LMFDB - Embedded newform 4016.2.a.j.1.3

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.53364$ 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)

$f(q)$	$=$	$q-2.53364 q^{3} +2.70770 q^{5} -1.66309 q^{7} +3.41934 q^{9} +O(q^{10})$	$q-2.53364 q^{3} +2.70770 q^{5} -1.66309 q^{7} +3.41934 q^{9} -3.22692 q^{11} -5.30349 q^{13} -6.86033 q^{15} +3.60034 q^{17} +2.49017 q^{19} +4.21368 q^{21} +9.44085 q^{23} +2.33163 q^{25} -1.06245 q^{27} -0.668304 q^{29} -1.89493 q^{31} +8.17586 q^{33} -4.50315 q^{35} -4.26565 q^{37} +13.4371 q^{39} +5.63994 q^{41} +7.45482 q^{43} +9.25853 q^{45} -4.38668 q^{47} -4.23412 q^{49} -9.12197 q^{51} -13.7249 q^{53} -8.73753 q^{55} -6.30920 q^{57} -10.5842 q^{59} +4.80848 q^{61} -5.68667 q^{63} -14.3602 q^{65} +8.11901 q^{67} -23.9197 q^{69} -8.73759 q^{71} +5.71642 q^{73} -5.90751 q^{75} +5.36667 q^{77} +2.24905 q^{79} -7.56615 q^{81} -2.30070 q^{83} +9.74864 q^{85} +1.69324 q^{87} +15.4266 q^{89} +8.82019 q^{91} +4.80106 q^{93} +6.74264 q^{95} +5.68637 q^{97} -11.0339 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	−2.53364	−1.46280	−0.731399	−	0.681950i	$-0.761132\pi$
$3$	−2.53364	−1.46280	−0.731399	+	0.681950i	$0.761132\pi$
$4$	0	0
$5$	2.70770	1.21092	0.605460	−	0.795876i	$-0.292989\pi$
$5$	2.70770	1.21092	0.605460	+	0.795876i	$0.292989\pi$
$6$	0	0
$7$	−1.66309	−0.628590	−0.314295	−	0.949325i	$-0.601768\pi$
$7$	−1.66309	−0.628590	−0.314295	+	0.949325i	$0.601768\pi$
$8$	0	0
$9$	3.41934	1.13978
$10$	0	0
$11$	−3.22692	−0.972954	−0.486477	−	0.873693i	$-0.661718\pi$
$11$	−3.22692	−0.972954	−0.486477	+	0.873693i	$0.661718\pi$
$12$	0	0
$13$	−5.30349	−1.47092	−0.735461	−	0.677567i	$-0.763034\pi$
$13$	−5.30349	−1.47092	−0.735461	+	0.677567i	$0.763034\pi$
$14$	0	0
$15$	−6.86033	−1.77133
$16$	0	0
$17$	3.60034	0.873211	0.436606	−	0.899653i	$-0.356181\pi$
$17$	3.60034	0.873211	0.436606	+	0.899653i	$0.356181\pi$
$18$	0	0
$19$	2.49017	0.571285	0.285642	−	0.958336i	$-0.407793\pi$
$19$	2.49017	0.571285	0.285642	+	0.958336i	$0.407793\pi$
$20$	0	0
$21$	4.21368	0.919500
$22$	0	0
$23$	9.44085	1.96855	0.984276	−	0.176636i	$-0.0565214\pi$
$23$	9.44085	1.96855	0.984276	+	0.176636i	$0.0565214\pi$
$24$	0	0
$25$	2.33163	0.466326
$26$	0	0
$27$	−1.06245	−0.204468
$28$	0	0
$29$	−0.668304	−0.124101	−0.0620505	−	0.998073i	$-0.519764\pi$
$29$	−0.668304	−0.124101	−0.0620505	+	0.998073i	$0.519764\pi$
$30$	0	0
$31$	−1.89493	−0.340339	−0.170169	−	0.985415i	$-0.554431\pi$
$31$	−1.89493	−0.340339	−0.170169	+	0.985415i	$0.554431\pi$
$32$	0	0
$33$	8.17586	1.42323
$34$	0	0
$35$	−4.50315	−0.761172
$36$	0	0
$37$	−4.26565	−0.701269	−0.350635	−	0.936512i	$-0.614034\pi$
$37$	−4.26565	−0.701269	−0.350635	+	0.936512i	$0.614034\pi$
$38$	0	0
$39$	13.4371	2.15166
$40$	0	0
$41$	5.63994	0.880811	0.440406	−	0.897799i	$-0.354835\pi$
$41$	5.63994	0.880811	0.440406	+	0.897799i	$0.354835\pi$
$42$	0	0
$43$	7.45482	1.13685	0.568424	−	0.822736i	$-0.307553\pi$
$43$	7.45482	1.13685	0.568424	+	0.822736i	$0.307553\pi$
$44$	0	0
$45$	9.25853	1.38018
$46$	0	0
$47$	−4.38668	−0.639863	−0.319932	−	0.947441i	$-0.603660\pi$
$47$	−4.38668	−0.639863	−0.319932	+	0.947441i	$0.603660\pi$
$48$	0	0
$49$	−4.23412	−0.604875
$50$	0	0
$51$	−9.12197	−1.27733
$52$	0	0
$53$	−13.7249	−1.88526	−0.942631	−	0.333837i	$-0.891657\pi$
$53$	−13.7249	−1.88526	−0.942631	+	0.333837i	$0.891657\pi$
$54$	0	0
$55$	−8.73753	−1.17817
$56$	0	0
$57$	−6.30920	−0.835674
$58$	0	0
$59$	−10.5842	−1.37795	−0.688973	−	0.724787i	$-0.741938\pi$
$59$	−10.5842	−1.37795	−0.688973	+	0.724787i	$0.741938\pi$
$60$	0	0
$61$	4.80848	0.615662	0.307831	−	0.951441i	$-0.400397\pi$
$61$	4.80848	0.615662	0.307831	+	0.951441i	$0.400397\pi$
$62$	0	0
$63$	−5.68667	−0.716453
$64$	0	0
$65$	−14.3602	−1.78117
$66$	0	0
$67$	8.11901	0.991895	0.495947	−	0.868353i	$-0.334821\pi$
$67$	8.11901	0.991895	0.495947	+	0.868353i	$0.334821\pi$
$68$	0	0
$69$	−23.9197	−2.87960
$70$	0	0
$71$	−8.73759	−1.03696	−0.518481	−	0.855089i	$-0.673502\pi$
$71$	−8.73759	−1.03696	−0.518481	+	0.855089i	$0.673502\pi$
$72$	0	0
$73$	5.71642	0.669056	0.334528	−	0.942386i	$-0.391423\pi$
$73$	5.71642	0.669056	0.334528	+	0.942386i	$0.391423\pi$
$74$	0	0
$75$	−5.90751	−0.682141
$76$	0	0
$77$	5.36667	0.611589
$78$	0	0
$79$	2.24905	0.253038	0.126519	−	0.991964i	$-0.459619\pi$
$79$	2.24905	0.253038	0.126519	+	0.991964i	$0.459619\pi$
$80$	0	0
$81$	−7.56615	−0.840683
$82$	0	0
$83$	−2.30070	−0.252534	−0.126267	−	0.991996i	$-0.540300\pi$
$83$	−2.30070	−0.252534	−0.126267	+	0.991996i	$0.540300\pi$
$84$	0	0
$85$	9.74864	1.05739
$86$	0	0
$87$	1.69324	0.181535
$88$	0	0
$89$	15.4266	1.63522	0.817608	−	0.575776i	$-0.195300\pi$
$89$	15.4266	1.63522	0.817608	+	0.575776i	$0.195300\pi$
$90$	0	0
$91$	8.82019	0.924607
$92$	0	0
$93$	4.80106	0.497847
$94$	0	0
$95$	6.74264	0.691780
$96$	0	0
$97$	5.68637	0.577364	0.288682	−	0.957425i	$-0.406783\pi$
$97$	5.68637	0.577364	0.288682	+	0.957425i	$0.406783\pi$
$98$	0	0
$99$	−11.0339	−1.10895
$100$	0	0
$101$	−5.12873	−0.510328	−0.255164	−	0.966898i	$-0.582129\pi$
$101$	−5.12873	−0.510328	−0.255164	+	0.966898i	$0.582129\pi$
$102$	0	0
$103$	−8.79517	−0.866614	−0.433307	−	0.901246i	$-0.642653\pi$
$103$	−8.79517	−0.866614	−0.433307	+	0.901246i	$0.642653\pi$
$104$	0	0
$105$	11.4094	1.11344
$106$	0	0
$107$	10.1074	0.977121	0.488560	−	0.872530i	$-0.337522\pi$
$107$	10.1074	0.977121	0.488560	+	0.872530i	$0.337522\pi$
$108$	0	0
$109$	7.87751	0.754528	0.377264	−	0.926106i	$-0.376865\pi$
$109$	7.87751	0.754528	0.377264	+	0.926106i	$0.376865\pi$
$110$	0	0
$111$	10.8076	1.02582
$112$	0	0
$113$	−5.94432	−0.559195	−0.279597	−	0.960117i	$-0.590201\pi$
$113$	−5.94432	−0.559195	−0.279597	+	0.960117i	$0.590201\pi$
$114$	0	0
$115$	25.5630	2.38376
$116$	0	0
$117$	−18.1344	−1.67653
$118$	0	0
$119$	−5.98770	−0.548892
$120$	0	0
$121$	−0.586971	−0.0533610
$122$	0	0
$123$	−14.2896	−1.28845
$124$	0	0
$125$	−7.22514	−0.646236
$126$	0	0
$127$	−6.49541	−0.576375	−0.288187	−	0.957574i	$-0.593053\pi$
$127$	−6.49541	−0.576375	−0.288187	+	0.957574i	$0.593053\pi$
$128$	0	0
$129$	−18.8878	−1.66298
$130$	0	0
$131$	−10.5751	−0.923949	−0.461974	−	0.886893i	$-0.652859\pi$
$131$	−10.5751	−0.923949	−0.461974	+	0.886893i	$0.652859\pi$
$132$	0	0
$133$	−4.14139	−0.359104
$134$	0	0
$135$	−2.87678	−0.247594
$136$	0	0
$137$	−15.8302	−1.35246	−0.676232	−	0.736689i	$-0.736388\pi$
$137$	−15.8302	−1.35246	−0.676232	+	0.736689i	$0.736388\pi$
$138$	0	0
$139$	−16.3374	−1.38572	−0.692861	−	0.721071i	$-0.743650\pi$
$139$	−16.3374	−1.38572	−0.692861	+	0.721071i	$0.743650\pi$
$140$	0	0
$141$	11.1143	0.935991
$142$	0	0
$143$	17.1139	1.43114
$144$	0	0
$145$	−1.80957	−0.150276
$146$	0	0
$147$	10.7277	0.884809
$148$	0	0
$149$	17.9391	1.46963	0.734813	−	0.678270i	$-0.237270\pi$
$149$	17.9391	1.46963	0.734813	+	0.678270i	$0.237270\pi$
$150$	0	0
$151$	−6.28510	−0.511474	−0.255737	−	0.966746i	$-0.582318\pi$
$151$	−6.28510	−0.511474	−0.255737	+	0.966746i	$0.582318\pi$
$152$	0	0
$153$	12.3108	0.995267
$154$	0	0
$155$	−5.13089	−0.412123
$156$	0	0
$157$	15.7510	1.25707	0.628534	−	0.777782i	$-0.283655\pi$
$157$	15.7510	1.25707	0.628534	+	0.777782i	$0.283655\pi$
$158$	0	0
$159$	34.7740	2.75776
$160$	0	0
$161$	−15.7010	−1.23741
$162$	0	0
$163$	−2.20823	−0.172962	−0.0864810	−	0.996253i	$-0.527562\pi$
$163$	−2.20823	−0.172962	−0.0864810	+	0.996253i	$0.527562\pi$
$164$	0	0
$165$	22.1378	1.72342
$166$	0	0
$167$	−19.3764	−1.49939	−0.749696	−	0.661783i	$-0.769800\pi$
$167$	−19.3764	−1.49939	−0.749696	+	0.661783i	$0.769800\pi$
$168$	0	0
$169$	15.1270	1.16361
$170$	0	0
$171$	8.51474	0.651138
$172$	0	0
$173$	−13.2171	−1.00487	−0.502437	−	0.864614i	$-0.667563\pi$
$173$	−13.2171	−1.00487	−0.502437	+	0.864614i	$0.667563\pi$
$174$	0	0
$175$	−3.87772	−0.293128
$176$	0	0
$177$	26.8166	2.01566
$178$	0	0
$179$	−11.8228	−0.883674	−0.441837	−	0.897095i	$-0.645673\pi$
$179$	−11.8228	−0.883674	−0.441837	+	0.897095i	$0.645673\pi$
$180$	0	0
$181$	−16.3008	−1.21163	−0.605816	−	0.795605i	$-0.707153\pi$
$181$	−16.3008	−1.21163	−0.605816	+	0.795605i	$0.707153\pi$
$182$	0	0
$183$	−12.1830	−0.900590
$184$	0	0
$185$	−11.5501	−0.849180
$186$	0	0
$187$	−11.6180	−0.849594
$188$	0	0
$189$	1.76695	0.128526
$190$	0	0
$191$	12.8529	0.930004	0.465002	−	0.885310i	$-0.346054\pi$
$191$	12.8529	0.930004	0.465002	+	0.885310i	$0.346054\pi$
$192$	0	0
$193$	0.578921	0.0416717	0.0208358	−	0.999783i	$-0.493367\pi$
$193$	0.578921	0.0416717	0.0208358	+	0.999783i	$0.493367\pi$
$194$	0	0
$195$	36.3837	2.60549
$196$	0	0
$197$	−22.4759	−1.60134	−0.800672	−	0.599104i	$-0.795524\pi$
$197$	−22.4759	−1.60134	−0.800672	+	0.599104i	$0.795524\pi$
$198$	0	0
$199$	1.84324	0.130664	0.0653319	−	0.997864i	$-0.479189\pi$
$199$	1.84324	0.130664	0.0653319	+	0.997864i	$0.479189\pi$
$200$	0	0
$201$	−20.5707	−1.45094
$202$	0	0
$203$	1.11145	0.0780086
$204$	0	0
$205$	15.2713	1.06659
$206$	0	0
$207$	32.2814	2.24371
$208$	0	0
$209$	−8.03559	−0.555834
$210$	0	0
$211$	−24.3880	−1.67894	−0.839470	−	0.543406i	$-0.817134\pi$
$211$	−24.3880	−1.67894	−0.839470	+	0.543406i	$0.817134\pi$
$212$	0	0
$213$	22.1379	1.51687
$214$	0	0
$215$	20.1854	1.37663
$216$	0	0
$217$	3.15144	0.213934
$218$	0	0
$219$	−14.4833	−0.978694
$220$	0	0
$221$	−19.0944	−1.28443
$222$	0	0
$223$	13.7244	0.919055	0.459527	−	0.888164i	$-0.348019\pi$
$223$	13.7244	0.919055	0.459527	+	0.888164i	$0.348019\pi$
$224$	0	0
$225$	7.97263	0.531508
$226$	0	0
$227$	−18.0337	−1.19694	−0.598471	−	0.801145i	$-0.704225\pi$
$227$	−18.0337	−1.19694	−0.598471	+	0.801145i	$0.704225\pi$
$228$	0	0
$229$	−13.9194	−0.919818	−0.459909	−	0.887966i	$-0.652118\pi$
$229$	−13.9194	−0.919818	−0.459909	+	0.887966i	$0.652118\pi$
$230$	0	0
$231$	−13.5972	−0.894631
$232$	0	0
$233$	−6.55318	−0.429313	−0.214657	−	0.976690i	$-0.568863\pi$
$233$	−6.55318	−0.429313	−0.214657	+	0.976690i	$0.568863\pi$
$234$	0	0
$235$	−11.8778	−0.774823
$236$	0	0
$237$	−5.69829	−0.370144
$238$	0	0
$239$	19.4466	1.25790	0.628948	−	0.777448i	$-0.283486\pi$
$239$	19.4466	1.25790	0.628948	+	0.777448i	$0.283486\pi$
$240$	0	0
$241$	−7.99253	−0.514844	−0.257422	−	0.966299i	$-0.582873\pi$
$241$	−7.99253	−0.514844	−0.257422	+	0.966299i	$0.582873\pi$
$242$	0	0
$243$	22.3572	1.43422
$244$	0	0
$245$	−11.4647	−0.732454
$246$	0	0
$247$	−13.2066	−0.840315
$248$	0	0
$249$	5.82914	0.369407
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−30.4649	−1.91531
$254$	0	0
$255$	−24.6996	−1.54675
$256$	0	0
$257$	−10.5459	−0.657838	−0.328919	−	0.944358i	$-0.606684\pi$
$257$	−10.5459	−0.657838	−0.328919	+	0.944358i	$0.606684\pi$
$258$	0	0
$259$	7.09418	0.440811
$260$	0	0
$261$	−2.28516	−0.141448
$262$	0	0
$263$	23.7679	1.46559	0.732796	−	0.680449i	$-0.238215\pi$
$263$	23.7679	1.46559	0.732796	+	0.680449i	$0.238215\pi$
$264$	0	0
$265$	−37.1629	−2.28290
$266$	0	0
$267$	−39.0854	−2.39199
$268$	0	0
$269$	−24.9594	−1.52180	−0.760902	−	0.648867i	$-0.775243\pi$
$269$	−24.9594	−1.52180	−0.760902	+	0.648867i	$0.775243\pi$
$270$	0	0
$271$	−21.5226	−1.30740	−0.653702	−	0.756752i	$-0.726785\pi$
$271$	−21.5226	−1.30740	−0.653702	+	0.756752i	$0.726785\pi$
$272$	0	0
$273$	−22.3472	−1.35251
$274$	0	0
$275$	−7.52399	−0.453714
$276$	0	0
$277$	−23.9187	−1.43714	−0.718569	−	0.695456i	$-0.755202\pi$
$277$	−23.9187	−1.43714	−0.718569	+	0.695456i	$0.755202\pi$
$278$	0	0
$279$	−6.47939	−0.387911
$280$	0	0
$281$	−8.08897	−0.482548	−0.241274	−	0.970457i	$-0.577565\pi$
$281$	−8.08897	−0.482548	−0.241274	+	0.970457i	$0.577565\pi$
$282$	0	0
$283$	−5.21337	−0.309903	−0.154951	−	0.987922i	$-0.549522\pi$
$283$	−5.21337	−0.309903	−0.154951	+	0.987922i	$0.549522\pi$
$284$	0	0
$285$	−17.0834	−1.01193
$286$	0	0
$287$	−9.37975	−0.553669
$288$	0	0
$289$	−4.03754	−0.237502
$290$	0	0
$291$	−14.4072	−0.844566
$292$	0	0
$293$	11.4390	0.668274	0.334137	−	0.942525i	$-0.391555\pi$
$293$	11.4390	0.668274	0.334137	+	0.942525i	$0.391555\pi$
$294$	0	0
$295$	−28.6588	−1.66858
$296$	0	0
$297$	3.42843	0.198938
$298$	0	0
$299$	−50.0694	−2.89559
$300$	0	0
$301$	−12.3980	−0.714612
$302$	0	0
$303$	12.9944	0.746507
$304$	0	0
$305$	13.0199	0.745518
$306$	0	0
$307$	1.47936	0.0844315	0.0422157	−	0.999109i	$-0.486558\pi$
$307$	1.47936	0.0844315	0.0422157	+	0.999109i	$0.486558\pi$
$308$	0	0
$309$	22.2838	1.26768
$310$	0	0
$311$	20.0241	1.13546	0.567731	−	0.823214i	$-0.307821\pi$
$311$	20.0241	1.13546	0.567731	+	0.823214i	$0.307821\pi$
$312$	0	0
$313$	27.2154	1.53830	0.769152	−	0.639066i	$-0.220679\pi$
$313$	27.2154	1.53830	0.769152	+	0.639066i	$0.220679\pi$
$314$	0	0
$315$	−15.3978	−0.867567
$316$	0	0
$317$	30.0766	1.68927	0.844636	−	0.535340i	$-0.179817\pi$
$317$	30.0766	1.68927	0.844636	+	0.535340i	$0.179817\pi$
$318$	0	0
$319$	2.15656	0.120744
$320$	0	0
$321$	−25.6086	−1.42933
$322$	0	0
$323$	8.96547	0.498852
$324$	0	0
$325$	−12.3658	−0.685929
$326$	0	0
$327$	−19.9588	−1.10372
$328$	0	0
$329$	7.29546	0.402212
$330$	0	0
$331$	14.7923	0.813058	0.406529	−	0.913638i	$-0.366739\pi$
$331$	14.7923	0.813058	0.406529	+	0.913638i	$0.366739\pi$
$332$	0	0
$333$	−14.5857	−0.799291
$334$	0	0
$335$	21.9838	1.20110
$336$	0	0
$337$	−11.0697	−0.603003	−0.301501	−	0.953466i	$-0.597488\pi$
$337$	−11.0697	−0.603003	−0.301501	+	0.953466i	$0.597488\pi$
$338$	0	0
$339$	15.0608	0.817989
$340$	0	0
$341$	6.11478	0.331134
$342$	0	0
$343$	18.6834	1.00881
$344$	0	0
$345$	−64.7674	−3.48696
$346$	0	0
$347$	−13.2353	−0.710506	−0.355253	−	0.934770i	$-0.615605\pi$
$347$	−13.2353	−0.710506	−0.355253	+	0.934770i	$0.615605\pi$
$348$	0	0
$349$	−15.8843	−0.850269	−0.425135	−	0.905130i	$-0.639773\pi$
$349$	−15.8843	−0.850269	−0.425135	+	0.905130i	$0.639773\pi$
$350$	0	0
$351$	5.63467	0.300756
$352$	0	0
$353$	1.29827	0.0690999	0.0345500	−	0.999403i	$-0.489000\pi$
$353$	1.29827	0.0690999	0.0345500	+	0.999403i	$0.489000\pi$
$354$	0	0
$355$	−23.6588	−1.25568
$356$	0	0
$357$	15.1707	0.802918
$358$	0	0
$359$	−35.5797	−1.87782	−0.938912	−	0.344158i	$-0.888164\pi$
$359$	−35.5797	−1.87782	−0.938912	+	0.344158i	$0.888164\pi$
$360$	0	0
$361$	−12.7990	−0.673634
$362$	0	0
$363$	1.48717	0.0780564
$364$	0	0
$365$	15.4783	0.810173
$366$	0	0
$367$	20.5916	1.07487	0.537435	−	0.843305i	$-0.319393\pi$
$367$	20.5916	1.07487	0.537435	+	0.843305i	$0.319393\pi$
$368$	0	0
$369$	19.2849	1.00393
$370$	0	0
$371$	22.8258	1.18506
$372$	0	0
$373$	7.99992	0.414220	0.207110	−	0.978318i	$-0.433594\pi$
$373$	7.99992	0.414220	0.207110	+	0.978318i	$0.433594\pi$
$374$	0	0
$375$	18.3059	0.945313
$376$	0	0
$377$	3.54434	0.182543
$378$	0	0
$379$	−16.1733	−0.830767	−0.415384	−	0.909646i	$-0.636353\pi$
$379$	−16.1733	−0.830767	−0.415384	+	0.909646i	$0.636353\pi$
$380$	0	0
$381$	16.4570	0.843120
$382$	0	0
$383$	−20.5230	−1.04868	−0.524338	−	0.851510i	$-0.675687\pi$
$383$	−20.5230	−1.04868	−0.524338	+	0.851510i	$0.675687\pi$
$384$	0	0
$385$	14.5313	0.740585
$386$	0	0
$387$	25.4905	1.29576
$388$	0	0
$389$	−19.2861	−0.977843	−0.488922	−	0.872328i	$-0.662610\pi$
$389$	−19.2861	−0.977843	−0.488922	+	0.872328i	$0.662610\pi$
$390$	0	0
$391$	33.9903	1.71896
$392$	0	0
$393$	26.7934	1.35155
$394$	0	0
$395$	6.08976	0.306409
$396$	0	0
$397$	13.4004	0.672547	0.336273	−	0.941764i	$-0.390833\pi$
$397$	13.4004	0.672547	0.336273	+	0.941764i	$0.390833\pi$
$398$	0	0
$399$	10.4928	0.525297
$400$	0	0
$401$	−9.72700	−0.485743	−0.242872	−	0.970058i	$-0.578089\pi$
$401$	−9.72700	−0.485743	−0.242872	+	0.970058i	$0.578089\pi$
$402$	0	0
$403$	10.0497	0.500612
$404$	0	0
$405$	−20.4869	−1.01800
$406$	0	0
$407$	13.7649	0.682302
$408$	0	0
$409$	10.2040	0.504554	0.252277	−	0.967655i	$-0.418821\pi$
$409$	10.2040	0.504554	0.252277	+	0.967655i	$0.418821\pi$
$410$	0	0
$411$	40.1080	1.97838
$412$	0	0
$413$	17.6025	0.866163
$414$	0	0
$415$	−6.22960	−0.305799
$416$	0	0
$417$	41.3932	2.02703
$418$	0	0
$419$	−9.93843	−0.485524	−0.242762	−	0.970086i	$-0.578053\pi$
$419$	−9.93843	−0.485524	−0.242762	+	0.970086i	$0.578053\pi$
$420$	0	0
$421$	−10.4701	−0.510284	−0.255142	−	0.966904i	$-0.582122\pi$
$421$	−10.4701	−0.510284	−0.255142	+	0.966904i	$0.582122\pi$
$422$	0	0
$423$	−14.9995	−0.729302
$424$	0	0
$425$	8.39467	0.407201
$426$	0	0
$427$	−7.99694	−0.386999
$428$	0	0
$429$	−43.3606	−2.09347
$430$	0	0
$431$	21.9003	1.05490	0.527451	−	0.849586i	$-0.323148\pi$
$431$	21.9003	1.05490	0.527451	+	0.849586i	$0.323148\pi$
$432$	0	0
$433$	33.6836	1.61873	0.809364	−	0.587307i	$-0.199812\pi$
$433$	33.6836	1.61873	0.809364	+	0.587307i	$0.199812\pi$
$434$	0	0
$435$	4.58479	0.219824
$436$	0	0
$437$	23.5093	1.12460
$438$	0	0
$439$	−28.7355	−1.37147	−0.685736	−	0.727851i	$-0.740519\pi$
$439$	−28.7355	−1.37147	−0.685736	+	0.727851i	$0.740519\pi$
$440$	0	0
$441$	−14.4779	−0.689423
$442$	0	0
$443$	11.4373	0.543400	0.271700	−	0.962382i	$-0.412414\pi$
$443$	11.4373	0.543400	0.271700	+	0.962382i	$0.412414\pi$
$444$	0	0
$445$	41.7706	1.98011
$446$	0	0
$447$	−45.4512	−2.14977
$448$	0	0
$449$	−20.4037	−0.962912	−0.481456	−	0.876470i	$-0.659892\pi$
$449$	−20.4037	−0.962912	−0.481456	+	0.876470i	$0.659892\pi$
$450$	0	0
$451$	−18.1997	−0.856988
$452$	0	0
$453$	15.9242	0.748183
$454$	0	0
$455$	23.8824	1.11962
$456$	0	0
$457$	29.6209	1.38560	0.692802	−	0.721127i	$-0.256376\pi$
$457$	29.6209	1.38560	0.692802	+	0.721127i	$0.256376\pi$
$458$	0	0
$459$	−3.82517	−0.178544
$460$	0	0
$461$	−19.9499	−0.929162	−0.464581	−	0.885531i	$-0.653795\pi$
$461$	−19.9499	−0.929162	−0.464581	+	0.885531i	$0.653795\pi$
$462$	0	0
$463$	−33.4171	−1.55302	−0.776511	−	0.630103i	$-0.783012\pi$
$463$	−33.4171	−1.55302	−0.776511	+	0.630103i	$0.783012\pi$
$464$	0	0
$465$	12.9998	0.602853
$466$	0	0
$467$	3.04101	0.140721	0.0703607	−	0.997522i	$-0.477585\pi$
$467$	3.04101	0.140721	0.0703607	+	0.997522i	$0.477585\pi$
$468$	0	0
$469$	−13.5027	−0.623495
$470$	0	0
$471$	−39.9075	−1.83884
$472$	0	0
$473$	−24.0561	−1.10610
$474$	0	0
$475$	5.80616	0.266405
$476$	0	0
$477$	−46.9301	−2.14878
$478$	0	0
$479$	7.64655	0.349380	0.174690	−	0.984623i	$-0.444108\pi$
$479$	7.64655	0.349380	0.174690	+	0.984623i	$0.444108\pi$
$480$	0	0
$481$	22.6228	1.03151
$482$	0	0
$483$	39.7807	1.81008
$484$	0	0
$485$	15.3970	0.699141
$486$	0	0
$487$	−21.7583	−0.985961	−0.492981	−	0.870040i	$-0.664093\pi$
$487$	−21.7583	−0.985961	−0.492981	+	0.870040i	$0.664093\pi$
$488$	0	0
$489$	5.59487	0.253009
$490$	0	0
$491$	−11.4315	−0.515897	−0.257949	−	0.966159i	$-0.583047\pi$
$491$	−11.4315	−0.515897	−0.257949	+	0.966159i	$0.583047\pi$
$492$	0	0
$493$	−2.40612	−0.108366
$494$	0	0
$495$	−29.8766	−1.34285
$496$	0	0
$497$	14.5314	0.651824
$498$	0	0
$499$	−32.0439	−1.43448	−0.717242	−	0.696824i	$-0.754596\pi$
$499$	−32.0439	−1.43448	−0.717242	+	0.696824i	$0.754596\pi$
$500$	0	0
$501$	49.0929	2.19331
$502$	0	0
$503$	−7.07414	−0.315420	−0.157710	−	0.987485i	$-0.550411\pi$
$503$	−7.07414	−0.315420	−0.157710	+	0.987485i	$0.550411\pi$
$504$	0	0
$505$	−13.8871	−0.617966
$506$	0	0
$507$	−38.3263	−1.70213
$508$	0	0
$509$	2.36779	0.104951	0.0524753	−	0.998622i	$-0.483289\pi$
$509$	2.36779	0.104951	0.0524753	+	0.998622i	$0.483289\pi$
$510$	0	0
$511$	−9.50693	−0.420562
$512$	0	0
$513$	−2.64567	−0.116809
$514$	0	0
$515$	−23.8147	−1.04940
$516$	0	0
$517$	14.1555	0.622557
$518$	0	0
$519$	33.4873	1.46993
$520$	0	0
$521$	−11.0997	−0.486289	−0.243144	−	0.969990i	$-0.578179\pi$
$521$	−11.0997	−0.486289	−0.243144	+	0.969990i	$0.578179\pi$
$522$	0	0
$523$	−14.2177	−0.621698	−0.310849	−	0.950459i	$-0.600613\pi$
$523$	−14.2177	−0.621698	−0.310849	+	0.950459i	$0.600613\pi$
$524$	0	0
$525$	9.82474	0.428787
$526$	0	0
$527$	−6.82238	−0.297188
$528$	0	0
$529$	66.1296	2.87520
$530$	0	0
$531$	−36.1909	−1.57055
$532$	0	0
$533$	−29.9114	−1.29560
$534$	0	0
$535$	27.3678	1.18321
$536$	0	0
$537$	29.9546	1.29264
$538$	0	0
$539$	13.6632	0.588515
$540$	0	0
$541$	−4.20232	−0.180672	−0.0903360	−	0.995911i	$-0.528794\pi$
$541$	−4.20232	−0.180672	−0.0903360	+	0.995911i	$0.528794\pi$
$542$	0	0
$543$	41.3005	1.77237
$544$	0	0
$545$	21.3299	0.913673
$546$	0	0
$547$	−25.7693	−1.10182	−0.550909	−	0.834566i	$-0.685719\pi$
$547$	−25.7693	−1.10182	−0.550909	+	0.834566i	$0.685719\pi$
$548$	0	0
$549$	16.4418	0.701719
$550$	0	0
$551$	−1.66419	−0.0708970
$552$	0	0
$553$	−3.74039	−0.159057
$554$	0	0
$555$	29.2638	1.24218
$556$	0	0
$557$	37.8997	1.60586	0.802931	−	0.596072i	$-0.203273\pi$
$557$	37.8997	1.60586	0.802931	+	0.596072i	$0.203273\pi$
$558$	0	0
$559$	−39.5365	−1.67222
$560$	0	0
$561$	29.4359	1.24278
$562$	0	0
$563$	−5.59019	−0.235599	−0.117799	−	0.993037i	$-0.537584\pi$
$563$	−5.59019	−0.235599	−0.117799	+	0.993037i	$0.537584\pi$
$564$	0	0
$565$	−16.0954	−0.677140
$566$	0	0
$567$	12.5832	0.528445
$568$	0	0
$569$	−35.1997	−1.47565	−0.737824	−	0.674993i	$-0.764147\pi$
$569$	−35.1997	−1.47565	−0.737824	+	0.674993i	$0.764147\pi$
$570$	0	0
$571$	32.7949	1.37242	0.686212	−	0.727401i	$-0.259272\pi$
$571$	32.7949	1.37242	0.686212	+	0.727401i	$0.259272\pi$
$572$	0	0
$573$	−32.5647	−1.36041
$574$	0	0
$575$	22.0126	0.917987
$576$	0	0
$577$	12.2433	0.509695	0.254847	−	0.966981i	$-0.417975\pi$
$577$	12.2433	0.509695	0.254847	+	0.966981i	$0.417975\pi$
$578$	0	0
$579$	−1.46678	−0.0609572
$580$	0	0
$581$	3.82628	0.158741
$582$	0	0
$583$	44.2892	1.83427
$584$	0	0
$585$	−49.1025	−2.03014
$586$	0	0
$587$	19.3012	0.796647	0.398324	−	0.917245i	$-0.369592\pi$
$587$	19.3012	0.796647	0.398324	+	0.917245i	$0.369592\pi$
$588$	0	0
$589$	−4.71869	−0.194430
$590$	0	0
$591$	56.9459	2.34244
$592$	0	0
$593$	40.7775	1.67453	0.837266	−	0.546796i	$-0.184153\pi$
$593$	40.7775	1.67453	0.837266	+	0.546796i	$0.184153\pi$
$594$	0	0
$595$	−16.2129	−0.664664
$596$	0	0
$597$	−4.67011	−0.191135
$598$	0	0
$599$	−16.1323	−0.659147	−0.329574	−	0.944130i	$-0.606905\pi$
$599$	−16.1323	−0.659147	−0.329574	+	0.944130i	$0.606905\pi$
$600$	0	0
$601$	44.8646	1.83007	0.915033	−	0.403380i	$-0.132165\pi$
$601$	44.8646	1.83007	0.915033	+	0.403380i	$0.132165\pi$
$602$	0	0
$603$	27.7616	1.13054
$604$	0	0
$605$	−1.58934	−0.0646159
$606$	0	0
$607$	−12.0278	−0.488192	−0.244096	−	0.969751i	$-0.578491\pi$
$607$	−12.0278	−0.488192	−0.244096	+	0.969751i	$0.578491\pi$
$608$	0	0
$609$	−2.81602	−0.114111
$610$	0	0
$611$	23.2647	0.941189
$612$	0	0
$613$	2.73064	0.110289	0.0551447	−	0.998478i	$-0.482438\pi$
$613$	2.73064	0.110289	0.0551447	+	0.998478i	$0.482438\pi$
$614$	0	0
$615$	−38.6919	−1.56021
$616$	0	0
$617$	−17.6543	−0.710734	−0.355367	−	0.934727i	$-0.615644\pi$
$617$	−17.6543	−0.710734	−0.355367	+	0.934727i	$0.615644\pi$
$618$	0	0
$619$	−37.6642	−1.51385	−0.756927	−	0.653500i	$-0.773300\pi$
$619$	−37.6642	−1.51385	−0.756927	+	0.653500i	$0.773300\pi$
$620$	0	0
$621$	−10.0304	−0.402506
$622$	0	0
$623$	−25.6559	−1.02788
$624$	0	0
$625$	−31.2217	−1.24887
$626$	0	0
$627$	20.3593	0.813072
$628$	0	0
$629$	−15.3578	−0.612356
$630$	0	0
$631$	−45.3088	−1.80371	−0.901857	−	0.432034i	$-0.857796\pi$
$631$	−45.3088	−1.80371	−0.901857	+	0.432034i	$0.857796\pi$
$632$	0	0
$633$	61.7905	2.45595
$634$	0	0
$635$	−17.5876	−0.697943
$636$	0	0
$637$	22.4556	0.889723
$638$	0	0
$639$	−29.8768	−1.18191
$640$	0	0
$641$	−1.43529	−0.0566904	−0.0283452	−	0.999598i	$-0.509024\pi$
$641$	−1.43529	−0.0566904	−0.0283452	+	0.999598i	$0.509024\pi$
$642$	0	0
$643$	36.9682	1.45788	0.728941	−	0.684577i	$-0.240013\pi$
$643$	36.9682	1.45788	0.728941	+	0.684577i	$0.240013\pi$
$644$	0	0
$645$	−51.1425	−2.01374
$646$	0	0
$647$	−1.37923	−0.0542233	−0.0271116	−	0.999632i	$-0.508631\pi$
$647$	−1.37923	−0.0542233	−0.0271116	+	0.999632i	$0.508631\pi$
$648$	0	0
$649$	34.1544	1.34068
$650$	0	0
$651$	−7.98461	−0.312942
$652$	0	0
$653$	−15.7010	−0.614426	−0.307213	−	0.951641i	$-0.599396\pi$
$653$	−15.7010	−0.614426	−0.307213	+	0.951641i	$0.599396\pi$
$654$	0	0
$655$	−28.6341	−1.11883
$656$	0	0
$657$	19.5463	0.762576
$658$	0	0
$659$	20.3353	0.792152	0.396076	−	0.918218i	$-0.370372\pi$
$659$	20.3353	0.792152	0.396076	+	0.918218i	$0.370372\pi$
$660$	0	0
$661$	−9.03223	−0.351313	−0.175657	−	0.984452i	$-0.556205\pi$
$661$	−9.03223	−0.351313	−0.175657	+	0.984452i	$0.556205\pi$
$662$	0	0
$663$	48.3783	1.87886
$664$	0	0
$665$	−11.2136	−0.434846
$666$	0	0
$667$	−6.30935	−0.244299
$668$	0	0
$669$	−34.7727	−1.34439
$670$	0	0
$671$	−15.5166	−0.599011
$672$	0	0
$673$	35.9791	1.38689	0.693447	−	0.720508i	$-0.256091\pi$
$673$	35.9791	1.38689	0.693447	+	0.720508i	$0.256091\pi$
$674$	0	0
$675$	−2.47723	−0.0953487
$676$	0	0
$677$	−14.1471	−0.543718	−0.271859	−	0.962337i	$-0.587638\pi$
$677$	−14.1471	−0.543718	−0.271859	+	0.962337i	$0.587638\pi$
$678$	0	0
$679$	−9.45697	−0.362925
$680$	0	0
$681$	45.6910	1.75088
$682$	0	0
$683$	−3.98547	−0.152500	−0.0762499	−	0.997089i	$-0.524295\pi$
$683$	−3.98547	−0.152500	−0.0762499	+	0.997089i	$0.524295\pi$
$684$	0	0
$685$	−42.8633	−1.63772
$686$	0	0
$687$	35.2667	1.34551
$688$	0	0
$689$	72.7899	2.77307
$690$	0	0
$691$	−1.11084	−0.0422582	−0.0211291	−	0.999777i	$-0.506726\pi$
$691$	−1.11084	−0.0422582	−0.0211291	+	0.999777i	$0.506726\pi$
$692$	0	0
$693$	18.3505	0.697076
$694$	0	0
$695$	−44.2368	−1.67800
$696$	0	0
$697$	20.3057	0.769134
$698$	0	0
$699$	16.6034	0.627999
$700$	0	0
$701$	36.0636	1.36210	0.681051	−	0.732236i	$-0.261523\pi$
$701$	36.0636	1.36210	0.681051	+	0.732236i	$0.261523\pi$
$702$	0	0
$703$	−10.6222	−0.400624
$704$	0	0
$705$	30.0941	1.13341
$706$	0	0
$707$	8.52956	0.320787
$708$	0	0
$709$	−23.6901	−0.889701	−0.444850	−	0.895605i	$-0.646743\pi$
$709$	−23.6901	−0.889701	−0.444850	+	0.895605i	$0.646743\pi$
$710$	0	0
$711$	7.69027	0.288408
$712$	0	0
$713$	−17.8897	−0.669975
$714$	0	0
$715$	46.3394	1.73299
$716$	0	0
$717$	−49.2707	−1.84005
$718$	0	0
$719$	12.6927	0.473357	0.236679	−	0.971588i	$-0.423941\pi$
$719$	12.6927	0.473357	0.236679	+	0.971588i	$0.423941\pi$
$720$	0	0
$721$	14.6272	0.544745
$722$	0	0
$723$	20.2502	0.753113
$724$	0	0
$725$	−1.55824	−0.0578715
$726$	0	0
$727$	9.13718	0.338879	0.169440	−	0.985541i	$-0.445804\pi$
$727$	9.13718	0.338879	0.169440	+	0.985541i	$0.445804\pi$
$728$	0	0
$729$	−33.9468	−1.25729
$730$	0	0
$731$	26.8399	0.992709
$732$	0	0
$733$	−41.0549	−1.51640	−0.758199	−	0.652023i	$-0.773920\pi$
$733$	−41.0549	−1.51640	−0.758199	+	0.652023i	$0.773920\pi$
$734$	0	0
$735$	29.0475	1.07143
$736$	0	0
$737$	−26.1994	−0.965068
$738$	0	0
$739$	27.7604	1.02118	0.510591	−	0.859824i	$-0.329427\pi$
$739$	27.7604	1.02118	0.510591	+	0.859824i	$0.329427\pi$
$740$	0	0
$741$	33.4608	1.22921
$742$	0	0
$743$	17.1039	0.627483	0.313741	−	0.949508i	$-0.398417\pi$
$743$	17.1039	0.627483	0.313741	+	0.949508i	$0.398417\pi$
$744$	0	0
$745$	48.5736	1.77960
$746$	0	0
$747$	−7.86686	−0.287833
$748$	0	0
$749$	−16.8096	−0.614208
$750$	0	0
$751$	4.17031	0.152177	0.0760884	−	0.997101i	$-0.475757\pi$
$751$	4.17031	0.152177	0.0760884	+	0.997101i	$0.475757\pi$
$752$	0	0
$753$	−2.53364	−0.0923310
$754$	0	0
$755$	−17.0181	−0.619354
$756$	0	0
$757$	−36.0596	−1.31061	−0.655304	−	0.755365i	$-0.727459\pi$
$757$	−36.0596	−1.31061	−0.655304	+	0.755365i	$0.727459\pi$
$758$	0	0
$759$	77.1871	2.80171
$760$	0	0
$761$	20.4591	0.741641	0.370820	−	0.928705i	$-0.379077\pi$
$761$	20.4591	0.741641	0.370820	+	0.928705i	$0.379077\pi$
$762$	0	0
$763$	−13.1010	−0.474289
$764$	0	0
$765$	33.3339	1.20519
$766$	0	0
$767$	56.1332	2.02685
$768$	0	0
$769$	−55.0624	−1.98560	−0.992800	−	0.119782i	$-0.961780\pi$
$769$	−55.0624	−1.98560	−0.992800	+	0.119782i	$0.961780\pi$
$770$	0	0
$771$	26.7196	0.962284
$772$	0	0
$773$	−30.6721	−1.10320	−0.551599	−	0.834109i	$-0.685982\pi$
$773$	−30.6721	−1.10320	−0.551599	+	0.834109i	$0.685982\pi$
$774$	0	0
$775$	−4.41827	−0.158709
$776$	0	0
$777$	−17.9741	−0.644817
$778$	0	0
$779$	14.0444	0.503194
$780$	0	0
$781$	28.1955	1.00892
$782$	0	0
$783$	0.710037	0.0253746
$784$	0	0
$785$	42.6490	1.52221
$786$	0	0
$787$	−26.3280	−0.938492	−0.469246	−	0.883067i	$-0.655474\pi$
$787$	−26.3280	−0.938492	−0.469246	+	0.883067i	$0.655474\pi$
$788$	0	0
$789$	−60.2193	−2.14386
$790$	0	0
$791$	9.88596	0.351504
$792$	0	0
$793$	−25.5017	−0.905592
$794$	0	0
$795$	94.1575	3.33942
$796$	0	0
$797$	8.58289	0.304022	0.152011	−	0.988379i	$-0.451425\pi$
$797$	8.58289	0.304022	0.152011	+	0.988379i	$0.451425\pi$
$798$	0	0
$799$	−15.7936	−0.558736
$800$	0	0
$801$	52.7487	1.86378
$802$	0	0
$803$	−18.4464	−0.650960
$804$	0	0
$805$	−42.5136	−1.49841
$806$	0	0
$807$	63.2383	2.22609
$808$	0	0
$809$	−8.79502	−0.309216	−0.154608	−	0.987976i	$-0.549412\pi$
$809$	−8.79502	−0.309216	−0.154608	+	0.987976i	$0.549412\pi$
$810$	0	0
$811$	44.0607	1.54718	0.773591	−	0.633686i	$-0.218459\pi$
$811$	44.0607	1.54718	0.773591	+	0.633686i	$0.218459\pi$
$812$	0	0
$813$	54.5305	1.91247
$814$	0	0
$815$	−5.97923	−0.209443
$816$	0	0
$817$	18.5638	0.649464
$818$	0	0
$819$	30.1592	1.05385
$820$	0	0
$821$	−4.27900	−0.149338	−0.0746690	−	0.997208i	$-0.523790\pi$
$821$	−4.27900	−0.149338	−0.0746690	+	0.997208i	$0.523790\pi$
$822$	0	0
$823$	28.2812	0.985821	0.492911	−	0.870080i	$-0.335933\pi$
$823$	28.2812	0.985821	0.492911	+	0.870080i	$0.335933\pi$
$824$	0	0
$825$	19.0631	0.663692
$826$	0	0
$827$	−27.4581	−0.954810	−0.477405	−	0.878683i	$-0.658423\pi$
$827$	−27.4581	−0.954810	−0.477405	+	0.878683i	$0.658423\pi$
$828$	0	0
$829$	49.3263	1.71317	0.856586	−	0.516004i	$-0.172581\pi$
$829$	49.3263	1.71317	0.856586	+	0.516004i	$0.172581\pi$
$830$	0	0
$831$	60.6015	2.10224
$832$	0	0
$833$	−15.2443	−0.528183
$834$	0	0
$835$	−52.4655	−1.81564
$836$	0	0
$837$	2.01326	0.0695883
$838$	0	0
$839$	−33.8210	−1.16763	−0.583815	−	0.811887i	$-0.698441\pi$
$839$	−33.8210	−1.16763	−0.583815	+	0.811887i	$0.698441\pi$
$840$	0	0
$841$	−28.5534	−0.984599
$842$	0	0
$843$	20.4946	0.705870
$844$	0	0
$845$	40.9592	1.40904
$846$	0	0
$847$	0.976188	0.0335422
$848$	0	0
$849$	13.2088	0.453325
$850$	0	0
$851$	−40.2714	−1.38049
$852$	0	0
$853$	5.24511	0.179589	0.0897946	−	0.995960i	$-0.471379\pi$
$853$	5.24511	0.179589	0.0897946	+	0.995960i	$0.471379\pi$
$854$	0	0
$855$	23.0553	0.788476
$856$	0	0
$857$	−27.1229	−0.926499	−0.463250	−	0.886228i	$-0.653317\pi$
$857$	−27.1229	−0.926499	−0.463250	+	0.886228i	$0.653317\pi$
$858$	0	0
$859$	33.1399	1.13072	0.565360	−	0.824845i	$-0.308737\pi$
$859$	33.1399	1.13072	0.565360	+	0.824845i	$0.308737\pi$
$860$	0	0
$861$	23.7649	0.809906
$862$	0	0
$863$	37.4403	1.27448	0.637241	−	0.770665i	$-0.280076\pi$
$863$	37.4403	1.27448	0.637241	+	0.770665i	$0.280076\pi$
$864$	0	0
$865$	−35.7878	−1.21682
$866$	0	0
$867$	10.2297	0.347418
$868$	0	0
$869$	−7.25752	−0.246195
$870$	0	0
$871$	−43.0590	−1.45900
$872$	0	0
$873$	19.4436	0.658067
$874$	0	0
$875$	12.0161	0.406218
$876$	0	0
$877$	−22.4431	−0.757850	−0.378925	−	0.925427i	$-0.623706\pi$
$877$	−22.4431	−0.757850	−0.378925	+	0.925427i	$0.623706\pi$
$878$	0	0
$879$	−28.9823	−0.977550
$880$	0	0
$881$	47.9906	1.61684	0.808422	−	0.588604i	$-0.200322\pi$
$881$	47.9906	1.61684	0.808422	+	0.588604i	$0.200322\pi$
$882$	0	0
$883$	33.5960	1.13060	0.565298	−	0.824887i	$-0.308761\pi$
$883$	33.5960	1.13060	0.565298	+	0.824887i	$0.308761\pi$
$884$	0	0
$885$	72.6112	2.44080
$886$	0	0
$887$	28.9340	0.971508	0.485754	−	0.874096i	$-0.338545\pi$
$887$	28.9340	0.971508	0.485754	+	0.874096i	$0.338545\pi$
$888$	0	0
$889$	10.8025	0.362303
$890$	0	0
$891$	24.4154	0.817946
$892$	0	0
$893$	−10.9236	−0.365544
$894$	0	0
$895$	−32.0124	−1.07006
$896$	0	0
$897$	126.858	4.23566
$898$	0	0
$899$	1.26639	0.0422364
$900$	0	0
$901$	−49.4144	−1.64623
$902$	0	0
$903$	31.4122	1.04533
$904$	0	0
$905$	−44.1378	−1.46719
$906$	0	0
$907$	−6.62901	−0.220113	−0.110056	−	0.993925i	$-0.535103\pi$
$907$	−6.62901	−0.220113	−0.110056	+	0.993925i	$0.535103\pi$
$908$	0	0
$909$	−17.5369	−0.581661
$910$	0	0
$911$	24.9716	0.827345	0.413673	−	0.910426i	$-0.364246\pi$
$911$	24.9716	0.827345	0.413673	+	0.910426i	$0.364246\pi$
$912$	0	0
$913$	7.42418	0.245704
$914$	0	0
$915$	−32.9878	−1.09054
$916$	0	0
$917$	17.5873	0.580785
$918$	0	0
$919$	−47.4074	−1.56383	−0.781914	−	0.623387i	$-0.785756\pi$
$919$	−47.4074	−1.56383	−0.781914	+	0.623387i	$0.785756\pi$
$920$	0	0
$921$	−3.74816	−0.123506
$922$	0	0
$923$	46.3397	1.52529
$924$	0	0
$925$	−9.94593	−0.327020
$926$	0	0
$927$	−30.0736	−0.987748
$928$	0	0
$929$	12.5106	0.410461	0.205231	−	0.978714i	$-0.434206\pi$
$929$	12.5106	0.410461	0.205231	+	0.978714i	$0.434206\pi$
$930$	0	0
$931$	−10.5437	−0.345556
$932$	0	0
$933$	−50.7339	−1.66095
$934$	0	0
$935$	−31.4581	−1.02879
$936$	0	0
$937$	22.6107	0.738659	0.369329	−	0.929299i	$-0.379587\pi$
$937$	22.6107	0.738659	0.369329	+	0.929299i	$0.379587\pi$
$938$	0	0
$939$	−68.9540	−2.25023
$940$	0	0
$941$	0.200768	0.00654485	0.00327242	−	0.999995i	$-0.498958\pi$
$941$	0.200768	0.00654485	0.00327242	+	0.999995i	$0.498958\pi$
$942$	0	0
$943$	53.2458	1.73392
$944$	0	0
$945$	4.78436	0.155635
$946$	0	0
$947$	28.2447	0.917829	0.458914	−	0.888481i	$-0.348238\pi$
$947$	28.2447	0.917829	0.458914	+	0.888481i	$0.348238\pi$
$948$	0	0
$949$	−30.3169	−0.984129
$950$	0	0
$951$	−76.2034	−2.47107
$952$	0	0
$953$	28.5010	0.923238	0.461619	−	0.887078i	$-0.347269\pi$
$953$	28.5010	0.923238	0.461619	+	0.887078i	$0.347269\pi$
$954$	0	0
$955$	34.8018	1.12616
$956$	0	0
$957$	−5.46396	−0.176625
$958$	0	0
$959$	26.3271	0.850145
$960$	0	0
$961$	−27.4093	−0.884170
$962$	0	0
$963$	34.5606	1.11370
$964$	0	0
$965$	1.56754	0.0504610
$966$	0	0
$967$	−25.3970	−0.816713	−0.408356	−	0.912823i	$-0.633898\pi$
$967$	−25.3970	−0.816713	−0.408356	+	0.912823i	$0.633898\pi$
$968$	0	0
$969$	−22.7153	−0.729720
$970$	0	0
$971$	−61.2093	−1.96430	−0.982149	−	0.188103i	$-0.939766\pi$
$971$	−61.2093	−1.96430	−0.982149	+	0.188103i	$0.939766\pi$
$972$	0	0
$973$	27.1707	0.871052
$974$	0	0
$975$	31.3304	1.00338
$976$	0	0
$977$	54.3713	1.73949	0.869746	−	0.493500i	$-0.164283\pi$
$977$	54.3713	1.73949	0.869746	+	0.493500i	$0.164283\pi$
$978$	0	0
$979$	−49.7804	−1.59099
$980$	0	0
$981$	26.9358	0.859995
$982$	0	0
$983$	16.1072	0.513741	0.256870	−	0.966446i	$-0.417309\pi$
$983$	16.1072	0.513741	0.256870	+	0.966446i	$0.417309\pi$
$984$	0	0
$985$	−60.8580	−1.93910
$986$	0	0
$987$	−18.4841	−0.588354
$988$	0	0
$989$	70.3798	2.23795
$990$	0	0
$991$	31.5652	1.00270	0.501351	−	0.865244i	$-0.332836\pi$
$991$	31.5652	1.00270	0.501351	+	0.865244i	$0.332836\pi$
$992$	0	0
$993$	−37.4784	−1.18934
$994$	0	0
$995$	4.99094	0.158223
$996$	0	0
$997$	2.89921	0.0918189	0.0459094	−	0.998946i	$-0.485381\pi$
$997$	2.89921	0.0918189	0.0459094	+	0.998946i	$0.485381\pi$
$998$	0	0
$999$	4.53203	0.143387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.3		14
4.3	odd	2		1004.2.a.b.1.12	✓	14
12.11	even	2		9036.2.a.m.1.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.12	✓	14	4.3	odd	2
4016.2.a.j.1.3		14	1.1	even	1	trivial
9036.2.a.m.1.4		14	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-2.53364 q^{3} +2.70770 q^{5} -1.66309 q^{7} +3.41934 q^{9} +O(q^{10})\)	\(q-2.53364 q^{3} +2.70770 q^{5} -1.66309 q^{7} +3.41934 q^{9} -3.22692 q^{11} -5.30349 q^{13} -6.86033 q^{15} +3.60034 q^{17} +2.49017 q^{19} +4.21368 q^{21} +9.44085 q^{23} +2.33163 q^{25} -1.06245 q^{27} -0.668304 q^{29} -1.89493 q^{31} +8.17586 q^{33} -4.50315 q^{35} -4.26565 q^{37} +13.4371 q^{39} +5.63994 q^{41} +7.45482 q^{43} +9.25853 q^{45} -4.38668 q^{47} -4.23412 q^{49} -9.12197 q^{51} -13.7249 q^{53} -8.73753 q^{55} -6.30920 q^{57} -10.5842 q^{59} +4.80848 q^{61} -5.68667 q^{63} -14.3602 q^{65} +8.11901 q^{67} -23.9197 q^{69} -8.73759 q^{71} +5.71642 q^{73} -5.90751 q^{75} +5.36667 q^{77} +2.24905 q^{79} -7.56615 q^{81} -2.30070 q^{83} +9.74864 q^{85} +1.69324 q^{87} +15.4266 q^{89} +8.82019 q^{91} +4.80106 q^{93} +6.74264 q^{95} +5.68637 q^{97} -11.0339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

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