LMFDB - Embedded newform 4016.2.a.f.1.2

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:	$6$
Coefficient field:	6.6.60853001.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 $ x^6 - 3x^5 - 8x^4 + 15x^3 + 20x^2 - 12*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.76567$ 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)

$f(q)$	$=$	$q-2.70836 q^{3} -2.61208 q^{5} -1.43044 q^{7} +4.33524 q^{9} +O(q^{10})$	$q-2.70836 q^{3} -2.61208 q^{5} -1.43044 q^{7} +4.33524 q^{9} -3.64894 q^{11} -2.91926 q^{13} +7.07447 q^{15} +0.0204488 q^{17} +1.79476 q^{19} +3.87414 q^{21} +2.66448 q^{23} +1.82298 q^{25} -3.61630 q^{27} +6.24938 q^{29} +4.99864 q^{31} +9.88267 q^{33} +3.73642 q^{35} +0.458495 q^{37} +7.90642 q^{39} -1.83373 q^{41} -0.880289 q^{43} -11.3240 q^{45} +9.50177 q^{47} -4.95385 q^{49} -0.0553828 q^{51} +2.12892 q^{53} +9.53134 q^{55} -4.86087 q^{57} +1.13525 q^{59} -13.6489 q^{61} -6.20128 q^{63} +7.62535 q^{65} +4.26011 q^{67} -7.21639 q^{69} -10.1048 q^{71} +14.1773 q^{73} -4.93730 q^{75} +5.21958 q^{77} +8.58537 q^{79} -3.21144 q^{81} -5.09226 q^{83} -0.0534139 q^{85} -16.9256 q^{87} +2.58208 q^{89} +4.17582 q^{91} -13.5381 q^{93} -4.68807 q^{95} +11.5624 q^{97} -15.8190 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * 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
$3$	−2.70836	−1.56367	−0.781837	−	0.623482i	$-0.785717\pi$
$3$	−2.70836	−1.56367	−0.781837	+	0.623482i	$0.785717\pi$
$4$	0	0
$5$	−2.61208	−1.16816	−0.584080	−	0.811696i	$-0.698544\pi$
$5$	−2.61208	−1.16816	−0.584080	+	0.811696i	$0.698544\pi$
$6$	0	0
$7$	−1.43044	−0.540654	−0.270327	−	0.962769i	$-0.587132\pi$
$7$	−1.43044	−0.540654	−0.270327	+	0.962769i	$0.587132\pi$
$8$	0	0
$9$	4.33524	1.44508
$10$	0	0
$11$	−3.64894	−1.10020	−0.550099	−	0.835100i	$-0.685410\pi$
$11$	−3.64894	−1.10020	−0.550099	+	0.835100i	$0.685410\pi$
$12$	0	0
$13$	−2.91926	−0.809657	−0.404829	−	0.914393i	$-0.632669\pi$
$13$	−2.91926	−0.809657	−0.404829	+	0.914393i	$0.632669\pi$
$14$	0	0
$15$	7.07447	1.82662
$16$	0	0
$17$	0.0204488	0.00495956	0.00247978	−	0.999997i	$-0.499211\pi$
$17$	0.0204488	0.00495956	0.00247978	+	0.999997i	$0.499211\pi$
$18$	0	0
$19$	1.79476	0.411747	0.205874	−	0.978579i	$-0.433996\pi$
$19$	1.79476	0.411747	0.205874	+	0.978579i	$0.433996\pi$
$20$	0	0
$21$	3.87414	0.845408
$22$	0	0
$23$	2.66448	0.555583	0.277792	−	0.960641i	$-0.410397\pi$
$23$	2.66448	0.555583	0.277792	+	0.960641i	$0.410397\pi$
$24$	0	0
$25$	1.82298	0.364596
$26$	0	0
$27$	−3.61630	−0.695958
$28$	0	0
$29$	6.24938	1.16048	0.580240	−	0.814445i	$-0.302959\pi$
$29$	6.24938	1.16048	0.580240	+	0.814445i	$0.302959\pi$
$30$	0	0
$31$	4.99864	0.897782	0.448891	−	0.893586i	$-0.351819\pi$
$31$	4.99864	0.897782	0.448891	+	0.893586i	$0.351819\pi$
$32$	0	0
$33$	9.88267	1.72035
$34$	0	0
$35$	3.73642	0.631571
$36$	0	0
$37$	0.458495	0.0753761	0.0376880	−	0.999290i	$-0.488001\pi$
$37$	0.458495	0.0753761	0.0376880	+	0.999290i	$0.488001\pi$
$38$	0	0
$39$	7.90642	1.26604
$40$	0	0
$41$	−1.83373	−0.286381	−0.143190	−	0.989695i	$-0.545736\pi$
$41$	−1.83373	−0.286381	−0.143190	+	0.989695i	$0.545736\pi$
$42$	0	0
$43$	−0.880289	−0.134243	−0.0671214	−	0.997745i	$-0.521381\pi$
$43$	−0.880289	−0.134243	−0.0671214	+	0.997745i	$0.521381\pi$
$44$	0	0
$45$	−11.3240	−1.68808
$46$	0	0
$47$	9.50177	1.38598	0.692988	−	0.720949i	$-0.256294\pi$
$47$	9.50177	1.38598	0.692988	+	0.720949i	$0.256294\pi$
$48$	0	0
$49$	−4.95385	−0.707693
$50$	0	0
$51$	−0.0553828	−0.00775514
$52$	0	0
$53$	2.12892	0.292430	0.146215	−	0.989253i	$-0.453291\pi$
$53$	2.12892	0.292430	0.146215	+	0.989253i	$0.453291\pi$
$54$	0	0
$55$	9.53134	1.28521
$56$	0	0
$57$	−4.86087	−0.643839
$58$	0	0
$59$	1.13525	0.147796	0.0738982	−	0.997266i	$-0.476456\pi$
$59$	1.13525	0.147796	0.0738982	+	0.997266i	$0.476456\pi$
$60$	0	0
$61$	−13.6489	−1.74757	−0.873784	−	0.486314i	$-0.838341\pi$
$61$	−13.6489	−1.74757	−0.873784	+	0.486314i	$0.838341\pi$
$62$	0	0
$63$	−6.20128	−0.781288
$64$	0	0
$65$	7.62535	0.945809
$66$	0	0
$67$	4.26011	0.520456	0.260228	−	0.965547i	$-0.416202\pi$
$67$	4.26011	0.520456	0.260228	+	0.965547i	$0.416202\pi$
$68$	0	0
$69$	−7.21639	−0.868752
$70$	0	0
$71$	−10.1048	−1.19922	−0.599610	−	0.800293i	$-0.704677\pi$
$71$	−10.1048	−1.19922	−0.599610	+	0.800293i	$0.704677\pi$
$72$	0	0
$73$	14.1773	1.65933	0.829665	−	0.558261i	$-0.188531\pi$
$73$	14.1773	1.65933	0.829665	+	0.558261i	$0.188531\pi$
$74$	0	0
$75$	−4.93730	−0.570110
$76$	0	0
$77$	5.21958	0.594827
$78$	0	0
$79$	8.58537	0.965929	0.482965	−	0.875640i	$-0.339560\pi$
$79$	8.58537	0.965929	0.482965	+	0.875640i	$0.339560\pi$
$80$	0	0
$81$	−3.21144	−0.356827
$82$	0	0
$83$	−5.09226	−0.558949	−0.279474	−	0.960153i	$-0.590160\pi$
$83$	−5.09226	−0.558949	−0.279474	+	0.960153i	$0.590160\pi$
$84$	0	0
$85$	−0.0534139	−0.00579355
$86$	0	0
$87$	−16.9256	−1.81461
$88$	0	0
$89$	2.58208	0.273700	0.136850	−	0.990592i	$-0.456302\pi$
$89$	2.58208	0.273700	0.136850	+	0.990592i	$0.456302\pi$
$90$	0	0
$91$	4.17582	0.437745
$92$	0	0
$93$	−13.5381	−1.40384
$94$	0	0
$95$	−4.68807	−0.480986
$96$	0	0
$97$	11.5624	1.17399	0.586993	−	0.809592i	$-0.300311\pi$
$97$	11.5624	1.17399	0.586993	+	0.809592i	$0.300311\pi$
$98$	0	0
$99$	−15.8190	−1.58987
$100$	0	0
$101$	17.0280	1.69435	0.847177	−	0.531311i	$-0.178300\pi$
$101$	17.0280	1.69435	0.847177	+	0.531311i	$0.178300\pi$
$102$	0	0
$103$	14.6090	1.43947	0.719734	−	0.694250i	$-0.244264\pi$
$103$	14.6090	1.43947	0.719734	+	0.694250i	$0.244264\pi$
$104$	0	0
$105$	−10.1196	−0.987571
$106$	0	0
$107$	−7.25239	−0.701115	−0.350557	−	0.936541i	$-0.614008\pi$
$107$	−7.25239	−0.701115	−0.350557	+	0.936541i	$0.614008\pi$
$108$	0	0
$109$	−6.27437	−0.600976	−0.300488	−	0.953786i	$-0.597150\pi$
$109$	−6.27437	−0.600976	−0.300488	+	0.953786i	$0.597150\pi$
$110$	0	0
$111$	−1.24177	−0.117864
$112$	0	0
$113$	5.05926	0.475935	0.237967	−	0.971273i	$-0.423519\pi$
$113$	5.05926	0.475935	0.237967	+	0.971273i	$0.423519\pi$
$114$	0	0
$115$	−6.95986	−0.649010
$116$	0	0
$117$	−12.6557	−1.17002
$118$	0	0
$119$	−0.0292507	−0.00268141
$120$	0	0
$121$	2.31479	0.210435
$122$	0	0
$123$	4.96641	0.447806
$124$	0	0
$125$	8.29864	0.742253
$126$	0	0
$127$	−9.01047	−0.799550	−0.399775	−	0.916613i	$-0.630912\pi$
$127$	−9.01047	−0.799550	−0.399775	+	0.916613i	$0.630912\pi$
$128$	0	0
$129$	2.38414	0.209912
$130$	0	0
$131$	7.47089	0.652735	0.326367	−	0.945243i	$-0.394175\pi$
$131$	7.47089	0.652735	0.326367	+	0.945243i	$0.394175\pi$
$132$	0	0
$133$	−2.56730	−0.222613
$134$	0	0
$135$	9.44608	0.812989
$136$	0	0
$137$	18.5940	1.58860	0.794298	−	0.607529i	$-0.207839\pi$
$137$	18.5940	1.58860	0.794298	+	0.607529i	$0.207839\pi$
$138$	0	0
$139$	7.23757	0.613883	0.306942	−	0.951728i	$-0.400694\pi$
$139$	7.23757	0.613883	0.306942	+	0.951728i	$0.400694\pi$
$140$	0	0
$141$	−25.7342	−2.16721
$142$	0	0
$143$	10.6522	0.890783
$144$	0	0
$145$	−16.3239	−1.35563
$146$	0	0
$147$	13.4168	1.10660
$148$	0	0
$149$	−21.3197	−1.74658	−0.873288	−	0.487204i	$-0.838017\pi$
$149$	−21.3197	−1.74658	−0.873288	+	0.487204i	$0.838017\pi$
$150$	0	0
$151$	−19.0098	−1.54700	−0.773499	−	0.633798i	$-0.781495\pi$
$151$	−19.0098	−1.54700	−0.773499	+	0.633798i	$0.781495\pi$
$152$	0	0
$153$	0.0886503	0.00716695
$154$	0	0
$155$	−13.0569	−1.04875
$156$	0	0
$157$	−23.4057	−1.86798	−0.933988	−	0.357305i	$-0.883696\pi$
$157$	−23.4057	−1.86798	−0.933988	+	0.357305i	$0.883696\pi$
$158$	0	0
$159$	−5.76589	−0.457265
$160$	0	0
$161$	−3.81138	−0.300379
$162$	0	0
$163$	−13.3392	−1.04480	−0.522402	−	0.852699i	$-0.674964\pi$
$163$	−13.3392	−1.04480	−0.522402	+	0.852699i	$0.674964\pi$
$164$	0	0
$165$	−25.8143	−2.00964
$166$	0	0
$167$	−1.60778	−0.124414	−0.0622069	−	0.998063i	$-0.519814\pi$
$167$	−1.60778	−0.124414	−0.0622069	+	0.998063i	$0.519814\pi$
$168$	0	0
$169$	−4.47792	−0.344455
$170$	0	0
$171$	7.78072	0.595007
$172$	0	0
$173$	−9.41401	−0.715734	−0.357867	−	0.933773i	$-0.616496\pi$
$173$	−9.41401	−0.715734	−0.357867	+	0.933773i	$0.616496\pi$
$174$	0	0
$175$	−2.60766	−0.197121
$176$	0	0
$177$	−3.07466	−0.231106
$178$	0	0
$179$	9.84423	0.735792	0.367896	−	0.929867i	$-0.380078\pi$
$179$	9.84423	0.735792	0.367896	+	0.929867i	$0.380078\pi$
$180$	0	0
$181$	14.6815	1.09127	0.545633	−	0.838024i	$-0.316289\pi$
$181$	14.6815	1.09127	0.545633	+	0.838024i	$0.316289\pi$
$182$	0	0
$183$	36.9663	2.73263
$184$	0	0
$185$	−1.19763	−0.0880513
$186$	0	0
$187$	−0.0746165	−0.00545650
$188$	0	0
$189$	5.17289	0.376273
$190$	0	0
$191$	−26.0556	−1.88531	−0.942657	−	0.333763i	$-0.891681\pi$
$191$	−26.0556	−1.88531	−0.942657	+	0.333763i	$0.891681\pi$
$192$	0	0
$193$	16.5143	1.18873	0.594363	−	0.804197i	$-0.297404\pi$
$193$	16.5143	1.18873	0.594363	+	0.804197i	$0.297404\pi$
$194$	0	0
$195$	−20.6522	−1.47894
$196$	0	0
$197$	0.618365	0.0440567	0.0220283	−	0.999757i	$-0.492988\pi$
$197$	0.618365	0.0440567	0.0220283	+	0.999757i	$0.492988\pi$
$198$	0	0
$199$	−13.3420	−0.945791	−0.472896	−	0.881118i	$-0.656791\pi$
$199$	−13.3420	−0.945791	−0.472896	+	0.881118i	$0.656791\pi$
$200$	0	0
$201$	−11.5379	−0.813823
$202$	0	0
$203$	−8.93935	−0.627419
$204$	0	0
$205$	4.78986	0.334538
$206$	0	0
$207$	11.5512	0.802862
$208$	0	0
$209$	−6.54899	−0.453003
$210$	0	0
$211$	−20.4210	−1.40584	−0.702921	−	0.711268i	$-0.748121\pi$
$211$	−20.4210	−1.40584	−0.702921	+	0.711268i	$0.748121\pi$
$212$	0	0
$213$	27.3675	1.87519
$214$	0	0
$215$	2.29939	0.156817
$216$	0	0
$217$	−7.15024	−0.485390
$218$	0	0
$219$	−38.3974	−2.59465
$220$	0	0
$221$	−0.0596953	−0.00401554
$222$	0	0
$223$	−19.4722	−1.30395	−0.651977	−	0.758238i	$-0.726060\pi$
$223$	−19.4722	−1.30395	−0.651977	+	0.758238i	$0.726060\pi$
$224$	0	0
$225$	7.90305	0.526870
$226$	0	0
$227$	−4.57279	−0.303507	−0.151753	−	0.988418i	$-0.548492\pi$
$227$	−4.57279	−0.303507	−0.151753	+	0.988418i	$0.548492\pi$
$228$	0	0
$229$	−12.3743	−0.817718	−0.408859	−	0.912598i	$-0.634073\pi$
$229$	−12.3743	−0.817718	−0.408859	+	0.912598i	$0.634073\pi$
$230$	0	0
$231$	−14.1365	−0.930116
$232$	0	0
$233$	−1.88013	−0.123171	−0.0615857	−	0.998102i	$-0.519616\pi$
$233$	−1.88013	−0.123171	−0.0615857	+	0.998102i	$0.519616\pi$
$234$	0	0
$235$	−24.8194	−1.61904
$236$	0	0
$237$	−23.2523	−1.51040
$238$	0	0
$239$	20.5391	1.32856	0.664282	−	0.747482i	$-0.268737\pi$
$239$	20.5391	1.32856	0.664282	+	0.747482i	$0.268737\pi$
$240$	0	0
$241$	9.70894	0.625408	0.312704	−	0.949851i	$-0.398765\pi$
$241$	9.70894	0.625408	0.312704	+	0.949851i	$0.398765\pi$
$242$	0	0
$243$	19.5467	1.25392
$244$	0	0
$245$	12.9399	0.826698
$246$	0	0
$247$	−5.23938	−0.333374
$248$	0	0
$249$	13.7917	0.874014
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−9.72255	−0.611252
$254$	0	0
$255$	0.144664	0.00905923
$256$	0	0
$257$	8.29736	0.517575	0.258788	−	0.965934i	$-0.416677\pi$
$257$	8.29736	0.517575	0.258788	+	0.965934i	$0.416677\pi$
$258$	0	0
$259$	−0.655848	−0.0407524
$260$	0	0
$261$	27.0925	1.67699
$262$	0	0
$263$	−7.04431	−0.434371	−0.217185	−	0.976130i	$-0.569688\pi$
$263$	−7.04431	−0.434371	−0.217185	+	0.976130i	$0.569688\pi$
$264$	0	0
$265$	−5.56092	−0.341605
$266$	0	0
$267$	−6.99321	−0.427978
$268$	0	0
$269$	31.5210	1.92187	0.960934	−	0.276779i	$-0.0892669\pi$
$269$	31.5210	1.92187	0.960934	+	0.276779i	$0.0892669\pi$
$270$	0	0
$271$	−26.0328	−1.58138	−0.790691	−	0.612215i	$-0.790279\pi$
$271$	−26.0328	−1.58138	−0.790691	+	0.612215i	$0.790279\pi$
$272$	0	0
$273$	−11.3096	−0.684490
$274$	0	0
$275$	−6.65195	−0.401128
$276$	0	0
$277$	−28.6739	−1.72285	−0.861423	−	0.507889i	$-0.830426\pi$
$277$	−28.6739	−1.72285	−0.861423	+	0.507889i	$0.830426\pi$
$278$	0	0
$279$	21.6703	1.29737
$280$	0	0
$281$	−22.2475	−1.32717	−0.663586	−	0.748100i	$-0.730966\pi$
$281$	−22.2475	−1.32717	−0.663586	+	0.748100i	$0.730966\pi$
$282$	0	0
$283$	28.6340	1.70211	0.851056	−	0.525075i	$-0.175963\pi$
$283$	28.6340	1.70211	0.851056	+	0.525075i	$0.175963\pi$
$284$	0	0
$285$	12.6970	0.752106
$286$	0	0
$287$	2.62304	0.154833
$288$	0	0
$289$	−16.9996	−0.999975
$290$	0	0
$291$	−31.3153	−1.83573
$292$	0	0
$293$	−3.06657	−0.179151	−0.0895754	−	0.995980i	$-0.528551\pi$
$293$	−3.06657	−0.179151	−0.0895754	+	0.995980i	$0.528551\pi$
$294$	0	0
$295$	−2.96536	−0.172650
$296$	0	0
$297$	13.1957	0.765691
$298$	0	0
$299$	−7.77833	−0.449832
$300$	0	0
$301$	1.25920	0.0725790
$302$	0	0
$303$	−46.1181	−2.64942
$304$	0	0
$305$	35.6522	2.04144
$306$	0	0
$307$	−8.70247	−0.496676	−0.248338	−	0.968673i	$-0.579884\pi$
$307$	−8.70247	−0.496676	−0.248338	+	0.968673i	$0.579884\pi$
$308$	0	0
$309$	−39.5665	−2.25086
$310$	0	0
$311$	−14.7271	−0.835099	−0.417549	−	0.908654i	$-0.637111\pi$
$311$	−14.7271	−0.835099	−0.417549	+	0.908654i	$0.637111\pi$
$312$	0	0
$313$	−23.0794	−1.30453	−0.652263	−	0.757993i	$-0.726180\pi$
$313$	−23.0794	−1.30453	−0.652263	+	0.757993i	$0.726180\pi$
$314$	0	0
$315$	16.1983	0.912669
$316$	0	0
$317$	16.5442	0.929214	0.464607	−	0.885517i	$-0.346196\pi$
$317$	16.5442	0.929214	0.464607	+	0.885517i	$0.346196\pi$
$318$	0	0
$319$	−22.8036	−1.27676
$320$	0	0
$321$	19.6421	1.09632
$322$	0	0
$323$	0.0367007	0.00204208
$324$	0	0
$325$	−5.32176	−0.295198
$326$	0	0
$327$	16.9933	0.939731
$328$	0	0
$329$	−13.5917	−0.749334
$330$	0	0
$331$	8.24392	0.453127	0.226563	−	0.973996i	$-0.427251\pi$
$331$	8.24392	0.453127	0.226563	+	0.973996i	$0.427251\pi$
$332$	0	0
$333$	1.98768	0.108924
$334$	0	0
$335$	−11.1278	−0.607975
$336$	0	0
$337$	2.66698	0.145279	0.0726397	−	0.997358i	$-0.476858\pi$
$337$	2.66698	0.145279	0.0726397	+	0.997358i	$0.476858\pi$
$338$	0	0
$339$	−13.7023	−0.744207
$340$	0	0
$341$	−18.2398	−0.987738
$342$	0	0
$343$	17.0992	0.923272
$344$	0	0
$345$	18.8498	1.01484
$346$	0	0
$347$	−9.18295	−0.492967	−0.246483	−	0.969147i	$-0.579275\pi$
$347$	−9.18295	−0.492967	−0.246483	+	0.969147i	$0.579275\pi$
$348$	0	0
$349$	−16.8937	−0.904299	−0.452149	−	0.891942i	$-0.649343\pi$
$349$	−16.8937	−0.904299	−0.452149	+	0.891942i	$0.649343\pi$
$350$	0	0
$351$	10.5569	0.563487
$352$	0	0
$353$	22.5418	1.19978	0.599890	−	0.800083i	$-0.295211\pi$
$353$	22.5418	1.19978	0.599890	+	0.800083i	$0.295211\pi$
$354$	0	0
$355$	26.3946	1.40088
$356$	0	0
$357$	0.0792216	0.00419285
$358$	0	0
$359$	2.34025	0.123514	0.0617568	−	0.998091i	$-0.480330\pi$
$359$	2.34025	0.123514	0.0617568	+	0.998091i	$0.480330\pi$
$360$	0	0
$361$	−15.7788	−0.830464
$362$	0	0
$363$	−6.26928	−0.329052
$364$	0	0
$365$	−37.0324	−1.93836
$366$	0	0
$367$	18.8335	0.983098	0.491549	−	0.870850i	$-0.336431\pi$
$367$	18.8335	0.983098	0.491549	+	0.870850i	$0.336431\pi$
$368$	0	0
$369$	−7.94965	−0.413842
$370$	0	0
$371$	−3.04529	−0.158103
$372$	0	0
$373$	16.0394	0.830489	0.415244	−	0.909710i	$-0.363696\pi$
$373$	16.0394	0.830489	0.415244	+	0.909710i	$0.363696\pi$
$374$	0	0
$375$	−22.4757	−1.16064
$376$	0	0
$377$	−18.2436	−0.939592
$378$	0	0
$379$	28.6519	1.47175	0.735874	−	0.677118i	$-0.236771\pi$
$379$	28.6519	1.47175	0.735874	+	0.677118i	$0.236771\pi$
$380$	0	0
$381$	24.4036	1.25024
$382$	0	0
$383$	−19.0790	−0.974891	−0.487446	−	0.873153i	$-0.662071\pi$
$383$	−19.0790	−0.974891	−0.487446	+	0.873153i	$0.662071\pi$
$384$	0	0
$385$	−13.6340	−0.694852
$386$	0	0
$387$	−3.81626	−0.193991
$388$	0	0
$389$	11.2852	0.572183	0.286091	−	0.958202i	$-0.407644\pi$
$389$	11.2852	0.572183	0.286091	+	0.958202i	$0.407644\pi$
$390$	0	0
$391$	0.0544855	0.00275545
$392$	0	0
$393$	−20.2339	−1.02066
$394$	0	0
$395$	−22.4257	−1.12836
$396$	0	0
$397$	19.6595	0.986682	0.493341	−	0.869836i	$-0.335776\pi$
$397$	19.6595	0.986682	0.493341	+	0.869836i	$0.335776\pi$
$398$	0	0
$399$	6.95318	0.348094
$400$	0	0
$401$	−11.9725	−0.597877	−0.298939	−	0.954272i	$-0.596633\pi$
$401$	−11.9725	−0.597877	−0.298939	+	0.954272i	$0.596633\pi$
$402$	0	0
$403$	−14.5923	−0.726896
$404$	0	0
$405$	8.38856	0.416831
$406$	0	0
$407$	−1.67302	−0.0829286
$408$	0	0
$409$	8.77105	0.433700	0.216850	−	0.976205i	$-0.430422\pi$
$409$	8.77105	0.433700	0.216850	+	0.976205i	$0.430422\pi$
$410$	0	0
$411$	−50.3594	−2.48405
$412$	0	0
$413$	−1.62390	−0.0799068
$414$	0	0
$415$	13.3014	0.652941
$416$	0	0
$417$	−19.6020	−0.959913
$418$	0	0
$419$	−5.83664	−0.285139	−0.142569	−	0.989785i	$-0.545536\pi$
$419$	−5.83664	−0.285139	−0.142569	+	0.989785i	$0.545536\pi$
$420$	0	0
$421$	18.1974	0.886888	0.443444	−	0.896302i	$-0.353757\pi$
$421$	18.1974	0.886888	0.443444	+	0.896302i	$0.353757\pi$
$422$	0	0
$423$	41.1924	2.00284
$424$	0	0
$425$	0.0372777	0.00180824
$426$	0	0
$427$	19.5240	0.944830
$428$	0	0
$429$	−28.8501	−1.39290
$430$	0	0
$431$	23.9637	1.15429	0.577145	−	0.816642i	$-0.304167\pi$
$431$	23.9637	1.15429	0.577145	+	0.816642i	$0.304167\pi$
$432$	0	0
$433$	27.2657	1.31031	0.655153	−	0.755497i	$-0.272604\pi$
$433$	27.2657	1.31031	0.655153	+	0.755497i	$0.272604\pi$
$434$	0	0
$435$	44.2111	2.11976
$436$	0	0
$437$	4.78212	0.228760
$438$	0	0
$439$	−32.0561	−1.52995	−0.764977	−	0.644058i	$-0.777250\pi$
$439$	−32.0561	−1.52995	−0.764977	+	0.644058i	$0.777250\pi$
$440$	0	0
$441$	−21.4761	−1.02267
$442$	0	0
$443$	15.6035	0.741346	0.370673	−	0.928763i	$-0.379127\pi$
$443$	15.6035	0.741346	0.370673	+	0.928763i	$0.379127\pi$
$444$	0	0
$445$	−6.74461	−0.319725
$446$	0	0
$447$	57.7415	2.73108
$448$	0	0
$449$	−7.16703	−0.338233	−0.169117	−	0.985596i	$-0.554091\pi$
$449$	−7.16703	−0.338233	−0.169117	+	0.985596i	$0.554091\pi$
$450$	0	0
$451$	6.69118	0.315075
$452$	0	0
$453$	51.4855	2.41900
$454$	0	0
$455$	−10.9076	−0.511356
$456$	0	0
$457$	−22.4850	−1.05180	−0.525902	−	0.850545i	$-0.676272\pi$
$457$	−22.4850	−1.05180	−0.525902	+	0.850545i	$0.676272\pi$
$458$	0	0
$459$	−0.0739490	−0.00345164
$460$	0	0
$461$	23.7558	1.10642	0.553209	−	0.833042i	$-0.313403\pi$
$461$	23.7558	1.10642	0.553209	+	0.833042i	$0.313403\pi$
$462$	0	0
$463$	10.7863	0.501281	0.250640	−	0.968080i	$-0.419359\pi$
$463$	10.7863	0.501281	0.250640	+	0.968080i	$0.419359\pi$
$464$	0	0
$465$	35.3628	1.63991
$466$	0	0
$467$	33.9660	1.57176	0.785880	−	0.618379i	$-0.212210\pi$
$467$	33.9660	1.57176	0.785880	+	0.618379i	$0.212210\pi$
$468$	0	0
$469$	−6.09382	−0.281387
$470$	0	0
$471$	63.3911	2.92091
$472$	0	0
$473$	3.21212	0.147694
$474$	0	0
$475$	3.27182	0.150121
$476$	0	0
$477$	9.22937	0.422584
$478$	0	0
$479$	14.0751	0.643110	0.321555	−	0.946891i	$-0.395795\pi$
$479$	14.0751	0.643110	0.321555	+	0.946891i	$0.395795\pi$
$480$	0	0
$481$	−1.33847	−0.0610288
$482$	0	0
$483$	10.3226	0.469694
$484$	0	0
$485$	−30.2020	−1.37140
$486$	0	0
$487$	−0.402725	−0.0182492	−0.00912461	−	0.999958i	$-0.502904\pi$
$487$	−0.402725	−0.0182492	−0.00912461	+	0.999958i	$0.502904\pi$
$488$	0	0
$489$	36.1273	1.63373
$490$	0	0
$491$	−5.66875	−0.255827	−0.127914	−	0.991785i	$-0.540828\pi$
$491$	−5.66875	−0.255827	−0.127914	+	0.991785i	$0.540828\pi$
$492$	0	0
$493$	0.127792	0.00575547
$494$	0	0
$495$	41.3206	1.85722
$496$	0	0
$497$	14.4543	0.648363
$498$	0	0
$499$	−2.77578	−0.124261	−0.0621305	−	0.998068i	$-0.519789\pi$
$499$	−2.77578	−0.124261	−0.0621305	+	0.998068i	$0.519789\pi$
$500$	0	0
$501$	4.35446	0.194543
$502$	0	0
$503$	24.7823	1.10499	0.552493	−	0.833517i	$-0.313677\pi$
$503$	24.7823	1.10499	0.552493	+	0.833517i	$0.313677\pi$
$504$	0	0
$505$	−44.4787	−1.97927
$506$	0	0
$507$	12.1278	0.538616
$508$	0	0
$509$	−38.1365	−1.69037	−0.845185	−	0.534473i	$-0.820510\pi$
$509$	−38.1365	−1.69037	−0.845185	+	0.534473i	$0.820510\pi$
$510$	0	0
$511$	−20.2798	−0.897125
$512$	0	0
$513$	−6.49041	−0.286559
$514$	0	0
$515$	−38.1599	−1.68153
$516$	0	0
$517$	−34.6714	−1.52485
$518$	0	0
$519$	25.4966	1.11918
$520$	0	0
$521$	41.1800	1.80413	0.902064	−	0.431602i	$-0.142052\pi$
$521$	41.1800	1.80413	0.902064	+	0.431602i	$0.142052\pi$
$522$	0	0
$523$	17.1313	0.749102	0.374551	−	0.927206i	$-0.377797\pi$
$523$	17.1313	0.749102	0.374551	+	0.927206i	$0.377797\pi$
$524$	0	0
$525$	7.06249	0.308232
$526$	0	0
$527$	0.102216	0.00445261
$528$	0	0
$529$	−15.9005	−0.691327
$530$	0	0
$531$	4.92156	0.213578
$532$	0	0
$533$	5.35314	0.231870
$534$	0	0
$535$	18.9438	0.819014
$536$	0	0
$537$	−26.6617	−1.15054
$538$	0	0
$539$	18.0763	0.778602
$540$	0	0
$541$	0.224405	0.00964791	0.00482395	−	0.999988i	$-0.498464\pi$
$541$	0.224405	0.00964791	0.00482395	+	0.999988i	$0.498464\pi$
$542$	0	0
$543$	−39.7628	−1.70639
$544$	0	0
$545$	16.3892	0.702036
$546$	0	0
$547$	16.4048	0.701417	0.350708	−	0.936485i	$-0.385941\pi$
$547$	16.4048	0.701417	0.350708	+	0.936485i	$0.385941\pi$
$548$	0	0
$549$	−59.1714	−2.52537
$550$	0	0
$551$	11.2162	0.477825
$552$	0	0
$553$	−12.2808	−0.522234
$554$	0	0
$555$	3.24361	0.137684
$556$	0	0
$557$	7.27617	0.308301	0.154151	−	0.988047i	$-0.450736\pi$
$557$	7.27617	0.308301	0.154151	+	0.988047i	$0.450736\pi$
$558$	0	0
$559$	2.56979	0.108691
$560$	0	0
$561$	0.202089	0.00853218
$562$	0	0
$563$	14.0505	0.592158	0.296079	−	0.955163i	$-0.404321\pi$
$563$	14.0505	0.592158	0.296079	+	0.955163i	$0.404321\pi$
$564$	0	0
$565$	−13.2152	−0.555968
$566$	0	0
$567$	4.59377	0.192920
$568$	0	0
$569$	−7.37699	−0.309260	−0.154630	−	0.987972i	$-0.549418\pi$
$569$	−7.37699	−0.309260	−0.154630	+	0.987972i	$0.549418\pi$
$570$	0	0
$571$	−23.8412	−0.997724	−0.498862	−	0.866681i	$-0.666249\pi$
$571$	−23.8412	−0.997724	−0.498862	+	0.866681i	$0.666249\pi$
$572$	0	0
$573$	70.5679	2.94802
$574$	0	0
$575$	4.85730	0.202564
$576$	0	0
$577$	9.56963	0.398389	0.199195	−	0.979960i	$-0.436167\pi$
$577$	9.56963	0.398389	0.199195	+	0.979960i	$0.436167\pi$
$578$	0	0
$579$	−44.7268	−1.85878
$580$	0	0
$581$	7.28416	0.302198
$582$	0	0
$583$	−7.76831	−0.321731
$584$	0	0
$585$	33.0577	1.36677
$586$	0	0
$587$	−21.9615	−0.906450	−0.453225	−	0.891396i	$-0.649727\pi$
$587$	−21.9615	−0.906450	−0.453225	+	0.891396i	$0.649727\pi$
$588$	0	0
$589$	8.97138	0.369659
$590$	0	0
$591$	−1.67476	−0.0688903
$592$	0	0
$593$	−2.28740	−0.0939321	−0.0469660	−	0.998896i	$-0.514955\pi$
$593$	−2.28740	−0.0939321	−0.0469660	+	0.998896i	$0.514955\pi$
$594$	0	0
$595$	0.0764053	0.00313231
$596$	0	0
$597$	36.1351	1.47891
$598$	0	0
$599$	8.23248	0.336370	0.168185	−	0.985755i	$-0.446209\pi$
$599$	8.23248	0.336370	0.168185	+	0.985755i	$0.446209\pi$
$600$	0	0
$601$	−24.5955	−1.00327	−0.501637	−	0.865078i	$-0.667269\pi$
$601$	−24.5955	−1.00327	−0.501637	+	0.865078i	$0.667269\pi$
$602$	0	0
$603$	18.4686	0.752099
$604$	0	0
$605$	−6.04642	−0.245822
$606$	0	0
$607$	−3.93733	−0.159811	−0.0799056	−	0.996802i	$-0.525462\pi$
$607$	−3.93733	−0.159811	−0.0799056	+	0.996802i	$0.525462\pi$
$608$	0	0
$609$	24.2110	0.981079
$610$	0	0
$611$	−27.7381	−1.12217
$612$	0	0
$613$	−38.8300	−1.56833	−0.784165	−	0.620552i	$-0.786909\pi$
$613$	−38.8300	−1.56833	−0.784165	+	0.620552i	$0.786909\pi$
$614$	0	0
$615$	−12.9727	−0.523109
$616$	0	0
$617$	14.9901	0.603478	0.301739	−	0.953391i	$-0.402433\pi$
$617$	14.9901	0.603478	0.301739	+	0.953391i	$0.402433\pi$
$618$	0	0
$619$	20.5438	0.825724	0.412862	−	0.910794i	$-0.364529\pi$
$619$	20.5438	0.825724	0.412862	+	0.910794i	$0.364529\pi$
$620$	0	0
$621$	−9.63558	−0.386663
$622$	0	0
$623$	−3.69350	−0.147977
$624$	0	0
$625$	−30.7916	−1.23167
$626$	0	0
$627$	17.7371	0.708350
$628$	0	0
$629$	0.00937566	0.000373832 0
$630$	0	0
$631$	7.63228	0.303836	0.151918	−	0.988393i	$-0.451455\pi$
$631$	7.63228	0.303836	0.151918	+	0.988393i	$0.451455\pi$
$632$	0	0
$633$	55.3076	2.19828
$634$	0	0
$635$	23.5361	0.934002
$636$	0	0
$637$	14.4616	0.572989
$638$	0	0
$639$	−43.8067	−1.73297
$640$	0	0
$641$	−29.8466	−1.17887	−0.589434	−	0.807816i	$-0.700649\pi$
$641$	−29.8466	−1.17887	−0.589434	+	0.807816i	$0.700649\pi$
$642$	0	0
$643$	−47.2507	−1.86338	−0.931692	−	0.363248i	$-0.881668\pi$
$643$	−47.2507	−1.86338	−0.931692	+	0.363248i	$0.881668\pi$
$644$	0	0
$645$	−6.22758	−0.245211
$646$	0	0
$647$	12.7143	0.499851	0.249925	−	0.968265i	$-0.419594\pi$
$647$	12.7143	0.499851	0.249925	+	0.968265i	$0.419594\pi$
$648$	0	0
$649$	−4.14245	−0.162605
$650$	0	0
$651$	19.3655	0.758992
$652$	0	0
$653$	33.5784	1.31402	0.657012	−	0.753880i	$-0.271820\pi$
$653$	33.5784	1.31402	0.657012	+	0.753880i	$0.271820\pi$
$654$	0	0
$655$	−19.5146	−0.762498
$656$	0	0
$657$	61.4620	2.39786
$658$	0	0
$659$	31.0654	1.21014	0.605068	−	0.796174i	$-0.293146\pi$
$659$	31.0654	1.21014	0.605068	+	0.796174i	$0.293146\pi$
$660$	0	0
$661$	−41.6998	−1.62194	−0.810968	−	0.585091i	$-0.801059\pi$
$661$	−41.6998	−1.62194	−0.810968	+	0.585091i	$0.801059\pi$
$662$	0	0
$663$	0.161677	0.00627900
$664$	0	0
$665$	6.70600	0.260047
$666$	0	0
$667$	16.6514	0.644744
$668$	0	0
$669$	52.7378	2.03896
$670$	0	0
$671$	49.8042	1.92267
$672$	0	0
$673$	30.5148	1.17626	0.588130	−	0.808766i	$-0.299864\pi$
$673$	30.5148	1.17626	0.588130	+	0.808766i	$0.299864\pi$
$674$	0	0
$675$	−6.59245	−0.253743
$676$	0	0
$677$	−48.1917	−1.85216	−0.926078	−	0.377331i	$-0.876842\pi$
$677$	−48.1917	−1.85216	−0.926078	+	0.377331i	$0.876842\pi$
$678$	0	0
$679$	−16.5393	−0.634721
$680$	0	0
$681$	12.3848	0.474586
$682$	0	0
$683$	−11.1052	−0.424929	−0.212465	−	0.977169i	$-0.568149\pi$
$683$	−11.1052	−0.424929	−0.212465	+	0.977169i	$0.568149\pi$
$684$	0	0
$685$	−48.5692	−1.85573
$686$	0	0
$687$	33.5141	1.27864
$688$	0	0
$689$	−6.21488	−0.236768
$690$	0	0
$691$	37.8539	1.44003	0.720016	−	0.693958i	$-0.244135\pi$
$691$	37.8539	1.44003	0.720016	+	0.693958i	$0.244135\pi$
$692$	0	0
$693$	22.6281	0.859571
$694$	0	0
$695$	−18.9051	−0.717113
$696$	0	0
$697$	−0.0374976	−0.00142032
$698$	0	0
$699$	5.09208	0.192600
$700$	0	0
$701$	−25.5516	−0.965070	−0.482535	−	0.875877i	$-0.660284\pi$
$701$	−25.5516	−0.965070	−0.482535	+	0.875877i	$0.660284\pi$
$702$	0	0
$703$	0.822890	0.0310359
$704$	0	0
$705$	67.2200	2.53165
$706$	0	0
$707$	−24.3575	−0.916060
$708$	0	0
$709$	−6.68191	−0.250944	−0.125472	−	0.992097i	$-0.540045\pi$
$709$	−6.68191	−0.250944	−0.125472	+	0.992097i	$0.540045\pi$
$710$	0	0
$711$	37.2196	1.39584
$712$	0	0
$713$	13.3188	0.498793
$714$	0	0
$715$	−27.8245	−1.04058
$716$	0	0
$717$	−55.6273	−2.07744
$718$	0	0
$719$	−18.7636	−0.699765	−0.349882	−	0.936794i	$-0.613778\pi$
$719$	−18.7636	−0.699765	−0.349882	+	0.936794i	$0.613778\pi$
$720$	0	0
$721$	−20.8973	−0.778255
$722$	0	0
$723$	−26.2953	−0.977934
$724$	0	0
$725$	11.3925	0.423107
$726$	0	0
$727$	−23.8466	−0.884422	−0.442211	−	0.896911i	$-0.645806\pi$
$727$	−23.8466	−0.884422	−0.442211	+	0.896911i	$0.645806\pi$
$728$	0	0
$729$	−43.3051	−1.60389
$730$	0	0
$731$	−0.0180008	−0.000665785 0
$732$	0	0
$733$	5.17716	0.191223	0.0956114	−	0.995419i	$-0.469519\pi$
$733$	5.17716	0.191223	0.0956114	+	0.995419i	$0.469519\pi$
$734$	0	0
$735$	−35.0459	−1.29269
$736$	0	0
$737$	−15.5449	−0.572604
$738$	0	0
$739$	−4.42122	−0.162637	−0.0813186	−	0.996688i	$-0.525913\pi$
$739$	−4.42122	−0.162637	−0.0813186	+	0.996688i	$0.525913\pi$
$740$	0	0
$741$	14.1902	0.521289
$742$	0	0
$743$	4.81600	0.176682	0.0883409	−	0.996090i	$-0.471844\pi$
$743$	4.81600	0.176682	0.0883409	+	0.996090i	$0.471844\pi$
$744$	0	0
$745$	55.6888	2.04028
$746$	0	0
$747$	−22.0762	−0.807725
$748$	0	0
$749$	10.3741	0.379061
$750$	0	0
$751$	−9.92000	−0.361986	−0.180993	−	0.983484i	$-0.557931\pi$
$751$	−9.92000	−0.361986	−0.180993	+	0.983484i	$0.557931\pi$
$752$	0	0
$753$	−2.70836	−0.0986983
$754$	0	0
$755$	49.6552	1.80714
$756$	0	0
$757$	17.0114	0.618290	0.309145	−	0.951015i	$-0.399957\pi$
$757$	17.0114	0.618290	0.309145	+	0.951015i	$0.399957\pi$
$758$	0	0
$759$	26.3322	0.955799
$760$	0	0
$761$	−15.1421	−0.548902	−0.274451	−	0.961601i	$-0.588496\pi$
$761$	−15.1421	−0.548902	−0.274451	+	0.961601i	$0.588496\pi$
$762$	0	0
$763$	8.97510	0.324920
$764$	0	0
$765$	−0.231562	−0.00837214
$766$	0	0
$767$	−3.31408	−0.119665
$768$	0	0
$769$	21.0669	0.759690	0.379845	−	0.925050i	$-0.375977\pi$
$769$	21.0669	0.759690	0.379845	+	0.925050i	$0.375977\pi$
$770$	0	0
$771$	−22.4723	−0.809319
$772$	0	0
$773$	−29.2396	−1.05167	−0.525837	−	0.850585i	$-0.676248\pi$
$773$	−29.2396	−1.05167	−0.525837	+	0.850585i	$0.676248\pi$
$774$	0	0
$775$	9.11243	0.327328
$776$	0	0
$777$	1.77628	0.0637235
$778$	0	0
$779$	−3.29111	−0.117916
$780$	0	0
$781$	36.8718	1.31938
$782$	0	0
$783$	−22.5996	−0.807646
$784$	0	0
$785$	61.1376	2.18209
$786$	0	0
$787$	−30.7517	−1.09618	−0.548089	−	0.836420i	$-0.684644\pi$
$787$	−30.7517	−1.09618	−0.548089	+	0.836420i	$0.684644\pi$
$788$	0	0
$789$	19.0786	0.679214
$790$	0	0
$791$	−7.23695	−0.257316
$792$	0	0
$793$	39.8448	1.41493
$794$	0	0
$795$	15.0610	0.534158
$796$	0	0
$797$	−14.9467	−0.529440	−0.264720	−	0.964325i	$-0.585279\pi$
$797$	−14.9467	−0.529440	−0.264720	+	0.964325i	$0.585279\pi$
$798$	0	0
$799$	0.194300	0.00687383
$800$	0	0
$801$	11.1939	0.395518
$802$	0	0
$803$	−51.7323	−1.82559
$804$	0	0
$805$	9.95564	0.350890
$806$	0	0
$807$	−85.3703	−3.00518
$808$	0	0
$809$	−6.57345	−0.231110	−0.115555	−	0.993301i	$-0.536865\pi$
$809$	−6.57345	−0.231110	−0.115555	+	0.993301i	$0.536865\pi$
$810$	0	0
$811$	13.5096	0.474388	0.237194	−	0.971462i	$-0.423772\pi$
$811$	13.5096	0.474388	0.237194	+	0.971462i	$0.423772\pi$
$812$	0	0
$813$	70.5064	2.47277
$814$	0	0
$815$	34.8430	1.22050
$816$	0	0
$817$	−1.57991	−0.0552741
$818$	0	0
$819$	18.1032	0.632576
$820$	0	0
$821$	34.1260	1.19101	0.595503	−	0.803353i	$-0.296953\pi$
$821$	34.1260	1.19101	0.595503	+	0.803353i	$0.296953\pi$
$822$	0	0
$823$	−29.6211	−1.03252	−0.516262	−	0.856430i	$-0.672677\pi$
$823$	−29.6211	−1.03252	−0.516262	+	0.856430i	$0.672677\pi$
$824$	0	0
$825$	18.0159	0.627233
$826$	0	0
$827$	−28.4554	−0.989492	−0.494746	−	0.869038i	$-0.664739\pi$
$827$	−28.4554	−0.989492	−0.494746	+	0.869038i	$0.664739\pi$
$828$	0	0
$829$	−24.9323	−0.865934	−0.432967	−	0.901410i	$-0.642533\pi$
$829$	−24.9323	−0.865934	−0.432967	+	0.901410i	$0.642533\pi$
$830$	0	0
$831$	77.6592	2.69397
$832$	0	0
$833$	−0.101300	−0.00350984
$834$	0	0
$835$	4.19966	0.145335
$836$	0	0
$837$	−18.0766	−0.624819
$838$	0	0
$839$	−26.1061	−0.901284	−0.450642	−	0.892705i	$-0.648805\pi$
$839$	−26.1061	−0.901284	−0.450642	+	0.892705i	$0.648805\pi$
$840$	0	0
$841$	10.0547	0.346715
$842$	0	0
$843$	60.2542	2.07527
$844$	0	0
$845$	11.6967	0.402378
$846$	0	0
$847$	−3.31116	−0.113773
$848$	0	0
$849$	−77.5512	−2.66155
$850$	0	0
$851$	1.22165	0.0418777
$852$	0	0
$853$	−36.0264	−1.23352	−0.616761	−	0.787151i	$-0.711555\pi$
$853$	−36.0264	−1.23352	−0.616761	+	0.787151i	$0.711555\pi$
$854$	0	0
$855$	−20.3239	−0.695063
$856$	0	0
$857$	−39.7690	−1.35848	−0.679242	−	0.733915i	$-0.737691\pi$
$857$	−39.7690	−1.35848	−0.679242	+	0.733915i	$0.737691\pi$
$858$	0	0
$859$	−19.0750	−0.650831	−0.325415	−	0.945571i	$-0.605504\pi$
$859$	−19.0750	−0.650831	−0.325415	+	0.945571i	$0.605504\pi$
$860$	0	0
$861$	−7.10414	−0.242108
$862$	0	0
$863$	−34.2795	−1.16689	−0.583443	−	0.812154i	$-0.698295\pi$
$863$	−34.2795	−1.16689	−0.583443	+	0.812154i	$0.698295\pi$
$864$	0	0
$865$	24.5902	0.836091
$866$	0	0
$867$	46.0411	1.56364
$868$	0	0
$869$	−31.3275	−1.06271
$870$	0	0
$871$	−12.4364	−0.421391
$872$	0	0
$873$	50.1258	1.69650
$874$	0	0
$875$	−11.8707	−0.401302
$876$	0	0
$877$	−8.98239	−0.303314	−0.151657	−	0.988433i	$-0.548461\pi$
$877$	−8.98239	−0.303314	−0.151657	+	0.988433i	$0.548461\pi$
$878$	0	0
$879$	8.30538	0.280134
$880$	0	0
$881$	31.4670	1.06015	0.530075	−	0.847951i	$-0.322164\pi$
$881$	31.4670	1.06015	0.530075	+	0.847951i	$0.322164\pi$
$882$	0	0
$883$	−17.1158	−0.575994	−0.287997	−	0.957631i	$-0.592989\pi$
$883$	−17.1158	−0.575994	−0.287997	+	0.957631i	$0.592989\pi$
$884$	0	0
$885$	8.03127	0.269968
$886$	0	0
$887$	34.1137	1.14542	0.572712	−	0.819757i	$-0.305891\pi$
$887$	34.1137	1.14542	0.572712	+	0.819757i	$0.305891\pi$
$888$	0	0
$889$	12.8889	0.432280
$890$	0	0
$891$	11.7184	0.392580
$892$	0	0
$893$	17.0534	0.570671
$894$	0	0
$895$	−25.7139	−0.859522
$896$	0	0
$897$	21.0665	0.703391
$898$	0	0
$899$	31.2384	1.04186
$900$	0	0
$901$	0.0435338	0.00145032
$902$	0	0
$903$	−3.41037	−0.113490
$904$	0	0
$905$	−38.3493	−1.27477
$906$	0	0
$907$	24.0874	0.799807	0.399904	−	0.916557i	$-0.369044\pi$
$907$	24.0874	0.799807	0.399904	+	0.916557i	$0.369044\pi$
$908$	0	0
$909$	73.8206	2.44847
$910$	0	0
$911$	−4.51168	−0.149479	−0.0747393	−	0.997203i	$-0.523812\pi$
$911$	−4.51168	−0.149479	−0.0747393	+	0.997203i	$0.523812\pi$
$912$	0	0
$913$	18.5814	0.614954
$914$	0	0
$915$	−96.5591	−3.19214
$916$	0	0
$917$	−10.6866	−0.352904
$918$	0	0
$919$	11.2574	0.371348	0.185674	−	0.982611i	$-0.440553\pi$
$919$	11.2574	0.371348	0.185674	+	0.982611i	$0.440553\pi$
$920$	0	0
$921$	23.5694	0.776640
$922$	0	0
$923$	29.4986	0.970957
$924$	0	0
$925$	0.835827	0.0274818
$926$	0	0
$927$	63.3335	2.08014
$928$	0	0
$929$	−7.52639	−0.246933	−0.123466	−	0.992349i	$-0.539401\pi$
$929$	−7.52639	−0.246933	−0.123466	+	0.992349i	$0.539401\pi$
$930$	0	0
$931$	−8.89099	−0.291390
$932$	0	0
$933$	39.8864	1.30582
$934$	0	0
$935$	0.194904	0.00637406
$936$	0	0
$937$	−28.0715	−0.917057	−0.458529	−	0.888680i	$-0.651623\pi$
$937$	−28.0715	−0.917057	−0.458529	+	0.888680i	$0.651623\pi$
$938$	0	0
$939$	62.5075	2.03985
$940$	0	0
$941$	8.07819	0.263341	0.131671	−	0.991294i	$-0.457966\pi$
$941$	8.07819	0.263341	0.131671	+	0.991294i	$0.457966\pi$
$942$	0	0
$943$	−4.88595	−0.159108
$944$	0	0
$945$	−13.5120	−0.439546
$946$	0	0
$947$	35.0516	1.13902	0.569512	−	0.821983i	$-0.307132\pi$
$947$	35.0516	1.13902	0.569512	+	0.821983i	$0.307132\pi$
$948$	0	0
$949$	−41.3873	−1.34349
$950$	0	0
$951$	−44.8077	−1.45299
$952$	0	0
$953$	44.8319	1.45225	0.726124	−	0.687563i	$-0.241320\pi$
$953$	44.8319	1.45225	0.726124	+	0.687563i	$0.241320\pi$
$954$	0	0
$955$	68.0593	2.20235
$956$	0	0
$957$	61.7605	1.99643
$958$	0	0
$959$	−26.5976	−0.858881
$960$	0	0
$961$	−6.01358	−0.193987
$962$	0	0
$963$	−31.4408	−1.01317
$964$	0	0
$965$	−43.1368	−1.38862
$966$	0	0
$967$	−41.5845	−1.33727	−0.668634	−	0.743592i	$-0.733121\pi$
$967$	−41.5845	−1.33727	−0.668634	+	0.743592i	$0.733121\pi$
$968$	0	0
$969$	−0.0993990	−0.00319316
$970$	0	0
$971$	52.1117	1.67234	0.836172	−	0.548468i	$-0.184789\pi$
$971$	52.1117	1.67234	0.836172	+	0.548468i	$0.184789\pi$
$972$	0	0
$973$	−10.3529	−0.331899
$974$	0	0
$975$	14.4133	0.461594
$976$	0	0
$977$	−0.439092	−0.0140478	−0.00702390	−	0.999975i	$-0.502236\pi$
$977$	−0.439092	−0.0140478	−0.00702390	+	0.999975i	$0.502236\pi$
$978$	0	0
$979$	−9.42186	−0.301124
$980$	0	0
$981$	−27.2009	−0.868457
$982$	0	0
$983$	−10.9620	−0.349634	−0.174817	−	0.984601i	$-0.555933\pi$
$983$	−10.9620	−0.349634	−0.174817	+	0.984601i	$0.555933\pi$
$984$	0	0
$985$	−1.61522	−0.0514652
$986$	0	0
$987$	36.8112	1.17171
$988$	0	0
$989$	−2.34552	−0.0745831
$990$	0	0
$991$	20.5987	0.654341	0.327170	−	0.944965i	$-0.393905\pi$
$991$	20.5987	0.654341	0.327170	+	0.944965i	$0.393905\pi$
$992$	0	0
$993$	−22.3275	−0.708543
$994$	0	0
$995$	34.8505	1.10484
$996$	0	0
$997$	−44.7370	−1.41683	−0.708417	−	0.705794i	$-0.750590\pi$
$997$	−44.7370	−1.41683	−0.708417	+	0.705794i	$0.750590\pi$
$998$	0	0
$999$	−1.65806	−0.0524586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.f.1.2		6
4.3	odd	2		502.2.a.e.1.5	✓	6
12.11	even	2		4518.2.a.x.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.e.1.5	✓	6	4.3	odd	2
4016.2.a.f.1.2		6	1.1	even	1	trivial
4518.2.a.x.1.5		6	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)

\(f(q)\)	\(=\)	\(q-2.70836 q^{3} -2.61208 q^{5} -1.43044 q^{7} +4.33524 q^{9} +O(q^{10})\)	\(q-2.70836 q^{3} -2.61208 q^{5} -1.43044 q^{7} +4.33524 q^{9} -3.64894 q^{11} -2.91926 q^{13} +7.07447 q^{15} +0.0204488 q^{17} +1.79476 q^{19} +3.87414 q^{21} +2.66448 q^{23} +1.82298 q^{25} -3.61630 q^{27} +6.24938 q^{29} +4.99864 q^{31} +9.88267 q^{33} +3.73642 q^{35} +0.458495 q^{37} +7.90642 q^{39} -1.83373 q^{41} -0.880289 q^{43} -11.3240 q^{45} +9.50177 q^{47} -4.95385 q^{49} -0.0553828 q^{51} +2.12892 q^{53} +9.53134 q^{55} -4.86087 q^{57} +1.13525 q^{59} -13.6489 q^{61} -6.20128 q^{63} +7.62535 q^{65} +4.26011 q^{67} -7.21639 q^{69} -10.1048 q^{71} +14.1773 q^{73} -4.93730 q^{75} +5.21958 q^{77} +8.58537 q^{79} -3.21144 q^{81} -5.09226 q^{83} -0.0534139 q^{85} -16.9256 q^{87} +2.58208 q^{89} +4.17582 q^{91} -13.5381 q^{93} -4.68807 q^{95} +11.5624 q^{97} -15.8190 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * q^99

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