LMFDB - Embedded newform 2005.2.a.g.1.14

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:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-0.728237 q^{2} -2.86948 q^{3} -1.46967 q^{4} +1.00000 q^{5} +2.08966 q^{6} +1.10622 q^{7} +2.52674 q^{8} +5.23391 q^{9} +O(q^{10})$	\(q-0.728237 q^{2} -2.86948 q^{3} -1.46967 q^{4} +1.00000 q^{5} +2.08966 q^{6} +1.10622 q^{7} +2.52674 q^{8} +5.23391 q^{9} -0.728237 q^{10} +1.34088 q^{11} +4.21719 q^{12} +2.88650 q^{13} -0.805587 q^{14} -2.86948 q^{15} +1.09928 q^{16} +0.885986 q^{17} -3.81152 q^{18} +5.92676 q^{19} -1.46967 q^{20} -3.17426 q^{21} -0.976479 q^{22} +7.33990 q^{23} -7.25043 q^{24} +1.00000 q^{25} -2.10206 q^{26} -6.41015 q^{27} -1.62577 q^{28} -7.65690 q^{29} +2.08966 q^{30} -2.22349 q^{31} -5.85402 q^{32} -3.84763 q^{33} -0.645208 q^{34} +1.10622 q^{35} -7.69212 q^{36} -0.0300697 q^{37} -4.31608 q^{38} -8.28275 q^{39} +2.52674 q^{40} -0.301555 q^{41} +2.31161 q^{42} -9.11444 q^{43} -1.97066 q^{44} +5.23391 q^{45} -5.34518 q^{46} +10.3895 q^{47} -3.15435 q^{48} -5.77629 q^{49} -0.728237 q^{50} -2.54232 q^{51} -4.24221 q^{52} +9.47173 q^{53} +4.66811 q^{54} +1.34088 q^{55} +2.79512 q^{56} -17.0067 q^{57} +5.57604 q^{58} +3.16905 q^{59} +4.21719 q^{60} +10.4572 q^{61} +1.61922 q^{62} +5.78983 q^{63} +2.06456 q^{64} +2.88650 q^{65} +2.80199 q^{66} +8.25211 q^{67} -1.30211 q^{68} -21.0617 q^{69} -0.805587 q^{70} -4.88854 q^{71} +13.2247 q^{72} -11.7878 q^{73} +0.0218978 q^{74} -2.86948 q^{75} -8.71039 q^{76} +1.48330 q^{77} +6.03180 q^{78} -5.48242 q^{79} +1.09928 q^{80} +2.69206 q^{81} +0.219604 q^{82} -6.53369 q^{83} +4.66512 q^{84} +0.885986 q^{85} +6.63747 q^{86} +21.9713 q^{87} +3.38806 q^{88} -14.2319 q^{89} -3.81152 q^{90} +3.19309 q^{91} -10.7872 q^{92} +6.38025 q^{93} -7.56604 q^{94} +5.92676 q^{95} +16.7980 q^{96} +15.6720 q^{97} +4.20650 q^{98} +7.01805 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9}+O(q^{10}) $ 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9} + 11 q^{10} + 38 q^{11} + 24 q^{12} + 17 q^{13} + 14 q^{14} + 13 q^{15} + 47 q^{16} + 22 q^{17} + 18 q^{18} + 6 q^{19} + 43 q^{20} + 4 q^{21} + 18 q^{22} + 45 q^{23} - 19 q^{24} + 37 q^{25} - q^{26} + 43 q^{27} + 46 q^{28} + 17 q^{29} - 2 q^{30} - 13 q^{31} + 50 q^{32} + 12 q^{33} - 30 q^{34} + 34 q^{35} + 43 q^{36} + 9 q^{37} + q^{38} - 7 q^{39} + 27 q^{40} + 12 q^{41} - 38 q^{42} + 53 q^{43} + 36 q^{44} + 50 q^{45} - 15 q^{46} + 39 q^{47} - 6 q^{48} + 37 q^{49} + 11 q^{50} + 38 q^{51} + 17 q^{52} + 19 q^{53} - 62 q^{54} + 38 q^{55} + 18 q^{56} + 6 q^{57} - 13 q^{58} + 33 q^{59} + 24 q^{60} - 11 q^{61} + q^{62} + 98 q^{63} + 15 q^{64} + 17 q^{65} - 70 q^{66} + 49 q^{67} + 32 q^{68} - 36 q^{69} + 14 q^{70} + 19 q^{71} + 5 q^{72} + 39 q^{73} + 21 q^{74} + 13 q^{75} - 33 q^{76} + 34 q^{77} - 22 q^{78} - 9 q^{79} + 47 q^{80} + 49 q^{81} + 14 q^{82} + 114 q^{83} - 50 q^{84} + 22 q^{85} + 9 q^{86} + 56 q^{87} + 14 q^{88} + 2 q^{89} + 18 q^{90} - 29 q^{91} + 47 q^{92} - 7 q^{93} - 52 q^{94} + 6 q^{95} - 89 q^{96} + 4 q^{97} + 14 q^{98} + 57 q^{99}+O(q^{100}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9 + 11 * q^10 + 38 * q^11 + 24 * q^12 + 17 * q^13 + 14 * q^14 + 13 * q^15 + 47 * q^16 + 22 * q^17 + 18 * q^18 + 6 * q^19 + 43 * q^20 + 4 * q^21 + 18 * q^22 + 45 * q^23 - 19 * q^24 + 37 * q^25 - q^26 + 43 * q^27 + 46 * q^28 + 17 * q^29 - 2 * q^30 - 13 * q^31 + 50 * q^32 + 12 * q^33 - 30 * q^34 + 34 * q^35 + 43 * q^36 + 9 * q^37 + q^38 - 7 * q^39 + 27 * q^40 + 12 * q^41 - 38 * q^42 + 53 * q^43 + 36 * q^44 + 50 * q^45 - 15 * q^46 + 39 * q^47 - 6 * q^48 + 37 * q^49 + 11 * q^50 + 38 * q^51 + 17 * q^52 + 19 * q^53 - 62 * q^54 + 38 * q^55 + 18 * q^56 + 6 * q^57 - 13 * q^58 + 33 * q^59 + 24 * q^60 - 11 * q^61 + q^62 + 98 * q^63 + 15 * q^64 + 17 * q^65 - 70 * q^66 + 49 * q^67 + 32 * q^68 - 36 * q^69 + 14 * q^70 + 19 * q^71 + 5 * q^72 + 39 * q^73 + 21 * q^74 + 13 * q^75 - 33 * q^76 + 34 * q^77 - 22 * q^78 - 9 * q^79 + 47 * q^80 + 49 * q^81 + 14 * q^82 + 114 * q^83 - 50 * q^84 + 22 * q^85 + 9 * q^86 + 56 * q^87 + 14 * q^88 + 2 * q^89 + 18 * q^90 - 29 * q^91 + 47 * q^92 - 7 * q^93 - 52 * q^94 + 6 * q^95 - 89 * q^96 + 4 * q^97 + 14 * q^98 + 57 * 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 $

$2$	−0.728237	−0.514941	−0.257471	−	0.966286i	$-0.582889\pi$
$2$	−0.728237	−0.514941	−0.257471	+	0.966286i	$0.582889\pi$
$3$	−2.86948	−1.65669	−0.828347	−	0.560215i	$-0.810718\pi$
$3$	−2.86948	−1.65669	−0.828347	+	0.560215i	$0.810718\pi$
$4$	−1.46967	−0.734836
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.08966	0.853100
$7$	1.10622	0.418110	0.209055	−	0.977904i	$-0.432961\pi$
$7$	1.10622	0.418110	0.209055	+	0.977904i	$0.432961\pi$
$8$	2.52674	0.893338
$9$	5.23391	1.74464
$10$	−0.728237	−0.230289
$11$	1.34088	0.404291	0.202145	−	0.979356i	$-0.435209\pi$
$11$	1.34088	0.404291	0.202145	+	0.979356i	$0.435209\pi$
$12$	4.21719	1.21740
$13$	2.88650	0.800571	0.400286	−	0.916390i	$-0.368911\pi$
$13$	2.88650	0.800571	0.400286	+	0.916390i	$0.368911\pi$
$14$	−0.805587	−0.215302
$15$	−2.86948	−0.740896
$16$	1.09928	0.274819
$17$	0.885986	0.214883	0.107442	−	0.994211i	$-0.465734\pi$
$17$	0.885986	0.214883	0.107442	+	0.994211i	$0.465734\pi$
$18$	−3.81152	−0.898385
$19$	5.92676	1.35969	0.679846	−	0.733355i	$-0.262047\pi$
$19$	5.92676	1.35969	0.679846	+	0.733355i	$0.262047\pi$
$20$	−1.46967	−0.328629
$21$	−3.17426	−0.692681
$22$	−0.976479	−0.208186
$23$	7.33990	1.53048	0.765238	−	0.643748i	$-0.222622\pi$
$23$	7.33990	1.53048	0.765238	+	0.643748i	$0.222622\pi$
$24$	−7.25043	−1.47999
$25$	1.00000	0.200000
$26$	−2.10206	−0.412247
$27$	−6.41015	−1.23363
$28$	−1.62577	−0.307242
$29$	−7.65690	−1.42185	−0.710926	−	0.703267i	$-0.751724\pi$
$29$	−7.65690	−1.42185	−0.710926	+	0.703267i	$0.751724\pi$
$30$	2.08966	0.381518
$31$	−2.22349	−0.399350	−0.199675	−	0.979862i	$-0.563989\pi$
$31$	−2.22349	−0.399350	−0.199675	+	0.979862i	$0.563989\pi$
$32$	−5.85402	−1.03485
$33$	−3.84763	−0.669787
$34$	−0.645208	−0.110652
$35$	1.10622	0.186985
$36$	−7.69212	−1.28202
$37$	−0.0300697	−0.00494343	−0.00247171	−	0.999997i	$-0.500787\pi$
$37$	−0.0300697	−0.00494343	−0.00247171	+	0.999997i	$0.500787\pi$
$38$	−4.31608	−0.700161
$39$	−8.28275	−1.32630
$40$	2.52674	0.399513
$41$	−0.301555	−0.0470950	−0.0235475	−	0.999723i	$-0.507496\pi$
$41$	−0.301555	−0.0470950	−0.0235475	+	0.999723i	$0.507496\pi$
$42$	2.31161	0.356690
$43$	−9.11444	−1.38994	−0.694969	−	0.719039i	$-0.744582\pi$
$43$	−9.11444	−1.38994	−0.694969	+	0.719039i	$0.744582\pi$
$44$	−1.97066	−0.297087
$45$	5.23391	0.780225
$46$	−5.34518	−0.788104
$47$	10.3895	1.51547	0.757734	−	0.652563i	$-0.226306\pi$
$47$	10.3895	1.51547	0.757734	+	0.652563i	$0.226306\pi$
$48$	−3.15435	−0.455292
$49$	−5.77629	−0.825184
$50$	−0.728237	−0.102988
$51$	−2.54232	−0.355996
$52$	−4.24221	−0.588288
$53$	9.47173	1.30104	0.650521	−	0.759488i	$-0.274551\pi$
$53$	9.47173	1.30104	0.650521	+	0.759488i	$0.274551\pi$
$54$	4.66811	0.635249
$55$	1.34088	0.180804
$56$	2.79512	0.373514
$57$	−17.0067	−2.25259
$58$	5.57604	0.732170
$59$	3.16905	0.412575	0.206288	−	0.978491i	$-0.433862\pi$
$59$	3.16905	0.412575	0.206288	+	0.978491i	$0.433862\pi$
$60$	4.21719	0.544437
$61$	10.4572	1.33890	0.669451	−	0.742856i	$-0.266530\pi$
$61$	10.4572	1.33890	0.669451	+	0.742856i	$0.266530\pi$
$62$	1.61922	0.205642
$63$	5.78983	0.729450
$64$	2.06456	0.258069
$65$	2.88650	0.358026
$66$	2.80199	0.344901
$67$	8.25211	1.00816	0.504078	−	0.863658i	$-0.331833\pi$
$67$	8.25211	1.00816	0.504078	+	0.863658i	$0.331833\pi$
$68$	−1.30211	−0.157904
$69$	−21.0617	−2.53553
$70$	−0.805587	−0.0962860
$71$	−4.88854	−0.580163	−0.290081	−	0.957002i	$-0.593682\pi$
$71$	−4.88854	−0.580163	−0.290081	+	0.957002i	$0.593682\pi$
$72$	13.2247	1.55855
$73$	−11.7878	−1.37965	−0.689826	−	0.723975i	$-0.742313\pi$
$73$	−11.7878	−1.37965	−0.689826	+	0.723975i	$0.742313\pi$
$74$	0.0218978	0.00254557
$75$	−2.86948	−0.331339
$76$	−8.71039	−0.999150
$77$	1.48330	0.169038
$78$	6.03180	0.682967
$79$	−5.48242	−0.616821	−0.308410	−	0.951253i	$-0.599797\pi$
$79$	−5.48242	−0.616821	−0.308410	+	0.951253i	$0.599797\pi$
$80$	1.09928	0.122903
$81$	2.69206	0.299118
$82$	0.219604	0.0242512
$83$	−6.53369	−0.717165	−0.358583	−	0.933498i	$-0.616740\pi$
$83$	−6.53369	−0.717165	−0.358583	+	0.933498i	$0.616740\pi$
$84$	4.66512	0.509007
$85$	0.885986	0.0960987
$86$	6.63747	0.715737
$87$	21.9713	2.35557
$88$	3.38806	0.361169
$89$	−14.2319	−1.50857	−0.754287	−	0.656545i	$-0.772017\pi$
$89$	−14.2319	−1.50857	−0.754287	+	0.656545i	$0.772017\pi$
$90$	−3.81152	−0.401770
$91$	3.19309	0.334727
$92$	−10.7872	−1.12465
$93$	6.38025	0.661601
$94$	−7.56604	−0.780377
$95$	5.92676	0.608073
$96$	16.7980	1.71444
$97$	15.6720	1.59125	0.795626	−	0.605789i	$-0.207142\pi$
$97$	15.6720	1.59125	0.795626	+	0.605789i	$0.207142\pi$
$98$	4.20650	0.424921
$99$	7.01805	0.705341
$100$	−1.46967	−0.146967
$101$	14.1254	1.40553	0.702765	−	0.711422i	$-0.251949\pi$
$101$	14.1254	1.40553	0.702765	+	0.711422i	$0.251949\pi$
$102$	1.85141	0.183317
$103$	15.0204	1.48000	0.740002	−	0.672604i	$-0.234824\pi$
$103$	15.0204	1.48000	0.740002	+	0.672604i	$0.234824\pi$
$104$	7.29344	0.715181
$105$	−3.17426	−0.309776
$106$	−6.89766	−0.669960
$107$	−7.15826	−0.692015	−0.346007	−	0.938232i	$-0.612463\pi$
$107$	−7.15826	−0.692015	−0.346007	+	0.938232i	$0.612463\pi$
$108$	9.42081	0.906518
$109$	−5.72405	−0.548264	−0.274132	−	0.961692i	$-0.588391\pi$
$109$	−5.72405	−0.548264	−0.274132	+	0.961692i	$0.588391\pi$
$110$	−0.976479	−0.0931036
$111$	0.0862843	0.00818975
$112$	1.21604	0.114905
$113$	−6.47459	−0.609078	−0.304539	−	0.952500i	$-0.598502\pi$
$113$	−6.47459	−0.609078	−0.304539	+	0.952500i	$0.598502\pi$
$114$	12.3849	1.15995
$115$	7.33990	0.684449
$116$	11.2531	1.04483
$117$	15.1077	1.39671
$118$	−2.30782	−0.212452
$119$	0.980092	0.0898449
$120$	−7.25043	−0.661871
$121$	−9.20204	−0.836549
$122$	−7.61528	−0.689455
$123$	0.865307	0.0780221
$124$	3.26780	0.293457
$125$	1.00000	0.0894427
$126$	−4.21637	−0.375624
$127$	21.1572	1.87740	0.938699	−	0.344738i	$-0.112032\pi$
$127$	21.1572	1.87740	0.938699	+	0.344738i	$0.112032\pi$
$128$	10.2045	0.901963
$129$	26.1537	2.30270
$130$	−2.10206	−0.184362
$131$	7.87593	0.688123	0.344061	−	0.938947i	$-0.388197\pi$
$131$	7.87593	0.688123	0.344061	+	0.938947i	$0.388197\pi$
$132$	5.65475	0.492183
$133$	6.55628	0.568501
$134$	−6.00949	−0.519140
$135$	−6.41015	−0.551698
$136$	2.23866	0.191963
$137$	−8.06767	−0.689267	−0.344634	−	0.938737i	$-0.611997\pi$
$137$	−8.06767	−0.689267	−0.344634	+	0.938737i	$0.611997\pi$
$138$	15.3379	1.30565
$139$	−13.7782	−1.16865	−0.584325	−	0.811520i	$-0.698640\pi$
$139$	−13.7782	−1.16865	−0.584325	+	0.811520i	$0.698640\pi$
$140$	−1.62577	−0.137403
$141$	−29.8125	−2.51067
$142$	3.56001	0.298750
$143$	3.87046	0.323664
$144$	5.75351	0.479460
$145$	−7.65690	−0.635871
$146$	8.58427	0.710439
$147$	16.5749	1.36708
$148$	0.0441926	0.00363261
$149$	2.40633	0.197134	0.0985671	−	0.995130i	$-0.468574\pi$
$149$	2.40633	0.197134	0.0985671	+	0.995130i	$0.468574\pi$
$150$	2.08966	0.170620
$151$	17.8710	1.45432	0.727160	−	0.686468i	$-0.240840\pi$
$151$	17.8710	1.45432	0.727160	+	0.686468i	$0.240840\pi$
$152$	14.9754	1.21466
$153$	4.63717	0.374893
$154$	−1.08020	−0.0870447
$155$	−2.22349	−0.178595
$156$	12.1729	0.974614
$157$	14.4499	1.15323	0.576613	−	0.817018i	$-0.304374\pi$
$157$	14.4499	1.15323	0.576613	+	0.817018i	$0.304374\pi$
$158$	3.99250	0.317626
$159$	−27.1789	−2.15543
$160$	−5.85402	−0.462801
$161$	8.11951	0.639907
$162$	−1.96046	−0.154028
$163$	14.1568	1.10885	0.554424	−	0.832234i	$-0.312939\pi$
$163$	14.1568	1.10885	0.554424	+	0.832234i	$0.312939\pi$
$164$	0.443187	0.0346071
$165$	−3.84763	−0.299538
$166$	4.75807	0.369298
$167$	20.3130	1.57187	0.785934	−	0.618310i	$-0.212183\pi$
$167$	20.3130	1.57187	0.785934	+	0.618310i	$0.212183\pi$
$168$	−8.02054	−0.618798
$169$	−4.66811	−0.359085
$170$	−0.645208	−0.0494852
$171$	31.0201	2.37217
$172$	13.3952	1.02138
$173$	−20.0122	−1.52150	−0.760748	−	0.649047i	$-0.775168\pi$
$173$	−20.0122	−1.52150	−0.760748	+	0.649047i	$0.775168\pi$
$174$	−16.0003	−1.21298
$175$	1.10622	0.0836221
$176$	1.47400	0.111107
$177$	−9.09352	−0.683511
$178$	10.3642	0.776827
$179$	3.56869	0.266736	0.133368	−	0.991067i	$-0.457421\pi$
$179$	3.56869	0.266736	0.133368	+	0.991067i	$0.457421\pi$
$180$	−7.69212	−0.573337
$181$	17.6702	1.31342	0.656708	−	0.754145i	$-0.271948\pi$
$181$	17.6702	1.31342	0.656708	+	0.754145i	$0.271948\pi$
$182$	−2.32533	−0.172365
$183$	−30.0066	−2.21815
$184$	18.5460	1.36723
$185$	−0.0300697	−0.00221077
$186$	−4.64633	−0.340685
$187$	1.18800	0.0868753
$188$	−15.2692	−1.11362
$189$	−7.09101	−0.515795
$190$	−4.31608	−0.313122
$191$	−5.42118	−0.392263	−0.196132	−	0.980578i	$-0.562838\pi$
$191$	−5.42118	−0.392263	−0.196132	+	0.980578i	$0.562838\pi$
$192$	−5.92420	−0.427542
$193$	−1.25311	−0.0902009	−0.0451004	−	0.998982i	$-0.514361\pi$
$193$	−1.25311	−0.0902009	−0.0451004	+	0.998982i	$0.514361\pi$
$194$	−11.4129	−0.819401
$195$	−8.28275	−0.593140
$196$	8.48924	0.606375
$197$	15.8430	1.12877	0.564385	−	0.825512i	$-0.309113\pi$
$197$	15.8430	1.12877	0.564385	+	0.825512i	$0.309113\pi$
$198$	−5.11080	−0.363209
$199$	−12.4418	−0.881972	−0.440986	−	0.897514i	$-0.645371\pi$
$199$	−12.4418	−0.881972	−0.440986	+	0.897514i	$0.645371\pi$
$200$	2.52674	0.178668
$201$	−23.6792	−1.67020
$202$	−10.2866	−0.723765
$203$	−8.47019	−0.594491
$204$	3.73637	0.261598
$205$	−0.301555	−0.0210615
$206$	−10.9384	−0.762115
$207$	38.4164	2.67012
$208$	3.17306	0.220012
$209$	7.94708	0.549711
$210$	2.31161	0.159517
$211$	−1.30624	−0.0899250	−0.0449625	−	0.998989i	$-0.514317\pi$
$211$	−1.30624	−0.0899250	−0.0449625	+	0.998989i	$0.514317\pi$
$212$	−13.9203	−0.956052
$213$	14.0276	0.961152
$214$	5.21290	0.356347
$215$	−9.11444	−0.621600
$216$	−16.1968	−1.10205
$217$	−2.45966	−0.166972
$218$	4.16846	0.282324
$219$	33.8247	2.28566
$220$	−1.97066	−0.132862
$221$	2.55740	0.172029
$222$	−0.0628354	−0.00421724
$223$	−20.9013	−1.39965	−0.699827	−	0.714313i	$-0.746739\pi$
$223$	−20.9013	−1.39965	−0.699827	+	0.714313i	$0.746739\pi$
$224$	−6.47581	−0.432683
$225$	5.23391	0.348927
$226$	4.71503	0.313639
$227$	−22.1560	−1.47055	−0.735273	−	0.677771i	$-0.762946\pi$
$227$	−22.1560	−1.47055	−0.735273	+	0.677771i	$0.762946\pi$
$228$	24.9943	1.65529
$229$	−10.6279	−0.702312	−0.351156	−	0.936317i	$-0.614211\pi$
$229$	−10.6279	−0.702312	−0.351156	+	0.936317i	$0.614211\pi$
$230$	−5.34518	−0.352451
$231$	−4.25631	−0.280045
$232$	−19.3470	−1.27019
$233$	3.64155	0.238566	0.119283	−	0.992860i	$-0.461940\pi$
$233$	3.64155	0.238566	0.119283	+	0.992860i	$0.461940\pi$
$234$	−11.0020	−0.719221
$235$	10.3895	0.677738
$236$	−4.65746	−0.303175
$237$	15.7317	1.02188
$238$	−0.713739	−0.0462648
$239$	21.4558	1.38786	0.693931	−	0.720042i	$-0.255877\pi$
$239$	21.4558	1.38786	0.693931	+	0.720042i	$0.255877\pi$
$240$	−3.15435	−0.203613
$241$	−21.8214	−1.40564	−0.702821	−	0.711366i	$-0.748077\pi$
$241$	−21.8214	−1.40564	−0.702821	+	0.711366i	$0.748077\pi$
$242$	6.70126	0.430773
$243$	11.5056	0.738086
$244$	−15.3686	−0.983873
$245$	−5.77629	−0.369033
$246$	−0.630148	−0.0401768
$247$	17.1076	1.08853
$248$	−5.61818	−0.356755
$249$	18.7483	1.18812
$250$	−0.728237	−0.0460577
$251$	16.5669	1.04570	0.522848	−	0.852426i	$-0.324870\pi$
$251$	16.5669	1.04570	0.522848	+	0.852426i	$0.324870\pi$
$252$	−8.50915	−0.536026
$253$	9.84194	0.618757
$254$	−15.4074	−0.966749
$255$	−2.54232	−0.159206
$256$	−11.5604	−0.722527
$257$	−10.6878	−0.666687	−0.333343	−	0.942806i	$-0.608177\pi$
$257$	−10.6878	−0.666687	−0.333343	+	0.942806i	$0.608177\pi$
$258$	−19.0461	−1.18576
$259$	−0.0332636	−0.00206690
$260$	−4.24221	−0.263091
$261$	−40.0755	−2.48061
$262$	−5.73554	−0.354343
$263$	19.1121	1.17850	0.589250	−	0.807951i	$-0.299423\pi$
$263$	19.1121	1.17850	0.589250	+	0.807951i	$0.299423\pi$
$264$	−9.72197	−0.598346
$265$	9.47173	0.581844
$266$	−4.77452	−0.292745
$267$	40.8380	2.49925
$268$	−12.1279	−0.740828
$269$	−29.4509	−1.79565	−0.897826	−	0.440351i	$-0.854854\pi$
$269$	−29.4509	−1.79565	−0.897826	+	0.440351i	$0.854854\pi$
$270$	4.66811	0.284092
$271$	−15.5891	−0.946971	−0.473486	−	0.880802i	$-0.657004\pi$
$271$	−15.5891	−0.946971	−0.473486	+	0.880802i	$0.657004\pi$
$272$	0.973944	0.0590540
$273$	−9.16251	−0.554540
$274$	5.87517	0.354932
$275$	1.34088	0.0808582
$276$	30.9538	1.86320
$277$	−3.19909	−0.192215	−0.0961073	−	0.995371i	$-0.530639\pi$
$277$	−3.19909	−0.192215	−0.0961073	+	0.995371i	$0.530639\pi$
$278$	10.0338	0.601785
$279$	−11.6375	−0.696720
$280$	2.79512	0.167040
$281$	20.1930	1.20461	0.602307	−	0.798264i	$-0.294248\pi$
$281$	20.1930	1.20461	0.602307	+	0.798264i	$0.294248\pi$
$282$	21.7106	1.29285
$283$	14.3660	0.853971	0.426986	−	0.904258i	$-0.359575\pi$
$283$	14.3660	0.853971	0.426986	+	0.904258i	$0.359575\pi$
$284$	7.18454	0.426324
$285$	−17.0067	−1.00739
$286$	−2.81861	−0.166668
$287$	−0.333585	−0.0196909
$288$	−30.6394	−1.80544
$289$	−16.2150	−0.953825
$290$	5.57604	0.327436
$291$	−44.9705	−2.63622
$292$	17.3241	1.01382
$293$	5.07863	0.296697	0.148349	−	0.988935i	$-0.452604\pi$
$293$	5.07863	0.296697	0.148349	+	0.988935i	$0.452604\pi$
$294$	−12.0705	−0.703964
$295$	3.16905	0.184509
$296$	−0.0759783	−0.00441615
$297$	−8.59525	−0.498747
$298$	−1.75238	−0.101513
$299$	21.1866	1.22525
$300$	4.21719	0.243480
$301$	−10.0825	−0.581148
$302$	−13.0143	−0.748889
$303$	−40.5325	−2.32853
$304$	6.51515	0.373670
$305$	10.4572	0.598775
$306$	−3.37696	−0.193048
$307$	−11.6623	−0.665601	−0.332800	−	0.942997i	$-0.607994\pi$
$307$	−11.6623	−0.665601	−0.332800	+	0.942997i	$0.607994\pi$
$308$	−2.17997	−0.124215
$309$	−43.1007	−2.45192
$310$	1.61922	0.0919658
$311$	15.9699	0.905572	0.452786	−	0.891619i	$-0.350430\pi$
$311$	15.9699	0.905572	0.452786	+	0.891619i	$0.350430\pi$
$312$	−20.9284	−1.18484
$313$	1.03136	0.0582958	0.0291479	−	0.999575i	$-0.490721\pi$
$313$	1.03136	0.0582958	0.0291479	+	0.999575i	$0.490721\pi$
$314$	−10.5229	−0.593843
$315$	5.78983	0.326220
$316$	8.05736	0.453262
$317$	12.5846	0.706824	0.353412	−	0.935468i	$-0.385021\pi$
$317$	12.5846	0.706824	0.353412	+	0.935468i	$0.385021\pi$
$318$	19.7927	1.10992
$319$	−10.2670	−0.574842
$320$	2.06456	0.115412
$321$	20.5405	1.14646
$322$	−5.91293	−0.329515
$323$	5.25103	0.292175
$324$	−3.95645	−0.219803
$325$	2.88650	0.160114
$326$	−10.3095	−0.570992
$327$	16.4250	0.908306
$328$	−0.761952	−0.0420718
$329$	11.4931	0.633633
$330$	2.80199	0.154244
$331$	−21.1309	−1.16146	−0.580728	−	0.814097i	$-0.697232\pi$
$331$	−21.1309	−1.16146	−0.580728	+	0.814097i	$0.697232\pi$
$332$	9.60237	0.526999
$333$	−0.157382	−0.00862448
$334$	−14.7927	−0.809419
$335$	8.25211	0.450861
$336$	−3.48939	−0.190362
$337$	−32.7098	−1.78182	−0.890909	−	0.454183i	$-0.849931\pi$
$337$	−32.7098	−1.78182	−0.890909	+	0.454183i	$0.849931\pi$
$338$	3.39949	0.184908
$339$	18.5787	1.00906
$340$	−1.30211	−0.0706167
$341$	−2.98143	−0.161454
$342$	−22.5900	−1.22153
$343$	−14.1333	−0.763128
$344$	−23.0298	−1.24169
$345$	−21.0617	−1.13392
$346$	14.5736	0.783481
$347$	26.0761	1.39984	0.699920	−	0.714222i	$-0.253219\pi$
$347$	26.0761	1.39984	0.699920	+	0.714222i	$0.253219\pi$
$348$	−32.2906	−1.73096
$349$	1.30714	0.0699698	0.0349849	−	0.999388i	$-0.488862\pi$
$349$	1.30714	0.0699698	0.0349849	+	0.999388i	$0.488862\pi$
$350$	−0.805587	−0.0430604
$351$	−18.5029	−0.987612
$352$	−7.84954	−0.418382
$353$	−2.05558	−0.109408	−0.0547038	−	0.998503i	$-0.517421\pi$
$353$	−2.05558	−0.109408	−0.0547038	+	0.998503i	$0.517421\pi$
$354$	6.62224	0.351968
$355$	−4.88854	−0.259457
$356$	20.9162	1.10855
$357$	−2.81235	−0.148845
$358$	−2.59885	−0.137353
$359$	−0.611521	−0.0322748	−0.0161374	−	0.999870i	$-0.505137\pi$
$359$	−0.611521	−0.0322748	−0.0161374	+	0.999870i	$0.505137\pi$
$360$	13.2247	0.697005
$361$	16.1265	0.848762
$362$	−12.8681	−0.676332
$363$	26.4050	1.38591
$364$	−4.69280	−0.245969
$365$	−11.7878	−0.616999
$366$	21.8519	1.14222
$367$	−27.5178	−1.43642	−0.718208	−	0.695828i	$-0.755037\pi$
$367$	−27.5178	−1.43642	−0.718208	+	0.695828i	$0.755037\pi$
$368$	8.06858	0.420604
$369$	−1.57831	−0.0821637
$370$	0.0218978	0.00113841
$371$	10.4778	0.543979
$372$	−9.37687	−0.486168
$373$	26.5440	1.37440	0.687198	−	0.726471i	$-0.258841\pi$
$373$	26.5440	1.37440	0.687198	+	0.726471i	$0.258841\pi$
$374$	−0.865147	−0.0447357
$375$	−2.86948	−0.148179
$376$	26.2517	1.35383
$377$	−22.1017	−1.13829
$378$	5.16393	0.265604
$379$	−9.17558	−0.471318	−0.235659	−	0.971836i	$-0.575725\pi$
$379$	−9.17558	−0.471318	−0.235659	+	0.971836i	$0.575725\pi$
$380$	−8.71039	−0.446834
$381$	−60.7101	−3.11027
$382$	3.94790	0.201992
$383$	16.5557	0.845956	0.422978	−	0.906140i	$-0.360985\pi$
$383$	16.5557	0.845956	0.422978	+	0.906140i	$0.360985\pi$
$384$	−29.2817	−1.49428
$385$	1.48330	0.0755962
$386$	0.912561	0.0464481
$387$	−47.7041	−2.42494
$388$	−23.0327	−1.16931
$389$	−14.5858	−0.739530	−0.369765	−	0.929125i	$-0.620562\pi$
$389$	−14.5858	−0.739530	−0.369765	+	0.929125i	$0.620562\pi$
$390$	6.03180	0.305432
$391$	6.50305	0.328873
$392$	−14.5952	−0.737168
$393$	−22.5998	−1.14001
$394$	−11.5375	−0.581250
$395$	−5.48242	−0.275851
$396$	−10.3142	−0.518309
$397$	−8.45748	−0.424469	−0.212234	−	0.977219i	$-0.568074\pi$
$397$	−8.45748	−0.424469	−0.212234	+	0.977219i	$0.568074\pi$
$398$	9.06054	0.454164
$399$	−18.8131	−0.941833
$400$	1.09928	0.0549639
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	17.2441	0.860057
$403$	−6.41810	−0.319708
$404$	−20.7597	−1.03283
$405$	2.69206	0.133770
$406$	6.16830	0.306128
$407$	−0.0403199	−0.00199858
$408$	−6.42378	−0.318025
$409$	−13.6152	−0.673226	−0.336613	−	0.941643i	$-0.609281\pi$
$409$	−13.6152	−0.673226	−0.336613	+	0.941643i	$0.609281\pi$
$410$	0.219604	0.0108454
$411$	23.1500	1.14191
$412$	−22.0751	−1.08756
$413$	3.50565	0.172502
$414$	−27.9762	−1.37496
$415$	−6.53369	−0.320726
$416$	−16.8976	−0.828474
$417$	39.5362	1.93609
$418$	−5.78736	−0.283069
$419$	19.5044	0.952850	0.476425	−	0.879215i	$-0.341932\pi$
$419$	19.5044	0.952850	0.476425	+	0.879215i	$0.341932\pi$
$420$	4.66512	0.227635
$421$	8.31944	0.405465	0.202732	−	0.979234i	$-0.435018\pi$
$421$	8.31944	0.405465	0.202732	+	0.979234i	$0.435018\pi$
$422$	0.951249	0.0463061
$423$	54.3778	2.64394
$424$	23.9326	1.16227
$425$	0.885986	0.0429766
$426$	−10.2154	−0.494937
$427$	11.5679	0.559808
$428$	10.5203	0.508517
$429$	−11.1062	−0.536212
$430$	6.63747	0.320087
$431$	12.0781	0.581784	0.290892	−	0.956756i	$-0.406048\pi$
$431$	12.0781	0.581784	0.290892	+	0.956756i	$0.406048\pi$
$432$	−7.04653	−0.339026
$433$	−17.5226	−0.842083	−0.421042	−	0.907041i	$-0.638335\pi$
$433$	−17.5226	−0.842083	−0.421042	+	0.907041i	$0.638335\pi$
$434$	1.79121	0.0859809
$435$	21.9713	1.05344
$436$	8.41247	0.402884
$437$	43.5018	2.08097
$438$	−24.6324	−1.17698
$439$	39.0996	1.86612	0.933060	−	0.359720i	$-0.117128\pi$
$439$	39.0996	1.86612	0.933060	+	0.359720i	$0.117128\pi$
$440$	3.38806	0.161519
$441$	−30.2325	−1.43965
$442$	−1.86239	−0.0885850
$443$	30.6476	1.45611	0.728056	−	0.685518i	$-0.240424\pi$
$443$	30.6476	1.45611	0.728056	+	0.685518i	$0.240424\pi$
$444$	−0.126810	−0.00601812
$445$	−14.2319	−0.674655
$446$	15.2211	0.720739
$447$	−6.90491	−0.326591
$448$	2.28384	0.107901
$449$	−0.720969	−0.0340246	−0.0170123	−	0.999855i	$-0.505415\pi$
$449$	−0.720969	−0.0340246	−0.0170123	+	0.999855i	$0.505415\pi$
$450$	−3.81152	−0.179677
$451$	−0.404350	−0.0190401
$452$	9.51552	0.447573
$453$	−51.2804	−2.40936
$454$	16.1348	0.757244
$455$	3.19309	0.149695
$456$	−42.9716	−2.01233
$457$	12.3129	0.575972	0.287986	−	0.957635i	$-0.407014\pi$
$457$	12.3129	0.575972	0.287986	+	0.957635i	$0.407014\pi$
$458$	7.73963	0.361649
$459$	−5.67930	−0.265087
$460$	−10.7872	−0.502958
$461$	22.3746	1.04209	0.521045	−	0.853529i	$-0.325542\pi$
$461$	22.3746	1.04209	0.521045	+	0.853529i	$0.325542\pi$
$462$	3.09960	0.144206
$463$	2.21665	0.103017	0.0515083	−	0.998673i	$-0.483597\pi$
$463$	2.21665	0.103017	0.0515083	+	0.998673i	$0.483597\pi$
$464$	−8.41706	−0.390752
$465$	6.38025	0.295877
$466$	−2.65191	−0.122847
$467$	3.97855	0.184105	0.0920527	−	0.995754i	$-0.470657\pi$
$467$	3.97855	0.184105	0.0920527	+	0.995754i	$0.470657\pi$
$468$	−22.2033	−1.02635
$469$	9.12861	0.421520
$470$	−7.56604	−0.348995
$471$	−41.4636	−1.91054
$472$	8.00737	0.368569
$473$	−12.2214	−0.561940
$474$	−11.4564	−0.526210
$475$	5.92676	0.271938
$476$	−1.44041	−0.0660212
$477$	49.5742	2.26984
$478$	−15.6249	−0.714667
$479$	29.3441	1.34076	0.670382	−	0.742016i	$-0.266130\pi$
$479$	29.3441	1.34076	0.670382	+	0.742016i	$0.266130\pi$
$480$	16.7980	0.766719
$481$	−0.0867962	−0.00395757
$482$	15.8912	0.723823
$483$	−23.2988	−1.06013
$484$	13.5240	0.614726
$485$	15.6720	0.711629
$486$	−8.37882	−0.380071
$487$	−33.8217	−1.53261	−0.766303	−	0.642479i	$-0.777906\pi$
$487$	−33.8217	−1.53261	−0.766303	+	0.642479i	$0.777906\pi$
$488$	26.4225	1.19609
$489$	−40.6227	−1.83702
$490$	4.20650	0.190030
$491$	15.0285	0.678229	0.339114	−	0.940745i	$-0.389873\pi$
$491$	15.0285	0.678229	0.339114	+	0.940745i	$0.389873\pi$
$492$	−1.27172	−0.0573334
$493$	−6.78391	−0.305532
$494$	−12.4584	−0.560529
$495$	7.01805	0.315438
$496$	−2.44423	−0.109749
$497$	−5.40778	−0.242572
$498$	−13.6532	−0.611814
$499$	30.3754	1.35979	0.679896	−	0.733309i	$-0.262025\pi$
$499$	30.3754	1.35979	0.679896	+	0.733309i	$0.262025\pi$
$500$	−1.46967	−0.0657257
$501$	−58.2878	−2.60410
$502$	−12.0646	−0.538471
$503$	33.9067	1.51182	0.755912	−	0.654674i	$-0.227194\pi$
$503$	33.9067	1.51182	0.755912	+	0.654674i	$0.227194\pi$
$504$	14.6294	0.651646
$505$	14.1254	0.628572
$506$	−7.16726	−0.318623
$507$	13.3950	0.594895
$508$	−31.0941	−1.37958
$509$	3.67267	0.162788	0.0813942	−	0.996682i	$-0.474063\pi$
$509$	3.67267	0.162788	0.0813942	+	0.996682i	$0.474063\pi$
$510$	1.85141	0.0819818
$511$	−13.0398	−0.576847
$512$	−11.9904	−0.529904
$513$	−37.9914	−1.67736
$514$	7.78325	0.343304
$515$	15.0204	0.661878
$516$	−38.4373	−1.69211
$517$	13.9311	0.612690
$518$	0.0242237	0.00106433
$519$	57.4245	2.52065
$520$	7.29344	0.319839
$521$	8.99335	0.394006	0.197003	−	0.980403i	$-0.436879\pi$
$521$	8.99335	0.394006	0.197003	+	0.980403i	$0.436879\pi$
$522$	29.1845	1.27737
$523$	6.22558	0.272226	0.136113	−	0.990693i	$-0.456539\pi$
$523$	6.22558	0.272226	0.136113	+	0.990693i	$0.456539\pi$
$524$	−11.5750	−0.505657
$525$	−3.17426	−0.138536
$526$	−13.9181	−0.606858
$527$	−1.96998	−0.0858136
$528$	−4.22961	−0.184070
$529$	30.8741	1.34235
$530$	−6.89766	−0.299615
$531$	16.5865	0.719794
$532$	−9.63557	−0.417755
$533$	−0.870440	−0.0377029
$534$	−29.7397	−1.28696
$535$	−7.15826	−0.309478
$536$	20.8509	0.900623
$537$	−10.2403	−0.441901
$538$	21.4472	0.924654
$539$	−7.74532	−0.333614
$540$	9.42081	0.405407
$541$	−7.37680	−0.317153	−0.158577	−	0.987347i	$-0.550691\pi$
$541$	−7.37680	−0.317153	−0.158577	+	0.987347i	$0.550691\pi$
$542$	11.3526	0.487634
$543$	−50.7043	−2.17593
$544$	−5.18658	−0.222373
$545$	−5.72405	−0.245191
$546$	6.67248	0.285556
$547$	33.6128	1.43718	0.718591	−	0.695433i	$-0.244788\pi$
$547$	33.6128	1.43718	0.718591	+	0.695433i	$0.244788\pi$
$548$	11.8568	0.506498
$549$	54.7318	2.33590
$550$	−0.976479	−0.0416372
$551$	−45.3806	−1.93328
$552$	−53.2174	−2.26508
$553$	−6.06474	−0.257899
$554$	2.32969	0.0989792
$555$	0.0862843	0.00366257
$556$	20.2494	0.858765
$557$	14.2935	0.605634	0.302817	−	0.953049i	$-0.402073\pi$
$557$	14.2935	0.605634	0.302817	+	0.953049i	$0.402073\pi$
$558$	8.47487	0.358770
$559$	−26.3088	−1.11275
$560$	1.21604	0.0513870
$561$	−3.40895	−0.143926
$562$	−14.7053	−0.620305
$563$	1.09662	0.0462169	0.0231084	−	0.999733i	$-0.492644\pi$
$563$	1.09662	0.0462169	0.0231084	+	0.999733i	$0.492644\pi$
$564$	43.8146	1.84493
$565$	−6.47459	−0.272388
$566$	−10.4619	−0.439745
$567$	2.97800	0.125064
$568$	−12.3521	−0.518281
$569$	17.0641	0.715363	0.357681	−	0.933844i	$-0.383567\pi$
$569$	17.0641	0.715363	0.357681	+	0.933844i	$0.383567\pi$
$570$	12.3849	0.518747
$571$	−16.4090	−0.686694	−0.343347	−	0.939209i	$-0.611561\pi$
$571$	−16.4090	−0.686694	−0.343347	+	0.939209i	$0.611561\pi$
$572$	−5.68830	−0.237840
$573$	15.5560	0.649860
$574$	0.242929	0.0101397
$575$	7.33990	0.306095
$576$	10.8057	0.450237
$577$	46.7720	1.94714	0.973572	−	0.228379i	$-0.0733426\pi$
$577$	46.7720	1.94714	0.973572	+	0.228379i	$0.0733426\pi$
$578$	11.8084	0.491164
$579$	3.59577	0.149435
$580$	11.2531	0.467261
$581$	−7.22767	−0.299854
$582$	32.7492	1.35750
$583$	12.7005	0.526000
$584$	−29.7846	−1.23250
$585$	15.1077	0.624626
$586$	−3.69845	−0.152781
$587$	−33.2278	−1.37146	−0.685729	−	0.727857i	$-0.740516\pi$
$587$	−33.2278	−1.37146	−0.685729	+	0.727857i	$0.740516\pi$
$588$	−24.3597	−1.00458
$589$	−13.1781	−0.542993
$590$	−2.30782	−0.0950114
$591$	−45.4613	−1.87003
$592$	−0.0330549	−0.00135855
$593$	−25.5647	−1.04981	−0.524907	−	0.851159i	$-0.675900\pi$
$593$	−25.5647	−1.04981	−0.524907	+	0.851159i	$0.675900\pi$
$594$	6.25938	0.256825
$595$	0.980092	0.0401798
$596$	−3.53652	−0.144861
$597$	35.7013	1.46116
$598$	−15.4289	−0.630934
$599$	13.7634	0.562357	0.281179	−	0.959655i	$-0.409275\pi$
$599$	13.7634	0.562357	0.281179	+	0.959655i	$0.409275\pi$
$600$	−7.25043	−0.295998
$601$	−22.6543	−0.924086	−0.462043	−	0.886857i	$-0.652884\pi$
$601$	−22.6543	−0.924086	−0.462043	+	0.886857i	$0.652884\pi$
$602$	7.34247	0.299257
$603$	43.1908	1.75886
$604$	−26.2645	−1.06869
$605$	−9.20204	−0.374116
$606$	29.5173	1.19906
$607$	−19.0689	−0.773983	−0.386992	−	0.922083i	$-0.626486\pi$
$607$	−19.0689	−0.773983	−0.386992	+	0.922083i	$0.626486\pi$
$608$	−34.6954	−1.40708
$609$	24.3050	0.984889
$610$	−7.61528	−0.308334
$611$	29.9894	1.21324
$612$	−6.81512	−0.275485
$613$	31.0994	1.25609	0.628046	−	0.778176i	$-0.283855\pi$
$613$	31.0994	1.25609	0.628046	+	0.778176i	$0.283855\pi$
$614$	8.49289	0.342745
$615$	0.865307	0.0348925
$616$	3.74793	0.151008
$617$	25.5954	1.03043	0.515216	−	0.857061i	$-0.327712\pi$
$617$	25.5954	1.03043	0.515216	+	0.857061i	$0.327712\pi$
$618$	31.3875	1.26259
$619$	23.2866	0.935966	0.467983	−	0.883738i	$-0.344981\pi$
$619$	23.2866	0.935966	0.467983	+	0.883738i	$0.344981\pi$
$620$	3.26780	0.131238
$621$	−47.0499	−1.88805
$622$	−11.6299	−0.466316
$623$	−15.7435	−0.630750
$624$	−9.10504	−0.364493
$625$	1.00000	0.0400000
$626$	−0.751073	−0.0300189
$627$	−22.8040	−0.910703
$628$	−21.2366	−0.847431
$629$	−0.0266413	−0.00106226
$630$	−4.21637	−0.167984
$631$	−8.59416	−0.342128	−0.171064	−	0.985260i	$-0.554720\pi$
$631$	−8.59416	−0.342128	−0.171064	+	0.985260i	$0.554720\pi$
$632$	−13.8527	−0.551030
$633$	3.74822	0.148978
$634$	−9.16460	−0.363973
$635$	21.1572	0.839598
$636$	39.9441	1.58389
$637$	−16.6733	−0.660619
$638$	7.47681	0.296010
$639$	−25.5861	−1.01217
$640$	10.2045	0.403370
$641$	5.75223	0.227200	0.113600	−	0.993527i	$-0.463762\pi$
$641$	5.75223	0.227200	0.113600	+	0.993527i	$0.463762\pi$
$642$	−14.9583	−0.590358
$643$	28.8244	1.13672	0.568362	−	0.822779i	$-0.307578\pi$
$643$	28.8244	1.13672	0.568362	+	0.822779i	$0.307578\pi$
$644$	−11.9330	−0.470227
$645$	26.1537	1.02980
$646$	−3.82399	−0.150453
$647$	6.67101	0.262265	0.131132	−	0.991365i	$-0.458139\pi$
$647$	6.67101	0.262265	0.131132	+	0.991365i	$0.458139\pi$
$648$	6.80215	0.267214
$649$	4.24932	0.166800
$650$	−2.10206	−0.0824494
$651$	7.05793	0.276622
$652$	−20.8059	−0.814822
$653$	−7.80061	−0.305261	−0.152631	−	0.988283i	$-0.548775\pi$
$653$	−7.80061	−0.305261	−0.152631	+	0.988283i	$0.548775\pi$
$654$	−11.9613	−0.467724
$655$	7.87593	0.307738
$656$	−0.331493	−0.0129426
$657$	−61.6960	−2.40699
$658$	−8.36967	−0.326284
$659$	2.27824	0.0887477	0.0443738	−	0.999015i	$-0.485871\pi$
$659$	2.27824	0.0887477	0.0443738	+	0.999015i	$0.485871\pi$
$660$	5.65475	0.220111
$661$	21.2759	0.827536	0.413768	−	0.910382i	$-0.364212\pi$
$661$	21.2759	0.827536	0.413768	+	0.910382i	$0.364212\pi$
$662$	15.3883	0.598082
$663$	−7.33840	−0.285000
$664$	−16.5089	−0.640671
$665$	6.55628	0.254241
$666$	0.114611	0.00444110
$667$	−56.2009	−2.17611
$668$	−29.8535	−1.15506
$669$	59.9758	2.31880
$670$	−6.00949	−0.232167
$671$	14.0218	0.541306
$672$	18.5822	0.716823
$673$	19.8233	0.764134	0.382067	−	0.924135i	$-0.375212\pi$
$673$	19.8233	0.764134	0.382067	+	0.924135i	$0.375212\pi$
$674$	23.8205	0.917531
$675$	−6.41015	−0.246727
$676$	6.86059	0.263869
$677$	20.0693	0.771325	0.385662	−	0.922640i	$-0.373973\pi$
$677$	20.0693	0.771325	0.385662	+	0.922640i	$0.373973\pi$
$678$	−13.5297	−0.519605
$679$	17.3366	0.665319
$680$	2.23866	0.0858486
$681$	63.5762	2.43624
$682$	2.17119	0.0831391
$683$	−12.8816	−0.492900	−0.246450	−	0.969156i	$-0.579264\pi$
$683$	−12.8816	−0.492900	−0.246450	+	0.969156i	$0.579264\pi$
$684$	−45.5894	−1.74315
$685$	−8.06767	−0.308250
$686$	10.2924	0.392966
$687$	30.4965	1.16352
$688$	−10.0193	−0.381982
$689$	27.3402	1.04158
$690$	15.3379	0.583904
$691$	−17.2239	−0.655227	−0.327614	−	0.944812i	$-0.606244\pi$
$691$	−17.2239	−0.655227	−0.327614	+	0.944812i	$0.606244\pi$
$692$	29.4113	1.11805
$693$	7.76348	0.294910
$694$	−18.9896	−0.720835
$695$	−13.7782	−0.522636
$696$	55.5159	2.10432
$697$	−0.267174	−0.0101199
$698$	−0.951910	−0.0360303
$699$	−10.4493	−0.395231
$700$	−1.62577	−0.0614485
$701$	15.5608	0.587722	0.293861	−	0.955848i	$-0.405060\pi$
$701$	15.5608	0.587722	0.293861	+	0.955848i	$0.405060\pi$
$702$	13.4745	0.508562
$703$	−0.178216	−0.00672154
$704$	2.76832	0.104335
$705$	−29.8125	−1.12280
$706$	1.49695	0.0563385
$707$	15.6257	0.587666
$708$	13.3645	0.502268
$709$	43.8889	1.64828	0.824141	−	0.566385i	$-0.191658\pi$
$709$	43.8889	1.64828	0.824141	+	0.566385i	$0.191658\pi$
$710$	3.56001	0.133605
$711$	−28.6945	−1.07613
$712$	−35.9602	−1.34767
$713$	−16.3202	−0.611195
$714$	2.04806	0.0766466
$715$	3.87046	0.144747
$716$	−5.24480	−0.196007
$717$	−61.5670	−2.29926
$718$	0.445332	0.0166196
$719$	27.1761	1.01350	0.506748	−	0.862094i	$-0.330847\pi$
$719$	27.1761	1.01350	0.506748	+	0.862094i	$0.330847\pi$
$720$	5.75351	0.214421
$721$	16.6158	0.618805
$722$	−11.7439	−0.437063
$723$	62.6162	2.32872
$724$	−25.9694	−0.965145
$725$	−7.65690	−0.284370
$726$	−19.2291	−0.713660
$727$	−25.8079	−0.957160	−0.478580	−	0.878044i	$-0.658848\pi$
$727$	−25.8079	−0.957160	−0.478580	+	0.878044i	$0.658848\pi$
$728$	8.06812	0.299024
$729$	−41.0913	−1.52190
$730$	8.58427	0.317718
$731$	−8.07527	−0.298674
$732$	44.0998	1.62998
$733$	−3.90301	−0.144161	−0.0720806	−	0.997399i	$-0.522964\pi$
$733$	−3.90301	−0.144161	−0.0720806	+	0.997399i	$0.522964\pi$
$734$	20.0395	0.739670
$735$	16.5749	0.611376
$736$	−42.9679	−1.58382
$737$	11.0651	0.407588
$738$	1.14938	0.0423094
$739$	−8.23391	−0.302889	−0.151445	−	0.988466i	$-0.548393\pi$
$739$	−8.23391	−0.302889	−0.151445	+	0.988466i	$0.548393\pi$
$740$	0.0441926	0.00162455
$741$	−49.0899	−1.80336
$742$	−7.63030	−0.280117
$743$	−46.2799	−1.69785	−0.848923	−	0.528517i	$-0.822748\pi$
$743$	−46.2799	−1.69785	−0.848923	+	0.528517i	$0.822748\pi$
$744$	16.1212	0.591033
$745$	2.40633	0.0881611
$746$	−19.3303	−0.707732
$747$	−34.1967	−1.25119
$748$	−1.74597	−0.0638391
$749$	−7.91857	−0.289338
$750$	2.08966	0.0763036
$751$	−34.7528	−1.26815	−0.634073	−	0.773273i	$-0.718618\pi$
$751$	−34.7528	−1.26815	−0.634073	+	0.773273i	$0.718618\pi$
$752$	11.4210	0.416480
$753$	−47.5384	−1.73240
$754$	16.0952	0.586154
$755$	17.8710	0.650392
$756$	10.4215	0.379025
$757$	31.6463	1.15020	0.575102	−	0.818082i	$-0.304962\pi$
$757$	31.6463	1.15020	0.575102	+	0.818082i	$0.304962\pi$
$758$	6.68200	0.242701
$759$	−28.2412	−1.02509
$760$	14.9754	0.543215
$761$	32.5565	1.18017	0.590086	−	0.807340i	$-0.299094\pi$
$761$	32.5565	1.18017	0.590086	+	0.807340i	$0.299094\pi$
$762$	44.2113	1.60161
$763$	−6.33203	−0.229235
$764$	7.96736	0.288249
$765$	4.63717	0.167657
$766$	−12.0565	−0.435617
$767$	9.14747	0.330296
$768$	33.1724	1.19701
$769$	−40.2781	−1.45247	−0.726233	−	0.687448i	$-0.758731\pi$
$769$	−40.2781	−1.45247	−0.726233	+	0.687448i	$0.758731\pi$
$770$	−1.08020	−0.0389276
$771$	30.6684	1.10450
$772$	1.84166	0.0662828
$773$	−29.5946	−1.06444	−0.532222	−	0.846605i	$-0.678643\pi$
$773$	−29.5946	−1.06444	−0.532222	+	0.846605i	$0.678643\pi$
$774$	34.7399	1.24870
$775$	−2.22349	−0.0798700
$776$	39.5991	1.42153
$777$	0.0954491	0.00342422
$778$	10.6219	0.380814
$779$	−1.78725	−0.0640347
$780$	12.1729	0.435861
$781$	−6.55495	−0.234555
$782$	−4.73576	−0.169350
$783$	49.0819	1.75404
$784$	−6.34974	−0.226776
$785$	14.4499	0.515738
$786$	16.4580	0.587037
$787$	−37.7515	−1.34569	−0.672847	−	0.739782i	$-0.734929\pi$
$787$	−37.7515	−1.34569	−0.672847	+	0.739782i	$0.734929\pi$
$788$	−23.2841	−0.829461
$789$	−54.8416	−1.95241
$790$	3.99250	0.142047
$791$	−7.16230	−0.254662
$792$	17.7328	0.630108
$793$	30.1846	1.07189
$794$	6.15905	0.218576
$795$	−27.1789	−0.963937
$796$	18.2853	0.648105
$797$	49.5205	1.75411	0.877053	−	0.480393i	$-0.159506\pi$
$797$	49.5205	1.75411	0.877053	+	0.480393i	$0.159506\pi$
$798$	13.7004	0.484988
$799$	9.20498	0.325649
$800$	−5.85402	−0.206971
$801$	−74.4882	−2.63191
$802$	0.728237	0.0257149
$803$	−15.8060	−0.557781
$804$	34.8007	1.22733
$805$	8.11951	0.286175
$806$	4.67389	0.164631
$807$	84.5086	2.97484
$808$	35.6912	1.25561
$809$	−20.0037	−0.703293	−0.351646	−	0.936133i	$-0.614378\pi$
$809$	−20.0037	−0.703293	−0.351646	+	0.936133i	$0.614378\pi$
$810$	−1.96046	−0.0688835
$811$	−32.0250	−1.12455	−0.562275	−	0.826950i	$-0.690074\pi$
$811$	−32.0250	−1.12455	−0.562275	+	0.826950i	$0.690074\pi$
$812$	12.4484	0.436853
$813$	44.7326	1.56884
$814$	0.0293624	0.00102915
$815$	14.1568	0.495892
$816$	−2.79471	−0.0978345
$817$	−54.0191	−1.88989
$818$	9.91506	0.346672
$819$	16.7124	0.583977
$820$	0.443187	0.0154768
$821$	21.2425	0.741370	0.370685	−	0.928759i	$-0.379123\pi$
$821$	21.2425	0.741370	0.370685	+	0.928759i	$0.379123\pi$
$822$	−16.8587	−0.588014
$823$	−13.0318	−0.454260	−0.227130	−	0.973864i	$-0.572934\pi$
$823$	−13.0318	−0.454260	−0.227130	+	0.973864i	$0.572934\pi$
$824$	37.9527	1.32214
$825$	−3.84763	−0.133957
$826$	−2.55295	−0.0888283
$827$	9.04835	0.314642	0.157321	−	0.987548i	$-0.449714\pi$
$827$	9.04835	0.314642	0.157321	+	0.987548i	$0.449714\pi$
$828$	−56.4594	−1.96210
$829$	24.2850	0.843454	0.421727	−	0.906723i	$-0.361424\pi$
$829$	24.2850	0.843454	0.421727	+	0.906723i	$0.361424\pi$
$830$	4.75807	0.165155
$831$	9.17971	0.318441
$832$	5.95934	0.206603
$833$	−5.11771	−0.177318
$834$	−28.7917	−0.996974
$835$	20.3130	0.702961
$836$	−11.6796	−0.403947
$837$	14.2529	0.492652
$838$	−14.2038	−0.490662
$839$	−15.1343	−0.522494	−0.261247	−	0.965272i	$-0.584134\pi$
$839$	−15.1343	−0.522494	−0.261247	+	0.965272i	$0.584134\pi$
$840$	−8.02054	−0.276735
$841$	29.6282	1.02166
$842$	−6.05852	−0.208790
$843$	−57.9434	−1.99568
$844$	1.91974	0.0660801
$845$	−4.66811	−0.160588
$846$	−39.5999	−1.36147
$847$	−10.1794	−0.349770
$848$	10.4121	0.357551
$849$	−41.2230	−1.41477
$850$	−0.645208	−0.0221304
$851$	−0.220709	−0.00756579
$852$	−20.6159	−0.706289
$853$	−38.4153	−1.31531	−0.657657	−	0.753317i	$-0.728452\pi$
$853$	−38.4153	−1.31531	−0.657657	+	0.753317i	$0.728452\pi$
$854$	−8.42415	−0.288268
$855$	31.0201	1.06087
$856$	−18.0871	−0.618203
$857$	28.8245	0.984627	0.492314	−	0.870418i	$-0.336151\pi$
$857$	28.8245	0.984627	0.492314	+	0.870418i	$0.336151\pi$
$858$	8.08793	0.276118
$859$	0.982248	0.0335139	0.0167569	−	0.999860i	$-0.494666\pi$
$859$	0.982248	0.0335139	0.0167569	+	0.999860i	$0.494666\pi$
$860$	13.3952	0.456774
$861$	0.957216	0.0326218
$862$	−8.79575	−0.299584
$863$	2.07718	0.0707081	0.0353541	−	0.999375i	$-0.488744\pi$
$863$	2.07718	0.0707081	0.0353541	+	0.999375i	$0.488744\pi$
$864$	37.5251	1.27663
$865$	−20.0122	−0.680434
$866$	12.7606	0.433623
$867$	46.5287	1.58020
$868$	3.61489	0.122697
$869$	−7.35128	−0.249375
$870$	−16.0003	−0.542462
$871$	23.8197	0.807100
$872$	−14.4632	−0.489785
$873$	82.0258	2.77615
$874$	−31.6796	−1.07158
$875$	1.10622	0.0373969
$876$	−49.7112	−1.67959
$877$	−1.18925	−0.0401582	−0.0200791	−	0.999798i	$-0.506392\pi$
$877$	−1.18925	−0.0401582	−0.0200791	+	0.999798i	$0.506392\pi$
$878$	−28.4737	−0.960942
$879$	−14.5730	−0.491536
$880$	1.47400	0.0496885
$881$	5.06893	0.170777	0.0853883	−	0.996348i	$-0.472787\pi$
$881$	5.06893	0.170777	0.0853883	+	0.996348i	$0.472787\pi$
$882$	22.0164	0.741332
$883$	−43.7710	−1.47301	−0.736506	−	0.676431i	$-0.763526\pi$
$883$	−43.7710	−1.47301	−0.736506	+	0.676431i	$0.763526\pi$
$884$	−3.75854	−0.126413
$885$	−9.09352	−0.305675
$886$	−22.3187	−0.749812
$887$	−42.9085	−1.44073	−0.720363	−	0.693597i	$-0.756025\pi$
$887$	−42.9085	−1.44073	−0.720363	+	0.693597i	$0.756025\pi$
$888$	0.218018	0.00731621
$889$	23.4044	0.784959
$890$	10.3642	0.347407
$891$	3.60974	0.120931
$892$	30.7180	1.02852
$893$	61.5762	2.06057
$894$	5.02841	0.168175
$895$	3.56869	0.119288
$896$	11.2884	0.377120
$897$	−60.7946	−2.02987
$898$	0.525036	0.0175207
$899$	17.0250	0.567816
$900$	−7.69212	−0.256404
$901$	8.39182	0.279572
$902$	0.294462	0.00980453
$903$	28.9316	0.962784
$904$	−16.3596	−0.544113
$905$	17.6702	0.587377
$906$	37.3443	1.24068
$907$	59.6477	1.98057	0.990284	−	0.139058i	$-0.0444073\pi$
$907$	59.6477	1.98057	0.990284	+	0.139058i	$0.0444073\pi$
$908$	32.5620	1.08061
$909$	73.9310	2.45214
$910$	−2.32533	−0.0770838
$911$	−40.0494	−1.32689	−0.663447	−	0.748223i	$-0.730907\pi$
$911$	−40.0494	−1.32689	−0.663447	+	0.748223i	$0.730907\pi$
$912$	−18.6951	−0.619056
$913$	−8.76090	−0.289944
$914$	−8.96669	−0.296592
$915$	−30.0066	−0.991987
$916$	15.6195	0.516084
$917$	8.71247	0.287711
$918$	4.13588	0.136504
$919$	−8.62225	−0.284422	−0.142211	−	0.989836i	$-0.545421\pi$
$919$	−8.62225	−0.284422	−0.142211	+	0.989836i	$0.545421\pi$
$920$	18.5460	0.611445
$921$	33.4646	1.10270
$922$	−16.2940	−0.536615
$923$	−14.1108	−0.464462
$924$	6.25538	0.205787
$925$	−0.0300697	−0.000988685 0
$926$	−1.61425	−0.0530475
$927$	78.6154	2.58207
$928$	44.8236	1.47141
$929$	7.48792	0.245671	0.122835	−	0.992427i	$-0.460801\pi$
$929$	7.48792	0.245671	0.122835	+	0.992427i	$0.460801\pi$
$930$	−4.64633	−0.152359
$931$	−34.2347	−1.12200
$932$	−5.35188	−0.175307
$933$	−45.8254	−1.50026
$934$	−2.89733	−0.0948034
$935$	1.18800	0.0388518
$936$	38.1732	1.24773
$937$	−54.4441	−1.77861	−0.889306	−	0.457314i	$-0.848812\pi$
$937$	−54.4441	−1.77861	−0.889306	+	0.457314i	$0.848812\pi$
$938$	−6.64779	−0.217058
$939$	−2.95946	−0.0965784
$940$	−15.2692	−0.498026
$941$	−25.9735	−0.846713	−0.423357	−	0.905963i	$-0.639148\pi$
$941$	−25.9735	−0.846713	−0.423357	+	0.905963i	$0.639148\pi$
$942$	30.1953	0.983816
$943$	−2.21339	−0.0720778
$944$	3.48367	0.113384
$945$	−7.09101	−0.230671
$946$	8.90006	0.289366
$947$	−25.4893	−0.828290	−0.414145	−	0.910211i	$-0.635919\pi$
$947$	−25.4893	−0.828290	−0.414145	+	0.910211i	$0.635919\pi$
$948$	−23.1204	−0.750917
$949$	−34.0254	−1.10451
$950$	−4.31608	−0.140032
$951$	−36.1114	−1.17099
$952$	2.47644	0.0802618
$953$	21.0205	0.680922	0.340461	−	0.940259i	$-0.389417\pi$
$953$	21.0205	0.680922	0.340461	+	0.940259i	$0.389417\pi$
$954$	−36.1017	−1.16884
$955$	−5.42118	−0.175425
$956$	−31.5330	−1.01985
$957$	29.4609	0.952337
$958$	−21.3694	−0.690414
$959$	−8.92458	−0.288190
$960$	−5.92420	−0.191203
$961$	−26.0561	−0.840520
$962$	0.0632082	0.00203791
$963$	−37.4656	−1.20731
$964$	32.0704	1.03292
$965$	−1.25311	−0.0403391
$966$	16.9670	0.545905
$967$	−27.1412	−0.872803	−0.436402	−	0.899752i	$-0.643747\pi$
$967$	−27.1412	−0.872803	−0.436402	+	0.899752i	$0.643747\pi$
$968$	−23.2512	−0.747321
$969$	−15.0677	−0.484045
$970$	−11.4129	−0.366447
$971$	11.2841	0.362125	0.181063	−	0.983472i	$-0.442046\pi$
$971$	11.2841	0.362125	0.181063	+	0.983472i	$0.442046\pi$
$972$	−16.9095	−0.542372
$973$	−15.2416	−0.488624
$974$	24.6302	0.789202
$975$	−8.28275	−0.265260
$976$	11.4953	0.367956
$977$	−45.1682	−1.44506	−0.722529	−	0.691340i	$-0.757021\pi$
$977$	−45.1682	−1.44506	−0.722529	+	0.691340i	$0.757021\pi$
$978$	29.5830	0.945958
$979$	−19.0832	−0.609903
$980$	8.48924	0.271179
$981$	−29.9591	−0.956521
$982$	−10.9443	−0.349248
$983$	−43.1732	−1.37701	−0.688506	−	0.725231i	$-0.741733\pi$
$983$	−43.1732	−1.37701	−0.688506	+	0.725231i	$0.741733\pi$
$984$	2.18641	0.0697001
$985$	15.8430	0.504801
$986$	4.94029	0.157331
$987$	−32.9791	−1.04974
$988$	−25.1426	−0.799891
$989$	−66.8991	−2.12727
$990$	−5.11080	−0.162432
$991$	42.1115	1.33771	0.668857	−	0.743391i	$-0.266784\pi$
$991$	42.1115	1.33771	0.668857	+	0.743391i	$0.266784\pi$
$992$	13.0163	0.413269
$993$	60.6345	1.92418
$994$	3.93814	0.124910
$995$	−12.4418	−0.394430
$996$	−27.5538	−0.873076
$997$	−0.481768	−0.0152577	−0.00762887	−	0.999971i	$-0.502428\pi$
$997$	−0.481768	−0.0152577	−0.00762887	+	0.999971i	$0.502428\pi$
$998$	−22.1205	−0.700213
$999$	0.192751	0.00609838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.g.1.14	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.g.1.14	✓	37	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)

\(f(q)\)	\(=\)	\(q-0.728237 q^{2} -2.86948 q^{3} -1.46967 q^{4} +1.00000 q^{5} +2.08966 q^{6} +1.10622 q^{7} +2.52674 q^{8} +5.23391 q^{9} +O(q^{10})\)	\(q-0.728237 q^{2} -2.86948 q^{3} -1.46967 q^{4} +1.00000 q^{5} +2.08966 q^{6} +1.10622 q^{7} +2.52674 q^{8} +5.23391 q^{9} -0.728237 q^{10} +1.34088 q^{11} +4.21719 q^{12} +2.88650 q^{13} -0.805587 q^{14} -2.86948 q^{15} +1.09928 q^{16} +0.885986 q^{17} -3.81152 q^{18} +5.92676 q^{19} -1.46967 q^{20} -3.17426 q^{21} -0.976479 q^{22} +7.33990 q^{23} -7.25043 q^{24} +1.00000 q^{25} -2.10206 q^{26} -6.41015 q^{27} -1.62577 q^{28} -7.65690 q^{29} +2.08966 q^{30} -2.22349 q^{31} -5.85402 q^{32} -3.84763 q^{33} -0.645208 q^{34} +1.10622 q^{35} -7.69212 q^{36} -0.0300697 q^{37} -4.31608 q^{38} -8.28275 q^{39} +2.52674 q^{40} -0.301555 q^{41} +2.31161 q^{42} -9.11444 q^{43} -1.97066 q^{44} +5.23391 q^{45} -5.34518 q^{46} +10.3895 q^{47} -3.15435 q^{48} -5.77629 q^{49} -0.728237 q^{50} -2.54232 q^{51} -4.24221 q^{52} +9.47173 q^{53} +4.66811 q^{54} +1.34088 q^{55} +2.79512 q^{56} -17.0067 q^{57} +5.57604 q^{58} +3.16905 q^{59} +4.21719 q^{60} +10.4572 q^{61} +1.61922 q^{62} +5.78983 q^{63} +2.06456 q^{64} +2.88650 q^{65} +2.80199 q^{66} +8.25211 q^{67} -1.30211 q^{68} -21.0617 q^{69} -0.805587 q^{70} -4.88854 q^{71} +13.2247 q^{72} -11.7878 q^{73} +0.0218978 q^{74} -2.86948 q^{75} -8.71039 q^{76} +1.48330 q^{77} +6.03180 q^{78} -5.48242 q^{79} +1.09928 q^{80} +2.69206 q^{81} +0.219604 q^{82} -6.53369 q^{83} +4.66512 q^{84} +0.885986 q^{85} +6.63747 q^{86} +21.9713 q^{87} +3.38806 q^{88} -14.2319 q^{89} -3.81152 q^{90} +3.19309 q^{91} -10.7872 q^{92} +6.38025 q^{93} -7.56604 q^{94} +5.92676 q^{95} +16.7980 q^{96} +15.6720 q^{97} +4.20650 q^{98} +7.01805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9}+O(q^{10}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9} + 11 q^{10} + 38 q^{11} + 24 q^{12} + 17 q^{13} + 14 q^{14} + 13 q^{15} + 47 q^{16} + 22 q^{17} + 18 q^{18} + 6 q^{19} + 43 q^{20} + 4 q^{21} + 18 q^{22} + 45 q^{23} - 19 q^{24} + 37 q^{25} - q^{26} + 43 q^{27} + 46 q^{28} + 17 q^{29} - 2 q^{30} - 13 q^{31} + 50 q^{32} + 12 q^{33} - 30 q^{34} + 34 q^{35} + 43 q^{36} + 9 q^{37} + q^{38} - 7 q^{39} + 27 q^{40} + 12 q^{41} - 38 q^{42} + 53 q^{43} + 36 q^{44} + 50 q^{45} - 15 q^{46} + 39 q^{47} - 6 q^{48} + 37 q^{49} + 11 q^{50} + 38 q^{51} + 17 q^{52} + 19 q^{53} - 62 q^{54} + 38 q^{55} + 18 q^{56} + 6 q^{57} - 13 q^{58} + 33 q^{59} + 24 q^{60} - 11 q^{61} + q^{62} + 98 q^{63} + 15 q^{64} + 17 q^{65} - 70 q^{66} + 49 q^{67} + 32 q^{68} - 36 q^{69} + 14 q^{70} + 19 q^{71} + 5 q^{72} + 39 q^{73} + 21 q^{74} + 13 q^{75} - 33 q^{76} + 34 q^{77} - 22 q^{78} - 9 q^{79} + 47 q^{80} + 49 q^{81} + 14 q^{82} + 114 q^{83} - 50 q^{84} + 22 q^{85} + 9 q^{86} + 56 q^{87} + 14 q^{88} + 2 q^{89} + 18 q^{90} - 29 q^{91} + 47 q^{92} - 7 q^{93} - 52 q^{94} + 6 q^{95} - 89 q^{96} + 4 q^{97} + 14 q^{98} + 57 q^{99}+O(q^{100}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9 + 11 * q^10 + 38 * q^11 + 24 * q^12 + 17 * q^13 + 14 * q^14 + 13 * q^15 + 47 * q^16 + 22 * q^17 + 18 * q^18 + 6 * q^19 + 43 * q^20 + 4 * q^21 + 18 * q^22 + 45 * q^23 - 19 * q^24 + 37 * q^25 - q^26 + 43 * q^27 + 46 * q^28 + 17 * q^29 - 2 * q^30 - 13 * q^31 + 50 * q^32 + 12 * q^33 - 30 * q^34 + 34 * q^35 + 43 * q^36 + 9 * q^37 + q^38 - 7 * q^39 + 27 * q^40 + 12 * q^41 - 38 * q^42 + 53 * q^43 + 36 * q^44 + 50 * q^45 - 15 * q^46 + 39 * q^47 - 6 * q^48 + 37 * q^49 + 11 * q^50 + 38 * q^51 + 17 * q^52 + 19 * q^53 - 62 * q^54 + 38 * q^55 + 18 * q^56 + 6 * q^57 - 13 * q^58 + 33 * q^59 + 24 * q^60 - 11 * q^61 + q^62 + 98 * q^63 + 15 * q^64 + 17 * q^65 - 70 * q^66 + 49 * q^67 + 32 * q^68 - 36 * q^69 + 14 * q^70 + 19 * q^71 + 5 * q^72 + 39 * q^73 + 21 * q^74 + 13 * q^75 - 33 * q^76 + 34 * q^77 - 22 * q^78 - 9 * q^79 + 47 * q^80 + 49 * q^81 + 14 * q^82 + 114 * q^83 - 50 * q^84 + 22 * q^85 + 9 * q^86 + 56 * q^87 + 14 * q^88 + 2 * q^89 + 18 * q^90 - 29 * q^91 + 47 * q^92 - 7 * q^93 - 52 * q^94 + 6 * q^95 - 89 * q^96 + 4 * q^97 + 14 * q^98 + 57 * q^99

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