LMFDB - Embedded newform 445.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, 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("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.162147 q^{2} +1.19353 q^{3} -1.97371 q^{4} -1.00000 q^{5} +0.193527 q^{6} -0.475281 q^{7} -0.644326 q^{8} -1.57549 q^{9} +O(q^{10})$	\(q+0.162147 q^{2} +1.19353 q^{3} -1.97371 q^{4} -1.00000 q^{5} +0.193527 q^{6} -0.475281 q^{7} -0.644326 q^{8} -1.57549 q^{9} -0.162147 q^{10} -5.19353 q^{11} -2.35567 q^{12} -1.30490 q^{13} -0.0770654 q^{14} -1.19353 q^{15} +3.84294 q^{16} -7.40330 q^{17} -0.255462 q^{18} +6.83604 q^{19} +1.97371 q^{20} -0.567260 q^{21} -0.842116 q^{22} +2.69195 q^{23} -0.769020 q^{24} +1.00000 q^{25} -0.211586 q^{26} -5.46097 q^{27} +0.938065 q^{28} -0.00689789 q^{29} -0.193527 q^{30} -8.54230 q^{31} +1.91177 q^{32} -6.19862 q^{33} -1.20042 q^{34} +0.475281 q^{35} +3.10956 q^{36} -4.56119 q^{37} +1.10845 q^{38} -1.55743 q^{39} +0.644326 q^{40} +3.41020 q^{41} -0.0919797 q^{42} +9.87858 q^{43} +10.2505 q^{44} +1.57549 q^{45} +0.436493 q^{46} -4.62493 q^{47} +4.58665 q^{48} -6.77411 q^{49} +0.162147 q^{50} -8.83604 q^{51} +2.57549 q^{52} +2.86526 q^{53} -0.885482 q^{54} +5.19353 q^{55} +0.306236 q^{56} +8.15900 q^{57} -0.00111847 q^{58} -1.13767 q^{59} +2.35567 q^{60} +7.30684 q^{61} -1.38511 q^{62} +0.748801 q^{63} -7.37589 q^{64} +1.30490 q^{65} -1.00509 q^{66} -1.79531 q^{67} +14.6120 q^{68} +3.21292 q^{69} +0.0770654 q^{70} +2.91492 q^{71} +1.01513 q^{72} -2.26369 q^{73} -0.739584 q^{74} +1.19353 q^{75} -13.4924 q^{76} +2.46838 q^{77} -0.252534 q^{78} -10.5104 q^{79} -3.84294 q^{80} -1.79134 q^{81} +0.552955 q^{82} +6.35650 q^{83} +1.11961 q^{84} +7.40330 q^{85} +1.60178 q^{86} -0.00823281 q^{87} +3.34632 q^{88} -1.00000 q^{89} +0.255462 q^{90} +0.620194 q^{91} -5.31313 q^{92} -10.1955 q^{93} -0.749920 q^{94} -6.83604 q^{95} +2.28175 q^{96} +18.3113 q^{97} -1.09840 q^{98} +8.18237 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * 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 $

$2$	0.162147	0.114655	0.0573277	−	0.998355i	$-0.481742\pi$
$2$	0.162147	0.114655	0.0573277	+	0.998355i	$0.481742\pi$
$3$	1.19353	0.689083	0.344542	−	0.938771i	$-0.388034\pi$
$3$	1.19353	0.689083	0.344542	+	0.938771i	$0.388034\pi$
$4$	−1.97371	−0.986854
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.193527	0.0790071
$7$	−0.475281	−0.179639	−0.0898196	−	0.995958i	$-0.528629\pi$
$7$	−0.475281	−0.179639	−0.0898196	+	0.995958i	$0.528629\pi$
$8$	−0.644326	−0.227804
$9$	−1.57549	−0.525164
$10$	−0.162147	−0.0512754
$11$	−5.19353	−1.56591	−0.782954	−	0.622080i	$-0.786288\pi$
$11$	−5.19353	−1.56591	−0.782954	+	0.622080i	$0.786288\pi$
$12$	−2.35567	−0.680025
$13$	−1.30490	−0.361914	−0.180957	−	0.983491i	$-0.557920\pi$
$13$	−1.30490	−0.361914	−0.180957	+	0.983491i	$0.557920\pi$
$14$	−0.0770654	−0.0205966
$15$	−1.19353	−0.308167
$16$	3.84294	0.960735
$17$	−7.40330	−1.79556	−0.897782	−	0.440439i	$-0.854823\pi$
$17$	−7.40330	−1.79556	−0.897782	+	0.440439i	$0.854823\pi$
$18$	−0.255462	−0.0602129
$19$	6.83604	1.56830	0.784148	−	0.620574i	$-0.213100\pi$
$19$	6.83604	1.56830	0.784148	+	0.620574i	$0.213100\pi$
$20$	1.97371	0.441335
$21$	−0.567260	−0.123786
$22$	−0.842116	−0.179540
$23$	2.69195	0.561311	0.280656	−	0.959809i	$-0.409448\pi$
$23$	2.69195	0.561311	0.280656	+	0.959809i	$0.409448\pi$
$24$	−0.769020	−0.156976
$25$	1.00000	0.200000
$26$	−0.211586	−0.0414954
$27$	−5.46097	−1.05097
$28$	0.938065	0.177278
$29$	−0.00689789	−0.00128091	−0.000640453	−	1.00000i	$-0.500204\pi$
$29$	−0.00689789	−0.00128091	−0.000640453		1.00000i	$0.500204\pi$
$30$	−0.193527	−0.0353331
$31$	−8.54230	−1.53424	−0.767121	−	0.641502i	$-0.778312\pi$
$31$	−8.54230	−1.53424	−0.767121	+	0.641502i	$0.778312\pi$
$32$	1.91177	0.337957
$33$	−6.19862	−1.07904
$34$	−1.20042	−0.205871
$35$	0.475281	0.0803371
$36$	3.10956	0.518261
$37$	−4.56119	−0.749855	−0.374927	−	0.927054i	$-0.622332\pi$
$37$	−4.56119	−0.749855	−0.374927	+	0.927054i	$0.622332\pi$
$38$	1.10845	0.179814
$39$	−1.55743	−0.249389
$40$	0.644326	0.101877
$41$	3.41020	0.532584	0.266292	−	0.963892i	$-0.414201\pi$
$41$	3.41020	0.532584	0.266292	+	0.963892i	$0.414201\pi$
$42$	−0.0919797	−0.0141928
$43$	9.87858	1.50647	0.753235	−	0.657752i	$-0.228492\pi$
$43$	9.87858	1.50647	0.753235	+	0.657752i	$0.228492\pi$
$44$	10.2505	1.54532
$45$	1.57549	0.234861
$46$	0.436493	0.0643574
$47$	−4.62493	−0.674616	−0.337308	−	0.941394i	$-0.609516\pi$
$47$	−4.62493	−0.674616	−0.337308	+	0.941394i	$0.609516\pi$
$48$	4.58665	0.662026
$49$	−6.77411	−0.967730
$50$	0.162147	0.0229311
$51$	−8.83604	−1.23729
$52$	2.57549	0.357157
$53$	2.86526	0.393574	0.196787	−	0.980446i	$-0.436949\pi$
$53$	2.86526	0.393574	0.196787	+	0.980446i	$0.436949\pi$
$54$	−0.885482	−0.120499
$55$	5.19353	0.700295
$56$	0.306236	0.0409224
$57$	8.15900	1.08069
$58$	−0.00111847	−0.000146863 0
$59$	−1.13767	−0.148111	−0.0740557	−	0.997254i	$-0.523594\pi$
$59$	−1.13767	−0.148111	−0.0740557	+	0.997254i	$0.523594\pi$
$60$	2.35567	0.304116
$61$	7.30684	0.935546	0.467773	−	0.883849i	$-0.345057\pi$
$61$	7.30684	0.935546	0.467773	+	0.883849i	$0.345057\pi$
$62$	−1.38511	−0.175909
$63$	0.748801	0.0943401
$64$	−7.37589	−0.921987
$65$	1.30490	0.161853
$66$	−1.00509	−0.123718
$67$	−1.79531	−0.219332	−0.109666	−	0.993968i	$-0.534978\pi$
$67$	−1.79531	−0.219332	−0.109666	+	0.993968i	$0.534978\pi$
$68$	14.6120	1.77196
$69$	3.21292	0.386790
$70$	0.0770654	0.00921108
$71$	2.91492	0.345937	0.172969	−	0.984927i	$-0.444664\pi$
$71$	2.91492	0.345937	0.172969	+	0.984927i	$0.444664\pi$
$72$	1.01513	0.119634
$73$	−2.26369	−0.264945	−0.132473	−	0.991187i	$-0.542292\pi$
$73$	−2.26369	−0.264945	−0.132473	+	0.991187i	$0.542292\pi$
$74$	−0.739584	−0.0859749
$75$	1.19353	0.137817
$76$	−13.4924	−1.54768
$77$	2.46838	0.281298
$78$	−0.252534	−0.0285938
$79$	−10.5104	−1.18251	−0.591257	−	0.806483i	$-0.701368\pi$
$79$	−10.5104	−1.18251	−0.591257	+	0.806483i	$0.701368\pi$
$80$	−3.84294	−0.429654
$81$	−1.79134	−0.199038
$82$	0.552955	0.0610636
$83$	6.35650	0.697716	0.348858	−	0.937176i	$-0.386569\pi$
$83$	6.35650	0.697716	0.348858	+	0.937176i	$0.386569\pi$
$84$	1.11961	0.122159
$85$	7.40330	0.803001
$86$	1.60178	0.172725
$87$	−0.00823281	−0.000882650 0
$88$	3.34632	0.356719
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0.255462	0.0269280
$91$	0.620194	0.0650140
$92$	−5.31313	−0.553932
$93$	−10.1955	−1.05722
$94$	−0.749920	−0.0773483
$95$	−6.83604	−0.701363
$96$	2.28175	0.232880
$97$	18.3113	1.85923	0.929617	−	0.368528i	$-0.120138\pi$
$97$	18.3113	1.85923	0.929617	+	0.368528i	$0.120138\pi$
$98$	−1.09840	−0.110955
$99$	8.18237	0.822359
$100$	−1.97371	−0.197371
$101$	−14.3545	−1.42832	−0.714162	−	0.699981i	$-0.753192\pi$
$101$	−14.3545	−1.42832	−0.714162	+	0.699981i	$0.753192\pi$
$102$	−1.43274	−0.141862
$103$	5.43964	0.535983	0.267992	−	0.963421i	$-0.413640\pi$
$103$	5.43964	0.535983	0.267992	+	0.963421i	$0.413640\pi$
$104$	0.840781	0.0824454
$105$	0.567260	0.0553589
$106$	0.464594	0.0451254
$107$	−7.27486	−0.703287	−0.351643	−	0.936134i	$-0.614377\pi$
$107$	−7.27486	−0.703287	−0.351643	+	0.936134i	$0.614377\pi$
$108$	10.7784	1.03715
$109$	3.92293	0.375749	0.187874	−	0.982193i	$-0.439840\pi$
$109$	3.92293	0.375749	0.187874	+	0.982193i	$0.439840\pi$
$110$	0.842116	0.0802926
$111$	−5.44390	−0.516712
$112$	−1.82648	−0.172586
$113$	−15.2119	−1.43102	−0.715509	−	0.698603i	$-0.753805\pi$
$113$	−15.2119	−1.43102	−0.715509	+	0.698603i	$0.753805\pi$
$114$	1.32296	0.123907
$115$	−2.69195	−0.251026
$116$	0.0136144	0.00126407
$117$	2.05586	0.190064
$118$	−0.184469	−0.0169818
$119$	3.51865	0.322554
$120$	0.769020	0.0702016
$121$	15.9727	1.45207
$122$	1.18478	0.107265
$123$	4.07017	0.366995
$124$	16.8600	1.51407
$125$	−1.00000	−0.0894427
$126$	0.121416	0.0108166
$127$	14.3082	1.26965	0.634823	−	0.772658i	$-0.281073\pi$
$127$	14.3082	1.26965	0.634823	+	0.772658i	$0.281073\pi$
$128$	−5.01953	−0.443668
$129$	11.7904	1.03808
$130$	0.211586	0.0185573
$131$	−5.16529	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$131$	−5.16529	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$132$	12.2343	1.06486
$133$	−3.24904	−0.281727
$134$	−0.291105	−0.0251476
$135$	5.46097	0.470006
$136$	4.77014	0.409036
$137$	−10.1033	−0.863181	−0.431590	−	0.902070i	$-0.642047\pi$
$137$	−10.1033	−0.863181	−0.431590	+	0.902070i	$0.642047\pi$
$138$	0.520966	0.0443476
$139$	−21.0062	−1.78172	−0.890862	−	0.454275i	$-0.849898\pi$
$139$	−21.0062	−1.78172	−0.890862	+	0.454275i	$0.849898\pi$
$140$	−0.938065	−0.0792810
$141$	−5.51998	−0.464866
$142$	0.472646	0.0396636
$143$	6.77704	0.566724
$144$	−6.05453	−0.504544
$145$	0.00689789	0.000572838 0
$146$	−0.367052	−0.0303774
$147$	−8.08508	−0.666846
$148$	9.00245	0.739997
$149$	−4.55954	−0.373532	−0.186766	−	0.982404i	$-0.559801\pi$
$149$	−4.55954	−0.373532	−0.186766	+	0.982404i	$0.559801\pi$
$150$	0.193527	0.0158014
$151$	3.64071	0.296276	0.148138	−	0.988967i	$-0.452672\pi$
$151$	3.64071	0.296276	0.148138	+	0.988967i	$0.452672\pi$
$152$	−4.40464	−0.357263
$153$	11.6639	0.942967
$154$	0.400241	0.0322524
$155$	8.54230	0.686134
$156$	3.07392	0.246111
$157$	17.3667	1.38601	0.693006	−	0.720932i	$-0.256286\pi$
$157$	17.3667	1.38601	0.693006	+	0.720932i	$0.256286\pi$
$158$	−1.70423	−0.135582
$159$	3.41977	0.271205
$160$	−1.91177	−0.151139
$161$	−1.27943	−0.100834
$162$	−0.290461	−0.0228208
$163$	−9.06918	−0.710353	−0.355177	−	0.934799i	$-0.615579\pi$
$163$	−9.06918	−0.710353	−0.355177	+	0.934799i	$0.615579\pi$
$164$	−6.73074	−0.525583
$165$	6.19862	0.482562
$166$	1.03069	0.0799970
$167$	4.29059	0.332016	0.166008	−	0.986124i	$-0.446912\pi$
$167$	4.29059	0.332016	0.166008	+	0.986124i	$0.446912\pi$
$168$	0.365500	0.0281990
$169$	−11.2972	−0.869018
$170$	1.20042	0.0920684
$171$	−10.7701	−0.823613
$172$	−19.4974	−1.48667
$173$	21.1969	1.61157	0.805787	−	0.592206i	$-0.201743\pi$
$173$	21.1969	1.61157	0.805787	+	0.592206i	$0.201743\pi$
$174$	−0.00133493	−0.000101201 0
$175$	−0.475281	−0.0359278
$176$	−19.9584	−1.50442
$177$	−1.35783	−0.102061
$178$	−0.162147	−0.0121534
$179$	0.468991	0.0350541	0.0175270	−	0.999846i	$-0.494421\pi$
$179$	0.468991	0.0350541	0.0175270	+	0.999846i	$0.494421\pi$
$180$	−3.10956	−0.231773
$181$	−22.2732	−1.65555	−0.827776	−	0.561058i	$-0.810394\pi$
$181$	−22.2732	−1.65555	−0.827776	+	0.561058i	$0.810394\pi$
$182$	0.100563	0.00745420
$183$	8.72092	0.644669
$184$	−1.73450	−0.127869
$185$	4.56119	0.335345
$186$	−1.65317	−0.121216
$187$	38.4493	2.81169
$188$	9.12827	0.665747
$189$	2.59550	0.188795
$190$	−1.10845	−0.0804151
$191$	−21.3072	−1.54173	−0.770867	−	0.636996i	$-0.780177\pi$
$191$	−21.3072	−1.54173	−0.770867	+	0.636996i	$0.780177\pi$
$192$	−8.80333	−0.635325
$193$	−1.33822	−0.0963275	−0.0481637	−	0.998839i	$-0.515337\pi$
$193$	−1.33822	−0.0963275	−0.0481637	+	0.998839i	$0.515337\pi$
$194$	2.96913	0.213171
$195$	1.55743	0.111530
$196$	13.3701	0.955008
$197$	−9.65959	−0.688217	−0.344109	−	0.938930i	$-0.611819\pi$
$197$	−9.65959	−0.688217	−0.344109	+	0.938930i	$0.611819\pi$
$198$	1.32675	0.0942879
$199$	9.85738	0.698771	0.349386	−	0.936979i	$-0.386390\pi$
$199$	9.85738	0.698771	0.349386	+	0.936979i	$0.386390\pi$
$200$	−0.644326	−0.0455607
$201$	−2.14275	−0.151138
$202$	−2.32754	−0.163765
$203$	0.00327843	0.000230101 0
$204$	17.4398	1.22103
$205$	−3.41020	−0.238179
$206$	0.882022	0.0614534
$207$	−4.24116	−0.294781
$208$	−5.01466	−0.347704
$209$	−35.5032	−2.45581
$210$	0.0919797	0.00634720
$211$	−12.2510	−0.843396	−0.421698	−	0.906736i	$-0.638566\pi$
$211$	−12.2510	−0.843396	−0.421698	+	0.906736i	$0.638566\pi$
$212$	−5.65519	−0.388400
$213$	3.47903	0.238379
$214$	−1.17960	−0.0806356
$215$	−9.87858	−0.673714
$216$	3.51865	0.239414
$217$	4.05999	0.275610
$218$	0.636093	0.0430816
$219$	−2.70178	−0.182569
$220$	−10.2505	−0.691089
$221$	9.66057	0.649841
$222$	−0.882713	−0.0592438
$223$	−22.4242	−1.50163	−0.750817	−	0.660511i	$-0.770340\pi$
$223$	−22.4242	−1.50163	−0.750817	+	0.660511i	$0.770340\pi$
$224$	−0.908629	−0.0607103
$225$	−1.57549	−0.105033
$226$	−2.46657	−0.164074
$227$	−23.1665	−1.53762	−0.768809	−	0.639479i	$-0.779150\pi$
$227$	−23.1665	−1.53762	−0.768809	+	0.639479i	$0.779150\pi$
$228$	−16.1035	−1.06648
$229$	4.15788	0.274761	0.137380	−	0.990518i	$-0.456132\pi$
$229$	4.15788	0.274761	0.137380	+	0.990518i	$0.456132\pi$
$230$	−0.436493	−0.0287815
$231$	2.94608	0.193838
$232$	0.00444449	0.000291795 0
$233$	−11.7721	−0.771215	−0.385607	−	0.922663i	$-0.626008\pi$
$233$	−11.7721	−0.771215	−0.385607	+	0.922663i	$0.626008\pi$
$234$	0.333352	0.0217919
$235$	4.62493	0.301697
$236$	2.24542	0.146164
$237$	−12.5445	−0.814850
$238$	0.570539	0.0369825
$239$	−20.3834	−1.31849	−0.659246	−	0.751927i	$-0.729125\pi$
$239$	−20.3834	−1.31849	−0.659246	+	0.751927i	$0.729125\pi$
$240$	−4.58665	−0.296067
$241$	−11.3468	−0.730908	−0.365454	−	0.930829i	$-0.619086\pi$
$241$	−11.3468	−0.730908	−0.365454	+	0.930829i	$0.619086\pi$
$242$	2.58993	0.166487
$243$	14.2449	0.913811
$244$	−14.4216	−0.923247
$245$	6.77411	0.432782
$246$	0.659966	0.0420779
$247$	−8.92036	−0.567589
$248$	5.50403	0.349506
$249$	7.58665	0.480785
$250$	−0.162147	−0.0102551
$251$	−16.0015	−1.01000	−0.505002	−	0.863118i	$-0.668508\pi$
$251$	−16.0015	−1.01000	−0.505002	+	0.863118i	$0.668508\pi$
$252$	−1.47792	−0.0930999
$253$	−13.9807	−0.878962
$254$	2.32003	0.145572
$255$	8.83604	0.553334
$256$	13.9379	0.871118
$257$	9.71532	0.606025	0.303012	−	0.952987i	$-0.402008\pi$
$257$	9.71532	0.606025	0.303012	+	0.952987i	$0.402008\pi$
$258$	1.91177	0.119022
$259$	2.16784	0.134703
$260$	−2.57549	−0.159725
$261$	0.0108676	0.000672686 0
$262$	−0.837538	−0.0517432
$263$	23.1738	1.42896	0.714479	−	0.699657i	$-0.246664\pi$
$263$	23.1738	1.42896	0.714479	+	0.699657i	$0.246664\pi$
$264$	3.99393	0.245809
$265$	−2.86526	−0.176012
$266$	−0.526823	−0.0323016
$267$	−1.19353	−0.0730427
$268$	3.54342	0.216449
$269$	−4.92771	−0.300448	−0.150224	−	0.988652i	$-0.547999\pi$
$269$	−4.92771	−0.300448	−0.150224	+	0.988652i	$0.547999\pi$
$270$	0.885482	0.0538887
$271$	16.5952	1.00809	0.504044	−	0.863678i	$-0.331845\pi$
$271$	16.5952	1.00809	0.504044	+	0.863678i	$0.331845\pi$
$272$	−28.4505	−1.72506
$273$	0.740218	0.0448000
$274$	−1.63822	−0.0989684
$275$	−5.19353	−0.313181
$276$	−6.34137	−0.381706
$277$	−12.9860	−0.780256	−0.390128	−	0.920761i	$-0.627569\pi$
$277$	−12.9860	−0.780256	−0.390128	+	0.920761i	$0.627569\pi$
$278$	−3.40610	−0.204284
$279$	13.4583	0.805730
$280$	−0.306236	−0.0183011
$281$	29.4497	1.75682	0.878412	−	0.477904i	$-0.158603\pi$
$281$	29.4497	1.75682	0.878412	+	0.477904i	$0.158603\pi$
$282$	−0.895050	−0.0532994
$283$	−14.8999	−0.885708	−0.442854	−	0.896594i	$-0.646034\pi$
$283$	−14.8999	−0.885708	−0.442854	+	0.896594i	$0.646034\pi$
$284$	−5.75320	−0.341390
$285$	−8.15900	−0.483298
$286$	1.09888	0.0649780
$287$	−1.62080	−0.0956729
$288$	−3.01199	−0.177483
$289$	37.8089	2.22405
$290$	0.00111847	6.56790e−5 0
$291$	21.8551	1.28117
$292$	4.46787	0.261462
$293$	−2.70575	−0.158072	−0.0790358	−	0.996872i	$-0.525184\pi$
$293$	−2.70575	−0.158072	−0.0790358	+	0.996872i	$0.525184\pi$
$294$	−1.31097	−0.0764575
$295$	1.13767	0.0662374
$296$	2.93889	0.170820
$297$	28.3617	1.64571
$298$	−0.739316	−0.0428274
$299$	−3.51273	−0.203147
$300$	−2.35567	−0.136005
$301$	−4.69510	−0.270621
$302$	0.590330	0.0339697
$303$	−17.1325	−0.984234
$304$	26.2705	1.50672
$305$	−7.30684	−0.418389
$306$	1.89126	0.108116
$307$	−15.4273	−0.880481	−0.440241	−	0.897880i	$-0.645107\pi$
$307$	−15.4273	−0.880481	−0.440241	+	0.897880i	$0.645107\pi$
$308$	−4.87187	−0.277600
$309$	6.49235	0.369337
$310$	1.38511	0.0786690
$311$	−9.28894	−0.526728	−0.263364	−	0.964697i	$-0.584832\pi$
$311$	−9.28894	−0.526728	−0.263364	+	0.964697i	$0.584832\pi$
$312$	1.00349	0.0568117
$313$	−5.56675	−0.314651	−0.157326	−	0.987547i	$-0.550287\pi$
$313$	−5.56675	−0.314651	−0.157326	+	0.987547i	$0.550287\pi$
$314$	2.81596	0.158914
$315$	−0.748801	−0.0421902
$316$	20.7445	1.16697
$317$	30.9404	1.73778	0.868892	−	0.495002i	$-0.164833\pi$
$317$	30.9404	1.73778	0.868892	+	0.495002i	$0.164833\pi$
$318$	0.554506	0.0310952
$319$	0.0358244	0.00200578
$320$	7.37589	0.412325
$321$	−8.68274	−0.484623
$322$	−0.207457	−0.0115611
$323$	−50.6093	−2.81598
$324$	3.53559	0.196421
$325$	−1.30490	−0.0723829
$326$	−1.47054	−0.0814458
$327$	4.68213	0.258922
$328$	−2.19728	−0.121325
$329$	2.19814	0.121187
$330$	1.00509	0.0553283
$331$	0.0734460	0.00403696	0.00201848	−	0.999998i	$-0.499357\pi$
$331$	0.0734460	0.00403696	0.00201848	+	0.999998i	$0.499357\pi$
$332$	−12.5459	−0.688544
$333$	7.18612	0.393797
$334$	0.695708	0.0380674
$335$	1.79531	0.0980884
$336$	−2.17995	−0.118926
$337$	16.9619	0.923971	0.461986	−	0.886887i	$-0.347137\pi$
$337$	16.9619	0.923971	0.461986	+	0.886887i	$0.347137\pi$
$338$	−1.83182	−0.0996376
$339$	−18.1559	−0.986091
$340$	−14.6120	−0.792445
$341$	44.3647	2.40248
$342$	−1.74635	−0.0944317
$343$	6.54657	0.353481
$344$	−6.36503	−0.343179
$345$	−3.21292	−0.172978
$346$	3.43702	0.184776
$347$	−21.2221	−1.13926	−0.569630	−	0.821901i	$-0.692913\pi$
$347$	−21.2221	−1.13926	−0.569630	+	0.821901i	$0.692913\pi$
$348$	0.0162492	0.000871047 0
$349$	7.48298	0.400555	0.200277	−	0.979739i	$-0.435816\pi$
$349$	7.48298	0.400555	0.200277	+	0.979739i	$0.435816\pi$
$350$	−0.0770654	−0.00411932
$351$	7.12603	0.380359
$352$	−9.92885	−0.529209
$353$	2.09475	0.111492	0.0557461	−	0.998445i	$-0.482246\pi$
$353$	2.09475	0.111492	0.0557461	+	0.998445i	$0.482246\pi$
$354$	−0.220169	−0.0117019
$355$	−2.91492	−0.154708
$356$	1.97371	0.104606
$357$	4.19960	0.222266
$358$	0.0760456	0.00401914
$359$	−1.57842	−0.0833059	−0.0416529	−	0.999132i	$-0.513262\pi$
$359$	−1.57842	−0.0833059	−0.0416529	+	0.999132i	$0.513262\pi$
$360$	−1.01513	−0.0535021
$361$	27.7315	1.45955
$362$	−3.61153	−0.189818
$363$	19.0639	1.00059
$364$	−1.22408	−0.0641593
$365$	2.26369	0.118487
$366$	1.41407	0.0739147
$367$	34.6898	1.81079	0.905397	−	0.424567i	$-0.139574\pi$
$367$	34.6898	1.81079	0.905397	+	0.424567i	$0.139574\pi$
$368$	10.3450	0.539272
$369$	−5.37275	−0.279694
$370$	0.739584	0.0384491
$371$	−1.36180	−0.0707013
$372$	20.1229	1.04332
$373$	28.3861	1.46977	0.734887	−	0.678189i	$-0.237235\pi$
$373$	28.3861	1.46977	0.734887	+	0.678189i	$0.237235\pi$
$374$	6.23444	0.322375
$375$	−1.19353	−0.0616335
$376$	2.97996	0.153680
$377$	0.00900106	0.000463578 0
$378$	0.420852	0.0216463
$379$	−15.3669	−0.789344	−0.394672	−	0.918822i	$-0.629142\pi$
$379$	−15.3669	−0.789344	−0.394672	+	0.918822i	$0.629142\pi$
$380$	13.4924	0.692143
$381$	17.0772	0.874891
$382$	−3.45490	−0.176768
$383$	−23.8489	−1.21862	−0.609311	−	0.792932i	$-0.708554\pi$
$383$	−23.8489	−1.21862	−0.609311	+	0.792932i	$0.708554\pi$
$384$	−5.99094	−0.305724
$385$	−2.46838	−0.125800
$386$	−0.216989	−0.0110445
$387$	−15.5636	−0.791144
$388$	−36.1412	−1.83479
$389$	11.7669	0.596603	0.298302	−	0.954472i	$-0.403580\pi$
$389$	11.7669	0.596603	0.298302	+	0.954472i	$0.403580\pi$
$390$	0.252534	0.0127875
$391$	−19.9294	−1.00787
$392$	4.36473	0.220452
$393$	−6.16492	−0.310979
$394$	−1.56628	−0.0789078
$395$	10.5104	0.528836
$396$	−16.1496	−0.811548
$397$	−11.2778	−0.566019	−0.283009	−	0.959117i	$-0.591333\pi$
$397$	−11.2778	−0.566019	−0.283009	+	0.959117i	$0.591333\pi$
$398$	1.59835	0.0801179
$399$	−3.87782	−0.194134
$400$	3.84294	0.192147
$401$	12.9916	0.648770	0.324385	−	0.945925i	$-0.394843\pi$
$401$	12.9916	0.648770	0.324385	+	0.945925i	$0.394843\pi$
$402$	−0.347441	−0.0173288
$403$	11.1469	0.555264
$404$	28.3315	1.40955
$405$	1.79134	0.0890125
$406$	0.000531589 0	2.63823e−5 0
$407$	23.6887	1.17420
$408$	5.69329	0.281860
$409$	−14.2198	−0.703124	−0.351562	−	0.936165i	$-0.614349\pi$
$409$	−14.2198	−0.703124	−0.351562	+	0.936165i	$0.614349\pi$
$410$	−0.552955	−0.0273085
$411$	−12.0585	−0.594804
$412$	−10.7363	−0.528937
$413$	0.540710	0.0266066
$414$	−0.687692	−0.0337982
$415$	−6.35650	−0.312028
$416$	−2.49467	−0.122311
$417$	−25.0715	−1.22776
$418$	−5.75674	−0.281571
$419$	−32.8278	−1.60374	−0.801871	−	0.597497i	$-0.796162\pi$
$419$	−32.8278	−1.60374	−0.801871	+	0.597497i	$0.796162\pi$
$420$	−1.11961	−0.0546312
$421$	35.9329	1.75126	0.875630	−	0.482982i	$-0.160447\pi$
$421$	35.9329	1.75126	0.875630	+	0.482982i	$0.160447\pi$
$422$	−1.98647	−0.0966999
$423$	7.28655	0.354284
$424$	−1.84616	−0.0896576
$425$	−7.40330	−0.359113
$426$	0.564116	0.0273315
$427$	−3.47280	−0.168061
$428$	14.3584	0.694042
$429$	8.08858	0.390520
$430$	−1.60178	−0.0772449
$431$	−24.7729	−1.19327	−0.596634	−	0.802513i	$-0.703496\pi$
$431$	−24.7729	−1.19327	−0.596634	+	0.802513i	$0.703496\pi$
$432$	−20.9862	−1.00970
$433$	28.1739	1.35395	0.676976	−	0.736005i	$-0.263290\pi$
$433$	28.1739	1.35395	0.676976	+	0.736005i	$0.263290\pi$
$434$	0.658316	0.0316002
$435$	0.00823281	0.000394733 0
$436$	−7.74273	−0.370809
$437$	18.4023	0.880302
$438$	−0.438086	−0.0209326
$439$	−0.128231	−0.00612014	−0.00306007	−	0.999995i	$-0.500974\pi$
$439$	−0.128231	−0.00612014	−0.00306007	+	0.999995i	$0.500974\pi$
$440$	−3.34632	−0.159530
$441$	10.6726	0.508217
$442$	1.56644	0.0745077
$443$	3.96327	0.188301	0.0941504	−	0.995558i	$-0.469987\pi$
$443$	3.96327	0.188301	0.0941504	+	0.995558i	$0.469987\pi$
$444$	10.7447	0.509920
$445$	1.00000	0.0474045
$446$	−3.63602	−0.172170
$447$	−5.44193	−0.257395
$448$	3.50562	0.165625
$449$	−5.39593	−0.254650	−0.127325	−	0.991861i	$-0.540639\pi$
$449$	−5.39593	−0.254650	−0.127325	+	0.991861i	$0.540639\pi$
$450$	−0.255462	−0.0120426
$451$	−17.7110	−0.833977
$452$	30.0239	1.41221
$453$	4.34528	0.204159
$454$	−3.75639	−0.176296
$455$	−0.620194	−0.0290751
$456$	−5.25706	−0.246184
$457$	0.665823	0.0311459	0.0155729	−	0.999879i	$-0.495043\pi$
$457$	0.665823	0.0311459	0.0155729	+	0.999879i	$0.495043\pi$
$458$	0.674189	0.0315028
$459$	40.4293	1.88708
$460$	5.31313	0.247726
$461$	42.6998	1.98873	0.994365	−	0.106010i	$-0.0338076\pi$
$461$	42.6998	1.98873	0.994365	+	0.106010i	$0.0338076\pi$
$462$	0.477699	0.0222246
$463$	7.05789	0.328008	0.164004	−	0.986460i	$-0.447559\pi$
$463$	7.05789	0.328008	0.164004	+	0.986460i	$0.447559\pi$
$464$	−0.0265082	−0.00123061
$465$	10.1955	0.472804
$466$	−1.90881	−0.0884239
$467$	−30.6233	−1.41708	−0.708540	−	0.705671i	$-0.750646\pi$
$467$	−30.6233	−1.41708	−0.708540	+	0.705671i	$0.750646\pi$
$468$	−4.05767	−0.187566
$469$	0.853277	0.0394007
$470$	0.749920	0.0345912
$471$	20.7276	0.955077
$472$	0.733027	0.0337403
$473$	−51.3047	−2.35899
$474$	−2.03405	−0.0934270
$475$	6.83604	0.313659
$476$	−6.94478	−0.318314
$477$	−4.51420	−0.206691
$478$	−3.30511	−0.151172
$479$	24.7778	1.13213	0.566063	−	0.824362i	$-0.308466\pi$
$479$	24.7778	1.13213	0.566063	+	0.824362i	$0.308466\pi$
$480$	−2.28175	−0.104147
$481$	5.95190	0.271383
$482$	−1.83984	−0.0838026
$483$	−1.52704	−0.0694827
$484$	−31.5255	−1.43298
$485$	−18.3113	−0.831474
$486$	2.30977	0.104773
$487$	41.0479	1.86006	0.930029	−	0.367486i	$-0.119782\pi$
$487$	41.0479	1.86006	0.930029	+	0.367486i	$0.119782\pi$
$488$	−4.70799	−0.213121
$489$	−10.8243	−0.489492
$490$	1.09840	0.0496208
$491$	32.1154	1.44935	0.724674	−	0.689092i	$-0.241990\pi$
$491$	32.1154	1.44935	0.724674	+	0.689092i	$0.241990\pi$
$492$	−8.03332	−0.362170
$493$	0.0510671	0.00229995
$494$	−1.44641	−0.0650771
$495$	−8.18237	−0.367770
$496$	−32.8276	−1.47400
$497$	−1.38540	−0.0621439
$498$	1.23015	0.0551246
$499$	−29.5346	−1.32215	−0.661074	−	0.750321i	$-0.729899\pi$
$499$	−29.5346	−1.32215	−0.661074	+	0.750321i	$0.729899\pi$
$500$	1.97371	0.0882669
$501$	5.12094	0.228787
$502$	−2.59459	−0.115802
$503$	17.9883	0.802058	0.401029	−	0.916065i	$-0.368653\pi$
$503$	17.9883	0.802058	0.401029	+	0.916065i	$0.368653\pi$
$504$	−0.482472	−0.0214910
$505$	14.3545	0.638766
$506$	−2.26694	−0.100778
$507$	−13.4836	−0.598826
$508$	−28.2402	−1.25295
$509$	28.2936	1.25409	0.627045	−	0.778983i	$-0.284264\pi$
$509$	28.2936	1.25409	0.627045	+	0.778983i	$0.284264\pi$
$510$	1.43274	0.0634428
$511$	1.07589	0.0475946
$512$	12.2990	0.543546
$513$	−37.3315	−1.64822
$514$	1.57531	0.0694840
$515$	−5.43964	−0.239699
$516$	−23.2707	−1.02444
$517$	24.0197	1.05639
$518$	0.351510	0.0154445
$519$	25.2991	1.11051
$520$	−0.840781	−0.0368707
$521$	−13.2784	−0.581739	−0.290870	−	0.956763i	$-0.593945\pi$
$521$	−13.2784	−0.581739	−0.290870	+	0.956763i	$0.593945\pi$
$522$	0.00176215	7.71271e−5 0
$523$	−23.7559	−1.03877	−0.519386	−	0.854540i	$-0.673839\pi$
$523$	−23.7559	−1.03877	−0.519386	+	0.854540i	$0.673839\pi$
$524$	10.1948	0.445361
$525$	−0.567260	−0.0247573
$526$	3.75757	0.163838
$527$	63.2413	2.75483
$528$	−23.8209	−1.03667
$529$	−15.7534	−0.684930
$530$	−0.464594	−0.0201807
$531$	1.79238	0.0777828
$532$	6.41265	0.278024
$533$	−4.44997	−0.192750
$534$	−0.193527	−0.00837474
$535$	7.27486	0.314519
$536$	1.15677	0.0499647
$537$	0.559754	0.0241552
$538$	−0.799014	−0.0344479
$539$	35.1815	1.51538
$540$	−10.7784	−0.463827
$541$	−12.9184	−0.555404	−0.277702	−	0.960667i	$-0.589573\pi$
$541$	−12.9184	−0.555404	−0.277702	+	0.960667i	$0.589573\pi$
$542$	2.69087	0.115583
$543$	−26.5836	−1.14081
$544$	−14.1534	−0.606824
$545$	−3.92293	−0.168040
$546$	0.120024	0.00513657
$547$	39.0481	1.66958	0.834788	−	0.550571i	$-0.185590\pi$
$547$	39.0481	1.66958	0.834788	+	0.550571i	$0.185590\pi$
$548$	19.9409	0.851834
$549$	−11.5119	−0.491315
$550$	−0.842116	−0.0359079
$551$	−0.0471543	−0.00200884
$552$	−2.07017	−0.0881122
$553$	4.99540	0.212426
$554$	−2.10565	−0.0894605
$555$	5.44390	0.231081
$556$	41.4601	1.75830
$557$	5.51252	0.233573	0.116786	−	0.993157i	$-0.462741\pi$
$557$	5.51252	0.233573	0.116786	+	0.993157i	$0.462741\pi$
$558$	2.18223	0.0923813
$559$	−12.8906	−0.545213
$560$	1.82648	0.0771827
$561$	45.8902	1.93749
$562$	4.77519	0.201429
$563$	−37.2064	−1.56806	−0.784031	−	0.620722i	$-0.786840\pi$
$563$	−37.2064	−1.56806	−0.784031	+	0.620722i	$0.786840\pi$
$564$	10.8948	0.458755
$565$	15.2119	0.639971
$566$	−2.41598	−0.101551
$567$	0.851390	0.0357550
$568$	−1.87816	−0.0788057
$569$	−27.0351	−1.13337	−0.566685	−	0.823935i	$-0.691774\pi$
$569$	−27.0351	−1.13337	−0.566685	+	0.823935i	$0.691774\pi$
$570$	−1.32296	−0.0554127
$571$	32.8494	1.37471	0.687353	−	0.726324i	$-0.258773\pi$
$571$	32.8494	1.37471	0.687353	+	0.726324i	$0.258773\pi$
$572$	−13.3759	−0.559274
$573$	−25.4307	−1.06238
$574$	−0.262809	−0.0109694
$575$	2.69195	0.112262
$576$	11.6207	0.484195
$577$	35.2109	1.46585	0.732925	−	0.680309i	$-0.238155\pi$
$577$	35.2109	1.46585	0.732925	+	0.680309i	$0.238155\pi$
$578$	6.13061	0.255000
$579$	−1.59721	−0.0663776
$580$	−0.0136144	−0.000565308 0
$581$	−3.02112	−0.125337
$582$	3.54374	0.146893
$583$	−14.8808	−0.616301
$584$	1.45856	0.0603555
$585$	−2.05586	−0.0849994
$586$	−0.438730	−0.0181238
$587$	40.2094	1.65962	0.829811	−	0.558045i	$-0.188448\pi$
$587$	40.2094	1.65962	0.829811	+	0.558045i	$0.188448\pi$
$588$	15.9576	0.658080
$589$	−58.3956	−2.40615
$590$	0.184469	0.00759448
$591$	−11.5290	−0.474239
$592$	−17.5284	−0.720412
$593$	29.5937	1.21527	0.607633	−	0.794218i	$-0.292119\pi$
$593$	29.5937	1.21527	0.607633	+	0.794218i	$0.292119\pi$
$594$	4.59877	0.188690
$595$	−3.51865	−0.144250
$596$	8.99920	0.368621
$597$	11.7651	0.481511
$598$	−0.569580	−0.0232919
$599$	−8.74518	−0.357318	−0.178659	−	0.983911i	$-0.557176\pi$
$599$	−8.74518	−0.357318	−0.178659	+	0.983911i	$0.557176\pi$
$600$	−0.769020	−0.0313951
$601$	3.42482	0.139701	0.0698507	−	0.997557i	$-0.477748\pi$
$601$	3.42482	0.139701	0.0698507	+	0.997557i	$0.477748\pi$
$602$	−0.761297	−0.0310282
$603$	2.82850	0.115185
$604$	−7.18569	−0.292382
$605$	−15.9727	−0.649384
$606$	−2.77798	−0.112848
$607$	−20.5908	−0.835756	−0.417878	−	0.908503i	$-0.637226\pi$
$607$	−20.5908	−0.835756	−0.417878	+	0.908503i	$0.637226\pi$
$608$	13.0690	0.530017
$609$	0.00391290	0.000158559 0
$610$	−1.18478	−0.0479705
$611$	6.03508	0.244153
$612$	−23.0210	−0.930571
$613$	−26.4547	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$613$	−26.4547	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$614$	−2.50149	−0.100952
$615$	−4.07017	−0.164125
$616$	−1.59044	−0.0640808
$617$	−43.8302	−1.76454	−0.882268	−	0.470747i	$-0.843985\pi$
$617$	−43.8302	−1.76454	−0.882268	+	0.470747i	$0.843985\pi$
$618$	1.05272	0.0423465
$619$	−0.0432321	−0.00173765	−0.000868823	−	1.00000i	$-0.500277\pi$
$619$	−0.0432321	−0.00173765	−0.000868823		1.00000i	$0.500277\pi$
$620$	−16.8600	−0.677115
$621$	−14.7007	−0.589919
$622$	−1.50618	−0.0603922
$623$	0.475281	0.0190417
$624$	−5.98513	−0.239597
$625$	1.00000	0.0400000
$626$	−0.902633	−0.0360765
$627$	−42.3740	−1.69225
$628$	−34.2768	−1.36779
$629$	33.7679	1.34641
$630$	−0.121416	−0.00483733
$631$	−29.9042	−1.19047	−0.595233	−	0.803553i	$-0.702940\pi$
$631$	−29.9042	−1.19047	−0.595233	+	0.803553i	$0.702940\pi$
$632$	6.77213	0.269381
$633$	−14.6219	−0.581170
$634$	5.01689	0.199246
$635$	−14.3082	−0.567803
$636$	−6.74963	−0.267640
$637$	8.83954	0.350235
$638$	0.00580882	0.000229973 0
$639$	−4.59243	−0.181674
$640$	5.01953	0.198414
$641$	−16.3131	−0.644330	−0.322165	−	0.946684i	$-0.604411\pi$
$641$	−16.3131	−0.644330	−0.322165	+	0.946684i	$0.604411\pi$
$642$	−1.40788	−0.0555647
$643$	17.2134	0.678832	0.339416	−	0.940636i	$-0.389771\pi$
$643$	17.2134	0.678832	0.339416	+	0.940636i	$0.389771\pi$
$644$	2.52523	0.0995080
$645$	−11.7904	−0.464245
$646$	−8.20616	−0.322867
$647$	28.3178	1.11329	0.556645	−	0.830751i	$-0.312088\pi$
$647$	28.3178	1.11329	0.556645	+	0.830751i	$0.312088\pi$
$648$	1.15421	0.0453416
$649$	5.90850	0.231929
$650$	−0.211586	−0.00829909
$651$	4.84571	0.189918
$652$	17.8999	0.701015
$653$	27.2771	1.06743	0.533717	−	0.845663i	$-0.320795\pi$
$653$	27.2771	1.06743	0.533717	+	0.845663i	$0.320795\pi$
$654$	0.759194	0.0296868
$655$	5.16529	0.201825
$656$	13.1052	0.511672
$657$	3.56644	0.139140
$658$	0.356422	0.0138948
$659$	−36.6207	−1.42654	−0.713270	−	0.700889i	$-0.752787\pi$
$659$	−36.6207	−1.42654	−0.713270	+	0.700889i	$0.752787\pi$
$660$	−12.2343	−0.476218
$661$	−39.6430	−1.54193	−0.770967	−	0.636875i	$-0.780227\pi$
$661$	−39.6430	−1.54193	−0.770967	+	0.636875i	$0.780227\pi$
$662$	0.0119091	0.000462859 0
$663$	11.5302	0.447794
$664$	−4.09566	−0.158942
$665$	3.24904	0.125992
$666$	1.16521	0.0451509
$667$	−0.0185688	−0.000718987 0
$668$	−8.46838	−0.327652
$669$	−26.7638	−1.03475
$670$	0.291105	0.0112464
$671$	−37.9483	−1.46498
$672$	−1.08447	−0.0418345
$673$	7.70607	0.297047	0.148524	−	0.988909i	$-0.452548\pi$
$673$	7.70607	0.297047	0.148524	+	0.988909i	$0.452548\pi$
$674$	2.75032	0.105938
$675$	−5.46097	−0.210193
$676$	22.2974	0.857594
$677$	−16.9048	−0.649706	−0.324853	−	0.945765i	$-0.605315\pi$
$677$	−16.9048	−0.649706	−0.324853	+	0.945765i	$0.605315\pi$
$678$	−2.94392	−0.113061
$679$	−8.70302	−0.333991
$680$	−4.77014	−0.182926
$681$	−27.6499	−1.05955
$682$	7.19361	0.275458
$683$	−7.11167	−0.272120	−0.136060	−	0.990701i	$-0.543444\pi$
$683$	−7.11167	−0.272120	−0.136060	+	0.990701i	$0.543444\pi$
$684$	21.2571	0.812786
$685$	10.1033	0.386026
$686$	1.06151	0.0405285
$687$	4.96255	0.189333
$688$	37.9628	1.44732
$689$	−3.73888	−0.142440
$690$	−0.520966	−0.0198328
$691$	18.0632	0.687156	0.343578	−	0.939124i	$-0.388361\pi$
$691$	18.0632	0.687156	0.343578	+	0.939124i	$0.388361\pi$
$692$	−41.8366	−1.59039
$693$	−3.88892	−0.147728
$694$	−3.44110	−0.130622
$695$	21.0062	0.796811
$696$	0.00530461	0.000201071 0
$697$	−25.2468	−0.956289
$698$	1.21334	0.0459258
$699$	−14.0503	−0.531431
$700$	0.938065	0.0354555
$701$	14.5894	0.551034	0.275517	−	0.961296i	$-0.411151\pi$
$701$	14.5894	0.551034	0.275517	+	0.961296i	$0.411151\pi$
$702$	1.15547	0.0436102
$703$	−31.1805	−1.17599
$704$	38.3069	1.44375
$705$	5.51998	0.207895
$706$	0.339658	0.0127832
$707$	6.82240	0.256583
$708$	2.67997	0.100719
$709$	19.9577	0.749527	0.374763	−	0.927120i	$-0.377724\pi$
$709$	19.9577	0.749527	0.374763	+	0.927120i	$0.377724\pi$
$710$	−0.472646	−0.0177381
$711$	16.5591	0.621014
$712$	0.644326	0.0241471
$713$	−22.9955	−0.861188
$714$	0.680953	0.0254840
$715$	−6.77704	−0.253447
$716$	−0.925652	−0.0345932
$717$	−24.3281	−0.908551
$718$	−0.255937	−0.00955147
$719$	−1.62038	−0.0604298	−0.0302149	−	0.999543i	$-0.509619\pi$
$719$	−1.62038	−0.0604298	−0.0302149	+	0.999543i	$0.509619\pi$
$720$	6.05453	0.225639
$721$	−2.58535	−0.0962836
$722$	4.49658	0.167345
$723$	−13.5427	−0.503657
$724$	43.9608	1.63379
$725$	−0.00689789	−0.000256181 0
$726$	3.09115	0.114724
$727$	5.61596	0.208284	0.104142	−	0.994562i	$-0.466790\pi$
$727$	5.61596	0.208284	0.104142	+	0.994562i	$0.466790\pi$
$728$	−0.399607	−0.0148104
$729$	22.3757	0.828730
$730$	0.367052	0.0135852
$731$	−73.1342	−2.70496
$732$	−17.2125	−0.636194
$733$	48.1350	1.77791	0.888953	−	0.457998i	$-0.151433\pi$
$733$	48.1350	1.77791	0.888953	+	0.457998i	$0.151433\pi$
$734$	5.62485	0.207617
$735$	8.08508	0.298223
$736$	5.14641	0.189699
$737$	9.32400	0.343454
$738$	−0.871176	−0.0320684
$739$	−8.98910	−0.330669	−0.165335	−	0.986238i	$-0.552870\pi$
$739$	−8.98910	−0.330669	−0.165335	+	0.986238i	$0.552870\pi$
$740$	−9.00245	−0.330937
$741$	−10.6467	−0.391116
$742$	−0.220813	−0.00810629
$743$	2.20072	0.0807365	0.0403683	−	0.999185i	$-0.487147\pi$
$743$	2.20072	0.0807365	0.0403683	+	0.999185i	$0.487147\pi$
$744$	6.56920	0.240839
$745$	4.55954	0.167049
$746$	4.60272	0.168518
$747$	−10.0146	−0.366416
$748$	−75.8876	−2.77473
$749$	3.45760	0.126338
$750$	−0.193527	−0.00706661
$751$	−22.7671	−0.830781	−0.415391	−	0.909643i	$-0.636355\pi$
$751$	−22.7671	−0.830781	−0.415391	+	0.909643i	$0.636355\pi$
$752$	−17.7733	−0.648127
$753$	−19.0982	−0.695977
$754$	0.00145950	5.31517e−5 0
$755$	−3.64071	−0.132499
$756$	−5.12275	−0.186313
$757$	−30.2022	−1.09772	−0.548859	−	0.835915i	$-0.684937\pi$
$757$	−30.2022	−1.09772	−0.548859	+	0.835915i	$0.684937\pi$
$758$	−2.49170	−0.0905026
$759$	−16.6864	−0.605678
$760$	4.40464	0.159773
$761$	17.4802	0.633656	0.316828	−	0.948483i	$-0.397382\pi$
$761$	17.4802	0.633656	0.316828	+	0.948483i	$0.397382\pi$
$762$	2.76902	0.100311
$763$	−1.86449	−0.0674992
$764$	42.0542	1.52147
$765$	−11.6639	−0.421708
$766$	−3.86703	−0.139722
$767$	1.48454	0.0536036
$768$	16.6352	0.600273
$769$	−28.0588	−1.01183	−0.505913	−	0.862585i	$-0.668844\pi$
$769$	−28.0588	−1.01183	−0.505913	+	0.862585i	$0.668844\pi$
$770$	−0.400241	−0.0144237
$771$	11.5955	0.417602
$772$	2.64126	0.0950612
$773$	−2.19223	−0.0788489	−0.0394245	−	0.999223i	$-0.512552\pi$
$773$	−2.19223	−0.0788489	−0.0394245	+	0.999223i	$0.512552\pi$
$774$	−2.52360	−0.0907090
$775$	−8.54230	−0.306849
$776$	−11.7985	−0.423540
$777$	2.58738	0.0928218
$778$	1.90796	0.0684038
$779$	23.3123	0.835249
$780$	−3.07392	−0.110064
$781$	−15.1387	−0.541706
$782$	−3.23149	−0.115558
$783$	0.0376692	0.00134619
$784$	−26.0325	−0.929732
$785$	−17.3667	−0.619843
$786$	−0.999624	−0.0356554
$787$	−36.7899	−1.31142	−0.655709	−	0.755013i	$-0.727630\pi$
$787$	−36.7899	−1.31142	−0.655709	+	0.755013i	$0.727630\pi$
$788$	19.0652	0.679170
$789$	27.6585	0.984670
$790$	1.70423	0.0606339
$791$	7.22994	0.257067
$792$	−5.27211	−0.187336
$793$	−9.53471	−0.338587
$794$	−1.82867	−0.0648971
$795$	−3.41977	−0.121287
$796$	−19.4556	−0.689585
$797$	−42.4563	−1.50388	−0.751941	−	0.659231i	$-0.770882\pi$
$797$	−42.4563	−1.50388	−0.751941	+	0.659231i	$0.770882\pi$
$798$	−0.628777	−0.0222585
$799$	34.2398	1.21132
$800$	1.91177	0.0675914
$801$	1.57549	0.0556673
$802$	2.10655	0.0743850
$803$	11.7566	0.414880
$804$	4.22917	0.149151
$805$	1.27943	0.0450941
$806$	1.80743	0.0636641
$807$	−5.88135	−0.207033
$808$	9.24896	0.325377
$809$	28.4853	1.00149	0.500744	−	0.865595i	$-0.333060\pi$
$809$	28.4853	1.00149	0.500744	+	0.865595i	$0.333060\pi$
$810$	0.290461	0.0102058
$811$	−17.7783	−0.624282	−0.312141	−	0.950036i	$-0.601046\pi$
$811$	−17.7783	−0.624282	−0.312141	+	0.950036i	$0.601046\pi$
$812$	−0.00647067	−0.000227076 0
$813$	19.8069	0.694657
$814$	3.84105	0.134629
$815$	9.06918	0.317680
$816$	−33.9564	−1.18871
$817$	67.5304	2.36259
$818$	−2.30570	−0.0806169
$819$	−0.977111	−0.0341430
$820$	6.73074	0.235048
$821$	31.1566	1.08737	0.543687	−	0.839288i	$-0.317028\pi$
$821$	31.1566	1.08737	0.543687	+	0.839288i	$0.317028\pi$
$822$	−1.95526	−0.0681974
$823$	20.4644	0.713346	0.356673	−	0.934229i	$-0.383911\pi$
$823$	20.4644	0.713346	0.356673	+	0.934229i	$0.383911\pi$
$824$	−3.50490	−0.122099
$825$	−6.19862	−0.215808
$826$	0.0876747	0.00305059
$827$	−16.0701	−0.558812	−0.279406	−	0.960173i	$-0.590138\pi$
$827$	−16.0701	−0.558812	−0.279406	+	0.960173i	$0.590138\pi$
$828$	8.37080	0.290906
$829$	−34.1508	−1.18611	−0.593053	−	0.805163i	$-0.702078\pi$
$829$	−34.1508	−1.18611	−0.593053	+	0.805163i	$0.702078\pi$
$830$	−1.03069	−0.0357757
$831$	−15.4992	−0.537661
$832$	9.62481	0.333680
$833$	50.1508	1.73762
$834$	−4.06527	−0.140769
$835$	−4.29059	−0.148482
$836$	70.0729	2.42352
$837$	46.6493	1.61244
$838$	−5.32293	−0.183878
$839$	−2.87004	−0.0990846	−0.0495423	−	0.998772i	$-0.515776\pi$
$839$	−2.87004	−0.0990846	−0.0495423	+	0.998772i	$0.515776\pi$
$840$	−0.365500	−0.0126110
$841$	−29.0000	−0.999998
$842$	5.82641	0.200791
$843$	35.1491	1.21060
$844$	24.1800	0.832309
$845$	11.2972	0.388637
$846$	1.18149	0.0406206
$847$	−7.59153	−0.260848
$848$	11.0110	0.378121
$849$	−17.7835	−0.610327
$850$	−1.20042	−0.0411742
$851$	−12.2785	−0.420902
$852$	−6.86660	−0.235246
$853$	22.6448	0.775344	0.387672	−	0.921797i	$-0.373279\pi$
$853$	22.6448	0.775344	0.387672	+	0.921797i	$0.373279\pi$
$854$	−0.563105	−0.0192691
$855$	10.7701	0.368331
$856$	4.68738	0.160211
$857$	22.5002	0.768594	0.384297	−	0.923210i	$-0.374444\pi$
$857$	22.5002	0.768594	0.384297	+	0.923210i	$0.374444\pi$
$858$	1.31154	0.0447752
$859$	−38.7537	−1.32226	−0.661129	−	0.750272i	$-0.729923\pi$
$859$	−38.7537	−1.32226	−0.661129	+	0.750272i	$0.729923\pi$
$860$	19.4974	0.664857
$861$	−1.93447	−0.0659266
$862$	−4.01686	−0.136815
$863$	29.5580	1.00617	0.503083	−	0.864238i	$-0.332199\pi$
$863$	29.5580	1.00617	0.503083	+	0.864238i	$0.332199\pi$
$864$	−10.4401	−0.355181
$865$	−21.1969	−0.720718
$866$	4.56832	0.155238
$867$	45.1259	1.53256
$868$	−8.01324	−0.271987
$869$	54.5861	1.85171
$870$	0.00133493	4.52583e−5 0
$871$	2.34270	0.0793795
$872$	−2.52765	−0.0855969
$873$	−28.8494	−0.976403
$874$	2.98388	0.100931
$875$	0.475281	0.0160674
$876$	5.33253	0.180169
$877$	−0.306586	−0.0103527	−0.00517633	−	0.999987i	$-0.501648\pi$
$877$	−0.306586	−0.0103527	−0.00517633	+	0.999987i	$0.501648\pi$
$878$	−0.0207923	−0.000701707 0
$879$	−3.22939	−0.108925
$880$	19.9584	0.672798
$881$	4.86851	0.164024	0.0820121	−	0.996631i	$-0.473865\pi$
$881$	4.86851	0.164024	0.0820121	+	0.996631i	$0.473865\pi$
$882$	1.73053	0.0582698
$883$	54.1155	1.82113	0.910565	−	0.413366i	$-0.135647\pi$
$883$	54.1155	1.82113	0.910565	+	0.413366i	$0.135647\pi$
$884$	−19.0672	−0.641298
$885$	1.35783	0.0456431
$886$	0.642634	0.0215897
$887$	−42.0954	−1.41343	−0.706713	−	0.707500i	$-0.749823\pi$
$887$	−42.0954	−1.41343	−0.706713	+	0.707500i	$0.749823\pi$
$888$	3.50765	0.117709
$889$	−6.80040	−0.228078
$890$	0.162147	0.00543519
$891$	9.30338	0.311675
$892$	44.2588	1.48189
$893$	−31.6162	−1.05800
$894$	−0.882394	−0.0295117
$895$	−0.468991	−0.0156767
$896$	2.38568	0.0797001
$897$	−4.19254	−0.139985
$898$	−0.874935	−0.0291970
$899$	0.0589238	0.00196522
$900$	3.10956	0.103652
$901$	−21.2124	−0.706688
$902$	−2.87178	−0.0956200
$903$	−5.60373	−0.186480
$904$	9.80144	0.325991
$905$	22.2732	0.740386
$906$	0.704575	0.0234079
$907$	−3.59534	−0.119381	−0.0596907	−	0.998217i	$-0.519011\pi$
$907$	−3.59534	−0.119381	−0.0596907	+	0.998217i	$0.519011\pi$
$908$	45.7240	1.51740
$909$	22.6154	0.750105
$910$	−0.100563	−0.00333362
$911$	51.3001	1.69965	0.849823	−	0.527068i	$-0.176709\pi$
$911$	51.3001	1.69965	0.849823	+	0.527068i	$0.176709\pi$
$912$	31.3546	1.03825
$913$	−33.0127	−1.09256
$914$	0.107961	0.00357104
$915$	−8.72092	−0.288305
$916$	−8.20645	−0.271149
$917$	2.45496	0.0810700
$918$	6.55549	0.216363
$919$	−24.6752	−0.813959	−0.406979	−	0.913437i	$-0.633418\pi$
$919$	−24.6752	−0.813959	−0.406979	+	0.913437i	$0.633418\pi$
$920$	1.73450	0.0571846
$921$	−18.4129	−0.606725
$922$	6.92366	0.228019
$923$	−3.80368	−0.125200
$924$	−5.81471	−0.191290
$925$	−4.56119	−0.149971
$926$	1.14442	0.0376079
$927$	−8.57011	−0.281479
$928$	−0.0131872	−0.000432891 0
$929$	−6.23375	−0.204523	−0.102261	−	0.994758i	$-0.532608\pi$
$929$	−6.23375	−0.204523	−0.102261	+	0.994758i	$0.532608\pi$
$930$	1.65317	0.0542095
$931$	−46.3081	−1.51769
$932$	23.2347	0.761076
$933$	−11.0866	−0.362959
$934$	−4.96549	−0.162476
$935$	−38.4493	−1.25743
$936$	−1.32464	−0.0432974
$937$	−10.7283	−0.350479	−0.175240	−	0.984526i	$-0.556070\pi$
$937$	−10.7283	−0.350479	−0.175240	+	0.984526i	$0.556070\pi$
$938$	0.138356	0.00451750
$939$	−6.64407	−0.216821
$940$	−9.12827	−0.297731
$941$	11.6152	0.378644	0.189322	−	0.981915i	$-0.439371\pi$
$941$	11.6152	0.378644	0.189322	+	0.981915i	$0.439371\pi$
$942$	3.36092	0.109505
$943$	9.18011	0.298945
$944$	−4.37198	−0.142296
$945$	−2.59550	−0.0844315
$946$	−8.31891	−0.270471
$947$	9.67739	0.314473	0.157236	−	0.987561i	$-0.449742\pi$
$947$	9.67739	0.314473	0.157236	+	0.987561i	$0.449742\pi$
$948$	24.7591	0.804139
$949$	2.95390	0.0958875
$950$	1.10845	0.0359627
$951$	36.9282	1.19748
$952$	−2.26715	−0.0734789
$953$	18.4382	0.597273	0.298636	−	0.954367i	$-0.403468\pi$
$953$	18.4382	0.597273	0.298636	+	0.954367i	$0.403468\pi$
$954$	−0.731965	−0.0236983
$955$	21.3072	0.689485
$956$	40.2309	1.30116
$957$	0.0427573	0.00138215
$958$	4.01765	0.129804
$959$	4.80189	0.155061
$960$	8.80333	0.284126
$961$	41.9709	1.35390
$962$	0.965083	0.0311155
$963$	11.4615	0.369341
$964$	22.3952	0.721300
$965$	1.33822	0.0430790
$966$	−0.247605	−0.00796656
$967$	−45.7382	−1.47084	−0.735420	−	0.677611i	$-0.763015\pi$
$967$	−45.7382	−1.47084	−0.735420	+	0.677611i	$0.763015\pi$
$968$	−10.2916	−0.330786
$969$	−60.4036	−1.94044
$970$	−2.96913	−0.0953330
$971$	−18.8604	−0.605259	−0.302629	−	0.953108i	$-0.597864\pi$
$971$	−18.8604	−0.605259	−0.302629	+	0.953108i	$0.597864\pi$
$972$	−28.1153	−0.901799
$973$	9.98384	0.320067
$974$	6.65580	0.213266
$975$	−1.55743	−0.0498778
$976$	28.0798	0.898812
$977$	−5.89342	−0.188547	−0.0942736	−	0.995546i	$-0.530053\pi$
$977$	−5.89342	−0.188547	−0.0942736	+	0.995546i	$0.530053\pi$
$978$	−1.75513	−0.0561229
$979$	5.19353	0.165986
$980$	−13.3701	−0.427093
$981$	−6.18056	−0.197330
$982$	5.20742	0.166175
$983$	2.88660	0.0920683	0.0460341	−	0.998940i	$-0.485342\pi$
$983$	2.88660	0.0920683	0.0460341	+	0.998940i	$0.485342\pi$
$984$	−2.62251	−0.0836027
$985$	9.65959	0.307780
$986$	0.00828040	0.000263702 0
$987$	2.62354	0.0835082
$988$	17.6062	0.560127
$989$	26.5927	0.845599
$990$	−1.32675	−0.0421668
$991$	15.8725	0.504206	0.252103	−	0.967700i	$-0.418878\pi$
$991$	15.8725	0.504206	0.252103	+	0.967700i	$0.418878\pi$
$992$	−16.3309	−0.518508
$993$	0.0876598	0.00278180
$994$	−0.224639	−0.00712513
$995$	−9.85738	−0.312500
$996$	−14.9738	−0.474464
$997$	−28.4316	−0.900437	−0.450218	−	0.892918i	$-0.648654\pi$
$997$	−28.4316	−0.900437	−0.450218	+	0.892918i	$0.648654\pi$
$998$	−4.78895	−0.151591
$999$	24.9085	0.788071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.d.1.2	✓	4
3.2	odd	2		4005.2.a.l.1.3		4
4.3	odd	2		7120.2.a.bc.1.1		4
5.4	even	2		2225.2.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.2	✓	4	1.1	even	1	trivial
2225.2.a.i.1.3		4	5.4	even	2
4005.2.a.l.1.3		4	3.2	odd	2
7120.2.a.bc.1.1		4	4.3	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:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.162147 q^{2} +1.19353 q^{3} -1.97371 q^{4} -1.00000 q^{5} +0.193527 q^{6} -0.475281 q^{7} -0.644326 q^{8} -1.57549 q^{9} +O(q^{10})\)	\(q+0.162147 q^{2} +1.19353 q^{3} -1.97371 q^{4} -1.00000 q^{5} +0.193527 q^{6} -0.475281 q^{7} -0.644326 q^{8} -1.57549 q^{9} -0.162147 q^{10} -5.19353 q^{11} -2.35567 q^{12} -1.30490 q^{13} -0.0770654 q^{14} -1.19353 q^{15} +3.84294 q^{16} -7.40330 q^{17} -0.255462 q^{18} +6.83604 q^{19} +1.97371 q^{20} -0.567260 q^{21} -0.842116 q^{22} +2.69195 q^{23} -0.769020 q^{24} +1.00000 q^{25} -0.211586 q^{26} -5.46097 q^{27} +0.938065 q^{28} -0.00689789 q^{29} -0.193527 q^{30} -8.54230 q^{31} +1.91177 q^{32} -6.19862 q^{33} -1.20042 q^{34} +0.475281 q^{35} +3.10956 q^{36} -4.56119 q^{37} +1.10845 q^{38} -1.55743 q^{39} +0.644326 q^{40} +3.41020 q^{41} -0.0919797 q^{42} +9.87858 q^{43} +10.2505 q^{44} +1.57549 q^{45} +0.436493 q^{46} -4.62493 q^{47} +4.58665 q^{48} -6.77411 q^{49} +0.162147 q^{50} -8.83604 q^{51} +2.57549 q^{52} +2.86526 q^{53} -0.885482 q^{54} +5.19353 q^{55} +0.306236 q^{56} +8.15900 q^{57} -0.00111847 q^{58} -1.13767 q^{59} +2.35567 q^{60} +7.30684 q^{61} -1.38511 q^{62} +0.748801 q^{63} -7.37589 q^{64} +1.30490 q^{65} -1.00509 q^{66} -1.79531 q^{67} +14.6120 q^{68} +3.21292 q^{69} +0.0770654 q^{70} +2.91492 q^{71} +1.01513 q^{72} -2.26369 q^{73} -0.739584 q^{74} +1.19353 q^{75} -13.4924 q^{76} +2.46838 q^{77} -0.252534 q^{78} -10.5104 q^{79} -3.84294 q^{80} -1.79134 q^{81} +0.552955 q^{82} +6.35650 q^{83} +1.11961 q^{84} +7.40330 q^{85} +1.60178 q^{86} -0.00823281 q^{87} +3.34632 q^{88} -1.00000 q^{89} +0.255462 q^{90} +0.620194 q^{91} -5.31313 q^{92} -10.1955 q^{93} -0.749920 q^{94} -6.83604 q^{95} +2.28175 q^{96} +18.3113 q^{97} -1.09840 q^{98} +8.18237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * q^99

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