LMFDB - Embedded newform 8016.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.585786 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.585786 q^{5} +2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{13} -0.585786 q^{15} -2.58579 q^{17} -6.82843 q^{19} +2.82843 q^{21} +5.65685 q^{23} -4.65685 q^{25} +1.00000 q^{27} +2.00000 q^{29} -1.17157 q^{31} -2.00000 q^{33} -1.65685 q^{35} -4.00000 q^{37} +2.82843 q^{39} -2.58579 q^{41} +8.24264 q^{43} -0.585786 q^{45} +7.17157 q^{47} +1.00000 q^{49} -2.58579 q^{51} +2.24264 q^{53} +1.17157 q^{55} -6.82843 q^{57} +12.8284 q^{59} +14.1421 q^{61} +2.82843 q^{63} -1.65685 q^{65} -1.41421 q^{67} +5.65685 q^{69} +12.4853 q^{71} +12.1421 q^{73} -4.65685 q^{75} -5.65685 q^{77} +10.2426 q^{79} +1.00000 q^{81} +11.6569 q^{83} +1.51472 q^{85} +2.00000 q^{87} -13.3137 q^{89} +8.00000 q^{91} -1.17157 q^{93} +4.00000 q^{95} -17.3137 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{15} - 8 q^{17} - 8 q^{19} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 4 q^{33} + 8 q^{35} - 8 q^{37} - 8 q^{41} + 8 q^{43} - 4 q^{45} + 20 q^{47} + 2 q^{49} - 8 q^{51} - 4 q^{53} + 8 q^{55} - 8 q^{57} + 20 q^{59} + 8 q^{65} + 8 q^{71} - 4 q^{73} + 2 q^{75} + 12 q^{79} + 2 q^{81} + 12 q^{83} + 20 q^{85} + 4 q^{87} - 4 q^{89} + 16 q^{91} - 8 q^{93} + 8 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 - 4 * q^11 - 4 * q^15 - 8 * q^17 - 8 * q^19 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 4 * q^33 + 8 * q^35 - 8 * q^37 - 8 * q^41 + 8 * q^43 - 4 * q^45 + 20 * q^47 + 2 * q^49 - 8 * q^51 - 4 * q^53 + 8 * q^55 - 8 * q^57 + 20 * q^59 + 8 * q^65 + 8 * q^71 - 4 * q^73 + 2 * q^75 + 12 * q^79 + 2 * q^81 + 12 * q^83 + 20 * q^85 + 4 * q^87 - 4 * q^89 + 16 * q^91 - 8 * q^93 + 8 * q^95 - 12 * q^97 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	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
$15$	−0.585786	−0.151249
$16$	0	0
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−1.65685	−0.280059
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−2.58579	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$41$	−2.58579	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$42$	0	0
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	0	0
$47$	7.17157	1.04608	0.523041	−	0.852308i	$-0.324798\pi$
$47$	7.17157	1.04608	0.523041	+	0.852308i	$0.324798\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.58579	−0.362083
$52$	0	0
$53$	2.24264	0.308050	0.154025	−	0.988067i	$-0.450776\pi$
$53$	2.24264	0.308050	0.154025	+	0.988067i	$0.450776\pi$
$54$	0	0
$55$	1.17157	0.157975
$56$	0	0
$57$	−6.82843	−0.904447
$58$	0	0
$59$	12.8284	1.67012	0.835059	−	0.550160i	$-0.185433\pi$
$59$	12.8284	1.67012	0.835059	+	0.550160i	$0.185433\pi$
$60$	0	0
$61$	14.1421	1.81071	0.905357	−	0.424650i	$-0.139603\pi$
$61$	14.1421	1.81071	0.905357	+	0.424650i	$0.139603\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	−1.65685	−0.205507
$66$	0	0
$67$	−1.41421	−0.172774	−0.0863868	−	0.996262i	$-0.527532\pi$
$67$	−1.41421	−0.172774	−0.0863868	+	0.996262i	$0.527532\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	0	0
$73$	12.1421	1.42113	0.710565	−	0.703632i	$-0.248440\pi$
$73$	12.1421	1.42113	0.710565	+	0.703632i	$0.248440\pi$
$74$	0	0
$75$	−4.65685	−0.537727
$76$	0	0
$77$	−5.65685	−0.644658
$78$	0	0
$79$	10.2426	1.15239	0.576194	−	0.817313i	$-0.304537\pi$
$79$	10.2426	1.15239	0.576194	+	0.817313i	$0.304537\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.6569	1.27951	0.639753	−	0.768581i	$-0.279037\pi$
$83$	11.6569	1.27951	0.639753	+	0.768581i	$0.279037\pi$
$84$	0	0
$85$	1.51472	0.164294
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−1.17157	−0.121486
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−17.3137	−1.75794	−0.878970	−	0.476876i	$-0.841769\pi$
$97$	−17.3137	−1.75794	−0.878970	+	0.476876i	$0.841769\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	1.07107	0.106575	0.0532876	−	0.998579i	$-0.483030\pi$
$101$	1.07107	0.106575	0.0532876	+	0.998579i	$0.483030\pi$
$102$	0	0
$103$	−9.07107	−0.893799	−0.446899	−	0.894584i	$-0.647472\pi$
$103$	−9.07107	−0.893799	−0.446899	+	0.894584i	$0.647472\pi$
$104$	0	0
$105$	−1.65685	−0.161692
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	4.48528	0.429612	0.214806	−	0.976657i	$-0.431088\pi$
$109$	4.48528	0.429612	0.214806	+	0.976657i	$0.431088\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−0.242641	−0.0228257	−0.0114129	−	0.999935i	$-0.503633\pi$
$113$	−0.242641	−0.0228257	−0.0114129	+	0.999935i	$0.503633\pi$
$114$	0	0
$115$	−3.31371	−0.309005
$116$	0	0
$117$	2.82843	0.261488
$118$	0	0
$119$	−7.31371	−0.670447
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.58579	−0.233153
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	8.24264	0.725724
$130$	0	0
$131$	19.6569	1.71743	0.858714	−	0.512456i	$-0.171264\pi$
$131$	19.6569	1.71743	0.858714	+	0.512456i	$0.171264\pi$
$132$	0	0
$133$	−19.3137	−1.67471
$134$	0	0
$135$	−0.585786	−0.0504165
$136$	0	0
$137$	15.1716	1.29619	0.648097	−	0.761557i	$-0.275565\pi$
$137$	15.1716	1.29619	0.648097	+	0.761557i	$0.275565\pi$
$138$	0	0
$139$	5.89949	0.500389	0.250194	−	0.968196i	$-0.419505\pi$
$139$	5.89949	0.500389	0.250194	+	0.968196i	$0.419505\pi$
$140$	0	0
$141$	7.17157	0.603955
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	−1.17157	−0.0972938
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−2.24264	−0.183724	−0.0918621	−	0.995772i	$-0.529282\pi$
$149$	−2.24264	−0.183724	−0.0918621	+	0.995772i	$0.529282\pi$
$150$	0	0
$151$	14.2426	1.15905	0.579525	−	0.814955i	$-0.303238\pi$
$151$	14.2426	1.15905	0.579525	+	0.814955i	$0.303238\pi$
$152$	0	0
$153$	−2.58579	−0.209048
$154$	0	0
$155$	0.686292	0.0551243
$156$	0	0
$157$	−14.9706	−1.19478	−0.597390	−	0.801950i	$-0.703796\pi$
$157$	−14.9706	−1.19478	−0.597390	+	0.801950i	$0.703796\pi$
$158$	0	0
$159$	2.24264	0.177853
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	−5.89949	−0.462084	−0.231042	−	0.972944i	$-0.574213\pi$
$163$	−5.89949	−0.462084	−0.231042	+	0.972944i	$0.574213\pi$
$164$	0	0
$165$	1.17157	0.0912068
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−6.82843	−0.522183
$172$	0	0
$173$	20.1421	1.53138	0.765689	−	0.643211i	$-0.222398\pi$
$173$	20.1421	1.53138	0.765689	+	0.643211i	$0.222398\pi$
$174$	0	0
$175$	−13.1716	−0.995677
$176$	0	0
$177$	12.8284	0.964244
$178$	0	0
$179$	−23.3137	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$179$	−23.3137	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$180$	0	0
$181$	6.82843	0.507553	0.253776	−	0.967263i	$-0.418327\pi$
$181$	6.82843	0.507553	0.253776	+	0.967263i	$0.418327\pi$
$182$	0	0
$183$	14.1421	1.04542
$184$	0	0
$185$	2.34315	0.172272
$186$	0	0
$187$	5.17157	0.378183
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	12.1421	0.878574	0.439287	−	0.898347i	$-0.355231\pi$
$191$	12.1421	0.878574	0.439287	+	0.898347i	$0.355231\pi$
$192$	0	0
$193$	6.97056	0.501752	0.250876	−	0.968019i	$-0.419281\pi$
$193$	6.97056	0.501752	0.250876	+	0.968019i	$0.419281\pi$
$194$	0	0
$195$	−1.65685	−0.118650
$196$	0	0
$197$	17.5563	1.25084	0.625419	−	0.780289i	$-0.284928\pi$
$197$	17.5563	1.25084	0.625419	+	0.780289i	$0.284928\pi$
$198$	0	0
$199$	−17.6569	−1.25166	−0.625831	−	0.779959i	$-0.715240\pi$
$199$	−17.6569	−1.25166	−0.625831	+	0.779959i	$0.715240\pi$
$200$	0	0
$201$	−1.41421	−0.0997509
$202$	0	0
$203$	5.65685	0.397033
$204$	0	0
$205$	1.51472	0.105793
$206$	0	0
$207$	5.65685	0.393179
$208$	0	0
$209$	13.6569	0.944664
$210$	0	0
$211$	−14.1421	−0.973585	−0.486792	−	0.873518i	$-0.661833\pi$
$211$	−14.1421	−0.973585	−0.486792	+	0.873518i	$0.661833\pi$
$212$	0	0
$213$	12.4853	0.855477
$214$	0	0
$215$	−4.82843	−0.329296
$216$	0	0
$217$	−3.31371	−0.224949
$218$	0	0
$219$	12.1421	0.820489
$220$	0	0
$221$	−7.31371	−0.491973
$222$	0	0
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	0.828427	0.0549846	0.0274923	−	0.999622i	$-0.491248\pi$
$227$	0.828427	0.0549846	0.0274923	+	0.999622i	$0.491248\pi$
$228$	0	0
$229$	−2.14214	−0.141556	−0.0707782	−	0.997492i	$-0.522548\pi$
$229$	−2.14214	−0.141556	−0.0707782	+	0.997492i	$0.522548\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	1.31371	0.0860639	0.0430320	−	0.999074i	$-0.486298\pi$
$233$	1.31371	0.0860639	0.0430320	+	0.999074i	$0.486298\pi$
$234$	0	0
$235$	−4.20101	−0.274044
$236$	0	0
$237$	10.2426	0.665331
$238$	0	0
$239$	−26.4853	−1.71319	−0.856595	−	0.515989i	$-0.827425\pi$
$239$	−26.4853	−1.71319	−0.856595	+	0.515989i	$0.827425\pi$
$240$	0	0
$241$	−13.7990	−0.888871	−0.444436	−	0.895811i	$-0.646596\pi$
$241$	−13.7990	−0.888871	−0.444436	+	0.895811i	$0.646596\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.585786	−0.0374245
$246$	0	0
$247$	−19.3137	−1.22890
$248$	0	0
$249$	11.6569	0.738723
$250$	0	0
$251$	10.3431	0.652854	0.326427	−	0.945222i	$-0.394155\pi$
$251$	10.3431	0.652854	0.326427	+	0.945222i	$0.394155\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	1.51472	0.0948554
$256$	0	0
$257$	−7.07107	−0.441081	−0.220541	−	0.975378i	$-0.570782\pi$
$257$	−7.07107	−0.441081	−0.220541	+	0.975378i	$0.570782\pi$
$258$	0	0
$259$	−11.3137	−0.703000
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	11.3137	0.697633	0.348817	−	0.937191i	$-0.386584\pi$
$263$	11.3137	0.697633	0.348817	+	0.937191i	$0.386584\pi$
$264$	0	0
$265$	−1.31371	−0.0807005
$266$	0	0
$267$	−13.3137	−0.814786
$268$	0	0
$269$	1.75736	0.107148	0.0535740	−	0.998564i	$-0.482939\pi$
$269$	1.75736	0.107148	0.0535740	+	0.998564i	$0.482939\pi$
$270$	0	0
$271$	17.5563	1.06647	0.533236	−	0.845966i	$-0.320976\pi$
$271$	17.5563	1.06647	0.533236	+	0.845966i	$0.320976\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	9.31371	0.561638
$276$	0	0
$277$	6.82843	0.410280	0.205140	−	0.978733i	$-0.434235\pi$
$277$	6.82843	0.410280	0.205140	+	0.978733i	$0.434235\pi$
$278$	0	0
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	1.31371	0.0783693	0.0391846	−	0.999232i	$-0.487524\pi$
$281$	1.31371	0.0783693	0.0391846	+	0.999232i	$0.487524\pi$
$282$	0	0
$283$	26.1421	1.55399	0.776994	−	0.629508i	$-0.216743\pi$
$283$	26.1421	1.55399	0.776994	+	0.629508i	$0.216743\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−7.31371	−0.431715
$288$	0	0
$289$	−10.3137	−0.606689
$290$	0	0
$291$	−17.3137	−1.01495
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−7.51472	−0.437524
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	23.3137	1.34378
$302$	0	0
$303$	1.07107	0.0615312
$304$	0	0
$305$	−8.28427	−0.474356
$306$	0	0
$307$	31.5563	1.80102	0.900508	−	0.434839i	$-0.143195\pi$
$307$	31.5563	1.80102	0.900508	+	0.434839i	$0.143195\pi$
$308$	0	0
$309$	−9.07107	−0.516035
$310$	0	0
$311$	−27.3137	−1.54882	−0.774409	−	0.632685i	$-0.781953\pi$
$311$	−27.3137	−1.54882	−0.774409	+	0.632685i	$0.781953\pi$
$312$	0	0
$313$	6.97056	0.394000	0.197000	−	0.980404i	$-0.436880\pi$
$313$	6.97056	0.394000	0.197000	+	0.980404i	$0.436880\pi$
$314$	0	0
$315$	−1.65685	−0.0933532
$316$	0	0
$317$	−13.5147	−0.759062	−0.379531	−	0.925179i	$-0.623915\pi$
$317$	−13.5147	−0.759062	−0.379531	+	0.925179i	$0.623915\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	17.6569	0.982454
$324$	0	0
$325$	−13.1716	−0.730627
$326$	0	0
$327$	4.48528	0.248037
$328$	0	0
$329$	20.2843	1.11831
$330$	0	0
$331$	13.4142	0.737312	0.368656	−	0.929566i	$-0.379818\pi$
$331$	13.4142	0.737312	0.368656	+	0.929566i	$0.379818\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	0.828427	0.0452618
$336$	0	0
$337$	11.6569	0.634989	0.317495	−	0.948260i	$-0.397158\pi$
$337$	11.6569	0.634989	0.317495	+	0.948260i	$0.397158\pi$
$338$	0	0
$339$	−0.242641	−0.0131784
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−3.31371	−0.178404
$346$	0	0
$347$	4.82843	0.259204	0.129602	−	0.991566i	$-0.458630\pi$
$347$	4.82843	0.259204	0.129602	+	0.991566i	$0.458630\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	0	0
$353$	2.68629	0.142977	0.0714884	−	0.997441i	$-0.477225\pi$
$353$	2.68629	0.142977	0.0714884	+	0.997441i	$0.477225\pi$
$354$	0	0
$355$	−7.31371	−0.388171
$356$	0	0
$357$	−7.31371	−0.387083
$358$	0	0
$359$	−18.3431	−0.968114	−0.484057	−	0.875036i	$-0.660837\pi$
$359$	−18.3431	−0.968114	−0.484057	+	0.875036i	$0.660837\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−7.11270	−0.372296
$366$	0	0
$367$	14.8284	0.774038	0.387019	−	0.922072i	$-0.373505\pi$
$367$	14.8284	0.774038	0.387019	+	0.922072i	$0.373505\pi$
$368$	0	0
$369$	−2.58579	−0.134611
$370$	0	0
$371$	6.34315	0.329320
$372$	0	0
$373$	−32.9706	−1.70715	−0.853576	−	0.520969i	$-0.825571\pi$
$373$	−32.9706	−1.70715	−0.853576	+	0.520969i	$0.825571\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	5.65685	0.291343
$378$	0	0
$379$	−0.443651	−0.0227888	−0.0113944	−	0.999935i	$-0.503627\pi$
$379$	−0.443651	−0.0227888	−0.0113944	+	0.999935i	$0.503627\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	15.1716	0.775231	0.387616	−	0.921821i	$-0.373299\pi$
$383$	15.1716	0.775231	0.387616	+	0.921821i	$0.373299\pi$
$384$	0	0
$385$	3.31371	0.168882
$386$	0	0
$387$	8.24264	0.418997
$388$	0	0
$389$	−1.55635	−0.0789100	−0.0394550	−	0.999221i	$-0.512562\pi$
$389$	−1.55635	−0.0789100	−0.0394550	+	0.999221i	$0.512562\pi$
$390$	0	0
$391$	−14.6274	−0.739740
$392$	0	0
$393$	19.6569	0.991557
$394$	0	0
$395$	−6.00000	−0.301893
$396$	0	0
$397$	24.4853	1.22888	0.614441	−	0.788963i	$-0.289382\pi$
$397$	24.4853	1.22888	0.614441	+	0.788963i	$0.289382\pi$
$398$	0	0
$399$	−19.3137	−0.966895
$400$	0	0
$401$	9.41421	0.470123	0.235062	−	0.971980i	$-0.424471\pi$
$401$	9.41421	0.470123	0.235062	+	0.971980i	$0.424471\pi$
$402$	0	0
$403$	−3.31371	−0.165068
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−11.3137	−0.559427	−0.279713	−	0.960084i	$-0.590239\pi$
$409$	−11.3137	−0.559427	−0.279713	+	0.960084i	$0.590239\pi$
$410$	0	0
$411$	15.1716	0.748359
$412$	0	0
$413$	36.2843	1.78543
$414$	0	0
$415$	−6.82843	−0.335194
$416$	0	0
$417$	5.89949	0.288900
$418$	0	0
$419$	−27.9411	−1.36501	−0.682507	−	0.730879i	$-0.739110\pi$
$419$	−27.9411	−1.36501	−0.682507	+	0.730879i	$0.739110\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	7.17157	0.348694
$424$	0	0
$425$	12.0416	0.584105
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	−5.65685	−0.273115
$430$	0	0
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	0	0
$433$	11.3137	0.543702	0.271851	−	0.962339i	$-0.412364\pi$
$433$	11.3137	0.543702	0.271851	+	0.962339i	$0.412364\pi$
$434$	0	0
$435$	−1.17157	−0.0561726
$436$	0	0
$437$	−38.6274	−1.84780
$438$	0	0
$439$	−9.07107	−0.432938	−0.216469	−	0.976289i	$-0.569454\pi$
$439$	−9.07107	−0.432938	−0.216469	+	0.976289i	$0.569454\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	13.5147	0.642104	0.321052	−	0.947062i	$-0.395964\pi$
$443$	13.5147	0.642104	0.321052	+	0.947062i	$0.395964\pi$
$444$	0	0
$445$	7.79899	0.369708
$446$	0	0
$447$	−2.24264	−0.106073
$448$	0	0
$449$	27.1716	1.28231	0.641153	−	0.767413i	$-0.278456\pi$
$449$	27.1716	1.28231	0.641153	+	0.767413i	$0.278456\pi$
$450$	0	0
$451$	5.17157	0.243520
$452$	0	0
$453$	14.2426	0.669178
$454$	0	0
$455$	−4.68629	−0.219697
$456$	0	0
$457$	19.4558	0.910106	0.455053	−	0.890464i	$-0.349620\pi$
$457$	19.4558	0.910106	0.455053	+	0.890464i	$0.349620\pi$
$458$	0	0
$459$	−2.58579	−0.120694
$460$	0	0
$461$	−21.3137	−0.992678	−0.496339	−	0.868129i	$-0.665323\pi$
$461$	−21.3137	−0.992678	−0.496339	+	0.868129i	$0.665323\pi$
$462$	0	0
$463$	−14.2426	−0.661912	−0.330956	−	0.943646i	$-0.607371\pi$
$463$	−14.2426	−0.661912	−0.330956	+	0.943646i	$0.607371\pi$
$464$	0	0
$465$	0.686292	0.0318260
$466$	0	0
$467$	−7.31371	−0.338438	−0.169219	−	0.985578i	$-0.554125\pi$
$467$	−7.31371	−0.338438	−0.169219	+	0.985578i	$0.554125\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−14.9706	−0.689807
$472$	0	0
$473$	−16.4853	−0.757994
$474$	0	0
$475$	31.7990	1.45904
$476$	0	0
$477$	2.24264	0.102683
$478$	0	0
$479$	32.4853	1.48429	0.742145	−	0.670239i	$-0.233808\pi$
$479$	32.4853	1.48429	0.742145	+	0.670239i	$0.233808\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	10.1421	0.460531
$486$	0	0
$487$	−20.5858	−0.932831	−0.466416	−	0.884566i	$-0.654455\pi$
$487$	−20.5858	−0.932831	−0.466416	+	0.884566i	$0.654455\pi$
$488$	0	0
$489$	−5.89949	−0.266784
$490$	0	0
$491$	−30.9706	−1.39768	−0.698841	−	0.715277i	$-0.746300\pi$
$491$	−30.9706	−1.39768	−0.698841	+	0.715277i	$0.746300\pi$
$492$	0	0
$493$	−5.17157	−0.232916
$494$	0	0
$495$	1.17157	0.0526583
$496$	0	0
$497$	35.3137	1.58404
$498$	0	0
$499$	15.2721	0.683672	0.341836	−	0.939760i	$-0.388951\pi$
$499$	15.2721	0.683672	0.341836	+	0.939760i	$0.388951\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	29.1127	1.29807	0.649036	−	0.760758i	$-0.275173\pi$
$503$	29.1127	1.29807	0.649036	+	0.760758i	$0.275173\pi$
$504$	0	0
$505$	−0.627417	−0.0279197
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	−13.3137	−0.590120	−0.295060	−	0.955479i	$-0.595340\pi$
$509$	−13.3137	−0.590120	−0.295060	+	0.955479i	$0.595340\pi$
$510$	0	0
$511$	34.3431	1.51925
$512$	0	0
$513$	−6.82843	−0.301482
$514$	0	0
$515$	5.31371	0.234150
$516$	0	0
$517$	−14.3431	−0.630811
$518$	0	0
$519$	20.1421	0.884142
$520$	0	0
$521$	−40.0416	−1.75426	−0.877128	−	0.480257i	$-0.840543\pi$
$521$	−40.0416	−1.75426	−0.877128	+	0.480257i	$0.840543\pi$
$522$	0	0
$523$	41.4558	1.81274	0.906369	−	0.422488i	$-0.138843\pi$
$523$	41.4558	1.81274	0.906369	+	0.422488i	$0.138843\pi$
$524$	0	0
$525$	−13.1716	−0.574855
$526$	0	0
$527$	3.02944	0.131964
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	12.8284	0.556706
$532$	0	0
$533$	−7.31371	−0.316792
$534$	0	0
$535$	−5.85786	−0.253258
$536$	0	0
$537$	−23.3137	−1.00606
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−13.1716	−0.566290	−0.283145	−	0.959077i	$-0.591378\pi$
$541$	−13.1716	−0.566290	−0.283145	+	0.959077i	$0.591378\pi$
$542$	0	0
$543$	6.82843	0.293036
$544$	0	0
$545$	−2.62742	−0.112546
$546$	0	0
$547$	36.5269	1.56178	0.780889	−	0.624670i	$-0.214766\pi$
$547$	36.5269	1.56178	0.780889	+	0.624670i	$0.214766\pi$
$548$	0	0
$549$	14.1421	0.603572
$550$	0	0
$551$	−13.6569	−0.581802
$552$	0	0
$553$	28.9706	1.23195
$554$	0	0
$555$	2.34315	0.0994610
$556$	0	0
$557$	35.4558	1.50231	0.751156	−	0.660125i	$-0.229497\pi$
$557$	35.4558	1.50231	0.751156	+	0.660125i	$0.229497\pi$
$558$	0	0
$559$	23.3137	0.986065
$560$	0	0
$561$	5.17157	0.218344
$562$	0	0
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	0.142136	0.00597969
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	1.61522	0.0677137	0.0338568	−	0.999427i	$-0.489221\pi$
$569$	1.61522	0.0677137	0.0338568	+	0.999427i	$0.489221\pi$
$570$	0	0
$571$	−41.2132	−1.72472	−0.862359	−	0.506297i	$-0.831014\pi$
$571$	−41.2132	−1.72472	−0.862359	+	0.506297i	$0.831014\pi$
$572$	0	0
$573$	12.1421	0.507245
$574$	0	0
$575$	−26.3431	−1.09859
$576$	0	0
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	0	0
$579$	6.97056	0.289687
$580$	0	0
$581$	32.9706	1.36785
$582$	0	0
$583$	−4.48528	−0.185761
$584$	0	0
$585$	−1.65685	−0.0685025
$586$	0	0
$587$	−9.51472	−0.392714	−0.196357	−	0.980532i	$-0.562911\pi$
$587$	−9.51472	−0.392714	−0.196357	+	0.980532i	$0.562911\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	17.5563	0.722172
$592$	0	0
$593$	−33.2132	−1.36390	−0.681951	−	0.731397i	$-0.738868\pi$
$593$	−33.2132	−1.36390	−0.681951	+	0.731397i	$0.738868\pi$
$594$	0	0
$595$	4.28427	0.175638
$596$	0	0
$597$	−17.6569	−0.722647
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−32.6274	−1.33090	−0.665450	−	0.746442i	$-0.731760\pi$
$601$	−32.6274	−1.33090	−0.665450	+	0.746442i	$0.731760\pi$
$602$	0	0
$603$	−1.41421	−0.0575912
$604$	0	0
$605$	4.10051	0.166709
$606$	0	0
$607$	22.0416	0.894642	0.447321	−	0.894373i	$-0.352378\pi$
$607$	22.0416	0.894642	0.447321	+	0.894373i	$0.352378\pi$
$608$	0	0
$609$	5.65685	0.229227
$610$	0	0
$611$	20.2843	0.820614
$612$	0	0
$613$	16.4853	0.665834	0.332917	−	0.942956i	$-0.391967\pi$
$613$	16.4853	0.665834	0.332917	+	0.942956i	$0.391967\pi$
$614$	0	0
$615$	1.51472	0.0610794
$616$	0	0
$617$	29.1127	1.17203	0.586017	−	0.810299i	$-0.300695\pi$
$617$	29.1127	1.17203	0.586017	+	0.810299i	$0.300695\pi$
$618$	0	0
$619$	−45.2132	−1.81727	−0.908636	−	0.417589i	$-0.862875\pi$
$619$	−45.2132	−1.81727	−0.908636	+	0.417589i	$0.862875\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	−37.6569	−1.50869
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	13.6569	0.545402
$628$	0	0
$629$	10.3431	0.412408
$630$	0	0
$631$	−16.2843	−0.648267	−0.324133	−	0.946011i	$-0.605073\pi$
$631$	−16.2843	−0.648267	−0.324133	+	0.946011i	$0.605073\pi$
$632$	0	0
$633$	−14.1421	−0.562099
$634$	0	0
$635$	2.34315	0.0929849
$636$	0	0
$637$	2.82843	0.112066
$638$	0	0
$639$	12.4853	0.493910
$640$	0	0
$641$	−3.27208	−0.129239	−0.0646197	−	0.997910i	$-0.520583\pi$
$641$	−3.27208	−0.129239	−0.0646197	+	0.997910i	$0.520583\pi$
$642$	0	0
$643$	27.3553	1.07879	0.539395	−	0.842053i	$-0.318653\pi$
$643$	27.3553	1.07879	0.539395	+	0.842053i	$0.318653\pi$
$644$	0	0
$645$	−4.82843	−0.190119
$646$	0	0
$647$	−24.4853	−0.962616	−0.481308	−	0.876552i	$-0.659838\pi$
$647$	−24.4853	−0.962616	−0.481308	+	0.876552i	$0.659838\pi$
$648$	0	0
$649$	−25.6569	−1.00712
$650$	0	0
$651$	−3.31371	−0.129874
$652$	0	0
$653$	−19.4558	−0.761366	−0.380683	−	0.924706i	$-0.624311\pi$
$653$	−19.4558	−0.761366	−0.380683	+	0.924706i	$0.624311\pi$
$654$	0	0
$655$	−11.5147	−0.449917
$656$	0	0
$657$	12.1421	0.473710
$658$	0	0
$659$	17.5147	0.682277	0.341138	−	0.940013i	$-0.389188\pi$
$659$	17.5147	0.682277	0.341138	+	0.940013i	$0.389188\pi$
$660$	0	0
$661$	17.6569	0.686772	0.343386	−	0.939194i	$-0.388426\pi$
$661$	17.6569	0.686772	0.343386	+	0.939194i	$0.388426\pi$
$662$	0	0
$663$	−7.31371	−0.284041
$664$	0	0
$665$	11.3137	0.438727
$666$	0	0
$667$	11.3137	0.438069
$668$	0	0
$669$	−14.1421	−0.546767
$670$	0	0
$671$	−28.2843	−1.09190
$672$	0	0
$673$	2.97056	0.114507	0.0572534	−	0.998360i	$-0.481766\pi$
$673$	2.97056	0.114507	0.0572534	+	0.998360i	$0.481766\pi$
$674$	0	0
$675$	−4.65685	−0.179242
$676$	0	0
$677$	−1.02944	−0.0395645	−0.0197822	−	0.999804i	$-0.506297\pi$
$677$	−1.02944	−0.0395645	−0.0197822	+	0.999804i	$0.506297\pi$
$678$	0	0
$679$	−48.9706	−1.87932
$680$	0	0
$681$	0.828427	0.0317454
$682$	0	0
$683$	−6.68629	−0.255844	−0.127922	−	0.991784i	$-0.540831\pi$
$683$	−6.68629	−0.255844	−0.127922	+	0.991784i	$0.540831\pi$
$684$	0	0
$685$	−8.88730	−0.339566
$686$	0	0
$687$	−2.14214	−0.0817276
$688$	0	0
$689$	6.34315	0.241655
$690$	0	0
$691$	32.9289	1.25268	0.626338	−	0.779552i	$-0.284553\pi$
$691$	32.9289	1.25268	0.626338	+	0.779552i	$0.284553\pi$
$692$	0	0
$693$	−5.65685	−0.214886
$694$	0	0
$695$	−3.45584	−0.131088
$696$	0	0
$697$	6.68629	0.253261
$698$	0	0
$699$	1.31371	0.0496890
$700$	0	0
$701$	−43.9411	−1.65963	−0.829817	−	0.558036i	$-0.811555\pi$
$701$	−43.9411	−1.65963	−0.829817	+	0.558036i	$0.811555\pi$
$702$	0	0
$703$	27.3137	1.03016
$704$	0	0
$705$	−4.20101	−0.158219
$706$	0	0
$707$	3.02944	0.113934
$708$	0	0
$709$	13.4558	0.505345	0.252672	−	0.967552i	$-0.418691\pi$
$709$	13.4558	0.505345	0.252672	+	0.967552i	$0.418691\pi$
$710$	0	0
$711$	10.2426	0.384129
$712$	0	0
$713$	−6.62742	−0.248199
$714$	0	0
$715$	3.31371	0.123926
$716$	0	0
$717$	−26.4853	−0.989111
$718$	0	0
$719$	11.7990	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$719$	11.7990	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$720$	0	0
$721$	−25.6569	−0.955511
$722$	0	0
$723$	−13.7990	−0.513190
$724$	0	0
$725$	−9.31371	−0.345902
$726$	0	0
$727$	−7.89949	−0.292976	−0.146488	−	0.989212i	$-0.546797\pi$
$727$	−7.89949	−0.292976	−0.146488	+	0.989212i	$0.546797\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.3137	−0.788316
$732$	0	0
$733$	−23.7990	−0.879036	−0.439518	−	0.898234i	$-0.644851\pi$
$733$	−23.7990	−0.879036	−0.439518	+	0.898234i	$0.644851\pi$
$734$	0	0
$735$	−0.585786	−0.0216071
$736$	0	0
$737$	2.82843	0.104186
$738$	0	0
$739$	−3.75736	−0.138217	−0.0691083	−	0.997609i	$-0.522015\pi$
$739$	−3.75736	−0.138217	−0.0691083	+	0.997609i	$0.522015\pi$
$740$	0	0
$741$	−19.3137	−0.709507
$742$	0	0
$743$	26.3431	0.966436	0.483218	−	0.875500i	$-0.339468\pi$
$743$	26.3431	0.966436	0.483218	+	0.875500i	$0.339468\pi$
$744$	0	0
$745$	1.31371	0.0481306
$746$	0	0
$747$	11.6569	0.426502
$748$	0	0
$749$	28.2843	1.03348
$750$	0	0
$751$	−0.786797	−0.0287106	−0.0143553	−	0.999897i	$-0.504570\pi$
$751$	−0.786797	−0.0287106	−0.0143553	+	0.999897i	$0.504570\pi$
$752$	0	0
$753$	10.3431	0.376925
$754$	0	0
$755$	−8.34315	−0.303638
$756$	0	0
$757$	−47.6569	−1.73212	−0.866059	−	0.499942i	$-0.833355\pi$
$757$	−47.6569	−1.73212	−0.866059	+	0.499942i	$0.833355\pi$
$758$	0	0
$759$	−11.3137	−0.410662
$760$	0	0
$761$	−11.8579	−0.429847	−0.214924	−	0.976631i	$-0.568950\pi$
$761$	−11.8579	−0.429847	−0.214924	+	0.976631i	$0.568950\pi$
$762$	0	0
$763$	12.6863	0.459275
$764$	0	0
$765$	1.51472	0.0547648
$766$	0	0
$767$	36.2843	1.31015
$768$	0	0
$769$	23.6569	0.853088	0.426544	−	0.904467i	$-0.359731\pi$
$769$	23.6569	0.853088	0.426544	+	0.904467i	$0.359731\pi$
$770$	0	0
$771$	−7.07107	−0.254658
$772$	0	0
$773$	33.5563	1.20694	0.603469	−	0.797386i	$-0.293785\pi$
$773$	33.5563	1.20694	0.603469	+	0.797386i	$0.293785\pi$
$774$	0	0
$775$	5.45584	0.195980
$776$	0	0
$777$	−11.3137	−0.405877
$778$	0	0
$779$	17.6569	0.632622
$780$	0	0
$781$	−24.9706	−0.893517
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	8.76955	0.312999
$786$	0	0
$787$	−44.0416	−1.56991	−0.784957	−	0.619550i	$-0.787315\pi$
$787$	−44.0416	−1.56991	−0.784957	+	0.619550i	$0.787315\pi$
$788$	0	0
$789$	11.3137	0.402779
$790$	0	0
$791$	−0.686292	−0.0244017
$792$	0	0
$793$	40.0000	1.42044
$794$	0	0
$795$	−1.31371	−0.0465924
$796$	0	0
$797$	5.75736	0.203936	0.101968	−	0.994788i	$-0.467486\pi$
$797$	5.75736	0.203936	0.101968	+	0.994788i	$0.467486\pi$
$798$	0	0
$799$	−18.5442	−0.656045
$800$	0	0
$801$	−13.3137	−0.470417
$802$	0	0
$803$	−24.2843	−0.856973
$804$	0	0
$805$	−9.37258	−0.330340
$806$	0	0
$807$	1.75736	0.0618620
$808$	0	0
$809$	32.1421	1.13006	0.565029	−	0.825071i	$-0.308865\pi$
$809$	32.1421	1.13006	0.565029	+	0.825071i	$0.308865\pi$
$810$	0	0
$811$	−17.6152	−0.618554	−0.309277	−	0.950972i	$-0.600087\pi$
$811$	−17.6152	−0.618554	−0.309277	+	0.950972i	$0.600087\pi$
$812$	0	0
$813$	17.5563	0.615728
$814$	0	0
$815$	3.45584	0.121053
$816$	0	0
$817$	−56.2843	−1.96914
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−9.55635	−0.333519	−0.166759	−	0.985998i	$-0.553330\pi$
$821$	−9.55635	−0.333519	−0.166759	+	0.985998i	$0.553330\pi$
$822$	0	0
$823$	8.38478	0.292275	0.146137	−	0.989264i	$-0.453316\pi$
$823$	8.38478	0.292275	0.146137	+	0.989264i	$0.453316\pi$
$824$	0	0
$825$	9.31371	0.324262
$826$	0	0
$827$	−2.48528	−0.0864217	−0.0432109	−	0.999066i	$-0.513759\pi$
$827$	−2.48528	−0.0864217	−0.0432109	+	0.999066i	$0.513759\pi$
$828$	0	0
$829$	29.1716	1.01317	0.506585	−	0.862190i	$-0.330908\pi$
$829$	29.1716	1.01317	0.506585	+	0.862190i	$0.330908\pi$
$830$	0	0
$831$	6.82843	0.236876
$832$	0	0
$833$	−2.58579	−0.0895922
$834$	0	0
$835$	−0.585786	−0.0202720
$836$	0	0
$837$	−1.17157	−0.0404955
$838$	0	0
$839$	48.4264	1.67187	0.835933	−	0.548832i	$-0.184927\pi$
$839$	48.4264	1.67187	0.835933	+	0.548832i	$0.184927\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	1.31371	0.0452465
$844$	0	0
$845$	2.92893	0.100758
$846$	0	0
$847$	−19.7990	−0.680301
$848$	0	0
$849$	26.1421	0.897196
$850$	0	0
$851$	−22.6274	−0.775658
$852$	0	0
$853$	57.4558	1.96725	0.983625	−	0.180226i	$-0.0576828\pi$
$853$	57.4558	1.96725	0.983625	+	0.180226i	$0.0576828\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	15.4558	0.527962	0.263981	−	0.964528i	$-0.414964\pi$
$857$	15.4558	0.527962	0.263981	+	0.964528i	$0.414964\pi$
$858$	0	0
$859$	−45.2548	−1.54408	−0.772038	−	0.635577i	$-0.780762\pi$
$859$	−45.2548	−1.54408	−0.772038	+	0.635577i	$0.780762\pi$
$860$	0	0
$861$	−7.31371	−0.249251
$862$	0	0
$863$	−45.2548	−1.54049	−0.770246	−	0.637747i	$-0.779867\pi$
$863$	−45.2548	−1.54049	−0.770246	+	0.637747i	$0.779867\pi$
$864$	0	0
$865$	−11.7990	−0.401178
$866$	0	0
$867$	−10.3137	−0.350272
$868$	0	0
$869$	−20.4853	−0.694916
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−17.3137	−0.585980
$874$	0	0
$875$	16.0000	0.540899
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	32.7279	1.10263	0.551316	−	0.834297i	$-0.314126\pi$
$881$	32.7279	1.10263	0.551316	+	0.834297i	$0.314126\pi$
$882$	0	0
$883$	11.1127	0.373972	0.186986	−	0.982363i	$-0.440128\pi$
$883$	11.1127	0.373972	0.186986	+	0.982363i	$0.440128\pi$
$884$	0	0
$885$	−7.51472	−0.252605
$886$	0	0
$887$	20.7696	0.697373	0.348687	−	0.937239i	$-0.386628\pi$
$887$	20.7696	0.697373	0.348687	+	0.937239i	$0.386628\pi$
$888$	0	0
$889$	−11.3137	−0.379450
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−48.9706	−1.63874
$894$	0	0
$895$	13.6569	0.456498
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	−5.79899	−0.193192
$902$	0	0
$903$	23.3137	0.775832
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−56.9706	−1.89168	−0.945838	−	0.324638i	$-0.894757\pi$
$907$	−56.9706	−1.89168	−0.945838	+	0.324638i	$0.894757\pi$
$908$	0	0
$909$	1.07107	0.0355251
$910$	0	0
$911$	12.6863	0.420316	0.210158	−	0.977667i	$-0.432602\pi$
$911$	12.6863	0.420316	0.210158	+	0.977667i	$0.432602\pi$
$912$	0	0
$913$	−23.3137	−0.771571
$914$	0	0
$915$	−8.28427	−0.273870
$916$	0	0
$917$	55.5980	1.83601
$918$	0	0
$919$	−31.7990	−1.04895	−0.524476	−	0.851425i	$-0.675739\pi$
$919$	−31.7990	−1.04895	−0.524476	+	0.851425i	$0.675739\pi$
$920$	0	0
$921$	31.5563	1.03982
$922$	0	0
$923$	35.3137	1.16236
$924$	0	0
$925$	18.6274	0.612466
$926$	0	0
$927$	−9.07107	−0.297933
$928$	0	0
$929$	−14.2010	−0.465920	−0.232960	−	0.972486i	$-0.574841\pi$
$929$	−14.2010	−0.465920	−0.232960	+	0.972486i	$0.574841\pi$
$930$	0	0
$931$	−6.82843	−0.223793
$932$	0	0
$933$	−27.3137	−0.894211
$934$	0	0
$935$	−3.02944	−0.0990732
$936$	0	0
$937$	6.97056	0.227718	0.113859	−	0.993497i	$-0.463679\pi$
$937$	6.97056	0.227718	0.113859	+	0.993497i	$0.463679\pi$
$938$	0	0
$939$	6.97056	0.227476
$940$	0	0
$941$	21.0711	0.686897	0.343449	−	0.939171i	$-0.388405\pi$
$941$	21.0711	0.686897	0.343449	+	0.939171i	$0.388405\pi$
$942$	0	0
$943$	−14.6274	−0.476334
$944$	0	0
$945$	−1.65685	−0.0538975
$946$	0	0
$947$	−13.0294	−0.423400	−0.211700	−	0.977335i	$-0.567900\pi$
$947$	−13.0294	−0.423400	−0.211700	+	0.977335i	$0.567900\pi$
$948$	0	0
$949$	34.3431	1.11483
$950$	0	0
$951$	−13.5147	−0.438245
$952$	0	0
$953$	11.5563	0.374347	0.187173	−	0.982327i	$-0.440067\pi$
$953$	11.5563	0.374347	0.187173	+	0.982327i	$0.440067\pi$
$954$	0	0
$955$	−7.11270	−0.230162
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	42.9117	1.38569
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	−4.08326	−0.131445
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	0	0
$969$	17.6569	0.567220
$970$	0	0
$971$	21.0294	0.674867	0.337433	−	0.941349i	$-0.390441\pi$
$971$	21.0294	0.674867	0.337433	+	0.941349i	$0.390441\pi$
$972$	0	0
$973$	16.6863	0.534938
$974$	0	0
$975$	−13.1716	−0.421828
$976$	0	0
$977$	24.5269	0.784685	0.392343	−	0.919819i	$-0.371665\pi$
$977$	24.5269	0.784685	0.392343	+	0.919819i	$0.371665\pi$
$978$	0	0
$979$	26.6274	0.851016
$980$	0	0
$981$	4.48528	0.143204
$982$	0	0
$983$	−37.4558	−1.19466	−0.597328	−	0.801997i	$-0.703771\pi$
$983$	−37.4558	−1.19466	−0.597328	+	0.801997i	$0.703771\pi$
$984$	0	0
$985$	−10.2843	−0.327684
$986$	0	0
$987$	20.2843	0.645655
$988$	0	0
$989$	46.6274	1.48267
$990$	0	0
$991$	27.8995	0.886257	0.443128	−	0.896458i	$-0.353869\pi$
$991$	27.8995	0.886257	0.443128	+	0.896458i	$0.353869\pi$
$992$	0	0
$993$	13.4142	0.425687
$994$	0	0
$995$	10.3431	0.327900
$996$	0	0
$997$	−47.1127	−1.49207	−0.746037	−	0.665904i	$-0.768046\pi$
$997$	−47.1127	−1.49207	−0.746037	+	0.665904i	$0.768046\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.k.1.2		2
4.3	odd	2		1002.2.a.f.1.2	✓	2
12.11	even	2		3006.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.f.1.2	✓	2	4.3	odd	2
3006.2.a.n.1.1		2	12.11	even	2
8016.2.a.k.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.585786 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.585786 q^{5} +2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{13} -0.585786 q^{15} -2.58579 q^{17} -6.82843 q^{19} +2.82843 q^{21} +5.65685 q^{23} -4.65685 q^{25} +1.00000 q^{27} +2.00000 q^{29} -1.17157 q^{31} -2.00000 q^{33} -1.65685 q^{35} -4.00000 q^{37} +2.82843 q^{39} -2.58579 q^{41} +8.24264 q^{43} -0.585786 q^{45} +7.17157 q^{47} +1.00000 q^{49} -2.58579 q^{51} +2.24264 q^{53} +1.17157 q^{55} -6.82843 q^{57} +12.8284 q^{59} +14.1421 q^{61} +2.82843 q^{63} -1.65685 q^{65} -1.41421 q^{67} +5.65685 q^{69} +12.4853 q^{71} +12.1421 q^{73} -4.65685 q^{75} -5.65685 q^{77} +10.2426 q^{79} +1.00000 q^{81} +11.6569 q^{83} +1.51472 q^{85} +2.00000 q^{87} -13.3137 q^{89} +8.00000 q^{91} -1.17157 q^{93} +4.00000 q^{95} -17.3137 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{15} - 8 q^{17} - 8 q^{19} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 8 q^{31} - 4 q^{33} + 8 q^{35} - 8 q^{37} - 8 q^{41} + 8 q^{43} - 4 q^{45} + 20 q^{47} + 2 q^{49} - 8 q^{51} - 4 q^{53} + 8 q^{55} - 8 q^{57} + 20 q^{59} + 8 q^{65} + 8 q^{71} - 4 q^{73} + 2 q^{75} + 12 q^{79} + 2 q^{81} + 12 q^{83} + 20 q^{85} + 4 q^{87} - 4 q^{89} + 16 q^{91} - 8 q^{93} + 8 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 - 4 * q^11 - 4 * q^15 - 8 * q^17 - 8 * q^19 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 8 * q^31 - 4 * q^33 + 8 * q^35 - 8 * q^37 - 8 * q^41 + 8 * q^43 - 4 * q^45 + 20 * q^47 + 2 * q^49 - 8 * q^51 - 4 * q^53 + 8 * q^55 - 8 * q^57 + 20 * q^59 + 8 * q^65 + 8 * q^71 - 4 * q^73 + 2 * q^75 + 12 * q^79 + 2 * q^81 + 12 * q^83 + 20 * q^85 + 4 * q^87 - 4 * q^89 + 16 * q^91 - 8 * q^93 + 8 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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