LMFDB - Embedded newform 4030.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 6x^{5} + 54x^{4} + 46x^{3} - 32x^{2} - 43x - 11 $ x^8 - 14x^6 - 6x^5 + 54x^4 + 46x^3 - 32x^2 - 43x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.92608$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.43624 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.43624 q^{6} -2.74822 q^{7} -1.00000 q^{8} -0.937215 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.43624 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.43624 q^{6} -2.74822 q^{7} -1.00000 q^{8} -0.937215 q^{9} -1.00000 q^{10} -1.64899 q^{11} -1.43624 q^{12} -1.00000 q^{13} +2.74822 q^{14} -1.43624 q^{15} +1.00000 q^{16} +4.89350 q^{17} +0.937215 q^{18} -4.21937 q^{19} +1.00000 q^{20} +3.94710 q^{21} +1.64899 q^{22} -0.713951 q^{23} +1.43624 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.65479 q^{27} -2.74822 q^{28} -5.35761 q^{29} +1.43624 q^{30} -1.00000 q^{31} -1.00000 q^{32} +2.36834 q^{33} -4.89350 q^{34} -2.74822 q^{35} -0.937215 q^{36} +2.40168 q^{37} +4.21937 q^{38} +1.43624 q^{39} -1.00000 q^{40} -5.05241 q^{41} -3.94710 q^{42} -10.0960 q^{43} -1.64899 q^{44} -0.937215 q^{45} +0.713951 q^{46} -7.42361 q^{47} -1.43624 q^{48} +0.552707 q^{49} -1.00000 q^{50} -7.02824 q^{51} -1.00000 q^{52} -1.87100 q^{53} -5.65479 q^{54} -1.64899 q^{55} +2.74822 q^{56} +6.06003 q^{57} +5.35761 q^{58} -6.59656 q^{59} -1.43624 q^{60} +5.44337 q^{61} +1.00000 q^{62} +2.57567 q^{63} +1.00000 q^{64} -1.00000 q^{65} -2.36834 q^{66} +6.50071 q^{67} +4.89350 q^{68} +1.02540 q^{69} +2.74822 q^{70} +5.91227 q^{71} +0.937215 q^{72} +16.4812 q^{73} -2.40168 q^{74} -1.43624 q^{75} -4.21937 q^{76} +4.53177 q^{77} -1.43624 q^{78} -2.80072 q^{79} +1.00000 q^{80} -5.30998 q^{81} +5.05241 q^{82} +6.84106 q^{83} +3.94710 q^{84} +4.89350 q^{85} +10.0960 q^{86} +7.69482 q^{87} +1.64899 q^{88} +4.57817 q^{89} +0.937215 q^{90} +2.74822 q^{91} -0.713951 q^{92} +1.43624 q^{93} +7.42361 q^{94} -4.21937 q^{95} +1.43624 q^{96} +16.3044 q^{97} -0.552707 q^{98} +1.54545 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * q^98 + 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.43624	−0.829213	−0.414607	−	0.910001i	$-0.636081\pi$
$3$	−1.43624	−0.829213	−0.414607	+	0.910001i	$0.636081\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.43624	0.586342
$7$	−2.74822	−1.03873	−0.519365	−	0.854553i	$-0.673831\pi$
$7$	−2.74822	−1.03873	−0.519365	+	0.854553i	$0.673831\pi$
$8$	−1.00000	−0.353553
$9$	−0.937215	−0.312405
$10$	−1.00000	−0.316228
$11$	−1.64899	−0.497188	−0.248594	−	0.968608i	$-0.579968\pi$
$11$	−1.64899	−0.497188	−0.248594	+	0.968608i	$0.579968\pi$
$12$	−1.43624	−0.414607
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.74822	0.734492
$15$	−1.43624	−0.370836
$16$	1.00000	0.250000
$17$	4.89350	1.18685	0.593424	−	0.804890i	$-0.297776\pi$
$17$	4.89350	1.18685	0.593424	+	0.804890i	$0.297776\pi$
$18$	0.937215	0.220904
$19$	−4.21937	−0.967990	−0.483995	−	0.875071i	$-0.660815\pi$
$19$	−4.21937	−0.967990	−0.483995	+	0.875071i	$0.660815\pi$
$20$	1.00000	0.223607
$21$	3.94710	0.861328
$22$	1.64899	0.351565
$23$	−0.713951	−0.148869	−0.0744345	−	0.997226i	$-0.523715\pi$
$23$	−0.713951	−0.148869	−0.0744345	+	0.997226i	$0.523715\pi$
$24$	1.43624	0.293171
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	5.65479	1.08826
$28$	−2.74822	−0.519365
$29$	−5.35761	−0.994884	−0.497442	−	0.867497i	$-0.665727\pi$
$29$	−5.35761	−0.994884	−0.497442	+	0.867497i	$0.665727\pi$
$30$	1.43624	0.262220
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	2.36834	0.412275
$34$	−4.89350	−0.839228
$35$	−2.74822	−0.464534
$36$	−0.937215	−0.156203
$37$	2.40168	0.394833	0.197417	−	0.980320i	$-0.436745\pi$
$37$	2.40168	0.394833	0.197417	+	0.980320i	$0.436745\pi$
$38$	4.21937	0.684473
$39$	1.43624	0.229982
$40$	−1.00000	−0.158114
$41$	−5.05241	−0.789054	−0.394527	−	0.918884i	$-0.629092\pi$
$41$	−5.05241	−0.789054	−0.394527	+	0.918884i	$0.629092\pi$
$42$	−3.94710	−0.609051
$43$	−10.0960	−1.53963	−0.769815	−	0.638267i	$-0.779651\pi$
$43$	−10.0960	−1.53963	−0.769815	+	0.638267i	$0.779651\pi$
$44$	−1.64899	−0.248594
$45$	−0.937215	−0.139712
$46$	0.713951	0.105266
$47$	−7.42361	−1.08285	−0.541423	−	0.840751i	$-0.682114\pi$
$47$	−7.42361	−1.08285	−0.541423	+	0.840751i	$0.682114\pi$
$48$	−1.43624	−0.207303
$49$	0.552707	0.0789582
$50$	−1.00000	−0.141421
$51$	−7.02824	−0.984150
$52$	−1.00000	−0.138675
$53$	−1.87100	−0.257002	−0.128501	−	0.991709i	$-0.541017\pi$
$53$	−1.87100	−0.257002	−0.128501	+	0.991709i	$0.541017\pi$
$54$	−5.65479	−0.769519
$55$	−1.64899	−0.222349
$56$	2.74822	0.367246
$57$	6.06003	0.802671
$58$	5.35761	0.703489
$59$	−6.59656	−0.858799	−0.429399	−	0.903115i	$-0.641275\pi$
$59$	−6.59656	−0.858799	−0.429399	+	0.903115i	$0.641275\pi$
$60$	−1.43624	−0.185418
$61$	5.44337	0.696952	0.348476	−	0.937318i	$-0.386699\pi$
$61$	5.44337	0.696952	0.348476	+	0.937318i	$0.386699\pi$
$62$	1.00000	0.127000
$63$	2.57567	0.324504
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−2.36834	−0.291522
$67$	6.50071	0.794188	0.397094	−	0.917778i	$-0.370019\pi$
$67$	6.50071	0.794188	0.397094	+	0.917778i	$0.370019\pi$
$68$	4.89350	0.593424
$69$	1.02540	0.123444
$70$	2.74822	0.328475
$71$	5.91227	0.701657	0.350829	−	0.936440i	$-0.385900\pi$
$71$	5.91227	0.701657	0.350829	+	0.936440i	$0.385900\pi$
$72$	0.937215	0.110452
$73$	16.4812	1.92898	0.964488	−	0.264126i	$-0.0850835\pi$
$73$	16.4812	1.92898	0.964488	+	0.264126i	$0.0850835\pi$
$74$	−2.40168	−0.279189
$75$	−1.43624	−0.165843
$76$	−4.21937	−0.483995
$77$	4.53177	0.516443
$78$	−1.43624	−0.162622
$79$	−2.80072	−0.315106	−0.157553	−	0.987511i	$-0.550361\pi$
$79$	−2.80072	−0.315106	−0.157553	+	0.987511i	$0.550361\pi$
$80$	1.00000	0.111803
$81$	−5.30998	−0.589998
$82$	5.05241	0.557946
$83$	6.84106	0.750904	0.375452	−	0.926842i	$-0.377488\pi$
$83$	6.84106	0.750904	0.375452	+	0.926842i	$0.377488\pi$
$84$	3.94710	0.430664
$85$	4.89350	0.530775
$86$	10.0960	1.08868
$87$	7.69482	0.824971
$88$	1.64899	0.175782
$89$	4.57817	0.485285	0.242643	−	0.970116i	$-0.421986\pi$
$89$	4.57817	0.485285	0.242643	+	0.970116i	$0.421986\pi$
$90$	0.937215	0.0987912
$91$	2.74822	0.288092
$92$	−0.713951	−0.0744345
$93$	1.43624	0.148931
$94$	7.42361	0.765687
$95$	−4.21937	−0.432898
$96$	1.43624	0.146586
$97$	16.3044	1.65546	0.827731	−	0.561125i	$-0.189631\pi$
$97$	16.3044	1.65546	0.827731	+	0.561125i	$0.189631\pi$
$98$	−0.552707	−0.0558319
$99$	1.54545	0.155324
$100$	1.00000	0.100000
$101$	−18.3652	−1.82741	−0.913705	−	0.406379i	$-0.866791\pi$
$101$	−18.3652	−1.82741	−0.913705	+	0.406379i	$0.866791\pi$
$102$	7.02824	0.695899
$103$	−3.90207	−0.384482	−0.192241	−	0.981348i	$-0.561576\pi$
$103$	−3.90207	−0.384482	−0.192241	+	0.981348i	$0.561576\pi$
$104$	1.00000	0.0980581
$105$	3.94710	0.385198
$106$	1.87100	0.181728
$107$	−12.3139	−1.19043	−0.595217	−	0.803565i	$-0.702934\pi$
$107$	−12.3139	−1.19043	−0.595217	+	0.803565i	$0.702934\pi$
$108$	5.65479	0.544132
$109$	−2.37117	−0.227117	−0.113559	−	0.993531i	$-0.536225\pi$
$109$	−2.37117	−0.227117	−0.113559	+	0.993531i	$0.536225\pi$
$110$	1.64899	0.157225
$111$	−3.44938	−0.327401
$112$	−2.74822	−0.259682
$113$	−13.0884	−1.23126	−0.615628	−	0.788037i	$-0.711098\pi$
$113$	−13.0884	−1.23126	−0.615628	+	0.788037i	$0.711098\pi$
$114$	−6.06003	−0.567574
$115$	−0.713951	−0.0665763
$116$	−5.35761	−0.497442
$117$	0.937215	0.0866456
$118$	6.59656	0.607262
$119$	−13.4484	−1.23281
$120$	1.43624	0.131110
$121$	−8.28085	−0.752804
$122$	−5.44337	−0.492819
$123$	7.25647	0.654294
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−2.57567	−0.229459
$127$	3.39354	0.301128	0.150564	−	0.988600i	$-0.451891\pi$
$127$	3.39354	0.301128	0.150564	+	0.988600i	$0.451891\pi$
$128$	−1.00000	−0.0883883
$129$	14.5003	1.27668
$130$	1.00000	0.0877058
$131$	17.7271	1.54883	0.774414	−	0.632679i	$-0.218045\pi$
$131$	17.7271	1.54883	0.774414	+	0.632679i	$0.218045\pi$
$132$	2.36834	0.206137
$133$	11.5958	1.00548
$134$	−6.50071	−0.561576
$135$	5.65479	0.486686
$136$	−4.89350	−0.419614
$137$	5.22084	0.446046	0.223023	−	0.974813i	$-0.428407\pi$
$137$	5.22084	0.446046	0.223023	+	0.974813i	$0.428407\pi$
$138$	−1.02540	−0.0872882
$139$	2.99992	0.254450	0.127225	−	0.991874i	$-0.459393\pi$
$139$	2.99992	0.254450	0.127225	+	0.991874i	$0.459393\pi$
$140$	−2.74822	−0.232267
$141$	10.6621	0.897910
$142$	−5.91227	−0.496147
$143$	1.64899	0.137895
$144$	−0.937215	−0.0781013
$145$	−5.35761	−0.444925
$146$	−16.4812	−1.36399
$147$	−0.793820	−0.0654732
$148$	2.40168	0.197417
$149$	7.47059	0.612014	0.306007	−	0.952029i	$-0.401007\pi$
$149$	7.47059	0.612014	0.306007	+	0.952029i	$0.401007\pi$
$150$	1.43624	0.117268
$151$	−16.6877	−1.35803	−0.679013	−	0.734126i	$-0.737592\pi$
$151$	−16.6877	−1.35803	−0.679013	+	0.734126i	$0.737592\pi$
$152$	4.21937	0.342236
$153$	−4.58626	−0.370777
$154$	−4.53177	−0.365181
$155$	−1.00000	−0.0803219
$156$	1.43624	0.114991
$157$	−0.401664	−0.0320563	−0.0160281	−	0.999872i	$-0.505102\pi$
$157$	−0.401664	−0.0320563	−0.0160281	+	0.999872i	$0.505102\pi$
$158$	2.80072	0.222814
$159$	2.68721	0.213109
$160$	−1.00000	−0.0790569
$161$	1.96209	0.154635
$162$	5.30998	0.417192
$163$	−14.5685	−1.14110	−0.570548	−	0.821264i	$-0.693269\pi$
$163$	−14.5685	−1.14110	−0.570548	+	0.821264i	$0.693269\pi$
$164$	−5.05241	−0.394527
$165$	2.36834	0.184375
$166$	−6.84106	−0.530969
$167$	24.1771	1.87088	0.935441	−	0.353482i	$-0.115002\pi$
$167$	24.1771	1.87088	0.935441	+	0.353482i	$0.115002\pi$
$168$	−3.94710	−0.304525
$169$	1.00000	0.0769231
$170$	−4.89350	−0.375314
$171$	3.95446	0.302405
$172$	−10.0960	−0.769815
$173$	16.4727	1.25239	0.626197	−	0.779665i	$-0.284611\pi$
$173$	16.4727	1.25239	0.626197	+	0.779665i	$0.284611\pi$
$174$	−7.69482	−0.583342
$175$	−2.74822	−0.207746
$176$	−1.64899	−0.124297
$177$	9.47424	0.712127
$178$	−4.57817	−0.343149
$179$	4.31164	0.322267	0.161134	−	0.986933i	$-0.448485\pi$
$179$	4.31164	0.322267	0.161134	+	0.986933i	$0.448485\pi$
$180$	−0.937215	−0.0698559
$181$	−0.524825	−0.0390099	−0.0195050	−	0.999810i	$-0.506209\pi$
$181$	−0.524825	−0.0390099	−0.0195050	+	0.999810i	$0.506209\pi$
$182$	−2.74822	−0.203712
$183$	−7.81798	−0.577922
$184$	0.713951	0.0526331
$185$	2.40168	0.176575
$186$	−1.43624	−0.105310
$187$	−8.06931	−0.590086
$188$	−7.42361	−0.541423
$189$	−15.5406	−1.13041
$190$	4.21937	0.306105
$191$	−3.93960	−0.285059	−0.142530	−	0.989791i	$-0.545524\pi$
$191$	−3.93960	−0.285059	−0.142530	+	0.989791i	$0.545524\pi$
$192$	−1.43624	−0.103652
$193$	17.2505	1.24172	0.620861	−	0.783921i	$-0.286783\pi$
$193$	17.2505	1.24172	0.620861	+	0.783921i	$0.286783\pi$
$194$	−16.3044	−1.17059
$195$	1.43624	0.102851
$196$	0.552707	0.0394791
$197$	11.2690	0.802884	0.401442	−	0.915884i	$-0.368509\pi$
$197$	11.2690	0.802884	0.401442	+	0.915884i	$0.368509\pi$
$198$	−1.54545	−0.109831
$199$	20.0876	1.42397	0.711985	−	0.702195i	$-0.247796\pi$
$199$	20.0876	1.42397	0.711985	+	0.702195i	$0.247796\pi$
$200$	−1.00000	−0.0707107
$201$	−9.33658	−0.658552
$202$	18.3652	1.29217
$203$	14.7239	1.03341
$204$	−7.02824	−0.492075
$205$	−5.05241	−0.352876
$206$	3.90207	0.271870
$207$	0.669125	0.0465074
$208$	−1.00000	−0.0693375
$209$	6.95768	0.481273
$210$	−3.94710	−0.272376
$211$	−28.4397	−1.95787	−0.978935	−	0.204171i	$-0.934550\pi$
$211$	−28.4397	−1.95787	−0.978935	+	0.204171i	$0.934550\pi$
$212$	−1.87100	−0.128501
$213$	−8.49144	−0.581824
$214$	12.3139	0.841764
$215$	−10.0960	−0.688543
$216$	−5.65479	−0.384759
$217$	2.74822	0.186561
$218$	2.37117	0.160596
$219$	−23.6709	−1.59953
$220$	−1.64899	−0.111175
$221$	−4.89350	−0.329172
$222$	3.44938	0.231507
$223$	−5.70670	−0.382149	−0.191074	−	0.981576i	$-0.561197\pi$
$223$	−5.70670	−0.382149	−0.191074	+	0.981576i	$0.561197\pi$
$224$	2.74822	0.183623
$225$	−0.937215	−0.0624810
$226$	13.0884	0.870629
$227$	24.9586	1.65656	0.828282	−	0.560312i	$-0.189319\pi$
$227$	24.9586	1.65656	0.828282	+	0.560312i	$0.189319\pi$
$228$	6.06003	0.401335
$229$	2.38067	0.157319	0.0786595	−	0.996902i	$-0.474936\pi$
$229$	2.38067	0.157319	0.0786595	+	0.996902i	$0.474936\pi$
$230$	0.713951	0.0470765
$231$	−6.50871	−0.428242
$232$	5.35761	0.351744
$233$	26.9022	1.76242	0.881211	−	0.472724i	$-0.156729\pi$
$233$	26.9022	1.76242	0.881211	+	0.472724i	$0.156729\pi$
$234$	−0.937215	−0.0612677
$235$	−7.42361	−0.484263
$236$	−6.59656	−0.429399
$237$	4.02251	0.261290
$238$	13.4484	0.871731
$239$	24.8744	1.60899	0.804496	−	0.593958i	$-0.202435\pi$
$239$	24.8744	1.60899	0.804496	+	0.593958i	$0.202435\pi$
$240$	−1.43624	−0.0927089
$241$	0.940565	0.0605871	0.0302936	−	0.999541i	$-0.490356\pi$
$241$	0.940565	0.0605871	0.0302936	+	0.999541i	$0.490356\pi$
$242$	8.28085	0.532313
$243$	−9.33795	−0.599030
$244$	5.44337	0.348476
$245$	0.552707	0.0353112
$246$	−7.25647	−0.462656
$247$	4.21937	0.268472
$248$	1.00000	0.0635001
$249$	−9.82540	−0.622659
$250$	−1.00000	−0.0632456
$251$	−2.34387	−0.147944	−0.0739720	−	0.997260i	$-0.523568\pi$
$251$	−2.34387	−0.147944	−0.0739720	+	0.997260i	$0.523568\pi$
$252$	2.57567	0.162252
$253$	1.17729	0.0740159
$254$	−3.39354	−0.212930
$255$	−7.02824	−0.440125
$256$	1.00000	0.0625000
$257$	−13.4562	−0.839372	−0.419686	−	0.907669i	$-0.637860\pi$
$257$	−13.4562	−0.839372	−0.419686	+	0.907669i	$0.637860\pi$
$258$	−14.5003	−0.902750
$259$	−6.60033	−0.410125
$260$	−1.00000	−0.0620174
$261$	5.02123	0.310807
$262$	−17.7271	−1.09519
$263$	−5.44730	−0.335895	−0.167947	−	0.985796i	$-0.553714\pi$
$263$	−5.44730	−0.335895	−0.167947	+	0.985796i	$0.553714\pi$
$264$	−2.36834	−0.145761
$265$	−1.87100	−0.114935
$266$	−11.5958	−0.710982
$267$	−6.57535	−0.402405
$268$	6.50071	0.397094
$269$	−16.0405	−0.978006	−0.489003	−	0.872282i	$-0.662639\pi$
$269$	−16.0405	−0.978006	−0.489003	+	0.872282i	$0.662639\pi$
$270$	−5.65479	−0.344139
$271$	−11.8305	−0.718651	−0.359326	−	0.933212i	$-0.616993\pi$
$271$	−11.8305	−0.718651	−0.359326	+	0.933212i	$0.616993\pi$
$272$	4.89350	0.296712
$273$	−3.94710	−0.238889
$274$	−5.22084	−0.315402
$275$	−1.64899	−0.0994376
$276$	1.02540	0.0617221
$277$	6.80363	0.408791	0.204395	−	0.978888i	$-0.434477\pi$
$277$	6.80363	0.408791	0.204395	+	0.978888i	$0.434477\pi$
$278$	−2.99992	−0.179924
$279$	0.937215	0.0561096
$280$	2.74822	0.164237
$281$	27.7172	1.65347	0.826736	−	0.562590i	$-0.190195\pi$
$281$	27.7172	1.65347	0.826736	+	0.562590i	$0.190195\pi$
$282$	−10.6621	−0.634918
$283$	11.6535	0.692728	0.346364	−	0.938100i	$-0.387416\pi$
$283$	11.6535	0.692728	0.346364	+	0.938100i	$0.387416\pi$
$284$	5.91227	0.350829
$285$	6.06003	0.358965
$286$	−1.64899	−0.0975065
$287$	13.8851	0.819614
$288$	0.937215	0.0552259
$289$	6.94635	0.408609
$290$	5.35761	0.314610
$291$	−23.4171	−1.37273
$292$	16.4812	0.964488
$293$	21.1512	1.23567	0.617833	−	0.786310i	$-0.288011\pi$
$293$	21.1512	1.23567	0.617833	+	0.786310i	$0.288011\pi$
$294$	0.793820	0.0462965
$295$	−6.59656	−0.384066
$296$	−2.40168	−0.139595
$297$	−9.32466	−0.541072
$298$	−7.47059	−0.432760
$299$	0.713951	0.0412888
$300$	−1.43624	−0.0829213
$301$	27.7461	1.59926
$302$	16.6877	0.960270
$303$	26.3769	1.51531
$304$	−4.21937	−0.241998
$305$	5.44337	0.311686
$306$	4.58626	0.262179
$307$	2.89749	0.165368	0.0826842	−	0.996576i	$-0.473651\pi$
$307$	2.89749	0.165368	0.0826842	+	0.996576i	$0.473651\pi$
$308$	4.53177	0.258222
$309$	5.60431	0.318818
$310$	1.00000	0.0567962
$311$	13.7875	0.781818	0.390909	−	0.920429i	$-0.372161\pi$
$311$	13.7875	0.781818	0.390909	+	0.920429i	$0.372161\pi$
$312$	−1.43624	−0.0813111
$313$	20.0590	1.13380	0.566902	−	0.823785i	$-0.308142\pi$
$313$	20.0590	1.13380	0.566902	+	0.823785i	$0.308142\pi$
$314$	0.401664	0.0226672
$315$	2.57567	0.145123
$316$	−2.80072	−0.157553
$317$	16.8042	0.943820	0.471910	−	0.881647i	$-0.343565\pi$
$317$	16.8042	0.943820	0.471910	+	0.881647i	$0.343565\pi$
$318$	−2.68721	−0.150691
$319$	8.83462	0.494644
$320$	1.00000	0.0559017
$321$	17.6858	0.987124
$322$	−1.96209	−0.109343
$323$	−20.6475	−1.14886
$324$	−5.30998	−0.294999
$325$	−1.00000	−0.0554700
$326$	14.5685	0.806876
$327$	3.40557	0.188328
$328$	5.05241	0.278973
$329$	20.4017	1.12478
$330$	−2.36834	−0.130373
$331$	8.51246	0.467887	0.233944	−	0.972250i	$-0.424837\pi$
$331$	8.51246	0.467887	0.233944	+	0.972250i	$0.424837\pi$
$332$	6.84106	0.375452
$333$	−2.25089	−0.123348
$334$	−24.1771	−1.32291
$335$	6.50071	0.355172
$336$	3.94710	0.215332
$337$	16.2731	0.886451	0.443225	−	0.896410i	$-0.353834\pi$
$337$	16.2731	0.886451	0.443225	+	0.896410i	$0.353834\pi$
$338$	−1.00000	−0.0543928
$339$	18.7981	1.02097
$340$	4.89350	0.265387
$341$	1.64899	0.0892976
$342$	−3.95446	−0.213833
$343$	17.7186	0.956713
$344$	10.0960	0.544341
$345$	1.02540	0.0552059
$346$	−16.4727	−0.885576
$347$	−14.1701	−0.760690	−0.380345	−	0.924845i	$-0.624195\pi$
$347$	−14.1701	−0.760690	−0.380345	+	0.924845i	$0.624195\pi$
$348$	7.69482	0.412485
$349$	−28.9399	−1.54912	−0.774558	−	0.632502i	$-0.782028\pi$
$349$	−28.9399	−1.54912	−0.774558	+	0.632502i	$0.782028\pi$
$350$	2.74822	0.146898
$351$	−5.65479	−0.301830
$352$	1.64899	0.0878912
$353$	−30.4264	−1.61943	−0.809716	−	0.586822i	$-0.800379\pi$
$353$	−30.4264	−1.61943	−0.809716	+	0.586822i	$0.800379\pi$
$354$	−9.47424	−0.503550
$355$	5.91227	0.313791
$356$	4.57817	0.242643
$357$	19.3151	1.02227
$358$	−4.31164	−0.227877
$359$	−30.1689	−1.59225	−0.796126	−	0.605130i	$-0.793121\pi$
$359$	−30.1689	−1.59225	−0.796126	+	0.605130i	$0.793121\pi$
$360$	0.937215	0.0493956
$361$	−1.19690	−0.0629947
$362$	0.524825	0.0275842
$363$	11.8933	0.624235
$364$	2.74822	0.144046
$365$	16.4812	0.862664
$366$	7.81798	0.408653
$367$	13.8740	0.724219	0.362109	−	0.932136i	$-0.382057\pi$
$367$	13.8740	0.724219	0.362109	+	0.932136i	$0.382057\pi$
$368$	−0.713951	−0.0372173
$369$	4.73520	0.246504
$370$	−2.40168	−0.124857
$371$	5.14192	0.266955
$372$	1.43624	0.0744656
$373$	5.95195	0.308180	0.154090	−	0.988057i	$-0.450755\pi$
$373$	5.95195	0.308180	0.154090	+	0.988057i	$0.450755\pi$
$374$	8.06931	0.417254
$375$	−1.43624	−0.0741671
$376$	7.42361	0.382844
$377$	5.35761	0.275931
$378$	15.5406	0.799322
$379$	21.6132	1.11020	0.555098	−	0.831785i	$-0.312681\pi$
$379$	21.6132	1.11020	0.555098	+	0.831785i	$0.312681\pi$
$380$	−4.21937	−0.216449
$381$	−4.87394	−0.249699
$382$	3.93960	0.201567
$383$	−28.7618	−1.46966	−0.734831	−	0.678250i	$-0.762738\pi$
$383$	−28.7618	−1.46966	−0.734831	+	0.678250i	$0.762738\pi$
$384$	1.43624	0.0732928
$385$	4.53177	0.230961
$386$	−17.2505	−0.878029
$387$	9.46215	0.480988
$388$	16.3044	0.827731
$389$	9.69466	0.491539	0.245769	−	0.969328i	$-0.420959\pi$
$389$	9.69466	0.491539	0.245769	+	0.969328i	$0.420959\pi$
$390$	−1.43624	−0.0727268
$391$	−3.49372	−0.176685
$392$	−0.552707	−0.0279159
$393$	−25.4604	−1.28431
$394$	−11.2690	−0.567725
$395$	−2.80072	−0.140920
$396$	1.54545	0.0776620
$397$	−23.7160	−1.19027	−0.595137	−	0.803624i	$-0.702902\pi$
$397$	−23.7160	−1.19027	−0.595137	+	0.803624i	$0.702902\pi$
$398$	−20.0876	−1.00690
$399$	−16.6543	−0.833757
$400$	1.00000	0.0500000
$401$	31.8176	1.58890	0.794448	−	0.607332i	$-0.207760\pi$
$401$	31.8176	1.58890	0.794448	+	0.607332i	$0.207760\pi$
$402$	9.33658	0.465666
$403$	1.00000	0.0498135
$404$	−18.3652	−0.913705
$405$	−5.30998	−0.263855
$406$	−14.7239	−0.730734
$407$	−3.96033	−0.196306
$408$	7.02824	0.347950
$409$	−9.02657	−0.446335	−0.223168	−	0.974780i	$-0.571640\pi$
$409$	−9.02657	−0.446335	−0.223168	+	0.974780i	$0.571640\pi$
$410$	5.05241	0.249521
$411$	−7.49838	−0.369868
$412$	−3.90207	−0.192241
$413$	18.1288	0.892059
$414$	−0.669125	−0.0328857
$415$	6.84106	0.335814
$416$	1.00000	0.0490290
$417$	−4.30861	−0.210994
$418$	−6.95768	−0.340311
$419$	31.8516	1.55605	0.778027	−	0.628231i	$-0.216221\pi$
$419$	31.8516	1.55605	0.778027	+	0.628231i	$0.216221\pi$
$420$	3.94710	0.192599
$421$	28.3408	1.38124	0.690622	−	0.723216i	$-0.257337\pi$
$421$	28.3408	1.38124	0.690622	+	0.723216i	$0.257337\pi$
$422$	28.4397	1.38442
$423$	6.95752	0.338286
$424$	1.87100	0.0908639
$425$	4.89350	0.237370
$426$	8.49144	0.411411
$427$	−14.9596	−0.723944
$428$	−12.3139	−0.595217
$429$	−2.36834	−0.114344
$430$	10.0960	0.486874
$431$	30.8581	1.48638	0.743190	−	0.669080i	$-0.233312\pi$
$431$	30.8581	1.48638	0.743190	+	0.669080i	$0.233312\pi$
$432$	5.65479	0.272066
$433$	−12.9071	−0.620277	−0.310139	−	0.950691i	$-0.600375\pi$
$433$	−12.9071	−0.620277	−0.310139	+	0.950691i	$0.600375\pi$
$434$	−2.74822	−0.131919
$435$	7.69482	0.368938
$436$	−2.37117	−0.113559
$437$	3.01242	0.144104
$438$	23.6709	1.13104
$439$	−7.11467	−0.339565	−0.169782	−	0.985482i	$-0.554306\pi$
$439$	−7.11467	−0.339565	−0.169782	+	0.985482i	$0.554306\pi$
$440$	1.64899	0.0786123
$441$	−0.518006	−0.0246669
$442$	4.89350	0.232760
$443$	−10.2059	−0.484898	−0.242449	−	0.970164i	$-0.577951\pi$
$443$	−10.2059	−0.484898	−0.242449	+	0.970164i	$0.577951\pi$
$444$	−3.44938	−0.163700
$445$	4.57817	0.217026
$446$	5.70670	0.270220
$447$	−10.7296	−0.507491
$448$	−2.74822	−0.129841
$449$	11.7303	0.553588	0.276794	−	0.960929i	$-0.410728\pi$
$449$	11.7303	0.553588	0.276794	+	0.960929i	$0.410728\pi$
$450$	0.937215	0.0441807
$451$	8.33135	0.392308
$452$	−13.0884	−0.615628
$453$	23.9675	1.12609
$454$	−24.9586	−1.17137
$455$	2.74822	0.128838
$456$	−6.06003	−0.283787
$457$	10.2959	0.481624	0.240812	−	0.970572i	$-0.422586\pi$
$457$	10.2959	0.481624	0.240812	+	0.970572i	$0.422586\pi$
$458$	−2.38067	−0.111241
$459$	27.6717	1.29160
$460$	−0.713951	−0.0332881
$461$	14.4810	0.674446	0.337223	−	0.941425i	$-0.390512\pi$
$461$	14.4810	0.674446	0.337223	+	0.941425i	$0.390512\pi$
$462$	6.50871	0.302813
$463$	−37.3404	−1.73536	−0.867678	−	0.497127i	$-0.834388\pi$
$463$	−37.3404	−1.73536	−0.867678	+	0.497127i	$0.834388\pi$
$464$	−5.35761	−0.248721
$465$	1.43624	0.0666040
$466$	−26.9022	−1.24622
$467$	35.6819	1.65116	0.825580	−	0.564285i	$-0.190848\pi$
$467$	35.6819	1.65116	0.825580	+	0.564285i	$0.190848\pi$
$468$	0.937215	0.0433228
$469$	−17.8654	−0.824947
$470$	7.42361	0.342426
$471$	0.576886	0.0265815
$472$	6.59656	0.303631
$473$	16.6482	0.765485
$474$	−4.02251	−0.184760
$475$	−4.21937	−0.193598
$476$	−13.4484	−0.616407
$477$	1.75353	0.0802887
$478$	−24.8744	−1.13773
$479$	33.0588	1.51049	0.755247	−	0.655440i	$-0.227517\pi$
$479$	33.0588	1.51049	0.755247	+	0.655440i	$0.227517\pi$
$480$	1.43624	0.0655551
$481$	−2.40168	−0.109507
$482$	−0.940565	−0.0428416
$483$	−2.81804	−0.128225
$484$	−8.28085	−0.376402
$485$	16.3044	0.740345
$486$	9.33795	0.423578
$487$	28.6962	1.30035	0.650174	−	0.759785i	$-0.274696\pi$
$487$	28.6962	1.30035	0.650174	+	0.759785i	$0.274696\pi$
$488$	−5.44337	−0.246410
$489$	20.9239	0.946212
$490$	−0.552707	−0.0249688
$491$	32.6122	1.47177	0.735883	−	0.677109i	$-0.236767\pi$
$491$	32.6122	1.47177	0.735883	+	0.677109i	$0.236767\pi$
$492$	7.25647	0.327147
$493$	−26.2175	−1.18078
$494$	−4.21937	−0.189839
$495$	1.54545	0.0694630
$496$	−1.00000	−0.0449013
$497$	−16.2482	−0.728832
$498$	9.82540	0.440287
$499$	−14.4104	−0.645097	−0.322549	−	0.946553i	$-0.604540\pi$
$499$	−14.4104	−0.645097	−0.322549	+	0.946553i	$0.604540\pi$
$500$	1.00000	0.0447214
$501$	−34.7242	−1.55136
$502$	2.34387	0.104612
$503$	−25.8760	−1.15375	−0.576877	−	0.816831i	$-0.695729\pi$
$503$	−25.8760	−1.15375	−0.576877	+	0.816831i	$0.695729\pi$
$504$	−2.57567	−0.114730
$505$	−18.3652	−0.817242
$506$	−1.17729	−0.0523371
$507$	−1.43624	−0.0637857
$508$	3.39354	0.150564
$509$	−31.1500	−1.38070	−0.690350	−	0.723476i	$-0.742543\pi$
$509$	−31.1500	−1.38070	−0.690350	+	0.723476i	$0.742543\pi$
$510$	7.02824	0.311216
$511$	−45.2939	−2.00368
$512$	−1.00000	−0.0441942
$513$	−23.8596	−1.05343
$514$	13.4562	0.593525
$515$	−3.90207	−0.171946
$516$	14.5003	0.638341
$517$	12.2414	0.538377
$518$	6.60033	0.290002
$519$	−23.6587	−1.03850
$520$	1.00000	0.0438529
$521$	25.3713	1.11154	0.555769	−	0.831337i	$-0.312424\pi$
$521$	25.3713	1.11154	0.555769	+	0.831337i	$0.312424\pi$
$522$	−5.02123	−0.219773
$523$	10.4505	0.456969	0.228485	−	0.973548i	$-0.426623\pi$
$523$	10.4505	0.456969	0.228485	+	0.973548i	$0.426623\pi$
$524$	17.7271	0.774414
$525$	3.94710	0.172266
$526$	5.44730	0.237513
$527$	−4.89350	−0.213164
$528$	2.36834	0.103069
$529$	−22.4903	−0.977838
$530$	1.87100	0.0812711
$531$	6.18239	0.268293
$532$	11.5958	0.502740
$533$	5.05241	0.218844
$534$	6.57535	0.284543
$535$	−12.3139	−0.532378
$536$	−6.50071	−0.280788
$537$	−6.19255	−0.267228
$538$	16.0405	0.691555
$539$	−0.911406	−0.0392570
$540$	5.65479	0.243343
$541$	26.7989	1.15217	0.576086	−	0.817389i	$-0.304579\pi$
$541$	26.7989	1.15217	0.576086	+	0.817389i	$0.304579\pi$
$542$	11.8305	0.508163
$543$	0.753775	0.0323476
$544$	−4.89350	−0.209807
$545$	−2.37117	−0.101570
$546$	3.94710	0.168920
$547$	−21.6199	−0.924399	−0.462200	−	0.886776i	$-0.652940\pi$
$547$	−21.6199	−0.924399	−0.462200	+	0.886776i	$0.652940\pi$
$548$	5.22084	0.223023
$549$	−5.10161	−0.217731
$550$	1.64899	0.0703130
$551$	22.6058	0.963038
$552$	−1.02540	−0.0436441
$553$	7.69701	0.327310
$554$	−6.80363	−0.289059
$555$	−3.44938	−0.146418
$556$	2.99992	0.127225
$557$	39.5889	1.67744	0.838718	−	0.544566i	$-0.183306\pi$
$557$	39.5889	1.67744	0.838718	+	0.544566i	$0.183306\pi$
$558$	−0.937215	−0.0396755
$559$	10.0960	0.427016
$560$	−2.74822	−0.116133
$561$	11.5895	0.489308
$562$	−27.7172	−1.16918
$563$	15.4987	0.653191	0.326596	−	0.945164i	$-0.394098\pi$
$563$	15.4987	0.653191	0.326596	+	0.945164i	$0.394098\pi$
$564$	10.6621	0.448955
$565$	−13.0884	−0.550634
$566$	−11.6535	−0.489833
$567$	14.5930	0.612848
$568$	−5.91227	−0.248073
$569$	3.98560	0.167085	0.0835425	−	0.996504i	$-0.473377\pi$
$569$	3.98560	0.167085	0.0835425	+	0.996504i	$0.473377\pi$
$570$	−6.06003	−0.253827
$571$	21.0013	0.878879	0.439439	−	0.898272i	$-0.355177\pi$
$571$	21.0013	0.878879	0.439439	+	0.898272i	$0.355177\pi$
$572$	1.64899	0.0689475
$573$	5.65821	0.236375
$574$	−13.8851	−0.579554
$575$	−0.713951	−0.0297738
$576$	−0.937215	−0.0390506
$577$	38.9452	1.62131	0.810655	−	0.585523i	$-0.199111\pi$
$577$	38.9452	1.62131	0.810655	+	0.585523i	$0.199111\pi$
$578$	−6.94635	−0.288930
$579$	−24.7759	−1.02965
$580$	−5.35761	−0.222463
$581$	−18.8007	−0.779985
$582$	23.4171	0.970668
$583$	3.08525	0.127778
$584$	−16.4812	−0.681996
$585$	0.937215	0.0387491
$586$	−21.1512	−0.873747
$587$	−23.5896	−0.973646	−0.486823	−	0.873501i	$-0.661844\pi$
$587$	−23.5896	−0.973646	−0.486823	+	0.873501i	$0.661844\pi$
$588$	−0.793820	−0.0327366
$589$	4.21937	0.173856
$590$	6.59656	0.271576
$591$	−16.1850	−0.665762
$592$	2.40168	0.0987083
$593$	3.41757	0.140343	0.0701715	−	0.997535i	$-0.477645\pi$
$593$	3.41757	0.140343	0.0701715	+	0.997535i	$0.477645\pi$
$594$	9.32466	0.382595
$595$	−13.4484	−0.551331
$596$	7.47059	0.306007
$597$	−28.8506	−1.18077
$598$	−0.713951	−0.0291956
$599$	−37.4964	−1.53206	−0.766031	−	0.642804i	$-0.777771\pi$
$599$	−37.4964	−1.53206	−0.766031	+	0.642804i	$0.777771\pi$
$600$	1.43624	0.0586342
$601$	−38.0492	−1.55206	−0.776029	−	0.630698i	$-0.782769\pi$
$601$	−38.0492	−1.55206	−0.776029	+	0.630698i	$0.782769\pi$
$602$	−27.7461	−1.13085
$603$	−6.09257	−0.248108
$604$	−16.6877	−0.679013
$605$	−8.28085	−0.336664
$606$	−26.3769	−1.07149
$607$	0.117501	0.00476920	0.00238460	−	0.999997i	$-0.499241\pi$
$607$	0.117501	0.00476920	0.00238460	+	0.999997i	$0.499241\pi$
$608$	4.21937	0.171118
$609$	−21.1470	−0.856921
$610$	−5.44337	−0.220396
$611$	7.42361	0.300327
$612$	−4.58626	−0.185389
$613$	1.54114	0.0622461	0.0311231	−	0.999516i	$-0.490092\pi$
$613$	1.54114	0.0622461	0.0311231	+	0.999516i	$0.490092\pi$
$614$	−2.89749	−0.116933
$615$	7.25647	0.292609
$616$	−4.53177	−0.182590
$617$	−34.0912	−1.37246	−0.686231	−	0.727384i	$-0.740736\pi$
$617$	−34.0912	−1.37246	−0.686231	+	0.727384i	$0.740736\pi$
$618$	−5.60431	−0.225438
$619$	−14.7252	−0.591856	−0.295928	−	0.955210i	$-0.595629\pi$
$619$	−14.7252	−0.591856	−0.295928	+	0.955210i	$0.595629\pi$
$620$	−1.00000	−0.0401610
$621$	−4.03724	−0.162009
$622$	−13.7875	−0.552829
$623$	−12.5818	−0.504080
$624$	1.43624	0.0574956
$625$	1.00000	0.0400000
$626$	−20.0590	−0.801721
$627$	−9.99290	−0.399078
$628$	−0.401664	−0.0160281
$629$	11.7526	0.468607
$630$	−2.57567	−0.102617
$631$	29.6457	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$631$	29.6457	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$632$	2.80072	0.111407
$633$	40.8462	1.62349
$634$	−16.8042	−0.667381
$635$	3.39354	0.134669
$636$	2.68721	0.106555
$637$	−0.552707	−0.0218991
$638$	−8.83462	−0.349766
$639$	−5.54107	−0.219201
$640$	−1.00000	−0.0395285
$641$	3.36819	0.133036	0.0665178	−	0.997785i	$-0.478811\pi$
$641$	3.36819	0.133036	0.0665178	+	0.997785i	$0.478811\pi$
$642$	−17.6858	−0.698002
$643$	34.2250	1.34970	0.674851	−	0.737954i	$-0.264208\pi$
$643$	34.2250	1.34970	0.674851	+	0.737954i	$0.264208\pi$
$644$	1.96209	0.0773173
$645$	14.5003	0.570949
$646$	20.6475	0.812365
$647$	36.7915	1.44642	0.723212	−	0.690626i	$-0.242665\pi$
$647$	36.7915	1.44642	0.723212	+	0.690626i	$0.242665\pi$
$648$	5.30998	0.208596
$649$	10.8776	0.426984
$650$	1.00000	0.0392232
$651$	−3.94710	−0.154699
$652$	−14.5685	−0.570548
$653$	−18.0207	−0.705203	−0.352601	−	0.935774i	$-0.614703\pi$
$653$	−18.0207	−0.705203	−0.352601	+	0.935774i	$0.614703\pi$
$654$	−3.40557	−0.133168
$655$	17.7271	0.692657
$656$	−5.05241	−0.197264
$657$	−15.4464	−0.602622
$658$	−20.4017	−0.795342
$659$	20.4367	0.796102	0.398051	−	0.917363i	$-0.369687\pi$
$659$	20.4367	0.796102	0.398051	+	0.917363i	$0.369687\pi$
$660$	2.36834	0.0921874
$661$	14.3873	0.559600	0.279800	−	0.960058i	$-0.409732\pi$
$661$	14.3873	0.559600	0.279800	+	0.960058i	$0.409732\pi$
$662$	−8.51246	−0.330846
$663$	7.02824	0.272954
$664$	−6.84106	−0.265484
$665$	11.5958	0.449664
$666$	2.25089	0.0872201
$667$	3.82507	0.148107
$668$	24.1771	0.935441
$669$	8.19619	0.316883
$670$	−6.50071	−0.251144
$671$	−8.97604	−0.346516
$672$	−3.94710	−0.152263
$673$	−19.8770	−0.766201	−0.383100	−	0.923707i	$-0.625144\pi$
$673$	−19.8770	−0.766201	−0.383100	+	0.923707i	$0.625144\pi$
$674$	−16.2731	−0.626815
$675$	5.65479	0.217653
$676$	1.00000	0.0384615
$677$	8.77539	0.337266	0.168633	−	0.985679i	$-0.446065\pi$
$677$	8.77539	0.337266	0.168633	+	0.985679i	$0.446065\pi$
$678$	−18.7981	−0.721938
$679$	−44.8081	−1.71958
$680$	−4.89350	−0.187657
$681$	−35.8466	−1.37364
$682$	−1.64899	−0.0631429
$683$	−43.4360	−1.66203	−0.831016	−	0.556249i	$-0.812240\pi$
$683$	−43.4360	−1.66203	−0.831016	+	0.556249i	$0.812240\pi$
$684$	3.95446	0.151203
$685$	5.22084	0.199478
$686$	−17.7186	−0.676498
$687$	−3.41921	−0.130451
$688$	−10.0960	−0.384907
$689$	1.87100	0.0712795
$690$	−1.02540	−0.0390365
$691$	−47.9848	−1.82543	−0.912714	−	0.408598i	$-0.866018\pi$
$691$	−47.9848	−1.82543	−0.912714	+	0.408598i	$0.866018\pi$
$692$	16.4727	0.626197
$693$	−4.24725	−0.161340
$694$	14.1701	0.537889
$695$	2.99992	0.113794
$696$	−7.69482	−0.291671
$697$	−24.7240	−0.936487
$698$	28.9399	1.09539
$699$	−38.6380	−1.46142
$700$	−2.74822	−0.103873
$701$	−39.3730	−1.48710	−0.743549	−	0.668682i	$-0.766859\pi$
$701$	−39.3730	−1.48710	−0.743549	+	0.668682i	$0.766859\pi$
$702$	5.65479	0.213426
$703$	−10.1336	−0.382195
$704$	−1.64899	−0.0621485
$705$	10.6621	0.401558
$706$	30.4264	1.14511
$707$	50.4717	1.89818
$708$	9.47424	0.356064
$709$	−14.3837	−0.540190	−0.270095	−	0.962834i	$-0.587055\pi$
$709$	−14.3837	−0.540190	−0.270095	+	0.962834i	$0.587055\pi$
$710$	−5.91227	−0.221884
$711$	2.62488	0.0984408
$712$	−4.57817	−0.171574
$713$	0.713951	0.0267377
$714$	−19.3151	−0.722851
$715$	1.64899	0.0616686
$716$	4.31164	0.161134
$717$	−35.7256	−1.33420
$718$	30.1689	1.12589
$719$	10.7772	0.401921	0.200961	−	0.979599i	$-0.435594\pi$
$719$	10.7772	0.401921	0.200961	+	0.979599i	$0.435594\pi$
$720$	−0.937215	−0.0349279
$721$	10.7237	0.399373
$722$	1.19690	0.0445439
$723$	−1.35088	−0.0502397
$724$	−0.524825	−0.0195050
$725$	−5.35761	−0.198977
$726$	−11.8933	−0.441401
$727$	−35.8407	−1.32926	−0.664628	−	0.747174i	$-0.731410\pi$
$727$	−35.8407	−1.32926	−0.664628	+	0.747174i	$0.731410\pi$
$728$	−2.74822	−0.101856
$729$	29.3415	1.08672
$730$	−16.4812	−0.609996
$731$	−49.4049	−1.82731
$732$	−7.81798	−0.288961
$733$	18.0773	0.667700	0.333850	−	0.942626i	$-0.391652\pi$
$733$	18.0773	0.667700	0.333850	+	0.942626i	$0.391652\pi$
$734$	−13.8740	−0.512100
$735$	−0.793820	−0.0292805
$736$	0.713951	0.0263166
$737$	−10.7196	−0.394861
$738$	−4.73520	−0.174305
$739$	24.9597	0.918158	0.459079	−	0.888395i	$-0.348179\pi$
$739$	24.9597	0.918158	0.459079	+	0.888395i	$0.348179\pi$
$740$	2.40168	0.0882874
$741$	−6.06003	−0.222621
$742$	−5.14192	−0.188766
$743$	−6.04673	−0.221833	−0.110916	−	0.993830i	$-0.535379\pi$
$743$	−6.04673	−0.221833	−0.110916	+	0.993830i	$0.535379\pi$
$744$	−1.43624	−0.0526551
$745$	7.47059	0.273701
$746$	−5.95195	−0.217916
$747$	−6.41154	−0.234586
$748$	−8.06931	−0.295043
$749$	33.8414	1.23654
$750$	1.43624	0.0524441
$751$	−21.5041	−0.784694	−0.392347	−	0.919817i	$-0.628337\pi$
$751$	−21.5041	−0.784694	−0.392347	+	0.919817i	$0.628337\pi$
$752$	−7.42361	−0.270711
$753$	3.36636	0.122677
$754$	−5.35761	−0.195113
$755$	−16.6877	−0.607328
$756$	−15.5406	−0.565206
$757$	−21.0198	−0.763978	−0.381989	−	0.924167i	$-0.624761\pi$
$757$	−21.0198	−0.763978	−0.381989	+	0.924167i	$0.624761\pi$
$758$	−21.6132	−0.785026
$759$	−1.69088	−0.0613749
$760$	4.21937	0.153053
$761$	4.09850	0.148570	0.0742852	−	0.997237i	$-0.476332\pi$
$761$	4.09850	0.148570	0.0742852	+	0.997237i	$0.476332\pi$
$762$	4.87394	0.176564
$763$	6.51650	0.235913
$764$	−3.93960	−0.142530
$765$	−4.58626	−0.165817
$766$	28.7618	1.03921
$767$	6.59656	0.238188
$768$	−1.43624	−0.0518258
$769$	18.0167	0.649700	0.324850	−	0.945765i	$-0.394686\pi$
$769$	18.0167	0.649700	0.324850	+	0.945765i	$0.394686\pi$
$770$	−4.53177	−0.163314
$771$	19.3263	0.696018
$772$	17.2505	0.620861
$773$	−52.2323	−1.87867	−0.939333	−	0.343006i	$-0.888555\pi$
$773$	−52.2323	−1.87867	−0.939333	+	0.343006i	$0.888555\pi$
$774$	−9.46215	−0.340110
$775$	−1.00000	−0.0359211
$776$	−16.3044	−0.585294
$777$	9.47966	0.340081
$778$	−9.69466	−0.347570
$779$	21.3180	0.763797
$780$	1.43624	0.0514256
$781$	−9.74925	−0.348855
$782$	3.49372	0.124935
$783$	−30.2961	−1.08270
$784$	0.552707	0.0197395
$785$	−0.401664	−0.0143360
$786$	25.4604	0.908144
$787$	−16.9591	−0.604527	−0.302263	−	0.953224i	$-0.597742\pi$
$787$	−16.9591	−0.604527	−0.302263	+	0.953224i	$0.597742\pi$
$788$	11.2690	0.401442
$789$	7.82362	0.278528
$790$	2.80072	0.0996453
$791$	35.9699	1.27894
$792$	−1.54545	−0.0549153
$793$	−5.44337	−0.193300
$794$	23.7160	0.841651
$795$	2.68721	0.0953054
$796$	20.0876	0.711985
$797$	43.4395	1.53871	0.769353	−	0.638824i	$-0.220579\pi$
$797$	43.4395	1.53871	0.769353	+	0.638824i	$0.220579\pi$
$798$	16.6543	0.589555
$799$	−36.3275	−1.28517
$800$	−1.00000	−0.0353553
$801$	−4.29073	−0.151606
$802$	−31.8176	−1.12352
$803$	−27.1772	−0.959063
$804$	−9.33658	−0.329276
$805$	1.96209	0.0691547
$806$	−1.00000	−0.0352235
$807$	23.0380	0.810976
$808$	18.3652	0.646087
$809$	30.2386	1.06313	0.531567	−	0.847016i	$-0.321603\pi$
$809$	30.2386	1.06313	0.531567	+	0.847016i	$0.321603\pi$
$810$	5.30998	0.186574
$811$	−40.0121	−1.40502	−0.702508	−	0.711676i	$-0.747937\pi$
$811$	−40.0121	−1.40502	−0.702508	+	0.711676i	$0.747937\pi$
$812$	14.7239	0.516707
$813$	16.9914	0.595915
$814$	3.96033	0.138809
$815$	−14.5685	−0.510313
$816$	−7.02824	−0.246038
$817$	42.5989	1.49035
$818$	9.02657	0.315607
$819$	−2.57567	−0.0900013
$820$	−5.05241	−0.176438
$821$	−14.3507	−0.500843	−0.250421	−	0.968137i	$-0.580569\pi$
$821$	−14.3507	−0.500843	−0.250421	+	0.968137i	$0.580569\pi$
$822$	7.49838	0.261536
$823$	−21.1438	−0.737025	−0.368513	−	0.929623i	$-0.620133\pi$
$823$	−21.1438	−0.737025	−0.368513	+	0.929623i	$0.620133\pi$
$824$	3.90207	0.135935
$825$	2.36834	0.0824550
$826$	−18.1288	−0.630781
$827$	13.8892	0.482976	0.241488	−	0.970404i	$-0.422365\pi$
$827$	13.8892	0.482976	0.241488	+	0.970404i	$0.422365\pi$
$828$	0.669125	0.0232537
$829$	−9.59334	−0.333191	−0.166595	−	0.986025i	$-0.553277\pi$
$829$	−9.59334	−0.333191	−0.166595	+	0.986025i	$0.553277\pi$
$830$	−6.84106	−0.237457
$831$	−9.77165	−0.338975
$832$	−1.00000	−0.0346688
$833$	2.70467	0.0937114
$834$	4.30861	0.149195
$835$	24.1771	0.836684
$836$	6.95768	0.240636
$837$	−5.65479	−0.195458
$838$	−31.8516	−1.10030
$839$	−34.4058	−1.18782	−0.593911	−	0.804531i	$-0.702417\pi$
$839$	−34.4058	−1.18782	−0.593911	+	0.804531i	$0.702417\pi$
$840$	−3.94710	−0.136188
$841$	−0.295997	−0.0102068
$842$	−28.3408	−0.976687
$843$	−39.8086	−1.37108
$844$	−28.4397	−0.978935
$845$	1.00000	0.0344010
$846$	−6.95752	−0.239205
$847$	22.7576	0.781960
$848$	−1.87100	−0.0642505
$849$	−16.7372	−0.574420
$850$	−4.89350	−0.167846
$851$	−1.71468	−0.0587784
$852$	−8.49144	−0.290912
$853$	−5.61802	−0.192357	−0.0961786	−	0.995364i	$-0.530662\pi$
$853$	−5.61802	−0.192357	−0.0961786	+	0.995364i	$0.530662\pi$
$854$	14.9596	0.511906
$855$	3.95446	0.135240
$856$	12.3139	0.420882
$857$	53.0913	1.81357	0.906783	−	0.421599i	$-0.138531\pi$
$857$	53.0913	1.81357	0.906783	+	0.421599i	$0.138531\pi$
$858$	2.36834	0.0808537
$859$	44.3835	1.51435	0.757173	−	0.653214i	$-0.226580\pi$
$859$	44.3835	1.51435	0.757173	+	0.653214i	$0.226580\pi$
$860$	−10.0960	−0.344272
$861$	−19.9424	−0.679635
$862$	−30.8581	−1.05103
$863$	−13.0064	−0.442744	−0.221372	−	0.975189i	$-0.571053\pi$
$863$	−13.0064	−0.442744	−0.221372	+	0.975189i	$0.571053\pi$
$864$	−5.65479	−0.192380
$865$	16.4727	0.560087
$866$	12.9071	0.438602
$867$	−9.97662	−0.338824
$868$	2.74822	0.0932806
$869$	4.61835	0.156667
$870$	−7.69482	−0.260879
$871$	−6.50071	−0.220268
$872$	2.37117	0.0802980
$873$	−15.2807	−0.517175
$874$	−3.01242	−0.101897
$875$	−2.74822	−0.0929068
$876$	−23.6709	−0.799767
$877$	−11.7109	−0.395447	−0.197724	−	0.980258i	$-0.563355\pi$
$877$	−11.7109	−0.395447	−0.197724	+	0.980258i	$0.563355\pi$
$878$	7.11467	0.240109
$879$	−30.3782	−1.02463
$880$	−1.64899	−0.0555873
$881$	−32.3689	−1.09054	−0.545268	−	0.838262i	$-0.683572\pi$
$881$	−32.3689	−1.09054	−0.545268	+	0.838262i	$0.683572\pi$
$882$	0.518006	0.0174422
$883$	−23.6522	−0.795961	−0.397980	−	0.917394i	$-0.630289\pi$
$883$	−23.6522	−0.795961	−0.397980	+	0.917394i	$0.630289\pi$
$884$	−4.89350	−0.164586
$885$	9.47424	0.318473
$886$	10.2059	0.342875
$887$	29.0830	0.976511	0.488256	−	0.872701i	$-0.337633\pi$
$887$	29.0830	0.976511	0.488256	+	0.872701i	$0.337633\pi$
$888$	3.44938	0.115754
$889$	−9.32619	−0.312790
$890$	−4.57817	−0.153461
$891$	8.75608	0.293340
$892$	−5.70670	−0.191074
$893$	31.3230	1.04818
$894$	10.7296	0.358850
$895$	4.31164	0.144122
$896$	2.74822	0.0918116
$897$	−1.02540	−0.0342373
$898$	−11.7303	−0.391446
$899$	5.35761	0.178686
$900$	−0.937215	−0.0312405
$901$	−9.15575	−0.305022
$902$	−8.33135	−0.277404
$903$	−39.8500	−1.32613
$904$	13.0884	0.435315
$905$	−0.524825	−0.0174458
$906$	−23.9675	−0.796268
$907$	−57.7940	−1.91902	−0.959508	−	0.281680i	$-0.909108\pi$
$907$	−57.7940	−1.91902	−0.959508	+	0.281680i	$0.909108\pi$
$908$	24.9586	0.828282
$909$	17.2122	0.570892
$910$	−2.74822	−0.0911026
$911$	12.5848	0.416952	0.208476	−	0.978028i	$-0.433150\pi$
$911$	12.5848	0.416952	0.208476	+	0.978028i	$0.433150\pi$
$912$	6.06003	0.200668
$913$	−11.2808	−0.373340
$914$	−10.2959	−0.340559
$915$	−7.81798	−0.258455
$916$	2.38067	0.0786595
$917$	−48.7181	−1.60881
$918$	−27.6717	−0.913302
$919$	34.7201	1.14531	0.572655	−	0.819796i	$-0.305913\pi$
$919$	34.7201	1.14531	0.572655	+	0.819796i	$0.305913\pi$
$920$	0.713951	0.0235383
$921$	−4.16149	−0.137126
$922$	−14.4810	−0.476905
$923$	−5.91227	−0.194605
$924$	−6.50871	−0.214121
$925$	2.40168	0.0789666
$926$	37.3404	1.22708
$927$	3.65708	0.120114
$928$	5.35761	0.175872
$929$	25.5331	0.837713	0.418857	−	0.908052i	$-0.362431\pi$
$929$	25.5331	0.837713	0.418857	+	0.908052i	$0.362431\pi$
$930$	−1.43624	−0.0470962
$931$	−2.33208	−0.0764308
$932$	26.9022	0.881211
$933$	−19.8022	−0.648294
$934$	−35.6819	−1.16755
$935$	−8.06931	−0.263895
$936$	−0.937215	−0.0306338
$937$	−30.2646	−0.988701	−0.494350	−	0.869263i	$-0.664594\pi$
$937$	−30.2646	−0.988701	−0.494350	+	0.869263i	$0.664594\pi$
$938$	17.8654	0.583325
$939$	−28.8096	−0.940166
$940$	−7.42361	−0.242132
$941$	11.1492	0.363453	0.181727	−	0.983349i	$-0.441831\pi$
$941$	11.1492	0.363453	0.181727	+	0.983349i	$0.441831\pi$
$942$	−0.576886	−0.0187960
$943$	3.60717	0.117466
$944$	−6.59656	−0.214700
$945$	−15.5406	−0.505535
$946$	−16.6482	−0.541280
$947$	−36.8427	−1.19723	−0.598614	−	0.801038i	$-0.704282\pi$
$947$	−36.8427	−1.19723	−0.598614	+	0.801038i	$0.704282\pi$
$948$	4.02251	0.130645
$949$	−16.4812	−0.535002
$950$	4.21937	0.136895
$951$	−24.1349	−0.782628
$952$	13.4484	0.435865
$953$	−30.6204	−0.991893	−0.495947	−	0.868353i	$-0.665179\pi$
$953$	−30.6204	−0.991893	−0.495947	+	0.868353i	$0.665179\pi$
$954$	−1.75353	−0.0567727
$955$	−3.93960	−0.127482
$956$	24.8744	0.804496
$957$	−12.6886	−0.410165
$958$	−33.0588	−1.06808
$959$	−14.3480	−0.463321
$960$	−1.43624	−0.0463544
$961$	1.00000	0.0322581
$962$	2.40168	0.0774332
$963$	11.5408	0.371898
$964$	0.940565	0.0302936
$965$	17.2505	0.555315
$966$	2.81804	0.0906688
$967$	−22.1790	−0.713227	−0.356614	−	0.934252i	$-0.616069\pi$
$967$	−22.1790	−0.713227	−0.356614	+	0.934252i	$0.616069\pi$
$968$	8.28085	0.266157
$969$	29.6548	0.952648
$970$	−16.3044	−0.523503
$971$	18.2549	0.585828	0.292914	−	0.956139i	$-0.405375\pi$
$971$	18.2549	0.585828	0.292914	+	0.956139i	$0.405375\pi$
$972$	−9.33795	−0.299515
$973$	−8.24445	−0.264305
$974$	−28.6962	−0.919486
$975$	1.43624	0.0459965
$976$	5.44337	0.174238
$977$	−29.4891	−0.943441	−0.471721	−	0.881748i	$-0.656367\pi$
$977$	−29.4891	−0.943441	−0.471721	+	0.881748i	$0.656367\pi$
$978$	−20.9239	−0.669073
$979$	−7.54934	−0.241278
$980$	0.552707	0.0176556
$981$	2.22230	0.0709525
$982$	−32.6122	−1.04070
$983$	0.266680	0.00850578	0.00425289	−	0.999991i	$-0.498646\pi$
$983$	0.266680	0.00850578	0.00425289	+	0.999991i	$0.498646\pi$
$984$	−7.25647	−0.231328
$985$	11.2690	0.359061
$986$	26.2175	0.834934
$987$	−29.3018	−0.932685
$988$	4.21937	0.134236
$989$	7.20807	0.229203
$990$	−1.54545	−0.0491178
$991$	−39.2234	−1.24597	−0.622987	−	0.782232i	$-0.714081\pi$
$991$	−39.2234	−1.24597	−0.622987	+	0.782232i	$0.714081\pi$
$992$	1.00000	0.0317500
$993$	−12.2259	−0.387978
$994$	16.2482	0.515362
$995$	20.0876	0.636819
$996$	−9.82540	−0.311330
$997$	30.6389	0.970343	0.485171	−	0.874419i	$-0.338757\pi$
$997$	30.6389	0.970343	0.485171	+	0.874419i	$0.338757\pi$
$998$	14.4104	0.456153
$999$	13.5810	0.429683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.l.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.l.1.3	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.43624 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.43624 q^{6} -2.74822 q^{7} -1.00000 q^{8} -0.937215 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.43624 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.43624 q^{6} -2.74822 q^{7} -1.00000 q^{8} -0.937215 q^{9} -1.00000 q^{10} -1.64899 q^{11} -1.43624 q^{12} -1.00000 q^{13} +2.74822 q^{14} -1.43624 q^{15} +1.00000 q^{16} +4.89350 q^{17} +0.937215 q^{18} -4.21937 q^{19} +1.00000 q^{20} +3.94710 q^{21} +1.64899 q^{22} -0.713951 q^{23} +1.43624 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.65479 q^{27} -2.74822 q^{28} -5.35761 q^{29} +1.43624 q^{30} -1.00000 q^{31} -1.00000 q^{32} +2.36834 q^{33} -4.89350 q^{34} -2.74822 q^{35} -0.937215 q^{36} +2.40168 q^{37} +4.21937 q^{38} +1.43624 q^{39} -1.00000 q^{40} -5.05241 q^{41} -3.94710 q^{42} -10.0960 q^{43} -1.64899 q^{44} -0.937215 q^{45} +0.713951 q^{46} -7.42361 q^{47} -1.43624 q^{48} +0.552707 q^{49} -1.00000 q^{50} -7.02824 q^{51} -1.00000 q^{52} -1.87100 q^{53} -5.65479 q^{54} -1.64899 q^{55} +2.74822 q^{56} +6.06003 q^{57} +5.35761 q^{58} -6.59656 q^{59} -1.43624 q^{60} +5.44337 q^{61} +1.00000 q^{62} +2.57567 q^{63} +1.00000 q^{64} -1.00000 q^{65} -2.36834 q^{66} +6.50071 q^{67} +4.89350 q^{68} +1.02540 q^{69} +2.74822 q^{70} +5.91227 q^{71} +0.937215 q^{72} +16.4812 q^{73} -2.40168 q^{74} -1.43624 q^{75} -4.21937 q^{76} +4.53177 q^{77} -1.43624 q^{78} -2.80072 q^{79} +1.00000 q^{80} -5.30998 q^{81} +5.05241 q^{82} +6.84106 q^{83} +3.94710 q^{84} +4.89350 q^{85} +10.0960 q^{86} +7.69482 q^{87} +1.64899 q^{88} +4.57817 q^{89} +0.937215 q^{90} +2.74822 q^{91} -0.713951 q^{92} +1.43624 q^{93} +7.42361 q^{94} -4.21937 q^{95} +1.43624 q^{96} +16.3044 q^{97} -0.552707 q^{98} +1.54545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * q^98 + 23 * q^99

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