LMFDB - Embedded newform 4009.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62936 q^{2} +3.14041 q^{3} +4.91352 q^{4} +2.46626 q^{5} -8.25726 q^{6} +1.15889 q^{7} -7.66068 q^{8} +6.86218 q^{9} +O(q^{10})$	\(q-2.62936 q^{2} +3.14041 q^{3} +4.91352 q^{4} +2.46626 q^{5} -8.25726 q^{6} +1.15889 q^{7} -7.66068 q^{8} +6.86218 q^{9} -6.48468 q^{10} +5.06391 q^{11} +15.4305 q^{12} +3.51750 q^{13} -3.04714 q^{14} +7.74507 q^{15} +10.3156 q^{16} -0.875800 q^{17} -18.0431 q^{18} +1.00000 q^{19} +12.1180 q^{20} +3.63940 q^{21} -13.3148 q^{22} -5.03844 q^{23} -24.0577 q^{24} +1.08243 q^{25} -9.24877 q^{26} +12.1288 q^{27} +5.69425 q^{28} +7.39407 q^{29} -20.3645 q^{30} -3.14194 q^{31} -11.8021 q^{32} +15.9028 q^{33} +2.30279 q^{34} +2.85813 q^{35} +33.7175 q^{36} -1.68116 q^{37} -2.62936 q^{38} +11.0464 q^{39} -18.8932 q^{40} +8.51936 q^{41} -9.56929 q^{42} +3.26605 q^{43} +24.8816 q^{44} +16.9239 q^{45} +13.2479 q^{46} -9.92344 q^{47} +32.3953 q^{48} -5.65697 q^{49} -2.84610 q^{50} -2.75037 q^{51} +17.2833 q^{52} -5.61561 q^{53} -31.8911 q^{54} +12.4889 q^{55} -8.87792 q^{56} +3.14041 q^{57} -19.4416 q^{58} +1.08087 q^{59} +38.0555 q^{60} -12.6746 q^{61} +8.26127 q^{62} +7.95254 q^{63} +10.4007 q^{64} +8.67507 q^{65} -41.8140 q^{66} -13.6436 q^{67} -4.30326 q^{68} -15.8228 q^{69} -7.51505 q^{70} +4.76934 q^{71} -52.5690 q^{72} -4.97631 q^{73} +4.42038 q^{74} +3.39928 q^{75} +4.91352 q^{76} +5.86853 q^{77} -29.0449 q^{78} +1.01863 q^{79} +25.4410 q^{80} +17.5030 q^{81} -22.4004 q^{82} -15.3391 q^{83} +17.8823 q^{84} -2.15995 q^{85} -8.58762 q^{86} +23.2204 q^{87} -38.7930 q^{88} +9.78565 q^{89} -44.4990 q^{90} +4.07641 q^{91} -24.7565 q^{92} -9.86697 q^{93} +26.0923 q^{94} +2.46626 q^{95} -37.0635 q^{96} -12.5231 q^{97} +14.8742 q^{98} +34.7495 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.62936	−1.85924	−0.929618	−	0.368524i	$-0.879863\pi$
$2$	−2.62936	−1.85924	−0.929618	+	0.368524i	$0.879863\pi$
$3$	3.14041	1.81312	0.906559	−	0.422080i	$-0.138700\pi$
$3$	3.14041	1.81312	0.906559	+	0.422080i	$0.138700\pi$
$4$	4.91352	2.45676
$5$	2.46626	1.10294	0.551472	−	0.834193i	$-0.314066\pi$
$5$	2.46626	1.10294	0.551472	+	0.834193i	$0.314066\pi$
$6$	−8.25726	−3.37101
$7$	1.15889	0.438021	0.219010	−	0.975723i	$-0.429717\pi$
$7$	1.15889	0.438021	0.219010	+	0.975723i	$0.429717\pi$
$8$	−7.66068	−2.70846
$9$	6.86218	2.28739
$10$	−6.48468	−2.05063
$11$	5.06391	1.52683	0.763413	−	0.645911i	$-0.223522\pi$
$11$	5.06391	1.52683	0.763413	+	0.645911i	$0.223522\pi$
$12$	15.4305	4.45439
$13$	3.51750	0.975579	0.487790	−	0.872961i	$-0.337803\pi$
$13$	3.51750	0.975579	0.487790	+	0.872961i	$0.337803\pi$
$14$	−3.04714	−0.814384
$15$	7.74507	1.99977
$16$	10.3156	2.57891
$17$	−0.875800	−0.212413	−0.106206	−	0.994344i	$-0.533870\pi$
$17$	−0.875800	−0.212413	−0.106206	+	0.994344i	$0.533870\pi$
$18$	−18.0431	−4.25281
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	12.1180	2.70967
$21$	3.63940	0.794183
$22$	−13.3148	−2.83873
$23$	−5.03844	−1.05059	−0.525294	−	0.850921i	$-0.676045\pi$
$23$	−5.03844	−1.05059	−0.525294	+	0.850921i	$0.676045\pi$
$24$	−24.0577	−4.91076
$25$	1.08243	0.216487
$26$	−9.24877	−1.81383
$27$	12.1288	2.33420
$28$	5.69425	1.07611
$29$	7.39407	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$29$	7.39407	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$30$	−20.3645	−3.71804
$31$	−3.14194	−0.564308	−0.282154	−	0.959369i	$-0.591049\pi$
$31$	−3.14194	−0.564308	−0.282154	+	0.959369i	$0.591049\pi$
$32$	−11.8021	−2.08634
$33$	15.9028	2.76831
$34$	2.30279	0.394926
$35$	2.85813	0.483112
$36$	33.7175	5.61958
$37$	−1.68116	−0.276382	−0.138191	−	0.990406i	$-0.544129\pi$
$37$	−1.68116	−0.276382	−0.138191	+	0.990406i	$0.544129\pi$
$38$	−2.62936	−0.426538
$39$	11.0464	1.76884
$40$	−18.8932	−2.98728
$41$	8.51936	1.33050	0.665250	−	0.746621i	$-0.268325\pi$
$41$	8.51936	1.33050	0.665250	+	0.746621i	$0.268325\pi$
$42$	−9.56929	−1.47657
$43$	3.26605	0.498068	0.249034	−	0.968495i	$-0.419887\pi$
$43$	3.26605	0.498068	0.249034	+	0.968495i	$0.419887\pi$
$44$	24.8816	3.75104
$45$	16.9239	2.52287
$46$	13.2479	1.95329
$47$	−9.92344	−1.44748	−0.723741	−	0.690071i	$-0.757579\pi$
$47$	−9.92344	−1.44748	−0.723741	+	0.690071i	$0.757579\pi$
$48$	32.3953	4.67586
$49$	−5.65697	−0.808138
$50$	−2.84610	−0.402500
$51$	−2.75037	−0.385129
$52$	17.2833	2.39676
$53$	−5.61561	−0.771364	−0.385682	−	0.922632i	$-0.626034\pi$
$53$	−5.61561	−0.771364	−0.385682	+	0.922632i	$0.626034\pi$
$54$	−31.8911	−4.33983
$55$	12.4889	1.68400
$56$	−8.87792	−1.18636
$57$	3.14041	0.415958
$58$	−19.4416	−2.55281
$59$	1.08087	0.140718	0.0703589	−	0.997522i	$-0.477586\pi$
$59$	1.08087	0.140718	0.0703589	+	0.997522i	$0.477586\pi$
$60$	38.0555	4.91295
$61$	−12.6746	−1.62282	−0.811410	−	0.584477i	$-0.801300\pi$
$61$	−12.6746	−1.62282	−0.811410	+	0.584477i	$0.801300\pi$
$62$	8.26127	1.04918
$63$	7.95254	1.00193
$64$	10.4007	1.30009
$65$	8.67507	1.07601
$66$	−41.8140	−5.14695
$67$	−13.6436	−1.66683	−0.833417	−	0.552644i	$-0.813619\pi$
$67$	−13.6436	−1.66683	−0.833417	+	0.552644i	$0.813619\pi$
$68$	−4.30326	−0.521847
$69$	−15.8228	−1.90484
$70$	−7.51505	−0.898220
$71$	4.76934	0.566016	0.283008	−	0.959117i	$-0.408668\pi$
$71$	4.76934	0.566016	0.283008	+	0.959117i	$0.408668\pi$
$72$	−52.5690	−6.19532
$73$	−4.97631	−0.582433	−0.291216	−	0.956657i	$-0.594060\pi$
$73$	−4.97631	−0.582433	−0.291216	+	0.956657i	$0.594060\pi$
$74$	4.42038	0.513859
$75$	3.39928	0.392516
$76$	4.91352	0.563619
$77$	5.86853	0.668781
$78$	−29.0449	−3.28869
$79$	1.01863	0.114605	0.0573026	−	0.998357i	$-0.481750\pi$
$79$	1.01863	0.114605	0.0573026	+	0.998357i	$0.481750\pi$
$80$	25.4410	2.84439
$81$	17.5030	1.94478
$82$	−22.4004	−2.47371
$83$	−15.3391	−1.68368	−0.841842	−	0.539723i	$-0.818529\pi$
$83$	−15.3391	−1.68368	−0.841842	+	0.539723i	$0.818529\pi$
$84$	17.8823	1.95112
$85$	−2.15995	−0.234280
$86$	−8.58762	−0.926026
$87$	23.2204	2.48949
$88$	−38.7930	−4.13535
$89$	9.78565	1.03728	0.518638	−	0.854994i	$-0.326439\pi$
$89$	9.78565	1.03728	0.518638	+	0.854994i	$0.326439\pi$
$90$	−44.4990	−4.69061
$91$	4.07641	0.427324
$92$	−24.7565	−2.58104
$93$	−9.86697	−1.02316
$94$	26.0923	2.69121
$95$	2.46626	0.253033
$96$	−37.0635	−3.78278
$97$	−12.5231	−1.27153	−0.635765	−	0.771883i	$-0.719315\pi$
$97$	−12.5231	−1.27153	−0.635765	+	0.771883i	$0.719315\pi$
$98$	14.8742	1.50252
$99$	34.7495	3.49245
$100$	5.31856	0.531856
$101$	11.6359	1.15782	0.578908	−	0.815393i	$-0.303479\pi$
$101$	11.6359	1.15782	0.578908	+	0.815393i	$0.303479\pi$
$102$	7.23172	0.716047
$103$	9.63666	0.949528	0.474764	−	0.880113i	$-0.342533\pi$
$103$	9.63666	0.949528	0.474764	+	0.880113i	$0.342533\pi$
$104$	−26.9465	−2.64232
$105$	8.97571	0.875939
$106$	14.7655	1.43415
$107$	1.53142	0.148048	0.0740241	−	0.997256i	$-0.476416\pi$
$107$	1.53142	0.148048	0.0740241	+	0.997256i	$0.476416\pi$
$108$	59.5953	5.73456
$109$	−11.6232	−1.11330	−0.556652	−	0.830746i	$-0.687914\pi$
$109$	−11.6232	−1.11330	−0.556652	+	0.830746i	$0.687914\pi$
$110$	−32.8378	−3.13096
$111$	−5.27954	−0.501112
$112$	11.9547	1.12961
$113$	−13.8677	−1.30456	−0.652282	−	0.757976i	$-0.726188\pi$
$113$	−13.8677	−1.30456	−0.652282	+	0.757976i	$0.726188\pi$
$114$	−8.25726	−0.773364
$115$	−12.4261	−1.15874
$116$	36.3309	3.37324
$117$	24.1377	2.23153
$118$	−2.84200	−0.261627
$119$	−1.01496	−0.0930412
$120$	−59.3325	−5.41629
$121$	14.6432	1.33120
$122$	33.3261	3.01721
$123$	26.7543	2.41235
$124$	−15.4380	−1.38637
$125$	−9.66173	−0.864172
$126$	−20.9101	−1.86282
$127$	18.5977	1.65028	0.825139	−	0.564930i	$-0.191097\pi$
$127$	18.5977	1.65028	0.825139	+	0.564930i	$0.191097\pi$
$128$	−3.74297	−0.330835
$129$	10.2567	0.903056
$130$	−22.8099	−2.00056
$131$	−4.32901	−0.378228	−0.189114	−	0.981955i	$-0.560562\pi$
$131$	−4.32901	−0.378228	−0.189114	+	0.981955i	$0.560562\pi$
$132$	78.1385	6.80108
$133$	1.15889	0.100489
$134$	35.8740	3.09904
$135$	29.9129	2.57449
$136$	6.70923	0.575312
$137$	2.79847	0.239090	0.119545	−	0.992829i	$-0.461856\pi$
$137$	2.79847	0.239090	0.119545	+	0.992829i	$0.461856\pi$
$138$	41.6038	3.54155
$139$	−21.8071	−1.84966	−0.924828	−	0.380386i	$-0.875791\pi$
$139$	−21.8071	−1.84966	−0.924828	+	0.380386i	$0.875791\pi$
$140$	14.0435	1.18689
$141$	−31.1637	−2.62446
$142$	−12.5403	−1.05236
$143$	17.8123	1.48954
$144$	70.7878	5.89898
$145$	18.2357	1.51439
$146$	13.0845	1.08288
$147$	−17.7652	−1.46525
$148$	−8.26043	−0.679003
$149$	−9.62813	−0.788767	−0.394384	−	0.918946i	$-0.629042\pi$
$149$	−9.62813	−0.788767	−0.394384	+	0.918946i	$0.629042\pi$
$150$	−8.93793	−0.729779
$151$	6.39543	0.520453	0.260226	−	0.965548i	$-0.416203\pi$
$151$	6.39543	0.520453	0.260226	+	0.965548i	$0.416203\pi$
$152$	−7.66068	−0.621363
$153$	−6.00990	−0.485872
$154$	−15.4305	−1.24342
$155$	−7.74883	−0.622401
$156$	54.2767	4.34561
$157$	1.29166	0.103085	0.0515427	−	0.998671i	$-0.483586\pi$
$157$	1.29166	0.103085	0.0515427	+	0.998671i	$0.483586\pi$
$158$	−2.67835	−0.213078
$159$	−17.6353	−1.39857
$160$	−29.1071	−2.30112
$161$	−5.83902	−0.460179
$162$	−46.0217	−3.61581
$163$	−6.59970	−0.516929	−0.258464	−	0.966021i	$-0.583216\pi$
$163$	−6.59970	−0.516929	−0.258464	+	0.966021i	$0.583216\pi$
$164$	41.8600	3.26872
$165$	39.2203	3.05330
$166$	40.3320	3.13037
$167$	12.2755	0.949909	0.474955	−	0.880010i	$-0.342464\pi$
$167$	12.2755	0.949909	0.474955	+	0.880010i	$0.342464\pi$
$168$	−27.8803	−2.15101
$169$	−0.627188	−0.0482453
$170$	5.67928	0.435581
$171$	6.86218	0.524764
$172$	16.0478	1.22363
$173$	−23.8377	−1.81235	−0.906175	−	0.422903i	$-0.861011\pi$
$173$	−23.8377	−1.81235	−0.906175	+	0.422903i	$0.861011\pi$
$174$	−61.0547	−4.62855
$175$	1.25442	0.0948256
$176$	52.2374	3.93754
$177$	3.39439	0.255138
$178$	−25.7300	−1.92854
$179$	−4.02650	−0.300955	−0.150477	−	0.988613i	$-0.548081\pi$
$179$	−4.02650	−0.300955	−0.150477	+	0.988613i	$0.548081\pi$
$180$	83.1560	6.19808
$181$	−5.47099	−0.406656	−0.203328	−	0.979111i	$-0.565176\pi$
$181$	−5.47099	−0.406656	−0.203328	+	0.979111i	$0.565176\pi$
$182$	−10.7183	−0.794496
$183$	−39.8036	−2.94236
$184$	38.5979	2.84548
$185$	−4.14618	−0.304833
$186$	25.9438	1.90229
$187$	−4.43497	−0.324317
$188$	−48.7590	−3.55612
$189$	14.0560	1.02243
$190$	−6.48468	−0.470448
$191$	15.0317	1.08765	0.543827	−	0.839197i	$-0.316975\pi$
$191$	15.0317	1.08765	0.543827	+	0.839197i	$0.316975\pi$
$192$	32.6625	2.35722
$193$	−0.544465	−0.0391914	−0.0195957	−	0.999808i	$-0.506238\pi$
$193$	−0.544465	−0.0391914	−0.0195957	+	0.999808i	$0.506238\pi$
$194$	32.9278	2.36407
$195$	27.2433	1.95093
$196$	−27.7956	−1.98540
$197$	3.04495	0.216944	0.108472	−	0.994100i	$-0.465404\pi$
$197$	3.04495	0.216944	0.108472	+	0.994100i	$0.465404\pi$
$198$	−91.3688	−6.49330
$199$	−4.98333	−0.353259	−0.176630	−	0.984277i	$-0.556519\pi$
$199$	−4.98333	−0.353259	−0.176630	+	0.984277i	$0.556519\pi$
$200$	−8.29218	−0.586345
$201$	−42.8466	−3.02217
$202$	−30.5950	−2.15266
$203$	8.56893	0.601421
$204$	−13.5140	−0.946170
$205$	21.0109	1.46747
$206$	−25.3382	−1.76540
$207$	−34.5747	−2.40311
$208$	36.2853	2.51593
$209$	5.06391	0.350278
$210$	−23.6003	−1.62858
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−27.5924	−1.89506
$213$	14.9777	1.02625
$214$	−4.02665	−0.275256
$215$	8.05493	0.549342
$216$	−92.9153	−6.32208
$217$	−3.64117	−0.247179
$218$	30.5616	2.06989
$219$	−15.6277	−1.05602
$220$	61.3645	4.13719
$221$	−3.08063	−0.207226
$222$	13.8818	0.931686
$223$	23.5308	1.57574	0.787871	−	0.615841i	$-0.211184\pi$
$223$	23.5308	1.57574	0.787871	+	0.615841i	$0.211184\pi$
$224$	−13.6774	−0.913860
$225$	7.42785	0.495190
$226$	36.4632	2.42549
$227$	−25.6181	−1.70033	−0.850165	−	0.526516i	$-0.823498\pi$
$227$	−25.6181	−1.70033	−0.850165	+	0.526516i	$0.823498\pi$
$228$	15.4305	1.02191
$229$	−14.4439	−0.954481	−0.477240	−	0.878773i	$-0.658363\pi$
$229$	−14.4439	−0.954481	−0.477240	+	0.878773i	$0.658363\pi$
$230$	32.6727	2.15437
$231$	18.4296	1.21258
$232$	−56.6436	−3.71883
$233$	14.0263	0.918893	0.459446	−	0.888206i	$-0.348048\pi$
$233$	14.0263	0.918893	0.459446	+	0.888206i	$0.348048\pi$
$234$	−63.4667	−4.14895
$235$	−24.4738	−1.59649
$236$	5.31089	0.345710
$237$	3.19893	0.207793
$238$	2.66869	0.172986
$239$	28.8729	1.86763	0.933815	−	0.357756i	$-0.116458\pi$
$239$	28.8729	1.86763	0.933815	+	0.357756i	$0.116458\pi$
$240$	79.8953	5.15722
$241$	−18.7948	−1.21068	−0.605341	−	0.795967i	$-0.706963\pi$
$241$	−18.7948	−1.21068	−0.605341	+	0.795967i	$0.706963\pi$
$242$	−38.5021	−2.47501
$243$	18.5801	1.19192
$244$	−62.2771	−3.98688
$245$	−13.9515	−0.891331
$246$	−70.3466	−4.48513
$247$	3.51750	0.223813
$248$	24.0694	1.52841
$249$	−48.1711	−3.05272
$250$	25.4042	1.60670
$251$	−7.17240	−0.452718	−0.226359	−	0.974044i	$-0.572682\pi$
$251$	−7.17240	−0.452718	−0.226359	+	0.974044i	$0.572682\pi$
$252$	39.0750	2.46149
$253$	−25.5142	−1.60407
$254$	−48.9000	−3.06826
$255$	−6.78313	−0.424776
$256$	−10.9598	−0.684989
$257$	4.57332	0.285276	0.142638	−	0.989775i	$-0.454442\pi$
$257$	4.57332	0.285276	0.142638	+	0.989775i	$0.454442\pi$
$258$	−26.9686	−1.67899
$259$	−1.94829	−0.121061
$260$	42.6251	2.64350
$261$	50.7394	3.14069
$262$	11.3825	0.703215
$263$	−22.6952	−1.39945	−0.699724	−	0.714413i	$-0.746694\pi$
$263$	−22.6952	−1.39945	−0.699724	+	0.714413i	$0.746694\pi$
$264$	−121.826	−7.49787
$265$	−13.8496	−0.850771
$266$	−3.04714	−0.186832
$267$	30.7310	1.88070
$268$	−67.0382	−4.09501
$269$	4.80764	0.293127	0.146563	−	0.989201i	$-0.453179\pi$
$269$	4.80764	0.293127	0.146563	+	0.989201i	$0.453179\pi$
$270$	−78.6516	−4.78659
$271$	15.2856	0.928535	0.464267	−	0.885695i	$-0.346318\pi$
$271$	15.2856	0.928535	0.464267	+	0.885695i	$0.346318\pi$
$272$	−9.03444	−0.547793
$273$	12.8016	0.774788
$274$	−7.35818	−0.444524
$275$	5.48134	0.330537
$276$	−77.7456	−4.67973
$277$	10.6750	0.641401	0.320701	−	0.947181i	$-0.396082\pi$
$277$	10.6750	0.641401	0.320701	+	0.947181i	$0.396082\pi$
$278$	57.3387	3.43895
$279$	−21.5605	−1.29080
$280$	−21.8952	−1.30849
$281$	23.1657	1.38195	0.690975	−	0.722879i	$-0.257181\pi$
$281$	23.1657	1.38195	0.690975	+	0.722879i	$0.257181\pi$
$282$	81.9405	4.87948
$283$	6.61719	0.393351	0.196675	−	0.980469i	$-0.436985\pi$
$283$	6.61719	0.393351	0.196675	+	0.980469i	$0.436985\pi$
$284$	23.4342	1.39057
$285$	7.74507	0.458778
$286$	−46.8349	−2.76941
$287$	9.87303	0.582786
$288$	−80.9883	−4.77228
$289$	−16.2330	−0.954881
$290$	−47.9481	−2.81561
$291$	−39.3277	−2.30543
$292$	−24.4512	−1.43090
$293$	7.22492	0.422085	0.211042	−	0.977477i	$-0.432314\pi$
$293$	7.22492	0.422085	0.211042	+	0.977477i	$0.432314\pi$
$294$	46.7111	2.72424
$295$	2.66571	0.155204
$296$	12.8789	0.748568
$297$	61.4194	3.56391
$298$	25.3158	1.46650
$299$	−17.7227	−1.02493
$300$	16.7025	0.964317
$301$	3.78501	0.218164
$302$	−16.8159	−0.967644
$303$	36.5416	2.09926
$304$	10.3156	0.591642
$305$	−31.2589	−1.78988
$306$	15.8022	0.903351
$307$	−4.66418	−0.266199	−0.133099	−	0.991103i	$-0.542493\pi$
$307$	−4.66418	−0.266199	−0.133099	+	0.991103i	$0.542493\pi$
$308$	28.8351	1.64303
$309$	30.2631	1.72161
$310$	20.3744	1.15719
$311$	−0.00212840	−0.000120691 0	−6.03453e−5	−	1.00000i	$-0.500019\pi$
$311$	−0.00212840	−0.000120691 0	−6.03453e−5		1.00000i	$0.500019\pi$
$312$	−84.6230	−4.79083
$313$	−31.6653	−1.78983	−0.894913	−	0.446241i	$-0.852763\pi$
$313$	−31.6653	−1.78983	−0.894913	+	0.446241i	$0.852763\pi$
$314$	−3.39623	−0.191660
$315$	19.6130	1.10507
$316$	5.00508	0.281558
$317$	−0.126049	−0.00707960	−0.00353980	−	0.999994i	$-0.501127\pi$
$317$	−0.126049	−0.00707960	−0.00353980	+	0.999994i	$0.501127\pi$
$318$	46.3696	2.60028
$319$	37.4429	2.09640
$320$	25.6509	1.43393
$321$	4.80929	0.268429
$322$	15.3529	0.855582
$323$	−0.875800	−0.0487308
$324$	86.0014	4.77786
$325$	3.80746	0.211200
$326$	17.3530	0.961092
$327$	−36.5017	−2.01855
$328$	−65.2641	−3.60361
$329$	−11.5002	−0.634027
$330$	−103.124	−5.67680
$331$	8.31125	0.456828	0.228414	−	0.973564i	$-0.426646\pi$
$331$	8.31125	0.456828	0.228414	+	0.973564i	$0.426646\pi$
$332$	−75.3690	−4.13641
$333$	−11.5365	−0.632194
$334$	−32.2768	−1.76611
$335$	−33.6487	−1.83843
$336$	37.5427	2.04812
$337$	19.2991	1.05129	0.525645	−	0.850704i	$-0.323824\pi$
$337$	19.2991	1.05129	0.525645	+	0.850704i	$0.323824\pi$
$338$	1.64910	0.0896993
$339$	−43.5503	−2.36533
$340$	−10.6130	−0.575569
$341$	−15.9105	−0.861601
$342$	−18.0431	−0.975661
$343$	−14.6681	−0.792002
$344$	−25.0202	−1.34900
$345$	−39.0231	−2.10093
$346$	62.6779	3.36959
$347$	−29.3541	−1.57581	−0.787906	−	0.615796i	$-0.788835\pi$
$347$	−29.3541	−1.57581	−0.787906	+	0.615796i	$0.788835\pi$
$348$	114.094	6.11608
$349$	−10.5247	−0.563374	−0.281687	−	0.959506i	$-0.590894\pi$
$349$	−10.5247	−0.563374	−0.281687	+	0.959506i	$0.590894\pi$
$350$	−3.29833	−0.176303
$351$	42.6632	2.27719
$352$	−59.7649	−3.18548
$353$	27.8614	1.48291	0.741455	−	0.671002i	$-0.234136\pi$
$353$	27.8614	1.48291	0.741455	+	0.671002i	$0.234136\pi$
$354$	−8.92506	−0.474361
$355$	11.7624	0.624285
$356$	48.0820	2.54834
$357$	−3.18739	−0.168695
$358$	10.5871	0.559546
$359$	1.56976	0.0828485	0.0414243	−	0.999142i	$-0.486810\pi$
$359$	1.56976	0.0828485	0.0414243	+	0.999142i	$0.486810\pi$
$360$	−129.649	−6.83309
$361$	1.00000	0.0526316
$362$	14.3852	0.756069
$363$	45.9856	2.41362
$364$	20.0295	1.04983
$365$	−12.2729	−0.642391
$366$	104.658	5.47055
$367$	−9.50924	−0.496378	−0.248189	−	0.968712i	$-0.579835\pi$
$367$	−9.50924	−0.496378	−0.248189	+	0.968712i	$0.579835\pi$
$368$	−51.9748	−2.70937
$369$	58.4614	3.04338
$370$	10.9018	0.566758
$371$	−6.50790	−0.337873
$372$	−48.4816	−2.51365
$373$	17.2074	0.890966	0.445483	−	0.895290i	$-0.353032\pi$
$373$	17.2074	0.890966	0.445483	+	0.895290i	$0.353032\pi$
$374$	11.6611	0.602983
$375$	−30.3418	−1.56684
$376$	76.0203	3.92045
$377$	26.0086	1.33951
$378$	−36.9584	−1.90093
$379$	−28.3086	−1.45411	−0.727057	−	0.686577i	$-0.759112\pi$
$379$	−28.3086	−1.45411	−0.727057	+	0.686577i	$0.759112\pi$
$380$	12.1180	0.621641
$381$	58.4044	2.99215
$382$	−39.5237	−2.02221
$383$	15.6265	0.798476	0.399238	−	0.916847i	$-0.369275\pi$
$383$	15.6265	0.798476	0.399238	+	0.916847i	$0.369275\pi$
$384$	−11.7545	−0.599843
$385$	14.4733	0.737628
$386$	1.43159	0.0728661
$387$	22.4122	1.13928
$388$	−61.5326	−3.12384
$389$	19.8082	1.00431	0.502157	−	0.864776i	$-0.332540\pi$
$389$	19.8082	1.00431	0.502157	+	0.864776i	$0.332540\pi$
$390$	−71.6323	−3.62724
$391$	4.41267	0.223158
$392$	43.3362	2.18881
$393$	−13.5949	−0.685771
$394$	−8.00627	−0.403350
$395$	2.51222	0.126403
$396$	170.742	8.58012
$397$	−32.4558	−1.62891	−0.814456	−	0.580225i	$-0.802965\pi$
$397$	−32.4558	−1.62891	−0.814456	+	0.580225i	$0.802965\pi$
$398$	13.1030	0.656792
$399$	3.63940	0.182198
$400$	11.1660	0.558299
$401$	25.7885	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$401$	25.7885	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$402$	112.659	5.61892
$403$	−11.0518	−0.550527
$404$	57.1733	2.84448
$405$	43.1670	2.14498
$406$	−22.5308	−1.11818
$407$	−8.51326	−0.421986
$408$	21.0697	1.04311
$409$	5.95473	0.294443	0.147221	−	0.989104i	$-0.452967\pi$
$409$	5.95473	0.294443	0.147221	+	0.989104i	$0.452967\pi$
$410$	−55.2453	−2.72837
$411$	8.78836	0.433498
$412$	47.3499	2.33276
$413$	1.25262	0.0616372
$414$	90.9093	4.46795
$415$	−37.8302	−1.85701
$416$	−41.5140	−2.03539
$417$	−68.4833	−3.35364
$418$	−13.3148	−0.651249
$419$	−30.9450	−1.51176	−0.755882	−	0.654708i	$-0.772792\pi$
$419$	−30.9450	−1.51176	−0.755882	+	0.654708i	$0.772792\pi$
$420$	44.1023	2.15197
$421$	18.6221	0.907587	0.453794	−	0.891107i	$-0.350070\pi$
$421$	18.6221	0.907587	0.453794	+	0.891107i	$0.350070\pi$
$422$	2.62936	0.127995
$423$	−68.0965	−3.31096
$424$	43.0194	2.08921
$425$	−0.947995	−0.0459845
$426$	−39.3817	−1.90805
$427$	−14.6885	−0.710829
$428$	7.52467	0.363719
$429$	55.9380	2.70071
$430$	−21.1793	−1.02136
$431$	31.0677	1.49648	0.748239	−	0.663429i	$-0.230900\pi$
$431$	31.0677	1.49648	0.748239	+	0.663429i	$0.230900\pi$
$432$	125.117	6.01968
$433$	−18.1466	−0.872070	−0.436035	−	0.899930i	$-0.643618\pi$
$433$	−18.1466	−0.872070	−0.436035	+	0.899930i	$0.643618\pi$
$434$	9.57393	0.459564
$435$	57.2675	2.74577
$436$	−57.1110	−2.73512
$437$	−5.03844	−0.241021
$438$	41.0907	1.96339
$439$	−4.66298	−0.222552	−0.111276	−	0.993790i	$-0.535494\pi$
$439$	−4.66298	−0.222552	−0.111276	+	0.993790i	$0.535494\pi$
$440$	−95.6736	−4.56106
$441$	−38.8191	−1.84853
$442$	8.10007	0.385281
$443$	28.4716	1.35273	0.676364	−	0.736568i	$-0.263555\pi$
$443$	28.4716	1.35273	0.676364	+	0.736568i	$0.263555\pi$
$444$	−25.9411	−1.23111
$445$	24.1339	1.14406
$446$	−61.8710	−2.92968
$447$	−30.2363	−1.43013
$448$	12.0533	0.569466
$449$	9.73307	0.459332	0.229666	−	0.973270i	$-0.426237\pi$
$449$	9.73307	0.459332	0.229666	+	0.973270i	$0.426237\pi$
$450$	−19.5305	−0.920676
$451$	43.1412	2.03144
$452$	−68.1393	−3.20500
$453$	20.0843	0.943642
$454$	67.3590	3.16132
$455$	10.0535	0.471314
$456$	−24.0577	−1.12660
$457$	−24.3525	−1.13916	−0.569580	−	0.821936i	$-0.692894\pi$
$457$	−24.3525	−1.13916	−0.569580	+	0.821936i	$0.692894\pi$
$458$	37.9782	1.77461
$459$	−10.6225	−0.495814
$460$	−61.0559	−2.84675
$461$	13.3073	0.619783	0.309892	−	0.950772i	$-0.399707\pi$
$461$	13.3073	0.619783	0.309892	+	0.950772i	$0.399707\pi$
$462$	−48.4580	−2.25447
$463$	2.14520	0.0996961	0.0498480	−	0.998757i	$-0.484126\pi$
$463$	2.14520	0.0996961	0.0498480	+	0.998757i	$0.484126\pi$
$464$	76.2745	3.54095
$465$	−24.3345	−1.12849
$466$	−36.8801	−1.70844
$467$	35.0892	1.62373	0.811867	−	0.583842i	$-0.198451\pi$
$467$	35.0892	1.62373	0.811867	+	0.583842i	$0.198451\pi$
$468$	118.601	5.48234
$469$	−15.8115	−0.730108
$470$	64.3503	2.96826
$471$	4.05633	0.186906
$472$	−8.28023	−0.381128
$473$	16.5390	0.760463
$474$	−8.41113	−0.386336
$475$	1.08243	0.0496654
$476$	−4.98702	−0.228580
$477$	−38.5354	−1.76441
$478$	−75.9171	−3.47237
$479$	−3.71082	−0.169552	−0.0847758	−	0.996400i	$-0.527017\pi$
$479$	−3.71082	−0.169552	−0.0847758	+	0.996400i	$0.527017\pi$
$480$	−91.4082	−4.17220
$481$	−5.91349	−0.269632
$482$	49.4183	2.25094
$483$	−18.3369	−0.834359
$484$	71.9495	3.27043
$485$	−30.8853	−1.40243
$486$	−48.8538	−2.21605
$487$	−12.0754	−0.547190	−0.273595	−	0.961845i	$-0.588213\pi$
$487$	−12.0754	−0.547190	−0.273595	+	0.961845i	$0.588213\pi$
$488$	97.0963	4.39535
$489$	−20.7258	−0.937252
$490$	36.6836	1.65720
$491$	−4.99102	−0.225242	−0.112621	−	0.993638i	$-0.535925\pi$
$491$	−4.99102	−0.225242	−0.112621	+	0.993638i	$0.535925\pi$
$492$	131.458	5.92657
$493$	−6.47573	−0.291652
$494$	−9.24877	−0.416122
$495$	85.7012	3.85198
$496$	−32.4111	−1.45530
$497$	5.52716	0.247927
$498$	126.659	5.67572
$499$	34.5847	1.54822	0.774111	−	0.633050i	$-0.218197\pi$
$499$	34.5847	1.54822	0.774111	+	0.633050i	$0.218197\pi$
$500$	−47.4731	−2.12306
$501$	38.5502	1.72230
$502$	18.8588	0.841709
$503$	0.178687	0.00796724	0.00398362	−	0.999992i	$-0.498732\pi$
$503$	0.178687	0.00796724	0.00398362	+	0.999992i	$0.498732\pi$
$504$	−60.9219	−2.71368
$505$	28.6972	1.27701
$506$	67.0860	2.98234
$507$	−1.96963	−0.0874743
$508$	91.3801	4.05434
$509$	25.2244	1.11805	0.559026	−	0.829150i	$-0.311175\pi$
$509$	25.2244	1.11805	0.559026	+	0.829150i	$0.311175\pi$
$510$	17.8353	0.789760
$511$	−5.76701	−0.255118
$512$	36.3033	1.60439
$513$	12.1288	0.535502
$514$	−12.0249	−0.530395
$515$	23.7665	1.04728
$516$	50.3967	2.21859
$517$	−50.2514	−2.21005
$518$	5.12275	0.225081
$519$	−74.8603	−3.28600
$520$	−66.4569	−2.91433
$521$	−15.9617	−0.699294	−0.349647	−	0.936881i	$-0.613698\pi$
$521$	−15.9617	−0.699294	−0.349647	+	0.936881i	$0.613698\pi$
$522$	−133.412	−5.83929
$523$	18.2339	0.797313	0.398657	−	0.917100i	$-0.369477\pi$
$523$	18.2339	0.797313	0.398657	+	0.917100i	$0.369477\pi$
$524$	−21.2707	−0.929215
$525$	3.93941	0.171930
$526$	59.6739	2.60190
$527$	2.75171	0.119866
$528$	164.047	7.13923
$529$	2.38592	0.103736
$530$	36.4154	1.58178
$531$	7.41715	0.321877
$532$	5.69425	0.246877
$533$	29.9668	1.29801
$534$	−80.8027	−3.49667
$535$	3.77688	0.163289
$536$	104.520	4.51456
$537$	−12.6449	−0.545666
$538$	−12.6410	−0.544992
$539$	−28.6464	−1.23389
$540$	146.978	6.32490
$541$	1.15703	0.0497444	0.0248722	−	0.999691i	$-0.492082\pi$
$541$	1.15703	0.0497444	0.0248722	+	0.999691i	$0.492082\pi$
$542$	−40.1913	−1.72637
$543$	−17.1812	−0.737314
$544$	10.3363	0.443165
$545$	−28.6659	−1.22791
$546$	−33.6600	−1.44051
$547$	−11.2477	−0.480916	−0.240458	−	0.970660i	$-0.577298\pi$
$547$	−11.2477	−0.480916	−0.240458	+	0.970660i	$0.577298\pi$
$548$	13.7504	0.587386
$549$	−86.9757	−3.71203
$550$	−14.4124	−0.614547
$551$	7.39407	0.314998
$552$	121.213	5.15918
$553$	1.18049	0.0501995
$554$	−28.0685	−1.19252
$555$	−13.0207	−0.552699
$556$	−107.150	−4.54416
$557$	−4.64959	−0.197009	−0.0985047	−	0.995137i	$-0.531406\pi$
$557$	−4.64959	−0.197009	−0.0985047	+	0.995137i	$0.531406\pi$
$558$	56.6904	2.39989
$559$	11.4883	0.485905
$560$	29.4834	1.24590
$561$	−13.9276	−0.588026
$562$	−60.9109	−2.56937
$563$	25.5657	1.07747	0.538733	−	0.842476i	$-0.318903\pi$
$563$	25.5657	1.07747	0.538733	+	0.842476i	$0.318903\pi$
$564$	−153.123	−6.44766
$565$	−34.2014	−1.43886
$566$	−17.3989	−0.731332
$567$	20.2841	0.851854
$568$	−36.5364	−1.53303
$569$	36.1452	1.51528	0.757642	−	0.652670i	$-0.226351\pi$
$569$	36.1452	1.51528	0.757642	+	0.652670i	$0.226351\pi$
$570$	−20.3645	−0.852977
$571$	−3.67096	−0.153625	−0.0768124	−	0.997046i	$-0.524474\pi$
$571$	−3.67096	−0.153625	−0.0768124	+	0.997046i	$0.524474\pi$
$572$	87.5211	3.65944
$573$	47.2057	1.97204
$574$	−25.9597	−1.08354
$575$	−5.45378	−0.227438
$576$	71.3717	2.97382
$577$	−13.5562	−0.564354	−0.282177	−	0.959362i	$-0.591057\pi$
$577$	−13.5562	−0.564354	−0.282177	+	0.959362i	$0.591057\pi$
$578$	42.6823	1.77535
$579$	−1.70984	−0.0710587
$580$	89.6014	3.72049
$581$	−17.7764	−0.737488
$582$	103.407	4.28635
$583$	−28.4369	−1.17774
$584$	38.1219	1.57750
$585$	59.5299	2.46126
$586$	−18.9969	−0.784755
$587$	31.9339	1.31805	0.659026	−	0.752120i	$-0.270969\pi$
$587$	31.9339	1.31805	0.659026	+	0.752120i	$0.270969\pi$
$588$	−87.2897	−3.59976
$589$	−3.14194	−0.129461
$590$	−7.00911	−0.288561
$591$	9.56240	0.393345
$592$	−17.3423	−0.712763
$593$	32.9245	1.35205	0.676024	−	0.736880i	$-0.263702\pi$
$593$	32.9245	1.35205	0.676024	+	0.736880i	$0.263702\pi$
$594$	−161.493	−6.62616
$595$	−2.50315	−0.102619
$596$	−47.3080	−1.93781
$597$	−15.6497	−0.640500
$598$	46.5994	1.90559
$599$	−17.2400	−0.704406	−0.352203	−	0.935924i	$-0.614567\pi$
$599$	−17.2400	−0.704406	−0.352203	+	0.935924i	$0.614567\pi$
$600$	−26.0408	−1.06311
$601$	−4.60770	−0.187952	−0.0939761	−	0.995574i	$-0.529958\pi$
$601$	−4.60770	−0.187952	−0.0939761	+	0.995574i	$0.529958\pi$
$602$	−9.95213	−0.405619
$603$	−93.6251	−3.81271
$604$	31.4241	1.27863
$605$	36.1138	1.46824
$606$	−96.0808	−3.90302
$607$	44.1703	1.79282	0.896409	−	0.443227i	$-0.146167\pi$
$607$	44.1703	1.79282	0.896409	+	0.443227i	$0.146167\pi$
$608$	−11.8021	−0.478639
$609$	26.9100	1.09045
$610$	82.1909	3.32781
$611$	−34.9057	−1.41213
$612$	−29.5298	−1.19367
$613$	46.3855	1.87349	0.936746	−	0.350010i	$-0.113822\pi$
$613$	46.3855	1.87349	0.936746	+	0.350010i	$0.113822\pi$
$614$	12.2638	0.494927
$615$	65.9830	2.66069
$616$	−44.9570	−1.81137
$617$	30.1596	1.21418	0.607090	−	0.794633i	$-0.292337\pi$
$617$	30.1596	1.21418	0.607090	+	0.794633i	$0.292337\pi$
$618$	−79.5725	−3.20087
$619$	11.2011	0.450209	0.225105	−	0.974335i	$-0.427728\pi$
$619$	11.2011	0.450209	0.225105	+	0.974335i	$0.427728\pi$
$620$	−38.0740	−1.52909
$621$	−61.1105	−2.45228
$622$	0.00559633	0.000224392 0
$623$	11.3405	0.454349
$624$	113.951	4.56168
$625$	−29.2405	−1.16962
$626$	83.2593	3.32771
$627$	15.9028	0.635095
$628$	6.34658	0.253256
$629$	1.47236	0.0587070
$630$	−51.5696	−2.05458
$631$	42.6485	1.69781	0.848905	−	0.528545i	$-0.177262\pi$
$631$	42.6485	1.69781	0.848905	+	0.528545i	$0.177262\pi$
$632$	−7.80343	−0.310404
$633$	−3.14041	−0.124820
$634$	0.331427	0.0131627
$635$	45.8667	1.82016
$636$	−86.6516	−3.43596
$637$	−19.8984	−0.788403
$638$	−98.4507	−3.89770
$639$	32.7281	1.29470
$640$	−9.23114	−0.364893
$641$	−21.5803	−0.852371	−0.426186	−	0.904636i	$-0.640143\pi$
$641$	−21.5803	−0.852371	−0.426186	+	0.904636i	$0.640143\pi$
$642$	−12.6454	−0.499072
$643$	37.3507	1.47297	0.736484	−	0.676455i	$-0.236485\pi$
$643$	37.3507	1.47297	0.736484	+	0.676455i	$0.236485\pi$
$644$	−28.6901	−1.13055
$645$	25.2958	0.996021
$646$	2.30279	0.0906022
$647$	−45.7114	−1.79710	−0.898551	−	0.438870i	$-0.855379\pi$
$647$	−45.7114	−1.79710	−0.898551	+	0.438870i	$0.855379\pi$
$648$	−134.085	−5.26736
$649$	5.47344	0.214851
$650$	−10.0112	−0.392670
$651$	−11.4348	−0.448164
$652$	−32.4278	−1.26997
$653$	5.37300	0.210262	0.105131	−	0.994458i	$-0.466474\pi$
$653$	5.37300	0.210262	0.105131	+	0.994458i	$0.466474\pi$
$654$	95.9761	3.75296
$655$	−10.6765	−0.417164
$656$	87.8826	3.43124
$657$	−34.1483	−1.33225
$658$	30.2382	1.17881
$659$	15.9065	0.619630	0.309815	−	0.950797i	$-0.399733\pi$
$659$	15.9065	0.619630	0.309815	+	0.950797i	$0.399733\pi$
$660$	192.710	7.50122
$661$	−0.0774141	−0.00301106	−0.00150553	−	0.999999i	$-0.500479\pi$
$661$	−0.0774141	−0.00301106	−0.00150553	+	0.999999i	$0.500479\pi$
$662$	−21.8532	−0.849350
$663$	−9.67444	−0.375724
$664$	117.508	4.56019
$665$	2.85813	0.110834
$666$	30.3335	1.17540
$667$	−37.2546	−1.44250
$668$	60.3161	2.33370
$669$	73.8965	2.85700
$670$	88.4745	3.41807
$671$	−64.1832	−2.47776
$672$	−42.9527	−1.65693
$673$	31.5421	1.21586	0.607930	−	0.793991i	$-0.292000\pi$
$673$	31.5421	1.21586	0.607930	+	0.793991i	$0.292000\pi$
$674$	−50.7443	−1.95460
$675$	13.1287	0.505323
$676$	−3.08170	−0.118527
$677$	0.298376	0.0114675	0.00573377	−	0.999984i	$-0.498175\pi$
$677$	0.298376	0.0114675	0.00573377	+	0.999984i	$0.498175\pi$
$678$	114.509	4.39770
$679$	−14.5130	−0.556956
$680$	16.5467	0.634537
$681$	−80.4513	−3.08290
$682$	41.8343	1.60192
$683$	49.8171	1.90620	0.953098	−	0.302661i	$-0.0978750\pi$
$683$	49.8171	1.90620	0.953098	+	0.302661i	$0.0978750\pi$
$684$	33.7175	1.28922
$685$	6.90176	0.263703
$686$	38.5676	1.47252
$687$	−45.3598	−1.73059
$688$	33.6914	1.28447
$689$	−19.7529	−0.752526
$690$	102.606	3.90613
$691$	44.6232	1.69755	0.848773	−	0.528757i	$-0.177342\pi$
$691$	44.6232	1.69755	0.848773	+	0.528757i	$0.177342\pi$
$692$	−117.127	−4.45251
$693$	40.2709	1.52977
$694$	77.1825	2.92981
$695$	−53.7820	−2.04007
$696$	−177.884	−6.74268
$697$	−7.46126	−0.282615
$698$	27.6732	1.04745
$699$	44.0483	1.66606
$700$	6.16364	0.232964
$701$	−6.03030	−0.227761	−0.113881	−	0.993494i	$-0.536328\pi$
$701$	−6.03030	−0.227761	−0.113881	+	0.993494i	$0.536328\pi$
$702$	−112.177	−4.23384
$703$	−1.68116	−0.0634063
$704$	52.6683	1.98501
$705$	−76.8577	−2.89463
$706$	−73.2575	−2.75708
$707$	13.4848	0.507148
$708$	16.6784	0.626812
$709$	−18.7798	−0.705289	−0.352644	−	0.935757i	$-0.614717\pi$
$709$	−18.7798	−0.705289	−0.352644	+	0.935757i	$0.614717\pi$
$710$	−30.9276	−1.16069
$711$	6.99005	0.262148
$712$	−74.9648	−2.80942
$713$	15.8305	0.592856
$714$	8.38079	0.313643
$715$	43.9297	1.64288
$716$	−19.7843	−0.739373
$717$	90.6727	3.38623
$718$	−4.12745	−0.154035
$719$	32.3105	1.20498	0.602488	−	0.798128i	$-0.294176\pi$
$719$	32.3105	1.20498	0.602488	+	0.798128i	$0.294176\pi$
$720$	174.581	6.50625
$721$	11.1679	0.415913
$722$	−2.62936	−0.0978545
$723$	−59.0235	−2.19511
$724$	−26.8818	−0.999055
$725$	8.00358	0.297245
$726$	−120.913	−4.48748
$727$	32.8096	1.21684	0.608420	−	0.793615i	$-0.291804\pi$
$727$	32.8096	1.21684	0.608420	+	0.793615i	$0.291804\pi$
$728$	−31.2281	−1.15739
$729$	5.84023	0.216305
$730$	32.2697	1.19436
$731$	−2.86041	−0.105796
$732$	−195.576	−7.22868
$733$	3.98672	0.147253	0.0736265	−	0.997286i	$-0.476543\pi$
$733$	3.98672	0.147253	0.0736265	+	0.997286i	$0.476543\pi$
$734$	25.0032	0.922885
$735$	−43.8136	−1.61609
$736$	59.4643	2.19188
$737$	−69.0901	−2.54497
$738$	−153.716	−5.65836
$739$	−20.2946	−0.746549	−0.373274	−	0.927721i	$-0.621765\pi$
$739$	−20.2946	−0.746549	−0.373274	+	0.927721i	$0.621765\pi$
$740$	−20.3724	−0.748903
$741$	11.0464	0.405800
$742$	17.1116	0.628186
$743$	13.6513	0.500817	0.250409	−	0.968140i	$-0.419435\pi$
$743$	13.6513	0.500817	0.250409	+	0.968140i	$0.419435\pi$
$744$	75.5877	2.77118
$745$	−23.7455	−0.869967
$746$	−45.2445	−1.65652
$747$	−105.260	−3.85125
$748$	−21.7913	−0.796770
$749$	1.77475	0.0648481
$750$	79.7795	2.91314
$751$	−0.325726	−0.0118859	−0.00594296	−	0.999982i	$-0.501892\pi$
$751$	−0.325726	−0.0118859	−0.00594296	+	0.999982i	$0.501892\pi$
$752$	−102.367	−3.73293
$753$	−22.5243	−0.820831
$754$	−68.3860	−2.49047
$755$	15.7728	0.574030
$756$	69.0646	2.51186
$757$	−6.92799	−0.251802	−0.125901	−	0.992043i	$-0.540182\pi$
$757$	−6.92799	−0.251802	−0.125901	+	0.992043i	$0.540182\pi$
$758$	74.4333	2.70354
$759$	−80.1252	−2.90836
$760$	−18.8932	−0.685329
$761$	−8.04415	−0.291600	−0.145800	−	0.989314i	$-0.546576\pi$
$761$	−8.04415	−0.291600	−0.145800	+	0.989314i	$0.546576\pi$
$762$	−153.566	−5.56311
$763$	−13.4701	−0.487650
$764$	73.8585	2.67211
$765$	−14.8220	−0.535890
$766$	−41.0876	−1.48455
$767$	3.80197	0.137281
$768$	−34.4184	−1.24197
$769$	12.3664	0.445942	0.222971	−	0.974825i	$-0.428424\pi$
$769$	12.3664	0.445942	0.222971	+	0.974825i	$0.428424\pi$
$770$	−38.0555	−1.37143
$771$	14.3621	0.517239
$772$	−2.67524	−0.0962840
$773$	−18.4434	−0.663363	−0.331682	−	0.943391i	$-0.607616\pi$
$773$	−18.4434	−0.663363	−0.331682	+	0.943391i	$0.607616\pi$
$774$	−58.9298	−2.11819
$775$	−3.40093	−0.122165
$776$	95.9356	3.44389
$777$	−6.11843	−0.219497
$778$	−52.0828	−1.86726
$779$	8.51936	0.305238
$780$	133.860	4.79297
$781$	24.1515	0.864208
$782$	−11.6025	−0.414904
$783$	89.6815	3.20496
$784$	−58.3552	−2.08411
$785$	3.18556	0.113697
$786$	35.7458	1.27501
$787$	−29.5318	−1.05269	−0.526347	−	0.850270i	$-0.676439\pi$
$787$	−29.5318	−1.05269	−0.526347	+	0.850270i	$0.676439\pi$
$788$	14.9614	0.532979
$789$	−71.2724	−2.53736
$790$	−6.60551	−0.235014
$791$	−16.0712	−0.571426
$792$	−266.205	−9.45917
$793$	−44.5830	−1.58319
$794$	85.3380	3.02853
$795$	−43.4933	−1.54255
$796$	−24.4857	−0.867873
$797$	−10.8901	−0.385749	−0.192874	−	0.981223i	$-0.561781\pi$
$797$	−10.8901	−0.385749	−0.192874	+	0.981223i	$0.561781\pi$
$798$	−9.56929	−0.338749
$799$	8.69096	0.307464
$800$	−12.7750	−0.451665
$801$	67.1509	2.37266
$802$	−67.8072	−2.39435
$803$	−25.1996	−0.889274
$804$	−210.528	−7.42474
$805$	−14.4005	−0.507552
$806$	29.0590	1.02356
$807$	15.0980	0.531474
$808$	−89.1391	−3.13590
$809$	−19.9573	−0.701662	−0.350831	−	0.936439i	$-0.614101\pi$
$809$	−19.9573	−0.701662	−0.350831	+	0.936439i	$0.614101\pi$
$810$	−113.501	−3.98803
$811$	22.9044	0.804284	0.402142	−	0.915577i	$-0.368266\pi$
$811$	22.9044	0.804284	0.402142	+	0.915577i	$0.368266\pi$
$812$	42.1036	1.47755
$813$	48.0031	1.68354
$814$	22.3844	0.784573
$815$	−16.2766	−0.570143
$816$	−28.3719	−0.993213
$817$	3.26605	0.114265
$818$	−15.6571	−0.547438
$819$	27.9731	0.977458
$820$	103.238	3.60522
$821$	−5.74950	−0.200659	−0.100329	−	0.994954i	$-0.531990\pi$
$821$	−5.74950	−0.200659	−0.100329	+	0.994954i	$0.531990\pi$
$822$	−23.1077	−0.805975
$823$	30.9203	1.07781	0.538906	−	0.842366i	$-0.318838\pi$
$823$	30.9203	1.07781	0.538906	+	0.842366i	$0.318838\pi$
$824$	−73.8234	−2.57176
$825$	17.2137	0.599303
$826$	−3.29358	−0.114598
$827$	19.4079	0.674878	0.337439	−	0.941347i	$-0.390439\pi$
$827$	19.4079	0.674878	0.337439	+	0.941347i	$0.390439\pi$
$828$	−169.884	−5.90386
$829$	−50.4499	−1.75220	−0.876100	−	0.482130i	$-0.839863\pi$
$829$	−50.4499	−1.75220	−0.876100	+	0.482130i	$0.839863\pi$
$830$	99.4691	3.45262
$831$	33.5240	1.16294
$832$	36.5846	1.26834
$833$	4.95437	0.171659
$834$	180.067	6.23521
$835$	30.2746	1.04770
$836$	24.8816	0.860549
$837$	−38.1081	−1.31721
$838$	81.3656	2.81073
$839$	−48.5902	−1.67752	−0.838759	−	0.544502i	$-0.816719\pi$
$839$	−48.5902	−1.67752	−0.838759	+	0.544502i	$0.816719\pi$
$840$	−68.7601	−2.37245
$841$	25.6722	0.885248
$842$	−48.9642	−1.68742
$843$	72.7499	2.50564
$844$	−4.91352	−0.169130
$845$	−1.54681	−0.0532118
$846$	179.050	6.15587
$847$	16.9699	0.583092
$848$	−57.9286	−1.98928
$849$	20.7807	0.713191
$850$	2.49262	0.0854961
$851$	8.47045	0.290363
$852$	73.5932	2.52126
$853$	40.9090	1.40070	0.700348	−	0.713801i	$-0.253028\pi$
$853$	40.9090	1.40070	0.700348	+	0.713801i	$0.253028\pi$
$854$	38.6214	1.32160
$855$	16.9239	0.578786
$856$	−11.7317	−0.400982
$857$	44.5943	1.52331	0.761656	−	0.647981i	$-0.224387\pi$
$857$	44.5943	1.52331	0.761656	+	0.647981i	$0.224387\pi$
$858$	−147.081	−5.02126
$859$	18.6871	0.637594	0.318797	−	0.947823i	$-0.396721\pi$
$859$	18.6871	0.637594	0.318797	+	0.947823i	$0.396721\pi$
$860$	39.5780	1.34960
$861$	31.0054	1.05666
$862$	−81.6881	−2.78231
$863$	−20.5122	−0.698244	−0.349122	−	0.937077i	$-0.613520\pi$
$863$	−20.5122	−0.698244	−0.349122	+	0.937077i	$0.613520\pi$
$864$	−143.146	−4.86993
$865$	−58.7900	−1.99892
$866$	47.7139	1.62138
$867$	−50.9782	−1.73131
$868$	−17.8910	−0.607259
$869$	5.15827	0.174982
$870$	−150.577	−5.10503
$871$	−47.9915	−1.62613
$872$	89.0419	3.01534
$873$	−85.9359	−2.90849
$874$	13.2479	0.448116
$875$	−11.1969	−0.378525
$876$	−76.7868	−2.59439
$877$	10.2413	0.345825	0.172912	−	0.984937i	$-0.444682\pi$
$877$	10.2413	0.345825	0.172912	+	0.984937i	$0.444682\pi$
$878$	12.2606	0.413776
$879$	22.6892	0.765289
$880$	128.831	4.34289
$881$	47.1344	1.58800	0.793999	−	0.607919i	$-0.207996\pi$
$881$	47.1344	1.58800	0.793999	+	0.607919i	$0.207996\pi$
$882$	102.069	3.43686
$883$	25.5090	0.858446	0.429223	−	0.903199i	$-0.358787\pi$
$883$	25.5090	0.858446	0.429223	+	0.903199i	$0.358787\pi$
$884$	−15.1367	−0.509103
$885$	8.37144	0.281403
$886$	−74.8620	−2.51504
$887$	26.0441	0.874476	0.437238	−	0.899346i	$-0.355957\pi$
$887$	26.0441	0.874476	0.437238	+	0.899346i	$0.355957\pi$
$888$	40.4449	1.35724
$889$	21.5527	0.722855
$890$	−63.4568	−2.12708
$891$	88.6337	2.96934
$892$	115.619	3.87122
$893$	−9.92344	−0.332075
$894$	79.5020	2.65895
$895$	−9.93039	−0.331936
$896$	−4.33771	−0.144913
$897$	−55.6567	−1.85832
$898$	−25.5917	−0.854006
$899$	−23.2317	−0.774820
$900$	36.4969	1.21656
$901$	4.91816	0.163848
$902$	−113.434	−3.77693
$903$	11.8865	0.395557
$904$	106.236	3.53336
$905$	−13.4929	−0.448519
$906$	−52.8087	−1.75445
$907$	0.993472	0.0329877	0.0164939	−	0.999864i	$-0.494750\pi$
$907$	0.993472	0.0329877	0.0164939	+	0.999864i	$0.494750\pi$
$908$	−125.875	−4.17730
$909$	79.8478	2.64838
$910$	−26.4342	−0.876285
$911$	17.1649	0.568700	0.284350	−	0.958721i	$-0.408222\pi$
$911$	17.1649	0.568700	0.284350	+	0.958721i	$0.408222\pi$
$912$	32.3953	1.07272
$913$	−77.6758	−2.57069
$914$	64.0314	2.11797
$915$	−98.1659	−3.24526
$916$	−70.9705	−2.34493
$917$	−5.01687	−0.165672
$918$	27.9302	0.921835
$919$	23.9531	0.790140	0.395070	−	0.918651i	$-0.370720\pi$
$919$	23.9531	0.790140	0.395070	+	0.918651i	$0.370720\pi$
$920$	95.1925	3.13840
$921$	−14.6474	−0.482650
$922$	−34.9897	−1.15232
$923$	16.7762	0.552194
$924$	90.5542	2.97901
$925$	−1.81975	−0.0598329
$926$	−5.64051	−0.185359
$927$	66.1285	2.17195
$928$	−87.2657	−2.86464
$929$	47.4472	1.55669	0.778345	−	0.627836i	$-0.216059\pi$
$929$	47.4472	1.55669	0.778345	+	0.627836i	$0.216059\pi$
$930$	63.9841	2.09812
$931$	−5.65697	−0.185400
$932$	68.9184	2.25750
$933$	−0.00668406	−0.000218826 0
$934$	−92.2621	−3.01891
$935$	−10.9378	−0.357704
$936$	−184.912	−6.04402
$937$	49.2706	1.60960	0.804800	−	0.593546i	$-0.202272\pi$
$937$	49.2706	1.60960	0.804800	+	0.593546i	$0.202272\pi$
$938$	41.5741	1.35744
$939$	−99.4419	−3.24516
$940$	−120.252	−3.92220
$941$	−4.23670	−0.138112	−0.0690562	−	0.997613i	$-0.521999\pi$
$941$	−4.23670	−0.138112	−0.0690562	+	0.997613i	$0.521999\pi$
$942$	−10.6655	−0.347502
$943$	−42.9243	−1.39781
$944$	11.1499	0.362898
$945$	34.6658	1.12768
$946$	−43.4869	−1.41388
$947$	41.5205	1.34924	0.674618	−	0.738167i	$-0.264308\pi$
$947$	41.5205	1.34924	0.674618	+	0.738167i	$0.264308\pi$
$948$	15.7180	0.510497
$949$	−17.5042	−0.568209
$950$	−2.84610	−0.0923398
$951$	−0.395845	−0.0128361
$952$	7.77528	0.251998
$953$	0.103865	0.00336451	0.00168225	−	0.999999i	$-0.499465\pi$
$953$	0.103865	0.00336451	0.00168225	+	0.999999i	$0.499465\pi$
$954$	101.323	3.28046
$955$	37.0720	1.19962
$956$	141.867	4.58832
$957$	117.586	3.80102
$958$	9.75707	0.315237
$959$	3.24313	0.104726
$960$	80.5543	2.59988
$961$	−21.1282	−0.681556
$962$	15.5487	0.501310
$963$	10.5089	0.338644
$964$	−92.3487	−2.97435
$965$	−1.34279	−0.0432260
$966$	48.2143	1.55127
$967$	−16.1184	−0.518333	−0.259167	−	0.965833i	$-0.583448\pi$
$967$	−16.1184	−0.518333	−0.259167	+	0.965833i	$0.583448\pi$
$968$	−112.177	−3.60550
$969$	−2.75037	−0.0883547
$970$	81.2084	2.60744
$971$	−52.6303	−1.68899	−0.844494	−	0.535566i	$-0.820098\pi$
$971$	−52.6303	−1.68899	−0.844494	+	0.535566i	$0.820098\pi$
$972$	91.2939	2.92825
$973$	−25.2721	−0.810187
$974$	31.7506	1.01736
$975$	11.9570	0.382930
$976$	−130.747	−4.18511
$977$	0.0977514	0.00312734	0.00156367	−	0.999999i	$-0.499502\pi$
$977$	0.0977514	0.00312734	0.00156367	+	0.999999i	$0.499502\pi$
$978$	54.4955	1.74257
$979$	49.5536	1.58374
$980$	−68.5512	−2.18979
$981$	−79.7608	−2.54657
$982$	13.1232	0.418777
$983$	39.1169	1.24764	0.623818	−	0.781570i	$-0.285581\pi$
$983$	39.1169	1.24764	0.623818	+	0.781570i	$0.285581\pi$
$984$	−204.956	−6.53376
$985$	7.50964	0.239277
$986$	17.0270	0.542250
$987$	−36.1154	−1.14957
$988$	17.2833	0.549855
$989$	−16.4558	−0.523265
$990$	−225.339	−7.16175
$991$	−17.3529	−0.551231	−0.275616	−	0.961268i	$-0.588882\pi$
$991$	−17.3529	−0.551231	−0.275616	+	0.961268i	$0.588882\pi$
$992$	37.0815	1.17734
$993$	26.1007	0.828282
$994$	−14.5329	−0.460955
$995$	−12.2902	−0.389625
$996$	−236.690	−7.49979
$997$	43.2231	1.36889	0.684445	−	0.729064i	$-0.260045\pi$
$997$	43.2231	1.36889	0.684445	+	0.729064i	$0.260045\pi$
$998$	−90.9354	−2.87851
$999$	−20.3906	−0.645129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.3	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.3	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.62936 q^{2} +3.14041 q^{3} +4.91352 q^{4} +2.46626 q^{5} -8.25726 q^{6} +1.15889 q^{7} -7.66068 q^{8} +6.86218 q^{9} +O(q^{10})\)	\(q-2.62936 q^{2} +3.14041 q^{3} +4.91352 q^{4} +2.46626 q^{5} -8.25726 q^{6} +1.15889 q^{7} -7.66068 q^{8} +6.86218 q^{9} -6.48468 q^{10} +5.06391 q^{11} +15.4305 q^{12} +3.51750 q^{13} -3.04714 q^{14} +7.74507 q^{15} +10.3156 q^{16} -0.875800 q^{17} -18.0431 q^{18} +1.00000 q^{19} +12.1180 q^{20} +3.63940 q^{21} -13.3148 q^{22} -5.03844 q^{23} -24.0577 q^{24} +1.08243 q^{25} -9.24877 q^{26} +12.1288 q^{27} +5.69425 q^{28} +7.39407 q^{29} -20.3645 q^{30} -3.14194 q^{31} -11.8021 q^{32} +15.9028 q^{33} +2.30279 q^{34} +2.85813 q^{35} +33.7175 q^{36} -1.68116 q^{37} -2.62936 q^{38} +11.0464 q^{39} -18.8932 q^{40} +8.51936 q^{41} -9.56929 q^{42} +3.26605 q^{43} +24.8816 q^{44} +16.9239 q^{45} +13.2479 q^{46} -9.92344 q^{47} +32.3953 q^{48} -5.65697 q^{49} -2.84610 q^{50} -2.75037 q^{51} +17.2833 q^{52} -5.61561 q^{53} -31.8911 q^{54} +12.4889 q^{55} -8.87792 q^{56} +3.14041 q^{57} -19.4416 q^{58} +1.08087 q^{59} +38.0555 q^{60} -12.6746 q^{61} +8.26127 q^{62} +7.95254 q^{63} +10.4007 q^{64} +8.67507 q^{65} -41.8140 q^{66} -13.6436 q^{67} -4.30326 q^{68} -15.8228 q^{69} -7.51505 q^{70} +4.76934 q^{71} -52.5690 q^{72} -4.97631 q^{73} +4.42038 q^{74} +3.39928 q^{75} +4.91352 q^{76} +5.86853 q^{77} -29.0449 q^{78} +1.01863 q^{79} +25.4410 q^{80} +17.5030 q^{81} -22.4004 q^{82} -15.3391 q^{83} +17.8823 q^{84} -2.15995 q^{85} -8.58762 q^{86} +23.2204 q^{87} -38.7930 q^{88} +9.78565 q^{89} -44.4990 q^{90} +4.07641 q^{91} -24.7565 q^{92} -9.86697 q^{93} +26.0923 q^{94} +2.46626 q^{95} -37.0635 q^{96} -12.5231 q^{97} +14.8742 q^{98} +34.7495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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