LMFDB - Embedded newform 8042.2.a.a.1.54

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.54
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.60231 q^{3} +1.00000 q^{4} +2.12177 q^{5} +1.60231 q^{6} -3.00020 q^{7} +1.00000 q^{8} -0.432617 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.60231 q^{3} +1.00000 q^{4} +2.12177 q^{5} +1.60231 q^{6} -3.00020 q^{7} +1.00000 q^{8} -0.432617 q^{9} +2.12177 q^{10} -4.64620 q^{11} +1.60231 q^{12} +0.0810555 q^{13} -3.00020 q^{14} +3.39973 q^{15} +1.00000 q^{16} -0.657428 q^{17} -0.432617 q^{18} +6.08843 q^{19} +2.12177 q^{20} -4.80724 q^{21} -4.64620 q^{22} -2.73445 q^{23} +1.60231 q^{24} -0.498081 q^{25} +0.0810555 q^{26} -5.50010 q^{27} -3.00020 q^{28} -8.38979 q^{29} +3.39973 q^{30} +2.31316 q^{31} +1.00000 q^{32} -7.44464 q^{33} -0.657428 q^{34} -6.36575 q^{35} -0.432617 q^{36} -6.46245 q^{37} +6.08843 q^{38} +0.129876 q^{39} +2.12177 q^{40} +4.21403 q^{41} -4.80724 q^{42} -1.41184 q^{43} -4.64620 q^{44} -0.917916 q^{45} -2.73445 q^{46} -6.07438 q^{47} +1.60231 q^{48} +2.00122 q^{49} -0.498081 q^{50} -1.05340 q^{51} +0.0810555 q^{52} -9.29155 q^{53} -5.50010 q^{54} -9.85819 q^{55} -3.00020 q^{56} +9.75552 q^{57} -8.38979 q^{58} -7.38292 q^{59} +3.39973 q^{60} -14.6948 q^{61} +2.31316 q^{62} +1.29794 q^{63} +1.00000 q^{64} +0.171981 q^{65} -7.44464 q^{66} +7.07438 q^{67} -0.657428 q^{68} -4.38142 q^{69} -6.36575 q^{70} -11.3636 q^{71} -0.432617 q^{72} +15.5477 q^{73} -6.46245 q^{74} -0.798078 q^{75} +6.08843 q^{76} +13.9396 q^{77} +0.129876 q^{78} +4.09958 q^{79} +2.12177 q^{80} -7.51499 q^{81} +4.21403 q^{82} +13.3636 q^{83} -4.80724 q^{84} -1.39491 q^{85} -1.41184 q^{86} -13.4430 q^{87} -4.64620 q^{88} +0.942798 q^{89} -0.917916 q^{90} -0.243183 q^{91} -2.73445 q^{92} +3.70639 q^{93} -6.07438 q^{94} +12.9183 q^{95} +1.60231 q^{96} -4.19989 q^{97} +2.00122 q^{98} +2.01003 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	1.60231	0.925091	0.462546	−	0.886595i	$-0.346936\pi$
$3$	1.60231	0.925091	0.462546	+	0.886595i	$0.346936\pi$
$4$	1.00000	0.500000
$5$	2.12177	0.948886	0.474443	−	0.880286i	$-0.342650\pi$
$5$	2.12177	0.948886	0.474443	+	0.880286i	$0.342650\pi$
$6$	1.60231	0.654138
$7$	−3.00020	−1.13397	−0.566985	−	0.823728i	$-0.691890\pi$
$7$	−3.00020	−1.13397	−0.566985	+	0.823728i	$0.691890\pi$
$8$	1.00000	0.353553
$9$	−0.432617	−0.144206
$10$	2.12177	0.670963
$11$	−4.64620	−1.40088	−0.700442	−	0.713710i	$-0.747014\pi$
$11$	−4.64620	−1.40088	−0.700442	+	0.713710i	$0.747014\pi$
$12$	1.60231	0.462546
$13$	0.0810555	0.0224807	0.0112404	−	0.999937i	$-0.496422\pi$
$13$	0.0810555	0.0224807	0.0112404	+	0.999937i	$0.496422\pi$
$14$	−3.00020	−0.801838
$15$	3.39973	0.877806
$16$	1.00000	0.250000
$17$	−0.657428	−0.159450	−0.0797249	−	0.996817i	$-0.525404\pi$
$17$	−0.657428	−0.159450	−0.0797249	+	0.996817i	$0.525404\pi$
$18$	−0.432617	−0.101969
$19$	6.08843	1.39678	0.698390	−	0.715717i	$-0.253900\pi$
$19$	6.08843	1.39678	0.698390	+	0.715717i	$0.253900\pi$
$20$	2.12177	0.474443
$21$	−4.80724	−1.04903
$22$	−4.64620	−0.990574
$23$	−2.73445	−0.570172	−0.285086	−	0.958502i	$-0.592022\pi$
$23$	−2.73445	−0.570172	−0.285086	+	0.958502i	$0.592022\pi$
$24$	1.60231	0.327069
$25$	−0.498081	−0.0996162
$26$	0.0810555	0.0158963
$27$	−5.50010	−1.05850
$28$	−3.00020	−0.566985
$29$	−8.38979	−1.55795	−0.778973	−	0.627058i	$-0.784259\pi$
$29$	−8.38979	−1.55795	−0.778973	+	0.627058i	$0.784259\pi$
$30$	3.39973	0.620703
$31$	2.31316	0.415456	0.207728	−	0.978187i	$-0.433393\pi$
$31$	2.31316	0.415456	0.207728	+	0.978187i	$0.433393\pi$
$32$	1.00000	0.176777
$33$	−7.44464	−1.29595
$34$	−0.657428	−0.112748
$35$	−6.36575	−1.07601
$36$	−0.432617	−0.0721029
$37$	−6.46245	−1.06242	−0.531210	−	0.847240i	$-0.678263\pi$
$37$	−6.46245	−1.06242	−0.531210	+	0.847240i	$0.678263\pi$
$38$	6.08843	0.987673
$39$	0.129876	0.0207967
$40$	2.12177	0.335482
$41$	4.21403	0.658121	0.329060	−	0.944309i	$-0.393268\pi$
$41$	4.21403	0.658121	0.329060	+	0.944309i	$0.393268\pi$
$42$	−4.80724	−0.741774
$43$	−1.41184	−0.215304	−0.107652	−	0.994189i	$-0.534333\pi$
$43$	−1.41184	−0.215304	−0.107652	+	0.994189i	$0.534333\pi$
$44$	−4.64620	−0.700442
$45$	−0.917916	−0.136835
$46$	−2.73445	−0.403172
$47$	−6.07438	−0.886039	−0.443019	−	0.896512i	$-0.646093\pi$
$47$	−6.07438	−0.886039	−0.443019	+	0.896512i	$0.646093\pi$
$48$	1.60231	0.231273
$49$	2.00122	0.285889
$50$	−0.498081	−0.0704393
$51$	−1.05340	−0.147506
$52$	0.0810555	0.0112404
$53$	−9.29155	−1.27629	−0.638146	−	0.769915i	$-0.720298\pi$
$53$	−9.29155	−1.27629	−0.638146	+	0.769915i	$0.720298\pi$
$54$	−5.50010	−0.748469
$55$	−9.85819	−1.32928
$56$	−3.00020	−0.400919
$57$	9.75552	1.29215
$58$	−8.38979	−1.10163
$59$	−7.38292	−0.961174	−0.480587	−	0.876947i	$-0.659576\pi$
$59$	−7.38292	−0.961174	−0.480587	+	0.876947i	$0.659576\pi$
$60$	3.39973	0.438903
$61$	−14.6948	−1.88148	−0.940739	−	0.339132i	$-0.889867\pi$
$61$	−14.6948	−1.88148	−0.940739	+	0.339132i	$0.889867\pi$
$62$	2.31316	0.293772
$63$	1.29794	0.163525
$64$	1.00000	0.125000
$65$	0.171981	0.0213317
$66$	−7.44464	−0.916372
$67$	7.07438	0.864273	0.432136	−	0.901808i	$-0.357760\pi$
$67$	7.07438	0.864273	0.432136	+	0.901808i	$0.357760\pi$
$68$	−0.657428	−0.0797249
$69$	−4.38142	−0.527461
$70$	−6.36575	−0.760853
$71$	−11.3636	−1.34861	−0.674305	−	0.738453i	$-0.735556\pi$
$71$	−11.3636	−1.34861	−0.674305	+	0.738453i	$0.735556\pi$
$72$	−0.432617	−0.0509845
$73$	15.5477	1.81972	0.909859	−	0.414917i	$-0.136189\pi$
$73$	15.5477	1.81972	0.909859	+	0.414917i	$0.136189\pi$
$74$	−6.46245	−0.751245
$75$	−0.798078	−0.0921541
$76$	6.08843	0.698390
$77$	13.9396	1.58856
$78$	0.129876	0.0147055
$79$	4.09958	0.461238	0.230619	−	0.973044i	$-0.425925\pi$
$79$	4.09958	0.461238	0.230619	+	0.973044i	$0.425925\pi$
$80$	2.12177	0.237221
$81$	−7.51499	−0.834999
$82$	4.21403	0.465362
$83$	13.3636	1.46685	0.733425	−	0.679770i	$-0.237920\pi$
$83$	13.3636	1.46685	0.733425	+	0.679770i	$0.237920\pi$
$84$	−4.80724	−0.524513
$85$	−1.39491	−0.151300
$86$	−1.41184	−0.152243
$87$	−13.4430	−1.44124
$88$	−4.64620	−0.495287
$89$	0.942798	0.0999364	0.0499682	−	0.998751i	$-0.484088\pi$
$89$	0.942798	0.0999364	0.0499682	+	0.998751i	$0.484088\pi$
$90$	−0.917916	−0.0967568
$91$	−0.243183	−0.0254925
$92$	−2.73445	−0.285086
$93$	3.70639	0.384335
$94$	−6.07438	−0.626524
$95$	12.9183	1.32539
$96$	1.60231	0.163535
$97$	−4.19989	−0.426435	−0.213217	−	0.977005i	$-0.568394\pi$
$97$	−4.19989	−0.426435	−0.213217	+	0.977005i	$0.568394\pi$
$98$	2.00122	0.202154
$99$	2.01003	0.202016
$100$	−0.498081	−0.0498081
$101$	3.93478	0.391525	0.195763	−	0.980651i	$-0.437282\pi$
$101$	3.93478	0.391525	0.195763	+	0.980651i	$0.437282\pi$
$102$	−1.05340	−0.104302
$103$	−1.15135	−0.113446	−0.0567231	−	0.998390i	$-0.518065\pi$
$103$	−1.15135	−0.113446	−0.0567231	+	0.998390i	$0.518065\pi$
$104$	0.0810555	0.00794814
$105$	−10.1999	−0.995406
$106$	−9.29155	−0.902475
$107$	6.94041	0.670955	0.335478	−	0.942048i	$-0.391102\pi$
$107$	6.94041	0.670955	0.335478	+	0.942048i	$0.391102\pi$
$108$	−5.50010	−0.529248
$109$	−6.44900	−0.617702	−0.308851	−	0.951110i	$-0.599944\pi$
$109$	−6.44900	−0.617702	−0.308851	+	0.951110i	$0.599944\pi$
$110$	−9.85819	−0.939941
$111$	−10.3548	−0.982836
$112$	−3.00020	−0.283493
$113$	18.6455	1.75402	0.877008	−	0.480475i	$-0.159536\pi$
$113$	18.6455	1.75402	0.877008	+	0.480475i	$0.159536\pi$
$114$	9.75552	0.913688
$115$	−5.80188	−0.541028
$116$	−8.38979	−0.778973
$117$	−0.0350660	−0.00324185
$118$	−7.38292	−0.679653
$119$	1.97242	0.180811
$120$	3.39973	0.310351
$121$	10.5872	0.962474
$122$	−14.6948	−1.33041
$123$	6.75216	0.608822
$124$	2.31316	0.207728
$125$	−11.6657	−1.04341
$126$	1.29794	0.115630
$127$	−12.0844	−1.07232	−0.536160	−	0.844116i	$-0.680126\pi$
$127$	−12.0844	−1.07232	−0.536160	+	0.844116i	$0.680126\pi$
$128$	1.00000	0.0883883
$129$	−2.26220	−0.199176
$130$	0.171981	0.0150838
$131$	12.3889	1.08242	0.541211	−	0.840887i	$-0.317966\pi$
$131$	12.3889	1.08242	0.541211	+	0.840887i	$0.317966\pi$
$132$	−7.44464	−0.647973
$133$	−18.2665	−1.58391
$134$	7.07438	0.611133
$135$	−11.6700	−1.00439
$136$	−0.657428	−0.0563740
$137$	9.98896	0.853415	0.426707	−	0.904390i	$-0.359673\pi$
$137$	9.98896	0.853415	0.426707	+	0.904390i	$0.359673\pi$
$138$	−4.38142	−0.372971
$139$	0.934529	0.0792658	0.0396329	−	0.999214i	$-0.487381\pi$
$139$	0.934529	0.0792658	0.0396329	+	0.999214i	$0.487381\pi$
$140$	−6.36575	−0.538004
$141$	−9.73301	−0.819667
$142$	−11.3636	−0.953611
$143$	−0.376600	−0.0314929
$144$	−0.432617	−0.0360515
$145$	−17.8012	−1.47831
$146$	15.5477	1.28674
$147$	3.20657	0.264474
$148$	−6.46245	−0.531210
$149$	4.45814	0.365225	0.182613	−	0.983185i	$-0.441545\pi$
$149$	4.45814	0.365225	0.182613	+	0.983185i	$0.441545\pi$
$150$	−0.798078	−0.0651628
$151$	−12.8824	−1.04835	−0.524176	−	0.851610i	$-0.675627\pi$
$151$	−12.8824	−1.04835	−0.524176	+	0.851610i	$0.675627\pi$
$152$	6.08843	0.493837
$153$	0.284415	0.0229936
$154$	13.9396	1.12328
$155$	4.90801	0.394220
$156$	0.129876	0.0103984
$157$	−5.06896	−0.404547	−0.202274	−	0.979329i	$-0.564833\pi$
$157$	−5.06896	−0.404547	−0.202274	+	0.979329i	$0.564833\pi$
$158$	4.09958	0.326145
$159$	−14.8879	−1.18069
$160$	2.12177	0.167741
$161$	8.20390	0.646558
$162$	−7.51499	−0.590433
$163$	−5.74314	−0.449837	−0.224919	−	0.974378i	$-0.572212\pi$
$163$	−5.74314	−0.449837	−0.224919	+	0.974378i	$0.572212\pi$
$164$	4.21403	0.329060
$165$	−15.7958	−1.22970
$166$	13.3636	1.03722
$167$	17.4040	1.34676	0.673382	−	0.739295i	$-0.264841\pi$
$167$	17.4040	1.34676	0.673382	+	0.739295i	$0.264841\pi$
$168$	−4.80724	−0.370887
$169$	−12.9934	−0.999495
$170$	−1.39491	−0.106985
$171$	−2.63396	−0.201424
$172$	−1.41184	−0.107652
$173$	11.4070	0.867260	0.433630	−	0.901091i	$-0.357232\pi$
$173$	11.4070	0.867260	0.433630	+	0.901091i	$0.357232\pi$
$174$	−13.4430	−1.01911
$175$	1.49434	0.112962
$176$	−4.64620	−0.350221
$177$	−11.8297	−0.889174
$178$	0.942798	0.0706657
$179$	−11.4760	−0.857754	−0.428877	−	0.903363i	$-0.641091\pi$
$179$	−11.4760	−0.857754	−0.428877	+	0.903363i	$0.641091\pi$
$180$	−0.917916	−0.0684174
$181$	5.40532	0.401775	0.200887	−	0.979614i	$-0.435618\pi$
$181$	5.40532	0.401775	0.200887	+	0.979614i	$0.435618\pi$
$182$	−0.243183	−0.0180259
$183$	−23.5456	−1.74054
$184$	−2.73445	−0.201586
$185$	−13.7119	−1.00812
$186$	3.70639	0.271766
$187$	3.05454	0.223370
$188$	−6.07438	−0.443019
$189$	16.5014	1.20030
$190$	12.9183	0.937189
$191$	−3.46185	−0.250491	−0.125245	−	0.992126i	$-0.539972\pi$
$191$	−3.46185	−0.250491	−0.125245	+	0.992126i	$0.539972\pi$
$192$	1.60231	0.115636
$193$	14.1336	1.01736	0.508680	−	0.860955i	$-0.330133\pi$
$193$	14.1336	1.01736	0.508680	+	0.860955i	$0.330133\pi$
$194$	−4.19989	−0.301535
$195$	0.275567	0.0197337
$196$	2.00122	0.142945
$197$	20.7698	1.47979	0.739894	−	0.672723i	$-0.234876\pi$
$197$	20.7698	1.47979	0.739894	+	0.672723i	$0.234876\pi$
$198$	2.01003	0.142847
$199$	−5.32686	−0.377611	−0.188805	−	0.982015i	$-0.560462\pi$
$199$	−5.32686	−0.377611	−0.188805	+	0.982015i	$0.560462\pi$
$200$	−0.498081	−0.0352196
$201$	11.3353	0.799531
$202$	3.93478	0.276850
$203$	25.1711	1.76666
$204$	−1.05340	−0.0737528
$205$	8.94121	0.624481
$206$	−1.15135	−0.0802186
$207$	1.18297	0.0822221
$208$	0.0810555	0.00562019
$209$	−28.2881	−1.95673
$210$	−10.1999	−0.703858
$211$	−20.5837	−1.41704	−0.708522	−	0.705689i	$-0.750638\pi$
$211$	−20.5837	−1.41704	−0.708522	+	0.705689i	$0.750638\pi$
$212$	−9.29155	−0.638146
$213$	−18.2079	−1.24759
$214$	6.94041	0.474437
$215$	−2.99561	−0.204299
$216$	−5.50010	−0.374235
$217$	−6.93996	−0.471115
$218$	−6.44900	−0.436781
$219$	24.9121	1.68341
$220$	−9.85819	−0.664639
$221$	−0.0532881	−0.00358455
$222$	−10.3548	−0.694970
$223$	8.02720	0.537541	0.268771	−	0.963204i	$-0.413383\pi$
$223$	8.02720	0.537541	0.268771	+	0.963204i	$0.413383\pi$
$224$	−3.00020	−0.200460
$225$	0.215478	0.0143652
$226$	18.6455	1.24028
$227$	−3.89538	−0.258546	−0.129273	−	0.991609i	$-0.541264\pi$
$227$	−3.89538	−0.258546	−0.129273	+	0.991609i	$0.541264\pi$
$228$	9.75552	0.646075
$229$	−5.33089	−0.352275	−0.176138	−	0.984366i	$-0.556360\pi$
$229$	−5.33089	−0.352275	−0.176138	+	0.984366i	$0.556360\pi$
$230$	−5.80188	−0.382565
$231$	22.3354	1.46956
$232$	−8.38979	−0.550817
$233$	−3.79302	−0.248489	−0.124245	−	0.992252i	$-0.539651\pi$
$233$	−3.79302	−0.248489	−0.124245	+	0.992252i	$0.539651\pi$
$234$	−0.0350660	−0.00229234
$235$	−12.8884	−0.840749
$236$	−7.38292	−0.480587
$237$	6.56877	0.426688
$238$	1.97242	0.127853
$239$	−24.8096	−1.60480	−0.802400	−	0.596787i	$-0.796444\pi$
$239$	−24.8096	−1.60480	−0.802400	+	0.596787i	$0.796444\pi$
$240$	3.39973	0.219451
$241$	3.09832	0.199580	0.0997901	−	0.995009i	$-0.468183\pi$
$241$	3.09832	0.199580	0.0997901	+	0.995009i	$0.468183\pi$
$242$	10.5872	0.680572
$243$	4.45900	0.286045
$244$	−14.6948	−0.940739
$245$	4.24614	0.271276
$246$	6.75216	0.430502
$247$	0.493500	0.0314007
$248$	2.31316	0.146886
$249$	21.4126	1.35697
$250$	−11.6657	−0.737802
$251$	−3.41937	−0.215829	−0.107914	−	0.994160i	$-0.534417\pi$
$251$	−3.41937	−0.215829	−0.107914	+	0.994160i	$0.534417\pi$
$252$	1.29794	0.0817626
$253$	12.7048	0.798744
$254$	−12.0844	−0.758245
$255$	−2.23508	−0.139966
$256$	1.00000	0.0625000
$257$	−8.25349	−0.514838	−0.257419	−	0.966300i	$-0.582872\pi$
$257$	−8.25349	−0.514838	−0.257419	+	0.966300i	$0.582872\pi$
$258$	−2.26220	−0.140839
$259$	19.3887	1.20475
$260$	0.171981	0.0106658
$261$	3.62957	0.224665
$262$	12.3889	0.765388
$263$	−19.7906	−1.22034	−0.610170	−	0.792271i	$-0.708899\pi$
$263$	−19.7906	−1.22034	−0.610170	+	0.792271i	$0.708899\pi$
$264$	−7.44464	−0.458186
$265$	−19.7146	−1.21106
$266$	−18.2665	−1.11999
$267$	1.51065	0.0924503
$268$	7.07438	0.432136
$269$	−24.9245	−1.51967	−0.759837	−	0.650114i	$-0.774721\pi$
$269$	−24.9245	−1.51967	−0.759837	+	0.650114i	$0.774721\pi$
$270$	−11.6700	−0.710211
$271$	−18.2974	−1.11149	−0.555743	−	0.831354i	$-0.687566\pi$
$271$	−18.2974	−1.11149	−0.555743	+	0.831354i	$0.687566\pi$
$272$	−0.657428	−0.0398624
$273$	−0.389653	−0.0235829
$274$	9.98896	0.603455
$275$	2.31419	0.139551
$276$	−4.38142	−0.263731
$277$	−3.71024	−0.222927	−0.111463	−	0.993769i	$-0.535554\pi$
$277$	−3.71024	−0.222927	−0.111463	+	0.993769i	$0.535554\pi$
$278$	0.934529	0.0560494
$279$	−1.00071	−0.0599112
$280$	−6.36575	−0.380426
$281$	−6.62948	−0.395481	−0.197741	−	0.980254i	$-0.563360\pi$
$281$	−6.62948	−0.395481	−0.197741	+	0.980254i	$0.563360\pi$
$282$	−9.73301	−0.579592
$283$	−5.03350	−0.299211	−0.149605	−	0.988746i	$-0.547800\pi$
$283$	−5.03350	−0.299211	−0.149605	+	0.988746i	$0.547800\pi$
$284$	−11.3636	−0.674305
$285$	20.6990	1.22610
$286$	−0.376600	−0.0222688
$287$	−12.6429	−0.746290
$288$	−0.432617	−0.0254922
$289$	−16.5678	−0.974576
$290$	−17.8012	−1.04532
$291$	−6.72951	−0.394491
$292$	15.5477	0.909859
$293$	30.0900	1.75788	0.878939	−	0.476933i	$-0.158252\pi$
$293$	30.0900	1.75788	0.878939	+	0.476933i	$0.158252\pi$
$294$	3.20657	0.187011
$295$	−15.6649	−0.912044
$296$	−6.46245	−0.375622
$297$	25.5546	1.48283
$298$	4.45814	0.258253
$299$	−0.221642	−0.0128179
$300$	−0.798078	−0.0460770
$301$	4.23582	0.244148
$302$	−12.8824	−0.741296
$303$	6.30472	0.362197
$304$	6.08843	0.349195
$305$	−31.1791	−1.78531
$306$	0.284415	0.0162589
$307$	−13.7593	−0.785283	−0.392641	−	0.919692i	$-0.628439\pi$
$307$	−13.7593	−0.785283	−0.392641	+	0.919692i	$0.628439\pi$
$308$	13.9396	0.794280
$309$	−1.84482	−0.104948
$310$	4.90801	0.278756
$311$	9.12074	0.517190	0.258595	−	0.965986i	$-0.416741\pi$
$311$	9.12074	0.517190	0.258595	+	0.965986i	$0.416741\pi$
$312$	0.129876	0.00735276
$313$	−11.9404	−0.674912	−0.337456	−	0.941341i	$-0.609566\pi$
$313$	−11.9404	−0.674912	−0.337456	+	0.941341i	$0.609566\pi$
$314$	−5.06896	−0.286058
$315$	2.75393	0.155167
$316$	4.09958	0.230619
$317$	−32.1446	−1.80542	−0.902709	−	0.430251i	$-0.858425\pi$
$317$	−32.1446	−1.80542	−0.902709	+	0.430251i	$0.858425\pi$
$318$	−14.8879	−0.834872
$319$	38.9807	2.18250
$320$	2.12177	0.118611
$321$	11.1207	0.620695
$322$	8.20390	0.457186
$323$	−4.00270	−0.222716
$324$	−7.51499	−0.417499
$325$	−0.0403722	−0.00223945
$326$	−5.74314	−0.318083
$327$	−10.3333	−0.571431
$328$	4.21403	0.232681
$329$	18.2244	1.00474
$330$	−15.7958	−0.869532
$331$	16.4485	0.904092	0.452046	−	0.891995i	$-0.350694\pi$
$331$	16.4485	0.904092	0.452046	+	0.891995i	$0.350694\pi$
$332$	13.3636	0.733425
$333$	2.79577	0.153207
$334$	17.4040	0.952306
$335$	15.0102	0.820096
$336$	−4.80724	−0.262257
$337$	−30.9053	−1.68352	−0.841760	−	0.539852i	$-0.818480\pi$
$337$	−30.9053	−1.68352	−0.841760	+	0.539852i	$0.818480\pi$
$338$	−12.9934	−0.706749
$339$	29.8757	1.62263
$340$	−1.39491	−0.0756498
$341$	−10.7474	−0.582006
$342$	−2.63396	−0.142428
$343$	14.9973	0.809781
$344$	−1.41184	−0.0761215
$345$	−9.29638	−0.500500
$346$	11.4070	0.613246
$347$	14.5983	0.783677	0.391839	−	0.920034i	$-0.371839\pi$
$347$	14.5983	0.783677	0.391839	+	0.920034i	$0.371839\pi$
$348$	−13.4430	−0.720621
$349$	−5.62297	−0.300990	−0.150495	−	0.988611i	$-0.548087\pi$
$349$	−5.62297	−0.300990	−0.150495	+	0.988611i	$0.548087\pi$
$350$	1.49434	0.0798761
$351$	−0.445813	−0.0237958
$352$	−4.64620	−0.247644
$353$	−10.2011	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$353$	−10.2011	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$354$	−11.8297	−0.628741
$355$	−24.1109	−1.27968
$356$	0.942798	0.0499682
$357$	3.16042	0.167267
$358$	−11.4760	−0.606524
$359$	−10.2192	−0.539346	−0.269673	−	0.962952i	$-0.586916\pi$
$359$	−10.2192	−0.539346	−0.269673	+	0.962952i	$0.586916\pi$
$360$	−0.917916	−0.0483784
$361$	18.0689	0.950997
$362$	5.40532	0.284098
$363$	16.9639	0.890376
$364$	−0.243183	−0.0127463
$365$	32.9887	1.72670
$366$	−23.5456	−1.23075
$367$	−13.0175	−0.679506	−0.339753	−	0.940515i	$-0.610344\pi$
$367$	−13.0175	−0.679506	−0.339753	+	0.940515i	$0.610344\pi$
$368$	−2.73445	−0.142543
$369$	−1.82306	−0.0949049
$370$	−13.7119	−0.712845
$371$	27.8765	1.44728
$372$	3.70639	0.192168
$373$	−18.8242	−0.974681	−0.487341	−	0.873212i	$-0.662033\pi$
$373$	−18.8242	−0.974681	−0.487341	+	0.873212i	$0.662033\pi$
$374$	3.05454	0.157947
$375$	−18.6920	−0.965250
$376$	−6.07438	−0.313262
$377$	−0.680039	−0.0350238
$378$	16.5014	0.848742
$379$	27.4161	1.40827	0.704135	−	0.710066i	$-0.251335\pi$
$379$	27.4161	1.40827	0.704135	+	0.710066i	$0.251335\pi$
$380$	12.9183	0.662693
$381$	−19.3630	−0.991994
$382$	−3.46185	−0.177124
$383$	9.04559	0.462208	0.231104	−	0.972929i	$-0.425766\pi$
$383$	9.04559	0.462208	0.231104	+	0.972929i	$0.425766\pi$
$384$	1.60231	0.0817673
$385$	29.5766	1.50736
$386$	14.1336	0.719383
$387$	0.610788	0.0310481
$388$	−4.19989	−0.213217
$389$	23.6647	1.19985	0.599923	−	0.800058i	$-0.295198\pi$
$389$	23.6647	1.19985	0.599923	+	0.800058i	$0.295198\pi$
$390$	0.275567	0.0139539
$391$	1.79770	0.0909138
$392$	2.00122	0.101077
$393$	19.8508	1.00134
$394$	20.7698	1.04637
$395$	8.69837	0.437662
$396$	2.01003	0.101008
$397$	−1.74761	−0.0877102	−0.0438551	−	0.999038i	$-0.513964\pi$
$397$	−1.74761	−0.0877102	−0.0438551	+	0.999038i	$0.513964\pi$
$398$	−5.32686	−0.267011
$399$	−29.2685	−1.46526
$400$	−0.498081	−0.0249040
$401$	9.49461	0.474138	0.237069	−	0.971493i	$-0.423813\pi$
$401$	9.49461	0.474138	0.237069	+	0.971493i	$0.423813\pi$
$402$	11.3353	0.565354
$403$	0.187495	0.00933977
$404$	3.93478	0.195763
$405$	−15.9451	−0.792318
$406$	25.1711	1.24922
$407$	30.0259	1.48833
$408$	−1.05340	−0.0521511
$409$	37.6753	1.86292	0.931461	−	0.363842i	$-0.118535\pi$
$409$	37.6753	1.86292	0.931461	+	0.363842i	$0.118535\pi$
$410$	8.94121	0.441575
$411$	16.0054	0.789487
$412$	−1.15135	−0.0567231
$413$	22.1503	1.08994
$414$	1.18297	0.0581398
$415$	28.3546	1.39187
$416$	0.0810555	0.00397407
$417$	1.49740	0.0733281
$418$	−28.2881	−1.38361
$419$	11.6405	0.568678	0.284339	−	0.958724i	$-0.408226\pi$
$419$	11.6405	0.568678	0.284339	+	0.958724i	$0.408226\pi$
$420$	−10.1999	−0.497703
$421$	−3.38781	−0.165112	−0.0825559	−	0.996586i	$-0.526308\pi$
$421$	−3.38781	−0.165112	−0.0825559	+	0.996586i	$0.526308\pi$
$422$	−20.5837	−1.00200
$423$	2.62788	0.127772
$424$	−9.29155	−0.451238
$425$	0.327452	0.0158838
$426$	−18.2079	−0.882177
$427$	44.0874	2.13354
$428$	6.94041	0.335478
$429$	−0.603429	−0.0291338
$430$	−2.99561	−0.144461
$431$	11.4963	0.553758	0.276879	−	0.960905i	$-0.410700\pi$
$431$	11.4963	0.553758	0.276879	+	0.960905i	$0.410700\pi$
$432$	−5.50010	−0.264624
$433$	−36.4793	−1.75308	−0.876542	−	0.481326i	$-0.840155\pi$
$433$	−36.4793	−1.75308	−0.876542	+	0.481326i	$0.840155\pi$
$434$	−6.93996	−0.333129
$435$	−28.5230	−1.36757
$436$	−6.44900	−0.308851
$437$	−16.6485	−0.796405
$438$	24.9121	1.19035
$439$	25.2294	1.20413	0.602067	−	0.798445i	$-0.294344\pi$
$439$	25.2294	1.20413	0.602067	+	0.798445i	$0.294344\pi$
$440$	−9.85819	−0.469971
$441$	−0.865764	−0.0412269
$442$	−0.0532881	−0.00253466
$443$	21.4445	1.01886	0.509429	−	0.860513i	$-0.329857\pi$
$443$	21.4445	1.01886	0.509429	+	0.860513i	$0.329857\pi$
$444$	−10.3548	−0.491418
$445$	2.00040	0.0948282
$446$	8.02720	0.380099
$447$	7.14331	0.337867
$448$	−3.00020	−0.141746
$449$	25.5519	1.20587	0.602933	−	0.797792i	$-0.293999\pi$
$449$	25.5519	1.20587	0.602933	+	0.797792i	$0.293999\pi$
$450$	0.215478	0.0101578
$451$	−19.5792	−0.921950
$452$	18.6455	0.877008
$453$	−20.6415	−0.969821
$454$	−3.89538	−0.182819
$455$	−0.515979	−0.0241895
$456$	9.75552	0.456844
$457$	−30.2578	−1.41540	−0.707701	−	0.706512i	$-0.750267\pi$
$457$	−30.2578	−1.41540	−0.707701	+	0.706512i	$0.750267\pi$
$458$	−5.33089	−0.249096
$459$	3.61592	0.168777
$460$	−5.80188	−0.270514
$461$	9.32005	0.434078	0.217039	−	0.976163i	$-0.430360\pi$
$461$	9.32005	0.434078	0.217039	+	0.976163i	$0.430360\pi$
$462$	22.3354	1.03914
$463$	41.5381	1.93044	0.965219	−	0.261443i	$-0.0841984\pi$
$463$	41.5381	1.93044	0.965219	+	0.261443i	$0.0841984\pi$
$464$	−8.38979	−0.389486
$465$	7.86412	0.364690
$466$	−3.79302	−0.175708
$467$	−26.5499	−1.22858	−0.614291	−	0.789079i	$-0.710558\pi$
$467$	−26.5499	−1.22858	−0.614291	+	0.789079i	$0.710558\pi$
$468$	−0.0350660	−0.00162093
$469$	−21.2246	−0.980060
$470$	−12.8884	−0.594500
$471$	−8.12202	−0.374243
$472$	−7.38292	−0.339826
$473$	6.55971	0.301616
$474$	6.56877	0.301714
$475$	−3.03253	−0.139142
$476$	1.97242	0.0904056
$477$	4.01969	0.184049
$478$	−24.8096	−1.13476
$479$	−0.269799	−0.0123274	−0.00616371	−	0.999981i	$-0.501962\pi$
$479$	−0.269799	−0.0123274	−0.00616371	+	0.999981i	$0.501962\pi$
$480$	3.39973	0.155176
$481$	−0.523817	−0.0238840
$482$	3.09832	0.141124
$483$	13.1452	0.598125
$484$	10.5872	0.481237
$485$	−8.91122	−0.404638
$486$	4.45900	0.202264
$487$	40.4361	1.83234	0.916168	−	0.400794i	$-0.131266\pi$
$487$	40.4361	1.83234	0.916168	+	0.400794i	$0.131266\pi$
$488$	−14.6948	−0.665203
$489$	−9.20226	−0.416141
$490$	4.24614	0.191821
$491$	31.7283	1.43188	0.715939	−	0.698163i	$-0.245999\pi$
$491$	31.7283	1.43188	0.715939	+	0.698163i	$0.245999\pi$
$492$	6.75216	0.304411
$493$	5.51568	0.248414
$494$	0.493500	0.0222036
$495$	4.26482	0.191690
$496$	2.31316	0.103864
$497$	34.0931	1.52928
$498$	21.4126	0.959523
$499$	26.4353	1.18341	0.591704	−	0.806155i	$-0.298455\pi$
$499$	26.4353	1.18341	0.591704	+	0.806155i	$0.298455\pi$
$500$	−11.6657	−0.521705
$501$	27.8866	1.24588
$502$	−3.41937	−0.152614
$503$	8.19371	0.365339	0.182670	−	0.983174i	$-0.441526\pi$
$503$	8.19371	0.365339	0.182670	+	0.983174i	$0.441526\pi$
$504$	1.29794	0.0578149
$505$	8.34871	0.371513
$506$	12.7048	0.564798
$507$	−20.8194	−0.924624
$508$	−12.0844	−0.536160
$509$	14.4575	0.640815	0.320408	−	0.947280i	$-0.396180\pi$
$509$	14.4575	0.640815	0.320408	+	0.947280i	$0.396180\pi$
$510$	−2.23508	−0.0989708
$511$	−46.6462	−2.06351
$512$	1.00000	0.0441942
$513$	−33.4870	−1.47849
$514$	−8.25349	−0.364046
$515$	−2.44291	−0.107647
$516$	−2.26220	−0.0995880
$517$	28.2228	1.24124
$518$	19.3887	0.851889
$519$	18.2775	0.802295
$520$	0.171981	0.00754188
$521$	−34.5584	−1.51403	−0.757016	−	0.653396i	$-0.773344\pi$
$521$	−34.5584	−1.51403	−0.757016	+	0.653396i	$0.773344\pi$
$522$	3.62957	0.158862
$523$	43.2608	1.89166	0.945832	−	0.324657i	$-0.105249\pi$
$523$	43.2608	1.89166	0.945832	+	0.324657i	$0.105249\pi$
$524$	12.3889	0.541211
$525$	2.39440	0.104500
$526$	−19.7906	−0.862910
$527$	−1.52074	−0.0662444
$528$	−7.44464	−0.323986
$529$	−15.5228	−0.674904
$530$	−19.7146	−0.856346
$531$	3.19398	0.138607
$532$	−18.2665	−0.791954
$533$	0.341570	0.0147950
$534$	1.51065	0.0653722
$535$	14.7260	0.636660
$536$	7.07438	0.305566
$537$	−18.3880	−0.793501
$538$	−24.9245	−1.07457
$539$	−9.29809	−0.400497
$540$	−11.6700	−0.502195
$541$	−17.4314	−0.749435	−0.374717	−	0.927139i	$-0.622260\pi$
$541$	−17.4314	−0.749435	−0.374717	+	0.927139i	$0.622260\pi$
$542$	−18.2974	−0.785940
$543$	8.66098	0.371678
$544$	−0.657428	−0.0281870
$545$	−13.6833	−0.586128
$546$	−0.389653	−0.0166756
$547$	−18.0548	−0.771968	−0.385984	−	0.922505i	$-0.626138\pi$
$547$	−18.0548	−0.771968	−0.385984	+	0.922505i	$0.626138\pi$
$548$	9.98896	0.426707
$549$	6.35723	0.271320
$550$	2.31419	0.0986772
$551$	−51.0806	−2.17611
$552$	−4.38142	−0.186486
$553$	−12.2996	−0.523031
$554$	−3.71024	−0.157633
$555$	−21.9706	−0.932599
$556$	0.934529	0.0396329
$557$	10.2031	0.432318	0.216159	−	0.976358i	$-0.430647\pi$
$557$	10.2031	0.432318	0.216159	+	0.976358i	$0.430647\pi$
$558$	−1.00071	−0.0423636
$559$	−0.114438	−0.00484019
$560$	−6.36575	−0.269002
$561$	4.89431	0.206638
$562$	−6.62948	−0.279648
$563$	−27.7783	−1.17072	−0.585358	−	0.810775i	$-0.699046\pi$
$563$	−27.7783	−1.17072	−0.585358	+	0.810775i	$0.699046\pi$
$564$	−9.73301	−0.409833
$565$	39.5614	1.66436
$566$	−5.03350	−0.211574
$567$	22.5465	0.946864
$568$	−11.3636	−0.476805
$569$	−7.67365	−0.321696	−0.160848	−	0.986979i	$-0.551423\pi$
$569$	−7.67365	−0.321696	−0.160848	+	0.986979i	$0.551423\pi$
$570$	20.6990	0.866985
$571$	28.2798	1.18347	0.591737	−	0.806131i	$-0.298442\pi$
$571$	28.2798	1.18347	0.591737	+	0.806131i	$0.298442\pi$
$572$	−0.376600	−0.0157464
$573$	−5.54694	−0.231727
$574$	−12.6429	−0.527706
$575$	1.36198	0.0567984
$576$	−0.432617	−0.0180257
$577$	−4.52213	−0.188259	−0.0941294	−	0.995560i	$-0.530007\pi$
$577$	−4.52213	−0.188259	−0.0941294	+	0.995560i	$0.530007\pi$
$578$	−16.5678	−0.689129
$579$	22.6464	0.941152
$580$	−17.8012	−0.739156
$581$	−40.0937	−1.66337
$582$	−6.72951	−0.278947
$583$	43.1704	1.78794
$584$	15.5477	0.643368
$585$	−0.0744021	−0.00307615
$586$	30.0900	1.24301
$587$	1.98229	0.0818177	0.0409088	−	0.999163i	$-0.486975\pi$
$587$	1.98229	0.0818177	0.0409088	+	0.999163i	$0.486975\pi$
$588$	3.20657	0.132237
$589$	14.0835	0.580301
$590$	−15.6649	−0.644913
$591$	33.2796	1.36894
$592$	−6.46245	−0.265605
$593$	−3.09795	−0.127218	−0.0636088	−	0.997975i	$-0.520261\pi$
$593$	−3.09795	−0.127218	−0.0636088	+	0.997975i	$0.520261\pi$
$594$	25.5546	1.04852
$595$	4.18502	0.171569
$596$	4.45814	0.182613
$597$	−8.53525	−0.349325
$598$	−0.221642	−0.00906362
$599$	4.35533	0.177954	0.0889770	−	0.996034i	$-0.471640\pi$
$599$	4.35533	0.177954	0.0889770	+	0.996034i	$0.471640\pi$
$600$	−0.798078	−0.0325814
$601$	−22.4396	−0.915329	−0.457664	−	0.889125i	$-0.651314\pi$
$601$	−22.4396	−0.915329	−0.457664	+	0.889125i	$0.651314\pi$
$602$	4.23582	0.172639
$603$	−3.06050	−0.124633
$604$	−12.8824	−0.524176
$605$	22.4637	0.913277
$606$	6.30472	0.256112
$607$	7.43446	0.301755	0.150878	−	0.988552i	$-0.451790\pi$
$607$	7.43446	0.301755	0.150878	+	0.988552i	$0.451790\pi$
$608$	6.08843	0.246918
$609$	40.3318	1.63433
$610$	−31.1791	−1.26240
$611$	−0.492361	−0.0199188
$612$	0.284415	0.0114968
$613$	−34.3620	−1.38787	−0.693935	−	0.720038i	$-0.744124\pi$
$613$	−34.3620	−1.38787	−0.693935	+	0.720038i	$0.744124\pi$
$614$	−13.7593	−0.555279
$615$	14.3266	0.577702
$616$	13.9396	0.561641
$617$	−30.4679	−1.22659	−0.613296	−	0.789853i	$-0.710157\pi$
$617$	−30.4679	−1.22659	−0.613296	+	0.789853i	$0.710157\pi$
$618$	−1.84482	−0.0742095
$619$	−20.1700	−0.810699	−0.405350	−	0.914162i	$-0.632850\pi$
$619$	−20.1700	−0.810699	−0.405350	+	0.914162i	$0.632850\pi$
$620$	4.90801	0.197110
$621$	15.0397	0.603524
$622$	9.12074	0.365708
$623$	−2.82859	−0.113325
$624$	0.129876	0.00519919
$625$	−22.2615	−0.890460
$626$	−11.9404	−0.477235
$627$	−45.3261	−1.81015
$628$	−5.06896	−0.202274
$629$	4.24860	0.169403
$630$	2.75393	0.109719
$631$	39.6647	1.57903	0.789513	−	0.613734i	$-0.210333\pi$
$631$	39.6647	1.57903	0.789513	+	0.613734i	$0.210333\pi$
$632$	4.09958	0.163072
$633$	−32.9814	−1.31089
$634$	−32.1446	−1.27662
$635$	−25.6404	−1.01751
$636$	−14.8879	−0.590344
$637$	0.162210	0.00642700
$638$	38.9807	1.54326
$639$	4.91608	0.194477
$640$	2.12177	0.0838704
$641$	42.8828	1.69377	0.846884	−	0.531778i	$-0.178476\pi$
$641$	42.8828	1.69377	0.846884	+	0.531778i	$0.178476\pi$
$642$	11.1207	0.438897
$643$	−17.5400	−0.691711	−0.345856	−	0.938288i	$-0.612411\pi$
$643$	−17.5400	−0.691711	−0.345856	+	0.938288i	$0.612411\pi$
$644$	8.20390	0.323279
$645$	−4.79988	−0.188995
$646$	−4.00270	−0.157484
$647$	−1.66862	−0.0656003	−0.0328002	−	0.999462i	$-0.510442\pi$
$647$	−1.66862	−0.0656003	−0.0328002	+	0.999462i	$0.510442\pi$
$648$	−7.51499	−0.295217
$649$	34.3025	1.34649
$650$	−0.0403722	−0.00158353
$651$	−11.1199	−0.435825
$652$	−5.74314	−0.224919
$653$	26.2503	1.02726	0.513628	−	0.858013i	$-0.328301\pi$
$653$	26.2503	1.02726	0.513628	+	0.858013i	$0.328301\pi$
$654$	−10.3333	−0.404063
$655$	26.2864	1.02709
$656$	4.21403	0.164530
$657$	−6.72620	−0.262414
$658$	18.2244	0.710460
$659$	8.09664	0.315400	0.157700	−	0.987487i	$-0.449592\pi$
$659$	8.09664	0.315400	0.157700	+	0.987487i	$0.449592\pi$
$660$	−15.7958	−0.614852
$661$	12.9475	0.503599	0.251800	−	0.967779i	$-0.418978\pi$
$661$	12.9475	0.503599	0.251800	+	0.967779i	$0.418978\pi$
$662$	16.4485	0.639290
$663$	−0.0853839	−0.00331604
$664$	13.3636	0.518610
$665$	−38.7574	−1.50295
$666$	2.79577	0.108334
$667$	22.9415	0.888297
$668$	17.4040	0.673382
$669$	12.8620	0.497275
$670$	15.0102	0.579895
$671$	68.2751	2.63573
$672$	−4.80724	−0.185443
$673$	−34.6223	−1.33459	−0.667295	−	0.744793i	$-0.732548\pi$
$673$	−34.6223	−1.33459	−0.667295	+	0.744793i	$0.732548\pi$
$674$	−30.9053	−1.19043
$675$	2.73950	0.105443
$676$	−12.9934	−0.499747
$677$	−9.99535	−0.384153	−0.192076	−	0.981380i	$-0.561522\pi$
$677$	−9.99535	−0.384153	−0.192076	+	0.981380i	$0.561522\pi$
$678$	29.8757	1.14737
$679$	12.6005	0.483564
$680$	−1.39491	−0.0534925
$681$	−6.24159	−0.239178
$682$	−10.7474	−0.411540
$683$	−10.2500	−0.392205	−0.196102	−	0.980583i	$-0.562828\pi$
$683$	−10.2500	−0.392205	−0.196102	+	0.980583i	$0.562828\pi$
$684$	−2.63396	−0.100712
$685$	21.1943	0.809793
$686$	14.9973	0.572601
$687$	−8.54172	−0.325887
$688$	−1.41184	−0.0538260
$689$	−0.753131	−0.0286920
$690$	−9.29638	−0.353907
$691$	0.524027	0.0199349	0.00996747	−	0.999950i	$-0.496827\pi$
$691$	0.524027	0.0199349	0.00996747	+	0.999950i	$0.496827\pi$
$692$	11.4070	0.433630
$693$	−6.03050	−0.229080
$694$	14.5983	0.554144
$695$	1.98286	0.0752141
$696$	−13.4430	−0.509556
$697$	−2.77042	−0.104937
$698$	−5.62297	−0.212832
$699$	−6.07758	−0.229875
$700$	1.49434	0.0564809
$701$	−21.9843	−0.830337	−0.415169	−	0.909744i	$-0.636277\pi$
$701$	−21.9843	−0.830337	−0.415169	+	0.909744i	$0.636277\pi$
$702$	−0.445813	−0.0168261
$703$	−39.3462	−1.48397
$704$	−4.64620	−0.175110
$705$	−20.6512	−0.777770
$706$	−10.2011	−0.383922
$707$	−11.8051	−0.443978
$708$	−11.8297	−0.444587
$709$	2.33031	0.0875167	0.0437583	−	0.999042i	$-0.486067\pi$
$709$	2.33031	0.0875167	0.0437583	+	0.999042i	$0.486067\pi$
$710$	−24.1109	−0.904867
$711$	−1.77355	−0.0665132
$712$	0.942798	0.0353328
$713$	−6.32523	−0.236882
$714$	3.16042	0.118276
$715$	−0.799060	−0.0298832
$716$	−11.4760	−0.428877
$717$	−39.7525	−1.48459
$718$	−10.2192	−0.381376
$719$	−8.31109	−0.309952	−0.154976	−	0.987918i	$-0.549530\pi$
$719$	−8.31109	−0.309952	−0.154976	+	0.987918i	$0.549530\pi$
$720$	−0.917916	−0.0342087
$721$	3.45429	0.128645
$722$	18.0689	0.672456
$723$	4.96445	0.184630
$724$	5.40532	0.200887
$725$	4.17879	0.155197
$726$	16.9639	0.629591
$727$	−2.79900	−0.103809	−0.0519046	−	0.998652i	$-0.516529\pi$
$727$	−2.79900	−0.103809	−0.0519046	+	0.998652i	$0.516529\pi$
$728$	−0.243183	−0.00901296
$729$	29.6896	1.09962
$730$	32.9887	1.22096
$731$	0.928185	0.0343302
$732$	−23.5456	−0.870270
$733$	−42.8081	−1.58115	−0.790576	−	0.612363i	$-0.790219\pi$
$733$	−42.8081	−1.58115	−0.790576	+	0.612363i	$0.790219\pi$
$734$	−13.0175	−0.480484
$735$	6.80362	0.250955
$736$	−2.73445	−0.100793
$737$	−32.8690	−1.21074
$738$	−1.82306	−0.0671079
$739$	5.71589	0.210263	0.105131	−	0.994458i	$-0.466474\pi$
$739$	5.71589	0.210263	0.105131	+	0.994458i	$0.466474\pi$
$740$	−13.7119	−0.504058
$741$	0.790738	0.0290485
$742$	27.8765	1.02338
$743$	−1.89211	−0.0694146	−0.0347073	−	0.999398i	$-0.511050\pi$
$743$	−1.89211	−0.0694146	−0.0347073	+	0.999398i	$0.511050\pi$
$744$	3.70639	0.135883
$745$	9.45916	0.346557
$746$	−18.8242	−0.689204
$747$	−5.78135	−0.211528
$748$	3.05454	0.111685
$749$	−20.8227	−0.760843
$750$	−18.6920	−0.682535
$751$	−27.2840	−0.995608	−0.497804	−	0.867289i	$-0.665860\pi$
$751$	−27.2840	−0.995608	−0.497804	+	0.867289i	$0.665860\pi$
$752$	−6.07438	−0.221510
$753$	−5.47887	−0.199661
$754$	−0.680039	−0.0247655
$755$	−27.3334	−0.994766
$756$	16.5014	0.600151
$757$	−17.2439	−0.626739	−0.313369	−	0.949631i	$-0.601458\pi$
$757$	−17.2439	−0.626739	−0.313369	+	0.949631i	$0.601458\pi$
$758$	27.4161	0.995798
$759$	20.3570	0.738912
$760$	12.9183	0.468594
$761$	31.1412	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$761$	31.1412	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$762$	−19.3630	−0.701446
$763$	19.3483	0.700456
$764$	−3.46185	−0.125245
$765$	0.603464	0.0218183
$766$	9.04559	0.326830
$767$	−0.598426	−0.0216079
$768$	1.60231	0.0578182
$769$	−51.6499	−1.86254	−0.931271	−	0.364328i	$-0.881299\pi$
$769$	−51.6499	−1.86254	−0.931271	+	0.364328i	$0.881299\pi$
$770$	29.5766	1.06587
$771$	−13.2246	−0.476273
$772$	14.1336	0.508680
$773$	26.8132	0.964403	0.482201	−	0.876060i	$-0.339837\pi$
$773$	26.8132	0.964403	0.482201	+	0.876060i	$0.339837\pi$
$774$	0.610788	0.0219543
$775$	−1.15214	−0.0413862
$776$	−4.19989	−0.150767
$777$	31.0666	1.11451
$778$	23.6647	0.848419
$779$	25.6568	0.919251
$780$	0.275567	0.00986687
$781$	52.7975	1.88924
$782$	1.79770	0.0642857
$783$	46.1447	1.64908
$784$	2.00122	0.0714723
$785$	−10.7552	−0.383869
$786$	19.8508	0.708054
$787$	−14.3016	−0.509798	−0.254899	−	0.966968i	$-0.582042\pi$
$787$	−14.3016	−0.509798	−0.254899	+	0.966968i	$0.582042\pi$
$788$	20.7698	0.739894
$789$	−31.7106	−1.12893
$790$	8.69837	0.309474
$791$	−55.9402	−1.98900
$792$	2.01003	0.0714233
$793$	−1.19110	−0.0422970
$794$	−1.74761	−0.0620204
$795$	−31.5887	−1.12034
$796$	−5.32686	−0.188805
$797$	−17.7435	−0.628509	−0.314254	−	0.949339i	$-0.601754\pi$
$797$	−17.7435	−0.628509	−0.314254	+	0.949339i	$0.601754\pi$
$798$	−29.2685	−1.03610
$799$	3.99347	0.141279
$800$	−0.498081	−0.0176098
$801$	−0.407871	−0.0144114
$802$	9.49461	0.335266
$803$	−72.2377	−2.54921
$804$	11.3353	0.399766
$805$	17.4068	0.613510
$806$	0.187495	0.00660421
$807$	−39.9367	−1.40584
$808$	3.93478	0.138425
$809$	9.36290	0.329182	0.164591	−	0.986362i	$-0.447370\pi$
$809$	9.36290	0.329182	0.164591	+	0.986362i	$0.447370\pi$
$810$	−15.9451	−0.560254
$811$	−9.40731	−0.330335	−0.165168	−	0.986266i	$-0.552817\pi$
$811$	−9.40731	−0.330335	−0.165168	+	0.986266i	$0.552817\pi$
$812$	25.1711	0.883332
$813$	−29.3180	−1.02823
$814$	30.0259	1.05241
$815$	−12.1856	−0.426844
$816$	−1.05340	−0.0368764
$817$	−8.59590	−0.300733
$818$	37.6753	1.31728
$819$	0.105205	0.00367617
$820$	8.94121	0.312241
$821$	11.9822	0.418181	0.209091	−	0.977896i	$-0.432950\pi$
$821$	11.9822	0.418181	0.209091	+	0.977896i	$0.432950\pi$
$822$	16.0054	0.558251
$823$	41.9492	1.46226	0.731129	−	0.682239i	$-0.238994\pi$
$823$	41.9492	1.46226	0.731129	+	0.682239i	$0.238994\pi$
$824$	−1.15135	−0.0401093
$825$	3.70803	0.129097
$826$	22.1503	0.770706
$827$	51.5217	1.79158	0.895792	−	0.444473i	$-0.146609\pi$
$827$	51.5217	1.79158	0.895792	+	0.444473i	$0.146609\pi$
$828$	1.18297	0.0411111
$829$	−4.84634	−0.168320	−0.0841601	−	0.996452i	$-0.526821\pi$
$829$	−4.84634	−0.168320	−0.0841601	+	0.996452i	$0.526821\pi$
$830$	28.3546	0.984203
$831$	−5.94494	−0.206228
$832$	0.0810555	0.00281009
$833$	−1.31566	−0.0455849
$834$	1.49740	0.0518508
$835$	36.9274	1.27792
$836$	−28.2881	−0.978363
$837$	−12.7226	−0.439758
$838$	11.6405	0.402116
$839$	−24.6343	−0.850472	−0.425236	−	0.905083i	$-0.639809\pi$
$839$	−24.6343	−0.850472	−0.425236	+	0.905083i	$0.639809\pi$
$840$	−10.1999	−0.351929
$841$	41.3886	1.42719
$842$	−3.38781	−0.116752
$843$	−10.6224	−0.365856
$844$	−20.5837	−0.708522
$845$	−27.5691	−0.948406
$846$	2.62788	0.0903484
$847$	−31.7638	−1.09142
$848$	−9.29155	−0.319073
$849$	−8.06521	−0.276797
$850$	0.327452	0.0112315
$851$	17.6712	0.605762
$852$	−18.2079	−0.623793
$853$	19.2991	0.660787	0.330394	−	0.943843i	$-0.392818\pi$
$853$	19.2991	0.660787	0.330394	+	0.943843i	$0.392818\pi$
$854$	44.0874	1.50864
$855$	−5.58866	−0.191128
$856$	6.94041	0.237218
$857$	−18.9674	−0.647916	−0.323958	−	0.946072i	$-0.605014\pi$
$857$	−18.9674	−0.647916	−0.323958	+	0.946072i	$0.605014\pi$
$858$	−0.603429	−0.0206007
$859$	46.6040	1.59011	0.795055	−	0.606538i	$-0.207442\pi$
$859$	46.6040	1.59011	0.795055	+	0.606538i	$0.207442\pi$
$860$	−2.99561	−0.102149
$861$	−20.2579	−0.690386
$862$	11.4963	0.391566
$863$	3.93485	0.133944	0.0669720	−	0.997755i	$-0.478666\pi$
$863$	3.93485	0.133944	0.0669720	+	0.997755i	$0.478666\pi$
$864$	−5.50010	−0.187117
$865$	24.2031	0.822931
$866$	−36.4793	−1.23962
$867$	−26.5467	−0.901572
$868$	−6.93996	−0.235558
$869$	−19.0475	−0.646141
$870$	−28.5230	−0.967021
$871$	0.573417	0.0194295
$872$	−6.44900	−0.218391
$873$	1.81695	0.0614943
$874$	−16.6485	−0.563144
$875$	34.9994	1.18320
$876$	24.9121	0.841703
$877$	−28.9084	−0.976167	−0.488083	−	0.872797i	$-0.662304\pi$
$877$	−28.9084	−0.976167	−0.488083	+	0.872797i	$0.662304\pi$
$878$	25.2294	0.851451
$879$	48.2134	1.62620
$880$	−9.85819	−0.332319
$881$	13.7216	0.462294	0.231147	−	0.972919i	$-0.425752\pi$
$881$	13.7216	0.462294	0.231147	+	0.972919i	$0.425752\pi$
$882$	−0.865764	−0.0291518
$883$	6.85113	0.230559	0.115279	−	0.993333i	$-0.463224\pi$
$883$	6.85113	0.230559	0.115279	+	0.993333i	$0.463224\pi$
$884$	−0.0532881	−0.00179227
$885$	−25.0999	−0.843724
$886$	21.4445	0.720441
$887$	−5.79022	−0.194417	−0.0972083	−	0.995264i	$-0.530991\pi$
$887$	−5.79022	−0.194417	−0.0972083	+	0.995264i	$0.530991\pi$
$888$	−10.3548	−0.347485
$889$	36.2558	1.21598
$890$	2.00040	0.0670536
$891$	34.9162	1.16974
$892$	8.02720	0.268771
$893$	−36.9834	−1.23760
$894$	7.14331	0.238908
$895$	−24.3494	−0.813911
$896$	−3.00020	−0.100230
$897$	−0.355138	−0.0118577
$898$	25.5519	0.852676
$899$	−19.4070	−0.647258
$900$	0.215478	0.00718262
$901$	6.10853	0.203504
$902$	−19.5792	−0.651917
$903$	6.78707	0.225860
$904$	18.6455	0.620139
$905$	11.4689	0.381238
$906$	−20.6415	−0.685767
$907$	−40.9155	−1.35858	−0.679289	−	0.733871i	$-0.737712\pi$
$907$	−40.9155	−1.35858	−0.679289	+	0.733871i	$0.737712\pi$
$908$	−3.89538	−0.129273
$909$	−1.70226	−0.0564602
$910$	−0.515979	−0.0171045
$911$	−8.92699	−0.295764	−0.147882	−	0.989005i	$-0.547246\pi$
$911$	−8.92699	−0.295764	−0.147882	+	0.989005i	$0.547246\pi$
$912$	9.75552	0.323038
$913$	−62.0902	−2.05489
$914$	−30.2578	−1.00084
$915$	−49.9584	−1.65157
$916$	−5.33089	−0.176138
$917$	−37.1692	−1.22743
$918$	3.61592	0.119343
$919$	17.3463	0.572201	0.286101	−	0.958200i	$-0.407641\pi$
$919$	17.3463	0.572201	0.286101	+	0.958200i	$0.407641\pi$
$920$	−5.80188	−0.191282
$921$	−22.0465	−0.726458
$922$	9.32005	0.306940
$923$	−0.921081	−0.0303177
$924$	22.3354	0.734782
$925$	3.21882	0.105834
$926$	41.5381	1.36503
$927$	0.498096	0.0163596
$928$	−8.38979	−0.275408
$929$	2.30838	0.0757355	0.0378677	−	0.999283i	$-0.487943\pi$
$929$	2.30838	0.0757355	0.0378677	+	0.999283i	$0.487943\pi$
$930$	7.86412	0.257875
$931$	12.1843	0.399324
$932$	−3.79302	−0.124245
$933$	14.6142	0.478448
$934$	−26.5499	−0.868739
$935$	6.48105	0.211953
$936$	−0.0350660	−0.00114617
$937$	−35.2444	−1.15138	−0.575692	−	0.817667i	$-0.695267\pi$
$937$	−35.2444	−1.15138	−0.575692	+	0.817667i	$0.695267\pi$
$938$	−21.2246	−0.693007
$939$	−19.1322	−0.624356
$940$	−12.8884	−0.420375
$941$	−5.65125	−0.184225	−0.0921127	−	0.995749i	$-0.529362\pi$
$941$	−5.65125	−0.184225	−0.0921127	+	0.995749i	$0.529362\pi$
$942$	−8.12202	−0.264630
$943$	−11.5230	−0.375242
$944$	−7.38292	−0.240294
$945$	35.0123	1.13895
$946$	6.55971	0.213275
$947$	9.09985	0.295705	0.147853	−	0.989009i	$-0.452764\pi$
$947$	9.09985	0.295705	0.147853	+	0.989009i	$0.452764\pi$
$948$	6.56877	0.213344
$949$	1.26023	0.0409086
$950$	−3.03253	−0.0983882
$951$	−51.5054	−1.67018
$952$	1.97242	0.0639264
$953$	23.0536	0.746779	0.373389	−	0.927675i	$-0.378196\pi$
$953$	23.0536	0.746779	0.373389	+	0.927675i	$0.378196\pi$
$954$	4.01969	0.130142
$955$	−7.34526	−0.237687
$956$	−24.8096	−0.802400
$957$	62.4590	2.01901
$958$	−0.269799	−0.00871681
$959$	−29.9689	−0.967747
$960$	3.39973	0.109726
$961$	−25.6493	−0.827396
$962$	−0.523817	−0.0168885
$963$	−3.00254	−0.0967556
$964$	3.09832	0.0997901
$965$	29.9883	0.965359
$966$	13.1452	0.422939
$967$	−12.2962	−0.395420	−0.197710	−	0.980261i	$-0.563350\pi$
$967$	−12.2962	−0.395420	−0.197710	+	0.980261i	$0.563350\pi$
$968$	10.5872	0.340286
$969$	−6.41355	−0.206033
$970$	−8.91122	−0.286122
$971$	37.4735	1.20258	0.601291	−	0.799030i	$-0.294653\pi$
$971$	37.4735	1.20258	0.601291	+	0.799030i	$0.294653\pi$
$972$	4.45900	0.143022
$973$	−2.80378	−0.0898850
$974$	40.4361	1.29566
$975$	−0.0646886	−0.00207169
$976$	−14.6948	−0.470369
$977$	54.3301	1.73817	0.869087	−	0.494658i	$-0.164707\pi$
$977$	54.3301	1.73817	0.869087	+	0.494658i	$0.164707\pi$
$978$	−9.20226	−0.294256
$979$	−4.38043	−0.139999
$980$	4.24614	0.135638
$981$	2.78995	0.0890762
$982$	31.7283	1.01249
$983$	25.6363	0.817670	0.408835	−	0.912608i	$-0.365935\pi$
$983$	25.6363	0.817670	0.408835	+	0.912608i	$0.365935\pi$
$984$	6.75216	0.215251
$985$	44.0688	1.40415
$986$	5.51568	0.175655
$987$	29.2010	0.929478
$988$	0.493500	0.0157003
$989$	3.86061	0.122760
$990$	4.26482	0.135545
$991$	1.17793	0.0374182	0.0187091	−	0.999825i	$-0.494044\pi$
$991$	1.17793	0.0374182	0.0187091	+	0.999825i	$0.494044\pi$
$992$	2.31316	0.0734430
$993$	26.3555	0.836368
$994$	34.0931	1.08137
$995$	−11.3024	−0.358309
$996$	21.4126	0.678485
$997$	−35.5489	−1.12584	−0.562922	−	0.826510i	$-0.690323\pi$
$997$	−35.5489	−1.12584	−0.562922	+	0.826510i	$0.690323\pi$
$998$	26.4353	0.836796
$999$	35.5441	1.12457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.54	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.54	✓	67	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.60231 q^{3} +1.00000 q^{4} +2.12177 q^{5} +1.60231 q^{6} -3.00020 q^{7} +1.00000 q^{8} -0.432617 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.60231 q^{3} +1.00000 q^{4} +2.12177 q^{5} +1.60231 q^{6} -3.00020 q^{7} +1.00000 q^{8} -0.432617 q^{9} +2.12177 q^{10} -4.64620 q^{11} +1.60231 q^{12} +0.0810555 q^{13} -3.00020 q^{14} +3.39973 q^{15} +1.00000 q^{16} -0.657428 q^{17} -0.432617 q^{18} +6.08843 q^{19} +2.12177 q^{20} -4.80724 q^{21} -4.64620 q^{22} -2.73445 q^{23} +1.60231 q^{24} -0.498081 q^{25} +0.0810555 q^{26} -5.50010 q^{27} -3.00020 q^{28} -8.38979 q^{29} +3.39973 q^{30} +2.31316 q^{31} +1.00000 q^{32} -7.44464 q^{33} -0.657428 q^{34} -6.36575 q^{35} -0.432617 q^{36} -6.46245 q^{37} +6.08843 q^{38} +0.129876 q^{39} +2.12177 q^{40} +4.21403 q^{41} -4.80724 q^{42} -1.41184 q^{43} -4.64620 q^{44} -0.917916 q^{45} -2.73445 q^{46} -6.07438 q^{47} +1.60231 q^{48} +2.00122 q^{49} -0.498081 q^{50} -1.05340 q^{51} +0.0810555 q^{52} -9.29155 q^{53} -5.50010 q^{54} -9.85819 q^{55} -3.00020 q^{56} +9.75552 q^{57} -8.38979 q^{58} -7.38292 q^{59} +3.39973 q^{60} -14.6948 q^{61} +2.31316 q^{62} +1.29794 q^{63} +1.00000 q^{64} +0.171981 q^{65} -7.44464 q^{66} +7.07438 q^{67} -0.657428 q^{68} -4.38142 q^{69} -6.36575 q^{70} -11.3636 q^{71} -0.432617 q^{72} +15.5477 q^{73} -6.46245 q^{74} -0.798078 q^{75} +6.08843 q^{76} +13.9396 q^{77} +0.129876 q^{78} +4.09958 q^{79} +2.12177 q^{80} -7.51499 q^{81} +4.21403 q^{82} +13.3636 q^{83} -4.80724 q^{84} -1.39491 q^{85} -1.41184 q^{86} -13.4430 q^{87} -4.64620 q^{88} +0.942798 q^{89} -0.917916 q^{90} -0.243183 q^{91} -2.73445 q^{92} +3.70639 q^{93} -6.07438 q^{94} +12.9183 q^{95} +1.60231 q^{96} -4.19989 q^{97} +2.00122 q^{98} +2.01003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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