LMFDB - Embedded newform 1338.2.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1338,2,Mod(1,1338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1338, 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("1338.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,-3] 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:	$10.6839837904$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.473.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 5x - 1 $ x^3 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.201640$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.95934 q^{5} +1.00000 q^{6} +3.20164 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.95934 q^{10} +2.20164 q^{11} -1.00000 q^{12} +6.95934 q^{13} -3.20164 q^{14} -2.95934 q^{15} +1.00000 q^{16} -3.55606 q^{17} -1.00000 q^{18} +4.75770 q^{19} +2.95934 q^{20} -3.20164 q^{21} -2.20164 q^{22} +5.71704 q^{23} +1.00000 q^{24} +3.75770 q^{25} -6.95934 q^{26} -1.00000 q^{27} +3.20164 q^{28} -2.79836 q^{29} +2.95934 q^{30} -4.35442 q^{31} -1.00000 q^{32} -2.20164 q^{33} +3.55606 q^{34} +9.47474 q^{35} +1.00000 q^{36} -11.6357 q^{37} -4.75770 q^{38} -6.95934 q^{39} -2.95934 q^{40} +4.95114 q^{41} +3.20164 q^{42} +3.35442 q^{43} +2.20164 q^{44} +2.95934 q^{45} -5.71704 q^{46} -7.36262 q^{47} -1.00000 q^{48} +3.25050 q^{49} -3.75770 q^{50} +3.55606 q^{51} +6.95934 q^{52} -8.67638 q^{53} +1.00000 q^{54} +6.51540 q^{55} -3.20164 q^{56} -4.75770 q^{57} +2.79836 q^{58} -11.8780 q^{59} -2.95934 q^{60} -5.91048 q^{61} +4.35442 q^{62} +3.20164 q^{63} +1.00000 q^{64} +20.5951 q^{65} +2.20164 q^{66} -4.87802 q^{67} -3.55606 q^{68} -5.71704 q^{69} -9.47474 q^{70} -4.59672 q^{71} -1.00000 q^{72} +0.749503 q^{73} +11.6357 q^{74} -3.75770 q^{75} +4.75770 q^{76} +7.04886 q^{77} +6.95934 q^{78} +12.9105 q^{79} +2.95934 q^{80} +1.00000 q^{81} -4.95114 q^{82} +15.4829 q^{83} -3.20164 q^{84} -10.5236 q^{85} -3.35442 q^{86} +2.79836 q^{87} -2.20164 q^{88} -18.5544 q^{89} -2.95934 q^{90} +22.2813 q^{91} +5.71704 q^{92} +4.35442 q^{93} +7.36262 q^{94} +14.0797 q^{95} +1.00000 q^{96} +1.95114 q^{97} -3.25050 q^{98} +2.20164 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 9 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} + 6 q^{11} - 3 q^{12} + 11 q^{13} - 9 q^{14} + q^{15} + 3 q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} - q^{20}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 9 * q^7 - 3 * q^8 + 3 * q^9 + q^10 + 6 * q^11 - 3 * q^12 + 11 * q^13 - 9 * q^14 + q^15 + 3 * q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 - q^20 - 9 * q^21 - 6 * q^22 - 2 * q^23 + 3 * q^24 + 2 * q^25 - 11 * q^26 - 3 * q^27 + 9 * q^28 - 9 * q^29 - q^30 - 5 * q^31 - 3 * q^32 - 6 * q^33 + 2 * q^34 + 3 * q^36 + 4 * q^37 - 5 * q^38 - 11 * q^39 + q^40 + 8 * q^41 + 9 * q^42 + 2 * q^43 + 6 * q^44 - q^45 + 2 * q^46 - 11 * q^47 - 3 * q^48 + 16 * q^49 - 2 * q^50 + 2 * q^51 + 11 * q^52 + 3 * q^53 + 3 * q^54 + q^55 - 9 * q^56 - 5 * q^57 + 9 * q^58 - 6 * q^59 + q^60 - q^61 + 5 * q^62 + 9 * q^63 + 3 * q^64 + 13 * q^65 + 6 * q^66 + 15 * q^67 - 2 * q^68 + 2 * q^69 - 15 * q^71 - 3 * q^72 - 4 * q^73 - 4 * q^74 - 2 * q^75 + 5 * q^76 + 28 * q^77 + 11 * q^78 + 22 * q^79 - q^80 + 3 * q^81 - 8 * q^82 + 15 * q^83 - 9 * q^84 - 10 * q^85 - 2 * q^86 + 9 * q^87 - 6 * q^88 + 3 * q^89 + q^90 + 36 * q^91 - 2 * q^92 + 5 * q^93 + 11 * q^94 + 12 * q^95 + 3 * q^96 - q^97 - 16 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.95934	1.32346	0.661729	−	0.749743i	$-0.269823\pi$
$5$	2.95934	1.32346	0.661729	+	0.749743i	$0.269823\pi$
$6$	1.00000	0.408248
$7$	3.20164	1.21011	0.605053	−	0.796185i	$-0.293152\pi$
$7$	3.20164	1.21011	0.605053	+	0.796185i	$0.293152\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.95934	−0.935826
$11$	2.20164	0.663819	0.331910	−	0.943311i	$-0.392307\pi$
$11$	2.20164	0.663819	0.331910	+	0.943311i	$0.392307\pi$
$12$	−1.00000	−0.288675
$13$	6.95934	1.93017	0.965087	−	0.261929i	$-0.0843588\pi$
$13$	6.95934	1.93017	0.965087	+	0.261929i	$0.0843588\pi$
$14$	−3.20164	−0.855674
$15$	−2.95934	−0.764099
$16$	1.00000	0.250000
$17$	−3.55606	−0.862472	−0.431236	−	0.902239i	$-0.641922\pi$
$17$	−3.55606	−0.862472	−0.431236	+	0.902239i	$0.641922\pi$
$18$	−1.00000	−0.235702
$19$	4.75770	1.09149	0.545746	−	0.837951i	$-0.316246\pi$
$19$	4.75770	1.09149	0.545746	+	0.837951i	$0.316246\pi$
$20$	2.95934	0.661729
$21$	−3.20164	−0.698655
$22$	−2.20164	−0.469391
$23$	5.71704	1.19209	0.596043	−	0.802953i	$-0.296739\pi$
$23$	5.71704	1.19209	0.596043	+	0.802953i	$0.296739\pi$
$24$	1.00000	0.204124
$25$	3.75770	0.751540
$26$	−6.95934	−1.36484
$27$	−1.00000	−0.192450
$28$	3.20164	0.605053
$29$	−2.79836	−0.519642	−0.259821	−	0.965657i	$-0.583664\pi$
$29$	−2.79836	−0.519642	−0.259821	+	0.965657i	$0.583664\pi$
$30$	2.95934	0.540299
$31$	−4.35442	−0.782077	−0.391039	−	0.920374i	$-0.627884\pi$
$31$	−4.35442	−0.782077	−0.391039	+	0.920374i	$0.627884\pi$
$32$	−1.00000	−0.176777
$33$	−2.20164	−0.383256
$34$	3.55606	0.609860
$35$	9.47474	1.60152
$36$	1.00000	0.166667
$37$	−11.6357	−1.91290	−0.956451	−	0.291894i	$-0.905715\pi$
$37$	−11.6357	−1.91290	−0.956451	+	0.291894i	$0.905715\pi$
$38$	−4.75770	−0.771801
$39$	−6.95934	−1.11439
$40$	−2.95934	−0.467913
$41$	4.95114	0.773239	0.386619	−	0.922239i	$-0.373643\pi$
$41$	4.95114	0.773239	0.386619	+	0.922239i	$0.373643\pi$
$42$	3.20164	0.494024
$43$	3.35442	0.511545	0.255772	−	0.966737i	$-0.417670\pi$
$43$	3.35442	0.511545	0.255772	+	0.966737i	$0.417670\pi$
$44$	2.20164	0.331910
$45$	2.95934	0.441153
$46$	−5.71704	−0.842932
$47$	−7.36262	−1.07395	−0.536974	−	0.843599i	$-0.680433\pi$
$47$	−7.36262	−1.07395	−0.536974	+	0.843599i	$0.680433\pi$
$48$	−1.00000	−0.144338
$49$	3.25050	0.464357
$50$	−3.75770	−0.531419
$51$	3.55606	0.497948
$52$	6.95934	0.965087
$53$	−8.67638	−1.19179	−0.595897	−	0.803061i	$-0.703203\pi$
$53$	−8.67638	−1.19179	−0.595897	+	0.803061i	$0.703203\pi$
$54$	1.00000	0.136083
$55$	6.51540	0.878537
$56$	−3.20164	−0.427837
$57$	−4.75770	−0.630173
$58$	2.79836	0.367443
$59$	−11.8780	−1.54639	−0.773194	−	0.634170i	$-0.781342\pi$
$59$	−11.8780	−1.54639	−0.773194	+	0.634170i	$0.781342\pi$
$60$	−2.95934	−0.382049
$61$	−5.91048	−0.756760	−0.378380	−	0.925650i	$-0.623519\pi$
$61$	−5.91048	−0.756760	−0.378380	+	0.925650i	$0.623519\pi$
$62$	4.35442	0.553012
$63$	3.20164	0.403369
$64$	1.00000	0.125000
$65$	20.5951	2.55450
$66$	2.20164	0.271003
$67$	−4.87802	−0.595946	−0.297973	−	0.954574i	$-0.596310\pi$
$67$	−4.87802	−0.595946	−0.297973	+	0.954574i	$0.596310\pi$
$68$	−3.55606	−0.431236
$69$	−5.71704	−0.688251
$70$	−9.47474	−1.13245
$71$	−4.59672	−0.545530	−0.272765	−	0.962081i	$-0.587938\pi$
$71$	−4.59672	−0.545530	−0.272765	+	0.962081i	$0.587938\pi$
$72$	−1.00000	−0.117851
$73$	0.749503	0.0877227	0.0438614	−	0.999038i	$-0.486034\pi$
$73$	0.749503	0.0877227	0.0438614	+	0.999038i	$0.486034\pi$
$74$	11.6357	1.35263
$75$	−3.75770	−0.433902
$76$	4.75770	0.545746
$77$	7.04886	0.803292
$78$	6.95934	0.787990
$79$	12.9105	1.45254	0.726271	−	0.687408i	$-0.241252\pi$
$79$	12.9105	1.45254	0.726271	+	0.687408i	$0.241252\pi$
$80$	2.95934	0.330864
$81$	1.00000	0.111111
$82$	−4.95114	−0.546762
$83$	15.4829	1.69947	0.849737	−	0.527207i	$-0.176761\pi$
$83$	15.4829	1.69947	0.849737	+	0.527207i	$0.176761\pi$
$84$	−3.20164	−0.349328
$85$	−10.5236	−1.14144
$86$	−3.35442	−0.361717
$87$	2.79836	0.300016
$88$	−2.20164	−0.234696
$89$	−18.5544	−1.96676	−0.983382	−	0.181550i	$-0.941889\pi$
$89$	−18.5544	−1.96676	−0.983382	+	0.181550i	$0.941889\pi$
$90$	−2.95934	−0.311942
$91$	22.2813	2.33572
$92$	5.71704	0.596043
$93$	4.35442	0.451533
$94$	7.36262	0.759396
$95$	14.0797	1.44454
$96$	1.00000	0.102062
$97$	1.95114	0.198109	0.0990543	−	0.995082i	$-0.468418\pi$
$97$	1.95114	0.198109	0.0990543	+	0.995082i	$0.468418\pi$
$98$	−3.25050	−0.328350
$99$	2.20164	0.221273
$100$	3.75770	0.375770
$101$	−9.92688	−0.987762	−0.493881	−	0.869530i	$-0.664422\pi$
$101$	−9.92688	−0.987762	−0.493881	+	0.869530i	$0.664422\pi$
$102$	−3.55606	−0.352103
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−6.95934	−0.682420
$105$	−9.47474	−0.924640
$106$	8.67638	0.842725
$107$	−12.5951	−1.21761	−0.608806	−	0.793319i	$-0.708351\pi$
$107$	−12.5951	−1.21761	−0.608806	+	0.793319i	$0.708351\pi$
$108$	−1.00000	−0.0962250
$109$	11.4829	1.09987	0.549933	−	0.835209i	$-0.314653\pi$
$109$	11.4829	1.09987	0.549933	+	0.835209i	$0.314653\pi$
$110$	−6.51540	−0.621219
$111$	11.6357	1.10441
$112$	3.20164	0.302527
$113$	−3.60492	−0.339122	−0.169561	−	0.985520i	$-0.554235\pi$
$113$	−3.60492	−0.339122	−0.169561	+	0.985520i	$0.554235\pi$
$114$	4.75770	0.445600
$115$	16.9187	1.57768
$116$	−2.79836	−0.259821
$117$	6.95934	0.643391
$118$	11.8780	1.09346
$119$	−11.3852	−1.04368
$120$	2.95934	0.270150
$121$	−6.15278	−0.559344
$122$	5.91048	0.535110
$123$	−4.95114	−0.446430
$124$	−4.35442	−0.391039
$125$	−3.67638	−0.328826
$126$	−3.20164	−0.285225
$127$	16.7885	1.48974	0.744870	−	0.667210i	$-0.232512\pi$
$127$	16.7885	1.48974	0.744870	+	0.667210i	$0.232512\pi$
$128$	−1.00000	−0.0883883
$129$	−3.35442	−0.295340
$130$	−20.5951	−1.80631
$131$	−2.91868	−0.255007	−0.127503	−	0.991838i	$-0.540696\pi$
$131$	−2.91868	−0.255007	−0.127503	+	0.991838i	$0.540696\pi$
$132$	−2.20164	−0.191628
$133$	15.2324	1.32082
$134$	4.87802	0.421397
$135$	−2.95934	−0.254700
$136$	3.55606	0.304930
$137$	11.2895	0.964527	0.482264	−	0.876026i	$-0.339815\pi$
$137$	11.2895	0.964527	0.482264	+	0.876026i	$0.339815\pi$
$138$	5.71704	0.486667
$139$	16.3626	1.38786	0.693930	−	0.720043i	$-0.255878\pi$
$139$	16.3626	1.38786	0.693930	+	0.720043i	$0.255878\pi$
$140$	9.47474	0.800762
$141$	7.36262	0.620045
$142$	4.59672	0.385748
$143$	15.3220	1.28129
$144$	1.00000	0.0833333
$145$	−8.28130	−0.687725
$146$	−0.749503	−0.0620293
$147$	−3.25050	−0.268096
$148$	−11.6357	−0.956451
$149$	5.16098	0.422804	0.211402	−	0.977399i	$-0.432197\pi$
$149$	5.16098	0.422804	0.211402	+	0.977399i	$0.432197\pi$
$150$	3.75770	0.306815
$151$	−13.9675	−1.13666	−0.568331	−	0.822800i	$-0.692411\pi$
$151$	−13.9675	−1.13666	−0.568331	+	0.822800i	$0.692411\pi$
$152$	−4.75770	−0.385901
$153$	−3.55606	−0.287491
$154$	−7.04886	−0.568013
$155$	−12.8862	−1.03505
$156$	−6.95934	−0.557193
$157$	15.3302	1.22348	0.611740	−	0.791059i	$-0.290470\pi$
$157$	15.3302	1.22348	0.611740	+	0.791059i	$0.290470\pi$
$158$	−12.9105	−1.02710
$159$	8.67638	0.688082
$160$	−2.95934	−0.233956
$161$	18.3039	1.44255
$162$	−1.00000	−0.0785674
$163$	−22.8682	−1.79117	−0.895587	−	0.444887i	$-0.853244\pi$
$163$	−22.8682	−1.79117	−0.895587	+	0.444887i	$0.853244\pi$
$164$	4.95114	0.386619
$165$	−6.51540	−0.507223
$166$	−15.4829	−1.20171
$167$	6.32362	0.489336	0.244668	−	0.969607i	$-0.421321\pi$
$167$	6.32362	0.489336	0.244668	+	0.969607i	$0.421321\pi$
$168$	3.20164	0.247012
$169$	35.4324	2.72557
$170$	10.5236	0.807123
$171$	4.75770	0.363831
$172$	3.35442	0.255772
$173$	−7.16918	−0.545063	−0.272531	−	0.962147i	$-0.587861\pi$
$173$	−7.16918	−0.545063	−0.272531	+	0.962147i	$0.587861\pi$
$174$	−2.79836	−0.212143
$175$	12.0308	0.909444
$176$	2.20164	0.165955
$177$	11.8780	0.892807
$178$	18.5544	1.39071
$179$	5.42589	0.405550	0.202775	−	0.979225i	$-0.435004\pi$
$179$	5.42589	0.405550	0.202775	+	0.979225i	$0.435004\pi$
$180$	2.95934	0.220576
$181$	−7.36262	−0.547259	−0.273630	−	0.961835i	$-0.588224\pi$
$181$	−7.36262	−0.547259	−0.273630	+	0.961835i	$0.588224\pi$
$182$	−22.2813	−1.65160
$183$	5.91048	0.436916
$184$	−5.71704	−0.421466
$185$	−34.4341	−2.53164
$186$	−4.35442	−0.319282
$187$	−7.82917	−0.572525
$188$	−7.36262	−0.536974
$189$	−3.20164	−0.232885
$190$	−14.0797	−1.02145
$191$	21.7885	1.57656	0.788281	−	0.615316i	$-0.210972\pi$
$191$	21.7885	1.57656	0.788281	+	0.615316i	$0.210972\pi$
$192$	−1.00000	−0.0721688
$193$	−11.4423	−0.823634	−0.411817	−	0.911267i	$-0.635106\pi$
$193$	−11.4423	−0.823634	−0.411817	+	0.911267i	$0.635106\pi$
$194$	−1.95114	−0.140084
$195$	−20.5951	−1.47484
$196$	3.25050	0.232178
$197$	0.0406586	0.00289680	0.00144840	−	0.999999i	$-0.499539\pi$
$197$	0.0406586	0.00289680	0.00144840	+	0.999999i	$0.499539\pi$
$198$	−2.20164	−0.156464
$199$	−9.60492	−0.680875	−0.340437	−	0.940267i	$-0.610575\pi$
$199$	−9.60492	−0.680875	−0.340437	+	0.940267i	$0.610575\pi$
$200$	−3.75770	−0.265710
$201$	4.87802	0.344069
$202$	9.92688	0.698453
$203$	−8.95934	−0.628822
$204$	3.55606	0.248974
$205$	14.6521	1.02335
$206$	−14.0000	−0.975426
$207$	5.71704	0.397362
$208$	6.95934	0.482544
$209$	10.4747	0.724553
$210$	9.47474	0.653820
$211$	−4.51540	−0.310853	−0.155427	−	0.987847i	$-0.549675\pi$
$211$	−4.51540	−0.310853	−0.155427	+	0.987847i	$0.549675\pi$
$212$	−8.67638	−0.595897
$213$	4.59672	0.314962
$214$	12.5951	0.860981
$215$	9.92688	0.677008
$216$	1.00000	0.0680414
$217$	−13.9413	−0.946397
$218$	−11.4829	−0.777723
$219$	−0.749503	−0.0506467
$220$	6.51540	0.439268
$221$	−24.7479	−1.66472
$222$	−11.6357	−0.780939
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−3.20164	−0.213919
$225$	3.75770	0.250513
$226$	3.60492	0.239796
$227$	−10.8292	−0.718757	−0.359379	−	0.933192i	$-0.617011\pi$
$227$	−10.8292	−0.718757	−0.359379	+	0.933192i	$0.617011\pi$
$228$	−4.75770	−0.315086
$229$	−28.2082	−1.86405	−0.932025	−	0.362395i	$-0.881959\pi$
$229$	−28.2082	−1.86405	−0.932025	+	0.362395i	$0.881959\pi$
$230$	−16.9187	−1.11558
$231$	−7.04886	−0.463781
$232$	2.79836	0.183721
$233$	−23.9901	−1.57165	−0.785823	−	0.618451i	$-0.787761\pi$
$233$	−23.9901	−1.57165	−0.785823	+	0.618451i	$0.787761\pi$
$234$	−6.95934	−0.454946
$235$	−21.7885	−1.42133
$236$	−11.8780	−0.773194
$237$	−12.9105	−0.838626
$238$	11.3852	0.737995
$239$	11.7741	0.761603	0.380802	−	0.924657i	$-0.375648\pi$
$239$	11.7741	0.761603	0.380802	+	0.924657i	$0.375648\pi$
$240$	−2.95934	−0.191025
$241$	−23.7252	−1.52828	−0.764139	−	0.645052i	$-0.776836\pi$
$241$	−23.7252	−1.52828	−0.764139	+	0.645052i	$0.776836\pi$
$242$	6.15278	0.395516
$243$	−1.00000	−0.0641500
$244$	−5.91048	−0.378380
$245$	9.61933	0.614556
$246$	4.95114	0.315673
$247$	33.1105	2.10677
$248$	4.35442	0.276506
$249$	−15.4829	−0.981192
$250$	3.67638	0.232515
$251$	−14.6033	−0.921750	−0.460875	−	0.887465i	$-0.652464\pi$
$251$	−14.6033	−0.921750	−0.460875	+	0.887465i	$0.652464\pi$
$252$	3.20164	0.201684
$253$	12.5869	0.791330
$254$	−16.7885	−1.05340
$255$	10.5236	0.659014
$256$	1.00000	0.0625000
$257$	−0.234100	−0.0146027	−0.00730137	−	0.999973i	$-0.502324\pi$
$257$	−0.234100	−0.0146027	−0.00730137	+	0.999973i	$0.502324\pi$
$258$	3.35442	0.208837
$259$	−37.2534	−2.31481
$260$	20.5951	1.27725
$261$	−2.79836	−0.173214
$262$	2.91868	0.180317
$263$	−12.3138	−0.759299	−0.379650	−	0.925130i	$-0.623955\pi$
$263$	−12.3138	−0.759299	−0.379650	+	0.925130i	$0.623955\pi$
$264$	2.20164	0.135502
$265$	−25.6764	−1.57729
$266$	−15.2324	−0.933961
$267$	18.5544	1.13551
$268$	−4.87802	−0.297973
$269$	5.35442	0.326465	0.163232	−	0.986588i	$-0.447808\pi$
$269$	5.35442	0.326465	0.163232	+	0.986588i	$0.447808\pi$
$270$	2.95934	0.180100
$271$	−22.5462	−1.36958	−0.684792	−	0.728738i	$-0.740107\pi$
$271$	−22.5462	−1.36958	−0.684792	+	0.728738i	$0.740107\pi$
$272$	−3.55606	−0.215618
$273$	−22.2813	−1.34853
$274$	−11.2895	−0.682024
$275$	8.27311	0.498887
$276$	−5.71704	−0.344126
$277$	24.0308	1.44387	0.721936	−	0.691960i	$-0.243253\pi$
$277$	24.0308	1.44387	0.721936	+	0.691960i	$0.243253\pi$
$278$	−16.3626	−0.981365
$279$	−4.35442	−0.260692
$280$	−9.47474	−0.566224
$281$	1.81476	0.108259	0.0541297	−	0.998534i	$-0.482762\pi$
$281$	1.81476	0.108259	0.0541297	+	0.998534i	$0.482762\pi$
$282$	−7.36262	−0.438438
$283$	15.8374	0.941434	0.470717	−	0.882284i	$-0.343995\pi$
$283$	15.8374	0.941434	0.470717	+	0.882284i	$0.343995\pi$
$284$	−4.59672	−0.272765
$285$	−14.0797	−0.834007
$286$	−15.3220	−0.906007
$287$	15.8518	0.935701
$288$	−1.00000	−0.0589256
$289$	−4.35442	−0.256142
$290$	8.28130	0.486295
$291$	−1.95114	−0.114378
$292$	0.749503	0.0438614
$293$	17.9088	1.04625	0.523123	−	0.852257i	$-0.324767\pi$
$293$	17.9088	1.04625	0.523123	+	0.852257i	$0.324767\pi$
$294$	3.25050	0.189573
$295$	−35.1511	−2.04658
$296$	11.6357	0.676313
$297$	−2.20164	−0.127752
$298$	−5.16098	−0.298968
$299$	39.7869	2.30093
$300$	−3.75770	−0.216951
$301$	10.7397	0.619023
$302$	13.9675	0.803742
$303$	9.92688	0.570284
$304$	4.75770	0.272873
$305$	−17.4911	−1.00154
$306$	3.55606	0.203287
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	7.04886	0.401646
$309$	−14.0000	−0.796432
$310$	12.8862	0.731888
$311$	14.0065	0.794238	0.397119	−	0.917767i	$-0.370010\pi$
$311$	14.0065	0.794238	0.397119	+	0.917767i	$0.370010\pi$
$312$	6.95934	0.393995
$313$	14.4829	0.818624	0.409312	−	0.912394i	$-0.365769\pi$
$313$	14.4829	0.818624	0.409312	+	0.912394i	$0.365769\pi$
$314$	−15.3302	−0.865131
$315$	9.47474	0.533841
$316$	12.9105	0.726271
$317$	35.1511	1.97428	0.987142	−	0.159845i	$-0.0510996\pi$
$317$	35.1511	1.97428	0.987142	+	0.159845i	$0.0510996\pi$
$318$	−8.67638	−0.486547
$319$	−6.16098	−0.344949
$320$	2.95934	0.165432
$321$	12.5951	0.702988
$322$	−18.3039	−1.02004
$323$	−16.9187	−0.941381
$324$	1.00000	0.0555556
$325$	26.1511	1.45060
$326$	22.8682	1.26655
$327$	−11.4829	−0.635008
$328$	−4.95114	−0.273381
$329$	−23.5725	−1.29959
$330$	6.51540	0.358661
$331$	1.42589	0.0783739	0.0391869	−	0.999232i	$-0.487523\pi$
$331$	1.42589	0.0783739	0.0391869	+	0.999232i	$0.487523\pi$
$332$	15.4829	0.849737
$333$	−11.6357	−0.637634
$334$	−6.32362	−0.346013
$335$	−14.4357	−0.788709
$336$	−3.20164	−0.174664
$337$	−11.1220	−0.605853	−0.302926	−	0.953014i	$-0.597964\pi$
$337$	−11.1220	−0.605853	−0.302926	+	0.953014i	$0.597964\pi$
$338$	−35.4324	−1.92727
$339$	3.60492	0.195792
$340$	−10.5236	−0.570722
$341$	−9.58687	−0.519158
$342$	−4.75770	−0.257267
$343$	−12.0046	−0.648185
$344$	−3.35442	−0.180858
$345$	−16.9187	−0.910871
$346$	7.16918	0.385417
$347$	29.0226	1.55802	0.779008	−	0.627014i	$-0.215723\pi$
$347$	29.0226	1.55802	0.779008	+	0.627014i	$0.215723\pi$
$348$	2.79836	0.150008
$349$	−5.36427	−0.287143	−0.143571	−	0.989640i	$-0.545859\pi$
$349$	−5.36427	−0.287143	−0.143571	+	0.989640i	$0.545859\pi$
$350$	−12.0308	−0.643074
$351$	−6.95934	−0.371462
$352$	−2.20164	−0.117348
$353$	5.12032	0.272527	0.136264	−	0.990673i	$-0.456491\pi$
$353$	5.12032	0.272527	0.136264	+	0.990673i	$0.456491\pi$
$354$	−11.8780	−0.631310
$355$	−13.6033	−0.721986
$356$	−18.5544	−0.983382
$357$	11.3852	0.602570
$358$	−5.42589	−0.286767
$359$	−3.35277	−0.176952	−0.0884762	−	0.996078i	$-0.528200\pi$
$359$	−3.35277	−0.176952	−0.0884762	+	0.996078i	$0.528200\pi$
$360$	−2.95934	−0.155971
$361$	3.63573	0.191354
$362$	7.36262	0.386971
$363$	6.15278	0.322937
$364$	22.2813	1.16786
$365$	2.21804	0.116097
$366$	−5.91048	−0.308946
$367$	13.4928	0.704318	0.352159	−	0.935940i	$-0.385448\pi$
$367$	13.4928	0.704318	0.352159	+	0.935940i	$0.385448\pi$
$368$	5.71704	0.298021
$369$	4.95114	0.257746
$370$	34.4341	1.79014
$371$	−27.7787	−1.44220
$372$	4.35442	0.225766
$373$	1.37082	0.0709783	0.0354892	−	0.999370i	$-0.488701\pi$
$373$	1.37082	0.0709783	0.0354892	+	0.999370i	$0.488701\pi$
$374$	7.82917	0.404837
$375$	3.67638	0.189848
$376$	7.36262	0.379698
$377$	−19.4747	−1.00300
$378$	3.20164	0.164675
$379$	22.3934	1.15027	0.575137	−	0.818057i	$-0.304949\pi$
$379$	22.3934	1.15027	0.575137	+	0.818057i	$0.304949\pi$
$380$	14.0797	0.722272
$381$	−16.7885	−0.860101
$382$	−21.7885	−1.11480
$383$	−12.7577	−0.651888	−0.325944	−	0.945389i	$-0.605682\pi$
$383$	−12.7577	−0.651888	−0.325944	+	0.945389i	$0.605682\pi$
$384$	1.00000	0.0510310
$385$	20.8600	1.06312
$386$	11.4423	0.582397
$387$	3.35442	0.170515
$388$	1.95114	0.0990543
$389$	−36.5236	−1.85182	−0.925910	−	0.377744i	$-0.876700\pi$
$389$	−36.5236	−1.85182	−0.925910	+	0.377744i	$0.876700\pi$
$390$	20.5951	1.04287
$391$	−20.3302	−1.02814
$392$	−3.25050	−0.164175
$393$	2.91868	0.147228
$394$	−0.0406586	−0.00204835
$395$	38.2065	1.92238
$396$	2.20164	0.110637
$397$	23.5787	1.18338	0.591690	−	0.806166i	$-0.298461\pi$
$397$	23.5787	1.18338	0.591690	+	0.806166i	$0.298461\pi$
$398$	9.60492	0.481451
$399$	−15.2324	−0.762576
$400$	3.75770	0.187885
$401$	−12.5561	−0.627020	−0.313510	−	0.949585i	$-0.601505\pi$
$401$	−12.5561	−0.627020	−0.313510	+	0.949585i	$0.601505\pi$
$402$	−4.87802	−0.243294
$403$	−30.3039	−1.50955
$404$	−9.92688	−0.493881
$405$	2.95934	0.147051
$406$	8.95934	0.444645
$407$	−25.6177	−1.26982
$408$	−3.55606	−0.176051
$409$	−28.1187	−1.39038	−0.695189	−	0.718827i	$-0.744679\pi$
$409$	−28.1187	−1.39038	−0.695189	+	0.718827i	$0.744679\pi$
$410$	−14.6521	−0.723617
$411$	−11.2895	−0.556870
$412$	14.0000	0.689730
$413$	−38.0292	−1.87129
$414$	−5.71704	−0.280977
$415$	45.8193	2.24918
$416$	−6.95934	−0.341210
$417$	−16.3626	−0.801281
$418$	−10.4747	−0.512337
$419$	19.8924	0.971809	0.485905	−	0.874012i	$-0.338490\pi$
$419$	19.8924	0.971809	0.485905	+	0.874012i	$0.338490\pi$
$420$	−9.47474	−0.462320
$421$	4.64392	0.226331	0.113166	−	0.993576i	$-0.463901\pi$
$421$	4.64392	0.226331	0.113166	+	0.993576i	$0.463901\pi$
$422$	4.51540	0.219806
$423$	−7.36262	−0.357983
$424$	8.67638	0.421362
$425$	−13.3626	−0.648182
$426$	−4.59672	−0.222712
$427$	−18.9232	−0.915760
$428$	−12.5951	−0.608806
$429$	−15.3220	−0.739751
$430$	−9.92688	−0.478717
$431$	5.78031	0.278428	0.139214	−	0.990262i	$-0.455542\pi$
$431$	5.78031	0.278428	0.139214	+	0.990262i	$0.455542\pi$
$432$	−1.00000	−0.0481125
$433$	28.0862	1.34974	0.674869	−	0.737938i	$-0.264200\pi$
$433$	28.0862	1.34974	0.674869	+	0.737938i	$0.264200\pi$
$434$	13.9413	0.669203
$435$	8.28130	0.397058
$436$	11.4829	0.549933
$437$	27.2000	1.30115
$438$	0.749503	0.0358127
$439$	−31.4423	−1.50066	−0.750329	−	0.661064i	$-0.770105\pi$
$439$	−31.4423	−1.50066	−0.750329	+	0.661064i	$0.770105\pi$
$440$	−6.51540	−0.310610
$441$	3.25050	0.154786
$442$	24.7479	1.17714
$443$	2.34622	0.111472	0.0557362	−	0.998446i	$-0.482249\pi$
$443$	2.34622	0.111472	0.0557362	+	0.998446i	$0.482249\pi$
$444$	11.6357	0.552207
$445$	−54.9088	−2.60293
$446$	−1.00000	−0.0473514
$447$	−5.16098	−0.244106
$448$	3.20164	0.151263
$449$	18.1934	0.858602	0.429301	−	0.903162i	$-0.358760\pi$
$449$	18.1934	0.858602	0.429301	+	0.903162i	$0.358760\pi$
$450$	−3.75770	−0.177140
$451$	10.9006	0.513291
$452$	−3.60492	−0.169561
$453$	13.9675	0.656252
$454$	10.8292	0.508238
$455$	65.9380	3.09122
$456$	4.75770	0.222800
$457$	−35.1757	−1.64545	−0.822726	−	0.568439i	$-0.807548\pi$
$457$	−35.1757	−1.64545	−0.822726	+	0.568439i	$0.807548\pi$
$458$	28.2082	1.31808
$459$	3.55606	0.165983
$460$	16.9187	0.788838
$461$	−2.26491	−0.105487	−0.0527436	−	0.998608i	$-0.516797\pi$
$461$	−2.26491	−0.105487	−0.0527436	+	0.998608i	$0.516797\pi$
$462$	7.04886	0.327943
$463$	−11.6682	−0.542267	−0.271133	−	0.962542i	$-0.587398\pi$
$463$	−11.6682	−0.542267	−0.271133	+	0.962542i	$0.587398\pi$
$464$	−2.79836	−0.129911
$465$	12.8862	0.597584
$466$	23.9901	1.11132
$467$	13.3462	0.617590	0.308795	−	0.951129i	$-0.400074\pi$
$467$	13.3462	0.617590	0.308795	+	0.951129i	$0.400074\pi$
$468$	6.95934	0.321696
$469$	−15.6177	−0.721157
$470$	21.7885	1.00503
$471$	−15.3302	−0.706377
$472$	11.8780	0.546730
$473$	7.38523	0.339573
$474$	12.9105	0.592998
$475$	17.8780	0.820300
$476$	−11.3852	−0.521841
$477$	−8.67638	−0.397264
$478$	−11.7741	−0.538535
$479$	−21.4829	−0.981581	−0.490790	−	0.871278i	$-0.663292\pi$
$479$	−21.4829	−0.981581	−0.490790	+	0.871278i	$0.663292\pi$
$480$	2.95934	0.135075
$481$	−80.9770	−3.69223
$482$	23.7252	1.08066
$483$	−18.3039	−0.832857
$484$	−6.15278	−0.279672
$485$	5.77410	0.262188
$486$	1.00000	0.0453609
$487$	30.6033	1.38677	0.693383	−	0.720569i	$-0.256119\pi$
$487$	30.6033	1.38677	0.693383	+	0.720569i	$0.256119\pi$
$488$	5.91048	0.267555
$489$	22.8682	1.03413
$490$	−9.61933	−0.434557
$491$	−34.5770	−1.56044	−0.780219	−	0.625506i	$-0.784893\pi$
$491$	−34.5770	−1.56044	−0.780219	+	0.625506i	$0.784893\pi$
$492$	−4.95114	−0.223215
$493$	9.95114	0.448177
$494$	−33.1105	−1.48971
$495$	6.51540	0.292846
$496$	−4.35442	−0.195519
$497$	−14.7170	−0.660150
$498$	15.4829	0.693807
$499$	12.3380	0.552326	0.276163	−	0.961111i	$-0.410937\pi$
$499$	12.3380	0.552326	0.276163	+	0.961111i	$0.410937\pi$
$500$	−3.67638	−0.164413
$501$	−6.32362	−0.282518
$502$	14.6033	0.651776
$503$	7.41313	0.330535	0.165268	−	0.986249i	$-0.447151\pi$
$503$	7.41313	0.330535	0.165268	+	0.986249i	$0.447151\pi$
$504$	−3.20164	−0.142612
$505$	−29.3770	−1.30726
$506$	−12.5869	−0.559555
$507$	−35.4324	−1.57361
$508$	16.7885	0.744870
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−10.5236	−0.465993
$511$	2.39964	0.106154
$512$	−1.00000	−0.0441942
$513$	−4.75770	−0.210058
$514$	0.234100	0.0103257
$515$	41.4308	1.82566
$516$	−3.35442	−0.147670
$517$	−16.2098	−0.712908
$518$	37.2534	1.63682
$519$	7.16918	0.314692
$520$	−20.5951	−0.903153
$521$	5.51706	0.241707	0.120853	−	0.992670i	$-0.461437\pi$
$521$	5.51706	0.241707	0.120853	+	0.992670i	$0.461437\pi$
$522$	2.79836	0.122481
$523$	−13.6033	−0.594829	−0.297415	−	0.954748i	$-0.596124\pi$
$523$	−13.6033	−0.594829	−0.297415	+	0.954748i	$0.596124\pi$
$524$	−2.91868	−0.127503
$525$	−12.0308	−0.525067
$526$	12.3138	0.536906
$527$	15.4846	0.674520
$528$	−2.20164	−0.0958141
$529$	9.68458	0.421069
$530$	25.6764	1.11531
$531$	−11.8780	−0.515462
$532$	15.2324	0.660410
$533$	34.4567	1.49249
$534$	−18.5544	−0.802928
$535$	−37.2731	−1.61146
$536$	4.87802	0.210699
$537$	−5.42589	−0.234144
$538$	−5.35442	−0.230846
$539$	7.15642	0.308249
$540$	−2.95934	−0.127350
$541$	1.75935	0.0756406	0.0378203	−	0.999285i	$-0.487959\pi$
$541$	1.75935	0.0756406	0.0378203	+	0.999285i	$0.487959\pi$
$542$	22.5462	0.968443
$543$	7.36262	0.315960
$544$	3.55606	0.152465
$545$	33.9820	1.45563
$546$	22.2813	0.953552
$547$	41.4242	1.77117	0.885586	−	0.464475i	$-0.153757\pi$
$547$	41.4242	1.77117	0.885586	+	0.464475i	$0.153757\pi$
$548$	11.2895	0.482264
$549$	−5.91048	−0.252253
$550$	−8.27311	−0.352766
$551$	−13.3138	−0.567185
$552$	5.71704	0.243334
$553$	41.3347	1.75773
$554$	−24.0308	−1.02097
$555$	34.4341	1.46165
$556$	16.3626	0.693930
$557$	23.7659	1.00699	0.503497	−	0.863997i	$-0.332047\pi$
$557$	23.7659	1.00699	0.503497	+	0.863997i	$0.332047\pi$
$558$	4.35442	0.184337
$559$	23.3446	0.987370
$560$	9.47474	0.400381
$561$	7.82917	0.330548
$562$	−1.81476	−0.0765509
$563$	23.7088	0.999209	0.499604	−	0.866254i	$-0.333479\pi$
$563$	23.7088	0.999209	0.499604	+	0.866254i	$0.333479\pi$
$564$	7.36262	0.310022
$565$	−10.6682	−0.448814
$566$	−15.8374	−0.665694
$567$	3.20164	0.134456
$568$	4.59672	0.192874
$569$	14.2731	0.598360	0.299180	−	0.954197i	$-0.403287\pi$
$569$	14.2731	0.598360	0.299180	+	0.954197i	$0.403287\pi$
$570$	14.0797	0.589732
$571$	9.87968	0.413452	0.206726	−	0.978399i	$-0.433719\pi$
$571$	9.87968	0.413452	0.206726	+	0.978399i	$0.433719\pi$
$572$	15.3220	0.640643
$573$	−21.7885	−0.910228
$574$	−15.8518	−0.661640
$575$	21.4829	0.895901
$576$	1.00000	0.0416667
$577$	23.7741	0.989729	0.494864	−	0.868970i	$-0.335218\pi$
$577$	23.7741	0.989729	0.494864	+	0.868970i	$0.335218\pi$
$578$	4.35442	0.181120
$579$	11.4423	0.475525
$580$	−8.28130	−0.343862
$581$	49.5708	2.05654
$582$	1.95114	0.0808775
$583$	−19.1023	−0.791135
$584$	−0.749503	−0.0310147
$585$	20.5951	0.851501
$586$	−17.9088	−0.739807
$587$	−20.4567	−0.844338	−0.422169	−	0.906517i	$-0.638731\pi$
$587$	−20.4567	−0.844338	−0.422169	+	0.906517i	$0.638731\pi$
$588$	−3.25050	−0.134048
$589$	−20.7170	−0.853631
$590$	35.1511	1.44715
$591$	−0.0406586	−0.00167247
$592$	−11.6357	−0.478225
$593$	19.3626	0.795128	0.397564	−	0.917575i	$-0.369856\pi$
$593$	19.3626	0.795128	0.397564	+	0.917575i	$0.369856\pi$
$594$	2.20164	0.0903344
$595$	−33.6928	−1.38127
$596$	5.16098	0.211402
$597$	9.60492	0.393103
$598$	−39.7869	−1.62701
$599$	−39.8193	−1.62697	−0.813487	−	0.581584i	$-0.802433\pi$
$599$	−39.8193	−1.62697	−0.813487	+	0.581584i	$0.802433\pi$
$600$	3.75770	0.153408
$601$	−21.0226	−0.857530	−0.428765	−	0.903416i	$-0.641051\pi$
$601$	−21.0226	−0.857530	−0.428765	+	0.903416i	$0.641051\pi$
$602$	−10.7397	−0.437716
$603$	−4.87802	−0.198649
$604$	−13.9675	−0.568331
$605$	−18.2082	−0.740268
$606$	−9.92688	−0.403252
$607$	−22.5400	−0.914870	−0.457435	−	0.889243i	$-0.651232\pi$
$607$	−22.5400	−0.914870	−0.457435	+	0.889243i	$0.651232\pi$
$608$	−4.75770	−0.192950
$609$	8.95934	0.363051
$610$	17.4911	0.708196
$611$	−51.2390	−2.07291
$612$	−3.55606	−0.143745
$613$	11.7560	0.474822	0.237411	−	0.971409i	$-0.423701\pi$
$613$	11.7560	0.474822	0.237411	+	0.971409i	$0.423701\pi$
$614$	−7.00000	−0.282497
$615$	−14.6521	−0.590831
$616$	−7.04886	−0.284007
$617$	−12.4423	−0.500908	−0.250454	−	0.968129i	$-0.580580\pi$
$617$	−12.4423	−0.500908	−0.250454	+	0.968129i	$0.580580\pi$
$618$	14.0000	0.563163
$619$	−5.51375	−0.221616	−0.110808	−	0.993842i	$-0.535344\pi$
$619$	−5.51375	−0.221616	−0.110808	+	0.993842i	$0.535344\pi$
$620$	−12.8862	−0.517523
$621$	−5.71704	−0.229417
$622$	−14.0065	−0.561611
$623$	−59.4045	−2.37999
$624$	−6.95934	−0.278597
$625$	−29.6682	−1.18673
$626$	−14.4829	−0.578855
$627$	−10.4747	−0.418321
$628$	15.3302	0.611740
$629$	41.3774	1.64982
$630$	−9.47474	−0.377483
$631$	23.6911	0.943129	0.471564	−	0.881832i	$-0.343689\pi$
$631$	23.6911	0.943129	0.471564	+	0.881832i	$0.343689\pi$
$632$	−12.9105	−0.513551
$633$	4.51540	0.179471
$634$	−35.1511	−1.39603
$635$	49.6829	1.97161
$636$	8.67638	0.344041
$637$	22.6213	0.896289
$638$	6.16098	0.243916
$639$	−4.59672	−0.181843
$640$	−2.95934	−0.116978
$641$	14.3121	0.565294	0.282647	−	0.959224i	$-0.408787\pi$
$641$	14.3121	0.565294	0.282647	+	0.959224i	$0.408787\pi$
$642$	−12.5951	−0.497088
$643$	−9.78851	−0.386021	−0.193011	−	0.981197i	$-0.561825\pi$
$643$	−9.78851	−0.386021	−0.193011	+	0.981197i	$0.561825\pi$
$644$	18.3039	0.721275
$645$	−9.92688	−0.390871
$646$	16.9187	0.665657
$647$	−1.16098	−0.0456429	−0.0228214	−	0.999740i	$-0.507265\pi$
$647$	−1.16098	−0.0456429	−0.0228214	+	0.999740i	$0.507265\pi$
$648$	−1.00000	−0.0392837
$649$	−26.1511	−1.02652
$650$	−26.1511	−1.02573
$651$	13.9413	0.546402
$652$	−22.8682	−0.895587
$653$	−9.12687	−0.357162	−0.178581	−	0.983925i	$-0.557151\pi$
$653$	−9.12687	−0.357162	−0.178581	+	0.983925i	$0.557151\pi$
$654$	11.4829	0.449019
$655$	−8.63738	−0.337490
$656$	4.95114	0.193310
$657$	0.749503	0.0292409
$658$	23.5725	0.918950
$659$	12.8698	0.501337	0.250669	−	0.968073i	$-0.419350\pi$
$659$	12.8698	0.501337	0.250669	+	0.968073i	$0.419350\pi$
$660$	−6.51540	−0.253612
$661$	22.6341	0.880363	0.440182	−	0.897909i	$-0.354914\pi$
$661$	22.6341	0.880363	0.440182	+	0.897909i	$0.354914\pi$
$662$	−1.42589	−0.0554187
$663$	24.7479	0.961127
$664$	−15.4829	−0.600855
$665$	45.0780	1.74805
$666$	11.6357	0.450875
$667$	−15.9983	−0.619458
$668$	6.32362	0.244668
$669$	−1.00000	−0.0386622
$670$	14.4357	0.557701
$671$	−13.0128	−0.502352
$672$	3.20164	0.123506
$673$	−0.468533	−0.0180606	−0.00903031	−	0.999959i	$-0.502874\pi$
$673$	−0.468533	−0.0180606	−0.00903031	+	0.999959i	$0.502874\pi$
$674$	11.1220	0.428402
$675$	−3.75770	−0.144634
$676$	35.4324	1.36279
$677$	−49.4406	−1.90016	−0.950079	−	0.312010i	$-0.898998\pi$
$677$	−49.4406	−1.90016	−0.950079	+	0.312010i	$0.898998\pi$
$678$	−3.60492	−0.138446
$679$	6.24686	0.239732
$680$	10.5236	0.403562
$681$	10.8292	0.414975
$682$	9.58687	0.367100
$683$	16.8616	0.645192	0.322596	−	0.946537i	$-0.395444\pi$
$683$	16.8616	0.645192	0.322596	+	0.946537i	$0.395444\pi$
$684$	4.75770	0.181915
$685$	33.4095	1.27651
$686$	12.0046	0.458336
$687$	28.2082	1.07621
$688$	3.35442	0.127886
$689$	−60.3819	−2.30037
$690$	16.9187	0.644083
$691$	−14.9088	−0.567159	−0.283579	−	0.958949i	$-0.591522\pi$
$691$	−14.9088	−0.567159	−0.283579	+	0.958949i	$0.591522\pi$
$692$	−7.16918	−0.272531
$693$	7.04886	0.267764
$694$	−29.0226	−1.10168
$695$	48.4226	1.83677
$696$	−2.79836	−0.106072
$697$	−17.6066	−0.666896
$698$	5.36427	0.203041
$699$	23.9901	0.907391
$700$	12.0308	0.454722
$701$	−8.94328	−0.337783	−0.168891	−	0.985635i	$-0.554019\pi$
$701$	−8.94328	−0.337783	−0.168891	+	0.985635i	$0.554019\pi$
$702$	6.95934	0.262663
$703$	−55.3593	−2.08792
$704$	2.20164	0.0829774
$705$	21.7885	0.820603
$706$	−5.12032	−0.192706
$707$	−31.7823	−1.19530
$708$	11.8780	0.446403
$709$	−13.2341	−0.497017	−0.248509	−	0.968630i	$-0.579940\pi$
$709$	−13.2341	−0.497017	−0.248509	+	0.968630i	$0.579940\pi$
$710$	13.6033	0.510521
$711$	12.9105	0.484181
$712$	18.5544	0.695356
$713$	−24.8944	−0.932303
$714$	−11.3852	−0.426082
$715$	45.3429	1.69573
$716$	5.42589	0.202775
$717$	−11.7741	−0.439712
$718$	3.35277	0.125124
$719$	48.1675	1.79635	0.898173	−	0.439643i	$-0.144895\pi$
$719$	48.1675	1.79635	0.898173	+	0.439643i	$0.144895\pi$
$720$	2.95934	0.110288
$721$	44.8230	1.66929
$722$	−3.63573	−0.135308
$723$	23.7252	0.882351
$724$	−7.36262	−0.273630
$725$	−10.5154	−0.390532
$726$	−6.15278	−0.228351
$727$	1.84556	0.0684482	0.0342241	−	0.999414i	$-0.489104\pi$
$727$	1.84556	0.0684482	0.0342241	+	0.999414i	$0.489104\pi$
$728$	−22.2813	−0.825800
$729$	1.00000	0.0370370
$730$	−2.21804	−0.0820932
$731$	−11.9285	−0.441193
$732$	5.91048	0.218458
$733$	44.7137	1.65154	0.825770	−	0.564007i	$-0.190741\pi$
$733$	44.7137	1.65154	0.825770	+	0.564007i	$0.190741\pi$
$734$	−13.4928	−0.498028
$735$	−9.61933	−0.354814
$736$	−5.71704	−0.210733
$737$	−10.7397	−0.395600
$738$	−4.95114	−0.182254
$739$	51.6973	1.90172	0.950859	−	0.309625i	$-0.100204\pi$
$739$	51.6973	1.90172	0.950859	+	0.309625i	$0.100204\pi$
$740$	−34.4341	−1.26582
$741$	−33.1105	−1.21634
$742$	27.7787	1.01979
$743$	27.5364	1.01021	0.505106	−	0.863058i	$-0.331454\pi$
$743$	27.5364	1.01021	0.505106	+	0.863058i	$0.331454\pi$
$744$	−4.35442	−0.159641
$745$	15.2731	0.559563
$746$	−1.37082	−0.0501893
$747$	15.4829	0.566491
$748$	−7.82917	−0.286263
$749$	−40.3249	−1.47344
$750$	−3.67638	−0.134243
$751$	−20.5967	−0.751585	−0.375793	−	0.926704i	$-0.622629\pi$
$751$	−20.5967	−0.751585	−0.375793	+	0.926704i	$0.622629\pi$
$752$	−7.36262	−0.268487
$753$	14.6033	0.532173
$754$	19.4747	0.709228
$755$	−41.3347	−1.50432
$756$	−3.20164	−0.116443
$757$	−17.7252	−0.644235	−0.322117	−	0.946700i	$-0.604395\pi$
$757$	−17.7252	−0.644235	−0.322117	+	0.946700i	$0.604395\pi$
$758$	−22.3934	−0.813366
$759$	−12.5869	−0.456874
$760$	−14.0797	−0.510723
$761$	38.2554	1.38676	0.693378	−	0.720574i	$-0.256121\pi$
$761$	38.2554	1.38676	0.693378	+	0.720574i	$0.256121\pi$
$762$	16.7885	0.608184
$763$	36.7642	1.33096
$764$	21.7885	0.788281
$765$	−10.5236	−0.380482
$766$	12.7577	0.460955
$767$	−82.6632	−2.98480
$768$	−1.00000	−0.0360844
$769$	−9.86328	−0.355679	−0.177839	−	0.984060i	$-0.556911\pi$
$769$	−9.86328	−0.355679	−0.177839	+	0.984060i	$0.556911\pi$
$770$	−20.8600	−0.751741
$771$	0.234100	0.00843090
$772$	−11.4423	−0.411817
$773$	−7.89409	−0.283931	−0.141965	−	0.989872i	$-0.545342\pi$
$773$	−7.89409	−0.283931	−0.141965	+	0.989872i	$0.545342\pi$
$774$	−3.35442	−0.120572
$775$	−16.3626	−0.587763
$776$	−1.95114	−0.0700420
$777$	37.2534	1.33646
$778$	36.5236	1.30943
$779$	23.5561	0.843984
$780$	−20.5951	−0.737422
$781$	−10.1203	−0.362134
$782$	20.3302	0.727005
$783$	2.79836	0.100005
$784$	3.25050	0.116089
$785$	45.3672	1.61922
$786$	−2.91868	−0.104106
$787$	−1.58852	−0.0566247	−0.0283124	−	0.999599i	$-0.509013\pi$
$787$	−1.58852	−0.0566247	−0.0283124	+	0.999599i	$0.509013\pi$
$788$	0.0406586	0.00144840
$789$	12.3138	0.438382
$790$	−38.2065	−1.35933
$791$	−11.5417	−0.410374
$792$	−2.20164	−0.0782319
$793$	−41.1331	−1.46068
$794$	−23.5787	−0.836776
$795$	25.6764	0.910648
$796$	−9.60492	−0.340437
$797$	53.3901	1.89118	0.945588	−	0.325368i	$-0.105488\pi$
$797$	53.3901	1.89118	0.945588	+	0.325368i	$0.105488\pi$
$798$	15.2324	0.539223
$799$	26.1819	0.926250
$800$	−3.75770	−0.132855
$801$	−18.5544	−0.655588
$802$	12.5561	0.443370
$803$	1.65014	0.0582320
$804$	4.87802	0.172035
$805$	54.1675	1.90915
$806$	30.3039	1.06741
$807$	−5.35442	−0.188485
$808$	9.92688	0.349226
$809$	9.69079	0.340710	0.170355	−	0.985383i	$-0.445509\pi$
$809$	9.69079	0.340710	0.170355	+	0.985383i	$0.445509\pi$
$810$	−2.95934	−0.103981
$811$	−18.7967	−0.660042	−0.330021	−	0.943974i	$-0.607056\pi$
$811$	−18.7967	−0.660042	−0.330021	+	0.943974i	$0.607056\pi$
$812$	−8.95934	−0.314411
$813$	22.5462	0.790730
$814$	25.6177	0.897899
$815$	−67.6747	−2.37054
$816$	3.55606	0.124487
$817$	15.9593	0.558347
$818$	28.1187	0.983146
$819$	22.2813	0.778572
$820$	14.6521	0.511674
$821$	8.91703	0.311206	0.155603	−	0.987820i	$-0.450268\pi$
$821$	8.91703	0.311206	0.155603	+	0.987820i	$0.450268\pi$
$822$	11.2895	0.393767
$823$	1.10426	0.0384921	0.0192460	−	0.999815i	$-0.493873\pi$
$823$	1.10426	0.0384921	0.0192460	+	0.999815i	$0.493873\pi$
$824$	−14.0000	−0.487713
$825$	−8.27311	−0.288033
$826$	38.0292	1.32320
$827$	11.2308	0.390533	0.195266	−	0.980750i	$-0.437443\pi$
$827$	11.2308	0.390533	0.195266	+	0.980750i	$0.437443\pi$
$828$	5.71704	0.198681
$829$	−21.2406	−0.737718	−0.368859	−	0.929485i	$-0.620251\pi$
$829$	−21.2406	−0.737718	−0.368859	+	0.929485i	$0.620251\pi$
$830$	−45.8193	−1.59041
$831$	−24.0308	−0.833619
$832$	6.95934	0.241272
$833$	−11.5590	−0.400494
$834$	16.3626	0.566591
$835$	18.7137	0.647616
$836$	10.4747	0.362277
$837$	4.35442	0.150511
$838$	−19.8924	−0.687173
$839$	−27.7154	−0.956842	−0.478421	−	0.878131i	$-0.658791\pi$
$839$	−27.7154	−0.956842	−0.478421	+	0.878131i	$0.658791\pi$
$840$	9.47474	0.326910
$841$	−21.1692	−0.729972
$842$	−4.64392	−0.160040
$843$	−1.81476	−0.0625036
$844$	−4.51540	−0.155427
$845$	104.857	3.60718
$846$	7.36262	0.253132
$847$	−19.6990	−0.676865
$848$	−8.67638	−0.297948
$849$	−15.8374	−0.543537
$850$	13.3626	0.458334
$851$	−66.5219	−2.28034
$852$	4.59672	0.157481
$853$	35.9967	1.23250	0.616252	−	0.787549i	$-0.288650\pi$
$853$	35.9967	1.23250	0.616252	+	0.787549i	$0.288650\pi$
$854$	18.9232	0.647540
$855$	14.0797	0.481514
$856$	12.5951	0.430491
$857$	4.62132	0.157861	0.0789306	−	0.996880i	$-0.474849\pi$
$857$	4.62132	0.157861	0.0789306	+	0.996880i	$0.474849\pi$
$858$	15.3220	0.523083
$859$	−6.15113	−0.209874	−0.104937	−	0.994479i	$-0.533464\pi$
$859$	−6.15113	−0.209874	−0.104937	+	0.994479i	$0.533464\pi$
$860$	9.92688	0.338504
$861$	−15.8518	−0.540227
$862$	−5.78031	−0.196878
$863$	6.53147	0.222334	0.111167	−	0.993802i	$-0.464541\pi$
$863$	6.53147	0.222334	0.111167	+	0.993802i	$0.464541\pi$
$864$	1.00000	0.0340207
$865$	−21.2160	−0.721367
$866$	−28.0862	−0.954408
$867$	4.35442	0.147884
$868$	−13.9413	−0.473198
$869$	28.4242	0.964226
$870$	−8.28130	−0.280762
$871$	−33.9478	−1.15028
$872$	−11.4829	−0.388862
$873$	1.95114	0.0660362
$874$	−27.2000	−0.920053
$875$	−11.7705	−0.397914
$876$	−0.749503	−0.0253234
$877$	−28.6049	−0.965919	−0.482960	−	0.875643i	$-0.660438\pi$
$877$	−28.6049	−0.965919	−0.482960	+	0.875643i	$0.660438\pi$
$878$	31.4423	1.06113
$879$	−17.9088	−0.604050
$880$	6.51540	0.219634
$881$	17.1265	0.577008	0.288504	−	0.957479i	$-0.406842\pi$
$881$	17.1265	0.577008	0.288504	+	0.957479i	$0.406842\pi$
$882$	−3.25050	−0.109450
$883$	22.2485	0.748722	0.374361	−	0.927283i	$-0.377862\pi$
$883$	22.2485	0.748722	0.374361	+	0.927283i	$0.377862\pi$
$884$	−24.7479	−0.832360
$885$	35.1511	1.18159
$886$	−2.34622	−0.0788229
$887$	47.2045	1.58497	0.792487	−	0.609889i	$-0.208786\pi$
$887$	47.2045	1.58497	0.792487	+	0.609889i	$0.208786\pi$
$888$	−11.6357	−0.390469
$889$	53.7508	1.80274
$890$	54.9088	1.84055
$891$	2.20164	0.0737577
$892$	1.00000	0.0334825
$893$	−35.0292	−1.17221
$894$	5.16098	0.172609
$895$	16.0571	0.536728
$896$	−3.20164	−0.106959
$897$	−39.7869	−1.32844
$898$	−18.1934	−0.607123
$899$	12.1852	0.406401
$900$	3.75770	0.125257
$901$	30.8538	1.02789
$902$	−10.9006	−0.362951
$903$	−10.7397	−0.357393
$904$	3.60492	0.119898
$905$	−21.7885	−0.724275
$906$	−13.9675	−0.464040
$907$	−44.3836	−1.47373	−0.736866	−	0.676039i	$-0.763695\pi$
$907$	−44.3836	−1.47373	−0.736866	+	0.676039i	$0.763695\pi$
$908$	−10.8292	−0.359379
$909$	−9.92688	−0.329254
$910$	−65.9380	−2.18582
$911$	20.7462	0.687352	0.343676	−	0.939088i	$-0.388328\pi$
$911$	20.7462	0.687352	0.343676	+	0.939088i	$0.388328\pi$
$912$	−4.75770	−0.157543
$913$	34.0879	1.12814
$914$	35.1757	1.16351
$915$	17.4911	0.578239
$916$	−28.2082	−0.932025
$917$	−9.34457	−0.308585
$918$	−3.55606	−0.117368
$919$	−5.18723	−0.171111	−0.0855555	−	0.996333i	$-0.527266\pi$
$919$	−5.18723	−0.171111	−0.0855555	+	0.996333i	$0.527266\pi$
$920$	−16.9187	−0.557792
$921$	−7.00000	−0.230658
$922$	2.26491	0.0745907
$923$	−31.9901	−1.05297
$924$	−7.04886	−0.231890
$925$	−43.7236	−1.43762
$926$	11.6682	0.383440
$927$	14.0000	0.459820
$928$	2.79836	0.0918607
$929$	−14.7318	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$929$	−14.7318	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$930$	−12.8862	−0.422556
$931$	15.4649	0.506841
$932$	−23.9901	−0.785823
$933$	−14.0065	−0.458554
$934$	−13.3462	−0.436702
$935$	−23.1692	−0.757713
$936$	−6.95934	−0.227473
$937$	−52.5692	−1.71736	−0.858680	−	0.512513i	$-0.828715\pi$
$937$	−52.5692	−1.71736	−0.858680	+	0.512513i	$0.828715\pi$
$938$	15.6177	0.509935
$939$	−14.4829	−0.472633
$940$	−21.7885	−0.710663
$941$	37.9072	1.23574	0.617869	−	0.786281i	$-0.287996\pi$
$941$	37.9072	1.23574	0.617869	+	0.786281i	$0.287996\pi$
$942$	15.3302	0.499484
$943$	28.3059	0.921767
$944$	−11.8780	−0.386597
$945$	−9.47474	−0.308213
$946$	−7.38523	−0.240115
$947$	26.1006	0.848156	0.424078	−	0.905626i	$-0.360598\pi$
$947$	26.1006	0.848156	0.424078	+	0.905626i	$0.360598\pi$
$948$	−12.9105	−0.419313
$949$	5.21605	0.169320
$950$	−17.8780	−0.580040
$951$	−35.1511	−1.13985
$952$	11.3852	0.368997
$953$	12.5089	0.405202	0.202601	−	0.979261i	$-0.435061\pi$
$953$	12.5089	0.405202	0.202601	+	0.979261i	$0.435061\pi$
$954$	8.67638	0.280908
$955$	64.4796	2.08651
$956$	11.7741	0.380802
$957$	6.16098	0.199156
$958$	21.4829	0.694082
$959$	36.1449	1.16718
$960$	−2.95934	−0.0955123
$961$	−12.0390	−0.388355
$962$	80.9770	2.61080
$963$	−12.5951	−0.405870
$964$	−23.7252	−0.764139
$965$	−33.8616	−1.09004
$966$	18.3039	0.588919
$967$	−27.4242	−0.881904	−0.440952	−	0.897531i	$-0.645359\pi$
$967$	−27.4242	−0.881904	−0.440952	+	0.897531i	$0.645359\pi$
$968$	6.15278	0.197758
$969$	16.9187	0.543506
$970$	−5.77410	−0.185395
$971$	−49.0111	−1.57284	−0.786421	−	0.617691i	$-0.788068\pi$
$971$	−49.0111	−1.57284	−0.786421	+	0.617691i	$0.788068\pi$
$972$	−1.00000	−0.0320750
$973$	52.3872	1.67946
$974$	−30.6033	−0.980592
$975$	−26.1511	−0.837506
$976$	−5.91048	−0.189190
$977$	−50.3511	−1.61087	−0.805437	−	0.592681i	$-0.798070\pi$
$977$	−50.3511	−1.61087	−0.805437	+	0.592681i	$0.798070\pi$
$978$	−22.8682	−0.731244
$979$	−40.8501	−1.30558
$980$	9.61933	0.307278
$981$	11.4829	0.366622
$982$	34.5770	1.10340
$983$	−48.8649	−1.55855	−0.779274	−	0.626684i	$-0.784412\pi$
$983$	−48.8649	−1.55855	−0.779274	+	0.626684i	$0.784412\pi$
$984$	4.95114	0.157837
$985$	0.120323	0.00383380
$986$	−9.95114	−0.316909
$987$	23.5725	0.750320
$988$	33.1105	1.05338
$989$	19.1774	0.609805
$990$	−6.51540	−0.207073
$991$	−1.02591	−0.0325893	−0.0162946	−	0.999867i	$-0.505187\pi$
$991$	−1.02591	−0.0325893	−0.0162946	+	0.999867i	$0.505187\pi$
$992$	4.35442	0.138253
$993$	−1.42589	−0.0452492
$994$	14.7170	0.466796
$995$	−28.4242	−0.901109
$996$	−15.4829	−0.490596
$997$	−7.67017	−0.242917	−0.121458	−	0.992597i	$-0.538757\pi$
$997$	−7.67017	−0.242917	−0.121458	+	0.992597i	$0.538757\pi$
$998$	−12.3380	−0.390553
$999$	11.6357	0.368138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.c.1.3	✓	3
3.2	odd	2		4014.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.c.1.3	✓	3	1.1	even	1	trivial
4014.2.a.p.1.1		3	3.2	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

Level:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.95934 q^{5} +1.00000 q^{6} +3.20164 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.95934 q^{10} +2.20164 q^{11} -1.00000 q^{12} +6.95934 q^{13} -3.20164 q^{14} -2.95934 q^{15} +1.00000 q^{16} -3.55606 q^{17} -1.00000 q^{18} +4.75770 q^{19} +2.95934 q^{20} -3.20164 q^{21} -2.20164 q^{22} +5.71704 q^{23} +1.00000 q^{24} +3.75770 q^{25} -6.95934 q^{26} -1.00000 q^{27} +3.20164 q^{28} -2.79836 q^{29} +2.95934 q^{30} -4.35442 q^{31} -1.00000 q^{32} -2.20164 q^{33} +3.55606 q^{34} +9.47474 q^{35} +1.00000 q^{36} -11.6357 q^{37} -4.75770 q^{38} -6.95934 q^{39} -2.95934 q^{40} +4.95114 q^{41} +3.20164 q^{42} +3.35442 q^{43} +2.20164 q^{44} +2.95934 q^{45} -5.71704 q^{46} -7.36262 q^{47} -1.00000 q^{48} +3.25050 q^{49} -3.75770 q^{50} +3.55606 q^{51} +6.95934 q^{52} -8.67638 q^{53} +1.00000 q^{54} +6.51540 q^{55} -3.20164 q^{56} -4.75770 q^{57} +2.79836 q^{58} -11.8780 q^{59} -2.95934 q^{60} -5.91048 q^{61} +4.35442 q^{62} +3.20164 q^{63} +1.00000 q^{64} +20.5951 q^{65} +2.20164 q^{66} -4.87802 q^{67} -3.55606 q^{68} -5.71704 q^{69} -9.47474 q^{70} -4.59672 q^{71} -1.00000 q^{72} +0.749503 q^{73} +11.6357 q^{74} -3.75770 q^{75} +4.75770 q^{76} +7.04886 q^{77} +6.95934 q^{78} +12.9105 q^{79} +2.95934 q^{80} +1.00000 q^{81} -4.95114 q^{82} +15.4829 q^{83} -3.20164 q^{84} -10.5236 q^{85} -3.35442 q^{86} +2.79836 q^{87} -2.20164 q^{88} -18.5544 q^{89} -2.95934 q^{90} +22.2813 q^{91} +5.71704 q^{92} +4.35442 q^{93} +7.36262 q^{94} +14.0797 q^{95} +1.00000 q^{96} +1.95114 q^{97} -3.25050 q^{98} +2.20164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 9 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} + 6 q^{11} - 3 q^{12} + 11 q^{13} - 9 q^{14} + q^{15} + 3 q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} - q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 9 * q^7 - 3 * q^8 + 3 * q^9 + q^10 + 6 * q^11 - 3 * q^12 + 11 * q^13 - 9 * q^14 + q^15 + 3 * q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 - q^20 - 9 * q^21 - 6 * q^22 - 2 * q^23 + 3 * q^24 + 2 * q^25 - 11 * q^26 - 3 * q^27 + 9 * q^28 - 9 * q^29 - q^30 - 5 * q^31 - 3 * q^32 - 6 * q^33 + 2 * q^34 + 3 * q^36 + 4 * q^37 - 5 * q^38 - 11 * q^39 + q^40 + 8 * q^41 + 9 * q^42 + 2 * q^43 + 6 * q^44 - q^45 + 2 * q^46 - 11 * q^47 - 3 * q^48 + 16 * q^49 - 2 * q^50 + 2 * q^51 + 11 * q^52 + 3 * q^53 + 3 * q^54 + q^55 - 9 * q^56 - 5 * q^57 + 9 * q^58 - 6 * q^59 + q^60 - q^61 + 5 * q^62 + 9 * q^63 + 3 * q^64 + 13 * q^65 + 6 * q^66 + 15 * q^67 - 2 * q^68 + 2 * q^69 - 15 * q^71 - 3 * q^72 - 4 * q^73 - 4 * q^74 - 2 * q^75 + 5 * q^76 + 28 * q^77 + 11 * q^78 + 22 * q^79 - q^80 + 3 * q^81 - 8 * q^82 + 15 * q^83 - 9 * q^84 - 10 * q^85 - 2 * q^86 + 9 * q^87 - 6 * q^88 + 3 * q^89 + q^90 + 36 * q^91 - 2 * q^92 + 5 * q^93 + 11 * q^94 + 12 * q^95 + 3 * q^96 - q^97 - 16 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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