LMFDB - Embedded newform 4016.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.58102$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58102 q^{3} +3.46315 q^{5} -1.91212 q^{7} -0.500388 q^{9} +O(q^{10})$	$q-1.58102 q^{3} +3.46315 q^{5} -1.91212 q^{7} -0.500388 q^{9} -3.90998 q^{11} +4.04859 q^{13} -5.47530 q^{15} -2.55749 q^{17} -1.03817 q^{19} +3.02309 q^{21} +1.30855 q^{23} +6.99341 q^{25} +5.53417 q^{27} -1.37282 q^{29} +7.77236 q^{31} +6.18174 q^{33} -6.62196 q^{35} -6.42786 q^{37} -6.40088 q^{39} +2.76750 q^{41} -3.92606 q^{43} -1.73292 q^{45} +0.337788 q^{47} -3.34380 q^{49} +4.04343 q^{51} -5.06813 q^{53} -13.5408 q^{55} +1.64136 q^{57} +5.85317 q^{59} -8.72856 q^{61} +0.956801 q^{63} +14.0209 q^{65} +2.09185 q^{67} -2.06883 q^{69} -3.70752 q^{71} -0.740937 q^{73} -11.0567 q^{75} +7.47634 q^{77} -14.1573 q^{79} -7.24845 q^{81} -8.17518 q^{83} -8.85696 q^{85} +2.17045 q^{87} +1.81704 q^{89} -7.74139 q^{91} -12.2882 q^{93} -3.59533 q^{95} -9.54460 q^{97} +1.95650 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	−1.58102	−0.912800	−0.456400	−	0.889775i	$-0.650861\pi$
$3$	−1.58102	−0.912800	−0.456400	+	0.889775i	$0.650861\pi$
$4$	0	0
$5$	3.46315	1.54877	0.774384	−	0.632716i	$-0.218060\pi$
$5$	3.46315	1.54877	0.774384	+	0.632716i	$0.218060\pi$
$6$	0	0
$7$	−1.91212	−0.722713	−0.361357	−	0.932428i	$-0.617686\pi$
$7$	−1.91212	−0.722713	−0.361357	+	0.932428i	$0.617686\pi$
$8$	0	0
$9$	−0.500388	−0.166796
$10$	0	0
$11$	−3.90998	−1.17890	−0.589451	−	0.807804i	$-0.700656\pi$
$11$	−3.90998	−1.17890	−0.589451	+	0.807804i	$0.700656\pi$
$12$	0	0
$13$	4.04859	1.12288	0.561438	−	0.827519i	$-0.310248\pi$
$13$	4.04859	1.12288	0.561438	+	0.827519i	$0.310248\pi$
$14$	0	0
$15$	−5.47530	−1.41372
$16$	0	0
$17$	−2.55749	−0.620281	−0.310141	−	0.950691i	$-0.600376\pi$
$17$	−2.55749	−0.620281	−0.310141	+	0.950691i	$0.600376\pi$
$18$	0	0
$19$	−1.03817	−0.238172	−0.119086	−	0.992884i	$-0.537996\pi$
$19$	−1.03817	−0.238172	−0.119086	+	0.992884i	$0.537996\pi$
$20$	0	0
$21$	3.02309	0.659693
$22$	0	0
$23$	1.30855	0.272851	0.136425	−	0.990650i	$-0.456439\pi$
$23$	1.30855	0.272851	0.136425	+	0.990650i	$0.456439\pi$
$24$	0	0
$25$	6.99341	1.39868
$26$	0	0
$27$	5.53417	1.06505
$28$	0	0
$29$	−1.37282	−0.254927	−0.127463	−	0.991843i	$-0.540684\pi$
$29$	−1.37282	−0.254927	−0.127463	+	0.991843i	$0.540684\pi$
$30$	0	0
$31$	7.77236	1.39596	0.697979	−	0.716119i	$-0.254083\pi$
$31$	7.77236	1.39596	0.697979	+	0.716119i	$0.254083\pi$
$32$	0	0
$33$	6.18174	1.07610
$34$	0	0
$35$	−6.62196	−1.11932
$36$	0	0
$37$	−6.42786	−1.05673	−0.528367	−	0.849016i	$-0.677196\pi$
$37$	−6.42786	−1.05673	−0.528367	+	0.849016i	$0.677196\pi$
$38$	0	0
$39$	−6.40088	−1.02496
$40$	0	0
$41$	2.76750	0.432210	0.216105	−	0.976370i	$-0.430665\pi$
$41$	2.76750	0.432210	0.216105	+	0.976370i	$0.430665\pi$
$42$	0	0
$43$	−3.92606	−0.598718	−0.299359	−	0.954141i	$-0.596773\pi$
$43$	−3.92606	−0.598718	−0.299359	+	0.954141i	$0.596773\pi$
$44$	0	0
$45$	−1.73292	−0.258328
$46$	0	0
$47$	0.337788	0.0492714	0.0246357	−	0.999696i	$-0.492157\pi$
$47$	0.337788	0.0492714	0.0246357	+	0.999696i	$0.492157\pi$
$48$	0	0
$49$	−3.34380	−0.477686
$50$	0	0
$51$	4.04343	0.566193
$52$	0	0
$53$	−5.06813	−0.696161	−0.348080	−	0.937465i	$-0.613166\pi$
$53$	−5.06813	−0.696161	−0.348080	+	0.937465i	$0.613166\pi$
$54$	0	0
$55$	−13.5408	−1.82585
$56$	0	0
$57$	1.64136	0.217403
$58$	0	0
$59$	5.85317	0.762018	0.381009	−	0.924571i	$-0.375577\pi$
$59$	5.85317	0.762018	0.381009	+	0.924571i	$0.375577\pi$
$60$	0	0
$61$	−8.72856	−1.11758	−0.558789	−	0.829310i	$-0.688734\pi$
$61$	−8.72856	−1.11758	−0.558789	+	0.829310i	$0.688734\pi$
$62$	0	0
$63$	0.956801	0.120546
$64$	0	0
$65$	14.0209	1.73908
$66$	0	0
$67$	2.09185	0.255560	0.127780	−	0.991803i	$-0.459215\pi$
$67$	2.09185	0.255560	0.127780	+	0.991803i	$0.459215\pi$
$68$	0	0
$69$	−2.06883	−0.249058
$70$	0	0
$71$	−3.70752	−0.440002	−0.220001	−	0.975500i	$-0.570606\pi$
$71$	−3.70752	−0.440002	−0.220001	+	0.975500i	$0.570606\pi$
$72$	0	0
$73$	−0.740937	−0.0867201	−0.0433600	−	0.999060i	$-0.513806\pi$
$73$	−0.740937	−0.0867201	−0.0433600	+	0.999060i	$0.513806\pi$
$74$	0	0
$75$	−11.0567	−1.27672
$76$	0	0
$77$	7.47634	0.852008
$78$	0	0
$79$	−14.1573	−1.59282	−0.796408	−	0.604759i	$-0.793269\pi$
$79$	−14.1573	−1.59282	−0.796408	+	0.604759i	$0.793269\pi$
$80$	0	0
$81$	−7.24845	−0.805383
$82$	0	0
$83$	−8.17518	−0.897343	−0.448671	−	0.893697i	$-0.648103\pi$
$83$	−8.17518	−0.897343	−0.448671	+	0.893697i	$0.648103\pi$
$84$	0	0
$85$	−8.85696	−0.960672
$86$	0	0
$87$	2.17045	0.232697
$88$	0	0
$89$	1.81704	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$89$	1.81704	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$90$	0	0
$91$	−7.74139	−0.811518
$92$	0	0
$93$	−12.2882	−1.27423
$94$	0	0
$95$	−3.59533	−0.368873
$96$	0	0
$97$	−9.54460	−0.969107	−0.484554	−	0.874762i	$-0.661018\pi$
$97$	−9.54460	−0.969107	−0.484554	+	0.874762i	$0.661018\pi$
$98$	0	0
$99$	1.95650	0.196636
$100$	0	0
$101$	−17.8890	−1.78002	−0.890011	−	0.455940i	$-0.849303\pi$
$101$	−17.8890	−1.78002	−0.890011	+	0.455940i	$0.849303\pi$
$102$	0	0
$103$	0.0113054	0.00111395	0.000556976	−	1.00000i	$-0.499823\pi$
$103$	0.0113054	0.00111395	0.000556976		1.00000i	$0.499823\pi$
$104$	0	0
$105$	10.4694	1.02171
$106$	0	0
$107$	5.50799	0.532477	0.266239	−	0.963907i	$-0.414219\pi$
$107$	5.50799	0.532477	0.266239	+	0.963907i	$0.414219\pi$
$108$	0	0
$109$	10.1169	0.969025	0.484512	−	0.874784i	$-0.338997\pi$
$109$	10.1169	0.969025	0.484512	+	0.874784i	$0.338997\pi$
$110$	0	0
$111$	10.1626	0.964587
$112$	0	0
$113$	−3.53805	−0.332832	−0.166416	−	0.986056i	$-0.553219\pi$
$113$	−3.53805	−0.332832	−0.166416	+	0.986056i	$0.553219\pi$
$114$	0	0
$115$	4.53170	0.422583
$116$	0	0
$117$	−2.02586	−0.187291
$118$	0	0
$119$	4.89022	0.448285
$120$	0	0
$121$	4.28792	0.389811
$122$	0	0
$123$	−4.37546	−0.394522
$124$	0	0
$125$	6.90349	0.617467
$126$	0	0
$127$	−2.19338	−0.194631	−0.0973155	−	0.995254i	$-0.531026\pi$
$127$	−2.19338	−0.194631	−0.0973155	+	0.995254i	$0.531026\pi$
$128$	0	0
$129$	6.20716	0.546510
$130$	0	0
$131$	8.41774	0.735461	0.367731	−	0.929932i	$-0.380135\pi$
$131$	8.41774	0.735461	0.367731	+	0.929932i	$0.380135\pi$
$132$	0	0
$133$	1.98510	0.172130
$134$	0	0
$135$	19.1657	1.64952
$136$	0	0
$137$	15.7508	1.34568	0.672840	−	0.739788i	$-0.265074\pi$
$137$	15.7508	1.34568	0.672840	+	0.739788i	$0.265074\pi$
$138$	0	0
$139$	−8.49009	−0.720120	−0.360060	−	0.932929i	$-0.617244\pi$
$139$	−8.49009	−0.720120	−0.360060	+	0.932929i	$0.617244\pi$
$140$	0	0
$141$	−0.534048	−0.0449750
$142$	0	0
$143$	−15.8299	−1.32376
$144$	0	0
$145$	−4.75429	−0.394822
$146$	0	0
$147$	5.28660	0.436032
$148$	0	0
$149$	11.6354	0.953205	0.476603	−	0.879119i	$-0.341868\pi$
$149$	11.6354	0.953205	0.476603	+	0.879119i	$0.341868\pi$
$150$	0	0
$151$	14.1743	1.15349	0.576743	−	0.816926i	$-0.304323\pi$
$151$	14.1743	1.15349	0.576743	+	0.816926i	$0.304323\pi$
$152$	0	0
$153$	1.27973	0.103460
$154$	0	0
$155$	26.9169	2.16201
$156$	0	0
$157$	−10.6863	−0.852858	−0.426429	−	0.904521i	$-0.640229\pi$
$157$	−10.6863	−0.852858	−0.426429	+	0.904521i	$0.640229\pi$
$158$	0	0
$159$	8.01279	0.635456
$160$	0	0
$161$	−2.50210	−0.197193
$162$	0	0
$163$	−25.3767	−1.98766	−0.993828	−	0.110936i	$-0.964615\pi$
$163$	−25.3767	−1.98766	−0.993828	+	0.110936i	$0.964615\pi$
$164$	0	0
$165$	21.4083	1.66663
$166$	0	0
$167$	−20.7509	−1.60575	−0.802876	−	0.596146i	$-0.796698\pi$
$167$	−20.7509	−1.60575	−0.802876	+	0.596146i	$0.796698\pi$
$168$	0	0
$169$	3.39107	0.260852
$170$	0	0
$171$	0.519486	0.0397261
$172$	0	0
$173$	−14.5129	−1.10340	−0.551699	−	0.834043i	$-0.686020\pi$
$173$	−14.5129	−1.10340	−0.551699	+	0.834043i	$0.686020\pi$
$174$	0	0
$175$	−13.3722	−1.01085
$176$	0	0
$177$	−9.25396	−0.695571
$178$	0	0
$179$	−1.15155	−0.0860711	−0.0430355	−	0.999074i	$-0.513703\pi$
$179$	−1.15155	−0.0860711	−0.0430355	+	0.999074i	$0.513703\pi$
$180$	0	0
$181$	10.2386	0.761032	0.380516	−	0.924774i	$-0.375746\pi$
$181$	10.2386	0.761032	0.380516	+	0.924774i	$0.375746\pi$
$182$	0	0
$183$	13.8000	1.02012
$184$	0	0
$185$	−22.2607	−1.63664
$186$	0	0
$187$	9.99971	0.731251
$188$	0	0
$189$	−10.5820	−0.769727
$190$	0	0
$191$	−8.90611	−0.644424	−0.322212	−	0.946668i	$-0.604426\pi$
$191$	−8.90611	−0.644424	−0.322212	+	0.946668i	$0.604426\pi$
$192$	0	0
$193$	0.335602	0.0241572	0.0120786	−	0.999927i	$-0.496155\pi$
$193$	0.335602	0.0241572	0.0120786	+	0.999927i	$0.496155\pi$
$194$	0	0
$195$	−22.1672	−1.58743
$196$	0	0
$197$	6.33623	0.451438	0.225719	−	0.974192i	$-0.427527\pi$
$197$	6.33623	0.451438	0.225719	+	0.974192i	$0.427527\pi$
$198$	0	0
$199$	−7.00597	−0.496640	−0.248320	−	0.968678i	$-0.579878\pi$
$199$	−7.00597	−0.496640	−0.248320	+	0.968678i	$0.579878\pi$
$200$	0	0
$201$	−3.30725	−0.233275
$202$	0	0
$203$	2.62500	0.184239
$204$	0	0
$205$	9.58426	0.669393
$206$	0	0
$207$	−0.654781	−0.0455104
$208$	0	0
$209$	4.05921	0.280782
$210$	0	0
$211$	−5.27028	−0.362821	−0.181411	−	0.983407i	$-0.558066\pi$
$211$	−5.27028	−0.362821	−0.181411	+	0.983407i	$0.558066\pi$
$212$	0	0
$213$	5.86165	0.401634
$214$	0	0
$215$	−13.5965	−0.927275
$216$	0	0
$217$	−14.8617	−1.00888
$218$	0	0
$219$	1.17143	0.0791581
$220$	0	0
$221$	−10.3542	−0.696499
$222$	0	0
$223$	−2.82705	−0.189313	−0.0946565	−	0.995510i	$-0.530175\pi$
$223$	−2.82705	−0.189313	−0.0946565	+	0.995510i	$0.530175\pi$
$224$	0	0
$225$	−3.49942	−0.233294
$226$	0	0
$227$	18.5225	1.22938	0.614692	−	0.788767i	$-0.289280\pi$
$227$	18.5225	1.22938	0.614692	+	0.788767i	$0.289280\pi$
$228$	0	0
$229$	11.7593	0.777073	0.388537	−	0.921433i	$-0.372981\pi$
$229$	11.7593	0.777073	0.388537	+	0.921433i	$0.372981\pi$
$230$	0	0
$231$	−11.8202	−0.777713
$232$	0	0
$233$	−16.6755	−1.09245	−0.546226	−	0.837638i	$-0.683936\pi$
$233$	−16.6755	−1.09245	−0.546226	+	0.837638i	$0.683936\pi$
$234$	0	0
$235$	1.16981	0.0763100
$236$	0	0
$237$	22.3829	1.45392
$238$	0	0
$239$	−24.7409	−1.60035	−0.800177	−	0.599765i	$-0.795261\pi$
$239$	−24.7409	−1.60035	−0.800177	+	0.599765i	$0.795261\pi$
$240$	0	0
$241$	−3.83856	−0.247263	−0.123632	−	0.992328i	$-0.539454\pi$
$241$	−3.83856	−0.247263	−0.123632	+	0.992328i	$0.539454\pi$
$242$	0	0
$243$	−5.14259	−0.329897
$244$	0	0
$245$	−11.5801	−0.739824
$246$	0	0
$247$	−4.20311	−0.267438
$248$	0	0
$249$	12.9251	0.819094
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−5.11639	−0.321665
$254$	0	0
$255$	14.0030	0.876901
$256$	0	0
$257$	20.4240	1.27401	0.637006	−	0.770859i	$-0.280173\pi$
$257$	20.4240	1.27401	0.637006	+	0.770859i	$0.280173\pi$
$258$	0	0
$259$	12.2908	0.763716
$260$	0	0
$261$	0.686943	0.0425207
$262$	0	0
$263$	−21.7799	−1.34301	−0.671504	−	0.741001i	$-0.734351\pi$
$263$	−21.7799	−1.34301	−0.671504	+	0.741001i	$0.734351\pi$
$264$	0	0
$265$	−17.5517	−1.07819
$266$	0	0
$267$	−2.87276	−0.175810
$268$	0	0
$269$	−6.25077	−0.381116	−0.190558	−	0.981676i	$-0.561030\pi$
$269$	−6.25077	−0.381116	−0.190558	+	0.981676i	$0.561030\pi$
$270$	0	0
$271$	−31.8427	−1.93431	−0.967153	−	0.254194i	$-0.918190\pi$
$271$	−31.8427	−1.93431	−0.967153	+	0.254194i	$0.918190\pi$
$272$	0	0
$273$	12.2393	0.740753
$274$	0	0
$275$	−27.3441	−1.64891
$276$	0	0
$277$	16.6469	1.00022	0.500108	−	0.865963i	$-0.333294\pi$
$277$	16.6469	1.00022	0.500108	+	0.865963i	$0.333294\pi$
$278$	0	0
$279$	−3.88919	−0.232840
$280$	0	0
$281$	−18.2529	−1.08888	−0.544438	−	0.838801i	$-0.683257\pi$
$281$	−18.2529	−1.08888	−0.544438	+	0.838801i	$0.683257\pi$
$282$	0	0
$283$	7.79523	0.463378	0.231689	−	0.972790i	$-0.425575\pi$
$283$	7.79523	0.463378	0.231689	+	0.972790i	$0.425575\pi$
$284$	0	0
$285$	5.68428	0.336708
$286$	0	0
$287$	−5.29178	−0.312364
$288$	0	0
$289$	−10.4593	−0.615251
$290$	0	0
$291$	15.0902	0.884601
$292$	0	0
$293$	−27.0889	−1.58255	−0.791274	−	0.611462i	$-0.790582\pi$
$293$	−27.0889	−1.58255	−0.791274	+	0.611462i	$0.790582\pi$
$294$	0	0
$295$	20.2704	1.18019
$296$	0	0
$297$	−21.6385	−1.25559
$298$	0	0
$299$	5.29777	0.306378
$300$	0	0
$301$	7.50709	0.432701
$302$	0	0
$303$	28.2828	1.62480
$304$	0	0
$305$	−30.2283	−1.73087
$306$	0	0
$307$	14.2014	0.810516	0.405258	−	0.914202i	$-0.367182\pi$
$307$	14.2014	0.810516	0.405258	+	0.914202i	$0.367182\pi$
$308$	0	0
$309$	−0.0178740	−0.00101681
$310$	0	0
$311$	−2.52586	−0.143229	−0.0716143	−	0.997432i	$-0.522815\pi$
$311$	−2.52586	−0.143229	−0.0716143	+	0.997432i	$0.522815\pi$
$312$	0	0
$313$	0.233335	0.0131889	0.00659443	−	0.999978i	$-0.497901\pi$
$313$	0.233335	0.0131889	0.00659443	+	0.999978i	$0.497901\pi$
$314$	0	0
$315$	3.31355	0.186697
$316$	0	0
$317$	17.4987	0.982825	0.491412	−	0.870927i	$-0.336481\pi$
$317$	17.4987	0.982825	0.491412	+	0.870927i	$0.336481\pi$
$318$	0	0
$319$	5.36770	0.300534
$320$	0	0
$321$	−8.70822	−0.486045
$322$	0	0
$323$	2.65510	0.147734
$324$	0	0
$325$	28.3135	1.57055
$326$	0	0
$327$	−15.9950	−0.884526
$328$	0	0
$329$	−0.645891	−0.0356091
$330$	0	0
$331$	16.3333	0.897759	0.448879	−	0.893592i	$-0.351823\pi$
$331$	16.3333	0.897759	0.448879	+	0.893592i	$0.351823\pi$
$332$	0	0
$333$	3.21642	0.176259
$334$	0	0
$335$	7.24439	0.395804
$336$	0	0
$337$	6.38203	0.347651	0.173826	−	0.984776i	$-0.444387\pi$
$337$	6.38203	0.347651	0.173826	+	0.984776i	$0.444387\pi$
$338$	0	0
$339$	5.59372	0.303809
$340$	0	0
$341$	−30.3898	−1.64570
$342$	0	0
$343$	19.7786	1.06794
$344$	0	0
$345$	−7.16469	−0.385734
$346$	0	0
$347$	8.42527	0.452292	0.226146	−	0.974093i	$-0.427387\pi$
$347$	8.42527	0.452292	0.226146	+	0.974093i	$0.427387\pi$
$348$	0	0
$349$	−35.0504	−1.87621	−0.938103	−	0.346356i	$-0.887419\pi$
$349$	−35.0504	−1.87621	−0.938103	+	0.346356i	$0.887419\pi$
$350$	0	0
$351$	22.4056	1.19592
$352$	0	0
$353$	−16.4320	−0.874585	−0.437292	−	0.899319i	$-0.644062\pi$
$353$	−16.4320	−0.874585	−0.437292	+	0.899319i	$0.644062\pi$
$354$	0	0
$355$	−12.8397	−0.681461
$356$	0	0
$357$	−7.73151	−0.409195
$358$	0	0
$359$	−7.75140	−0.409103	−0.204552	−	0.978856i	$-0.565574\pi$
$359$	−7.75140	−0.409103	−0.204552	+	0.978856i	$0.565574\pi$
$360$	0	0
$361$	−17.9222	−0.943274
$362$	0	0
$363$	−6.77927	−0.355820
$364$	0	0
$365$	−2.56598	−0.134309
$366$	0	0
$367$	−3.52715	−0.184116	−0.0920579	−	0.995754i	$-0.529344\pi$
$367$	−3.52715	−0.184116	−0.0920579	+	0.995754i	$0.529344\pi$
$368$	0	0
$369$	−1.38482	−0.0720909
$370$	0	0
$371$	9.69086	0.503125
$372$	0	0
$373$	24.5597	1.27165	0.635826	−	0.771833i	$-0.280660\pi$
$373$	24.5597	1.27165	0.635826	+	0.771833i	$0.280660\pi$
$374$	0	0
$375$	−10.9145	−0.563624
$376$	0	0
$377$	−5.55799	−0.286251
$378$	0	0
$379$	0.399266	0.0205089	0.0102545	−	0.999947i	$-0.496736\pi$
$379$	0.399266	0.0205089	0.0102545	+	0.999947i	$0.496736\pi$
$380$	0	0
$381$	3.46777	0.177659
$382$	0	0
$383$	11.1333	0.568883	0.284441	−	0.958693i	$-0.408192\pi$
$383$	11.1333	0.568883	0.284441	+	0.958693i	$0.408192\pi$
$384$	0	0
$385$	25.8917	1.31956
$386$	0	0
$387$	1.96455	0.0998637
$388$	0	0
$389$	27.9465	1.41694	0.708472	−	0.705738i	$-0.249385\pi$
$389$	27.9465	1.41694	0.708472	+	0.705738i	$0.249385\pi$
$390$	0	0
$391$	−3.34659	−0.169244
$392$	0	0
$393$	−13.3086	−0.671329
$394$	0	0
$395$	−49.0287	−2.46690
$396$	0	0
$397$	−14.7323	−0.739393	−0.369696	−	0.929153i	$-0.620538\pi$
$397$	−14.7323	−0.739393	−0.369696	+	0.929153i	$0.620538\pi$
$398$	0	0
$399$	−3.13848	−0.157120
$400$	0	0
$401$	−6.74794	−0.336976	−0.168488	−	0.985704i	$-0.553888\pi$
$401$	−6.74794	−0.336976	−0.168488	+	0.985704i	$0.553888\pi$
$402$	0	0
$403$	31.4671	1.56749
$404$	0	0
$405$	−25.1025	−1.24735
$406$	0	0
$407$	25.1328	1.24579
$408$	0	0
$409$	−30.5221	−1.50922	−0.754611	−	0.656172i	$-0.772174\pi$
$409$	−30.5221	−1.50922	−0.754611	+	0.656172i	$0.772174\pi$
$410$	0	0
$411$	−24.9023	−1.22834
$412$	0	0
$413$	−11.1920	−0.550721
$414$	0	0
$415$	−28.3119	−1.38978
$416$	0	0
$417$	13.4230	0.657325
$418$	0	0
$419$	−17.3549	−0.847844	−0.423922	−	0.905699i	$-0.639347\pi$
$419$	−17.3549	−0.847844	−0.423922	+	0.905699i	$0.639347\pi$
$420$	0	0
$421$	−14.5356	−0.708421	−0.354210	−	0.935166i	$-0.615250\pi$
$421$	−14.5356	−0.708421	−0.354210	+	0.935166i	$0.615250\pi$
$422$	0	0
$423$	−0.169025	−0.00821827
$424$	0	0
$425$	−17.8855	−0.867577
$426$	0	0
$427$	16.6900	0.807688
$428$	0	0
$429$	25.0273	1.20833
$430$	0	0
$431$	−11.8937	−0.572898	−0.286449	−	0.958096i	$-0.592475\pi$
$431$	−11.8937	−0.572898	−0.286449	+	0.958096i	$0.592475\pi$
$432$	0	0
$433$	−15.5836	−0.748899	−0.374450	−	0.927247i	$-0.622168\pi$
$433$	−15.5836	−0.748899	−0.374450	+	0.927247i	$0.622168\pi$
$434$	0	0
$435$	7.51661	0.360394
$436$	0	0
$437$	−1.35849	−0.0649855
$438$	0	0
$439$	33.8185	1.61407	0.807034	−	0.590505i	$-0.201071\pi$
$439$	33.8185	1.61407	0.807034	+	0.590505i	$0.201071\pi$
$440$	0	0
$441$	1.67320	0.0796760
$442$	0	0
$443$	−12.1405	−0.576812	−0.288406	−	0.957508i	$-0.593125\pi$
$443$	−12.1405	−0.576812	−0.288406	+	0.957508i	$0.593125\pi$
$444$	0	0
$445$	6.29267	0.298301
$446$	0	0
$447$	−18.3957	−0.870086
$448$	0	0
$449$	19.6962	0.929519	0.464760	−	0.885437i	$-0.346141\pi$
$449$	19.6962	0.929519	0.464760	+	0.885437i	$0.346141\pi$
$450$	0	0
$451$	−10.8208	−0.509534
$452$	0	0
$453$	−22.4098	−1.05290
$454$	0	0
$455$	−26.8096	−1.25685
$456$	0	0
$457$	−6.78804	−0.317531	−0.158765	−	0.987316i	$-0.550751\pi$
$457$	−6.78804	−0.317531	−0.158765	+	0.987316i	$0.550751\pi$
$458$	0	0
$459$	−14.1536	−0.660631
$460$	0	0
$461$	25.0676	1.16751	0.583756	−	0.811929i	$-0.301582\pi$
$461$	25.0676	1.16751	0.583756	+	0.811929i	$0.301582\pi$
$462$	0	0
$463$	5.75424	0.267422	0.133711	−	0.991020i	$-0.457311\pi$
$463$	5.75424	0.267422	0.133711	+	0.991020i	$0.457311\pi$
$464$	0	0
$465$	−42.5560	−1.97349
$466$	0	0
$467$	−13.4784	−0.623708	−0.311854	−	0.950130i	$-0.600950\pi$
$467$	−13.4784	−0.623708	−0.311854	+	0.950130i	$0.600950\pi$
$468$	0	0
$469$	−3.99987	−0.184697
$470$	0	0
$471$	16.8952	0.778489
$472$	0	0
$473$	15.3508	0.705830
$474$	0	0
$475$	−7.26034	−0.333127
$476$	0	0
$477$	2.53603	0.116117
$478$	0	0
$479$	−33.4615	−1.52889	−0.764447	−	0.644687i	$-0.776988\pi$
$479$	−33.4615	−1.52889	−0.764447	+	0.644687i	$0.776988\pi$
$480$	0	0
$481$	−26.0238	−1.18658
$482$	0	0
$483$	3.95586	0.179998
$484$	0	0
$485$	−33.0544	−1.50092
$486$	0	0
$487$	22.0218	0.997903	0.498952	−	0.866630i	$-0.333719\pi$
$487$	22.0218	0.997903	0.498952	+	0.866630i	$0.333719\pi$
$488$	0	0
$489$	40.1209	1.81433
$490$	0	0
$491$	−5.78858	−0.261235	−0.130617	−	0.991433i	$-0.541696\pi$
$491$	−5.78858	−0.261235	−0.130617	+	0.991433i	$0.541696\pi$
$492$	0	0
$493$	3.51097	0.158126
$494$	0	0
$495$	6.77567	0.304544
$496$	0	0
$497$	7.08923	0.317995
$498$	0	0
$499$	2.65550	0.118876	0.0594381	−	0.998232i	$-0.481069\pi$
$499$	2.65550	0.118876	0.0594381	+	0.998232i	$0.481069\pi$
$500$	0	0
$501$	32.8075	1.46573
$502$	0	0
$503$	39.5890	1.76519	0.882594	−	0.470136i	$-0.155795\pi$
$503$	39.5890	1.76519	0.882594	+	0.470136i	$0.155795\pi$
$504$	0	0
$505$	−61.9523	−2.75684
$506$	0	0
$507$	−5.36134	−0.238105
$508$	0	0
$509$	0.530798	0.0235272	0.0117636	−	0.999931i	$-0.496255\pi$
$509$	0.530798	0.0235272	0.0117636	+	0.999931i	$0.496255\pi$
$510$	0	0
$511$	1.41676	0.0626737
$512$	0	0
$513$	−5.74540	−0.253665
$514$	0	0
$515$	0.0391522	0.00172525
$516$	0	0
$517$	−1.32074	−0.0580862
$518$	0	0
$519$	22.9452	1.00718
$520$	0	0
$521$	38.9002	1.70425	0.852125	−	0.523338i	$-0.175314\pi$
$521$	38.9002	1.70425	0.852125	+	0.523338i	$0.175314\pi$
$522$	0	0
$523$	18.7876	0.821526	0.410763	−	0.911742i	$-0.365262\pi$
$523$	18.7876	0.821526	0.410763	+	0.911742i	$0.365262\pi$
$524$	0	0
$525$	21.1417	0.922701
$526$	0	0
$527$	−19.8777	−0.865886
$528$	0	0
$529$	−21.2877	−0.925552
$530$	0	0
$531$	−2.92886	−0.127102
$532$	0	0
$533$	11.2045	0.485319
$534$	0	0
$535$	19.0750	0.824684
$536$	0	0
$537$	1.82062	0.0785657
$538$	0	0
$539$	13.0742	0.563145
$540$	0	0
$541$	−34.6674	−1.49047	−0.745234	−	0.666803i	$-0.767662\pi$
$541$	−34.6674	−1.49047	−0.745234	+	0.666803i	$0.767662\pi$
$542$	0	0
$543$	−16.1875	−0.694670
$544$	0	0
$545$	35.0364	1.50079
$546$	0	0
$547$	−7.91958	−0.338617	−0.169308	−	0.985563i	$-0.554153\pi$
$547$	−7.91958	−0.338617	−0.169308	+	0.985563i	$0.554153\pi$
$548$	0	0
$549$	4.36766	0.186407
$550$	0	0
$551$	1.42522	0.0607164
$552$	0	0
$553$	27.0704	1.15115
$554$	0	0
$555$	35.1945	1.49392
$556$	0	0
$557$	−25.0870	−1.06297	−0.531486	−	0.847067i	$-0.678366\pi$
$557$	−25.0870	−1.06297	−0.531486	+	0.847067i	$0.678366\pi$
$558$	0	0
$559$	−15.8950	−0.672286
$560$	0	0
$561$	−15.8097	−0.667486
$562$	0	0
$563$	8.06513	0.339905	0.169952	−	0.985452i	$-0.445639\pi$
$563$	8.06513	0.339905	0.169952	+	0.985452i	$0.445639\pi$
$564$	0	0
$565$	−12.2528	−0.515479
$566$	0	0
$567$	13.8599	0.582061
$568$	0	0
$569$	38.5906	1.61780	0.808901	−	0.587946i	$-0.200063\pi$
$569$	38.5906	1.61780	0.808901	+	0.587946i	$0.200063\pi$
$570$	0	0
$571$	37.6091	1.57389	0.786945	−	0.617023i	$-0.211662\pi$
$571$	37.6091	1.57389	0.786945	+	0.617023i	$0.211662\pi$
$572$	0	0
$573$	14.0807	0.588230
$574$	0	0
$575$	9.15121	0.381632
$576$	0	0
$577$	−28.0006	−1.16568	−0.582839	−	0.812587i	$-0.698058\pi$
$577$	−28.0006	−1.16568	−0.582839	+	0.812587i	$0.698058\pi$
$578$	0	0
$579$	−0.530592	−0.0220507
$580$	0	0
$581$	15.6319	0.648521
$582$	0	0
$583$	19.8163	0.820706
$584$	0	0
$585$	−7.01587	−0.290071
$586$	0	0
$587$	31.1472	1.28558	0.642792	−	0.766041i	$-0.277776\pi$
$587$	31.1472	1.28558	0.642792	+	0.766041i	$0.277776\pi$
$588$	0	0
$589$	−8.06902	−0.332478
$590$	0	0
$591$	−10.0177	−0.412072
$592$	0	0
$593$	−21.0432	−0.864141	−0.432070	−	0.901840i	$-0.642217\pi$
$593$	−21.0432	−0.864141	−0.432070	+	0.901840i	$0.642217\pi$
$594$	0	0
$595$	16.9356	0.694290
$596$	0	0
$597$	11.0766	0.453333
$598$	0	0
$599$	22.7695	0.930336	0.465168	−	0.885223i	$-0.345994\pi$
$599$	22.7695	0.930336	0.465168	+	0.885223i	$0.345994\pi$
$600$	0	0
$601$	29.3365	1.19666	0.598330	−	0.801250i	$-0.295831\pi$
$601$	29.3365	1.19666	0.598330	+	0.801250i	$0.295831\pi$
$602$	0	0
$603$	−1.04674	−0.0426264
$604$	0	0
$605$	14.8497	0.603727
$606$	0	0
$607$	−41.8220	−1.69750	−0.848752	−	0.528791i	$-0.822645\pi$
$607$	−41.8220	−1.69750	−0.848752	+	0.528791i	$0.822645\pi$
$608$	0	0
$609$	−4.15017	−0.168173
$610$	0	0
$611$	1.36756	0.0553257
$612$	0	0
$613$	27.4235	1.10762	0.553812	−	0.832641i	$-0.313172\pi$
$613$	27.4235	1.10762	0.553812	+	0.832641i	$0.313172\pi$
$614$	0	0
$615$	−15.1529	−0.611022
$616$	0	0
$617$	−23.0904	−0.929585	−0.464793	−	0.885420i	$-0.653871\pi$
$617$	−23.0904	−0.929585	−0.464793	+	0.885420i	$0.653871\pi$
$618$	0	0
$619$	−5.20475	−0.209197	−0.104598	−	0.994515i	$-0.533356\pi$
$619$	−5.20475	−0.209197	−0.104598	+	0.994515i	$0.533356\pi$
$620$	0	0
$621$	7.24172	0.290600
$622$	0	0
$623$	−3.47439	−0.139198
$624$	0	0
$625$	−11.0592	−0.442370
$626$	0	0
$627$	−6.41768	−0.256297
$628$	0	0
$629$	16.4392	0.655473
$630$	0	0
$631$	−23.5765	−0.938564	−0.469282	−	0.883048i	$-0.655487\pi$
$631$	−23.5765	−0.938564	−0.469282	+	0.883048i	$0.655487\pi$
$632$	0	0
$633$	8.33240	0.331183
$634$	0	0
$635$	−7.59601	−0.301438
$636$	0	0
$637$	−13.5377	−0.536382
$638$	0	0
$639$	1.85520	0.0733905
$640$	0	0
$641$	15.6930	0.619837	0.309919	−	0.950763i	$-0.399698\pi$
$641$	15.6930	0.619837	0.309919	+	0.950763i	$0.399698\pi$
$642$	0	0
$643$	−9.68620	−0.381987	−0.190993	−	0.981591i	$-0.561171\pi$
$643$	−9.68620	−0.381987	−0.190993	+	0.981591i	$0.561171\pi$
$644$	0	0
$645$	21.4963	0.846417
$646$	0	0
$647$	−5.01699	−0.197238	−0.0986192	−	0.995125i	$-0.531443\pi$
$647$	−5.01699	−0.197238	−0.0986192	+	0.995125i	$0.531443\pi$
$648$	0	0
$649$	−22.8858	−0.898345
$650$	0	0
$651$	23.4966	0.920903
$652$	0	0
$653$	29.8032	1.16629	0.583145	−	0.812368i	$-0.301822\pi$
$653$	29.8032	1.16629	0.583145	+	0.812368i	$0.301822\pi$
$654$	0	0
$655$	29.1519	1.13906
$656$	0	0
$657$	0.370755	0.0144645
$658$	0	0
$659$	1.13940	0.0443846	0.0221923	−	0.999754i	$-0.492935\pi$
$659$	1.13940	0.0443846	0.0221923	+	0.999754i	$0.492935\pi$
$660$	0	0
$661$	−3.61023	−0.140422	−0.0702109	−	0.997532i	$-0.522367\pi$
$661$	−3.61023	−0.140422	−0.0702109	+	0.997532i	$0.522367\pi$
$662$	0	0
$663$	16.3702	0.635765
$664$	0	0
$665$	6.87470	0.266590
$666$	0	0
$667$	−1.79640	−0.0695570
$668$	0	0
$669$	4.46961	0.172805
$670$	0	0
$671$	34.1285	1.31751
$672$	0	0
$673$	18.7500	0.722761	0.361381	−	0.932418i	$-0.382305\pi$
$673$	18.7500	0.722761	0.361381	+	0.932418i	$0.382305\pi$
$674$	0	0
$675$	38.7027	1.48967
$676$	0	0
$677$	−39.8403	−1.53119	−0.765593	−	0.643326i	$-0.777554\pi$
$677$	−39.8403	−1.53119	−0.765593	+	0.643326i	$0.777554\pi$
$678$	0	0
$679$	18.2504	0.700387
$680$	0	0
$681$	−29.2844	−1.12218
$682$	0	0
$683$	−1.63828	−0.0626870	−0.0313435	−	0.999509i	$-0.509979\pi$
$683$	−1.63828	−0.0626870	−0.0313435	+	0.999509i	$0.509979\pi$
$684$	0	0
$685$	54.5474	2.08415
$686$	0	0
$687$	−18.5916	−0.709313
$688$	0	0
$689$	−20.5188	−0.781703
$690$	0	0
$691$	6.90234	0.262577	0.131289	−	0.991344i	$-0.458089\pi$
$691$	6.90234	0.262577	0.131289	+	0.991344i	$0.458089\pi$
$692$	0	0
$693$	−3.74107	−0.142111
$694$	0	0
$695$	−29.4024	−1.11530
$696$	0	0
$697$	−7.07783	−0.268092
$698$	0	0
$699$	26.3643	0.997190
$700$	0	0
$701$	22.9499	0.866806	0.433403	−	0.901200i	$-0.357313\pi$
$701$	22.9499	0.866806	0.433403	+	0.901200i	$0.357313\pi$
$702$	0	0
$703$	6.67320	0.251685
$704$	0	0
$705$	−1.84949	−0.0696558
$706$	0	0
$707$	34.2059	1.28645
$708$	0	0
$709$	23.2424	0.872888	0.436444	−	0.899731i	$-0.356238\pi$
$709$	23.2424	0.872888	0.436444	+	0.899731i	$0.356238\pi$
$710$	0	0
$711$	7.08412	0.265675
$712$	0	0
$713$	10.1705	0.380888
$714$	0	0
$715$	−54.8213	−2.05020
$716$	0	0
$717$	39.1157	1.46080
$718$	0	0
$719$	−31.8250	−1.18687	−0.593436	−	0.804881i	$-0.702229\pi$
$719$	−31.8250	−1.18687	−0.593436	+	0.804881i	$0.702229\pi$
$720$	0	0
$721$	−0.0216172	−0.000805067 0
$722$	0	0
$723$	6.06882	0.225702
$724$	0	0
$725$	−9.60071	−0.356561
$726$	0	0
$727$	21.1358	0.783884	0.391942	−	0.919990i	$-0.371803\pi$
$727$	21.1358	0.783884	0.391942	+	0.919990i	$0.371803\pi$
$728$	0	0
$729$	29.8759	1.10651
$730$	0	0
$731$	10.0408	0.371374
$732$	0	0
$733$	8.95844	0.330887	0.165444	−	0.986219i	$-0.447094\pi$
$733$	8.95844	0.330887	0.165444	+	0.986219i	$0.447094\pi$
$734$	0	0
$735$	18.3083	0.675312
$736$	0	0
$737$	−8.17909	−0.301281
$738$	0	0
$739$	53.1219	1.95412	0.977061	−	0.212959i	$-0.0683101\pi$
$739$	53.1219	1.95412	0.977061	+	0.212959i	$0.0683101\pi$
$740$	0	0
$741$	6.64519	0.244117
$742$	0	0
$743$	28.9831	1.06329	0.531644	−	0.846968i	$-0.321575\pi$
$743$	28.9831	1.06329	0.531644	+	0.846968i	$0.321575\pi$
$744$	0	0
$745$	40.2950	1.47629
$746$	0	0
$747$	4.09076	0.149673
$748$	0	0
$749$	−10.5319	−0.384828
$750$	0	0
$751$	53.8673	1.96564	0.982822	−	0.184556i	$-0.0590848\pi$
$751$	53.8673	1.96564	0.982822	+	0.184556i	$0.0590848\pi$
$752$	0	0
$753$	1.58102	0.0576154
$754$	0	0
$755$	49.0876	1.78648
$756$	0	0
$757$	16.4588	0.598203	0.299102	−	0.954221i	$-0.403313\pi$
$757$	16.4588	0.598203	0.299102	+	0.954221i	$0.403313\pi$
$758$	0	0
$759$	8.08910	0.293616
$760$	0	0
$761$	24.0492	0.871784	0.435892	−	0.899999i	$-0.356433\pi$
$761$	24.0492	0.871784	0.435892	+	0.899999i	$0.356433\pi$
$762$	0	0
$763$	−19.3447	−0.700327
$764$	0	0
$765$	4.43191	0.160236
$766$	0	0
$767$	23.6971	0.855653
$768$	0	0
$769$	−3.39072	−0.122272	−0.0611362	−	0.998129i	$-0.519472\pi$
$769$	−3.39072	−0.122272	−0.0611362	+	0.998129i	$0.519472\pi$
$770$	0	0
$771$	−32.2906	−1.16292
$772$	0	0
$773$	45.7425	1.64524	0.822621	−	0.568590i	$-0.192511\pi$
$773$	45.7425	1.64524	0.822621	+	0.568590i	$0.192511\pi$
$774$	0	0
$775$	54.3553	1.95250
$776$	0	0
$777$	−19.4320	−0.697120
$778$	0	0
$779$	−2.87313	−0.102940
$780$	0	0
$781$	14.4963	0.518719
$782$	0	0
$783$	−7.59743	−0.271510
$784$	0	0
$785$	−37.0082	−1.32088
$786$	0	0
$787$	3.80234	0.135539	0.0677693	−	0.997701i	$-0.478412\pi$
$787$	3.80234	0.135539	0.0677693	+	0.997701i	$0.478412\pi$
$788$	0	0
$789$	34.4344	1.22590
$790$	0	0
$791$	6.76518	0.240542
$792$	0	0
$793$	−35.3383	−1.25490
$794$	0	0
$795$	27.7495	0.984174
$796$	0	0
$797$	44.5390	1.57765	0.788826	−	0.614617i	$-0.210689\pi$
$797$	44.5390	1.57765	0.788826	+	0.614617i	$0.210689\pi$
$798$	0	0
$799$	−0.863887	−0.0305621
$800$	0	0
$801$	−0.909222	−0.0321258
$802$	0	0
$803$	2.89705	0.102235
$804$	0	0
$805$	−8.66515	−0.305406
$806$	0	0
$807$	9.88257	0.347883
$808$	0	0
$809$	−49.0602	−1.72486	−0.862432	−	0.506172i	$-0.831060\pi$
$809$	−49.0602	−1.72486	−0.862432	+	0.506172i	$0.831060\pi$
$810$	0	0
$811$	2.00713	0.0704797	0.0352399	−	0.999379i	$-0.488780\pi$
$811$	2.00713	0.0704797	0.0352399	+	0.999379i	$0.488780\pi$
$812$	0	0
$813$	50.3438	1.76564
$814$	0	0
$815$	−87.8833	−3.07842
$816$	0	0
$817$	4.07591	0.142598
$818$	0	0
$819$	3.87369	0.135358
$820$	0	0
$821$	−7.05193	−0.246114	−0.123057	−	0.992400i	$-0.539270\pi$
$821$	−7.05193	−0.246114	−0.123057	+	0.992400i	$0.539270\pi$
$822$	0	0
$823$	−16.4056	−0.571864	−0.285932	−	0.958250i	$-0.592303\pi$
$823$	−16.4056	−0.571864	−0.285932	+	0.958250i	$0.592303\pi$
$824$	0	0
$825$	43.2314	1.50513
$826$	0	0
$827$	−17.9968	−0.625810	−0.312905	−	0.949784i	$-0.601302\pi$
$827$	−17.9968	−0.625810	−0.312905	+	0.949784i	$0.601302\pi$
$828$	0	0
$829$	20.8256	0.723303	0.361651	−	0.932313i	$-0.382213\pi$
$829$	20.8256	0.723303	0.361651	+	0.932313i	$0.382213\pi$
$830$	0	0
$831$	−26.3190	−0.912997
$832$	0	0
$833$	8.55172	0.296299
$834$	0	0
$835$	−71.8635	−2.48694
$836$	0	0
$837$	43.0136	1.48677
$838$	0	0
$839$	−10.0454	−0.346805	−0.173403	−	0.984851i	$-0.555476\pi$
$839$	−10.0454	−0.346805	−0.173403	+	0.984851i	$0.555476\pi$
$840$	0	0
$841$	−27.1154	−0.935012
$842$	0	0
$843$	28.8581	0.993926
$844$	0	0
$845$	11.7438	0.403999
$846$	0	0
$847$	−8.19902	−0.281722
$848$	0	0
$849$	−12.3244	−0.422972
$850$	0	0
$851$	−8.41116	−0.288331
$852$	0	0
$853$	−24.5633	−0.841030	−0.420515	−	0.907286i	$-0.638151\pi$
$853$	−24.5633	−0.841030	−0.420515	+	0.907286i	$0.638151\pi$
$854$	0	0
$855$	1.79906	0.0615265
$856$	0	0
$857$	47.2624	1.61445	0.807227	−	0.590241i	$-0.200967\pi$
$857$	47.2624	1.61445	0.807227	+	0.590241i	$0.200967\pi$
$858$	0	0
$859$	−29.1419	−0.994308	−0.497154	−	0.867662i	$-0.665622\pi$
$859$	−29.1419	−0.994308	−0.497154	+	0.867662i	$0.665622\pi$
$860$	0	0
$861$	8.36639	0.285126
$862$	0	0
$863$	13.0615	0.444618	0.222309	−	0.974976i	$-0.428641\pi$
$863$	13.0615	0.444618	0.222309	+	0.974976i	$0.428641\pi$
$864$	0	0
$865$	−50.2605	−1.70891
$866$	0	0
$867$	16.5363	0.561601
$868$	0	0
$869$	55.3546	1.87778
$870$	0	0
$871$	8.46904	0.286963
$872$	0	0
$873$	4.77600	0.161643
$874$	0	0
$875$	−13.2003	−0.446251
$876$	0	0
$877$	−9.29082	−0.313729	−0.156864	−	0.987620i	$-0.550139\pi$
$877$	−9.29082	−0.313729	−0.156864	+	0.987620i	$0.550139\pi$
$878$	0	0
$879$	42.8279	1.44455
$880$	0	0
$881$	−49.4088	−1.66462	−0.832312	−	0.554308i	$-0.812983\pi$
$881$	−49.4088	−1.66462	−0.832312	+	0.554308i	$0.812983\pi$
$882$	0	0
$883$	−37.4941	−1.26178	−0.630888	−	0.775874i	$-0.717309\pi$
$883$	−37.4941	−1.26178	−0.630888	+	0.775874i	$0.717309\pi$
$884$	0	0
$885$	−32.0479	−1.07728
$886$	0	0
$887$	1.41214	0.0474151	0.0237075	−	0.999719i	$-0.492453\pi$
$887$	1.41214	0.0474151	0.0237075	+	0.999719i	$0.492453\pi$
$888$	0	0
$889$	4.19401	0.140662
$890$	0	0
$891$	28.3413	0.949468
$892$	0	0
$893$	−0.350680	−0.0117351
$894$	0	0
$895$	−3.98800	−0.133304
$896$	0	0
$897$	−8.37586	−0.279662
$898$	0	0
$899$	−10.6701	−0.355867
$900$	0	0
$901$	12.9617	0.431816
$902$	0	0
$903$	−11.8688	−0.394970
$904$	0	0
$905$	35.4580	1.17866
$906$	0	0
$907$	24.0932	0.800000	0.400000	−	0.916515i	$-0.369010\pi$
$907$	24.0932	0.800000	0.400000	+	0.916515i	$0.369010\pi$
$908$	0	0
$909$	8.95143	0.296900
$910$	0	0
$911$	55.5178	1.83939	0.919693	−	0.392638i	$-0.128438\pi$
$911$	55.5178	1.83939	0.919693	+	0.392638i	$0.128438\pi$
$912$	0	0
$913$	31.9648	1.05788
$914$	0	0
$915$	47.7915	1.57994
$916$	0	0
$917$	−16.0957	−0.531528
$918$	0	0
$919$	−22.7377	−0.750048	−0.375024	−	0.927015i	$-0.622366\pi$
$919$	−22.7377	−0.750048	−0.375024	+	0.927015i	$0.622366\pi$
$920$	0	0
$921$	−22.4526	−0.739839
$922$	0	0
$923$	−15.0102	−0.494068
$924$	0	0
$925$	−44.9527	−1.47804
$926$	0	0
$927$	−0.00565707	−0.000185802 0
$928$	0	0
$929$	−26.3904	−0.865840	−0.432920	−	0.901432i	$-0.642517\pi$
$929$	−26.3904	−0.865840	−0.432920	+	0.901432i	$0.642517\pi$
$930$	0	0
$931$	3.47143	0.113771
$932$	0	0
$933$	3.99343	0.130739
$934$	0	0
$935$	34.6305	1.13254
$936$	0	0
$937$	28.3468	0.926049	0.463024	−	0.886346i	$-0.346764\pi$
$937$	28.3468	0.926049	0.463024	+	0.886346i	$0.346764\pi$
$938$	0	0
$939$	−0.368906	−0.0120388
$940$	0	0
$941$	36.8974	1.20282	0.601411	−	0.798940i	$-0.294605\pi$
$941$	36.8974	1.20282	0.601411	+	0.798940i	$0.294605\pi$
$942$	0	0
$943$	3.62140	0.117929
$944$	0	0
$945$	−36.6470	−1.19213
$946$	0	0
$947$	25.1399	0.816938	0.408469	−	0.912772i	$-0.366063\pi$
$947$	25.1399	0.816938	0.408469	+	0.912772i	$0.366063\pi$
$948$	0	0
$949$	−2.99975	−0.0973759
$950$	0	0
$951$	−27.6657	−0.897123
$952$	0	0
$953$	20.3606	0.659546	0.329773	−	0.944060i	$-0.393028\pi$
$953$	20.3606	0.659546	0.329773	+	0.944060i	$0.393028\pi$
$954$	0	0
$955$	−30.8432	−0.998063
$956$	0	0
$957$	−8.48642	−0.274327
$958$	0	0
$959$	−30.1174	−0.972541
$960$	0	0
$961$	29.4096	0.948696
$962$	0	0
$963$	−2.75613	−0.0888150
$964$	0	0
$965$	1.16224	0.0374138
$966$	0	0
$967$	22.6957	0.729845	0.364923	−	0.931038i	$-0.381095\pi$
$967$	22.6957	0.729845	0.364923	+	0.931038i	$0.381095\pi$
$968$	0	0
$969$	−4.19775	−0.134851
$970$	0	0
$971$	−45.8164	−1.47032	−0.735159	−	0.677895i	$-0.762892\pi$
$971$	−45.8164	−1.47032	−0.735159	+	0.677895i	$0.762892\pi$
$972$	0	0
$973$	16.2341	0.520440
$974$	0	0
$975$	−44.7640	−1.43360
$976$	0	0
$977$	45.1259	1.44371	0.721853	−	0.692047i	$-0.243291\pi$
$977$	45.1259	1.44371	0.721853	+	0.692047i	$0.243291\pi$
$978$	0	0
$979$	−7.10457	−0.227063
$980$	0	0
$981$	−5.06238	−0.161629
$982$	0	0
$983$	0.423126	0.0134956	0.00674782	−	0.999977i	$-0.497852\pi$
$983$	0.423126	0.0134956	0.00674782	+	0.999977i	$0.497852\pi$
$984$	0	0
$985$	21.9433	0.699172
$986$	0	0
$987$	1.02116	0.0325040
$988$	0	0
$989$	−5.13743	−0.163361
$990$	0	0
$991$	−5.36808	−0.170523	−0.0852613	−	0.996359i	$-0.527173\pi$
$991$	−5.36808	−0.170523	−0.0852613	+	0.996359i	$0.527173\pi$
$992$	0	0
$993$	−25.8232	−0.819475
$994$	0	0
$995$	−24.2627	−0.769181
$996$	0	0
$997$	−17.0653	−0.540463	−0.270231	−	0.962795i	$-0.587100\pi$
$997$	−17.0653	−0.540463	−0.270231	+	0.962795i	$0.587100\pi$
$998$	0	0
$999$	−35.5729	−1.12548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.4		12
4.3	odd	2		2008.2.a.b.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.9	✓	12	4.3	odd	2
4016.2.a.i.1.4		12	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58102 q^{3} +3.46315 q^{5} -1.91212 q^{7} -0.500388 q^{9} +O(q^{10})\)	\(q-1.58102 q^{3} +3.46315 q^{5} -1.91212 q^{7} -0.500388 q^{9} -3.90998 q^{11} +4.04859 q^{13} -5.47530 q^{15} -2.55749 q^{17} -1.03817 q^{19} +3.02309 q^{21} +1.30855 q^{23} +6.99341 q^{25} +5.53417 q^{27} -1.37282 q^{29} +7.77236 q^{31} +6.18174 q^{33} -6.62196 q^{35} -6.42786 q^{37} -6.40088 q^{39} +2.76750 q^{41} -3.92606 q^{43} -1.73292 q^{45} +0.337788 q^{47} -3.34380 q^{49} +4.04343 q^{51} -5.06813 q^{53} -13.5408 q^{55} +1.64136 q^{57} +5.85317 q^{59} -8.72856 q^{61} +0.956801 q^{63} +14.0209 q^{65} +2.09185 q^{67} -2.06883 q^{69} -3.70752 q^{71} -0.740937 q^{73} -11.0567 q^{75} +7.47634 q^{77} -14.1573 q^{79} -7.24845 q^{81} -8.17518 q^{83} -8.85696 q^{85} +2.17045 q^{87} +1.81704 q^{89} -7.74139 q^{91} -12.2882 q^{93} -3.59533 q^{95} -9.54460 q^{97} +1.95650 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

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