LMFDB - Embedded newform 446.2.a.f.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [446,2,Mod(1,446)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(446, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("446.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 446 = 2 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	446.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.56132793015$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 12x^{6} + 46x^{5} + 54x^{4} - 148x^{3} - 98x^{2} + 126x + 69 $ x^8 - 4x^7 - 12x^6 + 46x^5 + 54x^4 - 148x^3 - 98x^2 + 126*x + 69
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			1.5
Root			$-0.517925$ of defining polynomial
Character	$\chi$	$=$	446.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.51793 q^{3} +1.00000 q^{4} +2.10687 q^{5} -1.51793 q^{6} +2.35547 q^{7} -1.00000 q^{8} -0.695903 q^{9} -2.10687 q^{10} -1.36297 q^{11} +1.51793 q^{12} +6.92929 q^{13} -2.35547 q^{14} +3.19807 q^{15} +1.00000 q^{16} -0.171333 q^{17} +0.695903 q^{18} -6.80032 q^{19} +2.10687 q^{20} +3.57543 q^{21} +1.36297 q^{22} -1.85608 q^{23} -1.51793 q^{24} -0.561091 q^{25} -6.92929 q^{26} -5.61010 q^{27} +2.35547 q^{28} -1.03585 q^{29} -3.19807 q^{30} +7.57814 q^{31} -1.00000 q^{32} -2.06888 q^{33} +0.171333 q^{34} +4.96267 q^{35} -0.695903 q^{36} +0.829338 q^{37} +6.80032 q^{38} +10.5181 q^{39} -2.10687 q^{40} -4.45177 q^{41} -3.57543 q^{42} +2.68998 q^{43} -1.36297 q^{44} -1.46618 q^{45} +1.85608 q^{46} +9.80625 q^{47} +1.51793 q^{48} -1.45177 q^{49} +0.561091 q^{50} -0.260070 q^{51} +6.92929 q^{52} +4.58658 q^{53} +5.61010 q^{54} -2.87160 q^{55} -2.35547 q^{56} -10.3224 q^{57} +1.03585 q^{58} -8.54953 q^{59} +3.19807 q^{60} +4.85077 q^{61} -7.57814 q^{62} -1.63918 q^{63} +1.00000 q^{64} +14.5991 q^{65} +2.06888 q^{66} +10.1639 q^{67} -0.171333 q^{68} -2.81739 q^{69} -4.96267 q^{70} -13.3869 q^{71} +0.695903 q^{72} -1.03805 q^{73} -0.829338 q^{74} -0.851694 q^{75} -6.80032 q^{76} -3.21043 q^{77} -10.5181 q^{78} -12.7446 q^{79} +2.10687 q^{80} -6.42801 q^{81} +4.45177 q^{82} -7.16289 q^{83} +3.57543 q^{84} -0.360976 q^{85} -2.68998 q^{86} -1.57234 q^{87} +1.36297 q^{88} +5.27074 q^{89} +1.46618 q^{90} +16.3217 q^{91} -1.85608 q^{92} +11.5030 q^{93} -9.80625 q^{94} -14.3274 q^{95} -1.51793 q^{96} +12.5216 q^{97} +1.45177 q^{98} +0.948494 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 10 q^{13} - 4 q^{14} - 4 q^{15} + 8 q^{16} + 8 q^{17} - 16 q^{18} + 16 q^{19} + 4 q^{20}+ \cdots - 40 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 10 * q^13 - 4 * q^14 - 4 * q^15 + 8 * q^16 + 8 * q^17 - 16 * q^18 + 16 * q^19 + 4 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 + 24 * q^25 - 10 * q^26 + 22 * q^27 + 4 * q^28 + 8 * q^29 + 4 * q^30 + 12 * q^31 - 8 * q^32 - 8 * q^34 - 24 * q^35 + 16 * q^36 + 16 * q^37 - 16 * q^38 - 4 * q^40 + 16 * q^41 + 12 * q^43 + 2 * q^44 + 34 * q^45 + 4 * q^46 - 4 * q^47 + 4 * q^48 + 40 * q^49 - 24 * q^50 + 26 * q^51 + 10 * q^52 - 8 * q^53 - 22 * q^54 + 8 * q^55 - 4 * q^56 + 16 * q^57 - 8 * q^58 - 4 * q^60 + 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 - 24 * q^65 - 10 * q^67 + 8 * q^68 - 28 * q^69 + 24 * q^70 + 8 * q^71 - 16 * q^72 + 8 * q^73 - 16 * q^74 + 6 * q^75 + 16 * q^76 - 56 * q^77 + 16 * q^79 + 4 * q^80 + 24 * q^81 - 16 * q^82 - 44 * q^83 - 18 * q^85 - 12 * q^86 - 72 * q^87 - 2 * q^88 - 34 * q^90 + 4 * q^91 - 4 * q^92 - 62 * q^93 + 4 * q^94 - 48 * q^95 - 4 * q^96 + 12 * q^97 - 40 * q^98 - 40 * 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
$3$	1.51793	0.876375	0.438187	−	0.898884i	$-0.355621\pi$
$3$	1.51793	0.876375	0.438187	+	0.898884i	$0.355621\pi$
$4$	1.00000	0.500000
$5$	2.10687	0.942222	0.471111	−	0.882074i	$-0.343853\pi$
$5$	2.10687	0.942222	0.471111	+	0.882074i	$0.343853\pi$
$6$	−1.51793	−0.619690
$7$	2.35547	0.890284	0.445142	−	0.895460i	$-0.353153\pi$
$7$	2.35547	0.890284	0.445142	+	0.895460i	$0.353153\pi$
$8$	−1.00000	−0.353553
$9$	−0.695903	−0.231968
$10$	−2.10687	−0.666251
$11$	−1.36297	−0.410951	−0.205475	−	0.978662i	$-0.565874\pi$
$11$	−1.36297	−0.410951	−0.205475	+	0.978662i	$0.565874\pi$
$12$	1.51793	0.438187
$13$	6.92929	1.92184	0.960919	−	0.276830i	$-0.0892838\pi$
$13$	6.92929	1.92184	0.960919	+	0.276830i	$0.0892838\pi$
$14$	−2.35547	−0.629526
$15$	3.19807	0.825739
$16$	1.00000	0.250000
$17$	−0.171333	−0.0415543	−0.0207771	−	0.999784i	$-0.506614\pi$
$17$	−0.171333	−0.0415543	−0.0207771	+	0.999784i	$0.506614\pi$
$18$	0.695903	0.164026
$19$	−6.80032	−1.56010	−0.780050	−	0.625717i	$-0.784807\pi$
$19$	−6.80032	−1.56010	−0.780050	+	0.625717i	$0.784807\pi$
$20$	2.10687	0.471111
$21$	3.57543	0.780222
$22$	1.36297	0.290586
$23$	−1.85608	−0.387019	−0.193510	−	0.981098i	$-0.561987\pi$
$23$	−1.85608	−0.387019	−0.193510	+	0.981098i	$0.561987\pi$
$24$	−1.51793	−0.309845
$25$	−0.561091	−0.112218
$26$	−6.92929	−1.35894
$27$	−5.61010	−1.07967
$28$	2.35547	0.445142
$29$	−1.03585	−0.192353	−0.0961763	−	0.995364i	$-0.530661\pi$
$29$	−1.03585	−0.192353	−0.0961763	+	0.995364i	$0.530661\pi$
$30$	−3.19807	−0.583886
$31$	7.57814	1.36107	0.680537	−	0.732714i	$-0.261747\pi$
$31$	7.57814	1.36107	0.680537	+	0.732714i	$0.261747\pi$
$32$	−1.00000	−0.176777
$33$	−2.06888	−0.360147
$34$	0.171333	0.0293833
$35$	4.96267	0.838845
$36$	−0.695903	−0.115984
$37$	0.829338	0.136342	0.0681711	−	0.997674i	$-0.478284\pi$
$37$	0.829338	0.136342	0.0681711	+	0.997674i	$0.478284\pi$
$38$	6.80032	1.10316
$39$	10.5181	1.68425
$40$	−2.10687	−0.333126
$41$	−4.45177	−0.695249	−0.347624	−	0.937634i	$-0.613012\pi$
$41$	−4.45177	−0.695249	−0.347624	+	0.937634i	$0.613012\pi$
$42$	−3.57543	−0.551700
$43$	2.68998	0.410218	0.205109	−	0.978739i	$-0.434245\pi$
$43$	2.68998	0.410218	0.205109	+	0.978739i	$0.434245\pi$
$44$	−1.36297	−0.205475
$45$	−1.46618	−0.218565
$46$	1.85608	0.273664
$47$	9.80625	1.43039	0.715194	−	0.698926i	$-0.246338\pi$
$47$	9.80625	1.43039	0.715194	+	0.698926i	$0.246338\pi$
$48$	1.51793	0.219094
$49$	−1.45177	−0.207395
$50$	0.561091	0.0793503
$51$	−0.260070	−0.0364171
$52$	6.92929	0.960919
$53$	4.58658	0.630015	0.315007	−	0.949089i	$-0.397993\pi$
$53$	4.58658	0.630015	0.315007	+	0.949089i	$0.397993\pi$
$54$	5.61010	0.763439
$55$	−2.87160	−0.387207
$56$	−2.35547	−0.314763
$57$	−10.3224	−1.36723
$58$	1.03585	0.136014
$59$	−8.54953	−1.11305	−0.556527	−	0.830830i	$-0.687866\pi$
$59$	−8.54953	−1.11305	−0.556527	+	0.830830i	$0.687866\pi$
$60$	3.19807	0.412870
$61$	4.85077	0.621078	0.310539	−	0.950561i	$-0.399490\pi$
$61$	4.85077	0.621078	0.310539	+	0.950561i	$0.399490\pi$
$62$	−7.57814	−0.962425
$63$	−1.63918	−0.206517
$64$	1.00000	0.125000
$65$	14.5991	1.81080
$66$	2.06888	0.254662
$67$	10.1639	1.24172	0.620858	−	0.783923i	$-0.286784\pi$
$67$	10.1639	1.24172	0.620858	+	0.783923i	$0.286784\pi$
$68$	−0.171333	−0.0207771
$69$	−2.81739	−0.339174
$70$	−4.96267	−0.593153
$71$	−13.3869	−1.58873	−0.794366	−	0.607439i	$-0.792197\pi$
$71$	−13.3869	−1.58873	−0.794366	+	0.607439i	$0.792197\pi$
$72$	0.695903	0.0820130
$73$	−1.03805	−0.121494	−0.0607470	−	0.998153i	$-0.519348\pi$
$73$	−1.03805	−0.121494	−0.0607470	+	0.998153i	$0.519348\pi$
$74$	−0.829338	−0.0964085
$75$	−0.851694	−0.0983452
$76$	−6.80032	−0.780050
$77$	−3.21043	−0.365863
$78$	−10.5181	−1.19094
$79$	−12.7446	−1.43388	−0.716940	−	0.697135i	$-0.754458\pi$
$79$	−12.7446	−1.43388	−0.716940	+	0.697135i	$0.754458\pi$
$80$	2.10687	0.235555
$81$	−6.42801	−0.714223
$82$	4.45177	0.491615
$83$	−7.16289	−0.786230	−0.393115	−	0.919489i	$-0.628603\pi$
$83$	−7.16289	−0.786230	−0.393115	+	0.919489i	$0.628603\pi$
$84$	3.57543	0.390111
$85$	−0.360976	−0.0391533
$86$	−2.68998	−0.290068
$87$	−1.57234	−0.168573
$88$	1.36297	0.145293
$89$	5.27074	0.558697	0.279349	−	0.960190i	$-0.409881\pi$
$89$	5.27074	0.558697	0.279349	+	0.960190i	$0.409881\pi$
$90$	1.46618	0.154549
$91$	16.3217	1.71098
$92$	−1.85608	−0.193510
$93$	11.5030	1.19281
$94$	−9.80625	−1.01144
$95$	−14.3274	−1.46996
$96$	−1.51793	−0.154923
$97$	12.5216	1.27137	0.635687	−	0.771947i	$-0.280717\pi$
$97$	12.5216	1.27137	0.635687	+	0.771947i	$0.280717\pi$
$98$	1.45177	0.146650
$99$	0.948494	0.0953273
$100$	−0.561091	−0.0561091
$101$	10.6788	1.06258	0.531290	−	0.847190i	$-0.321707\pi$
$101$	10.6788	1.06258	0.531290	+	0.847190i	$0.321707\pi$
$102$	0.260070	0.0257508
$103$	−11.2884	−1.11228	−0.556140	−	0.831089i	$-0.687718\pi$
$103$	−11.2884	−1.11228	−0.556140	+	0.831089i	$0.687718\pi$
$104$	−6.92929	−0.679472
$105$	7.53296	0.735142
$106$	−4.58658	−0.445488
$107$	−15.8258	−1.52993	−0.764967	−	0.644069i	$-0.777245\pi$
$107$	−15.8258	−1.52993	−0.764967	+	0.644069i	$0.777245\pi$
$108$	−5.61010	−0.539833
$109$	−10.6746	−1.02244	−0.511221	−	0.859449i	$-0.670807\pi$
$109$	−10.6746	−1.02244	−0.511221	+	0.859449i	$0.670807\pi$
$110$	2.87160	0.273796
$111$	1.25887	0.119487
$112$	2.35547	0.222571
$113$	−10.1967	−0.959223	−0.479611	−	0.877481i	$-0.659222\pi$
$113$	−10.1967	−0.959223	−0.479611	+	0.877481i	$0.659222\pi$
$114$	10.3224	0.966779
$115$	−3.91052	−0.364658
$116$	−1.03585	−0.0961763
$117$	−4.82211	−0.445804
$118$	8.54953	0.787048
$119$	−0.403569	−0.0369951
$120$	−3.19807	−0.291943
$121$	−9.14232	−0.831120
$122$	−4.85077	−0.439169
$123$	−6.75745	−0.609298
$124$	7.57814	0.680537
$125$	−11.7165	−1.04796
$126$	1.63918	0.146030
$127$	−1.16477	−0.103357	−0.0516784	−	0.998664i	$-0.516457\pi$
$127$	−1.16477	−0.103357	−0.0516784	+	0.998664i	$0.516457\pi$
$128$	−1.00000	−0.0883883
$129$	4.08318	0.359504
$130$	−14.5991	−1.28043
$131$	9.67725	0.845505	0.422753	−	0.906245i	$-0.361064\pi$
$131$	9.67725	0.845505	0.422753	+	0.906245i	$0.361064\pi$
$132$	−2.06888	−0.180073
$133$	−16.0179	−1.38893
$134$	−10.1639	−0.878026
$135$	−11.8198	−1.01728
$136$	0.171333	0.0146916
$137$	−14.8925	−1.27236	−0.636178	−	0.771542i	$-0.719486\pi$
$137$	−14.8925	−1.27236	−0.636178	+	0.771542i	$0.719486\pi$
$138$	2.81739	0.239832
$139$	−5.37395	−0.455813	−0.227906	−	0.973683i	$-0.573188\pi$
$139$	−5.37395	−0.455813	−0.227906	+	0.973683i	$0.573188\pi$
$140$	4.96267	0.419422
$141$	14.8851	1.25356
$142$	13.3869	1.12340
$143$	−9.44440	−0.789780
$144$	−0.695903	−0.0579919
$145$	−2.18240	−0.181239
$146$	1.03805	0.0859093
$147$	−2.20367	−0.181756
$148$	0.829338	0.0681711
$149$	−3.70803	−0.303773	−0.151887	−	0.988398i	$-0.548535\pi$
$149$	−3.70803	−0.303773	−0.151887	+	0.988398i	$0.548535\pi$
$150$	0.851694	0.0695406
$151$	12.7944	1.04120	0.520598	−	0.853802i	$-0.325709\pi$
$151$	12.7944	1.04120	0.520598	+	0.853802i	$0.325709\pi$
$152$	6.80032	0.551579
$153$	0.119231	0.00963925
$154$	3.21043	0.258704
$155$	15.9662	1.28243
$156$	10.5181	0.842125
$157$	6.80948	0.543456	0.271728	−	0.962374i	$-0.412405\pi$
$157$	6.80948	0.543456	0.271728	+	0.962374i	$0.412405\pi$
$158$	12.7446	1.01391
$159$	6.96208	0.552129
$160$	−2.10687	−0.166563
$161$	−4.37194	−0.344557
$162$	6.42801	0.505032
$163$	−18.2483	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$163$	−18.2483	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$164$	−4.45177	−0.347624
$165$	−4.35888	−0.339338
$166$	7.16289	0.555948
$167$	−13.8195	−1.06938	−0.534691	−	0.845048i	$-0.679572\pi$
$167$	−13.8195	−1.06938	−0.534691	+	0.845048i	$0.679572\pi$
$168$	−3.57543	−0.275850
$169$	35.0150	2.69346
$170$	0.360976	0.0276856
$171$	4.73236	0.361893
$172$	2.68998	0.205109
$173$	7.58319	0.576539	0.288270	−	0.957549i	$-0.406920\pi$
$173$	7.58319	0.576539	0.288270	+	0.957549i	$0.406920\pi$
$174$	1.57234	0.119199
$175$	−1.32163	−0.0999061
$176$	−1.36297	−0.102738
$177$	−12.9775	−0.975452
$178$	−5.27074	−0.395059
$179$	−6.37060	−0.476161	−0.238080	−	0.971245i	$-0.576518\pi$
$179$	−6.37060	−0.476161	−0.238080	+	0.971245i	$0.576518\pi$
$180$	−1.46618	−0.109283
$181$	24.9674	1.85581	0.927905	−	0.372817i	$-0.121608\pi$
$181$	24.9674	1.85581	0.927905	+	0.372817i	$0.121608\pi$
$182$	−16.3217	−1.20985
$183$	7.36311	0.544297
$184$	1.85608	0.136832
$185$	1.74731	0.128465
$186$	−11.5030	−0.843444
$187$	0.233521	0.0170767
$188$	9.80625	0.715194
$189$	−13.2144	−0.961208
$190$	14.3274	1.03942
$191$	−6.57141	−0.475490	−0.237745	−	0.971328i	$-0.576408\pi$
$191$	−6.57141	−0.475490	−0.237745	+	0.971328i	$0.576408\pi$
$192$	1.51793	0.109547
$193$	25.1278	1.80874	0.904369	−	0.426751i	$-0.140342\pi$
$193$	25.1278	1.80874	0.904369	+	0.426751i	$0.140342\pi$
$194$	−12.5216	−0.898998
$195$	22.1604	1.58694
$196$	−1.45177	−0.103698
$197$	2.06999	0.147481	0.0737404	−	0.997277i	$-0.476506\pi$
$197$	2.06999	0.147481	0.0737404	+	0.997277i	$0.476506\pi$
$198$	−0.948494	−0.0674066
$199$	10.7023	0.758665	0.379332	−	0.925260i	$-0.376154\pi$
$199$	10.7023	0.758665	0.379332	+	0.925260i	$0.376154\pi$
$200$	0.561091	0.0396751
$201$	15.4280	1.08821
$202$	−10.6788	−0.751358
$203$	−2.43991	−0.171248
$204$	−0.260070	−0.0182085
$205$	−9.37930	−0.655079
$206$	11.2884	0.786501
$207$	1.29165	0.0897760
$208$	6.92929	0.480460
$209$	9.26863	0.641124
$210$	−7.53296	−0.519824
$211$	−3.80476	−0.261930	−0.130965	−	0.991387i	$-0.541808\pi$
$211$	−3.80476	−0.261930	−0.130965	+	0.991387i	$0.541808\pi$
$212$	4.58658	0.315007
$213$	−20.3203	−1.39232
$214$	15.8258	1.08183
$215$	5.66744	0.386516
$216$	5.61010	0.381719
$217$	17.8501	1.21174
$218$	10.6746	0.722976
$219$	−1.57568	−0.106474
$220$	−2.87160	−0.193603
$221$	−1.18721	−0.0798606
$222$	−1.25887	−0.0844900
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−2.35547	−0.157381
$225$	0.390465	0.0260310
$226$	10.1967	0.678273
$227$	−17.7192	−1.17607	−0.588033	−	0.808837i	$-0.700097\pi$
$227$	−17.7192	−1.17607	−0.588033	+	0.808837i	$0.700097\pi$
$228$	−10.3224	−0.683616
$229$	19.7131	1.30268	0.651340	−	0.758786i	$-0.274207\pi$
$229$	19.7131	1.30268	0.651340	+	0.758786i	$0.274207\pi$
$230$	3.91052	0.257852
$231$	−4.87319	−0.320633
$232$	1.03585	0.0680069
$233$	−14.8042	−0.969856	−0.484928	−	0.874554i	$-0.661154\pi$
$233$	−14.8042	−0.969856	−0.484928	+	0.874554i	$0.661154\pi$
$234$	4.82211	0.315231
$235$	20.6605	1.34774
$236$	−8.54953	−0.556527
$237$	−19.3454	−1.25662
$238$	0.403569	0.0261595
$239$	−0.421862	−0.0272880	−0.0136440	−	0.999907i	$-0.504343\pi$
$239$	−0.421862	−0.0272880	−0.0136440	+	0.999907i	$0.504343\pi$
$240$	3.19807	0.206435
$241$	−2.44571	−0.157542	−0.0787709	−	0.996893i	$-0.525100\pi$
$241$	−2.44571	−0.157542	−0.0787709	+	0.996893i	$0.525100\pi$
$242$	9.14232	0.587690
$243$	7.07308	0.453738
$244$	4.85077	0.310539
$245$	−3.05868	−0.195412
$246$	6.75745	0.430839
$247$	−47.1214	−2.99826
$248$	−7.57814	−0.481212
$249$	−10.8727	−0.689032
$250$	11.7165	0.741017
$251$	9.45022	0.596492	0.298246	−	0.954489i	$-0.403598\pi$
$251$	9.45022	0.596492	0.298246	+	0.954489i	$0.403598\pi$
$252$	−1.63918	−0.103259
$253$	2.52978	0.159046
$254$	1.16477	0.0730843
$255$	−0.547934	−0.0343130
$256$	1.00000	0.0625000
$257$	23.9764	1.49561	0.747805	−	0.663919i	$-0.231108\pi$
$257$	23.9764	1.49561	0.747805	+	0.663919i	$0.231108\pi$
$258$	−4.08318	−0.254208
$259$	1.95348	0.121383
$260$	14.5991	0.905399
$261$	0.720852	0.0446196
$262$	−9.67725	−0.597863
$263$	22.2382	1.37126	0.685632	−	0.727948i	$-0.259526\pi$
$263$	22.2382	1.37126	0.685632	+	0.727948i	$0.259526\pi$
$264$	2.06888	0.127331
$265$	9.66333	0.593614
$266$	16.0179	0.982123
$267$	8.00059	0.489628
$268$	10.1639	0.620858
$269$	21.3087	1.29921	0.649606	−	0.760271i	$-0.274934\pi$
$269$	21.3087	1.29921	0.649606	+	0.760271i	$0.274934\pi$
$270$	11.8198	0.719328
$271$	−20.1222	−1.22234	−0.611169	−	0.791500i	$-0.709301\pi$
$271$	−20.1222	−1.22234	−0.611169	+	0.791500i	$0.709301\pi$
$272$	−0.171333	−0.0103886
$273$	24.7751	1.49946
$274$	14.8925	0.899691
$275$	0.764750	0.0461161
$276$	−2.81739	−0.169587
$277$	−18.7305	−1.12541	−0.562703	−	0.826659i	$-0.690238\pi$
$277$	−18.7305	−1.12541	−0.562703	+	0.826659i	$0.690238\pi$
$278$	5.37395	0.322308
$279$	−5.27365	−0.315725
$280$	−4.96267	−0.296576
$281$	−2.49592	−0.148894	−0.0744471	−	0.997225i	$-0.523719\pi$
$281$	−2.49592	−0.148894	−0.0744471	+	0.997225i	$0.523719\pi$
$282$	−14.8851	−0.886398
$283$	−3.05165	−0.181402	−0.0907010	−	0.995878i	$-0.528911\pi$
$283$	−3.05165	−0.181402	−0.0907010	+	0.995878i	$0.528911\pi$
$284$	−13.3869	−0.794366
$285$	−21.7479	−1.28824
$286$	9.44440	0.558459
$287$	−10.4860	−0.618969
$288$	0.695903	0.0410065
$289$	−16.9706	−0.998273
$290$	2.18240	0.128155
$291$	19.0068	1.11420
$292$	−1.03805	−0.0607470
$293$	19.5632	1.14289	0.571447	−	0.820639i	$-0.306382\pi$
$293$	19.5632	1.14289	0.571447	+	0.820639i	$0.306382\pi$
$294$	2.20367	0.128521
$295$	−18.0128	−1.04874
$296$	−0.829338	−0.0482043
$297$	7.64640	0.443689
$298$	3.70803	0.214800
$299$	−12.8613	−0.743788
$300$	−0.851694	−0.0491726
$301$	6.33616	0.365210
$302$	−12.7944	−0.736237
$303$	16.2096	0.931218
$304$	−6.80032	−0.390025
$305$	10.2200	0.585193
$306$	−0.119231	−0.00681598
$307$	11.2140	0.640014	0.320007	−	0.947415i	$-0.396315\pi$
$307$	11.2140	0.640014	0.320007	+	0.947415i	$0.396315\pi$
$308$	−3.21043	−0.182931
$309$	−17.1350	−0.974774
$310$	−15.9662	−0.906817
$311$	10.4216	0.590955	0.295477	−	0.955350i	$-0.404521\pi$
$311$	10.4216	0.590955	0.295477	+	0.955350i	$0.404521\pi$
$312$	−10.5181	−0.595472
$313$	10.4591	0.591182	0.295591	−	0.955315i	$-0.404484\pi$
$313$	10.4591	0.591182	0.295591	+	0.955315i	$0.404484\pi$
$314$	−6.80948	−0.384281
$315$	−3.45354	−0.194585
$316$	−12.7446	−0.716940
$317$	−23.3705	−1.31262	−0.656308	−	0.754493i	$-0.727883\pi$
$317$	−23.3705	−1.31262	−0.656308	+	0.754493i	$0.727883\pi$
$318$	−6.96208	−0.390414
$319$	1.41183	0.0790474
$320$	2.10687	0.117778
$321$	−24.0223	−1.34080
$322$	4.37194	0.243638
$323$	1.16512	0.0648288
$324$	−6.42801	−0.357112
$325$	−3.88796	−0.215665
$326$	18.2483	1.01068
$327$	−16.2033	−0.896043
$328$	4.45177	0.245808
$329$	23.0983	1.27345
$330$	4.35888	0.239948
$331$	5.97727	0.328541	0.164270	−	0.986415i	$-0.447473\pi$
$331$	5.97727	0.328541	0.164270	+	0.986415i	$0.447473\pi$
$332$	−7.16289	−0.393115
$333$	−0.577139	−0.0316270
$334$	13.8195	0.756167
$335$	21.4140	1.16997
$336$	3.57543	0.195055
$337$	−31.8872	−1.73701	−0.868504	−	0.495683i	$-0.834918\pi$
$337$	−31.8872	−1.73701	−0.868504	+	0.495683i	$0.834918\pi$
$338$	−35.0150	−1.90456
$339$	−15.4778	−0.840638
$340$	−0.360976	−0.0195767
$341$	−10.3288	−0.559334
$342$	−4.73236	−0.255897
$343$	−19.9079	−1.07492
$344$	−2.68998	−0.145034
$345$	−5.93588	−0.319577
$346$	−7.58319	−0.407675
$347$	−3.95241	−0.212176	−0.106088	−	0.994357i	$-0.533833\pi$
$347$	−3.95241	−0.212176	−0.106088	+	0.994357i	$0.533833\pi$
$348$	−1.57234	−0.0842865
$349$	−21.7402	−1.16372	−0.581862	−	0.813287i	$-0.697676\pi$
$349$	−21.7402	−1.16372	−0.581862	+	0.813287i	$0.697676\pi$
$350$	1.32163	0.0706443
$351$	−38.8740	−2.07494
$352$	1.36297	0.0726465
$353$	33.5093	1.78352	0.891761	−	0.452508i	$-0.149470\pi$
$353$	33.5093	1.78352	0.891761	+	0.452508i	$0.149470\pi$
$354$	12.9775	0.689749
$355$	−28.2045	−1.49694
$356$	5.27074	0.279349
$357$	−0.612587	−0.0324215
$358$	6.37060	0.336696
$359$	17.7400	0.936280	0.468140	−	0.883654i	$-0.344924\pi$
$359$	17.7400	0.936280	0.468140	+	0.883654i	$0.344924\pi$
$360$	1.46618	0.0772744
$361$	27.2444	1.43391
$362$	−24.9674	−1.31226
$363$	−13.8774	−0.728372
$364$	16.3217	0.855490
$365$	−2.18703	−0.114474
$366$	−7.36311	−0.384876
$367$	15.3095	0.799149	0.399575	−	0.916701i	$-0.369158\pi$
$367$	15.3095	0.799149	0.399575	+	0.916701i	$0.369158\pi$
$368$	−1.85608	−0.0967548
$369$	3.09800	0.161275
$370$	−1.74731	−0.0908382
$371$	10.8035	0.560892
$372$	11.5030	0.596405
$373$	−1.62350	−0.0840619	−0.0420309	−	0.999116i	$-0.513383\pi$
$373$	−1.62350	−0.0840619	−0.0420309	+	0.999116i	$0.513383\pi$
$374$	−0.233521	−0.0120751
$375$	−17.7848	−0.918402
$376$	−9.80625	−0.505719
$377$	−7.17770	−0.369671
$378$	13.2144	0.679677
$379$	−7.67269	−0.394120	−0.197060	−	0.980391i	$-0.563139\pi$
$379$	−7.67269	−0.394120	−0.197060	+	0.980391i	$0.563139\pi$
$380$	−14.3274	−0.734980
$381$	−1.76804	−0.0905793
$382$	6.57141	0.336223
$383$	27.3936	1.39975	0.699874	−	0.714266i	$-0.253239\pi$
$383$	27.3936	1.39975	0.699874	+	0.714266i	$0.253239\pi$
$384$	−1.51793	−0.0774613
$385$	−6.76397	−0.344724
$386$	−25.1278	−1.27897
$387$	−1.87196	−0.0951573
$388$	12.5216	0.635687
$389$	23.7356	1.20344	0.601722	−	0.798706i	$-0.294482\pi$
$389$	23.7356	1.20344	0.601722	+	0.798706i	$0.294482\pi$
$390$	−22.1604	−1.12213
$391$	0.318007	0.0160823
$392$	1.45177	0.0733252
$393$	14.6893	0.740979
$394$	−2.06999	−0.104285
$395$	−26.8513	−1.35103
$396$	0.948494	0.0476636
$397$	15.7036	0.788142	0.394071	−	0.919080i	$-0.371066\pi$
$397$	15.7036	0.788142	0.394071	+	0.919080i	$0.371066\pi$
$398$	−10.7023	−0.536457
$399$	−24.3140	−1.21722
$400$	−0.561091	−0.0280546
$401$	5.05626	0.252498	0.126249	−	0.991999i	$-0.459706\pi$
$401$	5.05626	0.252498	0.126249	+	0.991999i	$0.459706\pi$
$402$	−15.4280	−0.769479
$403$	52.5111	2.61576
$404$	10.6788	0.531290
$405$	−13.5430	−0.672957
$406$	2.43991	0.121091
$407$	−1.13036	−0.0560299
$408$	0.260070	0.0128754
$409$	−35.8885	−1.77457	−0.887285	−	0.461221i	$-0.847411\pi$
$409$	−35.8885	−1.77457	−0.887285	+	0.461221i	$0.847411\pi$
$410$	9.37930	0.463211
$411$	−22.6058	−1.11506
$412$	−11.2884	−0.556140
$413$	−20.1381	−0.990934
$414$	−1.29165	−0.0634812
$415$	−15.0913	−0.740803
$416$	−6.92929	−0.339736
$417$	−8.15726	−0.399463
$418$	−9.26863	−0.453343
$419$	12.3568	0.603668	0.301834	−	0.953360i	$-0.402401\pi$
$419$	12.3568	0.603668	0.301834	+	0.953360i	$0.402401\pi$
$420$	7.53296	0.367571
$421$	14.9471	0.728477	0.364239	−	0.931306i	$-0.381329\pi$
$421$	14.9471	0.728477	0.364239	+	0.931306i	$0.381329\pi$
$422$	3.80476	0.185213
$423$	−6.82420	−0.331804
$424$	−4.58658	−0.222744
$425$	0.0961332	0.00466315
$426$	20.3203	0.984522
$427$	11.4258	0.552936
$428$	−15.8258	−0.764967
$429$	−14.3359	−0.692143
$430$	−5.66744	−0.273308
$431$	35.5998	1.71478	0.857391	−	0.514665i	$-0.172084\pi$
$431$	35.5998	1.71478	0.857391	+	0.514665i	$0.172084\pi$
$432$	−5.61010	−0.269916
$433$	11.8926	0.571522	0.285761	−	0.958301i	$-0.407754\pi$
$433$	11.8926	0.571522	0.285761	+	0.958301i	$0.407754\pi$
$434$	−17.8501	−0.856831
$435$	−3.31273	−0.158833
$436$	−10.6746	−0.511221
$437$	12.6219	0.603789
$438$	1.57568	0.0752887
$439$	−10.8627	−0.518449	−0.259224	−	0.965817i	$-0.583467\pi$
$439$	−10.8627	−0.518449	−0.259224	+	0.965817i	$0.583467\pi$
$440$	2.87160	0.136898
$441$	1.01029	0.0481090
$442$	1.18721	0.0564699
$443$	−28.7169	−1.36438	−0.682191	−	0.731174i	$-0.738973\pi$
$443$	−28.7169	−1.36438	−0.682191	+	0.731174i	$0.738973\pi$
$444$	1.25887	0.0597434
$445$	11.1048	0.526417
$446$	−1.00000	−0.0473514
$447$	−5.62851	−0.266219
$448$	2.35547	0.111285
$449$	19.7104	0.930190	0.465095	−	0.885261i	$-0.346020\pi$
$449$	19.7104	0.930190	0.465095	+	0.885261i	$0.346020\pi$
$450$	−0.390465	−0.0184067
$451$	6.06762	0.285713
$452$	−10.1967	−0.479611
$453$	19.4210	0.912478
$454$	17.7192	0.831604
$455$	34.3878	1.61212
$456$	10.3224	0.483390
$457$	21.5825	1.00959	0.504794	−	0.863240i	$-0.331568\pi$
$457$	21.5825	1.00959	0.504794	+	0.863240i	$0.331568\pi$
$458$	−19.7131	−0.921134
$459$	0.961194	0.0448647
$460$	−3.91052	−0.182329
$461$	1.17709	0.0548225	0.0274113	−	0.999624i	$-0.491274\pi$
$461$	1.17709	0.0548225	0.0274113	+	0.999624i	$0.491274\pi$
$462$	4.87319	0.226722
$463$	11.6295	0.540469	0.270235	−	0.962794i	$-0.412899\pi$
$463$	11.6295	0.540469	0.270235	+	0.962794i	$0.412899\pi$
$464$	−1.03585	−0.0480881
$465$	24.2354	1.12389
$466$	14.8042	0.685792
$467$	−18.0809	−0.836682	−0.418341	−	0.908290i	$-0.637388\pi$
$467$	−18.0809	−0.836682	−0.418341	+	0.908290i	$0.637388\pi$
$468$	−4.82211	−0.222902
$469$	23.9407	1.10548
$470$	−20.6605	−0.952998
$471$	10.3363	0.476271
$472$	8.54953	0.393524
$473$	−3.66636	−0.168579
$474$	19.3454	0.888562
$475$	3.81560	0.175072
$476$	−0.403569	−0.0184975
$477$	−3.19181	−0.146143
$478$	0.421862	0.0192955
$479$	−33.4794	−1.52971	−0.764856	−	0.644201i	$-0.777190\pi$
$479$	−33.4794	−1.52971	−0.764856	+	0.644201i	$0.777190\pi$
$480$	−3.19807	−0.145971
$481$	5.74672	0.262028
$482$	2.44571	0.111399
$483$	−6.63627	−0.301961
$484$	−9.14232	−0.415560
$485$	26.3814	1.19792
$486$	−7.07308	−0.320841
$487$	−14.4847	−0.656365	−0.328182	−	0.944614i	$-0.606436\pi$
$487$	−14.4847	−0.656365	−0.328182	+	0.944614i	$0.606436\pi$
$488$	−4.85077	−0.219584
$489$	−27.6995	−1.25262
$490$	3.05868	0.138177
$491$	35.7970	1.61550	0.807748	−	0.589528i	$-0.200686\pi$
$491$	35.7970	1.61550	0.807748	+	0.589528i	$0.200686\pi$
$492$	−6.75745	−0.304649
$493$	0.177475	0.00799307
$494$	47.1214	2.12009
$495$	1.99836	0.0898194
$496$	7.57814	0.340268
$497$	−31.5324	−1.41442
$498$	10.8727	0.487219
$499$	25.0571	1.12171	0.560854	−	0.827915i	$-0.310473\pi$
$499$	25.0571	1.12171	0.560854	+	0.827915i	$0.310473\pi$
$500$	−11.7165	−0.523978
$501$	−20.9769	−0.937179
$502$	−9.45022	−0.421784
$503$	21.9912	0.980541	0.490271	−	0.871570i	$-0.336898\pi$
$503$	21.9912	0.980541	0.490271	+	0.871570i	$0.336898\pi$
$504$	1.63918	0.0730148
$505$	22.4989	1.00119
$506$	−2.52978	−0.112462
$507$	53.1501	2.36048
$508$	−1.16477	−0.0516784
$509$	5.12448	0.227139	0.113569	−	0.993530i	$-0.463772\pi$
$509$	5.12448	0.227139	0.113569	+	0.993530i	$0.463772\pi$
$510$	0.547934	0.0242629
$511$	−2.44509	−0.108164
$512$	−1.00000	−0.0441942
$513$	38.1505	1.68439
$514$	−23.9764	−1.05756
$515$	−23.7832	−1.04801
$516$	4.08318	0.179752
$517$	−13.3656	−0.587819
$518$	−1.95348	−0.0858310
$519$	11.5107	0.505265
$520$	−14.5991	−0.640214
$521$	42.7902	1.87467	0.937336	−	0.348428i	$-0.113284\pi$
$521$	42.7902	1.87467	0.937336	+	0.348428i	$0.113284\pi$
$522$	−0.720852	−0.0315508
$523$	−17.6467	−0.771635	−0.385818	−	0.922575i	$-0.626081\pi$
$523$	−17.6467	−0.771635	−0.385818	+	0.922575i	$0.626081\pi$
$524$	9.67725	0.422753
$525$	−2.00614	−0.0875551
$526$	−22.2382	−0.969630
$527$	−1.29838	−0.0565584
$528$	−2.06888	−0.0900367
$529$	−19.5550	−0.850216
$530$	−9.66333	−0.419748
$531$	5.94964	0.258193
$532$	−16.0179	−0.694466
$533$	−30.8476	−1.33616
$534$	−8.00059	−0.346219
$535$	−33.3429	−1.44154
$536$	−10.1639	−0.439013
$537$	−9.67009	−0.417295
$538$	−21.3087	−0.918682
$539$	1.97871	0.0852291
$540$	−11.8198	−0.508642
$541$	−23.0560	−0.991254	−0.495627	−	0.868535i	$-0.665062\pi$
$541$	−23.0560	−0.991254	−0.495627	+	0.868535i	$0.665062\pi$
$542$	20.1222	0.864324
$543$	37.8986	1.62638
$544$	0.171333	0.00734582
$545$	−22.4901	−0.963368
$546$	−24.7751	−1.06028
$547$	−39.9822	−1.70952	−0.854758	−	0.519027i	$-0.826294\pi$
$547$	−39.9822	−1.70952	−0.854758	+	0.519027i	$0.826294\pi$
$548$	−14.8925	−0.636178
$549$	−3.37567	−0.144070
$550$	−0.764750	−0.0326090
$551$	7.04412	0.300089
$552$	2.81739	0.119916
$553$	−30.0195	−1.27656
$554$	18.7305	0.795782
$555$	2.65228	0.112583
$556$	−5.37395	−0.227906
$557$	−8.62860	−0.365606	−0.182803	−	0.983150i	$-0.558517\pi$
$557$	−8.62860	−0.365606	−0.182803	+	0.983150i	$0.558517\pi$
$558$	5.27365	0.223251
$559$	18.6396	0.788372
$560$	4.96267	0.209711
$561$	0.354467	0.0149656
$562$	2.49592	0.105284
$563$	26.8712	1.13249	0.566243	−	0.824239i	$-0.308397\pi$
$563$	26.8712	1.13249	0.566243	+	0.824239i	$0.308397\pi$
$564$	14.8851	0.626778
$565$	−21.4831	−0.903800
$566$	3.05165	0.128271
$567$	−15.1410	−0.635861
$568$	13.3869	0.561702
$569$	17.3129	0.725795	0.362897	−	0.931829i	$-0.381788\pi$
$569$	17.3129	0.725795	0.362897	+	0.931829i	$0.381788\pi$
$570$	21.7479	0.910920
$571$	7.10163	0.297194	0.148597	−	0.988898i	$-0.452524\pi$
$571$	7.10163	0.297194	0.148597	+	0.988898i	$0.452524\pi$
$572$	−9.44440	−0.394890
$573$	−9.97491	−0.416708
$574$	10.4860	0.437677
$575$	1.04143	0.0434306
$576$	−0.695903	−0.0289960
$577$	8.75333	0.364406	0.182203	−	0.983261i	$-0.441677\pi$
$577$	8.75333	0.364406	0.182203	+	0.983261i	$0.441677\pi$
$578$	16.9706	0.705886
$579$	38.1421	1.58513
$580$	−2.18240	−0.0906194
$581$	−16.8720	−0.699967
$582$	−19.0068	−0.787859
$583$	−6.25136	−0.258905
$584$	1.03805	0.0429547
$585$	−10.1596	−0.420047
$586$	−19.5632	−0.808148
$587$	−41.6533	−1.71922	−0.859608	−	0.510954i	$-0.829292\pi$
$587$	−41.6533	−1.71922	−0.859608	+	0.510954i	$0.829292\pi$
$588$	−2.20367	−0.0908779
$589$	−51.5338	−2.12341
$590$	18.0128	0.741574
$591$	3.14209	0.129248
$592$	0.829338	0.0340856
$593$	27.2431	1.11874	0.559369	−	0.828918i	$-0.311043\pi$
$593$	27.2431	1.11874	0.559369	+	0.828918i	$0.311043\pi$
$594$	−7.64640	−0.313735
$595$	−0.850267	−0.0348576
$596$	−3.70803	−0.151887
$597$	16.2453	0.664874
$598$	12.8613	0.525938
$599$	15.4627	0.631789	0.315895	−	0.948794i	$-0.397695\pi$
$599$	15.4627	0.631789	0.315895	+	0.948794i	$0.397695\pi$
$600$	0.851694	0.0347703
$601$	−22.1109	−0.901922	−0.450961	−	0.892544i	$-0.648919\pi$
$601$	−22.1109	−0.901922	−0.450961	+	0.892544i	$0.648919\pi$
$602$	−6.33616	−0.258243
$603$	−7.07308	−0.288038
$604$	12.7944	0.520598
$605$	−19.2617	−0.783099
$606$	−16.2096	−0.658471
$607$	19.9185	0.808466	0.404233	−	0.914656i	$-0.367538\pi$
$607$	19.9185	0.808466	0.404233	+	0.914656i	$0.367538\pi$
$608$	6.80032	0.275789
$609$	−3.70361	−0.150078
$610$	−10.2200	−0.413794
$611$	67.9503	2.74897
$612$	0.119231	0.00481962
$613$	25.1647	1.01639	0.508195	−	0.861242i	$-0.330313\pi$
$613$	25.1647	1.01639	0.508195	+	0.861242i	$0.330313\pi$
$614$	−11.2140	−0.452558
$615$	−14.2371	−0.574094
$616$	3.21043	0.129352
$617$	−3.01547	−0.121398	−0.0606991	−	0.998156i	$-0.519333\pi$
$617$	−3.01547	−0.121398	−0.0606991	+	0.998156i	$0.519333\pi$
$618$	17.1350	0.689269
$619$	−6.06667	−0.243840	−0.121920	−	0.992540i	$-0.538905\pi$
$619$	−6.06667	−0.243840	−0.121920	+	0.992540i	$0.538905\pi$
$620$	15.9662	0.641217
$621$	10.4128	0.417851
$622$	−10.4216	−0.417868
$623$	12.4151	0.497399
$624$	10.5181	0.421062
$625$	−21.8797	−0.875189
$626$	−10.4591	−0.418028
$627$	14.0691	0.561865
$628$	6.80948	0.271728
$629$	−0.142093	−0.00566560
$630$	3.45354	0.137592
$631$	4.12131	0.164067	0.0820334	−	0.996630i	$-0.473859\pi$
$631$	4.12131	0.164067	0.0820334	+	0.996630i	$0.473859\pi$
$632$	12.7446	0.506953
$633$	−5.77534	−0.229549
$634$	23.3705	0.928160
$635$	−2.45403	−0.0973850
$636$	6.96208	0.276064
$637$	−10.0597	−0.398580
$638$	−1.41183	−0.0558950
$639$	9.31598	0.368535
$640$	−2.10687	−0.0832814
$641$	8.87808	0.350663	0.175332	−	0.984509i	$-0.443900\pi$
$641$	8.87808	0.350663	0.175332	+	0.984509i	$0.443900\pi$
$642$	24.0223	0.948086
$643$	−6.90212	−0.272193	−0.136096	−	0.990696i	$-0.543456\pi$
$643$	−6.90212	−0.272193	−0.136096	+	0.990696i	$0.543456\pi$
$644$	−4.37194	−0.172278
$645$	8.60275	0.338733
$646$	−1.16512	−0.0458409
$647$	28.7014	1.12837	0.564184	−	0.825649i	$-0.309191\pi$
$647$	28.7014	1.12837	0.564184	+	0.825649i	$0.309191\pi$
$648$	6.42801	0.252516
$649$	11.6527	0.457410
$650$	3.88796	0.152498
$651$	27.0951	1.06194
$652$	−18.2483	−0.714658
$653$	−33.6638	−1.31736	−0.658682	−	0.752421i	$-0.728886\pi$
$653$	−33.6638	−1.31736	−0.658682	+	0.752421i	$0.728886\pi$
$654$	16.2033	0.633598
$655$	20.3887	0.796653
$656$	−4.45177	−0.173812
$657$	0.722379	0.0281827
$658$	−23.0983	−0.900466
$659$	5.39295	0.210079	0.105040	−	0.994468i	$-0.466503\pi$
$659$	5.39295	0.210079	0.105040	+	0.994468i	$0.466503\pi$
$660$	−4.35888	−0.169669
$661$	−40.3142	−1.56804	−0.784021	−	0.620734i	$-0.786835\pi$
$661$	−40.3142	−1.56804	−0.784021	+	0.620734i	$0.786835\pi$
$662$	−5.97727	−0.232313
$663$	−1.80210	−0.0699878
$664$	7.16289	0.277974
$665$	−33.7478	−1.30868
$666$	0.577139	0.0223637
$667$	1.92262	0.0744441
$668$	−13.8195	−0.534691
$669$	1.51793	0.0586864
$670$	−21.4140	−0.827295
$671$	−6.61146	−0.255232
$672$	−3.57543	−0.137925
$673$	12.0463	0.464353	0.232176	−	0.972674i	$-0.425415\pi$
$673$	12.0463	0.464353	0.232176	+	0.972674i	$0.425415\pi$
$674$	31.8872	1.22825
$675$	3.14778	0.121158
$676$	35.0150	1.34673
$677$	30.9776	1.19057	0.595283	−	0.803516i	$-0.297040\pi$
$677$	30.9776	1.19057	0.595283	+	0.803516i	$0.297040\pi$
$678$	15.4778	0.594421
$679$	29.4942	1.13188
$680$	0.360976	0.0138428
$681$	−26.8964	−1.03067
$682$	10.3288	0.395509
$683$	−27.6910	−1.05957	−0.529783	−	0.848133i	$-0.677727\pi$
$683$	−27.6910	−1.05957	−0.529783	+	0.848133i	$0.677727\pi$
$684$	4.73236	0.180946
$685$	−31.3767	−1.19884
$686$	19.9079	0.760086
$687$	29.9231	1.14164
$688$	2.68998	0.102554
$689$	31.7817	1.21079
$690$	5.93588	0.225975
$691$	27.7433	1.05540	0.527702	−	0.849430i	$-0.323054\pi$
$691$	27.7433	1.05540	0.527702	+	0.849430i	$0.323054\pi$
$692$	7.58319	0.288270
$693$	2.23415	0.0848683
$694$	3.95241	0.150031
$695$	−11.3222	−0.429477
$696$	1.57234	0.0595995
$697$	0.762732	0.0288906
$698$	21.7402	0.822878
$699$	−22.4717	−0.849957
$700$	−1.32163	−0.0499530
$701$	−34.1074	−1.28822	−0.644109	−	0.764934i	$-0.722772\pi$
$701$	−34.1074	−1.28822	−0.644109	+	0.764934i	$0.722772\pi$
$702$	38.8740	1.46721
$703$	−5.63976	−0.212708
$704$	−1.36297	−0.0513688
$705$	31.3611	1.18113
$706$	−33.5093	−1.26114
$707$	25.1536	0.945998
$708$	−12.9775	−0.487726
$709$	−5.02486	−0.188713	−0.0943564	−	0.995538i	$-0.530079\pi$
$709$	−5.02486	−0.188713	−0.0943564	+	0.995538i	$0.530079\pi$
$710$	28.2045	1.05850
$711$	8.86901	0.332614
$712$	−5.27074	−0.197529
$713$	−14.0656	−0.526762
$714$	0.612587	0.0229255
$715$	−19.8981	−0.744148
$716$	−6.37060	−0.238080
$717$	−0.640354	−0.0239145
$718$	−17.7400	−0.662050
$719$	29.7339	1.10889	0.554443	−	0.832222i	$-0.312931\pi$
$719$	29.7339	1.10889	0.554443	+	0.832222i	$0.312931\pi$
$720$	−1.46618	−0.0546413
$721$	−26.5895	−0.990244
$722$	−27.2444	−1.01393
$723$	−3.71240	−0.138066
$724$	24.9674	0.927905
$725$	0.581207	0.0215855
$726$	13.8774	0.515037
$727$	45.2439	1.67801	0.839003	−	0.544127i	$-0.183139\pi$
$727$	45.2439	1.67801	0.839003	+	0.544127i	$0.183139\pi$
$728$	−16.3217	−0.604923
$729$	30.0204	1.11187
$730$	2.18703	0.0809456
$731$	−0.460881	−0.0170463
$732$	7.36311	0.272148
$733$	−12.3998	−0.457997	−0.228999	−	0.973427i	$-0.573545\pi$
$733$	−12.3998	−0.457997	−0.228999	+	0.973427i	$0.573545\pi$
$734$	−15.3095	−0.565084
$735$	−4.64285	−0.171254
$736$	1.85608	0.0684160
$737$	−13.8531	−0.510284
$738$	−3.09800	−0.114039
$739$	14.9324	0.549298	0.274649	−	0.961545i	$-0.411438\pi$
$739$	14.9324	0.549298	0.274649	+	0.961545i	$0.411438\pi$
$740$	1.74731	0.0642323
$741$	−71.5267	−2.62760
$742$	−10.8035	−0.396610
$743$	−41.9629	−1.53947	−0.769735	−	0.638364i	$-0.779611\pi$
$743$	−41.9629	−1.53947	−0.769735	+	0.638364i	$0.779611\pi$
$744$	−11.5030	−0.421722
$745$	−7.81233	−0.286222
$746$	1.62350	0.0594407
$747$	4.98468	0.182380
$748$	0.233521	0.00853837
$749$	−37.2771	−1.36208
$750$	17.7848	0.649408
$751$	25.2863	0.922709	0.461355	−	0.887216i	$-0.347364\pi$
$751$	25.2863	0.922709	0.461355	+	0.887216i	$0.347364\pi$
$752$	9.80625	0.357597
$753$	14.3447	0.522751
$754$	7.17770	0.261397
$755$	26.9562	0.981037
$756$	−13.2144	−0.480604
$757$	31.2666	1.13641	0.568203	−	0.822889i	$-0.307639\pi$
$757$	31.2666	1.13641	0.568203	+	0.822889i	$0.307639\pi$
$758$	7.67269	0.278685
$759$	3.84001	0.139384
$760$	14.3274	0.519710
$761$	−50.2106	−1.82013	−0.910067	−	0.414461i	$-0.863970\pi$
$761$	−50.2106	−1.82013	−0.910067	+	0.414461i	$0.863970\pi$
$762$	1.76804	0.0640492
$763$	−25.1437	−0.910264
$764$	−6.57141	−0.237745
$765$	0.251204	0.00908231
$766$	−27.3936	−0.989771
$767$	−59.2421	−2.13911
$768$	1.51793	0.0547734
$769$	−22.6772	−0.817762	−0.408881	−	0.912588i	$-0.634081\pi$
$769$	−22.6772	−0.817762	−0.408881	+	0.912588i	$0.634081\pi$
$770$	6.76397	0.243756
$771$	36.3944	1.31071
$772$	25.1278	0.904369
$773$	12.4465	0.447669	0.223835	−	0.974627i	$-0.428142\pi$
$773$	12.4465	0.447669	0.223835	+	0.974627i	$0.428142\pi$
$774$	1.87196	0.0672863
$775$	−4.25203	−0.152737
$776$	−12.5216	−0.449499
$777$	2.96524	0.106377
$778$	−23.7356	−0.850963
$779$	30.2734	1.08466
$780$	22.1604	0.793468
$781$	18.2459	0.652891
$782$	−0.318007	−0.0113719
$783$	5.81123	0.207676
$784$	−1.45177	−0.0518488
$785$	14.3467	0.512056
$786$	−14.6893	−0.523951
$787$	−24.3081	−0.866489	−0.433244	−	0.901276i	$-0.642631\pi$
$787$	−24.3081	−0.866489	−0.433244	+	0.901276i	$0.642631\pi$
$788$	2.06999	0.0737404
$789$	33.7559	1.20174
$790$	26.8513	0.955325
$791$	−24.0180	−0.853980
$792$	−0.948494	−0.0337033
$793$	33.6124	1.19361
$794$	−15.7036	−0.557300
$795$	14.6682	0.520228
$796$	10.7023	0.379332
$797$	−1.06917	−0.0378721	−0.0189361	−	0.999821i	$-0.506028\pi$
$797$	−1.06917	−0.0378721	−0.0189361	+	0.999821i	$0.506028\pi$
$798$	24.3140	0.860708
$799$	−1.68013	−0.0594387
$800$	0.561091	0.0198376
$801$	−3.66792	−0.129600
$802$	−5.05626	−0.178543
$803$	1.41482	0.0499281
$804$	15.4280	0.544104
$805$	−9.21111	−0.324649
$806$	−52.5111	−1.84962
$807$	32.3450	1.13860
$808$	−10.6788	−0.375679
$809$	−2.16235	−0.0760243	−0.0380121	−	0.999277i	$-0.512103\pi$
$809$	−2.16235	−0.0760243	−0.0380121	+	0.999277i	$0.512103\pi$
$810$	13.5430	0.475852
$811$	−22.5152	−0.790615	−0.395307	−	0.918549i	$-0.629362\pi$
$811$	−22.5152	−0.790615	−0.395307	+	0.918549i	$0.629362\pi$
$812$	−2.43991	−0.0856242
$813$	−30.5440	−1.07123
$814$	1.13036	0.0396191
$815$	−38.4468	−1.34673
$816$	−0.260070	−0.00910427
$817$	−18.2927	−0.639981
$818$	35.8885	1.25481
$819$	−11.3583	−0.396892
$820$	−9.37930	−0.327539
$821$	−12.4958	−0.436108	−0.218054	−	0.975937i	$-0.569971\pi$
$821$	−12.4958	−0.436108	−0.218054	+	0.975937i	$0.569971\pi$
$822$	22.6058	0.788467
$823$	32.5601	1.13497	0.567487	−	0.823382i	$-0.307916\pi$
$823$	32.5601	1.13497	0.567487	+	0.823382i	$0.307916\pi$
$824$	11.2884	0.393250
$825$	1.16083	0.0404150
$826$	20.1381	0.700696
$827$	−8.19538	−0.284981	−0.142491	−	0.989796i	$-0.545511\pi$
$827$	−8.19538	−0.284981	−0.142491	+	0.989796i	$0.545511\pi$
$828$	1.29165	0.0448880
$829$	36.9953	1.28490	0.642449	−	0.766328i	$-0.277918\pi$
$829$	36.9953	1.28490	0.642449	+	0.766328i	$0.277918\pi$
$830$	15.0913	0.523827
$831$	−28.4315	−0.986276
$832$	6.92929	0.240230
$833$	0.248735	0.00861815
$834$	8.15726	0.282463
$835$	−29.1158	−1.00760
$836$	9.26863	0.320562
$837$	−42.5141	−1.46950
$838$	−12.3568	−0.426858
$839$	22.6063	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$839$	22.6063	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$840$	−7.53296	−0.259912
$841$	−27.9270	−0.963000
$842$	−14.9471	−0.515111
$843$	−3.78862	−0.130487
$844$	−3.80476	−0.130965
$845$	73.7721	2.53784
$846$	6.82420	0.234621
$847$	−21.5344	−0.739932
$848$	4.58658	0.157504
$849$	−4.63218	−0.158976
$850$	−0.0961332	−0.00329734
$851$	−1.53932	−0.0527671
$852$	−20.3203	−0.696162
$853$	21.9112	0.750224	0.375112	−	0.926979i	$-0.377604\pi$
$853$	21.9112	0.750224	0.375112	+	0.926979i	$0.377604\pi$
$854$	−11.4258	−0.390985
$855$	9.97049	0.340983
$856$	15.8258	0.540914
$857$	38.6483	1.32020	0.660100	−	0.751178i	$-0.270514\pi$
$857$	38.6483	1.32020	0.660100	+	0.751178i	$0.270514\pi$
$858$	14.3359	0.489419
$859$	0.629990	0.0214950	0.0107475	−	0.999942i	$-0.496579\pi$
$859$	0.629990	0.0214950	0.0107475	+	0.999942i	$0.496579\pi$
$860$	5.66744	0.193258
$861$	−15.9170	−0.542448
$862$	−35.5998	−1.21253
$863$	−21.4888	−0.731487	−0.365744	−	0.930716i	$-0.619185\pi$
$863$	−21.4888	−0.731487	−0.365744	+	0.930716i	$0.619185\pi$
$864$	5.61010	0.190860
$865$	15.9768	0.543228
$866$	−11.8926	−0.404127
$867$	−25.7602	−0.874861
$868$	17.8501	0.605871
$869$	17.3705	0.589254
$870$	3.31273	0.112312
$871$	70.4284	2.38638
$872$	10.6746	0.361488
$873$	−8.71381	−0.294918
$874$	−12.6219	−0.426943
$875$	−27.5979	−0.932978
$876$	−1.57568	−0.0532372
$877$	−50.2579	−1.69709	−0.848545	−	0.529123i	$-0.822521\pi$
$877$	−50.2579	−1.69709	−0.848545	+	0.529123i	$0.822521\pi$
$878$	10.8627	0.366599
$879$	29.6955	1.00160
$880$	−2.87160	−0.0968016
$881$	−11.8441	−0.399038	−0.199519	−	0.979894i	$-0.563938\pi$
$881$	−11.8441	−0.399038	−0.199519	+	0.979894i	$0.563938\pi$
$882$	−1.01029	−0.0340182
$883$	−8.70618	−0.292986	−0.146493	−	0.989212i	$-0.546799\pi$
$883$	−8.70618	−0.292986	−0.146493	+	0.989212i	$0.546799\pi$
$884$	−1.18721	−0.0399303
$885$	−27.3420	−0.919092
$886$	28.7169	0.964764
$887$	24.4541	0.821087	0.410544	−	0.911841i	$-0.365339\pi$
$887$	24.4541	0.821087	0.410544	+	0.911841i	$0.365339\pi$
$888$	−1.25887	−0.0422450
$889$	−2.74358	−0.0920169
$890$	−11.1048	−0.372233
$891$	8.76118	0.293510
$892$	1.00000	0.0334825
$893$	−66.6856	−2.23155
$894$	5.62851	0.188245
$895$	−13.4220	−0.448649
$896$	−2.35547	−0.0786907
$897$	−19.5225	−0.651837
$898$	−19.7104	−0.657744
$899$	−7.84982	−0.261806
$900$	0.390465	0.0130155
$901$	−0.785830	−0.0261798
$902$	−6.06762	−0.202030
$903$	9.61781	0.320061
$904$	10.1967	0.339136
$905$	52.6030	1.74858
$906$	−19.4210	−0.645219
$907$	15.5342	0.515806	0.257903	−	0.966171i	$-0.416969\pi$
$907$	15.5342	0.515806	0.257903	+	0.966171i	$0.416969\pi$
$908$	−17.7192	−0.588033
$909$	−7.43141	−0.246484
$910$	−34.3878	−1.13994
$911$	−11.4563	−0.379564	−0.189782	−	0.981826i	$-0.560778\pi$
$911$	−11.4563	−0.379564	−0.189782	+	0.981826i	$0.560778\pi$
$912$	−10.3224	−0.341808
$913$	9.76280	0.323102
$914$	−21.5825	−0.713887
$915$	15.5131	0.512848
$916$	19.7131	0.651340
$917$	22.7945	0.752740
$918$	−0.961194	−0.0317241
$919$	14.3652	0.473864	0.236932	−	0.971526i	$-0.423858\pi$
$919$	14.3652	0.473864	0.236932	+	0.971526i	$0.423858\pi$
$920$	3.91052	0.128926
$921$	17.0219	0.560892
$922$	−1.17709	−0.0387654
$923$	−92.7616	−3.05329
$924$	−4.87319	−0.160316
$925$	−0.465334	−0.0153001
$926$	−11.6295	−0.382170
$927$	7.85564	0.258013
$928$	1.03585	0.0340035
$929$	47.5784	1.56100	0.780498	−	0.625158i	$-0.214966\pi$
$929$	47.5784	1.56100	0.780498	+	0.625158i	$0.214966\pi$
$930$	−24.2354	−0.794712
$931$	9.87247	0.323557
$932$	−14.8042	−0.484928
$933$	15.8192	0.517898
$934$	18.0809	0.591624
$935$	0.491999	0.0160901
$936$	4.82211	0.157616
$937$	27.2472	0.890128	0.445064	−	0.895499i	$-0.353181\pi$
$937$	27.2472	0.890128	0.445064	+	0.895499i	$0.353181\pi$
$938$	−23.9407	−0.781692
$939$	15.8761	0.518096
$940$	20.6605	0.673871
$941$	1.00531	0.0327723	0.0163862	−	0.999866i	$-0.494784\pi$
$941$	1.00531	0.0327723	0.0163862	+	0.999866i	$0.494784\pi$
$942$	−10.3363	−0.336774
$943$	8.26283	0.269075
$944$	−8.54953	−0.278263
$945$	−27.8411	−0.905671
$946$	3.66636	0.119203
$947$	1.79575	0.0583539	0.0291770	−	0.999574i	$-0.490711\pi$
$947$	1.79575	0.0583539	0.0291770	+	0.999574i	$0.490711\pi$
$948$	−19.3454	−0.628308
$949$	−7.19292	−0.233492
$950$	−3.81560	−0.123794
$951$	−35.4746	−1.15034
$952$	0.403569	0.0130797
$953$	48.8428	1.58217	0.791086	−	0.611705i	$-0.209516\pi$
$953$	48.8428	1.58217	0.791086	+	0.611705i	$0.209516\pi$
$954$	3.19181	0.103339
$955$	−13.8451	−0.448017
$956$	−0.421862	−0.0136440
$957$	2.14306	0.0692751
$958$	33.4794	1.08167
$959$	−35.0789	−1.13276
$960$	3.19807	0.103217
$961$	26.4282	0.852522
$962$	−5.74672	−0.185282
$963$	11.0132	0.354895
$964$	−2.44571	−0.0787709
$965$	52.9410	1.70423
$966$	6.63627	0.213519
$967$	−8.74791	−0.281314	−0.140657	−	0.990058i	$-0.544921\pi$
$967$	−8.74791	−0.281314	−0.140657	+	0.990058i	$0.544921\pi$
$968$	9.14232	0.293845
$969$	1.76856	0.0568143
$970$	−26.3814	−0.847055
$971$	−47.0999	−1.51151	−0.755753	−	0.654857i	$-0.772729\pi$
$971$	−47.0999	−1.51151	−0.755753	+	0.654857i	$0.772729\pi$
$972$	7.07308	0.226869
$973$	−12.6582	−0.405803
$974$	14.4847	0.464120
$975$	−5.90163	−0.189004
$976$	4.85077	0.155270
$977$	25.4309	0.813607	0.406803	−	0.913516i	$-0.366643\pi$
$977$	25.4309	0.813607	0.406803	+	0.913516i	$0.366643\pi$
$978$	27.6995	0.885733
$979$	−7.18385	−0.229597
$980$	−3.05868	−0.0977061
$981$	7.42850	0.237174
$982$	−35.7970	−1.14233
$983$	−2.21783	−0.0707378	−0.0353689	−	0.999374i	$-0.511261\pi$
$983$	−2.21783	−0.0707378	−0.0353689	+	0.999374i	$0.511261\pi$
$984$	6.75745	0.215420
$985$	4.36121	0.138960
$986$	−0.177475	−0.00565195
$987$	35.0615	1.11602
$988$	−47.1214	−1.49913
$989$	−4.99281	−0.158762
$990$	−1.99836	−0.0635119
$991$	33.3991	1.06096	0.530479	−	0.847698i	$-0.322012\pi$
$991$	33.3991	1.06096	0.530479	+	0.847698i	$0.322012\pi$
$992$	−7.57814	−0.240606
$993$	9.07306	0.287925
$994$	31.5324	1.00015
$995$	22.5483	0.714830
$996$	−10.8727	−0.344516
$997$	−60.1287	−1.90430	−0.952148	−	0.305639i	$-0.901130\pi$
$997$	−60.1287	−1.90430	−0.952148	+	0.305639i	$0.901130\pi$
$998$	−25.0571	−0.793167
$999$	−4.65267	−0.147204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	446.2.a.f.1.5	✓	8
3.2	odd	2		4014.2.a.z.1.4		8
4.3	odd	2		3568.2.a.n.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.f.1.5	✓	8	1.1	even	1	trivial
3568.2.a.n.1.4		8	4.3	odd	2
4014.2.a.z.1.4		8	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:	\( N \)	\(=\)	\( 446 = 2 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	446.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.51793 q^{3} +1.00000 q^{4} +2.10687 q^{5} -1.51793 q^{6} +2.35547 q^{7} -1.00000 q^{8} -0.695903 q^{9} -2.10687 q^{10} -1.36297 q^{11} +1.51793 q^{12} +6.92929 q^{13} -2.35547 q^{14} +3.19807 q^{15} +1.00000 q^{16} -0.171333 q^{17} +0.695903 q^{18} -6.80032 q^{19} +2.10687 q^{20} +3.57543 q^{21} +1.36297 q^{22} -1.85608 q^{23} -1.51793 q^{24} -0.561091 q^{25} -6.92929 q^{26} -5.61010 q^{27} +2.35547 q^{28} -1.03585 q^{29} -3.19807 q^{30} +7.57814 q^{31} -1.00000 q^{32} -2.06888 q^{33} +0.171333 q^{34} +4.96267 q^{35} -0.695903 q^{36} +0.829338 q^{37} +6.80032 q^{38} +10.5181 q^{39} -2.10687 q^{40} -4.45177 q^{41} -3.57543 q^{42} +2.68998 q^{43} -1.36297 q^{44} -1.46618 q^{45} +1.85608 q^{46} +9.80625 q^{47} +1.51793 q^{48} -1.45177 q^{49} +0.561091 q^{50} -0.260070 q^{51} +6.92929 q^{52} +4.58658 q^{53} +5.61010 q^{54} -2.87160 q^{55} -2.35547 q^{56} -10.3224 q^{57} +1.03585 q^{58} -8.54953 q^{59} +3.19807 q^{60} +4.85077 q^{61} -7.57814 q^{62} -1.63918 q^{63} +1.00000 q^{64} +14.5991 q^{65} +2.06888 q^{66} +10.1639 q^{67} -0.171333 q^{68} -2.81739 q^{69} -4.96267 q^{70} -13.3869 q^{71} +0.695903 q^{72} -1.03805 q^{73} -0.829338 q^{74} -0.851694 q^{75} -6.80032 q^{76} -3.21043 q^{77} -10.5181 q^{78} -12.7446 q^{79} +2.10687 q^{80} -6.42801 q^{81} +4.45177 q^{82} -7.16289 q^{83} +3.57543 q^{84} -0.360976 q^{85} -2.68998 q^{86} -1.57234 q^{87} +1.36297 q^{88} +5.27074 q^{89} +1.46618 q^{90} +16.3217 q^{91} -1.85608 q^{92} +11.5030 q^{93} -9.80625 q^{94} -14.3274 q^{95} -1.51793 q^{96} +12.5216 q^{97} +1.45177 q^{98} +0.948494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 8 q^{8} + 16 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 10 q^{13} - 4 q^{14} - 4 q^{15} + 8 q^{16} + 8 q^{17} - 16 q^{18} + 16 q^{19} + 4 q^{20}+ \cdots - 40 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^5 - 4 * q^6 + 4 * q^7 - 8 * q^8 + 16 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 10 * q^13 - 4 * q^14 - 4 * q^15 + 8 * q^16 + 8 * q^17 - 16 * q^18 + 16 * q^19 + 4 * q^20 - 2 * q^22 - 4 * q^23 - 4 * q^24 + 24 * q^25 - 10 * q^26 + 22 * q^27 + 4 * q^28 + 8 * q^29 + 4 * q^30 + 12 * q^31 - 8 * q^32 - 8 * q^34 - 24 * q^35 + 16 * q^36 + 16 * q^37 - 16 * q^38 - 4 * q^40 + 16 * q^41 + 12 * q^43 + 2 * q^44 + 34 * q^45 + 4 * q^46 - 4 * q^47 + 4 * q^48 + 40 * q^49 - 24 * q^50 + 26 * q^51 + 10 * q^52 - 8 * q^53 - 22 * q^54 + 8 * q^55 - 4 * q^56 + 16 * q^57 - 8 * q^58 - 4 * q^60 + 26 * q^61 - 12 * q^62 - 12 * q^63 + 8 * q^64 - 24 * q^65 - 10 * q^67 + 8 * q^68 - 28 * q^69 + 24 * q^70 + 8 * q^71 - 16 * q^72 + 8 * q^73 - 16 * q^74 + 6 * q^75 + 16 * q^76 - 56 * q^77 + 16 * q^79 + 4 * q^80 + 24 * q^81 - 16 * q^82 - 44 * q^83 - 18 * q^85 - 12 * q^86 - 72 * q^87 - 2 * q^88 - 34 * q^90 + 4 * q^91 - 4 * q^92 - 62 * q^93 + 4 * q^94 - 48 * q^95 - 4 * q^96 + 12 * q^97 - 40 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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