LMFDB - Embedded newform 4009.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06463 q^{2} +1.41935 q^{3} +2.26270 q^{4} +1.21401 q^{5} -2.93043 q^{6} +2.08294 q^{7} -0.542368 q^{8} -0.985454 q^{9} +O(q^{10})$	\(q-2.06463 q^{2} +1.41935 q^{3} +2.26270 q^{4} +1.21401 q^{5} -2.93043 q^{6} +2.08294 q^{7} -0.542368 q^{8} -0.985454 q^{9} -2.50648 q^{10} +1.19570 q^{11} +3.21155 q^{12} +1.93375 q^{13} -4.30050 q^{14} +1.72310 q^{15} -3.40560 q^{16} -4.50653 q^{17} +2.03460 q^{18} -1.00000 q^{19} +2.74694 q^{20} +2.95641 q^{21} -2.46867 q^{22} -0.0131475 q^{23} -0.769808 q^{24} -3.52618 q^{25} -3.99248 q^{26} -5.65674 q^{27} +4.71306 q^{28} +3.03620 q^{29} -3.55757 q^{30} -8.38162 q^{31} +8.11604 q^{32} +1.69711 q^{33} +9.30431 q^{34} +2.52871 q^{35} -2.22978 q^{36} -3.18842 q^{37} +2.06463 q^{38} +2.74466 q^{39} -0.658441 q^{40} -5.67741 q^{41} -6.10390 q^{42} -3.92950 q^{43} +2.70549 q^{44} -1.19635 q^{45} +0.0271447 q^{46} -4.08419 q^{47} -4.83373 q^{48} -2.66136 q^{49} +7.28025 q^{50} -6.39633 q^{51} +4.37549 q^{52} -9.89964 q^{53} +11.6791 q^{54} +1.45159 q^{55} -1.12972 q^{56} -1.41935 q^{57} -6.26863 q^{58} -9.13096 q^{59} +3.89886 q^{60} +10.4529 q^{61} +17.3049 q^{62} -2.05264 q^{63} -9.94542 q^{64} +2.34760 q^{65} -3.50390 q^{66} +2.13659 q^{67} -10.1969 q^{68} -0.0186608 q^{69} -5.22085 q^{70} -10.3451 q^{71} +0.534479 q^{72} -10.6577 q^{73} +6.58291 q^{74} -5.00487 q^{75} -2.26270 q^{76} +2.49056 q^{77} -5.66671 q^{78} +2.14707 q^{79} -4.13444 q^{80} -5.07252 q^{81} +11.7217 q^{82} +1.50888 q^{83} +6.68946 q^{84} -5.47098 q^{85} +8.11295 q^{86} +4.30942 q^{87} -0.648507 q^{88} -4.64639 q^{89} +2.47002 q^{90} +4.02788 q^{91} -0.0297488 q^{92} -11.8964 q^{93} +8.43233 q^{94} -1.21401 q^{95} +11.5195 q^{96} +11.9003 q^{97} +5.49473 q^{98} -1.17830 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * 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 $

$2$	−2.06463	−1.45991	−0.729957	−	0.683493i	$-0.760460\pi$
$2$	−2.06463	−1.45991	−0.729957	+	0.683493i	$0.760460\pi$
$3$	1.41935	0.819460	0.409730	−	0.912207i	$-0.365623\pi$
$3$	1.41935	0.819460	0.409730	+	0.912207i	$0.365623\pi$
$4$	2.26270	1.13135
$5$	1.21401	0.542923	0.271461	−	0.962449i	$-0.412493\pi$
$5$	1.21401	0.542923	0.271461	+	0.962449i	$0.412493\pi$
$6$	−2.93043	−1.19634
$7$	2.08294	0.787277	0.393639	−	0.919265i	$-0.371216\pi$
$7$	2.08294	0.787277	0.393639	+	0.919265i	$0.371216\pi$
$8$	−0.542368	−0.191756
$9$	−0.985454	−0.328485
$10$	−2.50648	−0.792620
$11$	1.19570	0.360516	0.180258	−	0.983619i	$-0.442307\pi$
$11$	1.19570	0.360516	0.180258	+	0.983619i	$0.442307\pi$
$12$	3.21155	0.927095
$13$	1.93375	0.536326	0.268163	−	0.963374i	$-0.413583\pi$
$13$	1.93375	0.536326	0.268163	+	0.963374i	$0.413583\pi$
$14$	−4.30050	−1.14936
$15$	1.72310	0.444904
$16$	−3.40560	−0.851400
$17$	−4.50653	−1.09299	−0.546497	−	0.837461i	$-0.684039\pi$
$17$	−4.50653	−1.09299	−0.546497	+	0.837461i	$0.684039\pi$
$18$	2.03460	0.479559
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.74694	0.614234
$21$	2.95641	0.645142
$22$	−2.46867	−0.526322
$23$	−0.0131475	−0.00274144	−0.00137072	−	0.999999i	$-0.500436\pi$
$23$	−0.0131475	−0.00274144	−0.00137072	+	0.999999i	$0.500436\pi$
$24$	−0.769808	−0.157136
$25$	−3.52618	−0.705235
$26$	−3.99248	−0.782989
$27$	−5.65674	−1.08864
$28$	4.71306	0.890684
$29$	3.03620	0.563808	0.281904	−	0.959443i	$-0.409034\pi$
$29$	3.03620	0.563808	0.281904	+	0.959443i	$0.409034\pi$
$30$	−3.55757	−0.649521
$31$	−8.38162	−1.50538	−0.752692	−	0.658373i	$-0.771245\pi$
$31$	−8.38162	−1.50538	−0.752692	+	0.658373i	$0.771245\pi$
$32$	8.11604	1.43473
$33$	1.69711	0.295428
$34$	9.30431	1.59568
$35$	2.52871	0.427430
$36$	−2.22978	−0.371630
$37$	−3.18842	−0.524174	−0.262087	−	0.965044i	$-0.584411\pi$
$37$	−3.18842	−0.524174	−0.262087	+	0.965044i	$0.584411\pi$
$38$	2.06463	0.334927
$39$	2.74466	0.439498
$40$	−0.658441	−0.104109
$41$	−5.67741	−0.886662	−0.443331	−	0.896358i	$-0.646203\pi$
$41$	−5.67741	−0.886662	−0.443331	+	0.896358i	$0.646203\pi$
$42$	−6.10390	−0.941852
$43$	−3.92950	−0.599243	−0.299621	−	0.954058i	$-0.596860\pi$
$43$	−3.92950	−0.599243	−0.299621	+	0.954058i	$0.596860\pi$
$44$	2.70549	0.407869
$45$	−1.19635	−0.178342
$46$	0.0271447	0.00400227
$47$	−4.08419	−0.595740	−0.297870	−	0.954606i	$-0.596276\pi$
$47$	−4.08419	−0.595740	−0.297870	+	0.954606i	$0.596276\pi$
$48$	−4.83373	−0.697689
$49$	−2.66136	−0.380195
$50$	7.28025	1.02958
$51$	−6.39633	−0.895665
$52$	4.37549	0.606771
$53$	−9.89964	−1.35982	−0.679910	−	0.733296i	$-0.737981\pi$
$53$	−9.89964	−1.35982	−0.679910	+	0.733296i	$0.737981\pi$
$54$	11.6791	1.58932
$55$	1.45159	0.195732
$56$	−1.12972	−0.150965
$57$	−1.41935	−0.187997
$58$	−6.26863	−0.823112
$59$	−9.13096	−1.18875	−0.594375	−	0.804188i	$-0.702600\pi$
$59$	−9.13096	−1.18875	−0.594375	+	0.804188i	$0.702600\pi$
$60$	3.89886	0.503341
$61$	10.4529	1.33835	0.669176	−	0.743104i	$-0.266647\pi$
$61$	10.4529	1.33835	0.669176	+	0.743104i	$0.266647\pi$
$62$	17.3049	2.19773
$63$	−2.05264	−0.258608
$64$	−9.94542	−1.24318
$65$	2.34760	0.291183
$66$	−3.50390	−0.431300
$67$	2.13659	0.261026	0.130513	−	0.991447i	$-0.458338\pi$
$67$	2.13659	0.261026	0.130513	+	0.991447i	$0.458338\pi$
$68$	−10.1969	−1.23656
$69$	−0.0186608	−0.00224650
$70$	−5.22085	−0.624012
$71$	−10.3451	−1.22774	−0.613869	−	0.789408i	$-0.710388\pi$
$71$	−10.3451	−1.22774	−0.613869	+	0.789408i	$0.710388\pi$
$72$	0.534479	0.0629889
$73$	−10.6577	−1.24739	−0.623696	−	0.781667i	$-0.714370\pi$
$73$	−10.6577	−1.24739	−0.623696	+	0.781667i	$0.714370\pi$
$74$	6.58291	0.765248
$75$	−5.00487	−0.577912
$76$	−2.26270	−0.259549
$77$	2.49056	0.283826
$78$	−5.66671	−0.641629
$79$	2.14707	0.241564	0.120782	−	0.992679i	$-0.461460\pi$
$79$	2.14707	0.241564	0.120782	+	0.992679i	$0.461460\pi$
$80$	−4.13444	−0.462244
$81$	−5.07252	−0.563613
$82$	11.7217	1.29445
$83$	1.50888	0.165621	0.0828107	−	0.996565i	$-0.473610\pi$
$83$	1.50888	0.165621	0.0828107	+	0.996565i	$0.473610\pi$
$84$	6.68946	0.729880
$85$	−5.47098	−0.593411
$86$	8.11295	0.874842
$87$	4.30942	0.462019
$88$	−0.648507	−0.0691311
$89$	−4.64639	−0.492516	−0.246258	−	0.969204i	$-0.579201\pi$
$89$	−4.64639	−0.492516	−0.246258	+	0.969204i	$0.579201\pi$
$90$	2.47002	0.260363
$91$	4.02788	0.422237
$92$	−0.0297488	−0.00310152
$93$	−11.8964	−1.23360
$94$	8.43233	0.869729
$95$	−1.21401	−0.124555
$96$	11.5195	1.17570
$97$	11.9003	1.20829	0.604144	−	0.796875i	$-0.293515\pi$
$97$	11.9003	1.20829	0.604144	+	0.796875i	$0.293515\pi$
$98$	5.49473	0.555052
$99$	−1.17830	−0.118424
$100$	−7.97866	−0.797866
$101$	−2.86367	−0.284946	−0.142473	−	0.989799i	$-0.545505\pi$
$101$	−2.86367	−0.284946	−0.142473	+	0.989799i	$0.545505\pi$
$102$	13.2060	1.30759
$103$	−6.02505	−0.593666	−0.296833	−	0.954929i	$-0.595930\pi$
$103$	−6.02505	−0.593666	−0.296833	+	0.954929i	$0.595930\pi$
$104$	−1.04880	−0.102844
$105$	3.58912	0.350262
$106$	20.4391	1.98522
$107$	3.17639	0.307074	0.153537	−	0.988143i	$-0.450934\pi$
$107$	3.17639	0.307074	0.153537	+	0.988143i	$0.450934\pi$
$108$	−12.7995	−1.23163
$109$	13.3952	1.28303	0.641516	−	0.767110i	$-0.278306\pi$
$109$	13.3952	1.28303	0.641516	+	0.767110i	$0.278306\pi$
$110$	−2.99699	−0.285752
$111$	−4.52548	−0.429540
$112$	−7.09366	−0.670288
$113$	−13.8456	−1.30248	−0.651242	−	0.758870i	$-0.725752\pi$
$113$	−13.8456	−1.30248	−0.651242	+	0.758870i	$0.725752\pi$
$114$	2.93043	0.274460
$115$	−0.0159612	−0.00148839
$116$	6.87000	0.637863
$117$	−1.90562	−0.176175
$118$	18.8520	1.73547
$119$	−9.38683	−0.860489
$120$	−0.934556	−0.0853129
$121$	−9.57031	−0.870028
$122$	−21.5813	−1.95388
$123$	−8.05821	−0.726584
$124$	−18.9650	−1.70311
$125$	−10.3509	−0.925811
$126$	4.23794	0.377546
$127$	11.4012	1.01169	0.505846	−	0.862624i	$-0.331181\pi$
$127$	11.4012	1.01169	0.505846	+	0.862624i	$0.331181\pi$
$128$	4.30152	0.380204
$129$	−5.57732	−0.491056
$130$	−4.84691	−0.425102
$131$	−3.84651	−0.336071	−0.168035	−	0.985781i	$-0.553742\pi$
$131$	−3.84651	−0.336071	−0.168035	+	0.985781i	$0.553742\pi$
$132$	3.84003	0.334232
$133$	−2.08294	−0.180614
$134$	−4.41127	−0.381076
$135$	−6.86735	−0.591047
$136$	2.44420	0.209588
$137$	6.09143	0.520426	0.260213	−	0.965551i	$-0.416207\pi$
$137$	6.09143	0.520426	0.260213	+	0.965551i	$0.416207\pi$
$138$	0.0385277	0.00327970
$139$	−13.8667	−1.17615	−0.588077	−	0.808805i	$-0.700115\pi$
$139$	−13.8667	−1.17615	−0.588077	+	0.808805i	$0.700115\pi$
$140$	5.72171	0.483572
$141$	−5.79688	−0.488185
$142$	21.3588	1.79239
$143$	2.31218	0.193354
$144$	3.35606	0.279672
$145$	3.68598	0.306104
$146$	22.0042	1.82108
$147$	−3.77740	−0.311555
$148$	−7.21443	−0.593023
$149$	23.7262	1.94373	0.971863	−	0.235548i	$-0.0756886\pi$
$149$	23.7262	1.94373	0.971863	+	0.235548i	$0.0756886\pi$
$150$	10.3332	0.843702
$151$	4.65078	0.378475	0.189238	−	0.981931i	$-0.439398\pi$
$151$	4.65078	0.378475	0.189238	+	0.981931i	$0.439398\pi$
$152$	0.542368	0.0439919
$153$	4.44098	0.359032
$154$	−5.14209	−0.414361
$155$	−10.1754	−0.817307
$156$	6.21033	0.497225
$157$	1.78804	0.142701	0.0713505	−	0.997451i	$-0.477269\pi$
$157$	1.78804	0.142701	0.0713505	+	0.997451i	$0.477269\pi$
$158$	−4.43291	−0.352663
$159$	−14.0510	−1.11432
$160$	9.85297	0.778946
$161$	−0.0273854	−0.00215827
$162$	10.4729	0.822827
$163$	10.5911	0.829559	0.414780	−	0.909922i	$-0.363859\pi$
$163$	10.5911	0.829559	0.414780	+	0.909922i	$0.363859\pi$
$164$	−12.8462	−1.00312
$165$	2.06031	0.160395
$166$	−3.11528	−0.241793
$167$	24.5110	1.89672	0.948359	−	0.317199i	$-0.102742\pi$
$167$	24.5110	1.89672	0.948359	+	0.317199i	$0.102742\pi$
$168$	−1.60346	−0.123710
$169$	−9.26061	−0.712355
$170$	11.2955	0.866329
$171$	0.985454	0.0753595
$172$	−8.89125	−0.677952
$173$	−13.7838	−1.04796	−0.523981	−	0.851730i	$-0.675554\pi$
$173$	−13.7838	−1.04796	−0.523981	+	0.851730i	$0.675554\pi$
$174$	−8.89736	−0.674507
$175$	−7.34481	−0.555215
$176$	−4.07206	−0.306943
$177$	−12.9600	−0.974133
$178$	9.59307	0.719031
$179$	−17.4832	−1.30675	−0.653376	−	0.757033i	$-0.726648\pi$
$179$	−17.4832	−1.30675	−0.653376	+	0.757033i	$0.726648\pi$
$180$	−2.70698	−0.201766
$181$	16.3271	1.21359	0.606793	−	0.794860i	$-0.292456\pi$
$181$	16.3271	1.21359	0.606793	+	0.794860i	$0.292456\pi$
$182$	−8.31609	−0.616429
$183$	14.8362	1.09673
$184$	0.00713078	0.000525688 0
$185$	−3.87078	−0.284586
$186$	24.5617	1.80095
$187$	−5.38844	−0.394041
$188$	−9.24127	−0.673989
$189$	−11.7827	−0.857062
$190$	2.50648	0.181839
$191$	25.3490	1.83419	0.917093	−	0.398674i	$-0.130529\pi$
$191$	25.3490	1.83419	0.917093	+	0.398674i	$0.130529\pi$
$192$	−14.1160	−1.01873
$193$	0.776019	0.0558591	0.0279295	−	0.999610i	$-0.491109\pi$
$193$	0.776019	0.0558591	0.0279295	+	0.999610i	$0.491109\pi$
$194$	−24.5696	−1.76400
$195$	3.33205	0.238613
$196$	−6.02185	−0.430132
$197$	−23.2427	−1.65598	−0.827988	−	0.560745i	$-0.810515\pi$
$197$	−23.2427	−1.65598	−0.827988	+	0.560745i	$0.810515\pi$
$198$	2.43276	0.172889
$199$	23.7578	1.68415	0.842073	−	0.539364i	$-0.181335\pi$
$199$	23.7578	1.68415	0.842073	+	0.539364i	$0.181335\pi$
$200$	1.91248	0.135233
$201$	3.03256	0.213901
$202$	5.91242	0.415997
$203$	6.32422	0.443873
$204$	−14.4729	−1.01331
$205$	−6.89244	−0.481389
$206$	12.4395	0.866701
$207$	0.0129562	0.000900521 0
$208$	−6.58558	−0.456628
$209$	−1.19570	−0.0827080
$210$	−7.41020	−0.511353
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−22.3999	−1.53843
$213$	−14.6833	−1.00608
$214$	−6.55808	−0.448301
$215$	−4.77045	−0.325342
$216$	3.06804	0.208753
$217$	−17.4584	−1.18515
$218$	−27.6562	−1.87311
$219$	−15.1270	−1.02219
$220$	3.28450	0.221441
$221$	−8.71450	−0.586201
$222$	9.34344	0.627091
$223$	18.1385	1.21464	0.607321	−	0.794457i	$-0.292244\pi$
$223$	18.1385	1.21464	0.607321	+	0.794457i	$0.292244\pi$
$224$	16.9052	1.12953
$225$	3.47488	0.231659
$226$	28.5860	1.90152
$227$	−12.0240	−0.798063	−0.399032	−	0.916937i	$-0.630654\pi$
$227$	−12.0240	−0.798063	−0.399032	+	0.916937i	$0.630654\pi$
$228$	−3.21155	−0.212690
$229$	23.2521	1.53654	0.768271	−	0.640124i	$-0.221117\pi$
$229$	23.2521	1.53654	0.768271	+	0.640124i	$0.221117\pi$
$230$	0.0329540	0.00217292
$231$	3.53497	0.232584
$232$	−1.64674	−0.108114
$233$	28.1509	1.84423	0.922115	−	0.386916i	$-0.126460\pi$
$233$	28.1509	1.84423	0.922115	+	0.386916i	$0.126460\pi$
$234$	3.93440	0.257200
$235$	−4.95825	−0.323441
$236$	−20.6606	−1.34489
$237$	3.04744	0.197952
$238$	19.3803	1.25624
$239$	3.04798	0.197157	0.0985786	−	0.995129i	$-0.468570\pi$
$239$	3.04798	0.197157	0.0985786	+	0.995129i	$0.468570\pi$
$240$	−5.86820	−0.378791
$241$	16.8298	1.08410	0.542051	−	0.840345i	$-0.317648\pi$
$241$	16.8298	1.08410	0.542051	+	0.840345i	$0.317648\pi$
$242$	19.7592	1.27017
$243$	9.77056	0.626782
$244$	23.6517	1.51414
$245$	−3.23093	−0.206416
$246$	16.6372	1.06075
$247$	−1.93375	−0.123042
$248$	4.54592	0.288666
$249$	2.14163	0.135720
$250$	21.3707	1.35160
$251$	−11.3227	−0.714684	−0.357342	−	0.933974i	$-0.616317\pi$
$251$	−11.3227	−0.714684	−0.357342	+	0.933974i	$0.616317\pi$
$252$	−4.64450	−0.292576
$253$	−0.0157204	−0.000988332 0
$254$	−23.5392	−1.47698
$255$	−7.76522	−0.486277
$256$	11.0098	0.688112
$257$	−30.2626	−1.88773	−0.943866	−	0.330329i	$-0.892840\pi$
$257$	−30.2626	−1.88773	−0.943866	+	0.330329i	$0.892840\pi$
$258$	11.5151	0.716899
$259$	−6.64129	−0.412670
$260$	5.31189	0.329430
$261$	−2.99204	−0.185202
$262$	7.94161	0.490634
$263$	−17.9809	−1.10875	−0.554374	−	0.832268i	$-0.687042\pi$
$263$	−17.9809	−1.10875	−0.554374	+	0.832268i	$0.687042\pi$
$264$	−0.920456	−0.0566502
$265$	−12.0183	−0.738277
$266$	4.30050	0.263680
$267$	−6.59484	−0.403598
$268$	4.83445	0.295311
$269$	16.9081	1.03091	0.515454	−	0.856918i	$-0.327623\pi$
$269$	16.9081	1.03091	0.515454	+	0.856918i	$0.327623\pi$
$270$	14.1785	0.862878
$271$	24.7640	1.50431	0.752154	−	0.658988i	$-0.229015\pi$
$271$	24.7640	1.50431	0.752154	+	0.658988i	$0.229015\pi$
$272$	15.3474	0.930575
$273$	5.71696	0.346006
$274$	−12.5765	−0.759776
$275$	−4.21623	−0.254248
$276$	−0.0422238	−0.00254157
$277$	−9.15682	−0.550180	−0.275090	−	0.961418i	$-0.588708\pi$
$277$	−9.15682	−0.550180	−0.275090	+	0.961418i	$0.588708\pi$
$278$	28.6295	1.71708
$279$	8.25970	0.494495
$280$	−1.37149	−0.0819624
$281$	13.2365	0.789623	0.394811	−	0.918762i	$-0.370810\pi$
$281$	13.2365	0.789623	0.394811	+	0.918762i	$0.370810\pi$
$282$	11.9684	0.712708
$283$	14.9040	0.885950	0.442975	−	0.896534i	$-0.353923\pi$
$283$	14.9040	0.885950	0.442975	+	0.896534i	$0.353923\pi$
$284$	−23.4078	−1.38900
$285$	−1.72310	−0.102068
$286$	−4.77379	−0.282280
$287$	−11.8257	−0.698048
$288$	−7.99798	−0.471286
$289$	3.30880	0.194635
$290$	−7.61019	−0.446886
$291$	16.8906	0.990144
$292$	−24.1152	−1.41123
$293$	−1.21149	−0.0707759	−0.0353880	−	0.999374i	$-0.511267\pi$
$293$	−1.21149	−0.0707759	−0.0353880	+	0.999374i	$0.511267\pi$
$294$	7.79893	0.454843
$295$	−11.0851	−0.645399
$296$	1.72930	0.100513
$297$	−6.76374	−0.392472
$298$	−48.9858	−2.83767
$299$	−0.0254240	−0.00147031
$300$	−11.3245	−0.653820
$301$	−8.18490	−0.471770
$302$	−9.60214	−0.552541
$303$	−4.06454	−0.233502
$304$	3.40560	0.195325
$305$	12.6899	0.726622
$306$	−9.16897	−0.524155
$307$	−25.0324	−1.42867	−0.714337	−	0.699802i	$-0.753271\pi$
$307$	−25.0324	−1.42867	−0.714337	+	0.699802i	$0.753271\pi$
$308$	5.63538	0.321106
$309$	−8.55164	−0.486486
$310$	21.0084	1.19320
$311$	−17.8492	−1.01213	−0.506066	−	0.862494i	$-0.668901\pi$
$311$	−17.8492	−1.01213	−0.506066	+	0.862494i	$0.668901\pi$
$312$	−1.48862	−0.0842763
$313$	2.85785	0.161535	0.0807676	−	0.996733i	$-0.474263\pi$
$313$	2.85785	0.161535	0.0807676	+	0.996733i	$0.474263\pi$
$314$	−3.69163	−0.208331
$315$	−2.49193	−0.140404
$316$	4.85817	0.273293
$317$	22.8771	1.28491	0.642453	−	0.766325i	$-0.277917\pi$
$317$	22.8771	1.28491	0.642453	+	0.766325i	$0.277917\pi$
$318$	29.0101	1.62681
$319$	3.63037	0.203262
$320$	−12.0739	−0.674949
$321$	4.50841	0.251635
$322$	0.0565407	0.00315089
$323$	4.50653	0.250750
$324$	−11.4776	−0.637642
$325$	−6.81874	−0.378236
$326$	−21.8667	−1.21109
$327$	19.0125	1.05139
$328$	3.07924	0.170023
$329$	−8.50711	−0.469012
$330$	−4.25377	−0.234162
$331$	29.2529	1.60789	0.803943	−	0.594707i	$-0.202732\pi$
$331$	29.2529	1.60789	0.803943	+	0.594707i	$0.202732\pi$
$332$	3.41414	0.187375
$333$	3.14205	0.172183
$334$	−50.6061	−2.76904
$335$	2.59385	0.141717
$336$	−10.0684	−0.549274
$337$	14.6633	0.798759	0.399379	−	0.916786i	$-0.369226\pi$
$337$	14.6633	0.798759	0.399379	+	0.916786i	$0.369226\pi$
$338$	19.1197	1.03998
$339$	−19.6517	−1.06733
$340$	−12.3792	−0.671354
$341$	−10.0219	−0.542714
$342$	−2.03460	−0.110018
$343$	−20.1240	−1.08660
$344$	2.13123	0.114908
$345$	−0.0226545	−0.00121968
$346$	28.4584	1.52994
$347$	31.7614	1.70504	0.852521	−	0.522692i	$-0.175072\pi$
$347$	31.7614	1.70504	0.852521	+	0.522692i	$0.175072\pi$
$348$	9.75091	0.522704
$349$	−21.3330	−1.14193	−0.570964	−	0.820975i	$-0.693430\pi$
$349$	−21.3330	−1.14193	−0.570964	+	0.820975i	$0.693430\pi$
$350$	15.1643	0.810567
$351$	−10.9387	−0.583866
$352$	9.70431	0.517242
$353$	−24.7879	−1.31933	−0.659663	−	0.751561i	$-0.729301\pi$
$353$	−24.7879	−1.31933	−0.659663	+	0.751561i	$0.729301\pi$
$354$	26.7576	1.42215
$355$	−12.5591	−0.666567
$356$	−10.5134	−0.557207
$357$	−13.3232	−0.705137
$358$	36.0962	1.90775
$359$	−18.6287	−0.983186	−0.491593	−	0.870825i	$-0.663585\pi$
$359$	−18.6287	−0.983186	−0.491593	+	0.870825i	$0.663585\pi$
$360$	0.648863	0.0341981
$361$	1.00000	0.0526316
$362$	−33.7095	−1.77173
$363$	−13.5836	−0.712954
$364$	9.11387	0.477697
$365$	−12.9386	−0.677237
$366$	−30.6314	−1.60113
$367$	−22.5163	−1.17534	−0.587671	−	0.809100i	$-0.699955\pi$
$367$	−22.5163	−1.17534	−0.587671	+	0.809100i	$0.699955\pi$
$368$	0.0447751	0.00233406
$369$	5.59482	0.291255
$370$	7.99174	0.415471
$371$	−20.6203	−1.07055
$372$	−26.9180	−1.39563
$373$	−28.1656	−1.45836	−0.729180	−	0.684322i	$-0.760098\pi$
$373$	−28.1656	−1.45836	−0.729180	+	0.684322i	$0.760098\pi$
$374$	11.1251	0.575266
$375$	−14.6915	−0.758665
$376$	2.21513	0.114237
$377$	5.87125	0.302385
$378$	24.3268	1.25124
$379$	19.6214	1.00788	0.503941	−	0.863738i	$-0.331883\pi$
$379$	19.6214	1.00788	0.503941	+	0.863738i	$0.331883\pi$
$380$	−2.74694	−0.140915
$381$	16.1822	0.829041
$382$	−52.3362	−2.67775
$383$	19.9255	1.01814	0.509072	−	0.860724i	$-0.329989\pi$
$383$	19.9255	1.01814	0.509072	+	0.860724i	$0.329989\pi$
$384$	6.10534	0.311562
$385$	3.02357	0.154095
$386$	−1.60219	−0.0815494
$387$	3.87234	0.196842
$388$	26.9267	1.36699
$389$	−19.8914	−1.00854	−0.504268	−	0.863547i	$-0.668238\pi$
$389$	−19.8914	−1.00854	−0.504268	+	0.863547i	$0.668238\pi$
$390$	−6.87945	−0.348355
$391$	0.0592495	0.00299638
$392$	1.44344	0.0729047
$393$	−5.45953	−0.275397
$394$	47.9876	2.41758
$395$	2.60657	0.131151
$396$	−2.66614	−0.133979
$397$	−15.0353	−0.754600	−0.377300	−	0.926091i	$-0.623147\pi$
$397$	−15.0353	−0.754600	−0.377300	+	0.926091i	$0.623147\pi$
$398$	−49.0510	−2.45871
$399$	−2.95641	−0.148006
$400$	12.0087	0.600437
$401$	28.0364	1.40007	0.700036	−	0.714107i	$-0.253167\pi$
$401$	28.0364	1.40007	0.700036	+	0.714107i	$0.253167\pi$
$402$	−6.26112	−0.312276
$403$	−16.2080	−0.807376
$404$	−6.47962	−0.322373
$405$	−6.15810	−0.305998
$406$	−13.0572	−0.648017
$407$	−3.81238	−0.188973
$408$	3.46916	0.171749
$409$	−11.5137	−0.569315	−0.284657	−	0.958629i	$-0.591880\pi$
$409$	−11.5137	−0.569315	−0.284657	+	0.958629i	$0.591880\pi$
$410$	14.2303	0.702786
$411$	8.64585	0.426468
$412$	−13.6329	−0.671642
$413$	−19.0192	−0.935875
$414$	−0.0267498	−0.00131468
$415$	1.83180	0.0899196
$416$	15.6944	0.769481
$417$	−19.6816	−0.963812
$418$	2.46867	0.120746
$419$	−10.3570	−0.505973	−0.252987	−	0.967470i	$-0.581413\pi$
$419$	−10.3570	−0.505973	−0.252987	+	0.967470i	$0.581413\pi$
$420$	8.12109	0.396268
$421$	3.29482	0.160580	0.0802898	−	0.996772i	$-0.474415\pi$
$421$	3.29482	0.160580	0.0802898	+	0.996772i	$0.474415\pi$
$422$	2.06463	0.100505
$423$	4.02478	0.195691
$424$	5.36925	0.260754
$425$	15.8908	0.770818
$426$	30.3156	1.46879
$427$	21.7727	1.05365
$428$	7.18721	0.347407
$429$	3.28178	0.158446
$430$	9.84922	0.474972
$431$	7.37468	0.355226	0.177613	−	0.984100i	$-0.443162\pi$
$431$	7.37468	0.355226	0.177613	+	0.984100i	$0.443162\pi$
$432$	19.2646	0.926869
$433$	5.09639	0.244917	0.122458	−	0.992474i	$-0.460922\pi$
$433$	5.09639	0.244917	0.122458	+	0.992474i	$0.460922\pi$
$434$	36.0451	1.73022
$435$	5.23169	0.250840
$436$	30.3093	1.45155
$437$	0.0131475	0.000628930 0
$438$	31.2317	1.49231
$439$	22.5806	1.07771	0.538856	−	0.842398i	$-0.318857\pi$
$439$	22.5806	1.07771	0.538856	+	0.842398i	$0.318857\pi$
$440$	−0.787295	−0.0375328
$441$	2.62265	0.124888
$442$	17.9922	0.855802
$443$	−7.00795	−0.332958	−0.166479	−	0.986045i	$-0.553240\pi$
$443$	−7.00795	−0.332958	−0.166479	+	0.986045i	$0.553240\pi$
$444$	−10.2398	−0.485959
$445$	−5.64077	−0.267398
$446$	−37.4492	−1.77327
$447$	33.6757	1.59281
$448$	−20.7157	−0.978725
$449$	−24.9739	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$449$	−24.9739	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$450$	−7.17435	−0.338202
$451$	−6.78845	−0.319655
$452$	−31.3284	−1.47356
$453$	6.60107	0.310145
$454$	24.8252	1.16510
$455$	4.88990	0.229242
$456$	0.769808	0.0360496
$457$	−28.5379	−1.33495	−0.667473	−	0.744634i	$-0.732624\pi$
$457$	−28.5379	−1.33495	−0.667473	+	0.744634i	$0.732624\pi$
$458$	−48.0070	−2.24322
$459$	25.4923	1.18988
$460$	−0.0361153	−0.00168389
$461$	7.79441	0.363022	0.181511	−	0.983389i	$-0.441901\pi$
$461$	7.79441	0.363022	0.181511	+	0.983389i	$0.441901\pi$
$462$	−7.29840	−0.339552
$463$	−12.8440	−0.596913	−0.298457	−	0.954423i	$-0.596472\pi$
$463$	−12.8440	−0.596913	−0.298457	+	0.954423i	$0.596472\pi$
$464$	−10.3401	−0.480027
$465$	−14.4424	−0.669750
$466$	−58.1213	−2.69242
$467$	−34.0989	−1.57791	−0.788953	−	0.614453i	$-0.789377\pi$
$467$	−34.0989	−1.57791	−0.788953	+	0.614453i	$0.789377\pi$
$468$	−4.31184	−0.199315
$469$	4.45039	0.205500
$470$	10.2370	0.472195
$471$	2.53785	0.116938
$472$	4.95234	0.227950
$473$	−4.69848	−0.216036
$474$	−6.29183	−0.288993
$475$	3.52618	0.161792
$476$	−21.2395	−0.973512
$477$	9.75563	0.446680
$478$	−6.29294	−0.287832
$479$	17.5731	0.802937	0.401469	−	0.915873i	$-0.368500\pi$
$479$	17.5731	0.802937	0.401469	+	0.915873i	$0.368500\pi$
$480$	13.9848	0.638315
$481$	−6.16561	−0.281128
$482$	−34.7473	−1.58270
$483$	−0.0388694	−0.00176862
$484$	−21.6547	−0.984305
$485$	14.4471	0.656007
$486$	−20.1726	−0.915047
$487$	39.1482	1.77397	0.886986	−	0.461795i	$-0.152795\pi$
$487$	39.1482	1.77397	0.886986	+	0.461795i	$0.152795\pi$
$488$	−5.66930	−0.256637
$489$	15.0325	0.679791
$490$	6.67067	0.301350
$491$	−14.5573	−0.656963	−0.328481	−	0.944510i	$-0.606537\pi$
$491$	−14.5573	−0.656963	−0.328481	+	0.944510i	$0.606537\pi$
$492$	−18.2333	−0.822019
$493$	−13.6827	−0.616239
$494$	3.99248	0.179630
$495$	−1.43047	−0.0642950
$496$	28.5445	1.28168
$497$	−21.5482	−0.966570
$498$	−4.42167	−0.198140
$499$	−42.0548	−1.88263	−0.941316	−	0.337527i	$-0.890410\pi$
$499$	−42.0548	−1.88263	−0.941316	+	0.337527i	$0.890410\pi$
$500$	−23.4209	−1.04741
$501$	34.7896	1.55429
$502$	23.3772	1.04338
$503$	−38.3995	−1.71215	−0.856074	−	0.516854i	$-0.827103\pi$
$503$	−38.3995	−1.71215	−0.856074	+	0.516854i	$0.827103\pi$
$504$	1.11329	0.0495897
$505$	−3.47653	−0.154704
$506$	0.0324568	0.00144288
$507$	−13.1440	−0.583746
$508$	25.7974	1.14457
$509$	−13.5554	−0.600832	−0.300416	−	0.953808i	$-0.597126\pi$
$509$	−13.5554	−0.600832	−0.300416	+	0.953808i	$0.597126\pi$
$510$	16.0323	0.709922
$511$	−22.1994	−0.982043
$512$	−31.3342	−1.38479
$513$	5.65674	0.249751
$514$	62.4811	2.75593
$515$	−7.31448	−0.322315
$516$	−12.6198	−0.555554
$517$	−4.88344	−0.214774
$518$	13.7118	0.602462
$519$	−19.5640	−0.858764
$520$	−1.27326	−0.0558362
$521$	−21.4588	−0.940128	−0.470064	−	0.882632i	$-0.655769\pi$
$521$	−21.4588	−0.940128	−0.470064	+	0.882632i	$0.655769\pi$
$522$	6.17745	0.270380
$523$	24.2487	1.06032	0.530161	−	0.847897i	$-0.322132\pi$
$523$	24.2487	1.06032	0.530161	+	0.847897i	$0.322132\pi$
$524$	−8.70347	−0.380213
$525$	−10.4248	−0.454977
$526$	37.1239	1.61868
$527$	37.7720	1.64537
$528$	−5.77967	−0.251528
$529$	−22.9998	−0.999992
$530$	24.8133	1.07782
$531$	8.99814	0.390486
$532$	−4.71306	−0.204337
$533$	−10.9787	−0.475539
$534$	13.6159	0.589218
$535$	3.85618	0.166717
$536$	−1.15882	−0.0500533
$537$	−24.8147	−1.07083
$538$	−34.9090	−1.50504
$539$	−3.18218	−0.137066
$540$	−15.5387	−0.668680
$541$	−37.9160	−1.63014	−0.815068	−	0.579365i	$-0.803300\pi$
$541$	−37.9160	−1.63014	−0.815068	+	0.579365i	$0.803300\pi$
$542$	−51.1285	−2.19616
$543$	23.1739	0.994485
$544$	−36.5752	−1.56815
$545$	16.2620	0.696587
$546$	−11.8034	−0.505140
$547$	23.7804	1.01678	0.508389	−	0.861128i	$-0.330241\pi$
$547$	23.7804	1.01678	0.508389	+	0.861128i	$0.330241\pi$
$548$	13.7830	0.588782
$549$	−10.3008	−0.439628
$550$	8.70496	0.371181
$551$	−3.03620	−0.129347
$552$	0.0101210	0.000430780 0
$553$	4.47222	0.190178
$554$	18.9054	0.803215
$555$	−5.49399	−0.233207
$556$	−31.3760	−1.33064
$557$	3.13115	0.132671	0.0663356	−	0.997797i	$-0.478869\pi$
$557$	3.13115	0.132671	0.0663356	+	0.997797i	$0.478869\pi$
$558$	−17.0532	−0.721920
$559$	−7.59866	−0.321389
$560$	−8.61179	−0.363914
$561$	−7.64806	−0.322901
$562$	−27.3285	−1.15278
$563$	−33.7049	−1.42049	−0.710247	−	0.703952i	$-0.751417\pi$
$563$	−33.7049	−1.42049	−0.710247	+	0.703952i	$0.751417\pi$
$564$	−13.1166	−0.552307
$565$	−16.8087	−0.707148
$566$	−30.7712	−1.29341
$567$	−10.5657	−0.443720
$568$	5.61086	0.235426
$569$	−46.6246	−1.95461	−0.977303	−	0.211846i	$-0.932052\pi$
$569$	−46.6246	−1.95461	−0.977303	+	0.211846i	$0.932052\pi$
$570$	3.55757	0.149010
$571$	−21.0106	−0.879265	−0.439632	−	0.898178i	$-0.644891\pi$
$571$	−21.0106	−0.879265	−0.439632	+	0.898178i	$0.644891\pi$
$572$	5.23175	0.218750
$573$	35.9790	1.50304
$574$	24.4157	1.01909
$575$	0.0463603	0.00193336
$576$	9.80075	0.408365
$577$	34.2980	1.42784	0.713922	−	0.700225i	$-0.246917\pi$
$577$	34.2980	1.42784	0.713922	+	0.700225i	$0.246917\pi$
$578$	−6.83145	−0.284151
$579$	1.10144	0.0457743
$580$	8.34026	0.346310
$581$	3.14291	0.130390
$582$	−34.8728	−1.44553
$583$	−11.8369	−0.490236
$584$	5.78041	0.239195
$585$	−2.31345	−0.0956493
$586$	2.50128	0.103327
$587$	25.9153	1.06964	0.534820	−	0.844966i	$-0.320379\pi$
$587$	25.9153	1.06964	0.534820	+	0.844966i	$0.320379\pi$
$588$	−8.54710	−0.352477
$589$	8.38162	0.345359
$590$	22.8866	0.942226
$591$	−32.9895	−1.35701
$592$	10.8585	0.446282
$593$	22.0890	0.907087	0.453543	−	0.891234i	$-0.350160\pi$
$593$	22.0890	0.907087	0.453543	+	0.891234i	$0.350160\pi$
$594$	13.9646	0.572975
$595$	−11.3957	−0.467179
$596$	53.6851	2.19903
$597$	33.7206	1.38009
$598$	0.0524910	0.00214652
$599$	−14.4037	−0.588519	−0.294260	−	0.955725i	$-0.595073\pi$
$599$	−14.4037	−0.588519	−0.294260	+	0.955725i	$0.595073\pi$
$600$	2.71448	0.110818
$601$	4.68850	0.191248	0.0956239	−	0.995418i	$-0.469515\pi$
$601$	4.68850	0.191248	0.0956239	+	0.995418i	$0.469515\pi$
$602$	16.8988	0.688743
$603$	−2.10551	−0.0857431
$604$	10.5233	0.428187
$605$	−11.6185	−0.472358
$606$	8.39178	0.340893
$607$	8.02361	0.325668	0.162834	−	0.986653i	$-0.447936\pi$
$607$	8.02361	0.325668	0.162834	+	0.986653i	$0.447936\pi$
$608$	−8.11604	−0.329149
$609$	8.97627	0.363737
$610$	−26.1999	−1.06080
$611$	−7.89780	−0.319511
$612$	10.0486	0.406190
$613$	−22.6260	−0.913856	−0.456928	−	0.889504i	$-0.651050\pi$
$613$	−22.6260	−0.913856	−0.456928	+	0.889504i	$0.651050\pi$
$614$	51.6826	2.08574
$615$	−9.78276	−0.394479
$616$	−1.35080	−0.0544253
$617$	−43.8265	−1.76439	−0.882195	−	0.470885i	$-0.843935\pi$
$617$	−43.8265	−1.76439	−0.882195	+	0.470885i	$0.843935\pi$
$618$	17.6560	0.710227
$619$	−14.1324	−0.568029	−0.284015	−	0.958820i	$-0.591666\pi$
$619$	−14.1324	−0.568029	−0.284015	+	0.958820i	$0.591666\pi$
$620$	−23.0238	−0.924658
$621$	0.0743719	0.00298444
$622$	36.8519	1.47763
$623$	−9.67815	−0.387747
$624$	−9.34722	−0.374188
$625$	5.06479	0.202592
$626$	−5.90040	−0.235827
$627$	−1.69711	−0.0677759
$628$	4.04578	0.161444
$629$	14.3687	0.572919
$630$	5.14491	0.204978
$631$	−22.5785	−0.898837	−0.449419	−	0.893321i	$-0.648369\pi$
$631$	−22.5785	−0.898837	−0.449419	+	0.893321i	$0.648369\pi$
$632$	−1.16450	−0.0463214
$633$	−1.41935	−0.0564140
$634$	−47.2327	−1.87585
$635$	13.8412	0.549270
$636$	−31.7932	−1.26068
$637$	−5.14641	−0.203908
$638$	−7.49537	−0.296745
$639$	10.1946	0.403293
$640$	5.22209	0.206421
$641$	14.7476	0.582494	0.291247	−	0.956648i	$-0.405930\pi$
$641$	14.7476	0.582494	0.291247	+	0.956648i	$0.405930\pi$
$642$	−9.30819	−0.367365
$643$	−14.8680	−0.586338	−0.293169	−	0.956061i	$-0.594710\pi$
$643$	−14.8680	−0.586338	−0.293169	+	0.956061i	$0.594710\pi$
$644$	−0.0619649	−0.00244176
$645$	−6.77093	−0.266605
$646$	−9.30431	−0.366073
$647$	32.8119	1.28997	0.644985	−	0.764195i	$-0.276864\pi$
$647$	32.8119	1.28997	0.644985	+	0.764195i	$0.276864\pi$
$648$	2.75117	0.108076
$649$	−10.9178	−0.428563
$650$	14.0782	0.552191
$651$	−24.7795	−0.971187
$652$	23.9644	0.938520
$653$	−6.46244	−0.252895	−0.126447	−	0.991973i	$-0.540357\pi$
$653$	−6.46244	−0.252895	−0.126447	+	0.991973i	$0.540357\pi$
$654$	−39.2538	−1.53494
$655$	−4.66970	−0.182460
$656$	19.3350	0.754904
$657$	10.5027	0.409749
$658$	17.5640	0.684718
$659$	−32.0053	−1.24675	−0.623374	−	0.781924i	$-0.714239\pi$
$659$	−32.0053	−1.24675	−0.623374	+	0.781924i	$0.714239\pi$
$660$	4.66185	0.181462
$661$	21.9626	0.854246	0.427123	−	0.904193i	$-0.359527\pi$
$661$	21.9626	0.854246	0.427123	+	0.904193i	$0.359527\pi$
$662$	−60.3964	−2.34737
$663$	−12.3689	−0.480368
$664$	−0.818370	−0.0317589
$665$	−2.52871	−0.0980593
$666$	−6.48716	−0.251372
$667$	−0.0399184	−0.00154565
$668$	55.4609	2.14585
$669$	25.7448	0.995351
$670$	−5.35533	−0.206895
$671$	12.4984	0.482497
$672$	23.9944	0.925603
$673$	−20.4485	−0.788233	−0.394117	−	0.919060i	$-0.628949\pi$
$673$	−20.4485	−0.788233	−0.394117	+	0.919060i	$0.628949\pi$
$674$	−30.2742	−1.16612
$675$	19.9467	0.767748
$676$	−20.9539	−0.805921
$677$	30.7766	1.18284	0.591420	−	0.806364i	$-0.298568\pi$
$677$	30.7766	1.18284	0.591420	+	0.806364i	$0.298568\pi$
$678$	40.5735	1.55822
$679$	24.7875	0.951258
$680$	2.96728	0.113790
$681$	−17.0663	−0.653981
$682$	20.6914	0.792316
$683$	−38.7712	−1.48354	−0.741770	−	0.670654i	$-0.766013\pi$
$683$	−38.7712	−1.48354	−0.741770	+	0.670654i	$0.766013\pi$
$684$	2.22978	0.0852578
$685$	7.39506	0.282551
$686$	41.5487	1.58634
$687$	33.0028	1.25914
$688$	13.3823	0.510195
$689$	−19.1434	−0.729306
$690$	0.0467731	0.00178062
$691$	−23.0585	−0.877186	−0.438593	−	0.898686i	$-0.644523\pi$
$691$	−23.0585	−0.877186	−0.438593	+	0.898686i	$0.644523\pi$
$692$	−31.1885	−1.18561
$693$	−2.45433	−0.0932324
$694$	−65.5756	−2.48922
$695$	−16.8343	−0.638561
$696$	−2.33729	−0.0885949
$697$	25.5854	0.969116
$698$	44.0447	1.66712
$699$	39.9560	1.51127
$700$	−16.6191	−0.628142
$701$	10.1471	0.383251	0.191626	−	0.981468i	$-0.438624\pi$
$701$	10.1471	0.383251	0.191626	+	0.981468i	$0.438624\pi$
$702$	22.5844	0.852394
$703$	3.18842	0.120254
$704$	−11.8917	−0.448185
$705$	−7.03748	−0.265047
$706$	51.1778	1.92610
$707$	−5.96486	−0.224331
$708$	−29.3245	−1.10208
$709$	−2.66158	−0.0999577	−0.0499789	−	0.998750i	$-0.515915\pi$
$709$	−2.66158	−0.0999577	−0.0499789	+	0.998750i	$0.515915\pi$
$710$	25.9299	0.973130
$711$	−2.11584	−0.0793502
$712$	2.52005	0.0944430
$713$	0.110197	0.00412692
$714$	27.5074	1.02944
$715$	2.80701	0.104976
$716$	−39.5591	−1.47839
$717$	4.32614	0.161563
$718$	38.4614	1.43537
$719$	−3.99727	−0.149073	−0.0745364	−	0.997218i	$-0.523748\pi$
$719$	−3.99727	−0.149073	−0.0745364	+	0.997218i	$0.523748\pi$
$720$	4.07430	0.151840
$721$	−12.5498	−0.467379
$722$	−2.06463	−0.0768376
$723$	23.8873	0.888379
$724$	36.9433	1.37299
$725$	−10.7062	−0.397617
$726$	28.0451	1.04085
$727$	−4.55523	−0.168944	−0.0844720	−	0.996426i	$-0.526920\pi$
$727$	−4.55523	−0.168944	−0.0844720	+	0.996426i	$0.526920\pi$
$728$	−2.18460	−0.0809665
$729$	29.0854	1.07724
$730$	26.7134	0.988708
$731$	17.7084	0.654968
$732$	33.5699	1.24078
$733$	36.9304	1.36405	0.682027	−	0.731327i	$-0.261099\pi$
$733$	36.9304	1.36405	0.682027	+	0.731327i	$0.261099\pi$
$734$	46.4878	1.71590
$735$	−4.58581	−0.169150
$736$	−0.106706	−0.00393322
$737$	2.55471	0.0941040
$738$	−11.5512	−0.425207
$739$	−36.3798	−1.33825	−0.669126	−	0.743149i	$-0.733331\pi$
$739$	−36.3798	−1.33825	−0.669126	+	0.743149i	$0.733331\pi$
$740$	−8.75840	−0.321965
$741$	−2.74466	−0.100828
$742$	42.5734	1.56292
$743$	18.0879	0.663580	0.331790	−	0.943353i	$-0.392348\pi$
$743$	18.0879	0.663580	0.331790	+	0.943353i	$0.392348\pi$
$744$	6.45224	0.236551
$745$	28.8039	1.05529
$746$	58.1516	2.12908
$747$	−1.48693	−0.0544041
$748$	−12.1924	−0.445798
$749$	6.61624	0.241752
$750$	30.3325	1.10759
$751$	−35.9411	−1.31151	−0.655755	−	0.754973i	$-0.727650\pi$
$751$	−35.9411	−1.31151	−0.655755	+	0.754973i	$0.727650\pi$
$752$	13.9091	0.507213
$753$	−16.0709	−0.585655
$754$	−12.1220	−0.441456
$755$	5.64610	0.205483
$756$	−26.6605	−0.969635
$757$	16.4821	0.599053	0.299526	−	0.954088i	$-0.403171\pi$
$757$	16.4821	0.599053	0.299526	+	0.954088i	$0.403171\pi$
$758$	−40.5109	−1.47142
$759$	−0.0223127	−0.000809899 0
$760$	0.658441	0.0238842
$761$	−18.1031	−0.656238	−0.328119	−	0.944636i	$-0.606415\pi$
$761$	−18.1031	−0.656238	−0.328119	+	0.944636i	$0.606415\pi$
$762$	−33.4103	−1.21033
$763$	27.9015	1.01010
$764$	57.3569	2.07510
$765$	5.39140	0.194926
$766$	−41.1387	−1.48640
$767$	−17.6570	−0.637557
$768$	15.6267	0.563881
$769$	19.2769	0.695142	0.347571	−	0.937654i	$-0.387007\pi$
$769$	19.2769	0.695142	0.347571	+	0.937654i	$0.387007\pi$
$770$	−6.24255	−0.224966
$771$	−42.9532	−1.54692
$772$	1.75589	0.0631960
$773$	33.3596	1.19986	0.599931	−	0.800052i	$-0.295195\pi$
$773$	33.3596	1.19986	0.599931	+	0.800052i	$0.295195\pi$
$774$	−7.99494	−0.287372
$775$	29.5551	1.06165
$776$	−6.45432	−0.231697
$777$	−9.42630	−0.338167
$778$	41.0685	1.47238
$779$	5.67741	0.203414
$780$	7.53942	0.269954
$781$	−12.3696	−0.442619
$782$	−0.122328	−0.00437445
$783$	−17.1750	−0.613785
$784$	9.06354	0.323698
$785$	2.17070	0.0774755
$786$	11.2719	0.402055
$787$	31.3937	1.11906	0.559532	−	0.828809i	$-0.310981\pi$
$787$	31.3937	1.11906	0.559532	+	0.828809i	$0.310981\pi$
$788$	−52.5912	−1.87348
$789$	−25.5211	−0.908575
$790$	−5.38160	−0.191469
$791$	−28.8396	−1.02542
$792$	0.639074	0.0227085
$793$	20.2132	0.717793
$794$	31.0423	1.10165
$795$	−17.0581	−0.604988
$796$	53.7566	1.90535
$797$	−6.62502	−0.234670	−0.117335	−	0.993092i	$-0.537435\pi$
$797$	−6.62502	−0.234670	−0.117335	+	0.993092i	$0.537435\pi$
$798$	6.10390	0.216076
$799$	18.4055	0.651140
$800$	−28.6186	−1.01182
$801$	4.57880	0.161784
$802$	−57.8848	−2.04398
$803$	−12.7434	−0.449704
$804$	6.86177	0.241996
$805$	−0.0332462	−0.00117178
$806$	33.4634	1.17870
$807$	23.9985	0.844787
$808$	1.55316	0.0546401
$809$	23.6987	0.833203	0.416602	−	0.909089i	$-0.363221\pi$
$809$	23.6987	0.833203	0.416602	+	0.909089i	$0.363221\pi$
$810$	12.7142	0.446731
$811$	1.76283	0.0619015	0.0309507	−	0.999521i	$-0.490146\pi$
$811$	1.76283	0.0619015	0.0309507	+	0.999521i	$0.490146\pi$
$812$	14.3098	0.502175
$813$	35.1487	1.23272
$814$	7.87116	0.275884
$815$	12.8577	0.450387
$816$	21.7833	0.762569
$817$	3.92950	0.137476
$818$	23.7715	0.831150
$819$	−3.96929	−0.138698
$820$	−15.5955	−0.544618
$821$	2.45246	0.0855916	0.0427958	−	0.999084i	$-0.486374\pi$
$821$	2.45246	0.0855916	0.0427958	+	0.999084i	$0.486374\pi$
$822$	−17.8505	−0.622607
$823$	32.1205	1.11965	0.559826	−	0.828610i	$-0.310868\pi$
$823$	32.1205	1.11965	0.559826	+	0.828610i	$0.310868\pi$
$824$	3.26779	0.113839
$825$	−5.98430	−0.208346
$826$	39.2677	1.36630
$827$	−18.5422	−0.644775	−0.322387	−	0.946608i	$-0.604485\pi$
$827$	−18.5422	−0.644775	−0.322387	+	0.946608i	$0.604485\pi$
$828$	0.0293160	0.00101880
$829$	38.0192	1.32046	0.660231	−	0.751063i	$-0.270458\pi$
$829$	38.0192	1.32046	0.660231	+	0.751063i	$0.270458\pi$
$830$	−3.78199	−0.131275
$831$	−12.9967	−0.450851
$832$	−19.2319	−0.666748
$833$	11.9935	0.415551
$834$	40.6352	1.40708
$835$	29.7566	1.02977
$836$	−2.70549	−0.0935715
$837$	47.4127	1.63882
$838$	21.3834	0.738677
$839$	40.1099	1.38475	0.692374	−	0.721539i	$-0.256565\pi$
$839$	40.1099	1.38475	0.692374	+	0.721539i	$0.256565\pi$
$840$	−1.94662	−0.0671649
$841$	−19.7815	−0.682120
$842$	−6.80258	−0.234432
$843$	18.7872	0.647065
$844$	−2.26270	−0.0778852
$845$	−11.2425	−0.386753
$846$	−8.30968	−0.285693
$847$	−19.9344	−0.684953
$848$	33.7142	1.15775
$849$	21.1539	0.726001
$850$	−32.8086	−1.12533
$851$	0.0419198	0.00143699
$852$	−33.2238	−1.13823
$853$	−30.0102	−1.02753	−0.513765	−	0.857931i	$-0.671750\pi$
$853$	−30.0102	−1.02753	−0.513765	+	0.857931i	$0.671750\pi$
$854$	−44.9525	−1.53824
$855$	1.19635	0.0409144
$856$	−1.72277	−0.0588832
$857$	−14.8183	−0.506185	−0.253092	−	0.967442i	$-0.581448\pi$
$857$	−14.8183	−0.506185	−0.253092	+	0.967442i	$0.581448\pi$
$858$	−6.77566	−0.231317
$859$	6.50287	0.221875	0.110938	−	0.993827i	$-0.464615\pi$
$859$	6.50287	0.221875	0.110938	+	0.993827i	$0.464615\pi$
$860$	−10.7941	−0.368075
$861$	−16.7848	−0.572023
$862$	−15.2260	−0.518599
$863$	52.6162	1.79108	0.895538	−	0.444984i	$-0.146791\pi$
$863$	52.6162	1.79108	0.895538	+	0.444984i	$0.146791\pi$
$864$	−45.9104	−1.56190
$865$	−16.7337	−0.568963
$866$	−10.5222	−0.357557
$867$	4.69633	0.159496
$868$	−39.5030	−1.34082
$869$	2.56724	0.0870878
$870$	−10.8015	−0.366205
$871$	4.13163	0.139995
$872$	−7.26515	−0.246029
$873$	−11.7272	−0.396904
$874$	−0.0271447	−0.000918183 0
$875$	−21.5602	−0.728869
$876$	−34.2278	−1.15645
$877$	47.0186	1.58770	0.793852	−	0.608111i	$-0.208072\pi$
$877$	47.0186	1.58770	0.793852	+	0.608111i	$0.208072\pi$
$878$	−46.6205	−1.57337
$879$	−1.71952	−0.0579981
$880$	−4.94353	−0.166646
$881$	−23.8262	−0.802725	−0.401363	−	0.915919i	$-0.631463\pi$
$881$	−23.8262	−0.802725	−0.401363	+	0.915919i	$0.631463\pi$
$882$	−5.41480	−0.182326
$883$	−22.3346	−0.751618	−0.375809	−	0.926697i	$-0.622635\pi$
$883$	−22.3346	−0.751618	−0.375809	+	0.926697i	$0.622635\pi$
$884$	−19.7183	−0.663197
$885$	−15.7336	−0.528879
$886$	14.4688	0.486090
$887$	16.7914	0.563798	0.281899	−	0.959444i	$-0.409036\pi$
$887$	16.7914	0.563798	0.281899	+	0.959444i	$0.409036\pi$
$888$	2.45448	0.0823668
$889$	23.7480	0.796481
$890$	11.6461	0.390378
$891$	−6.06519	−0.203191
$892$	41.0418	1.37418
$893$	4.08419	0.136672
$894$	−69.5278	−2.32536
$895$	−21.2248	−0.709465
$896$	8.95980	0.299326
$897$	−0.0360854	−0.00120486
$898$	51.5618	1.72064
$899$	−25.4483	−0.848748
$900$	7.86260	0.262087
$901$	44.6130	1.48627
$902$	14.0156	0.466669
$903$	−11.6172	−0.386597
$904$	7.50941	0.249759
$905$	19.8213	0.658883
$906$	−13.6288	−0.452786
$907$	1.16401	0.0386502	0.0193251	−	0.999813i	$-0.493848\pi$
$907$	1.16401	0.0386502	0.0193251	+	0.999813i	$0.493848\pi$
$908$	−27.2067	−0.902887
$909$	2.82202	0.0936004
$910$	−10.0958	−0.334673
$911$	33.2915	1.10300	0.551499	−	0.834176i	$-0.314056\pi$
$911$	33.2915	1.10300	0.551499	+	0.834176i	$0.314056\pi$
$912$	4.83373	0.160061
$913$	1.80416	0.0597091
$914$	58.9201	1.94890
$915$	18.0114	0.595438
$916$	52.6124	1.73836
$917$	−8.01204	−0.264581
$918$	−52.6321	−1.73712
$919$	−15.9285	−0.525434	−0.262717	−	0.964873i	$-0.584619\pi$
$919$	−15.9285	−0.525434	−0.262717	+	0.964873i	$0.584619\pi$
$920$	0.00865685	0.000285408 0
$921$	−35.5296	−1.17074
$922$	−16.0926	−0.529980
$923$	−20.0049	−0.658468
$924$	7.99856	0.263133
$925$	11.2429	0.369666
$926$	26.5182	0.871442
$927$	5.93741	0.195010
$928$	24.6419	0.808911
$929$	8.03345	0.263569	0.131784	−	0.991278i	$-0.457929\pi$
$929$	8.03345	0.263569	0.131784	+	0.991278i	$0.457929\pi$
$930$	29.8182	0.977778
$931$	2.66136	0.0872227
$932$	63.6970	2.08647
$933$	−25.3342	−0.829403
$934$	70.4015	2.30361
$935$	−6.54162	−0.213934
$936$	1.03355	0.0337826
$937$	−58.3477	−1.90614	−0.953068	−	0.302757i	$-0.902093\pi$
$937$	−58.3477	−1.90614	−0.953068	+	0.302757i	$0.902093\pi$
$938$	−9.18840	−0.300012
$939$	4.05628	0.132372
$940$	−11.2190	−0.365924
$941$	−8.16754	−0.266254	−0.133127	−	0.991099i	$-0.542502\pi$
$941$	−8.16754	−0.266254	−0.133127	+	0.991099i	$0.542502\pi$
$942$	−5.23971	−0.170719
$943$	0.0746436	0.00243073
$944$	31.0964	1.01210
$945$	−14.3043	−0.465318
$946$	9.70062	0.315394
$947$	29.3696	0.954384	0.477192	−	0.878799i	$-0.341655\pi$
$947$	29.3696	0.954384	0.477192	+	0.878799i	$0.341655\pi$
$948$	6.89543	0.223953
$949$	−20.6094	−0.669008
$950$	−7.28025	−0.236202
$951$	32.4706	1.05293
$952$	5.09111	0.165004
$953$	−49.4909	−1.60317	−0.801583	−	0.597883i	$-0.796009\pi$
$953$	−49.4909	−1.60317	−0.801583	+	0.597883i	$0.796009\pi$
$954$	−20.1418	−0.652114
$955$	30.7739	0.995821
$956$	6.89664	0.223053
$957$	5.15276	0.166565
$958$	−36.2820	−1.17222
$959$	12.6881	0.409719
$960$	−17.1370	−0.553094
$961$	39.2515	1.26618
$962$	12.7297	0.410422
$963$	−3.13019	−0.100869
$964$	38.0807	1.22650
$965$	0.942096	0.0303271
$966$	0.0802509	0.00258203
$967$	35.7837	1.15073	0.575364	−	0.817898i	$-0.304860\pi$
$967$	35.7837	1.15073	0.575364	+	0.817898i	$0.304860\pi$
$968$	5.19063	0.166833
$969$	6.39633	0.205480
$970$	−29.8278	−0.957713
$971$	−42.1535	−1.35277	−0.676384	−	0.736549i	$-0.736454\pi$
$971$	−42.1535	−1.35277	−0.676384	+	0.736549i	$0.736454\pi$
$972$	22.1078	0.709108
$973$	−28.8834	−0.925959
$974$	−80.8265	−2.58985
$975$	−9.67816	−0.309949
$976$	−35.5983	−1.13947
$977$	−22.2883	−0.713067	−0.356533	−	0.934283i	$-0.616041\pi$
$977$	−22.2883	−0.713067	−0.356533	+	0.934283i	$0.616041\pi$
$978$	−31.0364	−0.992436
$979$	−5.55567	−0.177560
$980$	−7.31060	−0.233529
$981$	−13.2004	−0.421456
$982$	30.0555	0.959109
$983$	−32.3603	−1.03213	−0.516067	−	0.856548i	$-0.672604\pi$
$983$	−32.3603	−1.03213	−0.516067	+	0.856548i	$0.672604\pi$
$984$	4.37051	0.139327
$985$	−28.2170	−0.899067
$986$	28.2498	0.899656
$987$	−12.0745	−0.384337
$988$	−4.37549	−0.139203
$989$	0.0516630	0.00164279
$990$	2.95340	0.0938651
$991$	48.0648	1.52683	0.763415	−	0.645909i	$-0.223521\pi$
$991$	48.0648	1.52683	0.763415	+	0.645909i	$0.223521\pi$
$992$	−68.0256	−2.15981
$993$	41.5200	1.31760
$994$	44.4891	1.41111
$995$	28.8422	0.914360
$996$	4.84585	0.153547
$997$	46.1847	1.46268	0.731342	−	0.682011i	$-0.238894\pi$
$997$	46.1847	1.46268	0.731342	+	0.682011i	$0.238894\pi$
$998$	86.8276	2.74848
$999$	18.0361	0.570637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.14	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.14	✓	75	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.06463 q^{2} +1.41935 q^{3} +2.26270 q^{4} +1.21401 q^{5} -2.93043 q^{6} +2.08294 q^{7} -0.542368 q^{8} -0.985454 q^{9} +O(q^{10})\)	\(q-2.06463 q^{2} +1.41935 q^{3} +2.26270 q^{4} +1.21401 q^{5} -2.93043 q^{6} +2.08294 q^{7} -0.542368 q^{8} -0.985454 q^{9} -2.50648 q^{10} +1.19570 q^{11} +3.21155 q^{12} +1.93375 q^{13} -4.30050 q^{14} +1.72310 q^{15} -3.40560 q^{16} -4.50653 q^{17} +2.03460 q^{18} -1.00000 q^{19} +2.74694 q^{20} +2.95641 q^{21} -2.46867 q^{22} -0.0131475 q^{23} -0.769808 q^{24} -3.52618 q^{25} -3.99248 q^{26} -5.65674 q^{27} +4.71306 q^{28} +3.03620 q^{29} -3.55757 q^{30} -8.38162 q^{31} +8.11604 q^{32} +1.69711 q^{33} +9.30431 q^{34} +2.52871 q^{35} -2.22978 q^{36} -3.18842 q^{37} +2.06463 q^{38} +2.74466 q^{39} -0.658441 q^{40} -5.67741 q^{41} -6.10390 q^{42} -3.92950 q^{43} +2.70549 q^{44} -1.19635 q^{45} +0.0271447 q^{46} -4.08419 q^{47} -4.83373 q^{48} -2.66136 q^{49} +7.28025 q^{50} -6.39633 q^{51} +4.37549 q^{52} -9.89964 q^{53} +11.6791 q^{54} +1.45159 q^{55} -1.12972 q^{56} -1.41935 q^{57} -6.26863 q^{58} -9.13096 q^{59} +3.89886 q^{60} +10.4529 q^{61} +17.3049 q^{62} -2.05264 q^{63} -9.94542 q^{64} +2.34760 q^{65} -3.50390 q^{66} +2.13659 q^{67} -10.1969 q^{68} -0.0186608 q^{69} -5.22085 q^{70} -10.3451 q^{71} +0.534479 q^{72} -10.6577 q^{73} +6.58291 q^{74} -5.00487 q^{75} -2.26270 q^{76} +2.49056 q^{77} -5.66671 q^{78} +2.14707 q^{79} -4.13444 q^{80} -5.07252 q^{81} +11.7217 q^{82} +1.50888 q^{83} +6.68946 q^{84} -5.47098 q^{85} +8.11295 q^{86} +4.30942 q^{87} -0.648507 q^{88} -4.64639 q^{89} +2.47002 q^{90} +4.02788 q^{91} -0.0297488 q^{92} -11.8964 q^{93} +8.43233 q^{94} -1.21401 q^{95} +11.5195 q^{96} +11.9003 q^{97} +5.49473 q^{98} -1.17830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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