LMFDB - Embedded newform 2005.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.879385 q^{2} -3.22668 q^{3} -1.22668 q^{4} +1.00000 q^{5} -2.83750 q^{6} -3.71688 q^{7} -2.83750 q^{8} +7.41147 q^{9} +O(q^{10})$	\(q+0.879385 q^{2} -3.22668 q^{3} -1.22668 q^{4} +1.00000 q^{5} -2.83750 q^{6} -3.71688 q^{7} -2.83750 q^{8} +7.41147 q^{9} +0.879385 q^{10} +1.18479 q^{11} +3.95811 q^{12} +1.57398 q^{13} -3.26857 q^{14} -3.22668 q^{15} -0.0418891 q^{16} +3.29086 q^{17} +6.51754 q^{18} +3.00000 q^{19} -1.22668 q^{20} +11.9932 q^{21} +1.04189 q^{22} -1.81521 q^{23} +9.15570 q^{24} +1.00000 q^{25} +1.38413 q^{26} -14.2344 q^{27} +4.55943 q^{28} -5.98545 q^{29} -2.83750 q^{30} +4.71688 q^{31} +5.63816 q^{32} -3.82295 q^{33} +2.89393 q^{34} -3.71688 q^{35} -9.09152 q^{36} +9.59627 q^{37} +2.63816 q^{38} -5.07873 q^{39} -2.83750 q^{40} -3.42602 q^{41} +10.5466 q^{42} -0.958111 q^{43} -1.45336 q^{44} +7.41147 q^{45} -1.59627 q^{46} -6.33275 q^{47} +0.135163 q^{48} +6.81521 q^{49} +0.879385 q^{50} -10.6186 q^{51} -1.93077 q^{52} -6.53209 q^{53} -12.5175 q^{54} +1.18479 q^{55} +10.5466 q^{56} -9.68004 q^{57} -5.26352 q^{58} +7.74422 q^{59} +3.95811 q^{60} -9.00774 q^{61} +4.14796 q^{62} -27.5476 q^{63} +5.04189 q^{64} +1.57398 q^{65} -3.36184 q^{66} -13.4534 q^{67} -4.03684 q^{68} +5.85710 q^{69} -3.26857 q^{70} -1.32770 q^{71} -21.0300 q^{72} -3.24897 q^{73} +8.43882 q^{74} -3.22668 q^{75} -3.68004 q^{76} -4.40373 q^{77} -4.46616 q^{78} +13.6655 q^{79} -0.0418891 q^{80} +23.6955 q^{81} -3.01279 q^{82} -14.1848 q^{83} -14.7118 q^{84} +3.29086 q^{85} -0.842549 q^{86} +19.3131 q^{87} -3.36184 q^{88} -2.32770 q^{89} +6.51754 q^{90} -5.85029 q^{91} +2.22668 q^{92} -15.2199 q^{93} -5.56893 q^{94} +3.00000 q^{95} -18.1925 q^{96} +4.84524 q^{97} +5.99319 q^{98} +8.78106 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 6 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 6 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 6 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{10} + 15 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} + 9 q^{19} + 3 q^{20} - 6 q^{21} - 9 q^{23} - 12 q^{24} + 3 q^{25} + 15 q^{26} - 12 q^{27} - 12 q^{28} - 6 q^{30} + 6 q^{31} + 9 q^{33} + 21 q^{34} - 3 q^{35} + 3 q^{36} + 15 q^{37} - 9 q^{38} - 24 q^{39} - 6 q^{40} - 18 q^{41} + 45 q^{42} - 6 q^{43} + 9 q^{44} + 12 q^{45} + 9 q^{46} + 24 q^{48} + 24 q^{49} - 3 q^{50} - 12 q^{51} - 30 q^{52} - 15 q^{53} - 15 q^{54} + 45 q^{56} - 9 q^{57} - 21 q^{58} - 6 q^{59} + 15 q^{60} - 3 q^{61} - 3 q^{62} - 30 q^{63} + 12 q^{64} - 3 q^{65} - 27 q^{66} - 27 q^{67} - 24 q^{68} + 18 q^{69} - 33 q^{72} + 3 q^{73} - 6 q^{74} - 3 q^{75} + 9 q^{76} - 27 q^{77} + 39 q^{78} + 3 q^{79} + 3 q^{80} + 3 q^{81} + 30 q^{82} - 39 q^{83} - 51 q^{84} - 6 q^{85} + 15 q^{86} + 36 q^{87} - 27 q^{88} - 3 q^{89} - 3 q^{90} + 24 q^{91} + 3 q^{93} - 24 q^{94} + 9 q^{95} - 27 q^{96} - 12 q^{97} - 24 q^{98} + 9 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 6 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9 - 3 * q^10 + 15 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 + 9 * q^19 + 3 * q^20 - 6 * q^21 - 9 * q^23 - 12 * q^24 + 3 * q^25 + 15 * q^26 - 12 * q^27 - 12 * q^28 - 6 * q^30 + 6 * q^31 + 9 * q^33 + 21 * q^34 - 3 * q^35 + 3 * q^36 + 15 * q^37 - 9 * q^38 - 24 * q^39 - 6 * q^40 - 18 * q^41 + 45 * q^42 - 6 * q^43 + 9 * q^44 + 12 * q^45 + 9 * q^46 + 24 * q^48 + 24 * q^49 - 3 * q^50 - 12 * q^51 - 30 * q^52 - 15 * q^53 - 15 * q^54 + 45 * q^56 - 9 * q^57 - 21 * q^58 - 6 * q^59 + 15 * q^60 - 3 * q^61 - 3 * q^62 - 30 * q^63 + 12 * q^64 - 3 * q^65 - 27 * q^66 - 27 * q^67 - 24 * q^68 + 18 * q^69 - 33 * q^72 + 3 * q^73 - 6 * q^74 - 3 * q^75 + 9 * q^76 - 27 * q^77 + 39 * q^78 + 3 * q^79 + 3 * q^80 + 3 * q^81 + 30 * q^82 - 39 * q^83 - 51 * q^84 - 6 * q^85 + 15 * q^86 + 36 * q^87 - 27 * q^88 - 3 * q^89 - 3 * q^90 + 24 * q^91 + 3 * q^93 - 24 * q^94 + 9 * q^95 - 27 * q^96 - 12 * q^97 - 24 * q^98 + 9 * 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 $

$2$	0.879385	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$2$	0.879385	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$3$	−3.22668	−1.86293	−0.931463	−	0.363837i	$-0.881467\pi$
$3$	−3.22668	−1.86293	−0.931463	+	0.363837i	$0.881467\pi$
$4$	−1.22668	−0.613341
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.83750	−1.15840
$7$	−3.71688	−1.40485	−0.702425	−	0.711758i	$-0.747899\pi$
$7$	−3.71688	−1.40485	−0.702425	+	0.711758i	$0.747899\pi$
$8$	−2.83750	−1.00321
$9$	7.41147	2.47049
$10$	0.879385	0.278086
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	3.95811	1.14261
$13$	1.57398	0.436543	0.218271	−	0.975888i	$-0.429958\pi$
$13$	1.57398	0.436543	0.218271	+	0.975888i	$0.429958\pi$
$14$	−3.26857	−0.873562
$15$	−3.22668	−0.833126
$16$	−0.0418891	−0.0104723
$17$	3.29086	0.798151	0.399075	−	0.916918i	$-0.369331\pi$
$17$	3.29086	0.798151	0.399075	+	0.916918i	$0.369331\pi$
$18$	6.51754	1.53620
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−1.22668	−0.274294
$21$	11.9932	2.61713
$22$	1.04189	0.222131
$23$	−1.81521	−0.378497	−0.189248	−	0.981929i	$-0.560605\pi$
$23$	−1.81521	−0.378497	−0.189248	+	0.981929i	$0.560605\pi$
$24$	9.15570	1.86890
$25$	1.00000	0.200000
$26$	1.38413	0.271451
$27$	−14.2344	−2.73942
$28$	4.55943	0.861651
$29$	−5.98545	−1.11147	−0.555735	−	0.831359i	$-0.687563\pi$
$29$	−5.98545	−1.11147	−0.555735	+	0.831359i	$0.687563\pi$
$30$	−2.83750	−0.518054
$31$	4.71688	0.847177	0.423588	−	0.905855i	$-0.360770\pi$
$31$	4.71688	0.847177	0.423588	+	0.905855i	$0.360770\pi$
$32$	5.63816	0.996695
$33$	−3.82295	−0.665490
$34$	2.89393	0.496305
$35$	−3.71688	−0.628268
$36$	−9.09152	−1.51525
$37$	9.59627	1.57762	0.788808	−	0.614639i	$-0.210698\pi$
$37$	9.59627	1.57762	0.788808	+	0.614639i	$0.210698\pi$
$38$	2.63816	0.427965
$39$	−5.07873	−0.813247
$40$	−2.83750	−0.448648
$41$	−3.42602	−0.535055	−0.267527	−	0.963550i	$-0.586207\pi$
$41$	−3.42602	−0.535055	−0.267527	+	0.963550i	$0.586207\pi$
$42$	10.5466	1.62738
$43$	−0.958111	−0.146111	−0.0730553	−	0.997328i	$-0.523275\pi$
$43$	−0.958111	−0.146111	−0.0730553	+	0.997328i	$0.523275\pi$
$44$	−1.45336	−0.219103
$45$	7.41147	1.10484
$46$	−1.59627	−0.235357
$47$	−6.33275	−0.923726	−0.461863	−	0.886951i	$-0.652819\pi$
$47$	−6.33275	−0.923726	−0.461863	+	0.886951i	$0.652819\pi$
$48$	0.135163	0.0195091
$49$	6.81521	0.973601
$50$	0.879385	0.124364
$51$	−10.6186	−1.48690
$52$	−1.93077	−0.267750
$53$	−6.53209	−0.897251	−0.448626	−	0.893720i	$-0.648086\pi$
$53$	−6.53209	−0.897251	−0.448626	+	0.893720i	$0.648086\pi$
$54$	−12.5175	−1.70342
$55$	1.18479	0.159757
$56$	10.5466	1.40935
$57$	−9.68004	−1.28215
$58$	−5.26352	−0.691134
$59$	7.74422	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$59$	7.74422	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$60$	3.95811	0.510990
$61$	−9.00774	−1.15332	−0.576662	−	0.816983i	$-0.695645\pi$
$61$	−9.00774	−1.15332	−0.576662	+	0.816983i	$0.695645\pi$
$62$	4.14796	0.526791
$63$	−27.5476	−3.47067
$64$	5.04189	0.630236
$65$	1.57398	0.195228
$66$	−3.36184	−0.413814
$67$	−13.4534	−1.64359	−0.821795	−	0.569783i	$-0.807027\pi$
$67$	−13.4534	−1.64359	−0.821795	+	0.569783i	$0.807027\pi$
$68$	−4.03684	−0.489538
$69$	5.85710	0.705112
$70$	−3.26857	−0.390669
$71$	−1.32770	−0.157569	−0.0787843	−	0.996892i	$-0.525104\pi$
$71$	−1.32770	−0.157569	−0.0787843	+	0.996892i	$0.525104\pi$
$72$	−21.0300	−2.47841
$73$	−3.24897	−0.380263	−0.190132	−	0.981759i	$-0.560891\pi$
$73$	−3.24897	−0.380263	−0.190132	+	0.981759i	$0.560891\pi$
$74$	8.43882	0.980992
$75$	−3.22668	−0.372585
$76$	−3.68004	−0.422130
$77$	−4.40373	−0.501852
$78$	−4.46616	−0.505693
$79$	13.6655	1.53749	0.768744	−	0.639556i	$-0.220882\pi$
$79$	13.6655	1.53749	0.768744	+	0.639556i	$0.220882\pi$
$80$	−0.0418891	−0.00468334
$81$	23.6955	2.63284
$82$	−3.01279	−0.332707
$83$	−14.1848	−1.55698	−0.778492	−	0.627655i	$-0.784015\pi$
$83$	−14.1848	−1.55698	−0.778492	+	0.627655i	$0.784015\pi$
$84$	−14.7118	−1.60519
$85$	3.29086	0.356944
$86$	−0.842549	−0.0908544
$87$	19.3131	2.07059
$88$	−3.36184	−0.358374
$89$	−2.32770	−0.246735	−0.123368	−	0.992361i	$-0.539369\pi$
$89$	−2.32770	−0.246735	−0.123368	+	0.992361i	$0.539369\pi$
$90$	6.51754	0.687009
$91$	−5.85029	−0.613277
$92$	2.22668	0.232148
$93$	−15.2199	−1.57823
$94$	−5.56893	−0.574391
$95$	3.00000	0.307794
$96$	−18.1925	−1.85677
$97$	4.84524	0.491959	0.245980	−	0.969275i	$-0.420890\pi$
$97$	4.84524	0.491959	0.245980	+	0.969275i	$0.420890\pi$
$98$	5.99319	0.605404
$99$	8.78106	0.882530
$100$	−1.22668	−0.122668
$101$	5.51754	0.549016	0.274508	−	0.961585i	$-0.411485\pi$
$101$	5.51754	0.549016	0.274508	+	0.961585i	$0.411485\pi$
$102$	−9.33780	−0.924580
$103$	−17.6878	−1.74283	−0.871415	−	0.490547i	$-0.836797\pi$
$103$	−17.6878	−1.74283	−0.871415	+	0.490547i	$0.836797\pi$
$104$	−4.46616	−0.437943
$105$	11.9932	1.17042
$106$	−5.74422	−0.557928
$107$	8.78106	0.848897	0.424449	−	0.905452i	$-0.360468\pi$
$107$	8.78106	0.848897	0.424449	+	0.905452i	$0.360468\pi$
$108$	17.4611	1.68020
$109$	9.47565	0.907603	0.453801	−	0.891103i	$-0.350068\pi$
$109$	9.47565	0.907603	0.453801	+	0.891103i	$0.350068\pi$
$110$	1.04189	0.0993402
$111$	−30.9641	−2.93898
$112$	0.155697	0.0147120
$113$	−5.36959	−0.505128	−0.252564	−	0.967580i	$-0.581274\pi$
$113$	−5.36959	−0.505128	−0.252564	+	0.967580i	$0.581274\pi$
$114$	−8.51249	−0.797268
$115$	−1.81521	−0.169269
$116$	7.34224	0.681710
$117$	11.6655	1.07848
$118$	6.81016	0.626926
$119$	−12.2317	−1.12128
$120$	9.15570	0.835797
$121$	−9.59627	−0.872388
$122$	−7.92127	−0.717158
$123$	11.0547	0.996767
$124$	−5.78611	−0.519608
$125$	1.00000	0.0894427
$126$	−24.2249	−2.15813
$127$	−10.6159	−0.942006	−0.471003	−	0.882132i	$-0.656108\pi$
$127$	−10.6159	−0.942006	−0.471003	+	0.882132i	$0.656108\pi$
$128$	−6.84255	−0.604802
$129$	3.09152	0.272193
$130$	1.38413	0.121396
$131$	−15.9067	−1.38978	−0.694889	−	0.719117i	$-0.744546\pi$
$131$	−15.9067	−1.38978	−0.694889	+	0.719117i	$0.744546\pi$
$132$	4.68954	0.408172
$133$	−11.1506	−0.966883
$134$	−11.8307	−1.02202
$135$	−14.2344	−1.22510
$136$	−9.33780	−0.800710
$137$	14.2344	1.21613	0.608064	−	0.793888i	$-0.291946\pi$
$137$	14.2344	1.21613	0.608064	+	0.793888i	$0.291946\pi$
$138$	5.15064	0.438452
$139$	−7.35235	−0.623618	−0.311809	−	0.950145i	$-0.600935\pi$
$139$	−7.35235	−0.623618	−0.311809	+	0.950145i	$0.600935\pi$
$140$	4.55943	0.385342
$141$	20.4338	1.72083
$142$	−1.16756	−0.0979791
$143$	1.86484	0.155946
$144$	−0.310460	−0.0258716
$145$	−5.98545	−0.497065
$146$	−2.85710	−0.236455
$147$	−21.9905	−1.81375
$148$	−11.7716	−0.967617
$149$	15.2959	1.25309	0.626545	−	0.779385i	$-0.284468\pi$
$149$	15.2959	1.25309	0.626545	+	0.779385i	$0.284468\pi$
$150$	−2.83750	−0.231681
$151$	−17.2199	−1.40133	−0.700667	−	0.713489i	$-0.747114\pi$
$151$	−17.2199	−1.40133	−0.700667	+	0.713489i	$0.747114\pi$
$152$	−8.51249	−0.690454
$153$	24.3901	1.97182
$154$	−3.87258	−0.312061
$155$	4.71688	0.378869
$156$	6.22998	0.498798
$157$	4.23442	0.337944	0.168972	−	0.985621i	$-0.445955\pi$
$157$	4.23442	0.337944	0.168972	+	0.985621i	$0.445955\pi$
$158$	12.0172	0.956040
$159$	21.0770	1.67151
$160$	5.63816	0.445735
$161$	6.74691	0.531731
$162$	20.8375	1.63715
$163$	0.440570	0.0345081	0.0172541	−	0.999851i	$-0.494508\pi$
$163$	0.440570	0.0345081	0.0172541	+	0.999851i	$0.494508\pi$
$164$	4.20264	0.328171
$165$	−3.82295	−0.297616
$166$	−12.4739	−0.968162
$167$	−0.389185	−0.0301161	−0.0150580	−	0.999887i	$-0.504793\pi$
$167$	−0.389185	−0.0301161	−0.0150580	+	0.999887i	$0.504793\pi$
$168$	−34.0306	−2.62552
$169$	−10.5226	−0.809430
$170$	2.89393	0.221955
$171$	22.2344	1.70031
$172$	1.17530	0.0896156
$173$	−7.55169	−0.574144	−0.287072	−	0.957909i	$-0.592682\pi$
$173$	−7.55169	−0.574144	−0.287072	+	0.957909i	$0.592682\pi$
$174$	16.9837	1.28753
$175$	−3.71688	−0.280970
$176$	−0.0496299	−0.00374099
$177$	−24.9881	−1.87822
$178$	−2.04694	−0.153425
$179$	−4.19934	−0.313873	−0.156937	−	0.987609i	$-0.550162\pi$
$179$	−4.19934	−0.313873	−0.156937	+	0.987609i	$0.550162\pi$
$180$	−9.09152	−0.677642
$181$	6.28405	0.467090	0.233545	−	0.972346i	$-0.424967\pi$
$181$	6.28405	0.467090	0.233545	+	0.972346i	$0.424967\pi$
$182$	−5.14466	−0.381347
$183$	29.0651	2.14855
$184$	5.15064	0.379711
$185$	9.59627	0.705532
$186$	−13.3841	−0.981372
$187$	3.89899	0.285122
$188$	7.76827	0.566559
$189$	52.9077	3.84847
$190$	2.63816	0.191392
$191$	−10.6527	−0.770803	−0.385401	−	0.922749i	$-0.625937\pi$
$191$	−10.6527	−0.770803	−0.385401	+	0.922749i	$0.625937\pi$
$192$	−16.2686	−1.17408
$193$	−20.0155	−1.44075	−0.720373	−	0.693587i	$-0.756029\pi$
$193$	−20.0155	−1.44075	−0.720373	+	0.693587i	$0.756029\pi$
$194$	4.26083	0.305910
$195$	−5.07873	−0.363695
$196$	−8.36009	−0.597149
$197$	0.182104	0.0129744	0.00648719	−	0.999979i	$-0.497935\pi$
$197$	0.182104	0.0129744	0.00648719	+	0.999979i	$0.497935\pi$
$198$	7.72193	0.548774
$199$	19.3824	1.37398	0.686990	−	0.726667i	$-0.258932\pi$
$199$	19.3824	1.37398	0.686990	+	0.726667i	$0.258932\pi$
$200$	−2.83750	−0.200641
$201$	43.4097	3.06189
$202$	4.85204	0.341389
$203$	22.2472	1.56145
$204$	13.0256	0.911973
$205$	−3.42602	−0.239284
$206$	−15.5544	−1.08372
$207$	−13.4534	−0.935073
$208$	−0.0659325	−0.00457159
$209$	3.55438	0.245861
$210$	10.5466	0.727787
$211$	10.6108	0.730479	0.365239	−	0.930914i	$-0.380987\pi$
$211$	10.6108	0.730479	0.365239	+	0.930914i	$0.380987\pi$
$212$	8.01279	0.550321
$213$	4.28405	0.293538
$214$	7.72193	0.527861
$215$	−0.958111	−0.0653426
$216$	40.3901	2.74820
$217$	−17.5321	−1.19016
$218$	8.33275	0.564365
$219$	10.4834	0.708402
$220$	−1.45336	−0.0979857
$221$	5.17974	0.348427
$222$	−27.2294	−1.82752
$223$	21.6459	1.44952	0.724758	−	0.689003i	$-0.241951\pi$
$223$	21.6459	1.44952	0.724758	+	0.689003i	$0.241951\pi$
$224$	−20.9564	−1.40021
$225$	7.41147	0.494098
$226$	−4.72193	−0.314098
$227$	−15.4730	−1.02698	−0.513488	−	0.858097i	$-0.671647\pi$
$227$	−15.4730	−1.02698	−0.513488	+	0.858097i	$0.671647\pi$
$228$	11.8743	0.786397
$229$	−5.86484	−0.387559	−0.193780	−	0.981045i	$-0.562075\pi$
$229$	−5.86484	−0.387559	−0.193780	+	0.981045i	$0.562075\pi$
$230$	−1.59627	−0.105255
$231$	14.2094	0.934913
$232$	16.9837	1.11503
$233$	21.1361	1.38467	0.692336	−	0.721575i	$-0.256582\pi$
$233$	21.1361	1.38467	0.692336	+	0.721575i	$0.256582\pi$
$234$	10.2585	0.670617
$235$	−6.33275	−0.413103
$236$	−9.49970	−0.618377
$237$	−44.0942	−2.86423
$238$	−10.7564	−0.697234
$239$	−28.1908	−1.82351	−0.911755	−	0.410735i	$-0.865272\pi$
$239$	−28.1908	−1.82351	−0.911755	+	0.410735i	$0.865272\pi$
$240$	0.135163	0.00872471
$241$	17.5243	1.12884	0.564421	−	0.825487i	$-0.309099\pi$
$241$	17.5243	1.12884	0.564421	+	0.825487i	$0.309099\pi$
$242$	−8.43882	−0.542468
$243$	−33.7547	−2.16536
$244$	11.0496	0.707380
$245$	6.81521	0.435408
$246$	9.72132	0.619809
$247$	4.72193	0.300449
$248$	−13.3841	−0.849893
$249$	45.7698	2.90054
$250$	0.879385	0.0556172
$251$	14.9118	0.941223	0.470612	−	0.882340i	$-0.344033\pi$
$251$	14.9118	0.941223	0.470612	+	0.882340i	$0.344033\pi$
$252$	33.7921	2.12870
$253$	−2.15064	−0.135210
$254$	−9.33544	−0.585757
$255$	−10.6186	−0.664960
$256$	−16.1010	−1.00631
$257$	9.95811	0.621170	0.310585	−	0.950546i	$-0.399475\pi$
$257$	9.95811	0.621170	0.310585	+	0.950546i	$0.399475\pi$
$258$	2.71864	0.169255
$259$	−35.6682	−2.21631
$260$	−1.93077	−0.119741
$261$	−44.3610	−2.74588
$262$	−13.9881	−0.864190
$263$	−9.39187	−0.579128	−0.289564	−	0.957159i	$-0.593510\pi$
$263$	−9.39187	−0.579128	−0.289564	+	0.957159i	$0.593510\pi$
$264$	10.8476	0.667624
$265$	−6.53209	−0.401263
$266$	−9.80571	−0.601227
$267$	7.51073	0.459649
$268$	16.5030	1.00808
$269$	−29.7178	−1.81193	−0.905964	−	0.423356i	$-0.860852\pi$
$269$	−29.7178	−1.81193	−0.905964	+	0.423356i	$0.860852\pi$
$270$	−12.5175	−0.761793
$271$	−2.14796	−0.130479	−0.0652395	−	0.997870i	$-0.520781\pi$
$271$	−2.14796	−0.130479	−0.0652395	+	0.997870i	$0.520781\pi$
$272$	−0.137851	−0.00835845
$273$	18.8770	1.14249
$274$	12.5175	0.756212
$275$	1.18479	0.0714457
$276$	−7.18479	−0.432474
$277$	−26.9222	−1.61760	−0.808799	−	0.588085i	$-0.799882\pi$
$277$	−26.9222	−1.61760	−0.808799	+	0.588085i	$0.799882\pi$
$278$	−6.46555	−0.387778
$279$	34.9590	2.09294
$280$	10.5466	0.630282
$281$	−8.71688	−0.520006	−0.260003	−	0.965608i	$-0.583723\pi$
$281$	−8.71688	−0.520006	−0.260003	+	0.965608i	$0.583723\pi$
$282$	17.9691	1.07005
$283$	−18.1061	−1.07629	−0.538147	−	0.842851i	$-0.680875\pi$
$283$	−18.1061	−1.07629	−0.538147	+	0.842851i	$0.680875\pi$
$284$	1.62866	0.0966432
$285$	−9.68004	−0.573396
$286$	1.63991	0.0969699
$287$	12.7341	0.751671
$288$	41.7870	2.46233
$289$	−6.17024	−0.362956
$290$	−5.26352	−0.309084
$291$	−15.6340	−0.916483
$292$	3.98545	0.233231
$293$	−10.3824	−0.606545	−0.303273	−	0.952904i	$-0.598079\pi$
$293$	−10.3824	−0.606545	−0.303273	+	0.952904i	$0.598079\pi$
$294$	−19.3381	−1.12782
$295$	7.74422	0.450886
$296$	−27.2294	−1.58267
$297$	−16.8648	−0.978597
$298$	13.4510	0.779195
$299$	−2.85710	−0.165230
$300$	3.95811	0.228522
$301$	3.56118	0.205263
$302$	−15.1429	−0.871376
$303$	−17.8033	−1.02278
$304$	−0.125667	−0.00720751
$305$	−9.00774	−0.515782
$306$	21.4483	1.22612
$307$	11.0104	0.628398	0.314199	−	0.949357i	$-0.398264\pi$
$307$	11.0104	0.628398	0.314199	+	0.949357i	$0.398264\pi$
$308$	5.40198	0.307806
$309$	57.0729	3.24676
$310$	4.14796	0.235588
$311$	−14.5645	−0.825876	−0.412938	−	0.910759i	$-0.635497\pi$
$311$	−14.5645	−0.825876	−0.412938	+	0.910759i	$0.635497\pi$
$312$	14.4109	0.815855
$313$	4.18984	0.236824	0.118412	−	0.992965i	$-0.462220\pi$
$313$	4.18984	0.236824	0.118412	+	0.992965i	$0.462220\pi$
$314$	3.72369	0.210140
$315$	−27.5476	−1.55213
$316$	−16.7632	−0.943004
$317$	−9.86215	−0.553913	−0.276957	−	0.960882i	$-0.589326\pi$
$317$	−9.86215	−0.553913	−0.276957	+	0.960882i	$0.589326\pi$
$318$	18.5348	1.03938
$319$	−7.09152	−0.397049
$320$	5.04189	0.281850
$321$	−28.3337	−1.58143
$322$	5.93313	0.330641
$323$	9.87258	0.549325
$324$	−29.0669	−1.61483
$325$	1.57398	0.0873086
$326$	0.387431	0.0214578
$327$	−30.5749	−1.69080
$328$	9.72132	0.536770
$329$	23.5381	1.29770
$330$	−3.36184	−0.185063
$331$	−31.3746	−1.72451	−0.862253	−	0.506478i	$-0.830947\pi$
$331$	−31.3746	−1.72451	−0.862253	+	0.506478i	$0.830947\pi$
$332$	17.4002	0.954961
$333$	71.1225	3.89749
$334$	−0.342244	−0.0187268
$335$	−13.4534	−0.735036
$336$	−0.502384	−0.0274073
$337$	−11.0966	−0.604469	−0.302234	−	0.953234i	$-0.597733\pi$
$337$	−11.0966	−0.604469	−0.302234	+	0.953234i	$0.597733\pi$
$338$	−9.25341	−0.503319
$339$	17.3259	0.941016
$340$	−4.03684	−0.218928
$341$	5.58853	0.302636
$342$	19.5526	1.05728
$343$	0.686852	0.0370865
$344$	2.71864	0.146579
$345$	5.85710	0.315335
$346$	−6.64084	−0.357014
$347$	−6.12742	−0.328937	−0.164469	−	0.986382i	$-0.552591\pi$
$347$	−6.12742	−0.328937	−0.164469	+	0.986382i	$0.552591\pi$
$348$	−23.6911	−1.26998
$349$	23.3054	1.24751	0.623755	−	0.781620i	$-0.285606\pi$
$349$	23.3054	1.24751	0.623755	+	0.781620i	$0.285606\pi$
$350$	−3.26857	−0.174712
$351$	−22.4047	−1.19587
$352$	6.68004	0.356048
$353$	28.3901	1.51105	0.755527	−	0.655118i	$-0.227381\pi$
$353$	28.3901	1.51105	0.755527	+	0.655118i	$0.227381\pi$
$354$	−21.9742	−1.16792
$355$	−1.32770	−0.0704668
$356$	2.85534	0.151333
$357$	39.4679	2.08886
$358$	−3.69284	−0.195173
$359$	4.50980	0.238018	0.119009	−	0.992893i	$-0.462028\pi$
$359$	4.50980	0.238018	0.119009	+	0.992893i	$0.462028\pi$
$360$	−21.0300	−1.10838
$361$	−10.0000	−0.526316
$362$	5.52610	0.290446
$363$	30.9641	1.62519
$364$	7.17644	0.376148
$365$	−3.24897	−0.170059
$366$	25.5594	1.33601
$367$	38.0283	1.98506	0.992530	−	0.122002i	$-0.0389313\pi$
$367$	38.0283	1.98506	0.992530	+	0.122002i	$0.0389313\pi$
$368$	0.0760373	0.00396372
$369$	−25.3919	−1.32185
$370$	8.43882	0.438713
$371$	24.2790	1.26050
$372$	18.6699	0.967991
$373$	−27.0009	−1.39806	−0.699028	−	0.715095i	$-0.746384\pi$
$373$	−27.0009	−1.39806	−0.699028	+	0.715095i	$0.746384\pi$
$374$	3.42871	0.177294
$375$	−3.22668	−0.166625
$376$	17.9691	0.926688
$377$	−9.42097	−0.485205
$378$	46.5262	2.39305
$379$	13.1506	0.675503	0.337752	−	0.941235i	$-0.390334\pi$
$379$	13.1506	0.675503	0.337752	+	0.941235i	$0.390334\pi$
$380$	−3.68004	−0.188782
$381$	34.2540	1.75489
$382$	−9.36783	−0.479300
$383$	8.27631	0.422900	0.211450	−	0.977389i	$-0.432181\pi$
$383$	8.27631	0.422900	0.211450	+	0.977389i	$0.432181\pi$
$384$	22.0787	1.12670
$385$	−4.40373	−0.224435
$386$	−17.6013	−0.895884
$387$	−7.10101	−0.360965
$388$	−5.94356	−0.301739
$389$	28.2371	1.43168	0.715839	−	0.698265i	$-0.246044\pi$
$389$	28.2371	1.43168	0.715839	+	0.698265i	$0.246044\pi$
$390$	−4.46616	−0.226153
$391$	−5.97359	−0.302098
$392$	−19.3381	−0.976723
$393$	51.3259	2.58905
$394$	0.160140	0.00806772
$395$	13.6655	0.687586
$396$	−10.7716	−0.541291
$397$	20.9145	1.04967	0.524834	−	0.851205i	$-0.324128\pi$
$397$	20.9145	1.04967	0.524834	+	0.851205i	$0.324128\pi$
$398$	17.0446	0.854367
$399$	35.9796	1.80123
$400$	−0.0418891	−0.00209445
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	38.1739	1.90394
$403$	7.42427	0.369829
$404$	−6.76827	−0.336734
$405$	23.6955	1.17744
$406$	19.5639	0.970939
$407$	11.3696	0.563569
$408$	30.1301	1.49166
$409$	−15.7733	−0.779940	−0.389970	−	0.920827i	$-0.627515\pi$
$409$	−15.7733	−0.779940	−0.389970	+	0.920827i	$0.627515\pi$
$410$	−3.01279	−0.148791
$411$	−45.9299	−2.26556
$412$	21.6973	1.06895
$413$	−28.7844	−1.41639
$414$	−11.8307	−0.581447
$415$	−14.1848	−0.696304
$416$	8.87433	0.435100
$417$	23.7237	1.16175
$418$	3.12567	0.152881
$419$	18.6810	0.912626	0.456313	−	0.889819i	$-0.349170\pi$
$419$	18.6810	0.912626	0.456313	+	0.889819i	$0.349170\pi$
$420$	−14.7118	−0.717864
$421$	−35.7101	−1.74040	−0.870201	−	0.492696i	$-0.836011\pi$
$421$	−35.7101	−1.74040	−0.870201	+	0.492696i	$0.836011\pi$
$422$	9.33099	0.454226
$423$	−46.9350	−2.28206
$424$	18.5348	0.900128
$425$	3.29086	0.159630
$426$	3.76733	0.182528
$427$	33.4807	1.62024
$428$	−10.7716	−0.520663
$429$	−6.01724	−0.290515
$430$	−0.842549	−0.0406313
$431$	25.0310	1.20570	0.602850	−	0.797855i	$-0.294032\pi$
$431$	25.0310	1.20570	0.602850	+	0.797855i	$0.294032\pi$
$432$	0.596267	0.0286879
$433$	−27.6827	−1.33035	−0.665174	−	0.746689i	$-0.731642\pi$
$433$	−27.6827	−1.33035	−0.665174	+	0.746689i	$0.731642\pi$
$434$	−15.4175	−0.740062
$435$	19.3131	0.925995
$436$	−11.6236	−0.556670
$437$	−5.44562	−0.260499
$438$	9.21894	0.440498
$439$	31.0405	1.48148	0.740740	−	0.671792i	$-0.234475\pi$
$439$	31.0405	1.48148	0.740740	+	0.671792i	$0.234475\pi$
$440$	−3.36184	−0.160270
$441$	50.5107	2.40527
$442$	4.55499	0.216659
$443$	−7.12061	−0.338311	−0.169155	−	0.985589i	$-0.554104\pi$
$443$	−7.12061	−0.338311	−0.169155	+	0.985589i	$0.554104\pi$
$444$	37.9831	1.80260
$445$	−2.32770	−0.110343
$446$	19.0351	0.901337
$447$	−49.3550	−2.33441
$448$	−18.7401	−0.885387
$449$	−32.6905	−1.54276	−0.771379	−	0.636376i	$-0.780433\pi$
$449$	−32.6905	−1.54276	−0.771379	+	0.636376i	$0.780433\pi$
$450$	6.51754	0.307240
$451$	−4.05913	−0.191137
$452$	6.58677	0.309816
$453$	55.5631	2.61058
$454$	−13.6067	−0.638594
$455$	−5.85029	−0.274266
$456$	27.4671	1.28626
$457$	−16.0770	−0.752049	−0.376024	−	0.926610i	$-0.622709\pi$
$457$	−16.0770	−0.752049	−0.376024	+	0.926610i	$0.622709\pi$
$458$	−5.15745	−0.240992
$459$	−46.8435	−2.18647
$460$	2.22668	0.103820
$461$	10.4260	0.485588	0.242794	−	0.970078i	$-0.421936\pi$
$461$	10.4260	0.485588	0.242794	+	0.970078i	$0.421936\pi$
$462$	12.4956	0.581347
$463$	18.0155	0.837250	0.418625	−	0.908159i	$-0.362512\pi$
$463$	18.0155	0.837250	0.418625	+	0.908159i	$0.362512\pi$
$464$	0.250725	0.0116396
$465$	−15.2199	−0.705805
$466$	18.5868	0.861016
$467$	−5.29591	−0.245066	−0.122533	−	0.992464i	$-0.539102\pi$
$467$	−5.29591	−0.245066	−0.122533	+	0.992464i	$0.539102\pi$
$468$	−14.3099	−0.661473
$469$	50.0046	2.30900
$470$	−5.56893	−0.256875
$471$	−13.6631	−0.629564
$472$	−21.9742	−1.01144
$473$	−1.13516	−0.0521948
$474$	−38.7758	−1.78103
$475$	3.00000	0.137649
$476$	15.0044	0.687728
$477$	−48.4124	−2.21665
$478$	−24.7906	−1.13389
$479$	−18.6091	−0.850270	−0.425135	−	0.905130i	$-0.639773\pi$
$479$	−18.6091	−0.850270	−0.425135	+	0.905130i	$0.639773\pi$
$480$	−18.1925	−0.830372
$481$	15.1043	0.688697
$482$	15.4107	0.701936
$483$	−21.7701	−0.990575
$484$	11.7716	0.535071
$485$	4.84524	0.220011
$486$	−29.6833	−1.34646
$487$	−40.6560	−1.84230	−0.921150	−	0.389209i	$-0.872749\pi$
$487$	−40.6560	−1.84230	−0.921150	+	0.389209i	$0.872749\pi$
$488$	25.5594	1.15702
$489$	−1.42158	−0.0642860
$490$	5.99319	0.270745
$491$	−23.2499	−1.04925	−0.524627	−	0.851332i	$-0.675795\pi$
$491$	−23.2499	−1.04925	−0.524627	+	0.851332i	$0.675795\pi$
$492$	−13.5606	−0.611358
$493$	−19.6973	−0.887121
$494$	4.15240	0.186825
$495$	8.78106	0.394679
$496$	−0.197586	−0.00887186
$497$	4.93489	0.221360
$498$	40.2493	1.80361
$499$	−14.9709	−0.670190	−0.335095	−	0.942184i	$-0.608768\pi$
$499$	−14.9709	−0.670190	−0.335095	+	0.942184i	$0.608768\pi$
$500$	−1.22668	−0.0548589
$501$	1.25578	0.0561040
$502$	13.1132	0.585271
$503$	8.57491	0.382336	0.191168	−	0.981557i	$-0.438772\pi$
$503$	8.57491	0.382336	0.191168	+	0.981557i	$0.438772\pi$
$504$	78.1661	3.48180
$505$	5.51754	0.245527
$506$	−1.89124	−0.0840761
$507$	33.9531	1.50791
$508$	13.0223	0.577771
$509$	−31.7425	−1.40696	−0.703480	−	0.710715i	$-0.748372\pi$
$509$	−31.7425	−1.40696	−0.703480	+	0.710715i	$0.748372\pi$
$510$	−9.33780	−0.413485
$511$	12.0760	0.534212
$512$	−0.473897	−0.0209435
$513$	−42.7033	−1.88540
$514$	8.75702	0.386255
$515$	−17.6878	−0.779417
$516$	−3.79231	−0.166947
$517$	−7.50299	−0.329981
$518$	−31.3661	−1.37815
$519$	24.3669	1.06959
$520$	−4.46616	−0.195854
$521$	10.5202	0.460900	0.230450	−	0.973084i	$-0.425980\pi$
$521$	10.5202	0.460900	0.230450	+	0.973084i	$0.425980\pi$
$522$	−39.0104	−1.70744
$523$	−18.0155	−0.787762	−0.393881	−	0.919161i	$-0.628868\pi$
$523$	−18.0155	−0.787762	−0.393881	+	0.919161i	$0.628868\pi$
$524$	19.5125	0.852407
$525$	11.9932	0.523426
$526$	−8.25908	−0.360113
$527$	15.5226	0.676175
$528$	0.160140	0.00696919
$529$	−19.7050	−0.856740
$530$	−5.74422	−0.249513
$531$	57.3961	2.49078
$532$	13.6783	0.593029
$533$	−5.39248	−0.233574
$534$	6.60483	0.285819
$535$	8.78106	0.379638
$536$	38.1739	1.64886
$537$	13.5499	0.584723
$538$	−26.1334	−1.12669
$539$	8.07461	0.347798
$540$	17.4611	0.751406
$541$	−33.4100	−1.43641	−0.718205	−	0.695832i	$-0.755036\pi$
$541$	−33.4100	−1.43641	−0.718205	+	0.695832i	$0.755036\pi$
$542$	−1.88888	−0.0811344
$543$	−20.2766	−0.870154
$544$	18.5544	0.795512
$545$	9.47565	0.405892
$546$	16.6002	0.710422
$547$	35.7962	1.53054	0.765268	−	0.643712i	$-0.222607\pi$
$547$	35.7962	1.53054	0.765268	+	0.643712i	$0.222607\pi$
$548$	−17.4611	−0.745901
$549$	−66.7606	−2.84927
$550$	1.04189	0.0444263
$551$	−17.9564	−0.764966
$552$	−16.6195	−0.707372
$553$	−50.7930	−2.15994
$554$	−23.6750	−1.00585
$555$	−30.9641	−1.31435
$556$	9.01899	0.382490
$557$	−26.0966	−1.10575	−0.552874	−	0.833265i	$-0.686469\pi$
$557$	−26.0966	−1.10575	−0.552874	+	0.833265i	$0.686469\pi$
$558$	30.7425	1.30143
$559$	−1.50805	−0.0637835
$560$	0.155697	0.00657939
$561$	−12.5808	−0.531161
$562$	−7.66550	−0.323349
$563$	6.58347	0.277460	0.138730	−	0.990330i	$-0.455698\pi$
$563$	6.58347	0.277460	0.138730	+	0.990330i	$0.455698\pi$
$564$	−25.0657	−1.05546
$565$	−5.36959	−0.225900
$566$	−15.9222	−0.669260
$567$	−88.0735	−3.69874
$568$	3.76733	0.158074
$569$	−32.0283	−1.34270	−0.671348	−	0.741143i	$-0.734284\pi$
$569$	−32.0283	−1.34270	−0.671348	+	0.741143i	$0.734284\pi$
$570$	−8.51249	−0.356549
$571$	−10.4192	−0.436031	−0.218015	−	0.975945i	$-0.569958\pi$
$571$	−10.4192	−0.436031	−0.218015	+	0.975945i	$0.569958\pi$
$572$	−2.28756	−0.0956478
$573$	34.3729	1.43595
$574$	11.1982	0.467404
$575$	−1.81521	−0.0756994
$576$	37.3678	1.55699
$577$	3.33181	0.138705	0.0693526	−	0.997592i	$-0.477907\pi$
$577$	3.33181	0.138705	0.0693526	+	0.997592i	$0.477907\pi$
$578$	−5.42602	−0.225693
$579$	64.5836	2.68400
$580$	7.34224	0.304870
$581$	52.7232	2.18733
$582$	−13.7483	−0.569887
$583$	−7.73917	−0.320524
$584$	9.21894	0.381482
$585$	11.6655	0.482309
$586$	−9.13011	−0.377161
$587$	−24.5149	−1.01184	−0.505918	−	0.862582i	$-0.668846\pi$
$587$	−24.5149	−1.01184	−0.505918	+	0.862582i	$0.668846\pi$
$588$	26.9753	1.11244
$589$	14.1506	0.583067
$590$	6.81016	0.280370
$591$	−0.587592	−0.0241703
$592$	−0.401979	−0.0165212
$593$	−25.9017	−1.06365	−0.531827	−	0.846853i	$-0.678494\pi$
$593$	−25.9017	−1.06365	−0.531827	+	0.846853i	$0.678494\pi$
$594$	−14.8307	−0.608511
$595$	−12.2317	−0.501452
$596$	−18.7632	−0.768571
$597$	−62.5408	−2.55962
$598$	−2.51249	−0.102743
$599$	−6.63722	−0.271190	−0.135595	−	0.990764i	$-0.543295\pi$
$599$	−6.63722	−0.271190	−0.135595	+	0.990764i	$0.543295\pi$
$600$	9.15570	0.373780
$601$	43.2199	1.76298	0.881488	−	0.472207i	$-0.156542\pi$
$601$	43.2199	1.76298	0.881488	+	0.472207i	$0.156542\pi$
$602$	3.13165	0.127637
$603$	−99.7093	−4.06047
$604$	21.1233	0.859495
$605$	−9.59627	−0.390144
$606$	−15.6560	−0.635982
$607$	−6.51518	−0.264443	−0.132221	−	0.991220i	$-0.542211\pi$
$607$	−6.51518	−0.264443	−0.132221	+	0.991220i	$0.542211\pi$
$608$	16.9145	0.685972
$609$	−71.7847	−2.90886
$610$	−7.92127	−0.320723
$611$	−9.96761	−0.403246
$612$	−29.9189	−1.20940
$613$	25.8702	1.04489	0.522444	−	0.852674i	$-0.325020\pi$
$613$	25.8702	1.04489	0.522444	+	0.852674i	$0.325020\pi$
$614$	9.68241	0.390750
$615$	11.0547	0.445768
$616$	12.4956	0.503461
$617$	−17.9881	−0.724175	−0.362088	−	0.932144i	$-0.617936\pi$
$617$	−17.9881	−0.724175	−0.362088	+	0.932144i	$0.617936\pi$
$618$	50.1890	2.01890
$619$	−17.5672	−0.706084	−0.353042	−	0.935607i	$-0.614853\pi$
$619$	−17.5672	−0.706084	−0.353042	+	0.935607i	$0.614853\pi$
$620$	−5.78611	−0.232376
$621$	25.8384	1.03686
$622$	−12.8078	−0.513546
$623$	8.65177	0.346626
$624$	0.212743	0.00851654
$625$	1.00000	0.0400000
$626$	3.68449	0.147262
$627$	−11.4688	−0.458022
$628$	−5.19429	−0.207275
$629$	31.5800	1.25918
$630$	−24.2249	−0.965144
$631$	28.9914	1.15413	0.577065	−	0.816698i	$-0.304198\pi$
$631$	28.9914	1.15413	0.577065	+	0.816698i	$0.304198\pi$
$632$	−38.7758	−1.54242
$633$	−34.2377	−1.36083
$634$	−8.67263	−0.344434
$635$	−10.6159	−0.421278
$636$	−25.8547	−1.02521
$637$	10.7270	0.425019
$638$	−6.23618	−0.246893
$639$	−9.84018	−0.389272
$640$	−6.84255	−0.270475
$641$	−23.8830	−0.943322	−0.471661	−	0.881780i	$-0.656345\pi$
$641$	−23.8830	−0.943322	−0.471661	+	0.881780i	$0.656345\pi$
$642$	−24.9162	−0.983365
$643$	7.09564	0.279825	0.139912	−	0.990164i	$-0.455318\pi$
$643$	7.09564	0.279825	0.139912	+	0.990164i	$0.455318\pi$
$644$	−8.27631	−0.326132
$645$	3.09152	0.121728
$646$	8.68180	0.341581
$647$	14.4757	0.569097	0.284548	−	0.958662i	$-0.408156\pi$
$647$	14.4757	0.569097	0.284548	+	0.958662i	$0.408156\pi$
$648$	−67.2360	−2.64128
$649$	9.17530	0.360162
$650$	1.38413	0.0542902
$651$	56.5705	2.21717
$652$	−0.540439	−0.0211652
$653$	−18.6851	−0.731204	−0.365602	−	0.930771i	$-0.619137\pi$
$653$	−18.6851	−0.731204	−0.365602	+	0.930771i	$0.619137\pi$
$654$	−26.8871	−1.05137
$655$	−15.9067	−0.621527
$656$	0.143513	0.00560324
$657$	−24.0797	−0.939437
$658$	20.6990	0.806932
$659$	−47.6418	−1.85586	−0.927930	−	0.372754i	$-0.878413\pi$
$659$	−47.6418	−1.85586	−0.927930	+	0.372754i	$0.878413\pi$
$660$	4.68954	0.182540
$661$	−19.7466	−0.768053	−0.384027	−	0.923322i	$-0.625463\pi$
$661$	−19.7466	−0.768053	−0.384027	+	0.923322i	$0.625463\pi$
$662$	−27.5904	−1.07233
$663$	−16.7134	−0.649094
$664$	40.2493	1.56198
$665$	−11.1506	−0.432403
$666$	62.5441	2.42353
$667$	10.8648	0.420688
$668$	0.477407	0.0184714
$669$	−69.8444	−2.70034
$670$	−11.8307	−0.457059
$671$	−10.6723	−0.412000
$672$	67.6195	2.60848
$673$	−5.00538	−0.192943	−0.0964715	−	0.995336i	$-0.530756\pi$
$673$	−5.00538	−0.192943	−0.0964715	+	0.995336i	$0.530756\pi$
$674$	−9.75816	−0.375870
$675$	−14.2344	−0.547883
$676$	12.9079	0.496457
$677$	28.0847	1.07938	0.539692	−	0.841863i	$-0.318541\pi$
$677$	28.0847	1.07938	0.539692	+	0.841863i	$0.318541\pi$
$678$	15.2362	0.585142
$679$	−18.0092	−0.691129
$680$	−9.33780	−0.358088
$681$	49.9263	1.91318
$682$	4.91447	0.188185
$683$	−28.8384	−1.10347	−0.551736	−	0.834019i	$-0.686034\pi$
$683$	−28.8384	−1.10347	−0.551736	+	0.834019i	$0.686034\pi$
$684$	−27.2746	−1.04287
$685$	14.2344	0.543869
$686$	0.604007	0.0230611
$687$	18.9240	0.721994
$688$	0.0401344	0.00153011
$689$	−10.2814	−0.391689
$690$	5.15064	0.196082
$691$	−14.4466	−0.549573	−0.274786	−	0.961505i	$-0.588607\pi$
$691$	−14.4466	−0.549573	−0.274786	+	0.961505i	$0.588607\pi$
$692$	9.26352	0.352146
$693$	−32.6382	−1.23982
$694$	−5.38836	−0.204540
$695$	−7.35235	−0.278890
$696$	−54.8010	−2.07723
$697$	−11.2746	−0.427054
$698$	20.4944	0.775726
$699$	−68.1995	−2.57954
$700$	4.55943	0.172330
$701$	40.7187	1.53793	0.768963	−	0.639293i	$-0.220773\pi$
$701$	40.7187	1.53793	0.768963	+	0.639293i	$0.220773\pi$
$702$	−19.7023	−0.743617
$703$	28.7888	1.08579
$704$	5.97359	0.225138
$705$	20.4338	0.769580
$706$	24.9659	0.939602
$707$	−20.5080	−0.771284
$708$	30.6525	1.15199
$709$	48.3928	1.81743	0.908715	−	0.417417i	$-0.137065\pi$
$709$	48.3928	1.81743	0.908715	+	0.417417i	$0.137065\pi$
$710$	−1.16756	−0.0438176
$711$	101.281	3.79835
$712$	6.60483	0.247526
$713$	−8.56212	−0.320654
$714$	34.7075	1.29890
$715$	1.86484	0.0697410
$716$	5.15125	0.192511
$717$	90.9627	3.39706
$718$	3.96585	0.148004
$719$	23.3027	0.869045	0.434522	−	0.900661i	$-0.356917\pi$
$719$	23.3027	0.869045	0.434522	+	0.900661i	$0.356917\pi$
$720$	−0.310460	−0.0115702
$721$	65.7434	2.44841
$722$	−8.79385	−0.327273
$723$	−56.5455	−2.10295
$724$	−7.70853	−0.286485
$725$	−5.98545	−0.222294
$726$	27.2294	1.01058
$727$	−28.4780	−1.05619	−0.528096	−	0.849185i	$-0.677094\pi$
$727$	−28.4780	−1.05619	−0.528096	+	0.849185i	$0.677094\pi$
$728$	16.6002	0.615243
$729$	37.8289	1.40107
$730$	−2.85710	−0.105746
$731$	−3.15301	−0.116618
$732$	−35.6536	−1.31780
$733$	−26.1712	−0.966655	−0.483327	−	0.875440i	$-0.660572\pi$
$733$	−26.1712	−0.966655	−0.483327	+	0.875440i	$0.660572\pi$
$734$	33.4415	1.23435
$735$	−21.9905	−0.811132
$736$	−10.2344	−0.377246
$737$	−15.9394	−0.587137
$738$	−22.3292	−0.821951
$739$	−41.5749	−1.52936	−0.764679	−	0.644411i	$-0.777102\pi$
$739$	−41.5749	−1.52936	−0.764679	+	0.644411i	$0.777102\pi$
$740$	−11.7716	−0.432731
$741$	−15.2362	−0.559715
$742$	21.3506	0.783805
$743$	−27.3396	−1.00299	−0.501496	−	0.865160i	$-0.667217\pi$
$743$	−27.3396	−1.00299	−0.501496	+	0.865160i	$0.667217\pi$
$744$	43.1863	1.58329
$745$	15.2959	0.560399
$746$	−23.7442	−0.869338
$747$	−105.130	−3.84651
$748$	−4.78281	−0.174877
$749$	−32.6382	−1.19257
$750$	−2.83750	−0.103611
$751$	−9.30304	−0.339473	−0.169736	−	0.985489i	$-0.554292\pi$
$751$	−9.30304	−0.339473	−0.169736	+	0.985489i	$0.554292\pi$
$752$	0.265273	0.00967351
$753$	−48.1156	−1.75343
$754$	−8.28466	−0.301710
$755$	−17.2199	−0.626695
$756$	−64.9009	−2.36042
$757$	27.2199	0.989323	0.494662	−	0.869086i	$-0.335292\pi$
$757$	27.2199	0.989323	0.494662	+	0.869086i	$0.335292\pi$
$758$	11.5645	0.420041
$759$	6.93944	0.251886
$760$	−8.51249	−0.308780
$761$	−16.4293	−0.595562	−0.297781	−	0.954634i	$-0.596247\pi$
$761$	−16.4293	−0.595562	−0.297781	+	0.954634i	$0.596247\pi$
$762$	30.1225	1.09122
$763$	−35.2199	−1.27505
$764$	13.0675	0.472765
$765$	24.3901	0.881827
$766$	7.27807	0.262967
$767$	12.1892	0.440128
$768$	51.9528	1.87469
$769$	−10.2995	−0.371411	−0.185705	−	0.982605i	$-0.559457\pi$
$769$	−10.2995	−0.371411	−0.185705	+	0.982605i	$0.559457\pi$
$770$	−3.87258	−0.139558
$771$	−32.1317	−1.15719
$772$	24.5526	0.883668
$773$	−4.65270	−0.167346	−0.0836731	−	0.996493i	$-0.526665\pi$
$773$	−4.65270	−0.167346	−0.0836731	+	0.996493i	$0.526665\pi$
$774$	−6.24453	−0.224455
$775$	4.71688	0.169435
$776$	−13.7483	−0.493537
$777$	115.090	4.12883
$778$	24.8313	0.890245
$779$	−10.2781	−0.368250
$780$	6.22998	0.223069
$781$	−1.57304	−0.0562880
$782$	−5.25309	−0.187850
$783$	85.1995	3.04478
$784$	−0.285483	−0.0101958
$785$	4.23442	0.151133
$786$	45.1353	1.60992
$787$	41.7315	1.48757	0.743784	−	0.668420i	$-0.233029\pi$
$787$	41.7315	1.48757	0.743784	+	0.668420i	$0.233029\pi$
$788$	−0.223384	−0.00795772
$789$	30.3046	1.07887
$790$	12.0172	0.427554
$791$	19.9581	0.709629
$792$	−24.9162	−0.885359
$793$	−14.1780	−0.503475
$794$	18.3919	0.652703
$795$	21.0770	0.747523
$796$	−23.7760	−0.842718
$797$	−38.0242	−1.34688	−0.673442	−	0.739240i	$-0.735185\pi$
$797$	−38.0242	−1.34688	−0.673442	+	0.739240i	$0.735185\pi$
$798$	31.6399	1.12004
$799$	−20.8402	−0.737273
$800$	5.63816	0.199339
$801$	−17.2517	−0.609557
$802$	0.879385	0.0310522
$803$	−3.84936	−0.135841
$804$	−53.2499	−1.87798
$805$	6.74691	0.237797
$806$	6.52879	0.229967
$807$	95.8899	3.37549
$808$	−15.6560	−0.550776
$809$	−13.4442	−0.472673	−0.236336	−	0.971671i	$-0.575947\pi$
$809$	−13.4442	−0.472673	−0.236336	+	0.971671i	$0.575947\pi$
$810$	20.8375	0.732155
$811$	−19.8821	−0.698154	−0.349077	−	0.937094i	$-0.613505\pi$
$811$	−19.8821	−0.698154	−0.349077	+	0.937094i	$0.613505\pi$
$812$	−27.2902	−0.957700
$813$	6.93077	0.243073
$814$	9.99825	0.350438
$815$	0.440570	0.0154325
$816$	0.444801	0.0155712
$817$	−2.87433	−0.100560
$818$	−13.8708	−0.484982
$819$	−43.3593	−1.51510
$820$	4.20264	0.146762
$821$	−45.4415	−1.58592	−0.792960	−	0.609274i	$-0.791461\pi$
$821$	−45.4415	−1.58592	−0.792960	+	0.609274i	$0.791461\pi$
$822$	−40.3901	−1.40877
$823$	29.1388	1.01571	0.507857	−	0.861441i	$-0.330438\pi$
$823$	29.1388	1.01571	0.507857	+	0.861441i	$0.330438\pi$
$824$	50.1890	1.74842
$825$	−3.82295	−0.133098
$826$	−25.3125	−0.880736
$827$	25.3583	0.881796	0.440898	−	0.897557i	$-0.354660\pi$
$827$	25.3583	0.881796	0.440898	+	0.897557i	$0.354660\pi$
$828$	16.5030	0.573519
$829$	−43.3569	−1.50585	−0.752924	−	0.658108i	$-0.771357\pi$
$829$	−43.3569	−1.50585	−0.752924	+	0.658108i	$0.771357\pi$
$830$	−12.4739	−0.432975
$831$	86.8694	3.01347
$832$	7.93582	0.275125
$833$	22.4279	0.777080
$834$	20.8623	0.722401
$835$	−0.389185	−0.0134683
$836$	−4.36009	−0.150797
$837$	−67.1421	−2.32077
$838$	16.4278	0.567488
$839$	−6.03602	−0.208386	−0.104193	−	0.994557i	$-0.533226\pi$
$839$	−6.03602	−0.208386	−0.104193	+	0.994557i	$0.533226\pi$
$840$	−34.0306	−1.17417
$841$	6.82564	0.235367
$842$	−31.4029	−1.08222
$843$	28.1266	0.968732
$844$	−13.0161	−0.448032
$845$	−10.5226	−0.361988
$846$	−41.2739	−1.41903
$847$	35.6682	1.22557
$848$	0.273623	0.00939626
$849$	58.4225	2.00506
$850$	2.89393	0.0992611
$851$	−17.4192	−0.597123
$852$	−5.25517	−0.180039
$853$	−9.86390	−0.337734	−0.168867	−	0.985639i	$-0.554011\pi$
$853$	−9.86390	−0.337734	−0.168867	+	0.985639i	$0.554011\pi$
$854$	29.4424	1.00750
$855$	22.2344	0.760401
$856$	−24.9162	−0.851619
$857$	4.67768	0.159787	0.0798933	−	0.996803i	$-0.474542\pi$
$857$	4.67768	0.159787	0.0798933	+	0.996803i	$0.474542\pi$
$858$	−5.29147	−0.180648
$859$	−23.3541	−0.796832	−0.398416	−	0.917205i	$-0.630440\pi$
$859$	−23.3541	−0.796832	−0.398416	+	0.917205i	$0.630440\pi$
$860$	1.17530	0.0400773
$861$	−41.0889	−1.40031
$862$	22.0119	0.749727
$863$	23.4557	0.798442	0.399221	−	0.916855i	$-0.369281\pi$
$863$	23.4557	0.798442	0.399221	+	0.916855i	$0.369281\pi$
$864$	−80.2559	−2.73036
$865$	−7.55169	−0.256765
$866$	−24.3438	−0.827236
$867$	19.9094	0.676159
$868$	21.5063	0.729971
$869$	16.1908	0.549235
$870$	16.9837	0.575801
$871$	−21.1753	−0.717498
$872$	−26.8871	−0.910513
$873$	35.9103	1.21538
$874$	−4.78880	−0.161984
$875$	−3.71688	−0.125654
$876$	−12.8598	−0.434492
$877$	−9.13072	−0.308322	−0.154161	−	0.988046i	$-0.549268\pi$
$877$	−9.13072	−0.308322	−0.154161	+	0.988046i	$0.549268\pi$
$878$	27.2965	0.921213
$879$	33.5006	1.12995
$880$	−0.0496299	−0.00167302
$881$	−23.2513	−0.783357	−0.391679	−	0.920102i	$-0.628106\pi$
$881$	−23.2513	−0.783357	−0.391679	+	0.920102i	$0.628106\pi$
$882$	44.4184	1.49565
$883$	−2.88175	−0.0969786	−0.0484893	−	0.998824i	$-0.515441\pi$
$883$	−2.88175	−0.0969786	−0.0484893	+	0.998824i	$0.515441\pi$
$884$	−6.35389	−0.213705
$885$	−24.9881	−0.839967
$886$	−6.26176	−0.210368
$887$	37.0983	1.24564	0.622820	−	0.782365i	$-0.285987\pi$
$887$	37.0983	1.24564	0.622820	+	0.782365i	$0.285987\pi$
$888$	87.8605	2.94841
$889$	39.4579	1.32338
$890$	−2.04694	−0.0686136
$891$	28.0743	0.940524
$892$	−26.5526	−0.889048
$893$	−18.9982	−0.635752
$894$	−43.4021	−1.45158
$895$	−4.19934	−0.140368
$896$	25.4329	0.849655
$897$	9.21894	0.307811
$898$	−28.7475	−0.959317
$899$	−28.2327	−0.941612
$900$	−9.09152	−0.303051
$901$	−21.4962	−0.716142
$902$	−3.56953	−0.118853
$903$	−11.4908	−0.382390
$904$	15.2362	0.506748
$905$	6.28405	0.208889
$906$	48.8613	1.62331
$907$	23.9573	0.795489	0.397744	−	0.917496i	$-0.369793\pi$
$907$	23.9573	0.795489	0.397744	+	0.917496i	$0.369793\pi$
$908$	18.9804	0.629887
$909$	40.8931	1.35634
$910$	−5.14466	−0.170544
$911$	33.4884	1.10952	0.554761	−	0.832010i	$-0.312810\pi$
$911$	33.4884	1.10952	0.554761	+	0.832010i	$0.312810\pi$
$912$	0.405488	0.0134271
$913$	−16.8060	−0.556199
$914$	−14.1379	−0.467639
$915$	29.0651	0.960863
$916$	7.19429	0.237706
$917$	59.1234	1.95243
$918$	−41.1935	−1.35959
$919$	45.9067	1.51432	0.757162	−	0.653228i	$-0.226586\pi$
$919$	45.9067	1.51432	0.757162	+	0.653228i	$0.226586\pi$
$920$	5.15064	0.169812
$921$	−35.5271	−1.17066
$922$	9.16849	0.301948
$923$	−2.08976	−0.0687854
$924$	−17.4305	−0.573420
$925$	9.59627	0.315523
$926$	15.8425	0.520618
$927$	−131.093	−4.30564
$928$	−33.7469	−1.10780
$929$	5.71513	0.187507	0.0937536	−	0.995595i	$-0.470113\pi$
$929$	5.71513	0.187507	0.0937536	+	0.995595i	$0.470113\pi$
$930$	−13.3841	−0.438883
$931$	20.4456	0.670078
$932$	−25.9273	−0.849276
$933$	46.9949	1.53855
$934$	−4.65715	−0.152386
$935$	3.89899	0.127510
$936$	−33.1008	−1.08193
$937$	13.4216	0.438464	0.219232	−	0.975673i	$-0.429645\pi$
$937$	13.4216	0.438464	0.219232	+	0.975673i	$0.429645\pi$
$938$	43.9733	1.43578
$939$	−13.5193	−0.441186
$940$	7.76827	0.253373
$941$	3.08109	0.100441	0.0502203	−	0.998738i	$-0.484008\pi$
$941$	3.08109	0.100441	0.0502203	+	0.998738i	$0.484008\pi$
$942$	−12.0152	−0.391475
$943$	6.21894	0.202517
$944$	−0.324398	−0.0105583
$945$	52.9077	1.72109
$946$	−0.998245	−0.0324558
$947$	13.6922	0.444938	0.222469	−	0.974940i	$-0.428588\pi$
$947$	13.6922	0.444938	0.222469	+	0.974940i	$0.428588\pi$
$948$	54.0896	1.75675
$949$	−5.11381	−0.166001
$950$	2.63816	0.0855931
$951$	31.8220	1.03190
$952$	34.7075	1.12488
$953$	−7.08915	−0.229640	−0.114820	−	0.993386i	$-0.536629\pi$
$953$	−7.08915	−0.229640	−0.114820	+	0.993386i	$0.536629\pi$
$954$	−42.5732	−1.37836
$955$	−10.6527	−0.344713
$956$	34.5811	1.11843
$957$	22.8821	0.739672
$958$	−16.3645	−0.528714
$959$	−52.9077	−1.70848
$960$	−16.2686	−0.525066
$961$	−8.75103	−0.282291
$962$	13.2825	0.428245
$963$	65.0806	2.09719
$964$	−21.4968	−0.692365
$965$	−20.0155	−0.644321
$966$	−19.1443	−0.615959
$967$	−48.3732	−1.55558	−0.777789	−	0.628526i	$-0.783659\pi$
$967$	−48.3732	−1.55558	−0.777789	+	0.628526i	$0.783659\pi$
$968$	27.2294	0.875185
$969$	−31.8557	−1.02335
$970$	4.26083	0.136807
$971$	41.4466	1.33008	0.665042	−	0.746806i	$-0.268414\pi$
$971$	41.4466	1.33008	0.665042	+	0.746806i	$0.268414\pi$
$972$	41.4062	1.32810
$973$	27.3278	0.876089
$974$	−35.7523	−1.14558
$975$	−5.07873	−0.162649
$976$	0.377326	0.0120779
$977$	−33.8563	−1.08316	−0.541579	−	0.840650i	$-0.682173\pi$
$977$	−33.8563	−1.08316	−0.541579	+	0.840650i	$0.682173\pi$
$978$	−1.25012	−0.0399743
$979$	−2.75784	−0.0881408
$980$	−8.36009	−0.267053
$981$	70.2285	2.24223
$982$	−20.4456	−0.652446
$983$	12.1010	0.385962	0.192981	−	0.981202i	$-0.438184\pi$
$983$	12.1010	0.385962	0.192981	+	0.981202i	$0.438184\pi$
$984$	−31.3676	−0.999963
$985$	0.182104	0.00580232
$986$	−17.3215	−0.551629
$987$	−75.9499	−2.41751
$988$	−5.79231	−0.184278
$989$	1.73917	0.0553024
$990$	7.72193	0.245419
$991$	62.0479	1.97102	0.985508	−	0.169630i	$-0.0542572\pi$
$991$	62.0479	1.97102	0.985508	+	0.169630i	$0.0542572\pi$
$992$	26.5945	0.844377
$993$	101.236	3.21263
$994$	4.33967	0.137646
$995$	19.3824	0.614463
$996$	−56.1450	−1.77902
$997$	10.8871	0.344799	0.172399	−	0.985027i	$-0.444848\pi$
$997$	10.8871	0.344799	0.172399	+	0.985027i	$0.444848\pi$
$998$	−13.1652	−0.416737
$999$	−136.597	−4.32175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.c.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.c.1.3	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.879385 q^{2} -3.22668 q^{3} -1.22668 q^{4} +1.00000 q^{5} -2.83750 q^{6} -3.71688 q^{7} -2.83750 q^{8} +7.41147 q^{9} +O(q^{10})\)	\(q+0.879385 q^{2} -3.22668 q^{3} -1.22668 q^{4} +1.00000 q^{5} -2.83750 q^{6} -3.71688 q^{7} -2.83750 q^{8} +7.41147 q^{9} +0.879385 q^{10} +1.18479 q^{11} +3.95811 q^{12} +1.57398 q^{13} -3.26857 q^{14} -3.22668 q^{15} -0.0418891 q^{16} +3.29086 q^{17} +6.51754 q^{18} +3.00000 q^{19} -1.22668 q^{20} +11.9932 q^{21} +1.04189 q^{22} -1.81521 q^{23} +9.15570 q^{24} +1.00000 q^{25} +1.38413 q^{26} -14.2344 q^{27} +4.55943 q^{28} -5.98545 q^{29} -2.83750 q^{30} +4.71688 q^{31} +5.63816 q^{32} -3.82295 q^{33} +2.89393 q^{34} -3.71688 q^{35} -9.09152 q^{36} +9.59627 q^{37} +2.63816 q^{38} -5.07873 q^{39} -2.83750 q^{40} -3.42602 q^{41} +10.5466 q^{42} -0.958111 q^{43} -1.45336 q^{44} +7.41147 q^{45} -1.59627 q^{46} -6.33275 q^{47} +0.135163 q^{48} +6.81521 q^{49} +0.879385 q^{50} -10.6186 q^{51} -1.93077 q^{52} -6.53209 q^{53} -12.5175 q^{54} +1.18479 q^{55} +10.5466 q^{56} -9.68004 q^{57} -5.26352 q^{58} +7.74422 q^{59} +3.95811 q^{60} -9.00774 q^{61} +4.14796 q^{62} -27.5476 q^{63} +5.04189 q^{64} +1.57398 q^{65} -3.36184 q^{66} -13.4534 q^{67} -4.03684 q^{68} +5.85710 q^{69} -3.26857 q^{70} -1.32770 q^{71} -21.0300 q^{72} -3.24897 q^{73} +8.43882 q^{74} -3.22668 q^{75} -3.68004 q^{76} -4.40373 q^{77} -4.46616 q^{78} +13.6655 q^{79} -0.0418891 q^{80} +23.6955 q^{81} -3.01279 q^{82} -14.1848 q^{83} -14.7118 q^{84} +3.29086 q^{85} -0.842549 q^{86} +19.3131 q^{87} -3.36184 q^{88} -2.32770 q^{89} +6.51754 q^{90} -5.85029 q^{91} +2.22668 q^{92} -15.2199 q^{93} -5.56893 q^{94} +3.00000 q^{95} -18.1925 q^{96} +4.84524 q^{97} +5.99319 q^{98} +8.78106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 6 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 6 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 6 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{10} + 15 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} + 9 q^{19} + 3 q^{20} - 6 q^{21} - 9 q^{23} - 12 q^{24} + 3 q^{25} + 15 q^{26} - 12 q^{27} - 12 q^{28} - 6 q^{30} + 6 q^{31} + 9 q^{33} + 21 q^{34} - 3 q^{35} + 3 q^{36} + 15 q^{37} - 9 q^{38} - 24 q^{39} - 6 q^{40} - 18 q^{41} + 45 q^{42} - 6 q^{43} + 9 q^{44} + 12 q^{45} + 9 q^{46} + 24 q^{48} + 24 q^{49} - 3 q^{50} - 12 q^{51} - 30 q^{52} - 15 q^{53} - 15 q^{54} + 45 q^{56} - 9 q^{57} - 21 q^{58} - 6 q^{59} + 15 q^{60} - 3 q^{61} - 3 q^{62} - 30 q^{63} + 12 q^{64} - 3 q^{65} - 27 q^{66} - 27 q^{67} - 24 q^{68} + 18 q^{69} - 33 q^{72} + 3 q^{73} - 6 q^{74} - 3 q^{75} + 9 q^{76} - 27 q^{77} + 39 q^{78} + 3 q^{79} + 3 q^{80} + 3 q^{81} + 30 q^{82} - 39 q^{83} - 51 q^{84} - 6 q^{85} + 15 q^{86} + 36 q^{87} - 27 q^{88} - 3 q^{89} - 3 q^{90} + 24 q^{91} + 3 q^{93} - 24 q^{94} + 9 q^{95} - 27 q^{96} - 12 q^{97} - 24 q^{98} + 9 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 6 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9 - 3 * q^10 + 15 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 + 9 * q^19 + 3 * q^20 - 6 * q^21 - 9 * q^23 - 12 * q^24 + 3 * q^25 + 15 * q^26 - 12 * q^27 - 12 * q^28 - 6 * q^30 + 6 * q^31 + 9 * q^33 + 21 * q^34 - 3 * q^35 + 3 * q^36 + 15 * q^37 - 9 * q^38 - 24 * q^39 - 6 * q^40 - 18 * q^41 + 45 * q^42 - 6 * q^43 + 9 * q^44 + 12 * q^45 + 9 * q^46 + 24 * q^48 + 24 * q^49 - 3 * q^50 - 12 * q^51 - 30 * q^52 - 15 * q^53 - 15 * q^54 + 45 * q^56 - 9 * q^57 - 21 * q^58 - 6 * q^59 + 15 * q^60 - 3 * q^61 - 3 * q^62 - 30 * q^63 + 12 * q^64 - 3 * q^65 - 27 * q^66 - 27 * q^67 - 24 * q^68 + 18 * q^69 - 33 * q^72 + 3 * q^73 - 6 * q^74 - 3 * q^75 + 9 * q^76 - 27 * q^77 + 39 * q^78 + 3 * q^79 + 3 * q^80 + 3 * q^81 + 30 * q^82 - 39 * q^83 - 51 * q^84 - 6 * q^85 + 15 * q^86 + 36 * q^87 - 27 * q^88 - 3 * q^89 - 3 * q^90 + 24 * q^91 + 3 * q^93 - 24 * q^94 + 9 * q^95 - 27 * q^96 - 12 * q^97 - 24 * q^98 + 9 * q^99

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