LMFDB - Embedded newform 8038.2.a.a.1.12

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			1.12
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.54835 q^{3} +1.00000 q^{4} -2.91194 q^{5} -2.54835 q^{6} -2.80949 q^{7} +1.00000 q^{8} +3.49408 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.54835 q^{3} +1.00000 q^{4} -2.91194 q^{5} -2.54835 q^{6} -2.80949 q^{7} +1.00000 q^{8} +3.49408 q^{9} -2.91194 q^{10} -5.24844 q^{11} -2.54835 q^{12} +3.44810 q^{13} -2.80949 q^{14} +7.42065 q^{15} +1.00000 q^{16} +3.68555 q^{17} +3.49408 q^{18} +0.758216 q^{19} -2.91194 q^{20} +7.15957 q^{21} -5.24844 q^{22} -9.11047 q^{23} -2.54835 q^{24} +3.47941 q^{25} +3.44810 q^{26} -1.25909 q^{27} -2.80949 q^{28} -0.770972 q^{29} +7.42065 q^{30} +6.57099 q^{31} +1.00000 q^{32} +13.3749 q^{33} +3.68555 q^{34} +8.18108 q^{35} +3.49408 q^{36} -9.67014 q^{37} +0.758216 q^{38} -8.78696 q^{39} -2.91194 q^{40} +7.66312 q^{41} +7.15957 q^{42} +8.88867 q^{43} -5.24844 q^{44} -10.1746 q^{45} -9.11047 q^{46} +2.87603 q^{47} -2.54835 q^{48} +0.893244 q^{49} +3.47941 q^{50} -9.39207 q^{51} +3.44810 q^{52} +12.5683 q^{53} -1.25909 q^{54} +15.2832 q^{55} -2.80949 q^{56} -1.93220 q^{57} -0.770972 q^{58} -7.00747 q^{59} +7.42065 q^{60} -4.56717 q^{61} +6.57099 q^{62} -9.81660 q^{63} +1.00000 q^{64} -10.0407 q^{65} +13.3749 q^{66} +15.4899 q^{67} +3.68555 q^{68} +23.2166 q^{69} +8.18108 q^{70} +4.78809 q^{71} +3.49408 q^{72} +7.38903 q^{73} -9.67014 q^{74} -8.86675 q^{75} +0.758216 q^{76} +14.7454 q^{77} -8.78696 q^{78} -10.3207 q^{79} -2.91194 q^{80} -7.27363 q^{81} +7.66312 q^{82} +4.17899 q^{83} +7.15957 q^{84} -10.7321 q^{85} +8.88867 q^{86} +1.96470 q^{87} -5.24844 q^{88} -13.0865 q^{89} -10.1746 q^{90} -9.68741 q^{91} -9.11047 q^{92} -16.7452 q^{93} +2.87603 q^{94} -2.20788 q^{95} -2.54835 q^{96} -4.12871 q^{97} +0.893244 q^{98} -18.3385 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.54835	−1.47129	−0.735645	−	0.677367i	$-0.763121\pi$
$3$	−2.54835	−1.47129	−0.735645	+	0.677367i	$0.763121\pi$
$4$	1.00000	0.500000
$5$	−2.91194	−1.30226	−0.651130	−	0.758966i	$-0.725705\pi$
$5$	−2.91194	−1.30226	−0.651130	+	0.758966i	$0.725705\pi$
$6$	−2.54835	−1.04036
$7$	−2.80949	−1.06189	−0.530944	−	0.847407i	$-0.678163\pi$
$7$	−2.80949	−1.06189	−0.530944	+	0.847407i	$0.678163\pi$
$8$	1.00000	0.353553
$9$	3.49408	1.16469
$10$	−2.91194	−0.920837
$11$	−5.24844	−1.58246	−0.791232	−	0.611516i	$-0.790560\pi$
$11$	−5.24844	−1.58246	−0.791232	+	0.611516i	$0.790560\pi$
$12$	−2.54835	−0.735645
$13$	3.44810	0.956331	0.478165	−	0.878270i	$-0.341302\pi$
$13$	3.44810	0.956331	0.478165	+	0.878270i	$0.341302\pi$
$14$	−2.80949	−0.750868
$15$	7.42065	1.91600
$16$	1.00000	0.250000
$17$	3.68555	0.893877	0.446939	−	0.894565i	$-0.352514\pi$
$17$	3.68555	0.893877	0.446939	+	0.894565i	$0.352514\pi$
$18$	3.49408	0.823563
$19$	0.758216	0.173947	0.0869733	−	0.996211i	$-0.472281\pi$
$19$	0.758216	0.173947	0.0869733	+	0.996211i	$0.472281\pi$
$20$	−2.91194	−0.651130
$21$	7.15957	1.56235
$22$	−5.24844	−1.11897
$23$	−9.11047	−1.89966	−0.949832	−	0.312762i	$-0.898746\pi$
$23$	−9.11047	−1.89966	−0.949832	+	0.312762i	$0.898746\pi$
$24$	−2.54835	−0.520180
$25$	3.47941	0.695882
$26$	3.44810	0.676228
$27$	−1.25909	−0.242313
$28$	−2.80949	−0.530944
$29$	−0.770972	−0.143166	−0.0715829	−	0.997435i	$-0.522805\pi$
$29$	−0.770972	−0.143166	−0.0715829	+	0.997435i	$0.522805\pi$
$30$	7.42065	1.35482
$31$	6.57099	1.18018	0.590092	−	0.807336i	$-0.299091\pi$
$31$	6.57099	1.18018	0.590092	+	0.807336i	$0.299091\pi$
$32$	1.00000	0.176777
$33$	13.3749	2.32826
$34$	3.68555	0.632066
$35$	8.18108	1.38285
$36$	3.49408	0.582347
$37$	−9.67014	−1.58976	−0.794881	−	0.606766i	$-0.792467\pi$
$37$	−9.67014	−1.58976	−0.794881	+	0.606766i	$0.792467\pi$
$38$	0.758216	0.122999
$39$	−8.78696	−1.40704
$40$	−2.91194	−0.460419
$41$	7.66312	1.19678	0.598389	−	0.801206i	$-0.295808\pi$
$41$	7.66312	1.19678	0.598389	+	0.801206i	$0.295808\pi$
$42$	7.15957	1.10474
$43$	8.88867	1.35551	0.677754	−	0.735288i	$-0.262953\pi$
$43$	8.88867	1.35551	0.677754	+	0.735288i	$0.262953\pi$
$44$	−5.24844	−0.791232
$45$	−10.1746	−1.51673
$46$	−9.11047	−1.34326
$47$	2.87603	0.419511	0.209756	−	0.977754i	$-0.432733\pi$
$47$	2.87603	0.419511	0.209756	+	0.977754i	$0.432733\pi$
$48$	−2.54835	−0.367822
$49$	0.893244	0.127606
$50$	3.47941	0.492063
$51$	−9.39207	−1.31515
$52$	3.44810	0.478165
$53$	12.5683	1.72638	0.863192	−	0.504875i	$-0.168461\pi$
$53$	12.5683	1.72638	0.863192	+	0.504875i	$0.168461\pi$
$54$	−1.25909	−0.171341
$55$	15.2832	2.06078
$56$	−2.80949	−0.375434
$57$	−1.93220	−0.255926
$58$	−0.770972	−0.101234
$59$	−7.00747	−0.912294	−0.456147	−	0.889904i	$-0.650771\pi$
$59$	−7.00747	−0.912294	−0.456147	+	0.889904i	$0.650771\pi$
$60$	7.42065	0.958001
$61$	−4.56717	−0.584766	−0.292383	−	0.956301i	$-0.594448\pi$
$61$	−4.56717	−0.584766	−0.292383	+	0.956301i	$0.594448\pi$
$62$	6.57099	0.834517
$63$	−9.81660	−1.23677
$64$	1.00000	0.125000
$65$	−10.0407	−1.24539
$66$	13.3749	1.64633
$67$	15.4899	1.89239	0.946194	−	0.323600i	$-0.104893\pi$
$67$	15.4899	1.89239	0.946194	+	0.323600i	$0.104893\pi$
$68$	3.68555	0.446939
$69$	23.2166	2.79496
$70$	8.18108	0.977826
$71$	4.78809	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$71$	4.78809	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$72$	3.49408	0.411782
$73$	7.38903	0.864820	0.432410	−	0.901677i	$-0.357663\pi$
$73$	7.38903	0.864820	0.432410	+	0.901677i	$0.357663\pi$
$74$	−9.67014	−1.12413
$75$	−8.86675	−1.02384
$76$	0.758216	0.0869733
$77$	14.7454	1.68040
$78$	−8.78696	−0.994928
$79$	−10.3207	−1.16117	−0.580585	−	0.814200i	$-0.697176\pi$
$79$	−10.3207	−1.16117	−0.580585	+	0.814200i	$0.697176\pi$
$80$	−2.91194	−0.325565
$81$	−7.27363	−0.808182
$82$	7.66312	0.846250
$83$	4.17899	0.458704	0.229352	−	0.973344i	$-0.426339\pi$
$83$	4.17899	0.458704	0.229352	+	0.973344i	$0.426339\pi$
$84$	7.15957	0.781173
$85$	−10.7321	−1.16406
$86$	8.88867	0.958490
$87$	1.96470	0.210638
$88$	−5.24844	−0.559485
$89$	−13.0865	−1.38717	−0.693585	−	0.720374i	$-0.743970\pi$
$89$	−13.0865	−1.38717	−0.693585	+	0.720374i	$0.743970\pi$
$90$	−10.1746	−1.07249
$91$	−9.68741	−1.01552
$92$	−9.11047	−0.949832
$93$	−16.7452	−1.73639
$94$	2.87603	0.296639
$95$	−2.20788	−0.226524
$96$	−2.54835	−0.260090
$97$	−4.12871	−0.419207	−0.209603	−	0.977787i	$-0.567217\pi$
$97$	−4.12871	−0.419207	−0.209603	+	0.977787i	$0.567217\pi$
$98$	0.893244	0.0902312
$99$	−18.3385	−1.84309
$100$	3.47941	0.347941
$101$	6.59938	0.656663	0.328331	−	0.944563i	$-0.393514\pi$
$101$	6.59938	0.656663	0.328331	+	0.944563i	$0.393514\pi$
$102$	−9.39207	−0.929953
$103$	13.9914	1.37861	0.689306	−	0.724470i	$-0.257915\pi$
$103$	13.9914	1.37861	0.689306	+	0.724470i	$0.257915\pi$
$104$	3.44810	0.338114
$105$	−20.8482	−2.03458
$106$	12.5683	1.22074
$107$	−11.7705	−1.13790	−0.568950	−	0.822372i	$-0.692650\pi$
$107$	−11.7705	−1.13790	−0.568950	+	0.822372i	$0.692650\pi$
$108$	−1.25909	−0.121156
$109$	17.6326	1.68890	0.844449	−	0.535635i	$-0.179928\pi$
$109$	17.6326	1.68890	0.844449	+	0.535635i	$0.179928\pi$
$110$	15.2832	1.45719
$111$	24.6429	2.33900
$112$	−2.80949	−0.265472
$113$	15.3246	1.44161	0.720807	−	0.693136i	$-0.243771\pi$
$113$	15.3246	1.44161	0.720807	+	0.693136i	$0.243771\pi$
$114$	−1.93220	−0.180967
$115$	26.5292	2.47386
$116$	−0.770972	−0.0715829
$117$	12.0479	1.11383
$118$	−7.00747	−0.645089
$119$	−10.3545	−0.949197
$120$	7.42065	0.677409
$121$	16.5461	1.50419
$122$	−4.56717	−0.413492
$123$	−19.5283	−1.76081
$124$	6.57099	0.590092
$125$	4.42788	0.396041
$126$	−9.81660	−0.874532
$127$	−13.0196	−1.15531	−0.577653	−	0.816282i	$-0.696031\pi$
$127$	−13.0196	−1.15531	−0.577653	+	0.816282i	$0.696031\pi$
$128$	1.00000	0.0883883
$129$	−22.6514	−1.99435
$130$	−10.0407	−0.880625
$131$	−2.21894	−0.193870	−0.0969350	−	0.995291i	$-0.530904\pi$
$131$	−2.21894	−0.193870	−0.0969350	+	0.995291i	$0.530904\pi$
$132$	13.3749	1.16413
$133$	−2.13020	−0.184712
$134$	15.4899	1.33812
$135$	3.66641	0.315554
$136$	3.68555	0.316033
$137$	−10.1062	−0.863427	−0.431714	−	0.902011i	$-0.642091\pi$
$137$	−10.1062	−0.863427	−0.431714	+	0.902011i	$0.642091\pi$
$138$	23.2166	1.97633
$139$	−4.42630	−0.375434	−0.187717	−	0.982223i	$-0.560109\pi$
$139$	−4.42630	−0.375434	−0.187717	+	0.982223i	$0.560109\pi$
$140$	8.18108	0.691427
$141$	−7.32912	−0.617223
$142$	4.78809	0.401808
$143$	−18.0971	−1.51336
$144$	3.49408	0.291174
$145$	2.24502	0.186439
$146$	7.38903	0.611520
$147$	−2.27630	−0.187746
$148$	−9.67014	−0.794881
$149$	4.28220	0.350812	0.175406	−	0.984496i	$-0.443876\pi$
$149$	4.28220	0.350812	0.175406	+	0.984496i	$0.443876\pi$
$150$	−8.86675	−0.723967
$151$	2.45987	0.200181	0.100091	−	0.994978i	$-0.468087\pi$
$151$	2.45987	0.200181	0.100091	+	0.994978i	$0.468087\pi$
$152$	0.758216	0.0614994
$153$	12.8776	1.04109
$154$	14.7454	1.18822
$155$	−19.1343	−1.53691
$156$	−8.78696	−0.703520
$157$	−2.68855	−0.214570	−0.107285	−	0.994228i	$-0.534216\pi$
$157$	−2.68855	−0.214570	−0.107285	+	0.994228i	$0.534216\pi$
$158$	−10.3207	−0.821071
$159$	−32.0283	−2.54001
$160$	−2.91194	−0.230209
$161$	25.5958	2.01723
$162$	−7.27363	−0.571471
$163$	−8.19021	−0.641507	−0.320754	−	0.947163i	$-0.603936\pi$
$163$	−8.19021	−0.641507	−0.320754	+	0.947163i	$0.603936\pi$
$164$	7.66312	0.598389
$165$	−38.9468	−3.03200
$166$	4.17899	0.324353
$167$	−25.2134	−1.95108	−0.975538	−	0.219832i	$-0.929449\pi$
$167$	−25.2134	−1.95108	−0.975538	+	0.219832i	$0.929449\pi$
$168$	7.15957	0.552372
$169$	−1.11060	−0.0854311
$170$	−10.7321	−0.823115
$171$	2.64927	0.202595
$172$	8.88867	0.677754
$173$	16.2824	1.23793	0.618963	−	0.785420i	$-0.287553\pi$
$173$	16.2824	1.23793	0.618963	+	0.785420i	$0.287553\pi$
$174$	1.96470	0.148944
$175$	−9.77537	−0.738948
$176$	−5.24844	−0.395616
$177$	17.8575	1.34225
$178$	−13.0865	−0.980878
$179$	−2.38997	−0.178635	−0.0893173	−	0.996003i	$-0.528469\pi$
$179$	−2.38997	−0.178635	−0.0893173	+	0.996003i	$0.528469\pi$
$180$	−10.1746	−0.758367
$181$	−6.15511	−0.457506	−0.228753	−	0.973485i	$-0.573465\pi$
$181$	−6.15511	−0.457506	−0.228753	+	0.973485i	$0.573465\pi$
$182$	−9.68741	−0.718079
$183$	11.6387	0.860361
$184$	−9.11047	−0.671632
$185$	28.1589	2.07028
$186$	−16.7452	−1.22782
$187$	−19.3434	−1.41453
$188$	2.87603	0.209756
$189$	3.53742	0.257309
$190$	−2.20788	−0.160176
$191$	−9.67821	−0.700291	−0.350145	−	0.936695i	$-0.613868\pi$
$191$	−9.67821	−0.700291	−0.350145	+	0.936695i	$0.613868\pi$
$192$	−2.54835	−0.183911
$193$	2.48869	0.179140	0.0895700	−	0.995981i	$-0.471451\pi$
$193$	2.48869	0.179140	0.0895700	+	0.995981i	$0.471451\pi$
$194$	−4.12871	−0.296424
$195$	25.5871	1.83233
$196$	0.893244	0.0638031
$197$	7.17015	0.510852	0.255426	−	0.966829i	$-0.417784\pi$
$197$	7.17015	0.510852	0.255426	+	0.966829i	$0.417784\pi$
$198$	−18.3385	−1.30326
$199$	−7.78506	−0.551868	−0.275934	−	0.961177i	$-0.588987\pi$
$199$	−7.78506	−0.551868	−0.275934	+	0.961177i	$0.588987\pi$
$200$	3.47941	0.246031
$201$	−39.4736	−2.78425
$202$	6.59938	0.464331
$203$	2.16604	0.152026
$204$	−9.39207	−0.657576
$205$	−22.3146	−1.55852
$206$	13.9914	0.974826
$207$	−31.8327	−2.21253
$208$	3.44810	0.239083
$209$	−3.97945	−0.275264
$210$	−20.8482	−1.43867
$211$	12.6525	0.871036	0.435518	−	0.900180i	$-0.356565\pi$
$211$	12.6525	0.871036	0.435518	+	0.900180i	$0.356565\pi$
$212$	12.5683	0.863192
$213$	−12.2017	−0.836049
$214$	−11.7705	−0.804616
$215$	−25.8833	−1.76523
$216$	−1.25909	−0.0856705
$217$	−18.4611	−1.25322
$218$	17.6326	1.19423
$219$	−18.8298	−1.27240
$220$	15.2832	1.03039
$221$	12.7081	0.854842
$222$	24.6429	1.65392
$223$	−17.4679	−1.16974	−0.584870	−	0.811127i	$-0.698854\pi$
$223$	−17.4679	−1.16974	−0.584870	+	0.811127i	$0.698854\pi$
$224$	−2.80949	−0.187717
$225$	12.1573	0.810489
$226$	15.3246	1.01938
$227$	−23.5458	−1.56279	−0.781394	−	0.624038i	$-0.785491\pi$
$227$	−23.5458	−1.56279	−0.781394	+	0.624038i	$0.785491\pi$
$228$	−1.93220	−0.127963
$229$	27.5986	1.82376	0.911882	−	0.410453i	$-0.134629\pi$
$229$	27.5986	1.82376	0.911882	+	0.410453i	$0.134629\pi$
$230$	26.5292	1.74928
$231$	−37.5765	−2.47235
$232$	−0.770972	−0.0506168
$233$	−2.80266	−0.183608	−0.0918041	−	0.995777i	$-0.529263\pi$
$233$	−2.80266	−0.183608	−0.0918041	+	0.995777i	$0.529263\pi$
$234$	12.0479	0.787599
$235$	−8.37482	−0.546313
$236$	−7.00747	−0.456147
$237$	26.3008	1.70842
$238$	−10.3545	−0.671184
$239$	1.11698	0.0722515	0.0361257	−	0.999347i	$-0.488498\pi$
$239$	1.11698	0.0722515	0.0361257	+	0.999347i	$0.488498\pi$
$240$	7.42065	0.479001
$241$	−12.6950	−0.817756	−0.408878	−	0.912589i	$-0.634080\pi$
$241$	−12.6950	−0.817756	−0.408878	+	0.912589i	$0.634080\pi$
$242$	16.5461	1.06362
$243$	22.3130	1.43138
$244$	−4.56717	−0.292383
$245$	−2.60107	−0.166176
$246$	−19.5283	−1.24508
$247$	2.61440	0.166350
$248$	6.57099	0.417258
$249$	−10.6495	−0.674887
$250$	4.42788	0.280043
$251$	6.57018	0.414706	0.207353	−	0.978266i	$-0.433515\pi$
$251$	6.57018	0.414706	0.207353	+	0.978266i	$0.433515\pi$
$252$	−9.81660	−0.618387
$253$	47.8157	3.00615
$254$	−13.0196	−0.816925
$255$	27.3492	1.71267
$256$	1.00000	0.0625000
$257$	−22.3663	−1.39517	−0.697586	−	0.716502i	$-0.745742\pi$
$257$	−22.3663	−1.39517	−0.697586	+	0.716502i	$0.745742\pi$
$258$	−22.6514	−1.41022
$259$	27.1682	1.68815
$260$	−10.0407	−0.622696
$261$	−2.69384	−0.166744
$262$	−2.21894	−0.137087
$263$	−22.0317	−1.35853	−0.679267	−	0.733891i	$-0.737702\pi$
$263$	−22.0317	−1.35853	−0.679267	+	0.733891i	$0.737702\pi$
$264$	13.3749	0.823165
$265$	−36.5981	−2.24820
$266$	−2.13020	−0.130611
$267$	33.3491	2.04093
$268$	15.4899	0.946194
$269$	7.80561	0.475916	0.237958	−	0.971275i	$-0.423522\pi$
$269$	7.80561	0.475916	0.237958	+	0.971275i	$0.423522\pi$
$270$	3.66641	0.223131
$271$	−8.87914	−0.539369	−0.269685	−	0.962949i	$-0.586919\pi$
$271$	−8.87914	−0.539369	−0.269685	+	0.962949i	$0.586919\pi$
$272$	3.68555	0.223469
$273$	24.6869	1.49412
$274$	−10.1062	−0.610535
$275$	−18.2615	−1.10121
$276$	23.2166	1.39748
$277$	−6.57444	−0.395020	−0.197510	−	0.980301i	$-0.563285\pi$
$277$	−6.57444	−0.395020	−0.197510	+	0.980301i	$0.563285\pi$
$278$	−4.42630	−0.265472
$279$	22.9596	1.37455
$280$	8.18108	0.488913
$281$	−2.39747	−0.143021	−0.0715104	−	0.997440i	$-0.522782\pi$
$281$	−2.39747	−0.143021	−0.0715104	+	0.997440i	$0.522782\pi$
$282$	−7.32912	−0.436443
$283$	−14.5571	−0.865327	−0.432663	−	0.901556i	$-0.642426\pi$
$283$	−14.5571	−0.865327	−0.432663	+	0.901556i	$0.642426\pi$
$284$	4.78809	0.284121
$285$	5.62645	0.333282
$286$	−18.0971	−1.07011
$287$	−21.5295	−1.27084
$288$	3.49408	0.205891
$289$	−3.41673	−0.200984
$290$	2.24502	0.131832
$291$	10.5214	0.616774
$292$	7.38903	0.432410
$293$	−5.43372	−0.317441	−0.158721	−	0.987324i	$-0.550737\pi$
$293$	−5.43372	−0.317441	−0.158721	+	0.987324i	$0.550737\pi$
$294$	−2.27630	−0.132756
$295$	20.4053	1.18804
$296$	−9.67014	−0.562065
$297$	6.60828	0.383451
$298$	4.28220	0.248061
$299$	−31.4138	−1.81671
$300$	−8.86675	−0.511922
$301$	−24.9726	−1.43940
$302$	2.45987	0.141550
$303$	−16.8175	−0.966142
$304$	0.758216	0.0434866
$305$	13.2993	0.761518
$306$	12.8776	0.736164
$307$	−22.6084	−1.29033	−0.645164	−	0.764044i	$-0.723211\pi$
$307$	−22.6084	−1.29033	−0.645164	+	0.764044i	$0.723211\pi$
$308$	14.7454	0.840200
$309$	−35.6549	−2.02834
$310$	−19.1343	−1.08676
$311$	−12.4102	−0.703719	−0.351859	−	0.936053i	$-0.614450\pi$
$311$	−12.4102	−0.703719	−0.351859	+	0.936053i	$0.614450\pi$
$312$	−8.78696	−0.497464
$313$	10.9788	0.620559	0.310279	−	0.950645i	$-0.399577\pi$
$313$	10.9788	0.620559	0.310279	+	0.950645i	$0.399577\pi$
$314$	−2.68855	−0.151724
$315$	28.5854	1.61060
$316$	−10.3207	−0.580585
$317$	−19.5518	−1.09814	−0.549068	−	0.835778i	$-0.685017\pi$
$317$	−19.5518	−1.09814	−0.549068	+	0.835778i	$0.685017\pi$
$318$	−32.0283	−1.79606
$319$	4.04640	0.226555
$320$	−2.91194	−0.162783
$321$	29.9954	1.67418
$322$	25.5958	1.42640
$323$	2.79444	0.155487
$324$	−7.27363	−0.404091
$325$	11.9973	0.665493
$326$	−8.19021	−0.453614
$327$	−44.9341	−2.48486
$328$	7.66312	0.423125
$329$	−8.08017	−0.445474
$330$	−38.9468	−2.14395
$331$	19.1227	1.05108	0.525540	−	0.850769i	$-0.323863\pi$
$331$	19.1227	1.05108	0.525540	+	0.850769i	$0.323863\pi$
$332$	4.17899	0.229352
$333$	−33.7883	−1.85159
$334$	−25.2134	−1.37962
$335$	−45.1056	−2.46438
$336$	7.15957	0.390586
$337$	19.7580	1.07628	0.538142	−	0.842854i	$-0.319126\pi$
$337$	19.7580	1.07628	0.538142	+	0.842854i	$0.319126\pi$
$338$	−1.11060	−0.0604089
$339$	−39.0524	−2.12103
$340$	−10.7321	−0.582030
$341$	−34.4874	−1.86760
$342$	2.64927	0.143256
$343$	17.1569	0.926385
$344$	8.88867	0.479245
$345$	−67.6055	−3.63976
$346$	16.2824	0.875345
$347$	−5.60003	−0.300625	−0.150313	−	0.988639i	$-0.548028\pi$
$347$	−5.60003	−0.300625	−0.150313	+	0.988639i	$0.548028\pi$
$348$	1.96470	0.105319
$349$	1.74609	0.0934662	0.0467331	−	0.998907i	$-0.485119\pi$
$349$	1.74609	0.0934662	0.0467331	+	0.998907i	$0.485119\pi$
$350$	−9.77537	−0.522515
$351$	−4.34148	−0.231731
$352$	−5.24844	−0.279743
$353$	−11.0457	−0.587904	−0.293952	−	0.955820i	$-0.594971\pi$
$353$	−11.0457	−0.587904	−0.293952	+	0.955820i	$0.594971\pi$
$354$	17.8575	0.949114
$355$	−13.9426	−0.739999
$356$	−13.0865	−0.693585
$357$	26.3869	1.39654
$358$	−2.38997	−0.126314
$359$	−6.15081	−0.324627	−0.162314	−	0.986739i	$-0.551896\pi$
$359$	−6.15081	−0.324627	−0.162314	+	0.986739i	$0.551896\pi$
$360$	−10.1746	−0.536247
$361$	−18.4251	−0.969743
$362$	−6.15511	−0.323505
$363$	−42.1653	−2.21310
$364$	−9.68741	−0.507758
$365$	−21.5164	−1.12622
$366$	11.6387	0.608367
$367$	23.1995	1.21101	0.605503	−	0.795843i	$-0.292972\pi$
$367$	23.1995	1.21101	0.605503	+	0.795843i	$0.292972\pi$
$368$	−9.11047	−0.474916
$369$	26.7756	1.39388
$370$	28.1589	1.46391
$371$	−35.3105	−1.83323
$372$	−16.7452	−0.868197
$373$	1.96223	0.101600	0.0508001	−	0.998709i	$-0.483823\pi$
$373$	1.96223	0.101600	0.0508001	+	0.998709i	$0.483823\pi$
$374$	−19.3434	−1.00022
$375$	−11.2838	−0.582691
$376$	2.87603	0.148320
$377$	−2.65839	−0.136914
$378$	3.53742	0.181945
$379$	25.4477	1.30716	0.653580	−	0.756857i	$-0.273266\pi$
$379$	25.4477	1.30716	0.653580	+	0.756857i	$0.273266\pi$
$380$	−2.20788	−0.113262
$381$	33.1786	1.69979
$382$	−9.67821	−0.495180
$383$	−28.4468	−1.45356	−0.726781	−	0.686870i	$-0.758984\pi$
$383$	−28.4468	−1.45356	−0.726781	+	0.686870i	$0.758984\pi$
$384$	−2.54835	−0.130045
$385$	−42.9379	−2.18832
$386$	2.48869	0.126671
$387$	31.0577	1.57875
$388$	−4.12871	−0.209603
$389$	−19.8980	−1.00887	−0.504433	−	0.863451i	$-0.668299\pi$
$389$	−19.8980	−1.00887	−0.504433	+	0.863451i	$0.668299\pi$
$390$	25.5871	1.29565
$391$	−33.5771	−1.69807
$392$	0.893244	0.0451156
$393$	5.65464	0.285239
$394$	7.17015	0.361227
$395$	30.0533	1.51215
$396$	−18.3385	−0.921543
$397$	28.8905	1.44997	0.724986	−	0.688764i	$-0.241846\pi$
$397$	28.8905	1.44997	0.724986	+	0.688764i	$0.241846\pi$
$398$	−7.78506	−0.390230
$399$	5.42849	0.271765
$400$	3.47941	0.173970
$401$	1.62547	0.0811723	0.0405862	−	0.999176i	$-0.487077\pi$
$401$	1.62547	0.0811723	0.0405862	+	0.999176i	$0.487077\pi$
$402$	−39.4736	−1.96876
$403$	22.6574	1.12865
$404$	6.59938	0.328331
$405$	21.1804	1.05246
$406$	2.16604	0.107499
$407$	50.7531	2.51574
$408$	−9.39207	−0.464977
$409$	−23.7408	−1.17391	−0.586955	−	0.809620i	$-0.699673\pi$
$409$	−23.7408	−1.17391	−0.586955	+	0.809620i	$0.699673\pi$
$410$	−22.3146	−1.10204
$411$	25.7540	1.27035
$412$	13.9914	0.689306
$413$	19.6874	0.968754
$414$	−31.8327	−1.56449
$415$	−12.1690	−0.597352
$416$	3.44810	0.169057
$417$	11.2798	0.552373
$418$	−3.97945	−0.194641
$419$	0.564623	0.0275836	0.0137918	−	0.999905i	$-0.495610\pi$
$419$	0.564623	0.0275836	0.0137918	+	0.999905i	$0.495610\pi$
$420$	−20.8482	−1.01729
$421$	5.50142	0.268123	0.134062	−	0.990973i	$-0.457198\pi$
$421$	5.50142	0.268123	0.134062	+	0.990973i	$0.457198\pi$
$422$	12.6525	0.615915
$423$	10.0491	0.488603
$424$	12.5683	0.610369
$425$	12.8235	0.622033
$426$	−12.2017	−0.591176
$427$	12.8314	0.620956
$428$	−11.7705	−0.568950
$429$	46.1178	2.22659
$430$	−25.8833	−1.24820
$431$	38.9800	1.87760	0.938801	−	0.344459i	$-0.111938\pi$
$431$	38.9800	1.87760	0.938801	+	0.344459i	$0.111938\pi$
$432$	−1.25909	−0.0605782
$433$	−16.3502	−0.785739	−0.392870	−	0.919594i	$-0.628518\pi$
$433$	−16.3502	−0.785739	−0.392870	+	0.919594i	$0.628518\pi$
$434$	−18.4611	−0.886163
$435$	−5.72111	−0.274306
$436$	17.6326	0.844449
$437$	−6.90770	−0.330440
$438$	−18.8298	−0.899724
$439$	35.8690	1.71193	0.855967	−	0.517030i	$-0.172963\pi$
$439$	35.8690	1.71193	0.855967	+	0.517030i	$0.172963\pi$
$440$	15.2832	0.728596
$441$	3.12107	0.148622
$442$	12.7081	0.604465
$443$	−16.3456	−0.776605	−0.388302	−	0.921532i	$-0.626938\pi$
$443$	−16.3456	−0.776605	−0.388302	+	0.921532i	$0.626938\pi$
$444$	24.6429	1.16950
$445$	38.1073	1.80646
$446$	−17.4679	−0.827131
$447$	−10.9125	−0.516146
$448$	−2.80949	−0.132736
$449$	9.56177	0.451248	0.225624	−	0.974215i	$-0.427558\pi$
$449$	9.56177	0.451248	0.225624	+	0.974215i	$0.427558\pi$
$450$	12.1573	0.573102
$451$	−40.2194	−1.89386
$452$	15.3246	0.720807
$453$	−6.26860	−0.294525
$454$	−23.5458	−1.10506
$455$	28.2092	1.32247
$456$	−1.93220	−0.0904834
$457$	13.9508	0.652590	0.326295	−	0.945268i	$-0.394200\pi$
$457$	13.9508	0.652590	0.326295	+	0.945268i	$0.394200\pi$
$458$	27.5986	1.28960
$459$	−4.64046	−0.216598
$460$	26.5292	1.23693
$461$	21.9703	1.02326	0.511628	−	0.859207i	$-0.329042\pi$
$461$	21.9703	1.02326	0.511628	+	0.859207i	$0.329042\pi$
$462$	−37.5765	−1.74822
$463$	−12.4914	−0.580525	−0.290263	−	0.956947i	$-0.593743\pi$
$463$	−12.4914	−0.580525	−0.290263	+	0.956947i	$0.593743\pi$
$464$	−0.770972	−0.0357915
$465$	48.7610	2.26124
$466$	−2.80266	−0.129831
$467$	33.0450	1.52914	0.764571	−	0.644539i	$-0.222951\pi$
$467$	33.0450	1.52914	0.764571	+	0.644539i	$0.222951\pi$
$468$	12.0479	0.556917
$469$	−43.5186	−2.00950
$470$	−8.37482	−0.386302
$471$	6.85136	0.315694
$472$	−7.00747	−0.322545
$473$	−46.6516	−2.14504
$474$	26.3008	1.20803
$475$	2.63814	0.121046
$476$	−10.3545	−0.474599
$477$	43.9146	2.01071
$478$	1.11698	0.0510895
$479$	16.9312	0.773606	0.386803	−	0.922162i	$-0.373579\pi$
$479$	16.9312	0.773606	0.386803	+	0.922162i	$0.373579\pi$
$480$	7.42065	0.338705
$481$	−33.3436	−1.52034
$482$	−12.6950	−0.578241
$483$	−65.2270	−2.96793
$484$	16.5461	0.752096
$485$	12.0226	0.545916
$486$	22.3130	1.01214
$487$	13.2230	0.599191	0.299596	−	0.954066i	$-0.403148\pi$
$487$	13.2230	0.599191	0.299596	+	0.954066i	$0.403148\pi$
$488$	−4.56717	−0.206746
$489$	20.8715	0.943843
$490$	−2.60107	−0.117505
$491$	4.84397	0.218605	0.109303	−	0.994009i	$-0.465138\pi$
$491$	4.84397	0.218605	0.109303	+	0.994009i	$0.465138\pi$
$492$	−19.5283	−0.880404
$493$	−2.84145	−0.127973
$494$	2.61440	0.117628
$495$	53.4006	2.40018
$496$	6.57099	0.295046
$497$	−13.4521	−0.603409
$498$	−10.6495	−0.477217
$499$	−14.5967	−0.653437	−0.326718	−	0.945122i	$-0.605943\pi$
$499$	−14.5967	−0.653437	−0.326718	+	0.945122i	$0.605943\pi$
$500$	4.42788	0.198021
$501$	64.2527	2.87060
$502$	6.57018	0.293241
$503$	6.80344	0.303350	0.151675	−	0.988430i	$-0.451533\pi$
$503$	6.80344	0.303350	0.151675	+	0.988430i	$0.451533\pi$
$504$	−9.81660	−0.437266
$505$	−19.2170	−0.855146
$506$	47.8157	2.12567
$507$	2.83021	0.125694
$508$	−13.0196	−0.577653
$509$	13.6860	0.606619	0.303310	−	0.952892i	$-0.401908\pi$
$509$	13.6860	0.606619	0.303310	+	0.952892i	$0.401908\pi$
$510$	27.3492	1.21104
$511$	−20.7594	−0.918343
$512$	1.00000	0.0441942
$513$	−0.954665	−0.0421495
$514$	−22.3663	−0.986535
$515$	−40.7421	−1.79531
$516$	−22.6514	−0.997173
$517$	−15.0946	−0.663862
$518$	27.1682	1.19370
$519$	−41.4931	−1.82135
$520$	−10.0407	−0.440312
$521$	−33.2813	−1.45808	−0.729041	−	0.684470i	$-0.760034\pi$
$521$	−33.2813	−1.45808	−0.729041	+	0.684470i	$0.760034\pi$
$522$	−2.69384	−0.117906
$523$	−0.694540	−0.0303701	−0.0151851	−	0.999885i	$-0.504834\pi$
$523$	−0.694540	−0.0303701	−0.0151851	+	0.999885i	$0.504834\pi$
$524$	−2.21894	−0.0969350
$525$	24.9111	1.08721
$526$	−22.0317	−0.960629
$527$	24.2177	1.05494
$528$	13.3749	0.582066
$529$	60.0006	2.60872
$530$	−36.5981	−1.58972
$531$	−24.4847	−1.06254
$532$	−2.13020	−0.0923559
$533$	26.4232	1.14452
$534$	33.3491	1.44316
$535$	34.2751	1.48184
$536$	15.4899	0.669060
$537$	6.09047	0.262823
$538$	7.80561	0.336524
$539$	−4.68813	−0.201932
$540$	3.66641	0.157777
$541$	−7.60342	−0.326897	−0.163448	−	0.986552i	$-0.552262\pi$
$541$	−7.60342	−0.326897	−0.163448	+	0.986552i	$0.552262\pi$
$542$	−8.87914	−0.381392
$543$	15.6854	0.673123
$544$	3.68555	0.158017
$545$	−51.3452	−2.19939
$546$	24.6869	1.05650
$547$	−10.3533	−0.442676	−0.221338	−	0.975197i	$-0.571042\pi$
$547$	−10.3533	−0.442676	−0.221338	+	0.975197i	$0.571042\pi$
$548$	−10.1062	−0.431714
$549$	−15.9581	−0.681074
$550$	−18.2615	−0.778671
$551$	−0.584563	−0.0249032
$552$	23.2166	0.988166
$553$	28.9959	1.23303
$554$	−6.57444	−0.279321
$555$	−71.7587	−3.04599
$556$	−4.42630	−0.187717
$557$	−7.43166	−0.314890	−0.157445	−	0.987528i	$-0.550326\pi$
$557$	−7.43166	−0.314890	−0.157445	+	0.987528i	$0.550326\pi$
$558$	22.9596	0.971957
$559$	30.6490	1.29632
$560$	8.18108	0.345714
$561$	49.2937	2.08118
$562$	−2.39747	−0.101131
$563$	−13.4018	−0.564819	−0.282409	−	0.959294i	$-0.591134\pi$
$563$	−13.4018	−0.564819	−0.282409	+	0.959294i	$0.591134\pi$
$564$	−7.32912	−0.308612
$565$	−44.6243	−1.87736
$566$	−14.5571	−0.611878
$567$	20.4352	0.858198
$568$	4.78809	0.200904
$569$	17.1088	0.717239	0.358619	−	0.933484i	$-0.383248\pi$
$569$	17.1088	0.717239	0.358619	+	0.933484i	$0.383248\pi$
$570$	5.62645	0.235666
$571$	28.1568	1.17832	0.589162	−	0.808015i	$-0.299458\pi$
$571$	28.1568	1.17832	0.589162	+	0.808015i	$0.299458\pi$
$572$	−18.0971	−0.756680
$573$	24.6635	1.03033
$574$	−21.5295	−0.898623
$575$	−31.6990	−1.32194
$576$	3.49408	0.145587
$577$	−13.3968	−0.557714	−0.278857	−	0.960333i	$-0.589956\pi$
$577$	−13.3968	−0.557714	−0.278857	+	0.960333i	$0.589956\pi$
$578$	−3.41673	−0.142117
$579$	−6.34205	−0.263567
$580$	2.24502	0.0932196
$581$	−11.7408	−0.487092
$582$	10.5214	0.436125
$583$	−65.9638	−2.73194
$584$	7.38903	0.305760
$585$	−35.0829	−1.45050
$586$	−5.43372	−0.224465
$587$	−27.0882	−1.11805	−0.559025	−	0.829151i	$-0.688824\pi$
$587$	−27.0882	−1.11805	−0.559025	+	0.829151i	$0.688824\pi$
$588$	−2.27630	−0.0938729
$589$	4.98223	0.205289
$590$	20.4053	0.840074
$591$	−18.2720	−0.751611
$592$	−9.67014	−0.397440
$593$	−27.7582	−1.13989	−0.569946	−	0.821682i	$-0.693036\pi$
$593$	−27.7582	−1.13989	−0.569946	+	0.821682i	$0.693036\pi$
$594$	6.60828	0.271141
$595$	30.1518	1.23610
$596$	4.28220	0.175406
$597$	19.8390	0.811958
$598$	−31.4138	−1.28461
$599$	14.4760	0.591471	0.295736	−	0.955270i	$-0.404435\pi$
$599$	14.4760	0.591471	0.295736	+	0.955270i	$0.404435\pi$
$600$	−8.86675	−0.361983
$601$	−2.51614	−0.102635	−0.0513177	−	0.998682i	$-0.516342\pi$
$601$	−2.51614	−0.102635	−0.0513177	+	0.998682i	$0.516342\pi$
$602$	−24.9726	−1.01781
$603$	54.1229	2.20405
$604$	2.45987	0.100091
$605$	−48.1813	−1.95885
$606$	−16.8175	−0.683165
$607$	19.2920	0.783038	0.391519	−	0.920170i	$-0.371950\pi$
$607$	19.2920	0.783038	0.391519	+	0.920170i	$0.371950\pi$
$608$	0.758216	0.0307497
$609$	−5.51982	−0.223674
$610$	13.2993	0.538474
$611$	9.91682	0.401192
$612$	12.8776	0.520547
$613$	29.1665	1.17802	0.589012	−	0.808124i	$-0.299517\pi$
$613$	29.1665	1.17802	0.589012	+	0.808124i	$0.299517\pi$
$614$	−22.6084	−0.912399
$615$	56.8653	2.29303
$616$	14.7454	0.594111
$617$	27.4441	1.10486	0.552429	−	0.833560i	$-0.313701\pi$
$617$	27.4441	1.10486	0.552429	+	0.833560i	$0.313701\pi$
$618$	−35.6549	−1.43425
$619$	−6.09562	−0.245004	−0.122502	−	0.992468i	$-0.539092\pi$
$619$	−6.09562	−0.245004	−0.122502	+	0.992468i	$0.539092\pi$
$620$	−19.1343	−0.768454
$621$	11.4709	0.460313
$622$	−12.4102	−0.497604
$623$	36.7665	1.47302
$624$	−8.78696	−0.351760
$625$	−30.2908	−1.21163
$626$	10.9788	0.438801
$627$	10.1410	0.404993
$628$	−2.68855	−0.107285
$629$	−35.6398	−1.42105
$630$	28.5854	1.13887
$631$	34.4017	1.36951	0.684754	−	0.728774i	$-0.259910\pi$
$631$	34.4017	1.36951	0.684754	+	0.728774i	$0.259910\pi$
$632$	−10.3207	−0.410536
$633$	−32.2431	−1.28155
$634$	−19.5518	−0.776500
$635$	37.9124	1.50451
$636$	−32.0283	−1.27001
$637$	3.07999	0.122034
$638$	4.04640	0.160198
$639$	16.7300	0.661828
$640$	−2.91194	−0.115105
$641$	−0.168567	−0.00665798	−0.00332899	−	0.999994i	$-0.501060\pi$
$641$	−0.168567	−0.00665798	−0.00332899	+	0.999994i	$0.501060\pi$
$642$	29.9954	1.18382
$643$	−48.3512	−1.90678	−0.953392	−	0.301734i	$-0.902435\pi$
$643$	−48.3512	−1.90678	−0.953392	+	0.301734i	$0.902435\pi$
$644$	25.5958	1.00861
$645$	65.9596	2.59716
$646$	2.79444	0.109946
$647$	−39.8041	−1.56486	−0.782430	−	0.622739i	$-0.786020\pi$
$647$	−39.8041	−1.56486	−0.782430	+	0.622739i	$0.786020\pi$
$648$	−7.27363	−0.285735
$649$	36.7783	1.44367
$650$	11.9973	0.470575
$651$	47.0454	1.84386
$652$	−8.19021	−0.320754
$653$	−5.48337	−0.214581	−0.107290	−	0.994228i	$-0.534217\pi$
$653$	−5.48337	−0.214581	−0.107290	+	0.994228i	$0.534217\pi$
$654$	−44.9341	−1.75706
$655$	6.46143	0.252469
$656$	7.66312	0.299195
$657$	25.8179	1.00725
$658$	−8.08017	−0.314998
$659$	−9.35737	−0.364511	−0.182256	−	0.983251i	$-0.558340\pi$
$659$	−9.35737	−0.364511	−0.182256	+	0.983251i	$0.558340\pi$
$660$	−38.9468	−1.51600
$661$	16.1211	0.627038	0.313519	−	0.949582i	$-0.398492\pi$
$661$	16.1211	0.627038	0.313519	+	0.949582i	$0.398492\pi$
$662$	19.1227	0.743226
$663$	−32.3848	−1.25772
$664$	4.17899	0.162176
$665$	6.20302	0.240543
$666$	−33.7883	−1.30927
$667$	7.02391	0.271967
$668$	−25.2134	−0.975538
$669$	44.5144	1.72103
$670$	−45.1056	−1.74258
$671$	23.9705	0.925371
$672$	7.15957	0.276186
$673$	1.65236	0.0636938	0.0318469	−	0.999493i	$-0.489861\pi$
$673$	1.65236	0.0636938	0.0318469	+	0.999493i	$0.489861\pi$
$674$	19.7580	0.761048
$675$	−4.38090	−0.168621
$676$	−1.11060	−0.0427156
$677$	−20.7556	−0.797703	−0.398852	−	0.917015i	$-0.630591\pi$
$677$	−20.7556	−0.797703	−0.398852	+	0.917015i	$0.630591\pi$
$678$	−39.0524	−1.49980
$679$	11.5996	0.445150
$680$	−10.7321	−0.411558
$681$	60.0029	2.29932
$682$	−34.4874	−1.32059
$683$	−30.5403	−1.16859	−0.584296	−	0.811540i	$-0.698629\pi$
$683$	−30.5403	−1.16859	−0.584296	+	0.811540i	$0.698629\pi$
$684$	2.64927	0.101297
$685$	29.4285	1.12441
$686$	17.1569	0.655053
$687$	−70.3307	−2.68329
$688$	8.88867	0.338877
$689$	43.3367	1.65100
$690$	−67.6055	−2.57370
$691$	−15.9453	−0.606588	−0.303294	−	0.952897i	$-0.598086\pi$
$691$	−15.9453	−0.606588	−0.303294	+	0.952897i	$0.598086\pi$
$692$	16.2824	0.618963
$693$	51.5218	1.95715
$694$	−5.60003	−0.212574
$695$	12.8891	0.488913
$696$	1.96470	0.0744719
$697$	28.2428	1.06977
$698$	1.74609	0.0660906
$699$	7.14215	0.270141
$700$	−9.77537	−0.369474
$701$	8.78170	0.331680	0.165840	−	0.986153i	$-0.446966\pi$
$701$	8.78170	0.331680	0.165840	+	0.986153i	$0.446966\pi$
$702$	−4.34148	−0.163859
$703$	−7.33205	−0.276534
$704$	−5.24844	−0.197808
$705$	21.3420	0.803785
$706$	−11.0457	−0.415711
$707$	−18.5409	−0.697303
$708$	17.8575	0.671125
$709$	24.8599	0.933634	0.466817	−	0.884354i	$-0.345401\pi$
$709$	24.8599	0.933634	0.466817	+	0.884354i	$0.345401\pi$
$710$	−13.9426	−0.523258
$711$	−36.0614	−1.35241
$712$	−13.0865	−0.490439
$713$	−59.8648	−2.24195
$714$	26.3869	0.987506
$715$	52.6978	1.97079
$716$	−2.38997	−0.0893173
$717$	−2.84646	−0.106303
$718$	−6.15081	−0.229546
$719$	−12.7970	−0.477248	−0.238624	−	0.971112i	$-0.576696\pi$
$719$	−12.7970	−0.477248	−0.238624	+	0.971112i	$0.576696\pi$
$720$	−10.1746	−0.379184
$721$	−39.3087	−1.46393
$722$	−18.4251	−0.685712
$723$	32.3513	1.20316
$724$	−6.15511	−0.228753
$725$	−2.68252	−0.0996265
$726$	−42.1653	−1.56490
$727$	44.9476	1.66702	0.833508	−	0.552508i	$-0.186329\pi$
$727$	44.9476	1.66702	0.833508	+	0.552508i	$0.186329\pi$
$728$	−9.68741	−0.359039
$729$	−35.0405	−1.29780
$730$	−21.5164	−0.796359
$731$	32.7596	1.21166
$732$	11.6387	0.430180
$733$	31.8283	1.17560	0.587802	−	0.809004i	$-0.299993\pi$
$733$	31.8283	1.17560	0.587802	+	0.809004i	$0.299993\pi$
$734$	23.1995	0.856310
$735$	6.62844	0.244494
$736$	−9.11047	−0.335816
$737$	−81.2976	−2.99464
$738$	26.7756	0.985622
$739$	−18.9772	−0.698088	−0.349044	−	0.937106i	$-0.613494\pi$
$739$	−18.9772	−0.698088	−0.349044	+	0.937106i	$0.613494\pi$
$740$	28.1589	1.03514
$741$	−6.66241	−0.244750
$742$	−35.3105	−1.29629
$743$	23.4377	0.859844	0.429922	−	0.902866i	$-0.358541\pi$
$743$	23.4377	0.859844	0.429922	+	0.902866i	$0.358541\pi$
$744$	−16.7452	−0.613908
$745$	−12.4695	−0.456848
$746$	1.96223	0.0718422
$747$	14.6017	0.534250
$748$	−19.3434	−0.707264
$749$	33.0692	1.20832
$750$	−11.2838	−0.412025
$751$	−19.2065	−0.700856	−0.350428	−	0.936590i	$-0.613964\pi$
$751$	−19.2065	−0.700856	−0.350428	+	0.936590i	$0.613964\pi$
$752$	2.87603	0.104878
$753$	−16.7431	−0.610153
$754$	−2.65839	−0.0968127
$755$	−7.16300	−0.260688
$756$	3.53742	0.128655
$757$	−45.8828	−1.66764	−0.833820	−	0.552036i	$-0.813851\pi$
$757$	−45.8828	−1.66764	−0.833820	+	0.552036i	$0.813851\pi$
$758$	25.4477	0.924302
$759$	−121.851	−4.42292
$760$	−2.20788	−0.0800882
$761$	−48.5095	−1.75847	−0.879233	−	0.476392i	$-0.841944\pi$
$761$	−48.5095	−1.75847	−0.879233	+	0.476392i	$0.841944\pi$
$762$	33.1786	1.20193
$763$	−49.5387	−1.79342
$764$	−9.67821	−0.350145
$765$	−37.4989	−1.35577
$766$	−28.4468	−1.02782
$767$	−24.1624	−0.872455
$768$	−2.54835	−0.0919556
$769$	−45.1038	−1.62648	−0.813242	−	0.581926i	$-0.802299\pi$
$769$	−45.1038	−1.62648	−0.813242	+	0.581926i	$0.802299\pi$
$770$	−42.9379	−1.54737
$771$	56.9971	2.05270
$772$	2.48869	0.0895700
$773$	−44.8458	−1.61299	−0.806495	−	0.591241i	$-0.798638\pi$
$773$	−44.8458	−1.61299	−0.806495	+	0.591241i	$0.798638\pi$
$774$	31.0577	1.11635
$775$	22.8632	0.821269
$776$	−4.12871	−0.148212
$777$	−69.2340	−2.48376
$778$	−19.8980	−0.713376
$779$	5.81030	0.208175
$780$	25.5871	0.916166
$781$	−25.1300	−0.899222
$782$	−33.5771	−1.20071
$783$	0.970726	0.0346909
$784$	0.893244	0.0319016
$785$	7.82890	0.279425
$786$	5.65464	0.201694
$787$	5.95956	0.212436	0.106218	−	0.994343i	$-0.466126\pi$
$787$	5.95956	0.212436	0.106218	+	0.994343i	$0.466126\pi$
$788$	7.17015	0.255426
$789$	56.1445	1.99880
$790$	30.0533	1.06925
$791$	−43.0543	−1.53083
$792$	−18.3385	−0.651629
$793$	−15.7481	−0.559230
$794$	28.8905	1.02528
$795$	93.2647	3.30776
$796$	−7.78506	−0.275934
$797$	10.1355	0.359018	0.179509	−	0.983756i	$-0.442549\pi$
$797$	10.1355	0.359018	0.179509	+	0.983756i	$0.442549\pi$
$798$	5.42849	0.192167
$799$	10.5997	0.374992
$800$	3.47941	0.123016
$801$	−45.7255	−1.61563
$802$	1.62547	0.0573975
$803$	−38.7809	−1.36855
$804$	−39.4736	−1.39213
$805$	−74.5334	−2.62696
$806$	22.6574	0.798074
$807$	−19.8914	−0.700211
$808$	6.59938	0.232165
$809$	34.3797	1.20873	0.604363	−	0.796709i	$-0.293428\pi$
$809$	34.3797	1.20873	0.604363	+	0.796709i	$0.293428\pi$
$810$	21.1804	0.744204
$811$	18.9300	0.664721	0.332360	−	0.943152i	$-0.392155\pi$
$811$	18.9300	0.664721	0.332360	+	0.943152i	$0.392155\pi$
$812$	2.16604	0.0760130
$813$	22.6271	0.793568
$814$	50.7531	1.77890
$815$	23.8494	0.835409
$816$	−9.39207	−0.328788
$817$	6.73952	0.235786
$818$	−23.7408	−0.830079
$819$	−33.8486	−1.18277
$820$	−22.3146	−0.779258
$821$	−10.0231	−0.349808	−0.174904	−	0.984586i	$-0.555961\pi$
$821$	−10.0231	−0.349808	−0.174904	+	0.984586i	$0.555961\pi$
$822$	25.7540	0.898274
$823$	1.87488	0.0653542	0.0326771	−	0.999466i	$-0.489597\pi$
$823$	1.87488	0.0653542	0.0326771	+	0.999466i	$0.489597\pi$
$824$	13.9914	0.487413
$825$	46.5366	1.62020
$826$	19.6874	0.685013
$827$	2.84136	0.0988039	0.0494020	−	0.998779i	$-0.484268\pi$
$827$	2.84136	0.0988039	0.0494020	+	0.998779i	$0.484268\pi$
$828$	−31.8327	−1.10626
$829$	−31.3529	−1.08893	−0.544465	−	0.838783i	$-0.683267\pi$
$829$	−31.3529	−1.08893	−0.544465	+	0.838783i	$0.683267\pi$
$830$	−12.1690	−0.422392
$831$	16.7540	0.581189
$832$	3.44810	0.119541
$833$	3.29209	0.114064
$834$	11.2798	0.390586
$835$	73.4201	2.54081
$836$	−3.97945	−0.137632
$837$	−8.27350	−0.285974
$838$	0.564623	0.0195046
$839$	−52.0210	−1.79596	−0.897981	−	0.440034i	$-0.854967\pi$
$839$	−52.0210	−1.79596	−0.897981	+	0.440034i	$0.854967\pi$
$840$	−20.8482	−0.719333
$841$	−28.4056	−0.979504
$842$	5.50142	0.189592
$843$	6.10958	0.210425
$844$	12.6525	0.435518
$845$	3.23402	0.111254
$846$	10.0491	0.345494
$847$	−46.4862	−1.59728
$848$	12.5683	0.431596
$849$	37.0964	1.27315
$850$	12.8235	0.439843
$851$	88.0995	3.02001
$852$	−12.2017	−0.418024
$853$	3.18496	0.109051	0.0545254	−	0.998512i	$-0.482635\pi$
$853$	3.18496	0.109051	0.0545254	+	0.998512i	$0.482635\pi$
$854$	12.8314	0.439082
$855$	−7.71451	−0.263831
$856$	−11.7705	−0.402308
$857$	−27.2205	−0.929835	−0.464918	−	0.885354i	$-0.653916\pi$
$857$	−27.2205	−0.929835	−0.464918	+	0.885354i	$0.653916\pi$
$858$	46.1178	1.57444
$859$	−25.6312	−0.874524	−0.437262	−	0.899334i	$-0.644052\pi$
$859$	−25.6312	−0.874524	−0.437262	+	0.899334i	$0.644052\pi$
$860$	−25.8833	−0.882613
$861$	54.8646	1.86978
$862$	38.9800	1.32767
$863$	−3.13407	−0.106685	−0.0533424	−	0.998576i	$-0.516987\pi$
$863$	−3.13407	−0.106685	−0.0533424	+	0.998576i	$0.516987\pi$
$864$	−1.25909	−0.0428353
$865$	−47.4133	−1.61210
$866$	−16.3502	−0.555602
$867$	8.70701	0.295706
$868$	−18.4611	−0.626612
$869$	54.1676	1.83751
$870$	−5.72111	−0.193964
$871$	53.4106	1.80975
$872$	17.6326	0.597116
$873$	−14.4260	−0.488247
$874$	−6.90770	−0.233656
$875$	−12.4401	−0.420551
$876$	−18.8298	−0.636201
$877$	49.3964	1.66800	0.833999	−	0.551766i	$-0.186046\pi$
$877$	49.3964	1.66800	0.833999	+	0.551766i	$0.186046\pi$
$878$	35.8690	1.21052
$879$	13.8470	0.467048
$880$	15.2832	0.515195
$881$	32.9505	1.11013	0.555066	−	0.831806i	$-0.312693\pi$
$881$	32.9505	1.11013	0.555066	+	0.831806i	$0.312693\pi$
$882$	3.12107	0.105092
$883$	13.2908	0.447272	0.223636	−	0.974673i	$-0.428207\pi$
$883$	13.2908	0.447272	0.223636	+	0.974673i	$0.428207\pi$
$884$	12.7081	0.427421
$885$	−51.9999	−1.74796
$886$	−16.3456	−0.549143
$887$	55.0831	1.84951	0.924754	−	0.380565i	$-0.124271\pi$
$887$	55.0831	1.84951	0.924754	+	0.380565i	$0.124271\pi$
$888$	24.6429	0.826961
$889$	36.5786	1.22681
$890$	38.1073	1.27736
$891$	38.1752	1.27892
$892$	−17.4679	−0.584870
$893$	2.18065	0.0729726
$894$	−10.9125	−0.364970
$895$	6.95945	0.232629
$896$	−2.80949	−0.0938585
$897$	80.0533	2.67290
$898$	9.56177	0.319080
$899$	−5.06605	−0.168962
$900$	12.1573	0.405245
$901$	46.3210	1.54318
$902$	−40.2194	−1.33916
$903$	63.6390	2.11777
$904$	15.3246	0.509688
$905$	17.9233	0.595791
$906$	−6.26860	−0.208260
$907$	−35.1048	−1.16564	−0.582819	−	0.812602i	$-0.698050\pi$
$907$	−35.1048	−1.16564	−0.582819	+	0.812602i	$0.698050\pi$
$908$	−23.5458	−0.781394
$909$	23.0588	0.764811
$910$	28.2092	0.935125
$911$	−38.3460	−1.27046	−0.635231	−	0.772323i	$-0.719095\pi$
$911$	−38.3460	−1.27046	−0.635231	+	0.772323i	$0.719095\pi$
$912$	−1.93220	−0.0639815
$913$	−21.9332	−0.725882
$914$	13.9508	0.461451
$915$	−33.8913	−1.12041
$916$	27.5986	0.911882
$917$	6.23410	0.205868
$918$	−4.64046	−0.153158
$919$	29.8298	0.983993	0.491997	−	0.870597i	$-0.336267\pi$
$919$	29.8298	0.983993	0.491997	+	0.870597i	$0.336267\pi$
$920$	26.5292	0.874640
$921$	57.6140	1.89845
$922$	21.9703	0.723552
$923$	16.5098	0.543427
$924$	−37.5765	−1.23618
$925$	−33.6464	−1.10629
$926$	−12.4914	−0.410493
$927$	48.8871	1.60566
$928$	−0.770972	−0.0253084
$929$	−35.1997	−1.15486	−0.577432	−	0.816439i	$-0.695945\pi$
$929$	−35.1997	−1.15486	−0.577432	+	0.816439i	$0.695945\pi$
$930$	48.7610	1.59894
$931$	0.677271	0.0221967
$932$	−2.80266	−0.0918041
$933$	31.6255	1.03537
$934$	33.0450	1.08127
$935$	56.3268	1.84208
$936$	12.0479	0.393799
$937$	3.51867	0.114950	0.0574749	−	0.998347i	$-0.481695\pi$
$937$	3.51867	0.114950	0.0574749	+	0.998347i	$0.481695\pi$
$938$	−43.5186	−1.42093
$939$	−27.9778	−0.913022
$940$	−8.37482	−0.273157
$941$	14.4497	0.471048	0.235524	−	0.971869i	$-0.424319\pi$
$941$	14.4497	0.471048	0.235524	+	0.971869i	$0.424319\pi$
$942$	6.85136	0.223229
$943$	−69.8146	−2.27348
$944$	−7.00747	−0.228074
$945$	−10.3008	−0.335083
$946$	−46.6516	−1.51677
$947$	−55.4450	−1.80172	−0.900861	−	0.434109i	$-0.857063\pi$
$947$	−55.4450	−1.80172	−0.900861	+	0.434109i	$0.857063\pi$
$948$	26.3008	0.854209
$949$	25.4781	0.827055
$950$	2.63814	0.0855926
$951$	49.8247	1.61568
$952$	−10.3545	−0.335592
$953$	16.5631	0.536533	0.268267	−	0.963345i	$-0.413549\pi$
$953$	16.5631	0.536533	0.268267	+	0.963345i	$0.413549\pi$
$954$	43.9146	1.42179
$955$	28.1824	0.911961
$956$	1.11698	0.0361257
$957$	−10.3116	−0.333328
$958$	16.9312	0.547022
$959$	28.3932	0.916863
$960$	7.42065	0.239500
$961$	12.1779	0.392836
$962$	−33.3436	−1.07504
$963$	−41.1272	−1.32530
$964$	−12.6950	−0.408878
$965$	−7.24693	−0.233287
$966$	−65.2270	−2.09864
$967$	−26.1798	−0.841885	−0.420942	−	0.907087i	$-0.638301\pi$
$967$	−26.1798	−0.841885	−0.420942	+	0.907087i	$0.638301\pi$
$968$	16.5461	0.531812
$969$	−7.12121	−0.228766
$970$	12.0226	0.386021
$971$	−13.7278	−0.440547	−0.220273	−	0.975438i	$-0.570695\pi$
$971$	−13.7278	−0.440547	−0.220273	+	0.975438i	$0.570695\pi$
$972$	22.3130	0.715691
$973$	12.4357	0.398669
$974$	13.2230	0.423692
$975$	−30.5734	−0.979133
$976$	−4.56717	−0.146192
$977$	3.05890	0.0978629	0.0489315	−	0.998802i	$-0.484418\pi$
$977$	3.05890	0.0978629	0.0489315	+	0.998802i	$0.484418\pi$
$978$	20.8715	0.667398
$979$	68.6839	2.19515
$980$	−2.60107	−0.0830882
$981$	61.6098	1.96705
$982$	4.84397	0.154577
$983$	−34.7157	−1.10726	−0.553629	−	0.832764i	$-0.686757\pi$
$983$	−34.7157	−1.10726	−0.553629	+	0.832764i	$0.686757\pi$
$984$	−19.5283	−0.622539
$985$	−20.8791	−0.665262
$986$	−2.84145	−0.0904903
$987$	20.5911	0.655422
$988$	2.61440	0.0831752
$989$	−80.9799	−2.57501
$990$	53.4006	1.69718
$991$	−30.8199	−0.979027	−0.489513	−	0.871996i	$-0.662826\pi$
$991$	−30.8199	−0.979027	−0.489513	+	0.871996i	$0.662826\pi$
$992$	6.57099	0.208629
$993$	−48.7314	−1.54644
$994$	−13.4521	−0.426675
$995$	22.6696	0.718676
$996$	−10.6495	−0.337443
$997$	−35.3092	−1.11825	−0.559126	−	0.829083i	$-0.688863\pi$
$997$	−35.3092	−1.11825	−0.559126	+	0.829083i	$0.688863\pi$
$998$	−14.5967	−0.462050
$999$	12.1756	0.385220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.12	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.54835 q^{3} +1.00000 q^{4} -2.91194 q^{5} -2.54835 q^{6} -2.80949 q^{7} +1.00000 q^{8} +3.49408 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.54835 q^{3} +1.00000 q^{4} -2.91194 q^{5} -2.54835 q^{6} -2.80949 q^{7} +1.00000 q^{8} +3.49408 q^{9} -2.91194 q^{10} -5.24844 q^{11} -2.54835 q^{12} +3.44810 q^{13} -2.80949 q^{14} +7.42065 q^{15} +1.00000 q^{16} +3.68555 q^{17} +3.49408 q^{18} +0.758216 q^{19} -2.91194 q^{20} +7.15957 q^{21} -5.24844 q^{22} -9.11047 q^{23} -2.54835 q^{24} +3.47941 q^{25} +3.44810 q^{26} -1.25909 q^{27} -2.80949 q^{28} -0.770972 q^{29} +7.42065 q^{30} +6.57099 q^{31} +1.00000 q^{32} +13.3749 q^{33} +3.68555 q^{34} +8.18108 q^{35} +3.49408 q^{36} -9.67014 q^{37} +0.758216 q^{38} -8.78696 q^{39} -2.91194 q^{40} +7.66312 q^{41} +7.15957 q^{42} +8.88867 q^{43} -5.24844 q^{44} -10.1746 q^{45} -9.11047 q^{46} +2.87603 q^{47} -2.54835 q^{48} +0.893244 q^{49} +3.47941 q^{50} -9.39207 q^{51} +3.44810 q^{52} +12.5683 q^{53} -1.25909 q^{54} +15.2832 q^{55} -2.80949 q^{56} -1.93220 q^{57} -0.770972 q^{58} -7.00747 q^{59} +7.42065 q^{60} -4.56717 q^{61} +6.57099 q^{62} -9.81660 q^{63} +1.00000 q^{64} -10.0407 q^{65} +13.3749 q^{66} +15.4899 q^{67} +3.68555 q^{68} +23.2166 q^{69} +8.18108 q^{70} +4.78809 q^{71} +3.49408 q^{72} +7.38903 q^{73} -9.67014 q^{74} -8.86675 q^{75} +0.758216 q^{76} +14.7454 q^{77} -8.78696 q^{78} -10.3207 q^{79} -2.91194 q^{80} -7.27363 q^{81} +7.66312 q^{82} +4.17899 q^{83} +7.15957 q^{84} -10.7321 q^{85} +8.88867 q^{86} +1.96470 q^{87} -5.24844 q^{88} -13.0865 q^{89} -10.1746 q^{90} -9.68741 q^{91} -9.11047 q^{92} -16.7452 q^{93} +2.87603 q^{94} -2.20788 q^{95} -2.54835 q^{96} -4.12871 q^{97} +0.893244 q^{98} -18.3385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 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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