LMFDB - Embedded newform 1045.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.55629$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.913732 q^{2} -2.55629 q^{3} -1.16509 q^{4} -1.00000 q^{5} -2.33576 q^{6} +3.50085 q^{7} -2.89205 q^{8} +3.53460 q^{9} +O(q^{10})$	\(q+0.913732 q^{2} -2.55629 q^{3} -1.16509 q^{4} -1.00000 q^{5} -2.33576 q^{6} +3.50085 q^{7} -2.89205 q^{8} +3.53460 q^{9} -0.913732 q^{10} +1.00000 q^{11} +2.97832 q^{12} +2.97274 q^{13} +3.19884 q^{14} +2.55629 q^{15} -0.312366 q^{16} -2.40252 q^{17} +3.22968 q^{18} +1.00000 q^{19} +1.16509 q^{20} -8.94919 q^{21} +0.913732 q^{22} -7.06629 q^{23} +7.39290 q^{24} +1.00000 q^{25} +2.71629 q^{26} -1.36660 q^{27} -4.07883 q^{28} -6.78021 q^{29} +2.33576 q^{30} -1.89901 q^{31} +5.49868 q^{32} -2.55629 q^{33} -2.19526 q^{34} -3.50085 q^{35} -4.11815 q^{36} +4.56477 q^{37} +0.913732 q^{38} -7.59919 q^{39} +2.89205 q^{40} -9.23914 q^{41} -8.17716 q^{42} -8.74630 q^{43} -1.16509 q^{44} -3.53460 q^{45} -6.45670 q^{46} -5.02260 q^{47} +0.798498 q^{48} +5.25598 q^{49} +0.913732 q^{50} +6.14154 q^{51} -3.46353 q^{52} -0.575060 q^{53} -1.24870 q^{54} -1.00000 q^{55} -10.1246 q^{56} -2.55629 q^{57} -6.19529 q^{58} -3.65915 q^{59} -2.97832 q^{60} -11.1471 q^{61} -1.73518 q^{62} +12.3741 q^{63} +5.64905 q^{64} -2.97274 q^{65} -2.33576 q^{66} -3.46923 q^{67} +2.79917 q^{68} +18.0635 q^{69} -3.19884 q^{70} -2.47964 q^{71} -10.2222 q^{72} +13.3455 q^{73} +4.17097 q^{74} -2.55629 q^{75} -1.16509 q^{76} +3.50085 q^{77} -6.94362 q^{78} +12.1367 q^{79} +0.312366 q^{80} -7.11039 q^{81} -8.44209 q^{82} -10.9897 q^{83} +10.4266 q^{84} +2.40252 q^{85} -7.99177 q^{86} +17.3321 q^{87} -2.89205 q^{88} +1.74968 q^{89} -3.22968 q^{90} +10.4071 q^{91} +8.23290 q^{92} +4.85441 q^{93} -4.58931 q^{94} -1.00000 q^{95} -14.0562 q^{96} -8.37309 q^{97} +4.80256 q^{98} +3.53460 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) $ 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9} + q^{10} + 5 q^{11} + 3 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} - 11 q^{16} - 11 q^{17} + 3 q^{18} + 5 q^{19} - q^{20} - 3 q^{21} - q^{22} + 5 q^{24} + 5 q^{25} - 8 q^{26} - 8 q^{28} - 15 q^{29} + 4 q^{30} - 9 q^{31} - 3 q^{33} - 14 q^{34} - 3 q^{35} - 7 q^{36} + 11 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 23 q^{41} - 15 q^{42} + 9 q^{43} + q^{44} + 4 q^{45} + 8 q^{46} - 6 q^{47} + 4 q^{48} - 12 q^{49} - q^{50} - 27 q^{52} - 13 q^{53} + 9 q^{54} - 5 q^{55} - 12 q^{56} - 3 q^{57} + 17 q^{58} - 21 q^{59} - 3 q^{60} - 31 q^{61} - 18 q^{62} + 10 q^{63} - q^{64} + 3 q^{65} - 4 q^{66} + 5 q^{68} - 2 q^{70} - 28 q^{71} - 20 q^{72} - 14 q^{73} + 21 q^{74} - 3 q^{75} + q^{76} + 3 q^{77} + 13 q^{78} + 3 q^{79} + 11 q^{80} - 3 q^{81} - 18 q^{82} - 33 q^{83} + 13 q^{84} + 11 q^{85} - 20 q^{86} + 22 q^{87} + 3 q^{88} - 10 q^{89} - 3 q^{90} + 14 q^{91} + 21 q^{92} + 30 q^{93} + 14 q^{94} - 5 q^{95} - 3 q^{96} - 10 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9 + q^10 + 5 * q^11 + 3 * q^12 - 3 * q^13 + 2 * q^14 + 3 * q^15 - 11 * q^16 - 11 * q^17 + 3 * q^18 + 5 * q^19 - q^20 - 3 * q^21 - q^22 + 5 * q^24 + 5 * q^25 - 8 * q^26 - 8 * q^28 - 15 * q^29 + 4 * q^30 - 9 * q^31 - 3 * q^33 - 14 * q^34 - 3 * q^35 - 7 * q^36 + 11 * q^37 - q^38 - 10 * q^39 - 3 * q^40 - 23 * q^41 - 15 * q^42 + 9 * q^43 + q^44 + 4 * q^45 + 8 * q^46 - 6 * q^47 + 4 * q^48 - 12 * q^49 - q^50 - 27 * q^52 - 13 * q^53 + 9 * q^54 - 5 * q^55 - 12 * q^56 - 3 * q^57 + 17 * q^58 - 21 * q^59 - 3 * q^60 - 31 * q^61 - 18 * q^62 + 10 * q^63 - q^64 + 3 * q^65 - 4 * q^66 + 5 * q^68 - 2 * q^70 - 28 * q^71 - 20 * q^72 - 14 * q^73 + 21 * q^74 - 3 * q^75 + q^76 + 3 * q^77 + 13 * q^78 + 3 * q^79 + 11 * q^80 - 3 * q^81 - 18 * q^82 - 33 * q^83 + 13 * q^84 + 11 * q^85 - 20 * q^86 + 22 * q^87 + 3 * q^88 - 10 * q^89 - 3 * q^90 + 14 * q^91 + 21 * q^92 + 30 * q^93 + 14 * q^94 - 5 * q^95 - 3 * q^96 - 10 * q^97 + 8 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.913732	0.646106	0.323053	−	0.946381i	$-0.395291\pi$
$2$	0.913732	0.646106	0.323053	+	0.946381i	$0.395291\pi$
$3$	−2.55629	−1.47587	−0.737936	−	0.674870i	$-0.764200\pi$
$3$	−2.55629	−1.47587	−0.737936	+	0.674870i	$0.764200\pi$
$4$	−1.16509	−0.582547
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.33576	−0.953570
$7$	3.50085	1.32320	0.661599	−	0.749858i	$-0.269878\pi$
$7$	3.50085	1.32320	0.661599	+	0.749858i	$0.269878\pi$
$8$	−2.89205	−1.02249
$9$	3.53460	1.17820
$10$	−0.913732	−0.288947
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.97832	0.859766
$13$	2.97274	0.824491	0.412245	−	0.911073i	$-0.364745\pi$
$13$	2.97274	0.824491	0.412245	+	0.911073i	$0.364745\pi$
$14$	3.19884	0.854926
$15$	2.55629	0.660030
$16$	−0.312366	−0.0780916
$17$	−2.40252	−0.582698	−0.291349	−	0.956617i	$-0.594104\pi$
$17$	−2.40252	−0.582698	−0.291349	+	0.956617i	$0.594104\pi$
$18$	3.22968	0.761242
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.16509	0.260523
$21$	−8.94919	−1.95287
$22$	0.913732	0.194808
$23$	−7.06629	−1.47342	−0.736712	−	0.676207i	$-0.763623\pi$
$23$	−7.06629	−1.47342	−0.736712	+	0.676207i	$0.763623\pi$
$24$	7.39290	1.50907
$25$	1.00000	0.200000
$26$	2.71629	0.532708
$27$	−1.36660	−0.263002
$28$	−4.07883	−0.770826
$29$	−6.78021	−1.25905	−0.629526	−	0.776979i	$-0.716751\pi$
$29$	−6.78021	−1.25905	−0.629526	+	0.776979i	$0.716751\pi$
$30$	2.33576	0.426450
$31$	−1.89901	−0.341072	−0.170536	−	0.985351i	$-0.554550\pi$
$31$	−1.89901	−0.341072	−0.170536	+	0.985351i	$0.554550\pi$
$32$	5.49868	0.972038
$33$	−2.55629	−0.444992
$34$	−2.19526	−0.376484
$35$	−3.50085	−0.591752
$36$	−4.11815	−0.686358
$37$	4.56477	0.750444	0.375222	−	0.926935i	$-0.377567\pi$
$37$	4.56477	0.750444	0.375222	+	0.926935i	$0.377567\pi$
$38$	0.913732	0.148227
$39$	−7.59919	−1.21684
$40$	2.89205	0.457273
$41$	−9.23914	−1.44291	−0.721456	−	0.692461i	$-0.756527\pi$
$41$	−9.23914	−1.44291	−0.721456	+	0.692461i	$0.756527\pi$
$42$	−8.17716	−1.26176
$43$	−8.74630	−1.33380	−0.666899	−	0.745148i	$-0.732379\pi$
$43$	−8.74630	−1.33380	−0.666899	+	0.745148i	$0.732379\pi$
$44$	−1.16509	−0.175645
$45$	−3.53460	−0.526907
$46$	−6.45670	−0.951988
$47$	−5.02260	−0.732622	−0.366311	−	0.930492i	$-0.619379\pi$
$47$	−5.02260	−0.732622	−0.366311	+	0.930492i	$0.619379\pi$
$48$	0.798498	0.115253
$49$	5.25598	0.750855
$50$	0.913732	0.129221
$51$	6.14154	0.859987
$52$	−3.46353	−0.480305
$53$	−0.575060	−0.0789906	−0.0394953	−	0.999220i	$-0.512575\pi$
$53$	−0.575060	−0.0789906	−0.0394953	+	0.999220i	$0.512575\pi$
$54$	−1.24870	−0.169927
$55$	−1.00000	−0.134840
$56$	−10.1246	−1.35296
$57$	−2.55629	−0.338588
$58$	−6.19529	−0.813481
$59$	−3.65915	−0.476381	−0.238190	−	0.971219i	$-0.576554\pi$
$59$	−3.65915	−0.476381	−0.238190	+	0.971219i	$0.576554\pi$
$60$	−2.97832	−0.384499
$61$	−11.1471	−1.42724	−0.713621	−	0.700532i	$-0.752946\pi$
$61$	−11.1471	−1.42724	−0.713621	+	0.700532i	$0.752946\pi$
$62$	−1.73518	−0.220369
$63$	12.3741	1.55899
$64$	5.64905	0.706131
$65$	−2.97274	−0.368723
$66$	−2.33576	−0.287512
$67$	−3.46923	−0.423834	−0.211917	−	0.977288i	$-0.567971\pi$
$67$	−3.46923	−0.423834	−0.211917	+	0.977288i	$0.567971\pi$
$68$	2.79917	0.339449
$69$	18.0635	2.17459
$70$	−3.19884	−0.382335
$71$	−2.47964	−0.294279	−0.147140	−	0.989116i	$-0.547007\pi$
$71$	−2.47964	−0.294279	−0.147140	+	0.989116i	$0.547007\pi$
$72$	−10.2222	−1.20470
$73$	13.3455	1.56197	0.780985	−	0.624549i	$-0.214717\pi$
$73$	13.3455	1.56197	0.780985	+	0.624549i	$0.214717\pi$
$74$	4.17097	0.484866
$75$	−2.55629	−0.295175
$76$	−1.16509	−0.133645
$77$	3.50085	0.398959
$78$	−6.94362	−0.786210
$79$	12.1367	1.36549	0.682743	−	0.730659i	$-0.260787\pi$
$79$	12.1367	1.36549	0.682743	+	0.730659i	$0.260787\pi$
$80$	0.312366	0.0349236
$81$	−7.11039	−0.790044
$82$	−8.44209	−0.932273
$83$	−10.9897	−1.20628	−0.603139	−	0.797636i	$-0.706083\pi$
$83$	−10.9897	−1.20628	−0.603139	+	0.797636i	$0.706083\pi$
$84$	10.4266	1.13764
$85$	2.40252	0.260590
$86$	−7.99177	−0.861775
$87$	17.3321	1.85820
$88$	−2.89205	−0.308293
$89$	1.74968	0.185465	0.0927327	−	0.995691i	$-0.470440\pi$
$89$	1.74968	0.185465	0.0927327	+	0.995691i	$0.470440\pi$
$90$	−3.22968	−0.340438
$91$	10.4071	1.09097
$92$	8.23290	0.858339
$93$	4.85441	0.503379
$94$	−4.58931	−0.473351
$95$	−1.00000	−0.102598
$96$	−14.0562	−1.43460
$97$	−8.37309	−0.850158	−0.425079	−	0.905156i	$-0.639754\pi$
$97$	−8.37309	−0.850158	−0.425079	+	0.905156i	$0.639754\pi$
$98$	4.80256	0.485132
$99$	3.53460	0.355241
$100$	−1.16509	−0.116509
$101$	−5.68441	−0.565620	−0.282810	−	0.959176i	$-0.591267\pi$
$101$	−5.68441	−0.565620	−0.282810	+	0.959176i	$0.591267\pi$
$102$	5.61172	0.555643
$103$	−2.15021	−0.211866	−0.105933	−	0.994373i	$-0.533783\pi$
$103$	−2.15021	−0.211866	−0.105933	+	0.994373i	$0.533783\pi$
$104$	−8.59732	−0.843036
$105$	8.94919	0.873351
$106$	−0.525451	−0.0510363
$107$	0.835402	0.0807614	0.0403807	−	0.999184i	$-0.487143\pi$
$107$	0.835402	0.0807614	0.0403807	+	0.999184i	$0.487143\pi$
$108$	1.59221	0.153211
$109$	−13.0946	−1.25423	−0.627117	−	0.778925i	$-0.715765\pi$
$109$	−13.0946	−1.25423	−0.627117	+	0.778925i	$0.715765\pi$
$110$	−0.913732	−0.0871209
$111$	−11.6689	−1.10756
$112$	−1.09355	−0.103331
$113$	16.2015	1.52411	0.762053	−	0.647515i	$-0.224191\pi$
$113$	16.2015	1.52411	0.762053	+	0.647515i	$0.224191\pi$
$114$	−2.33576	−0.218764
$115$	7.06629	0.658935
$116$	7.89958	0.733458
$117$	10.5075	0.971416
$118$	−3.34348	−0.307792
$119$	−8.41089	−0.771025
$120$	−7.39290	−0.674877
$121$	1.00000	0.0909091
$122$	−10.1855	−0.922149
$123$	23.6179	2.12955
$124$	2.21252	0.198691
$125$	−1.00000	−0.0894427
$126$	11.3066	1.00727
$127$	0.192351	0.0170683	0.00853417	−	0.999964i	$-0.497283\pi$
$127$	0.192351	0.0170683	0.00853417	+	0.999964i	$0.497283\pi$
$128$	−5.83564	−0.515802
$129$	22.3580	1.96852
$130$	−2.71629	−0.238234
$131$	−6.12245	−0.534921	−0.267460	−	0.963569i	$-0.586184\pi$
$131$	−6.12245	−0.534921	−0.267460	+	0.963569i	$0.586184\pi$
$132$	2.97832	0.259229
$133$	3.50085	0.303563
$134$	−3.16994	−0.273841
$135$	1.36660	0.117618
$136$	6.94821	0.595804
$137$	5.04153	0.430727	0.215363	−	0.976534i	$-0.430906\pi$
$137$	5.04153	0.430727	0.215363	+	0.976534i	$0.430906\pi$
$138$	16.5052	1.40501
$139$	−11.8558	−1.00560	−0.502798	−	0.864404i	$-0.667696\pi$
$139$	−11.8558	−1.00560	−0.502798	+	0.864404i	$0.667696\pi$
$140$	4.07883	0.344724
$141$	12.8392	1.08126
$142$	−2.26573	−0.190135
$143$	2.97274	0.248593
$144$	−1.10409	−0.0920076
$145$	6.78021	0.563065
$146$	12.1942	1.00920
$147$	−13.4358	−1.10817
$148$	−5.31839	−0.437169
$149$	−7.61147	−0.623556	−0.311778	−	0.950155i	$-0.600924\pi$
$149$	−7.61147	−0.623556	−0.311778	+	0.950155i	$0.600924\pi$
$150$	−2.33576	−0.190714
$151$	13.2179	1.07566	0.537828	−	0.843054i	$-0.319245\pi$
$151$	13.2179	1.07566	0.537828	+	0.843054i	$0.319245\pi$
$152$	−2.89205	−0.234576
$153$	−8.49196	−0.686535
$154$	3.19884	0.257770
$155$	1.89901	0.152532
$156$	8.85377	0.708869
$157$	11.4670	0.915170	0.457585	−	0.889166i	$-0.348715\pi$
$157$	11.4670	0.915170	0.457585	+	0.889166i	$0.348715\pi$
$158$	11.0897	0.882248
$159$	1.47002	0.116580
$160$	−5.49868	−0.434708
$161$	−24.7381	−1.94963
$162$	−6.49699	−0.510452
$163$	−20.1666	−1.57957	−0.789784	−	0.613385i	$-0.789808\pi$
$163$	−20.1666	−1.57957	−0.789784	+	0.613385i	$0.789808\pi$
$164$	10.7645	0.840564
$165$	2.55629	0.199007
$166$	−10.0416	−0.779383
$167$	3.77184	0.291874	0.145937	−	0.989294i	$-0.453380\pi$
$167$	3.77184	0.291874	0.145937	+	0.989294i	$0.453380\pi$
$168$	25.8815	1.99680
$169$	−4.16279	−0.320215
$170$	2.19526	0.168369
$171$	3.53460	0.270298
$172$	10.1903	0.777000
$173$	20.3744	1.54904	0.774518	−	0.632552i	$-0.217993\pi$
$173$	20.3744	1.54904	0.774518	+	0.632552i	$0.217993\pi$
$174$	15.8369	1.20059
$175$	3.50085	0.264640
$176$	−0.312366	−0.0235455
$177$	9.35383	0.703077
$178$	1.59874	0.119830
$179$	−19.7401	−1.47544	−0.737722	−	0.675105i	$-0.764098\pi$
$179$	−19.7401	−1.47544	−0.737722	+	0.675105i	$0.764098\pi$
$180$	4.11815	0.306948
$181$	−1.71764	−0.127671	−0.0638356	−	0.997960i	$-0.520333\pi$
$181$	−1.71764	−0.127671	−0.0638356	+	0.997960i	$0.520333\pi$
$182$	9.50934	0.704879
$183$	28.4952	2.10643
$184$	20.4361	1.50657
$185$	−4.56477	−0.335609
$186$	4.43563	0.325236
$187$	−2.40252	−0.175690
$188$	5.85181	0.426787
$189$	−4.78426	−0.348003
$190$	−0.913732	−0.0662891
$191$	−0.535136	−0.0387210	−0.0193605	−	0.999813i	$-0.506163\pi$
$191$	−0.535136	−0.0387210	−0.0193605	+	0.999813i	$0.506163\pi$
$192$	−14.4406	−1.04216
$193$	8.02309	0.577514	0.288757	−	0.957402i	$-0.406758\pi$
$193$	8.02309	0.577514	0.288757	+	0.957402i	$0.406758\pi$
$194$	−7.65076	−0.549292
$195$	7.59919	0.544189
$196$	−6.12372	−0.437408
$197$	4.30091	0.306427	0.153214	−	0.988193i	$-0.451038\pi$
$197$	4.30091	0.306427	0.153214	+	0.988193i	$0.451038\pi$
$198$	3.22968	0.229523
$199$	−9.66547	−0.685167	−0.342583	−	0.939487i	$-0.611302\pi$
$199$	−9.66547	−0.685167	−0.342583	+	0.939487i	$0.611302\pi$
$200$	−2.89205	−0.204499
$201$	8.86834	0.625525
$202$	−5.19403	−0.365451
$203$	−23.7365	−1.66598
$204$	−7.15547	−0.500983
$205$	9.23914	0.645289
$206$	−1.96471	−0.136888
$207$	−24.9765	−1.73599
$208$	−0.928585	−0.0643858
$209$	1.00000	0.0691714
$210$	8.17716	0.564277
$211$	−2.31547	−0.159403	−0.0797016	−	0.996819i	$-0.525397\pi$
$211$	−2.31547	−0.159403	−0.0797016	+	0.996819i	$0.525397\pi$
$212$	0.669999	0.0460157
$213$	6.33867	0.434319
$214$	0.763334	0.0521804
$215$	8.74630	0.596492
$216$	3.95226	0.268917
$217$	−6.64815	−0.451306
$218$	−11.9649	−0.810368
$219$	−34.1149	−2.30527
$220$	1.16509	0.0785506
$221$	−7.14209	−0.480429
$222$	−10.6622	−0.715601
$223$	−27.3769	−1.83329	−0.916647	−	0.399699i	$-0.869115\pi$
$223$	−27.3769	−1.83329	−0.916647	+	0.399699i	$0.869115\pi$
$224$	19.2501	1.28620
$225$	3.53460	0.235640
$226$	14.8038	0.984734
$227$	−1.67215	−0.110985	−0.0554923	−	0.998459i	$-0.517673\pi$
$227$	−1.67215	−0.110985	−0.0554923	+	0.998459i	$0.517673\pi$
$228$	2.97832	0.197244
$229$	−13.1463	−0.868735	−0.434368	−	0.900736i	$-0.643028\pi$
$229$	−13.1463	−0.868735	−0.434368	+	0.900736i	$0.643028\pi$
$230$	6.45670	0.425742
$231$	−8.94919	−0.588813
$232$	19.6087	1.28737
$233$	23.1988	1.51980	0.759901	−	0.650039i	$-0.225247\pi$
$233$	23.1988	1.51980	0.759901	+	0.650039i	$0.225247\pi$
$234$	9.60101	0.627637
$235$	5.02260	0.327639
$236$	4.26325	0.277514
$237$	−31.0249	−2.01528
$238$	−7.68529	−0.498164
$239$	−4.90363	−0.317189	−0.158595	−	0.987344i	$-0.550696\pi$
$239$	−4.90363	−0.317189	−0.158595	+	0.987344i	$0.550696\pi$
$240$	−0.798498	−0.0515428
$241$	7.00317	0.451114	0.225557	−	0.974230i	$-0.427580\pi$
$241$	7.00317	0.451114	0.225557	+	0.974230i	$0.427580\pi$
$242$	0.913732	0.0587369
$243$	22.2760	1.42901
$244$	12.9874	0.831435
$245$	−5.25598	−0.335792
$246$	21.5804	1.37592
$247$	2.97274	0.189151
$248$	5.49202	0.348744
$249$	28.0928	1.78031
$250$	−0.913732	−0.0577895
$251$	−18.4937	−1.16731	−0.583657	−	0.812001i	$-0.698379\pi$
$251$	−18.4937	−1.16731	−0.583657	+	0.812001i	$0.698379\pi$
$252$	−14.4170	−0.908187
$253$	−7.06629	−0.444254
$254$	0.175757	0.0110280
$255$	−6.14154	−0.384598
$256$	−16.6303	−1.03939
$257$	13.2282	0.825150	0.412575	−	0.910924i	$-0.364629\pi$
$257$	13.2282	0.825150	0.412575	+	0.910924i	$0.364629\pi$
$258$	20.4293	1.27187
$259$	15.9806	0.992986
$260$	3.46353	0.214799
$261$	−23.9653	−1.48342
$262$	−5.59427	−0.345615
$263$	28.7080	1.77021	0.885106	−	0.465389i	$-0.154085\pi$
$263$	28.7080	1.77021	0.885106	+	0.465389i	$0.154085\pi$
$264$	7.39290	0.455002
$265$	0.575060	0.0353257
$266$	3.19884	0.196134
$267$	−4.47268	−0.273723
$268$	4.04198	0.246903
$269$	−16.8208	−1.02558	−0.512792	−	0.858513i	$-0.671389\pi$
$269$	−16.8208	−1.02558	−0.512792	+	0.858513i	$0.671389\pi$
$270$	1.24870	0.0759936
$271$	−28.7042	−1.74365	−0.871827	−	0.489814i	$-0.837065\pi$
$271$	−28.7042	−1.74365	−0.871827	+	0.489814i	$0.837065\pi$
$272$	0.750467	0.0455038
$273$	−26.6036	−1.61013
$274$	4.60660	0.278295
$275$	1.00000	0.0603023
$276$	−21.0456	−1.26680
$277$	31.0460	1.86537	0.932686	−	0.360689i	$-0.117458\pi$
$277$	31.0460	1.86537	0.932686	+	0.360689i	$0.117458\pi$
$278$	−10.8330	−0.649722
$279$	−6.71224	−0.401851
$280$	10.1246	0.605063
$281$	19.3745	1.15579	0.577893	−	0.816113i	$-0.303875\pi$
$281$	19.3745	1.15579	0.577893	+	0.816113i	$0.303875\pi$
$282$	11.7316	0.698607
$283$	−13.8591	−0.823839	−0.411920	−	0.911220i	$-0.635142\pi$
$283$	−13.8591	−0.823839	−0.411920	+	0.911220i	$0.635142\pi$
$284$	2.88901	0.171431
$285$	2.55629	0.151421
$286$	2.71629	0.160618
$287$	−32.3449	−1.90926
$288$	19.4356	1.14526
$289$	−11.2279	−0.660464
$290$	6.19529	0.363800
$291$	21.4040	1.25473
$292$	−15.5487	−0.909922
$293$	−8.67643	−0.506883	−0.253441	−	0.967351i	$-0.581562\pi$
$293$	−8.67643	−0.506883	−0.253441	+	0.967351i	$0.581562\pi$
$294$	−12.2767	−0.715993
$295$	3.65915	0.213044
$296$	−13.2015	−0.767323
$297$	−1.36660	−0.0792980
$298$	−6.95484	−0.402883
$299$	−21.0063	−1.21482
$300$	2.97832	0.171953
$301$	−30.6195	−1.76488
$302$	12.0776	0.694988
$303$	14.5310	0.834784
$304$	−0.312366	−0.0179154
$305$	11.1471	0.638282
$306$	−7.75938	−0.443574
$307$	11.7172	0.668737	0.334368	−	0.942443i	$-0.391477\pi$
$307$	11.7172	0.668737	0.334368	+	0.942443i	$0.391477\pi$
$308$	−4.07883	−0.232413
$309$	5.49655	0.312688
$310$	1.73518	0.0985519
$311$	−1.50817	−0.0855203	−0.0427601	−	0.999085i	$-0.513615\pi$
$311$	−1.50817	−0.0855203	−0.0427601	+	0.999085i	$0.513615\pi$
$312$	21.9772	1.24421
$313$	−25.3038	−1.43026	−0.715128	−	0.698993i	$-0.753632\pi$
$313$	−25.3038	−1.43026	−0.715128	+	0.698993i	$0.753632\pi$
$314$	10.4778	0.591297
$315$	−12.3741	−0.697203
$316$	−14.1404	−0.795460
$317$	16.7125	0.938667	0.469333	−	0.883021i	$-0.344494\pi$
$317$	16.7125	0.938667	0.469333	+	0.883021i	$0.344494\pi$
$318$	1.34320	0.0753231
$319$	−6.78021	−0.379619
$320$	−5.64905	−0.315791
$321$	−2.13553	−0.119194
$322$	−22.6040	−1.25967
$323$	−2.40252	−0.133680
$324$	8.28428	0.460238
$325$	2.97274	0.164898
$326$	−18.4268	−1.02057
$327$	33.4735	1.85109
$328$	26.7200	1.47537
$329$	−17.5834	−0.969405
$330$	2.33576	0.128579
$331$	16.7507	0.920700	0.460350	−	0.887738i	$-0.347724\pi$
$331$	16.7507	0.920700	0.460350	+	0.887738i	$0.347724\pi$
$332$	12.8040	0.702713
$333$	16.1346	0.884173
$334$	3.44645	0.188582
$335$	3.46923	0.189544
$336$	2.79543	0.152503
$337$	13.5980	0.740729	0.370365	−	0.928886i	$-0.379233\pi$
$337$	13.5980	0.740729	0.370365	+	0.928886i	$0.379233\pi$
$338$	−3.80368	−0.206893
$339$	−41.4156	−2.24939
$340$	−2.79917	−0.151806
$341$	−1.89901	−0.102837
$342$	3.22968	0.174641
$343$	−6.10555	−0.329669
$344$	25.2947	1.36380
$345$	−18.0635	−0.972505
$346$	18.6167	1.00084
$347$	−34.8212	−1.86930	−0.934650	−	0.355570i	$-0.884287\pi$
$347$	−34.8212	−1.86930	−0.934650	+	0.355570i	$0.884287\pi$
$348$	−20.1936	−1.08249
$349$	21.7524	1.16438	0.582191	−	0.813052i	$-0.302196\pi$
$349$	21.7524	1.16438	0.582191	+	0.813052i	$0.302196\pi$
$350$	3.19884	0.170985
$351$	−4.06254	−0.216842
$352$	5.49868	0.293080
$353$	19.3292	1.02879	0.514394	−	0.857554i	$-0.328017\pi$
$353$	19.3292	1.02879	0.514394	+	0.857554i	$0.328017\pi$
$354$	8.54689	0.454262
$355$	2.47964	0.131606
$356$	−2.03854	−0.108042
$357$	21.5006	1.13793
$358$	−18.0371	−0.953292
$359$	−30.1461	−1.59105	−0.795526	−	0.605920i	$-0.792805\pi$
$359$	−30.1461	−1.59105	−0.795526	+	0.605920i	$0.792805\pi$
$360$	10.2222	0.538759
$361$	1.00000	0.0526316
$362$	−1.56946	−0.0824891
$363$	−2.55629	−0.134170
$364$	−12.1253	−0.635539
$365$	−13.3455	−0.698534
$366$	26.0370	1.36097
$367$	25.7004	1.34155	0.670775	−	0.741661i	$-0.265961\pi$
$367$	25.7004	1.34155	0.670775	+	0.741661i	$0.265961\pi$
$368$	2.20727	0.115062
$369$	−32.6567	−1.70004
$370$	−4.17097	−0.216839
$371$	−2.01320	−0.104520
$372$	−5.65585	−0.293242
$373$	−13.1534	−0.681059	−0.340529	−	0.940234i	$-0.610606\pi$
$373$	−13.1534	−0.681059	−0.340529	+	0.940234i	$0.610606\pi$
$374$	−2.19526	−0.113514
$375$	2.55629	0.132006
$376$	14.5256	0.749101
$377$	−20.1558	−1.03808
$378$	−4.37153	−0.224847
$379$	−22.1651	−1.13854	−0.569271	−	0.822150i	$-0.692775\pi$
$379$	−22.1651	−1.13854	−0.569271	+	0.822150i	$0.692775\pi$
$380$	1.16509	0.0597681
$381$	−0.491703	−0.0251907
$382$	−0.488970	−0.0250179
$383$	−10.8730	−0.555586	−0.277793	−	0.960641i	$-0.589603\pi$
$383$	−10.8730	−0.555586	−0.277793	+	0.960641i	$0.589603\pi$
$384$	14.9176	0.761259
$385$	−3.50085	−0.178420
$386$	7.33095	0.373135
$387$	−30.9147	−1.57148
$388$	9.75544	0.495257
$389$	12.6160	0.639656	0.319828	−	0.947476i	$-0.396375\pi$
$389$	12.6160	0.639656	0.319828	+	0.947476i	$0.396375\pi$
$390$	6.94362	0.351604
$391$	16.9769	0.858560
$392$	−15.2006	−0.767744
$393$	15.6507	0.789475
$394$	3.92988	0.197985
$395$	−12.1367	−0.610664
$396$	−4.11815	−0.206945
$397$	−33.9542	−1.70411	−0.852057	−	0.523449i	$-0.824645\pi$
$397$	−33.9542	−1.70411	−0.852057	+	0.523449i	$0.824645\pi$
$398$	−8.83164	−0.442690
$399$	−8.94919	−0.448020
$400$	−0.312366	−0.0156183
$401$	31.4825	1.57216	0.786082	−	0.618123i	$-0.212106\pi$
$401$	31.4825	1.57216	0.786082	+	0.618123i	$0.212106\pi$
$402$	8.10329	0.404155
$403$	−5.64527	−0.281211
$404$	6.62288	0.329501
$405$	7.11039	0.353318
$406$	−21.6888	−1.07640
$407$	4.56477	0.226267
$408$	−17.7616	−0.879331
$409$	16.2792	0.804953	0.402476	−	0.915430i	$-0.368150\pi$
$409$	16.2792	0.804953	0.402476	+	0.915430i	$0.368150\pi$
$410$	8.44209	0.416925
$411$	−12.8876	−0.635698
$412$	2.50520	0.123422
$413$	−12.8101	−0.630346
$414$	−22.8219	−1.12163
$415$	10.9897	0.539464
$416$	16.3462	0.801436
$417$	30.3069	1.48413
$418$	0.913732	0.0446921
$419$	31.5123	1.53948	0.769739	−	0.638359i	$-0.220386\pi$
$419$	31.5123	1.53948	0.769739	+	0.638359i	$0.220386\pi$
$420$	−10.4266	−0.508768
$421$	0.911055	0.0444021	0.0222011	−	0.999754i	$-0.492933\pi$
$421$	0.911055	0.0444021	0.0222011	+	0.999754i	$0.492933\pi$
$422$	−2.11571	−0.102991
$423$	−17.7529	−0.863176
$424$	1.66310	0.0807673
$425$	−2.40252	−0.116540
$426$	5.79185	0.280616
$427$	−39.0244	−1.88852
$428$	−0.973323	−0.0470473
$429$	−7.59919	−0.366892
$430$	7.99177	0.385397
$431$	30.1001	1.44987	0.724935	−	0.688817i	$-0.241870\pi$
$431$	30.1001	1.44987	0.724935	+	0.688817i	$0.241870\pi$
$432$	0.426879	0.0205382
$433$	−21.9412	−1.05443	−0.527215	−	0.849732i	$-0.676764\pi$
$433$	−21.9412	−1.05443	−0.527215	+	0.849732i	$0.676764\pi$
$434$	−6.07463	−0.291591
$435$	−17.3321	−0.831013
$436$	15.2564	0.730651
$437$	−7.06629	−0.338027
$438$	−31.1718	−1.48945
$439$	−14.6200	−0.697774	−0.348887	−	0.937165i	$-0.613440\pi$
$439$	−14.6200	−0.697774	−0.348887	+	0.937165i	$0.613440\pi$
$440$	2.89205	0.137873
$441$	18.5778	0.884658
$442$	−6.52595	−0.310408
$443$	7.83707	0.372350	0.186175	−	0.982517i	$-0.440391\pi$
$443$	7.83707	0.372350	0.186175	+	0.982517i	$0.440391\pi$
$444$	13.5953	0.645206
$445$	−1.74968	−0.0829427
$446$	−25.0151	−1.18450
$447$	19.4571	0.920289
$448$	19.7765	0.934351
$449$	30.2188	1.42611	0.713056	−	0.701107i	$-0.247311\pi$
$449$	30.2188	1.42611	0.713056	+	0.701107i	$0.247311\pi$
$450$	3.22968	0.152248
$451$	−9.23914	−0.435054
$452$	−18.8762	−0.887864
$453$	−33.7887	−1.58753
$454$	−1.52790	−0.0717078
$455$	−10.4071	−0.487894
$456$	7.39290	0.346204
$457$	−8.18577	−0.382914	−0.191457	−	0.981501i	$-0.561321\pi$
$457$	−8.18577	−0.382914	−0.191457	+	0.981501i	$0.561321\pi$
$458$	−12.0122	−0.561295
$459$	3.28328	0.153250
$460$	−8.23290	−0.383861
$461$	−26.9586	−1.25559	−0.627793	−	0.778381i	$-0.716041\pi$
$461$	−26.9586	−1.25559	−0.627793	+	0.778381i	$0.716041\pi$
$462$	−8.17716	−0.380436
$463$	37.1787	1.72784	0.863921	−	0.503627i	$-0.168001\pi$
$463$	37.1787	1.72784	0.863921	+	0.503627i	$0.168001\pi$
$464$	2.11791	0.0983214
$465$	−4.85441	−0.225118
$466$	21.1975	0.981953
$467$	−17.0308	−0.788092	−0.394046	−	0.919091i	$-0.628925\pi$
$467$	−17.0308	−0.788092	−0.394046	+	0.919091i	$0.628925\pi$
$468$	−12.2422	−0.565895
$469$	−12.1453	−0.560816
$470$	4.58931	0.211689
$471$	−29.3131	−1.35067
$472$	10.5824	0.487096
$473$	−8.74630	−0.402155
$474$	−28.3484	−1.30209
$475$	1.00000	0.0458831
$476$	9.79948	0.449158
$477$	−2.03261	−0.0930668
$478$	−4.48060	−0.204938
$479$	13.5101	0.617290	0.308645	−	0.951177i	$-0.400124\pi$
$479$	13.5101	0.617290	0.308645	+	0.951177i	$0.400124\pi$
$480$	14.0562	0.641574
$481$	13.5699	0.618734
$482$	6.39902	0.291467
$483$	63.2376	2.87741
$484$	−1.16509	−0.0529588
$485$	8.37309	0.380202
$486$	20.3543	0.923289
$487$	4.56091	0.206675	0.103337	−	0.994646i	$-0.467048\pi$
$487$	4.56091	0.206675	0.103337	+	0.994646i	$0.467048\pi$
$488$	32.2380	1.45934
$489$	51.5516	2.33124
$490$	−4.80256	−0.216957
$491$	10.5513	0.476173	0.238087	−	0.971244i	$-0.423480\pi$
$491$	10.5513	0.476173	0.238087	+	0.971244i	$0.423480\pi$
$492$	−27.5171	−1.24057
$493$	16.2896	0.733647
$494$	2.71629	0.122212
$495$	−3.53460	−0.158869
$496$	0.593186	0.0266349
$497$	−8.68086	−0.389390
$498$	25.6693	1.15027
$499$	24.1305	1.08023	0.540115	−	0.841591i	$-0.318381\pi$
$499$	24.1305	1.08023	0.540115	+	0.841591i	$0.318381\pi$
$500$	1.16509	0.0521046
$501$	−9.64191	−0.430769
$502$	−16.8983	−0.754208
$503$	6.74351	0.300678	0.150339	−	0.988634i	$-0.451963\pi$
$503$	6.74351	0.300678	0.150339	+	0.988634i	$0.451963\pi$
$504$	−35.7866	−1.59406
$505$	5.68441	0.252953
$506$	−6.45670	−0.287035
$507$	10.6413	0.472597
$508$	−0.224107	−0.00994312
$509$	2.78459	0.123425	0.0617123	−	0.998094i	$-0.480344\pi$
$509$	2.78459	0.123425	0.0617123	+	0.998094i	$0.480344\pi$
$510$	−5.61172	−0.248491
$511$	46.7206	2.06680
$512$	−3.52436	−0.155756
$513$	−1.36660	−0.0603367
$514$	12.0870	0.533135
$515$	2.15021	0.0947496
$516$	−26.0492	−1.14675
$517$	−5.02260	−0.220894
$518$	14.6020	0.641574
$519$	−52.0828	−2.28618
$520$	8.59732	0.377017
$521$	26.3641	1.15503	0.577517	−	0.816379i	$-0.304022\pi$
$521$	26.3641	1.15503	0.577517	+	0.816379i	$0.304022\pi$
$522$	−21.8979	−0.958444
$523$	25.4570	1.11316	0.556579	−	0.830795i	$-0.312114\pi$
$523$	25.4570	1.11316	0.556579	+	0.830795i	$0.312114\pi$
$524$	7.13323	0.311617
$525$	−8.94919	−0.390575
$526$	26.2314	1.14374
$527$	4.56241	0.198742
$528$	0.798498	0.0347502
$529$	26.9325	1.17098
$530$	0.525451	0.0228241
$531$	−12.9336	−0.561272
$532$	−4.07883	−0.176840
$533$	−27.4656	−1.18967
$534$	−4.08683	−0.176854
$535$	−0.835402	−0.0361176
$536$	10.0332	0.433367
$537$	50.4613	2.17757
$538$	−15.3697	−0.662636
$539$	5.25598	0.226391
$540$	−1.59221	−0.0685180
$541$	7.40334	0.318294	0.159147	−	0.987255i	$-0.449126\pi$
$541$	7.40334	0.318294	0.159147	+	0.987255i	$0.449126\pi$
$542$	−26.2279	−1.12659
$543$	4.39078	0.188426
$544$	−13.2107	−0.566404
$545$	13.0946	0.560911
$546$	−24.3086	−1.04031
$547$	16.7446	0.715946	0.357973	−	0.933732i	$-0.383468\pi$
$547$	16.7446	0.715946	0.357973	+	0.933732i	$0.383468\pi$
$548$	−5.87385	−0.250919
$549$	−39.4006	−1.68158
$550$	0.913732	0.0389617
$551$	−6.78021	−0.288846
$552$	−52.2404	−2.22350
$553$	42.4888	1.80681
$554$	28.3677	1.20523
$555$	11.6689	0.495316
$556$	13.8131	0.585808
$557$	−1.28778	−0.0545650	−0.0272825	−	0.999628i	$-0.508685\pi$
$557$	−1.28778	−0.0545650	−0.0272825	+	0.999628i	$0.508685\pi$
$558$	−6.13319	−0.259638
$559$	−26.0005	−1.09970
$560$	1.09355	0.0462109
$561$	6.14154	0.259296
$562$	17.7031	0.746760
$563$	−24.0602	−1.01402	−0.507008	−	0.861942i	$-0.669248\pi$
$563$	−24.0602	−1.01402	−0.507008	+	0.861942i	$0.669248\pi$
$564$	−14.9589	−0.629883
$565$	−16.2015	−0.681601
$566$	−12.6635	−0.532287
$567$	−24.8925	−1.04538
$568$	7.17124	0.300898
$569$	9.58205	0.401701	0.200850	−	0.979622i	$-0.435630\pi$
$569$	9.58205	0.401701	0.200850	+	0.979622i	$0.435630\pi$
$570$	2.33576	0.0978342
$571$	43.4416	1.81798	0.908988	−	0.416823i	$-0.136857\pi$
$571$	43.4416	1.81798	0.908988	+	0.416823i	$0.136857\pi$
$572$	−3.46353	−0.144817
$573$	1.36796	0.0571473
$574$	−29.5545	−1.23358
$575$	−7.06629	−0.294685
$576$	19.9671	0.831964
$577$	22.0679	0.918697	0.459349	−	0.888256i	$-0.348083\pi$
$577$	22.0679	0.918697	0.459349	+	0.888256i	$0.348083\pi$
$578$	−10.2593	−0.426729
$579$	−20.5093	−0.852338
$580$	−7.89958	−0.328012
$581$	−38.4734	−1.59614
$582$	19.5575	0.810686
$583$	−0.575060	−0.0238166
$584$	−38.5958	−1.59710
$585$	−10.5075	−0.434430
$586$	−7.92793	−0.327500
$587$	−0.547893	−0.0226140	−0.0113070	−	0.999936i	$-0.503599\pi$
$587$	−0.547893	−0.0226140	−0.0113070	+	0.999936i	$0.503599\pi$
$588$	15.6540	0.645559
$589$	−1.89901	−0.0782473
$590$	3.34348	0.137649
$591$	−10.9944	−0.452248
$592$	−1.42588	−0.0586033
$593$	4.08623	0.167801	0.0839007	−	0.996474i	$-0.473262\pi$
$593$	4.08623	0.167801	0.0839007	+	0.996474i	$0.473262\pi$
$594$	−1.24870	−0.0512349
$595$	8.41089	0.344813
$596$	8.86808	0.363251
$597$	24.7077	1.01122
$598$	−19.1941	−0.784905
$599$	−34.3363	−1.40294	−0.701471	−	0.712698i	$-0.747473\pi$
$599$	−34.3363	−1.40294	−0.701471	+	0.712698i	$0.747473\pi$
$600$	7.39290	0.301814
$601$	−32.0784	−1.30850	−0.654252	−	0.756276i	$-0.727017\pi$
$601$	−32.0784	−1.30850	−0.654252	+	0.756276i	$0.727017\pi$
$602$	−27.9780	−1.14030
$603$	−12.2623	−0.499361
$604$	−15.4001	−0.626621
$605$	−1.00000	−0.0406558
$606$	13.2774	0.539359
$607$	−30.3231	−1.23078	−0.615389	−	0.788224i	$-0.711001\pi$
$607$	−30.3231	−1.23078	−0.615389	+	0.788224i	$0.711001\pi$
$608$	5.49868	0.223001
$609$	60.6773	2.45877
$610$	10.1855	0.412398
$611$	−14.9309	−0.604040
$612$	9.89394	0.399939
$613$	45.2390	1.82718	0.913592	−	0.406631i	$-0.133297\pi$
$613$	45.2390	1.82718	0.913592	+	0.406631i	$0.133297\pi$
$614$	10.7064	0.432075
$615$	−23.6179	−0.952365
$616$	−10.1246	−0.407933
$617$	−28.2331	−1.13662	−0.568311	−	0.822814i	$-0.692403\pi$
$617$	−28.2331	−1.13662	−0.568311	+	0.822814i	$0.692403\pi$
$618$	5.02237	0.202030
$619$	1.92228	0.0772630	0.0386315	−	0.999254i	$-0.487700\pi$
$619$	1.92228	0.0772630	0.0386315	+	0.999254i	$0.487700\pi$
$620$	−2.21252	−0.0888571
$621$	9.65677	0.387513
$622$	−1.37806	−0.0552552
$623$	6.12537	0.245408
$624$	2.37373	0.0950252
$625$	1.00000	0.0400000
$626$	−23.1209	−0.924097
$627$	−2.55629	−0.102088
$628$	−13.3602	−0.533130
$629$	−10.9670	−0.437282
$630$	−11.3066	−0.450467
$631$	−23.0795	−0.918780	−0.459390	−	0.888235i	$-0.651932\pi$
$631$	−23.0795	−0.918780	−0.459390	+	0.888235i	$0.651932\pi$
$632$	−35.0999	−1.39620
$633$	5.91899	0.235259
$634$	15.2707	0.606478
$635$	−0.192351	−0.00763320
$636$	−1.71271	−0.0679134
$637$	15.6247	0.619073
$638$	−6.19529	−0.245274
$639$	−8.76454	−0.346720
$640$	5.83564	0.230674
$641$	31.7543	1.25422	0.627109	−	0.778932i	$-0.284238\pi$
$641$	31.7543	1.25422	0.627109	+	0.778932i	$0.284238\pi$
$642$	−1.95130	−0.0770117
$643$	−23.4624	−0.925268	−0.462634	−	0.886549i	$-0.653096\pi$
$643$	−23.4624	−0.925268	−0.462634	+	0.886549i	$0.653096\pi$
$644$	28.8222	1.13575
$645$	−22.3580	−0.880347
$646$	−2.19526	−0.0863714
$647$	9.97853	0.392296	0.196148	−	0.980574i	$-0.437157\pi$
$647$	9.97853	0.392296	0.196148	+	0.980574i	$0.437157\pi$
$648$	20.5636	0.807814
$649$	−3.65915	−0.143634
$650$	2.71629	0.106542
$651$	16.9946	0.666070
$652$	23.4960	0.920173
$653$	−40.9000	−1.60054	−0.800270	−	0.599639i	$-0.795311\pi$
$653$	−40.9000	−1.60054	−0.800270	+	0.599639i	$0.795311\pi$
$654$	30.5858	1.19600
$655$	6.12245	0.239224
$656$	2.88600	0.112679
$657$	47.1710	1.84031
$658$	−16.0665	−0.626338
$659$	−21.2802	−0.828960	−0.414480	−	0.910059i	$-0.636036\pi$
$659$	−21.2802	−0.828960	−0.414480	+	0.910059i	$0.636036\pi$
$660$	−2.97832	−0.115931
$661$	41.3961	1.61012	0.805061	−	0.593191i	$-0.202132\pi$
$661$	41.3961	1.61012	0.805061	+	0.593191i	$0.202132\pi$
$662$	15.3056	0.594870
$663$	18.2572	0.709052
$664$	31.7828	1.23341
$665$	−3.50085	−0.135757
$666$	14.7427	0.571269
$667$	47.9109	1.85512
$668$	−4.39455	−0.170030
$669$	69.9832	2.70571
$670$	3.16994	0.122466
$671$	−11.1471	−0.430329
$672$	−49.2087	−1.89827
$673$	11.4758	0.442359	0.221179	−	0.975233i	$-0.429009\pi$
$673$	11.4758	0.442359	0.221179	+	0.975233i	$0.429009\pi$
$674$	12.4249	0.478590
$675$	−1.36660	−0.0526003
$676$	4.85005	0.186540
$677$	−20.6749	−0.794601	−0.397300	−	0.917689i	$-0.630053\pi$
$677$	−20.6749	−0.794601	−0.397300	+	0.917689i	$0.630053\pi$
$678$	−37.8428	−1.45334
$679$	−29.3130	−1.12493
$680$	−6.94821	−0.266452
$681$	4.27450	0.163799
$682$	−1.73518	−0.0664436
$683$	6.82376	0.261104	0.130552	−	0.991441i	$-0.458325\pi$
$683$	6.82376	0.261104	0.130552	+	0.991441i	$0.458325\pi$
$684$	−4.11815	−0.157461
$685$	−5.04153	−0.192627
$686$	−5.57883	−0.213001
$687$	33.6058	1.28214
$688$	2.73205	0.104158
$689$	−1.70951	−0.0651270
$690$	−16.5052	−0.628341
$691$	−16.1784	−0.615456	−0.307728	−	0.951474i	$-0.599569\pi$
$691$	−16.1784	−0.615456	−0.307728	+	0.951474i	$0.599569\pi$
$692$	−23.7381	−0.902386
$693$	12.3741	0.470054
$694$	−31.8172	−1.20777
$695$	11.8558	0.449717
$696$	−50.1254	−1.90000
$697$	22.1972	0.840781
$698$	19.8759	0.752314
$699$	−59.3027	−2.24303
$700$	−4.07883	−0.154165
$701$	−41.1507	−1.55424	−0.777119	−	0.629353i	$-0.783320\pi$
$701$	−41.1507	−1.55424	−0.777119	+	0.629353i	$0.783320\pi$
$702$	−3.71207	−0.140103
$703$	4.56477	0.172164
$704$	5.64905	0.212906
$705$	−12.8392	−0.483553
$706$	17.6617	0.664706
$707$	−19.9003	−0.748428
$708$	−10.8981	−0.409576
$709$	−8.49548	−0.319054	−0.159527	−	0.987194i	$-0.550997\pi$
$709$	−8.49548	−0.319054	−0.159527	+	0.987194i	$0.550997\pi$
$710$	2.26573	0.0850312
$711$	42.8984	1.60882
$712$	−5.06015	−0.189637
$713$	13.4190	0.502544
$714$	19.6458	0.735226
$715$	−2.97274	−0.111174
$716$	22.9991	0.859515
$717$	12.5351	0.468131
$718$	−27.5455	−1.02799
$719$	−41.1753	−1.53558	−0.767789	−	0.640702i	$-0.778643\pi$
$719$	−41.1753	−1.53558	−0.767789	+	0.640702i	$0.778643\pi$
$720$	1.10409	0.0411470
$721$	−7.52757	−0.280341
$722$	0.913732	0.0340056
$723$	−17.9021	−0.665786
$724$	2.00121	0.0743745
$725$	−6.78021	−0.251811
$726$	−2.33576	−0.0866882
$727$	47.1635	1.74920	0.874599	−	0.484847i	$-0.161125\pi$
$727$	47.1635	1.74920	0.874599	+	0.484847i	$0.161125\pi$
$728$	−30.0980	−1.11550
$729$	−35.6127	−1.31899
$730$	−12.1942	−0.451327
$731$	21.0132	0.777201
$732$	−33.1996	−1.22709
$733$	27.9759	1.03331	0.516657	−	0.856193i	$-0.327176\pi$
$733$	27.9759	1.03331	0.516657	+	0.856193i	$0.327176\pi$
$734$	23.4833	0.866784
$735$	13.4358	0.495587
$736$	−38.8552	−1.43222
$737$	−3.46923	−0.127791
$738$	−29.8394	−1.09841
$739$	29.9803	1.10284	0.551422	−	0.834226i	$-0.314085\pi$
$739$	29.9803	1.10284	0.551422	+	0.834226i	$0.314085\pi$
$740$	5.31839	0.195508
$741$	−7.59919	−0.279163
$742$	−1.83953	−0.0675311
$743$	−49.0599	−1.79983	−0.899916	−	0.436063i	$-0.856372\pi$
$743$	−49.0599	−1.79983	−0.899916	+	0.436063i	$0.856372\pi$
$744$	−14.0392	−0.514701
$745$	7.61147	0.278863
$746$	−12.0187	−0.440036
$747$	−38.8443	−1.42124
$748$	2.79917	0.102348
$749$	2.92462	0.106863
$750$	2.33576	0.0852899
$751$	−45.0129	−1.64254	−0.821272	−	0.570537i	$-0.806735\pi$
$751$	−45.0129	−1.64254	−0.821272	+	0.570537i	$0.806735\pi$
$752$	1.56889	0.0572116
$753$	47.2753	1.72281
$754$	−18.4170	−0.670708
$755$	−13.2179	−0.481048
$756$	5.57411	0.202728
$757$	47.5976	1.72996	0.864981	−	0.501804i	$-0.167330\pi$
$757$	47.5976	1.72996	0.864981	+	0.501804i	$0.167330\pi$
$758$	−20.2529	−0.735619
$759$	18.0635	0.655662
$760$	2.89205	0.104906
$761$	−53.3503	−1.93395	−0.966973	−	0.254877i	$-0.917965\pi$
$761$	−53.3503	−1.93395	−0.966973	+	0.254877i	$0.917965\pi$
$762$	−0.449285	−0.0162759
$763$	−45.8423	−1.65960
$764$	0.623483	0.0225568
$765$	8.49196	0.307028
$766$	−9.93503	−0.358967
$767$	−10.8777	−0.392771
$768$	42.5118	1.53401
$769$	15.2089	0.548446	0.274223	−	0.961666i	$-0.411579\pi$
$769$	15.2089	0.548446	0.274223	+	0.961666i	$0.411579\pi$
$770$	−3.19884	−0.115278
$771$	−33.8150	−1.21782
$772$	−9.34765	−0.336429
$773$	−38.2122	−1.37440	−0.687198	−	0.726470i	$-0.741159\pi$
$773$	−38.2122	−1.37440	−0.687198	+	0.726470i	$0.741159\pi$
$774$	−28.2477	−1.01534
$775$	−1.89901	−0.0682144
$776$	24.2154	0.869281
$777$	−40.8510	−1.46552
$778$	11.5276	0.413286
$779$	−9.23914	−0.331026
$780$	−8.85377	−0.317016
$781$	−2.47964	−0.0887285
$782$	15.5124	0.554721
$783$	9.26580	0.331133
$784$	−1.64179	−0.0586354
$785$	−11.4670	−0.409277
$786$	14.3006	0.510084
$787$	−30.1184	−1.07360	−0.536802	−	0.843708i	$-0.680368\pi$
$787$	−30.1184	−1.07360	−0.536802	+	0.843708i	$0.680368\pi$
$788$	−5.01097	−0.178508
$789$	−73.3860	−2.61261
$790$	−11.0897	−0.394553
$791$	56.7190	2.01670
$792$	−10.2222	−0.363231
$793$	−33.1375	−1.17675
$794$	−31.0251	−1.10104
$795$	−1.47002	−0.0521362
$796$	11.2612	0.399142
$797$	−45.5833	−1.61464	−0.807322	−	0.590111i	$-0.799084\pi$
$797$	−45.5833	−1.61464	−0.807322	+	0.590111i	$0.799084\pi$
$798$	−8.17716	−0.289468
$799$	12.0669	0.426897
$800$	5.49868	0.194408
$801$	6.18442	0.218516
$802$	28.7666	1.01578
$803$	13.3455	0.470952
$804$	−10.3325	−0.364398
$805$	24.7381	0.871902
$806$	−5.15826	−0.181692
$807$	42.9989	1.51363
$808$	16.4396	0.578343
$809$	−20.4193	−0.717905	−0.358953	−	0.933356i	$-0.616866\pi$
$809$	−20.4193	−0.717905	−0.358953	+	0.933356i	$0.616866\pi$
$810$	6.49699	0.228281
$811$	−16.2242	−0.569710	−0.284855	−	0.958571i	$-0.591946\pi$
$811$	−16.2242	−0.569710	−0.284855	+	0.958571i	$0.591946\pi$
$812$	27.6553	0.970510
$813$	73.3761	2.57341
$814$	4.17097	0.146193
$815$	20.1666	0.706405
$816$	−1.91841	−0.0671578
$817$	−8.74630	−0.305994
$818$	14.8748	0.520085
$819$	36.7851	1.28538
$820$	−10.7645	−0.375912
$821$	45.1035	1.57412	0.787062	−	0.616873i	$-0.211601\pi$
$821$	45.1035	1.57412	0.787062	+	0.616873i	$0.211601\pi$
$822$	−11.7758	−0.410728
$823$	33.5853	1.17071	0.585356	−	0.810777i	$-0.300955\pi$
$823$	33.5853	1.17071	0.585356	+	0.810777i	$0.300955\pi$
$824$	6.21851	0.216632
$825$	−2.55629	−0.0889985
$826$	−11.7050	−0.407270
$827$	−6.21222	−0.216020	−0.108010	−	0.994150i	$-0.534448\pi$
$827$	−6.21222	−0.216020	−0.108010	+	0.994150i	$0.534448\pi$
$828$	29.1000	1.01130
$829$	44.3020	1.53867	0.769336	−	0.638845i	$-0.220587\pi$
$829$	44.3020	1.53867	0.769336	+	0.638845i	$0.220587\pi$
$830$	10.0416	0.348551
$831$	−79.3624	−2.75305
$832$	16.7932	0.582198
$833$	−12.6276	−0.437521
$834$	27.6923	0.958907
$835$	−3.77184	−0.130530
$836$	−1.16509	−0.0402956
$837$	2.59518	0.0897025
$838$	28.7938	0.994665
$839$	40.1131	1.38486	0.692429	−	0.721486i	$-0.256541\pi$
$839$	40.1131	1.38486	0.692429	+	0.721486i	$0.256541\pi$
$840$	−25.8815	−0.892996
$841$	16.9712	0.585213
$842$	0.832460	0.0286885
$843$	−49.5268	−1.70579
$844$	2.69774	0.0928599
$845$	4.16279	0.143204
$846$	−16.2214	−0.557703
$847$	3.50085	0.120291
$848$	0.179629	0.00616850
$849$	35.4279	1.21588
$850$	−2.19526	−0.0752969
$851$	−32.2560	−1.10572
$852$	−7.38515	−0.253011
$853$	−30.9036	−1.05812	−0.529060	−	0.848584i	$-0.677455\pi$
$853$	−30.9036	−1.05812	−0.529060	+	0.848584i	$0.677455\pi$
$854$	−35.6578	−1.22019
$855$	−3.53460	−0.120881
$856$	−2.41602	−0.0825780
$857$	−12.8388	−0.438566	−0.219283	−	0.975661i	$-0.570372\pi$
$857$	−12.8388	−0.438566	−0.219283	+	0.975661i	$0.570372\pi$
$858$	−6.94362	−0.237051
$859$	18.2071	0.621220	0.310610	−	0.950538i	$-0.399467\pi$
$859$	18.2071	0.621220	0.310610	+	0.950538i	$0.399467\pi$
$860$	−10.1903	−0.347485
$861$	82.6828	2.81782
$862$	27.5034	0.936770
$863$	−10.5637	−0.359593	−0.179797	−	0.983704i	$-0.557544\pi$
$863$	−10.5637	−0.359593	−0.179797	+	0.983704i	$0.557544\pi$
$864$	−7.51447	−0.255647
$865$	−20.3744	−0.692750
$866$	−20.0484	−0.681273
$867$	28.7017	0.974760
$868$	7.74573	0.262907
$869$	12.1367	0.411709
$870$	−15.8369	−0.536922
$871$	−10.3131	−0.349447
$872$	37.8702	1.28245
$873$	−29.5955	−1.00166
$874$	−6.45670	−0.218401
$875$	−3.50085	−0.118350
$876$	39.7471	1.34293
$877$	−21.0971	−0.712398	−0.356199	−	0.934410i	$-0.615927\pi$
$877$	−21.0971	−0.712398	−0.356199	+	0.934410i	$0.615927\pi$
$878$	−13.3587	−0.450836
$879$	22.1795	0.748094
$880$	0.312366	0.0105299
$881$	18.3281	0.617489	0.308744	−	0.951145i	$-0.400091\pi$
$881$	18.3281	0.617489	0.308744	+	0.951145i	$0.400091\pi$
$882$	16.9751	0.571582
$883$	−57.1617	−1.92365	−0.961823	−	0.273672i	$-0.911762\pi$
$883$	−57.1617	−1.92365	−0.961823	+	0.273672i	$0.911762\pi$
$884$	8.32121	0.279872
$885$	−9.35383	−0.314426
$886$	7.16098	0.240578
$887$	54.5774	1.83253	0.916264	−	0.400574i	$-0.131189\pi$
$887$	54.5774	1.83253	0.916264	+	0.400574i	$0.131189\pi$
$888$	33.7469	1.13247
$889$	0.673391	0.0225848
$890$	−1.59874	−0.0535898
$891$	−7.11039	−0.238207
$892$	31.8967	1.06798
$893$	−5.02260	−0.168075
$894$	17.7786	0.594604
$895$	19.7401	0.659838
$896$	−20.4297	−0.682509
$897$	53.6981	1.79293
$898$	27.6118	0.921419
$899$	12.8757	0.429428
$900$	−4.11815	−0.137272
$901$	1.38160	0.0460276
$902$	−8.44209	−0.281091
$903$	78.2723	2.60474
$904$	−46.8554	−1.55839
$905$	1.71764	0.0570963
$906$	−30.8738	−1.02571
$907$	50.9611	1.69213	0.846067	−	0.533076i	$-0.178964\pi$
$907$	50.9611	1.69213	0.846067	+	0.533076i	$0.178964\pi$
$908$	1.94822	0.0646538
$909$	−20.0921	−0.666414
$910$	−9.50934	−0.315231
$911$	8.68178	0.287640	0.143820	−	0.989604i	$-0.454061\pi$
$911$	8.68178	0.287640	0.143820	+	0.989604i	$0.454061\pi$
$912$	0.798498	0.0264409
$913$	−10.9897	−0.363706
$914$	−7.47959	−0.247403
$915$	−28.4952	−0.942023
$916$	15.3167	0.506079
$917$	−21.4338	−0.707806
$918$	3.00004	0.0990160
$919$	−36.9561	−1.21907	−0.609534	−	0.792760i	$-0.708644\pi$
$919$	−36.9561	−1.21907	−0.609534	+	0.792760i	$0.708644\pi$
$920$	−20.4361	−0.673757
$921$	−29.9525	−0.986970
$922$	−24.6329	−0.811241
$923$	−7.37134	−0.242630
$924$	10.4266	0.343012
$925$	4.56477	0.150089
$926$	33.9714	1.11637
$927$	−7.60014	−0.249621
$928$	−37.2821	−1.22385
$929$	3.03822	0.0996809	0.0498405	−	0.998757i	$-0.484129\pi$
$929$	3.03822	0.0996809	0.0498405	+	0.998757i	$0.484129\pi$
$930$	−4.43563	−0.145450
$931$	5.25598	0.172258
$932$	−27.0288	−0.885356
$933$	3.85531	0.126217
$934$	−15.5616	−0.509191
$935$	2.40252	0.0785709
$936$	−30.3881	−0.993266
$937$	19.5909	0.640007	0.320003	−	0.947416i	$-0.396316\pi$
$937$	19.5909	0.640007	0.320003	+	0.947416i	$0.396316\pi$
$938$	−11.0975	−0.362347
$939$	64.6838	2.11088
$940$	−5.85181	−0.190865
$941$	18.9384	0.617373	0.308687	−	0.951164i	$-0.400111\pi$
$941$	18.9384	0.617373	0.308687	+	0.951164i	$0.400111\pi$
$942$	−26.7843	−0.872679
$943$	65.2865	2.12602
$944$	1.14299	0.0372013
$945$	4.78426	0.155632
$946$	−7.99177	−0.259835
$947$	27.3288	0.888068	0.444034	−	0.896010i	$-0.353547\pi$
$947$	27.3288	0.888068	0.444034	+	0.896010i	$0.353547\pi$
$948$	36.1469	1.17400
$949$	39.6727	1.28783
$950$	0.913732	0.0296454
$951$	−42.7219	−1.38535
$952$	24.3247	0.788367
$953$	31.2114	1.01104	0.505518	−	0.862816i	$-0.331302\pi$
$953$	31.2114	1.01104	0.505518	+	0.862816i	$0.331302\pi$
$954$	−1.85726	−0.0601310
$955$	0.535136	0.0173166
$956$	5.71319	0.184778
$957$	17.3321	0.560269
$958$	12.3446	0.398835
$959$	17.6496	0.569937
$960$	14.4406	0.466068
$961$	−27.3938	−0.883670
$962$	12.3992	0.399768
$963$	2.95282	0.0951531
$964$	−8.15935	−0.262795
$965$	−8.02309	−0.258272
$966$	57.7822	1.85911
$967$	9.58738	0.308309	0.154155	−	0.988047i	$-0.450735\pi$
$967$	9.58738	0.308309	0.154155	+	0.988047i	$0.450735\pi$
$968$	−2.89205	−0.0929539
$969$	6.14154	0.197295
$970$	7.65076	0.245651
$971$	34.2274	1.09841	0.549205	−	0.835688i	$-0.314931\pi$
$971$	34.2274	1.09841	0.549205	+	0.835688i	$0.314931\pi$
$972$	−25.9536	−0.832463
$973$	−41.5055	−1.33060
$974$	4.16745	0.133534
$975$	−7.59919	−0.243369
$976$	3.48198	0.111456
$977$	11.8385	0.378748	0.189374	−	0.981905i	$-0.439354\pi$
$977$	11.8385	0.378748	0.189374	+	0.981905i	$0.439354\pi$
$978$	47.1043	1.50623
$979$	1.74968	0.0559200
$980$	6.12372	0.195615
$981$	−46.2842	−1.47774
$982$	9.64106	0.307658
$983$	19.1020	0.609258	0.304629	−	0.952471i	$-0.401467\pi$
$983$	19.1020	0.609258	0.304629	+	0.952471i	$0.401467\pi$
$984$	−68.3040	−2.17745
$985$	−4.30091	−0.137039
$986$	14.8843	0.474014
$987$	44.9482	1.43072
$988$	−3.46353	−0.110189
$989$	61.8039	1.96525
$990$	−3.22968	−0.102646
$991$	−7.06509	−0.224430	−0.112215	−	0.993684i	$-0.535795\pi$
$991$	−7.06509	−0.224430	−0.112215	+	0.993684i	$0.535795\pi$
$992$	−10.4420	−0.331535
$993$	−42.8195	−1.35884
$994$	−7.93198	−0.251587
$995$	9.66547	0.306416
$996$	−32.7308	−1.03712
$997$	−18.1117	−0.573603	−0.286802	−	0.957990i	$-0.592592\pi$
$997$	−18.1117	−0.573603	−0.286802	+	0.957990i	$0.592592\pi$
$998$	22.0488	0.697943
$999$	−6.23820	−0.197368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.e.1.4	✓	5
3.2	odd	2		9405.2.a.u.1.2		5
5.4	even	2		5225.2.a.i.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.e.1.4	✓	5	1.1	even	1	trivial
5225.2.a.i.1.2		5	5.4	even	2
9405.2.a.u.1.2		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.913732 q^{2} -2.55629 q^{3} -1.16509 q^{4} -1.00000 q^{5} -2.33576 q^{6} +3.50085 q^{7} -2.89205 q^{8} +3.53460 q^{9} +O(q^{10})\)	\(q+0.913732 q^{2} -2.55629 q^{3} -1.16509 q^{4} -1.00000 q^{5} -2.33576 q^{6} +3.50085 q^{7} -2.89205 q^{8} +3.53460 q^{9} -0.913732 q^{10} +1.00000 q^{11} +2.97832 q^{12} +2.97274 q^{13} +3.19884 q^{14} +2.55629 q^{15} -0.312366 q^{16} -2.40252 q^{17} +3.22968 q^{18} +1.00000 q^{19} +1.16509 q^{20} -8.94919 q^{21} +0.913732 q^{22} -7.06629 q^{23} +7.39290 q^{24} +1.00000 q^{25} +2.71629 q^{26} -1.36660 q^{27} -4.07883 q^{28} -6.78021 q^{29} +2.33576 q^{30} -1.89901 q^{31} +5.49868 q^{32} -2.55629 q^{33} -2.19526 q^{34} -3.50085 q^{35} -4.11815 q^{36} +4.56477 q^{37} +0.913732 q^{38} -7.59919 q^{39} +2.89205 q^{40} -9.23914 q^{41} -8.17716 q^{42} -8.74630 q^{43} -1.16509 q^{44} -3.53460 q^{45} -6.45670 q^{46} -5.02260 q^{47} +0.798498 q^{48} +5.25598 q^{49} +0.913732 q^{50} +6.14154 q^{51} -3.46353 q^{52} -0.575060 q^{53} -1.24870 q^{54} -1.00000 q^{55} -10.1246 q^{56} -2.55629 q^{57} -6.19529 q^{58} -3.65915 q^{59} -2.97832 q^{60} -11.1471 q^{61} -1.73518 q^{62} +12.3741 q^{63} +5.64905 q^{64} -2.97274 q^{65} -2.33576 q^{66} -3.46923 q^{67} +2.79917 q^{68} +18.0635 q^{69} -3.19884 q^{70} -2.47964 q^{71} -10.2222 q^{72} +13.3455 q^{73} +4.17097 q^{74} -2.55629 q^{75} -1.16509 q^{76} +3.50085 q^{77} -6.94362 q^{78} +12.1367 q^{79} +0.312366 q^{80} -7.11039 q^{81} -8.44209 q^{82} -10.9897 q^{83} +10.4266 q^{84} +2.40252 q^{85} -7.99177 q^{86} +17.3321 q^{87} -2.89205 q^{88} +1.74968 q^{89} -3.22968 q^{90} +10.4071 q^{91} +8.23290 q^{92} +4.85441 q^{93} -4.58931 q^{94} -1.00000 q^{95} -14.0562 q^{96} -8.37309 q^{97} +4.80256 q^{98} +3.53460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9} + q^{10} + 5 q^{11} + 3 q^{12} - 3 q^{13} + 2 q^{14} + 3 q^{15} - 11 q^{16} - 11 q^{17} + 3 q^{18} + 5 q^{19} - q^{20} - 3 q^{21} - q^{22} + 5 q^{24} + 5 q^{25} - 8 q^{26} - 8 q^{28} - 15 q^{29} + 4 q^{30} - 9 q^{31} - 3 q^{33} - 14 q^{34} - 3 q^{35} - 7 q^{36} + 11 q^{37} - q^{38} - 10 q^{39} - 3 q^{40} - 23 q^{41} - 15 q^{42} + 9 q^{43} + q^{44} + 4 q^{45} + 8 q^{46} - 6 q^{47} + 4 q^{48} - 12 q^{49} - q^{50} - 27 q^{52} - 13 q^{53} + 9 q^{54} - 5 q^{55} - 12 q^{56} - 3 q^{57} + 17 q^{58} - 21 q^{59} - 3 q^{60} - 31 q^{61} - 18 q^{62} + 10 q^{63} - q^{64} + 3 q^{65} - 4 q^{66} + 5 q^{68} - 2 q^{70} - 28 q^{71} - 20 q^{72} - 14 q^{73} + 21 q^{74} - 3 q^{75} + q^{76} + 3 q^{77} + 13 q^{78} + 3 q^{79} + 11 q^{80} - 3 q^{81} - 18 q^{82} - 33 q^{83} + 13 q^{84} + 11 q^{85} - 20 q^{86} + 22 q^{87} + 3 q^{88} - 10 q^{89} - 3 q^{90} + 14 q^{91} + 21 q^{92} + 30 q^{93} + 14 q^{94} - 5 q^{95} - 3 q^{96} - 10 q^{97} + 8 q^{98} - 4 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + q^4 - 5 * q^5 - 4 * q^6 + 3 * q^7 + 3 * q^8 - 4 * q^9 + q^10 + 5 * q^11 + 3 * q^12 - 3 * q^13 + 2 * q^14 + 3 * q^15 - 11 * q^16 - 11 * q^17 + 3 * q^18 + 5 * q^19 - q^20 - 3 * q^21 - q^22 + 5 * q^24 + 5 * q^25 - 8 * q^26 - 8 * q^28 - 15 * q^29 + 4 * q^30 - 9 * q^31 - 3 * q^33 - 14 * q^34 - 3 * q^35 - 7 * q^36 + 11 * q^37 - q^38 - 10 * q^39 - 3 * q^40 - 23 * q^41 - 15 * q^42 + 9 * q^43 + q^44 + 4 * q^45 + 8 * q^46 - 6 * q^47 + 4 * q^48 - 12 * q^49 - q^50 - 27 * q^52 - 13 * q^53 + 9 * q^54 - 5 * q^55 - 12 * q^56 - 3 * q^57 + 17 * q^58 - 21 * q^59 - 3 * q^60 - 31 * q^61 - 18 * q^62 + 10 * q^63 - q^64 + 3 * q^65 - 4 * q^66 + 5 * q^68 - 2 * q^70 - 28 * q^71 - 20 * q^72 - 14 * q^73 + 21 * q^74 - 3 * q^75 + q^76 + 3 * q^77 + 13 * q^78 + 3 * q^79 + 11 * q^80 - 3 * q^81 - 18 * q^82 - 33 * q^83 + 13 * q^84 + 11 * q^85 - 20 * q^86 + 22 * q^87 + 3 * q^88 - 10 * q^89 - 3 * q^90 + 14 * q^91 + 21 * q^92 + 30 * q^93 + 14 * q^94 - 5 * q^95 - 3 * q^96 - 10 * q^97 + 8 * q^98 - 4 * q^99

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