LMFDB - Embedded newform 4009.2.a.e.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.66851 q^{2} -1.79206 q^{3} +0.783934 q^{4} +2.47432 q^{5} +2.99007 q^{6} +0.566931 q^{7} +2.02902 q^{8} +0.211475 q^{9} +O(q^{10})$	\(q-1.66851 q^{2} -1.79206 q^{3} +0.783934 q^{4} +2.47432 q^{5} +2.99007 q^{6} +0.566931 q^{7} +2.02902 q^{8} +0.211475 q^{9} -4.12843 q^{10} -0.155885 q^{11} -1.40486 q^{12} +3.80664 q^{13} -0.945931 q^{14} -4.43412 q^{15} -4.95332 q^{16} +6.11789 q^{17} -0.352849 q^{18} +1.00000 q^{19} +1.93970 q^{20} -1.01597 q^{21} +0.260095 q^{22} +1.53841 q^{23} -3.63613 q^{24} +1.12224 q^{25} -6.35143 q^{26} +4.99720 q^{27} +0.444436 q^{28} +3.16630 q^{29} +7.39839 q^{30} +5.87936 q^{31} +4.20662 q^{32} +0.279354 q^{33} -10.2078 q^{34} +1.40277 q^{35} +0.165782 q^{36} +3.06214 q^{37} -1.66851 q^{38} -6.82173 q^{39} +5.02044 q^{40} -11.8272 q^{41} +1.69516 q^{42} +4.73780 q^{43} -0.122203 q^{44} +0.523256 q^{45} -2.56685 q^{46} +2.40797 q^{47} +8.87663 q^{48} -6.67859 q^{49} -1.87248 q^{50} -10.9636 q^{51} +2.98416 q^{52} +10.1388 q^{53} -8.33789 q^{54} -0.385708 q^{55} +1.15031 q^{56} -1.79206 q^{57} -5.28301 q^{58} +3.26691 q^{59} -3.47606 q^{60} +9.36498 q^{61} -9.80979 q^{62} +0.119892 q^{63} +2.88783 q^{64} +9.41884 q^{65} -0.466106 q^{66} +7.12472 q^{67} +4.79602 q^{68} -2.75692 q^{69} -2.34053 q^{70} +4.85174 q^{71} +0.429088 q^{72} +3.54192 q^{73} -5.10922 q^{74} -2.01112 q^{75} +0.783934 q^{76} -0.0883758 q^{77} +11.3821 q^{78} +15.1550 q^{79} -12.2561 q^{80} -9.58970 q^{81} +19.7338 q^{82} +0.683905 q^{83} -0.796456 q^{84} +15.1376 q^{85} -7.90507 q^{86} -5.67420 q^{87} -0.316293 q^{88} -14.6592 q^{89} -0.873060 q^{90} +2.15810 q^{91} +1.20601 q^{92} -10.5362 q^{93} -4.01773 q^{94} +2.47432 q^{95} -7.53852 q^{96} -7.85702 q^{97} +11.1433 q^{98} -0.0329657 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.66851	−1.17982	−0.589908	−	0.807470i	$-0.700836\pi$
$2$	−1.66851	−1.17982	−0.589908	+	0.807470i	$0.700836\pi$
$3$	−1.79206	−1.03465	−0.517323	−	0.855790i	$-0.673071\pi$
$3$	−1.79206	−1.03465	−0.517323	+	0.855790i	$0.673071\pi$
$4$	0.783934	0.391967
$5$	2.47432	1.10655	0.553274	−	0.832999i	$-0.313378\pi$
$5$	2.47432	1.10655	0.553274	+	0.832999i	$0.313378\pi$
$6$	2.99007	1.22069
$7$	0.566931	0.214280	0.107140	−	0.994244i	$-0.465831\pi$
$7$	0.566931	0.214280	0.107140	+	0.994244i	$0.465831\pi$
$8$	2.02902	0.717368
$9$	0.211475	0.0704917
$10$	−4.12843	−1.30552
$11$	−0.155885	−0.0470010	−0.0235005	−	0.999724i	$-0.507481\pi$
$11$	−0.155885	−0.0470010	−0.0235005	+	0.999724i	$0.507481\pi$
$12$	−1.40486	−0.405547
$13$	3.80664	1.05577	0.527886	−	0.849315i	$-0.322985\pi$
$13$	3.80664	1.05577	0.527886	+	0.849315i	$0.322985\pi$
$14$	−0.945931	−0.252811
$15$	−4.43412	−1.14489
$16$	−4.95332	−1.23833
$17$	6.11789	1.48381	0.741904	−	0.670507i	$-0.233923\pi$
$17$	6.11789	1.48381	0.741904	+	0.670507i	$0.233923\pi$
$18$	−0.352849	−0.0831673
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.93970	0.433730
$21$	−1.01597	−0.221704
$22$	0.260095	0.0554525
$23$	1.53841	0.320780	0.160390	−	0.987054i	$-0.448725\pi$
$23$	1.53841	0.320780	0.160390	+	0.987054i	$0.448725\pi$
$24$	−3.63613	−0.742221
$25$	1.12224	0.224449
$26$	−6.35143	−1.24562
$27$	4.99720	0.961712
$28$	0.444436	0.0839905
$29$	3.16630	0.587967	0.293984	−	0.955810i	$-0.405019\pi$
$29$	3.16630	0.587967	0.293984	+	0.955810i	$0.405019\pi$
$30$	7.39839	1.35075
$31$	5.87936	1.05596	0.527982	−	0.849255i	$-0.322949\pi$
$31$	5.87936	1.05596	0.527982	+	0.849255i	$0.322949\pi$
$32$	4.20662	0.743633
$33$	0.279354	0.0486293
$34$	−10.2078	−1.75062
$35$	1.40277	0.237111
$36$	0.165782	0.0276304
$37$	3.06214	0.503413	0.251707	−	0.967804i	$-0.419008\pi$
$37$	3.06214	0.503413	0.251707	+	0.967804i	$0.419008\pi$
$38$	−1.66851	−0.270668
$39$	−6.82173	−1.09235
$40$	5.02044	0.793802
$41$	−11.8272	−1.84709	−0.923546	−	0.383488i	$-0.874723\pi$
$41$	−11.8272	−1.84709	−0.923546	+	0.383488i	$0.874723\pi$
$42$	1.69516	0.261569
$43$	4.73780	0.722507	0.361254	−	0.932468i	$-0.382349\pi$
$43$	4.73780	0.722507	0.361254	+	0.932468i	$0.382349\pi$
$44$	−0.122203	−0.0184228
$45$	0.523256	0.0780025
$46$	−2.56685	−0.378462
$47$	2.40797	0.351239	0.175620	−	0.984458i	$-0.443807\pi$
$47$	2.40797	0.351239	0.175620	+	0.984458i	$0.443807\pi$
$48$	8.87663	1.28123
$49$	−6.67859	−0.954084
$50$	−1.87248	−0.264808
$51$	−10.9636	−1.53521
$52$	2.98416	0.413828
$53$	10.1388	1.39267	0.696333	−	0.717718i	$-0.254813\pi$
$53$	10.1388	1.39267	0.696333	+	0.717718i	$0.254813\pi$
$54$	−8.33789	−1.13464
$55$	−0.385708	−0.0520088
$56$	1.15031	0.153717
$57$	−1.79206	−0.237364
$58$	−5.28301	−0.693693
$59$	3.26691	0.425316	0.212658	−	0.977127i	$-0.431788\pi$
$59$	3.26691	0.425316	0.212658	+	0.977127i	$0.431788\pi$
$60$	−3.47606	−0.448757
$61$	9.36498	1.19906	0.599531	−	0.800351i	$-0.295354\pi$
$61$	9.36498	1.19906	0.599531	+	0.800351i	$0.295354\pi$
$62$	−9.80979	−1.24584
$63$	0.119892	0.0151049
$64$	2.88783	0.360978
$65$	9.41884	1.16826
$66$	−0.466106	−0.0573737
$67$	7.12472	0.870424	0.435212	−	0.900328i	$-0.356673\pi$
$67$	7.12472	0.870424	0.435212	+	0.900328i	$0.356673\pi$
$68$	4.79602	0.581603
$69$	−2.75692	−0.331894
$70$	−2.34053	−0.279747
$71$	4.85174	0.575796	0.287898	−	0.957661i	$-0.407044\pi$
$71$	4.85174	0.575796	0.287898	+	0.957661i	$0.407044\pi$
$72$	0.429088	0.0505685
$73$	3.54192	0.414551	0.207275	−	0.978283i	$-0.433540\pi$
$73$	3.54192	0.414551	0.207275	+	0.978283i	$0.433540\pi$
$74$	−5.10922	−0.593935
$75$	−2.01112	−0.232225
$76$	0.783934	0.0899234
$77$	−0.0883758	−0.0100714
$78$	11.3821	1.28877
$79$	15.1550	1.70507	0.852533	−	0.522673i	$-0.175065\pi$
$79$	15.1550	1.70507	0.852533	+	0.522673i	$0.175065\pi$
$80$	−12.2561	−1.37027
$81$	−9.58970	−1.06552
$82$	19.7338	2.17923
$83$	0.683905	0.0750683	0.0375342	−	0.999295i	$-0.488050\pi$
$83$	0.683905	0.0750683	0.0375342	+	0.999295i	$0.488050\pi$
$84$	−0.796456	−0.0869004
$85$	15.1376	1.64190
$86$	−7.90507	−0.852426
$87$	−5.67420	−0.608338
$88$	−0.316293	−0.0337170
$89$	−14.6592	−1.55387	−0.776934	−	0.629582i	$-0.783226\pi$
$89$	−14.6592	−1.55387	−0.776934	+	0.629582i	$0.783226\pi$
$90$	−0.873060	−0.0920286
$91$	2.15810	0.226231
$92$	1.20601	0.125735
$93$	−10.5362	−1.09255
$94$	−4.01773	−0.414398
$95$	2.47432	0.253860
$96$	−7.53852	−0.769397
$97$	−7.85702	−0.797760	−0.398880	−	0.917003i	$-0.630601\pi$
$97$	−7.85702	−0.797760	−0.398880	+	0.917003i	$0.630601\pi$
$98$	11.1433	1.12564
$99$	−0.0329657	−0.00331318
$100$	0.879764	0.0879764
$101$	−6.71787	−0.668453	−0.334226	−	0.942493i	$-0.608475\pi$
$101$	−6.71787	−0.668453	−0.334226	+	0.942493i	$0.608475\pi$
$102$	18.2929	1.81127
$103$	−4.44112	−0.437597	−0.218798	−	0.975770i	$-0.570214\pi$
$103$	−4.44112	−0.437597	−0.218798	+	0.975770i	$0.570214\pi$
$104$	7.72376	0.757377
$105$	−2.51384	−0.245326
$106$	−16.9167	−1.64309
$107$	6.39171	0.617910	0.308955	−	0.951077i	$-0.400021\pi$
$107$	6.39171	0.617910	0.308955	+	0.951077i	$0.400021\pi$
$108$	3.91747	0.376959
$109$	−13.5337	−1.29629	−0.648145	−	0.761517i	$-0.724455\pi$
$109$	−13.5337	−1.29629	−0.648145	+	0.761517i	$0.724455\pi$
$110$	0.643558	0.0613609
$111$	−5.48754	−0.520854
$112$	−2.80819	−0.265349
$113$	−18.5326	−1.74340	−0.871699	−	0.490042i	$-0.836981\pi$
$113$	−18.5326	−1.74340	−0.871699	+	0.490042i	$0.836981\pi$
$114$	2.99007	0.280046
$115$	3.80651	0.354959
$116$	2.48217	0.230464
$117$	0.805010	0.0744232
$118$	−5.45089	−0.501795
$119$	3.46842	0.317950
$120$	−8.99693	−0.821303
$121$	−10.9757	−0.997791
$122$	−15.6256	−1.41467
$123$	21.1950	1.91109
$124$	4.60903	0.413903
$125$	−9.59480	−0.858185
$126$	−0.200041	−0.0178211
$127$	−2.21175	−0.196261	−0.0981307	−	0.995174i	$-0.531286\pi$
$127$	−2.21175	−0.196261	−0.0981307	+	0.995174i	$0.531286\pi$
$128$	−13.2316	−1.16952
$129$	−8.49041	−0.747539
$130$	−15.7155	−1.37834
$131$	9.78619	0.855023	0.427512	−	0.904010i	$-0.359390\pi$
$131$	9.78619	0.855023	0.427512	+	0.904010i	$0.359390\pi$
$132$	0.218995	0.0190611
$133$	0.566931	0.0491591
$134$	−11.8877	−1.02694
$135$	12.3647	1.06418
$136$	12.4133	1.06444
$137$	−7.08171	−0.605032	−0.302516	−	0.953144i	$-0.597826\pi$
$137$	−7.08171	−0.605032	−0.302516	+	0.953144i	$0.597826\pi$
$138$	4.59995	0.391574
$139$	−4.69240	−0.398004	−0.199002	−	0.979999i	$-0.563770\pi$
$139$	−4.69240	−0.398004	−0.199002	+	0.979999i	$0.563770\pi$
$140$	1.09968	0.0929395
$141$	−4.31523	−0.363408
$142$	−8.09520	−0.679334
$143$	−0.593397	−0.0496223
$144$	−1.04750	−0.0872919
$145$	7.83443	0.650614
$146$	−5.90974	−0.489094
$147$	11.9684	0.987139
$148$	2.40052	0.197321
$149$	3.02384	0.247722	0.123861	−	0.992300i	$-0.460472\pi$
$149$	3.02384	0.247722	0.123861	+	0.992300i	$0.460472\pi$
$150$	3.35559	0.273982
$151$	−23.8466	−1.94061	−0.970305	−	0.241885i	$-0.922234\pi$
$151$	−23.8466	−1.94061	−0.970305	+	0.241885i	$0.922234\pi$
$152$	2.02902	0.164575
$153$	1.29378	0.104596
$154$	0.147456	0.0118823
$155$	14.5474	1.16848
$156$	−5.34778	−0.428165
$157$	−13.2611	−1.05835	−0.529175	−	0.848513i	$-0.677499\pi$
$157$	−13.2611	−1.05835	−0.529175	+	0.848513i	$0.677499\pi$
$158$	−25.2862	−2.01167
$159$	−18.1693	−1.44092
$160$	10.4085	0.822866
$161$	0.872171	0.0687367
$162$	16.0005	1.25712
$163$	9.20873	0.721284	0.360642	−	0.932704i	$-0.382558\pi$
$163$	9.20873	0.721284	0.360642	+	0.932704i	$0.382558\pi$
$164$	−9.27171	−0.723999
$165$	0.691211	0.0538107
$166$	−1.14110	−0.0885669
$167$	10.2702	0.794732	0.397366	−	0.917660i	$-0.369924\pi$
$167$	10.2702	0.794732	0.397366	+	0.917660i	$0.369924\pi$
$168$	−2.06143	−0.159043
$169$	1.49053	0.114656
$170$	−25.2573	−1.93715
$171$	0.211475	0.0161719
$172$	3.71412	0.283199
$173$	0.594107	0.0451691	0.0225846	−	0.999745i	$-0.492810\pi$
$173$	0.594107	0.0451691	0.0225846	+	0.999745i	$0.492810\pi$
$174$	9.46747	0.717727
$175$	0.636234	0.0480948
$176$	0.772145	0.0582027
$177$	−5.85450	−0.440051
$178$	24.4590	1.83328
$179$	13.0741	0.977205	0.488602	−	0.872507i	$-0.337507\pi$
$179$	13.0741	0.977205	0.488602	+	0.872507i	$0.337507\pi$
$180$	0.410198	0.0305744
$181$	16.5860	1.23283	0.616413	−	0.787423i	$-0.288585\pi$
$181$	16.5860	1.23283	0.616413	+	0.787423i	$0.288585\pi$
$182$	−3.60082	−0.266911
$183$	−16.7826	−1.24060
$184$	3.12146	0.230117
$185$	7.57671	0.557051
$186$	17.5797	1.28901
$187$	−0.953685	−0.0697404
$188$	1.88769	0.137674
$189$	2.83307	0.206075
$190$	−4.12843	−0.299508
$191$	2.92564	0.211692	0.105846	−	0.994383i	$-0.466245\pi$
$191$	2.92564	0.211692	0.105846	+	0.994383i	$0.466245\pi$
$192$	−5.17515	−0.373485
$193$	15.4007	1.10857	0.554284	−	0.832328i	$-0.312992\pi$
$193$	15.4007	1.10857	0.554284	+	0.832328i	$0.312992\pi$
$194$	13.1095	0.941210
$195$	−16.8791	−1.20874
$196$	−5.23557	−0.373969
$197$	−26.5721	−1.89318	−0.946591	−	0.322437i	$-0.895498\pi$
$197$	−26.5721	−1.89318	−0.946591	+	0.322437i	$0.895498\pi$
$198$	0.0550037	0.00390894
$199$	−1.16638	−0.0826828	−0.0413414	−	0.999145i	$-0.513163\pi$
$199$	−1.16638	−0.0826828	−0.0413414	+	0.999145i	$0.513163\pi$
$200$	2.27705	0.161012
$201$	−12.7679	−0.900580
$202$	11.2088	0.788652
$203$	1.79507	0.125989
$204$	−8.59475	−0.601753
$205$	−29.2641	−2.04390
$206$	7.41007	0.516284
$207$	0.325335	0.0226124
$208$	−18.8555	−1.30739
$209$	−0.155885	−0.0107828
$210$	4.19437	0.289439
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	7.94812	0.545879
$213$	−8.69461	−0.595745
$214$	−10.6647	−0.729020
$215$	11.7228	0.799489
$216$	10.1394	0.689901
$217$	3.33319	0.226272
$218$	22.5811	1.52938
$219$	−6.34733	−0.428913
$220$	−0.302369	−0.0203857
$221$	23.2886	1.56656
$222$	9.15603	0.614513
$223$	−4.47841	−0.299897	−0.149948	−	0.988694i	$-0.547911\pi$
$223$	−4.47841	−0.299897	−0.149948	+	0.988694i	$0.547911\pi$
$224$	2.38486	0.159345
$225$	0.237326	0.0158218
$226$	30.9218	2.05689
$227$	25.5258	1.69421	0.847103	−	0.531429i	$-0.178345\pi$
$227$	25.5258	1.69421	0.847103	+	0.531429i	$0.178345\pi$
$228$	−1.40486	−0.0930388
$229$	−13.6809	−0.904058	−0.452029	−	0.892003i	$-0.649300\pi$
$229$	−13.6809	−0.904058	−0.452029	+	0.892003i	$0.649300\pi$
$230$	−6.35121	−0.418786
$231$	0.158375	0.0104203
$232$	6.42449	0.421789
$233$	−7.15905	−0.469005	−0.234503	−	0.972116i	$-0.575346\pi$
$233$	−7.15905	−0.469005	−0.234503	+	0.972116i	$0.575346\pi$
$234$	−1.34317	−0.0878058
$235$	5.95809	0.388663
$236$	2.56104	0.166710
$237$	−27.1586	−1.76414
$238$	−5.78710	−0.375122
$239$	3.99283	0.258275	0.129137	−	0.991627i	$-0.458779\pi$
$239$	3.99283	0.258275	0.129137	+	0.991627i	$0.458779\pi$
$240$	21.9636	1.41774
$241$	7.68998	0.495355	0.247678	−	0.968842i	$-0.420333\pi$
$241$	7.68998	0.495355	0.247678	+	0.968842i	$0.420333\pi$
$242$	18.3131	1.17721
$243$	2.19371	0.140727
$244$	7.34152	0.469993
$245$	−16.5249	−1.05574
$246$	−35.3641	−2.25473
$247$	3.80664	0.242211
$248$	11.9294	0.757515
$249$	−1.22560	−0.0776691
$250$	16.0090	1.01250
$251$	2.17620	0.137360	0.0686802	−	0.997639i	$-0.478121\pi$
$251$	2.17620	0.137360	0.0686802	+	0.997639i	$0.478121\pi$
$252$	0.0939872	0.00592064
$253$	−0.239814	−0.0150770
$254$	3.69034	0.231552
$255$	−27.1275	−1.69879
$256$	16.3015	1.01884
$257$	16.0031	0.998249	0.499124	−	0.866530i	$-0.333655\pi$
$257$	16.0031	0.998249	0.499124	+	0.866530i	$0.333655\pi$
$258$	14.1664	0.881959
$259$	1.73602	0.107871
$260$	7.38375	0.457920
$261$	0.669594	0.0414468
$262$	−16.3284	−1.00877
$263$	−13.9237	−0.858569	−0.429285	−	0.903169i	$-0.641234\pi$
$263$	−13.9237	−0.858569	−0.429285	+	0.903169i	$0.641234\pi$
$264$	0.566816	0.0348851
$265$	25.0865	1.54105
$266$	−0.945931	−0.0579987
$267$	26.2701	1.60770
$268$	5.58531	0.341177
$269$	8.29834	0.505959	0.252979	−	0.967472i	$-0.418589\pi$
$269$	8.29834	0.505959	0.252979	+	0.967472i	$0.418589\pi$
$270$	−20.6306	−1.25554
$271$	−10.3552	−0.629035	−0.314517	−	0.949252i	$-0.601843\pi$
$271$	−10.3552	−0.629035	−0.314517	+	0.949252i	$0.601843\pi$
$272$	−30.3039	−1.83744
$273$	−3.86745	−0.234069
$274$	11.8159	0.713826
$275$	−0.174940	−0.0105493
$276$	−2.16124	−0.130091
$277$	23.4968	1.41179	0.705893	−	0.708318i	$-0.250546\pi$
$277$	23.4968	1.41179	0.705893	+	0.708318i	$0.250546\pi$
$278$	7.82933	0.469572
$279$	1.24334	0.0744368
$280$	2.84624	0.170096
$281$	23.6557	1.41118	0.705590	−	0.708621i	$-0.250682\pi$
$281$	23.6557	1.41118	0.705590	+	0.708621i	$0.250682\pi$
$282$	7.20002	0.428755
$283$	−22.5137	−1.33830	−0.669150	−	0.743127i	$-0.733342\pi$
$283$	−22.5137	−1.33830	−0.669150	+	0.743127i	$0.733342\pi$
$284$	3.80345	0.225693
$285$	−4.43412	−0.262655
$286$	0.990090	0.0585453
$287$	−6.70518	−0.395794
$288$	0.889597	0.0524200
$289$	20.4286	1.20168
$290$	−13.0718	−0.767605
$291$	14.0802	0.825399
$292$	2.77663	0.162490
$293$	22.8191	1.33310	0.666552	−	0.745458i	$-0.267769\pi$
$293$	22.8191	1.33310	0.666552	+	0.745458i	$0.267769\pi$
$294$	−19.9695	−1.16464
$295$	8.08338	0.470632
$296$	6.21316	0.361132
$297$	−0.778986	−0.0452014
$298$	−5.04531	−0.292267
$299$	5.85617	0.338671
$300$	−1.57659	−0.0910244
$301$	2.68600	0.154819
$302$	39.7884	2.28956
$303$	12.0388	0.691612
$304$	−4.95332	−0.284092
$305$	23.1719	1.32682
$306$	−2.15869	−0.123404
$307$	0.372073	0.0212353	0.0106177	−	0.999944i	$-0.496620\pi$
$307$	0.372073	0.0212353	0.0106177	+	0.999944i	$0.496620\pi$
$308$	−0.0692807	−0.00394764
$309$	7.95875	0.452757
$310$	−24.2725	−1.37859
$311$	−10.5265	−0.596902	−0.298451	−	0.954425i	$-0.596470\pi$
$311$	−10.5265	−0.596902	−0.298451	+	0.954425i	$0.596470\pi$
$312$	−13.8414	−0.783617
$313$	−4.81747	−0.272299	−0.136150	−	0.990688i	$-0.543473\pi$
$313$	−4.81747	−0.272299	−0.136150	+	0.990688i	$0.543473\pi$
$314$	22.1263	1.24866
$315$	0.296650	0.0167143
$316$	11.8805	0.668329
$317$	15.4372	0.867039	0.433520	−	0.901144i	$-0.357271\pi$
$317$	15.4372	0.867039	0.433520	+	0.901144i	$0.357271\pi$
$318$	30.3157	1.70002
$319$	−0.493577	−0.0276350
$320$	7.14539	0.399440
$321$	−11.4543	−0.639318
$322$	−1.45523	−0.0810967
$323$	6.11789	0.340409
$324$	−7.51769	−0.417650
$325$	4.27198	0.236967
$326$	−15.3649	−0.850982
$327$	24.2531	1.34120
$328$	−23.9976	−1.32504
$329$	1.36515	0.0752634
$330$	−1.15329	−0.0634868
$331$	27.9981	1.53892	0.769458	−	0.638697i	$-0.220526\pi$
$331$	27.9981	1.53892	0.769458	+	0.638697i	$0.220526\pi$
$332$	0.536136	0.0294243
$333$	0.647567	0.0354865
$334$	−17.1360	−0.937638
$335$	17.6288	0.963165
$336$	5.03244	0.274542
$337$	−6.25357	−0.340654	−0.170327	−	0.985388i	$-0.554482\pi$
$337$	−6.25357	−0.340654	−0.170327	+	0.985388i	$0.554482\pi$
$338$	−2.48697	−0.135273
$339$	33.2115	1.80380
$340$	11.8669	0.643572
$341$	−0.916502	−0.0496314
$342$	−0.352849	−0.0190799
$343$	−7.75481	−0.418721
$344$	9.61310	0.518303
$345$	−6.82149	−0.367257
$346$	−0.991276	−0.0532913
$347$	−26.2382	−1.40854	−0.704271	−	0.709931i	$-0.748726\pi$
$347$	−26.2382	−1.40854	−0.704271	+	0.709931i	$0.748726\pi$
$348$	−4.44819	−0.238448
$349$	−20.7827	−1.11247	−0.556236	−	0.831024i	$-0.687755\pi$
$349$	−20.7827	−1.11247	−0.556236	+	0.831024i	$0.687755\pi$
$350$	−1.06156	−0.0567430
$351$	19.0226	1.01535
$352$	−0.655748	−0.0349515
$353$	−3.50019	−0.186297	−0.0931483	−	0.995652i	$-0.529693\pi$
$353$	−3.50019	−0.186297	−0.0931483	+	0.995652i	$0.529693\pi$
$354$	9.76831	0.519180
$355$	12.0048	0.637146
$356$	−11.4918	−0.609065
$357$	−6.21562	−0.328965
$358$	−21.8143	−1.15292
$359$	13.1251	0.692714	0.346357	−	0.938103i	$-0.387419\pi$
$359$	13.1251	0.692714	0.346357	+	0.938103i	$0.387419\pi$
$360$	1.06170	0.0559564
$361$	1.00000	0.0526316
$362$	−27.6739	−1.45451
$363$	19.6691	1.03236
$364$	1.69181	0.0886749
$365$	8.76384	0.458720
$366$	28.0020	1.46369
$367$	21.3280	1.11331	0.556656	−	0.830743i	$-0.312084\pi$
$367$	21.3280	1.11331	0.556656	+	0.830743i	$0.312084\pi$
$368$	−7.62022	−0.397232
$369$	−2.50115	−0.130205
$370$	−12.6418	−0.657218
$371$	5.74798	0.298420
$372$	−8.25965	−0.428243
$373$	−3.39795	−0.175939	−0.0879696	−	0.996123i	$-0.528038\pi$
$373$	−3.39795	−0.175939	−0.0879696	+	0.996123i	$0.528038\pi$
$374$	1.59124	0.0822808
$375$	17.1944	0.887917
$376$	4.88583	0.251967
$377$	12.0530	0.620760
$378$	−4.72701	−0.243131
$379$	29.0717	1.49331	0.746656	−	0.665210i	$-0.231658\pi$
$379$	29.0717	1.49331	0.746656	+	0.665210i	$0.231658\pi$
$380$	1.93970	0.0995045
$381$	3.96359	0.203061
$382$	−4.88147	−0.249758
$383$	37.5762	1.92005	0.960026	−	0.279911i	$-0.0903048\pi$
$383$	37.5762	1.92005	0.960026	+	0.279911i	$0.0903048\pi$
$384$	23.7118	1.21004
$385$	−0.218670	−0.0111444
$386$	−25.6963	−1.30791
$387$	1.00193	0.0509308
$388$	−6.15939	−0.312695
$389$	33.9139	1.71950	0.859751	−	0.510714i	$-0.170619\pi$
$389$	33.9139	1.71950	0.859751	+	0.510714i	$0.170619\pi$
$390$	28.1630	1.42609
$391$	9.41182	0.475976
$392$	−13.5510	−0.684429
$393$	−17.5374	−0.884646
$394$	44.3358	2.23361
$395$	37.4982	1.88674
$396$	−0.0258429	−0.00129866
$397$	−6.94077	−0.348347	−0.174174	−	0.984715i	$-0.555725\pi$
$397$	−6.94077	−0.348347	−0.174174	+	0.984715i	$0.555725\pi$
$398$	1.94613	0.0975505
$399$	−1.01597	−0.0508623
$400$	−5.55882	−0.277941
$401$	−11.1219	−0.555399	−0.277700	−	0.960668i	$-0.589572\pi$
$401$	−11.1219	−0.555399	−0.277700	+	0.960668i	$0.589572\pi$
$402$	21.3034	1.06252
$403$	22.3806	1.11486
$404$	−5.26636	−0.262011
$405$	−23.7280	−1.17905
$406$	−2.99510	−0.148644
$407$	−0.477341	−0.0236609
$408$	−22.2454	−1.10131
$409$	11.0946	0.548594	0.274297	−	0.961645i	$-0.411555\pi$
$409$	11.0946	0.548594	0.274297	+	0.961645i	$0.411555\pi$
$410$	48.8276	2.41142
$411$	12.6908	0.625993
$412$	−3.48154	−0.171523
$413$	1.85211	0.0911365
$414$	−0.542826	−0.0266784
$415$	1.69220	0.0830667
$416$	16.0131	0.785108
$417$	8.40906	0.411794
$418$	0.260095	0.0127217
$419$	−28.9937	−1.41643	−0.708217	−	0.705995i	$-0.750500\pi$
$419$	−28.9937	−1.41643	−0.708217	+	0.705995i	$0.750500\pi$
$420$	−1.97068	−0.0961595
$421$	33.9502	1.65463	0.827316	−	0.561737i	$-0.189867\pi$
$421$	33.9502	1.65463	0.827316	+	0.561737i	$0.189867\pi$
$422$	1.66851	0.0812219
$423$	0.509227	0.0247594
$424$	20.5718	0.999054
$425$	6.86576	0.333038
$426$	14.5071	0.702870
$427$	5.30929	0.256935
$428$	5.01068	0.242200
$429$	1.06340	0.0513415
$430$	−19.5597	−0.943250
$431$	−18.0053	−0.867284	−0.433642	−	0.901085i	$-0.642772\pi$
$431$	−18.0053	−0.867284	−0.433642	+	0.901085i	$0.642772\pi$
$432$	−24.7527	−1.19092
$433$	−0.0208588	−0.00100241	−0.000501204	−	1.00000i	$-0.500160\pi$
$433$	−0.0208588	−0.00100241	−0.000501204		1.00000i	$0.500160\pi$
$434$	−5.56147	−0.266959
$435$	−14.0398	−0.673155
$436$	−10.6095	−0.508103
$437$	1.53841	0.0735921
$438$	10.5906	0.506039
$439$	8.16253	0.389576	0.194788	−	0.980845i	$-0.437598\pi$
$439$	8.16253	0.389576	0.194788	+	0.980845i	$0.437598\pi$
$440$	−0.782609	−0.0373094
$441$	−1.41236	−0.0672550
$442$	−38.8574	−1.84826
$443$	−4.83345	−0.229644	−0.114822	−	0.993386i	$-0.536630\pi$
$443$	−4.83345	−0.229644	−0.114822	+	0.993386i	$0.536630\pi$
$444$	−4.30187	−0.204158
$445$	−36.2714	−1.71943
$446$	7.47228	0.353823
$447$	−5.41889	−0.256305
$448$	1.63720	0.0773503
$449$	−11.9573	−0.564300	−0.282150	−	0.959370i	$-0.591048\pi$
$449$	−11.9573	−0.564300	−0.282150	+	0.959370i	$0.591048\pi$
$450$	−0.395982	−0.0186668
$451$	1.84367	0.0868151
$452$	−14.5283	−0.683354
$453$	42.7345	2.00784
$454$	−42.5901	−1.99885
$455$	5.33983	0.250335
$456$	−3.63613	−0.170277
$457$	6.20894	0.290442	0.145221	−	0.989399i	$-0.453611\pi$
$457$	6.20894	0.290442	0.145221	+	0.989399i	$0.453611\pi$
$458$	22.8267	1.06662
$459$	30.5723	1.42699
$460$	2.98405	0.139132
$461$	−8.47993	−0.394950	−0.197475	−	0.980308i	$-0.563274\pi$
$461$	−8.47993	−0.394950	−0.197475	+	0.980308i	$0.563274\pi$
$462$	−0.264250	−0.0122940
$463$	−38.8456	−1.80531	−0.902655	−	0.430365i	$-0.858385\pi$
$463$	−38.8456	−1.80531	−0.902655	+	0.430365i	$0.858385\pi$
$464$	−15.6837	−0.728097
$465$	−26.0698	−1.20896
$466$	11.9450	0.553340
$467$	−29.1208	−1.34755	−0.673775	−	0.738937i	$-0.735328\pi$
$467$	−29.1208	−1.34755	−0.673775	+	0.738937i	$0.735328\pi$
$468$	0.631075	0.0291714
$469$	4.03922	0.186514
$470$	−9.94115	−0.458551
$471$	23.7647	1.09502
$472$	6.62864	0.305108
$473$	−0.738550	−0.0339585
$474$	45.3144	2.08136
$475$	1.12224	0.0514920
$476$	2.71901	0.124626
$477$	2.14410	0.0981715
$478$	−6.66209	−0.304717
$479$	19.0132	0.868736	0.434368	−	0.900736i	$-0.356972\pi$
$479$	19.0132	0.868736	0.434368	+	0.900736i	$0.356972\pi$
$480$	−18.6527	−0.851375
$481$	11.6565	0.531490
$482$	−12.8308	−0.584428
$483$	−1.56298	−0.0711181
$484$	−8.60422	−0.391101
$485$	−19.4408	−0.882760
$486$	−3.66023	−0.166032
$487$	−16.0858	−0.728915	−0.364457	−	0.931220i	$-0.618746\pi$
$487$	−16.0858	−0.728915	−0.364457	+	0.931220i	$0.618746\pi$
$488$	19.0017	0.860169
$489$	−16.5026	−0.746273
$490$	27.5721	1.24558
$491$	−4.88256	−0.220347	−0.110173	−	0.993912i	$-0.535141\pi$
$491$	−4.88256	−0.220347	−0.110173	+	0.993912i	$0.535141\pi$
$492$	16.6154	0.749082
$493$	19.3711	0.872430
$494$	−6.35143	−0.285764
$495$	−0.0815676	−0.00366619
$496$	−29.1223	−1.30763
$497$	2.75060	0.123381
$498$	2.04493	0.0916353
$499$	17.4267	0.780124	0.390062	−	0.920789i	$-0.372454\pi$
$499$	17.4267	0.780124	0.390062	+	0.920789i	$0.372454\pi$
$500$	−7.52169	−0.336380
$501$	−18.4048	−0.822267
$502$	−3.63101	−0.162060
$503$	−1.64159	−0.0731949	−0.0365974	−	0.999330i	$-0.511652\pi$
$503$	−1.64159	−0.0731949	−0.0365974	+	0.999330i	$0.511652\pi$
$504$	0.243263	0.0108358
$505$	−16.6221	−0.739675
$506$	0.400133	0.0177881
$507$	−2.67112	−0.118629
$508$	−1.73387	−0.0769280
$509$	−20.3947	−0.903980	−0.451990	−	0.892023i	$-0.649286\pi$
$509$	−20.3947	−0.903980	−0.451990	+	0.892023i	$0.649286\pi$
$510$	45.2625	2.00426
$511$	2.00802	0.0888298
$512$	−0.735968	−0.0325255
$513$	4.99720	0.220632
$514$	−26.7014	−1.17775
$515$	−10.9887	−0.484222
$516$	−6.65592	−0.293011
$517$	−0.375366	−0.0165086
$518$	−2.89658	−0.127268
$519$	−1.06468	−0.0467341
$520$	19.1110	0.838074
$521$	23.8620	1.04541	0.522706	−	0.852513i	$-0.324923\pi$
$521$	23.8620	1.04541	0.522706	+	0.852513i	$0.324923\pi$
$522$	−1.11723	−0.0488996
$523$	−16.8674	−0.737558	−0.368779	−	0.929517i	$-0.620224\pi$
$523$	−16.8674	−0.737558	−0.368779	+	0.929517i	$0.620224\pi$
$524$	7.67172	0.335141
$525$	−1.14017	−0.0497610
$526$	23.2318	1.01295
$527$	35.9693	1.56685
$528$	−1.38373	−0.0602191
$529$	−20.6333	−0.897100
$530$	−41.8572	−1.81816
$531$	0.690871	0.0299812
$532$	0.444436	0.0192687
$533$	−45.0218	−1.95011
$534$	−43.8319	−1.89679
$535$	15.8151	0.683747
$536$	14.4562	0.624414
$537$	−23.4296	−1.01106
$538$	−13.8459	−0.596939
$539$	1.04109	0.0448429
$540$	9.69307	0.417123
$541$	−28.6306	−1.23093	−0.615463	−	0.788166i	$-0.711031\pi$
$541$	−28.6306	−1.23093	−0.615463	+	0.788166i	$0.711031\pi$
$542$	17.2778	0.742146
$543$	−29.7230	−1.27554
$544$	25.7357	1.10341
$545$	−33.4866	−1.43441
$546$	6.45288	0.276158
$547$	−26.2789	−1.12360	−0.561802	−	0.827272i	$-0.689892\pi$
$547$	−26.2789	−1.12360	−0.561802	+	0.827272i	$0.689892\pi$
$548$	−5.55159	−0.237152
$549$	1.98046	0.0845240
$550$	0.291890	0.0124462
$551$	3.16630	0.134889
$552$	−5.59385	−0.238090
$553$	8.59181	0.365361
$554$	−39.2047	−1.66565
$555$	−13.5779	−0.576350
$556$	−3.67853	−0.156005
$557$	15.1860	0.643453	0.321726	−	0.946833i	$-0.395737\pi$
$557$	15.1860	0.643453	0.321726	+	0.946833i	$0.395737\pi$
$558$	−2.07453	−0.0878217
$559$	18.0351	0.762804
$560$	−6.94834	−0.293621
$561$	1.70906	0.0721566
$562$	−39.4698	−1.66493
$563$	22.4681	0.946917	0.473458	−	0.880816i	$-0.343005\pi$
$563$	22.4681	0.946917	0.473458	+	0.880816i	$0.343005\pi$
$564$	−3.38285	−0.142444
$565$	−45.8554	−1.92915
$566$	37.5644	1.57895
$567$	−5.43670	−0.228320
$568$	9.84429	0.413057
$569$	23.7453	0.995454	0.497727	−	0.867334i	$-0.334168\pi$
$569$	23.7453	0.995454	0.497727	+	0.867334i	$0.334168\pi$
$570$	7.39839	0.309884
$571$	−35.5662	−1.48840	−0.744200	−	0.667957i	$-0.767169\pi$
$571$	−35.5662	−1.48840	−0.744200	+	0.667957i	$0.767169\pi$
$572$	−0.465184	−0.0194503
$573$	−5.24292	−0.219026
$574$	11.1877	0.466965
$575$	1.72647	0.0719987
$576$	0.610703	0.0254460
$577$	45.8205	1.90753	0.953767	−	0.300548i	$-0.0971694\pi$
$577$	45.8205	1.90753	0.953767	+	0.300548i	$0.0971694\pi$
$578$	−34.0854	−1.41777
$579$	−27.5990	−1.14697
$580$	6.14167	0.255019
$581$	0.387727	0.0160856
$582$	−23.4931	−0.973819
$583$	−1.58048	−0.0654567
$584$	7.18664	0.297385
$585$	1.99185	0.0823529
$586$	−38.0739	−1.57282
$587$	33.4407	1.38024	0.690122	−	0.723693i	$-0.257557\pi$
$587$	33.4407	1.38024	0.690122	+	0.723693i	$0.257557\pi$
$588$	9.38245	0.386926
$589$	5.87936	0.242255
$590$	−13.4872	−0.555260
$591$	47.6187	1.95877
$592$	−15.1678	−0.623391
$593$	35.8372	1.47166	0.735828	−	0.677169i	$-0.236793\pi$
$593$	35.8372	1.47166	0.735828	+	0.677169i	$0.236793\pi$
$594$	1.29975	0.0533293
$595$	8.58197	0.351827
$596$	2.37049	0.0970989
$597$	2.09023	0.0855474
$598$	−9.77110	−0.399570
$599$	22.3584	0.913541	0.456770	−	0.889585i	$-0.349006\pi$
$599$	22.3584	0.913541	0.456770	+	0.889585i	$0.349006\pi$
$600$	−4.08062	−0.166590
$601$	−28.5399	−1.16417	−0.582084	−	0.813129i	$-0.697762\pi$
$601$	−28.5399	−1.16417	−0.582084	+	0.813129i	$0.697762\pi$
$602$	−4.48163	−0.182658
$603$	1.50670	0.0613577
$604$	−18.6942	−0.760655
$605$	−27.1574	−1.10410
$606$	−20.0869	−0.815975
$607$	−11.4808	−0.465990	−0.232995	−	0.972478i	$-0.574853\pi$
$607$	−11.4808	−0.465990	−0.232995	+	0.972478i	$0.574853\pi$
$608$	4.20662	0.170601
$609$	−3.21688	−0.130354
$610$	−38.6626	−1.56540
$611$	9.16630	0.370829
$612$	1.01424	0.0409982
$613$	−3.21322	−0.129781	−0.0648904	−	0.997892i	$-0.520670\pi$
$613$	−3.21322	−0.129781	−0.0648904	+	0.997892i	$0.520670\pi$
$614$	−0.620809	−0.0250538
$615$	52.4431	2.11471
$616$	−0.179316	−0.00722486
$617$	−8.23823	−0.331659	−0.165829	−	0.986154i	$-0.553030\pi$
$617$	−8.23823	−0.331659	−0.165829	+	0.986154i	$0.553030\pi$
$618$	−13.2793	−0.534171
$619$	35.9740	1.44592	0.722958	−	0.690892i	$-0.242782\pi$
$619$	35.9740	1.44592	0.722958	+	0.690892i	$0.242782\pi$
$620$	11.4042	0.458004
$621$	7.68774	0.308498
$622$	17.5636	0.704235
$623$	−8.31073	−0.332962
$624$	33.7902	1.35269
$625$	−29.3518	−1.17407
$626$	8.03800	0.321263
$627$	0.279354	0.0111563
$628$	−10.3958	−0.414838
$629$	18.7339	0.746968
$630$	−0.494964	−0.0197199
$631$	18.6141	0.741017	0.370509	−	0.928829i	$-0.379183\pi$
$631$	18.6141	0.741017	0.370509	+	0.928829i	$0.379183\pi$
$632$	30.7497	1.22316
$633$	1.79206	0.0712279
$634$	−25.7572	−1.02295
$635$	−5.47258	−0.217173
$636$	−14.2435	−0.564792
$637$	−25.4230	−1.00730
$638$	0.823540	0.0326043
$639$	1.02602	0.0405889
$640$	−32.7392	−1.29413
$641$	−45.6729	−1.80397	−0.901986	−	0.431766i	$-0.857891\pi$
$641$	−45.6729	−1.80397	−0.901986	+	0.431766i	$0.857891\pi$
$642$	19.1117	0.754278
$643$	−30.1266	−1.18808	−0.594039	−	0.804436i	$-0.702467\pi$
$643$	−30.1266	−1.18808	−0.594039	+	0.804436i	$0.702467\pi$
$644$	0.683724	0.0269425
$645$	−21.0080	−0.827188
$646$	−10.2078	−0.401620
$647$	32.5820	1.28093	0.640465	−	0.767987i	$-0.278742\pi$
$647$	32.5820	1.28093	0.640465	+	0.767987i	$0.278742\pi$
$648$	−19.4577	−0.764371
$649$	−0.509261	−0.0199903
$650$	−7.12785	−0.279577
$651$	−5.97328	−0.234111
$652$	7.21904	0.282719
$653$	26.2552	1.02745	0.513723	−	0.857956i	$-0.328266\pi$
$653$	26.2552	1.02745	0.513723	+	0.857956i	$0.328266\pi$
$654$	−40.4667	−1.58237
$655$	24.2141	0.946124
$656$	58.5836	2.28731
$657$	0.749028	0.0292224
$658$	−2.27778	−0.0887970
$659$	32.3826	1.26145	0.630723	−	0.776008i	$-0.282758\pi$
$659$	32.3826	1.26145	0.630723	+	0.776008i	$0.282758\pi$
$660$	0.541864	0.0210920
$661$	21.7226	0.844910	0.422455	−	0.906384i	$-0.361168\pi$
$661$	21.7226	0.844910	0.422455	+	0.906384i	$0.361168\pi$
$662$	−46.7152	−1.81564
$663$	−41.7346	−1.62084
$664$	1.38766	0.0538516
$665$	1.40277	0.0543969
$666$	−1.08047	−0.0418675
$667$	4.87106	0.188608
$668$	8.05116	0.311509
$669$	8.02558	0.310287
$670$	−29.4139	−1.13636
$671$	−1.45986	−0.0563571
$672$	−4.27382	−0.164866
$673$	−15.3979	−0.593547	−0.296773	−	0.954948i	$-0.595911\pi$
$673$	−15.3979	−0.593547	−0.296773	+	0.954948i	$0.595911\pi$
$674$	10.4342	0.401909
$675$	5.60807	0.215855
$676$	1.16848	0.0449414
$677$	29.3403	1.12764	0.563820	−	0.825898i	$-0.309331\pi$
$677$	29.3403	1.12764	0.563820	+	0.825898i	$0.309331\pi$
$678$	−55.4137	−2.12815
$679$	−4.45439	−0.170944
$680$	30.7145	1.17785
$681$	−45.7437	−1.75290
$682$	1.52919	0.0585559
$683$	−23.8037	−0.910823	−0.455411	−	0.890281i	$-0.650508\pi$
$683$	−23.8037	−0.910823	−0.455411	+	0.890281i	$0.650508\pi$
$684$	0.165782	0.00633885
$685$	−17.5224	−0.669496
$686$	12.9390	0.494013
$687$	24.5169	0.935380
$688$	−23.4678	−0.894702
$689$	38.5947	1.47034
$690$	11.3817	0.433295
$691$	29.3214	1.11544	0.557719	−	0.830030i	$-0.311677\pi$
$691$	29.3214	1.11544	0.557719	+	0.830030i	$0.311677\pi$
$692$	0.465741	0.0177048
$693$	−0.0186893	−0.000709947 0
$694$	43.7788	1.66182
$695$	−11.6105	−0.440411
$696$	−11.5131	−0.436402
$697$	−72.3573	−2.74073
$698$	34.6762	1.31251
$699$	12.8294	0.485254
$700$	0.498765	0.0188515
$701$	41.5544	1.56949	0.784744	−	0.619820i	$-0.212794\pi$
$701$	41.5544	1.56949	0.784744	+	0.619820i	$0.212794\pi$
$702$	−31.7394	−1.19793
$703$	3.06214	0.115491
$704$	−0.450167	−0.0169663
$705$	−10.6772	−0.402128
$706$	5.84012	0.219796
$707$	−3.80857	−0.143236
$708$	−4.58954	−0.172485
$709$	−12.3574	−0.464091	−0.232046	−	0.972705i	$-0.574542\pi$
$709$	−12.3574	−0.464091	−0.232046	+	0.972705i	$0.574542\pi$
$710$	−20.0301	−0.751715
$711$	3.20490	0.120193
$712$	−29.7438	−1.11469
$713$	9.04486	0.338733
$714$	10.3708	0.388119
$715$	−1.46825	−0.0549095
$716$	10.2492	0.383032
$717$	−7.15539	−0.267223
$718$	−21.8993	−0.817275
$719$	34.3823	1.28224	0.641122	−	0.767439i	$-0.278469\pi$
$719$	34.3823	1.28224	0.641122	+	0.767439i	$0.278469\pi$
$720$	−2.59185	−0.0965927
$721$	−2.51781	−0.0937681
$722$	−1.66851	−0.0620956
$723$	−13.7809	−0.512517
$724$	13.0023	0.483227
$725$	3.55336	0.131968
$726$	−32.8181	−1.21800
$727$	19.5165	0.723828	0.361914	−	0.932212i	$-0.382123\pi$
$727$	19.5165	0.723828	0.361914	+	0.932212i	$0.382123\pi$
$728$	4.37884	0.162291
$729$	24.8379	0.919920
$730$	−14.6226	−0.541206
$731$	28.9853	1.07206
$732$	−13.1564	−0.486276
$733$	−16.2633	−0.600699	−0.300350	−	0.953829i	$-0.597103\pi$
$733$	−16.2633	−0.600699	−0.300350	+	0.953829i	$0.597103\pi$
$734$	−35.5860	−1.31350
$735$	29.6137	1.09232
$736$	6.47151	0.238543
$737$	−1.11063	−0.0409107
$738$	4.17320	0.153618
$739$	−32.3413	−1.18969	−0.594847	−	0.803839i	$-0.702787\pi$
$739$	−32.3413	−1.18969	−0.594847	+	0.803839i	$0.702787\pi$
$740$	5.93964	0.218346
$741$	−6.82173	−0.250602
$742$	−9.59057	−0.352081
$743$	−16.3304	−0.599103	−0.299551	−	0.954080i	$-0.596837\pi$
$743$	−16.3304	−0.599103	−0.299551	+	0.954080i	$0.596837\pi$
$744$	−21.3781	−0.783759
$745$	7.48193	0.274117
$746$	5.66952	0.207576
$747$	0.144629	0.00529170
$748$	−0.747626	−0.0273359
$749$	3.62366	0.132406
$750$	−28.6891	−1.04758
$751$	−22.2607	−0.812303	−0.406152	−	0.913806i	$-0.633129\pi$
$751$	−22.2607	−0.812303	−0.406152	+	0.913806i	$0.633129\pi$
$752$	−11.9275	−0.434949
$753$	−3.89987	−0.142119
$754$	−20.1105	−0.732383
$755$	−59.0041	−2.14738
$756$	2.22094	0.0807747
$757$	3.09842	0.112614	0.0563070	−	0.998414i	$-0.482067\pi$
$757$	3.09842	0.112614	0.0563070	+	0.998414i	$0.482067\pi$
$758$	−48.5065	−1.76184
$759$	0.429761	0.0155993
$760$	5.02044	0.182111
$761$	21.7427	0.788171	0.394086	−	0.919074i	$-0.371061\pi$
$761$	21.7427	0.788171	0.394086	+	0.919074i	$0.371061\pi$
$762$	−6.61331	−0.239575
$763$	−7.67265	−0.277769
$764$	2.29351	0.0829763
$765$	3.20123	0.115741
$766$	−62.6963	−2.26531
$767$	12.4360	0.449037
$768$	−29.2132	−1.05414
$769$	−40.5722	−1.46307	−0.731535	−	0.681803i	$-0.761196\pi$
$769$	−40.5722	−1.46307	−0.731535	+	0.681803i	$0.761196\pi$
$770$	0.364853	0.0131484
$771$	−28.6786	−1.03283
$772$	12.0731	0.434522
$773$	−1.46900	−0.0528362	−0.0264181	−	0.999651i	$-0.508410\pi$
$773$	−1.46900	−0.0528362	−0.0264181	+	0.999651i	$0.508410\pi$
$774$	−1.67173	−0.0600890
$775$	6.59807	0.237010
$776$	−15.9421	−0.572287
$777$	−3.11106	−0.111609
$778$	−56.5857	−2.02870
$779$	−11.8272	−0.423752
$780$	−13.2321	−0.473785
$781$	−0.756312	−0.0270630
$782$	−15.7037	−0.561564
$783$	15.8226	0.565455
$784$	33.0812	1.18147
$785$	−32.8121	−1.17112
$786$	29.2614	1.04372
$787$	−48.7971	−1.73943	−0.869715	−	0.493554i	$-0.835698\pi$
$787$	−48.7971	−1.73943	−0.869715	+	0.493554i	$0.835698\pi$
$788$	−20.8307	−0.742065
$789$	24.9520	0.888315
$790$	−62.5662	−2.22600
$791$	−10.5067	−0.373575
$792$	−0.0668881	−0.00237677
$793$	35.6491	1.26594
$794$	11.5808	0.410986
$795$	−44.9565	−1.59444
$796$	−0.914368	−0.0324089
$797$	−11.8251	−0.418867	−0.209433	−	0.977823i	$-0.567162\pi$
$797$	−11.8251	−0.418867	−0.209433	+	0.977823i	$0.567162\pi$
$798$	1.69516	0.0600082
$799$	14.7317	0.521171
$800$	4.72085	0.166907
$801$	−3.10005	−0.109535
$802$	18.5570	0.655269
$803$	−0.552131	−0.0194843
$804$	−10.0092	−0.352997
$805$	2.15803	0.0760605
$806$	−37.3424	−1.31533
$807$	−14.8711	−0.523488
$808$	−13.6307	−0.479526
$809$	39.6127	1.39271	0.696354	−	0.717698i	$-0.254804\pi$
$809$	39.6127	1.39271	0.696354	+	0.717698i	$0.254804\pi$
$810$	39.5904	1.39106
$811$	21.9744	0.771625	0.385813	−	0.922577i	$-0.373921\pi$
$811$	21.9744	0.771625	0.385813	+	0.922577i	$0.373921\pi$
$812$	1.40722	0.0493837
$813$	18.5572	0.650828
$814$	0.796449	0.0279155
$815$	22.7853	0.798135
$816$	54.3063	1.90110
$817$	4.73780	0.165755
$818$	−18.5115	−0.647240
$819$	0.456385	0.0159474
$820$	−22.9411	−0.801139
$821$	4.50420	0.157198	0.0785988	−	0.996906i	$-0.474955\pi$
$821$	4.50420	0.157198	0.0785988	+	0.996906i	$0.474955\pi$
$822$	−21.1748	−0.738557
$823$	−6.47063	−0.225552	−0.112776	−	0.993620i	$-0.535974\pi$
$823$	−6.47063	−0.225552	−0.112776	+	0.993620i	$0.535974\pi$
$824$	−9.01113	−0.313918
$825$	0.313503	0.0109148
$826$	−3.09027	−0.107524
$827$	−7.00511	−0.243591	−0.121796	−	0.992555i	$-0.538865\pi$
$827$	−7.00511	−0.243591	−0.121796	+	0.992555i	$0.538865\pi$
$828$	0.255041	0.00886329
$829$	5.67968	0.197264	0.0986318	−	0.995124i	$-0.468553\pi$
$829$	5.67968	0.197264	0.0986318	+	0.995124i	$0.468553\pi$
$830$	−2.82345	−0.0980035
$831$	−42.1077	−1.46070
$832$	10.9929	0.381111
$833$	−40.8589	−1.41568
$834$	−14.0306	−0.485841
$835$	25.4117	0.879410
$836$	−0.122203	−0.00422648
$837$	29.3804	1.01553
$838$	48.3763	1.67113
$839$	−44.1218	−1.52325	−0.761627	−	0.648016i	$-0.775599\pi$
$839$	−44.1218	−1.52325	−0.761627	+	0.648016i	$0.775599\pi$
$840$	−5.10064	−0.175989
$841$	−18.9745	−0.654295
$842$	−56.6463	−1.95216
$843$	−42.3924	−1.46007
$844$	−0.783934	−0.0269841
$845$	3.68804	0.126873
$846$	−0.849651	−0.0292116
$847$	−6.22246	−0.213806
$848$	−50.2205	−1.72458
$849$	40.3459	1.38467
$850$	−11.4556	−0.392924
$851$	4.71083	0.161485
$852$	−6.81600	−0.233512
$853$	−22.5372	−0.771660	−0.385830	−	0.922570i	$-0.626085\pi$
$853$	−22.5372	−0.771660	−0.385830	+	0.922570i	$0.626085\pi$
$854$	−8.85862	−0.303136
$855$	0.523256	0.0178950
$856$	12.9689	0.443269
$857$	46.5871	1.59138	0.795692	−	0.605701i	$-0.207107\pi$
$857$	46.5871	1.59138	0.795692	+	0.605701i	$0.207107\pi$
$858$	−1.77430	−0.0605736
$859$	10.0614	0.343291	0.171646	−	0.985159i	$-0.445092\pi$
$859$	10.0614	0.343291	0.171646	+	0.985159i	$0.445092\pi$
$860$	9.18991	0.313373
$861$	12.0161	0.409507
$862$	30.0420	1.02324
$863$	−25.8768	−0.880858	−0.440429	−	0.897787i	$-0.645174\pi$
$863$	−25.8768	−0.880858	−0.440429	+	0.897787i	$0.645174\pi$
$864$	21.0213	0.715161
$865$	1.47001	0.0499818
$866$	0.0348031	0.00118266
$867$	−36.6093	−1.24332
$868$	2.61300	0.0886910
$869$	−2.36242	−0.0801398
$870$	23.4255	0.794199
$871$	27.1213	0.918970
$872$	−27.4601	−0.929917
$873$	−1.66157	−0.0562355
$874$	−2.56685	−0.0868251
$875$	−5.43959	−0.183892
$876$	−4.97589	−0.168120
$877$	8.44841	0.285283	0.142641	−	0.989774i	$-0.454440\pi$
$877$	8.44841	0.285283	0.142641	+	0.989774i	$0.454440\pi$
$878$	−13.6193	−0.459629
$879$	−40.8931	−1.37929
$880$	1.91053	0.0644040
$881$	−2.88694	−0.0972634	−0.0486317	−	0.998817i	$-0.515486\pi$
$881$	−2.88694	−0.0972634	−0.0486317	+	0.998817i	$0.515486\pi$
$882$	2.35653	0.0793486
$883$	16.7533	0.563793	0.281896	−	0.959445i	$-0.409037\pi$
$883$	16.7533	0.563793	0.281896	+	0.959445i	$0.409037\pi$
$884$	18.2567	0.614041
$885$	−14.4859	−0.486938
$886$	8.06468	0.270938
$887$	4.26092	0.143068	0.0715339	−	0.997438i	$-0.477211\pi$
$887$	4.26092	0.143068	0.0715339	+	0.997438i	$0.477211\pi$
$888$	−11.1343	−0.373644
$889$	−1.25391	−0.0420548
$890$	60.5193	2.02861
$891$	1.49489	0.0500806
$892$	−3.51078	−0.117550
$893$	2.40797	0.0805798
$894$	9.04149	0.302393
$895$	32.3495	1.08132
$896$	−7.50141	−0.250605
$897$	−10.4946	−0.350405
$898$	19.9509	0.665771
$899$	18.6158	0.620873
$900$	0.186048	0.00620161
$901$	62.0279	2.06645
$902$	−3.07619	−0.102426
$903$	−4.81348	−0.160182
$904$	−37.6030	−1.25066
$905$	41.0390	1.36418
$906$	−71.3031	−2.36889
$907$	27.9914	0.929439	0.464719	−	0.885458i	$-0.346155\pi$
$907$	27.9914	0.929439	0.464719	+	0.885458i	$0.346155\pi$
$908$	20.0105	0.664072
$909$	−1.42066	−0.0471204
$910$	−8.90957	−0.295349
$911$	−45.0651	−1.49307	−0.746537	−	0.665344i	$-0.768285\pi$
$911$	−45.0651	−1.49307	−0.746537	+	0.665344i	$0.768285\pi$
$912$	8.87663	0.293935
$913$	−0.106610	−0.00352828
$914$	−10.3597	−0.342668
$915$	−41.5254	−1.37279
$916$	−10.7249	−0.354361
$917$	5.54809	0.183214
$918$	−51.0103	−1.68359
$919$	−36.0081	−1.18780	−0.593899	−	0.804540i	$-0.702412\pi$
$919$	−36.0081	−1.18780	−0.593899	+	0.804540i	$0.702412\pi$
$920$	7.72349	0.254636
$921$	−0.666777	−0.0219710
$922$	14.1489	0.465968
$923$	18.4689	0.607910
$924$	0.124155	0.00408440
$925$	3.43647	0.112990
$926$	64.8144	2.12993
$927$	−0.939187	−0.0308469
$928$	13.3194	0.437232
$929$	54.6003	1.79138	0.895690	−	0.444680i	$-0.146682\pi$
$929$	54.6003	1.79138	0.895690	+	0.444680i	$0.146682\pi$
$930$	43.4978	1.42635
$931$	−6.67859	−0.218882
$932$	−5.61222	−0.183834
$933$	18.8641	0.617582
$934$	48.5884	1.58986
$935$	−2.35972	−0.0771711
$936$	1.63338	0.0533888
$937$	17.0271	0.556252	0.278126	−	0.960545i	$-0.410287\pi$
$937$	17.0271	0.556252	0.278126	+	0.960545i	$0.410287\pi$
$938$	−6.73950	−0.220052
$939$	8.63318	0.281733
$940$	4.67075	0.152343
$941$	36.5758	1.19234	0.596169	−	0.802859i	$-0.296689\pi$
$941$	36.5758	1.19234	0.596169	+	0.802859i	$0.296689\pi$
$942$	−39.6516	−1.29192
$943$	−18.1950	−0.592511
$944$	−16.1821	−0.526681
$945$	7.00990	0.228032
$946$	1.23228	0.0400649
$947$	−35.2853	−1.14662	−0.573309	−	0.819339i	$-0.694340\pi$
$947$	−35.2853	−1.14662	−0.573309	+	0.819339i	$0.694340\pi$
$948$	−21.2905	−0.691484
$949$	13.4828	0.437671
$950$	−1.87248	−0.0607511
$951$	−27.6644	−0.897078
$952$	7.03750	0.228087
$953$	−38.0693	−1.23319	−0.616593	−	0.787282i	$-0.711487\pi$
$953$	−38.0693	−1.23319	−0.616593	+	0.787282i	$0.711487\pi$
$954$	−3.57745	−0.115824
$955$	7.23897	0.234247
$956$	3.13012	0.101235
$957$	0.884520	0.0285925
$958$	−31.7238	−1.02495
$959$	−4.01484	−0.129646
$960$	−12.8050	−0.413279
$961$	3.56690	0.115061
$962$	−19.4490	−0.627061
$963$	1.35169	0.0435575
$964$	6.02844	0.194163
$965$	38.1062	1.22668
$966$	2.60785	0.0839063
$967$	−2.99011	−0.0961554	−0.0480777	−	0.998844i	$-0.515310\pi$
$967$	−2.99011	−0.0961554	−0.0480777	+	0.998844i	$0.515310\pi$
$968$	−22.2699	−0.715783
$969$	−10.9636	−0.352202
$970$	32.4372	1.04149
$971$	39.7477	1.27557	0.637783	−	0.770216i	$-0.279852\pi$
$971$	39.7477	1.27557	0.637783	+	0.770216i	$0.279852\pi$
$972$	1.71972	0.0551602
$973$	−2.66027	−0.0852843
$974$	26.8393	0.859986
$975$	−7.65563	−0.245177
$976$	−46.3877	−1.48483
$977$	42.8727	1.37162	0.685810	−	0.727781i	$-0.259448\pi$
$977$	42.8727	1.37162	0.685810	+	0.727781i	$0.259448\pi$
$978$	27.5348	0.880465
$979$	2.28514	0.0730333
$980$	−12.9545	−0.413815
$981$	−2.86204	−0.0913777
$982$	8.14661	0.259969
$983$	42.9524	1.36997	0.684985	−	0.728558i	$-0.259809\pi$
$983$	42.9524	1.36997	0.684985	+	0.728558i	$0.259809\pi$
$984$	43.0050	1.37095
$985$	−65.7477	−2.09490
$986$	−32.3209	−1.02931
$987$	−2.44644	−0.0778709
$988$	2.98416	0.0949386
$989$	7.28867	0.231766
$990$	0.136097	0.00432543
$991$	−23.7772	−0.755309	−0.377654	−	0.925947i	$-0.623269\pi$
$991$	−23.7772	−0.755309	−0.377654	+	0.925947i	$0.623269\pi$
$992$	24.7323	0.785250
$993$	−50.1743	−1.59223
$994$	−4.58941	−0.145567
$995$	−2.88600	−0.0914925
$996$	−0.960788	−0.0304437
$997$	45.6591	1.44604	0.723019	−	0.690828i	$-0.242754\pi$
$997$	45.6591	1.44604	0.723019	+	0.690828i	$0.242754\pi$
$998$	−29.0766	−0.920403
$999$	15.3021	0.484139

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.17	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-1.66851 q^{2} -1.79206 q^{3} +0.783934 q^{4} +2.47432 q^{5} +2.99007 q^{6} +0.566931 q^{7} +2.02902 q^{8} +0.211475 q^{9} +O(q^{10})\)	\(q-1.66851 q^{2} -1.79206 q^{3} +0.783934 q^{4} +2.47432 q^{5} +2.99007 q^{6} +0.566931 q^{7} +2.02902 q^{8} +0.211475 q^{9} -4.12843 q^{10} -0.155885 q^{11} -1.40486 q^{12} +3.80664 q^{13} -0.945931 q^{14} -4.43412 q^{15} -4.95332 q^{16} +6.11789 q^{17} -0.352849 q^{18} +1.00000 q^{19} +1.93970 q^{20} -1.01597 q^{21} +0.260095 q^{22} +1.53841 q^{23} -3.63613 q^{24} +1.12224 q^{25} -6.35143 q^{26} +4.99720 q^{27} +0.444436 q^{28} +3.16630 q^{29} +7.39839 q^{30} +5.87936 q^{31} +4.20662 q^{32} +0.279354 q^{33} -10.2078 q^{34} +1.40277 q^{35} +0.165782 q^{36} +3.06214 q^{37} -1.66851 q^{38} -6.82173 q^{39} +5.02044 q^{40} -11.8272 q^{41} +1.69516 q^{42} +4.73780 q^{43} -0.122203 q^{44} +0.523256 q^{45} -2.56685 q^{46} +2.40797 q^{47} +8.87663 q^{48} -6.67859 q^{49} -1.87248 q^{50} -10.9636 q^{51} +2.98416 q^{52} +10.1388 q^{53} -8.33789 q^{54} -0.385708 q^{55} +1.15031 q^{56} -1.79206 q^{57} -5.28301 q^{58} +3.26691 q^{59} -3.47606 q^{60} +9.36498 q^{61} -9.80979 q^{62} +0.119892 q^{63} +2.88783 q^{64} +9.41884 q^{65} -0.466106 q^{66} +7.12472 q^{67} +4.79602 q^{68} -2.75692 q^{69} -2.34053 q^{70} +4.85174 q^{71} +0.429088 q^{72} +3.54192 q^{73} -5.10922 q^{74} -2.01112 q^{75} +0.783934 q^{76} -0.0883758 q^{77} +11.3821 q^{78} +15.1550 q^{79} -12.2561 q^{80} -9.58970 q^{81} +19.7338 q^{82} +0.683905 q^{83} -0.796456 q^{84} +15.1376 q^{85} -7.90507 q^{86} -5.67420 q^{87} -0.316293 q^{88} -14.6592 q^{89} -0.873060 q^{90} +2.15810 q^{91} +1.20601 q^{92} -10.5362 q^{93} -4.01773 q^{94} +2.47432 q^{95} -7.53852 q^{96} -7.85702 q^{97} +11.1433 q^{98} -0.0329657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 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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