LMFDB - Embedded newform 285.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.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:	$2.27573645761$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	285.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -2.24264 q^{11} +3.82843 q^{12} -3.41421 q^{13} -3.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} +1.17157 q^{17} +2.41421 q^{18} -1.00000 q^{19} -3.82843 q^{20} -1.41421 q^{21} -5.41421 q^{22} +7.65685 q^{23} +4.41421 q^{24} +1.00000 q^{25} -8.24264 q^{26} +1.00000 q^{27} -5.41421 q^{28} +1.41421 q^{29} -2.41421 q^{30} -3.17157 q^{31} -1.58579 q^{32} -2.24264 q^{33} +2.82843 q^{34} +1.41421 q^{35} +3.82843 q^{36} -3.41421 q^{37} -2.41421 q^{38} -3.41421 q^{39} -4.41421 q^{40} -0.242641 q^{41} -3.41421 q^{42} +12.2426 q^{43} -8.58579 q^{44} -1.00000 q^{45} +18.4853 q^{46} -7.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} +2.41421 q^{50} +1.17157 q^{51} -13.0711 q^{52} +8.00000 q^{53} +2.41421 q^{54} +2.24264 q^{55} -6.24264 q^{56} -1.00000 q^{57} +3.41421 q^{58} +12.4853 q^{59} -3.82843 q^{60} +7.31371 q^{61} -7.65685 q^{62} -1.41421 q^{63} -9.82843 q^{64} +3.41421 q^{65} -5.41421 q^{66} -9.65685 q^{67} +4.48528 q^{68} +7.65685 q^{69} +3.41421 q^{70} -10.8284 q^{71} +4.41421 q^{72} -7.65685 q^{73} -8.24264 q^{74} +1.00000 q^{75} -3.82843 q^{76} +3.17157 q^{77} -8.24264 q^{78} -3.00000 q^{80} +1.00000 q^{81} -0.585786 q^{82} +12.8284 q^{83} -5.41421 q^{84} -1.17157 q^{85} +29.5563 q^{86} +1.41421 q^{87} -9.89949 q^{88} +5.89949 q^{89} -2.41421 q^{90} +4.82843 q^{91} +29.3137 q^{92} -3.17157 q^{93} -18.4853 q^{94} +1.00000 q^{95} -1.58579 q^{96} -9.75736 q^{97} -12.0711 q^{98} -2.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 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^5 + 2 * q^6 + 6 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 8 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 8 q^{26} + 2 q^{27} - 8 q^{28} - 2 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 8 q^{41} - 4 q^{42} + 16 q^{43} - 20 q^{44} - 2 q^{45} + 20 q^{46} - 4 q^{47} + 6 q^{48} - 10 q^{49} + 2 q^{50} + 8 q^{51} - 12 q^{52} + 16 q^{53} + 2 q^{54} - 4 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} + 8 q^{59} - 2 q^{60} - 8 q^{61} - 4 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{66} - 8 q^{67} - 8 q^{68} + 4 q^{69} + 4 q^{70} - 16 q^{71} + 6 q^{72} - 4 q^{73} - 8 q^{74} + 2 q^{75} - 2 q^{76} + 12 q^{77} - 8 q^{78} - 6 q^{80} + 2 q^{81} - 4 q^{82} + 20 q^{83} - 8 q^{84} - 8 q^{85} + 28 q^{86} - 8 q^{89} - 2 q^{90} + 4 q^{91} + 36 q^{92} - 12 q^{93} - 20 q^{94} + 2 q^{95} - 6 q^{96} - 28 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^13 - 4 * q^14 - 2 * q^15 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 8 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 8 * q^26 + 2 * q^27 - 8 * q^28 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 8 * q^41 - 4 * q^42 + 16 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 - 4 * q^47 + 6 * q^48 - 10 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 + 16 * q^53 + 2 * q^54 - 4 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 + 8 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^66 - 8 * q^67 - 8 * q^68 + 4 * q^69 + 4 * q^70 - 16 * q^71 + 6 * q^72 - 4 * q^73 - 8 * q^74 + 2 * q^75 - 2 * q^76 + 12 * q^77 - 8 * q^78 - 6 * q^80 + 2 * q^81 - 4 * q^82 + 20 * q^83 - 8 * q^84 - 8 * q^85 + 28 * q^86 - 8 * q^89 - 2 * q^90 + 4 * q^91 + 36 * q^92 - 12 * q^93 - 20 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 10 * q^98 + 4 * 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 $

$2$	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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.41421	0.985599
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	4.41421	1.56066
$9$	1.00000	0.333333
$10$	−2.41421	−0.763441
$11$	−2.24264	−0.676182	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−2.24264	−0.676182	−0.338091	+	0.941113i	$0.609781\pi$
$12$	3.82843	1.10517
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	−3.41421	−0.912487
$15$	−1.00000	−0.258199
$16$	3.00000	0.750000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	2.41421	0.569036
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.82843	−0.856062
$21$	−1.41421	−0.308607
$22$	−5.41421	−1.15431
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	4.41421	0.901048
$25$	1.00000	0.200000
$26$	−8.24264	−1.61651
$27$	1.00000	0.192450
$28$	−5.41421	−1.02319
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	−2.41421	−0.440773
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	−1.58579	−0.280330
$33$	−2.24264	−0.390394
$34$	2.82843	0.485071
$35$	1.41421	0.239046
$36$	3.82843	0.638071
$37$	−3.41421	−0.561293	−0.280647	−	0.959811i	$-0.590549\pi$
$37$	−3.41421	−0.561293	−0.280647	+	0.959811i	$0.590549\pi$
$38$	−2.41421	−0.391637
$39$	−3.41421	−0.546712
$40$	−4.41421	−0.697948
$41$	−0.242641	−0.0378941	−0.0189471	−	0.999820i	$-0.506031\pi$
$41$	−0.242641	−0.0378941	−0.0189471	+	0.999820i	$0.506031\pi$
$42$	−3.41421	−0.526825
$43$	12.2426	1.86699	0.933493	−	0.358597i	$-0.116745\pi$
$43$	12.2426	1.86699	0.933493	+	0.358597i	$0.116745\pi$
$44$	−8.58579	−1.29436
$45$	−1.00000	−0.149071
$46$	18.4853	2.72551
$47$	−7.65685	−1.11687	−0.558433	−	0.829549i	$-0.688597\pi$
$47$	−7.65685	−1.11687	−0.558433	+	0.829549i	$0.688597\pi$
$48$	3.00000	0.433013
$49$	−5.00000	−0.714286
$50$	2.41421	0.341421
$51$	1.17157	0.164053
$52$	−13.0711	−1.81263
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	2.41421	0.328533
$55$	2.24264	0.302398
$56$	−6.24264	−0.834208
$57$	−1.00000	−0.132453
$58$	3.41421	0.448308
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	−3.82843	−0.494248
$61$	7.31371	0.936424	0.468212	−	0.883616i	$-0.344898\pi$
$61$	7.31371	0.936424	0.468212	+	0.883616i	$0.344898\pi$
$62$	−7.65685	−0.972421
$63$	−1.41421	−0.178174
$64$	−9.82843	−1.22855
$65$	3.41421	0.423481
$66$	−5.41421	−0.666444
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	4.48528	0.543920
$69$	7.65685	0.921777
$70$	3.41421	0.408077
$71$	−10.8284	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$71$	−10.8284	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$72$	4.41421	0.520220
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	−8.24264	−0.958188
$75$	1.00000	0.115470
$76$	−3.82843	−0.439151
$77$	3.17157	0.361434
$78$	−8.24264	−0.933295
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−0.585786	−0.0646893
$83$	12.8284	1.40810	0.704051	−	0.710149i	$-0.251372\pi$
$83$	12.8284	1.40810	0.704051	+	0.710149i	$0.251372\pi$
$84$	−5.41421	−0.590739
$85$	−1.17157	−0.127075
$86$	29.5563	3.18714
$87$	1.41421	0.151620
$88$	−9.89949	−1.05529
$89$	5.89949	0.625345	0.312673	−	0.949861i	$-0.398776\pi$
$89$	5.89949	0.625345	0.312673	+	0.949861i	$0.398776\pi$
$90$	−2.41421	−0.254480
$91$	4.82843	0.506157
$92$	29.3137	3.05617
$93$	−3.17157	−0.328877
$94$	−18.4853	−1.90661
$95$	1.00000	0.102598
$96$	−1.58579	−0.161849
$97$	−9.75736	−0.990710	−0.495355	−	0.868691i	$-0.664962\pi$
$97$	−9.75736	−0.990710	−0.495355	+	0.868691i	$0.664962\pi$
$98$	−12.0711	−1.21936
$99$	−2.24264	−0.225394
$100$	3.82843	0.382843
$101$	−3.17157	−0.315583	−0.157792	−	0.987472i	$-0.550437\pi$
$101$	−3.17157	−0.315583	−0.157792	+	0.987472i	$0.550437\pi$
$102$	2.82843	0.280056
$103$	−7.31371	−0.720641	−0.360321	−	0.932829i	$-0.617333\pi$
$103$	−7.31371	−0.720641	−0.360321	+	0.932829i	$0.617333\pi$
$104$	−15.0711	−1.47784
$105$	1.41421	0.138013
$106$	19.3137	1.87591
$107$	19.3137	1.86713	0.933563	−	0.358412i	$-0.116682\pi$
$107$	19.3137	1.86713	0.933563	+	0.358412i	$0.116682\pi$
$108$	3.82843	0.368391
$109$	−6.48528	−0.621177	−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528	−0.621177	−0.310589	+	0.950544i	$0.600526\pi$
$110$	5.41421	0.516225
$111$	−3.41421	−0.324063
$112$	−4.24264	−0.400892
$113$	10.1421	0.954092	0.477046	−	0.878878i	$-0.341708\pi$
$113$	10.1421	0.954092	0.477046	+	0.878878i	$0.341708\pi$
$114$	−2.41421	−0.226112
$115$	−7.65685	−0.714005
$116$	5.41421	0.502697
$117$	−3.41421	−0.315644
$118$	30.1421	2.77481
$119$	−1.65685	−0.151884
$120$	−4.41421	−0.402961
$121$	−5.97056	−0.542778
$122$	17.6569	1.59858
$123$	−0.242641	−0.0218782
$124$	−12.1421	−1.09040
$125$	−1.00000	−0.0894427
$126$	−3.41421	−0.304162
$127$	19.3137	1.71381	0.856907	−	0.515471i	$-0.172383\pi$
$127$	19.3137	1.71381	0.856907	+	0.515471i	$0.172383\pi$
$128$	−20.5563	−1.81694
$129$	12.2426	1.07790
$130$	8.24264	0.722927
$131$	−8.58579	−0.750144	−0.375072	−	0.926996i	$-0.622382\pi$
$131$	−8.58579	−0.750144	−0.375072	+	0.926996i	$0.622382\pi$
$132$	−8.58579	−0.747297
$133$	1.41421	0.122628
$134$	−23.3137	−2.01400
$135$	−1.00000	−0.0860663
$136$	5.17157	0.443459
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	18.4853	1.57357
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	5.41421	0.457585
$141$	−7.65685	−0.644823
$142$	−26.1421	−2.19380
$143$	7.65685	0.640298
$144$	3.00000	0.250000
$145$	−1.41421	−0.117444
$146$	−18.4853	−1.52985
$147$	−5.00000	−0.412393
$148$	−13.0711	−1.07444
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	2.41421	0.197120
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	−4.41421	−0.358040
$153$	1.17157	0.0947161
$154$	7.65685	0.617007
$155$	3.17157	0.254747
$156$	−13.0711	−1.04652
$157$	10.4853	0.836817	0.418408	−	0.908259i	$-0.362588\pi$
$157$	10.4853	0.836817	0.418408	+	0.908259i	$0.362588\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	1.58579	0.125367
$161$	−10.8284	−0.853400
$162$	2.41421	0.189679
$163$	21.8995	1.71530	0.857650	−	0.514233i	$-0.171923\pi$
$163$	21.8995	1.71530	0.857650	+	0.514233i	$0.171923\pi$
$164$	−0.928932	−0.0725374
$165$	2.24264	0.174589
$166$	30.9706	2.40378
$167$	−17.3137	−1.33977	−0.669887	−	0.742463i	$-0.733658\pi$
$167$	−17.3137	−1.33977	−0.669887	+	0.742463i	$0.733658\pi$
$168$	−6.24264	−0.481630
$169$	−1.34315	−0.103319
$170$	−2.82843	−0.216930
$171$	−1.00000	−0.0764719
$172$	46.8701	3.57381
$173$	−19.7990	−1.50529	−0.752645	−	0.658427i	$-0.771222\pi$
$173$	−19.7990	−1.50529	−0.752645	+	0.658427i	$0.771222\pi$
$174$	3.41421	0.258831
$175$	−1.41421	−0.106904
$176$	−6.72792	−0.507136
$177$	12.4853	0.938451
$178$	14.2426	1.06753
$179$	16.4853	1.23217	0.616084	−	0.787681i	$-0.288718\pi$
$179$	16.4853	1.23217	0.616084	+	0.787681i	$0.288718\pi$
$180$	−3.82843	−0.285354
$181$	−20.8284	−1.54816	−0.774082	−	0.633085i	$-0.781788\pi$
$181$	−20.8284	−1.54816	−0.774082	+	0.633085i	$0.781788\pi$
$182$	11.6569	0.864064
$183$	7.31371	0.540645
$184$	33.7990	2.49169
$185$	3.41421	0.251018
$186$	−7.65685	−0.561428
$187$	−2.62742	−0.192136
$188$	−29.3137	−2.13792
$189$	−1.41421	−0.102869
$190$	2.41421	0.175145
$191$	10.2426	0.741131	0.370566	−	0.928806i	$-0.379164\pi$
$191$	10.2426	0.741131	0.370566	+	0.928806i	$0.379164\pi$
$192$	−9.82843	−0.709306
$193$	5.07107	0.365023	0.182512	−	0.983204i	$-0.441577\pi$
$193$	5.07107	0.365023	0.182512	+	0.983204i	$0.441577\pi$
$194$	−23.5563	−1.69125
$195$	3.41421	0.244497
$196$	−19.1421	−1.36730
$197$	6.82843	0.486505	0.243253	−	0.969963i	$-0.421786\pi$
$197$	6.82843	0.486505	0.243253	+	0.969963i	$0.421786\pi$
$198$	−5.41421	−0.384771
$199$	18.1421	1.28606	0.643031	−	0.765840i	$-0.277677\pi$
$199$	18.1421	1.28606	0.643031	+	0.765840i	$0.277677\pi$
$200$	4.41421	0.312132
$201$	−9.65685	−0.681142
$202$	−7.65685	−0.538734
$203$	−2.00000	−0.140372
$204$	4.48528	0.314033
$205$	0.242641	0.0169468
$206$	−17.6569	−1.23021
$207$	7.65685	0.532188
$208$	−10.2426	−0.710199
$209$	2.24264	0.155127
$210$	3.41421	0.235603
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	30.6274	2.10350
$213$	−10.8284	−0.741952
$214$	46.6274	3.18738
$215$	−12.2426	−0.834941
$216$	4.41421	0.300349
$217$	4.48528	0.304481
$218$	−15.6569	−1.06042
$219$	−7.65685	−0.517402
$220$	8.58579	0.578854
$221$	−4.00000	−0.269069
$222$	−8.24264	−0.553210
$223$	18.6274	1.24738	0.623692	−	0.781670i	$-0.285632\pi$
$223$	18.6274	1.24738	0.623692	+	0.781670i	$0.285632\pi$
$224$	2.24264	0.149843
$225$	1.00000	0.0666667
$226$	24.4853	1.62874
$227$	14.9706	0.993631	0.496816	−	0.867856i	$-0.334503\pi$
$227$	14.9706	0.993631	0.496816	+	0.867856i	$0.334503\pi$
$228$	−3.82843	−0.253544
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	−18.4853	−1.21888
$231$	3.17157	0.208674
$232$	6.24264	0.409849
$233$	−0.343146	−0.0224802	−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146	−0.0224802	−0.0112401	+	0.999937i	$0.503578\pi$
$234$	−8.24264	−0.538838
$235$	7.65685	0.499478
$236$	47.7990	3.11145
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−26.7279	−1.72889	−0.864443	−	0.502731i	$-0.832329\pi$
$239$	−26.7279	−1.72889	−0.864443	+	0.502731i	$0.832329\pi$
$240$	−3.00000	−0.193649
$241$	−19.6569	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$241$	−19.6569	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$242$	−14.4142	−0.926581
$243$	1.00000	0.0641500
$244$	28.0000	1.79252
$245$	5.00000	0.319438
$246$	−0.585786	−0.0373484
$247$	3.41421	0.217241
$248$	−14.0000	−0.889001
$249$	12.8284	0.812969
$250$	−2.41421	−0.152688
$251$	−1.75736	−0.110924	−0.0554618	−	0.998461i	$-0.517663\pi$
$251$	−1.75736	−0.110924	−0.0554618	+	0.998461i	$0.517663\pi$
$252$	−5.41421	−0.341063
$253$	−17.1716	−1.07957
$254$	46.6274	2.92566
$255$	−1.17157	−0.0733667
$256$	−29.9706	−1.87316
$257$	−4.48528	−0.279784	−0.139892	−	0.990167i	$-0.544676\pi$
$257$	−4.48528	−0.279784	−0.139892	+	0.990167i	$0.544676\pi$
$258$	29.5563	1.84010
$259$	4.82843	0.300024
$260$	13.0711	0.810633
$261$	1.41421	0.0875376
$262$	−20.7279	−1.28058
$263$	−23.4558	−1.44635	−0.723175	−	0.690665i	$-0.757318\pi$
$263$	−23.4558	−1.44635	−0.723175	+	0.690665i	$0.757318\pi$
$264$	−9.89949	−0.609272
$265$	−8.00000	−0.491436
$266$	3.41421	0.209339
$267$	5.89949	0.361043
$268$	−36.9706	−2.25834
$269$	−3.07107	−0.187246	−0.0936232	−	0.995608i	$-0.529845\pi$
$269$	−3.07107	−0.187246	−0.0936232	+	0.995608i	$0.529845\pi$
$270$	−2.41421	−0.146924
$271$	−10.8284	−0.657780	−0.328890	−	0.944368i	$-0.606675\pi$
$271$	−10.8284	−0.657780	−0.328890	+	0.944368i	$0.606675\pi$
$272$	3.51472	0.213111
$273$	4.82843	0.292230
$274$	24.1421	1.45848
$275$	−2.24264	−0.135236
$276$	29.3137	1.76448
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−34.1421	−2.04771
$279$	−3.17157	−0.189877
$280$	6.24264	0.373069
$281$	−14.5858	−0.870115	−0.435058	−	0.900403i	$-0.643272\pi$
$281$	−14.5858	−0.870115	−0.435058	+	0.900403i	$0.643272\pi$
$282$	−18.4853	−1.10078
$283$	−10.3848	−0.617311	−0.308655	−	0.951174i	$-0.599879\pi$
$283$	−10.3848	−0.617311	−0.308655	+	0.951174i	$0.599879\pi$
$284$	−41.4558	−2.45995
$285$	1.00000	0.0592349
$286$	18.4853	1.09306
$287$	0.343146	0.0202553
$288$	−1.58579	−0.0934434
$289$	−15.6274	−0.919260
$290$	−3.41421	−0.200490
$291$	−9.75736	−0.571987
$292$	−29.3137	−1.71546
$293$	−11.5147	−0.672697	−0.336349	−	0.941738i	$-0.609192\pi$
$293$	−11.5147	−0.672697	−0.336349	+	0.941738i	$0.609192\pi$
$294$	−12.0711	−0.703999
$295$	−12.4853	−0.726921
$296$	−15.0711	−0.875988
$297$	−2.24264	−0.130131
$298$	−14.4853	−0.839110
$299$	−26.1421	−1.51184
$300$	3.82843	0.221034
$301$	−17.3137	−0.997946
$302$	−15.6569	−0.900951
$303$	−3.17157	−0.182202
$304$	−3.00000	−0.172062
$305$	−7.31371	−0.418782
$306$	2.82843	0.161690
$307$	21.1716	1.20833	0.604163	−	0.796861i	$-0.293508\pi$
$307$	21.1716	1.20833	0.604163	+	0.796861i	$0.293508\pi$
$308$	12.1421	0.691862
$309$	−7.31371	−0.416062
$310$	7.65685	0.434880
$311$	−6.24264	−0.353988	−0.176994	−	0.984212i	$-0.556637\pi$
$311$	−6.24264	−0.353988	−0.176994	+	0.984212i	$0.556637\pi$
$312$	−15.0711	−0.853231
$313$	5.79899	0.327778	0.163889	−	0.986479i	$-0.447596\pi$
$313$	5.79899	0.327778	0.163889	+	0.986479i	$0.447596\pi$
$314$	25.3137	1.42854
$315$	1.41421	0.0796819
$316$	0	0
$317$	−26.6274	−1.49554	−0.747772	−	0.663955i	$-0.768877\pi$
$317$	−26.6274	−1.49554	−0.747772	+	0.663955i	$0.768877\pi$
$318$	19.3137	1.08306
$319$	−3.17157	−0.177574
$320$	9.82843	0.549426
$321$	19.3137	1.07799
$322$	−26.1421	−1.45684
$323$	−1.17157	−0.0651881
$324$	3.82843	0.212690
$325$	−3.41421	−0.189386
$326$	52.8701	2.92820
$327$	−6.48528	−0.358637
$328$	−1.07107	−0.0591398
$329$	10.8284	0.596991
$330$	5.41421	0.298043
$331$	28.1421	1.54683	0.773416	−	0.633899i	$-0.218547\pi$
$331$	28.1421	1.54683	0.773416	+	0.633899i	$0.218547\pi$
$332$	49.1127	2.69541
$333$	−3.41421	−0.187098
$334$	−41.7990	−2.28714
$335$	9.65685	0.527610
$336$	−4.24264	−0.231455
$337$	−24.5858	−1.33927	−0.669637	−	0.742689i	$-0.733550\pi$
$337$	−24.5858	−1.33927	−0.669637	+	0.742689i	$0.733550\pi$
$338$	−3.24264	−0.176376
$339$	10.1421	0.550845
$340$	−4.48528	−0.243249
$341$	7.11270	0.385174
$342$	−2.41421	−0.130546
$343$	16.9706	0.916324
$344$	54.0416	2.91373
$345$	−7.65685	−0.412231
$346$	−47.7990	−2.56969
$347$	−27.4558	−1.47391	−0.736953	−	0.675943i	$-0.763736\pi$
$347$	−27.4558	−1.47391	−0.736953	+	0.675943i	$0.763736\pi$
$348$	5.41421	0.290232
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	−3.41421	−0.182497
$351$	−3.41421	−0.182237
$352$	3.55635	0.189554
$353$	3.65685	0.194635	0.0973174	−	0.995253i	$-0.468974\pi$
$353$	3.65685	0.194635	0.0973174	+	0.995253i	$0.468974\pi$
$354$	30.1421	1.60204
$355$	10.8284	0.574713
$356$	22.5858	1.19704
$357$	−1.65685	−0.0876900
$358$	39.7990	2.10344
$359$	24.8701	1.31259	0.656296	−	0.754504i	$-0.272122\pi$
$359$	24.8701	1.31259	0.656296	+	0.754504i	$0.272122\pi$
$360$	−4.41421	−0.232649
$361$	1.00000	0.0526316
$362$	−50.2843	−2.64288
$363$	−5.97056	−0.313373
$364$	18.4853	0.968892
$365$	7.65685	0.400778
$366$	17.6569	0.922939
$367$	−14.3848	−0.750879	−0.375440	−	0.926847i	$-0.622508\pi$
$367$	−14.3848	−0.750879	−0.375440	+	0.926847i	$0.622508\pi$
$368$	22.9706	1.19742
$369$	−0.242641	−0.0126314
$370$	8.24264	0.428514
$371$	−11.3137	−0.587378
$372$	−12.1421	−0.629540
$373$	0.585786	0.0303309	0.0151654	−	0.999885i	$-0.495173\pi$
$373$	0.585786	0.0303309	0.0151654	+	0.999885i	$0.495173\pi$
$374$	−6.34315	−0.327996
$375$	−1.00000	−0.0516398
$376$	−33.7990	−1.74305
$377$	−4.82843	−0.248677
$378$	−3.41421	−0.175608
$379$	3.17157	0.162913	0.0814564	−	0.996677i	$-0.474043\pi$
$379$	3.17157	0.162913	0.0814564	+	0.996677i	$0.474043\pi$
$380$	3.82843	0.196394
$381$	19.3137	0.989471
$382$	24.7279	1.26519
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	−20.5563	−1.04901
$385$	−3.17157	−0.161638
$386$	12.2426	0.623134
$387$	12.2426	0.622328
$388$	−37.3553	−1.89643
$389$	30.9706	1.57027	0.785135	−	0.619325i	$-0.212594\pi$
$389$	30.9706	1.57027	0.785135	+	0.619325i	$0.212594\pi$
$390$	8.24264	0.417382
$391$	8.97056	0.453661
$392$	−22.0711	−1.11476
$393$	−8.58579	−0.433096
$394$	16.4853	0.830516
$395$	0	0
$396$	−8.58579	−0.431452
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	43.7990	2.19544
$399$	1.41421	0.0707992
$400$	3.00000	0.150000
$401$	−7.75736	−0.387384	−0.193692	−	0.981062i	$-0.562046\pi$
$401$	−7.75736	−0.387384	−0.193692	+	0.981062i	$0.562046\pi$
$402$	−23.3137	−1.16278
$403$	10.8284	0.539402
$404$	−12.1421	−0.604094
$405$	−1.00000	−0.0496904
$406$	−4.82843	−0.239631
$407$	7.65685	0.379536
$408$	5.17157	0.256031
$409$	−12.8284	−0.634325	−0.317162	−	0.948371i	$-0.602730\pi$
$409$	−12.8284	−0.634325	−0.317162	+	0.948371i	$0.602730\pi$
$410$	0.585786	0.0289299
$411$	10.0000	0.493264
$412$	−28.0000	−1.37946
$413$	−17.6569	−0.868837
$414$	18.4853	0.908502
$415$	−12.8284	−0.629723
$416$	5.41421	0.265454
$417$	−14.1421	−0.692543
$418$	5.41421	0.264818
$419$	3.41421	0.166795	0.0833976	−	0.996516i	$-0.473423\pi$
$419$	3.41421	0.166795	0.0833976	+	0.996516i	$0.473423\pi$
$420$	5.41421	0.264187
$421$	9.31371	0.453922	0.226961	−	0.973904i	$-0.427121\pi$
$421$	9.31371	0.453922	0.226961	+	0.973904i	$0.427121\pi$
$422$	17.6569	0.859522
$423$	−7.65685	−0.372289
$424$	35.3137	1.71499
$425$	1.17157	0.0568296
$426$	−26.1421	−1.26659
$427$	−10.3431	−0.500540
$428$	73.9411	3.57408
$429$	7.65685	0.369676
$430$	−29.5563	−1.42533
$431$	−31.1127	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$431$	−31.1127	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$432$	3.00000	0.144338
$433$	−10.9289	−0.525211	−0.262605	−	0.964903i	$-0.584582\pi$
$433$	−10.9289	−0.525211	−0.262605	+	0.964903i	$0.584582\pi$
$434$	10.8284	0.519781
$435$	−1.41421	−0.0678064
$436$	−24.8284	−1.18907
$437$	−7.65685	−0.366277
$438$	−18.4853	−0.883261
$439$	−21.6569	−1.03363	−0.516813	−	0.856099i	$-0.672882\pi$
$439$	−21.6569	−1.03363	−0.516813	+	0.856099i	$0.672882\pi$
$440$	9.89949	0.471940
$441$	−5.00000	−0.238095
$442$	−9.65685	−0.459330
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	−13.0711	−0.620325
$445$	−5.89949	−0.279663
$446$	44.9706	2.12942
$447$	−6.00000	−0.283790
$448$	13.8995	0.656689
$449$	−26.8701	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$449$	−26.8701	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$450$	2.41421	0.113807
$451$	0.544156	0.0256233
$452$	38.8284	1.82634
$453$	−6.48528	−0.304705
$454$	36.1421	1.69623
$455$	−4.82843	−0.226360
$456$	−4.41421	−0.206714
$457$	32.8284	1.53565	0.767825	−	0.640660i	$-0.221339\pi$
$457$	32.8284	1.53565	0.767825	+	0.640660i	$0.221339\pi$
$458$	−54.6274	−2.55257
$459$	1.17157	0.0546843
$460$	−29.3137	−1.36676
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	7.65685	0.356229
$463$	24.2426	1.12665	0.563326	−	0.826235i	$-0.309522\pi$
$463$	24.2426	1.12665	0.563326	+	0.826235i	$0.309522\pi$
$464$	4.24264	0.196960
$465$	3.17157	0.147078
$466$	−0.828427	−0.0383761
$467$	35.6569	1.65000	0.825001	−	0.565131i	$-0.191174\pi$
$467$	35.6569	1.65000	0.825001	+	0.565131i	$0.191174\pi$
$468$	−13.0711	−0.604210
$469$	13.6569	0.630615
$470$	18.4853	0.852662
$471$	10.4853	0.483136
$472$	55.1127	2.53677
$473$	−27.4558	−1.26242
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−6.34315	−0.290738
$477$	8.00000	0.366295
$478$	−64.5269	−2.95139
$479$	17.0711	0.779997	0.389998	−	0.920815i	$-0.372476\pi$
$479$	17.0711	0.779997	0.389998	+	0.920815i	$0.372476\pi$
$480$	1.58579	0.0723809
$481$	11.6569	0.531507
$482$	−47.4558	−2.16155
$483$	−10.8284	−0.492710
$484$	−22.8579	−1.03899
$485$	9.75736	0.443059
$486$	2.41421	0.109511
$487$	20.4853	0.928277	0.464138	−	0.885763i	$-0.346364\pi$
$487$	20.4853	0.928277	0.464138	+	0.885763i	$0.346364\pi$
$488$	32.2843	1.46144
$489$	21.8995	0.990329
$490$	12.0711	0.545315
$491$	−1.75736	−0.0793085	−0.0396543	−	0.999213i	$-0.512626\pi$
$491$	−1.75736	−0.0793085	−0.0396543	+	0.999213i	$0.512626\pi$
$492$	−0.928932	−0.0418795
$493$	1.65685	0.0746210
$494$	8.24264	0.370854
$495$	2.24264	0.100799
$496$	−9.51472	−0.427223
$497$	15.3137	0.686914
$498$	30.9706	1.38782
$499$	−5.17157	−0.231511	−0.115756	−	0.993278i	$-0.536929\pi$
$499$	−5.17157	−0.231511	−0.115756	+	0.993278i	$0.536929\pi$
$500$	−3.82843	−0.171212
$501$	−17.3137	−0.773519
$502$	−4.24264	−0.189358
$503$	4.82843	0.215289	0.107644	−	0.994189i	$-0.465669\pi$
$503$	4.82843	0.215289	0.107644	+	0.994189i	$0.465669\pi$
$504$	−6.24264	−0.278069
$505$	3.17157	0.141133
$506$	−41.4558	−1.84294
$507$	−1.34315	−0.0596512
$508$	73.9411	3.28061
$509$	−0.727922	−0.0322646	−0.0161323	−	0.999870i	$-0.505135\pi$
$509$	−0.727922	−0.0322646	−0.0161323	+	0.999870i	$0.505135\pi$
$510$	−2.82843	−0.125245
$511$	10.8284	0.479021
$512$	−31.2426	−1.38074
$513$	−1.00000	−0.0441511
$514$	−10.8284	−0.477621
$515$	7.31371	0.322281
$516$	46.8701	2.06334
$517$	17.1716	0.755205
$518$	11.6569	0.512173
$519$	−19.7990	−0.869079
$520$	15.0711	0.660910
$521$	7.75736	0.339856	0.169928	−	0.985456i	$-0.445646\pi$
$521$	7.75736	0.339856	0.169928	+	0.985456i	$0.445646\pi$
$522$	3.41421	0.149436
$523$	−15.5147	−0.678411	−0.339206	−	0.940712i	$-0.610158\pi$
$523$	−15.5147	−0.678411	−0.339206	+	0.940712i	$0.610158\pi$
$524$	−32.8701	−1.43594
$525$	−1.41421	−0.0617213
$526$	−56.6274	−2.46907
$527$	−3.71573	−0.161860
$528$	−6.72792	−0.292795
$529$	35.6274	1.54902
$530$	−19.3137	−0.838934
$531$	12.4853	0.541815
$532$	5.41421	0.234736
$533$	0.828427	0.0358832
$534$	14.2426	0.616339
$535$	−19.3137	−0.835004
$536$	−42.6274	−1.84122
$537$	16.4853	0.711392
$538$	−7.41421	−0.319649
$539$	11.2132	0.482987
$540$	−3.82843	−0.164749
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−26.1421	−1.12290
$543$	−20.8284	−0.893833
$544$	−1.85786	−0.0796553
$545$	6.48528	0.277799
$546$	11.6569	0.498867
$547$	24.4853	1.04692	0.523458	−	0.852052i	$-0.324642\pi$
$547$	24.4853	1.04692	0.523458	+	0.852052i	$0.324642\pi$
$548$	38.2843	1.63542
$549$	7.31371	0.312141
$550$	−5.41421	−0.230863
$551$	−1.41421	−0.0602475
$552$	33.7990	1.43858
$553$	0	0
$554$	−53.1127	−2.25654
$555$	3.41421	0.144925
$556$	−54.1421	−2.29614
$557$	−25.3137	−1.07258	−0.536288	−	0.844035i	$-0.680174\pi$
$557$	−25.3137	−1.07258	−0.536288	+	0.844035i	$0.680174\pi$
$558$	−7.65685	−0.324140
$559$	−41.7990	−1.76791
$560$	4.24264	0.179284
$561$	−2.62742	−0.110930
$562$	−35.2132	−1.48538
$563$	−30.2843	−1.27633	−0.638165	−	0.769900i	$-0.720306\pi$
$563$	−30.2843	−1.27633	−0.638165	+	0.769900i	$0.720306\pi$
$564$	−29.3137	−1.23433
$565$	−10.1421	−0.426683
$566$	−25.0711	−1.05382
$567$	−1.41421	−0.0593914
$568$	−47.7990	−2.00560
$569$	31.0711	1.30257	0.651283	−	0.758835i	$-0.274231\pi$
$569$	31.0711	1.30257	0.651283	+	0.758835i	$0.274231\pi$
$570$	2.41421	0.101120
$571$	9.17157	0.383818	0.191909	−	0.981413i	$-0.438532\pi$
$571$	9.17157	0.383818	0.191909	+	0.981413i	$0.438532\pi$
$572$	29.3137	1.22567
$573$	10.2426	0.427892
$574$	0.828427	0.0345779
$575$	7.65685	0.319313
$576$	−9.82843	−0.409518
$577$	19.4558	0.809957	0.404979	−	0.914326i	$-0.367279\pi$
$577$	19.4558	0.809957	0.404979	+	0.914326i	$0.367279\pi$
$578$	−37.7279	−1.56927
$579$	5.07107	0.210746
$580$	−5.41421	−0.224813
$581$	−18.1421	−0.752663
$582$	−23.5563	−0.976442
$583$	−17.9411	−0.743045
$584$	−33.7990	−1.39861
$585$	3.41421	0.141160
$586$	−27.7990	−1.14837
$587$	−12.3431	−0.509456	−0.254728	−	0.967013i	$-0.581986\pi$
$587$	−12.3431	−0.509456	−0.254728	+	0.967013i	$0.581986\pi$
$588$	−19.1421	−0.789408
$589$	3.17157	0.130682
$590$	−30.1421	−1.24093
$591$	6.82843	0.280884
$592$	−10.2426	−0.420970
$593$	36.6274	1.50411	0.752054	−	0.659102i	$-0.229063\pi$
$593$	36.6274	1.50411	0.752054	+	0.659102i	$0.229063\pi$
$594$	−5.41421	−0.222148
$595$	1.65685	0.0679244
$596$	−22.9706	−0.940911
$597$	18.1421	0.742508
$598$	−63.1127	−2.58087
$599$	−3.02944	−0.123779	−0.0618897	−	0.998083i	$-0.519713\pi$
$599$	−3.02944	−0.123779	−0.0618897	+	0.998083i	$0.519713\pi$
$600$	4.41421	0.180210
$601$	8.14214	0.332125	0.166062	−	0.986115i	$-0.446895\pi$
$601$	8.14214	0.332125	0.166062	+	0.986115i	$0.446895\pi$
$602$	−41.7990	−1.70360
$603$	−9.65685	−0.393258
$604$	−24.8284	−1.01025
$605$	5.97056	0.242738
$606$	−7.65685	−0.311038
$607$	16.4853	0.669117	0.334558	−	0.942375i	$-0.391413\pi$
$607$	16.4853	0.669117	0.334558	+	0.942375i	$0.391413\pi$
$608$	1.58579	0.0643121
$609$	−2.00000	−0.0810441
$610$	−17.6569	−0.714905
$611$	26.1421	1.05760
$612$	4.48528	0.181307
$613$	−10.4853	−0.423497	−0.211748	−	0.977324i	$-0.567916\pi$
$613$	−10.4853	−0.423497	−0.211748	+	0.977324i	$0.567916\pi$
$614$	51.1127	2.06274
$615$	0.242641	0.00978422
$616$	14.0000	0.564076
$617$	28.4853	1.14677	0.573387	−	0.819285i	$-0.305629\pi$
$617$	28.4853	1.14677	0.573387	+	0.819285i	$0.305629\pi$
$618$	−17.6569	−0.710263
$619$	−20.4853	−0.823373	−0.411686	−	0.911326i	$-0.635060\pi$
$619$	−20.4853	−0.823373	−0.411686	+	0.911326i	$0.635060\pi$
$620$	12.1421	0.487640
$621$	7.65685	0.307259
$622$	−15.0711	−0.604295
$623$	−8.34315	−0.334261
$624$	−10.2426	−0.410034
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	2.24264	0.0895624
$628$	40.1421	1.60185
$629$	−4.00000	−0.159490
$630$	3.41421	0.136026
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	7.31371	0.290694
$634$	−64.2843	−2.55305
$635$	−19.3137	−0.766441
$636$	30.6274	1.21446
$637$	17.0711	0.676380
$638$	−7.65685	−0.303138
$639$	−10.8284	−0.428366
$640$	20.5563	0.812561
$641$	−31.3553	−1.23846	−0.619231	−	0.785209i	$-0.712555\pi$
$641$	−31.3553	−1.23846	−0.619231	+	0.785209i	$0.712555\pi$
$642$	46.6274	1.84024
$643$	42.3848	1.67149	0.835746	−	0.549116i	$-0.185035\pi$
$643$	42.3848	1.67149	0.835746	+	0.549116i	$0.185035\pi$
$644$	−41.4558	−1.63359
$645$	−12.2426	−0.482054
$646$	−2.82843	−0.111283
$647$	−36.1421	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$647$	−36.1421	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$648$	4.41421	0.173407
$649$	−28.0000	−1.09910
$650$	−8.24264	−0.323303
$651$	4.48528	0.175792
$652$	83.8406	3.28345
$653$	42.8284	1.67601	0.838003	−	0.545666i	$-0.183723\pi$
$653$	42.8284	1.67601	0.838003	+	0.545666i	$0.183723\pi$
$654$	−15.6569	−0.612231
$655$	8.58579	0.335474
$656$	−0.727922	−0.0284206
$657$	−7.65685	−0.298722
$658$	26.1421	1.01913
$659$	−18.6274	−0.725621	−0.362811	−	0.931863i	$-0.618183\pi$
$659$	−18.6274	−0.725621	−0.362811	+	0.931863i	$0.618183\pi$
$660$	8.58579	0.334201
$661$	7.45584	0.289999	0.144999	−	0.989432i	$-0.453682\pi$
$661$	7.45584	0.289999	0.144999	+	0.989432i	$0.453682\pi$
$662$	67.9411	2.64061
$663$	−4.00000	−0.155347
$664$	56.6274	2.19757
$665$	−1.41421	−0.0548408
$666$	−8.24264	−0.319396
$667$	10.8284	0.419278
$668$	−66.2843	−2.56462
$669$	18.6274	0.720178
$670$	23.3137	0.900687
$671$	−16.4020	−0.633193
$672$	2.24264	0.0865117
$673$	−44.1838	−1.70316	−0.851580	−	0.524225i	$-0.824355\pi$
$673$	−44.1838	−1.70316	−0.851580	+	0.524225i	$0.824355\pi$
$674$	−59.3553	−2.28628
$675$	1.00000	0.0384900
$676$	−5.14214	−0.197774
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	24.4853	0.940352
$679$	13.7990	0.529557
$680$	−5.17157	−0.198321
$681$	14.9706	0.573673
$682$	17.1716	0.657534
$683$	18.3431	0.701881	0.350940	−	0.936398i	$-0.385862\pi$
$683$	18.3431	0.701881	0.350940	+	0.936398i	$0.385862\pi$
$684$	−3.82843	−0.146384
$685$	−10.0000	−0.382080
$686$	40.9706	1.56426
$687$	−22.6274	−0.863290
$688$	36.7279	1.40024
$689$	−27.3137	−1.04057
$690$	−18.4853	−0.703723
$691$	46.8284	1.78144	0.890719	−	0.454555i	$-0.150202\pi$
$691$	46.8284	1.78144	0.890719	+	0.454555i	$0.150202\pi$
$692$	−75.7990	−2.88145
$693$	3.17157	0.120478
$694$	−66.2843	−2.51612
$695$	14.1421	0.536442
$696$	6.24264	0.236627
$697$	−0.284271	−0.0107675
$698$	43.4558	1.64483
$699$	−0.343146	−0.0129790
$700$	−5.41421	−0.204638
$701$	6.68629	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$701$	6.68629	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$702$	−8.24264	−0.311098
$703$	3.41421	0.128770
$704$	22.0416	0.830725
$705$	7.65685	0.288374
$706$	8.82843	0.332262
$707$	4.48528	0.168686
$708$	47.7990	1.79640
$709$	28.9706	1.08801	0.544006	−	0.839081i	$-0.316907\pi$
$709$	28.9706	1.08801	0.544006	+	0.839081i	$0.316907\pi$
$710$	26.1421	0.981097
$711$	0	0
$712$	26.0416	0.975951
$713$	−24.2843	−0.909453
$714$	−4.00000	−0.149696
$715$	−7.65685	−0.286350
$716$	63.1127	2.35863
$717$	−26.7279	−0.998173
$718$	60.0416	2.24073
$719$	−1.07107	−0.0399441	−0.0199720	−	0.999801i	$-0.506358\pi$
$719$	−1.07107	−0.0399441	−0.0199720	+	0.999801i	$0.506358\pi$
$720$	−3.00000	−0.111803
$721$	10.3431	0.385199
$722$	2.41421	0.0898477
$723$	−19.6569	−0.731046
$724$	−79.7401	−2.96352
$725$	1.41421	0.0525226
$726$	−14.4142	−0.534962
$727$	47.3553	1.75631	0.878156	−	0.478374i	$-0.158774\pi$
$727$	47.3553	1.75631	0.878156	+	0.478374i	$0.158774\pi$
$728$	21.3137	0.789939
$729$	1.00000	0.0370370
$730$	18.4853	0.684171
$731$	14.3431	0.530500
$732$	28.0000	1.03491
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−34.7279	−1.28183
$735$	5.00000	0.184428
$736$	−12.1421	−0.447565
$737$	21.6569	0.797740
$738$	−0.585786	−0.0215631
$739$	14.3431	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$739$	14.3431	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$740$	13.0711	0.480502
$741$	3.41421	0.125424
$742$	−27.3137	−1.00272
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	−14.0000	−0.513265
$745$	6.00000	0.219823
$746$	1.41421	0.0517780
$747$	12.8284	0.469368
$748$	−10.0589	−0.367789
$749$	−27.3137	−0.998021
$750$	−2.41421	−0.0881546
$751$	−24.1421	−0.880959	−0.440480	−	0.897763i	$-0.645192\pi$
$751$	−24.1421	−0.880959	−0.440480	+	0.897763i	$0.645192\pi$
$752$	−22.9706	−0.837650
$753$	−1.75736	−0.0640417
$754$	−11.6569	−0.424518
$755$	6.48528	0.236024
$756$	−5.41421	−0.196913
$757$	−32.4264	−1.17856	−0.589279	−	0.807930i	$-0.700588\pi$
$757$	−32.4264	−1.17856	−0.589279	+	0.807930i	$0.700588\pi$
$758$	7.65685	0.278109
$759$	−17.1716	−0.623289
$760$	4.41421	0.160120
$761$	43.9411	1.59286	0.796432	−	0.604728i	$-0.206718\pi$
$761$	43.9411	1.59286	0.796432	+	0.604728i	$0.206718\pi$
$762$	46.6274	1.68913
$763$	9.17157	0.332033
$764$	39.2132	1.41868
$765$	−1.17157	−0.0423583
$766$	67.5980	2.44241
$767$	−42.6274	−1.53919
$768$	−29.9706	−1.08147
$769$	42.9706	1.54956	0.774779	−	0.632232i	$-0.217861\pi$
$769$	42.9706	1.54956	0.774779	+	0.632232i	$0.217861\pi$
$770$	−7.65685	−0.275934
$771$	−4.48528	−0.161533
$772$	19.4142	0.698733
$773$	−25.6569	−0.922813	−0.461406	−	0.887189i	$-0.652655\pi$
$773$	−25.6569	−0.922813	−0.461406	+	0.887189i	$0.652655\pi$
$774$	29.5563	1.06238
$775$	−3.17157	−0.113926
$776$	−43.0711	−1.54616
$777$	4.82843	0.173219
$778$	74.7696	2.68062
$779$	0.242641	0.00869350
$780$	13.0711	0.468019
$781$	24.2843	0.868960
$782$	21.6569	0.774448
$783$	1.41421	0.0505399
$784$	−15.0000	−0.535714
$785$	−10.4853	−0.374236
$786$	−20.7279	−0.739340
$787$	−42.8284	−1.52667	−0.763334	−	0.646004i	$-0.776439\pi$
$787$	−42.8284	−1.52667	−0.763334	+	0.646004i	$0.776439\pi$
$788$	26.1421	0.931275
$789$	−23.4558	−0.835050
$790$	0	0
$791$	−14.3431	−0.509984
$792$	−9.89949	−0.351763
$793$	−24.9706	−0.886731
$794$	69.1127	2.45272
$795$	−8.00000	−0.283731
$796$	69.4558	2.46180
$797$	−14.1421	−0.500940	−0.250470	−	0.968124i	$-0.580585\pi$
$797$	−14.1421	−0.500940	−0.250470	+	0.968124i	$0.580585\pi$
$798$	3.41421	0.120862
$799$	−8.97056	−0.317356
$800$	−1.58579	−0.0560660
$801$	5.89949	0.208448
$802$	−18.7279	−0.661306
$803$	17.1716	0.605972
$804$	−36.9706	−1.30385
$805$	10.8284	0.381652
$806$	26.1421	0.920817
$807$	−3.07107	−0.108107
$808$	−14.0000	−0.492518
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	−2.41421	−0.0848268
$811$	8.68629	0.305017	0.152508	−	0.988302i	$-0.451265\pi$
$811$	8.68629	0.305017	0.152508	+	0.988302i	$0.451265\pi$
$812$	−7.65685	−0.268703
$813$	−10.8284	−0.379770
$814$	18.4853	0.647909
$815$	−21.8995	−0.767106
$816$	3.51472	0.123040
$817$	−12.2426	−0.428316
$818$	−30.9706	−1.08286
$819$	4.82843	0.168719
$820$	0.928932	0.0324397
$821$	14.4853	0.505540	0.252770	−	0.967526i	$-0.418658\pi$
$821$	14.4853	0.505540	0.252770	+	0.967526i	$0.418658\pi$
$822$	24.1421	0.842054
$823$	−25.0122	−0.871870	−0.435935	−	0.899978i	$-0.643582\pi$
$823$	−25.0122	−0.871870	−0.435935	+	0.899978i	$0.643582\pi$
$824$	−32.2843	−1.12468
$825$	−2.24264	−0.0780787
$826$	−42.6274	−1.48320
$827$	−2.68629	−0.0934115	−0.0467058	−	0.998909i	$-0.514872\pi$
$827$	−2.68629	−0.0934115	−0.0467058	+	0.998909i	$0.514872\pi$
$828$	29.3137	1.01872
$829$	46.4853	1.61450	0.807250	−	0.590209i	$-0.200955\pi$
$829$	46.4853	1.61450	0.807250	+	0.590209i	$0.200955\pi$
$830$	−30.9706	−1.07500
$831$	−22.0000	−0.763172
$832$	33.5563	1.16336
$833$	−5.85786	−0.202963
$834$	−34.1421	−1.18225
$835$	17.3137	0.599166
$836$	8.58579	0.296946
$837$	−3.17157	−0.109626
$838$	8.24264	0.284737
$839$	−9.85786	−0.340331	−0.170166	−	0.985415i	$-0.554430\pi$
$839$	−9.85786	−0.340331	−0.170166	+	0.985415i	$0.554430\pi$
$840$	6.24264	0.215392
$841$	−27.0000	−0.931034
$842$	22.4853	0.774894
$843$	−14.5858	−0.502361
$844$	28.0000	0.963800
$845$	1.34315	0.0462056
$846$	−18.4853	−0.635537
$847$	8.44365	0.290127
$848$	24.0000	0.824163
$849$	−10.3848	−0.356405
$850$	2.82843	0.0970143
$851$	−26.1421	−0.896141
$852$	−41.4558	−1.42025
$853$	−16.8284	−0.576194	−0.288097	−	0.957601i	$-0.593023\pi$
$853$	−16.8284	−0.576194	−0.288097	+	0.957601i	$0.593023\pi$
$854$	−24.9706	−0.854475
$855$	1.00000	0.0341993
$856$	85.2548	2.91395
$857$	12.6863	0.433355	0.216678	−	0.976243i	$-0.430478\pi$
$857$	12.6863	0.433355	0.216678	+	0.976243i	$0.430478\pi$
$858$	18.4853	0.631077
$859$	−49.9411	−1.70397	−0.851985	−	0.523567i	$-0.824601\pi$
$859$	−49.9411	−1.70397	−0.851985	+	0.523567i	$0.824601\pi$
$860$	−46.8701	−1.59826
$861$	0.343146	0.0116944
$862$	−75.1127	−2.55835
$863$	8.68629	0.295685	0.147842	−	0.989011i	$-0.452767\pi$
$863$	8.68629	0.295685	0.147842	+	0.989011i	$0.452767\pi$
$864$	−1.58579	−0.0539496
$865$	19.7990	0.673186
$866$	−26.3848	−0.896591
$867$	−15.6274	−0.530735
$868$	17.1716	0.582841
$869$	0	0
$870$	−3.41421	−0.115753
$871$	32.9706	1.11716
$872$	−28.6274	−0.969447
$873$	−9.75736	−0.330237
$874$	−18.4853	−0.625274
$875$	1.41421	0.0478091
$876$	−29.3137	−0.990418
$877$	−34.9289	−1.17947	−0.589733	−	0.807598i	$-0.700767\pi$
$877$	−34.9289	−1.17947	−0.589733	+	0.807598i	$0.700767\pi$
$878$	−52.2843	−1.76451
$879$	−11.5147	−0.388382
$880$	6.72792	0.226798
$881$	−5.79899	−0.195373	−0.0976865	−	0.995217i	$-0.531144\pi$
$881$	−5.79899	−0.195373	−0.0976865	+	0.995217i	$0.531144\pi$
$882$	−12.0711	−0.406454
$883$	−12.0416	−0.405233	−0.202617	−	0.979258i	$-0.564945\pi$
$883$	−12.0416	−0.405233	−0.202617	+	0.979258i	$0.564945\pi$
$884$	−15.3137	−0.515056
$885$	−12.4853	−0.419688
$886$	43.4558	1.45993
$887$	25.9411	0.871018	0.435509	−	0.900184i	$-0.356568\pi$
$887$	25.9411	0.871018	0.435509	+	0.900184i	$0.356568\pi$
$888$	−15.0711	−0.505752
$889$	−27.3137	−0.916072
$890$	−14.2426	−0.477414
$891$	−2.24264	−0.0751313
$892$	71.3137	2.38776
$893$	7.65685	0.256227
$894$	−14.4853	−0.484460
$895$	−16.4853	−0.551042
$896$	29.0711	0.971196
$897$	−26.1421	−0.872861
$898$	−64.8701	−2.16474
$899$	−4.48528	−0.149593
$900$	3.82843	0.127614
$901$	9.37258	0.312246
$902$	1.31371	0.0437417
$903$	−17.3137	−0.576164
$904$	44.7696	1.48901
$905$	20.8284	0.692360
$906$	−15.6569	−0.520164
$907$	−18.1421	−0.602400	−0.301200	−	0.953561i	$-0.597387\pi$
$907$	−18.1421	−0.602400	−0.301200	+	0.953561i	$0.597387\pi$
$908$	57.3137	1.90202
$909$	−3.17157	−0.105194
$910$	−11.6569	−0.386421
$911$	8.68629	0.287790	0.143895	−	0.989593i	$-0.454037\pi$
$911$	8.68629	0.287790	0.143895	+	0.989593i	$0.454037\pi$
$912$	−3.00000	−0.0993399
$913$	−28.7696	−0.952133
$914$	79.2548	2.62152
$915$	−7.31371	−0.241784
$916$	−86.6274	−2.86225
$917$	12.1421	0.400969
$918$	2.82843	0.0933520
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	−33.7990	−1.11432
$921$	21.1716	0.697627
$922$	47.4558	1.56287
$923$	36.9706	1.21690
$924$	12.1421	0.399447
$925$	−3.41421	−0.112259
$926$	58.5269	1.92331
$927$	−7.31371	−0.240214
$928$	−2.24264	−0.0736183
$929$	33.1127	1.08639	0.543196	−	0.839606i	$-0.317214\pi$
$929$	33.1127	1.08639	0.543196	+	0.839606i	$0.317214\pi$
$930$	7.65685	0.251078
$931$	5.00000	0.163868
$932$	−1.31371	−0.0430320
$933$	−6.24264	−0.204375
$934$	86.0833	2.81673
$935$	2.62742	0.0859257
$936$	−15.0711	−0.492613
$937$	−29.7990	−0.973491	−0.486745	−	0.873544i	$-0.661816\pi$
$937$	−29.7990	−0.973491	−0.486745	+	0.873544i	$0.661816\pi$
$938$	32.9706	1.07653
$939$	5.79899	0.189243
$940$	29.3137	0.956108
$941$	22.8701	0.745543	0.372771	−	0.927923i	$-0.378408\pi$
$941$	22.8701	0.745543	0.372771	+	0.927923i	$0.378408\pi$
$942$	25.3137	0.824765
$943$	−1.85786	−0.0605004
$944$	37.4558	1.21908
$945$	1.41421	0.0460044
$946$	−66.2843	−2.15509
$947$	−47.1716	−1.53287	−0.766435	−	0.642322i	$-0.777971\pi$
$947$	−47.1716	−1.53287	−0.766435	+	0.642322i	$0.777971\pi$
$948$	0	0
$949$	26.1421	0.848610
$950$	−2.41421	−0.0783274
$951$	−26.6274	−0.863453
$952$	−7.31371	−0.237039
$953$	−53.4558	−1.73160	−0.865802	−	0.500386i	$-0.833191\pi$
$953$	−53.4558	−1.73160	−0.865802	+	0.500386i	$0.833191\pi$
$954$	19.3137	0.625304
$955$	−10.2426	−0.331444
$956$	−102.326	−3.30946
$957$	−3.17157	−0.102522
$958$	41.2132	1.33154
$959$	−14.1421	−0.456673
$960$	9.82843	0.317211
$961$	−20.9411	−0.675520
$962$	28.1421	0.907339
$963$	19.3137	0.622376
$964$	−75.2548	−2.42379
$965$	−5.07107	−0.163243
$966$	−26.1421	−0.841109
$967$	8.04163	0.258601	0.129301	−	0.991605i	$-0.458727\pi$
$967$	8.04163	0.258601	0.129301	+	0.991605i	$0.458727\pi$
$968$	−26.3553	−0.847093
$969$	−1.17157	−0.0376363
$970$	23.5563	0.756349
$971$	22.3431	0.717026	0.358513	−	0.933525i	$-0.383284\pi$
$971$	22.3431	0.717026	0.358513	+	0.933525i	$0.383284\pi$
$972$	3.82843	0.122797
$973$	20.0000	0.641171
$974$	49.4558	1.58467
$975$	−3.41421	−0.109342
$976$	21.9411	0.702318
$977$	32.2843	1.03287	0.516433	−	0.856328i	$-0.327260\pi$
$977$	32.2843	1.03287	0.516433	+	0.856328i	$0.327260\pi$
$978$	52.8701	1.69060
$979$	−13.2304	−0.422847
$980$	19.1421	0.611473
$981$	−6.48528	−0.207059
$982$	−4.24264	−0.135388
$983$	−24.6274	−0.785493	−0.392746	−	0.919647i	$-0.628475\pi$
$983$	−24.6274	−0.785493	−0.392746	+	0.919647i	$0.628475\pi$
$984$	−1.07107	−0.0341444
$985$	−6.82843	−0.217572
$986$	4.00000	0.127386
$987$	10.8284	0.344673
$988$	13.0711	0.415846
$989$	93.7401	2.98076
$990$	5.41421	0.172075
$991$	2.34315	0.0744325	0.0372162	−	0.999307i	$-0.488151\pi$
$991$	2.34315	0.0744325	0.0372162	+	0.999307i	$0.488151\pi$
$992$	5.02944	0.159685
$993$	28.1421	0.893064
$994$	36.9706	1.17264
$995$	−18.1421	−0.575144
$996$	49.1127	1.55620
$997$	−38.0833	−1.20611	−0.603054	−	0.797700i	$-0.706050\pi$
$997$	−38.0833	−1.20611	−0.603054	+	0.797700i	$0.706050\pi$
$998$	−12.4853	−0.395215
$999$	−3.41421	−0.108021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.a.g.1.2	✓	2
3.2	odd	2		855.2.a.d.1.1		2
4.3	odd	2		4560.2.a.bf.1.2		2
5.2	odd	4		1425.2.c.l.799.4		4
5.3	odd	4		1425.2.c.l.799.1		4
5.4	even	2		1425.2.a.k.1.1		2
15.14	odd	2		4275.2.a.y.1.2		2
19.18	odd	2		5415.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.g.1.2	✓	2	1.1	even	1	trivial
855.2.a.d.1.1		2	3.2	odd	2
1425.2.a.k.1.1		2	5.4	even	2
1425.2.c.l.799.1		4	5.3	odd	4
1425.2.c.l.799.4		4	5.2	odd	4
4275.2.a.y.1.2		2	15.14	odd	2
4560.2.a.bf.1.2		2	4.3	odd	2
5415.2.a.n.1.1		2	19.18	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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} -1.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -2.24264 q^{11} +3.82843 q^{12} -3.41421 q^{13} -3.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} +1.17157 q^{17} +2.41421 q^{18} -1.00000 q^{19} -3.82843 q^{20} -1.41421 q^{21} -5.41421 q^{22} +7.65685 q^{23} +4.41421 q^{24} +1.00000 q^{25} -8.24264 q^{26} +1.00000 q^{27} -5.41421 q^{28} +1.41421 q^{29} -2.41421 q^{30} -3.17157 q^{31} -1.58579 q^{32} -2.24264 q^{33} +2.82843 q^{34} +1.41421 q^{35} +3.82843 q^{36} -3.41421 q^{37} -2.41421 q^{38} -3.41421 q^{39} -4.41421 q^{40} -0.242641 q^{41} -3.41421 q^{42} +12.2426 q^{43} -8.58579 q^{44} -1.00000 q^{45} +18.4853 q^{46} -7.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} +2.41421 q^{50} +1.17157 q^{51} -13.0711 q^{52} +8.00000 q^{53} +2.41421 q^{54} +2.24264 q^{55} -6.24264 q^{56} -1.00000 q^{57} +3.41421 q^{58} +12.4853 q^{59} -3.82843 q^{60} +7.31371 q^{61} -7.65685 q^{62} -1.41421 q^{63} -9.82843 q^{64} +3.41421 q^{65} -5.41421 q^{66} -9.65685 q^{67} +4.48528 q^{68} +7.65685 q^{69} +3.41421 q^{70} -10.8284 q^{71} +4.41421 q^{72} -7.65685 q^{73} -8.24264 q^{74} +1.00000 q^{75} -3.82843 q^{76} +3.17157 q^{77} -8.24264 q^{78} -3.00000 q^{80} +1.00000 q^{81} -0.585786 q^{82} +12.8284 q^{83} -5.41421 q^{84} -1.17157 q^{85} +29.5563 q^{86} +1.41421 q^{87} -9.89949 q^{88} +5.89949 q^{89} -2.41421 q^{90} +4.82843 q^{91} +29.3137 q^{92} -3.17157 q^{93} -18.4853 q^{94} +1.00000 q^{95} -1.58579 q^{96} -9.75736 q^{97} -12.0711 q^{98} -2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 8 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 8 q^{26} + 2 q^{27} - 8 q^{28} - 2 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 8 q^{41} - 4 q^{42} + 16 q^{43} - 20 q^{44} - 2 q^{45} + 20 q^{46} - 4 q^{47} + 6 q^{48} - 10 q^{49} + 2 q^{50} + 8 q^{51} - 12 q^{52} + 16 q^{53} + 2 q^{54} - 4 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} + 8 q^{59} - 2 q^{60} - 8 q^{61} - 4 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{66} - 8 q^{67} - 8 q^{68} + 4 q^{69} + 4 q^{70} - 16 q^{71} + 6 q^{72} - 4 q^{73} - 8 q^{74} + 2 q^{75} - 2 q^{76} + 12 q^{77} - 8 q^{78} - 6 q^{80} + 2 q^{81} - 4 q^{82} + 20 q^{83} - 8 q^{84} - 8 q^{85} + 28 q^{86} - 8 q^{89} - 2 q^{90} + 4 q^{91} + 36 q^{92} - 12 q^{93} - 20 q^{94} + 2 q^{95} - 6 q^{96} - 28 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^13 - 4 * q^14 - 2 * q^15 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 8 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 8 * q^26 + 2 * q^27 - 8 * q^28 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 8 * q^41 - 4 * q^42 + 16 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 - 4 * q^47 + 6 * q^48 - 10 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 + 16 * q^53 + 2 * q^54 - 4 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 + 8 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^66 - 8 * q^67 - 8 * q^68 + 4 * q^69 + 4 * q^70 - 16 * q^71 + 6 * q^72 - 4 * q^73 - 8 * q^74 + 2 * q^75 - 2 * q^76 + 12 * q^77 - 8 * q^78 - 6 * q^80 + 2 * q^81 - 4 * q^82 + 20 * q^83 - 8 * q^84 - 8 * q^85 + 28 * q^86 - 8 * q^89 - 2 * q^90 + 4 * q^91 + 36 * q^92 - 12 * q^93 - 20 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 10 * q^98 + 4 * q^99

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