LMFDB - Embedded newform 1045.2.a.e.1.3

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			1.3
Root			$1.33419$ 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)

$f(q)$	$=$	$q-0.584664 q^{2} +0.334185 q^{3} -1.65817 q^{4} -1.00000 q^{5} -0.195386 q^{6} +1.85355 q^{7} +2.13880 q^{8} -2.88832 q^{9} +O(q^{10})$	\(q-0.584664 q^{2} +0.334185 q^{3} -1.65817 q^{4} -1.00000 q^{5} -0.195386 q^{6} +1.85355 q^{7} +2.13880 q^{8} -2.88832 q^{9} +0.584664 q^{10} +1.00000 q^{11} -0.554135 q^{12} +2.56681 q^{13} -1.08371 q^{14} -0.334185 q^{15} +2.06586 q^{16} -7.56490 q^{17} +1.68870 q^{18} +1.00000 q^{19} +1.65817 q^{20} +0.619430 q^{21} -0.584664 q^{22} -1.73764 q^{23} +0.714755 q^{24} +1.00000 q^{25} -1.50072 q^{26} -1.96779 q^{27} -3.07350 q^{28} +9.32369 q^{29} +0.195386 q^{30} -4.31802 q^{31} -5.48543 q^{32} +0.334185 q^{33} +4.42292 q^{34} -1.85355 q^{35} +4.78932 q^{36} -8.96941 q^{37} -0.584664 q^{38} +0.857791 q^{39} -2.13880 q^{40} -10.6138 q^{41} -0.362159 q^{42} +11.5068 q^{43} -1.65817 q^{44} +2.88832 q^{45} +1.01593 q^{46} -6.50003 q^{47} +0.690379 q^{48} -3.56434 q^{49} -0.584664 q^{50} -2.52808 q^{51} -4.25621 q^{52} -8.73423 q^{53} +1.15050 q^{54} -1.00000 q^{55} +3.96438 q^{56} +0.334185 q^{57} -5.45123 q^{58} -3.97991 q^{59} +0.554135 q^{60} +0.649027 q^{61} +2.52459 q^{62} -5.35366 q^{63} -0.924581 q^{64} -2.56681 q^{65} -0.195386 q^{66} -0.319933 q^{67} +12.5439 q^{68} -0.580692 q^{69} +1.08371 q^{70} -9.93130 q^{71} -6.17754 q^{72} -15.3646 q^{73} +5.24409 q^{74} +0.334185 q^{75} -1.65817 q^{76} +1.85355 q^{77} -0.501520 q^{78} -10.2604 q^{79} -2.06586 q^{80} +8.00735 q^{81} +6.20553 q^{82} +10.7036 q^{83} -1.02712 q^{84} +7.56490 q^{85} -6.72762 q^{86} +3.11584 q^{87} +2.13880 q^{88} -13.3892 q^{89} -1.68870 q^{90} +4.75773 q^{91} +2.88129 q^{92} -1.44302 q^{93} +3.80033 q^{94} -1.00000 q^{95} -1.83315 q^{96} -3.79273 q^{97} +2.08394 q^{98} -2.88832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.584664	−0.413420	−0.206710	−	0.978402i	$-0.566276\pi$
$2$	−0.584664	−0.413420	−0.206710	+	0.978402i	$0.566276\pi$
$3$	0.334185	0.192942	0.0964709	−	0.995336i	$-0.469245\pi$
$3$	0.334185	0.192942	0.0964709	+	0.995336i	$0.469245\pi$
$4$	−1.65817	−0.829084
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.195386	−0.0797660
$7$	1.85355	0.700578	0.350289	−	0.936642i	$-0.386083\pi$
$7$	1.85355	0.700578	0.350289	+	0.936642i	$0.386083\pi$
$8$	2.13880	0.756180
$9$	−2.88832	−0.962773
$10$	0.584664	0.184887
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.554135	−0.159965
$13$	2.56681	0.711906	0.355953	−	0.934504i	$-0.384156\pi$
$13$	2.56681	0.711906	0.355953	+	0.934504i	$0.384156\pi$
$14$	−1.08371	−0.289633
$15$	−0.334185	−0.0862862
$16$	2.06586	0.516464
$17$	−7.56490	−1.83476	−0.917379	−	0.398015i	$-0.869699\pi$
$17$	−7.56490	−1.83476	−0.917379	+	0.398015i	$0.869699\pi$
$18$	1.68870	0.398030
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.65817	0.370778
$21$	0.619430	0.135171
$22$	−0.584664	−0.124651
$23$	−1.73764	−0.362322	−0.181161	−	0.983453i	$-0.557986\pi$
$23$	−1.73764	−0.362322	−0.181161	+	0.983453i	$0.557986\pi$
$24$	0.714755	0.145899
$25$	1.00000	0.200000
$26$	−1.50072	−0.294316
$27$	−1.96779	−0.378701
$28$	−3.07350	−0.580838
$29$	9.32369	1.73137	0.865683	−	0.500593i	$-0.166885\pi$
$29$	9.32369	1.73137	0.865683	+	0.500593i	$0.166885\pi$
$30$	0.195386	0.0356724
$31$	−4.31802	−0.775539	−0.387769	−	0.921756i	$-0.626754\pi$
$31$	−4.31802	−0.775539	−0.387769	+	0.921756i	$0.626754\pi$
$32$	−5.48543	−0.969696
$33$	0.334185	0.0581742
$34$	4.42292	0.758525
$35$	−1.85355	−0.313308
$36$	4.78932	0.798220
$37$	−8.96941	−1.47456	−0.737281	−	0.675586i	$-0.763891\pi$
$37$	−8.96941	−1.47456	−0.737281	+	0.675586i	$0.763891\pi$
$38$	−0.584664	−0.0948450
$39$	0.857791	0.137357
$40$	−2.13880	−0.338174
$41$	−10.6138	−1.65760	−0.828802	−	0.559543i	$-0.810977\pi$
$41$	−10.6138	−1.65760	−0.828802	+	0.559543i	$0.810977\pi$
$42$	−0.362159	−0.0558823
$43$	11.5068	1.75477	0.877387	−	0.479784i	$-0.159285\pi$
$43$	11.5068	1.75477	0.877387	+	0.479784i	$0.159285\pi$
$44$	−1.65817	−0.249978
$45$	2.88832	0.430565
$46$	1.01593	0.149791
$47$	−6.50003	−0.948126	−0.474063	−	0.880491i	$-0.657213\pi$
$47$	−6.50003	−0.948126	−0.474063	+	0.880491i	$0.657213\pi$
$48$	0.690379	0.0996476
$49$	−3.56434	−0.509191
$50$	−0.584664	−0.0826840
$51$	−2.52808	−0.354002
$52$	−4.25621	−0.590230
$53$	−8.73423	−1.19974	−0.599869	−	0.800098i	$-0.704781\pi$
$53$	−8.73423	−1.19974	−0.599869	+	0.800098i	$0.704781\pi$
$54$	1.15050	0.156563
$55$	−1.00000	−0.134840
$56$	3.96438	0.529763
$57$	0.334185	0.0442639
$58$	−5.45123	−0.715781
$59$	−3.97991	−0.518140	−0.259070	−	0.965859i	$-0.583416\pi$
$59$	−3.97991	−0.518140	−0.259070	+	0.965859i	$0.583416\pi$
$60$	0.554135	0.0715385
$61$	0.649027	0.0830994	0.0415497	−	0.999136i	$-0.486771\pi$
$61$	0.649027	0.0830994	0.0415497	+	0.999136i	$0.486771\pi$
$62$	2.52459	0.320623
$63$	−5.35366	−0.674497
$64$	−0.924581	−0.115573
$65$	−2.56681	−0.318374
$66$	−0.195386	−0.0240504
$67$	−0.319933	−0.0390861	−0.0195430	−	0.999809i	$-0.506221\pi$
$67$	−0.319933	−0.0390861	−0.0195430	+	0.999809i	$0.506221\pi$
$68$	12.5439	1.52117
$69$	−0.580692	−0.0699072
$70$	1.08371	0.129528
$71$	−9.93130	−1.17863	−0.589314	−	0.807904i	$-0.700602\pi$
$71$	−9.93130	−1.17863	−0.589314	+	0.807904i	$0.700602\pi$
$72$	−6.17754	−0.728030
$73$	−15.3646	−1.79829	−0.899146	−	0.437649i	$-0.855811\pi$
$73$	−15.3646	−1.79829	−0.899146	+	0.437649i	$0.855811\pi$
$74$	5.24409	0.609613
$75$	0.334185	0.0385884
$76$	−1.65817	−0.190205
$77$	1.85355	0.211232
$78$	−0.501520	−0.0567859
$79$	−10.2604	−1.15438	−0.577192	−	0.816608i	$-0.695852\pi$
$79$	−10.2604	−1.15438	−0.577192	+	0.816608i	$0.695852\pi$
$80$	−2.06586	−0.230970
$81$	8.00735	0.889706
$82$	6.20553	0.685286
$83$	10.7036	1.17488	0.587439	−	0.809269i	$-0.300136\pi$
$83$	10.7036	1.17488	0.587439	+	0.809269i	$0.300136\pi$
$84$	−1.02712	−0.112068
$85$	7.56490	0.820529
$86$	−6.72762	−0.725458
$87$	3.11584	0.334053
$88$	2.13880	0.227997
$89$	−13.3892	−1.41925	−0.709627	−	0.704577i	$-0.751137\pi$
$89$	−13.3892	−1.41925	−0.709627	+	0.704577i	$0.751137\pi$
$90$	−1.68870	−0.178004
$91$	4.75773	0.498746
$92$	2.88129	0.300396
$93$	−1.44302	−0.149634
$94$	3.80033	0.391974
$95$	−1.00000	−0.102598
$96$	−1.83315	−0.187095
$97$	−3.79273	−0.385093	−0.192547	−	0.981288i	$-0.561675\pi$
$97$	−3.79273	−0.385093	−0.192547	+	0.981288i	$0.561675\pi$
$98$	2.08394	0.210510
$99$	−2.88832	−0.290287
$100$	−1.65817	−0.165817
$101$	−11.8733	−1.18143	−0.590717	−	0.806879i	$-0.701155\pi$
$101$	−11.8733	−1.18143	−0.590717	+	0.806879i	$0.701155\pi$
$102$	1.47808	0.146351
$103$	2.36415	0.232947	0.116473	−	0.993194i	$-0.462841\pi$
$103$	2.36415	0.232947	0.116473	+	0.993194i	$0.462841\pi$
$104$	5.48990	0.538329
$105$	−0.619430	−0.0604502
$106$	5.10659	0.495996
$107$	18.8479	1.82209	0.911046	−	0.412305i	$-0.135276\pi$
$107$	18.8479	1.82209	0.911046	+	0.412305i	$0.135276\pi$
$108$	3.26293	0.313975
$109$	4.97557	0.476573	0.238286	−	0.971195i	$-0.423414\pi$
$109$	4.97557	0.476573	0.238286	+	0.971195i	$0.423414\pi$
$110$	0.584664	0.0557455
$111$	−2.99744	−0.284505
$112$	3.82918	0.361823
$113$	6.56929	0.617987	0.308994	−	0.951064i	$-0.400008\pi$
$113$	6.56929	0.617987	0.308994	+	0.951064i	$0.400008\pi$
$114$	−0.195386	−0.0182996
$115$	1.73764	0.162035
$116$	−15.4602	−1.43545
$117$	−7.41378	−0.685405
$118$	2.32691	0.214209
$119$	−14.0220	−1.28539
$120$	−0.714755	−0.0652479
$121$	1.00000	0.0909091
$122$	−0.379463	−0.0343549
$123$	−3.54699	−0.319821
$124$	7.16000	0.642987
$125$	−1.00000	−0.0894427
$126$	3.13009	0.278851
$127$	1.09135	0.0968419	0.0484210	−	0.998827i	$-0.484581\pi$
$127$	1.09135	0.0968419	0.0484210	+	0.998827i	$0.484581\pi$
$128$	11.5114	1.01748
$129$	3.84541	0.338569
$130$	1.50072	0.131622
$131$	−0.274169	−0.0239543	−0.0119771	−	0.999928i	$-0.503813\pi$
$131$	−0.274169	−0.0239543	−0.0119771	+	0.999928i	$0.503813\pi$
$132$	−0.554135	−0.0482313
$133$	1.85355	0.160724
$134$	0.187053	0.0161590
$135$	1.96779	0.169360
$136$	−16.1798	−1.38741
$137$	12.3503	1.05516	0.527579	−	0.849506i	$-0.323100\pi$
$137$	12.3503	1.05516	0.527579	+	0.849506i	$0.323100\pi$
$138$	0.339510	0.0289010
$139$	1.44278	0.122375	0.0611877	−	0.998126i	$-0.480511\pi$
$139$	1.44278	0.122375	0.0611877	+	0.998126i	$0.480511\pi$
$140$	3.07350	0.259758
$141$	−2.17221	−0.182933
$142$	5.80647	0.487268
$143$	2.56681	0.214648
$144$	−5.96686	−0.497238
$145$	−9.32369	−0.774290
$146$	8.98313	0.743449
$147$	−1.19115	−0.0982443
$148$	14.8728	1.22254
$149$	−15.3857	−1.26044	−0.630222	−	0.776415i	$-0.717036\pi$
$149$	−15.3857	−1.26044	−0.630222	+	0.776415i	$0.717036\pi$
$150$	−0.195386	−0.0159532
$151$	13.9513	1.13534	0.567668	−	0.823257i	$-0.307846\pi$
$151$	13.9513	1.13534	0.567668	+	0.823257i	$0.307846\pi$
$152$	2.13880	0.173480
$153$	21.8499	1.76646
$154$	−1.08371	−0.0873275
$155$	4.31802	0.346832
$156$	−1.42236	−0.113880
$157$	−1.01905	−0.0813291	−0.0406646	−	0.999173i	$-0.512948\pi$
$157$	−1.01905	−0.0813291	−0.0406646	+	0.999173i	$0.512948\pi$
$158$	5.99888	0.477245
$159$	−2.91885	−0.231480
$160$	5.48543	0.433661
$161$	−3.22080	−0.253835
$162$	−4.68161	−0.367822
$163$	7.17421	0.561927	0.280964	−	0.959718i	$-0.409346\pi$
$163$	7.17421	0.561927	0.280964	+	0.959718i	$0.409346\pi$
$164$	17.5995	1.37429
$165$	−0.334185	−0.0260163
$166$	−6.25803	−0.485718
$167$	−12.7807	−0.989003	−0.494502	−	0.869177i	$-0.664649\pi$
$167$	−12.7807	−0.989003	−0.494502	+	0.869177i	$0.664649\pi$
$168$	1.32484	0.102213
$169$	−6.41146	−0.493189
$170$	−4.42292	−0.339223
$171$	−2.88832	−0.220875
$172$	−19.0802	−1.45485
$173$	−9.26126	−0.704121	−0.352060	−	0.935977i	$-0.614519\pi$
$173$	−9.26126	−0.704121	−0.352060	+	0.935977i	$0.614519\pi$
$174$	−1.82172	−0.138104
$175$	1.85355	0.140116
$176$	2.06586	0.155720
$177$	−1.33003	−0.0999709
$178$	7.82819	0.586748
$179$	1.36521	0.102041	0.0510204	−	0.998698i	$-0.483753\pi$
$179$	1.36521	0.102041	0.0510204	+	0.998698i	$0.483753\pi$
$180$	−4.78932	−0.356975
$181$	11.9402	0.887511	0.443755	−	0.896148i	$-0.353646\pi$
$181$	11.9402	0.887511	0.443755	+	0.896148i	$0.353646\pi$
$182$	−2.78167	−0.206191
$183$	0.216895	0.0160334
$184$	−3.71646	−0.273981
$185$	8.96941	0.659444
$186$	0.843680	0.0618616
$187$	−7.56490	−0.553200
$188$	10.7781	0.786076
$189$	−3.64740	−0.265310
$190$	0.584664	0.0424160
$191$	19.9632	1.44449	0.722243	−	0.691639i	$-0.243111\pi$
$191$	19.9632	1.44449	0.722243	+	0.691639i	$0.243111\pi$
$192$	−0.308981	−0.0222988
$193$	4.55437	0.327831	0.163915	−	0.986474i	$-0.447588\pi$
$193$	4.55437	0.327831	0.163915	+	0.986474i	$0.447588\pi$
$194$	2.21747	0.159205
$195$	−0.857791	−0.0614277
$196$	5.91027	0.422162
$197$	4.46917	0.318415	0.159208	−	0.987245i	$-0.449106\pi$
$197$	4.46917	0.318415	0.159208	+	0.987245i	$0.449106\pi$
$198$	1.68870	0.120010
$199$	−13.4574	−0.953968	−0.476984	−	0.878912i	$-0.658270\pi$
$199$	−13.4574	−0.953968	−0.476984	+	0.878912i	$0.658270\pi$
$200$	2.13880	0.151236
$201$	−0.106917	−0.00754134
$202$	6.94187	0.488428
$203$	17.2820	1.21296
$204$	4.19198	0.293497
$205$	10.6138	0.741303
$206$	−1.38223	−0.0963048
$207$	5.01885	0.348834
$208$	5.30267	0.367674
$209$	1.00000	0.0691714
$210$	0.362159	0.0249913
$211$	−7.21902	−0.496978	−0.248489	−	0.968635i	$-0.579934\pi$
$211$	−7.21902	−0.496978	−0.248489	+	0.968635i	$0.579934\pi$
$212$	14.4828	0.994684
$213$	−3.31889	−0.227407
$214$	−11.0197	−0.753289
$215$	−11.5068	−0.784759
$216$	−4.20871	−0.286366
$217$	−8.00368	−0.543325
$218$	−2.90903	−0.197025
$219$	−5.13462	−0.346966
$220$	1.65817	0.111794
$221$	−19.4177	−1.30618
$222$	1.75250	0.117620
$223$	0.918786	0.0615265	0.0307632	−	0.999527i	$-0.490206\pi$
$223$	0.918786	0.0615265	0.0307632	+	0.999527i	$0.490206\pi$
$224$	−10.1675	−0.679347
$225$	−2.88832	−0.192555
$226$	−3.84083	−0.255488
$227$	−18.9477	−1.25760	−0.628800	−	0.777567i	$-0.716454\pi$
$227$	−18.9477	−1.25760	−0.628800	+	0.777567i	$0.716454\pi$
$228$	−0.554135	−0.0366985
$229$	−8.46391	−0.559311	−0.279655	−	0.960100i	$-0.590220\pi$
$229$	−8.46391	−0.559311	−0.279655	+	0.960100i	$0.590220\pi$
$230$	−1.01593	−0.0669887
$231$	0.619430	0.0407555
$232$	19.9415	1.30922
$233$	−10.8413	−0.710238	−0.355119	−	0.934821i	$-0.615560\pi$
$233$	−10.8413	−0.710238	−0.355119	+	0.934821i	$0.615560\pi$
$234$	4.33457	0.283360
$235$	6.50003	0.424015
$236$	6.59935	0.429581
$237$	−3.42887	−0.222729
$238$	8.19813	0.531406
$239$	3.48915	0.225694	0.112847	−	0.993612i	$-0.464003\pi$
$239$	3.48915	0.225694	0.112847	+	0.993612i	$0.464003\pi$
$240$	−0.690379	−0.0445638
$241$	12.3218	0.793718	0.396859	−	0.917880i	$-0.370100\pi$
$241$	12.3218	0.793718	0.396859	+	0.917880i	$0.370100\pi$
$242$	−0.584664	−0.0375836
$243$	8.57931	0.550363
$244$	−1.07620	−0.0688964
$245$	3.56434	0.227717
$246$	2.07380	0.132220
$247$	2.56681	0.163323
$248$	−9.23537	−0.586447
$249$	3.57700	0.226683
$250$	0.584664	0.0369774
$251$	−18.0493	−1.13926	−0.569630	−	0.821901i	$-0.692913\pi$
$251$	−18.0493	−1.13926	−0.569630	+	0.821901i	$0.692913\pi$
$252$	8.87726	0.559215
$253$	−1.73764	−0.109244
$254$	−0.638075	−0.0400364
$255$	2.52808	0.158314
$256$	−4.88116	−0.305072
$257$	12.2032	0.761215	0.380608	−	0.924737i	$-0.375715\pi$
$257$	12.2032	0.761215	0.380608	+	0.924737i	$0.375715\pi$
$258$	−2.24827	−0.139971
$259$	−16.6253	−1.03305
$260$	4.25621	0.263959
$261$	−26.9298	−1.66691
$262$	0.160297	0.00990318
$263$	−14.3679	−0.885964	−0.442982	−	0.896531i	$-0.646079\pi$
$263$	−14.3679	−0.885964	−0.442982	+	0.896531i	$0.646079\pi$
$264$	0.714755	0.0439901
$265$	8.73423	0.536539
$266$	−1.08371	−0.0664463
$267$	−4.47448	−0.273834
$268$	0.530503	0.0324056
$269$	26.3664	1.60759	0.803793	−	0.594910i	$-0.202812\pi$
$269$	26.3664	1.60759	0.803793	+	0.594910i	$0.202812\pi$
$270$	−1.15050	−0.0700169
$271$	14.5399	0.883233	0.441617	−	0.897204i	$-0.354405\pi$
$271$	14.5399	0.883233	0.441617	+	0.897204i	$0.354405\pi$
$272$	−15.6280	−0.947587
$273$	1.58996	0.0962289
$274$	−7.22078	−0.436223
$275$	1.00000	0.0603023
$276$	0.962886	0.0579589
$277$	18.6181	1.11866	0.559328	−	0.828947i	$-0.311059\pi$
$277$	18.6181	1.11866	0.559328	+	0.828947i	$0.311059\pi$
$278$	−0.843544	−0.0505924
$279$	12.4718	0.746668
$280$	−3.96438	−0.236917
$281$	−21.3535	−1.27385	−0.636923	−	0.770928i	$-0.719793\pi$
$281$	−21.3535	−1.27385	−0.636923	+	0.770928i	$0.719793\pi$
$282$	1.27001	0.0756282
$283$	24.0828	1.43157	0.715787	−	0.698319i	$-0.246068\pi$
$283$	24.0828	1.43157	0.715787	+	0.698319i	$0.246068\pi$
$284$	16.4678	0.977182
$285$	−0.334185	−0.0197954
$286$	−1.50072	−0.0887397
$287$	−19.6733	−1.16128
$288$	15.8437	0.933598
$289$	40.2277	2.36634
$290$	5.45123	0.320107
$291$	−1.26747	−0.0743006
$292$	25.4771	1.49093
$293$	−1.69328	−0.0989223	−0.0494612	−	0.998776i	$-0.515750\pi$
$293$	−1.69328	−0.0989223	−0.0494612	+	0.998776i	$0.515750\pi$
$294$	0.696422	0.0406161
$295$	3.97991	0.231719
$296$	−19.1838	−1.11503
$297$	−1.96779	−0.114183
$298$	8.99545	0.521093
$299$	−4.46019	−0.257940
$300$	−0.554135	−0.0319930
$301$	21.3285	1.22935
$302$	−8.15679	−0.469371
$303$	−3.96787	−0.227948
$304$	2.06586	0.118485
$305$	−0.649027	−0.0371632
$306$	−12.7748	−0.730288
$307$	5.44517	0.310772	0.155386	−	0.987854i	$-0.450338\pi$
$307$	5.44517	0.310772	0.155386	+	0.987854i	$0.450338\pi$
$308$	−3.07350	−0.175129
$309$	0.790064	0.0449452
$310$	−2.52459	−0.143387
$311$	−3.18004	−0.180324	−0.0901618	−	0.995927i	$-0.528738\pi$
$311$	−3.18004	−0.180324	−0.0901618	+	0.995927i	$0.528738\pi$
$312$	1.83464	0.103866
$313$	13.7658	0.778087	0.389044	−	0.921219i	$-0.372805\pi$
$313$	13.7658	0.778087	0.389044	+	0.921219i	$0.372805\pi$
$314$	0.595803	0.0336231
$315$	5.35366	0.301644
$316$	17.0134	0.957081
$317$	27.9418	1.56937	0.784684	−	0.619896i	$-0.212825\pi$
$317$	27.9418	1.56937	0.784684	+	0.619896i	$0.212825\pi$
$318$	1.70655	0.0956983
$319$	9.32369	0.522026
$320$	0.924581	0.0516856
$321$	6.29867	0.351558
$322$	1.88309	0.104940
$323$	−7.56490	−0.420922
$324$	−13.2775	−0.737641
$325$	2.56681	0.142381
$326$	−4.19450	−0.232312
$327$	1.66276	0.0919508
$328$	−22.7009	−1.25345
$329$	−12.0481	−0.664236
$330$	0.195386	0.0107556
$331$	25.4481	1.39875	0.699377	−	0.714753i	$-0.253461\pi$
$331$	25.4481	1.39875	0.699377	+	0.714753i	$0.253461\pi$
$332$	−17.7484	−0.974072
$333$	25.9065	1.41967
$334$	7.47244	0.408874
$335$	0.319933	0.0174798
$336$	1.27965	0.0698109
$337$	9.36359	0.510067	0.255034	−	0.966932i	$-0.417913\pi$
$337$	9.36359	0.510067	0.255034	+	0.966932i	$0.417913\pi$
$338$	3.74855	0.203894
$339$	2.19536	0.119236
$340$	−12.5439	−0.680287
$341$	−4.31802	−0.233834
$342$	1.68870	0.0913143
$343$	−19.5816	−1.05731
$344$	24.6108	1.32692
$345$	0.580692	0.0312634
$346$	5.41473	0.291098
$347$	−15.3578	−0.824450	−0.412225	−	0.911082i	$-0.635248\pi$
$347$	−15.3578	−0.824450	−0.412225	+	0.911082i	$0.635248\pi$
$348$	−5.16658	−0.276958
$349$	25.1922	1.34851	0.674253	−	0.738501i	$-0.264466\pi$
$349$	25.1922	1.34851	0.674253	+	0.738501i	$0.264466\pi$
$350$	−1.08371	−0.0579265
$351$	−5.05095	−0.269600
$352$	−5.48543	−0.292374
$353$	−2.98436	−0.158842	−0.0794208	−	0.996841i	$-0.525307\pi$
$353$	−2.98436	−0.158842	−0.0794208	+	0.996841i	$0.525307\pi$
$354$	0.777618	0.0413299
$355$	9.93130	0.527098
$356$	22.2016	1.17668
$357$	−4.68593	−0.248006
$358$	−0.798191	−0.0421857
$359$	−18.5020	−0.976496	−0.488248	−	0.872705i	$-0.662364\pi$
$359$	−18.5020	−0.976496	−0.488248	+	0.872705i	$0.662364\pi$
$360$	6.17754	0.325585
$361$	1.00000	0.0526316
$362$	−6.98103	−0.366915
$363$	0.334185	0.0175402
$364$	−7.88912	−0.413502
$365$	15.3646	0.804220
$366$	−0.126811	−0.00662851
$367$	−11.2265	−0.586021	−0.293010	−	0.956109i	$-0.594657\pi$
$367$	−11.2265	−0.586021	−0.293010	+	0.956109i	$0.594657\pi$
$368$	−3.58971	−0.187127
$369$	30.6562	1.59590
$370$	−5.24409	−0.272627
$371$	−16.1894	−0.840510
$372$	2.39277	0.124059
$373$	−4.82845	−0.250007	−0.125004	−	0.992156i	$-0.539894\pi$
$373$	−4.82845	−0.250007	−0.125004	+	0.992156i	$0.539894\pi$
$374$	4.42292	0.228704
$375$	−0.334185	−0.0172572
$376$	−13.9022	−0.716954
$377$	23.9322	1.23257
$378$	2.13251	0.109684
$379$	−12.9178	−0.663545	−0.331772	−	0.943359i	$-0.607647\pi$
$379$	−12.9178	−0.663545	−0.331772	+	0.943359i	$0.607647\pi$
$380$	1.65817	0.0850622
$381$	0.364714	0.0186849
$382$	−11.6718	−0.597179
$383$	−6.70039	−0.342374	−0.171187	−	0.985239i	$-0.554760\pi$
$383$	−6.70039	−0.342374	−0.171187	+	0.985239i	$0.554760\pi$
$384$	3.84695	0.196314
$385$	−1.85355	−0.0944659
$386$	−2.66278	−0.135532
$387$	−33.2354	−1.68945
$388$	6.28898	0.319275
$389$	−26.0532	−1.32095	−0.660476	−	0.750847i	$-0.729646\pi$
$389$	−26.0532	−1.32095	−0.660476	+	0.750847i	$0.729646\pi$
$390$	0.501520	0.0253954
$391$	13.1450	0.664774
$392$	−7.62340	−0.385040
$393$	−0.0916233	−0.00462178
$394$	−2.61296	−0.131639
$395$	10.2604	0.516256
$396$	4.78932	0.240672
$397$	24.5643	1.23285	0.616423	−	0.787415i	$-0.288581\pi$
$397$	24.5643	1.23285	0.616423	+	0.787415i	$0.288581\pi$
$398$	7.86805	0.394389
$399$	0.619430	0.0310103
$400$	2.06586	0.103293
$401$	−0.303645	−0.0151633	−0.00758165	−	0.999971i	$-0.502413\pi$
$401$	−0.303645	−0.0151633	−0.00758165	+	0.999971i	$0.502413\pi$
$402$	0.0625105	0.00311774
$403$	−11.0836	−0.552111
$404$	19.6879	0.979508
$405$	−8.00735	−0.397889
$406$	−10.1041	−0.501460
$407$	−8.96941	−0.444597
$408$	−5.40705	−0.267689
$409$	−16.8426	−0.832812	−0.416406	−	0.909179i	$-0.636710\pi$
$409$	−16.8426	−0.832812	−0.416406	+	0.909179i	$0.636710\pi$
$410$	−6.20553	−0.306469
$411$	4.12729	0.203584
$412$	−3.92016	−0.193132
$413$	−7.37697	−0.362997
$414$	−2.93434	−0.144215
$415$	−10.7036	−0.525421
$416$	−14.0801	−0.690333
$417$	0.482157	0.0236113
$418$	−0.584664	−0.0285969
$419$	−36.4464	−1.78052	−0.890262	−	0.455449i	$-0.849479\pi$
$419$	−36.4464	−1.78052	−0.890262	+	0.455449i	$0.849479\pi$
$420$	1.02712	0.0501183
$421$	−28.4297	−1.38558	−0.692789	−	0.721141i	$-0.743618\pi$
$421$	−28.4297	−1.38558	−0.692789	+	0.721141i	$0.743618\pi$
$422$	4.22070	0.205460
$423$	18.7742	0.912831
$424$	−18.6808	−0.907218
$425$	−7.56490	−0.366952
$426$	1.94044	0.0940144
$427$	1.20301	0.0582176
$428$	−31.2529	−1.51067
$429$	0.857791	0.0414146
$430$	6.72762	0.324435
$431$	−20.8541	−1.00451	−0.502254	−	0.864720i	$-0.667496\pi$
$431$	−20.8541	−1.00451	−0.502254	+	0.864720i	$0.667496\pi$
$432$	−4.06517	−0.195586
$433$	8.63662	0.415049	0.207525	−	0.978230i	$-0.433459\pi$
$433$	8.63662	0.415049	0.207525	+	0.978230i	$0.433459\pi$
$434$	4.67946	0.224621
$435$	−3.11584	−0.149393
$436$	−8.25032	−0.395119
$437$	−1.73764	−0.0831224
$438$	3.00203	0.143443
$439$	−27.9016	−1.33167	−0.665834	−	0.746100i	$-0.731924\pi$
$439$	−27.9016	−1.33167	−0.665834	+	0.746100i	$0.731924\pi$
$440$	−2.13880	−0.101963
$441$	10.2949	0.490236
$442$	11.3528	0.539999
$443$	2.53769	0.120569	0.0602846	−	0.998181i	$-0.480799\pi$
$443$	2.53769	0.120569	0.0602846	+	0.998181i	$0.480799\pi$
$444$	4.97027	0.235878
$445$	13.3892	0.634710
$446$	−0.537181	−0.0254363
$447$	−5.14167	−0.243192
$448$	−1.71376	−0.0809676
$449$	13.2827	0.626848	0.313424	−	0.949613i	$-0.398524\pi$
$449$	13.2827	0.626848	0.313424	+	0.949613i	$0.398524\pi$
$450$	1.68870	0.0796059
$451$	−10.6138	−0.499786
$452$	−10.8930	−0.512363
$453$	4.66230	0.219054
$454$	11.0780	0.519917
$455$	−4.75773	−0.223046
$456$	0.714755	0.0334715
$457$	−31.2328	−1.46101	−0.730505	−	0.682907i	$-0.760715\pi$
$457$	−31.2328	−1.46101	−0.730505	+	0.682907i	$0.760715\pi$
$458$	4.94854	0.231230
$459$	14.8861	0.694825
$460$	−2.88129	−0.134341
$461$	5.04910	0.235160	0.117580	−	0.993063i	$-0.462486\pi$
$461$	5.04910	0.235160	0.117580	+	0.993063i	$0.462486\pi$
$462$	−0.362159	−0.0168491
$463$	23.0184	1.06976	0.534878	−	0.844929i	$-0.320357\pi$
$463$	23.0184	1.06976	0.534878	+	0.844929i	$0.320357\pi$
$464$	19.2614	0.894189
$465$	1.44302	0.0669183
$466$	6.33853	0.293627
$467$	19.1144	0.884510	0.442255	−	0.896889i	$-0.354179\pi$
$467$	19.1144	0.884510	0.442255	+	0.896889i	$0.354179\pi$
$468$	12.2933	0.568258
$469$	−0.593014	−0.0273828
$470$	−3.80033	−0.175296
$471$	−0.340552	−0.0156918
$472$	−8.51222	−0.391807
$473$	11.5068	0.529084
$474$	2.00474	0.0920806
$475$	1.00000	0.0458831
$476$	23.2507	1.06570
$477$	25.2272	1.15508
$478$	−2.03998	−0.0933066
$479$	−29.7781	−1.36059	−0.680297	−	0.732936i	$-0.738149\pi$
$479$	−29.7781	−1.36059	−0.680297	+	0.732936i	$0.738149\pi$
$480$	1.83315	0.0836714
$481$	−23.0228	−1.04975
$482$	−7.20413	−0.328139
$483$	−1.07634	−0.0489754
$484$	−1.65817	−0.0753713
$485$	3.79273	0.172219
$486$	−5.01601	−0.227531
$487$	−5.27518	−0.239041	−0.119521	−	0.992832i	$-0.538136\pi$
$487$	−5.27518	−0.239041	−0.119521	+	0.992832i	$0.538136\pi$
$488$	1.38814	0.0628381
$489$	2.39751	0.108419
$490$	−2.08394	−0.0941428
$491$	−44.1380	−1.99192	−0.995960	−	0.0897969i	$-0.971378\pi$
$491$	−44.1380	−1.99192	−0.995960	+	0.0897969i	$0.971378\pi$
$492$	5.88150	0.265159
$493$	−70.5328	−3.17664
$494$	−1.50072	−0.0675208
$495$	2.88832	0.129820
$496$	−8.92041	−0.400538
$497$	−18.4082	−0.825720
$498$	−2.09134	−0.0937153
$499$	−10.8222	−0.484467	−0.242233	−	0.970218i	$-0.577880\pi$
$499$	−10.8222	−0.484467	−0.242233	+	0.970218i	$0.577880\pi$
$500$	1.65817	0.0741555
$501$	−4.27113	−0.190820
$502$	10.5528	0.470993
$503$	35.5134	1.58346	0.791731	−	0.610869i	$-0.209180\pi$
$503$	35.5134	1.58346	0.791731	+	0.610869i	$0.209180\pi$
$504$	−11.4504	−0.510041
$505$	11.8733	0.528353
$506$	1.01593	0.0451638
$507$	−2.14262	−0.0951569
$508$	−1.80965	−0.0802901
$509$	−29.8008	−1.32090	−0.660448	−	0.750871i	$-0.729634\pi$
$509$	−29.8008	−1.32090	−0.660448	+	0.750871i	$0.729634\pi$
$510$	−1.47808	−0.0654503
$511$	−28.4791	−1.25984
$512$	−20.1690	−0.891353
$513$	−1.96779	−0.0868800
$514$	−7.13478	−0.314701
$515$	−2.36415	−0.104177
$516$	−6.37633	−0.280702
$517$	−6.50003	−0.285871
$518$	9.72021	0.427081
$519$	−3.09498	−0.135854
$520$	−5.48990	−0.240748
$521$	2.21855	0.0971963	0.0485982	−	0.998818i	$-0.484525\pi$
$521$	2.21855	0.0971963	0.0485982	+	0.998818i	$0.484525\pi$
$522$	15.7449	0.689135
$523$	−13.4641	−0.588745	−0.294373	−	0.955691i	$-0.595111\pi$
$523$	−13.4641	−0.588745	−0.294373	+	0.955691i	$0.595111\pi$
$524$	0.454619	0.0198601
$525$	0.619430	0.0270342
$526$	8.40040	0.366275
$527$	32.6654	1.42293
$528$	0.690379	0.0300449
$529$	−19.9806	−0.868723
$530$	−5.10659	−0.221816
$531$	11.4952	0.498851
$532$	−3.07350	−0.133253
$533$	−27.2438	−1.18006
$534$	2.61607	0.113208
$535$	−18.8479	−0.814864
$536$	−0.684273	−0.0295561
$537$	0.456234	0.0196880
$538$	−15.4155	−0.664608
$539$	−3.56434	−0.153527
$540$	−3.26293	−0.140414
$541$	17.2003	0.739497	0.369749	−	0.929132i	$-0.379444\pi$
$541$	17.2003	0.739497	0.369749	+	0.929132i	$0.379444\pi$
$542$	−8.50093	−0.365146
$543$	3.99025	0.171238
$544$	41.4967	1.77916
$545$	−4.97557	−0.213130
$546$	−0.929594	−0.0397829
$547$	17.3636	0.742414	0.371207	−	0.928550i	$-0.378944\pi$
$547$	17.3636	0.742414	0.371207	+	0.928550i	$0.378944\pi$
$548$	−20.4789	−0.874815
$549$	−1.87460	−0.0800059
$550$	−0.584664	−0.0249302
$551$	9.32369	0.397203
$552$	−1.24198	−0.0528624
$553$	−19.0182	−0.808736
$554$	−10.8854	−0.462474
$555$	2.99744	0.127234
$556$	−2.39238	−0.101459
$557$	14.6832	0.622149	0.311075	−	0.950385i	$-0.399311\pi$
$557$	14.6832	0.622149	0.311075	+	0.950385i	$0.399311\pi$
$558$	−7.29182	−0.308688
$559$	29.5359	1.24923
$560$	−3.82918	−0.161812
$561$	−2.52808	−0.106735
$562$	12.4846	0.526633
$563$	12.2619	0.516778	0.258389	−	0.966041i	$-0.416808\pi$
$563$	12.2619	0.516778	0.258389	+	0.966041i	$0.416808\pi$
$564$	3.60189	0.151667
$565$	−6.56929	−0.276372
$566$	−14.0803	−0.591841
$567$	14.8421	0.623308
$568$	−21.2410	−0.891254
$569$	−39.1105	−1.63960	−0.819799	−	0.572651i	$-0.805915\pi$
$569$	−39.1105	−1.63960	−0.819799	+	0.572651i	$0.805915\pi$
$570$	0.195386	0.00818382
$571$	12.2964	0.514589	0.257294	−	0.966333i	$-0.417169\pi$
$571$	12.2964	0.514589	0.257294	+	0.966333i	$0.417169\pi$
$572$	−4.25621	−0.177961
$573$	6.67141	0.278702
$574$	11.5023	0.480096
$575$	−1.73764	−0.0724645
$576$	2.67048	0.111270
$577$	−10.4836	−0.436439	−0.218220	−	0.975900i	$-0.570025\pi$
$577$	−10.4836	−0.436439	−0.218220	+	0.975900i	$0.570025\pi$
$578$	−23.5197	−0.978290
$579$	1.52200	0.0632523
$580$	15.4602	0.641952
$581$	19.8398	0.823093
$582$	0.741046	0.0307174
$583$	−8.73423	−0.361735
$584$	−32.8618	−1.35983
$585$	7.41378	0.306522
$586$	0.989998	0.0408965
$587$	−33.6559	−1.38913	−0.694563	−	0.719432i	$-0.744402\pi$
$587$	−33.6559	−1.38913	−0.694563	+	0.719432i	$0.744402\pi$
$588$	1.97512	0.0814528
$589$	−4.31802	−0.177921
$590$	−2.32691	−0.0957973
$591$	1.49353	0.0614356
$592$	−18.5295	−0.761559
$593$	38.1656	1.56727	0.783636	−	0.621220i	$-0.213363\pi$
$593$	38.1656	1.56727	0.783636	+	0.621220i	$0.213363\pi$
$594$	1.15050	0.0472054
$595$	14.0220	0.574844
$596$	25.5120	1.04501
$597$	−4.49726	−0.184060
$598$	2.60771	0.106637
$599$	40.9180	1.67186	0.835932	−	0.548832i	$-0.184927\pi$
$599$	40.9180	1.67186	0.835932	+	0.548832i	$0.184927\pi$
$600$	0.714755	0.0291797
$601$	18.0055	0.734458	0.367229	−	0.930131i	$-0.380307\pi$
$601$	18.0055	0.734458	0.367229	+	0.930131i	$0.380307\pi$
$602$	−12.4700	−0.508240
$603$	0.924070	0.0376310
$604$	−23.1335	−0.941290
$605$	−1.00000	−0.0406558
$606$	2.31987	0.0942382
$607$	24.9470	1.01257	0.506283	−	0.862367i	$-0.331019\pi$
$607$	24.9470	1.01257	0.506283	+	0.862367i	$0.331019\pi$
$608$	−5.48543	−0.222464
$609$	5.77538	0.234030
$610$	0.379463	0.0153640
$611$	−16.6844	−0.674977
$612$	−36.2307	−1.46454
$613$	33.8227	1.36608	0.683042	−	0.730379i	$-0.260657\pi$
$613$	33.8227	1.36608	0.683042	+	0.730379i	$0.260657\pi$
$614$	−3.18359	−0.128479
$615$	3.54699	0.143028
$616$	3.96438	0.159729
$617$	−15.7001	−0.632063	−0.316031	−	0.948749i	$-0.602350\pi$
$617$	−15.7001	−0.632063	−0.316031	+	0.948749i	$0.602350\pi$
$618$	−0.461922	−0.0185812
$619$	15.6715	0.629892	0.314946	−	0.949110i	$-0.398014\pi$
$619$	15.6715	0.629892	0.314946	+	0.949110i	$0.398014\pi$
$620$	−7.16000	−0.287552
$621$	3.41930	0.137212
$622$	1.85926	0.0745494
$623$	−24.8176	−0.994298
$624$	1.77207	0.0709398
$625$	1.00000	0.0400000
$626$	−8.04835	−0.321677
$627$	0.334185	0.0133461
$628$	1.68976	0.0674287
$629$	67.8527	2.70546
$630$	−3.13009	−0.124706
$631$	−41.2356	−1.64156	−0.820781	−	0.571243i	$-0.806461\pi$
$631$	−41.2356	−1.64156	−0.820781	+	0.571243i	$0.806461\pi$
$632$	−21.9449	−0.872922
$633$	−2.41249	−0.0958878
$634$	−16.3366	−0.648808
$635$	−1.09135	−0.0433090
$636$	4.83994	0.191916
$637$	−9.14899	−0.362496
$638$	−5.45123	−0.215816
$639$	28.6848	1.13475
$640$	−11.5114	−0.455029
$641$	38.3845	1.51610	0.758049	−	0.652198i	$-0.226153\pi$
$641$	38.3845	1.51610	0.758049	+	0.652198i	$0.226153\pi$
$642$	−3.68261	−0.145341
$643$	−17.9219	−0.706772	−0.353386	−	0.935478i	$-0.614970\pi$
$643$	−17.9219	−0.706772	−0.353386	+	0.935478i	$0.614970\pi$
$644$	5.34063	0.210450
$645$	−3.84541	−0.151413
$646$	4.42292	0.174018
$647$	−25.4790	−1.00168	−0.500842	−	0.865539i	$-0.666976\pi$
$647$	−25.4790	−1.00168	−0.500842	+	0.865539i	$0.666976\pi$
$648$	17.1261	0.672778
$649$	−3.97991	−0.156225
$650$	−1.50072	−0.0588632
$651$	−2.67471	−0.104830
$652$	−11.8960	−0.465885
$653$	−48.6094	−1.90223	−0.951117	−	0.308832i	$-0.900062\pi$
$653$	−48.6094	−1.90223	−0.951117	+	0.308832i	$0.900062\pi$
$654$	−0.972156	−0.0380143
$655$	0.274169	0.0107127
$656$	−21.9267	−0.856093
$657$	44.3779	1.73135
$658$	7.04412	0.274608
$659$	29.3918	1.14494	0.572470	−	0.819925i	$-0.305985\pi$
$659$	29.3918	1.14494	0.572470	+	0.819925i	$0.305985\pi$
$660$	0.554135	0.0215697
$661$	−32.8576	−1.27801	−0.639006	−	0.769202i	$-0.720654\pi$
$661$	−32.8576	−1.27801	−0.639006	+	0.769202i	$0.720654\pi$
$662$	−14.8786	−0.578273
$663$	−6.48911	−0.252016
$664$	22.8929	0.888418
$665$	−1.85355	−0.0718777
$666$	−15.1466	−0.586920
$667$	−16.2012	−0.627313
$668$	21.1926	0.819967
$669$	0.307045	0.0118710
$670$	−0.187053	−0.00722651
$671$	0.649027	0.0250554
$672$	−3.39784	−0.131075
$673$	−23.5619	−0.908244	−0.454122	−	0.890940i	$-0.650047\pi$
$673$	−23.5619	−0.908244	−0.454122	+	0.890940i	$0.650047\pi$
$674$	−5.47456	−0.210872
$675$	−1.96779	−0.0757402
$676$	10.6313	0.408895
$677$	10.2123	0.392490	0.196245	−	0.980555i	$-0.437125\pi$
$677$	10.2123	0.392490	0.196245	+	0.980555i	$0.437125\pi$
$678$	−1.28355	−0.0492944
$679$	−7.03003	−0.269788
$680$	16.1798	0.620467
$681$	−6.33202	−0.242644
$682$	2.52459	0.0966715
$683$	−15.2708	−0.584319	−0.292160	−	0.956370i	$-0.594374\pi$
$683$	−15.2708	−0.584319	−0.292160	+	0.956370i	$0.594374\pi$
$684$	4.78932	0.183124
$685$	−12.3503	−0.471881
$686$	11.4486	0.437111
$687$	−2.82851	−0.107914
$688$	23.7714	0.906278
$689$	−22.4191	−0.854101
$690$	−0.339510	−0.0129249
$691$	−30.0322	−1.14248	−0.571239	−	0.820783i	$-0.693537\pi$
$691$	−30.0322	−1.14248	−0.571239	+	0.820783i	$0.693537\pi$
$692$	15.3567	0.583775
$693$	−5.35366	−0.203369
$694$	8.97915	0.340844
$695$	−1.44278	−0.0547279
$696$	6.66415	0.252604
$697$	80.2926	3.04130
$698$	−14.7289	−0.557499
$699$	−3.62301	−0.137035
$700$	−3.07350	−0.116168
$701$	5.90462	0.223015	0.111507	−	0.993764i	$-0.464432\pi$
$701$	5.90462	0.223015	0.111507	+	0.993764i	$0.464432\pi$
$702$	2.95311	0.111458
$703$	−8.96941	−0.338288
$704$	−0.924581	−0.0348464
$705$	2.17221	0.0818102
$706$	1.74485	0.0656683
$707$	−22.0077	−0.827686
$708$	2.20541	0.0828842
$709$	17.5476	0.659013	0.329507	−	0.944153i	$-0.393118\pi$
$709$	17.5476	0.659013	0.329507	+	0.944153i	$0.393118\pi$
$710$	−5.80647	−0.217913
$711$	29.6353	1.11141
$712$	−28.6369	−1.07321
$713$	7.50315	0.280995
$714$	2.73969	0.102530
$715$	−2.56681	−0.0959934
$716$	−2.26375	−0.0846004
$717$	1.16602	0.0435459
$718$	10.8174	0.403703
$719$	3.99061	0.148825	0.0744124	−	0.997228i	$-0.476292\pi$
$719$	3.99061	0.148825	0.0744124	+	0.997228i	$0.476292\pi$
$720$	5.96686	0.222372
$721$	4.38208	0.163197
$722$	−0.584664	−0.0217589
$723$	4.11777	0.153142
$724$	−19.7989	−0.735821
$725$	9.32369	0.346273
$726$	−0.195386	−0.00725146
$727$	46.2696	1.71604	0.858021	−	0.513614i	$-0.171694\pi$
$727$	46.2696	1.71604	0.858021	+	0.513614i	$0.171694\pi$
$728$	10.1758	0.377141
$729$	−21.1550	−0.783518
$730$	−8.98313	−0.332481
$731$	−87.0479	−3.21958
$732$	−0.359649	−0.0132930
$733$	−5.99820	−0.221548	−0.110774	−	0.993846i	$-0.535333\pi$
$733$	−5.99820	−0.221548	−0.110774	+	0.993846i	$0.535333\pi$
$734$	6.56376	0.242273
$735$	1.19115	0.0439362
$736$	9.53169	0.351343
$737$	−0.319933	−0.0117849
$738$	−17.9236	−0.659775
$739$	5.41614	0.199236	0.0996179	−	0.995026i	$-0.468238\pi$
$739$	5.41614	0.199236	0.0996179	+	0.995026i	$0.468238\pi$
$740$	−14.8728	−0.546735
$741$	0.857791	0.0315118
$742$	9.46534	0.347483
$743$	51.8360	1.90168	0.950838	−	0.309687i	$-0.100224\pi$
$743$	51.8360	1.90168	0.950838	+	0.309687i	$0.100224\pi$
$744$	−3.08632	−0.113150
$745$	15.3857	0.563688
$746$	2.82302	0.103358
$747$	−30.9155	−1.13114
$748$	12.5439	0.458649
$749$	34.9355	1.27652
$750$	0.195386	0.00713449
$751$	50.2801	1.83475	0.917373	−	0.398029i	$-0.130306\pi$
$751$	50.2801	1.83475	0.917373	+	0.398029i	$0.130306\pi$
$752$	−13.4281	−0.489673
$753$	−6.03180	−0.219811
$754$	−13.9923	−0.509569
$755$	−13.9513	−0.507738
$756$	6.04801	0.219964
$757$	19.0283	0.691597	0.345798	−	0.938309i	$-0.387608\pi$
$757$	19.0283	0.691597	0.345798	+	0.938309i	$0.387608\pi$
$758$	7.55260	0.274323
$759$	−0.580692	−0.0210778
$760$	−2.13880	−0.0775824
$761$	4.38531	0.158967	0.0794837	−	0.996836i	$-0.474673\pi$
$761$	4.38531	0.158967	0.0794837	+	0.996836i	$0.474673\pi$
$762$	−0.213235	−0.00772469
$763$	9.22248	0.333876
$764$	−33.1023	−1.19760
$765$	−21.8499	−0.789983
$766$	3.91748	0.141544
$767$	−10.2157	−0.368867
$768$	−1.63121	−0.0588612
$769$	−11.9368	−0.430453	−0.215227	−	0.976564i	$-0.569049\pi$
$769$	−11.9368	−0.430453	−0.215227	+	0.976564i	$0.569049\pi$
$770$	1.08371	0.0390541
$771$	4.07813	0.146870
$772$	−7.55191	−0.271799
$773$	50.2033	1.80569	0.902843	−	0.429971i	$-0.141476\pi$
$773$	50.2033	1.80569	0.902843	+	0.429971i	$0.141476\pi$
$774$	19.4315	0.698452
$775$	−4.31802	−0.155108
$776$	−8.11189	−0.291200
$777$	−5.55593	−0.199318
$778$	15.2324	0.546108
$779$	−10.6138	−0.380280
$780$	1.42236	0.0509287
$781$	−9.93130	−0.355370
$782$	−7.68544	−0.274831
$783$	−18.3471	−0.655670
$784$	−7.36341	−0.262979
$785$	1.01905	0.0363715
$786$	0.0535688	0.00191074
$787$	4.84005	0.172529	0.0862647	−	0.996272i	$-0.472507\pi$
$787$	4.84005	0.172529	0.0862647	+	0.996272i	$0.472507\pi$
$788$	−7.41063	−0.263993
$789$	−4.80154	−0.170939
$790$	−5.99888	−0.213431
$791$	12.1765	0.432948
$792$	−6.17754	−0.219509
$793$	1.66593	0.0591590
$794$	−14.3618	−0.509683
$795$	2.91885	0.103521
$796$	22.3146	0.790920
$797$	26.3359	0.932865	0.466432	−	0.884557i	$-0.345539\pi$
$797$	26.3359	0.932865	0.466432	+	0.884557i	$0.345539\pi$
$798$	−0.362159	−0.0128203
$799$	49.1720	1.73958
$800$	−5.48543	−0.193939
$801$	38.6724	1.36642
$802$	0.177530	0.00626881
$803$	−15.3646	−0.542205
$804$	0.177286	0.00625241
$805$	3.22080	0.113518
$806$	6.48015	0.228254
$807$	8.81125	0.310171
$808$	−25.3945	−0.893376
$809$	4.63056	0.162802	0.0814009	−	0.996681i	$-0.474061\pi$
$809$	4.63056	0.162802	0.0814009	+	0.996681i	$0.474061\pi$
$810$	4.68161	0.164495
$811$	0.583214	0.0204794	0.0102397	−	0.999948i	$-0.496741\pi$
$811$	0.583214	0.0204794	0.0102397	+	0.999948i	$0.496741\pi$
$812$	−28.6564	−1.00564
$813$	4.85900	0.170413
$814$	5.24409	0.183805
$815$	−7.17421	−0.251301
$816$	−5.22265	−0.182829
$817$	11.5068	0.402573
$818$	9.84725	0.344301
$819$	−13.7418	−0.480179
$820$	−17.5995	−0.614602
$821$	35.4319	1.23658	0.618290	−	0.785950i	$-0.287826\pi$
$821$	35.4319	1.23658	0.618290	+	0.785950i	$0.287826\pi$
$822$	−2.41308	−0.0841658
$823$	−35.3477	−1.23214	−0.616072	−	0.787690i	$-0.711277\pi$
$823$	−35.3477	−1.23214	−0.616072	+	0.787690i	$0.711277\pi$
$824$	5.05645	0.176150
$825$	0.334185	0.0116348
$826$	4.31305	0.150070
$827$	−16.7617	−0.582861	−0.291430	−	0.956592i	$-0.594131\pi$
$827$	−16.7617	−0.582861	−0.291430	+	0.956592i	$0.594131\pi$
$828$	−8.32210	−0.289213
$829$	−19.2674	−0.669186	−0.334593	−	0.942363i	$-0.608599\pi$
$829$	−19.2674	−0.669186	−0.334593	+	0.942363i	$0.608599\pi$
$830$	6.25803	0.217219
$831$	6.22191	0.215836
$832$	−2.37323	−0.0822769
$833$	26.9639	0.934242
$834$	−0.281900	−0.00976139
$835$	12.7807	0.442296
$836$	−1.65817	−0.0573489
$837$	8.49695	0.293698
$838$	21.3089	0.736104
$839$	44.5687	1.53868	0.769341	−	0.638839i	$-0.220585\pi$
$839$	44.5687	1.53868	0.769341	+	0.638839i	$0.220585\pi$
$840$	−1.32484	−0.0457112
$841$	57.9312	1.99763
$842$	16.6218	0.572825
$843$	−7.13604	−0.245778
$844$	11.9703	0.412036
$845$	6.41146	0.220561
$846$	−10.9766	−0.377382
$847$	1.85355	0.0636889
$848$	−18.0437	−0.619622
$849$	8.04811	0.276210
$850$	4.42292	0.151705
$851$	15.5856	0.534267
$852$	5.50328	0.188539
$853$	−1.06714	−0.0365381	−0.0182690	−	0.999833i	$-0.505816\pi$
$853$	−1.06714	−0.0365381	−0.0182690	+	0.999833i	$0.505816\pi$
$854$	−0.703355	−0.0240683
$855$	2.88832	0.0987785
$856$	40.3118	1.37783
$857$	−24.2621	−0.828777	−0.414388	−	0.910100i	$-0.636004\pi$
$857$	−24.2621	−0.828777	−0.414388	+	0.910100i	$0.636004\pi$
$858$	−0.501520	−0.0171216
$859$	−2.20953	−0.0753882	−0.0376941	−	0.999289i	$-0.512001\pi$
$859$	−2.20953	−0.0753882	−0.0376941	+	0.999289i	$0.512001\pi$
$860$	19.0802	0.650631
$861$	−6.57453	−0.224059
$862$	12.1927	0.415283
$863$	−6.31080	−0.214822	−0.107411	−	0.994215i	$-0.534256\pi$
$863$	−6.31080	−0.214822	−0.107411	+	0.994215i	$0.534256\pi$
$864$	10.7942	0.367225
$865$	9.26126	0.314892
$866$	−5.04952	−0.171590
$867$	13.4435	0.456565
$868$	13.2714	0.450462
$869$	−10.2604	−0.348060
$870$	1.82172	0.0617621
$871$	−0.821210	−0.0278256
$872$	10.6417	0.360375
$873$	10.9546	0.370758
$874$	1.01593	0.0343645
$875$	−1.85355	−0.0626616
$876$	8.51407	0.287664
$877$	1.50724	0.0508959	0.0254479	−	0.999676i	$-0.491899\pi$
$877$	1.50724	0.0508959	0.0254479	+	0.999676i	$0.491899\pi$
$878$	16.3130	0.550538
$879$	−0.565868	−0.0190863
$880$	−2.06586	−0.0696400
$881$	21.1890	0.713877	0.356938	−	0.934128i	$-0.383821\pi$
$881$	21.1890	0.713877	0.356938	+	0.934128i	$0.383821\pi$
$882$	−6.01909	−0.202673
$883$	−16.0935	−0.541589	−0.270794	−	0.962637i	$-0.587286\pi$
$883$	−16.0935	−0.541589	−0.270794	+	0.962637i	$0.587286\pi$
$884$	32.1978	1.08293
$885$	1.33003	0.0447083
$886$	−1.48369	−0.0498457
$887$	26.3625	0.885166	0.442583	−	0.896728i	$-0.354062\pi$
$887$	26.3625	0.885166	0.442583	+	0.896728i	$0.354062\pi$
$888$	−6.41093	−0.215137
$889$	2.02288	0.0678453
$890$	−7.82819	−0.262402
$891$	8.00735	0.268256
$892$	−1.52350	−0.0510106
$893$	−6.50003	−0.217515
$894$	3.00615	0.100541
$895$	−1.36521	−0.0456340
$896$	21.3371	0.712821
$897$	−1.49053	−0.0497674
$898$	−7.76589	−0.259151
$899$	−40.2599	−1.34274
$900$	4.78932	0.159644
$901$	66.0736	2.20123
$902$	6.20553	0.206622
$903$	7.12767	0.237194
$904$	14.0504	0.467309
$905$	−11.9402	−0.396907
$906$	−2.72588	−0.0905613
$907$	37.8709	1.25748	0.628742	−	0.777614i	$-0.283570\pi$
$907$	37.8709	1.25748	0.628742	+	0.777614i	$0.283570\pi$
$908$	31.4184	1.04266
$909$	34.2938	1.13745
$910$	2.78167	0.0922116
$911$	−36.2441	−1.20082	−0.600410	−	0.799692i	$-0.704996\pi$
$911$	−36.2441	−1.20082	−0.600410	+	0.799692i	$0.704996\pi$
$912$	0.690379	0.0228607
$913$	10.7036	0.354239
$914$	18.2607	0.604011
$915$	−0.216895	−0.00717033
$916$	14.0346	0.463716
$917$	−0.508188	−0.0167818
$918$	−8.70338	−0.287254
$919$	−57.6780	−1.90262	−0.951310	−	0.308235i	$-0.900262\pi$
$919$	−57.6780	−1.90262	−0.951310	+	0.308235i	$0.900262\pi$
$920$	3.71646	0.122528
$921$	1.81969	0.0599609
$922$	−2.95202	−0.0972198
$923$	−25.4918	−0.839073
$924$	−1.02712	−0.0337897
$925$	−8.96941	−0.294912
$926$	−13.4580	−0.442259
$927$	−6.82843	−0.224275
$928$	−51.1445	−1.67890
$929$	45.2673	1.48517	0.742586	−	0.669751i	$-0.233599\pi$
$929$	45.2673	1.48517	0.742586	+	0.669751i	$0.233599\pi$
$930$	−0.843680	−0.0276654
$931$	−3.56434	−0.116816
$932$	17.9767	0.588847
$933$	−1.06272	−0.0347920
$934$	−11.1755	−0.365674
$935$	7.56490	0.247399
$936$	−15.8566	−0.518289
$937$	−49.0343	−1.60188	−0.800940	−	0.598745i	$-0.795666\pi$
$937$	−49.0343	−1.60188	−0.800940	+	0.598745i	$0.795666\pi$
$938$	0.346714	0.0113206
$939$	4.60032	0.150126
$940$	−10.7781	−0.351544
$941$	5.61799	0.183141	0.0915706	−	0.995799i	$-0.470811\pi$
$941$	5.61799	0.183141	0.0915706	+	0.995799i	$0.470811\pi$
$942$	0.199108	0.00648730
$943$	18.4430	0.600587
$944$	−8.22192	−0.267601
$945$	3.64740	0.118650
$946$	−6.72762	−0.218734
$947$	−46.5630	−1.51309	−0.756547	−	0.653939i	$-0.773115\pi$
$947$	−46.5630	−1.51309	−0.756547	+	0.653939i	$0.773115\pi$
$948$	5.68564	0.184661
$949$	−39.4381	−1.28022
$950$	−0.584664	−0.0189690
$951$	9.33774	0.302797
$952$	−29.9901	−0.971986
$953$	−13.6432	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$953$	−13.6432	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$954$	−14.7495	−0.477531
$955$	−19.9632	−0.645994
$956$	−5.78560	−0.187120
$957$	3.11584	0.100721
$958$	17.4102	0.562497
$959$	22.8920	0.739220
$960$	0.308981	0.00997232
$961$	−12.3547	−0.398539
$962$	13.4606	0.433988
$963$	−54.4387	−1.75426
$964$	−20.4316	−0.658059
$965$	−4.55437	−0.146610
$966$	0.629300	0.0202474
$967$	7.80756	0.251074	0.125537	−	0.992089i	$-0.459935\pi$
$967$	7.80756	0.251074	0.125537	+	0.992089i	$0.459935\pi$
$968$	2.13880	0.0687436
$969$	−2.52808	−0.0812135
$970$	−2.21747	−0.0711988
$971$	8.59548	0.275842	0.137921	−	0.990443i	$-0.455958\pi$
$971$	8.59548	0.275842	0.137921	+	0.990443i	$0.455958\pi$
$972$	−14.2259	−0.456297
$973$	2.67428	0.0857334
$974$	3.08421	0.0988243
$975$	0.857791	0.0274713
$976$	1.34080	0.0429179
$977$	46.8731	1.49960	0.749801	−	0.661663i	$-0.230149\pi$
$977$	46.8731	1.49960	0.749801	+	0.661663i	$0.230149\pi$
$978$	−1.40174	−0.0448227
$979$	−13.3892	−0.427921
$980$	−5.91027	−0.188797
$981$	−14.3710	−0.458832
$982$	25.8059	0.823499
$983$	−15.9451	−0.508569	−0.254284	−	0.967130i	$-0.581840\pi$
$983$	−15.9451	−0.508569	−0.254284	+	0.967130i	$0.581840\pi$
$984$	−7.58629	−0.241842
$985$	−4.46917	−0.142400
$986$	41.2380	1.31328
$987$	−4.02631	−0.128159
$988$	−4.25621	−0.135408
$989$	−19.9947	−0.635794
$990$	−1.68870	−0.0536703
$991$	−5.95902	−0.189295	−0.0946473	−	0.995511i	$-0.530172\pi$
$991$	−5.95902	−0.189295	−0.0946473	+	0.995511i	$0.530172\pi$
$992$	23.6862	0.752037
$993$	8.50438	0.269878
$994$	10.7626	0.341369
$995$	13.4574	0.426628
$996$	−5.93126	−0.187939
$997$	29.3619	0.929902	0.464951	−	0.885337i	$-0.346072\pi$
$997$	29.3619	0.929902	0.464951	+	0.885337i	$0.346072\pi$
$998$	6.32733	0.200288
$999$	17.6499	0.558419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.e.1.3	✓	5
3.2	odd	2		9405.2.a.u.1.3		5
5.4	even	2		5225.2.a.i.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.e.1.3	✓	5	1.1	even	1	trivial
5225.2.a.i.1.3		5	5.4	even	2
9405.2.a.u.1.3		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)

\(f(q)\)	\(=\)	\(q-0.584664 q^{2} +0.334185 q^{3} -1.65817 q^{4} -1.00000 q^{5} -0.195386 q^{6} +1.85355 q^{7} +2.13880 q^{8} -2.88832 q^{9} +O(q^{10})\)	\(q-0.584664 q^{2} +0.334185 q^{3} -1.65817 q^{4} -1.00000 q^{5} -0.195386 q^{6} +1.85355 q^{7} +2.13880 q^{8} -2.88832 q^{9} +0.584664 q^{10} +1.00000 q^{11} -0.554135 q^{12} +2.56681 q^{13} -1.08371 q^{14} -0.334185 q^{15} +2.06586 q^{16} -7.56490 q^{17} +1.68870 q^{18} +1.00000 q^{19} +1.65817 q^{20} +0.619430 q^{21} -0.584664 q^{22} -1.73764 q^{23} +0.714755 q^{24} +1.00000 q^{25} -1.50072 q^{26} -1.96779 q^{27} -3.07350 q^{28} +9.32369 q^{29} +0.195386 q^{30} -4.31802 q^{31} -5.48543 q^{32} +0.334185 q^{33} +4.42292 q^{34} -1.85355 q^{35} +4.78932 q^{36} -8.96941 q^{37} -0.584664 q^{38} +0.857791 q^{39} -2.13880 q^{40} -10.6138 q^{41} -0.362159 q^{42} +11.5068 q^{43} -1.65817 q^{44} +2.88832 q^{45} +1.01593 q^{46} -6.50003 q^{47} +0.690379 q^{48} -3.56434 q^{49} -0.584664 q^{50} -2.52808 q^{51} -4.25621 q^{52} -8.73423 q^{53} +1.15050 q^{54} -1.00000 q^{55} +3.96438 q^{56} +0.334185 q^{57} -5.45123 q^{58} -3.97991 q^{59} +0.554135 q^{60} +0.649027 q^{61} +2.52459 q^{62} -5.35366 q^{63} -0.924581 q^{64} -2.56681 q^{65} -0.195386 q^{66} -0.319933 q^{67} +12.5439 q^{68} -0.580692 q^{69} +1.08371 q^{70} -9.93130 q^{71} -6.17754 q^{72} -15.3646 q^{73} +5.24409 q^{74} +0.334185 q^{75} -1.65817 q^{76} +1.85355 q^{77} -0.501520 q^{78} -10.2604 q^{79} -2.06586 q^{80} +8.00735 q^{81} +6.20553 q^{82} +10.7036 q^{83} -1.02712 q^{84} +7.56490 q^{85} -6.72762 q^{86} +3.11584 q^{87} +2.13880 q^{88} -13.3892 q^{89} -1.68870 q^{90} +4.75773 q^{91} +2.88129 q^{92} -1.44302 q^{93} +3.80033 q^{94} -1.00000 q^{95} -1.83315 q^{96} -3.79273 q^{97} +2.08394 q^{98} -2.88832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 3 q^{3} + q^{4} - 5 q^{5} - 4 q^{6} + 3 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 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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