LMFDB - Embedded newform 1006.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1006,2,Mod(1,1006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1006 = 2 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1006.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:	$8.03295044334$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.49390$ of defining polynomial
Character	$\chi$	$=$	1006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.49390 q^{3} +1.00000 q^{4} +1.94778 q^{5} +2.49390 q^{6} -1.87622 q^{7} +1.00000 q^{8} +3.21954 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.49390 q^{3} +1.00000 q^{4} +1.94778 q^{5} +2.49390 q^{6} -1.87622 q^{7} +1.00000 q^{8} +3.21954 q^{9} +1.94778 q^{10} +5.83443 q^{11} +2.49390 q^{12} -0.298003 q^{13} -1.87622 q^{14} +4.85757 q^{15} +1.00000 q^{16} -7.33964 q^{17} +3.21954 q^{18} -0.888571 q^{19} +1.94778 q^{20} -4.67910 q^{21} +5.83443 q^{22} -8.64250 q^{23} +2.49390 q^{24} -1.20615 q^{25} -0.298003 q^{26} +0.547512 q^{27} -1.87622 q^{28} +0.804767 q^{29} +4.85757 q^{30} -5.43347 q^{31} +1.00000 q^{32} +14.5505 q^{33} -7.33964 q^{34} -3.65446 q^{35} +3.21954 q^{36} +4.16366 q^{37} -0.888571 q^{38} -0.743190 q^{39} +1.94778 q^{40} +2.14268 q^{41} -4.67910 q^{42} +5.11414 q^{43} +5.83443 q^{44} +6.27096 q^{45} -8.64250 q^{46} +10.2919 q^{47} +2.49390 q^{48} -3.47981 q^{49} -1.20615 q^{50} -18.3043 q^{51} -0.298003 q^{52} +9.43805 q^{53} +0.547512 q^{54} +11.3642 q^{55} -1.87622 q^{56} -2.21601 q^{57} +0.804767 q^{58} -14.1014 q^{59} +4.85757 q^{60} +14.1214 q^{61} -5.43347 q^{62} -6.04056 q^{63} +1.00000 q^{64} -0.580445 q^{65} +14.5505 q^{66} -6.24714 q^{67} -7.33964 q^{68} -21.5535 q^{69} -3.65446 q^{70} +4.03233 q^{71} +3.21954 q^{72} -8.07158 q^{73} +4.16366 q^{74} -3.00801 q^{75} -0.888571 q^{76} -10.9467 q^{77} -0.743190 q^{78} +15.7749 q^{79} +1.94778 q^{80} -8.29318 q^{81} +2.14268 q^{82} +3.93865 q^{83} -4.67910 q^{84} -14.2960 q^{85} +5.11414 q^{86} +2.00701 q^{87} +5.83443 q^{88} -4.09320 q^{89} +6.27096 q^{90} +0.559119 q^{91} -8.64250 q^{92} -13.5505 q^{93} +10.2919 q^{94} -1.73074 q^{95} +2.49390 q^{96} -8.46702 q^{97} -3.47981 q^{98} +18.7842 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 + 5 * q^3 + 12 * q^4 + 5 * q^5 + 5 * q^6 + 8 * q^7 + 12 * q^8 + 15 * q^9	\( 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9} + 5 q^{10} + 18 q^{11} + 5 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{15} + 12 q^{16} - 2 q^{17} + 15 q^{18} + 6 q^{19} + 5 q^{20} + q^{21} + 18 q^{22} + 13 q^{23} + 5 q^{24} + q^{25} + 4 q^{26} + 8 q^{27} + 8 q^{28} + 20 q^{29} + 2 q^{30} + 7 q^{31} + 12 q^{32} - 8 q^{33} - 2 q^{34} + q^{35} + 15 q^{36} + 10 q^{37} + 6 q^{38} + 7 q^{39} + 5 q^{40} + 2 q^{41} + q^{42} + 8 q^{43} + 18 q^{44} - 7 q^{45} + 13 q^{46} + 12 q^{47} + 5 q^{48} + 4 q^{49} + q^{50} + 2 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 8 q^{55} + 8 q^{56} - 10 q^{57} + 20 q^{58} + 6 q^{59} + 2 q^{60} - 10 q^{61} + 7 q^{62} + 7 q^{63} + 12 q^{64} + 4 q^{65} - 8 q^{66} + 7 q^{67} - 2 q^{68} - 12 q^{69} + q^{70} + 22 q^{71} + 15 q^{72} - 23 q^{73} + 10 q^{74} - 34 q^{75} + 6 q^{76} - 19 q^{77} + 7 q^{78} + 13 q^{79} + 5 q^{80} - 28 q^{81} + 2 q^{82} + q^{83} + q^{84} - 28 q^{85} + 8 q^{86} - 22 q^{87} + 18 q^{88} - 3 q^{89} - 7 q^{90} - 21 q^{91} + 13 q^{92} - 33 q^{93} + 12 q^{94} - 2 q^{95} + 5 q^{96} - 70 q^{97} + 4 q^{98} - 6 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 5 * q^3 + 12 * q^4 + 5 * q^5 + 5 * q^6 + 8 * q^7 + 12 * q^8 + 15 * q^9 + 5 * q^10 + 18 * q^11 + 5 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^15 + 12 * q^16 - 2 * q^17 + 15 * q^18 + 6 * q^19 + 5 * q^20 + q^21 + 18 * q^22 + 13 * q^23 + 5 * q^24 + q^25 + 4 * q^26 + 8 * q^27 + 8 * q^28 + 20 * q^29 + 2 * q^30 + 7 * q^31 + 12 * q^32 - 8 * q^33 - 2 * q^34 + q^35 + 15 * q^36 + 10 * q^37 + 6 * q^38 + 7 * q^39 + 5 * q^40 + 2 * q^41 + q^42 + 8 * q^43 + 18 * q^44 - 7 * q^45 + 13 * q^46 + 12 * q^47 + 5 * q^48 + 4 * q^49 + q^50 + 2 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 8 * q^55 + 8 * q^56 - 10 * q^57 + 20 * q^58 + 6 * q^59 + 2 * q^60 - 10 * q^61 + 7 * q^62 + 7 * q^63 + 12 * q^64 + 4 * q^65 - 8 * q^66 + 7 * q^67 - 2 * q^68 - 12 * q^69 + q^70 + 22 * q^71 + 15 * q^72 - 23 * q^73 + 10 * q^74 - 34 * q^75 + 6 * q^76 - 19 * q^77 + 7 * q^78 + 13 * q^79 + 5 * q^80 - 28 * q^81 + 2 * q^82 + q^83 + q^84 - 28 * q^85 + 8 * q^86 - 22 * q^87 + 18 * q^88 - 3 * q^89 - 7 * q^90 - 21 * q^91 + 13 * q^92 - 33 * q^93 + 12 * q^94 - 2 * q^95 + 5 * q^96 - 70 * q^97 + 4 * q^98 - 6 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.49390	1.43985	0.719927	−	0.694050i	$-0.244175\pi$
$3$	2.49390	1.43985	0.719927	+	0.694050i	$0.244175\pi$
$4$	1.00000	0.500000
$5$	1.94778	0.871075	0.435537	−	0.900171i	$-0.356558\pi$
$5$	1.94778	0.871075	0.435537	+	0.900171i	$0.356558\pi$
$6$	2.49390	1.01813
$7$	−1.87622	−0.709144	−0.354572	−	0.935029i	$-0.615373\pi$
$7$	−1.87622	−0.709144	−0.354572	+	0.935029i	$0.615373\pi$
$8$	1.00000	0.353553
$9$	3.21954	1.07318
$10$	1.94778	0.615943
$11$	5.83443	1.75915	0.879574	−	0.475763i	$-0.157828\pi$
$11$	5.83443	1.75915	0.879574	+	0.475763i	$0.157828\pi$
$12$	2.49390	0.719927
$13$	−0.298003	−0.0826512	−0.0413256	−	0.999146i	$-0.513158\pi$
$13$	−0.298003	−0.0826512	−0.0413256	+	0.999146i	$0.513158\pi$
$14$	−1.87622	−0.501440
$15$	4.85757	1.25422
$16$	1.00000	0.250000
$17$	−7.33964	−1.78012	−0.890062	−	0.455840i	$-0.849339\pi$
$17$	−7.33964	−1.78012	−0.890062	+	0.455840i	$0.849339\pi$
$18$	3.21954	0.758853
$19$	−0.888571	−0.203852	−0.101926	−	0.994792i	$-0.532501\pi$
$19$	−0.888571	−0.203852	−0.101926	+	0.994792i	$0.532501\pi$
$20$	1.94778	0.435537
$21$	−4.67910	−1.02106
$22$	5.83443	1.24391
$23$	−8.64250	−1.80209	−0.901043	−	0.433730i	$-0.857197\pi$
$23$	−8.64250	−1.80209	−0.901043	+	0.433730i	$0.857197\pi$
$24$	2.49390	0.509065
$25$	−1.20615	−0.241229
$26$	−0.298003	−0.0584432
$27$	0.547512	0.105369
$28$	−1.87622	−0.354572
$29$	0.804767	0.149441	0.0747207	−	0.997204i	$-0.476193\pi$
$29$	0.804767	0.149441	0.0747207	+	0.997204i	$0.476193\pi$
$30$	4.85757	0.886868
$31$	−5.43347	−0.975880	−0.487940	−	0.872877i	$-0.662252\pi$
$31$	−5.43347	−0.975880	−0.487940	+	0.872877i	$0.662252\pi$
$32$	1.00000	0.176777
$33$	14.5505	2.53292
$34$	−7.33964	−1.25874
$35$	−3.65446	−0.617717
$36$	3.21954	0.536590
$37$	4.16366	0.684501	0.342251	−	0.939609i	$-0.388811\pi$
$37$	4.16366	0.684501	0.342251	+	0.939609i	$0.388811\pi$
$38$	−0.888571	−0.144145
$39$	−0.743190	−0.119006
$40$	1.94778	0.307971
$41$	2.14268	0.334630	0.167315	−	0.985903i	$-0.446490\pi$
$41$	2.14268	0.334630	0.167315	+	0.985903i	$0.446490\pi$
$42$	−4.67910	−0.722001
$43$	5.11414	0.779899	0.389950	−	0.920836i	$-0.372492\pi$
$43$	5.11414	0.779899	0.389950	+	0.920836i	$0.372492\pi$
$44$	5.83443	0.879574
$45$	6.27096	0.934820
$46$	−8.64250	−1.27427
$47$	10.2919	1.50123	0.750617	−	0.660738i	$-0.229757\pi$
$47$	10.2919	1.50123	0.750617	+	0.660738i	$0.229757\pi$
$48$	2.49390	0.359964
$49$	−3.47981	−0.497115
$50$	−1.20615	−0.170575
$51$	−18.3043	−2.56312
$52$	−0.298003	−0.0413256
$53$	9.43805	1.29642	0.648208	−	0.761464i	$-0.275519\pi$
$53$	9.43805	1.29642	0.648208	+	0.761464i	$0.275519\pi$
$54$	0.547512	0.0745069
$55$	11.3642	1.53235
$56$	−1.87622	−0.250720
$57$	−2.21601	−0.293517
$58$	0.804767	0.105671
$59$	−14.1014	−1.83585	−0.917924	−	0.396756i	$-0.870136\pi$
$59$	−14.1014	−1.83585	−0.917924	+	0.396756i	$0.870136\pi$
$60$	4.85757	0.627110
$61$	14.1214	1.80806	0.904030	−	0.427469i	$-0.140595\pi$
$61$	14.1214	1.80806	0.904030	+	0.427469i	$0.140595\pi$
$62$	−5.43347	−0.690051
$63$	−6.04056	−0.761039
$64$	1.00000	0.125000
$65$	−0.580445	−0.0719953
$66$	14.5505	1.79104
$67$	−6.24714	−0.763210	−0.381605	−	0.924326i	$-0.624629\pi$
$67$	−6.24714	−0.763210	−0.381605	+	0.924326i	$0.624629\pi$
$68$	−7.33964	−0.890062
$69$	−21.5535	−2.59474
$70$	−3.65446	−0.436792
$71$	4.03233	0.478549	0.239275	−	0.970952i	$-0.423090\pi$
$71$	4.03233	0.478549	0.239275	+	0.970952i	$0.423090\pi$
$72$	3.21954	0.379426
$73$	−8.07158	−0.944707	−0.472354	−	0.881409i	$-0.656595\pi$
$73$	−8.07158	−0.944707	−0.472354	+	0.881409i	$0.656595\pi$
$74$	4.16366	0.484016
$75$	−3.00801	−0.347335
$76$	−0.888571	−0.101926
$77$	−10.9467	−1.24749
$78$	−0.743190	−0.0841497
$79$	15.7749	1.77482	0.887409	−	0.460982i	$-0.152503\pi$
$79$	15.7749	1.77482	0.887409	+	0.460982i	$0.152503\pi$
$80$	1.94778	0.217769
$81$	−8.29318	−0.921465
$82$	2.14268	0.236619
$83$	3.93865	0.432323	0.216162	−	0.976358i	$-0.430646\pi$
$83$	3.93865	0.432323	0.216162	+	0.976358i	$0.430646\pi$
$84$	−4.67910	−0.510532
$85$	−14.2960	−1.55062
$86$	5.11414	0.551472
$87$	2.00701	0.215174
$88$	5.83443	0.621953
$89$	−4.09320	−0.433878	−0.216939	−	0.976185i	$-0.569607\pi$
$89$	−4.09320	−0.433878	−0.216939	+	0.976185i	$0.569607\pi$
$90$	6.27096	0.661017
$91$	0.559119	0.0586116
$92$	−8.64250	−0.901043
$93$	−13.5505	−1.40513
$94$	10.2919	1.06153
$95$	−1.73074	−0.177570
$96$	2.49390	0.254533
$97$	−8.46702	−0.859696	−0.429848	−	0.902901i	$-0.641433\pi$
$97$	−8.46702	−0.859696	−0.429848	+	0.902901i	$0.641433\pi$
$98$	−3.47981	−0.351513
$99$	18.7842	1.88788
$100$	−1.20615	−0.120615
$101$	−6.91486	−0.688054	−0.344027	−	0.938960i	$-0.611791\pi$
$101$	−6.91486	−0.688054	−0.344027	+	0.938960i	$0.611791\pi$
$102$	−18.3043	−1.81240
$103$	5.31246	0.523452	0.261726	−	0.965142i	$-0.415708\pi$
$103$	5.31246	0.523452	0.261726	+	0.965142i	$0.415708\pi$
$104$	−0.298003	−0.0292216
$105$	−9.11387	−0.889423
$106$	9.43805	0.916704
$107$	−10.3616	−1.00169	−0.500846	−	0.865536i	$-0.666978\pi$
$107$	−10.3616	−1.00169	−0.500846	+	0.865536i	$0.666978\pi$
$108$	0.547512	0.0526843
$109$	11.0031	1.05390	0.526952	−	0.849895i	$-0.323335\pi$
$109$	11.0031	1.05390	0.526952	+	0.849895i	$0.323335\pi$
$110$	11.3642	1.08353
$111$	10.3838	0.985582
$112$	−1.87622	−0.177286
$113$	−5.93637	−0.558447	−0.279223	−	0.960226i	$-0.590077\pi$
$113$	−5.93637	−0.558447	−0.279223	+	0.960226i	$0.590077\pi$
$114$	−2.21601	−0.207548
$115$	−16.8337	−1.56975
$116$	0.804767	0.0747207
$117$	−0.959433	−0.0886996
$118$	−14.1014	−1.29814
$119$	13.7708	1.26236
$120$	4.85757	0.443434
$121$	23.0406	2.09460
$122$	14.1214	1.27849
$123$	5.34363	0.481819
$124$	−5.43347	−0.487940
$125$	−12.0882	−1.08120
$126$	−6.04056	−0.538136
$127$	4.22816	0.375188	0.187594	−	0.982247i	$-0.439931\pi$
$127$	4.22816	0.375188	0.187594	+	0.982247i	$0.439931\pi$
$128$	1.00000	0.0883883
$129$	12.7542	1.12294
$130$	−0.580445	−0.0509084
$131$	−12.9405	−1.13061	−0.565307	−	0.824881i	$-0.691242\pi$
$131$	−12.9405	−1.13061	−0.565307	+	0.824881i	$0.691242\pi$
$132$	14.5505	1.26646
$133$	1.66715	0.144560
$134$	−6.24714	−0.539671
$135$	1.06643	0.0917840
$136$	−7.33964	−0.629369
$137$	3.32249	0.283860	0.141930	−	0.989877i	$-0.454669\pi$
$137$	3.32249	0.283860	0.141930	+	0.989877i	$0.454669\pi$
$138$	−21.5535	−1.83476
$139$	17.4440	1.47958	0.739792	−	0.672836i	$-0.234924\pi$
$139$	17.4440	1.47958	0.739792	+	0.672836i	$0.234924\pi$
$140$	−3.65446	−0.308859
$141$	25.6671	2.16156
$142$	4.03233	0.338385
$143$	−1.73868	−0.145396
$144$	3.21954	0.268295
$145$	1.56751	0.130175
$146$	−8.07158	−0.668009
$147$	−8.67829	−0.715773
$148$	4.16366	0.342251
$149$	−2.15279	−0.176363	−0.0881817	−	0.996104i	$-0.528106\pi$
$149$	−2.15279	−0.176363	−0.0881817	+	0.996104i	$0.528106\pi$
$150$	−3.00801	−0.245603
$151$	8.86908	0.721756	0.360878	−	0.932613i	$-0.382477\pi$
$151$	8.86908	0.721756	0.360878	+	0.932613i	$0.382477\pi$
$152$	−0.888571	−0.0720726
$153$	−23.6303	−1.91039
$154$	−10.9467	−0.882108
$155$	−10.5832	−0.850064
$156$	−0.743190	−0.0595028
$157$	−6.74771	−0.538526	−0.269263	−	0.963067i	$-0.586780\pi$
$157$	−6.74771	−0.538526	−0.269263	+	0.963067i	$0.586780\pi$
$158$	15.7749	1.25499
$159$	23.5375	1.86665
$160$	1.94778	0.153986
$161$	16.2152	1.27794
$162$	−8.29318	−0.651574
$163$	6.34872	0.497270	0.248635	−	0.968597i	$-0.420018\pi$
$163$	6.34872	0.497270	0.248635	+	0.968597i	$0.420018\pi$
$164$	2.14268	0.167315
$165$	28.3412	2.20636
$166$	3.93865	0.305699
$167$	−22.9525	−1.77612	−0.888060	−	0.459727i	$-0.847947\pi$
$167$	−22.9525	−1.77612	−0.888060	+	0.459727i	$0.847947\pi$
$168$	−4.67910	−0.361001
$169$	−12.9112	−0.993169
$170$	−14.2960	−1.09645
$171$	−2.86079	−0.218770
$172$	5.11414	0.389950
$173$	24.5497	1.86648	0.933238	−	0.359258i	$-0.116970\pi$
$173$	24.5497	1.86648	0.933238	+	0.359258i	$0.116970\pi$
$174$	2.00701	0.152151
$175$	2.26299	0.171066
$176$	5.83443	0.439787
$177$	−35.1675	−2.64335
$178$	−4.09320	−0.306798
$179$	9.50528	0.710458	0.355229	−	0.934779i	$-0.384403\pi$
$179$	9.50528	0.710458	0.355229	+	0.934779i	$0.384403\pi$
$180$	6.27096	0.467410
$181$	−20.8706	−1.55130	−0.775651	−	0.631162i	$-0.782578\pi$
$181$	−20.8706	−1.55130	−0.775651	+	0.631162i	$0.782578\pi$
$182$	0.559119	0.0414446
$183$	35.2174	2.60334
$184$	−8.64250	−0.637133
$185$	8.10990	0.596252
$186$	−13.5505	−0.993574
$187$	−42.8226	−3.13150
$188$	10.2919	0.750617
$189$	−1.02725	−0.0747215
$190$	−1.73074	−0.125561
$191$	19.8473	1.43610	0.718049	−	0.695993i	$-0.245035\pi$
$191$	19.8473	1.43610	0.718049	+	0.695993i	$0.245035\pi$
$192$	2.49390	0.179982
$193$	−14.6490	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$193$	−14.6490	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$194$	−8.46702	−0.607897
$195$	−1.44757	−0.103663
$196$	−3.47981	−0.248558
$197$	−9.37280	−0.667785	−0.333892	−	0.942611i	$-0.608362\pi$
$197$	−9.37280	−0.667785	−0.333892	+	0.942611i	$0.608362\pi$
$198$	18.7842	1.33493
$199$	−14.6360	−1.03752	−0.518760	−	0.854920i	$-0.673606\pi$
$199$	−14.6360	−1.03752	−0.518760	+	0.854920i	$0.673606\pi$
$200$	−1.20615	−0.0852874
$201$	−15.5798	−1.09891
$202$	−6.91486	−0.486528
$203$	−1.50992	−0.105975
$204$	−18.3043	−1.28156
$205$	4.17347	0.291488
$206$	5.31246	0.370137
$207$	−27.8249	−1.93396
$208$	−0.298003	−0.0206628
$209$	−5.18431	−0.358606
$210$	−9.11387	−0.628917
$211$	3.94613	0.271663	0.135831	−	0.990732i	$-0.456629\pi$
$211$	3.94613	0.271663	0.135831	+	0.990732i	$0.456629\pi$
$212$	9.43805	0.648208
$213$	10.0562	0.689041
$214$	−10.3616	−0.708304
$215$	9.96123	0.679350
$216$	0.547512	0.0372535
$217$	10.1944	0.692039
$218$	11.0031	0.745223
$219$	−20.1297	−1.36024
$220$	11.3642	0.766174
$221$	2.18723	0.147129
$222$	10.3838	0.696912
$223$	−4.32975	−0.289941	−0.144971	−	0.989436i	$-0.546309\pi$
$223$	−4.32975	−0.289941	−0.144971	+	0.989436i	$0.546309\pi$
$224$	−1.87622	−0.125360
$225$	−3.88324	−0.258882
$226$	−5.93637	−0.394881
$227$	1.56410	0.103813	0.0519064	−	0.998652i	$-0.483470\pi$
$227$	1.56410	0.103813	0.0519064	+	0.998652i	$0.483470\pi$
$228$	−2.21601	−0.146759
$229$	−23.6879	−1.56534	−0.782672	−	0.622435i	$-0.786144\pi$
$229$	−23.6879	−1.56534	−0.782672	+	0.622435i	$0.786144\pi$
$230$	−16.8337	−1.10998
$231$	−27.2999	−1.79620
$232$	0.804767	0.0528355
$233$	19.7694	1.29513	0.647567	−	0.762008i	$-0.275787\pi$
$233$	19.7694	1.29513	0.647567	+	0.762008i	$0.275787\pi$
$234$	−0.959433	−0.0627201
$235$	20.0464	1.30769
$236$	−14.1014	−0.917924
$237$	39.3411	2.55548
$238$	13.7708	0.892626
$239$	−2.12172	−0.137243	−0.0686214	−	0.997643i	$-0.521860\pi$
$239$	−2.12172	−0.137243	−0.0686214	+	0.997643i	$0.521860\pi$
$240$	4.85757	0.313555
$241$	20.8089	1.34042	0.670210	−	0.742172i	$-0.266204\pi$
$241$	20.8089	1.34042	0.670210	+	0.742172i	$0.266204\pi$
$242$	23.0406	1.48111
$243$	−22.3249	−1.43214
$244$	14.1214	0.904030
$245$	−6.77790	−0.433024
$246$	5.34363	0.340697
$247$	0.264797	0.0168486
$248$	−5.43347	−0.345026
$249$	9.82260	0.622482
$250$	−12.0882	−0.764526
$251$	19.6095	1.23774	0.618870	−	0.785494i	$-0.287591\pi$
$251$	19.6095	1.23774	0.618870	+	0.785494i	$0.287591\pi$
$252$	−6.04056	−0.380520
$253$	−50.4241	−3.17013
$254$	4.22816	0.265298
$255$	−35.6528	−2.23267
$256$	1.00000	0.0625000
$257$	4.20578	0.262349	0.131175	−	0.991359i	$-0.458125\pi$
$257$	4.20578	0.262349	0.131175	+	0.991359i	$0.458125\pi$
$258$	12.7542	0.794039
$259$	−7.81193	−0.485410
$260$	−0.580445	−0.0359977
$261$	2.59098	0.160378
$262$	−12.9405	−0.799465
$263$	4.95237	0.305376	0.152688	−	0.988274i	$-0.451207\pi$
$263$	4.95237	0.305376	0.152688	+	0.988274i	$0.451207\pi$
$264$	14.5505	0.895521
$265$	18.3833	1.12927
$266$	1.66715	0.102220
$267$	−10.2080	−0.624721
$268$	−6.24714	−0.381605
$269$	17.6754	1.07769	0.538843	−	0.842406i	$-0.318862\pi$
$269$	17.6754	1.07769	0.538843	+	0.842406i	$0.318862\pi$
$270$	1.06643	0.0649011
$271$	6.71122	0.407678	0.203839	−	0.979004i	$-0.434658\pi$
$271$	6.71122	0.407678	0.203839	+	0.979004i	$0.434658\pi$
$272$	−7.33964	−0.445031
$273$	1.39439	0.0843921
$274$	3.32249	0.200719
$275$	−7.03718	−0.424358
$276$	−21.5535	−1.29737
$277$	−22.4543	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$277$	−22.4543	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$278$	17.4440	1.04622
$279$	−17.4933	−1.04730
$280$	−3.65446	−0.218396
$281$	20.1623	1.20278	0.601391	−	0.798955i	$-0.294613\pi$
$281$	20.1623	1.20278	0.601391	+	0.798955i	$0.294613\pi$
$282$	25.6671	1.52845
$283$	26.3840	1.56837	0.784183	−	0.620530i	$-0.213082\pi$
$283$	26.3840	1.56837	0.784183	+	0.620530i	$0.213082\pi$
$284$	4.03233	0.239275
$285$	−4.31630	−0.255675
$286$	−1.73868	−0.102810
$287$	−4.02013	−0.237301
$288$	3.21954	0.189713
$289$	36.8703	2.16884
$290$	1.56751	0.0920474
$291$	−21.1159	−1.23784
$292$	−8.07158	−0.472354
$293$	−7.77069	−0.453969	−0.226984	−	0.973898i	$-0.572887\pi$
$293$	−7.77069	−0.453969	−0.226984	+	0.973898i	$0.572887\pi$
$294$	−8.67829	−0.506128
$295$	−27.4665	−1.59916
$296$	4.16366	0.242008
$297$	3.19442	0.185359
$298$	−2.15279	−0.124708
$299$	2.57549	0.148945
$300$	−3.00801	−0.173667
$301$	−9.59524	−0.553061
$302$	8.86908	0.510358
$303$	−17.2450	−0.990697
$304$	−0.888571	−0.0509630
$305$	27.5054	1.57496
$306$	−23.6303	−1.35085
$307$	27.3128	1.55883	0.779413	−	0.626510i	$-0.215517\pi$
$307$	27.3128	1.55883	0.779413	+	0.626510i	$0.215517\pi$
$308$	−10.9467	−0.623744
$309$	13.2488	0.753695
$310$	−10.5832	−0.601086
$311$	10.3916	0.589252	0.294626	−	0.955613i	$-0.404805\pi$
$311$	10.3916	0.589252	0.294626	+	0.955613i	$0.404805\pi$
$312$	−0.743190	−0.0420749
$313$	−8.62733	−0.487645	−0.243823	−	0.969820i	$-0.578401\pi$
$313$	−8.62733	−0.487645	−0.243823	+	0.969820i	$0.578401\pi$
$314$	−6.74771	−0.380796
$315$	−11.7657	−0.662922
$316$	15.7749	0.887409
$317$	23.7917	1.33628	0.668138	−	0.744037i	$-0.267092\pi$
$317$	23.7917	1.33628	0.668138	+	0.744037i	$0.267092\pi$
$318$	23.5375	1.31992
$319$	4.69536	0.262890
$320$	1.94778	0.108884
$321$	−25.8408	−1.44229
$322$	16.2152	0.903639
$323$	6.52179	0.362882
$324$	−8.29318	−0.460732
$325$	0.359435	0.0199379
$326$	6.34872	0.351623
$327$	27.4406	1.51747
$328$	2.14268	0.118310
$329$	−19.3099	−1.06459
$330$	28.3412	1.56013
$331$	−16.6716	−0.916353	−0.458177	−	0.888861i	$-0.651497\pi$
$331$	−16.6716	−0.916353	−0.458177	+	0.888861i	$0.651497\pi$
$332$	3.93865	0.216162
$333$	13.4051	0.734593
$334$	−22.9525	−1.25591
$335$	−12.1681	−0.664813
$336$	−4.67910	−0.255266
$337$	−11.6612	−0.635225	−0.317612	−	0.948221i	$-0.602881\pi$
$337$	−11.6612	−0.635225	−0.317612	+	0.948221i	$0.602881\pi$
$338$	−12.9112	−0.702276
$339$	−14.8047	−0.804082
$340$	−14.2960	−0.775310
$341$	−31.7012	−1.71672
$342$	−2.86079	−0.154694
$343$	19.6624	1.06167
$344$	5.11414	0.275736
$345$	−41.9816	−2.26021
$346$	24.5497	1.31980
$347$	−13.8126	−0.741500	−0.370750	−	0.928733i	$-0.620899\pi$
$347$	−13.8126	−0.741500	−0.370750	+	0.928733i	$0.620899\pi$
$348$	2.00701	0.107587
$349$	13.7647	0.736807	0.368404	−	0.929666i	$-0.379904\pi$
$349$	13.7647	0.736807	0.368404	+	0.929666i	$0.379904\pi$
$350$	2.26299	0.120962
$351$	−0.163160	−0.00870885
$352$	5.83443	0.310976
$353$	7.48060	0.398152	0.199076	−	0.979984i	$-0.436206\pi$
$353$	7.48060	0.398152	0.199076	+	0.979984i	$0.436206\pi$
$354$	−35.1675	−1.86913
$355$	7.85410	0.416852
$356$	−4.09320	−0.216939
$357$	34.3429	1.81762
$358$	9.50528	0.502370
$359$	13.8737	0.732227	0.366113	−	0.930570i	$-0.380688\pi$
$359$	13.8737	0.732227	0.366113	+	0.930570i	$0.380688\pi$
$360$	6.27096	0.330509
$361$	−18.2104	−0.958444
$362$	−20.8706	−1.09694
$363$	57.4610	3.01592
$364$	0.559119	0.0293058
$365$	−15.7217	−0.822910
$366$	35.2174	1.84084
$367$	−10.7012	−0.558599	−0.279300	−	0.960204i	$-0.590102\pi$
$367$	−10.7012	−0.558599	−0.279300	+	0.960204i	$0.590102\pi$
$368$	−8.64250	−0.450521
$369$	6.89844	0.359119
$370$	8.10990	0.421614
$371$	−17.7078	−0.919345
$372$	−13.5505	−0.702563
$373$	−27.4619	−1.42192	−0.710962	−	0.703230i	$-0.751740\pi$
$373$	−27.4619	−1.42192	−0.710962	+	0.703230i	$0.751740\pi$
$374$	−42.8226	−2.21430
$375$	−30.1468	−1.55677
$376$	10.2919	0.530766
$377$	−0.239823	−0.0123515
$378$	−1.02725	−0.0528361
$379$	−17.8711	−0.917978	−0.458989	−	0.888442i	$-0.651788\pi$
$379$	−17.8711	−0.917978	−0.458989	+	0.888442i	$0.651788\pi$
$380$	−1.73074	−0.0887852
$381$	10.5446	0.540217
$382$	19.8473	1.01547
$383$	10.8778	0.555832	0.277916	−	0.960605i	$-0.410356\pi$
$383$	10.8778	0.555832	0.277916	+	0.960605i	$0.410356\pi$
$384$	2.49390	0.127266
$385$	−21.3217	−1.08666
$386$	−14.6490	−0.745614
$387$	16.4652	0.836972
$388$	−8.46702	−0.429848
$389$	−4.52471	−0.229412	−0.114706	−	0.993399i	$-0.536593\pi$
$389$	−4.52471	−0.229412	−0.114706	+	0.993399i	$0.536593\pi$
$390$	−1.44757	−0.0733007
$391$	63.4328	3.20794
$392$	−3.47981	−0.175757
$393$	−32.2722	−1.62792
$394$	−9.37280	−0.472195
$395$	30.7261	1.54600
$396$	18.7842	0.943941
$397$	−27.6450	−1.38746	−0.693731	−	0.720234i	$-0.744034\pi$
$397$	−27.6450	−1.38746	−0.693731	+	0.720234i	$0.744034\pi$
$398$	−14.6360	−0.733637
$399$	4.15771	0.208146
$400$	−1.20615	−0.0603073
$401$	−1.76900	−0.0883397	−0.0441699	−	0.999024i	$-0.514064\pi$
$401$	−1.76900	−0.0883397	−0.0441699	+	0.999024i	$0.514064\pi$
$402$	−15.5798	−0.777047
$403$	1.61919	0.0806577
$404$	−6.91486	−0.344027
$405$	−16.1533	−0.802664
$406$	−1.50992	−0.0749360
$407$	24.2926	1.20414
$408$	−18.3043	−0.906199
$409$	−26.0238	−1.28679	−0.643396	−	0.765533i	$-0.722475\pi$
$409$	−26.0238	−1.28679	−0.643396	+	0.765533i	$0.722475\pi$
$410$	4.17347	0.206113
$411$	8.28596	0.408716
$412$	5.31246	0.261726
$413$	26.4573	1.30188
$414$	−27.8249	−1.36752
$415$	7.67163	0.376586
$416$	−0.298003	−0.0146108
$417$	43.5037	2.13038
$418$	−5.18431	−0.253573
$419$	4.90699	0.239722	0.119861	−	0.992791i	$-0.461755\pi$
$419$	4.90699	0.239722	0.119861	+	0.992791i	$0.461755\pi$
$420$	−9.11387	−0.444711
$421$	7.00976	0.341635	0.170817	−	0.985303i	$-0.445359\pi$
$421$	7.00976	0.341635	0.170817	+	0.985303i	$0.445359\pi$
$422$	3.94613	0.192095
$423$	33.1353	1.61109
$424$	9.43805	0.458352
$425$	8.85267	0.429418
$426$	10.0562	0.487226
$427$	−26.4948	−1.28217
$428$	−10.3616	−0.500846
$429$	−4.33609	−0.209348
$430$	9.96123	0.480373
$431$	16.6838	0.803628	0.401814	−	0.915721i	$-0.368380\pi$
$431$	16.6838	0.803628	0.401814	+	0.915721i	$0.368380\pi$
$432$	0.547512	0.0263422
$433$	−16.3926	−0.787777	−0.393889	−	0.919158i	$-0.628870\pi$
$433$	−16.3926	−0.787777	−0.393889	+	0.919158i	$0.628870\pi$
$434$	10.1944	0.489346
$435$	3.90921	0.187433
$436$	11.0031	0.526952
$437$	7.67947	0.367359
$438$	−20.1297	−0.961835
$439$	−10.9201	−0.521188	−0.260594	−	0.965448i	$-0.583918\pi$
$439$	−10.9201	−0.521188	−0.260594	+	0.965448i	$0.583918\pi$
$440$	11.3642	0.541767
$441$	−11.2034	−0.533494
$442$	2.18723	0.104036
$443$	−8.44935	−0.401441	−0.200720	−	0.979649i	$-0.564328\pi$
$443$	−8.44935	−0.401441	−0.200720	+	0.979649i	$0.564328\pi$
$444$	10.3838	0.492791
$445$	−7.97266	−0.377940
$446$	−4.32975	−0.205019
$447$	−5.36884	−0.253938
$448$	−1.87622	−0.0886430
$449$	12.1220	0.572071	0.286036	−	0.958219i	$-0.407662\pi$
$449$	12.1220	0.572071	0.286036	+	0.958219i	$0.407662\pi$
$450$	−3.88324	−0.183057
$451$	12.5013	0.588664
$452$	−5.93637	−0.279223
$453$	22.1186	1.03922
$454$	1.56410	0.0734067
$455$	1.08904	0.0510551
$456$	−2.21601	−0.103774
$457$	−37.0355	−1.73245	−0.866223	−	0.499657i	$-0.833459\pi$
$457$	−37.0355	−1.73245	−0.866223	+	0.499657i	$0.833459\pi$
$458$	−23.6879	−1.10686
$459$	−4.01854	−0.187569
$460$	−16.8337	−0.784875
$461$	−3.06027	−0.142531	−0.0712656	−	0.997457i	$-0.522704\pi$
$461$	−3.06027	−0.142531	−0.0712656	+	0.997457i	$0.522704\pi$
$462$	−27.2999	−1.27011
$463$	7.69735	0.357726	0.178863	−	0.983874i	$-0.442758\pi$
$463$	7.69735	0.357726	0.178863	+	0.983874i	$0.442758\pi$
$464$	0.804767	0.0373604
$465$	−26.3935	−1.22397
$466$	19.7694	0.915798
$467$	−22.8952	−1.05946	−0.529732	−	0.848165i	$-0.677707\pi$
$467$	−22.8952	−1.05946	−0.529732	+	0.848165i	$0.677707\pi$
$468$	−0.959433	−0.0443498
$469$	11.7210	0.541226
$470$	20.0464	0.924674
$471$	−16.8281	−0.775399
$472$	−14.1014	−0.649070
$473$	29.8381	1.37196
$474$	39.3411	1.80700
$475$	1.07175	0.0491751
$476$	13.7708	0.631182
$477$	30.3862	1.39129
$478$	−2.12172	−0.0970454
$479$	25.5092	1.16555	0.582773	−	0.812635i	$-0.301968\pi$
$479$	25.5092	1.16555	0.582773	+	0.812635i	$0.301968\pi$
$480$	4.85757	0.221717
$481$	−1.24078	−0.0565749
$482$	20.8089	0.947820
$483$	40.4391	1.84004
$484$	23.0406	1.04730
$485$	−16.4919	−0.748859
$486$	−22.3249	−1.01268
$487$	27.0841	1.22730	0.613648	−	0.789580i	$-0.289701\pi$
$487$	27.0841	1.22730	0.613648	+	0.789580i	$0.289701\pi$
$488$	14.1214	0.639246
$489$	15.8331	0.715997
$490$	−6.77790	−0.306194
$491$	36.9457	1.66734	0.833669	−	0.552265i	$-0.186236\pi$
$491$	36.9457	1.66734	0.833669	+	0.552265i	$0.186236\pi$
$492$	5.34363	0.240909
$493$	−5.90670	−0.266024
$494$	0.264797	0.0119138
$495$	36.5875	1.64449
$496$	−5.43347	−0.243970
$497$	−7.56553	−0.339360
$498$	9.82260	0.440161
$499$	43.5308	1.94871	0.974353	−	0.225027i	$-0.0722470\pi$
$499$	43.5308	1.94871	0.974353	+	0.225027i	$0.0722470\pi$
$500$	−12.0882	−0.540602
$501$	−57.2413	−2.55735
$502$	19.6095	0.875214
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	−6.04056	−0.269068
$505$	−13.4686	−0.599346
$506$	−50.4241	−2.24162
$507$	−32.1992	−1.43002
$508$	4.22816	0.187594
$509$	−8.75597	−0.388101	−0.194051	−	0.980992i	$-0.562163\pi$
$509$	−8.75597	−0.388101	−0.194051	+	0.980992i	$0.562163\pi$
$510$	−35.6528	−1.57873
$511$	15.1441	0.669933
$512$	1.00000	0.0441942
$513$	−0.486503	−0.0214796
$514$	4.20578	0.185509
$515$	10.3475	0.455966
$516$	12.7542	0.561470
$517$	60.0476	2.64089
$518$	−7.81193	−0.343237
$519$	61.2244	2.68745
$520$	−0.580445	−0.0254542
$521$	23.4843	1.02887	0.514433	−	0.857530i	$-0.328002\pi$
$521$	23.4843	1.02887	0.514433	+	0.857530i	$0.328002\pi$
$522$	2.59098	0.113404
$523$	−16.0353	−0.701177	−0.350588	−	0.936530i	$-0.614018\pi$
$523$	−16.0353	−0.701177	−0.350588	+	0.936530i	$0.614018\pi$
$524$	−12.9405	−0.565307
$525$	5.64368	0.246310
$526$	4.95237	0.215934
$527$	39.8797	1.73719
$528$	14.5505	0.633229
$529$	51.6928	2.24751
$530$	18.3833	0.798518
$531$	−45.4001	−1.97020
$532$	1.66715	0.0722802
$533$	−0.638525	−0.0276576
$534$	−10.2080	−0.441745
$535$	−20.1821	−0.872549
$536$	−6.24714	−0.269835
$537$	23.7052	1.02296
$538$	17.6754	0.762039
$539$	−20.3027	−0.874499
$540$	1.06643	0.0458920
$541$	20.8582	0.896765	0.448382	−	0.893842i	$-0.352000\pi$
$541$	20.8582	0.896765	0.448382	+	0.893842i	$0.352000\pi$
$542$	6.71122	0.288272
$543$	−52.0493	−2.23365
$544$	−7.33964	−0.314684
$545$	21.4316	0.918029
$546$	1.39439	0.0596742
$547$	0.852077	0.0364322	0.0182161	−	0.999834i	$-0.494201\pi$
$547$	0.852077	0.0364322	0.0182161	+	0.999834i	$0.494201\pi$
$548$	3.32249	0.141930
$549$	45.4644	1.94037
$550$	−7.03718	−0.300066
$551$	−0.715092	−0.0304640
$552$	−21.5535	−0.917379
$553$	−29.5972	−1.25860
$554$	−22.4543	−0.953994
$555$	20.2253	0.858516
$556$	17.4440	0.739792
$557$	−35.0509	−1.48515	−0.742576	−	0.669762i	$-0.766396\pi$
$557$	−35.0509	−1.48515	−0.742576	+	0.669762i	$0.766396\pi$
$558$	−17.4933	−0.740550
$559$	−1.52403	−0.0644596
$560$	−3.65446	−0.154429
$561$	−106.795	−4.50890
$562$	20.1623	0.850495
$563$	32.9907	1.39039	0.695196	−	0.718820i	$-0.255317\pi$
$563$	32.9907	1.39039	0.695196	+	0.718820i	$0.255317\pi$
$564$	25.6671	1.08078
$565$	−11.5628	−0.486449
$566$	26.3840	1.10900
$567$	15.5598	0.653451
$568$	4.03233	0.169193
$569$	−18.1590	−0.761263	−0.380632	−	0.924727i	$-0.624293\pi$
$569$	−18.1590	−0.761263	−0.380632	+	0.924727i	$0.624293\pi$
$570$	−4.31630	−0.180790
$571$	−29.0313	−1.21492	−0.607462	−	0.794349i	$-0.707812\pi$
$571$	−29.0313	−1.21492	−0.607462	+	0.794349i	$0.707812\pi$
$572$	−1.73868	−0.0726978
$573$	49.4971	2.06777
$574$	−4.02013	−0.167797
$575$	10.4241	0.434716
$576$	3.21954	0.134148
$577$	15.0405	0.626145	0.313073	−	0.949729i	$-0.398642\pi$
$577$	15.0405	0.626145	0.313073	+	0.949729i	$0.398642\pi$
$578$	36.8703	1.53360
$579$	−36.5331	−1.51826
$580$	1.56751	0.0650873
$581$	−7.38977	−0.306579
$582$	−21.1159	−0.875283
$583$	55.0656	2.28059
$584$	−8.07158	−0.334004
$585$	−1.86877	−0.0772640
$586$	−7.77069	−0.321004
$587$	27.1709	1.12146	0.560732	−	0.827997i	$-0.310520\pi$
$587$	27.1709	1.12146	0.560732	+	0.827997i	$0.310520\pi$
$588$	−8.67829	−0.357887
$589$	4.82802	0.198935
$590$	−27.4665	−1.13078
$591$	−23.3748	−0.961512
$592$	4.16366	0.171125
$593$	12.0238	0.493757	0.246878	−	0.969046i	$-0.420595\pi$
$593$	12.0238	0.493757	0.246878	+	0.969046i	$0.420595\pi$
$594$	3.19442	0.131069
$595$	26.8224	1.09961
$596$	−2.15279	−0.0881817
$597$	−36.5008	−1.49388
$598$	2.57549	0.105320
$599$	15.3739	0.628160	0.314080	−	0.949396i	$-0.398304\pi$
$599$	15.3739	0.628160	0.314080	+	0.949396i	$0.398304\pi$
$600$	−3.00801	−0.122801
$601$	−16.1170	−0.657427	−0.328713	−	0.944430i	$-0.606615\pi$
$601$	−16.1170	−0.657427	−0.328713	+	0.944430i	$0.606615\pi$
$602$	−9.59524	−0.391073
$603$	−20.1129	−0.819062
$604$	8.86908	0.360878
$605$	44.8781	1.82455
$606$	−17.2450	−0.700529
$607$	42.5003	1.72503	0.862517	−	0.506029i	$-0.168887\pi$
$607$	42.5003	1.72503	0.862517	+	0.506029i	$0.168887\pi$
$608$	−0.888571	−0.0360363
$609$	−3.76559	−0.152589
$610$	27.5054	1.11366
$611$	−3.06703	−0.124079
$612$	−23.6303	−0.955197
$613$	7.01602	0.283374	0.141687	−	0.989911i	$-0.454747\pi$
$613$	7.01602	0.283374	0.141687	+	0.989911i	$0.454747\pi$
$614$	27.3128	1.10226
$615$	10.4082	0.419700
$616$	−10.9467	−0.441054
$617$	−14.1457	−0.569483	−0.284742	−	0.958604i	$-0.591908\pi$
$617$	−14.1457	−0.569483	−0.284742	+	0.958604i	$0.591908\pi$
$618$	13.2488	0.532943
$619$	9.37865	0.376960	0.188480	−	0.982077i	$-0.439644\pi$
$619$	9.37865	0.376960	0.188480	+	0.982077i	$0.439644\pi$
$620$	−10.5832	−0.425032
$621$	−4.73187	−0.189883
$622$	10.3916	0.416664
$623$	7.67973	0.307682
$624$	−0.743190	−0.0297514
$625$	−17.5145	−0.700579
$626$	−8.62733	−0.344817
$627$	−12.9291	−0.516340
$628$	−6.74771	−0.269263
$629$	−30.5598	−1.21850
$630$	−11.7657	−0.468756
$631$	−8.90810	−0.354626	−0.177313	−	0.984155i	$-0.556740\pi$
$631$	−8.90810	−0.354626	−0.177313	+	0.984155i	$0.556740\pi$
$632$	15.7749	0.627493
$633$	9.84126	0.391155
$634$	23.7917	0.944890
$635$	8.23553	0.326817
$636$	23.5375	0.933324
$637$	1.03699	0.0410871
$638$	4.69536	0.185891
$639$	12.9822	0.513570
$640$	1.94778	0.0769928
$641$	−31.0208	−1.22525	−0.612625	−	0.790374i	$-0.709886\pi$
$641$	−31.0208	−1.22525	−0.612625	+	0.790374i	$0.709886\pi$
$642$	−25.8408	−1.01985
$643$	−36.4005	−1.43550	−0.717749	−	0.696302i	$-0.754827\pi$
$643$	−36.4005	−1.43550	−0.717749	+	0.696302i	$0.754827\pi$
$644$	16.2152	0.638969
$645$	24.8423	0.978165
$646$	6.52179	0.256596
$647$	6.09466	0.239606	0.119803	−	0.992798i	$-0.461774\pi$
$647$	6.09466	0.239606	0.119803	+	0.992798i	$0.461774\pi$
$648$	−8.29318	−0.325787
$649$	−82.2738	−3.22953
$650$	0.359435	0.0140982
$651$	25.4238	0.996436
$652$	6.34872	0.248635
$653$	−25.9660	−1.01613	−0.508063	−	0.861320i	$-0.669638\pi$
$653$	−25.9660	−1.01613	−0.508063	+	0.861320i	$0.669638\pi$
$654$	27.4406	1.07301
$655$	−25.2052	−0.984849
$656$	2.14268	0.0836576
$657$	−25.9868	−1.01384
$658$	−19.3099	−0.752779
$659$	−39.3845	−1.53420	−0.767102	−	0.641525i	$-0.778302\pi$
$659$	−39.3845	−1.53420	−0.767102	+	0.641525i	$0.778302\pi$
$660$	28.3412	1.10318
$661$	−3.77429	−0.146803	−0.0734015	−	0.997302i	$-0.523385\pi$
$661$	−3.77429	−0.146803	−0.0734015	+	0.997302i	$0.523385\pi$
$662$	−16.6716	−0.647960
$663$	5.45475	0.211845
$664$	3.93865	0.152849
$665$	3.24725	0.125923
$666$	13.4051	0.519436
$667$	−6.95520	−0.269306
$668$	−22.9525	−0.888060
$669$	−10.7980	−0.417473
$670$	−12.1681	−0.470094
$671$	82.3904	3.18064
$672$	−4.67910	−0.180500
$673$	0.143299	0.00552375	0.00276188	−	0.999996i	$-0.499121\pi$
$673$	0.143299	0.00552375	0.00276188	+	0.999996i	$0.499121\pi$
$674$	−11.6612	−0.449172
$675$	−0.660379	−0.0254180
$676$	−12.9112	−0.496584
$677$	47.2278	1.81511	0.907556	−	0.419931i	$-0.137946\pi$
$677$	47.2278	1.81511	0.907556	+	0.419931i	$0.137946\pi$
$678$	−14.8047	−0.568572
$679$	15.8860	0.609648
$680$	−14.2960	−0.548227
$681$	3.90070	0.149475
$682$	−31.7012	−1.21390
$683$	30.6265	1.17189	0.585945	−	0.810351i	$-0.300724\pi$
$683$	30.6265	1.17189	0.585945	+	0.810351i	$0.300724\pi$
$684$	−2.86079	−0.109385
$685$	6.47149	0.247263
$686$	19.6624	0.750714
$687$	−59.0754	−2.25387
$688$	5.11414	0.194975
$689$	−2.81257	−0.107150
$690$	−41.9816	−1.59821
$691$	16.3359	0.621448	0.310724	−	0.950500i	$-0.399429\pi$
$691$	16.3359	0.621448	0.310724	+	0.950500i	$0.399429\pi$
$692$	24.5497	0.933238
$693$	−35.2432	−1.33878
$694$	−13.8126	−0.524320
$695$	33.9772	1.28883
$696$	2.00701	0.0760755
$697$	−15.7265	−0.595683
$698$	13.7647	0.521001
$699$	49.3028	1.86480
$700$	2.26299	0.0855331
$701$	−26.1216	−0.986599	−0.493300	−	0.869859i	$-0.664209\pi$
$701$	−26.1216	−0.986599	−0.493300	+	0.869859i	$0.664209\pi$
$702$	−0.163160	−0.00615808
$703$	−3.69971	−0.139537
$704$	5.83443	0.219893
$705$	49.9938	1.88288
$706$	7.48060	0.281536
$707$	12.9738	0.487929
$708$	−35.1675	−1.32168
$709$	−8.13805	−0.305631	−0.152815	−	0.988255i	$-0.548834\pi$
$709$	−8.13805	−0.305631	−0.152815	+	0.988255i	$0.548834\pi$
$710$	7.85410	0.294759
$711$	50.7880	1.90470
$712$	−4.09320	−0.153399
$713$	46.9588	1.75862
$714$	34.3429	1.28525
$715$	−3.38657	−0.126650
$716$	9.50528	0.355229
$717$	−5.29137	−0.197610
$718$	13.8737	0.517762
$719$	−5.26201	−0.196240	−0.0981199	−	0.995175i	$-0.531283\pi$
$719$	−5.26201	−0.196240	−0.0981199	+	0.995175i	$0.531283\pi$
$720$	6.27096	0.233705
$721$	−9.96734	−0.371203
$722$	−18.2104	−0.677722
$723$	51.8953	1.93001
$724$	−20.8706	−0.775651
$725$	−0.970666	−0.0360496
$726$	57.4610	2.13258
$727$	−8.93001	−0.331196	−0.165598	−	0.986193i	$-0.552955\pi$
$727$	−8.93001	−0.331196	−0.165598	+	0.986193i	$0.552955\pi$
$728$	0.559119	0.0207223
$729$	−30.7966	−1.14061
$730$	−15.7217	−0.581886
$731$	−37.5359	−1.38832
$732$	35.2174	1.30167
$733$	−22.6911	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$733$	−22.6911	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$734$	−10.7012	−0.394989
$735$	−16.9034	−0.623492
$736$	−8.64250	−0.318567
$737$	−36.4485	−1.34260
$738$	6.89844	0.253935
$739$	−28.2017	−1.03742	−0.518709	−	0.854951i	$-0.673587\pi$
$739$	−28.2017	−1.03742	−0.518709	+	0.854951i	$0.673587\pi$
$740$	8.10990	0.298126
$741$	0.660377	0.0242596
$742$	−17.7078	−0.650075
$743$	−9.58989	−0.351819	−0.175910	−	0.984406i	$-0.556287\pi$
$743$	−9.58989	−0.351819	−0.175910	+	0.984406i	$0.556287\pi$
$744$	−13.5505	−0.496787
$745$	−4.19316	−0.153626
$746$	−27.4619	−1.00545
$747$	12.6806	0.463961
$748$	−42.8226	−1.56575
$749$	19.4406	0.710344
$750$	−30.1468	−1.10081
$751$	−6.78458	−0.247573	−0.123786	−	0.992309i	$-0.539504\pi$
$751$	−6.78458	−0.247573	−0.123786	+	0.992309i	$0.539504\pi$
$752$	10.2919	0.375308
$753$	48.9041	1.78216
$754$	−0.239823	−0.00873384
$755$	17.2750	0.628703
$756$	−1.02725	−0.0373608
$757$	13.5670	0.493102	0.246551	−	0.969130i	$-0.420703\pi$
$757$	13.5670	0.493102	0.246551	+	0.969130i	$0.420703\pi$
$758$	−17.8711	−0.649108
$759$	−125.753	−4.56453
$760$	−1.73074	−0.0627806
$761$	12.5351	0.454396	0.227198	−	0.973849i	$-0.427043\pi$
$761$	12.5351	0.454396	0.227198	+	0.973849i	$0.427043\pi$
$762$	10.5446	0.381991
$763$	−20.6442	−0.747369
$764$	19.8473	0.718049
$765$	−46.0266	−1.66409
$766$	10.8778	0.393032
$767$	4.20227	0.151735
$768$	2.49390	0.0899909
$769$	−7.08694	−0.255562	−0.127781	−	0.991802i	$-0.540785\pi$
$769$	−7.08694	−0.255562	−0.127781	+	0.991802i	$0.540785\pi$
$770$	−21.3217	−0.768381
$771$	10.4888	0.377744
$772$	−14.6490	−0.527228
$773$	−30.5161	−1.09759	−0.548793	−	0.835958i	$-0.684912\pi$
$773$	−30.5161	−1.09759	−0.548793	+	0.835958i	$0.684912\pi$
$774$	16.4652	0.591829
$775$	6.55356	0.235411
$776$	−8.46702	−0.303948
$777$	−19.4822	−0.698920
$778$	−4.52471	−0.162219
$779$	−1.90392	−0.0682151
$780$	−1.44757	−0.0518314
$781$	23.5263	0.841839
$782$	63.4328	2.26835
$783$	0.440619	0.0157464
$784$	−3.47981	−0.124279
$785$	−13.1431	−0.469096
$786$	−32.2722	−1.15111
$787$	−9.83380	−0.350537	−0.175269	−	0.984521i	$-0.556079\pi$
$787$	−9.83380	−0.350537	−0.175269	+	0.984521i	$0.556079\pi$
$788$	−9.37280	−0.333892
$789$	12.3507	0.439697
$790$	30.7261	1.09319
$791$	11.1379	0.396019
$792$	18.7842	0.667467
$793$	−4.20822	−0.149438
$794$	−27.6450	−0.981084
$795$	45.8460	1.62599
$796$	−14.6360	−0.518760
$797$	−32.6934	−1.15806	−0.579030	−	0.815307i	$-0.696568\pi$
$797$	−32.6934	−1.15806	−0.579030	+	0.815307i	$0.696568\pi$
$798$	4.15771	0.147181
$799$	−75.5391	−2.67238
$800$	−1.20615	−0.0426437
$801$	−13.1782	−0.465629
$802$	−1.76900	−0.0624656
$803$	−47.0931	−1.66188
$804$	−15.5798	−0.549455
$805$	31.5837	1.11318
$806$	1.61919	0.0570336
$807$	44.0806	1.55171
$808$	−6.91486	−0.243264
$809$	23.3454	0.820780	0.410390	−	0.911910i	$-0.365392\pi$
$809$	23.3454	0.820780	0.410390	+	0.911910i	$0.365392\pi$
$810$	−16.1533	−0.567569
$811$	−29.0983	−1.02178	−0.510889	−	0.859646i	$-0.670684\pi$
$811$	−29.0983	−1.02178	−0.510889	+	0.859646i	$0.670684\pi$
$812$	−1.50992	−0.0529877
$813$	16.7371	0.586997
$814$	24.2926	0.851455
$815$	12.3659	0.433159
$816$	−18.3043	−0.640780
$817$	−4.54428	−0.158984
$818$	−26.0238	−0.909900
$819$	1.80011	0.0629008
$820$	4.17347	0.145744
$821$	−12.7977	−0.446644	−0.223322	−	0.974745i	$-0.571690\pi$
$821$	−12.7977	−0.446644	−0.223322	+	0.974745i	$0.571690\pi$
$822$	8.28596	0.289006
$823$	17.5980	0.613428	0.306714	−	0.951802i	$-0.400771\pi$
$823$	17.5980	0.613428	0.306714	+	0.951802i	$0.400771\pi$
$824$	5.31246	0.185068
$825$	−17.5500	−0.611013
$826$	26.4573	0.920568
$827$	7.92016	0.275411	0.137706	−	0.990473i	$-0.456027\pi$
$827$	7.92016	0.275411	0.137706	+	0.990473i	$0.456027\pi$
$828$	−27.8249	−0.966981
$829$	8.92241	0.309888	0.154944	−	0.987923i	$-0.450480\pi$
$829$	8.92241	0.309888	0.154944	+	0.987923i	$0.450480\pi$
$830$	7.67163	0.266286
$831$	−55.9989	−1.94258
$832$	−0.298003	−0.0103314
$833$	25.5405	0.884926
$834$	43.5037	1.50641
$835$	−44.7065	−1.54713
$836$	−5.18431	−0.179303
$837$	−2.97489	−0.102827
$838$	4.90699	0.169509
$839$	−25.0092	−0.863415	−0.431707	−	0.902014i	$-0.642089\pi$
$839$	−25.0092	−0.863415	−0.431707	+	0.902014i	$0.642089\pi$
$840$	−9.11387	−0.314458
$841$	−28.3524	−0.977667
$842$	7.00976	0.241572
$843$	50.2828	1.73183
$844$	3.94613	0.135831
$845$	−25.1482	−0.865124
$846$	33.1353	1.13922
$847$	−43.2292	−1.48537
$848$	9.43805	0.324104
$849$	65.7991	2.25822
$850$	8.85267	0.303644
$851$	−35.9844	−1.23353
$852$	10.0562	0.344521
$853$	49.3810	1.69077	0.845386	−	0.534156i	$-0.179370\pi$
$853$	49.3810	1.69077	0.845386	+	0.534156i	$0.179370\pi$
$854$	−26.4948	−0.906634
$855$	−5.57219	−0.190565
$856$	−10.3616	−0.354152
$857$	−19.6134	−0.669981	−0.334990	−	0.942222i	$-0.608733\pi$
$857$	−19.6134	−0.669981	−0.334990	+	0.942222i	$0.608733\pi$
$858$	−4.33609	−0.148032
$859$	49.7275	1.69668	0.848340	−	0.529452i	$-0.177602\pi$
$859$	49.7275	1.69668	0.848340	+	0.529452i	$0.177602\pi$
$860$	9.96123	0.339675
$861$	−10.0258	−0.341679
$862$	16.6838	0.568251
$863$	−25.8661	−0.880493	−0.440247	−	0.897877i	$-0.645109\pi$
$863$	−25.8661	−0.880493	−0.440247	+	0.897877i	$0.645109\pi$
$864$	0.547512	0.0186267
$865$	47.8174	1.62584
$866$	−16.3926	−0.557043
$867$	91.9508	3.12281
$868$	10.1944	0.346020
$869$	92.0378	3.12217
$870$	3.90921	0.132535
$871$	1.86167	0.0630802
$872$	11.0031	0.372611
$873$	−27.2599	−0.922609
$874$	7.67947	0.259762
$875$	22.6801	0.766728
$876$	−20.1297	−0.680120
$877$	12.2543	0.413798	0.206899	−	0.978362i	$-0.433663\pi$
$877$	12.2543	0.413798	0.206899	+	0.978362i	$0.433663\pi$
$878$	−10.9201	−0.368535
$879$	−19.3793	−0.653649
$880$	11.3642	0.383087
$881$	29.4082	0.990787	0.495393	−	0.868669i	$-0.335024\pi$
$881$	29.4082	0.990787	0.495393	+	0.868669i	$0.335024\pi$
$882$	−11.2034	−0.377237
$883$	28.0532	0.944064	0.472032	−	0.881581i	$-0.343521\pi$
$883$	28.0532	0.944064	0.472032	+	0.881581i	$0.343521\pi$
$884$	2.18723	0.0735647
$885$	−68.4987	−2.30256
$886$	−8.44935	−0.283861
$887$	−14.5827	−0.489640	−0.244820	−	0.969569i	$-0.578729\pi$
$887$	−14.5827	−0.489640	−0.244820	+	0.969569i	$0.578729\pi$
$888$	10.3838	0.348456
$889$	−7.93295	−0.266063
$890$	−7.97266	−0.267244
$891$	−48.3860	−1.62099
$892$	−4.32975	−0.144971
$893$	−9.14511	−0.306030
$894$	−5.36884	−0.179561
$895$	18.5142	0.618862
$896$	−1.87622	−0.0626800
$897$	6.42302	0.214458
$898$	12.1220	0.404515
$899$	−4.37268	−0.145837
$900$	−3.88324	−0.129441
$901$	−69.2718	−2.30778
$902$	12.5013	0.416248
$903$	−23.9296	−0.796327
$904$	−5.93637	−0.197441
$905$	−40.6514	−1.35130
$906$	22.1186	0.734842
$907$	−34.1779	−1.13486	−0.567430	−	0.823422i	$-0.692062\pi$
$907$	−34.1779	−1.13486	−0.567430	+	0.823422i	$0.692062\pi$
$908$	1.56410	0.0519064
$909$	−22.2627	−0.738406
$910$	1.08904	0.0361014
$911$	25.0912	0.831309	0.415654	−	0.909523i	$-0.363553\pi$
$911$	25.0912	0.831309	0.415654	+	0.909523i	$0.363553\pi$
$912$	−2.21601	−0.0733793
$913$	22.9798	0.760520
$914$	−37.0355	−1.22502
$915$	68.5958	2.26771
$916$	−23.6879	−0.782672
$917$	24.2791	0.801768
$918$	−4.01854	−0.132632
$919$	16.8082	0.554452	0.277226	−	0.960805i	$-0.410585\pi$
$919$	16.8082	0.554452	0.277226	+	0.960805i	$0.410585\pi$
$920$	−16.8337	−0.554991
$921$	68.1155	2.24448
$922$	−3.06027	−0.100785
$923$	−1.20165	−0.0395527
$924$	−27.2999	−0.898101
$925$	−5.02198	−0.165122
$926$	7.69735	0.252951
$927$	17.1037	0.561759
$928$	0.804767	0.0264178
$929$	7.62631	0.250211	0.125105	−	0.992143i	$-0.460073\pi$
$929$	7.62631	0.250211	0.125105	+	0.992143i	$0.460073\pi$
$930$	−26.3935	−0.865477
$931$	3.09205	0.101338
$932$	19.7694	0.647567
$933$	25.9155	0.848437
$934$	−22.8952	−0.749154
$935$	−83.4091	−2.72777
$936$	−0.959433	−0.0313600
$937$	38.6313	1.26203	0.631015	−	0.775771i	$-0.282639\pi$
$937$	38.6313	1.26203	0.631015	+	0.775771i	$0.282639\pi$
$938$	11.7210	0.382704
$939$	−21.5157	−0.702138
$940$	20.0464	0.653843
$941$	25.9587	0.846230	0.423115	−	0.906076i	$-0.360937\pi$
$941$	25.9587	0.846230	0.423115	+	0.906076i	$0.360937\pi$
$942$	−16.8281	−0.548290
$943$	−18.5181	−0.603032
$944$	−14.1014	−0.458962
$945$	−2.00086	−0.0650880
$946$	29.8381	0.970120
$947$	−42.9918	−1.39705	−0.698523	−	0.715588i	$-0.746159\pi$
$947$	−42.9918	−1.39705	−0.698523	+	0.715588i	$0.746159\pi$
$948$	39.3411	1.27774
$949$	2.40536	0.0780812
$950$	1.07175	0.0347720
$951$	59.3342	1.92404
$952$	13.7708	0.446313
$953$	48.4871	1.57065	0.785326	−	0.619083i	$-0.212496\pi$
$953$	48.4871	1.57065	0.785326	+	0.619083i	$0.212496\pi$
$954$	30.3862	0.983789
$955$	38.6581	1.25095
$956$	−2.12172	−0.0686214
$957$	11.7098	0.378523
$958$	25.5092	0.824166
$959$	−6.23372	−0.201297
$960$	4.85757	0.156778
$961$	−1.47740	−0.0476579
$962$	−1.24078	−0.0400045
$963$	−33.3596	−1.07500
$964$	20.8089	0.670210
$965$	−28.5330	−0.918511
$966$	40.4391	1.30111
$967$	−32.6038	−1.04847	−0.524234	−	0.851574i	$-0.675648\pi$
$967$	−32.6038	−1.04847	−0.524234	+	0.851574i	$0.675648\pi$
$968$	23.0406	0.740553
$969$	16.2647	0.522497
$970$	−16.4919	−0.529523
$971$	−21.5755	−0.692390	−0.346195	−	0.938163i	$-0.612526\pi$
$971$	−21.5755	−0.692390	−0.346195	+	0.938163i	$0.612526\pi$
$972$	−22.3249	−0.716072
$973$	−32.7288	−1.04924
$974$	27.0841	0.867829
$975$	0.896396	0.0287076
$976$	14.1214	0.452015
$977$	3.84983	0.123167	0.0615835	−	0.998102i	$-0.480385\pi$
$977$	3.84983	0.123167	0.0615835	+	0.998102i	$0.480385\pi$
$978$	15.8331	0.506286
$979$	−23.8815	−0.763256
$980$	−6.77790	−0.216512
$981$	35.4249	1.13103
$982$	36.9457	1.17899
$983$	−16.1448	−0.514939	−0.257470	−	0.966286i	$-0.582889\pi$
$983$	−16.1448	−0.514939	−0.257470	+	0.966286i	$0.582889\pi$
$984$	5.34363	0.170349
$985$	−18.2562	−0.581690
$986$	−5.90670	−0.188108
$987$	−48.1570	−1.53285
$988$	0.264797	0.00842431
$989$	−44.1990	−1.40544
$990$	36.5875	1.16283
$991$	11.3906	0.361834	0.180917	−	0.983498i	$-0.442093\pi$
$991$	11.3906	0.361834	0.180917	+	0.983498i	$0.442093\pi$
$992$	−5.43347	−0.172513
$993$	−41.5773	−1.31942
$994$	−7.56553	−0.239964
$995$	−28.5078	−0.903757
$996$	9.82260	0.311241
$997$	−47.5526	−1.50601	−0.753003	−	0.658017i	$-0.771396\pi$
$997$	−47.5526	−1.50601	−0.753003	+	0.658017i	$0.771396\pi$
$998$	43.5308	1.37794
$999$	2.27965	0.0721250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1006.2.a.j.1.11	✓	12
3.2	odd	2		9054.2.a.bi.1.4		12
4.3	odd	2		8048.2.a.q.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.11	✓	12	1.1	even	1	trivial
8048.2.a.q.1.2		12	4.3	odd	2
9054.2.a.bi.1.4		12	3.2	odd	2

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:	\( N \)	\(=\)	\( 1006 = 2 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.49390 q^{3} +1.00000 q^{4} +1.94778 q^{5} +2.49390 q^{6} -1.87622 q^{7} +1.00000 q^{8} +3.21954 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.49390 q^{3} +1.00000 q^{4} +1.94778 q^{5} +2.49390 q^{6} -1.87622 q^{7} +1.00000 q^{8} +3.21954 q^{9} +1.94778 q^{10} +5.83443 q^{11} +2.49390 q^{12} -0.298003 q^{13} -1.87622 q^{14} +4.85757 q^{15} +1.00000 q^{16} -7.33964 q^{17} +3.21954 q^{18} -0.888571 q^{19} +1.94778 q^{20} -4.67910 q^{21} +5.83443 q^{22} -8.64250 q^{23} +2.49390 q^{24} -1.20615 q^{25} -0.298003 q^{26} +0.547512 q^{27} -1.87622 q^{28} +0.804767 q^{29} +4.85757 q^{30} -5.43347 q^{31} +1.00000 q^{32} +14.5505 q^{33} -7.33964 q^{34} -3.65446 q^{35} +3.21954 q^{36} +4.16366 q^{37} -0.888571 q^{38} -0.743190 q^{39} +1.94778 q^{40} +2.14268 q^{41} -4.67910 q^{42} +5.11414 q^{43} +5.83443 q^{44} +6.27096 q^{45} -8.64250 q^{46} +10.2919 q^{47} +2.49390 q^{48} -3.47981 q^{49} -1.20615 q^{50} -18.3043 q^{51} -0.298003 q^{52} +9.43805 q^{53} +0.547512 q^{54} +11.3642 q^{55} -1.87622 q^{56} -2.21601 q^{57} +0.804767 q^{58} -14.1014 q^{59} +4.85757 q^{60} +14.1214 q^{61} -5.43347 q^{62} -6.04056 q^{63} +1.00000 q^{64} -0.580445 q^{65} +14.5505 q^{66} -6.24714 q^{67} -7.33964 q^{68} -21.5535 q^{69} -3.65446 q^{70} +4.03233 q^{71} +3.21954 q^{72} -8.07158 q^{73} +4.16366 q^{74} -3.00801 q^{75} -0.888571 q^{76} -10.9467 q^{77} -0.743190 q^{78} +15.7749 q^{79} +1.94778 q^{80} -8.29318 q^{81} +2.14268 q^{82} +3.93865 q^{83} -4.67910 q^{84} -14.2960 q^{85} +5.11414 q^{86} +2.00701 q^{87} +5.83443 q^{88} -4.09320 q^{89} +6.27096 q^{90} +0.559119 q^{91} -8.64250 q^{92} -13.5505 q^{93} +10.2919 q^{94} -1.73074 q^{95} +2.49390 q^{96} -8.46702 q^{97} -3.47981 q^{98} +18.7842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 + 5 * q^3 + 12 * q^4 + 5 * q^5 + 5 * q^6 + 8 * q^7 + 12 * q^8 + 15 * q^9	\( 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9} + 5 q^{10} + 18 q^{11} + 5 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{15} + 12 q^{16} - 2 q^{17} + 15 q^{18} + 6 q^{19} + 5 q^{20} + q^{21} + 18 q^{22} + 13 q^{23} + 5 q^{24} + q^{25} + 4 q^{26} + 8 q^{27} + 8 q^{28} + 20 q^{29} + 2 q^{30} + 7 q^{31} + 12 q^{32} - 8 q^{33} - 2 q^{34} + q^{35} + 15 q^{36} + 10 q^{37} + 6 q^{38} + 7 q^{39} + 5 q^{40} + 2 q^{41} + q^{42} + 8 q^{43} + 18 q^{44} - 7 q^{45} + 13 q^{46} + 12 q^{47} + 5 q^{48} + 4 q^{49} + q^{50} + 2 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 8 q^{55} + 8 q^{56} - 10 q^{57} + 20 q^{58} + 6 q^{59} + 2 q^{60} - 10 q^{61} + 7 q^{62} + 7 q^{63} + 12 q^{64} + 4 q^{65} - 8 q^{66} + 7 q^{67} - 2 q^{68} - 12 q^{69} + q^{70} + 22 q^{71} + 15 q^{72} - 23 q^{73} + 10 q^{74} - 34 q^{75} + 6 q^{76} - 19 q^{77} + 7 q^{78} + 13 q^{79} + 5 q^{80} - 28 q^{81} + 2 q^{82} + q^{83} + q^{84} - 28 q^{85} + 8 q^{86} - 22 q^{87} + 18 q^{88} - 3 q^{89} - 7 q^{90} - 21 q^{91} + 13 q^{92} - 33 q^{93} + 12 q^{94} - 2 q^{95} + 5 q^{96} - 70 q^{97} + 4 q^{98} - 6 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 5 * q^3 + 12 * q^4 + 5 * q^5 + 5 * q^6 + 8 * q^7 + 12 * q^8 + 15 * q^9 + 5 * q^10 + 18 * q^11 + 5 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^15 + 12 * q^16 - 2 * q^17 + 15 * q^18 + 6 * q^19 + 5 * q^20 + q^21 + 18 * q^22 + 13 * q^23 + 5 * q^24 + q^25 + 4 * q^26 + 8 * q^27 + 8 * q^28 + 20 * q^29 + 2 * q^30 + 7 * q^31 + 12 * q^32 - 8 * q^33 - 2 * q^34 + q^35 + 15 * q^36 + 10 * q^37 + 6 * q^38 + 7 * q^39 + 5 * q^40 + 2 * q^41 + q^42 + 8 * q^43 + 18 * q^44 - 7 * q^45 + 13 * q^46 + 12 * q^47 + 5 * q^48 + 4 * q^49 + q^50 + 2 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 8 * q^55 + 8 * q^56 - 10 * q^57 + 20 * q^58 + 6 * q^59 + 2 * q^60 - 10 * q^61 + 7 * q^62 + 7 * q^63 + 12 * q^64 + 4 * q^65 - 8 * q^66 + 7 * q^67 - 2 * q^68 - 12 * q^69 + q^70 + 22 * q^71 + 15 * q^72 - 23 * q^73 + 10 * q^74 - 34 * q^75 + 6 * q^76 - 19 * q^77 + 7 * q^78 + 13 * q^79 + 5 * q^80 - 28 * q^81 + 2 * q^82 + q^83 + q^84 - 28 * q^85 + 8 * q^86 - 22 * q^87 + 18 * q^88 - 3 * q^89 - 7 * q^90 - 21 * q^91 + 13 * q^92 - 33 * q^93 + 12 * q^94 - 2 * q^95 + 5 * q^96 - 70 * q^97 + 4 * q^98 - 6 * q^99

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