LMFDB - Embedded newform 4719.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4719,2,Mod(1,4719)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4719, 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("4719.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4719 = 3 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4719.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:	$37.6814047138$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4719.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +6.82843 q^{10} +3.82843 q^{12} +1.00000 q^{13} +6.82843 q^{14} +2.82843 q^{15} +3.00000 q^{16} +3.65685 q^{17} +2.41421 q^{18} -2.82843 q^{19} +10.8284 q^{20} +2.82843 q^{21} -4.00000 q^{23} +4.41421 q^{24} +3.00000 q^{25} +2.41421 q^{26} +1.00000 q^{27} +10.8284 q^{28} -2.00000 q^{29} +6.82843 q^{30} -6.82843 q^{31} -1.58579 q^{32} +8.82843 q^{34} +8.00000 q^{35} +3.82843 q^{36} +3.65685 q^{37} -6.82843 q^{38} +1.00000 q^{39} +12.4853 q^{40} -10.8284 q^{41} +6.82843 q^{42} -9.65685 q^{43} +2.82843 q^{45} -9.65685 q^{46} -0.343146 q^{47} +3.00000 q^{48} +1.00000 q^{49} +7.24264 q^{50} +3.65685 q^{51} +3.82843 q^{52} -2.00000 q^{53} +2.41421 q^{54} +12.4853 q^{56} -2.82843 q^{57} -4.82843 q^{58} -3.65685 q^{59} +10.8284 q^{60} +9.31371 q^{61} -16.4853 q^{62} +2.82843 q^{63} -9.82843 q^{64} +2.82843 q^{65} +1.17157 q^{67} +14.0000 q^{68} -4.00000 q^{69} +19.3137 q^{70} +2.00000 q^{71} +4.41421 q^{72} -11.6569 q^{73} +8.82843 q^{74} +3.00000 q^{75} -10.8284 q^{76} +2.41421 q^{78} -11.3137 q^{79} +8.48528 q^{80} +1.00000 q^{81} -26.1421 q^{82} +7.65685 q^{83} +10.8284 q^{84} +10.3431 q^{85} -23.3137 q^{86} -2.00000 q^{87} +9.17157 q^{89} +6.82843 q^{90} +2.82843 q^{91} -15.3137 q^{92} -6.82843 q^{93} -0.828427 q^{94} -8.00000 q^{95} -1.58579 q^{96} -7.65685 q^{97} +2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{12} + 2 q^{13} + 8 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 16 q^{20} - 8 q^{23} + 6 q^{24} + 6 q^{25} + 2 q^{26} + 2 q^{27} + 16 q^{28} - 4 q^{29} + 8 q^{30} - 8 q^{31} - 6 q^{32} + 12 q^{34} + 16 q^{35} + 2 q^{36} - 4 q^{37} - 8 q^{38} + 2 q^{39} + 8 q^{40} - 16 q^{41} + 8 q^{42} - 8 q^{43} - 8 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} + 6 q^{50} - 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{56} - 4 q^{58} + 4 q^{59} + 16 q^{60} - 4 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{67} + 28 q^{68} - 8 q^{69} + 16 q^{70} + 4 q^{71} + 6 q^{72} - 12 q^{73} + 12 q^{74} + 6 q^{75} - 16 q^{76} + 2 q^{78} + 2 q^{81} - 24 q^{82} + 4 q^{83} + 16 q^{84} + 32 q^{85} - 24 q^{86} - 4 q^{87} + 24 q^{89} + 8 q^{90} - 8 q^{92} - 8 q^{93} + 4 q^{94} - 16 q^{95} - 6 q^{96} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^12 + 2 * q^13 + 8 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 16 * q^20 - 8 * q^23 + 6 * q^24 + 6 * q^25 + 2 * q^26 + 2 * q^27 + 16 * q^28 - 4 * q^29 + 8 * q^30 - 8 * q^31 - 6 * q^32 + 12 * q^34 + 16 * q^35 + 2 * q^36 - 4 * q^37 - 8 * q^38 + 2 * q^39 + 8 * q^40 - 16 * q^41 + 8 * q^42 - 8 * q^43 - 8 * q^46 - 12 * q^47 + 6 * q^48 + 2 * q^49 + 6 * q^50 - 4 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^56 - 4 * q^58 + 4 * q^59 + 16 * q^60 - 4 * q^61 - 16 * q^62 - 14 * q^64 + 8 * q^67 + 28 * q^68 - 8 * q^69 + 16 * q^70 + 4 * q^71 + 6 * q^72 - 12 * q^73 + 12 * q^74 + 6 * q^75 - 16 * q^76 + 2 * q^78 + 2 * q^81 - 24 * q^82 + 4 * q^83 + 16 * q^84 + 32 * q^85 - 24 * q^86 - 4 * q^87 + 24 * q^89 + 8 * q^90 - 8 * q^92 - 8 * q^93 + 4 * q^94 - 16 * q^95 - 6 * q^96 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.82843	1.91421
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	2.41421	0.985599
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	4.41421	1.56066
$9$	1.00000	0.333333
$10$	6.82843	2.15934
$11$	0	0
$11$	0	0
$12$	3.82843	1.10517
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.82843	1.82497
$15$	2.82843	0.730297
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	2.41421	0.569036
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	10.8284	2.42131
$21$	2.82843	0.617213
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	4.41421	0.901048
$25$	3.00000	0.600000
$26$	2.41421	0.473466
$27$	1.00000	0.192450
$28$	10.8284	2.04638
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	6.82843	1.24669
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	8.82843	1.51406
$35$	8.00000	1.35225
$36$	3.82843	0.638071
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	−6.82843	−1.10772
$39$	1.00000	0.160128
$40$	12.4853	1.97410
$41$	−10.8284	−1.69112	−0.845558	−	0.533883i	$-0.820732\pi$
$41$	−10.8284	−1.69112	−0.845558	+	0.533883i	$0.820732\pi$
$42$	6.82843	1.05365
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	−9.65685	−1.42383
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	3.00000	0.433013
$49$	1.00000	0.142857
$50$	7.24264	1.02426
$51$	3.65685	0.512062
$52$	3.82843	0.530907
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	2.41421	0.328533
$55$	0	0
$56$	12.4853	1.66842
$57$	−2.82843	−0.374634
$58$	−4.82843	−0.634004
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	10.8284	1.39794
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	−16.4853	−2.09363
$63$	2.82843	0.356348
$64$	−9.82843	−1.22855
$65$	2.82843	0.350823
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	14.0000	1.69775
$69$	−4.00000	−0.481543
$70$	19.3137	2.30843
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	4.41421	0.520220
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	8.82843	1.02628
$75$	3.00000	0.346410
$76$	−10.8284	−1.24211
$77$	0	0
$78$	2.41421	0.273356
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	8.48528	0.948683
$81$	1.00000	0.111111
$82$	−26.1421	−2.88692
$83$	7.65685	0.840449	0.420224	−	0.907420i	$-0.361951\pi$
$83$	7.65685	0.840449	0.420224	+	0.907420i	$0.361951\pi$
$84$	10.8284	1.18148
$85$	10.3431	1.12187
$86$	−23.3137	−2.51398
$87$	−2.00000	−0.214423
$88$	0	0
$89$	9.17157	0.972185	0.486092	−	0.873907i	$-0.338422\pi$
$89$	9.17157	0.972185	0.486092	+	0.873907i	$0.338422\pi$
$90$	6.82843	0.719779
$91$	2.82843	0.296500
$92$	−15.3137	−1.59656
$93$	−6.82843	−0.708075
$94$	−0.828427	−0.0854457
$95$	−8.00000	−0.820783
$96$	−1.58579	−0.161849
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	2.41421	0.243872
$99$	0	0
$100$	11.4853	1.14853
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	8.82843	0.874145
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	4.41421	0.432849
$105$	8.00000	0.780720
$106$	−4.82843	−0.468978
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	3.82843	0.368391
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	3.65685	0.347093
$112$	8.48528	0.801784
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	−6.82843	−0.639541
$115$	−11.3137	−1.05501
$116$	−7.65685	−0.710921
$117$	1.00000	0.0924500
$118$	−8.82843	−0.812723
$119$	10.3431	0.948155
$120$	12.4853	1.13975
$121$	0	0
$122$	22.4853	2.03572
$123$	−10.8284	−0.976366
$124$	−26.1421	−2.34763
$125$	−5.65685	−0.505964
$126$	6.82843	0.608325
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	−20.5563	−1.81694
$129$	−9.65685	−0.850239
$130$	6.82843	0.598893
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	2.82843	0.244339
$135$	2.82843	0.243432
$136$	16.1421	1.38418
$137$	−5.17157	−0.441837	−0.220919	−	0.975292i	$-0.570906\pi$
$137$	−5.17157	−0.441837	−0.220919	+	0.975292i	$0.570906\pi$
$138$	−9.65685	−0.822046
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	30.6274	2.58849
$141$	−0.343146	−0.0288981
$142$	4.82843	0.405193
$143$	0	0
$144$	3.00000	0.250000
$145$	−5.65685	−0.469776
$146$	−28.1421	−2.32906
$147$	1.00000	0.0824786
$148$	14.0000	1.15079
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	7.24264	0.591359
$151$	20.4853	1.66707	0.833534	−	0.552468i	$-0.186314\pi$
$151$	20.4853	1.66707	0.833534	+	0.552468i	$0.186314\pi$
$152$	−12.4853	−1.01269
$153$	3.65685	0.295639
$154$	0	0
$155$	−19.3137	−1.55131
$156$	3.82843	0.306519
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−27.3137	−2.17296
$159$	−2.00000	−0.158610
$160$	−4.48528	−0.354593
$161$	−11.3137	−0.891645
$162$	2.41421	0.189679
$163$	13.1716	1.03168	0.515839	−	0.856686i	$-0.327480\pi$
$163$	13.1716	1.03168	0.515839	+	0.856686i	$0.327480\pi$
$164$	−41.4558	−3.23716
$165$	0	0
$166$	18.4853	1.43474
$167$	−7.65685	−0.592505	−0.296253	−	0.955110i	$-0.595737\pi$
$167$	−7.65685	−0.592505	−0.296253	+	0.955110i	$0.595737\pi$
$168$	12.4853	0.963260
$169$	1.00000	0.0769231
$170$	24.9706	1.91515
$171$	−2.82843	−0.216295
$172$	−36.9706	−2.81898
$173$	0.343146	0.0260889	0.0130444	−	0.999915i	$-0.495848\pi$
$173$	0.343146	0.0260889	0.0130444	+	0.999915i	$0.495848\pi$
$174$	−4.82843	−0.366042
$175$	8.48528	0.641427
$176$	0	0
$177$	−3.65685	−0.274866
$178$	22.1421	1.65962
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	10.8284	0.807103
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	6.82843	0.506157
$183$	9.31371	0.688489
$184$	−17.6569	−1.30168
$185$	10.3431	0.760443
$186$	−16.4853	−1.20876
$187$	0	0
$188$	−1.31371	−0.0958120
$189$	2.82843	0.205738
$190$	−19.3137	−1.40116
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	−9.82843	−0.709306
$193$	17.3137	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$193$	17.3137	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$194$	−18.4853	−1.32717
$195$	2.82843	0.202548
$196$	3.82843	0.273459
$197$	16.4853	1.17453	0.587264	−	0.809396i	$-0.300205\pi$
$197$	16.4853	1.17453	0.587264	+	0.809396i	$0.300205\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	13.2426	0.936396
$201$	1.17157	0.0826364
$202$	8.82843	0.621166
$203$	−5.65685	−0.397033
$204$	14.0000	0.980196
$205$	−30.6274	−2.13911
$206$	32.9706	2.29717
$207$	−4.00000	−0.278019
$208$	3.00000	0.208013
$209$	0	0
$210$	19.3137	1.33277
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−7.65685	−0.525875
$213$	2.00000	0.137038
$214$	−27.3137	−1.86713
$215$	−27.3137	−1.86278
$216$	4.41421	0.300349
$217$	−19.3137	−1.31110
$218$	41.7990	2.83098
$219$	−11.6569	−0.787697
$220$	0	0
$221$	3.65685	0.245987
$222$	8.82843	0.592525
$223$	4.48528	0.300357	0.150178	−	0.988659i	$-0.452015\pi$
$223$	4.48528	0.300357	0.150178	+	0.988659i	$0.452015\pi$
$224$	−4.48528	−0.299685
$225$	3.00000	0.200000
$226$	41.7990	2.78043
$227$	−5.31371	−0.352683	−0.176342	−	0.984329i	$-0.556426\pi$
$227$	−5.31371	−0.352683	−0.176342	+	0.984329i	$0.556426\pi$
$228$	−10.8284	−0.717130
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	−27.3137	−1.80101
$231$	0	0
$232$	−8.82843	−0.579615
$233$	26.9706	1.76690	0.883450	−	0.468525i	$-0.155214\pi$
$233$	26.9706	1.76690	0.883450	+	0.468525i	$0.155214\pi$
$234$	2.41421	0.157822
$235$	−0.970563	−0.0633125
$236$	−14.0000	−0.911322
$237$	−11.3137	−0.734904
$238$	24.9706	1.61860
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	8.48528	0.547723
$241$	−11.6569	−0.750884	−0.375442	−	0.926846i	$-0.622509\pi$
$241$	−11.6569	−0.750884	−0.375442	+	0.926846i	$0.622509\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	35.6569	2.28270
$245$	2.82843	0.180702
$246$	−26.1421	−1.66676
$247$	−2.82843	−0.179969
$248$	−30.1421	−1.91403
$249$	7.65685	0.485233
$250$	−13.6569	−0.863735
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	10.8284	0.682127
$253$	0	0
$254$	13.6569	0.856907
$255$	10.3431	0.647713
$256$	−29.9706	−1.87316
$257$	−15.6569	−0.976648	−0.488324	−	0.872662i	$-0.662392\pi$
$257$	−15.6569	−0.976648	−0.488324	+	0.872662i	$0.662392\pi$
$258$	−23.3137	−1.45145
$259$	10.3431	0.642692
$260$	10.8284	0.671551
$261$	−2.00000	−0.123797
$262$	19.3137	1.19320
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	−19.3137	−1.18420
$267$	9.17157	0.561291
$268$	4.48528	0.273982
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	6.82843	0.415565
$271$	−11.7990	−0.716738	−0.358369	−	0.933580i	$-0.616667\pi$
$271$	−11.7990	−0.716738	−0.358369	+	0.933580i	$0.616667\pi$
$272$	10.9706	0.665188
$273$	2.82843	0.171184
$274$	−12.4853	−0.754263
$275$	0	0
$276$	−15.3137	−0.921777
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−36.9706	−2.21735
$279$	−6.82843	−0.408807
$280$	35.3137	2.11040
$281$	−26.8284	−1.60045	−0.800225	−	0.599700i	$-0.795287\pi$
$281$	−26.8284	−1.60045	−0.800225	+	0.599700i	$0.795287\pi$
$282$	−0.828427	−0.0493321
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	7.65685	0.454351
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−30.6274	−1.80788
$288$	−1.58579	−0.0934434
$289$	−3.62742	−0.213377
$290$	−13.6569	−0.801958
$291$	−7.65685	−0.448853
$292$	−44.6274	−2.61162
$293$	26.1421	1.52724	0.763620	−	0.645666i	$-0.223420\pi$
$293$	26.1421	1.52724	0.763620	+	0.645666i	$0.223420\pi$
$294$	2.41421	0.140800
$295$	−10.3431	−0.602201
$296$	16.1421	0.938243
$297$	0	0
$298$	35.7990	2.07378
$299$	−4.00000	−0.231326
$300$	11.4853	0.663103
$301$	−27.3137	−1.57434
$302$	49.4558	2.84586
$303$	3.65685	0.210081
$304$	−8.48528	−0.486664
$305$	26.3431	1.50840
$306$	8.82843	0.504688
$307$	17.1716	0.980033	0.490017	−	0.871713i	$-0.336991\pi$
$307$	17.1716	0.980033	0.490017	+	0.871713i	$0.336991\pi$
$308$	0	0
$309$	13.6569	0.776911
$310$	−46.6274	−2.64826
$311$	34.6274	1.96354	0.981770	−	0.190071i	$-0.0608718\pi$
$311$	34.6274	1.96354	0.981770	+	0.190071i	$0.0608718\pi$
$312$	4.41421	0.249906
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−24.1421	−1.36242
$315$	8.00000	0.450749
$316$	−43.3137	−2.43659
$317$	−8.48528	−0.476581	−0.238290	−	0.971194i	$-0.576587\pi$
$317$	−8.48528	−0.476581	−0.238290	+	0.971194i	$0.576587\pi$
$318$	−4.82843	−0.270765
$319$	0	0
$320$	−27.7990	−1.55401
$321$	−11.3137	−0.631470
$322$	−27.3137	−1.52213
$323$	−10.3431	−0.575508
$324$	3.82843	0.212690
$325$	3.00000	0.166410
$326$	31.7990	1.76118
$327$	17.3137	0.957450
$328$	−47.7990	−2.63926
$329$	−0.970563	−0.0535089
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	29.3137	1.60880
$333$	3.65685	0.200394
$334$	−18.4853	−1.01147
$335$	3.31371	0.181047
$336$	8.48528	0.462910
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	2.41421	0.131316
$339$	17.3137	0.940352
$340$	39.5980	2.14750
$341$	0	0
$342$	−6.82843	−0.369239
$343$	−16.9706	−0.916324
$344$	−42.6274	−2.29832
$345$	−11.3137	−0.609110
$346$	0.828427	0.0445365
$347$	31.3137	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$347$	31.3137	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$348$	−7.65685	−0.410450
$349$	7.65685	0.409862	0.204931	−	0.978776i	$-0.434303\pi$
$349$	7.65685	0.409862	0.204931	+	0.978776i	$0.434303\pi$
$350$	20.4853	1.09498
$351$	1.00000	0.0533761
$352$	0	0
$353$	17.4558	0.929081	0.464540	−	0.885552i	$-0.346220\pi$
$353$	17.4558	0.929081	0.464540	+	0.885552i	$0.346220\pi$
$354$	−8.82843	−0.469226
$355$	5.65685	0.300235
$356$	35.1127	1.86097
$357$	10.3431	0.547417
$358$	−1.65685	−0.0875675
$359$	−1.02944	−0.0543316	−0.0271658	−	0.999631i	$-0.508648\pi$
$359$	−1.02944	−0.0543316	−0.0271658	+	0.999631i	$0.508648\pi$
$360$	12.4853	0.658032
$361$	−11.0000	−0.578947
$362$	33.7990	1.77644
$363$	0	0
$364$	10.8284	0.567564
$365$	−32.9706	−1.72576
$366$	22.4853	1.17532
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−12.0000	−0.625543
$369$	−10.8284	−0.563705
$370$	24.9706	1.29816
$371$	−5.65685	−0.293689
$372$	−26.1421	−1.35541
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	−1.51472	−0.0781156
$377$	−2.00000	−0.103005
$378$	6.82843	0.351216
$379$	−16.4853	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$379$	−16.4853	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$380$	−30.6274	−1.57115
$381$	5.65685	0.289809
$382$	−46.6274	−2.38567
$383$	−2.97056	−0.151789	−0.0758943	−	0.997116i	$-0.524181\pi$
$383$	−2.97056	−0.151789	−0.0758943	+	0.997116i	$0.524181\pi$
$384$	−20.5563	−1.04901
$385$	0	0
$386$	41.7990	2.12751
$387$	−9.65685	−0.490885
$388$	−29.3137	−1.48818
$389$	6.97056	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$389$	6.97056	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$390$	6.82843	0.345771
$391$	−14.6274	−0.739740
$392$	4.41421	0.222951
$393$	8.00000	0.403547
$394$	39.7990	2.00504
$395$	−32.0000	−1.61009
$396$	0	0
$397$	−2.97056	−0.149088	−0.0745441	−	0.997218i	$-0.523750\pi$
$397$	−2.97056	−0.149088	−0.0745441	+	0.997218i	$0.523750\pi$
$398$	24.9706	1.25166
$399$	−8.00000	−0.400501
$400$	9.00000	0.450000
$401$	−2.14214	−0.106973	−0.0534866	−	0.998569i	$-0.517033\pi$
$401$	−2.14214	−0.106973	−0.0534866	+	0.998569i	$0.517033\pi$
$402$	2.82843	0.141069
$403$	−6.82843	−0.340148
$404$	14.0000	0.696526
$405$	2.82843	0.140546
$406$	−13.6569	−0.677778
$407$	0	0
$408$	16.1421	0.799155
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	−73.9411	−3.65169
$411$	−5.17157	−0.255095
$412$	52.2843	2.57586
$413$	−10.3431	−0.508953
$414$	−9.65685	−0.474608
$415$	21.6569	1.06309
$416$	−1.58579	−0.0777496
$417$	−15.3137	−0.749916
$418$	0	0
$419$	−30.6274	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$419$	−30.6274	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$420$	30.6274	1.49446
$421$	14.6863	0.715766	0.357883	−	0.933766i	$-0.383499\pi$
$421$	14.6863	0.715766	0.357883	+	0.933766i	$0.383499\pi$
$422$	28.9706	1.41026
$423$	−0.343146	−0.0166843
$424$	−8.82843	−0.428746
$425$	10.9706	0.532150
$426$	4.82843	0.233938
$427$	26.3431	1.27483
$428$	−43.3137	−2.09365
$429$	0	0
$430$	−65.9411	−3.17996
$431$	−19.6569	−0.946837	−0.473419	−	0.880838i	$-0.656980\pi$
$431$	−19.6569	−0.946837	−0.473419	+	0.880838i	$0.656980\pi$
$432$	3.00000	0.144338
$433$	1.31371	0.0631328	0.0315664	−	0.999502i	$-0.489950\pi$
$433$	1.31371	0.0631328	0.0315664	+	0.999502i	$0.489950\pi$
$434$	−46.6274	−2.23819
$435$	−5.65685	−0.271225
$436$	66.2843	3.17444
$437$	11.3137	0.541208
$438$	−28.1421	−1.34468
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	8.82843	0.419925
$443$	41.9411	1.99268	0.996342	−	0.0854611i	$-0.0272364\pi$
$443$	41.9411	1.99268	0.996342	+	0.0854611i	$0.0272364\pi$
$444$	14.0000	0.664411
$445$	25.9411	1.22973
$446$	10.8284	0.512741
$447$	14.8284	0.701361
$448$	−27.7990	−1.31338
$449$	7.79899	0.368057	0.184029	−	0.982921i	$-0.441086\pi$
$449$	7.79899	0.368057	0.184029	+	0.982921i	$0.441086\pi$
$450$	7.24264	0.341421
$451$	0	0
$452$	66.2843	3.11775
$453$	20.4853	0.962482
$454$	−12.8284	−0.602068
$455$	8.00000	0.375046
$456$	−12.4853	−0.584677
$457$	−3.65685	−0.171060	−0.0855302	−	0.996336i	$-0.527258\pi$
$457$	−3.65685	−0.171060	−0.0855302	+	0.996336i	$0.527258\pi$
$458$	51.4558	2.40437
$459$	3.65685	0.170687
$460$	−43.3137	−2.01951
$461$	−10.8284	−0.504330	−0.252165	−	0.967684i	$-0.581143\pi$
$461$	−10.8284	−0.504330	−0.252165	+	0.967684i	$0.581143\pi$
$462$	0	0
$463$	−7.51472	−0.349239	−0.174619	−	0.984636i	$-0.555869\pi$
$463$	−7.51472	−0.349239	−0.174619	+	0.984636i	$0.555869\pi$
$464$	−6.00000	−0.278543
$465$	−19.3137	−0.895652
$466$	65.1127	3.01629
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	3.82843	0.176969
$469$	3.31371	0.153013
$470$	−2.34315	−0.108081
$471$	−10.0000	−0.460776
$472$	−16.1421	−0.743002
$473$	0	0
$474$	−27.3137	−1.25456
$475$	−8.48528	−0.389331
$476$	39.5980	1.81497
$477$	−2.00000	−0.0915737
$478$	−4.82843	−0.220847
$479$	−2.68629	−0.122740	−0.0613699	−	0.998115i	$-0.519547\pi$
$479$	−2.68629	−0.122740	−0.0613699	+	0.998115i	$0.519547\pi$
$480$	−4.48528	−0.204724
$481$	3.65685	0.166738
$482$	−28.1421	−1.28184
$483$	−11.3137	−0.514792
$484$	0	0
$485$	−21.6569	−0.983387
$486$	2.41421	0.109511
$487$	31.7990	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$487$	31.7990	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$488$	41.1127	1.86108
$489$	13.1716	0.595639
$490$	6.82843	0.308477
$491$	14.6274	0.660126	0.330063	−	0.943959i	$-0.392930\pi$
$491$	14.6274	0.660126	0.330063	+	0.943959i	$0.392930\pi$
$492$	−41.4558	−1.86897
$493$	−7.31371	−0.329393
$494$	−6.82843	−0.307225
$495$	0	0
$496$	−20.4853	−0.919816
$497$	5.65685	0.253745
$498$	18.4853	0.828345
$499$	−2.14214	−0.0958952	−0.0479476	−	0.998850i	$-0.515268\pi$
$499$	−2.14214	−0.0958952	−0.0479476	+	0.998850i	$0.515268\pi$
$500$	−21.6569	−0.968524
$501$	−7.65685	−0.342083
$502$	0	0
$503$	−15.3137	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$503$	−15.3137	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$504$	12.4853	0.556139
$505$	10.3431	0.460264
$506$	0	0
$507$	1.00000	0.0444116
$508$	21.6569	0.960868
$509$	−27.7990	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$509$	−27.7990	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$510$	24.9706	1.10572
$511$	−32.9706	−1.45853
$512$	−31.2426	−1.38074
$513$	−2.82843	−0.124878
$514$	−37.7990	−1.66724
$515$	38.6274	1.70213
$516$	−36.9706	−1.62754
$517$	0	0
$518$	24.9706	1.09714
$519$	0.343146	0.0150624
$520$	12.4853	0.547516
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	−4.82843	−0.211335
$523$	−7.31371	−0.319806	−0.159903	−	0.987133i	$-0.551118\pi$
$523$	−7.31371	−0.319806	−0.159903	+	0.987133i	$0.551118\pi$
$524$	30.6274	1.33796
$525$	8.48528	0.370328
$526$	−28.9706	−1.26318
$527$	−24.9706	−1.08773
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−13.6569	−0.593216
$531$	−3.65685	−0.158694
$532$	−30.6274	−1.32787
$533$	−10.8284	−0.469031
$534$	22.1421	0.958184
$535$	−32.0000	−1.38348
$536$	5.17157	0.223378
$537$	−0.686292	−0.0296157
$538$	43.4558	1.87351
$539$	0	0
$540$	10.8284	0.465981
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−28.4853	−1.22355
$543$	14.0000	0.600798
$544$	−5.79899	−0.248630
$545$	48.9706	2.09767
$546$	6.82843	0.292230
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	−19.7990	−0.845771
$549$	9.31371	0.397499
$550$	0	0
$551$	5.65685	0.240990
$552$	−17.6569	−0.751526
$553$	−32.0000	−1.36078
$554$	4.82843	0.205140
$555$	10.3431	0.439042
$556$	−58.6274	−2.48636
$557$	−31.7990	−1.34737	−0.673683	−	0.739020i	$-0.735289\pi$
$557$	−31.7990	−1.34737	−0.673683	+	0.739020i	$0.735289\pi$
$558$	−16.4853	−0.697878
$559$	−9.65685	−0.408441
$560$	24.0000	1.01419
$561$	0	0
$562$	−64.7696	−2.73214
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−1.31371	−0.0553171
$565$	48.9706	2.06021
$566$	12.0000	0.504398
$567$	2.82843	0.118783
$568$	8.82843	0.370433
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	−19.3137	−0.808962
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	−19.3137	−0.806842
$574$	−73.9411	−3.08624
$575$	−12.0000	−0.500435
$576$	−9.82843	−0.409518
$577$	35.9411	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$577$	35.9411	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$578$	−8.75736	−0.364258
$579$	17.3137	0.719533
$580$	−21.6569	−0.899252
$581$	21.6569	0.898478
$582$	−18.4853	−0.766240
$583$	0	0
$584$	−51.4558	−2.12926
$585$	2.82843	0.116941
$586$	63.1127	2.60716
$587$	22.9706	0.948097	0.474048	−	0.880499i	$-0.342792\pi$
$587$	22.9706	0.948097	0.474048	+	0.880499i	$0.342792\pi$
$588$	3.82843	0.157882
$589$	19.3137	0.795807
$590$	−24.9706	−1.02802
$591$	16.4853	0.678114
$592$	10.9706	0.450887
$593$	3.51472	0.144332	0.0721661	−	0.997393i	$-0.477009\pi$
$593$	3.51472	0.144332	0.0721661	+	0.997393i	$0.477009\pi$
$594$	0	0
$595$	29.2548	1.19933
$596$	56.7696	2.32537
$597$	10.3431	0.423317
$598$	−9.65685	−0.394898
$599$	−0.686292	−0.0280411	−0.0140206	−	0.999902i	$-0.504463\pi$
$599$	−0.686292	−0.0280411	−0.0140206	+	0.999902i	$0.504463\pi$
$600$	13.2426	0.540629
$601$	−44.6274	−1.82039	−0.910195	−	0.414180i	$-0.864069\pi$
$601$	−44.6274	−1.82039	−0.910195	+	0.414180i	$0.864069\pi$
$602$	−65.9411	−2.68756
$603$	1.17157	0.0477101
$604$	78.4264	3.19113
$605$	0	0
$606$	8.82843	0.358630
$607$	25.9411	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$607$	25.9411	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$608$	4.48528	0.181902
$609$	−5.65685	−0.229227
$610$	63.5980	2.57501
$611$	−0.343146	−0.0138822
$612$	14.0000	0.565916
$613$	36.3431	1.46789	0.733943	−	0.679211i	$-0.237678\pi$
$613$	36.3431	1.46789	0.733943	+	0.679211i	$0.237678\pi$
$614$	41.4558	1.67302
$615$	−30.6274	−1.23502
$616$	0	0
$617$	−29.1716	−1.17440	−0.587202	−	0.809441i	$-0.699770\pi$
$617$	−29.1716	−1.17440	−0.587202	+	0.809441i	$0.699770\pi$
$618$	32.9706	1.32627
$619$	−15.7990	−0.635015	−0.317508	−	0.948256i	$-0.602846\pi$
$619$	−15.7990	−0.635015	−0.317508	+	0.948256i	$0.602846\pi$
$620$	−73.9411	−2.96955
$621$	−4.00000	−0.160514
$622$	83.5980	3.35197
$623$	25.9411	1.03931
$624$	3.00000	0.120096
$625$	−31.0000	−1.24000
$626$	14.4853	0.578948
$627$	0	0
$628$	−38.2843	−1.52771
$629$	13.3726	0.533200
$630$	19.3137	0.769477
$631$	−19.1127	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$631$	−19.1127	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$632$	−49.9411	−1.98655
$633$	12.0000	0.476957
$634$	−20.4853	−0.813574
$635$	16.0000	0.634941
$636$	−7.65685	−0.303614
$637$	1.00000	0.0396214
$638$	0	0
$639$	2.00000	0.0791188
$640$	−58.1421	−2.29827
$641$	−26.2843	−1.03817	−0.519083	−	0.854724i	$-0.673727\pi$
$641$	−26.2843	−1.03817	−0.519083	+	0.854724i	$0.673727\pi$
$642$	−27.3137	−1.07799
$643$	17.1716	0.677181	0.338590	−	0.940934i	$-0.390050\pi$
$643$	17.1716	0.677181	0.338590	+	0.940934i	$0.390050\pi$
$644$	−43.3137	−1.70680
$645$	−27.3137	−1.07548
$646$	−24.9706	−0.982454
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	4.41421	0.173407
$649$	0	0
$650$	7.24264	0.284080
$651$	−19.3137	−0.756964
$652$	50.4264	1.97485
$653$	−2.68629	−0.105123	−0.0525614	−	0.998618i	$-0.516739\pi$
$653$	−2.68629	−0.105123	−0.0525614	+	0.998618i	$0.516739\pi$
$654$	41.7990	1.63447
$655$	22.6274	0.884126
$656$	−32.4853	−1.26834
$657$	−11.6569	−0.454777
$658$	−2.34315	−0.0913453
$659$	24.6863	0.961641	0.480821	−	0.876819i	$-0.340339\pi$
$659$	24.6863	0.961641	0.480821	+	0.876819i	$0.340339\pi$
$660$	0	0
$661$	−1.02944	−0.0400405	−0.0200202	−	0.999800i	$-0.506373\pi$
$661$	−1.02944	−0.0400405	−0.0200202	+	0.999800i	$0.506373\pi$
$662$	−5.17157	−0.200999
$663$	3.65685	0.142020
$664$	33.7990	1.31166
$665$	−22.6274	−0.877454
$666$	8.82843	0.342095
$667$	8.00000	0.309761
$668$	−29.3137	−1.13418
$669$	4.48528	0.173411
$670$	8.00000	0.309067
$671$	0	0
$672$	−4.48528	−0.173023
$673$	28.6274	1.10351	0.551753	−	0.834008i	$-0.313959\pi$
$673$	28.6274	1.10351	0.551753	+	0.834008i	$0.313959\pi$
$674$	32.1421	1.23807
$675$	3.00000	0.115470
$676$	3.82843	0.147247
$677$	−49.3137	−1.89528	−0.947640	−	0.319341i	$-0.896538\pi$
$677$	−49.3137	−1.89528	−0.947640	+	0.319341i	$0.896538\pi$
$678$	41.7990	1.60528
$679$	−21.6569	−0.831114
$680$	45.6569	1.75086
$681$	−5.31371	−0.203622
$682$	0	0
$683$	−19.9411	−0.763026	−0.381513	−	0.924363i	$-0.624597\pi$
$683$	−19.9411	−0.763026	−0.381513	+	0.924363i	$0.624597\pi$
$684$	−10.8284	−0.414035
$685$	−14.6274	−0.558885
$686$	−40.9706	−1.56426
$687$	21.3137	0.813169
$688$	−28.9706	−1.10449
$689$	−2.00000	−0.0761939
$690$	−27.3137	−1.03982
$691$	−34.1421	−1.29883	−0.649414	−	0.760435i	$-0.724986\pi$
$691$	−34.1421	−1.29883	−0.649414	+	0.760435i	$0.724986\pi$
$692$	1.31371	0.0499397
$693$	0	0
$694$	75.5980	2.86966
$695$	−43.3137	−1.64298
$696$	−8.82843	−0.334641
$697$	−39.5980	−1.49988
$698$	18.4853	0.699678
$699$	26.9706	1.02012
$700$	32.4853	1.22783
$701$	−38.9706	−1.47190	−0.735949	−	0.677037i	$-0.763264\pi$
$701$	−38.9706	−1.47190	−0.735949	+	0.677037i	$0.763264\pi$
$702$	2.41421	0.0911186
$703$	−10.3431	−0.390099
$704$	0	0
$705$	−0.970563	−0.0365535
$706$	42.1421	1.58604
$707$	10.3431	0.388994
$708$	−14.0000	−0.526152
$709$	40.6274	1.52579	0.762897	−	0.646520i	$-0.223776\pi$
$709$	40.6274	1.52579	0.762897	+	0.646520i	$0.223776\pi$
$710$	13.6569	0.512533
$711$	−11.3137	−0.424297
$712$	40.4853	1.51725
$713$	27.3137	1.02291
$714$	24.9706	0.934500
$715$	0	0
$716$	−2.62742	−0.0981912
$717$	−2.00000	−0.0746914
$718$	−2.48528	−0.0927499
$719$	37.9411	1.41497	0.707483	−	0.706731i	$-0.249831\pi$
$719$	37.9411	1.41497	0.707483	+	0.706731i	$0.249831\pi$
$720$	8.48528	0.316228
$721$	38.6274	1.43856
$722$	−26.5563	−0.988325
$723$	−11.6569	−0.433523
$724$	53.5980	1.99195
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−21.6569	−0.803208	−0.401604	−	0.915813i	$-0.631547\pi$
$727$	−21.6569	−0.803208	−0.401604	+	0.915813i	$0.631547\pi$
$728$	12.4853	0.462735
$729$	1.00000	0.0370370
$730$	−79.5980	−2.94605
$731$	−35.3137	−1.30612
$732$	35.6569	1.31792
$733$	−8.62742	−0.318661	−0.159330	−	0.987225i	$-0.550934\pi$
$733$	−8.62742	−0.318661	−0.159330	+	0.987225i	$0.550934\pi$
$734$	−57.9411	−2.13865
$735$	2.82843	0.104328
$736$	6.34315	0.233811
$737$	0	0
$738$	−26.1421	−0.962305
$739$	−10.1421	−0.373084	−0.186542	−	0.982447i	$-0.559728\pi$
$739$	−10.1421	−0.373084	−0.186542	+	0.982447i	$0.559728\pi$
$740$	39.5980	1.45565
$741$	−2.82843	−0.103905
$742$	−13.6569	−0.501359
$743$	−2.00000	−0.0733729	−0.0366864	−	0.999327i	$-0.511680\pi$
$743$	−2.00000	−0.0733729	−0.0366864	+	0.999327i	$0.511680\pi$
$744$	−30.1421	−1.10506
$745$	41.9411	1.53660
$746$	−24.1421	−0.883906
$747$	7.65685	0.280150
$748$	0	0
$749$	−32.0000	−1.16925
$750$	−13.6569	−0.498678
$751$	32.9706	1.20311	0.601556	−	0.798830i	$-0.294547\pi$
$751$	32.9706	1.20311	0.601556	+	0.798830i	$0.294547\pi$
$752$	−1.02944	−0.0375397
$753$	0	0
$754$	−4.82843	−0.175841
$755$	57.9411	2.10869
$756$	10.8284	0.393826
$757$	−15.9411	−0.579390	−0.289695	−	0.957119i	$-0.593554\pi$
$757$	−15.9411	−0.579390	−0.289695	+	0.957119i	$0.593554\pi$
$758$	−39.7990	−1.44556
$759$	0	0
$760$	−35.3137	−1.28096
$761$	−15.5147	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$761$	−15.5147	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$762$	13.6569	0.494736
$763$	48.9706	1.77285
$764$	−73.9411	−2.67510
$765$	10.3431	0.373957
$766$	−7.17157	−0.259119
$767$	−3.65685	−0.132041
$768$	−29.9706	−1.08147
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−15.6569	−0.563868
$772$	66.2843	2.38562
$773$	5.85786	0.210693	0.105346	−	0.994436i	$-0.466405\pi$
$773$	5.85786	0.210693	0.105346	+	0.994436i	$0.466405\pi$
$774$	−23.3137	−0.837994
$775$	−20.4853	−0.735853
$776$	−33.7990	−1.21331
$777$	10.3431	0.371058
$778$	16.8284	0.603328
$779$	30.6274	1.09734
$780$	10.8284	0.387720
$781$	0	0
$782$	−35.3137	−1.26282
$783$	−2.00000	−0.0714742
$784$	3.00000	0.107143
$785$	−28.2843	−1.00951
$786$	19.3137	0.688897
$787$	−32.7696	−1.16811	−0.584054	−	0.811715i	$-0.698534\pi$
$787$	−32.7696	−1.16811	−0.584054	+	0.811715i	$0.698534\pi$
$788$	63.1127	2.24830
$789$	−12.0000	−0.427211
$790$	−77.2548	−2.74860
$791$	48.9706	1.74119
$792$	0	0
$793$	9.31371	0.330739
$794$	−7.17157	−0.254510
$795$	−5.65685	−0.200628
$796$	39.5980	1.40351
$797$	35.6569	1.26303	0.631515	−	0.775363i	$-0.282433\pi$
$797$	35.6569	1.26303	0.631515	+	0.775363i	$0.282433\pi$
$798$	−19.3137	−0.683698
$799$	−1.25483	−0.0443928
$800$	−4.75736	−0.168198
$801$	9.17157	0.324062
$802$	−5.17157	−0.182615
$803$	0	0
$804$	4.48528	0.158184
$805$	−32.0000	−1.12785
$806$	−16.4853	−0.580669
$807$	18.0000	0.633630
$808$	16.1421	0.567878
$809$	−41.3137	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$809$	−41.3137	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$810$	6.82843	0.239926
$811$	1.85786	0.0652384	0.0326192	−	0.999468i	$-0.489615\pi$
$811$	1.85786	0.0652384	0.0326192	+	0.999468i	$0.489615\pi$
$812$	−21.6569	−0.760007
$813$	−11.7990	−0.413809
$814$	0	0
$815$	37.2548	1.30498
$816$	10.9706	0.384047
$817$	27.3137	0.955586
$818$	2.48528	0.0868958
$819$	2.82843	0.0988332
$820$	−117.255	−4.09472
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	−12.4853	−0.435474
$823$	48.9706	1.70701	0.853503	−	0.521088i	$-0.174473\pi$
$823$	48.9706	1.70701	0.853503	+	0.521088i	$0.174473\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	−24.9706	−0.868837
$827$	26.0000	0.904109	0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000	0.904109	0.452054	+	0.891990i	$0.350691\pi$
$828$	−15.3137	−0.532188
$829$	5.31371	0.184553	0.0922764	−	0.995733i	$-0.470586\pi$
$829$	5.31371	0.184553	0.0922764	+	0.995733i	$0.470586\pi$
$830$	52.2843	1.81481
$831$	2.00000	0.0693792
$832$	−9.82843	−0.340739
$833$	3.65685	0.126702
$834$	−36.9706	−1.28019
$835$	−21.6569	−0.749466
$836$	0	0
$837$	−6.82843	−0.236025
$838$	−73.9411	−2.55425
$839$	−47.2548	−1.63142	−0.815709	−	0.578462i	$-0.803653\pi$
$839$	−47.2548	−1.63142	−0.815709	+	0.578462i	$0.803653\pi$
$840$	35.3137	1.21844
$841$	−25.0000	−0.862069
$842$	35.4558	1.22189
$843$	−26.8284	−0.924020
$844$	45.9411	1.58136
$845$	2.82843	0.0973009
$846$	−0.828427	−0.0284819
$847$	0	0
$848$	−6.00000	−0.206041
$849$	4.97056	0.170589
$850$	26.4853	0.908438
$851$	−14.6274	−0.501421
$852$	7.65685	0.262320
$853$	7.65685	0.262166	0.131083	−	0.991371i	$-0.458155\pi$
$853$	7.65685	0.262166	0.131083	+	0.991371i	$0.458155\pi$
$854$	63.5980	2.17628
$855$	−8.00000	−0.273594
$856$	−49.9411	−1.70695
$857$	−29.5980	−1.01105	−0.505524	−	0.862813i	$-0.668701\pi$
$857$	−29.5980	−1.01105	−0.505524	+	0.862813i	$0.668701\pi$
$858$	0	0
$859$	−23.3137	−0.795453	−0.397727	−	0.917504i	$-0.630201\pi$
$859$	−23.3137	−0.795453	−0.397727	+	0.917504i	$0.630201\pi$
$860$	−104.569	−3.56576
$861$	−30.6274	−1.04378
$862$	−47.4558	−1.61635
$863$	−39.6569	−1.34994	−0.674968	−	0.737847i	$-0.735842\pi$
$863$	−39.6569	−1.34994	−0.674968	+	0.737847i	$0.735842\pi$
$864$	−1.58579	−0.0539496
$865$	0.970563	0.0330001
$866$	3.17157	0.107774
$867$	−3.62742	−0.123194
$868$	−73.9411	−2.50973
$869$	0	0
$870$	−13.6569	−0.463011
$871$	1.17157	0.0396972
$872$	76.4264	2.58812
$873$	−7.65685	−0.259145
$874$	27.3137	0.923900
$875$	−16.0000	−0.540899
$876$	−44.6274	−1.50782
$877$	14.2843	0.482346	0.241173	−	0.970482i	$-0.422468\pi$
$877$	14.2843	0.482346	0.241173	+	0.970482i	$0.422468\pi$
$878$	40.9706	1.38269
$879$	26.1421	0.881752
$880$	0	0
$881$	53.5980	1.80576	0.902881	−	0.429891i	$-0.141448\pi$
$881$	53.5980	1.80576	0.902881	+	0.429891i	$0.141448\pi$
$882$	2.41421	0.0812908
$883$	−51.5980	−1.73641	−0.868205	−	0.496205i	$-0.834726\pi$
$883$	−51.5980	−1.73641	−0.868205	+	0.496205i	$0.834726\pi$
$884$	14.0000	0.470871
$885$	−10.3431	−0.347681
$886$	101.255	3.40172
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	16.1421	0.541695
$889$	16.0000	0.536623
$890$	62.6274	2.09928
$891$	0	0
$892$	17.1716	0.574947
$893$	0.970563	0.0324786
$894$	35.7990	1.19730
$895$	−1.94113	−0.0648847
$896$	−58.1421	−1.94239
$897$	−4.00000	−0.133556
$898$	18.8284	0.628313
$899$	13.6569	0.455482
$900$	11.4853	0.382843
$901$	−7.31371	−0.243655
$902$	0	0
$903$	−27.3137	−0.908943
$904$	76.4264	2.54190
$905$	39.5980	1.31628
$906$	49.4558	1.64306
$907$	20.9706	0.696316	0.348158	−	0.937436i	$-0.386807\pi$
$907$	20.9706	0.696316	0.348158	+	0.937436i	$0.386807\pi$
$908$	−20.3431	−0.675111
$909$	3.65685	0.121290
$910$	19.3137	0.640243
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−8.48528	−0.280976
$913$	0	0
$914$	−8.82843	−0.292018
$915$	26.3431	0.870878
$916$	81.5980	2.69607
$917$	22.6274	0.747223
$918$	8.82843	0.291382
$919$	19.3137	0.637100	0.318550	−	0.947906i	$-0.396804\pi$
$919$	19.3137	0.637100	0.318550	+	0.947906i	$0.396804\pi$
$920$	−49.9411	−1.64651
$921$	17.1716	0.565823
$922$	−26.1421	−0.860945
$923$	2.00000	0.0658308
$924$	0	0
$925$	10.9706	0.360710
$926$	−18.1421	−0.596188
$927$	13.6569	0.448550
$928$	3.17157	0.104112
$929$	−27.7990	−0.912055	−0.456028	−	0.889966i	$-0.650728\pi$
$929$	−27.7990	−0.912055	−0.456028	+	0.889966i	$0.650728\pi$
$930$	−46.6274	−1.52897
$931$	−2.82843	−0.0926980
$932$	103.255	3.38222
$933$	34.6274	1.13365
$934$	−19.3137	−0.631964
$935$	0	0
$936$	4.41421	0.144283
$937$	−1.31371	−0.0429170	−0.0214585	−	0.999770i	$-0.506831\pi$
$937$	−1.31371	−0.0429170	−0.0214585	+	0.999770i	$0.506831\pi$
$938$	8.00000	0.261209
$939$	6.00000	0.195803
$940$	−3.71573	−0.121194
$941$	−5.85786	−0.190961	−0.0954805	−	0.995431i	$-0.530439\pi$
$941$	−5.85786	−0.190961	−0.0954805	+	0.995431i	$0.530439\pi$
$942$	−24.1421	−0.786593
$943$	43.3137	1.41049
$944$	−10.9706	−0.357061
$945$	8.00000	0.260240
$946$	0	0
$947$	−54.9706	−1.78630	−0.893152	−	0.449756i	$-0.851511\pi$
$947$	−54.9706	−1.78630	−0.893152	+	0.449756i	$0.851511\pi$
$948$	−43.3137	−1.40676
$949$	−11.6569	−0.378398
$950$	−20.4853	−0.664630
$951$	−8.48528	−0.275154
$952$	45.6569	1.47975
$953$	−51.6569	−1.67333	−0.836665	−	0.547715i	$-0.815498\pi$
$953$	−51.6569	−1.67333	−0.836665	+	0.547715i	$0.815498\pi$
$954$	−4.82843	−0.156326
$955$	−54.6274	−1.76770
$956$	−7.65685	−0.247640
$957$	0	0
$958$	−6.48528	−0.209530
$959$	−14.6274	−0.472344
$960$	−27.7990	−0.897209
$961$	15.6274	0.504110
$962$	8.82843	0.284640
$963$	−11.3137	−0.364579
$964$	−44.6274	−1.43735
$965$	48.9706	1.57642
$966$	−27.3137	−0.878804
$967$	10.1421	0.326149	0.163075	−	0.986614i	$-0.447859\pi$
$967$	10.1421	0.326149	0.163075	+	0.986614i	$0.447859\pi$
$968$	0	0
$969$	−10.3431	−0.332270
$970$	−52.2843	−1.67875
$971$	7.31371	0.234708	0.117354	−	0.993090i	$-0.462559\pi$
$971$	7.31371	0.234708	0.117354	+	0.993090i	$0.462559\pi$
$972$	3.82843	0.122797
$973$	−43.3137	−1.38857
$974$	76.7696	2.45986
$975$	3.00000	0.0960769
$976$	27.9411	0.894374
$977$	13.8579	0.443352	0.221676	−	0.975120i	$-0.428847\pi$
$977$	13.8579	0.443352	0.221676	+	0.975120i	$0.428847\pi$
$978$	31.7990	1.01682
$979$	0	0
$980$	10.8284	0.345901
$981$	17.3137	0.552784
$982$	35.3137	1.12691
$983$	2.68629	0.0856794	0.0428397	−	0.999082i	$-0.486360\pi$
$983$	2.68629	0.0856794	0.0428397	+	0.999082i	$0.486360\pi$
$984$	−47.7990	−1.52378
$985$	46.6274	1.48567
$986$	−17.6569	−0.562309
$987$	−0.970563	−0.0308934
$988$	−10.8284	−0.344498
$989$	38.6274	1.22828
$990$	0	0
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	10.8284	0.343803
$993$	−2.14214	−0.0679786
$994$	13.6569	0.433169
$995$	29.2548	0.927441
$996$	29.3137	0.928840
$997$	−51.2548	−1.62326	−0.811628	−	0.584174i	$-0.801419\pi$
$997$	−51.2548	−1.62326	−0.811628	+	0.584174i	$0.801419\pi$
$998$	−5.17157	−0.163703
$999$	3.65685	0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4719.2.a.p.1.2		2
11.10	odd	2		39.2.a.b.1.1	✓	2
33.32	even	2		117.2.a.c.1.2		2
44.43	even	2		624.2.a.k.1.2		2
55.32	even	4		975.2.c.h.274.1		4
55.43	even	4		975.2.c.h.274.4		4
55.54	odd	2		975.2.a.l.1.2		2
77.76	even	2		1911.2.a.h.1.1		2
88.21	odd	2		2496.2.a.bf.1.1		2
88.43	even	2		2496.2.a.bi.1.1		2
99.32	even	6		1053.2.e.e.703.1		4
99.43	odd	6		1053.2.e.m.352.2		4
99.65	even	6		1053.2.e.e.352.1		4
99.76	odd	6		1053.2.e.m.703.2		4
132.131	odd	2		1872.2.a.w.1.1		2
143.10	odd	6		507.2.e.d.22.1		4
143.21	even	4		507.2.b.e.337.1		4
143.32	even	12		507.2.j.f.361.4		8
143.43	odd	6		507.2.e.d.484.1		4
143.54	even	12		507.2.j.f.316.1		8
143.76	even	12		507.2.j.f.316.4		8
143.87	odd	6		507.2.e.h.484.2		4
143.98	even	12		507.2.j.f.361.1		8
143.109	even	4		507.2.b.e.337.4		4
143.120	odd	6		507.2.e.h.22.2		4
143.142	odd	2		507.2.a.h.1.2		2
165.32	odd	4		2925.2.c.u.2224.4		4
165.98	odd	4		2925.2.c.u.2224.1		4
165.164	even	2		2925.2.a.v.1.1		2
231.230	odd	2		5733.2.a.u.1.2		2
264.131	odd	2		7488.2.a.co.1.2		2
264.197	even	2		7488.2.a.cl.1.2		2
429.164	odd	4		1521.2.b.j.1351.4		4
429.395	odd	4		1521.2.b.j.1351.1		4
429.428	even	2		1521.2.a.f.1.1		2
572.571	even	2		8112.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	11.10	odd	2
117.2.a.c.1.2		2	33.32	even	2
507.2.a.h.1.2		2	143.142	odd	2
507.2.b.e.337.1		4	143.21	even	4
507.2.b.e.337.4		4	143.109	even	4
507.2.e.d.22.1		4	143.10	odd	6
507.2.e.d.484.1		4	143.43	odd	6
507.2.e.h.22.2		4	143.120	odd	6
507.2.e.h.484.2		4	143.87	odd	6
507.2.j.f.316.1		8	143.54	even	12
507.2.j.f.316.4		8	143.76	even	12
507.2.j.f.361.1		8	143.98	even	12
507.2.j.f.361.4		8	143.32	even	12
624.2.a.k.1.2		2	44.43	even	2
975.2.a.l.1.2		2	55.54	odd	2
975.2.c.h.274.1		4	55.32	even	4
975.2.c.h.274.4		4	55.43	even	4
1053.2.e.e.352.1		4	99.65	even	6
1053.2.e.e.703.1		4	99.32	even	6
1053.2.e.m.352.2		4	99.43	odd	6
1053.2.e.m.703.2		4	99.76	odd	6
1521.2.a.f.1.1		2	429.428	even	2
1521.2.b.j.1351.1		4	429.395	odd	4
1521.2.b.j.1351.4		4	429.164	odd	4
1872.2.a.w.1.1		2	132.131	odd	2
1911.2.a.h.1.1		2	77.76	even	2
2496.2.a.bf.1.1		2	88.21	odd	2
2496.2.a.bi.1.1		2	88.43	even	2
2925.2.a.v.1.1		2	165.164	even	2
2925.2.c.u.2224.1		4	165.98	odd	4
2925.2.c.u.2224.4		4	165.32	odd	4
4719.2.a.p.1.2		2	1.1	even	1	trivial
5733.2.a.u.1.2		2	231.230	odd	2
7488.2.a.cl.1.2		2	264.197	even	2
7488.2.a.co.1.2		2	264.131	odd	2
8112.2.a.bm.1.1		2	572.571	even	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:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.82843 q^{5} +2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +6.82843 q^{10} +3.82843 q^{12} +1.00000 q^{13} +6.82843 q^{14} +2.82843 q^{15} +3.00000 q^{16} +3.65685 q^{17} +2.41421 q^{18} -2.82843 q^{19} +10.8284 q^{20} +2.82843 q^{21} -4.00000 q^{23} +4.41421 q^{24} +3.00000 q^{25} +2.41421 q^{26} +1.00000 q^{27} +10.8284 q^{28} -2.00000 q^{29} +6.82843 q^{30} -6.82843 q^{31} -1.58579 q^{32} +8.82843 q^{34} +8.00000 q^{35} +3.82843 q^{36} +3.65685 q^{37} -6.82843 q^{38} +1.00000 q^{39} +12.4853 q^{40} -10.8284 q^{41} +6.82843 q^{42} -9.65685 q^{43} +2.82843 q^{45} -9.65685 q^{46} -0.343146 q^{47} +3.00000 q^{48} +1.00000 q^{49} +7.24264 q^{50} +3.65685 q^{51} +3.82843 q^{52} -2.00000 q^{53} +2.41421 q^{54} +12.4853 q^{56} -2.82843 q^{57} -4.82843 q^{58} -3.65685 q^{59} +10.8284 q^{60} +9.31371 q^{61} -16.4853 q^{62} +2.82843 q^{63} -9.82843 q^{64} +2.82843 q^{65} +1.17157 q^{67} +14.0000 q^{68} -4.00000 q^{69} +19.3137 q^{70} +2.00000 q^{71} +4.41421 q^{72} -11.6569 q^{73} +8.82843 q^{74} +3.00000 q^{75} -10.8284 q^{76} +2.41421 q^{78} -11.3137 q^{79} +8.48528 q^{80} +1.00000 q^{81} -26.1421 q^{82} +7.65685 q^{83} +10.8284 q^{84} +10.3431 q^{85} -23.3137 q^{86} -2.00000 q^{87} +9.17157 q^{89} +6.82843 q^{90} +2.82843 q^{91} -15.3137 q^{92} -6.82843 q^{93} -0.828427 q^{94} -8.00000 q^{95} -1.58579 q^{96} -7.65685 q^{97} +2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{12} + 2 q^{13} + 8 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 16 q^{20} - 8 q^{23} + 6 q^{24} + 6 q^{25} + 2 q^{26} + 2 q^{27} + 16 q^{28} - 4 q^{29} + 8 q^{30} - 8 q^{31} - 6 q^{32} + 12 q^{34} + 16 q^{35} + 2 q^{36} - 4 q^{37} - 8 q^{38} + 2 q^{39} + 8 q^{40} - 16 q^{41} + 8 q^{42} - 8 q^{43} - 8 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} + 6 q^{50} - 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{56} - 4 q^{58} + 4 q^{59} + 16 q^{60} - 4 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{67} + 28 q^{68} - 8 q^{69} + 16 q^{70} + 4 q^{71} + 6 q^{72} - 12 q^{73} + 12 q^{74} + 6 q^{75} - 16 q^{76} + 2 q^{78} + 2 q^{81} - 24 q^{82} + 4 q^{83} + 16 q^{84} + 32 q^{85} - 24 q^{86} - 4 q^{87} + 24 q^{89} + 8 q^{90} - 8 q^{92} - 8 q^{93} + 4 q^{94} - 16 q^{95} - 6 q^{96} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^12 + 2 * q^13 + 8 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 16 * q^20 - 8 * q^23 + 6 * q^24 + 6 * q^25 + 2 * q^26 + 2 * q^27 + 16 * q^28 - 4 * q^29 + 8 * q^30 - 8 * q^31 - 6 * q^32 + 12 * q^34 + 16 * q^35 + 2 * q^36 - 4 * q^37 - 8 * q^38 + 2 * q^39 + 8 * q^40 - 16 * q^41 + 8 * q^42 - 8 * q^43 - 8 * q^46 - 12 * q^47 + 6 * q^48 + 2 * q^49 + 6 * q^50 - 4 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^56 - 4 * q^58 + 4 * q^59 + 16 * q^60 - 4 * q^61 - 16 * q^62 - 14 * q^64 + 8 * q^67 + 28 * q^68 - 8 * q^69 + 16 * q^70 + 4 * q^71 + 6 * q^72 - 12 * q^73 + 12 * q^74 + 6 * q^75 - 16 * q^76 + 2 * q^78 + 2 * q^81 - 24 * q^82 + 4 * q^83 + 16 * q^84 + 32 * q^85 - 24 * q^86 - 4 * q^87 + 24 * q^89 + 8 * q^90 - 8 * q^92 - 8 * q^93 + 4 * q^94 - 16 * q^95 - 6 * q^96 - 4 * q^97 + 2 * q^98

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