LMFDB - Embedded newform 512.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [512,2,Mod(1,512)] 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("512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.08834058349$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -2.82843 q^{5} +4.00000 q^{7} -1.00000 q^{9} -1.41421 q^{11} +2.82843 q^{13} +4.00000 q^{15} -4.00000 q^{17} +7.07107 q^{19} -5.65685 q^{21} +4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} +8.48528 q^{29} +8.00000 q^{31} +2.00000 q^{33} -11.3137 q^{35} +2.82843 q^{37} -4.00000 q^{39} +2.00000 q^{41} -4.24264 q^{43} +2.82843 q^{45} +9.00000 q^{49} +5.65685 q^{51} +2.82843 q^{53} +4.00000 q^{55} -10.0000 q^{57} -4.24264 q^{59} -8.48528 q^{61} -4.00000 q^{63} -8.00000 q^{65} -4.24264 q^{67} -5.65685 q^{69} -4.00000 q^{71} -4.00000 q^{73} -4.24264 q^{75} -5.65685 q^{77} -8.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +11.3137 q^{85} -12.0000 q^{87} +12.0000 q^{89} +11.3137 q^{91} -11.3137 q^{93} -20.0000 q^{95} -4.00000 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9} + 8 q^{15} - 8 q^{17} + 8 q^{23} + 6 q^{25} + 16 q^{31} + 4 q^{33} - 8 q^{39} + 4 q^{41} + 18 q^{49} + 8 q^{55} - 20 q^{57} - 8 q^{63} - 16 q^{65} - 8 q^{71} - 8 q^{73} - 16 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 + 8 * q^15 - 8 * q^17 + 8 * q^23 + 6 * q^25 + 16 * q^31 + 4 * q^33 - 8 * q^39 + 4 * q^41 + 18 * q^49 + 8 * q^55 - 20 * q^57 - 8 * q^63 - 16 * q^65 - 8 * q^71 - 8 * q^73 - 16 * q^79 - 10 * q^81 - 24 * q^87 + 24 * q^89 - 40 * q^95 - 8 * q^97

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	7.07107	1.62221	0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107	1.62221	0.811107	+	0.584898i	$0.198865\pi$
$20$	0	0
$21$	−5.65685	−1.23443
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−11.3137	−1.91237
$36$	0	0
$37$	2.82843	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.24264	−0.646997	−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264	−0.646997	−0.323498	+	0.946229i	$0.604859\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−10.0000	−1.32453
$58$	0	0
$59$	−4.24264	−0.552345	−0.276172	−	0.961108i	$-0.589066\pi$
$59$	−4.24264	−0.552345	−0.276172	+	0.961108i	$0.589066\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−4.24264	−0.489898
$76$	0	0
$77$	−5.65685	−0.644658
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	11.3137	1.22714
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	11.3137	1.18600
$92$	0	0
$93$	−11.3137	−1.17318
$94$	0	0
$95$	−20.0000	−2.05196
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	2.82843	0.281439	0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843	0.281439	0.140720	+	0.990050i	$0.455058\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	16.0000	1.56144
$106$	0	0
$107$	9.89949	0.957020	0.478510	−	0.878082i	$-0.341177\pi$
$107$	9.89949	0.957020	0.478510	+	0.878082i	$0.341177\pi$
$108$	0	0
$109$	−14.1421	−1.35457	−0.677285	−	0.735720i	$-0.736844\pi$
$109$	−14.1421	−1.35457	−0.677285	+	0.735720i	$0.736844\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−2.82843	−0.255031
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	7.07107	0.617802	0.308901	−	0.951094i	$-0.400039\pi$
$131$	7.07107	0.617802	0.308901	+	0.951094i	$0.400039\pi$
$132$	0	0
$133$	28.2843	2.45256
$134$	0	0
$135$	−16.0000	−1.37706
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	−12.7279	−1.04978
$148$	0	0
$149$	−2.82843	−0.231714	−0.115857	−	0.993266i	$-0.536961\pi$
$149$	−2.82843	−0.231714	−0.115857	+	0.993266i	$0.536961\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−22.6274	−1.81748
$156$	0	0
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	9.89949	0.775388	0.387694	−	0.921788i	$-0.373272\pi$
$163$	9.89949	0.775388	0.387694	+	0.921788i	$0.373272\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−7.07107	−0.540738
$172$	0	0
$173$	14.1421	1.07521	0.537603	−	0.843198i	$-0.319330\pi$
$173$	14.1421	1.07521	0.537603	+	0.843198i	$0.319330\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	−1.41421	−0.105703	−0.0528516	−	0.998602i	$-0.516831\pi$
$179$	−1.41421	−0.105703	−0.0528516	+	0.998602i	$0.516831\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	5.65685	0.413670
$188$	0	0
$189$	22.6274	1.64590
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	11.3137	0.810191
$196$	0	0
$197$	−14.1421	−1.00759	−0.503793	−	0.863825i	$-0.668062\pi$
$197$	−14.1421	−1.00759	−0.503793	+	0.863825i	$0.668062\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	33.9411	2.38220
$204$	0	0
$205$	−5.65685	−0.395092
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−24.0416	−1.65509	−0.827547	−	0.561396i	$-0.810264\pi$
$211$	−24.0416	−1.65509	−0.827547	+	0.561396i	$0.810264\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	32.0000	2.17230
$218$	0	0
$219$	5.65685	0.382255
$220$	0	0
$221$	−11.3137	−0.761042
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	21.2132	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$227$	21.2132	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$228$	0	0
$229$	−25.4558	−1.68217	−0.841085	−	0.540903i	$-0.818082\pi$
$229$	−25.4558	−1.68217	−0.841085	+	0.540903i	$0.818082\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.3137	0.734904
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−25.4558	−1.62631
$246$	0	0
$247$	20.0000	1.27257
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	−4.24264	−0.267793	−0.133897	−	0.990995i	$-0.542749\pi$
$251$	−4.24264	−0.267793	−0.133897	+	0.990995i	$0.542749\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	−16.0000	−1.00196
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	11.3137	0.703000
$260$	0	0
$261$	−8.48528	−0.525226
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	−16.9706	−1.03858
$268$	0	0
$269$	19.7990	1.20717	0.603583	−	0.797300i	$-0.293739\pi$
$269$	19.7990	1.20717	0.603583	+	0.797300i	$0.293739\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	−8.48528	−0.509831	−0.254916	−	0.966963i	$-0.582048\pi$
$277$	−8.48528	−0.509831	−0.254916	+	0.966963i	$0.582048\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−24.0416	−1.42913	−0.714563	−	0.699571i	$-0.753375\pi$
$283$	−24.0416	−1.42913	−0.714563	+	0.699571i	$0.753375\pi$
$284$	0	0
$285$	28.2843	1.67542
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	5.65685	0.331611
$292$	0	0
$293$	8.48528	0.495715	0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528	0.495715	0.247858	+	0.968796i	$0.420273\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	11.3137	0.654289
$300$	0	0
$301$	−16.9706	−0.978167
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	7.07107	0.403567	0.201784	−	0.979430i	$-0.435326\pi$
$307$	7.07107	0.403567	0.201784	+	0.979430i	$0.435326\pi$
$308$	0	0
$309$	−16.9706	−0.965422
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	11.3137	0.637455
$316$	0	0
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−14.0000	−0.781404
$322$	0	0
$323$	−28.2843	−1.57378
$324$	0	0
$325$	8.48528	0.470679
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.7279	−0.699590	−0.349795	−	0.936826i	$-0.613749\pi$
$331$	−12.7279	−0.699590	−0.349795	+	0.936826i	$0.613749\pi$
$332$	0	0
$333$	−2.82843	−0.154997
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	−19.7990	−1.07533
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	−15.5563	−0.835109	−0.417554	−	0.908652i	$-0.637113\pi$
$347$	−15.5563	−0.835109	−0.417554	+	0.908652i	$0.637113\pi$
$348$	0	0
$349$	19.7990	1.05982	0.529908	−	0.848055i	$-0.322227\pi$
$349$	19.7990	1.05982	0.529908	+	0.848055i	$0.322227\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	11.3137	0.600469
$356$	0	0
$357$	22.6274	1.19757
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	12.7279	0.668043
$364$	0	0
$365$	11.3137	0.592187
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	11.3137	0.587378
$372$	0	0
$373$	36.7696	1.90386	0.951928	−	0.306323i	$-0.0990988\pi$
$373$	36.7696	1.90386	0.951928	+	0.306323i	$0.0990988\pi$
$374$	0	0
$375$	−8.00000	−0.413118
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	29.6985	1.52551	0.762754	−	0.646688i	$-0.223847\pi$
$379$	29.6985	1.52551	0.762754	+	0.646688i	$0.223847\pi$
$380$	0	0
$381$	−11.3137	−0.579619
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	4.24264	0.215666
$388$	0	0
$389$	−19.7990	−1.00385	−0.501924	−	0.864912i	$-0.667374\pi$
$389$	−19.7990	−1.00385	−0.501924	+	0.864912i	$0.667374\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−10.0000	−0.504433
$394$	0	0
$395$	22.6274	1.13851
$396$	0	0
$397$	−2.82843	−0.141955	−0.0709773	−	0.997478i	$-0.522612\pi$
$397$	−2.82843	−0.141955	−0.0709773	+	0.997478i	$0.522612\pi$
$398$	0	0
$399$	−40.0000	−2.00250
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	22.6274	1.12715
$404$	0	0
$405$	14.1421	0.702728
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	25.4558	1.25564
$412$	0	0
$413$	−16.9706	−0.835067
$414$	0	0
$415$	−28.0000	−1.37447
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	−26.8701	−1.31269	−0.656344	−	0.754462i	$-0.727898\pi$
$419$	−26.8701	−1.31269	−0.656344	+	0.754462i	$0.727898\pi$
$420$	0	0
$421$	8.48528	0.413547	0.206774	−	0.978389i	$-0.433704\pi$
$421$	8.48528	0.413547	0.206774	+	0.978389i	$0.433704\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−33.9411	−1.64253
$428$	0	0
$429$	5.65685	0.273115
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	33.9411	1.62735
$436$	0	0
$437$	28.2843	1.35302
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	−35.3553	−1.67978	−0.839891	−	0.542754i	$-0.817381\pi$
$443$	−35.3553	−1.67978	−0.839891	+	0.542754i	$0.817381\pi$
$444$	0	0
$445$	−33.9411	−1.60896
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−2.82843	−0.133185
$452$	0	0
$453$	−5.65685	−0.265782
$454$	0	0
$455$	−32.0000	−1.50018
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	−22.6274	−1.05616
$460$	0	0
$461$	−14.1421	−0.658665	−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421	−0.658665	−0.329332	+	0.944214i	$0.606824\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	32.0000	1.48396
$466$	0	0
$467$	18.3848	0.850746	0.425373	−	0.905018i	$-0.360143\pi$
$467$	18.3848	0.850746	0.425373	+	0.905018i	$0.360143\pi$
$468$	0	0
$469$	−16.9706	−0.783628
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	21.2132	0.973329
$476$	0	0
$477$	−2.82843	−0.129505
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	−22.6274	−1.02958
$484$	0	0
$485$	11.3137	0.513729
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−15.5563	−0.702048	−0.351024	−	0.936366i	$-0.614166\pi$
$491$	−15.5563	−0.702048	−0.351024	+	0.936366i	$0.614166\pi$
$492$	0	0
$493$	−33.9411	−1.52863
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−12.7279	−0.569780	−0.284890	−	0.958560i	$-0.591957\pi$
$499$	−12.7279	−0.569780	−0.284890	+	0.958560i	$0.591957\pi$
$500$	0	0
$501$	16.9706	0.758189
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	7.07107	0.314037
$508$	0	0
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	−33.9411	−1.49562
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−20.0000	−0.877903
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	9.89949	0.432875	0.216437	−	0.976297i	$-0.430556\pi$
$523$	9.89949	0.432875	0.216437	+	0.976297i	$0.430556\pi$
$524$	0	0
$525$	−16.9706	−0.740656
$526$	0	0
$527$	−32.0000	−1.39394
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.24264	0.184115
$532$	0	0
$533$	5.65685	0.245026
$534$	0	0
$535$	−28.0000	−1.21055
$536$	0	0
$537$	2.00000	0.0863064
$538$	0	0
$539$	−12.7279	−0.548230
$540$	0	0
$541$	−8.48528	−0.364811	−0.182405	−	0.983223i	$-0.558388\pi$
$541$	−8.48528	−0.364811	−0.182405	+	0.983223i	$0.558388\pi$
$542$	0	0
$543$	−12.0000	−0.514969
$544$	0	0
$545$	40.0000	1.71341
$546$	0	0
$547$	−26.8701	−1.14888	−0.574440	−	0.818546i	$-0.694780\pi$
$547$	−26.8701	−1.14888	−0.574440	+	0.818546i	$0.694780\pi$
$548$	0	0
$549$	8.48528	0.362143
$550$	0	0
$551$	60.0000	2.55609
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	11.3137	0.480240
$556$	0	0
$557$	−31.1127	−1.31829	−0.659144	−	0.752017i	$-0.729081\pi$
$557$	−31.1127	−1.31829	−0.659144	+	0.752017i	$0.729081\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	7.07107	0.298010	0.149005	−	0.988836i	$-0.452393\pi$
$563$	7.07107	0.298010	0.149005	+	0.988836i	$0.452393\pi$
$564$	0	0
$565$	−39.5980	−1.66590
$566$	0	0
$567$	−20.0000	−0.839921
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	32.5269	1.36121	0.680604	−	0.732651i	$-0.261717\pi$
$571$	32.5269	1.36121	0.680604	+	0.732651i	$0.261717\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	5.65685	0.235091
$580$	0	0
$581$	39.5980	1.64280
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	−1.41421	−0.0583708	−0.0291854	−	0.999574i	$-0.509291\pi$
$587$	−1.41421	−0.0583708	−0.0291854	+	0.999574i	$0.509291\pi$
$588$	0	0
$589$	56.5685	2.33087
$590$	0	0
$591$	20.0000	0.822690
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	45.2548	1.85527
$596$	0	0
$597$	−5.65685	−0.231520
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	4.24264	0.172774
$604$	0	0
$605$	25.4558	1.03493
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.48528	0.342717	0.171359	−	0.985209i	$-0.445184\pi$
$613$	8.48528	0.342717	0.171359	+	0.985209i	$0.445184\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	0	0
$619$	−1.41421	−0.0568420	−0.0284210	−	0.999596i	$-0.509048\pi$
$619$	−1.41421	−0.0568420	−0.0284210	+	0.999596i	$0.509048\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	48.0000	1.92308
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	14.1421	0.564782
$628$	0	0
$629$	−11.3137	−0.451107
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	34.0000	1.35138
$634$	0	0
$635$	−22.6274	−0.897942
$636$	0	0
$637$	25.4558	1.00860
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	−4.24264	−0.167313	−0.0836567	−	0.996495i	$-0.526660\pi$
$643$	−4.24264	−0.167313	−0.0836567	+	0.996495i	$0.526660\pi$
$644$	0	0
$645$	−16.9706	−0.668215
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−45.2548	−1.77368
$652$	0	0
$653$	−48.0833	−1.88164	−0.940822	−	0.338902i	$-0.889945\pi$
$653$	−48.0833	−1.88164	−0.940822	+	0.338902i	$0.889945\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	32.5269	1.26707	0.633534	−	0.773715i	$-0.281604\pi$
$659$	32.5269	1.26707	0.633534	+	0.773715i	$0.281604\pi$
$660$	0	0
$661$	31.1127	1.21014	0.605072	−	0.796171i	$-0.293144\pi$
$661$	31.1127	1.21014	0.605072	+	0.796171i	$0.293144\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	−80.0000	−3.10227
$666$	0	0
$667$	33.9411	1.31421
$668$	0	0
$669$	−33.9411	−1.31224
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	−42.4264	−1.63058	−0.815290	−	0.579053i	$-0.803422\pi$
$677$	−42.4264	−1.63058	−0.815290	+	0.579053i	$0.803422\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	29.6985	1.13638	0.568190	−	0.822897i	$-0.307644\pi$
$683$	29.6985	1.13638	0.568190	+	0.822897i	$0.307644\pi$
$684$	0	0
$685$	50.9117	1.94524
$686$	0	0
$687$	36.0000	1.37349
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	−46.6690	−1.77537	−0.887687	−	0.460447i	$-0.847689\pi$
$691$	−46.6690	−1.77537	−0.887687	+	0.460447i	$0.847689\pi$
$692$	0	0
$693$	5.65685	0.214886
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	−16.9706	−0.641886
$700$	0	0
$701$	48.0833	1.81608	0.908040	−	0.418884i	$-0.137579\pi$
$701$	48.0833	1.81608	0.908040	+	0.418884i	$0.137579\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.3137	0.425496
$708$	0	0
$709$	−8.48528	−0.318671	−0.159336	−	0.987224i	$-0.550935\pi$
$709$	−8.48528	−0.318671	−0.159336	+	0.987224i	$0.550935\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	11.3137	0.423109
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	−16.9706	−0.631142
$724$	0	0
$725$	25.4558	0.945406
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	16.9706	0.627679
$732$	0	0
$733$	19.7990	0.731292	0.365646	−	0.930754i	$-0.380848\pi$
$733$	19.7990	0.731292	0.365646	+	0.930754i	$0.380848\pi$
$734$	0	0
$735$	36.0000	1.32788
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	−24.0416	−0.884386	−0.442193	−	0.896920i	$-0.645799\pi$
$739$	−24.0416	−0.884386	−0.442193	+	0.896920i	$0.645799\pi$
$740$	0	0
$741$	−28.2843	−1.03905
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	−9.89949	−0.362204
$748$	0	0
$749$	39.5980	1.44688
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	0	0
$755$	−11.3137	−0.411748
$756$	0	0
$757$	−25.4558	−0.925208	−0.462604	−	0.886565i	$-0.653085\pi$
$757$	−25.4558	−0.925208	−0.462604	+	0.886565i	$0.653085\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−56.5685	−2.04792
$764$	0	0
$765$	−11.3137	−0.409048
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	2.82843	0.101863
$772$	0	0
$773$	−8.48528	−0.305194	−0.152597	−	0.988288i	$-0.548764\pi$
$773$	−8.48528	−0.305194	−0.152597	+	0.988288i	$0.548764\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	14.1421	0.506695
$780$	0	0
$781$	5.65685	0.202418
$782$	0	0
$783$	48.0000	1.71538
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	−46.6690	−1.66357	−0.831786	−	0.555097i	$-0.812681\pi$
$787$	−46.6690	−1.66357	−0.831786	+	0.555097i	$0.812681\pi$
$788$	0	0
$789$	−39.5980	−1.40973
$790$	0	0
$791$	56.0000	1.99113
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	11.3137	0.401256
$796$	0	0
$797$	−19.7990	−0.701316	−0.350658	−	0.936504i	$-0.614042\pi$
$797$	−19.7990	−0.701316	−0.350658	+	0.936504i	$0.614042\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	−45.2548	−1.59502
$806$	0	0
$807$	−28.0000	−0.985647
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−24.0416	−0.844216	−0.422108	−	0.906546i	$-0.638710\pi$
$811$	−24.0416	−0.844216	−0.422108	+	0.906546i	$0.638710\pi$
$812$	0	0
$813$	−45.2548	−1.58716
$814$	0	0
$815$	−28.0000	−0.980797
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	−11.3137	−0.395333
$820$	0	0
$821$	−8.48528	−0.296138	−0.148069	−	0.988977i	$-0.547306\pi$
$821$	−8.48528	−0.296138	−0.148069	+	0.988977i	$0.547306\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	21.2132	0.737655	0.368828	−	0.929498i	$-0.379759\pi$
$827$	21.2132	0.737655	0.368828	+	0.929498i	$0.379759\pi$
$828$	0	0
$829$	25.4558	0.884118	0.442059	−	0.896986i	$-0.354248\pi$
$829$	25.4558	0.884118	0.442059	+	0.896986i	$0.354248\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	−36.0000	−1.24733
$834$	0	0
$835$	33.9411	1.17458
$836$	0	0
$837$	45.2548	1.56424
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	0	0
$843$	28.2843	0.974162
$844$	0	0
$845$	14.1421	0.486504
$846$	0	0
$847$	−36.0000	−1.23697
$848$	0	0
$849$	34.0000	1.16688
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	0	0
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	−38.1838	−1.30281	−0.651407	−	0.758729i	$-0.725821\pi$
$859$	−38.1838	−1.30281	−0.651407	+	0.758729i	$0.725821\pi$
$860$	0	0
$861$	−11.3137	−0.385570
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−40.0000	−1.36004
$866$	0	0
$867$	1.41421	0.0480292
$868$	0	0
$869$	11.3137	0.383791
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	22.6274	0.764946
$876$	0	0
$877$	8.48528	0.286528	0.143264	−	0.989685i	$-0.454240\pi$
$877$	8.48528	0.286528	0.143264	+	0.989685i	$0.454240\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	0	0
$883$	41.0122	1.38017	0.690085	−	0.723728i	$-0.257573\pi$
$883$	41.0122	1.38017	0.690085	+	0.723728i	$0.257573\pi$
$884$	0	0
$885$	−16.9706	−0.570459
$886$	0	0
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	67.8823	2.26400
$900$	0	0
$901$	−11.3137	−0.376914
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	7.07107	0.234791	0.117395	−	0.993085i	$-0.462545\pi$
$907$	7.07107	0.234791	0.117395	+	0.993085i	$0.462545\pi$
$908$	0	0
$909$	−2.82843	−0.0938130
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	0	0
$915$	−33.9411	−1.12206
$916$	0	0
$917$	28.2843	0.934029
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	−11.3137	−0.372395
$924$	0	0
$925$	8.48528	0.278994
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	60.0000	1.96854	0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000	1.96854	0.984268	+	0.176682i	$0.0565363\pi$
$930$	0	0
$931$	63.6396	2.08570
$932$	0	0
$933$	28.2843	0.925985
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−20.0000	−0.653372	−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000	−0.653372	−0.326686	+	0.945133i	$0.605932\pi$
$938$	0	0
$939$	19.7990	0.646116
$940$	0	0
$941$	−36.7696	−1.19865	−0.599327	−	0.800505i	$-0.704565\pi$
$941$	−36.7696	−1.19865	−0.599327	+	0.800505i	$0.704565\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	−64.0000	−2.08192
$946$	0	0
$947$	−60.8112	−1.97610	−0.988049	−	0.154140i	$-0.950739\pi$
$947$	−60.8112	−1.97610	−0.988049	+	0.154140i	$0.950739\pi$
$948$	0	0
$949$	−11.3137	−0.367259
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	−22.6274	−0.732206
$956$	0	0
$957$	16.9706	0.548580
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−9.89949	−0.319007
$964$	0	0
$965$	11.3137	0.364201
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	40.0000	1.28499
$970$	0	0
$971$	−15.5563	−0.499227	−0.249614	−	0.968346i	$-0.580304\pi$
$971$	−15.5563	−0.499227	−0.249614	+	0.968346i	$0.580304\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−16.9706	−0.542382
$980$	0	0
$981$	14.1421	0.451524
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	40.0000	1.27451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	18.0000	0.571213
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	14.1421	0.447886	0.223943	−	0.974602i	$-0.428107\pi$
$997$	14.1421	0.447886	0.223943	+	0.974602i	$0.428107\pi$
$998$	0	0
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.2.a.d.1.1	yes	2
3.2	odd	2		4608.2.a.m.1.2		2
4.3	odd	2		512.2.a.c.1.2	yes	2
8.3	odd	2		512.2.a.c.1.1	✓	2
8.5	even	2	inner	512.2.a.d.1.2	yes	2
12.11	even	2		4608.2.a.f.1.2		2
16.3	odd	4		512.2.b.b.257.2		2
16.5	even	4		512.2.b.a.257.2		2
16.11	odd	4		512.2.b.b.257.1		2
16.13	even	4		512.2.b.a.257.1		2
24.5	odd	2		4608.2.a.m.1.1		2
24.11	even	2		4608.2.a.f.1.1		2
32.3	odd	8		1024.2.e.e.257.1		2
32.5	even	8		1024.2.e.d.769.1		2
32.11	odd	8		1024.2.e.e.769.1		2
32.13	even	8		1024.2.e.d.257.1		2
32.19	odd	8		1024.2.e.b.257.1		2
32.21	even	8		1024.2.e.c.769.1		2
32.27	odd	8		1024.2.e.b.769.1		2
32.29	even	8		1024.2.e.c.257.1		2
48.5	odd	4		4608.2.d.a.2305.2		2
48.11	even	4		4608.2.d.b.2305.2		2
48.29	odd	4		4608.2.d.a.2305.1		2
48.35	even	4		4608.2.d.b.2305.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.c.1.1	✓	2	8.3	odd	2
512.2.a.c.1.2	yes	2	4.3	odd	2
512.2.a.d.1.1	yes	2	1.1	even	1	trivial
512.2.a.d.1.2	yes	2	8.5	even	2	inner
512.2.b.a.257.1		2	16.13	even	4
512.2.b.a.257.2		2	16.5	even	4
512.2.b.b.257.1		2	16.11	odd	4
512.2.b.b.257.2		2	16.3	odd	4
1024.2.e.b.257.1		2	32.19	odd	8
1024.2.e.b.769.1		2	32.27	odd	8
1024.2.e.c.257.1		2	32.29	even	8
1024.2.e.c.769.1		2	32.21	even	8
1024.2.e.d.257.1		2	32.13	even	8
1024.2.e.d.769.1		2	32.5	even	8
1024.2.e.e.257.1		2	32.3	odd	8
1024.2.e.e.769.1		2	32.11	odd	8
4608.2.a.f.1.1		2	24.11	even	2
4608.2.a.f.1.2		2	12.11	even	2
4608.2.a.m.1.1		2	24.5	odd	2
4608.2.a.m.1.2		2	3.2	odd	2
4608.2.d.a.2305.1		2	48.29	odd	4
4608.2.d.a.2305.2		2	48.5	odd	4
4608.2.d.b.2305.1		2	48.35	even	4
4608.2.d.b.2305.2		2	48.11	even	4

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:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -2.82843 q^{5} +4.00000 q^{7} -1.00000 q^{9} -1.41421 q^{11} +2.82843 q^{13} +4.00000 q^{15} -4.00000 q^{17} +7.07107 q^{19} -5.65685 q^{21} +4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} +8.48528 q^{29} +8.00000 q^{31} +2.00000 q^{33} -11.3137 q^{35} +2.82843 q^{37} -4.00000 q^{39} +2.00000 q^{41} -4.24264 q^{43} +2.82843 q^{45} +9.00000 q^{49} +5.65685 q^{51} +2.82843 q^{53} +4.00000 q^{55} -10.0000 q^{57} -4.24264 q^{59} -8.48528 q^{61} -4.00000 q^{63} -8.00000 q^{65} -4.24264 q^{67} -5.65685 q^{69} -4.00000 q^{71} -4.00000 q^{73} -4.24264 q^{75} -5.65685 q^{77} -8.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +11.3137 q^{85} -12.0000 q^{87} +12.0000 q^{89} +11.3137 q^{91} -11.3137 q^{93} -20.0000 q^{95} -4.00000 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9} + 8 q^{15} - 8 q^{17} + 8 q^{23} + 6 q^{25} + 16 q^{31} + 4 q^{33} - 8 q^{39} + 4 q^{41} + 18 q^{49} + 8 q^{55} - 20 q^{57} - 8 q^{63} - 16 q^{65} - 8 q^{71} - 8 q^{73} - 16 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 + 8 * q^15 - 8 * q^17 + 8 * q^23 + 6 * q^25 + 16 * q^31 + 4 * q^33 - 8 * q^39 + 4 * q^41 + 18 * q^49 + 8 * q^55 - 20 * q^57 - 8 * q^63 - 16 * q^65 - 8 * q^71 - 8 * q^73 - 16 * q^79 - 10 * q^81 - 24 * q^87 + 24 * q^89 - 40 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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