LMFDB - Embedded newform 71.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,2,Mod(1,71)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("71.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	71.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:	$0.566937854351$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
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			1.3
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	71.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83424 q^{2} -0.364448 q^{3} +1.36445 q^{4} -2.83424 q^{5} -0.668486 q^{6} +4.39738 q^{7} -1.16576 q^{8} -2.86718 q^{9} +O(q^{10})$	\(q+1.83424 q^{2} -0.364448 q^{3} +1.36445 q^{4} -2.83424 q^{5} -0.668486 q^{6} +4.39738 q^{7} -1.16576 q^{8} -2.86718 q^{9} -5.19869 q^{10} -4.39738 q^{11} -0.497270 q^{12} +4.00000 q^{13} +8.06587 q^{14} +1.03293 q^{15} -4.86718 q^{16} +4.39738 q^{17} -5.25910 q^{18} +1.80131 q^{19} -3.86718 q^{20} -1.60262 q^{21} -8.06587 q^{22} +2.72890 q^{23} +0.424858 q^{24} +3.03293 q^{25} +7.33697 q^{26} +2.13828 q^{27} +6.00000 q^{28} +2.03293 q^{29} +1.89465 q^{30} -5.66849 q^{31} -6.59607 q^{32} +1.60262 q^{33} +8.06587 q^{34} -12.4633 q^{35} -3.91211 q^{36} +1.07241 q^{37} +3.30404 q^{38} -1.45779 q^{39} +3.30404 q^{40} +8.39738 q^{41} -2.93959 q^{42} -11.2316 q^{43} -6.00000 q^{44} +8.12628 q^{45} +5.00546 q^{46} -3.27110 q^{47} +1.77383 q^{48} +12.3370 q^{49} +5.56314 q^{50} -1.60262 q^{51} +5.45779 q^{52} -3.66849 q^{53} +3.92213 q^{54} +12.4633 q^{55} -5.12628 q^{56} -0.656483 q^{57} +3.72890 q^{58} -3.60262 q^{59} +1.40939 q^{60} -8.46325 q^{61} -10.3974 q^{62} -12.6081 q^{63} -2.36445 q^{64} -11.3370 q^{65} +2.93959 q^{66} +3.33697 q^{67} +6.00000 q^{68} -0.994541 q^{69} -22.8606 q^{70} +1.00000 q^{71} +3.34243 q^{72} +2.83424 q^{73} +1.96707 q^{74} -1.10535 q^{75} +2.45779 q^{76} -19.3370 q^{77} -2.67395 q^{78} -10.5686 q^{79} +13.7948 q^{80} +7.82224 q^{81} +15.4028 q^{82} +5.80131 q^{83} -2.18669 q^{84} -12.4633 q^{85} -20.6015 q^{86} -0.740899 q^{87} +5.12628 q^{88} +0.509273 q^{89} +14.9056 q^{90} +17.5895 q^{91} +3.72344 q^{92} +2.06587 q^{93} -6.00000 q^{94} -5.10535 q^{95} +2.40393 q^{96} -6.06587 q^{97} +22.6290 q^{98} +12.6081 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 4 q^{4} - 3 q^{5} + 9 q^{6} + 2 q^{7} - 9 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q - q^3 + 4 * q^4 - 3 * q^5 + 9 * q^6 + 2 * q^7 - 9 * q^8 + 8 * q^9	\( 3 q - q^{3} + 4 q^{4} - 3 q^{5} + 9 q^{6} + 2 q^{7} - 9 q^{8} + 8 q^{9} - 10 q^{10} - 2 q^{11} - 18 q^{12} + 12 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} + 2 q^{17} - 21 q^{18} + 11 q^{19} + 5 q^{20} - 16 q^{21} - 2 q^{22} + 8 q^{23} + 12 q^{24} - 2 q^{25} - 10 q^{27} + 18 q^{28} - 5 q^{29} + 11 q^{30} - 6 q^{31} - 3 q^{32} + 16 q^{33} + 2 q^{34} - 4 q^{35} + 21 q^{36} + 9 q^{37} - q^{38} - 4 q^{39} - q^{40} + 14 q^{41} + 2 q^{42} - 17 q^{43} - 18 q^{44} + 13 q^{45} - 18 q^{46} - 10 q^{47} - 11 q^{48} + 15 q^{49} + 11 q^{50} - 16 q^{51} + 16 q^{52} + 39 q^{54} + 4 q^{55} - 4 q^{56} + 4 q^{57} + 11 q^{58} - 22 q^{59} - 12 q^{60} + 8 q^{61} - 20 q^{62} - 16 q^{63} - 7 q^{64} - 12 q^{65} - 2 q^{66} - 12 q^{67} + 18 q^{68} - 36 q^{69} - 24 q^{70} + 3 q^{71} - 45 q^{72} + 3 q^{73} + 17 q^{74} + 2 q^{75} + 7 q^{76} - 36 q^{77} + 36 q^{78} + 7 q^{79} + 19 q^{80} + 23 q^{81} + 2 q^{82} + 23 q^{83} - 6 q^{84} - 4 q^{85} - 12 q^{86} + 3 q^{87} + 4 q^{88} + 13 q^{89} - 16 q^{90} + 8 q^{91} + 44 q^{92} - 16 q^{93} - 18 q^{94} - 10 q^{95} + 24 q^{96} + 4 q^{97} + 40 q^{98} + 16 q^{99}+O(q^{100}) \) 3 * q - q^3 + 4 * q^4 - 3 * q^5 + 9 * q^6 + 2 * q^7 - 9 * q^8 + 8 * q^9 - 10 * q^10 - 2 * q^11 - 18 * q^12 + 12 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 + 2 * q^17 - 21 * q^18 + 11 * q^19 + 5 * q^20 - 16 * q^21 - 2 * q^22 + 8 * q^23 + 12 * q^24 - 2 * q^25 - 10 * q^27 + 18 * q^28 - 5 * q^29 + 11 * q^30 - 6 * q^31 - 3 * q^32 + 16 * q^33 + 2 * q^34 - 4 * q^35 + 21 * q^36 + 9 * q^37 - q^38 - 4 * q^39 - q^40 + 14 * q^41 + 2 * q^42 - 17 * q^43 - 18 * q^44 + 13 * q^45 - 18 * q^46 - 10 * q^47 - 11 * q^48 + 15 * q^49 + 11 * q^50 - 16 * q^51 + 16 * q^52 + 39 * q^54 + 4 * q^55 - 4 * q^56 + 4 * q^57 + 11 * q^58 - 22 * q^59 - 12 * q^60 + 8 * q^61 - 20 * q^62 - 16 * q^63 - 7 * q^64 - 12 * q^65 - 2 * q^66 - 12 * q^67 + 18 * q^68 - 36 * q^69 - 24 * q^70 + 3 * q^71 - 45 * q^72 + 3 * q^73 + 17 * q^74 + 2 * q^75 + 7 * q^76 - 36 * q^77 + 36 * q^78 + 7 * q^79 + 19 * q^80 + 23 * q^81 + 2 * q^82 + 23 * q^83 - 6 * q^84 - 4 * q^85 - 12 * q^86 + 3 * q^87 + 4 * q^88 + 13 * q^89 - 16 * q^90 + 8 * q^91 + 44 * q^92 - 16 * q^93 - 18 * q^94 - 10 * q^95 + 24 * q^96 + 4 * q^97 + 40 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.83424	1.29701	0.648503	−	0.761212i	$-0.275395\pi$
$2$	1.83424	1.29701	0.648503	+	0.761212i	$0.275395\pi$
$3$	−0.364448	−0.210414	−0.105207	−	0.994450i	$-0.533551\pi$
$3$	−0.364448	−0.210414	−0.105207	+	0.994450i	$0.533551\pi$
$4$	1.36445	0.682224
$5$	−2.83424	−1.26751	−0.633756	−	0.773533i	$-0.718488\pi$
$5$	−2.83424	−1.26751	−0.633756	+	0.773533i	$0.718488\pi$
$6$	−0.668486	−0.272908
$7$	4.39738	1.66205	0.831027	−	0.556232i	$-0.187753\pi$
$7$	4.39738	1.66205	0.831027	+	0.556232i	$0.187753\pi$
$8$	−1.16576	−0.412157
$9$	−2.86718	−0.955726
$10$	−5.19869	−1.64397
$11$	−4.39738	−1.32586	−0.662930	−	0.748681i	$-0.730687\pi$
$11$	−4.39738	−1.32586	−0.662930	+	0.748681i	$0.730687\pi$
$12$	−0.497270	−0.143550
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	8.06587	2.15569
$15$	1.03293	0.266703
$16$	−4.86718	−1.21679
$17$	4.39738	1.06652	0.533261	−	0.845951i	$-0.320966\pi$
$17$	4.39738	1.06652	0.533261	+	0.845951i	$0.320966\pi$
$18$	−5.25910	−1.23958
$19$	1.80131	0.413249	0.206624	−	0.978420i	$-0.433752\pi$
$19$	1.80131	0.413249	0.206624	+	0.978420i	$0.433752\pi$
$20$	−3.86718	−0.864727
$21$	−1.60262	−0.349720
$22$	−8.06587	−1.71965
$23$	2.72890	0.569014	0.284507	−	0.958674i	$-0.408170\pi$
$23$	2.72890	0.569014	0.284507	+	0.958674i	$0.408170\pi$
$24$	0.424858	0.0867237
$25$	3.03293	0.606587
$26$	7.33697	1.43890
$27$	2.13828	0.411512
$28$	6.00000	1.13389
$29$	2.03293	0.377506	0.188753	−	0.982025i	$-0.439555\pi$
$29$	2.03293	0.377506	0.188753	+	0.982025i	$0.439555\pi$
$30$	1.89465	0.345915
$31$	−5.66849	−1.01809	−0.509045	−	0.860740i	$-0.670001\pi$
$31$	−5.66849	−1.01809	−0.509045	+	0.860740i	$0.670001\pi$
$32$	−6.59607	−1.16603
$33$	1.60262	0.278980
$34$	8.06587	1.38329
$35$	−12.4633	−2.10667
$36$	−3.91211	−0.652019
$37$	1.07241	0.176304	0.0881518	−	0.996107i	$-0.471904\pi$
$37$	1.07241	0.176304	0.0881518	+	0.996107i	$0.471904\pi$
$38$	3.30404	0.535986
$39$	−1.45779	−0.233434
$40$	3.30404	0.522414
$41$	8.39738	1.31145	0.655725	−	0.754999i	$-0.272363\pi$
$41$	8.39738	1.31145	0.655725	+	0.754999i	$0.272363\pi$
$42$	−2.93959	−0.453589
$43$	−11.2316	−1.71281	−0.856403	−	0.516307i	$-0.827306\pi$
$43$	−11.2316	−1.71281	−0.856403	+	0.516307i	$0.827306\pi$
$44$	−6.00000	−0.904534
$45$	8.12628	1.21139
$46$	5.00546	0.738015
$47$	−3.27110	−0.477140	−0.238570	−	0.971125i	$-0.576679\pi$
$47$	−3.27110	−0.477140	−0.238570	+	0.971125i	$0.576679\pi$
$48$	1.77383	0.256031
$49$	12.3370	1.76242
$50$	5.56314	0.786747
$51$	−1.60262	−0.224411
$52$	5.45779	0.756860
$53$	−3.66849	−0.503905	−0.251953	−	0.967740i	$-0.581073\pi$
$53$	−3.66849	−0.503905	−0.251953	+	0.967740i	$0.581073\pi$
$54$	3.92213	0.533734
$55$	12.4633	1.68054
$56$	−5.12628	−0.685028
$57$	−0.656483	−0.0869534
$58$	3.72890	0.489628
$59$	−3.60262	−0.469021	−0.234510	−	0.972114i	$-0.575349\pi$
$59$	−3.60262	−0.469021	−0.234510	+	0.972114i	$0.575349\pi$
$60$	1.40939	0.181951
$61$	−8.46325	−1.08361	−0.541804	−	0.840505i	$-0.682259\pi$
$61$	−8.46325	−1.08361	−0.541804	+	0.840505i	$0.682259\pi$
$62$	−10.3974	−1.32047
$63$	−12.6081	−1.58847
$64$	−2.36445	−0.295556
$65$	−11.3370	−1.40618
$66$	2.93959	0.361839
$67$	3.33697	0.407676	0.203838	−	0.979005i	$-0.434658\pi$
$67$	3.33697	0.407676	0.203838	+	0.979005i	$0.434658\pi$
$68$	6.00000	0.727607
$69$	−0.994541	−0.119729
$70$	−22.8606	−2.73237
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	3.34243	0.393909
$73$	2.83424	0.331723	0.165862	−	0.986149i	$-0.446960\pi$
$73$	2.83424	0.331723	0.165862	+	0.986149i	$0.446960\pi$
$74$	1.96707	0.228667
$75$	−1.10535	−0.127634
$76$	2.45779	0.281928
$77$	−19.3370	−2.20365
$78$	−2.67395	−0.302765
$79$	−10.5686	−1.18906	−0.594530	−	0.804073i	$-0.702662\pi$
$79$	−10.5686	−1.18906	−0.594530	+	0.804073i	$0.702662\pi$
$80$	13.7948	1.54230
$81$	7.82224	0.869138
$82$	15.4028	1.70096
$83$	5.80131	0.636776	0.318388	−	0.947960i	$-0.396858\pi$
$83$	5.80131	0.636776	0.318388	+	0.947960i	$0.396858\pi$
$84$	−2.18669	−0.238587
$85$	−12.4633	−1.35183
$86$	−20.6015	−2.22152
$87$	−0.740899	−0.0794327
$88$	5.12628	0.546463
$89$	0.509273	0.0539829	0.0269914	−	0.999636i	$-0.491407\pi$
$89$	0.509273	0.0539829	0.0269914	+	0.999636i	$0.491407\pi$
$90$	14.9056	1.57119
$91$	17.5895	1.84388
$92$	3.72344	0.388195
$93$	2.06587	0.214221
$94$	−6.00000	−0.618853
$95$	−5.10535	−0.523798
$96$	2.40393	0.245350
$97$	−6.06587	−0.615896	−0.307948	−	0.951403i	$-0.599642\pi$
$97$	−6.06587	−0.615896	−0.307948	+	0.951403i	$0.599642\pi$
$98$	22.6290	2.28588
$99$	12.6081	1.26716
$100$	4.13828	0.413828
$101$	−10.2646	−1.02136	−0.510681	−	0.859770i	$-0.670607\pi$
$101$	−10.2646	−1.02136	−0.510681	+	0.859770i	$0.670607\pi$
$102$	−2.93959	−0.291063
$103$	−2.90666	−0.286401	−0.143201	−	0.989694i	$-0.545739\pi$
$103$	−2.90666	−0.286401	−0.143201	+	0.989694i	$0.545739\pi$
$104$	−4.66303	−0.457247
$105$	4.54221	0.443274
$106$	−6.72890	−0.653568
$107$	3.33697	0.322597	0.161299	−	0.986906i	$-0.448432\pi$
$107$	3.33697	0.322597	0.161299	+	0.986906i	$0.448432\pi$
$108$	2.91757	0.280744
$109$	17.3699	1.66374	0.831868	−	0.554974i	$-0.187272\pi$
$109$	17.3699	1.66374	0.831868	+	0.554974i	$0.187272\pi$
$110$	22.8606	2.17968
$111$	−0.390839	−0.0370968
$112$	−21.4028	−2.02238
$113$	18.5291	1.74307	0.871537	−	0.490331i	$-0.163124\pi$
$113$	18.5291	1.74307	0.871537	+	0.490331i	$0.163124\pi$
$114$	−1.20415	−0.112779
$115$	−7.73436	−0.721232
$116$	2.77383	0.257544
$117$	−11.4687	−1.06028
$118$	−6.60808	−0.608323
$119$	19.3370	1.77262
$120$	−1.20415	−0.109923
$121$	8.33697	0.757907
$122$	−15.5237	−1.40545
$123$	−3.06041	−0.275948
$124$	−7.73436	−0.694566
$125$	5.57514	0.498656
$126$	−23.1263	−2.06025
$127$	6.13174	0.544104	0.272052	−	0.962283i	$-0.412298\pi$
$127$	6.13174	0.544104	0.272052	+	0.962283i	$0.412298\pi$
$128$	8.85517	0.782694
$129$	4.09334	0.360399
$130$	−20.7948	−1.82382
$131$	4.90011	0.428125	0.214062	−	0.976820i	$-0.431330\pi$
$131$	4.90011	0.428125	0.214062	+	0.976820i	$0.431330\pi$
$132$	2.18669	0.190327
$133$	7.92104	0.686842
$134$	6.12082	0.528758
$135$	−6.06041	−0.521597
$136$	−5.12628	−0.439575
$137$	7.40284	0.632467	0.316234	−	0.948681i	$-0.397582\pi$
$137$	7.40284	0.632467	0.316234	+	0.948681i	$0.397582\pi$
$138$	−1.82423	−0.155289
$139$	8.72890	0.740375	0.370188	−	0.928957i	$-0.379293\pi$
$139$	8.72890	0.740375	0.370188	+	0.928957i	$0.379293\pi$
$140$	−17.0055	−1.43722
$141$	1.19215	0.100397
$142$	1.83424	0.153926
$143$	−17.5895	−1.47091
$144$	13.9551	1.16292
$145$	−5.76183	−0.478494
$146$	5.19869	0.430247
$147$	−4.49619	−0.370839
$148$	1.46325	0.120279
$149$	−18.1317	−1.48541	−0.742705	−	0.669619i	$-0.766457\pi$
$149$	−18.1317	−1.48541	−0.742705	+	0.669619i	$0.766457\pi$
$150$	−2.02748	−0.165543
$151$	−16.4303	−1.33708	−0.668540	−	0.743676i	$-0.733080\pi$
$151$	−16.4303	−1.33708	−0.668540	+	0.743676i	$0.733080\pi$
$152$	−2.09989	−0.170323
$153$	−12.6081	−1.01930
$154$	−35.4687	−2.85815
$155$	16.0659	1.29044
$156$	−1.98908	−0.159254
$157$	1.30404	0.104074	0.0520368	−	0.998645i	$-0.483429\pi$
$157$	1.30404	0.104074	0.0520368	+	0.998645i	$0.483429\pi$
$158$	−19.3854	−1.54222
$159$	1.33697	0.106029
$160$	18.6949	1.47796
$161$	12.0000	0.945732
$162$	14.3479	1.12728
$163$	−7.73436	−0.605801	−0.302901	−	0.953022i	$-0.597955\pi$
$163$	−7.73436	−0.605801	−0.302901	+	0.953022i	$0.597955\pi$
$164$	11.4578	0.894703
$165$	−4.54221	−0.353610
$166$	10.6410	0.825903
$167$	12.7684	0.988046	0.494023	−	0.869449i	$-0.335526\pi$
$167$	12.7684	0.988046	0.494023	+	0.869449i	$0.335526\pi$
$168$	1.86826	0.144140
$169$	3.00000	0.230769
$170$	−22.8606	−1.75333
$171$	−5.16467	−0.394952
$172$	−15.3250	−1.16852
$173$	−2.18669	−0.166251	−0.0831254	−	0.996539i	$-0.526490\pi$
$173$	−2.18669	−0.166251	−0.0831254	+	0.996539i	$0.526490\pi$
$174$	−1.35899	−0.103025
$175$	13.3370	1.00818
$176$	21.4028	1.61330
$177$	1.31297	0.0986886
$178$	0.934131	0.0700161
$179$	−2.64755	−0.197888	−0.0989438	−	0.995093i	$-0.531546\pi$
$179$	−2.64755	−0.197888	−0.0989438	+	0.995093i	$0.531546\pi$
$180$	11.0879	0.826442
$181$	23.7344	1.76416	0.882080	−	0.471099i	$-0.156143\pi$
$181$	23.7344	1.76416	0.882080	+	0.471099i	$0.156143\pi$
$182$	32.2635	2.39153
$183$	3.08442	0.228007
$184$	−3.18123	−0.234523
$185$	−3.03948	−0.223467
$186$	3.78931	0.277845
$187$	−19.3370	−1.41406
$188$	−4.46325	−0.325516
$189$	9.40284	0.683956
$190$	−9.36445	−0.679368
$191$	7.03839	0.509280	0.254640	−	0.967036i	$-0.418043\pi$
$191$	7.03839	0.509280	0.254640	+	0.967036i	$0.418043\pi$
$192$	0.861719	0.0621892
$193$	−11.5477	−0.831219	−0.415610	−	0.909543i	$-0.636432\pi$
$193$	−11.5477	−0.831219	−0.415610	+	0.909543i	$0.636432\pi$
$194$	−11.1263	−0.798820
$195$	4.13174	0.295880
$196$	16.8332	1.20237
$197$	−7.33697	−0.522738	−0.261369	−	0.965239i	$-0.584174\pi$
$197$	−7.33697	−0.522738	−0.261369	+	0.965239i	$0.584174\pi$
$198$	23.1263	1.64351
$199$	−2.86172	−0.202862	−0.101431	−	0.994843i	$-0.532342\pi$
$199$	−2.86172	−0.202862	−0.101431	+	0.994843i	$0.532342\pi$
$200$	−3.53566	−0.250009
$201$	−1.21615	−0.0857808
$202$	−18.8277	−1.32471
$203$	8.93959	0.627436
$204$	−2.18669	−0.153099
$205$	−23.8002	−1.66228
$206$	−5.33151	−0.371464
$207$	−7.82423	−0.543822
$208$	−19.4687	−1.34991
$209$	−7.92104	−0.547910
$210$	8.33151	0.574929
$211$	4.54221	0.312698	0.156349	−	0.987702i	$-0.450027\pi$
$211$	4.54221	0.312698	0.156349	+	0.987702i	$0.450027\pi$
$212$	−5.00546	−0.343776
$213$	−0.364448	−0.0249716
$214$	6.12082	0.418411
$215$	31.8332	2.17100
$216$	−2.49272	−0.169608
$217$	−24.9265	−1.69212
$218$	31.8606	2.15787
$219$	−1.03293	−0.0697992
$220$	17.0055	1.14651
$221$	17.5895	1.18320
$222$	−0.716893	−0.0481147
$223$	−18.7278	−1.25411	−0.627054	−	0.778976i	$-0.715739\pi$
$223$	−18.7278	−1.25411	−0.627054	+	0.778976i	$0.715739\pi$
$224$	−29.0055	−1.93801
$225$	−8.69596	−0.579731
$226$	33.9869	2.26078
$227$	19.9869	1.32658	0.663289	−	0.748363i	$-0.269160\pi$
$227$	19.9869	1.32658	0.663289	+	0.748363i	$0.269160\pi$
$228$	−0.895738	−0.0593217
$229$	24.4238	1.61397	0.806984	−	0.590573i	$-0.201098\pi$
$229$	24.4238	1.61397	0.806984	+	0.590573i	$0.201098\pi$
$230$	−14.1867	−0.935443
$231$	7.04732	0.463680
$232$	−2.36991	−0.155592
$233$	−27.4567	−1.79875	−0.899374	−	0.437179i	$-0.855978\pi$
$233$	−27.4567	−1.79875	−0.899374	+	0.437179i	$0.855978\pi$
$234$	−21.0364	−1.37519
$235$	9.27110	0.604780
$236$	−4.91558	−0.319977
$237$	3.85171	0.250195
$238$	35.4687	2.29910
$239$	−18.9396	−1.22510	−0.612550	−	0.790432i	$-0.709856\pi$
$239$	−18.9396	−1.22510	−0.612550	+	0.790432i	$0.709856\pi$
$240$	−5.02748	−0.324522
$241$	20.7289	1.33527	0.667633	−	0.744491i	$-0.267308\pi$
$241$	20.7289	1.33527	0.667633	+	0.744491i	$0.267308\pi$
$242$	15.2920	0.983009
$243$	−9.26564	−0.594391
$244$	−11.5477	−0.739264
$245$	−34.9660	−2.23389
$246$	−5.61354	−0.357906
$247$	7.20524	0.458458
$248$	6.60808	0.419613
$249$	−2.11428	−0.133987
$250$	10.2262	0.646760
$251$	−14.3304	−0.904529	−0.452264	−	0.891884i	$-0.649384\pi$
$251$	−14.3304	−0.904529	−0.452264	+	0.891884i	$0.649384\pi$
$252$	−17.2031	−1.08369
$253$	−12.0000	−0.754434
$254$	11.2471	0.705706
$255$	4.54221	0.284444
$256$	20.9714	1.31072
$257$	−0.518202	−0.0323246	−0.0161623	−	0.999869i	$-0.505145\pi$
$257$	−0.518202	−0.0323246	−0.0161623	+	0.999869i	$0.505145\pi$
$258$	7.50819	0.467439
$259$	4.71581	0.293026
$260$	−15.4687	−0.959329
$261$	−5.82878	−0.360793
$262$	8.98800	0.555280
$263$	−21.6949	−1.33776	−0.668882	−	0.743369i	$-0.733227\pi$
$263$	−21.6949	−1.33776	−0.668882	+	0.743369i	$0.733227\pi$
$264$	−1.86826	−0.114984
$265$	10.3974	0.638706
$266$	14.5291	0.890838
$267$	−0.185604	−0.0113588
$268$	4.55313	0.278126
$269$	−19.1921	−1.17017	−0.585083	−	0.810973i	$-0.698938\pi$
$269$	−19.1921	−1.17017	−0.585083	+	0.810973i	$0.698938\pi$
$270$	−11.1163	−0.676514
$271$	7.63555	0.463827	0.231913	−	0.972736i	$-0.425501\pi$
$271$	7.63555	0.463827	0.231913	+	0.972736i	$0.425501\pi$
$272$	−21.4028	−1.29774
$273$	−6.41047	−0.387979
$274$	13.5786	0.820314
$275$	−13.3370	−0.804250
$276$	−1.35700	−0.0816818
$277$	19.4567	1.16904	0.584520	−	0.811379i	$-0.301283\pi$
$277$	19.4567	1.16904	0.584520	+	0.811379i	$0.301283\pi$
$278$	16.0109	0.960271
$279$	16.2526	0.973015
$280$	14.5291	0.868281
$281$	−0.794765	−0.0474117	−0.0237059	−	0.999719i	$-0.507547\pi$
$281$	−0.794765	−0.0474117	−0.0237059	+	0.999719i	$0.507547\pi$
$282$	2.18669	0.130215
$283$	−29.0055	−1.72420	−0.862098	−	0.506742i	$-0.830850\pi$
$283$	−29.0055	−1.72420	−0.862098	+	0.506742i	$0.830850\pi$
$284$	1.36445	0.0809651
$285$	1.86063	0.110214
$286$	−32.2635	−1.90778
$287$	36.9265	2.17970
$288$	18.9121	1.11441
$289$	2.33697	0.137469
$290$	−10.5686	−0.620609
$291$	2.21069	0.129593
$292$	3.86718	0.226309
$293$	4.25256	0.248437	0.124219	−	0.992255i	$-0.460358\pi$
$293$	4.25256	0.248437	0.124219	+	0.992255i	$0.460358\pi$
$294$	−8.24710	−0.480981
$295$	10.2107	0.594490
$296$	−1.25017	−0.0726648
$297$	−9.40284	−0.545608
$298$	−33.2580	−1.92659
$299$	10.9156	0.631265
$300$	−1.50819	−0.0870753
$301$	−49.3898	−2.84678
$302$	−30.1372	−1.73420
$303$	3.74090	0.214909
$304$	−8.76729	−0.502839
$305$	23.9869	1.37349
$306$	−23.1263	−1.32204
$307$	−6.91558	−0.394693	−0.197347	−	0.980334i	$-0.563232\pi$
$307$	−6.91558	−0.394693	−0.197347	+	0.980334i	$0.563232\pi$
$308$	−26.3843	−1.50338
$309$	1.05933	0.0602629
$310$	29.4687	1.67371
$311$	−28.1856	−1.59826	−0.799129	−	0.601159i	$-0.794706\pi$
$311$	−28.1856	−1.59826	−0.799129	+	0.601159i	$0.794706\pi$
$312$	1.69943	0.0962113
$313$	30.7937	1.74056	0.870281	−	0.492556i	$-0.163937\pi$
$313$	30.7937	1.74056	0.870281	+	0.492556i	$0.163937\pi$
$314$	2.39192	0.134984
$315$	35.7344	2.01340
$316$	−14.4203	−0.811205
$317$	−9.66849	−0.543036	−0.271518	−	0.962433i	$-0.587526\pi$
$317$	−9.66849	−0.543036	−0.271518	+	0.962433i	$0.587526\pi$
$318$	2.45233	0.137520
$319$	−8.93959	−0.500521
$320$	6.70142	0.374621
$321$	−1.21615	−0.0678791
$322$	22.0109	1.22662
$323$	7.92104	0.440739
$324$	10.6730	0.592947
$325$	12.1317	0.672948
$326$	−14.1867	−0.785728
$327$	−6.33043	−0.350074
$328$	−9.78931	−0.540524
$329$	−14.3843	−0.793032
$330$	−8.33151	−0.458635
$331$	−16.3843	−0.900562	−0.450281	−	0.892887i	$-0.648676\pi$
$331$	−16.3843	−0.900562	−0.450281	+	0.892887i	$0.648676\pi$
$332$	7.91558	0.434424
$333$	−3.07480	−0.168498
$334$	23.4203	1.28150
$335$	−9.45779	−0.516734
$336$	7.80022	0.425537
$337$	7.53675	0.410553	0.205276	−	0.978704i	$-0.434191\pi$
$337$	7.53675	0.410553	0.205276	+	0.978704i	$0.434191\pi$
$338$	5.50273	0.299309
$339$	−6.75290	−0.366767
$340$	−17.0055	−0.922251
$341$	24.9265	1.34985
$342$	−9.47326	−0.512255
$343$	23.4687	1.26719
$344$	13.0933	0.705946
$345$	2.81877	0.151758
$346$	−4.01092	−0.215628
$347$	15.9210	0.854686	0.427343	−	0.904089i	$-0.359450\pi$
$347$	15.9210	0.854686	0.427343	+	0.904089i	$0.359450\pi$
$348$	−1.01092	−0.0541909
$349$	14.2766	0.764207	0.382103	−	0.924120i	$-0.375200\pi$
$349$	14.2766	0.764207	0.382103	+	0.924120i	$0.375200\pi$
$350$	24.4633	1.30762
$351$	8.55313	0.456532
$352$	29.0055	1.54600
$353$	−3.20524	−0.170597	−0.0852987	−	0.996355i	$-0.527184\pi$
$353$	−3.20524	−0.170597	−0.0852987	+	0.996355i	$0.527184\pi$
$354$	2.40830	0.128000
$355$	−2.83424	−0.150426
$356$	0.694877	0.0368284
$357$	−7.04732	−0.372984
$358$	−4.85626	−0.256661
$359$	−4.97907	−0.262785	−0.131393	−	0.991330i	$-0.541945\pi$
$359$	−4.97907	−0.262785	−0.131393	+	0.991330i	$0.541945\pi$
$360$	−9.47326	−0.499285
$361$	−15.7553	−0.829226
$362$	43.5346	2.28813
$363$	−3.03839	−0.159474
$364$	24.0000	1.25794
$365$	−8.03293	−0.420463
$366$	5.65757	0.295726
$367$	21.8266	1.13934	0.569670	−	0.821874i	$-0.307071\pi$
$367$	21.8266	1.13934	0.569670	+	0.821874i	$0.307071\pi$
$368$	−13.2820	−0.692373
$369$	−24.0768	−1.25339
$370$	−5.57514	−0.289838
$371$	−16.1317	−0.837518
$372$	2.81877	0.146146
$373$	−18.0923	−0.936782	−0.468391	−	0.883521i	$-0.655166\pi$
$373$	−18.0923	−0.936782	−0.468391	+	0.883521i	$0.655166\pi$
$374$	−35.4687	−1.83404
$375$	−2.03185	−0.104924
$376$	3.81331	0.196657
$377$	8.13174	0.418806
$378$	17.2471	0.887095
$379$	4.36445	0.224187	0.112093	−	0.993698i	$-0.464244\pi$
$379$	4.36445	0.224187	0.112093	+	0.993698i	$0.464244\pi$
$380$	−6.96598	−0.357347
$381$	−2.23470	−0.114487
$382$	12.9101	0.660539
$383$	18.7398	0.957560	0.478780	−	0.877935i	$-0.341079\pi$
$383$	18.7398	0.957560	0.478780	+	0.877935i	$0.341079\pi$
$384$	−3.22725	−0.164690
$385$	54.8057	2.79316
$386$	−21.1812	−1.07810
$387$	32.2031	1.63697
$388$	−8.27656	−0.420179
$389$	−5.93413	−0.300872	−0.150436	−	0.988620i	$-0.548068\pi$
$389$	−5.93413	−0.300872	−0.150436	+	0.988620i	$0.548068\pi$
$390$	7.57861	0.383758
$391$	12.0000	0.606866
$392$	−14.3819	−0.726396
$393$	−1.78584	−0.0900835
$394$	−13.4578	−0.677994
$395$	29.9540	1.50715
$396$	17.2031	0.864487
$397$	−24.1976	−1.21444	−0.607222	−	0.794533i	$-0.707716\pi$
$397$	−24.1976	−1.21444	−0.607222	+	0.794533i	$0.707716\pi$
$398$	−5.24909	−0.263113
$399$	−2.88681	−0.144521
$400$	−14.7618	−0.738092
$401$	38.7267	1.93392	0.966960	−	0.254927i	$-0.0820515\pi$
$401$	38.7267	1.93392	0.966960	+	0.254927i	$0.0820515\pi$
$402$	−2.23072	−0.111258
$403$	−22.6739	−1.12947
$404$	−14.0055	−0.696798
$405$	−22.1701	−1.10164
$406$	16.3974	0.813788
$407$	−4.71581	−0.233754
$408$	1.86826	0.0924928
$409$	13.9012	0.687370	0.343685	−	0.939085i	$-0.388325\pi$
$409$	13.9012	0.687370	0.343685	+	0.939085i	$0.388325\pi$
$410$	−43.6554	−2.15599
$411$	−2.69795	−0.133080
$412$	−3.96598	−0.195390
$413$	−15.8421	−0.779538
$414$	−14.3515	−0.705340
$415$	−16.4423	−0.807122
$416$	−26.3843	−1.29360
$417$	−3.18123	−0.155785
$418$	−14.5291	−0.710642
$419$	−8.43032	−0.411848	−0.205924	−	0.978568i	$-0.566020\pi$
$419$	−8.43032	−0.411848	−0.205924	+	0.978568i	$0.566020\pi$
$420$	6.19761	0.302412
$421$	−25.2820	−1.23217	−0.616085	−	0.787680i	$-0.711282\pi$
$421$	−25.2820	−1.23217	−0.616085	+	0.787680i	$0.711282\pi$
$422$	8.33151	0.405572
$423$	9.37884	0.456015
$424$	4.27656	0.207688
$425$	13.3370	0.646938
$426$	−0.668486	−0.0323883
$427$	−37.2162	−1.80102
$428$	4.55313	0.220084
$429$	6.41047	0.309500
$430$	58.3898	2.81580
$431$	−0.0538660	−0.00259463	−0.00129732	−	0.999999i	$-0.500413\pi$
$431$	−0.0538660	−0.00259463	−0.00129732	+	0.999999i	$0.500413\pi$
$432$	−10.4074	−0.500726
$433$	−24.5531	−1.17995	−0.589974	−	0.807422i	$-0.700862\pi$
$433$	−24.5531	−1.17995	−0.589974	+	0.807422i	$0.700862\pi$
$434$	−45.7213	−2.19469
$435$	2.09989	0.100682
$436$	23.7003	1.13504
$437$	4.91558	0.235144
$438$	−1.89465	−0.0905300
$439$	−27.9210	−1.33260	−0.666299	−	0.745684i	$-0.732123\pi$
$439$	−27.9210	−1.33260	−0.666299	+	0.745684i	$0.732123\pi$
$440$	−14.5291	−0.692649
$441$	−35.3723	−1.68439
$442$	32.2635	1.53462
$443$	33.8661	1.60903	0.804513	−	0.593935i	$-0.202426\pi$
$443$	33.8661	1.60903	0.804513	+	0.593935i	$0.202426\pi$
$444$	−0.533279	−0.0253083
$445$	−1.44340	−0.0684239
$446$	−34.3514	−1.62658
$447$	6.60808	0.312551
$448$	−10.3974	−0.491230
$449$	−8.93959	−0.421885	−0.210943	−	0.977498i	$-0.567653\pi$
$449$	−8.93959	−0.421885	−0.210943	+	0.977498i	$0.567653\pi$
$450$	−15.9505	−0.751914
$451$	−36.9265	−1.73880
$452$	25.2820	1.18917
$453$	5.98800	0.281341
$454$	36.6609	1.72058
$455$	−49.8530	−2.33714
$456$	0.765300	0.0358385
$457$	20.6609	0.966474	0.483237	−	0.875490i	$-0.339461\pi$
$457$	20.6609	0.966474	0.483237	+	0.875490i	$0.339461\pi$
$458$	44.7991	2.09333
$459$	9.40284	0.438887
$460$	−10.5531	−0.492042
$461$	8.60808	0.400918	0.200459	−	0.979702i	$-0.435757\pi$
$461$	8.60808	0.400918	0.200459	+	0.979702i	$0.435757\pi$
$462$	12.9265	0.601395
$463$	−20.3435	−0.945443	−0.472722	−	0.881212i	$-0.656728\pi$
$463$	−20.3435	−0.945443	−0.472722	+	0.881212i	$0.656728\pi$
$464$	−9.89465	−0.459348
$465$	−5.85517	−0.271527
$466$	−50.3623	−2.33299
$467$	6.47634	0.299689	0.149845	−	0.988710i	$-0.452123\pi$
$467$	6.47634	0.299689	0.149845	+	0.988710i	$0.452123\pi$
$468$	−15.6485	−0.723350
$469$	14.6739	0.677580
$470$	17.0055	0.784403
$471$	−0.475254	−0.0218986
$472$	4.19978	0.193310
$473$	49.3898	2.27094
$474$	7.06496	0.324505
$475$	5.46325	0.250671
$476$	26.3843	1.20932
$477$	10.5182	0.481595
$478$	−34.7398	−1.58896
$479$	−12.1976	−0.557323	−0.278661	−	0.960389i	$-0.589891\pi$
$479$	−12.1976	−0.557323	−0.278661	+	0.960389i	$0.589891\pi$
$480$	−6.81331	−0.310984
$481$	4.28965	0.195591
$482$	38.0218	1.73185
$483$	−4.37338	−0.198996
$484$	11.3754	0.517062
$485$	17.1921	0.780655
$486$	−16.9954	−0.770929
$487$	29.1372	1.32033	0.660166	−	0.751120i	$-0.270486\pi$
$487$	29.1372	1.32033	0.660166	+	0.751120i	$0.270486\pi$
$488$	9.86609	0.446617
$489$	2.81877	0.127469
$490$	−64.1361	−2.89737
$491$	5.00546	0.225893	0.112947	−	0.993601i	$-0.463971\pi$
$491$	5.00546	0.225893	0.112947	+	0.993601i	$0.463971\pi$
$492$	−4.17577	−0.188258
$493$	8.93959	0.402619
$494$	13.2162	0.594623
$495$	−35.7344	−1.60614
$496$	27.5895	1.23881
$497$	4.39738	0.197250
$498$	−3.87810	−0.173782
$499$	16.5621	0.741419	0.370710	−	0.928749i	$-0.379114\pi$
$499$	16.5621	0.741419	0.370710	+	0.928749i	$0.379114\pi$
$500$	7.60699	0.340195
$501$	−4.65341	−0.207899
$502$	−26.2855	−1.17318
$503$	33.3568	1.48731	0.743654	−	0.668565i	$-0.233091\pi$
$503$	33.3568	1.48731	0.743654	+	0.668565i	$0.233091\pi$
$504$	14.6980	0.654699
$505$	29.0923	1.29459
$506$	−22.0109	−0.978505
$507$	−1.09334	−0.0485571
$508$	8.36644	0.371201
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	8.33151	0.368926
$511$	12.4633	0.551342
$512$	20.7564	0.917311
$513$	3.85171	0.170057
$514$	−0.950508	−0.0419251
$515$	8.23817	0.363017
$516$	5.58516	0.245873
$517$	14.3843	0.632621
$518$	8.64994	0.380057
$519$	0.796934	0.0349815
$520$	13.2162	0.579567
$521$	31.7607	1.39146	0.695732	−	0.718302i	$-0.255080\pi$
$521$	31.7607	1.39146	0.695732	+	0.718302i	$0.255080\pi$
$522$	−10.6914	−0.467950
$523$	15.7344	0.688015	0.344008	−	0.938967i	$-0.388215\pi$
$523$	15.7344	0.688015	0.344008	+	0.938967i	$0.388215\pi$
$524$	6.68595	0.292077
$525$	−4.86063	−0.212135
$526$	−39.7937	−1.73509
$527$	−24.9265	−1.08582
$528$	−7.80022	−0.339461
$529$	−15.5531	−0.676223
$530$	19.0713	0.828406
$531$	10.3293	0.448255
$532$	10.8079	0.468580
$533$	33.5895	1.45492
$534$	−0.340442	−0.0147324
$535$	−9.45779	−0.408896
$536$	−3.89010	−0.168027
$537$	0.964896	0.0416383
$538$	−35.2031	−1.51771
$539$	−54.2504	−2.33673
$540$	−8.26911	−0.355846
$541$	12.4633	0.535837	0.267919	−	0.963442i	$-0.413664\pi$
$541$	12.4633	0.535837	0.267919	+	0.963442i	$0.413664\pi$
$542$	14.0055	0.601586
$543$	−8.64994	−0.371204
$544$	−29.0055	−1.24360
$545$	−49.2305	−2.10880
$546$	−11.7584	−0.503211
$547$	−7.65648	−0.327368	−0.163684	−	0.986513i	$-0.552338\pi$
$547$	−7.65648	−0.327368	−0.163684	+	0.986513i	$0.552338\pi$
$548$	10.1008	0.431484
$549$	24.2656	1.03563
$550$	−24.4633	−1.04312
$551$	3.66194	0.156004
$552$	1.15939	0.0493470
$553$	−46.4742	−1.97628
$554$	35.6883	1.51625
$555$	1.10773	0.0470206
$556$	11.9101	0.505102
$557$	−34.3623	−1.45598	−0.727988	−	0.685590i	$-0.759544\pi$
$557$	−34.3623	−1.45598	−0.727988	+	0.685590i	$0.759544\pi$
$558$	29.8111	1.26201
$559$	−44.9265	−1.90019
$560$	60.6609	2.56339
$561$	7.04732	0.297538
$562$	−1.45779	−0.0614933
$563$	−35.2689	−1.48641	−0.743204	−	0.669065i	$-0.766695\pi$
$563$	−35.2689	−1.48641	−0.743204	+	0.669065i	$0.766695\pi$
$564$	1.62662	0.0684932
$565$	−52.5160	−2.20937
$566$	−53.2031	−2.23629
$567$	34.3974	1.44455
$568$	−1.16576	−0.0489141
$569$	−13.7498	−0.576423	−0.288211	−	0.957567i	$-0.593061\pi$
$569$	−13.7498	−0.576423	−0.288211	+	0.957567i	$0.593061\pi$
$570$	3.41285	0.142949
$571$	38.3304	1.60408	0.802039	−	0.597271i	$-0.203748\pi$
$571$	38.3304	1.60408	0.802039	+	0.597271i	$0.203748\pi$
$572$	−24.0000	−1.00349
$573$	−2.56513	−0.107160
$574$	67.7322	2.82709
$575$	8.27656	0.345157
$576$	6.77929	0.282471
$577$	29.8816	1.24399	0.621993	−	0.783023i	$-0.286323\pi$
$577$	29.8816	1.24399	0.621993	+	0.783023i	$0.286323\pi$
$578$	4.28658	0.178298
$579$	4.20852	0.174900
$580$	−7.86172	−0.326440
$581$	25.5106	1.05836
$582$	4.05495	0.168083
$583$	16.1317	0.668108
$584$	−3.30404	−0.136722
$585$	32.5051	1.34392
$586$	7.80022	0.322224
$587$	22.6170	0.933504	0.466752	−	0.884388i	$-0.345424\pi$
$587$	22.6170	0.933504	0.466752	+	0.884388i	$0.345424\pi$
$588$	−6.13481	−0.252995
$589$	−10.2107	−0.420724
$590$	18.7289	0.771056
$591$	2.67395	0.109991
$592$	−5.21962	−0.214525
$593$	1.96707	0.0807777	0.0403889	−	0.999184i	$-0.487140\pi$
$593$	1.96707	0.0807777	0.0403889	+	0.999184i	$0.487140\pi$
$594$	−17.2471	−0.707657
$595$	−54.8057	−2.24681
$596$	−24.7398	−1.01338
$597$	1.04295	0.0426850
$598$	20.0218	0.818754
$599$	6.39738	0.261390	0.130695	−	0.991423i	$-0.458279\pi$
$599$	6.39738	0.261390	0.130695	+	0.991423i	$0.458279\pi$
$600$	1.28857	0.0526055
$601$	31.4028	1.28095	0.640474	−	0.767980i	$-0.278738\pi$
$601$	31.4028	1.28095	0.640474	+	0.767980i	$0.278738\pi$
$602$	−90.5928	−3.69229
$603$	−9.56769	−0.389627
$604$	−22.4183	−0.912188
$605$	−23.6290	−0.960656
$606$	6.86172	0.278738
$607$	−17.3370	−0.703686	−0.351843	−	0.936059i	$-0.614445\pi$
$607$	−17.3370	−0.703686	−0.351843	+	0.936059i	$0.614445\pi$
$608$	−11.8816	−0.481861
$609$	−3.25802	−0.132021
$610$	43.9978	1.78142
$611$	−13.0844	−0.529339
$612$	−17.2031	−0.695393
$613$	13.7433	0.555086	0.277543	−	0.960713i	$-0.410480\pi$
$613$	13.7433	0.555086	0.277543	+	0.960713i	$0.410480\pi$
$614$	−12.6849	−0.511919
$615$	8.67395	0.349767
$616$	22.5422	0.908251
$617$	−3.83970	−0.154581	−0.0772903	−	0.997009i	$-0.524627\pi$
$617$	−3.83970	−0.154581	−0.0772903	+	0.997009i	$0.524627\pi$
$618$	1.94306	0.0781613
$619$	−15.3788	−0.618128	−0.309064	−	0.951041i	$-0.600016\pi$
$619$	−15.3788	−0.618128	−0.309064	+	0.951041i	$0.600016\pi$
$620$	21.9210	0.880370
$621$	5.83515	0.234156
$622$	−51.6993	−2.07295
$623$	2.23947	0.0897225
$624$	7.09533	0.284041
$625$	−30.9660	−1.23864
$626$	56.4831	2.25752
$627$	2.88681	0.115288
$628$	1.77929	0.0710015
$629$	4.71581	0.188032
$630$	65.5455	2.61140
$631$	2.46325	0.0980605	0.0490302	−	0.998797i	$-0.484387\pi$
$631$	2.46325	0.0980605	0.0490302	+	0.998797i	$0.484387\pi$
$632$	12.3204	0.490080
$633$	−1.65540	−0.0657962
$634$	−17.7344	−0.704321
$635$	−17.3788	−0.689658
$636$	1.82423	0.0723354
$637$	49.3479	1.95523
$638$	−16.3974	−0.649179
$639$	−2.86718	−0.113424
$640$	−25.0977	−0.992074
$641$	−19.4567	−0.768494	−0.384247	−	0.923230i	$-0.625539\pi$
$641$	−19.4567	−0.768494	−0.384247	+	0.923230i	$0.625539\pi$
$642$	−2.23072	−0.0880395
$643$	−21.0844	−0.831488	−0.415744	−	0.909482i	$-0.636479\pi$
$643$	−21.0844	−0.831488	−0.415744	+	0.909482i	$0.636479\pi$
$644$	16.3734	0.645201
$645$	−11.6015	−0.456810
$646$	14.5291	0.571641
$647$	−20.1317	−0.791460	−0.395730	−	0.918367i	$-0.629508\pi$
$647$	−20.1317	−0.791460	−0.395730	+	0.918367i	$0.629508\pi$
$648$	−9.11883	−0.358221
$649$	15.8421	0.621856
$650$	22.2526	0.872817
$651$	9.08442	0.356046
$652$	−10.5531	−0.413292
$653$	−9.46871	−0.370539	−0.185270	−	0.982688i	$-0.559316\pi$
$653$	−9.46871	−0.370539	−0.185270	+	0.982688i	$0.559316\pi$
$654$	−11.6115	−0.454047
$655$	−13.8881	−0.542653
$656$	−40.8716	−1.59577
$657$	−8.12628	−0.317036
$658$	−26.3843	−1.02857
$659$	30.8805	1.20293	0.601466	−	0.798898i	$-0.294583\pi$
$659$	30.8805	1.20293	0.601466	+	0.798898i	$0.294583\pi$
$660$	−6.19761	−0.241242
$661$	7.80022	0.303394	0.151697	−	0.988427i	$-0.451526\pi$
$661$	7.80022	0.303394	0.151697	+	0.988427i	$0.451526\pi$
$662$	−30.0528	−1.16803
$663$	−6.41047	−0.248962
$664$	−6.76292	−0.262452
$665$	−22.4502	−0.870580
$666$	−5.63993	−0.218543
$667$	5.54767	0.214807
$668$	17.4218	0.674069
$669$	6.82531	0.263882
$670$	−17.3479	−0.670207
$671$	37.2162	1.43671
$672$	10.5710	0.407785
$673$	−44.8475	−1.72875	−0.864373	−	0.502851i	$-0.832284\pi$
$673$	−44.8475	−1.72875	−0.864373	+	0.502851i	$0.832284\pi$
$674$	13.8242	0.532489
$675$	6.48527	0.249618
$676$	4.09334	0.157436
$677$	−14.0329	−0.539329	−0.269665	−	0.962954i	$-0.586913\pi$
$677$	−14.0329	−0.539329	−0.269665	+	0.962954i	$0.586913\pi$
$678$	−12.3865	−0.475699
$679$	−26.6739	−1.02365
$680$	14.5291	0.557166
$681$	−7.28419	−0.279131
$682$	45.7213	1.75076
$683$	23.7474	0.908671	0.454335	−	0.890831i	$-0.349877\pi$
$683$	23.7474	0.908671	0.454335	+	0.890831i	$0.349877\pi$
$684$	−7.04693	−0.269446
$685$	−20.9815	−0.801660
$686$	43.0473	1.64355
$687$	−8.90120	−0.339602
$688$	54.6663	2.08413
$689$	−14.6739	−0.559033
$690$	5.17031	0.196830
$691$	−42.9505	−1.63391	−0.816957	−	0.576698i	$-0.804341\pi$
$691$	−42.9505	−1.63391	−0.816957	+	0.576698i	$0.804341\pi$
$692$	−2.98362	−0.113420
$693$	55.4425	2.10609
$694$	29.2031	1.10853
$695$	−24.7398	−0.938435
$696$	0.863708	0.0327388
$697$	36.9265	1.39869
$698$	26.1867	0.991181
$699$	10.0065	0.378482
$700$	18.1976	0.687805
$701$	−12.4873	−0.471637	−0.235819	−	0.971797i	$-0.575777\pi$
$701$	−12.4873	−0.471637	−0.235819	+	0.971797i	$0.575777\pi$
$702$	15.6885	0.592125
$703$	1.93175	0.0728572
$704$	10.3974	0.391866
$705$	−3.37884	−0.127254
$706$	−5.87918	−0.221266
$707$	−45.1372	−1.69756
$708$	1.79148	0.0673278
$709$	31.2711	1.17441	0.587205	−	0.809438i	$-0.300228\pi$
$709$	31.2711	1.17441	0.587205	+	0.809438i	$0.300228\pi$
$710$	−5.19869	−0.195103
$711$	30.3020	1.13642
$712$	−0.593689	−0.0222494
$713$	−15.4687	−0.579308
$714$	−12.9265	−0.483762
$715$	49.8530	1.86440
$716$	−3.61245	−0.135004
$717$	6.90250	0.257778
$718$	−9.13282	−0.340834
$719$	−18.6201	−0.694412	−0.347206	−	0.937789i	$-0.612870\pi$
$719$	−18.6201	−0.694412	−0.347206	+	0.937789i	$0.612870\pi$
$720$	−39.5520	−1.47402
$721$	−12.7817	−0.476015
$722$	−28.8990	−1.07551
$723$	−7.55461	−0.280959
$724$	32.3843	1.20355
$725$	6.16576	0.228990
$726$	−5.57315	−0.206839
$727$	−20.5531	−0.762273	−0.381137	−	0.924519i	$-0.624467\pi$
$727$	−20.5531	−0.762273	−0.381137	+	0.924519i	$0.624467\pi$
$728$	−20.5051	−0.759970
$729$	−20.0899	−0.744069
$730$	−14.7344	−0.545343
$731$	−49.3898	−1.82675
$732$	4.20852	0.155552
$733$	−2.99454	−0.110606	−0.0553029	−	0.998470i	$-0.517612\pi$
$733$	−2.99454	−0.110606	−0.0553029	+	0.998470i	$0.517612\pi$
$734$	40.0353	1.47773
$735$	12.7433	0.470043
$736$	−18.0000	−0.663489
$737$	−14.6739	−0.540522
$738$	−44.1627	−1.62565
$739$	46.4323	1.70804	0.854020	−	0.520240i	$-0.174158\pi$
$739$	46.4323	1.70804	0.854020	+	0.520240i	$0.174158\pi$
$740$	−4.14721	−0.152454
$741$	−2.62593	−0.0964661
$742$	−29.5895	−1.08627
$743$	−50.7817	−1.86300	−0.931500	−	0.363743i	$-0.881499\pi$
$743$	−50.7817	−1.86300	−0.931500	+	0.363743i	$0.881499\pi$
$744$	−2.40830	−0.0882926
$745$	51.3898	1.88277
$746$	−33.1856	−1.21501
$747$	−16.6334	−0.608584
$748$	−26.3843	−0.964705
$749$	14.6739	0.536174
$750$	−3.72691	−0.136087
$751$	−2.06587	−0.0753846	−0.0376923	−	0.999289i	$-0.512001\pi$
$751$	−2.06587	−0.0753846	−0.0376923	+	0.999289i	$0.512001\pi$
$752$	15.9210	0.580581
$753$	5.22270	0.190326
$754$	14.9156	0.543194
$755$	46.5675	1.69477
$756$	12.8297	0.466611
$757$	−21.1921	−0.770242	−0.385121	−	0.922866i	$-0.625840\pi$
$757$	−21.1921	−0.770242	−0.385121	+	0.922866i	$0.625840\pi$
$758$	8.00546	0.290771
$759$	4.37338	0.158744
$760$	5.95159	0.215887
$761$	−20.7948	−0.753810	−0.376905	−	0.926252i	$-0.623012\pi$
$761$	−20.7948	−0.753810	−0.376905	+	0.926252i	$0.623012\pi$
$762$	−4.09898	−0.148490
$763$	76.3821	2.76522
$764$	9.60352	0.347443
$765$	35.7344	1.29198
$766$	34.3734	1.24196
$767$	−14.4105	−0.520332
$768$	−7.64300	−0.275793
$769$	−22.1976	−0.800466	−0.400233	−	0.916413i	$-0.631071\pi$
$769$	−22.1976	−0.800466	−0.400233	+	0.916413i	$0.631071\pi$
$770$	100.527	3.62274
$771$	0.188858	0.00680154
$772$	−15.7562	−0.567078
$773$	13.6135	0.489645	0.244822	−	0.969568i	$-0.421270\pi$
$773$	13.6135	0.489645	0.244822	+	0.969568i	$0.421270\pi$
$774$	59.0683	2.12316
$775$	−17.1921	−0.617560
$776$	7.07133	0.253846
$777$	−1.71867	−0.0616568
$778$	−10.8846	−0.390233
$779$	15.1263	0.541955
$780$	5.63754	0.201856
$781$	−4.39738	−0.157351
$782$	22.0109	0.787109
$783$	4.34699	0.155349
$784$	−60.0462	−2.14451
$785$	−3.69596	−0.131915
$786$	−3.27566	−0.116839
$787$	5.58060	0.198927	0.0994635	−	0.995041i	$-0.468287\pi$
$787$	5.58060	0.198927	0.0994635	+	0.995041i	$0.468287\pi$
$788$	−10.0109	−0.356624
$789$	7.90666	0.281484
$790$	54.9429	1.95478
$791$	81.4796	2.89708
$792$	−14.6980	−0.522269
$793$	−33.8530	−1.20216
$794$	−44.3843	−1.57514
$795$	−3.78931	−0.134393
$796$	−3.90467	−0.138397
$797$	48.1437	1.70534	0.852669	−	0.522451i	$-0.174982\pi$
$797$	48.1437	1.70534	0.852669	+	0.522451i	$0.174982\pi$
$798$	−5.29511	−0.187445
$799$	−14.3843	−0.508880
$800$	−20.0055	−0.707300
$801$	−1.46018	−0.0515928
$802$	71.0342	2.50831
$803$	−12.4633	−0.439819
$804$	−1.65938	−0.0585217
$805$	−34.0109	−1.19873
$806$	−41.5895	−1.46493
$807$	6.99454	0.246220
$808$	11.9660	0.420962
$809$	−19.1921	−0.674760	−0.337380	−	0.941369i	$-0.609541\pi$
$809$	−19.1921	−0.674760	−0.337380	+	0.941369i	$0.609541\pi$
$810$	−40.6654	−1.42884
$811$	53.7477	1.88734	0.943668	−	0.330894i	$-0.107350\pi$
$811$	53.7477	1.88734	0.943668	+	0.330894i	$0.107350\pi$
$812$	12.1976	0.428052
$813$	−2.78276	−0.0975957
$814$	−8.64994	−0.303180
$815$	21.9210	0.767861
$816$	7.80022	0.273062
$817$	−20.2316	−0.707815
$818$	25.4982	0.891523
$819$	−50.4323	−1.76225
$820$	−32.4742	−1.13405
$821$	26.1328	0.912042	0.456021	−	0.889969i	$-0.349274\pi$
$821$	26.1328	0.912042	0.456021	+	0.889969i	$0.349274\pi$
$822$	−4.94870	−0.172606
$823$	23.6554	0.824575	0.412288	−	0.911054i	$-0.364730\pi$
$823$	23.6554	0.824575	0.412288	+	0.911054i	$0.364730\pi$
$824$	3.38845	0.118042
$825$	4.86063	0.169226
$826$	−29.0582	−1.01107
$827$	−28.8606	−1.00358	−0.501791	−	0.864989i	$-0.667325\pi$
$827$	−28.8606	−1.00358	−0.501791	+	0.864989i	$0.667325\pi$
$828$	−10.6758	−0.371008
$829$	−23.3250	−0.810110	−0.405055	−	0.914292i	$-0.632748\pi$
$829$	−23.3250	−0.810110	−0.405055	+	0.914292i	$0.632748\pi$
$830$	−30.1592	−1.04684
$831$	−7.09096	−0.245983
$832$	−9.45779	−0.327890
$833$	54.2504	1.87966
$834$	−5.83515	−0.202055
$835$	−36.1887	−1.25236
$836$	−10.8079	−0.373797
$837$	−12.1208	−0.418957
$838$	−15.4633	−0.534169
$839$	−18.6201	−0.642837	−0.321418	−	0.946937i	$-0.604160\pi$
$839$	−18.6201	−0.642837	−0.321418	+	0.946937i	$0.604160\pi$
$840$	−5.29511	−0.182699
$841$	−24.8672	−0.857489
$842$	−46.3734	−1.59813
$843$	0.289651	0.00997609
$844$	6.19761	0.213330
$845$	−8.50273	−0.292503
$846$	17.2031	0.591454
$847$	36.6609	1.25968
$848$	17.8552	0.613149
$849$	10.5710	0.362795
$850$	24.4633	0.839083
$851$	2.92650	0.100319
$852$	−0.497270	−0.0170362
$853$	8.99653	0.308035	0.154018	−	0.988068i	$-0.450779\pi$
$853$	8.99653	0.308035	0.154018	+	0.988068i	$0.450779\pi$
$854$	−68.2635	−2.33593
$855$	14.6379	0.500607
$856$	−3.89010	−0.132961
$857$	−40.2196	−1.37388	−0.686938	−	0.726716i	$-0.741046\pi$
$857$	−40.2196	−1.37388	−0.686938	+	0.726716i	$0.741046\pi$
$858$	11.7584	0.401424
$859$	−7.32389	−0.249888	−0.124944	−	0.992164i	$-0.539875\pi$
$859$	−7.32389	−0.249888	−0.124944	+	0.992164i	$0.539875\pi$
$860$	43.4347	1.48111
$861$	−13.4578	−0.458640
$862$	−0.0988033	−0.00336525
$863$	−12.7267	−0.433223	−0.216611	−	0.976258i	$-0.569500\pi$
$863$	−12.7267	−0.433223	−0.216611	+	0.976258i	$0.569500\pi$
$864$	−14.1043	−0.479837
$865$	6.19761	0.210725
$866$	−45.0364	−1.53040
$867$	−0.851705	−0.0289254
$868$	−34.0109	−1.15441
$869$	46.4742	1.57653
$870$	3.85171	0.130585
$871$	13.3479	0.452276
$872$	−20.2491	−0.685721
$873$	17.3919	0.588627
$874$	9.01638	0.304984
$875$	24.5160	0.828793
$876$	−1.40939	−0.0476187
$877$	23.5762	0.796113	0.398056	−	0.917361i	$-0.369685\pi$
$877$	23.5762	0.796113	0.398056	+	0.917361i	$0.369685\pi$
$878$	−51.2140	−1.72839
$879$	−1.54984	−0.0522747
$880$	−60.6609	−2.04488
$881$	21.2975	0.717531	0.358765	−	0.933428i	$-0.383198\pi$
$881$	21.2975	0.717531	0.358765	+	0.933428i	$0.383198\pi$
$882$	−64.8814	−2.18467
$883$	16.3734	0.551008	0.275504	−	0.961300i	$-0.411155\pi$
$883$	16.3734	0.551008	0.275504	+	0.961300i	$0.411155\pi$
$884$	24.0000	0.807207
$885$	−3.72127	−0.125089
$886$	62.1187	2.08692
$887$	−23.1791	−0.778277	−0.389138	−	0.921179i	$-0.627227\pi$
$887$	−23.1791	−0.778277	−0.389138	+	0.921179i	$0.627227\pi$
$888$	0.455623	0.0152897
$889$	26.9636	0.904330
$890$	−2.64755	−0.0887462
$891$	−34.3974	−1.15236
$892$	−25.5531	−0.855582
$893$	−5.89227	−0.197177
$894$	12.1208	0.405381
$895$	7.50381	0.250825
$896$	38.9396	1.30088
$897$	−3.97816	−0.132827
$898$	−16.3974	−0.547188
$899$	−11.5237	−0.384336
$900$	−11.8652	−0.395506
$901$	−16.1317	−0.537426
$902$	−67.7322	−2.25524
$903$	18.0000	0.599002
$904$	−21.6004	−0.718420
$905$	−67.2689	−2.23610
$906$	10.9834	0.364900
$907$	8.34460	0.277078	0.138539	−	0.990357i	$-0.455759\pi$
$907$	8.34460	0.277078	0.138539	+	0.990357i	$0.455759\pi$
$908$	27.2711	0.905023
$909$	29.4303	0.976142
$910$	−91.4425	−3.03129
$911$	7.50580	0.248678	0.124339	−	0.992240i	$-0.460319\pi$
$911$	7.50580	0.248678	0.124339	+	0.992240i	$0.460319\pi$
$912$	3.19522	0.105804
$913$	−25.5106	−0.844277
$914$	37.8970	1.25352
$915$	−8.74198	−0.289001
$916$	33.3250	1.10109
$917$	21.5477	0.711567
$918$	17.2471	0.569239
$919$	12.1976	0.402362	0.201181	−	0.979554i	$-0.435522\pi$
$919$	12.1976	0.402362	0.201181	+	0.979554i	$0.435522\pi$
$920$	9.01638	0.297261
$921$	2.52037	0.0830491
$922$	15.7893	0.519993
$923$	4.00000	0.131662
$924$	9.61571	0.316333
$925$	3.25256	0.106943
$926$	−37.3150	−1.22625
$927$	8.33390	0.273721
$928$	−13.4094	−0.440185
$929$	−14.9756	−0.491333	−0.245667	−	0.969354i	$-0.579007\pi$
$929$	−14.9756	−0.491333	−0.245667	+	0.969354i	$0.579007\pi$
$930$	−10.7398	−0.352172
$931$	22.2227	0.728319
$932$	−37.4633	−1.22715
$933$	10.2722	0.336296
$934$	11.8792	0.388699
$935$	54.8057	1.79234
$936$	13.3697	0.437003
$937$	29.9869	0.979630	0.489815	−	0.871826i	$-0.337064\pi$
$937$	29.9869	0.979630	0.489815	+	0.871826i	$0.337064\pi$
$938$	26.9156	0.878825
$939$	−11.2227	−0.366239
$940$	12.6499	0.412596
$941$	60.9145	1.98576	0.992878	−	0.119136i	$-0.0380124\pi$
$941$	60.9145	1.98576	0.992878	+	0.119136i	$0.0380124\pi$
$942$	−0.871732	−0.0284026
$943$	22.9156	0.746234
$944$	17.5346	0.570702
$945$	−26.6499	−0.866923
$946$	90.5928	2.94543
$947$	−0.0263906	−0.000857581 0	−0.000428790	−	1.00000i	$-0.500136\pi$
$947$	−0.0263906	−0.000857581 0	−0.000428790		1.00000i	$0.500136\pi$
$948$	5.25545	0.170689
$949$	11.3370	0.368014
$950$	10.0209	0.325122
$951$	3.52366	0.114263
$952$	−22.5422	−0.730597
$953$	−11.5762	−0.374991	−0.187495	−	0.982265i	$-0.560037\pi$
$953$	−11.5762	−0.374991	−0.187495	+	0.982265i	$0.560037\pi$
$954$	19.2929	0.624632
$955$	−19.9485	−0.645519
$956$	−25.8421	−0.835793
$957$	3.25802	0.105317
$958$	−22.3734	−0.722851
$959$	32.5531	1.05120
$960$	−2.44232	−0.0788255
$961$	1.13174	0.0365077
$962$	7.86826	0.253683
$963$	−9.56769	−0.308315
$964$	28.2835	0.910950
$965$	32.7289	1.05358
$966$	−8.02184	−0.258098
$967$	−46.7398	−1.50305	−0.751526	−	0.659704i	$-0.770682\pi$
$967$	−46.7398	−1.50305	−0.751526	+	0.659704i	$0.770682\pi$
$968$	−9.71888	−0.312377
$969$	−2.88681	−0.0927377
$970$	31.5346	1.01251
$971$	32.4214	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$971$	32.4214	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$972$	−12.6425	−0.405508
$973$	38.3843	1.23054
$974$	53.4447	1.71248
$975$	−4.42139	−0.141598
$976$	41.1921	1.31853
$977$	−47.2491	−1.51163	−0.755816	−	0.654784i	$-0.772760\pi$
$977$	−47.2491	−1.51163	−0.755816	+	0.654784i	$0.772760\pi$
$978$	5.17031	0.165328
$979$	−2.23947	−0.0715738
$980$	−47.7093	−1.52402
$981$	−49.8026	−1.59007
$982$	9.18123	0.292985
$983$	5.16229	0.164651	0.0823257	−	0.996605i	$-0.473765\pi$
$983$	5.16229	0.164651	0.0823257	+	0.996605i	$0.473765\pi$
$984$	3.56769	0.113734
$985$	20.7948	0.662576
$986$	16.3974	0.522199
$987$	5.24233	0.166865
$988$	9.83117	0.312771
$989$	−30.6499	−0.974611
$990$	−65.5455	−2.08317
$991$	−16.2656	−0.516695	−0.258348	−	0.966052i	$-0.583178\pi$
$991$	−16.2656	−0.516695	−0.258348	+	0.966052i	$0.583178\pi$
$992$	37.3898	1.18713
$993$	5.97122	0.189491
$994$	8.06587	0.255834
$995$	8.11081	0.257130
$996$	−2.88482	−0.0914090
$997$	−41.8026	−1.32390	−0.661951	−	0.749547i	$-0.730271\pi$
$997$	−41.8026	−1.32390	−0.661951	+	0.749547i	$0.730271\pi$
$998$	30.3788	0.961625
$999$	2.29312	0.0725511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.a.b.1.3	✓	3
3.2	odd	2		639.2.a.g.1.1		3
4.3	odd	2		1136.2.a.i.1.2		3
5.4	even	2		1775.2.a.e.1.1		3
7.6	odd	2		3479.2.a.l.1.3		3
8.3	odd	2		4544.2.a.t.1.2		3
8.5	even	2		4544.2.a.w.1.2		3
11.10	odd	2		8591.2.a.f.1.1		3
71.70	odd	2		5041.2.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.a.b.1.3	✓	3	1.1	even	1	trivial
639.2.a.g.1.1		3	3.2	odd	2
1136.2.a.i.1.2		3	4.3	odd	2
1775.2.a.e.1.1		3	5.4	even	2
3479.2.a.l.1.3		3	7.6	odd	2
4544.2.a.t.1.2		3	8.3	odd	2
4544.2.a.w.1.2		3	8.5	even	2
5041.2.a.b.1.3		3	71.70	odd	2
8591.2.a.f.1.1		3	11.10	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:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	71.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83424 q^{2} -0.364448 q^{3} +1.36445 q^{4} -2.83424 q^{5} -0.668486 q^{6} +4.39738 q^{7} -1.16576 q^{8} -2.86718 q^{9} +O(q^{10})\)	\(q+1.83424 q^{2} -0.364448 q^{3} +1.36445 q^{4} -2.83424 q^{5} -0.668486 q^{6} +4.39738 q^{7} -1.16576 q^{8} -2.86718 q^{9} -5.19869 q^{10} -4.39738 q^{11} -0.497270 q^{12} +4.00000 q^{13} +8.06587 q^{14} +1.03293 q^{15} -4.86718 q^{16} +4.39738 q^{17} -5.25910 q^{18} +1.80131 q^{19} -3.86718 q^{20} -1.60262 q^{21} -8.06587 q^{22} +2.72890 q^{23} +0.424858 q^{24} +3.03293 q^{25} +7.33697 q^{26} +2.13828 q^{27} +6.00000 q^{28} +2.03293 q^{29} +1.89465 q^{30} -5.66849 q^{31} -6.59607 q^{32} +1.60262 q^{33} +8.06587 q^{34} -12.4633 q^{35} -3.91211 q^{36} +1.07241 q^{37} +3.30404 q^{38} -1.45779 q^{39} +3.30404 q^{40} +8.39738 q^{41} -2.93959 q^{42} -11.2316 q^{43} -6.00000 q^{44} +8.12628 q^{45} +5.00546 q^{46} -3.27110 q^{47} +1.77383 q^{48} +12.3370 q^{49} +5.56314 q^{50} -1.60262 q^{51} +5.45779 q^{52} -3.66849 q^{53} +3.92213 q^{54} +12.4633 q^{55} -5.12628 q^{56} -0.656483 q^{57} +3.72890 q^{58} -3.60262 q^{59} +1.40939 q^{60} -8.46325 q^{61} -10.3974 q^{62} -12.6081 q^{63} -2.36445 q^{64} -11.3370 q^{65} +2.93959 q^{66} +3.33697 q^{67} +6.00000 q^{68} -0.994541 q^{69} -22.8606 q^{70} +1.00000 q^{71} +3.34243 q^{72} +2.83424 q^{73} +1.96707 q^{74} -1.10535 q^{75} +2.45779 q^{76} -19.3370 q^{77} -2.67395 q^{78} -10.5686 q^{79} +13.7948 q^{80} +7.82224 q^{81} +15.4028 q^{82} +5.80131 q^{83} -2.18669 q^{84} -12.4633 q^{85} -20.6015 q^{86} -0.740899 q^{87} +5.12628 q^{88} +0.509273 q^{89} +14.9056 q^{90} +17.5895 q^{91} +3.72344 q^{92} +2.06587 q^{93} -6.00000 q^{94} -5.10535 q^{95} +2.40393 q^{96} -6.06587 q^{97} +22.6290 q^{98} +12.6081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 4 q^{4} - 3 q^{5} + 9 q^{6} + 2 q^{7} - 9 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q - q^3 + 4 * q^4 - 3 * q^5 + 9 * q^6 + 2 * q^7 - 9 * q^8 + 8 * q^9	\( 3 q - q^{3} + 4 q^{4} - 3 q^{5} + 9 q^{6} + 2 q^{7} - 9 q^{8} + 8 q^{9} - 10 q^{10} - 2 q^{11} - 18 q^{12} + 12 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} + 2 q^{17} - 21 q^{18} + 11 q^{19} + 5 q^{20} - 16 q^{21} - 2 q^{22} + 8 q^{23} + 12 q^{24} - 2 q^{25} - 10 q^{27} + 18 q^{28} - 5 q^{29} + 11 q^{30} - 6 q^{31} - 3 q^{32} + 16 q^{33} + 2 q^{34} - 4 q^{35} + 21 q^{36} + 9 q^{37} - q^{38} - 4 q^{39} - q^{40} + 14 q^{41} + 2 q^{42} - 17 q^{43} - 18 q^{44} + 13 q^{45} - 18 q^{46} - 10 q^{47} - 11 q^{48} + 15 q^{49} + 11 q^{50} - 16 q^{51} + 16 q^{52} + 39 q^{54} + 4 q^{55} - 4 q^{56} + 4 q^{57} + 11 q^{58} - 22 q^{59} - 12 q^{60} + 8 q^{61} - 20 q^{62} - 16 q^{63} - 7 q^{64} - 12 q^{65} - 2 q^{66} - 12 q^{67} + 18 q^{68} - 36 q^{69} - 24 q^{70} + 3 q^{71} - 45 q^{72} + 3 q^{73} + 17 q^{74} + 2 q^{75} + 7 q^{76} - 36 q^{77} + 36 q^{78} + 7 q^{79} + 19 q^{80} + 23 q^{81} + 2 q^{82} + 23 q^{83} - 6 q^{84} - 4 q^{85} - 12 q^{86} + 3 q^{87} + 4 q^{88} + 13 q^{89} - 16 q^{90} + 8 q^{91} + 44 q^{92} - 16 q^{93} - 18 q^{94} - 10 q^{95} + 24 q^{96} + 4 q^{97} + 40 q^{98} + 16 q^{99}+O(q^{100}) \) 3 * q - q^3 + 4 * q^4 - 3 * q^5 + 9 * q^6 + 2 * q^7 - 9 * q^8 + 8 * q^9 - 10 * q^10 - 2 * q^11 - 18 * q^12 + 12 * q^13 + 2 * q^14 - 8 * q^15 + 2 * q^16 + 2 * q^17 - 21 * q^18 + 11 * q^19 + 5 * q^20 - 16 * q^21 - 2 * q^22 + 8 * q^23 + 12 * q^24 - 2 * q^25 - 10 * q^27 + 18 * q^28 - 5 * q^29 + 11 * q^30 - 6 * q^31 - 3 * q^32 + 16 * q^33 + 2 * q^34 - 4 * q^35 + 21 * q^36 + 9 * q^37 - q^38 - 4 * q^39 - q^40 + 14 * q^41 + 2 * q^42 - 17 * q^43 - 18 * q^44 + 13 * q^45 - 18 * q^46 - 10 * q^47 - 11 * q^48 + 15 * q^49 + 11 * q^50 - 16 * q^51 + 16 * q^52 + 39 * q^54 + 4 * q^55 - 4 * q^56 + 4 * q^57 + 11 * q^58 - 22 * q^59 - 12 * q^60 + 8 * q^61 - 20 * q^62 - 16 * q^63 - 7 * q^64 - 12 * q^65 - 2 * q^66 - 12 * q^67 + 18 * q^68 - 36 * q^69 - 24 * q^70 + 3 * q^71 - 45 * q^72 + 3 * q^73 + 17 * q^74 + 2 * q^75 + 7 * q^76 - 36 * q^77 + 36 * q^78 + 7 * q^79 + 19 * q^80 + 23 * q^81 + 2 * q^82 + 23 * q^83 - 6 * q^84 - 4 * q^85 - 12 * q^86 + 3 * q^87 + 4 * q^88 + 13 * q^89 - 16 * q^90 + 8 * q^91 + 44 * q^92 - 16 * q^93 - 18 * q^94 - 10 * q^95 + 24 * q^96 + 4 * q^97 + 40 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more