LMFDB - Embedded newform 667.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [667,2,Mod(1,667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(667, 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("667.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 667 = 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	667.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:	$5.32602181482$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.67549$ of defining polynomial
Character	$\chi$	$=$	667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67549 q^{2} -1.95508 q^{3} +5.15827 q^{4} +1.93239 q^{5} +5.23080 q^{6} +0.721979 q^{7} -8.44992 q^{8} +0.822337 q^{9} +O(q^{10})$	\(q-2.67549 q^{2} -1.95508 q^{3} +5.15827 q^{4} +1.93239 q^{5} +5.23080 q^{6} +0.721979 q^{7} -8.44992 q^{8} +0.822337 q^{9} -5.17011 q^{10} -1.21766 q^{11} -10.0848 q^{12} +0.759191 q^{13} -1.93165 q^{14} -3.77799 q^{15} +12.2912 q^{16} -7.43542 q^{17} -2.20016 q^{18} +0.975858 q^{19} +9.96781 q^{20} -1.41153 q^{21} +3.25785 q^{22} +1.00000 q^{23} +16.5203 q^{24} -1.26585 q^{25} -2.03121 q^{26} +4.25751 q^{27} +3.72416 q^{28} +1.00000 q^{29} +10.1080 q^{30} +4.89136 q^{31} -15.9851 q^{32} +2.38063 q^{33} +19.8934 q^{34} +1.39515 q^{35} +4.24183 q^{36} -7.11770 q^{37} -2.61090 q^{38} -1.48428 q^{39} -16.3286 q^{40} +6.93952 q^{41} +3.77653 q^{42} +4.20028 q^{43} -6.28103 q^{44} +1.58908 q^{45} -2.67549 q^{46} -6.78394 q^{47} -24.0302 q^{48} -6.47875 q^{49} +3.38678 q^{50} +14.5368 q^{51} +3.91611 q^{52} -7.29810 q^{53} -11.3909 q^{54} -2.35301 q^{55} -6.10066 q^{56} -1.90788 q^{57} -2.67549 q^{58} -6.54924 q^{59} -19.4879 q^{60} -7.08057 q^{61} -13.0868 q^{62} +0.593710 q^{63} +18.1857 q^{64} +1.46706 q^{65} -6.36936 q^{66} +1.32547 q^{67} -38.3539 q^{68} -1.95508 q^{69} -3.73271 q^{70} -12.3498 q^{71} -6.94868 q^{72} -4.80670 q^{73} +19.0434 q^{74} +2.47484 q^{75} +5.03373 q^{76} -0.879128 q^{77} +3.97118 q^{78} +14.4789 q^{79} +23.7514 q^{80} -10.7908 q^{81} -18.5667 q^{82} -8.13628 q^{83} -7.28103 q^{84} -14.3682 q^{85} -11.2378 q^{86} -1.95508 q^{87} +10.2892 q^{88} -0.456964 q^{89} -4.25157 q^{90} +0.548120 q^{91} +5.15827 q^{92} -9.56300 q^{93} +18.1504 q^{94} +1.88574 q^{95} +31.2522 q^{96} -6.57680 q^{97} +17.3338 q^{98} -1.00133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} - 9 q^{3} + 9 q^{4} - 10 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 7 q^{9}+O(q^{10}) $ 10 * q - 3 * q^2 - 9 * q^3 + 9 * q^4 - 10 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 7 * q^9	\( 10 q - 3 q^{2} - 9 q^{3} + 9 q^{4} - 10 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 7 q^{9} - 6 q^{10} - 17 q^{12} - 13 q^{13} - 12 q^{14} + 2 q^{15} - 5 q^{16} - 22 q^{17} + 12 q^{18} - 2 q^{19} + 3 q^{20} - 7 q^{21} + 3 q^{22} + 10 q^{23} - 6 q^{24} + 10 q^{25} - 25 q^{26} - 24 q^{27} + 19 q^{28} + 10 q^{29} - 3 q^{30} - 22 q^{31} - 31 q^{32} - 9 q^{33} + 13 q^{34} - 15 q^{35} + 19 q^{36} - 9 q^{37} - 10 q^{38} + 4 q^{39} - 6 q^{40} - 25 q^{41} - 34 q^{42} + 3 q^{43} - 27 q^{44} - 28 q^{45} - 3 q^{46} - 17 q^{47} - 3 q^{48} + 17 q^{49} + 2 q^{50} + 38 q^{51} - 18 q^{52} - 43 q^{53} - 47 q^{54} - 11 q^{55} - 7 q^{56} + 18 q^{57} - 3 q^{58} - 7 q^{59} - 21 q^{60} - 6 q^{61} + 3 q^{62} + 11 q^{63} + 33 q^{64} + 11 q^{65} + 55 q^{66} + 11 q^{67} - 51 q^{68} - 9 q^{69} + 34 q^{70} - 17 q^{71} + 34 q^{72} - 44 q^{73} + 9 q^{74} + q^{75} + 24 q^{76} - 71 q^{77} + 38 q^{78} + 5 q^{79} + 38 q^{80} + 18 q^{81} + 33 q^{82} - 32 q^{83} + 14 q^{84} + 16 q^{85} - 9 q^{86} - 9 q^{87} + 18 q^{88} - 10 q^{89} - 9 q^{90} - 3 q^{91} + 9 q^{92} - 8 q^{93} + 47 q^{94} - 8 q^{95} + 60 q^{96} + 6 q^{97} - 73 q^{98} + 26 q^{99}+O(q^{100}) \) 10 * q - 3 * q^2 - 9 * q^3 + 9 * q^4 - 10 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 7 * q^9 - 6 * q^10 - 17 * q^12 - 13 * q^13 - 12 * q^14 + 2 * q^15 - 5 * q^16 - 22 * q^17 + 12 * q^18 - 2 * q^19 + 3 * q^20 - 7 * q^21 + 3 * q^22 + 10 * q^23 - 6 * q^24 + 10 * q^25 - 25 * q^26 - 24 * q^27 + 19 * q^28 + 10 * q^29 - 3 * q^30 - 22 * q^31 - 31 * q^32 - 9 * q^33 + 13 * q^34 - 15 * q^35 + 19 * q^36 - 9 * q^37 - 10 * q^38 + 4 * q^39 - 6 * q^40 - 25 * q^41 - 34 * q^42 + 3 * q^43 - 27 * q^44 - 28 * q^45 - 3 * q^46 - 17 * q^47 - 3 * q^48 + 17 * q^49 + 2 * q^50 + 38 * q^51 - 18 * q^52 - 43 * q^53 - 47 * q^54 - 11 * q^55 - 7 * q^56 + 18 * q^57 - 3 * q^58 - 7 * q^59 - 21 * q^60 - 6 * q^61 + 3 * q^62 + 11 * q^63 + 33 * q^64 + 11 * q^65 + 55 * q^66 + 11 * q^67 - 51 * q^68 - 9 * q^69 + 34 * q^70 - 17 * q^71 + 34 * q^72 - 44 * q^73 + 9 * q^74 + q^75 + 24 * q^76 - 71 * q^77 + 38 * q^78 + 5 * q^79 + 38 * q^80 + 18 * q^81 + 33 * q^82 - 32 * q^83 + 14 * q^84 + 16 * q^85 - 9 * q^86 - 9 * q^87 + 18 * q^88 - 10 * q^89 - 9 * q^90 - 3 * q^91 + 9 * q^92 - 8 * q^93 + 47 * q^94 - 8 * q^95 + 60 * q^96 + 6 * q^97 - 73 * q^98 + 26 * 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 $

$2$	−2.67549	−1.89186	−0.945930	−	0.324371i	$-0.894847\pi$
$2$	−2.67549	−1.89186	−0.945930	+	0.324371i	$0.894847\pi$
$3$	−1.95508	−1.12877	−0.564383	−	0.825513i	$-0.690886\pi$
$3$	−1.95508	−1.12877	−0.564383	+	0.825513i	$0.690886\pi$
$4$	5.15827	2.57913
$5$	1.93239	0.864193	0.432097	−	0.901827i	$-0.357774\pi$
$5$	1.93239	0.864193	0.432097	+	0.901827i	$0.357774\pi$
$6$	5.23080	2.13547
$7$	0.721979	0.272882	0.136441	−	0.990648i	$-0.456434\pi$
$7$	0.721979	0.272882	0.136441	+	0.990648i	$0.456434\pi$
$8$	−8.44992	−2.98750
$9$	0.822337	0.274112
$10$	−5.17011	−1.63493
$11$	−1.21766	−0.367139	−0.183570	−	0.983007i	$-0.558765\pi$
$11$	−1.21766	−0.367139	−0.183570	+	0.983007i	$0.558765\pi$
$12$	−10.0848	−2.91124
$13$	0.759191	0.210562	0.105281	−	0.994443i	$-0.466426\pi$
$13$	0.759191	0.210562	0.105281	+	0.994443i	$0.466426\pi$
$14$	−1.93165	−0.516255
$15$	−3.77799	−0.975472
$16$	12.2912	3.07279
$17$	−7.43542	−1.80335	−0.901677	−	0.432409i	$-0.857664\pi$
$17$	−7.43542	−1.80335	−0.901677	+	0.432409i	$0.857664\pi$
$18$	−2.20016	−0.518582
$19$	0.975858	0.223877	0.111939	−	0.993715i	$-0.464294\pi$
$19$	0.975858	0.223877	0.111939	+	0.993715i	$0.464294\pi$
$20$	9.96781	2.22887
$21$	−1.41153	−0.308020
$22$	3.25785	0.694576
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	16.5203	3.37219
$25$	−1.26585	−0.253170
$26$	−2.03121	−0.398353
$27$	4.25751	0.819357
$28$	3.72416	0.703800
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	10.1080	1.84546
$31$	4.89136	0.878515	0.439257	−	0.898361i	$-0.355242\pi$
$31$	4.89136	0.878515	0.439257	+	0.898361i	$0.355242\pi$
$32$	−15.9851	−2.82580
$33$	2.38063	0.414414
$34$	19.8934	3.41169
$35$	1.39515	0.235823
$36$	4.24183	0.706972
$37$	−7.11770	−1.17014	−0.585071	−	0.810982i	$-0.698933\pi$
$37$	−7.11770	−1.17014	−0.585071	+	0.810982i	$0.698933\pi$
$38$	−2.61090	−0.423544
$39$	−1.48428	−0.237675
$40$	−16.3286	−2.58178
$41$	6.93952	1.08377	0.541886	−	0.840452i	$-0.317711\pi$
$41$	6.93952	1.08377	0.541886	+	0.840452i	$0.317711\pi$
$42$	3.77653	0.582731
$43$	4.20028	0.640537	0.320268	−	0.947327i	$-0.396227\pi$
$43$	4.20028	0.640537	0.320268	+	0.947327i	$0.396227\pi$
$44$	−6.28103	−0.946901
$45$	1.58908	0.236886
$46$	−2.67549	−0.394480
$47$	−6.78394	−0.989540	−0.494770	−	0.869024i	$-0.664748\pi$
$47$	−6.78394	−0.989540	−0.494770	+	0.869024i	$0.664748\pi$
$48$	−24.0302	−3.46846
$49$	−6.47875	−0.925535
$50$	3.38678	0.478963
$51$	14.5368	2.03557
$52$	3.91611	0.543067
$53$	−7.29810	−1.00247	−0.501236	−	0.865311i	$-0.667121\pi$
$53$	−7.29810	−1.00247	−0.501236	+	0.865311i	$0.667121\pi$
$54$	−11.3909	−1.55011
$55$	−2.35301	−0.317279
$56$	−6.10066	−0.815236
$57$	−1.90788	−0.252705
$58$	−2.67549	−0.351310
$59$	−6.54924	−0.852638	−0.426319	−	0.904573i	$-0.640190\pi$
$59$	−6.54924	−0.852638	−0.426319	+	0.904573i	$0.640190\pi$
$60$	−19.4879	−2.51587
$61$	−7.08057	−0.906574	−0.453287	−	0.891365i	$-0.649749\pi$
$61$	−7.08057	−0.906574	−0.453287	+	0.891365i	$0.649749\pi$
$62$	−13.0868	−1.66203
$63$	0.593710	0.0748004
$64$	18.1857	2.27322
$65$	1.46706	0.181966
$66$	−6.36936	−0.784014
$67$	1.32547	0.161932	0.0809660	−	0.996717i	$-0.474199\pi$
$67$	1.32547	0.161932	0.0809660	+	0.996717i	$0.474199\pi$
$68$	−38.3539	−4.65109
$69$	−1.95508	−0.235364
$70$	−3.73271	−0.446144
$71$	−12.3498	−1.46565	−0.732825	−	0.680417i	$-0.761799\pi$
$71$	−12.3498	−1.46565	−0.732825	+	0.680417i	$0.761799\pi$
$72$	−6.94868	−0.818910
$73$	−4.80670	−0.562581	−0.281291	−	0.959623i	$-0.590762\pi$
$73$	−4.80670	−0.562581	−0.281291	+	0.959623i	$0.590762\pi$
$74$	19.0434	2.21374
$75$	2.47484	0.285770
$76$	5.03373	0.577409
$77$	−0.879128	−0.100186
$78$	3.97118	0.449648
$79$	14.4789	1.62901	0.814504	−	0.580159i	$-0.197009\pi$
$79$	14.4789	1.62901	0.814504	+	0.580159i	$0.197009\pi$
$80$	23.7514	2.65549
$81$	−10.7908	−1.19897
$82$	−18.5667	−2.05034
$83$	−8.13628	−0.893073	−0.446536	−	0.894765i	$-0.647343\pi$
$83$	−8.13628	−0.893073	−0.446536	+	0.894765i	$0.647343\pi$
$84$	−7.28103	−0.794425
$85$	−14.3682	−1.55845
$86$	−11.2378	−1.21181
$87$	−1.95508	−0.209607
$88$	10.2892	1.09683
$89$	−0.456964	−0.0484381	−0.0242191	−	0.999707i	$-0.507710\pi$
$89$	−0.456964	−0.0484381	−0.0242191	+	0.999707i	$0.507710\pi$
$90$	−4.25157	−0.448155
$91$	0.548120	0.0574586
$92$	5.15827	0.537786
$93$	−9.56300	−0.991637
$94$	18.1504	1.87207
$95$	1.88574	0.193473
$96$	31.2522	3.18966
$97$	−6.57680	−0.667773	−0.333886	−	0.942613i	$-0.608360\pi$
$97$	−6.57680	−0.667773	−0.333886	+	0.942613i	$0.608360\pi$
$98$	17.3338	1.75098
$99$	−1.00133	−0.100637
$100$	−6.52960	−0.652960
$101$	8.50531	0.846310	0.423155	−	0.906057i	$-0.360923\pi$
$101$	8.50531	0.846310	0.423155	+	0.906057i	$0.360923\pi$
$102$	−38.8932	−3.85100
$103$	6.73914	0.664027	0.332013	−	0.943275i	$-0.392272\pi$
$103$	6.73914	0.664027	0.332013	+	0.943275i	$0.392272\pi$
$104$	−6.41511	−0.629053
$105$	−2.72763	−0.266189
$106$	19.5260	1.89654
$107$	10.7615	1.04036	0.520178	−	0.854058i	$-0.325866\pi$
$107$	10.7615	1.04036	0.520178	+	0.854058i	$0.325866\pi$
$108$	21.9613	2.11323
$109$	1.68415	0.161312	0.0806562	−	0.996742i	$-0.474298\pi$
$109$	1.68415	0.161312	0.0806562	+	0.996742i	$0.474298\pi$
$110$	6.29546	0.600248
$111$	13.9157	1.32082
$112$	8.87397	0.838511
$113$	−9.47905	−0.891714	−0.445857	−	0.895104i	$-0.647101\pi$
$113$	−9.47905	−0.891714	−0.445857	+	0.895104i	$0.647101\pi$
$114$	5.10452	0.478082
$115$	1.93239	0.180197
$116$	5.15827	0.478933
$117$	0.624311	0.0577176
$118$	17.5224	1.61307
$119$	−5.36822	−0.492104
$120$	31.9237	2.91422
$121$	−9.51729	−0.865209
$122$	18.9440	1.71511
$123$	−13.5673	−1.22332
$124$	25.2309	2.26581
$125$	−12.1081	−1.08298
$126$	−1.58847	−0.141512
$127$	−7.40313	−0.656922	−0.328461	−	0.944518i	$-0.606530\pi$
$127$	−7.40313	−0.656922	−0.328461	+	0.944518i	$0.606530\pi$
$128$	−16.6856	−1.47481
$129$	−8.21188	−0.723016
$130$	−3.92510	−0.344254
$131$	−13.1875	−1.15220	−0.576099	−	0.817380i	$-0.695426\pi$
$131$	−13.1875	−1.15220	−0.576099	+	0.817380i	$0.695426\pi$
$132$	12.2799	1.06883
$133$	0.704549	0.0610921
$134$	−3.54629	−0.306353
$135$	8.22718	0.708083
$136$	62.8287	5.38752
$137$	−14.7027	−1.25614	−0.628069	−	0.778158i	$-0.716155\pi$
$137$	−14.7027	−1.25614	−0.628069	+	0.778158i	$0.716155\pi$
$138$	5.23080	0.445276
$139$	−2.25950	−0.191649	−0.0958243	−	0.995398i	$-0.530549\pi$
$139$	−2.25950	−0.191649	−0.0958243	+	0.995398i	$0.530549\pi$
$140$	7.19655	0.608219
$141$	13.2632	1.11696
$142$	33.0418	2.77280
$143$	−0.924440	−0.0773055
$144$	10.1075	0.842290
$145$	1.93239	0.160477
$146$	12.8603	1.06432
$147$	12.6665	1.04471
$148$	−36.7150	−3.01795
$149$	−8.31996	−0.681598	−0.340799	−	0.940136i	$-0.610697\pi$
$149$	−8.31996	−0.681598	−0.340799	+	0.940136i	$0.610697\pi$
$150$	−6.62142	−0.540636
$151$	−2.15704	−0.175537	−0.0877685	−	0.996141i	$-0.527974\pi$
$151$	−2.15704	−0.175537	−0.0877685	+	0.996141i	$0.527974\pi$
$152$	−8.24592	−0.668833
$153$	−6.11442	−0.494322
$154$	2.35210	0.189538
$155$	9.45204	0.759206
$156$	−7.65631	−0.612995
$157$	13.5196	1.07898	0.539489	−	0.841993i	$-0.318617\pi$
$157$	13.5196	1.07898	0.539489	+	0.841993i	$0.318617\pi$
$158$	−38.7383	−3.08185
$159$	14.2684	1.13156
$160$	−30.8896	−2.44203
$161$	0.721979	0.0568999
$162$	28.8706	2.26829
$163$	−16.2609	−1.27365	−0.636827	−	0.771007i	$-0.719754\pi$
$163$	−16.2609	−1.27365	−0.636827	+	0.771007i	$0.719754\pi$
$164$	35.7959	2.79519
$165$	4.60032	0.358134
$166$	21.7686	1.68957
$167$	−0.0282352	−0.00218490	−0.00109245	−	0.999999i	$-0.500348\pi$
$167$	−0.0282352	−0.00218490	−0.00109245	+	0.999999i	$0.500348\pi$
$168$	11.9273	0.920210
$169$	−12.4236	−0.955664
$170$	38.4419	2.94836
$171$	0.802484	0.0613675
$172$	21.6662	1.65203
$173$	−4.70262	−0.357534	−0.178767	−	0.983891i	$-0.557211\pi$
$173$	−4.70262	−0.357534	−0.178767	+	0.983891i	$0.557211\pi$
$174$	5.23080	0.396546
$175$	−0.913918	−0.0690857
$176$	−14.9665	−1.12814
$177$	12.8043	0.962429
$178$	1.22260	0.0916381
$179$	−14.7323	−1.10114	−0.550570	−	0.834789i	$-0.685590\pi$
$179$	−14.7323	−1.10114	−0.550570	+	0.834789i	$0.685590\pi$
$180$	8.19689	0.610960
$181$	−9.13899	−0.679296	−0.339648	−	0.940553i	$-0.610308\pi$
$181$	−9.13899	−0.679296	−0.339648	+	0.940553i	$0.610308\pi$
$182$	−1.46649	−0.108704
$183$	13.8431	1.02331
$184$	−8.44992	−0.622936
$185$	−13.7542	−1.01123
$186$	25.5858	1.87604
$187$	9.05385	0.662083
$188$	−34.9934	−2.55216
$189$	3.07383	0.223588
$190$	−5.04529	−0.366024
$191$	8.88101	0.642608	0.321304	−	0.946976i	$-0.395879\pi$
$191$	8.88101	0.642608	0.321304	+	0.946976i	$0.395879\pi$
$192$	−35.5546	−2.56593
$193$	−22.3157	−1.60632	−0.803160	−	0.595763i	$-0.796850\pi$
$193$	−22.3157	−1.60632	−0.803160	+	0.595763i	$0.796850\pi$
$194$	17.5962	1.26333
$195$	−2.86821	−0.205397
$196$	−33.4191	−2.38708
$197$	14.0126	0.998354	0.499177	−	0.866500i	$-0.333636\pi$
$197$	14.0126	0.998354	0.499177	+	0.866500i	$0.333636\pi$
$198$	2.67905	0.190392
$199$	−26.3908	−1.87079	−0.935397	−	0.353600i	$-0.884957\pi$
$199$	−26.3908	−1.87079	−0.935397	+	0.353600i	$0.884957\pi$
$200$	10.6963	0.756345
$201$	−2.59140	−0.182783
$202$	−22.7559	−1.60110
$203$	0.721979	0.0506730
$204$	74.9849	5.24999
$205$	13.4099	0.936588
$206$	−18.0305	−1.25625
$207$	0.822337	0.0571563
$208$	9.33135	0.647013
$209$	−1.18827	−0.0821941
$210$	7.29775	0.503592
$211$	9.86160	0.678900	0.339450	−	0.940624i	$-0.389759\pi$
$211$	9.86160	0.678900	0.339450	+	0.940624i	$0.389759\pi$
$212$	−37.6456	−2.58551
$213$	24.1448	1.65438
$214$	−28.7924	−1.96821
$215$	8.11660	0.553547
$216$	−35.9756	−2.44783
$217$	3.53146	0.239731
$218$	−4.50593	−0.305180
$219$	9.39747	0.635022
$220$	−12.1374	−0.818306
$221$	−5.64491	−0.379718
$222$	−37.2313	−2.49880
$223$	24.6137	1.64825	0.824126	−	0.566406i	$-0.191667\pi$
$223$	24.6137	1.64825	0.824126	+	0.566406i	$0.191667\pi$
$224$	−11.5409	−0.771110
$225$	−1.04096	−0.0693970
$226$	25.3611	1.68700
$227$	−12.2803	−0.815069	−0.407535	−	0.913190i	$-0.633611\pi$
$227$	−12.2803	−0.815069	−0.407535	+	0.913190i	$0.633611\pi$
$228$	−9.84135	−0.651759
$229$	4.04487	0.267293	0.133646	−	0.991029i	$-0.457331\pi$
$229$	4.04487	0.267293	0.133646	+	0.991029i	$0.457331\pi$
$230$	−5.17011	−0.340907
$231$	1.71876	0.113086
$232$	−8.44992	−0.554764
$233$	17.1248	1.12188	0.560942	−	0.827855i	$-0.310439\pi$
$233$	17.1248	1.12188	0.560942	+	0.827855i	$0.310439\pi$
$234$	−1.67034	−0.109194
$235$	−13.1093	−0.855154
$236$	−33.7827	−2.19907
$237$	−28.3075	−1.83877
$238$	14.3626	0.930991
$239$	24.7785	1.60279	0.801393	−	0.598138i	$-0.204093\pi$
$239$	24.7785	1.60279	0.801393	+	0.598138i	$0.204093\pi$
$240$	−46.4359	−2.99742
$241$	19.2639	1.24090	0.620450	−	0.784246i	$-0.286950\pi$
$241$	19.2639	1.24090	0.620450	+	0.784246i	$0.286950\pi$
$242$	25.4635	1.63685
$243$	8.32430	0.534004
$244$	−36.5235	−2.33817
$245$	−12.5195	−0.799841
$246$	36.2993	2.31436
$247$	0.740863	0.0471400
$248$	−41.3316	−2.62456
$249$	15.9071	1.00807
$250$	32.3951	2.04885
$251$	−6.40532	−0.404300	−0.202150	−	0.979355i	$-0.564793\pi$
$251$	−6.40532	−0.404300	−0.202150	+	0.979355i	$0.564793\pi$
$252$	3.06251	0.192920
$253$	−1.21766	−0.0765539
$254$	19.8070	1.24280
$255$	28.0909	1.75912
$256$	8.27069	0.516918
$257$	11.5345	0.719501	0.359751	−	0.933049i	$-0.382862\pi$
$257$	11.5345	0.719501	0.359751	+	0.933049i	$0.382862\pi$
$258$	21.9708	1.36784
$259$	−5.13883	−0.319311
$260$	7.56747	0.469315
$261$	0.822337	0.0509014
$262$	35.2831	2.17980
$263$	29.4709	1.81725	0.908626	−	0.417610i	$-0.137132\pi$
$263$	29.4709	1.81725	0.908626	+	0.417610i	$0.137132\pi$
$264$	−20.1161	−1.23806
$265$	−14.1028	−0.866329
$266$	−1.88502	−0.115578
$267$	0.893401	0.0546753
$268$	6.83713	0.417644
$269$	7.08827	0.432180	0.216090	−	0.976373i	$-0.430670\pi$
$269$	7.08827	0.432180	0.216090	+	0.976373i	$0.430670\pi$
$270$	−22.0118	−1.33959
$271$	−15.1303	−0.919101	−0.459550	−	0.888152i	$-0.651989\pi$
$271$	−15.1303	−0.919101	−0.459550	+	0.888152i	$0.651989\pi$
$272$	−91.3901	−5.54134
$273$	−1.07162	−0.0648573
$274$	39.3370	2.37644
$275$	1.54138	0.0929488
$276$	−10.0848	−0.607035
$277$	−23.8949	−1.43571	−0.717854	−	0.696194i	$-0.754875\pi$
$277$	−23.8949	−1.43571	−0.717854	+	0.696194i	$0.754875\pi$
$278$	6.04529	0.362572
$279$	4.02235	0.240812
$280$	−11.7889	−0.704521
$281$	−6.56691	−0.391749	−0.195874	−	0.980629i	$-0.562754\pi$
$281$	−6.56691	−0.391749	−0.195874	+	0.980629i	$0.562754\pi$
$282$	−35.4855	−2.11313
$283$	−26.1297	−1.55325	−0.776624	−	0.629965i	$-0.783069\pi$
$283$	−26.1297	−1.55325	−0.776624	+	0.629965i	$0.783069\pi$
$284$	−63.7035	−3.78011
$285$	−3.68678	−0.218386
$286$	2.47333	0.146251
$287$	5.01019	0.295742
$288$	−13.1452	−0.774585
$289$	38.2855	2.25209
$290$	−5.17011	−0.303599
$291$	12.8582	0.753759
$292$	−24.7942	−1.45097
$293$	23.9338	1.39823	0.699114	−	0.715010i	$-0.253578\pi$
$293$	23.9338	1.39823	0.699114	+	0.715010i	$0.253578\pi$
$294$	−33.8890	−1.97645
$295$	−12.6557	−0.736844
$296$	60.1440	3.49580
$297$	−5.18421	−0.300818
$298$	22.2600	1.28949
$299$	0.759191	0.0439052
$300$	12.7659	0.737038
$301$	3.03251	0.174791
$302$	5.77113	0.332091
$303$	−16.6286	−0.955286
$304$	11.9944	0.687928
$305$	−13.6825	−0.783455
$306$	16.3591	0.935187
$307$	15.7887	0.901109	0.450554	−	0.892749i	$-0.351226\pi$
$307$	15.7887	0.901109	0.450554	+	0.892749i	$0.351226\pi$
$308$	−4.53477	−0.258393
$309$	−13.1756	−0.749531
$310$	−25.2889	−1.43631
$311$	−10.0741	−0.571249	−0.285624	−	0.958342i	$-0.592201\pi$
$311$	−10.0741	−0.571249	−0.285624	+	0.958342i	$0.592201\pi$
$312$	12.5420	0.710053
$313$	15.6828	0.886447	0.443223	−	0.896411i	$-0.353835\pi$
$313$	15.6828	0.886447	0.443223	+	0.896411i	$0.353835\pi$
$314$	−36.1715	−2.04128
$315$	1.14728	0.0646420
$316$	74.6862	4.20143
$317$	−1.15227	−0.0647178	−0.0323589	−	0.999476i	$-0.510302\pi$
$317$	−1.15227	−0.0647178	−0.0323589	+	0.999476i	$0.510302\pi$
$318$	−38.1750	−2.14074
$319$	−1.21766	−0.0681761
$320$	35.1420	1.96450
$321$	−21.0396	−1.17432
$322$	−1.93165	−0.107647
$323$	−7.25592	−0.403730
$324$	−55.6617	−3.09232
$325$	−0.961023	−0.0533080
$326$	43.5060	2.40958
$327$	−3.29265	−0.182084
$328$	−58.6384	−3.23777
$329$	−4.89786	−0.270028
$330$	−12.3081	−0.677540
$331$	−32.9192	−1.80941	−0.904703	−	0.426044i	$-0.859907\pi$
$331$	−32.9192	−1.80941	−0.904703	+	0.426044i	$0.859907\pi$
$332$	−41.9691	−2.30335
$333$	−5.85314	−0.320750
$334$	0.0755431	0.00413353
$335$	2.56133	0.139941
$336$	−17.3493	−0.946483
$337$	−28.4389	−1.54916	−0.774582	−	0.632474i	$-0.782040\pi$
$337$	−28.4389	−1.54916	−0.774582	+	0.632474i	$0.782040\pi$
$338$	33.2393	1.80798
$339$	18.5323	1.00654
$340$	−74.1148	−4.01944
$341$	−5.95603	−0.322537
$342$	−2.14704	−0.116099
$343$	−9.73137	−0.525445
$344$	−35.4920	−1.91360
$345$	−3.77799	−0.203400
$346$	12.5818	0.676404
$347$	21.2888	1.14284	0.571421	−	0.820657i	$-0.306392\pi$
$347$	21.2888	1.14284	0.571421	+	0.820657i	$0.306392\pi$
$348$	−10.0848	−0.540603
$349$	27.4019	1.46679	0.733395	−	0.679803i	$-0.237935\pi$
$349$	27.4019	1.46679	0.733395	+	0.679803i	$0.237935\pi$
$350$	2.44518	0.130700
$351$	3.23226	0.172525
$352$	19.4645	1.03746
$353$	19.2067	1.02227	0.511135	−	0.859501i	$-0.329225\pi$
$353$	19.2067	1.02227	0.511135	+	0.859501i	$0.329225\pi$
$354$	−34.2578	−1.82078
$355$	−23.8647	−1.26660
$356$	−2.35714	−0.124928
$357$	10.4953	0.555470
$358$	39.4161	2.08320
$359$	26.0413	1.37440	0.687202	−	0.726466i	$-0.258839\pi$
$359$	26.0413	1.37440	0.687202	+	0.726466i	$0.258839\pi$
$360$	−13.4276	−0.707696
$361$	−18.0477	−0.949879
$362$	24.4513	1.28513
$363$	18.6071	0.976618
$364$	2.82735	0.148193
$365$	−9.28843	−0.486179
$366$	−37.0371	−1.93596
$367$	−17.8240	−0.930403	−0.465201	−	0.885205i	$-0.654018\pi$
$367$	−17.8240	−0.930403	−0.465201	+	0.885205i	$0.654018\pi$
$368$	12.2912	0.640722
$369$	5.70662	0.297075
$370$	36.7993	1.91310
$371$	−5.26908	−0.273557
$372$	−49.3285	−2.55756
$373$	25.1002	1.29964	0.649819	−	0.760089i	$-0.274845\pi$
$373$	25.1002	1.29964	0.649819	+	0.760089i	$0.274845\pi$
$374$	−24.2235	−1.25257
$375$	23.6723	1.22243
$376$	57.3238	2.95625
$377$	0.759191	0.0391003
$378$	−8.22401	−0.422997
$379$	29.1521	1.49744	0.748720	−	0.662886i	$-0.230669\pi$
$379$	29.1521	1.49744	0.748720	+	0.662886i	$0.230669\pi$
$380$	9.72716	0.498993
$381$	14.4737	0.741511
$382$	−23.7611	−1.21572
$383$	20.5470	1.04990	0.524952	−	0.851132i	$-0.324083\pi$
$383$	20.5470	1.04990	0.524952	+	0.851132i	$0.324083\pi$
$384$	32.6216	1.66472
$385$	−1.69882	−0.0865800
$386$	59.7056	3.03893
$387$	3.45404	0.175579
$388$	−33.9249	−1.72227
$389$	0.146769	0.00744146	0.00372073	−	0.999993i	$-0.498816\pi$
$389$	0.146769	0.00744146	0.00372073	+	0.999993i	$0.498816\pi$
$390$	7.67389	0.388582
$391$	−7.43542	−0.376025
$392$	54.7449	2.76503
$393$	25.7826	1.30056
$394$	−37.4905	−1.88875
$395$	27.9790	1.40778
$396$	−5.16512	−0.259557
$397$	−33.1878	−1.66565	−0.832823	−	0.553539i	$-0.813277\pi$
$397$	−33.1878	−1.66565	−0.832823	+	0.553539i	$0.813277\pi$
$398$	70.6084	3.53928
$399$	−1.37745	−0.0689587
$400$	−15.5588	−0.777940
$401$	21.7592	1.08660	0.543302	−	0.839537i	$-0.317174\pi$
$401$	21.7592	1.08660	0.543302	+	0.839537i	$0.317174\pi$
$402$	6.93328	0.345801
$403$	3.71348	0.184982
$404$	43.8727	2.18275
$405$	−20.8520	−1.03615
$406$	−1.93165	−0.0958662
$407$	8.66696	0.429605
$408$	−122.835	−6.08125
$409$	−29.6837	−1.46777	−0.733883	−	0.679276i	$-0.762294\pi$
$409$	−29.6837	−1.46777	−0.733883	+	0.679276i	$0.762294\pi$
$410$	−35.8781	−1.77189
$411$	28.7450	1.41789
$412$	34.7623	1.71261
$413$	−4.72841	−0.232670
$414$	−2.20016	−0.108132
$415$	−15.7225	−0.771787
$416$	−12.1358	−0.595005
$417$	4.41751	0.216326
$418$	3.17920	0.155500
$419$	−18.6118	−0.909244	−0.454622	−	0.890684i	$-0.650226\pi$
$419$	−18.6118	−0.909244	−0.454622	+	0.890684i	$0.650226\pi$
$420$	−14.0698	−0.686537
$421$	−9.48608	−0.462323	−0.231162	−	0.972915i	$-0.574253\pi$
$421$	−9.48608	−0.462323	−0.231162	+	0.972915i	$0.574253\pi$
$422$	−26.3846	−1.28438
$423$	−5.57869	−0.271245
$424$	61.6684	2.99488
$425$	9.41214	0.456556
$426$	−64.5993	−3.12985
$427$	−5.11202	−0.247388
$428$	55.5108	2.68322
$429$	1.80735	0.0872598
$430$	−21.7159	−1.04723
$431$	26.2928	1.26648	0.633240	−	0.773956i	$-0.281725\pi$
$431$	26.2928	1.26648	0.633240	+	0.773956i	$0.281725\pi$
$432$	52.3297	2.51772
$433$	33.6618	1.61768	0.808842	−	0.588026i	$-0.200095\pi$
$433$	33.6618	1.61768	0.808842	+	0.588026i	$0.200095\pi$
$434$	−9.44840	−0.453538
$435$	−3.77799	−0.181141
$436$	8.68729	0.416046
$437$	0.975858	0.0466816
$438$	−25.1429	−1.20137
$439$	−20.1034	−0.959481	−0.479741	−	0.877410i	$-0.659269\pi$
$439$	−20.1034	−0.959481	−0.479741	+	0.877410i	$0.659269\pi$
$440$	19.8827	0.947872
$441$	−5.32771	−0.253701
$442$	15.1029	0.718372
$443$	12.9472	0.615142	0.307571	−	0.951525i	$-0.400484\pi$
$443$	12.9472	0.615142	0.307571	+	0.951525i	$0.400484\pi$
$444$	71.7807	3.40656
$445$	−0.883035	−0.0418599
$446$	−65.8537	−3.11826
$447$	16.2662	0.769364
$448$	13.1297	0.620321
$449$	15.5981	0.736119	0.368059	−	0.929802i	$-0.380022\pi$
$449$	15.5981	0.736119	0.368059	+	0.929802i	$0.380022\pi$
$450$	2.78507	0.131289
$451$	−8.45001	−0.397895
$452$	−48.8954	−2.29985
$453$	4.21718	0.198140
$454$	32.8558	1.54200
$455$	1.05918	0.0496553
$456$	16.1214	0.754955
$457$	−16.3556	−0.765084	−0.382542	−	0.923938i	$-0.624951\pi$
$457$	−16.3556	−0.765084	−0.382542	+	0.923938i	$0.624951\pi$
$458$	−10.8220	−0.505680
$459$	−31.6564	−1.47759
$460$	9.96781	0.464751
$461$	−40.7099	−1.89605	−0.948025	−	0.318196i	$-0.896923\pi$
$461$	−40.7099	−1.89605	−0.948025	+	0.318196i	$0.896923\pi$
$462$	−4.59854	−0.213944
$463$	34.9631	1.62487	0.812437	−	0.583049i	$-0.198140\pi$
$463$	34.9631	1.62487	0.812437	+	0.583049i	$0.198140\pi$
$464$	12.2912	0.570603
$465$	−18.4795	−0.856966
$466$	−45.8174	−2.12245
$467$	7.18417	0.332444	0.166222	−	0.986088i	$-0.446843\pi$
$467$	7.18417	0.332444	0.166222	+	0.986088i	$0.446843\pi$
$468$	3.22036	0.148861
$469$	0.956962	0.0441884
$470$	35.0737	1.61783
$471$	−26.4318	−1.21791
$472$	55.3405	2.54725
$473$	−5.11453	−0.235166
$474$	75.7364	3.47869
$475$	−1.23529	−0.0566790
$476$	−27.6907	−1.26920
$477$	−6.00150	−0.274790
$478$	−66.2946	−3.03225
$479$	−14.9608	−0.683577	−0.341788	−	0.939777i	$-0.611033\pi$
$479$	−14.9608	−0.683577	−0.341788	+	0.939777i	$0.611033\pi$
$480$	60.3916	2.75648
$481$	−5.40369	−0.246387
$482$	−51.5406	−2.34761
$483$	−1.41153	−0.0642267
$484$	−49.0927	−2.23149
$485$	−12.7090	−0.577084
$486$	−22.2716	−1.01026
$487$	33.3897	1.51303	0.756515	−	0.653976i	$-0.226900\pi$
$487$	33.3897	1.51303	0.756515	+	0.653976i	$0.226900\pi$
$488$	59.8302	2.70839
$489$	31.7914	1.43766
$490$	33.4958	1.51319
$491$	−14.9913	−0.676548	−0.338274	−	0.941048i	$-0.609843\pi$
$491$	−14.9913	−0.676548	−0.338274	+	0.941048i	$0.609843\pi$
$492$	−69.9839	−3.15512
$493$	−7.43542	−0.334875
$494$	−1.98217	−0.0891822
$495$	−1.93496	−0.0869702
$496$	60.1206	2.69949
$497$	−8.91629	−0.399950
$498$	−42.5593	−1.90713
$499$	11.3162	0.506584	0.253292	−	0.967390i	$-0.418487\pi$
$499$	11.3162	0.506584	0.253292	+	0.967390i	$0.418487\pi$
$500$	−62.4568	−2.79315
$501$	0.0552020	0.00246625
$502$	17.1374	0.764879
$503$	28.5521	1.27307	0.636537	−	0.771246i	$-0.280366\pi$
$503$	28.5521	1.27307	0.636537	+	0.771246i	$0.280366\pi$
$504$	−5.01680	−0.223466
$505$	16.4356	0.731376
$506$	3.25785	0.144829
$507$	24.2892	1.07872
$508$	−38.1873	−1.69429
$509$	6.23210	0.276233	0.138116	−	0.990416i	$-0.455895\pi$
$509$	6.23210	0.276233	0.138116	+	0.990416i	$0.455895\pi$
$510$	−75.1571	−3.32801
$511$	−3.47033	−0.153518
$512$	11.2430	0.496874
$513$	4.15472	0.183435
$514$	−30.8604	−1.36120
$515$	13.0227	0.573848
$516$	−42.3591	−1.86475
$517$	8.26056	0.363299
$518$	13.7489	0.604092
$519$	9.19400	0.403572
$520$	−12.3965	−0.543623
$521$	−13.0252	−0.570645	−0.285322	−	0.958432i	$-0.592101\pi$
$521$	−13.0252	−0.570645	−0.285322	+	0.958432i	$0.592101\pi$
$522$	−2.20016	−0.0962982
$523$	−32.9039	−1.43879	−0.719394	−	0.694602i	$-0.755580\pi$
$523$	−32.9039	−1.43879	−0.719394	+	0.694602i	$0.755580\pi$
$524$	−68.0247	−2.97167
$525$	1.78678	0.0779816
$526$	−78.8492	−3.43799
$527$	−36.3693	−1.58427
$528$	29.2607	1.27341
$529$	1.00000	0.0434783
$530$	37.7320	1.63897
$531$	−5.38568	−0.233719
$532$	3.63425	0.157565
$533$	5.26843	0.228201
$534$	−2.39029	−0.103438
$535$	20.7955	0.899069
$536$	−11.2001	−0.483772
$537$	28.8027	1.24293
$538$	−18.9646	−0.817623
$539$	7.88894	0.339801
$540$	42.4380	1.82624
$541$	−17.4943	−0.752140	−0.376070	−	0.926591i	$-0.622725\pi$
$541$	−17.4943	−0.752140	−0.376070	+	0.926591i	$0.622725\pi$
$542$	40.4810	1.73881
$543$	17.8674	0.766766
$544$	118.856	5.09591
$545$	3.25444	0.139405
$546$	2.86711	0.122701
$547$	2.73202	0.116813	0.0584063	−	0.998293i	$-0.481398\pi$
$547$	2.73202	0.116813	0.0584063	+	0.998293i	$0.481398\pi$
$548$	−75.8406	−3.23975
$549$	−5.82261	−0.248503
$550$	−4.12396	−0.175846
$551$	0.975858	0.0415729
$552$	16.5203	0.703149
$553$	10.4535	0.444527
$554$	63.9307	2.71616
$555$	26.8906	1.14144
$556$	−11.6551	−0.494287
$557$	−26.0332	−1.10306	−0.551531	−	0.834155i	$-0.685956\pi$
$557$	−26.0332	−1.10306	−0.551531	+	0.834155i	$0.685956\pi$
$558$	−10.7618	−0.455582
$559$	3.18882	0.134873
$560$	17.1480	0.724636
$561$	−17.7010	−0.747336
$562$	17.5697	0.741134
$563$	−33.1978	−1.39912	−0.699561	−	0.714573i	$-0.746621\pi$
$563$	−33.1978	−1.39912	−0.699561	+	0.714573i	$0.746621\pi$
$564$	68.4149	2.88079
$565$	−18.3173	−0.770613
$566$	69.9098	2.93853
$567$	−7.79071	−0.327179
$568$	104.355	4.37863
$569$	33.6879	1.41227	0.706135	−	0.708077i	$-0.250437\pi$
$569$	33.6879	1.41227	0.706135	+	0.708077i	$0.250437\pi$
$570$	9.86395	0.413155
$571$	−31.9158	−1.33564	−0.667818	−	0.744324i	$-0.732772\pi$
$571$	−31.9158	−1.33564	−0.667818	+	0.744324i	$0.732772\pi$
$572$	−4.76851	−0.199381
$573$	−17.3631	−0.725353
$574$	−13.4047	−0.559503
$575$	−1.26585	−0.0527896
$576$	14.9548	0.623117
$577$	12.9808	0.540397	0.270198	−	0.962805i	$-0.412911\pi$
$577$	12.9808	0.540397	0.270198	+	0.962805i	$0.412911\pi$
$578$	−102.433	−4.26064
$579$	43.6290	1.81316
$580$	9.96781	0.413891
$581$	−5.87422	−0.243704
$582$	−34.4019	−1.42601
$583$	8.88664	0.368047
$584$	40.6162	1.68071
$585$	1.20641	0.0498791
$586$	−64.0347	−2.64525
$587$	−34.4468	−1.42177	−0.710885	−	0.703308i	$-0.751705\pi$
$587$	−34.4468	−1.42177	−0.710885	+	0.703308i	$0.751705\pi$
$588$	65.3370	2.69445
$589$	4.77327	0.196679
$590$	33.8603	1.39401
$591$	−27.3957	−1.12691
$592$	−87.4849	−3.59561
$593$	31.9913	1.31373	0.656863	−	0.754010i	$-0.271883\pi$
$593$	31.9913	1.31373	0.656863	+	0.754010i	$0.271883\pi$
$594$	13.8703	0.569106
$595$	−10.3735	−0.425273
$596$	−42.9166	−1.75793
$597$	51.5961	2.11169
$598$	−2.03121	−0.0830624
$599$	5.18325	0.211782	0.105891	−	0.994378i	$-0.466231\pi$
$599$	5.18325	0.211782	0.105891	+	0.994378i	$0.466231\pi$
$600$	−20.9122	−0.853737
$601$	28.5632	1.16512	0.582560	−	0.812788i	$-0.302051\pi$
$601$	28.5632	1.16512	0.582560	+	0.812788i	$0.302051\pi$
$602$	−8.11347	−0.330680
$603$	1.08998	0.0443876
$604$	−11.1266	−0.452733
$605$	−18.3912	−0.747707
$606$	44.4896	1.80727
$607$	37.5625	1.52461	0.762307	−	0.647215i	$-0.224066\pi$
$607$	37.5625	1.52461	0.762307	+	0.647215i	$0.224066\pi$
$608$	−15.5992	−0.632631
$609$	−1.41153	−0.0571979
$610$	36.6073	1.48219
$611$	−5.15031	−0.208359
$612$	−31.5398	−1.27492
$613$	21.8684	0.883257	0.441628	−	0.897198i	$-0.354401\pi$
$613$	21.8684	0.883257	0.441628	+	0.897198i	$0.354401\pi$
$614$	−42.2426	−1.70477
$615$	−26.2174	−1.05719
$616$	7.42856	0.299305
$617$	−6.11377	−0.246131	−0.123066	−	0.992399i	$-0.539273\pi$
$617$	−6.11377	−0.246131	−0.123066	+	0.992399i	$0.539273\pi$
$618$	35.2511	1.41801
$619$	16.0179	0.643812	0.321906	−	0.946772i	$-0.395677\pi$
$619$	16.0179	0.643812	0.321906	+	0.946772i	$0.395677\pi$
$620$	48.7561	1.95809
$621$	4.25751	0.170848
$622$	26.9531	1.08072
$623$	−0.329918	−0.0132179
$624$	−18.2435	−0.730326
$625$	−17.0684	−0.682735
$626$	−41.9594	−1.67703
$627$	2.32316	0.0927779
$628$	69.7375	2.78283
$629$	52.9231	2.11018
$630$	−3.06954	−0.122294
$631$	22.0014	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$631$	22.0014	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$632$	−122.346	−4.86666
$633$	−19.2802	−0.766319
$634$	3.08289	0.122437
$635$	−14.3058	−0.567707
$636$	73.6001	2.91843
$637$	−4.91861	−0.194882
$638$	3.25785	0.128980
$639$	−10.1557	−0.401753
$640$	−32.2431	−1.27452
$641$	−34.4264	−1.35976	−0.679881	−	0.733322i	$-0.737969\pi$
$641$	−34.4264	−1.35976	−0.679881	+	0.733322i	$0.737969\pi$
$642$	56.2914	2.22165
$643$	21.9206	0.864464	0.432232	−	0.901762i	$-0.357726\pi$
$643$	21.9206	0.864464	0.432232	+	0.901762i	$0.357726\pi$
$644$	3.72416	0.146752
$645$	−15.8686	−0.624825
$646$	19.4132	0.763800
$647$	23.9747	0.942542	0.471271	−	0.881988i	$-0.343795\pi$
$647$	23.9747	0.942542	0.471271	+	0.881988i	$0.343795\pi$
$648$	91.1812	3.58193
$649$	7.97477	0.313037
$650$	2.57121	0.100851
$651$	−6.90429	−0.270600
$652$	−83.8782	−3.28492
$653$	30.8057	1.20552	0.602760	−	0.797922i	$-0.294068\pi$
$653$	30.8057	1.20552	0.602760	+	0.797922i	$0.294068\pi$
$654$	8.80946	0.344477
$655$	−25.4835	−0.995722
$656$	85.2949	3.33021
$657$	−3.95272	−0.154210
$658$	13.1042	0.510855
$659$	−18.1149	−0.705656	−0.352828	−	0.935688i	$-0.614780\pi$
$659$	−18.1149	−0.705656	−0.352828	+	0.935688i	$0.614780\pi$
$660$	23.7297	0.923676
$661$	30.7408	1.19568	0.597840	−	0.801616i	$-0.296026\pi$
$661$	30.7408	1.19568	0.597840	+	0.801616i	$0.296026\pi$
$662$	88.0752	3.42314
$663$	11.0362	0.428612
$664$	68.7509	2.66805
$665$	1.36147	0.0527954
$666$	15.6600	0.606815
$667$	1.00000	0.0387202
$668$	−0.145645	−0.00563516
$669$	−48.1217	−1.86049
$670$	−6.85283	−0.264748
$671$	8.62175	0.332839
$672$	22.5634	0.870403
$673$	−13.7528	−0.530132	−0.265066	−	0.964230i	$-0.585394\pi$
$673$	−13.7528	−0.530132	−0.265066	+	0.964230i	$0.585394\pi$
$674$	76.0880	2.93080
$675$	−5.38937	−0.207437
$676$	−64.0844	−2.46478
$677$	22.5089	0.865086	0.432543	−	0.901613i	$-0.357616\pi$
$677$	22.5089	0.865086	0.432543	+	0.901613i	$0.357616\pi$
$678$	−49.5830	−1.90422
$679$	−4.74831	−0.182223
$680$	121.410	4.65586
$681$	24.0089	0.920022
$682$	15.9353	0.610195
$683$	7.72190	0.295470	0.147735	−	0.989027i	$-0.452802\pi$
$683$	7.72190	0.295470	0.147735	+	0.989027i	$0.452802\pi$
$684$	4.13942	0.158275
$685$	−28.4115	−1.08555
$686$	26.0362	0.994068
$687$	−7.90804	−0.301711
$688$	51.6264	1.96824
$689$	−5.54066	−0.211082
$690$	10.1080	0.384804
$691$	9.51693	0.362041	0.181020	−	0.983479i	$-0.442060\pi$
$691$	9.51693	0.362041	0.181020	+	0.983479i	$0.442060\pi$
$692$	−24.2574	−0.922127
$693$	−0.722939	−0.0274622
$694$	−56.9580	−2.16210
$695$	−4.36625	−0.165621
$696$	16.5203	0.626199
$697$	−51.5983	−1.95442
$698$	−73.3136	−2.77496
$699$	−33.4804	−1.26634
$700$	−4.71423	−0.178181
$701$	4.42170	0.167005	0.0835026	−	0.996508i	$-0.473389\pi$
$701$	4.42170	0.167005	0.0835026	+	0.996508i	$0.473389\pi$
$702$	−8.64789	−0.326394
$703$	−6.94586	−0.261968
$704$	−22.1441	−0.834588
$705$	25.6296	0.965268
$706$	−51.3874	−1.93399
$707$	6.14066	0.230943
$708$	66.0479	2.48223
$709$	−0.791672	−0.0297319	−0.0148659	−	0.999889i	$-0.504732\pi$
$709$	−0.791672	−0.0297319	−0.0148659	+	0.999889i	$0.504732\pi$
$710$	63.8497	2.39624
$711$	11.9066	0.446531
$712$	3.86131	0.144709
$713$	4.89136	0.183183
$714$	−28.0801	−1.05087
$715$	−1.78638	−0.0668069
$716$	−75.9929	−2.83999
$717$	−48.4439	−1.80917
$718$	−69.6732	−2.60018
$719$	−19.2556	−0.718112	−0.359056	−	0.933316i	$-0.616901\pi$
$719$	−19.2556	−0.718112	−0.359056	+	0.933316i	$0.616901\pi$
$720$	19.5316	0.727902
$721$	4.86551	0.181201
$722$	48.2865	1.79704
$723$	−37.6625	−1.40069
$724$	−47.1413	−1.75199
$725$	−1.26585	−0.0470125
$726$	−49.7831	−1.84762
$727$	−13.2064	−0.489800	−0.244900	−	0.969548i	$-0.578755\pi$
$727$	−13.2064	−0.489800	−0.244900	+	0.969548i	$0.578755\pi$
$728$	−4.63157	−0.171657
$729$	16.0976	0.596209
$730$	24.8511	0.919782
$731$	−31.2309	−1.15511
$732$	71.4063	2.63925
$733$	−10.3235	−0.381309	−0.190654	−	0.981657i	$-0.561061\pi$
$733$	−10.3235	−0.381309	−0.190654	+	0.981657i	$0.561061\pi$
$734$	47.6879	1.76019
$735$	24.4766	0.902833
$736$	−15.9851	−0.589219
$737$	−1.61398	−0.0594517
$738$	−15.2680	−0.562024
$739$	−18.6857	−0.687363	−0.343682	−	0.939086i	$-0.611674\pi$
$739$	−18.6857	−0.687363	−0.343682	+	0.939086i	$0.611674\pi$
$740$	−70.9478	−2.60809
$741$	−1.44845	−0.0532100
$742$	14.0974	0.517531
$743$	−13.6337	−0.500171	−0.250085	−	0.968224i	$-0.580459\pi$
$743$	−13.6337	−0.500171	−0.250085	+	0.968224i	$0.580459\pi$
$744$	80.8066	2.96251
$745$	−16.0774	−0.589032
$746$	−67.1554	−2.45873
$747$	−6.69076	−0.244802
$748$	46.7021	1.70760
$749$	7.76960	0.283895
$750$	−63.3351	−2.31267
$751$	1.12990	0.0412307	0.0206153	−	0.999787i	$-0.493437\pi$
$751$	1.12990	0.0412307	0.0206153	+	0.999787i	$0.493437\pi$
$752$	−83.3826	−3.04065
$753$	12.5229	0.456360
$754$	−2.03121	−0.0739724
$755$	−4.16824	−0.151698
$756$	15.8556	0.576664
$757$	−18.0367	−0.655554	−0.327777	−	0.944755i	$-0.606300\pi$
$757$	−18.0367	−0.655554	−0.327777	+	0.944755i	$0.606300\pi$
$758$	−77.9962	−2.83295
$759$	2.38063	0.0864114
$760$	−15.9344	−0.578000
$761$	41.9255	1.51980	0.759899	−	0.650041i	$-0.225248\pi$
$761$	41.9255	1.51980	0.759899	+	0.650041i	$0.225248\pi$
$762$	−38.7243	−1.40283
$763$	1.21592	0.0440193
$764$	45.8106	1.65737
$765$	−11.8155	−0.427189
$766$	−54.9735	−1.98627
$767$	−4.97212	−0.179533
$768$	−16.1699	−0.583480
$769$	−19.9423	−0.719138	−0.359569	−	0.933118i	$-0.617076\pi$
$769$	−19.9423	−0.719138	−0.359569	+	0.933118i	$0.617076\pi$
$770$	4.54519	0.163797
$771$	−22.5508	−0.812148
$772$	−115.110	−4.14291
$773$	−53.2127	−1.91393	−0.956964	−	0.290206i	$-0.906276\pi$
$773$	−53.2127	−1.91393	−0.956964	+	0.290206i	$0.906276\pi$
$774$	−9.24127	−0.332171
$775$	−6.19174	−0.222414
$776$	55.5734	1.99497
$777$	10.0468	0.360428
$778$	−0.392678	−0.0140782
$779$	6.77199	0.242632
$780$	−14.7950	−0.529746
$781$	15.0379	0.538098
$782$	19.8934	0.711387
$783$	4.25751	0.152151
$784$	−79.6314	−2.84398
$785$	26.1251	0.932446
$786$	−68.9813	−2.46048
$787$	−34.6938	−1.23670	−0.618351	−	0.785902i	$-0.712199\pi$
$787$	−34.6938	−1.23670	−0.618351	+	0.785902i	$0.712199\pi$
$788$	72.2805	2.57489
$789$	−57.6179	−2.05125
$790$	−74.8577	−2.66332
$791$	−6.84367	−0.243333
$792$	8.46116	0.300654
$793$	−5.37551	−0.190890
$794$	88.7936	3.15117
$795$	27.5721	0.977883
$796$	−136.131	−4.82502
$797$	−25.4424	−0.901218	−0.450609	−	0.892721i	$-0.648793\pi$
$797$	−25.4424	−0.901218	−0.450609	+	0.892721i	$0.648793\pi$
$798$	3.68536	0.130460
$799$	50.4415	1.78449
$800$	20.2348	0.715408
$801$	−0.375778	−0.0132775
$802$	−58.2167	−2.05570
$803$	5.85294	0.206546
$804$	−13.3671	−0.471423
$805$	1.39515	0.0491725
$806$	−9.93539	−0.349959
$807$	−13.8581	−0.487830
$808$	−71.8692	−2.52835
$809$	5.25710	0.184830	0.0924149	−	0.995721i	$-0.470541\pi$
$809$	5.25710	0.184830	0.0924149	+	0.995721i	$0.470541\pi$
$810$	55.7895	1.96024
$811$	23.1032	0.811261	0.405631	−	0.914037i	$-0.367052\pi$
$811$	23.1032	0.811261	0.405631	+	0.914037i	$0.367052\pi$
$812$	3.72416	0.130692
$813$	29.5810	1.03745
$814$	−23.1884	−0.812753
$815$	−31.4225	−1.10068
$816$	178.675	6.25487
$817$	4.09888	0.143402
$818$	79.4186	2.77681
$819$	0.450739	0.0157501
$820$	69.1718	2.41558
$821$	−23.2924	−0.812912	−0.406456	−	0.913670i	$-0.633235\pi$
$821$	−23.2924	−0.812912	−0.406456	+	0.913670i	$0.633235\pi$
$822$	−76.9070	−2.68244
$823$	50.0863	1.74590	0.872950	−	0.487810i	$-0.162204\pi$
$823$	50.0863	1.74590	0.872950	+	0.487810i	$0.162204\pi$
$824$	−56.9452	−1.98378
$825$	−3.01352	−0.104917
$826$	12.6508	0.440179
$827$	19.9497	0.693719	0.346859	−	0.937917i	$-0.387248\pi$
$827$	19.9497	0.693719	0.346859	+	0.937917i	$0.387248\pi$
$828$	4.24183	0.147414
$829$	16.1619	0.561325	0.280663	−	0.959806i	$-0.409446\pi$
$829$	16.1619	0.561325	0.280663	+	0.959806i	$0.409446\pi$
$830$	42.0655	1.46011
$831$	46.7165	1.62058
$832$	13.8065	0.478653
$833$	48.1722	1.66907
$834$	−11.8190	−0.409259
$835$	−0.0545615	−0.00188818
$836$	−6.12940	−0.211990
$837$	20.8250	0.719817
$838$	49.7957	1.72016
$839$	8.99321	0.310480	0.155240	−	0.987877i	$-0.450385\pi$
$839$	8.99321	0.310480	0.155240	+	0.987877i	$0.450385\pi$
$840$	23.0482	0.795239
$841$	1.00000	0.0344828
$842$	25.3799	0.874650
$843$	12.8388	0.442193
$844$	50.8687	1.75097
$845$	−24.0074	−0.825878
$846$	14.9257	0.513157
$847$	−6.87129	−0.236100
$848$	−89.7023	−3.08039
$849$	51.0856	1.75325
$850$	−25.1821	−0.863739
$851$	−7.11770	−0.243992
$852$	124.545	4.26685
$853$	−14.6568	−0.501841	−0.250920	−	0.968008i	$-0.580733\pi$
$853$	−14.6568	−0.501841	−0.250920	+	0.968008i	$0.580733\pi$
$854$	13.6772	0.468024
$855$	1.55072	0.0530333
$856$	−90.9340	−3.10806
$857$	−49.1741	−1.67976	−0.839878	−	0.542775i	$-0.817373\pi$
$857$	−49.1741	−1.67976	−0.839878	+	0.542775i	$0.817373\pi$
$858$	−4.83556	−0.165083
$859$	−49.7842	−1.69862	−0.849308	−	0.527898i	$-0.822980\pi$
$859$	−49.7842	−1.69862	−0.849308	+	0.527898i	$0.822980\pi$
$860$	41.8676	1.42767
$861$	−9.79532	−0.333824
$862$	−70.3462	−2.39600
$863$	19.2285	0.654546	0.327273	−	0.944930i	$-0.393870\pi$
$863$	19.2285	0.654546	0.327273	+	0.944930i	$0.393870\pi$
$864$	−68.0567	−2.31534
$865$	−9.08732	−0.308978
$866$	−90.0620	−3.06043
$867$	−74.8512	−2.54208
$868$	18.2162	0.618299
$869$	−17.6305	−0.598073
$870$	10.1080	0.342692
$871$	1.00629	0.0340967
$872$	−14.2309	−0.481920
$873$	−5.40834	−0.183045
$874$	−2.61090	−0.0883151
$875$	−8.74179	−0.295526
$876$	48.4747	1.63781
$877$	−11.6483	−0.393333	−0.196667	−	0.980470i	$-0.563012\pi$
$877$	−11.6483	−0.393333	−0.196667	+	0.980470i	$0.563012\pi$
$878$	53.7864	1.81520
$879$	−46.7925	−1.57827
$880$	−28.9212	−0.974934
$881$	−57.7298	−1.94497	−0.972483	−	0.232975i	$-0.925154\pi$
$881$	−57.7298	−1.94497	−0.972483	+	0.232975i	$0.925154\pi$
$882$	14.2543	0.479966
$883$	−57.9128	−1.94892	−0.974460	−	0.224561i	$-0.927905\pi$
$883$	−57.9128	−1.94892	−0.974460	+	0.224561i	$0.927905\pi$
$884$	−29.1179	−0.979342
$885$	24.7429	0.831724
$886$	−34.6403	−1.16376
$887$	2.99750	0.100646	0.0503231	−	0.998733i	$-0.483975\pi$
$887$	2.99750	0.100646	0.0503231	+	0.998733i	$0.483975\pi$
$888$	−117.586	−3.94594
$889$	−5.34491	−0.179262
$890$	2.36255	0.0791930
$891$	13.1395	0.440191
$892$	126.964	4.25106
$893$	−6.62016	−0.221535
$894$	−43.5201	−1.45553
$895$	−28.4685	−0.951598
$896$	−12.0466	−0.402450
$897$	−1.48428	−0.0495586
$898$	−41.7325	−1.39263
$899$	4.89136	0.163136
$900$	−5.36953	−0.178984
$901$	54.2645	1.80781
$902$	22.6079	0.752762
$903$	−5.92881	−0.197298
$904$	80.0972	2.66399
$905$	−17.6601	−0.587043
$906$	−11.2830	−0.374853
$907$	3.99187	0.132548	0.0662740	−	0.997801i	$-0.478889\pi$
$907$	3.99187	0.132548	0.0662740	+	0.997801i	$0.478889\pi$
$908$	−63.3448	−2.10217
$909$	6.99423	0.231984
$910$	−2.83384	−0.0939409
$911$	−40.9059	−1.35527	−0.677637	−	0.735397i	$-0.736996\pi$
$911$	−40.9059	−1.35527	−0.677637	+	0.735397i	$0.736996\pi$
$912$	−23.4501	−0.776510
$913$	9.90726	0.327882
$914$	43.7594	1.44743
$915$	26.7503	0.884337
$916$	20.8645	0.689383
$917$	−9.52111	−0.314415
$918$	84.6964	2.79540
$919$	49.8939	1.64585	0.822925	−	0.568150i	$-0.192341\pi$
$919$	49.8939	1.64585	0.822925	+	0.568150i	$0.192341\pi$
$920$	−16.3286	−0.538337
$921$	−30.8682	−1.01714
$922$	108.919	3.58706
$923$	−9.37585	−0.308610
$924$	8.86584	0.291665
$925$	9.00994	0.296245
$926$	−93.5436	−3.07403
$927$	5.54184	0.182018
$928$	−15.9851	−0.524737
$929$	7.25862	0.238148	0.119074	−	0.992885i	$-0.462007\pi$
$929$	7.25862	0.238148	0.119074	+	0.992885i	$0.462007\pi$
$930$	49.4418	1.62126
$931$	−6.32234	−0.207206
$932$	88.3344	2.89349
$933$	19.6956	0.644806
$934$	−19.2212	−0.628937
$935$	17.4956	0.572167
$936$	−5.27538	−0.172431
$937$	−18.9134	−0.617872	−0.308936	−	0.951083i	$-0.599973\pi$
$937$	−18.9134	−0.617872	−0.308936	+	0.951083i	$0.599973\pi$
$938$	−2.56035	−0.0835983
$939$	−30.6612	−1.00059
$940$	−67.6210	−2.20555
$941$	21.8323	0.711712	0.355856	−	0.934541i	$-0.384189\pi$
$941$	21.8323	0.711712	0.355856	+	0.934541i	$0.384189\pi$
$942$	70.7182	2.30412
$943$	6.93952	0.225982
$944$	−80.4978	−2.61998
$945$	5.93985	0.193223
$946$	13.6839	0.444902
$947$	7.36345	0.239280	0.119640	−	0.992817i	$-0.461826\pi$
$947$	7.36345	0.239280	0.119640	+	0.992817i	$0.461826\pi$
$948$	−146.017	−4.74243
$949$	−3.64920	−0.118458
$950$	3.30501	0.107229
$951$	2.25278	0.0730512
$952$	45.3610	1.47016
$953$	11.8527	0.383945	0.191973	−	0.981400i	$-0.438512\pi$
$953$	11.8527	0.383945	0.191973	+	0.981400i	$0.438512\pi$
$954$	16.0570	0.519864
$955$	17.1616	0.555337
$956$	127.814	4.13380
$957$	2.38063	0.0769548
$958$	40.0275	1.29323
$959$	−10.6151	−0.342778
$960$	−68.7055	−2.21746
$961$	−7.07458	−0.228212
$962$	14.4575	0.466130
$963$	8.84960	0.285174
$964$	99.3686	3.20045
$965$	−43.1228	−1.38817
$966$	3.77653	0.121508
$967$	14.5412	0.467614	0.233807	−	0.972283i	$-0.424882\pi$
$967$	14.5412	0.467614	0.233807	+	0.972283i	$0.424882\pi$
$968$	80.4204	2.58481
$969$	14.1859	0.455717
$970$	34.0028	1.09176
$971$	−0.384819	−0.0123494	−0.00617471	−	0.999981i	$-0.501965\pi$
$971$	−0.384819	−0.0123494	−0.00617471	+	0.999981i	$0.501965\pi$
$972$	42.9390	1.37727
$973$	−1.63131	−0.0522975
$974$	−89.3338	−2.86244
$975$	1.87888	0.0601722
$976$	−87.0285	−2.78572
$977$	38.9574	1.24636	0.623179	−	0.782079i	$-0.285841\pi$
$977$	38.9574	1.24636	0.623179	+	0.782079i	$0.285841\pi$
$978$	−85.0577	−2.71985
$979$	0.556429	0.0177835
$980$	−64.5789	−2.06290
$981$	1.38494	0.0442177
$982$	40.1091	1.27993
$983$	−52.8096	−1.68436	−0.842182	−	0.539193i	$-0.818729\pi$
$983$	−52.8096	−1.68436	−0.842182	+	0.539193i	$0.818729\pi$
$984$	114.643	3.65468
$985$	27.0778	0.862771
$986$	19.8934	0.633536
$987$	9.57572	0.304798
$988$	3.82157	0.121580
$989$	4.20028	0.133561
$990$	5.17698	0.164535
$991$	−12.9556	−0.411547	−0.205773	−	0.978600i	$-0.565971\pi$
$991$	−12.9556	−0.411547	−0.205773	+	0.978600i	$0.565971\pi$
$992$	−78.1890	−2.48250
$993$	64.3597	2.04239
$994$	23.8555	0.756649
$995$	−50.9974	−1.61673
$996$	82.0529	2.59995
$997$	31.5432	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$997$	31.5432	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$998$	−30.2765	−0.958386
$999$	−30.3036	−0.958765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.a.a.1.1	✓	10
3.2	odd	2		6003.2.a.l.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.1	✓	10	1.1	even	1	trivial
6003.2.a.l.1.10		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67549 q^{2} -1.95508 q^{3} +5.15827 q^{4} +1.93239 q^{5} +5.23080 q^{6} +0.721979 q^{7} -8.44992 q^{8} +0.822337 q^{9} +O(q^{10})\)	\(q-2.67549 q^{2} -1.95508 q^{3} +5.15827 q^{4} +1.93239 q^{5} +5.23080 q^{6} +0.721979 q^{7} -8.44992 q^{8} +0.822337 q^{9} -5.17011 q^{10} -1.21766 q^{11} -10.0848 q^{12} +0.759191 q^{13} -1.93165 q^{14} -3.77799 q^{15} +12.2912 q^{16} -7.43542 q^{17} -2.20016 q^{18} +0.975858 q^{19} +9.96781 q^{20} -1.41153 q^{21} +3.25785 q^{22} +1.00000 q^{23} +16.5203 q^{24} -1.26585 q^{25} -2.03121 q^{26} +4.25751 q^{27} +3.72416 q^{28} +1.00000 q^{29} +10.1080 q^{30} +4.89136 q^{31} -15.9851 q^{32} +2.38063 q^{33} +19.8934 q^{34} +1.39515 q^{35} +4.24183 q^{36} -7.11770 q^{37} -2.61090 q^{38} -1.48428 q^{39} -16.3286 q^{40} +6.93952 q^{41} +3.77653 q^{42} +4.20028 q^{43} -6.28103 q^{44} +1.58908 q^{45} -2.67549 q^{46} -6.78394 q^{47} -24.0302 q^{48} -6.47875 q^{49} +3.38678 q^{50} +14.5368 q^{51} +3.91611 q^{52} -7.29810 q^{53} -11.3909 q^{54} -2.35301 q^{55} -6.10066 q^{56} -1.90788 q^{57} -2.67549 q^{58} -6.54924 q^{59} -19.4879 q^{60} -7.08057 q^{61} -13.0868 q^{62} +0.593710 q^{63} +18.1857 q^{64} +1.46706 q^{65} -6.36936 q^{66} +1.32547 q^{67} -38.3539 q^{68} -1.95508 q^{69} -3.73271 q^{70} -12.3498 q^{71} -6.94868 q^{72} -4.80670 q^{73} +19.0434 q^{74} +2.47484 q^{75} +5.03373 q^{76} -0.879128 q^{77} +3.97118 q^{78} +14.4789 q^{79} +23.7514 q^{80} -10.7908 q^{81} -18.5667 q^{82} -8.13628 q^{83} -7.28103 q^{84} -14.3682 q^{85} -11.2378 q^{86} -1.95508 q^{87} +10.2892 q^{88} -0.456964 q^{89} -4.25157 q^{90} +0.548120 q^{91} +5.15827 q^{92} -9.56300 q^{93} +18.1504 q^{94} +1.88574 q^{95} +31.2522 q^{96} -6.57680 q^{97} +17.3338 q^{98} -1.00133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} - 9 q^{3} + 9 q^{4} - 10 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 7 q^{9}+O(q^{10}) \) 10 * q - 3 * q^2 - 9 * q^3 + 9 * q^4 - 10 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 7 * q^9	\( 10 q - 3 q^{2} - 9 q^{3} + 9 q^{4} - 10 q^{5} + 4 q^{6} + q^{7} - 9 q^{8} + 7 q^{9} - 6 q^{10} - 17 q^{12} - 13 q^{13} - 12 q^{14} + 2 q^{15} - 5 q^{16} - 22 q^{17} + 12 q^{18} - 2 q^{19} + 3 q^{20} - 7 q^{21} + 3 q^{22} + 10 q^{23} - 6 q^{24} + 10 q^{25} - 25 q^{26} - 24 q^{27} + 19 q^{28} + 10 q^{29} - 3 q^{30} - 22 q^{31} - 31 q^{32} - 9 q^{33} + 13 q^{34} - 15 q^{35} + 19 q^{36} - 9 q^{37} - 10 q^{38} + 4 q^{39} - 6 q^{40} - 25 q^{41} - 34 q^{42} + 3 q^{43} - 27 q^{44} - 28 q^{45} - 3 q^{46} - 17 q^{47} - 3 q^{48} + 17 q^{49} + 2 q^{50} + 38 q^{51} - 18 q^{52} - 43 q^{53} - 47 q^{54} - 11 q^{55} - 7 q^{56} + 18 q^{57} - 3 q^{58} - 7 q^{59} - 21 q^{60} - 6 q^{61} + 3 q^{62} + 11 q^{63} + 33 q^{64} + 11 q^{65} + 55 q^{66} + 11 q^{67} - 51 q^{68} - 9 q^{69} + 34 q^{70} - 17 q^{71} + 34 q^{72} - 44 q^{73} + 9 q^{74} + q^{75} + 24 q^{76} - 71 q^{77} + 38 q^{78} + 5 q^{79} + 38 q^{80} + 18 q^{81} + 33 q^{82} - 32 q^{83} + 14 q^{84} + 16 q^{85} - 9 q^{86} - 9 q^{87} + 18 q^{88} - 10 q^{89} - 9 q^{90} - 3 q^{91} + 9 q^{92} - 8 q^{93} + 47 q^{94} - 8 q^{95} + 60 q^{96} + 6 q^{97} - 73 q^{98} + 26 q^{99}+O(q^{100}) \) 10 * q - 3 * q^2 - 9 * q^3 + 9 * q^4 - 10 * q^5 + 4 * q^6 + q^7 - 9 * q^8 + 7 * q^9 - 6 * q^10 - 17 * q^12 - 13 * q^13 - 12 * q^14 + 2 * q^15 - 5 * q^16 - 22 * q^17 + 12 * q^18 - 2 * q^19 + 3 * q^20 - 7 * q^21 + 3 * q^22 + 10 * q^23 - 6 * q^24 + 10 * q^25 - 25 * q^26 - 24 * q^27 + 19 * q^28 + 10 * q^29 - 3 * q^30 - 22 * q^31 - 31 * q^32 - 9 * q^33 + 13 * q^34 - 15 * q^35 + 19 * q^36 - 9 * q^37 - 10 * q^38 + 4 * q^39 - 6 * q^40 - 25 * q^41 - 34 * q^42 + 3 * q^43 - 27 * q^44 - 28 * q^45 - 3 * q^46 - 17 * q^47 - 3 * q^48 + 17 * q^49 + 2 * q^50 + 38 * q^51 - 18 * q^52 - 43 * q^53 - 47 * q^54 - 11 * q^55 - 7 * q^56 + 18 * q^57 - 3 * q^58 - 7 * q^59 - 21 * q^60 - 6 * q^61 + 3 * q^62 + 11 * q^63 + 33 * q^64 + 11 * q^65 + 55 * q^66 + 11 * q^67 - 51 * q^68 - 9 * q^69 + 34 * q^70 - 17 * q^71 + 34 * q^72 - 44 * q^73 + 9 * q^74 + q^75 + 24 * q^76 - 71 * q^77 + 38 * q^78 + 5 * q^79 + 38 * q^80 + 18 * q^81 + 33 * q^82 - 32 * q^83 + 14 * q^84 + 16 * q^85 - 9 * q^86 - 9 * q^87 + 18 * q^88 - 10 * q^89 - 9 * q^90 - 3 * q^91 + 9 * q^92 - 8 * q^93 + 47 * q^94 - 8 * q^95 + 60 * q^96 + 6 * q^97 - 73 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more