LMFDB - Embedded newform 6038.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$1$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.21264 q^{3} +1.00000 q^{4} +1.98344 q^{5} +1.21264 q^{6} -0.999136 q^{7} -1.00000 q^{8} -1.52951 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.21264 q^{3} +1.00000 q^{4} +1.98344 q^{5} +1.21264 q^{6} -0.999136 q^{7} -1.00000 q^{8} -1.52951 q^{9} -1.98344 q^{10} -0.227591 q^{11} -1.21264 q^{12} -1.23911 q^{13} +0.999136 q^{14} -2.40519 q^{15} +1.00000 q^{16} +5.85559 q^{17} +1.52951 q^{18} -5.06265 q^{19} +1.98344 q^{20} +1.21159 q^{21} +0.227591 q^{22} +1.60709 q^{23} +1.21264 q^{24} -1.06597 q^{25} +1.23911 q^{26} +5.49265 q^{27} -0.999136 q^{28} -6.10457 q^{29} +2.40519 q^{30} +2.35672 q^{31} -1.00000 q^{32} +0.275985 q^{33} -5.85559 q^{34} -1.98172 q^{35} -1.52951 q^{36} +10.4299 q^{37} +5.06265 q^{38} +1.50260 q^{39} -1.98344 q^{40} -8.77517 q^{41} -1.21159 q^{42} +9.86721 q^{43} -0.227591 q^{44} -3.03370 q^{45} -1.60709 q^{46} -8.53042 q^{47} -1.21264 q^{48} -6.00173 q^{49} +1.06597 q^{50} -7.10070 q^{51} -1.23911 q^{52} +10.4646 q^{53} -5.49265 q^{54} -0.451413 q^{55} +0.999136 q^{56} +6.13915 q^{57} +6.10457 q^{58} +2.93833 q^{59} -2.40519 q^{60} +2.80348 q^{61} -2.35672 q^{62} +1.52819 q^{63} +1.00000 q^{64} -2.45771 q^{65} -0.275985 q^{66} +7.43810 q^{67} +5.85559 q^{68} -1.94881 q^{69} +1.98172 q^{70} +12.1770 q^{71} +1.52951 q^{72} -14.7467 q^{73} -10.4299 q^{74} +1.29264 q^{75} -5.06265 q^{76} +0.227395 q^{77} -1.50260 q^{78} -1.74986 q^{79} +1.98344 q^{80} -2.07204 q^{81} +8.77517 q^{82} +5.36275 q^{83} +1.21159 q^{84} +11.6142 q^{85} -9.86721 q^{86} +7.40262 q^{87} +0.227591 q^{88} -11.3690 q^{89} +3.03370 q^{90} +1.23804 q^{91} +1.60709 q^{92} -2.85785 q^{93} +8.53042 q^{94} -10.0415 q^{95} +1.21264 q^{96} -16.4609 q^{97} +6.00173 q^{98} +0.348104 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) $ 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * 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
$3$	−1.21264	−0.700116	−0.350058	−	0.936728i	$-0.613838\pi$
$3$	−1.21264	−0.700116	−0.350058	+	0.936728i	$0.613838\pi$
$4$	1.00000	0.500000
$5$	1.98344	0.887020	0.443510	−	0.896269i	$-0.353733\pi$
$5$	1.98344	0.887020	0.443510	+	0.896269i	$0.353733\pi$
$6$	1.21264	0.495056
$7$	−0.999136	−0.377638	−0.188819	−	0.982012i	$-0.560466\pi$
$7$	−0.999136	−0.377638	−0.188819	+	0.982012i	$0.560466\pi$
$8$	−1.00000	−0.353553
$9$	−1.52951	−0.509838
$10$	−1.98344	−0.627218
$11$	−0.227591	−0.0686214	−0.0343107	−	0.999411i	$-0.510924\pi$
$11$	−0.227591	−0.0686214	−0.0343107	+	0.999411i	$0.510924\pi$
$12$	−1.21264	−0.350058
$13$	−1.23911	−0.343669	−0.171834	−	0.985126i	$-0.554969\pi$
$13$	−1.23911	−0.343669	−0.171834	+	0.985126i	$0.554969\pi$
$14$	0.999136	0.267030
$15$	−2.40519	−0.621017
$16$	1.00000	0.250000
$17$	5.85559	1.42019	0.710095	−	0.704106i	$-0.248652\pi$
$17$	5.85559	1.42019	0.710095	+	0.704106i	$0.248652\pi$
$18$	1.52951	0.360510
$19$	−5.06265	−1.16145	−0.580726	−	0.814099i	$-0.697231\pi$
$19$	−5.06265	−1.16145	−0.580726	+	0.814099i	$0.697231\pi$
$20$	1.98344	0.443510
$21$	1.21159	0.264390
$22$	0.227591	0.0485226
$23$	1.60709	0.335101	0.167550	−	0.985864i	$-0.446414\pi$
$23$	1.60709	0.335101	0.167550	+	0.985864i	$0.446414\pi$
$24$	1.21264	0.247528
$25$	−1.06597	−0.213195
$26$	1.23911	0.243010
$27$	5.49265	1.05706
$28$	−0.999136	−0.188819
$29$	−6.10457	−1.13359	−0.566795	−	0.823859i	$-0.691817\pi$
$29$	−6.10457	−1.13359	−0.566795	+	0.823859i	$0.691817\pi$
$30$	2.40519	0.439125
$31$	2.35672	0.423280	0.211640	−	0.977348i	$-0.432120\pi$
$31$	2.35672	0.423280	0.211640	+	0.977348i	$0.432120\pi$
$32$	−1.00000	−0.176777
$33$	0.275985	0.0480429
$34$	−5.85559	−1.00423
$35$	−1.98172	−0.334972
$36$	−1.52951	−0.254919
$37$	10.4299	1.71466	0.857330	−	0.514766i	$-0.172121\pi$
$37$	10.4299	1.71466	0.857330	+	0.514766i	$0.172121\pi$
$38$	5.06265	0.821270
$39$	1.50260	0.240608
$40$	−1.98344	−0.313609
$41$	−8.77517	−1.37045	−0.685226	−	0.728331i	$-0.740296\pi$
$41$	−8.77517	−1.37045	−0.685226	+	0.728331i	$0.740296\pi$
$42$	−1.21159	−0.186952
$43$	9.86721	1.50473	0.752367	−	0.658744i	$-0.228912\pi$
$43$	9.86721	1.50473	0.752367	+	0.658744i	$0.228912\pi$
$44$	−0.227591	−0.0343107
$45$	−3.03370	−0.452237
$46$	−1.60709	−0.236952
$47$	−8.53042	−1.24429	−0.622145	−	0.782902i	$-0.713739\pi$
$47$	−8.53042	−1.24429	−0.622145	+	0.782902i	$0.713739\pi$
$48$	−1.21264	−0.175029
$49$	−6.00173	−0.857390
$50$	1.06597	0.150752
$51$	−7.10070	−0.994297
$52$	−1.23911	−0.171834
$53$	10.4646	1.43743	0.718713	−	0.695307i	$-0.244732\pi$
$53$	10.4646	1.43743	0.718713	+	0.695307i	$0.244732\pi$
$54$	−5.49265	−0.747455
$55$	−0.451413	−0.0608685
$56$	0.999136	0.133515
$57$	6.13915	0.813151
$58$	6.10457	0.801569
$59$	2.93833	0.382538	0.191269	−	0.981538i	$-0.438740\pi$
$59$	2.93833	0.382538	0.191269	+	0.981538i	$0.438740\pi$
$60$	−2.40519	−0.310508
$61$	2.80348	0.358949	0.179475	−	0.983763i	$-0.442560\pi$
$61$	2.80348	0.358949	0.179475	+	0.983763i	$0.442560\pi$
$62$	−2.35672	−0.299304
$63$	1.52819	0.192534
$64$	1.00000	0.125000
$65$	−2.45771	−0.304841
$66$	−0.275985	−0.0339714
$67$	7.43810	0.908709	0.454354	−	0.890821i	$-0.349870\pi$
$67$	7.43810	0.908709	0.454354	+	0.890821i	$0.349870\pi$
$68$	5.85559	0.710095
$69$	−1.94881	−0.234609
$70$	1.98172	0.236861
$71$	12.1770	1.44514	0.722569	−	0.691298i	$-0.242961\pi$
$71$	12.1770	1.44514	0.722569	+	0.691298i	$0.242961\pi$
$72$	1.52951	0.180255
$73$	−14.7467	−1.72597	−0.862985	−	0.505230i	$-0.831408\pi$
$73$	−14.7467	−1.72597	−0.862985	+	0.505230i	$0.831408\pi$
$74$	−10.4299	−1.21245
$75$	1.29264	0.149261
$76$	−5.06265	−0.580726
$77$	0.227395	0.0259140
$78$	−1.50260	−0.170135
$79$	−1.74986	−0.196874	−0.0984371	−	0.995143i	$-0.531384\pi$
$79$	−1.74986	−0.196874	−0.0984371	+	0.995143i	$0.531384\pi$
$80$	1.98344	0.221755
$81$	−2.07204	−0.230227
$82$	8.77517	0.969055
$83$	5.36275	0.588638	0.294319	−	0.955707i	$-0.404907\pi$
$83$	5.36275	0.588638	0.294319	+	0.955707i	$0.404907\pi$
$84$	1.21159	0.132195
$85$	11.6142	1.25974
$86$	−9.86721	−1.06401
$87$	7.40262	0.793644
$88$	0.227591	0.0242613
$89$	−11.3690	−1.20512	−0.602558	−	0.798075i	$-0.705852\pi$
$89$	−11.3690	−1.20512	−0.602558	+	0.798075i	$0.705852\pi$
$90$	3.03370	0.319780
$91$	1.23804	0.129782
$92$	1.60709	0.167550
$93$	−2.85785	−0.296345
$94$	8.53042	0.879846
$95$	−10.0415	−1.03023
$96$	1.21264	0.123764
$97$	−16.4609	−1.67135	−0.835674	−	0.549226i	$-0.814923\pi$
$97$	−16.4609	−1.67135	−0.835674	+	0.549226i	$0.814923\pi$
$98$	6.00173	0.606266
$99$	0.348104	0.0349858
$100$	−1.06597	−0.106597
$101$	9.71529	0.966708	0.483354	−	0.875425i	$-0.339418\pi$
$101$	9.71529	0.966708	0.483354	+	0.875425i	$0.339418\pi$
$102$	7.10070	0.703074
$103$	−2.95375	−0.291042	−0.145521	−	0.989355i	$-0.546486\pi$
$103$	−2.95375	−0.291042	−0.145521	+	0.989355i	$0.546486\pi$
$104$	1.23911	0.121505
$105$	2.40311	0.234519
$106$	−10.4646	−1.01641
$107$	−14.3959	−1.39171	−0.695853	−	0.718184i	$-0.744974\pi$
$107$	−14.3959	−1.39171	−0.695853	+	0.718184i	$0.744974\pi$
$108$	5.49265	0.528531
$109$	5.45841	0.522821	0.261410	−	0.965228i	$-0.415812\pi$
$109$	5.45841	0.522821	0.261410	+	0.965228i	$0.415812\pi$
$110$	0.451413	0.0430406
$111$	−12.6476	−1.20046
$112$	−0.999136	−0.0944094
$113$	−19.3971	−1.82473	−0.912364	−	0.409381i	$-0.865745\pi$
$113$	−19.3971	−1.82473	−0.912364	+	0.409381i	$0.865745\pi$
$114$	−6.13915	−0.574984
$115$	3.18755	0.297241
$116$	−6.10457	−0.566795
$117$	1.89524	0.175215
$118$	−2.93833	−0.270495
$119$	−5.85053	−0.536317
$120$	2.40519	0.219563
$121$	−10.9482	−0.995291
$122$	−2.80348	−0.253815
$123$	10.6411	0.959474
$124$	2.35672	0.211640
$125$	−12.0315	−1.07613
$126$	−1.52819	−0.136142
$127$	15.1158	1.34131	0.670657	−	0.741768i	$-0.266012\pi$
$127$	15.1158	1.34131	0.670657	+	0.741768i	$0.266012\pi$
$128$	−1.00000	−0.0883883
$129$	−11.9653	−1.05349
$130$	2.45771	0.215555
$131$	−9.31315	−0.813694	−0.406847	−	0.913496i	$-0.633372\pi$
$131$	−9.31315	−0.813694	−0.406847	+	0.913496i	$0.633372\pi$
$132$	0.275985	0.0240214
$133$	5.05828	0.438608
$134$	−7.43810	−0.642554
$135$	10.8943	0.937635
$136$	−5.85559	−0.502113
$137$	−13.4184	−1.14641	−0.573206	−	0.819411i	$-0.694301\pi$
$137$	−13.4184	−1.14641	−0.573206	+	0.819411i	$0.694301\pi$
$138$	1.94881	0.165894
$139$	−19.8290	−1.68188	−0.840938	−	0.541132i	$-0.817996\pi$
$139$	−19.8290	−1.68188	−0.840938	+	0.541132i	$0.817996\pi$
$140$	−1.98172	−0.167486
$141$	10.3443	0.871147
$142$	−12.1770	−1.02187
$143$	0.282012	0.0235830
$144$	−1.52951	−0.127460
$145$	−12.1080	−1.00552
$146$	14.7467	1.22044
$147$	7.27791	0.600272
$148$	10.4299	0.857330
$149$	16.5793	1.35823	0.679115	−	0.734032i	$-0.262364\pi$
$149$	16.5793	1.35823	0.679115	+	0.734032i	$0.262364\pi$
$150$	−1.29264	−0.105544
$151$	19.8991	1.61937	0.809684	−	0.586867i	$-0.199639\pi$
$151$	19.8991	1.61937	0.809684	+	0.586867i	$0.199639\pi$
$152$	5.06265	0.410635
$153$	−8.95621	−0.724067
$154$	−0.227395	−0.0183240
$155$	4.67441	0.375458
$156$	1.50260	0.120304
$157$	8.08808	0.645499	0.322750	−	0.946484i	$-0.395393\pi$
$157$	8.08808	0.645499	0.322750	+	0.946484i	$0.395393\pi$
$158$	1.74986	0.139211
$159$	−12.6898	−1.00636
$160$	−1.98344	−0.156805
$161$	−1.60570	−0.126547
$162$	2.07204	0.162795
$163$	16.9199	1.32527	0.662636	−	0.748942i	$-0.269438\pi$
$163$	16.9199	1.32527	0.662636	+	0.748942i	$0.269438\pi$
$164$	−8.77517	−0.685226
$165$	0.547400	0.0426150
$166$	−5.36275	−0.416230
$167$	−16.5516	−1.28080	−0.640402	−	0.768040i	$-0.721232\pi$
$167$	−16.5516	−1.28080	−0.640402	+	0.768040i	$0.721232\pi$
$168$	−1.21159	−0.0934760
$169$	−11.4646	−0.881892
$170$	−11.6142	−0.890768
$171$	7.74340	0.592152
$172$	9.86721	0.752367
$173$	6.64768	0.505414	0.252707	−	0.967543i	$-0.418679\pi$
$173$	6.64768	0.505414	0.252707	+	0.967543i	$0.418679\pi$
$174$	−7.40262	−0.561191
$175$	1.06505	0.0805105
$176$	−0.227591	−0.0171553
$177$	−3.56312	−0.267821
$178$	11.3690	0.852145
$179$	−0.387984	−0.0289993	−0.0144997	−	0.999895i	$-0.504616\pi$
$179$	−0.387984	−0.0289993	−0.0144997	+	0.999895i	$0.504616\pi$
$180$	−3.03370	−0.226118
$181$	−17.3705	−1.29114	−0.645569	−	0.763702i	$-0.723380\pi$
$181$	−17.3705	−1.29114	−0.645569	+	0.763702i	$0.723380\pi$
$182$	−1.23804	−0.0917699
$183$	−3.39960	−0.251306
$184$	−1.60709	−0.118476
$185$	20.6870	1.52094
$186$	2.85785	0.209547
$187$	−1.33268	−0.0974553
$188$	−8.53042	−0.622145
$189$	−5.48790	−0.399186
$190$	10.0415	0.728484
$191$	−13.3864	−0.968605	−0.484303	−	0.874900i	$-0.660927\pi$
$191$	−13.3864	−0.968605	−0.484303	+	0.874900i	$0.660927\pi$
$192$	−1.21264	−0.0875145
$193$	13.2131	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$193$	13.2131	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$194$	16.4609	1.18182
$195$	2.98030	0.213424
$196$	−6.00173	−0.428695
$197$	−0.365279	−0.0260251	−0.0130125	−	0.999915i	$-0.504142\pi$
$197$	−0.365279	−0.0260251	−0.0130125	+	0.999915i	$0.504142\pi$
$198$	−0.348104	−0.0247387
$199$	3.07430	0.217931	0.108966	−	0.994046i	$-0.465246\pi$
$199$	3.07430	0.217931	0.108966	+	0.994046i	$0.465246\pi$
$200$	1.06597	0.0753758
$201$	−9.01971	−0.636201
$202$	−9.71529	−0.683565
$203$	6.09929	0.428086
$204$	−7.10070	−0.497148
$205$	−17.4050	−1.21562
$206$	2.95375	0.205798
$207$	−2.45806	−0.170847
$208$	−1.23911	−0.0859172
$209$	1.15222	0.0797004
$210$	−2.40311	−0.165830
$211$	2.47039	0.170068	0.0850342	−	0.996378i	$-0.472900\pi$
$211$	2.47039	0.170068	0.0850342	+	0.996378i	$0.472900\pi$
$212$	10.4646	0.718713
$213$	−14.7662	−1.01176
$214$	14.3959	0.984085
$215$	19.5710	1.33473
$216$	−5.49265	−0.373728
$217$	−2.35469	−0.159846
$218$	−5.45841	−0.369690
$219$	17.8824	1.20838
$220$	−0.451413	−0.0304343
$221$	−7.25575	−0.488075
$222$	12.6476	0.848854
$223$	−0.971481	−0.0650552	−0.0325276	−	0.999471i	$-0.510356\pi$
$223$	−0.971481	−0.0650552	−0.0325276	+	0.999471i	$0.510356\pi$
$224$	0.999136	0.0667576
$225$	1.63042	0.108695
$226$	19.3971	1.29028
$227$	−13.7887	−0.915186	−0.457593	−	0.889162i	$-0.651288\pi$
$227$	−13.7887	−0.915186	−0.457593	+	0.889162i	$0.651288\pi$
$228$	6.13915	0.406575
$229$	−2.13143	−0.140849	−0.0704244	−	0.997517i	$-0.522435\pi$
$229$	−2.13143	−0.140849	−0.0704244	+	0.997517i	$0.522435\pi$
$230$	−3.18755	−0.210181
$231$	−0.275747	−0.0181428
$232$	6.10457	0.400785
$233$	5.89571	0.386241	0.193120	−	0.981175i	$-0.438139\pi$
$233$	5.89571	0.386241	0.193120	+	0.981175i	$0.438139\pi$
$234$	−1.89524	−0.123896
$235$	−16.9196	−1.10371
$236$	2.93833	0.191269
$237$	2.12194	0.137835
$238$	5.85053	0.379234
$239$	12.9244	0.836011	0.418005	−	0.908445i	$-0.362729\pi$
$239$	12.9244	0.836011	0.418005	+	0.908445i	$0.362729\pi$
$240$	−2.40519	−0.155254
$241$	7.99949	0.515292	0.257646	−	0.966239i	$-0.417053\pi$
$241$	7.99949	0.515292	0.257646	+	0.966239i	$0.417053\pi$
$242$	10.9482	0.703777
$243$	−13.9653	−0.895876
$244$	2.80348	0.179475
$245$	−11.9041	−0.760522
$246$	−10.6411	−0.678451
$247$	6.27321	0.399155
$248$	−2.35672	−0.149652
$249$	−6.50306	−0.412115
$250$	12.0315	0.760938
$251$	29.0042	1.83073	0.915364	−	0.402627i	$-0.131903\pi$
$251$	29.0042	1.83073	0.915364	+	0.402627i	$0.131903\pi$
$252$	1.52819	0.0962671
$253$	−0.365759	−0.0229951
$254$	−15.1158	−0.948452
$255$	−14.0838	−0.881961
$256$	1.00000	0.0625000
$257$	−11.9998	−0.748526	−0.374263	−	0.927323i	$-0.622104\pi$
$257$	−11.9998	−0.748526	−0.374263	+	0.927323i	$0.622104\pi$
$258$	11.9653	0.744929
$259$	−10.4209	−0.647521
$260$	−2.45771	−0.152421
$261$	9.33703	0.577947
$262$	9.31315	0.575368
$263$	9.46548	0.583667	0.291833	−	0.956469i	$-0.405735\pi$
$263$	9.46548	0.583667	0.291833	+	0.956469i	$0.405735\pi$
$264$	−0.275985	−0.0169857
$265$	20.7559	1.27503
$266$	−5.05828	−0.310143
$267$	13.7865	0.843720
$268$	7.43810	0.454354
$269$	−28.1928	−1.71894	−0.859472	−	0.511182i	$-0.829208\pi$
$269$	−28.1928	−1.71894	−0.859472	+	0.511182i	$0.829208\pi$
$270$	−10.8943	−0.663008
$271$	−1.42206	−0.0863838	−0.0431919	−	0.999067i	$-0.513753\pi$
$271$	−1.42206	−0.0863838	−0.0431919	+	0.999067i	$0.513753\pi$
$272$	5.85559	0.355047
$273$	−1.50130	−0.0908626
$274$	13.4184	0.810636
$275$	0.242607	0.0146297
$276$	−1.94881	−0.117305
$277$	−25.4533	−1.52934	−0.764669	−	0.644423i	$-0.777097\pi$
$277$	−25.4533	−1.52934	−0.764669	+	0.644423i	$0.777097\pi$
$278$	19.8290	1.18927
$279$	−3.60464	−0.215804
$280$	1.98172	0.118431
$281$	4.59537	0.274137	0.137068	−	0.990562i	$-0.456232\pi$
$281$	4.59537	0.274137	0.137068	+	0.990562i	$0.456232\pi$
$282$	−10.3443	−0.615994
$283$	−16.7651	−0.996584	−0.498292	−	0.867009i	$-0.666039\pi$
$283$	−16.7651	−0.996584	−0.498292	+	0.867009i	$0.666039\pi$
$284$	12.1770	0.722569
$285$	12.1766	0.721281
$286$	−0.282012	−0.0166757
$287$	8.76759	0.517534
$288$	1.52951	0.0901275
$289$	17.2879	1.01694
$290$	12.1080	0.711008
$291$	19.9610	1.17014
$292$	−14.7467	−0.862985
$293$	9.11436	0.532467	0.266233	−	0.963909i	$-0.414221\pi$
$293$	9.11436	0.532467	0.266233	+	0.963909i	$0.414221\pi$
$294$	−7.27791	−0.424456
$295$	5.82800	0.339319
$296$	−10.4299	−0.606224
$297$	−1.25008	−0.0725370
$298$	−16.5793	−0.960414
$299$	−1.99136	−0.115164
$300$	1.29264	0.0746306
$301$	−9.85868	−0.568245
$302$	−19.8991	−1.14507
$303$	−11.7811	−0.676807
$304$	−5.06265	−0.290363
$305$	5.56053	0.318395
$306$	8.95621	0.511992
$307$	−29.0280	−1.65672	−0.828358	−	0.560199i	$-0.810724\pi$
$307$	−29.0280	−1.65672	−0.828358	+	0.560199i	$0.810724\pi$
$308$	0.227395	0.0129570
$309$	3.58182	0.203763
$310$	−4.67441	−0.265489
$311$	1.42037	0.0805418	0.0402709	−	0.999189i	$-0.487178\pi$
$311$	1.42037	0.0805418	0.0402709	+	0.999189i	$0.487178\pi$
$312$	−1.50260	−0.0850677
$313$	−30.0373	−1.69781	−0.848903	−	0.528549i	$-0.822736\pi$
$313$	−30.0373	−1.69781	−0.848903	+	0.528549i	$0.822736\pi$
$314$	−8.08808	−0.456437
$315$	3.03107	0.170782
$316$	−1.74986	−0.0984371
$317$	−23.2574	−1.30627	−0.653133	−	0.757243i	$-0.726546\pi$
$317$	−23.2574	−1.30627	−0.653133	+	0.757243i	$0.726546\pi$
$318$	12.6898	0.711607
$319$	1.38935	0.0777885
$320$	1.98344	0.110878
$321$	17.4570	0.974355
$322$	1.60570	0.0894820
$323$	−29.6448	−1.64948
$324$	−2.07204	−0.115113
$325$	1.32087	0.0732684
$326$	−16.9199	−0.937108
$327$	−6.61906	−0.366035
$328$	8.77517	0.484528
$329$	8.52305	0.469891
$330$	−0.547400	−0.0301334
$331$	33.8421	1.86013	0.930064	−	0.367397i	$-0.119751\pi$
$331$	33.8421	1.86013	0.930064	+	0.367397i	$0.119751\pi$
$332$	5.36275	0.294319
$333$	−15.9526	−0.874200
$334$	16.5516	0.905665
$335$	14.7530	0.806043
$336$	1.21159	0.0660975
$337$	24.3438	1.32609	0.663047	−	0.748578i	$-0.269263\pi$
$337$	24.3438	1.32609	0.663047	+	0.748578i	$0.269263\pi$
$338$	11.4646	0.623592
$339$	23.5216	1.27752
$340$	11.6142	0.629868
$341$	−0.536370	−0.0290460
$342$	−7.74340	−0.418715
$343$	12.9905	0.701421
$344$	−9.86721	−0.532004
$345$	−3.86534	−0.208103
$346$	−6.64768	−0.357381
$347$	−9.56681	−0.513573	−0.256787	−	0.966468i	$-0.582664\pi$
$347$	−9.56681	−0.513573	−0.256787	+	0.966468i	$0.582664\pi$
$348$	7.40262	0.396822
$349$	−9.19296	−0.492088	−0.246044	−	0.969259i	$-0.579131\pi$
$349$	−9.19296	−0.492088	−0.246044	+	0.969259i	$0.579131\pi$
$350$	−1.06505	−0.0569295
$351$	−6.80603	−0.363279
$352$	0.227591	0.0121307
$353$	15.6770	0.834401	0.417200	−	0.908815i	$-0.363011\pi$
$353$	15.6770	0.834401	0.417200	+	0.908815i	$0.363011\pi$
$354$	3.56312	0.189378
$355$	24.1522	1.28187
$356$	−11.3690	−0.602558
$357$	7.09456	0.375484
$358$	0.387984	0.0205056
$359$	4.80690	0.253698	0.126849	−	0.991922i	$-0.459514\pi$
$359$	4.80690	0.253698	0.126849	+	0.991922i	$0.459514\pi$
$360$	3.03370	0.159890
$361$	6.63043	0.348970
$362$	17.3705	0.912973
$363$	13.2762	0.696819
$364$	1.23804	0.0648911
$365$	−29.2491	−1.53097
$366$	3.39960	0.177700
$367$	6.71124	0.350324	0.175162	−	0.984540i	$-0.443955\pi$
$367$	6.71124	0.350324	0.175162	+	0.984540i	$0.443955\pi$
$368$	1.60709	0.0837751
$369$	13.4217	0.698708
$370$	−20.6870	−1.07547
$371$	−10.4556	−0.542826
$372$	−2.85785	−0.148172
$373$	−5.01248	−0.259537	−0.129768	−	0.991544i	$-0.541423\pi$
$373$	−5.01248	−0.259537	−0.129768	+	0.991544i	$0.541423\pi$
$374$	1.33268	0.0689113
$375$	14.5898	0.753414
$376$	8.53042	0.439923
$377$	7.56426	0.389579
$378$	5.48790	0.282267
$379$	−35.0218	−1.79895	−0.899474	−	0.436975i	$-0.856050\pi$
$379$	−35.0218	−1.79895	−0.899474	+	0.436975i	$0.856050\pi$
$380$	−10.0415	−0.515116
$381$	−18.3300	−0.939074
$382$	13.3864	0.684907
$383$	−1.66790	−0.0852259	−0.0426130	−	0.999092i	$-0.513568\pi$
$383$	−1.66790	−0.0852259	−0.0426130	+	0.999092i	$0.513568\pi$
$384$	1.21264	0.0618821
$385$	0.451023	0.0229863
$386$	−13.2131	−0.672530
$387$	−15.0920	−0.767171
$388$	−16.4609	−0.835674
$389$	14.6360	0.742074	0.371037	−	0.928618i	$-0.379002\pi$
$389$	14.6360	0.742074	0.371037	+	0.928618i	$0.379002\pi$
$390$	−2.98030	−0.150914
$391$	9.41044	0.475906
$392$	6.00173	0.303133
$393$	11.2935	0.569680
$394$	0.365279	0.0184025
$395$	−3.47073	−0.174631
$396$	0.348104	0.0174929
$397$	−24.3791	−1.22355	−0.611775	−	0.791032i	$-0.709544\pi$
$397$	−24.3791	−1.22355	−0.611775	+	0.791032i	$0.709544\pi$
$398$	−3.07430	−0.154101
$399$	−6.13385	−0.307076
$400$	−1.06597	−0.0532987
$401$	−14.0121	−0.699730	−0.349865	−	0.936800i	$-0.613773\pi$
$401$	−14.0121	−0.699730	−0.349865	+	0.936800i	$0.613773\pi$
$402$	9.01971	0.449862
$403$	−2.92025	−0.145468
$404$	9.71529	0.483354
$405$	−4.10977	−0.204216
$406$	−6.09929	−0.302703
$407$	−2.37375	−0.117662
$408$	7.10070	0.351537
$409$	−15.8003	−0.781275	−0.390637	−	0.920545i	$-0.627745\pi$
$409$	−15.8003	−0.781275	−0.390637	+	0.920545i	$0.627745\pi$
$410$	17.4050	0.859572
$411$	16.2716	0.802621
$412$	−2.95375	−0.145521
$413$	−2.93579	−0.144461
$414$	2.45806	0.120807
$415$	10.6367	0.522134
$416$	1.23911	0.0607526
$417$	24.0454	1.17751
$418$	−1.15222	−0.0563567
$419$	−13.8525	−0.676739	−0.338369	−	0.941013i	$-0.609875\pi$
$419$	−13.8525	−0.676739	−0.338369	+	0.941013i	$0.609875\pi$
$420$	2.40311	0.117260
$421$	−9.28131	−0.452343	−0.226172	−	0.974087i	$-0.572621\pi$
$421$	−9.28131	−0.452343	−0.226172	+	0.974087i	$0.572621\pi$
$422$	−2.47039	−0.120257
$423$	13.0474	0.634386
$424$	−10.4646	−0.508207
$425$	−6.24191	−0.302777
$426$	14.7662	0.715425
$427$	−2.80106	−0.135553
$428$	−14.3959	−0.695853
$429$	−0.341978	−0.0165108
$430$	−19.5710	−0.943797
$431$	−2.76703	−0.133283	−0.0666416	−	0.997777i	$-0.521228\pi$
$431$	−2.76703	−0.133283	−0.0666416	+	0.997777i	$0.521228\pi$
$432$	5.49265	0.264265
$433$	6.11458	0.293848	0.146924	−	0.989148i	$-0.453063\pi$
$433$	6.11458	0.293848	0.146924	+	0.989148i	$0.453063\pi$
$434$	2.35469	0.113029
$435$	14.6826	0.703979
$436$	5.45841	0.261410
$437$	−8.13611	−0.389203
$438$	−17.8824	−0.854452
$439$	−31.6427	−1.51022	−0.755112	−	0.655596i	$-0.772417\pi$
$439$	−31.6427	−1.51022	−0.755112	+	0.655596i	$0.772417\pi$
$440$	0.451413	0.0215203
$441$	9.17973	0.437130
$442$	7.25575	0.345121
$443$	6.56430	0.311879	0.155940	−	0.987767i	$-0.450159\pi$
$443$	6.56430	0.311879	0.155940	+	0.987767i	$0.450159\pi$
$444$	−12.6476	−0.600230
$445$	−22.5498	−1.06896
$446$	0.971481	0.0460009
$447$	−20.1047	−0.950918
$448$	−0.999136	−0.0472047
$449$	−11.8871	−0.560988	−0.280494	−	0.959856i	$-0.590498\pi$
$449$	−11.8871	−0.560988	−0.280494	+	0.959856i	$0.590498\pi$
$450$	−1.63042	−0.0768589
$451$	1.99715	0.0940422
$452$	−19.3971	−0.912364
$453$	−24.1304	−1.13374
$454$	13.7887	0.647135
$455$	2.45558	0.115120
$456$	−6.13915	−0.287492
$457$	−6.63876	−0.310548	−0.155274	−	0.987871i	$-0.549626\pi$
$457$	−6.63876	−0.310548	−0.155274	+	0.987871i	$0.549626\pi$
$458$	2.13143	0.0995952
$459$	32.1627	1.50123
$460$	3.18755	0.148620
$461$	12.2812	0.571991	0.285996	−	0.958231i	$-0.407676\pi$
$461$	12.2812	0.571991	0.285996	+	0.958231i	$0.407676\pi$
$462$	0.275747	0.0128289
$463$	−11.5692	−0.537666	−0.268833	−	0.963187i	$-0.586638\pi$
$463$	−11.5692	−0.537666	−0.268833	+	0.963187i	$0.586638\pi$
$464$	−6.10457	−0.283398
$465$	−5.66836	−0.262864
$466$	−5.89571	−0.273113
$467$	−4.00673	−0.185409	−0.0927046	−	0.995694i	$-0.529551\pi$
$467$	−4.00673	−0.185409	−0.0927046	+	0.995694i	$0.529551\pi$
$468$	1.89524	0.0876077
$469$	−7.43167	−0.343163
$470$	16.9196	0.780441
$471$	−9.80790	−0.451924
$472$	−2.93833	−0.135248
$473$	−2.24569	−0.103257
$474$	−2.12194	−0.0974638
$475$	5.39666	0.247616
$476$	−5.85053	−0.268159
$477$	−16.0058	−0.732854
$478$	−12.9244	−0.591149
$479$	−26.0637	−1.19088	−0.595441	−	0.803399i	$-0.703022\pi$
$479$	−26.0637	−1.19088	−0.595441	+	0.803399i	$0.703022\pi$
$480$	2.40519	0.109781
$481$	−12.9238	−0.589275
$482$	−7.99949	−0.364367
$483$	1.94713	0.0885973
$484$	−10.9482	−0.497646
$485$	−32.6491	−1.48252
$486$	13.9653	0.633480
$487$	−35.7988	−1.62220	−0.811100	−	0.584908i	$-0.801131\pi$
$487$	−35.7988	−1.62220	−0.811100	+	0.584908i	$0.801131\pi$
$488$	−2.80348	−0.126908
$489$	−20.5177	−0.927843
$490$	11.9041	0.537770
$491$	−18.2267	−0.822560	−0.411280	−	0.911509i	$-0.634918\pi$
$491$	−18.2267	−0.822560	−0.411280	+	0.911509i	$0.634918\pi$
$492$	10.6411	0.479737
$493$	−35.7459	−1.60991
$494$	−6.27321	−0.282245
$495$	0.690443	0.0310331
$496$	2.35672	0.105820
$497$	−12.1664	−0.545739
$498$	6.50306	0.291409
$499$	−11.1227	−0.497922	−0.248961	−	0.968514i	$-0.580089\pi$
$499$	−11.1227	−0.497922	−0.248961	+	0.968514i	$0.580089\pi$
$500$	−12.0315	−0.538064
$501$	20.0711	0.896711
$502$	−29.0042	−1.29452
$503$	12.4173	0.553658	0.276829	−	0.960919i	$-0.410716\pi$
$503$	12.4173	0.553658	0.276829	+	0.960919i	$0.410716\pi$
$504$	−1.52819	−0.0680711
$505$	19.2697	0.857489
$506$	0.365759	0.0162600
$507$	13.9024	0.617426
$508$	15.1158	0.670657
$509$	16.8637	0.747472	0.373736	−	0.927535i	$-0.378077\pi$
$509$	16.8637	0.747472	0.373736	+	0.927535i	$0.378077\pi$
$510$	14.0838	0.623641
$511$	14.7339	0.651791
$512$	−1.00000	−0.0441942
$513$	−27.8074	−1.22773
$514$	11.9998	0.529288
$515$	−5.85858	−0.258160
$516$	−11.9653	−0.526744
$517$	1.94145	0.0853849
$518$	10.4209	0.457866
$519$	−8.06121	−0.353848
$520$	2.45771	0.107778
$521$	−9.07413	−0.397545	−0.198773	−	0.980046i	$-0.563695\pi$
$521$	−9.07413	−0.397545	−0.198773	+	0.980046i	$0.563695\pi$
$522$	−9.33703	−0.408671
$523$	−14.0710	−0.615281	−0.307640	−	0.951503i	$-0.599539\pi$
$523$	−14.0710	−0.615281	−0.307640	+	0.951503i	$0.599539\pi$
$524$	−9.31315	−0.406847
$525$	−1.29152	−0.0563666
$526$	−9.46548	−0.412715
$527$	13.8000	0.601138
$528$	0.275985	0.0120107
$529$	−20.4173	−0.887708
$530$	−20.7559	−0.901579
$531$	−4.49422	−0.195032
$532$	5.05828	0.219304
$533$	10.8734	0.470981
$534$	−13.7865	−0.596600
$535$	−28.5534	−1.23447
$536$	−7.43810	−0.321277
$537$	0.470484	0.0203029
$538$	28.1928	1.21548
$539$	1.36594	0.0588352
$540$	10.8943	0.468817
$541$	3.82237	0.164337	0.0821683	−	0.996618i	$-0.473815\pi$
$541$	3.82237	0.164337	0.0821683	+	0.996618i	$0.473815\pi$
$542$	1.42206	0.0610825
$543$	21.0641	0.903946
$544$	−5.85559	−0.251056
$545$	10.8264	0.463752
$546$	1.50130	0.0642496
$547$	5.07379	0.216939	0.108470	−	0.994100i	$-0.465405\pi$
$547$	5.07379	0.216939	0.108470	+	0.994100i	$0.465405\pi$
$548$	−13.4184	−0.573206
$549$	−4.28797	−0.183006
$550$	−0.242607	−0.0103448
$551$	30.9053	1.31661
$552$	1.94881	0.0829469
$553$	1.74834	0.0743471
$554$	25.4533	1.08141
$555$	−25.0858	−1.06483
$556$	−19.8290	−0.840938
$557$	−17.8660	−0.757005	−0.378502	−	0.925600i	$-0.623561\pi$
$557$	−17.8660	−0.757005	−0.378502	+	0.925600i	$0.623561\pi$
$558$	3.60464	0.152597
$559$	−12.2266	−0.517130
$560$	−1.98172	−0.0837431
$561$	1.61606	0.0682300
$562$	−4.59537	−0.193844
$563$	−33.6420	−1.41784	−0.708921	−	0.705288i	$-0.750818\pi$
$563$	−33.6420	−1.41784	−0.708921	+	0.705288i	$0.750818\pi$
$564$	10.3443	0.435573
$565$	−38.4730	−1.61857
$566$	16.7651	0.704691
$567$	2.07025	0.0869424
$568$	−12.1770	−0.510934
$569$	16.9922	0.712351	0.356175	−	0.934419i	$-0.384081\pi$
$569$	16.9922	0.712351	0.356175	+	0.934419i	$0.384081\pi$
$570$	−12.1766	−0.510023
$571$	6.02664	0.252207	0.126104	−	0.992017i	$-0.459753\pi$
$571$	6.02664	0.252207	0.126104	+	0.992017i	$0.459753\pi$
$572$	0.282012	0.0117915
$573$	16.2328	0.678136
$574$	−8.76759	−0.365952
$575$	−1.71311	−0.0714417
$576$	−1.52951	−0.0637298
$577$	−8.61100	−0.358481	−0.179240	−	0.983805i	$-0.557364\pi$
$577$	−8.61100	−0.358481	−0.179240	+	0.983805i	$0.557364\pi$
$578$	−17.2879	−0.719084
$579$	−16.0227	−0.665881
$580$	−12.1080	−0.502759
$581$	−5.35812	−0.222292
$582$	−19.9610	−0.827411
$583$	−2.38166	−0.0986381
$584$	14.7467	0.610222
$585$	3.75910	0.155420
$586$	−9.11436	−0.376511
$587$	−22.7018	−0.937005	−0.468503	−	0.883462i	$-0.655206\pi$
$587$	−22.7018	−0.937005	−0.468503	+	0.883462i	$0.655206\pi$
$588$	7.27791	0.300136
$589$	−11.9313	−0.491619
$590$	−5.82800	−0.239935
$591$	0.442951	0.0182206
$592$	10.4299	0.428665
$593$	38.4017	1.57697	0.788485	−	0.615054i	$-0.210866\pi$
$593$	38.4017	1.57697	0.788485	+	0.615054i	$0.210866\pi$
$594$	1.25008	0.0512914
$595$	−11.6042	−0.475724
$596$	16.5793	0.679115
$597$	−3.72800	−0.152577
$598$	1.99136	0.0814329
$599$	37.3211	1.52490	0.762450	−	0.647047i	$-0.223996\pi$
$599$	37.3211	1.52490	0.762450	+	0.647047i	$0.223996\pi$
$600$	−1.29264	−0.0527718
$601$	−15.8725	−0.647454	−0.323727	−	0.946150i	$-0.604936\pi$
$601$	−15.8725	−0.647454	−0.323727	+	0.946150i	$0.604936\pi$
$602$	9.85868	0.401810
$603$	−11.3767	−0.463294
$604$	19.8991	0.809684
$605$	−21.7151	−0.882843
$606$	11.7811	0.478575
$607$	−10.8186	−0.439115	−0.219558	−	0.975600i	$-0.570461\pi$
$607$	−10.8186	−0.439115	−0.219558	+	0.975600i	$0.570461\pi$
$608$	5.06265	0.205318
$609$	−7.39622	−0.299710
$610$	−5.56053	−0.225139
$611$	10.5702	0.427623
$612$	−8.95621	−0.362033
$613$	15.1482	0.611831	0.305916	−	0.952059i	$-0.401037\pi$
$613$	15.1482	0.611831	0.305916	+	0.952059i	$0.401037\pi$
$614$	29.0280	1.17147
$615$	21.1059	0.851073
$616$	−0.227395	−0.00916199
$617$	−46.6538	−1.87821	−0.939105	−	0.343631i	$-0.888343\pi$
$617$	−46.6538	−1.87821	−0.939105	+	0.343631i	$0.888343\pi$
$618$	−3.58182	−0.144082
$619$	−16.1121	−0.647600	−0.323800	−	0.946126i	$-0.604960\pi$
$619$	−16.1121	−0.647600	−0.323800	+	0.946126i	$0.604960\pi$
$620$	4.67441	0.187729
$621$	8.82716	0.354222
$622$	−1.42037	−0.0569516
$623$	11.3592	0.455097
$624$	1.50260	0.0601519
$625$	−18.5338	−0.741353
$626$	30.0373	1.20053
$627$	−1.39722	−0.0557995
$628$	8.08808	0.322750
$629$	61.0731	2.43514
$630$	−3.03107	−0.120761
$631$	−8.66084	−0.344782	−0.172391	−	0.985029i	$-0.555149\pi$
$631$	−8.66084	−0.344782	−0.172391	+	0.985029i	$0.555149\pi$
$632$	1.74986	0.0696055
$633$	−2.99568	−0.119068
$634$	23.2574	0.923670
$635$	29.9813	1.18977
$636$	−12.6898	−0.503182
$637$	7.43683	0.294658
$638$	−1.38935	−0.0550048
$639$	−18.6248	−0.736787
$640$	−1.98344	−0.0784023
$641$	−36.9499	−1.45943	−0.729716	−	0.683751i	$-0.760348\pi$
$641$	−36.9499	−1.45943	−0.729716	+	0.683751i	$0.760348\pi$
$642$	−17.4570	−0.688973
$643$	12.5475	0.494825	0.247413	−	0.968910i	$-0.420420\pi$
$643$	12.5475	0.494825	0.247413	+	0.968910i	$0.420420\pi$
$644$	−1.60570	−0.0632733
$645$	−23.7325	−0.934466
$646$	29.6448	1.16636
$647$	34.6417	1.36191	0.680953	−	0.732327i	$-0.261566\pi$
$647$	34.6417	1.36191	0.680953	+	0.732327i	$0.261566\pi$
$648$	2.07204	0.0813975
$649$	−0.668738	−0.0262503
$650$	−1.32087	−0.0518086
$651$	2.85538	0.111911
$652$	16.9199	0.662636
$653$	−18.0747	−0.707318	−0.353659	−	0.935374i	$-0.615063\pi$
$653$	−18.0747	−0.707318	−0.353659	+	0.935374i	$0.615063\pi$
$654$	6.61906	0.258826
$655$	−18.4721	−0.721763
$656$	−8.77517	−0.342613
$657$	22.5553	0.879965
$658$	−8.52305	−0.332263
$659$	18.0792	0.704266	0.352133	−	0.935950i	$-0.385457\pi$
$659$	18.0792	0.704266	0.352133	+	0.935950i	$0.385457\pi$
$660$	0.547400	0.0213075
$661$	−0.817116	−0.0317821	−0.0158911	−	0.999874i	$-0.505058\pi$
$661$	−0.817116	−0.0317821	−0.0158911	+	0.999874i	$0.505058\pi$
$662$	−33.8421	−1.31531
$663$	8.79858	0.341709
$664$	−5.36275	−0.208115
$665$	10.0328	0.389054
$666$	15.9526	0.618152
$667$	−9.81057	−0.379867
$668$	−16.5516	−0.640402
$669$	1.17805	0.0455461
$670$	−14.7530	−0.569958
$671$	−0.638048	−0.0246316
$672$	−1.21159	−0.0467380
$673$	−12.9572	−0.499462	−0.249731	−	0.968315i	$-0.580342\pi$
$673$	−12.9572	−0.499462	−0.249731	+	0.968315i	$0.580342\pi$
$674$	−24.3438	−0.937690
$675$	−5.85503	−0.225360
$676$	−11.4646	−0.440946
$677$	16.3775	0.629437	0.314718	−	0.949185i	$-0.398090\pi$
$677$	16.3775	0.629437	0.314718	+	0.949185i	$0.398090\pi$
$678$	−23.5216	−0.903343
$679$	16.4466	0.631164
$680$	−11.6142	−0.445384
$681$	16.7206	0.640736
$682$	0.536370	0.0205387
$683$	−2.79903	−0.107102	−0.0535509	−	0.998565i	$-0.517054\pi$
$683$	−2.79903	−0.107102	−0.0535509	+	0.998565i	$0.517054\pi$
$684$	7.74340	0.296076
$685$	−26.6146	−1.01689
$686$	−12.9905	−0.495979
$687$	2.58465	0.0986105
$688$	9.86721	0.376184
$689$	−12.9669	−0.493998
$690$	3.86534	0.147151
$691$	−9.62219	−0.366045	−0.183023	−	0.983109i	$-0.558588\pi$
$691$	−9.62219	−0.366045	−0.183023	+	0.983109i	$0.558588\pi$
$692$	6.64768	0.252707
$693$	−0.347803	−0.0132120
$694$	9.56681	0.363151
$695$	−39.3296	−1.49186
$696$	−7.40262	−0.280596
$697$	−51.3838	−1.94630
$698$	9.19296	0.347959
$699$	−7.14935	−0.270413
$700$	1.06505	0.0402552
$701$	35.7969	1.35203	0.676016	−	0.736887i	$-0.263705\pi$
$701$	35.7969	1.35203	0.676016	+	0.736887i	$0.263705\pi$
$702$	6.80603	0.256877
$703$	−52.8028	−1.99150
$704$	−0.227591	−0.00857767
$705$	20.5173	0.772725
$706$	−15.6770	−0.590010
$707$	−9.70689	−0.365065
$708$	−3.56312	−0.133910
$709$	−12.5437	−0.471088	−0.235544	−	0.971864i	$-0.575687\pi$
$709$	−12.5437	−0.471088	−0.235544	+	0.971864i	$0.575687\pi$
$710$	−24.1522	−0.906417
$711$	2.67643	0.100374
$712$	11.3690	0.426073
$713$	3.78746	0.141841
$714$	−7.09456	−0.265507
$715$	0.559353	0.0209186
$716$	−0.387984	−0.0144997
$717$	−15.6726	−0.585304
$718$	−4.80690	−0.179392
$719$	−2.33190	−0.0869652	−0.0434826	−	0.999054i	$-0.513845\pi$
$719$	−2.33190	−0.0869652	−0.0434826	+	0.999054i	$0.513845\pi$
$720$	−3.03370	−0.113059
$721$	2.95120	0.109908
$722$	−6.63043	−0.246759
$723$	−9.70046	−0.360764
$724$	−17.3705	−0.645569
$725$	6.50732	0.241676
$726$	−13.2762	−0.492725
$727$	−36.4878	−1.35326	−0.676629	−	0.736324i	$-0.736560\pi$
$727$	−36.4878	−1.35326	−0.676629	+	0.736324i	$0.736560\pi$
$728$	−1.23804	−0.0458850
$729$	23.1510	0.857444
$730$	29.2491	1.08256
$731$	57.7783	2.13701
$732$	−3.39960	−0.125653
$733$	−23.4274	−0.865309	−0.432655	−	0.901560i	$-0.642423\pi$
$733$	−23.4274	−0.865309	−0.432655	+	0.901560i	$0.642423\pi$
$734$	−6.71124	−0.247716
$735$	14.4353	0.532453
$736$	−1.60709	−0.0592380
$737$	−1.69285	−0.0623568
$738$	−13.4217	−0.494061
$739$	7.00808	0.257797	0.128898	−	0.991658i	$-0.458856\pi$
$739$	7.00808	0.257797	0.128898	+	0.991658i	$0.458856\pi$
$740$	20.6870	0.760470
$741$	−7.60711	−0.279454
$742$	10.4556	0.383836
$743$	−2.39275	−0.0877814	−0.0438907	−	0.999036i	$-0.513975\pi$
$743$	−2.39275	−0.0877814	−0.0438907	+	0.999036i	$0.513975\pi$
$744$	2.85785	0.104774
$745$	32.8840	1.20478
$746$	5.01248	0.183520
$747$	−8.20241	−0.300110
$748$	−1.33268	−0.0487277
$749$	14.3835	0.525561
$750$	−14.5898	−0.532744
$751$	−27.6111	−1.00755	−0.503773	−	0.863836i	$-0.668055\pi$
$751$	−27.6111	−1.00755	−0.503773	+	0.863836i	$0.668055\pi$
$752$	−8.53042	−0.311072
$753$	−35.1715	−1.28172
$754$	−7.56426	−0.275474
$755$	39.4687	1.43641
$756$	−5.48790	−0.199593
$757$	−39.0037	−1.41761	−0.708807	−	0.705402i	$-0.750766\pi$
$757$	−39.0037	−1.41761	−0.708807	+	0.705402i	$0.750766\pi$
$758$	35.0218	1.27205
$759$	0.443532	0.0160992
$760$	10.0415	0.364242
$761$	−46.0236	−1.66835	−0.834177	−	0.551497i	$-0.814057\pi$
$761$	−46.0236	−1.66835	−0.834177	+	0.551497i	$0.814057\pi$
$762$	18.3300	0.664026
$763$	−5.45369	−0.197437
$764$	−13.3864	−0.484303
$765$	−17.7641	−0.642262
$766$	1.66790	0.0602638
$767$	−3.64093	−0.131466
$768$	−1.21264	−0.0437572
$769$	24.4933	0.883250	0.441625	−	0.897200i	$-0.354402\pi$
$769$	24.4933	0.883250	0.441625	+	0.897200i	$0.354402\pi$
$770$	−0.451023	−0.0162537
$771$	14.5514	0.524055
$772$	13.2131	0.475551
$773$	−5.72031	−0.205745	−0.102873	−	0.994695i	$-0.532803\pi$
$773$	−5.72031	−0.205745	−0.102873	+	0.994695i	$0.532803\pi$
$774$	15.0920	0.542472
$775$	−2.51221	−0.0902411
$776$	16.4609	0.590911
$777$	12.6367	0.453339
$778$	−14.6360	−0.524726
$779$	44.4256	1.59171
$780$	2.98030	0.106712
$781$	−2.77137	−0.0991674
$782$	−9.41044	−0.336517
$783$	−33.5303	−1.19827
$784$	−6.00173	−0.214347
$785$	16.0422	0.572571
$786$	−11.2935	−0.402824
$787$	−20.9157	−0.745563	−0.372782	−	0.927919i	$-0.621596\pi$
$787$	−20.9157	−0.745563	−0.372782	+	0.927919i	$0.621596\pi$
$788$	−0.365279	−0.0130125
$789$	−11.4782	−0.408634
$790$	3.47073	0.123483
$791$	19.3804	0.689086
$792$	−0.348104	−0.0123693
$793$	−3.47384	−0.123360
$794$	24.3791	0.865180
$795$	−25.1694	−0.892665
$796$	3.07430	0.108966
$797$	1.52768	0.0541131	0.0270566	−	0.999634i	$-0.491387\pi$
$797$	1.52768	0.0541131	0.0270566	+	0.999634i	$0.491387\pi$
$798$	6.13385	0.217136
$799$	−49.9507	−1.76713
$800$	1.06597	0.0376879
$801$	17.3891	0.614414
$802$	14.0121	0.494784
$803$	3.35622	0.118438
$804$	−9.01971	−0.318101
$805$	−3.18480	−0.112249
$806$	2.92025	0.102861
$807$	34.1876	1.20346
$808$	−9.71529	−0.341783
$809$	44.6949	1.57139	0.785694	−	0.618615i	$-0.212306\pi$
$809$	44.6949	1.57139	0.785694	+	0.618615i	$0.212306\pi$
$810$	4.10977	0.144403
$811$	12.8395	0.450857	0.225429	−	0.974260i	$-0.427622\pi$
$811$	12.8395	0.450857	0.225429	+	0.974260i	$0.427622\pi$
$812$	6.09929	0.214043
$813$	1.72444	0.0604786
$814$	2.37375	0.0831999
$815$	33.5596	1.17554
$816$	−7.10070	−0.248574
$817$	−49.9542	−1.74768
$818$	15.8003	0.552445
$819$	−1.89361	−0.0661679
$820$	−17.4050	−0.607809
$821$	41.6434	1.45336	0.726682	−	0.686974i	$-0.241061\pi$
$821$	41.6434	1.45336	0.726682	+	0.686974i	$0.241061\pi$
$822$	−16.2716	−0.567539
$823$	36.4170	1.26942	0.634709	−	0.772752i	$-0.281120\pi$
$823$	36.4170	1.26942	0.634709	+	0.772752i	$0.281120\pi$
$824$	2.95375	0.102899
$825$	−0.294193	−0.0102425
$826$	2.93579	0.102149
$827$	−30.1543	−1.04857	−0.524284	−	0.851544i	$-0.675667\pi$
$827$	−30.1543	−1.04857	−0.524284	+	0.851544i	$0.675667\pi$
$828$	−2.45806	−0.0854235
$829$	48.4143	1.68150	0.840748	−	0.541426i	$-0.182115\pi$
$829$	48.4143	1.68150	0.840748	+	0.541426i	$0.182115\pi$
$830$	−10.6367	−0.369205
$831$	30.8655	1.07071
$832$	−1.23911	−0.0429586
$833$	−35.1437	−1.21766
$834$	−24.0454	−0.832623
$835$	−32.8291	−1.13610
$836$	1.15222	0.0398502
$837$	12.9447	0.447433
$838$	13.8525	0.478527
$839$	32.3445	1.11666	0.558329	−	0.829620i	$-0.311443\pi$
$839$	32.3445	1.11666	0.558329	+	0.829620i	$0.311443\pi$
$840$	−2.40311	−0.0829151
$841$	8.26577	0.285027
$842$	9.28131	0.319855
$843$	−5.57251	−0.191928
$844$	2.47039	0.0850342
$845$	−22.7393	−0.782256
$846$	−13.0474	−0.448579
$847$	10.9387	0.375860
$848$	10.4646	0.359356
$849$	20.3300	0.697724
$850$	6.24191	0.214096
$851$	16.7617	0.574584
$852$	−14.7662	−0.505882
$853$	−21.8933	−0.749613	−0.374806	−	0.927103i	$-0.622291\pi$
$853$	−21.8933	−0.749613	−0.374806	+	0.927103i	$0.622291\pi$
$854$	2.80106	0.0958503
$855$	15.3585	0.525251
$856$	14.3959	0.492043
$857$	−12.0077	−0.410176	−0.205088	−	0.978744i	$-0.565748\pi$
$857$	−12.0077	−0.410176	−0.205088	+	0.978744i	$0.565748\pi$
$858$	0.341978	0.0116749
$859$	−16.0557	−0.547812	−0.273906	−	0.961757i	$-0.588316\pi$
$859$	−16.0557	−0.547812	−0.273906	+	0.961757i	$0.588316\pi$
$860$	19.5710	0.667365
$861$	−10.6319	−0.362334
$862$	2.76703	0.0942455
$863$	13.6897	0.466003	0.233002	−	0.972476i	$-0.425145\pi$
$863$	13.6897	0.466003	0.233002	+	0.972476i	$0.425145\pi$
$864$	−5.49265	−0.186864
$865$	13.1853	0.448312
$866$	−6.11458	−0.207782
$867$	−20.9640	−0.711974
$868$	−2.35469	−0.0799232
$869$	0.398252	0.0135098
$870$	−14.6826	−0.497788
$871$	−9.21666	−0.312295
$872$	−5.45841	−0.184845
$873$	25.1771	0.852117
$874$	8.13611	0.275208
$875$	12.0211	0.406387
$876$	17.8824	0.604189
$877$	10.5174	0.355147	0.177573	−	0.984108i	$-0.443175\pi$
$877$	10.5174	0.355147	0.177573	+	0.984108i	$0.443175\pi$
$878$	31.6427	1.06789
$879$	−11.0524	−0.372788
$880$	−0.451413	−0.0152171
$881$	−18.2671	−0.615434	−0.307717	−	0.951478i	$-0.599565\pi$
$881$	−18.2671	−0.615434	−0.307717	+	0.951478i	$0.599565\pi$
$882$	−9.17973	−0.309098
$883$	−35.0034	−1.17796	−0.588980	−	0.808148i	$-0.700470\pi$
$883$	−35.0034	−1.17796	−0.588980	+	0.808148i	$0.700470\pi$
$884$	−7.25575	−0.244037
$885$	−7.06724	−0.237563
$886$	−6.56430	−0.220532
$887$	−7.67268	−0.257623	−0.128812	−	0.991669i	$-0.541116\pi$
$887$	−7.67268	−0.257623	−0.128812	+	0.991669i	$0.541116\pi$
$888$	12.6476	0.424427
$889$	−15.1028	−0.506531
$890$	22.5498	0.755870
$891$	0.471579	0.0157985
$892$	−0.971481	−0.0325276
$893$	43.1865	1.44518
$894$	20.1047	0.672401
$895$	−0.769543	−0.0257230
$896$	0.999136	0.0333788
$897$	2.41480	0.0806278
$898$	11.8871	0.396679
$899$	−14.3868	−0.479826
$900$	1.63042	0.0543475
$901$	61.2765	2.04142
$902$	−1.99715	−0.0664979
$903$	11.9550	0.397837
$904$	19.3971	0.645139
$905$	−34.4533	−1.14527
$906$	24.1304	0.801678
$907$	12.9264	0.429215	0.214608	−	0.976700i	$-0.431153\pi$
$907$	12.9264	0.429215	0.214608	+	0.976700i	$0.431153\pi$
$908$	−13.7887	−0.457593
$909$	−14.8597	−0.492864
$910$	−2.45558	−0.0814018
$911$	30.0475	0.995519	0.497760	−	0.867315i	$-0.334156\pi$
$911$	30.0475	0.995519	0.497760	+	0.867315i	$0.334156\pi$
$912$	6.13915	0.203288
$913$	−1.22052	−0.0403932
$914$	6.63876	0.219591
$915$	−6.74290	−0.222913
$916$	−2.13143	−0.0704244
$917$	9.30510	0.307282
$918$	−32.1627	−1.06153
$919$	36.5369	1.20524	0.602621	−	0.798027i	$-0.294123\pi$
$919$	36.5369	1.20524	0.602621	+	0.798027i	$0.294123\pi$
$920$	−3.18755	−0.105091
$921$	35.2004	1.15989
$922$	−12.2812	−0.404459
$923$	−15.0886	−0.496649
$924$	−0.275747	−0.00907140
$925$	−11.1180	−0.365557
$926$	11.5692	0.380188
$927$	4.51780	0.148384
$928$	6.10457	0.200392
$929$	−17.2144	−0.564785	−0.282392	−	0.959299i	$-0.591128\pi$
$929$	−17.2144	−0.564785	−0.282392	+	0.959299i	$0.591128\pi$
$930$	5.66836	0.185873
$931$	30.3847	0.995817
$932$	5.89571	0.193120
$933$	−1.72239	−0.0563886
$934$	4.00673	0.131104
$935$	−2.64329	−0.0864448
$936$	−1.89524	−0.0619480
$937$	−20.1372	−0.657855	−0.328928	−	0.944355i	$-0.606687\pi$
$937$	−20.1372	−0.657855	−0.328928	+	0.944355i	$0.606687\pi$
$938$	7.43167	0.242653
$939$	36.4242	1.18866
$940$	−16.9196	−0.551855
$941$	6.66508	0.217275	0.108638	−	0.994081i	$-0.465351\pi$
$941$	6.66508	0.217275	0.108638	+	0.994081i	$0.465351\pi$
$942$	9.80790	0.319559
$943$	−14.1025	−0.459239
$944$	2.93833	0.0956345
$945$	−10.8849	−0.354086
$946$	2.24569	0.0730137
$947$	−40.0261	−1.30067	−0.650336	−	0.759646i	$-0.725372\pi$
$947$	−40.0261	−1.30067	−0.650336	+	0.759646i	$0.725372\pi$
$948$	2.12194	0.0689173
$949$	18.2728	0.593161
$950$	−5.39666	−0.175091
$951$	28.2028	0.914538
$952$	5.85053	0.189617
$953$	11.8409	0.383564	0.191782	−	0.981438i	$-0.438573\pi$
$953$	11.8409	0.383564	0.191782	+	0.981438i	$0.438573\pi$
$954$	16.0058	0.518206
$955$	−26.5511	−0.859173
$956$	12.9244	0.418005
$957$	−1.68477	−0.0544609
$958$	26.0637	0.842080
$959$	13.4068	0.432928
$960$	−2.40519	−0.0776271
$961$	−25.4459	−0.820834
$962$	12.9238	0.416680
$963$	22.0188	0.709545
$964$	7.99949	0.257646
$965$	26.2074	0.843646
$966$	−1.94713	−0.0626477
$967$	23.0605	0.741575	0.370787	−	0.928718i	$-0.379088\pi$
$967$	23.0605	0.741575	0.370787	+	0.928718i	$0.379088\pi$
$968$	10.9482	0.351889
$969$	35.9484	1.15483
$970$	32.6491	1.04830
$971$	4.27747	0.137271	0.0686353	−	0.997642i	$-0.478136\pi$
$971$	4.27747	0.137271	0.0686353	+	0.997642i	$0.478136\pi$
$972$	−13.9653	−0.447938
$973$	19.8119	0.635140
$974$	35.7988	1.14707
$975$	−1.60173	−0.0512964
$976$	2.80348	0.0897373
$977$	17.0485	0.545430	0.272715	−	0.962095i	$-0.412078\pi$
$977$	17.0485	0.545430	0.272715	+	0.962095i	$0.412078\pi$
$978$	20.5177	0.656084
$979$	2.58749	0.0826967
$980$	−11.9041	−0.380261
$981$	−8.34871	−0.266554
$982$	18.2267	0.581638
$983$	−19.7536	−0.630042	−0.315021	−	0.949085i	$-0.602012\pi$
$983$	−19.7536	−0.630042	−0.315021	+	0.949085i	$0.602012\pi$
$984$	−10.6411	−0.339225
$985$	−0.724509	−0.0230848
$986$	35.7459	1.13838
$987$	−10.3354	−0.328978
$988$	6.27321	0.199577
$989$	15.8574	0.504238
$990$	−0.690443	−0.0219437
$991$	−26.9628	−0.856501	−0.428251	−	0.903660i	$-0.640870\pi$
$991$	−26.9628	−0.856501	−0.428251	+	0.903660i	$0.640870\pi$
$992$	−2.35672	−0.0748260
$993$	−41.0381	−1.30231
$994$	12.1664	0.385896
$995$	6.09767	0.193309
$996$	−6.50306	−0.206057
$997$	40.8484	1.29368	0.646840	−	0.762626i	$-0.276090\pi$
$997$	40.8484	1.29368	0.646840	+	0.762626i	$0.276090\pi$
$998$	11.1227	0.352084
$999$	57.2877	1.81250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.c.1.20	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.c.1.20	✓	57	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.21264 q^{3} +1.00000 q^{4} +1.98344 q^{5} +1.21264 q^{6} -0.999136 q^{7} -1.00000 q^{8} -1.52951 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.21264 q^{3} +1.00000 q^{4} +1.98344 q^{5} +1.21264 q^{6} -0.999136 q^{7} -1.00000 q^{8} -1.52951 q^{9} -1.98344 q^{10} -0.227591 q^{11} -1.21264 q^{12} -1.23911 q^{13} +0.999136 q^{14} -2.40519 q^{15} +1.00000 q^{16} +5.85559 q^{17} +1.52951 q^{18} -5.06265 q^{19} +1.98344 q^{20} +1.21159 q^{21} +0.227591 q^{22} +1.60709 q^{23} +1.21264 q^{24} -1.06597 q^{25} +1.23911 q^{26} +5.49265 q^{27} -0.999136 q^{28} -6.10457 q^{29} +2.40519 q^{30} +2.35672 q^{31} -1.00000 q^{32} +0.275985 q^{33} -5.85559 q^{34} -1.98172 q^{35} -1.52951 q^{36} +10.4299 q^{37} +5.06265 q^{38} +1.50260 q^{39} -1.98344 q^{40} -8.77517 q^{41} -1.21159 q^{42} +9.86721 q^{43} -0.227591 q^{44} -3.03370 q^{45} -1.60709 q^{46} -8.53042 q^{47} -1.21264 q^{48} -6.00173 q^{49} +1.06597 q^{50} -7.10070 q^{51} -1.23911 q^{52} +10.4646 q^{53} -5.49265 q^{54} -0.451413 q^{55} +0.999136 q^{56} +6.13915 q^{57} +6.10457 q^{58} +2.93833 q^{59} -2.40519 q^{60} +2.80348 q^{61} -2.35672 q^{62} +1.52819 q^{63} +1.00000 q^{64} -2.45771 q^{65} -0.275985 q^{66} +7.43810 q^{67} +5.85559 q^{68} -1.94881 q^{69} +1.98172 q^{70} +12.1770 q^{71} +1.52951 q^{72} -14.7467 q^{73} -10.4299 q^{74} +1.29264 q^{75} -5.06265 q^{76} +0.227395 q^{77} -1.50260 q^{78} -1.74986 q^{79} +1.98344 q^{80} -2.07204 q^{81} +8.77517 q^{82} +5.36275 q^{83} +1.21159 q^{84} +11.6142 q^{85} -9.86721 q^{86} +7.40262 q^{87} +0.227591 q^{88} -11.3690 q^{89} +3.03370 q^{90} +1.23804 q^{91} +1.60709 q^{92} -2.85785 q^{93} +8.53042 q^{94} -10.0415 q^{95} +1.21264 q^{96} -16.4609 q^{97} +6.00173 q^{98} +0.348104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

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