LMFDB - Embedded newform 429.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(1,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("429.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.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:	$3.42558224671$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	429.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} -4.73205 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.46410 q^{14} +2.73205 q^{15} -5.00000 q^{16} +3.26795 q^{17} +1.73205 q^{18} -7.46410 q^{19} -2.73205 q^{20} +2.00000 q^{21} -1.73205 q^{22} -2.00000 q^{23} +1.73205 q^{24} +2.46410 q^{25} -1.73205 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.19615 q^{29} +4.73205 q^{30} +2.19615 q^{31} -5.19615 q^{32} +1.00000 q^{33} +5.66025 q^{34} +5.46410 q^{35} +1.00000 q^{36} -2.00000 q^{37} -12.9282 q^{38} +1.00000 q^{39} +4.73205 q^{40} +1.46410 q^{41} +3.46410 q^{42} +4.19615 q^{43} -1.00000 q^{44} -2.73205 q^{45} -3.46410 q^{46} +5.46410 q^{47} +5.00000 q^{48} -3.00000 q^{49} +4.26795 q^{50} -3.26795 q^{51} -1.00000 q^{52} +2.00000 q^{53} -1.73205 q^{54} +2.73205 q^{55} +3.46410 q^{56} +7.46410 q^{57} +10.7321 q^{58} -6.53590 q^{59} +2.73205 q^{60} -8.92820 q^{61} +3.80385 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.73205 q^{65} +1.73205 q^{66} -1.80385 q^{67} +3.26795 q^{68} +2.00000 q^{69} +9.46410 q^{70} -6.92820 q^{71} -1.73205 q^{72} -10.9282 q^{73} -3.46410 q^{74} -2.46410 q^{75} -7.46410 q^{76} +2.00000 q^{77} +1.73205 q^{78} +13.6603 q^{79} +13.6603 q^{80} +1.00000 q^{81} +2.53590 q^{82} -13.8564 q^{83} +2.00000 q^{84} -8.92820 q^{85} +7.26795 q^{86} -6.19615 q^{87} +1.73205 q^{88} +1.66025 q^{89} -4.73205 q^{90} +2.00000 q^{91} -2.00000 q^{92} -2.19615 q^{93} +9.46410 q^{94} +20.3923 q^{95} +5.19615 q^{96} -4.92820 q^{97} -5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{9} - 6 q^{10} - 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{15} - 10 q^{16} + 10 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{21} - 4 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{28} + 2 q^{29} + 6 q^{30} - 6 q^{31} + 2 q^{33} - 6 q^{34} + 4 q^{35} + 2 q^{36} - 4 q^{37} - 12 q^{38} + 2 q^{39} + 6 q^{40} - 4 q^{41} - 2 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{47} + 10 q^{48} - 6 q^{49} + 12 q^{50} - 10 q^{51} - 2 q^{52} + 4 q^{53} + 2 q^{55} + 8 q^{57} + 18 q^{58} - 20 q^{59} + 2 q^{60} - 4 q^{61} + 18 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 14 q^{67} + 10 q^{68} + 4 q^{69} + 12 q^{70} - 8 q^{73} + 2 q^{75} - 8 q^{76} + 4 q^{77} + 10 q^{79} + 10 q^{80} + 2 q^{81} + 12 q^{82} + 4 q^{84} - 4 q^{85} + 18 q^{86} - 2 q^{87} - 14 q^{89} - 6 q^{90} + 4 q^{91} - 4 q^{92} + 6 q^{93} + 12 q^{94} + 20 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^9 - 6 * q^10 - 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^15 - 10 * q^16 + 10 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^21 - 4 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^28 + 2 * q^29 + 6 * q^30 - 6 * q^31 + 2 * q^33 - 6 * q^34 + 4 * q^35 + 2 * q^36 - 4 * q^37 - 12 * q^38 + 2 * q^39 + 6 * q^40 - 4 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^45 + 4 * q^47 + 10 * q^48 - 6 * q^49 + 12 * q^50 - 10 * q^51 - 2 * q^52 + 4 * q^53 + 2 * q^55 + 8 * q^57 + 18 * q^58 - 20 * q^59 + 2 * q^60 - 4 * q^61 + 18 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 14 * q^67 + 10 * q^68 + 4 * q^69 + 12 * q^70 - 8 * q^73 + 2 * q^75 - 8 * q^76 + 4 * q^77 + 10 * q^79 + 10 * q^80 + 2 * q^81 + 12 * q^82 + 4 * q^84 - 4 * q^85 + 18 * q^86 - 2 * q^87 - 14 * q^89 - 6 * q^90 + 4 * q^91 - 4 * q^92 + 6 * q^93 + 12 * q^94 + 20 * q^95 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	−1.73205	−0.707107
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	−4.73205	−1.49641
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.46410	−0.925820
$15$	2.73205	0.705412
$16$	−5.00000	−1.25000
$17$	3.26795	0.792594	0.396297	−	0.918122i	$-0.370295\pi$
$17$	3.26795	0.792594	0.396297	+	0.918122i	$0.370295\pi$
$18$	1.73205	0.408248
$19$	−7.46410	−1.71238	−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410	−1.71238	−0.856191	+	0.516659i	$0.827175\pi$
$20$	−2.73205	−0.610905
$21$	2.00000	0.436436
$22$	−1.73205	−0.369274
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	1.73205	0.353553
$25$	2.46410	0.492820
$26$	−1.73205	−0.339683
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	6.19615	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$29$	6.19615	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$30$	4.73205	0.863950
$31$	2.19615	0.394441	0.197220	−	0.980359i	$-0.436809\pi$
$31$	2.19615	0.394441	0.197220	+	0.980359i	$0.436809\pi$
$32$	−5.19615	−0.918559
$33$	1.00000	0.174078
$34$	5.66025	0.970726
$35$	5.46410	0.923602
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−12.9282	−2.09723
$39$	1.00000	0.160128
$40$	4.73205	0.748203
$41$	1.46410	0.228654	0.114327	−	0.993443i	$-0.463529\pi$
$41$	1.46410	0.228654	0.114327	+	0.993443i	$0.463529\pi$
$42$	3.46410	0.534522
$43$	4.19615	0.639907	0.319954	−	0.947433i	$-0.396333\pi$
$43$	4.19615	0.639907	0.319954	+	0.947433i	$0.396333\pi$
$44$	−1.00000	−0.150756
$45$	−2.73205	−0.407270
$46$	−3.46410	−0.510754
$47$	5.46410	0.797021	0.398511	−	0.917164i	$-0.369527\pi$
$47$	5.46410	0.797021	0.398511	+	0.917164i	$0.369527\pi$
$48$	5.00000	0.721688
$49$	−3.00000	−0.428571
$50$	4.26795	0.603579
$51$	−3.26795	−0.457604
$52$	−1.00000	−0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.73205	−0.235702
$55$	2.73205	0.368390
$56$	3.46410	0.462910
$57$	7.46410	0.988644
$58$	10.7321	1.40919
$59$	−6.53590	−0.850901	−0.425451	−	0.904982i	$-0.639884\pi$
$59$	−6.53590	−0.850901	−0.425451	+	0.904982i	$0.639884\pi$
$60$	2.73205	0.352706
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	3.80385	0.483089
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	2.73205	0.338869
$66$	1.73205	0.213201
$67$	−1.80385	−0.220375	−0.110188	−	0.993911i	$-0.535145\pi$
$67$	−1.80385	−0.220375	−0.110188	+	0.993911i	$0.535145\pi$
$68$	3.26795	0.396297
$69$	2.00000	0.240772
$70$	9.46410	1.13118
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	−1.73205	−0.204124
$73$	−10.9282	−1.27905	−0.639525	−	0.768771i	$-0.720869\pi$
$73$	−10.9282	−1.27905	−0.639525	+	0.768771i	$0.720869\pi$
$74$	−3.46410	−0.402694
$75$	−2.46410	−0.284530
$76$	−7.46410	−0.856191
$77$	2.00000	0.227921
$78$	1.73205	0.196116
$79$	13.6603	1.53690	0.768449	−	0.639911i	$-0.221029\pi$
$79$	13.6603	1.53690	0.768449	+	0.639911i	$0.221029\pi$
$80$	13.6603	1.52726
$81$	1.00000	0.111111
$82$	2.53590	0.280043
$83$	−13.8564	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$84$	2.00000	0.218218
$85$	−8.92820	−0.968400
$86$	7.26795	0.783723
$87$	−6.19615	−0.664297
$88$	1.73205	0.184637
$89$	1.66025	0.175987	0.0879933	−	0.996121i	$-0.471955\pi$
$89$	1.66025	0.175987	0.0879933	+	0.996121i	$0.471955\pi$
$90$	−4.73205	−0.498802
$91$	2.00000	0.209657
$92$	−2.00000	−0.208514
$93$	−2.19615	−0.227730
$94$	9.46410	0.976148
$95$	20.3923	2.09221
$96$	5.19615	0.530330
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	−5.19615	−0.524891
$99$	−1.00000	−0.100504
$100$	2.46410	0.246410
$101$	−2.19615	−0.218525	−0.109263	−	0.994013i	$-0.534849\pi$
$101$	−2.19615	−0.218525	−0.109263	+	0.994013i	$0.534849\pi$
$102$	−5.66025	−0.560449
$103$	−15.3205	−1.50957	−0.754787	−	0.655970i	$-0.772260\pi$
$103$	−15.3205	−1.50957	−0.754787	+	0.655970i	$0.772260\pi$
$104$	1.73205	0.169842
$105$	−5.46410	−0.533242
$106$	3.46410	0.336463
$107$	12.3923	1.19801	0.599005	−	0.800746i	$-0.295563\pi$
$107$	12.3923	1.19801	0.599005	+	0.800746i	$0.295563\pi$
$108$	−1.00000	−0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	4.73205	0.451183
$111$	2.00000	0.189832
$112$	10.0000	0.944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	12.9282	1.21084
$115$	5.46410	0.509530
$116$	6.19615	0.575298
$117$	−1.00000	−0.0924500
$118$	−11.3205	−1.04214
$119$	−6.53590	−0.599145
$120$	−4.73205	−0.431975
$121$	1.00000	0.0909091
$122$	−15.4641	−1.40005
$123$	−1.46410	−0.132014
$124$	2.19615	0.197220
$125$	6.92820	0.619677
$126$	−3.46410	−0.308607
$127$	−11.1244	−0.987127	−0.493563	−	0.869710i	$-0.664306\pi$
$127$	−11.1244	−0.987127	−0.493563	+	0.869710i	$0.664306\pi$
$128$	12.1244	1.07165
$129$	−4.19615	−0.369451
$130$	4.73205	0.415028
$131$	−18.9282	−1.65376	−0.826882	−	0.562375i	$-0.809888\pi$
$131$	−18.9282	−1.65376	−0.826882	+	0.562375i	$0.809888\pi$
$132$	1.00000	0.0870388
$133$	14.9282	1.29444
$134$	−3.12436	−0.269903
$135$	2.73205	0.235137
$136$	−5.66025	−0.485363
$137$	16.5885	1.41725	0.708624	−	0.705587i	$-0.249316\pi$
$137$	16.5885	1.41725	0.708624	+	0.705587i	$0.249316\pi$
$138$	3.46410	0.294884
$139$	−20.1962	−1.71302	−0.856508	−	0.516134i	$-0.827371\pi$
$139$	−20.1962	−1.71302	−0.856508	+	0.516134i	$0.827371\pi$
$140$	5.46410	0.461801
$141$	−5.46410	−0.460160
$142$	−12.0000	−1.00702
$143$	1.00000	0.0836242
$144$	−5.00000	−0.416667
$145$	−16.9282	−1.40581
$146$	−18.9282	−1.56651
$147$	3.00000	0.247436
$148$	−2.00000	−0.164399
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	−4.26795	−0.348477
$151$	−11.4641	−0.932935	−0.466468	−	0.884538i	$-0.654474\pi$
$151$	−11.4641	−0.932935	−0.466468	+	0.884538i	$0.654474\pi$
$152$	12.9282	1.04862
$153$	3.26795	0.264198
$154$	3.46410	0.279145
$155$	−6.00000	−0.481932
$156$	1.00000	0.0800641
$157$	19.3205	1.54194	0.770972	−	0.636869i	$-0.219771\pi$
$157$	19.3205	1.54194	0.770972	+	0.636869i	$0.219771\pi$
$158$	23.6603	1.88231
$159$	−2.00000	−0.158610
$160$	14.1962	1.12230
$161$	4.00000	0.315244
$162$	1.73205	0.136083
$163$	−20.7321	−1.62386	−0.811930	−	0.583755i	$-0.801583\pi$
$163$	−20.7321	−1.62386	−0.811930	+	0.583755i	$0.801583\pi$
$164$	1.46410	0.114327
$165$	−2.73205	−0.212690
$166$	−24.0000	−1.86276
$167$	17.8564	1.38177	0.690885	−	0.722965i	$-0.257221\pi$
$167$	17.8564	1.38177	0.690885	+	0.722965i	$0.257221\pi$
$168$	−3.46410	−0.267261
$169$	1.00000	0.0769231
$170$	−15.4641	−1.18604
$171$	−7.46410	−0.570794
$172$	4.19615	0.319954
$173$	−5.12436	−0.389598	−0.194799	−	0.980843i	$-0.562405\pi$
$173$	−5.12436	−0.389598	−0.194799	+	0.980843i	$0.562405\pi$
$174$	−10.7321	−0.813595
$175$	−4.92820	−0.372537
$176$	5.00000	0.376889
$177$	6.53590	0.491268
$178$	2.87564	0.215539
$179$	−19.8564	−1.48414	−0.742069	−	0.670324i	$-0.766155\pi$
$179$	−19.8564	−1.48414	−0.742069	+	0.670324i	$0.766155\pi$
$180$	−2.73205	−0.203635
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	3.46410	0.256776
$183$	8.92820	0.659992
$184$	3.46410	0.255377
$185$	5.46410	0.401729
$186$	−3.80385	−0.278912
$187$	−3.26795	−0.238976
$188$	5.46410	0.398511
$189$	2.00000	0.145479
$190$	35.3205	2.56242
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	−1.00000	−0.0721688
$193$	6.92820	0.498703	0.249351	−	0.968413i	$-0.419783\pi$
$193$	6.92820	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$194$	−8.53590	−0.612842
$195$	−2.73205	−0.195646
$196$	−3.00000	−0.214286
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−1.73205	−0.123091
$199$	17.4641	1.23800	0.618999	−	0.785392i	$-0.287539\pi$
$199$	17.4641	1.23800	0.618999	+	0.785392i	$0.287539\pi$
$200$	−4.26795	−0.301790
$201$	1.80385	0.127234
$202$	−3.80385	−0.267638
$203$	−12.3923	−0.869769
$204$	−3.26795	−0.228802
$205$	−4.00000	−0.279372
$206$	−26.5359	−1.84884
$207$	−2.00000	−0.139010
$208$	5.00000	0.346688
$209$	7.46410	0.516303
$210$	−9.46410	−0.653085
$211$	22.7321	1.56494	0.782469	−	0.622689i	$-0.213960\pi$
$211$	22.7321	1.56494	0.782469	+	0.622689i	$0.213960\pi$
$212$	2.00000	0.137361
$213$	6.92820	0.474713
$214$	21.4641	1.46726
$215$	−11.4641	−0.781845
$216$	1.73205	0.117851
$217$	−4.39230	−0.298169
$218$	−13.8564	−0.938474
$219$	10.9282	0.738460
$220$	2.73205	0.184195
$221$	−3.26795	−0.219826
$222$	3.46410	0.232495
$223$	21.5167	1.44086	0.720431	−	0.693527i	$-0.243944\pi$
$223$	21.5167	1.44086	0.720431	+	0.693527i	$0.243944\pi$
$224$	10.3923	0.694365
$225$	2.46410	0.164273
$226$	17.3205	1.15214
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	7.46410	0.494322
$229$	−8.53590	−0.564068	−0.282034	−	0.959404i	$-0.591009\pi$
$229$	−8.53590	−0.564068	−0.282034	+	0.959404i	$0.591009\pi$
$230$	9.46410	0.624044
$231$	−2.00000	−0.131590
$232$	−10.7321	−0.704594
$233$	−6.19615	−0.405923	−0.202962	−	0.979187i	$-0.565057\pi$
$233$	−6.19615	−0.405923	−0.202962	+	0.979187i	$0.565057\pi$
$234$	−1.73205	−0.113228
$235$	−14.9282	−0.973809
$236$	−6.53590	−0.425451
$237$	−13.6603	−0.887329
$238$	−11.3205	−0.733800
$239$	1.60770	0.103993	0.0519966	−	0.998647i	$-0.483442\pi$
$239$	1.60770	0.103993	0.0519966	+	0.998647i	$0.483442\pi$
$240$	−13.6603	−0.881766
$241$	22.7846	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$241$	22.7846	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$242$	1.73205	0.111340
$243$	−1.00000	−0.0641500
$244$	−8.92820	−0.571570
$245$	8.19615	0.523633
$246$	−2.53590	−0.161683
$247$	7.46410	0.474929
$248$	−3.80385	−0.241545
$249$	13.8564	0.878114
$250$	12.0000	0.758947
$251$	−30.9282	−1.95217	−0.976085	−	0.217387i	$-0.930247\pi$
$251$	−30.9282	−1.95217	−0.976085	+	0.217387i	$0.930247\pi$
$252$	−2.00000	−0.125988
$253$	2.00000	0.125739
$254$	−19.2679	−1.20898
$255$	8.92820	0.559106
$256$	19.0000	1.18750
$257$	−17.3205	−1.08042	−0.540212	−	0.841529i	$-0.681656\pi$
$257$	−17.3205	−1.08042	−0.540212	+	0.841529i	$0.681656\pi$
$258$	−7.26795	−0.452483
$259$	4.00000	0.248548
$260$	2.73205	0.169435
$261$	6.19615	0.383532
$262$	−32.7846	−2.02544
$263$	2.53590	0.156370	0.0781851	−	0.996939i	$-0.475087\pi$
$263$	2.53590	0.156370	0.0781851	+	0.996939i	$0.475087\pi$
$264$	−1.73205	−0.106600
$265$	−5.46410	−0.335657
$266$	25.8564	1.58536
$267$	−1.66025	−0.101606
$268$	−1.80385	−0.110188
$269$	17.3205	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$269$	17.3205	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$270$	4.73205	0.287983
$271$	−25.3205	−1.53811	−0.769056	−	0.639182i	$-0.779273\pi$
$271$	−25.3205	−1.53811	−0.769056	+	0.639182i	$0.779273\pi$
$272$	−16.3397	−0.990743
$273$	−2.00000	−0.121046
$274$	28.7321	1.73577
$275$	−2.46410	−0.148591
$276$	2.00000	0.120386
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−34.9808	−2.09801
$279$	2.19615	0.131480
$280$	−9.46410	−0.565588
$281$	−11.3205	−0.675325	−0.337662	−	0.941267i	$-0.609636\pi$
$281$	−11.3205	−0.675325	−0.337662	+	0.941267i	$0.609636\pi$
$282$	−9.46410	−0.563579
$283$	20.1962	1.20054	0.600268	−	0.799799i	$-0.295060\pi$
$283$	20.1962	1.20054	0.600268	+	0.799799i	$0.295060\pi$
$284$	−6.92820	−0.411113
$285$	−20.3923	−1.20794
$286$	1.73205	0.102418
$287$	−2.92820	−0.172846
$288$	−5.19615	−0.306186
$289$	−6.32051	−0.371795
$290$	−29.3205	−1.72176
$291$	4.92820	0.288896
$292$	−10.9282	−0.639525
$293$	26.9282	1.57316	0.786581	−	0.617487i	$-0.211849\pi$
$293$	26.9282	1.57316	0.786581	+	0.617487i	$0.211849\pi$
$294$	5.19615	0.303046
$295$	17.8564	1.03964
$296$	3.46410	0.201347
$297$	1.00000	0.0580259
$298$	13.8564	0.802680
$299$	2.00000	0.115663
$300$	−2.46410	−0.142265
$301$	−8.39230	−0.483724
$302$	−19.8564	−1.14261
$303$	2.19615	0.126166
$304$	37.3205	2.14048
$305$	24.3923	1.39670
$306$	5.66025	0.323575
$307$	−27.4641	−1.56746	−0.783730	−	0.621102i	$-0.786685\pi$
$307$	−27.4641	−1.56746	−0.783730	+	0.621102i	$0.786685\pi$
$308$	2.00000	0.113961
$309$	15.3205	0.871553
$310$	−10.3923	−0.590243
$311$	−16.9282	−0.959910	−0.479955	−	0.877293i	$-0.659347\pi$
$311$	−16.9282	−0.959910	−0.479955	+	0.877293i	$0.659347\pi$
$312$	−1.73205	−0.0980581
$313$	−2.53590	−0.143337	−0.0716687	−	0.997428i	$-0.522832\pi$
$313$	−2.53590	−0.143337	−0.0716687	+	0.997428i	$0.522832\pi$
$314$	33.4641	1.88849
$315$	5.46410	0.307867
$316$	13.6603	0.768449
$317$	−7.12436	−0.400144	−0.200072	−	0.979781i	$-0.564118\pi$
$317$	−7.12436	−0.400144	−0.200072	+	0.979781i	$0.564118\pi$
$318$	−3.46410	−0.194257
$319$	−6.19615	−0.346918
$320$	−2.73205	−0.152726
$321$	−12.3923	−0.691671
$322$	6.92820	0.386094
$323$	−24.3923	−1.35722
$324$	1.00000	0.0555556
$325$	−2.46410	−0.136684
$326$	−35.9090	−1.98881
$327$	8.00000	0.442401
$328$	−2.53590	−0.140022
$329$	−10.9282	−0.602491
$330$	−4.73205	−0.260491
$331$	−21.5167	−1.18266	−0.591331	−	0.806429i	$-0.701397\pi$
$331$	−21.5167	−1.18266	−0.591331	+	0.806429i	$0.701397\pi$
$332$	−13.8564	−0.760469
$333$	−2.00000	−0.109599
$334$	30.9282	1.69232
$335$	4.92820	0.269257
$336$	−10.0000	−0.545545
$337$	−25.3205	−1.37930	−0.689648	−	0.724145i	$-0.742235\pi$
$337$	−25.3205	−1.37930	−0.689648	+	0.724145i	$0.742235\pi$
$338$	1.73205	0.0942111
$339$	−10.0000	−0.543125
$340$	−8.92820	−0.484200
$341$	−2.19615	−0.118928
$342$	−12.9282	−0.699077
$343$	20.0000	1.07990
$344$	−7.26795	−0.391862
$345$	−5.46410	−0.294177
$346$	−8.87564	−0.477158
$347$	16.3923	0.879985	0.439993	−	0.898001i	$-0.354981\pi$
$347$	16.3923	0.879985	0.439993	+	0.898001i	$0.354981\pi$
$348$	−6.19615	−0.332149
$349$	23.8564	1.27700	0.638502	−	0.769620i	$-0.279554\pi$
$349$	23.8564	1.27700	0.638502	+	0.769620i	$0.279554\pi$
$350$	−8.53590	−0.456263
$351$	1.00000	0.0533761
$352$	5.19615	0.276956
$353$	−28.9808	−1.54249	−0.771245	−	0.636538i	$-0.780366\pi$
$353$	−28.9808	−1.54249	−0.771245	+	0.636538i	$0.780366\pi$
$354$	11.3205	0.601678
$355$	18.9282	1.00460
$356$	1.66025	0.0879933
$357$	6.53590	0.345916
$358$	−34.3923	−1.81769
$359$	−23.4641	−1.23839	−0.619194	−	0.785238i	$-0.712541\pi$
$359$	−23.4641	−1.23839	−0.619194	+	0.785238i	$0.712541\pi$
$360$	4.73205	0.249401
$361$	36.7128	1.93225
$362$	−21.4641	−1.12813
$363$	−1.00000	−0.0524864
$364$	2.00000	0.104828
$365$	29.8564	1.56276
$366$	15.4641	0.808322
$367$	17.0718	0.891141	0.445570	−	0.895247i	$-0.353001\pi$
$367$	17.0718	0.891141	0.445570	+	0.895247i	$0.353001\pi$
$368$	10.0000	0.521286
$369$	1.46410	0.0762181
$370$	9.46410	0.492015
$371$	−4.00000	−0.207670
$372$	−2.19615	−0.113865
$373$	32.2487	1.66977	0.834887	−	0.550421i	$-0.185533\pi$
$373$	32.2487	1.66977	0.834887	+	0.550421i	$0.185533\pi$
$374$	−5.66025	−0.292685
$375$	−6.92820	−0.357771
$376$	−9.46410	−0.488074
$377$	−6.19615	−0.319118
$378$	3.46410	0.178174
$379$	1.80385	0.0926574	0.0463287	−	0.998926i	$-0.485248\pi$
$379$	1.80385	0.0926574	0.0463287	+	0.998926i	$0.485248\pi$
$380$	20.3923	1.04610
$381$	11.1244	0.569918
$382$	−8.78461	−0.449460
$383$	3.60770	0.184345	0.0921723	−	0.995743i	$-0.470619\pi$
$383$	3.60770	0.184345	0.0921723	+	0.995743i	$0.470619\pi$
$384$	−12.1244	−0.618718
$385$	−5.46410	−0.278476
$386$	12.0000	0.610784
$387$	4.19615	0.213302
$388$	−4.92820	−0.250192
$389$	15.4641	0.784061	0.392031	−	0.919952i	$-0.371773\pi$
$389$	15.4641	0.784061	0.392031	+	0.919952i	$0.371773\pi$
$390$	−4.73205	−0.239617
$391$	−6.53590	−0.330535
$392$	5.19615	0.262445
$393$	18.9282	0.954802
$394$	13.8564	0.698076
$395$	−37.3205	−1.87780
$396$	−1.00000	−0.0502519
$397$	11.0718	0.555678	0.277839	−	0.960628i	$-0.410382\pi$
$397$	11.0718	0.555678	0.277839	+	0.960628i	$0.410382\pi$
$398$	30.2487	1.51623
$399$	−14.9282	−0.747345
$400$	−12.3205	−0.616025
$401$	15.8038	0.789206	0.394603	−	0.918852i	$-0.370882\pi$
$401$	15.8038	0.789206	0.394603	+	0.918852i	$0.370882\pi$
$402$	3.12436	0.155829
$403$	−2.19615	−0.109398
$404$	−2.19615	−0.109263
$405$	−2.73205	−0.135757
$406$	−21.4641	−1.06525
$407$	2.00000	0.0991363
$408$	5.66025	0.280224
$409$	27.7128	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$409$	27.7128	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$410$	−6.92820	−0.342160
$411$	−16.5885	−0.818248
$412$	−15.3205	−0.754787
$413$	13.0718	0.643221
$414$	−3.46410	−0.170251
$415$	37.8564	1.85830
$416$	5.19615	0.254762
$417$	20.1962	0.989010
$418$	12.9282	0.632339
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−5.46410	−0.266621
$421$	−4.92820	−0.240186	−0.120093	−	0.992763i	$-0.538319\pi$
$421$	−4.92820	−0.240186	−0.120093	+	0.992763i	$0.538319\pi$
$422$	39.3731	1.91665
$423$	5.46410	0.265674
$424$	−3.46410	−0.168232
$425$	8.05256	0.390606
$426$	12.0000	0.581402
$427$	17.8564	0.864132
$428$	12.3923	0.599005
$429$	−1.00000	−0.0482805
$430$	−19.8564	−0.957561
$431$	28.7846	1.38651	0.693253	−	0.720694i	$-0.256177\pi$
$431$	28.7846	1.38651	0.693253	+	0.720694i	$0.256177\pi$
$432$	5.00000	0.240563
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−7.60770	−0.365181
$435$	16.9282	0.811645
$436$	−8.00000	−0.383131
$437$	14.9282	0.714113
$438$	18.9282	0.904425
$439$	35.9090	1.71384	0.856921	−	0.515448i	$-0.172375\pi$
$439$	35.9090	1.71384	0.856921	+	0.515448i	$0.172375\pi$
$440$	−4.73205	−0.225592
$441$	−3.00000	−0.142857
$442$	−5.66025	−0.269231
$443$	36.7846	1.74769	0.873845	−	0.486205i	$-0.161619\pi$
$443$	36.7846	1.74769	0.873845	+	0.486205i	$0.161619\pi$
$444$	2.00000	0.0949158
$445$	−4.53590	−0.215022
$446$	37.2679	1.76469
$447$	−8.00000	−0.378387
$448$	−2.00000	−0.0944911
$449$	8.19615	0.386800	0.193400	−	0.981120i	$-0.438048\pi$
$449$	8.19615	0.386800	0.193400	+	0.981120i	$0.438048\pi$
$450$	4.26795	0.201193
$451$	−1.46410	−0.0689419
$452$	10.0000	0.470360
$453$	11.4641	0.538630
$454$	18.0000	0.844782
$455$	−5.46410	−0.256161
$456$	−12.9282	−0.605419
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	−14.7846	−0.690839
$459$	−3.26795	−0.152535
$460$	5.46410	0.254765
$461$	−10.5359	−0.490706	−0.245353	−	0.969434i	$-0.578904\pi$
$461$	−10.5359	−0.490706	−0.245353	+	0.969434i	$0.578904\pi$
$462$	−3.46410	−0.161165
$463$	−25.5167	−1.18586	−0.592930	−	0.805254i	$-0.702029\pi$
$463$	−25.5167	−1.18586	−0.592930	+	0.805254i	$0.702029\pi$
$464$	−30.9808	−1.43825
$465$	6.00000	0.278243
$466$	−10.7321	−0.497153
$467$	−19.0718	−0.882538	−0.441269	−	0.897375i	$-0.645471\pi$
$467$	−19.0718	−0.882538	−0.441269	+	0.897375i	$0.645471\pi$
$468$	−1.00000	−0.0462250
$469$	3.60770	0.166588
$470$	−25.8564	−1.19267
$471$	−19.3205	−0.890242
$472$	11.3205	0.521069
$473$	−4.19615	−0.192939
$474$	−23.6603	−1.08675
$475$	−18.3923	−0.843897
$476$	−6.53590	−0.299572
$477$	2.00000	0.0915737
$478$	2.78461	0.127365
$479$	27.4641	1.25487	0.627433	−	0.778670i	$-0.284105\pi$
$479$	27.4641	1.25487	0.627433	+	0.778670i	$0.284105\pi$
$480$	−14.1962	−0.647963
$481$	2.00000	0.0911922
$482$	39.4641	1.79754
$483$	−4.00000	−0.182006
$484$	1.00000	0.0454545
$485$	13.4641	0.611373
$486$	−1.73205	−0.0785674
$487$	−8.33975	−0.377910	−0.188955	−	0.981986i	$-0.560510\pi$
$487$	−8.33975	−0.377910	−0.188955	+	0.981986i	$0.560510\pi$
$488$	15.4641	0.700027
$489$	20.7321	0.937536
$490$	14.1962	0.641317
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	−1.46410	−0.0660068
$493$	20.2487	0.911956
$494$	12.9282	0.581667
$495$	2.73205	0.122797
$496$	−10.9808	−0.493051
$497$	13.8564	0.621545
$498$	24.0000	1.07547
$499$	−17.8038	−0.797010	−0.398505	−	0.917166i	$-0.630471\pi$
$499$	−17.8038	−0.797010	−0.398505	+	0.917166i	$0.630471\pi$
$500$	6.92820	0.309839
$501$	−17.8564	−0.797765
$502$	−53.5692	−2.39091
$503$	−21.4641	−0.957037	−0.478518	−	0.878077i	$-0.658826\pi$
$503$	−21.4641	−0.957037	−0.478518	+	0.878077i	$0.658826\pi$
$504$	3.46410	0.154303
$505$	6.00000	0.266996
$506$	3.46410	0.153998
$507$	−1.00000	−0.0444116
$508$	−11.1244	−0.493563
$509$	−41.6603	−1.84656	−0.923279	−	0.384130i	$-0.874502\pi$
$509$	−41.6603	−1.84656	−0.923279	+	0.384130i	$0.874502\pi$
$510$	15.4641	0.684762
$511$	21.8564	0.966870
$512$	8.66025	0.382733
$513$	7.46410	0.329548
$514$	−30.0000	−1.32324
$515$	41.8564	1.84441
$516$	−4.19615	−0.184725
$517$	−5.46410	−0.240311
$518$	6.92820	0.304408
$519$	5.12436	0.224934
$520$	−4.73205	−0.207514
$521$	−40.9282	−1.79310	−0.896549	−	0.442945i	$-0.853934\pi$
$521$	−40.9282	−1.79310	−0.896549	+	0.442945i	$0.853934\pi$
$522$	10.7321	0.469729
$523$	−21.2679	−0.929982	−0.464991	−	0.885315i	$-0.653943\pi$
$523$	−21.2679	−0.929982	−0.464991	+	0.885315i	$0.653943\pi$
$524$	−18.9282	−0.826882
$525$	4.92820	0.215084
$526$	4.39230	0.191514
$527$	7.17691	0.312631
$528$	−5.00000	−0.217597
$529$	−19.0000	−0.826087
$530$	−9.46410	−0.411094
$531$	−6.53590	−0.283634
$532$	14.9282	0.647220
$533$	−1.46410	−0.0634173
$534$	−2.87564	−0.124441
$535$	−33.8564	−1.46374
$536$	3.12436	0.134952
$537$	19.8564	0.856867
$538$	30.0000	1.29339
$539$	3.00000	0.129219
$540$	2.73205	0.117569
$541$	−14.7846	−0.635640	−0.317820	−	0.948151i	$-0.602951\pi$
$541$	−14.7846	−0.635640	−0.317820	+	0.948151i	$0.602951\pi$
$542$	−43.8564	−1.88379
$543$	12.3923	0.531805
$544$	−16.9808	−0.728044
$545$	21.8564	0.936226
$546$	−3.46410	−0.148250
$547$	12.9808	0.555017	0.277509	−	0.960723i	$-0.410491\pi$
$547$	12.9808	0.555017	0.277509	+	0.960723i	$0.410491\pi$
$548$	16.5885	0.708624
$549$	−8.92820	−0.381046
$550$	−4.26795	−0.181986
$551$	−46.2487	−1.97026
$552$	−3.46410	−0.147442
$553$	−27.3205	−1.16179
$554$	−17.3205	−0.735878
$555$	−5.46410	−0.231938
$556$	−20.1962	−0.856508
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	3.80385	0.161030
$559$	−4.19615	−0.177478
$560$	−27.3205	−1.15450
$561$	3.26795	0.137973
$562$	−19.6077	−0.827101
$563$	−39.7128	−1.67370	−0.836848	−	0.547436i	$-0.815604\pi$
$563$	−39.7128	−1.67370	−0.836848	+	0.547436i	$0.815604\pi$
$564$	−5.46410	−0.230080
$565$	−27.3205	−1.14938
$566$	34.9808	1.47035
$567$	−2.00000	−0.0839921
$568$	12.0000	0.503509
$569$	15.6603	0.656512	0.328256	−	0.944589i	$-0.393539\pi$
$569$	15.6603	0.656512	0.328256	+	0.944589i	$0.393539\pi$
$570$	−35.3205	−1.47941
$571$	16.5885	0.694205	0.347103	−	0.937827i	$-0.387166\pi$
$571$	16.5885	0.694205	0.347103	+	0.937827i	$0.387166\pi$
$572$	1.00000	0.0418121
$573$	5.07180	0.211877
$574$	−5.07180	−0.211693
$575$	−4.92820	−0.205520
$576$	1.00000	0.0416667
$577$	−8.92820	−0.371686	−0.185843	−	0.982579i	$-0.559502\pi$
$577$	−8.92820	−0.371686	−0.185843	+	0.982579i	$0.559502\pi$
$578$	−10.9474	−0.455354
$579$	−6.92820	−0.287926
$580$	−16.9282	−0.702905
$581$	27.7128	1.14972
$582$	8.53590	0.353824
$583$	−2.00000	−0.0828315
$584$	18.9282	0.783255
$585$	2.73205	0.112956
$586$	46.6410	1.92672
$587$	2.53590	0.104668	0.0523339	−	0.998630i	$-0.483334\pi$
$587$	2.53590	0.104668	0.0523339	+	0.998630i	$0.483334\pi$
$588$	3.00000	0.123718
$589$	−16.3923	−0.675433
$590$	30.9282	1.27329
$591$	−8.00000	−0.329076
$592$	10.0000	0.410997
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	1.73205	0.0710669
$595$	17.8564	0.732041
$596$	8.00000	0.327693
$597$	−17.4641	−0.714758
$598$	3.46410	0.141658
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	4.26795	0.174238
$601$	0.143594	0.00585730	0.00292865	−	0.999996i	$-0.499068\pi$
$601$	0.143594	0.00585730	0.00292865	+	0.999996i	$0.499068\pi$
$602$	−14.5359	−0.592439
$603$	−1.80385	−0.0734584
$604$	−11.4641	−0.466468
$605$	−2.73205	−0.111074
$606$	3.80385	0.154521
$607$	23.8038	0.966168	0.483084	−	0.875574i	$-0.339517\pi$
$607$	23.8038	0.966168	0.483084	+	0.875574i	$0.339517\pi$
$608$	38.7846	1.57292
$609$	12.3923	0.502162
$610$	42.2487	1.71060
$611$	−5.46410	−0.221054
$612$	3.26795	0.132099
$613$	−9.85641	−0.398097	−0.199048	−	0.979990i	$-0.563785\pi$
$613$	−9.85641	−0.398097	−0.199048	+	0.979990i	$0.563785\pi$
$614$	−47.5692	−1.91974
$615$	4.00000	0.161296
$616$	−3.46410	−0.139573
$617$	6.33975	0.255229	0.127614	−	0.991824i	$-0.459268\pi$
$617$	6.33975	0.255229	0.127614	+	0.991824i	$0.459268\pi$
$618$	26.5359	1.06743
$619$	2.58846	0.104039	0.0520194	−	0.998646i	$-0.483434\pi$
$619$	2.58846	0.104039	0.0520194	+	0.998646i	$0.483434\pi$
$620$	−6.00000	−0.240966
$621$	2.00000	0.0802572
$622$	−29.3205	−1.17565
$623$	−3.32051	−0.133033
$624$	−5.00000	−0.200160
$625$	−31.2487	−1.24995
$626$	−4.39230	−0.175552
$627$	−7.46410	−0.298088
$628$	19.3205	0.770972
$629$	−6.53590	−0.260603
$630$	9.46410	0.377059
$631$	−23.6603	−0.941900	−0.470950	−	0.882160i	$-0.656089\pi$
$631$	−23.6603	−0.941900	−0.470950	+	0.882160i	$0.656089\pi$
$632$	−23.6603	−0.941154
$633$	−22.7321	−0.903518
$634$	−12.3397	−0.490074
$635$	30.3923	1.20608
$636$	−2.00000	−0.0793052
$637$	3.00000	0.118864
$638$	−10.7321	−0.424886
$639$	−6.92820	−0.274075
$640$	−33.1244	−1.30936
$641$	−7.85641	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$641$	−7.85641	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$642$	−21.4641	−0.847121
$643$	−8.05256	−0.317562	−0.158781	−	0.987314i	$-0.550756\pi$
$643$	−8.05256	−0.317562	−0.158781	+	0.987314i	$0.550756\pi$
$644$	4.00000	0.157622
$645$	11.4641	0.451399
$646$	−42.2487	−1.66225
$647$	18.9282	0.744144	0.372072	−	0.928204i	$-0.378647\pi$
$647$	18.9282	0.744144	0.372072	+	0.928204i	$0.378647\pi$
$648$	−1.73205	−0.0680414
$649$	6.53590	0.256556
$650$	−4.26795	−0.167403
$651$	4.39230	0.172148
$652$	−20.7321	−0.811930
$653$	2.67949	0.104857	0.0524283	−	0.998625i	$-0.483304\pi$
$653$	2.67949	0.104857	0.0524283	+	0.998625i	$0.483304\pi$
$654$	13.8564	0.541828
$655$	51.7128	2.02059
$656$	−7.32051	−0.285818
$657$	−10.9282	−0.426350
$658$	−18.9282	−0.737898
$659$	−38.9282	−1.51643	−0.758214	−	0.652006i	$-0.773928\pi$
$659$	−38.9282	−1.51643	−0.758214	+	0.652006i	$0.773928\pi$
$660$	−2.73205	−0.106345
$661$	−11.4641	−0.445902	−0.222951	−	0.974830i	$-0.571569\pi$
$661$	−11.4641	−0.445902	−0.222951	+	0.974830i	$0.571569\pi$
$662$	−37.2679	−1.44846
$663$	3.26795	0.126917
$664$	24.0000	0.931381
$665$	−40.7846	−1.58156
$666$	−3.46410	−0.134231
$667$	−12.3923	−0.479832
$668$	17.8564	0.690885
$669$	−21.5167	−0.831882
$670$	8.53590	0.329771
$671$	8.92820	0.344669
$672$	−10.3923	−0.400892
$673$	−16.5359	−0.637412	−0.318706	−	0.947854i	$-0.603248\pi$
$673$	−16.5359	−0.637412	−0.318706	+	0.947854i	$0.603248\pi$
$674$	−43.8564	−1.68929
$675$	−2.46410	−0.0948433
$676$	1.00000	0.0384615
$677$	46.3013	1.77950	0.889751	−	0.456446i	$-0.150878\pi$
$677$	46.3013	1.77950	0.889751	+	0.456446i	$0.150878\pi$
$678$	−17.3205	−0.665190
$679$	9.85641	0.378254
$680$	15.4641	0.593021
$681$	−10.3923	−0.398234
$682$	−3.80385	−0.145657
$683$	16.7846	0.642245	0.321123	−	0.947038i	$-0.395940\pi$
$683$	16.7846	0.642245	0.321123	+	0.947038i	$0.395940\pi$
$684$	−7.46410	−0.285397
$685$	−45.3205	−1.73161
$686$	34.6410	1.32260
$687$	8.53590	0.325665
$688$	−20.9808	−0.799884
$689$	−2.00000	−0.0761939
$690$	−9.46410	−0.360292
$691$	−32.4449	−1.23426	−0.617130	−	0.786861i	$-0.711705\pi$
$691$	−32.4449	−1.23426	−0.617130	+	0.786861i	$0.711705\pi$
$692$	−5.12436	−0.194799
$693$	2.00000	0.0759737
$694$	28.3923	1.07776
$695$	55.1769	2.09298
$696$	10.7321	0.406797
$697$	4.78461	0.181230
$698$	41.3205	1.56400
$699$	6.19615	0.234360
$700$	−4.92820	−0.186269
$701$	−44.0526	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$701$	−44.0526	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$702$	1.73205	0.0653720
$703$	14.9282	0.563028
$704$	−1.00000	−0.0376889
$705$	14.9282	0.562229
$706$	−50.1962	−1.88916
$707$	4.39230	0.165190
$708$	6.53590	0.245634
$709$	−11.4641	−0.430543	−0.215272	−	0.976554i	$-0.569064\pi$
$709$	−11.4641	−0.430543	−0.215272	+	0.976554i	$0.569064\pi$
$710$	32.7846	1.23038
$711$	13.6603	0.512300
$712$	−2.87564	−0.107769
$713$	−4.39230	−0.164493
$714$	11.3205	0.423659
$715$	−2.73205	−0.102173
$716$	−19.8564	−0.742069
$717$	−1.60770	−0.0600405
$718$	−40.6410	−1.51671
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	13.6603	0.509088
$721$	30.6410	1.14113
$722$	63.5885	2.36652
$723$	−22.7846	−0.847369
$724$	−12.3923	−0.460556
$725$	15.2679	0.567037
$726$	−1.73205	−0.0642824
$727$	4.67949	0.173553	0.0867764	−	0.996228i	$-0.472343\pi$
$727$	4.67949	0.173553	0.0867764	+	0.996228i	$0.472343\pi$
$728$	−3.46410	−0.128388
$729$	1.00000	0.0370370
$730$	51.7128	1.91398
$731$	13.7128	0.507187
$732$	8.92820	0.329996
$733$	−9.07180	−0.335074	−0.167537	−	0.985866i	$-0.553581\pi$
$733$	−9.07180	−0.335074	−0.167537	+	0.985866i	$0.553581\pi$
$734$	29.5692	1.09142
$735$	−8.19615	−0.302320
$736$	10.3923	0.383065
$737$	1.80385	0.0664456
$738$	2.53590	0.0933477
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	5.46410	0.200864
$741$	−7.46410	−0.274201
$742$	−6.92820	−0.254342
$743$	10.1436	0.372132	0.186066	−	0.982537i	$-0.440426\pi$
$743$	10.1436	0.372132	0.186066	+	0.982537i	$0.440426\pi$
$744$	3.80385	0.139456
$745$	−21.8564	−0.800757
$746$	55.8564	2.04505
$747$	−13.8564	−0.506979
$748$	−3.26795	−0.119488
$749$	−24.7846	−0.905610
$750$	−12.0000	−0.438178
$751$	−44.7846	−1.63421	−0.817107	−	0.576486i	$-0.804423\pi$
$751$	−44.7846	−1.63421	−0.817107	+	0.576486i	$0.804423\pi$
$752$	−27.3205	−0.996276
$753$	30.9282	1.12709
$754$	−10.7321	−0.390838
$755$	31.3205	1.13987
$756$	2.00000	0.0727393
$757$	−2.53590	−0.0921688	−0.0460844	−	0.998938i	$-0.514674\pi$
$757$	−2.53590	−0.0921688	−0.0460844	+	0.998938i	$0.514674\pi$
$758$	3.12436	0.113482
$759$	−2.00000	−0.0725954
$760$	−35.3205	−1.28121
$761$	−14.6410	−0.530736	−0.265368	−	0.964147i	$-0.585494\pi$
$761$	−14.6410	−0.530736	−0.265368	+	0.964147i	$0.585494\pi$
$762$	19.2679	0.698004
$763$	16.0000	0.579239
$764$	−5.07180	−0.183491
$765$	−8.92820	−0.322800
$766$	6.24871	0.225775
$767$	6.53590	0.235998
$768$	−19.0000	−0.685603
$769$	43.8564	1.58150	0.790751	−	0.612138i	$-0.209690\pi$
$769$	43.8564	1.58150	0.790751	+	0.612138i	$0.209690\pi$
$770$	−9.46410	−0.341063
$771$	17.3205	0.623783
$772$	6.92820	0.249351
$773$	−4.48334	−0.161255	−0.0806273	−	0.996744i	$-0.525692\pi$
$773$	−4.48334	−0.161255	−0.0806273	+	0.996744i	$0.525692\pi$
$774$	7.26795	0.261241
$775$	5.41154	0.194388
$776$	8.53590	0.306421
$777$	−4.00000	−0.143499
$778$	26.7846	0.960275
$779$	−10.9282	−0.391544
$780$	−2.73205	−0.0978231
$781$	6.92820	0.247911
$782$	−11.3205	−0.404821
$783$	−6.19615	−0.221432
$784$	15.0000	0.535714
$785$	−52.7846	−1.88396
$786$	32.7846	1.16939
$787$	−23.0718	−0.822421	−0.411210	−	0.911540i	$-0.634894\pi$
$787$	−23.0718	−0.822421	−0.411210	+	0.911540i	$0.634894\pi$
$788$	8.00000	0.284988
$789$	−2.53590	−0.0902804
$790$	−64.6410	−2.29982
$791$	−20.0000	−0.711118
$792$	1.73205	0.0615457
$793$	8.92820	0.317050
$794$	19.1769	0.680563
$795$	5.46410	0.193792
$796$	17.4641	0.618999
$797$	−29.6077	−1.04876	−0.524379	−	0.851485i	$-0.675703\pi$
$797$	−29.6077	−1.04876	−0.524379	+	0.851485i	$0.675703\pi$
$798$	−25.8564	−0.915307
$799$	17.8564	0.631714
$800$	−12.8038	−0.452684
$801$	1.66025	0.0586622
$802$	27.3731	0.966577
$803$	10.9282	0.385648
$804$	1.80385	0.0636168
$805$	−10.9282	−0.385169
$806$	−3.80385	−0.133985
$807$	−17.3205	−0.609711
$808$	3.80385	0.133819
$809$	−36.8372	−1.29513	−0.647563	−	0.762012i	$-0.724212\pi$
$809$	−36.8372	−1.29513	−0.647563	+	0.762012i	$0.724212\pi$
$810$	−4.73205	−0.166267
$811$	1.60770	0.0564538	0.0282269	−	0.999602i	$-0.491014\pi$
$811$	1.60770	0.0564538	0.0282269	+	0.999602i	$0.491014\pi$
$812$	−12.3923	−0.434885
$813$	25.3205	0.888029
$814$	3.46410	0.121417
$815$	56.6410	1.98405
$816$	16.3397	0.572006
$817$	−31.3205	−1.09577
$818$	48.0000	1.67828
$819$	2.00000	0.0698857
$820$	−4.00000	−0.139686
$821$	−47.3205	−1.65150	−0.825749	−	0.564038i	$-0.809247\pi$
$821$	−47.3205	−1.65150	−0.825749	+	0.564038i	$0.809247\pi$
$822$	−28.7321	−1.00215
$823$	32.3923	1.12912	0.564562	−	0.825390i	$-0.309045\pi$
$823$	32.3923	1.12912	0.564562	+	0.825390i	$0.309045\pi$
$824$	26.5359	0.924422
$825$	2.46410	0.0857890
$826$	22.6410	0.787782
$827$	−28.2487	−0.982304	−0.491152	−	0.871074i	$-0.663424\pi$
$827$	−28.2487	−0.982304	−0.491152	+	0.871074i	$0.663424\pi$
$828$	−2.00000	−0.0695048
$829$	−20.1436	−0.699616	−0.349808	−	0.936821i	$-0.613753\pi$
$829$	−20.1436	−0.699616	−0.349808	+	0.936821i	$0.613753\pi$
$830$	65.5692	2.27594
$831$	10.0000	0.346896
$832$	−1.00000	−0.0346688
$833$	−9.80385	−0.339683
$834$	34.9808	1.21128
$835$	−48.7846	−1.68826
$836$	7.46410	0.258151
$837$	−2.19615	−0.0759101
$838$	45.0333	1.55565
$839$	−31.7128	−1.09485	−0.547424	−	0.836855i	$-0.684391\pi$
$839$	−31.7128	−1.09485	−0.547424	+	0.836855i	$0.684391\pi$
$840$	9.46410	0.326543
$841$	9.39230	0.323873
$842$	−8.53590	−0.294166
$843$	11.3205	0.389899
$844$	22.7321	0.782469
$845$	−2.73205	−0.0939854
$846$	9.46410	0.325383
$847$	−2.00000	−0.0687208
$848$	−10.0000	−0.343401
$849$	−20.1962	−0.693130
$850$	13.9474	0.478393
$851$	4.00000	0.137118
$852$	6.92820	0.237356
$853$	−39.8564	−1.36466	−0.682329	−	0.731046i	$-0.739033\pi$
$853$	−39.8564	−1.36466	−0.682329	+	0.731046i	$0.739033\pi$
$854$	30.9282	1.05834
$855$	20.3923	0.697402
$856$	−21.4641	−0.733628
$857$	42.1962	1.44139	0.720697	−	0.693251i	$-0.243822\pi$
$857$	42.1962	1.44139	0.720697	+	0.693251i	$0.243822\pi$
$858$	−1.73205	−0.0591312
$859$	−53.9615	−1.84114	−0.920572	−	0.390574i	$-0.872277\pi$
$859$	−53.9615	−1.84114	−0.920572	+	0.390574i	$0.872277\pi$
$860$	−11.4641	−0.390923
$861$	2.92820	0.0997929
$862$	49.8564	1.69812
$863$	−23.3205	−0.793839	−0.396920	−	0.917853i	$-0.629921\pi$
$863$	−23.3205	−0.793839	−0.396920	+	0.917853i	$0.629921\pi$
$864$	5.19615	0.176777
$865$	14.0000	0.476014
$866$	−10.3923	−0.353145
$867$	6.32051	0.214656
$868$	−4.39230	−0.149085
$869$	−13.6603	−0.463392
$870$	29.3205	0.994058
$871$	1.80385	0.0611210
$872$	13.8564	0.469237
$873$	−4.92820	−0.166794
$874$	25.8564	0.874606
$875$	−13.8564	−0.468432
$876$	10.9282	0.369230
$877$	−11.8564	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$877$	−11.8564	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$878$	62.1962	2.09902
$879$	−26.9282	−0.908266
$880$	−13.6603	−0.460487
$881$	40.6410	1.36923	0.684615	−	0.728905i	$-0.259970\pi$
$881$	40.6410	1.36923	0.684615	+	0.728905i	$0.259970\pi$
$882$	−5.19615	−0.174964
$883$	46.2487	1.55639	0.778197	−	0.628021i	$-0.216135\pi$
$883$	46.2487	1.55639	0.778197	+	0.628021i	$0.216135\pi$
$884$	−3.26795	−0.109913
$885$	−17.8564	−0.600237
$886$	63.7128	2.14047
$887$	10.6410	0.357290	0.178645	−	0.983914i	$-0.442829\pi$
$887$	10.6410	0.357290	0.178645	+	0.983914i	$0.442829\pi$
$888$	−3.46410	−0.116248
$889$	22.2487	0.746198
$890$	−7.85641	−0.263347
$891$	−1.00000	−0.0335013
$892$	21.5167	0.720431
$893$	−40.7846	−1.36480
$894$	−13.8564	−0.463428
$895$	54.2487	1.81333
$896$	−24.2487	−0.810093
$897$	−2.00000	−0.0667781
$898$	14.1962	0.473732
$899$	13.6077	0.453842
$900$	2.46410	0.0821367
$901$	6.53590	0.217742
$902$	−2.53590	−0.0844362
$903$	8.39230	0.279278
$904$	−17.3205	−0.576072
$905$	33.8564	1.12543
$906$	19.8564	0.659685
$907$	10.9282	0.362865	0.181433	−	0.983403i	$-0.441927\pi$
$907$	10.9282	0.362865	0.181433	+	0.983403i	$0.441927\pi$
$908$	10.3923	0.344881
$909$	−2.19615	−0.0728418
$910$	−9.46410	−0.313732
$911$	2.14359	0.0710204	0.0355102	−	0.999369i	$-0.488694\pi$
$911$	2.14359	0.0710204	0.0355102	+	0.999369i	$0.488694\pi$
$912$	−37.3205	−1.23581
$913$	13.8564	0.458580
$914$	18.6795	0.617863
$915$	−24.3923	−0.806385
$916$	−8.53590	−0.282034
$917$	37.8564	1.25013
$918$	−5.66025	−0.186816
$919$	0.875644	0.0288848	0.0144424	−	0.999896i	$-0.495403\pi$
$919$	0.875644	0.0288848	0.0144424	+	0.999896i	$0.495403\pi$
$920$	−9.46410	−0.312022
$921$	27.4641	0.904973
$922$	−18.2487	−0.600989
$923$	6.92820	0.228045
$924$	−2.00000	−0.0657952
$925$	−4.92820	−0.162038
$926$	−44.1962	−1.45238
$927$	−15.3205	−0.503192
$928$	−32.1962	−1.05689
$929$	−51.1244	−1.67734	−0.838668	−	0.544643i	$-0.816665\pi$
$929$	−51.1244	−1.67734	−0.838668	+	0.544643i	$0.816665\pi$
$930$	10.3923	0.340777
$931$	22.3923	0.733878
$932$	−6.19615	−0.202962
$933$	16.9282	0.554204
$934$	−33.0333	−1.08088
$935$	8.92820	0.291983
$936$	1.73205	0.0566139
$937$	33.0333	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$937$	33.0333	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$938$	6.24871	0.204028
$939$	2.53590	0.0827559
$940$	−14.9282	−0.486904
$941$	−16.3923	−0.534374	−0.267187	−	0.963645i	$-0.586094\pi$
$941$	−16.3923	−0.534374	−0.267187	+	0.963645i	$0.586094\pi$
$942$	−33.4641	−1.09032
$943$	−2.92820	−0.0953554
$944$	32.6795	1.06363
$945$	−5.46410	−0.177747
$946$	−7.26795	−0.236301
$947$	0.679492	0.0220805	0.0110403	−	0.999939i	$-0.496486\pi$
$947$	0.679492	0.0220805	0.0110403	+	0.999939i	$0.496486\pi$
$948$	−13.6603	−0.443664
$949$	10.9282	0.354744
$950$	−31.8564	−1.03356
$951$	7.12436	0.231023
$952$	11.3205	0.366900
$953$	7.26795	0.235432	0.117716	−	0.993047i	$-0.462443\pi$
$953$	7.26795	0.235432	0.117716	+	0.993047i	$0.462443\pi$
$954$	3.46410	0.112154
$955$	13.8564	0.448383
$956$	1.60770	0.0519966
$957$	6.19615	0.200293
$958$	47.5692	1.53689
$959$	−33.1769	−1.07134
$960$	2.73205	0.0881766
$961$	−26.1769	−0.844417
$962$	3.46410	0.111687
$963$	12.3923	0.399336
$964$	22.7846	0.733843
$965$	−18.9282	−0.609320
$966$	−6.92820	−0.222911
$967$	27.0718	0.870570	0.435285	−	0.900293i	$-0.356648\pi$
$967$	27.0718	0.870570	0.435285	+	0.900293i	$0.356648\pi$
$968$	−1.73205	−0.0556702
$969$	24.3923	0.783594
$970$	23.3205	0.748776
$971$	3.85641	0.123758	0.0618790	−	0.998084i	$-0.480291\pi$
$971$	3.85641	0.123758	0.0618790	+	0.998084i	$0.480291\pi$
$972$	−1.00000	−0.0320750
$973$	40.3923	1.29492
$974$	−14.4449	−0.462843
$975$	2.46410	0.0789144
$976$	44.6410	1.42892
$977$	−17.6603	−0.565002	−0.282501	−	0.959267i	$-0.591164\pi$
$977$	−17.6603	−0.565002	−0.282501	+	0.959267i	$0.591164\pi$
$978$	35.9090	1.14824
$979$	−1.66025	−0.0530619
$980$	8.19615	0.261816
$981$	−8.00000	−0.255420
$982$	4.39230	0.140164
$983$	−33.0718	−1.05483	−0.527413	−	0.849609i	$-0.676838\pi$
$983$	−33.0718	−1.05483	−0.527413	+	0.849609i	$0.676838\pi$
$984$	2.53590	0.0808415
$985$	−21.8564	−0.696403
$986$	35.0718	1.11691
$987$	10.9282	0.347849
$988$	7.46410	0.237465
$989$	−8.39230	−0.266860
$990$	4.73205	0.150394
$991$	−5.17691	−0.164450	−0.0822251	−	0.996614i	$-0.526203\pi$
$991$	−5.17691	−0.164450	−0.0822251	+	0.996614i	$0.526203\pi$
$992$	−11.4115	−0.362317
$993$	21.5167	0.682811
$994$	24.0000	0.761234
$995$	−47.7128	−1.51260
$996$	13.8564	0.439057
$997$	41.3205	1.30863	0.654317	−	0.756221i	$-0.272956\pi$
$997$	41.3205	1.30863	0.654317	+	0.756221i	$0.272956\pi$
$998$	−30.8372	−0.976134
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.a.d.1.2	✓	2
3.2	odd	2		1287.2.a.f.1.1		2
4.3	odd	2		6864.2.a.bk.1.1		2
11.10	odd	2		4719.2.a.n.1.1		2
13.12	even	2		5577.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.d.1.2	✓	2	1.1	even	1	trivial
1287.2.a.f.1.1		2	3.2	odd	2
4719.2.a.n.1.1		2	11.10	odd	2
5577.2.a.h.1.1		2	13.12	even	2
6864.2.a.bk.1.1		2	4.3	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.73205 q^{5} -1.73205 q^{6} -2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} -4.73205 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.46410 q^{14} +2.73205 q^{15} -5.00000 q^{16} +3.26795 q^{17} +1.73205 q^{18} -7.46410 q^{19} -2.73205 q^{20} +2.00000 q^{21} -1.73205 q^{22} -2.00000 q^{23} +1.73205 q^{24} +2.46410 q^{25} -1.73205 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.19615 q^{29} +4.73205 q^{30} +2.19615 q^{31} -5.19615 q^{32} +1.00000 q^{33} +5.66025 q^{34} +5.46410 q^{35} +1.00000 q^{36} -2.00000 q^{37} -12.9282 q^{38} +1.00000 q^{39} +4.73205 q^{40} +1.46410 q^{41} +3.46410 q^{42} +4.19615 q^{43} -1.00000 q^{44} -2.73205 q^{45} -3.46410 q^{46} +5.46410 q^{47} +5.00000 q^{48} -3.00000 q^{49} +4.26795 q^{50} -3.26795 q^{51} -1.00000 q^{52} +2.00000 q^{53} -1.73205 q^{54} +2.73205 q^{55} +3.46410 q^{56} +7.46410 q^{57} +10.7321 q^{58} -6.53590 q^{59} +2.73205 q^{60} -8.92820 q^{61} +3.80385 q^{62} -2.00000 q^{63} +1.00000 q^{64} +2.73205 q^{65} +1.73205 q^{66} -1.80385 q^{67} +3.26795 q^{68} +2.00000 q^{69} +9.46410 q^{70} -6.92820 q^{71} -1.73205 q^{72} -10.9282 q^{73} -3.46410 q^{74} -2.46410 q^{75} -7.46410 q^{76} +2.00000 q^{77} +1.73205 q^{78} +13.6603 q^{79} +13.6603 q^{80} +1.00000 q^{81} +2.53590 q^{82} -13.8564 q^{83} +2.00000 q^{84} -8.92820 q^{85} +7.26795 q^{86} -6.19615 q^{87} +1.73205 q^{88} +1.66025 q^{89} -4.73205 q^{90} +2.00000 q^{91} -2.00000 q^{92} -2.19615 q^{93} +9.46410 q^{94} +20.3923 q^{95} +5.19615 q^{96} -4.92820 q^{97} -5.19615 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{9} - 6 q^{10} - 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{15} - 10 q^{16} + 10 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{21} - 4 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{28} + 2 q^{29} + 6 q^{30} - 6 q^{31} + 2 q^{33} - 6 q^{34} + 4 q^{35} + 2 q^{36} - 4 q^{37} - 12 q^{38} + 2 q^{39} + 6 q^{40} - 4 q^{41} - 2 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{47} + 10 q^{48} - 6 q^{49} + 12 q^{50} - 10 q^{51} - 2 q^{52} + 4 q^{53} + 2 q^{55} + 8 q^{57} + 18 q^{58} - 20 q^{59} + 2 q^{60} - 4 q^{61} + 18 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 14 q^{67} + 10 q^{68} + 4 q^{69} + 12 q^{70} - 8 q^{73} + 2 q^{75} - 8 q^{76} + 4 q^{77} + 10 q^{79} + 10 q^{80} + 2 q^{81} + 12 q^{82} + 4 q^{84} - 4 q^{85} + 18 q^{86} - 2 q^{87} - 14 q^{89} - 6 q^{90} + 4 q^{91} - 4 q^{92} + 6 q^{93} + 12 q^{94} + 20 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^9 - 6 * q^10 - 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^15 - 10 * q^16 + 10 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^21 - 4 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^28 + 2 * q^29 + 6 * q^30 - 6 * q^31 + 2 * q^33 - 6 * q^34 + 4 * q^35 + 2 * q^36 - 4 * q^37 - 12 * q^38 + 2 * q^39 + 6 * q^40 - 4 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^45 + 4 * q^47 + 10 * q^48 - 6 * q^49 + 12 * q^50 - 10 * q^51 - 2 * q^52 + 4 * q^53 + 2 * q^55 + 8 * q^57 + 18 * q^58 - 20 * q^59 + 2 * q^60 - 4 * q^61 + 18 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 14 * q^67 + 10 * q^68 + 4 * q^69 + 12 * q^70 - 8 * q^73 + 2 * q^75 - 8 * q^76 + 4 * q^77 + 10 * q^79 + 10 * q^80 + 2 * q^81 + 12 * q^82 + 4 * q^84 - 4 * q^85 + 18 * q^86 - 2 * q^87 - 14 * q^89 - 6 * q^90 + 4 * q^91 - 4 * q^92 + 6 * q^93 + 12 * q^94 + 20 * q^95 + 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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