LMFDB - Embedded newform 4009.2.a.e.1.12

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:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.09995 q^{2} -2.73690 q^{3} +2.40979 q^{4} +2.29135 q^{5} +5.74735 q^{6} -1.74327 q^{7} -0.860544 q^{8} +4.49062 q^{9} +O(q^{10})$	\(q-2.09995 q^{2} -2.73690 q^{3} +2.40979 q^{4} +2.29135 q^{5} +5.74735 q^{6} -1.74327 q^{7} -0.860544 q^{8} +4.49062 q^{9} -4.81171 q^{10} +2.05408 q^{11} -6.59536 q^{12} +0.301723 q^{13} +3.66079 q^{14} -6.27119 q^{15} -3.01249 q^{16} -4.31891 q^{17} -9.43008 q^{18} +1.00000 q^{19} +5.52167 q^{20} +4.77117 q^{21} -4.31346 q^{22} -3.85580 q^{23} +2.35522 q^{24} +0.250269 q^{25} -0.633603 q^{26} -4.07968 q^{27} -4.20093 q^{28} -4.33443 q^{29} +13.1692 q^{30} +8.81166 q^{31} +8.04716 q^{32} -5.62180 q^{33} +9.06949 q^{34} -3.99444 q^{35} +10.8215 q^{36} -7.93044 q^{37} -2.09995 q^{38} -0.825785 q^{39} -1.97180 q^{40} +12.5249 q^{41} -10.0192 q^{42} -0.430093 q^{43} +4.94990 q^{44} +10.2896 q^{45} +8.09699 q^{46} -0.109721 q^{47} +8.24487 q^{48} -3.96100 q^{49} -0.525552 q^{50} +11.8204 q^{51} +0.727089 q^{52} -5.26169 q^{53} +8.56713 q^{54} +4.70660 q^{55} +1.50016 q^{56} -2.73690 q^{57} +9.10208 q^{58} +5.61368 q^{59} -15.1123 q^{60} +3.59162 q^{61} -18.5040 q^{62} -7.82838 q^{63} -10.8737 q^{64} +0.691351 q^{65} +11.8055 q^{66} +13.9798 q^{67} -10.4077 q^{68} +10.5529 q^{69} +8.38813 q^{70} -12.7291 q^{71} -3.86438 q^{72} +0.490187 q^{73} +16.6535 q^{74} -0.684960 q^{75} +2.40979 q^{76} -3.58082 q^{77} +1.73411 q^{78} +13.3836 q^{79} -6.90265 q^{80} -2.30618 q^{81} -26.3016 q^{82} +0.974571 q^{83} +11.4975 q^{84} -9.89612 q^{85} +0.903175 q^{86} +11.8629 q^{87} -1.76762 q^{88} +11.7013 q^{89} -21.6076 q^{90} -0.525985 q^{91} -9.29168 q^{92} -24.1166 q^{93} +0.230410 q^{94} +2.29135 q^{95} -22.0243 q^{96} +0.170536 q^{97} +8.31790 q^{98} +9.22408 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−2.09995	−1.48489	−0.742445	−	0.669907i	$-0.766334\pi$
$2$	−2.09995	−1.48489	−0.742445	+	0.669907i	$0.766334\pi$
$3$	−2.73690	−1.58015	−0.790075	−	0.613010i	$-0.789958\pi$
$3$	−2.73690	−1.58015	−0.790075	+	0.613010i	$0.789958\pi$
$4$	2.40979	1.20490
$5$	2.29135	1.02472	0.512361	−	0.858770i	$-0.328771\pi$
$5$	2.29135	1.02472	0.512361	+	0.858770i	$0.328771\pi$
$6$	5.74735	2.34635
$7$	−1.74327	−0.658895	−0.329448	−	0.944174i	$-0.606863\pi$
$7$	−1.74327	−0.658895	−0.329448	+	0.944174i	$0.606863\pi$
$8$	−0.860544	−0.304248
$9$	4.49062	1.49687
$10$	−4.81171	−1.52160
$11$	2.05408	0.619328	0.309664	−	0.950846i	$-0.399783\pi$
$11$	2.05408	0.619328	0.309664	+	0.950846i	$0.399783\pi$
$12$	−6.59536	−1.90392
$13$	0.301723	0.0836828	0.0418414	−	0.999124i	$-0.486678\pi$
$13$	0.301723	0.0836828	0.0418414	+	0.999124i	$0.486678\pi$
$14$	3.66079	0.978387
$15$	−6.27119	−1.61921
$16$	−3.01249	−0.753121
$17$	−4.31891	−1.04749	−0.523745	−	0.851875i	$-0.675465\pi$
$17$	−4.31891	−1.04749	−0.523745	+	0.851875i	$0.675465\pi$
$18$	−9.43008	−2.22269
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	5.52167	1.23468
$21$	4.77117	1.04115
$22$	−4.31346	−0.919633
$23$	−3.85580	−0.803990	−0.401995	−	0.915642i	$-0.631683\pi$
$23$	−3.85580	−0.803990	−0.401995	+	0.915642i	$0.631683\pi$
$24$	2.35522	0.480758
$25$	0.250269	0.0500537
$26$	−0.633603	−0.124260
$27$	−4.07968	−0.785135
$28$	−4.20093	−0.793901
$29$	−4.33443	−0.804883	−0.402441	−	0.915446i	$-0.631838\pi$
$29$	−4.33443	−0.804883	−0.402441	+	0.915446i	$0.631838\pi$
$30$	13.1692	2.40435
$31$	8.81166	1.58262	0.791310	−	0.611415i	$-0.209399\pi$
$31$	8.81166	1.58262	0.791310	+	0.611415i	$0.209399\pi$
$32$	8.04716	1.42255
$33$	−5.62180	−0.978631
$34$	9.06949	1.55541
$35$	−3.99444	−0.675184
$36$	10.8215	1.80358
$37$	−7.93044	−1.30376	−0.651878	−	0.758324i	$-0.726019\pi$
$37$	−7.93044	−1.30376	−0.651878	+	0.758324i	$0.726019\pi$
$38$	−2.09995	−0.340657
$39$	−0.825785	−0.132231
$40$	−1.97180	−0.311770
$41$	12.5249	1.95606	0.978028	−	0.208476i	$-0.0668503\pi$
$41$	12.5249	1.95606	0.978028	+	0.208476i	$0.0668503\pi$
$42$	−10.0192	−1.54600
$43$	−0.430093	−0.0655886	−0.0327943	−	0.999462i	$-0.510441\pi$
$43$	−0.430093	−0.0655886	−0.0327943	+	0.999462i	$0.510441\pi$
$44$	4.94990	0.746226
$45$	10.2896	1.53388
$46$	8.09699	1.19384
$47$	−0.109721	−0.0160045	−0.00800226	−	0.999968i	$-0.502547\pi$
$47$	−0.109721	−0.0160045	−0.00800226	+	0.999968i	$0.502547\pi$
$48$	8.24487	1.19004
$49$	−3.96100	−0.565857
$50$	−0.525552	−0.0743242
$51$	11.8204	1.65519
$52$	0.727089	0.100829
$53$	−5.26169	−0.722749	−0.361375	−	0.932421i	$-0.617692\pi$
$53$	−5.26169	−0.722749	−0.361375	+	0.932421i	$0.617692\pi$
$54$	8.56713	1.16584
$55$	4.70660	0.634638
$56$	1.50016	0.200468
$57$	−2.73690	−0.362511
$58$	9.10208	1.19516
$59$	5.61368	0.730838	0.365419	−	0.930843i	$-0.380926\pi$
$59$	5.61368	0.730838	0.365419	+	0.930843i	$0.380926\pi$
$60$	−15.1123	−1.95098
$61$	3.59162	0.459860	0.229930	−	0.973207i	$-0.426150\pi$
$61$	3.59162	0.459860	0.229930	+	0.973207i	$0.426150\pi$
$62$	−18.5040	−2.35002
$63$	−7.82838	−0.986283
$64$	−10.8737	−1.35921
$65$	0.691351	0.0857515
$66$	11.8055	1.45316
$67$	13.9798	1.70790	0.853950	−	0.520356i	$-0.174201\pi$
$67$	13.9798	1.70790	0.853950	+	0.520356i	$0.174201\pi$
$68$	−10.4077	−1.26212
$69$	10.5529	1.27042
$70$	8.38813	1.00257
$71$	−12.7291	−1.51066	−0.755332	−	0.655342i	$-0.772525\pi$
$71$	−12.7291	−1.51066	−0.755332	+	0.655342i	$0.772525\pi$
$72$	−3.86438	−0.455421
$73$	0.490187	0.0573721	0.0286860	−	0.999588i	$-0.490868\pi$
$73$	0.490187	0.0573721	0.0286860	+	0.999588i	$0.490868\pi$
$74$	16.6535	1.93593
$75$	−0.684960	−0.0790924
$76$	2.40979	0.276422
$77$	−3.58082	−0.408072
$78$	1.73411	0.196349
$79$	13.3836	1.50577	0.752887	−	0.658149i	$-0.228661\pi$
$79$	13.3836	1.50577	0.752887	+	0.658149i	$0.228661\pi$
$80$	−6.90265	−0.771739
$81$	−2.30618	−0.256242
$82$	−26.3016	−2.90453
$83$	0.974571	0.106973	0.0534865	−	0.998569i	$-0.482967\pi$
$83$	0.974571	0.106973	0.0534865	+	0.998569i	$0.482967\pi$
$84$	11.4975	1.25448
$85$	−9.89612	−1.07338
$86$	0.903175	0.0973918
$87$	11.8629	1.27184
$88$	−1.76762	−0.188429
$89$	11.7013	1.24034	0.620170	−	0.784468i	$-0.287064\pi$
$89$	11.7013	1.24034	0.620170	+	0.784468i	$0.287064\pi$
$90$	−21.6076	−2.27764
$91$	−0.525985	−0.0551382
$92$	−9.29168	−0.968724
$93$	−24.1166	−2.50078
$94$	0.230410	0.0237649
$95$	2.29135	0.235087
$96$	−22.0243	−2.24784
$97$	0.170536	0.0173153	0.00865763	−	0.999963i	$-0.497244\pi$
$97$	0.170536	0.0173153	0.00865763	+	0.999963i	$0.497244\pi$
$98$	8.31790	0.840235
$99$	9.22408	0.927055
$100$	0.603095	0.0603095
$101$	−16.0116	−1.59322	−0.796609	−	0.604495i	$-0.793375\pi$
$101$	−16.0116	−1.59322	−0.796609	+	0.604495i	$0.793375\pi$
$102$	−24.8223	−2.45777
$103$	6.88896	0.678790	0.339395	−	0.940644i	$-0.389778\pi$
$103$	6.88896	0.678790	0.339395	+	0.940644i	$0.389778\pi$
$104$	−0.259646	−0.0254603
$105$	10.9324	1.06689
$106$	11.0493	1.07320
$107$	−0.976265	−0.0943791	−0.0471896	−	0.998886i	$-0.515026\pi$
$107$	−0.976265	−0.0943791	−0.0471896	+	0.998886i	$0.515026\pi$
$108$	−9.83119	−0.946007
$109$	−7.46780	−0.715286	−0.357643	−	0.933858i	$-0.616420\pi$
$109$	−7.46780	−0.715286	−0.357643	+	0.933858i	$0.616420\pi$
$110$	−9.88363	−0.942367
$111$	21.7048	2.06013
$112$	5.25159	0.496228
$113$	0.916926	0.0862571	0.0431286	−	0.999070i	$-0.486267\pi$
$113$	0.916926	0.0862571	0.0431286	+	0.999070i	$0.486267\pi$
$114$	5.74735	0.538289
$115$	−8.83497	−0.823865
$116$	−10.4451	−0.969800
$117$	1.35492	0.125263
$118$	−11.7884	−1.08521
$119$	7.52904	0.690186
$120$	5.39663	0.492643
$121$	−6.78077	−0.616433
$122$	−7.54222	−0.682841
$123$	−34.2793	−3.09086
$124$	21.2343	1.90689
$125$	−10.8833	−0.973430
$126$	16.4392	1.46452
$127$	−4.08483	−0.362470	−0.181235	−	0.983440i	$-0.558010\pi$
$127$	−4.08483	−0.362470	−0.181235	+	0.983440i	$0.558010\pi$
$128$	6.73984	0.595723
$129$	1.17712	0.103640
$130$	−1.45180	−0.127332
$131$	−5.06579	−0.442600	−0.221300	−	0.975206i	$-0.571030\pi$
$131$	−5.06579	−0.442600	−0.221300	+	0.975206i	$0.571030\pi$
$132$	−13.5474	−1.17915
$133$	−1.74327	−0.151161
$134$	−29.3568	−2.53604
$135$	−9.34797	−0.804545
$136$	3.71661	0.318697
$137$	8.73847	0.746578	0.373289	−	0.927715i	$-0.378230\pi$
$137$	8.73847	0.746578	0.373289	+	0.927715i	$0.378230\pi$
$138$	−22.1606	−1.88644
$139$	−5.11193	−0.433588	−0.216794	−	0.976217i	$-0.569560\pi$
$139$	−5.11193	−0.433588	−0.216794	+	0.976217i	$0.569560\pi$
$140$	−9.62578	−0.813527
$141$	0.300297	0.0252895
$142$	26.7305	2.24317
$143$	0.619762	0.0518271
$144$	−13.5279	−1.12733
$145$	−9.93167	−0.824780
$146$	−1.02937	−0.0851912
$147$	10.8409	0.894139
$148$	−19.1107	−1.57089
$149$	5.88070	0.481765	0.240883	−	0.970554i	$-0.422563\pi$
$149$	5.88070	0.481765	0.240883	+	0.970554i	$0.422563\pi$
$150$	1.43838	0.117443
$151$	5.62225	0.457532	0.228766	−	0.973481i	$-0.426531\pi$
$151$	5.62225	0.457532	0.228766	+	0.973481i	$0.426531\pi$
$152$	−0.860544	−0.0697993
$153$	−19.3946	−1.56796
$154$	7.51954	0.605942
$155$	20.1906	1.62175
$156$	−1.98997	−0.159325
$157$	4.84794	0.386907	0.193454	−	0.981109i	$-0.438031\pi$
$157$	4.84794	0.386907	0.193454	+	0.981109i	$0.438031\pi$
$158$	−28.1049	−2.23591
$159$	14.4007	1.14205
$160$	18.4388	1.45772
$161$	6.72171	0.529745
$162$	4.84287	0.380491
$163$	−9.77552	−0.765677	−0.382839	−	0.923815i	$-0.625054\pi$
$163$	−9.77552	−0.765677	−0.382839	+	0.923815i	$0.625054\pi$
$164$	30.1823	2.35684
$165$	−12.8815	−1.00282
$166$	−2.04655	−0.158843
$167$	5.04942	0.390736	0.195368	−	0.980730i	$-0.437410\pi$
$167$	5.04942	0.390736	0.195368	+	0.980730i	$0.437410\pi$
$168$	−4.10580	−0.316769
$169$	−12.9090	−0.992997
$170$	20.7814	1.59386
$171$	4.49062	0.343406
$172$	−1.03644	−0.0790275
$173$	−10.9298	−0.830977	−0.415488	−	0.909599i	$-0.636389\pi$
$173$	−10.9298	−0.830977	−0.415488	+	0.909599i	$0.636389\pi$
$174$	−24.9115	−1.88853
$175$	−0.436287	−0.0329802
$176$	−6.18788	−0.466429
$177$	−15.3641	−1.15483
$178$	−24.5722	−1.84177
$179$	−2.00851	−0.150123	−0.0750617	−	0.997179i	$-0.523915\pi$
$179$	−2.00851	−0.150123	−0.0750617	+	0.997179i	$0.523915\pi$
$180$	24.7957	1.84816
$181$	−4.30148	−0.319727	−0.159863	−	0.987139i	$-0.551105\pi$
$181$	−4.30148	−0.319727	−0.159863	+	0.987139i	$0.551105\pi$
$182$	1.10454	0.0818741
$183$	−9.82990	−0.726647
$184$	3.31808	0.244612
$185$	−18.1714	−1.33599
$186$	50.6437	3.71338
$187$	−8.87137	−0.648739
$188$	−0.264406	−0.0192838
$189$	7.11200	0.517322
$190$	−4.81171	−0.349078
$191$	−2.91517	−0.210934	−0.105467	−	0.994423i	$-0.533634\pi$
$191$	−2.91517	−0.210934	−0.105467	+	0.994423i	$0.533634\pi$
$192$	29.7601	2.14775
$193$	−22.0623	−1.58808	−0.794038	−	0.607868i	$-0.792025\pi$
$193$	−22.0623	−1.58808	−0.794038	+	0.607868i	$0.792025\pi$
$194$	−0.358116	−0.0257112
$195$	−1.89216	−0.135500
$196$	−9.54518	−0.681799
$197$	18.7967	1.33921	0.669605	−	0.742718i	$-0.266464\pi$
$197$	18.7967	1.33921	0.669605	+	0.742718i	$0.266464\pi$
$198$	−19.3701	−1.37657
$199$	22.0561	1.56352	0.781759	−	0.623580i	$-0.214323\pi$
$199$	22.0561	1.56352	0.781759	+	0.623580i	$0.214323\pi$
$200$	−0.215367	−0.0152288
$201$	−38.2612	−2.69874
$202$	33.6237	2.36575
$203$	7.55609	0.530333
$204$	28.4848	1.99433
$205$	28.6988	2.00441
$206$	−14.4665	−1.00793
$207$	−17.3149	−1.20347
$208$	−0.908935	−0.0630233
$209$	2.05408	0.142084
$210$	−22.9575	−1.58422
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−12.6796	−0.870838
$213$	34.8382	2.38708
$214$	2.05011	0.140143
$215$	−0.985493	−0.0672101
$216$	3.51075	0.238876
$217$	−15.3611	−1.04278
$218$	15.6820	1.06212
$219$	−1.34159	−0.0906565
$220$	11.3419	0.764673
$221$	−1.30311	−0.0876568
$222$	−45.5790	−3.05907
$223$	−24.2819	−1.62604	−0.813018	−	0.582239i	$-0.802177\pi$
$223$	−24.2819	−1.62604	−0.813018	+	0.582239i	$0.802177\pi$
$224$	−14.0284	−0.937312
$225$	1.12386	0.0749241
$226$	−1.92550	−0.128082
$227$	28.4364	1.88739	0.943695	−	0.330817i	$-0.107324\pi$
$227$	28.4364	1.88739	0.943695	+	0.330817i	$0.107324\pi$
$228$	−6.59536	−0.436788
$229$	12.5338	0.828255	0.414128	−	0.910219i	$-0.364087\pi$
$229$	12.5338	0.828255	0.414128	+	0.910219i	$0.364087\pi$
$230$	18.5530	1.22335
$231$	9.80034	0.644815
$232$	3.72996	0.244884
$233$	−5.25178	−0.344056	−0.172028	−	0.985092i	$-0.555032\pi$
$233$	−5.25178	−0.344056	−0.172028	+	0.985092i	$0.555032\pi$
$234$	−2.84527	−0.186001
$235$	−0.251410	−0.0164002
$236$	13.5278	0.880584
$237$	−36.6296	−2.37935
$238$	−15.8106	−1.02485
$239$	7.50283	0.485317	0.242659	−	0.970112i	$-0.421980\pi$
$239$	7.50283	0.485317	0.242659	+	0.970112i	$0.421980\pi$
$240$	18.8919	1.21946
$241$	0.417180	0.0268729	0.0134364	−	0.999910i	$-0.495723\pi$
$241$	0.417180	0.0268729	0.0134364	+	0.999910i	$0.495723\pi$
$242$	14.2393	0.915335
$243$	18.5508	1.19004
$244$	8.65506	0.554083
$245$	−9.07602	−0.579845
$246$	71.9848	4.58959
$247$	0.301723	0.0191982
$248$	−7.58282	−0.481509
$249$	−2.66730	−0.169033
$250$	22.8544	1.44544
$251$	18.7789	1.18531	0.592656	−	0.805456i	$-0.298079\pi$
$251$	18.7789	1.18531	0.592656	+	0.805456i	$0.298079\pi$
$252$	−18.8648	−1.18837
$253$	−7.92011	−0.497933
$254$	8.57795	0.538228
$255$	27.0847	1.69611
$256$	7.59400	0.474625
$257$	13.5438	0.844842	0.422421	−	0.906400i	$-0.361180\pi$
$257$	13.5438	0.844842	0.422421	+	0.906400i	$0.361180\pi$
$258$	−2.47190	−0.153894
$259$	13.8249	0.859039
$260$	1.66601	0.103322
$261$	−19.4643	−1.20481
$262$	10.6379	0.657212
$263$	−0.122115	−0.00752994	−0.00376497	−	0.999993i	$-0.501198\pi$
$263$	−0.122115	−0.00752994	−0.00376497	+	0.999993i	$0.501198\pi$
$264$	4.83781	0.297747
$265$	−12.0564	−0.740617
$266$	3.66079	0.224457
$267$	−32.0254	−1.95992
$268$	33.6883	2.05784
$269$	1.06494	0.0649307	0.0324653	−	0.999473i	$-0.489664\pi$
$269$	1.06494	0.0649307	0.0324653	+	0.999473i	$0.489664\pi$
$270$	19.6303	1.19466
$271$	−19.4468	−1.18131	−0.590655	−	0.806924i	$-0.701131\pi$
$271$	−19.4468	−1.18131	−0.590655	+	0.806924i	$0.701131\pi$
$272$	13.0106	0.788886
$273$	1.43957	0.0871267
$274$	−18.3504	−1.10859
$275$	0.514071	0.0309997
$276$	25.4304	1.53073
$277$	12.2422	0.735563	0.367782	−	0.929912i	$-0.380117\pi$
$277$	12.2422	0.735563	0.367782	+	0.929912i	$0.380117\pi$
$278$	10.7348	0.643830
$279$	39.5698	2.36898
$280$	3.43739	0.205424
$281$	−8.36507	−0.499018	−0.249509	−	0.968372i	$-0.580269\pi$
$281$	−8.36507	−0.499018	−0.249509	+	0.968372i	$0.580269\pi$
$282$	−0.630608	−0.0375522
$283$	23.9529	1.42385	0.711925	−	0.702255i	$-0.247823\pi$
$283$	23.9529	1.42385	0.711925	+	0.702255i	$0.247823\pi$
$284$	−30.6745	−1.82019
$285$	−6.27119	−0.371473
$286$	−1.30147	−0.0769575
$287$	−21.8343	−1.28884
$288$	36.1367	2.12938
$289$	1.65297	0.0972335
$290$	20.8560	1.22471
$291$	−0.466739	−0.0273607
$292$	1.18125	0.0691274
$293$	23.0568	1.34699	0.673496	−	0.739191i	$-0.264792\pi$
$293$	23.0568	1.34699	0.673496	+	0.739191i	$0.264792\pi$
$294$	−22.7653	−1.32770
$295$	12.8629	0.748906
$296$	6.82449	0.396665
$297$	−8.37998	−0.486256
$298$	−12.3492	−0.715368
$299$	−1.16338	−0.0672801
$300$	−1.65061	−0.0952981
$301$	0.749770	0.0432160
$302$	−11.8064	−0.679385
$303$	43.8223	2.51752
$304$	−3.01249	−0.172778
$305$	8.22964	0.471228
$306$	40.7277	2.32825
$307$	−30.4658	−1.73877	−0.869387	−	0.494132i	$-0.835486\pi$
$307$	−30.4658	−1.73877	−0.869387	+	0.494132i	$0.835486\pi$
$308$	−8.62903	−0.491685
$309$	−18.8544	−1.07259
$310$	−42.3992	−2.40811
$311$	−4.36477	−0.247503	−0.123752	−	0.992313i	$-0.539493\pi$
$311$	−4.36477	−0.247503	−0.123752	+	0.992313i	$0.539493\pi$
$312$	0.710624	0.0402312
$313$	−16.5523	−0.935590	−0.467795	−	0.883837i	$-0.654952\pi$
$313$	−16.5523	−0.935590	−0.467795	+	0.883837i	$0.654952\pi$
$314$	−10.1804	−0.574515
$315$	−17.9375	−1.01067
$316$	32.2517	1.81430
$317$	10.5903	0.594809	0.297404	−	0.954752i	$-0.403879\pi$
$317$	10.5903	0.594809	0.297404	+	0.954752i	$0.403879\pi$
$318$	−30.2408	−1.69582
$319$	−8.90324	−0.498486
$320$	−24.9153	−1.39281
$321$	2.67194	0.149133
$322$	−14.1153	−0.786613
$323$	−4.31891	−0.240310
$324$	−5.55742	−0.308745
$325$	0.0755117	0.00418864
$326$	20.5281	1.13695
$327$	20.4386	1.13026
$328$	−10.7782	−0.595126
$329$	0.191274	0.0105453
$330$	27.0505	1.48908
$331$	30.5391	1.67858	0.839291	−	0.543683i	$-0.182971\pi$
$331$	30.5391	1.67858	0.839291	+	0.543683i	$0.182971\pi$
$332$	2.34851	0.128891
$333$	−35.6126	−1.95156
$334$	−10.6035	−0.580199
$335$	32.0325	1.75012
$336$	−14.3731	−0.784115
$337$	−16.0430	−0.873918	−0.436959	−	0.899481i	$-0.643945\pi$
$337$	−16.0430	−0.873918	−0.436959	+	0.899481i	$0.643945\pi$
$338$	27.1082	1.47449
$339$	−2.50953	−0.136299
$340$	−23.8476	−1.29332
$341$	18.0998	0.980161
$342$	−9.43008	−0.509921
$343$	19.1080	1.03174
$344$	0.370114	0.0199552
$345$	24.1804	1.30183
$346$	22.9520	1.23391
$347$	−24.9698	−1.34045	−0.670224	−	0.742159i	$-0.733802\pi$
$347$	−24.9698	−1.34045	−0.670224	+	0.742159i	$0.733802\pi$
$348$	28.5871	1.53243
$349$	4.91088	0.262873	0.131437	−	0.991325i	$-0.458041\pi$
$349$	4.91088	0.262873	0.131437	+	0.991325i	$0.458041\pi$
$350$	0.916180	0.0489719
$351$	−1.23093	−0.0657023
$352$	16.5295	0.881025
$353$	−1.24366	−0.0661931	−0.0330965	−	0.999452i	$-0.510537\pi$
$353$	−1.24366	−0.0661931	−0.0330965	+	0.999452i	$0.510537\pi$
$354$	32.2638	1.71480
$355$	−29.1667	−1.54801
$356$	28.1978	1.49448
$357$	−20.6062	−1.09060
$358$	4.21778	0.222917
$359$	22.7777	1.20216	0.601082	−	0.799187i	$-0.294737\pi$
$359$	22.7777	1.20216	0.601082	+	0.799187i	$0.294737\pi$
$360$	−8.85463	−0.466680
$361$	1.00000	0.0526316
$362$	9.03290	0.474759
$363$	18.5583	0.974057
$364$	−1.26751	−0.0664358
$365$	1.12319	0.0587904
$366$	20.6423	1.07899
$367$	26.4990	1.38324	0.691619	−	0.722263i	$-0.256898\pi$
$367$	26.4990	1.38324	0.691619	+	0.722263i	$0.256898\pi$
$368$	11.6155	0.605502
$369$	56.2444	2.92797
$370$	38.1590	1.98379
$371$	9.17257	0.476216
$372$	−58.1161	−3.01318
$373$	18.5527	0.960624	0.480312	−	0.877098i	$-0.340523\pi$
$373$	18.5527	0.960624	0.480312	+	0.877098i	$0.340523\pi$
$374$	18.6294	0.963306
$375$	29.7865	1.53817
$376$	0.0944201	0.00486934
$377$	−1.30779	−0.0673548
$378$	−14.9349	−0.768166
$379$	−32.3687	−1.66267	−0.831335	−	0.555771i	$-0.812423\pi$
$379$	−32.3687	−1.66267	−0.831335	+	0.555771i	$0.812423\pi$
$380$	5.52167	0.283256
$381$	11.1798	0.572758
$382$	6.12170	0.313214
$383$	−17.3178	−0.884899	−0.442449	−	0.896793i	$-0.645890\pi$
$383$	−17.3178	−0.884899	−0.442449	+	0.896793i	$0.645890\pi$
$384$	−18.4463	−0.941332
$385$	−8.20490	−0.418160
$386$	46.3297	2.35812
$387$	−1.93139	−0.0981779
$388$	0.410955	0.0208631
$389$	27.0382	1.37089	0.685447	−	0.728123i	$-0.259607\pi$
$389$	27.0382	1.37089	0.685447	+	0.728123i	$0.259607\pi$
$390$	3.97344	0.201203
$391$	16.6528	0.842171
$392$	3.40861	0.172161
$393$	13.8646	0.699374
$394$	−39.4721	−1.98858
$395$	30.6665	1.54300
$396$	22.2281	1.11701
$397$	6.51197	0.326826	0.163413	−	0.986558i	$-0.447750\pi$
$397$	6.51197	0.326826	0.163413	+	0.986558i	$0.447750\pi$
$398$	−46.3168	−2.32165
$399$	4.77117	0.238857
$400$	−0.753931	−0.0376965
$401$	38.5785	1.92652	0.963259	−	0.268576i	$-0.0865530\pi$
$401$	38.5785	1.92652	0.963259	+	0.268576i	$0.0865530\pi$
$402$	80.3466	4.00733
$403$	2.65868	0.132438
$404$	−38.5847	−1.91966
$405$	−5.28426	−0.262577
$406$	−15.8674	−0.787486
$407$	−16.2897	−0.807452
$408$	−10.1720	−0.503588
$409$	5.46553	0.270253	0.135126	−	0.990828i	$-0.456856\pi$
$409$	5.46553	0.270253	0.135126	+	0.990828i	$0.456856\pi$
$410$	−60.2661	−2.97633
$411$	−23.9163	−1.17971
$412$	16.6010	0.817871
$413$	−9.78617	−0.481546
$414$	36.3605	1.78702
$415$	2.23308	0.109618
$416$	2.42801	0.119043
$417$	13.9908	0.685134
$418$	−4.31346	−0.210978
$419$	29.1462	1.42389	0.711943	−	0.702237i	$-0.247815\pi$
$419$	29.1462	1.42389	0.711943	+	0.702237i	$0.247815\pi$
$420$	26.3448	1.28549
$421$	−23.5077	−1.14570	−0.572848	−	0.819661i	$-0.694162\pi$
$421$	−23.5077	−1.14570	−0.572848	+	0.819661i	$0.694162\pi$
$422$	2.09995	0.102224
$423$	−0.492717	−0.0239567
$424$	4.52792	0.219895
$425$	−1.08089	−0.0524307
$426$	−73.1586	−3.54454
$427$	−6.26117	−0.302999
$428$	−2.35260	−0.113717
$429$	−1.69623	−0.0818945
$430$	2.06949	0.0997995
$431$	−22.8544	−1.10086	−0.550429	−	0.834882i	$-0.685536\pi$
$431$	−22.8544	−1.10086	−0.550429	+	0.834882i	$0.685536\pi$
$432$	12.2900	0.591302
$433$	0.506625	0.0243468	0.0121734	−	0.999926i	$-0.496125\pi$
$433$	0.506625	0.0243468	0.0121734	+	0.999926i	$0.496125\pi$
$434$	32.2576	1.54842
$435$	27.1820	1.30328
$436$	−17.9959	−0.861845
$437$	−3.85580	−0.184448
$438$	2.81728	0.134615
$439$	−6.49705	−0.310087	−0.155044	−	0.987908i	$-0.549552\pi$
$439$	−6.49705	−0.310087	−0.155044	+	0.987908i	$0.549552\pi$
$440$	−4.05024	−0.193088
$441$	−17.7873	−0.847016
$442$	2.73647	0.130161
$443$	36.6055	1.73918	0.869591	−	0.493773i	$-0.164383\pi$
$443$	36.6055	1.73918	0.869591	+	0.493773i	$0.164383\pi$
$444$	52.3041	2.48224
$445$	26.8118	1.27100
$446$	50.9908	2.41448
$447$	−16.0949	−0.761262
$448$	18.9558	0.895576
$449$	14.4062	0.679870	0.339935	−	0.940449i	$-0.389595\pi$
$449$	14.4062	0.679870	0.339935	+	0.940449i	$0.389595\pi$
$450$	−2.36005	−0.111254
$451$	25.7270	1.21144
$452$	2.20960	0.103931
$453$	−15.3875	−0.722970
$454$	−59.7150	−2.80257
$455$	−1.20521	−0.0565013
$456$	2.35522	0.110293
$457$	−28.6659	−1.34094	−0.670468	−	0.741939i	$-0.733907\pi$
$457$	−28.6659	−1.34094	−0.670468	+	0.741939i	$0.733907\pi$
$458$	−26.3203	−1.22987
$459$	17.6198	0.822421
$460$	−21.2904	−0.992672
$461$	29.1025	1.35544	0.677720	−	0.735320i	$-0.262968\pi$
$461$	29.1025	1.35544	0.677720	+	0.735320i	$0.262968\pi$
$462$	−20.5802	−0.957479
$463$	14.3711	0.667882	0.333941	−	0.942594i	$-0.391621\pi$
$463$	14.3711	0.667882	0.333941	+	0.942594i	$0.391621\pi$
$464$	13.0574	0.606174
$465$	−55.2596	−2.56260
$466$	11.0285	0.510885
$467$	22.6674	1.04892	0.524461	−	0.851434i	$-0.324267\pi$
$467$	22.6674	1.04892	0.524461	+	0.851434i	$0.324267\pi$
$468$	3.26508	0.150928
$469$	−24.3705	−1.12533
$470$	0.527948	0.0243524
$471$	−13.2683	−0.611372
$472$	−4.83081	−0.222356
$473$	−0.883445	−0.0406208
$474$	76.9204	3.53307
$475$	0.250269	0.0114831
$476$	18.1434	0.831602
$477$	−23.6283	−1.08186
$478$	−15.7556	−0.720643
$479$	6.92802	0.316549	0.158275	−	0.987395i	$-0.449407\pi$
$479$	6.92802	0.316549	0.158275	+	0.987395i	$0.449407\pi$
$480$	−50.4652	−2.30341
$481$	−2.39279	−0.109102
$482$	−0.876056	−0.0399033
$483$	−18.3967	−0.837077
$484$	−16.3402	−0.742738
$485$	0.390756	0.0177433
$486$	−38.9558	−1.76707
$487$	24.7907	1.12337	0.561686	−	0.827350i	$-0.310153\pi$
$487$	24.7907	1.12337	0.561686	+	0.827350i	$0.310153\pi$
$488$	−3.09074	−0.139911
$489$	26.7546	1.20989
$490$	19.0592	0.861006
$491$	−25.0606	−1.13097	−0.565484	−	0.824759i	$-0.691311\pi$
$491$	−25.0606	−1.13097	−0.565484	+	0.824759i	$0.691311\pi$
$492$	−82.6060	−3.72417
$493$	18.7200	0.843106
$494$	−0.633603	−0.0285071
$495$	21.1356	0.949973
$496$	−26.5450	−1.19191
$497$	22.1903	0.995370
$498$	5.60121	0.250996
$499$	28.0232	1.25449	0.627246	−	0.778821i	$-0.284182\pi$
$499$	28.0232	1.25449	0.627246	+	0.778821i	$0.284182\pi$
$500$	−26.2264	−1.17288
$501$	−13.8198	−0.617421
$502$	−39.4347	−1.76006
$503$	−4.96246	−0.221265	−0.110633	−	0.993861i	$-0.535288\pi$
$503$	−4.96246	−0.221265	−0.110633	+	0.993861i	$0.535288\pi$
$504$	6.73667	0.300075
$505$	−36.6882	−1.63260
$506$	16.6318	0.739375
$507$	35.3305	1.56908
$508$	−9.84360	−0.436739
$509$	8.34399	0.369841	0.184920	−	0.982754i	$-0.440797\pi$
$509$	8.34399	0.369841	0.184920	+	0.982754i	$0.440797\pi$
$510$	−56.8765	−2.51853
$511$	−0.854530	−0.0378022
$512$	−29.4267	−1.30049
$513$	−4.07968	−0.180122
$514$	−28.4414	−1.25450
$515$	15.7850	0.695570
$516$	2.83662	0.124875
$517$	−0.225376	−0.00991204
$518$	−29.0317	−1.27558
$519$	29.9137	1.31307
$520$	−0.594938	−0.0260897
$521$	13.6639	0.598625	0.299312	−	0.954155i	$-0.403243\pi$
$521$	13.6639	0.598625	0.299312	+	0.954155i	$0.403243\pi$
$522$	40.8740	1.78901
$523$	35.0282	1.53168	0.765838	−	0.643034i	$-0.222325\pi$
$523$	35.0282	1.53168	0.765838	+	0.643034i	$0.222325\pi$
$524$	−12.2075	−0.533287
$525$	1.19407	0.0521136
$526$	0.256436	0.0111811
$527$	−38.0568	−1.65778
$528$	16.9356	0.737028
$529$	−8.13281	−0.353601
$530$	25.3178	1.09973
$531$	25.2089	1.09397
$532$	−4.20093	−0.182133
$533$	3.77903	0.163688
$534$	67.2517	2.91027
$535$	−2.23696	−0.0967123
$536$	−12.0302	−0.519625
$537$	5.49710	0.237217
$538$	−2.23633	−0.0964149
$539$	−8.13620	−0.350451
$540$	−22.5267	−0.969393
$541$	32.9832	1.41806	0.709030	−	0.705179i	$-0.249133\pi$
$541$	32.9832	1.41806	0.709030	+	0.705179i	$0.249133\pi$
$542$	40.8374	1.75412
$543$	11.7727	0.505216
$544$	−34.7549	−1.49011
$545$	−17.1113	−0.732969
$546$	−3.02302	−0.129373
$547$	42.0507	1.79796	0.898978	−	0.437993i	$-0.144311\pi$
$547$	42.0507	1.79796	0.898978	+	0.437993i	$0.144311\pi$
$548$	21.0579	0.899549
$549$	16.1286	0.688352
$550$	−1.07952	−0.0460311
$551$	−4.33443	−0.184653
$552$	−9.08126	−0.386524
$553$	−23.3313	−0.992148
$554$	−25.7080	−1.09223
$555$	49.7333	2.11106
$556$	−12.3187	−0.522428
$557$	2.12434	0.0900114	0.0450057	−	0.998987i	$-0.485669\pi$
$557$	2.12434	0.0900114	0.0450057	+	0.998987i	$0.485669\pi$
$558$	−83.0947	−3.51768
$559$	−0.129769	−0.00548864
$560$	12.0332	0.508496
$561$	24.2801	1.02510
$562$	17.5662	0.740987
$563$	0.678022	0.0285752	0.0142876	−	0.999898i	$-0.495452\pi$
$563$	0.678022	0.0285752	0.0142876	+	0.999898i	$0.495452\pi$
$564$	0.723652	0.0304713
$565$	2.10099	0.0883895
$566$	−50.2999	−2.11426
$567$	4.02030	0.168837
$568$	10.9539	0.459617
$569$	6.97967	0.292603	0.146301	−	0.989240i	$-0.453263\pi$
$569$	6.97967	0.292603	0.146301	+	0.989240i	$0.453263\pi$
$570$	13.1692	0.551596
$571$	−5.88624	−0.246331	−0.123166	−	0.992386i	$-0.539305\pi$
$571$	−5.88624	−0.246331	−0.123166	+	0.992386i	$0.539305\pi$
$572$	1.49350	0.0624462
$573$	7.97852	0.333307
$574$	45.8509	1.91378
$575$	−0.964985	−0.0402427
$576$	−48.8295	−2.03456
$577$	18.0139	0.749928	0.374964	−	0.927039i	$-0.377655\pi$
$577$	18.0139	0.749928	0.374964	+	0.927039i	$0.377655\pi$
$578$	−3.47116	−0.144381
$579$	60.3822	2.50940
$580$	−23.9333	−0.993775
$581$	−1.69894	−0.0704841
$582$	0.980128	0.0406276
$583$	−10.8079	−0.447619
$584$	−0.421827	−0.0174553
$585$	3.10460	0.128359
$586$	−48.4181	−2.00013
$587$	11.4059	0.470770	0.235385	−	0.971902i	$-0.424365\pi$
$587$	11.4059	0.470770	0.235385	+	0.971902i	$0.424365\pi$
$588$	26.1242	1.07734
$589$	8.81166	0.363078
$590$	−27.0114	−1.11204
$591$	−51.4447	−2.11615
$592$	23.8903	0.981887
$593$	10.4267	0.428173	0.214086	−	0.976815i	$-0.431323\pi$
$593$	10.4267	0.428173	0.214086	+	0.976815i	$0.431323\pi$
$594$	17.5976	0.722036
$595$	17.2516	0.707248
$596$	14.1713	0.580477
$597$	−60.3655	−2.47059
$598$	2.44304	0.0999035
$599$	21.9003	0.894820	0.447410	−	0.894329i	$-0.352346\pi$
$599$	21.9003	0.894820	0.447410	+	0.894329i	$0.352346\pi$
$600$	0.589438	0.0240637
$601$	18.5986	0.758654	0.379327	−	0.925263i	$-0.376156\pi$
$601$	18.5986	0.758654	0.379327	+	0.925263i	$0.376156\pi$
$602$	−1.57448	−0.0641710
$603$	62.7778	2.55651
$604$	13.5485	0.551279
$605$	−15.5371	−0.631672
$606$	−92.0246	−3.73824
$607$	26.8838	1.09118	0.545590	−	0.838052i	$-0.316306\pi$
$607$	26.8838	1.09118	0.545590	+	0.838052i	$0.316306\pi$
$608$	8.04716	0.326355
$609$	−20.6803	−0.838006
$610$	−17.2818	−0.699721
$611$	−0.0331054	−0.00133930
$612$	−46.7369	−1.88923
$613$	−24.2213	−0.978287	−0.489144	−	0.872203i	$-0.662691\pi$
$613$	−24.2213	−0.978287	−0.489144	+	0.872203i	$0.662691\pi$
$614$	63.9766	2.58189
$615$	−78.5457	−3.16727
$616$	3.08145	0.124155
$617$	−35.6384	−1.43475	−0.717375	−	0.696687i	$-0.754656\pi$
$617$	−35.6384	−1.43475	−0.717375	+	0.696687i	$0.754656\pi$
$618$	39.5933	1.59268
$619$	−18.6952	−0.751422	−0.375711	−	0.926737i	$-0.622601\pi$
$619$	−18.6952	−0.751422	−0.375711	+	0.926737i	$0.622601\pi$
$620$	48.6551	1.95403
$621$	15.7304	0.631241
$622$	9.16580	0.367515
$623$	−20.3986	−0.817254
$624$	2.48766	0.0995863
$625$	−26.1887	−1.04755
$626$	34.7590	1.38925
$627$	−5.62180	−0.224513
$628$	11.6825	0.466183
$629$	34.2508	1.36567
$630$	37.6679	1.50073
$631$	−8.95144	−0.356351	−0.178176	−	0.983999i	$-0.557019\pi$
$631$	−8.95144	−0.356351	−0.178176	+	0.983999i	$0.557019\pi$
$632$	−11.5172	−0.458129
$633$	2.73690	0.108782
$634$	−22.2390	−0.883225
$635$	−9.35977	−0.371431
$636$	34.7028	1.37605
$637$	−1.19512	−0.0473525
$638$	18.6964	0.740197
$639$	−57.1615	−2.26127
$640$	15.4433	0.610450
$641$	2.50423	0.0989110	0.0494555	−	0.998776i	$-0.484251\pi$
$641$	2.50423	0.0989110	0.0494555	+	0.998776i	$0.484251\pi$
$642$	−5.61094	−0.221446
$643$	−47.7114	−1.88155	−0.940777	−	0.339026i	$-0.889902\pi$
$643$	−47.7114	−1.88155	−0.940777	+	0.339026i	$0.889902\pi$
$644$	16.1979	0.638288
$645$	2.69720	0.106202
$646$	9.06949	0.356834
$647$	−42.0168	−1.65185	−0.825925	−	0.563780i	$-0.809347\pi$
$647$	−42.0168	−1.65185	−0.825925	+	0.563780i	$0.809347\pi$
$648$	1.98457	0.0779612
$649$	11.5309	0.452628
$650$	−0.158571	−0.00621966
$651$	42.0419	1.64775
$652$	−23.5570	−0.922562
$653$	27.9125	1.09230	0.546151	−	0.837687i	$-0.316093\pi$
$653$	27.9125	1.09230	0.546151	+	0.837687i	$0.316093\pi$
$654$	−42.9201	−1.67831
$655$	−11.6075	−0.453541
$656$	−37.7310	−1.47315
$657$	2.20125	0.0858788
$658$	−0.401667	−0.0156586
$659$	−35.2624	−1.37363	−0.686815	−	0.726832i	$-0.740992\pi$
$659$	−35.2624	−1.37363	−0.686815	+	0.726832i	$0.740992\pi$
$660$	−31.0417	−1.20830
$661$	−31.3615	−1.21982	−0.609910	−	0.792471i	$-0.708794\pi$
$661$	−31.3615	−1.21982	−0.609910	+	0.792471i	$0.708794\pi$
$662$	−64.1306	−2.49251
$663$	3.56649	0.138511
$664$	−0.838661	−0.0325464
$665$	−3.99444	−0.154898
$666$	74.7847	2.89785
$667$	16.7127	0.647117
$668$	12.1681	0.470796
$669$	66.4571	2.56938
$670$	−67.2666	−2.59874
$671$	7.37746	0.284804
$672$	38.3943	1.48109
$673$	−31.3147	−1.20709	−0.603546	−	0.797328i	$-0.706246\pi$
$673$	−31.3147	−1.20709	−0.603546	+	0.797328i	$0.706246\pi$
$674$	33.6895	1.29767
$675$	−1.02102	−0.0392989
$676$	−31.1079	−1.19646
$677$	−0.349229	−0.0134220	−0.00671098	−	0.999977i	$-0.502136\pi$
$677$	−0.349229	−0.0134220	−0.00671098	+	0.999977i	$0.502136\pi$
$678$	5.26990	0.202389
$679$	−0.297290	−0.0114089
$680$	8.51604	0.326575
$681$	−77.8276	−2.98236
$682$	−38.0087	−1.45543
$683$	−1.46159	−0.0559262	−0.0279631	−	0.999609i	$-0.508902\pi$
$683$	−1.46159	−0.0559262	−0.0279631	+	0.999609i	$0.508902\pi$
$684$	10.8215	0.413769
$685$	20.0229	0.765035
$686$	−40.1259	−1.53201
$687$	−34.3037	−1.30877
$688$	1.29565	0.0493962
$689$	−1.58757	−0.0604817
$690$	−50.7777	−1.93307
$691$	−38.5259	−1.46559	−0.732796	−	0.680448i	$-0.761785\pi$
$691$	−38.5259	−1.46559	−0.732796	+	0.680448i	$0.761785\pi$
$692$	−26.3385	−1.00124
$693$	−16.0801	−0.610833
$694$	52.4353	1.99042
$695$	−11.7132	−0.444307
$696$	−10.2085	−0.386953
$697$	−54.0937	−2.04895
$698$	−10.3126	−0.390338
$699$	14.3736	0.543660
$700$	−1.05136	−0.0397377
$701$	21.9891	0.830515	0.415258	−	0.909704i	$-0.363691\pi$
$701$	21.9891	0.830515	0.415258	+	0.909704i	$0.363691\pi$
$702$	2.58490	0.0975607
$703$	−7.93044	−0.299102
$704$	−22.3353	−0.841795
$705$	0.688083	0.0259147
$706$	2.61161	0.0982894
$707$	27.9127	1.04976
$708$	−37.0242	−1.39146
$709$	−13.9438	−0.523671	−0.261835	−	0.965113i	$-0.584328\pi$
$709$	−13.9438	−0.523671	−0.261835	+	0.965113i	$0.584328\pi$
$710$	61.2487	2.29862
$711$	60.1008	2.25395
$712$	−10.0695	−0.377371
$713$	−33.9760	−1.27241
$714$	43.2721	1.61942
$715$	1.42009	0.0531083
$716$	−4.84010	−0.180883
$717$	−20.5345	−0.766874
$718$	−47.8321	−1.78508
$719$	35.3513	1.31838	0.659191	−	0.751976i	$-0.270899\pi$
$719$	35.3513	1.31838	0.659191	+	0.751976i	$0.270899\pi$
$720$	−30.9972	−1.15520
$721$	−12.0093	−0.447251
$722$	−2.09995	−0.0781521
$723$	−1.14178	−0.0424632
$724$	−10.3657	−0.385238
$725$	−1.08477	−0.0402874
$726$	−38.9715	−1.44637
$727$	−16.3176	−0.605187	−0.302593	−	0.953120i	$-0.597852\pi$
$727$	−16.3176	−0.605187	−0.302593	+	0.953120i	$0.597852\pi$
$728$	0.452633	0.0167757
$729$	−43.8532	−1.62419
$730$	−2.35864	−0.0872972
$731$	1.85753	0.0687034
$732$	−23.6880	−0.875534
$733$	31.0247	1.14592	0.572962	−	0.819582i	$-0.305794\pi$
$733$	31.0247	1.14592	0.572962	+	0.819582i	$0.305794\pi$
$734$	−55.6466	−2.05396
$735$	24.8402	0.916243
$736$	−31.0282	−1.14372
$737$	28.7155	1.05775
$738$	−118.111	−4.34771
$739$	10.5273	0.387255	0.193627	−	0.981075i	$-0.437975\pi$
$739$	10.5273	0.387255	0.193627	+	0.981075i	$0.437975\pi$
$740$	−43.7893	−1.60973
$741$	−0.825785	−0.0303360
$742$	−19.2619	−0.707128
$743$	12.8303	0.470700	0.235350	−	0.971911i	$-0.424376\pi$
$743$	12.8303	0.470700	0.235350	+	0.971911i	$0.424376\pi$
$744$	20.7534	0.760857
$745$	13.4747	0.493675
$746$	−38.9598	−1.42642
$747$	4.37643	0.160125
$748$	−21.3782	−0.781663
$749$	1.70190	0.0621860
$750$	−62.5501	−2.28401
$751$	45.0187	1.64276	0.821379	−	0.570383i	$-0.193205\pi$
$751$	45.0187	1.64276	0.821379	+	0.570383i	$0.193205\pi$
$752$	0.330534	0.0120533
$753$	−51.3959	−1.87297
$754$	2.74630	0.100014
$755$	12.8825	0.468843
$756$	17.1385	0.623319
$757$	−46.9250	−1.70552	−0.852759	−	0.522304i	$-0.825073\pi$
$757$	−46.9250	−1.70552	−0.852759	+	0.522304i	$0.825073\pi$
$758$	67.9728	2.46888
$759$	21.6765	0.786809
$760$	−1.97180	−0.0715248
$761$	−0.378353	−0.0137153	−0.00685765	−	0.999976i	$-0.502183\pi$
$761$	−0.378353	−0.0137153	−0.00685765	+	0.999976i	$0.502183\pi$
$762$	−23.4770	−0.850482
$763$	13.0184	0.471299
$764$	−7.02494	−0.254154
$765$	−44.4397	−1.60672
$766$	36.3665	1.31398
$767$	1.69377	0.0611586
$768$	−20.7840	−0.749978
$769$	−4.96615	−0.179084	−0.0895420	−	0.995983i	$-0.528540\pi$
$769$	−4.96615	−0.179084	−0.0895420	+	0.995983i	$0.528540\pi$
$770$	17.2299	0.620922
$771$	−37.0682	−1.33498
$772$	−53.1655	−1.91347
$773$	27.8844	1.00293	0.501465	−	0.865178i	$-0.332794\pi$
$773$	27.8844	1.00293	0.501465	+	0.865178i	$0.332794\pi$
$774$	4.05582	0.145783
$775$	2.20528	0.0792161
$776$	−0.146753	−0.00526813
$777$	−37.8374	−1.35741
$778$	−56.7790	−2.03562
$779$	12.5249	0.448750
$780$	−4.55971	−0.163264
$781$	−26.1465	−0.935596
$782$	−34.9701	−1.25053
$783$	17.6831	0.631942
$784$	11.9324	0.426159
$785$	11.1083	0.396472
$786$	−29.1149	−1.03849
$787$	−11.5693	−0.412402	−0.206201	−	0.978510i	$-0.566110\pi$
$787$	−11.5693	−0.412402	−0.206201	+	0.978510i	$0.566110\pi$
$788$	45.2961	1.61361
$789$	0.334217	0.0118984
$790$	−64.3981	−2.29118
$791$	−1.59845	−0.0568344
$792$	−7.93773	−0.282055
$793$	1.08367	0.0384823
$794$	−13.6748	−0.485301
$795$	32.9971	1.17029
$796$	53.1507	1.88388
$797$	16.0739	0.569365	0.284683	−	0.958622i	$-0.408112\pi$
$797$	16.0739	0.569365	0.284683	+	0.958622i	$0.408112\pi$
$798$	−10.0192	−0.354676
$799$	0.473877	0.0167646
$800$	2.01395	0.0712039
$801$	52.5463	1.85663
$802$	−81.0129	−2.86066
$803$	1.00688	0.0355321
$804$	−92.2015	−3.25170
$805$	15.4018	0.542841
$806$	−5.58309	−0.196656
$807$	−2.91464	−0.102600
$808$	13.7787	0.484734
$809$	−18.2500	−0.641635	−0.320817	−	0.947141i	$-0.603958\pi$
$809$	−18.2500	−0.641635	−0.320817	+	0.947141i	$0.603958\pi$
$810$	11.0967	0.389898
$811$	−3.13746	−0.110171	−0.0550856	−	0.998482i	$-0.517543\pi$
$811$	−3.13746	−0.110171	−0.0550856	+	0.998482i	$0.517543\pi$
$812$	18.2086	0.638997
$813$	53.2240	1.86665
$814$	34.2076	1.19898
$815$	−22.3991	−0.784606
$816$	−35.6088	−1.24656
$817$	−0.430093	−0.0150471
$818$	−11.4773	−0.401296
$819$	−2.36200	−0.0825350
$820$	69.1582	2.41511
$821$	6.56380	0.229078	0.114539	−	0.993419i	$-0.463461\pi$
$821$	6.56380	0.229078	0.114539	+	0.993419i	$0.463461\pi$
$822$	50.2231	1.75173
$823$	−11.6546	−0.406254	−0.203127	−	0.979152i	$-0.565110\pi$
$823$	−11.6546	−0.406254	−0.203127	+	0.979152i	$0.565110\pi$
$824$	−5.92825	−0.206520
$825$	−1.40696	−0.0489841
$826$	20.5505	0.715043
$827$	−21.1062	−0.733934	−0.366967	−	0.930234i	$-0.619604\pi$
$827$	−21.1062	−0.733934	−0.366967	+	0.930234i	$0.619604\pi$
$828$	−41.7254	−1.45006
$829$	−47.3325	−1.64393	−0.821963	−	0.569541i	$-0.807121\pi$
$829$	−47.3325	−1.64393	−0.821963	+	0.569541i	$0.807121\pi$
$830$	−4.68936	−0.162770
$831$	−33.5057	−1.16230
$832$	−3.28083	−0.113742
$833$	17.1072	0.592729
$834$	−29.3801	−1.01735
$835$	11.5700	0.400395
$836$	4.94990	0.171196
$837$	−35.9488	−1.24257
$838$	−61.2056	−2.11431
$839$	28.9221	0.998503	0.499251	−	0.866457i	$-0.333608\pi$
$839$	28.9221	0.998503	0.499251	+	0.866457i	$0.333608\pi$
$840$	−9.40780	−0.324600
$841$	−10.2128	−0.352164
$842$	49.3651	1.70123
$843$	22.8944	0.788524
$844$	−2.40979	−0.0829485
$845$	−29.5789	−1.01755
$846$	1.03468	0.0355731
$847$	11.8207	0.406165
$848$	15.8508	0.544318
$849$	−65.5566	−2.24990
$850$	2.26981	0.0778538
$851$	30.5782	1.04821
$852$	83.9529	2.87618
$853$	−41.1167	−1.40781	−0.703905	−	0.710294i	$-0.748562\pi$
$853$	−41.1167	−1.40781	−0.703905	+	0.710294i	$0.748562\pi$
$854$	13.1482	0.449921
$855$	10.2896	0.351896
$856$	0.840119	0.0287147
$857$	23.2707	0.794911	0.397455	−	0.917622i	$-0.369893\pi$
$857$	23.2707	0.794911	0.397455	+	0.917622i	$0.369893\pi$
$858$	3.56199	0.121604
$859$	1.20396	0.0410786	0.0205393	−	0.999789i	$-0.493462\pi$
$859$	1.20396	0.0410786	0.0205393	+	0.999789i	$0.493462\pi$
$860$	−2.37483	−0.0809811
$861$	59.7582	2.03655
$862$	47.9931	1.63465
$863$	47.8993	1.63051	0.815256	−	0.579101i	$-0.196596\pi$
$863$	47.8993	1.63051	0.815256	+	0.579101i	$0.196596\pi$
$864$	−32.8299	−1.11689
$865$	−25.0439	−0.851519
$866$	−1.06389	−0.0361523
$867$	−4.52401	−0.153644
$868$	−37.0171	−1.25644
$869$	27.4910	0.932568
$870$	−57.0808	−1.93522
$871$	4.21801	0.142922
$872$	6.42637	0.217624
$873$	0.765810	0.0259188
$874$	8.09699	0.273885
$875$	18.9725	0.641389
$876$	−3.23296	−0.109232
$877$	48.8270	1.64877	0.824385	−	0.566030i	$-0.191521\pi$
$877$	48.8270	1.64877	0.824385	+	0.566030i	$0.191521\pi$
$878$	13.6435	0.460445
$879$	−63.1041	−2.12845
$880$	−14.1786	−0.477960
$881$	36.6437	1.23456	0.617278	−	0.786745i	$-0.288235\pi$
$881$	36.6437	1.23456	0.617278	+	0.786745i	$0.288235\pi$
$882$	37.3525	1.25773
$883$	−35.7204	−1.20209	−0.601044	−	0.799216i	$-0.705248\pi$
$883$	−35.7204	−1.20209	−0.601044	+	0.799216i	$0.705248\pi$
$884$	−3.14023	−0.105617
$885$	−35.2044	−1.18338
$886$	−76.8698	−2.58249
$887$	52.9558	1.77808	0.889040	−	0.457829i	$-0.151373\pi$
$887$	52.9558	1.77808	0.889040	+	0.457829i	$0.151373\pi$
$888$	−18.6779	−0.626791
$889$	7.12098	0.238830
$890$	−56.3035	−1.88730
$891$	−4.73707	−0.158698
$892$	−58.5143	−1.95920
$893$	−0.109721	−0.00367169
$894$	33.7984	1.13039
$895$	−4.60220	−0.153835
$896$	−11.7494	−0.392519
$897$	3.18406	0.106313
$898$	−30.2523	−1.00953
$899$	−38.1935	−1.27382
$900$	2.70827	0.0902758
$901$	22.7248	0.757072
$902$	−54.0255	−1.79885
$903$	−2.05205	−0.0682878
$904$	−0.789055	−0.0262436
$905$	−9.85619	−0.327631
$906$	32.3131	1.07353
$907$	−37.5392	−1.24647	−0.623234	−	0.782035i	$-0.714182\pi$
$907$	−37.5392	−1.24647	−0.623234	+	0.782035i	$0.714182\pi$
$908$	68.5258	2.27411
$909$	−71.9022	−2.38485
$910$	2.53089	0.0838982
$911$	15.5661	0.515727	0.257864	−	0.966181i	$-0.416981\pi$
$911$	15.5661	0.515727	0.257864	+	0.966181i	$0.416981\pi$
$912$	8.24487	0.273015
$913$	2.00184	0.0662514
$914$	60.1970	1.99114
$915$	−22.5237	−0.744611
$916$	30.2038	0.997962
$917$	8.83105	0.291627
$918$	−37.0007	−1.22120
$919$	10.5301	0.347355	0.173677	−	0.984803i	$-0.444435\pi$
$919$	10.5301	0.347355	0.173677	+	0.984803i	$0.444435\pi$
$920$	7.60288	0.250660
$921$	83.3818	2.74752
$922$	−61.1139	−2.01268
$923$	−3.84065	−0.126417
$924$	23.6168	0.776935
$925$	−1.98474	−0.0652578
$926$	−30.1786	−0.991731
$927$	30.9357	1.01606
$928$	−34.8798	−1.14499
$929$	2.30587	0.0756532	0.0378266	−	0.999284i	$-0.487957\pi$
$929$	2.30587	0.0756532	0.0378266	+	0.999284i	$0.487957\pi$
$930$	116.042	3.80518
$931$	−3.96100	−0.129816
$932$	−12.6557	−0.414551
$933$	11.9459	0.391093
$934$	−47.6004	−1.55753
$935$	−20.3274	−0.664777
$936$	−1.16597	−0.0381109
$937$	−12.0337	−0.393125	−0.196563	−	0.980491i	$-0.562978\pi$
$937$	−12.0337	−0.393125	−0.196563	+	0.980491i	$0.562978\pi$
$938$	51.1769	1.67099
$939$	45.3019	1.47837
$940$	−0.605845	−0.0197605
$941$	4.99397	0.162799	0.0813994	−	0.996682i	$-0.474061\pi$
$941$	4.99397	0.162799	0.0813994	+	0.996682i	$0.474061\pi$
$942$	27.8628	0.907819
$943$	−48.2934	−1.57265
$944$	−16.9111	−0.550410
$945$	16.2961	0.530111
$946$	1.85519	0.0603175
$947$	10.6327	0.345516	0.172758	−	0.984964i	$-0.444732\pi$
$947$	10.6327	0.345516	0.172758	+	0.984964i	$0.444732\pi$
$948$	−88.2698	−2.86687
$949$	0.147901	0.00480106
$950$	−0.525552	−0.0170511
$951$	−28.9845	−0.939887
$952$	−6.47907	−0.209988
$953$	50.8777	1.64809	0.824045	−	0.566525i	$-0.191712\pi$
$953$	50.8777	1.64809	0.824045	+	0.566525i	$0.191712\pi$
$954$	49.6182	1.60645
$955$	−6.67965	−0.216149
$956$	18.0803	0.584757
$957$	24.3673	0.787683
$958$	−14.5485	−0.470041
$959$	−15.2336	−0.491917
$960$	68.1908	2.20085
$961$	46.6453	1.50469
$962$	5.02475	0.162004
$963$	−4.38404	−0.141274
$964$	1.00532	0.0323790
$965$	−50.5523	−1.62734
$966$	38.6321	1.24297
$967$	36.2463	1.16560	0.582801	−	0.812615i	$-0.301957\pi$
$967$	36.2463	1.16560	0.582801	+	0.812615i	$0.301957\pi$
$968$	5.83515	0.187549
$969$	11.8204	0.379727
$970$	−0.820568	−0.0263469
$971$	20.1283	0.645947	0.322973	−	0.946408i	$-0.395318\pi$
$971$	20.1283	0.645947	0.322973	+	0.946408i	$0.395318\pi$
$972$	44.7037	1.43387
$973$	8.91149	0.285689
$974$	−52.0592	−1.66808
$975$	−0.206668	−0.00661867
$976$	−10.8197	−0.346330
$977$	35.1146	1.12341	0.561707	−	0.827336i	$-0.310145\pi$
$977$	35.1146	1.12341	0.561707	+	0.827336i	$0.310145\pi$
$978$	−56.1834	−1.79655
$979$	24.0355	0.768176
$980$	−21.8713	−0.698654
$981$	−33.5351	−1.07069
$982$	52.6260	1.67936
$983$	−25.2953	−0.806794	−0.403397	−	0.915025i	$-0.632171\pi$
$983$	−25.2953	−0.806794	−0.403397	+	0.915025i	$0.632171\pi$
$984$	29.4988	0.940389
$985$	43.0697	1.37232
$986$	−39.3110	−1.25192
$987$	−0.523499	−0.0166632
$988$	0.727089	0.0231318
$989$	1.65835	0.0527326
$990$	−44.3837	−1.41061
$991$	−27.8093	−0.883392	−0.441696	−	0.897165i	$-0.645623\pi$
$991$	−27.8093	−0.883392	−0.441696	+	0.897165i	$0.645623\pi$
$992$	70.9088	2.25136
$993$	−83.5825	−2.65241
$994$	−46.5985	−1.47801
$995$	50.5383	1.60217
$996$	−6.42765	−0.203668
$997$	−32.3536	−1.02465	−0.512325	−	0.858792i	$-0.671216\pi$
$997$	−32.3536	−1.02465	−0.512325	+	0.858792i	$0.671216\pi$
$998$	−58.8474	−1.86278
$999$	32.3537	1.02363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.12	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.12	✓	82	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)

\(f(q)\)	\(=\)	\(q-2.09995 q^{2} -2.73690 q^{3} +2.40979 q^{4} +2.29135 q^{5} +5.74735 q^{6} -1.74327 q^{7} -0.860544 q^{8} +4.49062 q^{9} +O(q^{10})\)	\(q-2.09995 q^{2} -2.73690 q^{3} +2.40979 q^{4} +2.29135 q^{5} +5.74735 q^{6} -1.74327 q^{7} -0.860544 q^{8} +4.49062 q^{9} -4.81171 q^{10} +2.05408 q^{11} -6.59536 q^{12} +0.301723 q^{13} +3.66079 q^{14} -6.27119 q^{15} -3.01249 q^{16} -4.31891 q^{17} -9.43008 q^{18} +1.00000 q^{19} +5.52167 q^{20} +4.77117 q^{21} -4.31346 q^{22} -3.85580 q^{23} +2.35522 q^{24} +0.250269 q^{25} -0.633603 q^{26} -4.07968 q^{27} -4.20093 q^{28} -4.33443 q^{29} +13.1692 q^{30} +8.81166 q^{31} +8.04716 q^{32} -5.62180 q^{33} +9.06949 q^{34} -3.99444 q^{35} +10.8215 q^{36} -7.93044 q^{37} -2.09995 q^{38} -0.825785 q^{39} -1.97180 q^{40} +12.5249 q^{41} -10.0192 q^{42} -0.430093 q^{43} +4.94990 q^{44} +10.2896 q^{45} +8.09699 q^{46} -0.109721 q^{47} +8.24487 q^{48} -3.96100 q^{49} -0.525552 q^{50} +11.8204 q^{51} +0.727089 q^{52} -5.26169 q^{53} +8.56713 q^{54} +4.70660 q^{55} +1.50016 q^{56} -2.73690 q^{57} +9.10208 q^{58} +5.61368 q^{59} -15.1123 q^{60} +3.59162 q^{61} -18.5040 q^{62} -7.82838 q^{63} -10.8737 q^{64} +0.691351 q^{65} +11.8055 q^{66} +13.9798 q^{67} -10.4077 q^{68} +10.5529 q^{69} +8.38813 q^{70} -12.7291 q^{71} -3.86438 q^{72} +0.490187 q^{73} +16.6535 q^{74} -0.684960 q^{75} +2.40979 q^{76} -3.58082 q^{77} +1.73411 q^{78} +13.3836 q^{79} -6.90265 q^{80} -2.30618 q^{81} -26.3016 q^{82} +0.974571 q^{83} +11.4975 q^{84} -9.89612 q^{85} +0.903175 q^{86} +11.8629 q^{87} -1.76762 q^{88} +11.7013 q^{89} -21.6076 q^{90} -0.525985 q^{91} -9.29168 q^{92} -24.1166 q^{93} +0.230410 q^{94} +2.29135 q^{95} -22.0243 q^{96} +0.170536 q^{97} +8.31790 q^{98} +9.22408 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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