LMFDB - Embedded newform 4016.2.a.h.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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.49215$ 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)

$f(q)$	$=$	$q-1.49215 q^{3} +0.0991275 q^{5} +1.31645 q^{7} -0.773492 q^{9} +O(q^{10})$	$q-1.49215 q^{3} +0.0991275 q^{5} +1.31645 q^{7} -0.773492 q^{9} -2.16946 q^{11} -3.30066 q^{13} -0.147913 q^{15} -0.622502 q^{17} +7.82545 q^{19} -1.96434 q^{21} +4.06476 q^{23} -4.99017 q^{25} +5.63061 q^{27} +0.0908665 q^{29} +2.93245 q^{31} +3.23715 q^{33} +0.130496 q^{35} +3.54754 q^{37} +4.92508 q^{39} +1.35578 q^{41} -2.82384 q^{43} -0.0766743 q^{45} -12.2437 q^{47} -5.26696 q^{49} +0.928866 q^{51} +4.57135 q^{53} -0.215053 q^{55} -11.6767 q^{57} +4.01028 q^{59} -5.72724 q^{61} -1.01826 q^{63} -0.327186 q^{65} +0.767539 q^{67} -6.06523 q^{69} +5.28186 q^{71} -7.81375 q^{73} +7.44608 q^{75} -2.85598 q^{77} -9.59809 q^{79} -6.08123 q^{81} -0.137472 q^{83} -0.0617071 q^{85} -0.135586 q^{87} -6.77769 q^{89} -4.34515 q^{91} -4.37566 q^{93} +0.775718 q^{95} +4.66268 q^{97} +1.67806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

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
$3$	−1.49215	−0.861493	−0.430746	−	0.902473i	$-0.641750\pi$
$3$	−1.49215	−0.861493	−0.430746	+	0.902473i	$0.641750\pi$
$4$	0	0
$5$	0.0991275	0.0443312	0.0221656	−	0.999754i	$-0.492944\pi$
$5$	0.0991275	0.0443312	0.0221656	+	0.999754i	$0.492944\pi$
$6$	0	0
$7$	1.31645	0.497571	0.248786	−	0.968559i	$-0.419969\pi$
$7$	1.31645	0.497571	0.248786	+	0.968559i	$0.419969\pi$
$8$	0	0
$9$	−0.773492	−0.257831
$10$	0	0
$11$	−2.16946	−0.654115	−0.327058	−	0.945004i	$-0.606057\pi$
$11$	−2.16946	−0.654115	−0.327058	+	0.945004i	$0.606057\pi$
$12$	0	0
$13$	−3.30066	−0.915439	−0.457719	−	0.889097i	$-0.651334\pi$
$13$	−3.30066	−0.915439	−0.457719	+	0.889097i	$0.651334\pi$
$14$	0	0
$15$	−0.147913	−0.0381910
$16$	0	0
$17$	−0.622502	−0.150979	−0.0754895	−	0.997147i	$-0.524052\pi$
$17$	−0.622502	−0.150979	−0.0754895	+	0.997147i	$0.524052\pi$
$18$	0	0
$19$	7.82545	1.79528	0.897641	−	0.440727i	$-0.145279\pi$
$19$	7.82545	1.79528	0.897641	+	0.440727i	$0.145279\pi$
$20$	0	0
$21$	−1.96434	−0.428654
$22$	0	0
$23$	4.06476	0.847561	0.423781	−	0.905765i	$-0.360703\pi$
$23$	4.06476	0.847561	0.423781	+	0.905765i	$0.360703\pi$
$24$	0	0
$25$	−4.99017	−0.998035
$26$	0	0
$27$	5.63061	1.08361
$28$	0	0
$29$	0.0908665	0.0168735	0.00843674	−	0.999964i	$-0.497314\pi$
$29$	0.0908665	0.0168735	0.00843674	+	0.999964i	$0.497314\pi$
$30$	0	0
$31$	2.93245	0.526684	0.263342	−	0.964702i	$-0.415175\pi$
$31$	2.93245	0.526684	0.263342	+	0.964702i	$0.415175\pi$
$32$	0	0
$33$	3.23715	0.563516
$34$	0	0
$35$	0.130496	0.0220579
$36$	0	0
$37$	3.54754	0.583212	0.291606	−	0.956538i	$-0.405810\pi$
$37$	3.54754	0.583212	0.291606	+	0.956538i	$0.405810\pi$
$38$	0	0
$39$	4.92508	0.788644
$40$	0	0
$41$	1.35578	0.211738	0.105869	−	0.994380i	$-0.466238\pi$
$41$	1.35578	0.211738	0.105869	+	0.994380i	$0.466238\pi$
$42$	0	0
$43$	−2.82384	−0.430632	−0.215316	−	0.976544i	$-0.569078\pi$
$43$	−2.82384	−0.430632	−0.215316	+	0.976544i	$0.569078\pi$
$44$	0	0
$45$	−0.0766743	−0.0114299
$46$	0	0
$47$	−12.2437	−1.78593	−0.892964	−	0.450128i	$-0.851379\pi$
$47$	−12.2437	−1.78593	−0.892964	+	0.450128i	$0.851379\pi$
$48$	0	0
$49$	−5.26696	−0.752423
$50$	0	0
$51$	0.928866	0.130067
$52$	0	0
$53$	4.57135	0.627924	0.313962	−	0.949436i	$-0.398344\pi$
$53$	4.57135	0.627924	0.313962	+	0.949436i	$0.398344\pi$
$54$	0	0
$55$	−0.215053	−0.0289977
$56$	0	0
$57$	−11.6767	−1.54662
$58$	0	0
$59$	4.01028	0.522093	0.261047	−	0.965326i	$-0.415932\pi$
$59$	4.01028	0.522093	0.261047	+	0.965326i	$0.415932\pi$
$60$	0	0
$61$	−5.72724	−0.733297	−0.366649	−	0.930359i	$-0.619495\pi$
$61$	−5.72724	−0.733297	−0.366649	+	0.930359i	$0.619495\pi$
$62$	0	0
$63$	−1.01826	−0.128289
$64$	0	0
$65$	−0.327186	−0.0405825
$66$	0	0
$67$	0.767539	0.0937698	0.0468849	−	0.998900i	$-0.485071\pi$
$67$	0.767539	0.0937698	0.0468849	+	0.998900i	$0.485071\pi$
$68$	0	0
$69$	−6.06523	−0.730168
$70$	0	0
$71$	5.28186	0.626842	0.313421	−	0.949614i	$-0.398525\pi$
$71$	5.28186	0.626842	0.313421	+	0.949614i	$0.398525\pi$
$72$	0	0
$73$	−7.81375	−0.914530	−0.457265	−	0.889331i	$-0.651171\pi$
$73$	−7.81375	−0.914530	−0.457265	+	0.889331i	$0.651171\pi$
$74$	0	0
$75$	7.44608	0.859799
$76$	0	0
$77$	−2.85598	−0.325469
$78$	0	0
$79$	−9.59809	−1.07987	−0.539935	−	0.841707i	$-0.681551\pi$
$79$	−9.59809	−1.07987	−0.539935	+	0.841707i	$0.681551\pi$
$80$	0	0
$81$	−6.08123	−0.675693
$82$	0	0
$83$	−0.137472	−0.0150895	−0.00754474	−	0.999972i	$-0.502402\pi$
$83$	−0.137472	−0.0150895	−0.00754474	+	0.999972i	$0.502402\pi$
$84$	0	0
$85$	−0.0617071	−0.00669307
$86$	0	0
$87$	−0.135586	−0.0145364
$88$	0	0
$89$	−6.77769	−0.718434	−0.359217	−	0.933254i	$-0.616956\pi$
$89$	−6.77769	−0.718434	−0.359217	+	0.933254i	$0.616956\pi$
$90$	0	0
$91$	−4.34515	−0.455496
$92$	0	0
$93$	−4.37566	−0.453735
$94$	0	0
$95$	0.775718	0.0795870
$96$	0	0
$97$	4.66268	0.473423	0.236712	−	0.971580i	$-0.423930\pi$
$97$	4.66268	0.473423	0.236712	+	0.971580i	$0.423930\pi$
$98$	0	0
$99$	1.67806	0.168651
$100$	0	0
$101$	6.14653	0.611603	0.305801	−	0.952095i	$-0.401076\pi$
$101$	6.14653	0.611603	0.305801	+	0.952095i	$0.401076\pi$
$102$	0	0
$103$	−4.19366	−0.413214	−0.206607	−	0.978424i	$-0.566242\pi$
$103$	−4.19366	−0.413214	−0.206607	+	0.978424i	$0.566242\pi$
$104$	0	0
$105$	−0.194720	−0.0190027
$106$	0	0
$107$	10.2881	0.994590	0.497295	−	0.867581i	$-0.334327\pi$
$107$	10.2881	0.994590	0.497295	+	0.867581i	$0.334327\pi$
$108$	0	0
$109$	−6.06916	−0.581320	−0.290660	−	0.956826i	$-0.593875\pi$
$109$	−6.06916	−0.581320	−0.290660	+	0.956826i	$0.593875\pi$
$110$	0	0
$111$	−5.29346	−0.502433
$112$	0	0
$113$	1.59294	0.149851	0.0749256	−	0.997189i	$-0.476128\pi$
$113$	1.59294	0.149851	0.0749256	+	0.997189i	$0.476128\pi$
$114$	0	0
$115$	0.402930	0.0375734
$116$	0	0
$117$	2.55304	0.236028
$118$	0	0
$119$	−0.819492	−0.0751228
$120$	0	0
$121$	−6.29346	−0.572133
$122$	0	0
$123$	−2.02303	−0.182411
$124$	0	0
$125$	−0.990301	−0.0885752
$126$	0	0
$127$	3.89721	0.345821	0.172911	−	0.984938i	$-0.444683\pi$
$127$	3.89721	0.345821	0.172911	+	0.984938i	$0.444683\pi$
$128$	0	0
$129$	4.21359	0.370986
$130$	0	0
$131$	−5.76092	−0.503334	−0.251667	−	0.967814i	$-0.580979\pi$
$131$	−5.76092	−0.503334	−0.251667	+	0.967814i	$0.580979\pi$
$132$	0	0
$133$	10.3018	0.893281
$134$	0	0
$135$	0.558148	0.0480378
$136$	0	0
$137$	−0.872393	−0.0745335	−0.0372668	−	0.999305i	$-0.511865\pi$
$137$	−0.872393	−0.0745335	−0.0372668	+	0.999305i	$0.511865\pi$
$138$	0	0
$139$	−8.27742	−0.702082	−0.351041	−	0.936360i	$-0.614172\pi$
$139$	−8.27742	−0.702082	−0.351041	+	0.936360i	$0.614172\pi$
$140$	0	0
$141$	18.2694	1.53856
$142$	0	0
$143$	7.16064	0.598803
$144$	0	0
$145$	0.00900737	0.000748021 0
$146$	0	0
$147$	7.85909	0.648207
$148$	0	0
$149$	−14.5742	−1.19396	−0.596981	−	0.802256i	$-0.703633\pi$
$149$	−14.5742	−1.19396	−0.596981	+	0.802256i	$0.703633\pi$
$150$	0	0
$151$	−12.1462	−0.988443	−0.494222	−	0.869336i	$-0.664547\pi$
$151$	−12.1462	−0.988443	−0.494222	+	0.869336i	$0.664547\pi$
$152$	0	0
$153$	0.481500	0.0389270
$154$	0	0
$155$	0.290687	0.0233485
$156$	0	0
$157$	−12.9610	−1.03440	−0.517202	−	0.855863i	$-0.673026\pi$
$157$	−12.9610	−1.03440	−0.517202	+	0.855863i	$0.673026\pi$
$158$	0	0
$159$	−6.82114	−0.540952
$160$	0	0
$161$	5.35105	0.421722
$162$	0	0
$163$	8.14020	0.637589	0.318795	−	0.947824i	$-0.396722\pi$
$163$	8.14020	0.637589	0.318795	+	0.947824i	$0.396722\pi$
$164$	0	0
$165$	0.320891	0.0249813
$166$	0	0
$167$	−19.9874	−1.54667	−0.773335	−	0.633998i	$-0.781413\pi$
$167$	−19.9874	−1.54667	−0.773335	+	0.633998i	$0.781413\pi$
$168$	0	0
$169$	−2.10563	−0.161972
$170$	0	0
$171$	−6.05293	−0.462879
$172$	0	0
$173$	13.6320	1.03642	0.518212	−	0.855252i	$-0.326598\pi$
$173$	13.6320	1.03642	0.518212	+	0.855252i	$0.326598\pi$
$174$	0	0
$175$	−6.56931	−0.496593
$176$	0	0
$177$	−5.98393	−0.449780
$178$	0	0
$179$	15.4332	1.15353	0.576765	−	0.816910i	$-0.304315\pi$
$179$	15.4332	1.15353	0.576765	+	0.816910i	$0.304315\pi$
$180$	0	0
$181$	−9.63583	−0.716226	−0.358113	−	0.933678i	$-0.616580\pi$
$181$	−9.63583	−0.716226	−0.358113	+	0.933678i	$0.616580\pi$
$182$	0	0
$183$	8.54589	0.631730
$184$	0	0
$185$	0.351659	0.0258545
$186$	0	0
$187$	1.35049	0.0987576
$188$	0	0
$189$	7.41242	0.539174
$190$	0	0
$191$	−11.3187	−0.818994	−0.409497	−	0.912311i	$-0.634296\pi$
$191$	−11.3187	−0.818994	−0.409497	+	0.912311i	$0.634296\pi$
$192$	0	0
$193$	5.50061	0.395943	0.197971	−	0.980208i	$-0.436565\pi$
$193$	5.50061	0.395943	0.197971	+	0.980208i	$0.436565\pi$
$194$	0	0
$195$	0.488211	0.0349615
$196$	0	0
$197$	−15.6224	−1.11305	−0.556523	−	0.830832i	$-0.687865\pi$
$197$	−15.6224	−1.11305	−0.556523	+	0.830832i	$0.687865\pi$
$198$	0	0
$199$	−4.31806	−0.306099	−0.153050	−	0.988218i	$-0.548909\pi$
$199$	−4.31806	−0.306099	−0.153050	+	0.988218i	$0.548909\pi$
$200$	0	0
$201$	−1.14528	−0.0807820
$202$	0	0
$203$	0.119621	0.00839576
$204$	0	0
$205$	0.134396	0.00938659
$206$	0	0
$207$	−3.14406	−0.218527
$208$	0	0
$209$	−16.9770	−1.17432
$210$	0	0
$211$	−17.9714	−1.23721	−0.618603	−	0.785704i	$-0.712301\pi$
$211$	−17.9714	−1.23721	−0.618603	+	0.785704i	$0.712301\pi$
$212$	0	0
$213$	−7.88132	−0.540019
$214$	0	0
$215$	−0.279920	−0.0190904
$216$	0	0
$217$	3.86043	0.262063
$218$	0	0
$219$	11.6593	0.787861
$220$	0	0
$221$	2.05467	0.138212
$222$	0	0
$223$	19.0052	1.27268	0.636341	−	0.771408i	$-0.280447\pi$
$223$	19.0052	1.27268	0.636341	+	0.771408i	$0.280447\pi$
$224$	0	0
$225$	3.85986	0.257324
$226$	0	0
$227$	11.8420	0.785978	0.392989	−	0.919543i	$-0.371441\pi$
$227$	11.8420	0.785978	0.392989	+	0.919543i	$0.371441\pi$
$228$	0	0
$229$	12.2570	0.809963	0.404982	−	0.914325i	$-0.367278\pi$
$229$	12.2570	0.809963	0.404982	+	0.914325i	$0.367278\pi$
$230$	0	0
$231$	4.26155	0.280389
$232$	0	0
$233$	−10.2888	−0.674041	−0.337020	−	0.941497i	$-0.609419\pi$
$233$	−10.2888	−0.674041	−0.337020	+	0.941497i	$0.609419\pi$
$234$	0	0
$235$	−1.21369	−0.0791723
$236$	0	0
$237$	14.3218	0.930299
$238$	0	0
$239$	−24.9270	−1.61239	−0.806197	−	0.591647i	$-0.798478\pi$
$239$	−24.9270	−1.61239	−0.806197	+	0.591647i	$0.798478\pi$
$240$	0	0
$241$	−0.847306	−0.0545798	−0.0272899	−	0.999628i	$-0.508688\pi$
$241$	−0.847306	−0.0545798	−0.0272899	+	0.999628i	$0.508688\pi$
$242$	0	0
$243$	−7.81773	−0.501507
$244$	0	0
$245$	−0.522101	−0.0333558
$246$	0	0
$247$	−25.8292	−1.64347
$248$	0	0
$249$	0.205128	0.0129995
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−8.81832	−0.554403
$254$	0	0
$255$	0.0920761	0.00576603
$256$	0	0
$257$	−15.5570	−0.970418	−0.485209	−	0.874398i	$-0.661256\pi$
$257$	−15.5570	−0.970418	−0.485209	+	0.874398i	$0.661256\pi$
$258$	0	0
$259$	4.67016	0.290190
$260$	0	0
$261$	−0.0702845	−0.00435050
$262$	0	0
$263$	21.9615	1.35420	0.677101	−	0.735890i	$-0.263236\pi$
$263$	21.9615	1.35420	0.677101	+	0.735890i	$0.263236\pi$
$264$	0	0
$265$	0.453147	0.0278366
$266$	0	0
$267$	10.1133	0.618925
$268$	0	0
$269$	4.29910	0.262121	0.131060	−	0.991374i	$-0.458162\pi$
$269$	4.29910	0.262121	0.131060	+	0.991374i	$0.458162\pi$
$270$	0	0
$271$	−16.9512	−1.02971	−0.514857	−	0.857276i	$-0.672155\pi$
$271$	−16.9512	−1.02971	−0.514857	+	0.857276i	$0.672155\pi$
$272$	0	0
$273$	6.48362	0.392406
$274$	0	0
$275$	10.8260	0.652830
$276$	0	0
$277$	−6.21282	−0.373292	−0.186646	−	0.982427i	$-0.559762\pi$
$277$	−6.21282	−0.373292	−0.186646	+	0.982427i	$0.559762\pi$
$278$	0	0
$279$	−2.26823	−0.135795
$280$	0	0
$281$	−0.962388	−0.0574113	−0.0287056	−	0.999588i	$-0.509139\pi$
$281$	−0.962388	−0.0574113	−0.0287056	+	0.999588i	$0.509139\pi$
$282$	0	0
$283$	−27.2389	−1.61918	−0.809591	−	0.586994i	$-0.800311\pi$
$283$	−27.2389	−1.61918	−0.809591	+	0.586994i	$0.800311\pi$
$284$	0	0
$285$	−1.15749	−0.0685636
$286$	0	0
$287$	1.78482	0.105355
$288$	0	0
$289$	−16.6125	−0.977205
$290$	0	0
$291$	−6.95741	−0.407851
$292$	0	0
$293$	−12.3993	−0.724374	−0.362187	−	0.932106i	$-0.617970\pi$
$293$	−12.3993	−0.724374	−0.362187	+	0.932106i	$0.617970\pi$
$294$	0	0
$295$	0.397529	0.0231450
$296$	0	0
$297$	−12.2154	−0.708807
$298$	0	0
$299$	−13.4164	−0.775891
$300$	0	0
$301$	−3.71745	−0.214270
$302$	0	0
$303$	−9.17154	−0.526891
$304$	0	0
$305$	−0.567727	−0.0325079
$306$	0	0
$307$	−30.1169	−1.71886	−0.859431	−	0.511252i	$-0.829182\pi$
$307$	−30.1169	−1.71886	−0.859431	+	0.511252i	$0.829182\pi$
$308$	0	0
$309$	6.25756	0.355980
$310$	0	0
$311$	−1.16420	−0.0660155	−0.0330078	−	0.999455i	$-0.510509\pi$
$311$	−1.16420	−0.0660155	−0.0330078	+	0.999455i	$0.510509\pi$
$312$	0	0
$313$	−20.4970	−1.15856	−0.579278	−	0.815130i	$-0.696666\pi$
$313$	−20.4970	−1.15856	−0.579278	+	0.815130i	$0.696666\pi$
$314$	0	0
$315$	−0.100938	−0.00568721
$316$	0	0
$317$	−24.7700	−1.39122	−0.695611	−	0.718419i	$-0.744866\pi$
$317$	−24.7700	−1.39122	−0.695611	+	0.718419i	$0.744866\pi$
$318$	0	0
$319$	−0.197131	−0.0110372
$320$	0	0
$321$	−15.3514	−0.856832
$322$	0	0
$323$	−4.87136	−0.271050
$324$	0	0
$325$	16.4709	0.913640
$326$	0	0
$327$	9.05609	0.500803
$328$	0	0
$329$	−16.1182	−0.888627
$330$	0	0
$331$	18.0154	0.990214	0.495107	−	0.868832i	$-0.335129\pi$
$331$	18.0154	0.990214	0.495107	+	0.868832i	$0.335129\pi$
$332$	0	0
$333$	−2.74400	−0.150370
$334$	0	0
$335$	0.0760842	0.00415693
$336$	0	0
$337$	−12.2595	−0.667818	−0.333909	−	0.942605i	$-0.608368\pi$
$337$	−12.2595	−0.667818	−0.333909	+	0.942605i	$0.608368\pi$
$338$	0	0
$339$	−2.37690	−0.129096
$340$	0	0
$341$	−6.36183	−0.344512
$342$	0	0
$343$	−16.1488	−0.871955
$344$	0	0
$345$	−0.601231	−0.0323692
$346$	0	0
$347$	−15.1054	−0.810899	−0.405449	−	0.914117i	$-0.632885\pi$
$347$	−15.1054	−0.810899	−0.405449	+	0.914117i	$0.632885\pi$
$348$	0	0
$349$	20.5183	1.09832	0.549160	−	0.835717i	$-0.314948\pi$
$349$	20.5183	1.09832	0.549160	+	0.835717i	$0.314948\pi$
$350$	0	0
$351$	−18.5847	−0.991980
$352$	0	0
$353$	15.1384	0.805734	0.402867	−	0.915259i	$-0.368014\pi$
$353$	15.1384	0.805734	0.402867	+	0.915259i	$0.368014\pi$
$354$	0	0
$355$	0.523578	0.0277886
$356$	0	0
$357$	1.22280	0.0647177
$358$	0	0
$359$	12.9374	0.682807	0.341404	−	0.939917i	$-0.389098\pi$
$359$	12.9374	0.682807	0.341404	+	0.939917i	$0.389098\pi$
$360$	0	0
$361$	42.2377	2.22304
$362$	0	0
$363$	9.39078	0.492888
$364$	0	0
$365$	−0.774557	−0.0405422
$366$	0	0
$367$	−13.5958	−0.709695	−0.354848	−	0.934924i	$-0.615467\pi$
$367$	−13.5958	−0.709695	−0.354848	+	0.934924i	$0.615467\pi$
$368$	0	0
$369$	−1.04869	−0.0545925
$370$	0	0
$371$	6.01796	0.312437
$372$	0	0
$373$	−4.12158	−0.213407	−0.106704	−	0.994291i	$-0.534030\pi$
$373$	−4.12158	−0.213407	−0.106704	+	0.994291i	$0.534030\pi$
$374$	0	0
$375$	1.47768	0.0763069
$376$	0	0
$377$	−0.299919	−0.0154466
$378$	0	0
$379$	−24.2751	−1.24693	−0.623465	−	0.781852i	$-0.714275\pi$
$379$	−24.2751	−1.24693	−0.623465	+	0.781852i	$0.714275\pi$
$380$	0	0
$381$	−5.81522	−0.297923
$382$	0	0
$383$	−0.144661	−0.00739181	−0.00369590	−	0.999993i	$-0.501176\pi$
$383$	−0.144661	−0.00739181	−0.00369590	+	0.999993i	$0.501176\pi$
$384$	0	0
$385$	−0.283106	−0.0144284
$386$	0	0
$387$	2.18422	0.111030
$388$	0	0
$389$	−1.53188	−0.0776692	−0.0388346	−	0.999246i	$-0.512365\pi$
$389$	−1.53188	−0.0776692	−0.0388346	+	0.999246i	$0.512365\pi$
$390$	0	0
$391$	−2.53032	−0.127964
$392$	0	0
$393$	8.59615	0.433618
$394$	0	0
$395$	−0.951434	−0.0478719
$396$	0	0
$397$	2.45238	0.123082	0.0615408	−	0.998105i	$-0.480399\pi$
$397$	2.45238	0.123082	0.0615408	+	0.998105i	$0.480399\pi$
$398$	0	0
$399$	−15.3718	−0.769555
$400$	0	0
$401$	−11.4613	−0.572350	−0.286175	−	0.958177i	$-0.592384\pi$
$401$	−11.4613	−0.572350	−0.286175	+	0.958177i	$0.592384\pi$
$402$	0	0
$403$	−9.67904	−0.482147
$404$	0	0
$405$	−0.602818	−0.0299542
$406$	0	0
$407$	−7.69623	−0.381488
$408$	0	0
$409$	−23.4084	−1.15747	−0.578736	−	0.815515i	$-0.696454\pi$
$409$	−23.4084	−1.15747	−0.578736	+	0.815515i	$0.696454\pi$
$410$	0	0
$411$	1.30174	0.0642101
$412$	0	0
$413$	5.27933	0.259779
$414$	0	0
$415$	−0.0136272	−0.000668934 0
$416$	0	0
$417$	12.3511	0.604838
$418$	0	0
$419$	−8.82573	−0.431165	−0.215582	−	0.976486i	$-0.569165\pi$
$419$	−8.82573	−0.431165	−0.215582	+	0.976486i	$0.569165\pi$
$420$	0	0
$421$	34.3495	1.67409	0.837046	−	0.547133i	$-0.184281\pi$
$421$	34.3495	1.67409	0.837046	+	0.547133i	$0.184281\pi$
$422$	0	0
$423$	9.47041	0.460467
$424$	0	0
$425$	3.10639	0.150682
$426$	0	0
$427$	−7.53962	−0.364868
$428$	0	0
$429$	−10.6847	−0.515864
$430$	0	0
$431$	22.9097	1.10352	0.551760	−	0.834003i	$-0.313957\pi$
$431$	22.9097	1.10352	0.551760	+	0.834003i	$0.313957\pi$
$432$	0	0
$433$	−13.0701	−0.628109	−0.314054	−	0.949405i	$-0.601687\pi$
$433$	−13.0701	−0.628109	−0.314054	+	0.949405i	$0.601687\pi$
$434$	0	0
$435$	−0.0134403	−0.000644414 0
$436$	0	0
$437$	31.8086	1.52161
$438$	0	0
$439$	28.6805	1.36885	0.684424	−	0.729084i	$-0.260054\pi$
$439$	28.6805	1.36885	0.684424	+	0.729084i	$0.260054\pi$
$440$	0	0
$441$	4.07395	0.193998
$442$	0	0
$443$	20.4090	0.969661	0.484831	−	0.874608i	$-0.338881\pi$
$443$	20.4090	0.969661	0.484831	+	0.874608i	$0.338881\pi$
$444$	0	0
$445$	−0.671856	−0.0318490
$446$	0	0
$447$	21.7468	1.02859
$448$	0	0
$449$	5.79147	0.273316	0.136658	−	0.990618i	$-0.456364\pi$
$449$	5.79147	0.273316	0.136658	+	0.990618i	$0.456364\pi$
$450$	0	0
$451$	−2.94131	−0.138501
$452$	0	0
$453$	18.1239	0.851536
$454$	0	0
$455$	−0.430724	−0.0201927
$456$	0	0
$457$	16.4022	0.767261	0.383630	−	0.923487i	$-0.374674\pi$
$457$	16.4022	0.767261	0.383630	+	0.923487i	$0.374674\pi$
$458$	0	0
$459$	−3.50507	−0.163603
$460$	0	0
$461$	34.3033	1.59766	0.798832	−	0.601554i	$-0.205452\pi$
$461$	34.3033	1.59766	0.798832	+	0.601554i	$0.205452\pi$
$462$	0	0
$463$	23.5032	1.09229	0.546144	−	0.837692i	$-0.316095\pi$
$463$	23.5032	1.09229	0.546144	+	0.837692i	$0.316095\pi$
$464$	0	0
$465$	−0.433748	−0.0201146
$466$	0	0
$467$	−13.0188	−0.602437	−0.301218	−	0.953555i	$-0.597393\pi$
$467$	−13.0188	−0.602437	−0.301218	+	0.953555i	$0.597393\pi$
$468$	0	0
$469$	1.01043	0.0466572
$470$	0	0
$471$	19.3398	0.891131
$472$	0	0
$473$	6.12620	0.281683
$474$	0	0
$475$	−39.0504	−1.79175
$476$	0	0
$477$	−3.53591	−0.161898
$478$	0	0
$479$	−2.13392	−0.0975012	−0.0487506	−	0.998811i	$-0.515524\pi$
$479$	−2.13392	−0.0975012	−0.0487506	+	0.998811i	$0.515524\pi$
$480$	0	0
$481$	−11.7092	−0.533895
$482$	0	0
$483$	−7.98457	−0.363310
$484$	0	0
$485$	0.462200	0.0209874
$486$	0	0
$487$	−12.0297	−0.545117	−0.272558	−	0.962139i	$-0.587870\pi$
$487$	−12.0297	−0.545117	−0.272558	+	0.962139i	$0.587870\pi$
$488$	0	0
$489$	−12.1464	−0.549278
$490$	0	0
$491$	2.86419	0.129259	0.0646295	−	0.997909i	$-0.479413\pi$
$491$	2.86419	0.129259	0.0646295	+	0.997909i	$0.479413\pi$
$492$	0	0
$493$	−0.0565646	−0.00254754
$494$	0	0
$495$	0.166342	0.00747650
$496$	0	0
$497$	6.95331	0.311898
$498$	0	0
$499$	−2.10979	−0.0944474	−0.0472237	−	0.998884i	$-0.515037\pi$
$499$	−2.10979	−0.0944474	−0.0472237	+	0.998884i	$0.515037\pi$
$500$	0	0
$501$	29.8241	1.33244
$502$	0	0
$503$	5.17136	0.230579	0.115290	−	0.993332i	$-0.463220\pi$
$503$	5.17136	0.230579	0.115290	+	0.993332i	$0.463220\pi$
$504$	0	0
$505$	0.609290	0.0271131
$506$	0	0
$507$	3.14191	0.139537
$508$	0	0
$509$	−19.2428	−0.852924	−0.426462	−	0.904505i	$-0.640240\pi$
$509$	−19.2428	−0.852924	−0.426462	+	0.904505i	$0.640240\pi$
$510$	0	0
$511$	−10.2864	−0.455044
$512$	0	0
$513$	44.0621	1.94539
$514$	0	0
$515$	−0.415707	−0.0183182
$516$	0	0
$517$	26.5622	1.16820
$518$	0	0
$519$	−20.3410	−0.892872
$520$	0	0
$521$	40.3494	1.76774	0.883869	−	0.467734i	$-0.154930\pi$
$521$	40.3494	1.76774	0.883869	+	0.467734i	$0.154930\pi$
$522$	0	0
$523$	−1.84142	−0.0805198	−0.0402599	−	0.999189i	$-0.512819\pi$
$523$	−1.84142	−0.0805198	−0.0402599	+	0.999189i	$0.512819\pi$
$524$	0	0
$525$	9.80239	0.427811
$526$	0	0
$527$	−1.82546	−0.0795182
$528$	0	0
$529$	−6.47772	−0.281640
$530$	0	0
$531$	−3.10192	−0.134612
$532$	0	0
$533$	−4.47499	−0.193833
$534$	0	0
$535$	1.01984	0.0440914
$536$	0	0
$537$	−23.0286	−0.993757
$538$	0	0
$539$	11.4264	0.492171
$540$	0	0
$541$	−26.0860	−1.12152	−0.560762	−	0.827977i	$-0.689492\pi$
$541$	−26.0860	−1.12152	−0.560762	+	0.827977i	$0.689492\pi$
$542$	0	0
$543$	14.3781	0.617023
$544$	0	0
$545$	−0.601620	−0.0257706
$546$	0	0
$547$	23.8620	1.02027	0.510134	−	0.860095i	$-0.329596\pi$
$547$	23.8620	1.02027	0.510134	+	0.860095i	$0.329596\pi$
$548$	0	0
$549$	4.42997	0.189067
$550$	0	0
$551$	0.711071	0.0302927
$552$	0	0
$553$	−12.6354	−0.537312
$554$	0	0
$555$	−0.524727	−0.0222734
$556$	0	0
$557$	39.8241	1.68740	0.843700	−	0.536814i	$-0.180372\pi$
$557$	39.8241	1.68740	0.843700	+	0.536814i	$0.180372\pi$
$558$	0	0
$559$	9.32055	0.394217
$560$	0	0
$561$	−2.01513	−0.0850790
$562$	0	0
$563$	33.6139	1.41666	0.708329	−	0.705882i	$-0.249449\pi$
$563$	33.6139	1.41666	0.708329	+	0.705882i	$0.249449\pi$
$564$	0	0
$565$	0.157904	0.00664307
$566$	0	0
$567$	−8.00564	−0.336205
$568$	0	0
$569$	−25.9416	−1.08753	−0.543763	−	0.839239i	$-0.683001\pi$
$569$	−25.9416	−1.08753	−0.543763	+	0.839239i	$0.683001\pi$
$570$	0	0
$571$	−9.57657	−0.400767	−0.200384	−	0.979718i	$-0.564219\pi$
$571$	−9.57657	−0.400767	−0.200384	+	0.979718i	$0.564219\pi$
$572$	0	0
$573$	16.8892	0.705558
$574$	0	0
$575$	−20.2839	−0.845896
$576$	0	0
$577$	−29.8536	−1.24282	−0.621412	−	0.783484i	$-0.713441\pi$
$577$	−29.8536	−1.24282	−0.621412	+	0.783484i	$0.713441\pi$
$578$	0	0
$579$	−8.20773	−0.341102
$580$	0	0
$581$	−0.180975	−0.00750809
$582$	0	0
$583$	−9.91735	−0.410735
$584$	0	0
$585$	0.253076	0.0104634
$586$	0	0
$587$	−23.2793	−0.960839	−0.480420	−	0.877039i	$-0.659516\pi$
$587$	−23.2793	−0.960839	−0.480420	+	0.877039i	$0.659516\pi$
$588$	0	0
$589$	22.9478	0.945547
$590$	0	0
$591$	23.3109	0.958881
$592$	0	0
$593$	−12.2367	−0.502500	−0.251250	−	0.967922i	$-0.580842\pi$
$593$	−12.2367	−0.502500	−0.251250	+	0.967922i	$0.580842\pi$
$594$	0	0
$595$	−0.0812342	−0.00333028
$596$	0	0
$597$	6.44319	0.263702
$598$	0	0
$599$	−14.8696	−0.607556	−0.303778	−	0.952743i	$-0.598248\pi$
$599$	−14.8696	−0.607556	−0.303778	+	0.952743i	$0.598248\pi$
$600$	0	0
$601$	28.4766	1.16158	0.580792	−	0.814052i	$-0.302743\pi$
$601$	28.4766	1.16158	0.580792	+	0.814052i	$0.302743\pi$
$602$	0	0
$603$	−0.593685	−0.0241767
$604$	0	0
$605$	−0.623855	−0.0253633
$606$	0	0
$607$	−31.6533	−1.28477	−0.642385	−	0.766382i	$-0.722055\pi$
$607$	−31.6533	−1.28477	−0.642385	+	0.766382i	$0.722055\pi$
$608$	0	0
$609$	−0.178493	−0.00723288
$610$	0	0
$611$	40.4123	1.63491
$612$	0	0
$613$	−0.775028	−0.0313031	−0.0156515	−	0.999878i	$-0.504982\pi$
$613$	−0.775028	−0.0313031	−0.0156515	+	0.999878i	$0.504982\pi$
$614$	0	0
$615$	−0.200538	−0.00808648
$616$	0	0
$617$	−27.7994	−1.11916	−0.559580	−	0.828776i	$-0.689038\pi$
$617$	−27.7994	−1.11916	−0.559580	+	0.828776i	$0.689038\pi$
$618$	0	0
$619$	30.3962	1.22173	0.610863	−	0.791736i	$-0.290822\pi$
$619$	30.3962	1.22173	0.610863	+	0.791736i	$0.290822\pi$
$620$	0	0
$621$	22.8871	0.918427
$622$	0	0
$623$	−8.92249	−0.357472
$624$	0	0
$625$	24.8527	0.994108
$626$	0	0
$627$	25.3322	1.01167
$628$	0	0
$629$	−2.20835	−0.0880528
$630$	0	0
$631$	32.3935	1.28956	0.644782	−	0.764366i	$-0.276948\pi$
$631$	32.3935	1.28956	0.644782	+	0.764366i	$0.276948\pi$
$632$	0	0
$633$	26.8161	1.06584
$634$	0	0
$635$	0.386321	0.0153307
$636$	0	0
$637$	17.3845	0.688797
$638$	0	0
$639$	−4.08548	−0.161619
$640$	0	0
$641$	−14.6822	−0.579912	−0.289956	−	0.957040i	$-0.593641\pi$
$641$	−14.6822	−0.579912	−0.289956	+	0.957040i	$0.593641\pi$
$642$	0	0
$643$	−20.5239	−0.809385	−0.404692	−	0.914453i	$-0.632621\pi$
$643$	−20.5239	−0.809385	−0.404692	+	0.914453i	$0.632621\pi$
$644$	0	0
$645$	0.417683	0.0164463
$646$	0	0
$647$	27.2760	1.07233	0.536165	−	0.844113i	$-0.319873\pi$
$647$	27.2760	1.07233	0.536165	+	0.844113i	$0.319873\pi$
$648$	0	0
$649$	−8.70011	−0.341509
$650$	0	0
$651$	−5.76033	−0.225765
$652$	0	0
$653$	−15.7746	−0.617307	−0.308654	−	0.951174i	$-0.599878\pi$
$653$	−15.7746	−0.617307	−0.308654	+	0.951174i	$0.599878\pi$
$654$	0	0
$655$	−0.571066	−0.0223134
$656$	0	0
$657$	6.04387	0.235794
$658$	0	0
$659$	10.0576	0.391787	0.195894	−	0.980625i	$-0.437239\pi$
$659$	10.0576	0.391787	0.195894	+	0.980625i	$0.437239\pi$
$660$	0	0
$661$	45.7588	1.77981	0.889906	−	0.456145i	$-0.150770\pi$
$661$	45.7588	1.77981	0.889906	+	0.456145i	$0.150770\pi$
$662$	0	0
$663$	−3.06587	−0.119069
$664$	0	0
$665$	1.02119	0.0396002
$666$	0	0
$667$	0.369350	0.0143013
$668$	0	0
$669$	−28.3586	−1.09641
$670$	0	0
$671$	12.4250	0.479661
$672$	0	0
$673$	−18.3661	−0.707963	−0.353981	−	0.935252i	$-0.615172\pi$
$673$	−18.3661	−0.707963	−0.353981	+	0.935252i	$0.615172\pi$
$674$	0	0
$675$	−28.0977	−1.08148
$676$	0	0
$677$	16.2373	0.624050	0.312025	−	0.950074i	$-0.398993\pi$
$677$	16.2373	0.624050	0.312025	+	0.950074i	$0.398993\pi$
$678$	0	0
$679$	6.13818	0.235562
$680$	0	0
$681$	−17.6700	−0.677114
$682$	0	0
$683$	28.5387	1.09200	0.546001	−	0.837785i	$-0.316150\pi$
$683$	28.5387	1.09200	0.546001	+	0.837785i	$0.316150\pi$
$684$	0	0
$685$	−0.0864781	−0.00330416
$686$	0	0
$687$	−18.2892	−0.697777
$688$	0	0
$689$	−15.0885	−0.574826
$690$	0	0
$691$	−32.4071	−1.23282	−0.616411	−	0.787424i	$-0.711414\pi$
$691$	−32.4071	−1.23282	−0.616411	+	0.787424i	$0.711414\pi$
$692$	0	0
$693$	2.20908	0.0839159
$694$	0	0
$695$	−0.820520	−0.0311241
$696$	0	0
$697$	−0.843979	−0.0319680
$698$	0	0
$699$	15.3524	0.580681
$700$	0	0
$701$	−11.9810	−0.452518	−0.226259	−	0.974067i	$-0.572650\pi$
$701$	−11.9810	−0.452518	−0.226259	+	0.974067i	$0.572650\pi$
$702$	0	0
$703$	27.7611	1.04703
$704$	0	0
$705$	1.81100	0.0682063
$706$	0	0
$707$	8.09160	0.304316
$708$	0	0
$709$	22.1255	0.830940	0.415470	−	0.909607i	$-0.363617\pi$
$709$	22.1255	0.830940	0.415470	+	0.909607i	$0.363617\pi$
$710$	0	0
$711$	7.42404	0.278423
$712$	0	0
$713$	11.9197	0.446397
$714$	0	0
$715$	0.709816	0.0265456
$716$	0	0
$717$	37.1948	1.38907
$718$	0	0
$719$	−21.1253	−0.787839	−0.393920	−	0.919145i	$-0.628881\pi$
$719$	−21.1253	−0.787839	−0.393920	+	0.919145i	$0.628881\pi$
$720$	0	0
$721$	−5.52074	−0.205603
$722$	0	0
$723$	1.26431	0.0470201
$724$	0	0
$725$	−0.453439	−0.0168403
$726$	0	0
$727$	16.2680	0.603348	0.301674	−	0.953411i	$-0.402455\pi$
$727$	16.2680	0.603348	0.301674	+	0.953411i	$0.402455\pi$
$728$	0	0
$729$	29.9089	1.10774
$730$	0	0
$731$	1.75785	0.0650164
$732$	0	0
$733$	−20.0170	−0.739346	−0.369673	−	0.929162i	$-0.620530\pi$
$733$	−20.0170	−0.739346	−0.369673	+	0.929162i	$0.620530\pi$
$734$	0	0
$735$	0.779052	0.0287358
$736$	0	0
$737$	−1.66514	−0.0613363
$738$	0	0
$739$	−25.2601	−0.929207	−0.464603	−	0.885519i	$-0.653803\pi$
$739$	−25.2601	−0.929207	−0.464603	+	0.885519i	$0.653803\pi$
$740$	0	0
$741$	38.5410	1.41584
$742$	0	0
$743$	−22.1139	−0.811280	−0.405640	−	0.914033i	$-0.632951\pi$
$743$	−22.1139	−0.811280	−0.405640	+	0.914033i	$0.632951\pi$
$744$	0	0
$745$	−1.44470	−0.0529297
$746$	0	0
$747$	0.106333	0.00389053
$748$	0	0
$749$	13.5438	0.494880
$750$	0	0
$751$	−25.4085	−0.927168	−0.463584	−	0.886053i	$-0.653437\pi$
$751$	−25.4085	−0.927168	−0.463584	+	0.886053i	$0.653437\pi$
$752$	0	0
$753$	1.49215	0.0543769
$754$	0	0
$755$	−1.20402	−0.0438188
$756$	0	0
$757$	33.8548	1.23047	0.615237	−	0.788343i	$-0.289060\pi$
$757$	33.8548	1.23047	0.615237	+	0.788343i	$0.289060\pi$
$758$	0	0
$759$	13.1582	0.477614
$760$	0	0
$761$	4.78450	0.173438	0.0867190	−	0.996233i	$-0.472362\pi$
$761$	4.78450	0.173438	0.0867190	+	0.996233i	$0.472362\pi$
$762$	0	0
$763$	−7.98974	−0.289248
$764$	0	0
$765$	0.0477299	0.00172568
$766$	0	0
$767$	−13.2366	−0.477945
$768$	0	0
$769$	−48.4640	−1.74766	−0.873829	−	0.486233i	$-0.838371\pi$
$769$	−48.4640	−1.74766	−0.873829	+	0.486233i	$0.838371\pi$
$770$	0	0
$771$	23.2133	0.836008
$772$	0	0
$773$	14.6257	0.526051	0.263025	−	0.964789i	$-0.415280\pi$
$773$	14.6257	0.526051	0.263025	+	0.964789i	$0.415280\pi$
$774$	0	0
$775$	−14.6335	−0.525649
$776$	0	0
$777$	−6.96857	−0.249996
$778$	0	0
$779$	10.6096	0.380129
$780$	0	0
$781$	−11.4588	−0.410027
$782$	0	0
$783$	0.511634	0.0182843
$784$	0	0
$785$	−1.28480	−0.0458563
$786$	0	0
$787$	49.6196	1.76875	0.884374	−	0.466778i	$-0.154585\pi$
$787$	49.6196	1.76875	0.884374	+	0.466778i	$0.154585\pi$
$788$	0	0
$789$	−32.7698	−1.16664
$790$	0	0
$791$	2.09702	0.0745616
$792$	0	0
$793$	18.9037	0.671289
$794$	0	0
$795$	−0.676163	−0.0239810
$796$	0	0
$797$	27.4373	0.971878	0.485939	−	0.873993i	$-0.338478\pi$
$797$	27.4373	0.971878	0.485939	+	0.873993i	$0.338478\pi$
$798$	0	0
$799$	7.62173	0.269638
$800$	0	0
$801$	5.24249	0.185234
$802$	0	0
$803$	16.9516	0.598208
$804$	0	0
$805$	0.530436	0.0186954
$806$	0	0
$807$	−6.41490	−0.225815
$808$	0	0
$809$	−6.38028	−0.224319	−0.112159	−	0.993690i	$-0.535777\pi$
$809$	−6.38028	−0.224319	−0.112159	+	0.993690i	$0.535777\pi$
$810$	0	0
$811$	−40.8909	−1.43587	−0.717937	−	0.696109i	$-0.754913\pi$
$811$	−40.8909	−1.43587	−0.717937	+	0.696109i	$0.754913\pi$
$812$	0	0
$813$	25.2938	0.887091
$814$	0	0
$815$	0.806917	0.0282651
$816$	0	0
$817$	−22.0979	−0.773106
$818$	0	0
$819$	3.36094	0.117441
$820$	0	0
$821$	−35.9730	−1.25547	−0.627733	−	0.778429i	$-0.716017\pi$
$821$	−35.9730	−1.25547	−0.627733	+	0.778429i	$0.716017\pi$
$822$	0	0
$823$	7.54165	0.262885	0.131443	−	0.991324i	$-0.458039\pi$
$823$	7.54165	0.262885	0.131443	+	0.991324i	$0.458039\pi$
$824$	0	0
$825$	−16.1539	−0.562408
$826$	0	0
$827$	14.3869	0.500283	0.250142	−	0.968209i	$-0.419523\pi$
$827$	14.3869	0.500283	0.250142	+	0.968209i	$0.419523\pi$
$828$	0	0
$829$	−2.01778	−0.0700805	−0.0350403	−	0.999386i	$-0.511156\pi$
$829$	−2.01778	−0.0700805	−0.0350403	+	0.999386i	$0.511156\pi$
$830$	0	0
$831$	9.27045	0.321588
$832$	0	0
$833$	3.27869	0.113600
$834$	0	0
$835$	−1.98130	−0.0685656
$836$	0	0
$837$	16.5115	0.570721
$838$	0	0
$839$	34.2583	1.18273	0.591365	−	0.806404i	$-0.298589\pi$
$839$	34.2583	1.18273	0.591365	+	0.806404i	$0.298589\pi$
$840$	0	0
$841$	−28.9917	−0.999715
$842$	0	0
$843$	1.43603	0.0494594
$844$	0	0
$845$	−0.208726	−0.00718039
$846$	0	0
$847$	−8.28503	−0.284677
$848$	0	0
$849$	40.6444	1.39491
$850$	0	0
$851$	14.4199	0.494308
$852$	0	0
$853$	10.6965	0.366243	0.183121	−	0.983090i	$-0.441380\pi$
$853$	10.6965	0.366243	0.183121	+	0.983090i	$0.441380\pi$
$854$	0	0
$855$	−0.600011	−0.0205200
$856$	0	0
$857$	38.0288	1.29904	0.649519	−	0.760345i	$-0.274970\pi$
$857$	38.0288	1.29904	0.649519	+	0.760345i	$0.274970\pi$
$858$	0	0
$859$	9.16129	0.312579	0.156290	−	0.987711i	$-0.450047\pi$
$859$	9.16129	0.312579	0.156290	+	0.987711i	$0.450047\pi$
$860$	0	0
$861$	−2.66322	−0.0907623
$862$	0	0
$863$	50.3239	1.71305	0.856523	−	0.516109i	$-0.172620\pi$
$863$	50.3239	1.71305	0.856523	+	0.516109i	$0.172620\pi$
$864$	0	0
$865$	1.35131	0.0459459
$866$	0	0
$867$	24.7883	0.841855
$868$	0	0
$869$	20.8226	0.706359
$870$	0	0
$871$	−2.53339	−0.0858405
$872$	0	0
$873$	−3.60655	−0.122063
$874$	0	0
$875$	−1.30368	−0.0440725
$876$	0	0
$877$	8.87320	0.299627	0.149813	−	0.988714i	$-0.452133\pi$
$877$	8.87320	0.299627	0.149813	+	0.988714i	$0.452133\pi$
$878$	0	0
$879$	18.5016	0.624043
$880$	0	0
$881$	−4.28001	−0.144197	−0.0720986	−	0.997398i	$-0.522970\pi$
$881$	−4.28001	−0.144197	−0.0720986	+	0.997398i	$0.522970\pi$
$882$	0	0
$883$	20.6695	0.695584	0.347792	−	0.937572i	$-0.386932\pi$
$883$	20.6695	0.695584	0.347792	+	0.937572i	$0.386932\pi$
$884$	0	0
$885$	−0.593172	−0.0199393
$886$	0	0
$887$	12.8799	0.432465	0.216232	−	0.976342i	$-0.430623\pi$
$887$	12.8799	0.432465	0.216232	+	0.976342i	$0.430623\pi$
$888$	0	0
$889$	5.13048	0.172071
$890$	0	0
$891$	13.1930	0.441981
$892$	0	0
$893$	−95.8126	−3.20625
$894$	0	0
$895$	1.52985	0.0511373
$896$	0	0
$897$	20.0193	0.668424
$898$	0	0
$899$	0.266462	0.00888700
$900$	0	0
$901$	−2.84568	−0.0948032
$902$	0	0
$903$	5.54698	0.184592
$904$	0	0
$905$	−0.955176	−0.0317511
$906$	0	0
$907$	52.7469	1.75143	0.875716	−	0.482827i	$-0.160390\pi$
$907$	52.7469	1.75143	0.875716	+	0.482827i	$0.160390\pi$
$908$	0	0
$909$	−4.75429	−0.157690
$910$	0	0
$911$	−0.616700	−0.0204322	−0.0102161	−	0.999948i	$-0.503252\pi$
$911$	−0.616700	−0.0204322	−0.0102161	+	0.999948i	$0.503252\pi$
$912$	0	0
$913$	0.298239	0.00987026
$914$	0	0
$915$	0.847133	0.0280053
$916$	0	0
$917$	−7.58396	−0.250444
$918$	0	0
$919$	−4.59507	−0.151577	−0.0757886	−	0.997124i	$-0.524147\pi$
$919$	−4.59507	−0.151577	−0.0757886	+	0.997124i	$0.524147\pi$
$920$	0	0
$921$	44.9389	1.48079
$922$	0	0
$923$	−17.4336	−0.573835
$924$	0	0
$925$	−17.7029	−0.582066
$926$	0	0
$927$	3.24376	0.106539
$928$	0	0
$929$	−44.0349	−1.44474	−0.722369	−	0.691508i	$-0.756947\pi$
$929$	−44.0349	−1.44474	−0.722369	+	0.691508i	$0.756947\pi$
$930$	0	0
$931$	−41.2164	−1.35081
$932$	0	0
$933$	1.73715	0.0568719
$934$	0	0
$935$	0.133871	0.00437804
$936$	0	0
$937$	15.4772	0.505619	0.252810	−	0.967516i	$-0.418645\pi$
$937$	15.4772	0.505619	0.252810	+	0.967516i	$0.418645\pi$
$938$	0	0
$939$	30.5845	0.998088
$940$	0	0
$941$	−49.3847	−1.60990	−0.804948	−	0.593345i	$-0.797807\pi$
$941$	−49.3847	−1.60990	−0.804948	+	0.593345i	$0.797807\pi$
$942$	0	0
$943$	5.51094	0.179461
$944$	0	0
$945$	0.734774	0.0239022
$946$	0	0
$947$	14.6064	0.474644	0.237322	−	0.971431i	$-0.423730\pi$
$947$	14.6064	0.474644	0.237322	+	0.971431i	$0.423730\pi$
$948$	0	0
$949$	25.7905	0.837196
$950$	0	0
$951$	36.9605	1.19853
$952$	0	0
$953$	25.9344	0.840096	0.420048	−	0.907502i	$-0.362013\pi$
$953$	25.9344	0.840096	0.420048	+	0.907502i	$0.362013\pi$
$954$	0	0
$955$	−1.12200	−0.0363070
$956$	0	0
$957$	0.294148	0.00950847
$958$	0	0
$959$	−1.14846	−0.0370857
$960$	0	0
$961$	−22.4007	−0.722604
$962$	0	0
$963$	−7.95778	−0.256436
$964$	0	0
$965$	0.545262	0.0175526
$966$	0	0
$967$	22.3365	0.718293	0.359147	−	0.933281i	$-0.383068\pi$
$967$	22.3365	0.718293	0.359147	+	0.933281i	$0.383068\pi$
$968$	0	0
$969$	7.26880	0.233507
$970$	0	0
$971$	9.15377	0.293758	0.146879	−	0.989154i	$-0.453077\pi$
$971$	9.15377	0.293758	0.146879	+	0.989154i	$0.453077\pi$
$972$	0	0
$973$	−10.8968	−0.349336
$974$	0	0
$975$	−24.5770	−0.787094
$976$	0	0
$977$	−53.6543	−1.71655	−0.858277	−	0.513187i	$-0.828465\pi$
$977$	−53.6543	−1.71655	−0.858277	+	0.513187i	$0.828465\pi$
$978$	0	0
$979$	14.7039	0.469939
$980$	0	0
$981$	4.69444	0.149882
$982$	0	0
$983$	42.6896	1.36159	0.680794	−	0.732475i	$-0.261635\pi$
$983$	42.6896	1.36159	0.680794	+	0.732475i	$0.261635\pi$
$984$	0	0
$985$	−1.54860	−0.0493426
$986$	0	0
$987$	24.0508	0.765545
$988$	0	0
$989$	−11.4782	−0.364987
$990$	0	0
$991$	−4.94545	−0.157097	−0.0785486	−	0.996910i	$-0.525029\pi$
$991$	−4.94545	−0.157097	−0.0785486	+	0.996910i	$0.525029\pi$
$992$	0	0
$993$	−26.8816	−0.853062
$994$	0	0
$995$	−0.428039	−0.0135697
$996$	0	0
$997$	−1.55093	−0.0491184	−0.0245592	−	0.999698i	$-0.507818\pi$
$997$	−1.55093	−0.0491184	−0.0245592	+	0.999698i	$0.507818\pi$
$998$	0	0
$999$	19.9748	0.631976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.2		9
4.3	odd	2		2008.2.a.a.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.8	✓	9	4.3	odd	2
4016.2.a.h.1.2		9	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)

\(f(q)\)	\(=\)	\(q-1.49215 q^{3} +0.0991275 q^{5} +1.31645 q^{7} -0.773492 q^{9} +O(q^{10})\)	\(q-1.49215 q^{3} +0.0991275 q^{5} +1.31645 q^{7} -0.773492 q^{9} -2.16946 q^{11} -3.30066 q^{13} -0.147913 q^{15} -0.622502 q^{17} +7.82545 q^{19} -1.96434 q^{21} +4.06476 q^{23} -4.99017 q^{25} +5.63061 q^{27} +0.0908665 q^{29} +2.93245 q^{31} +3.23715 q^{33} +0.130496 q^{35} +3.54754 q^{37} +4.92508 q^{39} +1.35578 q^{41} -2.82384 q^{43} -0.0766743 q^{45} -12.2437 q^{47} -5.26696 q^{49} +0.928866 q^{51} +4.57135 q^{53} -0.215053 q^{55} -11.6767 q^{57} +4.01028 q^{59} -5.72724 q^{61} -1.01826 q^{63} -0.327186 q^{65} +0.767539 q^{67} -6.06523 q^{69} +5.28186 q^{71} -7.81375 q^{73} +7.44608 q^{75} -2.85598 q^{77} -9.59809 q^{79} -6.08123 q^{81} -0.137472 q^{83} -0.0617071 q^{85} -0.135586 q^{87} -6.77769 q^{89} -4.34515 q^{91} -4.37566 q^{93} +0.775718 q^{95} +4.66268 q^{97} +1.67806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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