LMFDB - Embedded newform 8016.2.a.k.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -3.41421 q^{15} -5.41421 q^{17} -1.17157 q^{19} -2.82843 q^{21} -5.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +2.00000 q^{29} -6.82843 q^{31} -2.00000 q^{33} +9.65685 q^{35} -4.00000 q^{37} -2.82843 q^{39} -5.41421 q^{41} -0.242641 q^{43} -3.41421 q^{45} +12.8284 q^{47} +1.00000 q^{49} -5.41421 q^{51} -6.24264 q^{53} +6.82843 q^{55} -1.17157 q^{57} +7.17157 q^{59} -14.1421 q^{61} -2.82843 q^{63} +9.65685 q^{65} +1.41421 q^{67} -5.65685 q^{69} -4.48528 q^{71} -16.1421 q^{73} +6.65685 q^{75} +5.65685 q^{77} +1.75736 q^{79} +1.00000 q^{81} +0.343146 q^{83} +18.4853 q^{85} +2.00000 q^{87} +9.31371 q^{89} +8.00000 q^{91} -6.82843 q^{93} +4.00000 q^{95} +5.31371 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	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	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	−5.41421	−1.31314	−0.656570	−	0.754265i	$-0.727993\pi$
$17$	−5.41421	−1.31314	−0.656570	+	0.754265i	$0.727993\pi$
$18$	0	0
$19$	−1.17157	−0.268777	−0.134389	−	0.990929i	$-0.542907\pi$
$19$	−1.17157	−0.268777	−0.134389	+	0.990929i	$0.542907\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	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
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	9.65685	1.63231
$36$	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
$41$	−5.41421	−0.845558	−0.422779	−	0.906233i	$-0.638945\pi$
$41$	−5.41421	−0.845558	−0.422779	+	0.906233i	$0.638945\pi$
$42$	0	0
$43$	−0.242641	−0.0370024	−0.0185012	−	0.999829i	$-0.505889\pi$
$43$	−0.242641	−0.0370024	−0.0185012	+	0.999829i	$0.505889\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	0	0
$47$	12.8284	1.87122	0.935609	−	0.353037i	$-0.114851\pi$
$47$	12.8284	1.87122	0.935609	+	0.353037i	$0.114851\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.41421	−0.758142
$52$	0	0
$53$	−6.24264	−0.857493	−0.428746	−	0.903425i	$-0.641045\pi$
$53$	−6.24264	−0.857493	−0.428746	+	0.903425i	$0.641045\pi$
$54$	0	0
$55$	6.82843	0.920745
$56$	0	0
$57$	−1.17157	−0.155179
$58$	0	0
$59$	7.17157	0.933659	0.466830	−	0.884347i	$-0.345396\pi$
$59$	7.17157	0.933659	0.466830	+	0.884347i	$0.345396\pi$
$60$	0	0
$61$	−14.1421	−1.81071	−0.905357	−	0.424650i	$-0.860397\pi$
$61$	−14.1421	−1.81071	−0.905357	+	0.424650i	$0.860397\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	9.65685	1.19779
$66$	0	0
$67$	1.41421	0.172774	0.0863868	−	0.996262i	$-0.472468\pi$
$67$	1.41421	0.172774	0.0863868	+	0.996262i	$0.472468\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	0	0
$73$	−16.1421	−1.88929	−0.944647	−	0.328088i	$-0.893596\pi$
$73$	−16.1421	−1.88929	−0.944647	+	0.328088i	$0.893596\pi$
$74$	0	0
$75$	6.65685	0.768667
$76$	0	0
$77$	5.65685	0.644658
$78$	0	0
$79$	1.75736	0.197718	0.0988592	−	0.995101i	$-0.468481\pi$
$79$	1.75736	0.197718	0.0988592	+	0.995101i	$0.468481\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.343146	0.0376651	0.0188326	−	0.999823i	$-0.494005\pi$
$83$	0.343146	0.0376651	0.0188326	+	0.999823i	$0.494005\pi$
$84$	0	0
$85$	18.4853	2.00501
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	9.31371	0.987251	0.493626	−	0.869675i	$-0.335671\pi$
$89$	9.31371	0.987251	0.493626	+	0.869675i	$0.335671\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−6.82843	−0.708075
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	5.31371	0.539525	0.269763	−	0.962927i	$-0.413055\pi$
$97$	5.31371	0.539525	0.269763	+	0.962927i	$0.413055\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−13.0711	−1.30062	−0.650310	−	0.759669i	$-0.725361\pi$
$101$	−13.0711	−1.30062	−0.650310	+	0.759669i	$0.725361\pi$
$102$	0	0
$103$	5.07107	0.499667	0.249834	−	0.968289i	$-0.419624\pi$
$103$	5.07107	0.499667	0.249834	+	0.968289i	$0.419624\pi$
$104$	0	0
$105$	9.65685	0.942412
$106$	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
$109$	−12.4853	−1.19587	−0.597937	−	0.801543i	$-0.704013\pi$
$109$	−12.4853	−1.19587	−0.597937	+	0.801543i	$0.704013\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	8.24264	0.775402	0.387701	−	0.921785i	$-0.373269\pi$
$113$	8.24264	0.775402	0.387701	+	0.921785i	$0.373269\pi$
$114$	0	0
$115$	19.3137	1.80101
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	15.3137	1.40381
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−5.41421	−0.488183
$124$	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
$129$	−0.242641	−0.0213633
$130$	0	0
$131$	8.34315	0.728944	0.364472	−	0.931214i	$-0.381249\pi$
$131$	8.34315	0.728944	0.364472	+	0.931214i	$0.381249\pi$
$132$	0	0
$133$	3.31371	0.287335
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	20.8284	1.77949	0.889746	−	0.456455i	$-0.150881\pi$
$137$	20.8284	1.77949	0.889746	+	0.456455i	$0.150881\pi$
$138$	0	0
$139$	−13.8995	−1.17894	−0.589470	−	0.807790i	$-0.700663\pi$
$139$	−13.8995	−1.17894	−0.589470	+	0.807790i	$0.700663\pi$
$140$	0	0
$141$	12.8284	1.08035
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−6.82843	−0.567070
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	6.24264	0.511417	0.255709	−	0.966754i	$-0.417691\pi$
$149$	6.24264	0.511417	0.255709	+	0.966754i	$0.417691\pi$
$150$	0	0
$151$	5.75736	0.468527	0.234264	−	0.972173i	$-0.424732\pi$
$151$	5.75736	0.468527	0.234264	+	0.972173i	$0.424732\pi$
$152$	0	0
$153$	−5.41421	−0.437713
$154$	0	0
$155$	23.3137	1.87260
$156$	0	0
$157$	18.9706	1.51402	0.757008	−	0.653406i	$-0.226660\pi$
$157$	18.9706	1.51402	0.757008	+	0.653406i	$0.226660\pi$
$158$	0	0
$159$	−6.24264	−0.495074
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	13.8995	1.08869	0.544346	−	0.838861i	$-0.316778\pi$
$163$	13.8995	1.08869	0.544346	+	0.838861i	$0.316778\pi$
$164$	0	0
$165$	6.82843	0.531592
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−1.17157	−0.0895924
$172$	0	0
$173$	−8.14214	−0.619035	−0.309518	−	0.950894i	$-0.600168\pi$
$173$	−8.14214	−0.619035	−0.309518	+	0.950894i	$0.600168\pi$
$174$	0	0
$175$	−18.8284	−1.42330
$176$	0	0
$177$	7.17157	0.539048
$178$	0	0
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	1.17157	0.0870823	0.0435412	−	0.999052i	$-0.486136\pi$
$181$	1.17157	0.0870823	0.0435412	+	0.999052i	$0.486136\pi$
$182$	0	0
$183$	−14.1421	−1.04542
$184$	0	0
$185$	13.6569	1.00407
$186$	0	0
$187$	10.8284	0.791853
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−16.1421	−1.16800	−0.584002	−	0.811752i	$-0.698514\pi$
$191$	−16.1421	−1.16800	−0.584002	+	0.811752i	$0.698514\pi$
$192$	0	0
$193$	−26.9706	−1.94138	−0.970692	−	0.240328i	$-0.922745\pi$
$193$	−26.9706	−1.94138	−0.970692	+	0.240328i	$0.922745\pi$
$194$	0	0
$195$	9.65685	0.691542
$196$	0	0
$197$	−13.5563	−0.965850	−0.482925	−	0.875662i	$-0.660426\pi$
$197$	−13.5563	−0.965850	−0.482925	+	0.875662i	$0.660426\pi$
$198$	0	0
$199$	−6.34315	−0.449654	−0.224827	−	0.974399i	$-0.572182\pi$
$199$	−6.34315	−0.449654	−0.224827	+	0.974399i	$0.572182\pi$
$200$	0	0
$201$	1.41421	0.0997509
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	18.4853	1.29107
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	2.34315	0.162079
$210$	0	0
$211$	14.1421	0.973585	0.486792	−	0.873518i	$-0.338167\pi$
$211$	14.1421	0.973585	0.486792	+	0.873518i	$0.338167\pi$
$212$	0	0
$213$	−4.48528	−0.307326
$214$	0	0
$215$	0.828427	0.0564983
$216$	0	0
$217$	19.3137	1.31110
$218$	0	0
$219$	−16.1421	−1.09078
$220$	0	0
$221$	15.3137	1.03011
$222$	0	0
$223$	14.1421	0.947027	0.473514	−	0.880786i	$-0.342985\pi$
$223$	14.1421	0.947027	0.473514	+	0.880786i	$0.342985\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	−4.82843	−0.320474	−0.160237	−	0.987079i	$-0.551226\pi$
$227$	−4.82843	−0.320474	−0.160237	+	0.987079i	$0.551226\pi$
$228$	0	0
$229$	26.1421	1.72752	0.863760	−	0.503903i	$-0.168103\pi$
$229$	26.1421	1.72752	0.863760	+	0.503903i	$0.168103\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	−21.3137	−1.39631	−0.698154	−	0.715948i	$-0.745995\pi$
$233$	−21.3137	−1.39631	−0.698154	+	0.715948i	$0.745995\pi$
$234$	0	0
$235$	−43.7990	−2.85713
$236$	0	0
$237$	1.75736	0.114153
$238$	0	0
$239$	−9.51472	−0.615456	−0.307728	−	0.951474i	$-0.599569\pi$
$239$	−9.51472	−0.615456	−0.307728	+	0.951474i	$0.599569\pi$
$240$	0	0
$241$	25.7990	1.66186	0.830930	−	0.556378i	$-0.187809\pi$
$241$	25.7990	1.66186	0.830930	+	0.556378i	$0.187809\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.41421	−0.218126
$246$	0	0
$247$	3.31371	0.210846
$248$	0	0
$249$	0.343146	0.0217460
$250$	0	0
$251$	21.6569	1.36697	0.683484	−	0.729965i	$-0.260464\pi$
$251$	21.6569	1.36697	0.683484	+	0.729965i	$0.260464\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	18.4853	1.15759
$256$	0	0
$257$	7.07107	0.441081	0.220541	−	0.975378i	$-0.429218\pi$
$257$	7.07107	0.441081	0.220541	+	0.975378i	$0.429218\pi$
$258$	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	$-0.613416\pi$
$263$	−11.3137	−0.697633	−0.348817	+	0.937191i	$0.613416\pi$
$264$	0	0
$265$	21.3137	1.30929
$266$	0	0
$267$	9.31371	0.569990
$268$	0	0
$269$	10.2426	0.624505	0.312252	−	0.949999i	$-0.398917\pi$
$269$	10.2426	0.624505	0.312252	+	0.949999i	$0.398917\pi$
$270$	0	0
$271$	−13.5563	−0.823490	−0.411745	−	0.911299i	$-0.635080\pi$
$271$	−13.5563	−0.823490	−0.411745	+	0.911299i	$0.635080\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	−13.3137	−0.802847
$276$	0	0
$277$	1.17157	0.0703930	0.0351965	−	0.999380i	$-0.488794\pi$
$277$	1.17157	0.0703930	0.0351965	+	0.999380i	$0.488794\pi$
$278$	0	0
$279$	−6.82843	−0.408807
$280$	0	0
$281$	−21.3137	−1.27147	−0.635735	−	0.771908i	$-0.719303\pi$
$281$	−21.3137	−1.27147	−0.635735	+	0.771908i	$0.719303\pi$
$282$	0	0
$283$	−2.14214	−0.127337	−0.0636684	−	0.997971i	$-0.520280\pi$
$283$	−2.14214	−0.127337	−0.0636684	+	0.997971i	$0.520280\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	15.3137	0.903940
$288$	0	0
$289$	12.3137	0.724336
$290$	0	0
$291$	5.31371	0.311495
$292$	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
$295$	−24.4853	−1.42559
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	0.686292	0.0395572
$302$	0	0
$303$	−13.0711	−0.750913
$304$	0	0
$305$	48.2843	2.76475
$306$	0	0
$307$	0.443651	0.0253205	0.0126602	−	0.999920i	$-0.495970\pi$
$307$	0.443651	0.0253205	0.0126602	+	0.999920i	$0.495970\pi$
$308$	0	0
$309$	5.07107	0.288483
$310$	0	0
$311$	−4.68629	−0.265735	−0.132868	−	0.991134i	$-0.542419\pi$
$311$	−4.68629	−0.265735	−0.132868	+	0.991134i	$0.542419\pi$
$312$	0	0
$313$	−26.9706	−1.52447	−0.762233	−	0.647303i	$-0.775897\pi$
$313$	−26.9706	−1.52447	−0.762233	+	0.647303i	$0.775897\pi$
$314$	0	0
$315$	9.65685	0.544102
$316$	0	0
$317$	−30.4853	−1.71222	−0.856112	−	0.516790i	$-0.827127\pi$
$317$	−30.4853	−1.71222	−0.856112	+	0.516790i	$0.827127\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	6.34315	0.352942
$324$	0	0
$325$	−18.8284	−1.04441
$326$	0	0
$327$	−12.4853	−0.690438
$328$	0	0
$329$	−36.2843	−2.00042
$330$	0	0
$331$	10.5858	0.581847	0.290924	−	0.956746i	$-0.406037\pi$
$331$	10.5858	0.581847	0.290924	+	0.956746i	$0.406037\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−4.82843	−0.263805
$336$	0	0
$337$	0.343146	0.0186923	0.00934617	−	0.999956i	$-0.497025\pi$
$337$	0.343146	0.0186923	0.00934617	+	0.999956i	$0.497025\pi$
$338$	0	0
$339$	8.24264	0.447679
$340$	0	0
$341$	13.6569	0.739560
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	19.3137	1.03982
$346$	0	0
$347$	−0.828427	−0.0444723	−0.0222361	−	0.999753i	$-0.507079\pi$
$347$	−0.828427	−0.0444723	−0.0222361	+	0.999753i	$0.507079\pi$
$348$	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
$353$	25.3137	1.34731	0.673656	−	0.739045i	$-0.264723\pi$
$353$	25.3137	1.34731	0.673656	+	0.739045i	$0.264723\pi$
$354$	0	0
$355$	15.3137	0.812767
$356$	0	0
$357$	15.3137	0.810487
$358$	0	0
$359$	−29.6569	−1.56523	−0.782614	−	0.622507i	$-0.786114\pi$
$359$	−29.6569	−1.56523	−0.782614	+	0.622507i	$0.786114\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	55.1127	2.88473
$366$	0	0
$367$	9.17157	0.478752	0.239376	−	0.970927i	$-0.423057\pi$
$367$	9.17157	0.478752	0.239376	+	0.970927i	$0.423057\pi$
$368$	0	0
$369$	−5.41421	−0.281853
$370$	0	0
$371$	17.6569	0.916698
$372$	0	0
$373$	0.970563	0.0502538	0.0251269	−	0.999684i	$-0.492001\pi$
$373$	0.970563	0.0502538	0.0251269	+	0.999684i	$0.492001\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−5.65685	−0.291343
$378$	0	0
$379$	−31.5563	−1.62094	−0.810470	−	0.585780i	$-0.800788\pi$
$379$	−31.5563	−1.62094	−0.810470	+	0.585780i	$0.800788\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	20.8284	1.06428	0.532141	−	0.846655i	$-0.321387\pi$
$383$	20.8284	1.06428	0.532141	+	0.846655i	$0.321387\pi$
$384$	0	0
$385$	−19.3137	−0.984318
$386$	0	0
$387$	−0.242641	−0.0123341
$388$	0	0
$389$	29.5563	1.49857	0.749283	−	0.662250i	$-0.230398\pi$
$389$	29.5563	1.49857	0.749283	+	0.662250i	$0.230398\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	8.34315	0.420856
$394$	0	0
$395$	−6.00000	−0.301893
$396$	0	0
$397$	7.51472	0.377153	0.188576	−	0.982059i	$-0.439613\pi$
$397$	7.51472	0.377153	0.188576	+	0.982059i	$0.439613\pi$
$398$	0	0
$399$	3.31371	0.165893
$400$	0	0
$401$	6.58579	0.328878	0.164439	−	0.986387i	$-0.447419\pi$
$401$	6.58579	0.328878	0.164439	+	0.986387i	$0.447419\pi$
$402$	0	0
$403$	19.3137	0.962084
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	11.3137	0.559427	0.279713	−	0.960084i	$-0.409761\pi$
$409$	11.3137	0.559427	0.279713	+	0.960084i	$0.409761\pi$
$410$	0	0
$411$	20.8284	1.02739
$412$	0	0
$413$	−20.2843	−0.998124
$414$	0	0
$415$	−1.17157	−0.0575103
$416$	0	0
$417$	−13.8995	−0.680661
$418$	0	0
$419$	39.9411	1.95125	0.975626	−	0.219441i	$-0.0704233\pi$
$419$	39.9411	1.95125	0.975626	+	0.219441i	$0.0704233\pi$
$420$	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
$423$	12.8284	0.623739
$424$	0	0
$425$	−36.0416	−1.74828
$426$	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	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	0	0
$433$	−11.3137	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$433$	−11.3137	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$434$	0	0
$435$	−6.82843	−0.327398
$436$	0	0
$437$	6.62742	0.317032
$438$	0	0
$439$	5.07107	0.242029	0.121014	−	0.992651i	$-0.461385\pi$
$439$	5.07107	0.242029	0.121014	+	0.992651i	$0.461385\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	30.4853	1.44840	0.724200	−	0.689590i	$-0.242209\pi$
$443$	30.4853	1.44840	0.724200	+	0.689590i	$0.242209\pi$
$444$	0	0
$445$	−31.7990	−1.50742
$446$	0	0
$447$	6.24264	0.295267
$448$	0	0
$449$	32.8284	1.54927	0.774635	−	0.632409i	$-0.217934\pi$
$449$	32.8284	1.54927	0.774635	+	0.632409i	$0.217934\pi$
$450$	0	0
$451$	10.8284	0.509891
$452$	0	0
$453$	5.75736	0.270504
$454$	0	0
$455$	−27.3137	−1.28049
$456$	0	0
$457$	−31.4558	−1.47144	−0.735721	−	0.677285i	$-0.763157\pi$
$457$	−31.4558	−1.47144	−0.735721	+	0.677285i	$0.763157\pi$
$458$	0	0
$459$	−5.41421	−0.252714
$460$	0	0
$461$	1.31371	0.0611855	0.0305928	−	0.999532i	$-0.490261\pi$
$461$	1.31371	0.0611855	0.0305928	+	0.999532i	$0.490261\pi$
$462$	0	0
$463$	−5.75736	−0.267567	−0.133784	−	0.991011i	$-0.542713\pi$
$463$	−5.75736	−0.267567	−0.133784	+	0.991011i	$0.542713\pi$
$464$	0	0
$465$	23.3137	1.08115
$466$	0	0
$467$	15.3137	0.708634	0.354317	−	0.935125i	$-0.384713\pi$
$467$	15.3137	0.708634	0.354317	+	0.935125i	$0.384713\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	18.9706	0.874117
$472$	0	0
$473$	0.485281	0.0223133
$474$	0	0
$475$	−7.79899	−0.357842
$476$	0	0
$477$	−6.24264	−0.285831
$478$	0	0
$479$	15.5147	0.708886	0.354443	−	0.935078i	$-0.384671\pi$
$479$	15.5147	0.708886	0.354443	+	0.935078i	$0.384671\pi$
$480$	0	0
$481$	11.3137	0.515861
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	−18.1421	−0.823792
$486$	0	0
$487$	−23.4142	−1.06100	−0.530500	−	0.847685i	$-0.677996\pi$
$487$	−23.4142	−1.06100	−0.530500	+	0.847685i	$0.677996\pi$
$488$	0	0
$489$	13.8995	0.628557
$490$	0	0
$491$	2.97056	0.134060	0.0670298	−	0.997751i	$-0.478648\pi$
$491$	2.97056	0.134060	0.0670298	+	0.997751i	$0.478648\pi$
$492$	0	0
$493$	−10.8284	−0.487688
$494$	0	0
$495$	6.82843	0.306915
$496$	0	0
$497$	12.6863	0.569058
$498$	0	0
$499$	40.7279	1.82323	0.911616	−	0.411043i	$-0.134835\pi$
$499$	40.7279	1.82323	0.911616	+	0.411043i	$0.134835\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−33.1127	−1.47642	−0.738211	−	0.674570i	$-0.764329\pi$
$503$	−33.1127	−1.47642	−0.738211	+	0.674570i	$0.764329\pi$
$504$	0	0
$505$	44.6274	1.98589
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	9.31371	0.412823	0.206411	−	0.978465i	$-0.433821\pi$
$509$	9.31371	0.412823	0.206411	+	0.978465i	$0.433821\pi$
$510$	0	0
$511$	45.6569	2.01974
$512$	0	0
$513$	−1.17157	−0.0517262
$514$	0	0
$515$	−17.3137	−0.762933
$516$	0	0
$517$	−25.6569	−1.12839
$518$	0	0
$519$	−8.14214	−0.357400
$520$	0	0
$521$	8.04163	0.352310	0.176155	−	0.984362i	$-0.443634\pi$
$521$	8.04163	0.352310	0.176155	+	0.984362i	$0.443634\pi$
$522$	0	0
$523$	−9.45584	−0.413475	−0.206738	−	0.978396i	$-0.566285\pi$
$523$	−9.45584	−0.413475	−0.206738	+	0.978396i	$0.566285\pi$
$524$	0	0
$525$	−18.8284	−0.821740
$526$	0	0
$527$	36.9706	1.61046
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	7.17157	0.311220
$532$	0	0
$533$	15.3137	0.663310
$534$	0	0
$535$	−34.1421	−1.47609
$536$	0	0
$537$	−0.686292	−0.0296157
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−18.8284	−0.809497	−0.404749	−	0.914428i	$-0.632641\pi$
$541$	−18.8284	−0.809497	−0.404749	+	0.914428i	$0.632641\pi$
$542$	0	0
$543$	1.17157	0.0502770
$544$	0	0
$545$	42.6274	1.82596
$546$	0	0
$547$	−28.5269	−1.21972	−0.609861	−	0.792508i	$-0.708775\pi$
$547$	−28.5269	−1.21972	−0.609861	+	0.792508i	$0.708775\pi$
$548$	0	0
$549$	−14.1421	−0.603572
$550$	0	0
$551$	−2.34315	−0.0998214
$552$	0	0
$553$	−4.97056	−0.211370
$554$	0	0
$555$	13.6569	0.579701
$556$	0	0
$557$	−15.4558	−0.654885	−0.327443	−	0.944871i	$-0.606187\pi$
$557$	−15.4558	−0.654885	−0.327443	+	0.944871i	$0.606187\pi$
$558$	0	0
$559$	0.686292	0.0290270
$560$	0	0
$561$	10.8284	0.457177
$562$	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
$565$	−28.1421	−1.18395
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	38.3848	1.60917	0.804587	−	0.593835i	$-0.202387\pi$
$569$	38.3848	1.60917	0.804587	+	0.593835i	$0.202387\pi$
$570$	0	0
$571$	1.21320	0.0507710	0.0253855	−	0.999678i	$-0.491919\pi$
$571$	1.21320	0.0507710	0.0253855	+	0.999678i	$0.491919\pi$
$572$	0	0
$573$	−16.1421	−0.674347
$574$	0	0
$575$	−37.6569	−1.57040
$576$	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
$579$	−26.9706	−1.12086
$580$	0	0
$581$	−0.970563	−0.0402657
$582$	0	0
$583$	12.4853	0.517088
$584$	0	0
$585$	9.65685	0.399262
$586$	0	0
$587$	−26.4853	−1.09316	−0.546582	−	0.837405i	$-0.684071\pi$
$587$	−26.4853	−1.09316	−0.546582	+	0.837405i	$0.684071\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−13.5563	−0.557634
$592$	0	0
$593$	9.21320	0.378341	0.189170	−	0.981944i	$-0.439420\pi$
$593$	9.21320	0.378341	0.189170	+	0.981944i	$0.439420\pi$
$594$	0	0
$595$	−52.2843	−2.14345
$596$	0	0
$597$	−6.34315	−0.259608
$598$	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
$601$	12.6274	0.515083	0.257542	−	0.966267i	$-0.417088\pi$
$601$	12.6274	0.515083	0.257542	+	0.966267i	$0.417088\pi$
$602$	0	0
$603$	1.41421	0.0575912
$604$	0	0
$605$	23.8995	0.971653
$606$	0	0
$607$	−26.0416	−1.05700	−0.528499	−	0.848934i	$-0.677245\pi$
$607$	−26.0416	−1.05700	−0.528499	+	0.848934i	$0.677245\pi$
$608$	0	0
$609$	−5.65685	−0.229227
$610$	0	0
$611$	−36.2843	−1.46790
$612$	0	0
$613$	−0.485281	−0.0196003	−0.00980017	−	0.999952i	$-0.503120\pi$
$613$	−0.485281	−0.0196003	−0.00980017	+	0.999952i	$0.503120\pi$
$614$	0	0
$615$	18.4853	0.745398
$616$	0	0
$617$	−33.1127	−1.33307	−0.666534	−	0.745475i	$-0.732223\pi$
$617$	−33.1127	−1.33307	−0.666534	+	0.745475i	$0.732223\pi$
$618$	0	0
$619$	−2.78680	−0.112011	−0.0560054	−	0.998430i	$-0.517836\pi$
$619$	−2.78680	−0.112011	−0.0560054	+	0.998430i	$0.517836\pi$
$620$	0	0
$621$	−5.65685	−0.227002
$622$	0	0
$623$	−26.3431	−1.05542
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	2.34315	0.0935762
$628$	0	0
$629$	21.6569	0.863515
$630$	0	0
$631$	40.2843	1.60369	0.801846	−	0.597531i	$-0.203852\pi$
$631$	40.2843	1.60369	0.801846	+	0.597531i	$0.203852\pi$
$632$	0	0
$633$	14.1421	0.562099
$634$	0	0
$635$	13.6569	0.541956
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	−4.48528	−0.177435
$640$	0	0
$641$	−28.7279	−1.13468	−0.567342	−	0.823482i	$-0.692028\pi$
$641$	−28.7279	−1.13468	−0.567342	+	0.823482i	$0.692028\pi$
$642$	0	0
$643$	−43.3553	−1.70977	−0.854884	−	0.518819i	$-0.826372\pi$
$643$	−43.3553	−1.70977	−0.854884	+	0.518819i	$0.826372\pi$
$644$	0	0
$645$	0.828427	0.0326193
$646$	0	0
$647$	−7.51472	−0.295434	−0.147717	−	0.989030i	$-0.547192\pi$
$647$	−7.51472	−0.295434	−0.147717	+	0.989030i	$0.547192\pi$
$648$	0	0
$649$	−14.3431	−0.563018
$650$	0	0
$651$	19.3137	0.756964
$652$	0	0
$653$	31.4558	1.23096	0.615481	−	0.788152i	$-0.288962\pi$
$653$	31.4558	1.23096	0.615481	+	0.788152i	$0.288962\pi$
$654$	0	0
$655$	−28.4853	−1.11301
$656$	0	0
$657$	−16.1421	−0.629765
$658$	0	0
$659$	34.4853	1.34336	0.671678	−	0.740843i	$-0.265574\pi$
$659$	34.4853	1.34336	0.671678	+	0.740843i	$0.265574\pi$
$660$	0	0
$661$	6.34315	0.246720	0.123360	−	0.992362i	$-0.460633\pi$
$661$	6.34315	0.246720	0.123360	+	0.992362i	$0.460633\pi$
$662$	0	0
$663$	15.3137	0.594735
$664$	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
$673$	−30.9706	−1.19383	−0.596914	−	0.802305i	$-0.703607\pi$
$673$	−30.9706	−1.19383	−0.596914	+	0.802305i	$0.703607\pi$
$674$	0	0
$675$	6.65685	0.256222
$676$	0	0
$677$	−34.9706	−1.34403	−0.672014	−	0.740538i	$-0.734571\pi$
$677$	−34.9706	−1.34403	−0.672014	+	0.740538i	$0.734571\pi$
$678$	0	0
$679$	−15.0294	−0.576777
$680$	0	0
$681$	−4.82843	−0.185026
$682$	0	0
$683$	−29.3137	−1.12166	−0.560829	−	0.827932i	$-0.689517\pi$
$683$	−29.3137	−1.12166	−0.560829	+	0.827932i	$0.689517\pi$
$684$	0	0
$685$	−71.1127	−2.71708
$686$	0	0
$687$	26.1421	0.997385
$688$	0	0
$689$	17.6569	0.672673
$690$	0	0
$691$	47.0711	1.79067	0.895334	−	0.445396i	$-0.146937\pi$
$691$	47.0711	1.79067	0.895334	+	0.445396i	$0.146937\pi$
$692$	0	0
$693$	5.65685	0.214886
$694$	0	0
$695$	47.4558	1.80010
$696$	0	0
$697$	29.3137	1.11034
$698$	0	0
$699$	−21.3137	−0.806158
$700$	0	0
$701$	23.9411	0.904244	0.452122	−	0.891956i	$-0.350667\pi$
$701$	23.9411	0.904244	0.452122	+	0.891956i	$0.350667\pi$
$702$	0	0
$703$	4.68629	0.176747
$704$	0	0
$705$	−43.7990	−1.64957
$706$	0	0
$707$	36.9706	1.39042
$708$	0	0
$709$	−37.4558	−1.40668	−0.703342	−	0.710852i	$-0.748310\pi$
$709$	−37.4558	−1.40668	−0.703342	+	0.710852i	$0.748310\pi$
$710$	0	0
$711$	1.75736	0.0659061
$712$	0	0
$713$	38.6274	1.44661
$714$	0	0
$715$	−19.3137	−0.722292
$716$	0	0
$717$	−9.51472	−0.355334
$718$	0	0
$719$	−27.7990	−1.03673	−0.518364	−	0.855160i	$-0.673459\pi$
$719$	−27.7990	−1.03673	−0.518364	+	0.855160i	$0.673459\pi$
$720$	0	0
$721$	−14.3431	−0.534167
$722$	0	0
$723$	25.7990	0.959475
$724$	0	0
$725$	13.3137	0.494459
$726$	0	0
$727$	11.8995	0.441328	0.220664	−	0.975350i	$-0.429178\pi$
$727$	11.8995	0.441328	0.220664	+	0.975350i	$0.429178\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.31371	0.0485893
$732$	0	0
$733$	15.7990	0.583549	0.291775	−	0.956487i	$-0.405754\pi$
$733$	15.7990	0.583549	0.291775	+	0.956487i	$0.405754\pi$
$734$	0	0
$735$	−3.41421	−0.125935
$736$	0	0
$737$	−2.82843	−0.104186
$738$	0	0
$739$	−12.2426	−0.450353	−0.225176	−	0.974318i	$-0.572296\pi$
$739$	−12.2426	−0.450353	−0.225176	+	0.974318i	$0.572296\pi$
$740$	0	0
$741$	3.31371	0.121732
$742$	0	0
$743$	37.6569	1.38150	0.690748	−	0.723096i	$-0.257281\pi$
$743$	37.6569	1.38150	0.690748	+	0.723096i	$0.257281\pi$
$744$	0	0
$745$	−21.3137	−0.780874
$746$	0	0
$747$	0.343146	0.0125550
$748$	0	0
$749$	−28.2843	−1.03348
$750$	0	0
$751$	−43.2132	−1.57687	−0.788436	−	0.615117i	$-0.789109\pi$
$751$	−43.2132	−1.57687	−0.788436	+	0.615117i	$0.789109\pi$
$752$	0	0
$753$	21.6569	0.789220
$754$	0	0
$755$	−19.6569	−0.715386
$756$	0	0
$757$	−36.3431	−1.32091	−0.660457	−	0.750864i	$-0.729637\pi$
$757$	−36.3431	−1.32091	−0.660457	+	0.750864i	$0.729637\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	−40.1421	−1.45515	−0.727576	−	0.686027i	$-0.759353\pi$
$761$	−40.1421	−1.45515	−0.727576	+	0.686027i	$0.759353\pi$
$762$	0	0
$763$	35.3137	1.27844
$764$	0	0
$765$	18.4853	0.668337
$766$	0	0
$767$	−20.2843	−0.732423
$768$	0	0
$769$	12.3431	0.445105	0.222553	−	0.974921i	$-0.428561\pi$
$769$	12.3431	0.445105	0.222553	+	0.974921i	$0.428561\pi$
$770$	0	0
$771$	7.07107	0.254658
$772$	0	0
$773$	2.44365	0.0878920	0.0439460	−	0.999034i	$-0.486007\pi$
$773$	2.44365	0.0878920	0.0439460	+	0.999034i	$0.486007\pi$
$774$	0	0
$775$	−45.4558	−1.63282
$776$	0	0
$777$	11.3137	0.405877
$778$	0	0
$779$	6.34315	0.227267
$780$	0	0
$781$	8.97056	0.320992
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−64.7696	−2.31172
$786$	0	0
$787$	4.04163	0.144069	0.0720343	−	0.997402i	$-0.477051\pi$
$787$	4.04163	0.144069	0.0720343	+	0.997402i	$0.477051\pi$
$788$	0	0
$789$	−11.3137	−0.402779
$790$	0	0
$791$	−23.3137	−0.828940
$792$	0	0
$793$	40.0000	1.42044
$794$	0	0
$795$	21.3137	0.755919
$796$	0	0
$797$	14.2426	0.504500	0.252250	−	0.967662i	$-0.418829\pi$
$797$	14.2426	0.504500	0.252250	+	0.967662i	$0.418829\pi$
$798$	0	0
$799$	−69.4558	−2.45717
$800$	0	0
$801$	9.31371	0.329084
$802$	0	0
$803$	32.2843	1.13929
$804$	0	0
$805$	−54.6274	−1.92536
$806$	0	0
$807$	10.2426	0.360558
$808$	0	0
$809$	3.85786	0.135635	0.0678176	−	0.997698i	$-0.478396\pi$
$809$	3.85786	0.135635	0.0678176	+	0.997698i	$0.478396\pi$
$810$	0	0
$811$	−54.3848	−1.90971	−0.954854	−	0.297076i	$-0.903989\pi$
$811$	−54.3848	−1.90971	−0.954854	+	0.297076i	$0.903989\pi$
$812$	0	0
$813$	−13.5563	−0.475442
$814$	0	0
$815$	−47.4558	−1.66231
$816$	0	0
$817$	0.284271	0.00994539
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	21.5563	0.752322	0.376161	−	0.926554i	$-0.377244\pi$
$821$	21.5563	0.752322	0.376161	+	0.926554i	$0.377244\pi$
$822$	0	0
$823$	−28.3848	−0.989431	−0.494716	−	0.869055i	$-0.664728\pi$
$823$	−28.3848	−0.989431	−0.494716	+	0.869055i	$0.664728\pi$
$824$	0	0
$825$	−13.3137	−0.463524
$826$	0	0
$827$	14.4853	0.503703	0.251851	−	0.967766i	$-0.418961\pi$
$827$	14.4853	0.503703	0.251851	+	0.967766i	$0.418961\pi$
$828$	0	0
$829$	34.8284	1.20964	0.604821	−	0.796362i	$-0.293245\pi$
$829$	34.8284	1.20964	0.604821	+	0.796362i	$0.293245\pi$
$830$	0	0
$831$	1.17157	0.0406414
$832$	0	0
$833$	−5.41421	−0.187591
$834$	0	0
$835$	−3.41421	−0.118154
$836$	0	0
$837$	−6.82843	−0.236025
$838$	0	0
$839$	−36.4264	−1.25758	−0.628790	−	0.777575i	$-0.716449\pi$
$839$	−36.4264	−1.25758	−0.628790	+	0.777575i	$0.716449\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−21.3137	−0.734083
$844$	0	0
$845$	17.0711	0.587263
$846$	0	0
$847$	19.7990	0.680301
$848$	0	0
$849$	−2.14214	−0.0735179
$850$	0	0
$851$	22.6274	0.775658
$852$	0	0
$853$	6.54416	0.224068	0.112034	−	0.993704i	$-0.464264\pi$
$853$	6.54416	0.224068	0.112034	+	0.993704i	$0.464264\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−35.4558	−1.21115	−0.605574	−	0.795789i	$-0.707057\pi$
$857$	−35.4558	−1.21115	−0.605574	+	0.795789i	$0.707057\pi$
$858$	0	0
$859$	45.2548	1.54408	0.772038	−	0.635577i	$-0.219238\pi$
$859$	45.2548	1.54408	0.772038	+	0.635577i	$0.219238\pi$
$860$	0	0
$861$	15.3137	0.521890
$862$	0	0
$863$	45.2548	1.54049	0.770246	−	0.637747i	$-0.220133\pi$
$863$	45.2548	1.54049	0.770246	+	0.637747i	$0.220133\pi$
$864$	0	0
$865$	27.7990	0.945194
$866$	0	0
$867$	12.3137	0.418195
$868$	0	0
$869$	−3.51472	−0.119229
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	5.31371	0.179842
$874$	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
$881$	7.27208	0.245003	0.122501	−	0.992468i	$-0.460908\pi$
$881$	7.27208	0.245003	0.122501	+	0.992468i	$0.460908\pi$
$882$	0	0
$883$	−51.1127	−1.72008	−0.860040	−	0.510227i	$-0.829561\pi$
$883$	−51.1127	−1.72008	−0.860040	+	0.510227i	$0.829561\pi$
$884$	0	0
$885$	−24.4853	−0.823064
$886$	0	0
$887$	−52.7696	−1.77183	−0.885914	−	0.463849i	$-0.846468\pi$
$887$	−52.7696	−1.77183	−0.885914	+	0.463849i	$0.846468\pi$
$888$	0	0
$889$	11.3137	0.379450
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−15.0294	−0.502941
$894$	0	0
$895$	2.34315	0.0783227
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−13.6569	−0.455482
$900$	0	0
$901$	33.7990	1.12601
$902$	0	0
$903$	0.686292	0.0228384
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	−23.0294	−0.764680	−0.382340	−	0.924022i	$-0.624882\pi$
$907$	−23.0294	−0.764680	−0.382340	+	0.924022i	$0.624882\pi$
$908$	0	0
$909$	−13.0711	−0.433540
$910$	0	0
$911$	35.3137	1.17000	0.584998	−	0.811035i	$-0.301095\pi$
$911$	35.3137	1.17000	0.584998	+	0.811035i	$0.301095\pi$
$912$	0	0
$913$	−0.686292	−0.0227129
$914$	0	0
$915$	48.2843	1.59623
$916$	0	0
$917$	−23.5980	−0.779274
$918$	0	0
$919$	7.79899	0.257265	0.128632	−	0.991692i	$-0.458941\pi$
$919$	7.79899	0.257265	0.128632	+	0.991692i	$0.458941\pi$
$920$	0	0
$921$	0.443651	0.0146188
$922$	0	0
$923$	12.6863	0.417574
$924$	0	0
$925$	−26.6274	−0.875504
$926$	0	0
$927$	5.07107	0.166556
$928$	0	0
$929$	−53.7990	−1.76509	−0.882544	−	0.470230i	$-0.844171\pi$
$929$	−53.7990	−1.76509	−0.882544	+	0.470230i	$0.844171\pi$
$930$	0	0
$931$	−1.17157	−0.0383968
$932$	0	0
$933$	−4.68629	−0.153422
$934$	0	0
$935$	−36.9706	−1.20907
$936$	0	0
$937$	−26.9706	−0.881090	−0.440545	−	0.897731i	$-0.645215\pi$
$937$	−26.9706	−0.881090	−0.440545	+	0.897731i	$0.645215\pi$
$938$	0	0
$939$	−26.9706	−0.880151
$940$	0	0
$941$	6.92893	0.225877	0.112938	−	0.993602i	$-0.463974\pi$
$941$	6.92893	0.225877	0.112938	+	0.993602i	$0.463974\pi$
$942$	0	0
$943$	30.6274	0.997366
$944$	0	0
$945$	9.65685	0.314137
$946$	0	0
$947$	−46.9706	−1.52634	−0.763169	−	0.646199i	$-0.776358\pi$
$947$	−46.9706	−1.52634	−0.763169	+	0.646199i	$0.776358\pi$
$948$	0	0
$949$	45.6569	1.48208
$950$	0	0
$951$	−30.4853	−0.988553
$952$	0	0
$953$	−19.5563	−0.633492	−0.316746	−	0.948510i	$-0.602590\pi$
$953$	−19.5563	−0.633492	−0.316746	+	0.948510i	$0.602590\pi$
$954$	0	0
$955$	55.1127	1.78341
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	−58.9117	−1.90236
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	92.0833	2.96427
$966$	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
$969$	6.34315	0.203771
$970$	0	0
$971$	54.9706	1.76409	0.882045	−	0.471166i	$-0.156167\pi$
$971$	54.9706	1.76409	0.882045	+	0.471166i	$0.156167\pi$
$972$	0	0
$973$	39.3137	1.26034
$974$	0	0
$975$	−18.8284	−0.602992
$976$	0	0
$977$	−40.5269	−1.29657	−0.648285	−	0.761397i	$-0.724514\pi$
$977$	−40.5269	−1.29657	−0.648285	+	0.761397i	$0.724514\pi$
$978$	0	0
$979$	−18.6274	−0.595335
$980$	0	0
$981$	−12.4853	−0.398624
$982$	0	0
$983$	13.4558	0.429175	0.214587	−	0.976705i	$-0.431159\pi$
$983$	13.4558	0.429175	0.214587	+	0.976705i	$0.431159\pi$
$984$	0	0
$985$	46.2843	1.47474
$986$	0	0
$987$	−36.2843	−1.15494
$988$	0	0
$989$	1.37258	0.0436456
$990$	0	0
$991$	8.10051	0.257321	0.128661	−	0.991689i	$-0.458932\pi$
$991$	8.10051	0.257321	0.128661	+	0.991689i	$0.458932\pi$
$992$	0	0
$993$	10.5858	0.335930
$994$	0	0
$995$	21.6569	0.686568
$996$	0	0
$997$	15.1127	0.478624	0.239312	−	0.970943i	$-0.423078\pi$
$997$	15.1127	0.478624	0.239312	+	0.970943i	$0.423078\pi$
$998$	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.1.1		2
4.3	odd	2		1002.2.a.f.1.1	✓	2
12.11	even	2		3006.2.a.n.1.2		2

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -3.41421 q^{15} -5.41421 q^{17} -1.17157 q^{19} -2.82843 q^{21} -5.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +2.00000 q^{29} -6.82843 q^{31} -2.00000 q^{33} +9.65685 q^{35} -4.00000 q^{37} -2.82843 q^{39} -5.41421 q^{41} -0.242641 q^{43} -3.41421 q^{45} +12.8284 q^{47} +1.00000 q^{49} -5.41421 q^{51} -6.24264 q^{53} +6.82843 q^{55} -1.17157 q^{57} +7.17157 q^{59} -14.1421 q^{61} -2.82843 q^{63} +9.65685 q^{65} +1.41421 q^{67} -5.65685 q^{69} -4.48528 q^{71} -16.1421 q^{73} +6.65685 q^{75} +5.65685 q^{77} +1.75736 q^{79} +1.00000 q^{81} +0.343146 q^{83} +18.4853 q^{85} +2.00000 q^{87} +9.31371 q^{89} +8.00000 q^{91} -6.82843 q^{93} +4.00000 q^{95} +5.31371 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

Properties

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