LMFDB - Embedded newform 4018.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4018,2,Mod(1,4018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4018, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4018 = 2 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4018.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:	$32.0838915322$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 574)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} -2.73205 q^{11} -2.00000 q^{12} +5.46410 q^{13} +5.46410 q^{15} +1.00000 q^{16} +1.00000 q^{18} +3.46410 q^{19} -2.73205 q^{20} -2.73205 q^{22} -2.00000 q^{23} -2.00000 q^{24} +2.46410 q^{25} +5.46410 q^{26} +4.00000 q^{27} -7.66025 q^{29} +5.46410 q^{30} -1.46410 q^{31} +1.00000 q^{32} +5.46410 q^{33} +1.00000 q^{36} +4.92820 q^{37} +3.46410 q^{38} -10.9282 q^{39} -2.73205 q^{40} -1.00000 q^{41} +12.3923 q^{43} -2.73205 q^{44} -2.73205 q^{45} -2.00000 q^{46} -0.535898 q^{47} -2.00000 q^{48} +2.46410 q^{50} +5.46410 q^{52} +7.66025 q^{53} +4.00000 q^{54} +7.46410 q^{55} -6.92820 q^{57} -7.66025 q^{58} -13.1244 q^{59} +5.46410 q^{60} -6.73205 q^{61} -1.46410 q^{62} +1.00000 q^{64} -14.9282 q^{65} +5.46410 q^{66} +0.196152 q^{67} +4.00000 q^{69} +2.53590 q^{71} +1.00000 q^{72} +8.92820 q^{73} +4.92820 q^{74} -4.92820 q^{75} +3.46410 q^{76} -10.9282 q^{78} -2.73205 q^{80} -11.0000 q^{81} -1.00000 q^{82} -4.73205 q^{83} +12.3923 q^{86} +15.3205 q^{87} -2.73205 q^{88} -12.9282 q^{89} -2.73205 q^{90} -2.00000 q^{92} +2.92820 q^{93} -0.535898 q^{94} -9.46410 q^{95} -2.00000 q^{96} -2.00000 q^{97} -2.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 2 q^{20} - 2 q^{22} - 4 q^{23} - 4 q^{24} - 2 q^{25} + 4 q^{26} + 8 q^{27} + 2 q^{29} + 4 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} - 4 q^{37} - 8 q^{39} - 2 q^{40} - 2 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} - 4 q^{46} - 8 q^{47} - 4 q^{48} - 2 q^{50} + 4 q^{52} - 2 q^{53} + 8 q^{54} + 8 q^{55} + 2 q^{58} - 2 q^{59} + 4 q^{60} - 10 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} + 4 q^{66} - 10 q^{67} + 8 q^{69} + 12 q^{71} + 2 q^{72} + 4 q^{73} - 4 q^{74} + 4 q^{75} - 8 q^{78} - 2 q^{80} - 22 q^{81} - 2 q^{82} - 6 q^{83} + 4 q^{86} - 4 q^{87} - 2 q^{88} - 12 q^{89} - 2 q^{90} - 4 q^{92} - 8 q^{93} - 8 q^{94} - 12 q^{95} - 4 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^18 - 2 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 - 2 * q^25 + 4 * q^26 + 8 * q^27 + 2 * q^29 + 4 * q^30 + 4 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 8 * q^39 - 2 * q^40 - 2 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^45 - 4 * q^46 - 8 * q^47 - 4 * q^48 - 2 * q^50 + 4 * q^52 - 2 * q^53 + 8 * q^54 + 8 * q^55 + 2 * q^58 - 2 * q^59 + 4 * q^60 - 10 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 + 4 * q^66 - 10 * q^67 + 8 * q^69 + 12 * q^71 + 2 * q^72 + 4 * q^73 - 4 * q^74 + 4 * q^75 - 8 * q^78 - 2 * q^80 - 22 * q^81 - 2 * q^82 - 6 * q^83 + 4 * q^86 - 4 * q^87 - 2 * q^88 - 12 * q^89 - 2 * q^90 - 4 * q^92 - 8 * q^93 - 8 * q^94 - 12 * q^95 - 4 * q^96 - 4 * q^97 - 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.73205	−0.863950
$11$	−2.73205	−0.823744	−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205	−0.823744	−0.411872	+	0.911242i	$0.635125\pi$
$12$	−2.00000	−0.577350
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	5.46410	1.41082
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	−2.73205	−0.610905
$21$	0	0
$22$	−2.73205	−0.582475
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−2.00000	−0.408248
$25$	2.46410	0.492820
$26$	5.46410	1.07160
$27$	4.00000	0.769800
$28$	0	0
$29$	−7.66025	−1.42247	−0.711237	−	0.702953i	$-0.751865\pi$
$29$	−7.66025	−1.42247	−0.711237	+	0.702953i	$0.751865\pi$
$30$	5.46410	0.997604
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	1.00000	0.176777
$33$	5.46410	0.951178
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	3.46410	0.561951
$39$	−10.9282	−1.74991
$40$	−2.73205	−0.431975
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	12.3923	1.88981	0.944904	−	0.327346i	$-0.106154\pi$
$43$	12.3923	1.88981	0.944904	+	0.327346i	$0.106154\pi$
$44$	−2.73205	−0.411872
$45$	−2.73205	−0.407270
$46$	−2.00000	−0.294884
$47$	−0.535898	−0.0781688	−0.0390844	−	0.999236i	$-0.512444\pi$
$47$	−0.535898	−0.0781688	−0.0390844	+	0.999236i	$0.512444\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	2.46410	0.348477
$51$	0	0
$52$	5.46410	0.757735
$53$	7.66025	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$53$	7.66025	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$54$	4.00000	0.544331
$55$	7.46410	1.00646
$56$	0	0
$57$	−6.92820	−0.917663
$58$	−7.66025	−1.00584
$59$	−13.1244	−1.70865	−0.854323	−	0.519743i	$-0.826028\pi$
$59$	−13.1244	−1.70865	−0.854323	+	0.519743i	$0.826028\pi$
$60$	5.46410	0.705412
$61$	−6.73205	−0.861951	−0.430975	−	0.902364i	$-0.641830\pi$
$61$	−6.73205	−0.861951	−0.430975	+	0.902364i	$0.641830\pi$
$62$	−1.46410	−0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	−14.9282	−1.85162
$66$	5.46410	0.672584
$67$	0.196152	0.0239638	0.0119819	−	0.999928i	$-0.496186\pi$
$67$	0.196152	0.0239638	0.0119819	+	0.999928i	$0.496186\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	1.00000	0.117851
$73$	8.92820	1.04497	0.522484	−	0.852649i	$-0.325006\pi$
$73$	8.92820	1.04497	0.522484	+	0.852649i	$0.325006\pi$
$74$	4.92820	0.572892
$75$	−4.92820	−0.569060
$76$	3.46410	0.397360
$77$	0	0
$78$	−10.9282	−1.23738
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.73205	−0.305453
$81$	−11.0000	−1.22222
$82$	−1.00000	−0.110432
$83$	−4.73205	−0.519410	−0.259705	−	0.965688i	$-0.583625\pi$
$83$	−4.73205	−0.519410	−0.259705	+	0.965688i	$0.583625\pi$
$84$	0	0
$85$	0	0
$86$	12.3923	1.33630
$87$	15.3205	1.64253
$88$	−2.73205	−0.291238
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	−2.73205	−0.287983
$91$	0	0
$92$	−2.00000	−0.208514
$93$	2.92820	0.303641
$94$	−0.535898	−0.0552737
$95$	−9.46410	−0.970996
$96$	−2.00000	−0.204124
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.73205	−0.274581
$100$	2.46410	0.246410
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−5.46410	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$103$	−5.46410	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$104$	5.46410	0.535799
$105$	0	0
$106$	7.66025	0.744030
$107$	−2.92820	−0.283080	−0.141540	−	0.989933i	$-0.545205\pi$
$107$	−2.92820	−0.283080	−0.141540	+	0.989933i	$0.545205\pi$
$108$	4.00000	0.384900
$109$	−15.6603	−1.49998	−0.749990	−	0.661449i	$-0.769942\pi$
$109$	−15.6603	−1.49998	−0.749990	+	0.661449i	$0.769942\pi$
$110$	7.46410	0.711674
$111$	−9.85641	−0.935529
$112$	0	0
$113$	−9.46410	−0.890308	−0.445154	−	0.895454i	$-0.646851\pi$
$113$	−9.46410	−0.890308	−0.445154	+	0.895454i	$0.646851\pi$
$114$	−6.92820	−0.648886
$115$	5.46410	0.509530
$116$	−7.66025	−0.711237
$117$	5.46410	0.505156
$118$	−13.1244	−1.20819
$119$	0	0
$120$	5.46410	0.498802
$121$	−3.53590	−0.321445
$122$	−6.73205	−0.609491
$123$	2.00000	0.180334
$124$	−1.46410	−0.131480
$125$	6.92820	0.619677
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	1.00000	0.0883883
$129$	−24.7846	−2.18216
$130$	−14.9282	−1.30929
$131$	9.12436	0.797199	0.398599	−	0.917125i	$-0.369496\pi$
$131$	9.12436	0.797199	0.398599	+	0.917125i	$0.369496\pi$
$132$	5.46410	0.475589
$133$	0	0
$134$	0.196152	0.0169450
$135$	−10.9282	−0.940550
$136$	0	0
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	4.00000	0.340503
$139$	−6.19615	−0.525551	−0.262775	−	0.964857i	$-0.584638\pi$
$139$	−6.19615	−0.525551	−0.262775	+	0.964857i	$0.584638\pi$
$140$	0	0
$141$	1.07180	0.0902616
$142$	2.53590	0.212808
$143$	−14.9282	−1.24836
$144$	1.00000	0.0833333
$145$	20.9282	1.73799
$146$	8.92820	0.738903
$147$	0	0
$148$	4.92820	0.405096
$149$	−19.6603	−1.61063	−0.805315	−	0.592847i	$-0.798004\pi$
$149$	−19.6603	−1.61063	−0.805315	+	0.592847i	$0.798004\pi$
$150$	−4.92820	−0.402386
$151$	−17.4641	−1.42121	−0.710604	−	0.703592i	$-0.751578\pi$
$151$	−17.4641	−1.42121	−0.710604	+	0.703592i	$0.751578\pi$
$152$	3.46410	0.280976
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	−10.9282	−0.874957
$157$	−21.4641	−1.71302	−0.856511	−	0.516129i	$-0.827373\pi$
$157$	−21.4641	−1.71302	−0.856511	+	0.516129i	$0.827373\pi$
$158$	0	0
$159$	−15.3205	−1.21500
$160$	−2.73205	−0.215988
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−17.4641	−1.36789	−0.683947	−	0.729532i	$-0.739738\pi$
$163$	−17.4641	−1.36789	−0.683947	+	0.729532i	$0.739738\pi$
$164$	−1.00000	−0.0780869
$165$	−14.9282	−1.16216
$166$	−4.73205	−0.367278
$167$	−14.9282	−1.15518	−0.577590	−	0.816327i	$-0.696007\pi$
$167$	−14.9282	−1.15518	−0.577590	+	0.816327i	$0.696007\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	3.46410	0.264906
$172$	12.3923	0.944904
$173$	10.7321	0.815943	0.407971	−	0.912995i	$-0.366236\pi$
$173$	10.7321	0.815943	0.407971	+	0.912995i	$0.366236\pi$
$174$	15.3205	1.16144
$175$	0	0
$176$	−2.73205	−0.205936
$177$	26.2487	1.97297
$178$	−12.9282	−0.969010
$179$	−5.26795	−0.393745	−0.196873	−	0.980429i	$-0.563079\pi$
$179$	−5.26795	−0.393745	−0.196873	+	0.980429i	$0.563079\pi$
$180$	−2.73205	−0.203635
$181$	−14.9282	−1.10960	−0.554802	−	0.831982i	$-0.687206\pi$
$181$	−14.9282	−1.10960	−0.554802	+	0.831982i	$0.687206\pi$
$182$	0	0
$183$	13.4641	0.995295
$184$	−2.00000	−0.147442
$185$	−13.4641	−0.989900
$186$	2.92820	0.214706
$187$	0	0
$188$	−0.535898	−0.0390844
$189$	0	0
$190$	−9.46410	−0.686598
$191$	12.3923	0.896676	0.448338	−	0.893864i	$-0.352016\pi$
$191$	12.3923	0.896676	0.448338	+	0.893864i	$0.352016\pi$
$192$	−2.00000	−0.144338
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	29.8564	2.13806
$196$	0	0
$197$	2.39230	0.170445	0.0852223	−	0.996362i	$-0.472840\pi$
$197$	2.39230	0.170445	0.0852223	+	0.996362i	$0.472840\pi$
$198$	−2.73205	−0.194158
$199$	24.2487	1.71895	0.859473	−	0.511182i	$-0.170792\pi$
$199$	24.2487	1.71895	0.859473	+	0.511182i	$0.170792\pi$
$200$	2.46410	0.174238
$201$	−0.392305	−0.0276711
$202$	−8.00000	−0.562878
$203$	0	0
$204$	0	0
$205$	2.73205	0.190815
$206$	−5.46410	−0.380702
$207$	−2.00000	−0.139010
$208$	5.46410	0.378867
$209$	−9.46410	−0.654646
$210$	0	0
$211$	−10.0526	−0.692047	−0.346023	−	0.938226i	$-0.612468\pi$
$211$	−10.0526	−0.692047	−0.346023	+	0.938226i	$0.612468\pi$
$212$	7.66025	0.526108
$213$	−5.07180	−0.347514
$214$	−2.92820	−0.200168
$215$	−33.8564	−2.30899
$216$	4.00000	0.272166
$217$	0	0
$218$	−15.6603	−1.06065
$219$	−17.8564	−1.20662
$220$	7.46410	0.503230
$221$	0	0
$222$	−9.85641	−0.661519
$223$	12.7846	0.856121	0.428060	−	0.903750i	$-0.359197\pi$
$223$	12.7846	0.856121	0.428060	+	0.903750i	$0.359197\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	−9.46410	−0.629543
$227$	−15.4641	−1.02639	−0.513194	−	0.858272i	$-0.671538\pi$
$227$	−15.4641	−1.02639	−0.513194	+	0.858272i	$0.671538\pi$
$228$	−6.92820	−0.458831
$229$	18.9282	1.25081	0.625405	−	0.780300i	$-0.284934\pi$
$229$	18.9282	1.25081	0.625405	+	0.780300i	$0.284934\pi$
$230$	5.46410	0.360292
$231$	0	0
$232$	−7.66025	−0.502920
$233$	−28.9282	−1.89515	−0.947575	−	0.319534i	$-0.896474\pi$
$233$	−28.9282	−1.89515	−0.947575	+	0.319534i	$0.896474\pi$
$234$	5.46410	0.357199
$235$	1.46410	0.0955075
$236$	−13.1244	−0.854323
$237$	0	0
$238$	0	0
$239$	25.4641	1.64714	0.823568	−	0.567218i	$-0.191980\pi$
$239$	25.4641	1.64714	0.823568	+	0.567218i	$0.191980\pi$
$240$	5.46410	0.352706
$241$	20.9282	1.34810	0.674052	−	0.738684i	$-0.264552\pi$
$241$	20.9282	1.34810	0.674052	+	0.738684i	$0.264552\pi$
$242$	−3.53590	−0.227296
$243$	10.0000	0.641500
$244$	−6.73205	−0.430975
$245$	0	0
$246$	2.00000	0.127515
$247$	18.9282	1.20437
$248$	−1.46410	−0.0929705
$249$	9.46410	0.599763
$250$	6.92820	0.438178
$251$	−16.7321	−1.05612	−0.528059	−	0.849208i	$-0.677080\pi$
$251$	−16.7321	−1.05612	−0.528059	+	0.849208i	$0.677080\pi$
$252$	0	0
$253$	5.46410	0.343525
$254$	10.0000	0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.7846	1.42126	0.710632	−	0.703563i	$-0.248409\pi$
$257$	22.7846	1.42126	0.710632	+	0.703563i	$0.248409\pi$
$258$	−24.7846	−1.54302
$259$	0	0
$260$	−14.9282	−0.925808
$261$	−7.66025	−0.474158
$262$	9.12436	0.563705
$263$	26.9282	1.66046	0.830232	−	0.557418i	$-0.188208\pi$
$263$	26.9282	1.66046	0.830232	+	0.557418i	$0.188208\pi$
$264$	5.46410	0.336292
$265$	−20.9282	−1.28561
$266$	0	0
$267$	25.8564	1.58239
$268$	0.196152	0.0119819
$269$	−7.80385	−0.475809	−0.237904	−	0.971289i	$-0.576461\pi$
$269$	−7.80385	−0.475809	−0.237904	+	0.971289i	$0.576461\pi$
$270$	−10.9282	−0.665069
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−3.46410	−0.209274
$275$	−6.73205	−0.405958
$276$	4.00000	0.240772
$277$	−0.535898	−0.0321990	−0.0160995	−	0.999870i	$-0.505125\pi$
$277$	−0.535898	−0.0321990	−0.0160995	+	0.999870i	$0.505125\pi$
$278$	−6.19615	−0.371621
$279$	−1.46410	−0.0876535
$280$	0	0
$281$	20.9282	1.24847	0.624236	−	0.781236i	$-0.285410\pi$
$281$	20.9282	1.24847	0.624236	+	0.781236i	$0.285410\pi$
$282$	1.07180	0.0638246
$283$	−7.66025	−0.455355	−0.227677	−	0.973737i	$-0.573113\pi$
$283$	−7.66025	−0.455355	−0.227677	+	0.973737i	$0.573113\pi$
$284$	2.53590	0.150478
$285$	18.9282	1.12121
$286$	−14.9282	−0.882723
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	20.9282	1.22895
$291$	4.00000	0.234484
$292$	8.92820	0.522484
$293$	−21.4641	−1.25395	−0.626973	−	0.779041i	$-0.715706\pi$
$293$	−21.4641	−1.25395	−0.626973	+	0.779041i	$0.715706\pi$
$294$	0	0
$295$	35.8564	2.08764
$296$	4.92820	0.286446
$297$	−10.9282	−0.634119
$298$	−19.6603	−1.13889
$299$	−10.9282	−0.631994
$300$	−4.92820	−0.284530
$301$	0	0
$302$	−17.4641	−1.00495
$303$	16.0000	0.919176
$304$	3.46410	0.198680
$305$	18.3923	1.05314
$306$	0	0
$307$	−9.80385	−0.559535	−0.279768	−	0.960068i	$-0.590257\pi$
$307$	−9.80385	−0.559535	−0.279768	+	0.960068i	$0.590257\pi$
$308$	0	0
$309$	10.9282	0.621684
$310$	4.00000	0.227185
$311$	−7.46410	−0.423250	−0.211625	−	0.977351i	$-0.567876\pi$
$311$	−7.46410	−0.423250	−0.211625	+	0.977351i	$0.567876\pi$
$312$	−10.9282	−0.618688
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−21.4641	−1.21129
$315$	0	0
$316$	0	0
$317$	4.33975	0.243744	0.121872	−	0.992546i	$-0.461110\pi$
$317$	4.33975	0.243744	0.121872	+	0.992546i	$0.461110\pi$
$318$	−15.3205	−0.859131
$319$	20.9282	1.17175
$320$	−2.73205	−0.152726
$321$	5.85641	0.326873
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	13.4641	0.746854
$326$	−17.4641	−0.967247
$327$	31.3205	1.73203
$328$	−1.00000	−0.0552158
$329$	0	0
$330$	−14.9282	−0.821771
$331$	−12.5885	−0.691924	−0.345962	−	0.938248i	$-0.612447\pi$
$331$	−12.5885	−0.691924	−0.345962	+	0.938248i	$0.612447\pi$
$332$	−4.73205	−0.259705
$333$	4.92820	0.270064
$334$	−14.9282	−0.816835
$335$	−0.535898	−0.0292793
$336$	0	0
$337$	−22.7846	−1.24116	−0.620578	−	0.784144i	$-0.713102\pi$
$337$	−22.7846	−1.24116	−0.620578	+	0.784144i	$0.713102\pi$
$338$	16.8564	0.916868
$339$	18.9282	1.02804
$340$	0	0
$341$	4.00000	0.216612
$342$	3.46410	0.187317
$343$	0	0
$344$	12.3923	0.668148
$345$	−10.9282	−0.588355
$346$	10.7321	0.576959
$347$	32.1962	1.72838	0.864190	−	0.503166i	$-0.167831\pi$
$347$	32.1962	1.72838	0.864190	+	0.503166i	$0.167831\pi$
$348$	15.3205	0.821265
$349$	−7.80385	−0.417730	−0.208865	−	0.977944i	$-0.566977\pi$
$349$	−7.80385	−0.417730	−0.208865	+	0.977944i	$0.566977\pi$
$350$	0	0
$351$	21.8564	1.16661
$352$	−2.73205	−0.145619
$353$	29.3205	1.56057	0.780287	−	0.625422i	$-0.215073\pi$
$353$	29.3205	1.56057	0.780287	+	0.625422i	$0.215073\pi$
$354$	26.2487	1.39510
$355$	−6.92820	−0.367711
$356$	−12.9282	−0.685193
$357$	0	0
$358$	−5.26795	−0.278420
$359$	2.92820	0.154545	0.0772723	−	0.997010i	$-0.475379\pi$
$359$	2.92820	0.154545	0.0772723	+	0.997010i	$0.475379\pi$
$360$	−2.73205	−0.143992
$361$	−7.00000	−0.368421
$362$	−14.9282	−0.784609
$363$	7.07180	0.371173
$364$	0	0
$365$	−24.3923	−1.27675
$366$	13.4641	0.703780
$367$	−25.8564	−1.34969	−0.674847	−	0.737958i	$-0.735790\pi$
$367$	−25.8564	−1.34969	−0.674847	+	0.737958i	$0.735790\pi$
$368$	−2.00000	−0.104257
$369$	−1.00000	−0.0520579
$370$	−13.4641	−0.699965
$371$	0	0
$372$	2.92820	0.151820
$373$	−32.9282	−1.70496	−0.852479	−	0.522762i	$-0.824902\pi$
$373$	−32.9282	−1.70496	−0.852479	+	0.522762i	$0.824902\pi$
$374$	0	0
$375$	−13.8564	−0.715542
$376$	−0.535898	−0.0276368
$377$	−41.8564	−2.15571
$378$	0	0
$379$	−16.7846	−0.862167	−0.431084	−	0.902312i	$-0.641869\pi$
$379$	−16.7846	−0.862167	−0.431084	+	0.902312i	$0.641869\pi$
$380$	−9.46410	−0.485498
$381$	−20.0000	−1.02463
$382$	12.3923	0.634045
$383$	−25.8564	−1.32120	−0.660600	−	0.750738i	$-0.729698\pi$
$383$	−25.8564	−1.32120	−0.660600	+	0.750738i	$0.729698\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−10.0000	−0.508987
$387$	12.3923	0.629936
$388$	−2.00000	−0.101535
$389$	−26.7846	−1.35803	−0.679017	−	0.734123i	$-0.737594\pi$
$389$	−26.7846	−1.35803	−0.679017	+	0.734123i	$0.737594\pi$
$390$	29.8564	1.51184
$391$	0	0
$392$	0	0
$393$	−18.2487	−0.920526
$394$	2.39230	0.120523
$395$	0	0
$396$	−2.73205	−0.137291
$397$	−4.78461	−0.240133	−0.120066	−	0.992766i	$-0.538311\pi$
$397$	−4.78461	−0.240133	−0.120066	+	0.992766i	$0.538311\pi$
$398$	24.2487	1.21548
$399$	0	0
$400$	2.46410	0.123205
$401$	27.3205	1.36432	0.682161	−	0.731202i	$-0.261041\pi$
$401$	27.3205	1.36432	0.682161	+	0.731202i	$0.261041\pi$
$402$	−0.392305	−0.0195664
$403$	−8.00000	−0.398508
$404$	−8.00000	−0.398015
$405$	30.0526	1.49332
$406$	0	0
$407$	−13.4641	−0.667391
$408$	0	0
$409$	15.0718	0.745252	0.372626	−	0.927982i	$-0.378457\pi$
$409$	15.0718	0.745252	0.372626	+	0.927982i	$0.378457\pi$
$410$	2.73205	0.134926
$411$	6.92820	0.341743
$412$	−5.46410	−0.269197
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	12.9282	0.634621
$416$	5.46410	0.267900
$417$	12.3923	0.606854
$418$	−9.46410	−0.462904
$419$	−20.4449	−0.998797	−0.499398	−	0.866372i	$-0.666446\pi$
$419$	−20.4449	−0.998797	−0.499398	+	0.866372i	$0.666446\pi$
$420$	0	0
$421$	18.5885	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$421$	18.5885	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$422$	−10.0526	−0.489351
$423$	−0.535898	−0.0260563
$424$	7.66025	0.372015
$425$	0	0
$426$	−5.07180	−0.245729
$427$	0	0
$428$	−2.92820	−0.141540
$429$	29.8564	1.44148
$430$	−33.8564	−1.63270
$431$	−25.7128	−1.23854	−0.619271	−	0.785177i	$-0.712572\pi$
$431$	−25.7128	−1.23854	−0.619271	+	0.785177i	$0.712572\pi$
$432$	4.00000	0.192450
$433$	−29.7128	−1.42791	−0.713953	−	0.700193i	$-0.753097\pi$
$433$	−29.7128	−1.42791	−0.713953	+	0.700193i	$0.753097\pi$
$434$	0	0
$435$	−41.8564	−2.00686
$436$	−15.6603	−0.749990
$437$	−6.92820	−0.331421
$438$	−17.8564	−0.853212
$439$	31.4641	1.50170	0.750850	−	0.660473i	$-0.229644\pi$
$439$	31.4641	1.50170	0.750850	+	0.660473i	$0.229644\pi$
$440$	7.46410	0.355837
$441$	0	0
$442$	0	0
$443$	21.4641	1.01979	0.509895	−	0.860237i	$-0.329684\pi$
$443$	21.4641	1.01979	0.509895	+	0.860237i	$0.329684\pi$
$444$	−9.85641	−0.467764
$445$	35.3205	1.67435
$446$	12.7846	0.605369
$447$	39.3205	1.85980
$448$	0	0
$449$	−4.92820	−0.232576	−0.116288	−	0.993216i	$-0.537100\pi$
$449$	−4.92820	−0.232576	−0.116288	+	0.993216i	$0.537100\pi$
$450$	2.46410	0.116159
$451$	2.73205	0.128647
$452$	−9.46410	−0.445154
$453$	34.9282	1.64107
$454$	−15.4641	−0.725766
$455$	0	0
$456$	−6.92820	−0.324443
$457$	−11.8564	−0.554619	−0.277310	−	0.960781i	$-0.589443\pi$
$457$	−11.8564	−0.554619	−0.277310	+	0.960781i	$0.589443\pi$
$458$	18.9282	0.884457
$459$	0	0
$460$	5.46410	0.254765
$461$	−0.875644	−0.0407828	−0.0203914	−	0.999792i	$-0.506491\pi$
$461$	−0.875644	−0.0407828	−0.0203914	+	0.999792i	$0.506491\pi$
$462$	0	0
$463$	13.0718	0.607498	0.303749	−	0.952752i	$-0.401762\pi$
$463$	13.0718	0.607498	0.303749	+	0.952752i	$0.401762\pi$
$464$	−7.66025	−0.355618
$465$	−8.00000	−0.370991
$466$	−28.9282	−1.34007
$467$	20.4449	0.946075	0.473038	−	0.881042i	$-0.343157\pi$
$467$	20.4449	0.946075	0.473038	+	0.881042i	$0.343157\pi$
$468$	5.46410	0.252578
$469$	0	0
$470$	1.46410	0.0675340
$471$	42.9282	1.97803
$472$	−13.1244	−0.604097
$473$	−33.8564	−1.55672
$474$	0	0
$475$	8.53590	0.391654
$476$	0	0
$477$	7.66025	0.350739
$478$	25.4641	1.16470
$479$	33.3205	1.52245	0.761226	−	0.648486i	$-0.224598\pi$
$479$	33.3205	1.52245	0.761226	+	0.648486i	$0.224598\pi$
$480$	5.46410	0.249401
$481$	26.9282	1.22782
$482$	20.9282	0.953254
$483$	0	0
$484$	−3.53590	−0.160723
$485$	5.46410	0.248112
$486$	10.0000	0.453609
$487$	10.9282	0.495204	0.247602	−	0.968862i	$-0.420357\pi$
$487$	10.9282	0.495204	0.247602	+	0.968862i	$0.420357\pi$
$488$	−6.73205	−0.304746
$489$	34.9282	1.57951
$490$	0	0
$491$	−11.6077	−0.523848	−0.261924	−	0.965089i	$-0.584357\pi$
$491$	−11.6077	−0.523848	−0.261924	+	0.965089i	$0.584357\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	18.9282	0.851620
$495$	7.46410	0.335486
$496$	−1.46410	−0.0657401
$497$	0	0
$498$	9.46410	0.424097
$499$	5.66025	0.253388	0.126694	−	0.991942i	$-0.459563\pi$
$499$	5.66025	0.253388	0.126694	+	0.991942i	$0.459563\pi$
$500$	6.92820	0.309839
$501$	29.8564	1.33389
$502$	−16.7321	−0.746788
$503$	2.14359	0.0955781	0.0477891	−	0.998857i	$-0.484782\pi$
$503$	2.14359	0.0955781	0.0477891	+	0.998857i	$0.484782\pi$
$504$	0	0
$505$	21.8564	0.972597
$506$	5.46410	0.242909
$507$	−33.7128	−1.49724
$508$	10.0000	0.443678
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	13.8564	0.611775
$514$	22.7846	1.00499
$515$	14.9282	0.657815
$516$	−24.7846	−1.09108
$517$	1.46410	0.0643911
$518$	0	0
$519$	−21.4641	−0.942169
$520$	−14.9282	−0.654645
$521$	−8.78461	−0.384861	−0.192430	−	0.981311i	$-0.561637\pi$
$521$	−8.78461	−0.384861	−0.192430	+	0.981311i	$0.561637\pi$
$522$	−7.66025	−0.335280
$523$	14.1962	0.620754	0.310377	−	0.950613i	$-0.399545\pi$
$523$	14.1962	0.620754	0.310377	+	0.950613i	$0.399545\pi$
$524$	9.12436	0.398599
$525$	0	0
$526$	26.9282	1.17413
$527$	0	0
$528$	5.46410	0.237795
$529$	−19.0000	−0.826087
$530$	−20.9282	−0.909063
$531$	−13.1244	−0.569549
$532$	0	0
$533$	−5.46410	−0.236677
$534$	25.8564	1.11892
$535$	8.00000	0.345870
$536$	0.196152	0.00847249
$537$	10.5359	0.454658
$538$	−7.80385	−0.336448
$539$	0	0
$540$	−10.9282	−0.470275
$541$	−18.7846	−0.807613	−0.403807	−	0.914844i	$-0.632313\pi$
$541$	−18.7846	−0.807613	−0.403807	+	0.914844i	$0.632313\pi$
$542$	−8.00000	−0.343629
$543$	29.8564	1.28126
$544$	0	0
$545$	42.7846	1.83269
$546$	0	0
$547$	−19.1244	−0.817698	−0.408849	−	0.912602i	$-0.634070\pi$
$547$	−19.1244	−0.817698	−0.408849	+	0.912602i	$0.634070\pi$
$548$	−3.46410	−0.147979
$549$	−6.73205	−0.287317
$550$	−6.73205	−0.287056
$551$	−26.5359	−1.13047
$552$	4.00000	0.170251
$553$	0	0
$554$	−0.535898	−0.0227681
$555$	26.9282	1.14304
$556$	−6.19615	−0.262775
$557$	7.26795	0.307953	0.153976	−	0.988075i	$-0.450792\pi$
$557$	7.26795	0.307953	0.153976	+	0.988075i	$0.450792\pi$
$558$	−1.46410	−0.0619804
$559$	67.7128	2.86395
$560$	0	0
$561$	0	0
$562$	20.9282	0.882803
$563$	28.5359	1.20264	0.601322	−	0.799007i	$-0.294641\pi$
$563$	28.5359	1.20264	0.601322	+	0.799007i	$0.294641\pi$
$564$	1.07180	0.0451308
$565$	25.8564	1.08779
$566$	−7.66025	−0.321984
$567$	0	0
$568$	2.53590	0.106404
$569$	11.3205	0.474580	0.237290	−	0.971439i	$-0.423741\pi$
$569$	11.3205	0.474580	0.237290	+	0.971439i	$0.423741\pi$
$570$	18.9282	0.792815
$571$	−22.7321	−0.951307	−0.475653	−	0.879633i	$-0.657788\pi$
$571$	−22.7321	−0.951307	−0.475653	+	0.879633i	$0.657788\pi$
$572$	−14.9282	−0.624180
$573$	−24.7846	−1.03539
$574$	0	0
$575$	−4.92820	−0.205520
$576$	1.00000	0.0416667
$577$	25.8564	1.07642	0.538208	−	0.842812i	$-0.319101\pi$
$577$	25.8564	1.07642	0.538208	+	0.842812i	$0.319101\pi$
$578$	−17.0000	−0.707107
$579$	20.0000	0.831172
$580$	20.9282	0.868996
$581$	0	0
$582$	4.00000	0.165805
$583$	−20.9282	−0.866758
$584$	8.92820	0.369452
$585$	−14.9282	−0.617205
$586$	−21.4641	−0.886674
$587$	36.9282	1.52419	0.762095	−	0.647465i	$-0.224171\pi$
$587$	36.9282	1.52419	0.762095	+	0.647465i	$0.224171\pi$
$588$	0	0
$589$	−5.07180	−0.208980
$590$	35.8564	1.47618
$591$	−4.78461	−0.196813
$592$	4.92820	0.202548
$593$	−6.78461	−0.278611	−0.139305	−	0.990249i	$-0.544487\pi$
$593$	−6.78461	−0.278611	−0.139305	+	0.990249i	$0.544487\pi$
$594$	−10.9282	−0.448390
$595$	0	0
$596$	−19.6603	−0.805315
$597$	−48.4974	−1.98487
$598$	−10.9282	−0.446887
$599$	32.7846	1.33954	0.669771	−	0.742567i	$-0.266392\pi$
$599$	32.7846	1.33954	0.669771	+	0.742567i	$0.266392\pi$
$600$	−4.92820	−0.201193
$601$	−14.9282	−0.608934	−0.304467	−	0.952523i	$-0.598478\pi$
$601$	−14.9282	−0.608934	−0.304467	+	0.952523i	$0.598478\pi$
$602$	0	0
$603$	0.196152	0.00798794
$604$	−17.4641	−0.710604
$605$	9.66025	0.392745
$606$	16.0000	0.649956
$607$	−35.7128	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$607$	−35.7128	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$608$	3.46410	0.140488
$609$	0	0
$610$	18.3923	0.744683
$611$	−2.92820	−0.118462
$612$	0	0
$613$	−13.6077	−0.549610	−0.274805	−	0.961500i	$-0.588613\pi$
$613$	−13.6077	−0.549610	−0.274805	+	0.961500i	$0.588613\pi$
$614$	−9.80385	−0.395651
$615$	−5.46410	−0.220334
$616$	0	0
$617$	−26.5359	−1.06830	−0.534148	−	0.845391i	$-0.679367\pi$
$617$	−26.5359	−1.06830	−0.534148	+	0.845391i	$0.679367\pi$
$618$	10.9282	0.439597
$619$	33.5167	1.34715	0.673574	−	0.739120i	$-0.264758\pi$
$619$	33.5167	1.34715	0.673574	+	0.739120i	$0.264758\pi$
$620$	4.00000	0.160644
$621$	−8.00000	−0.321029
$622$	−7.46410	−0.299283
$623$	0	0
$624$	−10.9282	−0.437478
$625$	−31.2487	−1.24995
$626$	20.0000	0.799361
$627$	18.9282	0.755920
$628$	−21.4641	−0.856511
$629$	0	0
$630$	0	0
$631$	28.6410	1.14018	0.570090	−	0.821582i	$-0.306908\pi$
$631$	28.6410	1.14018	0.570090	+	0.821582i	$0.306908\pi$
$632$	0	0
$633$	20.1051	0.799107
$634$	4.33975	0.172353
$635$	−27.3205	−1.08418
$636$	−15.3205	−0.607498
$637$	0	0
$638$	20.9282	0.828556
$639$	2.53590	0.100319
$640$	−2.73205	−0.107994
$641$	31.8564	1.25825	0.629126	−	0.777303i	$-0.283413\pi$
$641$	31.8564	1.25825	0.629126	+	0.777303i	$0.283413\pi$
$642$	5.85641	0.231134
$643$	1.60770	0.0634013	0.0317007	−	0.999497i	$-0.489908\pi$
$643$	1.60770	0.0634013	0.0317007	+	0.999497i	$0.489908\pi$
$644$	0	0
$645$	67.7128	2.66619
$646$	0	0
$647$	5.46410	0.214816	0.107408	−	0.994215i	$-0.465745\pi$
$647$	5.46410	0.214816	0.107408	+	0.994215i	$0.465745\pi$
$648$	−11.0000	−0.432121
$649$	35.8564	1.40749
$650$	13.4641	0.528106
$651$	0	0
$652$	−17.4641	−0.683947
$653$	−9.80385	−0.383654	−0.191827	−	0.981429i	$-0.561441\pi$
$653$	−9.80385	−0.383654	−0.191827	+	0.981429i	$0.561441\pi$
$654$	31.3205	1.22473
$655$	−24.9282	−0.974025
$656$	−1.00000	−0.0390434
$657$	8.92820	0.348322
$658$	0	0
$659$	−6.73205	−0.262243	−0.131122	−	0.991366i	$-0.541858\pi$
$659$	−6.73205	−0.262243	−0.131122	+	0.991366i	$0.541858\pi$
$660$	−14.9282	−0.581080
$661$	−39.1244	−1.52176	−0.760881	−	0.648892i	$-0.775233\pi$
$661$	−39.1244	−1.52176	−0.760881	+	0.648892i	$0.775233\pi$
$662$	−12.5885	−0.489264
$663$	0	0
$664$	−4.73205	−0.183639
$665$	0	0
$666$	4.92820	0.190964
$667$	15.3205	0.593212
$668$	−14.9282	−0.577590
$669$	−25.5692	−0.988563
$670$	−0.535898	−0.0207036
$671$	18.3923	0.710027
$672$	0	0
$673$	−28.5359	−1.09998	−0.549989	−	0.835172i	$-0.685368\pi$
$673$	−28.5359	−1.09998	−0.549989	+	0.835172i	$0.685368\pi$
$674$	−22.7846	−0.877630
$675$	9.85641	0.379373
$676$	16.8564	0.648323
$677$	−6.33975	−0.243656	−0.121828	−	0.992551i	$-0.538876\pi$
$677$	−6.33975	−0.243656	−0.121828	+	0.992551i	$0.538876\pi$
$678$	18.9282	0.726933
$679$	0	0
$680$	0	0
$681$	30.9282	1.18517
$682$	4.00000	0.153168
$683$	−14.0526	−0.537706	−0.268853	−	0.963181i	$-0.586645\pi$
$683$	−14.0526	−0.537706	−0.268853	+	0.963181i	$0.586645\pi$
$684$	3.46410	0.132453
$685$	9.46410	0.361605
$686$	0	0
$687$	−37.8564	−1.44431
$688$	12.3923	0.472452
$689$	41.8564	1.59460
$690$	−10.9282	−0.416030
$691$	−36.9282	−1.40482	−0.702408	−	0.711775i	$-0.747892\pi$
$691$	−36.9282	−1.40482	−0.702408	+	0.711775i	$0.747892\pi$
$692$	10.7321	0.407971
$693$	0	0
$694$	32.1962	1.22215
$695$	16.9282	0.642123
$696$	15.3205	0.580722
$697$	0	0
$698$	−7.80385	−0.295380
$699$	57.8564	2.18833
$700$	0	0
$701$	3.07180	0.116020	0.0580101	−	0.998316i	$-0.481524\pi$
$701$	3.07180	0.116020	0.0580101	+	0.998316i	$0.481524\pi$
$702$	21.8564	0.824917
$703$	17.0718	0.643875
$704$	−2.73205	−0.102968
$705$	−2.92820	−0.110283
$706$	29.3205	1.10349
$707$	0	0
$708$	26.2487	0.986487
$709$	47.3731	1.77913	0.889566	−	0.456806i	$-0.151007\pi$
$709$	47.3731	1.77913	0.889566	+	0.456806i	$0.151007\pi$
$710$	−6.92820	−0.260011
$711$	0	0
$712$	−12.9282	−0.484505
$713$	2.92820	0.109662
$714$	0	0
$715$	40.7846	1.52526
$716$	−5.26795	−0.196873
$717$	−50.9282	−1.90195
$718$	2.92820	0.109280
$719$	−23.7128	−0.884339	−0.442169	−	0.896932i	$-0.645791\pi$
$719$	−23.7128	−0.884339	−0.442169	+	0.896932i	$0.645791\pi$
$720$	−2.73205	−0.101818
$721$	0	0
$722$	−7.00000	−0.260513
$723$	−41.8564	−1.55666
$724$	−14.9282	−0.554802
$725$	−18.8756	−0.701024
$726$	7.07180	0.262459
$727$	41.8564	1.55237	0.776184	−	0.630506i	$-0.217153\pi$
$727$	41.8564	1.55237	0.776184	+	0.630506i	$0.217153\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−24.3923	−0.902800
$731$	0	0
$732$	13.4641	0.497648
$733$	−13.2679	−0.490063	−0.245031	−	0.969515i	$-0.578798\pi$
$733$	−13.2679	−0.490063	−0.245031	+	0.969515i	$0.578798\pi$
$734$	−25.8564	−0.954377
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	−0.535898	−0.0197401
$738$	−1.00000	−0.0368105
$739$	14.5359	0.534712	0.267356	−	0.963598i	$-0.413850\pi$
$739$	14.5359	0.534712	0.267356	+	0.963598i	$0.413850\pi$
$740$	−13.4641	−0.494950
$741$	−37.8564	−1.39069
$742$	0	0
$743$	−27.7128	−1.01668	−0.508342	−	0.861155i	$-0.669742\pi$
$743$	−27.7128	−1.01668	−0.508342	+	0.861155i	$0.669742\pi$
$744$	2.92820	0.107353
$745$	53.7128	1.96789
$746$	−32.9282	−1.20559
$747$	−4.73205	−0.173137
$748$	0	0
$749$	0	0
$750$	−13.8564	−0.505964
$751$	−3.32051	−0.121167	−0.0605835	−	0.998163i	$-0.519296\pi$
$751$	−3.32051	−0.121167	−0.0605835	+	0.998163i	$0.519296\pi$
$752$	−0.535898	−0.0195422
$753$	33.4641	1.21950
$754$	−41.8564	−1.52432
$755$	47.7128	1.73645
$756$	0	0
$757$	−35.3731	−1.28566	−0.642828	−	0.766011i	$-0.722239\pi$
$757$	−35.3731	−1.28566	−0.642828	+	0.766011i	$0.722239\pi$
$758$	−16.7846	−0.609644
$759$	−10.9282	−0.396669
$760$	−9.46410	−0.343299
$761$	15.1769	0.550163	0.275081	−	0.961421i	$-0.411295\pi$
$761$	15.1769	0.550163	0.275081	+	0.961421i	$0.411295\pi$
$762$	−20.0000	−0.724524
$763$	0	0
$764$	12.3923	0.448338
$765$	0	0
$766$	−25.8564	−0.934230
$767$	−71.7128	−2.58940
$768$	−2.00000	−0.0721688
$769$	−15.1769	−0.547294	−0.273647	−	0.961830i	$-0.588230\pi$
$769$	−15.1769	−0.547294	−0.273647	+	0.961830i	$0.588230\pi$
$770$	0	0
$771$	−45.5692	−1.64114
$772$	−10.0000	−0.359908
$773$	−41.1769	−1.48103	−0.740515	−	0.672039i	$-0.765419\pi$
$773$	−41.1769	−1.48103	−0.740515	+	0.672039i	$0.765419\pi$
$774$	12.3923	0.445432
$775$	−3.60770	−0.129592
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−26.7846	−0.960275
$779$	−3.46410	−0.124114
$780$	29.8564	1.06903
$781$	−6.92820	−0.247911
$782$	0	0
$783$	−30.6410	−1.09502
$784$	0	0
$785$	58.6410	2.09299
$786$	−18.2487	−0.650910
$787$	−18.1962	−0.648623	−0.324311	−	0.945950i	$-0.605133\pi$
$787$	−18.1962	−0.648623	−0.324311	+	0.945950i	$0.605133\pi$
$788$	2.39230	0.0852223
$789$	−53.8564	−1.91734
$790$	0	0
$791$	0	0
$792$	−2.73205	−0.0970792
$793$	−36.7846	−1.30626
$794$	−4.78461	−0.169799
$795$	41.8564	1.48449
$796$	24.2487	0.859473
$797$	−44.3013	−1.56923	−0.784616	−	0.619982i	$-0.787140\pi$
$797$	−44.3013	−1.56923	−0.784616	+	0.619982i	$0.787140\pi$
$798$	0	0
$799$	0	0
$800$	2.46410	0.0871191
$801$	−12.9282	−0.456796
$802$	27.3205	0.964721
$803$	−24.3923	−0.860786
$804$	−0.392305	−0.0138355
$805$	0	0
$806$	−8.00000	−0.281788
$807$	15.6077	0.549417
$808$	−8.00000	−0.281439
$809$	54.7846	1.92612	0.963062	−	0.269279i	$-0.0867854\pi$
$809$	54.7846	1.92612	0.963062	+	0.269279i	$0.0867854\pi$
$810$	30.0526	1.05594
$811$	−46.3013	−1.62586	−0.812929	−	0.582363i	$-0.802128\pi$
$811$	−46.3013	−1.62586	−0.812929	+	0.582363i	$0.802128\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	−13.4641	−0.471917
$815$	47.7128	1.67131
$816$	0	0
$817$	42.9282	1.50187
$818$	15.0718	0.526973
$819$	0	0
$820$	2.73205	0.0954074
$821$	16.5359	0.577107	0.288553	−	0.957464i	$-0.406826\pi$
$821$	16.5359	0.577107	0.288553	+	0.957464i	$0.406826\pi$
$822$	6.92820	0.241649
$823$	−21.1769	−0.738181	−0.369090	−	0.929393i	$-0.620331\pi$
$823$	−21.1769	−0.738181	−0.369090	+	0.929393i	$0.620331\pi$
$824$	−5.46410	−0.190351
$825$	13.4641	0.468760
$826$	0	0
$827$	0.588457	0.0204627	0.0102313	−	0.999948i	$-0.496743\pi$
$827$	0.588457	0.0204627	0.0102313	+	0.999948i	$0.496743\pi$
$828$	−2.00000	−0.0695048
$829$	−33.3731	−1.15909	−0.579547	−	0.814939i	$-0.696771\pi$
$829$	−33.3731	−1.15909	−0.579547	+	0.814939i	$0.696771\pi$
$830$	12.9282	0.448744
$831$	1.07180	0.0371802
$832$	5.46410	0.189434
$833$	0	0
$834$	12.3923	0.429110
$835$	40.7846	1.41141
$836$	−9.46410	−0.327323
$837$	−5.85641	−0.202427
$838$	−20.4449	−0.706256
$839$	−0.535898	−0.0185013	−0.00925063	−	0.999957i	$-0.502945\pi$
$839$	−0.535898	−0.0185013	−0.00925063	+	0.999957i	$0.502945\pi$
$840$	0	0
$841$	29.6795	1.02343
$842$	18.5885	0.640601
$843$	−41.8564	−1.44161
$844$	−10.0526	−0.346023
$845$	−46.0526	−1.58426
$846$	−0.535898	−0.0184246
$847$	0	0
$848$	7.66025	0.263054
$849$	15.3205	0.525798
$850$	0	0
$851$	−9.85641	−0.337873
$852$	−5.07180	−0.173757
$853$	22.4449	0.768497	0.384249	−	0.923230i	$-0.374461\pi$
$853$	22.4449	0.768497	0.384249	+	0.923230i	$0.374461\pi$
$854$	0	0
$855$	−9.46410	−0.323665
$856$	−2.92820	−0.100084
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	29.8564	1.01928
$859$	24.4449	0.834048	0.417024	−	0.908895i	$-0.363073\pi$
$859$	24.4449	0.834048	0.417024	+	0.908895i	$0.363073\pi$
$860$	−33.8564	−1.15449
$861$	0	0
$862$	−25.7128	−0.875782
$863$	−46.6410	−1.58768	−0.793839	−	0.608128i	$-0.791921\pi$
$863$	−46.6410	−1.58768	−0.793839	+	0.608128i	$0.791921\pi$
$864$	4.00000	0.136083
$865$	−29.3205	−0.996927
$866$	−29.7128	−1.00968
$867$	34.0000	1.15470
$868$	0	0
$869$	0	0
$870$	−41.8564	−1.41907
$871$	1.07180	0.0363164
$872$	−15.6603	−0.530323
$873$	−2.00000	−0.0676897
$874$	−6.92820	−0.234350
$875$	0	0
$876$	−17.8564	−0.603312
$877$	56.2487	1.89938	0.949692	−	0.313185i	$-0.101396\pi$
$877$	56.2487	1.89938	0.949692	+	0.313185i	$0.101396\pi$
$878$	31.4641	1.06186
$879$	42.9282	1.44793
$880$	7.46410	0.251615
$881$	−21.6077	−0.727982	−0.363991	−	0.931403i	$-0.618586\pi$
$881$	−21.6077	−0.727982	−0.363991	+	0.931403i	$0.618586\pi$
$882$	0	0
$883$	50.0526	1.68440	0.842201	−	0.539163i	$-0.181259\pi$
$883$	50.0526	1.68440	0.842201	+	0.539163i	$0.181259\pi$
$884$	0	0
$885$	−71.7128	−2.41060
$886$	21.4641	0.721101
$887$	−35.1769	−1.18113	−0.590563	−	0.806992i	$-0.701094\pi$
$887$	−35.1769	−1.18113	−0.590563	+	0.806992i	$0.701094\pi$
$888$	−9.85641	−0.330759
$889$	0	0
$890$	35.3205	1.18395
$891$	30.0526	1.00680
$892$	12.7846	0.428060
$893$	−1.85641	−0.0621223
$894$	39.3205	1.31507
$895$	14.3923	0.481082
$896$	0	0
$897$	21.8564	0.729764
$898$	−4.92820	−0.164456
$899$	11.2154	0.374054
$900$	2.46410	0.0821367
$901$	0	0
$902$	2.73205	0.0909673
$903$	0	0
$904$	−9.46410	−0.314771
$905$	40.7846	1.35573
$906$	34.9282	1.16041
$907$	10.5359	0.349839	0.174919	−	0.984583i	$-0.444034\pi$
$907$	10.5359	0.349839	0.174919	+	0.984583i	$0.444034\pi$
$908$	−15.4641	−0.513194
$909$	−8.00000	−0.265343
$910$	0	0
$911$	4.92820	0.163279	0.0816393	−	0.996662i	$-0.473984\pi$
$911$	4.92820	0.163279	0.0816393	+	0.996662i	$0.473984\pi$
$912$	−6.92820	−0.229416
$913$	12.9282	0.427861
$914$	−11.8564	−0.392175
$915$	−36.7846	−1.21606
$916$	18.9282	0.625405
$917$	0	0
$918$	0	0
$919$	0.392305	0.0129409	0.00647047	−	0.999979i	$-0.497940\pi$
$919$	0.392305	0.0129409	0.00647047	+	0.999979i	$0.497940\pi$
$920$	5.46410	0.180146
$921$	19.6077	0.646096
$922$	−0.875644	−0.0288378
$923$	13.8564	0.456089
$924$	0	0
$925$	12.1436	0.399279
$926$	13.0718	0.429566
$927$	−5.46410	−0.179465
$928$	−7.66025	−0.251460
$929$	−8.14359	−0.267183	−0.133591	−	0.991037i	$-0.542651\pi$
$929$	−8.14359	−0.267183	−0.133591	+	0.991037i	$0.542651\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	−28.9282	−0.947575
$933$	14.9282	0.488727
$934$	20.4449	0.668976
$935$	0	0
$936$	5.46410	0.178600
$937$	12.9282	0.422346	0.211173	−	0.977449i	$-0.432272\pi$
$937$	12.9282	0.422346	0.211173	+	0.977449i	$0.432272\pi$
$938$	0	0
$939$	−40.0000	−1.30535
$940$	1.46410	0.0477537
$941$	28.9808	0.944746	0.472373	−	0.881399i	$-0.343398\pi$
$941$	28.9808	0.944746	0.472373	+	0.881399i	$0.343398\pi$
$942$	42.9282	1.39868
$943$	2.00000	0.0651290
$944$	−13.1244	−0.427161
$945$	0	0
$946$	−33.8564	−1.10077
$947$	22.6410	0.735734	0.367867	−	0.929878i	$-0.380088\pi$
$947$	22.6410	0.735734	0.367867	+	0.929878i	$0.380088\pi$
$948$	0	0
$949$	48.7846	1.58362
$950$	8.53590	0.276941
$951$	−8.67949	−0.281452
$952$	0	0
$953$	18.2487	0.591134	0.295567	−	0.955322i	$-0.404491\pi$
$953$	18.2487	0.591134	0.295567	+	0.955322i	$0.404491\pi$
$954$	7.66025	0.248010
$955$	−33.8564	−1.09557
$956$	25.4641	0.823568
$957$	−41.8564	−1.35303
$958$	33.3205	1.07654
$959$	0	0
$960$	5.46410	0.176353
$961$	−28.8564	−0.930852
$962$	26.9282	0.868200
$963$	−2.92820	−0.0943600
$964$	20.9282	0.674052
$965$	27.3205	0.879478
$966$	0	0
$967$	36.7846	1.18291	0.591457	−	0.806337i	$-0.298553\pi$
$967$	36.7846	1.18291	0.591457	+	0.806337i	$0.298553\pi$
$968$	−3.53590	−0.113648
$969$	0	0
$970$	5.46410	0.175442
$971$	30.1051	0.966119	0.483060	−	0.875587i	$-0.339525\pi$
$971$	30.1051	0.966119	0.483060	+	0.875587i	$0.339525\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	10.9282	0.350162
$975$	−26.9282	−0.862393
$976$	−6.73205	−0.215488
$977$	47.1769	1.50932	0.754662	−	0.656114i	$-0.227801\pi$
$977$	47.1769	1.50932	0.754662	+	0.656114i	$0.227801\pi$
$978$	34.9282	1.11688
$979$	35.3205	1.12885
$980$	0	0
$981$	−15.6603	−0.499993
$982$	−11.6077	−0.370416
$983$	37.4641	1.19492	0.597460	−	0.801899i	$-0.296177\pi$
$983$	37.4641	1.19492	0.597460	+	0.801899i	$0.296177\pi$
$984$	2.00000	0.0637577
$985$	−6.53590	−0.208251
$986$	0	0
$987$	0	0
$988$	18.9282	0.602186
$989$	−24.7846	−0.788105
$990$	7.46410	0.237225
$991$	7.32051	0.232544	0.116272	−	0.993217i	$-0.462906\pi$
$991$	7.32051	0.232544	0.116272	+	0.993217i	$0.462906\pi$
$992$	−1.46410	−0.0464853
$993$	25.1769	0.798965
$994$	0	0
$995$	−66.2487	−2.10023
$996$	9.46410	0.299882
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	5.66025	0.179172
$999$	19.7128	0.623686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4018.2.a.x.1.1		2
7.6	odd	2		574.2.a.k.1.2	✓	2
21.20	even	2		5166.2.a.bo.1.1		2
28.27	even	2		4592.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.a.k.1.2	✓	2	7.6	odd	2
4018.2.a.x.1.1		2	1.1	even	1	trivial
4592.2.a.m.1.2		2	28.27	even	2
5166.2.a.bo.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4018.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} -2.73205 q^{11} -2.00000 q^{12} +5.46410 q^{13} +5.46410 q^{15} +1.00000 q^{16} +1.00000 q^{18} +3.46410 q^{19} -2.73205 q^{20} -2.73205 q^{22} -2.00000 q^{23} -2.00000 q^{24} +2.46410 q^{25} +5.46410 q^{26} +4.00000 q^{27} -7.66025 q^{29} +5.46410 q^{30} -1.46410 q^{31} +1.00000 q^{32} +5.46410 q^{33} +1.00000 q^{36} +4.92820 q^{37} +3.46410 q^{38} -10.9282 q^{39} -2.73205 q^{40} -1.00000 q^{41} +12.3923 q^{43} -2.73205 q^{44} -2.73205 q^{45} -2.00000 q^{46} -0.535898 q^{47} -2.00000 q^{48} +2.46410 q^{50} +5.46410 q^{52} +7.66025 q^{53} +4.00000 q^{54} +7.46410 q^{55} -6.92820 q^{57} -7.66025 q^{58} -13.1244 q^{59} +5.46410 q^{60} -6.73205 q^{61} -1.46410 q^{62} +1.00000 q^{64} -14.9282 q^{65} +5.46410 q^{66} +0.196152 q^{67} +4.00000 q^{69} +2.53590 q^{71} +1.00000 q^{72} +8.92820 q^{73} +4.92820 q^{74} -4.92820 q^{75} +3.46410 q^{76} -10.9282 q^{78} -2.73205 q^{80} -11.0000 q^{81} -1.00000 q^{82} -4.73205 q^{83} +12.3923 q^{86} +15.3205 q^{87} -2.73205 q^{88} -12.9282 q^{89} -2.73205 q^{90} -2.00000 q^{92} +2.92820 q^{93} -0.535898 q^{94} -9.46410 q^{95} -2.00000 q^{96} -2.00000 q^{97} -2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 2 q^{20} - 2 q^{22} - 4 q^{23} - 4 q^{24} - 2 q^{25} + 4 q^{26} + 8 q^{27} + 2 q^{29} + 4 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{36} - 4 q^{37} - 8 q^{39} - 2 q^{40} - 2 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} - 4 q^{46} - 8 q^{47} - 4 q^{48} - 2 q^{50} + 4 q^{52} - 2 q^{53} + 8 q^{54} + 8 q^{55} + 2 q^{58} - 2 q^{59} + 4 q^{60} - 10 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} + 4 q^{66} - 10 q^{67} + 8 q^{69} + 12 q^{71} + 2 q^{72} + 4 q^{73} - 4 q^{74} + 4 q^{75} - 8 q^{78} - 2 q^{80} - 22 q^{81} - 2 q^{82} - 6 q^{83} + 4 q^{86} - 4 q^{87} - 2 q^{88} - 12 q^{89} - 2 q^{90} - 4 q^{92} - 8 q^{93} - 8 q^{94} - 12 q^{95} - 4 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^18 - 2 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 - 2 * q^25 + 4 * q^26 + 8 * q^27 + 2 * q^29 + 4 * q^30 + 4 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 8 * q^39 - 2 * q^40 - 2 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^45 - 4 * q^46 - 8 * q^47 - 4 * q^48 - 2 * q^50 + 4 * q^52 - 2 * q^53 + 8 * q^54 + 8 * q^55 + 2 * q^58 - 2 * q^59 + 4 * q^60 - 10 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 + 4 * q^66 - 10 * q^67 + 8 * q^69 + 12 * q^71 + 2 * q^72 + 4 * q^73 - 4 * q^74 + 4 * q^75 - 8 * q^78 - 2 * q^80 - 22 * q^81 - 2 * q^82 - 6 * q^83 + 4 * q^86 - 4 * q^87 - 2 * q^88 - 12 * q^89 - 2 * q^90 - 4 * q^92 - 8 * q^93 - 8 * q^94 - 12 * q^95 - 4 * q^96 - 4 * q^97 - 2 * q^99

Introduction

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