LMFDB - Embedded newform 8512.2.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.11491 q^{3} +0.357926 q^{5} -1.00000 q^{7} -1.75698 q^{9} +O(q^{10})$	$q+1.11491 q^{3} +0.357926 q^{5} -1.00000 q^{7} -1.75698 q^{9} -0.527166 q^{11} +0.357926 q^{13} +0.399055 q^{15} -3.70265 q^{17} -1.00000 q^{19} -1.11491 q^{21} +8.58774 q^{23} -4.87189 q^{25} -5.30359 q^{27} +4.04113 q^{29} -1.24302 q^{31} -0.587741 q^{33} -0.357926 q^{35} -3.30359 q^{37} +0.399055 q^{39} +12.1623 q^{41} +0.715853 q^{43} -0.628870 q^{45} +10.8176 q^{47} +1.00000 q^{49} -4.12811 q^{51} +6.06058 q^{53} -0.188687 q^{55} -1.11491 q^{57} +7.87189 q^{59} -5.53341 q^{61} +1.75698 q^{63} +0.128111 q^{65} -8.64832 q^{67} +9.57454 q^{69} -7.53341 q^{71} -9.00624 q^{73} -5.43171 q^{75} +0.527166 q^{77} -9.17548 q^{79} -0.642074 q^{81} -16.0606 q^{83} -1.32528 q^{85} +4.50548 q^{87} -14.7981 q^{89} -0.357926 q^{91} -1.38585 q^{93} -0.357926 q^{95} -9.07378 q^{97} +0.926221 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9	$ 3 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} - 7 q^{15} + 7 q^{17} - 3 q^{19} + 3 q^{21} + 14 q^{23} - q^{25} - 6 q^{27} + 3 q^{29} - 11 q^{31} + 10 q^{33} - 2 q^{35} - 7 q^{39} - 7 q^{41} + 4 q^{43} + 19 q^{45} + 8 q^{47} + 3 q^{49} - 26 q^{51} + q^{53} + 3 q^{55} + 3 q^{57} + 10 q^{59} + 6 q^{61} - 2 q^{63} + 14 q^{65} + 3 q^{67} - 3 q^{69} + q^{73} - 20 q^{75} + 7 q^{77} - 4 q^{79} - q^{81} - 31 q^{83} + 7 q^{85} + 20 q^{87} - 28 q^{89} - 2 q^{91} + 24 q^{93} - 2 q^{95} - 30 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9 - 7 * q^11 + 2 * q^13 - 7 * q^15 + 7 * q^17 - 3 * q^19 + 3 * q^21 + 14 * q^23 - q^25 - 6 * q^27 + 3 * q^29 - 11 * q^31 + 10 * q^33 - 2 * q^35 - 7 * q^39 - 7 * q^41 + 4 * q^43 + 19 * q^45 + 8 * q^47 + 3 * q^49 - 26 * q^51 + q^53 + 3 * q^55 + 3 * q^57 + 10 * q^59 + 6 * q^61 - 2 * q^63 + 14 * q^65 + 3 * q^67 - 3 * q^69 + q^73 - 20 * q^75 + 7 * q^77 - 4 * q^79 - q^81 - 31 * q^83 + 7 * q^85 + 20 * q^87 - 28 * q^89 - 2 * q^91 + 24 * q^93 - 2 * q^95 - 30 * q^97

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
$3$	1.11491	0.643692	0.321846	−	0.946792i	$-0.395697\pi$
$3$	1.11491	0.643692	0.321846	+	0.946792i	$0.395697\pi$
$4$	0	0
$5$	0.357926	0.160070	0.0800348	−	0.996792i	$-0.474497\pi$
$5$	0.357926	0.160070	0.0800348	+	0.996792i	$0.474497\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.75698	−0.585660
$10$	0	0
$11$	−0.527166	−0.158947	−0.0794733	−	0.996837i	$-0.525324\pi$
$11$	−0.527166	−0.158947	−0.0794733	+	0.996837i	$0.525324\pi$
$12$	0	0
$13$	0.357926	0.0992709	0.0496355	−	0.998767i	$-0.484194\pi$
$13$	0.357926	0.0992709	0.0496355	+	0.998767i	$0.484194\pi$
$14$	0	0
$15$	0.399055	0.103036
$16$	0	0
$17$	−3.70265	−0.898024	−0.449012	−	0.893526i	$-0.648224\pi$
$17$	−3.70265	−0.898024	−0.449012	+	0.893526i	$0.648224\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.11491	−0.243293
$22$	0	0
$23$	8.58774	1.79067	0.895334	−	0.445395i	$-0.146937\pi$
$23$	8.58774	1.79067	0.895334	+	0.445395i	$0.146937\pi$
$24$	0	0
$25$	−4.87189	−0.974378
$26$	0	0
$27$	−5.30359	−1.02068
$28$	0	0
$29$	4.04113	0.750419	0.375209	−	0.926940i	$-0.377571\pi$
$29$	4.04113	0.750419	0.375209	+	0.926940i	$0.377571\pi$
$30$	0	0
$31$	−1.24302	−0.223253	−0.111626	−	0.993750i	$-0.535606\pi$
$31$	−1.24302	−0.223253	−0.111626	+	0.993750i	$0.535606\pi$
$32$	0	0
$33$	−0.587741	−0.102313
$34$	0	0
$35$	−0.357926	−0.0605006
$36$	0	0
$37$	−3.30359	−0.543108	−0.271554	−	0.962423i	$-0.587537\pi$
$37$	−3.30359	−0.543108	−0.271554	+	0.962423i	$0.587537\pi$
$38$	0	0
$39$	0.399055	0.0638999
$40$	0	0
$41$	12.1623	1.89943	0.949714	−	0.313117i	$-0.101373\pi$
$41$	12.1623	1.89943	0.949714	+	0.313117i	$0.101373\pi$
$42$	0	0
$43$	0.715853	0.109167	0.0545833	−	0.998509i	$-0.482617\pi$
$43$	0.715853	0.109167	0.0545833	+	0.998509i	$0.482617\pi$
$44$	0	0
$45$	−0.628870	−0.0937464
$46$	0	0
$47$	10.8176	1.57790	0.788951	−	0.614456i	$-0.210624\pi$
$47$	10.8176	1.57790	0.788951	+	0.614456i	$0.210624\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.12811	−0.578051
$52$	0	0
$53$	6.06058	0.832484	0.416242	−	0.909254i	$-0.363347\pi$
$53$	6.06058	0.832484	0.416242	+	0.909254i	$0.363347\pi$
$54$	0	0
$55$	−0.188687	−0.0254425
$56$	0	0
$57$	−1.11491	−0.147673
$58$	0	0
$59$	7.87189	1.02483	0.512416	−	0.858737i	$-0.328750\pi$
$59$	7.87189	1.02483	0.512416	+	0.858737i	$0.328750\pi$
$60$	0	0
$61$	−5.53341	−0.708480	−0.354240	−	0.935154i	$-0.615260\pi$
$61$	−5.53341	−0.708480	−0.354240	+	0.935154i	$0.615260\pi$
$62$	0	0
$63$	1.75698	0.221359
$64$	0	0
$65$	0.128111	0.0158902
$66$	0	0
$67$	−8.64832	−1.05656	−0.528280	−	0.849070i	$-0.677163\pi$
$67$	−8.64832	−1.05656	−0.528280	+	0.849070i	$0.677163\pi$
$68$	0	0
$69$	9.57454	1.15264
$70$	0	0
$71$	−7.53341	−0.894051	−0.447026	−	0.894521i	$-0.647517\pi$
$71$	−7.53341	−0.894051	−0.447026	+	0.894521i	$0.647517\pi$
$72$	0	0
$73$	−9.00624	−1.05410	−0.527051	−	0.849834i	$-0.676702\pi$
$73$	−9.00624	−1.05410	−0.527051	+	0.849834i	$0.676702\pi$
$74$	0	0
$75$	−5.43171	−0.627199
$76$	0	0
$77$	0.527166	0.0600762
$78$	0	0
$79$	−9.17548	−1.03232	−0.516161	−	0.856491i	$-0.672639\pi$
$79$	−9.17548	−1.03232	−0.516161	+	0.856491i	$0.672639\pi$
$80$	0	0
$81$	−0.642074	−0.0713415
$82$	0	0
$83$	−16.0606	−1.76288	−0.881439	−	0.472299i	$-0.843424\pi$
$83$	−16.0606	−1.76288	−0.881439	+	0.472299i	$0.843424\pi$
$84$	0	0
$85$	−1.32528	−0.143746
$86$	0	0
$87$	4.50548	0.483039
$88$	0	0
$89$	−14.7981	−1.56860	−0.784298	−	0.620384i	$-0.786977\pi$
$89$	−14.7981	−1.56860	−0.784298	+	0.620384i	$0.786977\pi$
$90$	0	0
$91$	−0.357926	−0.0375209
$92$	0	0
$93$	−1.38585	−0.143706
$94$	0	0
$95$	−0.357926	−0.0367225
$96$	0	0
$97$	−9.07378	−0.921303	−0.460651	−	0.887581i	$-0.652384\pi$
$97$	−9.07378	−0.921303	−0.460651	+	0.887581i	$0.652384\pi$
$98$	0	0
$99$	0.926221	0.0930887
$100$	0	0
$101$	−0.0194469	−0.00193504	−0.000967520	−	1.00000i	$-0.500308\pi$
$101$	−0.0194469	−0.00193504	−0.000967520		1.00000i	$0.500308\pi$
$102$	0	0
$103$	−19.6157	−1.93279	−0.966395	−	0.257064i	$-0.917245\pi$
$103$	−19.6157	−1.93279	−0.966395	+	0.257064i	$0.917245\pi$
$104$	0	0
$105$	−0.399055	−0.0389438
$106$	0	0
$107$	−13.6421	−1.31883	−0.659415	−	0.751780i	$-0.729196\pi$
$107$	−13.6421	−1.31883	−0.659415	+	0.751780i	$0.729196\pi$
$108$	0	0
$109$	−1.07378	−0.102849	−0.0514247	−	0.998677i	$-0.516376\pi$
$109$	−1.07378	−0.102849	−0.0514247	+	0.998677i	$0.516376\pi$
$110$	0	0
$111$	−3.68320	−0.349594
$112$	0	0
$113$	11.1972	1.05334	0.526670	−	0.850070i	$-0.323440\pi$
$113$	11.1972	1.05334	0.526670	+	0.850070i	$0.323440\pi$
$114$	0	0
$115$	3.07378	0.286631
$116$	0	0
$117$	−0.628870	−0.0581390
$118$	0	0
$119$	3.70265	0.339421
$120$	0	0
$121$	−10.7221	−0.974736
$122$	0	0
$123$	13.5598	1.22265
$124$	0	0
$125$	−3.53341	−0.316038
$126$	0	0
$127$	−12.8370	−1.13910	−0.569550	−	0.821957i	$-0.692882\pi$
$127$	−12.8370	−1.13910	−0.569550	+	0.821957i	$0.692882\pi$
$128$	0	0
$129$	0.798110	0.0702696
$130$	0	0
$131$	1.70265	0.148761	0.0743806	−	0.997230i	$-0.476302\pi$
$131$	1.70265	0.148761	0.0743806	+	0.997230i	$0.476302\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−1.89830	−0.163379
$136$	0	0
$137$	3.43171	0.293190	0.146595	−	0.989197i	$-0.453169\pi$
$137$	3.43171	0.293190	0.146595	+	0.989197i	$0.453169\pi$
$138$	0	0
$139$	4.96511	0.421136	0.210568	−	0.977579i	$-0.432469\pi$
$139$	4.96511	0.421136	0.210568	+	0.977579i	$0.432469\pi$
$140$	0	0
$141$	12.0606	1.01568
$142$	0	0
$143$	−0.188687	−0.0157788
$144$	0	0
$145$	1.44643	0.120119
$146$	0	0
$147$	1.11491	0.0919560
$148$	0	0
$149$	0.735300	0.0602381	0.0301190	−	0.999546i	$-0.490411\pi$
$149$	0.735300	0.0602381	0.0301190	+	0.999546i	$0.490411\pi$
$150$	0	0
$151$	1.24302	0.101155	0.0505777	−	0.998720i	$-0.483894\pi$
$151$	1.24302	0.101155	0.0505777	+	0.998720i	$0.483894\pi$
$152$	0	0
$153$	6.50548	0.525937
$154$	0	0
$155$	−0.444909	−0.0357360
$156$	0	0
$157$	15.8238	1.26288	0.631438	−	0.775426i	$-0.282465\pi$
$157$	15.8238	1.26288	0.631438	+	0.775426i	$0.282465\pi$
$158$	0	0
$159$	6.75698	0.535863
$160$	0	0
$161$	−8.58774	−0.676809
$162$	0	0
$163$	−20.5202	−1.60727	−0.803633	−	0.595125i	$-0.797103\pi$
$163$	−20.5202	−1.60727	−0.803633	+	0.595125i	$0.797103\pi$
$164$	0	0
$165$	−0.210368	−0.0163771
$166$	0	0
$167$	−7.27719	−0.563126	−0.281563	−	0.959543i	$-0.590853\pi$
$167$	−7.27719	−0.563126	−0.281563	+	0.959543i	$0.590853\pi$
$168$	0	0
$169$	−12.8719	−0.990145
$170$	0	0
$171$	1.75698	0.134360
$172$	0	0
$173$	−4.68945	−0.356532	−0.178266	−	0.983982i	$-0.557049\pi$
$173$	−4.68945	−0.356532	−0.178266	+	0.983982i	$0.557049\pi$
$174$	0	0
$175$	4.87189	0.368280
$176$	0	0
$177$	8.77643	0.659677
$178$	0	0
$179$	−11.5551	−0.863668	−0.431834	−	0.901953i	$-0.642133\pi$
$179$	−11.5551	−0.863668	−0.431834	+	0.901953i	$0.642133\pi$
$180$	0	0
$181$	−7.36417	−0.547374	−0.273687	−	0.961819i	$-0.588243\pi$
$181$	−7.36417	−0.547374	−0.273687	+	0.961819i	$0.588243\pi$
$182$	0	0
$183$	−6.16924	−0.456043
$184$	0	0
$185$	−1.18244	−0.0869350
$186$	0	0
$187$	1.95191	0.142738
$188$	0	0
$189$	5.30359	0.385780
$190$	0	0
$191$	9.95191	0.720095	0.360048	−	0.932934i	$-0.382760\pi$
$191$	9.95191	0.720095	0.360048	+	0.932934i	$0.382760\pi$
$192$	0	0
$193$	−9.09546	−0.654706	−0.327353	−	0.944902i	$-0.606157\pi$
$193$	−9.09546	−0.654706	−0.327353	+	0.944902i	$0.606157\pi$
$194$	0	0
$195$	0.142832	0.0102284
$196$	0	0
$197$	−22.8976	−1.63138	−0.815692	−	0.578486i	$-0.803644\pi$
$197$	−22.8976	−1.63138	−0.815692	+	0.578486i	$0.803644\pi$
$198$	0	0
$199$	−3.75074	−0.265883	−0.132941	−	0.991124i	$-0.542442\pi$
$199$	−3.75074	−0.265883	−0.132941	+	0.991124i	$0.542442\pi$
$200$	0	0
$201$	−9.64207	−0.680099
$202$	0	0
$203$	−4.04113	−0.283632
$204$	0	0
$205$	4.35320	0.304041
$206$	0	0
$207$	−15.0885	−1.04872
$208$	0	0
$209$	0.527166	0.0364648
$210$	0	0
$211$	13.9519	0.960489	0.480245	−	0.877135i	$-0.340548\pi$
$211$	13.9519	0.960489	0.480245	+	0.877135i	$0.340548\pi$
$212$	0	0
$213$	−8.39905	−0.575494
$214$	0	0
$215$	0.256223	0.0174742
$216$	0	0
$217$	1.24302	0.0843816
$218$	0	0
$219$	−10.0411	−0.678517
$220$	0	0
$221$	−1.32528	−0.0891477
$222$	0	0
$223$	10.8176	0.724397	0.362199	−	0.932101i	$-0.382026\pi$
$223$	10.8176	0.724397	0.362199	+	0.932101i	$0.382026\pi$
$224$	0	0
$225$	8.55982	0.570654
$226$	0	0
$227$	−10.0800	−0.669035	−0.334517	−	0.942390i	$-0.608573\pi$
$227$	−10.0800	−0.669035	−0.334517	+	0.942390i	$0.608573\pi$
$228$	0	0
$229$	−17.7827	−1.17511	−0.587556	−	0.809184i	$-0.699910\pi$
$229$	−17.7827	−1.17511	−0.587556	+	0.809184i	$0.699910\pi$
$230$	0	0
$231$	0.587741	0.0386705
$232$	0	0
$233$	13.2166	0.865849	0.432924	−	0.901430i	$-0.357482\pi$
$233$	13.2166	0.865849	0.432924	+	0.901430i	$0.357482\pi$
$234$	0	0
$235$	3.87189	0.252574
$236$	0	0
$237$	−10.2298	−0.664498
$238$	0	0
$239$	−0.587741	−0.0380178	−0.0190089	−	0.999819i	$-0.506051\pi$
$239$	−0.587741	−0.0380178	−0.0190089	+	0.999819i	$0.506051\pi$
$240$	0	0
$241$	12.4860	0.804296	0.402148	−	0.915575i	$-0.368264\pi$
$241$	12.4860	0.804296	0.402148	+	0.915575i	$0.368264\pi$
$242$	0	0
$243$	15.1949	0.974755
$244$	0	0
$245$	0.357926	0.0228671
$246$	0	0
$247$	−0.357926	−0.0227743
$248$	0	0
$249$	−17.9061	−1.13475
$250$	0	0
$251$	−25.4464	−1.60616	−0.803082	−	0.595868i	$-0.796808\pi$
$251$	−25.4464	−1.60616	−0.803082	+	0.595868i	$0.796808\pi$
$252$	0	0
$253$	−4.52717	−0.284620
$254$	0	0
$255$	−1.47756	−0.0925284
$256$	0	0
$257$	−15.1538	−0.945268	−0.472634	−	0.881259i	$-0.656697\pi$
$257$	−15.1538	−0.945268	−0.472634	+	0.881259i	$0.656697\pi$
$258$	0	0
$259$	3.30359	0.205275
$260$	0	0
$261$	−7.10019	−0.439491
$262$	0	0
$263$	−6.16924	−0.380412	−0.190206	−	0.981744i	$-0.560916\pi$
$263$	−6.16924	−0.380412	−0.190206	+	0.981744i	$0.560916\pi$
$264$	0	0
$265$	2.16924	0.133255
$266$	0	0
$267$	−16.4985	−1.00969
$268$	0	0
$269$	13.0062	0.793005	0.396502	−	0.918034i	$-0.370224\pi$
$269$	13.0062	0.793005	0.396502	+	0.918034i	$0.370224\pi$
$270$	0	0
$271$	7.67624	0.466298	0.233149	−	0.972441i	$-0.425097\pi$
$271$	7.67624	0.466298	0.233149	+	0.972441i	$0.425097\pi$
$272$	0	0
$273$	−0.399055	−0.0241519
$274$	0	0
$275$	2.56829	0.154874
$276$	0	0
$277$	−28.4332	−1.70839	−0.854193	−	0.519955i	$-0.825949\pi$
$277$	−28.4332	−1.70839	−0.854193	+	0.519955i	$0.825949\pi$
$278$	0	0
$279$	2.18396	0.130750
$280$	0	0
$281$	10.2104	0.609099	0.304550	−	0.952496i	$-0.401494\pi$
$281$	10.2104	0.609099	0.304550	+	0.952496i	$0.401494\pi$
$282$	0	0
$283$	−19.0451	−1.13212	−0.566058	−	0.824365i	$-0.691532\pi$
$283$	−19.0451	−1.13212	−0.566058	+	0.824365i	$0.691532\pi$
$284$	0	0
$285$	−0.399055	−0.0236380
$286$	0	0
$287$	−12.1623	−0.717917
$288$	0	0
$289$	−3.29039	−0.193552
$290$	0	0
$291$	−10.1164	−0.593035
$292$	0	0
$293$	27.5070	1.60698	0.803488	−	0.595321i	$-0.202975\pi$
$293$	27.5070	1.60698	0.803488	+	0.595321i	$0.202975\pi$
$294$	0	0
$295$	2.81756	0.164044
$296$	0	0
$297$	2.79588	0.162233
$298$	0	0
$299$	3.07378	0.177761
$300$	0	0
$301$	−0.715853	−0.0412611
$302$	0	0
$303$	−0.0216815	−0.00124557
$304$	0	0
$305$	−1.98055	−0.113406
$306$	0	0
$307$	10.7570	0.613933	0.306967	−	0.951720i	$-0.400686\pi$
$307$	10.7570	0.613933	0.306967	+	0.951720i	$0.400686\pi$
$308$	0	0
$309$	−21.8697	−1.24412
$310$	0	0
$311$	−26.6608	−1.51180	−0.755898	−	0.654690i	$-0.772799\pi$
$311$	−26.6608	−1.51180	−0.755898	+	0.654690i	$0.772799\pi$
$312$	0	0
$313$	23.5140	1.32909	0.664544	−	0.747249i	$-0.268626\pi$
$313$	23.5140	1.32909	0.664544	+	0.747249i	$0.268626\pi$
$314$	0	0
$315$	0.628870	0.0354328
$316$	0	0
$317$	−2.45963	−0.138147	−0.0690733	−	0.997612i	$-0.522004\pi$
$317$	−2.45963	−0.138147	−0.0690733	+	0.997612i	$0.522004\pi$
$318$	0	0
$319$	−2.13035	−0.119276
$320$	0	0
$321$	−15.2097	−0.848920
$322$	0	0
$323$	3.70265	0.206021
$324$	0	0
$325$	−1.74378	−0.0967274
$326$	0	0
$327$	−1.19716	−0.0662033
$328$	0	0
$329$	−10.8176	−0.596391
$330$	0	0
$331$	2.29039	0.125891	0.0629456	−	0.998017i	$-0.479951\pi$
$331$	2.29039	0.125891	0.0629456	+	0.998017i	$0.479951\pi$
$332$	0	0
$333$	5.80435	0.318077
$334$	0	0
$335$	−3.09546	−0.169123
$336$	0	0
$337$	30.6483	1.66952	0.834760	−	0.550614i	$-0.185606\pi$
$337$	30.6483	1.66952	0.834760	+	0.550614i	$0.185606\pi$
$338$	0	0
$339$	12.4838	0.678027
$340$	0	0
$341$	0.655277	0.0354853
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.42698	0.184502
$346$	0	0
$347$	−26.5466	−1.42510	−0.712548	−	0.701623i	$-0.752459\pi$
$347$	−26.5466	−1.42510	−0.712548	+	0.701623i	$0.752459\pi$
$348$	0	0
$349$	15.6134	0.835768	0.417884	−	0.908500i	$-0.362772\pi$
$349$	15.6134	0.835768	0.417884	+	0.908500i	$0.362772\pi$
$350$	0	0
$351$	−1.89830	−0.101324
$352$	0	0
$353$	27.3642	1.45645	0.728224	−	0.685339i	$-0.240346\pi$
$353$	27.3642	1.45645	0.728224	+	0.685339i	$0.240346\pi$
$354$	0	0
$355$	−2.69641	−0.143110
$356$	0	0
$357$	4.12811	0.218483
$358$	0	0
$359$	−5.58150	−0.294580	−0.147290	−	0.989093i	$-0.547055\pi$
$359$	−5.58150	−0.294580	−0.147290	+	0.989093i	$0.547055\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.9541	−0.627430
$364$	0	0
$365$	−3.22357	−0.168729
$366$	0	0
$367$	4.62664	0.241508	0.120754	−	0.992682i	$-0.461469\pi$
$367$	4.62664	0.241508	0.120754	+	0.992682i	$0.461469\pi$
$368$	0	0
$369$	−21.3689	−1.11242
$370$	0	0
$371$	−6.06058	−0.314649
$372$	0	0
$373$	30.1817	1.56275	0.781375	−	0.624061i	$-0.214519\pi$
$373$	30.1817	1.56275	0.781375	+	0.624061i	$0.214519\pi$
$374$	0	0
$375$	−3.93942	−0.203431
$376$	0	0
$377$	1.44643	0.0744948
$378$	0	0
$379$	21.5070	1.10474	0.552370	−	0.833599i	$-0.313723\pi$
$379$	21.5070	1.10474	0.552370	+	0.833599i	$0.313723\pi$
$380$	0	0
$381$	−14.3121	−0.733230
$382$	0	0
$383$	−3.87189	−0.197844	−0.0989221	−	0.995095i	$-0.531539\pi$
$383$	−3.87189	−0.197844	−0.0989221	+	0.995095i	$0.531539\pi$
$384$	0	0
$385$	0.188687	0.00961636
$386$	0	0
$387$	−1.25774	−0.0639345
$388$	0	0
$389$	15.9450	0.808441	0.404221	−	0.914662i	$-0.367543\pi$
$389$	15.9450	0.808441	0.404221	+	0.914662i	$0.367543\pi$
$390$	0	0
$391$	−31.7974	−1.60806
$392$	0	0
$393$	1.89830	0.0957564
$394$	0	0
$395$	−3.28415	−0.165243
$396$	0	0
$397$	34.2857	1.72075	0.860374	−	0.509663i	$-0.170230\pi$
$397$	34.2857	1.72075	0.860374	+	0.509663i	$0.170230\pi$
$398$	0	0
$399$	1.11491	0.0558152
$400$	0	0
$401$	−16.4185	−0.819901	−0.409950	−	0.912108i	$-0.634454\pi$
$401$	−16.4185	−0.819901	−0.409950	+	0.912108i	$0.634454\pi$
$402$	0	0
$403$	−0.444909	−0.0221625
$404$	0	0
$405$	−0.229815	−0.0114196
$406$	0	0
$407$	1.74154	0.0863251
$408$	0	0
$409$	11.1538	0.551520	0.275760	−	0.961227i	$-0.411071\pi$
$409$	11.1538	0.551520	0.275760	+	0.961227i	$0.411071\pi$
$410$	0	0
$411$	3.82603	0.188724
$412$	0	0
$413$	−7.87189	−0.387350
$414$	0	0
$415$	−5.74850	−0.282183
$416$	0	0
$417$	5.53564	0.271082
$418$	0	0
$419$	−2.10170	−0.102675	−0.0513375	−	0.998681i	$-0.516348\pi$
$419$	−2.10170	−0.102675	−0.0513375	+	0.998681i	$0.516348\pi$
$420$	0	0
$421$	−20.8998	−1.01859	−0.509297	−	0.860591i	$-0.670095\pi$
$421$	−20.8998	−1.01859	−0.509297	+	0.860591i	$0.670095\pi$
$422$	0	0
$423$	−19.0062	−0.924115
$424$	0	0
$425$	18.0389	0.875015
$426$	0	0
$427$	5.53341	0.267780
$428$	0	0
$429$	−0.210368	−0.0101567
$430$	0	0
$431$	−15.9861	−0.770022	−0.385011	−	0.922912i	$-0.625802\pi$
$431$	−15.9861	−0.770022	−0.385011	+	0.922912i	$0.625802\pi$
$432$	0	0
$433$	21.6157	1.03878	0.519391	−	0.854537i	$-0.326159\pi$
$433$	21.6157	1.03878	0.519391	+	0.854537i	$0.326159\pi$
$434$	0	0
$435$	1.61263	0.0773198
$436$	0	0
$437$	−8.58774	−0.410807
$438$	0	0
$439$	−16.9193	−0.807512	−0.403756	−	0.914867i	$-0.632296\pi$
$439$	−16.9193	−0.807512	−0.403756	+	0.914867i	$0.632296\pi$
$440$	0	0
$441$	−1.75698	−0.0836658
$442$	0	0
$443$	27.3308	1.29853	0.649263	−	0.760564i	$-0.275078\pi$
$443$	27.3308	1.29853	0.649263	+	0.760564i	$0.275078\pi$
$444$	0	0
$445$	−5.29663	−0.251085
$446$	0	0
$447$	0.819791	0.0387748
$448$	0	0
$449$	−3.49228	−0.164811	−0.0824055	−	0.996599i	$-0.526260\pi$
$449$	−3.49228	−0.164811	−0.0824055	+	0.996599i	$0.526260\pi$
$450$	0	0
$451$	−6.41154	−0.301908
$452$	0	0
$453$	1.38585	0.0651130
$454$	0	0
$455$	−0.128111	−0.00600595
$456$	0	0
$457$	−22.0995	−1.03377	−0.516885	−	0.856055i	$-0.672908\pi$
$457$	−22.0995	−1.03377	−0.516885	+	0.856055i	$0.672908\pi$
$458$	0	0
$459$	19.6373	0.916593
$460$	0	0
$461$	12.8851	0.600119	0.300059	−	0.953921i	$-0.402994\pi$
$461$	12.8851	0.600119	0.300059	+	0.953921i	$0.402994\pi$
$462$	0	0
$463$	27.0599	1.25758	0.628789	−	0.777576i	$-0.283551\pi$
$463$	27.0599	1.25758	0.628789	+	0.777576i	$0.283551\pi$
$464$	0	0
$465$	−0.496033	−0.0230030
$466$	0	0
$467$	−28.3991	−1.31415	−0.657076	−	0.753825i	$-0.728207\pi$
$467$	−28.3991	−1.31415	−0.657076	+	0.753825i	$0.728207\pi$
$468$	0	0
$469$	8.64832	0.399342
$470$	0	0
$471$	17.6421	0.812904
$472$	0	0
$473$	−0.377373	−0.0173516
$474$	0	0
$475$	4.87189	0.223538
$476$	0	0
$477$	−10.6483	−0.487553
$478$	0	0
$479$	12.5132	0.571745	0.285872	−	0.958268i	$-0.407717\pi$
$479$	12.5132	0.571745	0.285872	+	0.958268i	$0.407717\pi$
$480$	0	0
$481$	−1.18244	−0.0539148
$482$	0	0
$483$	−9.57454	−0.435657
$484$	0	0
$485$	−3.24774	−0.147472
$486$	0	0
$487$	−10.3968	−0.471125	−0.235562	−	0.971859i	$-0.575693\pi$
$487$	−10.3968	−0.471125	−0.235562	+	0.971859i	$0.575693\pi$
$488$	0	0
$489$	−22.8781	−1.03458
$490$	0	0
$491$	17.1366	0.773363	0.386682	−	0.922213i	$-0.373621\pi$
$491$	17.1366	0.773363	0.386682	+	0.922213i	$0.373621\pi$
$492$	0	0
$493$	−14.9629	−0.673894
$494$	0	0
$495$	0.331519	0.0149007
$496$	0	0
$497$	7.53341	0.337920
$498$	0	0
$499$	20.2640	0.907140	0.453570	−	0.891221i	$-0.350150\pi$
$499$	20.2640	0.907140	0.453570	+	0.891221i	$0.350150\pi$
$500$	0	0
$501$	−8.11339	−0.362480
$502$	0	0
$503$	−23.4876	−1.04726	−0.523629	−	0.851946i	$-0.675422\pi$
$503$	−23.4876	−1.04726	−0.523629	+	0.851946i	$0.675422\pi$
$504$	0	0
$505$	−0.00696057	−0.000309741 0
$506$	0	0
$507$	−14.3510	−0.637349
$508$	0	0
$509$	29.1057	1.29009	0.645044	−	0.764145i	$-0.276839\pi$
$509$	29.1057	1.29009	0.645044	+	0.764145i	$0.276839\pi$
$510$	0	0
$511$	9.00624	0.398413
$512$	0	0
$513$	5.30359	0.234159
$514$	0	0
$515$	−7.02096	−0.309381
$516$	0	0
$517$	−5.70265	−0.250802
$518$	0	0
$519$	−5.22830	−0.229497
$520$	0	0
$521$	−36.2229	−1.58695	−0.793476	−	0.608601i	$-0.791731\pi$
$521$	−36.2229	−1.58695	−0.793476	+	0.608601i	$0.791731\pi$
$522$	0	0
$523$	−27.2772	−1.19275	−0.596374	−	0.802707i	$-0.703392\pi$
$523$	−27.2772	−1.19275	−0.596374	+	0.802707i	$0.703392\pi$
$524$	0	0
$525$	5.43171	0.237059
$526$	0	0
$527$	4.60246	0.200486
$528$	0	0
$529$	50.7493	2.20649
$530$	0	0
$531$	−13.8308	−0.600204
$532$	0	0
$533$	4.35320	0.188558
$534$	0	0
$535$	−4.88286	−0.211104
$536$	0	0
$537$	−12.8829	−0.555936
$538$	0	0
$539$	−0.527166	−0.0227067
$540$	0	0
$541$	10.9193	0.469456	0.234728	−	0.972061i	$-0.424580\pi$
$541$	10.9193	0.469456	0.234728	+	0.972061i	$0.424580\pi$
$542$	0	0
$543$	−8.21037	−0.352341
$544$	0	0
$545$	−0.384334	−0.0164631
$546$	0	0
$547$	19.7655	0.845110	0.422555	−	0.906337i	$-0.361133\pi$
$547$	19.7655	0.845110	0.422555	+	0.906337i	$0.361133\pi$
$548$	0	0
$549$	9.72210	0.414929
$550$	0	0
$551$	−4.04113	−0.172158
$552$	0	0
$553$	9.17548	0.390181
$554$	0	0
$555$	−1.31832	−0.0559594
$556$	0	0
$557$	−34.7695	−1.47323	−0.736615	−	0.676313i	$-0.763577\pi$
$557$	−34.7695	−1.47323	−0.736615	+	0.676313i	$0.763577\pi$
$558$	0	0
$559$	0.256223	0.0108371
$560$	0	0
$561$	2.17620	0.0918792
$562$	0	0
$563$	−14.9526	−0.630178	−0.315089	−	0.949062i	$-0.602034\pi$
$563$	−14.9526	−0.630178	−0.315089	+	0.949062i	$0.602034\pi$
$564$	0	0
$565$	4.00776	0.168608
$566$	0	0
$567$	0.642074	0.0269646
$568$	0	0
$569$	24.7214	1.03637	0.518187	−	0.855267i	$-0.326607\pi$
$569$	24.7214	1.03637	0.518187	+	0.855267i	$0.326607\pi$
$570$	0	0
$571$	34.3440	1.43725	0.718626	−	0.695397i	$-0.244771\pi$
$571$	34.3440	1.43725	0.718626	+	0.695397i	$0.244771\pi$
$572$	0	0
$573$	11.0955	0.463520
$574$	0	0
$575$	−41.8385	−1.74479
$576$	0	0
$577$	13.3058	0.553929	0.276964	−	0.960880i	$-0.410672\pi$
$577$	13.3058	0.553929	0.276964	+	0.960880i	$0.410672\pi$
$578$	0	0
$579$	−10.1406	−0.421429
$580$	0	0
$581$	16.0606	0.666305
$582$	0	0
$583$	−3.19493	−0.132320
$584$	0	0
$585$	−0.225089	−0.00930629
$586$	0	0
$587$	−0.880366	−0.0363366	−0.0181683	−	0.999835i	$-0.505783\pi$
$587$	−0.880366	−0.0363366	−0.0181683	+	0.999835i	$0.505783\pi$
$588$	0	0
$589$	1.24302	0.0512177
$590$	0	0
$591$	−25.5287	−1.05011
$592$	0	0
$593$	−19.0474	−0.782182	−0.391091	−	0.920352i	$-0.627902\pi$
$593$	−19.0474	−0.782182	−0.391091	+	0.920352i	$0.627902\pi$
$594$	0	0
$595$	1.32528	0.0543310
$596$	0	0
$597$	−4.18173	−0.171147
$598$	0	0
$599$	15.2097	0.621449	0.310725	−	0.950500i	$-0.399428\pi$
$599$	15.2097	0.621449	0.310725	+	0.950500i	$0.399428\pi$
$600$	0	0
$601$	8.71113	0.355334	0.177667	−	0.984091i	$-0.443145\pi$
$601$	8.71113	0.355334	0.177667	+	0.984091i	$0.443145\pi$
$602$	0	0
$603$	15.1949	0.618785
$604$	0	0
$605$	−3.83772	−0.156026
$606$	0	0
$607$	−20.6266	−0.837209	−0.418605	−	0.908169i	$-0.637481\pi$
$607$	−20.6266	−0.837209	−0.418605	+	0.908169i	$0.637481\pi$
$608$	0	0
$609$	−4.50548	−0.182571
$610$	0	0
$611$	3.87189	0.156640
$612$	0	0
$613$	−27.4006	−1.10670	−0.553349	−	0.832949i	$-0.686651\pi$
$613$	−27.4006	−1.10670	−0.553349	+	0.832949i	$0.686651\pi$
$614$	0	0
$615$	4.85342	0.195709
$616$	0	0
$617$	6.22758	0.250713	0.125356	−	0.992112i	$-0.459993\pi$
$617$	6.22758	0.250713	0.125356	+	0.992112i	$0.459993\pi$
$618$	0	0
$619$	−4.02168	−0.161645	−0.0808225	−	0.996729i	$-0.525755\pi$
$619$	−4.02168	−0.161645	−0.0808225	+	0.996729i	$0.525755\pi$
$620$	0	0
$621$	−45.5459	−1.82769
$622$	0	0
$623$	14.7981	0.592874
$624$	0	0
$625$	23.0947	0.923790
$626$	0	0
$627$	0.587741	0.0234721
$628$	0	0
$629$	12.2320	0.487724
$630$	0	0
$631$	18.3051	0.728715	0.364357	−	0.931259i	$-0.381289\pi$
$631$	18.3051	0.728715	0.364357	+	0.931259i	$0.381289\pi$
$632$	0	0
$633$	15.5551	0.618259
$634$	0	0
$635$	−4.59470	−0.182335
$636$	0	0
$637$	0.357926	0.0141816
$638$	0	0
$639$	13.2361	0.523610
$640$	0	0
$641$	12.5830	0.496999	0.248500	−	0.968632i	$-0.420063\pi$
$641$	12.5830	0.496999	0.248500	+	0.968632i	$0.420063\pi$
$642$	0	0
$643$	−22.0055	−0.867813	−0.433907	−	0.900958i	$-0.642865\pi$
$643$	−22.0055	−0.867813	−0.433907	+	0.900958i	$0.642865\pi$
$644$	0	0
$645$	0.285664	0.0112480
$646$	0	0
$647$	−38.3440	−1.50746	−0.753729	−	0.657185i	$-0.771747\pi$
$647$	−38.3440	−1.50746	−0.753729	+	0.657185i	$0.771747\pi$
$648$	0	0
$649$	−4.14979	−0.162894
$650$	0	0
$651$	1.38585	0.0543158
$652$	0	0
$653$	20.1406	0.788162	0.394081	−	0.919076i	$-0.371063\pi$
$653$	20.1406	0.788162	0.394081	+	0.919076i	$0.371063\pi$
$654$	0	0
$655$	0.609423	0.0238121
$656$	0	0
$657$	15.8238	0.617345
$658$	0	0
$659$	−33.7026	−1.31287	−0.656434	−	0.754383i	$-0.727936\pi$
$659$	−33.7026	−1.31287	−0.656434	+	0.754383i	$0.727936\pi$
$660$	0	0
$661$	−49.9472	−1.94272	−0.971360	−	0.237612i	$-0.923635\pi$
$661$	−49.9472	−1.94272	−0.971360	+	0.237612i	$0.923635\pi$
$662$	0	0
$663$	−1.47756	−0.0573837
$664$	0	0
$665$	0.357926	0.0138798
$666$	0	0
$667$	34.7042	1.34375
$668$	0	0
$669$	12.0606	0.466289
$670$	0	0
$671$	2.91703	0.112611
$672$	0	0
$673$	34.7625	1.34000	0.669998	−	0.742363i	$-0.266295\pi$
$673$	34.7625	1.34000	0.669998	+	0.742363i	$0.266295\pi$
$674$	0	0
$675$	25.8385	0.994525
$676$	0	0
$677$	−20.6678	−0.794327	−0.397163	−	0.917748i	$-0.630005\pi$
$677$	−20.6678	−0.794327	−0.397163	+	0.917748i	$0.630005\pi$
$678$	0	0
$679$	9.07378	0.348220
$680$	0	0
$681$	−11.2383	−0.430652
$682$	0	0
$683$	1.42474	0.0545163	0.0272582	−	0.999628i	$-0.491322\pi$
$683$	1.42474	0.0545163	0.0272582	+	0.999628i	$0.491322\pi$
$684$	0	0
$685$	1.22830	0.0469308
$686$	0	0
$687$	−19.8260	−0.756410
$688$	0	0
$689$	2.16924	0.0826415
$690$	0	0
$691$	−7.66152	−0.291458	−0.145729	−	0.989325i	$-0.546553\pi$
$691$	−7.66152	−0.291458	−0.145729	+	0.989325i	$0.546553\pi$
$692$	0	0
$693$	−0.926221	−0.0351842
$694$	0	0
$695$	1.77715	0.0674110
$696$	0	0
$697$	−45.0327	−1.70573
$698$	0	0
$699$	14.7353	0.557340
$700$	0	0
$701$	8.90230	0.336235	0.168118	−	0.985767i	$-0.446231\pi$
$701$	8.90230	0.336235	0.168118	+	0.985767i	$0.446231\pi$
$702$	0	0
$703$	3.30359	0.124597
$704$	0	0
$705$	4.31680	0.162580
$706$	0	0
$707$	0.0194469	0.000731377 0
$708$	0	0
$709$	−8.56133	−0.321528	−0.160764	−	0.986993i	$-0.551396\pi$
$709$	−8.56133	−0.321528	−0.160764	+	0.986993i	$0.551396\pi$
$710$	0	0
$711$	16.1212	0.604590
$712$	0	0
$713$	−10.6747	−0.399772
$714$	0	0
$715$	−0.0675359	−0.00252570
$716$	0	0
$717$	−0.655277	−0.0244718
$718$	0	0
$719$	4.45963	0.166316	0.0831581	−	0.996536i	$-0.473499\pi$
$719$	4.45963	0.166316	0.0831581	+	0.996536i	$0.473499\pi$
$720$	0	0
$721$	19.6157	0.730526
$722$	0	0
$723$	13.9208	0.517719
$724$	0	0
$725$	−19.6879	−0.731191
$726$	0	0
$727$	41.7632	1.54891	0.774456	−	0.632628i	$-0.218024\pi$
$727$	41.7632	1.54891	0.774456	+	0.632628i	$0.218024\pi$
$728$	0	0
$729$	18.8672	0.698784
$730$	0	0
$731$	−2.65055	−0.0980342
$732$	0	0
$733$	34.9053	1.28926	0.644629	−	0.764496i	$-0.277012\pi$
$733$	34.9053	1.28926	0.644629	+	0.764496i	$0.277012\pi$
$734$	0	0
$735$	0.399055	0.0147194
$736$	0	0
$737$	4.55910	0.167937
$738$	0	0
$739$	−31.4372	−1.15644	−0.578219	−	0.815882i	$-0.696252\pi$
$739$	−31.4372	−1.15644	−0.578219	+	0.815882i	$0.696252\pi$
$740$	0	0
$741$	−0.399055	−0.0146596
$742$	0	0
$743$	10.8176	0.396858	0.198429	−	0.980115i	$-0.436416\pi$
$743$	10.8176	0.396858	0.198429	+	0.980115i	$0.436416\pi$
$744$	0	0
$745$	0.263183	0.00964228
$746$	0	0
$747$	28.2181	1.03245
$748$	0	0
$749$	13.6421	0.498471
$750$	0	0
$751$	2.75002	0.100350	0.0501748	−	0.998740i	$-0.484022\pi$
$751$	2.75002	0.100350	0.0501748	+	0.998740i	$0.484022\pi$
$752$	0	0
$753$	−28.3704	−1.03388
$754$	0	0
$755$	0.444909	0.0161919
$756$	0	0
$757$	7.37889	0.268190	0.134095	−	0.990968i	$-0.457187\pi$
$757$	7.37889	0.268190	0.134095	+	0.990968i	$0.457187\pi$
$758$	0	0
$759$	−5.04737	−0.183208
$760$	0	0
$761$	10.9651	0.397485	0.198743	−	0.980052i	$-0.436314\pi$
$761$	10.9651	0.397485	0.198743	+	0.980052i	$0.436314\pi$
$762$	0	0
$763$	1.07378	0.0388734
$764$	0	0
$765$	2.32848	0.0841865
$766$	0	0
$767$	2.81756	0.101736
$768$	0	0
$769$	−8.09474	−0.291904	−0.145952	−	0.989292i	$-0.546624\pi$
$769$	−8.09474	−0.291904	−0.145952	+	0.989292i	$0.546624\pi$
$770$	0	0
$771$	−16.8951	−0.608462
$772$	0	0
$773$	24.9457	0.897233	0.448617	−	0.893724i	$-0.351917\pi$
$773$	24.9457	0.897233	0.448617	+	0.893724i	$0.351917\pi$
$774$	0	0
$775$	6.05585	0.217533
$776$	0	0
$777$	3.68320	0.132134
$778$	0	0
$779$	−12.1623	−0.435759
$780$	0	0
$781$	3.97136	0.142106
$782$	0	0
$783$	−21.4325	−0.765935
$784$	0	0
$785$	5.66376	0.202148
$786$	0	0
$787$	−15.7827	−0.562591	−0.281296	−	0.959621i	$-0.590764\pi$
$787$	−15.7827	−0.562591	−0.281296	+	0.959621i	$0.590764\pi$
$788$	0	0
$789$	−6.87813	−0.244868
$790$	0	0
$791$	−11.1972	−0.398125
$792$	0	0
$793$	−1.98055	−0.0703315
$794$	0	0
$795$	2.41850	0.0857754
$796$	0	0
$797$	29.5940	1.04827	0.524136	−	0.851634i	$-0.324388\pi$
$797$	29.5940	1.04827	0.524136	+	0.851634i	$0.324388\pi$
$798$	0	0
$799$	−40.0536	−1.41700
$800$	0	0
$801$	26.0000	0.918665
$802$	0	0
$803$	4.74779	0.167546
$804$	0	0
$805$	−3.07378	−0.108336
$806$	0	0
$807$	14.5008	0.510451
$808$	0	0
$809$	−23.3859	−0.822203	−0.411101	−	0.911590i	$-0.634856\pi$
$809$	−23.3859	−0.822203	−0.411101	+	0.911590i	$0.634856\pi$
$810$	0	0
$811$	−34.3899	−1.20759	−0.603796	−	0.797139i	$-0.706346\pi$
$811$	−34.3899	−1.20759	−0.603796	+	0.797139i	$0.706346\pi$
$812$	0	0
$813$	8.55830	0.300153
$814$	0	0
$815$	−7.34472	−0.257274
$816$	0	0
$817$	−0.715853	−0.0250445
$818$	0	0
$819$	0.628870	0.0219745
$820$	0	0
$821$	26.6630	0.930546	0.465273	−	0.885167i	$-0.345956\pi$
$821$	26.6630	0.930546	0.465273	+	0.885167i	$0.345956\pi$
$822$	0	0
$823$	−49.9597	−1.74148	−0.870742	−	0.491740i	$-0.836361\pi$
$823$	−49.9597	−1.74148	−0.870742	+	0.491740i	$0.836361\pi$
$824$	0	0
$825$	2.86341	0.0996912
$826$	0	0
$827$	49.1546	1.70927	0.854636	−	0.519227i	$-0.173780\pi$
$827$	49.1546	1.70927	0.854636	+	0.519227i	$0.173780\pi$
$828$	0	0
$829$	44.7717	1.55499	0.777493	−	0.628892i	$-0.216491\pi$
$829$	44.7717	1.55499	0.777493	+	0.628892i	$0.216491\pi$
$830$	0	0
$831$	−31.7004	−1.09968
$832$	0	0
$833$	−3.70265	−0.128289
$834$	0	0
$835$	−2.60470	−0.0901393
$836$	0	0
$837$	6.59247	0.227869
$838$	0	0
$839$	16.1645	0.558061	0.279030	−	0.960282i	$-0.409987\pi$
$839$	16.1645	0.558061	0.279030	+	0.960282i	$0.409987\pi$
$840$	0	0
$841$	−12.6693	−0.436872
$842$	0	0
$843$	11.3836	0.392073
$844$	0	0
$845$	−4.60719	−0.158492
$846$	0	0
$847$	10.7221	0.368416
$848$	0	0
$849$	−21.2336	−0.728734
$850$	0	0
$851$	−28.3704	−0.972525
$852$	0	0
$853$	10.9409	0.374611	0.187305	−	0.982302i	$-0.440025\pi$
$853$	10.9409	0.374611	0.187305	+	0.982302i	$0.440025\pi$
$854$	0	0
$855$	0.628870	0.0215069
$856$	0	0
$857$	18.6872	0.638343	0.319171	−	0.947697i	$-0.396595\pi$
$857$	18.6872	0.638343	0.319171	+	0.947697i	$0.396595\pi$
$858$	0	0
$859$	−56.8906	−1.94108	−0.970541	−	0.240934i	$-0.922546\pi$
$859$	−56.8906	−1.94108	−0.970541	+	0.240934i	$0.922546\pi$
$860$	0	0
$861$	−13.5598	−0.462117
$862$	0	0
$863$	2.75002	0.0936118	0.0468059	−	0.998904i	$-0.485096\pi$
$863$	2.75002	0.0936118	0.0468059	+	0.998904i	$0.485096\pi$
$864$	0	0
$865$	−1.67848	−0.0570699
$866$	0	0
$867$	−3.66848	−0.124588
$868$	0	0
$869$	4.83700	0.164084
$870$	0	0
$871$	−3.09546	−0.104886
$872$	0	0
$873$	15.9425	0.539570
$874$	0	0
$875$	3.53341	0.119451
$876$	0	0
$877$	28.9846	0.978739	0.489370	−	0.872077i	$-0.337227\pi$
$877$	28.9846	0.978739	0.489370	+	0.872077i	$0.337227\pi$
$878$	0	0
$879$	30.6678	1.03440
$880$	0	0
$881$	43.4395	1.46351	0.731756	−	0.681566i	$-0.238701\pi$
$881$	43.4395	1.46351	0.731756	+	0.681566i	$0.238701\pi$
$882$	0	0
$883$	14.6964	0.494573	0.247287	−	0.968942i	$-0.420461\pi$
$883$	14.6964	0.494573	0.247287	+	0.968942i	$0.420461\pi$
$884$	0	0
$885$	3.14132	0.105594
$886$	0	0
$887$	54.8565	1.84190	0.920950	−	0.389682i	$-0.127415\pi$
$887$	54.8565	1.84190	0.920950	+	0.389682i	$0.127415\pi$
$888$	0	0
$889$	12.8370	0.430539
$890$	0	0
$891$	0.338479	0.0113395
$892$	0	0
$893$	−10.8176	−0.361996
$894$	0	0
$895$	−4.13587	−0.138247
$896$	0	0
$897$	3.42698	0.114424
$898$	0	0
$899$	−5.02320	−0.167533
$900$	0	0
$901$	−22.4402	−0.747591
$902$	0	0
$903$	−0.798110	−0.0265594
$904$	0	0
$905$	−2.63583	−0.0876180
$906$	0	0
$907$	−40.1142	−1.33197	−0.665985	−	0.745965i	$-0.731988\pi$
$907$	−40.1142	−1.33197	−0.665985	+	0.745965i	$0.731988\pi$
$908$	0	0
$909$	0.0341679	0.00113328
$910$	0	0
$911$	−16.2562	−0.538593	−0.269296	−	0.963057i	$-0.586791\pi$
$911$	−16.2562	−0.538593	−0.269296	+	0.963057i	$0.586791\pi$
$912$	0	0
$913$	8.46659	0.280203
$914$	0	0
$915$	−2.20813	−0.0729986
$916$	0	0
$917$	−1.70265	−0.0562264
$918$	0	0
$919$	−31.2633	−1.03128	−0.515640	−	0.856805i	$-0.672446\pi$
$919$	−31.2633	−1.03128	−0.515640	+	0.856805i	$0.672446\pi$
$920$	0	0
$921$	11.9930	0.395184
$922$	0	0
$923$	−2.69641	−0.0887533
$924$	0	0
$925$	16.0947	0.529192
$926$	0	0
$927$	34.4644	1.13196
$928$	0	0
$929$	−20.2904	−0.665706	−0.332853	−	0.942979i	$-0.608011\pi$
$929$	−20.2904	−0.665706	−0.332853	+	0.942979i	$0.608011\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−29.7243	−0.973131
$934$	0	0
$935$	0.698640	0.0228480
$936$	0	0
$937$	−48.4046	−1.58131	−0.790654	−	0.612263i	$-0.790259\pi$
$937$	−48.4046	−1.58131	−0.790654	+	0.612263i	$0.790259\pi$
$938$	0	0
$939$	26.2159	0.855523
$940$	0	0
$941$	3.43171	0.111870	0.0559352	−	0.998434i	$-0.482186\pi$
$941$	3.43171	0.111870	0.0559352	+	0.998434i	$0.482186\pi$
$942$	0	0
$943$	104.447	3.40125
$944$	0	0
$945$	1.89830	0.0617516
$946$	0	0
$947$	−44.7764	−1.45504	−0.727519	−	0.686087i	$-0.759327\pi$
$947$	−44.7764	−1.45504	−0.727519	+	0.686087i	$0.759327\pi$
$948$	0	0
$949$	−3.22357	−0.104642
$950$	0	0
$951$	−2.74226	−0.0889239
$952$	0	0
$953$	−13.7680	−0.445988	−0.222994	−	0.974820i	$-0.571583\pi$
$953$	−13.7680	−0.445988	−0.222994	+	0.974820i	$0.571583\pi$
$954$	0	0
$955$	3.56205	0.115265
$956$	0	0
$957$	−2.37514	−0.0767773
$958$	0	0
$959$	−3.43171	−0.110816
$960$	0	0
$961$	−29.4549	−0.950158
$962$	0	0
$963$	23.9689	0.772386
$964$	0	0
$965$	−3.25551	−0.104798
$966$	0	0
$967$	−24.2181	−0.778803	−0.389401	−	0.921068i	$-0.627318\pi$
$967$	−24.2181	−0.778803	−0.389401	+	0.921068i	$0.627318\pi$
$968$	0	0
$969$	4.12811	0.132614
$970$	0	0
$971$	−49.9527	−1.60306	−0.801529	−	0.597955i	$-0.795980\pi$
$971$	−49.9527	−1.60306	−0.801529	+	0.597955i	$0.795980\pi$
$972$	0	0
$973$	−4.96511	−0.159174
$974$	0	0
$975$	−1.94415	−0.0622626
$976$	0	0
$977$	−35.6785	−1.14146	−0.570728	−	0.821139i	$-0.693339\pi$
$977$	−35.6785	−1.14146	−0.570728	+	0.821139i	$0.693339\pi$
$978$	0	0
$979$	7.80106	0.249323
$980$	0	0
$981$	1.88661	0.0602348
$982$	0	0
$983$	−7.83299	−0.249834	−0.124917	−	0.992167i	$-0.539866\pi$
$983$	−7.83299	−0.249834	−0.124917	+	0.992167i	$0.539866\pi$
$984$	0	0
$985$	−8.19565	−0.261135
$986$	0	0
$987$	−12.0606	−0.383892
$988$	0	0
$989$	6.14756	0.195481
$990$	0	0
$991$	−34.4721	−1.09504	−0.547521	−	0.836792i	$-0.684429\pi$
$991$	−34.4721	−1.09504	−0.547521	+	0.836792i	$0.684429\pi$
$992$	0	0
$993$	2.55357	0.0810352
$994$	0	0
$995$	−1.34249	−0.0425597
$996$	0	0
$997$	−46.3073	−1.46657	−0.733284	−	0.679922i	$-0.762013\pi$
$997$	−46.3073	−1.46657	−0.733284	+	0.679922i	$0.762013\pi$
$998$	0	0
$999$	17.5209	0.554337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bi.1.3		3
4.3	odd	2		8512.2.a.bp.1.1		3
8.3	odd	2		2128.2.a.p.1.3		3
8.5	even	2		133.2.a.d.1.3	✓	3
24.5	odd	2		1197.2.a.k.1.1		3
40.29	even	2		3325.2.a.r.1.1		3
56.5	odd	6		931.2.f.m.704.1		6
56.13	odd	2		931.2.a.k.1.3		3
56.37	even	6		931.2.f.l.704.1		6
56.45	odd	6		931.2.f.m.324.1		6
56.53	even	6		931.2.f.l.324.1		6
152.37	odd	2		2527.2.a.f.1.1		3
168.125	even	2		8379.2.a.bo.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.d.1.3	✓	3	8.5	even	2
931.2.a.k.1.3		3	56.13	odd	2
931.2.f.l.324.1		6	56.53	even	6
931.2.f.l.704.1		6	56.37	even	6
931.2.f.m.324.1		6	56.45	odd	6
931.2.f.m.704.1		6	56.5	odd	6
1197.2.a.k.1.1		3	24.5	odd	2
2128.2.a.p.1.3		3	8.3	odd	2
2527.2.a.f.1.1		3	152.37	odd	2
3325.2.a.r.1.1		3	40.29	even	2
8379.2.a.bo.1.1		3	168.125	even	2
8512.2.a.bi.1.3		3	1.1	even	1	trivial
8512.2.a.bp.1.1		3	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+1.11491 q^{3} +0.357926 q^{5} -1.00000 q^{7} -1.75698 q^{9} +O(q^{10})\)	\(q+1.11491 q^{3} +0.357926 q^{5} -1.00000 q^{7} -1.75698 q^{9} -0.527166 q^{11} +0.357926 q^{13} +0.399055 q^{15} -3.70265 q^{17} -1.00000 q^{19} -1.11491 q^{21} +8.58774 q^{23} -4.87189 q^{25} -5.30359 q^{27} +4.04113 q^{29} -1.24302 q^{31} -0.587741 q^{33} -0.357926 q^{35} -3.30359 q^{37} +0.399055 q^{39} +12.1623 q^{41} +0.715853 q^{43} -0.628870 q^{45} +10.8176 q^{47} +1.00000 q^{49} -4.12811 q^{51} +6.06058 q^{53} -0.188687 q^{55} -1.11491 q^{57} +7.87189 q^{59} -5.53341 q^{61} +1.75698 q^{63} +0.128111 q^{65} -8.64832 q^{67} +9.57454 q^{69} -7.53341 q^{71} -9.00624 q^{73} -5.43171 q^{75} +0.527166 q^{77} -9.17548 q^{79} -0.642074 q^{81} -16.0606 q^{83} -1.32528 q^{85} +4.50548 q^{87} -14.7981 q^{89} -0.357926 q^{91} -1.38585 q^{93} -0.357926 q^{95} -9.07378 q^{97} +0.926221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9	\( 3 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} - 7 q^{15} + 7 q^{17} - 3 q^{19} + 3 q^{21} + 14 q^{23} - q^{25} - 6 q^{27} + 3 q^{29} - 11 q^{31} + 10 q^{33} - 2 q^{35} - 7 q^{39} - 7 q^{41} + 4 q^{43} + 19 q^{45} + 8 q^{47} + 3 q^{49} - 26 q^{51} + q^{53} + 3 q^{55} + 3 q^{57} + 10 q^{59} + 6 q^{61} - 2 q^{63} + 14 q^{65} + 3 q^{67} - 3 q^{69} + q^{73} - 20 q^{75} + 7 q^{77} - 4 q^{79} - q^{81} - 31 q^{83} + 7 q^{85} + 20 q^{87} - 28 q^{89} - 2 q^{91} + 24 q^{93} - 2 q^{95} - 30 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 2 * q^9 - 7 * q^11 + 2 * q^13 - 7 * q^15 + 7 * q^17 - 3 * q^19 + 3 * q^21 + 14 * q^23 - q^25 - 6 * q^27 + 3 * q^29 - 11 * q^31 + 10 * q^33 - 2 * q^35 - 7 * q^39 - 7 * q^41 + 4 * q^43 + 19 * q^45 + 8 * q^47 + 3 * q^49 - 26 * q^51 + q^53 + 3 * q^55 + 3 * q^57 + 10 * q^59 + 6 * q^61 - 2 * q^63 + 14 * q^65 + 3 * q^67 - 3 * q^69 + q^73 - 20 * q^75 + 7 * q^77 - 4 * q^79 - q^81 - 31 * q^83 + 7 * q^85 + 20 * q^87 - 28 * q^89 - 2 * q^91 + 24 * q^93 - 2 * q^95 - 30 * q^97

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