LMFDB - Embedded newform 4009.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71565 q^{2} -0.772476 q^{3} +0.943456 q^{4} -1.16427 q^{5} +1.32530 q^{6} -0.902518 q^{7} +1.81266 q^{8} -2.40328 q^{9} +O(q^{10})$	\(q-1.71565 q^{2} -0.772476 q^{3} +0.943456 q^{4} -1.16427 q^{5} +1.32530 q^{6} -0.902518 q^{7} +1.81266 q^{8} -2.40328 q^{9} +1.99748 q^{10} +1.44765 q^{11} -0.728797 q^{12} -0.547294 q^{13} +1.54841 q^{14} +0.899371 q^{15} -4.99680 q^{16} +0.974885 q^{17} +4.12319 q^{18} -1.00000 q^{19} -1.09844 q^{20} +0.697173 q^{21} -2.48367 q^{22} -7.06973 q^{23} -1.40024 q^{24} -3.64447 q^{25} +0.938965 q^{26} +4.17390 q^{27} -0.851486 q^{28} -0.0224442 q^{29} -1.54301 q^{30} +2.19393 q^{31} +4.94744 q^{32} -1.11828 q^{33} -1.67256 q^{34} +1.05078 q^{35} -2.26739 q^{36} +3.73592 q^{37} +1.71565 q^{38} +0.422771 q^{39} -2.11043 q^{40} -6.48604 q^{41} -1.19611 q^{42} +12.4524 q^{43} +1.36580 q^{44} +2.79807 q^{45} +12.1292 q^{46} -8.03568 q^{47} +3.85991 q^{48} -6.18546 q^{49} +6.25264 q^{50} -0.753075 q^{51} -0.516348 q^{52} +9.55827 q^{53} -7.16096 q^{54} -1.68546 q^{55} -1.63596 q^{56} +0.772476 q^{57} +0.0385064 q^{58} +0.643670 q^{59} +0.848517 q^{60} -12.3924 q^{61} -3.76401 q^{62} +2.16900 q^{63} +1.50552 q^{64} +0.637199 q^{65} +1.91857 q^{66} -9.75624 q^{67} +0.919761 q^{68} +5.46119 q^{69} -1.80276 q^{70} -3.82523 q^{71} -4.35633 q^{72} +1.08581 q^{73} -6.40953 q^{74} +2.81527 q^{75} -0.943456 q^{76} -1.30653 q^{77} -0.725328 q^{78} -2.21929 q^{79} +5.81763 q^{80} +3.98561 q^{81} +11.1278 q^{82} -15.2726 q^{83} +0.657752 q^{84} -1.13503 q^{85} -21.3640 q^{86} +0.0173376 q^{87} +2.62410 q^{88} +15.0121 q^{89} -4.80051 q^{90} +0.493943 q^{91} -6.66998 q^{92} -1.69476 q^{93} +13.7864 q^{94} +1.16427 q^{95} -3.82178 q^{96} -6.76410 q^{97} +10.6121 q^{98} -3.47912 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−1.71565	−1.21315	−0.606574	−	0.795027i	$-0.707457\pi$
$2$	−1.71565	−1.21315	−0.606574	+	0.795027i	$0.707457\pi$
$3$	−0.772476	−0.445989	−0.222995	−	0.974820i	$-0.571583\pi$
$3$	−0.772476	−0.445989	−0.222995	+	0.974820i	$0.571583\pi$
$4$	0.943456	0.471728
$5$	−1.16427	−0.520678	−0.260339	−	0.965517i	$-0.583834\pi$
$5$	−1.16427	−0.520678	−0.260339	+	0.965517i	$0.583834\pi$
$6$	1.32530	0.541051
$7$	−0.902518	−0.341120	−0.170560	−	0.985347i	$-0.554558\pi$
$7$	−0.902518	−0.341120	−0.170560	+	0.985347i	$0.554558\pi$
$8$	1.81266	0.640872
$9$	−2.40328	−0.801094
$10$	1.99748	0.631659
$11$	1.44765	0.436484	0.218242	−	0.975895i	$-0.429968\pi$
$11$	1.44765	0.436484	0.218242	+	0.975895i	$0.429968\pi$
$12$	−0.728797	−0.210385
$13$	−0.547294	−0.151792	−0.0758960	−	0.997116i	$-0.524182\pi$
$13$	−0.547294	−0.151792	−0.0758960	+	0.997116i	$0.524182\pi$
$14$	1.54841	0.413829
$15$	0.899371	0.232217
$16$	−4.99680	−1.24920
$17$	0.974885	0.236444	0.118222	−	0.992987i	$-0.462281\pi$
$17$	0.974885	0.236444	0.118222	+	0.992987i	$0.462281\pi$
$18$	4.12319	0.971845
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.09844	−0.245618
$21$	0.697173	0.152136
$22$	−2.48367	−0.529519
$23$	−7.06973	−1.47414	−0.737070	−	0.675816i	$-0.763791\pi$
$23$	−7.06973	−1.47414	−0.737070	+	0.675816i	$0.763791\pi$
$24$	−1.40024	−0.285822
$25$	−3.64447	−0.728895
$26$	0.938965	0.184146
$27$	4.17390	0.803268
$28$	−0.851486	−0.160916
$29$	−0.0224442	−0.00416778	−0.00208389	−	0.999998i	$-0.500663\pi$
$29$	−0.0224442	−0.00416778	−0.00208389	+	0.999998i	$0.500663\pi$
$30$	−1.54301	−0.281713
$31$	2.19393	0.394041	0.197020	−	0.980399i	$-0.436873\pi$
$31$	2.19393	0.394041	0.197020	+	0.980399i	$0.436873\pi$
$32$	4.94744	0.874593
$33$	−1.11828	−0.194667
$34$	−1.67256	−0.286842
$35$	1.05078	0.177613
$36$	−2.26739	−0.377898
$37$	3.73592	0.614181	0.307091	−	0.951680i	$-0.400645\pi$
$37$	3.73592	0.614181	0.307091	+	0.951680i	$0.400645\pi$
$38$	1.71565	0.278315
$39$	0.422771	0.0676976
$40$	−2.11043	−0.333688
$41$	−6.48604	−1.01295	−0.506475	−	0.862255i	$-0.669052\pi$
$41$	−6.48604	−1.01295	−0.506475	+	0.862255i	$0.669052\pi$
$42$	−1.19611	−0.184563
$43$	12.4524	1.89898	0.949488	−	0.313803i	$-0.101603\pi$
$43$	12.4524	1.89898	0.949488	+	0.313803i	$0.101603\pi$
$44$	1.36580	0.205902
$45$	2.79807	0.417112
$46$	12.1292	1.78835
$47$	−8.03568	−1.17212	−0.586062	−	0.810266i	$-0.699323\pi$
$47$	−8.03568	−1.17212	−0.586062	+	0.810266i	$0.699323\pi$
$48$	3.85991	0.557130
$49$	−6.18546	−0.883637
$50$	6.25264	0.884257
$51$	−0.753075	−0.105452
$52$	−0.516348	−0.0716046
$53$	9.55827	1.31293	0.656465	−	0.754357i	$-0.272051\pi$
$53$	9.55827	1.31293	0.656465	+	0.754357i	$0.272051\pi$
$54$	−7.16096	−0.974483
$55$	−1.68546	−0.227267
$56$	−1.63596	−0.218614
$57$	0.772476	0.102317
$58$	0.0385064	0.00505613
$59$	0.643670	0.0837987	0.0418994	−	0.999122i	$-0.486659\pi$
$59$	0.643670	0.0837987	0.0418994	+	0.999122i	$0.486659\pi$
$60$	0.848517	0.109543
$61$	−12.3924	−1.58668	−0.793340	−	0.608779i	$-0.791660\pi$
$61$	−12.3924	−1.58668	−0.793340	+	0.608779i	$0.791660\pi$
$62$	−3.76401	−0.478030
$63$	2.16900	0.273269
$64$	1.50552	0.188190
$65$	0.637199	0.0790348
$66$	1.91857	0.236160
$67$	−9.75624	−1.19191	−0.595957	−	0.803016i	$-0.703227\pi$
$67$	−9.75624	−1.19191	−0.595957	+	0.803016i	$0.703227\pi$
$68$	0.919761	0.111537
$69$	5.46119	0.657450
$70$	−1.80276	−0.215471
$71$	−3.82523	−0.453971	−0.226986	−	0.973898i	$-0.572887\pi$
$71$	−3.82523	−0.453971	−0.226986	+	0.973898i	$0.572887\pi$
$72$	−4.35633	−0.513399
$73$	1.08581	0.127084	0.0635420	−	0.997979i	$-0.479760\pi$
$73$	1.08581	0.127084	0.0635420	+	0.997979i	$0.479760\pi$
$74$	−6.40953	−0.745093
$75$	2.81527	0.325079
$76$	−0.943456	−0.108222
$77$	−1.30653	−0.148893
$78$	−0.725328	−0.0821272
$79$	−2.21929	−0.249689	−0.124845	−	0.992176i	$-0.539843\pi$
$79$	−2.21929	−0.249689	−0.124845	+	0.992176i	$0.539843\pi$
$80$	5.81763	0.650431
$81$	3.98561	0.442845
$82$	11.1278	1.22886
$83$	−15.2726	−1.67638	−0.838190	−	0.545378i	$-0.816386\pi$
$83$	−15.2726	−1.67638	−0.838190	+	0.545378i	$0.816386\pi$
$84$	0.657752	0.0717666
$85$	−1.13503	−0.123111
$86$	−21.3640	−2.30374
$87$	0.0173376	0.00185878
$88$	2.62410	0.279730
$89$	15.0121	1.59128	0.795639	−	0.605772i	$-0.207135\pi$
$89$	15.0121	1.59128	0.795639	+	0.605772i	$0.207135\pi$
$90$	−4.80051	−0.506018
$91$	0.493943	0.0517793
$92$	−6.66998	−0.695393
$93$	−1.69476	−0.175738
$94$	13.7864	1.42196
$95$	1.16427	0.119452
$96$	−3.82178	−0.390059
$97$	−6.76410	−0.686790	−0.343395	−	0.939191i	$-0.611577\pi$
$97$	−6.76410	−0.686790	−0.343395	+	0.939191i	$0.611577\pi$
$98$	10.6121	1.07198
$99$	−3.47912	−0.349665
$100$	−3.43840	−0.343840
$101$	−14.0567	−1.39870	−0.699349	−	0.714780i	$-0.746527\pi$
$101$	−14.0567	−1.39870	−0.699349	+	0.714780i	$0.746527\pi$
$102$	1.29201	0.127928
$103$	−18.2097	−1.79426	−0.897128	−	0.441772i	$-0.854350\pi$
$103$	−18.2097	−1.79426	−0.897128	+	0.441772i	$0.854350\pi$
$104$	−0.992059	−0.0972793
$105$	−0.811698	−0.0792137
$106$	−16.3987	−1.59278
$107$	10.0885	0.975297	0.487648	−	0.873040i	$-0.337855\pi$
$107$	10.0885	0.975297	0.487648	+	0.873040i	$0.337855\pi$
$108$	3.93789	0.378924
$109$	4.49837	0.430865	0.215433	−	0.976519i	$-0.430884\pi$
$109$	4.49837	0.430865	0.215433	+	0.976519i	$0.430884\pi$
$110$	2.89166	0.275709
$111$	−2.88591	−0.273918
$112$	4.50970	0.426127
$113$	0.673754	0.0633814	0.0316907	−	0.999498i	$-0.489911\pi$
$113$	0.673754	0.0633814	0.0316907	+	0.999498i	$0.489911\pi$
$114$	−1.32530	−0.124126
$115$	8.23108	0.767552
$116$	−0.0211751	−0.00196606
$117$	1.31530	0.121600
$118$	−1.10431	−0.101660
$119$	−0.879851	−0.0806558
$120$	1.63025	0.148821
$121$	−8.90430	−0.809482
$122$	21.2610	1.92488
$123$	5.01031	0.451764
$124$	2.06987	0.185880
$125$	10.0645	0.900197
$126$	−3.72125	−0.331516
$127$	3.23864	0.287383	0.143691	−	0.989623i	$-0.454103\pi$
$127$	3.23864	0.287383	0.143691	+	0.989623i	$0.454103\pi$
$128$	−12.4778	−1.10290
$129$	−9.61919	−0.846923
$130$	−1.09321	−0.0958809
$131$	−10.7960	−0.943252	−0.471626	−	0.881799i	$-0.656333\pi$
$131$	−10.7960	−0.943252	−0.471626	+	0.881799i	$0.656333\pi$
$132$	−1.05504	−0.0918299
$133$	0.902518	0.0782582
$134$	16.7383	1.44597
$135$	−4.85955	−0.418244
$136$	1.76714	0.151531
$137$	0.884784	0.0755922	0.0377961	−	0.999285i	$-0.487966\pi$
$137$	0.884784	0.0755922	0.0377961	+	0.999285i	$0.487966\pi$
$138$	−9.36950	−0.797585
$139$	8.00649	0.679102	0.339551	−	0.940588i	$-0.389725\pi$
$139$	8.00649	0.679102	0.339551	+	0.940588i	$0.389725\pi$
$140$	0.991360	0.0837852
$141$	6.20737	0.522755
$142$	6.56276	0.550734
$143$	−0.792292	−0.0662548
$144$	12.0087	1.00073
$145$	0.0261311	0.00217007
$146$	−1.86286	−0.154172
$147$	4.77812	0.394093
$148$	3.52467	0.289726
$149$	15.4384	1.26476	0.632382	−	0.774657i	$-0.282077\pi$
$149$	15.4384	1.26476	0.632382	+	0.774657i	$0.282077\pi$
$150$	−4.83001	−0.394369
$151$	17.8472	1.45238	0.726190	−	0.687494i	$-0.241289\pi$
$151$	17.8472	1.45238	0.726190	+	0.687494i	$0.241289\pi$
$152$	−1.81266	−0.147026
$153$	−2.34292	−0.189414
$154$	2.24155	0.180630
$155$	−2.55433	−0.205168
$156$	0.398866	0.0319348
$157$	3.04010	0.242627	0.121313	−	0.992614i	$-0.461289\pi$
$157$	3.04010	0.242627	0.121313	+	0.992614i	$0.461289\pi$
$158$	3.80752	0.302910
$159$	−7.38353	−0.585552
$160$	−5.76017	−0.455381
$161$	6.38056	0.502858
$162$	−6.83790	−0.537237
$163$	1.46993	0.115134	0.0575669	−	0.998342i	$-0.481666\pi$
$163$	1.46993	0.115134	0.0575669	+	0.998342i	$0.481666\pi$
$164$	−6.11929	−0.477837
$165$	1.30198	0.101359
$166$	26.2024	2.03370
$167$	−12.6995	−0.982714	−0.491357	−	0.870958i	$-0.663499\pi$
$167$	−12.6995	−0.982714	−0.491357	+	0.870958i	$0.663499\pi$
$168$	1.26374	0.0974995
$169$	−12.7005	−0.976959
$170$	1.94731	0.149352
$171$	2.40328	0.183784
$172$	11.7483	0.895800
$173$	11.0077	0.836899	0.418450	−	0.908240i	$-0.362574\pi$
$173$	11.0077	0.836899	0.418450	+	0.908240i	$0.362574\pi$
$174$	−0.0297452	−0.00225498
$175$	3.28920	0.248640
$176$	−7.23364	−0.545256
$177$	−0.497220	−0.0373733
$178$	−25.7555	−1.93045
$179$	2.74039	0.204826	0.102413	−	0.994742i	$-0.467344\pi$
$179$	2.74039	0.204826	0.102413	+	0.994742i	$0.467344\pi$
$180$	2.63986	0.196763
$181$	13.7008	1.01837	0.509186	−	0.860657i	$-0.329947\pi$
$181$	13.7008	1.01837	0.509186	+	0.860657i	$0.329947\pi$
$182$	−0.847433	−0.0628159
$183$	9.57280	0.707642
$184$	−12.8150	−0.944736
$185$	−4.34962	−0.319791
$186$	2.90761	0.213196
$187$	1.41130	0.103204
$188$	−7.58131	−0.552924
$189$	−3.76702	−0.274011
$190$	−1.99748	−0.144913
$191$	−14.0431	−1.01612	−0.508062	−	0.861321i	$-0.669638\pi$
$191$	−14.0431	−1.01612	−0.508062	+	0.861321i	$0.669638\pi$
$192$	−1.16298	−0.0839307
$193$	−10.2485	−0.737706	−0.368853	−	0.929488i	$-0.620249\pi$
$193$	−10.2485	−0.737706	−0.368853	+	0.929488i	$0.620249\pi$
$194$	11.6048	0.833178
$195$	−0.492220	−0.0352486
$196$	−5.83571	−0.416836
$197$	5.86674	0.417988	0.208994	−	0.977917i	$-0.432981\pi$
$197$	5.86674	0.417988	0.208994	+	0.977917i	$0.432981\pi$
$198$	5.96895	0.424195
$199$	−8.83427	−0.626245	−0.313122	−	0.949713i	$-0.601375\pi$
$199$	−8.83427	−0.626245	−0.313122	+	0.949713i	$0.601375\pi$
$200$	−6.60619	−0.467128
$201$	7.53646	0.531581
$202$	24.1165	1.69683
$203$	0.0202563	0.00142171
$204$	−0.710493	−0.0497444
$205$	7.55151	0.527420
$206$	31.2415	2.17670
$207$	16.9905	1.18092
$208$	2.73472	0.189619
$209$	−1.44765	−0.100136
$210$	1.39259	0.0960979
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	9.01781	0.619346
$213$	2.95490	0.202466
$214$	−17.3084	−1.18318
$215$	−14.4980	−0.988755
$216$	7.56587	0.514792
$217$	−1.98006	−0.134415
$218$	−7.71762	−0.522703
$219$	−0.838759	−0.0566781
$220$	−1.59016	−0.107208
$221$	−0.533549	−0.0358904
$222$	4.95121	0.332303
$223$	−7.50091	−0.502298	−0.251149	−	0.967948i	$-0.580808\pi$
$223$	−7.50091	−0.502298	−0.251149	+	0.967948i	$0.580808\pi$
$224$	−4.46516	−0.298341
$225$	8.75869	0.583913
$226$	−1.15593	−0.0768911
$227$	5.15435	0.342106	0.171053	−	0.985262i	$-0.445283\pi$
$227$	5.15435	0.342106	0.171053	+	0.985262i	$0.445283\pi$
$228$	0.728797	0.0482657
$229$	−5.50466	−0.363758	−0.181879	−	0.983321i	$-0.558218\pi$
$229$	−5.50466	−0.363758	−0.181879	+	0.983321i	$0.558218\pi$
$230$	−14.1217	−0.931154
$231$	1.00926	0.0664048
$232$	−0.0406837	−0.00267101
$233$	−10.5900	−0.693774	−0.346887	−	0.937907i	$-0.612761\pi$
$233$	−10.5900	−0.693774	−0.346887	+	0.937907i	$0.612761\pi$
$234$	−2.25660	−0.147518
$235$	9.35571	0.610299
$236$	0.607274	0.0395302
$237$	1.71435	0.111359
$238$	1.50952	0.0978474
$239$	16.0256	1.03661	0.518304	−	0.855197i	$-0.326564\pi$
$239$	16.0256	1.03661	0.518304	+	0.855197i	$0.326564\pi$
$240$	−4.49398	−0.290085
$241$	0.225197	0.0145062	0.00725310	−	0.999974i	$-0.497691\pi$
$241$	0.225197	0.0145062	0.00725310	+	0.999974i	$0.497691\pi$
$242$	15.2767	0.982021
$243$	−15.6005	−1.00077
$244$	−11.6916	−0.748481
$245$	7.20155	0.460090
$246$	−8.59594	−0.548057
$247$	0.547294	0.0348235
$248$	3.97685	0.252530
$249$	11.7977	0.747647
$250$	−17.2672	−1.09207
$251$	23.2216	1.46573	0.732867	−	0.680372i	$-0.238182\pi$
$251$	23.2216	1.46573	0.732867	+	0.680372i	$0.238182\pi$
$252$	2.04636	0.128909
$253$	−10.2345	−0.643438
$254$	−5.55637	−0.348638
$255$	0.876783	0.0549063
$256$	18.3966	1.14979
$257$	−2.71959	−0.169644	−0.0848218	−	0.996396i	$-0.527032\pi$
$257$	−2.71959	−0.169644	−0.0848218	+	0.996396i	$0.527032\pi$
$258$	16.5032	1.02744
$259$	−3.37173	−0.209509
$260$	0.601169	0.0372829
$261$	0.0539397	0.00333878
$262$	18.5222	1.14430
$263$	14.5400	0.896575	0.448288	−	0.893889i	$-0.352034\pi$
$263$	14.5400	0.896575	0.448288	+	0.893889i	$0.352034\pi$
$264$	−2.02706	−0.124757
$265$	−11.1284	−0.683613
$266$	−1.54841	−0.0949388
$267$	−11.5965	−0.709692
$268$	−9.20458	−0.562259
$269$	24.3514	1.48473	0.742364	−	0.669996i	$-0.233704\pi$
$269$	24.3514	1.48473	0.742364	+	0.669996i	$0.233704\pi$
$270$	8.33730	0.507392
$271$	26.0558	1.58278	0.791389	−	0.611312i	$-0.209358\pi$
$271$	26.0558	1.58278	0.791389	+	0.611312i	$0.209358\pi$
$272$	−4.87131	−0.295366
$273$	−0.381559	−0.0230930
$274$	−1.51798	−0.0917045
$275$	−5.27593	−0.318151
$276$	5.15239	0.310138
$277$	25.8293	1.55193	0.775965	−	0.630776i	$-0.217263\pi$
$277$	25.8293	1.55193	0.775965	+	0.630776i	$0.217263\pi$
$278$	−13.7363	−0.823851
$279$	−5.27262	−0.315664
$280$	1.90470	0.113828
$281$	16.3597	0.975937	0.487969	−	0.872861i	$-0.337738\pi$
$281$	16.3597	0.975937	0.487969	+	0.872861i	$0.337738\pi$
$282$	−10.6497	−0.634179
$283$	7.07431	0.420524	0.210262	−	0.977645i	$-0.432568\pi$
$283$	7.07431	0.420524	0.210262	+	0.977645i	$0.432568\pi$
$284$	−3.60893	−0.214151
$285$	−0.899371	−0.0532741
$286$	1.35930	0.0803769
$287$	5.85377	0.345537
$288$	−11.8901	−0.700631
$289$	−16.0496	−0.944094
$290$	−0.0448318	−0.00263262
$291$	5.22510	0.306301
$292$	1.02441	0.0599491
$293$	−10.8022	−0.631070	−0.315535	−	0.948914i	$-0.602184\pi$
$293$	−10.8022	−0.631070	−0.315535	+	0.948914i	$0.602184\pi$
$294$	−8.19758	−0.478093
$295$	−0.749407	−0.0436321
$296$	6.77195	0.393612
$297$	6.04236	0.350614
$298$	−26.4869	−1.53435
$299$	3.86922	0.223763
$300$	2.65608	0.153349
$301$	−11.2385	−0.647778
$302$	−30.6195	−1.76195
$303$	10.8585	0.623804
$304$	4.99680	0.286586
$305$	14.4281	0.826149
$306$	4.01964	0.229787
$307$	−25.9637	−1.48183	−0.740914	−	0.671600i	$-0.765607\pi$
$307$	−25.9637	−1.48183	−0.740914	+	0.671600i	$0.765607\pi$
$308$	−1.23266	−0.0702371
$309$	14.0666	0.800218
$310$	4.38233	0.248900
$311$	7.37427	0.418157	0.209078	−	0.977899i	$-0.432954\pi$
$311$	7.37427	0.418157	0.209078	+	0.977899i	$0.432954\pi$
$312$	0.766341	0.0433855
$313$	−28.0809	−1.58723	−0.793614	−	0.608421i	$-0.791803\pi$
$313$	−28.0809	−1.58723	−0.793614	+	0.608421i	$0.791803\pi$
$314$	−5.21575	−0.294342
$315$	−2.52531	−0.142285
$316$	−2.09380	−0.117785
$317$	24.5652	1.37972	0.689860	−	0.723942i	$-0.257672\pi$
$317$	24.5652	1.37972	0.689860	+	0.723942i	$0.257672\pi$
$318$	12.6676	0.710362
$319$	−0.0324914	−0.00181917
$320$	−1.75283	−0.0979864
$321$	−7.79316	−0.434972
$322$	−10.9468	−0.610042
$323$	−0.974885	−0.0542440
$324$	3.76024	0.208902
$325$	1.99460	0.110640
$326$	−2.52188	−0.139674
$327$	−3.47488	−0.192161
$328$	−11.7570	−0.649171
$329$	7.25235	0.399835
$330$	−2.23374	−0.122963
$331$	25.5205	1.40273	0.701367	−	0.712801i	$-0.252574\pi$
$331$	25.5205	1.40273	0.701367	+	0.712801i	$0.252574\pi$
$332$	−14.4090	−0.790795
$333$	−8.97846	−0.492017
$334$	21.7878	1.19218
$335$	11.3589	0.620603
$336$	−3.48364	−0.190048
$337$	25.8156	1.40627	0.703133	−	0.711058i	$-0.251784\pi$
$337$	25.8156	1.40627	0.703133	+	0.711058i	$0.251784\pi$
$338$	21.7896	1.18520
$339$	−0.520459	−0.0282674
$340$	−1.07085	−0.0580750
$341$	3.17605	0.171993
$342$	−4.12319	−0.222957
$343$	11.9001	0.642546
$344$	22.5720	1.21700
$345$	−6.35831	−0.342320
$346$	−18.8853	−1.01528
$347$	24.7246	1.32728	0.663642	−	0.748051i	$-0.269010\pi$
$347$	24.7246	1.32728	0.663642	+	0.748051i	$0.269010\pi$
$348$	0.0163572	0.000876840 0
$349$	−23.1973	−1.24172	−0.620862	−	0.783920i	$-0.713217\pi$
$349$	−23.1973	−1.24172	−0.620862	+	0.783920i	$0.713217\pi$
$350$	−5.64312	−0.301637
$351$	−2.28435	−0.121930
$352$	7.16218	0.381746
$353$	19.9062	1.05950	0.529749	−	0.848154i	$-0.322286\pi$
$353$	19.9062	1.05950	0.529749	+	0.848154i	$0.322286\pi$
$354$	0.853055	0.0453394
$355$	4.45360	0.236373
$356$	14.1632	0.750650
$357$	0.679664	0.0359716
$358$	−4.70155	−0.248485
$359$	9.26895	0.489196	0.244598	−	0.969625i	$-0.421344\pi$
$359$	9.26895	0.489196	0.244598	+	0.969625i	$0.421344\pi$
$360$	5.07195	0.267315
$361$	1.00000	0.0526316
$362$	−23.5058	−1.23544
$363$	6.87836	0.361020
$364$	0.466013	0.0244257
$365$	−1.26417	−0.0661698
$366$	−16.4236	−0.858474
$367$	−3.72760	−0.194579	−0.0972895	−	0.995256i	$-0.531017\pi$
$367$	−3.72760	−0.194579	−0.0972895	+	0.995256i	$0.531017\pi$
$368$	35.3260	1.84150
$369$	15.5878	0.811468
$370$	7.46243	0.387953
$371$	−8.62651	−0.447866
$372$	−1.59893	−0.0829005
$373$	−9.83907	−0.509448	−0.254724	−	0.967014i	$-0.581985\pi$
$373$	−9.83907	−0.509448	−0.254724	+	0.967014i	$0.581985\pi$
$374$	−2.42129	−0.125202
$375$	−7.77459	−0.401478
$376$	−14.5660	−0.751182
$377$	0.0122836	0.000632636 0
$378$	6.46289	0.332415
$379$	2.63509	0.135356	0.0676778	−	0.997707i	$-0.478441\pi$
$379$	2.63509	0.135356	0.0676778	+	0.997707i	$0.478441\pi$
$380$	1.09844	0.0563487
$381$	−2.50177	−0.128170
$382$	24.0931	1.23271
$383$	30.0752	1.53677	0.768386	−	0.639986i	$-0.221060\pi$
$383$	30.0752	1.53677	0.768386	+	0.639986i	$0.221060\pi$
$384$	9.63883	0.491879
$385$	1.52116	0.0775254
$386$	17.5829	0.894946
$387$	−29.9267	−1.52126
$388$	−6.38163	−0.323978
$389$	−1.42694	−0.0723487	−0.0361743	−	0.999345i	$-0.511517\pi$
$389$	−1.42694	−0.0723487	−0.0361743	+	0.999345i	$0.511517\pi$
$390$	0.844478	0.0427618
$391$	−6.89217	−0.348552
$392$	−11.2121	−0.566299
$393$	8.33966	0.420680
$394$	−10.0653	−0.507082
$395$	2.58385	0.130008
$396$	−3.28239	−0.164946
$397$	18.5651	0.931753	0.465877	−	0.884850i	$-0.345739\pi$
$397$	18.5651	0.931753	0.465877	+	0.884850i	$0.345739\pi$
$398$	15.1565	0.759727
$399$	−0.697173	−0.0349023
$400$	18.2107	0.910536
$401$	13.3385	0.666093	0.333047	−	0.942910i	$-0.391923\pi$
$401$	13.3385	0.666093	0.333047	+	0.942910i	$0.391923\pi$
$402$	−12.9299	−0.644886
$403$	−1.20072	−0.0598123
$404$	−13.2619	−0.659805
$405$	−4.64032	−0.230580
$406$	−0.0347527	−0.00172475
$407$	5.40832	0.268080
$408$	−1.36507	−0.0675810
$409$	13.0464	0.645101	0.322550	−	0.946552i	$-0.395460\pi$
$409$	13.0464	0.645101	0.322550	+	0.946552i	$0.395460\pi$
$410$	−12.9558	−0.639839
$411$	−0.683474	−0.0337133
$412$	−17.1800	−0.846400
$413$	−0.580924	−0.0285854
$414$	−29.1498	−1.43264
$415$	17.7814	0.872854
$416$	−2.70771	−0.132756
$417$	−6.18482	−0.302872
$418$	2.48367	0.121480
$419$	34.0603	1.66395	0.831976	−	0.554811i	$-0.187210\pi$
$419$	34.0603	1.66395	0.831976	+	0.554811i	$0.187210\pi$
$420$	−0.765802	−0.0373673
$421$	3.61866	0.176363	0.0881814	−	0.996104i	$-0.471894\pi$
$421$	3.61866	0.176363	0.0881814	+	0.996104i	$0.471894\pi$
$422$	−1.71565	−0.0835165
$423$	19.3120	0.938982
$424$	17.3259	0.841420
$425$	−3.55294	−0.172343
$426$	−5.06957	−0.245621
$427$	11.1843	0.541248
$428$	9.51810	0.460075
$429$	0.612026	0.0295489
$430$	24.8735	1.19951
$431$	−25.1811	−1.21293	−0.606464	−	0.795111i	$-0.707413\pi$
$431$	−25.1811	−1.21293	−0.606464	+	0.795111i	$0.707413\pi$
$432$	−20.8562	−1.00344
$433$	−26.7762	−1.28678	−0.643391	−	0.765538i	$-0.722473\pi$
$433$	−26.7762	−1.28678	−0.643391	+	0.765538i	$0.722473\pi$
$434$	3.39709	0.163065
$435$	−0.0201856	−0.000967827 0
$436$	4.24401	0.203251
$437$	7.06973	0.338191
$438$	1.43902	0.0687589
$439$	19.2513	0.918815	0.459408	−	0.888226i	$-0.348062\pi$
$439$	19.2513	0.918815	0.459408	+	0.888226i	$0.348062\pi$
$440$	−3.05517	−0.145649
$441$	14.8654	0.707876
$442$	0.915383	0.0435403
$443$	21.3989	1.01669	0.508346	−	0.861153i	$-0.330257\pi$
$443$	21.3989	1.01669	0.508346	+	0.861153i	$0.330257\pi$
$444$	−2.72272	−0.129215
$445$	−17.4781	−0.828543
$446$	12.8689	0.609362
$447$	−11.9258	−0.564071
$448$	−1.35876	−0.0641954
$449$	−5.82417	−0.274859	−0.137430	−	0.990512i	$-0.543884\pi$
$449$	−5.82417	−0.274859	−0.137430	+	0.990512i	$0.543884\pi$
$450$	−15.0269	−0.708373
$451$	−9.38954	−0.442136
$452$	0.635657	0.0298988
$453$	−13.7865	−0.647746
$454$	−8.84306	−0.415025
$455$	−0.575083	−0.0269603
$456$	1.40024	0.0655721
$457$	−38.5202	−1.80190	−0.900949	−	0.433925i	$-0.857128\pi$
$457$	−38.5202	−1.80190	−0.900949	+	0.433925i	$0.857128\pi$
$458$	9.44408	0.441293
$459$	4.06908	0.189928
$460$	7.76566	0.362076
$461$	25.0870	1.16842	0.584208	−	0.811604i	$-0.301405\pi$
$461$	25.0870	1.16842	0.584208	+	0.811604i	$0.301405\pi$
$462$	−1.73155	−0.0805588
$463$	26.0074	1.20867	0.604334	−	0.796731i	$-0.293439\pi$
$463$	26.0074	1.20867	0.604334	+	0.796731i	$0.293439\pi$
$464$	0.112149	0.00520639
$465$	1.97315	0.0915029
$466$	18.1687	0.841651
$467$	38.2936	1.77202	0.886008	−	0.463671i	$-0.153468\pi$
$467$	38.2936	1.77202	0.886008	+	0.463671i	$0.153468\pi$
$468$	1.24093	0.0573620
$469$	8.80518	0.406586
$470$	−16.0511	−0.740384
$471$	−2.34840	−0.108209
$472$	1.16676	0.0537043
$473$	18.0268	0.828873
$474$	−2.94122	−0.135095
$475$	3.64447	0.167220
$476$	−0.830101	−0.0380476
$477$	−22.9712	−1.05178
$478$	−27.4943	−1.25756
$479$	10.5081	0.480129	0.240064	−	0.970757i	$-0.422831\pi$
$479$	10.5081	0.480129	0.240064	+	0.970757i	$0.422831\pi$
$480$	4.44959	0.203095
$481$	−2.04465	−0.0932279
$482$	−0.386359	−0.0175982
$483$	−4.92882	−0.224269
$484$	−8.40081	−0.381855
$485$	7.87525	0.357597
$486$	26.7650	1.21408
$487$	−25.5707	−1.15872	−0.579359	−	0.815072i	$-0.696697\pi$
$487$	−25.5707	−1.15872	−0.579359	+	0.815072i	$0.696697\pi$
$488$	−22.4632	−1.01686
$489$	−1.13548	−0.0513484
$490$	−12.3553	−0.558158
$491$	0.767463	0.0346351	0.0173176	−	0.999850i	$-0.494487\pi$
$491$	0.767463	0.0346351	0.0173176	+	0.999850i	$0.494487\pi$
$492$	4.72701	0.213110
$493$	−0.0218805	−0.000985448 0
$494$	−0.938965	−0.0422460
$495$	4.05064	0.182063
$496$	−10.9626	−0.492236
$497$	3.45234	0.154859
$498$	−20.2407	−0.907007
$499$	−24.7625	−1.10852	−0.554260	−	0.832343i	$-0.686999\pi$
$499$	−24.7625	−1.10852	−0.554260	+	0.832343i	$0.686999\pi$
$500$	9.49542	0.424648
$501$	9.81002	0.438280
$502$	−39.8401	−1.77815
$503$	−27.1480	−1.21047	−0.605235	−	0.796047i	$-0.706921\pi$
$503$	−27.1480	−1.21047	−0.605235	+	0.796047i	$0.706921\pi$
$504$	3.93167	0.175130
$505$	16.3659	0.728271
$506$	17.5588	0.780586
$507$	9.81080	0.435713
$508$	3.05551	0.135566
$509$	26.0949	1.15664	0.578319	−	0.815811i	$-0.303709\pi$
$509$	26.0949	1.15664	0.578319	+	0.815811i	$0.303709\pi$
$510$	−1.50425	−0.0666095
$511$	−0.979960	−0.0433509
$512$	−6.60639	−0.291964
$513$	−4.17390	−0.184282
$514$	4.66587	0.205803
$515$	21.2010	0.934229
$516$	−9.07528	−0.399517
$517$	−11.6329	−0.511614
$518$	5.78472	0.254166
$519$	−8.50317	−0.373248
$520$	1.15502	0.0506512
$521$	−25.0465	−1.09731	−0.548654	−	0.836050i	$-0.684859\pi$
$521$	−25.0465	−1.09731	−0.548654	+	0.836050i	$0.684859\pi$
$522$	−0.0925416	−0.00405044
$523$	19.4187	0.849119	0.424560	−	0.905400i	$-0.360429\pi$
$523$	19.4187	0.849119	0.424560	+	0.905400i	$0.360429\pi$
$524$	−10.1856	−0.444958
$525$	−2.54083	−0.110891
$526$	−24.9456	−1.08768
$527$	2.13883	0.0931687
$528$	5.58781	0.243178
$529$	26.9811	1.17309
$530$	19.0925	0.829324
$531$	−1.54692	−0.0671307
$532$	0.851486	0.0369166
$533$	3.54977	0.153758
$534$	19.8955	0.860962
$535$	−11.7458	−0.507815
$536$	−17.6848	−0.763865
$537$	−2.11688	−0.0913503
$538$	−41.7784	−1.80120
$539$	−8.95440	−0.385693
$540$	−4.58477	−0.197297
$541$	−5.27891	−0.226958	−0.113479	−	0.993540i	$-0.536200\pi$
$541$	−5.27891	−0.226958	−0.113479	+	0.993540i	$0.536200\pi$
$542$	−44.7027	−1.92015
$543$	−10.5835	−0.454183
$544$	4.82319	0.206793
$545$	−5.23732	−0.224342
$546$	0.654621	0.0280152
$547$	24.4824	1.04679	0.523396	−	0.852089i	$-0.324665\pi$
$547$	24.4824	1.04679	0.523396	+	0.852089i	$0.324665\pi$
$548$	0.834754	0.0356589
$549$	29.7823	1.27108
$550$	9.05166	0.385964
$551$	0.0224442	0.000956154 0
$552$	9.89929	0.421342
$553$	2.00295	0.0851739
$554$	−44.3140	−1.88272
$555$	3.35998	0.142623
$556$	7.55377	0.320351
$557$	−26.3993	−1.11857	−0.559286	−	0.828975i	$-0.688925\pi$
$557$	−26.3993	−1.11857	−0.559286	+	0.828975i	$0.688925\pi$
$558$	9.04598	0.382947
$559$	−6.81514	−0.288250
$560$	−5.25052	−0.221875
$561$	−1.09019	−0.0460279
$562$	−28.0675	−1.18396
$563$	8.02352	0.338151	0.169076	−	0.985603i	$-0.445922\pi$
$563$	8.02352	0.338151	0.169076	+	0.985603i	$0.445922\pi$
$564$	5.85638	0.246598
$565$	−0.784432	−0.0330013
$566$	−12.1370	−0.510158
$567$	−3.59708	−0.151063
$568$	−6.93384	−0.290938
$569$	−14.4973	−0.607760	−0.303880	−	0.952710i	$-0.598282\pi$
$569$	−14.4973	−0.607760	−0.303880	+	0.952710i	$0.598282\pi$
$570$	1.54301	0.0646294
$571$	−23.5341	−0.984873	−0.492436	−	0.870348i	$-0.663894\pi$
$571$	−23.5341	−0.984873	−0.492436	+	0.870348i	$0.663894\pi$
$572$	−0.747493	−0.0312542
$573$	10.8480	0.453180
$574$	−10.0430	−0.419188
$575$	25.7654	1.07449
$576$	−3.61819	−0.150758
$577$	−15.5190	−0.646066	−0.323033	−	0.946388i	$-0.604703\pi$
$577$	−15.5190	−0.646066	−0.323033	+	0.946388i	$0.604703\pi$
$578$	27.5355	1.14533
$579$	7.91674	0.329009
$580$	0.0246535	0.00102368
$581$	13.7838	0.571846
$582$	−8.96445	−0.371588
$583$	13.8371	0.573073
$584$	1.96820	0.0814446
$585$	−1.53137	−0.0633143
$586$	18.5328	0.765581
$587$	18.3510	0.757426	0.378713	−	0.925514i	$-0.376367\pi$
$587$	18.3510	0.757426	0.378713	+	0.925514i	$0.376367\pi$
$588$	4.50794	0.185904
$589$	−2.19393	−0.0903992
$590$	1.28572	0.0529322
$591$	−4.53192	−0.186418
$592$	−18.6677	−0.767236
$593$	−25.2192	−1.03563	−0.517814	−	0.855493i	$-0.673254\pi$
$593$	−25.2192	−1.03563	−0.517814	+	0.855493i	$0.673254\pi$
$594$	−10.3666	−0.425346
$595$	1.02439	0.0419957
$596$	14.5655	0.596624
$597$	6.82426	0.279298
$598$	−6.63823	−0.271457
$599$	35.9084	1.46718	0.733588	−	0.679594i	$-0.237844\pi$
$599$	35.9084	1.46718	0.733588	+	0.679594i	$0.237844\pi$
$600$	5.10312	0.208334
$601$	12.7390	0.519634	0.259817	−	0.965658i	$-0.416338\pi$
$601$	12.7390	0.519634	0.259817	+	0.965658i	$0.416338\pi$
$602$	19.2814	0.785851
$603$	23.4470	0.954835
$604$	16.8380	0.685128
$605$	10.3670	0.421479
$606$	−18.6294	−0.756767
$607$	−26.4198	−1.07235	−0.536173	−	0.844108i	$-0.680130\pi$
$607$	−26.4198	−1.07235	−0.536173	+	0.844108i	$0.680130\pi$
$608$	−4.94744	−0.200645
$609$	−0.0156475	−0.000634068 0
$610$	−24.7535	−1.00224
$611$	4.39788	0.177919
$612$	−2.21044	−0.0893519
$613$	−34.5990	−1.39744	−0.698720	−	0.715395i	$-0.746247\pi$
$613$	−34.5990	−1.39744	−0.698720	+	0.715395i	$0.746247\pi$
$614$	44.5447	1.79768
$615$	−5.83336	−0.235224
$616$	−2.36830	−0.0954216
$617$	43.7136	1.75984	0.879922	−	0.475118i	$-0.157595\pi$
$617$	43.7136	1.75984	0.879922	+	0.475118i	$0.157595\pi$
$618$	−24.1333	−0.970783
$619$	26.2507	1.05510	0.527552	−	0.849523i	$-0.323110\pi$
$619$	26.2507	1.05510	0.527552	+	0.849523i	$0.323110\pi$
$620$	−2.40989	−0.0967836
$621$	−29.5084	−1.18413
$622$	−12.6517	−0.507286
$623$	−13.5487	−0.542816
$624$	−2.11251	−0.0845679
$625$	6.50455	0.260182
$626$	48.1771	1.92554
$627$	1.11828	0.0446597
$628$	2.86820	0.114454
$629$	3.64209	0.145220
$630$	4.33255	0.172613
$631$	1.39314	0.0554600	0.0277300	−	0.999615i	$-0.491172\pi$
$631$	1.39314	0.0554600	0.0277300	+	0.999615i	$0.491172\pi$
$632$	−4.02281	−0.160019
$633$	−0.772476	−0.0307032
$634$	−42.1453	−1.67381
$635$	−3.77065	−0.149634
$636$	−6.96604	−0.276221
$637$	3.38527	0.134129
$638$	0.0557438	0.00220692
$639$	9.19310	0.363674
$640$	14.5276	0.574253
$641$	41.9909	1.65854	0.829271	−	0.558847i	$-0.188756\pi$
$641$	41.9909	1.65854	0.829271	+	0.558847i	$0.188756\pi$
$642$	13.3703	0.527685
$643$	9.91518	0.391016	0.195508	−	0.980702i	$-0.437364\pi$
$643$	9.91518	0.391016	0.195508	+	0.980702i	$0.437364\pi$
$644$	6.01977	0.237212
$645$	11.1993	0.440974
$646$	1.67256	0.0658061
$647$	40.1862	1.57988	0.789942	−	0.613182i	$-0.210111\pi$
$647$	40.1862	1.57988	0.789942	+	0.613182i	$0.210111\pi$
$648$	7.22455	0.283807
$649$	0.931811	0.0365768
$650$	−3.42203	−0.134223
$651$	1.52955	0.0599477
$652$	1.38681	0.0543118
$653$	24.3435	0.952633	0.476317	−	0.879274i	$-0.341972\pi$
$653$	24.3435	0.952633	0.476317	+	0.879274i	$0.341972\pi$
$654$	5.96168	0.233120
$655$	12.5695	0.491130
$656$	32.4095	1.26538
$657$	−2.60950	−0.101806
$658$	−12.4425	−0.485059
$659$	−5.76917	−0.224735	−0.112367	−	0.993667i	$-0.535843\pi$
$659$	−5.76917	−0.224735	−0.112367	+	0.993667i	$0.535843\pi$
$660$	1.22836	0.0478138
$661$	−9.58647	−0.372870	−0.186435	−	0.982467i	$-0.559693\pi$
$661$	−9.58647	−0.372870	−0.186435	+	0.982467i	$0.559693\pi$
$662$	−43.7843	−1.70172
$663$	0.412153	0.0160067
$664$	−27.6840	−1.07435
$665$	−1.05078	−0.0407473
$666$	15.4039	0.596889
$667$	0.158674	0.00614389
$668$	−11.9814	−0.463573
$669$	5.79427	0.224020
$670$	−19.4879	−0.752884
$671$	−17.9399	−0.692560
$672$	3.44923	0.133057
$673$	6.45480	0.248814	0.124407	−	0.992231i	$-0.460297\pi$
$673$	6.45480	0.248814	0.124407	+	0.992231i	$0.460297\pi$
$674$	−44.2906	−1.70601
$675$	−15.2117	−0.585498
$676$	−11.9823	−0.460859
$677$	−1.94256	−0.0746587	−0.0373293	−	0.999303i	$-0.511885\pi$
$677$	−1.94256	−0.0746587	−0.0373293	+	0.999303i	$0.511885\pi$
$678$	0.892925	0.0342926
$679$	6.10472	0.234278
$680$	−2.05742	−0.0788986
$681$	−3.98161	−0.152576
$682$	−5.44898	−0.208652
$683$	−28.7003	−1.09819	−0.549093	−	0.835761i	$-0.685027\pi$
$683$	−28.7003	−1.09819	−0.549093	+	0.835761i	$0.685027\pi$
$684$	2.26739	0.0866958
$685$	−1.03013	−0.0393592
$686$	−20.4164	−0.779503
$687$	4.25222	0.162232
$688$	−62.2223	−2.37220
$689$	−5.23119	−0.199292
$690$	10.9086	0.415285
$691$	−41.0881	−1.56307	−0.781533	−	0.623865i	$-0.785562\pi$
$691$	−41.0881	−1.56307	−0.781533	+	0.623865i	$0.785562\pi$
$692$	10.3853	0.394789
$693$	3.13997	0.119277
$694$	−42.4187	−1.61019
$695$	−9.32172	−0.353593
$696$	0.0314271	0.00119124
$697$	−6.32314	−0.239506
$698$	39.7985	1.50640
$699$	8.18052	0.309416
$700$	3.10322	0.117291
$701$	32.2575	1.21835	0.609174	−	0.793036i	$-0.291501\pi$
$701$	32.2575	1.21835	0.609174	+	0.793036i	$0.291501\pi$
$702$	3.91915	0.147919
$703$	−3.73592	−0.140903
$704$	2.17947	0.0821419
$705$	−7.22706	−0.272187
$706$	−34.1520	−1.28533
$707$	12.6865	0.477124
$708$	−0.469105	−0.0176300
$709$	29.3153	1.10096	0.550479	−	0.834849i	$-0.314445\pi$
$709$	29.3153	1.10096	0.550479	+	0.834849i	$0.314445\pi$
$710$	−7.64083	−0.286755
$711$	5.33357	0.200025
$712$	27.2118	1.01981
$713$	−15.5105	−0.580872
$714$	−1.16606	−0.0436389
$715$	0.922443	0.0344974
$716$	2.58543	0.0966222
$717$	−12.3794	−0.462315
$718$	−15.9023	−0.593467
$719$	19.2477	0.717818	0.358909	−	0.933373i	$-0.383149\pi$
$719$	19.2477	0.717818	0.358909	+	0.933373i	$0.383149\pi$
$720$	−13.9814	−0.521056
$721$	16.4346	0.612056
$722$	−1.71565	−0.0638499
$723$	−0.173959	−0.00646961
$724$	12.9261	0.480394
$725$	0.0817972	0.00303787
$726$	−11.8009	−0.437971
$727$	−31.2342	−1.15841	−0.579207	−	0.815181i	$-0.696638\pi$
$727$	−31.2342	−1.15841	−0.579207	+	0.815181i	$0.696638\pi$
$728$	0.895351	0.0331839
$729$	0.0941854	0.00348835
$730$	2.16888	0.0802738
$731$	12.1397	0.449002
$732$	9.03151	0.333814
$733$	39.1870	1.44740	0.723702	−	0.690112i	$-0.242439\pi$
$733$	39.1870	1.44740	0.723702	+	0.690112i	$0.242439\pi$
$734$	6.39525	0.236053
$735$	−5.56302	−0.205195
$736$	−34.9771	−1.28927
$737$	−14.1237	−0.520251
$738$	−26.7432	−0.984430
$739$	−4.85238	−0.178498	−0.0892489	−	0.996009i	$-0.528447\pi$
$739$	−4.85238	−0.178498	−0.0892489	+	0.996009i	$0.528447\pi$
$740$	−4.10368	−0.150854
$741$	−0.422771	−0.0155309
$742$	14.8001	0.543328
$743$	−47.2671	−1.73406	−0.867030	−	0.498255i	$-0.833974\pi$
$743$	−47.2671	−1.73406	−0.867030	+	0.498255i	$0.833974\pi$
$744$	−3.07202	−0.112626
$745$	−17.9745	−0.658534
$746$	16.8804	0.618036
$747$	36.7042	1.34294
$748$	1.33149	0.0486843
$749$	−9.10509	−0.332693
$750$	13.3385	0.487052
$751$	−17.0864	−0.623493	−0.311747	−	0.950165i	$-0.600914\pi$
$751$	−17.0864	−0.623493	−0.311747	+	0.950165i	$0.600914\pi$
$752$	40.1527	1.46422
$753$	−17.9381	−0.653701
$754$	−0.0210743	−0.000767481 0
$755$	−20.7789	−0.756222
$756$	−3.55402	−0.129258
$757$	40.9146	1.48707	0.743533	−	0.668699i	$-0.233149\pi$
$757$	40.9146	1.48707	0.743533	+	0.668699i	$0.233149\pi$
$758$	−4.52090	−0.164206
$759$	7.90591	0.286967
$760$	2.11043	0.0765533
$761$	0.368261	0.0133495	0.00667473	−	0.999978i	$-0.497875\pi$
$761$	0.368261	0.0133495	0.00667473	+	0.999978i	$0.497875\pi$
$762$	4.29216	0.155489
$763$	−4.05986	−0.146977
$764$	−13.2491	−0.479334
$765$	2.72780	0.0986237
$766$	−51.5986	−1.86433
$767$	−0.352277	−0.0127200
$768$	−14.2109	−0.512792
$769$	39.6874	1.43116	0.715582	−	0.698528i	$-0.246161\pi$
$769$	39.6874	1.43116	0.715582	+	0.698528i	$0.246161\pi$
$770$	−2.60978	−0.0940498
$771$	2.10082	0.0756592
$772$	−9.66904	−0.347996
$773$	−17.1139	−0.615545	−0.307772	−	0.951460i	$-0.599584\pi$
$773$	−17.1139	−0.615545	−0.307772	+	0.951460i	$0.599584\pi$
$774$	51.3437	1.84551
$775$	−7.99571	−0.287214
$776$	−12.2610	−0.440145
$777$	2.60458	0.0934389
$778$	2.44813	0.0877696
$779$	6.48604	0.232387
$780$	−0.464388	−0.0166278
$781$	−5.53761	−0.198151
$782$	11.8246	0.422845
$783$	−0.0936798	−0.00334784
$784$	30.9075	1.10384
$785$	−3.53950	−0.126330
$786$	−14.3079	−0.510347
$787$	31.1845	1.11161	0.555805	−	0.831313i	$-0.312410\pi$
$787$	31.1845	1.11161	0.555805	+	0.831313i	$0.312410\pi$
$788$	5.53501	0.197177
$789$	−11.2318	−0.399863
$790$	−4.43299	−0.157719
$791$	−0.608075	−0.0216207
$792$	−6.30646	−0.224090
$793$	6.78227	0.240845
$794$	−31.8511	−1.13035
$795$	8.59643	0.304884
$796$	−8.33474	−0.295417
$797$	−16.1180	−0.570929	−0.285464	−	0.958389i	$-0.592148\pi$
$797$	−16.1180	−0.570929	−0.285464	+	0.958389i	$0.592148\pi$
$798$	1.19611	0.0423417
$799$	−7.83387	−0.277142
$800$	−18.0308	−0.637486
$801$	−36.0783	−1.27476
$802$	−22.8842	−0.808070
$803$	1.57187	0.0554701
$804$	7.11031	0.250761
$805$	−7.42870	−0.261827
$806$	2.06002	0.0725612
$807$	−18.8108	−0.662173
$808$	−25.4801	−0.896387
$809$	−14.9209	−0.524590	−0.262295	−	0.964988i	$-0.584479\pi$
$809$	−14.9209	−0.524590	−0.262295	+	0.964988i	$0.584479\pi$
$810$	7.96117	0.279727
$811$	−18.2925	−0.642338	−0.321169	−	0.947022i	$-0.604076\pi$
$811$	−18.2925	−0.642338	−0.321169	+	0.947022i	$0.604076\pi$
$812$	0.0191109	0.000670661 0
$813$	−20.1275	−0.705902
$814$	−9.27878	−0.325221
$815$	−1.71140	−0.0599476
$816$	3.76297	0.131730
$817$	−12.4524	−0.435655
$818$	−22.3830	−0.782603
$819$	−1.18708	−0.0414801
$820$	7.12452	0.248799
$821$	−13.8187	−0.482276	−0.241138	−	0.970491i	$-0.577521\pi$
$821$	−13.8187	−0.482276	−0.241138	+	0.970491i	$0.577521\pi$
$822$	1.17260	0.0408992
$823$	−20.9097	−0.728865	−0.364432	−	0.931230i	$-0.618737\pi$
$823$	−20.9097	−0.728865	−0.364432	+	0.931230i	$0.618737\pi$
$824$	−33.0080	−1.14989
$825$	4.07553	0.141892
$826$	0.996662	0.0346783
$827$	−39.6382	−1.37836	−0.689178	−	0.724592i	$-0.742028\pi$
$827$	−39.6382	−1.37836	−0.689178	+	0.724592i	$0.742028\pi$
$828$	16.0298	0.557075
$829$	30.2736	1.05145	0.525724	−	0.850655i	$-0.323795\pi$
$829$	30.2736	1.05145	0.525724	+	0.850655i	$0.323795\pi$
$830$	−30.5066	−1.05890
$831$	−19.9525	−0.692144
$832$	−0.823963	−0.0285658
$833$	−6.03011	−0.208931
$834$	10.6110	0.367428
$835$	14.7856	0.511677
$836$	−1.36580	−0.0472371
$837$	9.15724	0.316521
$838$	−58.4355	−2.01862
$839$	19.6006	0.676688	0.338344	−	0.941023i	$-0.390133\pi$
$839$	19.6006	0.676688	0.338344	+	0.941023i	$0.390133\pi$
$840$	−1.47133	−0.0507658
$841$	−28.9995	−0.999983
$842$	−6.20836	−0.213954
$843$	−12.6375	−0.435257
$844$	0.943456	0.0324751
$845$	14.7868	0.508681
$846$	−33.1327	−1.13912
$847$	8.03629	0.276130
$848$	−47.7608	−1.64011
$849$	−5.46473	−0.187549
$850$	6.09561	0.209078
$851$	−26.4119	−0.905389
$852$	2.78781	0.0955089
$853$	−22.2502	−0.761834	−0.380917	−	0.924609i	$-0.624392\pi$
$853$	−22.2502	−0.761834	−0.380917	+	0.924609i	$0.624392\pi$
$854$	−19.1884	−0.656614
$855$	−2.79807	−0.0956920
$856$	18.2871	0.625041
$857$	8.64887	0.295440	0.147720	−	0.989029i	$-0.452807\pi$
$857$	8.64887	0.295440	0.147720	+	0.989029i	$0.452807\pi$
$858$	−1.05002	−0.0358472
$859$	−4.70086	−0.160391	−0.0801957	−	0.996779i	$-0.525555\pi$
$859$	−4.70086	−0.160391	−0.0801957	+	0.996779i	$0.525555\pi$
$860$	−13.6782	−0.466423
$861$	−4.52189	−0.154106
$862$	43.2019	1.47146
$863$	−40.8727	−1.39132	−0.695661	−	0.718370i	$-0.744888\pi$
$863$	−40.8727	−1.39132	−0.695661	+	0.718370i	$0.744888\pi$
$864$	20.6502	0.702533
$865$	−12.8159	−0.435755
$866$	45.9386	1.56106
$867$	12.3979	0.421056
$868$	−1.86810	−0.0634074
$869$	−3.21276	−0.108985
$870$	0.0346315	0.00117412
$871$	5.33953	0.180923
$872$	8.15401	0.276130
$873$	16.2560	0.550183
$874$	−12.1292	−0.410276
$875$	−9.08340	−0.307075
$876$	−0.791332	−0.0267366
$877$	−0.532057	−0.0179663	−0.00898314	−	0.999960i	$-0.502859\pi$
$877$	−0.532057	−0.0179663	−0.00898314	+	0.999960i	$0.502859\pi$
$878$	−33.0285	−1.11466
$879$	8.34442	0.281450
$880$	8.42191	0.283903
$881$	40.4299	1.36212	0.681058	−	0.732229i	$-0.261520\pi$
$881$	40.4299	1.36212	0.681058	+	0.732229i	$0.261520\pi$
$882$	−25.5038	−0.858759
$883$	−9.90083	−0.333189	−0.166595	−	0.986025i	$-0.553277\pi$
$883$	−9.90083	−0.333189	−0.166595	+	0.986025i	$0.553277\pi$
$884$	−0.503380	−0.0169305
$885$	0.578898	0.0194595
$886$	−36.7130	−1.23340
$887$	−47.5619	−1.59697	−0.798485	−	0.602014i	$-0.794365\pi$
$887$	−47.5619	−1.59697	−0.798485	+	0.602014i	$0.794365\pi$
$888$	−5.23117	−0.175547
$889$	−2.92293	−0.0980319
$890$	29.9864	1.00514
$891$	5.76977	0.193295
$892$	−7.07678	−0.236948
$893$	8.03568	0.268904
$894$	20.4605	0.684301
$895$	−3.19055	−0.106648
$896$	11.2615	0.376219
$897$	−2.98888	−0.0997958
$898$	9.99223	0.333445
$899$	−0.0492409	−0.00164228
$900$	8.26344	0.275448
$901$	9.31822	0.310435
$902$	16.1092	0.536377
$903$	8.68149	0.288902
$904$	1.22129	0.0406194
$905$	−15.9514	−0.530244
$906$	23.6528	0.785811
$907$	16.7199	0.555175	0.277588	−	0.960700i	$-0.410465\pi$
$907$	16.7199	0.555175	0.277588	+	0.960700i	$0.410465\pi$
$908$	4.86290	0.161381
$909$	33.7823	1.12049
$910$	0.986642	0.0327069
$911$	−8.39091	−0.278003	−0.139002	−	0.990292i	$-0.544389\pi$
$911$	−8.39091	−0.278003	−0.139002	+	0.990292i	$0.544389\pi$
$912$	−3.85991	−0.127814
$913$	−22.1094	−0.731713
$914$	66.0872	2.18597
$915$	−11.1453	−0.368453
$916$	−5.19341	−0.171595
$917$	9.74359	0.321762
$918$	−6.98111	−0.230411
$919$	5.08203	0.167641	0.0838204	−	0.996481i	$-0.473288\pi$
$919$	5.08203	0.167641	0.0838204	+	0.996481i	$0.473288\pi$
$920$	14.9202	0.491903
$921$	20.0564	0.660879
$922$	−43.0404	−1.41746
$923$	2.09353	0.0689092
$924$	0.952197	0.0313250
$925$	−13.6155	−0.447673
$926$	−44.6197	−1.46629
$927$	43.7630	1.43737
$928$	−0.111041	−0.00364511
$929$	−40.5038	−1.32889	−0.664444	−	0.747338i	$-0.731331\pi$
$929$	−40.5038	−1.32889	−0.664444	+	0.747338i	$0.731331\pi$
$930$	−3.38524	−0.111006
$931$	6.18546	0.202720
$932$	−9.99120	−0.327273
$933$	−5.69644	−0.186493
$934$	−65.6984	−2.14972
$935$	−1.64313	−0.0537361
$936$	2.38420	0.0779299
$937$	−40.0953	−1.30986	−0.654928	−	0.755691i	$-0.727301\pi$
$937$	−40.0953	−1.30986	−0.654928	+	0.755691i	$0.727301\pi$
$938$	−15.1066	−0.493248
$939$	21.6918	0.707886
$940$	8.82670	0.287895
$941$	−16.5216	−0.538588	−0.269294	−	0.963058i	$-0.586790\pi$
$941$	−16.5216	−0.538588	−0.269294	+	0.963058i	$0.586790\pi$
$942$	4.02904	0.131273
$943$	45.8546	1.49323
$944$	−3.21629	−0.104681
$945$	4.38584	0.142671
$946$	−30.9277	−1.00554
$947$	5.48332	0.178184	0.0890920	−	0.996023i	$-0.471603\pi$
$947$	5.48332	0.178184	0.0890920	+	0.996023i	$0.471603\pi$
$948$	1.61741	0.0525310
$949$	−0.594256	−0.0192904
$950$	−6.25264	−0.202862
$951$	−18.9760	−0.615340
$952$	−1.59487	−0.0516901
$953$	−58.8965	−1.90785	−0.953923	−	0.300051i	$-0.902996\pi$
$953$	−58.8965	−1.90785	−0.953923	+	0.300051i	$0.902996\pi$
$954$	39.4106	1.27596
$955$	16.3500	0.529073
$956$	15.1194	0.488997
$957$	0.0250988	0.000811329 0
$958$	−18.0283	−0.582467
$959$	−0.798533	−0.0257860
$960$	1.35402	0.0437009
$961$	−26.1867	−0.844732
$962$	3.50790	0.113099
$963$	−24.2456	−0.781304
$964$	0.212463	0.00684298
$965$	11.9321	0.384107
$966$	8.45614	0.272072
$967$	3.77433	0.121374	0.0606871	−	0.998157i	$-0.480671\pi$
$967$	3.77433	0.121374	0.0606871	+	0.998157i	$0.480671\pi$
$968$	−16.1405	−0.518774
$969$	0.753075	0.0241922
$970$	−13.5112	−0.433817
$971$	−3.10681	−0.0997024	−0.0498512	−	0.998757i	$-0.515875\pi$
$971$	−3.10681	−0.0997024	−0.0498512	+	0.998757i	$0.515875\pi$
$972$	−14.7184	−0.472092
$973$	−7.22600	−0.231655
$974$	43.8704	1.40570
$975$	−1.54078	−0.0493444
$976$	61.9222	1.98208
$977$	20.7114	0.662618	0.331309	−	0.943522i	$-0.392510\pi$
$977$	20.7114	0.662618	0.331309	+	0.943522i	$0.392510\pi$
$978$	1.94809	0.0622932
$979$	21.7323	0.694567
$980$	6.79435	0.217037
$981$	−10.8108	−0.345164
$982$	−1.31670	−0.0420175
$983$	−22.1077	−0.705126	−0.352563	−	0.935788i	$-0.614690\pi$
$983$	−22.1077	−0.705126	−0.352563	+	0.935788i	$0.614690\pi$
$984$	9.08199	0.289523
$985$	−6.83048	−0.217637
$986$	0.0375393	0.00119549
$987$	−5.60226	−0.178322
$988$	0.516348	0.0164272
$989$	−88.0352	−2.79936
$990$	−6.94947	−0.220869
$991$	−6.05672	−0.192398	−0.0961990	−	0.995362i	$-0.530669\pi$
$991$	−6.05672	−0.192398	−0.0961990	+	0.995362i	$0.530669\pi$
$992$	10.8543	0.344625
$993$	−19.7140	−0.625604
$994$	−5.92301	−0.187866
$995$	10.2855	0.326072
$996$	11.1306	0.352686
$997$	38.6809	1.22503	0.612517	−	0.790457i	$-0.290157\pi$
$997$	38.6809	1.22503	0.612517	+	0.790457i	$0.290157\pi$
$998$	42.4837	1.34480
$999$	15.5934	0.493352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.18	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.18	✓	83	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71565 q^{2} -0.772476 q^{3} +0.943456 q^{4} -1.16427 q^{5} +1.32530 q^{6} -0.902518 q^{7} +1.81266 q^{8} -2.40328 q^{9} +O(q^{10})\)	\(q-1.71565 q^{2} -0.772476 q^{3} +0.943456 q^{4} -1.16427 q^{5} +1.32530 q^{6} -0.902518 q^{7} +1.81266 q^{8} -2.40328 q^{9} +1.99748 q^{10} +1.44765 q^{11} -0.728797 q^{12} -0.547294 q^{13} +1.54841 q^{14} +0.899371 q^{15} -4.99680 q^{16} +0.974885 q^{17} +4.12319 q^{18} -1.00000 q^{19} -1.09844 q^{20} +0.697173 q^{21} -2.48367 q^{22} -7.06973 q^{23} -1.40024 q^{24} -3.64447 q^{25} +0.938965 q^{26} +4.17390 q^{27} -0.851486 q^{28} -0.0224442 q^{29} -1.54301 q^{30} +2.19393 q^{31} +4.94744 q^{32} -1.11828 q^{33} -1.67256 q^{34} +1.05078 q^{35} -2.26739 q^{36} +3.73592 q^{37} +1.71565 q^{38} +0.422771 q^{39} -2.11043 q^{40} -6.48604 q^{41} -1.19611 q^{42} +12.4524 q^{43} +1.36580 q^{44} +2.79807 q^{45} +12.1292 q^{46} -8.03568 q^{47} +3.85991 q^{48} -6.18546 q^{49} +6.25264 q^{50} -0.753075 q^{51} -0.516348 q^{52} +9.55827 q^{53} -7.16096 q^{54} -1.68546 q^{55} -1.63596 q^{56} +0.772476 q^{57} +0.0385064 q^{58} +0.643670 q^{59} +0.848517 q^{60} -12.3924 q^{61} -3.76401 q^{62} +2.16900 q^{63} +1.50552 q^{64} +0.637199 q^{65} +1.91857 q^{66} -9.75624 q^{67} +0.919761 q^{68} +5.46119 q^{69} -1.80276 q^{70} -3.82523 q^{71} -4.35633 q^{72} +1.08581 q^{73} -6.40953 q^{74} +2.81527 q^{75} -0.943456 q^{76} -1.30653 q^{77} -0.725328 q^{78} -2.21929 q^{79} +5.81763 q^{80} +3.98561 q^{81} +11.1278 q^{82} -15.2726 q^{83} +0.657752 q^{84} -1.13503 q^{85} -21.3640 q^{86} +0.0173376 q^{87} +2.62410 q^{88} +15.0121 q^{89} -4.80051 q^{90} +0.493943 q^{91} -6.66998 q^{92} -1.69476 q^{93} +13.7864 q^{94} +1.16427 q^{95} -3.82178 q^{96} -6.76410 q^{97} +10.6121 q^{98} -3.47912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

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