LMFDB - Embedded newform 8043.2.a.t.1.40

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77087 q^{2} -1.00000 q^{3} +1.13598 q^{4} -1.21221 q^{5} -1.77087 q^{6} +1.00000 q^{7} -1.53006 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.77087 q^{2} -1.00000 q^{3} +1.13598 q^{4} -1.21221 q^{5} -1.77087 q^{6} +1.00000 q^{7} -1.53006 q^{8} +1.00000 q^{9} -2.14666 q^{10} +0.420116 q^{11} -1.13598 q^{12} -4.68326 q^{13} +1.77087 q^{14} +1.21221 q^{15} -4.98151 q^{16} -1.47613 q^{17} +1.77087 q^{18} +0.623303 q^{19} -1.37705 q^{20} -1.00000 q^{21} +0.743972 q^{22} +6.61561 q^{23} +1.53006 q^{24} -3.53056 q^{25} -8.29346 q^{26} -1.00000 q^{27} +1.13598 q^{28} +8.96966 q^{29} +2.14666 q^{30} -4.10556 q^{31} -5.76148 q^{32} -0.420116 q^{33} -2.61403 q^{34} -1.21221 q^{35} +1.13598 q^{36} -0.108692 q^{37} +1.10379 q^{38} +4.68326 q^{39} +1.85475 q^{40} -8.41060 q^{41} -1.77087 q^{42} +4.87673 q^{43} +0.477245 q^{44} -1.21221 q^{45} +11.7154 q^{46} +4.58782 q^{47} +4.98151 q^{48} +1.00000 q^{49} -6.25216 q^{50} +1.47613 q^{51} -5.32011 q^{52} -12.2746 q^{53} -1.77087 q^{54} -0.509268 q^{55} -1.53006 q^{56} -0.623303 q^{57} +15.8841 q^{58} +6.29726 q^{59} +1.37705 q^{60} -4.87908 q^{61} -7.27041 q^{62} +1.00000 q^{63} -0.239827 q^{64} +5.67708 q^{65} -0.743972 q^{66} -7.73109 q^{67} -1.67686 q^{68} -6.61561 q^{69} -2.14666 q^{70} +0.996207 q^{71} -1.53006 q^{72} +10.0810 q^{73} -0.192479 q^{74} +3.53056 q^{75} +0.708061 q^{76} +0.420116 q^{77} +8.29346 q^{78} +10.5616 q^{79} +6.03861 q^{80} +1.00000 q^{81} -14.8941 q^{82} +14.9617 q^{83} -1.13598 q^{84} +1.78937 q^{85} +8.63605 q^{86} -8.96966 q^{87} -0.642804 q^{88} -1.47668 q^{89} -2.14666 q^{90} -4.68326 q^{91} +7.51522 q^{92} +4.10556 q^{93} +8.12444 q^{94} -0.755571 q^{95} +5.76148 q^{96} -16.6828 q^{97} +1.77087 q^{98} +0.420116 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	1.77087	1.25219	0.626097	−	0.779745i	$-0.284651\pi$
$2$	1.77087	1.25219	0.626097	+	0.779745i	$0.284651\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.13598	0.567992
$5$	−1.21221	−0.542115	−0.271057	−	0.962563i	$-0.587373\pi$
$5$	−1.21221	−0.542115	−0.271057	+	0.962563i	$0.587373\pi$
$6$	−1.77087	−0.722955
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.53006	−0.540959
$9$	1.00000	0.333333
$10$	−2.14666	−0.678833
$11$	0.420116	0.126670	0.0633349	−	0.997992i	$-0.479826\pi$
$11$	0.420116	0.126670	0.0633349	+	0.997992i	$0.479826\pi$
$12$	−1.13598	−0.327930
$13$	−4.68326	−1.29890	−0.649452	−	0.760403i	$-0.725002\pi$
$13$	−4.68326	−1.29890	−0.649452	+	0.760403i	$0.725002\pi$
$14$	1.77087	0.473285
$15$	1.21221	0.312990
$16$	−4.98151	−1.24538
$17$	−1.47613	−0.358014	−0.179007	−	0.983848i	$-0.557288\pi$
$17$	−1.47613	−0.358014	−0.179007	+	0.983848i	$0.557288\pi$
$18$	1.77087	0.417398
$19$	0.623303	0.142995	0.0714977	−	0.997441i	$-0.477222\pi$
$19$	0.623303	0.142995	0.0714977	+	0.997441i	$0.477222\pi$
$20$	−1.37705	−0.307917
$21$	−1.00000	−0.218218
$22$	0.743972	0.158615
$23$	6.61561	1.37945	0.689725	−	0.724071i	$-0.257731\pi$
$23$	6.61561	1.37945	0.689725	+	0.724071i	$0.257731\pi$
$24$	1.53006	0.312323
$25$	−3.53056	−0.706112
$26$	−8.29346	−1.62648
$27$	−1.00000	−0.192450
$28$	1.13598	0.214681
$29$	8.96966	1.66562	0.832812	−	0.553556i	$-0.186730\pi$
$29$	8.96966	1.66562	0.832812	+	0.553556i	$0.186730\pi$
$30$	2.14666	0.391925
$31$	−4.10556	−0.737380	−0.368690	−	0.929552i	$-0.620194\pi$
$31$	−4.10556	−0.737380	−0.368690	+	0.929552i	$0.620194\pi$
$32$	−5.76148	−1.01850
$33$	−0.420116	−0.0731329
$34$	−2.61403	−0.448303
$35$	−1.21221	−0.204900
$36$	1.13598	0.189331
$37$	−0.108692	−0.0178688	−0.00893440	−	0.999960i	$-0.502844\pi$
$37$	−0.108692	−0.0178688	−0.00893440	+	0.999960i	$0.502844\pi$
$38$	1.10379	0.179058
$39$	4.68326	0.749922
$40$	1.85475	0.293262
$41$	−8.41060	−1.31351	−0.656757	−	0.754102i	$-0.728072\pi$
$41$	−8.41060	−1.31351	−0.656757	+	0.754102i	$0.728072\pi$
$42$	−1.77087	−0.273251
$43$	4.87673	0.743694	0.371847	−	0.928294i	$-0.378725\pi$
$43$	4.87673	0.743694	0.371847	+	0.928294i	$0.378725\pi$
$44$	0.477245	0.0719474
$45$	−1.21221	−0.180705
$46$	11.7154	1.72734
$47$	4.58782	0.669203	0.334602	−	0.942360i	$-0.391398\pi$
$47$	4.58782	0.669203	0.334602	+	0.942360i	$0.391398\pi$
$48$	4.98151	0.719019
$49$	1.00000	0.142857
$50$	−6.25216	−0.884189
$51$	1.47613	0.206699
$52$	−5.32011	−0.737767
$53$	−12.2746	−1.68604	−0.843020	−	0.537883i	$-0.819224\pi$
$53$	−12.2746	−1.68604	−0.843020	+	0.537883i	$0.819224\pi$
$54$	−1.77087	−0.240985
$55$	−0.509268	−0.0686696
$56$	−1.53006	−0.204463
$57$	−0.623303	−0.0825585
$58$	15.8841	2.08568
$59$	6.29726	0.819833	0.409917	−	0.912123i	$-0.365558\pi$
$59$	6.29726	0.819833	0.409917	+	0.912123i	$0.365558\pi$
$60$	1.37705	0.177776
$61$	−4.87908	−0.624703	−0.312351	−	0.949967i	$-0.601117\pi$
$61$	−4.87908	−0.624703	−0.312351	+	0.949967i	$0.601117\pi$
$62$	−7.27041	−0.923343
$63$	1.00000	0.125988
$64$	−0.239827	−0.0299784
$65$	5.67708	0.704155
$66$	−0.743972	−0.0915766
$67$	−7.73109	−0.944503	−0.472251	−	0.881464i	$-0.656559\pi$
$67$	−7.73109	−0.944503	−0.472251	+	0.881464i	$0.656559\pi$
$68$	−1.67686	−0.203349
$69$	−6.61561	−0.796426
$70$	−2.14666	−0.256575
$71$	0.996207	0.118228	0.0591140	−	0.998251i	$-0.481172\pi$
$71$	0.996207	0.118228	0.0591140	+	0.998251i	$0.481172\pi$
$72$	−1.53006	−0.180320
$73$	10.0810	1.17989	0.589945	−	0.807444i	$-0.299150\pi$
$73$	10.0810	1.17989	0.589945	+	0.807444i	$0.299150\pi$
$74$	−0.192479	−0.0223752
$75$	3.53056	0.407674
$76$	0.708061	0.0812202
$77$	0.420116	0.0478767
$78$	8.29346	0.939049
$79$	10.5616	1.18827	0.594134	−	0.804366i	$-0.297495\pi$
$79$	10.5616	1.18827	0.594134	+	0.804366i	$0.297495\pi$
$80$	6.03861	0.675137
$81$	1.00000	0.111111
$82$	−14.8941	−1.64478
$83$	14.9617	1.64226	0.821128	−	0.570744i	$-0.193345\pi$
$83$	14.9617	1.64226	0.821128	+	0.570744i	$0.193345\pi$
$84$	−1.13598	−0.123946
$85$	1.78937	0.194084
$86$	8.63605	0.931249
$87$	−8.96966	−0.961648
$88$	−0.642804	−0.0685232
$89$	−1.47668	−0.156528	−0.0782639	−	0.996933i	$-0.524938\pi$
$89$	−1.47668	−0.156528	−0.0782639	+	0.996933i	$0.524938\pi$
$90$	−2.14666	−0.226278
$91$	−4.68326	−0.490940
$92$	7.51522	0.783516
$93$	4.10556	0.425726
$94$	8.12444	0.837973
$95$	−0.755571	−0.0775200
$96$	5.76148	0.588029
$97$	−16.6828	−1.69388	−0.846939	−	0.531690i	$-0.821557\pi$
$97$	−16.6828	−1.69388	−0.846939	+	0.531690i	$0.821557\pi$
$98$	1.77087	0.178885
$99$	0.420116	0.0422233
$100$	−4.01065	−0.401065
$101$	1.40508	0.139811	0.0699055	−	0.997554i	$-0.477730\pi$
$101$	1.40508	0.139811	0.0699055	+	0.997554i	$0.477730\pi$
$102$	2.61403	0.258828
$103$	7.85599	0.774074	0.387037	−	0.922064i	$-0.373499\pi$
$103$	7.85599	0.774074	0.387037	+	0.922064i	$0.373499\pi$
$104$	7.16568	0.702653
$105$	1.21221	0.118299
$106$	−21.7366	−2.11125
$107$	17.0277	1.64613	0.823063	−	0.567950i	$-0.192263\pi$
$107$	17.0277	1.64613	0.823063	+	0.567950i	$0.192263\pi$
$108$	−1.13598	−0.109310
$109$	0.158297	0.0151621	0.00758104	−	0.999971i	$-0.497587\pi$
$109$	0.158297	0.0151621	0.00758104	+	0.999971i	$0.497587\pi$
$110$	−0.901847	−0.0859877
$111$	0.108692	0.0103166
$112$	−4.98151	−0.470708
$113$	16.5420	1.55614	0.778071	−	0.628176i	$-0.216198\pi$
$113$	16.5420	1.55614	0.778071	+	0.628176i	$0.216198\pi$
$114$	−1.10379	−0.103379
$115$	−8.01948	−0.747820
$116$	10.1894	0.946060
$117$	−4.68326	−0.432968
$118$	11.1516	1.02659
$119$	−1.47613	−0.135316
$120$	−1.85475	−0.169315
$121$	−10.8235	−0.983955
$122$	−8.64023	−0.782249
$123$	8.41060	0.758358
$124$	−4.66384	−0.418826
$125$	10.3408	0.924908
$126$	1.77087	0.157762
$127$	17.5682	1.55892	0.779462	−	0.626450i	$-0.215493\pi$
$127$	17.5682	1.55892	0.779462	+	0.626450i	$0.215493\pi$
$128$	11.0983	0.980957
$129$	−4.87673	−0.429372
$130$	10.0534	0.881739
$131$	6.82416	0.596229	0.298115	−	0.954530i	$-0.403642\pi$
$131$	6.82416	0.596229	0.298115	+	0.954530i	$0.403642\pi$
$132$	−0.477245	−0.0415389
$133$	0.623303	0.0540472
$134$	−13.6908	−1.18270
$135$	1.21221	0.104330
$136$	2.25857	0.193671
$137$	13.1077	1.11986	0.559931	−	0.828539i	$-0.310828\pi$
$137$	13.1077	1.11986	0.559931	+	0.828539i	$0.310828\pi$
$138$	−11.7154	−0.997280
$139$	14.7253	1.24898	0.624492	−	0.781031i	$-0.285306\pi$
$139$	14.7253	1.24898	0.624492	+	0.781031i	$0.285306\pi$
$140$	−1.37705	−0.116382
$141$	−4.58782	−0.386365
$142$	1.76415	0.148044
$143$	−1.96752	−0.164532
$144$	−4.98151	−0.415126
$145$	−10.8731	−0.902959
$146$	17.8521	1.47745
$147$	−1.00000	−0.0824786
$148$	−0.123472	−0.0101493
$149$	−1.65147	−0.135293	−0.0676467	−	0.997709i	$-0.521549\pi$
$149$	−1.65147	−0.135293	−0.0676467	+	0.997709i	$0.521549\pi$
$150$	6.25216	0.510487
$151$	−3.53254	−0.287474	−0.143737	−	0.989616i	$-0.545912\pi$
$151$	−3.53254	−0.287474	−0.143737	+	0.989616i	$0.545912\pi$
$152$	−0.953692	−0.0773546
$153$	−1.47613	−0.119338
$154$	0.743972	0.0599510
$155$	4.97678	0.399745
$156$	5.32011	0.425950
$157$	−7.90679	−0.631030	−0.315515	−	0.948921i	$-0.602177\pi$
$157$	−7.90679	−0.631030	−0.315515	+	0.948921i	$0.602177\pi$
$158$	18.7032	1.48794
$159$	12.2746	0.973435
$160$	6.98410	0.552142
$161$	6.61561	0.521383
$162$	1.77087	0.139133
$163$	6.64592	0.520548	0.260274	−	0.965535i	$-0.416187\pi$
$163$	6.64592	0.520548	0.260274	+	0.965535i	$0.416187\pi$
$164$	−9.55430	−0.746065
$165$	0.509268	0.0396464
$166$	26.4952	2.05642
$167$	−14.1173	−1.09243	−0.546214	−	0.837646i	$-0.683931\pi$
$167$	−14.1173	−1.09243	−0.546214	+	0.837646i	$0.683931\pi$
$168$	1.53006	0.118047
$169$	8.93297	0.687151
$170$	3.16874	0.243032
$171$	0.623303	0.0476651
$172$	5.53988	0.422412
$173$	−17.1313	−1.30247	−0.651235	−	0.758876i	$-0.725749\pi$
$173$	−17.1313	−1.30247	−0.651235	+	0.758876i	$0.725749\pi$
$174$	−15.8841	−1.20417
$175$	−3.53056	−0.266885
$176$	−2.09281	−0.157752
$177$	−6.29726	−0.473331
$178$	−2.61501	−0.196003
$179$	−11.4984	−0.859430	−0.429715	−	0.902964i	$-0.641386\pi$
$179$	−11.4984	−0.859430	−0.429715	+	0.902964i	$0.641386\pi$
$180$	−1.37705	−0.102639
$181$	14.3265	1.06488	0.532442	−	0.846467i	$-0.321275\pi$
$181$	14.3265	1.06488	0.532442	+	0.846467i	$0.321275\pi$
$182$	−8.29346	−0.614752
$183$	4.87908	0.360672
$184$	−10.1223	−0.746225
$185$	0.131757	0.00968694
$186$	7.27041	0.533092
$187$	−0.620146	−0.0453495
$188$	5.21169	0.380102
$189$	−1.00000	−0.0727393
$190$	−1.33802	−0.0970701
$191$	10.2576	0.742213	0.371107	−	0.928590i	$-0.378978\pi$
$191$	10.2576	0.742213	0.371107	+	0.928590i	$0.378978\pi$
$192$	0.239827	0.0173080
$193$	19.2605	1.38640	0.693200	−	0.720745i	$-0.256200\pi$
$193$	19.2605	1.38640	0.693200	+	0.720745i	$0.256200\pi$
$194$	−29.5430	−2.12107
$195$	−5.67708	−0.406544
$196$	1.13598	0.0811417
$197$	10.3996	0.740940	0.370470	−	0.928844i	$-0.379197\pi$
$197$	10.3996	0.740940	0.370470	+	0.928844i	$0.379197\pi$
$198$	0.743972	0.0528718
$199$	6.05043	0.428903	0.214452	−	0.976735i	$-0.431204\pi$
$199$	6.05043	0.428903	0.214452	+	0.976735i	$0.431204\pi$
$200$	5.40197	0.381977
$201$	7.73109	0.545309
$202$	2.48822	0.175071
$203$	8.96966	0.629546
$204$	1.67686	0.117403
$205$	10.1954	0.712076
$206$	13.9119	0.969291
$207$	6.61561	0.459817
$208$	23.3297	1.61763
$209$	0.261860	0.0181132
$210$	2.14666	0.148134
$211$	19.7409	1.35902	0.679509	−	0.733667i	$-0.262193\pi$
$211$	19.7409	1.35902	0.679509	+	0.733667i	$0.262193\pi$
$212$	−13.9437	−0.957656
$213$	−0.996207	−0.0682589
$214$	30.1538	2.06127
$215$	−5.91159	−0.403167
$216$	1.53006	0.104108
$217$	−4.10556	−0.278703
$218$	0.280323	0.0189859
$219$	−10.0810	−0.681210
$220$	−0.578519	−0.0390038
$221$	6.91310	0.465025
$222$	0.192479	0.0129183
$223$	−19.0009	−1.27240	−0.636198	−	0.771526i	$-0.719494\pi$
$223$	−19.0009	−1.27240	−0.636198	+	0.771526i	$0.719494\pi$
$224$	−5.76148	−0.384955
$225$	−3.53056	−0.235371
$226$	29.2938	1.94859
$227$	−12.6916	−0.842370	−0.421185	−	0.906975i	$-0.638386\pi$
$227$	−12.6916	−0.842370	−0.421185	+	0.906975i	$0.638386\pi$
$228$	−0.708061	−0.0468925
$229$	1.33425	0.0881697	0.0440848	−	0.999028i	$-0.485963\pi$
$229$	1.33425	0.0881697	0.0440848	+	0.999028i	$0.485963\pi$
$230$	−14.2015	−0.936417
$231$	−0.420116	−0.0276416
$232$	−13.7241	−0.901033
$233$	−29.9922	−1.96485	−0.982427	−	0.186648i	$-0.940238\pi$
$233$	−29.9922	−1.96485	−0.982427	+	0.186648i	$0.940238\pi$
$234$	−8.29346	−0.542160
$235$	−5.56139	−0.362785
$236$	7.15358	0.465658
$237$	−10.5616	−0.686047
$238$	−2.61403	−0.169443
$239$	7.06786	0.457182	0.228591	−	0.973523i	$-0.426588\pi$
$239$	7.06786	0.457182	0.228591	+	0.973523i	$0.426588\pi$
$240$	−6.03861	−0.389791
$241$	2.30503	0.148480	0.0742401	−	0.997240i	$-0.476347\pi$
$241$	2.30503	0.148480	0.0742401	+	0.997240i	$0.476347\pi$
$242$	−19.1670	−1.23210
$243$	−1.00000	−0.0641500
$244$	−5.54256	−0.354826
$245$	−1.21221	−0.0774450
$246$	14.8941	0.949612
$247$	−2.91909	−0.185737
$248$	6.28176	0.398892
$249$	−14.9617	−0.948157
$250$	18.3122	1.15817
$251$	8.13438	0.513438	0.256719	−	0.966486i	$-0.417359\pi$
$251$	8.13438	0.513438	0.256719	+	0.966486i	$0.417359\pi$
$252$	1.13598	0.0715602
$253$	2.77933	0.174735
$254$	31.1110	1.95208
$255$	−1.78937	−0.112055
$256$	20.1332	1.25833
$257$	−4.31496	−0.269160	−0.134580	−	0.990903i	$-0.542969\pi$
$257$	−4.31496	−0.269160	−0.134580	+	0.990903i	$0.542969\pi$
$258$	−8.63605	−0.537657
$259$	−0.108692	−0.00675377
$260$	6.44907	0.399954
$261$	8.96966	0.555208
$262$	12.0847	0.746595
$263$	−4.96863	−0.306379	−0.153189	−	0.988197i	$-0.548954\pi$
$263$	−4.96863	−0.306379	−0.153189	+	0.988197i	$0.548954\pi$
$264$	0.642804	0.0395619
$265$	14.8793	0.914027
$266$	1.10379	0.0676776
$267$	1.47668	0.0903714
$268$	−8.78239	−0.536470
$269$	6.96796	0.424844	0.212422	−	0.977178i	$-0.431865\pi$
$269$	6.96796	0.424844	0.212422	+	0.977178i	$0.431865\pi$
$270$	2.14666	0.130642
$271$	26.6849	1.62099	0.810495	−	0.585746i	$-0.199198\pi$
$271$	26.6849	1.62099	0.810495	+	0.585746i	$0.199198\pi$
$272$	7.35334	0.445862
$273$	4.68326	0.283444
$274$	23.2120	1.40229
$275$	−1.48325	−0.0894431
$276$	−7.51522	−0.452363
$277$	2.56608	0.154181	0.0770904	−	0.997024i	$-0.475437\pi$
$277$	2.56608	0.154181	0.0770904	+	0.997024i	$0.475437\pi$
$278$	26.0766	1.56397
$279$	−4.10556	−0.245793
$280$	1.85475	0.110842
$281$	−3.78768	−0.225954	−0.112977	−	0.993598i	$-0.536039\pi$
$281$	−3.78768	−0.225954	−0.112977	+	0.993598i	$0.536039\pi$
$282$	−8.12444	−0.483804
$283$	1.80094	0.107055	0.0535274	−	0.998566i	$-0.482954\pi$
$283$	1.80094	0.107055	0.0535274	+	0.998566i	$0.482954\pi$
$284$	1.13167	0.0671525
$285$	0.755571	0.0447562
$286$	−3.48422	−0.206026
$287$	−8.41060	−0.496462
$288$	−5.76148	−0.339499
$289$	−14.8210	−0.871826
$290$	−19.2548	−1.13068
$291$	16.6828	0.977961
$292$	11.4518	0.670167
$293$	24.5449	1.43393	0.716964	−	0.697110i	$-0.245531\pi$
$293$	24.5449	1.43393	0.716964	+	0.697110i	$0.245531\pi$
$294$	−1.77087	−0.103279
$295$	−7.63357	−0.444444
$296$	0.166305	0.00966628
$297$	−0.420116	−0.0243776
$298$	−2.92453	−0.169414
$299$	−30.9826	−1.79177
$300$	4.01065	0.231555
$301$	4.87673	0.281090
$302$	−6.25567	−0.359973
$303$	−1.40508	−0.0807200
$304$	−3.10499	−0.178083
$305$	5.91445	0.338661
$306$	−2.61403	−0.149434
$307$	13.4231	0.766097	0.383049	−	0.923728i	$-0.374874\pi$
$307$	13.4231	0.766097	0.383049	+	0.923728i	$0.374874\pi$
$308$	0.477245	0.0271936
$309$	−7.85599	−0.446912
$310$	8.81323	0.500558
$311$	2.83039	0.160497	0.0802483	−	0.996775i	$-0.474429\pi$
$311$	2.83039	0.160497	0.0802483	+	0.996775i	$0.474429\pi$
$312$	−7.16568	−0.405677
$313$	−0.795693	−0.0449753	−0.0224876	−	0.999747i	$-0.507159\pi$
$313$	−0.795693	−0.0449753	−0.0224876	+	0.999747i	$0.507159\pi$
$314$	−14.0019	−0.790173
$315$	−1.21221	−0.0683000
$316$	11.9978	0.674927
$317$	8.32951	0.467832	0.233916	−	0.972257i	$-0.424846\pi$
$317$	8.32951	0.467832	0.233916	+	0.972257i	$0.424846\pi$
$318$	21.7366	1.21893
$319$	3.76830	0.210984
$320$	0.290719	0.0162517
$321$	−17.0277	−0.950391
$322$	11.7154	0.652873
$323$	−0.920075	−0.0511943
$324$	1.13598	0.0631102
$325$	16.5345	0.917171
$326$	11.7691	0.651828
$327$	−0.158297	−0.00875383
$328$	12.8687	0.710557
$329$	4.58782	0.252935
$330$	0.901847	0.0496450
$331$	−12.8910	−0.708551	−0.354276	−	0.935141i	$-0.615273\pi$
$331$	−12.8910	−0.708551	−0.354276	+	0.935141i	$0.615273\pi$
$332$	16.9962	0.932788
$333$	−0.108692	−0.00595626
$334$	−24.9999	−1.36793
$335$	9.37167	0.512029
$336$	4.98151	0.271764
$337$	22.8525	1.24486	0.622428	−	0.782677i	$-0.286147\pi$
$337$	22.8525	1.24486	0.622428	+	0.782677i	$0.286147\pi$
$338$	15.8191	0.860447
$339$	−16.5420	−0.898439
$340$	2.03270	0.110238
$341$	−1.72481	−0.0934038
$342$	1.10379	0.0596860
$343$	1.00000	0.0539949
$344$	−7.46169	−0.402307
$345$	8.01948	0.431754
$346$	−30.3373	−1.63095
$347$	−13.2286	−0.710148	−0.355074	−	0.934838i	$-0.615544\pi$
$347$	−13.2286	−0.710148	−0.355074	+	0.934838i	$0.615544\pi$
$348$	−10.1894	−0.546208
$349$	7.83818	0.419568	0.209784	−	0.977748i	$-0.432724\pi$
$349$	7.83818	0.419568	0.209784	+	0.977748i	$0.432724\pi$
$350$	−6.25216	−0.334192
$351$	4.68326	0.249974
$352$	−2.42049	−0.129013
$353$	8.70079	0.463097	0.231548	−	0.972823i	$-0.425621\pi$
$353$	8.70079	0.463097	0.231548	+	0.972823i	$0.425621\pi$
$354$	−11.1516	−0.592702
$355$	−1.20761	−0.0640931
$356$	−1.67748	−0.0889065
$357$	1.47613	0.0781250
$358$	−20.3622	−1.07617
$359$	29.4248	1.55298	0.776490	−	0.630130i	$-0.216998\pi$
$359$	29.4248	1.55298	0.776490	+	0.630130i	$0.216998\pi$
$360$	1.85475	0.0977539
$361$	−18.6115	−0.979552
$362$	25.3704	1.33344
$363$	10.8235	0.568087
$364$	−5.32011	−0.278850
$365$	−12.2202	−0.639636
$366$	8.64023	0.451632
$367$	−21.8582	−1.14099	−0.570494	−	0.821302i	$-0.693248\pi$
$367$	−21.8582	−1.14099	−0.570494	+	0.821302i	$0.693248\pi$
$368$	−32.9557	−1.71794
$369$	−8.41060	−0.437838
$370$	0.233324	0.0121299
$371$	−12.2746	−0.637263
$372$	4.66384	0.241809
$373$	14.1149	0.730841	0.365421	−	0.930843i	$-0.380925\pi$
$373$	14.1149	0.730841	0.365421	+	0.930843i	$0.380925\pi$
$374$	−1.09820	−0.0567865
$375$	−10.3408	−0.533996
$376$	−7.01966	−0.362011
$377$	−42.0073	−2.16348
$378$	−1.77087	−0.0910838
$379$	−10.5898	−0.543962	−0.271981	−	0.962303i	$-0.587679\pi$
$379$	−10.5898	−0.543962	−0.271981	+	0.962303i	$0.587679\pi$
$380$	−0.858316	−0.0440307
$381$	−17.5682	−0.900045
$382$	18.1649	0.929396
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−11.0983	−0.566356
$385$	−0.509268	−0.0259547
$386$	34.1078	1.73604
$387$	4.87673	0.247898
$388$	−18.9513	−0.962109
$389$	−31.9477	−1.61981	−0.809905	−	0.586560i	$-0.800482\pi$
$389$	−31.9477	−1.61981	−0.809905	+	0.586560i	$0.800482\pi$
$390$	−10.0534	−0.509072
$391$	−9.76549	−0.493862
$392$	−1.53006	−0.0772798
$393$	−6.82416	−0.344233
$394$	18.4163	0.927801
$395$	−12.8028	−0.644178
$396$	0.477245	0.0239825
$397$	20.0119	1.00437	0.502183	−	0.864761i	$-0.332530\pi$
$397$	20.0119	1.00437	0.502183	+	0.864761i	$0.332530\pi$
$398$	10.7145	0.537071
$399$	−0.623303	−0.0312042
$400$	17.5875	0.879375
$401$	−6.36180	−0.317693	−0.158847	−	0.987303i	$-0.550778\pi$
$401$	−6.36180	−0.317693	−0.158847	+	0.987303i	$0.550778\pi$
$402$	13.6908	0.682833
$403$	19.2274	0.957786
$404$	1.59615	0.0794115
$405$	−1.21221	−0.0602350
$406$	15.8841	0.788315
$407$	−0.0456631	−0.00226344
$408$	−2.25857	−0.111816
$409$	3.13588	0.155059	0.0775295	−	0.996990i	$-0.475297\pi$
$409$	3.13588	0.155059	0.0775295	+	0.996990i	$0.475297\pi$
$410$	18.0547	0.891658
$411$	−13.1077	−0.646553
$412$	8.92428	0.439668
$413$	6.29726	0.309868
$414$	11.7154	0.575780
$415$	−18.1366	−0.890291
$416$	26.9826	1.32293
$417$	−14.7253	−0.721101
$418$	0.463720	0.0226813
$419$	17.5353	0.856655	0.428328	−	0.903623i	$-0.359103\pi$
$419$	17.5353	0.856655	0.428328	+	0.903623i	$0.359103\pi$
$420$	1.37705	0.0671929
$421$	−4.21920	−0.205631	−0.102816	−	0.994700i	$-0.532785\pi$
$421$	−4.21920	−0.205631	−0.102816	+	0.994700i	$0.532785\pi$
$422$	34.9585	1.70176
$423$	4.58782	0.223068
$424$	18.7808	0.912078
$425$	5.21155	0.252798
$426$	−1.76415	−0.0854735
$427$	−4.87908	−0.236115
$428$	19.3431	0.934986
$429$	1.96752	0.0949926
$430$	−10.4687	−0.504844
$431$	−0.633916	−0.0305347	−0.0152673	−	0.999883i	$-0.504860\pi$
$431$	−0.633916	−0.0305347	−0.0152673	+	0.999883i	$0.504860\pi$
$432$	4.98151	0.239673
$433$	−2.61229	−0.125539	−0.0627693	−	0.998028i	$-0.519993\pi$
$433$	−2.61229	−0.125539	−0.0627693	+	0.998028i	$0.519993\pi$
$434$	−7.27041	−0.348991
$435$	10.8731	0.521324
$436$	0.179822	0.00861193
$437$	4.12353	0.197255
$438$	−17.8521	−0.853007
$439$	18.7863	0.896621	0.448310	−	0.893878i	$-0.352026\pi$
$439$	18.7863	0.896621	0.448310	+	0.893878i	$0.352026\pi$
$440$	0.779211	0.0371474
$441$	1.00000	0.0476190
$442$	12.2422	0.582302
$443$	6.06208	0.288018	0.144009	−	0.989576i	$-0.454001\pi$
$443$	6.06208	0.288018	0.144009	+	0.989576i	$0.454001\pi$
$444$	0.123472	0.00585972
$445$	1.79004	0.0848560
$446$	−33.6482	−1.59329
$447$	1.65147	0.0781117
$448$	−0.239827	−0.0113308
$449$	11.6760	0.551024	0.275512	−	0.961298i	$-0.411153\pi$
$449$	11.6760	0.551024	0.275512	+	0.961298i	$0.411153\pi$
$450$	−6.25216	−0.294730
$451$	−3.53343	−0.166383
$452$	18.7915	0.883876
$453$	3.53254	0.165973
$454$	−22.4752	−1.05481
$455$	5.67708	0.266146
$456$	0.953692	0.0446607
$457$	−2.22357	−0.104014	−0.0520071	−	0.998647i	$-0.516562\pi$
$457$	−2.22357	−0.104014	−0.0520071	+	0.998647i	$0.516562\pi$
$458$	2.36278	0.110406
$459$	1.47613	0.0688998
$460$	−9.10999	−0.424756
$461$	−20.0323	−0.932999	−0.466499	−	0.884522i	$-0.654485\pi$
$461$	−20.0323	−0.932999	−0.466499	+	0.884522i	$0.654485\pi$
$462$	−0.743972	−0.0346127
$463$	26.2859	1.22161	0.610804	−	0.791782i	$-0.290846\pi$
$463$	26.2859	1.22161	0.610804	+	0.791782i	$0.290846\pi$
$464$	−44.6824	−2.07433
$465$	−4.97678	−0.230793
$466$	−53.1123	−2.46038
$467$	22.0433	1.02004	0.510020	−	0.860162i	$-0.329638\pi$
$467$	22.0433	1.02004	0.510020	+	0.860162i	$0.329638\pi$
$468$	−5.32011	−0.245922
$469$	−7.73109	−0.356988
$470$	−9.84850	−0.454277
$471$	7.90679	0.364326
$472$	−9.63519	−0.443496
$473$	2.04879	0.0942036
$474$	−18.7032	−0.859065
$475$	−2.20061	−0.100971
$476$	−1.67686	−0.0768586
$477$	−12.2746	−0.562013
$478$	12.5163	0.572481
$479$	−0.550027	−0.0251314	−0.0125657	−	0.999921i	$-0.504000\pi$
$479$	−0.550027	−0.0251314	−0.0125657	+	0.999921i	$0.504000\pi$
$480$	−6.98410	−0.318779
$481$	0.509032	0.0232098
$482$	4.08191	0.185926
$483$	−6.61561	−0.301021
$484$	−12.2953	−0.558878
$485$	20.2229	0.918276
$486$	−1.77087	−0.0803283
$487$	25.6534	1.16247	0.581234	−	0.813737i	$-0.302570\pi$
$487$	25.6534	1.16247	0.581234	+	0.813737i	$0.302570\pi$
$488$	7.46530	0.337938
$489$	−6.64592	−0.300539
$490$	−2.14666	−0.0969762
$491$	−19.0917	−0.861597	−0.430799	−	0.902448i	$-0.641768\pi$
$491$	−19.0917	−0.861597	−0.430799	+	0.902448i	$0.641768\pi$
$492$	9.55430	0.430741
$493$	−13.2404	−0.596316
$494$	−5.16933	−0.232579
$495$	−0.509268	−0.0228899
$496$	20.4519	0.918316
$497$	0.996207	0.0446860
$498$	−26.4952	−1.18728
$499$	29.8401	1.33583	0.667914	−	0.744238i	$-0.267187\pi$
$499$	29.8401	1.33583	0.667914	+	0.744238i	$0.267187\pi$
$500$	11.7470	0.525340
$501$	14.1173	0.630714
$502$	14.4049	0.642924
$503$	−18.6596	−0.831991	−0.415996	−	0.909367i	$-0.636567\pi$
$503$	−18.6596	−0.831991	−0.415996	+	0.909367i	$0.636567\pi$
$504$	−1.53006	−0.0681544
$505$	−1.70325	−0.0757937
$506$	4.92183	0.218802
$507$	−8.93297	−0.396727
$508$	19.9572	0.885455
$509$	15.5755	0.690370	0.345185	−	0.938535i	$-0.387816\pi$
$509$	15.5755	0.690370	0.345185	+	0.938535i	$0.387816\pi$
$510$	−3.16874	−0.140314
$511$	10.0810	0.445956
$512$	13.4568	0.594714
$513$	−0.623303	−0.0275195
$514$	−7.64124	−0.337041
$515$	−9.52308	−0.419637
$516$	−5.53988	−0.243880
$517$	1.92742	0.0847679
$518$	−0.192479	−0.00845703
$519$	17.1313	0.751981
$520$	−8.68628	−0.380919
$521$	36.5917	1.60311	0.801555	−	0.597921i	$-0.204006\pi$
$521$	36.5917	1.60311	0.801555	+	0.597921i	$0.204006\pi$
$522$	15.8841	0.695228
$523$	23.4310	1.02457	0.512283	−	0.858817i	$-0.328800\pi$
$523$	23.4310	1.02457	0.512283	+	0.858817i	$0.328800\pi$
$524$	7.75213	0.338653
$525$	3.53056	0.154086
$526$	−8.79880	−0.383646
$527$	6.06033	0.263992
$528$	2.09281	0.0910780
$529$	20.7663	0.902882
$530$	26.3493	1.14454
$531$	6.29726	0.273278
$532$	0.708061	0.0306984
$533$	39.3891	1.70613
$534$	2.61501	0.113163
$535$	−20.6410	−0.892389
$536$	11.8290	0.510937
$537$	11.4984	0.496192
$538$	12.3394	0.531987
$539$	0.420116	0.0180957
$540$	1.37705	0.0592586
$541$	−13.6708	−0.587754	−0.293877	−	0.955843i	$-0.594946\pi$
$541$	−13.6708	−0.587754	−0.293877	+	0.955843i	$0.594946\pi$
$542$	47.2554	2.02979
$543$	−14.3265	−0.614811
$544$	8.50469	0.364635
$545$	−0.191888	−0.00821959
$546$	8.29346	0.354927
$547$	40.1320	1.71592	0.857960	−	0.513716i	$-0.171732\pi$
$547$	40.1320	1.71592	0.857960	+	0.513716i	$0.171732\pi$
$548$	14.8901	0.636073
$549$	−4.87908	−0.208234
$550$	−2.62664	−0.112000
$551$	5.59081	0.238177
$552$	10.1223	0.430833
$553$	10.5616	0.449123
$554$	4.54420	0.193064
$555$	−0.131757	−0.00559276
$556$	16.7277	0.709412
$557$	−4.54533	−0.192592	−0.0962959	−	0.995353i	$-0.530700\pi$
$557$	−4.54533	−0.192592	−0.0962959	+	0.995353i	$0.530700\pi$
$558$	−7.27041	−0.307781
$559$	−22.8390	−0.965986
$560$	6.03861	0.255178
$561$	0.620146	0.0261826
$562$	−6.70749	−0.282938
$563$	−27.9107	−1.17629	−0.588147	−	0.808754i	$-0.700142\pi$
$563$	−27.9107	−1.17629	−0.588147	+	0.808754i	$0.700142\pi$
$564$	−5.21169	−0.219452
$565$	−20.0523	−0.843608
$566$	3.18923	0.134053
$567$	1.00000	0.0419961
$568$	−1.52426	−0.0639564
$569$	−41.8818	−1.75578	−0.877888	−	0.478867i	$-0.841048\pi$
$569$	−41.8818	−1.75578	−0.877888	+	0.478867i	$0.841048\pi$
$570$	1.33802	0.0560434
$571$	32.3647	1.35442	0.677210	−	0.735790i	$-0.263189\pi$
$571$	32.3647	1.35442	0.677210	+	0.735790i	$0.263189\pi$
$572$	−2.23507	−0.0934528
$573$	−10.2576	−0.428517
$574$	−14.8941	−0.621667
$575$	−23.3568	−0.974046
$576$	−0.239827	−0.00999278
$577$	−4.60787	−0.191828	−0.0959141	−	0.995390i	$-0.530577\pi$
$577$	−4.60787	−0.191828	−0.0959141	+	0.995390i	$0.530577\pi$
$578$	−26.2462	−1.09170
$579$	−19.2605	−0.800438
$580$	−12.3516	−0.512873
$581$	14.9617	0.620714
$582$	29.5430	1.22460
$583$	−5.15674	−0.213570
$584$	−15.4245	−0.638271
$585$	5.67708	0.234718
$586$	43.4658	1.79556
$587$	−12.5096	−0.516325	−0.258162	−	0.966102i	$-0.583117\pi$
$587$	−12.5096	−0.516325	−0.258162	+	0.966102i	$0.583117\pi$
$588$	−1.13598	−0.0468472
$589$	−2.55901	−0.105442
$590$	−13.5181	−0.556530
$591$	−10.3996	−0.427782
$592$	0.541448	0.0222534
$593$	−15.3520	−0.630432	−0.315216	−	0.949020i	$-0.602077\pi$
$593$	−15.3520	−0.630432	−0.315216	+	0.949020i	$0.602077\pi$
$594$	−0.743972	−0.0305255
$595$	1.78937	0.0733570
$596$	−1.87604	−0.0768455
$597$	−6.05043	−0.247627
$598$	−54.8663	−2.24365
$599$	−4.18739	−0.171092	−0.0855460	−	0.996334i	$-0.527263\pi$
$599$	−4.18739	−0.171092	−0.0855460	+	0.996334i	$0.527263\pi$
$600$	−5.40197	−0.220535
$601$	2.43864	0.0994741	0.0497371	−	0.998762i	$-0.484162\pi$
$601$	2.43864	0.0994741	0.0497371	+	0.998762i	$0.484162\pi$
$602$	8.63605	0.351979
$603$	−7.73109	−0.314834
$604$	−4.01291	−0.163283
$605$	13.1203	0.533416
$606$	−2.48822	−0.101077
$607$	25.5161	1.03567	0.517833	−	0.855482i	$-0.326739\pi$
$607$	25.5161	1.03567	0.517833	+	0.855482i	$0.326739\pi$
$608$	−3.59115	−0.145640
$609$	−8.96966	−0.363469
$610$	10.4737	0.424069
$611$	−21.4860	−0.869230
$612$	−1.67686	−0.0677829
$613$	−10.0245	−0.404887	−0.202444	−	0.979294i	$-0.564888\pi$
$613$	−10.0245	−0.404887	−0.202444	+	0.979294i	$0.564888\pi$
$614$	23.7706	0.959303
$615$	−10.1954	−0.411117
$616$	−0.642804	−0.0258993
$617$	−13.3854	−0.538875	−0.269437	−	0.963018i	$-0.586838\pi$
$617$	−13.3854	−0.538875	−0.269437	+	0.963018i	$0.586838\pi$
$618$	−13.9119	−0.559621
$619$	7.75583	0.311733	0.155867	−	0.987778i	$-0.450183\pi$
$619$	7.75583	0.311733	0.155867	+	0.987778i	$0.450183\pi$
$620$	5.65354	0.227052
$621$	−6.61561	−0.265475
$622$	5.01225	0.200973
$623$	−1.47668	−0.0591619
$624$	−23.3297	−0.933936
$625$	5.11762	0.204705
$626$	−1.40907	−0.0563178
$627$	−0.261860	−0.0104577
$628$	−8.98198	−0.358420
$629$	0.160443	0.00639727
$630$	−2.14666	−0.0855250
$631$	−40.0509	−1.59440	−0.797200	−	0.603715i	$-0.793686\pi$
$631$	−40.0509	−1.59440	−0.797200	+	0.603715i	$0.793686\pi$
$632$	−16.1598	−0.642804
$633$	−19.7409	−0.784629
$634$	14.7505	0.585817
$635$	−21.2962	−0.845115
$636$	13.9437	0.552903
$637$	−4.68326	−0.185558
$638$	6.67317	0.264193
$639$	0.996207	0.0394093
$640$	−13.4534	−0.531792
$641$	−3.08906	−0.122011	−0.0610053	−	0.998137i	$-0.519431\pi$
$641$	−3.08906	−0.122011	−0.0610053	+	0.998137i	$0.519431\pi$
$642$	−30.1538	−1.19007
$643$	−14.0632	−0.554600	−0.277300	−	0.960783i	$-0.589440\pi$
$643$	−14.0632	−0.554600	−0.277300	+	0.960783i	$0.589440\pi$
$644$	7.51522	0.296141
$645$	5.91159	0.232769
$646$	−1.62933	−0.0641053
$647$	−40.4010	−1.58833	−0.794164	−	0.607704i	$-0.792091\pi$
$647$	−40.4010	−1.58833	−0.794164	+	0.607704i	$0.792091\pi$
$648$	−1.53006	−0.0601065
$649$	2.64558	0.103848
$650$	29.2805	1.14848
$651$	4.10556	0.160909
$652$	7.54965	0.295667
$653$	29.6697	1.16106	0.580532	−	0.814237i	$-0.302844\pi$
$653$	29.6697	1.16106	0.580532	+	0.814237i	$0.302844\pi$
$654$	−0.280323	−0.0109615
$655$	−8.27228	−0.323225
$656$	41.8975	1.63582
$657$	10.0810	0.393296
$658$	8.12444	0.316724
$659$	−6.81234	−0.265371	−0.132686	−	0.991158i	$-0.542360\pi$
$659$	−6.81234	−0.265371	−0.132686	+	0.991158i	$0.542360\pi$
$660$	0.578519	0.0225188
$661$	−10.8723	−0.422885	−0.211442	−	0.977390i	$-0.567816\pi$
$661$	−10.8723	−0.422885	−0.211442	+	0.977390i	$0.567816\pi$
$662$	−22.8282	−0.887244
$663$	−6.91310	−0.268482
$664$	−22.8923	−0.888393
$665$	−0.755571	−0.0292998
$666$	−0.192479	−0.00745840
$667$	59.3397	2.29764
$668$	−16.0370	−0.620490
$669$	19.0009	0.734618
$670$	16.5960	0.641160
$671$	−2.04978	−0.0791310
$672$	5.76148	0.222254
$673$	−11.9064	−0.458957	−0.229479	−	0.973314i	$-0.573702\pi$
$673$	−11.9064	−0.458957	−0.229479	+	0.973314i	$0.573702\pi$
$674$	40.4688	1.55880
$675$	3.53056	0.135891
$676$	10.1477	0.390296
$677$	11.6203	0.446604	0.223302	−	0.974749i	$-0.428316\pi$
$677$	11.6203	0.446604	0.223302	+	0.974749i	$0.428316\pi$
$678$	−29.2938	−1.12502
$679$	−16.6828	−0.640226
$680$	−2.73785	−0.104992
$681$	12.6916	0.486343
$682$	−3.05442	−0.116960
$683$	30.8933	1.18210	0.591050	−	0.806635i	$-0.298714\pi$
$683$	30.8933	1.18210	0.591050	+	0.806635i	$0.298714\pi$
$684$	0.708061	0.0270734
$685$	−15.8892	−0.607094
$686$	1.77087	0.0676122
$687$	−1.33425	−0.0509048
$688$	−24.2934	−0.926179
$689$	57.4850	2.19000
$690$	14.2015	0.540640
$691$	1.29456	0.0492472	0.0246236	−	0.999697i	$-0.492161\pi$
$691$	1.29456	0.0492472	0.0246236	+	0.999697i	$0.492161\pi$
$692$	−19.4609	−0.739792
$693$	0.420116	0.0159589
$694$	−23.4261	−0.889243
$695$	−17.8501	−0.677092
$696$	13.7241	0.520212
$697$	12.4151	0.470256
$698$	13.8804	0.525381
$699$	29.9922	1.13441
$700$	−4.01065	−0.151588
$701$	13.3104	0.502728	0.251364	−	0.967893i	$-0.419121\pi$
$701$	13.3104	0.502728	0.251364	+	0.967893i	$0.419121\pi$
$702$	8.29346	0.313016
$703$	−0.0677478	−0.00255516
$704$	−0.100755	−0.00379735
$705$	5.56139	0.209454
$706$	15.4080	0.579887
$707$	1.40508	0.0528436
$708$	−7.15358	−0.268848
$709$	12.7574	0.479116	0.239558	−	0.970882i	$-0.422998\pi$
$709$	12.7574	0.479116	0.239558	+	0.970882i	$0.422998\pi$
$710$	−2.13852	−0.0802571
$711$	10.5616	0.396090
$712$	2.25941	0.0846750
$713$	−27.1608	−1.01718
$714$	2.61403	0.0978277
$715$	2.38503	0.0891952
$716$	−13.0620	−0.488149
$717$	−7.06786	−0.263954
$718$	52.1075	1.94463
$719$	−18.9785	−0.707780	−0.353890	−	0.935287i	$-0.615141\pi$
$719$	−18.9785	−0.707780	−0.353890	+	0.935287i	$0.615141\pi$
$720$	6.03861	0.225046
$721$	7.85599	0.292572
$722$	−32.9586	−1.22659
$723$	−2.30503	−0.0857250
$724$	16.2747	0.604845
$725$	−31.6679	−1.17612
$726$	19.1670	0.711355
$727$	−23.8514	−0.884600	−0.442300	−	0.896867i	$-0.645837\pi$
$727$	−23.8514	−0.884600	−0.442300	+	0.896867i	$0.645837\pi$
$728$	7.16568	0.265578
$729$	1.00000	0.0370370
$730$	−21.6404	−0.800948
$731$	−7.19867	−0.266252
$732$	5.54256	0.204859
$733$	−43.5777	−1.60958	−0.804789	−	0.593560i	$-0.797722\pi$
$733$	−43.5777	−1.60958	−0.804789	+	0.593560i	$0.797722\pi$
$734$	−38.7080	−1.42874
$735$	1.21221	0.0447129
$736$	−38.1157	−1.40496
$737$	−3.24796	−0.119640
$738$	−14.8941	−0.548259
$739$	−11.2610	−0.414243	−0.207121	−	0.978315i	$-0.566410\pi$
$739$	−11.2610	−0.414243	−0.207121	+	0.978315i	$0.566410\pi$
$740$	0.149673	0.00550210
$741$	2.91909	0.107236
$742$	−21.7366	−0.797977
$743$	41.1665	1.51025	0.755126	−	0.655580i	$-0.227576\pi$
$743$	41.1665	1.51025	0.755126	+	0.655580i	$0.227576\pi$
$744$	−6.28176	−0.230300
$745$	2.00192	0.0733446
$746$	24.9956	0.915155
$747$	14.9617	0.547419
$748$	−0.704475	−0.0257582
$749$	17.0277	0.622177
$750$	−18.3122	−0.668667
$751$	27.4947	1.00330	0.501648	−	0.865072i	$-0.332727\pi$
$751$	27.4947	1.00330	0.501648	+	0.865072i	$0.332727\pi$
$752$	−22.8543	−0.833410
$753$	−8.13438	−0.296433
$754$	−74.3895	−2.70910
$755$	4.28216	0.155844
$756$	−1.13598	−0.0413153
$757$	27.6925	1.00650	0.503251	−	0.864140i	$-0.332137\pi$
$757$	27.6925	1.00650	0.503251	+	0.864140i	$0.332137\pi$
$758$	−18.7532	−0.681146
$759$	−2.77933	−0.100883
$760$	1.15607	0.0419351
$761$	−23.6029	−0.855603	−0.427802	−	0.903873i	$-0.640712\pi$
$761$	−23.6029	−0.855603	−0.427802	+	0.903873i	$0.640712\pi$
$762$	−31.1110	−1.12703
$763$	0.158297	0.00573073
$764$	11.6525	0.421571
$765$	1.78937	0.0646948
$766$	−1.77087	−0.0639842
$767$	−29.4917	−1.06488
$768$	−20.1332	−0.726496
$769$	15.3808	0.554646	0.277323	−	0.960777i	$-0.410553\pi$
$769$	15.3808	0.554646	0.277323	+	0.960777i	$0.410553\pi$
$770$	−0.901847	−0.0325003
$771$	4.31496	0.155400
$772$	21.8796	0.787464
$773$	−26.2890	−0.945548	−0.472774	−	0.881184i	$-0.656747\pi$
$773$	−26.2890	−0.945548	−0.472774	+	0.881184i	$0.656747\pi$
$774$	8.63605	0.310416
$775$	14.4949	0.520672
$776$	25.5257	0.916318
$777$	0.108692	0.00389929
$778$	−56.5752	−2.02832
$779$	−5.24235	−0.187827
$780$	−6.44907	−0.230914
$781$	0.418523	0.0149759
$782$	−17.2934	−0.618411
$783$	−8.96966	−0.320549
$784$	−4.98151	−0.177911
$785$	9.58465	0.342091
$786$	−12.0847	−0.431047
$787$	−42.1216	−1.50147	−0.750737	−	0.660602i	$-0.770301\pi$
$787$	−42.1216	−1.50147	−0.750737	+	0.660602i	$0.770301\pi$
$788$	11.8138	0.420848
$789$	4.96863	0.176888
$790$	−22.6721	−0.806637
$791$	16.5420	0.588166
$792$	−0.642804	−0.0228411
$793$	22.8500	0.811429
$794$	35.4384	1.25766
$795$	−14.8793	−0.527714
$796$	6.87318	0.243614
$797$	−20.7629	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$797$	−20.7629	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$798$	−1.10379	−0.0390737
$799$	−6.77222	−0.239584
$800$	20.3413	0.719172
$801$	−1.47668	−0.0521759
$802$	−11.2659	−0.397814
$803$	4.23519	0.149456
$804$	8.78239	0.309731
$805$	−8.01948	−0.282650
$806$	34.0493	1.19933
$807$	−6.96796	−0.245284
$808$	−2.14987	−0.0756320
$809$	−34.9375	−1.22834	−0.614168	−	0.789175i	$-0.710508\pi$
$809$	−34.9375	−1.22834	−0.614168	+	0.789175i	$0.710508\pi$
$810$	−2.14666	−0.0754259
$811$	−18.6071	−0.653384	−0.326692	−	0.945131i	$-0.605934\pi$
$811$	−18.6071	−0.653384	−0.326692	+	0.945131i	$0.605934\pi$
$812$	10.1894	0.357577
$813$	−26.6849	−0.935879
$814$	−0.0808635	−0.00283427
$815$	−8.05622	−0.282197
$816$	−7.35334	−0.257419
$817$	3.03968	0.106345
$818$	5.55323	0.194164
$819$	−4.68326	−0.163647
$820$	11.5818	0.404453
$821$	−20.1560	−0.703449	−0.351724	−	0.936104i	$-0.614405\pi$
$821$	−20.1560	−0.703449	−0.351724	+	0.936104i	$0.614405\pi$
$822$	−23.2120	−0.809610
$823$	21.7386	0.757758	0.378879	−	0.925446i	$-0.376310\pi$
$823$	21.7386	0.757758	0.378879	+	0.925446i	$0.376310\pi$
$824$	−12.0202	−0.418742
$825$	1.48325	0.0516400
$826$	11.1516	0.388015
$827$	46.7285	1.62491	0.812455	−	0.583024i	$-0.198131\pi$
$827$	46.7285	1.62491	0.812455	+	0.583024i	$0.198131\pi$
$828$	7.51522	0.261172
$829$	−47.5431	−1.65124	−0.825620	−	0.564227i	$-0.809174\pi$
$829$	−47.5431	−1.65124	−0.825620	+	0.564227i	$0.809174\pi$
$830$	−32.1176	−1.11482
$831$	−2.56608	−0.0890163
$832$	1.12317	0.0389390
$833$	−1.47613	−0.0511448
$834$	−26.0766	−0.902959
$835$	17.1130	0.592221
$836$	0.297468	0.0102882
$837$	4.10556	0.141909
$838$	31.0527	1.07270
$839$	−10.8235	−0.373668	−0.186834	−	0.982392i	$-0.559823\pi$
$839$	−10.8235	−0.373668	−0.186834	+	0.982392i	$0.559823\pi$
$840$	−1.85475	−0.0639949
$841$	51.4547	1.77430
$842$	−7.47165	−0.257490
$843$	3.78768	0.130455
$844$	22.4253	0.771911
$845$	−10.8286	−0.372515
$846$	8.12444	0.279324
$847$	−10.8235	−0.371900
$848$	61.1458	2.09976
$849$	−1.80094	−0.0618081
$850$	9.22899	0.316552
$851$	−0.719061	−0.0246491
$852$	−1.13167	−0.0387705
$853$	−32.6636	−1.11838	−0.559191	−	0.829039i	$-0.688888\pi$
$853$	−32.6636	−1.11838	−0.559191	+	0.829039i	$0.688888\pi$
$854$	−8.64023	−0.295663
$855$	−0.755571	−0.0258400
$856$	−26.0534	−0.890486
$857$	4.08173	0.139429	0.0697146	−	0.997567i	$-0.477791\pi$
$857$	4.08173	0.139429	0.0697146	+	0.997567i	$0.477791\pi$
$858$	3.48422	0.118949
$859$	−22.1876	−0.757031	−0.378516	−	0.925595i	$-0.623565\pi$
$859$	−22.1876	−0.757031	−0.378516	+	0.925595i	$0.623565\pi$
$860$	−6.71547	−0.228996
$861$	8.41060	0.286632
$862$	−1.12258	−0.0382354
$863$	−48.0254	−1.63480	−0.817401	−	0.576069i	$-0.804586\pi$
$863$	−48.0254	−1.63480	−0.817401	+	0.576069i	$0.804586\pi$
$864$	5.76148	0.196010
$865$	20.7667	0.706088
$866$	−4.62603	−0.157199
$867$	14.8210	0.503349
$868$	−4.66384	−0.158301
$869$	4.43709	0.150518
$870$	19.2548	0.652799
$871$	36.2067	1.22682
$872$	−0.242204	−0.00820205
$873$	−16.6828	−0.564626
$874$	7.30223	0.247002
$875$	10.3408	0.349582
$876$	−11.4518	−0.386921
$877$	33.4154	1.12836	0.564179	−	0.825653i	$-0.309193\pi$
$877$	33.4154	1.12836	0.564179	+	0.825653i	$0.309193\pi$
$878$	33.2681	1.12274
$879$	−24.5449	−0.827878
$880$	2.53692	0.0855196
$881$	−21.9971	−0.741101	−0.370551	−	0.928812i	$-0.620831\pi$
$881$	−21.9971	−0.741101	−0.370551	+	0.928812i	$0.620831\pi$
$882$	1.77087	0.0596283
$883$	−40.8425	−1.37446	−0.687229	−	0.726441i	$-0.741173\pi$
$883$	−40.8425	−1.37446	−0.687229	+	0.726441i	$0.741173\pi$
$884$	7.85316	0.264130
$885$	7.63357	0.256600
$886$	10.7352	0.360655
$887$	11.9150	0.400067	0.200034	−	0.979789i	$-0.435895\pi$
$887$	11.9150	0.400067	0.200034	+	0.979789i	$0.435895\pi$
$888$	−0.166305	−0.00558083
$889$	17.5682	0.589218
$890$	3.16993	0.106256
$891$	0.420116	0.0140744
$892$	−21.5847	−0.722710
$893$	2.85960	0.0956930
$894$	2.92453	0.0978110
$895$	13.9384	0.465910
$896$	11.0983	0.370767
$897$	30.9826	1.03448
$898$	20.6767	0.689989
$899$	−36.8254	−1.22820
$900$	−4.01065	−0.133688
$901$	18.1188	0.603625
$902$	−6.25725	−0.208344
$903$	−4.87673	−0.162287
$904$	−25.3103	−0.841808
$905$	−17.3667	−0.577289
$906$	6.25567	0.207831
$907$	35.5729	1.18118	0.590590	−	0.806972i	$-0.298895\pi$
$907$	35.5729	1.18118	0.590590	+	0.806972i	$0.298895\pi$
$908$	−14.4174	−0.478459
$909$	1.40508	0.0466037
$910$	10.0534	0.333266
$911$	−26.2178	−0.868636	−0.434318	−	0.900760i	$-0.643011\pi$
$911$	−26.2178	−0.868636	−0.434318	+	0.900760i	$0.643011\pi$
$912$	3.10499	0.102816
$913$	6.28564	0.208024
$914$	−3.93766	−0.130246
$915$	−5.91445	−0.195526
$916$	1.51568	0.0500796
$917$	6.82416	0.225353
$918$	2.61403	0.0862759
$919$	−6.34639	−0.209348	−0.104674	−	0.994507i	$-0.533380\pi$
$919$	−6.34639	−0.209348	−0.104674	+	0.994507i	$0.533380\pi$
$920$	12.2703	0.404540
$921$	−13.4231	−0.442306
$922$	−35.4747	−1.16830
$923$	−4.66550	−0.153567
$924$	−0.477245	−0.0157002
$925$	0.383742	0.0126174
$926$	46.5489	1.52969
$927$	7.85599	0.258025
$928$	−51.6785	−1.69643
$929$	−22.0667	−0.723986	−0.361993	−	0.932181i	$-0.617904\pi$
$929$	−22.0667	−0.723986	−0.361993	+	0.932181i	$0.617904\pi$
$930$	−8.81323	−0.288997
$931$	0.623303	0.0204279
$932$	−34.0706	−1.11602
$933$	−2.83039	−0.0926627
$934$	39.0358	1.27729
$935$	0.751744	0.0245847
$936$	7.16568	0.234218
$937$	−43.9204	−1.43482	−0.717408	−	0.696653i	$-0.754672\pi$
$937$	−43.9204	−1.43482	−0.717408	+	0.696653i	$0.754672\pi$
$938$	−13.6908	−0.447019
$939$	0.795693	0.0259665
$940$	−6.31764	−0.206059
$941$	33.7864	1.10141	0.550703	−	0.834701i	$-0.314360\pi$
$941$	33.7864	1.10141	0.550703	+	0.834701i	$0.314360\pi$
$942$	14.0019	0.456207
$943$	−55.6412	−1.81193
$944$	−31.3698	−1.02100
$945$	1.21221	0.0394331
$946$	3.62815	0.117961
$947$	34.0742	1.10726	0.553632	−	0.832762i	$-0.313242\pi$
$947$	34.0742	1.10726	0.553632	+	0.832762i	$0.313242\pi$
$948$	−11.9978	−0.389669
$949$	−47.2119	−1.53256
$950$	−3.89699	−0.126435
$951$	−8.32951	−0.270103
$952$	2.25857	0.0732006
$953$	−39.1843	−1.26930	−0.634652	−	0.772798i	$-0.718856\pi$
$953$	−39.1843	−1.26930	−0.634652	+	0.772798i	$0.718856\pi$
$954$	−21.7366	−0.703750
$955$	−12.4343	−0.402365
$956$	8.02897	0.259676
$957$	−3.76830	−0.121812
$958$	−0.974027	−0.0314694
$959$	13.1077	0.423268
$960$	−0.290719	−0.00938293
$961$	−14.1444	−0.456271
$962$	0.901429	0.0290632
$963$	17.0277	0.548709
$964$	2.61848	0.0843355
$965$	−23.3477	−0.751588
$966$	−11.7154	−0.376936
$967$	−17.7790	−0.571735	−0.285868	−	0.958269i	$-0.592282\pi$
$967$	−17.7790	−0.571735	−0.285868	+	0.958269i	$0.592282\pi$
$968$	16.5606	0.532279
$969$	0.920075	0.0295571
$970$	35.8122	1.14986
$971$	1.39112	0.0446432	0.0223216	−	0.999751i	$-0.492894\pi$
$971$	1.39112	0.0446432	0.0223216	+	0.999751i	$0.492894\pi$
$972$	−1.13598	−0.0364367
$973$	14.7253	0.472071
$974$	45.4289	1.45564
$975$	−16.5345	−0.529529
$976$	24.3052	0.777990
$977$	5.37901	0.172090	0.0860449	−	0.996291i	$-0.472577\pi$
$977$	5.37901	0.172090	0.0860449	+	0.996291i	$0.472577\pi$
$978$	−11.7691	−0.376333
$979$	−0.620378	−0.0198274
$980$	−1.37705	−0.0439881
$981$	0.158297	0.00505403
$982$	−33.8090	−1.07889
$983$	−25.2715	−0.806036	−0.403018	−	0.915192i	$-0.632039\pi$
$983$	−25.2715	−0.806036	−0.403018	+	0.915192i	$0.632039\pi$
$984$	−12.8687	−0.410240
$985$	−12.6064	−0.401674
$986$	−23.4470	−0.746704
$987$	−4.58782	−0.146032
$988$	−3.31604	−0.105497
$989$	32.2625	1.02589
$990$	−0.901847	−0.0286626
$991$	26.5498	0.843381	0.421691	−	0.906740i	$-0.361437\pi$
$991$	26.5498	0.843381	0.421691	+	0.906740i	$0.361437\pi$
$992$	23.6541	0.751019
$993$	12.8910	0.409082
$994$	1.76415	0.0559555
$995$	−7.33436	−0.232515
$996$	−16.9962	−0.538545
$997$	−4.88810	−0.154808	−0.0774038	−	0.997000i	$-0.524663\pi$
$997$	−4.88810	−0.154808	−0.0774038	+	0.997000i	$0.524663\pi$
$998$	52.8430	1.67272
$999$	0.108692	0.00343885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.40	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.40	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77087 q^{2} -1.00000 q^{3} +1.13598 q^{4} -1.21221 q^{5} -1.77087 q^{6} +1.00000 q^{7} -1.53006 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.77087 q^{2} -1.00000 q^{3} +1.13598 q^{4} -1.21221 q^{5} -1.77087 q^{6} +1.00000 q^{7} -1.53006 q^{8} +1.00000 q^{9} -2.14666 q^{10} +0.420116 q^{11} -1.13598 q^{12} -4.68326 q^{13} +1.77087 q^{14} +1.21221 q^{15} -4.98151 q^{16} -1.47613 q^{17} +1.77087 q^{18} +0.623303 q^{19} -1.37705 q^{20} -1.00000 q^{21} +0.743972 q^{22} +6.61561 q^{23} +1.53006 q^{24} -3.53056 q^{25} -8.29346 q^{26} -1.00000 q^{27} +1.13598 q^{28} +8.96966 q^{29} +2.14666 q^{30} -4.10556 q^{31} -5.76148 q^{32} -0.420116 q^{33} -2.61403 q^{34} -1.21221 q^{35} +1.13598 q^{36} -0.108692 q^{37} +1.10379 q^{38} +4.68326 q^{39} +1.85475 q^{40} -8.41060 q^{41} -1.77087 q^{42} +4.87673 q^{43} +0.477245 q^{44} -1.21221 q^{45} +11.7154 q^{46} +4.58782 q^{47} +4.98151 q^{48} +1.00000 q^{49} -6.25216 q^{50} +1.47613 q^{51} -5.32011 q^{52} -12.2746 q^{53} -1.77087 q^{54} -0.509268 q^{55} -1.53006 q^{56} -0.623303 q^{57} +15.8841 q^{58} +6.29726 q^{59} +1.37705 q^{60} -4.87908 q^{61} -7.27041 q^{62} +1.00000 q^{63} -0.239827 q^{64} +5.67708 q^{65} -0.743972 q^{66} -7.73109 q^{67} -1.67686 q^{68} -6.61561 q^{69} -2.14666 q^{70} +0.996207 q^{71} -1.53006 q^{72} +10.0810 q^{73} -0.192479 q^{74} +3.53056 q^{75} +0.708061 q^{76} +0.420116 q^{77} +8.29346 q^{78} +10.5616 q^{79} +6.03861 q^{80} +1.00000 q^{81} -14.8941 q^{82} +14.9617 q^{83} -1.13598 q^{84} +1.78937 q^{85} +8.63605 q^{86} -8.96966 q^{87} -0.642804 q^{88} -1.47668 q^{89} -2.14666 q^{90} -4.68326 q^{91} +7.51522 q^{92} +4.10556 q^{93} +8.12444 q^{94} -0.755571 q^{95} +5.76148 q^{96} -16.6828 q^{97} +1.77087 q^{98} +0.420116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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