LMFDB - Embedded newform 429.2.a.c.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:	$\Q(\zeta_{8})^+$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.242641 q^{10} +1.00000 q^{11} -1.82843 q^{12} -1.00000 q^{13} -2.00000 q^{14} -0.585786 q^{15} +3.00000 q^{16} -5.41421 q^{17} +0.414214 q^{18} -6.00000 q^{19} +1.07107 q^{20} -4.82843 q^{21} +0.414214 q^{22} -0.828427 q^{23} -1.58579 q^{24} -4.65685 q^{25} -0.414214 q^{26} +1.00000 q^{27} +8.82843 q^{28} +4.24264 q^{29} -0.242641 q^{30} +4.24264 q^{31} +4.41421 q^{32} +1.00000 q^{33} -2.24264 q^{34} +2.82843 q^{35} -1.82843 q^{36} +7.65685 q^{37} -2.48528 q^{38} -1.00000 q^{39} +0.928932 q^{40} -12.0000 q^{41} -2.00000 q^{42} +5.07107 q^{43} -1.82843 q^{44} -0.585786 q^{45} -0.343146 q^{46} -8.48528 q^{47} +3.00000 q^{48} +16.3137 q^{49} -1.92893 q^{50} -5.41421 q^{51} +1.82843 q^{52} +13.3137 q^{53} +0.414214 q^{54} -0.585786 q^{55} +7.65685 q^{56} -6.00000 q^{57} +1.75736 q^{58} -10.8284 q^{59} +1.07107 q^{60} -2.00000 q^{61} +1.75736 q^{62} -4.82843 q^{63} -4.17157 q^{64} +0.585786 q^{65} +0.414214 q^{66} -3.07107 q^{67} +9.89949 q^{68} -0.828427 q^{69} +1.17157 q^{70} -5.65685 q^{71} -1.58579 q^{72} +12.4853 q^{73} +3.17157 q^{74} -4.65685 q^{75} +10.9706 q^{76} -4.82843 q^{77} -0.414214 q^{78} -9.07107 q^{79} -1.75736 q^{80} +1.00000 q^{81} -4.97056 q^{82} +11.3137 q^{83} +8.82843 q^{84} +3.17157 q^{85} +2.10051 q^{86} +4.24264 q^{87} -1.58579 q^{88} -15.4142 q^{89} -0.242641 q^{90} +4.82843 q^{91} +1.51472 q^{92} +4.24264 q^{93} -3.51472 q^{94} +3.51472 q^{95} +4.41421 q^{96} +10.0000 q^{97} +6.75736 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 12 q^{19} - 12 q^{20} - 4 q^{21} - 2 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 12 q^{28} + 8 q^{30} + 6 q^{32} + 2 q^{33} + 4 q^{34} + 2 q^{36} + 4 q^{37} + 12 q^{38} - 2 q^{39} + 16 q^{40} - 24 q^{41} - 4 q^{42} - 4 q^{43} + 2 q^{44} - 4 q^{45} - 12 q^{46} + 6 q^{48} + 10 q^{49} - 18 q^{50} - 8 q^{51} - 2 q^{52} + 4 q^{53} - 2 q^{54} - 4 q^{55} + 4 q^{56} - 12 q^{57} + 12 q^{58} - 16 q^{59} - 12 q^{60} - 4 q^{61} + 12 q^{62} - 4 q^{63} - 14 q^{64} + 4 q^{65} - 2 q^{66} + 8 q^{67} + 4 q^{69} + 8 q^{70} - 6 q^{72} + 8 q^{73} + 12 q^{74} + 2 q^{75} - 12 q^{76} - 4 q^{77} + 2 q^{78} - 4 q^{79} - 12 q^{80} + 2 q^{81} + 24 q^{82} + 12 q^{84} + 12 q^{85} + 24 q^{86} - 6 q^{88} - 28 q^{89} + 8 q^{90} + 4 q^{91} + 20 q^{92} - 24 q^{94} + 24 q^{95} + 6 q^{96} + 20 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 12 * q^19 - 12 * q^20 - 4 * q^21 - 2 * q^22 + 4 * q^23 - 6 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 + 12 * q^28 + 8 * q^30 + 6 * q^32 + 2 * q^33 + 4 * q^34 + 2 * q^36 + 4 * q^37 + 12 * q^38 - 2 * q^39 + 16 * q^40 - 24 * q^41 - 4 * q^42 - 4 * q^43 + 2 * q^44 - 4 * q^45 - 12 * q^46 + 6 * q^48 + 10 * q^49 - 18 * q^50 - 8 * q^51 - 2 * q^52 + 4 * q^53 - 2 * q^54 - 4 * q^55 + 4 * q^56 - 12 * q^57 + 12 * q^58 - 16 * q^59 - 12 * q^60 - 4 * q^61 + 12 * q^62 - 4 * q^63 - 14 * q^64 + 4 * q^65 - 2 * q^66 + 8 * q^67 + 4 * q^69 + 8 * q^70 - 6 * q^72 + 8 * q^73 + 12 * q^74 + 2 * q^75 - 12 * q^76 - 4 * q^77 + 2 * q^78 - 4 * q^79 - 12 * q^80 + 2 * q^81 + 24 * q^82 + 12 * q^84 + 12 * q^85 + 24 * q^86 - 6 * q^88 - 28 * q^89 + 8 * q^90 + 4 * q^91 + 20 * q^92 - 24 * q^94 + 24 * q^95 + 6 * q^96 + 20 * q^97 + 22 * q^98 + 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82843	−0.914214
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0.414214	0.169102
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	−0.242641	−0.0767297
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.82843	−0.527821
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.00000	−0.534522
$15$	−0.585786	−0.151249
$16$	3.00000	0.750000
$17$	−5.41421	−1.31314	−0.656570	−	0.754265i	$-0.727993\pi$
$17$	−5.41421	−1.31314	−0.656570	+	0.754265i	$0.727993\pi$
$18$	0.414214	0.0976311
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.07107	0.239498
$21$	−4.82843	−1.05365
$22$	0.414214	0.0883106
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	−1.58579	−0.323697
$25$	−4.65685	−0.931371
$26$	−0.414214	−0.0812340
$27$	1.00000	0.192450
$28$	8.82843	1.66842
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	−0.242641	−0.0442999
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	4.41421	0.780330
$33$	1.00000	0.174078
$34$	−2.24264	−0.384610
$35$	2.82843	0.478091
$36$	−1.82843	−0.304738
$37$	7.65685	1.25878	0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685	1.25878	0.629390	+	0.777090i	$0.283305\pi$
$38$	−2.48528	−0.403166
$39$	−1.00000	−0.160128
$40$	0.928932	0.146877
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	−2.00000	−0.308607
$43$	5.07107	0.773331	0.386665	−	0.922220i	$-0.373627\pi$
$43$	5.07107	0.773331	0.386665	+	0.922220i	$0.373627\pi$
$44$	−1.82843	−0.275646
$45$	−0.585786	−0.0873239
$46$	−0.343146	−0.0505941
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	3.00000	0.433013
$49$	16.3137	2.33053
$50$	−1.92893	−0.272792
$51$	−5.41421	−0.758142
$52$	1.82843	0.253557
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	0.414214	0.0563673
$55$	−0.585786	−0.0789874
$56$	7.65685	1.02319
$57$	−6.00000	−0.794719
$58$	1.75736	0.230753
$59$	−10.8284	−1.40974	−0.704871	−	0.709336i	$-0.748995\pi$
$59$	−10.8284	−1.40974	−0.704871	+	0.709336i	$0.748995\pi$
$60$	1.07107	0.138274
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	1.75736	0.223185
$63$	−4.82843	−0.608325
$64$	−4.17157	−0.521447
$65$	0.585786	0.0726579
$66$	0.414214	0.0509862
$67$	−3.07107	−0.375191	−0.187595	−	0.982246i	$-0.560069\pi$
$67$	−3.07107	−0.375191	−0.187595	+	0.982246i	$0.560069\pi$
$68$	9.89949	1.20049
$69$	−0.828427	−0.0997309
$70$	1.17157	0.140030
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	−1.58579	−0.186887
$73$	12.4853	1.46129	0.730646	−	0.682757i	$-0.239219\pi$
$73$	12.4853	1.46129	0.730646	+	0.682757i	$0.239219\pi$
$74$	3.17157	0.368688
$75$	−4.65685	−0.537727
$76$	10.9706	1.25841
$77$	−4.82843	−0.550250
$78$	−0.414214	−0.0469005
$79$	−9.07107	−1.02057	−0.510287	−	0.860004i	$-0.670461\pi$
$79$	−9.07107	−1.02057	−0.510287	+	0.860004i	$0.670461\pi$
$80$	−1.75736	−0.196479
$81$	1.00000	0.111111
$82$	−4.97056	−0.548907
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	8.82843	0.963260
$85$	3.17157	0.344005
$86$	2.10051	0.226503
$87$	4.24264	0.454859
$88$	−1.58579	−0.169045
$89$	−15.4142	−1.63390	−0.816952	−	0.576706i	$-0.804338\pi$
$89$	−15.4142	−1.63390	−0.816952	+	0.576706i	$0.804338\pi$
$90$	−0.242641	−0.0255766
$91$	4.82843	0.506157
$92$	1.51472	0.157920
$93$	4.24264	0.439941
$94$	−3.51472	−0.362516
$95$	3.51472	0.360603
$96$	4.41421	0.450524
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	6.75736	0.682596
$99$	1.00000	0.100504
$100$	8.51472	0.851472
$101$	−9.89949	−0.985037	−0.492518	−	0.870302i	$-0.663924\pi$
$101$	−9.89949	−0.985037	−0.492518	+	0.870302i	$0.663924\pi$
$102$	−2.24264	−0.222055
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	1.58579	0.155499
$105$	2.82843	0.276026
$106$	5.51472	0.535637
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	−1.82843	−0.175940
$109$	−6.82843	−0.654045	−0.327022	−	0.945017i	$-0.606045\pi$
$109$	−6.82843	−0.654045	−0.327022	+	0.945017i	$0.606045\pi$
$110$	−0.242641	−0.0231349
$111$	7.65685	0.726756
$112$	−14.4853	−1.36873
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	−2.48528	−0.232768
$115$	0.485281	0.0452527
$116$	−7.75736	−0.720253
$117$	−1.00000	−0.0924500
$118$	−4.48528	−0.412904
$119$	26.1421	2.39645
$120$	0.928932	0.0847995
$121$	1.00000	0.0909091
$122$	−0.828427	−0.0750023
$123$	−12.0000	−1.08200
$124$	−7.75736	−0.696631
$125$	5.65685	0.505964
$126$	−2.00000	−0.178174
$127$	14.2426	1.26383	0.631915	−	0.775038i	$-0.282269\pi$
$127$	14.2426	1.26383	0.631915	+	0.775038i	$0.282269\pi$
$128$	−10.5563	−0.933058
$129$	5.07107	0.446483
$130$	0.242641	0.0212810
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	−1.82843	−0.159144
$133$	28.9706	2.51207
$134$	−1.27208	−0.109891
$135$	−0.585786	−0.0504165
$136$	8.58579	0.736225
$137$	−6.72792	−0.574805	−0.287403	−	0.957810i	$-0.592792\pi$
$137$	−6.72792	−0.574805	−0.287403	+	0.957810i	$0.592792\pi$
$138$	−0.343146	−0.0292105
$139$	−11.4142	−0.968141	−0.484070	−	0.875029i	$-0.660842\pi$
$139$	−11.4142	−0.968141	−0.484070	+	0.875029i	$0.660842\pi$
$140$	−5.17157	−0.437078
$141$	−8.48528	−0.714590
$142$	−2.34315	−0.196632
$143$	−1.00000	−0.0836242
$144$	3.00000	0.250000
$145$	−2.48528	−0.206391
$146$	5.17157	0.428002
$147$	16.3137	1.34553
$148$	−14.0000	−1.15079
$149$	3.51472	0.287937	0.143968	−	0.989582i	$-0.454014\pi$
$149$	3.51472	0.287937	0.143968	+	0.989582i	$0.454014\pi$
$150$	−1.92893	−0.157497
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	9.51472	0.771746
$153$	−5.41421	−0.437713
$154$	−2.00000	−0.161165
$155$	−2.48528	−0.199623
$156$	1.82843	0.146391
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−3.75736	−0.298919
$159$	13.3137	1.05585
$160$	−2.58579	−0.204424
$161$	4.00000	0.315244
$162$	0.414214	0.0325437
$163$	−12.7279	−0.996928	−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279	−0.996928	−0.498464	+	0.866910i	$0.666102\pi$
$164$	21.9411	1.71331
$165$	−0.585786	−0.0456034
$166$	4.68629	0.363727
$167$	−17.6569	−1.36633	−0.683164	−	0.730265i	$-0.739397\pi$
$167$	−17.6569	−1.36633	−0.683164	+	0.730265i	$0.739397\pi$
$168$	7.65685	0.590739
$169$	1.00000	0.0769231
$170$	1.31371	0.100757
$171$	−6.00000	−0.458831
$172$	−9.27208	−0.706989
$173$	23.0711	1.75406	0.877030	−	0.480435i	$-0.159521\pi$
$173$	23.0711	1.75406	0.877030	+	0.480435i	$0.159521\pi$
$174$	1.75736	0.133225
$175$	22.4853	1.69973
$176$	3.00000	0.226134
$177$	−10.8284	−0.813914
$178$	−6.38478	−0.478559
$179$	4.14214	0.309598	0.154799	−	0.987946i	$-0.450527\pi$
$179$	4.14214	0.309598	0.154799	+	0.987946i	$0.450527\pi$
$180$	1.07107	0.0798327
$181$	8.97056	0.666777	0.333388	−	0.942790i	$-0.391808\pi$
$181$	8.97056	0.666777	0.333388	+	0.942790i	$0.391808\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	1.31371	0.0968479
$185$	−4.48528	−0.329764
$186$	1.75736	0.128856
$187$	−5.41421	−0.395927
$188$	15.5147	1.13153
$189$	−4.82843	−0.351216
$190$	1.45584	0.105618
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	−4.17157	−0.301057
$193$	−13.1716	−0.948111	−0.474055	−	0.880495i	$-0.657210\pi$
$193$	−13.1716	−0.948111	−0.474055	+	0.880495i	$0.657210\pi$
$194$	4.14214	0.297388
$195$	0.585786	0.0419490
$196$	−29.8284	−2.13060
$197$	−14.8284	−1.05648	−0.528241	−	0.849095i	$-0.677148\pi$
$197$	−14.8284	−1.05648	−0.528241	+	0.849095i	$0.677148\pi$
$198$	0.414214	0.0294369
$199$	−10.8284	−0.767607	−0.383803	−	0.923415i	$-0.625386\pi$
$199$	−10.8284	−0.767607	−0.383803	+	0.923415i	$0.625386\pi$
$200$	7.38478	0.522183
$201$	−3.07107	−0.216616
$202$	−4.10051	−0.288511
$203$	−20.4853	−1.43778
$204$	9.89949	0.693103
$205$	7.02944	0.490957
$206$	−3.51472	−0.244882
$207$	−0.828427	−0.0575797
$208$	−3.00000	−0.208013
$209$	−6.00000	−0.415029
$210$	1.17157	0.0808462
$211$	−18.7279	−1.28928	−0.644642	−	0.764485i	$-0.722993\pi$
$211$	−18.7279	−1.28928	−0.644642	+	0.764485i	$0.722993\pi$
$212$	−24.3431	−1.67189
$213$	−5.65685	−0.387601
$214$	3.51472	0.240261
$215$	−2.97056	−0.202591
$216$	−1.58579	−0.107899
$217$	−20.4853	−1.39063
$218$	−2.82843	−0.191565
$219$	12.4853	0.843677
$220$	1.07107	0.0722114
$221$	5.41421	0.364199
$222$	3.17157	0.212862
$223$	−2.58579	−0.173157	−0.0865785	−	0.996245i	$-0.527593\pi$
$223$	−2.58579	−0.173157	−0.0865785	+	0.996245i	$0.527593\pi$
$224$	−21.3137	−1.42408
$225$	−4.65685	−0.310457
$226$	−4.82843	−0.321182
$227$	−10.9706	−0.728142	−0.364071	−	0.931371i	$-0.618613\pi$
$227$	−10.9706	−0.728142	−0.364071	+	0.931371i	$0.618613\pi$
$228$	10.9706	0.726543
$229$	9.51472	0.628750	0.314375	−	0.949299i	$-0.398205\pi$
$229$	9.51472	0.628750	0.314375	+	0.949299i	$0.398205\pi$
$230$	0.201010	0.0132542
$231$	−4.82843	−0.317687
$232$	−6.72792	−0.441710
$233$	−4.24264	−0.277945	−0.138972	−	0.990296i	$-0.544380\pi$
$233$	−4.24264	−0.277945	−0.138972	+	0.990296i	$0.544380\pi$
$234$	−0.414214	−0.0270780
$235$	4.97056	0.324244
$236$	19.7990	1.28880
$237$	−9.07107	−0.589229
$238$	10.8284	0.701903
$239$	−4.34315	−0.280935	−0.140467	−	0.990085i	$-0.544861\pi$
$239$	−4.34315	−0.280935	−0.140467	+	0.990085i	$0.544861\pi$
$240$	−1.75736	−0.113437
$241$	−21.3137	−1.37294	−0.686468	−	0.727160i	$-0.740840\pi$
$241$	−21.3137	−1.37294	−0.686468	+	0.727160i	$0.740840\pi$
$242$	0.414214	0.0266267
$243$	1.00000	0.0641500
$244$	3.65685	0.234106
$245$	−9.55635	−0.610533
$246$	−4.97056	−0.316912
$247$	6.00000	0.381771
$248$	−6.72792	−0.427223
$249$	11.3137	0.716977
$250$	2.34315	0.148194
$251$	15.3137	0.966593	0.483296	−	0.875457i	$-0.339439\pi$
$251$	15.3137	0.966593	0.483296	+	0.875457i	$0.339439\pi$
$252$	8.82843	0.556139
$253$	−0.828427	−0.0520828
$254$	5.89949	0.370167
$255$	3.17157	0.198612
$256$	3.97056	0.248160
$257$	−15.1716	−0.946377	−0.473188	−	0.880961i	$-0.656897\pi$
$257$	−15.1716	−0.946377	−0.473188	+	0.880961i	$0.656897\pi$
$258$	2.10051	0.130772
$259$	−36.9706	−2.29724
$260$	−1.07107	−0.0664248
$261$	4.24264	0.262613
$262$	−7.02944	−0.434280
$263$	−7.79899	−0.480906	−0.240453	−	0.970661i	$-0.577296\pi$
$263$	−7.79899	−0.480906	−0.240453	+	0.970661i	$0.577296\pi$
$264$	−1.58579	−0.0975984
$265$	−7.79899	−0.479088
$266$	12.0000	0.735767
$267$	−15.4142	−0.943335
$268$	5.61522	0.343004
$269$	16.1421	0.984203	0.492102	−	0.870538i	$-0.336229\pi$
$269$	16.1421	0.984203	0.492102	+	0.870538i	$0.336229\pi$
$270$	−0.242641	−0.0147666
$271$	−15.6569	−0.951086	−0.475543	−	0.879692i	$-0.657748\pi$
$271$	−15.6569	−0.951086	−0.475543	+	0.879692i	$0.657748\pi$
$272$	−16.2426	−0.984855
$273$	4.82843	0.292230
$274$	−2.78680	−0.168357
$275$	−4.65685	−0.280819
$276$	1.51472	0.0911753
$277$	19.6569	1.18107	0.590533	−	0.807014i	$-0.298918\pi$
$277$	19.6569	1.18107	0.590533	+	0.807014i	$0.298918\pi$
$278$	−4.72792	−0.283562
$279$	4.24264	0.254000
$280$	−4.48528	−0.268047
$281$	10.3431	0.617020	0.308510	−	0.951221i	$-0.400170\pi$
$281$	10.3431	0.617020	0.308510	+	0.951221i	$0.400170\pi$
$282$	−3.51472	−0.209298
$283$	−1.27208	−0.0756172	−0.0378086	−	0.999285i	$-0.512038\pi$
$283$	−1.27208	−0.0756172	−0.0378086	+	0.999285i	$0.512038\pi$
$284$	10.3431	0.613753
$285$	3.51472	0.208194
$286$	−0.414214	−0.0244930
$287$	57.9411	3.42016
$288$	4.41421	0.260110
$289$	12.3137	0.724336
$290$	−1.02944	−0.0604506
$291$	10.0000	0.586210
$292$	−22.8284	−1.33593
$293$	14.8284	0.866286	0.433143	−	0.901325i	$-0.357405\pi$
$293$	14.8284	0.866286	0.433143	+	0.901325i	$0.357405\pi$
$294$	6.75736	0.394097
$295$	6.34315	0.369312
$296$	−12.1421	−0.705747
$297$	1.00000	0.0580259
$298$	1.45584	0.0843348
$299$	0.828427	0.0479092
$300$	8.51472	0.491598
$301$	−24.4853	−1.41131
$302$	2.48528	0.143012
$303$	−9.89949	−0.568711
$304$	−18.0000	−1.03237
$305$	1.17157	0.0670841
$306$	−2.24264	−0.128203
$307$	26.2843	1.50012	0.750061	−	0.661368i	$-0.230024\pi$
$307$	26.2843	1.50012	0.750061	+	0.661368i	$0.230024\pi$
$308$	8.82843	0.503046
$309$	−8.48528	−0.482711
$310$	−1.02944	−0.0584681
$311$	−4.82843	−0.273795	−0.136897	−	0.990585i	$-0.543713\pi$
$311$	−4.82843	−0.273795	−0.136897	+	0.990585i	$0.543713\pi$
$312$	1.58579	0.0897775
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	1.65685	0.0935017
$315$	2.82843	0.159364
$316$	16.5858	0.933023
$317$	−15.4142	−0.865748	−0.432874	−	0.901454i	$-0.642501\pi$
$317$	−15.4142	−0.865748	−0.432874	+	0.901454i	$0.642501\pi$
$318$	5.51472	0.309250
$319$	4.24264	0.237542
$320$	2.44365	0.136604
$321$	8.48528	0.473602
$322$	1.65685	0.0923329
$323$	32.4853	1.80753
$324$	−1.82843	−0.101579
$325$	4.65685	0.258316
$326$	−5.27208	−0.291993
$327$	−6.82843	−0.377613
$328$	19.0294	1.05072
$329$	40.9706	2.25878
$330$	−0.242641	−0.0133569
$331$	−11.7574	−0.646243	−0.323121	−	0.946358i	$-0.604732\pi$
$331$	−11.7574	−0.646243	−0.323121	+	0.946358i	$0.604732\pi$
$332$	−20.6863	−1.13531
$333$	7.65685	0.419593
$334$	−7.31371	−0.400188
$335$	1.79899	0.0982893
$336$	−14.4853	−0.790237
$337$	−19.4558	−1.05983	−0.529914	−	0.848052i	$-0.677776\pi$
$337$	−19.4558	−1.05983	−0.529914	+	0.848052i	$0.677776\pi$
$338$	0.414214	0.0225302
$339$	−11.6569	−0.633113
$340$	−5.79899	−0.314494
$341$	4.24264	0.229752
$342$	−2.48528	−0.134389
$343$	−44.9706	−2.42818
$344$	−8.04163	−0.433576
$345$	0.485281	0.0261267
$346$	9.55635	0.513753
$347$	8.48528	0.455514	0.227757	−	0.973718i	$-0.426861\pi$
$347$	8.48528	0.455514	0.227757	+	0.973718i	$0.426861\pi$
$348$	−7.75736	−0.415838
$349$	18.9706	1.01547	0.507735	−	0.861513i	$-0.330483\pi$
$349$	18.9706	1.01547	0.507735	+	0.861513i	$0.330483\pi$
$350$	9.31371	0.497839
$351$	−1.00000	−0.0533761
$352$	4.41421	0.235278
$353$	−2.44365	−0.130062	−0.0650312	−	0.997883i	$-0.520715\pi$
$353$	−2.44365	−0.130062	−0.0650312	+	0.997883i	$0.520715\pi$
$354$	−4.48528	−0.238390
$355$	3.31371	0.175873
$356$	28.1838	1.49374
$357$	26.1421	1.38359
$358$	1.71573	0.0906791
$359$	−8.34315	−0.440334	−0.220167	−	0.975462i	$-0.570660\pi$
$359$	−8.34315	−0.440334	−0.220167	+	0.975462i	$0.570660\pi$
$360$	0.928932	0.0489590
$361$	17.0000	0.894737
$362$	3.71573	0.195294
$363$	1.00000	0.0524864
$364$	−8.82843	−0.462735
$365$	−7.31371	−0.382817
$366$	−0.828427	−0.0433026
$367$	−4.97056	−0.259461	−0.129731	−	0.991549i	$-0.541411\pi$
$367$	−4.97056	−0.259461	−0.129731	+	0.991549i	$0.541411\pi$
$368$	−2.48528	−0.129554
$369$	−12.0000	−0.624695
$370$	−1.85786	−0.0965858
$371$	−64.2843	−3.33747
$372$	−7.75736	−0.402200
$373$	14.4853	0.750019	0.375010	−	0.927021i	$-0.377639\pi$
$373$	14.4853	0.750019	0.375010	+	0.927021i	$0.377639\pi$
$374$	−2.24264	−0.115964
$375$	5.65685	0.292119
$376$	13.4558	0.693932
$377$	−4.24264	−0.218507
$378$	−2.00000	−0.102869
$379$	23.0711	1.18508	0.592541	−	0.805541i	$-0.298125\pi$
$379$	23.0711	1.18508	0.592541	+	0.805541i	$0.298125\pi$
$380$	−6.42641	−0.329668
$381$	14.2426	0.729673
$382$	1.37258	0.0702275
$383$	−2.82843	−0.144526	−0.0722629	−	0.997386i	$-0.523022\pi$
$383$	−2.82843	−0.144526	−0.0722629	+	0.997386i	$0.523022\pi$
$384$	−10.5563	−0.538701
$385$	2.82843	0.144150
$386$	−5.45584	−0.277695
$387$	5.07107	0.257777
$388$	−18.2843	−0.928243
$389$	14.4853	0.734433	0.367216	−	0.930136i	$-0.380311\pi$
$389$	14.4853	0.734433	0.367216	+	0.930136i	$0.380311\pi$
$390$	0.242641	0.0122866
$391$	4.48528	0.226830
$392$	−25.8701	−1.30664
$393$	−16.9706	−0.856052
$394$	−6.14214	−0.309436
$395$	5.31371	0.267362
$396$	−1.82843	−0.0918819
$397$	24.6274	1.23601	0.618007	−	0.786172i	$-0.287940\pi$
$397$	24.6274	1.23601	0.618007	+	0.786172i	$0.287940\pi$
$398$	−4.48528	−0.224827
$399$	28.9706	1.45034
$400$	−13.9706	−0.698528
$401$	−13.0711	−0.652738	−0.326369	−	0.945242i	$-0.605825\pi$
$401$	−13.0711	−0.652738	−0.326369	+	0.945242i	$0.605825\pi$
$402$	−1.27208	−0.0634455
$403$	−4.24264	−0.211341
$404$	18.1005	0.900534
$405$	−0.585786	−0.0291080
$406$	−8.48528	−0.421117
$407$	7.65685	0.379536
$408$	8.58579	0.425060
$409$	11.5147	0.569366	0.284683	−	0.958622i	$-0.408112\pi$
$409$	11.5147	0.569366	0.284683	+	0.958622i	$0.408112\pi$
$410$	2.91169	0.143798
$411$	−6.72792	−0.331864
$412$	15.5147	0.764355
$413$	52.2843	2.57274
$414$	−0.343146	−0.0168647
$415$	−6.62742	−0.325327
$416$	−4.41421	−0.216425
$417$	−11.4142	−0.558956
$418$	−2.48528	−0.121559
$419$	−3.85786	−0.188469	−0.0942345	−	0.995550i	$-0.530040\pi$
$419$	−3.85786	−0.188469	−0.0942345	+	0.995550i	$0.530040\pi$
$420$	−5.17157	−0.252347
$421$	−8.62742	−0.420475	−0.210237	−	0.977650i	$-0.567424\pi$
$421$	−8.62742	−0.420475	−0.210237	+	0.977650i	$0.567424\pi$
$422$	−7.75736	−0.377622
$423$	−8.48528	−0.412568
$424$	−21.1127	−1.02532
$425$	25.2132	1.22302
$426$	−2.34315	−0.113526
$427$	9.65685	0.467328
$428$	−15.5147	−0.749932
$429$	−1.00000	−0.0482805
$430$	−1.23045	−0.0593374
$431$	0.686292	0.0330575	0.0165287	−	0.999863i	$-0.494738\pi$
$431$	0.686292	0.0330575	0.0165287	+	0.999863i	$0.494738\pi$
$432$	3.00000	0.144338
$433$	37.3137	1.79318	0.896591	−	0.442859i	$-0.146036\pi$
$433$	37.3137	1.79318	0.896591	+	0.442859i	$0.146036\pi$
$434$	−8.48528	−0.407307
$435$	−2.48528	−0.119160
$436$	12.4853	0.597937
$437$	4.97056	0.237774
$438$	5.17157	0.247107
$439$	−6.92893	−0.330700	−0.165350	−	0.986235i	$-0.552875\pi$
$439$	−6.92893	−0.330700	−0.165350	+	0.986235i	$0.552875\pi$
$440$	0.928932	0.0442851
$441$	16.3137	0.776843
$442$	2.24264	0.106672
$443$	−14.3431	−0.681463	−0.340732	−	0.940161i	$-0.610675\pi$
$443$	−14.3431	−0.681463	−0.340732	+	0.940161i	$0.610675\pi$
$444$	−14.0000	−0.664411
$445$	9.02944	0.428036
$446$	−1.07107	−0.0507165
$447$	3.51472	0.166240
$448$	20.1421	0.951626
$449$	7.41421	0.349898	0.174949	−	0.984577i	$-0.444024\pi$
$449$	7.41421	0.349898	0.174949	+	0.984577i	$0.444024\pi$
$450$	−1.92893	−0.0909307
$451$	−12.0000	−0.565058
$452$	21.3137	1.00251
$453$	6.00000	0.281905
$454$	−4.54416	−0.213268
$455$	−2.82843	−0.132599
$456$	9.51472	0.445568
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	3.94113	0.184157
$459$	−5.41421	−0.252714
$460$	−0.887302	−0.0413707
$461$	3.31371	0.154335	0.0771674	−	0.997018i	$-0.475412\pi$
$461$	3.31371	0.154335	0.0771674	+	0.997018i	$0.475412\pi$
$462$	−2.00000	−0.0930484
$463$	24.2426	1.12665	0.563326	−	0.826235i	$-0.309522\pi$
$463$	24.2426	1.12665	0.563326	+	0.826235i	$0.309522\pi$
$464$	12.7279	0.590879
$465$	−2.48528	−0.115252
$466$	−1.75736	−0.0814081
$467$	−13.7990	−0.638541	−0.319271	−	0.947664i	$-0.603438\pi$
$467$	−13.7990	−0.638541	−0.319271	+	0.947664i	$0.603438\pi$
$468$	1.82843	0.0845191
$469$	14.8284	0.684713
$470$	2.05887	0.0949688
$471$	4.00000	0.184310
$472$	17.1716	0.790386
$473$	5.07107	0.233168
$474$	−3.75736	−0.172581
$475$	27.9411	1.28203
$476$	−47.7990	−2.19086
$477$	13.3137	0.609593
$478$	−1.79899	−0.0822839
$479$	27.9411	1.27666	0.638331	−	0.769762i	$-0.279625\pi$
$479$	27.9411	1.27666	0.638331	+	0.769762i	$0.279625\pi$
$480$	−2.58579	−0.118024
$481$	−7.65685	−0.349123
$482$	−8.82843	−0.402124
$483$	4.00000	0.182006
$484$	−1.82843	−0.0831103
$485$	−5.85786	−0.265992
$486$	0.414214	0.0187891
$487$	4.72792	0.214243	0.107121	−	0.994246i	$-0.465837\pi$
$487$	4.72792	0.214243	0.107121	+	0.994246i	$0.465837\pi$
$488$	3.17157	0.143570
$489$	−12.7279	−0.575577
$490$	−3.95837	−0.178821
$491$	−10.8284	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$491$	−10.8284	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$492$	21.9411	0.989182
$493$	−22.9706	−1.03454
$494$	2.48528	0.111818
$495$	−0.585786	−0.0263291
$496$	12.7279	0.571501
$497$	27.3137	1.22519
$498$	4.68629	0.209998
$499$	−14.3848	−0.643951	−0.321976	−	0.946748i	$-0.604347\pi$
$499$	−14.3848	−0.643951	−0.321976	+	0.946748i	$0.604347\pi$
$500$	−10.3431	−0.462560
$501$	−17.6569	−0.788850
$502$	6.34315	0.283108
$503$	3.51472	0.156714	0.0783568	−	0.996925i	$-0.475033\pi$
$503$	3.51472	0.156714	0.0783568	+	0.996925i	$0.475033\pi$
$504$	7.65685	0.341063
$505$	5.79899	0.258052
$506$	−0.343146	−0.0152547
$507$	1.00000	0.0444116
$508$	−26.0416	−1.15541
$509$	−3.89949	−0.172842	−0.0864210	−	0.996259i	$-0.527543\pi$
$509$	−3.89949	−0.172842	−0.0864210	+	0.996259i	$0.527543\pi$
$510$	1.31371	0.0581720
$511$	−60.2843	−2.66682
$512$	22.7574	1.00574
$513$	−6.00000	−0.264906
$514$	−6.28427	−0.277187
$515$	4.97056	0.219029
$516$	−9.27208	−0.408180
$517$	−8.48528	−0.373182
$518$	−15.3137	−0.672846
$519$	23.0711	1.01271
$520$	−0.928932	−0.0407364
$521$	1.02944	0.0451005	0.0225502	−	0.999746i	$-0.492821\pi$
$521$	1.02944	0.0451005	0.0225502	+	0.999746i	$0.492821\pi$
$522$	1.75736	0.0769175
$523$	2.24264	0.0980638	0.0490319	−	0.998797i	$-0.484386\pi$
$523$	2.24264	0.0980638	0.0490319	+	0.998797i	$0.484386\pi$
$524$	31.0294	1.35553
$525$	22.4853	0.981338
$526$	−3.23045	−0.140854
$527$	−22.9706	−1.00061
$528$	3.00000	0.130558
$529$	−22.3137	−0.970161
$530$	−3.23045	−0.140322
$531$	−10.8284	−0.469914
$532$	−52.9706	−2.29657
$533$	12.0000	0.519778
$534$	−6.38478	−0.276296
$535$	−4.97056	−0.214896
$536$	4.87006	0.210354
$537$	4.14214	0.178746
$538$	6.68629	0.288266
$539$	16.3137	0.702681
$540$	1.07107	0.0460914
$541$	1.02944	0.0442590	0.0221295	−	0.999755i	$-0.492955\pi$
$541$	1.02944	0.0442590	0.0221295	+	0.999755i	$0.492955\pi$
$542$	−6.48528	−0.278567
$543$	8.97056	0.384964
$544$	−23.8995	−1.02468
$545$	4.00000	0.171341
$546$	2.00000	0.0855921
$547$	5.75736	0.246167	0.123083	−	0.992396i	$-0.460722\pi$
$547$	5.75736	0.246167	0.123083	+	0.992396i	$0.460722\pi$
$548$	12.3015	0.525495
$549$	−2.00000	−0.0853579
$550$	−1.92893	−0.0822499
$551$	−25.4558	−1.08446
$552$	1.31371	0.0559151
$553$	43.7990	1.86252
$554$	8.14214	0.345926
$555$	−4.48528	−0.190390
$556$	20.8701	0.885088
$557$	22.8284	0.967272	0.483636	−	0.875269i	$-0.339316\pi$
$557$	22.8284	0.967272	0.483636	+	0.875269i	$0.339316\pi$
$558$	1.75736	0.0743950
$559$	−5.07107	−0.214483
$560$	8.48528	0.358569
$561$	−5.41421	−0.228588
$562$	4.28427	0.180721
$563$	−17.6569	−0.744148	−0.372074	−	0.928203i	$-0.621353\pi$
$563$	−17.6569	−0.744148	−0.372074	+	0.928203i	$0.621353\pi$
$564$	15.5147	0.653288
$565$	6.82843	0.287274
$566$	−0.526912	−0.0221478
$567$	−4.82843	−0.202775
$568$	8.97056	0.376396
$569$	0.727922	0.0305161	0.0152580	−	0.999884i	$-0.495143\pi$
$569$	0.727922	0.0305161	0.0152580	+	0.999884i	$0.495143\pi$
$570$	1.45584	0.0609786
$571$	26.2426	1.09822	0.549110	−	0.835750i	$-0.314967\pi$
$571$	26.2426	1.09822	0.549110	+	0.835750i	$0.314967\pi$
$572$	1.82843	0.0764504
$573$	3.31371	0.138432
$574$	24.0000	1.00174
$575$	3.85786	0.160884
$576$	−4.17157	−0.173816
$577$	7.65685	0.318759	0.159380	−	0.987217i	$-0.449051\pi$
$577$	7.65685	0.318759	0.159380	+	0.987217i	$0.449051\pi$
$578$	5.10051	0.212153
$579$	−13.1716	−0.547392
$580$	4.54416	0.188686
$581$	−54.6274	−2.26633
$582$	4.14214	0.171697
$583$	13.3137	0.551397
$584$	−19.7990	−0.819288
$585$	0.585786	0.0242193
$586$	6.14214	0.253729
$587$	−42.1421	−1.73939	−0.869696	−	0.493588i	$-0.835685\pi$
$587$	−42.1421	−1.73939	−0.869696	+	0.493588i	$0.835685\pi$
$588$	−29.8284	−1.23010
$589$	−25.4558	−1.04889
$590$	2.62742	0.108169
$591$	−14.8284	−0.609960
$592$	22.9706	0.944084
$593$	−44.7696	−1.83847	−0.919233	−	0.393715i	$-0.871190\pi$
$593$	−44.7696	−1.83847	−0.919233	+	0.393715i	$0.871190\pi$
$594$	0.414214	0.0169954
$595$	−15.3137	−0.627801
$596$	−6.42641	−0.263236
$597$	−10.8284	−0.443178
$598$	0.343146	0.0140323
$599$	−29.6569	−1.21175	−0.605873	−	0.795561i	$-0.707176\pi$
$599$	−29.6569	−1.21175	−0.605873	+	0.795561i	$0.707176\pi$
$600$	7.38478	0.301482
$601$	38.9706	1.58964	0.794821	−	0.606844i	$-0.207565\pi$
$601$	38.9706	1.58964	0.794821	+	0.606844i	$0.207565\pi$
$602$	−10.1421	−0.413363
$603$	−3.07107	−0.125064
$604$	−10.9706	−0.446386
$605$	−0.585786	−0.0238156
$606$	−4.10051	−0.166572
$607$	−23.4142	−0.950354	−0.475177	−	0.879890i	$-0.657616\pi$
$607$	−23.4142	−0.950354	−0.475177	+	0.879890i	$0.657616\pi$
$608$	−26.4853	−1.07412
$609$	−20.4853	−0.830105
$610$	0.485281	0.0196485
$611$	8.48528	0.343278
$612$	9.89949	0.400163
$613$	−3.79899	−0.153440	−0.0767199	−	0.997053i	$-0.524445\pi$
$613$	−3.79899	−0.153440	−0.0767199	+	0.997053i	$0.524445\pi$
$614$	10.8873	0.439376
$615$	7.02944	0.283454
$616$	7.65685	0.308503
$617$	35.0122	1.40954	0.704769	−	0.709437i	$-0.251051\pi$
$617$	35.0122	1.40954	0.704769	+	0.709437i	$0.251051\pi$
$618$	−3.51472	−0.141383
$619$	−21.2132	−0.852631	−0.426315	−	0.904575i	$-0.640189\pi$
$619$	−21.2132	−0.852631	−0.426315	+	0.904575i	$0.640189\pi$
$620$	4.54416	0.182498
$621$	−0.828427	−0.0332436
$622$	−2.00000	−0.0801927
$623$	74.4264	2.98183
$624$	−3.00000	−0.120096
$625$	19.9706	0.798823
$626$	8.28427	0.331106
$627$	−6.00000	−0.239617
$628$	−7.31371	−0.291849
$629$	−41.4558	−1.65295
$630$	1.17157	0.0466766
$631$	48.5269	1.93183	0.965913	−	0.258867i	$-0.0833492\pi$
$631$	48.5269	1.93183	0.965913	+	0.258867i	$0.0833492\pi$
$632$	14.3848	0.572196
$633$	−18.7279	−0.744368
$634$	−6.38478	−0.253572
$635$	−8.34315	−0.331088
$636$	−24.3431	−0.965269
$637$	−16.3137	−0.646373
$638$	1.75736	0.0695745
$639$	−5.65685	−0.223782
$640$	6.18377	0.244435
$641$	27.6569	1.09238	0.546190	−	0.837661i	$-0.316078\pi$
$641$	27.6569	1.09238	0.546190	+	0.837661i	$0.316078\pi$
$642$	3.51472	0.138715
$643$	4.72792	0.186451	0.0932255	−	0.995645i	$-0.470282\pi$
$643$	4.72792	0.186451	0.0932255	+	0.995645i	$0.470282\pi$
$644$	−7.31371	−0.288200
$645$	−2.97056	−0.116966
$646$	13.4558	0.529413
$647$	−6.62742	−0.260551	−0.130275	−	0.991478i	$-0.541586\pi$
$647$	−6.62742	−0.260551	−0.130275	+	0.991478i	$0.541586\pi$
$648$	−1.58579	−0.0622956
$649$	−10.8284	−0.425053
$650$	1.92893	0.0756589
$651$	−20.4853	−0.802881
$652$	23.2721	0.911405
$653$	−27.1716	−1.06331	−0.531653	−	0.846962i	$-0.678429\pi$
$653$	−27.1716	−1.06331	−0.531653	+	0.846962i	$0.678429\pi$
$654$	−2.82843	−0.110600
$655$	9.94113	0.388432
$656$	−36.0000	−1.40556
$657$	12.4853	0.487097
$658$	16.9706	0.661581
$659$	−28.2843	−1.10180	−0.550899	−	0.834572i	$-0.685715\pi$
$659$	−28.2843	−1.10180	−0.550899	+	0.834572i	$0.685715\pi$
$660$	1.07107	0.0416913
$661$	−11.4558	−0.445581	−0.222790	−	0.974866i	$-0.571517\pi$
$661$	−11.4558	−0.445581	−0.222790	+	0.974866i	$0.571517\pi$
$662$	−4.87006	−0.189280
$663$	5.41421	0.210271
$664$	−17.9411	−0.696251
$665$	−16.9706	−0.658090
$666$	3.17157	0.122896
$667$	−3.51472	−0.136090
$668$	32.2843	1.24912
$669$	−2.58579	−0.0999723
$670$	0.745166	0.0287883
$671$	−2.00000	−0.0772091
$672$	−21.3137	−0.822194
$673$	18.4853	0.712555	0.356278	−	0.934380i	$-0.384046\pi$
$673$	18.4853	0.712555	0.356278	+	0.934380i	$0.384046\pi$
$674$	−8.05887	−0.310416
$675$	−4.65685	−0.179242
$676$	−1.82843	−0.0703241
$677$	42.3848	1.62898	0.814490	−	0.580178i	$-0.197017\pi$
$677$	42.3848	1.62898	0.814490	+	0.580178i	$0.197017\pi$
$678$	−4.82843	−0.185435
$679$	−48.2843	−1.85298
$680$	−5.02944	−0.192870
$681$	−10.9706	−0.420393
$682$	1.75736	0.0672928
$683$	−42.6274	−1.63109	−0.815546	−	0.578692i	$-0.803563\pi$
$683$	−42.6274	−1.63109	−0.815546	+	0.578692i	$0.803563\pi$
$684$	10.9706	0.419470
$685$	3.94113	0.150583
$686$	−18.6274	−0.711198
$687$	9.51472	0.363009
$688$	15.2132	0.579998
$689$	−13.3137	−0.507212
$690$	0.201010	0.00765232
$691$	−22.1005	−0.840743	−0.420371	−	0.907352i	$-0.638100\pi$
$691$	−22.1005	−0.840743	−0.420371	+	0.907352i	$0.638100\pi$
$692$	−42.1838	−1.60359
$693$	−4.82843	−0.183417
$694$	3.51472	0.133417
$695$	6.68629	0.253625
$696$	−6.72792	−0.255021
$697$	64.9706	2.46094
$698$	7.85786	0.297425
$699$	−4.24264	−0.160471
$700$	−41.1127	−1.55391
$701$	−28.9289	−1.09263	−0.546315	−	0.837580i	$-0.683970\pi$
$701$	−28.9289	−1.09263	−0.546315	+	0.837580i	$0.683970\pi$
$702$	−0.414214	−0.0156335
$703$	−45.9411	−1.73270
$704$	−4.17157	−0.157222
$705$	4.97056	0.187202
$706$	−1.01219	−0.0380944
$707$	47.7990	1.79767
$708$	19.7990	0.744092
$709$	2.48528	0.0933367	0.0466684	−	0.998910i	$-0.485140\pi$
$709$	2.48528	0.0933367	0.0466684	+	0.998910i	$0.485140\pi$
$710$	1.37258	0.0515121
$711$	−9.07107	−0.340192
$712$	24.4437	0.916065
$713$	−3.51472	−0.131627
$714$	10.8284	0.405244
$715$	0.585786	0.0219072
$716$	−7.57359	−0.283038
$717$	−4.34315	−0.162198
$718$	−3.45584	−0.128971
$719$	−45.2548	−1.68772	−0.843860	−	0.536563i	$-0.819722\pi$
$719$	−45.2548	−1.68772	−0.843860	+	0.536563i	$0.819722\pi$
$720$	−1.75736	−0.0654929
$721$	40.9706	1.52582
$722$	7.04163	0.262062
$723$	−21.3137	−0.792665
$724$	−16.4020	−0.609576
$725$	−19.7574	−0.733770
$726$	0.414214	0.0153729
$727$	6.82843	0.253252	0.126626	−	0.991951i	$-0.459585\pi$
$727$	6.82843	0.253252	0.126626	+	0.991951i	$0.459585\pi$
$728$	−7.65685	−0.283782
$729$	1.00000	0.0370370
$730$	−3.02944	−0.112125
$731$	−27.4558	−1.01549
$732$	3.65685	0.135161
$733$	−15.5147	−0.573049	−0.286525	−	0.958073i	$-0.592500\pi$
$733$	−15.5147	−0.573049	−0.286525	+	0.958073i	$0.592500\pi$
$734$	−2.05887	−0.0759944
$735$	−9.55635	−0.352491
$736$	−3.65685	−0.134793
$737$	−3.07107	−0.113124
$738$	−4.97056	−0.182969
$739$	2.20101	0.0809655	0.0404827	−	0.999180i	$-0.487110\pi$
$739$	2.20101	0.0809655	0.0404827	+	0.999180i	$0.487110\pi$
$740$	8.20101	0.301475
$741$	6.00000	0.220416
$742$	−26.6274	−0.977523
$743$	12.2843	0.450666	0.225333	−	0.974282i	$-0.427653\pi$
$743$	12.2843	0.450666	0.225333	+	0.974282i	$0.427653\pi$
$744$	−6.72792	−0.246658
$745$	−2.05887	−0.0754313
$746$	6.00000	0.219676
$747$	11.3137	0.413947
$748$	9.89949	0.361961
$749$	−40.9706	−1.49703
$750$	2.34315	0.0855596
$751$	44.9706	1.64100	0.820500	−	0.571647i	$-0.193695\pi$
$751$	44.9706	1.64100	0.820500	+	0.571647i	$0.193695\pi$
$752$	−25.4558	−0.928279
$753$	15.3137	0.558063
$754$	−1.75736	−0.0639993
$755$	−3.51472	−0.127914
$756$	8.82843	0.321087
$757$	−5.37258	−0.195270	−0.0976349	−	0.995222i	$-0.531128\pi$
$757$	−5.37258	−0.195270	−0.0976349	+	0.995222i	$0.531128\pi$
$758$	9.55635	0.347102
$759$	−0.828427	−0.0300700
$760$	−5.57359	−0.202175
$761$	−13.4558	−0.487774	−0.243887	−	0.969804i	$-0.578423\pi$
$761$	−13.4558	−0.487774	−0.243887	+	0.969804i	$0.578423\pi$
$762$	5.89949	0.213716
$763$	32.9706	1.19361
$764$	−6.05887	−0.219202
$765$	3.17157	0.114668
$766$	−1.17157	−0.0423306
$767$	10.8284	0.390992
$768$	3.97056	0.143275
$769$	31.9411	1.15183	0.575913	−	0.817511i	$-0.304647\pi$
$769$	31.9411	1.15183	0.575913	+	0.817511i	$0.304647\pi$
$770$	1.17157	0.0422206
$771$	−15.1716	−0.546391
$772$	24.0833	0.866776
$773$	−42.0416	−1.51213	−0.756066	−	0.654495i	$-0.772881\pi$
$773$	−42.0416	−1.51213	−0.756066	+	0.654495i	$0.772881\pi$
$774$	2.10051	0.0755011
$775$	−19.7574	−0.709705
$776$	−15.8579	−0.569264
$777$	−36.9706	−1.32631
$778$	6.00000	0.215110
$779$	72.0000	2.57967
$780$	−1.07107	−0.0383504
$781$	−5.65685	−0.202418
$782$	1.85786	0.0664371
$783$	4.24264	0.151620
$784$	48.9411	1.74790
$785$	−2.34315	−0.0836305
$786$	−7.02944	−0.250732
$787$	−6.20101	−0.221042	−0.110521	−	0.993874i	$-0.535252\pi$
$787$	−6.20101	−0.221042	−0.110521	+	0.993874i	$0.535252\pi$
$788$	27.1127	0.965850
$789$	−7.79899	−0.277651
$790$	2.20101	0.0783084
$791$	56.2843	2.00124
$792$	−1.58579	−0.0563485
$793$	2.00000	0.0710221
$794$	10.2010	0.362020
$795$	−7.79899	−0.276602
$796$	19.7990	0.701757
$797$	−38.7696	−1.37329	−0.686644	−	0.726994i	$-0.740917\pi$
$797$	−38.7696	−1.37329	−0.686644	+	0.726994i	$0.740917\pi$
$798$	12.0000	0.424795
$799$	45.9411	1.62528
$800$	−20.5563	−0.726777
$801$	−15.4142	−0.544634
$802$	−5.41421	−0.191183
$803$	12.4853	0.440596
$804$	5.61522	0.198034
$805$	−2.34315	−0.0825850
$806$	−1.75736	−0.0619003
$807$	16.1421	0.568230
$808$	15.6985	0.552271
$809$	19.0711	0.670503	0.335252	−	0.942129i	$-0.391179\pi$
$809$	19.0711	0.670503	0.335252	+	0.942129i	$0.391179\pi$
$810$	−0.242641	−0.00852552
$811$	12.6274	0.443409	0.221704	−	0.975114i	$-0.428838\pi$
$811$	12.6274	0.443409	0.221704	+	0.975114i	$0.428838\pi$
$812$	37.4558	1.31444
$813$	−15.6569	−0.549110
$814$	3.17157	0.111164
$815$	7.45584	0.261167
$816$	−16.2426	−0.568606
$817$	−30.4264	−1.06449
$818$	4.76955	0.166763
$819$	4.82843	0.168719
$820$	−12.8528	−0.448840
$821$	4.97056	0.173474	0.0867369	−	0.996231i	$-0.472356\pi$
$821$	4.97056	0.173474	0.0867369	+	0.996231i	$0.472356\pi$
$822$	−2.78680	−0.0972007
$823$	19.5147	0.680240	0.340120	−	0.940382i	$-0.389532\pi$
$823$	19.5147	0.680240	0.340120	+	0.940382i	$0.389532\pi$
$824$	13.4558	0.468757
$825$	−4.65685	−0.162131
$826$	21.6569	0.753538
$827$	17.3137	0.602057	0.301028	−	0.953615i	$-0.402670\pi$
$827$	17.3137	0.602057	0.301028	+	0.953615i	$0.402670\pi$
$828$	1.51472	0.0526401
$829$	−14.6863	−0.510076	−0.255038	−	0.966931i	$-0.582088\pi$
$829$	−14.6863	−0.510076	−0.255038	+	0.966931i	$0.582088\pi$
$830$	−2.74517	−0.0952861
$831$	19.6569	0.681889
$832$	4.17157	0.144623
$833$	−88.3259	−3.06031
$834$	−4.72792	−0.163715
$835$	10.3431	0.357939
$836$	10.9706	0.379425
$837$	4.24264	0.146647
$838$	−1.59798	−0.0552013
$839$	26.6274	0.919281	0.459640	−	0.888105i	$-0.347978\pi$
$839$	26.6274	0.919281	0.459640	+	0.888105i	$0.347978\pi$
$840$	−4.48528	−0.154757
$841$	−11.0000	−0.379310
$842$	−3.57359	−0.123154
$843$	10.3431	0.356237
$844$	34.2426	1.17868
$845$	−0.585786	−0.0201517
$846$	−3.51472	−0.120839
$847$	−4.82843	−0.165907
$848$	39.9411	1.37158
$849$	−1.27208	−0.0436576
$850$	10.4437	0.358214
$851$	−6.34315	−0.217440
$852$	10.3431	0.354350
$853$	46.9706	1.60824	0.804121	−	0.594466i	$-0.202637\pi$
$853$	46.9706	1.60824	0.804121	+	0.594466i	$0.202637\pi$
$854$	4.00000	0.136877
$855$	3.51472	0.120201
$856$	−13.4558	−0.459911
$857$	−7.75736	−0.264986	−0.132493	−	0.991184i	$-0.542298\pi$
$857$	−7.75736	−0.264986	−0.132493	+	0.991184i	$0.542298\pi$
$858$	−0.414214	−0.0141410
$859$	16.4853	0.562471	0.281235	−	0.959639i	$-0.409256\pi$
$859$	16.4853	0.562471	0.281235	+	0.959639i	$0.409256\pi$
$860$	5.43146	0.185211
$861$	57.9411	1.97463
$862$	0.284271	0.00968232
$863$	20.7696	0.707004	0.353502	−	0.935434i	$-0.384991\pi$
$863$	20.7696	0.707004	0.353502	+	0.935434i	$0.384991\pi$
$864$	4.41421	0.150175
$865$	−13.5147	−0.459514
$866$	15.4558	0.525211
$867$	12.3137	0.418195
$868$	37.4558	1.27133
$869$	−9.07107	−0.307715
$870$	−1.02944	−0.0349012
$871$	3.07107	0.104059
$872$	10.8284	0.366697
$873$	10.0000	0.338449
$874$	2.05887	0.0696425
$875$	−27.3137	−0.923372
$876$	−22.8284	−0.771301
$877$	−46.9706	−1.58608	−0.793042	−	0.609167i	$-0.791504\pi$
$877$	−46.9706	−1.58608	−0.793042	+	0.609167i	$0.791504\pi$
$878$	−2.87006	−0.0968598
$879$	14.8284	0.500150
$880$	−1.75736	−0.0592406
$881$	−10.2843	−0.346486	−0.173243	−	0.984879i	$-0.555425\pi$
$881$	−10.2843	−0.346486	−0.173243	+	0.984879i	$0.555425\pi$
$882$	6.75736	0.227532
$883$	11.5147	0.387501	0.193751	−	0.981051i	$-0.437935\pi$
$883$	11.5147	0.387501	0.193751	+	0.981051i	$0.437935\pi$
$884$	−9.89949	−0.332956
$885$	6.34315	0.213223
$886$	−5.94113	−0.199596
$887$	31.5980	1.06096	0.530478	−	0.847699i	$-0.322012\pi$
$887$	31.5980	1.06096	0.530478	+	0.847699i	$0.322012\pi$
$888$	−12.1421	−0.407463
$889$	−68.7696	−2.30646
$890$	3.74012	0.125369
$891$	1.00000	0.0335013
$892$	4.72792	0.158303
$893$	50.9117	1.70369
$894$	1.45584	0.0486907
$895$	−2.42641	−0.0811058
$896$	50.9706	1.70281
$897$	0.828427	0.0276604
$898$	3.07107	0.102483
$899$	18.0000	0.600334
$900$	8.51472	0.283824
$901$	−72.0833	−2.40144
$902$	−4.97056	−0.165502
$903$	−24.4853	−0.814819
$904$	18.4853	0.614811
$905$	−5.25483	−0.174677
$906$	2.48528	0.0825679
$907$	2.34315	0.0778029	0.0389014	−	0.999243i	$-0.487614\pi$
$907$	2.34315	0.0778029	0.0389014	+	0.999243i	$0.487614\pi$
$908$	20.0589	0.665677
$909$	−9.89949	−0.328346
$910$	−1.17157	−0.0388373
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	−18.0000	−0.596040
$913$	11.3137	0.374429
$914$	−5.79899	−0.191814
$915$	1.17157	0.0387310
$916$	−17.3970	−0.574812
$917$	81.9411	2.70593
$918$	−2.24264	−0.0740182
$919$	−46.7279	−1.54141	−0.770706	−	0.637191i	$-0.780096\pi$
$919$	−46.7279	−1.54141	−0.770706	+	0.637191i	$0.780096\pi$
$920$	−0.769553	−0.0253714
$921$	26.2843	0.866096
$922$	1.37258	0.0452036
$923$	5.65685	0.186198
$924$	8.82843	0.290434
$925$	−35.6569	−1.17239
$926$	10.0416	0.329988
$927$	−8.48528	−0.278693
$928$	18.7279	0.614774
$929$	15.8995	0.521646	0.260823	−	0.965387i	$-0.416006\pi$
$929$	15.8995	0.521646	0.260823	+	0.965387i	$0.416006\pi$
$930$	−1.02944	−0.0337566
$931$	−97.8823	−3.20796
$932$	7.75736	0.254101
$933$	−4.82843	−0.158076
$934$	−5.71573	−0.187024
$935$	3.17157	0.103722
$936$	1.58579	0.0518331
$937$	−16.8284	−0.549761	−0.274880	−	0.961478i	$-0.588638\pi$
$937$	−16.8284	−0.549761	−0.274880	+	0.961478i	$0.588638\pi$
$938$	6.14214	0.200548
$939$	20.0000	0.652675
$940$	−9.08831	−0.296428
$941$	38.6274	1.25922	0.629609	−	0.776912i	$-0.283215\pi$
$941$	38.6274	1.25922	0.629609	+	0.776912i	$0.283215\pi$
$942$	1.65685	0.0539832
$943$	9.94113	0.323728
$944$	−32.4853	−1.05731
$945$	2.82843	0.0920087
$946$	2.10051	0.0682933
$947$	−51.1127	−1.66094	−0.830470	−	0.557064i	$-0.811928\pi$
$947$	−51.1127	−1.66094	−0.830470	+	0.557064i	$0.811928\pi$
$948$	16.5858	0.538681
$949$	−12.4853	−0.405289
$950$	11.5736	0.375497
$951$	−15.4142	−0.499840
$952$	−41.4558	−1.34359
$953$	−26.1005	−0.845478	−0.422739	−	0.906251i	$-0.638931\pi$
$953$	−26.1005	−0.845478	−0.422739	+	0.906251i	$0.638931\pi$
$954$	5.51472	0.178546
$955$	−1.94113	−0.0628133
$956$	7.94113	0.256834
$957$	4.24264	0.137145
$958$	11.5736	0.373926
$959$	32.4853	1.04900
$960$	2.44365	0.0788685
$961$	−13.0000	−0.419355
$962$	−3.17157	−0.102256
$963$	8.48528	0.273434
$964$	38.9706	1.25516
$965$	7.71573	0.248378
$966$	1.65685	0.0533084
$967$	7.85786	0.252692	0.126346	−	0.991986i	$-0.459675\pi$
$967$	7.85786	0.252692	0.126346	+	0.991986i	$0.459675\pi$
$968$	−1.58579	−0.0509691
$969$	32.4853	1.04358
$970$	−2.42641	−0.0779072
$971$	−3.17157	−0.101781	−0.0508903	−	0.998704i	$-0.516206\pi$
$971$	−3.17157	−0.101781	−0.0508903	+	0.998704i	$0.516206\pi$
$972$	−1.82843	−0.0586468
$973$	55.1127	1.76683
$974$	1.95837	0.0627502
$975$	4.65685	0.149139
$976$	−6.00000	−0.192055
$977$	59.6985	1.90992	0.954962	−	0.296729i	$-0.0958957\pi$
$977$	59.6985	1.90992	0.954962	+	0.296729i	$0.0958957\pi$
$978$	−5.27208	−0.168582
$979$	−15.4142	−0.492640
$980$	17.4731	0.558157
$981$	−6.82843	−0.218015
$982$	−4.48528	−0.143131
$983$	4.97056	0.158536	0.0792682	−	0.996853i	$-0.474742\pi$
$983$	4.97056	0.158536	0.0792682	+	0.996853i	$0.474742\pi$
$984$	19.0294	0.606636
$985$	8.68629	0.276768
$986$	−9.51472	−0.303010
$987$	40.9706	1.30411
$988$	−10.9706	−0.349020
$989$	−4.20101	−0.133584
$990$	−0.242641	−0.00771163
$991$	−49.4558	−1.57102	−0.785508	−	0.618851i	$-0.787598\pi$
$991$	−49.4558	−1.57102	−0.785508	+	0.618851i	$0.787598\pi$
$992$	18.7279	0.594612
$993$	−11.7574	−0.373109
$994$	11.3137	0.358849
$995$	6.34315	0.201091
$996$	−20.6863	−0.655470
$997$	−34.7696	−1.10116	−0.550581	−	0.834781i	$-0.685594\pi$
$997$	−34.7696	−1.10116	−0.550581	+	0.834781i	$0.685594\pi$
$998$	−5.95837	−0.188609
$999$	7.65685	0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.a.c.1.2	✓	2
3.2	odd	2		1287.2.a.g.1.1		2
4.3	odd	2		6864.2.a.bc.1.2		2
11.10	odd	2		4719.2.a.o.1.1		2
13.12	even	2		5577.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.c.1.2	✓	2	1.1	even	1	trivial
1287.2.a.g.1.1		2	3.2	odd	2
4719.2.a.o.1.1		2	11.10	odd	2
5577.2.a.i.1.1		2	13.12	even	2
6864.2.a.bc.1.2		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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.242641 q^{10} +1.00000 q^{11} -1.82843 q^{12} -1.00000 q^{13} -2.00000 q^{14} -0.585786 q^{15} +3.00000 q^{16} -5.41421 q^{17} +0.414214 q^{18} -6.00000 q^{19} +1.07107 q^{20} -4.82843 q^{21} +0.414214 q^{22} -0.828427 q^{23} -1.58579 q^{24} -4.65685 q^{25} -0.414214 q^{26} +1.00000 q^{27} +8.82843 q^{28} +4.24264 q^{29} -0.242641 q^{30} +4.24264 q^{31} +4.41421 q^{32} +1.00000 q^{33} -2.24264 q^{34} +2.82843 q^{35} -1.82843 q^{36} +7.65685 q^{37} -2.48528 q^{38} -1.00000 q^{39} +0.928932 q^{40} -12.0000 q^{41} -2.00000 q^{42} +5.07107 q^{43} -1.82843 q^{44} -0.585786 q^{45} -0.343146 q^{46} -8.48528 q^{47} +3.00000 q^{48} +16.3137 q^{49} -1.92893 q^{50} -5.41421 q^{51} +1.82843 q^{52} +13.3137 q^{53} +0.414214 q^{54} -0.585786 q^{55} +7.65685 q^{56} -6.00000 q^{57} +1.75736 q^{58} -10.8284 q^{59} +1.07107 q^{60} -2.00000 q^{61} +1.75736 q^{62} -4.82843 q^{63} -4.17157 q^{64} +0.585786 q^{65} +0.414214 q^{66} -3.07107 q^{67} +9.89949 q^{68} -0.828427 q^{69} +1.17157 q^{70} -5.65685 q^{71} -1.58579 q^{72} +12.4853 q^{73} +3.17157 q^{74} -4.65685 q^{75} +10.9706 q^{76} -4.82843 q^{77} -0.414214 q^{78} -9.07107 q^{79} -1.75736 q^{80} +1.00000 q^{81} -4.97056 q^{82} +11.3137 q^{83} +8.82843 q^{84} +3.17157 q^{85} +2.10051 q^{86} +4.24264 q^{87} -1.58579 q^{88} -15.4142 q^{89} -0.242641 q^{90} +4.82843 q^{91} +1.51472 q^{92} +4.24264 q^{93} -3.51472 q^{94} +3.51472 q^{95} +4.41421 q^{96} +10.0000 q^{97} +6.75736 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 12 q^{19} - 12 q^{20} - 4 q^{21} - 2 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 12 q^{28} + 8 q^{30} + 6 q^{32} + 2 q^{33} + 4 q^{34} + 2 q^{36} + 4 q^{37} + 12 q^{38} - 2 q^{39} + 16 q^{40} - 24 q^{41} - 4 q^{42} - 4 q^{43} + 2 q^{44} - 4 q^{45} - 12 q^{46} + 6 q^{48} + 10 q^{49} - 18 q^{50} - 8 q^{51} - 2 q^{52} + 4 q^{53} - 2 q^{54} - 4 q^{55} + 4 q^{56} - 12 q^{57} + 12 q^{58} - 16 q^{59} - 12 q^{60} - 4 q^{61} + 12 q^{62} - 4 q^{63} - 14 q^{64} + 4 q^{65} - 2 q^{66} + 8 q^{67} + 4 q^{69} + 8 q^{70} - 6 q^{72} + 8 q^{73} + 12 q^{74} + 2 q^{75} - 12 q^{76} - 4 q^{77} + 2 q^{78} - 4 q^{79} - 12 q^{80} + 2 q^{81} + 24 q^{82} + 12 q^{84} + 12 q^{85} + 24 q^{86} - 6 q^{88} - 28 q^{89} + 8 q^{90} + 4 q^{91} + 20 q^{92} - 24 q^{94} + 24 q^{95} + 6 q^{96} + 20 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 12 * q^19 - 12 * q^20 - 4 * q^21 - 2 * q^22 + 4 * q^23 - 6 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 + 12 * q^28 + 8 * q^30 + 6 * q^32 + 2 * q^33 + 4 * q^34 + 2 * q^36 + 4 * q^37 + 12 * q^38 - 2 * q^39 + 16 * q^40 - 24 * q^41 - 4 * q^42 - 4 * q^43 + 2 * q^44 - 4 * q^45 - 12 * q^46 + 6 * q^48 + 10 * q^49 - 18 * q^50 - 8 * q^51 - 2 * q^52 + 4 * q^53 - 2 * q^54 - 4 * q^55 + 4 * q^56 - 12 * q^57 + 12 * q^58 - 16 * q^59 - 12 * q^60 - 4 * q^61 + 12 * q^62 - 4 * q^63 - 14 * q^64 + 4 * q^65 - 2 * q^66 + 8 * q^67 + 4 * q^69 + 8 * q^70 - 6 * q^72 + 8 * q^73 + 12 * q^74 + 2 * q^75 - 12 * q^76 - 4 * q^77 + 2 * q^78 - 4 * q^79 - 12 * q^80 + 2 * q^81 + 24 * q^82 + 12 * q^84 + 12 * q^85 + 24 * q^86 - 6 * q^88 - 28 * q^89 + 8 * q^90 + 4 * q^91 + 20 * q^92 - 24 * q^94 + 24 * q^95 + 6 * q^96 + 20 * q^97 + 22 * q^98 + 2 * q^99

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