LMFDB - Embedded newform 6040.2.a.s.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03886 q^{3} +1.00000 q^{5} +1.67649 q^{7} +1.15693 q^{9} +O(q^{10})$	$q-2.03886 q^{3} +1.00000 q^{5} +1.67649 q^{7} +1.15693 q^{9} -5.06888 q^{11} +1.01327 q^{13} -2.03886 q^{15} +5.26784 q^{17} -7.35236 q^{19} -3.41811 q^{21} -2.57030 q^{23} +1.00000 q^{25} +3.75775 q^{27} +1.36035 q^{29} -2.47089 q^{31} +10.3347 q^{33} +1.67649 q^{35} -4.47186 q^{37} -2.06591 q^{39} +0.813460 q^{41} +7.43339 q^{43} +1.15693 q^{45} -4.02065 q^{47} -4.18939 q^{49} -10.7404 q^{51} +4.55687 q^{53} -5.06888 q^{55} +14.9904 q^{57} -7.80075 q^{59} +0.00886299 q^{61} +1.93958 q^{63} +1.01327 q^{65} +1.63247 q^{67} +5.24048 q^{69} +10.6828 q^{71} +0.0298735 q^{73} -2.03886 q^{75} -8.49791 q^{77} +10.8638 q^{79} -11.1323 q^{81} +8.24573 q^{83} +5.26784 q^{85} -2.77355 q^{87} +11.7289 q^{89} +1.69873 q^{91} +5.03779 q^{93} -7.35236 q^{95} +8.92711 q^{97} -5.86435 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * 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
$3$	−2.03886	−1.17713	−0.588567	−	0.808448i	$-0.700308\pi$
$3$	−2.03886	−1.17713	−0.588567	+	0.808448i	$0.700308\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.67649	0.633652	0.316826	−	0.948484i	$-0.397383\pi$
$7$	1.67649	0.633652	0.316826	+	0.948484i	$0.397383\pi$
$8$	0	0
$9$	1.15693	0.385644
$10$	0	0
$11$	−5.06888	−1.52832	−0.764162	−	0.645024i	$-0.776847\pi$
$11$	−5.06888	−1.52832	−0.764162	+	0.645024i	$0.776847\pi$
$12$	0	0
$13$	1.01327	0.281030	0.140515	−	0.990079i	$-0.455124\pi$
$13$	1.01327	0.281030	0.140515	+	0.990079i	$0.455124\pi$
$14$	0	0
$15$	−2.03886	−0.526430
$16$	0	0
$17$	5.26784	1.27764	0.638819	−	0.769357i	$-0.279423\pi$
$17$	5.26784	1.27764	0.638819	+	0.769357i	$0.279423\pi$
$18$	0	0
$19$	−7.35236	−1.68675	−0.843373	−	0.537328i	$-0.819434\pi$
$19$	−7.35236	−1.68675	−0.843373	+	0.537328i	$0.819434\pi$
$20$	0	0
$21$	−3.41811	−0.745894
$22$	0	0
$23$	−2.57030	−0.535945	−0.267973	−	0.963427i	$-0.586354\pi$
$23$	−2.57030	−0.535945	−0.267973	+	0.963427i	$0.586354\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.75775	0.723179
$28$	0	0
$29$	1.36035	0.252610	0.126305	−	0.991991i	$-0.459688\pi$
$29$	1.36035	0.252610	0.126305	+	0.991991i	$0.459688\pi$
$30$	0	0
$31$	−2.47089	−0.443785	−0.221892	−	0.975071i	$-0.571223\pi$
$31$	−2.47089	−0.443785	−0.221892	+	0.975071i	$0.571223\pi$
$32$	0	0
$33$	10.3347	1.79904
$34$	0	0
$35$	1.67649	0.283378
$36$	0	0
$37$	−4.47186	−0.735170	−0.367585	−	0.929990i	$-0.619815\pi$
$37$	−4.47186	−0.735170	−0.367585	+	0.929990i	$0.619815\pi$
$38$	0	0
$39$	−2.06591	−0.330810
$40$	0	0
$41$	0.813460	0.127041	0.0635206	−	0.997981i	$-0.479767\pi$
$41$	0.813460	0.127041	0.0635206	+	0.997981i	$0.479767\pi$
$42$	0	0
$43$	7.43339	1.13358	0.566791	−	0.823862i	$-0.308185\pi$
$43$	7.43339	1.13358	0.566791	+	0.823862i	$0.308185\pi$
$44$	0	0
$45$	1.15693	0.172465
$46$	0	0
$47$	−4.02065	−0.586473	−0.293236	−	0.956040i	$-0.594732\pi$
$47$	−4.02065	−0.586473	−0.293236	+	0.956040i	$0.594732\pi$
$48$	0	0
$49$	−4.18939	−0.598485
$50$	0	0
$51$	−10.7404	−1.50395
$52$	0	0
$53$	4.55687	0.625934	0.312967	−	0.949764i	$-0.398677\pi$
$53$	4.55687	0.625934	0.312967	+	0.949764i	$0.398677\pi$
$54$	0	0
$55$	−5.06888	−0.683488
$56$	0	0
$57$	14.9904	1.98553
$58$	0	0
$59$	−7.80075	−1.01557	−0.507785	−	0.861484i	$-0.669536\pi$
$59$	−7.80075	−1.01557	−0.507785	+	0.861484i	$0.669536\pi$
$60$	0	0
$61$	0.00886299	0.00113479	0.000567395	−	1.00000i	$-0.499819\pi$
$61$	0.00886299	0.00113479	0.000567395		1.00000i	$0.499819\pi$
$62$	0	0
$63$	1.93958	0.244364
$64$	0	0
$65$	1.01327	0.125680
$66$	0	0
$67$	1.63247	0.199438	0.0997190	−	0.995016i	$-0.468206\pi$
$67$	1.63247	0.199438	0.0997190	+	0.995016i	$0.468206\pi$
$68$	0	0
$69$	5.24048	0.630879
$70$	0	0
$71$	10.6828	1.26781	0.633907	−	0.773410i	$-0.281450\pi$
$71$	10.6828	1.26781	0.633907	+	0.773410i	$0.281450\pi$
$72$	0	0
$73$	0.0298735	0.00349643	0.00174821	−	0.999998i	$-0.499444\pi$
$73$	0.0298735	0.00349643	0.00174821	+	0.999998i	$0.499444\pi$
$74$	0	0
$75$	−2.03886	−0.235427
$76$	0	0
$77$	−8.49791	−0.968427
$78$	0	0
$79$	10.8638	1.22228	0.611138	−	0.791524i	$-0.290712\pi$
$79$	10.8638	1.22228	0.611138	+	0.791524i	$0.290712\pi$
$80$	0	0
$81$	−11.1323	−1.23692
$82$	0	0
$83$	8.24573	0.905086	0.452543	−	0.891742i	$-0.350517\pi$
$83$	8.24573	0.905086	0.452543	+	0.891742i	$0.350517\pi$
$84$	0	0
$85$	5.26784	0.571377
$86$	0	0
$87$	−2.77355	−0.297356
$88$	0	0
$89$	11.7289	1.24326	0.621631	−	0.783310i	$-0.286470\pi$
$89$	11.7289	1.24326	0.621631	+	0.783310i	$0.286470\pi$
$90$	0	0
$91$	1.69873	0.178075
$92$	0	0
$93$	5.03779	0.522394
$94$	0	0
$95$	−7.35236	−0.754336
$96$	0	0
$97$	8.92711	0.906411	0.453205	−	0.891406i	$-0.350280\pi$
$97$	8.92711	0.906411	0.453205	+	0.891406i	$0.350280\pi$
$98$	0	0
$99$	−5.86435	−0.589389
$100$	0	0
$101$	−15.0782	−1.50034	−0.750168	−	0.661248i	$-0.770027\pi$
$101$	−15.0782	−1.50034	−0.750168	+	0.661248i	$0.770027\pi$
$102$	0	0
$103$	0.496343	0.0489061	0.0244531	−	0.999701i	$-0.492216\pi$
$103$	0.496343	0.0489061	0.0244531	+	0.999701i	$0.492216\pi$
$104$	0	0
$105$	−3.41811	−0.333574
$106$	0	0
$107$	−16.4973	−1.59485	−0.797426	−	0.603417i	$-0.793806\pi$
$107$	−16.4973	−1.59485	−0.797426	+	0.603417i	$0.793806\pi$
$108$	0	0
$109$	12.1206	1.16095	0.580473	−	0.814279i	$-0.302868\pi$
$109$	12.1206	1.16095	0.580473	+	0.814279i	$0.302868\pi$
$110$	0	0
$111$	9.11748	0.865393
$112$	0	0
$113$	20.3410	1.91352	0.956761	−	0.290877i	$-0.0939470\pi$
$113$	20.3410	1.91352	0.956761	+	0.290877i	$0.0939470\pi$
$114$	0	0
$115$	−2.57030	−0.239682
$116$	0	0
$117$	1.17228	0.108378
$118$	0	0
$119$	8.83146	0.809579
$120$	0	0
$121$	14.6935	1.33578
$122$	0	0
$123$	−1.65853	−0.149544
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.30616	−0.648317	−0.324158	−	0.946003i	$-0.605081\pi$
$127$	−7.30616	−0.648317	−0.324158	+	0.946003i	$0.605081\pi$
$128$	0	0
$129$	−15.1556	−1.33438
$130$	0	0
$131$	−3.31976	−0.290049	−0.145024	−	0.989428i	$-0.546326\pi$
$131$	−3.31976	−0.290049	−0.145024	+	0.989428i	$0.546326\pi$
$132$	0	0
$133$	−12.3261	−1.06881
$134$	0	0
$135$	3.75775	0.323416
$136$	0	0
$137$	−13.0894	−1.11830	−0.559150	−	0.829066i	$-0.688873\pi$
$137$	−13.0894	−1.11830	−0.559150	+	0.829066i	$0.688873\pi$
$138$	0	0
$139$	18.6530	1.58212	0.791062	−	0.611737i	$-0.209529\pi$
$139$	18.6530	1.58212	0.791062	+	0.611737i	$0.209529\pi$
$140$	0	0
$141$	8.19753	0.690357
$142$	0	0
$143$	−5.13613	−0.429505
$144$	0	0
$145$	1.36035	0.112971
$146$	0	0
$147$	8.54157	0.704497
$148$	0	0
$149$	3.59029	0.294128	0.147064	−	0.989127i	$-0.453018\pi$
$149$	3.59029	0.294128	0.147064	+	0.989127i	$0.453018\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	6.09453	0.492714
$154$	0	0
$155$	−2.47089	−0.198467
$156$	0	0
$157$	−6.23207	−0.497374	−0.248687	−	0.968584i	$-0.579999\pi$
$157$	−6.23207	−0.497374	−0.248687	+	0.968584i	$0.579999\pi$
$158$	0	0
$159$	−9.29080	−0.736808
$160$	0	0
$161$	−4.30908	−0.339603
$162$	0	0
$163$	−3.95973	−0.310150	−0.155075	−	0.987903i	$-0.549562\pi$
$163$	−3.95973	−0.310150	−0.155075	+	0.987903i	$0.549562\pi$
$164$	0	0
$165$	10.3347	0.804556
$166$	0	0
$167$	−11.3535	−0.878563	−0.439281	−	0.898350i	$-0.644767\pi$
$167$	−11.3535	−0.878563	−0.439281	+	0.898350i	$0.644767\pi$
$168$	0	0
$169$	−11.9733	−0.921022
$170$	0	0
$171$	−8.50618	−0.650484
$172$	0	0
$173$	−5.62843	−0.427922	−0.213961	−	0.976842i	$-0.568637\pi$
$173$	−5.62843	−0.427922	−0.213961	+	0.976842i	$0.568637\pi$
$174$	0	0
$175$	1.67649	0.126730
$176$	0	0
$177$	15.9046	1.19546
$178$	0	0
$179$	−13.6392	−1.01944	−0.509719	−	0.860341i	$-0.670251\pi$
$179$	−13.6392	−1.01944	−0.509719	+	0.860341i	$0.670251\pi$
$180$	0	0
$181$	−18.5162	−1.37630	−0.688149	−	0.725569i	$-0.741577\pi$
$181$	−18.5162	−1.37630	−0.688149	+	0.725569i	$0.741577\pi$
$182$	0	0
$183$	−0.0180704	−0.00133580
$184$	0	0
$185$	−4.47186	−0.328778
$186$	0	0
$187$	−26.7020	−1.95265
$188$	0	0
$189$	6.29981	0.458244
$190$	0	0
$191$	8.08352	0.584903	0.292452	−	0.956280i	$-0.405529\pi$
$191$	8.08352	0.584903	0.292452	+	0.956280i	$0.405529\pi$
$192$	0	0
$193$	−7.78137	−0.560115	−0.280057	−	0.959983i	$-0.590354\pi$
$193$	−7.78137	−0.560115	−0.280057	+	0.959983i	$0.590354\pi$
$194$	0	0
$195$	−2.06591	−0.147943
$196$	0	0
$197$	1.01817	0.0725413	0.0362707	−	0.999342i	$-0.488452\pi$
$197$	1.01817	0.0725413	0.0362707	+	0.999342i	$0.488452\pi$
$198$	0	0
$199$	26.1668	1.85491	0.927457	−	0.373929i	$-0.121990\pi$
$199$	26.1668	1.85491	0.927457	+	0.373929i	$0.121990\pi$
$200$	0	0
$201$	−3.32837	−0.234765
$202$	0	0
$203$	2.28061	0.160067
$204$	0	0
$205$	0.813460	0.0568145
$206$	0	0
$207$	−2.97367	−0.206684
$208$	0	0
$209$	37.2682	2.57790
$210$	0	0
$211$	−17.4539	−1.20157	−0.600787	−	0.799410i	$-0.705146\pi$
$211$	−17.4539	−1.20157	−0.600787	+	0.799410i	$0.705146\pi$
$212$	0	0
$213$	−21.7807	−1.49239
$214$	0	0
$215$	7.43339	0.506953
$216$	0	0
$217$	−4.14241	−0.281205
$218$	0	0
$219$	−0.0609077	−0.00411576
$220$	0	0
$221$	5.33773	0.359055
$222$	0	0
$223$	12.6666	0.848220	0.424110	−	0.905611i	$-0.360587\pi$
$223$	12.6666	0.848220	0.424110	+	0.905611i	$0.360587\pi$
$224$	0	0
$225$	1.15693	0.0771288
$226$	0	0
$227$	16.7984	1.11495	0.557473	−	0.830195i	$-0.311771\pi$
$227$	16.7984	1.11495	0.557473	+	0.830195i	$0.311771\pi$
$228$	0	0
$229$	20.1525	1.33171	0.665856	−	0.746080i	$-0.268066\pi$
$229$	20.1525	1.33171	0.665856	+	0.746080i	$0.268066\pi$
$230$	0	0
$231$	17.3260	1.13997
$232$	0	0
$233$	6.01638	0.394146	0.197073	−	0.980389i	$-0.436856\pi$
$233$	6.01638	0.394146	0.197073	+	0.980389i	$0.436856\pi$
$234$	0	0
$235$	−4.02065	−0.262279
$236$	0	0
$237$	−22.1498	−1.43878
$238$	0	0
$239$	7.84122	0.507207	0.253603	−	0.967308i	$-0.418384\pi$
$239$	7.84122	0.507207	0.253603	+	0.967308i	$0.418384\pi$
$240$	0	0
$241$	−23.8898	−1.53888	−0.769438	−	0.638722i	$-0.779464\pi$
$241$	−23.8898	−1.53888	−0.769438	+	0.638722i	$0.779464\pi$
$242$	0	0
$243$	11.4239	0.732844
$244$	0	0
$245$	−4.18939	−0.267651
$246$	0	0
$247$	−7.44991	−0.474026
$248$	0	0
$249$	−16.8119	−1.06541
$250$	0	0
$251$	18.8474	1.18964	0.594818	−	0.803860i	$-0.297224\pi$
$251$	18.8474	1.18964	0.594818	+	0.803860i	$0.297224\pi$
$252$	0	0
$253$	13.0286	0.819098
$254$	0	0
$255$	−10.7404	−0.672588
$256$	0	0
$257$	10.3423	0.645136	0.322568	−	0.946546i	$-0.395454\pi$
$257$	10.3423	0.645136	0.322568	+	0.946546i	$0.395454\pi$
$258$	0	0
$259$	−7.49702	−0.465842
$260$	0	0
$261$	1.57383	0.0974177
$262$	0	0
$263$	12.6987	0.783037	0.391519	−	0.920170i	$-0.371950\pi$
$263$	12.6987	0.783037	0.391519	+	0.920170i	$0.371950\pi$
$264$	0	0
$265$	4.55687	0.279926
$266$	0	0
$267$	−23.9136	−1.46349
$268$	0	0
$269$	11.3638	0.692864	0.346432	−	0.938075i	$-0.387393\pi$
$269$	11.3638	0.692864	0.346432	+	0.938075i	$0.387393\pi$
$270$	0	0
$271$	23.2919	1.41488	0.707441	−	0.706772i	$-0.249849\pi$
$271$	23.2919	1.41488	0.707441	+	0.706772i	$0.249849\pi$
$272$	0	0
$273$	−3.46346	−0.209618
$274$	0	0
$275$	−5.06888	−0.305665
$276$	0	0
$277$	16.0616	0.965048	0.482524	−	0.875883i	$-0.339720\pi$
$277$	16.0616	0.965048	0.482524	+	0.875883i	$0.339720\pi$
$278$	0	0
$279$	−2.85865	−0.171143
$280$	0	0
$281$	29.6823	1.77070	0.885349	−	0.464927i	$-0.153919\pi$
$281$	29.6823	1.77070	0.885349	+	0.464927i	$0.153919\pi$
$282$	0	0
$283$	11.0741	0.658288	0.329144	−	0.944280i	$-0.393240\pi$
$283$	11.0741	0.658288	0.329144	+	0.944280i	$0.393240\pi$
$284$	0	0
$285$	14.9904	0.887954
$286$	0	0
$287$	1.36375	0.0804999
$288$	0	0
$289$	10.7501	0.632361
$290$	0	0
$291$	−18.2011	−1.06697
$292$	0	0
$293$	29.2783	1.71046	0.855229	−	0.518251i	$-0.173417\pi$
$293$	29.2783	1.71046	0.855229	+	0.518251i	$0.173417\pi$
$294$	0	0
$295$	−7.80075	−0.454177
$296$	0	0
$297$	−19.0476	−1.10525
$298$	0	0
$299$	−2.60441	−0.150617
$300$	0	0
$301$	12.4620	0.718296
$302$	0	0
$303$	30.7422	1.76610
$304$	0	0
$305$	0.00886299	0.000507493 0
$306$	0	0
$307$	3.80479	0.217151	0.108575	−	0.994088i	$-0.465371\pi$
$307$	3.80479	0.217151	0.108575	+	0.994088i	$0.465371\pi$
$308$	0	0
$309$	−1.01197	−0.0575691
$310$	0	0
$311$	6.52833	0.370188	0.185094	−	0.982721i	$-0.440741\pi$
$311$	6.52833	0.370188	0.185094	+	0.982721i	$0.440741\pi$
$312$	0	0
$313$	−14.9893	−0.847245	−0.423622	−	0.905839i	$-0.639242\pi$
$313$	−14.9893	−0.847245	−0.423622	+	0.905839i	$0.639242\pi$
$314$	0	0
$315$	1.93958	0.109283
$316$	0	0
$317$	27.4200	1.54006	0.770031	−	0.638006i	$-0.220241\pi$
$317$	27.4200	1.54006	0.770031	+	0.638006i	$0.220241\pi$
$318$	0	0
$319$	−6.89544	−0.386071
$320$	0	0
$321$	33.6356	1.87735
$322$	0	0
$323$	−38.7310	−2.15505
$324$	0	0
$325$	1.01327	0.0562060
$326$	0	0
$327$	−24.7122	−1.36659
$328$	0	0
$329$	−6.74057	−0.371620
$330$	0	0
$331$	−9.99696	−0.549483	−0.274741	−	0.961518i	$-0.588592\pi$
$331$	−9.99696	−0.549483	−0.274741	+	0.961518i	$0.588592\pi$
$332$	0	0
$333$	−5.17364	−0.283514
$334$	0	0
$335$	1.63247	0.0891914
$336$	0	0
$337$	16.8913	0.920128	0.460064	−	0.887886i	$-0.347827\pi$
$337$	16.8913	0.920128	0.460064	+	0.887886i	$0.347827\pi$
$338$	0	0
$339$	−41.4724	−2.25247
$340$	0	0
$341$	12.5246	0.678248
$342$	0	0
$343$	−18.7589	−1.01288
$344$	0	0
$345$	5.24048	0.282138
$346$	0	0
$347$	−23.4133	−1.25689	−0.628445	−	0.777854i	$-0.716308\pi$
$347$	−23.4133	−1.25689	−0.628445	+	0.777854i	$0.716308\pi$
$348$	0	0
$349$	2.17328	0.116333	0.0581666	−	0.998307i	$-0.481475\pi$
$349$	2.17328	0.116333	0.0581666	+	0.998307i	$0.481475\pi$
$350$	0	0
$351$	3.80761	0.203235
$352$	0	0
$353$	31.8815	1.69688	0.848440	−	0.529292i	$-0.177542\pi$
$353$	31.8815	1.69688	0.848440	+	0.529292i	$0.177542\pi$
$354$	0	0
$355$	10.6828	0.566983
$356$	0	0
$357$	−18.0061	−0.952982
$358$	0	0
$359$	34.4767	1.81961	0.909805	−	0.415035i	$-0.136231\pi$
$359$	34.4767	1.81961	0.909805	+	0.415035i	$0.136231\pi$
$360$	0	0
$361$	35.0572	1.84511
$362$	0	0
$363$	−29.9580	−1.57239
$364$	0	0
$365$	0.0298735	0.00156365
$366$	0	0
$367$	23.6927	1.23675	0.618374	−	0.785884i	$-0.287792\pi$
$367$	23.6927	1.23675	0.618374	+	0.785884i	$0.287792\pi$
$368$	0	0
$369$	0.941118	0.0489927
$370$	0	0
$371$	7.63953	0.396625
$372$	0	0
$373$	10.6604	0.551972	0.275986	−	0.961162i	$-0.410996\pi$
$373$	10.6604	0.551972	0.275986	+	0.961162i	$0.410996\pi$
$374$	0	0
$375$	−2.03886	−0.105286
$376$	0	0
$377$	1.37840	0.0709911
$378$	0	0
$379$	−4.71580	−0.242235	−0.121117	−	0.992638i	$-0.538648\pi$
$379$	−4.71580	−0.242235	−0.121117	+	0.992638i	$0.538648\pi$
$380$	0	0
$381$	14.8962	0.763156
$382$	0	0
$383$	−28.7361	−1.46835	−0.734174	−	0.678962i	$-0.762430\pi$
$383$	−28.7361	−1.46835	−0.734174	+	0.678962i	$0.762430\pi$
$384$	0	0
$385$	−8.49791	−0.433094
$386$	0	0
$387$	8.59993	0.437159
$388$	0	0
$389$	12.8066	0.649322	0.324661	−	0.945830i	$-0.394750\pi$
$389$	12.8066	0.649322	0.324661	+	0.945830i	$0.394750\pi$
$390$	0	0
$391$	−13.5399	−0.684744
$392$	0	0
$393$	6.76852	0.341426
$394$	0	0
$395$	10.8638	0.546619
$396$	0	0
$397$	1.44315	0.0724298	0.0362149	−	0.999344i	$-0.488470\pi$
$397$	1.44315	0.0724298	0.0362149	+	0.999344i	$0.488470\pi$
$398$	0	0
$399$	25.1312	1.25813
$400$	0	0
$401$	−8.69148	−0.434032	−0.217016	−	0.976168i	$-0.569632\pi$
$401$	−8.69148	−0.434032	−0.217016	+	0.976168i	$0.569632\pi$
$402$	0	0
$403$	−2.50367	−0.124717
$404$	0	0
$405$	−11.1323	−0.553169
$406$	0	0
$407$	22.6673	1.12358
$408$	0	0
$409$	32.2672	1.59551	0.797756	−	0.602980i	$-0.206020\pi$
$409$	32.2672	1.59551	0.797756	+	0.602980i	$0.206020\pi$
$410$	0	0
$411$	26.6873	1.31639
$412$	0	0
$413$	−13.0778	−0.643519
$414$	0	0
$415$	8.24573	0.404767
$416$	0	0
$417$	−38.0307	−1.86237
$418$	0	0
$419$	−14.1500	−0.691272	−0.345636	−	0.938369i	$-0.612337\pi$
$419$	−14.1500	−0.691272	−0.345636	+	0.938369i	$0.612337\pi$
$420$	0	0
$421$	−40.7107	−1.98412	−0.992059	−	0.125773i	$-0.959859\pi$
$421$	−40.7107	−1.98412	−0.992059	+	0.125773i	$0.959859\pi$
$422$	0	0
$423$	−4.65162	−0.226170
$424$	0	0
$425$	5.26784	0.255528
$426$	0	0
$427$	0.0148587	0.000719062 0
$428$	0	0
$429$	10.4718	0.505585
$430$	0	0
$431$	37.6885	1.81539	0.907696	−	0.419629i	$-0.137840\pi$
$431$	37.6885	1.81539	0.907696	+	0.419629i	$0.137840\pi$
$432$	0	0
$433$	14.4650	0.695142	0.347571	−	0.937654i	$-0.387007\pi$
$433$	14.4650	0.695142	0.347571	+	0.937654i	$0.387007\pi$
$434$	0	0
$435$	−2.77355	−0.132982
$436$	0	0
$437$	18.8978	0.904004
$438$	0	0
$439$	−4.33199	−0.206755	−0.103377	−	0.994642i	$-0.532965\pi$
$439$	−4.33199	−0.206755	−0.103377	+	0.994642i	$0.532965\pi$
$440$	0	0
$441$	−4.84684	−0.230802
$442$	0	0
$443$	2.25857	0.107308	0.0536539	−	0.998560i	$-0.482913\pi$
$443$	2.25857	0.107308	0.0536539	+	0.998560i	$0.482913\pi$
$444$	0	0
$445$	11.7289	0.556004
$446$	0	0
$447$	−7.32008	−0.346228
$448$	0	0
$449$	−12.6052	−0.594877	−0.297438	−	0.954741i	$-0.596132\pi$
$449$	−12.6052	−0.594877	−0.297438	+	0.954741i	$0.596132\pi$
$450$	0	0
$451$	−4.12333	−0.194160
$452$	0	0
$453$	−2.03886	−0.0957938
$454$	0	0
$455$	1.69873	0.0796377
$456$	0	0
$457$	−13.2604	−0.620294	−0.310147	−	0.950689i	$-0.600378\pi$
$457$	−13.2604	−0.620294	−0.310147	+	0.950689i	$0.600378\pi$
$458$	0	0
$459$	19.7952	0.923962
$460$	0	0
$461$	42.5382	1.98120	0.990600	−	0.136789i	$-0.0436784\pi$
$461$	42.5382	1.98120	0.990600	+	0.136789i	$0.0436784\pi$
$462$	0	0
$463$	3.75716	0.174610	0.0873050	−	0.996182i	$-0.472175\pi$
$463$	3.75716	0.174610	0.0873050	+	0.996182i	$0.472175\pi$
$464$	0	0
$465$	5.03779	0.233622
$466$	0	0
$467$	−19.2589	−0.891194	−0.445597	−	0.895234i	$-0.647009\pi$
$467$	−19.2589	−0.891194	−0.445597	+	0.895234i	$0.647009\pi$
$468$	0	0
$469$	2.73682	0.126374
$470$	0	0
$471$	12.7063	0.585475
$472$	0	0
$473$	−37.6790	−1.73248
$474$	0	0
$475$	−7.35236	−0.337349
$476$	0	0
$477$	5.27199	0.241388
$478$	0	0
$479$	−6.25127	−0.285628	−0.142814	−	0.989750i	$-0.545615\pi$
$479$	−6.25127	−0.285628	−0.142814	+	0.989750i	$0.545615\pi$
$480$	0	0
$481$	−4.53120	−0.206605
$482$	0	0
$483$	8.78559	0.399758
$484$	0	0
$485$	8.92711	0.405359
$486$	0	0
$487$	−19.9144	−0.902409	−0.451204	−	0.892421i	$-0.649005\pi$
$487$	−19.9144	−0.902409	−0.451204	+	0.892421i	$0.649005\pi$
$488$	0	0
$489$	8.07332	0.365088
$490$	0	0
$491$	25.2557	1.13977	0.569887	−	0.821723i	$-0.306987\pi$
$491$	25.2557	1.13977	0.569887	+	0.821723i	$0.306987\pi$
$492$	0	0
$493$	7.16610	0.322745
$494$	0	0
$495$	−5.86435	−0.263583
$496$	0	0
$497$	17.9095	0.803353
$498$	0	0
$499$	13.4264	0.601049	0.300524	−	0.953774i	$-0.402838\pi$
$499$	13.4264	0.601049	0.300524	+	0.953774i	$0.402838\pi$
$500$	0	0
$501$	23.1482	1.03419
$502$	0	0
$503$	0.0869372	0.00387634	0.00193817	−	0.999998i	$-0.499383\pi$
$503$	0.0869372	0.00387634	0.00193817	+	0.999998i	$0.499383\pi$
$504$	0	0
$505$	−15.0782	−0.670970
$506$	0	0
$507$	24.4118	1.08417
$508$	0	0
$509$	6.30709	0.279557	0.139779	−	0.990183i	$-0.455361\pi$
$509$	6.30709	0.279557	0.139779	+	0.990183i	$0.455361\pi$
$510$	0	0
$511$	0.0500825	0.00221552
$512$	0	0
$513$	−27.6283	−1.21982
$514$	0	0
$515$	0.496343	0.0218715
$516$	0	0
$517$	20.3802	0.896321
$518$	0	0
$519$	11.4756	0.503721
$520$	0	0
$521$	7.94085	0.347895	0.173947	−	0.984755i	$-0.444348\pi$
$521$	7.94085	0.347895	0.173947	+	0.984755i	$0.444348\pi$
$522$	0	0
$523$	33.3911	1.46009	0.730046	−	0.683398i	$-0.239499\pi$
$523$	33.3911	1.46009	0.730046	+	0.683398i	$0.239499\pi$
$524$	0	0
$525$	−3.41811	−0.149179
$526$	0	0
$527$	−13.0163	−0.566997
$528$	0	0
$529$	−16.3935	−0.712763
$530$	0	0
$531$	−9.02494	−0.391649
$532$	0	0
$533$	0.824253	0.0357024
$534$	0	0
$535$	−16.4973	−0.713239
$536$	0	0
$537$	27.8083	1.20002
$538$	0	0
$539$	21.2355	0.914679
$540$	0	0
$541$	6.14871	0.264354	0.132177	−	0.991226i	$-0.457803\pi$
$541$	6.14871	0.264354	0.132177	+	0.991226i	$0.457803\pi$
$542$	0	0
$543$	37.7519	1.62009
$544$	0	0
$545$	12.1206	0.519191
$546$	0	0
$547$	−30.5057	−1.30433	−0.652164	−	0.758078i	$-0.726139\pi$
$547$	−30.5057	−1.30433	−0.652164	+	0.758078i	$0.726139\pi$
$548$	0	0
$549$	0.0102539	0.000437625 0
$550$	0	0
$551$	−10.0018	−0.426090
$552$	0	0
$553$	18.2131	0.774499
$554$	0	0
$555$	9.11748	0.387016
$556$	0	0
$557$	35.3820	1.49918	0.749591	−	0.661901i	$-0.230250\pi$
$557$	35.3820	1.49918	0.749591	+	0.661901i	$0.230250\pi$
$558$	0	0
$559$	7.53201	0.318570
$560$	0	0
$561$	54.4416	2.29853
$562$	0	0
$563$	8.87879	0.374197	0.187098	−	0.982341i	$-0.440092\pi$
$563$	8.87879	0.374197	0.187098	+	0.982341i	$0.440092\pi$
$564$	0	0
$565$	20.3410	0.855753
$566$	0	0
$567$	−18.6632	−0.783779
$568$	0	0
$569$	−21.9653	−0.920834	−0.460417	−	0.887703i	$-0.652300\pi$
$569$	−21.9653	−0.920834	−0.460417	+	0.887703i	$0.652300\pi$
$570$	0	0
$571$	−3.48453	−0.145823	−0.0729115	−	0.997338i	$-0.523229\pi$
$571$	−3.48453	−0.145823	−0.0729115	+	0.997338i	$0.523229\pi$
$572$	0	0
$573$	−16.4811	−0.688509
$574$	0	0
$575$	−2.57030	−0.107189
$576$	0	0
$577$	−0.282068	−0.0117427	−0.00587133	−	0.999983i	$-0.501869\pi$
$577$	−0.282068	−0.0117427	−0.00587133	+	0.999983i	$0.501869\pi$
$578$	0	0
$579$	15.8651	0.659330
$580$	0	0
$581$	13.8239	0.573510
$582$	0	0
$583$	−23.0982	−0.956631
$584$	0	0
$585$	1.17228	0.0484679
$586$	0	0
$587$	−8.19584	−0.338279	−0.169139	−	0.985592i	$-0.554099\pi$
$587$	−8.19584	−0.338279	−0.169139	+	0.985592i	$0.554099\pi$
$588$	0	0
$589$	18.1669	0.748553
$590$	0	0
$591$	−2.07589	−0.0853908
$592$	0	0
$593$	17.3431	0.712195	0.356098	−	0.934449i	$-0.384107\pi$
$593$	17.3431	0.712195	0.356098	+	0.934449i	$0.384107\pi$
$594$	0	0
$595$	8.83146	0.362055
$596$	0	0
$597$	−53.3503	−2.18348
$598$	0	0
$599$	−13.1475	−0.537192	−0.268596	−	0.963253i	$-0.586560\pi$
$599$	−13.1475	−0.537192	−0.268596	+	0.963253i	$0.586560\pi$
$600$	0	0
$601$	15.2803	0.623296	0.311648	−	0.950198i	$-0.399119\pi$
$601$	15.2803	0.623296	0.311648	+	0.950198i	$0.399119\pi$
$602$	0	0
$603$	1.88866	0.0769121
$604$	0	0
$605$	14.6935	0.597378
$606$	0	0
$607$	−40.9579	−1.66243	−0.831215	−	0.555951i	$-0.812354\pi$
$607$	−40.9579	−1.66243	−0.831215	+	0.555951i	$0.812354\pi$
$608$	0	0
$609$	−4.64983	−0.188420
$610$	0	0
$611$	−4.07400	−0.164816
$612$	0	0
$613$	−13.5865	−0.548753	−0.274377	−	0.961622i	$-0.588472\pi$
$613$	−13.5865	−0.548753	−0.274377	+	0.961622i	$0.588472\pi$
$614$	0	0
$615$	−1.65853	−0.0668783
$616$	0	0
$617$	15.8989	0.640068	0.320034	−	0.947406i	$-0.396306\pi$
$617$	15.8989	0.640068	0.320034	+	0.947406i	$0.396306\pi$
$618$	0	0
$619$	−18.6586	−0.749952	−0.374976	−	0.927035i	$-0.622349\pi$
$619$	−18.6586	−0.749952	−0.374976	+	0.927035i	$0.622349\pi$
$620$	0	0
$621$	−9.65855	−0.387584
$622$	0	0
$623$	19.6634	0.787796
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−75.9845	−3.03453
$628$	0	0
$629$	−23.5571	−0.939281
$630$	0	0
$631$	28.3628	1.12910	0.564552	−	0.825397i	$-0.309049\pi$
$631$	28.3628	1.12910	0.564552	+	0.825397i	$0.309049\pi$
$632$	0	0
$633$	35.5859	1.41441
$634$	0	0
$635$	−7.30616	−0.289936
$636$	0	0
$637$	−4.24498	−0.168192
$638$	0	0
$639$	12.3593	0.488925
$640$	0	0
$641$	37.1701	1.46813	0.734066	−	0.679078i	$-0.237620\pi$
$641$	37.1701	1.46813	0.734066	+	0.679078i	$0.237620\pi$
$642$	0	0
$643$	6.45222	0.254451	0.127225	−	0.991874i	$-0.459393\pi$
$643$	6.45222	0.254451	0.127225	+	0.991874i	$0.459393\pi$
$644$	0	0
$645$	−15.1556	−0.596751
$646$	0	0
$647$	9.64566	0.379210	0.189605	−	0.981860i	$-0.439279\pi$
$647$	9.64566	0.379210	0.189605	+	0.981860i	$0.439279\pi$
$648$	0	0
$649$	39.5411	1.55212
$650$	0	0
$651$	8.44578	0.331016
$652$	0	0
$653$	−2.29199	−0.0896925	−0.0448462	−	0.998994i	$-0.514280\pi$
$653$	−2.29199	−0.0896925	−0.0448462	+	0.998994i	$0.514280\pi$
$654$	0	0
$655$	−3.31976	−0.129714
$656$	0	0
$657$	0.0345616	0.00134838
$658$	0	0
$659$	19.4554	0.757874	0.378937	−	0.925422i	$-0.376290\pi$
$659$	19.4554	0.757874	0.378937	+	0.925422i	$0.376290\pi$
$660$	0	0
$661$	−9.28319	−0.361074	−0.180537	−	0.983568i	$-0.557784\pi$
$661$	−9.28319	−0.361074	−0.180537	+	0.983568i	$0.557784\pi$
$662$	0	0
$663$	−10.8829	−0.422655
$664$	0	0
$665$	−12.3261	−0.477987
$666$	0	0
$667$	−3.49651	−0.135385
$668$	0	0
$669$	−25.8254	−0.998469
$670$	0	0
$671$	−0.0449254	−0.00173433
$672$	0	0
$673$	−38.3309	−1.47755	−0.738774	−	0.673953i	$-0.764595\pi$
$673$	−38.3309	−1.47755	−0.738774	+	0.673953i	$0.764595\pi$
$674$	0	0
$675$	3.75775	0.144636
$676$	0	0
$677$	−28.8680	−1.10949	−0.554743	−	0.832022i	$-0.687183\pi$
$677$	−28.8680	−1.10949	−0.554743	+	0.832022i	$0.687183\pi$
$678$	0	0
$679$	14.9662	0.574349
$680$	0	0
$681$	−34.2494	−1.31244
$682$	0	0
$683$	30.3957	1.16306	0.581529	−	0.813526i	$-0.302455\pi$
$683$	30.3957	1.16306	0.581529	+	0.813526i	$0.302455\pi$
$684$	0	0
$685$	−13.0894	−0.500119
$686$	0	0
$687$	−41.0880	−1.56760
$688$	0	0
$689$	4.61733	0.175906
$690$	0	0
$691$	38.8076	1.47631	0.738156	−	0.674630i	$-0.235697\pi$
$691$	38.8076	1.47631	0.738156	+	0.674630i	$0.235697\pi$
$692$	0	0
$693$	−9.83151	−0.373468
$694$	0	0
$695$	18.6530	0.707547
$696$	0	0
$697$	4.28518	0.162313
$698$	0	0
$699$	−12.2665	−0.463963
$700$	0	0
$701$	26.9364	1.01737	0.508686	−	0.860952i	$-0.330131\pi$
$701$	26.9364	1.01737	0.508686	+	0.860952i	$0.330131\pi$
$702$	0	0
$703$	32.8787	1.24005
$704$	0	0
$705$	8.19753	0.308737
$706$	0	0
$707$	−25.2784	−0.950691
$708$	0	0
$709$	41.7453	1.56778	0.783889	−	0.620901i	$-0.213233\pi$
$709$	41.7453	1.56778	0.783889	+	0.620901i	$0.213233\pi$
$710$	0	0
$711$	12.5687	0.471364
$712$	0	0
$713$	6.35094	0.237844
$714$	0	0
$715$	−5.13613	−0.192081
$716$	0	0
$717$	−15.9871	−0.597050
$718$	0	0
$719$	1.66632	0.0621431	0.0310716	−	0.999517i	$-0.490108\pi$
$719$	1.66632	0.0621431	0.0310716	+	0.999517i	$0.490108\pi$
$720$	0	0
$721$	0.832113	0.0309895
$722$	0	0
$723$	48.7078	1.81146
$724$	0	0
$725$	1.36035	0.0505221
$726$	0	0
$727$	0.702481	0.0260536	0.0130268	−	0.999915i	$-0.495853\pi$
$727$	0.702481	0.0260536	0.0130268	+	0.999915i	$0.495853\pi$
$728$	0	0
$729$	10.1052	0.374267
$730$	0	0
$731$	39.1579	1.44831
$732$	0	0
$733$	19.3376	0.714249	0.357124	−	0.934057i	$-0.383757\pi$
$733$	19.3376	0.714249	0.357124	+	0.934057i	$0.383757\pi$
$734$	0	0
$735$	8.54157	0.315060
$736$	0	0
$737$	−8.27480	−0.304806
$738$	0	0
$739$	−45.6374	−1.67880	−0.839400	−	0.543515i	$-0.817093\pi$
$739$	−45.6374	−1.67880	−0.839400	+	0.543515i	$0.817093\pi$
$740$	0	0
$741$	15.1893	0.557992
$742$	0	0
$743$	−10.5896	−0.388494	−0.194247	−	0.980953i	$-0.562226\pi$
$743$	−10.5896	−0.388494	−0.194247	+	0.980953i	$0.562226\pi$
$744$	0	0
$745$	3.59029	0.131538
$746$	0	0
$747$	9.53975	0.349041
$748$	0	0
$749$	−27.6575	−1.01058
$750$	0	0
$751$	−14.3829	−0.524838	−0.262419	−	0.964954i	$-0.584520\pi$
$751$	−14.3829	−0.524838	−0.262419	+	0.964954i	$0.584520\pi$
$752$	0	0
$753$	−38.4271	−1.40036
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−36.1180	−1.31273	−0.656366	−	0.754442i	$-0.727907\pi$
$757$	−36.1180	−1.31273	−0.656366	+	0.754442i	$0.727907\pi$
$758$	0	0
$759$	−26.5634	−0.964189
$760$	0	0
$761$	−42.5080	−1.54091	−0.770456	−	0.637493i	$-0.779972\pi$
$761$	−42.5080	−1.54091	−0.770456	+	0.637493i	$0.779972\pi$
$762$	0	0
$763$	20.3201	0.735636
$764$	0	0
$765$	6.09453	0.220348
$766$	0	0
$767$	−7.90425	−0.285406
$768$	0	0
$769$	7.98201	0.287839	0.143919	−	0.989589i	$-0.454029\pi$
$769$	7.98201	0.287839	0.143919	+	0.989589i	$0.454029\pi$
$770$	0	0
$771$	−21.0865	−0.759412
$772$	0	0
$773$	−11.6051	−0.417405	−0.208703	−	0.977979i	$-0.566924\pi$
$773$	−11.6051	−0.417405	−0.208703	+	0.977979i	$0.566924\pi$
$774$	0	0
$775$	−2.47089	−0.0887570
$776$	0	0
$777$	15.2853	0.548358
$778$	0	0
$779$	−5.98085	−0.214286
$780$	0	0
$781$	−54.1498	−1.93763
$782$	0	0
$783$	5.11185	0.182683
$784$	0	0
$785$	−6.23207	−0.222432
$786$	0	0
$787$	−36.7770	−1.31096	−0.655479	−	0.755213i	$-0.727533\pi$
$787$	−36.7770	−1.31096	−0.655479	+	0.755213i	$0.727533\pi$
$788$	0	0
$789$	−25.8909	−0.921739
$790$	0	0
$791$	34.1014	1.21251
$792$	0	0
$793$	0.00898058	0.000318910 0
$794$	0	0
$795$	−9.29080	−0.329511
$796$	0	0
$797$	23.6535	0.837851	0.418926	−	0.908021i	$-0.362407\pi$
$797$	23.6535	0.837851	0.418926	+	0.908021i	$0.362407\pi$
$798$	0	0
$799$	−21.1802	−0.749300
$800$	0	0
$801$	13.5696	0.479457
$802$	0	0
$803$	−0.151425	−0.00534368
$804$	0	0
$805$	−4.30908	−0.151875
$806$	0	0
$807$	−23.1692	−0.815594
$808$	0	0
$809$	−12.7813	−0.449365	−0.224682	−	0.974432i	$-0.572134\pi$
$809$	−12.7813	−0.449365	−0.224682	+	0.974432i	$0.572134\pi$
$810$	0	0
$811$	−33.4072	−1.17308	−0.586542	−	0.809919i	$-0.699511\pi$
$811$	−33.4072	−1.17308	−0.586542	+	0.809919i	$0.699511\pi$
$812$	0	0
$813$	−47.4888	−1.66551
$814$	0	0
$815$	−3.95973	−0.138703
$816$	0	0
$817$	−54.6529	−1.91206
$818$	0	0
$819$	1.96532	0.0686737
$820$	0	0
$821$	−42.6401	−1.48815	−0.744074	−	0.668097i	$-0.767109\pi$
$821$	−42.6401	−1.48815	−0.744074	+	0.668097i	$0.767109\pi$
$822$	0	0
$823$	43.7253	1.52417	0.762083	−	0.647479i	$-0.224177\pi$
$823$	43.7253	1.52417	0.762083	+	0.647479i	$0.224177\pi$
$824$	0	0
$825$	10.3347	0.359809
$826$	0	0
$827$	−28.7889	−1.00109	−0.500544	−	0.865711i	$-0.666867\pi$
$827$	−28.7889	−1.00109	−0.500544	+	0.865711i	$0.666867\pi$
$828$	0	0
$829$	−43.5485	−1.51250	−0.756251	−	0.654282i	$-0.772971\pi$
$829$	−43.5485	−1.51250	−0.756251	+	0.654282i	$0.772971\pi$
$830$	0	0
$831$	−32.7473	−1.13599
$832$	0	0
$833$	−22.0691	−0.764647
$834$	0	0
$835$	−11.3535	−0.392905
$836$	0	0
$837$	−9.28498	−0.320936
$838$	0	0
$839$	−29.3460	−1.01313	−0.506567	−	0.862200i	$-0.669086\pi$
$839$	−29.3460	−1.01313	−0.506567	+	0.862200i	$0.669086\pi$
$840$	0	0
$841$	−27.1495	−0.936188
$842$	0	0
$843$	−60.5179	−2.08435
$844$	0	0
$845$	−11.9733	−0.411894
$846$	0	0
$847$	24.6335	0.846418
$848$	0	0
$849$	−22.5785	−0.774893
$850$	0	0
$851$	11.4940	0.394011
$852$	0	0
$853$	23.3647	0.799993	0.399997	−	0.916517i	$-0.369011\pi$
$853$	23.3647	0.799993	0.399997	+	0.916517i	$0.369011\pi$
$854$	0	0
$855$	−8.50618	−0.290905
$856$	0	0
$857$	8.65448	0.295632	0.147816	−	0.989015i	$-0.452776\pi$
$857$	8.65448	0.295632	0.147816	+	0.989015i	$0.452776\pi$
$858$	0	0
$859$	30.7685	1.04981	0.524904	−	0.851161i	$-0.324101\pi$
$859$	30.7685	1.04981	0.524904	+	0.851161i	$0.324101\pi$
$860$	0	0
$861$	−2.78050	−0.0947591
$862$	0	0
$863$	41.9758	1.42887	0.714436	−	0.699701i	$-0.246683\pi$
$863$	41.9758	1.42887	0.714436	+	0.699701i	$0.246683\pi$
$864$	0	0
$865$	−5.62843	−0.191372
$866$	0	0
$867$	−21.9180	−0.744373
$868$	0	0
$869$	−55.0675	−1.86804
$870$	0	0
$871$	1.65413	0.0560481
$872$	0	0
$873$	10.3281	0.349552
$874$	0	0
$875$	1.67649	0.0566756
$876$	0	0
$877$	−37.5330	−1.26740	−0.633700	−	0.773579i	$-0.718465\pi$
$877$	−37.5330	−1.26740	−0.633700	+	0.773579i	$0.718465\pi$
$878$	0	0
$879$	−59.6943	−2.01344
$880$	0	0
$881$	0.370838	0.0124938	0.00624692	−	0.999980i	$-0.498012\pi$
$881$	0.370838	0.0124938	0.00624692	+	0.999980i	$0.498012\pi$
$882$	0	0
$883$	54.9459	1.84908	0.924538	−	0.381090i	$-0.124451\pi$
$883$	54.9459	1.84908	0.924538	+	0.381090i	$0.124451\pi$
$884$	0	0
$885$	15.9046	0.534627
$886$	0	0
$887$	2.42671	0.0814808	0.0407404	−	0.999170i	$-0.487028\pi$
$887$	2.42671	0.0814808	0.0407404	+	0.999170i	$0.487028\pi$
$888$	0	0
$889$	−12.2487	−0.410807
$890$	0	0
$891$	56.4283	1.89042
$892$	0	0
$893$	29.5613	0.989231
$894$	0	0
$895$	−13.6392	−0.455907
$896$	0	0
$897$	5.31001	0.177296
$898$	0	0
$899$	−3.36127	−0.112105
$900$	0	0
$901$	24.0049	0.799718
$902$	0	0
$903$	−25.4082	−0.845531
$904$	0	0
$905$	−18.5162	−0.615500
$906$	0	0
$907$	49.8483	1.65518	0.827592	−	0.561329i	$-0.189710\pi$
$907$	49.8483	1.65518	0.827592	+	0.561329i	$0.189710\pi$
$908$	0	0
$909$	−17.4444	−0.578595
$910$	0	0
$911$	13.9844	0.463323	0.231662	−	0.972796i	$-0.425584\pi$
$911$	13.9844	0.463323	0.231662	+	0.972796i	$0.425584\pi$
$912$	0	0
$913$	−41.7966	−1.38327
$914$	0	0
$915$	−0.0180704	−0.000597388 0
$916$	0	0
$917$	−5.56554	−0.183790
$918$	0	0
$919$	−22.0528	−0.727454	−0.363727	−	0.931506i	$-0.618496\pi$
$919$	−22.0528	−0.727454	−0.363727	+	0.931506i	$0.618496\pi$
$920$	0	0
$921$	−7.75742	−0.255616
$922$	0	0
$923$	10.8245	0.356294
$924$	0	0
$925$	−4.47186	−0.147034
$926$	0	0
$927$	0.574235	0.0188604
$928$	0	0
$929$	−30.9226	−1.01454	−0.507270	−	0.861787i	$-0.669345\pi$
$929$	−30.9226	−1.01454	−0.507270	+	0.861787i	$0.669345\pi$
$930$	0	0
$931$	30.8019	1.00949
$932$	0	0
$933$	−13.3103	−0.435761
$934$	0	0
$935$	−26.7020	−0.873250
$936$	0	0
$937$	−49.2714	−1.60963	−0.804814	−	0.593528i	$-0.797735\pi$
$937$	−49.2714	−1.60963	−0.804814	+	0.593528i	$0.797735\pi$
$938$	0	0
$939$	30.5610	0.997321
$940$	0	0
$941$	29.1647	0.950741	0.475371	−	0.879786i	$-0.342314\pi$
$941$	29.1647	0.950741	0.475371	+	0.879786i	$0.342314\pi$
$942$	0	0
$943$	−2.09084	−0.0680871
$944$	0	0
$945$	6.29981	0.204933
$946$	0	0
$947$	14.0430	0.456335	0.228168	−	0.973622i	$-0.426727\pi$
$947$	14.0430	0.456335	0.228168	+	0.973622i	$0.426727\pi$
$948$	0	0
$949$	0.0302699	0.000982601 0
$950$	0	0
$951$	−55.9055	−1.81286
$952$	0	0
$953$	17.4898	0.566550	0.283275	−	0.959039i	$-0.408579\pi$
$953$	17.4898	0.566550	0.283275	+	0.959039i	$0.408579\pi$
$954$	0	0
$955$	8.08352	0.261577
$956$	0	0
$957$	14.0588	0.454457
$958$	0	0
$959$	−21.9442	−0.708614
$960$	0	0
$961$	−24.8947	−0.803055
$962$	0	0
$963$	−19.0862	−0.615045
$964$	0	0
$965$	−7.78137	−0.250491
$966$	0	0
$967$	3.61039	0.116102	0.0580512	−	0.998314i	$-0.481511\pi$
$967$	3.61039	0.116102	0.0580512	+	0.998314i	$0.481511\pi$
$968$	0	0
$969$	78.9670	2.53679
$970$	0	0
$971$	−11.7645	−0.377542	−0.188771	−	0.982021i	$-0.560450\pi$
$971$	−11.7645	−0.377542	−0.188771	+	0.982021i	$0.560450\pi$
$972$	0	0
$973$	31.2714	1.00252
$974$	0	0
$975$	−2.06591	−0.0661620
$976$	0	0
$977$	32.5561	1.04156	0.520780	−	0.853691i	$-0.325641\pi$
$977$	32.5561	1.04156	0.520780	+	0.853691i	$0.325641\pi$
$978$	0	0
$979$	−59.4524	−1.90011
$980$	0	0
$981$	14.0228	0.447712
$982$	0	0
$983$	−0.666882	−0.0212702	−0.0106351	−	0.999943i	$-0.503385\pi$
$983$	−0.666882	−0.0212702	−0.0106351	+	0.999943i	$0.503385\pi$
$984$	0	0
$985$	1.01817	0.0324415
$986$	0	0
$987$	13.7431	0.437446
$988$	0	0
$989$	−19.1061	−0.607537
$990$	0	0
$991$	6.27051	0.199189	0.0995946	−	0.995028i	$-0.468245\pi$
$991$	6.27051	0.199189	0.0995946	+	0.995028i	$0.468245\pi$
$992$	0	0
$993$	20.3824	0.646815
$994$	0	0
$995$	26.1668	0.829543
$996$	0	0
$997$	−2.72198	−0.0862059	−0.0431030	−	0.999071i	$-0.513724\pi$
$997$	−2.72198	−0.0862059	−0.0431030	+	0.999071i	$0.513724\pi$
$998$	0	0
$999$	−16.8041	−0.531659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.7	✓	24	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03886 q^{3} +1.00000 q^{5} +1.67649 q^{7} +1.15693 q^{9} +O(q^{10})\)	\(q-2.03886 q^{3} +1.00000 q^{5} +1.67649 q^{7} +1.15693 q^{9} -5.06888 q^{11} +1.01327 q^{13} -2.03886 q^{15} +5.26784 q^{17} -7.35236 q^{19} -3.41811 q^{21} -2.57030 q^{23} +1.00000 q^{25} +3.75775 q^{27} +1.36035 q^{29} -2.47089 q^{31} +10.3347 q^{33} +1.67649 q^{35} -4.47186 q^{37} -2.06591 q^{39} +0.813460 q^{41} +7.43339 q^{43} +1.15693 q^{45} -4.02065 q^{47} -4.18939 q^{49} -10.7404 q^{51} +4.55687 q^{53} -5.06888 q^{55} +14.9904 q^{57} -7.80075 q^{59} +0.00886299 q^{61} +1.93958 q^{63} +1.01327 q^{65} +1.63247 q^{67} +5.24048 q^{69} +10.6828 q^{71} +0.0298735 q^{73} -2.03886 q^{75} -8.49791 q^{77} +10.8638 q^{79} -11.1323 q^{81} +8.24573 q^{83} +5.26784 q^{85} -2.77355 q^{87} +11.7289 q^{89} +1.69873 q^{91} +5.03779 q^{93} -7.35236 q^{95} +8.92711 q^{97} -5.86435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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