LMFDB - Embedded newform 4016.2.a.l.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:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.99202$ 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)

$f(q)$	$=$	$q-1.99202 q^{3} +2.61608 q^{5} +4.58459 q^{7} +0.968146 q^{9} +O(q^{10})$	$q-1.99202 q^{3} +2.61608 q^{5} +4.58459 q^{7} +0.968146 q^{9} +4.22135 q^{11} -4.74601 q^{13} -5.21129 q^{15} -3.73618 q^{17} -3.23893 q^{19} -9.13260 q^{21} +6.76346 q^{23} +1.84388 q^{25} +4.04750 q^{27} +7.19117 q^{29} +1.31451 q^{31} -8.40901 q^{33} +11.9937 q^{35} -3.53391 q^{37} +9.45415 q^{39} +7.91005 q^{41} -2.99922 q^{43} +2.53275 q^{45} +4.75392 q^{47} +14.0185 q^{49} +7.44256 q^{51} +2.40706 q^{53} +11.0434 q^{55} +6.45201 q^{57} +7.15519 q^{59} -8.22898 q^{61} +4.43855 q^{63} -12.4160 q^{65} -0.0714429 q^{67} -13.4730 q^{69} +3.10885 q^{71} -10.8201 q^{73} -3.67305 q^{75} +19.3532 q^{77} -4.94584 q^{79} -10.9671 q^{81} -12.3058 q^{83} -9.77416 q^{85} -14.3250 q^{87} +18.2011 q^{89} -21.7585 q^{91} -2.61853 q^{93} -8.47330 q^{95} -2.58258 q^{97} +4.08688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * 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
$3$	−1.99202	−1.15009	−0.575047	−	0.818121i	$-0.695016\pi$
$3$	−1.99202	−1.15009	−0.575047	+	0.818121i	$0.695016\pi$
$4$	0	0
$5$	2.61608	1.16995	0.584974	−	0.811052i	$-0.301105\pi$
$5$	2.61608	1.16995	0.584974	+	0.811052i	$0.301105\pi$
$6$	0	0
$7$	4.58459	1.73281	0.866406	−	0.499340i	$-0.166424\pi$
$7$	4.58459	1.73281	0.866406	+	0.499340i	$0.166424\pi$
$8$	0	0
$9$	0.968146	0.322715
$10$	0	0
$11$	4.22135	1.27278	0.636392	−	0.771366i	$-0.280426\pi$
$11$	4.22135	1.27278	0.636392	+	0.771366i	$0.280426\pi$
$12$	0	0
$13$	−4.74601	−1.31631	−0.658153	−	0.752884i	$-0.728662\pi$
$13$	−4.74601	−1.31631	−0.658153	+	0.752884i	$0.728662\pi$
$14$	0	0
$15$	−5.21129	−1.34555
$16$	0	0
$17$	−3.73618	−0.906158	−0.453079	−	0.891470i	$-0.649674\pi$
$17$	−3.73618	−0.906158	−0.453079	+	0.891470i	$0.649674\pi$
$18$	0	0
$19$	−3.23893	−0.743061	−0.371531	−	0.928421i	$-0.621167\pi$
$19$	−3.23893	−0.743061	−0.371531	+	0.928421i	$0.621167\pi$
$20$	0	0
$21$	−9.13260	−1.99290
$22$	0	0
$23$	6.76346	1.41028	0.705140	−	0.709068i	$-0.250884\pi$
$23$	6.76346	1.41028	0.705140	+	0.709068i	$0.250884\pi$
$24$	0	0
$25$	1.84388	0.368777
$26$	0	0
$27$	4.04750	0.778941
$28$	0	0
$29$	7.19117	1.33537	0.667684	−	0.744445i	$-0.267286\pi$
$29$	7.19117	1.33537	0.667684	+	0.744445i	$0.267286\pi$
$30$	0	0
$31$	1.31451	0.236093	0.118046	−	0.993008i	$-0.462337\pi$
$31$	1.31451	0.236093	0.118046	+	0.993008i	$0.462337\pi$
$32$	0	0
$33$	−8.40901	−1.46382
$34$	0	0
$35$	11.9937	2.02730
$36$	0	0
$37$	−3.53391	−0.580971	−0.290485	−	0.956879i	$-0.593817\pi$
$37$	−3.53391	−0.580971	−0.290485	+	0.956879i	$0.593817\pi$
$38$	0	0
$39$	9.45415	1.51388
$40$	0	0
$41$	7.91005	1.23534	0.617671	−	0.786437i	$-0.288076\pi$
$41$	7.91005	1.23534	0.617671	+	0.786437i	$0.288076\pi$
$42$	0	0
$43$	−2.99922	−0.457377	−0.228689	−	0.973500i	$-0.573444\pi$
$43$	−2.99922	−0.457377	−0.228689	+	0.973500i	$0.573444\pi$
$44$	0	0
$45$	2.53275	0.377560
$46$	0	0
$47$	4.75392	0.693431	0.346715	−	0.937970i	$-0.387297\pi$
$47$	4.75392	0.693431	0.346715	+	0.937970i	$0.387297\pi$
$48$	0	0
$49$	14.0185	2.00264
$50$	0	0
$51$	7.44256	1.04217
$52$	0	0
$53$	2.40706	0.330635	0.165317	−	0.986240i	$-0.447135\pi$
$53$	2.40706	0.330635	0.165317	+	0.986240i	$0.447135\pi$
$54$	0	0
$55$	11.0434	1.48909
$56$	0	0
$57$	6.45201	0.854590
$58$	0	0
$59$	7.15519	0.931526	0.465763	−	0.884909i	$-0.345780\pi$
$59$	7.15519	0.931526	0.465763	+	0.884909i	$0.345780\pi$
$60$	0	0
$61$	−8.22898	−1.05361	−0.526807	−	0.849985i	$-0.676611\pi$
$61$	−8.22898	−1.05361	−0.526807	+	0.849985i	$0.676611\pi$
$62$	0	0
$63$	4.43855	0.559205
$64$	0	0
$65$	−12.4160	−1.54001
$66$	0	0
$67$	−0.0714429	−0.00872814	−0.00436407	−	0.999990i	$-0.501389\pi$
$67$	−0.0714429	−0.00872814	−0.00436407	+	0.999990i	$0.501389\pi$
$68$	0	0
$69$	−13.4730	−1.62195
$70$	0	0
$71$	3.10885	0.368953	0.184476	−	0.982837i	$-0.440941\pi$
$71$	3.10885	0.368953	0.184476	+	0.982837i	$0.440941\pi$
$72$	0	0
$73$	−10.8201	−1.26640	−0.633201	−	0.773987i	$-0.718259\pi$
$73$	−10.8201	−1.26640	−0.633201	+	0.773987i	$0.718259\pi$
$74$	0	0
$75$	−3.67305	−0.424128
$76$	0	0
$77$	19.3532	2.20550
$78$	0	0
$79$	−4.94584	−0.556451	−0.278225	−	0.960516i	$-0.589746\pi$
$79$	−4.94584	−0.556451	−0.278225	+	0.960516i	$0.589746\pi$
$80$	0	0
$81$	−10.9671	−1.21857
$82$	0	0
$83$	−12.3058	−1.35073	−0.675367	−	0.737482i	$-0.736015\pi$
$83$	−12.3058	−1.35073	−0.675367	+	0.737482i	$0.736015\pi$
$84$	0	0
$85$	−9.77416	−1.06016
$86$	0	0
$87$	−14.3250	−1.53580
$88$	0	0
$89$	18.2011	1.92931	0.964655	−	0.263515i	$-0.0848819\pi$
$89$	18.2011	1.92931	0.964655	+	0.263515i	$0.0848819\pi$
$90$	0	0
$91$	−21.7585	−2.28091
$92$	0	0
$93$	−2.61853	−0.271529
$94$	0	0
$95$	−8.47330	−0.869342
$96$	0	0
$97$	−2.58258	−0.262222	−0.131111	−	0.991368i	$-0.541854\pi$
$97$	−2.58258	−0.262222	−0.131111	+	0.991368i	$0.541854\pi$
$98$	0	0
$99$	4.08688	0.410747
$100$	0	0
$101$	16.5975	1.65152	0.825758	−	0.564024i	$-0.190747\pi$
$101$	16.5975	1.65152	0.825758	+	0.564024i	$0.190747\pi$
$102$	0	0
$103$	10.1249	0.997639	0.498819	−	0.866706i	$-0.333767\pi$
$103$	10.1249	0.997639	0.498819	+	0.866706i	$0.333767\pi$
$104$	0	0
$105$	−23.8916	−2.33158
$106$	0	0
$107$	3.08856	0.298582	0.149291	−	0.988793i	$-0.452301\pi$
$107$	3.08856	0.298582	0.149291	+	0.988793i	$0.452301\pi$
$108$	0	0
$109$	18.9145	1.81168	0.905840	−	0.423620i	$-0.139241\pi$
$109$	18.9145	1.81168	0.905840	+	0.423620i	$0.139241\pi$
$110$	0	0
$111$	7.03961	0.668171
$112$	0	0
$113$	−17.2958	−1.62705	−0.813526	−	0.581529i	$-0.802455\pi$
$113$	−17.2958	−1.62705	−0.813526	+	0.581529i	$0.802455\pi$
$114$	0	0
$115$	17.6938	1.64995
$116$	0	0
$117$	−4.59483	−0.424792
$118$	0	0
$119$	−17.1289	−1.57020
$120$	0	0
$121$	6.81978	0.619980
$122$	0	0
$123$	−15.7570	−1.42076
$124$	0	0
$125$	−8.25666	−0.738498
$126$	0	0
$127$	10.6612	0.946024	0.473012	−	0.881056i	$-0.343167\pi$
$127$	10.6612	0.946024	0.473012	+	0.881056i	$0.343167\pi$
$128$	0	0
$129$	5.97451	0.526027
$130$	0	0
$131$	−14.6650	−1.28129	−0.640645	−	0.767837i	$-0.721333\pi$
$131$	−14.6650	−1.28129	−0.640645	+	0.767837i	$0.721333\pi$
$132$	0	0
$133$	−14.8492	−1.28759
$134$	0	0
$135$	10.5886	0.911320
$136$	0	0
$137$	5.47918	0.468118	0.234059	−	0.972222i	$-0.424799\pi$
$137$	5.47918	0.468118	0.234059	+	0.972222i	$0.424799\pi$
$138$	0	0
$139$	−11.2323	−0.952710	−0.476355	−	0.879253i	$-0.658042\pi$
$139$	−11.2323	−0.952710	−0.476355	+	0.879253i	$0.658042\pi$
$140$	0	0
$141$	−9.46991	−0.797510
$142$	0	0
$143$	−20.0346	−1.67537
$144$	0	0
$145$	18.8127	1.56231
$146$	0	0
$147$	−27.9251	−2.30322
$148$	0	0
$149$	−18.8250	−1.54221	−0.771103	−	0.636710i	$-0.780295\pi$
$149$	−18.8250	−1.54221	−0.771103	+	0.636710i	$0.780295\pi$
$150$	0	0
$151$	2.00490	0.163157	0.0815783	−	0.996667i	$-0.474004\pi$
$151$	2.00490	0.163157	0.0815783	+	0.996667i	$0.474004\pi$
$152$	0	0
$153$	−3.61717	−0.292431
$154$	0	0
$155$	3.43886	0.276216
$156$	0	0
$157$	8.62339	0.688221	0.344111	−	0.938929i	$-0.388180\pi$
$157$	8.62339	0.688221	0.344111	+	0.938929i	$0.388180\pi$
$158$	0	0
$159$	−4.79491	−0.380261
$160$	0	0
$161$	31.0077	2.44375
$162$	0	0
$163$	7.33477	0.574503	0.287252	−	0.957855i	$-0.407258\pi$
$163$	7.33477	0.574503	0.287252	+	0.957855i	$0.407258\pi$
$164$	0	0
$165$	−21.9987	−1.71259
$166$	0	0
$167$	17.5277	1.35633	0.678167	−	0.734908i	$-0.262775\pi$
$167$	17.5277	1.35633	0.678167	+	0.734908i	$0.262775\pi$
$168$	0	0
$169$	9.52462	0.732663
$170$	0	0
$171$	−3.13575	−0.239797
$172$	0	0
$173$	6.79348	0.516499	0.258250	−	0.966078i	$-0.416854\pi$
$173$	6.79348	0.516499	0.258250	+	0.966078i	$0.416854\pi$
$174$	0	0
$175$	8.45345	0.639021
$176$	0	0
$177$	−14.2533	−1.07134
$178$	0	0
$179$	−22.5551	−1.68585	−0.842924	−	0.538033i	$-0.819168\pi$
$179$	−22.5551	−1.68585	−0.842924	+	0.538033i	$0.819168\pi$
$180$	0	0
$181$	−18.3794	−1.36613	−0.683065	−	0.730358i	$-0.739353\pi$
$181$	−18.3794	−1.36613	−0.683065	+	0.730358i	$0.739353\pi$
$182$	0	0
$183$	16.3923	1.21175
$184$	0	0
$185$	−9.24499	−0.679705
$186$	0	0
$187$	−15.7717	−1.15334
$188$	0	0
$189$	18.5561	1.34976
$190$	0	0
$191$	21.3252	1.54304	0.771518	−	0.636208i	$-0.219498\pi$
$191$	21.3252	1.54304	0.771518	+	0.636208i	$0.219498\pi$
$192$	0	0
$193$	3.72303	0.267990	0.133995	−	0.990982i	$-0.457219\pi$
$193$	3.72303	0.267990	0.133995	+	0.990982i	$0.457219\pi$
$194$	0	0
$195$	24.7328	1.77115
$196$	0	0
$197$	23.5637	1.67885	0.839423	−	0.543479i	$-0.182893\pi$
$197$	23.5637	1.67885	0.839423	+	0.543479i	$0.182893\pi$
$198$	0	0
$199$	12.4113	0.879811	0.439906	−	0.898044i	$-0.355012\pi$
$199$	12.4113	0.879811	0.439906	+	0.898044i	$0.355012\pi$
$200$	0	0
$201$	0.142316	0.0100382
$202$	0	0
$203$	32.9686	2.31394
$204$	0	0
$205$	20.6933	1.44528
$206$	0	0
$207$	6.54802	0.455119
$208$	0	0
$209$	−13.6726	−0.945756
$210$	0	0
$211$	11.9863	0.825171	0.412585	−	0.910919i	$-0.364626\pi$
$211$	11.9863	0.825171	0.412585	+	0.910919i	$0.364626\pi$
$212$	0	0
$213$	−6.19290	−0.424330
$214$	0	0
$215$	−7.84621	−0.535107
$216$	0	0
$217$	6.02648	0.409104
$218$	0	0
$219$	21.5540	1.45648
$220$	0	0
$221$	17.7320	1.19278
$222$	0	0
$223$	20.3984	1.36598	0.682990	−	0.730428i	$-0.260679\pi$
$223$	20.3984	1.36598	0.682990	+	0.730428i	$0.260679\pi$
$224$	0	0
$225$	1.78515	0.119010
$226$	0	0
$227$	6.64485	0.441034	0.220517	−	0.975383i	$-0.429226\pi$
$227$	6.64485	0.441034	0.220517	+	0.975383i	$0.429226\pi$
$228$	0	0
$229$	−12.9271	−0.854245	−0.427122	−	0.904194i	$-0.640473\pi$
$229$	−12.9271	−0.854245	−0.427122	+	0.904194i	$0.640473\pi$
$230$	0	0
$231$	−38.5519	−2.53653
$232$	0	0
$233$	−26.9392	−1.76485	−0.882423	−	0.470456i	$-0.844089\pi$
$233$	−26.9392	−1.76485	−0.882423	+	0.470456i	$0.844089\pi$
$234$	0	0
$235$	12.4366	0.811277
$236$	0	0
$237$	9.85222	0.639970
$238$	0	0
$239$	24.1979	1.56523	0.782616	−	0.622505i	$-0.213885\pi$
$239$	24.1979	1.56523	0.782616	+	0.622505i	$0.213885\pi$
$240$	0	0
$241$	−2.82422	−0.181924	−0.0909619	−	0.995854i	$-0.528994\pi$
$241$	−2.82422	−0.181924	−0.0909619	+	0.995854i	$0.528994\pi$
$242$	0	0
$243$	9.70426	0.622529
$244$	0	0
$245$	36.6735	2.34298
$246$	0	0
$247$	15.3720	0.978096
$248$	0	0
$249$	24.5134	1.55347
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	28.5509	1.79498
$254$	0	0
$255$	19.4703	1.21928
$256$	0	0
$257$	−19.0301	−1.18706	−0.593532	−	0.804811i	$-0.702267\pi$
$257$	−19.0301	−1.18706	−0.593532	+	0.804811i	$0.702267\pi$
$258$	0	0
$259$	−16.2015	−1.00671
$260$	0	0
$261$	6.96211	0.430943
$262$	0	0
$263$	0.0505579	0.00311754	0.00155877	−	0.999999i	$-0.499504\pi$
$263$	0.0505579	0.00311754	0.00155877	+	0.999999i	$0.499504\pi$
$264$	0	0
$265$	6.29706	0.386826
$266$	0	0
$267$	−36.2569	−2.21889
$268$	0	0
$269$	28.2684	1.72355	0.861776	−	0.507289i	$-0.169352\pi$
$269$	28.2684	1.72355	0.861776	+	0.507289i	$0.169352\pi$
$270$	0	0
$271$	8.95702	0.544100	0.272050	−	0.962283i	$-0.412298\pi$
$271$	8.95702	0.544100	0.272050	+	0.962283i	$0.412298\pi$
$272$	0	0
$273$	43.3434	2.62326
$274$	0	0
$275$	7.78367	0.469373
$276$	0	0
$277$	−17.2645	−1.03732	−0.518662	−	0.854979i	$-0.673570\pi$
$277$	−17.2645	−1.03732	−0.518662	+	0.854979i	$0.673570\pi$
$278$	0	0
$279$	1.27264	0.0761907
$280$	0	0
$281$	11.5375	0.688272	0.344136	−	0.938920i	$-0.388172\pi$
$281$	11.5375	0.688272	0.344136	+	0.938920i	$0.388172\pi$
$282$	0	0
$283$	−7.84379	−0.466265	−0.233132	−	0.972445i	$-0.574898\pi$
$283$	−7.84379	−0.466265	−0.233132	+	0.972445i	$0.574898\pi$
$284$	0	0
$285$	16.8790	0.999825
$286$	0	0
$287$	36.2643	2.14062
$288$	0	0
$289$	−3.04092	−0.178878
$290$	0	0
$291$	5.14456	0.301579
$292$	0	0
$293$	−22.2737	−1.30125	−0.650623	−	0.759401i	$-0.725492\pi$
$293$	−22.2737	−1.30125	−0.650623	+	0.759401i	$0.725492\pi$
$294$	0	0
$295$	18.7186	1.08984
$296$	0	0
$297$	17.0859	0.991424
$298$	0	0
$299$	−32.0995	−1.85636
$300$	0	0
$301$	−13.7502	−0.792549
$302$	0	0
$303$	−33.0626	−1.89940
$304$	0	0
$305$	−21.5277	−1.23267
$306$	0	0
$307$	−18.5512	−1.05877	−0.529387	−	0.848381i	$-0.677578\pi$
$307$	−18.5512	−1.05877	−0.529387	+	0.848381i	$0.677578\pi$
$308$	0	0
$309$	−20.1691	−1.14738
$310$	0	0
$311$	−9.68250	−0.549044	−0.274522	−	0.961581i	$-0.588520\pi$
$311$	−9.68250	−0.549044	−0.274522	+	0.961581i	$0.588520\pi$
$312$	0	0
$313$	−4.09050	−0.231209	−0.115604	−	0.993295i	$-0.536880\pi$
$313$	−4.09050	−0.231209	−0.115604	+	0.993295i	$0.536880\pi$
$314$	0	0
$315$	11.6116	0.654240
$316$	0	0
$317$	−10.2984	−0.578416	−0.289208	−	0.957266i	$-0.593392\pi$
$317$	−10.2984	−0.578416	−0.289208	+	0.957266i	$0.593392\pi$
$318$	0	0
$319$	30.3564	1.69963
$320$	0	0
$321$	−6.15247	−0.343398
$322$	0	0
$323$	12.1012	0.673331
$324$	0	0
$325$	−8.75109	−0.485423
$326$	0	0
$327$	−37.6781	−2.08360
$328$	0	0
$329$	21.7948	1.20159
$330$	0	0
$331$	−4.82456	−0.265182	−0.132591	−	0.991171i	$-0.542330\pi$
$331$	−4.82456	−0.265182	−0.132591	+	0.991171i	$0.542330\pi$
$332$	0	0
$333$	−3.42134	−0.187488
$334$	0	0
$335$	−0.186901	−0.0102115
$336$	0	0
$337$	30.4333	1.65781	0.828903	−	0.559393i	$-0.188966\pi$
$337$	30.4333	1.65781	0.828903	+	0.559393i	$0.188966\pi$
$338$	0	0
$339$	34.4536	1.87126
$340$	0	0
$341$	5.54899	0.300495
$342$	0	0
$343$	32.1768	1.73739
$344$	0	0
$345$	−35.2464	−1.89760
$346$	0	0
$347$	4.99220	0.267995	0.133998	−	0.990982i	$-0.457219\pi$
$347$	4.99220	0.267995	0.133998	+	0.990982i	$0.457219\pi$
$348$	0	0
$349$	−11.6691	−0.624631	−0.312315	−	0.949978i	$-0.601105\pi$
$349$	−11.6691	−0.624631	−0.312315	+	0.949978i	$0.601105\pi$
$350$	0	0
$351$	−19.2095	−1.02532
$352$	0	0
$353$	−22.1005	−1.17629	−0.588146	−	0.808755i	$-0.700142\pi$
$353$	−22.1005	−1.17629	−0.588146	+	0.808755i	$0.700142\pi$
$354$	0	0
$355$	8.13301	0.431655
$356$	0	0
$357$	34.1211	1.80588
$358$	0	0
$359$	15.8215	0.835029	0.417514	−	0.908670i	$-0.362901\pi$
$359$	15.8215	0.835029	0.417514	+	0.908670i	$0.362901\pi$
$360$	0	0
$361$	−8.50934	−0.447860
$362$	0	0
$363$	−13.5851	−0.713035
$364$	0	0
$365$	−28.3064	−1.48162
$366$	0	0
$367$	1.39366	0.0727485	0.0363742	−	0.999338i	$-0.488419\pi$
$367$	1.39366	0.0727485	0.0363742	+	0.999338i	$0.488419\pi$
$368$	0	0
$369$	7.65808	0.398664
$370$	0	0
$371$	11.0354	0.572928
$372$	0	0
$373$	37.8015	1.95729	0.978645	−	0.205558i	$-0.0659009\pi$
$373$	37.8015	1.95729	0.978645	+	0.205558i	$0.0659009\pi$
$374$	0	0
$375$	16.4474	0.849342
$376$	0	0
$377$	−34.1294	−1.75775
$378$	0	0
$379$	21.0314	1.08031	0.540155	−	0.841566i	$-0.318366\pi$
$379$	21.0314	1.08031	0.540155	+	0.841566i	$0.318366\pi$
$380$	0	0
$381$	−21.2372	−1.08802
$382$	0	0
$383$	−17.9958	−0.919542	−0.459771	−	0.888037i	$-0.652069\pi$
$383$	−17.9958	−0.919542	−0.459771	+	0.888037i	$0.652069\pi$
$384$	0	0
$385$	50.6294	2.58031
$386$	0	0
$387$	−2.90369	−0.147603
$388$	0	0
$389$	−17.5105	−0.887817	−0.443908	−	0.896072i	$-0.646408\pi$
$389$	−17.5105	−0.887817	−0.443908	+	0.896072i	$0.646408\pi$
$390$	0	0
$391$	−25.2696	−1.27794
$392$	0	0
$393$	29.2131	1.47360
$394$	0	0
$395$	−12.9387	−0.651018
$396$	0	0
$397$	−7.12384	−0.357535	−0.178768	−	0.983891i	$-0.557211\pi$
$397$	−7.12384	−0.357535	−0.178768	+	0.983891i	$0.557211\pi$
$398$	0	0
$399$	29.5798	1.48084
$400$	0	0
$401$	32.7190	1.63391	0.816954	−	0.576703i	$-0.195661\pi$
$401$	32.7190	1.63391	0.816954	+	0.576703i	$0.195661\pi$
$402$	0	0
$403$	−6.23867	−0.310770
$404$	0	0
$405$	−28.6909	−1.42566
$406$	0	0
$407$	−14.9178	−0.739450
$408$	0	0
$409$	0.552711	0.0273298	0.0136649	−	0.999907i	$-0.495650\pi$
$409$	0.552711	0.0273298	0.0136649	+	0.999907i	$0.495650\pi$
$410$	0	0
$411$	−10.9146	−0.538379
$412$	0	0
$413$	32.8036	1.61416
$414$	0	0
$415$	−32.1929	−1.58029
$416$	0	0
$417$	22.3749	1.09571
$418$	0	0
$419$	−23.3171	−1.13911	−0.569557	−	0.821952i	$-0.692885\pi$
$419$	−23.3171	−1.13911	−0.569557	+	0.821952i	$0.692885\pi$
$420$	0	0
$421$	−4.16019	−0.202755	−0.101378	−	0.994848i	$-0.532325\pi$
$421$	−4.16019	−0.202755	−0.101378	+	0.994848i	$0.532325\pi$
$422$	0	0
$423$	4.60249	0.223781
$424$	0	0
$425$	−6.88909	−0.334170
$426$	0	0
$427$	−37.7265	−1.82571
$428$	0	0
$429$	39.9093	1.92684
$430$	0	0
$431$	−3.68916	−0.177701	−0.0888504	−	0.996045i	$-0.528319\pi$
$431$	−3.68916	−0.177701	−0.0888504	+	0.996045i	$0.528319\pi$
$432$	0	0
$433$	24.1302	1.15962	0.579811	−	0.814751i	$-0.303127\pi$
$433$	24.1302	1.15962	0.579811	+	0.814751i	$0.303127\pi$
$434$	0	0
$435$	−37.4753	−1.79680
$436$	0	0
$437$	−21.9064	−1.04792
$438$	0	0
$439$	10.1344	0.483687	0.241843	−	0.970315i	$-0.422248\pi$
$439$	10.1344	0.483687	0.241843	+	0.970315i	$0.422248\pi$
$440$	0	0
$441$	13.5719	0.646282
$442$	0	0
$443$	21.3684	1.01524	0.507622	−	0.861580i	$-0.330525\pi$
$443$	21.3684	1.01524	0.507622	+	0.861580i	$0.330525\pi$
$444$	0	0
$445$	47.6155	2.25719
$446$	0	0
$447$	37.4998	1.77368
$448$	0	0
$449$	−14.4429	−0.681601	−0.340800	−	0.940136i	$-0.610698\pi$
$449$	−14.4429	−0.681601	−0.340800	+	0.940136i	$0.610698\pi$
$450$	0	0
$451$	33.3911	1.57232
$452$	0	0
$453$	−3.99380	−0.187645
$454$	0	0
$455$	−56.9221	−2.66855
$456$	0	0
$457$	28.0449	1.31189	0.655943	−	0.754810i	$-0.272271\pi$
$457$	28.0449	1.31189	0.655943	+	0.754810i	$0.272271\pi$
$458$	0	0
$459$	−15.1222	−0.705843
$460$	0	0
$461$	−26.5165	−1.23500	−0.617499	−	0.786572i	$-0.711854\pi$
$461$	−26.5165	−1.23500	−0.617499	+	0.786572i	$0.711854\pi$
$462$	0	0
$463$	−14.2530	−0.662391	−0.331196	−	0.943562i	$-0.607452\pi$
$463$	−14.2530	−0.662391	−0.331196	+	0.943562i	$0.607452\pi$
$464$	0	0
$465$	−6.85028	−0.317674
$466$	0	0
$467$	7.73837	0.358089	0.179044	−	0.983841i	$-0.442699\pi$
$467$	7.73837	0.358089	0.179044	+	0.983841i	$0.442699\pi$
$468$	0	0
$469$	−0.327537	−0.0151242
$470$	0	0
$471$	−17.1780	−0.791519
$472$	0	0
$473$	−12.6608	−0.582143
$474$	0	0
$475$	−5.97220	−0.274024
$476$	0	0
$477$	2.33038	0.106701
$478$	0	0
$479$	34.8845	1.59391	0.796957	−	0.604035i	$-0.206441\pi$
$479$	34.8845	1.59391	0.796957	+	0.604035i	$0.206441\pi$
$480$	0	0
$481$	16.7720	0.764735
$482$	0	0
$483$	−61.7680	−2.81054
$484$	0	0
$485$	−6.75625	−0.306785
$486$	0	0
$487$	−36.4439	−1.65143	−0.825716	−	0.564086i	$-0.809229\pi$
$487$	−36.4439	−1.65143	−0.825716	+	0.564086i	$0.809229\pi$
$488$	0	0
$489$	−14.6110	−0.660733
$490$	0	0
$491$	10.8494	0.489628	0.244814	−	0.969570i	$-0.421273\pi$
$491$	10.8494	0.489628	0.244814	+	0.969570i	$0.421273\pi$
$492$	0	0
$493$	−26.8676	−1.21005
$494$	0	0
$495$	10.6916	0.480552
$496$	0	0
$497$	14.2528	0.639326
$498$	0	0
$499$	19.6632	0.880244	0.440122	−	0.897938i	$-0.354935\pi$
$499$	19.6632	0.880244	0.440122	+	0.897938i	$0.354935\pi$
$500$	0	0
$501$	−34.9155	−1.55991
$502$	0	0
$503$	−5.27745	−0.235310	−0.117655	−	0.993055i	$-0.537538\pi$
$503$	−5.27745	−0.235310	−0.117655	+	0.993055i	$0.537538\pi$
$504$	0	0
$505$	43.4205	1.93219
$506$	0	0
$507$	−18.9732	−0.842631
$508$	0	0
$509$	−31.1142	−1.37911	−0.689557	−	0.724231i	$-0.742195\pi$
$509$	−31.1142	−1.37911	−0.689557	+	0.724231i	$0.742195\pi$
$510$	0	0
$511$	−49.6059	−2.19444
$512$	0	0
$513$	−13.1095	−0.578801
$514$	0	0
$515$	26.4876	1.16718
$516$	0	0
$517$	20.0680	0.882588
$518$	0	0
$519$	−13.5328	−0.594022
$520$	0	0
$521$	11.0809	0.485464	0.242732	−	0.970093i	$-0.421957\pi$
$521$	11.0809	0.485464	0.242732	+	0.970093i	$0.421957\pi$
$522$	0	0
$523$	−3.98058	−0.174059	−0.0870293	−	0.996206i	$-0.527737\pi$
$523$	−3.98058	−0.174059	−0.0870293	+	0.996206i	$0.527737\pi$
$524$	0	0
$525$	−16.8394	−0.734934
$526$	0	0
$527$	−4.91124	−0.213937
$528$	0	0
$529$	22.7445	0.988889
$530$	0	0
$531$	6.92727	0.300618
$532$	0	0
$533$	−37.5412	−1.62609
$534$	0	0
$535$	8.07992	0.349325
$536$	0	0
$537$	44.9302	1.93888
$538$	0	0
$539$	59.1768	2.54893
$540$	0	0
$541$	41.4211	1.78083	0.890416	−	0.455147i	$-0.150413\pi$
$541$	41.4211	1.78083	0.890416	+	0.455147i	$0.150413\pi$
$542$	0	0
$543$	36.6121	1.57118
$544$	0	0
$545$	49.4819	2.11957
$546$	0	0
$547$	29.3705	1.25579	0.627895	−	0.778298i	$-0.283917\pi$
$547$	29.3705	1.25579	0.627895	+	0.778298i	$0.283917\pi$
$548$	0	0
$549$	−7.96686	−0.340017
$550$	0	0
$551$	−23.2917	−0.992260
$552$	0	0
$553$	−22.6747	−0.964225
$554$	0	0
$555$	18.4162	0.781724
$556$	0	0
$557$	14.8530	0.629341	0.314670	−	0.949201i	$-0.398106\pi$
$557$	14.8530	0.629341	0.314670	+	0.949201i	$0.398106\pi$
$558$	0	0
$559$	14.2343	0.602049
$560$	0	0
$561$	31.4176	1.32645
$562$	0	0
$563$	16.1288	0.679747	0.339873	−	0.940471i	$-0.389616\pi$
$563$	16.1288	0.679747	0.339873	+	0.940471i	$0.389616\pi$
$564$	0	0
$565$	−45.2472	−1.90357
$566$	0	0
$567$	−50.2798	−2.11155
$568$	0	0
$569$	27.9824	1.17308	0.586541	−	0.809920i	$-0.300489\pi$
$569$	27.9824	1.17308	0.586541	+	0.809920i	$0.300489\pi$
$570$	0	0
$571$	−45.8688	−1.91955	−0.959775	−	0.280769i	$-0.909410\pi$
$571$	−45.8688	−1.91955	−0.959775	+	0.280769i	$0.909410\pi$
$572$	0	0
$573$	−42.4802	−1.77464
$574$	0	0
$575$	12.4710	0.520078
$576$	0	0
$577$	−12.8898	−0.536608	−0.268304	−	0.963334i	$-0.586463\pi$
$577$	−12.8898	−0.536608	−0.268304	+	0.963334i	$0.586463\pi$
$578$	0	0
$579$	−7.41635	−0.308213
$580$	0	0
$581$	−56.4169	−2.34057
$582$	0	0
$583$	10.1610	0.420827
$584$	0	0
$585$	−12.0205	−0.496984
$586$	0	0
$587$	−6.68196	−0.275794	−0.137897	−	0.990447i	$-0.544034\pi$
$587$	−6.68196	−0.275794	−0.137897	+	0.990447i	$0.544034\pi$
$588$	0	0
$589$	−4.25760	−0.175431
$590$	0	0
$591$	−46.9394	−1.93083
$592$	0	0
$593$	−6.71014	−0.275552	−0.137776	−	0.990463i	$-0.543995\pi$
$593$	−6.71014	−0.275552	−0.137776	+	0.990463i	$0.543995\pi$
$594$	0	0
$595$	−44.8105	−1.83705
$596$	0	0
$597$	−24.7235	−1.01187
$598$	0	0
$599$	1.35781	0.0554788	0.0277394	−	0.999615i	$-0.491169\pi$
$599$	1.35781	0.0554788	0.0277394	+	0.999615i	$0.491169\pi$
$600$	0	0
$601$	2.40399	0.0980607	0.0490304	−	0.998797i	$-0.484387\pi$
$601$	2.40399	0.0980607	0.0490304	+	0.998797i	$0.484387\pi$
$602$	0	0
$603$	−0.0691672	−0.00281671
$604$	0	0
$605$	17.8411	0.725344
$606$	0	0
$607$	−25.8772	−1.05032	−0.525161	−	0.851003i	$-0.675995\pi$
$607$	−25.8772	−1.05032	−0.525161	+	0.851003i	$0.675995\pi$
$608$	0	0
$609$	−65.6741	−2.66125
$610$	0	0
$611$	−22.5622	−0.912767
$612$	0	0
$613$	−25.9910	−1.04977	−0.524884	−	0.851174i	$-0.675891\pi$
$613$	−25.9910	−1.04977	−0.524884	+	0.851174i	$0.675891\pi$
$614$	0	0
$615$	−41.2215	−1.66221
$616$	0	0
$617$	−5.54161	−0.223097	−0.111548	−	0.993759i	$-0.535581\pi$
$617$	−5.54161	−0.223097	−0.111548	+	0.993759i	$0.535581\pi$
$618$	0	0
$619$	−12.4016	−0.498463	−0.249231	−	0.968444i	$-0.580178\pi$
$619$	−12.4016	−0.498463	−0.249231	+	0.968444i	$0.580178\pi$
$620$	0	0
$621$	27.3751	1.09852
$622$	0	0
$623$	83.4445	3.34313
$624$	0	0
$625$	−30.8195	−1.23278
$626$	0	0
$627$	27.2362	1.08771
$628$	0	0
$629$	13.2033	0.526451
$630$	0	0
$631$	4.62345	0.184057	0.0920284	−	0.995756i	$-0.470665\pi$
$631$	4.62345	0.184057	0.0920284	+	0.995756i	$0.470665\pi$
$632$	0	0
$633$	−23.8769	−0.949024
$634$	0	0
$635$	27.8904	1.10680
$636$	0	0
$637$	−66.5318	−2.63609
$638$	0	0
$639$	3.00982	0.119067
$640$	0	0
$641$	−24.2961	−0.959636	−0.479818	−	0.877368i	$-0.659297\pi$
$641$	−24.2961	−0.959636	−0.479818	+	0.877368i	$0.659297\pi$
$642$	0	0
$643$	−38.7845	−1.52951	−0.764756	−	0.644320i	$-0.777141\pi$
$643$	−38.7845	−1.52951	−0.764756	+	0.644320i	$0.777141\pi$
$644$	0	0
$645$	15.6298	0.615423
$646$	0	0
$647$	−23.1556	−0.910340	−0.455170	−	0.890405i	$-0.650422\pi$
$647$	−23.1556	−0.910340	−0.455170	+	0.890405i	$0.650422\pi$
$648$	0	0
$649$	30.2045	1.18563
$650$	0	0
$651$	−12.0049	−0.470508
$652$	0	0
$653$	27.5047	1.07634	0.538171	−	0.842836i	$-0.319116\pi$
$653$	27.5047	1.07634	0.538171	+	0.842836i	$0.319116\pi$
$654$	0	0
$655$	−38.3649	−1.49904
$656$	0	0
$657$	−10.4755	−0.408687
$658$	0	0
$659$	21.9518	0.855121	0.427560	−	0.903987i	$-0.359373\pi$
$659$	21.9518	0.855121	0.427560	+	0.903987i	$0.359373\pi$
$660$	0	0
$661$	33.2637	1.29381	0.646904	−	0.762571i	$-0.276063\pi$
$661$	33.2637	1.29381	0.646904	+	0.762571i	$0.276063\pi$
$662$	0	0
$663$	−35.3225	−1.37181
$664$	0	0
$665$	−38.8466	−1.50641
$666$	0	0
$667$	48.6373	1.88324
$668$	0	0
$669$	−40.6341	−1.57100
$670$	0	0
$671$	−34.7374	−1.34102
$672$	0	0
$673$	15.0797	0.581281	0.290640	−	0.956832i	$-0.406132\pi$
$673$	15.0797	0.581281	0.290640	+	0.956832i	$0.406132\pi$
$674$	0	0
$675$	7.46311	0.287255
$676$	0	0
$677$	19.3345	0.743086	0.371543	−	0.928416i	$-0.378829\pi$
$677$	19.3345	0.743086	0.371543	+	0.928416i	$0.378829\pi$
$678$	0	0
$679$	−11.8401	−0.454381
$680$	0	0
$681$	−13.2367	−0.507230
$682$	0	0
$683$	−17.3222	−0.662816	−0.331408	−	0.943487i	$-0.607524\pi$
$683$	−17.3222	−0.662816	−0.331408	+	0.943487i	$0.607524\pi$
$684$	0	0
$685$	14.3340	0.547673
$686$	0	0
$687$	25.7510	0.982462
$688$	0	0
$689$	−11.4239	−0.435217
$690$	0	0
$691$	−23.0314	−0.876157	−0.438078	−	0.898937i	$-0.644341\pi$
$691$	−23.0314	−0.876157	−0.438078	+	0.898937i	$0.644341\pi$
$692$	0	0
$693$	18.7367	0.711747
$694$	0	0
$695$	−29.3846	−1.11462
$696$	0	0
$697$	−29.5534	−1.11941
$698$	0	0
$699$	53.6635	2.02974
$700$	0	0
$701$	−13.9511	−0.526926	−0.263463	−	0.964670i	$-0.584865\pi$
$701$	−13.9511	−0.526926	−0.263463	+	0.964670i	$0.584865\pi$
$702$	0	0
$703$	11.4461	0.431697
$704$	0	0
$705$	−24.7741	−0.933045
$706$	0	0
$707$	76.0929	2.86177
$708$	0	0
$709$	−38.8376	−1.45858	−0.729288	−	0.684207i	$-0.760149\pi$
$709$	−38.8376	−1.45858	−0.729288	+	0.684207i	$0.760149\pi$
$710$	0	0
$711$	−4.78830	−0.179575
$712$	0	0
$713$	8.89063	0.332957
$714$	0	0
$715$	−52.4120	−1.96010
$716$	0	0
$717$	−48.2027	−1.80016
$718$	0	0
$719$	9.67830	0.360940	0.180470	−	0.983581i	$-0.442238\pi$
$719$	9.67830	0.360940	0.180470	+	0.983581i	$0.442238\pi$
$720$	0	0
$721$	46.4186	1.72872
$722$	0	0
$723$	5.62590	0.209229
$724$	0	0
$725$	13.2597	0.492452
$726$	0	0
$727$	33.1857	1.23079	0.615394	−	0.788219i	$-0.288997\pi$
$727$	33.1857	1.23079	0.615394	+	0.788219i	$0.288997\pi$
$728$	0	0
$729$	13.5703	0.502604
$730$	0	0
$731$	11.2057	0.414456
$732$	0	0
$733$	20.0095	0.739067	0.369534	−	0.929217i	$-0.379517\pi$
$733$	20.0095	0.739067	0.369534	+	0.929217i	$0.379517\pi$
$734$	0	0
$735$	−73.0543	−2.69465
$736$	0	0
$737$	−0.301585	−0.0111090
$738$	0	0
$739$	10.2285	0.376262	0.188131	−	0.982144i	$-0.439757\pi$
$739$	10.2285	0.376262	0.188131	+	0.982144i	$0.439757\pi$
$740$	0	0
$741$	−30.6213	−1.12490
$742$	0	0
$743$	6.90145	0.253190	0.126595	−	0.991955i	$-0.459595\pi$
$743$	6.90145	0.253190	0.126595	+	0.991955i	$0.459595\pi$
$744$	0	0
$745$	−49.2478	−1.80430
$746$	0	0
$747$	−11.9138	−0.435903
$748$	0	0
$749$	14.1598	0.517387
$750$	0	0
$751$	−5.29953	−0.193383	−0.0966914	−	0.995314i	$-0.530826\pi$
$751$	−5.29953	−0.193383	−0.0966914	+	0.995314i	$0.530826\pi$
$752$	0	0
$753$	−1.99202	−0.0725933
$754$	0	0
$755$	5.24499	0.190885
$756$	0	0
$757$	27.2750	0.991326	0.495663	−	0.868515i	$-0.334925\pi$
$757$	27.2750	0.991326	0.495663	+	0.868515i	$0.334925\pi$
$758$	0	0
$759$	−56.8740	−2.06440
$760$	0	0
$761$	−8.35502	−0.302869	−0.151435	−	0.988467i	$-0.548389\pi$
$761$	−8.35502	−0.302869	−0.151435	+	0.988467i	$0.548389\pi$
$762$	0	0
$763$	86.7152	3.13930
$764$	0	0
$765$	−9.46282	−0.342129
$766$	0	0
$767$	−33.9586	−1.22617
$768$	0	0
$769$	5.78615	0.208654	0.104327	−	0.994543i	$-0.466731\pi$
$769$	5.78615	0.208654	0.104327	+	0.994543i	$0.466731\pi$
$770$	0	0
$771$	37.9083	1.36523
$772$	0	0
$773$	−18.3573	−0.660267	−0.330133	−	0.943934i	$-0.607094\pi$
$773$	−18.3573	−0.660267	−0.330133	+	0.943934i	$0.607094\pi$
$774$	0	0
$775$	2.42380	0.0870654
$776$	0	0
$777$	32.2737	1.15781
$778$	0	0
$779$	−25.6201	−0.917935
$780$	0	0
$781$	13.1235	0.469597
$782$	0	0
$783$	29.1062	1.04017
$784$	0	0
$785$	22.5595	0.805183
$786$	0	0
$787$	−41.3289	−1.47321	−0.736607	−	0.676321i	$-0.763573\pi$
$787$	−41.3289	−1.47321	−0.736607	+	0.676321i	$0.763573\pi$
$788$	0	0
$789$	−0.100712	−0.00358546
$790$	0	0
$791$	−79.2942	−2.81938
$792$	0	0
$793$	39.0549	1.38688
$794$	0	0
$795$	−12.5439	−0.444886
$796$	0	0
$797$	−27.4290	−0.971585	−0.485792	−	0.874074i	$-0.661469\pi$
$797$	−27.4290	−0.971585	−0.485792	+	0.874074i	$0.661469\pi$
$798$	0	0
$799$	−17.7615	−0.628358
$800$	0	0
$801$	17.6213	0.622618
$802$	0	0
$803$	−45.6756	−1.61186
$804$	0	0
$805$	81.1187	2.85906
$806$	0	0
$807$	−56.3111	−1.98225
$808$	0	0
$809$	−16.3492	−0.574806	−0.287403	−	0.957810i	$-0.592792\pi$
$809$	−16.3492	−0.574806	−0.287403	+	0.957810i	$0.592792\pi$
$810$	0	0
$811$	10.1997	0.358159	0.179079	−	0.983835i	$-0.442688\pi$
$811$	10.1997	0.358159	0.179079	+	0.983835i	$0.442688\pi$
$812$	0	0
$813$	−17.8426	−0.625766
$814$	0	0
$815$	19.1884	0.672139
$816$	0	0
$817$	9.71427	0.339859
$818$	0	0
$819$	−21.0654	−0.736085
$820$	0	0
$821$	−6.63902	−0.231703	−0.115852	−	0.993267i	$-0.536960\pi$
$821$	−6.63902	−0.231703	−0.115852	+	0.993267i	$0.536960\pi$
$822$	0	0
$823$	−19.4021	−0.676316	−0.338158	−	0.941089i	$-0.609804\pi$
$823$	−19.4021	−0.676316	−0.338158	+	0.941089i	$0.609804\pi$
$824$	0	0
$825$	−15.5052	−0.539823
$826$	0	0
$827$	40.4814	1.40767	0.703837	−	0.710361i	$-0.251468\pi$
$827$	40.4814	1.40767	0.703837	+	0.710361i	$0.251468\pi$
$828$	0	0
$829$	−20.3939	−0.708309	−0.354154	−	0.935187i	$-0.615231\pi$
$829$	−20.3939	−0.708309	−0.354154	+	0.935187i	$0.615231\pi$
$830$	0	0
$831$	34.3913	1.19302
$832$	0	0
$833$	−52.3756	−1.81471
$834$	0	0
$835$	45.8539	1.58684
$836$	0	0
$837$	5.32046	0.183902
$838$	0	0
$839$	44.5562	1.53825	0.769126	−	0.639097i	$-0.220692\pi$
$839$	44.5562	1.53825	0.769126	+	0.639097i	$0.220692\pi$
$840$	0	0
$841$	22.7130	0.783207
$842$	0	0
$843$	−22.9830	−0.791577
$844$	0	0
$845$	24.9172	0.857177
$846$	0	0
$847$	31.2659	1.07431
$848$	0	0
$849$	15.6250	0.536248
$850$	0	0
$851$	−23.9014	−0.819331
$852$	0	0
$853$	−5.21114	−0.178426	−0.0892131	−	0.996013i	$-0.528435\pi$
$853$	−5.21114	−0.178426	−0.0892131	+	0.996013i	$0.528435\pi$
$854$	0	0
$855$	−8.20339	−0.280550
$856$	0	0
$857$	−4.23643	−0.144714	−0.0723569	−	0.997379i	$-0.523052\pi$
$857$	−4.23643	−0.144714	−0.0723569	+	0.997379i	$0.523052\pi$
$858$	0	0
$859$	−29.5722	−1.00899	−0.504496	−	0.863414i	$-0.668322\pi$
$859$	−29.5722	−1.00899	−0.504496	+	0.863414i	$0.668322\pi$
$860$	0	0
$861$	−72.2393	−2.46191
$862$	0	0
$863$	−32.2093	−1.09642	−0.548209	−	0.836342i	$-0.684690\pi$
$863$	−32.2093	−1.09642	−0.548209	+	0.836342i	$0.684690\pi$
$864$	0	0
$865$	17.7723	0.604277
$866$	0	0
$867$	6.05758	0.205726
$868$	0	0
$869$	−20.8781	−0.708242
$870$	0	0
$871$	0.339069	0.0114889
$872$	0	0
$873$	−2.50032	−0.0846229
$874$	0	0
$875$	−37.8534	−1.27968
$876$	0	0
$877$	−45.7438	−1.54466	−0.772330	−	0.635222i	$-0.780909\pi$
$877$	−45.7438	−1.54466	−0.772330	+	0.635222i	$0.780909\pi$
$878$	0	0
$879$	44.3697	1.49655
$880$	0	0
$881$	−32.1922	−1.08458	−0.542291	−	0.840191i	$-0.682443\pi$
$881$	−32.1922	−1.08458	−0.542291	+	0.840191i	$0.682443\pi$
$882$	0	0
$883$	−42.5335	−1.43137	−0.715683	−	0.698426i	$-0.753884\pi$
$883$	−42.5335	−1.43137	−0.715683	+	0.698426i	$0.753884\pi$
$884$	0	0
$885$	−37.2877	−1.25341
$886$	0	0
$887$	−40.6177	−1.36381	−0.681905	−	0.731441i	$-0.738848\pi$
$887$	−40.6177	−1.36381	−0.681905	+	0.731441i	$0.738848\pi$
$888$	0	0
$889$	48.8770	1.63928
$890$	0	0
$891$	−46.2961	−1.55098
$892$	0	0
$893$	−15.3976	−0.515261
$894$	0	0
$895$	−59.0060	−1.97235
$896$	0	0
$897$	63.9428	2.13499
$898$	0	0
$899$	9.45285	0.315270
$900$	0	0
$901$	−8.99322	−0.299608
$902$	0	0
$903$	27.3907	0.911505
$904$	0	0
$905$	−48.0820	−1.59830
$906$	0	0
$907$	−25.9430	−0.861425	−0.430712	−	0.902489i	$-0.641738\pi$
$907$	−25.9430	−0.861425	−0.430712	+	0.902489i	$0.641738\pi$
$908$	0	0
$909$	16.0688	0.532970
$910$	0	0
$911$	9.34910	0.309749	0.154875	−	0.987934i	$-0.450503\pi$
$911$	9.34910	0.309749	0.154875	+	0.987934i	$0.450503\pi$
$912$	0	0
$913$	−51.9470	−1.71919
$914$	0	0
$915$	42.8836	1.41769
$916$	0	0
$917$	−67.2332	−2.22024
$918$	0	0
$919$	34.1844	1.12764	0.563820	−	0.825897i	$-0.309331\pi$
$919$	34.1844	1.12764	0.563820	+	0.825897i	$0.309331\pi$
$920$	0	0
$921$	36.9544	1.21769
$922$	0	0
$923$	−14.7546	−0.485655
$924$	0	0
$925$	−6.51611	−0.214248
$926$	0	0
$927$	9.80240	0.321953
$928$	0	0
$929$	40.7349	1.33647	0.668235	−	0.743950i	$-0.267050\pi$
$929$	40.7349	1.33647	0.668235	+	0.743950i	$0.267050\pi$
$930$	0	0
$931$	−45.4048	−1.48808
$932$	0	0
$933$	19.2877	0.631452
$934$	0	0
$935$	−41.2601	−1.34935
$936$	0	0
$937$	31.4094	1.02610	0.513049	−	0.858359i	$-0.328516\pi$
$937$	31.4094	1.02610	0.513049	+	0.858359i	$0.328516\pi$
$938$	0	0
$939$	8.14835	0.265912
$940$	0	0
$941$	−20.3324	−0.662818	−0.331409	−	0.943487i	$-0.607524\pi$
$941$	−20.3324	−0.662818	−0.331409	+	0.943487i	$0.607524\pi$
$942$	0	0
$943$	53.4993	1.74218
$944$	0	0
$945$	48.5443	1.57915
$946$	0	0
$947$	−10.1704	−0.330493	−0.165247	−	0.986252i	$-0.552842\pi$
$947$	−10.1704	−0.330493	−0.165247	+	0.986252i	$0.552842\pi$
$948$	0	0
$949$	51.3525	1.66697
$950$	0	0
$951$	20.5146	0.665232
$952$	0	0
$953$	29.6541	0.960591	0.480295	−	0.877107i	$-0.340529\pi$
$953$	29.6541	0.960591	0.480295	+	0.877107i	$0.340529\pi$
$954$	0	0
$955$	55.7884	1.80527
$956$	0	0
$957$	−60.4707	−1.95474
$958$	0	0
$959$	25.1198	0.811160
$960$	0	0
$961$	−29.2721	−0.944260
$962$	0	0
$963$	2.99018	0.0963570
$964$	0	0
$965$	9.73975	0.313534
$966$	0	0
$967$	−14.1618	−0.455413	−0.227707	−	0.973730i	$-0.573123\pi$
$967$	−14.1618	−0.455413	−0.227707	+	0.973730i	$0.573123\pi$
$968$	0	0
$969$	−24.1059	−0.774393
$970$	0	0
$971$	9.26803	0.297425	0.148713	−	0.988880i	$-0.452487\pi$
$971$	9.26803	0.297425	0.148713	+	0.988880i	$0.452487\pi$
$972$	0	0
$973$	−51.4954	−1.65087
$974$	0	0
$975$	17.4323	0.558282
$976$	0	0
$977$	−53.6029	−1.71491	−0.857455	−	0.514559i	$-0.827956\pi$
$977$	−53.6029	−1.71491	−0.857455	+	0.514559i	$0.827956\pi$
$978$	0	0
$979$	76.8331	2.45560
$980$	0	0
$981$	18.3120	0.584657
$982$	0	0
$983$	36.2820	1.15722	0.578608	−	0.815606i	$-0.303596\pi$
$983$	36.2820	1.15722	0.578608	+	0.815606i	$0.303596\pi$
$984$	0	0
$985$	61.6446	1.96416
$986$	0	0
$987$	−43.4157	−1.38194
$988$	0	0
$989$	−20.2851	−0.645030
$990$	0	0
$991$	−35.8569	−1.13903	−0.569515	−	0.821981i	$-0.692869\pi$
$991$	−35.8569	−1.13903	−0.569515	+	0.821981i	$0.692869\pi$
$992$	0	0
$993$	9.61061	0.304984
$994$	0	0
$995$	32.4689	1.02933
$996$	0	0
$997$	−19.4580	−0.616240	−0.308120	−	0.951347i	$-0.599700\pi$
$997$	−19.4580	−0.616240	−0.308120	+	0.951347i	$0.599700\pi$
$998$	0	0
$999$	−14.3035	−0.452542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.4		19
4.3	odd	2		2008.2.a.c.1.16	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.16	✓	19	4.3	odd	2
4016.2.a.l.1.4		19	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)

\(f(q)\)	\(=\)	\(q-1.99202 q^{3} +2.61608 q^{5} +4.58459 q^{7} +0.968146 q^{9} +O(q^{10})\)	\(q-1.99202 q^{3} +2.61608 q^{5} +4.58459 q^{7} +0.968146 q^{9} +4.22135 q^{11} -4.74601 q^{13} -5.21129 q^{15} -3.73618 q^{17} -3.23893 q^{19} -9.13260 q^{21} +6.76346 q^{23} +1.84388 q^{25} +4.04750 q^{27} +7.19117 q^{29} +1.31451 q^{31} -8.40901 q^{33} +11.9937 q^{35} -3.53391 q^{37} +9.45415 q^{39} +7.91005 q^{41} -2.99922 q^{43} +2.53275 q^{45} +4.75392 q^{47} +14.0185 q^{49} +7.44256 q^{51} +2.40706 q^{53} +11.0434 q^{55} +6.45201 q^{57} +7.15519 q^{59} -8.22898 q^{61} +4.43855 q^{63} -12.4160 q^{65} -0.0714429 q^{67} -13.4730 q^{69} +3.10885 q^{71} -10.8201 q^{73} -3.67305 q^{75} +19.3532 q^{77} -4.94584 q^{79} -10.9671 q^{81} -12.3058 q^{83} -9.77416 q^{85} -14.3250 q^{87} +18.2011 q^{89} -21.7585 q^{91} -2.61853 q^{93} -8.47330 q^{95} -2.58258 q^{97} +4.08688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more