LMFDB - Embedded newform 8049.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$104$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53392 q^{2} -1.00000 q^{3} +4.42075 q^{4} -2.15927 q^{5} +2.53392 q^{6} -0.210471 q^{7} -6.13399 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.53392 q^{2} -1.00000 q^{3} +4.42075 q^{4} -2.15927 q^{5} +2.53392 q^{6} -0.210471 q^{7} -6.13399 q^{8} +1.00000 q^{9} +5.47142 q^{10} +4.95415 q^{11} -4.42075 q^{12} +3.17017 q^{13} +0.533318 q^{14} +2.15927 q^{15} +6.70153 q^{16} -2.47909 q^{17} -2.53392 q^{18} +0.0375390 q^{19} -9.54559 q^{20} +0.210471 q^{21} -12.5534 q^{22} +3.57941 q^{23} +6.13399 q^{24} -0.337553 q^{25} -8.03296 q^{26} -1.00000 q^{27} -0.930441 q^{28} -8.44408 q^{29} -5.47142 q^{30} +8.41917 q^{31} -4.71317 q^{32} -4.95415 q^{33} +6.28180 q^{34} +0.454464 q^{35} +4.42075 q^{36} +8.81744 q^{37} -0.0951208 q^{38} -3.17017 q^{39} +13.2449 q^{40} -8.62513 q^{41} -0.533318 q^{42} +0.695413 q^{43} +21.9011 q^{44} -2.15927 q^{45} -9.06994 q^{46} +7.17461 q^{47} -6.70153 q^{48} -6.95570 q^{49} +0.855333 q^{50} +2.47909 q^{51} +14.0145 q^{52} +1.95072 q^{53} +2.53392 q^{54} -10.6974 q^{55} +1.29103 q^{56} -0.0375390 q^{57} +21.3966 q^{58} -4.82543 q^{59} +9.54559 q^{60} +2.29965 q^{61} -21.3335 q^{62} -0.210471 q^{63} -1.46027 q^{64} -6.84525 q^{65} +12.5534 q^{66} -10.2531 q^{67} -10.9594 q^{68} -3.57941 q^{69} -1.15158 q^{70} -4.97845 q^{71} -6.13399 q^{72} -0.0679022 q^{73} -22.3427 q^{74} +0.337553 q^{75} +0.165950 q^{76} -1.04271 q^{77} +8.03296 q^{78} -3.39674 q^{79} -14.4704 q^{80} +1.00000 q^{81} +21.8554 q^{82} -8.96886 q^{83} +0.930441 q^{84} +5.35302 q^{85} -1.76212 q^{86} +8.44408 q^{87} -30.3887 q^{88} -12.8332 q^{89} +5.47142 q^{90} -0.667230 q^{91} +15.8237 q^{92} -8.41917 q^{93} -18.1799 q^{94} -0.0810568 q^{95} +4.71317 q^{96} -7.80085 q^{97} +17.6252 q^{98} +4.95415 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) $ 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * 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 $

$2$	−2.53392	−1.79175	−0.895876	−	0.444304i	$-0.853451\pi$
$2$	−2.53392	−1.79175	−0.895876	+	0.444304i	$0.853451\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.42075	2.21038
$5$	−2.15927	−0.965655	−0.482827	−	0.875716i	$-0.660390\pi$
$5$	−2.15927	−0.965655	−0.482827	+	0.875716i	$0.660390\pi$
$6$	2.53392	1.03447
$7$	−0.210471	−0.0795507	−0.0397753	−	0.999209i	$-0.512664\pi$
$7$	−0.210471	−0.0795507	−0.0397753	+	0.999209i	$0.512664\pi$
$8$	−6.13399	−2.16869
$9$	1.00000	0.333333
$10$	5.47142	1.73021
$11$	4.95415	1.49373	0.746866	−	0.664974i	$-0.231557\pi$
$11$	4.95415	1.49373	0.746866	+	0.664974i	$0.231557\pi$
$12$	−4.42075	−1.27616
$13$	3.17017	0.879247	0.439623	−	0.898182i	$-0.355112\pi$
$13$	3.17017	0.879247	0.439623	+	0.898182i	$0.355112\pi$
$14$	0.533318	0.142535
$15$	2.15927	0.557521
$16$	6.70153	1.67538
$17$	−2.47909	−0.601267	−0.300633	−	0.953740i	$-0.597198\pi$
$17$	−2.47909	−0.601267	−0.300633	+	0.953740i	$0.597198\pi$
$18$	−2.53392	−0.597251
$19$	0.0375390	0.00861203	0.00430602	−	0.999991i	$-0.498629\pi$
$19$	0.0375390	0.00861203	0.00430602	+	0.999991i	$0.498629\pi$
$20$	−9.54559	−2.13446
$21$	0.210471	0.0459286
$22$	−12.5534	−2.67640
$23$	3.57941	0.746358	0.373179	−	0.927759i	$-0.378268\pi$
$23$	3.57941	0.746358	0.373179	+	0.927759i	$0.378268\pi$
$24$	6.13399	1.25209
$25$	−0.337553	−0.0675107
$26$	−8.03296	−1.57539
$27$	−1.00000	−0.192450
$28$	−0.930441	−0.175837
$29$	−8.44408	−1.56803	−0.784013	−	0.620744i	$-0.786830\pi$
$29$	−8.44408	−1.56803	−0.784013	+	0.620744i	$0.786830\pi$
$30$	−5.47142	−0.998940
$31$	8.41917	1.51213	0.756063	−	0.654498i	$-0.227120\pi$
$31$	8.41917	1.51213	0.756063	+	0.654498i	$0.227120\pi$
$32$	−4.71317	−0.833179
$33$	−4.95415	−0.862407
$34$	6.28180	1.07732
$35$	0.454464	0.0768185
$36$	4.42075	0.736792
$37$	8.81744	1.44958	0.724789	−	0.688971i	$-0.241937\pi$
$37$	8.81744	1.44958	0.724789	+	0.688971i	$0.241937\pi$
$38$	−0.0951208	−0.0154306
$39$	−3.17017	−0.507633
$40$	13.2449	2.09421
$41$	−8.62513	−1.34702	−0.673510	−	0.739178i	$-0.735214\pi$
$41$	−8.62513	−1.34702	−0.673510	+	0.739178i	$0.735214\pi$
$42$	−0.533318	−0.0822927
$43$	0.695413	0.106049	0.0530247	−	0.998593i	$-0.483114\pi$
$43$	0.695413	0.106049	0.0530247	+	0.998593i	$0.483114\pi$
$44$	21.9011	3.30171
$45$	−2.15927	−0.321885
$46$	−9.06994	−1.33729
$47$	7.17461	1.04652	0.523262	−	0.852172i	$-0.324715\pi$
$47$	7.17461	1.04652	0.523262	+	0.852172i	$0.324715\pi$
$48$	−6.70153	−0.967283
$49$	−6.95570	−0.993672
$50$	0.855333	0.120962
$51$	2.47909	0.347141
$52$	14.0145	1.94347
$53$	1.95072	0.267953	0.133976	−	0.990985i	$-0.457225\pi$
$53$	1.95072	0.267953	0.133976	+	0.990985i	$0.457225\pi$
$54$	2.53392	0.344823
$55$	−10.6974	−1.44243
$56$	1.29103	0.172521
$57$	−0.0375390	−0.00497216
$58$	21.3966	2.80951
$59$	−4.82543	−0.628218	−0.314109	−	0.949387i	$-0.601706\pi$
$59$	−4.82543	−0.628218	−0.314109	+	0.949387i	$0.601706\pi$
$60$	9.54559	1.23233
$61$	2.29965	0.294440	0.147220	−	0.989104i	$-0.452967\pi$
$61$	2.29965	0.294440	0.147220	+	0.989104i	$0.452967\pi$
$62$	−21.3335	−2.70936
$63$	−0.210471	−0.0265169
$64$	−1.46027	−0.182533
$65$	−6.84525	−0.849049
$66$	12.5534	1.54522
$67$	−10.2531	−1.25262	−0.626310	−	0.779574i	$-0.715436\pi$
$67$	−10.2531	−1.25262	−0.626310	+	0.779574i	$0.715436\pi$
$68$	−10.9594	−1.32902
$69$	−3.57941	−0.430910
$70$	−1.15158	−0.137640
$71$	−4.97845	−0.590833	−0.295416	−	0.955369i	$-0.595458\pi$
$71$	−4.97845	−0.590833	−0.295416	+	0.955369i	$0.595458\pi$
$72$	−6.13399	−0.722897
$73$	−0.0679022	−0.00794735	−0.00397367	−	0.999992i	$-0.501265\pi$
$73$	−0.0679022	−0.00794735	−0.00397367	+	0.999992i	$0.501265\pi$
$74$	−22.3427	−2.59728
$75$	0.337553	0.0389773
$76$	0.165950	0.0190358
$77$	−1.04271	−0.118827
$78$	8.03296	0.909553
$79$	−3.39674	−0.382163	−0.191081	−	0.981574i	$-0.561199\pi$
$79$	−3.39674	−0.382163	−0.191081	+	0.981574i	$0.561199\pi$
$80$	−14.4704	−1.61784
$81$	1.00000	0.111111
$82$	21.8554	2.41353
$83$	−8.96886	−0.984461	−0.492230	−	0.870465i	$-0.663818\pi$
$83$	−8.96886	−0.984461	−0.492230	+	0.870465i	$0.663818\pi$
$84$	0.930441	0.101519
$85$	5.35302	0.580616
$86$	−1.76212	−0.190014
$87$	8.44408	0.905300
$88$	−30.3887	−3.23945
$89$	−12.8332	−1.36032	−0.680159	−	0.733065i	$-0.738089\pi$
$89$	−12.8332	−1.36032	−0.680159	+	0.733065i	$0.738089\pi$
$90$	5.47142	0.576738
$91$	−0.667230	−0.0699447
$92$	15.8237	1.64973
$93$	−8.41917	−0.873027
$94$	−18.1799	−1.87511
$95$	−0.0810568	−0.00831625
$96$	4.71317	0.481036
$97$	−7.80085	−0.792056	−0.396028	−	0.918238i	$-0.629612\pi$
$97$	−7.80085	−0.792056	−0.396028	+	0.918238i	$0.629612\pi$
$98$	17.6252	1.78041
$99$	4.95415	0.497911
$100$	−1.49224	−0.149224
$101$	−2.67098	−0.265773	−0.132886	−	0.991131i	$-0.542425\pi$
$101$	−2.67098	−0.265773	−0.132886	+	0.991131i	$0.542425\pi$
$102$	−6.28180	−0.621991
$103$	14.7181	1.45022	0.725109	−	0.688634i	$-0.241789\pi$
$103$	14.7181	1.45022	0.725109	+	0.688634i	$0.241789\pi$
$104$	−19.4458	−1.90682
$105$	−0.454464	−0.0443512
$106$	−4.94298	−0.480105
$107$	−9.87828	−0.954969	−0.477485	−	0.878640i	$-0.658451\pi$
$107$	−9.87828	−0.954969	−0.477485	+	0.878640i	$0.658451\pi$
$108$	−4.42075	−0.425387
$109$	0.382364	0.0366238	0.0183119	−	0.999832i	$-0.494171\pi$
$109$	0.382364	0.0366238	0.0183119	+	0.999832i	$0.494171\pi$
$110$	27.1062	2.58448
$111$	−8.81744	−0.836914
$112$	−1.41048	−0.133278
$113$	−14.7871	−1.39105	−0.695525	−	0.718502i	$-0.744828\pi$
$113$	−14.7871	−1.39105	−0.695525	+	0.718502i	$0.744828\pi$
$114$	0.0951208	0.00890888
$115$	−7.72891	−0.720725
$116$	−37.3292	−3.46593
$117$	3.17017	0.293082
$118$	12.2273	1.12561
$119$	0.521777	0.0478312
$120$	−13.2449	−1.20909
$121$	13.5436	1.23124
$122$	−5.82713	−0.527564
$123$	8.62513	0.777702
$124$	37.2190	3.34237
$125$	11.5252	1.03085
$126$	0.533318	0.0475117
$127$	9.21534	0.817729	0.408865	−	0.912595i	$-0.365925\pi$
$127$	9.21534	0.817729	0.408865	+	0.912595i	$0.365925\pi$
$128$	13.1265	1.16023
$129$	−0.695413	−0.0612277
$130$	17.3453	1.52129
$131$	4.93059	0.430788	0.215394	−	0.976527i	$-0.430896\pi$
$131$	4.93059	0.430788	0.215394	+	0.976527i	$0.430896\pi$
$132$	−21.9011	−1.90624
$133$	−0.00790088	−0.000685093 0
$134$	25.9806	2.24439
$135$	2.15927	0.185840
$136$	15.2067	1.30396
$137$	8.84217	0.755437	0.377719	−	0.925920i	$-0.376709\pi$
$137$	8.84217	0.755437	0.377719	+	0.925920i	$0.376709\pi$
$138$	9.06994	0.772084
$139$	−10.5942	−0.898587	−0.449293	−	0.893384i	$-0.648324\pi$
$139$	−10.5942	−0.898587	−0.449293	+	0.893384i	$0.648324\pi$
$140$	2.00907	0.169798
$141$	−7.17461	−0.604211
$142$	12.6150	1.05863
$143$	15.7055	1.31336
$144$	6.70153	0.558461
$145$	18.2330	1.51417
$146$	0.172059	0.0142397
$147$	6.95570	0.573697
$148$	38.9797	3.20411
$149$	−4.04104	−0.331055	−0.165527	−	0.986205i	$-0.552933\pi$
$149$	−4.04104	−0.331055	−0.165527	+	0.986205i	$0.552933\pi$
$150$	−0.855333	−0.0698377
$151$	11.1533	0.907643	0.453821	−	0.891093i	$-0.350060\pi$
$151$	11.1533	0.907643	0.453821	+	0.891093i	$0.350060\pi$
$152$	−0.230264	−0.0186768
$153$	−2.47909	−0.200422
$154$	2.64214	0.212909
$155$	−18.1793	−1.46019
$156$	−14.0145	−1.12206
$157$	−16.7613	−1.33770	−0.668850	−	0.743397i	$-0.733213\pi$
$157$	−16.7613	−1.33770	−0.668850	+	0.743397i	$0.733213\pi$
$158$	8.60706	0.684741
$159$	−1.95072	−0.154703
$160$	10.1770	0.804563
$161$	−0.753363	−0.0593733
$162$	−2.53392	−0.199084
$163$	5.32158	0.416818	0.208409	−	0.978042i	$-0.433171\pi$
$163$	5.32158	0.416818	0.208409	+	0.978042i	$0.433171\pi$
$164$	−38.1296	−2.97742
$165$	10.6974	0.832788
$166$	22.7264	1.76391
$167$	12.0542	0.932784	0.466392	−	0.884578i	$-0.345554\pi$
$167$	12.0542	0.932784	0.466392	+	0.884578i	$0.345554\pi$
$168$	−1.29103	−0.0996050
$169$	−2.95002	−0.226925
$170$	−13.5641	−1.04032
$171$	0.0375390	0.00287068
$172$	3.07425	0.234409
$173$	0.377943	0.0287345	0.0143672	−	0.999897i	$-0.495427\pi$
$173$	0.377943	0.0287345	0.0143672	+	0.999897i	$0.495427\pi$
$174$	−21.3966	−1.62207
$175$	0.0710453	0.00537052
$176$	33.2004	2.50257
$177$	4.82543	0.362702
$178$	32.5183	2.43735
$179$	10.0801	0.753424	0.376712	−	0.926330i	$-0.377055\pi$
$179$	10.0801	0.753424	0.376712	+	0.926330i	$0.377055\pi$
$180$	−9.54559	−0.711486
$181$	−19.7835	−1.47049	−0.735247	−	0.677799i	$-0.762934\pi$
$181$	−19.7835	−1.47049	−0.735247	+	0.677799i	$0.762934\pi$
$182$	1.69071	0.125324
$183$	−2.29965	−0.169995
$184$	−21.9561	−1.61862
$185$	−19.0392	−1.39979
$186$	21.3335	1.56425
$187$	−12.2818	−0.898132
$188$	31.7171	2.31321
$189$	0.210471	0.0153095
$190$	0.205391	0.0149007
$191$	−17.1941	−1.24412	−0.622059	−	0.782970i	$-0.713704\pi$
$191$	−17.1941	−1.24412	−0.622059	+	0.782970i	$0.713704\pi$
$192$	1.46027	0.105386
$193$	−11.3932	−0.820104	−0.410052	−	0.912062i	$-0.634489\pi$
$193$	−11.3932	−0.820104	−0.410052	+	0.912062i	$0.634489\pi$
$194$	19.7667	1.41917
$195$	6.84525	0.490199
$196$	−30.7494	−2.19639
$197$	−1.89490	−0.135006	−0.0675029	−	0.997719i	$-0.521503\pi$
$197$	−1.89490	−0.135006	−0.0675029	+	0.997719i	$0.521503\pi$
$198$	−12.5534	−0.892133
$199$	−1.04415	−0.0740178	−0.0370089	−	0.999315i	$-0.511783\pi$
$199$	−1.04415	−0.0740178	−0.0370089	+	0.999315i	$0.511783\pi$
$200$	2.07055	0.146410
$201$	10.2531	0.723201
$202$	6.76805	0.476199
$203$	1.77724	0.124738
$204$	10.9594	0.767313
$205$	18.6240	1.30076
$206$	−37.2945	−2.59843
$207$	3.57941	0.248786
$208$	21.2450	1.47308
$209$	0.185974	0.0128641
$210$	1.15158	0.0794663
$211$	11.4434	0.787794	0.393897	−	0.919155i	$-0.371127\pi$
$211$	11.4434	0.787794	0.393897	+	0.919155i	$0.371127\pi$
$212$	8.62367	0.592276
$213$	4.97845	0.341118
$214$	25.0308	1.71107
$215$	−1.50158	−0.102407
$216$	6.13399	0.417365
$217$	−1.77199	−0.120291
$218$	−0.968879	−0.0656208
$219$	0.0679022	0.00458840
$220$	−47.2903	−3.18831
$221$	−7.85912	−0.528662
$222$	22.3427	1.49954
$223$	23.2045	1.55389	0.776943	−	0.629570i	$-0.216769\pi$
$223$	23.2045	1.55389	0.776943	+	0.629570i	$0.216769\pi$
$224$	0.991987	0.0662799
$225$	−0.337553	−0.0225036
$226$	37.4692	2.49242
$227$	16.5453	1.09815	0.549077	−	0.835772i	$-0.314980\pi$
$227$	16.5453	1.09815	0.549077	+	0.835772i	$0.314980\pi$
$228$	−0.165950	−0.0109903
$229$	1.07105	0.0707772	0.0353886	−	0.999374i	$-0.488733\pi$
$229$	1.07105	0.0707772	0.0353886	+	0.999374i	$0.488733\pi$
$230$	19.5844	1.29136
$231$	1.04271	0.0686051
$232$	51.7959	3.40057
$233$	−21.0847	−1.38130	−0.690651	−	0.723188i	$-0.742676\pi$
$233$	−21.0847	−1.38130	−0.690651	+	0.723188i	$0.742676\pi$
$234$	−8.03296	−0.525131
$235$	−15.4919	−1.01058
$236$	−21.3320	−1.38860
$237$	3.39674	0.220642
$238$	−1.32214	−0.0857016
$239$	−7.76787	−0.502462	−0.251231	−	0.967927i	$-0.580835\pi$
$239$	−7.76787	−0.502462	−0.251231	+	0.967927i	$0.580835\pi$
$240$	14.4704	0.934061
$241$	11.5929	0.746766	0.373383	−	0.927677i	$-0.378198\pi$
$241$	11.5929	0.746766	0.373383	+	0.927677i	$0.378198\pi$
$242$	−34.3184	−2.20607
$243$	−1.00000	−0.0641500
$244$	10.1662	0.650823
$245$	15.0192	0.959544
$246$	−21.8554	−1.39345
$247$	0.119005	0.00757210
$248$	−51.6431	−3.27934
$249$	8.96886	0.568379
$250$	−29.2040	−1.84702
$251$	9.12711	0.576098	0.288049	−	0.957616i	$-0.406993\pi$
$251$	9.12711	0.576098	0.288049	+	0.957616i	$0.406993\pi$
$252$	−0.930441	−0.0586123
$253$	17.7329	1.11486
$254$	−23.3509	−1.46517
$255$	−5.35302	−0.335219
$256$	−30.3411	−1.89632
$257$	15.8882	0.991081	0.495541	−	0.868585i	$-0.334970\pi$
$257$	15.8882	0.991081	0.495541	+	0.868585i	$0.334970\pi$
$258$	1.76212	0.109705
$259$	−1.85582	−0.115315
$260$	−30.2612	−1.87672
$261$	−8.44408	−0.522675
$262$	−12.4937	−0.771865
$263$	14.1588	0.873067	0.436534	−	0.899688i	$-0.356206\pi$
$263$	14.1588	0.873067	0.436534	+	0.899688i	$0.356206\pi$
$264$	30.3887	1.87030
$265$	−4.21214	−0.258750
$266$	0.0200202	0.00122752
$267$	12.8332	0.785380
$268$	−45.3266	−2.76876
$269$	−27.0050	−1.64652	−0.823262	−	0.567661i	$-0.807848\pi$
$269$	−27.0050	−1.64652	−0.823262	+	0.567661i	$0.807848\pi$
$270$	−5.47142	−0.332980
$271$	19.6144	1.19149	0.595744	−	0.803174i	$-0.296857\pi$
$271$	19.6144	1.19149	0.595744	+	0.803174i	$0.296857\pi$
$272$	−16.6137	−1.00735
$273$	0.667230	0.0403826
$274$	−22.4053	−1.35356
$275$	−1.67229	−0.100843
$276$	−15.8237	−0.952473
$277$	20.8464	1.25254	0.626269	−	0.779607i	$-0.284581\pi$
$277$	20.8464	1.25254	0.626269	+	0.779607i	$0.284581\pi$
$278$	26.8448	1.61004
$279$	8.41917	0.504042
$280$	−2.78768	−0.166596
$281$	−6.85051	−0.408667	−0.204333	−	0.978901i	$-0.565503\pi$
$281$	−6.85051	−0.408667	−0.204333	+	0.978901i	$0.565503\pi$
$282$	18.1799	1.08260
$283$	−8.02191	−0.476853	−0.238427	−	0.971161i	$-0.576632\pi$
$283$	−8.02191	−0.476853	−0.238427	+	0.971161i	$0.576632\pi$
$284$	−22.0085	−1.30596
$285$	0.0810568	0.00480139
$286$	−39.7965	−2.35322
$287$	1.81534	0.107156
$288$	−4.71317	−0.277726
$289$	−10.8541	−0.638478
$290$	−46.2011	−2.71302
$291$	7.80085	0.457294
$292$	−0.300179	−0.0175666
$293$	−14.0544	−0.821066	−0.410533	−	0.911846i	$-0.634657\pi$
$293$	−14.0544	−0.821066	−0.410533	+	0.911846i	$0.634657\pi$
$294$	−17.6252	−1.02792
$295$	10.4194	0.606642
$296$	−54.0860	−3.14369
$297$	−4.95415	−0.287469
$298$	10.2397	0.593168
$299$	11.3473	0.656233
$300$	1.49224	0.0861544
$301$	−0.146364	−0.00843631
$302$	−28.2616	−1.62627
$303$	2.67098	0.153444
$304$	0.251569	0.0144285
$305$	−4.96557	−0.284328
$306$	6.28180	0.359107
$307$	24.1636	1.37909	0.689546	−	0.724242i	$-0.257810\pi$
$307$	24.1636	1.37909	0.689546	+	0.724242i	$0.257810\pi$
$308$	−4.60955	−0.262653
$309$	−14.7181	−0.837284
$310$	46.0648	2.61630
$311$	6.32510	0.358664	0.179332	−	0.983789i	$-0.442606\pi$
$311$	6.32510	0.358664	0.179332	+	0.983789i	$0.442606\pi$
$312$	19.4458	1.10090
$313$	−6.03813	−0.341295	−0.170648	−	0.985332i	$-0.554586\pi$
$313$	−6.03813	−0.341295	−0.170648	+	0.985332i	$0.554586\pi$
$314$	42.4719	2.39683
$315$	0.454464	0.0256062
$316$	−15.0161	−0.844723
$317$	9.06757	0.509285	0.254643	−	0.967035i	$-0.418042\pi$
$317$	9.06757	0.509285	0.254643	+	0.967035i	$0.418042\pi$
$318$	4.94298	0.277189
$319$	−41.8332	−2.34221
$320$	3.15311	0.176264
$321$	9.87828	0.551352
$322$	1.90896	0.106382
$323$	−0.0930624	−0.00517813
$324$	4.42075	0.245597
$325$	−1.07010	−0.0593585
$326$	−13.4845	−0.746835
$327$	−0.382364	−0.0211448
$328$	52.9065	2.92127
$329$	−1.51005	−0.0832517
$330$	−27.1062	−1.49215
$331$	−1.35314	−0.0743750	−0.0371875	−	0.999308i	$-0.511840\pi$
$331$	−1.35314	−0.0743750	−0.0371875	+	0.999308i	$0.511840\pi$
$332$	−39.6491	−2.17603
$333$	8.81744	0.483192
$334$	−30.5444	−1.67132
$335$	22.1393	1.20960
$336$	1.41048	0.0769480
$337$	−16.3185	−0.888927	−0.444464	−	0.895797i	$-0.646606\pi$
$337$	−16.3185	−0.888927	−0.444464	+	0.895797i	$0.646606\pi$
$338$	7.47512	0.406593
$339$	14.7871	0.803123
$340$	23.6643	1.28338
$341$	41.7098	2.25871
$342$	−0.0951208	−0.00514354
$343$	2.93728	0.158598
$344$	−4.26565	−0.229989
$345$	7.72891	0.416111
$346$	−0.957677	−0.0514850
$347$	18.7931	1.00887	0.504434	−	0.863451i	$-0.331701\pi$
$347$	18.7931	1.00887	0.504434	+	0.863451i	$0.331701\pi$
$348$	37.3292	2.00105
$349$	−3.78013	−0.202346	−0.101173	−	0.994869i	$-0.532260\pi$
$349$	−3.78013	−0.202346	−0.101173	+	0.994869i	$0.532260\pi$
$350$	−0.180023	−0.00962264
$351$	−3.17017	−0.169211
$352$	−23.3498	−1.24455
$353$	−26.2945	−1.39952	−0.699759	−	0.714379i	$-0.746709\pi$
$353$	−26.2945	−1.39952	−0.699759	+	0.714379i	$0.746709\pi$
$354$	−12.2273	−0.649871
$355$	10.7498	0.570541
$356$	−56.7324	−3.00681
$357$	−0.521777	−0.0276153
$358$	−25.5422	−1.34995
$359$	−23.3845	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$359$	−23.3845	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$360$	13.2449	0.698069
$361$	−18.9986	−0.999926
$362$	50.1297	2.63476
$363$	−13.5436	−0.710856
$364$	−2.94966	−0.154604
$365$	0.146619	0.00767439
$366$	5.82713	0.304589
$367$	−21.9042	−1.14339	−0.571695	−	0.820466i	$-0.693714\pi$
$367$	−21.9042	−1.14339	−0.571695	+	0.820466i	$0.693714\pi$
$368$	23.9875	1.25044
$369$	−8.62513	−0.449007
$370$	48.2439	2.50808
$371$	−0.410572	−0.0213158
$372$	−37.2190	−1.92972
$373$	36.9726	1.91437	0.957185	−	0.289477i	$-0.0934813\pi$
$373$	36.9726	1.91437	0.957185	+	0.289477i	$0.0934813\pi$
$374$	31.1210	1.60923
$375$	−11.5252	−0.595160
$376$	−44.0089	−2.26959
$377$	−26.7692	−1.37868
$378$	−0.533318	−0.0274309
$379$	−3.06793	−0.157589	−0.0787946	−	0.996891i	$-0.525107\pi$
$379$	−3.06793	−0.157589	−0.0787946	+	0.996891i	$0.525107\pi$
$380$	−0.358332	−0.0183820
$381$	−9.21534	−0.472116
$382$	43.5684	2.22915
$383$	−7.40703	−0.378482	−0.189241	−	0.981931i	$-0.560603\pi$
$383$	−7.40703	−0.378482	−0.189241	+	0.981931i	$0.560603\pi$
$384$	−13.1265	−0.669861
$385$	2.25149	0.114746
$386$	28.8696	1.46942
$387$	0.695413	0.0353498
$388$	−34.4856	−1.75074
$389$	−15.2650	−0.773965	−0.386983	−	0.922087i	$-0.626483\pi$
$389$	−15.2650	−0.773965	−0.386983	+	0.922087i	$0.626483\pi$
$390$	−17.3453	−0.878314
$391$	−8.87366	−0.448760
$392$	42.6662	2.15497
$393$	−4.93059	−0.248715
$394$	4.80152	0.241897
$395$	7.33447	0.369037
$396$	21.9011	1.10057
$397$	16.3579	0.820981	0.410490	−	0.911865i	$-0.365358\pi$
$397$	16.3579	0.820981	0.410490	+	0.911865i	$0.365358\pi$
$398$	2.64579	0.132621
$399$	0.00790088	0.000395539 0
$400$	−2.26212	−0.113106
$401$	6.30226	0.314720	0.157360	−	0.987541i	$-0.449702\pi$
$401$	6.30226	0.314720	0.157360	+	0.987541i	$0.449702\pi$
$402$	−25.9806	−1.29580
$403$	26.6902	1.32953
$404$	−11.8077	−0.587457
$405$	−2.15927	−0.107295
$406$	−4.50338	−0.223499
$407$	43.6829	2.16528
$408$	−15.2067	−0.752843
$409$	6.72900	0.332727	0.166364	−	0.986064i	$-0.446797\pi$
$409$	6.72900	0.332727	0.166364	+	0.986064i	$0.446797\pi$
$410$	−47.1917	−2.33063
$411$	−8.84217	−0.436152
$412$	65.0651	3.20553
$413$	1.01562	0.0499752
$414$	−9.06994	−0.445763
$415$	19.3662	0.950649
$416$	−14.9416	−0.732570
$417$	10.5942	0.518799
$418$	−0.471243	−0.0230492
$419$	19.1610	0.936076	0.468038	−	0.883708i	$-0.344961\pi$
$419$	19.1610	0.936076	0.468038	+	0.883708i	$0.344961\pi$
$420$	−2.00907	−0.0980328
$421$	9.01348	0.439290	0.219645	−	0.975580i	$-0.429510\pi$
$421$	9.01348	0.439290	0.219645	+	0.975580i	$0.429510\pi$
$422$	−28.9966	−1.41153
$423$	7.17461	0.348841
$424$	−11.9657	−0.581107
$425$	0.836824	0.0405919
$426$	−12.6150	−0.611198
$427$	−0.484011	−0.0234229
$428$	−43.6694	−2.11084
$429$	−15.7055	−0.758269
$430$	3.80489	0.183488
$431$	27.5787	1.32842	0.664209	−	0.747547i	$-0.268769\pi$
$431$	27.5787	1.32842	0.664209	+	0.747547i	$0.268769\pi$
$432$	−6.70153	−0.322428
$433$	9.80373	0.471137	0.235569	−	0.971858i	$-0.424305\pi$
$433$	9.80373	0.471137	0.235569	+	0.971858i	$0.424305\pi$
$434$	4.49009	0.215531
$435$	−18.2330	−0.874208
$436$	1.69034	0.0809524
$437$	0.134367	0.00642766
$438$	−0.172059	−0.00822128
$439$	−11.2785	−0.538294	−0.269147	−	0.963099i	$-0.586742\pi$
$439$	−11.2785	−0.538294	−0.269147	+	0.963099i	$0.586742\pi$
$440$	65.6174	3.12819
$441$	−6.95570	−0.331224
$442$	19.9144	0.947231
$443$	−17.6993	−0.840917	−0.420458	−	0.907312i	$-0.638131\pi$
$443$	−17.6993	−0.840917	−0.420458	+	0.907312i	$0.638131\pi$
$444$	−38.9797	−1.84989
$445$	27.7104	1.31360
$446$	−58.7983	−2.78418
$447$	4.04104	0.191135
$448$	0.307344	0.0145207
$449$	−1.71388	−0.0808831	−0.0404416	−	0.999182i	$-0.512876\pi$
$449$	−1.71388	−0.0808831	−0.0404416	+	0.999182i	$0.512876\pi$
$450$	0.855333	0.0403208
$451$	−42.7302	−2.01209
$452$	−65.3699	−3.07474
$453$	−11.1533	−0.524028
$454$	−41.9246	−1.96762
$455$	1.44073	0.0675424
$456$	0.230264	0.0107831
$457$	10.6836	0.499758	0.249879	−	0.968277i	$-0.419609\pi$
$457$	10.6836	0.499758	0.249879	+	0.968277i	$0.419609\pi$
$458$	−2.71396	−0.126815
$459$	2.47909	0.115714
$460$	−34.1676	−1.59307
$461$	29.1405	1.35721	0.678604	−	0.734504i	$-0.262585\pi$
$461$	29.1405	1.35721	0.678604	+	0.734504i	$0.262585\pi$
$462$	−2.64214	−0.122923
$463$	−22.4720	−1.04436	−0.522181	−	0.852834i	$-0.674882\pi$
$463$	−22.4720	−1.04436	−0.522181	+	0.852834i	$0.674882\pi$
$464$	−56.5883	−2.62704
$465$	18.1793	0.843043
$466$	53.4268	2.47495
$467$	−22.6846	−1.04972	−0.524859	−	0.851189i	$-0.675882\pi$
$467$	−22.6846	−1.04972	−0.524859	+	0.851189i	$0.675882\pi$
$468$	14.0145	0.647822
$469$	2.15799	0.0996469
$470$	39.2553	1.81071
$471$	16.7613	0.772322
$472$	29.5991	1.36241
$473$	3.44518	0.158410
$474$	−8.60706	−0.395335
$475$	−0.0126714	−0.000581404 0
$476$	2.30664	0.105725
$477$	1.95072	0.0893175
$478$	19.6832	0.900287
$479$	25.9342	1.18496	0.592482	−	0.805584i	$-0.298148\pi$
$479$	25.9342	1.18496	0.592482	+	0.805584i	$0.298148\pi$
$480$	−10.1770	−0.464515
$481$	27.9528	1.27454
$482$	−29.3755	−1.33802
$483$	0.753363	0.0342792
$484$	59.8730	2.72150
$485$	16.8441	0.764853
$486$	2.53392	0.114941
$487$	−9.17053	−0.415556	−0.207778	−	0.978176i	$-0.566623\pi$
$487$	−9.17053	−0.415556	−0.207778	+	0.978176i	$0.566623\pi$
$488$	−14.1060	−0.638550
$489$	−5.32158	−0.240650
$490$	−38.0575	−1.71926
$491$	−19.3991	−0.875467	−0.437734	−	0.899105i	$-0.644219\pi$
$491$	−19.3991	−0.875467	−0.437734	+	0.899105i	$0.644219\pi$
$492$	38.1296	1.71901
$493$	20.9336	0.942802
$494$	−0.301549	−0.0135673
$495$	−10.6974	−0.480810
$496$	56.4213	2.53339
$497$	1.04782	0.0470012
$498$	−22.7264	−1.01839
$499$	30.3713	1.35960	0.679802	−	0.733395i	$-0.262066\pi$
$499$	30.3713	1.35960	0.679802	+	0.733395i	$0.262066\pi$
$500$	50.9501	2.27856
$501$	−12.0542	−0.538543
$502$	−23.1274	−1.03222
$503$	36.0906	1.60920	0.804601	−	0.593816i	$-0.202379\pi$
$503$	36.0906	1.60920	0.804601	+	0.593816i	$0.202379\pi$
$504$	1.29103	0.0575070
$505$	5.76737	0.256645
$506$	−44.9338	−1.99755
$507$	2.95002	0.131015
$508$	40.7387	1.80749
$509$	−41.9871	−1.86105	−0.930523	−	0.366232i	$-0.880648\pi$
$509$	−41.9871	−1.86105	−0.930523	+	0.366232i	$0.880648\pi$
$510$	13.5641	0.600629
$511$	0.0142915	0.000632217 0
$512$	50.6288	2.23750
$513$	−0.0375390	−0.00165739
$514$	−40.2595	−1.77577
$515$	−31.7804	−1.40041
$516$	−3.07425	−0.135336
$517$	35.5441	1.56323
$518$	4.70249	0.206616
$519$	−0.377943	−0.0165899
$520$	41.9887	1.84133
$521$	−4.62364	−0.202565	−0.101283	−	0.994858i	$-0.532295\pi$
$521$	−4.62364	−0.202565	−0.101283	+	0.994858i	$0.532295\pi$
$522$	21.3966	0.936505
$523$	19.6205	0.857944	0.428972	−	0.903318i	$-0.358876\pi$
$523$	19.6205	0.857944	0.428972	+	0.903318i	$0.358876\pi$
$524$	21.7969	0.952202
$525$	−0.0710453	−0.00310067
$526$	−35.8772	−1.56432
$527$	−20.8718	−0.909191
$528$	−33.2004	−1.44486
$529$	−10.1878	−0.442949
$530$	10.6732	0.463615
$531$	−4.82543	−0.209406
$532$	−0.0349278	−0.00151431
$533$	−27.3431	−1.18436
$534$	−32.5183	−1.40721
$535$	21.3299	0.922170
$536$	62.8926	2.71655
$537$	−10.0801	−0.434990
$538$	68.4285	2.95016
$539$	−34.4596	−1.48428
$540$	9.54559	0.410777
$541$	−9.81009	−0.421769	−0.210884	−	0.977511i	$-0.567634\pi$
$541$	−9.81009	−0.421769	−0.210884	+	0.977511i	$0.567634\pi$
$542$	−49.7013	−2.13485
$543$	19.7835	0.848990
$544$	11.6844	0.500963
$545$	−0.825627	−0.0353660
$546$	−1.69071	−0.0723556
$547$	−24.0936	−1.03017	−0.515083	−	0.857140i	$-0.672239\pi$
$547$	−24.0936	−1.03017	−0.515083	+	0.857140i	$0.672239\pi$
$548$	39.0890	1.66980
$549$	2.29965	0.0981467
$550$	4.23745	0.180685
$551$	−0.316982	−0.0135039
$552$	21.9561	0.934512
$553$	0.714916	0.0304013
$554$	−52.8230	−2.24424
$555$	19.0392	0.808170
$556$	−46.8342	−1.98621
$557$	−11.2242	−0.475584	−0.237792	−	0.971316i	$-0.576424\pi$
$557$	−11.2242	−0.475584	−0.237792	+	0.971316i	$0.576424\pi$
$558$	−21.3335	−0.903119
$559$	2.20458	0.0932436
$560$	3.04561	0.128700
$561$	12.2818	0.518537
$562$	17.3586	0.732230
$563$	−14.4149	−0.607516	−0.303758	−	0.952749i	$-0.598241\pi$
$563$	−14.4149	−0.607516	−0.303758	+	0.952749i	$0.598241\pi$
$564$	−31.7171	−1.33553
$565$	31.9293	1.34327
$566$	20.3269	0.854403
$567$	−0.210471	−0.00883897
$568$	30.5377	1.28133
$569$	−20.9676	−0.879007	−0.439504	−	0.898241i	$-0.644846\pi$
$569$	−20.9676	−0.879007	−0.439504	+	0.898241i	$0.644846\pi$
$570$	−0.205391	−0.00860290
$571$	−1.73595	−0.0726473	−0.0363236	−	0.999340i	$-0.511565\pi$
$571$	−1.73595	−0.0726473	−0.0363236	+	0.999340i	$0.511565\pi$
$572$	69.4301	2.90302
$573$	17.1941	0.718292
$574$	−4.59994	−0.191998
$575$	−1.20824	−0.0503872
$576$	−1.46027	−0.0608444
$577$	−42.8431	−1.78358	−0.891791	−	0.452447i	$-0.850551\pi$
$577$	−42.8431	−1.78358	−0.891791	+	0.452447i	$0.850551\pi$
$578$	27.5035	1.14399
$579$	11.3932	0.473487
$580$	80.6037	3.34689
$581$	1.88769	0.0783145
$582$	−19.7667	−0.819357
$583$	9.66419	0.400250
$584$	0.416511	0.0172353
$585$	−6.84525	−0.283016
$586$	35.6127	1.47115
$587$	22.1667	0.914917	0.457459	−	0.889231i	$-0.348760\pi$
$587$	22.1667	0.914917	0.457459	+	0.889231i	$0.348760\pi$
$588$	30.7494	1.26808
$589$	0.316047	0.0130225
$590$	−26.4020	−1.08695
$591$	1.89490	0.0779457
$592$	59.0903	2.42860
$593$	19.1022	0.784435	0.392217	−	0.919873i	$-0.371708\pi$
$593$	19.1022	0.784435	0.392217	+	0.919873i	$0.371708\pi$
$594$	12.5534	0.515073
$595$	−1.12666	−0.0461884
$596$	−17.8644	−0.731755
$597$	1.04415	0.0427342
$598$	−28.7532	−1.17581
$599$	17.4288	0.712121	0.356060	−	0.934463i	$-0.384120\pi$
$599$	17.4288	0.712121	0.356060	+	0.934463i	$0.384120\pi$
$600$	−2.07055	−0.0845298
$601$	−30.4392	−1.24164	−0.620821	−	0.783952i	$-0.713201\pi$
$601$	−30.4392	−1.24164	−0.620821	+	0.783952i	$0.713201\pi$
$602$	0.370876	0.0151158
$603$	−10.2531	−0.417540
$604$	49.3059	2.00623
$605$	−29.2443	−1.18895
$606$	−6.76805	−0.274933
$607$	−27.9352	−1.13386	−0.566928	−	0.823767i	$-0.691868\pi$
$607$	−27.9352	−1.13386	−0.566928	+	0.823767i	$0.691868\pi$
$608$	−0.176928	−0.00717536
$609$	−1.77724	−0.0720173
$610$	12.5824	0.509445
$611$	22.7447	0.920153
$612$	−10.9594	−0.443008
$613$	−39.6225	−1.60034	−0.800168	−	0.599775i	$-0.795257\pi$
$613$	−39.6225	−1.60034	−0.800168	+	0.599775i	$0.795257\pi$
$614$	−61.2287	−2.47099
$615$	−18.6240	−0.750992
$616$	6.39595	0.257700
$617$	22.7275	0.914976	0.457488	−	0.889216i	$-0.348749\pi$
$617$	22.7275	0.914976	0.457488	+	0.889216i	$0.348749\pi$
$618$	37.2945	1.50021
$619$	−17.6532	−0.709541	−0.354770	−	0.934953i	$-0.615441\pi$
$619$	−17.6532	−0.709541	−0.354770	+	0.934953i	$0.615441\pi$
$620$	−80.3659	−3.22757
$621$	−3.57941	−0.143637
$622$	−16.0273	−0.642636
$623$	2.70102	0.108214
$624$	−21.2450	−0.850480
$625$	−23.1983	−0.927932
$626$	15.3001	0.611517
$627$	−0.185974	−0.00742708
$628$	−74.0977	−2.95682
$629$	−21.8592	−0.871583
$630$	−1.15158	−0.0458799
$631$	−34.0307	−1.35474	−0.677371	−	0.735641i	$-0.736881\pi$
$631$	−34.0307	−1.35474	−0.677371	+	0.735641i	$0.736881\pi$
$632$	20.8355	0.828794
$633$	−11.4434	−0.454833
$634$	−22.9765	−0.912513
$635$	−19.8984	−0.789644
$636$	−8.62367	−0.341951
$637$	−22.0508	−0.873683
$638$	106.002	4.19666
$639$	−4.97845	−0.196944
$640$	−28.3437	−1.12038
$641$	−21.9582	−0.867298	−0.433649	−	0.901082i	$-0.642774\pi$
$641$	−21.9582	−0.867298	−0.433649	+	0.901082i	$0.642774\pi$
$642$	−25.0308	−0.987885
$643$	−17.9590	−0.708232	−0.354116	−	0.935201i	$-0.615218\pi$
$643$	−17.9590	−0.708232	−0.354116	+	0.935201i	$0.615218\pi$
$644$	−3.33043	−0.131237
$645$	1.50158	0.0591248
$646$	0.235813	0.00927792
$647$	42.7940	1.68241	0.841203	−	0.540720i	$-0.181848\pi$
$647$	42.7940	1.68241	0.841203	+	0.540720i	$0.181848\pi$
$648$	−6.13399	−0.240966
$649$	−23.9059	−0.938390
$650$	2.71155	0.106356
$651$	1.77199	0.0694499
$652$	23.5254	0.921325
$653$	−27.2585	−1.06671	−0.533354	−	0.845892i	$-0.679069\pi$
$653$	−27.2585	−1.06671	−0.533354	+	0.845892i	$0.679069\pi$
$654$	0.968879	0.0378862
$655$	−10.6465	−0.415992
$656$	−57.8016	−2.25677
$657$	−0.0679022	−0.00264912
$658$	3.82634	0.149166
$659$	6.86983	0.267611	0.133805	−	0.991008i	$-0.457280\pi$
$659$	6.86983	0.267611	0.133805	+	0.991008i	$0.457280\pi$
$660$	47.2903	1.84077
$661$	23.7634	0.924288	0.462144	−	0.886805i	$-0.347080\pi$
$661$	23.7634	0.924288	0.462144	+	0.886805i	$0.347080\pi$
$662$	3.42874	0.133262
$663$	7.85912	0.305223
$664$	55.0149	2.13499
$665$	0.0170601	0.000661564 0
$666$	−22.3427	−0.865761
$667$	−30.2248	−1.17031
$668$	53.2887	2.06180
$669$	−23.2045	−0.897137
$670$	−56.0992	−2.16730
$671$	11.3928	0.439815
$672$	−0.991987	−0.0382667
$673$	40.0847	1.54515	0.772575	−	0.634924i	$-0.218968\pi$
$673$	40.0847	1.54515	0.772575	+	0.634924i	$0.218968\pi$
$674$	41.3499	1.59274
$675$	0.337553	0.0129924
$676$	−13.0413	−0.501589
$677$	−2.79725	−0.107507	−0.0537536	−	0.998554i	$-0.517119\pi$
$677$	−2.79725	−0.107507	−0.0537536	+	0.998554i	$0.517119\pi$
$678$	−37.4692	−1.43900
$679$	1.64186	0.0630086
$680$	−32.8353	−1.25918
$681$	−16.5453	−0.634019
$682$	−105.689	−4.04705
$683$	0.593121	0.0226952	0.0113476	−	0.999936i	$-0.496388\pi$
$683$	0.593121	0.0226952	0.0113476	+	0.999936i	$0.496388\pi$
$684$	0.165950	0.00634527
$685$	−19.0926	−0.729492
$686$	−7.44282	−0.284168
$687$	−1.07105	−0.0408632
$688$	4.66033	0.177673
$689$	6.18413	0.235597
$690$	−19.5844	−0.745567
$691$	−32.6376	−1.24159	−0.620796	−	0.783972i	$-0.713190\pi$
$691$	−32.6376	−1.24159	−0.620796	+	0.783972i	$0.713190\pi$
$692$	1.67079	0.0635139
$693$	−1.04271	−0.0396092
$694$	−47.6202	−1.80764
$695$	22.8757	0.867725
$696$	−51.7959	−1.96332
$697$	21.3824	0.809918
$698$	9.57855	0.362554
$699$	21.0847	0.797495
$700$	0.314074	0.0118709
$701$	42.6036	1.60911	0.804557	−	0.593875i	$-0.202403\pi$
$701$	42.6036	1.60911	0.804557	+	0.593875i	$0.202403\pi$
$702$	8.03296	0.303184
$703$	0.330998	0.0124838
$704$	−7.23438	−0.272656
$705$	15.4919	0.583459
$706$	66.6283	2.50759
$707$	0.562165	0.0211424
$708$	21.3320	0.801707
$709$	−26.3606	−0.989995	−0.494997	−	0.868895i	$-0.664831\pi$
$709$	−26.3606	−0.989995	−0.494997	+	0.868895i	$0.664831\pi$
$710$	−27.2392	−1.02227
$711$	−3.39674	−0.127388
$712$	78.7187	2.95011
$713$	30.1356	1.12859
$714$	1.32214	0.0494798
$715$	−33.9124	−1.26825
$716$	44.5617	1.66535
$717$	7.76787	0.290097
$718$	59.2543	2.21135
$719$	19.6618	0.733261	0.366630	−	0.930367i	$-0.380511\pi$
$719$	19.6618	0.733261	0.366630	+	0.930367i	$0.380511\pi$
$720$	−14.4704	−0.539281
$721$	−3.09774	−0.115366
$722$	48.1409	1.79162
$723$	−11.5929	−0.431146
$724$	−87.4578	−3.25034
$725$	2.85033	0.105858
$726$	34.3184	1.27368
$727$	−18.9700	−0.703557	−0.351778	−	0.936083i	$-0.614423\pi$
$727$	−18.9700	−0.703557	−0.351778	+	0.936083i	$0.614423\pi$
$728$	4.09278	0.151689
$729$	1.00000	0.0370370
$730$	−0.371521	−0.0137506
$731$	−1.72399	−0.0637640
$732$	−10.1662	−0.375753
$733$	15.5673	0.574990	0.287495	−	0.957782i	$-0.407177\pi$
$733$	15.5673	0.574990	0.287495	+	0.957782i	$0.407177\pi$
$734$	55.5035	2.04867
$735$	−15.0192	−0.553993
$736$	−16.8704	−0.621850
$737$	−50.7956	−1.87108
$738$	21.8554	0.804508
$739$	−39.2937	−1.44544	−0.722720	−	0.691140i	$-0.757109\pi$
$739$	−39.2937	−1.44544	−0.722720	+	0.691140i	$0.757109\pi$
$740$	−84.1677	−3.09406
$741$	−0.119005	−0.00437176
$742$	1.04036	0.0381927
$743$	−28.5843	−1.04866	−0.524329	−	0.851516i	$-0.675684\pi$
$743$	−28.5843	−1.04866	−0.524329	+	0.851516i	$0.675684\pi$
$744$	51.6431	1.89333
$745$	8.72569	0.319685
$746$	−93.6857	−3.43008
$747$	−8.96886	−0.328154
$748$	−54.2946	−1.98521
$749$	2.07909	0.0759684
$750$	29.2040	1.06638
$751$	−10.5026	−0.383245	−0.191623	−	0.981469i	$-0.561375\pi$
$751$	−10.5026	−0.383245	−0.191623	+	0.981469i	$0.561375\pi$
$752$	48.0808	1.75333
$753$	−9.12711	−0.332610
$754$	67.8309	2.47026
$755$	−24.0830	−0.876469
$756$	0.930441	0.0338398
$757$	−22.2104	−0.807251	−0.403625	−	0.914924i	$-0.632250\pi$
$757$	−22.2104	−0.807251	−0.403625	+	0.914924i	$0.632250\pi$
$758$	7.77390	0.282361
$759$	−17.7329	−0.643665
$760$	0.497201	0.0180354
$761$	16.4833	0.597520	0.298760	−	0.954328i	$-0.403427\pi$
$761$	16.4833	0.597520	0.298760	+	0.954328i	$0.403427\pi$
$762$	23.3509	0.845915
$763$	−0.0804766	−0.00291345
$764$	−76.0106	−2.74997
$765$	5.35302	0.193539
$766$	18.7688	0.678145
$767$	−15.2974	−0.552358
$768$	30.3411	1.09484
$769$	−21.6265	−0.779870	−0.389935	−	0.920842i	$-0.627503\pi$
$769$	−21.6265	−0.779870	−0.389935	+	0.920842i	$0.627503\pi$
$770$	−5.70509	−0.205597
$771$	−15.8882	−0.572201
$772$	−50.3667	−1.81274
$773$	53.3929	1.92041	0.960205	−	0.279298i	$-0.0901017\pi$
$773$	53.3929	1.92041	0.960205	+	0.279298i	$0.0901017\pi$
$774$	−1.76212	−0.0633381
$775$	−2.84192	−0.102085
$776$	47.8503	1.71773
$777$	1.85582	0.0665771
$778$	38.6802	1.38675
$779$	−0.323779	−0.0116006
$780$	30.2612	1.08352
$781$	−24.6640	−0.882547
$782$	22.4852	0.804067
$783$	8.44408	0.301767
$784$	−46.6139	−1.66478
$785$	36.1923	1.29176
$786$	12.4937	0.445636
$787$	10.7402	0.382846	0.191423	−	0.981508i	$-0.438690\pi$
$787$	10.7402	0.382846	0.191423	+	0.981508i	$0.438690\pi$
$788$	−8.37687	−0.298414
$789$	−14.1588	−0.504066
$790$	−18.5850	−0.661224
$791$	3.11225	0.110659
$792$	−30.3887	−1.07982
$793$	7.29029	0.258886
$794$	−41.4497	−1.47099
$795$	4.21214	0.149389
$796$	−4.61592	−0.163607
$797$	−38.2165	−1.35370	−0.676849	−	0.736122i	$-0.736655\pi$
$797$	−38.2165	−1.35370	−0.676849	+	0.736122i	$0.736655\pi$
$798$	−0.0200202	−0.000708707 0
$799$	−17.7865	−0.629240
$800$	1.59095	0.0562484
$801$	−12.8332	−0.453439
$802$	−15.9694	−0.563900
$803$	−0.336398	−0.0118712
$804$	45.3266	1.59855
$805$	1.62671	0.0573342
$806$	−67.6308	−2.38219
$807$	27.0050	0.950621
$808$	16.3838	0.576379
$809$	26.2313	0.922244	0.461122	−	0.887337i	$-0.347447\pi$
$809$	26.2313	0.922244	0.461122	+	0.887337i	$0.347447\pi$
$810$	5.47142	0.192246
$811$	−27.7577	−0.974703	−0.487352	−	0.873206i	$-0.662037\pi$
$811$	−27.7577	−0.974703	−0.487352	+	0.873206i	$0.662037\pi$
$812$	7.85672	0.275717
$813$	−19.6144	−0.687906
$814$	−110.689	−3.87965
$815$	−11.4907	−0.402503
$816$	16.6137	0.581595
$817$	0.0261051	0.000913301 0
$818$	−17.0507	−0.596165
$819$	−0.667230	−0.0233149
$820$	82.3320	2.87516
$821$	−16.5301	−0.576906	−0.288453	−	0.957494i	$-0.593141\pi$
$821$	−16.5301	−0.576906	−0.288453	+	0.957494i	$0.593141\pi$
$822$	22.4053	0.781476
$823$	−16.4143	−0.572168	−0.286084	−	0.958205i	$-0.592354\pi$
$823$	−16.4143	−0.572168	−0.286084	+	0.958205i	$0.592354\pi$
$824$	−90.2807	−3.14508
$825$	1.67229	0.0582217
$826$	−2.57349	−0.0895431
$827$	−7.50232	−0.260881	−0.130441	−	0.991456i	$-0.541639\pi$
$827$	−7.50232	−0.260881	−0.130441	+	0.991456i	$0.541639\pi$
$828$	15.8237	0.549911
$829$	0.308120	0.0107015	0.00535073	−	0.999986i	$-0.498297\pi$
$829$	0.308120	0.0107015	0.00535073	+	0.999986i	$0.498297\pi$
$830$	−49.0724	−1.70333
$831$	−20.8464	−0.723153
$832$	−4.62929	−0.160492
$833$	17.2438	0.597462
$834$	−26.8448	−0.929560
$835$	−26.0283	−0.900747
$836$	0.822144	0.0284344
$837$	−8.41917	−0.291009
$838$	−48.5524	−1.67722
$839$	19.2078	0.663126	0.331563	−	0.943433i	$-0.392424\pi$
$839$	19.2078	0.663126	0.331563	+	0.943433i	$0.392424\pi$
$840$	2.78768	0.0961841
$841$	42.3025	1.45871
$842$	−22.8394	−0.787099
$843$	6.85051	0.235944
$844$	50.5883	1.74132
$845$	6.36990	0.219131
$846$	−18.1799	−0.625037
$847$	−2.85054	−0.0979459
$848$	13.0728	0.448923
$849$	8.02191	0.275311
$850$	−2.12044	−0.0727306
$851$	31.5612	1.08190
$852$	22.0085	0.753998
$853$	−28.4047	−0.972559	−0.486280	−	0.873803i	$-0.661646\pi$
$853$	−28.4047	−0.972559	−0.486280	+	0.873803i	$0.661646\pi$
$854$	1.22644	0.0419681
$855$	−0.0810568	−0.00277208
$856$	60.5932	2.07103
$857$	−15.4285	−0.527027	−0.263513	−	0.964656i	$-0.584881\pi$
$857$	−15.4285	−0.527027	−0.263513	+	0.964656i	$0.584881\pi$
$858$	39.7965	1.35863
$859$	−56.3847	−1.92382	−0.961911	−	0.273363i	$-0.911864\pi$
$859$	−56.3847	−1.92382	−0.961911	+	0.273363i	$0.911864\pi$
$860$	−6.63813	−0.226358
$861$	−1.81534	−0.0618667
$862$	−69.8821	−2.38020
$863$	−41.7007	−1.41951	−0.709754	−	0.704450i	$-0.751194\pi$
$863$	−41.7007	−1.41951	−0.709754	+	0.704450i	$0.751194\pi$
$864$	4.71317	0.160345
$865$	−0.816081	−0.0277476
$866$	−24.8419	−0.844161
$867$	10.8541	0.368626
$868$	−7.83354	−0.265888
$869$	−16.8280	−0.570849
$870$	46.2011	1.56636
$871$	−32.5042	−1.10136
$872$	−2.34541	−0.0794258
$873$	−7.80085	−0.264019
$874$	−0.340476	−0.0115168
$875$	−2.42573	−0.0820046
$876$	0.300179	0.0101421
$877$	0.932797	0.0314983	0.0157492	−	0.999876i	$-0.494987\pi$
$877$	0.932797	0.0314983	0.0157492	+	0.999876i	$0.494987\pi$
$878$	28.5788	0.964489
$879$	14.0544	0.474043
$880$	−71.6886	−2.41662
$881$	41.2243	1.38888	0.694441	−	0.719550i	$-0.255652\pi$
$881$	41.2243	1.38888	0.694441	+	0.719550i	$0.255652\pi$
$882$	17.6252	0.593471
$883$	−32.1024	−1.08033	−0.540166	−	0.841558i	$-0.681639\pi$
$883$	−32.1024	−1.08033	−0.540166	+	0.841558i	$0.681639\pi$
$884$	−34.7432	−1.16854
$885$	−10.4194	−0.350245
$886$	44.8485	1.50671
$887$	19.6162	0.658646	0.329323	−	0.944217i	$-0.393180\pi$
$887$	19.6162	0.658646	0.329323	+	0.944217i	$0.393180\pi$
$888$	54.0860	1.81501
$889$	−1.93957	−0.0650509
$890$	−70.2158	−2.35364
$891$	4.95415	0.165970
$892$	102.581	3.43467
$893$	0.269327	0.00901270
$894$	−10.2397	−0.342466
$895$	−21.7657	−0.727548
$896$	−2.76276	−0.0922974
$897$	−11.3473	−0.378876
$898$	4.34284	0.144922
$899$	−71.0921	−2.37105
$900$	−1.49224	−0.0497413
$901$	−4.83601	−0.161111
$902$	108.275	3.60516
$903$	0.146364	0.00487070
$904$	90.7037	3.01676
$905$	42.7179	1.41999
$906$	28.2616	0.938928
$907$	−8.15862	−0.270903	−0.135451	−	0.990784i	$-0.543248\pi$
$907$	−8.15862	−0.270903	−0.135451	+	0.990784i	$0.543248\pi$
$908$	73.1428	2.42733
$909$	−2.67098	−0.0885909
$910$	−3.65069	−0.121019
$911$	0.920630	0.0305018	0.0152509	−	0.999884i	$-0.495145\pi$
$911$	0.920630	0.0305018	0.0152509	+	0.999884i	$0.495145\pi$
$912$	−0.251569	−0.00833027
$913$	−44.4331	−1.47052
$914$	−27.0714	−0.895442
$915$	4.96557	0.164157
$916$	4.73486	0.156444
$917$	−1.03775	−0.0342695
$918$	−6.28180	−0.207330
$919$	−50.0515	−1.65105	−0.825523	−	0.564368i	$-0.809120\pi$
$919$	−50.0515	−1.65105	−0.825523	+	0.564368i	$0.809120\pi$
$920$	47.4090	1.56303
$921$	−24.1636	−0.796219
$922$	−73.8397	−2.43178
$923$	−15.7825	−0.519488
$924$	4.60955	0.151643
$925$	−2.97635	−0.0978619
$926$	56.9423	1.87124
$927$	14.7181	0.483406
$928$	39.7984	1.30645
$929$	−20.9598	−0.687670	−0.343835	−	0.939030i	$-0.611726\pi$
$929$	−20.9598	−0.687670	−0.343835	+	0.939030i	$0.611726\pi$
$930$	−46.0648	−1.51052
$931$	−0.261110	−0.00855753
$932$	−93.2100	−3.05319
$933$	−6.32510	−0.207075
$934$	57.4809	1.88083
$935$	26.5197	0.867285
$936$	−19.4458	−0.635605
$937$	43.7590	1.42954	0.714771	−	0.699358i	$-0.246531\pi$
$937$	43.7590	1.42954	0.714771	+	0.699358i	$0.246531\pi$
$938$	−5.46818	−0.178542
$939$	6.03813	0.197047
$940$	−68.4859	−2.23376
$941$	−44.6654	−1.45605	−0.728025	−	0.685551i	$-0.759562\pi$
$941$	−44.6654	−1.45605	−0.728025	+	0.685551i	$0.759562\pi$
$942$	−42.4719	−1.38381
$943$	−30.8729	−1.00536
$944$	−32.3378	−1.05251
$945$	−0.454464	−0.0147837
$946$	−8.72981	−0.283831
$947$	−10.9123	−0.354602	−0.177301	−	0.984157i	$-0.556737\pi$
$947$	−10.9123	−0.354602	−0.177301	+	0.984157i	$0.556737\pi$
$948$	15.0161	0.487701
$949$	−0.215261	−0.00698768
$950$	0.0321083	0.00104173
$951$	−9.06757	−0.294036
$952$	−3.20057	−0.103731
$953$	−17.1127	−0.554335	−0.277167	−	0.960822i	$-0.589396\pi$
$953$	−17.1127	−0.554335	−0.277167	+	0.960822i	$0.589396\pi$
$954$	−4.94298	−0.160035
$955$	37.1266	1.20139
$956$	−34.3398	−1.11063
$957$	41.8332	1.35228
$958$	−65.7151	−2.12316
$959$	−1.86102	−0.0600956
$960$	−3.15311	−0.101766
$961$	39.8823	1.28653
$962$	−70.8301	−2.28365
$963$	−9.87828	−0.318323
$964$	51.2494	1.65063
$965$	24.6011	0.791937
$966$	−1.90896	−0.0614198
$967$	−8.21837	−0.264285	−0.132143	−	0.991231i	$-0.542186\pi$
$967$	−8.21837	−0.264285	−0.132143	+	0.991231i	$0.542186\pi$
$968$	−83.0764	−2.67018
$969$	0.0930624	0.00298959
$970$	−42.6817	−1.37043
$971$	−10.4353	−0.334886	−0.167443	−	0.985882i	$-0.553551\pi$
$971$	−10.4353	−0.334886	−0.167443	+	0.985882i	$0.553551\pi$
$972$	−4.42075	−0.141796
$973$	2.22977	0.0714832
$974$	23.2374	0.744574
$975$	1.07010	0.0342707
$976$	15.4112	0.493300
$977$	34.8985	1.11650	0.558251	−	0.829672i	$-0.311473\pi$
$977$	34.8985	1.11650	0.558251	+	0.829672i	$0.311473\pi$
$978$	13.4845	0.431185
$979$	−63.5777	−2.03195
$980$	66.3963	2.12095
$981$	0.382364	0.0122079
$982$	49.1557	1.56862
$983$	8.37931	0.267259	0.133629	−	0.991031i	$-0.457337\pi$
$983$	8.37931	0.267259	0.133629	+	0.991031i	$0.457337\pi$
$984$	−52.9065	−1.68660
$985$	4.09159	0.130369
$986$	−53.0441	−1.68927
$987$	1.51005	0.0480654
$988$	0.526091	0.0167372
$989$	2.48917	0.0791509
$990$	27.1062	0.861493
$991$	21.0678	0.669242	0.334621	−	0.942353i	$-0.391392\pi$
$991$	21.0678	0.669242	0.334621	+	0.942353i	$0.391392\pi$
$992$	−39.6810	−1.25987
$993$	1.35314	0.0429405
$994$	−2.65509	−0.0842144
$995$	2.25460	0.0714756
$996$	39.6491	1.25633
$997$	−55.6604	−1.76278	−0.881391	−	0.472387i	$-0.843392\pi$
$997$	−55.6604	−1.76278	−0.881391	+	0.472387i	$0.843392\pi$
$998$	−76.9584	−2.43607
$999$	−8.81744	−0.278971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.b.1.7	✓	104

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.b.1.7	✓	104	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53392 q^{2} -1.00000 q^{3} +4.42075 q^{4} -2.15927 q^{5} +2.53392 q^{6} -0.210471 q^{7} -6.13399 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.53392 q^{2} -1.00000 q^{3} +4.42075 q^{4} -2.15927 q^{5} +2.53392 q^{6} -0.210471 q^{7} -6.13399 q^{8} +1.00000 q^{9} +5.47142 q^{10} +4.95415 q^{11} -4.42075 q^{12} +3.17017 q^{13} +0.533318 q^{14} +2.15927 q^{15} +6.70153 q^{16} -2.47909 q^{17} -2.53392 q^{18} +0.0375390 q^{19} -9.54559 q^{20} +0.210471 q^{21} -12.5534 q^{22} +3.57941 q^{23} +6.13399 q^{24} -0.337553 q^{25} -8.03296 q^{26} -1.00000 q^{27} -0.930441 q^{28} -8.44408 q^{29} -5.47142 q^{30} +8.41917 q^{31} -4.71317 q^{32} -4.95415 q^{33} +6.28180 q^{34} +0.454464 q^{35} +4.42075 q^{36} +8.81744 q^{37} -0.0951208 q^{38} -3.17017 q^{39} +13.2449 q^{40} -8.62513 q^{41} -0.533318 q^{42} +0.695413 q^{43} +21.9011 q^{44} -2.15927 q^{45} -9.06994 q^{46} +7.17461 q^{47} -6.70153 q^{48} -6.95570 q^{49} +0.855333 q^{50} +2.47909 q^{51} +14.0145 q^{52} +1.95072 q^{53} +2.53392 q^{54} -10.6974 q^{55} +1.29103 q^{56} -0.0375390 q^{57} +21.3966 q^{58} -4.82543 q^{59} +9.54559 q^{60} +2.29965 q^{61} -21.3335 q^{62} -0.210471 q^{63} -1.46027 q^{64} -6.84525 q^{65} +12.5534 q^{66} -10.2531 q^{67} -10.9594 q^{68} -3.57941 q^{69} -1.15158 q^{70} -4.97845 q^{71} -6.13399 q^{72} -0.0679022 q^{73} -22.3427 q^{74} +0.337553 q^{75} +0.165950 q^{76} -1.04271 q^{77} +8.03296 q^{78} -3.39674 q^{79} -14.4704 q^{80} +1.00000 q^{81} +21.8554 q^{82} -8.96886 q^{83} +0.930441 q^{84} +5.35302 q^{85} -1.76212 q^{86} +8.44408 q^{87} -30.3887 q^{88} -12.8332 q^{89} +5.47142 q^{90} -0.667230 q^{91} +15.8237 q^{92} -8.41917 q^{93} -18.1799 q^{94} -0.0810568 q^{95} +4.71317 q^{96} -7.80085 q^{97} +17.6252 q^{98} +4.95415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

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