LMFDB - Embedded newform 4009.2.a.f.1.2

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:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.70959 q^{2} +1.36959 q^{3} +5.34187 q^{4} +3.33123 q^{5} -3.71102 q^{6} +3.43608 q^{7} -9.05510 q^{8} -1.12423 q^{9} +O(q^{10})$	\(q-2.70959 q^{2} +1.36959 q^{3} +5.34187 q^{4} +3.33123 q^{5} -3.71102 q^{6} +3.43608 q^{7} -9.05510 q^{8} -1.12423 q^{9} -9.02626 q^{10} -1.42878 q^{11} +7.31617 q^{12} +0.669511 q^{13} -9.31037 q^{14} +4.56242 q^{15} +13.8519 q^{16} +7.79873 q^{17} +3.04619 q^{18} -1.00000 q^{19} +17.7950 q^{20} +4.70602 q^{21} +3.87140 q^{22} +3.15155 q^{23} -12.4018 q^{24} +6.09709 q^{25} -1.81410 q^{26} -5.64849 q^{27} +18.3551 q^{28} +10.6434 q^{29} -12.3623 q^{30} +5.50316 q^{31} -19.4226 q^{32} -1.95684 q^{33} -21.1314 q^{34} +11.4464 q^{35} -6.00547 q^{36} -3.16721 q^{37} +2.70959 q^{38} +0.916955 q^{39} -30.1646 q^{40} -5.08920 q^{41} -12.7514 q^{42} -2.94120 q^{43} -7.63234 q^{44} -3.74505 q^{45} -8.53940 q^{46} +2.14077 q^{47} +18.9714 q^{48} +4.80666 q^{49} -16.5206 q^{50} +10.6811 q^{51} +3.57644 q^{52} +4.02001 q^{53} +15.3051 q^{54} -4.75958 q^{55} -31.1141 q^{56} -1.36959 q^{57} -28.8392 q^{58} -0.345194 q^{59} +24.3718 q^{60} -9.90686 q^{61} -14.9113 q^{62} -3.86293 q^{63} +24.9237 q^{64} +2.23029 q^{65} +5.30222 q^{66} -3.81158 q^{67} +41.6598 q^{68} +4.31632 q^{69} -31.0150 q^{70} +3.85278 q^{71} +10.1800 q^{72} -7.13054 q^{73} +8.58185 q^{74} +8.35051 q^{75} -5.34187 q^{76} -4.90939 q^{77} -2.48457 q^{78} -15.3752 q^{79} +46.1437 q^{80} -4.36344 q^{81} +13.7897 q^{82} +17.0117 q^{83} +25.1390 q^{84} +25.9794 q^{85} +7.96945 q^{86} +14.5770 q^{87} +12.9377 q^{88} -14.1980 q^{89} +10.1476 q^{90} +2.30049 q^{91} +16.8352 q^{92} +7.53707 q^{93} -5.80059 q^{94} -3.33123 q^{95} -26.6010 q^{96} +14.0037 q^{97} -13.0241 q^{98} +1.60627 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−2.70959	−1.91597	−0.957984	−	0.286820i	$-0.907402\pi$
$2$	−2.70959	−1.91597	−0.957984	+	0.286820i	$0.907402\pi$
$3$	1.36959	0.790733	0.395366	−	0.918524i	$-0.370618\pi$
$3$	1.36959	0.790733	0.395366	+	0.918524i	$0.370618\pi$
$4$	5.34187	2.67094
$5$	3.33123	1.48977	0.744886	−	0.667192i	$-0.232504\pi$
$5$	3.33123	1.48977	0.744886	+	0.667192i	$0.232504\pi$
$6$	−3.71102	−1.51502
$7$	3.43608	1.29872	0.649358	−	0.760482i	$-0.275038\pi$
$7$	3.43608	1.29872	0.649358	+	0.760482i	$0.275038\pi$
$8$	−9.05510	−3.20146
$9$	−1.12423	−0.374742
$10$	−9.02626	−2.85435
$11$	−1.42878	−0.430792	−0.215396	−	0.976527i	$-0.569104\pi$
$11$	−1.42878	−0.430792	−0.215396	+	0.976527i	$0.569104\pi$
$12$	7.31617	2.11200
$13$	0.669511	0.185689	0.0928444	−	0.995681i	$-0.470404\pi$
$13$	0.669511	0.185689	0.0928444	+	0.995681i	$0.470404\pi$
$14$	−9.31037	−2.48830
$15$	4.56242	1.17801
$16$	13.8519	3.46296
$17$	7.79873	1.89147	0.945735	−	0.324939i	$-0.105344\pi$
$17$	7.79873	1.89147	0.945735	+	0.324939i	$0.105344\pi$
$18$	3.04619	0.717993
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	17.7950	3.97908
$21$	4.70602	1.02694
$22$	3.87140	0.825384
$23$	3.15155	0.657143	0.328571	−	0.944479i	$-0.393433\pi$
$23$	3.15155	0.657143	0.328571	+	0.944479i	$0.393433\pi$
$24$	−12.4018	−2.53150
$25$	6.09709	1.21942
$26$	−1.81410	−0.355774
$27$	−5.64849	−1.08705
$28$	18.3551	3.46879
$29$	10.6434	1.97642	0.988212	−	0.153089i	$-0.0489221\pi$
$29$	10.6434	1.97642	0.988212	+	0.153089i	$0.0489221\pi$
$30$	−12.3623	−2.25703
$31$	5.50316	0.988397	0.494198	−	0.869349i	$-0.335462\pi$
$31$	5.50316	0.988397	0.494198	+	0.869349i	$0.335462\pi$
$32$	−19.4226	−3.43347
$33$	−1.95684	−0.340641
$34$	−21.1314	−3.62400
$35$	11.4464	1.93479
$36$	−6.00547	−1.00091
$37$	−3.16721	−0.520687	−0.260343	−	0.965516i	$-0.583836\pi$
$37$	−3.16721	−0.520687	−0.260343	+	0.965516i	$0.583836\pi$
$38$	2.70959	0.439553
$39$	0.916955	0.146830
$40$	−30.1646	−4.76944
$41$	−5.08920	−0.794800	−0.397400	−	0.917645i	$-0.630087\pi$
$41$	−5.08920	−0.794800	−0.397400	+	0.917645i	$0.630087\pi$
$42$	−12.7514	−1.96758
$43$	−2.94120	−0.448529	−0.224265	−	0.974528i	$-0.571998\pi$
$43$	−2.94120	−0.448529	−0.224265	+	0.974528i	$0.571998\pi$
$44$	−7.63234	−1.15062
$45$	−3.74505	−0.558279
$46$	−8.53940	−1.25907
$47$	2.14077	0.312263	0.156131	−	0.987736i	$-0.450098\pi$
$47$	2.14077	0.312263	0.156131	+	0.987736i	$0.450098\pi$
$48$	18.9714	2.73828
$49$	4.80666	0.686666
$50$	−16.5206	−2.33637
$51$	10.6811	1.49565
$52$	3.57644	0.495963
$53$	4.02001	0.552191	0.276095	−	0.961130i	$-0.410959\pi$
$53$	4.02001	0.552191	0.276095	+	0.961130i	$0.410959\pi$
$54$	15.3051	2.08276
$55$	−4.75958	−0.641782
$56$	−31.1141	−4.15779
$57$	−1.36959	−0.181407
$58$	−28.8392	−3.78677
$59$	−0.345194	−0.0449404	−0.0224702	−	0.999748i	$-0.507153\pi$
$59$	−0.345194	−0.0449404	−0.0224702	+	0.999748i	$0.507153\pi$
$60$	24.3718	3.14639
$61$	−9.90686	−1.26844	−0.634222	−	0.773151i	$-0.718679\pi$
$61$	−9.90686	−1.26844	−0.634222	+	0.773151i	$0.718679\pi$
$62$	−14.9113	−1.89374
$63$	−3.86293	−0.486683
$64$	24.9237	3.11546
$65$	2.23029	0.276634
$66$	5.30222	0.652658
$67$	−3.81158	−0.465658	−0.232829	−	0.972518i	$-0.574798\pi$
$67$	−3.81158	−0.465658	−0.232829	+	0.972518i	$0.574798\pi$
$68$	41.6598	5.05200
$69$	4.31632	0.519624
$70$	−31.0150	−3.70700
$71$	3.85278	0.457241	0.228620	−	0.973516i	$-0.426579\pi$
$71$	3.85278	0.457241	0.228620	+	0.973516i	$0.426579\pi$
$72$	10.1800	1.19972
$73$	−7.13054	−0.834567	−0.417284	−	0.908776i	$-0.637018\pi$
$73$	−7.13054	−0.834567	−0.417284	+	0.908776i	$0.637018\pi$
$74$	8.58185	0.997620
$75$	8.35051	0.964234
$76$	−5.34187	−0.612755
$77$	−4.90939	−0.559477
$78$	−2.48457	−0.281322
$79$	−15.3752	−1.72985	−0.864923	−	0.501905i	$-0.832633\pi$
$79$	−15.3752	−1.72985	−0.864923	+	0.501905i	$0.832633\pi$
$80$	46.1437	5.15902
$81$	−4.36344	−0.484827
$82$	13.7897	1.52281
$83$	17.0117	1.86728	0.933638	−	0.358218i	$-0.116616\pi$
$83$	17.0117	1.86728	0.933638	+	0.358218i	$0.116616\pi$
$84$	25.1390	2.74289
$85$	25.9794	2.81786
$86$	7.96945	0.859368
$87$	14.5770	1.56282
$88$	12.9377	1.37916
$89$	−14.1980	−1.50498	−0.752491	−	0.658603i	$-0.771148\pi$
$89$	−14.1980	−1.50498	−0.752491	+	0.658603i	$0.771148\pi$
$90$	10.1476	1.06965
$91$	2.30049	0.241157
$92$	16.8352	1.75519
$93$	7.53707	0.781558
$94$	−5.80059	−0.598286
$95$	−3.33123	−0.341777
$96$	−26.6010	−2.71496
$97$	14.0037	1.42186	0.710928	−	0.703265i	$-0.248275\pi$
$97$	14.0037	1.42186	0.710928	+	0.703265i	$0.248275\pi$
$98$	−13.0241	−1.31563
$99$	1.60627	0.161436
$100$	32.5699	3.25699
$101$	2.05555	0.204535	0.102268	−	0.994757i	$-0.467390\pi$
$101$	2.05555	0.204535	0.102268	+	0.994757i	$0.467390\pi$
$102$	−28.9413	−2.86561
$103$	3.00872	0.296458	0.148229	−	0.988953i	$-0.452643\pi$
$103$	3.00872	0.296458	0.148229	+	0.988953i	$0.452643\pi$
$104$	−6.06249	−0.594476
$105$	15.6768	1.52990
$106$	−10.8926	−1.05798
$107$	6.32993	0.611937	0.305969	−	0.952042i	$-0.401020\pi$
$107$	6.32993	0.611937	0.305969	+	0.952042i	$0.401020\pi$
$108$	−30.1735	−2.90345
$109$	1.06274	0.101792	0.0508962	−	0.998704i	$-0.483792\pi$
$109$	1.06274	0.101792	0.0508962	+	0.998704i	$0.483792\pi$
$110$	12.8965	1.22963
$111$	−4.33778	−0.411724
$112$	47.5961	4.49741
$113$	−13.2646	−1.24783	−0.623916	−	0.781492i	$-0.714459\pi$
$113$	−13.2646	−1.24783	−0.623916	+	0.781492i	$0.714459\pi$
$114$	3.71102	0.347569
$115$	10.4985	0.978992
$116$	56.8555	5.27890
$117$	−0.752681	−0.0695854
$118$	0.935333	0.0861044
$119$	26.7971	2.45648
$120$	−41.3131	−3.77136
$121$	−8.95860	−0.814418
$122$	26.8435	2.43030
$123$	−6.97012	−0.628475
$124$	29.3972	2.63994
$125$	3.65465	0.326882
$126$	10.4670	0.932470
$127$	−12.1379	−1.07707	−0.538533	−	0.842604i	$-0.681021\pi$
$127$	−12.1379	−1.07707	−0.538533	+	0.842604i	$0.681021\pi$
$128$	−28.6876	−2.53565
$129$	−4.02824	−0.354667
$130$	−6.04318	−0.530022
$131$	−4.99750	−0.436633	−0.218317	−	0.975878i	$-0.570057\pi$
$131$	−4.99750	−0.436633	−0.218317	+	0.975878i	$0.570057\pi$
$132$	−10.4532	−0.909832
$133$	−3.43608	−0.297946
$134$	10.3278	0.892186
$135$	−18.8164	−1.61946
$136$	−70.6183	−6.05547
$137$	−10.7973	−0.922472	−0.461236	−	0.887278i	$-0.652594\pi$
$137$	−10.7973	−0.922472	−0.461236	+	0.887278i	$0.652594\pi$
$138$	−11.6955	−0.995584
$139$	10.8455	0.919905	0.459953	−	0.887943i	$-0.347866\pi$
$139$	10.8455	0.919905	0.459953	+	0.887943i	$0.347866\pi$
$140$	61.1451	5.16770
$141$	2.93197	0.246916
$142$	−10.4394	−0.876059
$143$	−0.956581	−0.0799933
$144$	−15.5726	−1.29772
$145$	35.4555	2.94442
$146$	19.3208	1.59900
$147$	6.58315	0.542969
$148$	−16.9189	−1.39072
$149$	20.2298	1.65729	0.828643	−	0.559777i	$-0.189113\pi$
$149$	20.2298	1.65729	0.828643	+	0.559777i	$0.189113\pi$
$150$	−22.6264	−1.84744
$151$	−9.93519	−0.808514	−0.404257	−	0.914645i	$-0.632470\pi$
$151$	−9.93519	−0.808514	−0.404257	+	0.914645i	$0.632470\pi$
$152$	9.05510	0.734466
$153$	−8.76753	−0.708813
$154$	13.3024	1.07194
$155$	18.3323	1.47248
$156$	4.89826	0.392174
$157$	14.4684	1.15470	0.577351	−	0.816496i	$-0.304086\pi$
$157$	14.4684	1.15470	0.577351	+	0.816496i	$0.304086\pi$
$158$	41.6605	3.31433
$159$	5.50576	0.436635
$160$	−64.7013	−5.11508
$161$	10.8290	0.853443
$162$	11.8231	0.928913
$163$	4.07280	0.319006	0.159503	−	0.987197i	$-0.449011\pi$
$163$	4.07280	0.319006	0.159503	+	0.987197i	$0.449011\pi$
$164$	−27.1859	−2.12286
$165$	−6.51867	−0.507478
$166$	−46.0947	−3.57764
$167$	19.4857	1.50785	0.753925	−	0.656960i	$-0.228158\pi$
$167$	19.4857	1.50785	0.753925	+	0.656960i	$0.228158\pi$
$168$	−42.6135	−3.28770
$169$	−12.5518	−0.965520
$170$	−70.3934	−5.39893
$171$	1.12423	0.0859716
$172$	−15.7115	−1.19799
$173$	−19.9132	−1.51397	−0.756986	−	0.653431i	$-0.773329\pi$
$173$	−19.9132	−1.51397	−0.756986	+	0.653431i	$0.773329\pi$
$174$	−39.4978	−2.99432
$175$	20.9501	1.58368
$176$	−19.7912	−1.49182
$177$	−0.472774	−0.0355358
$178$	38.4706	2.88350
$179$	−9.52030	−0.711581	−0.355790	−	0.934566i	$-0.615788\pi$
$179$	−9.52030	−0.711581	−0.355790	+	0.934566i	$0.615788\pi$
$180$	−20.0056	−1.49113
$181$	2.18782	0.162619	0.0813096	−	0.996689i	$-0.474090\pi$
$181$	2.18782	0.162619	0.0813096	+	0.996689i	$0.474090\pi$
$182$	−6.23339	−0.462050
$183$	−13.5683	−1.00300
$184$	−28.5376	−2.10382
$185$	−10.5507	−0.775704
$186$	−20.4224	−1.49744
$187$	−11.1426	−0.814830
$188$	11.4357	0.834034
$189$	−19.4087	−1.41177
$190$	9.02626	0.654834
$191$	−13.1244	−0.949650	−0.474825	−	0.880080i	$-0.657489\pi$
$191$	−13.1244	−0.949650	−0.474825	+	0.880080i	$0.657489\pi$
$192$	34.1352	2.46349
$193$	−20.0545	−1.44356	−0.721778	−	0.692125i	$-0.756675\pi$
$193$	−20.0545	−1.44356	−0.721778	+	0.692125i	$0.756675\pi$
$194$	−37.9442	−2.72423
$195$	3.05459	0.218744
$196$	25.6766	1.83404
$197$	21.3251	1.51935	0.759674	−	0.650305i	$-0.225359\pi$
$197$	21.3251	1.51935	0.759674	+	0.650305i	$0.225359\pi$
$198$	−4.35232	−0.309306
$199$	−6.31009	−0.447310	−0.223655	−	0.974668i	$-0.571799\pi$
$199$	−6.31009	−0.447310	−0.223655	+	0.974668i	$0.571799\pi$
$200$	−55.2098	−3.90392
$201$	−5.22029	−0.368211
$202$	−5.56971	−0.391883
$203$	36.5715	2.56682
$204$	57.0569	3.99478
$205$	−16.9533	−1.18407
$206$	−8.15240	−0.568005
$207$	−3.54305	−0.246259
$208$	9.27397	0.643034
$209$	1.42878	0.0988305
$210$	−42.4778	−2.93125
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	21.4744	1.47487
$213$	5.27672	0.361555
$214$	−17.1515	−1.17245
$215$	−9.79782	−0.668206
$216$	51.1477	3.48016
$217$	18.9093	1.28365
$218$	−2.87960	−0.195031
$219$	−9.76592	−0.659919
$220$	−25.4251	−1.71416
$221$	5.22134	0.351225
$222$	11.7536	0.788850
$223$	−18.1905	−1.21813	−0.609065	−	0.793121i	$-0.708455\pi$
$223$	−18.1905	−1.21813	−0.609065	+	0.793121i	$0.708455\pi$
$224$	−66.7378	−4.45911
$225$	−6.85450	−0.456967
$226$	35.9417	2.39081
$227$	16.3836	1.08742	0.543710	−	0.839273i	$-0.317019\pi$
$227$	16.3836	1.08742	0.543710	+	0.839273i	$0.317019\pi$
$228$	−7.31617	−0.484525
$229$	11.2284	0.741994	0.370997	−	0.928634i	$-0.379016\pi$
$229$	11.2284	0.741994	0.370997	+	0.928634i	$0.379016\pi$
$230$	−28.4467	−1.87572
$231$	−6.72385	−0.442397
$232$	−96.3768	−6.32745
$233$	−15.2782	−1.00091	−0.500456	−	0.865762i	$-0.666834\pi$
$233$	−15.2782	−1.00091	−0.500456	+	0.865762i	$0.666834\pi$
$234$	2.03946	0.133323
$235$	7.13138	0.465200
$236$	−1.84398	−0.120033
$237$	−21.0577	−1.36785
$238$	−72.6091	−4.70655
$239$	21.8316	1.41217	0.706083	−	0.708129i	$-0.250460\pi$
$239$	21.8316	1.41217	0.706083	+	0.708129i	$0.250460\pi$
$240$	63.1979	4.07941
$241$	−3.43407	−0.221208	−0.110604	−	0.993865i	$-0.535279\pi$
$241$	−3.43407	−0.221208	−0.110604	+	0.993865i	$0.535279\pi$
$242$	24.2741	1.56040
$243$	10.9694	0.703685
$244$	−52.9212	−3.38793
$245$	16.0121	1.02297
$246$	18.8862	1.20414
$247$	−0.669511	−0.0426000
$248$	−49.8317	−3.16431
$249$	23.2990	1.47652
$250$	−9.90261	−0.626296
$251$	17.9957	1.13588	0.567940	−	0.823070i	$-0.307741\pi$
$251$	17.9957	1.13588	0.567940	+	0.823070i	$0.307741\pi$
$252$	−20.6353	−1.29990
$253$	−4.50285	−0.283092
$254$	32.8888	2.06362
$255$	35.5811	2.22817
$256$	27.8842	1.74276
$257$	15.2347	0.950312	0.475156	−	0.879902i	$-0.342392\pi$
$257$	15.2347	0.950312	0.475156	+	0.879902i	$0.342392\pi$
$258$	10.9149	0.679530
$259$	−10.8828	−0.676225
$260$	11.9139	0.738872
$261$	−11.9655	−0.740649
$262$	13.5412	0.836576
$263$	3.62055	0.223253	0.111626	−	0.993750i	$-0.464394\pi$
$263$	3.62055	0.223253	0.111626	+	0.993750i	$0.464394\pi$
$264$	17.7193	1.09055
$265$	13.3916	0.822638
$266$	9.31037	0.570855
$267$	−19.4454	−1.19004
$268$	−20.3610	−1.24374
$269$	−28.6194	−1.74496	−0.872478	−	0.488653i	$-0.837489\pi$
$269$	−28.6194	−1.74496	−0.872478	+	0.488653i	$0.837489\pi$
$270$	50.9848	3.10284
$271$	14.2760	0.867203	0.433602	−	0.901105i	$-0.357243\pi$
$271$	14.2760	0.867203	0.433602	+	0.901105i	$0.357243\pi$
$272$	108.027	6.55009
$273$	3.15073	0.190691
$274$	29.2561	1.76743
$275$	−8.71137	−0.525316
$276$	23.0573	1.38788
$277$	−5.25930	−0.316001	−0.158000	−	0.987439i	$-0.550505\pi$
$277$	−5.25930	−0.316001	−0.158000	+	0.987439i	$0.550505\pi$
$278$	−29.3869	−1.76251
$279$	−6.18679	−0.370393
$280$	−103.648	−6.19416
$281$	4.80104	0.286406	0.143203	−	0.989693i	$-0.454260\pi$
$281$	4.80104	0.286406	0.143203	+	0.989693i	$0.454260\pi$
$282$	−7.94443	−0.473084
$283$	22.4686	1.33562	0.667811	−	0.744331i	$-0.267231\pi$
$283$	22.4686	1.33562	0.667811	+	0.744331i	$0.267231\pi$
$284$	20.5811	1.22126
$285$	−4.56242	−0.270254
$286$	2.59194	0.153265
$287$	−17.4869	−1.03222
$288$	21.8354	1.28666
$289$	43.8202	2.57766
$290$	−96.0699	−5.64142
$291$	19.1793	1.12431
$292$	−38.0905	−2.22908
$293$	−17.8767	−1.04437	−0.522184	−	0.852833i	$-0.674883\pi$
$293$	−17.8767	−1.04437	−0.522184	+	0.852833i	$0.674883\pi$
$294$	−17.8376	−1.04031
$295$	−1.14992	−0.0669509
$296$	28.6794	1.66696
$297$	8.07043	0.468294
$298$	−54.8143	−3.17531
$299$	2.10999	0.122024
$300$	44.6073	2.57541
$301$	−10.1062	−0.582512
$302$	26.9203	1.54909
$303$	2.81527	0.161733
$304$	−13.8519	−0.794459
$305$	−33.0020	−1.88969
$306$	23.7564	1.35806
$307$	−9.08382	−0.518441	−0.259221	−	0.965818i	$-0.583466\pi$
$307$	−9.08382	−0.518441	−0.259221	+	0.965818i	$0.583466\pi$
$308$	−26.2253	−1.49433
$309$	4.12071	0.234419
$310$	−49.6730	−2.82123
$311$	1.61957	0.0918372	0.0459186	−	0.998945i	$-0.485379\pi$
$311$	1.61957	0.0918372	0.0459186	+	0.998945i	$0.485379\pi$
$312$	−8.30312	−0.470072
$313$	7.86544	0.444581	0.222291	−	0.974980i	$-0.428647\pi$
$313$	7.86544	0.444581	0.222291	+	0.974980i	$0.428647\pi$
$314$	−39.2033	−2.21237
$315$	−12.8683	−0.725047
$316$	−82.1324	−4.62031
$317$	−23.3092	−1.30918	−0.654588	−	0.755986i	$-0.727158\pi$
$317$	−23.3092	−1.30918	−0.654588	+	0.755986i	$0.727158\pi$
$318$	−14.9184	−0.836580
$319$	−15.2070	−0.851428
$320$	83.0264	4.64132
$321$	8.66940	0.483879
$322$	−29.3421	−1.63517
$323$	−7.79873	−0.433933
$324$	−23.3090	−1.29494
$325$	4.08207	0.226432
$326$	−11.0356	−0.611206
$327$	1.45552	0.0804906
$328$	46.0833	2.54452
$329$	7.35584	0.405541
$330$	17.6629	0.972312
$331$	8.05409	0.442693	0.221346	−	0.975195i	$-0.428955\pi$
$331$	8.05409	0.442693	0.221346	+	0.975195i	$0.428955\pi$
$332$	90.8743	4.98737
$333$	3.56066	0.195123
$334$	−52.7983	−2.88899
$335$	−12.6972	−0.693724
$336$	65.1871	3.55625
$337$	11.1777	0.608891	0.304445	−	0.952530i	$-0.401529\pi$
$337$	11.1777	0.608891	0.304445	+	0.952530i	$0.401529\pi$
$338$	34.0101	1.84991
$339$	−18.1671	−0.986701
$340$	138.778	7.52632
$341$	−7.86278	−0.425794
$342$	−3.04619	−0.164719
$343$	−7.53650	−0.406933
$344$	26.6329	1.43595
$345$	14.3787	0.774121
$346$	53.9565	2.90072
$347$	−13.4155	−0.720183	−0.360091	−	0.932917i	$-0.617254\pi$
$347$	−13.4155	−0.720183	−0.360091	+	0.932917i	$0.617254\pi$
$348$	77.8687	4.17420
$349$	−19.1814	−1.02676	−0.513379	−	0.858162i	$-0.671606\pi$
$349$	−19.1814	−1.02676	−0.513379	+	0.858162i	$0.671606\pi$
$350$	−56.7661	−3.03428
$351$	−3.78173	−0.201854
$352$	27.7506	1.47911
$353$	−31.1192	−1.65631	−0.828154	−	0.560501i	$-0.810609\pi$
$353$	−31.1192	−1.65631	−0.828154	+	0.560501i	$0.810609\pi$
$354$	1.28102	0.0680856
$355$	12.8345	0.681184
$356$	−75.8437	−4.01971
$357$	36.7010	1.94242
$358$	25.7961	1.36337
$359$	32.8198	1.73216	0.866081	−	0.499903i	$-0.166631\pi$
$359$	32.8198	1.73216	0.866081	+	0.499903i	$0.166631\pi$
$360$	33.9118	1.78731
$361$	1.00000	0.0526316
$362$	−5.92809	−0.311573
$363$	−12.2696	−0.643987
$364$	12.2889	0.644116
$365$	−23.7535	−1.24331
$366$	36.7646	1.92172
$367$	−16.5671	−0.864794	−0.432397	−	0.901683i	$-0.642332\pi$
$367$	−16.5671	−0.864794	−0.432397	+	0.901683i	$0.642332\pi$
$368$	43.6548	2.27566
$369$	5.72141	0.297845
$370$	28.5881	1.48622
$371$	13.8131	0.717139
$372$	40.2621	2.08749
$373$	33.8052	1.75036	0.875182	−	0.483793i	$-0.160741\pi$
$373$	33.8052	1.75036	0.875182	+	0.483793i	$0.160741\pi$
$374$	30.1920	1.56119
$375$	5.00537	0.258476
$376$	−19.3848	−0.999697
$377$	7.12585	0.367000
$378$	52.5896	2.70492
$379$	30.0571	1.54393	0.771966	−	0.635664i	$-0.219274\pi$
$379$	30.0571	1.54393	0.771966	+	0.635664i	$0.219274\pi$
$380$	−17.7950	−0.912864
$381$	−16.6240	−0.851671
$382$	35.5618	1.81950
$383$	25.0743	1.28124	0.640619	−	0.767859i	$-0.278678\pi$
$383$	25.0743	1.28124	0.640619	+	0.767859i	$0.278678\pi$
$384$	−39.2902	−2.00502
$385$	−16.3543	−0.833493
$386$	54.3395	2.76581
$387$	3.30657	0.168083
$388$	74.8058	3.79769
$389$	−28.4404	−1.44198	−0.720992	−	0.692944i	$-0.756313\pi$
$389$	−28.4404	−1.44198	−0.720992	+	0.692944i	$0.756313\pi$
$390$	−8.27668	−0.419106
$391$	24.5781	1.24297
$392$	−43.5248	−2.19833
$393$	−6.84452	−0.345260
$394$	−57.7821	−2.91102
$395$	−51.2183	−2.57707
$396$	8.58047	0.431185
$397$	−19.0971	−0.958458	−0.479229	−	0.877690i	$-0.659084\pi$
$397$	−19.0971	−0.958458	−0.479229	+	0.877690i	$0.659084\pi$
$398$	17.0978	0.857033
$399$	−4.70602	−0.235596
$400$	84.4560	4.22280
$401$	−10.4496	−0.521826	−0.260913	−	0.965362i	$-0.584024\pi$
$401$	−10.4496	−0.521826	−0.260913	+	0.965362i	$0.584024\pi$
$402$	14.1449	0.705481
$403$	3.68443	0.183534
$404$	10.9805	0.546301
$405$	−14.5356	−0.722281
$406$	−99.0937	−4.91794
$407$	4.52524	0.224308
$408$	−96.7181	−4.78826
$409$	10.9891	0.543377	0.271688	−	0.962385i	$-0.412418\pi$
$409$	10.9891	0.543377	0.271688	+	0.962385i	$0.412418\pi$
$410$	45.9365	2.26864
$411$	−14.7878	−0.729429
$412$	16.0722	0.791821
$413$	−1.18611	−0.0583649
$414$	9.60020	0.471824
$415$	56.6699	2.78181
$416$	−13.0037	−0.637557
$417$	14.8539	0.727399
$418$	−3.87140	−0.189356
$419$	−20.3571	−0.994510	−0.497255	−	0.867604i	$-0.665659\pi$
$419$	−20.3571	−0.994510	−0.497255	+	0.867604i	$0.665659\pi$
$420$	83.7437	4.08627
$421$	8.58781	0.418544	0.209272	−	0.977857i	$-0.432891\pi$
$421$	8.58781	0.418544	0.209272	+	0.977857i	$0.432891\pi$
$422$	−2.70959	−0.131901
$423$	−2.40670	−0.117018
$424$	−36.4016	−1.76782
$425$	47.5496	2.30649
$426$	−14.2978	−0.692728
$427$	−34.0408	−1.64735
$428$	33.8137	1.63445
$429$	−1.31012	−0.0632533
$430$	26.5481	1.28026
$431$	8.22039	0.395962	0.197981	−	0.980206i	$-0.436561\pi$
$431$	8.22039	0.395962	0.197981	+	0.980206i	$0.436561\pi$
$432$	−78.2421	−3.76443
$433$	11.3761	0.546702	0.273351	−	0.961914i	$-0.411868\pi$
$433$	11.3761	0.546702	0.273351	+	0.961914i	$0.411868\pi$
$434$	−51.2365	−2.45943
$435$	48.5595	2.32825
$436$	5.67705	0.271881
$437$	−3.15155	−0.150759
$438$	26.4616	1.26439
$439$	−19.4410	−0.927868	−0.463934	−	0.885870i	$-0.653562\pi$
$439$	−19.4410	−0.927868	−0.463934	+	0.885870i	$0.653562\pi$
$440$	43.0985	2.05464
$441$	−5.40377	−0.257322
$442$	−14.1477	−0.672936
$443$	−5.28175	−0.250943	−0.125472	−	0.992097i	$-0.540044\pi$
$443$	−5.28175	−0.250943	−0.125472	+	0.992097i	$0.540044\pi$
$444$	−23.1719	−1.09969
$445$	−47.2967	−2.24208
$446$	49.2889	2.33390
$447$	27.7065	1.31047
$448$	85.6397	4.04610
$449$	7.92365	0.373940	0.186970	−	0.982366i	$-0.440133\pi$
$449$	7.92365	0.373940	0.186970	+	0.982366i	$0.440133\pi$
$450$	18.5729	0.875534
$451$	7.27133	0.342394
$452$	−70.8579	−3.33288
$453$	−13.6071	−0.639318
$454$	−44.3929	−2.08346
$455$	7.66347	0.359269
$456$	12.4018	0.580766
$457$	8.94872	0.418604	0.209302	−	0.977851i	$-0.432881\pi$
$457$	8.94872	0.418604	0.209302	+	0.977851i	$0.432881\pi$
$458$	−30.4243	−1.42164
$459$	−44.0511	−2.05613
$460$	56.0818	2.61483
$461$	29.7252	1.38444	0.692220	−	0.721687i	$-0.256633\pi$
$461$	29.7252	1.38444	0.692220	+	0.721687i	$0.256633\pi$
$462$	18.2189	0.847618
$463$	−9.68901	−0.450287	−0.225143	−	0.974326i	$-0.572285\pi$
$463$	−9.68901	−0.450287	−0.225143	+	0.974326i	$0.572285\pi$
$464$	147.430	6.84429
$465$	25.1077	1.16434
$466$	41.3978	1.91771
$467$	−16.8327	−0.778927	−0.389463	−	0.921042i	$-0.627340\pi$
$467$	−16.8327	−0.778927	−0.389463	+	0.921042i	$0.627340\pi$
$468$	−4.02073	−0.185858
$469$	−13.0969	−0.604758
$470$	−19.3231	−0.891309
$471$	19.8157	0.913061
$472$	3.12576	0.143875
$473$	4.20232	0.193223
$474$	57.0577	2.62075
$475$	−6.09709	−0.279754
$476$	143.147	6.56111
$477$	−4.51940	−0.206929
$478$	−59.1546	−2.70567
$479$	2.75728	0.125983	0.0629917	−	0.998014i	$-0.479936\pi$
$479$	2.75728	0.125983	0.0629917	+	0.998014i	$0.479936\pi$
$480$	−88.6142	−4.04466
$481$	−2.12048	−0.0966857
$482$	9.30492	0.423828
$483$	14.8312	0.674845
$484$	−47.8557	−2.17526
$485$	46.6494	2.11824
$486$	−29.7225	−1.34824
$487$	−26.5164	−1.20157	−0.600786	−	0.799410i	$-0.705146\pi$
$487$	−26.5164	−1.20157	−0.600786	+	0.799410i	$0.705146\pi$
$488$	89.7076	4.06087
$489$	5.57806	0.252249
$490$	−43.3862	−1.95999
$491$	4.25285	0.191928	0.0959641	−	0.995385i	$-0.469407\pi$
$491$	4.25285	0.191928	0.0959641	+	0.995385i	$0.469407\pi$
$492$	−37.2335	−1.67862
$493$	83.0048	3.73835
$494$	1.81410	0.0816202
$495$	5.35084	0.240502
$496$	76.2290	3.42278
$497$	13.2385	0.593826
$498$	−63.1308	−2.82896
$499$	4.16767	0.186570	0.0932852	−	0.995639i	$-0.470263\pi$
$499$	4.16767	0.186570	0.0932852	+	0.995639i	$0.470263\pi$
$500$	19.5227	0.873081
$501$	26.6874	1.19231
$502$	−48.7610	−2.17631
$503$	−26.4724	−1.18035	−0.590173	−	0.807277i	$-0.700941\pi$
$503$	−26.4724	−1.18035	−0.590173	+	0.807277i	$0.700941\pi$
$504$	34.9792	1.55810
$505$	6.84752	0.304711
$506$	12.2009	0.542395
$507$	−17.1907	−0.763468
$508$	−64.8392	−2.87677
$509$	32.0992	1.42277	0.711387	−	0.702801i	$-0.248067\pi$
$509$	32.0992	1.42277	0.711387	+	0.702801i	$0.248067\pi$
$510$	−96.4100	−4.26911
$511$	−24.5011	−1.08387
$512$	−18.1797	−0.803436
$513$	5.64849	0.249387
$514$	−41.2797	−1.82077
$515$	10.0227	0.441655
$516$	−21.5183	−0.947292
$517$	−3.05867	−0.134520
$518$	29.4879	1.29563
$519$	−27.2729	−1.19715
$520$	−20.1955	−0.885633
$521$	0.177512	0.00777694	0.00388847	−	0.999992i	$-0.498762\pi$
$521$	0.177512	0.00777694	0.00388847	+	0.999992i	$0.498762\pi$
$522$	32.4217	1.41906
$523$	11.0657	0.483869	0.241934	−	0.970293i	$-0.422218\pi$
$523$	11.0657	0.483869	0.241934	+	0.970293i	$0.422218\pi$
$524$	−26.6960	−1.16622
$525$	28.6930	1.25227
$526$	−9.81020	−0.427745
$527$	42.9177	1.86952
$528$	−27.1058	−1.17963
$529$	−13.0678	−0.568163
$530$	−36.2857	−1.57615
$531$	0.388076	0.0168410
$532$	−18.3551	−0.795795
$533$	−3.40728	−0.147586
$534$	52.6890	2.28008
$535$	21.0864	0.911647
$536$	34.5142	1.49079
$537$	−13.0389	−0.562670
$538$	77.5469	3.34328
$539$	−6.86764	−0.295810
$540$	−100.515	−4.32548
$541$	−8.18575	−0.351933	−0.175966	−	0.984396i	$-0.556305\pi$
$541$	−8.18575	−0.351933	−0.175966	+	0.984396i	$0.556305\pi$
$542$	−38.6820	−1.66153
$543$	2.99641	0.128588
$544$	−151.472	−6.49431
$545$	3.54025	0.151647
$546$	−8.53719	−0.365358
$547$	−13.9944	−0.598358	−0.299179	−	0.954197i	$-0.596713\pi$
$547$	−13.9944	−0.598358	−0.299179	+	0.954197i	$0.596713\pi$
$548$	−57.6776	−2.46386
$549$	11.1375	0.475339
$550$	23.6042	1.00649
$551$	−10.6434	−0.453423
$552$	−39.0848	−1.66356
$553$	−52.8305	−2.24658
$554$	14.2505	0.605447
$555$	−14.4501	−0.613375
$556$	57.9354	2.45701
$557$	−34.9346	−1.48023	−0.740113	−	0.672482i	$-0.765228\pi$
$557$	−34.9346	−1.48023	−0.740113	+	0.672482i	$0.765228\pi$
$558$	16.7637	0.709662
$559$	−1.96917	−0.0832869
$560$	158.554	6.70011
$561$	−15.2608	−0.644313
$562$	−13.0088	−0.548745
$563$	−15.7617	−0.664277	−0.332138	−	0.943231i	$-0.607770\pi$
$563$	−15.7617	−0.664277	−0.332138	+	0.943231i	$0.607770\pi$
$564$	15.6622	0.659498
$565$	−44.1875	−1.85898
$566$	−60.8808	−2.55901
$567$	−14.9931	−0.629653
$568$	−34.8873	−1.46384
$569$	28.5544	1.19706	0.598532	−	0.801099i	$-0.295751\pi$
$569$	28.5544	1.19706	0.598532	+	0.801099i	$0.295751\pi$
$570$	12.3623	0.517799
$571$	34.2729	1.43427	0.717137	−	0.696932i	$-0.245452\pi$
$571$	34.2729	1.43427	0.717137	+	0.696932i	$0.245452\pi$
$572$	−5.10993	−0.213657
$573$	−17.9751	−0.750919
$574$	47.3824	1.97770
$575$	19.2153	0.801332
$576$	−28.0198	−1.16749
$577$	−35.7089	−1.48658	−0.743290	−	0.668969i	$-0.766736\pi$
$577$	−35.7089	−1.48658	−0.743290	+	0.668969i	$0.766736\pi$
$578$	−118.735	−4.93871
$579$	−27.4665	−1.14147
$580$	189.399	7.86436
$581$	58.4536	2.42506
$582$	−51.9679	−2.15414
$583$	−5.74369	−0.237879
$584$	64.5678	2.67183
$585$	−2.50735	−0.103666
$586$	48.4386	2.00098
$587$	−36.7822	−1.51817	−0.759083	−	0.650994i	$-0.774352\pi$
$587$	−36.7822	−1.51817	−0.759083	+	0.650994i	$0.774352\pi$
$588$	35.1663	1.45024
$589$	−5.50316	−0.226754
$590$	3.11581	0.128276
$591$	29.2066	1.20140
$592$	−43.8718	−1.80312
$593$	27.0618	1.11130	0.555648	−	0.831418i	$-0.312470\pi$
$593$	27.0618	1.11130	0.555648	+	0.831418i	$0.312470\pi$
$594$	−21.8676	−0.897237
$595$	89.2672	3.65960
$596$	108.065	4.42651
$597$	−8.64223	−0.353703
$598$	−5.71722	−0.233794
$599$	6.72813	0.274904	0.137452	−	0.990508i	$-0.456109\pi$
$599$	6.72813	0.274904	0.137452	+	0.990508i	$0.456109\pi$
$600$	−75.6147	−3.08696
$601$	45.7952	1.86802	0.934012	−	0.357242i	$-0.116283\pi$
$601$	45.7952	1.86802	0.934012	+	0.357242i	$0.116283\pi$
$602$	27.3837	1.11608
$603$	4.28507	0.174502
$604$	−53.0725	−2.15949
$605$	−29.8431	−1.21330
$606$	−7.62821	−0.309875
$607$	−5.39332	−0.218908	−0.109454	−	0.993992i	$-0.534910\pi$
$607$	−5.39332	−0.218908	−0.109454	+	0.993992i	$0.534910\pi$
$608$	19.4226	0.787692
$609$	50.0879	2.02967
$610$	89.4219	3.62059
$611$	1.43327	0.0579837
$612$	−46.8350	−1.89319
$613$	−23.5103	−0.949572	−0.474786	−	0.880101i	$-0.657475\pi$
$613$	−23.5103	−0.949572	−0.474786	+	0.880101i	$0.657475\pi$
$614$	24.6134	0.993317
$615$	−23.2191	−0.936283
$616$	44.4550	1.79114
$617$	47.1639	1.89875	0.949373	−	0.314150i	$-0.101719\pi$
$617$	47.1639	1.89875	0.949373	+	0.314150i	$0.101719\pi$
$618$	−11.1654	−0.449140
$619$	6.86560	0.275952	0.137976	−	0.990436i	$-0.455940\pi$
$619$	6.86560	0.275952	0.137976	+	0.990436i	$0.455940\pi$
$620$	97.9287	3.93291
$621$	−17.8015	−0.714349
$622$	−4.38836	−0.175957
$623$	−48.7854	−1.95454
$624$	12.7015	0.508468
$625$	−18.3110	−0.732438
$626$	−21.3121	−0.851804
$627$	1.95684	0.0781485
$628$	77.2882	3.08414
$629$	−24.7002	−0.984863
$630$	34.8678	1.38917
$631$	11.2675	0.448553	0.224277	−	0.974526i	$-0.427998\pi$
$631$	11.2675	0.448553	0.224277	+	0.974526i	$0.427998\pi$
$632$	139.224	5.53803
$633$	1.36959	0.0544363
$634$	63.1584	2.50834
$635$	−40.4342	−1.60458
$636$	29.4111	1.16623
$637$	3.21811	0.127506
$638$	41.2047	1.63131
$639$	−4.33139	−0.171347
$640$	−95.5649	−3.77753
$641$	−36.1922	−1.42951	−0.714753	−	0.699377i	$-0.753461\pi$
$641$	−36.1922	−1.42951	−0.714753	+	0.699377i	$0.753461\pi$
$642$	−23.4905	−0.927097
$643$	−19.6164	−0.773597	−0.386798	−	0.922164i	$-0.626419\pi$
$643$	−19.6164	−0.773597	−0.386798	+	0.922164i	$0.626419\pi$
$644$	57.8470	2.27949
$645$	−13.4190	−0.528372
$646$	21.1314	0.831402
$647$	−20.3164	−0.798720	−0.399360	−	0.916794i	$-0.630768\pi$
$647$	−20.3164	−0.798720	−0.399360	+	0.916794i	$0.630768\pi$
$648$	39.5114	1.55215
$649$	0.493205	0.0193600
$650$	−11.0607	−0.433837
$651$	25.8980	1.01502
$652$	21.7564	0.852045
$653$	6.85726	0.268345	0.134173	−	0.990958i	$-0.457162\pi$
$653$	6.85726	0.268345	0.134173	+	0.990958i	$0.457162\pi$
$654$	−3.94387	−0.154218
$655$	−16.6478	−0.650484
$656$	−70.4949	−2.75236
$657$	8.01634	0.312747
$658$	−19.9313	−0.777004
$659$	26.0613	1.01520	0.507602	−	0.861592i	$-0.330532\pi$
$659$	26.0613	1.01520	0.507602	+	0.861592i	$0.330532\pi$
$660$	−34.8219	−1.35544
$661$	−34.6300	−1.34695	−0.673475	−	0.739210i	$-0.735199\pi$
$661$	−34.6300	−1.34695	−0.673475	+	0.739210i	$0.735199\pi$
$662$	−21.8233	−0.848186
$663$	7.15108	0.277725
$664$	−154.043	−5.97801
$665$	−11.4464	−0.443871
$666$	−9.64793	−0.373850
$667$	33.5431	1.29879
$668$	104.090	4.02737
$669$	−24.9136	−0.963215
$670$	34.4043	1.32915
$671$	14.1547	0.546436
$672$	−91.4033	−3.52596
$673$	−9.87163	−0.380523	−0.190262	−	0.981733i	$-0.560934\pi$
$673$	−9.87163	−0.380523	−0.190262	+	0.981733i	$0.560934\pi$
$674$	−30.2871	−1.16662
$675$	−34.4394	−1.32557
$676$	−67.0499	−2.57884
$677$	21.6896	0.833600	0.416800	−	0.908998i	$-0.363151\pi$
$677$	21.6896	0.833600	0.416800	+	0.908998i	$0.363151\pi$
$678$	49.2253	1.89049
$679$	48.1177	1.84659
$680$	−235.246	−9.02126
$681$	22.4389	0.859859
$682$	21.3049	0.815807
$683$	14.9357	0.571497	0.285749	−	0.958305i	$-0.407758\pi$
$683$	14.9357	0.571497	0.285749	+	0.958305i	$0.407758\pi$
$684$	6.00547	0.229625
$685$	−35.9681	−1.37427
$686$	20.4208	0.779670
$687$	15.3783	0.586719
$688$	−40.7411	−1.55324
$689$	2.69144	0.102536
$690$	−38.9603	−1.48319
$691$	18.1567	0.690713	0.345356	−	0.938472i	$-0.387758\pi$
$691$	18.1567	0.690713	0.345356	+	0.938472i	$0.387758\pi$
$692$	−106.374	−4.04372
$693$	5.51926	0.209659
$694$	36.3505	1.37985
$695$	36.1289	1.37045
$696$	−131.997	−5.00332
$697$	−39.6893	−1.50334
$698$	51.9738	1.96724
$699$	−20.9249	−0.791453
$700$	111.913	4.22990
$701$	25.1088	0.948344	0.474172	−	0.880432i	$-0.342747\pi$
$701$	25.1088	0.948344	0.474172	+	0.880432i	$0.342747\pi$
$702$	10.2469	0.386745
$703$	3.16721	0.119454
$704$	−35.6103	−1.34211
$705$	9.76706	0.367849
$706$	84.3202	3.17343
$707$	7.06305	0.265633
$708$	−2.52550	−0.0949140
$709$	5.31731	0.199696	0.0998478	−	0.995003i	$-0.468164\pi$
$709$	5.31731	0.199696	0.0998478	+	0.995003i	$0.468164\pi$
$710$	−34.7762	−1.30513
$711$	17.2852	0.648245
$712$	128.564	4.81814
$713$	17.3435	0.649518
$714$	−99.4446	−3.72162
$715$	−3.18659	−0.119172
$716$	−50.8562	−1.90059
$717$	29.9003	1.11665
$718$	−88.9282	−3.31877
$719$	−22.7895	−0.849903	−0.424952	−	0.905216i	$-0.639709\pi$
$719$	−22.7895	−0.849903	−0.424952	+	0.905216i	$0.639709\pi$
$720$	−51.8759	−1.93330
$721$	10.3382	0.385015
$722$	−2.70959	−0.100840
$723$	−4.70327	−0.174916
$724$	11.6870	0.434346
$725$	64.8936	2.41009
$726$	33.2456	1.23386
$727$	−28.7802	−1.06740	−0.533698	−	0.845675i	$-0.679198\pi$
$727$	−28.7802	−1.06740	−0.533698	+	0.845675i	$0.679198\pi$
$728$	−20.8312	−0.772056
$729$	28.1138	1.04125
$730$	64.3622	2.38215
$731$	−22.9376	−0.848379
$732$	−72.4803	−2.67895
$733$	44.5819	1.64667	0.823336	−	0.567555i	$-0.192110\pi$
$733$	44.5819	1.64667	0.823336	+	0.567555i	$0.192110\pi$
$734$	44.8899	1.65692
$735$	21.9300	0.808899
$736$	−61.2113	−2.25628
$737$	5.44589	0.200602
$738$	−15.5027	−0.570661
$739$	33.7119	1.24011	0.620055	−	0.784558i	$-0.287110\pi$
$739$	33.7119	1.24011	0.620055	+	0.784558i	$0.287110\pi$
$740$	−56.3606	−2.07186
$741$	−0.916955	−0.0336852
$742$	−37.4278	−1.37402
$743$	34.7178	1.27367	0.636836	−	0.770999i	$-0.280243\pi$
$743$	34.7178	1.27367	0.636836	+	0.770999i	$0.280243\pi$
$744$	−68.2489	−2.50213
$745$	67.3900	2.46898
$746$	−91.5981	−3.35364
$747$	−19.1250	−0.699746
$748$	−59.5226	−2.17636
$749$	21.7502	0.794733
$750$	−13.5625	−0.495233
$751$	−41.5977	−1.51792	−0.758960	−	0.651137i	$-0.774292\pi$
$751$	−41.5977	−1.51792	−0.758960	+	0.651137i	$0.774292\pi$
$752$	29.6536	1.08135
$753$	24.6467	0.898177
$754$	−19.3081	−0.703161
$755$	−33.0964	−1.20450
$756$	−103.679	−3.77076
$757$	20.8310	0.757115	0.378557	−	0.925578i	$-0.376420\pi$
$757$	20.8310	0.757115	0.378557	+	0.925578i	$0.376420\pi$
$758$	−81.4425	−2.95812
$759$	−6.16706	−0.223850
$760$	30.1646	1.09419
$761$	−26.2733	−0.952406	−0.476203	−	0.879335i	$-0.657987\pi$
$761$	−26.2733	−0.952406	−0.476203	+	0.879335i	$0.657987\pi$
$762$	45.0441	1.63178
$763$	3.65168	0.132200
$764$	−70.1090	−2.53645
$765$	−29.2067	−1.05597
$766$	−67.9411	−2.45481
$767$	−0.231111	−0.00834493
$768$	38.1900	1.37806
$769$	−30.6367	−1.10479	−0.552394	−	0.833583i	$-0.686286\pi$
$769$	−30.6367	−1.10479	−0.552394	+	0.833583i	$0.686286\pi$
$770$	44.3135	1.59695
$771$	20.8652	0.751443
$772$	−107.129	−3.85565
$773$	−39.2741	−1.41259	−0.706295	−	0.707918i	$-0.749635\pi$
$773$	−39.2741	−1.41259	−0.706295	+	0.707918i	$0.749635\pi$
$774$	−8.95945	−0.322041
$775$	33.5533	1.20527
$776$	−126.805	−4.55202
$777$	−14.9050	−0.534713
$778$	77.0617	2.76279
$779$	5.08920	0.182340
$780$	16.3172	0.584250
$781$	−5.50476	−0.196976
$782$	−66.5965	−2.38148
$783$	−60.1190	−2.14848
$784$	66.5812	2.37790
$785$	48.1975	1.72024
$786$	18.5458	0.661508
$787$	−53.4648	−1.90582	−0.952908	−	0.303260i	$-0.901925\pi$
$787$	−53.4648	−1.90582	−0.952908	+	0.303260i	$0.901925\pi$
$788$	113.916	4.05808
$789$	4.95866	0.176533
$790$	138.781	4.93759
$791$	−45.5783	−1.62058
$792$	−14.5449	−0.516831
$793$	−6.63275	−0.235536
$794$	51.7454	1.83638
$795$	18.3410	0.650487
$796$	−33.7077	−1.19474
$797$	−47.1286	−1.66938	−0.834690	−	0.550720i	$-0.814353\pi$
$797$	−47.1286	−1.66938	−0.834690	+	0.550720i	$0.814353\pi$
$798$	12.7514	0.451394
$799$	16.6953	0.590636
$800$	−118.422	−4.18683
$801$	15.9617	0.563979
$802$	28.3140	0.999802
$803$	10.1879	0.359525
$804$	−27.8861	−0.983468
$805$	36.0738	1.27143
$806$	−9.98328	−0.351646
$807$	−39.1968	−1.37979
$808$	−18.6133	−0.654812
$809$	−3.60840	−0.126864	−0.0634322	−	0.997986i	$-0.520205\pi$
$809$	−3.60840	−0.126864	−0.0634322	+	0.997986i	$0.520205\pi$
$810$	39.3856	1.38387
$811$	−8.67635	−0.304668	−0.152334	−	0.988329i	$-0.548679\pi$
$811$	−8.67635	−0.304668	−0.152334	+	0.988329i	$0.548679\pi$
$812$	195.360	6.85580
$813$	19.5522	0.685726
$814$	−12.2615	−0.429767
$815$	13.5674	0.475246
$816$	147.953	5.17937
$817$	2.94120	0.102900
$818$	−29.7760	−1.04109
$819$	−2.58627	−0.0903717
$820$	−90.5624	−3.16258
$821$	−39.4667	−1.37740	−0.688699	−	0.725047i	$-0.741818\pi$
$821$	−39.4667	−1.37740	−0.688699	+	0.725047i	$0.741818\pi$
$822$	40.0689	1.39756
$823$	−43.8477	−1.52844	−0.764218	−	0.644958i	$-0.776875\pi$
$823$	−43.8477	−1.52844	−0.764218	+	0.644958i	$0.776875\pi$
$824$	−27.2443	−0.949100
$825$	−11.9310	−0.415384
$826$	3.21388	0.111825
$827$	24.2751	0.844129	0.422064	−	0.906566i	$-0.361306\pi$
$827$	24.2751	0.844129	0.422064	+	0.906566i	$0.361306\pi$
$828$	−18.9265	−0.657742
$829$	14.9336	0.518665	0.259332	−	0.965788i	$-0.416498\pi$
$829$	14.9336	0.518665	0.259332	+	0.965788i	$0.416498\pi$
$830$	−153.552	−5.32987
$831$	−7.20308	−0.249872
$832$	16.6867	0.578506
$833$	37.4858	1.29881
$834$	−40.2480	−1.39367
$835$	64.9114	2.24635
$836$	7.63234	0.263970
$837$	−31.0846	−1.07444
$838$	55.1594	1.90545
$839$	49.8756	1.72190	0.860948	−	0.508693i	$-0.169871\pi$
$839$	49.8756	1.72190	0.860948	+	0.508693i	$0.169871\pi$
$840$	−141.955	−4.89792
$841$	84.2814	2.90625
$842$	−23.2694	−0.801918
$843$	6.57545	0.226470
$844$	5.34187	0.183875
$845$	−41.8128	−1.43840
$846$	6.52117	0.224203
$847$	−30.7825	−1.05770
$848$	55.6846	1.91222
$849$	30.7728	1.05612
$850$	−128.840	−4.41917
$851$	−9.98162	−0.342166
$852$	28.1876	0.965691
$853$	−15.4554	−0.529183	−0.264592	−	0.964361i	$-0.585237\pi$
$853$	−15.4554	−0.529183	−0.264592	+	0.964361i	$0.585237\pi$
$854$	92.2365	3.15627
$855$	3.74505	0.128078
$856$	−57.3182	−1.95909
$857$	20.8480	0.712153	0.356077	−	0.934457i	$-0.384114\pi$
$857$	20.8480	0.712153	0.356077	+	0.934457i	$0.384114\pi$
$858$	3.54989	0.121191
$859$	−44.9691	−1.53432	−0.767162	−	0.641453i	$-0.778332\pi$
$859$	−44.9691	−1.53432	−0.767162	+	0.641453i	$0.778332\pi$
$860$	−52.3387	−1.78473
$861$	−23.9499	−0.816210
$862$	−22.2739	−0.758652
$863$	6.54894	0.222929	0.111464	−	0.993768i	$-0.464446\pi$
$863$	6.54894	0.222929	0.111464	+	0.993768i	$0.464446\pi$
$864$	109.709	3.73236
$865$	−66.3354	−2.25547
$866$	−30.8246	−1.04746
$867$	60.0157	2.03824
$868$	101.011	3.42854
$869$	21.9677	0.745204
$870$	−131.576	−4.46085
$871$	−2.55189	−0.0864675
$872$	−9.62326	−0.325885
$873$	−15.7433	−0.532829
$874$	8.53940	0.288849
$875$	12.5577	0.424527
$876$	−52.1683	−1.76260
$877$	16.4219	0.554529	0.277264	−	0.960794i	$-0.410572\pi$
$877$	16.4219	0.554529	0.277264	+	0.960794i	$0.410572\pi$
$878$	52.6771	1.77777
$879$	−24.4838	−0.825817
$880$	−65.9290	−2.22247
$881$	15.1594	0.510733	0.255367	−	0.966844i	$-0.417804\pi$
$881$	15.1594	0.510733	0.255367	+	0.966844i	$0.417804\pi$
$882$	14.6420	0.493021
$883$	4.50288	0.151534	0.0757671	−	0.997126i	$-0.475859\pi$
$883$	4.50288	0.151534	0.0757671	+	0.997126i	$0.475859\pi$
$884$	27.8917	0.938100
$885$	−1.57492	−0.0529403
$886$	14.3114	0.480800
$887$	−14.3897	−0.483160	−0.241580	−	0.970381i	$-0.577666\pi$
$887$	−14.3897	−0.483160	−0.241580	+	0.970381i	$0.577666\pi$
$888$	39.2791	1.31812
$889$	−41.7069	−1.39880
$890$	128.155	4.29575
$891$	6.23438	0.208860
$892$	−97.1716	−3.25355
$893$	−2.14077	−0.0716380
$894$	−75.0731	−2.51082
$895$	−31.7143	−1.06009
$896$	−98.5729	−3.29309
$897$	2.88983	0.0964885
$898$	−21.4698	−0.716457
$899$	58.5722	1.95349
$900$	−36.6159	−1.22053
$901$	31.3510	1.04445
$902$	−19.7023	−0.656016
$903$	−13.8414	−0.460612
$904$	120.113	3.99488
$905$	7.28812	0.242265
$906$	36.8697	1.22491
$907$	−7.26274	−0.241155	−0.120578	−	0.992704i	$-0.538475\pi$
$907$	−7.26274	−0.241155	−0.120578	+	0.992704i	$0.538475\pi$
$908$	87.5193	2.90443
$909$	−2.31091	−0.0766479
$910$	−20.7649	−0.688349
$911$	30.9906	1.02676	0.513382	−	0.858160i	$-0.328392\pi$
$911$	30.9906	1.02676	0.513382	+	0.858160i	$0.328392\pi$
$912$	−18.9714	−0.628204
$913$	−24.3059	−0.804408
$914$	−24.2474	−0.802032
$915$	−45.1992	−1.49424
$916$	59.9807	1.98182
$917$	−17.1718	−0.567063
$918$	119.360	3.93948
$919$	2.97483	0.0981307	0.0490653	−	0.998796i	$-0.484376\pi$
$919$	2.97483	0.0981307	0.0490653	+	0.998796i	$0.484376\pi$
$920$	−95.0652	−3.13421
$921$	−12.4411	−0.409948
$922$	−80.5430	−2.65254
$923$	2.57948	0.0849045
$924$	−35.9179	−1.18161
$925$	−19.3108	−0.634935
$926$	26.2532	0.862735
$927$	−3.38248	−0.111095
$928$	−206.722	−6.78599
$929$	−23.2691	−0.763433	−0.381717	−	0.924279i	$-0.624667\pi$
$929$	−23.2691	−0.763433	−0.381717	+	0.924279i	$0.624667\pi$
$930$	−68.0316	−2.23084
$931$	−4.80666	−0.157532
$932$	−81.6144	−2.67337
$933$	2.21814	0.0726187
$934$	45.6098	1.49240
$935$	−37.1187	−1.21391
$936$	6.81560	0.222775
$937$	17.5971	0.574873	0.287437	−	0.957800i	$-0.407197\pi$
$937$	17.5971	0.574873	0.287437	+	0.957800i	$0.407197\pi$
$938$	35.4872	1.15870
$939$	10.7724	0.351545
$940$	38.0949	1.24252
$941$	23.7796	0.775192	0.387596	−	0.921829i	$-0.373306\pi$
$941$	23.7796	0.775192	0.387596	+	0.921829i	$0.373306\pi$
$942$	−53.6925	−1.74940
$943$	−16.0389	−0.522297
$944$	−4.78157	−0.155627
$945$	−64.6548	−2.10322
$946$	−11.3866	−0.370209
$947$	15.9343	0.517794	0.258897	−	0.965905i	$-0.416641\pi$
$947$	15.9343	0.517794	0.258897	+	0.965905i	$0.416641\pi$
$948$	−112.488	−3.65343
$949$	−4.77398	−0.154970
$950$	16.5206	0.535999
$951$	−31.9241	−1.03521
$952$	−242.650	−7.86434
$953$	−22.4943	−0.728662	−0.364331	−	0.931270i	$-0.618702\pi$
$953$	−22.4943	−0.728662	−0.364331	+	0.931270i	$0.618702\pi$
$954$	12.2457	0.396469
$955$	−43.7205	−1.41476
$956$	116.621	3.77181
$957$	−20.8273	−0.673252
$958$	−7.47110	−0.241380
$959$	−37.1003	−1.19803
$960$	113.712	3.67004
$961$	−0.715232	−0.0230720
$962$	5.74564	0.185247
$963$	−7.11627	−0.229318
$964$	−18.3444	−0.590833
$965$	−66.8062	−2.15057
$966$	−40.1866	−1.29298
$967$	7.53904	0.242439	0.121220	−	0.992626i	$-0.461319\pi$
$967$	7.53904	0.242439	0.121220	+	0.992626i	$0.461319\pi$
$968$	81.1210	2.60733
$969$	−10.6811	−0.343125
$970$	−126.401	−4.05848
$971$	−28.0659	−0.900679	−0.450339	−	0.892857i	$-0.648697\pi$
$971$	−28.0659	−0.900679	−0.450339	+	0.892857i	$0.648697\pi$
$972$	58.5969	1.87950
$973$	37.2661	1.19470
$974$	71.8485	2.30218
$975$	5.59076	0.179047
$976$	−137.228	−4.39258
$977$	4.03055	0.128949	0.0644744	−	0.997919i	$-0.479463\pi$
$977$	4.03055	0.128949	0.0644744	+	0.997919i	$0.479463\pi$
$978$	−15.1143	−0.483300
$979$	20.2857	0.648334
$980$	85.5345	2.73230
$981$	−1.19476	−0.0381459
$982$	−11.5235	−0.367729
$983$	37.7367	1.20361	0.601807	−	0.798641i	$-0.294448\pi$
$983$	37.7367	1.20361	0.601807	+	0.798641i	$0.294448\pi$
$984$	63.1151	2.01204
$985$	71.0386	2.26348
$986$	−224.909	−7.16256
$987$	10.0745	0.320674
$988$	−3.57644	−0.113782
$989$	−9.26934	−0.294748
$990$	−14.4986	−0.460795
$991$	−46.1025	−1.46449	−0.732247	−	0.681039i	$-0.761528\pi$
$991$	−46.1025	−1.46449	−0.732247	+	0.681039i	$0.761528\pi$
$992$	−106.886	−3.39363
$993$	11.0308	0.350052
$994$	−35.8708	−1.13775
$995$	−21.0204	−0.666390
$996$	124.460	3.94368
$997$	−38.4050	−1.21630	−0.608150	−	0.793822i	$-0.708088\pi$
$997$	−38.4050	−1.21630	−0.608150	+	0.793822i	$0.708088\pi$
$998$	−11.2927	−0.357463
$999$	17.8900	0.566014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.2	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.2	✓	83	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)

\(f(q)\)	\(=\)	\(q-2.70959 q^{2} +1.36959 q^{3} +5.34187 q^{4} +3.33123 q^{5} -3.71102 q^{6} +3.43608 q^{7} -9.05510 q^{8} -1.12423 q^{9} +O(q^{10})\)	\(q-2.70959 q^{2} +1.36959 q^{3} +5.34187 q^{4} +3.33123 q^{5} -3.71102 q^{6} +3.43608 q^{7} -9.05510 q^{8} -1.12423 q^{9} -9.02626 q^{10} -1.42878 q^{11} +7.31617 q^{12} +0.669511 q^{13} -9.31037 q^{14} +4.56242 q^{15} +13.8519 q^{16} +7.79873 q^{17} +3.04619 q^{18} -1.00000 q^{19} +17.7950 q^{20} +4.70602 q^{21} +3.87140 q^{22} +3.15155 q^{23} -12.4018 q^{24} +6.09709 q^{25} -1.81410 q^{26} -5.64849 q^{27} +18.3551 q^{28} +10.6434 q^{29} -12.3623 q^{30} +5.50316 q^{31} -19.4226 q^{32} -1.95684 q^{33} -21.1314 q^{34} +11.4464 q^{35} -6.00547 q^{36} -3.16721 q^{37} +2.70959 q^{38} +0.916955 q^{39} -30.1646 q^{40} -5.08920 q^{41} -12.7514 q^{42} -2.94120 q^{43} -7.63234 q^{44} -3.74505 q^{45} -8.53940 q^{46} +2.14077 q^{47} +18.9714 q^{48} +4.80666 q^{49} -16.5206 q^{50} +10.6811 q^{51} +3.57644 q^{52} +4.02001 q^{53} +15.3051 q^{54} -4.75958 q^{55} -31.1141 q^{56} -1.36959 q^{57} -28.8392 q^{58} -0.345194 q^{59} +24.3718 q^{60} -9.90686 q^{61} -14.9113 q^{62} -3.86293 q^{63} +24.9237 q^{64} +2.23029 q^{65} +5.30222 q^{66} -3.81158 q^{67} +41.6598 q^{68} +4.31632 q^{69} -31.0150 q^{70} +3.85278 q^{71} +10.1800 q^{72} -7.13054 q^{73} +8.58185 q^{74} +8.35051 q^{75} -5.34187 q^{76} -4.90939 q^{77} -2.48457 q^{78} -15.3752 q^{79} +46.1437 q^{80} -4.36344 q^{81} +13.7897 q^{82} +17.0117 q^{83} +25.1390 q^{84} +25.9794 q^{85} +7.96945 q^{86} +14.5770 q^{87} +12.9377 q^{88} -14.1980 q^{89} +10.1476 q^{90} +2.30049 q^{91} +16.8352 q^{92} +7.53707 q^{93} -5.80059 q^{94} -3.33123 q^{95} -26.6010 q^{96} +14.0037 q^{97} -13.0241 q^{98} +1.60627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

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