12 q^{51} - q^{53} + 4 q^{55} - 10 q^{57} - 33 q^{59} + 9 q^{61} - q^{65} - 25 q^{67} + 23 q^{69} - 3 q^{71} - 16 q^{75} - 28 q^{79} + 4 q^{81} + 5 q^{83} - 27 q^{85} - 47 q^{87} - 3 q^{89} + 38 q^{93} - 25 q^{95} - 20 q^{97} + 10 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 10 * q^9 + 4 * q^11 + 2 * q^13 + q^15 - 3 * q^17 - 13 * q^19 - 10 * q^23 + 2 * q^25 - 23 * q^27 + 4 * q^29 - q^31 - 2 * q^33 - 8 * q^37 - 22 * q^39 - 9 * q^41 + 14 * q^43 - 11 * q^45 - 11 * q^47 + 12 * q^51 - q^53 + 4 * q^55 - 10 * q^57 - 33 * q^59 + 9 * q^61 - q^65 - 25 * q^67 + 23 * q^69 - 3 * q^71 - 16 * q^75 - 28 * q^79 + 4 * q^81 + 5 * q^83 - 27 * q^85 - 47 * q^87 - 3 * q^89 + 38 * q^93 - 25 * q^95 - 20 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.40788	0.812838	0.406419	−	0.913687i	$-0.366777\pi$
$3$	1.40788	0.812838	0.406419	+	0.913687i	$0.366777\pi$
$4$	0	0
$5$	2.61001	1.16723	0.583615	−	0.812030i	$-0.301638\pi$
$5$	2.61001	1.16723	0.583615	+	0.812030i	$0.301638\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.01788	−0.339295
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.40788	−0.390475	−0.195237	−	0.980756i	$-0.562548\pi$
$13$	−1.40788	−0.390475	−0.195237	+	0.980756i	$0.562548\pi$
$14$	0	0
$15$	3.67457	0.948769
$16$	0	0
$17$	−7.09671	−1.72121	−0.860603	−	0.509277i	$-0.829913\pi$
$17$	−7.09671	−1.72121	−0.860603	+	0.509277i	$0.829913\pi$
$18$	0	0
$19$	−0.123301	−0.0282873	−0.0141436	−	0.999900i	$-0.504502\pi$
$19$	−0.123301	−0.0282873	−0.0141436	+	0.999900i	$0.504502\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.20213	−0.459176	−0.229588	−	0.973288i	$-0.573738\pi$
$23$	−2.20213	−0.459176	−0.229588	+	0.973288i	$0.573738\pi$
$24$	0	0
$25$	1.81214	0.362428
$26$	0	0
$27$	−5.65668	−1.08863
$28$	0	0
$29$	−4.42576	−0.821843	−0.410922	−	0.911671i	$-0.634793\pi$
$29$	−4.42576	−0.821843	−0.410922	+	0.911671i	$0.634793\pi$
$30$	0	0
$31$	−1.46882	−0.263808	−0.131904	−	0.991262i	$-0.542109\pi$
$31$	−1.46882	−0.263808	−0.131904	+	0.991262i	$0.542109\pi$
$32$	0	0
$33$	1.40788	0.245080
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.22001	0.200569	0.100285	−	0.994959i	$-0.468025\pi$
$37$	1.22001	0.200569	0.100285	+	0.994959i	$0.468025\pi$
$38$	0	0
$39$	−1.98212	−0.317393
$40$	0	0
$41$	5.70672	0.891240	0.445620	−	0.895222i	$-0.352983\pi$
$41$	5.70672	0.891240	0.445620	+	0.895222i	$0.352983\pi$
$42$	0	0
$43$	1.59212	0.242797	0.121398	−	0.992604i	$-0.461262\pi$
$43$	1.59212	0.242797	0.121398	+	0.992604i	$0.461262\pi$
$44$	0	0
$45$	−2.65668	−0.396035
$46$	0	0
$47$	−10.2882	−1.50069	−0.750343	−	0.661048i	$-0.770112\pi$
$47$	−10.2882	−1.50069	−0.750343	+	0.661048i	$0.770112\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.99130	−1.39906
$52$	0	0
$53$	1.26669	0.173993	0.0869967	−	0.996209i	$-0.472273\pi$
$53$	1.26669	0.173993	0.0869967	+	0.996209i	$0.472273\pi$
$54$	0	0
$55$	2.61001	0.351933
$56$	0	0
$57$	−0.173593	−0.0229929
$58$	0	0
$59$	0.302459	0.0393768	0.0196884	−	0.999806i	$-0.493733\pi$
$59$	0.302459	0.0393768	0.0196884	+	0.999806i	$0.493733\pi$
$60$	0	0
$61$	6.27031	0.802830	0.401415	−	0.915896i	$-0.368518\pi$
$61$	6.27031	0.802830	0.401415	+	0.915896i	$0.368518\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.67457	−0.455774
$66$	0	0
$67$	−12.2988	−1.50254	−0.751271	−	0.659993i	$-0.770559\pi$
$67$	−12.2988	−1.50254	−0.751271	+	0.659993i	$0.770559\pi$
$68$	0	0
$69$	−3.10033	−0.373236
$70$	0	0
$71$	−7.09671	−0.842225	−0.421112	−	0.907008i	$-0.638360\pi$
$71$	−7.09671	−0.842225	−0.421112	+	0.907008i	$0.638360\pi$
$72$	0	0
$73$	4.94126	0.578331	0.289165	−	0.957279i	$-0.406622\pi$
$73$	4.94126	0.578331	0.289165	+	0.957279i	$0.406622\pi$
$74$	0	0
$75$	2.55127	0.294595
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.98573	−0.673447	−0.336724	−	0.941604i	$-0.609319\pi$
$79$	−5.98573	−0.673447	−0.336724	+	0.941604i	$0.609319\pi$
$80$	0	0
$81$	−4.91026	−0.545585
$82$	0	0
$83$	−8.31673	−0.912879	−0.456440	−	0.889754i	$-0.650876\pi$
$83$	−8.31673	−0.912879	−0.456440	+	0.889754i	$0.650876\pi$
$84$	0	0
$85$	−18.5225	−2.00904
$86$	0	0
$87$	−6.23092	−0.668025
$88$	0	0
$89$	10.7246	1.13681	0.568403	−	0.822750i	$-0.307562\pi$
$89$	10.7246	1.13681	0.568403	+	0.822750i	$0.307562\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.06792	−0.214433
$94$	0	0
$95$	−0.321817	−0.0330177
$96$	0	0
$97$	−13.1613	−1.33632	−0.668162	−	0.744015i	$-0.732919\pi$
$97$	−13.1613	−1.33632	−0.668162	+	0.744015i	$0.732919\pi$
$98$	0	0
$99$	−1.01788	−0.102301
$100$	0	0
$101$	13.7282	1.36601	0.683004	−	0.730414i	$-0.260673\pi$
$101$	13.7282	1.36601	0.683004	+	0.730414i	$0.260673\pi$
$102$	0	0
$103$	−15.4579	−1.52311	−0.761557	−	0.648098i	$-0.775565\pi$
$103$	−15.4579	−1.52311	−0.761557	+	0.648098i	$0.775565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3634	1.00187	0.500934	−	0.865485i	$-0.332990\pi$
$107$	10.3634	1.00187	0.500934	+	0.865485i	$0.332990\pi$
$108$	0	0
$109$	−20.1526	−1.93027	−0.965133	−	0.261760i	$-0.915697\pi$
$109$	−20.1526	−1.93027	−0.965133	+	0.261760i	$0.915697\pi$
$110$	0	0
$111$	1.71763	0.163030
$112$	0	0
$113$	10.5513	0.992580	0.496290	−	0.868157i	$-0.334695\pi$
$113$	10.5513	0.992580	0.496290	+	0.868157i	$0.334695\pi$
$114$	0	0
$115$	−5.74758	−0.535964
$116$	0	0
$117$	1.43305	0.132486
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.03436	0.724434
$124$	0	0
$125$	−8.32034	−0.744194
$126$	0	0
$127$	5.89097	0.522739	0.261369	−	0.965239i	$-0.415826\pi$
$127$	5.89097	0.522739	0.261369	+	0.965239i	$0.415826\pi$
$128$	0	0
$129$	2.24151	0.197354
$130$	0	0
$131$	−12.7246	−1.11175	−0.555877	−	0.831265i	$-0.687617\pi$
$131$	−12.7246	−1.11175	−0.555877	+	0.831265i	$0.687617\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−14.7640	−1.27068
$136$	0	0
$137$	−10.7140	−0.915355	−0.457677	−	0.889118i	$-0.651318\pi$
$137$	−10.7140	−0.915355	−0.457677	+	0.889118i	$0.651318\pi$
$138$	0	0
$139$	−0.0502917	−0.00426569	−0.00213284	−	0.999998i	$-0.500679\pi$
$139$	−0.0502917	−0.00426569	−0.00213284	+	0.999998i	$0.500679\pi$
$140$	0	0
$141$	−14.4845	−1.21982
$142$	0	0
$143$	−1.40788	−0.117733
$144$	0	0
$145$	−11.5513	−0.959280
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.6480	−1.03616	−0.518081	−	0.855331i	$-0.673354\pi$
$149$	−12.6480	−1.03616	−0.518081	+	0.855331i	$0.673354\pi$
$150$	0	0
$151$	7.40788	0.602844	0.301422	−	0.953491i	$-0.402539\pi$
$151$	7.40788	0.602844	0.301422	+	0.953491i	$0.402539\pi$
$152$	0	0
$153$	7.22363	0.583996
$154$	0	0
$155$	−3.83364	−0.307925
$156$	0	0
$157$	8.85881	0.707010	0.353505	−	0.935433i	$-0.384990\pi$
$157$	8.85881	0.707010	0.353505	+	0.935433i	$0.384990\pi$
$158$	0	0
$159$	1.78335	0.141428
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.66004	−0.208351	−0.104175	−	0.994559i	$-0.533220\pi$
$163$	−2.66004	−0.208351	−0.104175	+	0.994559i	$0.533220\pi$
$164$	0	0
$165$	3.67457	0.286065
$166$	0	0
$167$	21.1883	1.63960	0.819801	−	0.572648i	$-0.194084\pi$
$167$	21.1883	1.63960	0.819801	+	0.572648i	$0.194084\pi$
$168$	0	0
$169$	−11.0179	−0.847530
$170$	0	0
$171$	0.125506	0.00959771
$172$	0	0
$173$	8.96785	0.681813	0.340906	−	0.940097i	$-0.389266\pi$
$173$	8.96785	0.681813	0.340906	+	0.940097i	$0.389266\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.425825	0.0320070
$178$	0	0
$179$	−15.4939	−1.15807	−0.579036	−	0.815302i	$-0.696571\pi$
$179$	−15.4939	−1.15807	−0.579036	+	0.815302i	$0.696571\pi$
$180$	0	0
$181$	9.79278	0.727892	0.363946	−	0.931420i	$-0.381429\pi$
$181$	9.79278	0.727892	0.363946	+	0.931420i	$0.381429\pi$
$182$	0	0
$183$	8.82782	0.652571
$184$	0	0
$185$	3.18425	0.234110
$186$	0	0
$187$	−7.09671	−0.518963
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.0880	1.67059	0.835295	−	0.549802i	$-0.185297\pi$
$191$	23.0880	1.67059	0.835295	+	0.549802i	$0.185297\pi$
$192$	0	0
$193$	24.5582	1.76774	0.883870	−	0.467732i	$-0.154929\pi$
$193$	24.5582	1.76774	0.883870	+	0.467732i	$0.154929\pi$
$194$	0	0
$195$	−5.17334	−0.370470
$196$	0	0
$197$	−21.3167	−1.51875	−0.759377	−	0.650651i	$-0.774496\pi$
$197$	−21.3167	−1.51875	−0.759377	+	0.650651i	$0.774496\pi$
$198$	0	0
$199$	−22.3489	−1.58427	−0.792135	−	0.610346i	$-0.791030\pi$
$199$	−22.3489	−1.58427	−0.792135	+	0.610346i	$0.791030\pi$
$200$	0	0
$201$	−17.3153	−1.22132
$202$	0	0
$203$	0	0
$204$	0	0
$205$	14.8946	1.04028
$206$	0	0
$207$	2.24151	0.155796
$208$	0	0
$209$	−0.123301	−0.00852893
$210$	0	0
$211$	5.62428	0.387191	0.193596	−	0.981081i	$-0.437985\pi$
$211$	5.62428	0.387191	0.193596	+	0.981081i	$0.437985\pi$
$212$	0	0
$213$	−9.99130	−0.684592
$214$	0	0
$215$	4.15545	0.283400
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.95668	0.470089
$220$	0	0
$221$	9.99130	0.672087
$222$	0	0
$223$	−15.2988	−1.02449	−0.512243	−	0.858840i	$-0.671185\pi$
$223$	−15.2988	−1.02449	−0.512243	+	0.858840i	$0.671185\pi$
$224$	0	0
$225$	−1.84455	−0.122970
$226$	0	0
$227$	−12.5742	−0.834582	−0.417291	−	0.908773i	$-0.637020\pi$
$227$	−12.5742	−0.834582	−0.417291	+	0.908773i	$0.637020\pi$
$228$	0	0
$229$	−12.1218	−0.801033	−0.400516	−	0.916290i	$-0.631169\pi$
$229$	−12.1218	−0.801033	−0.400516	+	0.916290i	$0.631169\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9110	1.69749	0.848743	−	0.528806i	$-0.177360\pi$
$233$	25.9110	1.69749	0.848743	+	0.528806i	$0.177360\pi$
$234$	0	0
$235$	−26.8523	−1.75165
$236$	0	0
$237$	−8.42717	−0.547404
$238$	0	0
$239$	−24.7841	−1.60315	−0.801574	−	0.597895i	$-0.796004\pi$
$239$	−24.7841	−1.60315	−0.801574	+	0.597895i	$0.796004\pi$
$240$	0	0
$241$	−17.1879	−1.10717	−0.553584	−	0.832794i	$-0.686740\pi$
$241$	−17.1879	−1.10717	−0.553584	+	0.832794i	$0.686740\pi$
$242$	0	0
$243$	10.0570	0.645158
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.173593	0.0110455
$248$	0	0
$249$	−11.7089	−0.742023
$250$	0	0
$251$	−17.5082	−1.10511	−0.552554	−	0.833477i	$-0.686347\pi$
$251$	−17.5082	−1.10511	−0.552554	+	0.833477i	$0.686347\pi$
$252$	0	0
$253$	−2.20213	−0.138447
$254$	0	0
$255$	−26.0774	−1.63303
$256$	0	0
$257$	29.5283	1.84192	0.920962	−	0.389652i	$-0.127405\pi$
$257$	29.5283	1.84192	0.920962	+	0.389652i	$0.127405\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.50491	0.278847
$262$	0	0
$263$	18.9482	1.16840	0.584199	−	0.811610i	$-0.301409\pi$
$263$	18.9482	1.16840	0.584199	+	0.811610i	$0.301409\pi$
$264$	0	0
$265$	3.30607	0.203091
$266$	0	0
$267$	15.0989	0.924039
$268$	0	0
$269$	10.2846	0.627062	0.313531	−	0.949578i	$-0.398488\pi$
$269$	10.2846	0.627062	0.313531	+	0.949578i	$0.398488\pi$
$270$	0	0
$271$	−7.28237	−0.442372	−0.221186	−	0.975232i	$-0.570993\pi$
$271$	−7.28237	−0.442372	−0.221186	+	0.975232i	$0.570993\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.81214	0.109276
$276$	0	0
$277$	30.7855	1.84972	0.924860	−	0.380307i	$-0.124182\pi$
$277$	30.7855	1.84972	0.924860	+	0.380307i	$0.124182\pi$
$278$	0	0
$279$	1.49509	0.0895087
$280$	0	0
$281$	−1.92338	−0.114739	−0.0573695	−	0.998353i	$-0.518271\pi$
$281$	−1.92338	−0.114739	−0.0573695	+	0.998353i	$0.518271\pi$
$282$	0	0
$283$	−3.42938	−0.203855	−0.101928	−	0.994792i	$-0.532501\pi$
$283$	−3.42938	−0.203855	−0.101928	+	0.994792i	$0.532501\pi$
$284$	0	0
$285$	−0.453079	−0.0268381
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.3633	1.96255
$290$	0	0
$291$	−18.5294	−1.08622
$292$	0	0
$293$	−14.6837	−0.857834	−0.428917	−	0.903344i	$-0.641105\pi$
$293$	−14.6837	−0.857834	−0.428917	+	0.903344i	$0.641105\pi$
$294$	0	0
$295$	0.789420	0.0459618
$296$	0	0
$297$	−5.65668	−0.328234
$298$	0	0
$299$	3.10033	0.179297
$300$	0	0
$301$	0	0
$302$	0	0
$303$	19.3276	1.11034
$304$	0	0
$305$	16.3655	0.937088
$306$	0	0
$307$	−2.21298	−0.126301	−0.0631506	−	0.998004i	$-0.520115\pi$
$307$	−2.21298	−0.126301	−0.0631506	+	0.998004i	$0.520115\pi$
$308$	0	0
$309$	−21.7628	−1.23804
$310$	0	0
$311$	−18.2057	−1.03235	−0.516177	−	0.856482i	$-0.672645\pi$
$311$	−18.2057	−1.03235	−0.516177	+	0.856482i	$0.672645\pi$
$312$	0	0
$313$	1.61362	0.0912073	0.0456037	−	0.998960i	$-0.485479\pi$
$313$	1.61362	0.0912073	0.0456037	+	0.998960i	$0.485479\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.79278	0.437686	0.218843	−	0.975760i	$-0.429772\pi$
$317$	7.79278	0.437686	0.218843	+	0.975760i	$0.429772\pi$
$318$	0	0
$319$	−4.42576	−0.247795
$320$	0	0
$321$	14.5904	0.814356
$322$	0	0
$323$	0.875034	0.0486882
$324$	0	0
$325$	−2.55127	−0.141519
$326$	0	0
$327$	−28.3723	−1.56899
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.4564	1.34425	0.672124	−	0.740439i	$-0.265382\pi$
$331$	24.4564	1.34425	0.672124	+	0.740439i	$0.265382\pi$
$332$	0	0
$333$	−1.24183	−0.0680520
$334$	0	0
$335$	−32.1001	−1.75381
$336$	0	0
$337$	7.37908	0.401964	0.200982	−	0.979595i	$-0.435587\pi$
$337$	7.37908	0.401964	0.200982	+	0.979595i	$0.435587\pi$
$338$	0	0
$339$	14.8549	0.806806
$340$	0	0
$341$	−1.46882	−0.0795412
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.09188	−0.435652
$346$	0	0
$347$	−34.8967	−1.87335	−0.936677	−	0.350194i	$-0.886116\pi$
$347$	−34.8967	−1.87335	−0.936677	+	0.350194i	$0.886116\pi$
$348$	0	0
$349$	−0.389738	−0.0208622	−0.0104311	−	0.999946i	$-0.503320\pi$
$349$	−0.389738	−0.0208622	−0.0104311	+	0.999946i	$0.503320\pi$
$350$	0	0
$351$	7.96391	0.425082
$352$	0	0
$353$	−35.5317	−1.89116	−0.945580	−	0.325391i	$-0.894504\pi$
$353$	−35.5317	−1.89116	−0.945580	+	0.325391i	$0.894504\pi$
$354$	0	0
$355$	−18.5225	−0.983071
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.1325	1.27366	0.636832	−	0.771003i	$-0.280245\pi$
$359$	24.1325	1.27366	0.636832	+	0.771003i	$0.280245\pi$
$360$	0	0
$361$	−18.9848	−0.999200
$362$	0	0
$363$	1.40788	0.0738944
$364$	0	0
$365$	12.8967	0.675045
$366$	0	0
$367$	33.6163	1.75476	0.877378	−	0.479799i	$-0.159290\pi$
$367$	33.6163	1.75476	0.877378	+	0.479799i	$0.159290\pi$
$368$	0	0
$369$	−5.80878	−0.302393
$370$	0	0
$371$	0	0
$372$	0	0
$373$	33.2292	1.72054	0.860271	−	0.509837i	$-0.170294\pi$
$373$	33.2292	1.72054	0.860271	+	0.509837i	$0.170294\pi$
$374$	0	0
$375$	−11.7140	−0.604909
$376$	0	0
$377$	6.23092	0.320909
$378$	0	0
$379$	−8.01788	−0.411851	−0.205926	−	0.978568i	$-0.566020\pi$
$379$	−8.01788	−0.411851	−0.205926	+	0.978568i	$0.566020\pi$
$380$	0	0
$381$	8.29375	0.424902
$382$	0	0
$383$	10.8910	0.556502	0.278251	−	0.960508i	$-0.410245\pi$
$383$	10.8910	0.556502	0.278251	+	0.960508i	$0.410245\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.62060	−0.0823796
$388$	0	0
$389$	−22.7961	−1.15581	−0.577905	−	0.816104i	$-0.696130\pi$
$389$	−22.7961	−1.15581	−0.577905	+	0.816104i	$0.696130\pi$
$390$	0	0
$391$	15.6279	0.790336
$392$	0	0
$393$	−17.9147	−0.903676
$394$	0	0
$395$	−15.6228	−0.786068
$396$	0	0
$397$	−10.5692	−0.530450	−0.265225	−	0.964186i	$-0.585446\pi$
$397$	−10.5692	−0.530450	−0.265225	+	0.964186i	$0.585446\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.83389	−0.0915802	−0.0457901	−	0.998951i	$-0.514581\pi$
$401$	−1.83389	−0.0915802	−0.0457901	+	0.998951i	$0.514581\pi$
$402$	0	0
$403$	2.06792	0.103010
$404$	0	0
$405$	−12.8158	−0.636823
$406$	0	0
$407$	1.22001	0.0604739
$408$	0	0
$409$	−3.14627	−0.155573	−0.0777866	−	0.996970i	$-0.524785\pi$
$409$	−3.14627	−0.155573	−0.0777866	+	0.996970i	$0.524785\pi$
$410$	0	0
$411$	−15.0839	−0.744035
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−21.7067	−1.06554
$416$	0	0
$417$	−0.0708045	−0.00346731
$418$	0	0
$419$	−28.5976	−1.39709	−0.698543	−	0.715568i	$-0.746168\pi$
$419$	−28.5976	−1.39709	−0.698543	+	0.715568i	$0.746168\pi$
$420$	0	0
$421$	20.0628	0.977803	0.488901	−	0.872339i	$-0.337398\pi$
$421$	20.0628	0.977803	0.488901	+	0.872339i	$0.337398\pi$
$422$	0	0
$423$	10.4722	0.509175
$424$	0	0
$425$	−12.8602	−0.623813
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.98212	−0.0956975
$430$	0	0
$431$	26.9541	1.29833	0.649166	−	0.760647i	$-0.275118\pi$
$431$	26.9541	1.29833	0.649166	+	0.760647i	$0.275118\pi$
$432$	0	0
$433$	−6.65809	−0.319968	−0.159984	−	0.987120i	$-0.551144\pi$
$433$	−6.65809	−0.319968	−0.159984	+	0.987120i	$0.551144\pi$
$434$	0	0
$435$	−16.2628	−0.779740
$436$	0	0
$437$	0.271526	0.0129888
$438$	0	0
$439$	9.35250	0.446370	0.223185	−	0.974776i	$-0.428355\pi$
$439$	9.35250	0.446370	0.223185	+	0.974776i	$0.428355\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.4688	−0.544897	−0.272449	−	0.962170i	$-0.587833\pi$
$443$	−11.4688	−0.544897	−0.272449	+	0.962170i	$0.587833\pi$
$444$	0	0
$445$	27.9913	1.32691
$446$	0	0
$447$	−17.8068	−0.842232
$448$	0	0
$449$	14.6194	0.689934	0.344967	−	0.938615i	$-0.387890\pi$
$449$	14.6194	0.689934	0.344967	+	0.938615i	$0.387890\pi$
$450$	0	0
$451$	5.70672	0.268719
$452$	0	0
$453$	10.4294	0.490015
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.5174	1.42754	0.713772	−	0.700379i	$-0.246985\pi$
$457$	30.5174	1.42754	0.713772	+	0.700379i	$0.246985\pi$
$458$	0	0
$459$	40.1439	1.87376
$460$	0	0
$461$	11.6566	0.542903	0.271451	−	0.962452i	$-0.412496\pi$
$461$	11.6566	0.542903	0.271451	+	0.962452i	$0.412496\pi$
$462$	0	0
$463$	−20.2672	−0.941895	−0.470948	−	0.882161i	$-0.656088\pi$
$463$	−20.2672	−0.941895	−0.470948	+	0.882161i	$0.656088\pi$
$464$	0	0
$465$	−5.39729	−0.250293
$466$	0	0
$467$	29.2901	1.35538	0.677692	−	0.735346i	$-0.262980\pi$
$467$	29.2901	1.35538	0.677692	+	0.735346i	$0.262980\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.4721	0.574685
$472$	0	0
$473$	1.59212	0.0732059
$474$	0	0
$475$	−0.223439	−0.0102521
$476$	0	0
$477$	−1.28934	−0.0590350
$478$	0	0
$479$	−4.66513	−0.213155	−0.106578	−	0.994304i	$-0.533989\pi$
$479$	−4.66513	−0.213155	−0.106578	+	0.994304i	$0.533989\pi$
$480$	0	0
$481$	−1.71763	−0.0783172
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−34.3510	−1.55980
$486$	0	0
$487$	14.5728	0.660355	0.330178	−	0.943919i	$-0.392891\pi$
$487$	14.5728	0.660355	0.330178	+	0.943919i	$0.392891\pi$
$488$	0	0
$489$	−3.74501	−0.169355
$490$	0	0
$491$	−6.01401	−0.271409	−0.135704	−	0.990749i	$-0.543330\pi$
$491$	−6.01401	−0.271409	−0.135704	+	0.990749i	$0.543330\pi$
$492$	0	0
$493$	31.4084	1.41456
$494$	0	0
$495$	−2.65668	−0.119409
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.4514	−0.960297	−0.480149	−	0.877187i	$-0.659417\pi$
$499$	−21.4514	−0.960297	−0.480149	+	0.877187i	$0.659417\pi$
$500$	0	0
$501$	29.8306	1.33273
$502$	0	0
$503$	20.2923	0.904791	0.452395	−	0.891817i	$-0.350570\pi$
$503$	20.2923	0.904791	0.452395	+	0.891817i	$0.350570\pi$
$504$	0	0
$505$	35.8308	1.59445
$506$	0	0
$507$	−15.5118	−0.688904
$508$	0	0
$509$	13.2543	0.587487	0.293743	−	0.955884i	$-0.405099\pi$
$509$	13.2543	0.587487	0.293743	+	0.955884i	$0.405099\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.697476	0.0307943
$514$	0	0
$515$	−40.3453	−1.77782
$516$	0	0
$517$	−10.2882	−0.452474
$518$	0	0
$519$	12.6256	0.554203
$520$	0	0
$521$	−20.6779	−0.905916	−0.452958	−	0.891532i	$-0.649631\pi$
$521$	−20.6779	−0.905916	−0.452958	+	0.891532i	$0.649631\pi$
$522$	0	0
$523$	11.1857	0.489115	0.244557	−	0.969635i	$-0.421357\pi$
$523$	11.1857	0.489115	0.244557	+	0.969635i	$0.421357\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.4238	0.454068
$528$	0	0
$529$	−18.1506	−0.789157
$530$	0	0
$531$	−0.307868	−0.0133603
$532$	0	0
$533$	−8.03436	−0.348007
$534$	0	0
$535$	27.0486	1.16941
$536$	0	0
$537$	−21.8135	−0.941324
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.5403	1.82895	0.914475	−	0.404642i	$-0.132604\pi$
$541$	42.5403	1.82895	0.914475	+	0.404642i	$0.132604\pi$
$542$	0	0
$543$	13.7870	0.591658
$544$	0	0
$545$	−52.5984	−2.25307
$546$	0	0
$547$	−31.3346	−1.33977	−0.669886	−	0.742464i	$-0.733657\pi$
$547$	−31.3346	−1.33977	−0.669886	+	0.742464i	$0.733657\pi$
$548$	0	0
$549$	−6.38244	−0.272396
$550$	0	0
$551$	0.545702	0.0232477
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.48303	0.190294
$556$	0	0
$557$	−7.26087	−0.307653	−0.153826	−	0.988098i	$-0.549160\pi$
$557$	−7.26087	−0.307653	−0.153826	+	0.988098i	$0.549160\pi$
$558$	0	0
$559$	−2.24151	−0.0948059
$560$	0	0
$561$	−9.99130	−0.421833
$562$	0	0
$563$	−24.8392	−1.04685	−0.523424	−	0.852073i	$-0.675346\pi$
$563$	−24.8392	−1.04685	−0.523424	+	0.852073i	$0.675346\pi$
$564$	0	0
$565$	27.5389	1.15857
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.13978	−0.215471	−0.107735	−	0.994180i	$-0.534360\pi$
$569$	−5.13978	−0.215471	−0.107735	+	0.994180i	$0.534360\pi$
$570$	0	0
$571$	46.2664	1.93619	0.968094	−	0.250589i	$-0.0806241\pi$
$571$	46.2664	1.93619	0.968094	+	0.250589i	$0.0806241\pi$
$572$	0	0
$573$	32.5051	1.35792
$574$	0	0
$575$	−3.99057	−0.166418
$576$	0	0
$577$	−17.2451	−0.717924	−0.358962	−	0.933352i	$-0.616869\pi$
$577$	−17.2451	−0.717924	−0.358962	+	0.933352i	$0.616869\pi$
$578$	0	0
$579$	34.5750	1.43689
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.26669	0.0524610
$584$	0	0
$585$	3.74028	0.154642
$586$	0	0
$587$	−37.3189	−1.54031	−0.770157	−	0.637854i	$-0.779822\pi$
$587$	−37.3189	−1.54031	−0.770157	+	0.637854i	$0.779822\pi$
$588$	0	0
$589$	0.181108	0.00746241
$590$	0	0
$591$	−30.0113	−1.23450
$592$	0	0
$593$	35.6047	1.46211	0.731054	−	0.682319i	$-0.239029\pi$
$593$	35.6047	1.46211	0.731054	+	0.682319i	$0.239029\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−31.4645	−1.28775
$598$	0	0
$599$	−11.7783	−0.481249	−0.240625	−	0.970618i	$-0.577352\pi$
$599$	−11.7783	−0.481249	−0.240625	+	0.970618i	$0.577352\pi$
$600$	0	0
$601$	−15.4185	−0.628935	−0.314467	−	0.949268i	$-0.601826\pi$
$601$	−15.4185	−0.628935	−0.314467	+	0.949268i	$0.601826\pi$
$602$	0	0
$603$	12.5188	0.509805
$604$	0	0
$605$	2.61001	0.106112
$606$	0	0
$607$	−15.5921	−0.632862	−0.316431	−	0.948616i	$-0.602485\pi$
$607$	−15.5921	−0.632862	−0.316431	+	0.948616i	$0.602485\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.4845	0.585980
$612$	0	0
$613$	−19.3849	−0.782949	−0.391474	−	0.920189i	$-0.628035\pi$
$613$	−19.3849	−0.782949	−0.391474	+	0.920189i	$0.628035\pi$
$614$	0	0
$615$	20.9697	0.845581
$616$	0	0
$617$	41.1146	1.65521	0.827606	−	0.561310i	$-0.189702\pi$
$617$	41.1146	1.65521	0.827606	+	0.561310i	$0.189702\pi$
$618$	0	0
$619$	−11.3564	−0.456451	−0.228225	−	0.973608i	$-0.573292\pi$
$619$	−11.3564	−0.456451	−0.228225	+	0.973608i	$0.573292\pi$
$620$	0	0
$621$	12.4568	0.499872
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.7768	−1.23107
$626$	0	0
$627$	−0.173593	−0.00693264
$628$	0	0
$629$	−8.65809	−0.345221
$630$	0	0
$631$	11.3455	0.451658	0.225829	−	0.974167i	$-0.427491\pi$
$631$	11.3455	0.451658	0.225829	+	0.974167i	$0.427491\pi$
$632$	0	0
$633$	7.91829	0.314724
$634$	0	0
$635$	15.3755	0.610157
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.22363	0.285762
$640$	0	0
$641$	−17.5670	−0.693855	−0.346928	−	0.937892i	$-0.612775\pi$
$641$	−17.5670	−0.693855	−0.346928	+	0.937892i	$0.612775\pi$
$642$	0	0
$643$	10.9395	0.431413	0.215706	−	0.976458i	$-0.430795\pi$
$643$	10.9395	0.431413	0.215706	+	0.976458i	$0.430795\pi$
$644$	0	0
$645$	5.85037	0.230358
$646$	0	0
$647$	4.78722	0.188205	0.0941024	−	0.995563i	$-0.470002\pi$
$647$	4.78722	0.188205	0.0941024	+	0.995563i	$0.470002\pi$
$648$	0	0
$649$	0.302459	0.0118726
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0611	0.393723	0.196861	−	0.980431i	$-0.436925\pi$
$653$	10.0611	0.393723	0.196861	+	0.980431i	$0.436925\pi$
$654$	0	0
$655$	−33.2113	−1.29767
$656$	0	0
$657$	−5.02963	−0.196224
$658$	0	0
$659$	29.4651	1.14780	0.573899	−	0.818926i	$-0.305430\pi$
$659$	29.4651	1.14780	0.573899	+	0.818926i	$0.305430\pi$
$660$	0	0
$661$	−43.8271	−1.70468	−0.852338	−	0.522992i	$-0.824816\pi$
$661$	−43.8271	−1.70468	−0.852338	+	0.522992i	$0.824816\pi$
$662$	0	0
$663$	14.0665	0.546298
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.74610	0.377371
$668$	0	0
$669$	−21.5389	−0.832741
$670$	0	0
$671$	6.27031	0.242062
$672$	0	0
$673$	6.81134	0.262558	0.131279	−	0.991345i	$-0.458092\pi$
$673$	6.81134	0.262558	0.131279	+	0.991345i	$0.458092\pi$
$674$	0	0
$675$	−10.2507	−0.394549
$676$	0	0
$677$	32.3312	1.24259	0.621294	−	0.783577i	$-0.286607\pi$
$677$	32.3312	1.24259	0.621294	+	0.783577i	$0.286607\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−17.7030	−0.678380
$682$	0	0
$683$	37.0723	1.41853	0.709267	−	0.704940i	$-0.249026\pi$
$683$	37.0723	1.41853	0.709267	+	0.704940i	$0.249026\pi$
$684$	0	0
$685$	−27.9635	−1.06843
$686$	0	0
$687$	−17.0660	−0.651110
$688$	0	0
$689$	−1.78335	−0.0679400
$690$	0	0
$691$	−16.3390	−0.621566	−0.310783	−	0.950481i	$-0.600591\pi$
$691$	−16.3390	−0.621566	−0.310783	+	0.950481i	$0.600591\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.131262	−0.00497904
$696$	0	0
$697$	−40.4990	−1.53401
$698$	0	0
$699$	36.4795	1.37978
$700$	0	0
$701$	−13.3956	−0.505943	−0.252972	−	0.967474i	$-0.581408\pi$
$701$	−13.3956	−0.505943	−0.252972	+	0.967474i	$0.581408\pi$
$702$	0	0
$703$	−0.150429	−0.00567355
$704$	0	0
$705$	−37.8047	−1.42381
$706$	0	0
$707$	0	0
$708$	0	0
$709$	13.8513	0.520195	0.260098	−	0.965582i	$-0.416245\pi$
$709$	13.8513	0.520195	0.260098	+	0.965582i	$0.416245\pi$
$710$	0	0
$711$	6.09278	0.228497
$712$	0	0
$713$	3.23454	0.121134
$714$	0	0
$715$	−3.67457	−0.137421
$716$	0	0
$717$	−34.8929	−1.30310
$718$	0	0
$719$	26.4349	0.985857	0.492928	−	0.870070i	$-0.335926\pi$
$719$	26.4349	0.985857	0.492928	+	0.870070i	$0.335926\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−24.1984	−0.899948
$724$	0	0
$725$	−8.02009	−0.297859
$726$	0	0
$727$	5.40401	0.200424	0.100212	−	0.994966i	$-0.468048\pi$
$727$	5.40401	0.200424	0.100212	+	0.994966i	$0.468048\pi$
$728$	0	0
$729$	28.8898	1.06999
$730$	0	0
$731$	−11.2988	−0.417903
$732$	0	0
$733$	−37.6896	−1.39210	−0.696048	−	0.717995i	$-0.745060\pi$
$733$	−37.6896	−1.39210	−0.696048	+	0.717995i	$0.745060\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.2988	−0.453034
$738$	0	0
$739$	−35.5529	−1.30783	−0.653917	−	0.756566i	$-0.726876\pi$
$739$	−35.5529	−1.30783	−0.653917	+	0.756566i	$0.726876\pi$
$740$	0	0
$741$	0.244397	0.00897816
$742$	0	0
$743$	8.38465	0.307603	0.153801	−	0.988102i	$-0.450848\pi$
$743$	8.38465	0.307603	0.153801	+	0.988102i	$0.450848\pi$
$744$	0	0
$745$	−33.0113	−1.20944
$746$	0	0
$747$	8.46546	0.309735
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.09844	−0.0765733	−0.0382866	−	0.999267i	$-0.512190\pi$
$751$	−2.09844	−0.0765733	−0.0382866	+	0.999267i	$0.512190\pi$
$752$	0	0
$753$	−24.6494	−0.898274
$754$	0	0
$755$	19.3346	0.703659
$756$	0	0
$757$	−32.7962	−1.19200	−0.595999	−	0.802985i	$-0.703244\pi$
$757$	−32.7962	−1.19200	−0.595999	+	0.802985i	$0.703244\pi$
$758$	0	0
$759$	−3.10033	−0.112535
$760$	0	0
$761$	−29.5009	−1.06941	−0.534704	−	0.845040i	$-0.679577\pi$
$761$	−29.5009	−1.06941	−0.534704	+	0.845040i	$0.679577\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	18.8537	0.681658
$766$	0	0
$767$	−0.425825	−0.0153756
$768$	0	0
$769$	−40.0043	−1.44259	−0.721297	−	0.692626i	$-0.756454\pi$
$769$	−40.0043	−1.44259	−0.721297	+	0.692626i	$0.756454\pi$
$770$	0	0
$771$	41.5722	1.49719
$772$	0	0
$773$	16.5991	0.597028	0.298514	−	0.954405i	$-0.403509\pi$
$773$	16.5991	0.597028	0.298514	+	0.954405i	$0.403509\pi$
$774$	0	0
$775$	−2.66171	−0.0956114
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.703646	−0.0252107
$780$	0	0
$781$	−7.09671	−0.253940
$782$	0	0
$783$	25.0351	0.894682
$784$	0	0
$785$	23.1216	0.825244
$786$	0	0
$787$	−0.153980	−0.00548878	−0.00274439	−	0.999996i	$-0.500874\pi$
$787$	−0.153980	−0.00548878	−0.00274439	+	0.999996i	$0.500874\pi$
$788$	0	0
$789$	26.6768	0.949718
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.82782	−0.313485
$794$	0	0
$795$	4.65454	0.165080
$796$	0	0
$797$	43.6156	1.54494	0.772471	−	0.635050i	$-0.219020\pi$
$797$	43.6156	1.54494	0.772471	+	0.635050i	$0.219020\pi$
$798$	0	0
$799$	73.0123	2.58299
$800$	0	0
$801$	−10.9164	−0.385712
$802$	0	0
$803$	4.94126	0.174373
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.4794	0.509699
$808$	0	0
$809$	−10.7838	−0.379139	−0.189569	−	0.981867i	$-0.560709\pi$
$809$	−10.7838	−0.379139	−0.189569	+	0.981867i	$0.560709\pi$
$810$	0	0
$811$	−29.0858	−1.02134	−0.510670	−	0.859777i	$-0.670603\pi$
$811$	−29.0858	−1.02134	−0.510670	+	0.859777i	$0.670603\pi$
$812$	0	0
$813$	−10.2527	−0.359577
$814$	0	0
$815$	−6.94273	−0.243193
$816$	0	0
$817$	−0.196311	−0.00686805
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.5761	−1.20672	−0.603358	−	0.797471i	$-0.706171\pi$
$821$	−34.5761	−1.20672	−0.603358	+	0.797471i	$0.706171\pi$
$822$	0	0
$823$	32.9253	1.14770	0.573851	−	0.818960i	$-0.305449\pi$
$823$	32.9253	1.14770	0.573851	+	0.818960i	$0.305449\pi$
$824$	0	0
$825$	2.55127	0.0888237
$826$	0	0
$827$	−2.55152	−0.0887251	−0.0443626	−	0.999015i	$-0.514126\pi$
$827$	−2.55152	−0.0887251	−0.0443626	+	0.999015i	$0.514126\pi$
$828$	0	0
$829$	34.1724	1.18686	0.593428	−	0.804887i	$-0.297774\pi$
$829$	34.1724	1.18686	0.593428	+	0.804887i	$0.297774\pi$
$830$	0	0
$831$	43.3422	1.50352
$832$	0	0
$833$	0	0
$834$	0	0
$835$	55.3017	1.91379
$836$	0	0
$837$	8.30866	0.287189
$838$	0	0
$839$	−37.4297	−1.29222	−0.646108	−	0.763246i	$-0.723604\pi$
$839$	−37.4297	−1.29222	−0.646108	+	0.763246i	$0.723604\pi$
$840$	0	0
$841$	−9.41265	−0.324574
$842$	0	0
$843$	−2.70788	−0.0932642
$844$	0	0
$845$	−28.7568	−0.989263
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.82814	−0.165701
$850$	0	0
$851$	−2.68663	−0.0920966
$852$	0	0
$853$	1.16684	0.0399518	0.0199759	−	0.999800i	$-0.493641\pi$
$853$	1.16684	0.0399518	0.0199759	+	0.999800i	$0.493641\pi$
$854$	0	0
$855$	0.327573	0.0112027
$856$	0	0
$857$	−38.4211	−1.31244	−0.656221	−	0.754569i	$-0.727846\pi$
$857$	−38.4211	−1.31244	−0.656221	+	0.754569i	$0.727846\pi$
$858$	0	0
$859$	1.04661	0.0357099	0.0178550	−	0.999841i	$-0.494316\pi$
$859$	1.04661	0.0357099	0.0178550	+	0.999841i	$0.494316\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.5477	−0.393087	−0.196543	−	0.980495i	$-0.562972\pi$
$863$	−11.5477	−0.393087	−0.196543	+	0.980495i	$0.562972\pi$
$864$	0	0
$865$	23.4061	0.795833
$866$	0	0
$867$	46.9715	1.59523
$868$	0	0
$869$	−5.98573	−0.203052
$870$	0	0
$871$	17.3153	0.586705
$872$	0	0
$873$	13.3966	0.453408
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.24010	0.0418753	0.0209377	−	0.999781i	$-0.493335\pi$
$877$	1.24010	0.0418753	0.0209377	+	0.999781i	$0.493335\pi$
$878$	0	0
$879$	−20.6729	−0.697280
$880$	0	0
$881$	34.2996	1.15558	0.577791	−	0.816185i	$-0.303915\pi$
$881$	34.2996	1.15558	0.577791	+	0.816185i	$0.303915\pi$
$882$	0	0
$883$	−43.5196	−1.46455	−0.732275	−	0.681009i	$-0.761542\pi$
$883$	−43.5196	−1.46455	−0.732275	+	0.681009i	$0.761542\pi$
$884$	0	0
$885$	1.11141	0.0373595
$886$	0	0
$887$	−35.8143	−1.20253	−0.601263	−	0.799051i	$-0.705336\pi$
$887$	−35.8143	−1.20253	−0.601263	+	0.799051i	$0.705336\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.91026	−0.164500
$892$	0	0
$893$	1.26855	0.0424503
$894$	0	0
$895$	−40.4393	−1.35174
$896$	0	0
$897$	4.36488	0.145739
$898$	0	0
$899$	6.50066	0.216809
$900$	0	0
$901$	−8.98935	−0.299479
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.5592	0.849617
$906$	0	0
$907$	40.4327	1.34255	0.671273	−	0.741210i	$-0.265748\pi$
$907$	40.4327	1.34255	0.671273	+	0.741210i	$0.265748\pi$
$908$	0	0
$909$	−13.9737	−0.463479
$910$	0	0
$911$	−36.1235	−1.19683	−0.598413	−	0.801188i	$-0.704202\pi$
$911$	−36.1235	−1.19683	−0.598413	+	0.801188i	$0.704202\pi$
$912$	0	0
$913$	−8.31673	−0.275244
$914$	0	0
$915$	23.0407	0.761701
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.16300	0.104338	0.0521689	−	0.998638i	$-0.483387\pi$
$919$	3.16300	0.104338	0.0521689	+	0.998638i	$0.483387\pi$
$920$	0	0
$921$	−3.11560	−0.102662
$922$	0	0
$923$	9.99130	0.328868
$924$	0	0
$925$	2.21083	0.0726918
$926$	0	0
$927$	15.7344	0.516784
$928$	0	0
$929$	−39.3963	−1.29255	−0.646275	−	0.763104i	$-0.723674\pi$
$929$	−39.3963	−1.29255	−0.646275	+	0.763104i	$0.723674\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−25.6314	−0.839136
$934$	0	0
$935$	−18.5225	−0.605750
$936$	0	0
$937$	−8.99974	−0.294009	−0.147004	−	0.989136i	$-0.546963\pi$
$937$	−8.99974	−0.294009	−0.147004	+	0.989136i	$0.546963\pi$
$938$	0	0
$939$	2.27178	0.0741368
$940$	0	0
$941$	−1.30051	−0.0423954	−0.0211977	−	0.999775i	$-0.506748\pi$
$941$	−1.30051	−0.0423954	−0.0211977	+	0.999775i	$0.506748\pi$
$942$	0	0
$943$	−12.5669	−0.409236
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.3532	1.31130	0.655652	−	0.755063i	$-0.272394\pi$
$947$	40.3532	1.31130	0.655652	+	0.755063i	$0.272394\pi$
$948$	0	0
$949$	−6.95668	−0.225823
$950$	0	0
$951$	10.9713	0.355768
$952$	0	0
$953$	30.9719	1.00328	0.501640	−	0.865077i	$-0.332730\pi$
$953$	30.9719	1.00328	0.501640	+	0.865077i	$0.332730\pi$
$954$	0	0
$955$	60.2599	1.94996
$956$	0	0
$957$	−6.23092	−0.201417
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.8426	−0.930405
$962$	0	0
$963$	−10.5487	−0.339928
$964$	0	0
$965$	64.0972	2.06336
$966$	0	0
$967$	32.0863	1.03183	0.515913	−	0.856641i	$-0.327453\pi$
$967$	32.0863	1.03183	0.515913	+	0.856641i	$0.327453\pi$
$968$	0	0
$969$	1.23194	0.0395756
$970$	0	0
$971$	−33.9748	−1.09030	−0.545152	−	0.838337i	$-0.683528\pi$
$971$	−33.9748	−1.09030	−0.545152	+	0.838337i	$0.683528\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.59187	−0.115032
$976$	0	0
$977$	52.0812	1.66622	0.833112	−	0.553104i	$-0.186557\pi$
$977$	52.0812	1.66622	0.833112	+	0.553104i	$0.186557\pi$
$978$	0	0
$979$	10.7246	0.342760
$980$	0	0
$981$	20.5130	0.654929
$982$	0	0
$983$	−31.2710	−0.997391	−0.498696	−	0.866777i	$-0.666187\pi$
$983$	−31.2710	−0.997391	−0.498696	+	0.866777i	$0.666187\pi$
$984$	0	0
$985$	−55.6368	−1.77274
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.50606	−0.111486
$990$	0	0
$991$	32.1779	1.02217	0.511083	−	0.859531i	$-0.329244\pi$
$991$	32.1779	1.02217	0.511083	+	0.859531i	$0.329244\pi$
$992$	0	0
$993$	34.4316	1.09266
$994$	0	0
$995$	−58.3307	−1.84921
$996$	0	0
$997$	−57.4069	−1.81810	−0.909048	−	0.416691i	$-0.863190\pi$
$997$	−57.4069	−1.81810	−0.909048	+	0.416691i	$0.863190\pi$
$998$	0	0
$999$	−6.90124	−0.218345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cr.1.3		4
4.3	odd	2		4312.2.a.bd.1.2		4
7.3	odd	6		1232.2.q.n.177.3		8
7.5	odd	6		1232.2.q.n.529.3		8
7.6	odd	2		8624.2.a.cz.1.2		4
28.3	even	6		616.2.q.d.177.2	✓	8
28.19	even	6		616.2.q.d.529.2	yes	8
28.27	even	2		4312.2.a.y.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.d.177.2	✓	8	28.3	even	6
616.2.q.d.529.2	yes	8	28.19	even	6
1232.2.q.n.177.3		8	7.3	odd	6
1232.2.q.n.529.3		8	7.5	odd	6
4312.2.a.y.1.3		4	28.27	even	2
4312.2.a.bd.1.2		4	4.3	odd	2
8624.2.a.cr.1.3		4	1.1	even	1	trivial
8624.2.a.cz.1.2		4	7.6	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40788 q^{3} +2.61001 q^{5} -1.01788 q^{9} +O(q^{10})\)	\(q+1.40788 q^{3} +2.61001 q^{5} -1.01788 q^{9} +1.00000 q^{11} -1.40788 q^{13} +3.67457 q^{15} -7.09671 q^{17} -0.123301 q^{19} -2.20213 q^{23} +1.81214 q^{25} -5.65668 q^{27} -4.42576 q^{29} -1.46882 q^{31} +1.40788 q^{33} +1.22001 q^{37} -1.98212 q^{39} +5.70672 q^{41} +1.59212 q^{43} -2.65668 q^{45} -10.2882 q^{47} -9.99130 q^{51} +1.26669 q^{53} +2.61001 q^{55} -0.173593 q^{57} +0.302459 q^{59} +6.27031 q^{61} -3.67457 q^{65} -12.2988 q^{67} -3.10033 q^{69} -7.09671 q^{71} +4.94126 q^{73} +2.55127 q^{75} -5.98573 q^{79} -4.91026 q^{81} -8.31673 q^{83} -18.5225 q^{85} -6.23092 q^{87} +10.7246 q^{89} -2.06792 q^{93} -0.321817 q^{95} -13.1613 q^{97} -1.01788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 10 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 10 q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 3 q^{17} - 13 q^{19} - 10 q^{23} + 2 q^{25} - 23 q^{27} + 4 q^{29} - q^{31} - 2 q^{33} - 8 q^{37} - 22 q^{39} - 9 q^{41} + 14 q^{43} - 11 q^{45} - 11 q^{47} + 12 q^{51} - q^{53} + 4 q^{55} - 10 q^{57} - 33 q^{59} + 9 q^{61} - q^{65} - 25 q^{67} + 23 q^{69} - 3 q^{71} - 16 q^{75} - 28 q^{79} + 4 q^{81} + 5 q^{83} - 27 q^{85} - 47 q^{87} - 3 q^{89} + 38 q^{93} - 25 q^{95} - 20 q^{97} + 10 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 10 * q^9 + 4 * q^11 + 2 * q^13 + q^15 - 3 * q^17 - 13 * q^19 - 10 * q^23 + 2 * q^25 - 23 * q^27 + 4 * q^29 - q^31 - 2 * q^33 - 8 * q^37 - 22 * q^39 - 9 * q^41 + 14 * q^43 - 11 * q^45 - 11 * q^47 + 12 * q^51 - q^53 + 4 * q^55 - 10 * q^57 - 33 * q^59 + 9 * q^61 - q^65 - 25 * q^67 + 23 * q^69 - 3 * q^71 - 16 * q^75 - 28 * q^79 + 4 * q^81 + 5 * q^83 - 27 * q^85 - 47 * q^87 - 3 * q^89 + 38 * q^93 - 25 * q^95 - 20 * q^97 + 10 * q^99

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