LMFDB - Embedded newform 1338.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1338,2,Mod(1,1338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.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:	$10.6839837904$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.356173.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 $ x^5 - 2x^4 - 7x^3 + 9x^2 + 14x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.253142$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.686428 q^{5} -1.00000 q^{6} +2.87549 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.686428 q^{5} -1.00000 q^{6} +2.87549 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.686428 q^{10} +5.51891 q^{11} +1.00000 q^{12} +0.329853 q^{13} -2.87549 q^{14} +0.686428 q^{15} +1.00000 q^{16} +4.44220 q^{17} -1.00000 q^{18} -1.53786 q^{19} +0.686428 q^{20} +2.87549 q^{21} -5.51891 q^{22} -3.17011 q^{23} -1.00000 q^{24} -4.52882 q^{25} -0.329853 q^{26} +1.00000 q^{27} +2.87549 q^{28} +7.03297 q^{29} -0.686428 q^{30} -7.43855 q^{31} -1.00000 q^{32} +5.51891 q^{33} -4.44220 q^{34} +1.97382 q^{35} +1.00000 q^{36} -7.58613 q^{37} +1.53786 q^{38} +0.329853 q^{39} -0.686428 q^{40} -2.51679 q^{41} -2.87549 q^{42} +6.97369 q^{43} +5.51891 q^{44} +0.686428 q^{45} +3.17011 q^{46} +6.00990 q^{47} +1.00000 q^{48} +1.26844 q^{49} +4.52882 q^{50} +4.44220 q^{51} +0.329853 q^{52} -0.840259 q^{53} -1.00000 q^{54} +3.78834 q^{55} -2.87549 q^{56} -1.53786 q^{57} -7.03297 q^{58} -3.14631 q^{59} +0.686428 q^{60} -2.68741 q^{61} +7.43855 q^{62} +2.87549 q^{63} +1.00000 q^{64} +0.226420 q^{65} -5.51891 q^{66} -13.9815 q^{67} +4.44220 q^{68} -3.17011 q^{69} -1.97382 q^{70} -6.67275 q^{71} -1.00000 q^{72} +11.9575 q^{73} +7.58613 q^{74} -4.52882 q^{75} -1.53786 q^{76} +15.8696 q^{77} -0.329853 q^{78} +12.1543 q^{79} +0.686428 q^{80} +1.00000 q^{81} +2.51679 q^{82} -5.22743 q^{83} +2.87549 q^{84} +3.04925 q^{85} -6.97369 q^{86} +7.03297 q^{87} -5.51891 q^{88} +8.29874 q^{89} -0.686428 q^{90} +0.948489 q^{91} -3.17011 q^{92} -7.43855 q^{93} -6.00990 q^{94} -1.05563 q^{95} -1.00000 q^{96} +4.45112 q^{97} -1.26844 q^{98} +5.51891 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 5 q^{10} + 9 q^{11} + 5 q^{12} + q^{14} + 5 q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} - 4 q^{19} + 5 q^{20} - q^{21} - 9 q^{22} + 16 q^{23} - 5 q^{24} + 8 q^{25} + 5 q^{27} - q^{28} + 8 q^{29} - 5 q^{30} - q^{31} - 5 q^{32} + 9 q^{33} - 6 q^{34} + 22 q^{35} + 5 q^{36} - 2 q^{37} + 4 q^{38} - 5 q^{40} + 4 q^{41} + q^{42} + 3 q^{43} + 9 q^{44} + 5 q^{45} - 16 q^{46} + 18 q^{47} + 5 q^{48} + 2 q^{49} - 8 q^{50} + 6 q^{51} + 26 q^{53} - 5 q^{54} + q^{55} + q^{56} - 4 q^{57} - 8 q^{58} + 21 q^{59} + 5 q^{60} - 20 q^{61} + q^{62} - q^{63} + 5 q^{64} - 3 q^{65} - 9 q^{66} - 5 q^{67} + 6 q^{68} + 16 q^{69} - 22 q^{70} + 17 q^{71} - 5 q^{72} + 5 q^{73} + 2 q^{74} + 8 q^{75} - 4 q^{76} + 2 q^{77} - 21 q^{79} + 5 q^{80} + 5 q^{81} - 4 q^{82} + 11 q^{83} - q^{84} - 12 q^{85} - 3 q^{86} + 8 q^{87} - 9 q^{88} - 5 q^{89} - 5 q^{90} - 10 q^{91} + 16 q^{92} - q^{93} - 18 q^{94} + 10 q^{95} - 5 q^{96} - 11 q^{97} - 2 q^{98} + 9 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 5 * q^10 + 9 * q^11 + 5 * q^12 + q^14 + 5 * q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 - 4 * q^19 + 5 * q^20 - q^21 - 9 * q^22 + 16 * q^23 - 5 * q^24 + 8 * q^25 + 5 * q^27 - q^28 + 8 * q^29 - 5 * q^30 - q^31 - 5 * q^32 + 9 * q^33 - 6 * q^34 + 22 * q^35 + 5 * q^36 - 2 * q^37 + 4 * q^38 - 5 * q^40 + 4 * q^41 + q^42 + 3 * q^43 + 9 * q^44 + 5 * q^45 - 16 * q^46 + 18 * q^47 + 5 * q^48 + 2 * q^49 - 8 * q^50 + 6 * q^51 + 26 * q^53 - 5 * q^54 + q^55 + q^56 - 4 * q^57 - 8 * q^58 + 21 * q^59 + 5 * q^60 - 20 * q^61 + q^62 - q^63 + 5 * q^64 - 3 * q^65 - 9 * q^66 - 5 * q^67 + 6 * q^68 + 16 * q^69 - 22 * q^70 + 17 * q^71 - 5 * q^72 + 5 * q^73 + 2 * q^74 + 8 * q^75 - 4 * q^76 + 2 * q^77 - 21 * q^79 + 5 * q^80 + 5 * q^81 - 4 * q^82 + 11 * q^83 - q^84 - 12 * q^85 - 3 * q^86 + 8 * q^87 - 9 * q^88 - 5 * q^89 - 5 * q^90 - 10 * q^91 + 16 * q^92 - q^93 - 18 * q^94 + 10 * q^95 - 5 * q^96 - 11 * q^97 - 2 * q^98 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.686428	0.306980	0.153490	−	0.988150i	$-0.450949\pi$
$5$	0.686428	0.306980	0.153490	+	0.988150i	$0.450949\pi$
$6$	−1.00000	−0.408248
$7$	2.87549	1.08683	0.543416	−	0.839463i	$-0.317130\pi$
$7$	2.87549	1.08683	0.543416	+	0.839463i	$0.317130\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.686428	−0.217068
$11$	5.51891	1.66402	0.832008	−	0.554764i	$-0.187192\pi$
$11$	5.51891	1.66402	0.832008	+	0.554764i	$0.187192\pi$
$12$	1.00000	0.288675
$13$	0.329853	0.0914848	0.0457424	−	0.998953i	$-0.485435\pi$
$13$	0.329853	0.0914848	0.0457424	+	0.998953i	$0.485435\pi$
$14$	−2.87549	−0.768507
$15$	0.686428	0.177235
$16$	1.00000	0.250000
$17$	4.44220	1.07739	0.538696	−	0.842500i	$-0.318917\pi$
$17$	4.44220	1.07739	0.538696	+	0.842500i	$0.318917\pi$
$18$	−1.00000	−0.235702
$19$	−1.53786	−0.352810	−0.176405	−	0.984318i	$-0.556447\pi$
$19$	−1.53786	−0.352810	−0.176405	+	0.984318i	$0.556447\pi$
$20$	0.686428	0.153490
$21$	2.87549	0.627483
$22$	−5.51891	−1.17664
$23$	−3.17011	−0.661014	−0.330507	−	0.943804i	$-0.607220\pi$
$23$	−3.17011	−0.661014	−0.330507	+	0.943804i	$0.607220\pi$
$24$	−1.00000	−0.204124
$25$	−4.52882	−0.905763
$26$	−0.329853	−0.0646895
$27$	1.00000	0.192450
$28$	2.87549	0.543416
$29$	7.03297	1.30599	0.652995	−	0.757362i	$-0.273512\pi$
$29$	7.03297	1.30599	0.652995	+	0.757362i	$0.273512\pi$
$30$	−0.686428	−0.125324
$31$	−7.43855	−1.33600	−0.668002	−	0.744160i	$-0.732850\pi$
$31$	−7.43855	−1.33600	−0.668002	+	0.744160i	$0.732850\pi$
$32$	−1.00000	−0.176777
$33$	5.51891	0.960720
$34$	−4.44220	−0.761832
$35$	1.97382	0.333636
$36$	1.00000	0.166667
$37$	−7.58613	−1.24715	−0.623576	−	0.781762i	$-0.714321\pi$
$37$	−7.58613	−1.24715	−0.623576	+	0.781762i	$0.714321\pi$
$38$	1.53786	0.249474
$39$	0.329853	0.0528188
$40$	−0.686428	−0.108534
$41$	−2.51679	−0.393056	−0.196528	−	0.980498i	$-0.562967\pi$
$41$	−2.51679	−0.393056	−0.196528	+	0.980498i	$0.562967\pi$
$42$	−2.87549	−0.443698
$43$	6.97369	1.06348	0.531739	−	0.846908i	$-0.321539\pi$
$43$	6.97369	1.06348	0.531739	+	0.846908i	$0.321539\pi$
$44$	5.51891	0.832008
$45$	0.686428	0.102327
$46$	3.17011	0.467408
$47$	6.00990	0.876634	0.438317	−	0.898820i	$-0.355575\pi$
$47$	6.00990	0.876634	0.438317	+	0.898820i	$0.355575\pi$
$48$	1.00000	0.144338
$49$	1.26844	0.181206
$50$	4.52882	0.640471
$51$	4.44220	0.622033
$52$	0.329853	0.0457424
$53$	−0.840259	−0.115419	−0.0577093	−	0.998333i	$-0.518380\pi$
$53$	−0.840259	−0.115419	−0.0577093	+	0.998333i	$0.518380\pi$
$54$	−1.00000	−0.136083
$55$	3.78834	0.510819
$56$	−2.87549	−0.384253
$57$	−1.53786	−0.203695
$58$	−7.03297	−0.923475
$59$	−3.14631	−0.409614	−0.204807	−	0.978802i	$-0.565657\pi$
$59$	−3.14631	−0.409614	−0.204807	+	0.978802i	$0.565657\pi$
$60$	0.686428	0.0886175
$61$	−2.68741	−0.344088	−0.172044	−	0.985089i	$-0.555037\pi$
$61$	−2.68741	−0.344088	−0.172044	+	0.985089i	$0.555037\pi$
$62$	7.43855	0.944697
$63$	2.87549	0.362278
$64$	1.00000	0.125000
$65$	0.226420	0.0280840
$66$	−5.51891	−0.679331
$67$	−13.9815	−1.70811	−0.854057	−	0.520179i	$-0.825865\pi$
$67$	−13.9815	−1.70811	−0.854057	+	0.520179i	$0.825865\pi$
$68$	4.44220	0.538696
$69$	−3.17011	−0.381637
$70$	−1.97382	−0.235916
$71$	−6.67275	−0.791909	−0.395955	−	0.918270i	$-0.629586\pi$
$71$	−6.67275	−0.791909	−0.395955	+	0.918270i	$0.629586\pi$
$72$	−1.00000	−0.117851
$73$	11.9575	1.39952	0.699758	−	0.714380i	$-0.253291\pi$
$73$	11.9575	1.39952	0.699758	+	0.714380i	$0.253291\pi$
$74$	7.58613	0.881870
$75$	−4.52882	−0.522943
$76$	−1.53786	−0.176405
$77$	15.8696	1.80851
$78$	−0.329853	−0.0373485
$79$	12.1543	1.36747	0.683733	−	0.729732i	$-0.260355\pi$
$79$	12.1543	1.36747	0.683733	+	0.729732i	$0.260355\pi$
$80$	0.686428	0.0767450
$81$	1.00000	0.111111
$82$	2.51679	0.277932
$83$	−5.22743	−0.573785	−0.286892	−	0.957963i	$-0.592622\pi$
$83$	−5.22743	−0.573785	−0.286892	+	0.957963i	$0.592622\pi$
$84$	2.87549	0.313742
$85$	3.04925	0.330738
$86$	−6.97369	−0.751992
$87$	7.03297	0.754014
$88$	−5.51891	−0.588318
$89$	8.29874	0.879665	0.439833	−	0.898080i	$-0.355038\pi$
$89$	8.29874	0.879665	0.439833	+	0.898080i	$0.355038\pi$
$90$	−0.686428	−0.0723559
$91$	0.948489	0.0994286
$92$	−3.17011	−0.330507
$93$	−7.43855	−0.771342
$94$	−6.00990	−0.619874
$95$	−1.05563	−0.108306
$96$	−1.00000	−0.102062
$97$	4.45112	0.451943	0.225971	−	0.974134i	$-0.427444\pi$
$97$	4.45112	0.451943	0.225971	+	0.974134i	$0.427444\pi$
$98$	−1.26844	−0.128132
$99$	5.51891	0.554672
$100$	−4.52882	−0.452882
$101$	−3.23632	−0.322026	−0.161013	−	0.986952i	$-0.551476\pi$
$101$	−3.23632	−0.322026	−0.161013	+	0.986952i	$0.551476\pi$
$102$	−4.44220	−0.439844
$103$	−12.7274	−1.25407	−0.627034	−	0.778992i	$-0.715731\pi$
$103$	−12.7274	−1.25407	−0.627034	+	0.778992i	$0.715731\pi$
$104$	−0.329853	−0.0323447
$105$	1.97382	0.192625
$106$	0.840259	0.0816132
$107$	4.52529	0.437477	0.218738	−	0.975784i	$-0.429806\pi$
$107$	4.52529	0.437477	0.218738	+	0.975784i	$0.429806\pi$
$108$	1.00000	0.0962250
$109$	−10.5371	−1.00927	−0.504637	−	0.863332i	$-0.668374\pi$
$109$	−10.5371	−1.00927	−0.504637	+	0.863332i	$0.668374\pi$
$110$	−3.78834	−0.361204
$111$	−7.58613	−0.720044
$112$	2.87549	0.271708
$113$	2.93690	0.276281	0.138140	−	0.990413i	$-0.455888\pi$
$113$	2.93690	0.276281	0.138140	+	0.990413i	$0.455888\pi$
$114$	1.53786	0.144034
$115$	−2.17605	−0.202918
$116$	7.03297	0.652995
$117$	0.329853	0.0304949
$118$	3.14631	0.289641
$119$	12.7735	1.17095
$120$	−0.686428	−0.0626620
$121$	19.4584	1.76895
$122$	2.68741	0.243307
$123$	−2.51679	−0.226931
$124$	−7.43855	−0.668002
$125$	−6.54085	−0.585031
$126$	−2.87549	−0.256169
$127$	4.04306	0.358764	0.179382	−	0.983780i	$-0.442590\pi$
$127$	4.04306	0.358764	0.179382	+	0.983780i	$0.442590\pi$
$128$	−1.00000	−0.0883883
$129$	6.97369	0.613999
$130$	−0.226420	−0.0198584
$131$	6.64749	0.580794	0.290397	−	0.956906i	$-0.406213\pi$
$131$	6.64749	0.580794	0.290397	+	0.956906i	$0.406213\pi$
$132$	5.51891	0.480360
$133$	−4.42211	−0.383446
$134$	13.9815	1.20782
$135$	0.686428	0.0590783
$136$	−4.44220	−0.380916
$137$	10.9737	0.937545	0.468773	−	0.883319i	$-0.344696\pi$
$137$	10.9737	0.937545	0.468773	+	0.883319i	$0.344696\pi$
$138$	3.17011	0.269858
$139$	−15.2142	−1.29045	−0.645225	−	0.763992i	$-0.723237\pi$
$139$	−15.2142	−1.29045	−0.645225	+	0.763992i	$0.723237\pi$
$140$	1.97382	0.166818
$141$	6.00990	0.506125
$142$	6.67275	0.559964
$143$	1.82043	0.152232
$144$	1.00000	0.0833333
$145$	4.82763	0.400913
$146$	−11.9575	−0.989607
$147$	1.26844	0.104619
$148$	−7.58613	−0.623576
$149$	2.90167	0.237714	0.118857	−	0.992911i	$-0.462077\pi$
$149$	2.90167	0.237714	0.118857	+	0.992911i	$0.462077\pi$
$150$	4.52882	0.369776
$151$	2.66399	0.216792	0.108396	−	0.994108i	$-0.465429\pi$
$151$	2.66399	0.216792	0.108396	+	0.994108i	$0.465429\pi$
$152$	1.53786	0.124737
$153$	4.44220	0.359131
$154$	−15.8696	−1.27881
$155$	−5.10603	−0.410126
$156$	0.329853	0.0264094
$157$	15.2048	1.21347	0.606736	−	0.794903i	$-0.292478\pi$
$157$	15.2048	1.21347	0.606736	+	0.794903i	$0.292478\pi$
$158$	−12.1543	−0.966944
$159$	−0.840259	−0.0666369
$160$	−0.686428	−0.0542669
$161$	−9.11562	−0.718412
$162$	−1.00000	−0.0785674
$163$	12.4990	0.978996	0.489498	−	0.872004i	$-0.337180\pi$
$163$	12.4990	0.978996	0.489498	+	0.872004i	$0.337180\pi$
$164$	−2.51679	−0.196528
$165$	3.78834	0.294922
$166$	5.22743	0.405727
$167$	6.88133	0.532493	0.266247	−	0.963905i	$-0.414216\pi$
$167$	6.88133	0.532493	0.266247	+	0.963905i	$0.414216\pi$
$168$	−2.87549	−0.221849
$169$	−12.8912	−0.991631
$170$	−3.04925	−0.233867
$171$	−1.53786	−0.117603
$172$	6.97369	0.531739
$173$	−3.65630	−0.277984	−0.138992	−	0.990294i	$-0.544386\pi$
$173$	−3.65630	−0.277984	−0.138992	+	0.990294i	$0.544386\pi$
$174$	−7.03297	−0.533168
$175$	−13.0226	−0.984413
$176$	5.51891	0.416004
$177$	−3.14631	−0.236491
$178$	−8.29874	−0.622017
$179$	11.6410	0.870090	0.435045	−	0.900409i	$-0.356733\pi$
$179$	11.6410	0.870090	0.435045	+	0.900409i	$0.356733\pi$
$180$	0.686428	0.0511633
$181$	−13.8301	−1.02798	−0.513990	−	0.857796i	$-0.671833\pi$
$181$	−13.8301	−1.02798	−0.513990	+	0.857796i	$0.671833\pi$
$182$	−0.948489	−0.0703067
$183$	−2.68741	−0.198659
$184$	3.17011	0.233704
$185$	−5.20733	−0.382851
$186$	7.43855	0.545421
$187$	24.5161	1.79280
$188$	6.00990	0.438317
$189$	2.87549	0.209161
$190$	1.05563	0.0765836
$191$	−15.4332	−1.11671	−0.558354	−	0.829603i	$-0.688567\pi$
$191$	−15.4332	−1.11671	−0.558354	+	0.829603i	$0.688567\pi$
$192$	1.00000	0.0721688
$193$	−1.86292	−0.134096	−0.0670479	−	0.997750i	$-0.521358\pi$
$193$	−1.86292	−0.134096	−0.0670479	+	0.997750i	$0.521358\pi$
$194$	−4.45112	−0.319572
$195$	0.226420	0.0162143
$196$	1.26844	0.0906028
$197$	−18.0327	−1.28477	−0.642387	−	0.766381i	$-0.722056\pi$
$197$	−18.0327	−1.28477	−0.642387	+	0.766381i	$0.722056\pi$
$198$	−5.51891	−0.392212
$199$	24.0118	1.70215	0.851077	−	0.525042i	$-0.175950\pi$
$199$	24.0118	1.70215	0.851077	+	0.525042i	$0.175950\pi$
$200$	4.52882	0.320236
$201$	−13.9815	−0.986180
$202$	3.23632	0.227707
$203$	20.2232	1.41939
$204$	4.44220	0.311016
$205$	−1.72759	−0.120660
$206$	12.7274	0.886760
$207$	−3.17011	−0.220338
$208$	0.329853	0.0228712
$209$	−8.48734	−0.587081
$210$	−1.97382	−0.136206
$211$	6.05163	0.416611	0.208306	−	0.978064i	$-0.433205\pi$
$211$	6.05163	0.416611	0.208306	+	0.978064i	$0.433205\pi$
$212$	−0.840259	−0.0577093
$213$	−6.67275	−0.457209
$214$	−4.52529	−0.309343
$215$	4.78694	0.326466
$216$	−1.00000	−0.0680414
$217$	−21.3895	−1.45201
$218$	10.5371	0.713664
$219$	11.9575	0.808011
$220$	3.78834	0.255410
$221$	1.46527	0.0985650
$222$	7.58613	0.509148
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−2.87549	−0.192127
$225$	−4.52882	−0.301921
$226$	−2.93690	−0.195360
$227$	24.3730	1.61769	0.808846	−	0.588021i	$-0.200093\pi$
$227$	24.3730	1.61769	0.808846	+	0.588021i	$0.200093\pi$
$228$	−1.53786	−0.101847
$229$	7.57496	0.500567	0.250284	−	0.968173i	$-0.419476\pi$
$229$	7.57496	0.500567	0.250284	+	0.968173i	$0.419476\pi$
$230$	2.17605	0.143485
$231$	15.8696	1.04414
$232$	−7.03297	−0.461737
$233$	1.91430	0.125410	0.0627050	−	0.998032i	$-0.480027\pi$
$233$	1.91430	0.125410	0.0627050	+	0.998032i	$0.480027\pi$
$234$	−0.329853	−0.0215632
$235$	4.12537	0.269109
$236$	−3.14631	−0.204807
$237$	12.1543	0.789507
$238$	−12.7735	−0.827984
$239$	−18.8018	−1.21619	−0.608094	−	0.793865i	$-0.708066\pi$
$239$	−18.8018	−1.21619	−0.608094	+	0.793865i	$0.708066\pi$
$240$	0.686428	0.0443087
$241$	−12.7202	−0.819378	−0.409689	−	0.912225i	$-0.634363\pi$
$241$	−12.7202	−0.819378	−0.409689	+	0.912225i	$0.634363\pi$
$242$	−19.4584	−1.25083
$243$	1.00000	0.0641500
$244$	−2.68741	−0.172044
$245$	0.870693	0.0556265
$246$	2.51679	0.160464
$247$	−0.507269	−0.0322767
$248$	7.43855	0.472349
$249$	−5.22743	−0.331275
$250$	6.54085	0.413680
$251$	8.40005	0.530206	0.265103	−	0.964220i	$-0.414594\pi$
$251$	8.40005	0.530206	0.265103	+	0.964220i	$0.414594\pi$
$252$	2.87549	0.181139
$253$	−17.4956	−1.09994
$254$	−4.04306	−0.253684
$255$	3.04925	0.190952
$256$	1.00000	0.0625000
$257$	−23.1611	−1.44475	−0.722375	−	0.691502i	$-0.756949\pi$
$257$	−23.1611	−1.44475	−0.722375	+	0.691502i	$0.756949\pi$
$258$	−6.97369	−0.434163
$259$	−21.8138	−1.35545
$260$	0.226420	0.0140420
$261$	7.03297	0.435330
$262$	−6.64749	−0.410683
$263$	6.61626	0.407976	0.203988	−	0.978973i	$-0.434610\pi$
$263$	6.61626	0.407976	0.203988	+	0.978973i	$0.434610\pi$
$264$	−5.51891	−0.339666
$265$	−0.576778	−0.0354312
$266$	4.42211	0.271137
$267$	8.29874	0.507875
$268$	−13.9815	−0.854057
$269$	−4.44005	−0.270715	−0.135357	−	0.990797i	$-0.543218\pi$
$269$	−4.44005	−0.270715	−0.135357	+	0.990797i	$0.543218\pi$
$270$	−0.686428	−0.0417747
$271$	−6.05643	−0.367902	−0.183951	−	0.982935i	$-0.558889\pi$
$271$	−6.05643	−0.367902	−0.183951	+	0.982935i	$0.558889\pi$
$272$	4.44220	0.269348
$273$	0.948489	0.0574052
$274$	−10.9737	−0.662945
$275$	−24.9942	−1.50720
$276$	−3.17011	−0.190818
$277$	−25.7454	−1.54689	−0.773446	−	0.633862i	$-0.781469\pi$
$277$	−25.7454	−1.54689	−0.773446	+	0.633862i	$0.781469\pi$
$278$	15.2142	0.912487
$279$	−7.43855	−0.445334
$280$	−1.97382	−0.117958
$281$	−14.7983	−0.882790	−0.441395	−	0.897313i	$-0.645516\pi$
$281$	−14.7983	−0.882790	−0.441395	+	0.897313i	$0.645516\pi$
$282$	−6.00990	−0.357884
$283$	−4.17776	−0.248342	−0.124171	−	0.992261i	$-0.539627\pi$
$283$	−4.17776	−0.248342	−0.124171	+	0.992261i	$0.539627\pi$
$284$	−6.67275	−0.395955
$285$	−1.05563	−0.0625303
$286$	−1.82043	−0.107644
$287$	−7.23699	−0.427186
$288$	−1.00000	−0.0589256
$289$	2.73318	0.160775
$290$	−4.82763	−0.283488
$291$	4.45112	0.260929
$292$	11.9575	0.699758
$293$	−12.6202	−0.737281	−0.368641	−	0.929572i	$-0.620177\pi$
$293$	−12.6202	−0.737281	−0.368641	+	0.929572i	$0.620177\pi$
$294$	−1.26844	−0.0739769
$295$	−2.15971	−0.125743
$296$	7.58613	0.440935
$297$	5.51891	0.320240
$298$	−2.90167	−0.168089
$299$	−1.04567	−0.0604727
$300$	−4.52882	−0.261471
$301$	20.0528	1.15582
$302$	−2.66399	−0.153295
$303$	−3.23632	−0.185922
$304$	−1.53786	−0.0882025
$305$	−1.84472	−0.105628
$306$	−4.44220	−0.253944
$307$	−32.7514	−1.86922	−0.934612	−	0.355669i	$-0.884253\pi$
$307$	−32.7514	−1.86922	−0.934612	+	0.355669i	$0.884253\pi$
$308$	15.8696	0.904253
$309$	−12.7274	−0.724036
$310$	5.10603	0.290003
$311$	−5.09711	−0.289031	−0.144515	−	0.989503i	$-0.546162\pi$
$311$	−5.09711	−0.289031	−0.144515	+	0.989503i	$0.546162\pi$
$312$	−0.329853	−0.0186742
$313$	3.70513	0.209426	0.104713	−	0.994502i	$-0.466608\pi$
$313$	3.70513	0.209426	0.104713	+	0.994502i	$0.466608\pi$
$314$	−15.2048	−0.858055
$315$	1.97382	0.111212
$316$	12.1543	0.683733
$317$	11.1944	0.628743	0.314371	−	0.949300i	$-0.398206\pi$
$317$	11.1944	0.628743	0.314371	+	0.949300i	$0.398206\pi$
$318$	0.840259	0.0471194
$319$	38.8144	2.17319
$320$	0.686428	0.0383725
$321$	4.52529	0.252577
$322$	9.11562	0.507994
$323$	−6.83150	−0.380115
$324$	1.00000	0.0555556
$325$	−1.49384	−0.0828635
$326$	−12.4990	−0.692255
$327$	−10.5371	−0.582705
$328$	2.51679	0.138966
$329$	17.2814	0.952755
$330$	−3.78834	−0.208541
$331$	−30.7807	−1.69186	−0.845930	−	0.533293i	$-0.820954\pi$
$331$	−30.7807	−1.69186	−0.845930	+	0.533293i	$0.820954\pi$
$332$	−5.22743	−0.286892
$333$	−7.58613	−0.415718
$334$	−6.88133	−0.376530
$335$	−9.59731	−0.524357
$336$	2.87549	0.156871
$337$	−28.0781	−1.52951	−0.764755	−	0.644322i	$-0.777140\pi$
$337$	−28.0781	−1.52951	−0.764755	+	0.644322i	$0.777140\pi$
$338$	12.8912	0.701189
$339$	2.93690	0.159511
$340$	3.04925	0.165369
$341$	−41.0527	−2.22313
$342$	1.53786	0.0831581
$343$	−16.4810	−0.889893
$344$	−6.97369	−0.375996
$345$	−2.17605	−0.117155
$346$	3.65630	0.196564
$347$	4.81474	0.258469	0.129234	−	0.991614i	$-0.458748\pi$
$347$	4.81474	0.258469	0.129234	+	0.991614i	$0.458748\pi$
$348$	7.03297	0.377007
$349$	20.7579	1.11114	0.555571	−	0.831469i	$-0.312500\pi$
$349$	20.7579	1.11114	0.555571	+	0.831469i	$0.312500\pi$
$350$	13.0226	0.696085
$351$	0.329853	0.0176063
$352$	−5.51891	−0.294159
$353$	−3.21876	−0.171317	−0.0856587	−	0.996325i	$-0.527299\pi$
$353$	−3.21876	−0.171317	−0.0856587	+	0.996325i	$0.527299\pi$
$354$	3.14631	0.167224
$355$	−4.58036	−0.243100
$356$	8.29874	0.439833
$357$	12.7735	0.676046
$358$	−11.6410	−0.615246
$359$	25.5107	1.34641	0.673203	−	0.739458i	$-0.264918\pi$
$359$	25.5107	1.34641	0.673203	+	0.739458i	$0.264918\pi$
$360$	−0.686428	−0.0361779
$361$	−16.6350	−0.875525
$362$	13.8301	0.726892
$363$	19.4584	1.02130
$364$	0.948489	0.0497143
$365$	8.20794	0.429623
$366$	2.68741	0.140473
$367$	−24.4579	−1.27669	−0.638346	−	0.769750i	$-0.720381\pi$
$367$	−24.4579	−1.27669	−0.638346	+	0.769750i	$0.720381\pi$
$368$	−3.17011	−0.165254
$369$	−2.51679	−0.131019
$370$	5.20733	0.270716
$371$	−2.41616	−0.125441
$372$	−7.43855	−0.385671
$373$	−9.79830	−0.507337	−0.253668	−	0.967291i	$-0.581637\pi$
$373$	−9.79830	−0.507337	−0.253668	+	0.967291i	$0.581637\pi$
$374$	−24.5161	−1.26770
$375$	−6.54085	−0.337768
$376$	−6.00990	−0.309937
$377$	2.31985	0.119478
$378$	−2.87549	−0.147899
$379$	23.3632	1.20009	0.600043	−	0.799968i	$-0.295150\pi$
$379$	23.3632	1.20009	0.600043	+	0.799968i	$0.295150\pi$
$380$	−1.05563	−0.0541528
$381$	4.04306	0.207132
$382$	15.4332	0.789632
$383$	26.1424	1.33582	0.667908	−	0.744244i	$-0.267190\pi$
$383$	26.1424	1.33582	0.667908	+	0.744244i	$0.267190\pi$
$384$	−1.00000	−0.0510310
$385$	10.8933	0.555175
$386$	1.86292	0.0948201
$387$	6.97369	0.354492
$388$	4.45112	0.225971
$389$	3.73549	0.189397	0.0946985	−	0.995506i	$-0.469811\pi$
$389$	3.73549	0.189397	0.0946985	+	0.995506i	$0.469811\pi$
$390$	−0.226420	−0.0114652
$391$	−14.0823	−0.712172
$392$	−1.26844	−0.0640659
$393$	6.64749	0.335321
$394$	18.0327	0.908472
$395$	8.34305	0.419785
$396$	5.51891	0.277336
$397$	17.6199	0.884316	0.442158	−	0.896937i	$-0.354213\pi$
$397$	17.6199	0.884316	0.442158	+	0.896937i	$0.354213\pi$
$398$	−24.0118	−1.20360
$399$	−4.42211	−0.221382
$400$	−4.52882	−0.226441
$401$	−22.5232	−1.12475	−0.562376	−	0.826881i	$-0.690113\pi$
$401$	−22.5232	−1.12475	−0.562376	+	0.826881i	$0.690113\pi$
$402$	13.9815	0.697335
$403$	−2.45363	−0.122224
$404$	−3.23632	−0.161013
$405$	0.686428	0.0341089
$406$	−20.2232	−1.00366
$407$	−41.8672	−2.07528
$408$	−4.44220	−0.219922
$409$	−12.4542	−0.615821	−0.307910	−	0.951415i	$-0.599630\pi$
$409$	−12.4542	−0.615821	−0.307910	+	0.951415i	$0.599630\pi$
$410$	1.72759	0.0853197
$411$	10.9737	0.541292
$412$	−12.7274	−0.627034
$413$	−9.04717	−0.445182
$414$	3.17011	0.155803
$415$	−3.58825	−0.176140
$416$	−0.329853	−0.0161724
$417$	−15.2142	−0.745042
$418$	8.48734	0.415129
$419$	−30.5170	−1.49086	−0.745428	−	0.666586i	$-0.767755\pi$
$419$	−30.5170	−1.49086	−0.745428	+	0.666586i	$0.767755\pi$
$420$	1.97382	0.0963124
$421$	5.17369	0.252150	0.126075	−	0.992021i	$-0.459762\pi$
$421$	5.17369	0.252150	0.126075	+	0.992021i	$0.459762\pi$
$422$	−6.05163	−0.294589
$423$	6.00990	0.292211
$424$	0.840259	0.0408066
$425$	−20.1179	−0.975863
$426$	6.67275	0.323296
$427$	−7.72762	−0.373966
$428$	4.52529	0.218738
$429$	1.82043	0.0878912
$430$	−4.78694	−0.230847
$431$	−11.8685	−0.571686	−0.285843	−	0.958277i	$-0.592274\pi$
$431$	−11.8685	−0.571686	−0.285843	+	0.958277i	$0.592274\pi$
$432$	1.00000	0.0481125
$433$	2.77072	0.133152	0.0665762	−	0.997781i	$-0.478792\pi$
$433$	2.77072	0.133152	0.0665762	+	0.997781i	$0.478792\pi$
$434$	21.3895	1.02673
$435$	4.82763	0.231467
$436$	−10.5371	−0.504637
$437$	4.87520	0.233212
$438$	−11.9575	−0.571350
$439$	−13.5404	−0.646248	−0.323124	−	0.946357i	$-0.604733\pi$
$439$	−13.5404	−0.646248	−0.323124	+	0.946357i	$0.604733\pi$
$440$	−3.78834	−0.180602
$441$	1.26844	0.0604019
$442$	−1.46527	−0.0696960
$443$	38.7298	1.84011	0.920054	−	0.391791i	$-0.128144\pi$
$443$	38.7298	1.84011	0.920054	+	0.391791i	$0.128144\pi$
$444$	−7.58613	−0.360022
$445$	5.69649	0.270040
$446$	1.00000	0.0473514
$447$	2.90167	0.137244
$448$	2.87549	0.135854
$449$	−35.2746	−1.66471	−0.832356	−	0.554241i	$-0.813008\pi$
$449$	−35.2746	−1.66471	−0.832356	+	0.554241i	$0.813008\pi$
$450$	4.52882	0.213490
$451$	−13.8899	−0.654051
$452$	2.93690	0.138140
$453$	2.66399	0.125165
$454$	−24.3730	−1.14388
$455$	0.651069	0.0305226
$456$	1.53786	0.0720170
$457$	−15.7179	−0.735251	−0.367626	−	0.929974i	$-0.619829\pi$
$457$	−15.7179	−0.735251	−0.367626	+	0.929974i	$0.619829\pi$
$458$	−7.57496	−0.353955
$459$	4.44220	0.207344
$460$	−2.17605	−0.101459
$461$	−15.4964	−0.721739	−0.360869	−	0.932616i	$-0.617520\pi$
$461$	−15.4964	−0.721739	−0.360869	+	0.932616i	$0.617520\pi$
$462$	−15.8696	−0.738320
$463$	34.0979	1.58466	0.792331	−	0.610091i	$-0.208867\pi$
$463$	34.0979	1.58466	0.792331	+	0.610091i	$0.208867\pi$
$464$	7.03297	0.326498
$465$	−5.10603	−0.236787
$466$	−1.91430	−0.0886783
$467$	5.76270	0.266666	0.133333	−	0.991071i	$-0.457432\pi$
$467$	5.76270	0.266666	0.133333	+	0.991071i	$0.457432\pi$
$468$	0.329853	0.0152475
$469$	−40.2037	−1.85644
$470$	−4.12537	−0.190289
$471$	15.2048	0.700599
$472$	3.14631	0.144821
$473$	38.4872	1.76964
$474$	−12.1543	−0.558266
$475$	6.96470	0.319562
$476$	12.7735	0.585473
$477$	−0.840259	−0.0384728
$478$	18.8018	0.859975
$479$	−0.892156	−0.0407637	−0.0203818	−	0.999792i	$-0.506488\pi$
$479$	−0.892156	−0.0407637	−0.0203818	+	0.999792i	$0.506488\pi$
$480$	−0.686428	−0.0313310
$481$	−2.50231	−0.114095
$482$	12.7202	0.579387
$483$	−9.11562	−0.414775
$484$	19.4584	0.884473
$485$	3.05538	0.138737
$486$	−1.00000	−0.0453609
$487$	4.83587	0.219134	0.109567	−	0.993979i	$-0.465054\pi$
$487$	4.83587	0.219134	0.109567	+	0.993979i	$0.465054\pi$
$488$	2.68741	0.121653
$489$	12.4990	0.565223
$490$	−0.870693	−0.0393339
$491$	−14.5280	−0.655640	−0.327820	−	0.944740i	$-0.606314\pi$
$491$	−14.5280	−0.655640	−0.327820	+	0.944740i	$0.606314\pi$
$492$	−2.51679	−0.113465
$493$	31.2419	1.40706
$494$	0.507269	0.0228231
$495$	3.78834	0.170273
$496$	−7.43855	−0.334001
$497$	−19.1874	−0.860673
$498$	5.22743	0.234247
$499$	42.7043	1.91171	0.955854	−	0.293842i	$-0.0949339\pi$
$499$	42.7043	1.91171	0.955854	+	0.293842i	$0.0949339\pi$
$500$	−6.54085	−0.292516
$501$	6.88133	0.307435
$502$	−8.40005	−0.374912
$503$	−28.7298	−1.28100	−0.640499	−	0.767959i	$-0.721273\pi$
$503$	−28.7298	−1.28100	−0.640499	+	0.767959i	$0.721273\pi$
$504$	−2.87549	−0.128084
$505$	−2.22150	−0.0988556
$506$	17.4956	0.777773
$507$	−12.8912	−0.572518
$508$	4.04306	0.179382
$509$	−31.9400	−1.41572	−0.707858	−	0.706355i	$-0.750338\pi$
$509$	−31.9400	−1.41572	−0.707858	+	0.706355i	$0.750338\pi$
$510$	−3.04925	−0.135023
$511$	34.3836	1.52104
$512$	−1.00000	−0.0441942
$513$	−1.53786	−0.0678983
$514$	23.1611	1.02159
$515$	−8.73644	−0.384974
$516$	6.97369	0.307000
$517$	33.1681	1.45873
$518$	21.8138	0.958445
$519$	−3.65630	−0.160494
$520$	−0.226420	−0.00992919
$521$	34.5500	1.51366	0.756832	−	0.653609i	$-0.226746\pi$
$521$	34.5500	1.51366	0.756832	+	0.653609i	$0.226746\pi$
$522$	−7.03297	−0.307825
$523$	20.3878	0.891498	0.445749	−	0.895158i	$-0.352937\pi$
$523$	20.3878	0.891498	0.445749	+	0.895158i	$0.352937\pi$
$524$	6.64749	0.290397
$525$	−13.0226	−0.568351
$526$	−6.61626	−0.288483
$527$	−33.0436	−1.43940
$528$	5.51891	0.240180
$529$	−12.9504	−0.563060
$530$	0.576778	0.0250536
$531$	−3.14631	−0.136538
$532$	−4.42211	−0.191723
$533$	−0.830169	−0.0359586
$534$	−8.29874	−0.359122
$535$	3.10629	0.134297
$536$	13.9815	0.603910
$537$	11.6410	0.502346
$538$	4.44005	0.191424
$539$	7.00041	0.301529
$540$	0.686428	0.0295392
$541$	−26.5035	−1.13948	−0.569738	−	0.821827i	$-0.692955\pi$
$541$	−26.5035	−1.13948	−0.569738	+	0.821827i	$0.692955\pi$
$542$	6.05643	0.260146
$543$	−13.8301	−0.593505
$544$	−4.44220	−0.190458
$545$	−7.23298	−0.309827
$546$	−0.948489	−0.0405916
$547$	−0.362415	−0.0154958	−0.00774788	−	0.999970i	$-0.502466\pi$
$547$	−0.362415	−0.0154958	−0.00774788	+	0.999970i	$0.502466\pi$
$548$	10.9737	0.468773
$549$	−2.68741	−0.114696
$550$	24.9942	1.06575
$551$	−10.8158	−0.460766
$552$	3.17011	0.134929
$553$	34.9496	1.48621
$554$	25.7454	1.09382
$555$	−5.20733	−0.221039
$556$	−15.2142	−0.645225
$557$	−17.2544	−0.731092	−0.365546	−	0.930793i	$-0.619118\pi$
$557$	−17.2544	−0.731092	−0.365546	+	0.930793i	$0.619118\pi$
$558$	7.43855	0.314899
$559$	2.30029	0.0972920
$560$	1.97382	0.0834090
$561$	24.5161	1.03507
$562$	14.7983	0.624227
$563$	38.1428	1.60753	0.803764	−	0.594948i	$-0.202827\pi$
$563$	38.1428	1.60753	0.803764	+	0.594948i	$0.202827\pi$
$564$	6.00990	0.253063
$565$	2.01597	0.0848126
$566$	4.17776	0.175604
$567$	2.87549	0.120759
$568$	6.67275	0.279982
$569$	−9.79988	−0.410832	−0.205416	−	0.978675i	$-0.565855\pi$
$569$	−9.79988	−0.410832	−0.205416	+	0.978675i	$0.565855\pi$
$570$	1.05563	0.0442156
$571$	28.2196	1.18095	0.590476	−	0.807055i	$-0.298940\pi$
$571$	28.2196	1.18095	0.590476	+	0.807055i	$0.298940\pi$
$572$	1.82043	0.0761160
$573$	−15.4332	−0.644732
$574$	7.23699	0.302066
$575$	14.3569	0.598722
$576$	1.00000	0.0416667
$577$	28.2068	1.17427	0.587133	−	0.809491i	$-0.300257\pi$
$577$	28.2068	1.17427	0.587133	+	0.809491i	$0.300257\pi$
$578$	−2.73318	−0.113685
$579$	−1.86292	−0.0774203
$580$	4.82763	0.200456
$581$	−15.0314	−0.623608
$582$	−4.45112	−0.184505
$583$	−4.63732	−0.192058
$584$	−11.9575	−0.494803
$585$	0.226420	0.00936133
$586$	12.6202	0.521337
$587$	−12.2413	−0.505251	−0.252626	−	0.967564i	$-0.581294\pi$
$587$	−12.2413	−0.505251	−0.252626	+	0.967564i	$0.581294\pi$
$588$	1.26844	0.0523096
$589$	11.4395	0.471355
$590$	2.15971	0.0889140
$591$	−18.0327	−0.741764
$592$	−7.58613	−0.311788
$593$	28.6547	1.17671	0.588353	−	0.808604i	$-0.299777\pi$
$593$	28.6547	1.17671	0.588353	+	0.808604i	$0.299777\pi$
$594$	−5.51891	−0.226444
$595$	8.76810	0.359457
$596$	2.90167	0.118857
$597$	24.0118	0.982739
$598$	1.04567	0.0427607
$599$	−6.02231	−0.246065	−0.123032	−	0.992403i	$-0.539262\pi$
$599$	−6.02231	−0.246065	−0.123032	+	0.992403i	$0.539262\pi$
$600$	4.52882	0.184888
$601$	33.1083	1.35052	0.675258	−	0.737582i	$-0.264032\pi$
$601$	33.1083	1.35052	0.675258	+	0.737582i	$0.264032\pi$
$602$	−20.0528	−0.817290
$603$	−13.9815	−0.569372
$604$	2.66399	0.108396
$605$	13.3568	0.543031
$606$	3.23632	0.131467
$607$	35.7642	1.45163	0.725813	−	0.687892i	$-0.241464\pi$
$607$	35.7642	1.45163	0.725813	+	0.687892i	$0.241464\pi$
$608$	1.53786	0.0623686
$609$	20.2232	0.819487
$610$	1.84472	0.0746903
$611$	1.98238	0.0801987
$612$	4.44220	0.179565
$613$	−23.7521	−0.959339	−0.479669	−	0.877449i	$-0.659243\pi$
$613$	−23.7521	−0.959339	−0.479669	+	0.877449i	$0.659243\pi$
$614$	32.7514	1.32174
$615$	−1.72759	−0.0696632
$616$	−15.8696	−0.639404
$617$	−37.5379	−1.51122	−0.755609	−	0.655023i	$-0.772659\pi$
$617$	−37.5379	−1.51122	−0.755609	+	0.655023i	$0.772659\pi$
$618$	12.7274	0.511971
$619$	46.6282	1.87415	0.937073	−	0.349132i	$-0.113524\pi$
$619$	46.6282	1.87415	0.937073	+	0.349132i	$0.113524\pi$
$620$	−5.10603	−0.205063
$621$	−3.17011	−0.127212
$622$	5.09711	0.204376
$623$	23.8630	0.956049
$624$	0.329853	0.0132047
$625$	18.1543	0.726170
$626$	−3.70513	−0.148087
$627$	−8.48734	−0.338952
$628$	15.2048	0.606736
$629$	−33.6991	−1.34367
$630$	−1.97382	−0.0786387
$631$	28.3815	1.12985	0.564926	−	0.825142i	$-0.308905\pi$
$631$	28.3815	1.12985	0.564926	+	0.825142i	$0.308905\pi$
$632$	−12.1543	−0.483472
$633$	6.05163	0.240531
$634$	−11.1944	−0.444588
$635$	2.77527	0.110133
$636$	−0.840259	−0.0333185
$637$	0.418399	0.0165776
$638$	−38.8144	−1.53668
$639$	−6.67275	−0.263970
$640$	−0.686428	−0.0271335
$641$	11.2437	0.444098	0.222049	−	0.975036i	$-0.428726\pi$
$641$	11.2437	0.444098	0.222049	+	0.975036i	$0.428726\pi$
$642$	−4.52529	−0.178599
$643$	−0.473139	−0.0186588	−0.00932940	−	0.999956i	$-0.502970\pi$
$643$	−0.473139	−0.0186588	−0.00932940	+	0.999956i	$0.502970\pi$
$644$	−9.11562	−0.359206
$645$	4.78694	0.188485
$646$	6.83150	0.268782
$647$	−44.1533	−1.73585	−0.867923	−	0.496699i	$-0.834545\pi$
$647$	−44.1533	−1.73585	−0.867923	+	0.496699i	$0.834545\pi$
$648$	−1.00000	−0.0392837
$649$	−17.3642	−0.681604
$650$	1.49384	0.0585934
$651$	−21.3895	−0.838320
$652$	12.4990	0.489498
$653$	17.7560	0.694847	0.347423	−	0.937708i	$-0.387057\pi$
$653$	17.7560	0.694847	0.347423	+	0.937708i	$0.387057\pi$
$654$	10.5371	0.412034
$655$	4.56302	0.178292
$656$	−2.51679	−0.0982640
$657$	11.9575	0.466505
$658$	−17.2814	−0.673700
$659$	28.1314	1.09584	0.547921	−	0.836530i	$-0.315419\pi$
$659$	28.1314	1.09584	0.547921	+	0.836530i	$0.315419\pi$
$660$	3.78834	0.147461
$661$	13.7180	0.533567	0.266784	−	0.963756i	$-0.414039\pi$
$661$	13.7180	0.533567	0.266784	+	0.963756i	$0.414039\pi$
$662$	30.7807	1.19633
$663$	1.46527	0.0569065
$664$	5.22743	0.202864
$665$	−3.03546	−0.117710
$666$	7.58613	0.293957
$667$	−22.2953	−0.863278
$668$	6.88133	0.266247
$669$	−1.00000	−0.0386622
$670$	9.59731	0.370776
$671$	−14.8316	−0.572567
$672$	−2.87549	−0.110924
$673$	12.1670	0.469002	0.234501	−	0.972116i	$-0.424654\pi$
$673$	12.1670	0.469002	0.234501	+	0.972116i	$0.424654\pi$
$674$	28.0781	1.08153
$675$	−4.52882	−0.174314
$676$	−12.8912	−0.495815
$677$	42.3323	1.62696	0.813481	−	0.581592i	$-0.197570\pi$
$677$	42.3323	1.62696	0.813481	+	0.581592i	$0.197570\pi$
$678$	−2.93690	−0.112791
$679$	12.7992	0.491186
$680$	−3.04925	−0.116934
$681$	24.3730	0.933974
$682$	41.0527	1.57199
$683$	32.6141	1.24794	0.623972	−	0.781446i	$-0.285518\pi$
$683$	32.6141	1.24794	0.623972	+	0.781446i	$0.285518\pi$
$684$	−1.53786	−0.0588017
$685$	7.53265	0.287808
$686$	16.4810	0.629249
$687$	7.57496	0.289003
$688$	6.97369	0.265869
$689$	−0.277162	−0.0105590
$690$	2.17605	0.0828410
$691$	6.87327	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$691$	6.87327	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$692$	−3.65630	−0.138992
$693$	15.8696	0.602836
$694$	−4.81474	−0.182765
$695$	−10.4434	−0.396143
$696$	−7.03297	−0.266584
$697$	−11.1801	−0.423475
$698$	−20.7579	−0.785697
$699$	1.91430	0.0724056
$700$	−13.0226	−0.492207
$701$	−30.3022	−1.14450	−0.572249	−	0.820080i	$-0.693929\pi$
$701$	−30.3022	−1.14450	−0.572249	+	0.820080i	$0.693929\pi$
$702$	−0.329853	−0.0124495
$703$	11.6664	0.440008
$704$	5.51891	0.208002
$705$	4.12537	0.155370
$706$	3.21876	0.121140
$707$	−9.30601	−0.349989
$708$	−3.14631	−0.118245
$709$	−45.6605	−1.71482	−0.857408	−	0.514638i	$-0.827926\pi$
$709$	−45.6605	−1.71482	−0.857408	+	0.514638i	$0.827926\pi$
$710$	4.58036	0.171898
$711$	12.1543	0.455822
$712$	−8.29874	−0.311009
$713$	23.5810	0.883117
$714$	−12.7735	−0.478037
$715$	1.24959	0.0467322
$716$	11.6410	0.435045
$717$	−18.8018	−0.702167
$718$	−25.5107	−0.952053
$719$	−4.43820	−0.165517	−0.0827584	−	0.996570i	$-0.526373\pi$
$719$	−4.43820	−0.165517	−0.0827584	+	0.996570i	$0.526373\pi$
$720$	0.686428	0.0255817
$721$	−36.5975	−1.36296
$722$	16.6350	0.619090
$723$	−12.7202	−0.473068
$724$	−13.8301	−0.513990
$725$	−31.8510	−1.18292
$726$	−19.4584	−0.722170
$727$	3.18844	0.118253	0.0591263	−	0.998251i	$-0.481169\pi$
$727$	3.18844	0.118253	0.0591263	+	0.998251i	$0.481169\pi$
$728$	−0.948489	−0.0351533
$729$	1.00000	0.0370370
$730$	−8.20794	−0.303790
$731$	30.9785	1.14578
$732$	−2.68741	−0.0993296
$733$	7.01336	0.259045	0.129522	−	0.991577i	$-0.458656\pi$
$733$	7.01336	0.259045	0.129522	+	0.991577i	$0.458656\pi$
$734$	24.4579	0.902758
$735$	0.870693	0.0321160
$736$	3.17011	0.116852
$737$	−77.1628	−2.84233
$738$	2.51679	0.0926442
$739$	−24.9587	−0.918122	−0.459061	−	0.888405i	$-0.651814\pi$
$739$	−24.9587	−0.918122	−0.459061	+	0.888405i	$0.651814\pi$
$740$	−5.20733	−0.191425
$741$	−0.507269	−0.0186350
$742$	2.41616	0.0886999
$743$	28.8093	1.05691	0.528456	−	0.848961i	$-0.322771\pi$
$743$	28.8093	1.05691	0.528456	+	0.848961i	$0.322771\pi$
$744$	7.43855	0.272711
$745$	1.99179	0.0729735
$746$	9.79830	0.358741
$747$	−5.22743	−0.191262
$748$	24.5161	0.896399
$749$	13.0124	0.475464
$750$	6.54085	0.238838
$751$	−51.0813	−1.86398	−0.931992	−	0.362479i	$-0.881931\pi$
$751$	−51.0813	−1.86398	−0.931992	+	0.362479i	$0.881931\pi$
$752$	6.00990	0.219159
$753$	8.40005	0.306115
$754$	−2.31985	−0.0844838
$755$	1.82864	0.0665509
$756$	2.87549	0.104581
$757$	15.0243	0.546068	0.273034	−	0.962004i	$-0.411973\pi$
$757$	15.0243	0.546068	0.273034	+	0.962004i	$0.411973\pi$
$758$	−23.3632	−0.848588
$759$	−17.4956	−0.635049
$760$	1.05563	0.0382918
$761$	6.31216	0.228816	0.114408	−	0.993434i	$-0.463503\pi$
$761$	6.31216	0.228816	0.114408	+	0.993434i	$0.463503\pi$
$762$	−4.04306	−0.146465
$763$	−30.2994	−1.09691
$764$	−15.4332	−0.558354
$765$	3.04925	0.110246
$766$	−26.1424	−0.944564
$767$	−1.03782	−0.0374735
$768$	1.00000	0.0360844
$769$	24.2449	0.874293	0.437147	−	0.899390i	$-0.355989\pi$
$769$	24.2449	0.874293	0.437147	+	0.899390i	$0.355989\pi$
$770$	−10.8933	−0.392568
$771$	−23.1611	−0.834126
$772$	−1.86292	−0.0670479
$773$	14.1903	0.510390	0.255195	−	0.966890i	$-0.417860\pi$
$773$	14.1903	0.510390	0.255195	+	0.966890i	$0.417860\pi$
$774$	−6.97369	−0.250664
$775$	33.6878	1.21010
$776$	−4.45112	−0.159786
$777$	−21.8138	−0.782567
$778$	−3.73549	−0.133924
$779$	3.87047	0.138674
$780$	0.226420	0.00810715
$781$	−36.8263	−1.31775
$782$	14.0823	0.503582
$783$	7.03297	0.251338
$784$	1.26844	0.0453014
$785$	10.4370	0.372512
$786$	−6.64749	−0.237108
$787$	−17.2608	−0.615280	−0.307640	−	0.951503i	$-0.599539\pi$
$787$	−17.2608	−0.615280	−0.307640	+	0.951503i	$0.599539\pi$
$788$	−18.0327	−0.642387
$789$	6.61626	0.235545
$790$	−8.34305	−0.296833
$791$	8.44503	0.300271
$792$	−5.51891	−0.196106
$793$	−0.886451	−0.0314788
$794$	−17.6199	−0.625306
$795$	−0.576778	−0.0204562
$796$	24.0118	0.851077
$797$	−13.2887	−0.470710	−0.235355	−	0.971909i	$-0.575625\pi$
$797$	−13.2887	−0.470710	−0.235355	+	0.971909i	$0.575625\pi$
$798$	4.42211	0.156541
$799$	26.6972	0.944479
$800$	4.52882	0.160118
$801$	8.29874	0.293222
$802$	22.5232	0.795320
$803$	65.9922	2.32882
$804$	−13.9815	−0.493090
$805$	−6.25722	−0.220538
$806$	2.45363	0.0864254
$807$	−4.44005	−0.156297
$808$	3.23632	0.113853
$809$	−9.08056	−0.319255	−0.159628	−	0.987177i	$-0.551029\pi$
$809$	−9.08056	−0.319255	−0.159628	+	0.987177i	$0.551029\pi$
$810$	−0.686428	−0.0241186
$811$	−49.9105	−1.75260	−0.876298	−	0.481770i	$-0.839994\pi$
$811$	−49.9105	−1.75260	−0.876298	+	0.481770i	$0.839994\pi$
$812$	20.2232	0.709697
$813$	−6.05643	−0.212408
$814$	41.8672	1.46745
$815$	8.57965	0.300532
$816$	4.44220	0.155508
$817$	−10.7246	−0.375206
$818$	12.4542	0.435451
$819$	0.948489	0.0331429
$820$	−1.72759	−0.0603301
$821$	9.08016	0.316900	0.158450	−	0.987367i	$-0.449350\pi$
$821$	9.08016	0.316900	0.158450	+	0.987367i	$0.449350\pi$
$822$	−10.9737	−0.382751
$823$	19.7767	0.689371	0.344686	−	0.938718i	$-0.387986\pi$
$823$	19.7767	0.689371	0.344686	+	0.938718i	$0.387986\pi$
$824$	12.7274	0.443380
$825$	−24.9942	−0.870185
$826$	9.04717	0.314791
$827$	37.8945	1.31772	0.658860	−	0.752266i	$-0.271039\pi$
$827$	37.8945	1.31772	0.658860	+	0.752266i	$0.271039\pi$
$828$	−3.17011	−0.110169
$829$	−6.13822	−0.213189	−0.106595	−	0.994303i	$-0.533995\pi$
$829$	−6.13822	−0.213189	−0.106595	+	0.994303i	$0.533995\pi$
$830$	3.58825	0.124550
$831$	−25.7454	−0.893099
$832$	0.329853	0.0114356
$833$	5.63467	0.195230
$834$	15.2142	0.526824
$835$	4.72354	0.163465
$836$	−8.48734	−0.293541
$837$	−7.43855	−0.257114
$838$	30.5170	1.05419
$839$	37.2877	1.28731	0.643657	−	0.765314i	$-0.277416\pi$
$839$	37.2877	1.28731	0.643657	+	0.765314i	$0.277416\pi$
$840$	−1.97382	−0.0681031
$841$	20.4627	0.705610
$842$	−5.17369	−0.178297
$843$	−14.7983	−0.509679
$844$	6.05163	0.208306
$845$	−8.84888	−0.304411
$846$	−6.00990	−0.206625
$847$	55.9525	1.92255
$848$	−0.840259	−0.0288546
$849$	−4.17776	−0.143380
$850$	20.1179	0.690039
$851$	24.0489	0.824385
$852$	−6.67275	−0.228604
$853$	−5.19329	−0.177815	−0.0889075	−	0.996040i	$-0.528338\pi$
$853$	−5.19329	−0.177815	−0.0889075	+	0.996040i	$0.528338\pi$
$854$	7.72762	0.264434
$855$	−1.05563	−0.0361019
$856$	−4.52529	−0.154671
$857$	33.5933	1.14753	0.573763	−	0.819022i	$-0.305483\pi$
$857$	33.5933	1.14753	0.573763	+	0.819022i	$0.305483\pi$
$858$	−1.82043	−0.0621485
$859$	10.4008	0.354872	0.177436	−	0.984132i	$-0.443220\pi$
$859$	10.4008	0.354872	0.177436	+	0.984132i	$0.443220\pi$
$860$	4.78694	0.163233
$861$	−7.23699	−0.246636
$862$	11.8685	0.404243
$863$	−37.4393	−1.27445	−0.637224	−	0.770679i	$-0.719917\pi$
$863$	−37.4393	−1.27445	−0.637224	+	0.770679i	$0.719917\pi$
$864$	−1.00000	−0.0340207
$865$	−2.50979	−0.0853354
$866$	−2.77072	−0.0941530
$867$	2.73318	0.0928235
$868$	−21.3895	−0.726006
$869$	67.0785	2.27548
$870$	−4.82763	−0.163672
$871$	−4.61185	−0.156266
$872$	10.5371	0.356832
$873$	4.45112	0.150648
$874$	−4.87520	−0.164906
$875$	−18.8081	−0.635831
$876$	11.9575	0.404005
$877$	49.1976	1.66129	0.830643	−	0.556805i	$-0.187973\pi$
$877$	49.1976	1.66129	0.830643	+	0.556805i	$0.187973\pi$
$878$	13.5404	0.456966
$879$	−12.6202	−0.425670
$880$	3.78834	0.127705
$881$	−42.7892	−1.44161	−0.720803	−	0.693140i	$-0.756227\pi$
$881$	−42.7892	−1.44161	−0.720803	+	0.693140i	$0.756227\pi$
$882$	−1.26844	−0.0427106
$883$	−12.3874	−0.416870	−0.208435	−	0.978036i	$-0.566837\pi$
$883$	−12.3874	−0.416870	−0.208435	+	0.978036i	$0.566837\pi$
$884$	1.46527	0.0492825
$885$	−2.15971	−0.0725980
$886$	−38.7298	−1.30115
$887$	−1.39064	−0.0466930	−0.0233465	−	0.999727i	$-0.507432\pi$
$887$	−1.39064	−0.0466930	−0.0233465	+	0.999727i	$0.507432\pi$
$888$	7.58613	0.254574
$889$	11.6258	0.389916
$890$	−5.69649	−0.190947
$891$	5.51891	0.184891
$892$	−1.00000	−0.0334825
$893$	−9.24241	−0.309285
$894$	−2.90167	−0.0970465
$895$	7.99071	0.267100
$896$	−2.87549	−0.0960634
$897$	−1.04567	−0.0349139
$898$	35.2746	1.17713
$899$	−52.3151	−1.74481
$900$	−4.52882	−0.150961
$901$	−3.73260	−0.124351
$902$	13.8899	0.462484
$903$	20.0528	0.667314
$904$	−2.93690	−0.0976799
$905$	−9.49335	−0.315570
$906$	−2.66399	−0.0885051
$907$	−16.8327	−0.558921	−0.279460	−	0.960157i	$-0.590156\pi$
$907$	−16.8327	−0.558921	−0.279460	+	0.960157i	$0.590156\pi$
$908$	24.3730	0.808846
$909$	−3.23632	−0.107342
$910$	−0.651069	−0.0215827
$911$	25.6813	0.850860	0.425430	−	0.904991i	$-0.360123\pi$
$911$	25.6813	0.850860	0.425430	+	0.904991i	$0.360123\pi$
$912$	−1.53786	−0.0509237
$913$	−28.8497	−0.954787
$914$	15.7179	0.519901
$915$	−1.84472	−0.0609844
$916$	7.57496	0.250284
$917$	19.1148	0.631226
$918$	−4.44220	−0.146615
$919$	−34.0182	−1.12216	−0.561079	−	0.827762i	$-0.689614\pi$
$919$	−34.0182	−1.12216	−0.561079	+	0.827762i	$0.689614\pi$
$920$	2.17605	0.0717424
$921$	−32.7514	−1.07920
$922$	15.4964	0.510346
$923$	−2.20103	−0.0724476
$924$	15.8696	0.522071
$925$	34.3562	1.12963
$926$	−34.0979	−1.12053
$927$	−12.7274	−0.418022
$928$	−7.03297	−0.230869
$929$	−16.0961	−0.528096	−0.264048	−	0.964510i	$-0.585058\pi$
$929$	−16.0961	−0.528096	−0.264048	+	0.964510i	$0.585058\pi$
$930$	5.10603	0.167433
$931$	−1.95069	−0.0639312
$932$	1.91430	0.0627050
$933$	−5.09711	−0.166872
$934$	−5.76270	−0.188561
$935$	16.8286	0.550353
$936$	−0.329853	−0.0107816
$937$	−11.1780	−0.365169	−0.182584	−	0.983190i	$-0.558446\pi$
$937$	−11.1780	−0.365169	−0.182584	+	0.983190i	$0.558446\pi$
$938$	40.2037	1.31270
$939$	3.70513	0.120912
$940$	4.12537	0.134555
$941$	−55.8606	−1.82101	−0.910503	−	0.413504i	$-0.864305\pi$
$941$	−55.8606	−1.82101	−0.910503	+	0.413504i	$0.864305\pi$
$942$	−15.2048	−0.495398
$943$	7.97849	0.259815
$944$	−3.14631	−0.102404
$945$	1.97382	0.0642083
$946$	−38.4872	−1.25133
$947$	33.1983	1.07880	0.539400	−	0.842050i	$-0.318651\pi$
$947$	33.1983	1.07880	0.539400	+	0.842050i	$0.318651\pi$
$948$	12.1543	0.394753
$949$	3.94421	0.128034
$950$	−6.96470	−0.225965
$951$	11.1944	0.363005
$952$	−12.7735	−0.413992
$953$	−52.1647	−1.68978	−0.844891	−	0.534939i	$-0.820335\pi$
$953$	−52.1647	−1.68978	−0.844891	+	0.534939i	$0.820335\pi$
$954$	0.840259	0.0272044
$955$	−10.5938	−0.342807
$956$	−18.8018	−0.608094
$957$	38.8144	1.25469
$958$	0.892156	0.0288243
$959$	31.5547	1.01896
$960$	0.686428	0.0221544
$961$	24.3321	0.784905
$962$	2.50231	0.0806777
$963$	4.52529	0.145826
$964$	−12.7202	−0.409689
$965$	−1.27876	−0.0411648
$966$	9.11562	0.293290
$967$	−6.32108	−0.203272	−0.101636	−	0.994822i	$-0.532408\pi$
$967$	−6.32108	−0.203272	−0.101636	+	0.994822i	$0.532408\pi$
$968$	−19.4584	−0.625417
$969$	−6.83150	−0.219459
$970$	−3.05538	−0.0981022
$971$	24.4643	0.785098	0.392549	−	0.919731i	$-0.371593\pi$
$971$	24.4643	0.785098	0.392549	+	0.919731i	$0.371593\pi$
$972$	1.00000	0.0320750
$973$	−43.7482	−1.40250
$974$	−4.83587	−0.154951
$975$	−1.49384	−0.0478413
$976$	−2.68741	−0.0860220
$977$	−23.1983	−0.742181	−0.371090	−	0.928597i	$-0.621016\pi$
$977$	−23.1983	−0.742181	−0.371090	+	0.928597i	$0.621016\pi$
$978$	−12.4990	−0.399673
$979$	45.8001	1.46378
$980$	0.870693	0.0278133
$981$	−10.5371	−0.336425
$982$	14.5280	0.463608
$983$	−44.9757	−1.43450	−0.717252	−	0.696814i	$-0.754600\pi$
$983$	−44.9757	−1.43450	−0.717252	+	0.696814i	$0.754600\pi$
$984$	2.51679	0.0802322
$985$	−12.3781	−0.394400
$986$	−31.2419	−0.994945
$987$	17.2814	0.550073
$988$	−0.507269	−0.0161384
$989$	−22.1074	−0.702974
$990$	−3.78834	−0.120401
$991$	−54.4523	−1.72973	−0.864867	−	0.502001i	$-0.832597\pi$
$991$	−54.4523	−1.72973	−0.864867	+	0.502001i	$0.832597\pi$
$992$	7.43855	0.236174
$993$	−30.7807	−0.976796
$994$	19.1874	0.608588
$995$	16.4824	0.522527
$996$	−5.22743	−0.165637
$997$	20.4748	0.648444	0.324222	−	0.945981i	$-0.394898\pi$
$997$	20.4748	0.648444	0.324222	+	0.945981i	$0.394898\pi$
$998$	−42.7043	−1.35178
$999$	−7.58613	−0.240015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.h.1.2	✓	5
3.2	odd	2		4014.2.a.r.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.h.1.2	✓	5	1.1	even	1	trivial
4014.2.a.r.1.4		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:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.686428 q^{5} -1.00000 q^{6} +2.87549 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.686428 q^{5} -1.00000 q^{6} +2.87549 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.686428 q^{10} +5.51891 q^{11} +1.00000 q^{12} +0.329853 q^{13} -2.87549 q^{14} +0.686428 q^{15} +1.00000 q^{16} +4.44220 q^{17} -1.00000 q^{18} -1.53786 q^{19} +0.686428 q^{20} +2.87549 q^{21} -5.51891 q^{22} -3.17011 q^{23} -1.00000 q^{24} -4.52882 q^{25} -0.329853 q^{26} +1.00000 q^{27} +2.87549 q^{28} +7.03297 q^{29} -0.686428 q^{30} -7.43855 q^{31} -1.00000 q^{32} +5.51891 q^{33} -4.44220 q^{34} +1.97382 q^{35} +1.00000 q^{36} -7.58613 q^{37} +1.53786 q^{38} +0.329853 q^{39} -0.686428 q^{40} -2.51679 q^{41} -2.87549 q^{42} +6.97369 q^{43} +5.51891 q^{44} +0.686428 q^{45} +3.17011 q^{46} +6.00990 q^{47} +1.00000 q^{48} +1.26844 q^{49} +4.52882 q^{50} +4.44220 q^{51} +0.329853 q^{52} -0.840259 q^{53} -1.00000 q^{54} +3.78834 q^{55} -2.87549 q^{56} -1.53786 q^{57} -7.03297 q^{58} -3.14631 q^{59} +0.686428 q^{60} -2.68741 q^{61} +7.43855 q^{62} +2.87549 q^{63} +1.00000 q^{64} +0.226420 q^{65} -5.51891 q^{66} -13.9815 q^{67} +4.44220 q^{68} -3.17011 q^{69} -1.97382 q^{70} -6.67275 q^{71} -1.00000 q^{72} +11.9575 q^{73} +7.58613 q^{74} -4.52882 q^{75} -1.53786 q^{76} +15.8696 q^{77} -0.329853 q^{78} +12.1543 q^{79} +0.686428 q^{80} +1.00000 q^{81} +2.51679 q^{82} -5.22743 q^{83} +2.87549 q^{84} +3.04925 q^{85} -6.97369 q^{86} +7.03297 q^{87} -5.51891 q^{88} +8.29874 q^{89} -0.686428 q^{90} +0.948489 q^{91} -3.17011 q^{92} -7.43855 q^{93} -6.00990 q^{94} -1.05563 q^{95} -1.00000 q^{96} +4.45112 q^{97} -1.26844 q^{98} +5.51891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 5 q^{10} + 9 q^{11} + 5 q^{12} + q^{14} + 5 q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} - 4 q^{19} + 5 q^{20} - q^{21} - 9 q^{22} + 16 q^{23} - 5 q^{24} + 8 q^{25} + 5 q^{27} - q^{28} + 8 q^{29} - 5 q^{30} - q^{31} - 5 q^{32} + 9 q^{33} - 6 q^{34} + 22 q^{35} + 5 q^{36} - 2 q^{37} + 4 q^{38} - 5 q^{40} + 4 q^{41} + q^{42} + 3 q^{43} + 9 q^{44} + 5 q^{45} - 16 q^{46} + 18 q^{47} + 5 q^{48} + 2 q^{49} - 8 q^{50} + 6 q^{51} + 26 q^{53} - 5 q^{54} + q^{55} + q^{56} - 4 q^{57} - 8 q^{58} + 21 q^{59} + 5 q^{60} - 20 q^{61} + q^{62} - q^{63} + 5 q^{64} - 3 q^{65} - 9 q^{66} - 5 q^{67} + 6 q^{68} + 16 q^{69} - 22 q^{70} + 17 q^{71} - 5 q^{72} + 5 q^{73} + 2 q^{74} + 8 q^{75} - 4 q^{76} + 2 q^{77} - 21 q^{79} + 5 q^{80} + 5 q^{81} - 4 q^{82} + 11 q^{83} - q^{84} - 12 q^{85} - 3 q^{86} + 8 q^{87} - 9 q^{88} - 5 q^{89} - 5 q^{90} - 10 q^{91} + 16 q^{92} - q^{93} - 18 q^{94} + 10 q^{95} - 5 q^{96} - 11 q^{97} - 2 q^{98} + 9 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 5 * q^10 + 9 * q^11 + 5 * q^12 + q^14 + 5 * q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 - 4 * q^19 + 5 * q^20 - q^21 - 9 * q^22 + 16 * q^23 - 5 * q^24 + 8 * q^25 + 5 * q^27 - q^28 + 8 * q^29 - 5 * q^30 - q^31 - 5 * q^32 + 9 * q^33 - 6 * q^34 + 22 * q^35 + 5 * q^36 - 2 * q^37 + 4 * q^38 - 5 * q^40 + 4 * q^41 + q^42 + 3 * q^43 + 9 * q^44 + 5 * q^45 - 16 * q^46 + 18 * q^47 + 5 * q^48 + 2 * q^49 - 8 * q^50 + 6 * q^51 + 26 * q^53 - 5 * q^54 + q^55 + q^56 - 4 * q^57 - 8 * q^58 + 21 * q^59 + 5 * q^60 - 20 * q^61 + q^62 - q^63 + 5 * q^64 - 3 * q^65 - 9 * q^66 - 5 * q^67 + 6 * q^68 + 16 * q^69 - 22 * q^70 + 17 * q^71 - 5 * q^72 + 5 * q^73 + 2 * q^74 + 8 * q^75 - 4 * q^76 + 2 * q^77 - 21 * q^79 + 5 * q^80 + 5 * q^81 - 4 * q^82 + 11 * q^83 - q^84 - 12 * q^85 - 3 * q^86 + 8 * q^87 - 9 * q^88 - 5 * q^89 - 5 * q^90 - 10 * q^91 + 16 * q^92 - q^93 - 18 * q^94 + 10 * q^95 - 5 * q^96 - 11 * q^97 - 2 * q^98 + 9 * q^99

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