LMFDB - Embedded newform 8038.2.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.32095 q^{3} +1.00000 q^{4} -3.93433 q^{5} -3.32095 q^{6} -4.89708 q^{7} +1.00000 q^{8} +8.02870 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.32095 q^{3} +1.00000 q^{4} -3.93433 q^{5} -3.32095 q^{6} -4.89708 q^{7} +1.00000 q^{8} +8.02870 q^{9} -3.93433 q^{10} +2.20507 q^{11} -3.32095 q^{12} -1.31491 q^{13} -4.89708 q^{14} +13.0657 q^{15} +1.00000 q^{16} -0.579578 q^{17} +8.02870 q^{18} -8.18045 q^{19} -3.93433 q^{20} +16.2630 q^{21} +2.20507 q^{22} -5.63211 q^{23} -3.32095 q^{24} +10.4790 q^{25} -1.31491 q^{26} -16.7001 q^{27} -4.89708 q^{28} +0.411997 q^{29} +13.0657 q^{30} -8.40478 q^{31} +1.00000 q^{32} -7.32291 q^{33} -0.579578 q^{34} +19.2668 q^{35} +8.02870 q^{36} +11.7737 q^{37} -8.18045 q^{38} +4.36674 q^{39} -3.93433 q^{40} +6.60418 q^{41} +16.2630 q^{42} -8.08889 q^{43} +2.20507 q^{44} -31.5876 q^{45} -5.63211 q^{46} +7.39565 q^{47} -3.32095 q^{48} +16.9814 q^{49} +10.4790 q^{50} +1.92475 q^{51} -1.31491 q^{52} +6.81078 q^{53} -16.7001 q^{54} -8.67546 q^{55} -4.89708 q^{56} +27.1669 q^{57} +0.411997 q^{58} +0.679444 q^{59} +13.0657 q^{60} +14.1318 q^{61} -8.40478 q^{62} -39.3172 q^{63} +1.00000 q^{64} +5.17328 q^{65} -7.32291 q^{66} +4.52461 q^{67} -0.579578 q^{68} +18.7039 q^{69} +19.2668 q^{70} -7.28007 q^{71} +8.02870 q^{72} -10.8209 q^{73} +11.7737 q^{74} -34.8001 q^{75} -8.18045 q^{76} -10.7984 q^{77} +4.36674 q^{78} +8.34809 q^{79} -3.93433 q^{80} +31.3739 q^{81} +6.60418 q^{82} +14.9263 q^{83} +16.2630 q^{84} +2.28025 q^{85} -8.08889 q^{86} -1.36822 q^{87} +2.20507 q^{88} +10.7415 q^{89} -31.5876 q^{90} +6.43921 q^{91} -5.63211 q^{92} +27.9118 q^{93} +7.39565 q^{94} +32.1846 q^{95} -3.32095 q^{96} -13.1856 q^{97} +16.9814 q^{98} +17.7038 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * 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
$3$	−3.32095	−1.91735	−0.958675	−	0.284503i	$-0.908172\pi$
$3$	−3.32095	−1.91735	−0.958675	+	0.284503i	$0.908172\pi$
$4$	1.00000	0.500000
$5$	−3.93433	−1.75949	−0.879743	−	0.475449i	$-0.842285\pi$
$5$	−3.93433	−1.75949	−0.879743	+	0.475449i	$0.842285\pi$
$6$	−3.32095	−1.35577
$7$	−4.89708	−1.85092	−0.925462	−	0.378840i	$-0.876323\pi$
$7$	−4.89708	−1.85092	−0.925462	+	0.378840i	$0.876323\pi$
$8$	1.00000	0.353553
$9$	8.02870	2.67623
$10$	−3.93433	−1.24414
$11$	2.20507	0.664853	0.332426	−	0.943129i	$-0.392133\pi$
$11$	2.20507	0.664853	0.332426	+	0.943129i	$0.392133\pi$
$12$	−3.32095	−0.958675
$13$	−1.31491	−0.364690	−0.182345	−	0.983235i	$-0.558369\pi$
$13$	−1.31491	−0.364690	−0.182345	+	0.983235i	$0.558369\pi$
$14$	−4.89708	−1.30880
$15$	13.0657	3.37355
$16$	1.00000	0.250000
$17$	−0.579578	−0.140568	−0.0702842	−	0.997527i	$-0.522391\pi$
$17$	−0.579578	−0.140568	−0.0702842	+	0.997527i	$0.522391\pi$
$18$	8.02870	1.89238
$19$	−8.18045	−1.87672	−0.938362	−	0.345654i	$-0.887657\pi$
$19$	−8.18045	−1.87672	−0.938362	+	0.345654i	$0.887657\pi$
$20$	−3.93433	−0.879743
$21$	16.2630	3.54887
$22$	2.20507	0.470122
$23$	−5.63211	−1.17438	−0.587188	−	0.809451i	$-0.699765\pi$
$23$	−5.63211	−1.17438	−0.587188	+	0.809451i	$0.699765\pi$
$24$	−3.32095	−0.677886
$25$	10.4790	2.09579
$26$	−1.31491	−0.257875
$27$	−16.7001	−3.21393
$28$	−4.89708	−0.925462
$29$	0.411997	0.0765058	0.0382529	−	0.999268i	$-0.487821\pi$
$29$	0.411997	0.0765058	0.0382529	+	0.999268i	$0.487821\pi$
$30$	13.0657	2.38546
$31$	−8.40478	−1.50954	−0.754772	−	0.655988i	$-0.772252\pi$
$31$	−8.40478	−1.50954	−0.754772	+	0.655988i	$0.772252\pi$
$32$	1.00000	0.176777
$33$	−7.32291	−1.27476
$34$	−0.579578	−0.0993969
$35$	19.2668	3.25668
$36$	8.02870	1.33812
$37$	11.7737	1.93558	0.967788	−	0.251765i	$-0.0810110\pi$
$37$	11.7737	1.93558	0.967788	+	0.251765i	$0.0810110\pi$
$38$	−8.18045	−1.32704
$39$	4.36674	0.699238
$40$	−3.93433	−0.622072
$41$	6.60418	1.03140	0.515700	−	0.856770i	$-0.327532\pi$
$41$	6.60418	1.03140	0.515700	+	0.856770i	$0.327532\pi$
$42$	16.2630	2.50943
$43$	−8.08889	−1.23354	−0.616772	−	0.787142i	$-0.711560\pi$
$43$	−8.08889	−1.23354	−0.616772	+	0.787142i	$0.711560\pi$
$44$	2.20507	0.332426
$45$	−31.5876	−4.70880
$46$	−5.63211	−0.830409
$47$	7.39565	1.07877	0.539383	−	0.842061i	$-0.318658\pi$
$47$	7.39565	1.07877	0.539383	+	0.842061i	$0.318658\pi$
$48$	−3.32095	−0.479338
$49$	16.9814	2.42592
$50$	10.4790	1.48195
$51$	1.92475	0.269519
$52$	−1.31491	−0.182345
$53$	6.81078	0.935532	0.467766	−	0.883852i	$-0.345059\pi$
$53$	6.81078	0.935532	0.467766	+	0.883852i	$0.345059\pi$
$54$	−16.7001	−2.27259
$55$	−8.67546	−1.16980
$56$	−4.89708	−0.654400
$57$	27.1669	3.59834
$58$	0.411997	0.0540978
$59$	0.679444	0.0884561	0.0442281	−	0.999021i	$-0.485917\pi$
$59$	0.679444	0.0884561	0.0442281	+	0.999021i	$0.485917\pi$
$60$	13.0657	1.68678
$61$	14.1318	1.80939	0.904693	−	0.426063i	$-0.140100\pi$
$61$	14.1318	1.80939	0.904693	+	0.426063i	$0.140100\pi$
$62$	−8.40478	−1.06741
$63$	−39.3172	−4.95350
$64$	1.00000	0.125000
$65$	5.17328	0.641666
$66$	−7.32291	−0.901388
$67$	4.52461	0.552768	0.276384	−	0.961047i	$-0.410864\pi$
$67$	4.52461	0.552768	0.276384	+	0.961047i	$0.410864\pi$
$68$	−0.579578	−0.0702842
$69$	18.7039	2.25169
$70$	19.2668	2.30282
$71$	−7.28007	−0.863985	−0.431993	−	0.901877i	$-0.642189\pi$
$71$	−7.28007	−0.863985	−0.431993	+	0.901877i	$0.642189\pi$
$72$	8.02870	0.946191
$73$	−10.8209	−1.26649	−0.633246	−	0.773951i	$-0.718278\pi$
$73$	−10.8209	−1.26649	−0.633246	+	0.773951i	$0.718278\pi$
$74$	11.7737	1.36866
$75$	−34.8001	−4.01837
$76$	−8.18045	−0.938362
$77$	−10.7984	−1.23059
$78$	4.36674	0.494436
$79$	8.34809	0.939233	0.469617	−	0.882870i	$-0.344392\pi$
$79$	8.34809	0.939233	0.469617	+	0.882870i	$0.344392\pi$
$80$	−3.93433	−0.439872
$81$	31.3739	3.48599
$82$	6.60418	0.729309
$83$	14.9263	1.63838	0.819189	−	0.573523i	$-0.194424\pi$
$83$	14.9263	1.63838	0.819189	+	0.573523i	$0.194424\pi$
$84$	16.2630	1.77444
$85$	2.28025	0.247328
$86$	−8.08889	−0.872248
$87$	−1.36822	−0.146689
$88$	2.20507	0.235061
$89$	10.7415	1.13860	0.569298	−	0.822131i	$-0.307215\pi$
$89$	10.7415	1.13860	0.569298	+	0.822131i	$0.307215\pi$
$90$	−31.5876	−3.32962
$91$	6.43921	0.675013
$92$	−5.63211	−0.587188
$93$	27.9118	2.89432
$94$	7.39565	0.762803
$95$	32.1846	3.30207
$96$	−3.32095	−0.338943
$97$	−13.1856	−1.33879	−0.669396	−	0.742906i	$-0.733447\pi$
$97$	−13.1856	−1.33879	−0.669396	+	0.742906i	$0.733447\pi$
$98$	16.9814	1.71538
$99$	17.7038	1.77930
$100$	10.4790	1.04790
$101$	3.58951	0.357169	0.178585	−	0.983925i	$-0.442848\pi$
$101$	3.58951	0.357169	0.178585	+	0.983925i	$0.442848\pi$
$102$	1.92475	0.190579
$103$	9.16782	0.903332	0.451666	−	0.892187i	$-0.350830\pi$
$103$	9.16782	0.903332	0.451666	+	0.892187i	$0.350830\pi$
$104$	−1.31491	−0.128937
$105$	−63.9839	−6.24419
$106$	6.81078	0.661521
$107$	−3.07953	−0.297710	−0.148855	−	0.988859i	$-0.547559\pi$
$107$	−3.07953	−0.297710	−0.148855	+	0.988859i	$0.547559\pi$
$108$	−16.7001	−1.60696
$109$	0.140930	0.0134986	0.00674930	−	0.999977i	$-0.497852\pi$
$109$	0.140930	0.0134986	0.00674930	+	0.999977i	$0.497852\pi$
$110$	−8.67546	−0.827173
$111$	−39.0997	−3.71118
$112$	−4.89708	−0.462731
$113$	−4.37866	−0.411910	−0.205955	−	0.978561i	$-0.566030\pi$
$113$	−4.37866	−0.411910	−0.205955	+	0.978561i	$0.566030\pi$
$114$	27.1669	2.54441
$115$	22.1586	2.06630
$116$	0.411997	0.0382529
$117$	−10.5570	−0.975995
$118$	0.679444	0.0625479
$119$	2.83824	0.260181
$120$	13.0657	1.19273
$121$	−6.13768	−0.557971
$122$	14.1318	1.27943
$123$	−21.9321	−1.97755
$124$	−8.40478	−0.754772
$125$	−21.5560	−1.92803
$126$	−39.3172	−3.50266
$127$	−12.4740	−1.10689	−0.553443	−	0.832887i	$-0.686686\pi$
$127$	−12.4740	−1.10689	−0.553443	+	0.832887i	$0.686686\pi$
$128$	1.00000	0.0883883
$129$	26.8628	2.36514
$130$	5.17328	0.453727
$131$	5.05526	0.441680	0.220840	−	0.975310i	$-0.429120\pi$
$131$	5.05526	0.441680	0.220840	+	0.975310i	$0.429120\pi$
$132$	−7.32291	−0.637378
$133$	40.0604	3.47367
$134$	4.52461	0.390866
$135$	65.7035	5.65486
$136$	−0.579578	−0.0496984
$137$	1.79324	0.153207	0.0766036	−	0.997062i	$-0.475592\pi$
$137$	1.79324	0.153207	0.0766036	+	0.997062i	$0.475592\pi$
$138$	18.7039	1.59219
$139$	7.33693	0.622310	0.311155	−	0.950359i	$-0.399284\pi$
$139$	7.33693	0.622310	0.311155	+	0.950359i	$0.399284\pi$
$140$	19.2668	1.62834
$141$	−24.5606	−2.06837
$142$	−7.28007	−0.610930
$143$	−2.89946	−0.242465
$144$	8.02870	0.669058
$145$	−1.62093	−0.134611
$146$	−10.8209	−0.895544
$147$	−56.3945	−4.65134
$148$	11.7737	0.967788
$149$	−1.82035	−0.149129	−0.0745645	−	0.997216i	$-0.523757\pi$
$149$	−1.82035	−0.149129	−0.0745645	+	0.997216i	$0.523757\pi$
$150$	−34.8001	−2.84141
$151$	−15.2074	−1.23756	−0.618781	−	0.785564i	$-0.712373\pi$
$151$	−15.2074	−1.23756	−0.618781	+	0.785564i	$0.712373\pi$
$152$	−8.18045	−0.663522
$153$	−4.65326	−0.376194
$154$	−10.7984	−0.870160
$155$	33.0672	2.65602
$156$	4.36674	0.349619
$157$	0.658206	0.0525306	0.0262653	−	0.999655i	$-0.491639\pi$
$157$	0.658206	0.0525306	0.0262653	+	0.999655i	$0.491639\pi$
$158$	8.34809	0.664138
$159$	−22.6182	−1.79374
$160$	−3.93433	−0.311036
$161$	27.5809	2.17368
$162$	31.3739	2.46497
$163$	4.06551	0.318435	0.159218	−	0.987243i	$-0.449103\pi$
$163$	4.06551	0.318435	0.159218	+	0.987243i	$0.449103\pi$
$164$	6.60418	0.515700
$165$	28.8108	2.24291
$166$	14.9263	1.15851
$167$	1.47752	0.114334	0.0571669	−	0.998365i	$-0.481793\pi$
$167$	1.47752	0.114334	0.0571669	+	0.998365i	$0.481793\pi$
$168$	16.2630	1.25472
$169$	−11.2710	−0.867001
$170$	2.28025	0.174887
$171$	−65.6784	−5.02255
$172$	−8.08889	−0.616772
$173$	−23.5854	−1.79316	−0.896581	−	0.442880i	$-0.853957\pi$
$173$	−23.5854	−1.79316	−0.896581	+	0.442880i	$0.853957\pi$
$174$	−1.36822	−0.103724
$175$	−51.3163	−3.87915
$176$	2.20507	0.166213
$177$	−2.25640	−0.169601
$178$	10.7415	0.805109
$179$	11.6143	0.868096	0.434048	−	0.900890i	$-0.357085\pi$
$179$	11.6143	0.868096	0.434048	+	0.900890i	$0.357085\pi$
$180$	−31.5876	−2.35440
$181$	9.71441	0.722066	0.361033	−	0.932553i	$-0.382424\pi$
$181$	9.71441	0.722066	0.361033	+	0.932553i	$0.382424\pi$
$182$	6.43921	0.477306
$183$	−46.9309	−3.46923
$184$	−5.63211	−0.415205
$185$	−46.3215	−3.40562
$186$	27.9118	2.04660
$187$	−1.27801	−0.0934573
$188$	7.39565	0.539383
$189$	81.7816	5.94873
$190$	32.1846	2.33492
$191$	3.68678	0.266766	0.133383	−	0.991065i	$-0.457416\pi$
$191$	3.68678	0.266766	0.133383	+	0.991065i	$0.457416\pi$
$192$	−3.32095	−0.239669
$193$	−16.6958	−1.20179	−0.600894	−	0.799329i	$-0.705189\pi$
$193$	−16.6958	−1.20179	−0.600894	+	0.799329i	$0.705189\pi$
$194$	−13.1856	−0.946669
$195$	−17.1802	−1.23030
$196$	16.9814	1.21296
$197$	21.3337	1.51997	0.759983	−	0.649943i	$-0.225207\pi$
$197$	21.3337	1.51997	0.759983	+	0.649943i	$0.225207\pi$
$198$	17.7038	1.25816
$199$	0.425841	0.0301870	0.0150935	−	0.999886i	$-0.495195\pi$
$199$	0.425841	0.0301870	0.0150935	+	0.999886i	$0.495195\pi$
$200$	10.4790	0.740974
$201$	−15.0260	−1.05985
$202$	3.58951	0.252557
$203$	−2.01758	−0.141607
$204$	1.92475	0.134759
$205$	−25.9830	−1.81473
$206$	9.16782	0.638752
$207$	−45.2185	−3.14290
$208$	−1.31491	−0.0911724
$209$	−18.0384	−1.24774
$210$	−63.9839	−4.41531
$211$	9.77032	0.672616	0.336308	−	0.941752i	$-0.390822\pi$
$211$	9.77032	0.672616	0.336308	+	0.941752i	$0.390822\pi$
$212$	6.81078	0.467766
$213$	24.1767	1.65656
$214$	−3.07953	−0.210513
$215$	31.8244	2.17040
$216$	−16.7001	−1.13629
$217$	41.1589	2.79405
$218$	0.140930	0.00954496
$219$	35.9357	2.42831
$220$	−8.67546	−0.584899
$221$	0.762092	0.0512639
$222$	−39.0997	−2.62420
$223$	22.6059	1.51380	0.756901	−	0.653529i	$-0.226712\pi$
$223$	22.6059	1.51380	0.756901	+	0.653529i	$0.226712\pi$
$224$	−4.89708	−0.327200
$225$	84.1324	5.60883
$226$	−4.37866	−0.291264
$227$	−9.62439	−0.638793	−0.319397	−	0.947621i	$-0.603480\pi$
$227$	−9.62439	−0.638793	−0.319397	+	0.947621i	$0.603480\pi$
$228$	27.1669	1.79917
$229$	−8.85423	−0.585104	−0.292552	−	0.956250i	$-0.594504\pi$
$229$	−8.85423	−0.585104	−0.292552	+	0.956250i	$0.594504\pi$
$230$	22.1586	1.46109
$231$	35.8609	2.35948
$232$	0.411997	0.0270489
$233$	−7.64044	−0.500542	−0.250271	−	0.968176i	$-0.580520\pi$
$233$	−7.64044	−0.500542	−0.250271	+	0.968176i	$0.580520\pi$
$234$	−10.5570	−0.690132
$235$	−29.0969	−1.89807
$236$	0.679444	0.0442281
$237$	−27.7236	−1.80084
$238$	2.83824	0.183976
$239$	29.9011	1.93414	0.967072	−	0.254504i	$-0.0819121\pi$
$239$	29.9011	1.93414	0.967072	+	0.254504i	$0.0819121\pi$
$240$	13.0657	0.843388
$241$	−9.28632	−0.598184	−0.299092	−	0.954224i	$-0.596684\pi$
$241$	−9.28632	−0.598184	−0.299092	+	0.954224i	$0.596684\pi$
$242$	−6.13768	−0.394545
$243$	−54.0910	−3.46994
$244$	14.1318	0.904693
$245$	−66.8106	−4.26837
$246$	−21.9321	−1.39834
$247$	10.7565	0.684422
$248$	−8.40478	−0.533704
$249$	−49.5696	−3.14135
$250$	−21.5560	−1.36332
$251$	−19.0315	−1.20126	−0.600628	−	0.799528i	$-0.705083\pi$
$251$	−19.0315	−1.20126	−0.600628	+	0.799528i	$0.705083\pi$
$252$	−39.3172	−2.47675
$253$	−12.4192	−0.780787
$254$	−12.4740	−0.782686
$255$	−7.57260	−0.474215
$256$	1.00000	0.0625000
$257$	−10.2334	−0.638343	−0.319172	−	0.947697i	$-0.603405\pi$
$257$	−10.2334	−0.638343	−0.319172	+	0.947697i	$0.603405\pi$
$258$	26.8628	1.67240
$259$	−57.6566	−3.58261
$260$	5.17328	0.320833
$261$	3.30780	0.204747
$262$	5.05526	0.312315
$263$	18.7509	1.15623	0.578116	−	0.815954i	$-0.303788\pi$
$263$	18.7509	1.15623	0.578116	+	0.815954i	$0.303788\pi$
$264$	−7.32291	−0.450694
$265$	−26.7958	−1.64606
$266$	40.0604	2.45626
$267$	−35.6719	−2.18309
$268$	4.52461	0.276384
$269$	21.8151	1.33009	0.665044	−	0.746804i	$-0.268413\pi$
$269$	21.8151	1.33009	0.665044	+	0.746804i	$0.268413\pi$
$270$	65.7035	3.99859
$271$	5.15102	0.312902	0.156451	−	0.987686i	$-0.449995\pi$
$271$	5.15102	0.312902	0.156451	+	0.987686i	$0.449995\pi$
$272$	−0.579578	−0.0351421
$273$	−21.3843	−1.29424
$274$	1.79324	0.108334
$275$	23.1068	1.39339
$276$	18.7039	1.12585
$277$	−1.35226	−0.0812495	−0.0406248	−	0.999174i	$-0.512935\pi$
$277$	−1.35226	−0.0812495	−0.0406248	+	0.999174i	$0.512935\pi$
$278$	7.33693	0.440040
$279$	−67.4795	−4.03989
$280$	19.2668	1.15141
$281$	−13.2660	−0.791385	−0.395692	−	0.918383i	$-0.629495\pi$
$281$	−13.2660	−0.791385	−0.395692	+	0.918383i	$0.629495\pi$
$282$	−24.5606	−1.46256
$283$	3.33400	0.198186	0.0990928	−	0.995078i	$-0.468406\pi$
$283$	3.33400	0.198186	0.0990928	+	0.995078i	$0.468406\pi$
$284$	−7.28007	−0.431993
$285$	−106.883	−6.33123
$286$	−2.89946	−0.171449
$287$	−32.3412	−1.90904
$288$	8.02870	0.473096
$289$	−16.6641	−0.980241
$290$	−1.62093	−0.0951843
$291$	43.7886	2.56693
$292$	−10.8209	−0.633246
$293$	−10.9160	−0.637722	−0.318861	−	0.947802i	$-0.603300\pi$
$293$	−10.9160	−0.637722	−0.318861	+	0.947802i	$0.603300\pi$
$294$	−56.3945	−3.28899
$295$	−2.67316	−0.155637
$296$	11.7737	0.684330
$297$	−36.8247	−2.13679
$298$	−1.82035	−0.105450
$299$	7.40570	0.428283
$300$	−34.8001	−2.00918
$301$	39.6120	2.28320
$302$	−15.2074	−0.875089
$303$	−11.9206	−0.684819
$304$	−8.18045	−0.469181
$305$	−55.5990	−3.18359
$306$	−4.65326	−0.266009
$307$	−23.8863	−1.36326	−0.681631	−	0.731696i	$-0.738729\pi$
$307$	−23.8863	−1.36326	−0.681631	+	0.731696i	$0.738729\pi$
$308$	−10.7984	−0.615296
$309$	−30.4459	−1.73200
$310$	33.0672	1.87809
$311$	15.5490	0.881703	0.440851	−	0.897580i	$-0.354677\pi$
$311$	15.5490	0.881703	0.440851	+	0.897580i	$0.354677\pi$
$312$	4.36674	0.247218
$313$	−20.4200	−1.15421	−0.577105	−	0.816670i	$-0.695818\pi$
$313$	−20.4200	−1.15421	−0.577105	+	0.816670i	$0.695818\pi$
$314$	0.658206	0.0371447
$315$	154.687	8.71562
$316$	8.34809	0.469617
$317$	−25.4150	−1.42745	−0.713725	−	0.700426i	$-0.752993\pi$
$317$	−25.4150	−1.42745	−0.713725	+	0.700426i	$0.752993\pi$
$318$	−22.6182	−1.26837
$319$	0.908480	0.0508651
$320$	−3.93433	−0.219936
$321$	10.2270	0.570814
$322$	27.5809	1.53702
$323$	4.74121	0.263808
$324$	31.3739	1.74300
$325$	−13.7789	−0.764314
$326$	4.06551	0.225168
$327$	−0.468020	−0.0258816
$328$	6.60418	0.364655
$329$	−36.2171	−1.99671
$330$	28.8108	1.58598
$331$	1.19850	0.0658757	0.0329379	−	0.999457i	$-0.489514\pi$
$331$	1.19850	0.0658757	0.0329379	+	0.999457i	$0.489514\pi$
$332$	14.9263	0.819189
$333$	94.5271	5.18006
$334$	1.47752	0.0808463
$335$	−17.8013	−0.972589
$336$	16.2630	0.887218
$337$	−25.6993	−1.39993	−0.699964	−	0.714178i	$-0.746801\pi$
$337$	−25.6993	−1.39993	−0.699964	+	0.714178i	$0.746801\pi$
$338$	−11.2710	−0.613063
$339$	14.5413	0.789776
$340$	2.28025	0.123664
$341$	−18.5331	−1.00362
$342$	−65.6784	−3.55148
$343$	−48.8799	−2.63927
$344$	−8.08889	−0.436124
$345$	−73.5875	−3.96182
$346$	−23.5854	−1.26796
$347$	3.65073	0.195981	0.0979906	−	0.995187i	$-0.468758\pi$
$347$	3.65073	0.195981	0.0979906	+	0.995187i	$0.468758\pi$
$348$	−1.36822	−0.0733443
$349$	34.4366	1.84335	0.921674	−	0.387965i	$-0.126822\pi$
$349$	34.4366	1.84335	0.921674	+	0.387965i	$0.126822\pi$
$350$	−51.3163	−2.74297
$351$	21.9590	1.17209
$352$	2.20507	0.117530
$353$	−3.63947	−0.193710	−0.0968548	−	0.995299i	$-0.530878\pi$
$353$	−3.63947	−0.193710	−0.0968548	+	0.995299i	$0.530878\pi$
$354$	−2.25640	−0.119926
$355$	28.6422	1.52017
$356$	10.7415	0.569298
$357$	−9.42567	−0.498859
$358$	11.6143	0.613837
$359$	−10.9806	−0.579532	−0.289766	−	0.957097i	$-0.593578\pi$
$359$	−10.9806	−0.579532	−0.289766	+	0.957097i	$0.593578\pi$
$360$	−31.5876	−1.66481
$361$	47.9198	2.52209
$362$	9.71441	0.510578
$363$	20.3829	1.06983
$364$	6.43921	0.337506
$365$	42.5730	2.22837
$366$	−46.9309	−2.45312
$367$	18.4995	0.965666	0.482833	−	0.875712i	$-0.339608\pi$
$367$	18.4995	0.965666	0.482833	+	0.875712i	$0.339608\pi$
$368$	−5.63211	−0.293594
$369$	53.0229	2.76026
$370$	−46.3215	−2.40814
$371$	−33.3529	−1.73160
$372$	27.9118	1.44716
$373$	−31.4944	−1.63072	−0.815359	−	0.578956i	$-0.803460\pi$
$373$	−31.4944	−1.63072	−0.815359	+	0.578956i	$0.803460\pi$
$374$	−1.27801	−0.0660843
$375$	71.5865	3.69671
$376$	7.39565	0.381401
$377$	−0.541737	−0.0279009
$378$	81.7816	4.20639
$379$	20.6367	1.06003	0.530017	−	0.847987i	$-0.322185\pi$
$379$	20.6367	1.06003	0.530017	+	0.847987i	$0.322185\pi$
$380$	32.1846	1.65103
$381$	41.4254	2.12229
$382$	3.68678	0.188632
$383$	0.271368	0.0138663	0.00693313	−	0.999976i	$-0.497793\pi$
$383$	0.271368	0.0138663	0.00693313	+	0.999976i	$0.497793\pi$
$384$	−3.32095	−0.169471
$385$	42.4845	2.16521
$386$	−16.6958	−0.849792
$387$	−64.9433	−3.30125
$388$	−13.1856	−0.669396
$389$	−5.62628	−0.285264	−0.142632	−	0.989776i	$-0.545556\pi$
$389$	−5.62628	−0.285264	−0.142632	+	0.989776i	$0.545556\pi$
$390$	−17.1802	−0.869953
$391$	3.26425	0.165080
$392$	16.9814	0.857692
$393$	−16.7883	−0.846855
$394$	21.3337	1.07478
$395$	−32.8441	−1.65257
$396$	17.7038	0.889650
$397$	−5.86700	−0.294456	−0.147228	−	0.989103i	$-0.547035\pi$
$397$	−5.86700	−0.294456	−0.147228	+	0.989103i	$0.547035\pi$
$398$	0.425841	0.0213455
$399$	−133.038	−6.66025
$400$	10.4790	0.523948
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−15.0260	−0.749428
$403$	11.0515	0.550515
$404$	3.58951	0.178585
$405$	−123.435	−6.13355
$406$	−2.01758	−0.100131
$407$	25.9617	1.28687
$408$	1.92475	0.0952893
$409$	11.7639	0.581689	0.290845	−	0.956770i	$-0.406064\pi$
$409$	11.7639	0.581689	0.290845	+	0.956770i	$0.406064\pi$
$410$	−25.9830	−1.28321
$411$	−5.95527	−0.293752
$412$	9.16782	0.451666
$413$	−3.32730	−0.163726
$414$	−45.2185	−2.22237
$415$	−58.7252	−2.88271
$416$	−1.31491	−0.0644686
$417$	−24.3656	−1.19319
$418$	−18.0384	−0.882289
$419$	−23.4228	−1.14428	−0.572140	−	0.820156i	$-0.693887\pi$
$419$	−23.4228	−1.14428	−0.572140	+	0.820156i	$0.693887\pi$
$420$	−63.9839	−3.12209
$421$	6.61967	0.322623	0.161312	−	0.986904i	$-0.448428\pi$
$421$	6.61967	0.322623	0.161312	+	0.986904i	$0.448428\pi$
$422$	9.77032	0.475612
$423$	59.3774	2.88703
$424$	6.81078	0.330761
$425$	−6.07338	−0.294602
$426$	24.1767	1.17137
$427$	−69.2044	−3.34904
$428$	−3.07953	−0.148855
$429$	9.62895	0.464890
$430$	31.8244	1.53471
$431$	−20.2901	−0.977340	−0.488670	−	0.872469i	$-0.662518\pi$
$431$	−20.2901	−0.977340	−0.488670	+	0.872469i	$0.662518\pi$
$432$	−16.7001	−0.803482
$433$	−11.6244	−0.558634	−0.279317	−	0.960199i	$-0.590108\pi$
$433$	−11.6244	−0.558634	−0.279317	+	0.960199i	$0.590108\pi$
$434$	41.1589	1.97569
$435$	5.38303	0.258096
$436$	0.140930	0.00674930
$437$	46.0732	2.20398
$438$	35.9357	1.71707
$439$	15.4482	0.737302	0.368651	−	0.929568i	$-0.379820\pi$
$439$	15.4482	0.737302	0.368651	+	0.929568i	$0.379820\pi$
$440$	−8.67546	−0.413586
$441$	136.339	6.49233
$442$	0.762092	0.0362490
$443$	−2.17969	−0.103560	−0.0517800	−	0.998659i	$-0.516489\pi$
$443$	−2.17969	−0.103560	−0.0517800	+	0.998659i	$0.516489\pi$
$444$	−39.0997	−1.85559
$445$	−42.2606	−2.00334
$446$	22.6059	1.07042
$447$	6.04529	0.285933
$448$	−4.89708	−0.231366
$449$	6.37723	0.300960	0.150480	−	0.988613i	$-0.451918\pi$
$449$	6.37723	0.300960	0.150480	+	0.988613i	$0.451918\pi$
$450$	84.1324	3.96604
$451$	14.5626	0.685728
$452$	−4.37866	−0.205955
$453$	50.5030	2.37284
$454$	−9.62439	−0.451695
$455$	−25.3340	−1.18768
$456$	27.1669	1.27220
$457$	33.0766	1.54726	0.773629	−	0.633638i	$-0.218439\pi$
$457$	33.0766	1.54726	0.773629	+	0.633638i	$0.218439\pi$
$458$	−8.85423	−0.413731
$459$	9.67899	0.451777
$460$	22.1586	1.03315
$461$	31.4728	1.46583	0.732917	−	0.680318i	$-0.238158\pi$
$461$	31.4728	1.46583	0.732917	+	0.680318i	$0.238158\pi$
$462$	35.8609	1.66840
$463$	0.128998	0.00599504	0.00299752	−	0.999996i	$-0.499046\pi$
$463$	0.128998	0.00599504	0.00299752	+	0.999996i	$0.499046\pi$
$464$	0.411997	0.0191265
$465$	−109.814	−5.09252
$466$	−7.64044	−0.353937
$467$	−20.0184	−0.926343	−0.463172	−	0.886269i	$-0.653289\pi$
$467$	−20.0184	−0.926343	−0.463172	+	0.886269i	$0.653289\pi$
$468$	−10.5570	−0.487997
$469$	−22.1574	−1.02313
$470$	−29.0969	−1.34214
$471$	−2.18587	−0.100720
$472$	0.679444	0.0312740
$473$	−17.8365	−0.820125
$474$	−27.7236	−1.27339
$475$	−85.7226	−3.93322
$476$	2.83824	0.130091
$477$	54.6817	2.50370
$478$	29.9011	1.36765
$479$	34.1993	1.56261	0.781303	−	0.624151i	$-0.214555\pi$
$479$	34.1993	1.56261	0.781303	+	0.624151i	$0.214555\pi$
$480$	13.0657	0.596365
$481$	−15.4813	−0.705885
$482$	−9.28632	−0.422980
$483$	−91.5948	−4.16771
$484$	−6.13768	−0.278986
$485$	51.8764	2.35559
$486$	−54.0910	−2.45362
$487$	18.7882	0.851375	0.425687	−	0.904870i	$-0.360032\pi$
$487$	18.7882	0.851375	0.425687	+	0.904870i	$0.360032\pi$
$488$	14.1318	0.639715
$489$	−13.5014	−0.610552
$490$	−66.8106	−3.01820
$491$	36.6498	1.65398	0.826990	−	0.562216i	$-0.190051\pi$
$491$	36.6498	1.65398	0.826990	+	0.562216i	$0.190051\pi$
$492$	−21.9321	−0.988777
$493$	−0.238784	−0.0107543
$494$	10.7565	0.483959
$495$	−69.6527	−3.13065
$496$	−8.40478	−0.377386
$497$	35.6511	1.59917
$498$	−49.5696	−2.22127
$499$	−34.6700	−1.55204	−0.776021	−	0.630708i	$-0.782765\pi$
$499$	−34.6700	−1.55204	−0.776021	+	0.630708i	$0.782765\pi$
$500$	−21.5560	−0.964015
$501$	−4.90677	−0.219218
$502$	−19.0315	−0.849417
$503$	−5.13327	−0.228881	−0.114441	−	0.993430i	$-0.536508\pi$
$503$	−5.13327	−0.228881	−0.114441	+	0.993430i	$0.536508\pi$
$504$	−39.3172	−1.75133
$505$	−14.1223	−0.628435
$506$	−12.4192	−0.552100
$507$	37.4305	1.66235
$508$	−12.4740	−0.553443
$509$	37.5911	1.66620	0.833099	−	0.553124i	$-0.186565\pi$
$509$	37.5911	1.66620	0.833099	+	0.553124i	$0.186565\pi$
$510$	−7.57260	−0.335321
$511$	52.9909	2.34418
$512$	1.00000	0.0441942
$513$	136.614	6.03165
$514$	−10.2334	−0.451377
$515$	−36.0692	−1.58940
$516$	26.8628	1.18257
$517$	16.3079	0.717220
$518$	−57.6566	−2.53328
$519$	78.3258	3.43812
$520$	5.17328	0.226863
$521$	−30.6644	−1.34343	−0.671716	−	0.740809i	$-0.734442\pi$
$521$	−30.6644	−1.34343	−0.671716	+	0.740809i	$0.734442\pi$
$522$	3.30780	0.144778
$523$	40.2903	1.76177	0.880885	−	0.473330i	$-0.156948\pi$
$523$	40.2903	1.76177	0.880885	+	0.473330i	$0.156948\pi$
$524$	5.05526	0.220840
$525$	170.419	7.43769
$526$	18.7509	0.817580
$527$	4.87123	0.212194
$528$	−7.32291	−0.318689
$529$	8.72067	0.379159
$530$	−26.7958	−1.16394
$531$	5.45506	0.236729
$532$	40.0604	1.73684
$533$	−8.68388	−0.376141
$534$	−35.6719	−1.54368
$535$	12.1159	0.523816
$536$	4.52461	0.195433
$537$	−38.5706	−1.66445
$538$	21.8151	0.940514
$539$	37.4452	1.61288
$540$	65.7035	2.82743
$541$	14.2498	0.612646	0.306323	−	0.951928i	$-0.400901\pi$
$541$	14.2498	0.612646	0.306323	+	0.951928i	$0.400901\pi$
$542$	5.15102	0.221255
$543$	−32.2610	−1.38445
$544$	−0.579578	−0.0248492
$545$	−0.554464	−0.0237506
$546$	−21.3843	−0.915163
$547$	−2.41814	−0.103392	−0.0516961	−	0.998663i	$-0.516463\pi$
$547$	−2.41814	−0.103392	−0.0516961	+	0.998663i	$0.516463\pi$
$548$	1.79324	0.0766036
$549$	113.460	4.84234
$550$	23.1068	0.985277
$551$	−3.37032	−0.143580
$552$	18.7039	0.796093
$553$	−40.8813	−1.73845
$554$	−1.35226	−0.0574521
$555$	153.831	6.52977
$556$	7.33693	0.311155
$557$	2.88920	0.122419	0.0612096	−	0.998125i	$-0.480504\pi$
$557$	2.88920	0.122419	0.0612096	+	0.998125i	$0.480504\pi$
$558$	−67.4795	−2.85663
$559$	10.6361	0.449861
$560$	19.2668	0.814169
$561$	4.24420	0.179190
$562$	−13.2660	−0.559594
$563$	10.9232	0.460360	0.230180	−	0.973148i	$-0.426069\pi$
$563$	10.9232	0.460360	0.230180	+	0.973148i	$0.426069\pi$
$564$	−24.5606	−1.03419
$565$	17.2271	0.724750
$566$	3.33400	0.140138
$567$	−153.641	−6.45230
$568$	−7.28007	−0.305465
$569$	−24.2773	−1.01776	−0.508878	−	0.860838i	$-0.669940\pi$
$569$	−24.2773	−1.01776	−0.508878	+	0.860838i	$0.669940\pi$
$570$	−106.883	−4.47685
$571$	−4.04440	−0.169253	−0.0846264	−	0.996413i	$-0.526970\pi$
$571$	−4.04440	−0.169253	−0.0846264	+	0.996413i	$0.526970\pi$
$572$	−2.89946	−0.121232
$573$	−12.2436	−0.511484
$574$	−32.3412	−1.34990
$575$	−59.0187	−2.46125
$576$	8.02870	0.334529
$577$	12.4067	0.516497	0.258249	−	0.966078i	$-0.416855\pi$
$577$	12.4067	0.516497	0.258249	+	0.966078i	$0.416855\pi$
$578$	−16.6641	−0.693135
$579$	55.4458	2.30425
$580$	−1.62093	−0.0673055
$581$	−73.0956	−3.03252
$582$	43.7886	1.81510
$583$	15.0182	0.621991
$584$	−10.8209	−0.447772
$585$	41.5347	1.71725
$586$	−10.9160	−0.450937
$587$	1.32239	0.0545809	0.0272904	−	0.999628i	$-0.491312\pi$
$587$	1.32239	0.0545809	0.0272904	+	0.999628i	$0.491312\pi$
$588$	−56.3945	−2.32567
$589$	68.7549	2.83300
$590$	−2.67316	−0.110052
$591$	−70.8482	−2.91431
$592$	11.7737	0.483894
$593$	0.224373	0.00921391	0.00460695	−	0.999989i	$-0.498534\pi$
$593$	0.224373	0.00921391	0.00460695	+	0.999989i	$0.498534\pi$
$594$	−36.8247	−1.51094
$595$	−11.1666	−0.457786
$596$	−1.82035	−0.0745645
$597$	−1.41419	−0.0578791
$598$	7.40570	0.302842
$599$	−34.8299	−1.42311	−0.711556	−	0.702629i	$-0.752009\pi$
$599$	−34.8299	−1.42311	−0.711556	+	0.702629i	$0.752009\pi$
$600$	−34.8001	−1.42071
$601$	−16.6303	−0.678365	−0.339182	−	0.940721i	$-0.610150\pi$
$601$	−16.6303	−0.678365	−0.339182	+	0.940721i	$0.610150\pi$
$602$	39.6120	1.61446
$603$	36.3267	1.47934
$604$	−15.2074	−0.618781
$605$	24.1477	0.981742
$606$	−11.9206	−0.484240
$607$	−35.6677	−1.44771	−0.723854	−	0.689953i	$-0.757631\pi$
$607$	−35.6677	−1.44771	−0.723854	+	0.689953i	$0.757631\pi$
$608$	−8.18045	−0.331761
$609$	6.70029	0.271509
$610$	−55.5990	−2.25114
$611$	−9.72459	−0.393415
$612$	−4.65326	−0.188097
$613$	−19.6832	−0.794998	−0.397499	−	0.917603i	$-0.630122\pi$
$613$	−19.6832	−0.794998	−0.397499	+	0.917603i	$0.630122\pi$
$614$	−23.8863	−0.963972
$615$	86.2883	3.47948
$616$	−10.7984	−0.435080
$617$	−3.88387	−0.156359	−0.0781794	−	0.996939i	$-0.524911\pi$
$617$	−3.88387	−0.156359	−0.0781794	+	0.996939i	$0.524911\pi$
$618$	−30.4459	−1.22471
$619$	−6.24067	−0.250834	−0.125417	−	0.992104i	$-0.540027\pi$
$619$	−6.24067	−0.250834	−0.125417	+	0.992104i	$0.540027\pi$
$620$	33.0672	1.32801
$621$	94.0565	3.77436
$622$	15.5490	0.623458
$623$	−52.6020	−2.10745
$624$	4.36674	0.174809
$625$	32.4138	1.29655
$626$	−20.4200	−0.816149
$627$	59.9047	2.39236
$628$	0.658206	0.0262653
$629$	−6.82376	−0.272081
$630$	154.687	6.16288
$631$	−5.58258	−0.222239	−0.111120	−	0.993807i	$-0.535444\pi$
$631$	−5.58258	−0.222239	−0.111120	+	0.993807i	$0.535444\pi$
$632$	8.34809	0.332069
$633$	−32.4467	−1.28964
$634$	−25.4150	−1.00936
$635$	49.0767	1.94755
$636$	−22.6182	−0.896871
$637$	−22.3290	−0.884708
$638$	0.908480	0.0359671
$639$	−58.4495	−2.31223
$640$	−3.93433	−0.155518
$641$	4.13679	0.163394	0.0816968	−	0.996657i	$-0.473966\pi$
$641$	4.13679	0.163394	0.0816968	+	0.996657i	$0.473966\pi$
$642$	10.2270	0.403627
$643$	39.9079	1.57381	0.786906	−	0.617072i	$-0.211681\pi$
$643$	39.9079	1.57381	0.786906	+	0.617072i	$0.211681\pi$
$644$	27.5809	1.08684
$645$	−105.687	−4.16143
$646$	4.74121	0.186541
$647$	10.3949	0.408664	0.204332	−	0.978902i	$-0.434498\pi$
$647$	10.3949	0.408664	0.204332	+	0.978902i	$0.434498\pi$
$648$	31.3739	1.23248
$649$	1.49822	0.0588103
$650$	−13.7789	−0.540451
$651$	−136.687	−5.35717
$652$	4.06551	0.159218
$653$	12.1091	0.473865	0.236932	−	0.971526i	$-0.423858\pi$
$653$	12.1091	0.473865	0.236932	+	0.971526i	$0.423858\pi$
$654$	−0.468020	−0.0183010
$655$	−19.8891	−0.777130
$656$	6.60418	0.257850
$657$	−86.8778	−3.38943
$658$	−36.2171	−1.41189
$659$	16.8234	0.655345	0.327673	−	0.944791i	$-0.393736\pi$
$659$	16.8234	0.655345	0.327673	+	0.944791i	$0.393736\pi$
$660$	28.8108	1.12146
$661$	17.5333	0.681968	0.340984	−	0.940069i	$-0.389240\pi$
$661$	17.5333	0.681968	0.340984	+	0.940069i	$0.389240\pi$
$662$	1.19850	0.0465812
$663$	−2.53087	−0.0982908
$664$	14.9263	0.579254
$665$	−157.611	−6.11188
$666$	94.5271	3.66285
$667$	−2.32041	−0.0898466
$668$	1.47752	0.0571669
$669$	−75.0730	−2.90249
$670$	−17.8013	−0.687724
$671$	31.1615	1.20298
$672$	16.2630	0.627358
$673$	37.5370	1.44694	0.723472	−	0.690353i	$-0.242545\pi$
$673$	37.5370	1.44694	0.723472	+	0.690353i	$0.242545\pi$
$674$	−25.6993	−0.989899
$675$	−174.999	−6.73572
$676$	−11.2710	−0.433501
$677$	16.0259	0.615927	0.307963	−	0.951398i	$-0.400353\pi$
$677$	16.0259	0.615927	0.307963	+	0.951398i	$0.400353\pi$
$678$	14.5413	0.558456
$679$	64.5708	2.47800
$680$	2.28025	0.0874437
$681$	31.9621	1.22479
$682$	−18.5331	−0.709669
$683$	23.0371	0.881491	0.440745	−	0.897632i	$-0.354714\pi$
$683$	23.0371	0.881491	0.440745	+	0.897632i	$0.354714\pi$
$684$	−65.6784	−2.51128
$685$	−7.05521	−0.269566
$686$	−48.8799	−1.86625
$687$	29.4044	1.12185
$688$	−8.08889	−0.308386
$689$	−8.95554	−0.341179
$690$	−73.5875	−2.80143
$691$	−1.60678	−0.0611248	−0.0305624	−	0.999533i	$-0.509730\pi$
$691$	−1.60678	−0.0611248	−0.0305624	+	0.999533i	$0.509730\pi$
$692$	−23.5854	−0.896581
$693$	−86.6971	−3.29335
$694$	3.65073	0.138580
$695$	−28.8659	−1.09495
$696$	−1.36822	−0.0518622
$697$	−3.82764	−0.144982
$698$	34.4366	1.30344
$699$	25.3735	0.959714
$700$	−51.3163	−1.93958
$701$	−22.0578	−0.833113	−0.416556	−	0.909110i	$-0.636763\pi$
$701$	−22.0578	−0.833113	−0.416556	+	0.909110i	$0.636763\pi$
$702$	21.9590	0.828790
$703$	−96.3138	−3.63254
$704$	2.20507	0.0831066
$705$	96.6294	3.63927
$706$	−3.63947	−0.136973
$707$	−17.5781	−0.661093
$708$	−2.25640	−0.0848007
$709$	43.3629	1.62853	0.814265	−	0.580494i	$-0.197140\pi$
$709$	43.3629	1.62853	0.814265	+	0.580494i	$0.197140\pi$
$710$	28.6422	1.07492
$711$	67.0243	2.51361
$712$	10.7415	0.402554
$713$	47.3367	1.77277
$714$	−9.42567	−0.352747
$715$	11.4074	0.426614
$716$	11.6143	0.434048
$717$	−99.3001	−3.70843
$718$	−10.9806	−0.409791
$719$	−4.00375	−0.149315	−0.0746574	−	0.997209i	$-0.523786\pi$
$719$	−4.00375	−0.149315	−0.0746574	+	0.997209i	$0.523786\pi$
$720$	−31.5876	−1.17720
$721$	−44.8956	−1.67200
$722$	47.9198	1.78339
$723$	30.8394	1.14693
$724$	9.71441	0.361033
$725$	4.31730	0.160340
$726$	20.3829	0.756481
$727$	3.90785	0.144934	0.0724671	−	0.997371i	$-0.476913\pi$
$727$	3.90785	0.144934	0.0724671	+	0.997371i	$0.476913\pi$
$728$	6.43921	0.238653
$729$	85.5117	3.16710
$730$	42.5730	1.57570
$731$	4.68815	0.173397
$732$	−46.9309	−1.73461
$733$	−11.3044	−0.417537	−0.208769	−	0.977965i	$-0.566946\pi$
$733$	−11.3044	−0.417537	−0.208769	+	0.977965i	$0.566946\pi$
$734$	18.4995	0.682829
$735$	221.875	8.18397
$736$	−5.63211	−0.207602
$737$	9.97706	0.367510
$738$	53.0229	1.95180
$739$	6.86474	0.252524	0.126262	−	0.991997i	$-0.459702\pi$
$739$	6.86474	0.252524	0.126262	+	0.991997i	$0.459702\pi$
$740$	−46.3215	−1.70281
$741$	−35.7219	−1.31228
$742$	−33.3529	−1.22443
$743$	16.5180	0.605988	0.302994	−	0.952992i	$-0.402014\pi$
$743$	16.5180	0.605988	0.302994	+	0.952992i	$0.402014\pi$
$744$	27.9118	1.02330
$745$	7.16187	0.262390
$746$	−31.4944	−1.15309
$747$	119.839	4.38468
$748$	−1.27801	−0.0467286
$749$	15.0807	0.551038
$750$	71.5865	2.61397
$751$	−22.8918	−0.835335	−0.417667	−	0.908600i	$-0.637152\pi$
$751$	−22.8918	−0.835335	−0.417667	+	0.908600i	$0.637152\pi$
$752$	7.39565	0.269691
$753$	63.2026	2.30323
$754$	−0.541737	−0.0197289
$755$	59.8310	2.17747
$756$	81.7816	2.97437
$757$	−5.44374	−0.197856	−0.0989280	−	0.995095i	$-0.531541\pi$
$757$	−5.44374	−0.197856	−0.0989280	+	0.995095i	$0.531541\pi$
$758$	20.6367	0.749557
$759$	41.2435	1.49704
$760$	32.1846	1.16746
$761$	−13.4866	−0.488888	−0.244444	−	0.969663i	$-0.578605\pi$
$761$	−13.4866	−0.488888	−0.244444	+	0.969663i	$0.578605\pi$
$762$	41.4254	1.50068
$763$	−0.690144	−0.0249849
$764$	3.68678	0.133383
$765$	18.3075	0.661908
$766$	0.271368	0.00980492
$767$	−0.893407	−0.0322590
$768$	−3.32095	−0.119834
$769$	−28.0369	−1.01104	−0.505518	−	0.862816i	$-0.668698\pi$
$769$	−28.0369	−1.01104	−0.505518	+	0.862816i	$0.668698\pi$
$770$	42.4845	1.53103
$771$	33.9847	1.22393
$772$	−16.6958	−0.600894
$773$	1.85612	0.0667602	0.0333801	−	0.999443i	$-0.489373\pi$
$773$	1.85612	0.0667602	0.0333801	+	0.999443i	$0.489373\pi$
$774$	−64.9433	−2.33434
$775$	−88.0734	−3.16369
$776$	−13.1856	−0.473334
$777$	191.475	6.86911
$778$	−5.62628	−0.201712
$779$	−54.0251	−1.93565
$780$	−17.1802	−0.615150
$781$	−16.0530	−0.574423
$782$	3.26425	0.116729
$783$	−6.88036	−0.245884
$784$	16.9814	0.606480
$785$	−2.58960	−0.0924269
$786$	−16.7883	−0.598817
$787$	27.3194	0.973833	0.486916	−	0.873449i	$-0.338122\pi$
$787$	27.3194	0.973833	0.486916	+	0.873449i	$0.338122\pi$
$788$	21.3337	0.759983
$789$	−62.2709	−2.21690
$790$	−32.8441	−1.16854
$791$	21.4427	0.762414
$792$	17.7038	0.629078
$793$	−18.5820	−0.659865
$794$	−5.86700	−0.208212
$795$	88.9876	3.15607
$796$	0.425841	0.0150935
$797$	15.1568	0.536883	0.268442	−	0.963296i	$-0.413491\pi$
$797$	15.1568	0.536883	0.268442	+	0.963296i	$0.413491\pi$
$798$	−133.038	−4.70951
$799$	−4.28636	−0.151640
$800$	10.4790	0.370487
$801$	86.2402	3.04715
$802$	−22.0000	−0.776848
$803$	−23.8608	−0.842030
$804$	−15.0260	−0.529925
$805$	−108.512	−3.82456
$806$	11.0515	0.389273
$807$	−72.4467	−2.55024
$808$	3.58951	0.126278
$809$	−15.1847	−0.533864	−0.266932	−	0.963715i	$-0.586010\pi$
$809$	−15.1847	−0.533864	−0.266932	+	0.963715i	$0.586010\pi$
$810$	−123.435	−4.33708
$811$	−11.0006	−0.386282	−0.193141	−	0.981171i	$-0.561867\pi$
$811$	−11.0006	−0.386282	−0.193141	+	0.981171i	$0.561867\pi$
$812$	−2.01758	−0.0708033
$813$	−17.1063	−0.599944
$814$	25.9617	0.909957
$815$	−15.9951	−0.560283
$816$	1.92475	0.0673797
$817$	66.1708	2.31502
$818$	11.7639	0.411316
$819$	51.6985	1.80649
$820$	−25.9830	−0.907366
$821$	−3.93972	−0.137497	−0.0687485	−	0.997634i	$-0.521901\pi$
$821$	−3.93972	−0.137497	−0.0687485	+	0.997634i	$0.521901\pi$
$822$	−5.95527	−0.207714
$823$	1.27581	0.0444720	0.0222360	−	0.999753i	$-0.492921\pi$
$823$	1.27581	0.0444720	0.0222360	+	0.999753i	$0.492921\pi$
$824$	9.16782	0.319376
$825$	−76.7365	−2.67162
$826$	−3.32730	−0.115771
$827$	17.4357	0.606298	0.303149	−	0.952943i	$-0.401962\pi$
$827$	17.4357	0.606298	0.303149	+	0.952943i	$0.401962\pi$
$828$	−45.2185	−1.57145
$829$	−37.7928	−1.31260	−0.656300	−	0.754500i	$-0.727879\pi$
$829$	−37.7928	−1.31260	−0.656300	+	0.754500i	$0.727879\pi$
$830$	−58.7252	−2.03838
$831$	4.49079	0.155784
$832$	−1.31491	−0.0455862
$833$	−9.84208	−0.341008
$834$	−24.3656	−0.843710
$835$	−5.81305	−0.201169
$836$	−18.0384	−0.623872
$837$	140.360	4.85156
$838$	−23.4228	−0.809128
$839$	−6.47733	−0.223622	−0.111811	−	0.993729i	$-0.535665\pi$
$839$	−6.47733	−0.223622	−0.111811	+	0.993729i	$0.535665\pi$
$840$	−63.9839	−2.20765
$841$	−28.8303	−0.994147
$842$	6.61967	0.228129
$843$	44.0558	1.51736
$844$	9.77032	0.336308
$845$	44.3439	1.52548
$846$	59.3774	2.04144
$847$	30.0567	1.03276
$848$	6.81078	0.233883
$849$	−11.0720	−0.379991
$850$	−6.07338	−0.208315
$851$	−66.3105	−2.27310
$852$	24.1767	0.828281
$853$	4.62178	0.158247	0.0791234	−	0.996865i	$-0.474788\pi$
$853$	4.62178	0.158247	0.0791234	+	0.996865i	$0.474788\pi$
$854$	−69.2044	−2.36813
$855$	258.400	8.83711
$856$	−3.07953	−0.105256
$857$	13.8290	0.472389	0.236195	−	0.971706i	$-0.424100\pi$
$857$	13.8290	0.472389	0.236195	+	0.971706i	$0.424100\pi$
$858$	9.62895	0.328727
$859$	−38.1562	−1.30187	−0.650936	−	0.759132i	$-0.725624\pi$
$859$	−38.1562	−1.30187	−0.650936	+	0.759132i	$0.725624\pi$
$860$	31.8244	1.08520
$861$	107.403	3.66030
$862$	−20.2901	−0.691084
$863$	−3.22793	−0.109880	−0.0549400	−	0.998490i	$-0.517497\pi$
$863$	−3.22793	−0.109880	−0.0549400	+	0.998490i	$0.517497\pi$
$864$	−16.7001	−0.568147
$865$	92.7926	3.15504
$866$	−11.6244	−0.395014
$867$	55.3406	1.87946
$868$	41.1589	1.39703
$869$	18.4081	0.624452
$870$	5.38303	0.182502
$871$	−5.94944	−0.201589
$872$	0.140930	0.00477248
$873$	−105.863	−3.58292
$874$	46.0732	1.55845
$875$	105.562	3.56864
$876$	35.9357	1.21415
$877$	−35.0513	−1.18360	−0.591799	−	0.806086i	$-0.701582\pi$
$877$	−35.0513	−1.18360	−0.591799	+	0.806086i	$0.701582\pi$
$878$	15.4482	0.521351
$879$	36.2516	1.22274
$880$	−8.67546	−0.292450
$881$	11.0347	0.371768	0.185884	−	0.982572i	$-0.440485\pi$
$881$	11.0347	0.371768	0.185884	+	0.982572i	$0.440485\pi$
$882$	136.339	4.59077
$883$	−29.3223	−0.986775	−0.493388	−	0.869810i	$-0.664242\pi$
$883$	−29.3223	−0.986775	−0.493388	+	0.869810i	$0.664242\pi$
$884$	0.762092	0.0256319
$885$	8.87742	0.298411
$886$	−2.17969	−0.0732280
$887$	10.5221	0.353298	0.176649	−	0.984274i	$-0.443474\pi$
$887$	10.5221	0.353298	0.176649	+	0.984274i	$0.443474\pi$
$888$	−39.0997	−1.31210
$889$	61.0861	2.04876
$890$	−42.2606	−1.41658
$891$	69.1816	2.31767
$892$	22.6059	0.756901
$893$	−60.4997	−2.02455
$894$	6.04529	0.202185
$895$	−45.6947	−1.52740
$896$	−4.89708	−0.163600
$897$	−24.5940	−0.821168
$898$	6.37723	0.212811
$899$	−3.46274	−0.115489
$900$	84.1324	2.80441
$901$	−3.94738	−0.131506
$902$	14.5626	0.484883
$903$	−131.549	−4.37769
$904$	−4.37866	−0.145632
$905$	−38.2197	−1.27047
$906$	50.5030	1.67785
$907$	−8.10749	−0.269205	−0.134602	−	0.990900i	$-0.542976\pi$
$907$	−8.10749	−0.269205	−0.134602	+	0.990900i	$0.542976\pi$
$908$	−9.62439	−0.319397
$909$	28.8191	0.955869
$910$	−25.3340	−0.839814
$911$	10.4483	0.346168	0.173084	−	0.984907i	$-0.444627\pi$
$911$	10.4483	0.346168	0.173084	+	0.984907i	$0.444627\pi$
$912$	27.1669	0.899584
$913$	32.9136	1.08928
$914$	33.0766	1.09408
$915$	184.642	6.10406
$916$	−8.85423	−0.292552
$917$	−24.7560	−0.817516
$918$	9.67899	0.319454
$919$	33.2314	1.09620	0.548102	−	0.836412i	$-0.315351\pi$
$919$	33.2314	1.09620	0.548102	+	0.836412i	$0.315351\pi$
$920$	22.1586	0.730547
$921$	79.3251	2.61385
$922$	31.4728	1.03650
$923$	9.57262	0.315087
$924$	35.8609	1.17974
$925$	123.376	4.05657
$926$	0.128998	0.00423913
$927$	73.6057	2.41753
$928$	0.411997	0.0135245
$929$	−37.7761	−1.23939	−0.619696	−	0.784842i	$-0.712744\pi$
$929$	−37.7761	−1.23939	−0.619696	+	0.784842i	$0.712744\pi$
$930$	−109.814	−3.60096
$931$	−138.916	−4.55278
$932$	−7.64044	−0.250271
$933$	−51.6374	−1.69053
$934$	−20.0184	−0.655024
$935$	5.02811	0.164437
$936$	−10.5570	−0.345066
$937$	38.3128	1.25163	0.625813	−	0.779973i	$-0.284767\pi$
$937$	38.3128	1.25163	0.625813	+	0.779973i	$0.284767\pi$
$938$	−22.1574	−0.723464
$939$	67.8139	2.21302
$940$	−29.0969	−0.949037
$941$	30.8734	1.00644	0.503222	−	0.864157i	$-0.332148\pi$
$941$	30.8734	1.00644	0.503222	+	0.864157i	$0.332148\pi$
$942$	−2.18587	−0.0712195
$943$	−37.1954	−1.21125
$944$	0.679444	0.0221140
$945$	−321.756	−10.4667
$946$	−17.8365	−0.579916
$947$	11.2425	0.365331	0.182666	−	0.983175i	$-0.441527\pi$
$947$	11.2425	0.365331	0.182666	+	0.983175i	$0.441527\pi$
$948$	−27.7236	−0.900420
$949$	14.2285	0.461876
$950$	−85.7226	−2.78121
$951$	84.4020	2.73692
$952$	2.83824	0.0919880
$953$	−43.9184	−1.42266	−0.711329	−	0.702859i	$-0.751906\pi$
$953$	−43.9184	−1.42266	−0.711329	+	0.702859i	$0.751906\pi$
$954$	54.6817	1.77038
$955$	−14.5050	−0.469371
$956$	29.9011	0.967072
$957$	−3.01701	−0.0975262
$958$	34.1993	1.10493
$959$	−8.78167	−0.283575
$960$	13.0657	0.421694
$961$	39.6404	1.27872
$962$	−15.4813	−0.499136
$963$	−24.7247	−0.796741
$964$	−9.28632	−0.299092
$965$	65.6867	2.11453
$966$	−91.5948	−2.94702
$967$	−14.4483	−0.464626	−0.232313	−	0.972641i	$-0.574629\pi$
$967$	−14.4483	−0.464626	−0.232313	+	0.972641i	$0.574629\pi$
$968$	−6.13768	−0.197273
$969$	−15.7453	−0.505813
$970$	51.8764	1.66565
$971$	16.8237	0.539898	0.269949	−	0.962875i	$-0.412993\pi$
$971$	16.8237	0.539898	0.269949	+	0.962875i	$0.412993\pi$
$972$	−54.0910	−1.73497
$973$	−35.9295	−1.15185
$974$	18.7882	0.602013
$975$	45.7589	1.46546
$976$	14.1318	0.452347
$977$	34.8874	1.11615	0.558074	−	0.829791i	$-0.311541\pi$
$977$	34.8874	1.11615	0.558074	+	0.829791i	$0.311541\pi$
$978$	−13.5014	−0.431726
$979$	23.6857	0.756998
$980$	−66.8106	−2.13419
$981$	1.13148	0.0361254
$982$	36.6498	1.16954
$983$	−11.0553	−0.352608	−0.176304	−	0.984336i	$-0.556414\pi$
$983$	−11.0553	−0.352608	−0.176304	+	0.984336i	$0.556414\pi$
$984$	−21.9321	−0.699171
$985$	−83.9340	−2.67436
$986$	−0.238784	−0.00760444
$987$	120.275	3.82840
$988$	10.7565	0.342211
$989$	45.5575	1.44865
$990$	−69.6527	−2.21371
$991$	−33.8406	−1.07498	−0.537492	−	0.843269i	$-0.680628\pi$
$991$	−33.8406	−1.07498	−0.537492	+	0.843269i	$0.680628\pi$
$992$	−8.40478	−0.266852
$993$	−3.98017	−0.126307
$994$	35.6511	1.13078
$995$	−1.67540	−0.0531137
$996$	−49.5696	−1.57067
$997$	44.7664	1.41777	0.708884	−	0.705325i	$-0.249199\pi$
$997$	44.7664	1.41777	0.708884	+	0.705325i	$0.249199\pi$
$998$	−34.6700	−1.09746
$999$	−196.621	−6.22080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.3	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.3	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.32095 q^{3} +1.00000 q^{4} -3.93433 q^{5} -3.32095 q^{6} -4.89708 q^{7} +1.00000 q^{8} +8.02870 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.32095 q^{3} +1.00000 q^{4} -3.93433 q^{5} -3.32095 q^{6} -4.89708 q^{7} +1.00000 q^{8} +8.02870 q^{9} -3.93433 q^{10} +2.20507 q^{11} -3.32095 q^{12} -1.31491 q^{13} -4.89708 q^{14} +13.0657 q^{15} +1.00000 q^{16} -0.579578 q^{17} +8.02870 q^{18} -8.18045 q^{19} -3.93433 q^{20} +16.2630 q^{21} +2.20507 q^{22} -5.63211 q^{23} -3.32095 q^{24} +10.4790 q^{25} -1.31491 q^{26} -16.7001 q^{27} -4.89708 q^{28} +0.411997 q^{29} +13.0657 q^{30} -8.40478 q^{31} +1.00000 q^{32} -7.32291 q^{33} -0.579578 q^{34} +19.2668 q^{35} +8.02870 q^{36} +11.7737 q^{37} -8.18045 q^{38} +4.36674 q^{39} -3.93433 q^{40} +6.60418 q^{41} +16.2630 q^{42} -8.08889 q^{43} +2.20507 q^{44} -31.5876 q^{45} -5.63211 q^{46} +7.39565 q^{47} -3.32095 q^{48} +16.9814 q^{49} +10.4790 q^{50} +1.92475 q^{51} -1.31491 q^{52} +6.81078 q^{53} -16.7001 q^{54} -8.67546 q^{55} -4.89708 q^{56} +27.1669 q^{57} +0.411997 q^{58} +0.679444 q^{59} +13.0657 q^{60} +14.1318 q^{61} -8.40478 q^{62} -39.3172 q^{63} +1.00000 q^{64} +5.17328 q^{65} -7.32291 q^{66} +4.52461 q^{67} -0.579578 q^{68} +18.7039 q^{69} +19.2668 q^{70} -7.28007 q^{71} +8.02870 q^{72} -10.8209 q^{73} +11.7737 q^{74} -34.8001 q^{75} -8.18045 q^{76} -10.7984 q^{77} +4.36674 q^{78} +8.34809 q^{79} -3.93433 q^{80} +31.3739 q^{81} +6.60418 q^{82} +14.9263 q^{83} +16.2630 q^{84} +2.28025 q^{85} -8.08889 q^{86} -1.36822 q^{87} +2.20507 q^{88} +10.7415 q^{89} -31.5876 q^{90} +6.43921 q^{91} -5.63211 q^{92} +27.9118 q^{93} +7.39565 q^{94} +32.1846 q^{95} -3.32095 q^{96} -13.1856 q^{97} +16.9814 q^{98} +17.7038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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