LMFDB - Embedded newform 8043.2.a.t.1.45

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			1.45
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)

$f(q)$	$=$	$q+2.35796 q^{2} -1.00000 q^{3} +3.55996 q^{4} +1.43169 q^{5} -2.35796 q^{6} +1.00000 q^{7} +3.67830 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.35796 q^{2} -1.00000 q^{3} +3.55996 q^{4} +1.43169 q^{5} -2.35796 q^{6} +1.00000 q^{7} +3.67830 q^{8} +1.00000 q^{9} +3.37587 q^{10} -4.36512 q^{11} -3.55996 q^{12} +6.27913 q^{13} +2.35796 q^{14} -1.43169 q^{15} +1.55337 q^{16} +3.16581 q^{17} +2.35796 q^{18} +2.39131 q^{19} +5.09676 q^{20} -1.00000 q^{21} -10.2927 q^{22} -8.72942 q^{23} -3.67830 q^{24} -2.95026 q^{25} +14.8059 q^{26} -1.00000 q^{27} +3.55996 q^{28} +8.42255 q^{29} -3.37587 q^{30} -3.52801 q^{31} -3.69383 q^{32} +4.36512 q^{33} +7.46485 q^{34} +1.43169 q^{35} +3.55996 q^{36} +9.12536 q^{37} +5.63860 q^{38} -6.27913 q^{39} +5.26620 q^{40} +0.844792 q^{41} -2.35796 q^{42} +10.7642 q^{43} -15.5396 q^{44} +1.43169 q^{45} -20.5836 q^{46} +10.3940 q^{47} -1.55337 q^{48} +1.00000 q^{49} -6.95658 q^{50} -3.16581 q^{51} +22.3534 q^{52} -6.07467 q^{53} -2.35796 q^{54} -6.24950 q^{55} +3.67830 q^{56} -2.39131 q^{57} +19.8600 q^{58} +11.3611 q^{59} -5.09676 q^{60} -3.22662 q^{61} -8.31889 q^{62} +1.00000 q^{63} -11.8166 q^{64} +8.98978 q^{65} +10.2927 q^{66} +0.00565557 q^{67} +11.2702 q^{68} +8.72942 q^{69} +3.37587 q^{70} +7.39583 q^{71} +3.67830 q^{72} +12.9942 q^{73} +21.5172 q^{74} +2.95026 q^{75} +8.51295 q^{76} -4.36512 q^{77} -14.8059 q^{78} +12.5561 q^{79} +2.22395 q^{80} +1.00000 q^{81} +1.99198 q^{82} -9.65775 q^{83} -3.55996 q^{84} +4.53247 q^{85} +25.3816 q^{86} -8.42255 q^{87} -16.0562 q^{88} -3.78900 q^{89} +3.37587 q^{90} +6.27913 q^{91} -31.0764 q^{92} +3.52801 q^{93} +24.5085 q^{94} +3.42362 q^{95} +3.69383 q^{96} -5.91028 q^{97} +2.35796 q^{98} -4.36512 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.35796	1.66733	0.833663	−	0.552273i	$-0.186239\pi$
$2$	2.35796	1.66733	0.833663	+	0.552273i	$0.186239\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.55996	1.77998
$5$	1.43169	0.640272	0.320136	−	0.947372i	$-0.396271\pi$
$5$	1.43169	0.640272	0.320136	+	0.947372i	$0.396271\pi$
$6$	−2.35796	−0.962631
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.67830	1.30048
$9$	1.00000	0.333333
$10$	3.37587	1.06754
$11$	−4.36512	−1.31613	−0.658066	−	0.752960i	$-0.728625\pi$
$11$	−4.36512	−1.31613	−0.658066	+	0.752960i	$0.728625\pi$
$12$	−3.55996	−1.02767
$13$	6.27913	1.74152	0.870759	−	0.491710i	$-0.163628\pi$
$13$	6.27913	1.74152	0.870759	+	0.491710i	$0.163628\pi$
$14$	2.35796	0.630190
$15$	−1.43169	−0.369661
$16$	1.55337	0.388342
$17$	3.16581	0.767823	0.383911	−	0.923370i	$-0.374577\pi$
$17$	3.16581	0.767823	0.383911	+	0.923370i	$0.374577\pi$
$18$	2.35796	0.555775
$19$	2.39131	0.548604	0.274302	−	0.961644i	$-0.411553\pi$
$19$	2.39131	0.548604	0.274302	+	0.961644i	$0.411553\pi$
$20$	5.09676	1.13967
$21$	−1.00000	−0.218218
$22$	−10.2927	−2.19442
$23$	−8.72942	−1.82021	−0.910105	−	0.414377i	$-0.863999\pi$
$23$	−8.72942	−1.82021	−0.910105	+	0.414377i	$0.863999\pi$
$24$	−3.67830	−0.750831
$25$	−2.95026	−0.590052
$26$	14.8059	2.90368
$27$	−1.00000	−0.192450
$28$	3.55996	0.672768
$29$	8.42255	1.56403	0.782014	−	0.623260i	$-0.214192\pi$
$29$	8.42255	1.56403	0.782014	+	0.623260i	$0.214192\pi$
$30$	−3.37587	−0.616346
$31$	−3.52801	−0.633650	−0.316825	−	0.948484i	$-0.602617\pi$
$31$	−3.52801	−0.633650	−0.316825	+	0.948484i	$0.602617\pi$
$32$	−3.69383	−0.652984
$33$	4.36512	0.759869
$34$	7.46485	1.28021
$35$	1.43169	0.242000
$36$	3.55996	0.593326
$37$	9.12536	1.50020	0.750100	−	0.661325i	$-0.230005\pi$
$37$	9.12536	1.50020	0.750100	+	0.661325i	$0.230005\pi$
$38$	5.63860	0.914701
$39$	−6.27913	−1.00547
$40$	5.26620	0.832659
$41$	0.844792	0.131934	0.0659672	−	0.997822i	$-0.478987\pi$
$41$	0.844792	0.131934	0.0659672	+	0.997822i	$0.478987\pi$
$42$	−2.35796	−0.363840
$43$	10.7642	1.64153	0.820766	−	0.571264i	$-0.193547\pi$
$43$	10.7642	1.64153	0.820766	+	0.571264i	$0.193547\pi$
$44$	−15.5396	−2.34268
$45$	1.43169	0.213424
$46$	−20.5836	−3.03489
$47$	10.3940	1.51612	0.758059	−	0.652186i	$-0.226148\pi$
$47$	10.3940	1.51612	0.758059	+	0.652186i	$0.226148\pi$
$48$	−1.55337	−0.224210
$49$	1.00000	0.142857
$50$	−6.95658	−0.983809
$51$	−3.16581	−0.443303
$52$	22.3534	3.09986
$53$	−6.07467	−0.834421	−0.417210	−	0.908810i	$-0.636992\pi$
$53$	−6.07467	−0.834421	−0.417210	+	0.908810i	$0.636992\pi$
$54$	−2.35796	−0.320877
$55$	−6.24950	−0.842682
$56$	3.67830	0.491534
$57$	−2.39131	−0.316736
$58$	19.8600	2.60775
$59$	11.3611	1.47909	0.739547	−	0.673105i	$-0.235040\pi$
$59$	11.3611	1.47909	0.739547	+	0.673105i	$0.235040\pi$
$60$	−5.09676	−0.657989
$61$	−3.22662	−0.413126	−0.206563	−	0.978433i	$-0.566228\pi$
$61$	−3.22662	−0.413126	−0.206563	+	0.978433i	$0.566228\pi$
$62$	−8.31889	−1.05650
$63$	1.00000	0.125988
$64$	−11.8166	−1.47708
$65$	8.98978	1.11505
$66$	10.2927	1.26695
$67$	0.00565557	0.000690938 0	0.000345469	−	1.00000i	$-0.499890\pi$
$67$	0.00565557	0.000690938 0	0.000345469		1.00000i	$0.499890\pi$
$68$	11.2702	1.36671
$69$	8.72942	1.05090
$70$	3.37587	0.403493
$71$	7.39583	0.877723	0.438862	−	0.898555i	$-0.355382\pi$
$71$	7.39583	0.877723	0.438862	+	0.898555i	$0.355382\pi$
$72$	3.67830	0.433492
$73$	12.9942	1.52086	0.760431	−	0.649419i	$-0.224988\pi$
$73$	12.9942	1.52086	0.760431	+	0.649419i	$0.224988\pi$
$74$	21.5172	2.50132
$75$	2.95026	0.340666
$76$	8.51295	0.976502
$77$	−4.36512	−0.497451
$78$	−14.8059	−1.67644
$79$	12.5561	1.41267	0.706333	−	0.707879i	$-0.250348\pi$
$79$	12.5561	1.41267	0.706333	+	0.707879i	$0.250348\pi$
$80$	2.22395	0.248645
$81$	1.00000	0.111111
$82$	1.99198	0.219978
$83$	−9.65775	−1.06008	−0.530038	−	0.847974i	$-0.677822\pi$
$83$	−9.65775	−1.06008	−0.530038	+	0.847974i	$0.677822\pi$
$84$	−3.55996	−0.388423
$85$	4.53247	0.491615
$86$	25.3816	2.73697
$87$	−8.42255	−0.902993
$88$	−16.0562	−1.71160
$89$	−3.78900	−0.401633	−0.200817	−	0.979629i	$-0.564360\pi$
$89$	−3.78900	−0.401633	−0.200817	+	0.979629i	$0.564360\pi$
$90$	3.37587	0.355848
$91$	6.27913	0.658232
$92$	−31.0764	−3.23993
$93$	3.52801	0.365838
$94$	24.5085	2.52786
$95$	3.42362	0.351256
$96$	3.69383	0.377000
$97$	−5.91028	−0.600098	−0.300049	−	0.953924i	$-0.597003\pi$
$97$	−5.91028	−0.600098	−0.300049	+	0.953924i	$0.597003\pi$
$98$	2.35796	0.238189
$99$	−4.36512	−0.438711
$100$	−10.5028	−1.05028
$101$	−4.86006	−0.483594	−0.241797	−	0.970327i	$-0.577737\pi$
$101$	−4.86006	−0.483594	−0.241797	+	0.970327i	$0.577737\pi$
$102$	−7.46485	−0.739130
$103$	−12.2098	−1.20307	−0.601534	−	0.798847i	$-0.705443\pi$
$103$	−12.2098	−1.20307	−0.601534	+	0.798847i	$0.705443\pi$
$104$	23.0966	2.26480
$105$	−1.43169	−0.139719
$106$	−14.3238	−1.39125
$107$	4.40730	0.426070	0.213035	−	0.977045i	$-0.431665\pi$
$107$	4.40730	0.426070	0.213035	+	0.977045i	$0.431665\pi$
$108$	−3.55996	−0.342557
$109$	−12.9720	−1.24250	−0.621248	−	0.783614i	$-0.713374\pi$
$109$	−12.9720	−1.24250	−0.621248	+	0.783614i	$0.713374\pi$
$110$	−14.7360	−1.40503
$111$	−9.12536	−0.866141
$112$	1.55337	0.146780
$113$	10.4743	0.985343	0.492672	−	0.870215i	$-0.336020\pi$
$113$	10.4743	0.985343	0.492672	+	0.870215i	$0.336020\pi$
$114$	−5.63860	−0.528103
$115$	−12.4978	−1.16543
$116$	29.9839	2.78394
$117$	6.27913	0.580506
$118$	26.7891	2.46613
$119$	3.16581	0.290210
$120$	−5.26620	−0.480736
$121$	8.05423	0.732203
$122$	−7.60822	−0.688816
$123$	−0.844792	−0.0761724
$124$	−12.5596	−1.12788
$125$	−11.3823	−1.01807
$126$	2.35796	0.210063
$127$	−5.32100	−0.472162	−0.236081	−	0.971733i	$-0.575863\pi$
$127$	−5.32100	−0.472162	−0.236081	+	0.971733i	$0.575863\pi$
$128$	−20.4754	−1.80979
$129$	−10.7642	−0.947739
$130$	21.1975	1.85915
$131$	−17.1349	−1.49708	−0.748540	−	0.663089i	$-0.769245\pi$
$131$	−17.1349	−1.49708	−0.748540	+	0.663089i	$0.769245\pi$
$132$	15.5396	1.35255
$133$	2.39131	0.207353
$134$	0.0133356	0.00115202
$135$	−1.43169	−0.123220
$136$	11.6448	0.998536
$137$	−0.450551	−0.0384932	−0.0192466	−	0.999815i	$-0.506127\pi$
$137$	−0.450551	−0.0384932	−0.0192466	+	0.999815i	$0.506127\pi$
$138$	20.5836	1.75219
$139$	−10.3729	−0.879814	−0.439907	−	0.898043i	$-0.644989\pi$
$139$	−10.3729	−0.879814	−0.439907	+	0.898043i	$0.644989\pi$
$140$	5.09676	0.430755
$141$	−10.3940	−0.875331
$142$	17.4390	1.46345
$143$	−27.4091	−2.29207
$144$	1.55337	0.129447
$145$	12.0585	1.00140
$146$	30.6399	2.53577
$147$	−1.00000	−0.0824786
$148$	32.4859	2.67032
$149$	19.3145	1.58230	0.791151	−	0.611621i	$-0.209482\pi$
$149$	19.3145	1.58230	0.791151	+	0.611621i	$0.209482\pi$
$150$	6.95658	0.568002
$151$	7.29518	0.593673	0.296836	−	0.954928i	$-0.404068\pi$
$151$	7.29518	0.593673	0.296836	+	0.954928i	$0.404068\pi$
$152$	8.79596	0.713446
$153$	3.16581	0.255941
$154$	−10.2927	−0.829413
$155$	−5.05103	−0.405708
$156$	−22.3534	−1.78971
$157$	8.62254	0.688154	0.344077	−	0.938941i	$-0.388192\pi$
$157$	8.62254	0.688154	0.344077	+	0.938941i	$0.388192\pi$
$158$	29.6066	2.35538
$159$	6.07467	0.481753
$160$	−5.28843	−0.418087
$161$	−8.72942	−0.687975
$162$	2.35796	0.185258
$163$	5.99453	0.469528	0.234764	−	0.972052i	$-0.424568\pi$
$163$	5.99453	0.469528	0.234764	+	0.972052i	$0.424568\pi$
$164$	3.00742	0.234840
$165$	6.24950	0.486523
$166$	−22.7726	−1.76749
$167$	5.72338	0.442888	0.221444	−	0.975173i	$-0.428923\pi$
$167$	5.72338	0.442888	0.221444	+	0.975173i	$0.428923\pi$
$168$	−3.67830	−0.283787
$169$	26.4275	2.03289
$170$	10.6874	0.819683
$171$	2.39131	0.182868
$172$	38.3202	2.92189
$173$	12.1863	0.926509	0.463254	−	0.886225i	$-0.346682\pi$
$173$	12.1863	0.926509	0.463254	+	0.886225i	$0.346682\pi$
$174$	−19.8600	−1.50558
$175$	−2.95026	−0.223019
$176$	−6.78064	−0.511110
$177$	−11.3611	−0.853956
$178$	−8.93430	−0.669654
$179$	7.86630	0.587955	0.293978	−	0.955812i	$-0.405021\pi$
$179$	7.86630	0.587955	0.293978	+	0.955812i	$0.405021\pi$
$180$	5.09676	0.379890
$181$	0.251159	0.0186685	0.00933425	−	0.999956i	$-0.497029\pi$
$181$	0.251159	0.0186685	0.00933425	+	0.999956i	$0.497029\pi$
$182$	14.8059	1.09749
$183$	3.22662	0.238518
$184$	−32.1095	−2.36714
$185$	13.0647	0.960536
$186$	8.31889	0.609971
$187$	−13.8191	−1.01056
$188$	37.0021	2.69866
$189$	−1.00000	−0.0727393
$190$	8.07273	0.585658
$191$	−23.2271	−1.68065	−0.840326	−	0.542082i	$-0.817636\pi$
$191$	−23.2271	−1.68065	−0.840326	+	0.542082i	$0.817636\pi$
$192$	11.8166	0.852792
$193$	8.22253	0.591871	0.295935	−	0.955208i	$-0.404369\pi$
$193$	8.22253	0.591871	0.295935	+	0.955208i	$0.404369\pi$
$194$	−13.9362	−1.00056
$195$	−8.98978	−0.643772
$196$	3.55996	0.254283
$197$	−22.0586	−1.57161	−0.785805	−	0.618475i	$-0.787751\pi$
$197$	−22.0586	−1.57161	−0.785805	+	0.618475i	$0.787751\pi$
$198$	−10.2927	−0.731474
$199$	22.8099	1.61695	0.808476	−	0.588529i	$-0.200293\pi$
$199$	22.8099	1.61695	0.808476	+	0.588529i	$0.200293\pi$
$200$	−10.8519	−0.767349
$201$	−0.00565557	−0.000398913 0
$202$	−11.4598	−0.806309
$203$	8.42255	0.591147
$204$	−11.2702	−0.789069
$205$	1.20948	0.0844739
$206$	−28.7902	−2.00591
$207$	−8.72942	−0.606737
$208$	9.75382	0.676306
$209$	−10.4383	−0.722035
$210$	−3.37587	−0.232957
$211$	−12.4881	−0.859717	−0.429859	−	0.902896i	$-0.641437\pi$
$211$	−12.4881	−0.859717	−0.429859	+	0.902896i	$0.641437\pi$
$212$	−21.6256	−1.48525
$213$	−7.39583	−0.506754
$214$	10.3922	0.710397
$215$	15.4111	1.05103
$216$	−3.67830	−0.250277
$217$	−3.52801	−0.239497
$218$	−30.5875	−2.07165
$219$	−12.9942	−0.878070
$220$	−22.2479	−1.49996
$221$	19.8786	1.33718
$222$	−21.5172	−1.44414
$223$	18.4713	1.23693	0.618466	−	0.785811i	$-0.287754\pi$
$223$	18.4713	1.23693	0.618466	+	0.785811i	$0.287754\pi$
$224$	−3.69383	−0.246805
$225$	−2.95026	−0.196684
$226$	24.6980	1.64289
$227$	−15.0873	−1.00138	−0.500689	−	0.865627i	$-0.666920\pi$
$227$	−15.0873	−1.00138	−0.500689	+	0.865627i	$0.666920\pi$
$228$	−8.51295	−0.563784
$229$	15.6421	1.03366	0.516830	−	0.856088i	$-0.327112\pi$
$229$	15.6421	1.03366	0.516830	+	0.856088i	$0.327112\pi$
$230$	−29.4694	−1.94315
$231$	4.36512	0.287203
$232$	30.9807	2.03398
$233$	−2.89412	−0.189600	−0.0948000	−	0.995496i	$-0.530221\pi$
$233$	−2.89412	−0.189600	−0.0948000	+	0.995496i	$0.530221\pi$
$234$	14.8059	0.967893
$235$	14.8810	0.970728
$236$	40.4452	2.63276
$237$	−12.5561	−0.815604
$238$	7.46485	0.483874
$239$	13.5783	0.878310	0.439155	−	0.898411i	$-0.355278\pi$
$239$	13.5783	0.878310	0.439155	+	0.898411i	$0.355278\pi$
$240$	−2.22395	−0.143555
$241$	0.304916	0.0196414	0.00982070	−	0.999952i	$-0.496874\pi$
$241$	0.304916	0.0196414	0.00982070	+	0.999952i	$0.496874\pi$
$242$	18.9915	1.22082
$243$	−1.00000	−0.0641500
$244$	−11.4866	−0.735355
$245$	1.43169	0.0914674
$246$	−1.99198	−0.127004
$247$	15.0153	0.955403
$248$	−12.9771	−0.824047
$249$	9.65775	0.612035
$250$	−26.8390	−1.69745
$251$	−28.2439	−1.78274	−0.891369	−	0.453278i	$-0.850254\pi$
$251$	−28.2439	−1.78274	−0.891369	+	0.453278i	$0.850254\pi$
$252$	3.55996	0.224256
$253$	38.1049	2.39564
$254$	−12.5467	−0.787248
$255$	−4.53247	−0.283834
$256$	−24.6469	−1.54043
$257$	11.0519	0.689401	0.344701	−	0.938713i	$-0.387980\pi$
$257$	11.0519	0.689401	0.344701	+	0.938713i	$0.387980\pi$
$258$	−25.3816	−1.58019
$259$	9.12536	0.567022
$260$	32.0032	1.98476
$261$	8.42255	0.521343
$262$	−40.4033	−2.49612
$263$	−14.6634	−0.904185	−0.452093	−	0.891971i	$-0.649322\pi$
$263$	−14.6634	−0.904185	−0.452093	+	0.891971i	$0.649322\pi$
$264$	16.0562	0.988192
$265$	−8.69706	−0.534256
$266$	5.63860	0.345725
$267$	3.78900	0.231883
$268$	0.0201336	0.00122985
$269$	−19.6117	−1.19575	−0.597874	−	0.801590i	$-0.703988\pi$
$269$	−19.6117	−1.19575	−0.597874	+	0.801590i	$0.703988\pi$
$270$	−3.37587	−0.205449
$271$	−6.71460	−0.407883	−0.203941	−	0.978983i	$-0.565375\pi$
$271$	−6.71460	−0.407883	−0.203941	+	0.978983i	$0.565375\pi$
$272$	4.91768	0.298178
$273$	−6.27913	−0.380030
$274$	−1.06238	−0.0641807
$275$	12.8782	0.776586
$276$	31.0764	1.87058
$277$	−2.80019	−0.168247	−0.0841235	−	0.996455i	$-0.526809\pi$
$277$	−2.80019	−0.168247	−0.0841235	+	0.996455i	$0.526809\pi$
$278$	−24.4587	−1.46694
$279$	−3.52801	−0.211217
$280$	5.26620	0.314716
$281$	5.34685	0.318967	0.159483	−	0.987201i	$-0.449017\pi$
$281$	5.34685	0.318967	0.159483	+	0.987201i	$0.449017\pi$
$282$	−24.5085	−1.45946
$283$	26.6453	1.58390	0.791948	−	0.610589i	$-0.209067\pi$
$283$	26.6453	1.58390	0.791948	+	0.610589i	$0.209067\pi$
$284$	26.3288	1.56233
$285$	−3.42362	−0.202797
$286$	−64.6295	−3.82163
$287$	0.844792	0.0498665
$288$	−3.69383	−0.217661
$289$	−6.97762	−0.410449
$290$	28.4334	1.66967
$291$	5.91028	0.346467
$292$	46.2589	2.70710
$293$	−21.6035	−1.26209	−0.631046	−	0.775745i	$-0.717374\pi$
$293$	−21.6035	−1.26209	−0.631046	+	0.775745i	$0.717374\pi$
$294$	−2.35796	−0.137519
$295$	16.2657	0.947023
$296$	33.5658	1.95098
$297$	4.36512	0.253290
$298$	45.5426	2.63821
$299$	−54.8132	−3.16993
$300$	10.5028	0.606379
$301$	10.7642	0.620441
$302$	17.2017	0.989847
$303$	4.86006	0.279203
$304$	3.71458	0.213046
$305$	−4.61952	−0.264513
$306$	7.46485	0.426737
$307$	15.4644	0.882600	0.441300	−	0.897360i	$-0.354517\pi$
$307$	15.4644	0.882600	0.441300	+	0.897360i	$0.354517\pi$
$308$	−15.5396	−0.885452
$309$	12.2098	0.694591
$310$	−11.9101	−0.676448
$311$	−15.3382	−0.869749	−0.434875	−	0.900491i	$-0.643207\pi$
$311$	−15.3382	−0.869749	−0.434875	+	0.900491i	$0.643207\pi$
$312$	−23.0966	−1.30759
$313$	−9.21957	−0.521121	−0.260561	−	0.965458i	$-0.583907\pi$
$313$	−9.21957	−0.521121	−0.260561	+	0.965458i	$0.583907\pi$
$314$	20.3316	1.14738
$315$	1.43169	0.0806667
$316$	44.6990	2.51452
$317$	13.0864	0.735005	0.367502	−	0.930023i	$-0.380213\pi$
$317$	13.0864	0.735005	0.367502	+	0.930023i	$0.380213\pi$
$318$	14.3238	0.803240
$319$	−36.7654	−2.05847
$320$	−16.9178	−0.945733
$321$	−4.40730	−0.245991
$322$	−20.5836	−1.14708
$323$	7.57043	0.421230
$324$	3.55996	0.197775
$325$	−18.5251	−1.02759
$326$	14.1348	0.782856
$327$	12.9720	0.717355
$328$	3.10740	0.171578
$329$	10.3940	0.573039
$330$	14.7360	0.811193
$331$	0.953807	0.0524260	0.0262130	−	0.999656i	$-0.491655\pi$
$331$	0.953807	0.0524260	0.0262130	+	0.999656i	$0.491655\pi$
$332$	−34.3812	−1.88691
$333$	9.12536	0.500066
$334$	13.4955	0.738439
$335$	0.00809704	0.000442389 0
$336$	−1.55337	−0.0847433
$337$	10.0332	0.546541	0.273271	−	0.961937i	$-0.411895\pi$
$337$	10.0332	0.546541	0.273271	+	0.961937i	$0.411895\pi$
$338$	62.3149	3.38949
$339$	−10.4743	−0.568888
$340$	16.1354	0.875064
$341$	15.4002	0.833966
$342$	5.63860	0.304900
$343$	1.00000	0.0539949
$344$	39.5942	2.13478
$345$	12.4978	0.672861
$346$	28.7348	1.54479
$347$	−33.7742	−1.81309	−0.906546	−	0.422107i	$-0.861291\pi$
$347$	−33.7742	−1.81309	−0.906546	+	0.422107i	$0.861291\pi$
$348$	−29.9839	−1.60731
$349$	30.8933	1.65368	0.826839	−	0.562438i	$-0.190137\pi$
$349$	30.8933	1.65368	0.826839	+	0.562438i	$0.190137\pi$
$350$	−6.95658	−0.371845
$351$	−6.27913	−0.335155
$352$	16.1240	0.859412
$353$	−34.2916	−1.82516	−0.912578	−	0.408903i	$-0.865911\pi$
$353$	−34.2916	−1.82516	−0.912578	+	0.408903i	$0.865911\pi$
$354$	−26.7891	−1.42382
$355$	10.5885	0.561982
$356$	−13.4887	−0.714898
$357$	−3.16581	−0.167553
$358$	18.5484	0.980313
$359$	36.5397	1.92849	0.964246	−	0.265007i	$-0.0853744\pi$
$359$	36.5397	1.92849	0.964246	+	0.265007i	$0.0853744\pi$
$360$	5.26620	0.277553
$361$	−13.2816	−0.699034
$362$	0.592222	0.0311265
$363$	−8.05423	−0.422737
$364$	22.3534	1.17164
$365$	18.6038	0.973765
$366$	7.60822	0.397688
$367$	−12.5989	−0.657657	−0.328828	−	0.944390i	$-0.606654\pi$
$367$	−12.5989	−0.657657	−0.328828	+	0.944390i	$0.606654\pi$
$368$	−13.5600	−0.706865
$369$	0.844792	0.0439781
$370$	30.8060	1.60153
$371$	−6.07467	−0.315381
$372$	12.5596	0.651183
$373$	−6.35656	−0.329130	−0.164565	−	0.986366i	$-0.552622\pi$
$373$	−6.35656	−0.329130	−0.164565	+	0.986366i	$0.552622\pi$
$374$	−32.5849	−1.68493
$375$	11.3823	0.587780
$376$	38.2322	1.97168
$377$	52.8863	2.72379
$378$	−2.35796	−0.121280
$379$	−7.48695	−0.384579	−0.192289	−	0.981338i	$-0.561591\pi$
$379$	−7.48695	−0.384579	−0.192289	+	0.981338i	$0.561591\pi$
$380$	12.1879	0.625227
$381$	5.32100	0.272603
$382$	−54.7684	−2.80220
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	20.4754	1.04488
$385$	−6.24950	−0.318504
$386$	19.3884	0.986842
$387$	10.7642	0.547178
$388$	−21.0403	−1.06816
$389$	33.6092	1.70405	0.852026	−	0.523499i	$-0.175374\pi$
$389$	33.6092	1.70405	0.852026	+	0.523499i	$0.175374\pi$
$390$	−21.1975	−1.07338
$391$	−27.6357	−1.39760
$392$	3.67830	0.185782
$393$	17.1349	0.864340
$394$	−52.0132	−2.62039
$395$	17.9764	0.904491
$396$	−15.5396	−0.780895
$397$	17.9351	0.900137	0.450069	−	0.892994i	$-0.351400\pi$
$397$	17.9351	0.900137	0.450069	+	0.892994i	$0.351400\pi$
$398$	53.7848	2.69599
$399$	−2.39131	−0.119715
$400$	−4.58284	−0.229142
$401$	25.2975	1.26330	0.631649	−	0.775254i	$-0.282378\pi$
$401$	25.2975	1.26330	0.631649	+	0.775254i	$0.282378\pi$
$402$	−0.0133356	−0.000665119 0
$403$	−22.1529	−1.10351
$404$	−17.3016	−0.860786
$405$	1.43169	0.0711413
$406$	19.8600	0.985636
$407$	−39.8332	−1.97446
$408$	−11.6448	−0.576505
$409$	2.27955	0.112716	0.0563582	−	0.998411i	$-0.482051\pi$
$409$	2.27955	0.112716	0.0563582	+	0.998411i	$0.482051\pi$
$410$	2.85191	0.140846
$411$	0.450551	0.0222240
$412$	−43.4663	−2.14143
$413$	11.3611	0.559045
$414$	−20.5836	−1.01163
$415$	−13.8269	−0.678737
$416$	−23.1941	−1.13718
$417$	10.3729	0.507961
$418$	−24.6131	−1.20387
$419$	8.46522	0.413553	0.206776	−	0.978388i	$-0.433703\pi$
$419$	8.46522	0.413553	0.206776	+	0.978388i	$0.433703\pi$
$420$	−5.09676	−0.248696
$421$	−14.6432	−0.713664	−0.356832	−	0.934169i	$-0.616143\pi$
$421$	−14.6432	−0.713664	−0.356832	+	0.934169i	$0.616143\pi$
$422$	−29.4464	−1.43343
$423$	10.3940	0.505373
$424$	−22.3445	−1.08515
$425$	−9.33997	−0.453055
$426$	−17.4390	−0.844924
$427$	−3.22662	−0.156147
$428$	15.6898	0.758395
$429$	27.4091	1.32333
$430$	36.3387	1.75241
$431$	4.40023	0.211952	0.105976	−	0.994369i	$-0.466203\pi$
$431$	4.40023	0.211952	0.105976	+	0.994369i	$0.466203\pi$
$432$	−1.55337	−0.0747365
$433$	−10.8468	−0.521262	−0.260631	−	0.965438i	$-0.583931\pi$
$433$	−10.8468	−0.521262	−0.260631	+	0.965438i	$0.583931\pi$
$434$	−8.31889	−0.399320
$435$	−12.0585	−0.578161
$436$	−46.1798	−2.21161
$437$	−20.8747	−0.998574
$438$	−30.6399	−1.46403
$439$	9.67830	0.461920	0.230960	−	0.972963i	$-0.425813\pi$
$439$	9.67830	0.461920	0.230960	+	0.972963i	$0.425813\pi$
$440$	−22.9876	−1.09589
$441$	1.00000	0.0476190
$442$	46.8728	2.22951
$443$	7.56987	0.359656	0.179828	−	0.983698i	$-0.442446\pi$
$443$	7.56987	0.359656	0.179828	+	0.983698i	$0.442446\pi$
$444$	−32.4859	−1.54171
$445$	−5.42468	−0.257155
$446$	43.5546	2.06237
$447$	−19.3145	−0.913542
$448$	−11.8166	−0.558283
$449$	39.3902	1.85894	0.929470	−	0.368898i	$-0.120265\pi$
$449$	39.3902	1.85894	0.929470	+	0.368898i	$0.120265\pi$
$450$	−6.95658	−0.327936
$451$	−3.68762	−0.173643
$452$	37.2882	1.75389
$453$	−7.29518	−0.342757
$454$	−35.5751	−1.66962
$455$	8.98978	0.421448
$456$	−8.79596	−0.411908
$457$	−0.120475	−0.00563557	−0.00281779	−	0.999996i	$-0.500897\pi$
$457$	−0.120475	−0.00563557	−0.00281779	+	0.999996i	$0.500897\pi$
$458$	36.8834	1.72345
$459$	−3.16581	−0.147768
$460$	−44.4918	−2.07444
$461$	−21.3935	−0.996393	−0.498197	−	0.867064i	$-0.666004\pi$
$461$	−21.3935	−0.996393	−0.498197	+	0.867064i	$0.666004\pi$
$462$	10.2927	0.478862
$463$	−28.2271	−1.31182	−0.655911	−	0.754838i	$-0.727716\pi$
$463$	−28.2271	−1.31182	−0.655911	+	0.754838i	$0.727716\pi$
$464$	13.0833	0.607379
$465$	5.05103	0.234236
$466$	−6.82420	−0.316125
$467$	−6.84376	−0.316691	−0.158346	−	0.987384i	$-0.550616\pi$
$467$	−6.84376	−0.316691	−0.158346	+	0.987384i	$0.550616\pi$
$468$	22.3534	1.03329
$469$	0.00565557	0.000261150 0
$470$	35.0887	1.61852
$471$	−8.62254	−0.397306
$472$	41.7897	1.92353
$473$	−46.9872	−2.16047
$474$	−29.6066	−1.35988
$475$	−7.05498	−0.323704
$476$	11.2702	0.516567
$477$	−6.07467	−0.278140
$478$	32.0171	1.46443
$479$	−34.4176	−1.57258	−0.786289	−	0.617858i	$-0.788000\pi$
$479$	−34.4176	−1.57258	−0.786289	+	0.617858i	$0.788000\pi$
$480$	5.28843	0.241383
$481$	57.2993	2.61263
$482$	0.718979	0.0327486
$483$	8.72942	0.397203
$484$	28.6727	1.30330
$485$	−8.46169	−0.384226
$486$	−2.35796	−0.106959
$487$	−30.1182	−1.36479	−0.682393	−	0.730985i	$-0.739061\pi$
$487$	−30.1182	−1.36479	−0.682393	+	0.730985i	$0.739061\pi$
$488$	−11.8685	−0.537261
$489$	−5.99453	−0.271082
$490$	3.37587	0.152506
$491$	−18.7145	−0.844576	−0.422288	−	0.906462i	$-0.638773\pi$
$491$	−18.7145	−0.844576	−0.422288	+	0.906462i	$0.638773\pi$
$492$	−3.00742	−0.135585
$493$	26.6642	1.20090
$494$	35.4055	1.59297
$495$	−6.24950	−0.280894
$496$	−5.48031	−0.246073
$497$	7.39583	0.331748
$498$	22.7726	1.02046
$499$	−40.2777	−1.80308	−0.901539	−	0.432699i	$-0.857561\pi$
$499$	−40.2777	−1.80308	−0.901539	+	0.432699i	$0.857561\pi$
$500$	−40.5205	−1.81213
$501$	−5.72338	−0.255702
$502$	−66.5978	−2.97241
$503$	−10.4253	−0.464840	−0.232420	−	0.972616i	$-0.574664\pi$
$503$	−10.4253	−0.464840	−0.232420	+	0.972616i	$0.574664\pi$
$504$	3.67830	0.163845
$505$	−6.95810	−0.309632
$506$	89.8498	3.99431
$507$	−26.4275	−1.17369
$508$	−18.9425	−0.840438
$509$	−15.1562	−0.671785	−0.335893	−	0.941900i	$-0.609038\pi$
$509$	−15.1562	−0.671785	−0.335893	+	0.941900i	$0.609038\pi$
$510$	−10.6874	−0.473244
$511$	12.9942	0.574832
$512$	−17.1654	−0.758612
$513$	−2.39131	−0.105579
$514$	26.0600	1.14946
$515$	−17.4807	−0.770291
$516$	−38.3202	−1.68695
$517$	−45.3709	−1.99541
$518$	21.5172	0.945411
$519$	−12.1863	−0.534920
$520$	33.0672	1.45009
$521$	13.7355	0.601763	0.300882	−	0.953662i	$-0.402719\pi$
$521$	13.7355	0.601763	0.300882	+	0.953662i	$0.402719\pi$
$522$	19.8600	0.869249
$523$	−4.12892	−0.180545	−0.0902725	−	0.995917i	$-0.528774\pi$
$523$	−4.12892	−0.180545	−0.0902725	+	0.995917i	$0.528774\pi$
$524$	−60.9994	−2.66477
$525$	2.95026	0.128760
$526$	−34.5757	−1.50757
$527$	−11.1690	−0.486530
$528$	6.78064	0.295089
$529$	53.2028	2.31317
$530$	−20.5073	−0.890780
$531$	11.3611	0.493032
$532$	8.51295	0.369083
$533$	5.30456	0.229766
$534$	8.93430	0.386625
$535$	6.30989	0.272801
$536$	0.0208029	0.000898550 0
$537$	−7.86630	−0.339456
$538$	−46.2436	−1.99370
$539$	−4.36512	−0.188019
$540$	−5.09676	−0.219330
$541$	25.1511	1.08133	0.540665	−	0.841238i	$-0.318173\pi$
$541$	25.1511	1.08133	0.540665	+	0.841238i	$0.318173\pi$
$542$	−15.8327	−0.680074
$543$	−0.251159	−0.0107783
$544$	−11.6940	−0.501375
$545$	−18.5720	−0.795535
$546$	−14.8059	−0.633635
$547$	−39.0410	−1.66927	−0.834637	−	0.550800i	$-0.814323\pi$
$547$	−39.0410	−1.66927	−0.834637	+	0.550800i	$0.814323\pi$
$548$	−1.60394	−0.0685170
$549$	−3.22662	−0.137709
$550$	30.3663	1.29482
$551$	20.1409	0.858032
$552$	32.1095	1.36667
$553$	12.5561	0.533938
$554$	−6.60272	−0.280523
$555$	−13.0647	−0.554566
$556$	−36.9269	−1.56605
$557$	−25.0051	−1.05950	−0.529749	−	0.848154i	$-0.677714\pi$
$557$	−25.0051	−1.05950	−0.529749	+	0.848154i	$0.677714\pi$
$558$	−8.31889	−0.352167
$559$	67.5902	2.85876
$560$	2.22395	0.0939789
$561$	13.8191	0.583445
$562$	12.6076	0.531821
$563$	−19.5582	−0.824280	−0.412140	−	0.911121i	$-0.635219\pi$
$563$	−19.5582	−0.824280	−0.412140	+	0.911121i	$0.635219\pi$
$564$	−37.0021	−1.55807
$565$	14.9960	0.630888
$566$	62.8283	2.64087
$567$	1.00000	0.0419961
$568$	27.2041	1.14146
$569$	1.18862	0.0498294	0.0249147	−	0.999690i	$-0.492069\pi$
$569$	1.18862	0.0498294	0.0249147	+	0.999690i	$0.492069\pi$
$570$	−8.07273	−0.338130
$571$	−21.5906	−0.903538	−0.451769	−	0.892135i	$-0.649207\pi$
$571$	−21.5906	−0.903538	−0.451769	+	0.892135i	$0.649207\pi$
$572$	−97.5753	−4.07983
$573$	23.2271	0.970325
$574$	1.99198	0.0831438
$575$	25.7541	1.07402
$576$	−11.8166	−0.492360
$577$	−1.63152	−0.0679209	−0.0339604	−	0.999423i	$-0.510812\pi$
$577$	−1.63152	−0.0679209	−0.0339604	+	0.999423i	$0.510812\pi$
$578$	−16.4529	−0.684352
$579$	−8.22253	−0.341717
$580$	42.9277	1.78248
$581$	−9.65775	−0.400671
$582$	13.9362	0.577673
$583$	26.5167	1.09821
$584$	47.7968	1.97785
$585$	8.98978	0.371682
$586$	−50.9402	−2.10432
$587$	4.83586	0.199597	0.0997986	−	0.995008i	$-0.468180\pi$
$587$	4.83586	0.199597	0.0997986	+	0.995008i	$0.468180\pi$
$588$	−3.55996	−0.146810
$589$	−8.43656	−0.347622
$590$	38.3537	1.57900
$591$	22.0586	0.907369
$592$	14.1751	0.582591
$593$	−11.1365	−0.457322	−0.228661	−	0.973506i	$-0.573435\pi$
$593$	−11.1365	−0.457322	−0.228661	+	0.973506i	$0.573435\pi$
$594$	10.2927	0.422317
$595$	4.53247	0.185813
$596$	68.7586	2.81646
$597$	−22.8099	−0.933548
$598$	−129.247	−5.28531
$599$	−1.22994	−0.0502541	−0.0251270	−	0.999684i	$-0.507999\pi$
$599$	−1.22994	−0.0502541	−0.0251270	+	0.999684i	$0.507999\pi$
$600$	10.8519	0.443029
$601$	−39.1211	−1.59578	−0.797892	−	0.602801i	$-0.794051\pi$
$601$	−39.1211	−1.59578	−0.797892	+	0.602801i	$0.794051\pi$
$602$	25.3816	1.03448
$603$	0.00565557	0.000230313 0
$604$	25.9705	1.05672
$605$	11.5312	0.468809
$606$	11.4598	0.465522
$607$	−40.1687	−1.63040	−0.815199	−	0.579182i	$-0.803372\pi$
$607$	−40.1687	−1.63040	−0.815199	+	0.579182i	$0.803372\pi$
$608$	−8.83309	−0.358229
$609$	−8.42255	−0.341299
$610$	−10.8926	−0.441029
$611$	65.2652	2.64035
$612$	11.2702	0.455569
$613$	32.2727	1.30348	0.651742	−	0.758441i	$-0.274039\pi$
$613$	32.2727	1.30348	0.651742	+	0.758441i	$0.274039\pi$
$614$	36.4644	1.47158
$615$	−1.20948	−0.0487710
$616$	−16.0562	−0.646924
$617$	36.7284	1.47863	0.739316	−	0.673359i	$-0.235149\pi$
$617$	36.7284	1.47863	0.739316	+	0.673359i	$0.235149\pi$
$618$	28.7902	1.15811
$619$	−8.84864	−0.355657	−0.177828	−	0.984062i	$-0.556907\pi$
$619$	−8.84864	−0.355657	−0.177828	+	0.984062i	$0.556907\pi$
$620$	−17.9814	−0.722151
$621$	8.72942	0.350300
$622$	−36.1668	−1.45016
$623$	−3.78900	−0.151803
$624$	−9.75382	−0.390465
$625$	−1.54468	−0.0617873
$626$	−21.7393	−0.868879
$627$	10.4383	0.416867
$628$	30.6959	1.22490
$629$	28.8892	1.15189
$630$	3.37587	0.134498
$631$	−27.0343	−1.07622	−0.538110	−	0.842875i	$-0.680862\pi$
$631$	−27.0343	−1.07622	−0.538110	+	0.842875i	$0.680862\pi$
$632$	46.1850	1.83714
$633$	12.4881	0.496358
$634$	30.8571	1.22549
$635$	−7.61803	−0.302312
$636$	21.6256	0.857510
$637$	6.27913	0.248788
$638$	−86.6912	−3.43214
$639$	7.39583	0.292574
$640$	−29.3145	−1.15876
$641$	−49.6956	−1.96286	−0.981430	−	0.191821i	$-0.938561\pi$
$641$	−49.6956	−1.96286	−0.981430	+	0.191821i	$0.938561\pi$
$642$	−10.3922	−0.410148
$643$	16.7004	0.658599	0.329300	−	0.944225i	$-0.393187\pi$
$643$	16.7004	0.658599	0.329300	+	0.944225i	$0.393187\pi$
$644$	−31.0764	−1.22458
$645$	−15.4111	−0.606811
$646$	17.8507	0.702328
$647$	22.6541	0.890624	0.445312	−	0.895376i	$-0.353093\pi$
$647$	22.6541	0.890624	0.445312	+	0.895376i	$0.353093\pi$
$648$	3.67830	0.144497
$649$	−49.5927	−1.94668
$650$	−43.6813	−1.71332
$651$	3.52801	0.138274
$652$	21.3402	0.835748
$653$	−18.6236	−0.728798	−0.364399	−	0.931243i	$-0.618726\pi$
$653$	−18.6236	−0.728798	−0.364399	+	0.931243i	$0.618726\pi$
$654$	30.5875	1.19606
$655$	−24.5319	−0.958539
$656$	1.31228	0.0512357
$657$	12.9942	0.506954
$658$	24.5085	0.955443
$659$	−4.58187	−0.178484	−0.0892421	−	0.996010i	$-0.528444\pi$
$659$	−4.58187	−0.178484	−0.0892421	+	0.996010i	$0.528444\pi$
$660$	22.2479	0.866000
$661$	19.7352	0.767609	0.383805	−	0.923414i	$-0.374614\pi$
$661$	19.7352	0.767609	0.383805	+	0.923414i	$0.374614\pi$
$662$	2.24904	0.0874113
$663$	−19.8786	−0.772020
$664$	−35.5242	−1.37860
$665$	3.42362	0.132762
$666$	21.5172	0.833774
$667$	−73.5240	−2.84686
$668$	20.3750	0.788331
$669$	−18.4713	−0.714143
$670$	0.0190925	0.000737606 0
$671$	14.0846	0.543728
$672$	3.69383	0.142493
$673$	−9.54034	−0.367753	−0.183877	−	0.982949i	$-0.558865\pi$
$673$	−9.54034	−0.367753	−0.183877	+	0.982949i	$0.558865\pi$
$674$	23.6578	0.911263
$675$	2.95026	0.113555
$676$	94.0808	3.61849
$677$	4.83242	0.185725	0.0928626	−	0.995679i	$-0.470398\pi$
$677$	4.83242	0.185725	0.0928626	+	0.995679i	$0.470398\pi$
$678$	−24.6980	−0.948523
$679$	−5.91028	−0.226816
$680$	16.6718	0.639335
$681$	15.0873	0.578145
$682$	36.3129	1.39049
$683$	−33.7253	−1.29046	−0.645232	−	0.763986i	$-0.723239\pi$
$683$	−33.7253	−1.29046	−0.645232	+	0.763986i	$0.723239\pi$
$684$	8.51295	0.325501
$685$	−0.645050	−0.0246461
$686$	2.35796	0.0900272
$687$	−15.6421	−0.596784
$688$	16.7209	0.637477
$689$	−38.1437	−1.45316
$690$	29.4694	1.12188
$691$	−41.2895	−1.57073	−0.785364	−	0.619035i	$-0.787524\pi$
$691$	−41.2895	−1.57073	−0.785364	+	0.619035i	$0.787524\pi$
$692$	43.3827	1.64916
$693$	−4.36512	−0.165817
$694$	−79.6380	−3.02302
$695$	−14.8507	−0.563320
$696$	−30.9807	−1.17432
$697$	2.67446	0.101302
$698$	72.8449	2.75722
$699$	2.89412	0.109466
$700$	−10.5028	−0.396968
$701$	−24.2530	−0.916022	−0.458011	−	0.888947i	$-0.651438\pi$
$701$	−24.2530	−0.916022	−0.458011	+	0.888947i	$0.651438\pi$
$702$	−14.8059	−0.558813
$703$	21.8215	0.823015
$704$	51.5810	1.94403
$705$	−14.8810	−0.560450
$706$	−80.8580	−3.04313
$707$	−4.86006	−0.182781
$708$	−40.4452	−1.52002
$709$	−6.01257	−0.225807	−0.112903	−	0.993606i	$-0.536015\pi$
$709$	−6.01257	−0.225807	−0.112903	+	0.993606i	$0.536015\pi$
$710$	24.9673	0.937007
$711$	12.5561	0.470889
$712$	−13.9371	−0.522315
$713$	30.7975	1.15338
$714$	−7.46485	−0.279365
$715$	−39.2414	−1.46755
$716$	28.0037	1.04655
$717$	−13.5783	−0.507093
$718$	86.1590	3.21543
$719$	9.58924	0.357618	0.178809	−	0.983884i	$-0.442776\pi$
$719$	9.58924	0.357618	0.178809	+	0.983884i	$0.442776\pi$
$720$	2.22395	0.0828816
$721$	−12.2098	−0.454717
$722$	−31.3175	−1.16552
$723$	−0.304916	−0.0113400
$724$	0.894114	0.0332295
$725$	−24.8487	−0.922858
$726$	−18.9915	−0.704841
$727$	−16.6844	−0.618789	−0.309395	−	0.950934i	$-0.600126\pi$
$727$	−16.6844	−0.618789	−0.309395	+	0.950934i	$0.600126\pi$
$728$	23.0966	0.856016
$729$	1.00000	0.0370370
$730$	43.8668	1.62358
$731$	34.0776	1.26041
$732$	11.4866	0.424557
$733$	37.2954	1.37754	0.688768	−	0.724982i	$-0.258152\pi$
$733$	37.2954	1.37754	0.688768	+	0.724982i	$0.258152\pi$
$734$	−29.7076	−1.09653
$735$	−1.43169	−0.0528087
$736$	32.2450	1.18857
$737$	−0.0246872	−0.000909366 0
$738$	1.99198	0.0733259
$739$	42.5537	1.56536	0.782682	−	0.622421i	$-0.213851\pi$
$739$	42.5537	1.56536	0.782682	+	0.622421i	$0.213851\pi$
$740$	46.5097	1.70973
$741$	−15.0153	−0.551602
$742$	−14.3238	−0.525844
$743$	30.8722	1.13259	0.566295	−	0.824203i	$-0.308376\pi$
$743$	30.8722	1.13259	0.566295	+	0.824203i	$0.308376\pi$
$744$	12.9771	0.475764
$745$	27.6523	1.01310
$746$	−14.9885	−0.548768
$747$	−9.65775	−0.353359
$748$	−49.1955	−1.79877
$749$	4.40730	0.161039
$750$	26.8390	0.980022
$751$	33.3561	1.21718	0.608590	−	0.793485i	$-0.291735\pi$
$751$	33.3561	1.21718	0.608590	+	0.793485i	$0.291735\pi$
$752$	16.1457	0.588773
$753$	28.2439	1.02926
$754$	124.704	4.54144
$755$	10.4444	0.380112
$756$	−3.55996	−0.129474
$757$	−34.7387	−1.26260	−0.631300	−	0.775539i	$-0.717478\pi$
$757$	−34.7387	−1.26260	−0.631300	+	0.775539i	$0.717478\pi$
$758$	−17.6539	−0.641218
$759$	−38.1049	−1.38312
$760$	12.5931	0.456800
$761$	−7.70707	−0.279381	−0.139690	−	0.990195i	$-0.544611\pi$
$761$	−7.70707	−0.279381	−0.139690	+	0.990195i	$0.544611\pi$
$762$	12.5467	0.454518
$763$	−12.9720	−0.469619
$764$	−82.6873	−2.99152
$765$	4.53247	0.163872
$766$	−2.35796	−0.0851964
$767$	71.3381	2.57587
$768$	24.6469	0.889368
$769$	18.8981	0.681482	0.340741	−	0.940157i	$-0.389322\pi$
$769$	18.8981	0.681482	0.340741	+	0.940157i	$0.389322\pi$
$770$	−14.7360	−0.531050
$771$	−11.0519	−0.398026
$772$	29.2719	1.05352
$773$	−11.9315	−0.429147	−0.214573	−	0.976708i	$-0.568836\pi$
$773$	−11.9315	−0.429147	−0.214573	+	0.976708i	$0.568836\pi$
$774$	25.3816	0.912324
$775$	10.4085	0.373886
$776$	−21.7398	−0.780413
$777$	−9.12536	−0.327370
$778$	79.2489	2.84121
$779$	2.02016	0.0723797
$780$	−32.0032	−1.14590
$781$	−32.2836	−1.15520
$782$	−65.1638	−2.33025
$783$	−8.42255	−0.300998
$784$	1.55337	0.0554775
$785$	12.3448	0.440606
$786$	40.4033	1.44114
$787$	−20.7016	−0.737934	−0.368967	−	0.929443i	$-0.620288\pi$
$787$	−20.7016	−0.737934	−0.368967	+	0.929443i	$0.620288\pi$
$788$	−78.5276	−2.79743
$789$	14.6634	0.522032
$790$	42.3876	1.50808
$791$	10.4743	0.372425
$792$	−16.0562	−0.570533
$793$	−20.2604	−0.719466
$794$	42.2902	1.50082
$795$	8.69706	0.308453
$796$	81.2023	2.87814
$797$	−20.5348	−0.727380	−0.363690	−	0.931520i	$-0.618483\pi$
$797$	−20.5348	−0.727380	−0.363690	+	0.931520i	$0.618483\pi$
$798$	−5.63860	−0.199604
$799$	32.9054	1.16411
$800$	10.8978	0.385294
$801$	−3.78900	−0.133878
$802$	59.6505	2.10633
$803$	−56.7214	−2.00165
$804$	−0.0201336	−0.000710057 0
$805$	−12.4978	−0.440491
$806$	−52.2355	−1.83992
$807$	19.6117	0.690366
$808$	−17.8768	−0.628903
$809$	−10.3787	−0.364895	−0.182448	−	0.983216i	$-0.558402\pi$
$809$	−10.3787	−0.364895	−0.182448	+	0.983216i	$0.558402\pi$
$810$	3.37587	0.118616
$811$	45.5446	1.59929	0.799644	−	0.600475i	$-0.205022\pi$
$811$	45.5446	1.59929	0.799644	+	0.600475i	$0.205022\pi$
$812$	29.9839	1.05223
$813$	6.71460	0.235491
$814$	−93.9250	−3.29207
$815$	8.58231	0.300625
$816$	−4.91768	−0.172153
$817$	25.7406	0.900551
$818$	5.37508	0.187935
$819$	6.27913	0.219411
$820$	4.30570	0.150362
$821$	13.1971	0.460582	0.230291	−	0.973122i	$-0.426032\pi$
$821$	13.1971	0.460582	0.230291	+	0.973122i	$0.426032\pi$
$822$	1.06238	0.0370547
$823$	9.18432	0.320145	0.160073	−	0.987105i	$-0.448827\pi$
$823$	9.18432	0.320145	0.160073	+	0.987105i	$0.448827\pi$
$824$	−44.9114	−1.56456
$825$	−12.8782	−0.448362
$826$	26.7891	0.932111
$827$	−0.687392	−0.0239030	−0.0119515	−	0.999929i	$-0.503804\pi$
$827$	−0.687392	−0.0239030	−0.0119515	+	0.999929i	$0.503804\pi$
$828$	−31.0764	−1.07998
$829$	−23.8218	−0.827366	−0.413683	−	0.910421i	$-0.635758\pi$
$829$	−23.8218	−0.827366	−0.413683	+	0.910421i	$0.635758\pi$
$830$	−32.6033	−1.13168
$831$	2.80019	0.0971375
$832$	−74.1982	−2.57236
$833$	3.16581	0.109689
$834$	24.4587	0.846937
$835$	8.19411	0.283569
$836$	−37.1600	−1.28521
$837$	3.52801	0.121946
$838$	19.9606	0.689528
$839$	−29.6876	−1.02493	−0.512465	−	0.858708i	$-0.671268\pi$
$839$	−29.6876	−1.02493	−0.512465	+	0.858708i	$0.671268\pi$
$840$	−5.26620	−0.181701
$841$	41.9394	1.44619
$842$	−34.5279	−1.18991
$843$	−5.34685	−0.184155
$844$	−44.4571	−1.53028
$845$	37.8361	1.30160
$846$	24.5085	0.842621
$847$	8.05423	0.276747
$848$	−9.43622	−0.324041
$849$	−26.6453	−0.914463
$850$	−22.0232	−0.755391
$851$	−79.6591	−2.73068
$852$	−26.3288	−0.902010
$853$	10.6533	0.364762	0.182381	−	0.983228i	$-0.441620\pi$
$853$	10.6533	0.364762	0.182381	+	0.983228i	$0.441620\pi$
$854$	−7.60822	−0.260348
$855$	3.42362	0.117085
$856$	16.2114	0.554094
$857$	−52.8448	−1.80514	−0.902572	−	0.430538i	$-0.858324\pi$
$857$	−52.8448	−1.80514	−0.902572	+	0.430538i	$0.858324\pi$
$858$	64.6295	2.20642
$859$	−25.1690	−0.858756	−0.429378	−	0.903125i	$-0.641267\pi$
$859$	−25.1690	−0.858756	−0.429378	+	0.903125i	$0.641267\pi$
$860$	54.8628	1.87081
$861$	−0.844792	−0.0287904
$862$	10.3755	0.353392
$863$	−20.7287	−0.705614	−0.352807	−	0.935696i	$-0.614773\pi$
$863$	−20.7287	−0.705614	−0.352807	+	0.935696i	$0.614773\pi$
$864$	3.69383	0.125667
$865$	17.4471	0.593218
$866$	−25.5762	−0.869114
$867$	6.97762	0.236973
$868$	−12.5596	−0.426299
$869$	−54.8086	−1.85926
$870$	−28.4334	−0.963983
$871$	0.0355121	0.00120328
$872$	−47.7151	−1.61584
$873$	−5.91028	−0.200033
$874$	−49.2217	−1.66495
$875$	−11.3823	−0.384793
$876$	−46.2589	−1.56294
$877$	−29.2694	−0.988358	−0.494179	−	0.869360i	$-0.664531\pi$
$877$	−29.2694	−0.988358	−0.494179	+	0.869360i	$0.664531\pi$
$878$	22.8210	0.770171
$879$	21.6035	0.728670
$880$	−9.70778	−0.327249
$881$	−30.0684	−1.01303	−0.506515	−	0.862231i	$-0.669066\pi$
$881$	−30.0684	−1.01303	−0.506515	+	0.862231i	$0.669066\pi$
$882$	2.35796	0.0793965
$883$	−4.03992	−0.135954	−0.0679771	−	0.997687i	$-0.521654\pi$
$883$	−4.03992	−0.135954	−0.0679771	+	0.997687i	$0.521654\pi$
$884$	70.7668	2.38015
$885$	−16.2657	−0.546764
$886$	17.8494	0.599663
$887$	30.9608	1.03956	0.519781	−	0.854299i	$-0.326013\pi$
$887$	30.9608	1.03956	0.519781	+	0.854299i	$0.326013\pi$
$888$	−33.5658	−1.12640
$889$	−5.32100	−0.178460
$890$	−12.7912	−0.428761
$891$	−4.36512	−0.146237
$892$	65.7571	2.20171
$893$	24.8552	0.831748
$894$	−45.5426	−1.52317
$895$	11.2621	0.376451
$896$	−20.4754	−0.684036
$897$	54.8132	1.83016
$898$	92.8804	3.09946
$899$	−29.7149	−0.991046
$900$	−10.5028	−0.350093
$901$	−19.2313	−0.640687
$902$	−8.69524	−0.289520
$903$	−10.7642	−0.358212
$904$	38.5278	1.28142
$905$	0.359582	0.0119529
$906$	−17.2017	−0.571488
$907$	13.6220	0.452312	0.226156	−	0.974091i	$-0.427384\pi$
$907$	13.6220	0.452312	0.226156	+	0.974091i	$0.427384\pi$
$908$	−53.7100	−1.78243
$909$	−4.86006	−0.161198
$910$	21.1975	0.702691
$911$	24.1478	0.800053	0.400026	−	0.916504i	$-0.369001\pi$
$911$	24.1478	0.800053	0.400026	+	0.916504i	$0.369001\pi$
$912$	−3.71458	−0.123002
$913$	42.1572	1.39520
$914$	−0.284074	−0.00939634
$915$	4.61952	0.152717
$916$	55.6852	1.83989
$917$	−17.1349	−0.565843
$918$	−7.46485	−0.246377
$919$	24.1235	0.795761	0.397880	−	0.917437i	$-0.369746\pi$
$919$	24.1235	0.795761	0.397880	+	0.917437i	$0.369746\pi$
$920$	−45.9709	−1.51562
$921$	−15.4644	−0.509569
$922$	−50.4448	−1.66131
$923$	46.4394	1.52857
$924$	15.5396	0.511216
$925$	−26.9222	−0.885195
$926$	−66.5582	−2.18724
$927$	−12.2098	−0.401023
$928$	−31.1115	−1.02129
$929$	42.8777	1.40677	0.703386	−	0.710808i	$-0.251671\pi$
$929$	42.8777	1.40677	0.703386	+	0.710808i	$0.251671\pi$
$930$	11.9101	0.390547
$931$	2.39131	0.0783719
$932$	−10.3029	−0.337484
$933$	15.3382	0.502150
$934$	−16.1373	−0.528028
$935$	−19.7848	−0.647031
$936$	23.0966	0.754935
$937$	24.5459	0.801880	0.400940	−	0.916104i	$-0.368684\pi$
$937$	24.5459	0.801880	0.400940	+	0.916104i	$0.368684\pi$
$938$	0.0133356	0.000435423 0
$939$	9.21957	0.300869
$940$	52.9756	1.72787
$941$	−18.2239	−0.594081	−0.297040	−	0.954865i	$-0.596000\pi$
$941$	−18.2239	−0.594081	−0.297040	+	0.954865i	$0.596000\pi$
$942$	−20.3316	−0.662438
$943$	−7.37455	−0.240148
$944$	17.6481	0.574395
$945$	−1.43169	−0.0465729
$946$	−110.794	−3.60221
$947$	−12.0154	−0.390448	−0.195224	−	0.980759i	$-0.562543\pi$
$947$	−12.0154	−0.390448	−0.195224	+	0.980759i	$0.562543\pi$
$948$	−44.6990	−1.45176
$949$	81.5926	2.64861
$950$	−16.6353	−0.539721
$951$	−13.0864	−0.424355
$952$	11.6448	0.377411
$953$	−0.106442	−0.00344799	−0.00172400	−	0.999999i	$-0.500549\pi$
$953$	−0.106442	−0.00344799	−0.00172400	+	0.999999i	$0.500549\pi$
$954$	−14.3238	−0.463751
$955$	−33.2540	−1.07607
$956$	48.3383	1.56337
$957$	36.7654	1.18846
$958$	−81.1551	−2.62200
$959$	−0.450551	−0.0145491
$960$	16.9178	0.546019
$961$	−18.5531	−0.598488
$962$	135.109	4.35610
$963$	4.40730	0.142023
$964$	1.08549	0.0349612
$965$	11.7721	0.378958
$966$	20.5836	0.662266
$967$	−38.2788	−1.23096	−0.615481	−	0.788151i	$-0.711038\pi$
$967$	−38.2788	−1.23096	−0.615481	+	0.788151i	$0.711038\pi$
$968$	29.6259	0.952213
$969$	−7.57043	−0.243197
$970$	−19.9523	−0.640630
$971$	−9.26689	−0.297389	−0.148694	−	0.988883i	$-0.547507\pi$
$971$	−9.26689	−0.297389	−0.148694	+	0.988883i	$0.547507\pi$
$972$	−3.55996	−0.114186
$973$	−10.3729	−0.332538
$974$	−71.0174	−2.27554
$975$	18.5251	0.593277
$976$	−5.01213	−0.160434
$977$	21.2564	0.680054	0.340027	−	0.940416i	$-0.389564\pi$
$977$	21.2564	0.680054	0.340027	+	0.940416i	$0.389564\pi$
$978$	−14.1348	−0.451982
$979$	16.5394	0.528602
$980$	5.09676	0.162810
$981$	−12.9720	−0.414165
$982$	−44.1281	−1.40818
$983$	−32.5753	−1.03899	−0.519496	−	0.854473i	$-0.673880\pi$
$983$	−32.5753	−1.03899	−0.519496	+	0.854473i	$0.673880\pi$
$984$	−3.10740	−0.0990604
$985$	−31.5811	−1.00626
$986$	62.8731	2.00229
$987$	−10.3940	−0.330844
$988$	53.4539	1.70060
$989$	−93.9657	−2.98794
$990$	−14.7360	−0.468342
$991$	−30.5541	−0.970582	−0.485291	−	0.874353i	$-0.661286\pi$
$991$	−30.5541	−0.970582	−0.485291	+	0.874353i	$0.661286\pi$
$992$	13.0319	0.413763
$993$	−0.953807	−0.0302682
$994$	17.4390	0.553133
$995$	32.6568	1.03529
$996$	34.3812	1.08941
$997$	−16.8964	−0.535114	−0.267557	−	0.963542i	$-0.586216\pi$
$997$	−16.8964	−0.535114	−0.267557	+	0.963542i	$0.586216\pi$
$998$	−94.9730	−3.00632
$999$	−9.12536	−0.288714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.45	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.45	✓	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)

\(f(q)\)	\(=\)	\(q+2.35796 q^{2} -1.00000 q^{3} +3.55996 q^{4} +1.43169 q^{5} -2.35796 q^{6} +1.00000 q^{7} +3.67830 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.35796 q^{2} -1.00000 q^{3} +3.55996 q^{4} +1.43169 q^{5} -2.35796 q^{6} +1.00000 q^{7} +3.67830 q^{8} +1.00000 q^{9} +3.37587 q^{10} -4.36512 q^{11} -3.55996 q^{12} +6.27913 q^{13} +2.35796 q^{14} -1.43169 q^{15} +1.55337 q^{16} +3.16581 q^{17} +2.35796 q^{18} +2.39131 q^{19} +5.09676 q^{20} -1.00000 q^{21} -10.2927 q^{22} -8.72942 q^{23} -3.67830 q^{24} -2.95026 q^{25} +14.8059 q^{26} -1.00000 q^{27} +3.55996 q^{28} +8.42255 q^{29} -3.37587 q^{30} -3.52801 q^{31} -3.69383 q^{32} +4.36512 q^{33} +7.46485 q^{34} +1.43169 q^{35} +3.55996 q^{36} +9.12536 q^{37} +5.63860 q^{38} -6.27913 q^{39} +5.26620 q^{40} +0.844792 q^{41} -2.35796 q^{42} +10.7642 q^{43} -15.5396 q^{44} +1.43169 q^{45} -20.5836 q^{46} +10.3940 q^{47} -1.55337 q^{48} +1.00000 q^{49} -6.95658 q^{50} -3.16581 q^{51} +22.3534 q^{52} -6.07467 q^{53} -2.35796 q^{54} -6.24950 q^{55} +3.67830 q^{56} -2.39131 q^{57} +19.8600 q^{58} +11.3611 q^{59} -5.09676 q^{60} -3.22662 q^{61} -8.31889 q^{62} +1.00000 q^{63} -11.8166 q^{64} +8.98978 q^{65} +10.2927 q^{66} +0.00565557 q^{67} +11.2702 q^{68} +8.72942 q^{69} +3.37587 q^{70} +7.39583 q^{71} +3.67830 q^{72} +12.9942 q^{73} +21.5172 q^{74} +2.95026 q^{75} +8.51295 q^{76} -4.36512 q^{77} -14.8059 q^{78} +12.5561 q^{79} +2.22395 q^{80} +1.00000 q^{81} +1.99198 q^{82} -9.65775 q^{83} -3.55996 q^{84} +4.53247 q^{85} +25.3816 q^{86} -8.42255 q^{87} -16.0562 q^{88} -3.78900 q^{89} +3.37587 q^{90} +6.27913 q^{91} -31.0764 q^{92} +3.52801 q^{93} +24.5085 q^{94} +3.42362 q^{95} +3.69383 q^{96} -5.91028 q^{97} +2.35796 q^{98} -4.36512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 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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