LMFDB - Embedded newform 867.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(1,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$867 = 3 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	867.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 5x^{2} + 4x + 4$ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.63640$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.222191 q^{2} +1.00000 q^{3} -1.95063 q^{4} -0.636405 q^{5} +0.222191 q^{6} -1.72844 q^{7} -0.877796 q^{8} +1.00000 q^{9} -0.141404 q^{10} +4.95063 q^{11} -1.95063 q^{12} +2.50625 q^{13} -0.384044 q^{14} -0.636405 q^{15} +3.70622 q^{16} +0.222191 q^{18} -0.950631 q^{19} +1.24139 q^{20} -1.72844 q^{21} +1.09999 q^{22} -2.12220 q^{23} -0.877796 q^{24} -4.59499 q^{25} +0.556867 q^{26} +1.00000 q^{27} +3.37155 q^{28} +9.58704 q^{29} -0.141404 q^{30} +5.27281 q^{31} +2.57908 q^{32} +4.95063 q^{33} +1.09999 q^{35} -1.95063 q^{36} +8.48705 q^{37} -0.211222 q^{38} +2.50625 q^{39} +0.558634 q^{40} +6.92171 q^{41} -0.384044 q^{42} +7.15061 q^{43} -9.65685 q^{44} -0.636405 q^{45} -0.471535 q^{46} +8.10124 q^{47} +3.70622 q^{48} -4.01250 q^{49} -1.02097 q^{50} -4.88877 q^{52} -6.44438 q^{53} +0.222191 q^{54} -3.15061 q^{55} +1.51722 q^{56} -0.950631 q^{57} +2.13016 q^{58} -10.1125 q^{59} +1.24139 q^{60} +2.96983 q^{61} +1.17157 q^{62} -1.72844 q^{63} -6.83940 q^{64} -1.59499 q^{65} +1.09999 q^{66} +7.70129 q^{67} -2.12220 q^{69} +0.244408 q^{70} -0.384044 q^{71} -0.877796 q^{72} +6.51420 q^{73} +1.88575 q^{74} -4.59499 q^{75} +1.85433 q^{76} -8.55687 q^{77} +0.556867 q^{78} -2.37280 q^{79} -2.35866 q^{80} +1.00000 q^{81} +1.53794 q^{82} -13.3581 q^{83} +3.37155 q^{84} +1.58880 q^{86} +9.58704 q^{87} -4.34564 q^{88} -1.98875 q^{89} -0.141404 q^{90} -4.33190 q^{91} +4.13964 q^{92} +5.27281 q^{93} +1.80002 q^{94} +0.604986 q^{95} +2.57908 q^{96} +3.04142 q^{97} -0.891542 q^{98} +4.95063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 6 q^{11} + 6 q^{12} + 2 q^{13} + 4 q^{14} + 6 q^{15} + 6 q^{16} - 2 q^{18} + 10 q^{19} + 16 q^{20} + 4 q^{21}+ \cdots + 6 q^{99}+O(q^{100})$ 4 * q - 2 * q^2 + 4 * q^3 + 6 * q^4 + 6 * q^5 - 2 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 - 12 * q^10 + 6 * q^11 + 6 * q^12 + 2 * q^13 + 4 * q^14 + 6 * q^15 + 6 * q^16 - 2 * q^18 + 10 * q^19 + 16 * q^20 + 4 * q^21 + 4 * q^22 - 6 * q^23 - 6 * q^24 + 2 * q^25 - 20 * q^26 + 4 * q^27 + 24 * q^28 + 16 * q^29 - 12 * q^30 + 4 * q^31 - 14 * q^32 + 6 * q^33 + 4 * q^35 + 6 * q^36 + 12 * q^37 - 12 * q^38 + 2 * q^39 - 8 * q^40 - 14 * q^41 + 4 * q^42 + 14 * q^43 - 16 * q^44 + 6 * q^45 - 12 * q^46 + 4 * q^47 + 6 * q^48 - 30 * q^50 - 8 * q^52 - 20 * q^53 - 2 * q^54 + 2 * q^55 - 16 * q^56 + 10 * q^57 + 8 * q^58 - 24 * q^59 + 16 * q^60 + 12 * q^61 + 16 * q^62 + 4 * q^63 - 2 * q^64 + 14 * q^65 + 4 * q^66 + 4 * q^67 - 6 * q^69 - 4 * q^70 + 4 * q^71 - 6 * q^72 + 20 * q^73 + 12 * q^74 + 2 * q^75 + 40 * q^76 - 12 * q^77 - 20 * q^78 + 8 * q^79 + 4 * q^81 - 4 * q^82 - 4 * q^83 + 24 * q^84 - 4 * q^86 + 16 * q^87 + 16 * q^88 + 4 * q^89 - 12 * q^90 - 28 * q^91 + 16 * q^92 + 4 * q^93 + 8 * q^94 + 22 * q^95 - 14 * q^96 + 24 * q^97 + 38 * q^98 + 6 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0.222191	0.157113	0.0785565	−	0.996910i	$-0.474969\pi$
$2$	0.222191	0.157113	0.0785565	+	0.996910i	$0.474969\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.95063	−0.975315
$5$	−0.636405	−0.284609	−0.142304	−	0.989823i	$-0.545451\pi$
$5$	−0.636405	−0.284609	−0.142304	+	0.989823i	$0.545451\pi$
$6$	0.222191	0.0907092
$7$	−1.72844	−0.653289	−0.326644	−	0.945147i	$-0.605918\pi$
$7$	−1.72844	−0.653289	−0.326644	+	0.945147i	$0.605918\pi$
$8$	−0.877796	−0.310348
$9$	1.00000	0.333333
$10$	−0.141404	−0.0447158
$11$	4.95063	1.49267	0.746336	−	0.665570i	$-0.231811\pi$
$11$	4.95063	1.49267	0.746336	+	0.665570i	$0.231811\pi$
$12$	−1.95063	−0.563099
$13$	2.50625	0.695108	0.347554	−	0.937660i	$-0.387012\pi$
$13$	2.50625	0.695108	0.347554	+	0.937660i	$0.387012\pi$
$14$	−0.384044	−0.102640
$15$	−0.636405	−0.164319
$16$	3.70622	0.926556
$17$	0	0
$17$	0	0
$18$	0.222191	0.0523710
$19$	−0.950631	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$19$	−0.950631	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$20$	1.24139	0.277584
$21$	−1.72844	−0.377176
$22$	1.09999	0.234518
$23$	−2.12220	−0.442510	−0.221255	−	0.975216i	$-0.571015\pi$
$23$	−2.12220	−0.442510	−0.221255	+	0.975216i	$0.571015\pi$
$24$	−0.877796	−0.179179
$25$	−4.59499	−0.918998
$26$	0.556867	0.109211
$27$	1.00000	0.192450
$28$	3.37155	0.637163
$29$	9.58704	1.78027	0.890134	−	0.455699i	$-0.150611\pi$
$29$	9.58704	1.78027	0.890134	+	0.455699i	$0.150611\pi$
$30$	−0.141404	−0.0258167
$31$	5.27281	0.947025	0.473512	−	0.880787i	$-0.342986\pi$
$31$	5.27281	0.947025	0.473512	+	0.880787i	$0.342986\pi$
$32$	2.57908	0.455922
$33$	4.95063	0.861794
$34$	0	0
$35$	1.09999	0.185932
$36$	−1.95063	−0.325105
$37$	8.48705	1.39526	0.697631	−	0.716457i	$-0.254237\pi$
$37$	8.48705	1.39526	0.697631	+	0.716457i	$0.254237\pi$
$38$	−0.211222	−0.0342647
$39$	2.50625	0.401321
$40$	0.558634	0.0883278
$41$	6.92171	1.08099	0.540495	−	0.841347i	$-0.318237\pi$
$41$	6.92171	1.08099	0.540495	+	0.841347i	$0.318237\pi$
$42$	−0.384044	−0.0592593
$43$	7.15061	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$43$	7.15061	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$44$	−9.65685	−1.45583
$45$	−0.636405	−0.0948696
$46$	−0.471535	−0.0695241
$47$	8.10124	1.18169	0.590843	−	0.806786i	$-0.298795\pi$
$47$	8.10124	1.18169	0.590843	+	0.806786i	$0.298795\pi$
$48$	3.70622	0.534947
$49$	−4.01250	−0.573214
$50$	−1.02097	−0.144387
$51$	0	0
$52$	−4.88877	−0.677950
$53$	−6.44438	−0.885204	−0.442602	−	0.896718i	$-0.645945\pi$
$53$	−6.44438	−0.885204	−0.442602	+	0.896718i	$0.645945\pi$
$54$	0.222191	0.0302364
$55$	−3.15061	−0.424828
$56$	1.51722	0.202747
$57$	−0.950631	−0.125914
$58$	2.13016	0.279703
$59$	−10.1125	−1.31653	−0.658267	−	0.752785i	$-0.728710\pi$
$59$	−10.1125	−1.31653	−0.658267	+	0.752785i	$0.728710\pi$
$60$	1.24139	0.160263
$61$	2.96983	0.380248	0.190124	−	0.981760i	$-0.439111\pi$
$61$	2.96983	0.380248	0.190124	+	0.981760i	$0.439111\pi$
$62$	1.17157	0.148790
$63$	−1.72844	−0.217763
$64$	−6.83940	−0.854925
$65$	−1.59499	−0.197834
$66$	1.09999	0.135399
$67$	7.70129	0.940862	0.470431	−	0.882437i	$-0.344098\pi$
$67$	7.70129	0.940862	0.470431	+	0.882437i	$0.344098\pi$
$68$	0	0
$69$	−2.12220	−0.255483
$70$	0.244408	0.0292123
$71$	−0.384044	−0.0455777	−0.0227888	−	0.999740i	$-0.507255\pi$
$71$	−0.384044	−0.0455777	−0.0227888	+	0.999740i	$0.507255\pi$
$72$	−0.877796	−0.103449
$73$	6.51420	0.762430	0.381215	−	0.924487i	$-0.375506\pi$
$73$	6.51420	0.762430	0.381215	+	0.924487i	$0.375506\pi$
$74$	1.88575	0.219214
$75$	−4.59499	−0.530584
$76$	1.85433	0.212706
$77$	−8.55687	−0.975145
$78$	0.556867	0.0630527
$79$	−2.37280	−0.266961	−0.133480	−	0.991051i	$-0.542615\pi$
$79$	−2.37280	−0.266961	−0.133480	+	0.991051i	$0.542615\pi$
$80$	−2.35866	−0.263706
$81$	1.00000	0.111111
$82$	1.53794	0.169838
$83$	−13.3581	−1.46625	−0.733123	−	0.680096i	$-0.761938\pi$
$83$	−13.3581	−1.46625	−0.733123	+	0.680096i	$0.761938\pi$
$84$	3.37155	0.367866
$85$	0	0
$86$	1.58880	0.171325
$87$	9.58704	1.02784
$88$	−4.34564	−0.463247
$89$	−1.98875	−0.210807	−0.105404	−	0.994430i	$-0.533613\pi$
$89$	−1.98875	−0.210807	−0.105404	+	0.994430i	$0.533613\pi$
$90$	−0.141404	−0.0149053
$91$	−4.33190	−0.454106
$92$	4.13964	0.431587
$93$	5.27281	0.546765
$94$	1.80002	0.185658
$95$	0.604986	0.0620703
$96$	2.57908	0.263227
$97$	3.04142	0.308809	0.154405	−	0.988008i	$-0.450654\pi$
$97$	3.04142	0.308809	0.154405	+	0.988008i	$0.450654\pi$
$98$	−0.891542	−0.0900594
$99$	4.95063	0.497557
$100$	8.96313	0.896313
$101$	−18.5456	−1.84536	−0.922679	−	0.385569i	$-0.874005\pi$
$101$	−18.5456	−1.84536	−0.922679	+	0.385569i	$0.874005\pi$
$102$	0	0
$103$	3.39501	0.334521	0.167260	−	0.985913i	$-0.446508\pi$
$103$	3.39501	0.334521	0.167260	+	0.985913i	$0.446508\pi$
$104$	−2.19998	−0.215725
$105$	1.09999	0.107348
$106$	−1.43189	−0.139077
$107$	−13.1531	−1.27156	−0.635779	−	0.771871i	$-0.719321\pi$
$107$	−13.1531	−1.27156	−0.635779	+	0.771871i	$0.719321\pi$
$108$	−1.95063	−0.187700
$109$	−17.3896	−1.66562	−0.832809	−	0.553561i	$-0.813269\pi$
$109$	−17.3896	−1.66562	−0.832809	+	0.553561i	$0.813269\pi$
$110$	−0.700037	−0.0667460
$111$	8.48705	0.805555
$112$	−6.40598	−0.605309
$113$	0.776042	0.0730039	0.0365019	−	0.999334i	$-0.488378\pi$
$113$	0.776042	0.0730039	0.0365019	+	0.999334i	$0.488378\pi$
$114$	−0.211222	−0.0197828
$115$	1.35058	0.125942
$116$	−18.7008	−1.73632
$117$	2.50625	0.231703
$118$	−2.24691	−0.206845
$119$	0	0
$120$	0.558634	0.0509961
$121$	13.5087	1.22807
$122$	0.659871	0.0597419
$123$	6.92171	0.624110
$124$	−10.2853	−0.923648
$125$	6.10630	0.546164
$126$	−0.384044	−0.0342134
$127$	10.8519	0.962950	0.481475	−	0.876460i	$-0.340101\pi$
$127$	10.8519	0.962950	0.481475	+	0.876460i	$0.340101\pi$
$128$	−6.67782	−0.590242
$129$	7.15061	0.629576
$130$	−0.354393	−0.0310823
$131$	0.706223	0.0617030	0.0308515	−	0.999524i	$-0.490178\pi$
$131$	0.706223	0.0617030	0.0308515	+	0.999524i	$0.490178\pi$
$132$	−9.65685	−0.840521
$133$	1.64311	0.142476
$134$	1.71116	0.147822
$135$	−0.636405	−0.0547730
$136$	0	0
$137$	12.3125	1.05192	0.525962	−	0.850508i	$-0.323705\pi$
$137$	12.3125	1.05192	0.525962	+	0.850508i	$0.323705\pi$
$138$	−0.471535	−0.0401398
$139$	−13.3444	−1.13186	−0.565928	−	0.824454i	$-0.691482\pi$
$139$	−13.3444	−1.13186	−0.565928	+	0.824454i	$0.691482\pi$
$140$	−2.14567	−0.181342
$141$	8.10124	0.682247
$142$	−0.0853313	−0.00716085
$143$	12.4075	1.03757
$144$	3.70622	0.308852
$145$	−6.10124	−0.506680
$146$	1.44740	0.119788
$147$	−4.01250	−0.330945
$148$	−16.5551	−1.36082
$149$	0.356892	0.0292377	0.0146189	−	0.999893i	$-0.495347\pi$
$149$	0.356892	0.0292377	0.0146189	+	0.999893i	$0.495347\pi$
$150$	−1.02097	−0.0833616
$151$	9.13317	0.743247	0.371624	−	0.928384i	$-0.378801\pi$
$151$	9.13317	0.743247	0.371624	+	0.928384i	$0.378801\pi$
$152$	0.834460	0.0676837
$153$	0	0
$154$	−1.90126	−0.153208
$155$	−3.35564	−0.269532
$156$	−4.88877	−0.391414
$157$	17.5975	1.40443	0.702216	−	0.711964i	$-0.252194\pi$
$157$	17.5975	1.40443	0.702216	+	0.711964i	$0.252194\pi$
$158$	−0.527215	−0.0419430
$159$	−6.44438	−0.511073
$160$	−1.64134	−0.129759
$161$	3.66810	0.289087
$162$	0.222191	0.0174570
$163$	16.5901	1.29943	0.649717	−	0.760177i	$-0.274888\pi$
$163$	16.5901	1.29943	0.649717	+	0.760177i	$0.274888\pi$
$164$	−13.5017	−1.05431
$165$	−3.15061	−0.245274
$166$	−2.96806	−0.230366
$167$	−14.2234	−1.10064	−0.550321	−	0.834953i	$-0.685495\pi$
$167$	−14.2234	−1.10064	−0.550321	+	0.834953i	$0.685495\pi$
$168$	1.51722	0.117056
$169$	−6.71872	−0.516825
$170$	0	0
$171$	−0.950631	−0.0726966
$172$	−13.9482	−1.06354
$173$	−1.89331	−0.143946	−0.0719728	−	0.997407i	$-0.522929\pi$
$173$	−1.89331	−0.143946	−0.0719728	+	0.997407i	$0.522929\pi$
$174$	2.13016	0.161487
$175$	7.94216	0.600371
$176$	18.3481	1.38304
$177$	−10.1125	−0.760101
$178$	−0.441884	−0.0331206
$179$	−22.2469	−1.66281	−0.831406	−	0.555666i	$-0.812464\pi$
$179$	−22.2469	−1.66281	−0.831406	+	0.555666i	$0.812464\pi$
$180$	1.24139	0.0925278
$181$	−11.6142	−0.863276	−0.431638	−	0.902047i	$-0.642064\pi$
$181$	−11.6142	−0.863276	−0.431638	+	0.902047i	$0.642064\pi$
$182$	−0.962511	−0.0713460
$183$	2.96983	0.219536
$184$	1.86286	0.137332
$185$	−5.40120	−0.397104
$186$	1.17157	0.0859039
$187$	0	0
$188$	−15.8025	−1.15252
$189$	−1.72844	−0.125725
$190$	0.134423	0.00975205
$191$	6.63311	0.479955	0.239978	−	0.970778i	$-0.422860\pi$
$191$	6.63311	0.479955	0.239978	+	0.970778i	$0.422860\pi$
$192$	−6.83940	−0.493591
$193$	2.91769	0.210020	0.105010	−	0.994471i	$-0.466513\pi$
$193$	2.91769	0.210020	0.105010	+	0.994471i	$0.466513\pi$
$194$	0.675776	0.0485179
$195$	−1.59499	−0.114220
$196$	7.82690	0.559064
$197$	4.53517	0.323117	0.161559	−	0.986863i	$-0.448348\pi$
$197$	4.53517	0.323117	0.161559	+	0.986863i	$0.448348\pi$
$198$	1.09999	0.0781727
$199$	2.46029	0.174405	0.0872026	−	0.996191i	$-0.472207\pi$
$199$	2.46029	0.174405	0.0872026	+	0.996191i	$0.472207\pi$
$200$	4.03346	0.285209
$201$	7.70129	0.543207
$202$	−4.12068	−0.289930
$203$	−16.5706	−1.16303
$204$	0	0
$205$	−4.40501	−0.307659
$206$	0.754343	0.0525576
$207$	−2.12220	−0.147503
$208$	9.28872	0.644057
$209$	−4.70622	−0.325536
$210$	0.244408	0.0168657
$211$	19.9038	1.37023	0.685116	−	0.728434i	$-0.259752\pi$
$211$	19.9038	1.37023	0.685116	+	0.728434i	$0.259752\pi$
$212$	12.5706	0.863353
$213$	−0.384044	−0.0263143
$214$	−2.92251	−0.199778
$215$	−4.55068	−0.310354
$216$	−0.877796	−0.0597265
$217$	−9.11373	−0.618681
$218$	−3.86381	−0.261690
$219$	6.51420	0.440189
$220$	6.14567	0.414341
$221$	0	0
$222$	1.88575	0.126563
$223$	14.3631	0.961823	0.480911	−	0.876769i	$-0.340306\pi$
$223$	14.3631	0.961823	0.480911	+	0.876769i	$0.340306\pi$
$224$	−4.45779	−0.297849
$225$	−4.59499	−0.306333
$226$	0.172430	0.0114699
$227$	−10.8744	−0.721758	−0.360879	−	0.932613i	$-0.617523\pi$
$227$	−10.8744	−0.721758	−0.360879	+	0.932613i	$0.617523\pi$
$228$	1.85433	0.122806
$229$	18.9900	1.25489	0.627447	−	0.778659i	$-0.284100\pi$
$229$	18.9900	1.25489	0.627447	+	0.778659i	$0.284100\pi$
$230$	0.300087	0.0197872
$231$	−8.55687	−0.563000
$232$	−8.41546	−0.552502
$233$	7.08079	0.463878	0.231939	−	0.972730i	$-0.425493\pi$
$233$	7.08079	0.463878	0.231939	+	0.972730i	$0.425493\pi$
$234$	0.556867	0.0364035
$235$	−5.15567	−0.336319
$236$	19.7257	1.28404
$237$	−2.37280	−0.154130
$238$	0	0
$239$	−5.28872	−0.342099	−0.171049	−	0.985262i	$-0.554716\pi$
$239$	−5.28872	−0.342099	−0.171049	+	0.985262i	$0.554716\pi$
$240$	−2.35866	−0.152251
$241$	5.79826	0.373499	0.186749	−	0.982408i	$-0.440205\pi$
$241$	5.79826	0.373499	0.186749	+	0.982408i	$0.440205\pi$
$242$	3.00153	0.192945
$243$	1.00000	0.0641500
$244$	−5.79304	−0.370862
$245$	2.55357	0.163142
$246$	1.53794	0.0980558
$247$	−2.38252	−0.151596
$248$	−4.62845	−0.293907
$249$	−13.3581	−0.846538
$250$	1.35677	0.0858095
$251$	−8.33190	−0.525905	−0.262952	−	0.964809i	$-0.584696\pi$
$251$	−8.33190	−0.525905	−0.262952	+	0.964809i	$0.584696\pi$
$252$	3.37155	0.212388
$253$	−10.5062	−0.660522
$254$	2.41120	0.151292
$255$	0	0
$256$	12.1950	0.762190
$257$	−4.25565	−0.265460	−0.132730	−	0.991152i	$-0.542374\pi$
$257$	−4.25565	−0.265460	−0.132730	+	0.991152i	$0.542374\pi$
$258$	1.58880	0.0989146
$259$	−14.6694	−0.911509
$260$	3.11123	0.192951
$261$	9.58704	0.593423
$262$	0.156917	0.00969435
$263$	−9.09999	−0.561129	−0.280565	−	0.959835i	$-0.590522\pi$
$263$	−9.09999	−0.561129	−0.280565	+	0.959835i	$0.590522\pi$
$264$	−4.34564	−0.267456
$265$	4.10124	0.251937
$266$	0.365084	0.0223848
$267$	−1.98875	−0.121710
$268$	−15.0224	−0.917637
$269$	21.3820	1.30368	0.651842	−	0.758355i	$-0.273997\pi$
$269$	21.3820	1.30368	0.651842	+	0.758355i	$0.273997\pi$
$270$	−0.141404	−0.00860555
$271$	−19.3213	−1.17368	−0.586842	−	0.809702i	$-0.699629\pi$
$271$	−19.3213	−1.17368	−0.586842	+	0.809702i	$0.699629\pi$
$272$	0	0
$273$	−4.33190	−0.262178
$274$	2.73572	0.165271
$275$	−22.7481	−1.37176
$276$	4.13964	0.249177
$277$	3.34013	0.200689	0.100344	−	0.994953i	$-0.468006\pi$
$277$	3.34013	0.200689	0.100344	+	0.994953i	$0.468006\pi$
$278$	−2.96501	−0.177829
$279$	5.27281	0.315675
$280$	−0.965565	−0.0577035
$281$	4.67754	0.279039	0.139519	−	0.990219i	$-0.455444\pi$
$281$	4.67754	0.279039	0.139519	+	0.990219i	$0.455444\pi$
$282$	1.80002	0.107190
$283$	10.0582	0.597897	0.298948	−	0.954269i	$-0.403364\pi$
$283$	10.0582	0.597897	0.298948	+	0.954269i	$0.403364\pi$
$284$	0.749129	0.0444526
$285$	0.604986	0.0358363
$286$	2.75684	0.163015
$287$	−11.9638	−0.706198
$288$	2.57908	0.151974
$289$	0	0
$290$	−1.35564	−0.0796061
$291$	3.04142	0.178291
$292$	−12.7068	−0.743609
$293$	5.76934	0.337048	0.168524	−	0.985698i	$-0.446100\pi$
$293$	5.76934	0.337048	0.168524	+	0.985698i	$0.446100\pi$
$294$	−0.891542	−0.0519958
$295$	6.43563	0.374697
$296$	−7.44990	−0.433017
$297$	4.95063	0.287265
$298$	0.0792983	0.00459363
$299$	−5.31877	−0.307592
$300$	8.96313	0.517486
$301$	−12.3594	−0.712383
$302$	2.02931	0.116774
$303$	−18.5456	−1.06542
$304$	−3.52325	−0.202072
$305$	−1.89001	−0.108222
$306$	0	0
$307$	−13.2375	−0.755502	−0.377751	−	0.925907i	$-0.623302\pi$
$307$	−13.2375	−0.755502	−0.377751	+	0.925907i	$0.623302\pi$
$308$	16.6913	0.951074
$309$	3.39501	0.193136
$310$	−0.745595	−0.0423469
$311$	−1.29530	−0.0734499	−0.0367250	−	0.999325i	$-0.511693\pi$
$311$	−1.29530	−0.0734499	−0.0367250	+	0.999325i	$0.511693\pi$
$312$	−2.19998	−0.124549
$313$	−19.1007	−1.07964	−0.539818	−	0.841782i	$-0.681507\pi$
$313$	−19.1007	−1.07964	−0.539818	+	0.841782i	$0.681507\pi$
$314$	3.91001	0.220655
$315$	1.09999	0.0619773
$316$	4.62845	0.260371
$317$	15.1043	0.848339	0.424170	−	0.905583i	$-0.360566\pi$
$317$	15.1043	0.848339	0.424170	+	0.905583i	$0.360566\pi$
$318$	−1.43189	−0.0802962
$319$	47.4619	2.65735
$320$	4.35263	0.243319
$321$	−13.1531	−0.734135
$322$	0.815020	0.0454193
$323$	0	0
$324$	−1.95063	−0.108368
$325$	−11.5162	−0.638803
$326$	3.68617	0.204158
$327$	−17.3896	−0.961645
$328$	−6.07585	−0.335483
$329$	−14.0025	−0.771983
$330$	−0.700037	−0.0385358
$331$	−22.6769	−1.24644	−0.623218	−	0.782048i	$-0.714175\pi$
$331$	−22.6769	−1.24644	−0.623218	+	0.782048i	$0.714175\pi$
$332$	26.0568	1.43005
$333$	8.48705	0.465087
$334$	−3.16033	−0.172925
$335$	−4.90114	−0.267778
$336$	−6.40598	−0.349475
$337$	−1.61669	−0.0880666	−0.0440333	−	0.999030i	$-0.514021\pi$
$337$	−1.61669	−0.0880666	−0.0440333	+	0.999030i	$0.514021\pi$
$338$	−1.49284	−0.0811999
$339$	0.776042	0.0421488
$340$	0	0
$341$	26.1037	1.41360
$342$	−0.211222	−0.0114216
$343$	19.0344	1.02776
$344$	−6.27677	−0.338421
$345$	1.35058	0.0727128
$346$	−0.420677	−0.0226157
$347$	−15.3606	−0.824602	−0.412301	−	0.911048i	$-0.635275\pi$
$347$	−15.3606	−0.824602	−0.412301	+	0.911048i	$0.635275\pi$
$348$	−18.7008	−1.00247
$349$	−15.1087	−0.808749	−0.404374	−	0.914594i	$-0.632511\pi$
$349$	−15.1087	−0.808749	−0.404374	+	0.914594i	$0.632511\pi$
$350$	1.76468	0.0943261
$351$	2.50625	0.133774
$352$	12.7681	0.680541
$353$	−31.2482	−1.66317	−0.831586	−	0.555396i	$-0.812567\pi$
$353$	−31.2482	−1.66317	−0.831586	+	0.555396i	$0.812567\pi$
$354$	−2.24691	−0.119422
$355$	0.244408	0.0129718
$356$	3.87932	0.205604
$357$	0	0
$358$	−4.94307	−0.261249
$359$	2.72503	0.143822	0.0719108	−	0.997411i	$-0.477090\pi$
$359$	2.72503	0.143822	0.0719108	+	0.997411i	$0.477090\pi$
$360$	0.558634	0.0294426
$361$	−18.0963	−0.952437
$362$	−2.58057	−0.135632
$363$	13.5087	0.709025
$364$	8.44994	0.442897
$365$	−4.14567	−0.216994
$366$	0.659871	0.0344920
$367$	−18.4515	−0.963163	−0.481581	−	0.876401i	$-0.659937\pi$
$367$	−18.4515	−0.963163	−0.481581	+	0.876401i	$0.659937\pi$
$368$	−7.86536	−0.410010
$369$	6.92171	0.360330
$370$	−1.20010	−0.0623902
$371$	11.1387	0.578294
$372$	−10.2853	−0.533268
$373$	24.4049	1.26364	0.631820	−	0.775115i	$-0.282308\pi$
$373$	24.4049	1.26364	0.631820	+	0.775115i	$0.282308\pi$
$374$	0	0
$375$	6.10630	0.315328
$376$	−7.11123	−0.366734
$377$	24.0275	1.23748
$378$	−0.384044	−0.0197531
$379$	−5.89057	−0.302578	−0.151289	−	0.988490i	$-0.548342\pi$
$379$	−5.89057	−0.302578	−0.151289	+	0.988490i	$0.548342\pi$
$380$	−1.18010	−0.0605381
$381$	10.8519	0.555959
$382$	1.47382	0.0754072
$383$	30.7756	1.57256	0.786279	−	0.617871i	$-0.212005\pi$
$383$	30.7756	1.57256	0.786279	+	0.617871i	$0.212005\pi$
$384$	−6.67782	−0.340776
$385$	5.44563	0.277535
$386$	0.648284	0.0329968
$387$	7.15061	0.363486
$388$	−5.93268	−0.301186
$389$	19.5244	0.989925	0.494963	−	0.868914i	$-0.335182\pi$
$389$	19.5244	0.989925	0.494963	+	0.868914i	$0.335182\pi$
$390$	−0.354393	−0.0179454
$391$	0	0
$392$	3.52215	0.177896
$393$	0.706223	0.0356243
$394$	1.00768	0.0507659
$395$	1.51006	0.0759794
$396$	−9.65685	−0.485275
$397$	−0.904279	−0.0453844	−0.0226922	−	0.999742i	$-0.507224\pi$
$397$	−0.904279	−0.0453844	−0.0226922	+	0.999742i	$0.507224\pi$
$398$	0.546655	0.0274013
$399$	1.64311	0.0822583
$400$	−17.0301	−0.851503
$401$	1.26486	0.0631639	0.0315820	−	0.999501i	$-0.489945\pi$
$401$	1.26486	0.0631639	0.0315820	+	0.999501i	$0.489945\pi$
$402$	1.71116	0.0853449
$403$	13.2150	0.658285
$404$	36.1757	1.79981
$405$	−0.636405	−0.0316232
$406$	−3.68185	−0.182727
$407$	42.0162	2.08267
$408$	0	0
$409$	28.6350	1.41591	0.707954	−	0.706258i	$-0.249618\pi$
$409$	28.6350	1.41591	0.707954	+	0.706258i	$0.249618\pi$
$410$	−0.978756	−0.0483373
$411$	12.3125	0.607329
$412$	−6.62242	−0.326263
$413$	17.4788	0.860076
$414$	−0.471535	−0.0231747
$415$	8.50119	0.417307
$416$	6.46382	0.316915
$417$	−13.3444	−0.653478
$418$	−1.04568	−0.0511460
$419$	23.9581	1.17043	0.585214	−	0.810879i	$-0.301010\pi$
$419$	23.9581	1.17043	0.585214	+	0.810879i	$0.301010\pi$
$420$	−2.14567	−0.104698
$421$	−12.5312	−0.610735	−0.305368	−	0.952235i	$-0.598779\pi$
$421$	−12.5312	−0.610735	−0.305368	+	0.952235i	$0.598779\pi$
$422$	4.42244	0.215281
$423$	8.10124	0.393896
$424$	5.65685	0.274721
$425$	0	0
$426$	−0.0853313	−0.00413432
$427$	−5.13317	−0.248412
$428$	25.6569	1.24017
$429$	12.4075	0.599040
$430$	−1.01112	−0.0487606
$431$	−17.6728	−0.851267	−0.425633	−	0.904896i	$-0.639949\pi$
$431$	−17.6728	−0.851267	−0.425633	+	0.904896i	$0.639949\pi$
$432$	3.70622	0.178316
$433$	−14.0076	−0.673160	−0.336580	−	0.941655i	$-0.609270\pi$
$433$	−14.0076	−0.673160	−0.336580	+	0.941655i	$0.609270\pi$
$434$	−2.02499	−0.0972028
$435$	−6.10124	−0.292532
$436$	33.9206	1.62450
$437$	2.01743	0.0965069
$438$	1.44740	0.0691594
$439$	−11.5740	−0.552398	−0.276199	−	0.961100i	$-0.589075\pi$
$439$	−11.5740	−0.552398	−0.276199	+	0.961100i	$0.589075\pi$
$440$	2.76559	0.131844
$441$	−4.01250	−0.191071
$442$	0	0
$443$	−35.6174	−1.69223	−0.846117	−	0.532997i	$-0.821066\pi$
$443$	−35.6174	−1.69223	−0.846117	+	0.532997i	$0.821066\pi$
$444$	−16.5551	−0.785670
$445$	1.26565	0.0599977
$446$	3.19135	0.151115
$447$	0.356892	0.0168804
$448$	11.8215	0.558513
$449$	−22.1461	−1.04514	−0.522569	−	0.852597i	$-0.675026\pi$
$449$	−22.1461	−1.04514	−0.522569	+	0.852597i	$0.675026\pi$
$450$	−1.02097	−0.0481288
$451$	34.2668	1.61356
$452$	−1.51377	−0.0712018
$453$	9.13317	0.429114
$454$	−2.41619	−0.113398
$455$	2.75684	0.129243
$456$	0.834460	0.0390772
$457$	9.90632	0.463398	0.231699	−	0.972787i	$-0.425572\pi$
$457$	9.90632	0.463398	0.231699	+	0.972787i	$0.425572\pi$
$458$	4.21941	0.197160
$459$	0	0
$460$	−2.63449	−0.122834
$461$	−9.62367	−0.448219	−0.224109	−	0.974564i	$-0.571947\pi$
$461$	−9.62367	−0.448219	−0.224109	+	0.974564i	$0.571947\pi$
$462$	−1.90126	−0.0884547
$463$	15.3013	0.711113	0.355557	−	0.934655i	$-0.384291\pi$
$463$	15.3013	0.711113	0.355557	+	0.934655i	$0.384291\pi$
$464$	35.5317	1.64952
$465$	−3.35564	−0.155614
$466$	1.57329	0.0728812
$467$	−22.3150	−1.03261	−0.516307	−	0.856404i	$-0.672693\pi$
$467$	−22.3150	−1.03261	−0.516307	+	0.856404i	$0.672693\pi$
$468$	−4.88877	−0.225983
$469$	−13.3112	−0.614655
$470$	−1.14554	−0.0528400
$471$	17.5975	0.810849
$472$	8.87670	0.408583
$473$	35.4000	1.62769
$474$	−0.527215	−0.0242158
$475$	4.36814	0.200424
$476$	0	0
$477$	−6.44438	−0.295068
$478$	−1.17511	−0.0537481
$479$	−15.5571	−0.710824	−0.355412	−	0.934710i	$-0.615659\pi$
$479$	−15.5571	−0.710824	−0.355412	+	0.934710i	$0.615659\pi$
$480$	−1.64134	−0.0749166
$481$	21.2707	0.969858
$482$	1.28832	0.0586815
$483$	3.66810	0.166904
$484$	−26.3506	−1.19775
$485$	−1.93557	−0.0878898
$486$	0.222191	0.0100788
$487$	−0.0296516	−0.00134364	−0.000671822	−	1.00000i	$-0.500214\pi$
$487$	−0.0296516	−0.00134364	−0.000671822		1.00000i	$0.500214\pi$
$488$	−2.60691	−0.118009
$489$	16.5901	0.750228
$490$	0.567382	0.0256317
$491$	6.66935	0.300984	0.150492	−	0.988611i	$-0.451914\pi$
$491$	6.66935	0.300984	0.150492	+	0.988611i	$0.451914\pi$
$492$	−13.5017	−0.608704
$493$	0	0
$494$	−0.529375	−0.0238177
$495$	−3.15061	−0.141609
$496$	19.5422	0.877471
$497$	0.663798	0.0297754
$498$	−2.96806	−0.133002
$499$	−18.6469	−0.834748	−0.417374	−	0.908735i	$-0.637049\pi$
$499$	−18.6469	−0.834748	−0.417374	+	0.908735i	$0.637049\pi$
$500$	−11.9111	−0.532682
$501$	−14.2234	−0.635456
$502$	−1.85128	−0.0826265
$503$	25.7396	1.14767	0.573837	−	0.818970i	$-0.305454\pi$
$503$	25.7396	1.14767	0.573837	+	0.818970i	$0.305454\pi$
$504$	1.51722	0.0675822
$505$	11.8025	0.525205
$506$	−2.33440	−0.103777
$507$	−6.71872	−0.298389
$508$	−21.1680	−0.939180
$509$	41.1950	1.82594	0.912968	−	0.408032i	$-0.133785\pi$
$509$	41.1950	1.82594	0.912968	+	0.408032i	$0.133785\pi$
$510$	0	0
$511$	−11.2594	−0.498087
$512$	16.0653	0.709992
$513$	−0.950631	−0.0419714
$514$	−0.945570	−0.0417073
$515$	−2.16060	−0.0952076
$516$	−13.9482	−0.614035
$517$	40.1062	1.76387
$518$	−3.25940	−0.143210
$519$	−1.89331	−0.0831070
$520$	1.40007	0.0613973
$521$	8.98205	0.393511	0.196755	−	0.980453i	$-0.436960\pi$
$521$	8.98205	0.393511	0.196755	+	0.980453i	$0.436960\pi$
$522$	2.13016	0.0932344
$523$	−5.38995	−0.235686	−0.117843	−	0.993032i	$-0.537598\pi$
$523$	−5.38995	−0.235686	−0.117843	+	0.993032i	$0.537598\pi$
$524$	−1.37758	−0.0601799
$525$	7.94216	0.346624
$526$	−2.02194	−0.0881607
$527$	0	0
$528$	18.3481	0.798500
$529$	−18.4963	−0.804185
$530$	0.911259	0.0395826
$531$	−10.1125	−0.438844
$532$	−3.20510	−0.138959
$533$	17.3475	0.751405
$534$	−0.441884	−0.0191222
$535$	8.37070	0.361897
$536$	−6.76016	−0.291994
$537$	−22.2469	−0.960025
$538$	4.75090	0.204826
$539$	−19.8644	−0.855620
$540$	1.24139	0.0534210
$541$	1.22765	0.0527806	0.0263903	−	0.999652i	$-0.491599\pi$
$541$	1.22765	0.0527806	0.0263903	+	0.999652i	$0.491599\pi$
$542$	−4.29302	−0.184401
$543$	−11.6142	−0.498413
$544$	0	0
$545$	11.0668	0.474050
$546$	−0.962511	−0.0411916
$547$	13.5274	0.578391	0.289196	−	0.957270i	$-0.406612\pi$
$547$	13.5274	0.578391	0.289196	+	0.957270i	$0.406612\pi$
$548$	−24.0171	−1.02596
$549$	2.96983	0.126749
$550$	−5.05443	−0.215522
$551$	−9.11373	−0.388258
$552$	1.86286	0.0792887
$553$	4.10124	0.174402
$554$	0.742148	0.0315308
$555$	−5.40120	−0.229268
$556$	26.0300	1.10392
$557$	−18.3599	−0.777936	−0.388968	−	0.921251i	$-0.627168\pi$
$557$	−18.3599	−0.777936	−0.388968	+	0.921251i	$0.627168\pi$
$558$	1.17157	0.0495966
$559$	17.9212	0.757986
$560$	4.07680	0.172276
$561$	0	0
$562$	1.03931	0.0438406
$563$	3.33315	0.140475	0.0702377	−	0.997530i	$-0.477624\pi$
$563$	3.33315	0.140475	0.0702377	+	0.997530i	$0.477624\pi$
$564$	−15.8025	−0.665406
$565$	−0.493877	−0.0207775
$566$	2.23484	0.0939374
$567$	−1.72844	−0.0725876
$568$	0.337113	0.0141449
$569$	29.4369	1.23406	0.617029	−	0.786940i	$-0.288336\pi$
$569$	29.4369	1.23406	0.617029	+	0.786940i	$0.288336\pi$
$570$	0.134423	0.00563035
$571$	−25.8275	−1.08085	−0.540424	−	0.841393i	$-0.681736\pi$
$571$	−25.8275	−1.08085	−0.540424	+	0.841393i	$0.681736\pi$
$572$	−24.2025	−1.01196
$573$	6.63311	0.277102
$574$	−2.65824	−0.110953
$575$	9.75150	0.406666
$576$	−6.83940	−0.284975
$577$	−22.1850	−0.923575	−0.461788	−	0.886990i	$-0.652792\pi$
$577$	−22.1850	−0.923575	−0.461788	+	0.886990i	$0.652792\pi$
$578$	0	0
$579$	2.91769	0.121255
$580$	11.9013	0.494173
$581$	23.0887	0.957882
$582$	0.675776	0.0280118
$583$	−31.9038	−1.32132
$584$	−5.71814	−0.236618
$585$	−1.59499	−0.0659447
$586$	1.28190	0.0529547
$587$	32.1482	1.32690	0.663448	−	0.748222i	$-0.269092\pi$
$587$	32.1482	1.32690	0.663448	+	0.748222i	$0.269092\pi$
$588$	7.82690	0.322776
$589$	−5.01250	−0.206536
$590$	1.42994	0.0588698
$591$	4.53517	0.186552
$592$	31.4549	1.29279
$593$	−29.3406	−1.20488	−0.602438	−	0.798166i	$-0.705804\pi$
$593$	−29.3406	−1.20488	−0.602438	+	0.798166i	$0.705804\pi$
$594$	1.09999	0.0451330
$595$	0	0
$596$	−0.696164	−0.0285160
$597$	2.46029	0.100693
$598$	−1.18178	−0.0483268
$599$	−36.2632	−1.48167	−0.740836	−	0.671686i	$-0.765570\pi$
$599$	−36.2632	−1.48167	−0.740836	+	0.671686i	$0.765570\pi$
$600$	4.03346	0.164665
$601$	45.7383	1.86570	0.932851	−	0.360262i	$-0.117313\pi$
$601$	45.7383	1.86570	0.932851	+	0.360262i	$0.117313\pi$
$602$	−2.74615	−0.111925
$603$	7.70129	0.313621
$604$	−17.8155	−0.724900
$605$	−8.59703	−0.349519
$606$	−4.12068	−0.167391
$607$	−33.6640	−1.36638	−0.683190	−	0.730241i	$-0.739408\pi$
$607$	−33.6640	−1.36638	−0.683190	+	0.730241i	$0.739408\pi$
$608$	−2.45176	−0.0994318
$609$	−16.5706	−0.671475
$610$	−0.419945	−0.0170031
$611$	20.3037	0.821400
$612$	0	0
$613$	−5.89120	−0.237943	−0.118972	−	0.992898i	$-0.537960\pi$
$613$	−5.89120	−0.237943	−0.118972	+	0.992898i	$0.537960\pi$
$614$	−2.94125	−0.118699
$615$	−4.40501	−0.177627
$616$	7.51118	0.302634
$617$	−35.2179	−1.41782	−0.708909	−	0.705300i	$-0.750812\pi$
$617$	−35.2179	−1.41782	−0.708909	+	0.705300i	$0.750812\pi$
$618$	0.754343	0.0303441
$619$	−32.8288	−1.31950	−0.659750	−	0.751485i	$-0.729338\pi$
$619$	−32.8288	−1.31950	−0.659750	+	0.751485i	$0.729338\pi$
$620$	6.54562	0.262878
$621$	−2.12220	−0.0851611
$622$	−0.287805	−0.0115399
$623$	3.43744	0.137718
$624$	9.28872	0.371846
$625$	19.0889	0.763555
$626$	−4.24402	−0.169625
$627$	−4.70622	−0.187948
$628$	−34.3262	−1.36976
$629$	0	0
$630$	0.244408	0.00973744
$631$	10.7057	0.426186	0.213093	−	0.977032i	$-0.431646\pi$
$631$	10.7057	0.426186	0.213093	+	0.977032i	$0.431646\pi$
$632$	2.08283	0.0828506
$633$	19.9038	0.791103
$634$	3.35603	0.133285
$635$	−6.90620	−0.274064
$636$	12.5706	0.498457
$637$	−10.0563	−0.398446
$638$	10.5456	0.417505
$639$	−0.384044	−0.0151926
$640$	4.24980	0.167988
$641$	−36.9437	−1.45919	−0.729593	−	0.683881i	$-0.760291\pi$
$641$	−36.9437	−1.45919	−0.729593	+	0.683881i	$0.760291\pi$
$642$	−2.92251	−0.115342
$643$	−39.3137	−1.55038	−0.775191	−	0.631727i	$-0.782346\pi$
$643$	−39.3137	−1.55038	−0.775191	+	0.631727i	$0.782346\pi$
$644$	−7.15511	−0.281951
$645$	−4.55068	−0.179183
$646$	0	0
$647$	−14.8806	−0.585016	−0.292508	−	0.956263i	$-0.594490\pi$
$647$	−14.8806	−0.585016	−0.292508	+	0.956263i	$0.594490\pi$
$648$	−0.877796	−0.0344831
$649$	−50.0632	−1.96515
$650$	−2.55880	−0.100364
$651$	−9.11373	−0.357195
$652$	−32.3611	−1.26736
$653$	24.2908	0.950571	0.475285	−	0.879832i	$-0.342345\pi$
$653$	24.2908	0.950571	0.475285	+	0.879832i	$0.342345\pi$
$654$	−3.86381	−0.151087
$655$	−0.449444	−0.0175612
$656$	25.6534	1.00160
$657$	6.51420	0.254143
$658$	−3.11123	−0.121289
$659$	−9.14192	−0.356119	−0.178059	−	0.984020i	$-0.556982\pi$
$659$	−9.14192	−0.356119	−0.178059	+	0.984020i	$0.556982\pi$
$660$	6.14567	0.239220
$661$	−16.7951	−0.653253	−0.326627	−	0.945153i	$-0.605912\pi$
$661$	−16.7951	−0.653253	−0.326627	+	0.945153i	$0.605912\pi$
$662$	−5.03861	−0.195831
$663$	0	0
$664$	11.7257	0.455046
$665$	−1.04568	−0.0405498
$666$	1.88575	0.0730713
$667$	−20.3456	−0.787787
$668$	27.7447	1.07347
$669$	14.3631	0.555309
$670$	−1.08899	−0.0420714
$671$	14.7025	0.567585
$672$	−4.45779	−0.171963
$673$	18.6313	0.718186	0.359093	−	0.933302i	$-0.383086\pi$
$673$	18.6313	0.718186	0.359093	+	0.933302i	$0.383086\pi$
$674$	−0.359214	−0.0138364
$675$	−4.59499	−0.176861
$676$	13.1057	0.504067
$677$	10.8010	0.415117	0.207559	−	0.978223i	$-0.433448\pi$
$677$	10.8010	0.415117	0.207559	+	0.978223i	$0.433448\pi$
$678$	0.172430	0.00662212
$679$	−5.25690	−0.201741
$680$	0	0
$681$	−10.8744	−0.416707
$682$	5.80002	0.222094
$683$	−22.2199	−0.850221	−0.425111	−	0.905141i	$-0.639765\pi$
$683$	−22.2199	−0.850221	−0.425111	+	0.905141i	$0.639765\pi$
$684$	1.85433	0.0709021
$685$	−7.83571	−0.299387
$686$	4.22929	0.161475
$687$	18.9900	0.724514
$688$	26.5017	1.01037
$689$	−16.1512	−0.615313
$690$	0.300087	0.0114241
$691$	−38.5413	−1.46618	−0.733090	−	0.680132i	$-0.761923\pi$
$691$	−38.5413	−1.46618	−0.733090	+	0.680132i	$0.761923\pi$
$692$	3.69315	0.140392
$693$	−8.55687	−0.325048
$694$	−3.41300	−0.129556
$695$	8.49244	0.322137
$696$	−8.41546	−0.318987
$697$	0	0
$698$	−3.35702	−0.127065
$699$	7.08079	0.267820
$700$	−15.4922	−0.585551
$701$	18.0843	0.683034	0.341517	−	0.939876i	$-0.389059\pi$
$701$	18.0843	0.683034	0.341517	+	0.939876i	$0.389059\pi$
$702$	0.556867	0.0210176
$703$	−8.06805	−0.304292
$704$	−33.8593	−1.27612
$705$	−5.15567	−0.194174
$706$	−6.94307	−0.261306
$707$	32.0550	1.20555
$708$	19.7257	0.741338
$709$	52.2614	1.96272	0.981359	−	0.192185i	$-0.0615573\pi$
$709$	52.2614	1.96272	0.981359	+	0.192185i	$0.0615573\pi$
$710$	0.0543053	0.00203804
$711$	−2.37280	−0.0889869
$712$	1.74572	0.0654236
$713$	−11.1900	−0.419068
$714$	0	0
$715$	−7.89620	−0.295301
$716$	43.3955	1.62177
$717$	−5.28872	−0.197511
$718$	0.605478	0.0225962
$719$	2.25831	0.0842206	0.0421103	−	0.999113i	$-0.486592\pi$
$719$	2.25831	0.0842206	0.0421103	+	0.999113i	$0.486592\pi$
$720$	−2.35866	−0.0879020
$721$	−5.86808	−0.218539
$722$	−4.02084	−0.149640
$723$	5.79826	0.215640
$724$	22.6550	0.841966
$725$	−44.0523	−1.63606
$726$	3.00153	0.111397
$727$	−39.5601	−1.46720	−0.733601	−	0.679581i	$-0.762162\pi$
$727$	−39.5601	−1.46720	−0.733601	+	0.679581i	$0.762162\pi$
$728$	3.80252	0.140931
$729$	1.00000	0.0370370
$730$	−0.921132	−0.0340926
$731$	0	0
$732$	−5.79304	−0.214117
$733$	−13.5137	−0.499139	−0.249570	−	0.968357i	$-0.580289\pi$
$733$	−13.5137	−0.499139	−0.249570	+	0.968357i	$0.580289\pi$
$734$	−4.09977	−0.151325
$735$	2.55357	0.0941899
$736$	−5.47334	−0.201750
$737$	38.1262	1.40440
$738$	1.53794	0.0566125
$739$	−10.7282	−0.394642	−0.197321	−	0.980339i	$-0.563224\pi$
$739$	−10.7282	−0.394642	−0.197321	+	0.980339i	$0.563224\pi$
$740$	10.5357	0.387302
$741$	−2.38252	−0.0875240
$742$	2.47493	0.0908575
$743$	−43.9921	−1.61392	−0.806958	−	0.590609i	$-0.798887\pi$
$743$	−43.9921	−1.61392	−0.806958	+	0.590609i	$0.798887\pi$
$744$	−4.62845	−0.169687
$745$	−0.227128	−0.00832131
$746$	5.42257	0.198534
$747$	−13.3581	−0.488749
$748$	0	0
$749$	22.7343	0.830695
$750$	1.35677	0.0495421
$751$	−12.5047	−0.456304	−0.228152	−	0.973626i	$-0.573268\pi$
$751$	−12.5047	−0.456304	−0.228152	+	0.973626i	$0.573268\pi$
$752$	30.0250	1.09490
$753$	−8.33190	−0.303631
$754$	5.33870	0.194424
$755$	−5.81240	−0.211535
$756$	3.37155	0.122622
$757$	−35.8474	−1.30290	−0.651449	−	0.758693i	$-0.725838\pi$
$757$	−35.8474	−1.30290	−0.651449	+	0.758693i	$0.725838\pi$
$758$	−1.30883	−0.0475390
$759$	−10.5062	−0.381353
$760$	−0.531055	−0.0192634
$761$	−23.4468	−0.849944	−0.424972	−	0.905206i	$-0.639716\pi$
$761$	−23.4468	−0.849944	−0.424972	+	0.905206i	$0.639716\pi$
$762$	2.41120	0.0873485
$763$	30.0568	1.08813
$764$	−12.9388	−0.468108
$765$	0	0
$766$	6.83807	0.247070
$767$	−25.3444	−0.915133
$768$	12.1950	0.440051
$769$	22.8938	0.825573	0.412786	−	0.910828i	$-0.364556\pi$
$769$	22.8938	0.825573	0.412786	+	0.910828i	$0.364556\pi$
$770$	1.20997	0.0436044
$771$	−4.25565	−0.153264
$772$	−5.69133	−0.204835
$773$	34.1287	1.22753	0.613763	−	0.789491i	$-0.289655\pi$
$773$	34.1287	1.22753	0.613763	+	0.789491i	$0.289655\pi$
$774$	1.58880	0.0571083
$775$	−24.2285	−0.870313
$776$	−2.66974	−0.0958382
$777$	−14.6694	−0.526260
$778$	4.33815	0.155530
$779$	−6.57999	−0.235753
$780$	3.11123	0.111400
$781$	−1.90126	−0.0680325
$782$	0	0
$783$	9.58704	0.342613
$784$	−14.8712	−0.531115
$785$	−11.1991	−0.399714
$786$	0.156917	0.00559703
$787$	13.1730	0.469568	0.234784	−	0.972048i	$-0.424562\pi$
$787$	13.1730	0.469568	0.234784	+	0.972048i	$0.424562\pi$
$788$	−8.84644	−0.315141
$789$	−9.09999	−0.323968
$790$	0.335522	0.0119373
$791$	−1.34134	−0.0476926
$792$	−4.34564	−0.154416
$793$	7.44313	0.264313
$794$	−0.200923	−0.00713049
$795$	4.10124	0.145456
$796$	−4.79911	−0.170100
$797$	36.3506	1.28761	0.643803	−	0.765191i	$-0.277356\pi$
$797$	36.3506	1.28761	0.643803	+	0.765191i	$0.277356\pi$
$798$	0.365084	0.0129239
$799$	0	0
$800$	−11.8509	−0.418991
$801$	−1.98875	−0.0702691
$802$	0.281040	0.00992388
$803$	32.2494	1.13806
$804$	−15.0224	−0.529798
$805$	−2.33440	−0.0822767
$806$	2.93625	0.103425
$807$	21.3820	0.752682
$808$	16.2793	0.572703
$809$	−45.1267	−1.58657	−0.793285	−	0.608851i	$-0.791631\pi$
$809$	−45.1267	−1.58657	−0.793285	+	0.608851i	$0.791631\pi$
$810$	−0.141404	−0.00496842
$811$	−3.32815	−0.116867	−0.0584336	−	0.998291i	$-0.518611\pi$
$811$	−3.32815	−0.116867	−0.0584336	+	0.998291i	$0.518611\pi$
$812$	32.3232	1.13432
$813$	−19.3213	−0.677626
$814$	9.33565	0.327214
$815$	−10.5580	−0.369830
$816$	0	0
$817$	−6.79759	−0.237817
$818$	6.36244	0.222458
$819$	−4.33190	−0.151369
$820$	8.59255	0.300065
$821$	−7.46983	−0.260699	−0.130349	−	0.991468i	$-0.541610\pi$
$821$	−7.46983	−0.260699	−0.130349	+	0.991468i	$0.541610\pi$
$822$	2.73572	0.0954193
$823$	54.1834	1.88871	0.944357	−	0.328923i	$-0.106686\pi$
$823$	54.1834	1.88871	0.944357	+	0.328923i	$0.106686\pi$
$824$	−2.98013	−0.103818
$825$	−22.7481	−0.791987
$826$	3.88364	0.135129
$827$	−4.95063	−0.172150	−0.0860752	−	0.996289i	$-0.527433\pi$
$827$	−4.95063	−0.172150	−0.0860752	+	0.996289i	$0.527433\pi$
$828$	4.13964	0.143862
$829$	−0.0762440	−0.00264806	−0.00132403	−	0.999999i	$-0.500421\pi$
$829$	−0.0762440	−0.00264806	−0.00132403	+	0.999999i	$0.500421\pi$
$830$	1.88889	0.0655643
$831$	3.34013	0.115868
$832$	−17.1412	−0.594265
$833$	0	0
$834$	−2.96501	−0.102670
$835$	9.05187	0.313253
$836$	9.18010	0.317501
$837$	5.27281	0.182255
$838$	5.32328	0.183890
$839$	−36.4479	−1.25832	−0.629160	−	0.777276i	$-0.716601\pi$
$839$	−36.4479	−1.25832	−0.629160	+	0.777276i	$0.716601\pi$
$840$	−0.965565	−0.0333151
$841$	62.9113	2.16935
$842$	−2.78433	−0.0959545
$843$	4.67754	0.161103
$844$	−38.8249	−1.33641
$845$	4.27583	0.147093
$846$	1.80002	0.0618861
$847$	−23.3491	−0.802283
$848$	−23.8843	−0.820191
$849$	10.0582	0.345196
$850$	0	0
$851$	−18.0112	−0.617418
$852$	0.749129	0.0256647
$853$	39.9684	1.36849	0.684245	−	0.729252i	$-0.260132\pi$
$853$	39.9684	1.36849	0.684245	+	0.729252i	$0.260132\pi$
$854$	−1.14055	−0.0390287
$855$	0.604986	0.0206901
$856$	11.5457	0.394625
$857$	−2.95962	−0.101099	−0.0505493	−	0.998722i	$-0.516097\pi$
$857$	−2.95962	−0.101099	−0.0505493	+	0.998722i	$0.516097\pi$
$858$	2.75684	0.0941170
$859$	−22.4494	−0.765963	−0.382981	−	0.923756i	$-0.625103\pi$
$859$	−22.4494	−0.765963	−0.382981	+	0.923756i	$0.625103\pi$
$860$	8.87670	0.302693
$861$	−11.9638	−0.407724
$862$	−3.92673	−0.133745
$863$	−27.8981	−0.949661	−0.474831	−	0.880077i	$-0.657491\pi$
$863$	−27.8981	−0.949661	−0.474831	+	0.880077i	$0.657491\pi$
$864$	2.57908	0.0877422
$865$	1.20491	0.0409682
$866$	−3.11236	−0.105762
$867$	0	0
$868$	17.7775	0.603409
$869$	−11.7468	−0.398484
$870$	−1.35564	−0.0459606
$871$	19.3013	0.654001
$872$	15.2645	0.516921
$873$	3.04142	0.102936
$874$	0.448256	0.0151625
$875$	−10.5544	−0.356803
$876$	−12.7068	−0.429323
$877$	20.6961	0.698858	0.349429	−	0.936963i	$-0.386376\pi$
$877$	20.6961	0.698858	0.349429	+	0.936963i	$0.386376\pi$
$878$	−2.57165	−0.0867889
$879$	5.76934	0.194595
$880$	−11.6768	−0.393627
$881$	−22.2399	−0.749282	−0.374641	−	0.927170i	$-0.622234\pi$
$881$	−22.2399	−0.749282	−0.374641	+	0.927170i	$0.622234\pi$
$882$	−0.891542	−0.0300198
$883$	4.23253	0.142436	0.0712180	−	0.997461i	$-0.477311\pi$
$883$	4.23253	0.142436	0.0712180	+	0.997461i	$0.477311\pi$
$884$	0	0
$885$	6.43563	0.216331
$886$	−7.91388	−0.265872
$887$	−36.3047	−1.21899	−0.609496	−	0.792789i	$-0.708628\pi$
$887$	−36.3047	−1.21899	−0.609496	+	0.792789i	$0.708628\pi$
$888$	−7.44990	−0.250002
$889$	−18.7568	−0.629084
$890$	0.281217	0.00942642
$891$	4.95063	0.165852
$892$	−28.0171	−0.938081
$893$	−7.70129	−0.257714
$894$	0.0792983	0.00265213
$895$	14.1580	0.473251
$896$	11.5422	0.385598
$897$	−5.31877	−0.177589
$898$	−4.92066	−0.164205
$899$	50.5506	1.68596
$900$	8.96313	0.298771
$901$	0	0
$902$	7.61380	0.253512
$903$	−12.3594	−0.411295
$904$	−0.681206	−0.0226566
$905$	7.39133	0.245696
$906$	2.02931	0.0674194
$907$	24.3718	0.809251	0.404626	−	0.914482i	$-0.367402\pi$
$907$	24.3718	0.809251	0.404626	+	0.914482i	$0.367402\pi$
$908$	21.2119	0.703942
$909$	−18.5456	−0.615119
$910$	0.612546	0.0203057
$911$	26.9178	0.891826	0.445913	−	0.895076i	$-0.352879\pi$
$911$	26.9178	0.891826	0.445913	+	0.895076i	$0.352879\pi$
$912$	−3.52325	−0.116666
$913$	−66.1312	−2.18862
$914$	2.20110	0.0728059
$915$	−1.89001	−0.0624820
$916$	−37.0425	−1.22392
$917$	−1.22066	−0.0403099
$918$	0	0
$919$	−23.8000	−0.785088	−0.392544	−	0.919733i	$-0.628405\pi$
$919$	−23.8000	−0.785088	−0.392544	+	0.919733i	$0.628405\pi$
$920$	−1.18553	−0.0390859
$921$	−13.2375	−0.436189
$922$	−2.13830	−0.0704210
$923$	−0.962511	−0.0316814
$924$	16.6913	0.549103
$925$	−38.9979	−1.28224
$926$	3.39983	0.111725
$927$	3.39501	0.111507
$928$	24.7258	0.811663
$929$	−5.82742	−0.191191	−0.0955957	−	0.995420i	$-0.530476\pi$
$929$	−5.82742	−0.191191	−0.0955957	+	0.995420i	$0.530476\pi$
$930$	−0.745595	−0.0244490
$931$	3.81440	0.125012
$932$	−13.8120	−0.452427
$933$	−1.29530	−0.0424063
$934$	−4.95819	−0.162237
$935$	0	0
$936$	−2.19998	−0.0719084
$937$	55.4424	1.81123	0.905613	−	0.424106i	$-0.139411\pi$
$937$	55.4424	1.81123	0.905613	+	0.424106i	$0.139411\pi$
$938$	−2.95764	−0.0965702
$939$	−19.1007	−0.623328
$940$	10.0568	0.328017
$941$	−21.1696	−0.690109	−0.345054	−	0.938583i	$-0.612140\pi$
$941$	−21.1696	−0.690109	−0.345054	+	0.938583i	$0.612140\pi$
$942$	3.91001	0.127395
$943$	−14.6893	−0.478349
$944$	−37.4791	−1.21984
$945$	1.09999	0.0357826
$946$	7.86558	0.255732
$947$	37.5831	1.22129	0.610643	−	0.791906i	$-0.290911\pi$
$947$	37.5831	1.22129	0.610643	+	0.791906i	$0.290911\pi$
$948$	4.62845	0.150325
$949$	16.3262	0.529971
$950$	0.970563	0.0314892
$951$	15.1043	0.489789
$952$	0	0
$953$	21.2794	0.689307	0.344654	−	0.938730i	$-0.387996\pi$
$953$	21.2794	0.689307	0.344654	+	0.938730i	$0.387996\pi$
$954$	−1.43189	−0.0463590
$955$	−4.22134	−0.136599
$956$	10.3163	0.333654
$957$	47.4619	1.53422
$958$	−3.45666	−0.111680
$959$	−21.2813	−0.687210
$960$	4.35263	0.140480
$961$	−3.19748	−0.103144
$962$	4.72616	0.152377
$963$	−13.1531	−0.423853
$964$	−11.3103	−0.364279
$965$	−1.85683	−0.0597734
$966$	0.815020	0.0262229
$967$	−26.2320	−0.843563	−0.421782	−	0.906697i	$-0.638595\pi$
$967$	−26.2320	−0.843563	−0.421782	+	0.906697i	$0.638595\pi$
$968$	−11.8579	−0.381128
$969$	0	0
$970$	−0.430067	−0.0138086
$971$	−20.8243	−0.668285	−0.334142	−	0.942523i	$-0.608447\pi$
$971$	−20.8243	−0.668285	−0.334142	+	0.942523i	$0.608447\pi$
$972$	−1.95063	−0.0625665
$973$	23.0650	0.739429
$974$	−0.00658834	−0.000211104 0
$975$	−11.5162	−0.368813
$976$	11.0069	0.352321
$977$	16.2900	0.521162	0.260581	−	0.965452i	$-0.416086\pi$
$977$	16.2900	0.521162	0.260581	+	0.965452i	$0.416086\pi$
$978$	3.68617	0.117871
$979$	−9.84558	−0.314666
$980$	−4.98108	−0.159115
$981$	−17.3896	−0.555206
$982$	1.48187	0.0472884
$983$	0.931671	0.0297157	0.0148578	−	0.999890i	$-0.495270\pi$
$983$	0.931671	0.0297157	0.0148578	+	0.999890i	$0.495270\pi$
$984$	−6.07585	−0.193691
$985$	−2.88620	−0.0919621
$986$	0	0
$987$	−14.0025	−0.445704
$988$	4.64741	0.147854
$989$	−15.1750	−0.482538
$990$	−0.700037	−0.0222487
$991$	−21.6972	−0.689234	−0.344617	−	0.938743i	$-0.611991\pi$
$991$	−21.6972	−0.689234	−0.344617	+	0.938743i	$0.611991\pi$
$992$	13.5990	0.431769
$993$	−22.6769	−0.719630
$994$	0.147490	0.00467810
$995$	−1.56574	−0.0496373
$996$	26.0568	0.825641
$997$	−3.22709	−0.102203	−0.0511015	−	0.998693i	$-0.516273\pi$
$997$	−3.22709	−0.102203	−0.0511015	+	0.998693i	$0.516273\pi$
$998$	−4.14317	−0.131150
$999$	8.48705	0.268518

Display

a_p

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p

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a_n

instead) (See

a_n

instead) (See

a_n

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a_n

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up to: 50 250 1000 (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.a.l.1.3		4
3.2	odd	2		2601.2.a.be.1.2		4
17.2	even	8		867.2.e.g.616.3		8
17.3	odd	16		867.2.h.i.757.4		16
17.4	even	4		867.2.d.f.577.3		8
17.5	odd	16		867.2.h.k.688.1		16
17.6	odd	16		867.2.h.i.733.4		16
17.7	odd	16		867.2.h.k.712.1		16
17.8	even	8		51.2.e.a.13.2	yes	8
17.9	even	8		867.2.e.g.829.2		8
17.10	odd	16		867.2.h.k.712.2		16
17.11	odd	16		867.2.h.i.733.3		16
17.12	odd	16		867.2.h.k.688.2		16
17.13	even	4		867.2.d.f.577.4		8
17.14	odd	16		867.2.h.i.757.3		16
17.15	even	8		51.2.e.a.4.3	✓	8
17.16	even	2		867.2.a.k.1.3		4
51.8	odd	8		153.2.f.b.64.3		8
51.32	odd	8		153.2.f.b.55.2		8
51.50	odd	2		2601.2.a.bf.1.2		4
68.15	odd	8		816.2.bd.e.769.3		8
68.59	odd	8		816.2.bd.e.625.3		8
204.59	even	8		2448.2.be.x.1441.3		8
204.83	even	8		2448.2.be.x.1585.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.3	✓	8	17.15	even	8
51.2.e.a.13.2	yes	8	17.8	even	8
153.2.f.b.55.2		8	51.32	odd	8
153.2.f.b.64.3		8	51.8	odd	8
816.2.bd.e.625.3		8	68.59	odd	8
816.2.bd.e.769.3		8	68.15	odd	8
867.2.a.k.1.3		4	17.16	even	2
867.2.a.l.1.3		4	1.1	even	1	trivial
867.2.d.f.577.3		8	17.4	even	4
867.2.d.f.577.4		8	17.13	even	4
867.2.e.g.616.3		8	17.2	even	8
867.2.e.g.829.2		8	17.9	even	8
867.2.h.i.733.3		16	17.11	odd	16
867.2.h.i.733.4		16	17.6	odd	16
867.2.h.i.757.3		16	17.14	odd	16
867.2.h.i.757.4		16	17.3	odd	16
867.2.h.k.688.1		16	17.5	odd	16
867.2.h.k.688.2		16	17.12	odd	16
867.2.h.k.712.1		16	17.7	odd	16
867.2.h.k.712.2		16	17.10	odd	16
2448.2.be.x.1441.3		8	204.59	even	8
2448.2.be.x.1585.3		8	204.83	even	8
2601.2.a.be.1.2		4	3.2	odd	2
2601.2.a.bf.1.2		4	51.50	odd	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 $\F_{q}$
Belyi maps

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L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	867.2.a.l.1.3

Level	$867$
Weight	$2$
Character	867.1
Self dual	yes
Analytic conductor	$6.923$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$