LMFDB - Embedded newform 4009.2.a.e.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.27080 q^{2} -1.94834 q^{3} +3.15654 q^{4} +0.0772315 q^{5} +4.42430 q^{6} +2.91283 q^{7} -2.62627 q^{8} +0.796044 q^{9} +O(q^{10})$	\(q-2.27080 q^{2} -1.94834 q^{3} +3.15654 q^{4} +0.0772315 q^{5} +4.42430 q^{6} +2.91283 q^{7} -2.62627 q^{8} +0.796044 q^{9} -0.175377 q^{10} +5.63887 q^{11} -6.15003 q^{12} +0.905397 q^{13} -6.61446 q^{14} -0.150473 q^{15} -0.349335 q^{16} -4.24278 q^{17} -1.80766 q^{18} +1.00000 q^{19} +0.243784 q^{20} -5.67520 q^{21} -12.8047 q^{22} -4.07900 q^{23} +5.11688 q^{24} -4.99404 q^{25} -2.05598 q^{26} +4.29406 q^{27} +9.19447 q^{28} +7.65947 q^{29} +0.341695 q^{30} +0.845113 q^{31} +6.04582 q^{32} -10.9865 q^{33} +9.63451 q^{34} +0.224962 q^{35} +2.51275 q^{36} +9.64499 q^{37} -2.27080 q^{38} -1.76402 q^{39} -0.202831 q^{40} +4.95608 q^{41} +12.8873 q^{42} -7.04914 q^{43} +17.7993 q^{44} +0.0614797 q^{45} +9.26261 q^{46} +6.56056 q^{47} +0.680625 q^{48} +1.48459 q^{49} +11.3405 q^{50} +8.26639 q^{51} +2.85792 q^{52} +0.870056 q^{53} -9.75097 q^{54} +0.435498 q^{55} -7.64989 q^{56} -1.94834 q^{57} -17.3931 q^{58} +6.77985 q^{59} -0.474976 q^{60} +10.8522 q^{61} -1.91908 q^{62} +2.31874 q^{63} -13.0302 q^{64} +0.0699251 q^{65} +24.9481 q^{66} +0.548963 q^{67} -13.3925 q^{68} +7.94730 q^{69} -0.510845 q^{70} +5.15243 q^{71} -2.09063 q^{72} -5.31462 q^{73} -21.9019 q^{74} +9.73010 q^{75} +3.15654 q^{76} +16.4251 q^{77} +4.00575 q^{78} -14.4592 q^{79} -0.0269797 q^{80} -10.7544 q^{81} -11.2543 q^{82} -5.28507 q^{83} -17.9140 q^{84} -0.327676 q^{85} +16.0072 q^{86} -14.9233 q^{87} -14.8092 q^{88} -3.18174 q^{89} -0.139608 q^{90} +2.63727 q^{91} -12.8755 q^{92} -1.64657 q^{93} -14.8977 q^{94} +0.0772315 q^{95} -11.7793 q^{96} +10.1328 q^{97} -3.37121 q^{98} +4.48879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.27080	−1.60570	−0.802850	−	0.596182i	$-0.796684\pi$
$2$	−2.27080	−1.60570	−0.802850	+	0.596182i	$0.796684\pi$
$3$	−1.94834	−1.12488	−0.562438	−	0.826839i	$-0.690137\pi$
$3$	−1.94834	−1.12488	−0.562438	+	0.826839i	$0.690137\pi$
$4$	3.15654	1.57827
$5$	0.0772315	0.0345390	0.0172695	−	0.999851i	$-0.494503\pi$
$5$	0.0772315	0.0345390	0.0172695	+	0.999851i	$0.494503\pi$
$6$	4.42430	1.80621
$7$	2.91283	1.10095	0.550474	−	0.834853i	$-0.314447\pi$
$7$	2.91283	1.10095	0.550474	+	0.834853i	$0.314447\pi$
$8$	−2.62627	−0.928528
$9$	0.796044	0.265348
$10$	−0.175377	−0.0554592
$11$	5.63887	1.70018	0.850091	−	0.526636i	$-0.176547\pi$
$11$	5.63887	1.70018	0.850091	+	0.526636i	$0.176547\pi$
$12$	−6.15003	−1.77536
$13$	0.905397	0.251112	0.125556	−	0.992087i	$-0.459929\pi$
$13$	0.905397	0.251112	0.125556	+	0.992087i	$0.459929\pi$
$14$	−6.61446	−1.76779
$15$	−0.150473	−0.0388521
$16$	−0.349335	−0.0873338
$17$	−4.24278	−1.02902	−0.514512	−	0.857483i	$-0.672027\pi$
$17$	−4.24278	−1.02902	−0.514512	+	0.857483i	$0.672027\pi$
$18$	−1.80766	−0.426069
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.243784	0.0545118
$21$	−5.67520	−1.23843
$22$	−12.8047	−2.72998
$23$	−4.07900	−0.850531	−0.425265	−	0.905069i	$-0.639819\pi$
$23$	−4.07900	−0.850531	−0.425265	+	0.905069i	$0.639819\pi$
$24$	5.11688	1.04448
$25$	−4.99404	−0.998807
$26$	−2.05598	−0.403210
$27$	4.29406	0.826393
$28$	9.19447	1.73759
$29$	7.65947	1.42233	0.711164	−	0.703026i	$-0.248168\pi$
$29$	7.65947	1.42233	0.711164	+	0.703026i	$0.248168\pi$
$30$	0.341695	0.0623848
$31$	0.845113	0.151787	0.0758934	−	0.997116i	$-0.475819\pi$
$31$	0.845113	0.151787	0.0758934	+	0.997116i	$0.475819\pi$
$32$	6.04582	1.06876
$33$	−10.9865	−1.91250
$34$	9.63451	1.65230
$35$	0.224962	0.0380256
$36$	2.51275	0.418791
$37$	9.64499	1.58563	0.792814	−	0.609464i	$-0.208615\pi$
$37$	9.64499	1.58563	0.792814	+	0.609464i	$0.208615\pi$
$38$	−2.27080	−0.368373
$39$	−1.76402	−0.282470
$40$	−0.202831	−0.0320704
$41$	4.95608	0.774009	0.387005	−	0.922078i	$-0.373510\pi$
$41$	4.95608	0.774009	0.387005	+	0.922078i	$0.373510\pi$
$42$	12.8873	1.98855
$43$	−7.04914	−1.07498	−0.537492	−	0.843269i	$-0.680628\pi$
$43$	−7.04914	−1.07498	−0.537492	+	0.843269i	$0.680628\pi$
$44$	17.7993	2.68335
$45$	0.0614797	0.00916485
$46$	9.26261	1.36570
$47$	6.56056	0.956956	0.478478	−	0.878099i	$-0.341189\pi$
$47$	6.56056	0.956956	0.478478	+	0.878099i	$0.341189\pi$
$48$	0.680625	0.0982397
$49$	1.48459	0.212085
$50$	11.3405	1.60378
$51$	8.26639	1.15753
$52$	2.85792	0.396322
$53$	0.870056	0.119511	0.0597557	−	0.998213i	$-0.480968\pi$
$53$	0.870056	0.119511	0.0597557	+	0.998213i	$0.480968\pi$
$54$	−9.75097	−1.32694
$55$	0.435498	0.0587225
$56$	−7.64989	−1.02226
$57$	−1.94834	−0.258064
$58$	−17.3931	−2.28383
$59$	6.77985	0.882661	0.441330	−	0.897345i	$-0.354507\pi$
$59$	6.77985	0.882661	0.441330	+	0.897345i	$0.354507\pi$
$60$	−0.474976	−0.0613191
$61$	10.8522	1.38948	0.694738	−	0.719263i	$-0.255520\pi$
$61$	10.8522	1.38948	0.694738	+	0.719263i	$0.255520\pi$
$62$	−1.91908	−0.243724
$63$	2.31874	0.292134
$64$	−13.0302	−1.62877
$65$	0.0699251	0.00867314
$66$	24.9481	3.07089
$67$	0.548963	0.0670665	0.0335333	−	0.999438i	$-0.489324\pi$
$67$	0.548963	0.0670665	0.0335333	+	0.999438i	$0.489324\pi$
$68$	−13.3925	−1.62408
$69$	7.94730	0.956743
$70$	−0.510845	−0.0610576
$71$	5.15243	0.611481	0.305740	−	0.952115i	$-0.401096\pi$
$71$	5.15243	0.611481	0.305740	+	0.952115i	$0.401096\pi$
$72$	−2.09063	−0.246383
$73$	−5.31462	−0.622029	−0.311014	−	0.950405i	$-0.600669\pi$
$73$	−5.31462	−0.622029	−0.311014	+	0.950405i	$0.600669\pi$
$74$	−21.9019	−2.54604
$75$	9.73010	1.12353
$76$	3.15654	0.362080
$77$	16.4251	1.87181
$78$	4.00575	0.453562
$79$	−14.4592	−1.62679	−0.813394	−	0.581714i	$-0.802382\pi$
$79$	−14.4592	−1.62679	−0.813394	+	0.581714i	$0.802382\pi$
$80$	−0.0269797	−0.00301642
$81$	−10.7544	−1.19494
$82$	−11.2543	−1.24283
$83$	−5.28507	−0.580112	−0.290056	−	0.957010i	$-0.593674\pi$
$83$	−5.28507	−0.580112	−0.290056	+	0.957010i	$0.593674\pi$
$84$	−17.9140	−1.95458
$85$	−0.327676	−0.0355415
$86$	16.0072	1.72610
$87$	−14.9233	−1.59994
$88$	−14.8092	−1.57867
$89$	−3.18174	−0.337264	−0.168632	−	0.985679i	$-0.553935\pi$
$89$	−3.18174	−0.337264	−0.168632	+	0.985679i	$0.553935\pi$
$90$	−0.139608	−0.0147160
$91$	2.63727	0.276461
$92$	−12.8755	−1.34237
$93$	−1.64657	−0.170741
$94$	−14.8977	−1.53658
$95$	0.0772315	0.00792378
$96$	−11.7793	−1.20222
$97$	10.1328	1.02883	0.514417	−	0.857540i	$-0.328008\pi$
$97$	10.1328	1.02883	0.514417	+	0.857540i	$0.328008\pi$
$98$	−3.37121	−0.340544
$99$	4.48879	0.451140
$100$	−15.7639	−1.57639
$101$	15.7748	1.56965	0.784825	−	0.619718i	$-0.212753\pi$
$101$	15.7748	1.56965	0.784825	+	0.619718i	$0.212753\pi$
$102$	−18.7713	−1.85864
$103$	−10.1863	−1.00368	−0.501841	−	0.864960i	$-0.667344\pi$
$103$	−10.1863	−1.00368	−0.501841	+	0.864960i	$0.667344\pi$
$104$	−2.37782	−0.233164
$105$	−0.438304	−0.0427741
$106$	−1.97572	−0.191899
$107$	14.5194	1.40365	0.701823	−	0.712351i	$-0.252370\pi$
$107$	14.5194	1.40365	0.701823	+	0.712351i	$0.252370\pi$
$108$	13.5544	1.30427
$109$	9.40234	0.900580	0.450290	−	0.892882i	$-0.351321\pi$
$109$	9.40234	0.900580	0.450290	+	0.892882i	$0.351321\pi$
$110$	−0.988929	−0.0942907
$111$	−18.7918	−1.78364
$112$	−1.01755	−0.0961498
$113$	−10.2798	−0.967041	−0.483521	−	0.875333i	$-0.660642\pi$
$113$	−10.2798	−0.967041	−0.483521	+	0.875333i	$0.660642\pi$
$114$	4.42430	0.414374
$115$	−0.315027	−0.0293765
$116$	24.1774	2.24482
$117$	0.720736	0.0666320
$118$	−15.3957	−1.41729
$119$	−12.3585	−1.13290
$120$	0.395184	0.0360752
$121$	20.7968	1.89062
$122$	−24.6431	−2.23108
$123$	−9.65614	−0.870665
$124$	2.66763	0.239561
$125$	−0.771854	−0.0690367
$126$	−5.26541	−0.469080
$127$	−17.5299	−1.55553	−0.777763	−	0.628557i	$-0.783646\pi$
$127$	−17.5299	−1.55553	−0.777763	+	0.628557i	$0.783646\pi$
$128$	17.4973	1.54656
$129$	13.7342	1.20923
$130$	−0.158786	−0.0139265
$131$	−9.52241	−0.831977	−0.415989	−	0.909370i	$-0.636564\pi$
$131$	−9.52241	−0.831977	−0.415989	+	0.909370i	$0.636564\pi$
$132$	−34.6792	−3.01843
$133$	2.91283	0.252575
$134$	−1.24659	−0.107689
$135$	0.331637	0.0285428
$136$	11.1427	0.955478
$137$	−7.80931	−0.667194	−0.333597	−	0.942716i	$-0.608262\pi$
$137$	−7.80931	−0.667194	−0.333597	+	0.942716i	$0.608262\pi$
$138$	−18.0467	−1.53624
$139$	−9.90585	−0.840203	−0.420101	−	0.907477i	$-0.638006\pi$
$139$	−9.90585	−0.840203	−0.420101	+	0.907477i	$0.638006\pi$
$140$	0.710103	0.0600146
$141$	−12.7822	−1.07646
$142$	−11.7001	−0.981855
$143$	5.10541	0.426936
$144$	−0.278086	−0.0231738
$145$	0.591552	0.0491258
$146$	12.0684	0.998791
$147$	−2.89250	−0.238569
$148$	30.4448	2.50255
$149$	8.52986	0.698793	0.349397	−	0.936975i	$-0.386387\pi$
$149$	8.52986	0.698793	0.349397	+	0.936975i	$0.386387\pi$
$150$	−22.0951	−1.80406
$151$	7.04588	0.573385	0.286693	−	0.958023i	$-0.407444\pi$
$151$	7.04588	0.573385	0.286693	+	0.958023i	$0.407444\pi$
$152$	−2.62627	−0.213019
$153$	−3.37744	−0.273050
$154$	−37.2981	−3.00556
$155$	0.0652693	0.00524256
$156$	−5.56821	−0.445814
$157$	3.50443	0.279684	0.139842	−	0.990174i	$-0.455341\pi$
$157$	3.50443	0.279684	0.139842	+	0.990174i	$0.455341\pi$
$158$	32.8340	2.61213
$159$	−1.69517	−0.134436
$160$	0.466927	0.0369138
$161$	−11.8815	−0.936390
$162$	24.4212	1.91871
$163$	17.4763	1.36885	0.684424	−	0.729084i	$-0.260054\pi$
$163$	17.4763	1.36885	0.684424	+	0.729084i	$0.260054\pi$
$164$	15.6441	1.22160
$165$	−0.848500	−0.0660556
$166$	12.0013	0.931485
$167$	4.43978	0.343560	0.171780	−	0.985135i	$-0.445048\pi$
$167$	4.43978	0.343560	0.171780	+	0.985135i	$0.445048\pi$
$168$	14.9046	1.14992
$169$	−12.1803	−0.936943
$170$	0.744087	0.0570689
$171$	0.796044	0.0608750
$172$	−22.2509	−1.69662
$173$	0.528930	0.0402138	0.0201069	−	0.999798i	$-0.493599\pi$
$173$	0.528930	0.0402138	0.0201069	+	0.999798i	$0.493599\pi$
$174$	33.8878	2.56903
$175$	−14.5468	−1.09963
$176$	−1.96985	−0.148483
$177$	−13.2095	−0.992885
$178$	7.22510	0.541544
$179$	20.1389	1.50525	0.752627	−	0.658448i	$-0.228787\pi$
$179$	20.1389	1.50525	0.752627	+	0.658448i	$0.228787\pi$
$180$	0.194063	0.0144646
$181$	−9.47923	−0.704586	−0.352293	−	0.935890i	$-0.614598\pi$
$181$	−9.47923	−0.704586	−0.352293	+	0.935890i	$0.614598\pi$
$182$	−5.98871	−0.443913
$183$	−21.1437	−1.56299
$184$	10.7126	0.789742
$185$	0.744897	0.0547659
$186$	3.73904	0.274159
$187$	−23.9245	−1.74953
$188$	20.7087	1.51034
$189$	12.5079	0.909815
$190$	−0.175377	−0.0127232
$191$	0.601848	0.0435482	0.0217741	−	0.999763i	$-0.493069\pi$
$191$	0.601848	0.0435482	0.0217741	+	0.999763i	$0.493069\pi$
$192$	25.3873	1.83217
$193$	20.9751	1.50982	0.754909	−	0.655830i	$-0.227681\pi$
$193$	20.9751	1.50982	0.754909	+	0.655830i	$0.227681\pi$
$194$	−23.0097	−1.65200
$195$	−0.136238	−0.00975622
$196$	4.68617	0.334727
$197$	3.18455	0.226890	0.113445	−	0.993544i	$-0.463811\pi$
$197$	3.18455	0.226890	0.113445	+	0.993544i	$0.463811\pi$
$198$	−10.1931	−0.724395
$199$	−19.1182	−1.35525	−0.677626	−	0.735407i	$-0.736991\pi$
$199$	−19.1182	−1.35525	−0.677626	+	0.735407i	$0.736991\pi$
$200$	13.1157	0.927420
$201$	−1.06957	−0.0754416
$202$	−35.8214	−2.52039
$203$	22.3108	1.56591
$204$	26.0932	1.82689
$205$	0.382765	0.0267335
$206$	23.1310	1.61161
$207$	−3.24707	−0.225687
$208$	−0.316287	−0.0219305
$209$	5.63887	0.390049
$210$	0.995301	0.0686823
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	2.74637	0.188621
$213$	−10.0387	−0.687841
$214$	−32.9707	−2.25383
$215$	−0.544416	−0.0371288
$216$	−11.2774	−0.767329
$217$	2.46167	0.167109
$218$	−21.3508	−1.44606
$219$	10.3547	0.699706
$220$	1.37467	0.0926800
$221$	−3.84140	−0.258400
$222$	42.6724	2.86398
$223$	−22.9755	−1.53855	−0.769275	−	0.638918i	$-0.779382\pi$
$223$	−22.9755	−1.53855	−0.769275	+	0.638918i	$0.779382\pi$
$224$	17.6104	1.17665
$225$	−3.97547	−0.265031
$226$	23.3434	1.55278
$227$	−12.0670	−0.800918	−0.400459	−	0.916315i	$-0.631149\pi$
$227$	−12.0670	−0.800918	−0.400459	+	0.916315i	$0.631149\pi$
$228$	−6.15003	−0.407295
$229$	−17.8572	−1.18004	−0.590019	−	0.807389i	$-0.700880\pi$
$229$	−17.8572	−1.18004	−0.590019	+	0.807389i	$0.700880\pi$
$230$	0.715365	0.0471698
$231$	−32.0017	−2.10556
$232$	−20.1159	−1.32067
$233$	−17.4570	−1.14364	−0.571822	−	0.820378i	$-0.693763\pi$
$233$	−17.4570	−1.14364	−0.571822	+	0.820378i	$0.693763\pi$
$234$	−1.63665	−0.106991
$235$	0.506682	0.0330523
$236$	21.4009	1.39308
$237$	28.1715	1.82994
$238$	28.0637	1.81910
$239$	−7.32819	−0.474021	−0.237011	−	0.971507i	$-0.576168\pi$
$239$	−7.32819	−0.474021	−0.237011	+	0.971507i	$0.576168\pi$
$240$	0.0525657	0.00339310
$241$	6.18460	0.398385	0.199192	−	0.979960i	$-0.436168\pi$
$241$	6.18460	0.398385	0.199192	+	0.979960i	$0.436168\pi$
$242$	−47.2254	−3.03577
$243$	8.07117	0.517766
$244$	34.2553	2.19297
$245$	0.114657	0.00732518
$246$	21.9272	1.39803
$247$	0.905397	0.0576090
$248$	−2.21950	−0.140938
$249$	10.2971	0.652554
$250$	1.75273	0.110852
$251$	27.4610	1.73332	0.866660	−	0.498899i	$-0.166262\pi$
$251$	27.4610	1.73332	0.866660	+	0.498899i	$0.166262\pi$
$252$	7.31920	0.461067
$253$	−23.0010	−1.44606
$254$	39.8069	2.49771
$255$	0.638426	0.0399798
$256$	−13.6726	−0.854537
$257$	−23.2583	−1.45082	−0.725408	−	0.688320i	$-0.758349\pi$
$257$	−23.2583	−1.45082	−0.725408	+	0.688320i	$0.758349\pi$
$258$	−31.1875	−1.94165
$259$	28.0943	1.74569
$260$	0.220721	0.0136886
$261$	6.09728	0.377412
$262$	21.6235	1.33590
$263$	19.4244	1.19776	0.598881	−	0.800838i	$-0.295612\pi$
$263$	19.4244	1.19776	0.598881	+	0.800838i	$0.295612\pi$
$264$	28.8534	1.77581
$265$	0.0671957	0.00412780
$266$	−6.61446	−0.405559
$267$	6.19913	0.379380
$268$	1.73282	0.105849
$269$	11.4787	0.699871	0.349935	−	0.936774i	$-0.386204\pi$
$269$	11.4787	0.699871	0.349935	+	0.936774i	$0.386204\pi$
$270$	−0.753082	−0.0458311
$271$	15.4280	0.937182	0.468591	−	0.883415i	$-0.344762\pi$
$271$	15.4280	0.937182	0.468591	+	0.883415i	$0.344762\pi$
$272$	1.48215	0.0898686
$273$	−5.13831	−0.310984
$274$	17.7334	1.07131
$275$	−28.1607	−1.69815
$276$	25.0860	1.51000
$277$	−4.19261	−0.251910	−0.125955	−	0.992036i	$-0.540199\pi$
$277$	−4.19261	−0.251910	−0.125955	+	0.992036i	$0.540199\pi$
$278$	22.4942	1.34911
$279$	0.672747	0.0402763
$280$	−0.590812	−0.0353078
$281$	−5.42627	−0.323704	−0.161852	−	0.986815i	$-0.551747\pi$
$281$	−5.42627	−0.323704	−0.161852	+	0.986815i	$0.551747\pi$
$282$	29.0259	1.72847
$283$	16.2217	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$283$	16.2217	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$284$	16.2639	0.965082
$285$	−0.150473	−0.00891328
$286$	−11.5934	−0.685531
$287$	14.4362	0.852143
$288$	4.81274	0.283593
$289$	1.00116	0.0588920
$290$	−1.34330	−0.0788812
$291$	−19.7423	−1.15731
$292$	−16.7758	−0.981729
$293$	−22.4665	−1.31250	−0.656252	−	0.754542i	$-0.727859\pi$
$293$	−22.4665	−1.31250	−0.656252	+	0.754542i	$0.727859\pi$
$294$	6.56828	0.383070
$295$	0.523618	0.0304862
$296$	−25.3304	−1.47230
$297$	24.2137	1.40502
$298$	−19.3696	−1.12205
$299$	−3.69312	−0.213578
$300$	30.7134	1.77324
$301$	−20.5330	−1.18350
$302$	−15.9998	−0.920684
$303$	−30.7347	−1.76566
$304$	−0.349335	−0.0200357
$305$	0.838128	0.0479911
$306$	7.66949	0.438436
$307$	14.8576	0.847966	0.423983	−	0.905670i	$-0.360632\pi$
$307$	14.8576	0.847966	0.423983	+	0.905670i	$0.360632\pi$
$308$	51.8464	2.95422
$309$	19.8463	1.12902
$310$	−0.148214	−0.00841797
$311$	27.8058	1.57672	0.788361	−	0.615212i	$-0.210930\pi$
$311$	27.8058	1.57672	0.788361	+	0.615212i	$0.210930\pi$
$312$	4.63281	0.262281
$313$	0.199726	0.0112892	0.00564458	−	0.999984i	$-0.498203\pi$
$313$	0.199726	0.0112892	0.00564458	+	0.999984i	$0.498203\pi$
$314$	−7.95786	−0.449088
$315$	0.179080	0.0100900
$316$	−45.6410	−2.56751
$317$	−19.0297	−1.06881	−0.534406	−	0.845228i	$-0.679465\pi$
$317$	−19.0297	−1.06881	−0.534406	+	0.845228i	$0.679465\pi$
$318$	3.84939	0.215863
$319$	43.1907	2.41822
$320$	−1.00634	−0.0562561
$321$	−28.2888	−1.57893
$322$	26.9804	1.50356
$323$	−4.24278	−0.236074
$324$	−33.9468	−1.88594
$325$	−4.52158	−0.250812
$326$	−39.6852	−2.19796
$327$	−18.3190	−1.01304
$328$	−13.0160	−0.718689
$329$	19.1098	1.05356
$330$	1.92677	0.106065
$331$	−23.9361	−1.31565	−0.657824	−	0.753172i	$-0.728523\pi$
$331$	−23.9361	−1.31565	−0.657824	+	0.753172i	$0.728523\pi$
$332$	−16.6825	−0.915573
$333$	7.67784	0.420743
$334$	−10.0818	−0.551654
$335$	0.0423972	0.00231641
$336$	1.98255	0.108157
$337$	24.4635	1.33261	0.666305	−	0.745680i	$-0.267875\pi$
$337$	24.4635	1.33261	0.666305	+	0.745680i	$0.267875\pi$
$338$	27.6589	1.50445
$339$	20.0286	1.08780
$340$	−1.03432	−0.0560940
$341$	4.76548	0.258065
$342$	−1.80766	−0.0977470
$343$	−16.0655	−0.867453
$344$	18.5130	0.998153
$345$	0.613782	0.0330449
$346$	−1.20109	−0.0645713
$347$	13.3084	0.714430	0.357215	−	0.934022i	$-0.383726\pi$
$347$	13.3084	0.714430	0.357215	+	0.934022i	$0.383726\pi$
$348$	−47.1060	−2.52514
$349$	3.94971	0.211423	0.105711	−	0.994397i	$-0.466288\pi$
$349$	3.94971	0.211423	0.105711	+	0.994397i	$0.466288\pi$
$350$	33.0329	1.76568
$351$	3.88783	0.207517
$352$	34.0916	1.81709
$353$	7.45291	0.396678	0.198339	−	0.980133i	$-0.436445\pi$
$353$	7.45291	0.396678	0.198339	+	0.980133i	$0.436445\pi$
$354$	29.9961	1.59427
$355$	0.397930	0.0211199
$356$	−10.0433	−0.532294
$357$	24.0786	1.27438
$358$	−45.7315	−2.41698
$359$	3.03767	0.160322	0.0801609	−	0.996782i	$-0.474457\pi$
$359$	3.03767	0.160322	0.0801609	+	0.996782i	$0.474457\pi$
$360$	−0.161462	−0.00850981
$361$	1.00000	0.0526316
$362$	21.5255	1.13135
$363$	−40.5193	−2.12671
$364$	8.32464	0.436330
$365$	−0.410456	−0.0214842
$366$	48.0132	2.50969
$367$	3.00569	0.156896	0.0784479	−	0.996918i	$-0.475004\pi$
$367$	3.00569	0.156896	0.0784479	+	0.996918i	$0.475004\pi$
$368$	1.42494	0.0742801
$369$	3.94526	0.205382
$370$	−1.69151	−0.0879376
$371$	2.53433	0.131576
$372$	−5.19747	−0.269476
$373$	−17.0550	−0.883075	−0.441538	−	0.897243i	$-0.645567\pi$
$373$	−17.0550	−0.883075	−0.441538	+	0.897243i	$0.645567\pi$
$374$	54.3277	2.80922
$375$	1.50384	0.0776578
$376$	−17.2298	−0.888560
$377$	6.93486	0.357164
$378$	−28.4029	−1.46089
$379$	32.7714	1.68335	0.841677	−	0.539982i	$-0.181569\pi$
$379$	32.7714	1.68335	0.841677	+	0.539982i	$0.181569\pi$
$380$	0.243784	0.0125059
$381$	34.1543	1.74978
$382$	−1.36668	−0.0699253
$383$	31.7892	1.62435	0.812175	−	0.583413i	$-0.198283\pi$
$383$	31.7892	1.62435	0.812175	+	0.583413i	$0.198283\pi$
$384$	−34.0908	−1.73969
$385$	1.26853	0.0646504
$386$	−47.6302	−2.42431
$387$	−5.61143	−0.285245
$388$	31.9847	1.62378
$389$	−11.6367	−0.590003	−0.295002	−	0.955497i	$-0.595320\pi$
$389$	−11.6367	−0.590003	−0.295002	+	0.955497i	$0.595320\pi$
$390$	0.309370	0.0156656
$391$	17.3063	0.875217
$392$	−3.89894	−0.196926
$393$	18.5529	0.935872
$394$	−7.23149	−0.364317
$395$	−1.11671	−0.0561875
$396$	14.1690	0.712021
$397$	−6.18302	−0.310317	−0.155159	−	0.987890i	$-0.549589\pi$
$397$	−6.18302	−0.310317	−0.155159	+	0.987890i	$0.549589\pi$
$398$	43.4136	2.17613
$399$	−5.67520	−0.284115
$400$	1.74459	0.0872296
$401$	1.28770	0.0643045	0.0321523	−	0.999483i	$-0.489764\pi$
$401$	1.28770	0.0643045	0.0321523	+	0.999483i	$0.489764\pi$
$402$	2.42878	0.121136
$403$	0.765163	0.0381155
$404$	49.7937	2.47733
$405$	−0.830582	−0.0412719
$406$	−50.6633	−2.51438
$407$	54.3868	2.69586
$408$	−21.7098	−1.07480
$409$	16.7652	0.828987	0.414494	−	0.910052i	$-0.363959\pi$
$409$	16.7652	0.828987	0.414494	+	0.910052i	$0.363959\pi$
$410$	−0.869184	−0.0429259
$411$	15.2152	0.750511
$412$	−32.1533	−1.58408
$413$	19.7486	0.971763
$414$	7.37344	0.362385
$415$	−0.408174	−0.0200365
$416$	5.47386	0.268378
$417$	19.3000	0.945125
$418$	−12.8047	−0.626301
$419$	18.4380	0.900756	0.450378	−	0.892838i	$-0.351289\pi$
$419$	18.4380	0.900756	0.450378	+	0.892838i	$0.351289\pi$
$420$	−1.38352	−0.0675091
$421$	−5.14819	−0.250907	−0.125454	−	0.992099i	$-0.540039\pi$
$421$	−5.14819	−0.250907	−0.125454	+	0.992099i	$0.540039\pi$
$422$	2.27080	0.110541
$423$	5.22250	0.253926
$424$	−2.28500	−0.110970
$425$	21.1886	1.02780
$426$	22.7959	1.10447
$427$	31.6105	1.52974
$428$	45.8312	2.21533
$429$	−9.94710	−0.480250
$430$	1.23626	0.0596178
$431$	−7.18501	−0.346090	−0.173045	−	0.984914i	$-0.555361\pi$
$431$	−7.18501	−0.346090	−0.173045	+	0.984914i	$0.555361\pi$
$432$	−1.50007	−0.0721720
$433$	31.8243	1.52938	0.764690	−	0.644398i	$-0.222892\pi$
$433$	31.8243	1.52938	0.764690	+	0.644398i	$0.222892\pi$
$434$	−5.58997	−0.268327
$435$	−1.15255	−0.0552604
$436$	29.6788	1.42136
$437$	−4.07900	−0.195125
$438$	−23.5135	−1.12352
$439$	−3.60108	−0.171870	−0.0859350	−	0.996301i	$-0.527388\pi$
$439$	−3.60108	−0.171870	−0.0859350	+	0.996301i	$0.527388\pi$
$440$	−1.14374	−0.0545255
$441$	1.18180	0.0562762
$442$	8.72305	0.414913
$443$	11.7599	0.558730	0.279365	−	0.960185i	$-0.409876\pi$
$443$	11.7599	0.558730	0.279365	+	0.960185i	$0.409876\pi$
$444$	−59.3170	−2.81506
$445$	−0.245731	−0.0116487
$446$	52.1727	2.47045
$447$	−16.6191	−0.786056
$448$	−37.9547	−1.79319
$449$	35.2327	1.66274	0.831368	−	0.555723i	$-0.187558\pi$
$449$	35.2327	1.66274	0.831368	+	0.555723i	$0.187558\pi$
$450$	9.02751	0.425561
$451$	27.9467	1.31596
$452$	−32.4486	−1.52625
$453$	−13.7278	−0.644988
$454$	27.4019	1.28603
$455$	0.203680	0.00954867
$456$	5.11688	0.239620
$457$	−10.1524	−0.474908	−0.237454	−	0.971399i	$-0.576313\pi$
$457$	−10.1524	−0.474908	−0.237454	+	0.971399i	$0.576313\pi$
$458$	40.5502	1.89479
$459$	−18.2188	−0.850379
$460$	−0.994397	−0.0463640
$461$	−12.1349	−0.565179	−0.282590	−	0.959241i	$-0.591193\pi$
$461$	−12.1349	−0.565179	−0.282590	+	0.959241i	$0.591193\pi$
$462$	72.6695	3.38089
$463$	3.33810	0.155135	0.0775673	−	0.996987i	$-0.475285\pi$
$463$	3.33810	0.155135	0.0775673	+	0.996987i	$0.475285\pi$
$464$	−2.67572	−0.124217
$465$	−0.127167	−0.00589723
$466$	39.6413	1.83635
$467$	1.34265	0.0621307	0.0310653	−	0.999517i	$-0.490110\pi$
$467$	1.34265	0.0621307	0.0310653	+	0.999517i	$0.490110\pi$
$468$	2.27503	0.105163
$469$	1.59904	0.0738367
$470$	−1.15057	−0.0530720
$471$	−6.82783	−0.314610
$472$	−17.8057	−0.819575
$473$	−39.7492	−1.82767
$474$	−63.9719	−2.93833
$475$	−4.99404	−0.229142
$476$	−39.0101	−1.78803
$477$	0.692603	0.0317121
$478$	16.6409	0.761135
$479$	29.5003	1.34790	0.673952	−	0.738775i	$-0.264595\pi$
$479$	29.5003	1.34790	0.673952	+	0.738775i	$0.264595\pi$
$480$	−0.909735	−0.0415235
$481$	8.73255	0.398170
$482$	−14.0440	−0.639686
$483$	23.1492	1.05332
$484$	65.6460	2.98391
$485$	0.782574	0.0355349
$486$	−18.3280	−0.831376
$487$	−32.3826	−1.46739	−0.733697	−	0.679476i	$-0.762207\pi$
$487$	−32.3826	−1.46739	−0.733697	+	0.679476i	$0.762207\pi$
$488$	−28.5007	−1.29017
$489$	−34.0498	−1.53979
$490$	−0.260364	−0.0117620
$491$	−28.1990	−1.27260	−0.636302	−	0.771440i	$-0.719537\pi$
$491$	−28.1990	−1.27260	−0.636302	+	0.771440i	$0.719537\pi$
$492$	−30.4800	−1.37414
$493$	−32.4974	−1.46361
$494$	−2.05598	−0.0925027
$495$	0.346676	0.0155819
$496$	−0.295228	−0.0132561
$497$	15.0082	0.673208
$498$	−23.3827	−1.04781
$499$	22.7583	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$499$	22.7583	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$500$	−2.43639	−0.108959
$501$	−8.65021	−0.386463
$502$	−62.3584	−2.78319
$503$	0.966329	0.0430865	0.0215432	−	0.999768i	$-0.493142\pi$
$503$	0.966329	0.0430865	0.0215432	+	0.999768i	$0.493142\pi$
$504$	−6.08965	−0.271255
$505$	1.21831	0.0542141
$506$	52.2306	2.32193
$507$	23.7313	1.05395
$508$	−55.3338	−2.45504
$509$	20.3676	0.902778	0.451389	−	0.892327i	$-0.350929\pi$
$509$	20.3676	0.902778	0.451389	+	0.892327i	$0.350929\pi$
$510$	−1.44974	−0.0641955
$511$	−15.4806	−0.684821
$512$	−3.94691	−0.174431
$513$	4.29406	0.189588
$514$	52.8151	2.32957
$515$	−0.786699	−0.0346661
$516$	43.3524	1.90848
$517$	36.9941	1.62700
$518$	−63.7965	−2.80306
$519$	−1.03054	−0.0452356
$520$	−0.183642	−0.00805325
$521$	5.45548	0.239009	0.119504	−	0.992834i	$-0.461869\pi$
$521$	5.45548	0.239009	0.119504	+	0.992834i	$0.461869\pi$
$522$	−13.8457	−0.606010
$523$	−21.1402	−0.924398	−0.462199	−	0.886776i	$-0.652939\pi$
$523$	−21.1402	−0.924398	−0.462199	+	0.886776i	$0.652939\pi$
$524$	−30.0579	−1.31308
$525$	28.3421	1.23695
$526$	−44.1091	−1.92325
$527$	−3.58563	−0.156192
$528$	3.83795	0.167025
$529$	−6.36173	−0.276597
$530$	−0.152588	−0.00662800
$531$	5.39706	0.234212
$532$	9.19447	0.398631
$533$	4.48722	0.194363
$534$	−14.0770	−0.609171
$535$	1.12136	0.0484805
$536$	−1.44173	−0.0622731
$537$	−39.2375	−1.69322
$538$	−26.0659	−1.12378
$539$	8.37141	0.360582
$540$	1.04683	0.0450482
$541$	−5.90185	−0.253740	−0.126870	−	0.991919i	$-0.540493\pi$
$541$	−5.90185	−0.253740	−0.126870	+	0.991919i	$0.540493\pi$
$542$	−35.0338	−1.50483
$543$	18.4688	0.792572
$544$	−25.6511	−1.09978
$545$	0.726156	0.0311051
$546$	11.6681	0.499347
$547$	−14.6359	−0.625784	−0.312892	−	0.949789i	$-0.601298\pi$
$547$	−14.6359	−0.625784	−0.312892	+	0.949789i	$0.601298\pi$
$548$	−24.6504	−1.05301
$549$	8.63880	0.368695
$550$	63.9474	2.72672
$551$	7.65947	0.326305
$552$	−20.8718	−0.888362
$553$	−42.1172	−1.79101
$554$	9.52059	0.404491
$555$	−1.45132	−0.0616049
$556$	−31.2682	−1.32607
$557$	0.213211	0.00903402	0.00451701	−	0.999990i	$-0.498562\pi$
$557$	0.213211	0.00903402	0.00451701	+	0.999990i	$0.498562\pi$
$558$	−1.52768	−0.0646717
$559$	−6.38227	−0.269941
$560$	−0.0785872	−0.00332092
$561$	46.6131	1.96801
$562$	12.3220	0.519772
$563$	−1.74571	−0.0735729	−0.0367865	−	0.999323i	$-0.511712\pi$
$563$	−1.74571	−0.0735729	−0.0367865	+	0.999323i	$0.511712\pi$
$564$	−40.3476	−1.69894
$565$	−0.793923	−0.0334006
$566$	−36.8363	−1.54835
$567$	−31.3259	−1.31556
$568$	−13.5317	−0.567777
$569$	33.0334	1.38483	0.692417	−	0.721497i	$-0.256546\pi$
$569$	33.0334	1.38483	0.692417	+	0.721497i	$0.256546\pi$
$570$	0.341695	0.0143120
$571$	35.1680	1.47174	0.735868	−	0.677125i	$-0.236774\pi$
$571$	35.1680	1.47174	0.735868	+	0.677125i	$0.236774\pi$
$572$	16.1154	0.673820
$573$	−1.17261	−0.0489863
$574$	−32.7818	−1.36829
$575$	20.3707	0.849516
$576$	−10.3726	−0.432192
$577$	−9.68245	−0.403086	−0.201543	−	0.979480i	$-0.564596\pi$
$577$	−9.68245	−0.403086	−0.201543	+	0.979480i	$0.564596\pi$
$578$	−2.27345	−0.0945629
$579$	−40.8666	−1.69836
$580$	1.86726	0.0775337
$581$	−15.3945	−0.638672
$582$	44.8308	1.85829
$583$	4.90613	0.203191
$584$	13.9576	0.577571
$585$	0.0556635	0.00230140
$586$	51.0169	2.10749
$587$	13.9833	0.577151	0.288576	−	0.957457i	$-0.406818\pi$
$587$	13.9833	0.577151	0.288576	+	0.957457i	$0.406818\pi$
$588$	−9.13028	−0.376526
$589$	0.845113	0.0348223
$590$	−1.18903	−0.0489517
$591$	−6.20460	−0.255223
$592$	−3.36933	−0.138479
$593$	−38.0449	−1.56232	−0.781158	−	0.624334i	$-0.785371\pi$
$593$	−38.0449	−1.56232	−0.781158	+	0.624334i	$0.785371\pi$
$594$	−54.9844	−2.25604
$595$	−0.954465	−0.0391293
$596$	26.9248	1.10288
$597$	37.2488	1.52449
$598$	8.38633	0.342943
$599$	36.8229	1.50454	0.752271	−	0.658854i	$-0.228958\pi$
$599$	36.8229	1.50454	0.752271	+	0.658854i	$0.228958\pi$
$600$	−25.5539	−1.04323
$601$	−33.1157	−1.35082	−0.675408	−	0.737444i	$-0.736033\pi$
$601$	−33.1157	−1.35082	−0.675408	+	0.737444i	$0.736033\pi$
$602$	46.6263	1.90035
$603$	0.436999	0.0177960
$604$	22.2406	0.904957
$605$	1.60617	0.0653000
$606$	69.7924	2.83512
$607$	−31.6474	−1.28453	−0.642263	−	0.766484i	$-0.722004\pi$
$607$	−31.6474	−1.28453	−0.642263	+	0.766484i	$0.722004\pi$
$608$	6.04582	0.245190
$609$	−43.4690	−1.76145
$610$	−1.90322	−0.0770592
$611$	5.93991	0.240303
$612$	−10.6610	−0.430946
$613$	−11.0473	−0.446195	−0.223097	−	0.974796i	$-0.571617\pi$
$613$	−11.0473	−0.446195	−0.223097	+	0.974796i	$0.571617\pi$
$614$	−33.7386	−1.36158
$615$	−0.745758	−0.0300719
$616$	−43.1367	−1.73803
$617$	41.4491	1.66868	0.834340	−	0.551251i	$-0.185849\pi$
$617$	41.4491	1.66868	0.834340	+	0.551251i	$0.185849\pi$
$618$	−45.0671	−1.81286
$619$	37.7667	1.51797	0.758986	−	0.651107i	$-0.225695\pi$
$619$	37.7667	1.51797	0.758986	+	0.651107i	$0.225695\pi$
$620$	0.206025	0.00827417
$621$	−17.5155	−0.702873
$622$	−63.1415	−2.53174
$623$	−9.26788	−0.371310
$624$	0.616235	0.0246692
$625$	24.9106	0.996423
$626$	−0.453537	−0.0181270
$627$	−10.9865	−0.438757
$628$	11.0619	0.441417
$629$	−40.9216	−1.63165
$630$	−0.406655	−0.0162015
$631$	26.9751	1.07386	0.536931	−	0.843626i	$-0.319584\pi$
$631$	26.9751	1.07386	0.536931	+	0.843626i	$0.319584\pi$
$632$	37.9738	1.51052
$633$	1.94834	0.0774397
$634$	43.2126	1.71619
$635$	−1.35386	−0.0537263
$636$	−5.35086	−0.212176
$637$	1.34414	0.0532569
$638$	−98.0776	−3.88293
$639$	4.10156	0.162255
$640$	1.35134	0.0534166
$641$	30.9201	1.22127	0.610635	−	0.791912i	$-0.290914\pi$
$641$	30.9201	1.22127	0.610635	+	0.791912i	$0.290914\pi$
$642$	64.2383	2.53529
$643$	−0.402331	−0.0158664	−0.00793320	−	0.999969i	$-0.502525\pi$
$643$	−0.402331	−0.0158664	−0.00793320	+	0.999969i	$0.502525\pi$
$644$	−37.5043	−1.47788
$645$	1.06071	0.0417654
$646$	9.63451	0.379065
$647$	−9.52207	−0.374351	−0.187176	−	0.982326i	$-0.559933\pi$
$647$	−9.52207	−0.374351	−0.187176	+	0.982326i	$0.559933\pi$
$648$	28.2441	1.10953
$649$	38.2307	1.50068
$650$	10.2676	0.402729
$651$	−4.79619	−0.187977
$652$	55.1646	2.16041
$653$	10.9733	0.429418	0.214709	−	0.976678i	$-0.431120\pi$
$653$	10.9733	0.429418	0.214709	+	0.976678i	$0.431120\pi$
$654$	41.5988	1.62664
$655$	−0.735430	−0.0287356
$656$	−1.73133	−0.0675971
$657$	−4.23067	−0.165054
$658$	−43.3946	−1.69170
$659$	−18.4849	−0.720071	−0.360036	−	0.932939i	$-0.617235\pi$
$659$	−18.4849	−0.720071	−0.360036	+	0.932939i	$0.617235\pi$
$660$	−2.67832	−0.104254
$661$	11.5171	0.447964	0.223982	−	0.974593i	$-0.428094\pi$
$661$	11.5171	0.447964	0.223982	+	0.974593i	$0.428094\pi$
$662$	54.3542	2.11254
$663$	7.48436	0.290669
$664$	13.8800	0.538650
$665$	0.224962	0.00872366
$666$	−17.4349	−0.675587
$667$	−31.2430	−1.20973
$668$	14.0143	0.542231
$669$	44.7641	1.73068
$670$	−0.0962757	−0.00371945
$671$	61.1939	2.36236
$672$	−34.3112	−1.32358
$673$	13.6093	0.524600	0.262300	−	0.964986i	$-0.415519\pi$
$673$	13.6093	0.524600	0.262300	+	0.964986i	$0.415519\pi$
$674$	−55.5516	−2.13977
$675$	−21.4447	−0.825407
$676$	−38.4475	−1.47875
$677$	−27.0003	−1.03771	−0.518854	−	0.854863i	$-0.673641\pi$
$677$	−27.0003	−1.03771	−0.518854	+	0.854863i	$0.673641\pi$
$678$	−45.4809	−1.74668
$679$	29.5153	1.13269
$680$	0.860567	0.0330012
$681$	23.5108	0.900934
$682$	−10.8215	−0.414375
$683$	−27.9123	−1.06804	−0.534018	−	0.845473i	$-0.679319\pi$
$683$	−27.9123	−1.06804	−0.534018	+	0.845473i	$0.679319\pi$
$684$	2.51275	0.0960772
$685$	−0.603124	−0.0230442
$686$	36.4815	1.39287
$687$	34.7920	1.32740
$688$	2.46251	0.0938824
$689$	0.787746	0.0300107
$690$	−1.39378	−0.0530602
$691$	−21.0672	−0.801434	−0.400717	−	0.916202i	$-0.631239\pi$
$691$	−21.0672	−0.801434	−0.400717	+	0.916202i	$0.631239\pi$
$692$	1.66959	0.0634682
$693$	13.0751	0.496681
$694$	−30.2206	−1.14716
$695$	−0.765043	−0.0290197
$696$	39.1926	1.48559
$697$	−21.0275	−0.796475
$698$	−8.96900	−0.339482
$699$	34.0122	1.28646
$700$	−45.9175	−1.73552
$701$	16.1046	0.608262	0.304131	−	0.952630i	$-0.401634\pi$
$701$	16.1046	0.608262	0.304131	+	0.952630i	$0.401634\pi$
$702$	−8.82849	−0.333210
$703$	9.64499	0.363768
$704$	−73.4754	−2.76921
$705$	−0.987190	−0.0371797
$706$	−16.9241	−0.636946
$707$	45.9493	1.72810
$708$	−41.6962	−1.56704
$709$	−21.0919	−0.792122	−0.396061	−	0.918224i	$-0.629623\pi$
$709$	−21.0919	−0.792122	−0.396061	+	0.918224i	$0.629623\pi$
$710$	−0.903620	−0.0339122
$711$	−11.5102	−0.431665
$712$	8.35612	0.313159
$713$	−3.44722	−0.129099
$714$	−54.6777	−2.04626
$715$	0.394298	0.0147459
$716$	63.5693	2.37570
$717$	14.2778	0.533215
$718$	−6.89794	−0.257429
$719$	−9.75808	−0.363915	−0.181957	−	0.983306i	$-0.558243\pi$
$719$	−9.75808	−0.363915	−0.181957	+	0.983306i	$0.558243\pi$
$720$	−0.0214770	−0.000800400 0
$721$	−29.6708	−1.10500
$722$	−2.27080	−0.0845105
$723$	−12.0497	−0.448134
$724$	−29.9216	−1.11203
$725$	−38.2517	−1.42063
$726$	92.0114	3.41486
$727$	−12.3668	−0.458659	−0.229330	−	0.973349i	$-0.573653\pi$
$727$	−12.3668	−0.458659	−0.229330	+	0.973349i	$0.573653\pi$
$728$	−6.92619	−0.256702
$729$	16.5379	0.612516
$730$	0.932063	0.0344972
$731$	29.9080	1.10619
$732$	−66.7410	−2.46682
$733$	−28.1825	−1.04095	−0.520473	−	0.853878i	$-0.674244\pi$
$733$	−28.1825	−1.04095	−0.520473	+	0.853878i	$0.674244\pi$
$734$	−6.82533	−0.251927
$735$	−0.223392	−0.00823993
$736$	−24.6609	−0.909013
$737$	3.09553	0.114025
$738$	−8.95889	−0.329781
$739$	−15.3658	−0.565240	−0.282620	−	0.959232i	$-0.591203\pi$
$739$	−15.3658	−0.565240	−0.282620	+	0.959232i	$0.591203\pi$
$740$	2.35130	0.0864354
$741$	−1.76402	−0.0648030
$742$	−5.75495	−0.211271
$743$	−16.6442	−0.610616	−0.305308	−	0.952254i	$-0.598759\pi$
$743$	−16.6442	−0.610616	−0.305308	+	0.952254i	$0.598759\pi$
$744$	4.32435	0.158538
$745$	0.658773	0.0241356
$746$	38.7286	1.41795
$747$	−4.20715	−0.153931
$748$	−75.5185	−2.76123
$749$	42.2927	1.54534
$750$	−3.41492	−0.124695
$751$	−16.7314	−0.610536	−0.305268	−	0.952267i	$-0.598746\pi$
$751$	−16.7314	−0.610536	−0.305268	+	0.952267i	$0.598746\pi$
$752$	−2.29183	−0.0835746
$753$	−53.5034	−1.94977
$754$	−15.7477	−0.573497
$755$	0.544163	0.0198041
$756$	39.4817	1.43593
$757$	16.5641	0.602031	0.301016	−	0.953619i	$-0.402674\pi$
$757$	16.5641	0.602031	0.301016	+	0.953619i	$0.402674\pi$
$758$	−74.4173	−2.70296
$759$	44.8138	1.62664
$760$	−0.202831	−0.00735745
$761$	16.2701	0.589792	0.294896	−	0.955529i	$-0.404715\pi$
$761$	16.2701	0.589792	0.294896	+	0.955529i	$0.404715\pi$
$762$	−77.5576	−2.80961
$763$	27.3874	0.991491
$764$	1.89976	0.0687308
$765$	−0.260845	−0.00943085
$766$	−72.1869	−2.60822
$767$	6.13845	0.221647
$768$	26.6389	0.961248
$769$	−32.4951	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$769$	−32.4951	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$770$	−2.88059	−0.103809
$771$	45.3152	1.63199
$772$	66.2086	2.38290
$773$	13.0279	0.468580	0.234290	−	0.972167i	$-0.424723\pi$
$773$	13.0279	0.468580	0.234290	+	0.972167i	$0.424723\pi$
$774$	12.7424	0.458018
$775$	−4.22052	−0.151606
$776$	−26.6116	−0.955301
$777$	−54.7373	−1.96369
$778$	26.4246	0.947368
$779$	4.95608	0.177570
$780$	−0.430041	−0.0153979
$781$	29.0539	1.03963
$782$	−39.2992	−1.40534
$783$	32.8903	1.17540
$784$	−0.518620	−0.0185221
$785$	0.270652	0.00965999
$786$	−42.1300	−1.50273
$787$	43.5703	1.55311	0.776557	−	0.630048i	$-0.216965\pi$
$787$	43.5703	1.55311	0.776557	+	0.630048i	$0.216965\pi$
$788$	10.0522	0.358094
$789$	−37.8455	−1.34734
$790$	2.53582	0.0902203
$791$	−29.9433	−1.06466
$792$	−11.7888	−0.418896
$793$	9.82551	0.348914
$794$	14.0404	0.498276
$795$	−0.130920	−0.00464326
$796$	−60.3473	−2.13895
$797$	30.3419	1.07476	0.537382	−	0.843339i	$-0.319413\pi$
$797$	30.3419	1.07476	0.537382	+	0.843339i	$0.319413\pi$
$798$	12.8873	0.456204
$799$	−27.8350	−0.984732
$800$	−30.1930	−1.06748
$801$	−2.53281	−0.0894923
$802$	−2.92410	−0.103254
$803$	−29.9684	−1.05756
$804$	−3.37614	−0.119067
$805$	−0.917622	−0.0323419
$806$	−1.73753	−0.0612020
$807$	−22.3645	−0.787268
$808$	−41.4289	−1.45746
$809$	−17.7162	−0.622867	−0.311434	−	0.950268i	$-0.600809\pi$
$809$	−17.7162	−0.622867	−0.311434	+	0.950268i	$0.600809\pi$
$810$	1.88609	0.0662703
$811$	30.8099	1.08188	0.540940	−	0.841061i	$-0.318068\pi$
$811$	30.8099	1.08188	0.540940	+	0.841061i	$0.318068\pi$
$812$	70.4248	2.47143
$813$	−30.0590	−1.05421
$814$	−123.502	−4.32873
$815$	1.34972	0.0472786
$816$	−2.88774	−0.101091
$817$	−7.04914	−0.246618
$818$	−38.0705	−1.33110
$819$	2.09938	0.0733583
$820$	1.20821	0.0421926
$821$	26.8696	0.937754	0.468877	−	0.883263i	$-0.344659\pi$
$821$	26.8696	0.937754	0.468877	+	0.883263i	$0.344659\pi$
$822$	−34.5507	−1.20510
$823$	16.0582	0.559755	0.279877	−	0.960036i	$-0.409706\pi$
$823$	16.0582	0.559755	0.279877	+	0.960036i	$0.409706\pi$
$824$	26.7519	0.931946
$825$	54.8667	1.91021
$826$	−44.8451	−1.56036
$827$	51.8824	1.80413	0.902064	−	0.431603i	$-0.142052\pi$
$827$	51.8824	1.80413	0.902064	+	0.431603i	$0.142052\pi$
$828$	−10.2495	−0.356195
$829$	−33.0202	−1.14684	−0.573420	−	0.819262i	$-0.694383\pi$
$829$	−33.0202	−1.14684	−0.573420	+	0.819262i	$0.694383\pi$
$830$	0.926881	0.0321725
$831$	8.16865	0.283367
$832$	−11.7975	−0.409004
$833$	−6.29879	−0.218240
$834$	−43.8265	−1.51759
$835$	0.342890	0.0118662
$836$	17.7993	0.615602
$837$	3.62897	0.125436
$838$	−41.8691	−1.44634
$839$	23.2493	0.802656	0.401328	−	0.915935i	$-0.368549\pi$
$839$	23.2493	0.802656	0.401328	+	0.915935i	$0.368549\pi$
$840$	1.15111	0.0397169
$841$	29.6675	1.02302
$842$	11.6905	0.402882
$843$	10.5722	0.364127
$844$	−3.15654	−0.108653
$845$	−0.940699	−0.0323610
$846$	−11.8593	−0.407730
$847$	60.5776	2.08147
$848$	−0.303941	−0.0104374
$849$	−31.6055	−1.08470
$850$	−48.1151	−1.65033
$851$	−39.3420	−1.34863
$852$	−31.6876	−1.08560
$853$	9.00603	0.308361	0.154180	−	0.988043i	$-0.450726\pi$
$853$	9.00603	0.308361	0.154180	+	0.988043i	$0.450726\pi$
$854$	−71.7812	−2.45630
$855$	0.0614797	0.00210256
$856$	−38.1320	−1.30332
$857$	−16.0119	−0.546958	−0.273479	−	0.961878i	$-0.588174\pi$
$857$	−16.0119	−0.546958	−0.273479	+	0.961878i	$0.588174\pi$
$858$	22.5879	0.771138
$859$	18.4389	0.629127	0.314564	−	0.949236i	$-0.398142\pi$
$859$	18.4389	0.629127	0.314564	+	0.949236i	$0.398142\pi$
$860$	−1.71847	−0.0585993
$861$	−28.1267	−0.958556
$862$	16.3157	0.555716
$863$	47.7063	1.62394	0.811971	−	0.583697i	$-0.198394\pi$
$863$	47.7063	1.62394	0.811971	+	0.583697i	$0.198394\pi$
$864$	25.9611	0.883215
$865$	0.0408500	0.00138894
$866$	−72.2668	−2.45572
$867$	−1.95061	−0.0662463
$868$	7.77037	0.263744
$869$	−81.5335	−2.76583
$870$	2.61721	0.0887316
$871$	0.497029	0.0168412
$872$	−24.6931	−0.836214
$873$	8.06619	0.272999
$874$	9.26261	0.313312
$875$	−2.24828	−0.0760058
$876$	32.6850	1.10432
$877$	45.0864	1.52246	0.761229	−	0.648483i	$-0.224596\pi$
$877$	45.0864	1.52246	0.761229	+	0.648483i	$0.224596\pi$
$878$	8.17733	0.275972
$879$	43.7724	1.47641
$880$	−0.152135	−0.00512846
$881$	−3.28396	−0.110639	−0.0553196	−	0.998469i	$-0.517618\pi$
$881$	−3.28396	−0.110639	−0.0553196	+	0.998469i	$0.517618\pi$
$882$	−2.68363	−0.0903627
$883$	53.3541	1.79551	0.897755	−	0.440495i	$-0.145197\pi$
$883$	53.3541	1.79551	0.897755	+	0.440495i	$0.145197\pi$
$884$	−12.1255	−0.407825
$885$	−1.02019	−0.0342932
$886$	−26.7044	−0.897152
$887$	−27.9727	−0.939230	−0.469615	−	0.882871i	$-0.655607\pi$
$887$	−27.9727	−0.939230	−0.469615	+	0.882871i	$0.655607\pi$
$888$	49.3523	1.65616
$889$	−51.0617	−1.71255
$890$	0.558005	0.0187044
$891$	−60.6429	−2.03161
$892$	−72.5229	−2.42825
$893$	6.56056	0.219541
$894$	37.7387	1.26217
$895$	1.55536	0.0519899
$896$	50.9668	1.70268
$897$	7.19546	0.240249
$898$	−80.0065	−2.66985
$899$	6.47312	0.215891
$900$	−12.5487	−0.418291
$901$	−3.69145	−0.122980
$902$	−63.4613	−2.11303
$903$	40.0053	1.33129
$904$	26.9975	0.897925
$905$	−0.732095	−0.0243357
$906$	31.1731	1.03566
$907$	19.1502	0.635873	0.317937	−	0.948112i	$-0.397010\pi$
$907$	19.1502	0.635873	0.317937	+	0.948112i	$0.397010\pi$
$908$	−38.0901	−1.26406
$909$	12.5574	0.416503
$910$	−0.462517	−0.0153323
$911$	37.6174	1.24632	0.623159	−	0.782095i	$-0.285849\pi$
$911$	37.6174	1.24632	0.623159	+	0.782095i	$0.285849\pi$
$912$	0.680625	0.0225377
$913$	−29.8018	−0.986295
$914$	23.0540	0.762560
$915$	−1.63296	−0.0539841
$916$	−56.3670	−1.86242
$917$	−27.7372	−0.915963
$918$	41.3712	1.36545
$919$	34.3921	1.13449	0.567245	−	0.823549i	$-0.308009\pi$
$919$	34.3921	1.13449	0.567245	+	0.823549i	$0.308009\pi$
$920$	0.827348	0.0272769
$921$	−28.9476	−0.953857
$922$	27.5560	0.907508
$923$	4.66499	0.153550
$924$	−101.015	−3.32314
$925$	−48.1674	−1.58374
$926$	−7.58016	−0.249100
$927$	−8.10871	−0.266325
$928$	46.3078	1.52013
$929$	−8.68642	−0.284992	−0.142496	−	0.989795i	$-0.545513\pi$
$929$	−8.68642	−0.284992	−0.142496	+	0.989795i	$0.545513\pi$
$930$	0.288771	0.00946918
$931$	1.48459	0.0486555
$932$	−55.1036	−1.80498
$933$	−54.1753	−1.77362
$934$	−3.04890	−0.0997632
$935$	−1.84772	−0.0604269
$936$	−1.89285	−0.0618697
$937$	−13.8430	−0.452232	−0.226116	−	0.974100i	$-0.572603\pi$
$937$	−13.8430	−0.452232	−0.226116	+	0.974100i	$0.572603\pi$
$938$	−3.63110	−0.118560
$939$	−0.389134	−0.0126989
$940$	1.59936	0.0521654
$941$	23.6028	0.769430	0.384715	−	0.923035i	$-0.374300\pi$
$941$	23.6028	0.769430	0.384715	+	0.923035i	$0.374300\pi$
$942$	15.5047	0.505169
$943$	−20.2159	−0.658319
$944$	−2.36844	−0.0770861
$945$	0.966003	0.0314241
$946$	90.2625	2.93469
$947$	−58.2397	−1.89254	−0.946268	−	0.323384i	$-0.895180\pi$
$947$	−58.2397	−1.89254	−0.946268	+	0.323384i	$0.895180\pi$
$948$	88.9244	2.88813
$949$	−4.81184	−0.156199
$950$	11.3405	0.367933
$951$	37.0763	1.20228
$952$	32.4568	1.05193
$953$	−28.5847	−0.925950	−0.462975	−	0.886371i	$-0.653218\pi$
$953$	−28.5847	−0.925950	−0.462975	+	0.886371i	$0.653218\pi$
$954$	−1.57276	−0.0509201
$955$	0.0464816	0.00150411
$956$	−23.1317	−0.748133
$957$	−84.1504	−2.72020
$958$	−66.9894	−2.16433
$959$	−22.7472	−0.734545
$960$	1.96070	0.0632812
$961$	−30.2858	−0.976961
$962$	−19.8299	−0.639341
$963$	11.5581	0.372455
$964$	19.5219	0.628759
$965$	1.61993	0.0521475
$966$	−52.5671	−1.69132
$967$	−1.43008	−0.0459884	−0.0229942	−	0.999736i	$-0.507320\pi$
$967$	−1.43008	−0.0459884	−0.0229942	+	0.999736i	$0.507320\pi$
$968$	−54.6181	−1.75549
$969$	8.26639	0.265555
$970$	−1.77707	−0.0570583
$971$	−22.0844	−0.708721	−0.354360	−	0.935109i	$-0.615301\pi$
$971$	−22.0844	−0.708721	−0.354360	+	0.935109i	$0.615301\pi$
$972$	25.4770	0.817174
$973$	−28.8541	−0.925019
$974$	73.5344	2.35619
$975$	8.80960	0.282133
$976$	−3.79104	−0.121348
$977$	−2.26310	−0.0724031	−0.0362015	−	0.999345i	$-0.511526\pi$
$977$	−2.26310	−0.0724031	−0.0362015	+	0.999345i	$0.511526\pi$
$978$	77.3204	2.47243
$979$	−17.9414	−0.573410
$980$	0.361920	0.0115611
$981$	7.48467	0.238967
$982$	64.0344	2.04342
$983$	−17.6855	−0.564080	−0.282040	−	0.959403i	$-0.591011\pi$
$983$	−17.6855	−0.564080	−0.282040	+	0.959403i	$0.591011\pi$
$984$	25.3597	0.808437
$985$	0.245948	0.00783654
$986$	73.7952	2.35012
$987$	−37.2325	−1.18512
$988$	2.85792	0.0909226
$989$	28.7535	0.914307
$990$	−0.787231	−0.0250199
$991$	51.3248	1.63039	0.815193	−	0.579190i	$-0.196631\pi$
$991$	51.3248	1.63039	0.815193	+	0.579190i	$0.196631\pi$
$992$	5.10940	0.162224
$993$	46.6358	1.47994
$994$	−34.0806	−1.08097
$995$	−1.47653	−0.0468090
$996$	32.5033	1.02991
$997$	−27.0140	−0.855541	−0.427770	−	0.903887i	$-0.640701\pi$
$997$	−27.0140	−0.855541	−0.427770	+	0.903887i	$0.640701\pi$
$998$	−51.6795	−1.63589
$999$	41.4162	1.31035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.8	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.27080 q^{2} -1.94834 q^{3} +3.15654 q^{4} +0.0772315 q^{5} +4.42430 q^{6} +2.91283 q^{7} -2.62627 q^{8} +0.796044 q^{9} +O(q^{10})\)	\(q-2.27080 q^{2} -1.94834 q^{3} +3.15654 q^{4} +0.0772315 q^{5} +4.42430 q^{6} +2.91283 q^{7} -2.62627 q^{8} +0.796044 q^{9} -0.175377 q^{10} +5.63887 q^{11} -6.15003 q^{12} +0.905397 q^{13} -6.61446 q^{14} -0.150473 q^{15} -0.349335 q^{16} -4.24278 q^{17} -1.80766 q^{18} +1.00000 q^{19} +0.243784 q^{20} -5.67520 q^{21} -12.8047 q^{22} -4.07900 q^{23} +5.11688 q^{24} -4.99404 q^{25} -2.05598 q^{26} +4.29406 q^{27} +9.19447 q^{28} +7.65947 q^{29} +0.341695 q^{30} +0.845113 q^{31} +6.04582 q^{32} -10.9865 q^{33} +9.63451 q^{34} +0.224962 q^{35} +2.51275 q^{36} +9.64499 q^{37} -2.27080 q^{38} -1.76402 q^{39} -0.202831 q^{40} +4.95608 q^{41} +12.8873 q^{42} -7.04914 q^{43} +17.7993 q^{44} +0.0614797 q^{45} +9.26261 q^{46} +6.56056 q^{47} +0.680625 q^{48} +1.48459 q^{49} +11.3405 q^{50} +8.26639 q^{51} +2.85792 q^{52} +0.870056 q^{53} -9.75097 q^{54} +0.435498 q^{55} -7.64989 q^{56} -1.94834 q^{57} -17.3931 q^{58} +6.77985 q^{59} -0.474976 q^{60} +10.8522 q^{61} -1.91908 q^{62} +2.31874 q^{63} -13.0302 q^{64} +0.0699251 q^{65} +24.9481 q^{66} +0.548963 q^{67} -13.3925 q^{68} +7.94730 q^{69} -0.510845 q^{70} +5.15243 q^{71} -2.09063 q^{72} -5.31462 q^{73} -21.9019 q^{74} +9.73010 q^{75} +3.15654 q^{76} +16.4251 q^{77} +4.00575 q^{78} -14.4592 q^{79} -0.0269797 q^{80} -10.7544 q^{81} -11.2543 q^{82} -5.28507 q^{83} -17.9140 q^{84} -0.327676 q^{85} +16.0072 q^{86} -14.9233 q^{87} -14.8092 q^{88} -3.18174 q^{89} -0.139608 q^{90} +2.63727 q^{91} -12.8755 q^{92} -1.64657 q^{93} -14.8977 q^{94} +0.0772315 q^{95} -11.7793 q^{96} +10.1328 q^{97} -3.37121 q^{98} +4.48879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 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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