LMFDB - Embedded newform 6038.2.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.10359 q^{3} +1.00000 q^{4} +2.00437 q^{5} +2.10359 q^{6} +2.39594 q^{7} -1.00000 q^{8} +1.42511 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.10359 q^{3} +1.00000 q^{4} +2.00437 q^{5} +2.10359 q^{6} +2.39594 q^{7} -1.00000 q^{8} +1.42511 q^{9} -2.00437 q^{10} +5.58763 q^{11} -2.10359 q^{12} -5.42366 q^{13} -2.39594 q^{14} -4.21637 q^{15} +1.00000 q^{16} +0.676646 q^{17} -1.42511 q^{18} +2.24803 q^{19} +2.00437 q^{20} -5.04009 q^{21} -5.58763 q^{22} +3.76513 q^{23} +2.10359 q^{24} -0.982519 q^{25} +5.42366 q^{26} +3.31293 q^{27} +2.39594 q^{28} -8.76966 q^{29} +4.21637 q^{30} -5.65002 q^{31} -1.00000 q^{32} -11.7541 q^{33} -0.676646 q^{34} +4.80234 q^{35} +1.42511 q^{36} +2.99375 q^{37} -2.24803 q^{38} +11.4092 q^{39} -2.00437 q^{40} -2.44323 q^{41} +5.04009 q^{42} +3.33662 q^{43} +5.58763 q^{44} +2.85644 q^{45} -3.76513 q^{46} +10.0411 q^{47} -2.10359 q^{48} -1.25946 q^{49} +0.982519 q^{50} -1.42339 q^{51} -5.42366 q^{52} +2.40410 q^{53} -3.31293 q^{54} +11.1997 q^{55} -2.39594 q^{56} -4.72894 q^{57} +8.76966 q^{58} +9.48536 q^{59} -4.21637 q^{60} +3.38922 q^{61} +5.65002 q^{62} +3.41448 q^{63} +1.00000 q^{64} -10.8710 q^{65} +11.7541 q^{66} -0.473191 q^{67} +0.676646 q^{68} -7.92031 q^{69} -4.80234 q^{70} +9.77176 q^{71} -1.42511 q^{72} +8.13986 q^{73} -2.99375 q^{74} +2.06682 q^{75} +2.24803 q^{76} +13.3876 q^{77} -11.4092 q^{78} -5.29253 q^{79} +2.00437 q^{80} -11.2444 q^{81} +2.44323 q^{82} -13.2704 q^{83} -5.04009 q^{84} +1.35625 q^{85} -3.33662 q^{86} +18.4478 q^{87} -5.58763 q^{88} +14.0323 q^{89} -2.85644 q^{90} -12.9948 q^{91} +3.76513 q^{92} +11.8854 q^{93} -10.0411 q^{94} +4.50587 q^{95} +2.10359 q^{96} +17.4530 q^{97} +1.25946 q^{98} +7.96297 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−2.10359	−1.21451	−0.607255	−	0.794507i	$-0.707729\pi$
$3$	−2.10359	−1.21451	−0.607255	+	0.794507i	$0.707729\pi$
$4$	1.00000	0.500000
$5$	2.00437	0.896380	0.448190	−	0.893938i	$-0.352069\pi$
$5$	2.00437	0.896380	0.448190	+	0.893938i	$0.352069\pi$
$6$	2.10359	0.858789
$7$	2.39594	0.905581	0.452791	−	0.891617i	$-0.350429\pi$
$7$	2.39594	0.905581	0.452791	+	0.891617i	$0.350429\pi$
$8$	−1.00000	−0.353553
$9$	1.42511	0.475036
$10$	−2.00437	−0.633836
$11$	5.58763	1.68473	0.842367	−	0.538905i	$-0.181162\pi$
$11$	5.58763	1.68473	0.842367	+	0.538905i	$0.181162\pi$
$12$	−2.10359	−0.607255
$13$	−5.42366	−1.50425	−0.752126	−	0.659019i	$-0.770972\pi$
$13$	−5.42366	−1.50425	−0.752126	+	0.659019i	$0.770972\pi$
$14$	−2.39594	−0.640343
$15$	−4.21637	−1.08866
$16$	1.00000	0.250000
$17$	0.676646	0.164111	0.0820553	−	0.996628i	$-0.473852\pi$
$17$	0.676646	0.164111	0.0820553	+	0.996628i	$0.473852\pi$
$18$	−1.42511	−0.335901
$19$	2.24803	0.515733	0.257867	−	0.966181i	$-0.416981\pi$
$19$	2.24803	0.515733	0.257867	+	0.966181i	$0.416981\pi$
$20$	2.00437	0.448190
$21$	−5.04009	−1.09984
$22$	−5.58763	−1.19129
$23$	3.76513	0.785085	0.392542	−	0.919734i	$-0.371596\pi$
$23$	3.76513	0.785085	0.392542	+	0.919734i	$0.371596\pi$
$24$	2.10359	0.429394
$25$	−0.982519	−0.196504
$26$	5.42366	1.06367
$27$	3.31293	0.637574
$28$	2.39594	0.452791
$29$	−8.76966	−1.62849	−0.814243	−	0.580525i	$-0.802848\pi$
$29$	−8.76966	−1.62849	−0.814243	+	0.580525i	$0.802848\pi$
$30$	4.21637	0.769801
$31$	−5.65002	−1.01477	−0.507387	−	0.861718i	$-0.669389\pi$
$31$	−5.65002	−1.01477	−0.507387	+	0.861718i	$0.669389\pi$
$32$	−1.00000	−0.176777
$33$	−11.7541	−2.04613
$34$	−0.676646	−0.116044
$35$	4.80234	0.811744
$36$	1.42511	0.237518
$37$	2.99375	0.492170	0.246085	−	0.969248i	$-0.420856\pi$
$37$	2.99375	0.492170	0.246085	+	0.969248i	$0.420856\pi$
$38$	−2.24803	−0.364678
$39$	11.4092	1.82693
$40$	−2.00437	−0.316918
$41$	−2.44323	−0.381569	−0.190785	−	0.981632i	$-0.561103\pi$
$41$	−2.44323	−0.381569	−0.190785	+	0.981632i	$0.561103\pi$
$42$	5.04009	0.777703
$43$	3.33662	0.508829	0.254415	−	0.967095i	$-0.418117\pi$
$43$	3.33662	0.508829	0.254415	+	0.967095i	$0.418117\pi$
$44$	5.58763	0.842367
$45$	2.85644	0.425813
$46$	−3.76513	−0.555139
$47$	10.0411	1.46465	0.732324	−	0.680956i	$-0.238436\pi$
$47$	10.0411	1.46465	0.732324	+	0.680956i	$0.238436\pi$
$48$	−2.10359	−0.303628
$49$	−1.25946	−0.179923
$50$	0.982519	0.138949
$51$	−1.42339	−0.199314
$52$	−5.42366	−0.752126
$53$	2.40410	0.330229	0.165115	−	0.986274i	$-0.447201\pi$
$53$	2.40410	0.330229	0.165115	+	0.986274i	$0.447201\pi$
$54$	−3.31293	−0.450833
$55$	11.1997	1.51016
$56$	−2.39594	−0.320171
$57$	−4.72894	−0.626363
$58$	8.76966	1.15151
$59$	9.48536	1.23489	0.617444	−	0.786615i	$-0.288168\pi$
$59$	9.48536	1.23489	0.617444	+	0.786615i	$0.288168\pi$
$60$	−4.21637	−0.544331
$61$	3.38922	0.433945	0.216972	−	0.976178i	$-0.430382\pi$
$61$	3.38922	0.433945	0.216972	+	0.976178i	$0.430382\pi$
$62$	5.65002	0.717554
$63$	3.41448	0.430184
$64$	1.00000	0.125000
$65$	−10.8710	−1.34838
$66$	11.7541	1.44683
$67$	−0.473191	−0.0578095	−0.0289047	−	0.999582i	$-0.509202\pi$
$67$	−0.473191	−0.0578095	−0.0289047	+	0.999582i	$0.509202\pi$
$68$	0.676646	0.0820553
$69$	−7.92031	−0.953494
$70$	−4.80234	−0.573990
$71$	9.77176	1.15970	0.579848	−	0.814725i	$-0.303112\pi$
$71$	9.77176	1.15970	0.579848	+	0.814725i	$0.303112\pi$
$72$	−1.42511	−0.167951
$73$	8.13986	0.952699	0.476349	−	0.879256i	$-0.341960\pi$
$73$	8.13986	0.952699	0.476349	+	0.879256i	$0.341960\pi$
$74$	−2.99375	−0.348017
$75$	2.06682	0.238656
$76$	2.24803	0.257867
$77$	13.3876	1.52566
$78$	−11.4092	−1.29184
$79$	−5.29253	−0.595456	−0.297728	−	0.954651i	$-0.596229\pi$
$79$	−5.29253	−0.595456	−0.297728	+	0.954651i	$0.596229\pi$
$80$	2.00437	0.224095
$81$	−11.2444	−1.24938
$82$	2.44323	0.269810
$83$	−13.2704	−1.45662	−0.728308	−	0.685250i	$-0.759693\pi$
$83$	−13.2704	−1.45662	−0.728308	+	0.685250i	$0.759693\pi$
$84$	−5.04009	−0.549919
$85$	1.35625	0.147105
$86$	−3.33662	−0.359797
$87$	18.4478	1.97781
$88$	−5.58763	−0.595643
$89$	14.0323	1.48742	0.743710	−	0.668502i	$-0.233064\pi$
$89$	14.0323	1.48742	0.743710	+	0.668502i	$0.233064\pi$
$90$	−2.85644	−0.301095
$91$	−12.9948	−1.36222
$92$	3.76513	0.392542
$93$	11.8854	1.23245
$94$	−10.0411	−1.03566
$95$	4.50587	0.462293
$96$	2.10359	0.214697
$97$	17.4530	1.77208	0.886042	−	0.463605i	$-0.153444\pi$
$97$	17.4530	1.77208	0.886042	+	0.463605i	$0.153444\pi$
$98$	1.25946	0.127225
$99$	7.96297	0.800309
$100$	−0.982519	−0.0982519
$101$	5.39088	0.536412	0.268206	−	0.963362i	$-0.413569\pi$
$101$	5.39088	0.536412	0.268206	+	0.963362i	$0.413569\pi$
$102$	1.42339	0.140936
$103$	−12.5567	−1.23725	−0.618626	−	0.785686i	$-0.712310\pi$
$103$	−12.5567	−1.23725	−0.618626	+	0.785686i	$0.712310\pi$
$104$	5.42366	0.531834
$105$	−10.1022	−0.985872
$106$	−2.40410	−0.233507
$107$	10.6990	1.03431	0.517157	−	0.855891i	$-0.326990\pi$
$107$	10.6990	1.03431	0.517157	+	0.855891i	$0.326990\pi$
$108$	3.31293	0.318787
$109$	−15.4480	−1.47965	−0.739827	−	0.672797i	$-0.765093\pi$
$109$	−15.4480	−1.47965	−0.739827	+	0.672797i	$0.765093\pi$
$110$	−11.1997	−1.06784
$111$	−6.29764	−0.597746
$112$	2.39594	0.226395
$113$	−5.68055	−0.534381	−0.267191	−	0.963644i	$-0.586095\pi$
$113$	−5.68055	−0.534381	−0.267191	+	0.963644i	$0.586095\pi$
$114$	4.72894	0.442906
$115$	7.54670	0.703734
$116$	−8.76966	−0.814243
$117$	−7.72930	−0.714574
$118$	−9.48536	−0.873198
$119$	1.62120	0.148616
$120$	4.21637	0.384900
$121$	20.2216	1.83833
$122$	−3.38922	−0.306845
$123$	5.13957	0.463420
$124$	−5.65002	−0.507387
$125$	−11.9912	−1.07252
$126$	−3.41448	−0.304186
$127$	−6.77495	−0.601180	−0.300590	−	0.953753i	$-0.597184\pi$
$127$	−6.77495	−0.601180	−0.300590	+	0.953753i	$0.597184\pi$
$128$	−1.00000	−0.0883883
$129$	−7.01889	−0.617979
$130$	10.8710	0.953449
$131$	11.7803	1.02925	0.514624	−	0.857416i	$-0.327932\pi$
$131$	11.7803	1.02925	0.514624	+	0.857416i	$0.327932\pi$
$132$	−11.7541	−1.02306
$133$	5.38615	0.467038
$134$	0.473191	0.0408775
$135$	6.64033	0.571509
$136$	−0.676646	−0.0580219
$137$	1.56854	0.134010	0.0670048	−	0.997753i	$-0.478656\pi$
$137$	1.56854	0.134010	0.0670048	+	0.997753i	$0.478656\pi$
$138$	7.92031	0.674222
$139$	−1.39573	−0.118384	−0.0591922	−	0.998247i	$-0.518852\pi$
$139$	−1.39573	−0.118384	−0.0591922	+	0.998247i	$0.518852\pi$
$140$	4.80234	0.405872
$141$	−21.1224	−1.77883
$142$	−9.77176	−0.820028
$143$	−30.3054	−2.53426
$144$	1.42511	0.118759
$145$	−17.5776	−1.45974
$146$	−8.13986	−0.673660
$147$	2.64939	0.218518
$148$	2.99375	0.246085
$149$	4.83858	0.396392	0.198196	−	0.980162i	$-0.436492\pi$
$149$	4.83858	0.396392	0.198196	+	0.980162i	$0.436492\pi$
$150$	−2.06682	−0.168755
$151$	16.9409	1.37863	0.689316	−	0.724461i	$-0.257911\pi$
$151$	16.9409	1.37863	0.689316	+	0.724461i	$0.257911\pi$
$152$	−2.24803	−0.182339
$153$	0.964293	0.0779585
$154$	−13.3876	−1.07881
$155$	−11.3247	−0.909623
$156$	11.4092	0.913465
$157$	17.1074	1.36532	0.682661	−	0.730735i	$-0.260823\pi$
$157$	17.1074	1.36532	0.682661	+	0.730735i	$0.260823\pi$
$158$	5.29253	0.421051
$159$	−5.05726	−0.401067
$160$	−2.00437	−0.158459
$161$	9.02104	0.710958
$162$	11.2444	0.883443
$163$	20.0905	1.57361	0.786805	−	0.617201i	$-0.211734\pi$
$163$	20.0905	1.57361	0.786805	+	0.617201i	$0.211734\pi$
$164$	−2.44323	−0.190785
$165$	−23.5595	−1.83411
$166$	13.2704	1.02998
$167$	−15.8045	−1.22299	−0.611495	−	0.791248i	$-0.709432\pi$
$167$	−15.8045	−1.22299	−0.611495	+	0.791248i	$0.709432\pi$
$168$	5.04009	0.388851
$169$	16.4161	1.26278
$170$	−1.35625	−0.104019
$171$	3.20368	0.244992
$172$	3.33662	0.254415
$173$	13.6324	1.03645	0.518226	−	0.855244i	$-0.326592\pi$
$173$	13.6324	1.03645	0.518226	+	0.855244i	$0.326592\pi$
$174$	−18.4478	−1.39852
$175$	−2.35406	−0.177950
$176$	5.58763	0.421183
$177$	−19.9533	−1.49979
$178$	−14.0323	−1.05177
$179$	−15.3424	−1.14675	−0.573373	−	0.819294i	$-0.694366\pi$
$179$	−15.3424	−1.14675	−0.573373	+	0.819294i	$0.694366\pi$
$180$	2.85644	0.212906
$181$	−7.41681	−0.551287	−0.275644	−	0.961260i	$-0.588891\pi$
$181$	−7.41681	−0.551287	−0.275644	+	0.961260i	$0.588891\pi$
$182$	12.9948	0.963237
$183$	−7.12954	−0.527030
$184$	−3.76513	−0.277569
$185$	6.00058	0.441171
$186$	−11.8854	−0.871477
$187$	3.78084	0.276483
$188$	10.0411	0.732324
$189$	7.93760	0.577375
$190$	−4.50587	−0.326890
$191$	19.6094	1.41888	0.709441	−	0.704764i	$-0.248947\pi$
$191$	19.6094	1.41888	0.709441	+	0.704764i	$0.248947\pi$
$192$	−2.10359	−0.151814
$193$	−10.9791	−0.790291	−0.395145	−	0.918619i	$-0.629306\pi$
$193$	−10.9791	−0.790291	−0.395145	+	0.918619i	$0.629306\pi$
$194$	−17.4530	−1.25305
$195$	22.8682	1.63762
$196$	−1.25946	−0.0899614
$197$	19.8965	1.41757	0.708784	−	0.705426i	$-0.249244\pi$
$197$	19.8965	1.41757	0.708784	+	0.705426i	$0.249244\pi$
$198$	−7.96297	−0.565904
$199$	−9.85252	−0.698426	−0.349213	−	0.937043i	$-0.613551\pi$
$199$	−9.85252	−0.698426	−0.349213	+	0.937043i	$0.613551\pi$
$200$	0.982519	0.0694746
$201$	0.995402	0.0702102
$202$	−5.39088	−0.379301
$203$	−21.0116	−1.47473
$204$	−1.42339	−0.0996571
$205$	−4.89713	−0.342031
$206$	12.5567	0.874869
$207$	5.36572	0.372943
$208$	−5.42366	−0.376063
$209$	12.5612	0.868873
$210$	10.1022	0.697117
$211$	−10.1200	−0.696690	−0.348345	−	0.937366i	$-0.613256\pi$
$211$	−10.1200	−0.696690	−0.348345	+	0.937366i	$0.613256\pi$
$212$	2.40410	0.165115
$213$	−20.5558	−1.40846
$214$	−10.6990	−0.731371
$215$	6.68780	0.456104
$216$	−3.31293	−0.225417
$217$	−13.5371	−0.918960
$218$	15.4480	1.04627
$219$	−17.1230	−1.15706
$220$	11.1997	0.755080
$221$	−3.66990	−0.246864
$222$	6.29764	0.422670
$223$	−16.0708	−1.07618	−0.538092	−	0.842886i	$-0.680855\pi$
$223$	−16.0708	−1.07618	−0.538092	+	0.842886i	$0.680855\pi$
$224$	−2.39594	−0.160086
$225$	−1.40020	−0.0933464
$226$	5.68055	0.377865
$227$	−15.6310	−1.03747	−0.518734	−	0.854935i	$-0.673597\pi$
$227$	−15.6310	−1.03747	−0.518734	+	0.854935i	$0.673597\pi$
$228$	−4.72894	−0.313182
$229$	−20.8046	−1.37480	−0.687402	−	0.726277i	$-0.741249\pi$
$229$	−20.8046	−1.37480	−0.687402	+	0.726277i	$0.741249\pi$
$230$	−7.54670	−0.497615
$231$	−28.1622	−1.85293
$232$	8.76966	0.575757
$233$	7.20442	0.471977	0.235989	−	0.971756i	$-0.424167\pi$
$233$	7.20442	0.471977	0.235989	+	0.971756i	$0.424167\pi$
$234$	7.72930	0.505280
$235$	20.1261	1.31288
$236$	9.48536	0.617444
$237$	11.1333	0.723188
$238$	−1.62120	−0.105087
$239$	−24.3367	−1.57421	−0.787105	−	0.616819i	$-0.788421\pi$
$239$	−24.3367	−1.57421	−0.787105	+	0.616819i	$0.788421\pi$
$240$	−4.21637	−0.272166
$241$	0.933174	0.0601110	0.0300555	−	0.999548i	$-0.490432\pi$
$241$	0.933174	0.0601110	0.0300555	+	0.999548i	$0.490432\pi$
$242$	−20.2216	−1.29989
$243$	13.7148	0.879807
$244$	3.38922	0.216972
$245$	−2.52442	−0.161279
$246$	−5.13957	−0.327687
$247$	−12.1925	−0.775793
$248$	5.65002	0.358777
$249$	27.9155	1.76907
$250$	11.9912	0.758387
$251$	−28.5654	−1.80303	−0.901517	−	0.432744i	$-0.857545\pi$
$251$	−28.5654	−1.80303	−0.901517	+	0.432744i	$0.857545\pi$
$252$	3.41448	0.215092
$253$	21.0382	1.32266
$254$	6.77495	0.425098
$255$	−2.85299	−0.178661
$256$	1.00000	0.0625000
$257$	−5.99564	−0.373997	−0.186999	−	0.982360i	$-0.559876\pi$
$257$	−5.99564	−0.373997	−0.186999	+	0.982360i	$0.559876\pi$
$258$	7.01889	0.436977
$259$	7.17286	0.445700
$260$	−10.8710	−0.674191
$261$	−12.4977	−0.773589
$262$	−11.7803	−0.727788
$263$	−11.2459	−0.693452	−0.346726	−	0.937966i	$-0.612707\pi$
$263$	−11.2459	−0.693452	−0.346726	+	0.937966i	$0.612707\pi$
$264$	11.7541	0.723415
$265$	4.81870	0.296011
$266$	−5.38615	−0.330246
$267$	−29.5183	−1.80649
$268$	−0.473191	−0.0289047
$269$	12.8771	0.785129	0.392564	−	0.919725i	$-0.371588\pi$
$269$	12.8771	0.785129	0.392564	+	0.919725i	$0.371588\pi$
$270$	−6.64033	−0.404118
$271$	14.5283	0.882532	0.441266	−	0.897376i	$-0.354529\pi$
$271$	14.5283	0.882532	0.441266	+	0.897376i	$0.354529\pi$
$272$	0.676646	0.0410277
$273$	27.3357	1.65443
$274$	−1.56854	−0.0947591
$275$	−5.48995	−0.331056
$276$	−7.92031	−0.476747
$277$	−8.38991	−0.504101	−0.252051	−	0.967714i	$-0.581105\pi$
$277$	−8.38991	−0.504101	−0.252051	+	0.967714i	$0.581105\pi$
$278$	1.39573	0.0837104
$279$	−8.05189	−0.482054
$280$	−4.80234	−0.286995
$281$	−19.2619	−1.14907	−0.574535	−	0.818480i	$-0.694817\pi$
$281$	−19.2619	−1.14907	−0.574535	+	0.818480i	$0.694817\pi$
$282$	21.1224	1.25782
$283$	−9.37986	−0.557575	−0.278787	−	0.960353i	$-0.589932\pi$
$283$	−9.37986	−0.557575	−0.278787	+	0.960353i	$0.589932\pi$
$284$	9.77176	0.579848
$285$	−9.47852	−0.561459
$286$	30.3054	1.79200
$287$	−5.85385	−0.345542
$288$	−1.42511	−0.0839753
$289$	−16.5422	−0.973068
$290$	17.5776	1.03219
$291$	−36.7140	−2.15221
$292$	8.13986	0.476349
$293$	28.3070	1.65371	0.826857	−	0.562412i	$-0.190127\pi$
$293$	28.3070	1.65371	0.826857	+	0.562412i	$0.190127\pi$
$294$	−2.64939	−0.154516
$295$	19.0121	1.10693
$296$	−2.99375	−0.174008
$297$	18.5114	1.07414
$298$	−4.83858	−0.280292
$299$	−20.4208	−1.18097
$300$	2.06682	0.119328
$301$	7.99434	0.460786
$302$	−16.9409	−0.974840
$303$	−11.3402	−0.651478
$304$	2.24803	0.128933
$305$	6.79323	0.388979
$306$	−0.964293	−0.0551250
$307$	7.14414	0.407737	0.203869	−	0.978998i	$-0.434648\pi$
$307$	7.14414	0.407737	0.203869	+	0.978998i	$0.434648\pi$
$308$	13.3876	0.762831
$309$	26.4143	1.50266
$310$	11.3247	0.643200
$311$	14.0060	0.794209	0.397105	−	0.917773i	$-0.370015\pi$
$311$	14.0060	0.794209	0.397105	+	0.917773i	$0.370015\pi$
$312$	−11.4092	−0.645918
$313$	22.0714	1.24755	0.623773	−	0.781605i	$-0.285599\pi$
$313$	22.0714	1.24755	0.623773	+	0.781605i	$0.285599\pi$
$314$	−17.1074	−0.965428
$315$	6.84386	0.385608
$316$	−5.29253	−0.297728
$317$	29.1482	1.63713	0.818563	−	0.574417i	$-0.194771\pi$
$317$	29.1482	1.63713	0.818563	+	0.574417i	$0.194771\pi$
$318$	5.05726	0.283597
$319$	−49.0016	−2.74356
$320$	2.00437	0.112047
$321$	−22.5064	−1.25619
$322$	−9.02104	−0.502723
$323$	1.52112	0.0846373
$324$	−11.2444	−0.624688
$325$	5.32885	0.295591
$326$	−20.0905	−1.11271
$327$	32.4964	1.79706
$328$	2.44323	0.134905
$329$	24.0580	1.32636
$330$	23.5595	1.29691
$331$	24.1747	1.32876	0.664381	−	0.747394i	$-0.268695\pi$
$331$	24.1747	1.32876	0.664381	+	0.747394i	$0.268695\pi$
$332$	−13.2704	−0.728308
$333$	4.26642	0.233799
$334$	15.8045	0.864785
$335$	−0.948448	−0.0518192
$336$	−5.04009	−0.274959
$337$	17.9474	0.977655	0.488827	−	0.872380i	$-0.337425\pi$
$337$	17.9474	0.977655	0.488827	+	0.872380i	$0.337425\pi$
$338$	−16.4161	−0.892917
$339$	11.9496	0.649012
$340$	1.35625	0.0735527
$341$	−31.5702	−1.70962
$342$	−3.20368	−0.173235
$343$	−19.7892	−1.06852
$344$	−3.33662	−0.179898
$345$	−15.8752	−0.854692
$346$	−13.6324	−0.732883
$347$	12.7759	0.685844	0.342922	−	0.939364i	$-0.388583\pi$
$347$	12.7759	0.685844	0.342922	+	0.939364i	$0.388583\pi$
$348$	18.4478	0.988906
$349$	8.98793	0.481113	0.240556	−	0.970635i	$-0.422670\pi$
$349$	8.98793	0.481113	0.240556	+	0.970635i	$0.422670\pi$
$350$	2.35406	0.125830
$351$	−17.9682	−0.959073
$352$	−5.58763	−0.297822
$353$	15.5045	0.825222	0.412611	−	0.910907i	$-0.364617\pi$
$353$	15.5045	0.825222	0.412611	+	0.910907i	$0.364617\pi$
$354$	19.9533	1.06051
$355$	19.5862	1.03953
$356$	14.0323	0.743710
$357$	−3.41036	−0.180495
$358$	15.3424	0.810873
$359$	14.2565	0.752429	0.376214	−	0.926533i	$-0.377226\pi$
$359$	14.2565	0.752429	0.376214	+	0.926533i	$0.377226\pi$
$360$	−2.85644	−0.150547
$361$	−13.9464	−0.734019
$362$	7.41681	0.389819
$363$	−42.5380	−2.23267
$364$	−12.9948	−0.681111
$365$	16.3153	0.853979
$366$	7.12954	0.372667
$367$	20.8341	1.08753	0.543766	−	0.839237i	$-0.316998\pi$
$367$	20.8341	1.08753	0.543766	+	0.839237i	$0.316998\pi$
$368$	3.76513	0.196271
$369$	−3.48187	−0.181259
$370$	−6.00058	−0.311955
$371$	5.76010	0.299049
$372$	11.8854	0.616227
$373$	−5.00334	−0.259063	−0.129532	−	0.991575i	$-0.541347\pi$
$373$	−5.00334	−0.259063	−0.129532	+	0.991575i	$0.541347\pi$
$374$	−3.78084	−0.195503
$375$	25.2245	1.30259
$376$	−10.0411	−0.517831
$377$	47.5637	2.44965
$378$	−7.93760	−0.408266
$379$	18.9348	0.972615	0.486307	−	0.873788i	$-0.338344\pi$
$379$	18.9348	0.972615	0.486307	+	0.873788i	$0.338344\pi$
$380$	4.50587	0.231146
$381$	14.2517	0.730139
$382$	−19.6094	−1.00330
$383$	34.3554	1.75548	0.877738	−	0.479141i	$-0.159052\pi$
$383$	34.3554	1.75548	0.877738	+	0.479141i	$0.159052\pi$
$384$	2.10359	0.107349
$385$	26.8337	1.36757
$386$	10.9791	0.558820
$387$	4.75504	0.241712
$388$	17.4530	0.886042
$389$	−2.19845	−0.111466	−0.0557330	−	0.998446i	$-0.517750\pi$
$389$	−2.19845	−0.111466	−0.0557330	+	0.998446i	$0.517750\pi$
$390$	−22.8682	−1.15797
$391$	2.54766	0.128841
$392$	1.25946	0.0636123
$393$	−24.7809	−1.25003
$394$	−19.8965	−1.00237
$395$	−10.6082	−0.533755
$396$	7.96297	0.400155
$397$	5.73217	0.287689	0.143845	−	0.989600i	$-0.454053\pi$
$397$	5.73217	0.287689	0.143845	+	0.989600i	$0.454053\pi$
$398$	9.85252	0.493862
$399$	−11.3303	−0.567223
$400$	−0.982519	−0.0491259
$401$	−20.3947	−1.01846	−0.509232	−	0.860629i	$-0.670070\pi$
$401$	−20.3947	−1.01846	−0.509232	+	0.860629i	$0.670070\pi$
$402$	−0.995402	−0.0496461
$403$	30.6438	1.52648
$404$	5.39088	0.268206
$405$	−22.5379	−1.11992
$406$	21.0116	1.04279
$407$	16.7280	0.829176
$408$	1.42339	0.0704682
$409$	−18.3765	−0.908660	−0.454330	−	0.890833i	$-0.650121\pi$
$409$	−18.3765	−0.908660	−0.454330	+	0.890833i	$0.650121\pi$
$410$	4.89713	0.241852
$411$	−3.29958	−0.162756
$412$	−12.5567	−0.618626
$413$	22.7264	1.11829
$414$	−5.36572	−0.263711
$415$	−26.5987	−1.30568
$416$	5.42366	0.265917
$417$	2.93605	0.143779
$418$	−12.5612	−0.614386
$419$	−27.0216	−1.32009	−0.660045	−	0.751226i	$-0.729463\pi$
$419$	−27.0216	−1.32009	−0.660045	+	0.751226i	$0.729463\pi$
$420$	−10.1022	−0.492936
$421$	26.6447	1.29858	0.649291	−	0.760540i	$-0.275066\pi$
$421$	26.6447	1.29858	0.649291	+	0.760540i	$0.275066\pi$
$422$	10.1200	0.492634
$423$	14.3097	0.695761
$424$	−2.40410	−0.116754
$425$	−0.664817	−0.0322484
$426$	20.5558	0.995933
$427$	8.12037	0.392972
$428$	10.6990	0.517157
$429$	63.7503	3.07789
$430$	−6.68780	−0.322514
$431$	25.6330	1.23470	0.617350	−	0.786689i	$-0.288206\pi$
$431$	25.6330	1.23470	0.617350	+	0.786689i	$0.288206\pi$
$432$	3.31293	0.159394
$433$	4.06764	0.195478	0.0977391	−	0.995212i	$-0.468839\pi$
$433$	4.06764	0.195478	0.0977391	+	0.995212i	$0.468839\pi$
$434$	13.5371	0.649803
$435$	36.9762	1.77287
$436$	−15.4480	−0.739827
$437$	8.46413	0.404894
$438$	17.1230	0.818167
$439$	34.7285	1.65750	0.828751	−	0.559618i	$-0.189052\pi$
$439$	34.7285	1.65750	0.828751	+	0.559618i	$0.189052\pi$
$440$	−11.1997	−0.533922
$441$	−1.79487	−0.0854698
$442$	3.66990	0.174559
$443$	8.88503	0.422140	0.211070	−	0.977471i	$-0.432305\pi$
$443$	8.88503	0.422140	0.211070	+	0.977471i	$0.432305\pi$
$444$	−6.29764	−0.298873
$445$	28.1259	1.33329
$446$	16.0708	0.760977
$447$	−10.1784	−0.481423
$448$	2.39594	0.113198
$449$	14.3418	0.676832	0.338416	−	0.940997i	$-0.390109\pi$
$449$	14.3418	0.676832	0.338416	+	0.940997i	$0.390109\pi$
$450$	1.40020	0.0660058
$451$	−13.6519	−0.642842
$452$	−5.68055	−0.267191
$453$	−35.6368	−1.67436
$454$	15.6310	0.733601
$455$	−26.0463	−1.22107
$456$	4.72894	0.221453
$457$	15.8013	0.739155	0.369577	−	0.929200i	$-0.379502\pi$
$457$	15.8013	0.739155	0.369577	+	0.929200i	$0.379502\pi$
$458$	20.8046	0.972133
$459$	2.24168	0.104633
$460$	7.54670	0.351867
$461$	−5.55062	−0.258518	−0.129259	−	0.991611i	$-0.541260\pi$
$461$	−5.55062	−0.258518	−0.129259	+	0.991611i	$0.541260\pi$
$462$	28.1622	1.31022
$463$	20.6361	0.959039	0.479519	−	0.877531i	$-0.340811\pi$
$463$	20.6361	0.959039	0.479519	+	0.877531i	$0.340811\pi$
$464$	−8.76966	−0.407121
$465$	23.8226	1.10475
$466$	−7.20442	−0.333738
$467$	16.8784	0.781039	0.390520	−	0.920595i	$-0.372295\pi$
$467$	16.8784	0.781039	0.390520	+	0.920595i	$0.372295\pi$
$468$	−7.72930	−0.357287
$469$	−1.13374	−0.0523512
$470$	−20.1261	−0.928347
$471$	−35.9871	−1.65820
$472$	−9.48536	−0.436599
$473$	18.6438	0.857242
$474$	−11.1333	−0.511371
$475$	−2.20873	−0.101344
$476$	1.62120	0.0743078
$477$	3.42611	0.156871
$478$	24.3367	1.11313
$479$	−0.867337	−0.0396296	−0.0198148	−	0.999804i	$-0.506308\pi$
$479$	−0.867337	−0.0396296	−0.0198148	+	0.999804i	$0.506308\pi$
$480$	4.21637	0.192450
$481$	−16.2371	−0.740348
$482$	−0.933174	−0.0425049
$483$	−18.9766	−0.863466
$484$	20.2216	0.919164
$485$	34.9822	1.58846
$486$	−13.7148	−0.622117
$487$	29.5575	1.33938	0.669689	−	0.742641i	$-0.266427\pi$
$487$	29.5575	1.33938	0.669689	+	0.742641i	$0.266427\pi$
$488$	−3.38922	−0.153423
$489$	−42.2623	−1.91117
$490$	2.52442	0.114042
$491$	−16.9439	−0.764668	−0.382334	−	0.924024i	$-0.624880\pi$
$491$	−16.9439	−0.764668	−0.382334	+	0.924024i	$0.624880\pi$
$492$	5.13957	0.231710
$493$	−5.93395	−0.267252
$494$	12.1925	0.548568
$495$	15.9607	0.717381
$496$	−5.65002	−0.253694
$497$	23.4126	1.05020
$498$	−27.9155	−1.25092
$499$	39.0223	1.74688	0.873439	−	0.486933i	$-0.161885\pi$
$499$	39.0223	1.74688	0.873439	+	0.486933i	$0.161885\pi$
$500$	−11.9912	−0.536261
$501$	33.2463	1.48533
$502$	28.5654	1.27494
$503$	−8.97704	−0.400266	−0.200133	−	0.979769i	$-0.564137\pi$
$503$	−8.97704	−0.400266	−0.200133	+	0.979769i	$0.564137\pi$
$504$	−3.41448	−0.152093
$505$	10.8053	0.480829
$506$	−21.0382	−0.935261
$507$	−34.5328	−1.53365
$508$	−6.77495	−0.300590
$509$	−18.0649	−0.800712	−0.400356	−	0.916360i	$-0.631114\pi$
$509$	−18.0649	−0.800712	−0.400356	+	0.916360i	$0.631114\pi$
$510$	2.85299	0.126332
$511$	19.5026	0.862746
$512$	−1.00000	−0.0441942
$513$	7.44757	0.328818
$514$	5.99564	0.264456
$515$	−25.1683	−1.10905
$516$	−7.01889	−0.308989
$517$	56.1061	2.46754
$518$	−7.17286	−0.315158
$519$	−28.6770	−1.25878
$520$	10.8710	0.476725
$521$	−27.2818	−1.19524	−0.597619	−	0.801780i	$-0.703887\pi$
$521$	−27.2818	−1.19524	−0.597619	+	0.801780i	$0.703887\pi$
$522$	12.4977	0.547010
$523$	31.8182	1.39131	0.695657	−	0.718374i	$-0.255113\pi$
$523$	31.8182	1.39131	0.695657	+	0.718374i	$0.255113\pi$
$524$	11.7803	0.514624
$525$	4.95198	0.216122
$526$	11.2459	0.490344
$527$	−3.82306	−0.166535
$528$	−11.7541	−0.511532
$529$	−8.82377	−0.383642
$530$	−4.81870	−0.209311
$531$	13.5177	0.586617
$532$	5.38615	0.233519
$533$	13.2513	0.573976
$534$	29.5183	1.27738
$535$	21.4448	0.927138
$536$	0.473191	0.0204387
$537$	32.2742	1.39274
$538$	−12.8771	−0.555170
$539$	−7.03739	−0.303122
$540$	6.64033	0.285754
$541$	−0.781382	−0.0335943	−0.0167971	−	0.999859i	$-0.505347\pi$
$541$	−0.781382	−0.0335943	−0.0167971	+	0.999859i	$0.505347\pi$
$542$	−14.5283	−0.624045
$543$	15.6020	0.669544
$544$	−0.676646	−0.0290109
$545$	−30.9635	−1.32633
$546$	−27.3357	−1.16986
$547$	5.29135	0.226242	0.113121	−	0.993581i	$-0.463915\pi$
$547$	5.29135	0.226242	0.113121	+	0.993581i	$0.463915\pi$
$548$	1.56854	0.0670048
$549$	4.83000	0.206139
$550$	5.48995	0.234092
$551$	−19.7145	−0.839864
$552$	7.92031	0.337111
$553$	−12.6806	−0.539234
$554$	8.38991	0.356453
$555$	−12.6228	−0.535807
$556$	−1.39573	−0.0591922
$557$	2.94417	0.124748	0.0623742	−	0.998053i	$-0.480133\pi$
$557$	2.94417	0.124748	0.0623742	+	0.998053i	$0.480133\pi$
$558$	8.05189	0.340864
$559$	−18.0967	−0.765408
$560$	4.80234	0.202936
$561$	−7.95336	−0.335791
$562$	19.2619	0.812515
$563$	−12.0326	−0.507112	−0.253556	−	0.967321i	$-0.581600\pi$
$563$	−12.0326	−0.507112	−0.253556	+	0.967321i	$0.581600\pi$
$564$	−21.1224	−0.889416
$565$	−11.3859	−0.479008
$566$	9.37986	0.394265
$567$	−26.9409	−1.13141
$568$	−9.77176	−0.410014
$569$	−38.0333	−1.59444	−0.797219	−	0.603691i	$-0.793696\pi$
$569$	−38.0333	−1.59444	−0.797219	+	0.603691i	$0.793696\pi$
$570$	9.47852	0.397012
$571$	−35.8238	−1.49918	−0.749589	−	0.661904i	$-0.769749\pi$
$571$	−35.8238	−1.49918	−0.749589	+	0.661904i	$0.769749\pi$
$572$	−30.3054	−1.26713
$573$	−41.2501	−1.72325
$574$	5.85385	0.244335
$575$	−3.69931	−0.154272
$576$	1.42511	0.0593795
$577$	11.3047	0.470619	0.235310	−	0.971920i	$-0.424390\pi$
$577$	11.3047	0.470619	0.235310	+	0.971920i	$0.424390\pi$
$578$	16.5422	0.688063
$579$	23.0955	0.959817
$580$	−17.5776	−0.729870
$581$	−31.7951	−1.31908
$582$	36.7140	1.52185
$583$	13.4332	0.556348
$584$	−8.13986	−0.336830
$585$	−15.4923	−0.640530
$586$	−28.3070	−1.16935
$587$	−20.8291	−0.859711	−0.429855	−	0.902898i	$-0.641435\pi$
$587$	−20.8291	−0.859711	−0.429855	+	0.902898i	$0.641435\pi$
$588$	2.64939	0.109259
$589$	−12.7014	−0.523353
$590$	−19.0121	−0.782717
$591$	−41.8542	−1.72165
$592$	2.99375	0.123043
$593$	−20.9205	−0.859100	−0.429550	−	0.903043i	$-0.641328\pi$
$593$	−20.9205	−0.859100	−0.429550	+	0.903043i	$0.641328\pi$
$594$	−18.5114	−0.759534
$595$	3.24949	0.133216
$596$	4.83858	0.198196
$597$	20.7257	0.848246
$598$	20.4208	0.835069
$599$	10.8913	0.445005	0.222503	−	0.974932i	$-0.428577\pi$
$599$	10.8913	0.445005	0.222503	+	0.974932i	$0.428577\pi$
$600$	−2.06682	−0.0843776
$601$	14.1670	0.577884	0.288942	−	0.957347i	$-0.406697\pi$
$601$	14.1670	0.577884	0.288942	+	0.957347i	$0.406697\pi$
$602$	−7.99434	−0.325825
$603$	−0.674348	−0.0274616
$604$	16.9409	0.689316
$605$	40.5315	1.64784
$606$	11.3402	0.460665
$607$	39.1030	1.58714	0.793571	−	0.608477i	$-0.208219\pi$
$607$	39.1030	1.58714	0.793571	+	0.608477i	$0.208219\pi$
$608$	−2.24803	−0.0911696
$609$	44.1999	1.79107
$610$	−6.79323	−0.275050
$611$	−54.4596	−2.20320
$612$	0.964293	0.0389792
$613$	−37.5563	−1.51689	−0.758443	−	0.651739i	$-0.774040\pi$
$613$	−37.5563	−1.51689	−0.758443	+	0.651739i	$0.774040\pi$
$614$	−7.14414	−0.288314
$615$	10.3016	0.415400
$616$	−13.3876	−0.539403
$617$	−13.5761	−0.546553	−0.273277	−	0.961935i	$-0.588107\pi$
$617$	−13.5761	−0.546553	−0.273277	+	0.961935i	$0.588107\pi$
$618$	−26.4143	−1.06254
$619$	39.8098	1.60009	0.800046	−	0.599938i	$-0.204808\pi$
$619$	39.8098	1.60009	0.800046	+	0.599938i	$0.204808\pi$
$620$	−11.3247	−0.454811
$621$	12.4736	0.500550
$622$	−14.0060	−0.561591
$623$	33.6206	1.34698
$624$	11.4092	0.456733
$625$	−19.1221	−0.764883
$626$	−22.0714	−0.882149
$627$	−26.4236	−1.05526
$628$	17.1074	0.682661
$629$	2.02571	0.0807704
$630$	−6.84386	−0.272666
$631$	39.9061	1.58864	0.794318	−	0.607502i	$-0.207828\pi$
$631$	39.9061	1.58864	0.794318	+	0.607502i	$0.207828\pi$
$632$	5.29253	0.210526
$633$	21.2884	0.846137
$634$	−29.1482	−1.15762
$635$	−13.5795	−0.538885
$636$	−5.05726	−0.200533
$637$	6.83088	0.270649
$638$	49.0016	1.93999
$639$	13.9258	0.550897
$640$	−2.00437	−0.0792295
$641$	−9.57554	−0.378211	−0.189106	−	0.981957i	$-0.560559\pi$
$641$	−9.57554	−0.378211	−0.189106	+	0.981957i	$0.560559\pi$
$642$	22.5064	0.888257
$643$	−8.68499	−0.342503	−0.171251	−	0.985227i	$-0.554781\pi$
$643$	−8.68499	−0.342503	−0.171251	+	0.985227i	$0.554781\pi$
$644$	9.02104	0.355479
$645$	−14.0684	−0.553943
$646$	−1.52112	−0.0598476
$647$	38.6751	1.52048	0.760238	−	0.649645i	$-0.225082\pi$
$647$	38.6751	1.52048	0.760238	+	0.649645i	$0.225082\pi$
$648$	11.2444	0.441721
$649$	53.0007	2.08046
$650$	−5.32885	−0.209015
$651$	28.4766	1.11609
$652$	20.0905	0.786805
$653$	8.43251	0.329989	0.164995	−	0.986294i	$-0.447239\pi$
$653$	8.43251	0.329989	0.164995	+	0.986294i	$0.447239\pi$
$654$	−32.4964	−1.27071
$655$	23.6120	0.922596
$656$	−2.44323	−0.0953923
$657$	11.6002	0.452566
$658$	−24.0580	−0.937877
$659$	−37.6813	−1.46785	−0.733927	−	0.679228i	$-0.762315\pi$
$659$	−37.6813	−1.46785	−0.733927	+	0.679228i	$0.762315\pi$
$660$	−23.5595	−0.917053
$661$	22.2164	0.864119	0.432060	−	0.901845i	$-0.357787\pi$
$661$	22.2164	0.864119	0.432060	+	0.901845i	$0.357787\pi$
$662$	−24.1747	−0.939577
$663$	7.71997	0.299819
$664$	13.2704	0.514991
$665$	10.7958	0.418644
$666$	−4.26642	−0.165321
$667$	−33.0190	−1.27850
$668$	−15.8045	−0.611495
$669$	33.8065	1.30704
$670$	0.948448	0.0366417
$671$	18.9377	0.731081
$672$	5.04009	0.194426
$673$	24.8252	0.956941	0.478471	−	0.878104i	$-0.341191\pi$
$673$	24.8252	0.956941	0.478471	+	0.878104i	$0.341191\pi$
$674$	−17.9474	−0.691306
$675$	−3.25502	−0.125286
$676$	16.4161	0.631388
$677$	−0.563204	−0.0216457	−0.0108228	−	0.999941i	$-0.503445\pi$
$677$	−0.563204	−0.0216457	−0.0108228	+	0.999941i	$0.503445\pi$
$678$	−11.9496	−0.458921
$679$	41.8164	1.60477
$680$	−1.35625	−0.0520096
$681$	32.8814	1.26002
$682$	31.5702	1.20889
$683$	5.31280	0.203289	0.101644	−	0.994821i	$-0.467590\pi$
$683$	5.31280	0.203289	0.101644	+	0.994821i	$0.467590\pi$
$684$	3.20368	0.122496
$685$	3.14393	0.120123
$686$	19.7892	0.755555
$687$	43.7644	1.66971
$688$	3.33662	0.127207
$689$	−13.0390	−0.496748
$690$	15.8752	0.604359
$691$	4.59647	0.174858	0.0874290	−	0.996171i	$-0.472135\pi$
$691$	4.59647	0.174858	0.0874290	+	0.996171i	$0.472135\pi$
$692$	13.6324	0.518226
$693$	19.0788	0.724745
$694$	−12.7759	−0.484965
$695$	−2.79756	−0.106117
$696$	−18.4478	−0.699262
$697$	−1.65320	−0.0626195
$698$	−8.98793	−0.340198
$699$	−15.1552	−0.573222
$700$	−2.35406	−0.0889751
$701$	34.1515	1.28988	0.644942	−	0.764232i	$-0.276882\pi$
$701$	34.1515	1.28988	0.644942	+	0.764232i	$0.276882\pi$
$702$	17.9682	0.678167
$703$	6.73005	0.253828
$704$	5.58763	0.210592
$705$	−42.3371	−1.59451
$706$	−15.5045	−0.583520
$707$	12.9162	0.485765
$708$	−19.9533	−0.749893
$709$	−28.9204	−1.08613	−0.543064	−	0.839691i	$-0.682736\pi$
$709$	−28.9204	−1.08613	−0.543064	+	0.839691i	$0.682736\pi$
$710$	−19.5862	−0.735057
$711$	−7.54243	−0.282863
$712$	−14.0323	−0.525883
$713$	−21.2731	−0.796684
$714$	3.41036	0.127629
$715$	−60.7431	−2.27166
$716$	−15.3424	−0.573373
$717$	51.1945	1.91189
$718$	−14.2565	−0.532047
$719$	33.1503	1.23630	0.618149	−	0.786061i	$-0.287883\pi$
$719$	33.1503	1.23630	0.618149	+	0.786061i	$0.287883\pi$
$720$	2.85644	0.106453
$721$	−30.0852	−1.12043
$722$	13.9464	0.519030
$723$	−1.96302	−0.0730055
$724$	−7.41681	−0.275644
$725$	8.61636	0.320004
$726$	42.5380	1.57873
$727$	5.75524	0.213450	0.106725	−	0.994289i	$-0.465964\pi$
$727$	5.75524	0.213450	0.106725	+	0.994289i	$0.465964\pi$
$728$	12.9948	0.481618
$729$	4.88273	0.180842
$730$	−16.3153	−0.603855
$731$	2.25771	0.0835043
$732$	−7.12954	−0.263515
$733$	48.7619	1.80106	0.900531	−	0.434792i	$-0.143178\pi$
$733$	48.7619	1.80106	0.900531	+	0.434792i	$0.143178\pi$
$734$	−20.8341	−0.769001
$735$	5.31035	0.195875
$736$	−3.76513	−0.138785
$737$	−2.64402	−0.0973935
$738$	3.48187	0.128169
$739$	−14.8576	−0.546546	−0.273273	−	0.961937i	$-0.588106\pi$
$739$	−14.8576	−0.546546	−0.273273	+	0.961937i	$0.588106\pi$
$740$	6.00058	0.220586
$741$	25.6482	0.942209
$742$	−5.76010	−0.211460
$743$	−52.5995	−1.92969	−0.964845	−	0.262821i	$-0.915347\pi$
$743$	−52.5995	−1.92969	−0.964845	+	0.262821i	$0.915347\pi$
$744$	−11.8854	−0.435738
$745$	9.69829	0.355318
$746$	5.00334	0.183185
$747$	−18.9117	−0.691945
$748$	3.78084	0.138241
$749$	25.6343	0.936655
$750$	−25.2245	−0.921069
$751$	21.4858	0.784030	0.392015	−	0.919959i	$-0.371778\pi$
$751$	21.4858	0.784030	0.392015	+	0.919959i	$0.371778\pi$
$752$	10.0411	0.366162
$753$	60.0900	2.18980
$754$	−47.5637	−1.73217
$755$	33.9558	1.23578
$756$	7.93760	0.288688
$757$	−6.35034	−0.230807	−0.115403	−	0.993319i	$-0.536816\pi$
$757$	−6.35034	−0.230807	−0.115403	+	0.993319i	$0.536816\pi$
$758$	−18.9348	−0.687742
$759$	−44.2558	−1.60638
$760$	−4.50587	−0.163445
$761$	−32.8723	−1.19162	−0.595810	−	0.803126i	$-0.703169\pi$
$761$	−32.8723	−1.19162	−0.595810	+	0.803126i	$0.703169\pi$
$762$	−14.2517	−0.516286
$763$	−37.0126	−1.33995
$764$	19.6094	0.709441
$765$	1.93280	0.0698804
$766$	−34.3554	−1.24131
$767$	−51.4454	−1.85758
$768$	−2.10359	−0.0759069
$769$	−26.7512	−0.964674	−0.482337	−	0.875986i	$-0.660212\pi$
$769$	−26.7512	−0.964674	−0.482337	+	0.875986i	$0.660212\pi$
$770$	−26.8337	−0.967020
$771$	12.6124	0.454224
$772$	−10.9791	−0.395145
$773$	14.7802	0.531605	0.265803	−	0.964027i	$-0.414363\pi$
$773$	14.7802	0.531605	0.265803	+	0.964027i	$0.414363\pi$
$774$	−4.75504	−0.170916
$775$	5.55125	0.199407
$776$	−17.4530	−0.626526
$777$	−15.0888	−0.541307
$778$	2.19845	0.0788184
$779$	−5.49246	−0.196788
$780$	22.8682	0.818812
$781$	54.6010	1.95378
$782$	−2.54766	−0.0911042
$783$	−29.0533	−1.03828
$784$	−1.25946	−0.0449807
$785$	34.2895	1.22385
$786$	24.7809	0.883906
$787$	−36.5846	−1.30410	−0.652050	−	0.758176i	$-0.726091\pi$
$787$	−36.5846	−1.30410	−0.652050	+	0.758176i	$0.726091\pi$
$788$	19.8965	0.708784
$789$	23.6568	0.842205
$790$	10.6082	0.377422
$791$	−13.6103	−0.483926
$792$	−7.96297	−0.282952
$793$	−18.3820	−0.652762
$794$	−5.73217	−0.203427
$795$	−10.1366	−0.359508
$796$	−9.85252	−0.349213
$797$	−25.3111	−0.896564	−0.448282	−	0.893892i	$-0.647964\pi$
$797$	−25.3111	−0.896564	−0.448282	+	0.893892i	$0.647964\pi$
$798$	11.3303	0.401087
$799$	6.79428	0.240364
$800$	0.982519	0.0347373
$801$	19.9975	0.706578
$802$	20.3947	0.720163
$803$	45.4825	1.60504
$804$	0.995402	0.0351051
$805$	18.0815	0.637288
$806$	−30.6438	−1.07938
$807$	−27.0881	−0.953547
$808$	−5.39088	−0.189650
$809$	−49.1344	−1.72747	−0.863737	−	0.503943i	$-0.831882\pi$
$809$	−49.1344	−1.72747	−0.863737	+	0.503943i	$0.831882\pi$
$810$	22.5379	0.791900
$811$	8.67990	0.304792	0.152396	−	0.988319i	$-0.451301\pi$
$811$	8.67990	0.304792	0.152396	+	0.988319i	$0.451301\pi$
$812$	−21.0116	−0.737363
$813$	−30.5617	−1.07184
$814$	−16.7280	−0.586316
$815$	40.2687	1.41055
$816$	−1.42339	−0.0498285
$817$	7.50081	0.262420
$818$	18.3765	0.642520
$819$	−18.5190	−0.647105
$820$	−4.89713	−0.171015
$821$	10.8541	0.378812	0.189406	−	0.981899i	$-0.439344\pi$
$821$	10.8541	0.378812	0.189406	+	0.981899i	$0.439344\pi$
$822$	3.29958	0.115086
$823$	−27.1031	−0.944754	−0.472377	−	0.881397i	$-0.656604\pi$
$823$	−27.1031	−0.944754	−0.472377	+	0.881397i	$0.656604\pi$
$824$	12.5567	0.437435
$825$	11.5486	0.402072
$826$	−22.7264	−0.790752
$827$	47.1244	1.63868	0.819338	−	0.573311i	$-0.194341\pi$
$827$	47.1244	1.63868	0.819338	+	0.573311i	$0.194341\pi$
$828$	5.36572	0.186472
$829$	15.7437	0.546800	0.273400	−	0.961900i	$-0.411852\pi$
$829$	15.7437	0.546800	0.273400	+	0.961900i	$0.411852\pi$
$830$	26.5987	0.923255
$831$	17.6490	0.612236
$832$	−5.42366	−0.188032
$833$	−0.852207	−0.0295272
$834$	−2.93605	−0.101667
$835$	−31.6780	−1.09626
$836$	12.5612	0.434436
$837$	−18.7182	−0.646994
$838$	27.0216	0.933445
$839$	−30.9710	−1.06924	−0.534619	−	0.845093i	$-0.679545\pi$
$839$	−30.9710	−1.06924	−0.534619	+	0.845093i	$0.679545\pi$
$840$	10.1022	0.348558
$841$	47.9070	1.65196
$842$	−26.6447	−0.918236
$843$	40.5192	1.39556
$844$	−10.1200	−0.348345
$845$	32.9038	1.13193
$846$	−14.3097	−0.491977
$847$	48.4498	1.66475
$848$	2.40410	0.0825573
$849$	19.7314	0.677180
$850$	0.664817	0.0228030
$851$	11.2719	0.386395
$852$	−20.5558	−0.704231
$853$	−8.26444	−0.282969	−0.141484	−	0.989940i	$-0.545188\pi$
$853$	−8.26444	−0.282969	−0.141484	+	0.989940i	$0.545188\pi$
$854$	−8.12037	−0.277873
$855$	6.42135	0.219606
$856$	−10.6990	−0.365685
$857$	29.7512	1.01628	0.508141	−	0.861274i	$-0.330333\pi$
$857$	29.7512	1.01628	0.508141	+	0.861274i	$0.330333\pi$
$858$	−63.7503	−2.17640
$859$	48.2833	1.64741	0.823703	−	0.567022i	$-0.191905\pi$
$859$	48.2833	1.64741	0.823703	+	0.567022i	$0.191905\pi$
$860$	6.68780	0.228052
$861$	12.3141	0.419664
$862$	−25.6330	−0.873065
$863$	−9.13637	−0.311006	−0.155503	−	0.987835i	$-0.549700\pi$
$863$	−9.13637	−0.311006	−0.155503	+	0.987835i	$0.549700\pi$
$864$	−3.31293	−0.112708
$865$	27.3243	0.929055
$866$	−4.06764	−0.138224
$867$	34.7980	1.18180
$868$	−13.5371	−0.459480
$869$	−29.5727	−1.00319
$870$	−36.9762	−1.25361
$871$	2.56643	0.0869600
$872$	15.4480	0.523137
$873$	24.8724	0.841803
$874$	−8.46413	−0.286303
$875$	−28.7301	−0.971255
$876$	−17.1230	−0.578531
$877$	57.9285	1.95611	0.978054	−	0.208352i	$-0.0668098\pi$
$877$	57.9285	1.95611	0.978054	+	0.208352i	$0.0668098\pi$
$878$	−34.7285	−1.17203
$879$	−59.5465	−2.00845
$880$	11.1997	0.377540
$881$	−40.7556	−1.37309	−0.686546	−	0.727087i	$-0.740874\pi$
$881$	−40.7556	−1.37309	−0.686546	+	0.727087i	$0.740874\pi$
$882$	1.79487	0.0604363
$883$	−21.2360	−0.714649	−0.357325	−	0.933980i	$-0.616311\pi$
$883$	−21.2360	−0.714649	−0.357325	+	0.933980i	$0.616311\pi$
$884$	−3.66990	−0.123432
$885$	−39.9938	−1.34438
$886$	−8.88503	−0.298498
$887$	−44.1522	−1.48249	−0.741243	−	0.671236i	$-0.765764\pi$
$887$	−44.1522	−1.48249	−0.741243	+	0.671236i	$0.765764\pi$
$888$	6.29764	0.211335
$889$	−16.2324	−0.544417
$890$	−28.1259	−0.942781
$891$	−62.8295	−2.10487
$892$	−16.0708	−0.538092
$893$	22.5727	0.755368
$894$	10.1784	0.340417
$895$	−30.7518	−1.02792
$896$	−2.39594	−0.0800428
$897$	42.9571	1.43430
$898$	−14.3418	−0.478592
$899$	49.5488	1.65254
$900$	−1.40020	−0.0466732
$901$	1.62673	0.0541941
$902$	13.6519	0.454558
$903$	−16.8169	−0.559630
$904$	5.68055	0.188932
$905$	−14.8660	−0.494163
$906$	35.6368	1.18395
$907$	−17.6497	−0.586048	−0.293024	−	0.956105i	$-0.594662\pi$
$907$	−17.6497	−0.586048	−0.293024	+	0.956105i	$0.594662\pi$
$908$	−15.6310	−0.518734
$909$	7.68258	0.254815
$910$	26.0463	0.863426
$911$	−0.508002	−0.0168309	−0.00841543	−	0.999965i	$-0.502679\pi$
$911$	−0.508002	−0.0168309	−0.00841543	+	0.999965i	$0.502679\pi$
$912$	−4.72894	−0.156591
$913$	−74.1500	−2.45401
$914$	−15.8013	−0.522661
$915$	−14.2902	−0.472419
$916$	−20.8046	−0.687402
$917$	28.2249	0.932067
$918$	−2.24168	−0.0739865
$919$	26.0736	0.860090	0.430045	−	0.902808i	$-0.358498\pi$
$919$	26.0736	0.860090	0.430045	+	0.902808i	$0.358498\pi$
$920$	−7.54670	−0.248807
$921$	−15.0284	−0.495201
$922$	5.55062	0.182800
$923$	−52.9987	−1.74447
$924$	−28.1622	−0.926467
$925$	−2.94142	−0.0967133
$926$	−20.6361	−0.678143
$927$	−17.8947	−0.587739
$928$	8.76966	0.287878
$929$	48.0642	1.57693	0.788467	−	0.615077i	$-0.210875\pi$
$929$	48.0642	1.57693	0.788467	+	0.615077i	$0.210875\pi$
$930$	−23.8226	−0.781174
$931$	−2.83130	−0.0927921
$932$	7.20442	0.235989
$933$	−29.4630	−0.964576
$934$	−16.8784	−0.552278
$935$	7.57819	0.247833
$936$	7.72930	0.252640
$937$	−0.785784	−0.0256704	−0.0128352	−	0.999918i	$-0.504086\pi$
$937$	−0.785784	−0.0256704	−0.0128352	+	0.999918i	$0.504086\pi$
$938$	1.13374	0.0370179
$939$	−46.4292	−1.51516
$940$	20.1261	0.656440
$941$	−1.53999	−0.0502024	−0.0251012	−	0.999685i	$-0.507991\pi$
$941$	−1.53999	−0.0502024	−0.0251012	+	0.999685i	$0.507991\pi$
$942$	35.9871	1.17252
$943$	−9.19910	−0.299564
$944$	9.48536	0.308722
$945$	15.9098	0.517547
$946$	−18.6438	−0.606161
$947$	−10.4195	−0.338590	−0.169295	−	0.985565i	$-0.554149\pi$
$947$	−10.4195	−0.338590	−0.169295	+	0.985565i	$0.554149\pi$
$948$	11.1333	0.361594
$949$	−44.1478	−1.43310
$950$	2.20873	0.0716607
$951$	−61.3160	−1.98831
$952$	−1.62120	−0.0525435
$953$	11.2837	0.365514	0.182757	−	0.983158i	$-0.441498\pi$
$953$	11.2837	0.365514	0.182757	+	0.983158i	$0.441498\pi$
$954$	−3.42611	−0.110924
$955$	39.3043	1.27186
$956$	−24.3367	−0.787105
$957$	103.080	3.33209
$958$	0.867337	0.0280224
$959$	3.75814	0.121357
$960$	−4.21637	−0.136083
$961$	0.922765	0.0297666
$962$	16.2371	0.523505
$963$	15.2473	0.491336
$964$	0.933174	0.0300555
$965$	−22.0061	−0.708400
$966$	18.9766	0.610563
$967$	−56.6242	−1.82091	−0.910456	−	0.413606i	$-0.864269\pi$
$967$	−56.6242	−1.82091	−0.910456	+	0.413606i	$0.864269\pi$
$968$	−20.2216	−0.649947
$969$	−3.19982	−0.102793
$970$	−34.9822	−1.12321
$971$	−18.6502	−0.598514	−0.299257	−	0.954173i	$-0.596739\pi$
$971$	−18.6502	−0.598514	−0.299257	+	0.954173i	$0.596739\pi$
$972$	13.7148	0.439903
$973$	−3.34409	−0.107207
$974$	−29.5575	−0.947084
$975$	−11.2097	−0.358999
$976$	3.38922	0.108486
$977$	−45.2664	−1.44820	−0.724101	−	0.689694i	$-0.757745\pi$
$977$	−45.2664	−1.44820	−0.724101	+	0.689694i	$0.757745\pi$
$978$	42.2623	1.35140
$979$	78.4073	2.50591
$980$	−2.52442	−0.0806395
$981$	−22.0151	−0.702889
$982$	16.9439	0.540702
$983$	21.8392	0.696564	0.348282	−	0.937390i	$-0.386765\pi$
$983$	21.8392	0.696564	0.348282	+	0.937390i	$0.386765\pi$
$984$	−5.13957	−0.163844
$985$	39.8799	1.27068
$986$	5.93395	0.188976
$987$	−50.6082	−1.61088
$988$	−12.1925	−0.387896
$989$	12.5628	0.399474
$990$	−15.9607	−0.507265
$991$	−25.2246	−0.801287	−0.400643	−	0.916234i	$-0.631213\pi$
$991$	−25.2246	−0.801287	−0.400643	+	0.916234i	$0.631213\pi$
$992$	5.65002	0.179388
$993$	−50.8538	−1.61380
$994$	−23.4126	−0.742602
$995$	−19.7480	−0.626055
$996$	27.9155	0.884537
$997$	−28.7725	−0.911234	−0.455617	−	0.890176i	$-0.650581\pi$
$997$	−28.7725	−0.911234	−0.455617	+	0.890176i	$0.650581\pi$
$998$	−39.0223	−1.23523
$999$	9.91811	0.313795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.15	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.15	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.10359 q^{3} +1.00000 q^{4} +2.00437 q^{5} +2.10359 q^{6} +2.39594 q^{7} -1.00000 q^{8} +1.42511 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.10359 q^{3} +1.00000 q^{4} +2.00437 q^{5} +2.10359 q^{6} +2.39594 q^{7} -1.00000 q^{8} +1.42511 q^{9} -2.00437 q^{10} +5.58763 q^{11} -2.10359 q^{12} -5.42366 q^{13} -2.39594 q^{14} -4.21637 q^{15} +1.00000 q^{16} +0.676646 q^{17} -1.42511 q^{18} +2.24803 q^{19} +2.00437 q^{20} -5.04009 q^{21} -5.58763 q^{22} +3.76513 q^{23} +2.10359 q^{24} -0.982519 q^{25} +5.42366 q^{26} +3.31293 q^{27} +2.39594 q^{28} -8.76966 q^{29} +4.21637 q^{30} -5.65002 q^{31} -1.00000 q^{32} -11.7541 q^{33} -0.676646 q^{34} +4.80234 q^{35} +1.42511 q^{36} +2.99375 q^{37} -2.24803 q^{38} +11.4092 q^{39} -2.00437 q^{40} -2.44323 q^{41} +5.04009 q^{42} +3.33662 q^{43} +5.58763 q^{44} +2.85644 q^{45} -3.76513 q^{46} +10.0411 q^{47} -2.10359 q^{48} -1.25946 q^{49} +0.982519 q^{50} -1.42339 q^{51} -5.42366 q^{52} +2.40410 q^{53} -3.31293 q^{54} +11.1997 q^{55} -2.39594 q^{56} -4.72894 q^{57} +8.76966 q^{58} +9.48536 q^{59} -4.21637 q^{60} +3.38922 q^{61} +5.65002 q^{62} +3.41448 q^{63} +1.00000 q^{64} -10.8710 q^{65} +11.7541 q^{66} -0.473191 q^{67} +0.676646 q^{68} -7.92031 q^{69} -4.80234 q^{70} +9.77176 q^{71} -1.42511 q^{72} +8.13986 q^{73} -2.99375 q^{74} +2.06682 q^{75} +2.24803 q^{76} +13.3876 q^{77} -11.4092 q^{78} -5.29253 q^{79} +2.00437 q^{80} -11.2444 q^{81} +2.44323 q^{82} -13.2704 q^{83} -5.04009 q^{84} +1.35625 q^{85} -3.33662 q^{86} +18.4478 q^{87} -5.58763 q^{88} +14.0323 q^{89} -2.85644 q^{90} -12.9948 q^{91} +3.76513 q^{92} +11.8854 q^{93} -10.0411 q^{94} +4.50587 q^{95} +2.10359 q^{96} +17.4530 q^{97} +1.25946 q^{98} +7.96297 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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